LMFDB - Embedded newform 75.18.a.l.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,18,Mod(1,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 18, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$137.416565508$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3 x^{7} - 737726 x^{6} - 11148552 x^{5} + 153299147136 x^{4} + 3606301935360 x^{3} + \cdots + 56\!\cdots\!00 $ x^8 - 3x^7 - 737726x^6 - 11148552x^5 + 153299147136x^4 + 3606301935360x^3 - 8597697543219200x^2 + 321798165633024000*x + 56736819274792960000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{10}\cdot 5^{14} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-395.421$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+419.421 q^{2} -6561.00 q^{3} +44841.8 q^{4} -2.75182e6 q^{6} +1.07280e7 q^{7} -3.61668e7 q^{8} +4.30467e7 q^{9} +O(q^{10})$	\(q+419.421 q^{2} -6561.00 q^{3} +44841.8 q^{4} -2.75182e6 q^{6} +1.07280e7 q^{7} -3.61668e7 q^{8} +4.30467e7 q^{9} -3.34405e8 q^{11} -2.94207e8 q^{12} +1.71011e8 q^{13} +4.49954e9 q^{14} -2.10466e10 q^{16} -3.73698e9 q^{17} +1.80547e10 q^{18} +3.46862e10 q^{19} -7.03863e10 q^{21} -1.40256e11 q^{22} -5.95580e10 q^{23} +2.37290e11 q^{24} +7.17257e10 q^{26} -2.82430e11 q^{27} +4.81062e11 q^{28} -2.08236e12 q^{29} +5.85106e12 q^{31} -4.08693e12 q^{32} +2.19403e12 q^{33} -1.56737e12 q^{34} +1.93029e12 q^{36} -2.29506e13 q^{37} +1.45481e13 q^{38} -1.12201e12 q^{39} -3.39080e13 q^{41} -2.95215e13 q^{42} +7.65153e13 q^{43} -1.49953e13 q^{44} -2.49798e13 q^{46} +2.72333e14 q^{47} +1.38087e14 q^{48} -1.17541e14 q^{49} +2.45183e13 q^{51} +7.66845e12 q^{52} -4.91865e14 q^{53} -1.18457e14 q^{54} -3.87996e14 q^{56} -2.27576e14 q^{57} -8.73384e14 q^{58} +2.15693e15 q^{59} -1.43236e15 q^{61} +2.45405e15 q^{62} +4.61805e14 q^{63} +1.04448e15 q^{64} +9.20222e14 q^{66} -3.26290e15 q^{67} -1.67573e14 q^{68} +3.90760e14 q^{69} +1.54056e15 q^{71} -1.55686e15 q^{72} -1.33066e15 q^{73} -9.62597e15 q^{74} +1.55539e15 q^{76} -3.58749e15 q^{77} -4.70592e14 q^{78} +2.64034e16 q^{79} +1.85302e15 q^{81} -1.42217e16 q^{82} +3.48504e16 q^{83} -3.15625e15 q^{84} +3.20921e16 q^{86} +1.36623e16 q^{87} +1.20943e16 q^{88} +6.75463e16 q^{89} +1.83461e15 q^{91} -2.67068e15 q^{92} -3.83888e16 q^{93} +1.14222e17 q^{94} +2.68143e16 q^{96} +7.30588e16 q^{97} -4.92991e16 q^{98} -1.43950e16 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 189 q^{2} - 52488 q^{3} + 431349 q^{4} - 1240029 q^{6} + 1557468 q^{7} + 16708167 q^{8} + 344373768 q^{9}+O(q^{10}) $ 8 * q + 189 * q^2 - 52488 * q^3 + 431349 * q^4 - 1240029 * q^6 + 1557468 * q^7 + 16708167 * q^8 + 344373768 * q^9	$ 8 q + 189 q^{2} - 52488 q^{3} + 431349 q^{4} - 1240029 q^{6} + 1557468 q^{7} + 16708167 q^{8} + 344373768 q^{9} + 453693072 q^{11} - 2830080789 q^{12} + 3368969496 q^{13} - 19124164374 q^{14} + 32205319825 q^{16} + 48799816416 q^{17} + 8135830269 q^{18} - 22187938864 q^{19} - 10218547548 q^{21} + 787250677122 q^{22} + 502967181024 q^{23} - 109622283687 q^{24} + 1470111900726 q^{26} - 2259436291848 q^{27} - 2737813754934 q^{28} + 2674234302252 q^{29} - 8467982026112 q^{31} + 3075351773823 q^{32} - 2976680245392 q^{33} - 30424378840142 q^{34} + 18568160056629 q^{36} + 66039873947208 q^{37} + 48043709179344 q^{38} - 22103808863256 q^{39} - 2436672389592 q^{41} + 125473642457814 q^{42} + 99339976668240 q^{43} - 112299091156998 q^{44} - 521196437198212 q^{46} + 398557347904440 q^{47} - 211299103371825 q^{48} - 42787784806880 q^{49} - 320175595505376 q^{51} + 577920818057166 q^{52} + 12\!\cdots\!04 q^{53}+ \cdots + 19\!\cdots\!12 q^{99}+O(q^{100}) $ 8 * q + 189 * q^2 - 52488 * q^3 + 431349 * q^4 - 1240029 * q^6 + 1557468 * q^7 + 16708167 * q^8 + 344373768 * q^9 + 453693072 * q^11 - 2830080789 * q^12 + 3368969496 * q^13 - 19124164374 * q^14 + 32205319825 * q^16 + 48799816416 * q^17 + 8135830269 * q^18 - 22187938864 * q^19 - 10218547548 * q^21 + 787250677122 * q^22 + 502967181024 * q^23 - 109622283687 * q^24 + 1470111900726 * q^26 - 2259436291848 * q^27 - 2737813754934 * q^28 + 2674234302252 * q^29 - 8467982026112 * q^31 + 3075351773823 * q^32 - 2976680245392 * q^33 - 30424378840142 * q^34 + 18568160056629 * q^36 + 66039873947208 * q^37 + 48043709179344 * q^38 - 22103808863256 * q^39 - 2436672389592 * q^41 + 125473642457814 * q^42 + 99339976668240 * q^43 - 112299091156998 * q^44 - 521196437198212 * q^46 + 398557347904440 * q^47 - 211299103371825 * q^48 - 42787784806880 * q^49 - 320175595505376 * q^51 + 577920818057166 * q^52 + 1243174901396904 * q^53 - 53379182394909 * q^54 - 4446984070521330 * q^56 + 145575066886704 * q^57 - 1004924801698272 * q^58 - 1441181478551136 * q^59 - 3170473554101840 * q^61 - 4672982859855624 * q^62 + 67043890462428 * q^63 - 1269415330621439 * q^64 - 5165151692597442 * q^66 - 1714699776339144 * q^67 - 6455437732475622 * q^68 - 3299967674698464 * q^69 - 5869642874949504 * q^71 + 719231803270407 * q^72 + 10539831621925584 * q^73 + 3374513903609586 * q^74 - 16343894379552096 * q^76 - 700658937363744 * q^77 - 9645404180663286 * q^78 - 14327637370181920 * q^79 + 14824161510814728 * q^81 - 55062049219380774 * q^82 - 11585833046184528 * q^83 + 17962796046121974 * q^84 + 130829627044110732 * q^86 - 17545651257075372 * q^87 + 40969320608237742 * q^88 + 59540116605688656 * q^89 + 100956471302068128 * q^91 + 187424284036759308 * q^92 + 55558430073320832 * q^93 + 93327412597056904 * q^94 - 20177382988052703 * q^96 + 199357625739293136 * q^97 + 176696623500937929 * q^98 + 19529999090016912 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	419.421	1.15850	0.579249	−	0.815151i	$-0.303346\pi$
$2$	419.421	1.15850	0.579249	+	0.815151i	$0.303346\pi$
$3$	−6561.00	−0.577350
$3$	−6561.00	−0.577350
$4$	44841.8	0.342115
$5$	0	0
$5$	0	0
$6$	−2.75182e6	−0.668859
$7$	1.07280e7	0.703372	0.351686	−	0.936118i	$-0.385609\pi$
$7$	1.07280e7	0.703372	0.351686	+	0.936118i	$0.385609\pi$
$8$	−3.61668e7	−0.762157
$9$	4.30467e7	0.333333
$10$	0	0
$11$	−3.34405e8	−0.470365	−0.235182	−	0.971951i	$-0.575569\pi$
$11$	−3.34405e8	−0.470365	−0.235182	+	0.971951i	$0.575569\pi$
$12$	−2.94207e8	−0.197520
$13$	1.71011e8	0.0581442	0.0290721	−	0.999577i	$-0.490745\pi$
$13$	1.71011e8	0.0581442	0.0290721	+	0.999577i	$0.490745\pi$
$14$	4.49954e9	0.814854
$15$	0	0
$16$	−2.10466e10	−1.22507
$17$	−3.73698e9	−0.129929	−0.0649643	−	0.997888i	$-0.520693\pi$
$17$	−3.73698e9	−0.129929	−0.0649643	+	0.997888i	$0.520693\pi$
$18$	1.80547e10	0.386166
$19$	3.46862e10	0.468545	0.234272	−	0.972171i	$-0.424729\pi$
$19$	3.46862e10	0.468545	0.234272	+	0.972171i	$0.424729\pi$
$20$	0	0
$21$	−7.03863e10	−0.406092
$22$	−1.40256e11	−0.544916
$23$	−5.95580e10	−0.158582	−0.0792909	−	0.996852i	$-0.525266\pi$
$23$	−5.95580e10	−0.158582	−0.0792909	+	0.996852i	$0.525266\pi$
$24$	2.37290e11	0.440032
$25$	0	0
$26$	7.17257e10	0.0673599
$27$	−2.82430e11	−0.192450
$28$	4.81062e11	0.240634
$29$	−2.08236e12	−0.772987	−0.386494	−	0.922292i	$-0.626314\pi$
$29$	−2.08236e12	−0.772987	−0.386494	+	0.922292i	$0.626314\pi$
$30$	0	0
$31$	5.85106e12	1.23214	0.616070	−	0.787692i	$-0.288724\pi$
$31$	5.85106e12	1.23214	0.616070	+	0.787692i	$0.288724\pi$
$32$	−4.08693e12	−0.657086
$33$	2.19403e12	0.271565
$34$	−1.56737e12	−0.150522
$35$	0	0
$36$	1.93029e12	0.114038
$37$	−2.29506e13	−1.07419	−0.537093	−	0.843523i	$-0.680478\pi$
$37$	−2.29506e13	−1.07419	−0.537093	+	0.843523i	$0.680478\pi$
$38$	1.45481e13	0.542808
$39$	−1.12201e12	−0.0335696
$40$	0	0
$41$	−3.39080e13	−0.663193	−0.331596	−	0.943421i	$-0.607587\pi$
$41$	−3.39080e13	−0.663193	−0.331596	+	0.943421i	$0.607587\pi$
$42$	−2.95215e13	−0.470456
$43$	7.65153e13	0.998312	0.499156	−	0.866512i	$-0.333643\pi$
$43$	7.65153e13	0.998312	0.499156	+	0.866512i	$0.333643\pi$
$44$	−1.49953e13	−0.160919
$45$	0	0
$46$	−2.49798e13	−0.183716
$47$	2.72333e14	1.66828	0.834138	−	0.551556i	$-0.185966\pi$
$47$	2.72333e14	1.66828	0.834138	+	0.551556i	$0.185966\pi$
$48$	1.38087e14	0.707296
$49$	−1.17541e14	−0.505268
$50$	0	0
$51$	2.45183e13	0.0750144
$52$	7.66845e12	0.0198920
$53$	−4.91865e14	−1.08518	−0.542589	−	0.839998i	$-0.682556\pi$
$53$	−4.91865e14	−1.08518	−0.542589	+	0.839998i	$0.682556\pi$
$54$	−1.18457e14	−0.222953
$55$	0	0
$56$	−3.87996e14	−0.536080
$57$	−2.27576e14	−0.270514
$58$	−8.73384e14	−0.895503
$59$	2.15693e15	1.91247	0.956234	−	0.292604i	$-0.0945219\pi$
$59$	2.15693e15	1.91247	0.956234	+	0.292604i	$0.0945219\pi$
$60$	0	0
$61$	−1.43236e15	−0.956637	−0.478319	−	0.878186i	$-0.658754\pi$
$61$	−1.43236e15	−0.956637	−0.478319	+	0.878186i	$0.658754\pi$
$62$	2.45405e15	1.42743
$63$	4.61805e14	0.234457
$64$	1.04448e15	0.463841
$65$	0	0
$66$	9.20222e14	0.314607
$67$	−3.26290e15	−0.981676	−0.490838	−	0.871251i	$-0.663309\pi$
$67$	−3.26290e15	−0.981676	−0.490838	+	0.871251i	$0.663309\pi$
$68$	−1.67573e14	−0.0444506
$69$	3.90760e14	0.0915572
$70$	0	0
$71$	1.54056e15	0.283128	0.141564	−	0.989929i	$-0.454787\pi$
$71$	1.54056e15	0.283128	0.141564	+	0.989929i	$0.454787\pi$
$72$	−1.55686e15	−0.254052
$73$	−1.33066e15	−0.193118	−0.0965591	−	0.995327i	$-0.530784\pi$
$73$	−1.33066e15	−0.193118	−0.0965591	+	0.995327i	$0.530784\pi$
$74$	−9.62597e15	−1.24444
$75$	0	0
$76$	1.55539e15	0.160296
$77$	−3.58749e15	−0.330841
$78$	−4.70592e14	−0.0388903
$79$	2.64034e16	1.95808	0.979039	−	0.203672i	$-0.0652877\pi$
$79$	2.64034e16	1.95808	0.979039	+	0.203672i	$0.0652877\pi$
$80$	0	0
$81$	1.85302e15	0.111111
$82$	−1.42217e16	−0.768307
$83$	3.48504e16	1.69842	0.849209	−	0.528057i	$-0.177079\pi$
$83$	3.48504e16	1.69842	0.849209	+	0.528057i	$0.177079\pi$
$84$	−3.15625e15	−0.138930
$85$	0	0
$86$	3.20921e16	1.15654
$87$	1.36623e16	0.446284
$88$	1.20943e16	0.358492
$89$	6.75463e16	1.81881	0.909403	−	0.415917i	$-0.136539\pi$
$89$	6.75463e16	1.81881	0.909403	+	0.415917i	$0.136539\pi$
$90$	0	0
$91$	1.83461e15	0.0408970
$92$	−2.67068e15	−0.0542533
$93$	−3.83888e16	−0.711376
$94$	1.14222e17	1.93269
$95$	0	0
$96$	2.68143e16	0.379369
$97$	7.30588e16	0.946482	0.473241	−	0.880933i	$-0.343084\pi$
$97$	7.30588e16	0.946482	0.473241	+	0.880933i	$0.343084\pi$
$98$	−4.92991e16	−0.585352
$99$	−1.43950e16	−0.156788
$100$	0	0
$101$	1.78572e17	1.64090	0.820448	−	0.571722i	$-0.193724\pi$
$101$	1.78572e17	1.64090	0.820448	+	0.571722i	$0.193724\pi$
$102$	1.02835e16	0.0869039
$103$	1.71813e16	0.133641	0.0668205	−	0.997765i	$-0.478715\pi$
$103$	1.71813e16	0.133641	0.0668205	+	0.997765i	$0.478715\pi$
$104$	−6.18493e15	−0.0443150
$105$	0	0
$106$	−2.06298e17	−1.25718
$107$	2.34539e17	1.31963	0.659816	−	0.751427i	$-0.270634\pi$
$107$	2.34539e17	1.31963	0.659816	+	0.751427i	$0.270634\pi$
$108$	−1.26646e16	−0.0658402
$109$	1.61438e17	0.776032	0.388016	−	0.921653i	$-0.373161\pi$
$109$	1.61438e17	0.776032	0.388016	+	0.921653i	$0.373161\pi$
$110$	0	0
$111$	1.50579e17	0.620182
$112$	−2.25787e17	−0.861681
$113$	−2.96020e17	−1.04750	−0.523751	−	0.851872i	$-0.675468\pi$
$113$	−2.96020e17	−1.04750	−0.523751	+	0.851872i	$0.675468\pi$
$114$	−9.54501e16	−0.313390
$115$	0	0
$116$	−9.33766e16	−0.264451
$117$	7.36148e15	0.0193814
$118$	9.04661e17	2.21559
$119$	−4.00903e16	−0.0913882
$120$	0	0
$121$	−3.93620e17	−0.778757
$122$	−6.00760e17	−1.10826
$123$	2.22471e17	0.382895
$124$	2.62372e17	0.421534
$125$	0	0
$126$	1.93690e17	0.271618
$127$	1.16061e17	0.152180	0.0760898	−	0.997101i	$-0.475756\pi$
$127$	1.16061e17	0.152180	0.0760898	+	0.997101i	$0.475756\pi$
$128$	9.73757e17	1.19444
$129$	−5.02017e17	−0.576376
$130$	0	0
$131$	3.06668e17	0.308932	0.154466	−	0.987998i	$-0.450634\pi$
$131$	3.06668e17	0.308932	0.154466	+	0.987998i	$0.450634\pi$
$132$	9.83842e16	0.0929066
$133$	3.72113e17	0.329561
$134$	−1.36853e18	−1.13727
$135$	0	0
$136$	1.35154e17	0.0990261
$137$	1.12091e18	0.771695	0.385848	−	0.922563i	$-0.373909\pi$
$137$	1.12091e18	0.771695	0.385848	+	0.922563i	$0.373909\pi$
$138$	1.63893e17	0.106069
$139$	−1.86832e18	−1.13717	−0.568585	−	0.822624i	$-0.692509\pi$
$139$	−1.86832e18	−1.13717	−0.568585	+	0.822624i	$0.692509\pi$
$140$	0	0
$141$	−1.78677e18	−0.963179
$142$	6.46142e17	0.328002
$143$	−5.71870e16	−0.0273490
$144$	−9.05986e17	−0.408357
$145$	0	0
$146$	−5.58107e17	−0.223727
$147$	7.71185e17	0.291717
$148$	−1.02915e18	−0.367496
$149$	−3.24828e17	−0.109539	−0.0547696	−	0.998499i	$-0.517442\pi$
$149$	−3.24828e17	−0.109539	−0.0547696	+	0.998499i	$0.517442\pi$
$150$	0	0
$151$	−5.29857e18	−1.59535	−0.797673	−	0.603090i	$-0.793936\pi$
$151$	−5.29857e18	−1.59535	−0.797673	+	0.603090i	$0.793936\pi$
$152$	−1.25449e18	−0.357105
$153$	−1.60865e17	−0.0433096
$154$	−1.50467e18	−0.383279
$155$	0	0
$156$	−5.03127e16	−0.0114847
$157$	6.65127e18	1.43800	0.718998	−	0.695012i	$-0.244601\pi$
$157$	6.65127e18	1.43800	0.718998	+	0.695012i	$0.244601\pi$
$158$	1.10741e19	2.26843
$159$	3.22713e18	0.626528
$160$	0	0
$161$	−6.38937e17	−0.111542
$162$	7.77195e17	0.128722
$163$	2.57771e18	0.405172	0.202586	−	0.979265i	$-0.435065\pi$
$163$	2.57771e18	0.405172	0.202586	+	0.979265i	$0.435065\pi$
$164$	−1.52050e18	−0.226889
$165$	0	0
$166$	1.46170e19	1.96761
$167$	5.28879e18	0.676498	0.338249	−	0.941057i	$-0.390165\pi$
$167$	5.28879e18	0.676498	0.338249	+	0.941057i	$0.390165\pi$
$168$	2.54564e18	0.309506
$169$	−8.62117e18	−0.996619
$170$	0	0
$171$	1.49313e18	0.156182
$172$	3.43108e18	0.341538
$173$	5.73330e18	0.543266	0.271633	−	0.962401i	$-0.412436\pi$
$173$	5.73330e18	0.543266	0.271633	+	0.962401i	$0.412436\pi$
$174$	5.73027e18	0.517019
$175$	0	0
$176$	7.03808e18	0.576231
$177$	−1.41516e19	−1.10416
$178$	2.83303e19	2.10708
$179$	1.50499e19	1.06729	0.533645	−	0.845709i	$-0.320822\pi$
$179$	1.50499e19	1.06729	0.533645	+	0.845709i	$0.320822\pi$
$180$	0	0
$181$	1.29917e19	0.838296	0.419148	−	0.907918i	$-0.362329\pi$
$181$	1.29917e19	0.838296	0.419148	+	0.907918i	$0.362329\pi$
$182$	7.69473e17	0.0473791
$183$	9.39769e18	0.552315
$184$	2.15402e18	0.120864
$185$	0	0
$186$	−1.61011e19	−0.824127
$187$	1.24966e18	0.0611139
$188$	1.22119e19	0.570743
$189$	−3.02990e18	−0.135364
$190$	0	0
$191$	−2.59643e19	−1.06070	−0.530350	−	0.847779i	$-0.677939\pi$
$191$	−2.59643e19	−1.06070	−0.530350	+	0.847779i	$0.677939\pi$
$192$	−6.85281e18	−0.267799
$193$	2.61210e19	0.976681	0.488341	−	0.872653i	$-0.337602\pi$
$193$	2.61210e19	0.976681	0.488341	+	0.872653i	$0.337602\pi$
$194$	3.06424e19	1.09650
$195$	0	0
$196$	−5.27074e18	−0.172860
$197$	7.78588e18	0.244537	0.122268	−	0.992497i	$-0.460983\pi$
$197$	7.78588e18	0.244537	0.122268	+	0.992497i	$0.460983\pi$
$198$	−6.03757e18	−0.181639
$199$	6.77156e18	0.195181	0.0975905	−	0.995227i	$-0.468886\pi$
$199$	6.77156e18	0.195181	0.0975905	+	0.995227i	$0.468886\pi$
$200$	0	0
$201$	2.14079e19	0.566771
$202$	7.48967e19	1.90097
$203$	−2.23395e19	−0.543697
$204$	1.09945e18	0.0256636
$205$	0	0
$206$	7.20620e18	0.154823
$207$	−2.56377e18	−0.0528606
$208$	−3.59921e18	−0.0712309
$209$	−1.15992e19	−0.220387
$210$	0	0
$211$	2.18258e19	0.382445	0.191222	−	0.981547i	$-0.438755\pi$
$211$	2.18258e19	0.382445	0.191222	+	0.981547i	$0.438755\pi$
$212$	−2.20561e19	−0.371256
$213$	−1.01076e19	−0.163464
$214$	9.83705e19	1.52879
$215$	0	0
$216$	1.02146e19	0.146677
$217$	6.27701e19	0.866652
$218$	6.77103e19	0.899031
$219$	8.73047e18	0.111497
$220$	0	0
$221$	−6.39066e17	−0.00755460
$222$	6.31560e19	0.718479
$223$	1.29260e20	1.41538	0.707691	−	0.706522i	$-0.249737\pi$
$223$	1.29260e20	1.41538	0.707691	+	0.706522i	$0.249737\pi$
$224$	−4.38445e19	−0.462175
$225$	0	0
$226$	−1.24157e20	−1.21353
$227$	−8.45139e19	−0.795625	−0.397813	−	0.917467i	$-0.630231\pi$
$227$	−8.45139e19	−0.795625	−0.397813	+	0.917467i	$0.630231\pi$
$228$	−1.02049e19	−0.0925471
$229$	−1.99781e19	−0.174564	−0.0872818	−	0.996184i	$-0.527818\pi$
$229$	−1.99781e19	−0.174564	−0.0872818	+	0.996184i	$0.527818\pi$
$230$	0	0
$231$	2.35375e19	0.191011
$232$	7.53121e19	0.589138
$233$	1.10729e20	0.835096	0.417548	−	0.908655i	$-0.362890\pi$
$233$	1.10729e20	0.835096	0.417548	+	0.908655i	$0.362890\pi$
$234$	3.08756e18	0.0224533
$235$	0	0
$236$	9.67205e19	0.654285
$237$	−1.73233e20	−1.13050
$238$	−1.68147e19	−0.105873
$239$	−2.16598e20	−1.31605	−0.658024	−	0.752997i	$-0.728608\pi$
$239$	−2.16598e20	−1.31605	−0.658024	+	0.752997i	$0.728608\pi$
$240$	0	0
$241$	−1.37231e20	−0.776799	−0.388400	−	0.921491i	$-0.626972\pi$
$241$	−1.37231e20	−0.776799	−0.388400	+	0.921491i	$0.626972\pi$
$242$	−1.65093e20	−0.902188
$243$	−1.21577e19	−0.0641500
$244$	−6.42294e19	−0.327280
$245$	0	0
$246$	9.33088e19	0.443582
$247$	5.93173e18	0.0272432
$248$	−2.11614e20	−0.939084
$249$	−2.28654e20	−0.980582
$250$	0	0
$251$	−6.71142e19	−0.268898	−0.134449	−	0.990921i	$-0.542926\pi$
$251$	−6.71142e19	−0.268898	−0.134449	+	0.990921i	$0.542926\pi$
$252$	2.07081e19	0.0802114
$253$	1.99165e19	0.0745913
$254$	4.86786e19	0.176300
$255$	0	0
$256$	2.71512e20	0.919919
$257$	8.17440e19	0.267932	0.133966	−	0.990986i	$-0.457229\pi$
$257$	8.17440e19	0.267932	0.133966	+	0.990986i	$0.457229\pi$
$258$	−2.10556e20	−0.667729
$259$	−2.46214e20	−0.755553
$260$	0	0
$261$	−8.96387e19	−0.257662
$262$	1.28623e20	0.357896
$263$	−5.79577e20	−1.56130	−0.780652	−	0.624966i	$-0.785113\pi$
$263$	−5.79577e20	−1.56130	−0.780652	+	0.624966i	$0.785113\pi$
$264$	−7.93509e19	−0.206975
$265$	0	0
$266$	1.56072e20	0.381795
$267$	−4.43171e20	−1.05009
$268$	−1.46314e20	−0.335847
$269$	1.64818e20	0.366530	0.183265	−	0.983064i	$-0.441333\pi$
$269$	1.64818e20	0.366530	0.183265	+	0.983064i	$0.441333\pi$
$270$	0	0
$271$	−3.20470e20	−0.669187	−0.334594	−	0.942362i	$-0.608599\pi$
$271$	−3.20470e20	−0.669187	−0.334594	+	0.942362i	$0.608599\pi$
$272$	7.86507e19	0.159172
$273$	−1.20369e19	−0.0236119
$274$	4.70133e20	0.894007
$275$	0	0
$276$	1.75224e19	0.0313231
$277$	8.56736e20	1.48514	0.742572	−	0.669766i	$-0.233606\pi$
$277$	8.56736e20	1.48514	0.742572	+	0.669766i	$0.233606\pi$
$278$	−7.83613e20	−1.31741
$279$	2.51869e20	0.410713
$280$	0	0
$281$	−7.81806e20	−1.19976	−0.599881	−	0.800089i	$-0.704786\pi$
$281$	−7.81806e20	−1.19976	−0.599881	+	0.800089i	$0.704786\pi$
$282$	−7.49410e20	−1.11584
$283$	7.37354e18	0.0106535	0.00532674	−	0.999986i	$-0.498304\pi$
$283$	7.37354e18	0.0106535	0.00532674	+	0.999986i	$0.498304\pi$
$284$	6.90813e19	0.0968623
$285$	0	0
$286$	−2.39854e19	−0.0316837
$287$	−3.63765e20	−0.466471
$288$	−1.75929e20	−0.219029
$289$	−8.13275e20	−0.983119
$290$	0	0
$291$	−4.79338e20	−0.546452
$292$	−5.96692e19	−0.0660687
$293$	2.80055e20	0.301209	0.150605	−	0.988594i	$-0.451878\pi$
$293$	2.80055e20	0.301209	0.150605	+	0.988594i	$0.451878\pi$
$294$	3.23451e20	0.337953
$295$	0	0
$296$	8.30049e20	0.818699
$297$	9.44458e19	0.0905217
$298$	−1.36239e20	−0.126901
$299$	−1.01851e19	−0.00922061
$300$	0	0
$301$	8.20855e20	0.702184
$302$	−2.22233e21	−1.84820
$303$	−1.17161e21	−0.947371
$304$	−7.30026e20	−0.574001
$305$	0	0
$306$	−6.74700e19	−0.0501740
$307$	2.37978e21	1.72132	0.860660	−	0.509180i	$-0.170051\pi$
$307$	2.37978e21	1.72132	0.860660	+	0.509180i	$0.170051\pi$
$308$	−1.60869e20	−0.113186
$309$	−1.12727e20	−0.0771577
$310$	0	0
$311$	5.06135e20	0.327947	0.163973	−	0.986465i	$-0.447569\pi$
$311$	5.06135e20	0.327947	0.163973	+	0.986465i	$0.447569\pi$
$312$	4.05793e19	0.0255853
$313$	−2.02180e21	−1.24054	−0.620271	−	0.784388i	$-0.712977\pi$
$313$	−2.02180e21	−1.24054	−0.620271	+	0.784388i	$0.712977\pi$
$314$	2.78968e21	1.66591
$315$	0	0
$316$	1.18398e21	0.669889
$317$	−2.79743e21	−1.54083	−0.770415	−	0.637542i	$-0.779951\pi$
$317$	−2.79743e21	−1.54083	−0.770415	+	0.637542i	$0.779951\pi$
$318$	1.35352e21	0.725831
$319$	6.96350e20	0.363586
$320$	0	0
$321$	−1.53881e21	−0.761890
$322$	−2.67983e20	−0.129221
$323$	−1.29622e20	−0.0608774
$324$	8.30927e19	0.0380128
$325$	0	0
$326$	1.08114e21	0.469390
$327$	−1.05919e21	−0.448042
$328$	1.22634e21	0.505457
$329$	2.92158e21	1.17342
$330$	0	0
$331$	−1.11907e21	−0.426894	−0.213447	−	0.976955i	$-0.568469\pi$
$331$	−1.11907e21	−0.426894	−0.213447	+	0.976955i	$0.568469\pi$
$332$	1.56276e21	0.581055
$333$	−9.87949e20	−0.358062
$334$	2.21823e21	0.783721
$335$	0	0
$336$	1.48139e21	0.497492
$337$	3.91312e20	0.128135	0.0640677	−	0.997946i	$-0.479593\pi$
$337$	3.91312e20	0.128135	0.0640677	+	0.997946i	$0.479593\pi$
$338$	−3.61590e21	−1.15458
$339$	1.94219e21	0.604775
$340$	0	0
$341$	−1.95662e21	−0.579555
$342$	6.26248e20	0.180936
$343$	−3.75663e21	−1.05876
$344$	−2.76731e21	−0.760871
$345$	0	0
$346$	2.40466e21	0.629372
$347$	2.57931e21	0.658724	0.329362	−	0.944204i	$-0.393166\pi$
$347$	2.57931e21	0.658724	0.329362	+	0.944204i	$0.393166\pi$
$348$	6.12644e20	0.152681
$349$	−3.74101e21	−0.909856	−0.454928	−	0.890528i	$-0.650335\pi$
$349$	−3.74101e21	−0.909856	−0.454928	+	0.890528i	$0.650335\pi$
$350$	0	0
$351$	−4.82987e19	−0.0111899
$352$	1.36669e21	0.309070
$353$	2.90176e21	0.640585	0.320292	−	0.947319i	$-0.396219\pi$
$353$	2.90176e21	0.640585	0.320292	+	0.947319i	$0.396219\pi$
$354$	−5.93548e21	−1.27917
$355$	0	0
$356$	3.02889e21	0.622241
$357$	2.63032e20	0.0527630
$358$	6.31223e21	1.23645
$359$	−1.30144e21	−0.248955	−0.124477	−	0.992222i	$-0.539725\pi$
$359$	−1.30144e21	−0.248955	−0.124477	+	0.992222i	$0.539725\pi$
$360$	0	0
$361$	−4.27726e21	−0.780466
$362$	5.44898e21	0.971164
$363$	2.58254e21	0.449616
$364$	8.22670e19	0.0139915
$365$	0	0
$366$	3.94158e21	0.639855
$367$	−3.71583e21	−0.589379	−0.294689	−	0.955593i	$-0.595216\pi$
$367$	−3.71583e21	−0.589379	−0.294689	+	0.955593i	$0.595216\pi$
$368$	1.25349e21	0.194274
$369$	−1.45963e21	−0.221064
$370$	0	0
$371$	−5.27672e21	−0.763284
$372$	−1.72142e21	−0.243373
$373$	1.11016e22	1.53412	0.767060	−	0.641576i	$-0.221719\pi$
$373$	1.11016e22	1.53412	0.767060	+	0.641576i	$0.221719\pi$
$374$	5.24135e20	0.0708002
$375$	0	0
$376$	−9.84938e21	−1.27149
$377$	−3.56107e20	−0.0449447
$378$	−1.27080e21	−0.156819
$379$	−9.29176e20	−0.112115	−0.0560576	−	0.998428i	$-0.517853\pi$
$379$	−9.29176e20	−0.112115	−0.0560576	+	0.998428i	$0.517853\pi$
$380$	0	0
$381$	−7.61479e20	−0.0878609
$382$	−1.08900e22	−1.22882
$383$	9.75771e21	1.07686	0.538429	−	0.842671i	$-0.319018\pi$
$383$	9.75771e21	1.07686	0.538429	+	0.842671i	$0.319018\pi$
$384$	−6.38882e21	−0.689612
$385$	0	0
$386$	1.09557e22	1.13148
$387$	3.29373e21	0.332771
$388$	3.27608e21	0.323806
$389$	−9.36571e21	−0.905668	−0.452834	−	0.891595i	$-0.649587\pi$
$389$	−9.36571e21	−0.905668	−0.452834	+	0.891595i	$0.649587\pi$
$390$	0	0
$391$	2.22567e20	0.0206043
$392$	4.25107e21	0.385094
$393$	−2.01205e21	−0.178362
$394$	3.26556e21	0.283295
$395$	0	0
$396$	−6.45499e20	−0.0536397
$397$	4.74176e21	0.385674	0.192837	−	0.981231i	$-0.438231\pi$
$397$	4.74176e21	0.385674	0.192837	+	0.981231i	$0.438231\pi$
$398$	2.84013e21	0.226117
$399$	−2.44143e21	−0.190272
$400$	0	0
$401$	−5.99579e21	−0.447836	−0.223918	−	0.974608i	$-0.571885\pi$
$401$	−5.99579e21	−0.447836	−0.223918	+	0.974608i	$0.571885\pi$
$402$	8.97892e21	0.656603
$403$	1.00060e21	0.0716418
$404$	8.00747e21	0.561376
$405$	0	0
$406$	−9.36965e21	−0.629872
$407$	7.67480e21	0.505260
$408$	−8.86748e20	−0.0571727
$409$	−2.55749e22	−1.61498	−0.807489	−	0.589883i	$-0.799174\pi$
$409$	−2.55749e22	−1.61498	−0.807489	+	0.589883i	$0.799174\pi$
$410$	0	0
$411$	−7.35429e21	−0.445538
$412$	7.70440e20	0.0457207
$413$	2.31395e22	1.34518
$414$	−1.07530e21	−0.0612388
$415$	0	0
$416$	−6.98911e20	−0.0382057
$417$	1.22581e22	0.656546
$418$	−4.86496e21	−0.255318
$419$	2.84831e22	1.46477	0.732383	−	0.680893i	$-0.238408\pi$
$419$	2.84831e22	1.46477	0.732383	+	0.680893i	$0.238408\pi$
$420$	0	0
$421$	1.41976e22	0.701158	0.350579	−	0.936533i	$-0.385985\pi$
$421$	1.41976e22	0.701158	0.350579	+	0.936533i	$0.385985\pi$
$422$	9.15418e21	0.443061
$423$	1.17230e22	0.556092
$424$	1.77892e22	0.827076
$425$	0	0
$426$	−4.23934e21	−0.189372
$427$	−1.53663e22	−0.672872
$428$	1.05171e22	0.451467
$429$	3.75204e20	0.0157899
$430$	0	0
$431$	−4.51333e21	−0.182575	−0.0912873	−	0.995825i	$-0.529098\pi$
$431$	−4.51333e21	−0.182575	−0.0912873	+	0.995825i	$0.529098\pi$
$432$	5.94418e21	0.235765
$433$	−2.65630e22	−1.03307	−0.516536	−	0.856265i	$-0.672779\pi$
$433$	−2.65630e22	−1.03307	−0.516536	+	0.856265i	$0.672779\pi$
$434$	2.63271e22	1.00401
$435$	0	0
$436$	7.23915e21	0.265492
$437$	−2.06584e21	−0.0743026
$438$	3.66174e21	0.129169
$439$	1.24991e21	0.0432446	0.0216223	−	0.999766i	$-0.493117\pi$
$439$	1.24991e21	0.0432446	0.0216223	+	0.999766i	$0.493117\pi$
$440$	0	0
$441$	−5.05975e21	−0.168423
$442$	−2.68038e20	−0.00875198
$443$	−2.13227e22	−0.682984	−0.341492	−	0.939885i	$-0.610932\pi$
$443$	−2.13227e22	−0.682984	−0.341492	+	0.939885i	$0.610932\pi$
$444$	6.75223e21	0.212174
$445$	0	0
$446$	5.42145e22	1.63972
$447$	2.13119e21	0.0632425
$448$	1.12051e22	0.326252
$449$	4.98797e22	1.42505	0.712525	−	0.701647i	$-0.247552\pi$
$449$	4.98797e22	1.42505	0.712525	+	0.701647i	$0.247552\pi$
$450$	0	0
$451$	1.13390e22	0.311942
$452$	−1.32741e22	−0.358366
$453$	3.47639e22	0.921073
$454$	−3.54469e22	−0.921729
$455$	0	0
$456$	8.23069e21	0.206174
$457$	−5.82003e22	−1.43099	−0.715497	−	0.698616i	$-0.753800\pi$
$457$	−5.82003e22	−1.43099	−0.715497	+	0.698616i	$0.753800\pi$
$458$	−8.37925e21	−0.202231
$459$	1.05543e21	0.0250048
$460$	0	0
$461$	1.16830e22	0.266744	0.133372	−	0.991066i	$-0.457419\pi$
$461$	1.16830e22	0.266744	0.133372	+	0.991066i	$0.457419\pi$
$462$	9.87213e21	0.221286
$463$	−7.28447e21	−0.160310	−0.0801548	−	0.996782i	$-0.525541\pi$
$463$	−7.28447e21	−0.160310	−0.0801548	+	0.996782i	$0.525541\pi$
$464$	4.38265e22	0.946965
$465$	0	0
$466$	4.64421e22	0.967457
$467$	6.85242e22	1.40169	0.700843	−	0.713316i	$-0.252807\pi$
$467$	6.85242e22	1.40169	0.700843	+	0.713316i	$0.252807\pi$
$468$	3.30102e20	0.00663068
$469$	−3.50044e22	−0.690483
$470$	0	0
$471$	−4.36390e22	−0.830227
$472$	−7.80092e22	−1.45760
$473$	−2.55871e22	−0.469571
$474$	−7.26575e22	−1.30968
$475$	0	0
$476$	−1.79772e21	−0.0312653
$477$	−2.11732e22	−0.361726
$478$	−9.08456e22	−1.52464
$479$	1.05339e23	1.73676	0.868379	−	0.495901i	$-0.165162\pi$
$479$	1.05339e23	1.73676	0.868379	+	0.495901i	$0.165162\pi$
$480$	0	0
$481$	−3.92482e21	−0.0624577
$482$	−5.75577e22	−0.899920
$483$	4.19207e21	0.0643988
$484$	−1.76506e22	−0.266425
$485$	0	0
$486$	−5.09918e21	−0.0743176
$487$	1.06266e23	1.52195	0.760973	−	0.648783i	$-0.224722\pi$
$487$	1.06266e23	1.52195	0.760973	+	0.648783i	$0.224722\pi$
$488$	5.18037e22	0.729108
$489$	−1.69123e22	−0.233926
$490$	0	0
$491$	−1.66492e22	−0.222434	−0.111217	−	0.993796i	$-0.535475\pi$
$491$	−1.66492e22	−0.222434	−0.111217	+	0.993796i	$0.535475\pi$
$492$	9.97597e21	0.130994
$493$	7.78173e21	0.100433
$494$	2.48789e21	0.0315611
$495$	0	0
$496$	−1.23145e23	−1.50946
$497$	1.65271e22	0.199144
$498$	−9.59021e22	−1.13600
$499$	−8.56565e22	−0.997484	−0.498742	−	0.866751i	$-0.666204\pi$
$499$	−8.56565e22	−0.997484	−0.498742	+	0.866751i	$0.666204\pi$
$500$	0	0
$501$	−3.46998e22	−0.390576
$502$	−2.81491e22	−0.311518
$503$	−1.14040e23	−1.24087	−0.620437	−	0.784256i	$-0.713045\pi$
$503$	−1.14040e23	−1.24087	−0.620437	+	0.784256i	$0.713045\pi$
$504$	−1.67020e22	−0.178693
$505$	0	0
$506$	8.35338e21	0.0864138
$507$	5.65635e22	0.575398
$508$	5.20440e21	0.0520630
$509$	−8.98075e22	−0.883510	−0.441755	−	0.897136i	$-0.645644\pi$
$509$	−8.98075e22	−0.883510	−0.441755	+	0.897136i	$0.645644\pi$
$510$	0	0
$511$	−1.42753e22	−0.135834
$512$	−1.37544e22	−0.128720
$513$	−9.79640e21	−0.0901714
$514$	3.42851e22	0.310398
$515$	0	0
$516$	−2.25113e22	−0.197187
$517$	−9.10693e22	−0.784698
$518$	−1.03267e23	−0.875306
$519$	−3.76162e22	−0.313655
$520$	0	0
$521$	1.07887e23	0.870663	0.435331	−	0.900270i	$-0.356631\pi$
$521$	1.07887e23	0.870663	0.435331	+	0.900270i	$0.356631\pi$
$522$	−3.75963e22	−0.298501
$523$	−1.89182e23	−1.47780	−0.738899	−	0.673816i	$-0.764654\pi$
$523$	−1.89182e23	−1.47780	−0.738899	+	0.673816i	$0.764654\pi$
$524$	1.37515e22	0.105690
$525$	0	0
$526$	−2.43087e23	−1.80877
$527$	−2.18653e22	−0.160090
$528$	−4.61768e22	−0.332687
$529$	−1.37503e23	−0.974852
$530$	0	0
$531$	9.28488e22	0.637489
$532$	1.66862e22	0.112748
$533$	−5.79866e21	−0.0385608
$534$	−1.85875e23	−1.21652
$535$	0	0
$536$	1.18009e23	0.748192
$537$	−9.87423e22	−0.616200
$538$	6.91280e22	0.424624
$539$	3.93062e22	0.237660
$540$	0	0
$541$	2.66057e22	0.155883	0.0779413	−	0.996958i	$-0.475165\pi$
$541$	2.66057e22	0.155883	0.0779413	+	0.996958i	$0.475165\pi$
$542$	−1.34412e23	−0.775252
$543$	−8.52384e22	−0.483991
$544$	1.52728e22	0.0853743
$545$	0	0
$546$	−5.04851e21	−0.0273543
$547$	1.81383e23	0.967617	0.483809	−	0.875174i	$-0.339253\pi$
$547$	1.81383e23	0.967617	0.483809	+	0.875174i	$0.339253\pi$
$548$	5.02636e22	0.264009
$549$	−6.16582e22	−0.318879
$550$	0	0
$551$	−7.22290e22	−0.362179
$552$	−1.41325e22	−0.0697810
$553$	2.83256e23	1.37726
$554$	3.59333e23	1.72054
$555$	0	0
$556$	−8.37788e22	−0.389044
$557$	−3.44839e22	−0.157705	−0.0788527	−	0.996886i	$-0.525126\pi$
$557$	−3.44839e22	−0.157705	−0.0788527	+	0.996886i	$0.525126\pi$
$558$	1.05639e23	0.475810
$559$	1.30850e22	0.0580461
$560$	0	0
$561$	−8.19905e21	−0.0352841
$562$	−3.27906e23	−1.38992
$563$	−1.06909e22	−0.0446367	−0.0223184	−	0.999751i	$-0.507105\pi$
$563$	−1.06909e22	−0.0446367	−0.0223184	+	0.999751i	$0.507105\pi$
$564$	−8.01221e22	−0.329518
$565$	0	0
$566$	3.09261e21	0.0123420
$567$	1.98792e22	0.0781524
$568$	−5.57170e22	−0.215788
$569$	4.26503e23	1.62730	0.813651	−	0.581354i	$-0.197477\pi$
$569$	4.26503e23	1.62730	0.813651	+	0.581354i	$0.197477\pi$
$570$	0	0
$571$	3.00545e23	1.11302	0.556509	−	0.830842i	$-0.312141\pi$
$571$	3.00545e23	1.11302	0.556509	+	0.830842i	$0.312141\pi$
$572$	−2.56437e21	−0.00935651
$573$	1.70352e23	0.612395
$574$	−1.52571e23	−0.540405
$575$	0	0
$576$	4.49613e22	0.154614
$577$	4.83070e23	1.63687	0.818437	−	0.574596i	$-0.194841\pi$
$577$	4.83070e23	1.63687	0.818437	+	0.574596i	$0.194841\pi$
$578$	−3.41105e23	−1.13894
$579$	−1.71380e23	−0.563887
$580$	0	0
$581$	3.73875e23	1.19462
$582$	−2.01044e23	−0.633063
$583$	1.64482e23	0.510429
$584$	4.81257e22	0.147186
$585$	0	0
$586$	1.17461e23	0.348950
$587$	5.06179e22	0.148211	0.0741054	−	0.997250i	$-0.476390\pi$
$587$	5.06179e22	0.148211	0.0741054	+	0.997250i	$0.476390\pi$
$588$	3.45813e22	0.0998008
$589$	2.02951e23	0.577312
$590$	0	0
$591$	−5.10831e22	−0.141183
$592$	4.83032e23	1.31596
$593$	4.53419e23	1.21769	0.608843	−	0.793291i	$-0.291634\pi$
$593$	4.53419e23	1.21769	0.608843	+	0.793291i	$0.291634\pi$
$594$	3.96125e22	0.104869
$595$	0	0
$596$	−1.45658e22	−0.0374751
$597$	−4.44282e22	−0.112688
$598$	−4.27184e21	−0.0106821
$599$	3.08340e23	0.760154	0.380077	−	0.924955i	$-0.375898\pi$
$599$	3.08340e23	0.760154	0.380077	+	0.924955i	$0.375898\pi$
$600$	0	0
$601$	9.69990e22	0.232453	0.116226	−	0.993223i	$-0.462920\pi$
$601$	9.69990e22	0.232453	0.116226	+	0.993223i	$0.462920\pi$
$602$	3.44284e23	0.813479
$603$	−1.40457e23	−0.327225
$604$	−2.37597e23	−0.545792
$605$	0	0
$606$	−4.91397e23	−1.09753
$607$	6.98262e23	1.53785	0.768926	−	0.639338i	$-0.220792\pi$
$607$	6.98262e23	1.53785	0.768926	+	0.639338i	$0.220792\pi$
$608$	−1.41760e23	−0.307874
$609$	1.46569e23	0.313904
$610$	0	0
$611$	4.65720e22	0.0970006
$612$	−7.21346e21	−0.0148169
$613$	4.47675e23	0.906879	0.453439	−	0.891287i	$-0.350197\pi$
$613$	4.47675e23	0.906879	0.453439	+	0.891287i	$0.350197\pi$
$614$	9.98131e23	1.99414
$615$	0	0
$616$	1.29748e23	0.252153
$617$	−7.12716e23	−1.36613	−0.683066	−	0.730357i	$-0.739354\pi$
$617$	−7.12716e23	−1.36613	−0.683066	+	0.730357i	$0.739354\pi$
$618$	−4.72799e22	−0.0893870
$619$	2.79945e23	0.522038	0.261019	−	0.965334i	$-0.415941\pi$
$619$	2.79945e23	0.522038	0.261019	+	0.965334i	$0.415941\pi$
$620$	0	0
$621$	1.68209e22	0.0305191
$622$	2.12284e23	0.379925
$623$	7.24635e23	1.27930
$624$	2.36144e22	0.0411252
$625$	0	0
$626$	−8.47985e23	−1.43716
$627$	7.61025e22	0.127240
$628$	2.98255e23	0.491961
$629$	8.57660e22	0.139568
$630$	0	0
$631$	−5.21339e23	−0.825792	−0.412896	−	0.910778i	$-0.635483\pi$
$631$	−5.21339e23	−0.825792	−0.412896	+	0.910778i	$0.635483\pi$
$632$	−9.54926e23	−1.49236
$633$	−1.43199e23	−0.220804
$634$	−1.17330e24	−1.78505
$635$	0	0
$636$	1.44710e23	0.214345
$637$	−2.01008e22	−0.0293784
$638$	2.92064e23	0.421213
$639$	6.63160e22	0.0943759
$640$	0	0
$641$	8.46920e23	1.17368	0.586839	−	0.809703i	$-0.300372\pi$
$641$	8.46920e23	1.17368	0.586839	+	0.809703i	$0.300372\pi$
$642$	−6.45409e23	−0.882648
$643$	−4.26351e23	−0.575406	−0.287703	−	0.957720i	$-0.592892\pi$
$643$	−4.26351e23	−0.575406	−0.287703	+	0.957720i	$0.592892\pi$
$644$	−2.86511e22	−0.0381602
$645$	0	0
$646$	−5.43660e22	−0.0705263
$647$	1.01754e24	1.30276	0.651381	−	0.758751i	$-0.274190\pi$
$647$	1.01754e24	1.30276	0.651381	+	0.758751i	$0.274190\pi$
$648$	−6.70177e22	−0.0846841
$649$	−7.21288e23	−0.899557
$650$	0	0
$651$	−4.11834e23	−0.500362
$652$	1.15589e23	0.138615
$653$	8.08343e23	0.956827	0.478414	−	0.878135i	$-0.341212\pi$
$653$	8.08343e23	0.956827	0.478414	+	0.878135i	$0.341212\pi$
$654$	−4.44247e23	−0.519056
$655$	0	0
$656$	7.13648e23	0.812459
$657$	−5.72806e22	−0.0643727
$658$	1.22537e24	1.35940
$659$	−3.59315e23	−0.393504	−0.196752	−	0.980453i	$-0.563039\pi$
$659$	−3.59315e23	−0.393504	−0.196752	+	0.980453i	$0.563039\pi$
$660$	0	0
$661$	4.13902e23	0.441759	0.220879	−	0.975301i	$-0.429107\pi$
$661$	4.13902e23	0.441759	0.220879	+	0.975301i	$0.429107\pi$
$662$	−4.69362e23	−0.494555
$663$	4.19291e21	0.00436165
$664$	−1.26043e24	−1.29446
$665$	0	0
$666$	−4.14366e23	−0.414814
$667$	1.24021e23	0.122582
$668$	2.37159e23	0.231440
$669$	−8.48077e23	−0.817172
$670$	0	0
$671$	4.78987e23	0.449968
$672$	2.87664e23	0.266837
$673$	1.16691e24	1.06884	0.534418	−	0.845220i	$-0.320531\pi$
$673$	1.16691e24	1.06884	0.534418	+	0.845220i	$0.320531\pi$
$674$	1.64124e23	0.148444
$675$	0	0
$676$	−3.86588e23	−0.340959
$677$	1.51625e24	1.32059	0.660294	−	0.751007i	$-0.270432\pi$
$677$	1.51625e24	1.32059	0.660294	+	0.751007i	$0.270432\pi$
$678$	8.14594e23	0.700630
$679$	7.83773e23	0.665729
$680$	0	0
$681$	5.54496e23	0.459354
$682$	−8.20648e23	−0.671413
$683$	−1.21958e24	−0.985447	−0.492723	−	0.870186i	$-0.663999\pi$
$683$	−1.21958e24	−0.985447	−0.492723	+	0.870186i	$0.663999\pi$
$684$	6.69544e22	0.0534321
$685$	0	0
$686$	−1.57561e24	−1.22657
$687$	1.31077e23	0.100784
$688$	−1.61039e24	−1.22300
$689$	−8.41145e22	−0.0630968
$690$	0	0
$691$	−1.99985e24	−1.46364	−0.731820	−	0.681498i	$-0.761329\pi$
$691$	−1.99985e24	−1.46364	−0.731820	+	0.681498i	$0.761329\pi$
$692$	2.57091e23	0.185860
$693$	−1.54430e23	−0.110280
$694$	1.08182e24	0.763130
$695$	0	0
$696$	−4.94123e23	−0.340139
$697$	1.26714e23	0.0861678
$698$	−1.56906e24	−1.05407
$699$	−7.26494e23	−0.482143
$700$	0	0
$701$	1.54812e23	0.100277	0.0501387	−	0.998742i	$-0.484034\pi$
$701$	1.54812e23	0.100277	0.0501387	+	0.998742i	$0.484034\pi$
$702$	−2.02575e22	−0.0129634
$703$	−7.96070e23	−0.503304
$704$	−3.49278e23	−0.218174
$705$	0	0
$706$	1.21706e24	0.742116
$707$	1.91571e24	1.15416
$708$	−6.34584e23	−0.377751
$709$	1.48852e24	0.875508	0.437754	−	0.899095i	$-0.355774\pi$
$709$	1.48852e24	0.875508	0.437754	+	0.899095i	$0.355774\pi$
$710$	0	0
$711$	1.13658e24	0.652693
$712$	−2.44293e24	−1.38622
$713$	−3.48477e23	−0.195395
$714$	1.10321e23	0.0611258
$715$	0	0
$716$	6.74863e23	0.365136
$717$	1.42110e24	0.759821
$718$	−5.45850e23	−0.288414
$719$	−7.17786e23	−0.374800	−0.187400	−	0.982284i	$-0.560006\pi$
$719$	−7.17786e23	−0.374800	−0.187400	+	0.982284i	$0.560006\pi$
$720$	0	0
$721$	1.84321e23	0.0939993
$722$	−1.79397e24	−0.904168
$723$	9.00376e23	0.448485
$724$	5.82570e23	0.286794
$725$	0	0
$726$	1.08317e24	0.520878
$727$	−4.29185e23	−0.203987	−0.101993	−	0.994785i	$-0.532522\pi$
$727$	−4.29185e23	−0.203987	−0.101993	+	0.994785i	$0.532522\pi$
$728$	−6.63518e22	−0.0311699
$729$	7.97664e22	0.0370370
$730$	0	0
$731$	−2.85936e23	−0.129709
$732$	4.21409e23	0.188955
$733$	−3.92543e24	−1.73982	−0.869908	−	0.493213i	$-0.835822\pi$
$733$	−3.92543e24	−1.73982	−0.869908	+	0.493213i	$0.835822\pi$
$734$	−1.55850e24	−0.682794
$735$	0	0
$736$	2.43409e23	0.104202
$737$	1.09113e24	0.461746
$738$	−6.12199e23	−0.256102
$739$	2.27046e22	0.00938935	0.00469468	−	0.999989i	$-0.498506\pi$
$739$	2.27046e22	0.00938935	0.00469468	+	0.999989i	$0.498506\pi$
$740$	0	0
$741$	−3.89181e22	−0.0157288
$742$	−2.21317e24	−0.884262
$743$	−4.93045e23	−0.194752	−0.0973759	−	0.995248i	$-0.531045\pi$
$743$	−4.93045e23	−0.194752	−0.0973759	+	0.995248i	$0.531045\pi$
$744$	1.38840e24	0.542181
$745$	0	0
$746$	4.65623e24	1.77727
$747$	1.50020e24	0.566139
$748$	5.60372e22	0.0209080
$749$	2.51613e24	0.928192
$750$	0	0
$751$	−3.69489e24	−1.33248	−0.666242	−	0.745736i	$-0.732098\pi$
$751$	−3.69489e24	−1.33248	−0.666242	+	0.745736i	$0.732098\pi$
$752$	−5.73167e24	−2.04376
$753$	4.40336e23	0.155248
$754$	−1.49359e23	−0.0520683
$755$	0	0
$756$	−1.35866e23	−0.0463101
$757$	−4.49458e24	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$757$	−4.49458e24	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$758$	−3.89716e23	−0.129885
$759$	−1.30672e23	−0.0430653
$760$	0	0
$761$	−3.27781e24	−1.05637	−0.528183	−	0.849130i	$-0.677127\pi$
$761$	−3.27781e24	−1.05637	−0.528183	+	0.849130i	$0.677127\pi$
$762$	−3.19380e23	−0.101787
$763$	1.73190e24	0.545839
$764$	−1.16428e24	−0.362882
$765$	0	0
$766$	4.09259e24	1.24754
$767$	3.68860e23	0.111199
$768$	−1.78139e24	−0.531115
$769$	2.02659e24	0.597574	0.298787	−	0.954320i	$-0.403418\pi$
$769$	2.02659e24	0.597574	0.298787	+	0.954320i	$0.403418\pi$
$770$	0	0
$771$	−5.36323e23	−0.154691
$772$	1.17131e24	0.334138
$773$	1.38279e24	0.390149	0.195075	−	0.980788i	$-0.437505\pi$
$773$	1.38279e24	0.390149	0.195075	+	0.980788i	$0.437505\pi$
$774$	1.38146e24	0.385514
$775$	0	0
$776$	−2.64230e24	−0.721368
$777$	1.61541e24	0.436219
$778$	−3.92817e24	−1.04921
$779$	−1.17614e24	−0.310735
$780$	0	0
$781$	−5.15170e23	−0.133173
$782$	9.33492e22	0.0238700
$783$	5.88119e23	0.148761
$784$	2.47383e24	0.618990
$785$	0	0
$786$	−8.43895e23	−0.206632
$787$	3.07765e23	0.0745476	0.0372738	−	0.999305i	$-0.488133\pi$
$787$	3.07765e23	0.0745476	0.0372738	+	0.999305i	$0.488133\pi$
$788$	3.49132e23	0.0836599
$789$	3.80260e24	0.901419
$790$	0	0
$791$	−3.17570e24	−0.736783
$792$	5.20622e23	0.119497
$793$	−2.44949e23	−0.0556229
$794$	1.98879e24	0.446802
$795$	0	0
$796$	3.03649e23	0.0667745
$797$	8.31053e24	1.80814	0.904072	−	0.427379i	$-0.140563\pi$
$797$	8.31053e24	1.80814	0.904072	+	0.427379i	$0.140563\pi$
$798$	−1.02399e24	−0.220430
$799$	−1.01770e24	−0.216757
$800$	0	0
$801$	2.90764e24	0.606268
$802$	−2.51476e24	−0.518817
$803$	4.44979e23	0.0908360
$804$	9.59968e23	0.193901
$805$	0	0
$806$	4.19671e23	0.0829968
$807$	−1.08137e24	−0.211616
$808$	−6.45836e24	−1.25062
$809$	−7.72799e24	−1.48083	−0.740413	−	0.672152i	$-0.765370\pi$
$809$	−7.72799e24	−1.48083	−0.740413	+	0.672152i	$0.765370\pi$
$810$	0	0
$811$	7.39633e24	1.38784	0.693920	−	0.720052i	$-0.255882\pi$
$811$	7.39633e24	1.38784	0.693920	+	0.720052i	$0.255882\pi$
$812$	−1.00174e24	−0.186007
$813$	2.10260e24	0.386356
$814$	3.21897e24	0.585342
$815$	0	0
$816$	−5.16027e23	−0.0918980
$817$	2.65402e24	0.467754
$818$	−1.07267e25	−1.87095
$819$	7.89739e22	0.0136323
$820$	0	0
$821$	−7.64184e24	−1.29205	−0.646027	−	0.763314i	$-0.723571\pi$
$821$	−7.64184e24	−1.29205	−0.646027	+	0.763314i	$0.723571\pi$
$822$	−3.08454e24	−0.516155
$823$	−1.25140e24	−0.207252	−0.103626	−	0.994616i	$-0.533044\pi$
$823$	−1.25140e24	−0.207252	−0.103626	+	0.994616i	$0.533044\pi$
$824$	−6.21392e23	−0.101856
$825$	0	0
$826$	9.70519e24	1.55838
$827$	4.37351e24	0.695077	0.347539	−	0.937666i	$-0.387018\pi$
$827$	4.37351e24	0.695077	0.347539	+	0.937666i	$0.387018\pi$
$828$	−1.14964e23	−0.0180844
$829$	−7.31119e24	−1.13835	−0.569174	−	0.822217i	$-0.692737\pi$
$829$	−7.31119e24	−1.13835	−0.569174	+	0.822217i	$0.692737\pi$
$830$	0	0
$831$	−5.62104e24	−0.857449
$832$	1.78617e23	0.0269697
$833$	4.39248e23	0.0656488
$834$	5.14128e24	0.760606
$835$	0	0
$836$	−5.20130e23	−0.0753977
$837$	−1.65251e24	−0.237125
$838$	1.19464e25	1.69693
$839$	−7.56155e24	−1.06325	−0.531624	−	0.846981i	$-0.678418\pi$
$839$	−7.56155e24	−1.06325	−0.531624	+	0.846981i	$0.678418\pi$
$840$	0	0
$841$	−2.92094e24	−0.402491
$842$	5.95475e24	0.812289
$843$	5.12943e24	0.692683
$844$	9.78705e23	0.130840
$845$	0	0
$846$	4.91688e24	0.644231
$847$	−4.22276e24	−0.547756
$848$	1.03521e25	1.32942
$849$	−4.83778e22	−0.00615079
$850$	0	0
$851$	1.36689e24	0.170346
$852$	−4.53243e23	−0.0559235
$853$	−3.98595e23	−0.0486929	−0.0243464	−	0.999704i	$-0.507750\pi$
$853$	−3.98595e23	−0.0486929	−0.0243464	+	0.999704i	$0.507750\pi$
$854$	−6.44494e24	−0.779520
$855$	0	0
$856$	−8.48252e24	−1.00577
$857$	−6.59250e24	−0.773950	−0.386975	−	0.922090i	$-0.626480\pi$
$857$	−6.59250e24	−0.773950	−0.386975	+	0.922090i	$0.626480\pi$
$858$	1.57368e23	0.0182926
$859$	−2.27761e24	−0.262143	−0.131071	−	0.991373i	$-0.541842\pi$
$859$	−2.27761e24	−0.262143	−0.131071	+	0.991373i	$0.541842\pi$
$860$	0	0
$861$	2.38666e24	0.269317
$862$	−1.89298e24	−0.211512
$863$	1.31683e25	1.45693	0.728466	−	0.685082i	$-0.240234\pi$
$863$	1.31683e25	1.45693	0.728466	+	0.685082i	$0.240234\pi$
$864$	1.15427e24	0.126456
$865$	0	0
$866$	−1.11411e25	−1.19681
$867$	5.33590e24	0.567604
$868$	2.81472e24	0.296495
$869$	−8.82943e24	−0.921011
$870$	0	0
$871$	−5.57993e23	−0.0570788
$872$	−5.83867e24	−0.591458
$873$	3.14494e24	0.315494
$874$	−8.66455e23	−0.0860794
$875$	0	0
$876$	3.91489e23	0.0381448
$877$	1.30212e25	1.25648	0.628239	−	0.778020i	$-0.283776\pi$
$877$	1.30212e25	1.25648	0.628239	+	0.778020i	$0.283776\pi$
$878$	5.24240e23	0.0500987
$879$	−1.83744e24	−0.173903
$880$	0	0
$881$	1.89740e25	1.76142	0.880712	−	0.473651i	$-0.157064\pi$
$881$	1.89740e25	1.76142	0.880712	+	0.473651i	$0.157064\pi$
$882$	−2.12216e24	−0.195117
$883$	4.83659e24	0.440426	0.220213	−	0.975452i	$-0.429325\pi$
$883$	4.83659e24	0.440426	0.220213	+	0.975452i	$0.429325\pi$
$884$	−2.86569e22	−0.00258455
$885$	0	0
$886$	−8.94319e24	−0.791235
$887$	−1.40113e25	−1.22780	−0.613898	−	0.789385i	$-0.710399\pi$
$887$	−1.40113e25	−1.22780	−0.613898	+	0.789385i	$0.710399\pi$
$888$	−5.44595e24	−0.472676
$889$	1.24511e24	0.107039
$890$	0	0
$891$	−6.19659e23	−0.0522627
$892$	5.79626e24	0.484224
$893$	9.44618e24	0.781661
$894$	8.93867e23	0.0732663
$895$	0	0
$896$	1.04464e25	0.840138
$897$	6.68244e22	0.00532352
$898$	2.09206e25	1.65092
$899$	−1.21840e25	−0.952428
$900$	0	0
$901$	1.83809e24	0.140996
$902$	4.75582e24	0.361384
$903$	−5.38563e24	−0.405406
$904$	1.07061e25	0.798361
$905$	0	0
$906$	1.45807e25	1.06706
$907$	1.00093e25	0.725674	0.362837	−	0.931853i	$-0.381808\pi$
$907$	1.00093e25	0.725674	0.362837	+	0.931853i	$0.381808\pi$
$908$	−3.78975e24	−0.272196
$909$	7.68693e24	0.546965
$910$	0	0
$911$	−2.06992e25	−1.44560	−0.722800	−	0.691057i	$-0.757145\pi$
$911$	−2.06992e25	−1.44560	−0.722800	+	0.691057i	$0.757145\pi$
$912$	4.78970e24	0.331400
$913$	−1.16542e25	−0.798876
$914$	−2.44104e25	−1.65780
$915$	0	0
$916$	−8.95855e23	−0.0597209
$917$	3.28993e24	0.217294
$918$	4.42671e23	0.0289680
$919$	2.83913e25	1.84078	0.920392	−	0.390996i	$-0.127870\pi$
$919$	2.83913e25	1.84078	0.920392	+	0.390996i	$0.127870\pi$
$920$	0	0
$921$	−1.56138e25	−0.993804
$922$	4.90007e24	0.309022
$923$	2.63453e23	0.0164622
$924$	1.05546e24	0.0653479
$925$	0	0
$926$	−3.05526e24	−0.185718
$927$	7.39599e23	0.0445470
$928$	8.51044e24	0.507919
$929$	2.96280e25	1.75214	0.876069	−	0.482186i	$-0.160157\pi$
$929$	2.96280e25	1.75214	0.876069	+	0.482186i	$0.160157\pi$
$930$	0	0
$931$	−4.07704e24	−0.236741
$932$	4.96529e24	0.285699
$933$	−3.32075e24	−0.189340
$934$	2.87405e25	1.62385
$935$	0	0
$936$	−2.66241e23	−0.0147717
$937$	3.01741e25	1.65901	0.829503	−	0.558503i	$-0.188624\pi$
$937$	3.01741e25	1.65901	0.829503	+	0.558503i	$0.188624\pi$
$938$	−1.46816e25	−0.799923
$939$	1.32650e25	0.716227
$940$	0	0
$941$	−2.71509e24	−0.143970	−0.0719851	−	0.997406i	$-0.522933\pi$
$941$	−2.71509e24	−0.143970	−0.0719851	+	0.997406i	$0.522933\pi$
$942$	−1.83031e25	−0.961816
$943$	2.01949e24	0.105170
$944$	−4.53960e25	−2.34291
$945$	0	0
$946$	−1.07318e25	−0.543996
$947$	−1.64526e25	−0.826533	−0.413267	−	0.910610i	$-0.635612\pi$
$947$	−1.64526e25	−0.826533	−0.413267	+	0.910610i	$0.635612\pi$
$948$	−7.76807e24	−0.386761
$949$	−2.27558e23	−0.0112287
$950$	0	0
$951$	1.83539e25	0.889599
$952$	1.44994e24	0.0696522
$953$	−2.20027e25	−1.04758	−0.523789	−	0.851848i	$-0.675482\pi$
$953$	−2.20027e25	−1.04758	−0.523789	+	0.851848i	$0.675482\pi$
$954$	−8.88047e24	−0.419059
$955$	0	0
$956$	−9.71263e24	−0.450241
$957$	−4.56875e24	−0.209916
$958$	4.41816e25	2.01203
$959$	1.20251e25	0.542789
$960$	0	0
$961$	1.16848e25	0.518168
$962$	−1.64615e24	−0.0723571
$963$	1.00961e25	0.439878
$964$	−6.15370e24	−0.265755
$965$	0	0
$966$	1.75824e24	0.0746058
$967$	−1.65769e24	−0.0697235	−0.0348618	−	0.999392i	$-0.511099\pi$
$967$	−1.65769e24	−0.0697235	−0.0348618	+	0.999392i	$0.511099\pi$
$968$	1.42360e25	0.593535
$969$	8.50447e23	0.0351476
$970$	0	0
$971$	1.82700e25	0.741952	0.370976	−	0.928642i	$-0.379023\pi$
$971$	1.82700e25	0.741952	0.370976	+	0.928642i	$0.379023\pi$
$972$	−5.45171e23	−0.0219467
$973$	−2.00433e25	−0.799854
$974$	4.45703e25	1.76317
$975$	0	0
$976$	3.01462e25	1.17195
$977$	3.22266e25	1.24197	0.620984	−	0.783823i	$-0.286733\pi$
$977$	3.22266e25	1.24197	0.620984	+	0.783823i	$0.286733\pi$
$978$	−7.09339e24	−0.271003
$979$	−2.25878e25	−0.855502
$980$	0	0
$981$	6.94936e24	0.258677
$982$	−6.98303e24	−0.257689
$983$	1.74885e25	0.639804	0.319902	−	0.947451i	$-0.396350\pi$
$983$	1.74885e25	0.639804	0.319902	+	0.947451i	$0.396350\pi$
$984$	−8.04604e24	−0.291826
$985$	0	0
$986$	3.26382e24	0.116352
$987$	−1.91685e25	−0.677473
$988$	2.65989e23	0.00932031
$989$	−4.55709e24	−0.158314
$990$	0	0
$991$	2.45273e25	0.837575	0.418787	−	0.908084i	$-0.362455\pi$
$991$	2.45273e25	0.837575	0.418787	+	0.908084i	$0.362455\pi$
$992$	−2.39128e25	−0.809621
$993$	7.34223e24	0.246467
$994$	6.93180e24	0.230708
$995$	0	0
$996$	−1.02532e25	−0.335472
$997$	−2.80715e25	−0.910660	−0.455330	−	0.890323i	$-0.650479\pi$
$997$	−2.80715e25	−0.910660	−0.455330	+	0.890323i	$0.650479\pi$
$998$	−3.59261e25	−1.15558
$999$	6.48193e24	0.206727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.18.a.l.1.7		8
5.2	odd	4		15.18.b.a.4.13	yes	16
5.3	odd	4		15.18.b.a.4.4	✓	16
5.4	even	2		75.18.a.k.1.2		8
15.2	even	4		45.18.b.d.19.4		16
15.8	even	4		45.18.b.d.19.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.18.b.a.4.4	✓	16	5.3	odd	4
15.18.b.a.4.13	yes	16	5.2	odd	4
45.18.b.d.19.4		16	15.2	even	4
45.18.b.d.19.13		16	15.8	even	4
75.18.a.k.1.2		8	5.4	even	2
75.18.a.l.1.7		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+419.421 q^{2} -6561.00 q^{3} +44841.8 q^{4} -2.75182e6 q^{6} +1.07280e7 q^{7} -3.61668e7 q^{8} +4.30467e7 q^{9} +O(q^{10})\)	\(q+419.421 q^{2} -6561.00 q^{3} +44841.8 q^{4} -2.75182e6 q^{6} +1.07280e7 q^{7} -3.61668e7 q^{8} +4.30467e7 q^{9} -3.34405e8 q^{11} -2.94207e8 q^{12} +1.71011e8 q^{13} +4.49954e9 q^{14} -2.10466e10 q^{16} -3.73698e9 q^{17} +1.80547e10 q^{18} +3.46862e10 q^{19} -7.03863e10 q^{21} -1.40256e11 q^{22} -5.95580e10 q^{23} +2.37290e11 q^{24} +7.17257e10 q^{26} -2.82430e11 q^{27} +4.81062e11 q^{28} -2.08236e12 q^{29} +5.85106e12 q^{31} -4.08693e12 q^{32} +2.19403e12 q^{33} -1.56737e12 q^{34} +1.93029e12 q^{36} -2.29506e13 q^{37} +1.45481e13 q^{38} -1.12201e12 q^{39} -3.39080e13 q^{41} -2.95215e13 q^{42} +7.65153e13 q^{43} -1.49953e13 q^{44} -2.49798e13 q^{46} +2.72333e14 q^{47} +1.38087e14 q^{48} -1.17541e14 q^{49} +2.45183e13 q^{51} +7.66845e12 q^{52} -4.91865e14 q^{53} -1.18457e14 q^{54} -3.87996e14 q^{56} -2.27576e14 q^{57} -8.73384e14 q^{58} +2.15693e15 q^{59} -1.43236e15 q^{61} +2.45405e15 q^{62} +4.61805e14 q^{63} +1.04448e15 q^{64} +9.20222e14 q^{66} -3.26290e15 q^{67} -1.67573e14 q^{68} +3.90760e14 q^{69} +1.54056e15 q^{71} -1.55686e15 q^{72} -1.33066e15 q^{73} -9.62597e15 q^{74} +1.55539e15 q^{76} -3.58749e15 q^{77} -4.70592e14 q^{78} +2.64034e16 q^{79} +1.85302e15 q^{81} -1.42217e16 q^{82} +3.48504e16 q^{83} -3.15625e15 q^{84} +3.20921e16 q^{86} +1.36623e16 q^{87} +1.20943e16 q^{88} +6.75463e16 q^{89} +1.83461e15 q^{91} -2.67068e15 q^{92} -3.83888e16 q^{93} +1.14222e17 q^{94} +2.68143e16 q^{96} +7.30588e16 q^{97} -4.92991e16 q^{98} -1.43950e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 189 q^{2} - 52488 q^{3} + 431349 q^{4} - 1240029 q^{6} + 1557468 q^{7} + 16708167 q^{8} + 344373768 q^{9}+O(q^{10}) \) 8 * q + 189 * q^2 - 52488 * q^3 + 431349 * q^4 - 1240029 * q^6 + 1557468 * q^7 + 16708167 * q^8 + 344373768 * q^9	\( 8 q + 189 q^{2} - 52488 q^{3} + 431349 q^{4} - 1240029 q^{6} + 1557468 q^{7} + 16708167 q^{8} + 344373768 q^{9} + 453693072 q^{11} - 2830080789 q^{12} + 3368969496 q^{13} - 19124164374 q^{14} + 32205319825 q^{16} + 48799816416 q^{17} + 8135830269 q^{18} - 22187938864 q^{19} - 10218547548 q^{21} + 787250677122 q^{22} + 502967181024 q^{23} - 109622283687 q^{24} + 1470111900726 q^{26} - 2259436291848 q^{27} - 2737813754934 q^{28} + 2674234302252 q^{29} - 8467982026112 q^{31} + 3075351773823 q^{32} - 2976680245392 q^{33} - 30424378840142 q^{34} + 18568160056629 q^{36} + 66039873947208 q^{37} + 48043709179344 q^{38} - 22103808863256 q^{39} - 2436672389592 q^{41} + 125473642457814 q^{42} + 99339976668240 q^{43} - 112299091156998 q^{44} - 521196437198212 q^{46} + 398557347904440 q^{47} - 211299103371825 q^{48} - 42787784806880 q^{49} - 320175595505376 q^{51} + 577920818057166 q^{52} + 12\!\cdots\!04 q^{53}+ \cdots + 19\!\cdots\!12 q^{99}+O(q^{100}) \) 8 * q + 189 * q^2 - 52488 * q^3 + 431349 * q^4 - 1240029 * q^6 + 1557468 * q^7 + 16708167 * q^8 + 344373768 * q^9 + 453693072 * q^11 - 2830080789 * q^12 + 3368969496 * q^13 - 19124164374 * q^14 + 32205319825 * q^16 + 48799816416 * q^17 + 8135830269 * q^18 - 22187938864 * q^19 - 10218547548 * q^21 + 787250677122 * q^22 + 502967181024 * q^23 - 109622283687 * q^24 + 1470111900726 * q^26 - 2259436291848 * q^27 - 2737813754934 * q^28 + 2674234302252 * q^29 - 8467982026112 * q^31 + 3075351773823 * q^32 - 2976680245392 * q^33 - 30424378840142 * q^34 + 18568160056629 * q^36 + 66039873947208 * q^37 + 48043709179344 * q^38 - 22103808863256 * q^39 - 2436672389592 * q^41 + 125473642457814 * q^42 + 99339976668240 * q^43 - 112299091156998 * q^44 - 521196437198212 * q^46 + 398557347904440 * q^47 - 211299103371825 * q^48 - 42787784806880 * q^49 - 320175595505376 * q^51 + 577920818057166 * q^52 + 1243174901396904 * q^53 - 53379182394909 * q^54 - 4446984070521330 * q^56 + 145575066886704 * q^57 - 1004924801698272 * q^58 - 1441181478551136 * q^59 - 3170473554101840 * q^61 - 4672982859855624 * q^62 + 67043890462428 * q^63 - 1269415330621439 * q^64 - 5165151692597442 * q^66 - 1714699776339144 * q^67 - 6455437732475622 * q^68 - 3299967674698464 * q^69 - 5869642874949504 * q^71 + 719231803270407 * q^72 + 10539831621925584 * q^73 + 3374513903609586 * q^74 - 16343894379552096 * q^76 - 700658937363744 * q^77 - 9645404180663286 * q^78 - 14327637370181920 * q^79 + 14824161510814728 * q^81 - 55062049219380774 * q^82 - 11585833046184528 * q^83 + 17962796046121974 * q^84 + 130829627044110732 * q^86 - 17545651257075372 * q^87 + 40969320608237742 * q^88 + 59540116605688656 * q^89 + 100956471302068128 * q^91 + 187424284036759308 * q^92 + 55558430073320832 * q^93 + 93327412597056904 * q^94 - 20177382988052703 * q^96 + 199357625739293136 * q^97 + 176696623500937929 * q^98 + 19529999090016912 * q^99

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