LMFDB - Embedded newform 192.12.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,12,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	192.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:	$147.521890667$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1945}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 486 $ x^2 - x - 486
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$22.5511$ of defining polynomial
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-243.000 q^{3} -9000.71 q^{5} +33486.6 q^{7} +59049.0 q^{9} +O(q^{10})$	$q-243.000 q^{3} -9000.71 q^{5} +33486.6 q^{7} +59049.0 q^{9} +11409.4 q^{11} -424632. q^{13} +2.18717e6 q^{15} +7.76640e6 q^{17} -7.16619e6 q^{19} -8.13724e6 q^{21} -1.00865e7 q^{23} +3.21847e7 q^{25} -1.43489e7 q^{27} -3.12934e7 q^{29} -5.25994e6 q^{31} -2.77248e6 q^{33} -3.01403e8 q^{35} +5.58731e8 q^{37} +1.03186e8 q^{39} -2.54143e8 q^{41} -1.36088e9 q^{43} -5.31483e8 q^{45} -2.16161e9 q^{47} -8.55975e8 q^{49} -1.88723e9 q^{51} -3.93069e9 q^{53} -1.02693e8 q^{55} +1.74138e9 q^{57} +9.48166e9 q^{59} -9.90753e9 q^{61} +1.97735e9 q^{63} +3.82199e9 q^{65} +9.83448e8 q^{67} +2.45101e9 q^{69} +4.06177e8 q^{71} -5.83691e9 q^{73} -7.82087e9 q^{75} +3.82061e8 q^{77} +1.66139e10 q^{79} +3.48678e9 q^{81} +4.82664e10 q^{83} -6.99031e10 q^{85} +7.60429e9 q^{87} +4.93435e10 q^{89} -1.42195e10 q^{91} +1.27816e9 q^{93} +6.45008e10 q^{95} +1.09762e11 q^{97} +6.73712e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 486 q^{3} - 5300 q^{5} - 38872 q^{7} + 118098 q^{9}+O(q^{10}) $ 2 * q - 486 * q^3 - 5300 * q^5 - 38872 * q^7 + 118098 * q^9	$ 2 q - 486 q^{3} - 5300 q^{5} - 38872 q^{7} + 118098 q^{9} + 1047400 q^{11} + 22900 q^{13} + 1287900 q^{15} + 8733300 q^{17} - 7346600 q^{19} + 9445896 q^{21} + 6711744 q^{23} - 2948210 q^{25} - 28697814 q^{27} - 180180692 q^{29} + 211581400 q^{31} - 254518200 q^{33} - 569181200 q^{35} - 112222700 q^{37} - 5564700 q^{39} - 726961180 q^{41} - 216408856 q^{43} - 312959700 q^{45} - 2174779088 q^{47} + 2402462706 q^{49} - 2122191900 q^{51} + 112024700 q^{53} + 3731208560 q^{55} + 1785223800 q^{57} + 3243949400 q^{59} - 16526230620 q^{61} - 2295352728 q^{63} + 5478177080 q^{65} + 20772619112 q^{67} - 1630953792 q^{69} - 20637101600 q^{71} - 18548203500 q^{73} + 716415030 q^{75} - 74580754400 q^{77} + 28230083800 q^{79} + 6973568802 q^{81} - 7189282056 q^{83} - 66324859080 q^{85} + 43783908156 q^{87} + 103679180788 q^{89} - 46602268400 q^{91} - 51414280200 q^{93} + 63833162000 q^{95} + 199614486500 q^{97} + 61847922600 q^{99}+O(q^{100}) $ 2 * q - 486 * q^3 - 5300 * q^5 - 38872 * q^7 + 118098 * q^9 + 1047400 * q^11 + 22900 * q^13 + 1287900 * q^15 + 8733300 * q^17 - 7346600 * q^19 + 9445896 * q^21 + 6711744 * q^23 - 2948210 * q^25 - 28697814 * q^27 - 180180692 * q^29 + 211581400 * q^31 - 254518200 * q^33 - 569181200 * q^35 - 112222700 * q^37 - 5564700 * q^39 - 726961180 * q^41 - 216408856 * q^43 - 312959700 * q^45 - 2174779088 * q^47 + 2402462706 * q^49 - 2122191900 * q^51 + 112024700 * q^53 + 3731208560 * q^55 + 1785223800 * q^57 + 3243949400 * q^59 - 16526230620 * q^61 - 2295352728 * q^63 + 5478177080 * q^65 + 20772619112 * q^67 - 1630953792 * q^69 - 20637101600 * q^71 - 18548203500 * q^73 + 716415030 * q^75 - 74580754400 * q^77 + 28230083800 * q^79 + 6973568802 * q^81 - 7189282056 * q^83 - 66324859080 * q^85 + 43783908156 * q^87 + 103679180788 * q^89 - 46602268400 * q^91 - 51414280200 * q^93 + 63833162000 * q^95 + 199614486500 * q^97 + 61847922600 * 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 $

$2$	0	0
$2$	0	0
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	0	0
$5$	−9000.71	−1.28808	−0.644038	−	0.764993i	$-0.722742\pi$
$5$	−9000.71	−1.28808	−0.644038	+	0.764993i	$0.722742\pi$
$6$	0	0
$7$	33486.6	0.753064	0.376532	−	0.926404i	$-0.377117\pi$
$7$	33486.6	0.753064	0.376532	+	0.926404i	$0.377117\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	11409.4	0.0213600	0.0106800	−	0.999943i	$-0.496600\pi$
$11$	11409.4	0.0213600	0.0106800	+	0.999943i	$0.496600\pi$
$12$	0	0
$13$	−424632.	−0.317194	−0.158597	−	0.987343i	$-0.550697\pi$
$13$	−424632.	−0.317194	−0.158597	+	0.987343i	$0.550697\pi$
$14$	0	0
$15$	2.18717e6	0.743671
$16$	0	0
$17$	7.76640e6	1.32663	0.663316	−	0.748339i	$-0.269149\pi$
$17$	7.76640e6	1.32663	0.663316	+	0.748339i	$0.269149\pi$
$18$	0	0
$19$	−7.16619e6	−0.663962	−0.331981	−	0.943286i	$-0.607717\pi$
$19$	−7.16619e6	−0.663962	−0.331981	+	0.943286i	$0.607717\pi$
$20$	0	0
$21$	−8.13724e6	−0.434781
$22$	0	0
$23$	−1.00865e7	−0.326765	−0.163383	−	0.986563i	$-0.552241\pi$
$23$	−1.00865e7	−0.326765	−0.163383	+	0.986563i	$0.552241\pi$
$24$	0	0
$25$	3.21847e7	0.659142
$26$	0	0
$27$	−1.43489e7	−0.192450
$28$	0	0
$29$	−3.12934e7	−0.283311	−0.141655	−	0.989916i	$-0.545242\pi$
$29$	−3.12934e7	−0.283311	−0.141655	+	0.989916i	$0.545242\pi$
$30$	0	0
$31$	−5.25994e6	−0.0329983	−0.0164991	−	0.999864i	$-0.505252\pi$
$31$	−5.25994e6	−0.0329983	−0.0164991	+	0.999864i	$0.505252\pi$
$32$	0	0
$33$	−2.77248e6	−0.0123322
$34$	0	0
$35$	−3.01403e8	−0.970004
$36$	0	0
$37$	5.58731e8	1.32462	0.662312	−	0.749228i	$-0.269575\pi$
$37$	5.58731e8	1.32462	0.662312	+	0.749228i	$0.269575\pi$
$38$	0	0
$39$	1.03186e8	0.183132
$40$	0	0
$41$	−2.54143e8	−0.342583	−0.171292	−	0.985220i	$-0.554794\pi$
$41$	−2.54143e8	−0.342583	−0.171292	+	0.985220i	$0.554794\pi$
$42$	0	0
$43$	−1.36088e9	−1.41171	−0.705853	−	0.708359i	$-0.749436\pi$
$43$	−1.36088e9	−1.41171	−0.705853	+	0.708359i	$0.749436\pi$
$44$	0	0
$45$	−5.31483e8	−0.429359
$46$	0	0
$47$	−2.16161e9	−1.37480	−0.687400	−	0.726279i	$-0.741248\pi$
$47$	−2.16161e9	−1.37480	−0.687400	+	0.726279i	$0.741248\pi$
$48$	0	0
$49$	−8.55975e8	−0.432895
$50$	0	0
$51$	−1.88723e9	−0.765931
$52$	0	0
$53$	−3.93069e9	−1.29107	−0.645537	−	0.763729i	$-0.723366\pi$
$53$	−3.93069e9	−1.29107	−0.645537	+	0.763729i	$0.723366\pi$
$54$	0	0
$55$	−1.02693e8	−0.0275134
$56$	0	0
$57$	1.74138e9	0.383339
$58$	0	0
$59$	9.48166e9	1.72663	0.863313	−	0.504668i	$-0.168385\pi$
$59$	9.48166e9	1.72663	0.863313	+	0.504668i	$0.168385\pi$
$60$	0	0
$61$	−9.90753e9	−1.50193	−0.750967	−	0.660339i	$-0.770413\pi$
$61$	−9.90753e9	−1.50193	−0.750967	+	0.660339i	$0.770413\pi$
$62$	0	0
$63$	1.97735e9	0.251021
$64$	0	0
$65$	3.82199e9	0.408570
$66$	0	0
$67$	9.83448e8	0.0889898	0.0444949	−	0.999010i	$-0.485832\pi$
$67$	9.83448e8	0.0889898	0.0444949	+	0.999010i	$0.485832\pi$
$68$	0	0
$69$	2.45101e9	0.188658
$70$	0	0
$71$	4.06177e8	0.0267174	0.0133587	−	0.999911i	$-0.495748\pi$
$71$	4.06177e8	0.0267174	0.0133587	+	0.999911i	$0.495748\pi$
$72$	0	0
$73$	−5.83691e9	−0.329539	−0.164770	−	0.986332i	$-0.552688\pi$
$73$	−5.83691e9	−0.329539	−0.164770	+	0.986332i	$0.552688\pi$
$74$	0	0
$75$	−7.82087e9	−0.380556
$76$	0	0
$77$	3.82061e8	0.0160855
$78$	0	0
$79$	1.66139e10	0.607468	0.303734	−	0.952757i	$-0.401767\pi$
$79$	1.66139e10	0.607468	0.303734	+	0.952757i	$0.401767\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	4.82664e10	1.34498	0.672490	−	0.740107i	$-0.265225\pi$
$83$	4.82664e10	1.34498	0.672490	+	0.740107i	$0.265225\pi$
$84$	0	0
$85$	−6.99031e10	−1.70880
$86$	0	0
$87$	7.60429e9	0.163570
$88$	0	0
$89$	4.93435e10	0.936667	0.468334	−	0.883552i	$-0.344855\pi$
$89$	4.93435e10	0.936667	0.468334	+	0.883552i	$0.344855\pi$
$90$	0	0
$91$	−1.42195e10	−0.238867
$92$	0	0
$93$	1.27816e9	0.0190516
$94$	0	0
$95$	6.45008e10	0.855235
$96$	0	0
$97$	1.09762e11	1.29779	0.648897	−	0.760877i	$-0.275231\pi$
$97$	1.09762e11	1.29779	0.648897	+	0.760877i	$0.275231\pi$
$98$	0	0
$99$	6.73712e8	0.00712002
$100$	0	0
$101$	−1.42941e11	−1.35328	−0.676642	−	0.736312i	$-0.736566\pi$
$101$	−1.42941e11	−1.35328	−0.676642	+	0.736312i	$0.736566\pi$
$102$	0	0
$103$	−3.34275e10	−0.284118	−0.142059	−	0.989858i	$-0.545372\pi$
$103$	−3.34275e10	−0.284118	−0.142059	+	0.989858i	$0.545372\pi$
$104$	0	0
$105$	7.32409e10	0.560032
$106$	0	0
$107$	−1.86114e11	−1.28283	−0.641414	−	0.767195i	$-0.721652\pi$
$107$	−1.86114e11	−1.28283	−0.641414	+	0.767195i	$0.721652\pi$
$108$	0	0
$109$	−2.34144e11	−1.45760	−0.728799	−	0.684728i	$-0.759921\pi$
$109$	−2.34144e11	−1.45760	−0.728799	+	0.684728i	$0.759921\pi$
$110$	0	0
$111$	−1.35772e11	−0.764773
$112$	0	0
$113$	2.84940e11	1.45486	0.727432	−	0.686180i	$-0.240714\pi$
$113$	2.84940e11	1.45486	0.727432	+	0.686180i	$0.240714\pi$
$114$	0	0
$115$	9.07853e10	0.420899
$116$	0	0
$117$	−2.50741e10	−0.105731
$118$	0	0
$119$	2.60070e11	0.999038
$120$	0	0
$121$	−2.85181e11	−0.999544
$122$	0	0
$123$	6.17566e10	0.197791
$124$	0	0
$125$	1.49803e11	0.439052
$126$	0	0
$127$	−4.02713e11	−1.08162	−0.540810	−	0.841145i	$-0.681882\pi$
$127$	−4.02713e11	−1.08162	−0.540810	+	0.841145i	$0.681882\pi$
$128$	0	0
$129$	3.30694e11	0.815048
$130$	0	0
$131$	3.56436e11	0.807214	0.403607	−	0.914932i	$-0.367756\pi$
$131$	3.56436e11	0.807214	0.403607	+	0.914932i	$0.367756\pi$
$132$	0	0
$133$	−2.39971e11	−0.500006
$134$	0	0
$135$	1.29150e11	0.247890
$136$	0	0
$137$	6.54324e11	1.15832	0.579161	−	0.815213i	$-0.303380\pi$
$137$	6.54324e11	1.15832	0.579161	+	0.815213i	$0.303380\pi$
$138$	0	0
$139$	4.06208e11	0.663999	0.331999	−	0.943280i	$-0.392277\pi$
$139$	4.06208e11	0.663999	0.331999	+	0.943280i	$0.392277\pi$
$140$	0	0
$141$	5.25272e11	0.793741
$142$	0	0
$143$	−4.84479e9	−0.00677527
$144$	0	0
$145$	2.81662e11	0.364926
$146$	0	0
$147$	2.08002e11	0.249932
$148$	0	0
$149$	1.08177e11	0.120673	0.0603367	−	0.998178i	$-0.480783\pi$
$149$	1.08177e11	0.120673	0.0603367	+	0.998178i	$0.480783\pi$
$150$	0	0
$151$	1.86813e12	1.93657	0.968286	−	0.249845i	$-0.0803798\pi$
$151$	1.86813e12	1.93657	0.968286	+	0.249845i	$0.0803798\pi$
$152$	0	0
$153$	4.58598e11	0.442211
$154$	0	0
$155$	4.73432e10	0.0425043
$156$	0	0
$157$	−1.18804e12	−0.993993	−0.496996	−	0.867753i	$-0.665564\pi$
$157$	−1.18804e12	−0.993993	−0.496996	+	0.867753i	$0.665564\pi$
$158$	0	0
$159$	9.55157e11	0.745402
$160$	0	0
$161$	−3.37761e11	−0.246075
$162$	0	0
$163$	2.14333e12	1.45901	0.729505	−	0.683976i	$-0.239751\pi$
$163$	2.14333e12	1.45901	0.729505	+	0.683976i	$0.239751\pi$
$164$	0	0
$165$	2.49543e10	0.0158849
$166$	0	0
$167$	1.20449e12	0.717567	0.358784	−	0.933421i	$-0.383192\pi$
$167$	1.20449e12	0.717567	0.358784	+	0.933421i	$0.383192\pi$
$168$	0	0
$169$	−1.61185e12	−0.899388
$170$	0	0
$171$	−4.23156e11	−0.221321
$172$	0	0
$173$	1.56912e12	0.769844	0.384922	−	0.922949i	$-0.374228\pi$
$173$	1.56912e12	0.769844	0.384922	+	0.922949i	$0.374228\pi$
$174$	0	0
$175$	1.07775e12	0.496376
$176$	0	0
$177$	−2.30404e12	−0.996869
$178$	0	0
$179$	3.16535e12	1.28745	0.643724	−	0.765258i	$-0.277389\pi$
$179$	3.16535e12	1.28745	0.643724	+	0.765258i	$0.277389\pi$
$180$	0	0
$181$	3.20283e12	1.22547	0.612734	−	0.790289i	$-0.290070\pi$
$181$	3.20283e12	1.22547	0.612734	+	0.790289i	$0.290070\pi$
$182$	0	0
$183$	2.40753e12	0.867143
$184$	0	0
$185$	−5.02897e12	−1.70622
$186$	0	0
$187$	8.86098e10	0.0283369
$188$	0	0
$189$	−4.80496e11	−0.144927
$190$	0	0
$191$	−1.65351e12	−0.470677	−0.235338	−	0.971913i	$-0.575620\pi$
$191$	−1.65351e12	−0.470677	−0.235338	+	0.971913i	$0.575620\pi$
$192$	0	0
$193$	−2.64788e12	−0.711760	−0.355880	−	0.934532i	$-0.615819\pi$
$193$	−2.64788e12	−0.711760	−0.355880	+	0.934532i	$0.615819\pi$
$194$	0	0
$195$	−9.28744e11	−0.235888
$196$	0	0
$197$	1.30059e12	0.312302	0.156151	−	0.987733i	$-0.450091\pi$
$197$	1.30059e12	0.312302	0.156151	+	0.987733i	$0.450091\pi$
$198$	0	0
$199$	−9.06957e11	−0.206013	−0.103007	−	0.994681i	$-0.532846\pi$
$199$	−9.06957e11	−0.206013	−0.103007	+	0.994681i	$0.532846\pi$
$200$	0	0
$201$	−2.38978e11	−0.0513783
$202$	0	0
$203$	−1.04791e12	−0.213351
$204$	0	0
$205$	2.28746e12	0.441274
$206$	0	0
$207$	−5.95596e11	−0.108922
$208$	0	0
$209$	−8.17618e10	−0.0141823
$210$	0	0
$211$	9.60842e12	1.58161	0.790803	−	0.612070i	$-0.209663\pi$
$211$	9.60842e12	1.58161	0.790803	+	0.612070i	$0.209663\pi$
$212$	0	0
$213$	−9.87011e10	−0.0154253
$214$	0	0
$215$	1.22489e13	1.81838
$216$	0	0
$217$	−1.76137e11	−0.0248498
$218$	0	0
$219$	1.41837e12	0.190260
$220$	0	0
$221$	−3.29786e12	−0.420799
$222$	0	0
$223$	−5.21310e12	−0.633022	−0.316511	−	0.948589i	$-0.602511\pi$
$223$	−5.21310e12	−0.633022	−0.316511	+	0.948589i	$0.602511\pi$
$224$	0	0
$225$	1.90047e12	0.219714
$226$	0	0
$227$	1.78255e12	0.196291	0.0981453	−	0.995172i	$-0.468709\pi$
$227$	1.78255e12	0.196291	0.0981453	+	0.995172i	$0.468709\pi$
$228$	0	0
$229$	1.14181e13	1.19811	0.599057	−	0.800706i	$-0.295542\pi$
$229$	1.14181e13	1.19811	0.599057	+	0.800706i	$0.295542\pi$
$230$	0	0
$231$	−9.28408e10	−0.00928695
$232$	0	0
$233$	−1.73608e12	−0.165619	−0.0828097	−	0.996565i	$-0.526389\pi$
$233$	−1.73608e12	−0.165619	−0.0828097	+	0.996565i	$0.526389\pi$
$234$	0	0
$235$	1.94560e13	1.77085
$236$	0	0
$237$	−4.03718e12	−0.350722
$238$	0	0
$239$	2.19470e13	1.82048	0.910242	−	0.414077i	$-0.135896\pi$
$239$	2.19470e13	1.82048	0.910242	+	0.414077i	$0.135896\pi$
$240$	0	0
$241$	5.21970e12	0.413572	0.206786	−	0.978386i	$-0.433700\pi$
$241$	5.21970e12	0.413572	0.206786	+	0.978386i	$0.433700\pi$
$242$	0	0
$243$	−8.47289e11	−0.0641500
$244$	0	0
$245$	7.70439e12	0.557602
$246$	0	0
$247$	3.04299e12	0.210605
$248$	0	0
$249$	−1.17287e13	−0.776524
$250$	0	0
$251$	3.92349e12	0.248580	0.124290	−	0.992246i	$-0.460335\pi$
$251$	3.92349e12	0.248580	0.124290	+	0.992246i	$0.460335\pi$
$252$	0	0
$253$	−1.15080e11	−0.00697972
$254$	0	0
$255$	1.69865e13	0.986578
$256$	0	0
$257$	5.92822e10	0.00329832	0.00164916	−	0.999999i	$-0.499475\pi$
$257$	5.92822e10	0.00329832	0.00164916	+	0.999999i	$0.499475\pi$
$258$	0	0
$259$	1.87100e13	0.997527
$260$	0	0
$261$	−1.84784e12	−0.0944369
$262$	0	0
$263$	2.91910e13	1.43051	0.715256	−	0.698862i	$-0.246310\pi$
$263$	2.91910e13	1.43051	0.715256	+	0.698862i	$0.246310\pi$
$264$	0	0
$265$	3.53790e13	1.66300
$266$	0	0
$267$	−1.19905e13	−0.540785
$268$	0	0
$269$	−2.40962e13	−1.04306	−0.521532	−	0.853232i	$-0.674639\pi$
$269$	−2.40962e13	−1.04306	−0.521532	+	0.853232i	$0.674639\pi$
$270$	0	0
$271$	1.56884e12	0.0651999	0.0326000	−	0.999468i	$-0.489621\pi$
$271$	1.56884e12	0.0651999	0.0326000	+	0.999468i	$0.489621\pi$
$272$	0	0
$273$	3.45533e12	0.137910
$274$	0	0
$275$	3.67207e11	0.0140793
$276$	0	0
$277$	−1.76256e13	−0.649389	−0.324695	−	0.945819i	$-0.605262\pi$
$277$	−1.76256e13	−0.649389	−0.324695	+	0.945819i	$0.605262\pi$
$278$	0	0
$279$	−3.10594e11	−0.0109994
$280$	0	0
$281$	1.87069e13	0.636969	0.318484	−	0.947928i	$-0.396826\pi$
$281$	1.87069e13	0.636969	0.318484	+	0.947928i	$0.396826\pi$
$282$	0	0
$283$	2.92918e13	0.959225	0.479613	−	0.877480i	$-0.340777\pi$
$283$	2.92918e13	0.959225	0.479613	+	0.877480i	$0.340777\pi$
$284$	0	0
$285$	−1.56737e13	−0.493770
$286$	0	0
$287$	−8.51037e12	−0.257987
$288$	0	0
$289$	2.60450e13	0.759953
$290$	0	0
$291$	−2.66720e13	−0.749281
$292$	0	0
$293$	−1.29186e13	−0.349498	−0.174749	−	0.984613i	$-0.555911\pi$
$293$	−1.29186e13	−0.349498	−0.174749	+	0.984613i	$0.555911\pi$
$294$	0	0
$295$	−8.53417e13	−2.22403
$296$	0	0
$297$	−1.63712e11	−0.00411074
$298$	0	0
$299$	4.28304e12	0.103648
$300$	0	0
$301$	−4.55713e13	−1.06310
$302$	0	0
$303$	3.47346e13	0.781319
$304$	0	0
$305$	8.91748e13	1.93461
$306$	0	0
$307$	−1.46206e13	−0.305988	−0.152994	−	0.988227i	$-0.548892\pi$
$307$	−1.46206e13	−0.305988	−0.152994	+	0.988227i	$0.548892\pi$
$308$	0	0
$309$	8.12288e12	0.164036
$310$	0	0
$311$	−5.82400e13	−1.13511	−0.567557	−	0.823334i	$-0.692111\pi$
$311$	−5.82400e13	−1.13511	−0.567557	+	0.823334i	$0.692111\pi$
$312$	0	0
$313$	−4.93985e13	−0.929437	−0.464718	−	0.885459i	$-0.653844\pi$
$313$	−4.93985e13	−0.929437	−0.464718	+	0.885459i	$0.653844\pi$
$314$	0	0
$315$	−1.77975e13	−0.323335
$316$	0	0
$317$	3.82487e13	0.671105	0.335553	−	0.942021i	$-0.391077\pi$
$317$	3.82487e13	0.671105	0.335553	+	0.942021i	$0.391077\pi$
$318$	0	0
$319$	−3.57038e11	−0.00605153
$320$	0	0
$321$	4.52257e13	0.740641
$322$	0	0
$323$	−5.56555e13	−0.880834
$324$	0	0
$325$	−1.36666e13	−0.209076
$326$	0	0
$327$	5.68970e13	0.841544
$328$	0	0
$329$	−7.23850e13	−1.03531
$330$	0	0
$331$	1.14564e14	1.58487	0.792436	−	0.609955i	$-0.208812\pi$
$331$	1.14564e14	1.58487	0.792436	+	0.609955i	$0.208812\pi$
$332$	0	0
$333$	3.29925e13	0.441542
$334$	0	0
$335$	−8.85173e12	−0.114626
$336$	0	0
$337$	−1.29586e14	−1.62403	−0.812014	−	0.583638i	$-0.801629\pi$
$337$	−1.29586e14	−1.62403	−0.812014	+	0.583638i	$0.801629\pi$
$338$	0	0
$339$	−6.92405e13	−0.839966
$340$	0	0
$341$	−6.00126e10	−0.000704844 0
$342$	0	0
$343$	−9.48776e13	−1.07906
$344$	0	0
$345$	−2.20608e13	−0.243006
$346$	0	0
$347$	−1.51845e14	−1.62028	−0.810139	−	0.586238i	$-0.800609\pi$
$347$	−1.51845e14	−1.62028	−0.810139	+	0.586238i	$0.800609\pi$
$348$	0	0
$349$	−9.26556e13	−0.957925	−0.478963	−	0.877835i	$-0.658987\pi$
$349$	−9.26556e13	−0.957925	−0.478963	+	0.877835i	$0.658987\pi$
$350$	0	0
$351$	6.09301e12	0.0610439
$352$	0	0
$353$	5.63210e13	0.546902	0.273451	−	0.961886i	$-0.411835\pi$
$353$	5.63210e13	0.546902	0.273451	+	0.961886i	$0.411835\pi$
$354$	0	0
$355$	−3.65588e12	−0.0344141
$356$	0	0
$357$	−6.31970e13	−0.576795
$358$	0	0
$359$	1.19655e14	1.05904	0.529520	−	0.848297i	$-0.322372\pi$
$359$	1.19655e14	1.05904	0.529520	+	0.848297i	$0.322372\pi$
$360$	0	0
$361$	−6.51360e13	−0.559154
$362$	0	0
$363$	6.92991e13	0.577087
$364$	0	0
$365$	5.25363e13	0.424472
$366$	0	0
$367$	−1.87226e14	−1.46792	−0.733962	−	0.679190i	$-0.762331\pi$
$367$	−1.87226e14	−1.46792	−0.733962	+	0.679190i	$0.762331\pi$
$368$	0	0
$369$	−1.50069e13	−0.114194
$370$	0	0
$371$	−1.31625e14	−0.972261
$372$	0	0
$373$	6.38476e13	0.457874	0.228937	−	0.973441i	$-0.426475\pi$
$373$	6.38476e13	0.457874	0.228937	+	0.973441i	$0.426475\pi$
$374$	0	0
$375$	−3.64021e13	−0.253487
$376$	0	0
$377$	1.32882e13	0.0898644
$378$	0	0
$379$	9.43918e13	0.620038	0.310019	−	0.950730i	$-0.399665\pi$
$379$	9.43918e13	0.620038	0.310019	+	0.950730i	$0.399665\pi$
$380$	0	0
$381$	9.78592e13	0.624474
$382$	0	0
$383$	4.63711e13	0.287511	0.143755	−	0.989613i	$-0.454082\pi$
$383$	4.63711e13	0.287511	0.143755	+	0.989613i	$0.454082\pi$
$384$	0	0
$385$	−3.43882e12	−0.0207193
$386$	0	0
$387$	−8.03587e13	−0.470568
$388$	0	0
$389$	2.35478e14	1.34038	0.670190	−	0.742190i	$-0.266213\pi$
$389$	2.35478e14	1.34038	0.670190	+	0.742190i	$0.266213\pi$
$390$	0	0
$391$	−7.83355e13	−0.433497
$392$	0	0
$393$	−8.66138e13	−0.466045
$394$	0	0
$395$	−1.49537e14	−0.782465
$396$	0	0
$397$	1.93133e13	0.0982901	0.0491450	−	0.998792i	$-0.484350\pi$
$397$	1.93133e13	0.0982901	0.0491450	+	0.998792i	$0.484350\pi$
$398$	0	0
$399$	5.83130e13	0.288679
$400$	0	0
$401$	−2.42794e14	−1.16935	−0.584674	−	0.811269i	$-0.698777\pi$
$401$	−2.42794e14	−1.16935	−0.584674	+	0.811269i	$0.698777\pi$
$402$	0	0
$403$	2.23354e12	0.0104668
$404$	0	0
$405$	−3.13835e13	−0.143120
$406$	0	0
$407$	6.37477e12	0.0282940
$408$	0	0
$409$	−1.76114e13	−0.0760877	−0.0380439	−	0.999276i	$-0.512113\pi$
$409$	−1.76114e13	−0.0760877	−0.0380439	+	0.999276i	$0.512113\pi$
$410$	0	0
$411$	−1.59001e14	−0.668758
$412$	0	0
$413$	3.17509e14	1.30026
$414$	0	0
$415$	−4.34432e14	−1.73244
$416$	0	0
$417$	−9.87086e13	−0.383360
$418$	0	0
$419$	1.03722e14	0.392368	0.196184	−	0.980567i	$-0.437145\pi$
$419$	1.03722e14	0.392368	0.196184	+	0.980567i	$0.437145\pi$
$420$	0	0
$421$	4.24843e14	1.56559	0.782793	−	0.622282i	$-0.213794\pi$
$421$	4.24843e14	1.56559	0.782793	+	0.622282i	$0.213794\pi$
$422$	0	0
$423$	−1.27641e14	−0.458267
$424$	0	0
$425$	2.49959e14	0.874439
$426$	0	0
$427$	−3.31769e14	−1.13105
$428$	0	0
$429$	1.17728e12	0.00391170
$430$	0	0
$431$	−5.16724e14	−1.67353	−0.836766	−	0.547561i	$-0.815556\pi$
$431$	−5.16724e14	−1.67353	−0.836766	+	0.547561i	$0.815556\pi$
$432$	0	0
$433$	−2.55323e14	−0.806131	−0.403066	−	0.915171i	$-0.632055\pi$
$433$	−2.55323e14	−0.806131	−0.403066	+	0.915171i	$0.632055\pi$
$434$	0	0
$435$	−6.84440e13	−0.210690
$436$	0	0
$437$	7.22815e13	0.216960
$438$	0	0
$439$	−2.44953e14	−0.717016	−0.358508	−	0.933527i	$-0.616714\pi$
$439$	−2.44953e14	−0.717016	−0.358508	+	0.933527i	$0.616714\pi$
$440$	0	0
$441$	−5.05445e13	−0.144298
$442$	0	0
$443$	−2.64266e14	−0.735903	−0.367952	−	0.929845i	$-0.619941\pi$
$443$	−2.64266e14	−0.735903	−0.367952	+	0.929845i	$0.619941\pi$
$444$	0	0
$445$	−4.44127e14	−1.20650
$446$	0	0
$447$	−2.62871e13	−0.0696709
$448$	0	0
$449$	5.83254e14	1.50835	0.754177	−	0.656671i	$-0.228036\pi$
$449$	5.83254e14	1.50835	0.754177	+	0.656671i	$0.228036\pi$
$450$	0	0
$451$	−2.89961e12	−0.00731759
$452$	0	0
$453$	−4.53955e14	−1.11808
$454$	0	0
$455$	1.27985e14	0.307679
$456$	0	0
$457$	4.18044e14	0.981031	0.490515	−	0.871433i	$-0.336809\pi$
$457$	4.18044e14	0.981031	0.490515	+	0.871433i	$0.336809\pi$
$458$	0	0
$459$	−1.11439e14	−0.255310
$460$	0	0
$461$	−2.10916e13	−0.0471797	−0.0235898	−	0.999722i	$-0.507510\pi$
$461$	−2.10916e13	−0.0471797	−0.0235898	+	0.999722i	$0.507510\pi$
$462$	0	0
$463$	4.96784e14	1.08511	0.542553	−	0.840022i	$-0.317458\pi$
$463$	4.96784e14	1.08511	0.542553	+	0.840022i	$0.317458\pi$
$464$	0	0
$465$	−1.15044e13	−0.0245399
$466$	0	0
$467$	2.53978e14	0.529119	0.264559	−	0.964369i	$-0.414774\pi$
$467$	2.53978e14	0.529119	0.264559	+	0.964369i	$0.414774\pi$
$468$	0	0
$469$	3.29323e13	0.0670149
$470$	0	0
$471$	2.88694e14	0.573882
$472$	0	0
$473$	−1.55268e13	−0.0301541
$474$	0	0
$475$	−2.30641e14	−0.437645
$476$	0	0
$477$	−2.32103e14	−0.430358
$478$	0	0
$479$	−3.28286e14	−0.594849	−0.297424	−	0.954745i	$-0.596128\pi$
$479$	−3.28286e14	−0.594849	−0.297424	+	0.954745i	$0.596128\pi$
$480$	0	0
$481$	−2.37255e14	−0.420163
$482$	0	0
$483$	8.20760e13	0.142071
$484$	0	0
$485$	−9.87932e14	−1.67166
$486$	0	0
$487$	8.60547e14	1.42353	0.711763	−	0.702420i	$-0.247897\pi$
$487$	8.60547e14	1.42353	0.711763	+	0.702420i	$0.247897\pi$
$488$	0	0
$489$	−5.20830e14	−0.842360
$490$	0	0
$491$	7.83980e14	1.23981	0.619907	−	0.784675i	$-0.287170\pi$
$491$	7.83980e14	1.23981	0.619907	+	0.784675i	$0.287170\pi$
$492$	0	0
$493$	−2.43037e14	−0.375849
$494$	0	0
$495$	−6.06389e12	−0.00917113
$496$	0	0
$497$	1.36015e13	0.0201199
$498$	0	0
$499$	−7.10038e14	−1.02737	−0.513687	−	0.857977i	$-0.671721\pi$
$499$	−7.10038e14	−1.02737	−0.513687	+	0.857977i	$0.671721\pi$
$500$	0	0
$501$	−2.92691e14	−0.414288
$502$	0	0
$503$	6.82768e14	0.945474	0.472737	−	0.881204i	$-0.343266\pi$
$503$	6.82768e14	0.945474	0.472737	+	0.881204i	$0.343266\pi$
$504$	0	0
$505$	1.28657e15	1.74313
$506$	0	0
$507$	3.91679e14	0.519262
$508$	0	0
$509$	7.58395e14	0.983893	0.491947	−	0.870625i	$-0.336286\pi$
$509$	7.58395e14	0.983893	0.491947	+	0.870625i	$0.336286\pi$
$510$	0	0
$511$	−1.95458e14	−0.248164
$512$	0	0
$513$	1.02827e14	0.127780
$514$	0	0
$515$	3.00871e14	0.365966
$516$	0	0
$517$	−2.46626e13	−0.0293658
$518$	0	0
$519$	−3.81296e14	−0.444469
$520$	0	0
$521$	6.27290e14	0.715914	0.357957	−	0.933738i	$-0.383473\pi$
$521$	6.27290e14	0.715914	0.357957	+	0.933738i	$0.383473\pi$
$522$	0	0
$523$	1.36393e15	1.52417	0.762083	−	0.647480i	$-0.224177\pi$
$523$	1.36393e15	1.52417	0.762083	+	0.647480i	$0.224177\pi$
$524$	0	0
$525$	−2.61894e14	−0.286583
$526$	0	0
$527$	−4.08507e13	−0.0437766
$528$	0	0
$529$	−8.51073e14	−0.893224
$530$	0	0
$531$	5.59883e14	0.575542
$532$	0	0
$533$	1.07917e14	0.108665
$534$	0	0
$535$	1.67516e15	1.65238
$536$	0	0
$537$	−7.69179e14	−0.743308
$538$	0	0
$539$	−9.76615e12	−0.00924666
$540$	0	0
$541$	−1.18379e15	−1.09822	−0.549108	−	0.835751i	$-0.685033\pi$
$541$	−1.18379e15	−1.09822	−0.549108	+	0.835751i	$0.685033\pi$
$542$	0	0
$543$	−7.78289e14	−0.707525
$544$	0	0
$545$	2.10746e15	1.87750
$546$	0	0
$547$	9.02063e14	0.787601	0.393801	−	0.919196i	$-0.371160\pi$
$547$	9.02063e14	0.787601	0.393801	+	0.919196i	$0.371160\pi$
$548$	0	0
$549$	−5.85029e14	−0.500645
$550$	0	0
$551$	2.24254e14	0.188108
$552$	0	0
$553$	5.56343e14	0.457462
$554$	0	0
$555$	1.22204e15	0.985086
$556$	0	0
$557$	1.36673e15	1.08014	0.540069	−	0.841620i	$-0.318398\pi$
$557$	1.36673e15	1.08014	0.540069	+	0.841620i	$0.318398\pi$
$558$	0	0
$559$	5.77874e14	0.447784
$560$	0	0
$561$	−2.15322e13	−0.0163603
$562$	0	0
$563$	−1.48785e15	−1.10857	−0.554286	−	0.832326i	$-0.687008\pi$
$563$	−1.48785e15	−1.10857	−0.554286	+	0.832326i	$0.687008\pi$
$564$	0	0
$565$	−2.56466e15	−1.87398
$566$	0	0
$567$	1.16761e14	0.0836737
$568$	0	0
$569$	2.27343e14	0.159796	0.0798978	−	0.996803i	$-0.474541\pi$
$569$	2.27343e14	0.159796	0.0798978	+	0.996803i	$0.474541\pi$
$570$	0	0
$571$	−2.34051e15	−1.61366	−0.806831	−	0.590782i	$-0.798819\pi$
$571$	−2.34051e15	−1.61366	−0.806831	+	0.590782i	$0.798819\pi$
$572$	0	0
$573$	4.01803e14	0.271745
$574$	0	0
$575$	−3.24629e14	−0.215385
$576$	0	0
$577$	−2.77362e15	−1.80543	−0.902714	−	0.430242i	$-0.858428\pi$
$577$	−2.77362e15	−1.80543	−0.902714	+	0.430242i	$0.858428\pi$
$578$	0	0
$579$	6.43435e14	0.410935
$580$	0	0
$581$	1.61628e15	1.01285
$582$	0	0
$583$	−4.48467e13	−0.0275774
$584$	0	0
$585$	2.25685e14	0.136190
$586$	0	0
$587$	3.07408e15	1.82056	0.910282	−	0.413989i	$-0.135865\pi$
$587$	3.07408e15	1.82056	0.910282	+	0.413989i	$0.135865\pi$
$588$	0	0
$589$	3.76937e13	0.0219096
$590$	0	0
$591$	−3.16042e14	−0.180308
$592$	0	0
$593$	1.28285e15	0.718415	0.359207	−	0.933258i	$-0.383047\pi$
$593$	1.28285e15	0.718415	0.359207	+	0.933258i	$0.383047\pi$
$594$	0	0
$595$	−2.34082e15	−1.28684
$596$	0	0
$597$	2.20391e14	0.118942
$598$	0	0
$599$	−9.97600e14	−0.528578	−0.264289	−	0.964444i	$-0.585137\pi$
$599$	−9.97600e14	−0.528578	−0.264289	+	0.964444i	$0.585137\pi$
$600$	0	0
$601$	−2.80867e15	−1.46114	−0.730568	−	0.682840i	$-0.760744\pi$
$601$	−2.80867e15	−1.46114	−0.730568	+	0.682840i	$0.760744\pi$
$602$	0	0
$603$	5.80716e13	0.0296633
$604$	0	0
$605$	2.56684e15	1.28749
$606$	0	0
$607$	4.01624e15	1.97825	0.989126	−	0.147071i	$-0.0469845\pi$
$607$	4.01624e15	1.97825	0.989126	+	0.147071i	$0.0469845\pi$
$608$	0	0
$609$	2.54642e14	0.123178
$610$	0	0
$611$	9.17890e14	0.436078
$612$	0	0
$613$	2.80094e15	1.30699	0.653494	−	0.756932i	$-0.273303\pi$
$613$	2.80094e15	1.30699	0.653494	+	0.756932i	$0.273303\pi$
$614$	0	0
$615$	−5.55854e14	−0.254769
$616$	0	0
$617$	4.46965e14	0.201236	0.100618	−	0.994925i	$-0.467918\pi$
$617$	4.46965e14	0.201236	0.100618	+	0.994925i	$0.467918\pi$
$618$	0	0
$619$	3.69740e15	1.63530	0.817652	−	0.575713i	$-0.195276\pi$
$619$	3.69740e15	1.63530	0.817652	+	0.575713i	$0.195276\pi$
$620$	0	0
$621$	1.44730e14	0.0628860
$622$	0	0
$623$	1.65235e15	0.705370
$624$	0	0
$625$	−2.91985e15	−1.22467
$626$	0	0
$627$	1.98681e13	0.00818814
$628$	0	0
$629$	4.33932e15	1.75729
$630$	0	0
$631$	4.43356e15	1.76438	0.882188	−	0.470898i	$-0.156070\pi$
$631$	4.43356e15	1.76438	0.882188	+	0.470898i	$0.156070\pi$
$632$	0	0
$633$	−2.33485e15	−0.913141
$634$	0	0
$635$	3.62470e15	1.39321
$636$	0	0
$637$	3.63475e14	0.137312
$638$	0	0
$639$	2.39844e13	0.00890581
$640$	0	0
$641$	9.46029e14	0.345291	0.172645	−	0.984984i	$-0.444769\pi$
$641$	9.46029e14	0.345291	0.172645	+	0.984984i	$0.444769\pi$
$642$	0	0
$643$	−4.36837e15	−1.56732	−0.783662	−	0.621187i	$-0.786651\pi$
$643$	−4.36837e15	−1.56732	−0.783662	+	0.621187i	$0.786651\pi$
$644$	0	0
$645$	−2.97648e15	−1.04984
$646$	0	0
$647$	3.96702e14	0.137559	0.0687797	−	0.997632i	$-0.478089\pi$
$647$	3.96702e14	0.137559	0.0687797	+	0.997632i	$0.478089\pi$
$648$	0	0
$649$	1.08180e14	0.0368808
$650$	0	0
$651$	4.28014e13	0.0143470
$652$	0	0
$653$	2.05536e15	0.677430	0.338715	−	0.940889i	$-0.390008\pi$
$653$	2.05536e15	0.677430	0.338715	+	0.940889i	$0.390008\pi$
$654$	0	0
$655$	−3.20817e15	−1.03975
$656$	0	0
$657$	−3.44664e14	−0.109846
$658$	0	0
$659$	4.90805e14	0.153829	0.0769147	−	0.997038i	$-0.475493\pi$
$659$	4.90805e14	0.153829	0.0769147	+	0.997038i	$0.475493\pi$
$660$	0	0
$661$	−5.12959e14	−0.158116	−0.0790578	−	0.996870i	$-0.525191\pi$
$661$	−5.12959e14	−0.158116	−0.0790578	+	0.996870i	$0.525191\pi$
$662$	0	0
$663$	8.01380e14	0.242949
$664$	0	0
$665$	2.15991e15	0.644046
$666$	0	0
$667$	3.15639e14	0.0925761
$668$	0	0
$669$	1.26678e15	0.365476
$670$	0	0
$671$	−1.13039e14	−0.0320814
$672$	0	0
$673$	1.61703e15	0.451477	0.225738	−	0.974188i	$-0.427521\pi$
$673$	1.61703e15	0.451477	0.225738	+	0.974188i	$0.427521\pi$
$674$	0	0
$675$	−4.61815e14	−0.126852
$676$	0	0
$677$	−8.49167e14	−0.229486	−0.114743	−	0.993395i	$-0.536604\pi$
$677$	−8.49167e14	−0.229486	−0.114743	+	0.993395i	$0.536604\pi$
$678$	0	0
$679$	3.67554e15	0.977321
$680$	0	0
$681$	−4.33160e14	−0.113328
$682$	0	0
$683$	6.91094e15	1.77919	0.889596	−	0.456747i	$-0.150986\pi$
$683$	6.91094e15	1.77919	0.889596	+	0.456747i	$0.150986\pi$
$684$	0	0
$685$	−5.88938e15	−1.49201
$686$	0	0
$687$	−2.77460e15	−0.691732
$688$	0	0
$689$	1.66910e15	0.409520
$690$	0	0
$691$	−2.78992e15	−0.673693	−0.336846	−	0.941560i	$-0.609360\pi$
$691$	−2.78992e15	−0.673693	−0.336846	+	0.941560i	$0.609360\pi$
$692$	0	0
$693$	2.25603e13	0.00536182
$694$	0	0
$695$	−3.65616e15	−0.855281
$696$	0	0
$697$	−1.97377e15	−0.454482
$698$	0	0
$699$	4.21866e14	0.0956204
$700$	0	0
$701$	4.72938e15	1.05525	0.527624	−	0.849478i	$-0.323083\pi$
$701$	4.72938e15	1.05525	0.527624	+	0.849478i	$0.323083\pi$
$702$	0	0
$703$	−4.00397e15	−0.879501
$704$	0	0
$705$	−4.72782e15	−1.02240
$706$	0	0
$707$	−4.78660e15	−1.01911
$708$	0	0
$709$	2.22308e15	0.466016	0.233008	−	0.972475i	$-0.425143\pi$
$709$	2.22308e15	0.466016	0.233008	+	0.972475i	$0.425143\pi$
$710$	0	0
$711$	9.81035e14	0.202489
$712$	0	0
$713$	5.30542e13	0.0107827
$714$	0	0
$715$	4.36065e13	0.00872707
$716$	0	0
$717$	−5.33312e15	−1.05106
$718$	0	0
$719$	−4.63111e15	−0.898827	−0.449414	−	0.893324i	$-0.648367\pi$
$719$	−4.63111e15	−0.898827	−0.449414	+	0.893324i	$0.648367\pi$
$720$	0	0
$721$	−1.11937e15	−0.213959
$722$	0	0
$723$	−1.26839e15	−0.238776
$724$	0	0
$725$	−1.00717e15	−0.186742
$726$	0	0
$727$	−4.98413e15	−0.910228	−0.455114	−	0.890433i	$-0.650401\pi$
$727$	−4.98413e15	−0.910228	−0.455114	+	0.890433i	$0.650401\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−1.05691e16	−1.87281
$732$	0	0
$733$	−5.48833e15	−0.958006	−0.479003	−	0.877813i	$-0.659002\pi$
$733$	−5.48833e15	−0.958006	−0.479003	+	0.877813i	$0.659002\pi$
$734$	0	0
$735$	−1.87217e15	−0.321932
$736$	0	0
$737$	1.12205e13	0.00190083
$738$	0	0
$739$	3.36930e15	0.562336	0.281168	−	0.959659i	$-0.409278\pi$
$739$	3.36930e15	0.562336	0.281168	+	0.959659i	$0.409278\pi$
$740$	0	0
$741$	−7.39448e14	−0.121593
$742$	0	0
$743$	6.27159e15	1.01611	0.508054	−	0.861325i	$-0.330365\pi$
$743$	6.27159e15	1.01611	0.508054	+	0.861325i	$0.330365\pi$
$744$	0	0
$745$	−9.73673e14	−0.155437
$746$	0	0
$747$	2.85008e15	0.448326
$748$	0	0
$749$	−6.23232e15	−0.966051
$750$	0	0
$751$	−6.56083e15	−1.00216	−0.501082	−	0.865400i	$-0.667065\pi$
$751$	−6.56083e15	−1.00216	−0.501082	+	0.865400i	$0.667065\pi$
$752$	0	0
$753$	−9.53408e14	−0.143518
$754$	0	0
$755$	−1.68145e16	−2.49445
$756$	0	0
$757$	−2.80561e15	−0.410204	−0.205102	−	0.978741i	$-0.565753\pi$
$757$	−2.80561e15	−0.410204	−0.205102	+	0.978741i	$0.565753\pi$
$758$	0	0
$759$	2.79645e13	0.00402974
$760$	0	0
$761$	9.61478e15	1.36560	0.682800	−	0.730605i	$-0.260762\pi$
$761$	9.61478e15	1.36560	0.682800	+	0.730605i	$0.260762\pi$
$762$	0	0
$763$	−7.84069e15	−1.09766
$764$	0	0
$765$	−4.12771e15	−0.569601
$766$	0	0
$767$	−4.02622e15	−0.547675
$768$	0	0
$769$	9.75079e15	1.30751	0.653755	−	0.756706i	$-0.273193\pi$
$769$	9.75079e15	1.30751	0.653755	+	0.756706i	$0.273193\pi$
$770$	0	0
$771$	−1.44056e13	−0.00190428
$772$	0	0
$773$	6.72820e15	0.876823	0.438411	−	0.898774i	$-0.355541\pi$
$773$	6.72820e15	0.876823	0.438411	+	0.898774i	$0.355541\pi$
$774$	0	0
$775$	−1.69289e14	−0.0217505
$776$	0	0
$777$	−4.54652e15	−0.575922
$778$	0	0
$779$	1.82123e15	0.227462
$780$	0	0
$781$	4.63423e12	0.000570685 0
$782$	0	0
$783$	4.49025e14	0.0545232
$784$	0	0
$785$	1.06932e16	1.28034
$786$	0	0
$787$	2.10240e15	0.248230	0.124115	−	0.992268i	$-0.460391\pi$
$787$	2.10240e15	0.248230	0.124115	+	0.992268i	$0.460391\pi$
$788$	0	0
$789$	−7.09340e15	−0.825907
$790$	0	0
$791$	9.54167e15	1.09560
$792$	0	0
$793$	4.20705e15	0.476404
$794$	0	0
$795$	−8.59709e15	−0.960135
$796$	0	0
$797$	−1.27463e16	−1.40399	−0.701993	−	0.712184i	$-0.747706\pi$
$797$	−1.27463e16	−1.40399	−0.701993	+	0.712184i	$0.747706\pi$
$798$	0	0
$799$	−1.67879e16	−1.82385
$800$	0	0
$801$	2.91369e15	0.312222
$802$	0	0
$803$	−6.65955e13	−0.00703898
$804$	0	0
$805$	3.04009e15	0.316964
$806$	0	0
$807$	5.85537e15	0.602213
$808$	0	0
$809$	6.52250e15	0.661755	0.330878	−	0.943674i	$-0.392655\pi$
$809$	6.52250e15	0.661755	0.330878	+	0.943674i	$0.392655\pi$
$810$	0	0
$811$	1.79108e16	1.79267	0.896337	−	0.443374i	$-0.146219\pi$
$811$	1.79108e16	1.79267	0.896337	+	0.443374i	$0.146219\pi$
$812$	0	0
$813$	−3.81228e14	−0.0376432
$814$	0	0
$815$	−1.92915e16	−1.87932
$816$	0	0
$817$	9.75234e15	0.937319
$818$	0	0
$819$	−8.39646e14	−0.0796223
$820$	0	0
$821$	8.38301e15	0.784355	0.392178	−	0.919889i	$-0.371722\pi$
$821$	8.38301e15	0.784355	0.392178	+	0.919889i	$0.371722\pi$
$822$	0	0
$823$	7.01045e15	0.647213	0.323606	−	0.946192i	$-0.395105\pi$
$823$	7.01045e15	0.647213	0.323606	+	0.946192i	$0.395105\pi$
$824$	0	0
$825$	−8.92313e13	−0.00812869
$826$	0	0
$827$	1.88670e15	0.169598	0.0847991	−	0.996398i	$-0.472975\pi$
$827$	1.88670e15	0.169598	0.0847991	+	0.996398i	$0.472975\pi$
$828$	0	0
$829$	1.30038e16	1.15351	0.576754	−	0.816918i	$-0.304319\pi$
$829$	1.30038e16	1.15351	0.576754	+	0.816918i	$0.304319\pi$
$830$	0	0
$831$	4.28302e15	0.374925
$832$	0	0
$833$	−6.64784e15	−0.574293
$834$	0	0
$835$	−1.08413e16	−0.924282
$836$	0	0
$837$	7.54743e13	0.00635052
$838$	0	0
$839$	6.80392e15	0.565026	0.282513	−	0.959264i	$-0.408832\pi$
$839$	6.80392e15	0.565026	0.282513	+	0.959264i	$0.408832\pi$
$840$	0	0
$841$	−1.12212e16	−0.919735
$842$	0	0
$843$	−4.54579e15	−0.367754
$844$	0	0
$845$	1.45078e16	1.15848
$846$	0	0
$847$	−9.54975e15	−0.752720
$848$	0	0
$849$	−7.11791e15	−0.553809
$850$	0	0
$851$	−5.63562e15	−0.432841
$852$	0	0
$853$	−1.31430e16	−0.996493	−0.498247	−	0.867035i	$-0.666023\pi$
$853$	−1.31430e16	−0.996493	−0.498247	+	0.867035i	$0.666023\pi$
$854$	0	0
$855$	3.80871e15	0.285078
$856$	0	0
$857$	−6.93184e15	−0.512217	−0.256109	−	0.966648i	$-0.582440\pi$
$857$	−6.93184e15	−0.512217	−0.256109	+	0.966648i	$0.582440\pi$
$858$	0	0
$859$	1.87967e16	1.37126	0.685628	−	0.727952i	$-0.259528\pi$
$859$	1.87967e16	1.37126	0.685628	+	0.727952i	$0.259528\pi$
$860$	0	0
$861$	2.06802e15	0.148949
$862$	0	0
$863$	−8.94438e15	−0.636050	−0.318025	−	0.948082i	$-0.603020\pi$
$863$	−8.94438e15	−0.636050	−0.318025	+	0.948082i	$0.603020\pi$
$864$	0	0
$865$	−1.41232e16	−0.991618
$866$	0	0
$867$	−6.32894e15	−0.438759
$868$	0	0
$869$	1.89555e14	0.0129755
$870$	0	0
$871$	−4.17604e14	−0.0282270
$872$	0	0
$873$	6.48131e15	0.432598
$874$	0	0
$875$	5.01639e15	0.330634
$876$	0	0
$877$	4.43306e15	0.288540	0.144270	−	0.989538i	$-0.453917\pi$
$877$	4.43306e15	0.288540	0.144270	+	0.989538i	$0.453917\pi$
$878$	0	0
$879$	3.13923e15	0.201783
$880$	0	0
$881$	1.18254e16	0.750666	0.375333	−	0.926890i	$-0.377528\pi$
$881$	1.18254e16	0.750666	0.375333	+	0.926890i	$0.377528\pi$
$882$	0	0
$883$	−4.54771e15	−0.285108	−0.142554	−	0.989787i	$-0.545531\pi$
$883$	−4.54771e15	−0.285108	−0.142554	+	0.989787i	$0.545531\pi$
$884$	0	0
$885$	2.07380e16	1.28404
$886$	0	0
$887$	1.53394e16	0.938056	0.469028	−	0.883183i	$-0.344604\pi$
$887$	1.53394e16	0.938056	0.469028	+	0.883183i	$0.344604\pi$
$888$	0	0
$889$	−1.34855e16	−0.814529
$890$	0	0
$891$	3.97820e13	0.00237334
$892$	0	0
$893$	1.54905e16	0.912816
$894$	0	0
$895$	−2.84904e16	−1.65833
$896$	0	0
$897$	−1.04078e15	−0.0598411
$898$	0	0
$899$	1.64601e14	0.00934876
$900$	0	0
$901$	−3.05273e16	−1.71278
$902$	0	0
$903$	1.10738e16	0.613783
$904$	0	0
$905$	−2.88278e16	−1.57850
$906$	0	0
$907$	3.31822e15	0.179500	0.0897501	−	0.995964i	$-0.471393\pi$
$907$	3.31822e15	0.179500	0.0897501	+	0.995964i	$0.471393\pi$
$908$	0	0
$909$	−8.44052e15	−0.451095
$910$	0	0
$911$	6.57586e15	0.347217	0.173609	−	0.984815i	$-0.444457\pi$
$911$	6.57586e15	0.347217	0.173609	+	0.984815i	$0.444457\pi$
$912$	0	0
$913$	5.50690e14	0.0287288
$914$	0	0
$915$	−2.16695e16	−1.11695
$916$	0	0
$917$	1.19358e16	0.607884
$918$	0	0
$919$	1.25584e15	0.0631972	0.0315986	−	0.999501i	$-0.489940\pi$
$919$	1.25584e15	0.0631972	0.0315986	+	0.999501i	$0.489940\pi$
$920$	0	0
$921$	3.55281e15	0.176662
$922$	0	0
$923$	−1.72476e14	−0.00847460
$924$	0	0
$925$	1.79826e16	0.873116
$926$	0	0
$927$	−1.97386e15	−0.0947061
$928$	0	0
$929$	4.37281e15	0.207336	0.103668	−	0.994612i	$-0.466942\pi$
$929$	4.37281e15	0.207336	0.103668	+	0.994612i	$0.466942\pi$
$930$	0	0
$931$	6.13408e15	0.287426
$932$	0	0
$933$	1.41523e16	0.655358
$934$	0	0
$935$	−7.97551e14	−0.0365001
$936$	0	0
$937$	−1.64013e16	−0.741838	−0.370919	−	0.928665i	$-0.620957\pi$
$937$	−1.64013e16	−0.741838	−0.370919	+	0.928665i	$0.620957\pi$
$938$	0	0
$939$	1.20038e16	0.536610
$940$	0	0
$941$	−2.34547e16	−1.03630	−0.518152	−	0.855288i	$-0.673380\pi$
$941$	−2.34547e16	−1.03630	−0.518152	+	0.855288i	$0.673380\pi$
$942$	0	0
$943$	2.56340e15	0.111944
$944$	0	0
$945$	4.32480e15	0.186677
$946$	0	0
$947$	−1.89709e16	−0.809400	−0.404700	−	0.914450i	$-0.632624\pi$
$947$	−1.89709e16	−0.809400	−0.404700	+	0.914450i	$0.632624\pi$
$948$	0	0
$949$	2.47854e15	0.104528
$950$	0	0
$951$	−9.29443e15	−0.387463
$952$	0	0
$953$	2.69294e16	1.10973	0.554863	−	0.831942i	$-0.312771\pi$
$953$	2.69294e16	1.10973	0.554863	+	0.831942i	$0.312771\pi$
$954$	0	0
$955$	1.48828e16	0.606268
$956$	0	0
$957$	8.67602e13	0.00349385
$958$	0	0
$959$	2.19111e16	0.872290
$960$	0	0
$961$	−2.53808e16	−0.998911
$962$	0	0
$963$	−1.09898e16	−0.427609
$964$	0	0
$965$	2.38328e16	0.916801
$966$	0	0
$967$	−7.36986e15	−0.280294	−0.140147	−	0.990131i	$-0.544758\pi$
$967$	−7.36986e15	−0.280294	−0.140147	+	0.990131i	$0.544758\pi$
$968$	0	0
$969$	1.35243e16	0.508550
$970$	0	0
$971$	6.20619e15	0.230738	0.115369	−	0.993323i	$-0.463195\pi$
$971$	6.20619e15	0.230738	0.115369	+	0.993323i	$0.463195\pi$
$972$	0	0
$973$	1.36025e16	0.500033
$974$	0	0
$975$	3.32099e15	0.120710
$976$	0	0
$977$	−5.29889e16	−1.90443	−0.952214	−	0.305433i	$-0.901199\pi$
$977$	−5.29889e16	−1.90443	−0.952214	+	0.305433i	$0.901199\pi$
$978$	0	0
$979$	5.62979e14	0.0200073
$980$	0	0
$981$	−1.38260e16	−0.485866
$982$	0	0
$983$	5.48799e16	1.90708	0.953539	−	0.301269i	$-0.0974101\pi$
$983$	5.48799e16	1.90708	0.953539	+	0.301269i	$0.0974101\pi$
$984$	0	0
$985$	−1.17062e16	−0.402269
$986$	0	0
$987$	1.75896e16	0.597738
$988$	0	0
$989$	1.37265e16	0.461296
$990$	0	0
$991$	3.27494e16	1.08843	0.544213	−	0.838947i	$-0.316828\pi$
$991$	3.27494e16	1.08843	0.544213	+	0.838947i	$0.316828\pi$
$992$	0	0
$993$	−2.78391e16	−0.915027
$994$	0	0
$995$	8.16326e15	0.265361
$996$	0	0
$997$	9.58605e15	0.308188	0.154094	−	0.988056i	$-0.450754\pi$
$997$	9.58605e15	0.308188	0.154094	+	0.988056i	$0.450754\pi$
$998$	0	0
$999$	−8.01717e15	−0.254924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.12.a.u.1.1		2
4.3	odd	2		192.12.a.x.1.1		2
8.3	odd	2		96.12.a.d.1.2	✓	2
8.5	even	2		96.12.a.f.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.12.a.d.1.2	✓	2	8.3	odd	2
96.12.a.f.1.2	yes	2	8.5	even	2
192.12.a.u.1.1		2	1.1	even	1	trivial
192.12.a.x.1.1		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

\(f(q)\)	\(=\)	\(q-243.000 q^{3} -9000.71 q^{5} +33486.6 q^{7} +59049.0 q^{9} +O(q^{10})\)	\(q-243.000 q^{3} -9000.71 q^{5} +33486.6 q^{7} +59049.0 q^{9} +11409.4 q^{11} -424632. q^{13} +2.18717e6 q^{15} +7.76640e6 q^{17} -7.16619e6 q^{19} -8.13724e6 q^{21} -1.00865e7 q^{23} +3.21847e7 q^{25} -1.43489e7 q^{27} -3.12934e7 q^{29} -5.25994e6 q^{31} -2.77248e6 q^{33} -3.01403e8 q^{35} +5.58731e8 q^{37} +1.03186e8 q^{39} -2.54143e8 q^{41} -1.36088e9 q^{43} -5.31483e8 q^{45} -2.16161e9 q^{47} -8.55975e8 q^{49} -1.88723e9 q^{51} -3.93069e9 q^{53} -1.02693e8 q^{55} +1.74138e9 q^{57} +9.48166e9 q^{59} -9.90753e9 q^{61} +1.97735e9 q^{63} +3.82199e9 q^{65} +9.83448e8 q^{67} +2.45101e9 q^{69} +4.06177e8 q^{71} -5.83691e9 q^{73} -7.82087e9 q^{75} +3.82061e8 q^{77} +1.66139e10 q^{79} +3.48678e9 q^{81} +4.82664e10 q^{83} -6.99031e10 q^{85} +7.60429e9 q^{87} +4.93435e10 q^{89} -1.42195e10 q^{91} +1.27816e9 q^{93} +6.45008e10 q^{95} +1.09762e11 q^{97} +6.73712e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 486 q^{3} - 5300 q^{5} - 38872 q^{7} + 118098 q^{9}+O(q^{10}) \) 2 * q - 486 * q^3 - 5300 * q^5 - 38872 * q^7 + 118098 * q^9	\( 2 q - 486 q^{3} - 5300 q^{5} - 38872 q^{7} + 118098 q^{9} + 1047400 q^{11} + 22900 q^{13} + 1287900 q^{15} + 8733300 q^{17} - 7346600 q^{19} + 9445896 q^{21} + 6711744 q^{23} - 2948210 q^{25} - 28697814 q^{27} - 180180692 q^{29} + 211581400 q^{31} - 254518200 q^{33} - 569181200 q^{35} - 112222700 q^{37} - 5564700 q^{39} - 726961180 q^{41} - 216408856 q^{43} - 312959700 q^{45} - 2174779088 q^{47} + 2402462706 q^{49} - 2122191900 q^{51} + 112024700 q^{53} + 3731208560 q^{55} + 1785223800 q^{57} + 3243949400 q^{59} - 16526230620 q^{61} - 2295352728 q^{63} + 5478177080 q^{65} + 20772619112 q^{67} - 1630953792 q^{69} - 20637101600 q^{71} - 18548203500 q^{73} + 716415030 q^{75} - 74580754400 q^{77} + 28230083800 q^{79} + 6973568802 q^{81} - 7189282056 q^{83} - 66324859080 q^{85} + 43783908156 q^{87} + 103679180788 q^{89} - 46602268400 q^{91} - 51414280200 q^{93} + 63833162000 q^{95} + 199614486500 q^{97} + 61847922600 q^{99}+O(q^{100}) \) 2 * q - 486 * q^3 - 5300 * q^5 - 38872 * q^7 + 118098 * q^9 + 1047400 * q^11 + 22900 * q^13 + 1287900 * q^15 + 8733300 * q^17 - 7346600 * q^19 + 9445896 * q^21 + 6711744 * q^23 - 2948210 * q^25 - 28697814 * q^27 - 180180692 * q^29 + 211581400 * q^31 - 254518200 * q^33 - 569181200 * q^35 - 112222700 * q^37 - 5564700 * q^39 - 726961180 * q^41 - 216408856 * q^43 - 312959700 * q^45 - 2174779088 * q^47 + 2402462706 * q^49 - 2122191900 * q^51 + 112024700 * q^53 + 3731208560 * q^55 + 1785223800 * q^57 + 3243949400 * q^59 - 16526230620 * q^61 - 2295352728 * q^63 + 5478177080 * q^65 + 20772619112 * q^67 - 1630953792 * q^69 - 20637101600 * q^71 - 18548203500 * q^73 + 716415030 * q^75 - 74580754400 * q^77 + 28230083800 * q^79 + 6973568802 * q^81 - 7189282056 * q^83 - 66324859080 * q^85 + 43783908156 * q^87 + 103679180788 * q^89 - 46602268400 * q^91 - 51414280200 * q^93 + 63833162000 * q^95 + 199614486500 * q^97 + 61847922600 * q^99

Introduction

L-functions

Modular forms

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