LMFDB - Embedded newform 3381.2.a.bh.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 100x^{3} - 17x^{2} - 28x - 2 $ x^10 - 4x^9 - 6x^8 + 36x^7 + x^6 - 100x^5 + 26x^4 + 100x^3 - 17x^2 - 28x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.19951$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.19951 q^{2} +1.00000 q^{3} +2.83784 q^{4} -2.02197 q^{5} +2.19951 q^{6} +1.84284 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.19951 q^{2} +1.00000 q^{3} +2.83784 q^{4} -2.02197 q^{5} +2.19951 q^{6} +1.84284 q^{8} +1.00000 q^{9} -4.44733 q^{10} -0.770843 q^{11} +2.83784 q^{12} -5.34495 q^{13} -2.02197 q^{15} -1.62234 q^{16} -3.30778 q^{17} +2.19951 q^{18} -4.86616 q^{19} -5.73802 q^{20} -1.69548 q^{22} +1.00000 q^{23} +1.84284 q^{24} -0.911654 q^{25} -11.7563 q^{26} +1.00000 q^{27} -6.83725 q^{29} -4.44733 q^{30} -5.67211 q^{31} -7.25403 q^{32} -0.770843 q^{33} -7.27550 q^{34} +2.83784 q^{36} +2.63614 q^{37} -10.7032 q^{38} -5.34495 q^{39} -3.72616 q^{40} +12.0882 q^{41} +4.18627 q^{43} -2.18753 q^{44} -2.02197 q^{45} +2.19951 q^{46} +1.77768 q^{47} -1.62234 q^{48} -2.00519 q^{50} -3.30778 q^{51} -15.1681 q^{52} +9.28714 q^{53} +2.19951 q^{54} +1.55862 q^{55} -4.86616 q^{57} -15.0386 q^{58} -8.41464 q^{59} -5.73802 q^{60} +3.80472 q^{61} -12.4759 q^{62} -12.7106 q^{64} +10.8073 q^{65} -1.69548 q^{66} +13.5330 q^{67} -9.38696 q^{68} +1.00000 q^{69} +2.98469 q^{71} +1.84284 q^{72} -3.63418 q^{73} +5.79822 q^{74} -0.911654 q^{75} -13.8094 q^{76} -11.7563 q^{78} +3.46262 q^{79} +3.28031 q^{80} +1.00000 q^{81} +26.5881 q^{82} +7.39498 q^{83} +6.68822 q^{85} +9.20775 q^{86} -6.83725 q^{87} -1.42054 q^{88} -6.99456 q^{89} -4.44733 q^{90} +2.83784 q^{92} -5.67211 q^{93} +3.91001 q^{94} +9.83921 q^{95} -7.25403 q^{96} -10.3760 q^{97} -0.770843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{2} + 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} - 12 q^{8} + 10 q^{9}+O(q^{10}) $ 10 * q - 4 * q^2 + 10 * q^3 + 8 * q^4 - 4 * q^5 - 4 * q^6 - 12 * q^8 + 10 * q^9	$ 10 q - 4 q^{2} + 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} - 12 q^{8} + 10 q^{9} - 8 q^{10} - 2 q^{11} + 8 q^{12} - 16 q^{13} - 4 q^{15} + 4 q^{16} - 12 q^{17} - 4 q^{18} - 26 q^{19} - 8 q^{22} + 10 q^{23} - 12 q^{24} + 14 q^{25} + 12 q^{26} + 10 q^{27} - 16 q^{29} - 8 q^{30} - 20 q^{31} - 8 q^{32} - 2 q^{33} + 4 q^{34} + 8 q^{36} + 8 q^{37} + 8 q^{38} - 16 q^{39} + 12 q^{40} - 22 q^{41} - 4 q^{43} - 24 q^{44} - 4 q^{45} - 4 q^{46} - 6 q^{47} + 4 q^{48} - 48 q^{50} - 12 q^{51} - 24 q^{52} - 30 q^{53} - 4 q^{54} - 48 q^{55} - 26 q^{57} + 24 q^{58} - 42 q^{59} - 14 q^{61} - 40 q^{62} + 8 q^{64} - 44 q^{65} - 8 q^{66} - 8 q^{68} + 10 q^{69} + 8 q^{71} - 12 q^{72} - 24 q^{73} + 8 q^{74} + 14 q^{75} - 32 q^{76} + 12 q^{78} + 32 q^{79} - 28 q^{80} + 10 q^{81} + 64 q^{82} - 28 q^{83} - 4 q^{85} - 4 q^{86} - 16 q^{87} + 20 q^{88} - 8 q^{90} + 8 q^{92} - 20 q^{93} - 8 q^{94} - 16 q^{95} - 8 q^{96} + 12 q^{97} - 2 q^{99}+O(q^{100}) $ 10 * q - 4 * q^2 + 10 * q^3 + 8 * q^4 - 4 * q^5 - 4 * q^6 - 12 * q^8 + 10 * q^9 - 8 * q^10 - 2 * q^11 + 8 * q^12 - 16 * q^13 - 4 * q^15 + 4 * q^16 - 12 * q^17 - 4 * q^18 - 26 * q^19 - 8 * q^22 + 10 * q^23 - 12 * q^24 + 14 * q^25 + 12 * q^26 + 10 * q^27 - 16 * q^29 - 8 * q^30 - 20 * q^31 - 8 * q^32 - 2 * q^33 + 4 * q^34 + 8 * q^36 + 8 * q^37 + 8 * q^38 - 16 * q^39 + 12 * q^40 - 22 * q^41 - 4 * q^43 - 24 * q^44 - 4 * q^45 - 4 * q^46 - 6 * q^47 + 4 * q^48 - 48 * q^50 - 12 * q^51 - 24 * q^52 - 30 * q^53 - 4 * q^54 - 48 * q^55 - 26 * q^57 + 24 * q^58 - 42 * q^59 - 14 * q^61 - 40 * q^62 + 8 * q^64 - 44 * q^65 - 8 * q^66 - 8 * q^68 + 10 * q^69 + 8 * q^71 - 12 * q^72 - 24 * q^73 + 8 * q^74 + 14 * q^75 - 32 * q^76 + 12 * q^78 + 32 * q^79 - 28 * q^80 + 10 * q^81 + 64 * q^82 - 28 * q^83 - 4 * q^85 - 4 * q^86 - 16 * q^87 + 20 * q^88 - 8 * q^90 + 8 * q^92 - 20 * q^93 - 8 * q^94 - 16 * q^95 - 8 * q^96 + 12 * q^97 - 2 * 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 $

$2$	2.19951	1.55529	0.777644	−	0.628705i	$-0.216415\pi$
$2$	2.19951	1.55529	0.777644	+	0.628705i	$0.216415\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.83784	1.41892
$5$	−2.02197	−0.904251	−0.452125	−	0.891954i	$-0.649334\pi$
$5$	−2.02197	−0.904251	−0.452125	+	0.891954i	$0.649334\pi$
$6$	2.19951	0.897946
$7$	0	0
$7$	0	0
$8$	1.84284	0.651543
$9$	1.00000	0.333333
$10$	−4.44733	−1.40637
$11$	−0.770843	−0.232418	−0.116209	−	0.993225i	$-0.537074\pi$
$11$	−0.770843	−0.232418	−0.116209	+	0.993225i	$0.537074\pi$
$12$	2.83784	0.819215
$13$	−5.34495	−1.48242	−0.741211	−	0.671272i	$-0.765748\pi$
$13$	−5.34495	−1.48242	−0.741211	+	0.671272i	$0.765748\pi$
$14$	0	0
$15$	−2.02197	−0.522069
$16$	−1.62234	−0.405584
$17$	−3.30778	−0.802255	−0.401127	−	0.916022i	$-0.631382\pi$
$17$	−3.30778	−0.802255	−0.401127	+	0.916022i	$0.631382\pi$
$18$	2.19951	0.518429
$19$	−4.86616	−1.11637	−0.558187	−	0.829715i	$-0.688503\pi$
$19$	−4.86616	−1.11637	−0.558187	+	0.829715i	$0.688503\pi$
$20$	−5.73802	−1.28306
$21$	0	0
$22$	−1.69548	−0.361477
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.84284	0.376169
$25$	−0.911654	−0.182331
$26$	−11.7563	−2.30559
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.83725	−1.26965	−0.634823	−	0.772657i	$-0.718927\pi$
$29$	−6.83725	−1.26965	−0.634823	+	0.772657i	$0.718927\pi$
$30$	−4.44733	−0.811968
$31$	−5.67211	−1.01874	−0.509370	−	0.860548i	$-0.670122\pi$
$31$	−5.67211	−1.01874	−0.509370	+	0.860548i	$0.670122\pi$
$32$	−7.25403	−1.28234
$33$	−0.770843	−0.134187
$34$	−7.27550	−1.24774
$35$	0	0
$36$	2.83784	0.472974
$37$	2.63614	0.433379	0.216689	−	0.976241i	$-0.430474\pi$
$37$	2.63614	0.433379	0.216689	+	0.976241i	$0.430474\pi$
$38$	−10.7032	−1.73628
$39$	−5.34495	−0.855876
$40$	−3.72616	−0.589158
$41$	12.0882	1.88786	0.943930	−	0.330146i	$-0.107098\pi$
$41$	12.0882	1.88786	0.943930	+	0.330146i	$0.107098\pi$
$42$	0	0
$43$	4.18627	0.638401	0.319200	−	0.947687i	$-0.396586\pi$
$43$	4.18627	0.638401	0.319200	+	0.947687i	$0.396586\pi$
$44$	−2.18753	−0.329783
$45$	−2.02197	−0.301417
$46$	2.19951	0.324300
$47$	1.77768	0.259301	0.129650	−	0.991560i	$-0.458615\pi$
$47$	1.77768	0.259301	0.129650	+	0.991560i	$0.458615\pi$
$48$	−1.62234	−0.234164
$49$	0	0
$50$	−2.00519	−0.283577
$51$	−3.30778	−0.463182
$52$	−15.1681	−2.10344
$53$	9.28714	1.27569	0.637843	−	0.770166i	$-0.279827\pi$
$53$	9.28714	1.27569	0.637843	+	0.770166i	$0.279827\pi$
$54$	2.19951	0.299315
$55$	1.55862	0.210164
$56$	0	0
$57$	−4.86616	−0.644539
$58$	−15.0386	−1.97467
$59$	−8.41464	−1.09549	−0.547747	−	0.836644i	$-0.684514\pi$
$59$	−8.41464	−1.09549	−0.547747	+	0.836644i	$0.684514\pi$
$60$	−5.73802	−0.740775
$61$	3.80472	0.487144	0.243572	−	0.969883i	$-0.421681\pi$
$61$	3.80472	0.487144	0.243572	+	0.969883i	$0.421681\pi$
$62$	−12.4759	−1.58443
$63$	0	0
$64$	−12.7106	−1.58883
$65$	10.8073	1.34048
$66$	−1.69548	−0.208699
$67$	13.5330	1.65332	0.826661	−	0.562700i	$-0.190237\pi$
$67$	13.5330	1.65332	0.826661	+	0.562700i	$0.190237\pi$
$68$	−9.38696	−1.13834
$69$	1.00000	0.120386
$70$	0	0
$71$	2.98469	0.354218	0.177109	−	0.984191i	$-0.443326\pi$
$71$	2.98469	0.354218	0.177109	+	0.984191i	$0.443326\pi$
$72$	1.84284	0.217181
$73$	−3.63418	−0.425348	−0.212674	−	0.977123i	$-0.568217\pi$
$73$	−3.63418	−0.425348	−0.212674	+	0.977123i	$0.568217\pi$
$74$	5.79822	0.674029
$75$	−0.911654	−0.105269
$76$	−13.8094	−1.58405
$77$	0	0
$78$	−11.7563	−1.33113
$79$	3.46262	0.389575	0.194788	−	0.980845i	$-0.437598\pi$
$79$	3.46262	0.389575	0.194788	+	0.980845i	$0.437598\pi$
$80$	3.28031	0.366750
$81$	1.00000	0.111111
$82$	26.5881	2.93617
$83$	7.39498	0.811704	0.405852	−	0.913939i	$-0.366975\pi$
$83$	7.39498	0.811704	0.405852	+	0.913939i	$0.366975\pi$
$84$	0	0
$85$	6.68822	0.725439
$86$	9.20775	0.992897
$87$	−6.83725	−0.733031
$88$	−1.42054	−0.151430
$89$	−6.99456	−0.741422	−0.370711	−	0.928748i	$-0.620886\pi$
$89$	−6.99456	−0.741422	−0.370711	+	0.928748i	$0.620886\pi$
$90$	−4.44733	−0.468790
$91$	0	0
$92$	2.83784	0.295866
$93$	−5.67211	−0.588170
$94$	3.91001	0.403287
$95$	9.83921	1.00948
$96$	−7.25403	−0.740361
$97$	−10.3760	−1.05352	−0.526762	−	0.850013i	$-0.676594\pi$
$97$	−10.3760	−1.05352	−0.526762	+	0.850013i	$0.676594\pi$
$98$	0	0
$99$	−0.770843	−0.0774726
$100$	−2.58713	−0.258713
$101$	−0.635798	−0.0632642	−0.0316321	−	0.999500i	$-0.510070\pi$
$101$	−0.635798	−0.0632642	−0.0316321	+	0.999500i	$0.510070\pi$
$102$	−7.27550	−0.720381
$103$	−9.82705	−0.968288	−0.484144	−	0.874988i	$-0.660869\pi$
$103$	−9.82705	−0.968288	−0.484144	+	0.874988i	$0.660869\pi$
$104$	−9.84989	−0.965862
$105$	0	0
$106$	20.4271	1.98406
$107$	−11.0929	−1.07239	−0.536197	−	0.844093i	$-0.680140\pi$
$107$	−11.0929	−1.07239	−0.536197	+	0.844093i	$0.680140\pi$
$108$	2.83784	0.273072
$109$	−1.45283	−0.139156	−0.0695779	−	0.997577i	$-0.522165\pi$
$109$	−1.45283	−0.139156	−0.0695779	+	0.997577i	$0.522165\pi$
$110$	3.42819	0.326866
$111$	2.63614	0.250211
$112$	0	0
$113$	−1.64713	−0.154949	−0.0774745	−	0.996994i	$-0.524686\pi$
$113$	−1.64713	−0.154949	−0.0774745	+	0.996994i	$0.524686\pi$
$114$	−10.7032	−1.00244
$115$	−2.02197	−0.188549
$116$	−19.4031	−1.80153
$117$	−5.34495	−0.494141
$118$	−18.5081	−1.70381
$119$	0	0
$120$	−3.72616	−0.340151
$121$	−10.4058	−0.945982
$122$	8.36851	0.757650
$123$	12.0882	1.08996
$124$	−16.0965	−1.44551
$125$	11.9532	1.06912
$126$	0	0
$127$	−4.03711	−0.358236	−0.179118	−	0.983828i	$-0.557324\pi$
$127$	−4.03711	−0.358236	−0.179118	+	0.983828i	$0.557324\pi$
$128$	−13.4491	−1.18874
$129$	4.18627	0.368581
$130$	23.7708	2.08483
$131$	−15.5492	−1.35854	−0.679271	−	0.733887i	$-0.737704\pi$
$131$	−15.5492	−1.35854	−0.679271	+	0.733887i	$0.737704\pi$
$132$	−2.18753	−0.190400
$133$	0	0
$134$	29.7660	2.57139
$135$	−2.02197	−0.174023
$136$	−6.09572	−0.522704
$137$	−10.1938	−0.870913	−0.435457	−	0.900210i	$-0.643413\pi$
$137$	−10.1938	−0.870913	−0.435457	+	0.900210i	$0.643413\pi$
$138$	2.19951	0.187235
$139$	7.45070	0.631961	0.315980	−	0.948766i	$-0.397667\pi$
$139$	7.45070	0.631961	0.315980	+	0.948766i	$0.397667\pi$
$140$	0	0
$141$	1.77768	0.149707
$142$	6.56486	0.550911
$143$	4.12011	0.344541
$144$	−1.62234	−0.135195
$145$	13.8247	1.14808
$146$	−7.99341	−0.661539
$147$	0	0
$148$	7.48095	0.614930
$149$	−21.2827	−1.74355	−0.871773	−	0.489910i	$-0.837030\pi$
$149$	−21.2827	−1.74355	−0.871773	+	0.489910i	$0.837030\pi$
$150$	−2.00519	−0.163723
$151$	8.37306	0.681390	0.340695	−	0.940174i	$-0.389338\pi$
$151$	8.37306	0.681390	0.340695	+	0.940174i	$0.389338\pi$
$152$	−8.96756	−0.727365
$153$	−3.30778	−0.267418
$154$	0	0
$155$	11.4688	0.921197
$156$	−15.1681	−1.21442
$157$	9.96963	0.795663	0.397832	−	0.917458i	$-0.369763\pi$
$157$	9.96963	0.795663	0.397832	+	0.917458i	$0.369763\pi$
$158$	7.61607	0.605902
$159$	9.28714	0.736518
$160$	14.6674	1.15956
$161$	0	0
$162$	2.19951	0.172810
$163$	−0.853932	−0.0668851	−0.0334426	−	0.999441i	$-0.510647\pi$
$163$	−0.853932	−0.0668851	−0.0334426	+	0.999441i	$0.510647\pi$
$164$	34.3044	2.67872
$165$	1.55862	0.121338
$166$	16.2653	1.26243
$167$	−2.08307	−0.161193	−0.0805963	−	0.996747i	$-0.525682\pi$
$167$	−2.08307	−0.161193	−0.0805963	+	0.996747i	$0.525682\pi$
$168$	0	0
$169$	15.5685	1.19757
$170$	14.7108	1.12827
$171$	−4.86616	−0.372125
$172$	11.8800	0.905840
$173$	−15.8943	−1.20842	−0.604212	−	0.796824i	$-0.706512\pi$
$173$	−15.8943	−1.20842	−0.604212	+	0.796824i	$0.706512\pi$
$174$	−15.0386	−1.14007
$175$	0	0
$176$	1.25057	0.0942650
$177$	−8.41464	−0.632483
$178$	−15.3846	−1.15312
$179$	−22.2011	−1.65939	−0.829693	−	0.558220i	$-0.811484\pi$
$179$	−22.2011	−1.65939	−0.829693	+	0.558220i	$0.811484\pi$
$180$	−5.73802	−0.427687
$181$	22.0116	1.63611	0.818053	−	0.575143i	$-0.195054\pi$
$181$	22.0116	1.63611	0.818053	+	0.575143i	$0.195054\pi$
$182$	0	0
$183$	3.80472	0.281253
$184$	1.84284	0.135856
$185$	−5.33019	−0.391883
$186$	−12.4759	−0.914774
$187$	2.54978	0.186458
$188$	5.04476	0.367927
$189$	0	0
$190$	21.6414	1.57003
$191$	−4.64223	−0.335900	−0.167950	−	0.985796i	$-0.553715\pi$
$191$	−4.64223	−0.335900	−0.167950	+	0.985796i	$0.553715\pi$
$192$	−12.7106	−0.917311
$193$	0.451781	0.0325199	0.0162599	−	0.999868i	$-0.494824\pi$
$193$	0.451781	0.0325199	0.0162599	+	0.999868i	$0.494824\pi$
$194$	−22.8221	−1.63853
$195$	10.8073	0.773927
$196$	0	0
$197$	1.37160	0.0977222	0.0488611	−	0.998806i	$-0.484441\pi$
$197$	1.37160	0.0977222	0.0488611	+	0.998806i	$0.484441\pi$
$198$	−1.69548	−0.120492
$199$	17.5087	1.24116	0.620580	−	0.784143i	$-0.286897\pi$
$199$	17.5087	1.24116	0.620580	+	0.784143i	$0.286897\pi$
$200$	−1.68004	−0.118796
$201$	13.5330	0.954546
$202$	−1.39844	−0.0983941
$203$	0	0
$204$	−9.38696	−0.657219
$205$	−24.4419	−1.70710
$206$	−21.6147	−1.50597
$207$	1.00000	0.0695048
$208$	8.67130	0.601246
$209$	3.75104	0.259465
$210$	0	0
$211$	−5.72366	−0.394033	−0.197017	−	0.980400i	$-0.563125\pi$
$211$	−5.72366	−0.394033	−0.197017	+	0.980400i	$0.563125\pi$
$212$	26.3554	1.81010
$213$	2.98469	0.204508
$214$	−24.3990	−1.66788
$215$	−8.46450	−0.577274
$216$	1.84284	0.125390
$217$	0	0
$218$	−3.19551	−0.216427
$219$	−3.63418	−0.245575
$220$	4.42311	0.298206
$221$	17.6799	1.18928
$222$	5.79822	0.389151
$223$	−10.2090	−0.683646	−0.341823	−	0.939764i	$-0.611044\pi$
$223$	−10.2090	−0.683646	−0.341823	+	0.939764i	$0.611044\pi$
$224$	0	0
$225$	−0.911654	−0.0607770
$226$	−3.62288	−0.240990
$227$	−20.4888	−1.35989	−0.679946	−	0.733263i	$-0.737997\pi$
$227$	−20.4888	−1.35989	−0.679946	+	0.733263i	$0.737997\pi$
$228$	−13.8094	−0.914549
$229$	15.5450	1.02724	0.513622	−	0.858017i	$-0.328303\pi$
$229$	15.5450	1.02724	0.513622	+	0.858017i	$0.328303\pi$
$230$	−4.44733	−0.293248
$231$	0	0
$232$	−12.6000	−0.827229
$233$	−20.7860	−1.36174	−0.680868	−	0.732407i	$-0.738397\pi$
$233$	−20.7860	−1.36174	−0.680868	+	0.732407i	$0.738397\pi$
$234$	−11.7563	−0.768531
$235$	−3.59440	−0.234473
$236$	−23.8794	−1.55442
$237$	3.46262	0.224921
$238$	0	0
$239$	24.6126	1.59205	0.796027	−	0.605261i	$-0.206931\pi$
$239$	24.6126	1.59205	0.796027	+	0.605261i	$0.206931\pi$
$240$	3.28031	0.211743
$241$	−25.6895	−1.65481	−0.827404	−	0.561607i	$-0.810183\pi$
$241$	−25.6895	−1.65481	−0.827404	+	0.561607i	$0.810183\pi$
$242$	−22.8877	−1.47127
$243$	1.00000	0.0641500
$244$	10.7972	0.691219
$245$	0	0
$246$	26.5881	1.69520
$247$	26.0094	1.65494
$248$	−10.4528	−0.663753
$249$	7.39498	0.468638
$250$	26.2911	1.66279
$251$	20.6778	1.30517	0.652584	−	0.757716i	$-0.273685\pi$
$251$	20.6778	1.30517	0.652584	+	0.757716i	$0.273685\pi$
$252$	0	0
$253$	−0.770843	−0.0484625
$254$	−8.87967	−0.557160
$255$	6.68822	0.418833
$256$	−4.16016	−0.260010
$257$	−21.9669	−1.37026	−0.685129	−	0.728422i	$-0.740254\pi$
$257$	−21.9669	−1.37026	−0.685129	+	0.728422i	$0.740254\pi$
$258$	9.20775	0.573249
$259$	0	0
$260$	30.6694	1.90204
$261$	−6.83725	−0.423215
$262$	−34.2007	−2.11292
$263$	20.1274	1.24111	0.620553	−	0.784165i	$-0.286908\pi$
$263$	20.1274	1.24111	0.620553	+	0.784165i	$0.286908\pi$
$264$	−1.42054	−0.0874283
$265$	−18.7783	−1.15354
$266$	0	0
$267$	−6.99456	−0.428060
$268$	38.4046	2.34593
$269$	14.8051	0.902682	0.451341	−	0.892351i	$-0.350946\pi$
$269$	14.8051	0.902682	0.451341	+	0.892351i	$0.350946\pi$
$270$	−4.44733	−0.270656
$271$	7.72287	0.469131	0.234565	−	0.972100i	$-0.424633\pi$
$271$	7.72287	0.469131	0.234565	+	0.972100i	$0.424633\pi$
$272$	5.36633	0.325382
$273$	0	0
$274$	−22.4213	−1.35452
$275$	0.702742	0.0423770
$276$	2.83784	0.170818
$277$	−10.6917	−0.642403	−0.321202	−	0.947011i	$-0.604087\pi$
$277$	−10.6917	−0.642403	−0.321202	+	0.947011i	$0.604087\pi$
$278$	16.3879	0.982881
$279$	−5.67211	−0.339580
$280$	0	0
$281$	−6.00987	−0.358519	−0.179259	−	0.983802i	$-0.557370\pi$
$281$	−6.00987	−0.358519	−0.179259	+	0.983802i	$0.557370\pi$
$282$	3.91001	0.232838
$283$	7.93307	0.471572	0.235786	−	0.971805i	$-0.424234\pi$
$283$	7.93307	0.471572	0.235786	+	0.971805i	$0.424234\pi$
$284$	8.47009	0.502607
$285$	9.83921	0.582824
$286$	9.06223	0.535861
$287$	0	0
$288$	−7.25403	−0.427448
$289$	−6.05859	−0.356387
$290$	30.4076	1.78559
$291$	−10.3760	−0.608252
$292$	−10.3132	−0.603536
$293$	−5.46757	−0.319419	−0.159709	−	0.987164i	$-0.551056\pi$
$293$	−5.46757	−0.319419	−0.159709	+	0.987164i	$0.551056\pi$
$294$	0	0
$295$	17.0141	0.990600
$296$	4.85799	0.282365
$297$	−0.770843	−0.0447288
$298$	−46.8115	−2.71172
$299$	−5.34495	−0.309106
$300$	−2.58713	−0.149368
$301$	0	0
$302$	18.4166	1.05976
$303$	−0.635798	−0.0365256
$304$	7.89454	0.452783
$305$	−7.69301	−0.440500
$306$	−7.27550	−0.415912
$307$	8.78760	0.501535	0.250767	−	0.968047i	$-0.419317\pi$
$307$	8.78760	0.501535	0.250767	+	0.968047i	$0.419317\pi$
$308$	0	0
$309$	−9.82705	−0.559041
$310$	25.2257	1.43273
$311$	13.8568	0.785746	0.392873	−	0.919593i	$-0.371481\pi$
$311$	13.8568	0.785746	0.392873	+	0.919593i	$0.371481\pi$
$312$	−9.84989	−0.557640
$313$	32.0848	1.81354	0.906769	−	0.421627i	$-0.138541\pi$
$313$	32.0848	1.81354	0.906769	+	0.421627i	$0.138541\pi$
$314$	21.9283	1.23749
$315$	0	0
$316$	9.82637	0.552777
$317$	−32.0467	−1.79992	−0.899962	−	0.435969i	$-0.856406\pi$
$317$	−32.0467	−1.79992	−0.899962	+	0.435969i	$0.856406\pi$
$318$	20.4271	1.14550
$319$	5.27045	0.295088
$320$	25.7005	1.43670
$321$	−11.0929	−0.619147
$322$	0	0
$323$	16.0962	0.895616
$324$	2.83784	0.157658
$325$	4.87274	0.270291
$326$	−1.87823	−0.104026
$327$	−1.45283	−0.0803416
$328$	22.2766	1.23002
$329$	0	0
$330$	3.42819	0.188716
$331$	30.5694	1.68025	0.840124	−	0.542394i	$-0.182482\pi$
$331$	30.5694	1.68025	0.840124	+	0.542394i	$0.182482\pi$
$332$	20.9858	1.15174
$333$	2.63614	0.144460
$334$	−4.58173	−0.250701
$335$	−27.3633	−1.49502
$336$	0	0
$337$	−22.3749	−1.21884	−0.609419	−	0.792848i	$-0.708597\pi$
$337$	−22.3749	−1.21884	−0.609419	+	0.792848i	$0.708597\pi$
$338$	34.2430	1.86257
$339$	−1.64713	−0.0894598
$340$	18.9801	1.02934
$341$	4.37230	0.236773
$342$	−10.7032	−0.578761
$343$	0	0
$344$	7.71464	0.415946
$345$	−2.02197	−0.108859
$346$	−34.9597	−1.87945
$347$	−28.8004	−1.54609	−0.773043	−	0.634354i	$-0.781266\pi$
$347$	−28.8004	−1.54609	−0.773043	+	0.634354i	$0.781266\pi$
$348$	−19.4031	−1.04011
$349$	14.8464	0.794710	0.397355	−	0.917665i	$-0.369928\pi$
$349$	14.8464	0.794710	0.397355	+	0.917665i	$0.369928\pi$
$350$	0	0
$351$	−5.34495	−0.285292
$352$	5.59171	0.298039
$353$	−7.31260	−0.389210	−0.194605	−	0.980882i	$-0.562343\pi$
$353$	−7.31260	−0.389210	−0.194605	+	0.980882i	$0.562343\pi$
$354$	−18.5081	−0.983694
$355$	−6.03495	−0.320302
$356$	−19.8495	−1.05202
$357$	0	0
$358$	−48.8315	−2.58082
$359$	12.8145	0.676326	0.338163	−	0.941088i	$-0.390194\pi$
$359$	12.8145	0.676326	0.338163	+	0.941088i	$0.390194\pi$
$360$	−3.72616	−0.196386
$361$	4.67951	0.246290
$362$	48.4146	2.54462
$363$	−10.4058	−0.546163
$364$	0	0
$365$	7.34818	0.384621
$366$	8.36851	0.437429
$367$	−9.06544	−0.473212	−0.236606	−	0.971606i	$-0.576035\pi$
$367$	−9.06544	−0.473212	−0.236606	+	0.971606i	$0.576035\pi$
$368$	−1.62234	−0.0845701
$369$	12.0882	0.629287
$370$	−11.7238	−0.609491
$371$	0	0
$372$	−16.0965	−0.834567
$373$	19.5361	1.01154	0.505770	−	0.862669i	$-0.331209\pi$
$373$	19.5361	1.01154	0.505770	+	0.862669i	$0.331209\pi$
$374$	5.60826	0.289996
$375$	11.9532	0.617259
$376$	3.27598	0.168946
$377$	36.5448	1.88215
$378$	0	0
$379$	−11.5914	−0.595409	−0.297704	−	0.954658i	$-0.596221\pi$
$379$	−11.5914	−0.595409	−0.297704	+	0.954658i	$0.596221\pi$
$380$	27.9221	1.43237
$381$	−4.03711	−0.206828
$382$	−10.2106	−0.522421
$383$	21.9403	1.12110	0.560549	−	0.828121i	$-0.310590\pi$
$383$	21.9403	1.12110	0.560549	+	0.828121i	$0.310590\pi$
$384$	−13.4491	−0.686321
$385$	0	0
$386$	0.993696	0.0505778
$387$	4.18627	0.212800
$388$	−29.4455	−1.49487
$389$	−18.0471	−0.915026	−0.457513	−	0.889203i	$-0.651260\pi$
$389$	−18.0471	−0.915026	−0.457513	+	0.889203i	$0.651260\pi$
$390$	23.7708	1.20368
$391$	−3.30778	−0.167282
$392$	0	0
$393$	−15.5492	−0.784355
$394$	3.01684	0.151986
$395$	−7.00130	−0.352274
$396$	−2.18753	−0.109928
$397$	−22.6313	−1.13583	−0.567916	−	0.823087i	$-0.692250\pi$
$397$	−22.6313	−1.13583	−0.567916	+	0.823087i	$0.692250\pi$
$398$	38.5106	1.93036
$399$	0	0
$400$	1.47901	0.0739505
$401$	−16.2429	−0.811132	−0.405566	−	0.914066i	$-0.632925\pi$
$401$	−16.2429	−0.811132	−0.405566	+	0.914066i	$0.632925\pi$
$402$	29.7660	1.48459
$403$	30.3171	1.51020
$404$	−1.80429	−0.0897670
$405$	−2.02197	−0.100472
$406$	0	0
$407$	−2.03205	−0.100725
$408$	−6.09572	−0.301783
$409$	31.7328	1.56909	0.784544	−	0.620074i	$-0.212897\pi$
$409$	31.7328	1.56909	0.784544	+	0.620074i	$0.212897\pi$
$410$	−53.7602	−2.65503
$411$	−10.1938	−0.502822
$412$	−27.8876	−1.37392
$413$	0	0
$414$	2.19951	0.108100
$415$	−14.9524	−0.733984
$416$	38.7724	1.90097
$417$	7.45070	0.364863
$418$	8.25046	0.403543
$419$	−9.18679	−0.448804	−0.224402	−	0.974497i	$-0.572043\pi$
$419$	−9.18679	−0.448804	−0.224402	+	0.974497i	$0.572043\pi$
$420$	0	0
$421$	17.5886	0.857214	0.428607	−	0.903491i	$-0.359004\pi$
$421$	17.5886	0.857214	0.428607	+	0.903491i	$0.359004\pi$
$422$	−12.5893	−0.612835
$423$	1.77768	0.0864336
$424$	17.1147	0.831165
$425$	3.01555	0.146276
$426$	6.56486	0.318068
$427$	0	0
$428$	−31.4800	−1.52164
$429$	4.12011	0.198921
$430$	−18.6177	−0.897828
$431$	14.9538	0.720300	0.360150	−	0.932894i	$-0.382725\pi$
$431$	14.9538	0.720300	0.360150	+	0.932894i	$0.382725\pi$
$432$	−1.62234	−0.0780547
$433$	32.0055	1.53809	0.769043	−	0.639197i	$-0.220733\pi$
$433$	32.0055	1.53809	0.769043	+	0.639197i	$0.220733\pi$
$434$	0	0
$435$	13.8247	0.662843
$436$	−4.12290	−0.197451
$437$	−4.86616	−0.232780
$438$	−7.99341	−0.381940
$439$	0.565332	0.0269818	0.0134909	−	0.999909i	$-0.495706\pi$
$439$	0.565332	0.0269818	0.0134909	+	0.999909i	$0.495706\pi$
$440$	2.87229	0.136931
$441$	0	0
$442$	38.8871	1.84967
$443$	−8.49397	−0.403561	−0.201780	−	0.979431i	$-0.564673\pi$
$443$	−8.49397	−0.403561	−0.201780	+	0.979431i	$0.564673\pi$
$444$	7.48095	0.355030
$445$	14.1428	0.670431
$446$	−22.4548	−1.06327
$447$	−21.2827	−1.00664
$448$	0	0
$449$	20.5048	0.967681	0.483841	−	0.875156i	$-0.339241\pi$
$449$	20.5048	0.967681	0.483841	+	0.875156i	$0.339241\pi$
$450$	−2.00519	−0.0945257
$451$	−9.31810	−0.438772
$452$	−4.67430	−0.219860
$453$	8.37306	0.393401
$454$	−45.0654	−2.11502
$455$	0	0
$456$	−8.96756	−0.419945
$457$	0.899589	0.0420810	0.0210405	−	0.999779i	$-0.493302\pi$
$457$	0.899589	0.0420810	0.0210405	+	0.999779i	$0.493302\pi$
$458$	34.1914	1.59766
$459$	−3.30778	−0.154394
$460$	−5.73802	−0.267537
$461$	2.36074	0.109951	0.0549754	−	0.998488i	$-0.482492\pi$
$461$	2.36074	0.109951	0.0549754	+	0.998488i	$0.482492\pi$
$462$	0	0
$463$	−35.2799	−1.63960	−0.819799	−	0.572652i	$-0.805915\pi$
$463$	−35.2799	−1.63960	−0.819799	+	0.572652i	$0.805915\pi$
$464$	11.0923	0.514948
$465$	11.4688	0.531853
$466$	−45.7190	−2.11789
$467$	1.13070	0.0523226	0.0261613	−	0.999658i	$-0.491672\pi$
$467$	1.13070	0.0523226	0.0261613	+	0.999658i	$0.491672\pi$
$468$	−15.1681	−0.701146
$469$	0	0
$470$	−7.90592	−0.364673
$471$	9.96963	0.459376
$472$	−15.5069	−0.713761
$473$	−3.22696	−0.148376
$474$	7.61607	0.349818
$475$	4.43626	0.203549
$476$	0	0
$477$	9.28714	0.425229
$478$	54.1356	2.47610
$479$	−2.47140	−0.112921	−0.0564605	−	0.998405i	$-0.517981\pi$
$479$	−2.47140	−0.112921	−0.0564605	+	0.998405i	$0.517981\pi$
$480$	14.6674	0.669472
$481$	−14.0900	−0.642450
$482$	−56.5044	−2.57370
$483$	0	0
$484$	−29.5300	−1.34227
$485$	20.9799	0.952649
$486$	2.19951	0.0997718
$487$	10.8718	0.492647	0.246324	−	0.969188i	$-0.420777\pi$
$487$	10.8718	0.492647	0.246324	+	0.969188i	$0.420777\pi$
$488$	7.01149	0.317395
$489$	−0.853932	−0.0386161
$490$	0	0
$491$	26.3739	1.19024	0.595119	−	0.803638i	$-0.297105\pi$
$491$	26.3739	1.19024	0.595119	+	0.803638i	$0.297105\pi$
$492$	34.3044	1.54656
$493$	22.6161	1.01858
$494$	57.2078	2.57390
$495$	1.55862	0.0700547
$496$	9.20206	0.413185
$497$	0	0
$498$	16.2653	0.728866
$499$	−24.9552	−1.11715	−0.558575	−	0.829454i	$-0.688652\pi$
$499$	−24.9552	−1.11715	−0.558575	+	0.829454i	$0.688652\pi$
$500$	33.9212	1.51700
$501$	−2.08307	−0.0930646
$502$	45.4809	2.02991
$503$	21.7649	0.970449	0.485225	−	0.874390i	$-0.338738\pi$
$503$	21.7649	0.970449	0.485225	+	0.874390i	$0.338738\pi$
$504$	0	0
$505$	1.28556	0.0572067
$506$	−1.69548	−0.0753731
$507$	15.5685	0.691419
$508$	−11.4567	−0.508308
$509$	−27.5115	−1.21943	−0.609714	−	0.792622i	$-0.708716\pi$
$509$	−27.5115	−1.21943	−0.609714	+	0.792622i	$0.708716\pi$
$510$	14.7108	0.651405
$511$	0	0
$512$	17.7479	0.784353
$513$	−4.86616	−0.214846
$514$	−48.3164	−2.13115
$515$	19.8700	0.875575
$516$	11.8800	0.522987
$517$	−1.37031	−0.0602661
$518$	0	0
$519$	−15.8943	−0.697684
$520$	19.9161	0.873381
$521$	2.95749	0.129570	0.0647849	−	0.997899i	$-0.479364\pi$
$521$	2.95749	0.129570	0.0647849	+	0.997899i	$0.479364\pi$
$522$	−15.0386	−0.658222
$523$	−35.4543	−1.55031	−0.775154	−	0.631773i	$-0.782328\pi$
$523$	−35.4543	−1.55031	−0.775154	+	0.631773i	$0.782328\pi$
$524$	−44.1262	−1.92766
$525$	0	0
$526$	44.2703	1.93028
$527$	18.7621	0.817289
$528$	1.25057	0.0544239
$529$	1.00000	0.0434783
$530$	−41.3030	−1.79409
$531$	−8.41464	−0.365164
$532$	0	0
$533$	−64.6108	−2.79860
$534$	−15.3846	−0.665757
$535$	22.4295	0.969713
$536$	24.9392	1.07721
$537$	−22.2011	−0.958047
$538$	32.5639	1.40393
$539$	0	0
$540$	−5.73802	−0.246925
$541$	−37.1558	−1.59745	−0.798725	−	0.601696i	$-0.794492\pi$
$541$	−37.1558	−1.59745	−0.798725	+	0.601696i	$0.794492\pi$
$542$	16.9865	0.729634
$543$	22.0116	0.944606
$544$	23.9947	1.02877
$545$	2.93757	0.125832
$546$	0	0
$547$	4.27577	0.182819	0.0914093	−	0.995813i	$-0.470863\pi$
$547$	4.27577	0.182819	0.0914093	+	0.995813i	$0.470863\pi$
$548$	−28.9283	−1.23576
$549$	3.80472	0.162381
$550$	1.54569	0.0659084
$551$	33.2712	1.41740
$552$	1.84284	0.0784366
$553$	0	0
$554$	−23.5165	−0.999122
$555$	−5.33019	−0.226254
$556$	21.1439	0.896702
$557$	−36.0689	−1.52829	−0.764144	−	0.645046i	$-0.776838\pi$
$557$	−36.0689	−1.52829	−0.764144	+	0.645046i	$0.776838\pi$
$558$	−12.4759	−0.528145
$559$	−22.3754	−0.946379
$560$	0	0
$561$	2.54978	0.107652
$562$	−13.2188	−0.557600
$563$	−5.78229	−0.243695	−0.121847	−	0.992549i	$-0.538882\pi$
$563$	−5.78229	−0.243695	−0.121847	+	0.992549i	$0.538882\pi$
$564$	5.04476	0.212423
$565$	3.33044	0.140113
$566$	17.4489	0.733430
$567$	0	0
$568$	5.50032	0.230788
$569$	29.1545	1.22222	0.611110	−	0.791546i	$-0.290723\pi$
$569$	29.1545	1.22222	0.611110	+	0.791546i	$0.290723\pi$
$570$	21.6414	0.906460
$571$	−46.2706	−1.93636	−0.968181	−	0.250249i	$-0.919487\pi$
$571$	−46.2706	−1.93636	−0.968181	+	0.250249i	$0.919487\pi$
$572$	11.6922	0.488877
$573$	−4.64223	−0.193932
$574$	0	0
$575$	−0.911654	−0.0380186
$576$	−12.7106	−0.529610
$577$	−39.2126	−1.63244	−0.816222	−	0.577739i	$-0.803935\pi$
$577$	−39.2126	−1.63244	−0.816222	+	0.577739i	$0.803935\pi$
$578$	−13.3259	−0.554285
$579$	0.451781	0.0187754
$580$	39.2323	1.62903
$581$	0	0
$582$	−22.8221	−0.946007
$583$	−7.15892	−0.296492
$584$	−6.69721	−0.277133
$585$	10.8073	0.446827
$586$	−12.0260	−0.496789
$587$	−10.0803	−0.416060	−0.208030	−	0.978122i	$-0.566705\pi$
$587$	−10.0803	−0.416060	−0.208030	+	0.978122i	$0.566705\pi$
$588$	0	0
$589$	27.6014	1.13729
$590$	37.4227	1.54067
$591$	1.37160	0.0564199
$592$	−4.27671	−0.175772
$593$	5.99288	0.246098	0.123049	−	0.992401i	$-0.460733\pi$
$593$	5.99288	0.246098	0.123049	+	0.992401i	$0.460733\pi$
$594$	−1.69548	−0.0695662
$595$	0	0
$596$	−60.3969	−2.47395
$597$	17.5087	0.716584
$598$	−11.7563	−0.480749
$599$	−8.75046	−0.357534	−0.178767	−	0.983891i	$-0.557211\pi$
$599$	−8.75046	−0.357534	−0.178767	+	0.983891i	$0.557211\pi$
$600$	−1.68004	−0.0685871
$601$	37.5481	1.53162	0.765810	−	0.643067i	$-0.222338\pi$
$601$	37.5481	1.53162	0.765810	+	0.643067i	$0.222338\pi$
$602$	0	0
$603$	13.5330	0.551107
$604$	23.7614	0.966839
$605$	21.0402	0.855405
$606$	−1.39844	−0.0568079
$607$	−1.18412	−0.0480618	−0.0240309	−	0.999711i	$-0.507650\pi$
$607$	−1.18412	−0.0480618	−0.0240309	+	0.999711i	$0.507650\pi$
$608$	35.2993	1.43157
$609$	0	0
$610$	−16.9208	−0.685105
$611$	−9.50158	−0.384393
$612$	−9.38696	−0.379445
$613$	31.1986	1.26010	0.630051	−	0.776554i	$-0.283034\pi$
$613$	31.1986	1.26010	0.630051	+	0.776554i	$0.283034\pi$
$614$	19.3284	0.780031
$615$	−24.4419	−0.985594
$616$	0	0
$617$	−4.93570	−0.198704	−0.0993519	−	0.995052i	$-0.531677\pi$
$617$	−4.93570	−0.198704	−0.0993519	+	0.995052i	$0.531677\pi$
$618$	−21.6147	−0.869470
$619$	−44.2069	−1.77683	−0.888413	−	0.459044i	$-0.848192\pi$
$619$	−44.2069	−1.77683	−0.888413	+	0.459044i	$0.848192\pi$
$620$	32.5467	1.30711
$621$	1.00000	0.0401286
$622$	30.4781	1.22206
$623$	0	0
$624$	8.67130	0.347130
$625$	−19.6106	−0.784425
$626$	70.5708	2.82057
$627$	3.75104	0.149802
$628$	28.2922	1.12898
$629$	−8.71978	−0.347680
$630$	0	0
$631$	−26.7579	−1.06521	−0.532607	−	0.846363i	$-0.678788\pi$
$631$	−26.7579	−1.06521	−0.532607	+	0.846363i	$0.678788\pi$
$632$	6.38106	0.253825
$633$	−5.72366	−0.227495
$634$	−70.4871	−2.79940
$635$	8.16290	0.323935
$636$	26.3554	1.04506
$637$	0	0
$638$	11.5924	0.458948
$639$	2.98469	0.118073
$640$	27.1936	1.07492
$641$	2.59935	0.102668	0.0513342	−	0.998682i	$-0.483653\pi$
$641$	2.59935	0.102668	0.0513342	+	0.998682i	$0.483653\pi$
$642$	−24.3990	−0.962952
$643$	−8.05146	−0.317519	−0.158759	−	0.987317i	$-0.550749\pi$
$643$	−8.05146	−0.317519	−0.158759	+	0.987317i	$0.550749\pi$
$644$	0	0
$645$	−8.46450	−0.333289
$646$	35.4037	1.39294
$647$	−32.4658	−1.27636	−0.638182	−	0.769885i	$-0.720313\pi$
$647$	−32.4658	−1.27636	−0.638182	+	0.769885i	$0.720313\pi$
$648$	1.84284	0.0723937
$649$	6.48637	0.254612
$650$	10.7176	0.420381
$651$	0	0
$652$	−2.42332	−0.0949047
$653$	0.395275	0.0154683	0.00773414	−	0.999970i	$-0.497538\pi$
$653$	0.395275	0.0154683	0.00773414	+	0.999970i	$0.497538\pi$
$654$	−3.19551	−0.124954
$655$	31.4400	1.22846
$656$	−19.6111	−0.765685
$657$	−3.63418	−0.141783
$658$	0	0
$659$	−42.0291	−1.63722	−0.818610	−	0.574350i	$-0.805255\pi$
$659$	−42.0291	−1.63722	−0.818610	+	0.574350i	$0.805255\pi$
$660$	4.42311	0.172169
$661$	−42.0507	−1.63558	−0.817791	−	0.575516i	$-0.804801\pi$
$661$	−42.0507	−1.63558	−0.817791	+	0.575516i	$0.804801\pi$
$662$	67.2378	2.61327
$663$	17.6799	0.686631
$664$	13.6278	0.528860
$665$	0	0
$666$	5.79822	0.224676
$667$	−6.83725	−0.264740
$668$	−5.91142	−0.228720
$669$	−10.2090	−0.394703
$670$	−60.1859	−2.32518
$671$	−2.93284	−0.113221
$672$	0	0
$673$	32.5420	1.25440	0.627201	−	0.778858i	$-0.284201\pi$
$673$	32.5420	1.25440	0.627201	+	0.778858i	$0.284201\pi$
$674$	−49.2138	−1.89564
$675$	−0.911654	−0.0350896
$676$	44.1808	1.69926
$677$	−2.72400	−0.104692	−0.0523459	−	0.998629i	$-0.516670\pi$
$677$	−2.72400	−0.104692	−0.0523459	+	0.998629i	$0.516670\pi$
$678$	−3.62288	−0.139136
$679$	0	0
$680$	12.3253	0.472655
$681$	−20.4888	−0.785134
$682$	9.61692	0.368251
$683$	−34.4571	−1.31846	−0.659231	−	0.751940i	$-0.729118\pi$
$683$	−34.4571	−1.31846	−0.659231	+	0.751940i	$0.729118\pi$
$684$	−13.8094	−0.528015
$685$	20.6115	0.787524
$686$	0	0
$687$	15.5450	0.593080
$688$	−6.79154	−0.258925
$689$	−49.6393	−1.89111
$690$	−4.44733	−0.169307
$691$	8.21320	0.312445	0.156222	−	0.987722i	$-0.450068\pi$
$691$	8.21320	0.312445	0.156222	+	0.987722i	$0.450068\pi$
$692$	−45.1056	−1.71466
$693$	0	0
$694$	−63.3467	−2.40461
$695$	−15.0651	−0.571451
$696$	−12.6000	−0.477601
$697$	−39.9851	−1.51454
$698$	32.6548	1.23600
$699$	−20.7860	−0.786198
$700$	0	0
$701$	−26.1085	−0.986105	−0.493053	−	0.870000i	$-0.664119\pi$
$701$	−26.1085	−0.986105	−0.493053	+	0.870000i	$0.664119\pi$
$702$	−11.7563	−0.443712
$703$	−12.8279	−0.483813
$704$	9.79790	0.369272
$705$	−3.59440	−0.135373
$706$	−16.0841	−0.605334
$707$	0	0
$708$	−23.8794	−0.897444
$709$	9.51874	0.357484	0.178742	−	0.983896i	$-0.442797\pi$
$709$	9.51874	0.357484	0.178742	+	0.983896i	$0.442797\pi$
$710$	−13.2739	−0.498161
$711$	3.46262	0.129858
$712$	−12.8899	−0.483068
$713$	−5.67211	−0.212422
$714$	0	0
$715$	−8.33073	−0.311552
$716$	−63.0031	−2.35454
$717$	24.6126	0.919173
$718$	28.1857	1.05188
$719$	29.9224	1.11592	0.557958	−	0.829869i	$-0.311585\pi$
$719$	29.9224	1.11592	0.557958	+	0.829869i	$0.311585\pi$
$720$	3.28031	0.122250
$721$	0	0
$722$	10.2926	0.383052
$723$	−25.6895	−0.955404
$724$	62.4653	2.32151
$725$	6.23321	0.231496
$726$	−22.8877	−0.849441
$727$	17.6120	0.653192	0.326596	−	0.945164i	$-0.394098\pi$
$727$	17.6120	0.653192	0.326596	+	0.945164i	$0.394098\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	16.1624	0.598197
$731$	−13.8473	−0.512160
$732$	10.7972	0.399076
$733$	−44.1628	−1.63119	−0.815595	−	0.578623i	$-0.803590\pi$
$733$	−44.1628	−1.63119	−0.815595	+	0.578623i	$0.803590\pi$
$734$	−19.9395	−0.735981
$735$	0	0
$736$	−7.25403	−0.267387
$737$	−10.4318	−0.384262
$738$	26.5881	0.978722
$739$	33.5999	1.23599	0.617996	−	0.786181i	$-0.287945\pi$
$739$	33.5999	1.23599	0.617996	+	0.786181i	$0.287945\pi$
$740$	−15.1262	−0.556051
$741$	26.0094	0.955478
$742$	0	0
$743$	−25.0694	−0.919706	−0.459853	−	0.887995i	$-0.652098\pi$
$743$	−25.0694	−0.919706	−0.459853	+	0.887995i	$0.652098\pi$
$744$	−10.4528	−0.383218
$745$	43.0329	1.57660
$746$	42.9698	1.57323
$747$	7.39498	0.270568
$748$	7.23587	0.264570
$749$	0	0
$750$	26.2911	0.960015
$751$	−8.48633	−0.309671	−0.154835	−	0.987940i	$-0.549485\pi$
$751$	−8.48633	−0.309671	−0.154835	+	0.987940i	$0.549485\pi$
$752$	−2.88399	−0.105168
$753$	20.6778	0.753539
$754$	80.3806	2.92729
$755$	−16.9300	−0.616148
$756$	0	0
$757$	10.1697	0.369622	0.184811	−	0.982774i	$-0.440833\pi$
$757$	10.1697	0.369622	0.184811	+	0.982774i	$0.440833\pi$
$758$	−25.4953	−0.926032
$759$	−0.770843	−0.0279798
$760$	18.1321	0.657721
$761$	40.4234	1.46535	0.732674	−	0.680580i	$-0.238272\pi$
$761$	40.4234	1.46535	0.732674	+	0.680580i	$0.238272\pi$
$762$	−8.87967	−0.321676
$763$	0	0
$764$	−13.1739	−0.476616
$765$	6.68822	0.241813
$766$	48.2580	1.74363
$767$	44.9758	1.62398
$768$	−4.16016	−0.150117
$769$	−42.1358	−1.51946	−0.759728	−	0.650241i	$-0.774668\pi$
$769$	−42.1358	−1.51946	−0.759728	+	0.650241i	$0.774668\pi$
$770$	0	0
$771$	−21.9669	−0.791119
$772$	1.28208	0.0461432
$773$	23.2362	0.835750	0.417875	−	0.908505i	$-0.362775\pi$
$773$	23.2362	0.835750	0.417875	+	0.908505i	$0.362775\pi$
$774$	9.20775	0.330966
$775$	5.17100	0.185748
$776$	−19.1213	−0.686416
$777$	0	0
$778$	−39.6948	−1.42313
$779$	−58.8231	−2.10756
$780$	30.6694	1.09814
$781$	−2.30073	−0.0823265
$782$	−7.27550	−0.260171
$783$	−6.83725	−0.244344
$784$	0	0
$785$	−20.1583	−0.719479
$786$	−34.2007	−1.21990
$787$	52.8045	1.88228	0.941139	−	0.338020i	$-0.109757\pi$
$787$	52.8045	1.88228	0.941139	+	0.338020i	$0.109757\pi$
$788$	3.89237	0.138660
$789$	20.1274	0.716553
$790$	−15.3994	−0.547887
$791$	0	0
$792$	−1.42054	−0.0504768
$793$	−20.3360	−0.722153
$794$	−49.7777	−1.76654
$795$	−18.7783	−0.665997
$796$	49.6870	1.76111
$797$	5.29618	0.187600	0.0938002	−	0.995591i	$-0.470099\pi$
$797$	5.29618	0.187600	0.0938002	+	0.995591i	$0.470099\pi$
$798$	0	0
$799$	−5.88016	−0.208025
$800$	6.61317	0.233811
$801$	−6.99456	−0.247141
$802$	−35.7264	−1.26154
$803$	2.80138	0.0988585
$804$	38.4046	1.35443
$805$	0	0
$806$	66.6828	2.34880
$807$	14.8051	0.521164
$808$	−1.17168	−0.0412194
$809$	25.2540	0.887884	0.443942	−	0.896056i	$-0.353580\pi$
$809$	25.2540	0.887884	0.443942	+	0.896056i	$0.353580\pi$
$810$	−4.44733	−0.156263
$811$	−31.8443	−1.11821	−0.559103	−	0.829098i	$-0.688854\pi$
$811$	−31.8443	−1.11821	−0.559103	+	0.829098i	$0.688854\pi$
$812$	0	0
$813$	7.72287	0.270853
$814$	−4.46951	−0.156656
$815$	1.72662	0.0604809
$816$	5.36633	0.187859
$817$	−20.3711	−0.712694
$818$	69.7966	2.44038
$819$	0	0
$820$	−69.3623	−2.42224
$821$	−51.8137	−1.80831	−0.904155	−	0.427205i	$-0.859498\pi$
$821$	−51.8137	−1.80831	−0.904155	+	0.427205i	$0.859498\pi$
$822$	−22.4213	−0.782033
$823$	38.1457	1.32968	0.664839	−	0.746987i	$-0.268500\pi$
$823$	38.1457	1.32968	0.664839	+	0.746987i	$0.268500\pi$
$824$	−18.1097	−0.630881
$825$	0.702742	0.0244663
$826$	0	0
$827$	10.4356	0.362883	0.181441	−	0.983402i	$-0.441924\pi$
$827$	10.4356	0.362883	0.181441	+	0.983402i	$0.441924\pi$
$828$	2.83784	0.0986218
$829$	20.9420	0.727346	0.363673	−	0.931527i	$-0.381523\pi$
$829$	20.9420	0.727346	0.363673	+	0.931527i	$0.381523\pi$
$830$	−32.8879	−1.14156
$831$	−10.6917	−0.370892
$832$	67.9376	2.35531
$833$	0	0
$834$	16.3879	0.567466
$835$	4.21189	0.145759
$836$	10.6449	0.368161
$837$	−5.67211	−0.196057
$838$	−20.2064	−0.698020
$839$	24.6942	0.852538	0.426269	−	0.904596i	$-0.359828\pi$
$839$	24.6942	0.852538	0.426269	+	0.904596i	$0.359828\pi$
$840$	0	0
$841$	17.7481	0.612002
$842$	38.6862	1.33322
$843$	−6.00987	−0.206991
$844$	−16.2429	−0.559102
$845$	−31.4789	−1.08291
$846$	3.91001	0.134429
$847$	0	0
$848$	−15.0669	−0.517398
$849$	7.93307	0.272262
$850$	6.63274	0.227501
$851$	2.63614	0.0903657
$852$	8.47009	0.290180
$853$	37.0762	1.26947	0.634733	−	0.772731i	$-0.281110\pi$
$853$	37.0762	1.26947	0.634733	+	0.772731i	$0.281110\pi$
$854$	0	0
$855$	9.83921	0.336494
$856$	−20.4425	−0.698711
$857$	−53.2920	−1.82042	−0.910210	−	0.414146i	$-0.864080\pi$
$857$	−53.2920	−1.82042	−0.910210	+	0.414146i	$0.864080\pi$
$858$	9.06223	0.309379
$859$	−36.2889	−1.23816	−0.619080	−	0.785328i	$-0.712494\pi$
$859$	−36.2889	−1.23816	−0.619080	+	0.785328i	$0.712494\pi$
$860$	−24.0209	−0.819106
$861$	0	0
$862$	32.8911	1.12027
$863$	36.4294	1.24007	0.620036	−	0.784574i	$-0.287118\pi$
$863$	36.4294	1.24007	0.620036	+	0.784574i	$0.287118\pi$
$864$	−7.25403	−0.246787
$865$	32.1378	1.09272
$866$	70.3964	2.39217
$867$	−6.05859	−0.205760
$868$	0	0
$869$	−2.66914	−0.0905442
$870$	30.4076	1.03091
$871$	−72.3333	−2.45092
$872$	−2.67733	−0.0906660
$873$	−10.3760	−0.351175
$874$	−10.7032	−0.362040
$875$	0	0
$876$	−10.3132	−0.348451
$877$	32.5223	1.09820	0.549100	−	0.835757i	$-0.314971\pi$
$877$	32.5223	1.09820	0.549100	+	0.835757i	$0.314971\pi$
$878$	1.24345	0.0419645
$879$	−5.46757	−0.184417
$880$	−2.52860	−0.0852391
$881$	38.1367	1.28486	0.642428	−	0.766346i	$-0.277927\pi$
$881$	38.1367	1.28486	0.642428	+	0.766346i	$0.277927\pi$
$882$	0	0
$883$	−52.7834	−1.77630	−0.888151	−	0.459552i	$-0.848010\pi$
$883$	−52.7834	−1.77630	−0.888151	+	0.459552i	$0.848010\pi$
$884$	50.1728	1.68749
$885$	17.0141	0.571923
$886$	−18.6826	−0.627653
$887$	−42.0704	−1.41259	−0.706293	−	0.707920i	$-0.749634\pi$
$887$	−42.0704	−1.41259	−0.706293	+	0.707920i	$0.749634\pi$
$888$	4.85799	0.163024
$889$	0	0
$890$	31.1071	1.04271
$891$	−0.770843	−0.0258242
$892$	−28.9715	−0.970039
$893$	−8.65045	−0.289476
$894$	−46.8115	−1.56561
$895$	44.8898	1.50050
$896$	0	0
$897$	−5.34495	−0.178463
$898$	45.1005	1.50502
$899$	38.7816	1.29344
$900$	−2.58713	−0.0862377
$901$	−30.7198	−1.02343
$902$	−20.4953	−0.682417
$903$	0	0
$904$	−3.03540	−0.100956
$905$	−44.5066	−1.47945
$906$	18.4166	0.611852
$907$	59.0288	1.96002	0.980010	−	0.198950i	$-0.0637531\pi$
$907$	59.0288	1.96002	0.980010	+	0.198950i	$0.0637531\pi$
$908$	−58.1441	−1.92958
$909$	−0.635798	−0.0210881
$910$	0	0
$911$	−52.1773	−1.72871	−0.864355	−	0.502882i	$-0.832273\pi$
$911$	−52.1773	−1.72871	−0.864355	+	0.502882i	$0.832273\pi$
$912$	7.89454	0.261414
$913$	−5.70036	−0.188655
$914$	1.97865	0.0654481
$915$	−7.69301	−0.254323
$916$	44.1143	1.45758
$917$	0	0
$918$	−7.27550	−0.240127
$919$	−29.7064	−0.979924	−0.489962	−	0.871744i	$-0.662989\pi$
$919$	−29.7064	−0.979924	−0.489962	+	0.871744i	$0.662989\pi$
$920$	−3.72616	−0.122848
$921$	8.78760	0.289561
$922$	5.19248	0.171005
$923$	−15.9530	−0.525100
$924$	0	0
$925$	−2.40325	−0.0790183
$926$	−77.5985	−2.55005
$927$	−9.82705	−0.322763
$928$	49.5976	1.62812
$929$	−2.29948	−0.0754435	−0.0377217	−	0.999288i	$-0.512010\pi$
$929$	−2.29948	−0.0754435	−0.0377217	+	0.999288i	$0.512010\pi$
$930$	25.2257	0.827185
$931$	0	0
$932$	−58.9873	−1.93219
$933$	13.8568	0.453650
$934$	2.48699	0.0813767
$935$	−5.15557	−0.168605
$936$	−9.84989	−0.321954
$937$	29.6547	0.968776	0.484388	−	0.874853i	$-0.339042\pi$
$937$	29.6547	0.968776	0.484388	+	0.874853i	$0.339042\pi$
$938$	0	0
$939$	32.0848	1.04705
$940$	−10.2003	−0.332698
$941$	−48.0185	−1.56536	−0.782680	−	0.622425i	$-0.786148\pi$
$941$	−48.0185	−1.56536	−0.782680	+	0.622425i	$0.786148\pi$
$942$	21.9283	0.714463
$943$	12.0882	0.393646
$944$	13.6514	0.444314
$945$	0	0
$946$	−7.09773	−0.230767
$947$	28.9640	0.941204	0.470602	−	0.882346i	$-0.344037\pi$
$947$	28.9640	0.941204	0.470602	+	0.882346i	$0.344037\pi$
$948$	9.82637	0.319146
$949$	19.4245	0.630545
$950$	9.75759	0.316578
$951$	−32.0467	−1.03919
$952$	0	0
$953$	−42.1181	−1.36434	−0.682170	−	0.731193i	$-0.738964\pi$
$953$	−42.1181	−1.36434	−0.682170	+	0.731193i	$0.738964\pi$
$954$	20.4271	0.661353
$955$	9.38643	0.303738
$956$	69.8466	2.25900
$957$	5.27045	0.170369
$958$	−5.43586	−0.175625
$959$	0	0
$960$	25.7005	0.829479
$961$	1.17279	0.0378319
$962$	−30.9912	−0.999195
$963$	−11.0929	−0.357465
$964$	−72.9029	−2.34804
$965$	−0.913485	−0.0294061
$966$	0	0
$967$	48.8202	1.56995	0.784975	−	0.619527i	$-0.212676\pi$
$967$	48.8202	1.56995	0.784975	+	0.619527i	$0.212676\pi$
$968$	−19.1762	−0.616348
$969$	16.0962	0.517084
$970$	46.1456	1.48164
$971$	−0.872873	−0.0280118	−0.0140059	−	0.999902i	$-0.504458\pi$
$971$	−0.872873	−0.0280118	−0.0140059	+	0.999902i	$0.504458\pi$
$972$	2.83784	0.0910238
$973$	0	0
$974$	23.9126	0.766208
$975$	4.87274	0.156053
$976$	−6.17253	−0.197578
$977$	2.05665	0.0657981	0.0328991	−	0.999459i	$-0.489526\pi$
$977$	2.05665	0.0657981	0.0328991	+	0.999459i	$0.489526\pi$
$978$	−1.87823	−0.0600592
$979$	5.39171	0.172320
$980$	0	0
$981$	−1.45283	−0.0463852
$982$	58.0097	1.85116
$983$	37.0715	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$983$	37.0715	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$984$	22.2766	0.710153
$985$	−2.77332	−0.0883654
$986$	49.7444	1.58419
$987$	0	0
$988$	73.8105	2.34822
$989$	4.18627	0.133116
$990$	3.42819	0.108955
$991$	−8.40716	−0.267062	−0.133531	−	0.991045i	$-0.542632\pi$
$991$	−8.40716	−0.267062	−0.133531	+	0.991045i	$0.542632\pi$
$992$	41.1456	1.30637
$993$	30.5694	0.970092
$994$	0	0
$995$	−35.4020	−1.12232
$996$	20.9858	0.664960
$997$	−7.76086	−0.245789	−0.122895	−	0.992420i	$-0.539218\pi$
$997$	−7.76086	−0.245789	−0.122895	+	0.992420i	$0.539218\pi$
$998$	−54.8893	−1.73749
$999$	2.63614	0.0834038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bh.1.10	yes	10
7.6	odd	2		3381.2.a.bg.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3381.2.a.bg.1.10	✓	10	7.6	odd	2
3381.2.a.bh.1.10	yes	10	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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q+2.19951 q^{2} +1.00000 q^{3} +2.83784 q^{4} -2.02197 q^{5} +2.19951 q^{6} +1.84284 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.19951 q^{2} +1.00000 q^{3} +2.83784 q^{4} -2.02197 q^{5} +2.19951 q^{6} +1.84284 q^{8} +1.00000 q^{9} -4.44733 q^{10} -0.770843 q^{11} +2.83784 q^{12} -5.34495 q^{13} -2.02197 q^{15} -1.62234 q^{16} -3.30778 q^{17} +2.19951 q^{18} -4.86616 q^{19} -5.73802 q^{20} -1.69548 q^{22} +1.00000 q^{23} +1.84284 q^{24} -0.911654 q^{25} -11.7563 q^{26} +1.00000 q^{27} -6.83725 q^{29} -4.44733 q^{30} -5.67211 q^{31} -7.25403 q^{32} -0.770843 q^{33} -7.27550 q^{34} +2.83784 q^{36} +2.63614 q^{37} -10.7032 q^{38} -5.34495 q^{39} -3.72616 q^{40} +12.0882 q^{41} +4.18627 q^{43} -2.18753 q^{44} -2.02197 q^{45} +2.19951 q^{46} +1.77768 q^{47} -1.62234 q^{48} -2.00519 q^{50} -3.30778 q^{51} -15.1681 q^{52} +9.28714 q^{53} +2.19951 q^{54} +1.55862 q^{55} -4.86616 q^{57} -15.0386 q^{58} -8.41464 q^{59} -5.73802 q^{60} +3.80472 q^{61} -12.4759 q^{62} -12.7106 q^{64} +10.8073 q^{65} -1.69548 q^{66} +13.5330 q^{67} -9.38696 q^{68} +1.00000 q^{69} +2.98469 q^{71} +1.84284 q^{72} -3.63418 q^{73} +5.79822 q^{74} -0.911654 q^{75} -13.8094 q^{76} -11.7563 q^{78} +3.46262 q^{79} +3.28031 q^{80} +1.00000 q^{81} +26.5881 q^{82} +7.39498 q^{83} +6.68822 q^{85} +9.20775 q^{86} -6.83725 q^{87} -1.42054 q^{88} -6.99456 q^{89} -4.44733 q^{90} +2.83784 q^{92} -5.67211 q^{93} +3.91001 q^{94} +9.83921 q^{95} -7.25403 q^{96} -10.3760 q^{97} -0.770843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} + 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} - 12 q^{8} + 10 q^{9}+O(q^{10}) \) 10 * q - 4 * q^2 + 10 * q^3 + 8 * q^4 - 4 * q^5 - 4 * q^6 - 12 * q^8 + 10 * q^9	\( 10 q - 4 q^{2} + 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} - 12 q^{8} + 10 q^{9} - 8 q^{10} - 2 q^{11} + 8 q^{12} - 16 q^{13} - 4 q^{15} + 4 q^{16} - 12 q^{17} - 4 q^{18} - 26 q^{19} - 8 q^{22} + 10 q^{23} - 12 q^{24} + 14 q^{25} + 12 q^{26} + 10 q^{27} - 16 q^{29} - 8 q^{30} - 20 q^{31} - 8 q^{32} - 2 q^{33} + 4 q^{34} + 8 q^{36} + 8 q^{37} + 8 q^{38} - 16 q^{39} + 12 q^{40} - 22 q^{41} - 4 q^{43} - 24 q^{44} - 4 q^{45} - 4 q^{46} - 6 q^{47} + 4 q^{48} - 48 q^{50} - 12 q^{51} - 24 q^{52} - 30 q^{53} - 4 q^{54} - 48 q^{55} - 26 q^{57} + 24 q^{58} - 42 q^{59} - 14 q^{61} - 40 q^{62} + 8 q^{64} - 44 q^{65} - 8 q^{66} - 8 q^{68} + 10 q^{69} + 8 q^{71} - 12 q^{72} - 24 q^{73} + 8 q^{74} + 14 q^{75} - 32 q^{76} + 12 q^{78} + 32 q^{79} - 28 q^{80} + 10 q^{81} + 64 q^{82} - 28 q^{83} - 4 q^{85} - 4 q^{86} - 16 q^{87} + 20 q^{88} - 8 q^{90} + 8 q^{92} - 20 q^{93} - 8 q^{94} - 16 q^{95} - 8 q^{96} + 12 q^{97} - 2 q^{99}+O(q^{100}) \) 10 * q - 4 * q^2 + 10 * q^3 + 8 * q^4 - 4 * q^5 - 4 * q^6 - 12 * q^8 + 10 * q^9 - 8 * q^10 - 2 * q^11 + 8 * q^12 - 16 * q^13 - 4 * q^15 + 4 * q^16 - 12 * q^17 - 4 * q^18 - 26 * q^19 - 8 * q^22 + 10 * q^23 - 12 * q^24 + 14 * q^25 + 12 * q^26 + 10 * q^27 - 16 * q^29 - 8 * q^30 - 20 * q^31 - 8 * q^32 - 2 * q^33 + 4 * q^34 + 8 * q^36 + 8 * q^37 + 8 * q^38 - 16 * q^39 + 12 * q^40 - 22 * q^41 - 4 * q^43 - 24 * q^44 - 4 * q^45 - 4 * q^46 - 6 * q^47 + 4 * q^48 - 48 * q^50 - 12 * q^51 - 24 * q^52 - 30 * q^53 - 4 * q^54 - 48 * q^55 - 26 * q^57 + 24 * q^58 - 42 * q^59 - 14 * q^61 - 40 * q^62 + 8 * q^64 - 44 * q^65 - 8 * q^66 - 8 * q^68 + 10 * q^69 + 8 * q^71 - 12 * q^72 - 24 * q^73 + 8 * q^74 + 14 * q^75 - 32 * q^76 + 12 * q^78 + 32 * q^79 - 28 * q^80 + 10 * q^81 + 64 * q^82 - 28 * q^83 - 4 * q^85 - 4 * q^86 - 16 * q^87 + 20 * q^88 - 8 * q^90 + 8 * q^92 - 20 * q^93 - 8 * q^94 - 16 * q^95 - 8 * q^96 + 12 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

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Database

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