LMFDB - Embedded newform 3381.2.a.bk.1.8

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} + 108x^{3} - 33x^{2} - 36x + 14 $ x^10 - 4x^9 - 6x^8 + 36x^7 + x^6 - 100x^5 + 26x^4 + 108x^3 - 33x^2 - 36x + 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.94871$ 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)

$f(q)$	$=$	$q+1.94871 q^{2} -1.00000 q^{3} +1.79746 q^{4} -1.40111 q^{5} -1.94871 q^{6} -0.394699 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.94871 q^{2} -1.00000 q^{3} +1.79746 q^{4} -1.40111 q^{5} -1.94871 q^{6} -0.394699 q^{8} +1.00000 q^{9} -2.73035 q^{10} +4.53162 q^{11} -1.79746 q^{12} -2.35527 q^{13} +1.40111 q^{15} -4.36406 q^{16} -1.10811 q^{17} +1.94871 q^{18} +0.638333 q^{19} -2.51843 q^{20} +8.83079 q^{22} -1.00000 q^{23} +0.394699 q^{24} -3.03689 q^{25} -4.58973 q^{26} -1.00000 q^{27} +2.63096 q^{29} +2.73035 q^{30} -5.19267 q^{31} -7.71488 q^{32} -4.53162 q^{33} -2.15939 q^{34} +1.79746 q^{36} +4.53435 q^{37} +1.24392 q^{38} +2.35527 q^{39} +0.553017 q^{40} -1.45216 q^{41} +1.49386 q^{43} +8.14538 q^{44} -1.40111 q^{45} -1.94871 q^{46} -6.43916 q^{47} +4.36406 q^{48} -5.91801 q^{50} +1.10811 q^{51} -4.23350 q^{52} -11.1364 q^{53} -1.94871 q^{54} -6.34929 q^{55} -0.638333 q^{57} +5.12697 q^{58} -6.20679 q^{59} +2.51843 q^{60} -0.821232 q^{61} -10.1190 q^{62} -6.30591 q^{64} +3.29999 q^{65} -8.83079 q^{66} -3.33164 q^{67} -1.99179 q^{68} +1.00000 q^{69} -14.4996 q^{71} -0.394699 q^{72} -1.99401 q^{73} +8.83611 q^{74} +3.03689 q^{75} +1.14738 q^{76} +4.58973 q^{78} +4.18578 q^{79} +6.11453 q^{80} +1.00000 q^{81} -2.82984 q^{82} +8.69016 q^{83} +1.55259 q^{85} +2.91110 q^{86} -2.63096 q^{87} -1.78862 q^{88} -7.56109 q^{89} -2.73035 q^{90} -1.79746 q^{92} +5.19267 q^{93} -12.5480 q^{94} -0.894375 q^{95} +7.71488 q^{96} -9.73998 q^{97} +4.53162 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{15} + 4 q^{16} - 12 q^{17} + 4 q^{18} - 26 q^{19} - 24 q^{20} - 8 q^{22} - 10 q^{23} - 12 q^{24} - 2 q^{25} - 4 q^{26} - 10 q^{27} + 16 q^{29} + 8 q^{30} - 12 q^{31} + 8 q^{32} - 2 q^{33} - 28 q^{34} + 8 q^{36} - 8 q^{37} - 32 q^{38} - 4 q^{40} - 10 q^{41} - 4 q^{43} - 16 q^{44} - 4 q^{45} - 4 q^{46} - 2 q^{47} - 4 q^{48} - 8 q^{50} + 12 q^{51} - 24 q^{52} + 14 q^{53} - 4 q^{54} - 16 q^{55} + 26 q^{57} - 8 q^{58} - 38 q^{59} + 24 q^{60} - 14 q^{61} + 8 q^{62} + 8 q^{64} + 12 q^{65} + 8 q^{66} - 8 q^{68} + 10 q^{69} + 24 q^{71} + 12 q^{72} - 8 q^{73} - 8 q^{74} + 2 q^{75} - 64 q^{76} + 4 q^{78} - 16 q^{79} - 28 q^{80} + 10 q^{81} + 40 q^{82} - 28 q^{83} - 4 q^{85} + 20 q^{86} - 16 q^{87} - 68 q^{88} - 32 q^{89} - 8 q^{90} - 8 q^{92} + 12 q^{93} - 56 q^{94} + 8 q^{95} - 8 q^{96} - 4 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 + 4 * q^15 + 4 * q^16 - 12 * q^17 + 4 * q^18 - 26 * q^19 - 24 * q^20 - 8 * q^22 - 10 * q^23 - 12 * q^24 - 2 * q^25 - 4 * q^26 - 10 * q^27 + 16 * q^29 + 8 * q^30 - 12 * q^31 + 8 * q^32 - 2 * q^33 - 28 * q^34 + 8 * q^36 - 8 * q^37 - 32 * q^38 - 4 * q^40 - 10 * q^41 - 4 * q^43 - 16 * q^44 - 4 * q^45 - 4 * q^46 - 2 * q^47 - 4 * q^48 - 8 * q^50 + 12 * q^51 - 24 * q^52 + 14 * q^53 - 4 * q^54 - 16 * q^55 + 26 * q^57 - 8 * q^58 - 38 * q^59 + 24 * q^60 - 14 * q^61 + 8 * q^62 + 8 * q^64 + 12 * q^65 + 8 * q^66 - 8 * q^68 + 10 * q^69 + 24 * q^71 + 12 * q^72 - 8 * q^73 - 8 * q^74 + 2 * q^75 - 64 * q^76 + 4 * q^78 - 16 * q^79 - 28 * q^80 + 10 * q^81 + 40 * q^82 - 28 * q^83 - 4 * q^85 + 20 * q^86 - 16 * q^87 - 68 * q^88 - 32 * q^89 - 8 * q^90 - 8 * q^92 + 12 * q^93 - 56 * q^94 + 8 * q^95 - 8 * q^96 - 4 * 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 $

$2$	1.94871	1.37794	0.688972	−	0.724788i	$-0.258062\pi$
$2$	1.94871	1.37794	0.688972	+	0.724788i	$0.258062\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.79746	0.898728
$5$	−1.40111	−0.626595	−0.313298	−	0.949655i	$-0.601434\pi$
$5$	−1.40111	−0.626595	−0.313298	+	0.949655i	$0.601434\pi$
$6$	−1.94871	−0.795556
$7$	0	0
$7$	0	0
$8$	−0.394699	−0.139547
$9$	1.00000	0.333333
$10$	−2.73035	−0.863413
$11$	4.53162	1.36633	0.683167	−	0.730262i	$-0.260602\pi$
$11$	4.53162	1.36633	0.683167	+	0.730262i	$0.260602\pi$
$12$	−1.79746	−0.518881
$13$	−2.35527	−0.653235	−0.326617	−	0.945157i	$-0.605909\pi$
$13$	−2.35527	−0.653235	−0.326617	+	0.945157i	$0.605909\pi$
$14$	0	0
$15$	1.40111	0.361765
$16$	−4.36406	−1.09102
$17$	−1.10811	−0.268757	−0.134379	−	0.990930i	$-0.542904\pi$
$17$	−1.10811	−0.268757	−0.134379	+	0.990930i	$0.542904\pi$
$18$	1.94871	0.459314
$19$	0.638333	0.146444	0.0732218	−	0.997316i	$-0.476672\pi$
$19$	0.638333	0.146444	0.0732218	+	0.997316i	$0.476672\pi$
$20$	−2.51843	−0.563139
$21$	0	0
$22$	8.83079	1.88273
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0.394699	0.0805676
$25$	−3.03689	−0.607378
$26$	−4.58973	−0.900121
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.63096	0.488557	0.244278	−	0.969705i	$-0.421449\pi$
$29$	2.63096	0.488557	0.244278	+	0.969705i	$0.421449\pi$
$30$	2.73035	0.498492
$31$	−5.19267	−0.932632	−0.466316	−	0.884618i	$-0.654419\pi$
$31$	−5.19267	−0.932632	−0.466316	+	0.884618i	$0.654419\pi$
$32$	−7.71488	−1.36381
$33$	−4.53162	−0.788853
$34$	−2.15939	−0.370332
$35$	0	0
$36$	1.79746	0.299576
$37$	4.53435	0.745442	0.372721	−	0.927943i	$-0.378425\pi$
$37$	4.53435	0.745442	0.372721	+	0.927943i	$0.378425\pi$
$38$	1.24392	0.201791
$39$	2.35527	0.377145
$40$	0.553017	0.0874396
$41$	−1.45216	−0.226790	−0.113395	−	0.993550i	$-0.536173\pi$
$41$	−1.45216	−0.226790	−0.113395	+	0.993550i	$0.536173\pi$
$42$	0	0
$43$	1.49386	0.227812	0.113906	−	0.993492i	$-0.463664\pi$
$43$	1.49386	0.227812	0.113906	+	0.993492i	$0.463664\pi$
$44$	8.14538	1.22796
$45$	−1.40111	−0.208865
$46$	−1.94871	−0.287321
$47$	−6.43916	−0.939248	−0.469624	−	0.882866i	$-0.655611\pi$
$47$	−6.43916	−0.939248	−0.469624	+	0.882866i	$0.655611\pi$
$48$	4.36406	0.629898
$49$	0	0
$50$	−5.91801	−0.836933
$51$	1.10811	0.155167
$52$	−4.23350	−0.587080
$53$	−11.1364	−1.52970	−0.764849	−	0.644209i	$-0.777187\pi$
$53$	−11.1364	−1.52970	−0.764849	+	0.644209i	$0.777187\pi$
$54$	−1.94871	−0.265185
$55$	−6.34929	−0.856138
$56$	0	0
$57$	−0.638333	−0.0845493
$58$	5.12697	0.673204
$59$	−6.20679	−0.808056	−0.404028	−	0.914747i	$-0.632390\pi$
$59$	−6.20679	−0.808056	−0.404028	+	0.914747i	$0.632390\pi$
$60$	2.51843	0.325128
$61$	−0.821232	−0.105148	−0.0525740	−	0.998617i	$-0.516743\pi$
$61$	−0.821232	−0.105148	−0.0525740	+	0.998617i	$0.516743\pi$
$62$	−10.1190	−1.28511
$63$	0	0
$64$	−6.30591	−0.788239
$65$	3.29999	0.409314
$66$	−8.83079	−1.08699
$67$	−3.33164	−0.407025	−0.203513	−	0.979072i	$-0.565236\pi$
$67$	−3.33164	−0.407025	−0.203513	+	0.979072i	$0.565236\pi$
$68$	−1.99179	−0.241540
$69$	1.00000	0.120386
$70$	0	0
$71$	−14.4996	−1.72079	−0.860396	−	0.509627i	$-0.829784\pi$
$71$	−14.4996	−1.72079	−0.860396	+	0.509627i	$0.829784\pi$
$72$	−0.394699	−0.0465157
$73$	−1.99401	−0.233381	−0.116691	−	0.993168i	$-0.537229\pi$
$73$	−1.99401	−0.233381	−0.116691	+	0.993168i	$0.537229\pi$
$74$	8.83611	1.02718
$75$	3.03689	0.350670
$76$	1.14738	0.131613
$77$	0	0
$78$	4.58973	0.519685
$79$	4.18578	0.470938	0.235469	−	0.971882i	$-0.424337\pi$
$79$	4.18578	0.470938	0.235469	+	0.971882i	$0.424337\pi$
$80$	6.11453	0.683625
$81$	1.00000	0.111111
$82$	−2.82984	−0.312504
$83$	8.69016	0.953869	0.476934	−	0.878939i	$-0.341748\pi$
$83$	8.69016	0.953869	0.476934	+	0.878939i	$0.341748\pi$
$84$	0	0
$85$	1.55259	0.168402
$86$	2.91110	0.313911
$87$	−2.63096	−0.282068
$88$	−1.78862	−0.190668
$89$	−7.56109	−0.801474	−0.400737	−	0.916193i	$-0.631246\pi$
$89$	−7.56109	−0.801474	−0.400737	+	0.916193i	$0.631246\pi$
$90$	−2.73035	−0.287804
$91$	0	0
$92$	−1.79746	−0.187398
$93$	5.19267	0.538455
$94$	−12.5480	−1.29423
$95$	−0.894375	−0.0917609
$96$	7.71488	0.787397
$97$	−9.73998	−0.988945	−0.494473	−	0.869193i	$-0.664639\pi$
$97$	−9.73998	−0.988945	−0.494473	+	0.869193i	$0.664639\pi$
$98$	0	0
$99$	4.53162	0.455444
$100$	−5.45868	−0.545868
$101$	−9.80780	−0.975912	−0.487956	−	0.872868i	$-0.662257\pi$
$101$	−9.80780	−0.975912	−0.487956	+	0.872868i	$0.662257\pi$
$102$	2.15939	0.213812
$103$	14.8681	1.46499	0.732497	−	0.680770i	$-0.238355\pi$
$103$	14.8681	1.46499	0.732497	+	0.680770i	$0.238355\pi$
$104$	0.929623	0.0911571
$105$	0	0
$106$	−21.7015	−2.10784
$107$	−8.33893	−0.806155	−0.403077	−	0.915166i	$-0.632059\pi$
$107$	−8.33893	−0.806155	−0.403077	+	0.915166i	$0.632059\pi$
$108$	−1.79746	−0.172960
$109$	−12.4360	−1.19115	−0.595577	−	0.803298i	$-0.703077\pi$
$109$	−12.4360	−1.19115	−0.595577	+	0.803298i	$0.703077\pi$
$110$	−12.3729	−1.17971
$111$	−4.53435	−0.430381
$112$	0	0
$113$	−5.72393	−0.538462	−0.269231	−	0.963076i	$-0.586770\pi$
$113$	−5.72393	−0.538462	−0.269231	+	0.963076i	$0.586770\pi$
$114$	−1.24392	−0.116504
$115$	1.40111	0.130654
$116$	4.72903	0.439080
$117$	−2.35527	−0.217745
$118$	−12.0952	−1.11346
$119$	0	0
$120$	−0.553017	−0.0504833
$121$	9.53554	0.866867
$122$	−1.60034	−0.144888
$123$	1.45216	0.130937
$124$	−9.33360	−0.838182
$125$	11.2606	1.00718
$126$	0	0
$127$	3.10225	0.275281	0.137640	−	0.990482i	$-0.456048\pi$
$127$	3.10225	0.275281	0.137640	+	0.990482i	$0.456048\pi$
$128$	3.14140	0.277663
$129$	−1.49386	−0.131527
$130$	6.43072	0.564011
$131$	−5.54437	−0.484414	−0.242207	−	0.970225i	$-0.577871\pi$
$131$	−5.54437	−0.484414	−0.242207	+	0.970225i	$0.577871\pi$
$132$	−8.14538	−0.708964
$133$	0	0
$134$	−6.49240	−0.560858
$135$	1.40111	0.120588
$136$	0.437372	0.0375043
$137$	−8.50492	−0.726625	−0.363312	−	0.931667i	$-0.618354\pi$
$137$	−8.50492	−0.726625	−0.363312	+	0.931667i	$0.618354\pi$
$138$	1.94871	0.165885
$139$	−11.4089	−0.967686	−0.483843	−	0.875155i	$-0.660759\pi$
$139$	−11.4089	−0.967686	−0.483843	+	0.875155i	$0.660759\pi$
$140$	0	0
$141$	6.43916	0.542275
$142$	−28.2556	−2.37115
$143$	−10.6732	−0.892537
$144$	−4.36406	−0.363672
$145$	−3.68626	−0.306127
$146$	−3.88574	−0.321586
$147$	0	0
$148$	8.15029	0.669950
$149$	11.4888	0.941201	0.470601	−	0.882346i	$-0.344037\pi$
$149$	11.4888	0.941201	0.470601	+	0.882346i	$0.344037\pi$
$150$	5.91801	0.483204
$151$	−15.3689	−1.25070	−0.625352	−	0.780343i	$-0.715045\pi$
$151$	−15.3689	−1.25070	−0.625352	+	0.780343i	$0.715045\pi$
$152$	−0.251949	−0.0204358
$153$	−1.10811	−0.0895858
$154$	0	0
$155$	7.27551	0.584383
$156$	4.23350	0.338951
$157$	−2.60947	−0.208259	−0.104129	−	0.994564i	$-0.533206\pi$
$157$	−2.60947	−0.208259	−0.104129	+	0.994564i	$0.533206\pi$
$158$	8.15686	0.648925
$159$	11.1364	0.883172
$160$	10.8094	0.854558
$161$	0	0
$162$	1.94871	0.153105
$163$	20.0421	1.56982	0.784911	−	0.619609i	$-0.212709\pi$
$163$	20.0421	1.56982	0.784911	+	0.619609i	$0.212709\pi$
$164$	−2.61020	−0.203822
$165$	6.34929	0.494292
$166$	16.9346	1.31438
$167$	23.7143	1.83507	0.917535	−	0.397655i	$-0.130176\pi$
$167$	23.7143	1.83507	0.917535	+	0.397655i	$0.130176\pi$
$168$	0	0
$169$	−7.45269	−0.573284
$170$	3.02554	0.232049
$171$	0.638333	0.0488146
$172$	2.68515	0.204741
$173$	−0.0226019	−0.00171839	−0.000859195	−	1.00000i	$-0.500273\pi$
$173$	−0.0226019	−0.00171839	−0.000859195		1.00000i	$0.500273\pi$
$174$	−5.12697	−0.388674
$175$	0	0
$176$	−19.7763	−1.49069
$177$	6.20679	0.466531
$178$	−14.7343	−1.10439
$179$	3.86588	0.288949	0.144475	−	0.989509i	$-0.453851\pi$
$179$	3.86588	0.288949	0.144475	+	0.989509i	$0.453851\pi$
$180$	−2.51843	−0.187713
$181$	1.50794	0.112085	0.0560423	−	0.998428i	$-0.482152\pi$
$181$	1.50794	0.112085	0.0560423	+	0.998428i	$0.482152\pi$
$182$	0	0
$183$	0.821232	0.0607072
$184$	0.394699	0.0290976
$185$	−6.35312	−0.467091
$186$	10.1190	0.741961
$187$	−5.02155	−0.367212
$188$	−11.5741	−0.844129
$189$	0	0
$190$	−1.74287	−0.126441
$191$	18.4166	1.33258	0.666289	−	0.745693i	$-0.267882\pi$
$191$	18.4166	1.33258	0.666289	+	0.745693i	$0.267882\pi$
$192$	6.30591	0.455090
$193$	−13.4309	−0.966776	−0.483388	−	0.875406i	$-0.660594\pi$
$193$	−13.4309	−0.966776	−0.483388	+	0.875406i	$0.660594\pi$
$194$	−18.9804	−1.36271
$195$	−3.29999	−0.236317
$196$	0	0
$197$	2.56862	0.183007	0.0915033	−	0.995805i	$-0.470833\pi$
$197$	2.56862	0.183007	0.0915033	+	0.995805i	$0.470833\pi$
$198$	8.83079	0.627577
$199$	−16.5922	−1.17619	−0.588096	−	0.808791i	$-0.700122\pi$
$199$	−16.5922	−1.17619	−0.588096	+	0.808791i	$0.700122\pi$
$200$	1.19866	0.0847579
$201$	3.33164	0.234996
$202$	−19.1125	−1.34475
$203$	0	0
$204$	1.99179	0.139453
$205$	2.03464	0.142105
$206$	28.9735	2.01868
$207$	−1.00000	−0.0695048
$208$	10.2786	0.712690
$209$	2.89268	0.200091
$210$	0	0
$211$	−25.7245	−1.77095	−0.885474	−	0.464688i	$-0.846166\pi$
$211$	−25.7245	−1.77095	−0.885474	+	0.464688i	$0.846166\pi$
$212$	−20.0171	−1.37478
$213$	14.4996	0.993499
$214$	−16.2501	−1.11084
$215$	−2.09306	−0.142746
$216$	0.394699	0.0268559
$217$	0	0
$218$	−24.2341	−1.64134
$219$	1.99401	0.134743
$220$	−11.4126	−0.769435
$221$	2.60991	0.175562
$222$	−8.83611	−0.593041
$223$	18.2098	1.21942	0.609708	−	0.792626i	$-0.291287\pi$
$223$	18.2098	1.21942	0.609708	+	0.792626i	$0.291287\pi$
$224$	0	0
$225$	−3.03689	−0.202459
$226$	−11.1543	−0.741970
$227$	−2.07742	−0.137883	−0.0689416	−	0.997621i	$-0.521962\pi$
$227$	−2.07742	−0.137883	−0.0689416	+	0.997621i	$0.521962\pi$
$228$	−1.14738	−0.0759868
$229$	0.341004	0.0225342	0.0112671	−	0.999937i	$-0.496413\pi$
$229$	0.341004	0.0225342	0.0112671	+	0.999937i	$0.496413\pi$
$230$	2.73035	0.180034
$231$	0	0
$232$	−1.03844	−0.0681767
$233$	−5.84234	−0.382745	−0.191372	−	0.981518i	$-0.561294\pi$
$233$	−5.84234	−0.382745	−0.191372	+	0.981518i	$0.561294\pi$
$234$	−4.58973	−0.300040
$235$	9.02197	0.588528
$236$	−11.1564	−0.726222
$237$	−4.18578	−0.271896
$238$	0	0
$239$	−0.919771	−0.0594950	−0.0297475	−	0.999557i	$-0.509470\pi$
$239$	−0.919771	−0.0594950	−0.0297475	+	0.999557i	$0.509470\pi$
$240$	−6.11453	−0.394691
$241$	−18.4174	−1.18637	−0.593184	−	0.805067i	$-0.702129\pi$
$241$	−18.4174	−1.18637	−0.593184	+	0.805067i	$0.702129\pi$
$242$	18.5820	1.19449
$243$	−1.00000	−0.0641500
$244$	−1.47613	−0.0944994
$245$	0	0
$246$	2.82984	0.180424
$247$	−1.50345	−0.0956621
$248$	2.04954	0.130146
$249$	−8.69016	−0.550716
$250$	21.9435	1.38783
$251$	19.2274	1.21362	0.606812	−	0.794846i	$-0.292448\pi$
$251$	19.2274	1.21362	0.606812	+	0.794846i	$0.292448\pi$
$252$	0	0
$253$	−4.53162	−0.284900
$254$	6.04538	0.379321
$255$	−1.55259	−0.0972270
$256$	18.7335	1.17084
$257$	16.5661	1.03336	0.516681	−	0.856178i	$-0.327167\pi$
$257$	16.5661	1.03336	0.516681	+	0.856178i	$0.327167\pi$
$258$	−2.91110	−0.181237
$259$	0	0
$260$	5.93159	0.367862
$261$	2.63096	0.162852
$262$	−10.8044	−0.667495
$263$	28.5724	1.76185	0.880923	−	0.473259i	$-0.156923\pi$
$263$	28.5724	1.76185	0.880923	+	0.473259i	$0.156923\pi$
$264$	1.78862	0.110082
$265$	15.6033	0.958502
$266$	0	0
$267$	7.56109	0.462731
$268$	−5.98848	−0.365805
$269$	19.5613	1.19267	0.596336	−	0.802735i	$-0.296623\pi$
$269$	19.5613	1.19267	0.596336	+	0.802735i	$0.296623\pi$
$270$	2.73035	0.166164
$271$	3.41668	0.207549	0.103774	−	0.994601i	$-0.466908\pi$
$271$	3.41668	0.207549	0.103774	+	0.994601i	$0.466908\pi$
$272$	4.83588	0.293219
$273$	0	0
$274$	−16.5736	−1.00125
$275$	−13.7620	−0.829881
$276$	1.79746	0.108194
$277$	−3.33941	−0.200646	−0.100323	−	0.994955i	$-0.531988\pi$
$277$	−3.33941	−0.200646	−0.100323	+	0.994955i	$0.531988\pi$
$278$	−22.2325	−1.33342
$279$	−5.19267	−0.310877
$280$	0	0
$281$	−3.94689	−0.235451	−0.117726	−	0.993046i	$-0.537560\pi$
$281$	−3.94689	−0.235451	−0.117726	+	0.993046i	$0.537560\pi$
$282$	12.5480	0.747225
$283$	−5.85374	−0.347969	−0.173984	−	0.984748i	$-0.555664\pi$
$283$	−5.85374	−0.347969	−0.173984	+	0.984748i	$0.555664\pi$
$284$	−26.0625	−1.54652
$285$	0.894375	0.0529782
$286$	−20.7989	−1.22986
$287$	0	0
$288$	−7.71488	−0.454604
$289$	−15.7721	−0.927769
$290$	−7.18344	−0.421826
$291$	9.73998	0.570968
$292$	−3.58415	−0.209746
$293$	21.0256	1.22833	0.614165	−	0.789178i	$-0.289493\pi$
$293$	21.0256	1.22833	0.614165	+	0.789178i	$0.289493\pi$
$294$	0	0
$295$	8.69640	0.506324
$296$	−1.78970	−0.104024
$297$	−4.53162	−0.262951
$298$	22.3883	1.29692
$299$	2.35527	0.136209
$300$	5.45868	0.315157
$301$	0	0
$302$	−29.9495	−1.72340
$303$	9.80780	0.563443
$304$	−2.78573	−0.159772
$305$	1.15064	0.0658852
$306$	−2.15939	−0.123444
$307$	22.2208	1.26821	0.634105	−	0.773247i	$-0.281369\pi$
$307$	22.2208	1.26821	0.634105	+	0.773247i	$0.281369\pi$
$308$	0	0
$309$	−14.8681	−0.845815
$310$	14.1778	0.805246
$311$	33.3939	1.89360	0.946798	−	0.321828i	$-0.104297\pi$
$311$	33.3939	1.89360	0.946798	+	0.321828i	$0.104297\pi$
$312$	−0.929623	−0.0526296
$313$	6.16899	0.348692	0.174346	−	0.984684i	$-0.444219\pi$
$313$	6.16899	0.348692	0.174346	+	0.984684i	$0.444219\pi$
$314$	−5.08510	−0.286969
$315$	0	0
$316$	7.52376	0.423245
$317$	3.70665	0.208186	0.104093	−	0.994568i	$-0.466806\pi$
$317$	3.70665	0.208186	0.104093	+	0.994568i	$0.466806\pi$
$318$	21.7015	1.21696
$319$	11.9225	0.667532
$320$	8.83527	0.493906
$321$	8.33893	0.465434
$322$	0	0
$323$	−0.707346	−0.0393578
$324$	1.79746	0.0998587
$325$	7.15271	0.396761
$326$	39.0562	2.16313
$327$	12.4360	0.687713
$328$	0.573168	0.0316479
$329$	0	0
$330$	12.3729	0.681106
$331$	1.27030	0.0698219	0.0349109	−	0.999390i	$-0.488885\pi$
$331$	1.27030	0.0698219	0.0349109	+	0.999390i	$0.488885\pi$
$332$	15.6202	0.857269
$333$	4.53435	0.248481
$334$	46.2123	2.52862
$335$	4.66800	0.255040
$336$	0	0
$337$	−3.21513	−0.175139	−0.0875696	−	0.996158i	$-0.527910\pi$
$337$	−3.21513	−0.175139	−0.0875696	+	0.996158i	$0.527910\pi$
$338$	−14.5231	−0.789953
$339$	5.72393	0.310881
$340$	2.79071	0.151348
$341$	−23.5312	−1.27429
$342$	1.24392	0.0672637
$343$	0	0
$344$	−0.589625	−0.0317905
$345$	−1.40111	−0.0754332
$346$	−0.0440444	−0.00236784
$347$	−16.8860	−0.906489	−0.453245	−	0.891386i	$-0.649734\pi$
$347$	−16.8860	−0.906489	−0.453245	+	0.891386i	$0.649734\pi$
$348$	−4.72903	−0.253503
$349$	19.6002	1.04918	0.524589	−	0.851356i	$-0.324219\pi$
$349$	19.6002	1.04918	0.524589	+	0.851356i	$0.324219\pi$
$350$	0	0
$351$	2.35527	0.125715
$352$	−34.9609	−1.86342
$353$	26.9979	1.43696	0.718478	−	0.695550i	$-0.244839\pi$
$353$	26.9979	1.43696	0.718478	+	0.695550i	$0.244839\pi$
$354$	12.0952	0.642854
$355$	20.3156	1.07824
$356$	−13.5907	−0.720307
$357$	0	0
$358$	7.53346	0.398156
$359$	−19.2863	−1.01789	−0.508945	−	0.860799i	$-0.669964\pi$
$359$	−19.2863	−1.01789	−0.508945	+	0.860799i	$0.669964\pi$
$360$	0.553017	0.0291465
$361$	−18.5925	−0.978554
$362$	2.93854	0.154446
$363$	−9.53554	−0.500486
$364$	0	0
$365$	2.79383	0.146236
$366$	1.60034	0.0836511
$367$	6.96790	0.363722	0.181861	−	0.983324i	$-0.441788\pi$
$367$	6.96790	0.363722	0.181861	+	0.983324i	$0.441788\pi$
$368$	4.36406	0.227493
$369$	−1.45216	−0.0755966
$370$	−12.3804	−0.643624
$371$	0	0
$372$	9.33360	0.483925
$373$	19.6555	1.01772	0.508862	−	0.860848i	$-0.330066\pi$
$373$	19.6555	1.01772	0.508862	+	0.860848i	$0.330066\pi$
$374$	−9.78553	−0.505998
$375$	−11.2606	−0.581493
$376$	2.54153	0.131069
$377$	−6.19662	−0.319142
$378$	0	0
$379$	8.47505	0.435334	0.217667	−	0.976023i	$-0.430155\pi$
$379$	8.47505	0.435334	0.217667	+	0.976023i	$0.430155\pi$
$380$	−1.60760	−0.0824681
$381$	−3.10225	−0.158933
$382$	35.8885	1.83622
$383$	−31.1528	−1.59183	−0.795917	−	0.605405i	$-0.793011\pi$
$383$	−31.1528	−1.59183	−0.795917	+	0.605405i	$0.793011\pi$
$384$	−3.14140	−0.160309
$385$	0	0
$386$	−26.1728	−1.33216
$387$	1.49386	0.0759372
$388$	−17.5072	−0.888793
$389$	16.0452	0.813525	0.406762	−	0.913534i	$-0.366658\pi$
$389$	16.0452	0.813525	0.406762	+	0.913534i	$0.366658\pi$
$390$	−6.43072	−0.325632
$391$	1.10811	0.0560398
$392$	0	0
$393$	5.54437	0.279677
$394$	5.00549	0.252173
$395$	−5.86474	−0.295087
$396$	8.14538	0.409321
$397$	−11.7171	−0.588062	−0.294031	−	0.955796i	$-0.594997\pi$
$397$	−11.7171	−0.588062	−0.294031	+	0.955796i	$0.594997\pi$
$398$	−32.3334	−1.62072
$399$	0	0
$400$	13.2532	0.662660
$401$	26.4060	1.31865	0.659326	−	0.751857i	$-0.270842\pi$
$401$	26.4060	1.31865	0.659326	+	0.751857i	$0.270842\pi$
$402$	6.49240	0.323811
$403$	12.2302	0.609228
$404$	−17.6291	−0.877080
$405$	−1.40111	−0.0696217
$406$	0	0
$407$	20.5479	1.01852
$408$	−0.437372	−0.0216531
$409$	27.9543	1.38225	0.691125	−	0.722735i	$-0.257115\pi$
$409$	27.9543	1.38225	0.691125	+	0.722735i	$0.257115\pi$
$410$	3.96492	0.195813
$411$	8.50492	0.419517
$412$	26.7247	1.31663
$413$	0	0
$414$	−1.94871	−0.0957737
$415$	−12.1759	−0.597690
$416$	18.1706	0.890889
$417$	11.4089	0.558694
$418$	5.63698	0.275714
$419$	−34.2410	−1.67278	−0.836392	−	0.548132i	$-0.815339\pi$
$419$	−34.2410	−1.67278	−0.836392	+	0.548132i	$0.815339\pi$
$420$	0	0
$421$	−18.6345	−0.908190	−0.454095	−	0.890953i	$-0.650037\pi$
$421$	−18.6345	−0.908190	−0.454095	+	0.890953i	$0.650037\pi$
$422$	−50.1295	−2.44027
$423$	−6.43916	−0.313083
$424$	4.39552	0.213465
$425$	3.36523	0.163237
$426$	28.2556	1.36899
$427$	0	0
$428$	−14.9889	−0.724514
$429$	10.6732	0.515306
$430$	−4.07876	−0.196695
$431$	−2.25759	−0.108744	−0.0543721	−	0.998521i	$-0.517316\pi$
$431$	−2.25759	−0.108744	−0.0543721	+	0.998521i	$0.517316\pi$
$432$	4.36406	0.209966
$433$	−4.87028	−0.234051	−0.117025	−	0.993129i	$-0.537336\pi$
$433$	−4.87028	−0.234051	−0.117025	+	0.993129i	$0.537336\pi$
$434$	0	0
$435$	3.68626	0.176743
$436$	−22.3532	−1.07052
$437$	−0.638333	−0.0305356
$438$	3.88574	0.185668
$439$	−36.4355	−1.73897	−0.869487	−	0.493956i	$-0.835550\pi$
$439$	−36.4355	−1.73897	−0.869487	+	0.493956i	$0.835550\pi$
$440$	2.50606	0.119472
$441$	0	0
$442$	5.08595	0.241914
$443$	−8.20326	−0.389749	−0.194874	−	0.980828i	$-0.562430\pi$
$443$	−8.20326	−0.389749	−0.194874	+	0.980828i	$0.562430\pi$
$444$	−8.15029	−0.386796
$445$	10.5939	0.502200
$446$	35.4855	1.68029
$447$	−11.4888	−0.543403
$448$	0	0
$449$	9.75539	0.460385	0.230193	−	0.973145i	$-0.426064\pi$
$449$	9.75539	0.460385	0.230193	+	0.973145i	$0.426064\pi$
$450$	−5.91801	−0.278978
$451$	−6.58065	−0.309871
$452$	−10.2885	−0.483931
$453$	15.3689	0.722095
$454$	−4.04828	−0.189995
$455$	0	0
$456$	0.251949	0.0117986
$457$	−11.2789	−0.527603	−0.263801	−	0.964577i	$-0.584976\pi$
$457$	−11.2789	−0.527603	−0.263801	+	0.964577i	$0.584976\pi$
$458$	0.664517	0.0310508
$459$	1.10811	0.0517224
$460$	2.51843	0.117423
$461$	−7.90160	−0.368014	−0.184007	−	0.982925i	$-0.558907\pi$
$461$	−7.90160	−0.368014	−0.184007	+	0.982925i	$0.558907\pi$
$462$	0	0
$463$	9.71492	0.451490	0.225745	−	0.974186i	$-0.427518\pi$
$463$	9.71492	0.451490	0.225745	+	0.974186i	$0.427518\pi$
$464$	−11.4817	−0.533023
$465$	−7.27551	−0.337394
$466$	−11.3850	−0.527400
$467$	−0.0671295	−0.00310638	−0.00155319	−	0.999999i	$-0.500494\pi$
$467$	−0.0671295	−0.00310638	−0.00155319	+	0.999999i	$0.500494\pi$
$468$	−4.23350	−0.195693
$469$	0	0
$470$	17.5812	0.810959
$471$	2.60947	0.120238
$472$	2.44982	0.112762
$473$	6.76960	0.311267
$474$	−8.15686	−0.374657
$475$	−1.93855	−0.0889467
$476$	0	0
$477$	−11.1364	−0.509900
$478$	−1.79236	−0.0819808
$479$	−21.1614	−0.966890	−0.483445	−	0.875375i	$-0.660615\pi$
$479$	−21.1614	−0.966890	−0.483445	+	0.875375i	$0.660615\pi$
$480$	−10.8094	−0.493379
$481$	−10.6796	−0.486949
$482$	−35.8901	−1.63475
$483$	0	0
$484$	17.1397	0.779078
$485$	13.6468	0.619668
$486$	−1.94871	−0.0883951
$487$	12.9215	0.585527	0.292764	−	0.956185i	$-0.405425\pi$
$487$	12.9215	0.585527	0.292764	+	0.956185i	$0.405425\pi$
$488$	0.324139	0.0146731
$489$	−20.0421	−0.906337
$490$	0	0
$491$	4.57216	0.206338	0.103169	−	0.994664i	$-0.467102\pi$
$491$	4.57216	0.206338	0.103169	+	0.994664i	$0.467102\pi$
$492$	2.61020	0.117677
$493$	−2.91541	−0.131303
$494$	−2.92978	−0.131817
$495$	−6.34929	−0.285379
$496$	22.6612	1.01752
$497$	0	0
$498$	−16.9346	−0.758856
$499$	3.92492	0.175704	0.0878518	−	0.996134i	$-0.472000\pi$
$499$	3.92492	0.175704	0.0878518	+	0.996134i	$0.472000\pi$
$500$	20.2404	0.905177
$501$	−23.7143	−1.05948
$502$	37.4686	1.67230
$503$	17.8433	0.795594	0.397797	−	0.917473i	$-0.369775\pi$
$503$	17.8433	0.795594	0.397797	+	0.917473i	$0.369775\pi$
$504$	0	0
$505$	13.7418	0.611502
$506$	−8.83079	−0.392576
$507$	7.45269	0.330986
$508$	5.57617	0.247402
$509$	21.7044	0.962030	0.481015	−	0.876712i	$-0.340268\pi$
$509$	21.7044	0.962030	0.481015	+	0.876712i	$0.340268\pi$
$510$	−3.02554	−0.133973
$511$	0	0
$512$	30.2233	1.33569
$513$	−0.638333	−0.0281831
$514$	32.2824	1.42392
$515$	−20.8318	−0.917958
$516$	−2.68515	−0.118207
$517$	−29.1798	−1.28333
$518$	0	0
$519$	0.0226019	0.000992113 0
$520$	−1.30250	−0.0571186
$521$	−23.0131	−1.00822	−0.504112	−	0.863638i	$-0.668180\pi$
$521$	−23.0131	−1.00822	−0.504112	+	0.863638i	$0.668180\pi$
$522$	5.12697	0.224401
$523$	11.8608	0.518636	0.259318	−	0.965792i	$-0.416502\pi$
$523$	11.8608	0.518636	0.259318	+	0.965792i	$0.416502\pi$
$524$	−9.96577	−0.435357
$525$	0	0
$526$	55.6791	2.42772
$527$	5.75408	0.250652
$528$	19.7763	0.860651
$529$	1.00000	0.0434783
$530$	30.4062	1.32076
$531$	−6.20679	−0.269352
$532$	0	0
$533$	3.42024	0.148147
$534$	14.7343	0.637617
$535$	11.6838	0.505133
$536$	1.31500	0.0567992
$537$	−3.86588	−0.166825
$538$	38.1192	1.64343
$539$	0	0
$540$	2.51843	0.108376
$541$	−6.83384	−0.293810	−0.146905	−	0.989151i	$-0.546931\pi$
$541$	−6.83384	−0.293810	−0.146905	+	0.989151i	$0.546931\pi$
$542$	6.65811	0.285990
$543$	−1.50794	−0.0647120
$544$	8.54897	0.366534
$545$	17.4242	0.746372
$546$	0	0
$547$	−11.0185	−0.471117	−0.235558	−	0.971860i	$-0.575692\pi$
$547$	−11.0185	−0.471117	−0.235558	+	0.971860i	$0.575692\pi$
$548$	−15.2872	−0.653038
$549$	−0.821232	−0.0350493
$550$	−26.8181	−1.14353
$551$	1.67943	0.0715461
$552$	−0.394699	−0.0167995
$553$	0	0
$554$	−6.50753	−0.276479
$555$	6.35312	0.269675
$556$	−20.5069	−0.869687
$557$	5.27066	0.223325	0.111663	−	0.993746i	$-0.464382\pi$
$557$	5.27066	0.223325	0.111663	+	0.993746i	$0.464382\pi$
$558$	−10.1190	−0.428371
$559$	−3.51845	−0.148814
$560$	0	0
$561$	5.02155	0.212010
$562$	−7.69132	−0.324439
$563$	25.0844	1.05718	0.528591	−	0.848876i	$-0.322720\pi$
$563$	25.0844	1.05718	0.528591	+	0.848876i	$0.322720\pi$
$564$	11.5741	0.487358
$565$	8.01986	0.337398
$566$	−11.4072	−0.479481
$567$	0	0
$568$	5.72300	0.240132
$569$	15.5251	0.650848	0.325424	−	0.945568i	$-0.394493\pi$
$569$	15.5251	0.650848	0.325424	+	0.945568i	$0.394493\pi$
$570$	1.74287	0.0730009
$571$	−40.8737	−1.71051	−0.855256	−	0.518206i	$-0.826600\pi$
$571$	−40.8737	−1.71051	−0.855256	+	0.518206i	$0.826600\pi$
$572$	−19.1846	−0.802148
$573$	−18.4166	−0.769365
$574$	0	0
$575$	3.03689	0.126647
$576$	−6.30591	−0.262746
$577$	−13.0386	−0.542806	−0.271403	−	0.962466i	$-0.587488\pi$
$577$	−13.0386	−0.542806	−0.271403	+	0.962466i	$0.587488\pi$
$578$	−30.7352	−1.27841
$579$	13.4309	0.558168
$580$	−6.62589	−0.275125
$581$	0	0
$582$	18.9804	0.786761
$583$	−50.4658	−2.09008
$584$	0.787034	0.0325677
$585$	3.29999	0.136438
$586$	40.9728	1.69257
$587$	−25.7535	−1.06296	−0.531480	−	0.847071i	$-0.678364\pi$
$587$	−25.7535	−1.06296	−0.531480	+	0.847071i	$0.678364\pi$
$588$	0	0
$589$	−3.31466	−0.136578
$590$	16.9467	0.697686
$591$	−2.56862	−0.105659
$592$	−19.7882	−0.813289
$593$	−29.2054	−1.19932	−0.599662	−	0.800253i	$-0.704698\pi$
$593$	−29.2054	−1.19932	−0.599662	+	0.800253i	$0.704698\pi$
$594$	−8.83079	−0.362332
$595$	0	0
$596$	20.6507	0.845884
$597$	16.5922	0.679074
$598$	4.58973	0.187688
$599$	−10.2251	−0.417786	−0.208893	−	0.977938i	$-0.566986\pi$
$599$	−10.2251	−0.417786	−0.208893	+	0.977938i	$0.566986\pi$
$600$	−1.19866	−0.0489350
$601$	22.9043	0.934287	0.467144	−	0.884181i	$-0.345283\pi$
$601$	22.9043	0.934287	0.467144	+	0.884181i	$0.345283\pi$
$602$	0	0
$603$	−3.33164	−0.135675
$604$	−27.6249	−1.12404
$605$	−13.3603	−0.543175
$606$	19.1125	0.776393
$607$	−14.2730	−0.579324	−0.289662	−	0.957129i	$-0.593543\pi$
$607$	−14.2730	−0.579324	−0.289662	+	0.957129i	$0.593543\pi$
$608$	−4.92466	−0.199721
$609$	0	0
$610$	2.24225	0.0907861
$611$	15.1660	0.613550
$612$	−1.99179	−0.0805132
$613$	−27.2926	−1.10234	−0.551168	−	0.834394i	$-0.685818\pi$
$613$	−27.2926	−1.10234	−0.551168	+	0.834394i	$0.685818\pi$
$614$	43.3019	1.74752
$615$	−2.03464	−0.0820446
$616$	0	0
$617$	21.6783	0.872736	0.436368	−	0.899768i	$-0.356265\pi$
$617$	21.6783	0.872736	0.436368	+	0.899768i	$0.356265\pi$
$618$	−28.9735	−1.16548
$619$	−46.6222	−1.87391	−0.936953	−	0.349456i	$-0.886367\pi$
$619$	−46.6222	−1.87391	−0.936953	+	0.349456i	$0.886367\pi$
$620$	13.0774	0.525201
$621$	1.00000	0.0401286
$622$	65.0750	2.60927
$623$	0	0
$624$	−10.2786	−0.411472
$625$	−0.592827	−0.0237131
$626$	12.0216	0.480478
$627$	−2.89268	−0.115523
$628$	−4.69042	−0.187168
$629$	−5.02458	−0.200343
$630$	0	0
$631$	−48.1745	−1.91780	−0.958899	−	0.283749i	$-0.908422\pi$
$631$	−48.1745	−1.91780	−0.958899	+	0.283749i	$0.908422\pi$
$632$	−1.65212	−0.0657180
$633$	25.7245	1.02246
$634$	7.22318	0.286869
$635$	−4.34660	−0.172490
$636$	20.0171	0.793731
$637$	0	0
$638$	23.2334	0.919821
$639$	−14.4996	−0.573597
$640$	−4.40145	−0.173982
$641$	−18.1327	−0.716199	−0.358100	−	0.933683i	$-0.616575\pi$
$641$	−18.1327	−0.716199	−0.358100	+	0.933683i	$0.616575\pi$
$642$	16.2501	0.641341
$643$	20.1794	0.795797	0.397899	−	0.917429i	$-0.369740\pi$
$643$	20.1794	0.795797	0.397899	+	0.917429i	$0.369740\pi$
$644$	0	0
$645$	2.09306	0.0824143
$646$	−1.37841	−0.0542328
$647$	−13.4514	−0.528830	−0.264415	−	0.964409i	$-0.585179\pi$
$647$	−13.4514	−0.528830	−0.264415	+	0.964409i	$0.585179\pi$
$648$	−0.394699	−0.0155052
$649$	−28.1268	−1.10407
$650$	13.9385	0.546714
$651$	0	0
$652$	36.0249	1.41084
$653$	41.9956	1.64341	0.821707	−	0.569911i	$-0.193022\pi$
$653$	41.9956	1.64341	0.821707	+	0.569911i	$0.193022\pi$
$654$	24.2341	0.947630
$655$	7.76828	0.303532
$656$	6.33734	0.247431
$657$	−1.99401	−0.0777938
$658$	0	0
$659$	−39.4549	−1.53695	−0.768473	−	0.639882i	$-0.778983\pi$
$659$	−39.4549	−1.53695	−0.768473	+	0.639882i	$0.778983\pi$
$660$	11.4126	0.444234
$661$	28.1113	1.09340	0.546702	−	0.837328i	$-0.315883\pi$
$661$	28.1113	1.09340	0.546702	+	0.837328i	$0.315883\pi$
$662$	2.47544	0.0962106
$663$	−2.60991	−0.101361
$664$	−3.43000	−0.133110
$665$	0	0
$666$	8.83611	0.342392
$667$	−2.63096	−0.101871
$668$	42.6255	1.64923
$669$	−18.2098	−0.704031
$670$	9.09656	0.351431
$671$	−3.72151	−0.143667
$672$	0	0
$673$	47.0138	1.81225	0.906125	−	0.423009i	$-0.139026\pi$
$673$	47.0138	1.81225	0.906125	+	0.423009i	$0.139026\pi$
$674$	−6.26534	−0.241332
$675$	3.03689	0.116890
$676$	−13.3959	−0.515227
$677$	44.3570	1.70478	0.852390	−	0.522907i	$-0.175153\pi$
$677$	44.3570	1.70478	0.852390	+	0.522907i	$0.175153\pi$
$678$	11.1543	0.428377
$679$	0	0
$680$	−0.612806	−0.0235000
$681$	2.07742	0.0796069
$682$	−45.8554	−1.75589
$683$	−27.8672	−1.06631	−0.533154	−	0.846018i	$-0.678993\pi$
$683$	−27.8672	−1.06631	−0.533154	+	0.846018i	$0.678993\pi$
$684$	1.14738	0.0438710
$685$	11.9163	0.455299
$686$	0	0
$687$	−0.341004	−0.0130101
$688$	−6.51930	−0.248546
$689$	26.2292	0.999253
$690$	−2.73035	−0.103943
$691$	−43.1462	−1.64136	−0.820679	−	0.571390i	$-0.806404\pi$
$691$	−43.1462	−1.64136	−0.820679	+	0.571390i	$0.806404\pi$
$692$	−0.0406259	−0.00154436
$693$	0	0
$694$	−32.9059	−1.24909
$695$	15.9851	0.606348
$696$	1.03844	0.0393619
$697$	1.60916	0.0609514
$698$	38.1951	1.44571
$699$	5.84234	0.220978
$700$	0	0
$701$	−21.7309	−0.820764	−0.410382	−	0.911914i	$-0.634605\pi$
$701$	−21.7309	−0.820764	−0.410382	+	0.911914i	$0.634605\pi$
$702$	4.58973	0.173228
$703$	2.89442	0.109165
$704$	−28.5760	−1.07700
$705$	−9.02197	−0.339787
$706$	52.6111	1.98004
$707$	0	0
$708$	11.1564	0.419285
$709$	25.9385	0.974141	0.487070	−	0.873363i	$-0.338066\pi$
$709$	25.9385	0.974141	0.487070	+	0.873363i	$0.338066\pi$
$710$	39.5891	1.48575
$711$	4.18578	0.156979
$712$	2.98435	0.111843
$713$	5.19267	0.194467
$714$	0	0
$715$	14.9543	0.559259
$716$	6.94874	0.259687
$717$	0.919771	0.0343495
$718$	−37.5833	−1.40260
$719$	−29.6040	−1.10404	−0.552021	−	0.833831i	$-0.686143\pi$
$719$	−29.6040	−1.10404	−0.552021	+	0.833831i	$0.686143\pi$
$720$	6.11453	0.227875
$721$	0	0
$722$	−36.2314	−1.34839
$723$	18.4174	0.684950
$724$	2.71046	0.100734
$725$	−7.98994	−0.296739
$726$	−18.5820	−0.689641
$727$	−2.02104	−0.0749562	−0.0374781	−	0.999297i	$-0.511932\pi$
$727$	−2.02104	−0.0749562	−0.0374781	+	0.999297i	$0.511932\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	5.44435	0.201504
$731$	−1.65537	−0.0612260
$732$	1.47613	0.0545593
$733$	16.9934	0.627664	0.313832	−	0.949479i	$-0.398387\pi$
$733$	16.9934	0.627664	0.313832	+	0.949479i	$0.398387\pi$
$734$	13.5784	0.501188
$735$	0	0
$736$	7.71488	0.284374
$737$	−15.0977	−0.556132
$738$	−2.82984	−0.104168
$739$	4.19511	0.154319	0.0771597	−	0.997019i	$-0.475415\pi$
$739$	4.19511	0.154319	0.0771597	+	0.997019i	$0.475415\pi$
$740$	−11.4195	−0.419787
$741$	1.50345	0.0552305
$742$	0	0
$743$	50.7872	1.86320	0.931600	−	0.363485i	$-0.118413\pi$
$743$	50.7872	1.86320	0.931600	+	0.363485i	$0.118413\pi$
$744$	−2.04954	−0.0751399
$745$	−16.0971	−0.589752
$746$	38.3029	1.40237
$747$	8.69016	0.317956
$748$	−9.02602	−0.330024
$749$	0	0
$750$	−21.9435	−0.801265
$751$	37.9353	1.38428	0.692139	−	0.721764i	$-0.256669\pi$
$751$	37.9353	1.38428	0.692139	+	0.721764i	$0.256669\pi$
$752$	28.1009	1.02473
$753$	−19.2274	−0.700686
$754$	−12.0754	−0.439760
$755$	21.5335	0.783685
$756$	0	0
$757$	−53.9224	−1.95984	−0.979921	−	0.199388i	$-0.936105\pi$
$757$	−53.9224	−1.95984	−0.979921	+	0.199388i	$0.936105\pi$
$758$	16.5154	0.599866
$759$	4.53162	0.164487
$760$	0.353009	0.0128050
$761$	−19.0107	−0.689136	−0.344568	−	0.938761i	$-0.611975\pi$
$761$	−19.0107	−0.689136	−0.344568	+	0.938761i	$0.611975\pi$
$762$	−6.04538	−0.219001
$763$	0	0
$764$	33.1030	1.19763
$765$	1.55259	0.0561340
$766$	−60.7077	−2.19346
$767$	14.6187	0.527850
$768$	−18.7335	−0.675986
$769$	−16.6287	−0.599645	−0.299822	−	0.953995i	$-0.596927\pi$
$769$	−16.6287	−0.599645	−0.299822	+	0.953995i	$0.596927\pi$
$770$	0	0
$771$	−16.5661	−0.596612
$772$	−24.1414	−0.868869
$773$	−35.5289	−1.27788	−0.638942	−	0.769255i	$-0.720628\pi$
$773$	−35.5289	−1.27788	−0.638942	+	0.769255i	$0.720628\pi$
$774$	2.91110	0.104637
$775$	15.7696	0.566461
$776$	3.84436	0.138005
$777$	0	0
$778$	31.2674	1.12099
$779$	−0.926964	−0.0332119
$780$	−5.93159	−0.212385
$781$	−65.7068	−2.35117
$782$	2.15939	0.0772196
$783$	−2.63096	−0.0940228
$784$	0	0
$785$	3.65616	0.130494
$786$	10.8044	0.385379
$787$	−25.3486	−0.903579	−0.451790	−	0.892124i	$-0.649214\pi$
$787$	−25.3486	−0.903579	−0.451790	+	0.892124i	$0.649214\pi$
$788$	4.61698	0.164473
$789$	−28.5724	−1.01720
$790$	−11.4287	−0.406613
$791$	0	0
$792$	−1.78862	−0.0635560
$793$	1.93422	0.0686863
$794$	−22.8331	−0.810316
$795$	−15.6033	−0.553391
$796$	−29.8238	−1.05708
$797$	20.2350	0.716760	0.358380	−	0.933576i	$-0.383329\pi$
$797$	20.2350	0.716760	0.358380	+	0.933576i	$0.383329\pi$
$798$	0	0
$799$	7.13533	0.252430
$800$	23.4293	0.828349
$801$	−7.56109	−0.267158
$802$	51.4575	1.81703
$803$	−9.03609	−0.318877
$804$	5.98848	0.211198
$805$	0	0
$806$	23.8330	0.839481
$807$	−19.5613	−0.688590
$808$	3.87113	0.136186
$809$	27.0739	0.951866	0.475933	−	0.879481i	$-0.342110\pi$
$809$	27.0739	0.951866	0.475933	+	0.879481i	$0.342110\pi$
$810$	−2.73035	−0.0959348
$811$	3.11366	0.109335	0.0546677	−	0.998505i	$-0.482590\pi$
$811$	3.11366	0.109335	0.0546677	+	0.998505i	$0.482590\pi$
$812$	0	0
$813$	−3.41668	−0.119828
$814$	40.0419	1.40347
$815$	−28.0812	−0.983643
$816$	−4.83588	−0.169290
$817$	0.953581	0.0333616
$818$	54.4747	1.90466
$819$	0	0
$820$	3.65718	0.127714
$821$	43.0364	1.50198	0.750991	−	0.660313i	$-0.229576\pi$
$821$	43.0364	1.50198	0.750991	+	0.660313i	$0.229576\pi$
$822$	16.5736	0.578070
$823$	−30.0783	−1.04846	−0.524232	−	0.851575i	$-0.675648\pi$
$823$	−30.0783	−1.04846	−0.524232	+	0.851575i	$0.675648\pi$
$824$	−5.86841	−0.204436
$825$	13.7620	0.479132
$826$	0	0
$827$	45.9816	1.59894	0.799469	−	0.600707i	$-0.205114\pi$
$827$	45.9816	1.59894	0.799469	+	0.600707i	$0.205114\pi$
$828$	−1.79746	−0.0624659
$829$	55.1401	1.91510	0.957548	−	0.288274i	$-0.0930814\pi$
$829$	55.1401	1.91510	0.957548	+	0.288274i	$0.0930814\pi$
$830$	−23.7272	−0.823583
$831$	3.33941	0.115843
$832$	14.8521	0.514905
$833$	0	0
$834$	22.2325	0.769849
$835$	−33.2264	−1.14985
$836$	5.19946	0.179827
$837$	5.19267	0.179485
$838$	−66.7257	−2.30500
$839$	−29.3670	−1.01386	−0.506930	−	0.861987i	$-0.669220\pi$
$839$	−29.3670	−1.01386	−0.506930	+	0.861987i	$0.669220\pi$
$840$	0	0
$841$	−22.0781	−0.761312
$842$	−36.3132	−1.25143
$843$	3.94689	0.135938
$844$	−46.2387	−1.59160
$845$	10.4420	0.359217
$846$	−12.5480	−0.431410
$847$	0	0
$848$	48.5999	1.66893
$849$	5.85374	0.200900
$850$	6.55784	0.224932
$851$	−4.53435	−0.155435
$852$	26.0625	0.892886
$853$	14.0215	0.480086	0.240043	−	0.970762i	$-0.422838\pi$
$853$	14.0215	0.480086	0.240043	+	0.970762i	$0.422838\pi$
$854$	0	0
$855$	−0.894375	−0.0305870
$856$	3.29137	0.112497
$857$	10.4977	0.358596	0.179298	−	0.983795i	$-0.442617\pi$
$857$	10.4977	0.358596	0.179298	+	0.983795i	$0.442617\pi$
$858$	20.7989	0.710063
$859$	−23.0606	−0.786817	−0.393409	−	0.919364i	$-0.628704\pi$
$859$	−23.0606	−0.786817	−0.393409	+	0.919364i	$0.628704\pi$
$860$	−3.76219	−0.128290
$861$	0	0
$862$	−4.39937	−0.149843
$863$	5.74421	0.195535	0.0977676	−	0.995209i	$-0.468830\pi$
$863$	5.74421	0.195535	0.0977676	+	0.995209i	$0.468830\pi$
$864$	7.71488	0.262466
$865$	0.0316677	0.00107673
$866$	−9.49074	−0.322509
$867$	15.7721	0.535648
$868$	0	0
$869$	18.9684	0.643458
$870$	7.18344	0.243542
$871$	7.84693	0.265883
$872$	4.90848	0.166222
$873$	−9.73998	−0.329648
$874$	−1.24392	−0.0420763
$875$	0	0
$876$	3.58415	0.121097
$877$	−20.8315	−0.703429	−0.351715	−	0.936107i	$-0.614401\pi$
$877$	−20.8315	−0.703429	−0.351715	+	0.936107i	$0.614401\pi$
$878$	−71.0022	−2.39621
$879$	−21.0256	−0.709177
$880$	27.7087	0.934060
$881$	−14.7197	−0.495919	−0.247959	−	0.968770i	$-0.579760\pi$
$881$	−14.7197	−0.495919	−0.247959	+	0.968770i	$0.579760\pi$
$882$	0	0
$883$	−2.96791	−0.0998780	−0.0499390	−	0.998752i	$-0.515903\pi$
$883$	−2.96791	−0.0998780	−0.0499390	+	0.998752i	$0.515903\pi$
$884$	4.69120	0.157782
$885$	−8.69640	−0.292326
$886$	−15.9857	−0.537052
$887$	−15.3514	−0.515448	−0.257724	−	0.966219i	$-0.582973\pi$
$887$	−15.3514	−0.515448	−0.257724	+	0.966219i	$0.582973\pi$
$888$	1.78970	0.0600585
$889$	0	0
$890$	20.6444	0.692003
$891$	4.53162	0.151815
$892$	32.7313	1.09592
$893$	−4.11033	−0.137547
$894$	−22.3883	−0.748778
$895$	−5.41652	−0.181054
$896$	0	0
$897$	−2.35527	−0.0786402
$898$	19.0104	0.634385
$899$	−13.6617	−0.455644
$900$	−5.45868	−0.181956
$901$	12.3404	0.411118
$902$	−12.8237	−0.426984
$903$	0	0
$904$	2.25923	0.0751409
$905$	−2.11279	−0.0702316
$906$	29.9495	0.995005
$907$	17.0514	0.566182	0.283091	−	0.959093i	$-0.408640\pi$
$907$	17.0514	0.566182	0.283091	+	0.959093i	$0.408640\pi$
$908$	−3.73407	−0.123919
$909$	−9.80780	−0.325304
$910$	0	0
$911$	44.0199	1.45845	0.729223	−	0.684277i	$-0.239882\pi$
$911$	44.0199	1.45845	0.729223	+	0.684277i	$0.239882\pi$
$912$	2.78573	0.0922446
$913$	39.3805	1.30330
$914$	−21.9792	−0.727006
$915$	−1.15064	−0.0380388
$916$	0.612940	0.0202521
$917$	0	0
$918$	2.15939	0.0712705
$919$	−36.0193	−1.18817	−0.594083	−	0.804403i	$-0.702485\pi$
$919$	−36.0193	−1.18817	−0.594083	+	0.804403i	$0.702485\pi$
$920$	−0.553017	−0.0182324
$921$	−22.2208	−0.732202
$922$	−15.3979	−0.507103
$923$	34.1506	1.12408
$924$	0	0
$925$	−13.7703	−0.452766
$926$	18.9315	0.622128
$927$	14.8681	0.488331
$928$	−20.2975	−0.666299
$929$	24.2616	0.795996	0.397998	−	0.917386i	$-0.369705\pi$
$929$	24.2616	0.795996	0.397998	+	0.917386i	$0.369705\pi$
$930$	−14.1778	−0.464909
$931$	0	0
$932$	−10.5014	−0.343983
$933$	−33.3939	−1.09327
$934$	−0.130816	−0.00428042
$935$	7.03574	0.230093
$936$	0.929623	0.0303857
$937$	−16.1586	−0.527879	−0.263939	−	0.964539i	$-0.585022\pi$
$937$	−16.1586	−0.527879	−0.263939	+	0.964539i	$0.585022\pi$
$938$	0	0
$939$	−6.16899	−0.201318
$940$	16.2166	0.528927
$941$	17.7891	0.579908	0.289954	−	0.957041i	$-0.406360\pi$
$941$	17.7891	0.579908	0.289954	+	0.957041i	$0.406360\pi$
$942$	5.08510	0.165682
$943$	1.45216	0.0472890
$944$	27.0869	0.881602
$945$	0	0
$946$	13.1920	0.428908
$947$	−19.2463	−0.625420	−0.312710	−	0.949849i	$-0.601237\pi$
$947$	−19.2463	−0.625420	−0.312710	+	0.949849i	$0.601237\pi$
$948$	−7.52376	−0.244360
$949$	4.69644	0.152453
$950$	−3.77766	−0.122564
$951$	−3.70665	−0.120196
$952$	0	0
$953$	8.78877	0.284696	0.142348	−	0.989817i	$-0.454535\pi$
$953$	8.78877	0.284696	0.142348	+	0.989817i	$0.454535\pi$
$954$	−21.7015	−0.702613
$955$	−25.8037	−0.834987
$956$	−1.65325	−0.0534698
$957$	−11.9225	−0.385400
$958$	−41.2374	−1.33232
$959$	0	0
$960$	−8.83527	−0.285157
$961$	−4.03613	−0.130198
$962$	−20.8114	−0.670988
$963$	−8.33893	−0.268718
$964$	−33.1044	−1.06622
$965$	18.8181	0.605777
$966$	0	0
$967$	46.6382	1.49978	0.749892	−	0.661561i	$-0.230106\pi$
$967$	46.6382	1.49978	0.749892	+	0.661561i	$0.230106\pi$
$968$	−3.76367	−0.120969
$969$	0.707346	0.0227232
$970$	26.5936	0.853868
$971$	−53.8075	−1.72677	−0.863383	−	0.504550i	$-0.831658\pi$
$971$	−53.8075	−1.72677	−0.863383	+	0.504550i	$0.831658\pi$
$972$	−1.79746	−0.0576534
$973$	0	0
$974$	25.1801	0.806824
$975$	−7.15271	−0.229070
$976$	3.58391	0.114718
$977$	30.4644	0.974642	0.487321	−	0.873223i	$-0.337974\pi$
$977$	30.4644	0.974642	0.487321	+	0.873223i	$0.337974\pi$
$978$	−39.0562	−1.24888
$979$	−34.2640	−1.09508
$980$	0	0
$981$	−12.4360	−0.397051
$982$	8.90979	0.284323
$983$	9.50316	0.303104	0.151552	−	0.988449i	$-0.451573\pi$
$983$	9.50316	0.303104	0.151552	+	0.988449i	$0.451573\pi$
$984$	−0.573168	−0.0182719
$985$	−3.59892	−0.114671
$986$	−5.68127	−0.180928
$987$	0	0
$988$	−2.70238	−0.0859742
$989$	−1.49386	−0.0475020
$990$	−12.3729	−0.393237
$991$	4.93253	0.156687	0.0783435	−	0.996926i	$-0.475037\pi$
$991$	4.93253	0.156687	0.0783435	+	0.996926i	$0.475037\pi$
$992$	40.0609	1.27193
$993$	−1.27030	−0.0403117
$994$	0	0
$995$	23.2475	0.736996
$996$	−15.6202	−0.494944
$997$	19.3923	0.614159	0.307080	−	0.951684i	$-0.400648\pi$
$997$	19.3923	0.614159	0.307080	+	0.951684i	$0.400648\pi$
$998$	7.64852	0.242110
$999$	−4.53435	−0.143460

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bk.1.8	✓	10
7.6	odd	2		3381.2.a.bl.1.8	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3381.2.a.bk.1.8	✓	10	1.1	even	1	trivial
3381.2.a.bl.1.8	yes	10	7.6	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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q+1.94871 q^{2} -1.00000 q^{3} +1.79746 q^{4} -1.40111 q^{5} -1.94871 q^{6} -0.394699 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.94871 q^{2} -1.00000 q^{3} +1.79746 q^{4} -1.40111 q^{5} -1.94871 q^{6} -0.394699 q^{8} +1.00000 q^{9} -2.73035 q^{10} +4.53162 q^{11} -1.79746 q^{12} -2.35527 q^{13} +1.40111 q^{15} -4.36406 q^{16} -1.10811 q^{17} +1.94871 q^{18} +0.638333 q^{19} -2.51843 q^{20} +8.83079 q^{22} -1.00000 q^{23} +0.394699 q^{24} -3.03689 q^{25} -4.58973 q^{26} -1.00000 q^{27} +2.63096 q^{29} +2.73035 q^{30} -5.19267 q^{31} -7.71488 q^{32} -4.53162 q^{33} -2.15939 q^{34} +1.79746 q^{36} +4.53435 q^{37} +1.24392 q^{38} +2.35527 q^{39} +0.553017 q^{40} -1.45216 q^{41} +1.49386 q^{43} +8.14538 q^{44} -1.40111 q^{45} -1.94871 q^{46} -6.43916 q^{47} +4.36406 q^{48} -5.91801 q^{50} +1.10811 q^{51} -4.23350 q^{52} -11.1364 q^{53} -1.94871 q^{54} -6.34929 q^{55} -0.638333 q^{57} +5.12697 q^{58} -6.20679 q^{59} +2.51843 q^{60} -0.821232 q^{61} -10.1190 q^{62} -6.30591 q^{64} +3.29999 q^{65} -8.83079 q^{66} -3.33164 q^{67} -1.99179 q^{68} +1.00000 q^{69} -14.4996 q^{71} -0.394699 q^{72} -1.99401 q^{73} +8.83611 q^{74} +3.03689 q^{75} +1.14738 q^{76} +4.58973 q^{78} +4.18578 q^{79} +6.11453 q^{80} +1.00000 q^{81} -2.82984 q^{82} +8.69016 q^{83} +1.55259 q^{85} +2.91110 q^{86} -2.63096 q^{87} -1.78862 q^{88} -7.56109 q^{89} -2.73035 q^{90} -1.79746 q^{92} +5.19267 q^{93} -12.5480 q^{94} -0.894375 q^{95} +7.71488 q^{96} -9.73998 q^{97} +4.53162 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} - 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) 10 * q + 4 * q^2 - 10 * q^3 + 8 * q^4 - 4 * q^5 - 4 * q^6 + 12 * q^8 + 10 * q^9	\( 10 q + 4 q^{2} - 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{8} + 10 q^{9} - 8 q^{10} + 2 q^{11} - 8 q^{12} + 4 q^{15} + 4 q^{16} - 12 q^{17} + 4 q^{18} - 26 q^{19} - 24 q^{20} - 8 q^{22} - 10 q^{23} - 12 q^{24} - 2 q^{25} - 4 q^{26} - 10 q^{27} + 16 q^{29} + 8 q^{30} - 12 q^{31} + 8 q^{32} - 2 q^{33} - 28 q^{34} + 8 q^{36} - 8 q^{37} - 32 q^{38} - 4 q^{40} - 10 q^{41} - 4 q^{43} - 16 q^{44} - 4 q^{45} - 4 q^{46} - 2 q^{47} - 4 q^{48} - 8 q^{50} + 12 q^{51} - 24 q^{52} + 14 q^{53} - 4 q^{54} - 16 q^{55} + 26 q^{57} - 8 q^{58} - 38 q^{59} + 24 q^{60} - 14 q^{61} + 8 q^{62} + 8 q^{64} + 12 q^{65} + 8 q^{66} - 8 q^{68} + 10 q^{69} + 24 q^{71} + 12 q^{72} - 8 q^{73} - 8 q^{74} + 2 q^{75} - 64 q^{76} + 4 q^{78} - 16 q^{79} - 28 q^{80} + 10 q^{81} + 40 q^{82} - 28 q^{83} - 4 q^{85} + 20 q^{86} - 16 q^{87} - 68 q^{88} - 32 q^{89} - 8 q^{90} - 8 q^{92} + 12 q^{93} - 56 q^{94} + 8 q^{95} - 8 q^{96} - 4 q^{97} + 2 q^{99}+O(q^{100}) \) 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 + 4 * q^15 + 4 * q^16 - 12 * q^17 + 4 * q^18 - 26 * q^19 - 24 * q^20 - 8 * q^22 - 10 * q^23 - 12 * q^24 - 2 * q^25 - 4 * q^26 - 10 * q^27 + 16 * q^29 + 8 * q^30 - 12 * q^31 + 8 * q^32 - 2 * q^33 - 28 * q^34 + 8 * q^36 - 8 * q^37 - 32 * q^38 - 4 * q^40 - 10 * q^41 - 4 * q^43 - 16 * q^44 - 4 * q^45 - 4 * q^46 - 2 * q^47 - 4 * q^48 - 8 * q^50 + 12 * q^51 - 24 * q^52 + 14 * q^53 - 4 * q^54 - 16 * q^55 + 26 * q^57 - 8 * q^58 - 38 * q^59 + 24 * q^60 - 14 * q^61 + 8 * q^62 + 8 * q^64 + 12 * q^65 + 8 * q^66 - 8 * q^68 + 10 * q^69 + 24 * q^71 + 12 * q^72 - 8 * q^73 - 8 * q^74 + 2 * q^75 - 64 * q^76 + 4 * q^78 - 16 * q^79 - 28 * q^80 + 10 * q^81 + 40 * q^82 - 28 * q^83 - 4 * q^85 + 20 * q^86 - 16 * q^87 - 68 * q^88 - 32 * q^89 - 8 * q^90 - 8 * q^92 + 12 * q^93 - 56 * q^94 + 8 * q^95 - 8 * q^96 - 4 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more