LMFDB - Embedded newform 2011.2.a.a.1.26

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2011,2,Mod(1,2011)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2011, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2011.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2011.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:	$16.0579158465$
Analytic rank:	$1$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.26
Character	$\chi$	$=$	2011.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34536 q^{2} +2.37225 q^{3} -0.190019 q^{4} -0.880853 q^{5} -3.19152 q^{6} +0.181363 q^{7} +2.94635 q^{8} +2.62756 q^{9} +O(q^{10})$	\(q-1.34536 q^{2} +2.37225 q^{3} -0.190019 q^{4} -0.880853 q^{5} -3.19152 q^{6} +0.181363 q^{7} +2.94635 q^{8} +2.62756 q^{9} +1.18506 q^{10} -3.21107 q^{11} -0.450772 q^{12} +4.11946 q^{13} -0.243998 q^{14} -2.08960 q^{15} -3.58385 q^{16} -1.37412 q^{17} -3.53500 q^{18} -7.81119 q^{19} +0.167379 q^{20} +0.430239 q^{21} +4.32003 q^{22} +1.87559 q^{23} +6.98948 q^{24} -4.22410 q^{25} -5.54214 q^{26} -0.883520 q^{27} -0.0344625 q^{28} -7.97478 q^{29} +2.81126 q^{30} -0.656123 q^{31} -1.07115 q^{32} -7.61745 q^{33} +1.84868 q^{34} -0.159754 q^{35} -0.499287 q^{36} +5.23608 q^{37} +10.5088 q^{38} +9.77239 q^{39} -2.59530 q^{40} +7.56591 q^{41} -0.578824 q^{42} -4.03325 q^{43} +0.610165 q^{44} -2.31449 q^{45} -2.52334 q^{46} -1.49697 q^{47} -8.50179 q^{48} -6.96711 q^{49} +5.68291 q^{50} -3.25976 q^{51} -0.782777 q^{52} -1.60700 q^{53} +1.18865 q^{54} +2.82848 q^{55} +0.534361 q^{56} -18.5301 q^{57} +10.7289 q^{58} +1.99217 q^{59} +0.397064 q^{60} +9.10278 q^{61} +0.882718 q^{62} +0.476543 q^{63} +8.60879 q^{64} -3.62864 q^{65} +10.2482 q^{66} +8.59119 q^{67} +0.261109 q^{68} +4.44937 q^{69} +0.214927 q^{70} -14.9759 q^{71} +7.74172 q^{72} -2.06611 q^{73} -7.04439 q^{74} -10.0206 q^{75} +1.48427 q^{76} -0.582370 q^{77} -13.1473 q^{78} +11.9855 q^{79} +3.15685 q^{80} -9.97861 q^{81} -10.1788 q^{82} -12.0192 q^{83} -0.0817536 q^{84} +1.21040 q^{85} +5.42616 q^{86} -18.9182 q^{87} -9.46095 q^{88} +2.19716 q^{89} +3.11382 q^{90} +0.747120 q^{91} -0.356398 q^{92} -1.55649 q^{93} +2.01396 q^{94} +6.88051 q^{95} -2.54103 q^{96} -17.5954 q^{97} +9.37323 q^{98} -8.43728 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 13 q^{2} - 13 q^{3} + 67 q^{4} - 47 q^{5} - 20 q^{6} - 8 q^{7} - 33 q^{8} + 52 q^{9}+O(q^{10}) $ 77 * q - 13 * q^2 - 13 * q^3 + 67 * q^4 - 47 * q^5 - 20 * q^6 - 8 * q^7 - 33 * q^8 + 52 * q^9	\( 77 q - 13 q^{2} - 13 q^{3} + 67 q^{4} - 47 q^{5} - 20 q^{6} - 8 q^{7} - 33 q^{8} + 52 q^{9} - 21 q^{10} - 34 q^{11} - 36 q^{12} - 34 q^{13} - 49 q^{14} - 12 q^{15} + 47 q^{16} - 59 q^{17} - 24 q^{18} - 31 q^{19} - 82 q^{20} - 71 q^{21} - 3 q^{22} - 28 q^{23} - 50 q^{24} + 68 q^{25} - 54 q^{26} - 43 q^{27} - 2 q^{28} - 151 q^{29} + q^{30} - 37 q^{31} - 59 q^{32} - 35 q^{33} - q^{34} - 58 q^{35} + 19 q^{36} - 29 q^{37} - 22 q^{38} - 40 q^{39} - 41 q^{40} - 142 q^{41} + 16 q^{42} - 23 q^{43} - 89 q^{44} - 119 q^{45} - 6 q^{46} - 36 q^{47} - 46 q^{48} + 45 q^{49} - 29 q^{50} - 53 q^{51} - 11 q^{52} - 69 q^{53} - 50 q^{54} - 13 q^{55} - 122 q^{56} - 14 q^{57} + 31 q^{58} - 92 q^{59} + 20 q^{60} - 115 q^{61} - 66 q^{62} - 25 q^{63} + 37 q^{64} - 57 q^{65} - 17 q^{66} - 108 q^{68} - 160 q^{69} + 40 q^{70} - 67 q^{71} - 35 q^{72} - 36 q^{73} - 55 q^{74} - 51 q^{75} - 56 q^{76} - 116 q^{77} + 22 q^{78} - 42 q^{79} - 114 q^{80} + 37 q^{81} + 18 q^{82} - 42 q^{83} - 77 q^{84} - 18 q^{85} - 33 q^{86} - 7 q^{87} - 2 q^{88} - 93 q^{89} - 34 q^{90} - 37 q^{91} - 55 q^{92} - 8 q^{93} - 35 q^{94} - 64 q^{95} - 83 q^{96} - 16 q^{97} - 57 q^{98} - 65 q^{99}+O(q^{100}) \) 77 * q - 13 * q^2 - 13 * q^3 + 67 * q^4 - 47 * q^5 - 20 * q^6 - 8 * q^7 - 33 * q^8 + 52 * q^9 - 21 * q^10 - 34 * q^11 - 36 * q^12 - 34 * q^13 - 49 * q^14 - 12 * q^15 + 47 * q^16 - 59 * q^17 - 24 * q^18 - 31 * q^19 - 82 * q^20 - 71 * q^21 - 3 * q^22 - 28 * q^23 - 50 * q^24 + 68 * q^25 - 54 * q^26 - 43 * q^27 - 2 * q^28 - 151 * q^29 + q^30 - 37 * q^31 - 59 * q^32 - 35 * q^33 - q^34 - 58 * q^35 + 19 * q^36 - 29 * q^37 - 22 * q^38 - 40 * q^39 - 41 * q^40 - 142 * q^41 + 16 * q^42 - 23 * q^43 - 89 * q^44 - 119 * q^45 - 6 * q^46 - 36 * q^47 - 46 * q^48 + 45 * q^49 - 29 * q^50 - 53 * q^51 - 11 * q^52 - 69 * q^53 - 50 * q^54 - 13 * q^55 - 122 * q^56 - 14 * q^57 + 31 * q^58 - 92 * q^59 + 20 * q^60 - 115 * q^61 - 66 * q^62 - 25 * q^63 + 37 * q^64 - 57 * q^65 - 17 * q^66 - 108 * q^68 - 160 * q^69 + 40 * q^70 - 67 * q^71 - 35 * q^72 - 36 * q^73 - 55 * q^74 - 51 * q^75 - 56 * q^76 - 116 * q^77 + 22 * q^78 - 42 * q^79 - 114 * q^80 + 37 * q^81 + 18 * q^82 - 42 * q^83 - 77 * q^84 - 18 * q^85 - 33 * q^86 - 7 * q^87 - 2 * q^88 - 93 * q^89 - 34 * q^90 - 37 * q^91 - 55 * q^92 - 8 * q^93 - 35 * q^94 - 64 * q^95 - 83 * q^96 - 16 * q^97 - 57 * q^98 - 65 * 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 $

$2$	−1.34536	−0.951310	−0.475655	−	0.879632i	$-0.657789\pi$
$2$	−1.34536	−0.951310	−0.475655	+	0.879632i	$0.657789\pi$
$3$	2.37225	1.36962	0.684809	−	0.728723i	$-0.259886\pi$
$3$	2.37225	1.36962	0.684809	+	0.728723i	$0.259886\pi$
$4$	−0.190019	−0.0950096
$5$	−0.880853	−0.393929	−0.196965	−	0.980411i	$-0.563108\pi$
$5$	−0.880853	−0.393929	−0.196965	+	0.980411i	$0.563108\pi$
$6$	−3.19152	−1.30293
$7$	0.181363	0.0685489	0.0342745	−	0.999412i	$-0.489088\pi$
$7$	0.181363	0.0685489	0.0342745	+	0.999412i	$0.489088\pi$
$8$	2.94635	1.04169
$9$	2.62756	0.875853
$10$	1.18506	0.374749
$11$	−3.21107	−0.968174	−0.484087	−	0.875020i	$-0.660848\pi$
$11$	−3.21107	−0.968174	−0.484087	+	0.875020i	$0.660848\pi$
$12$	−0.450772	−0.130127
$13$	4.11946	1.14253	0.571267	−	0.820764i	$-0.306452\pi$
$13$	4.11946	1.14253	0.571267	+	0.820764i	$0.306452\pi$
$14$	−0.243998	−0.0652113
$15$	−2.08960	−0.539533
$16$	−3.58385	−0.895964
$17$	−1.37412	−0.333273	−0.166637	−	0.986018i	$-0.553291\pi$
$17$	−1.37412	−0.333273	−0.166637	+	0.986018i	$0.553291\pi$
$18$	−3.53500	−0.833208
$19$	−7.81119	−1.79201	−0.896005	−	0.444045i	$-0.853543\pi$
$19$	−7.81119	−1.79201	−0.896005	+	0.444045i	$0.853543\pi$
$20$	0.167379	0.0374271
$21$	0.430239	0.0938858
$22$	4.32003	0.921033
$23$	1.87559	0.391088	0.195544	−	0.980695i	$-0.437353\pi$
$23$	1.87559	0.391088	0.195544	+	0.980695i	$0.437353\pi$
$24$	6.98948	1.42672
$25$	−4.22410	−0.844820
$26$	−5.54214	−1.08690
$27$	−0.883520	−0.170033
$28$	−0.0344625	−0.00651280
$29$	−7.97478	−1.48088	−0.740440	−	0.672122i	$-0.765383\pi$
$29$	−7.97478	−1.48088	−0.740440	+	0.672122i	$0.765383\pi$
$30$	2.81126	0.513263
$31$	−0.656123	−0.117843	−0.0589215	−	0.998263i	$-0.518766\pi$
$31$	−0.656123	−0.117843	−0.0589215	+	0.998263i	$0.518766\pi$
$32$	−1.07115	−0.189354
$33$	−7.61745	−1.32603
$34$	1.84868	0.317046
$35$	−0.159754	−0.0270034
$36$	−0.499287	−0.0832144
$37$	5.23608	0.860806	0.430403	−	0.902637i	$-0.358371\pi$
$37$	5.23608	0.860806	0.430403	+	0.902637i	$0.358371\pi$
$38$	10.5088	1.70476
$39$	9.77239	1.56483
$40$	−2.59530	−0.410354
$41$	7.56591	1.18160	0.590798	−	0.806819i	$-0.298813\pi$
$41$	7.56591	1.18160	0.590798	+	0.806819i	$0.298813\pi$
$42$	−0.578824	−0.0893145
$43$	−4.03325	−0.615065	−0.307533	−	0.951538i	$-0.599503\pi$
$43$	−4.03325	−0.615065	−0.307533	+	0.951538i	$0.599503\pi$
$44$	0.610165	0.0919858
$45$	−2.31449	−0.345024
$46$	−2.52334	−0.372046
$47$	−1.49697	−0.218356	−0.109178	−	0.994022i	$-0.534822\pi$
$47$	−1.49697	−0.218356	−0.109178	+	0.994022i	$0.534822\pi$
$48$	−8.50179	−1.22713
$49$	−6.96711	−0.995301
$50$	5.68291	0.803685
$51$	−3.25976	−0.456457
$52$	−0.782777	−0.108552
$53$	−1.60700	−0.220739	−0.110369	−	0.993891i	$-0.535203\pi$
$53$	−1.60700	−0.220739	−0.110369	+	0.993891i	$0.535203\pi$
$54$	1.18865	0.161755
$55$	2.82848	0.381392
$56$	0.534361	0.0714070
$57$	−18.5301	−2.45437
$58$	10.7289	1.40878
$59$	1.99217	0.259358	0.129679	−	0.991556i	$-0.458605\pi$
$59$	1.99217	0.259358	0.129679	+	0.991556i	$0.458605\pi$
$60$	0.397064	0.0512608
$61$	9.10278	1.16549	0.582746	−	0.812655i	$-0.301978\pi$
$61$	9.10278	1.16549	0.582746	+	0.812655i	$0.301978\pi$
$62$	0.882718	0.112105
$63$	0.476543	0.0600388
$64$	8.60879	1.07610
$65$	−3.62864	−0.450078
$66$	10.2482	1.26146
$67$	8.59119	1.04958	0.524791	−	0.851231i	$-0.324144\pi$
$67$	8.59119	1.04958	0.524791	+	0.851231i	$0.324144\pi$
$68$	0.261109	0.0316642
$69$	4.44937	0.535641
$70$	0.214927	0.0256886
$71$	−14.9759	−1.77731	−0.888656	−	0.458574i	$-0.848360\pi$
$71$	−14.9759	−1.77731	−0.888656	+	0.458574i	$0.848360\pi$
$72$	7.74172	0.912371
$73$	−2.06611	−0.241820	−0.120910	−	0.992663i	$-0.538581\pi$
$73$	−2.06611	−0.241820	−0.120910	+	0.992663i	$0.538581\pi$
$74$	−7.04439	−0.818893
$75$	−10.0206	−1.15708
$76$	1.48427	0.170258
$77$	−0.582370	−0.0663673
$78$	−13.1473	−1.48864
$79$	11.9855	1.34847	0.674234	−	0.738517i	$-0.264474\pi$
$79$	11.9855	1.34847	0.674234	+	0.738517i	$0.264474\pi$
$80$	3.15685	0.352946
$81$	−9.97861	−1.10873
$82$	−10.1788	−1.12406
$83$	−12.0192	−1.31928	−0.659638	−	0.751584i	$-0.729290\pi$
$83$	−12.0192	−1.31928	−0.659638	+	0.751584i	$0.729290\pi$
$84$	−0.0817536	−0.00892005
$85$	1.21040	0.131286
$86$	5.42616	0.585117
$87$	−18.9182	−2.02824
$88$	−9.46095	−1.00854
$89$	2.19716	0.232898	0.116449	−	0.993197i	$-0.462849\pi$
$89$	2.19716	0.232898	0.116449	+	0.993197i	$0.462849\pi$
$90$	3.11382	0.328225
$91$	0.747120	0.0783194
$92$	−0.356398	−0.0371571
$93$	−1.55649	−0.161400
$94$	2.01396	0.207724
$95$	6.88051	0.705925
$96$	−2.54103	−0.259343
$97$	−17.5954	−1.78655	−0.893274	−	0.449513i	$-0.851597\pi$
$97$	−17.5954	−1.78655	−0.893274	+	0.449513i	$0.851597\pi$
$98$	9.37323	0.946840
$99$	−8.43728	−0.847978
$100$	0.802659	0.0802659
$101$	−14.3365	−1.42653	−0.713267	−	0.700892i	$-0.752785\pi$
$101$	−14.3365	−1.42653	−0.713267	+	0.700892i	$0.752785\pi$
$102$	4.38553	0.434232
$103$	6.11546	0.602574	0.301287	−	0.953533i	$-0.402584\pi$
$103$	6.11546	0.602574	0.301287	+	0.953533i	$0.402584\pi$
$104$	12.1374	1.19017
$105$	−0.378977	−0.0369844
$106$	2.16199	0.209991
$107$	−11.9712	−1.15730	−0.578652	−	0.815575i	$-0.696421\pi$
$107$	−11.9712	−1.15730	−0.578652	+	0.815575i	$0.696421\pi$
$108$	0.167886	0.0161548
$109$	−9.20982	−0.882141	−0.441070	−	0.897472i	$-0.645401\pi$
$109$	−9.20982	−0.882141	−0.441070	+	0.897472i	$0.645401\pi$
$110$	−3.80531	−0.362822
$111$	12.4213	1.17898
$112$	−0.649980	−0.0614173
$113$	−12.4996	−1.17587	−0.587934	−	0.808909i	$-0.700059\pi$
$113$	−12.4996	−1.17587	−0.587934	+	0.808909i	$0.700059\pi$
$114$	24.9295	2.33486
$115$	−1.65212	−0.154061
$116$	1.51536	0.140698
$117$	10.8241	1.00069
$118$	−2.68018	−0.246730
$119$	−0.249215	−0.0228455
$120$	−6.15671	−0.562028
$121$	−0.689035	−0.0626395
$122$	−12.2465	−1.10874
$123$	17.9482	1.61834
$124$	0.124676	0.0111962
$125$	8.12507	0.726729
$126$	−0.641120	−0.0571155
$127$	−4.44008	−0.393993	−0.196997	−	0.980404i	$-0.563119\pi$
$127$	−4.44008	−0.393993	−0.196997	+	0.980404i	$0.563119\pi$
$128$	−9.43958	−0.834349
$129$	−9.56787	−0.842404
$130$	4.88181	0.428163
$131$	9.87496	0.862779	0.431390	−	0.902166i	$-0.358023\pi$
$131$	9.87496	0.862779	0.431390	+	0.902166i	$0.358023\pi$
$132$	1.44746	0.125985
$133$	−1.41666	−0.122840
$134$	−11.5582	−0.998477
$135$	0.778251	0.0669812
$136$	−4.04865	−0.347169
$137$	−1.67848	−0.143402	−0.0717010	−	0.997426i	$-0.522843\pi$
$137$	−1.67848	−0.143402	−0.0717010	+	0.997426i	$0.522843\pi$
$138$	−5.98598	−0.509561
$139$	17.0054	1.44238	0.721190	−	0.692737i	$-0.243595\pi$
$139$	17.0054	1.44238	0.721190	+	0.692737i	$0.243595\pi$
$140$	0.0303564	0.00256558
$141$	−3.55119	−0.299064
$142$	20.1479	1.69077
$143$	−13.2279	−1.10617
$144$	−9.41679	−0.784733
$145$	7.02461	0.583362
$146$	2.77965	0.230046
$147$	−16.5277	−1.36318
$148$	−0.994955	−0.0817848
$149$	−6.87940	−0.563583	−0.281791	−	0.959476i	$-0.590929\pi$
$149$	−6.87940	−0.563583	−0.281791	+	0.959476i	$0.590929\pi$
$150$	13.4813	1.10074
$151$	−19.1844	−1.56120	−0.780602	−	0.625028i	$-0.785087\pi$
$151$	−19.1844	−1.56120	−0.780602	+	0.625028i	$0.785087\pi$
$152$	−23.0145	−1.86672
$153$	−3.61059	−0.291899
$154$	0.783495	0.0631358
$155$	0.577947	0.0464219
$156$	−1.85694	−0.148674
$157$	−17.5364	−1.39955	−0.699777	−	0.714361i	$-0.746717\pi$
$157$	−17.5364	−1.39955	−0.699777	+	0.714361i	$0.746717\pi$
$158$	−16.1247	−1.28281
$159$	−3.81221	−0.302328
$160$	0.943526	0.0745922
$161$	0.340164	0.0268087
$162$	13.4248	1.05475
$163$	15.5184	1.21550	0.607748	−	0.794130i	$-0.292073\pi$
$163$	15.5184	1.21550	0.607748	+	0.794130i	$0.292073\pi$
$164$	−1.43767	−0.112263
$165$	6.70985	0.522361
$166$	16.1701	1.25504
$167$	2.93952	0.227467	0.113734	−	0.993511i	$-0.463719\pi$
$167$	2.93952	0.227467	0.113734	+	0.993511i	$0.463719\pi$
$168$	1.26764	0.0978002
$169$	3.96998	0.305383
$170$	−1.62842	−0.124894
$171$	−20.5244	−1.56954
$172$	0.766395	0.0584371
$173$	3.82877	0.291096	0.145548	−	0.989351i	$-0.453505\pi$
$173$	3.82877	0.291096	0.145548	+	0.989351i	$0.453505\pi$
$174$	25.4516	1.92948
$175$	−0.766097	−0.0579115
$176$	11.5080	0.867448
$177$	4.72592	0.355222
$178$	−2.95596	−0.221559
$179$	9.11575	0.681343	0.340672	−	0.940182i	$-0.389346\pi$
$179$	9.11575	0.681343	0.340672	+	0.940182i	$0.389346\pi$
$180$	0.439798	0.0327806
$181$	2.36276	0.175623	0.0878113	−	0.996137i	$-0.472013\pi$
$181$	2.36276	0.175623	0.0878113	+	0.996137i	$0.472013\pi$
$182$	−1.00514	−0.0745061
$183$	21.5940	1.59628
$184$	5.52616	0.407394
$185$	−4.61222	−0.339097
$186$	2.09403	0.153541
$187$	4.41240	0.322667
$188$	0.284453	0.0207459
$189$	−0.160238	−0.0116556
$190$	−9.25673	−0.671554
$191$	11.4699	0.829933	0.414966	−	0.909837i	$-0.363793\pi$
$191$	11.4699	0.829933	0.414966	+	0.909837i	$0.363793\pi$
$192$	20.4222	1.47384
$193$	1.96520	0.141458	0.0707292	−	0.997496i	$-0.477467\pi$
$193$	1.96520	0.141458	0.0707292	+	0.997496i	$0.477467\pi$
$194$	23.6721	1.69956
$195$	−8.60804	−0.616434
$196$	1.32388	0.0945631
$197$	6.81215	0.485346	0.242673	−	0.970108i	$-0.421976\pi$
$197$	6.81215	0.485346	0.242673	+	0.970108i	$0.421976\pi$
$198$	11.3511	0.806690
$199$	28.0136	1.98583	0.992915	−	0.118828i	$-0.0379136\pi$
$199$	28.0136	1.98583	0.992915	+	0.118828i	$0.0379136\pi$
$200$	−12.4457	−0.880043
$201$	20.3804	1.43753
$202$	19.2877	1.35708
$203$	−1.44633	−0.101513
$204$	0.619416	0.0433678
$205$	−6.66445	−0.465466
$206$	−8.22747	−0.573235
$207$	4.92823	0.342536
$208$	−14.7636	−1.02367
$209$	25.0823	1.73498
$210$	0.509859	0.0351836
$211$	−9.97042	−0.686392	−0.343196	−	0.939264i	$-0.611510\pi$
$211$	−9.97042	−0.686392	−0.343196	+	0.939264i	$0.611510\pi$
$212$	0.305361	0.0209723
$213$	−35.5266	−2.43424
$214$	16.1056	1.10095
$215$	3.55270	0.242292
$216$	−2.60316	−0.177123
$217$	−0.118997	−0.00807802
$218$	12.3905	0.839189
$219$	−4.90132	−0.331201
$220$	−0.537465	−0.0362359
$221$	−5.66064	−0.380776
$222$	−16.7110	−1.12157
$223$	5.19051	0.347583	0.173791	−	0.984783i	$-0.444398\pi$
$223$	5.19051	0.347583	0.173791	+	0.984783i	$0.444398\pi$
$224$	−0.194267	−0.0129800
$225$	−11.0991	−0.739938
$226$	16.8165	1.11861
$227$	−14.4718	−0.960529	−0.480265	−	0.877124i	$-0.659459\pi$
$227$	−14.4718	−0.960529	−0.480265	+	0.877124i	$0.659459\pi$
$228$	3.52107	0.233188
$229$	7.40579	0.489388	0.244694	−	0.969600i	$-0.421312\pi$
$229$	7.40579	0.489388	0.244694	+	0.969600i	$0.421312\pi$
$230$	2.22269	0.146560
$231$	−1.38153	−0.0908978
$232$	−23.4965	−1.54262
$233$	−24.8778	−1.62980	−0.814901	−	0.579600i	$-0.803209\pi$
$233$	−24.8778	−1.62980	−0.814901	+	0.579600i	$0.803209\pi$
$234$	−14.5623	−0.951968
$235$	1.31861	0.0860168
$236$	−0.378550	−0.0246415
$237$	28.4325	1.84689
$238$	0.335283	0.0217332
$239$	13.7752	0.891044	0.445522	−	0.895271i	$-0.353018\pi$
$239$	13.7752	0.891044	0.445522	+	0.895271i	$0.353018\pi$
$240$	7.48883	0.483402
$241$	2.99541	0.192952	0.0964758	−	0.995335i	$-0.469243\pi$
$241$	2.99541	0.192952	0.0964758	+	0.995335i	$0.469243\pi$
$242$	0.926997	0.0595896
$243$	−21.0212	−1.34851
$244$	−1.72970	−0.110733
$245$	6.13700	0.392078
$246$	−24.1467	−1.53954
$247$	−32.1779	−2.04743
$248$	−1.93317	−0.122756
$249$	−28.5124	−1.80690
$250$	−10.9311	−0.691344
$251$	−11.1008	−0.700679	−0.350339	−	0.936623i	$-0.613934\pi$
$251$	−11.1008	−0.700679	−0.350339	+	0.936623i	$0.613934\pi$
$252$	−0.0905523	−0.00570426
$253$	−6.02265	−0.378641
$254$	5.97348	0.374810
$255$	2.87137	0.179812
$256$	−4.51799	−0.282374
$257$	−13.1784	−0.822045	−0.411022	−	0.911625i	$-0.634828\pi$
$257$	−13.1784	−0.822045	−0.411022	+	0.911625i	$0.634828\pi$
$258$	12.8722	0.801387
$259$	0.949633	0.0590073
$260$	0.689511	0.0427617
$261$	−20.9542	−1.29703
$262$	−13.2853	−0.820770
$263$	25.0252	1.54312	0.771559	−	0.636158i	$-0.219477\pi$
$263$	25.0252	1.54312	0.771559	+	0.636158i	$0.219477\pi$
$264$	−22.4437	−1.38131
$265$	1.41553	0.0869555
$266$	1.90592	0.116859
$267$	5.21221	0.318982
$268$	−1.63249	−0.0997203
$269$	0.292315	0.0178227	0.00891137	−	0.999960i	$-0.497163\pi$
$269$	0.292315	0.0178227	0.00891137	+	0.999960i	$0.497163\pi$
$270$	−1.04702	−0.0637199
$271$	3.70104	0.224822	0.112411	−	0.993662i	$-0.464143\pi$
$271$	3.70104	0.224822	0.112411	+	0.993662i	$0.464143\pi$
$272$	4.92465	0.298601
$273$	1.77235	0.107268
$274$	2.25815	0.136420
$275$	13.5639	0.817932
$276$	−0.845465	−0.0508910
$277$	−19.4721	−1.16997	−0.584984	−	0.811045i	$-0.698899\pi$
$277$	−19.4721	−1.16997	−0.584984	+	0.811045i	$0.698899\pi$
$278$	−22.8783	−1.37215
$279$	−1.72400	−0.103213
$280$	−0.470693	−0.0281293
$281$	−13.8421	−0.825749	−0.412875	−	0.910788i	$-0.635475\pi$
$281$	−13.8421	−0.825749	−0.412875	+	0.910788i	$0.635475\pi$
$282$	4.77761	0.284503
$283$	−1.65329	−0.0982776	−0.0491388	−	0.998792i	$-0.515648\pi$
$283$	−1.65329	−0.0982776	−0.0491388	+	0.998792i	$0.515648\pi$
$284$	2.84571	0.168862
$285$	16.3223	0.966848
$286$	17.7962	1.05231
$287$	1.37218	0.0809972
$288$	−2.81451	−0.165847
$289$	−15.1118	−0.888929
$290$	−9.45060	−0.554958
$291$	−41.7408	−2.44689
$292$	0.392600	0.0229752
$293$	0.245803	0.0143600	0.00717998	−	0.999974i	$-0.497715\pi$
$293$	0.245803	0.0143600	0.00717998	+	0.999974i	$0.497715\pi$
$294$	22.2356	1.29681
$295$	−1.75481	−0.102169
$296$	15.4273	0.896696
$297$	2.83704	0.164622
$298$	9.25524	0.536142
$299$	7.72643	0.446831
$300$	1.90411	0.109934
$301$	−0.731484	−0.0421620
$302$	25.8098	1.48519
$303$	−34.0097	−1.95381
$304$	27.9942	1.60558
$305$	−8.01821	−0.459121
$306$	4.85752	0.277686
$307$	−12.9070	−0.736643	−0.368321	−	0.929699i	$-0.620067\pi$
$307$	−12.9070	−0.736643	−0.368321	+	0.929699i	$0.620067\pi$
$308$	0.110662	0.00630552
$309$	14.5074	0.825297
$310$	−0.777545	−0.0441616
$311$	−30.0365	−1.70321	−0.851606	−	0.524183i	$-0.824371\pi$
$311$	−30.0365	−1.70321	−0.851606	+	0.524183i	$0.824371\pi$
$312$	28.7929	1.63008
$313$	17.9499	1.01459	0.507294	−	0.861773i	$-0.330646\pi$
$313$	17.9499	1.01459	0.507294	+	0.861773i	$0.330646\pi$
$314$	23.5926	1.33141
$315$	−0.419764	−0.0236510
$316$	−2.27747	−0.128117
$317$	−18.0129	−1.01170	−0.505852	−	0.862620i	$-0.668822\pi$
$317$	−18.0129	−1.01170	−0.505852	+	0.862620i	$0.668822\pi$
$318$	5.12877	0.287607
$319$	25.6076	1.43375
$320$	−7.58307	−0.423907
$321$	−28.3988	−1.58506
$322$	−0.457641	−0.0255033
$323$	10.7335	0.597229
$324$	1.89613	0.105340
$325$	−17.4010	−0.965235
$326$	−20.8778	−1.15631
$327$	−21.8480	−1.20820
$328$	22.2918	1.23086
$329$	−0.271496	−0.0149681
$330$	−9.02714	−0.496928
$331$	8.94698	0.491770	0.245885	−	0.969299i	$-0.420921\pi$
$331$	8.94698	0.491770	0.245885	+	0.969299i	$0.420921\pi$
$332$	2.28387	0.125344
$333$	13.7581	0.753940
$334$	−3.95470	−0.216392
$335$	−7.56758	−0.413461
$336$	−1.54191	−0.0841183
$337$	30.1604	1.64294	0.821471	−	0.570250i	$-0.193154\pi$
$337$	30.1604	1.64294	0.821471	+	0.570250i	$0.193154\pi$
$338$	−5.34104	−0.290514
$339$	−29.6523	−1.61049
$340$	−0.229999	−0.0124734
$341$	2.10685	0.114093
$342$	27.6126	1.49312
$343$	−2.53312	−0.136776
$344$	−11.8834	−0.640709
$345$	−3.91924	−0.211005
$346$	−5.15106	−0.276923
$347$	10.8692	0.583489	0.291744	−	0.956496i	$-0.405764\pi$
$347$	10.8692	0.583489	0.291744	+	0.956496i	$0.405764\pi$
$348$	3.59481	0.192702
$349$	−0.477247	−0.0255465	−0.0127732	−	0.999918i	$-0.504066\pi$
$349$	−0.477247	−0.0255465	−0.0127732	+	0.999918i	$0.504066\pi$
$350$	1.03067	0.0550918
$351$	−3.63963	−0.194269
$352$	3.43954	0.183328
$353$	30.7060	1.63431	0.817157	−	0.576416i	$-0.195549\pi$
$353$	30.7060	1.63431	0.817157	+	0.576416i	$0.195549\pi$
$354$	−6.35804	−0.337926
$355$	13.1916	0.700136
$356$	−0.417502	−0.0221276
$357$	−0.591200	−0.0312896
$358$	−12.2639	−0.648168
$359$	18.6779	0.985780	0.492890	−	0.870092i	$-0.335940\pi$
$359$	18.6779	0.985780	0.492890	+	0.870092i	$0.335940\pi$
$360$	−6.81932	−0.359410
$361$	42.0147	2.21130
$362$	−3.17875	−0.167072
$363$	−1.63456	−0.0857922
$364$	−0.141967	−0.00744110
$365$	1.81994	0.0952599
$366$	−29.0517	−1.51855
$367$	6.56786	0.342840	0.171420	−	0.985198i	$-0.445165\pi$
$367$	6.56786	0.342840	0.171420	+	0.985198i	$0.445165\pi$
$368$	−6.72185	−0.350401
$369$	19.8799	1.03491
$370$	6.20507	0.322586
$371$	−0.291451	−0.0151314
$372$	0.295762	0.0153345
$373$	21.1522	1.09522	0.547609	−	0.836734i	$-0.315538\pi$
$373$	21.1522	1.09522	0.547609	+	0.836734i	$0.315538\pi$
$374$	−5.93624	−0.306956
$375$	19.2747	0.995341
$376$	−4.41061	−0.227460
$377$	−32.8518	−1.69196
$378$	0.215577	0.0110881
$379$	−9.09191	−0.467020	−0.233510	−	0.972354i	$-0.575021\pi$
$379$	−9.09191	−0.467020	−0.233510	+	0.972354i	$0.575021\pi$
$380$	−1.30743	−0.0670696
$381$	−10.5330	−0.539620
$382$	−15.4311	−0.789523
$383$	−34.6872	−1.77243	−0.886217	−	0.463270i	$-0.846676\pi$
$383$	−34.6872	−1.77243	−0.886217	+	0.463270i	$0.846676\pi$
$384$	−22.3930	−1.14274
$385$	0.512983	0.0261440
$386$	−2.64389	−0.134571
$387$	−10.5976	−0.538707
$388$	3.34347	0.169739
$389$	−31.6237	−1.60338	−0.801692	−	0.597737i	$-0.796067\pi$
$389$	−31.6237	−1.60338	−0.801692	+	0.597737i	$0.796067\pi$
$390$	11.5809	0.586420
$391$	−2.57729	−0.130339
$392$	−20.5276	−1.03680
$393$	23.4259	1.18168
$394$	−9.16476	−0.461714
$395$	−10.5574	−0.531201
$396$	1.60324	0.0805660
$397$	23.3902	1.17392	0.586960	−	0.809616i	$-0.300325\pi$
$397$	23.3902	1.17392	0.586960	+	0.809616i	$0.300325\pi$
$398$	−37.6882	−1.88914
$399$	−3.36068	−0.168244
$400$	15.1386	0.756928
$401$	−8.95075	−0.446979	−0.223490	−	0.974706i	$-0.571745\pi$
$401$	−8.95075	−0.446979	−0.223490	+	0.974706i	$0.571745\pi$
$402$	−27.4189	−1.36753
$403$	−2.70287	−0.134640
$404$	2.72421	0.135534
$405$	8.78969	0.436763
$406$	1.94583	0.0965700
$407$	−16.8134	−0.833410
$408$	−9.60440	−0.475488
$409$	30.0996	1.48833	0.744165	−	0.667996i	$-0.232848\pi$
$409$	30.0996	1.48833	0.744165	+	0.667996i	$0.232848\pi$
$410$	8.96606	0.442802
$411$	−3.98176	−0.196406
$412$	−1.16205	−0.0572503
$413$	0.361307	0.0177787
$414$	−6.63022	−0.325858
$415$	10.5871	0.519701
$416$	−4.41256	−0.216344
$417$	40.3411	1.97551
$418$	−33.7446	−1.65050
$419$	−12.1481	−0.593472	−0.296736	−	0.954959i	$-0.595898\pi$
$419$	−12.1481	−0.593472	−0.296736	+	0.954959i	$0.595898\pi$
$420$	0.0720129	0.00351387
$421$	11.0239	0.537270	0.268635	−	0.963242i	$-0.413427\pi$
$421$	11.0239	0.537270	0.268635	+	0.963242i	$0.413427\pi$
$422$	13.4138	0.652971
$423$	−3.93338	−0.191248
$424$	−4.73480	−0.229942
$425$	5.80442	0.281556
$426$	47.7958	2.31572
$427$	1.65091	0.0798932
$428$	2.27477	0.109955
$429$	−31.3798	−1.51503
$430$	−4.77964	−0.230495
$431$	−5.07424	−0.244417	−0.122209	−	0.992504i	$-0.538998\pi$
$431$	−5.07424	−0.244417	−0.122209	+	0.992504i	$0.538998\pi$
$432$	3.16641	0.152344
$433$	22.1797	1.06589	0.532945	−	0.846150i	$-0.321085\pi$
$433$	22.1797	1.06589	0.532945	+	0.846150i	$0.321085\pi$
$434$	0.160093	0.00768470
$435$	16.6641	0.798983
$436$	1.75004	0.0838118
$437$	−14.6506	−0.700833
$438$	6.59402	0.315075
$439$	22.1857	1.05887	0.529434	−	0.848351i	$-0.322404\pi$
$439$	22.1857	1.05887	0.529434	+	0.848351i	$0.322404\pi$
$440$	8.33370	0.397294
$441$	−18.3065	−0.871738
$442$	7.61558	0.362236
$443$	0.620368	0.0294746	0.0147373	−	0.999891i	$-0.495309\pi$
$443$	0.620368	0.0294746	0.0147373	+	0.999891i	$0.495309\pi$
$444$	−2.36028	−0.112014
$445$	−1.93537	−0.0917455
$446$	−6.98309	−0.330659
$447$	−16.3196	−0.771893
$448$	1.56132	0.0737654
$449$	39.4786	1.86311	0.931555	−	0.363600i	$-0.118453\pi$
$449$	39.4786	1.86311	0.931555	+	0.363600i	$0.118453\pi$
$450$	14.9322	0.703910
$451$	−24.2947	−1.14399
$452$	2.37517	0.111719
$453$	−45.5101	−2.13825
$454$	19.4698	0.913761
$455$	−0.658103	−0.0308523
$456$	−54.5962	−2.55670
$457$	33.9376	1.58753	0.793766	−	0.608223i	$-0.208117\pi$
$457$	33.9376	1.58753	0.793766	+	0.608223i	$0.208117\pi$
$458$	−9.96342	−0.465560
$459$	1.21406	0.0566676
$460$	0.313934	0.0146373
$461$	11.4759	0.534486	0.267243	−	0.963629i	$-0.413887\pi$
$461$	11.4759	0.534486	0.267243	+	0.963629i	$0.413887\pi$
$462$	1.85864	0.0864720
$463$	5.21581	0.242399	0.121200	−	0.992628i	$-0.461326\pi$
$463$	5.21581	0.242399	0.121200	+	0.992628i	$0.461326\pi$
$464$	28.5805	1.32681
$465$	1.37103	0.0635802
$466$	33.4695	1.55045
$467$	32.8915	1.52204	0.761019	−	0.648729i	$-0.224699\pi$
$467$	32.8915	1.52204	0.761019	+	0.648729i	$0.224699\pi$
$468$	−2.05679	−0.0950753
$469$	1.55813	0.0719477
$470$	−1.77400	−0.0818286
$471$	−41.6006	−1.91685
$472$	5.86964	0.270172
$473$	12.9510	0.595490
$474$	−38.2518	−1.75696
$475$	32.9952	1.51392
$476$	0.0473557	0.00217054
$477$	−4.22249	−0.193335
$478$	−18.5325	−0.847659
$479$	30.7409	1.40459	0.702294	−	0.711887i	$-0.252159\pi$
$479$	30.7409	1.40459	0.702294	+	0.711887i	$0.252159\pi$
$480$	2.23828	0.102163
$481$	21.5698	0.983500
$482$	−4.02990	−0.183557
$483$	0.806953	0.0367176
$484$	0.130930	0.00595135
$485$	15.4990	0.703773
$486$	28.2809	1.28285
$487$	−18.5289	−0.839626	−0.419813	−	0.907611i	$-0.637904\pi$
$487$	−18.5289	−0.839626	−0.419813	+	0.907611i	$0.637904\pi$
$488$	26.8200	1.21408
$489$	36.8135	1.66476
$490$	−8.25644	−0.372988
$491$	−17.0751	−0.770588	−0.385294	−	0.922794i	$-0.625900\pi$
$491$	−17.0751	−0.770588	−0.385294	+	0.922794i	$0.625900\pi$
$492$	−3.41050	−0.153757
$493$	10.9583	0.493538
$494$	43.2907	1.94774
$495$	7.43200	0.334044
$496$	2.35145	0.105583
$497$	−2.71608	−0.121833
$498$	38.3594	1.71892
$499$	−0.583540	−0.0261228	−0.0130614	−	0.999915i	$-0.504158\pi$
$499$	−0.583540	−0.0261228	−0.0130614	+	0.999915i	$0.504158\pi$
$500$	−1.54392	−0.0690462
$501$	6.97327	0.311543
$502$	14.9346	0.666563
$503$	−31.6359	−1.41057	−0.705287	−	0.708922i	$-0.749182\pi$
$503$	−31.6359	−1.41057	−0.705287	+	0.708922i	$0.749182\pi$
$504$	1.40406	0.0625420
$505$	12.6283	0.561954
$506$	8.10261	0.360205
$507$	9.41778	0.418258
$508$	0.843700	0.0374331
$509$	17.6564	0.782605	0.391303	−	0.920262i	$-0.372025\pi$
$509$	17.6564	0.782605	0.391303	+	0.920262i	$0.372025\pi$
$510$	−3.86301	−0.171057
$511$	−0.374717	−0.0165765
$512$	24.9575	1.10297
$513$	6.90134	0.304702
$514$	17.7296	0.782019
$515$	−5.38682	−0.237372
$516$	1.81808	0.0800364
$517$	4.80688	0.211406
$518$	−1.27759	−0.0561342
$519$	9.08280	0.398691
$520$	−10.6913	−0.468843
$521$	−32.8506	−1.43921	−0.719606	−	0.694382i	$-0.755678\pi$
$521$	−32.8506	−1.43921	−0.719606	+	0.694382i	$0.755678\pi$
$522$	28.1909	1.23388
$523$	35.6292	1.55796	0.778978	−	0.627051i	$-0.215738\pi$
$523$	35.6292	1.55796	0.778978	+	0.627051i	$0.215738\pi$
$524$	−1.87643	−0.0819723
$525$	−1.81737	−0.0793166
$526$	−33.6677	−1.46798
$527$	0.901592	0.0392740
$528$	27.2998	1.18807
$529$	−19.4822	−0.847050
$530$	−1.90439	−0.0827216
$531$	5.23455	0.227160
$532$	0.269193	0.0116710
$533$	31.1675	1.35001
$534$	−7.01227	−0.303451
$535$	10.5449	0.455896
$536$	25.3127	1.09334
$537$	21.6248	0.933180
$538$	−0.393267	−0.0169550
$539$	22.3719	0.963624
$540$	−0.147883	−0.00636385
$541$	−3.18388	−0.136886	−0.0684428	−	0.997655i	$-0.521803\pi$
$541$	−3.18388	−0.136886	−0.0684428	+	0.997655i	$0.521803\pi$
$542$	−4.97921	−0.213875
$543$	5.60505	0.240536
$544$	1.47189	0.0631068
$545$	8.11250	0.347501
$546$	−2.38445	−0.102045
$547$	−39.1680	−1.67470	−0.837352	−	0.546665i	$-0.815897\pi$
$547$	−39.1680	−1.67470	−0.837352	+	0.546665i	$0.815897\pi$
$548$	0.318943	0.0136246
$549$	23.9181	1.02080
$550$	−18.2482	−0.778107
$551$	62.2925	2.65375
$552$	13.1094	0.557974
$553$	2.17372	0.0924361
$554$	26.1969	1.11300
$555$	−10.9413	−0.464433
$556$	−3.23135	−0.137040
$557$	−38.0807	−1.61353	−0.806766	−	0.590872i	$-0.798784\pi$
$557$	−38.0807	−1.61353	−0.806766	+	0.590872i	$0.798784\pi$
$558$	2.31939	0.0981878
$559$	−16.6148	−0.702733
$560$	0.572537	0.0241941
$561$	10.4673	0.441930
$562$	18.6225	0.785543
$563$	−12.5526	−0.529030	−0.264515	−	0.964382i	$-0.585212\pi$
$563$	−12.5526	−0.529030	−0.264515	+	0.964382i	$0.585212\pi$
$564$	0.674794	0.0284139
$565$	11.0104	0.463209
$566$	2.22426	0.0934925
$567$	−1.80975	−0.0760025
$568$	−44.1243	−1.85141
$569$	0.952403	0.0399268	0.0199634	−	0.999801i	$-0.493645\pi$
$569$	0.952403	0.0399268	0.0199634	+	0.999801i	$0.493645\pi$
$570$	−21.9593	−0.919772
$571$	17.2084	0.720147	0.360074	−	0.932924i	$-0.382752\pi$
$571$	17.2084	0.720147	0.360074	+	0.932924i	$0.382752\pi$
$572$	2.51355	0.105097
$573$	27.2094	1.13669
$574$	−1.84607	−0.0770534
$575$	−7.92268	−0.330399
$576$	22.6201	0.942504
$577$	33.3467	1.38824	0.694120	−	0.719859i	$-0.255794\pi$
$577$	33.3467	1.38824	0.694120	+	0.719859i	$0.255794\pi$
$578$	20.3307	0.845647
$579$	4.66195	0.193744
$580$	−1.33481	−0.0554250
$581$	−2.17984	−0.0904349
$582$	56.1562	2.32775
$583$	5.16019	0.213713
$584$	−6.08749	−0.251902
$585$	−9.53447	−0.394202
$586$	−0.330692	−0.0136608
$587$	40.0214	1.65186	0.825930	−	0.563772i	$-0.190650\pi$
$587$	40.0214	1.65186	0.825930	+	0.563772i	$0.190650\pi$
$588$	3.14058	0.129515
$589$	5.12510	0.211176
$590$	2.36084	0.0971943
$591$	16.1601	0.664738
$592$	−18.7653	−0.771251
$593$	−36.2037	−1.48671	−0.743353	−	0.668899i	$-0.766766\pi$
$593$	−36.2037	−1.48671	−0.743353	+	0.668899i	$0.766766\pi$
$594$	−3.81683	−0.156606
$595$	0.219522	0.00899952
$596$	1.30722	0.0535457
$597$	66.4551	2.71983
$598$	−10.3948	−0.425075
$599$	−9.95086	−0.406581	−0.203290	−	0.979118i	$-0.565164\pi$
$599$	−9.95086	−0.406581	−0.203290	+	0.979118i	$0.565164\pi$
$600$	−29.5243	−1.20532
$601$	22.4005	0.913736	0.456868	−	0.889535i	$-0.348971\pi$
$601$	22.4005	0.913736	0.456868	+	0.889535i	$0.348971\pi$
$602$	0.984106	0.0401092
$603$	22.5739	0.919279
$604$	3.64540	0.148329
$605$	0.606938	0.0246756
$606$	45.7552	1.85868
$607$	0.204633	0.00830581	0.00415290	−	0.999991i	$-0.498678\pi$
$607$	0.204633	0.00830581	0.00415290	+	0.999991i	$0.498678\pi$
$608$	8.36695	0.339325
$609$	−3.43106	−0.139034
$610$	10.7873	0.436767
$611$	−6.16672	−0.249479
$612$	0.686080	0.0277332
$613$	−11.1575	−0.450646	−0.225323	−	0.974284i	$-0.572344\pi$
$613$	−11.1575	−0.450646	−0.225323	+	0.974284i	$0.572344\pi$
$614$	17.3645	0.700776
$615$	−15.8097	−0.637510
$616$	−1.71587	−0.0691343
$617$	10.4107	0.419120	0.209560	−	0.977796i	$-0.432797\pi$
$617$	10.4107	0.419120	0.209560	+	0.977796i	$0.432797\pi$
$618$	−19.5176	−0.785113
$619$	11.2338	0.451523	0.225762	−	0.974183i	$-0.427513\pi$
$619$	11.2338	0.451523	0.225762	+	0.974183i	$0.427513\pi$
$620$	−0.109821	−0.00441052
$621$	−1.65712	−0.0664980
$622$	40.4097	1.62028
$623$	0.398484	0.0159649
$624$	−35.0228	−1.40203
$625$	13.9635	0.558540
$626$	−24.1490	−0.965187
$627$	59.5013	2.37625
$628$	3.33224	0.132971
$629$	−7.19501	−0.286884
$630$	0.564732	0.0224995
$631$	−14.2088	−0.565644	−0.282822	−	0.959172i	$-0.591270\pi$
$631$	−14.2088	−0.565644	−0.282822	+	0.959172i	$0.591270\pi$
$632$	35.3134	1.40469
$633$	−23.6523	−0.940095
$634$	24.2337	0.962444
$635$	3.91105	0.155205
$636$	0.724392	0.0287240
$637$	−28.7007	−1.13717
$638$	−34.4513	−1.36394
$639$	−39.3501	−1.55667
$640$	8.31488	0.328674
$641$	−19.8710	−0.784857	−0.392428	−	0.919783i	$-0.628365\pi$
$641$	−19.8710	−0.784857	−0.392428	+	0.919783i	$0.628365\pi$
$642$	38.2064	1.50789
$643$	−14.1793	−0.559179	−0.279589	−	0.960120i	$-0.590198\pi$
$643$	−14.1793	−0.559179	−0.279589	+	0.960120i	$0.590198\pi$
$644$	−0.0646376	−0.00254708
$645$	8.42789	0.331848
$646$	−14.4404	−0.568150
$647$	−40.7149	−1.60067	−0.800333	−	0.599555i	$-0.795344\pi$
$647$	−40.7149	−1.60067	−0.800333	+	0.599555i	$0.795344\pi$
$648$	−29.4005	−1.15496
$649$	−6.39700	−0.251104
$650$	23.4106	0.918237
$651$	−0.282289	−0.0110638
$652$	−2.94879	−0.115484
$653$	0.736708	0.0288296	0.0144148	−	0.999896i	$-0.495411\pi$
$653$	0.736708	0.0288296	0.0144148	+	0.999896i	$0.495411\pi$
$654$	29.3933	1.14937
$655$	−8.69839	−0.339874
$656$	−27.1151	−1.05867
$657$	−5.42883	−0.211799
$658$	0.365258	0.0142393
$659$	35.3468	1.37692	0.688458	−	0.725276i	$-0.258288\pi$
$659$	35.3468	1.37692	0.688458	+	0.725276i	$0.258288\pi$
$660$	−1.27500	−0.0496293
$661$	43.0472	1.67434	0.837172	−	0.546940i	$-0.184207\pi$
$661$	43.0472	1.67434	0.837172	+	0.546940i	$0.184207\pi$
$662$	−12.0369	−0.467826
$663$	−13.4284	−0.521518
$664$	−35.4127	−1.37428
$665$	1.24787	0.0483904
$666$	−18.5095	−0.717230
$667$	−14.9574	−0.579154
$668$	−0.558565	−0.0216115
$669$	12.3132	0.476055
$670$	10.1811	0.393329
$671$	−29.2296	−1.12840
$672$	−0.460850	−0.0177777
$673$	−10.7692	−0.415123	−0.207561	−	0.978222i	$-0.566553\pi$
$673$	−10.7692	−0.415123	−0.207561	+	0.978222i	$0.566553\pi$
$674$	−40.5765	−1.56295
$675$	3.73207	0.143648
$676$	−0.754373	−0.0290143
$677$	−32.0334	−1.23114	−0.615572	−	0.788081i	$-0.711075\pi$
$677$	−32.0334	−1.23114	−0.615572	+	0.788081i	$0.711075\pi$
$678$	39.8928	1.53208
$679$	−3.19117	−0.122466
$680$	3.56626	0.136760
$681$	−34.3308	−1.31556
$682$	−2.83447	−0.108537
$683$	25.3190	0.968804	0.484402	−	0.874845i	$-0.339037\pi$
$683$	25.3190	0.968804	0.484402	+	0.874845i	$0.339037\pi$
$684$	3.90002	0.149121
$685$	1.47849	0.0564903
$686$	3.40795	0.130116
$687$	17.5684	0.670275
$688$	14.4546	0.551076
$689$	−6.61999	−0.252201
$690$	5.27277	0.200731
$691$	−22.0454	−0.838647	−0.419323	−	0.907837i	$-0.637733\pi$
$691$	−22.0454	−0.838647	−0.419323	+	0.907837i	$0.637733\pi$
$692$	−0.727540	−0.0276569
$693$	−1.53021	−0.0581280
$694$	−14.6229	−0.555079
$695$	−14.9793	−0.568196
$696$	−55.7396	−2.11280
$697$	−10.3965	−0.393795
$698$	0.642067	0.0243026
$699$	−59.0164	−2.23221
$700$	0.145573	0.00550214
$701$	−17.2510	−0.651562	−0.325781	−	0.945445i	$-0.605627\pi$
$701$	−17.2510	−0.651562	−0.325781	+	0.945445i	$0.605627\pi$
$702$	4.89659	0.184810
$703$	−40.9000	−1.54257
$704$	−27.6434	−1.04185
$705$	3.12807	0.117810
$706$	−41.3104	−1.55474
$707$	−2.60012	−0.0977874
$708$	−0.898015	−0.0337495
$709$	−9.52826	−0.357841	−0.178921	−	0.983864i	$-0.557261\pi$
$709$	−9.52826	−0.357841	−0.178921	+	0.983864i	$0.557261\pi$
$710$	−17.7473	−0.666046
$711$	31.4925	1.18106
$712$	6.47361	0.242609
$713$	−1.23062	−0.0460870
$714$	0.795375	0.0297661
$715$	11.6518	0.435753
$716$	−1.73217	−0.0647341
$717$	32.6782	1.22039
$718$	−25.1284	−0.937783
$719$	41.2776	1.53939	0.769697	−	0.638409i	$-0.220407\pi$
$719$	41.2776	1.53939	0.769697	+	0.638409i	$0.220407\pi$
$720$	8.29481	0.309129
$721$	1.10912	0.0413058
$722$	−56.5246	−2.10363
$723$	7.10586	0.264270
$724$	−0.448970	−0.0166858
$725$	33.6863	1.25108
$726$	2.19907	0.0816150
$727$	−29.5478	−1.09587	−0.547933	−	0.836522i	$-0.684585\pi$
$727$	−29.5478	−1.09587	−0.547933	+	0.836522i	$0.684585\pi$
$728$	2.20128	0.0815848
$729$	−19.9316	−0.738208
$730$	−2.44846	−0.0906217
$731$	5.54218	0.204985
$732$	−4.10328	−0.151662
$733$	−3.11186	−0.114939	−0.0574696	−	0.998347i	$-0.518303\pi$
$733$	−3.11186	−0.114939	−0.0574696	+	0.998347i	$0.518303\pi$
$734$	−8.83611	−0.326147
$735$	14.5585	0.536998
$736$	−2.00904	−0.0740542
$737$	−27.5869	−1.01618
$738$	−26.7455	−0.984516
$739$	16.6928	0.614054	0.307027	−	0.951701i	$-0.400666\pi$
$739$	16.6928	0.614054	0.307027	+	0.951701i	$0.400666\pi$
$740$	0.876409	0.0322174
$741$	−76.3340	−2.80420
$742$	0.392106	0.0143946
$743$	36.9230	1.35457	0.677287	−	0.735719i	$-0.263155\pi$
$743$	36.9230	1.35457	0.677287	+	0.735719i	$0.263155\pi$
$744$	−4.58596	−0.168129
$745$	6.05974	0.222012
$746$	−28.4572	−1.04189
$747$	−31.5811	−1.15549
$748$	−0.838440	−0.0306564
$749$	−2.17115	−0.0793319
$750$	−25.9313	−0.946877
$751$	36.7471	1.34092	0.670461	−	0.741945i	$-0.266096\pi$
$751$	36.7471	1.34092	0.670461	+	0.741945i	$0.266096\pi$
$752$	5.36493	0.195639
$753$	−26.3339	−0.959662
$754$	44.1974	1.60957
$755$	16.8986	0.615004
$756$	0.0304483	0.00110739
$757$	−30.9380	−1.12446	−0.562230	−	0.826981i	$-0.690057\pi$
$757$	−30.9380	−1.12446	−0.562230	+	0.826981i	$0.690057\pi$
$758$	12.2319	0.444281
$759$	−14.2872	−0.518594
$760$	20.2724	0.735358
$761$	15.6400	0.566950	0.283475	−	0.958980i	$-0.408513\pi$
$761$	15.6400	0.566950	0.283475	+	0.958980i	$0.408513\pi$
$762$	14.1706	0.513346
$763$	−1.67032	−0.0604698
$764$	−2.17950	−0.0788516
$765$	3.18040	0.114987
$766$	46.6666	1.68613
$767$	8.20667	0.296326
$768$	−10.7178	−0.386745
$769$	35.2621	1.27158	0.635792	−	0.771860i	$-0.280673\pi$
$769$	35.2621	1.27158	0.635792	+	0.771860i	$0.280673\pi$
$770$	−0.690144	−0.0248711
$771$	−31.2624	−1.12589
$772$	−0.373426	−0.0134399
$773$	−17.2836	−0.621646	−0.310823	−	0.950468i	$-0.600605\pi$
$773$	−17.2836	−0.621646	−0.310823	+	0.950468i	$0.600605\pi$
$774$	14.2575	0.512477
$775$	2.77153	0.0995562
$776$	−51.8424	−1.86103
$777$	2.25276	0.0808175
$778$	42.5451	1.52532
$779$	−59.0987	−2.11743
$780$	1.63569	0.0585672
$781$	48.0887	1.72075
$782$	3.46737	0.123993
$783$	7.04588	0.251799
$784$	24.9691	0.891754
$785$	15.4470	0.551326
$786$	−31.5161	−1.12414
$787$	−24.9262	−0.888522	−0.444261	−	0.895897i	$-0.646534\pi$
$787$	−24.9262	−0.888522	−0.444261	+	0.895897i	$0.646534\pi$
$788$	−1.29444	−0.0461125
$789$	59.3659	2.11348
$790$	14.2035	0.505337
$791$	−2.26698	−0.0806045
$792$	−24.8592	−0.883333
$793$	37.4986	1.33161
$794$	−31.4681	−1.11676
$795$	3.35799	0.119096
$796$	−5.32312	−0.188673
$797$	20.1259	0.712897	0.356449	−	0.934315i	$-0.383987\pi$
$797$	20.1259	0.712897	0.356449	+	0.934315i	$0.383987\pi$
$798$	4.52130	0.160052
$799$	2.05702	0.0727722
$800$	4.52464	0.159970
$801$	5.77317	0.203985
$802$	12.0419	0.425216
$803$	6.63442	0.234124
$804$	−3.87267	−0.136579
$805$	−0.299634	−0.0105607
$806$	3.63632	0.128084
$807$	0.693443	0.0244104
$808$	−42.2404	−1.48601
$809$	−43.0354	−1.51304	−0.756521	−	0.653969i	$-0.773103\pi$
$809$	−43.0354	−1.51304	−0.756521	+	0.653969i	$0.773103\pi$
$810$	−11.8253	−0.415497
$811$	12.0740	0.423976	0.211988	−	0.977272i	$-0.432006\pi$
$811$	12.0740	0.423976	0.211988	+	0.977272i	$0.432006\pi$
$812$	0.274831	0.00964468
$813$	8.77977	0.307920
$814$	22.6200	0.792831
$815$	−13.6694	−0.478819
$816$	11.6825	0.408969
$817$	31.5045	1.10220
$818$	−40.4947	−1.41586
$819$	1.96310	0.0685963
$820$	1.26637	0.0442237
$821$	4.91791	0.171636	0.0858182	−	0.996311i	$-0.472650\pi$
$821$	4.91791	0.171636	0.0858182	+	0.996311i	$0.472650\pi$
$822$	5.35689	0.186843
$823$	28.6179	0.997558	0.498779	−	0.866729i	$-0.333782\pi$
$823$	28.6179	0.997558	0.498779	+	0.866729i	$0.333782\pi$
$824$	18.0183	0.627698
$825$	32.1769	1.12025
$826$	−0.486086	−0.0169131
$827$	−39.2638	−1.36534	−0.682668	−	0.730728i	$-0.739181\pi$
$827$	−39.2638	−1.36534	−0.682668	+	0.730728i	$0.739181\pi$
$828$	−0.936458	−0.0325442
$829$	−51.5571	−1.79065	−0.895327	−	0.445410i	$-0.853058\pi$
$829$	−51.5571	−1.79065	−0.895327	+	0.445410i	$0.853058\pi$
$830$	−14.2434	−0.494397
$831$	−46.1927	−1.60241
$832$	35.4636	1.22948
$833$	9.57365	0.331707
$834$	−54.2731	−1.87932
$835$	−2.58929	−0.0896060
$836$	−4.76611	−0.164839
$837$	0.579697	0.0200373
$838$	16.3435	0.564576
$839$	23.3706	0.806843	0.403422	−	0.915014i	$-0.367821\pi$
$839$	23.3706	0.806843	0.403422	+	0.915014i	$0.367821\pi$
$840$	−1.11660	−0.0385264
$841$	34.5972	1.19301
$842$	−14.8310	−0.511111
$843$	−32.8368	−1.13096
$844$	1.89457	0.0652138
$845$	−3.49697	−0.120299
$846$	5.29180	0.181936
$847$	−0.124966	−0.00429387
$848$	5.75926	0.197774
$849$	−3.92200	−0.134603
$850$	−7.80901	−0.267847
$851$	9.82075	0.336651
$852$	6.75072	0.231276
$853$	30.3388	1.03878	0.519391	−	0.854537i	$-0.326159\pi$
$853$	30.3388	1.03878	0.519391	+	0.854537i	$0.326159\pi$
$854$	−2.22106	−0.0760031
$855$	18.0789	0.618287
$856$	−35.2715	−1.20556
$857$	−26.2587	−0.896981	−0.448491	−	0.893788i	$-0.648038\pi$
$857$	−26.2587	−0.896981	−0.448491	+	0.893788i	$0.648038\pi$
$858$	42.2170	1.44126
$859$	−31.1234	−1.06192	−0.530958	−	0.847398i	$-0.678168\pi$
$859$	−31.1234	−1.06192	−0.530958	+	0.847398i	$0.678168\pi$
$860$	−0.675081	−0.0230201
$861$	3.25515	0.110935
$862$	6.82665	0.232517
$863$	12.6692	0.431263	0.215632	−	0.976475i	$-0.430819\pi$
$863$	12.6692	0.431263	0.215632	+	0.976475i	$0.430819\pi$
$864$	0.946382	0.0321966
$865$	−3.37259	−0.114671
$866$	−29.8396	−1.01399
$867$	−35.8489	−1.21749
$868$	0.0226116	0.000767489 0
$869$	−38.4861	−1.30555
$870$	−22.4192	−0.760081
$871$	35.3911	1.19918
$872$	−27.1354	−0.918920
$873$	−46.2331	−1.56475
$874$	19.7103	0.666710
$875$	1.47359	0.0498165
$876$	0.931345	0.0314672
$877$	−49.1125	−1.65841	−0.829205	−	0.558944i	$-0.811207\pi$
$877$	−49.1125	−1.65841	−0.829205	+	0.558944i	$0.811207\pi$
$878$	−29.8477	−1.00731
$879$	0.583106	0.0196677
$880$	−10.1369	−0.341713
$881$	−40.5314	−1.36554	−0.682769	−	0.730635i	$-0.739224\pi$
$881$	−40.5314	−1.36554	−0.682769	+	0.730635i	$0.739224\pi$
$882$	24.6287	0.829293
$883$	42.0719	1.41583	0.707915	−	0.706297i	$-0.249636\pi$
$883$	42.0719	1.41583	0.707915	+	0.706297i	$0.249636\pi$
$884$	1.07563	0.0361774
$885$	−4.16284	−0.139932
$886$	−0.834616	−0.0280395
$887$	−32.6919	−1.09769	−0.548843	−	0.835925i	$-0.684932\pi$
$887$	−32.6919	−1.09769	−0.548843	+	0.835925i	$0.684932\pi$
$888$	36.5975	1.22813
$889$	−0.805267	−0.0270078
$890$	2.60377	0.0872784
$891$	32.0420	1.07345
$892$	−0.986297	−0.0330237
$893$	11.6931	0.391296
$894$	21.9557	0.734309
$895$	−8.02963	−0.268401
$896$	−1.71199	−0.0571937
$897$	18.3290	0.611988
$898$	−53.1127	−1.77240
$899$	5.23243	0.174511
$900$	2.10904	0.0703012
$901$	2.20822	0.0735663
$902$	32.6849	1.08829
$903$	−1.73526	−0.0577459
$904$	−36.8284	−1.22489
$905$	−2.08124	−0.0691829
$906$	61.2273	2.03414
$907$	22.5549	0.748924	0.374462	−	0.927242i	$-0.377827\pi$
$907$	22.5549	0.748924	0.374462	+	0.927242i	$0.377827\pi$
$908$	2.74992	0.0912595
$909$	−37.6700	−1.24944
$910$	0.885382	0.0293501
$911$	−21.8956	−0.725434	−0.362717	−	0.931899i	$-0.618151\pi$
$911$	−21.8956	−0.725434	−0.362717	+	0.931899i	$0.618151\pi$
$912$	66.4091	2.19902
$913$	38.5944	1.27729
$914$	−45.6581	−1.51024
$915$	−19.0212	−0.628821
$916$	−1.40724	−0.0464966
$917$	1.79096	0.0591426
$918$	−1.63335	−0.0539085
$919$	−24.0357	−0.792866	−0.396433	−	0.918064i	$-0.629752\pi$
$919$	−24.0357	−0.792866	−0.396433	+	0.918064i	$0.629752\pi$
$920$	−4.86773	−0.160484
$921$	−30.6187	−1.00892
$922$	−15.4392	−0.508462
$923$	−61.6927	−2.03064
$924$	0.262517	0.00863616
$925$	−22.1177	−0.727226
$926$	−7.01712	−0.230597
$927$	16.0687	0.527767
$928$	8.54219	0.280411
$929$	34.1048	1.11894	0.559471	−	0.828850i	$-0.311004\pi$
$929$	34.1048	1.11894	0.559471	+	0.828850i	$0.311004\pi$
$930$	−1.84453	−0.0604845
$931$	54.4214	1.78359
$932$	4.72727	0.154847
$933$	−71.2539	−2.33275
$934$	−44.2508	−1.44793
$935$	−3.88667	−0.127108
$936$	31.8917	1.04241
$937$	5.86949	0.191748	0.0958740	−	0.995393i	$-0.469435\pi$
$937$	5.86949	0.191748	0.0958740	+	0.995393i	$0.469435\pi$
$938$	−2.09624	−0.0684445
$939$	42.5816	1.38960
$940$	−0.250561	−0.00817241
$941$	−15.7327	−0.512870	−0.256435	−	0.966561i	$-0.582548\pi$
$941$	−15.7327	−0.512870	−0.256435	+	0.966561i	$0.582548\pi$
$942$	55.9676	1.82352
$943$	14.1906	0.462108
$944$	−7.13965	−0.232376
$945$	0.141146	0.00459149
$946$	−17.4238	−0.566495
$947$	0.333875	0.0108495	0.00542475	−	0.999985i	$-0.498273\pi$
$947$	0.333875	0.0108495	0.00542475	+	0.999985i	$0.498273\pi$
$948$	−5.40271	−0.175472
$949$	−8.51126	−0.276287
$950$	−44.3903	−1.44021
$951$	−42.7310	−1.38565
$952$	−0.734276	−0.0237980
$953$	−44.8509	−1.45286	−0.726432	−	0.687239i	$-0.758823\pi$
$953$	−44.8509	−1.45286	−0.726432	+	0.687239i	$0.758823\pi$
$954$	5.68075	0.183921
$955$	−10.1033	−0.326935
$956$	−2.61755	−0.0846577
$957$	60.7475	1.96369
$958$	−41.3575	−1.33620
$959$	−0.304414	−0.00983005
$960$	−17.9889	−0.580590
$961$	−30.5695	−0.986113
$962$	−29.0191	−0.935613
$963$	−31.4552	−1.01363
$964$	−0.569186	−0.0183322
$965$	−1.73105	−0.0557246
$966$	−1.08564	−0.0349298
$967$	−36.5730	−1.17611	−0.588053	−	0.808822i	$-0.700106\pi$
$967$	−36.5730	−1.17611	−0.588053	+	0.808822i	$0.700106\pi$
$968$	−2.03014	−0.0652512
$969$	25.4626	0.817976
$970$	−20.8517	−0.669507
$971$	44.3154	1.42215	0.711074	−	0.703117i	$-0.248209\pi$
$971$	44.3154	1.42215	0.711074	+	0.703117i	$0.248209\pi$
$972$	3.99442	0.128121
$973$	3.08416	0.0988736
$974$	24.9280	0.798744
$975$	−41.2795	−1.32200
$976$	−32.6230	−1.04424
$977$	36.6696	1.17316	0.586582	−	0.809890i	$-0.300473\pi$
$977$	36.6696	1.17316	0.586582	+	0.809890i	$0.300473\pi$
$978$	−49.5272	−1.58371
$979$	−7.05523	−0.225486
$980$	−1.16615	−0.0372512
$981$	−24.1994	−0.772626
$982$	22.9720	0.733067
$983$	17.8047	0.567881	0.283941	−	0.958842i	$-0.408358\pi$
$983$	17.8047	0.567881	0.283941	+	0.958842i	$0.408358\pi$
$984$	52.8818	1.68581
$985$	−6.00050	−0.191192
$986$	−14.7428	−0.469507
$987$	−0.644055	−0.0205005
$988$	6.11442	0.194526
$989$	−7.56473	−0.240544
$990$	−9.99868	−0.317779
$991$	−15.3123	−0.486412	−0.243206	−	0.969975i	$-0.578199\pi$
$991$	−15.3123	−0.486412	−0.243206	+	0.969975i	$0.578199\pi$
$992$	0.702806	0.0223141
$993$	21.2244	0.673538
$994$	3.65409	0.115901
$995$	−24.6758	−0.782277
$996$	5.41791	0.171673
$997$	5.27673	0.167116	0.0835579	−	0.996503i	$-0.473372\pi$
$997$	5.27673	0.167116	0.0835579	+	0.996503i	$0.473372\pi$
$998$	0.785068	0.0248509
$999$	−4.62618	−0.146366

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2011.2.a.a.1.26	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2011.2.a.a.1.26	✓	77	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:	\( N \)	\(=\)	\( 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2011.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34536 q^{2} +2.37225 q^{3} -0.190019 q^{4} -0.880853 q^{5} -3.19152 q^{6} +0.181363 q^{7} +2.94635 q^{8} +2.62756 q^{9} +O(q^{10})\)	\(q-1.34536 q^{2} +2.37225 q^{3} -0.190019 q^{4} -0.880853 q^{5} -3.19152 q^{6} +0.181363 q^{7} +2.94635 q^{8} +2.62756 q^{9} +1.18506 q^{10} -3.21107 q^{11} -0.450772 q^{12} +4.11946 q^{13} -0.243998 q^{14} -2.08960 q^{15} -3.58385 q^{16} -1.37412 q^{17} -3.53500 q^{18} -7.81119 q^{19} +0.167379 q^{20} +0.430239 q^{21} +4.32003 q^{22} +1.87559 q^{23} +6.98948 q^{24} -4.22410 q^{25} -5.54214 q^{26} -0.883520 q^{27} -0.0344625 q^{28} -7.97478 q^{29} +2.81126 q^{30} -0.656123 q^{31} -1.07115 q^{32} -7.61745 q^{33} +1.84868 q^{34} -0.159754 q^{35} -0.499287 q^{36} +5.23608 q^{37} +10.5088 q^{38} +9.77239 q^{39} -2.59530 q^{40} +7.56591 q^{41} -0.578824 q^{42} -4.03325 q^{43} +0.610165 q^{44} -2.31449 q^{45} -2.52334 q^{46} -1.49697 q^{47} -8.50179 q^{48} -6.96711 q^{49} +5.68291 q^{50} -3.25976 q^{51} -0.782777 q^{52} -1.60700 q^{53} +1.18865 q^{54} +2.82848 q^{55} +0.534361 q^{56} -18.5301 q^{57} +10.7289 q^{58} +1.99217 q^{59} +0.397064 q^{60} +9.10278 q^{61} +0.882718 q^{62} +0.476543 q^{63} +8.60879 q^{64} -3.62864 q^{65} +10.2482 q^{66} +8.59119 q^{67} +0.261109 q^{68} +4.44937 q^{69} +0.214927 q^{70} -14.9759 q^{71} +7.74172 q^{72} -2.06611 q^{73} -7.04439 q^{74} -10.0206 q^{75} +1.48427 q^{76} -0.582370 q^{77} -13.1473 q^{78} +11.9855 q^{79} +3.15685 q^{80} -9.97861 q^{81} -10.1788 q^{82} -12.0192 q^{83} -0.0817536 q^{84} +1.21040 q^{85} +5.42616 q^{86} -18.9182 q^{87} -9.46095 q^{88} +2.19716 q^{89} +3.11382 q^{90} +0.747120 q^{91} -0.356398 q^{92} -1.55649 q^{93} +2.01396 q^{94} +6.88051 q^{95} -2.54103 q^{96} -17.5954 q^{97} +9.37323 q^{98} -8.43728 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 13 q^{2} - 13 q^{3} + 67 q^{4} - 47 q^{5} - 20 q^{6} - 8 q^{7} - 33 q^{8} + 52 q^{9}+O(q^{10}) \) 77 * q - 13 * q^2 - 13 * q^3 + 67 * q^4 - 47 * q^5 - 20 * q^6 - 8 * q^7 - 33 * q^8 + 52 * q^9	\( 77 q - 13 q^{2} - 13 q^{3} + 67 q^{4} - 47 q^{5} - 20 q^{6} - 8 q^{7} - 33 q^{8} + 52 q^{9} - 21 q^{10} - 34 q^{11} - 36 q^{12} - 34 q^{13} - 49 q^{14} - 12 q^{15} + 47 q^{16} - 59 q^{17} - 24 q^{18} - 31 q^{19} - 82 q^{20} - 71 q^{21} - 3 q^{22} - 28 q^{23} - 50 q^{24} + 68 q^{25} - 54 q^{26} - 43 q^{27} - 2 q^{28} - 151 q^{29} + q^{30} - 37 q^{31} - 59 q^{32} - 35 q^{33} - q^{34} - 58 q^{35} + 19 q^{36} - 29 q^{37} - 22 q^{38} - 40 q^{39} - 41 q^{40} - 142 q^{41} + 16 q^{42} - 23 q^{43} - 89 q^{44} - 119 q^{45} - 6 q^{46} - 36 q^{47} - 46 q^{48} + 45 q^{49} - 29 q^{50} - 53 q^{51} - 11 q^{52} - 69 q^{53} - 50 q^{54} - 13 q^{55} - 122 q^{56} - 14 q^{57} + 31 q^{58} - 92 q^{59} + 20 q^{60} - 115 q^{61} - 66 q^{62} - 25 q^{63} + 37 q^{64} - 57 q^{65} - 17 q^{66} - 108 q^{68} - 160 q^{69} + 40 q^{70} - 67 q^{71} - 35 q^{72} - 36 q^{73} - 55 q^{74} - 51 q^{75} - 56 q^{76} - 116 q^{77} + 22 q^{78} - 42 q^{79} - 114 q^{80} + 37 q^{81} + 18 q^{82} - 42 q^{83} - 77 q^{84} - 18 q^{85} - 33 q^{86} - 7 q^{87} - 2 q^{88} - 93 q^{89} - 34 q^{90} - 37 q^{91} - 55 q^{92} - 8 q^{93} - 35 q^{94} - 64 q^{95} - 83 q^{96} - 16 q^{97} - 57 q^{98} - 65 q^{99}+O(q^{100}) \) 77 * q - 13 * q^2 - 13 * q^3 + 67 * q^4 - 47 * q^5 - 20 * q^6 - 8 * q^7 - 33 * q^8 + 52 * q^9 - 21 * q^10 - 34 * q^11 - 36 * q^12 - 34 * q^13 - 49 * q^14 - 12 * q^15 + 47 * q^16 - 59 * q^17 - 24 * q^18 - 31 * q^19 - 82 * q^20 - 71 * q^21 - 3 * q^22 - 28 * q^23 - 50 * q^24 + 68 * q^25 - 54 * q^26 - 43 * q^27 - 2 * q^28 - 151 * q^29 + q^30 - 37 * q^31 - 59 * q^32 - 35 * q^33 - q^34 - 58 * q^35 + 19 * q^36 - 29 * q^37 - 22 * q^38 - 40 * q^39 - 41 * q^40 - 142 * q^41 + 16 * q^42 - 23 * q^43 - 89 * q^44 - 119 * q^45 - 6 * q^46 - 36 * q^47 - 46 * q^48 + 45 * q^49 - 29 * q^50 - 53 * q^51 - 11 * q^52 - 69 * q^53 - 50 * q^54 - 13 * q^55 - 122 * q^56 - 14 * q^57 + 31 * q^58 - 92 * q^59 + 20 * q^60 - 115 * q^61 - 66 * q^62 - 25 * q^63 + 37 * q^64 - 57 * q^65 - 17 * q^66 - 108 * q^68 - 160 * q^69 + 40 * q^70 - 67 * q^71 - 35 * q^72 - 36 * q^73 - 55 * q^74 - 51 * q^75 - 56 * q^76 - 116 * q^77 + 22 * q^78 - 42 * q^79 - 114 * q^80 + 37 * q^81 + 18 * q^82 - 42 * q^83 - 77 * q^84 - 18 * q^85 - 33 * q^86 - 7 * q^87 - 2 * q^88 - 93 * q^89 - 34 * q^90 - 37 * q^91 - 55 * q^92 - 8 * q^93 - 35 * q^94 - 64 * q^95 - 83 * q^96 - 16 * q^97 - 57 * q^98 - 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more