LMFDB - Embedded newform 4033.2.a.f.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4033,2,Mod(1,4033)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4033, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4033.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4033 = 37 \cdot 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4033.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:	$32.2036671352$
Analytic rank:	$0$
Dimension:	$85$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	4033.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11247 q^{2} +2.70421 q^{3} +2.46253 q^{4} +2.42834 q^{5} -5.71257 q^{6} +1.36575 q^{7} -0.977087 q^{8} +4.31276 q^{9} +O(q^{10})$	\(q-2.11247 q^{2} +2.70421 q^{3} +2.46253 q^{4} +2.42834 q^{5} -5.71257 q^{6} +1.36575 q^{7} -0.977087 q^{8} +4.31276 q^{9} -5.12979 q^{10} -5.83835 q^{11} +6.65921 q^{12} +3.94237 q^{13} -2.88511 q^{14} +6.56674 q^{15} -2.86100 q^{16} -0.443864 q^{17} -9.11057 q^{18} +4.49035 q^{19} +5.97986 q^{20} +3.69328 q^{21} +12.3333 q^{22} +5.11497 q^{23} -2.64225 q^{24} +0.896826 q^{25} -8.32814 q^{26} +3.54997 q^{27} +3.36320 q^{28} +8.05928 q^{29} -13.8720 q^{30} -3.45560 q^{31} +7.99795 q^{32} -15.7881 q^{33} +0.937649 q^{34} +3.31650 q^{35} +10.6203 q^{36} +1.00000 q^{37} -9.48573 q^{38} +10.6610 q^{39} -2.37270 q^{40} -0.332675 q^{41} -7.80194 q^{42} -3.72912 q^{43} -14.3771 q^{44} +10.4728 q^{45} -10.8052 q^{46} +2.07687 q^{47} -7.73674 q^{48} -5.13473 q^{49} -1.89452 q^{50} -1.20030 q^{51} +9.70821 q^{52} +2.70802 q^{53} -7.49920 q^{54} -14.1775 q^{55} -1.33446 q^{56} +12.1428 q^{57} -17.0250 q^{58} +2.65523 q^{59} +16.1708 q^{60} -2.31531 q^{61} +7.29986 q^{62} +5.89015 q^{63} -11.1734 q^{64} +9.57340 q^{65} +33.3520 q^{66} +13.8635 q^{67} -1.09303 q^{68} +13.8319 q^{69} -7.00602 q^{70} +1.12509 q^{71} -4.21394 q^{72} -6.62746 q^{73} -2.11247 q^{74} +2.42521 q^{75} +11.0576 q^{76} -7.97373 q^{77} -22.5210 q^{78} -3.29503 q^{79} -6.94747 q^{80} -3.33841 q^{81} +0.702766 q^{82} -0.226071 q^{83} +9.09481 q^{84} -1.07785 q^{85} +7.87765 q^{86} +21.7940 q^{87} +5.70458 q^{88} +18.1623 q^{89} -22.1235 q^{90} +5.38429 q^{91} +12.5958 q^{92} -9.34468 q^{93} -4.38732 q^{94} +10.9041 q^{95} +21.6281 q^{96} -12.4517 q^{97} +10.8470 q^{98} -25.1794 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9}+O(q^{10}) $ 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9	\( 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9} + 9 q^{10} + 37 q^{11} + 44 q^{12} + 14 q^{13} + 26 q^{14} + 27 q^{15} + 85 q^{16} + 34 q^{17} + 3 q^{18} + 15 q^{19} + 15 q^{20} + 17 q^{21} + q^{22} + 72 q^{23} + 15 q^{24} + 85 q^{25} + 33 q^{26} + 69 q^{27} + 7 q^{28} + 19 q^{29} - 9 q^{30} + 23 q^{31} + 51 q^{32} + 32 q^{33} + 49 q^{34} + 40 q^{35} + 121 q^{36} + 85 q^{37} + 84 q^{38} + 39 q^{39} + 22 q^{40} + 55 q^{41} - 28 q^{42} + 78 q^{44} + 28 q^{45} + 17 q^{46} + 184 q^{47} + 97 q^{48} + 88 q^{49} + 26 q^{50} + 27 q^{51} + 73 q^{52} + 64 q^{53} + 31 q^{54} + 39 q^{55} + 68 q^{56} - 33 q^{57} + 28 q^{58} + 60 q^{59} - 22 q^{60} + 7 q^{61} + 70 q^{62} + 28 q^{63} + 102 q^{64} + 17 q^{65} - 15 q^{66} + 82 q^{67} + 92 q^{68} + 22 q^{69} - 41 q^{70} + 113 q^{71} - 19 q^{73} + 11 q^{74} + 45 q^{75} + 34 q^{76} + 64 q^{77} + 29 q^{78} + 23 q^{79} + 54 q^{80} + 149 q^{81} + 4 q^{82} + 100 q^{83} - 49 q^{84} - 5 q^{85} - 24 q^{86} + 65 q^{87} + 14 q^{88} + 84 q^{89} - 21 q^{90} + 32 q^{91} + 95 q^{92} + 19 q^{93} - 47 q^{94} + 102 q^{95} + 29 q^{96} + 7 q^{97} + 26 q^{98} + 107 q^{99}+O(q^{100}) \) 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9 + 9 * q^10 + 37 * q^11 + 44 * q^12 + 14 * q^13 + 26 * q^14 + 27 * q^15 + 85 * q^16 + 34 * q^17 + 3 * q^18 + 15 * q^19 + 15 * q^20 + 17 * q^21 + q^22 + 72 * q^23 + 15 * q^24 + 85 * q^25 + 33 * q^26 + 69 * q^27 + 7 * q^28 + 19 * q^29 - 9 * q^30 + 23 * q^31 + 51 * q^32 + 32 * q^33 + 49 * q^34 + 40 * q^35 + 121 * q^36 + 85 * q^37 + 84 * q^38 + 39 * q^39 + 22 * q^40 + 55 * q^41 - 28 * q^42 + 78 * q^44 + 28 * q^45 + 17 * q^46 + 184 * q^47 + 97 * q^48 + 88 * q^49 + 26 * q^50 + 27 * q^51 + 73 * q^52 + 64 * q^53 + 31 * q^54 + 39 * q^55 + 68 * q^56 - 33 * q^57 + 28 * q^58 + 60 * q^59 - 22 * q^60 + 7 * q^61 + 70 * q^62 + 28 * q^63 + 102 * q^64 + 17 * q^65 - 15 * q^66 + 82 * q^67 + 92 * q^68 + 22 * q^69 - 41 * q^70 + 113 * q^71 - 19 * q^73 + 11 * q^74 + 45 * q^75 + 34 * q^76 + 64 * q^77 + 29 * q^78 + 23 * q^79 + 54 * q^80 + 149 * q^81 + 4 * q^82 + 100 * q^83 - 49 * q^84 - 5 * q^85 - 24 * q^86 + 65 * q^87 + 14 * q^88 + 84 * q^89 - 21 * q^90 + 32 * q^91 + 95 * q^92 + 19 * q^93 - 47 * q^94 + 102 * q^95 + 29 * q^96 + 7 * q^97 + 26 * q^98 + 107 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.11247	−1.49374	−0.746871	−	0.664969i	$-0.768445\pi$
$2$	−2.11247	−1.49374	−0.746871	+	0.664969i	$0.768445\pi$
$3$	2.70421	1.56128	0.780638	−	0.624983i	$-0.214894\pi$
$3$	2.70421	1.56128	0.780638	+	0.624983i	$0.214894\pi$
$4$	2.46253	1.23127
$5$	2.42834	1.08599	0.542993	−	0.839737i	$-0.317291\pi$
$5$	2.42834	1.08599	0.542993	+	0.839737i	$0.317291\pi$
$6$	−5.71257	−2.33215
$7$	1.36575	0.516205	0.258102	−	0.966118i	$-0.416903\pi$
$7$	1.36575	0.516205	0.258102	+	0.966118i	$0.416903\pi$
$8$	−0.977087	−0.345453
$9$	4.31276	1.43759
$10$	−5.12979	−1.62218
$11$	−5.83835	−1.76033	−0.880164	−	0.474669i	$-0.842568\pi$
$11$	−5.83835	−1.76033	−0.880164	+	0.474669i	$0.842568\pi$
$12$	6.65921	1.92235
$13$	3.94237	1.09342	0.546708	−	0.837323i	$-0.315881\pi$
$13$	3.94237	1.09342	0.546708	+	0.837323i	$0.315881\pi$
$14$	−2.88511	−0.771077
$15$	6.56674	1.69552
$16$	−2.86100	−0.715249
$17$	−0.443864	−0.107653	−0.0538264	−	0.998550i	$-0.517142\pi$
$17$	−0.443864	−0.107653	−0.0538264	+	0.998550i	$0.517142\pi$
$18$	−9.11057	−2.14738
$19$	4.49035	1.03016	0.515078	−	0.857143i	$-0.327763\pi$
$19$	4.49035	1.03016	0.515078	+	0.857143i	$0.327763\pi$
$20$	5.97986	1.33714
$21$	3.69328	0.805939
$22$	12.3333	2.62948
$23$	5.11497	1.06654	0.533272	−	0.845944i	$-0.320962\pi$
$23$	5.11497	1.06654	0.533272	+	0.845944i	$0.320962\pi$
$24$	−2.64225	−0.539347
$25$	0.896826	0.179365
$26$	−8.32814	−1.63328
$27$	3.54997	0.683192
$28$	3.36320	0.635586
$29$	8.05928	1.49657	0.748286	−	0.663377i	$-0.230877\pi$
$29$	8.05928	1.49657	0.748286	+	0.663377i	$0.230877\pi$
$30$	−13.8720	−2.53268
$31$	−3.45560	−0.620645	−0.310322	−	0.950631i	$-0.600437\pi$
$31$	−3.45560	−0.620645	−0.310322	+	0.950631i	$0.600437\pi$
$32$	7.99795	1.41385
$33$	−15.7881	−2.74836
$34$	0.937649	0.160806
$35$	3.31650	0.560591
$36$	10.6203	1.77005
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−9.48573	−1.53879
$39$	10.6610	1.70713
$40$	−2.37270	−0.375157
$41$	−0.332675	−0.0519551	−0.0259776	−	0.999663i	$-0.508270\pi$
$41$	−0.332675	−0.0519551	−0.0259776	+	0.999663i	$0.508270\pi$
$42$	−7.80194	−1.20387
$43$	−3.72912	−0.568685	−0.284342	−	0.958723i	$-0.591775\pi$
$43$	−3.72912	−0.568685	−0.284342	+	0.958723i	$0.591775\pi$
$44$	−14.3771	−2.16743
$45$	10.4728	1.56120
$46$	−10.8052	−1.59314
$47$	2.07687	0.302942	0.151471	−	0.988462i	$-0.451599\pi$
$47$	2.07687	0.302942	0.151471	+	0.988462i	$0.451599\pi$
$48$	−7.73674	−1.11670
$49$	−5.13473	−0.733532
$50$	−1.89452	−0.267925
$51$	−1.20030	−0.168076
$52$	9.70821	1.34629
$53$	2.70802	0.371975	0.185987	−	0.982552i	$-0.440452\pi$
$53$	2.70802	0.371975	0.185987	+	0.982552i	$0.440452\pi$
$54$	−7.49920	−1.02051
$55$	−14.1775	−1.91169
$56$	−1.33446	−0.178324
$57$	12.1428	1.60836
$58$	−17.0250	−2.23549
$59$	2.65523	0.345682	0.172841	−	0.984950i	$-0.444705\pi$
$59$	2.65523	0.345682	0.172841	+	0.984950i	$0.444705\pi$
$60$	16.1708	2.08764
$61$	−2.31531	−0.296445	−0.148222	−	0.988954i	$-0.547355\pi$
$61$	−2.31531	−0.296445	−0.148222	+	0.988954i	$0.547355\pi$
$62$	7.29986	0.927084
$63$	5.89015	0.742089
$64$	−11.1734	−1.39668
$65$	9.57340	1.18743
$66$	33.3520	4.10534
$67$	13.8635	1.69370	0.846851	−	0.531830i	$-0.178496\pi$
$67$	13.8635	1.69370	0.846851	+	0.531830i	$0.178496\pi$
$68$	−1.09303	−0.132549
$69$	13.8319	1.66517
$70$	−7.00602	−0.837379
$71$	1.12509	0.133524	0.0667618	−	0.997769i	$-0.478733\pi$
$71$	1.12509	0.133524	0.0667618	+	0.997769i	$0.478733\pi$
$72$	−4.21394	−0.496618
$73$	−6.62746	−0.775686	−0.387843	−	0.921726i	$-0.626780\pi$
$73$	−6.62746	−0.775686	−0.387843	+	0.921726i	$0.626780\pi$
$74$	−2.11247	−0.245570
$75$	2.42521	0.280039
$76$	11.0576	1.26840
$77$	−7.97373	−0.908690
$78$	−22.5210	−2.55001
$79$	−3.29503	−0.370720	−0.185360	−	0.982671i	$-0.559345\pi$
$79$	−3.29503	−0.370720	−0.185360	+	0.982671i	$0.559345\pi$
$80$	−6.94747	−0.776751
$81$	−3.33841	−0.370934
$82$	0.702766	0.0776075
$83$	−0.226071	−0.0248145	−0.0124073	−	0.999923i	$-0.503949\pi$
$83$	−0.226071	−0.0248145	−0.0124073	+	0.999923i	$0.503949\pi$
$84$	9.09481	0.992325
$85$	−1.07785	−0.116909
$86$	7.87765	0.849469
$87$	21.7940	2.33656
$88$	5.70458	0.608110
$89$	18.1623	1.92520	0.962599	−	0.270931i	$-0.0873315\pi$
$89$	18.1623	1.92520	0.962599	+	0.270931i	$0.0873315\pi$
$90$	−22.1235	−2.33203
$91$	5.38429	0.564427
$92$	12.5958	1.31320
$93$	−9.34468	−0.968998
$94$	−4.38732	−0.452518
$95$	10.9041	1.11874
$96$	21.6281	2.20741
$97$	−12.4517	−1.26428	−0.632139	−	0.774855i	$-0.717823\pi$
$97$	−12.4517	−1.26428	−0.632139	+	0.774855i	$0.717823\pi$
$98$	10.8470	1.09571
$99$	−25.1794	−2.53062
$100$	2.20846	0.220846
$101$	8.02699	0.798715	0.399357	−	0.916795i	$-0.369233\pi$
$101$	8.02699	0.798715	0.399357	+	0.916795i	$0.369233\pi$
$102$	2.53560	0.251062
$103$	−0.842268	−0.0829912	−0.0414956	−	0.999139i	$-0.513212\pi$
$103$	−0.842268	−0.0829912	−0.0414956	+	0.999139i	$0.513212\pi$
$104$	−3.85204	−0.377723
$105$	8.96852	0.875238
$106$	−5.72061	−0.555635
$107$	11.8826	1.14874	0.574368	−	0.818597i	$-0.305248\pi$
$107$	11.8826	1.14874	0.574368	+	0.818597i	$0.305248\pi$
$108$	8.74191	0.841191
$109$	−1.00000	−0.0957826
$109$	−1.00000	−0.0957826
$110$	29.9495	2.85558
$111$	2.70421	0.256672
$112$	−3.90741	−0.369215
$113$	9.90207	0.931508	0.465754	−	0.884914i	$-0.345783\pi$
$113$	9.90207	0.931508	0.465754	+	0.884914i	$0.345783\pi$
$114$	−25.6514	−2.40247
$115$	12.4209	1.15825
$116$	19.8462	1.84268
$117$	17.0025	1.57188
$118$	−5.60910	−0.516360
$119$	−0.606207	−0.0555709
$120$	−6.41628	−0.585723
$121$	23.0863	2.09876
$122$	4.89102	0.442812
$123$	−0.899623	−0.0811163
$124$	−8.50954	−0.764179
$125$	−9.96389	−0.891198
$126$	−12.4428	−1.10849
$127$	−3.41427	−0.302967	−0.151484	−	0.988460i	$-0.548405\pi$
$127$	−3.41427	−0.302967	−0.151484	+	0.988460i	$0.548405\pi$
$128$	7.60766	0.672429
$129$	−10.0843	−0.887875
$130$	−20.2235	−1.77372
$131$	10.6662	0.931912	0.465956	−	0.884808i	$-0.345710\pi$
$131$	10.6662	0.931912	0.465956	+	0.884808i	$0.345710\pi$
$132$	−38.8788	−3.38396
$133$	6.13269	0.531772
$134$	−29.2863	−2.52995
$135$	8.62052	0.741936
$136$	0.433694	0.0371889
$137$	−7.80770	−0.667057	−0.333529	−	0.942740i	$-0.608239\pi$
$137$	−7.80770	−0.667057	−0.333529	+	0.942740i	$0.608239\pi$
$138$	−29.2196	−2.48734
$139$	0.780164	0.0661727	0.0330863	−	0.999452i	$-0.489466\pi$
$139$	0.780164	0.0661727	0.0330863	+	0.999452i	$0.489466\pi$
$140$	8.16700	0.690237
$141$	5.61629	0.472977
$142$	−2.37672	−0.199450
$143$	−23.0169	−1.92477
$144$	−12.3388	−1.02823
$145$	19.5707	1.62526
$146$	14.0003	1.15867
$147$	−13.8854	−1.14525
$148$	2.46253	0.202419
$149$	−2.89956	−0.237541	−0.118771	−	0.992922i	$-0.537895\pi$
$149$	−2.89956	−0.237541	−0.118771	+	0.992922i	$0.537895\pi$
$150$	−5.12318	−0.418306
$151$	−15.3926	−1.25263	−0.626315	−	0.779570i	$-0.715438\pi$
$151$	−15.3926	−1.25263	−0.626315	+	0.779570i	$0.715438\pi$
$152$	−4.38746	−0.355870
$153$	−1.91428	−0.154760
$154$	16.8443	1.35735
$155$	−8.39138	−0.674012
$156$	26.2530	2.10193
$157$	−0.359790	−0.0287144	−0.0143572	−	0.999897i	$-0.504570\pi$
$157$	−0.359790	−0.0287144	−0.0143572	+	0.999897i	$0.504570\pi$
$158$	6.96065	0.553760
$159$	7.32305	0.580756
$160$	19.4217	1.53542
$161$	6.98576	0.550555
$162$	7.05228	0.554080
$163$	23.8185	1.86561	0.932805	−	0.360383i	$-0.117354\pi$
$163$	23.8185	1.86561	0.932805	+	0.360383i	$0.117354\pi$
$164$	−0.819223	−0.0639706
$165$	−38.3389	−2.98468
$166$	0.477568	0.0370665
$167$	−6.54347	−0.506349	−0.253175	−	0.967421i	$-0.581475\pi$
$167$	−6.54347	−0.506349	−0.253175	+	0.967421i	$0.581475\pi$
$168$	−3.60865	−0.278414
$169$	2.54226	0.195559
$170$	2.27693	0.174633
$171$	19.3658	1.48094
$172$	−9.18307	−0.700203
$173$	−19.2219	−1.46142	−0.730708	−	0.682691i	$-0.760810\pi$
$173$	−19.2219	−1.46142	−0.730708	+	0.682691i	$0.760810\pi$
$174$	−46.0392	−3.49022
$175$	1.22484	0.0925892
$176$	16.7035	1.25907
$177$	7.18031	0.539705
$178$	−38.3673	−2.87575
$179$	14.4784	1.08217	0.541084	−	0.840968i	$-0.318014\pi$
$179$	14.4784	1.08217	0.541084	+	0.840968i	$0.318014\pi$
$180$	25.7897	1.92225
$181$	−9.97548	−0.741471	−0.370736	−	0.928738i	$-0.620894\pi$
$181$	−9.97548	−0.741471	−0.370736	+	0.928738i	$0.620894\pi$
$182$	−11.3742	−0.843108
$183$	−6.26107	−0.462832
$184$	−4.99777	−0.368440
$185$	2.42834	0.178535
$186$	19.7404	1.44743
$187$	2.59143	0.189504
$188$	5.11436	0.373003
$189$	4.84837	0.352667
$190$	−23.0346	−1.67110
$191$	−3.05153	−0.220801	−0.110400	−	0.993887i	$-0.535213\pi$
$191$	−3.05153	−0.220801	−0.110400	+	0.993887i	$0.535213\pi$
$192$	−30.2153	−2.18060
$193$	−3.79710	−0.273321	−0.136661	−	0.990618i	$-0.543637\pi$
$193$	−3.79710	−0.273321	−0.136661	+	0.990618i	$0.543637\pi$
$194$	26.3039	1.88851
$195$	25.8885	1.85391
$196$	−12.6444	−0.903174
$197$	24.8818	1.77276	0.886378	−	0.462962i	$-0.153213\pi$
$197$	24.8818	1.77276	0.886378	+	0.462962i	$0.153213\pi$
$198$	53.1907	3.78010
$199$	7.27834	0.515948	0.257974	−	0.966152i	$-0.416945\pi$
$199$	7.27834	0.515948	0.257974	+	0.966152i	$0.416945\pi$
$200$	−0.876277	−0.0619622
$201$	37.4900	2.64434
$202$	−16.9568	−1.19307
$203$	11.0070	0.772537
$204$	−2.95578	−0.206946
$205$	−0.807847	−0.0564225
$206$	1.77927	0.123967
$207$	22.0596	1.53325
$208$	−11.2791	−0.782065
$209$	−26.2162	−1.81341
$210$	−18.9457	−1.30738
$211$	24.5592	1.69072	0.845362	−	0.534194i	$-0.179385\pi$
$211$	24.5592	1.69072	0.845362	+	0.534194i	$0.179385\pi$
$212$	6.66858	0.458000
$213$	3.04248	0.208467
$214$	−25.1017	−1.71592
$215$	−9.05556	−0.617584
$216$	−3.46863	−0.236010
$217$	−4.71949	−0.320380
$218$	2.11247	0.143075
$219$	−17.9220	−1.21106
$220$	−34.9125	−2.35380
$221$	−1.74987	−0.117709
$222$	−5.71257	−0.383402
$223$	−8.11629	−0.543507	−0.271753	−	0.962367i	$-0.587604\pi$
$223$	−8.11629	−0.543507	−0.271753	+	0.962367i	$0.587604\pi$
$224$	10.9232	0.729837
$225$	3.86779	0.257853
$226$	−20.9178	−1.39143
$227$	−16.5920	−1.10125	−0.550623	−	0.834754i	$-0.685610\pi$
$227$	−16.5920	−1.10125	−0.550623	+	0.834754i	$0.685610\pi$
$228$	29.9022	1.98032
$229$	8.05774	0.532470	0.266235	−	0.963908i	$-0.414220\pi$
$229$	8.05774	0.532470	0.266235	+	0.963908i	$0.414220\pi$
$230$	−26.2387	−1.73013
$231$	−21.5626	−1.41872
$232$	−7.87462	−0.516994
$233$	−2.42793	−0.159059	−0.0795295	−	0.996833i	$-0.525342\pi$
$233$	−2.42793	−0.159059	−0.0795295	+	0.996833i	$0.525342\pi$
$234$	−35.9172	−2.34798
$235$	5.04334	0.328991
$236$	6.53860	0.425626
$237$	−8.91046	−0.578796
$238$	1.28059	0.0830086
$239$	−11.8038	−0.763525	−0.381762	−	0.924261i	$-0.624683\pi$
$239$	−11.8038	−0.763525	−0.381762	+	0.924261i	$0.624683\pi$
$240$	−18.7874	−1.21272
$241$	10.7099	0.689886	0.344943	−	0.938624i	$-0.387898\pi$
$241$	10.7099	0.689886	0.344943	+	0.938624i	$0.387898\pi$
$242$	−48.7692	−3.13500
$243$	−19.6777	−1.26232
$244$	−5.70152	−0.365002
$245$	−12.4689	−0.796606
$246$	1.90043	0.121167
$247$	17.7026	1.12639
$248$	3.37643	0.214403
$249$	−0.611343	−0.0387423
$250$	21.0484	1.33122
$251$	−7.15304	−0.451496	−0.225748	−	0.974186i	$-0.572483\pi$
$251$	−7.15304	−0.451496	−0.225748	+	0.974186i	$0.572483\pi$
$252$	14.5047	0.913709
$253$	−29.8630	−1.87747
$254$	7.21254	0.452555
$255$	−2.91474	−0.182528
$256$	6.27591	0.392244
$257$	−20.0109	−1.24824	−0.624121	−	0.781327i	$-0.714543\pi$
$257$	−20.0109	−1.24824	−0.624121	+	0.781327i	$0.714543\pi$
$258$	21.3028	1.32626
$259$	1.36575	0.0848636
$260$	23.5748	1.46205
$261$	34.7577	2.15145
$262$	−22.5321	−1.39204
$263$	−7.90913	−0.487698	−0.243849	−	0.969813i	$-0.578410\pi$
$263$	−7.90913	−0.487698	−0.243849	+	0.969813i	$0.578410\pi$
$264$	15.4264	0.949428
$265$	6.57598	0.403959
$266$	−12.9551	−0.794330
$267$	49.1146	3.00577
$268$	34.1394	2.08540
$269$	8.60657	0.524752	0.262376	−	0.964966i	$-0.415494\pi$
$269$	8.60657	0.524752	0.262376	+	0.964966i	$0.415494\pi$
$270$	−18.2106	−1.10826
$271$	−0.196579	−0.0119413	−0.00597067	−	0.999982i	$-0.501901\pi$
$271$	−0.196579	−0.0119413	−0.00597067	+	0.999982i	$0.501901\pi$
$272$	1.26989	0.0769986
$273$	14.5603	0.881226
$274$	16.4935	0.996412
$275$	−5.23598	−0.315742
$276$	34.0616	2.05027
$277$	−19.8252	−1.19118	−0.595589	−	0.803289i	$-0.703081\pi$
$277$	−19.8252	−1.19118	−0.595589	+	0.803289i	$0.703081\pi$
$278$	−1.64807	−0.0988449
$279$	−14.9032	−0.892230
$280$	−3.24051	−0.193658
$281$	−15.6115	−0.931307	−0.465653	−	0.884967i	$-0.654181\pi$
$281$	−15.6115	−0.931307	−0.465653	+	0.884967i	$0.654181\pi$
$282$	−11.8642	−0.706506
$283$	−14.1811	−0.842976	−0.421488	−	0.906834i	$-0.638492\pi$
$283$	−14.1811	−0.842976	−0.421488	+	0.906834i	$0.638492\pi$
$284$	2.77057	0.164403
$285$	29.4869	1.74666
$286$	48.6226	2.87511
$287$	−0.454351	−0.0268195
$288$	34.4932	2.03253
$289$	−16.8030	−0.988411
$290$	−41.3425	−2.42771
$291$	−33.6720	−1.97389
$292$	−16.3203	−0.955076
$293$	−9.36224	−0.546948	−0.273474	−	0.961879i	$-0.588173\pi$
$293$	−9.36224	−0.546948	−0.273474	+	0.961879i	$0.588173\pi$
$294$	29.3325	1.71070
$295$	6.44780	0.375406
$296$	−0.977087	−0.0567921
$297$	−20.7260	−1.20264
$298$	6.12524	0.354825
$299$	20.1651	1.16618
$300$	5.97215	0.344802
$301$	−5.09304	−0.293558
$302$	32.5164	1.87111
$303$	21.7067	1.24701
$304$	−12.8469	−0.736819
$305$	−5.62235	−0.321935
$306$	4.04385	0.231172
$307$	7.92978	0.452576	0.226288	−	0.974060i	$-0.427341\pi$
$307$	7.92978	0.452576	0.226288	+	0.974060i	$0.427341\pi$
$308$	−19.6356	−1.11884
$309$	−2.27767	−0.129572
$310$	17.7265	1.00680
$311$	1.38182	0.0783558	0.0391779	−	0.999232i	$-0.487526\pi$
$311$	1.38182	0.0783558	0.0391779	+	0.999232i	$0.487526\pi$
$312$	−10.4167	−0.589731
$313$	4.57223	0.258438	0.129219	−	0.991616i	$-0.458753\pi$
$313$	4.57223	0.258438	0.129219	+	0.991616i	$0.458753\pi$
$314$	0.760046	0.0428919
$315$	14.3033	0.805898
$316$	−8.11412	−0.456455
$317$	−7.52627	−0.422717	−0.211359	−	0.977409i	$-0.567789\pi$
$317$	−7.52627	−0.422717	−0.211359	+	0.977409i	$0.567789\pi$
$318$	−15.4697	−0.867500
$319$	−47.0529	−2.63446
$320$	−27.1329	−1.51677
$321$	32.1331	1.79350
$322$	−14.7572	−0.822388
$323$	−1.99310	−0.110899
$324$	−8.22093	−0.456719
$325$	3.53562	0.196121
$326$	−50.3159	−2.78674
$327$	−2.70421	−0.149543
$328$	0.325053	0.0179480
$329$	2.83648	0.156380
$330$	80.9898	4.45834
$331$	−11.6500	−0.640340	−0.320170	−	0.947360i	$-0.603740\pi$
$331$	−11.6500	−0.640340	−0.320170	+	0.947360i	$0.603740\pi$
$332$	−0.556707	−0.0305533
$333$	4.31276	0.236338
$334$	13.8229	0.756355
$335$	33.6654	1.83934
$336$	−10.5665	−0.576447
$337$	−26.6134	−1.44972	−0.724862	−	0.688894i	$-0.758096\pi$
$337$	−26.6134	−1.44972	−0.724862	+	0.688894i	$0.758096\pi$
$338$	−5.37046	−0.292114
$339$	26.7773	1.45434
$340$	−2.65425	−0.143947
$341$	20.1750	1.09254
$342$	−40.9096	−2.21214
$343$	−16.5730	−0.894858
$344$	3.64367	0.196454
$345$	33.5886	1.80835
$346$	40.6057	2.18298
$347$	22.5875	1.21256	0.606279	−	0.795252i	$-0.292661\pi$
$347$	22.5875	1.21256	0.606279	+	0.795252i	$0.292661\pi$
$348$	53.6684	2.87693
$349$	−18.0850	−0.968065	−0.484033	−	0.875050i	$-0.660828\pi$
$349$	−18.0850	−0.968065	−0.484033	+	0.875050i	$0.660828\pi$
$350$	−2.58744	−0.138304
$351$	13.9953	0.747013
$352$	−46.6948	−2.48884
$353$	−26.1218	−1.39032	−0.695162	−	0.718854i	$-0.744667\pi$
$353$	−26.1218	−1.39032	−0.695162	+	0.718854i	$0.744667\pi$
$354$	−15.1682	−0.806180
$355$	2.73210	0.145005
$356$	44.7252	2.37043
$357$	−1.63931	−0.0867616
$358$	−30.5853	−1.61648
$359$	16.1626	0.853030	0.426515	−	0.904481i	$-0.359741\pi$
$359$	16.1626	0.853030	0.426515	+	0.904481i	$0.359741\pi$
$360$	−10.2329	−0.539320
$361$	1.16323	0.0612226
$362$	21.0729	1.10757
$363$	62.4303	3.27674
$364$	13.2590	0.694960
$365$	−16.0937	−0.842384
$366$	13.2263	0.691352
$367$	19.1446	0.999340	0.499670	−	0.866216i	$-0.333455\pi$
$367$	19.1446	0.999340	0.499670	+	0.866216i	$0.333455\pi$
$368$	−14.6339	−0.762845
$369$	−1.43475	−0.0746899
$370$	−5.12979	−0.266685
$371$	3.69848	0.192015
$372$	−23.0116	−1.19310
$373$	−31.1180	−1.61123	−0.805614	−	0.592441i	$-0.798164\pi$
$373$	−31.1180	−1.61123	−0.805614	+	0.592441i	$0.798164\pi$
$374$	−5.47433	−0.283071
$375$	−26.9445	−1.39141
$376$	−2.02928	−0.104652
$377$	31.7727	1.63637
$378$	−10.2420	−0.526794
$379$	−3.60303	−0.185075	−0.0925377	−	0.995709i	$-0.529498\pi$
$379$	−3.60303	−0.185075	−0.0925377	+	0.995709i	$0.529498\pi$
$380$	26.8517	1.37746
$381$	−9.23290	−0.473016
$382$	6.44626	0.329819
$383$	1.22529	0.0626095	0.0313048	−	0.999510i	$-0.490034\pi$
$383$	1.22529	0.0626095	0.0313048	+	0.999510i	$0.490034\pi$
$384$	20.5727	1.04985
$385$	−19.3629	−0.986825
$386$	8.02126	0.408272
$387$	−16.0828	−0.817533
$388$	−30.6627	−1.55666
$389$	−2.93083	−0.148599	−0.0742995	−	0.997236i	$-0.523672\pi$
$389$	−2.93083	−0.148599	−0.0742995	+	0.997236i	$0.523672\pi$
$390$	−54.6887	−2.76927
$391$	−2.27035	−0.114816
$392$	5.01708	0.253401
$393$	28.8437	1.45497
$394$	−52.5621	−2.64804
$395$	−8.00145	−0.402596
$396$	−62.0050	−3.11587
$397$	11.8536	0.594916	0.297458	−	0.954735i	$-0.403861\pi$
$397$	11.8536	0.594916	0.297458	+	0.954735i	$0.403861\pi$
$398$	−15.3753	−0.770693
$399$	16.5841	0.830243
$400$	−2.56582	−0.128291
$401$	5.00202	0.249789	0.124894	−	0.992170i	$-0.460141\pi$
$401$	5.00202	0.249789	0.124894	+	0.992170i	$0.460141\pi$
$402$	−79.1964	−3.94996
$403$	−13.6233	−0.678623
$404$	19.7667	0.983431
$405$	−8.10678	−0.402829
$406$	−23.2519	−1.15397
$407$	−5.83835	−0.289396
$408$	1.17280	0.0580622
$409$	−15.9656	−0.789448	−0.394724	−	0.918800i	$-0.629160\pi$
$409$	−15.9656	−0.789448	−0.394724	+	0.918800i	$0.629160\pi$
$410$	1.70655	0.0842807
$411$	−21.1137	−1.04146
$412$	−2.07411	−0.102184
$413$	3.62638	0.178443
$414$	−46.6003	−2.29028
$415$	−0.548977	−0.0269482
$416$	31.5309	1.54593
$417$	2.10973	0.103314
$418$	55.3810	2.70877
$419$	32.7296	1.59894	0.799472	−	0.600703i	$-0.205113\pi$
$419$	32.7296	1.59894	0.799472	+	0.600703i	$0.205113\pi$
$420$	22.0853	1.07765
$421$	11.0666	0.539355	0.269677	−	0.962951i	$-0.413083\pi$
$421$	11.0666	0.539355	0.269677	+	0.962951i	$0.413083\pi$
$422$	−51.8806	−2.52551
$423$	8.95702	0.435505
$424$	−2.64597	−0.128500
$425$	−0.398069	−0.0193092
$426$	−6.42715	−0.311396
$427$	−3.16213	−0.153026
$428$	29.2613	1.41440
$429$	−62.2426	−3.00510
$430$	19.1296	0.922511
$431$	−28.1460	−1.35574	−0.677871	−	0.735180i	$-0.737097\pi$
$431$	−28.1460	−1.35574	−0.677871	+	0.735180i	$0.737097\pi$
$432$	−10.1564	−0.488652
$433$	0.131642	0.00632632	0.00316316	−	0.999995i	$-0.498993\pi$
$433$	0.131642	0.00632632	0.00316316	+	0.999995i	$0.498993\pi$
$434$	9.96979	0.478565
$435$	52.9232	2.53747
$436$	−2.46253	−0.117934
$437$	22.9680	1.09871
$438$	37.8598	1.80901
$439$	29.4863	1.40730	0.703651	−	0.710546i	$-0.251552\pi$
$439$	29.4863	1.40730	0.703651	+	0.710546i	$0.251552\pi$
$440$	13.8526	0.660399
$441$	−22.1448	−1.05452
$442$	3.69656	0.175827
$443$	9.77181	0.464273	0.232136	−	0.972683i	$-0.425428\pi$
$443$	9.77181	0.464273	0.232136	+	0.972683i	$0.425428\pi$
$444$	6.65921	0.316032
$445$	44.1042	2.09074
$446$	17.1454	0.811859
$447$	−7.84102	−0.370868
$448$	−15.2601	−0.720973
$449$	26.7623	1.26299	0.631495	−	0.775380i	$-0.282442\pi$
$449$	26.7623	1.26299	0.631495	+	0.775380i	$0.282442\pi$
$450$	−8.17060	−0.385166
$451$	1.94227	0.0914580
$452$	24.3842	1.14693
$453$	−41.6248	−1.95570
$454$	35.0500	1.64498
$455$	13.0749	0.612960
$456$	−11.8646	−0.555612
$457$	−8.53641	−0.399317	−0.199658	−	0.979866i	$-0.563983\pi$
$457$	−8.53641	−0.399317	−0.199658	+	0.979866i	$0.563983\pi$
$458$	−17.0217	−0.795373
$459$	−1.57570	−0.0735475
$460$	30.5868	1.42612
$461$	16.4917	0.768096	0.384048	−	0.923313i	$-0.374530\pi$
$461$	16.4917	0.768096	0.384048	+	0.923313i	$0.374530\pi$
$462$	45.5504	2.11920
$463$	8.19071	0.380654	0.190327	−	0.981721i	$-0.439045\pi$
$463$	8.19071	0.380654	0.190327	+	0.981721i	$0.439045\pi$
$464$	−23.0576	−1.07042
$465$	−22.6920	−1.05232
$466$	5.12893	0.237593
$467$	34.2606	1.58539	0.792695	−	0.609618i	$-0.208677\pi$
$467$	34.2606	1.58539	0.792695	+	0.609618i	$0.208677\pi$
$468$	41.8691	1.93540
$469$	18.9341	0.874297
$470$	−10.6539	−0.491428
$471$	−0.972948	−0.0448311
$472$	−2.59439	−0.119417
$473$	21.7719	1.00107
$474$	18.8231	0.864573
$475$	4.02706	0.184774
$476$	−1.49280	−0.0684226
$477$	11.6790	0.534746
$478$	24.9352	1.14051
$479$	−29.1701	−1.33282	−0.666409	−	0.745586i	$-0.732170\pi$
$479$	−29.1701	−1.33282	−0.666409	+	0.745586i	$0.732170\pi$
$480$	52.5204	2.39722
$481$	3.94237	0.179756
$482$	−22.6244	−1.03051
$483$	18.8910	0.859569
$484$	56.8508	2.58413
$485$	−30.2369	−1.37299
$486$	41.5685	1.88558
$487$	−13.8088	−0.625736	−0.312868	−	0.949797i	$-0.601290\pi$
$487$	−13.8088	−0.625736	−0.312868	+	0.949797i	$0.601290\pi$
$488$	2.26226	0.102408
$489$	64.4102	2.91273
$490$	26.3401	1.18992
$491$	1.05407	0.0475694	0.0237847	−	0.999717i	$-0.492428\pi$
$491$	1.05407	0.0475694	0.0237847	+	0.999717i	$0.492428\pi$
$492$	−2.21535	−0.0998758
$493$	−3.57722	−0.161110
$494$	−37.3962	−1.68254
$495$	−61.1440	−2.74822
$496$	9.88648	0.443916
$497$	1.53659	0.0689255
$498$	1.29145	0.0578710
$499$	32.4643	1.45330	0.726650	−	0.687008i	$-0.241076\pi$
$499$	32.4643	1.45330	0.726650	+	0.687008i	$0.241076\pi$
$500$	−24.5364	−1.09730
$501$	−17.6949	−0.790551
$502$	15.1106	0.674418
$503$	−10.4898	−0.467717	−0.233858	−	0.972271i	$-0.575135\pi$
$503$	−10.4898	−0.467717	−0.233858	+	0.972271i	$0.575135\pi$
$504$	−5.75519	−0.256356
$505$	19.4922	0.867393
$506$	63.0846	2.80445
$507$	6.87482	0.305321
$508$	−8.40775	−0.373034
$509$	37.5494	1.66435	0.832174	−	0.554515i	$-0.187096\pi$
$509$	37.5494	1.66435	0.832174	+	0.554515i	$0.187096\pi$
$510$	6.15730	0.272650
$511$	−9.05145	−0.400413
$512$	−28.4730	−1.25834
$513$	15.9406	0.703794
$514$	42.2723	1.86455
$515$	−2.04531	−0.0901272
$516$	−24.8330	−1.09321
$517$	−12.1255	−0.533278
$518$	−2.88511	−0.126764
$519$	−51.9801	−2.28167
$520$	−9.35405	−0.410202
$521$	30.9278	1.35497	0.677486	−	0.735535i	$-0.263069\pi$
$521$	30.9278	1.35497	0.677486	+	0.735535i	$0.263069\pi$
$522$	−73.4247	−3.21371
$523$	2.02784	0.0886711	0.0443355	−	0.999017i	$-0.485883\pi$
$523$	2.02784	0.0886711	0.0443355	+	0.999017i	$0.485883\pi$
$524$	26.2659	1.14743
$525$	3.31223	0.144557
$526$	16.7078	0.728495
$527$	1.53382	0.0668142
$528$	45.1698	1.96576
$529$	3.16288	0.137516
$530$	−13.8916	−0.603411
$531$	11.4514	0.496947
$532$	15.1020	0.654753
$533$	−1.31153	−0.0568085
$534$	−103.753	−4.48984
$535$	28.8550	1.24751
$536$	−13.5459	−0.585094
$537$	39.1527	1.68957
$538$	−18.1811	−0.783844
$539$	29.9783	1.29126
$540$	21.2283	0.913521
$541$	−15.6952	−0.674789	−0.337394	−	0.941363i	$-0.609546\pi$
$541$	−15.6952	−0.674789	−0.337394	+	0.941363i	$0.609546\pi$
$542$	0.415268	0.0178373
$543$	−26.9758	−1.15764
$544$	−3.55000	−0.152205
$545$	−2.42834	−0.104019
$546$	−30.7581	−1.31633
$547$	−31.8968	−1.36381	−0.681904	−	0.731441i	$-0.738848\pi$
$547$	−31.8968	−1.36381	−0.681904	+	0.731441i	$0.738848\pi$
$548$	−19.2267	−0.821325
$549$	−9.98535	−0.426164
$550$	11.0609	0.471637
$551$	36.1890	1.54170
$552$	−13.5150	−0.575237
$553$	−4.50019	−0.191367
$554$	41.8801	1.77931
$555$	6.56674	0.278743
$556$	1.92118	0.0814762
$557$	14.2168	0.602386	0.301193	−	0.953563i	$-0.402615\pi$
$557$	14.2168	0.602386	0.301193	+	0.953563i	$0.402615\pi$
$558$	31.4825	1.33276
$559$	−14.7015	−0.621809
$560$	−9.48851	−0.400963
$561$	7.00778	0.295869
$562$	32.9789	1.39113
$563$	16.2007	0.682778	0.341389	−	0.939922i	$-0.389103\pi$
$563$	16.2007	0.682778	0.341389	+	0.939922i	$0.389103\pi$
$564$	13.8303	0.582360
$565$	24.0456	1.01160
$566$	29.9571	1.25919
$567$	−4.55943	−0.191478
$568$	−1.09931	−0.0461260
$569$	−44.2465	−1.85491	−0.927454	−	0.373937i	$-0.878007\pi$
$569$	−44.2465	−1.85491	−0.927454	+	0.373937i	$0.878007\pi$
$570$	−62.2903	−2.60905
$571$	46.5263	1.94707	0.973533	−	0.228546i	$-0.0733972\pi$
$571$	46.5263	1.94707	0.973533	+	0.228546i	$0.0733972\pi$
$572$	−56.6799	−2.36991
$573$	−8.25197	−0.344731
$574$	0.959803	0.0400614
$575$	4.58723	0.191301
$576$	−48.1883	−2.00785
$577$	−8.25627	−0.343713	−0.171857	−	0.985122i	$-0.554977\pi$
$577$	−8.25627	−0.343713	−0.171857	+	0.985122i	$0.554977\pi$
$578$	35.4958	1.47643
$579$	−10.2682	−0.426730
$580$	48.1934	2.00112
$581$	−0.308756	−0.0128094
$582$	71.1312	2.94848
$583$	−15.8104	−0.654798
$584$	6.47561	0.267963
$585$	41.2877	1.70704
$586$	19.7775	0.816999
$587$	−5.27724	−0.217815	−0.108908	−	0.994052i	$-0.534735\pi$
$587$	−5.27724	−0.217815	−0.108908	+	0.994052i	$0.534735\pi$
$588$	−34.1932	−1.41010
$589$	−15.5169	−0.639361
$590$	−13.6208	−0.560759
$591$	67.2857	2.76776
$592$	−2.86100	−0.117586
$593$	−37.1400	−1.52516	−0.762578	−	0.646896i	$-0.776067\pi$
$593$	−37.1400	−1.52516	−0.762578	+	0.646896i	$0.776067\pi$
$594$	43.7830	1.79644
$595$	−1.47208	−0.0603492
$596$	−7.14026	−0.292477
$597$	19.6822	0.805537
$598$	−42.5981	−1.74197
$599$	26.9329	1.10045	0.550223	−	0.835018i	$-0.314543\pi$
$599$	26.9329	1.10045	0.550223	+	0.835018i	$0.314543\pi$
$600$	−2.36964	−0.0967401
$601$	−37.0207	−1.51011	−0.755053	−	0.655664i	$-0.772389\pi$
$601$	−37.0207	−1.51011	−0.755053	+	0.655664i	$0.772389\pi$
$602$	10.7589	0.438500
$603$	59.7901	2.43484
$604$	−37.9047	−1.54232
$605$	56.0614	2.27922
$606$	−45.8547	−1.86272
$607$	25.2568	1.02514	0.512571	−	0.858645i	$-0.328693\pi$
$607$	25.2568	1.02514	0.512571	+	0.858645i	$0.328693\pi$
$608$	35.9136	1.45649
$609$	29.7652	1.20614
$610$	11.8770	0.480887
$611$	8.18778	0.331242
$612$	−4.71397	−0.190551
$613$	−11.4573	−0.462757	−0.231379	−	0.972864i	$-0.574324\pi$
$613$	−11.4573	−0.462757	−0.231379	+	0.972864i	$0.574324\pi$
$614$	−16.7514	−0.676033
$615$	−2.18459	−0.0880911
$616$	7.79103	0.313909
$617$	−25.9319	−1.04398	−0.521990	−	0.852952i	$-0.674810\pi$
$617$	−25.9319	−1.04398	−0.521990	+	0.852952i	$0.674810\pi$
$618$	4.81151	0.193547
$619$	12.0225	0.483225	0.241612	−	0.970373i	$-0.422324\pi$
$619$	12.0225	0.483225	0.241612	+	0.970373i	$0.422324\pi$
$620$	−20.6640	−0.829888
$621$	18.1580	0.728654
$622$	−2.91905	−0.117043
$623$	24.8051	0.993797
$624$	−30.5011	−1.22102
$625$	−28.6798	−1.14719
$626$	−9.65870	−0.386039
$627$	−70.8942	−2.83124
$628$	−0.885995	−0.0353551
$629$	−0.443864	−0.0176980
$630$	−30.2152	−1.20380
$631$	13.5994	0.541385	0.270693	−	0.962666i	$-0.412747\pi$
$631$	13.5994	0.541385	0.270693	+	0.962666i	$0.412747\pi$
$632$	3.21953	0.128066
$633$	66.4132	2.63969
$634$	15.8990	0.631431
$635$	−8.29100	−0.329018
$636$	18.0333	0.715065
$637$	−20.2430	−0.802056
$638$	99.3979	3.93520
$639$	4.85223	0.191951
$640$	18.4740	0.730248
$641$	10.5029	0.414840	0.207420	−	0.978252i	$-0.433493\pi$
$641$	10.5029	0.414840	0.207420	+	0.978252i	$0.433493\pi$
$642$	−67.8803	−2.67902
$643$	8.59087	0.338791	0.169395	−	0.985548i	$-0.445818\pi$
$643$	8.59087	0.338791	0.169395	+	0.985548i	$0.445818\pi$
$644$	17.2027	0.677880
$645$	−24.4881	−0.964219
$646$	4.21037	0.165655
$647$	−22.9431	−0.901986	−0.450993	−	0.892527i	$-0.648930\pi$
$647$	−22.9431	−0.901986	−0.450993	+	0.892527i	$0.648930\pi$
$648$	3.26191	0.128140
$649$	−15.5022	−0.608514
$650$	−7.46889	−0.292954
$651$	−12.7625	−0.500202
$652$	58.6538	2.29706
$653$	6.90598	0.270252	0.135126	−	0.990828i	$-0.456856\pi$
$653$	6.90598	0.270252	0.135126	+	0.990828i	$0.456856\pi$
$654$	5.71257	0.223379
$655$	25.9012	1.01204
$656$	0.951782	0.0371609
$657$	−28.5826	−1.11511
$658$	−5.99199	−0.233592
$659$	38.3691	1.49465	0.747324	−	0.664460i	$-0.231338\pi$
$659$	38.3691	1.49465	0.747324	+	0.664460i	$0.231338\pi$
$660$	−94.4108	−3.67494
$661$	−18.0755	−0.703055	−0.351527	−	0.936178i	$-0.614338\pi$
$661$	−18.0755	−0.703055	−0.351527	+	0.936178i	$0.614338\pi$
$662$	24.6102	0.956503
$663$	−4.73203	−0.183777
$664$	0.220891	0.00857224
$665$	14.8923	0.577497
$666$	−9.11057	−0.353027
$667$	41.2230	1.59616
$668$	−16.1135	−0.623451
$669$	−21.9482	−0.848565
$670$	−71.1171	−2.74749
$671$	13.5176	0.521840
$672$	29.5386	1.13948
$673$	−34.5285	−1.33098	−0.665488	−	0.746408i	$-0.731777\pi$
$673$	−34.5285	−1.33098	−0.665488	+	0.746408i	$0.731777\pi$
$674$	56.2200	2.16551
$675$	3.18370	0.122541
$676$	6.26041	0.240785
$677$	21.9742	0.844536	0.422268	−	0.906471i	$-0.361234\pi$
$677$	21.9742	0.844536	0.422268	+	0.906471i	$0.361234\pi$
$678$	−56.5662	−2.17241
$679$	−17.0059	−0.652627
$680$	1.05316	0.0403867
$681$	−44.8681	−1.71935
$682$	−42.6192	−1.63197
$683$	−29.6177	−1.13329	−0.566645	−	0.823962i	$-0.691759\pi$
$683$	−29.6177	−1.13329	−0.566645	+	0.823962i	$0.691759\pi$
$684$	47.6889	1.82343
$685$	−18.9597	−0.724415
$686$	35.0100	1.33669
$687$	21.7898	0.831333
$688$	10.6690	0.406752
$689$	10.6760	0.406723
$690$	−70.9550	−2.70121
$691$	15.0498	0.572521	0.286261	−	0.958152i	$-0.407588\pi$
$691$	15.0498	0.572521	0.286261	+	0.958152i	$0.407588\pi$
$692$	−47.3346	−1.79939
$693$	−34.3887	−1.30632
$694$	−47.7154	−1.81125
$695$	1.89450	0.0718626
$696$	−21.2946	−0.807171
$697$	0.147662	0.00559311
$698$	38.2039	1.44604
$699$	−6.56564	−0.248335
$700$	3.01621	0.114002
$701$	−1.80541	−0.0681896	−0.0340948	−	0.999419i	$-0.510855\pi$
$701$	−1.80541	−0.0681896	−0.0340948	+	0.999419i	$0.510855\pi$
$702$	−29.5646	−1.11584
$703$	4.49035	0.169357
$704$	65.2344	2.45862
$705$	13.6382	0.513646
$706$	55.1815	2.07678
$707$	10.9629	0.412301
$708$	17.6817	0.664521
$709$	−0.439093	−0.0164905	−0.00824524	−	0.999966i	$-0.502625\pi$
$709$	−0.439093	−0.0164905	−0.00824524	+	0.999966i	$0.502625\pi$
$710$	−5.77148	−0.216600
$711$	−14.2107	−0.532941
$712$	−17.7461	−0.665064
$713$	−17.6753	−0.661945
$714$	3.46300	0.129599
$715$	−55.8929	−2.09027
$716$	35.6536	1.33244
$717$	−31.9200	−1.19207
$718$	−34.1430	−1.27421
$719$	−18.9968	−0.708462	−0.354231	−	0.935158i	$-0.615257\pi$
$719$	−18.9968	−0.708462	−0.354231	+	0.935158i	$0.615257\pi$
$720$	−29.9627	−1.11665
$721$	−1.15033	−0.0428405
$722$	−2.45729	−0.0914507
$723$	28.9619	1.07710
$724$	−24.5649	−0.912949
$725$	7.22777	0.268433
$726$	−131.882	−4.89461
$727$	16.6450	0.617328	0.308664	−	0.951171i	$-0.400118\pi$
$727$	16.6450	0.617328	0.308664	+	0.951171i	$0.400118\pi$
$728$	−5.26092	−0.194983
$729$	−43.1973	−1.59990
$730$	33.9975	1.25830
$731$	1.65522	0.0612205
$732$	−15.4181	−0.569869
$733$	−46.7273	−1.72591	−0.862955	−	0.505280i	$-0.831389\pi$
$733$	−46.7273	−1.72591	−0.862955	+	0.505280i	$0.831389\pi$
$734$	−40.4424	−1.49276
$735$	−33.7184	−1.24372
$736$	40.9092	1.50793
$737$	−80.9402	−2.98147
$738$	3.03086	0.111567
$739$	−52.9090	−1.94629	−0.973144	−	0.230196i	$-0.926063\pi$
$739$	−52.9090	−1.94629	−0.973144	+	0.230196i	$0.926063\pi$
$740$	5.97986	0.219824
$741$	47.8716	1.75861
$742$	−7.81292	−0.286821
$743$	−12.5736	−0.461281	−0.230640	−	0.973039i	$-0.574082\pi$
$743$	−12.5736	−0.461281	−0.230640	+	0.973039i	$0.574082\pi$
$744$	9.13057	0.334743
$745$	−7.04111	−0.257966
$746$	65.7358	2.40676
$747$	−0.974989	−0.0356730
$748$	6.38149	0.233330
$749$	16.2287	0.592983
$750$	56.9194	2.07840
$751$	−34.4844	−1.25835	−0.629177	−	0.777262i	$-0.716608\pi$
$751$	−34.4844	−1.25835	−0.629177	+	0.777262i	$0.716608\pi$
$752$	−5.94191	−0.216679
$753$	−19.3433	−0.704910
$754$	−67.1188	−2.44432
$755$	−37.3784	−1.36034
$756$	11.9393	0.434227
$757$	27.7134	1.00726	0.503630	−	0.863920i	$-0.331998\pi$
$757$	27.7134	1.00726	0.503630	+	0.863920i	$0.331998\pi$
$758$	7.61130	0.276455
$759$	−80.7557	−2.93125
$760$	−10.6542	−0.386470
$761$	−45.3965	−1.64562	−0.822811	−	0.568315i	$-0.807595\pi$
$761$	−45.3965	−1.64562	−0.822811	+	0.568315i	$0.807595\pi$
$762$	19.5042	0.706564
$763$	−1.36575	−0.0494435
$764$	−7.51449	−0.271865
$765$	−4.64851	−0.168067
$766$	−2.58839	−0.0935225
$767$	10.4679	0.377974
$768$	16.9714	0.612402
$769$	9.03554	0.325830	0.162915	−	0.986640i	$-0.447910\pi$
$769$	9.03554	0.325830	0.162915	+	0.986640i	$0.447910\pi$
$770$	40.9036	1.47406
$771$	−54.1136	−1.94885
$772$	−9.35049	−0.336531
$773$	33.0170	1.18754	0.593770	−	0.804635i	$-0.297639\pi$
$773$	33.0170	1.18754	0.593770	+	0.804635i	$0.297639\pi$
$774$	33.9744	1.22118
$775$	−3.09908	−0.111322
$776$	12.1664	0.436748
$777$	3.69328	0.132496
$778$	6.19130	0.221969
$779$	−1.49383	−0.0535219
$780$	63.7513	2.28266
$781$	−6.56866	−0.235045
$782$	4.79605	0.171506
$783$	28.6102	1.02244
$784$	14.6904	0.524659
$785$	−0.873692	−0.0311834
$786$	−60.9315	−2.17336
$787$	−30.3462	−1.08172	−0.540862	−	0.841111i	$-0.681902\pi$
$787$	−30.3462	−1.08172	−0.540862	+	0.841111i	$0.681902\pi$
$788$	61.2723	2.18274
$789$	−21.3879	−0.761431
$790$	16.9028	0.601375
$791$	13.5237	0.480849
$792$	24.6024	0.874210
$793$	−9.12779	−0.324137
$794$	−25.0404	−0.888651
$795$	17.7828	0.630693
$796$	17.9232	0.635269
$797$	−32.3344	−1.14534	−0.572672	−	0.819785i	$-0.694093\pi$
$797$	−32.3344	−1.14534	−0.572672	+	0.819785i	$0.694093\pi$
$798$	−35.0334	−1.24017
$799$	−0.921847	−0.0326126
$800$	7.17277	0.253596
$801$	78.3295	2.76764
$802$	−10.5666	−0.373120
$803$	38.6934	1.36546
$804$	92.3202	3.25588
$805$	16.9638	0.597895
$806$	28.7787	1.01369
$807$	23.2740	0.819283
$808$	−7.84307	−0.275918
$809$	18.0109	0.633228	0.316614	−	0.948554i	$-0.397454\pi$
$809$	18.0109	0.633228	0.316614	+	0.948554i	$0.397454\pi$
$810$	17.1253	0.601723
$811$	−28.0217	−0.983976	−0.491988	−	0.870602i	$-0.663730\pi$
$811$	−28.0217	−0.983976	−0.491988	+	0.870602i	$0.663730\pi$
$812$	27.1050	0.951199
$813$	−0.531592	−0.0186437
$814$	12.3333	0.432283
$815$	57.8394	2.02602
$816$	3.43406	0.120216
$817$	−16.7450	−0.585835
$818$	33.7269	1.17923
$819$	23.2211	0.811412
$820$	−1.98935	−0.0694711
$821$	−21.5825	−0.753234	−0.376617	−	0.926369i	$-0.622913\pi$
$821$	−21.5825	−0.753234	−0.376617	+	0.926369i	$0.622913\pi$
$822$	44.6020	1.55567
$823$	24.1561	0.842028	0.421014	−	0.907054i	$-0.361674\pi$
$823$	24.1561	0.842028	0.421014	+	0.907054i	$0.361674\pi$
$824$	0.822970	0.0286695
$825$	−14.1592	−0.492960
$826$	−7.66063	−0.266547
$827$	19.6118	0.681969	0.340984	−	0.940069i	$-0.389240\pi$
$827$	19.6118	0.681969	0.340984	+	0.940069i	$0.389240\pi$
$828$	54.3225	1.88784
$829$	−40.4379	−1.40447	−0.702234	−	0.711946i	$-0.747814\pi$
$829$	−40.4379	−1.40447	−0.702234	+	0.711946i	$0.747814\pi$
$830$	1.15970	0.0402537
$831$	−53.6114	−1.85976
$832$	−44.0498	−1.52715
$833$	2.27912	0.0789668
$834$	−4.45674	−0.154324
$835$	−15.8898	−0.549888
$836$	−64.5583	−2.23280
$837$	−12.2673	−0.424019
$838$	−69.1403	−2.38841
$839$	−21.4299	−0.739844	−0.369922	−	0.929063i	$-0.620616\pi$
$839$	−21.4299	−0.739844	−0.369922	+	0.929063i	$0.620616\pi$
$840$	−8.76303	−0.302353
$841$	35.9520	1.23973
$842$	−23.3779	−0.805657
$843$	−42.2169	−1.45403
$844$	60.4778	2.08173
$845$	6.17347	0.212374
$846$	−18.9215	−0.650533
$847$	31.5301	1.08339
$848$	−7.74763	−0.266055
$849$	−38.3486	−1.31612
$850$	0.840908	0.0288429
$851$	5.11497	0.175339
$852$	7.49220	0.256679
$853$	45.5361	1.55913	0.779563	−	0.626323i	$-0.215441\pi$
$853$	45.5361	1.55913	0.779563	+	0.626323i	$0.215441\pi$
$854$	6.67991	0.228582
$855$	47.0267	1.60828
$856$	−11.6104	−0.396834
$857$	−11.2757	−0.385171	−0.192586	−	0.981280i	$-0.561687\pi$
$857$	−11.2757	−0.385171	−0.192586	+	0.981280i	$0.561687\pi$
$858$	131.486	4.48885
$859$	−1.68851	−0.0576112	−0.0288056	−	0.999585i	$-0.509170\pi$
$859$	−1.68851	−0.0576112	−0.0288056	+	0.999585i	$0.509170\pi$
$860$	−22.2996	−0.760410
$861$	−1.22866	−0.0418726
$862$	59.4575	2.02513
$863$	−5.22137	−0.177738	−0.0888688	−	0.996043i	$-0.528325\pi$
$863$	−5.22137	−0.177738	−0.0888688	+	0.996043i	$0.528325\pi$
$864$	28.3925	0.965931
$865$	−46.6773	−1.58708
$866$	−0.278090	−0.00944989
$867$	−45.4388	−1.54318
$868$	−11.6219	−0.394473
$869$	19.2375	0.652589
$870$	−111.799	−3.79033
$871$	54.6552	1.85192
$872$	0.977087	0.0330884
$873$	−53.7012	−1.81751
$874$	−48.5192	−1.64119
$875$	−13.6082	−0.460041
$876$	−44.1336	−1.49114
$877$	27.1859	0.918002	0.459001	−	0.888436i	$-0.348207\pi$
$877$	27.1859	0.918002	0.459001	+	0.888436i	$0.348207\pi$
$878$	−62.2889	−2.10215
$879$	−25.3175	−0.853937
$880$	40.5617	1.36734
$881$	−6.43419	−0.216773	−0.108387	−	0.994109i	$-0.534568\pi$
$881$	−6.43419	−0.216773	−0.108387	+	0.994109i	$0.534568\pi$
$882$	46.7803	1.57517
$883$	34.9601	1.17650	0.588251	−	0.808678i	$-0.299817\pi$
$883$	34.9601	1.17650	0.588251	+	0.808678i	$0.299817\pi$
$884$	−4.30912	−0.144932
$885$	17.4362	0.586112
$886$	−20.6427	−0.693504
$887$	56.0114	1.88068	0.940340	−	0.340237i	$-0.110507\pi$
$887$	56.0114	1.88068	0.940340	+	0.340237i	$0.110507\pi$
$888$	−2.64225	−0.0886681
$889$	−4.66304	−0.156393
$890$	−93.1687	−3.12302
$891$	19.4908	0.652966
$892$	−19.9866	−0.669202
$893$	9.32586	0.312078
$894$	16.5639	0.553981
$895$	35.1585	1.17522
$896$	10.3902	0.347111
$897$	54.5306	1.82072
$898$	−56.5345	−1.88658
$899$	−27.8497	−0.928839
$900$	9.52456	0.317485
$901$	−1.20199	−0.0400441
$902$	−4.10299	−0.136615
$903$	−13.7727	−0.458325
$904$	−9.67518	−0.321792
$905$	−24.2238	−0.805227
$906$	87.9311	2.92131
$907$	0.730391	0.0242522	0.0121261	−	0.999926i	$-0.496140\pi$
$907$	0.730391	0.0242522	0.0121261	+	0.999926i	$0.496140\pi$
$908$	−40.8582	−1.35593
$909$	34.6184	1.14822
$910$	−27.6203	−0.915604
$911$	−20.9578	−0.694364	−0.347182	−	0.937798i	$-0.612861\pi$
$911$	−20.9578	−0.694364	−0.347182	+	0.937798i	$0.612861\pi$
$912$	−34.7407	−1.15038
$913$	1.31988	0.0436817
$914$	18.0329	0.596476
$915$	−15.2040	−0.502629
$916$	19.8424	0.655613
$917$	14.5674	0.481058
$918$	3.32863	0.109861
$919$	−27.1161	−0.894478	−0.447239	−	0.894414i	$-0.647593\pi$
$919$	−27.1161	−0.894478	−0.447239	+	0.894414i	$0.647593\pi$
$920$	−12.1363	−0.400121
$921$	21.4438	0.706597
$922$	−34.8383	−1.14734
$923$	4.43552	0.145997
$924$	−53.0987	−1.74682
$925$	0.896826	0.0294875
$926$	−17.3026	−0.568600
$927$	−3.63250	−0.119307
$928$	64.4577	2.11593
$929$	33.1168	1.08653	0.543264	−	0.839562i	$-0.317188\pi$
$929$	33.1168	1.08653	0.543264	+	0.839562i	$0.317188\pi$
$930$	47.9363	1.57189
$931$	−23.0567	−0.755653
$932$	−5.97886	−0.195844
$933$	3.73673	0.122335
$934$	−72.3745	−2.36816
$935$	6.29287	0.205799
$936$	−16.6129	−0.543010
$937$	39.9188	1.30409	0.652045	−	0.758180i	$-0.273911\pi$
$937$	39.9188	1.30409	0.652045	+	0.758180i	$0.273911\pi$
$938$	−39.9978	−1.30597
$939$	12.3643	0.403493
$940$	12.4194	0.405076
$941$	−7.20902	−0.235007	−0.117504	−	0.993072i	$-0.537489\pi$
$941$	−7.20902	−0.235007	−0.117504	+	0.993072i	$0.537489\pi$
$942$	2.05532	0.0669661
$943$	−1.70162	−0.0554124
$944$	−7.59661	−0.247249
$945$	11.7735	0.382991
$946$	−45.9925	−1.49534
$947$	−13.6196	−0.442576	−0.221288	−	0.975208i	$-0.571026\pi$
$947$	−13.6196	−0.442576	−0.221288	+	0.975208i	$0.571026\pi$
$948$	−21.9423	−0.712652
$949$	−26.1279	−0.848147
$950$	−8.50705	−0.276005
$951$	−20.3526	−0.659979
$952$	0.592317	0.0191971
$953$	44.5830	1.44419	0.722093	−	0.691796i	$-0.243180\pi$
$953$	44.5830	1.44419	0.722093	+	0.691796i	$0.243180\pi$
$954$	−24.6716	−0.798772
$955$	−7.41014	−0.239787
$956$	−29.0673	−0.940102
$957$	−127.241	−4.11312
$958$	61.6211	1.99089
$959$	−10.6634	−0.344338
$960$	−73.3730	−2.36810
$961$	−19.0588	−0.614800
$962$	−8.32814	−0.268510
$963$	51.2468	1.65141
$964$	26.3735	0.849434
$965$	−9.22064	−0.296823
$966$	−39.9066	−1.28398
$967$	−35.9540	−1.15620	−0.578101	−	0.815965i	$-0.696206\pi$
$967$	−35.9540	−1.15620	−0.578101	+	0.815965i	$0.696206\pi$
$968$	−22.5574	−0.725021
$969$	−5.38977	−0.173144
$970$	63.8747	2.05089
$971$	−50.5357	−1.62177	−0.810884	−	0.585206i	$-0.801013\pi$
$971$	−50.5357	−1.62177	−0.810884	+	0.585206i	$0.801013\pi$
$972$	−48.4569	−1.55426
$973$	1.06551	0.0341587
$974$	29.1707	0.934688
$975$	9.56105	0.306199
$976$	6.62408	0.212032
$977$	−40.4478	−1.29404	−0.647020	−	0.762473i	$-0.723985\pi$
$977$	−40.4478	−1.29404	−0.647020	+	0.762473i	$0.723985\pi$
$978$	−136.065	−4.35087
$979$	−106.038	−3.38898
$980$	−30.7050	−0.980834
$981$	−4.31276	−0.137696
$982$	−2.22669	−0.0710564
$983$	42.8009	1.36514	0.682569	−	0.730822i	$-0.260863\pi$
$983$	42.8009	1.36514	0.682569	+	0.730822i	$0.260863\pi$
$984$	0.879011	0.0280218
$985$	60.4215	1.92519
$986$	7.55678	0.240657
$987$	7.67045	0.244153
$988$	43.5932	1.38689
$989$	−19.0743	−0.606528
$990$	129.165	4.10513
$991$	−51.5768	−1.63839	−0.819196	−	0.573514i	$-0.805580\pi$
$991$	−51.5768	−1.63839	−0.819196	+	0.573514i	$0.805580\pi$
$992$	−27.6377	−0.877499
$993$	−31.5040	−0.999748
$994$	−3.24600	−0.102957
$995$	17.6743	0.560312
$996$	−1.50545	−0.0477021
$997$	26.3663	0.835030	0.417515	−	0.908670i	$-0.362901\pi$
$997$	26.3663	0.835030	0.417515	+	0.908670i	$0.362901\pi$
$998$	−68.5798	−2.17086
$999$	3.54997	0.112316

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4033.2.a.f.1.12	✓	85

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4033.2.a.f.1.12	✓	85	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:	\( N \)	\(=\)	\( 4033 = 37 \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4033.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11247 q^{2} +2.70421 q^{3} +2.46253 q^{4} +2.42834 q^{5} -5.71257 q^{6} +1.36575 q^{7} -0.977087 q^{8} +4.31276 q^{9} +O(q^{10})\)	\(q-2.11247 q^{2} +2.70421 q^{3} +2.46253 q^{4} +2.42834 q^{5} -5.71257 q^{6} +1.36575 q^{7} -0.977087 q^{8} +4.31276 q^{9} -5.12979 q^{10} -5.83835 q^{11} +6.65921 q^{12} +3.94237 q^{13} -2.88511 q^{14} +6.56674 q^{15} -2.86100 q^{16} -0.443864 q^{17} -9.11057 q^{18} +4.49035 q^{19} +5.97986 q^{20} +3.69328 q^{21} +12.3333 q^{22} +5.11497 q^{23} -2.64225 q^{24} +0.896826 q^{25} -8.32814 q^{26} +3.54997 q^{27} +3.36320 q^{28} +8.05928 q^{29} -13.8720 q^{30} -3.45560 q^{31} +7.99795 q^{32} -15.7881 q^{33} +0.937649 q^{34} +3.31650 q^{35} +10.6203 q^{36} +1.00000 q^{37} -9.48573 q^{38} +10.6610 q^{39} -2.37270 q^{40} -0.332675 q^{41} -7.80194 q^{42} -3.72912 q^{43} -14.3771 q^{44} +10.4728 q^{45} -10.8052 q^{46} +2.07687 q^{47} -7.73674 q^{48} -5.13473 q^{49} -1.89452 q^{50} -1.20030 q^{51} +9.70821 q^{52} +2.70802 q^{53} -7.49920 q^{54} -14.1775 q^{55} -1.33446 q^{56} +12.1428 q^{57} -17.0250 q^{58} +2.65523 q^{59} +16.1708 q^{60} -2.31531 q^{61} +7.29986 q^{62} +5.89015 q^{63} -11.1734 q^{64} +9.57340 q^{65} +33.3520 q^{66} +13.8635 q^{67} -1.09303 q^{68} +13.8319 q^{69} -7.00602 q^{70} +1.12509 q^{71} -4.21394 q^{72} -6.62746 q^{73} -2.11247 q^{74} +2.42521 q^{75} +11.0576 q^{76} -7.97373 q^{77} -22.5210 q^{78} -3.29503 q^{79} -6.94747 q^{80} -3.33841 q^{81} +0.702766 q^{82} -0.226071 q^{83} +9.09481 q^{84} -1.07785 q^{85} +7.87765 q^{86} +21.7940 q^{87} +5.70458 q^{88} +18.1623 q^{89} -22.1235 q^{90} +5.38429 q^{91} +12.5958 q^{92} -9.34468 q^{93} -4.38732 q^{94} +10.9041 q^{95} +21.6281 q^{96} -12.4517 q^{97} +10.8470 q^{98} -25.1794 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9}+O(q^{10}) \) 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9	\( 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9} + 9 q^{10} + 37 q^{11} + 44 q^{12} + 14 q^{13} + 26 q^{14} + 27 q^{15} + 85 q^{16} + 34 q^{17} + 3 q^{18} + 15 q^{19} + 15 q^{20} + 17 q^{21} + q^{22} + 72 q^{23} + 15 q^{24} + 85 q^{25} + 33 q^{26} + 69 q^{27} + 7 q^{28} + 19 q^{29} - 9 q^{30} + 23 q^{31} + 51 q^{32} + 32 q^{33} + 49 q^{34} + 40 q^{35} + 121 q^{36} + 85 q^{37} + 84 q^{38} + 39 q^{39} + 22 q^{40} + 55 q^{41} - 28 q^{42} + 78 q^{44} + 28 q^{45} + 17 q^{46} + 184 q^{47} + 97 q^{48} + 88 q^{49} + 26 q^{50} + 27 q^{51} + 73 q^{52} + 64 q^{53} + 31 q^{54} + 39 q^{55} + 68 q^{56} - 33 q^{57} + 28 q^{58} + 60 q^{59} - 22 q^{60} + 7 q^{61} + 70 q^{62} + 28 q^{63} + 102 q^{64} + 17 q^{65} - 15 q^{66} + 82 q^{67} + 92 q^{68} + 22 q^{69} - 41 q^{70} + 113 q^{71} - 19 q^{73} + 11 q^{74} + 45 q^{75} + 34 q^{76} + 64 q^{77} + 29 q^{78} + 23 q^{79} + 54 q^{80} + 149 q^{81} + 4 q^{82} + 100 q^{83} - 49 q^{84} - 5 q^{85} - 24 q^{86} + 65 q^{87} + 14 q^{88} + 84 q^{89} - 21 q^{90} + 32 q^{91} + 95 q^{92} + 19 q^{93} - 47 q^{94} + 102 q^{95} + 29 q^{96} + 7 q^{97} + 26 q^{98} + 107 q^{99}+O(q^{100}) \) 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9 + 9 * q^10 + 37 * q^11 + 44 * q^12 + 14 * q^13 + 26 * q^14 + 27 * q^15 + 85 * q^16 + 34 * q^17 + 3 * q^18 + 15 * q^19 + 15 * q^20 + 17 * q^21 + q^22 + 72 * q^23 + 15 * q^24 + 85 * q^25 + 33 * q^26 + 69 * q^27 + 7 * q^28 + 19 * q^29 - 9 * q^30 + 23 * q^31 + 51 * q^32 + 32 * q^33 + 49 * q^34 + 40 * q^35 + 121 * q^36 + 85 * q^37 + 84 * q^38 + 39 * q^39 + 22 * q^40 + 55 * q^41 - 28 * q^42 + 78 * q^44 + 28 * q^45 + 17 * q^46 + 184 * q^47 + 97 * q^48 + 88 * q^49 + 26 * q^50 + 27 * q^51 + 73 * q^52 + 64 * q^53 + 31 * q^54 + 39 * q^55 + 68 * q^56 - 33 * q^57 + 28 * q^58 + 60 * q^59 - 22 * q^60 + 7 * q^61 + 70 * q^62 + 28 * q^63 + 102 * q^64 + 17 * q^65 - 15 * q^66 + 82 * q^67 + 92 * q^68 + 22 * q^69 - 41 * q^70 + 113 * q^71 - 19 * q^73 + 11 * q^74 + 45 * q^75 + 34 * q^76 + 64 * q^77 + 29 * q^78 + 23 * q^79 + 54 * q^80 + 149 * q^81 + 4 * q^82 + 100 * q^83 - 49 * q^84 - 5 * q^85 - 24 * q^86 + 65 * q^87 + 14 * q^88 + 84 * q^89 - 21 * q^90 + 32 * q^91 + 95 * q^92 + 19 * q^93 - 47 * q^94 + 102 * q^95 + 29 * q^96 + 7 * q^97 + 26 * q^98 + 107 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more