LMFDB - Embedded newform 8003.2.a.c.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8003,2,Mod(1,8003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8003, 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("8003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8003 = 53 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8003.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:	$63.9042767376$
Analytic rank:	$0$
Dimension:	$172$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53688 q^{2} -2.79023 q^{3} +4.43578 q^{4} +3.33172 q^{5} +7.07848 q^{6} -4.26650 q^{7} -6.17928 q^{8} +4.78538 q^{9} +O(q^{10})$	\(q-2.53688 q^{2} -2.79023 q^{3} +4.43578 q^{4} +3.33172 q^{5} +7.07848 q^{6} -4.26650 q^{7} -6.17928 q^{8} +4.78538 q^{9} -8.45219 q^{10} -4.42179 q^{11} -12.3768 q^{12} +6.67188 q^{13} +10.8236 q^{14} -9.29627 q^{15} +6.80456 q^{16} -2.89072 q^{17} -12.1399 q^{18} -7.22734 q^{19} +14.7788 q^{20} +11.9045 q^{21} +11.2176 q^{22} -3.14906 q^{23} +17.2416 q^{24} +6.10037 q^{25} -16.9258 q^{26} -4.98161 q^{27} -18.9252 q^{28} -9.62452 q^{29} +23.5835 q^{30} +8.77694 q^{31} -4.90381 q^{32} +12.3378 q^{33} +7.33341 q^{34} -14.2148 q^{35} +21.2269 q^{36} -0.615614 q^{37} +18.3349 q^{38} -18.6161 q^{39} -20.5876 q^{40} -5.40120 q^{41} -30.2004 q^{42} -5.31325 q^{43} -19.6141 q^{44} +15.9435 q^{45} +7.98881 q^{46} +5.48086 q^{47} -18.9863 q^{48} +11.2030 q^{49} -15.4759 q^{50} +8.06576 q^{51} +29.5949 q^{52} -1.00000 q^{53} +12.6378 q^{54} -14.7322 q^{55} +26.3639 q^{56} +20.1659 q^{57} +24.4163 q^{58} -8.73398 q^{59} -41.2362 q^{60} -9.66490 q^{61} -22.2661 q^{62} -20.4168 q^{63} -1.16872 q^{64} +22.2288 q^{65} -31.2996 q^{66} -5.02390 q^{67} -12.8226 q^{68} +8.78661 q^{69} +36.0613 q^{70} -15.3294 q^{71} -29.5702 q^{72} -2.68984 q^{73} +1.56174 q^{74} -17.0214 q^{75} -32.0589 q^{76} +18.8656 q^{77} +47.2268 q^{78} -0.747072 q^{79} +22.6709 q^{80} -0.456299 q^{81} +13.7022 q^{82} +2.15515 q^{83} +52.8057 q^{84} -9.63107 q^{85} +13.4791 q^{86} +26.8546 q^{87} +27.3235 q^{88} -12.3172 q^{89} -40.4469 q^{90} -28.4656 q^{91} -13.9685 q^{92} -24.4897 q^{93} -13.9043 q^{94} -24.0795 q^{95} +13.6828 q^{96} +12.0937 q^{97} -28.4208 q^{98} -21.1599 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9}+O(q^{10}) $ 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9} + 20 q^{10} - 3 q^{11} + 66 q^{12} + 121 q^{13} + 12 q^{14} + 30 q^{15} + 212 q^{16} + 8 q^{17} + 40 q^{18} + 41 q^{19} + 64 q^{20} + 56 q^{21} + 50 q^{22} + 28 q^{23} + 30 q^{24} + 231 q^{25} + 38 q^{26} + 100 q^{27} + 80 q^{28} + 26 q^{29} + 55 q^{30} + 66 q^{31} + 65 q^{32} + 99 q^{33} + 81 q^{34} + 36 q^{35} + 212 q^{36} + 153 q^{37} + q^{38} + 20 q^{39} + 59 q^{40} + 40 q^{41} + 50 q^{42} + 39 q^{43} - 51 q^{44} + 123 q^{45} + 59 q^{46} + 29 q^{47} + 128 q^{48} + 245 q^{49} + 19 q^{50} + 36 q^{51} + 215 q^{52} - 172 q^{53} + 40 q^{54} + 40 q^{55} + 15 q^{56} + 54 q^{57} + 44 q^{58} - 54 q^{60} + 100 q^{61} - 29 q^{62} + 92 q^{63} + 253 q^{64} + 77 q^{65} + 14 q^{66} + 126 q^{67} - 27 q^{68} + 47 q^{69} + 72 q^{70} + 38 q^{71} + 65 q^{72} + 185 q^{73} + 48 q^{74} + 75 q^{75} + 38 q^{76} + 120 q^{77} + 75 q^{78} + 79 q^{79} + 43 q^{80} + 232 q^{81} + 110 q^{82} + 90 q^{83} + 158 q^{84} + 115 q^{85} + 68 q^{86} + 61 q^{87} + 15 q^{88} - 36 q^{89} - 6 q^{90} + 33 q^{91} + 139 q^{92} + 103 q^{93} - 24 q^{94} - 45 q^{95} + 34 q^{96} + 159 q^{97} - 36 q^{98} + 27 q^{99}+O(q^{100}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9 + 20 * q^10 - 3 * q^11 + 66 * q^12 + 121 * q^13 + 12 * q^14 + 30 * q^15 + 212 * q^16 + 8 * q^17 + 40 * q^18 + 41 * q^19 + 64 * q^20 + 56 * q^21 + 50 * q^22 + 28 * q^23 + 30 * q^24 + 231 * q^25 + 38 * q^26 + 100 * q^27 + 80 * q^28 + 26 * q^29 + 55 * q^30 + 66 * q^31 + 65 * q^32 + 99 * q^33 + 81 * q^34 + 36 * q^35 + 212 * q^36 + 153 * q^37 + q^38 + 20 * q^39 + 59 * q^40 + 40 * q^41 + 50 * q^42 + 39 * q^43 - 51 * q^44 + 123 * q^45 + 59 * q^46 + 29 * q^47 + 128 * q^48 + 245 * q^49 + 19 * q^50 + 36 * q^51 + 215 * q^52 - 172 * q^53 + 40 * q^54 + 40 * q^55 + 15 * q^56 + 54 * q^57 + 44 * q^58 - 54 * q^60 + 100 * q^61 - 29 * q^62 + 92 * q^63 + 253 * q^64 + 77 * q^65 + 14 * q^66 + 126 * q^67 - 27 * q^68 + 47 * q^69 + 72 * q^70 + 38 * q^71 + 65 * q^72 + 185 * q^73 + 48 * q^74 + 75 * q^75 + 38 * q^76 + 120 * q^77 + 75 * q^78 + 79 * q^79 + 43 * q^80 + 232 * q^81 + 110 * q^82 + 90 * q^83 + 158 * q^84 + 115 * q^85 + 68 * q^86 + 61 * q^87 + 15 * q^88 - 36 * q^89 - 6 * q^90 + 33 * q^91 + 139 * q^92 + 103 * q^93 - 24 * q^94 - 45 * q^95 + 34 * q^96 + 159 * q^97 - 36 * q^98 + 27 * 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 $

$2$	−2.53688	−1.79385	−0.896924	−	0.442185i	$-0.854203\pi$
$2$	−2.53688	−1.79385	−0.896924	+	0.442185i	$0.854203\pi$
$3$	−2.79023	−1.61094	−0.805470	−	0.592637i	$-0.798087\pi$
$3$	−2.79023	−1.61094	−0.805470	+	0.592637i	$0.798087\pi$
$4$	4.43578	2.21789
$5$	3.33172	1.48999	0.744996	−	0.667069i	$-0.232452\pi$
$5$	3.33172	1.48999	0.744996	+	0.667069i	$0.232452\pi$
$6$	7.07848	2.88978
$7$	−4.26650	−1.61259	−0.806293	−	0.591517i	$-0.798529\pi$
$7$	−4.26650	−1.61259	−0.806293	+	0.591517i	$0.798529\pi$
$8$	−6.17928	−2.18471
$9$	4.78538	1.59513
$10$	−8.45219	−2.67282
$11$	−4.42179	−1.33322	−0.666610	−	0.745407i	$-0.732255\pi$
$11$	−4.42179	−1.33322	−0.666610	+	0.745407i	$0.732255\pi$
$12$	−12.3768	−3.57288
$13$	6.67188	1.85045	0.925223	−	0.379424i	$-0.123878\pi$
$13$	6.67188	1.85045	0.925223	+	0.379424i	$0.123878\pi$
$14$	10.8236	2.89273
$15$	−9.29627	−2.40029
$16$	6.80456	1.70114
$17$	−2.89072	−0.701102	−0.350551	−	0.936544i	$-0.614006\pi$
$17$	−2.89072	−0.701102	−0.350551	+	0.936544i	$0.614006\pi$
$18$	−12.1399	−2.86141
$19$	−7.22734	−1.65807	−0.829033	−	0.559200i	$-0.811108\pi$
$19$	−7.22734	−1.65807	−0.829033	+	0.559200i	$0.811108\pi$
$20$	14.7788	3.30463
$21$	11.9045	2.59778
$22$	11.2176	2.39159
$23$	−3.14906	−0.656625	−0.328313	−	0.944569i	$-0.606480\pi$
$23$	−3.14906	−0.656625	−0.328313	+	0.944569i	$0.606480\pi$
$24$	17.2416	3.51943
$25$	6.10037	1.22007
$26$	−16.9258	−3.31942
$27$	−4.98161	−0.958711
$28$	−18.9252	−3.57653
$29$	−9.62452	−1.78723	−0.893614	−	0.448837i	$-0.851838\pi$
$29$	−9.62452	−1.78723	−0.893614	+	0.448837i	$0.851838\pi$
$30$	23.5835	4.30575
$31$	8.77694	1.57639	0.788193	−	0.615428i	$-0.211017\pi$
$31$	8.77694	1.57639	0.788193	+	0.615428i	$0.211017\pi$
$32$	−4.90381	−0.866880
$33$	12.3378	2.14774
$34$	7.33341	1.25767
$35$	−14.2148	−2.40274
$36$	21.2269	3.53781
$37$	−0.615614	−0.101206	−0.0506031	−	0.998719i	$-0.516114\pi$
$37$	−0.615614	−0.101206	−0.0506031	+	0.998719i	$0.516114\pi$
$38$	18.3349	2.97432
$39$	−18.6161	−2.98096
$40$	−20.5876	−3.25519
$41$	−5.40120	−0.843526	−0.421763	−	0.906706i	$-0.638588\pi$
$41$	−5.40120	−0.843526	−0.421763	+	0.906706i	$0.638588\pi$
$42$	−30.2004	−4.66002
$43$	−5.31325	−0.810263	−0.405131	−	0.914258i	$-0.632774\pi$
$43$	−5.31325	−0.810263	−0.405131	+	0.914258i	$0.632774\pi$
$44$	−19.6141	−2.95693
$45$	15.9435	2.37672
$46$	7.98881	1.17789
$47$	5.48086	0.799465	0.399733	−	0.916632i	$-0.369103\pi$
$47$	5.48086	0.799465	0.399733	+	0.916632i	$0.369103\pi$
$48$	−18.9863	−2.74043
$49$	11.2030	1.60043
$50$	−15.4759	−2.18863
$51$	8.06576	1.12943
$52$	29.5949	4.10408
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	12.6378	1.71978
$55$	−14.7322	−1.98649
$56$	26.3639	3.52302
$57$	20.1659	2.67104
$58$	24.4163	3.20601
$59$	−8.73398	−1.13707	−0.568533	−	0.822660i	$-0.692489\pi$
$59$	−8.73398	−1.13707	−0.568533	+	0.822660i	$0.692489\pi$
$60$	−41.2362	−5.32357
$61$	−9.66490	−1.23746	−0.618732	−	0.785602i	$-0.712353\pi$
$61$	−9.66490	−1.23746	−0.618732	+	0.785602i	$0.712353\pi$
$62$	−22.2661	−2.82780
$63$	−20.4168	−2.57228
$64$	−1.16872	−0.146090
$65$	22.2288	2.75715
$66$	−31.2996	−3.85271
$67$	−5.02390	−0.613767	−0.306883	−	0.951747i	$-0.599286\pi$
$67$	−5.02390	−0.613767	−0.306883	+	0.951747i	$0.599286\pi$
$68$	−12.8226	−1.55497
$69$	8.78661	1.05778
$70$	36.0613	4.31015
$71$	−15.3294	−1.81926	−0.909630	−	0.415419i	$-0.863635\pi$
$71$	−15.3294	−1.81926	−0.909630	+	0.415419i	$0.863635\pi$
$72$	−29.5702	−3.48488
$73$	−2.68984	−0.314822	−0.157411	−	0.987533i	$-0.550315\pi$
$73$	−2.68984	−0.314822	−0.157411	+	0.987533i	$0.550315\pi$
$74$	1.56174	0.181549
$75$	−17.0214	−1.96547
$76$	−32.0589	−3.67740
$77$	18.8656	2.14993
$78$	47.2268	5.34738
$79$	−0.747072	−0.0840521	−0.0420261	−	0.999117i	$-0.513381\pi$
$79$	−0.747072	−0.0840521	−0.0420261	+	0.999117i	$0.513381\pi$
$80$	22.6709	2.53468
$81$	−0.456299	−0.0506999
$82$	13.7022	1.51316
$83$	2.15515	0.236559	0.118279	−	0.992980i	$-0.462262\pi$
$83$	2.15515	0.236559	0.118279	+	0.992980i	$0.462262\pi$
$84$	52.8057	5.76158
$85$	−9.63107	−1.04464
$86$	13.4791	1.45349
$87$	26.8546	2.87912
$88$	27.3235	2.91269
$89$	−12.3172	−1.30562	−0.652812	−	0.757520i	$-0.726411\pi$
$89$	−12.3172	−1.30562	−0.652812	+	0.757520i	$0.726411\pi$
$90$	−40.4469	−4.26348
$91$	−28.4656	−2.98400
$92$	−13.9685	−1.45632
$93$	−24.4897	−2.53946
$94$	−13.9043	−1.43412
$95$	−24.0795	−2.47050
$96$	13.6828	1.39649
$97$	12.0937	1.22793	0.613964	−	0.789334i	$-0.289574\pi$
$97$	12.0937	1.22793	0.613964	+	0.789334i	$0.289574\pi$
$98$	−28.4208	−2.87093
$99$	−21.1599	−2.12665
$100$	27.0599	2.70599
$101$	11.1599	1.11046	0.555228	−	0.831698i	$-0.312631\pi$
$101$	11.1599	1.11046	0.555228	+	0.831698i	$0.312631\pi$
$102$	−20.4619	−2.02603
$103$	4.73867	0.466915	0.233458	−	0.972367i	$-0.424996\pi$
$103$	4.73867	0.466915	0.233458	+	0.972367i	$0.424996\pi$
$104$	−41.2274	−4.04268
$105$	39.6625	3.87067
$106$	2.53688	0.246404
$107$	17.3994	1.68206	0.841031	−	0.540987i	$-0.181949\pi$
$107$	17.3994	1.68206	0.841031	+	0.540987i	$0.181949\pi$
$108$	−22.0973	−2.12631
$109$	−7.22806	−0.692323	−0.346161	−	0.938175i	$-0.612515\pi$
$109$	−7.22806	−0.692323	−0.346161	+	0.938175i	$0.612515\pi$
$110$	37.3738	3.56345
$111$	1.71770	0.163037
$112$	−29.0317	−2.74323
$113$	−16.3070	−1.53403	−0.767017	−	0.641627i	$-0.778260\pi$
$113$	−16.3070	−1.53403	−0.767017	+	0.641627i	$0.778260\pi$
$114$	−51.1586	−4.79144
$115$	−10.4918	−0.978366
$116$	−42.6922	−3.96387
$117$	31.9274	2.95169
$118$	22.1571	2.03972
$119$	12.3332	1.13059
$120$	57.4442	5.24392
$121$	8.55224	0.777476
$122$	24.5187	2.21982
$123$	15.0706	1.35887
$124$	38.9326	3.49625
$125$	3.66613	0.327909
$126$	51.7951	4.61427
$127$	−7.93101	−0.703763	−0.351881	−	0.936045i	$-0.614458\pi$
$127$	−7.93101	−0.703763	−0.351881	+	0.936045i	$0.614458\pi$
$128$	12.7725	1.12894
$129$	14.8252	1.30528
$130$	−56.3920	−4.94590
$131$	−1.73019	−0.151168	−0.0755838	−	0.997139i	$-0.524082\pi$
$131$	−1.73019	−0.151168	−0.0755838	+	0.997139i	$0.524082\pi$
$132$	54.7278	4.76344
$133$	30.8355	2.67377
$134$	12.7450	1.10100
$135$	−16.5973	−1.42847
$136$	17.8626	1.53170
$137$	1.56124	0.133386	0.0666928	−	0.997774i	$-0.478755\pi$
$137$	1.56124	0.133386	0.0666928	+	0.997774i	$0.478755\pi$
$138$	−22.2906	−1.89750
$139$	−0.245290	−0.0208053	−0.0104026	−	0.999946i	$-0.503311\pi$
$139$	−0.245290	−0.0208053	−0.0104026	+	0.999946i	$0.503311\pi$
$140$	−63.0536	−5.32900
$141$	−15.2929	−1.28789
$142$	38.8888	3.26348
$143$	−29.5016	−2.46705
$144$	32.5624	2.71353
$145$	−32.0662	−2.66295
$146$	6.82381	0.564743
$147$	−31.2590	−2.57820
$148$	−2.73073	−0.224464
$149$	14.9407	1.22399	0.611995	−	0.790862i	$-0.290367\pi$
$149$	14.9407	1.22399	0.611995	+	0.790862i	$0.290367\pi$
$150$	43.1814	3.52575
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	44.6598	3.62239
$153$	−13.8332	−1.11835
$154$	−47.8597	−3.85665
$155$	29.2423	2.34880
$156$	−82.5767	−6.61143
$157$	1.42369	0.113623	0.0568115	−	0.998385i	$-0.481907\pi$
$157$	1.42369	0.113623	0.0568115	+	0.998385i	$0.481907\pi$
$158$	1.89523	0.150777
$159$	2.79023	0.221280
$160$	−16.3381	−1.29164
$161$	13.4355	1.05886
$162$	1.15758	0.0909479
$163$	16.7984	1.31575	0.657876	−	0.753126i	$-0.271455\pi$
$163$	16.7984	1.31575	0.657876	+	0.753126i	$0.271455\pi$
$164$	−23.9585	−1.87085
$165$	41.1061	3.20011
$166$	−5.46737	−0.424350
$167$	2.09222	0.161901	0.0809503	−	0.996718i	$-0.474205\pi$
$167$	2.09222	0.161901	0.0809503	+	0.996718i	$0.474205\pi$
$168$	−73.5613	−5.67538
$169$	31.5139	2.42415
$170$	24.4329	1.87392
$171$	−34.5856	−2.64482
$172$	−23.5684	−1.79707
$173$	−8.43659	−0.641422	−0.320711	−	0.947177i	$-0.603922\pi$
$173$	−8.43659	−0.641422	−0.320711	+	0.947177i	$0.603922\pi$
$174$	−68.1270	−5.16469
$175$	−26.0272	−1.96747
$176$	−30.0883	−2.26799
$177$	24.3698	1.83175
$178$	31.2474	2.34209
$179$	−1.23546	−0.0923423	−0.0461711	−	0.998934i	$-0.514702\pi$
$179$	−1.23546	−0.0923423	−0.0461711	+	0.998934i	$0.514702\pi$
$180$	70.7220	5.27131
$181$	11.5450	0.858132	0.429066	−	0.903273i	$-0.358843\pi$
$181$	11.5450	0.858132	0.429066	+	0.903273i	$0.358843\pi$
$182$	72.2138	5.35284
$183$	26.9673	1.99348
$184$	19.4589	1.43453
$185$	−2.05105	−0.150797
$186$	62.1275	4.55541
$187$	12.7822	0.934723
$188$	24.3119	1.77313
$189$	21.2540	1.54600
$190$	61.0869	4.43171
$191$	16.2448	1.17544	0.587718	−	0.809066i	$-0.300027\pi$
$191$	16.2448	1.17544	0.587718	+	0.809066i	$0.300027\pi$
$192$	3.26100	0.235342
$193$	−18.7181	−1.34736	−0.673678	−	0.739025i	$-0.735287\pi$
$193$	−18.7181	−1.34736	−0.673678	+	0.739025i	$0.735287\pi$
$194$	−30.6803	−2.20271
$195$	−62.0235	−4.44160
$196$	49.6941	3.54958
$197$	7.49512	0.534005	0.267002	−	0.963696i	$-0.413967\pi$
$197$	7.49512	0.534005	0.267002	+	0.963696i	$0.413967\pi$
$198$	53.6803	3.81489
$199$	8.87690	0.629266	0.314633	−	0.949213i	$-0.398118\pi$
$199$	8.87690	0.629266	0.314633	+	0.949213i	$0.398118\pi$
$200$	−37.6959	−2.66550
$201$	14.0178	0.988741
$202$	−28.3115	−1.99199
$203$	41.0630	2.88206
$204$	35.7779	2.50496
$205$	−17.9953	−1.25685
$206$	−12.0215	−0.837575
$207$	−15.0695	−1.04740
$208$	45.3992	3.14787
$209$	31.9578	2.21057
$210$	−100.619	−6.94338
$211$	−10.3385	−0.711729	−0.355864	−	0.934538i	$-0.615813\pi$
$211$	−10.3385	−0.711729	−0.355864	+	0.934538i	$0.615813\pi$
$212$	−4.43578	−0.304650
$213$	42.7724	2.93072
$214$	−44.1402	−3.01736
$215$	−17.7023	−1.20728
$216$	30.7828	2.09450
$217$	−37.4468	−2.54206
$218$	18.3367	1.24192
$219$	7.50527	0.507159
$220$	−65.3486	−4.40581
$221$	−19.2865	−1.29735
$222$	−4.35761	−0.292464
$223$	7.98377	0.534633	0.267317	−	0.963609i	$-0.413863\pi$
$223$	7.98377	0.534633	0.267317	+	0.963609i	$0.413863\pi$
$224$	20.9221	1.39792
$225$	29.1926	1.94617
$226$	41.3690	2.75182
$227$	−13.1155	−0.870509	−0.435255	−	0.900307i	$-0.643342\pi$
$227$	−13.1155	−0.870509	−0.435255	+	0.900307i	$0.643342\pi$
$228$	89.4516	5.92408
$229$	−15.2888	−1.01031	−0.505156	−	0.863028i	$-0.668565\pi$
$229$	−15.2888	−1.01031	−0.505156	+	0.863028i	$0.668565\pi$
$230$	26.6165	1.75504
$231$	−52.6393	−3.46341
$232$	59.4726	3.90457
$233$	−19.3898	−1.27027	−0.635134	−	0.772402i	$-0.719055\pi$
$233$	−19.3898	−1.27027	−0.635134	+	0.772402i	$0.719055\pi$
$234$	−80.9962	−5.29489
$235$	18.2607	1.19120
$236$	−38.7420	−2.52189
$237$	2.08450	0.135403
$238$	−31.2880	−2.02810
$239$	3.16945	0.205015	0.102507	−	0.994732i	$-0.467313\pi$
$239$	3.16945	0.205015	0.102507	+	0.994732i	$0.467313\pi$
$240$	−63.2570	−4.08322
$241$	−20.0435	−1.29111	−0.645557	−	0.763712i	$-0.723375\pi$
$241$	−20.0435	−1.29111	−0.645557	+	0.763712i	$0.723375\pi$
$242$	−21.6960	−1.39467
$243$	16.2180	1.04039
$244$	−42.8713	−2.74456
$245$	37.3253	2.38463
$246$	−38.2323	−2.43760
$247$	−48.2199	−3.06816
$248$	−54.2352	−3.44394
$249$	−6.01337	−0.381082
$250$	−9.30055	−0.588218
$251$	−10.8561	−0.685232	−0.342616	−	0.939476i	$-0.611313\pi$
$251$	−10.8561	−0.685232	−0.342616	+	0.939476i	$0.611313\pi$
$252$	−90.5644	−5.70502
$253$	13.9245	0.875426
$254$	20.1200	1.26244
$255$	26.8729	1.68285
$256$	−30.0650	−1.87906
$257$	−1.26039	−0.0786211	−0.0393105	−	0.999227i	$-0.512516\pi$
$257$	−1.26039	−0.0786211	−0.0393105	+	0.999227i	$0.512516\pi$
$258$	−37.6097	−2.34148
$259$	2.62652	0.163204
$260$	98.6021	6.11505
$261$	−46.0569	−2.85085
$262$	4.38929	0.271171
$263$	−9.72372	−0.599590	−0.299795	−	0.954004i	$-0.596918\pi$
$263$	−9.72372	−0.599590	−0.299795	+	0.954004i	$0.596918\pi$
$264$	−76.2388	−4.69217
$265$	−3.33172	−0.204666
$266$	−78.2259	−4.79634
$267$	34.3679	2.10328
$268$	−22.2849	−1.36127
$269$	−1.87130	−0.114095	−0.0570476	−	0.998371i	$-0.518169\pi$
$269$	−1.87130	−0.114095	−0.0570476	+	0.998371i	$0.518169\pi$
$270$	42.1055	2.56246
$271$	0.909353	0.0552392	0.0276196	−	0.999619i	$-0.491207\pi$
$271$	0.909353	0.0552392	0.0276196	+	0.999619i	$0.491207\pi$
$272$	−19.6701	−1.19267
$273$	79.4254	4.80704
$274$	−3.96068	−0.239274
$275$	−26.9746	−1.62663
$276$	38.9754	2.34604
$277$	−18.9285	−1.13730	−0.568652	−	0.822578i	$-0.692535\pi$
$277$	−18.9285	−1.13730	−0.568652	+	0.822578i	$0.692535\pi$
$278$	0.622273	0.0373214
$279$	42.0010	2.51453
$280$	87.8372	5.24928
$281$	−25.1585	−1.50083	−0.750415	−	0.660967i	$-0.770146\pi$
$281$	−25.1585	−1.50083	−0.750415	+	0.660967i	$0.770146\pi$
$282$	38.7962	2.31028
$283$	11.5483	0.686473	0.343237	−	0.939249i	$-0.388477\pi$
$283$	11.5483	0.686473	0.343237	+	0.939249i	$0.388477\pi$
$284$	−67.9976	−4.03492
$285$	67.1873	3.97983
$286$	74.8422	4.42551
$287$	23.0442	1.36026
$288$	−23.4666	−1.38278
$289$	−8.64375	−0.508456
$290$	81.3482	4.77693
$291$	−33.7442	−1.97812
$292$	−11.9315	−0.698240
$293$	1.59756	0.0933307	0.0466653	−	0.998911i	$-0.485141\pi$
$293$	1.59756	0.0933307	0.0466653	+	0.998911i	$0.485141\pi$
$294$	79.3004	4.62489
$295$	−29.0992	−1.69422
$296$	3.80405	0.221106
$297$	22.0276	1.27817
$298$	−37.9028	−2.19565
$299$	−21.0102	−1.21505
$300$	−75.5033	−4.35918
$301$	22.6690	1.30662
$302$	−2.53688	−0.145981
$303$	−31.1388	−1.78888
$304$	−49.1789	−2.82060
$305$	−32.2008	−1.84381
$306$	35.0931	2.00614
$307$	2.22523	0.127000	0.0635002	−	0.997982i	$-0.479774\pi$
$307$	2.22523	0.127000	0.0635002	+	0.997982i	$0.479774\pi$
$308$	83.6835	4.76831
$309$	−13.2220	−0.752172
$310$	−74.1844	−4.21339
$311$	−30.8442	−1.74902	−0.874508	−	0.485010i	$-0.838816\pi$
$311$	−30.8442	−1.74902	−0.874508	+	0.485010i	$0.838816\pi$
$312$	115.034	6.51251
$313$	−26.0160	−1.47051	−0.735256	−	0.677789i	$-0.762938\pi$
$313$	−26.0160	−1.47051	−0.735256	+	0.677789i	$0.762938\pi$
$314$	−3.61174	−0.203822
$315$	−68.0231	−3.83267
$316$	−3.31384	−0.186418
$317$	−13.6480	−0.766548	−0.383274	−	0.923635i	$-0.625204\pi$
$317$	−13.6480	−0.766548	−0.383274	+	0.923635i	$0.625204\pi$
$318$	−7.07848	−0.396942
$319$	42.5576	2.38277
$320$	−3.89385	−0.217673
$321$	−48.5482	−2.70970
$322$	−34.0842	−1.89944
$323$	20.8922	1.16247
$324$	−2.02404	−0.112447
$325$	40.7009	2.25768
$326$	−42.6156	−2.36026
$327$	20.1679	1.11529
$328$	33.3755	1.84285
$329$	−23.3841	−1.28921
$330$	−104.281	−5.74051
$331$	4.00337	0.220045	0.110023	−	0.993929i	$-0.464908\pi$
$331$	4.00337	0.220045	0.110023	+	0.993929i	$0.464908\pi$
$332$	9.55978	0.524661
$333$	−2.94594	−0.161437
$334$	−5.30771	−0.290425
$335$	−16.7382	−0.914507
$336$	81.0049	4.41918
$337$	15.2092	0.828496	0.414248	−	0.910164i	$-0.364045\pi$
$337$	15.2092	0.828496	0.414248	+	0.910164i	$0.364045\pi$
$338$	−79.9472	−4.34855
$339$	45.5003	2.47124
$340$	−42.7213	−2.31689
$341$	−38.8098	−2.10167
$342$	87.7395	4.74441
$343$	−17.9322	−0.968247
$344$	32.8321	1.77019
$345$	29.2745	1.57609
$346$	21.4026	1.15061
$347$	24.8818	1.33572	0.667862	−	0.744285i	$-0.267210\pi$
$347$	24.8818	1.33572	0.667862	+	0.744285i	$0.267210\pi$
$348$	119.121	6.38556
$349$	−10.4576	−0.559784	−0.279892	−	0.960032i	$-0.590299\pi$
$349$	−10.4576	−0.559784	−0.279892	+	0.960032i	$0.590299\pi$
$350$	66.0281	3.52935
$351$	−33.2367	−1.77404
$352$	21.6836	1.15574
$353$	10.6149	0.564977	0.282488	−	0.959271i	$-0.408840\pi$
$353$	10.6149	0.564977	0.282488	+	0.959271i	$0.408840\pi$
$354$	−61.8233	−3.28587
$355$	−51.0732	−2.71068
$356$	−54.6365	−2.89573
$357$	−34.4126	−1.82131
$358$	3.13421	0.165648
$359$	18.3411	0.968006	0.484003	−	0.875066i	$-0.339182\pi$
$359$	18.3411	0.968006	0.484003	+	0.875066i	$0.339182\pi$
$360$	−98.5196	−5.19244
$361$	33.2345	1.74918
$362$	−29.2883	−1.53936
$363$	−23.8627	−1.25247
$364$	−126.267	−6.61818
$365$	−8.96180	−0.469082
$366$	−68.4129	−3.57600
$367$	−31.9156	−1.66598	−0.832991	−	0.553286i	$-0.813373\pi$
$367$	−31.9156	−1.66598	−0.832991	+	0.553286i	$0.813373\pi$
$368$	−21.4280	−1.11701
$369$	−25.8468	−1.34553
$370$	5.20329	0.270506
$371$	4.26650	0.221506
$372$	−108.631	−5.63224
$373$	1.91673	0.0992448	0.0496224	−	0.998768i	$-0.484198\pi$
$373$	1.91673	0.0992448	0.0496224	+	0.998768i	$0.484198\pi$
$374$	−32.4268	−1.67675
$375$	−10.2293	−0.528241
$376$	−33.8678	−1.74660
$377$	−64.2136	−3.30717
$378$	−53.9190	−2.77329
$379$	−31.9891	−1.64317	−0.821585	−	0.570086i	$-0.806910\pi$
$379$	−31.9891	−1.64317	−0.821585	+	0.570086i	$0.806910\pi$
$380$	−106.811	−5.47930
$381$	22.1293	1.13372
$382$	−41.2113	−2.10855
$383$	27.4791	1.40412	0.702058	−	0.712120i	$-0.252265\pi$
$383$	27.4791	1.40412	0.702058	+	0.712120i	$0.252265\pi$
$384$	−35.6383	−1.81866
$385$	62.8548	3.20338
$386$	47.4856	2.41695
$387$	−25.4259	−1.29247
$388$	53.6449	2.72341
$389$	32.4682	1.64620	0.823102	−	0.567893i	$-0.192241\pi$
$389$	32.4682	1.64620	0.823102	+	0.567893i	$0.192241\pi$
$390$	157.346	7.96755
$391$	9.10305	0.460361
$392$	−69.2266	−3.49647
$393$	4.82763	0.243522
$394$	−19.0142	−0.957923
$395$	−2.48904	−0.125237
$396$	−93.8607	−4.71668
$397$	24.3180	1.22049	0.610244	−	0.792214i	$-0.291071\pi$
$397$	24.3180	1.22049	0.610244	+	0.792214i	$0.291071\pi$
$398$	−22.5196	−1.12881
$399$	−86.0380	−4.30729
$400$	41.5103	2.07552
$401$	17.7472	0.886253	0.443127	−	0.896459i	$-0.353869\pi$
$401$	17.7472	0.886253	0.443127	+	0.896459i	$0.353869\pi$
$402$	−35.5616	−1.77365
$403$	58.5587	2.91702
$404$	49.5030	2.46287
$405$	−1.52026	−0.0755424
$406$	−104.172	−5.16997
$407$	2.72212	0.134930
$408$	−49.8406	−2.46748
$409$	−4.29402	−0.212325	−0.106163	−	0.994349i	$-0.533856\pi$
$409$	−4.29402	−0.212325	−0.106163	+	0.994349i	$0.533856\pi$
$410$	45.6520	2.25459
$411$	−4.35621	−0.214876
$412$	21.0197	1.03557
$413$	37.2635	1.83362
$414$	38.2294	1.87887
$415$	7.18037	0.352471
$416$	−32.7176	−1.60411
$417$	0.684416	0.0335160
$418$	−81.0732	−3.96542
$419$	−12.3248	−0.602108	−0.301054	−	0.953607i	$-0.597338\pi$
$419$	−12.3248	−0.602108	−0.301054	+	0.953607i	$0.597338\pi$
$420$	175.934	8.58470
$421$	23.5935	1.14988	0.574939	−	0.818196i	$-0.305026\pi$
$421$	23.5935	1.14988	0.574939	+	0.818196i	$0.305026\pi$
$422$	26.2274	1.27673
$423$	26.2280	1.27525
$424$	6.17928	0.300092
$425$	−17.6345	−0.855397
$426$	−108.509	−5.25726
$427$	41.2353	1.99552
$428$	77.1797	3.73062
$429$	82.3163	3.97427
$430$	44.9086	2.16568
$431$	−8.18146	−0.394087	−0.197044	−	0.980395i	$-0.563134\pi$
$431$	−8.18146	−0.394087	−0.197044	+	0.980395i	$0.563134\pi$
$432$	−33.8977	−1.63090
$433$	11.8709	0.570477	0.285239	−	0.958457i	$-0.407927\pi$
$433$	11.8709	0.570477	0.285239	+	0.958457i	$0.407927\pi$
$434$	94.9982	4.56006
$435$	89.4721	4.28986
$436$	−32.0621	−1.53549
$437$	22.7594	1.08873
$438$	−19.0400	−0.909766
$439$	38.5659	1.84065	0.920325	−	0.391154i	$-0.127924\pi$
$439$	38.5659	1.84065	0.920325	+	0.391154i	$0.127924\pi$
$440$	91.0343	4.33989
$441$	53.6107	2.55289
$442$	48.9276	2.32725
$443$	2.39972	0.114014	0.0570071	−	0.998374i	$-0.481844\pi$
$443$	2.39972	0.114014	0.0570071	+	0.998374i	$0.481844\pi$
$444$	7.61935	0.361598
$445$	−41.0376	−1.94537
$446$	−20.2539	−0.959050
$447$	−41.6880	−1.97177
$448$	4.98635	0.235583
$449$	−6.43202	−0.303546	−0.151773	−	0.988415i	$-0.548498\pi$
$449$	−6.43202	−0.303546	−0.151773	+	0.988415i	$0.548498\pi$
$450$	−74.0582	−3.49114
$451$	23.8830	1.12461
$452$	−72.3342	−3.40232
$453$	−2.79023	−0.131096
$454$	33.2726	1.56156
$455$	−94.8393	−4.44614
$456$	−124.611	−5.83544
$457$	−13.4364	−0.628531	−0.314265	−	0.949335i	$-0.601758\pi$
$457$	−13.4364	−0.628531	−0.314265	+	0.949335i	$0.601758\pi$
$458$	38.7859	1.81235
$459$	14.4004	0.672154
$460$	−46.5393	−2.16991
$461$	−35.2743	−1.64289	−0.821444	−	0.570289i	$-0.806831\pi$
$461$	−35.2743	−1.64289	−0.821444	+	0.570289i	$0.806831\pi$
$462$	133.540	6.21283
$463$	2.76050	0.128291	0.0641457	−	0.997941i	$-0.479568\pi$
$463$	2.76050	0.128291	0.0641457	+	0.997941i	$0.479568\pi$
$464$	−65.4906	−3.04032
$465$	−81.5928	−3.78378
$466$	49.1897	2.27867
$467$	−35.8710	−1.65991	−0.829956	−	0.557829i	$-0.811635\pi$
$467$	−35.8710	−1.65991	−0.829956	+	0.557829i	$0.811635\pi$
$468$	141.623	6.54652
$469$	21.4345	0.989752
$470$	−46.3253	−2.13682
$471$	−3.97243	−0.183040
$472$	53.9697	2.48416
$473$	23.4941	1.08026
$474$	−5.28814	−0.242892
$475$	−44.0895	−2.02296
$476$	54.7075	2.50752
$477$	−4.78538	−0.219107
$478$	−8.04053	−0.367765
$479$	−7.79711	−0.356259	−0.178129	−	0.984007i	$-0.557005\pi$
$479$	−7.79711	−0.356259	−0.178129	+	0.984007i	$0.557005\pi$
$480$	45.5871	2.08076
$481$	−4.10730	−0.187277
$482$	50.8480	2.31606
$483$	−37.4881	−1.70577
$484$	37.9358	1.72435
$485$	40.2928	1.82960
$486$	−41.1432	−1.86629
$487$	22.3134	1.01112	0.505558	−	0.862793i	$-0.331287\pi$
$487$	22.3134	1.01112	0.505558	+	0.862793i	$0.331287\pi$
$488$	59.7221	2.70349
$489$	−46.8714	−2.11960
$490$	−94.6900	−4.27766
$491$	0.762988	0.0344332	0.0172166	−	0.999852i	$-0.494520\pi$
$491$	0.762988	0.0344332	0.0172166	+	0.999852i	$0.494520\pi$
$492$	66.8497	3.01382
$493$	27.8218	1.25303
$494$	122.328	5.50381
$495$	−70.4990	−3.16870
$496$	59.7232	2.68165
$497$	65.4027	2.93371
$498$	15.2552	0.683603
$499$	−4.25179	−0.190336	−0.0951682	−	0.995461i	$-0.530339\pi$
$499$	−4.25179	−0.190336	−0.0951682	+	0.995461i	$0.530339\pi$
$500$	16.2621	0.727265
$501$	−5.83776	−0.260812
$502$	27.5407	1.22920
$503$	−31.9438	−1.42430	−0.712152	−	0.702026i	$-0.752279\pi$
$503$	−31.9438	−1.42430	−0.712152	+	0.702026i	$0.752279\pi$
$504$	126.161	5.61967
$505$	37.1818	1.65457
$506$	−35.3248	−1.57038
$507$	−87.9311	−3.90516
$508$	−35.1802	−1.56087
$509$	27.2916	1.20968	0.604839	−	0.796348i	$-0.293237\pi$
$509$	27.2916	1.20968	0.604839	+	0.796348i	$0.293237\pi$
$510$	−68.1734	−3.01877
$511$	11.4762	0.507677
$512$	50.7263	2.24181
$513$	36.0038	1.58961
$514$	3.19747	0.141034
$515$	15.7879	0.695700
$516$	65.7612	2.89497
$517$	−24.2352	−1.06586
$518$	−6.66317	−0.292763
$519$	23.5400	1.03329
$520$	−137.358	−6.02356
$521$	−29.1086	−1.27527	−0.637636	−	0.770338i	$-0.720088\pi$
$521$	−29.1086	−1.27527	−0.637636	+	0.770338i	$0.720088\pi$
$522$	116.841	5.11399
$523$	−18.0367	−0.788688	−0.394344	−	0.918963i	$-0.629028\pi$
$523$	−18.0367	−0.788688	−0.394344	+	0.918963i	$0.629028\pi$
$524$	−7.67474	−0.335273
$525$	72.6219	3.16948
$526$	24.6679	1.07557
$527$	−25.3717	−1.10521
$528$	83.9534	3.65360
$529$	−13.0834	−0.568844
$530$	8.45219	0.367140
$531$	−41.7954	−1.81376
$532$	136.779	5.93013
$533$	−36.0361	−1.56090
$534$	−87.1874	−3.77297
$535$	57.9699	2.50626
$536$	31.0441	1.34090
$537$	3.44720	0.148758
$538$	4.74727	0.204669
$539$	−49.5374	−2.13373
$540$	−73.6221	−3.16819
$541$	−15.7832	−0.678572	−0.339286	−	0.940683i	$-0.610185\pi$
$541$	−15.7832	−0.678572	−0.339286	+	0.940683i	$0.610185\pi$
$542$	−2.30692	−0.0990908
$543$	−32.2132	−1.38240
$544$	14.1755	0.607771
$545$	−24.0819	−1.03155
$546$	−201.493	−8.62310
$547$	13.5489	0.579309	0.289655	−	0.957131i	$-0.406460\pi$
$547$	13.5489	0.579309	0.289655	+	0.957131i	$0.406460\pi$
$548$	6.92531	0.295835
$549$	−46.2502	−1.97391
$550$	68.4313	2.91792
$551$	69.5597	2.96334
$552$	−54.2949	−2.31094
$553$	3.18738	0.135541
$554$	48.0194	2.04015
$555$	5.72291	0.242924
$556$	−1.08805	−0.0461437
$557$	−36.3645	−1.54081	−0.770407	−	0.637552i	$-0.779947\pi$
$557$	−36.3645	−1.54081	−0.770407	+	0.637552i	$0.779947\pi$
$558$	−106.552	−4.51069
$559$	−35.4493	−1.49935
$560$	−96.7254	−4.08739
$561$	−35.6651	−1.50578
$562$	63.8242	2.69226
$563$	35.1359	1.48080	0.740400	−	0.672166i	$-0.234636\pi$
$563$	35.1359	1.48080	0.740400	+	0.672166i	$0.234636\pi$
$564$	−67.8357	−2.85640
$565$	−54.3304	−2.28570
$566$	−29.2966	−1.23143
$567$	1.94680	0.0817579
$568$	94.7244	3.97455
$569$	13.5574	0.568355	0.284178	−	0.958772i	$-0.408279\pi$
$569$	13.5574	0.568355	0.284178	+	0.958772i	$0.408279\pi$
$570$	−170.446	−7.13921
$571$	21.6231	0.904900	0.452450	−	0.891790i	$-0.350550\pi$
$571$	21.6231	0.904900	0.452450	+	0.891790i	$0.350550\pi$
$572$	−130.863	−5.47164
$573$	−45.3268	−1.89356
$574$	−58.4605	−2.44009
$575$	−19.2105	−0.801131
$576$	−5.59277	−0.233032
$577$	−10.9765	−0.456959	−0.228479	−	0.973549i	$-0.573375\pi$
$577$	−10.9765	−0.456959	−0.228479	+	0.973549i	$0.573375\pi$
$578$	21.9282	0.912092
$579$	52.2277	2.17051
$580$	−142.239	−5.90613
$581$	−9.19496	−0.381471
$582$	85.6050	3.54844
$583$	4.42179	0.183132
$584$	16.6213	0.687793
$585$	106.373	4.39800
$586$	−4.05283	−0.167421
$587$	1.19047	0.0491361	0.0245681	−	0.999698i	$-0.492179\pi$
$587$	1.19047	0.0491361	0.0245681	+	0.999698i	$0.492179\pi$
$588$	−138.658	−5.71815
$589$	−63.4340	−2.61375
$590$	73.8212	3.03917
$591$	−20.9131	−0.860250
$592$	−4.18898	−0.172166
$593$	−28.8636	−1.18529	−0.592643	−	0.805465i	$-0.701915\pi$
$593$	−28.8636	−1.18529	−0.592643	+	0.805465i	$0.701915\pi$
$594$	−55.8815	−2.29285
$595$	41.0910	1.68456
$596$	66.2736	2.71467
$597$	−24.7686	−1.01371
$598$	53.3003	2.17961
$599$	45.0409	1.84032	0.920161	−	0.391540i	$-0.128058\pi$
$599$	45.0409	1.84032	0.920161	+	0.391540i	$0.128058\pi$
$600$	105.180	4.29396
$601$	−4.05588	−0.165443	−0.0827215	−	0.996573i	$-0.526361\pi$
$601$	−4.05588	−0.165443	−0.0827215	+	0.996573i	$0.526361\pi$
$602$	−57.5085	−2.34387
$603$	−24.0413	−0.979035
$604$	4.43578	0.180489
$605$	28.4937	1.15843
$606$	78.9954	3.20897
$607$	−20.2712	−0.822783	−0.411392	−	0.911459i	$-0.634957\pi$
$607$	−20.2712	−0.822783	−0.411392	+	0.911459i	$0.634957\pi$
$608$	35.4415	1.43734
$609$	−114.575	−4.64282
$610$	81.6896	3.30751
$611$	36.5676	1.47937
$612$	−61.3609	−2.48037
$613$	23.9420	0.967009	0.483504	−	0.875342i	$-0.339364\pi$
$613$	23.9420	0.967009	0.483504	+	0.875342i	$0.339364\pi$
$614$	−5.64514	−0.227819
$615$	50.2110	2.02470
$616$	−116.576	−4.69697
$617$	−2.09864	−0.0844881	−0.0422441	−	0.999107i	$-0.513451\pi$
$617$	−2.09864	−0.0844881	−0.0422441	+	0.999107i	$0.513451\pi$
$618$	33.5426	1.34928
$619$	−30.8519	−1.24004	−0.620022	−	0.784584i	$-0.712876\pi$
$619$	−30.8519	−1.24004	−0.620022	+	0.784584i	$0.712876\pi$
$620$	129.712	5.20938
$621$	15.6874	0.629514
$622$	78.2483	3.13747
$623$	52.5515	2.10543
$624$	−126.674	−5.07102
$625$	−18.2873	−0.731493
$626$	65.9996	2.63788
$627$	−89.1696	−3.56109
$628$	6.31518	0.252003
$629$	1.77957	0.0709559
$630$	172.567	6.87522
$631$	21.0748	0.838976	0.419488	−	0.907761i	$-0.362210\pi$
$631$	21.0748	0.838976	0.419488	+	0.907761i	$0.362210\pi$
$632$	4.61637	0.183629
$633$	28.8467	1.14655
$634$	34.6234	1.37507
$635$	−26.4239	−1.04860
$636$	12.3768	0.490773
$637$	74.7452	2.96151
$638$	−107.964	−4.27432
$639$	−73.3568	−2.90195
$640$	42.5545	1.68212
$641$	10.2635	0.405386	0.202693	−	0.979242i	$-0.435031\pi$
$641$	10.2635	0.405386	0.202693	+	0.979242i	$0.435031\pi$
$642$	123.161	4.86079
$643$	−15.6338	−0.616538	−0.308269	−	0.951299i	$-0.599750\pi$
$643$	−15.6338	−0.616538	−0.308269	+	0.951299i	$0.599750\pi$
$644$	59.5968	2.34844
$645$	49.3934	1.94486
$646$	−53.0011	−2.08530
$647$	23.3802	0.919170	0.459585	−	0.888134i	$-0.347998\pi$
$647$	23.3802	0.919170	0.459585	+	0.888134i	$0.347998\pi$
$648$	2.81960	0.110764
$649$	38.6198	1.51596
$650$	−103.253	−4.04993
$651$	104.485	4.09510
$652$	74.5139	2.91819
$653$	−3.16171	−0.123727	−0.0618637	−	0.998085i	$-0.519704\pi$
$653$	−3.16171	−0.123727	−0.0618637	+	0.998085i	$0.519704\pi$
$654$	−51.1637	−2.00066
$655$	−5.76452	−0.225238
$656$	−36.7528	−1.43495
$657$	−12.8719	−0.502181
$658$	59.3227	2.31264
$659$	−42.7875	−1.66676	−0.833382	−	0.552698i	$-0.813598\pi$
$659$	−42.7875	−1.66676	−0.833382	+	0.552698i	$0.813598\pi$
$660$	182.338	7.09748
$661$	34.9870	1.36084	0.680418	−	0.732824i	$-0.261798\pi$
$661$	34.9870	1.36084	0.680418	+	0.732824i	$0.261798\pi$
$662$	−10.1561	−0.394727
$663$	53.8138	2.08995
$664$	−13.3173	−0.516811
$665$	102.735	3.98390
$666$	7.47352	0.289593
$667$	30.3082	1.17354
$668$	9.28060	0.359077
$669$	−22.2766	−0.861261
$670$	42.4629	1.64049
$671$	42.7362	1.64981
$672$	−58.3775	−2.25196
$673$	−4.83381	−0.186330	−0.0931648	−	0.995651i	$-0.529698\pi$
$673$	−4.83381	−0.186330	−0.0931648	+	0.995651i	$0.529698\pi$
$674$	−38.5839	−1.48620
$675$	−30.3897	−1.16970
$676$	139.789	5.37649
$677$	−26.0103	−0.999657	−0.499828	−	0.866124i	$-0.666604\pi$
$677$	−26.0103	−0.999657	−0.499828	+	0.866124i	$0.666604\pi$
$678$	−115.429	−4.43302
$679$	−51.5977	−1.98014
$680$	59.5131	2.28222
$681$	36.5954	1.40234
$682$	98.4560	3.77007
$683$	−3.43112	−0.131288	−0.0656440	−	0.997843i	$-0.520910\pi$
$683$	−3.43112	−0.131288	−0.0656440	+	0.997843i	$0.520910\pi$
$684$	−153.414	−5.86592
$685$	5.20162	0.198744
$686$	45.4919	1.73689
$687$	42.6593	1.62755
$688$	−36.1543	−1.37837
$689$	−6.67188	−0.254178
$690$	−74.2661	−2.82726
$691$	3.54444	0.134837	0.0674185	−	0.997725i	$-0.478524\pi$
$691$	3.54444	0.134837	0.0674185	+	0.997725i	$0.478524\pi$
$692$	−37.4228	−1.42260
$693$	90.2789	3.42941
$694$	−63.1222	−2.39609
$695$	−0.817239	−0.0309996
$696$	−165.942	−6.29002
$697$	15.6133	0.591397
$698$	26.5298	1.00417
$699$	54.1020	2.04633
$700$	−115.451	−4.36364
$701$	14.3919	0.543576	0.271788	−	0.962357i	$-0.412385\pi$
$701$	14.3919	0.543576	0.271788	+	0.962357i	$0.412385\pi$
$702$	84.3176	3.18236
$703$	4.44925	0.167807
$704$	5.16784	0.194770
$705$	−50.9515	−1.91895
$706$	−26.9289	−1.01348
$707$	−47.6139	−1.79070
$708$	108.099	4.06261
$709$	−11.6729	−0.438385	−0.219193	−	0.975682i	$-0.570342\pi$
$709$	−11.6729	−0.438385	−0.219193	+	0.975682i	$0.570342\pi$
$710$	129.567	4.86255
$711$	−3.57502	−0.134074
$712$	76.1117	2.85241
$713$	−27.6391	−1.03509
$714$	87.3007	3.26715
$715$	−98.2913	−3.67588
$716$	−5.48020	−0.204805
$717$	−8.84349	−0.330266
$718$	−46.5292	−1.73645
$719$	−15.3409	−0.572118	−0.286059	−	0.958212i	$-0.592345\pi$
$719$	−15.3409	−0.572118	−0.286059	+	0.958212i	$0.592345\pi$
$720$	108.489	4.04314
$721$	−20.2175	−0.752941
$722$	−84.3120	−3.13777
$723$	55.9259	2.07991
$724$	51.2110	1.90324
$725$	−58.7131	−2.18055
$726$	60.5369	2.24673
$727$	12.0691	0.447618	0.223809	−	0.974633i	$-0.428151\pi$
$727$	12.0691	0.447618	0.223809	+	0.974633i	$0.428151\pi$
$728$	175.897	6.51916
$729$	−43.8831	−1.62530
$730$	22.7350	0.841462
$731$	15.3591	0.568077
$732$	119.621	4.42131
$733$	49.0519	1.81177	0.905886	−	0.423522i	$-0.139206\pi$
$733$	49.0519	1.81177	0.905886	+	0.423522i	$0.139206\pi$
$734$	80.9662	2.98852
$735$	−104.146	−3.84149
$736$	15.4424	0.569215
$737$	22.2146	0.818286
$738$	65.5702	2.41367
$739$	11.7857	0.433543	0.216772	−	0.976222i	$-0.430447\pi$
$739$	11.7857	0.433543	0.216772	+	0.976222i	$0.430447\pi$
$740$	−9.09802	−0.334450
$741$	134.545	4.94262
$742$	−10.8236	−0.397347
$743$	26.9068	0.987116	0.493558	−	0.869713i	$-0.335696\pi$
$743$	26.9068	0.987116	0.493558	+	0.869713i	$0.335696\pi$
$744$	151.329	5.54798
$745$	49.7783	1.82373
$746$	−4.86253	−0.178030
$747$	10.3132	0.377341
$748$	56.6988	2.07311
$749$	−74.2344	−2.71247
$750$	25.9507	0.947584
$751$	25.0729	0.914924	0.457462	−	0.889229i	$-0.348759\pi$
$751$	25.0729	0.914924	0.457462	+	0.889229i	$0.348759\pi$
$752$	37.2948	1.36000
$753$	30.2911	1.10387
$754$	162.902	5.93255
$755$	3.33172	0.121254
$756$	94.2782	3.42886
$757$	39.9659	1.45258	0.726292	−	0.687386i	$-0.241242\pi$
$757$	39.9659	1.45258	0.726292	+	0.687386i	$0.241242\pi$
$758$	81.1527	2.94760
$759$	−38.8525	−1.41026
$760$	148.794	5.39732
$761$	12.4607	0.451699	0.225850	−	0.974162i	$-0.427484\pi$
$761$	12.4607	0.451699	0.225850	+	0.974162i	$0.427484\pi$
$762$	−56.1395	−2.03372
$763$	30.8385	1.11643
$764$	72.0585	2.60698
$765$	−46.0883	−1.66633
$766$	−69.7112	−2.51877
$767$	−58.2720	−2.10408
$768$	83.8882	3.02705
$769$	28.3942	1.02392	0.511960	−	0.859009i	$-0.328920\pi$
$769$	28.3942	1.02392	0.511960	+	0.859009i	$0.328920\pi$
$770$	−159.455	−5.74637
$771$	3.51678	0.126654
$772$	−83.0292	−2.98829
$773$	9.31668	0.335098	0.167549	−	0.985864i	$-0.446415\pi$
$773$	9.31668	0.335098	0.167549	+	0.985864i	$0.446415\pi$
$774$	64.5025	2.31850
$775$	53.5426	1.92331
$776$	−74.7303	−2.68266
$777$	−7.32858	−0.262911
$778$	−82.3681	−2.95304
$779$	39.0363	1.39862
$780$	−275.123	−9.85097
$781$	67.7832	2.42547
$782$	−23.0934	−0.825818
$783$	47.9456	1.71344
$784$	76.2316	2.72256
$785$	4.74335	0.169297
$786$	−12.2471	−0.436841
$787$	45.2900	1.61442	0.807208	−	0.590268i	$-0.200978\pi$
$787$	45.2900	1.61442	0.807208	+	0.590268i	$0.200978\pi$
$788$	33.2467	1.18436
$789$	27.1314	0.965904
$790$	6.31439	0.224656
$791$	69.5738	2.47376
$792$	130.753	4.64611
$793$	−64.4830	−2.28986
$794$	−61.6920	−2.18937
$795$	9.29627	0.329705
$796$	39.3759	1.39564
$797$	−24.1518	−0.855499	−0.427749	−	0.903897i	$-0.640693\pi$
$797$	−24.1518	−0.855499	−0.427749	+	0.903897i	$0.640693\pi$
$798$	218.268	7.72661
$799$	−15.8436	−0.560507
$800$	−29.9151	−1.05766
$801$	−58.9426	−2.08264
$802$	−45.0226	−1.58980
$803$	11.8939	0.419727
$804$	62.1800	2.19292
$805$	44.7633	1.57770
$806$	−148.557	−5.23268
$807$	5.22135	0.183800
$808$	−68.9604	−2.42602
$809$	−5.47453	−0.192474	−0.0962372	−	0.995358i	$-0.530681\pi$
$809$	−5.47453	−0.192474	−0.0962372	+	0.995358i	$0.530681\pi$
$810$	3.85673	0.135512
$811$	−34.1169	−1.19801	−0.599004	−	0.800746i	$-0.704436\pi$
$811$	−34.1169	−1.19801	−0.599004	+	0.800746i	$0.704436\pi$
$812$	182.146	6.39208
$813$	−2.53730	−0.0889871
$814$	−6.90569	−0.242044
$815$	55.9676	1.96046
$816$	54.8840	1.92132
$817$	38.4007	1.34347
$818$	10.8934	0.380879
$819$	−136.218	−4.75986
$820$	−79.8231	−2.78754
$821$	1.64495	0.0574091	0.0287046	−	0.999588i	$-0.490862\pi$
$821$	1.64495	0.0574091	0.0287046	+	0.999588i	$0.490862\pi$
$822$	11.0512	0.385455
$823$	6.81136	0.237429	0.118715	−	0.992928i	$-0.462123\pi$
$823$	6.81136	0.237429	0.118715	+	0.992928i	$0.462123\pi$
$824$	−29.2816	−1.02007
$825$	75.2652	2.62040
$826$	−94.5332	−3.28923
$827$	−49.1504	−1.70913	−0.854563	−	0.519348i	$-0.826175\pi$
$827$	−49.1504	−1.70913	−0.854563	+	0.519348i	$0.826175\pi$
$828$	−66.8447	−2.32301
$829$	−45.9876	−1.59721	−0.798607	−	0.601853i	$-0.794429\pi$
$829$	−45.9876	−1.59721	−0.798607	+	0.601853i	$0.794429\pi$
$830$	−18.2158	−0.632278
$831$	52.8149	1.83213
$832$	−7.79756	−0.270332
$833$	−32.3848	−1.12207
$834$	−1.73628	−0.0601226
$835$	6.97068	0.241230
$836$	141.758	4.90279
$837$	−43.7233	−1.51130
$838$	31.2667	1.08009
$839$	−35.5172	−1.22619	−0.613095	−	0.790009i	$-0.710076\pi$
$839$	−35.5172	−1.22619	−0.613095	+	0.790009i	$0.710076\pi$
$840$	−245.086	−8.45626
$841$	63.6313	2.19418
$842$	−59.8540	−2.06270
$843$	70.1980	2.41775
$844$	−45.8591	−1.57853
$845$	104.996	3.61196
$846$	−66.5373	−2.28760
$847$	−36.4881	−1.25375
$848$	−6.80456	−0.233670
$849$	−32.2223	−1.10587
$850$	44.7365	1.53445
$851$	1.93861	0.0664546
$852$	189.729	6.50000
$853$	30.6390	1.04906	0.524529	−	0.851393i	$-0.324241\pi$
$853$	30.6390	1.04906	0.524529	+	0.851393i	$0.324241\pi$
$854$	−104.609	−3.57965
$855$	−115.229	−3.94076
$856$	−107.516	−3.67481
$857$	26.7959	0.915332	0.457666	−	0.889124i	$-0.348686\pi$
$857$	26.7959	0.915332	0.457666	+	0.889124i	$0.348686\pi$
$858$	−208.827	−7.12923
$859$	44.1114	1.50506	0.752531	−	0.658557i	$-0.228833\pi$
$859$	44.1114	1.50506	0.752531	+	0.658557i	$0.228833\pi$
$860$	−78.5233	−2.67762
$861$	−64.2986	−2.19129
$862$	20.7554	0.706932
$863$	−21.1381	−0.719548	−0.359774	−	0.933040i	$-0.617146\pi$
$863$	−21.1381	−0.719548	−0.359774	+	0.933040i	$0.617146\pi$
$864$	24.4289	0.831087
$865$	−28.1084	−0.955713
$866$	−30.1150	−1.02335
$867$	24.1180	0.819092
$868$	−166.106	−5.63800
$869$	3.30339	0.112060
$870$	−226.980	−7.69535
$871$	−33.5188	−1.13574
$872$	44.6642	1.51252
$873$	57.8728	1.95870
$874$	−57.7378	−1.95301
$875$	−15.6416	−0.528781
$876$	33.2917	1.12482
$877$	13.3385	0.450409	0.225205	−	0.974311i	$-0.427695\pi$
$877$	13.3385	0.450409	0.225205	+	0.974311i	$0.427695\pi$
$878$	−97.8372	−3.30185
$879$	−4.45757	−0.150350
$880$	−100.246	−3.37929
$881$	−25.9543	−0.874421	−0.437211	−	0.899359i	$-0.644034\pi$
$881$	−25.9543	−0.874421	−0.437211	+	0.899359i	$0.644034\pi$
$882$	−136.004	−4.57949
$883$	41.7160	1.40385	0.701927	−	0.712248i	$-0.252323\pi$
$883$	41.7160	1.40385	0.701927	+	0.712248i	$0.252323\pi$
$884$	−85.5506	−2.87738
$885$	81.1934	2.72929
$886$	−6.08782	−0.204524
$887$	−5.28684	−0.177515	−0.0887573	−	0.996053i	$-0.528290\pi$
$887$	−5.28684	−0.177515	−0.0887573	+	0.996053i	$0.528290\pi$
$888$	−10.6142	−0.356188
$889$	33.8376	1.13488
$890$	104.108	3.48970
$891$	2.01766	0.0675941
$892$	35.4142	1.18576
$893$	−39.6120	−1.32557
$894$	105.758	3.53706
$895$	−4.11619	−0.137589
$896$	−54.4940	−1.82052
$897$	58.6231	1.95737
$898$	16.3173	0.544515
$899$	−84.4738	−2.81736
$900$	129.492	4.31639
$901$	2.89072	0.0963038
$902$	−60.5883	−2.01737
$903$	−63.2516	−2.10488
$904$	100.766	3.35141
$905$	38.4647	1.27861
$906$	7.07848	0.235167
$907$	−7.12151	−0.236466	−0.118233	−	0.992986i	$-0.537723\pi$
$907$	−7.12151	−0.236466	−0.118233	+	0.992986i	$0.537723\pi$
$908$	−58.1776	−1.93069
$909$	53.4045	1.77132
$910$	240.596	7.97569
$911$	−55.5357	−1.83998	−0.919990	−	0.391941i	$-0.871804\pi$
$911$	−55.5357	−1.83998	−0.919990	+	0.391941i	$0.871804\pi$
$912$	137.220	4.54382
$913$	−9.52964	−0.315385
$914$	34.0867	1.12749
$915$	89.8475	2.97027
$916$	−67.8177	−2.24076
$917$	7.38186	0.243771
$918$	−36.5322	−1.20574
$919$	21.9205	0.723090	0.361545	−	0.932355i	$-0.382249\pi$
$919$	21.9205	0.723090	0.361545	+	0.932355i	$0.382249\pi$
$920$	64.8318	2.13744
$921$	−6.20889	−0.204590
$922$	89.4868	2.94709
$923$	−102.276	−3.36644
$924$	−233.496	−7.68145
$925$	−3.75547	−0.123479
$926$	−7.00307	−0.230135
$927$	22.6763	0.744788
$928$	47.1968	1.54931
$929$	44.6819	1.46597	0.732983	−	0.680247i	$-0.238128\pi$
$929$	44.6819	1.46597	0.732983	+	0.680247i	$0.238128\pi$
$930$	206.991	6.78752
$931$	−80.9681	−2.65362
$932$	−86.0089	−2.81731
$933$	86.0625	2.81756
$934$	91.0005	2.97763
$935$	42.5866	1.39273
$936$	−197.289	−6.44858
$937$	16.1922	0.528975	0.264488	−	0.964389i	$-0.414797\pi$
$937$	16.1922	0.528975	0.264488	+	0.964389i	$0.414797\pi$
$938$	−54.3767	−1.77546
$939$	72.5907	2.36891
$940$	81.0004	2.64194
$941$	17.6100	0.574069	0.287034	−	0.957920i	$-0.407331\pi$
$941$	17.6100	0.574069	0.287034	+	0.957920i	$0.407331\pi$
$942$	10.0776	0.328346
$943$	17.0087	0.553880
$944$	−59.4309	−1.93431
$945$	70.8125	2.30353
$946$	−59.6017	−1.93782
$947$	−37.0787	−1.20490	−0.602448	−	0.798158i	$-0.705808\pi$
$947$	−37.0787	−1.20490	−0.602448	+	0.798158i	$0.705808\pi$
$948$	9.24638	0.300308
$949$	−17.9463	−0.582561
$950$	111.850	3.62889
$951$	38.0810	1.23486
$952$	−76.2106	−2.47000
$953$	−14.9504	−0.484290	−0.242145	−	0.970240i	$-0.577851\pi$
$953$	−14.9504	−0.484290	−0.242145	+	0.970240i	$0.577851\pi$
$954$	12.1399	0.393045
$955$	54.1233	1.75139
$956$	14.0590	0.454700
$957$	−118.745	−3.83849
$958$	19.7804	0.639074
$959$	−6.66103	−0.215096
$960$	10.8647	0.350658
$961$	46.0347	1.48499
$962$	10.4197	0.335946
$963$	83.2626	2.68310
$964$	−88.9084	−2.86355
$965$	−62.3634	−2.00755
$966$	95.1028	3.05988
$967$	−51.6886	−1.66219	−0.831097	−	0.556128i	$-0.812286\pi$
$967$	−51.6886	−1.66219	−0.831097	+	0.556128i	$0.812286\pi$
$968$	−52.8467	−1.69856
$969$	−58.2940	−1.87267
$970$	−102.218	−3.28203
$971$	−10.3711	−0.332824	−0.166412	−	0.986056i	$-0.553218\pi$
$971$	−10.3711	−0.332824	−0.166412	+	0.986056i	$0.553218\pi$
$972$	71.9395	2.30746
$973$	1.04653	0.0335502
$974$	−56.6065	−1.81379
$975$	−113.565	−3.63699
$976$	−65.7654	−2.10510
$977$	45.5204	1.45633	0.728163	−	0.685403i	$-0.240374\pi$
$977$	45.5204	1.45633	0.728163	+	0.685403i	$0.240374\pi$
$978$	118.907	3.80223
$979$	54.4643	1.74069
$980$	165.567	5.28884
$981$	−34.5890	−1.10434
$982$	−1.93561	−0.0617679
$983$	3.73183	0.119027	0.0595134	−	0.998228i	$-0.481045\pi$
$983$	3.73183	0.119027	0.0595134	+	0.998228i	$0.481045\pi$
$984$	−93.1253	−2.96873
$985$	24.9716	0.795663
$986$	−70.5805	−2.24774
$987$	65.2469	2.07683
$988$	−213.893	−6.80484
$989$	16.7318	0.532039
$990$	178.848	5.68416
$991$	21.8638	0.694527	0.347264	−	0.937768i	$-0.387111\pi$
$991$	21.8638	0.694527	0.347264	+	0.937768i	$0.387111\pi$
$992$	−43.0405	−1.36654
$993$	−11.1703	−0.354479
$994$	−165.919	−5.26263
$995$	29.5753	0.937602
$996$	−26.6740	−0.845197
$997$	31.1409	0.986242	0.493121	−	0.869961i	$-0.335856\pi$
$997$	31.1409	0.986242	0.493121	+	0.869961i	$0.335856\pi$
$998$	10.7863	0.341434
$999$	3.06675	0.0970276

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8003.2.a.c.1.12	✓	172

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8003.2.a.c.1.12	✓	172	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:	\( N \)	\(=\)	\( 8003 = 53 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53688 q^{2} -2.79023 q^{3} +4.43578 q^{4} +3.33172 q^{5} +7.07848 q^{6} -4.26650 q^{7} -6.17928 q^{8} +4.78538 q^{9} +O(q^{10})\)	\(q-2.53688 q^{2} -2.79023 q^{3} +4.43578 q^{4} +3.33172 q^{5} +7.07848 q^{6} -4.26650 q^{7} -6.17928 q^{8} +4.78538 q^{9} -8.45219 q^{10} -4.42179 q^{11} -12.3768 q^{12} +6.67188 q^{13} +10.8236 q^{14} -9.29627 q^{15} +6.80456 q^{16} -2.89072 q^{17} -12.1399 q^{18} -7.22734 q^{19} +14.7788 q^{20} +11.9045 q^{21} +11.2176 q^{22} -3.14906 q^{23} +17.2416 q^{24} +6.10037 q^{25} -16.9258 q^{26} -4.98161 q^{27} -18.9252 q^{28} -9.62452 q^{29} +23.5835 q^{30} +8.77694 q^{31} -4.90381 q^{32} +12.3378 q^{33} +7.33341 q^{34} -14.2148 q^{35} +21.2269 q^{36} -0.615614 q^{37} +18.3349 q^{38} -18.6161 q^{39} -20.5876 q^{40} -5.40120 q^{41} -30.2004 q^{42} -5.31325 q^{43} -19.6141 q^{44} +15.9435 q^{45} +7.98881 q^{46} +5.48086 q^{47} -18.9863 q^{48} +11.2030 q^{49} -15.4759 q^{50} +8.06576 q^{51} +29.5949 q^{52} -1.00000 q^{53} +12.6378 q^{54} -14.7322 q^{55} +26.3639 q^{56} +20.1659 q^{57} +24.4163 q^{58} -8.73398 q^{59} -41.2362 q^{60} -9.66490 q^{61} -22.2661 q^{62} -20.4168 q^{63} -1.16872 q^{64} +22.2288 q^{65} -31.2996 q^{66} -5.02390 q^{67} -12.8226 q^{68} +8.78661 q^{69} +36.0613 q^{70} -15.3294 q^{71} -29.5702 q^{72} -2.68984 q^{73} +1.56174 q^{74} -17.0214 q^{75} -32.0589 q^{76} +18.8656 q^{77} +47.2268 q^{78} -0.747072 q^{79} +22.6709 q^{80} -0.456299 q^{81} +13.7022 q^{82} +2.15515 q^{83} +52.8057 q^{84} -9.63107 q^{85} +13.4791 q^{86} +26.8546 q^{87} +27.3235 q^{88} -12.3172 q^{89} -40.4469 q^{90} -28.4656 q^{91} -13.9685 q^{92} -24.4897 q^{93} -13.9043 q^{94} -24.0795 q^{95} +13.6828 q^{96} +12.0937 q^{97} -28.4208 q^{98} -21.1599 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9}+O(q^{10}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9} + 20 q^{10} - 3 q^{11} + 66 q^{12} + 121 q^{13} + 12 q^{14} + 30 q^{15} + 212 q^{16} + 8 q^{17} + 40 q^{18} + 41 q^{19} + 64 q^{20} + 56 q^{21} + 50 q^{22} + 28 q^{23} + 30 q^{24} + 231 q^{25} + 38 q^{26} + 100 q^{27} + 80 q^{28} + 26 q^{29} + 55 q^{30} + 66 q^{31} + 65 q^{32} + 99 q^{33} + 81 q^{34} + 36 q^{35} + 212 q^{36} + 153 q^{37} + q^{38} + 20 q^{39} + 59 q^{40} + 40 q^{41} + 50 q^{42} + 39 q^{43} - 51 q^{44} + 123 q^{45} + 59 q^{46} + 29 q^{47} + 128 q^{48} + 245 q^{49} + 19 q^{50} + 36 q^{51} + 215 q^{52} - 172 q^{53} + 40 q^{54} + 40 q^{55} + 15 q^{56} + 54 q^{57} + 44 q^{58} - 54 q^{60} + 100 q^{61} - 29 q^{62} + 92 q^{63} + 253 q^{64} + 77 q^{65} + 14 q^{66} + 126 q^{67} - 27 q^{68} + 47 q^{69} + 72 q^{70} + 38 q^{71} + 65 q^{72} + 185 q^{73} + 48 q^{74} + 75 q^{75} + 38 q^{76} + 120 q^{77} + 75 q^{78} + 79 q^{79} + 43 q^{80} + 232 q^{81} + 110 q^{82} + 90 q^{83} + 158 q^{84} + 115 q^{85} + 68 q^{86} + 61 q^{87} + 15 q^{88} - 36 q^{89} - 6 q^{90} + 33 q^{91} + 139 q^{92} + 103 q^{93} - 24 q^{94} - 45 q^{95} + 34 q^{96} + 159 q^{97} - 36 q^{98} + 27 q^{99}+O(q^{100}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9 + 20 * q^10 - 3 * q^11 + 66 * q^12 + 121 * q^13 + 12 * q^14 + 30 * q^15 + 212 * q^16 + 8 * q^17 + 40 * q^18 + 41 * q^19 + 64 * q^20 + 56 * q^21 + 50 * q^22 + 28 * q^23 + 30 * q^24 + 231 * q^25 + 38 * q^26 + 100 * q^27 + 80 * q^28 + 26 * q^29 + 55 * q^30 + 66 * q^31 + 65 * q^32 + 99 * q^33 + 81 * q^34 + 36 * q^35 + 212 * q^36 + 153 * q^37 + q^38 + 20 * q^39 + 59 * q^40 + 40 * q^41 + 50 * q^42 + 39 * q^43 - 51 * q^44 + 123 * q^45 + 59 * q^46 + 29 * q^47 + 128 * q^48 + 245 * q^49 + 19 * q^50 + 36 * q^51 + 215 * q^52 - 172 * q^53 + 40 * q^54 + 40 * q^55 + 15 * q^56 + 54 * q^57 + 44 * q^58 - 54 * q^60 + 100 * q^61 - 29 * q^62 + 92 * q^63 + 253 * q^64 + 77 * q^65 + 14 * q^66 + 126 * q^67 - 27 * q^68 + 47 * q^69 + 72 * q^70 + 38 * q^71 + 65 * q^72 + 185 * q^73 + 48 * q^74 + 75 * q^75 + 38 * q^76 + 120 * q^77 + 75 * q^78 + 79 * q^79 + 43 * q^80 + 232 * q^81 + 110 * q^82 + 90 * q^83 + 158 * q^84 + 115 * q^85 + 68 * q^86 + 61 * q^87 + 15 * q^88 - 36 * q^89 - 6 * q^90 + 33 * q^91 + 139 * q^92 + 103 * q^93 - 24 * q^94 - 45 * q^95 + 34 * q^96 + 159 * q^97 - 36 * q^98 + 27 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more