LMFDB - Embedded newform 4003.2.a.c.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4003,2,Mod(1,4003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4003, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4003.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:	$31.9641159291$
Analytic rank:	$0$
Dimension:	$179$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	4003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61424 q^{2} -1.76762 q^{3} +4.83424 q^{4} +0.635426 q^{5} +4.62098 q^{6} +1.21621 q^{7} -7.40938 q^{8} +0.124480 q^{9} +O(q^{10})$	\(q-2.61424 q^{2} -1.76762 q^{3} +4.83424 q^{4} +0.635426 q^{5} +4.62098 q^{6} +1.21621 q^{7} -7.40938 q^{8} +0.124480 q^{9} -1.66116 q^{10} +2.32539 q^{11} -8.54510 q^{12} +0.569099 q^{13} -3.17948 q^{14} -1.12319 q^{15} +9.70141 q^{16} +4.78556 q^{17} -0.325419 q^{18} +2.29795 q^{19} +3.07180 q^{20} -2.14981 q^{21} -6.07911 q^{22} -0.558266 q^{23} +13.0970 q^{24} -4.59623 q^{25} -1.48776 q^{26} +5.08283 q^{27} +5.87948 q^{28} +2.51766 q^{29} +2.93629 q^{30} -3.75396 q^{31} -10.5430 q^{32} -4.11040 q^{33} -12.5106 q^{34} +0.772815 q^{35} +0.601764 q^{36} -3.73653 q^{37} -6.00739 q^{38} -1.00595 q^{39} -4.70812 q^{40} +6.36538 q^{41} +5.62010 q^{42} +10.3933 q^{43} +11.2415 q^{44} +0.0790976 q^{45} +1.45944 q^{46} +11.6980 q^{47} -17.1484 q^{48} -5.52082 q^{49} +12.0156 q^{50} -8.45906 q^{51} +2.75116 q^{52} -4.60740 q^{53} -13.2877 q^{54} +1.47761 q^{55} -9.01140 q^{56} -4.06190 q^{57} -6.58176 q^{58} +1.77970 q^{59} -5.42978 q^{60} -2.96059 q^{61} +9.81375 q^{62} +0.151394 q^{63} +8.15916 q^{64} +0.361620 q^{65} +10.7456 q^{66} +6.25986 q^{67} +23.1346 q^{68} +0.986802 q^{69} -2.02032 q^{70} -14.4807 q^{71} -0.922317 q^{72} +11.9693 q^{73} +9.76817 q^{74} +8.12439 q^{75} +11.1088 q^{76} +2.82817 q^{77} +2.62979 q^{78} -2.61737 q^{79} +6.16453 q^{80} -9.35794 q^{81} -16.6406 q^{82} -4.81954 q^{83} -10.3927 q^{84} +3.04087 q^{85} -27.1705 q^{86} -4.45026 q^{87} -17.2297 q^{88} +7.97919 q^{89} -0.206780 q^{90} +0.692146 q^{91} -2.69879 q^{92} +6.63558 q^{93} -30.5815 q^{94} +1.46018 q^{95} +18.6361 q^{96} +15.5490 q^{97} +14.4327 q^{98} +0.289463 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 179 q + 22 q^{2} + 16 q^{3} + 196 q^{4} + 61 q^{5} + 7 q^{6} + 21 q^{7} + 60 q^{8} + 221 q^{9}+O(q^{10}) $ 179 * q + 22 * q^2 + 16 * q^3 + 196 * q^4 + 61 * q^5 + 7 * q^6 + 21 * q^7 + 60 * q^8 + 221 * q^9	\( 179 q + 22 q^{2} + 16 q^{3} + 196 q^{4} + 61 q^{5} + 7 q^{6} + 21 q^{7} + 60 q^{8} + 221 q^{9} + 9 q^{10} + 46 q^{11} + 33 q^{12} + 47 q^{13} + 22 q^{14} + 36 q^{15} + 222 q^{16} + 103 q^{17} + 43 q^{18} + 12 q^{19} + 102 q^{20} + 50 q^{21} + 39 q^{22} + 121 q^{23} - 3 q^{24} + 246 q^{25} + 52 q^{26} + 49 q^{27} + 41 q^{28} + 138 q^{29} + 28 q^{30} + 5 q^{31} + 137 q^{32} + 63 q^{33} + 2 q^{34} + 72 q^{35} + 279 q^{36} + 118 q^{37} + 123 q^{38} + q^{39} + 9 q^{40} + 50 q^{41} + 48 q^{42} + 48 q^{43} + 108 q^{44} + 158 q^{45} + 13 q^{46} + 85 q^{47} + 50 q^{48} + 230 q^{49} + 78 q^{50} + 15 q^{51} + 41 q^{52} + 399 q^{53} - 5 q^{54} + 24 q^{55} + 53 q^{56} + 45 q^{57} + 27 q^{58} + 48 q^{59} + 66 q^{60} + 46 q^{61} + 81 q^{62} + 78 q^{63} + 252 q^{64} + 153 q^{65} + 6 q^{66} + 70 q^{67} + 240 q^{68} + 120 q^{69} - 31 q^{70} + 86 q^{71} + 89 q^{72} + 45 q^{73} + 68 q^{74} + 17 q^{75} - 13 q^{76} + 362 q^{77} + 69 q^{78} + 31 q^{79} + 169 q^{80} + 303 q^{81} + 25 q^{82} + 106 q^{83} + 13 q^{84} + 115 q^{85} + 95 q^{86} + 32 q^{87} + 83 q^{88} + 105 q^{89} - 38 q^{90} + 3 q^{91} + 310 q^{92} + 298 q^{93} - 17 q^{94} + 102 q^{95} - 82 q^{96} + 34 q^{97} + 81 q^{98} + 58 q^{99}+O(q^{100}) \) 179 * q + 22 * q^2 + 16 * q^3 + 196 * q^4 + 61 * q^5 + 7 * q^6 + 21 * q^7 + 60 * q^8 + 221 * q^9 + 9 * q^10 + 46 * q^11 + 33 * q^12 + 47 * q^13 + 22 * q^14 + 36 * q^15 + 222 * q^16 + 103 * q^17 + 43 * q^18 + 12 * q^19 + 102 * q^20 + 50 * q^21 + 39 * q^22 + 121 * q^23 - 3 * q^24 + 246 * q^25 + 52 * q^26 + 49 * q^27 + 41 * q^28 + 138 * q^29 + 28 * q^30 + 5 * q^31 + 137 * q^32 + 63 * q^33 + 2 * q^34 + 72 * q^35 + 279 * q^36 + 118 * q^37 + 123 * q^38 + q^39 + 9 * q^40 + 50 * q^41 + 48 * q^42 + 48 * q^43 + 108 * q^44 + 158 * q^45 + 13 * q^46 + 85 * q^47 + 50 * q^48 + 230 * q^49 + 78 * q^50 + 15 * q^51 + 41 * q^52 + 399 * q^53 - 5 * q^54 + 24 * q^55 + 53 * q^56 + 45 * q^57 + 27 * q^58 + 48 * q^59 + 66 * q^60 + 46 * q^61 + 81 * q^62 + 78 * q^63 + 252 * q^64 + 153 * q^65 + 6 * q^66 + 70 * q^67 + 240 * q^68 + 120 * q^69 - 31 * q^70 + 86 * q^71 + 89 * q^72 + 45 * q^73 + 68 * q^74 + 17 * q^75 - 13 * q^76 + 362 * q^77 + 69 * q^78 + 31 * q^79 + 169 * q^80 + 303 * q^81 + 25 * q^82 + 106 * q^83 + 13 * q^84 + 115 * q^85 + 95 * q^86 + 32 * q^87 + 83 * q^88 + 105 * q^89 - 38 * q^90 + 3 * q^91 + 310 * q^92 + 298 * q^93 - 17 * q^94 + 102 * q^95 - 82 * q^96 + 34 * q^97 + 81 * q^98 + 58 * 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 $

$2$	−2.61424	−1.84855	−0.924273	−	0.381733i	$-0.875327\pi$
$2$	−2.61424	−1.84855	−0.924273	+	0.381733i	$0.875327\pi$
$3$	−1.76762	−1.02054	−0.510268	−	0.860016i	$-0.670454\pi$
$3$	−1.76762	−1.02054	−0.510268	+	0.860016i	$0.670454\pi$
$4$	4.83424	2.41712
$5$	0.635426	0.284171	0.142086	−	0.989854i	$-0.454619\pi$
$5$	0.635426	0.284171	0.142086	+	0.989854i	$0.454619\pi$
$6$	4.62098	1.88651
$7$	1.21621	0.459686	0.229843	−	0.973228i	$-0.426179\pi$
$7$	1.21621	0.459686	0.229843	+	0.973228i	$0.426179\pi$
$8$	−7.40938	−2.61961
$9$	0.124480	0.0414932
$10$	−1.66116	−0.525304
$11$	2.32539	0.701130	0.350565	−	0.936538i	$-0.385989\pi$
$11$	2.32539	0.701130	0.350565	+	0.936538i	$0.385989\pi$
$12$	−8.54510	−2.46676
$13$	0.569099	0.157840	0.0789198	−	0.996881i	$-0.474853\pi$
$13$	0.569099	0.157840	0.0789198	+	0.996881i	$0.474853\pi$
$14$	−3.17948	−0.849751
$15$	−1.12319	−0.290007
$16$	9.70141	2.42535
$17$	4.78556	1.16067	0.580335	−	0.814378i	$-0.302922\pi$
$17$	4.78556	1.16067	0.580335	+	0.814378i	$0.302922\pi$
$18$	−0.325419	−0.0767021
$19$	2.29795	0.527186	0.263593	−	0.964634i	$-0.415092\pi$
$19$	2.29795	0.527186	0.263593	+	0.964634i	$0.415092\pi$
$20$	3.07180	0.686876
$21$	−2.14981	−0.469126
$22$	−6.07911	−1.29607
$23$	−0.558266	−0.116406	−0.0582032	−	0.998305i	$-0.518537\pi$
$23$	−0.558266	−0.116406	−0.0582032	+	0.998305i	$0.518537\pi$
$24$	13.0970	2.67341
$25$	−4.59623	−0.919247
$26$	−1.48776	−0.291774
$27$	5.08283	0.978190
$28$	5.87948	1.11112
$29$	2.51766	0.467517	0.233759	−	0.972295i	$-0.424897\pi$
$29$	2.51766	0.467517	0.233759	+	0.972295i	$0.424897\pi$
$30$	2.93629	0.536091
$31$	−3.75396	−0.674231	−0.337116	−	0.941463i	$-0.609451\pi$
$31$	−3.75396	−0.674231	−0.337116	+	0.941463i	$0.609451\pi$
$32$	−10.5430	−1.86376
$33$	−4.11040	−0.715529
$34$	−12.5106	−2.14555
$35$	0.772815	0.130630
$36$	0.601764	0.100294
$37$	−3.73653	−0.614281	−0.307141	−	0.951664i	$-0.599372\pi$
$37$	−3.73653	−0.614281	−0.307141	+	0.951664i	$0.599372\pi$
$38$	−6.00739	−0.974527
$39$	−1.00595	−0.161081
$40$	−4.70812	−0.744419
$41$	6.36538	0.994105	0.497052	−	0.867720i	$-0.334416\pi$
$41$	6.36538	0.994105	0.497052	+	0.867720i	$0.334416\pi$
$42$	5.62010	0.867201
$43$	10.3933	1.58496	0.792480	−	0.609898i	$-0.208790\pi$
$43$	10.3933	1.58496	0.792480	+	0.609898i	$0.208790\pi$
$44$	11.2415	1.69472
$45$	0.0790976	0.0117912
$46$	1.45944	0.215183
$47$	11.6980	1.70634	0.853168	−	0.521637i	$-0.174678\pi$
$47$	11.6980	1.70634	0.853168	+	0.521637i	$0.174678\pi$
$48$	−17.1484	−2.47516
$49$	−5.52082	−0.788689
$50$	12.0156	1.69927
$51$	−8.45906	−1.18450
$52$	2.75116	0.381517
$53$	−4.60740	−0.632875	−0.316438	−	0.948613i	$-0.602487\pi$
$53$	−4.60740	−0.632875	−0.316438	+	0.948613i	$0.602487\pi$
$54$	−13.2877	−1.80823
$55$	1.47761	0.199241
$56$	−9.01140	−1.20420
$57$	−4.06190	−0.538012
$58$	−6.58176	−0.864227
$59$	1.77970	0.231697	0.115848	−	0.993267i	$-0.463041\pi$
$59$	1.77970	0.231697	0.115848	+	0.993267i	$0.463041\pi$
$60$	−5.42978	−0.700982
$61$	−2.96059	−0.379065	−0.189532	−	0.981874i	$-0.560697\pi$
$61$	−2.96059	−0.379065	−0.189532	+	0.981874i	$0.560697\pi$
$62$	9.81375	1.24635
$63$	0.151394	0.0190738
$64$	8.15916	1.01989
$65$	0.361620	0.0448535
$66$	10.7456	1.32269
$67$	6.25986	0.764764	0.382382	−	0.924004i	$-0.375104\pi$
$67$	6.25986	0.764764	0.382382	+	0.924004i	$0.375104\pi$
$68$	23.1346	2.80548
$69$	0.986802	0.118797
$70$	−2.02032	−0.241475
$71$	−14.4807	−1.71855	−0.859274	−	0.511515i	$-0.829084\pi$
$71$	−14.4807	−1.71855	−0.859274	+	0.511515i	$0.829084\pi$
$72$	−0.922317	−0.108696
$73$	11.9693	1.40090	0.700452	−	0.713699i	$-0.252982\pi$
$73$	11.9693	1.40090	0.700452	+	0.713699i	$0.252982\pi$
$74$	9.76817	1.13553
$75$	8.12439	0.938124
$76$	11.1088	1.27427
$77$	2.82817	0.322300
$78$	2.62979	0.297765
$79$	−2.61737	−0.294477	−0.147239	−	0.989101i	$-0.547038\pi$
$79$	−2.61737	−0.294477	−0.147239	+	0.989101i	$0.547038\pi$
$80$	6.16453	0.689215
$81$	−9.35794	−1.03977
$82$	−16.6406	−1.83765
$83$	−4.81954	−0.529014	−0.264507	−	0.964384i	$-0.585209\pi$
$83$	−4.81954	−0.529014	−0.264507	+	0.964384i	$0.585209\pi$
$84$	−10.3927	−1.13393
$85$	3.04087	0.329829
$86$	−27.1705	−2.92987
$87$	−4.45026	−0.477118
$88$	−17.2297	−1.83669
$89$	7.97919	0.845793	0.422896	−	0.906178i	$-0.361013\pi$
$89$	7.97919	0.845793	0.422896	+	0.906178i	$0.361013\pi$
$90$	−0.206780	−0.0217965
$91$	0.692146	0.0725566
$92$	−2.69879	−0.281369
$93$	6.63558	0.688077
$94$	−30.5815	−3.15424
$95$	1.46018	0.149811
$96$	18.6361	1.90203
$97$	15.5490	1.57876	0.789380	−	0.613905i	$-0.210402\pi$
$97$	15.5490	1.57876	0.789380	+	0.613905i	$0.210402\pi$
$98$	14.4327	1.45793
$99$	0.289463	0.0290921
$100$	−22.2193	−2.22193
$101$	13.9649	1.38956	0.694782	−	0.719221i	$-0.255501\pi$
$101$	13.9649	1.38956	0.694782	+	0.719221i	$0.255501\pi$
$102$	22.1140	2.18961
$103$	10.2346	1.00844	0.504221	−	0.863575i	$-0.331780\pi$
$103$	10.2346	1.00844	0.504221	+	0.863575i	$0.331780\pi$
$104$	−4.21667	−0.413478
$105$	−1.36604	−0.133312
$106$	12.0448	1.16990
$107$	17.7828	1.71913	0.859563	−	0.511030i	$-0.170736\pi$
$107$	17.7828	1.71913	0.859563	+	0.511030i	$0.170736\pi$
$108$	24.5716	2.36440
$109$	−2.53870	−0.243163	−0.121581	−	0.992581i	$-0.538797\pi$
$109$	−2.53870	−0.243163	−0.121581	+	0.992581i	$0.538797\pi$
$110$	−3.86283	−0.368306
$111$	6.60476	0.626896
$112$	11.7990	1.11490
$113$	9.84846	0.926466	0.463233	−	0.886237i	$-0.346689\pi$
$113$	9.84846	0.926466	0.463233	+	0.886237i	$0.346689\pi$
$114$	10.6188	0.994540
$115$	−0.354737	−0.0330794
$116$	12.1710	1.13005
$117$	0.0708412	0.00654927
$118$	−4.65255	−0.428302
$119$	5.82027	0.533544
$120$	8.32216	0.759706
$121$	−5.59258	−0.508416
$122$	7.73969	0.700718
$123$	−11.2516	−1.01452
$124$	−18.1476	−1.62970
$125$	−6.09770	−0.545395
$126$	−0.395780	−0.0352589
$127$	−11.3759	−1.00945	−0.504726	−	0.863280i	$-0.668407\pi$
$127$	−11.3759	−1.00945	−0.504726	+	0.863280i	$0.668407\pi$
$128$	−0.243942	−0.0215617
$129$	−18.3714	−1.61751
$130$	−0.945361	−0.0829137
$131$	−2.66845	−0.233144	−0.116572	−	0.993182i	$-0.537191\pi$
$131$	−2.66845	−0.233144	−0.116572	+	0.993182i	$0.537191\pi$
$132$	−19.8707	−1.72952
$133$	2.79480	0.242340
$134$	−16.3648	−1.41370
$135$	3.22976	0.277974
$136$	−35.4581	−3.04050
$137$	−12.9380	−1.10537	−0.552684	−	0.833391i	$-0.686397\pi$
$137$	−12.9380	−1.10537	−0.552684	+	0.833391i	$0.686397\pi$
$138$	−2.57973	−0.219602
$139$	−21.8624	−1.85435	−0.927174	−	0.374631i	$-0.877769\pi$
$139$	−21.8624	−1.85435	−0.927174	+	0.374631i	$0.877769\pi$
$140$	3.73597	0.315747
$141$	−20.6777	−1.74138
$142$	37.8561	3.17681
$143$	1.32337	0.110666
$144$	1.20763	0.100636
$145$	1.59979	0.132855
$146$	−31.2907	−2.58964
$147$	9.75871	0.804885
$148$	−18.0633	−1.48479
$149$	22.1066	1.81104	0.905521	−	0.424301i	$-0.139480\pi$
$149$	22.1066	1.81104	0.905521	+	0.424301i	$0.139480\pi$
$150$	−21.2391	−1.73417
$151$	−18.1451	−1.47663	−0.738313	−	0.674458i	$-0.764377\pi$
$151$	−18.1451	−1.47663	−0.738313	+	0.674458i	$0.764377\pi$
$152$	−17.0264	−1.38102
$153$	0.595705	0.0481599
$154$	−7.39351	−0.595786
$155$	−2.38537	−0.191597
$156$	−4.86300	−0.389352
$157$	−10.9297	−0.872287	−0.436143	−	0.899877i	$-0.643656\pi$
$157$	−10.9297	−0.872287	−0.436143	+	0.899877i	$0.643656\pi$
$158$	6.84243	0.544354
$159$	8.14413	0.645872
$160$	−6.69931	−0.529627
$161$	−0.678971	−0.0535104
$162$	24.4639	1.92206
$163$	−19.6291	−1.53747	−0.768735	−	0.639567i	$-0.779114\pi$
$163$	−19.6291	−1.53747	−0.768735	+	0.639567i	$0.779114\pi$
$164$	30.7718	2.40287
$165$	−2.61186	−0.203333
$166$	12.5994	0.977906
$167$	18.1017	1.40075	0.700377	−	0.713773i	$-0.253015\pi$
$167$	18.1017	1.40075	0.700377	+	0.713773i	$0.253015\pi$
$168$	15.9287	1.22893
$169$	−12.6761	−0.975087
$170$	−7.94957	−0.609704
$171$	0.286048	0.0218746
$172$	50.2436	3.83104
$173$	−20.4973	−1.55838	−0.779192	−	0.626786i	$-0.784370\pi$
$173$	−20.4973	−1.55838	−0.779192	+	0.626786i	$0.784370\pi$
$174$	11.6340	0.881974
$175$	−5.59001	−0.422565
$176$	22.5595	1.70049
$177$	−3.14583	−0.236455
$178$	−20.8595	−1.56349
$179$	−17.4580	−1.30487	−0.652436	−	0.757844i	$-0.726253\pi$
$179$	−17.4580	−1.30487	−0.652436	+	0.757844i	$0.726253\pi$
$180$	0.382377	0.0285007
$181$	−15.9102	−1.18260	−0.591298	−	0.806453i	$-0.701384\pi$
$181$	−15.9102	−1.18260	−0.591298	+	0.806453i	$0.701384\pi$
$182$	−1.80944	−0.134124
$183$	5.23320	0.386849
$184$	4.13640	0.304940
$185$	−2.37429	−0.174561
$186$	−17.3470	−1.27194
$187$	11.1283	0.813781
$188$	56.5512	4.12442
$189$	6.18181	0.449661
$190$	−3.81725	−0.276933
$191$	6.98269	0.505250	0.252625	−	0.967564i	$-0.418706\pi$
$191$	6.98269	0.505250	0.252625	+	0.967564i	$0.418706\pi$
$192$	−14.4223	−1.04084
$193$	−16.1536	−1.16276	−0.581379	−	0.813633i	$-0.697487\pi$
$193$	−16.1536	−1.16276	−0.581379	+	0.813633i	$0.697487\pi$
$194$	−40.6487	−2.91841
$195$	−0.639207	−0.0457746
$196$	−26.6890	−1.90636
$197$	−18.2648	−1.30131	−0.650656	−	0.759373i	$-0.725506\pi$
$197$	−18.2648	−1.30131	−0.650656	+	0.759373i	$0.725506\pi$
$198$	−0.756726	−0.0537782
$199$	18.8995	1.33975	0.669874	−	0.742474i	$-0.266348\pi$
$199$	18.8995	1.33975	0.669874	+	0.742474i	$0.266348\pi$
$200$	34.0552	2.40807
$201$	−11.0651	−0.780469
$202$	−36.5077	−2.56867
$203$	3.06201	0.214911
$204$	−40.8931	−2.86309
$205$	4.04473	0.282496
$206$	−26.7556	−1.86415
$207$	−0.0694927	−0.00483008
$208$	5.52106	0.382816
$209$	5.34362	0.369626
$210$	3.57116	0.246434
$211$	−3.25580	−0.224138	−0.112069	−	0.993700i	$-0.535748\pi$
$211$	−3.25580	−0.224138	−0.112069	+	0.993700i	$0.535748\pi$
$212$	−22.2733	−1.52974
$213$	25.5965	1.75384
$214$	−46.4884	−3.17788
$215$	6.60416	0.450400
$216$	−37.6606	−2.56248
$217$	−4.56562	−0.309935
$218$	6.63675	0.449498
$219$	−21.1572	−1.42967
$220$	7.14313	0.481590
$221$	2.72346	0.183200
$222$	−17.2664	−1.15885
$223$	−17.8718	−1.19678	−0.598391	−	0.801204i	$-0.704193\pi$
$223$	−17.8718	−1.19678	−0.598391	+	0.801204i	$0.704193\pi$
$224$	−12.8226	−0.856745
$225$	−0.572137	−0.0381425
$226$	−25.7462	−1.71261
$227$	−6.77533	−0.449694	−0.224847	−	0.974394i	$-0.572188\pi$
$227$	−6.77533	−0.449694	−0.224847	+	0.974394i	$0.572188\pi$
$228$	−19.6362	−1.30044
$229$	15.6402	1.03353	0.516767	−	0.856126i	$-0.327135\pi$
$229$	15.6402	1.03353	0.516767	+	0.856126i	$0.327135\pi$
$230$	0.927367	0.0611487
$231$	−4.99913	−0.328919
$232$	−18.6543	−1.22471
$233$	23.6859	1.55171	0.775856	−	0.630910i	$-0.217318\pi$
$233$	23.6859	1.55171	0.775856	+	0.630910i	$0.217318\pi$
$234$	−0.185196	−0.0121066
$235$	7.43325	0.484891
$236$	8.60348	0.560039
$237$	4.62651	0.300524
$238$	−15.2156	−0.986280
$239$	17.2769	1.11755	0.558775	−	0.829319i	$-0.311271\pi$
$239$	17.2769	1.11755	0.558775	+	0.829319i	$0.311271\pi$
$240$	−10.8965	−0.703369
$241$	15.4529	0.995411	0.497705	−	0.867346i	$-0.334176\pi$
$241$	15.4529	0.995411	0.497705	+	0.867346i	$0.334176\pi$
$242$	14.6203	0.939830
$243$	1.29281	0.0829335
$244$	−14.3122	−0.916245
$245$	−3.50808	−0.224123
$246$	29.4143	1.87539
$247$	1.30776	0.0832108
$248$	27.8145	1.76622
$249$	8.51912	0.539877
$250$	15.9408	1.00819
$251$	−23.9145	−1.50947	−0.754734	−	0.656031i	$-0.772234\pi$
$251$	−23.9145	−1.50947	−0.754734	+	0.656031i	$0.772234\pi$
$252$	0.731875	0.0461038
$253$	−1.29818	−0.0816161
$254$	29.7394	1.86602
$255$	−5.37511	−0.336602
$256$	−15.6806	−0.980037
$257$	−15.4765	−0.965400	−0.482700	−	0.875786i	$-0.660344\pi$
$257$	−15.4765	−0.965400	−0.482700	+	0.875786i	$0.660344\pi$
$258$	48.0271	2.99004
$259$	−4.54442	−0.282377
$260$	1.74816	0.108416
$261$	0.313397	0.0193988
$262$	6.97598	0.430977
$263$	23.0987	1.42433	0.712163	−	0.702014i	$-0.247716\pi$
$263$	23.0987	1.42433	0.712163	+	0.702014i	$0.247716\pi$
$264$	30.4555	1.87441
$265$	−2.92766	−0.179845
$266$	−7.30628	−0.447977
$267$	−14.1042	−0.863162
$268$	30.2617	1.84853
$269$	10.9497	0.667614	0.333807	−	0.942641i	$-0.391667\pi$
$269$	10.9497	0.667614	0.333807	+	0.942641i	$0.391667\pi$
$270$	−8.44337	−0.513847
$271$	27.8489	1.69170	0.845852	−	0.533418i	$-0.179093\pi$
$271$	27.8489	1.69170	0.845852	+	0.533418i	$0.179093\pi$
$272$	46.4267	2.81503
$273$	−1.22345	−0.0740466
$274$	33.8230	2.04332
$275$	−10.6880	−0.644512
$276$	4.77044	0.287147
$277$	16.1475	0.970210	0.485105	−	0.874456i	$-0.338781\pi$
$277$	16.1475	0.970210	0.485105	+	0.874456i	$0.338781\pi$
$278$	57.1536	3.42785
$279$	−0.467292	−0.0279760
$280$	−5.72608	−0.342199
$281$	20.3570	1.21439	0.607197	−	0.794551i	$-0.292294\pi$
$281$	20.3570	1.21439	0.607197	+	0.794551i	$0.292294\pi$
$282$	54.0564	3.21901
$283$	1.90046	0.112971	0.0564853	−	0.998403i	$-0.482011\pi$
$283$	1.90046	0.112971	0.0564853	+	0.998403i	$0.482011\pi$
$284$	−70.0034	−4.15394
$285$	−2.58104	−0.152888
$286$	−3.45962	−0.204571
$287$	7.74167	0.456976
$288$	−1.31239	−0.0773334
$289$	5.90161	0.347154
$290$	−4.18222	−0.245588
$291$	−27.4847	−1.61118
$292$	57.8627	3.38616
$293$	26.2495	1.53351	0.766757	−	0.641938i	$-0.221869\pi$
$293$	26.2495	1.53351	0.766757	+	0.641938i	$0.221869\pi$
$294$	−25.5116	−1.48787
$295$	1.13087	0.0658416
$296$	27.6854	1.60918
$297$	11.8195	0.685839
$298$	−57.7919	−3.34779
$299$	−0.317708	−0.0183735
$300$	39.2753	2.26756
$301$	12.6405	0.728584
$302$	47.4356	2.72961
$303$	−24.6847	−1.41810
$304$	22.2933	1.27861
$305$	−1.88124	−0.107719
$306$	−1.55731	−0.0890257
$307$	−15.1314	−0.863594	−0.431797	−	0.901971i	$-0.642120\pi$
$307$	−15.1314	−0.863594	−0.431797	+	0.901971i	$0.642120\pi$
$308$	13.6721	0.779038
$309$	−18.0908	−1.02915
$310$	6.23591	0.354176
$311$	21.5502	1.22200	0.610999	−	0.791632i	$-0.290768\pi$
$311$	21.5502	1.22200	0.610999	+	0.791632i	$0.290768\pi$
$312$	7.45347	0.421969
$313$	22.2632	1.25839	0.629194	−	0.777248i	$-0.283385\pi$
$313$	22.2632	1.25839	0.629194	+	0.777248i	$0.283385\pi$
$314$	28.5729	1.61246
$315$	0.0961997	0.00542024
$316$	−12.6530	−0.711787
$317$	−24.0174	−1.34895	−0.674476	−	0.738297i	$-0.735630\pi$
$317$	−24.0174	−1.34895	−0.674476	+	0.738297i	$0.735630\pi$
$318$	−21.2907	−1.19392
$319$	5.85453	0.327791
$320$	5.18455	0.289825
$321$	−31.4332	−1.75443
$322$	1.77499	0.0989165
$323$	10.9970	0.611889
$324$	−45.2386	−2.51325
$325$	−2.61571	−0.145093
$326$	51.3152	2.84209
$327$	4.48745	0.248156
$328$	−47.1635	−2.60417
$329$	14.2273	0.784378
$330$	6.82801	0.375870
$331$	−24.9804	−1.37305	−0.686523	−	0.727109i	$-0.740864\pi$
$331$	−24.9804	−1.37305	−0.686523	+	0.727109i	$0.740864\pi$
$332$	−23.2988	−1.27869
$333$	−0.465121	−0.0254885
$334$	−47.3223	−2.58936
$335$	3.97768	0.217324
$336$	−20.8561	−1.13780
$337$	−3.22461	−0.175656	−0.0878278	−	0.996136i	$-0.527993\pi$
$337$	−3.22461	−0.175656	−0.0878278	+	0.996136i	$0.527993\pi$
$338$	33.1384	1.80249
$339$	−17.4083	−0.945491
$340$	14.7003	0.797236
$341$	−8.72941	−0.472724
$342$	−0.747797	−0.0404363
$343$	−15.2280	−0.822235
$344$	−77.0077	−4.15198
$345$	0.627040	0.0337587
$346$	53.5849	2.88074
$347$	11.6525	0.625540	0.312770	−	0.949829i	$-0.398743\pi$
$347$	11.6525	0.625540	0.312770	+	0.949829i	$0.398743\pi$
$348$	−21.5136	−1.15325
$349$	20.8462	1.11587	0.557936	−	0.829884i	$-0.311594\pi$
$349$	20.8462	1.11587	0.557936	+	0.829884i	$0.311594\pi$
$350$	14.6136	0.781130
$351$	2.89263	0.154397
$352$	−24.5166	−1.30674
$353$	20.9571	1.11543	0.557717	−	0.830031i	$-0.311678\pi$
$353$	20.9571	1.11543	0.557717	+	0.830031i	$0.311678\pi$
$354$	8.22394	0.437097
$355$	−9.20145	−0.488362
$356$	38.5733	2.04438
$357$	−10.2880	−0.544500
$358$	45.6394	2.41212
$359$	−28.9982	−1.53047	−0.765233	−	0.643753i	$-0.777376\pi$
$359$	−28.9982	−1.53047	−0.765233	+	0.643753i	$0.777376\pi$
$360$	−0.586064	−0.0308883
$361$	−13.7194	−0.722075
$362$	41.5931	2.18608
$363$	9.88555	0.518857
$364$	3.34600	0.175378
$365$	7.60563	0.398097
$366$	−13.6808	−0.715108
$367$	20.0389	1.04602	0.523012	−	0.852326i	$-0.324808\pi$
$367$	20.0389	1.04602	0.523012	+	0.852326i	$0.324808\pi$
$368$	−5.41596	−0.282327
$369$	0.792360	0.0412486
$370$	6.20696	0.322684
$371$	−5.60359	−0.290924
$372$	32.0780	1.66317
$373$	22.2545	1.15229	0.576147	−	0.817346i	$-0.304556\pi$
$373$	22.2545	1.15229	0.576147	+	0.817346i	$0.304556\pi$
$374$	−29.0920	−1.50431
$375$	10.7784	0.556595
$376$	−86.6753	−4.46994
$377$	1.43280	0.0737927
$378$	−16.1607	−0.831218
$379$	−0.974799	−0.0500721	−0.0250360	−	0.999687i	$-0.507970\pi$
$379$	−0.974799	−0.0500721	−0.0250360	+	0.999687i	$0.507970\pi$
$380$	7.05885	0.362112
$381$	20.1084	1.03018
$382$	−18.2544	−0.933977
$383$	−10.1599	−0.519149	−0.259574	−	0.965723i	$-0.583582\pi$
$383$	−10.1599	−0.519149	−0.259574	+	0.965723i	$0.583582\pi$
$384$	0.431198	0.0220045
$385$	1.79709	0.0915884
$386$	42.2292	2.14941
$387$	1.29375	0.0657650
$388$	75.1675	3.81605
$389$	11.5605	0.586140	0.293070	−	0.956091i	$-0.405323\pi$
$389$	11.5605	0.586140	0.293070	+	0.956091i	$0.405323\pi$
$390$	1.67104	0.0846164
$391$	−2.67162	−0.135109
$392$	40.9059	2.06606
$393$	4.71681	0.237932
$394$	47.7485	2.40553
$395$	−1.66315	−0.0836819
$396$	1.39933	0.0703192
$397$	24.7766	1.24350	0.621752	−	0.783214i	$-0.286421\pi$
$397$	24.7766	1.24350	0.621752	+	0.783214i	$0.286421\pi$
$398$	−49.4078	−2.47659
$399$	−4.94015	−0.247317
$400$	−44.5899	−2.22950
$401$	−1.71878	−0.0858317	−0.0429159	−	0.999079i	$-0.513665\pi$
$401$	−1.71878	−0.0858317	−0.0429159	+	0.999079i	$0.513665\pi$
$402$	28.9267	1.44273
$403$	−2.13637	−0.106420
$404$	67.5099	3.35874
$405$	−5.94628	−0.295473
$406$	−8.00483	−0.397273
$407$	−8.68887	−0.430691
$408$	62.6764	3.10294
$409$	33.0250	1.63298	0.816491	−	0.577357i	$-0.195916\pi$
$409$	33.0250	1.63298	0.816491	+	0.577357i	$0.195916\pi$
$410$	−10.5739	−0.522207
$411$	22.8695	1.12807
$412$	49.4764	2.43753
$413$	2.16449	0.106508
$414$	0.181670	0.00892862
$415$	−3.06247	−0.150331
$416$	−6.00002	−0.294175
$417$	38.6445	1.89243
$418$	−13.9695	−0.683271
$419$	16.8697	0.824137	0.412069	−	0.911153i	$-0.364806\pi$
$419$	16.8697	0.824137	0.412069	+	0.911153i	$0.364806\pi$
$420$	−6.60378	−0.322232
$421$	34.8891	1.70039	0.850195	−	0.526467i	$-0.176484\pi$
$421$	34.8891	1.70039	0.850195	+	0.526467i	$0.176484\pi$
$422$	8.51143	0.414330
$423$	1.45617	0.0708013
$424$	34.1380	1.65789
$425$	−21.9956	−1.06694
$426$	−66.9152	−3.24205
$427$	−3.60071	−0.174251
$428$	85.9662	4.15533
$429$	−2.33922	−0.112939
$430$	−17.2648	−0.832585
$431$	−7.55076	−0.363707	−0.181854	−	0.983326i	$-0.558210\pi$
$431$	−7.55076	−0.363707	−0.181854	+	0.983326i	$0.558210\pi$
$432$	49.3106	2.37246
$433$	−6.37306	−0.306270	−0.153135	−	0.988205i	$-0.548937\pi$
$433$	−6.37306	−0.306270	−0.153135	+	0.988205i	$0.548937\pi$
$434$	11.9356	0.572928
$435$	−2.82781	−0.135583
$436$	−12.2727	−0.587754
$437$	−1.28287	−0.0613679
$438$	55.3101	2.64282
$439$	10.0360	0.478990	0.239495	−	0.970898i	$-0.423018\pi$
$439$	10.0360	0.478990	0.239495	+	0.970898i	$0.423018\pi$
$440$	−10.9482	−0.521935
$441$	−0.687230	−0.0327252
$442$	−7.11977	−0.338653
$443$	−37.9851	−1.80473	−0.902363	−	0.430976i	$-0.858169\pi$
$443$	−37.9851	−1.80473	−0.902363	+	0.430976i	$0.858169\pi$
$444$	31.9290	1.51528
$445$	5.07019	0.240350
$446$	46.7210	2.21231
$447$	−39.0760	−1.84823
$448$	9.92329	0.468832
$449$	−1.17432	−0.0554195	−0.0277097	−	0.999616i	$-0.508821\pi$
$449$	−1.17432	−0.0554195	−0.0277097	+	0.999616i	$0.508821\pi$
$450$	1.49570	0.0705081
$451$	14.8020	0.696997
$452$	47.6099	2.23938
$453$	32.0736	1.50695
$454$	17.7123	0.831280
$455$	0.439808	0.0206185
$456$	30.0962	1.40938
$457$	9.46386	0.442701	0.221350	−	0.975194i	$-0.428954\pi$
$457$	9.46386	0.442701	0.221350	+	0.975194i	$0.428954\pi$
$458$	−40.8872	−1.91054
$459$	24.3242	1.13536
$460$	−1.71488	−0.0799569
$461$	−16.6067	−0.773449	−0.386725	−	0.922195i	$-0.626394\pi$
$461$	−16.6067	−0.773449	−0.386725	+	0.922195i	$0.626394\pi$
$462$	13.0689	0.608021
$463$	−29.5498	−1.37330	−0.686648	−	0.726991i	$-0.740918\pi$
$463$	−29.5498	−1.37330	−0.686648	+	0.726991i	$0.740918\pi$
$464$	24.4248	1.13389
$465$	4.21642	0.195532
$466$	−61.9205	−2.86841
$467$	28.2613	1.30778	0.653888	−	0.756591i	$-0.273137\pi$
$467$	28.2613	1.30778	0.653888	+	0.756591i	$0.273137\pi$
$468$	0.342463	0.0158304
$469$	7.61334	0.351551
$470$	−19.4323	−0.896344
$471$	19.3196	0.890200
$472$	−13.1864	−0.606955
$473$	24.1684	1.11126
$474$	−12.0948	−0.555533
$475$	−10.5619	−0.484614
$476$	28.1366	1.28964
$477$	−0.573527	−0.0262600
$478$	−45.1660	−2.06584
$479$	10.7974	0.493348	0.246674	−	0.969099i	$-0.420662\pi$
$479$	10.7974	0.493348	0.246674	+	0.969099i	$0.420662\pi$
$480$	11.8418	0.540504
$481$	−2.12645	−0.0969579
$482$	−40.3977	−1.84006
$483$	1.20016	0.0546093
$484$	−27.0359	−1.22890
$485$	9.88023	0.448638
$486$	−3.37970	−0.153306
$487$	−18.5032	−0.838462	−0.419231	−	0.907880i	$-0.637700\pi$
$487$	−18.5032	−0.838462	−0.419231	+	0.907880i	$0.637700\pi$
$488$	21.9361	0.993002
$489$	34.6968	1.56904
$490$	9.17094	0.414301
$491$	8.86501	0.400072	0.200036	−	0.979789i	$-0.435894\pi$
$491$	8.86501	0.400072	0.200036	+	0.979789i	$0.435894\pi$
$492$	−54.3928	−2.45222
$493$	12.0484	0.542633
$494$	−3.41880	−0.153819
$495$	0.183933	0.00826715
$496$	−36.4187	−1.63525
$497$	−17.6117	−0.789993
$498$	−22.2710	−0.997988
$499$	30.8547	1.38124	0.690622	−	0.723216i	$-0.257337\pi$
$499$	30.8547	1.38124	0.690622	+	0.723216i	$0.257337\pi$
$500$	−29.4778	−1.31829
$501$	−31.9970	−1.42952
$502$	62.5181	2.79032
$503$	−23.8974	−1.06553	−0.532765	−	0.846263i	$-0.678847\pi$
$503$	−23.8974	−1.06553	−0.532765	+	0.846263i	$0.678847\pi$
$504$	−1.12174	−0.0499661
$505$	8.87369	0.394874
$506$	3.39376	0.150871
$507$	22.4066	0.995111
$508$	−54.9941	−2.43997
$509$	−12.7857	−0.566715	−0.283357	−	0.959014i	$-0.591448\pi$
$509$	−12.7857	−0.566715	−0.283357	+	0.959014i	$0.591448\pi$
$510$	14.0518	0.622225
$511$	14.5573	0.643976
$512$	41.4807	1.83321
$513$	11.6801	0.515688
$514$	40.4594	1.78459
$515$	6.50332	0.286570
$516$	−88.8116	−3.90971
$517$	27.2025	1.19636
$518$	11.8802	0.521986
$519$	36.2315	1.59039
$520$	−2.67938	−0.117499
$521$	−11.5015	−0.503891	−0.251945	−	0.967741i	$-0.581070\pi$
$521$	−11.5015	−0.503891	−0.251945	+	0.967741i	$0.581070\pi$
$522$	−0.819294	−0.0358595
$523$	20.0814	0.878099	0.439049	−	0.898463i	$-0.355315\pi$
$523$	20.0814	0.878099	0.439049	+	0.898463i	$0.355315\pi$
$524$	−12.9000	−0.563537
$525$	9.88101	0.431243
$526$	−60.3855	−2.63293
$527$	−17.9648	−0.782560
$528$	−39.8767	−1.73541
$529$	−22.6883	−0.986450
$530$	7.65361	0.332452
$531$	0.221536	0.00961384
$532$	13.5107	0.585765
$533$	3.62253	0.156909
$534$	36.8717	1.59559
$535$	11.2996	0.488526
$536$	−46.3817	−2.00338
$537$	30.8591	1.33167
$538$	−28.6251	−1.23411
$539$	−12.8380	−0.552974
$540$	15.6134	0.671896
$541$	6.25326	0.268849	0.134424	−	0.990924i	$-0.457081\pi$
$541$	6.25326	0.268849	0.134424	+	0.990924i	$0.457081\pi$
$542$	−72.8038	−3.12719
$543$	28.1232	1.20688
$544$	−50.4543	−2.16321
$545$	−1.61315	−0.0690999
$546$	3.19839	0.136879
$547$	31.0365	1.32702	0.663512	−	0.748166i	$-0.269065\pi$
$547$	31.0365	1.32702	0.663512	+	0.748166i	$0.269065\pi$
$548$	−62.5455	−2.67181
$549$	−0.368533	−0.0157286
$550$	27.9410	1.19141
$551$	5.78545	0.246468
$552$	−7.31159	−0.311202
$553$	−3.18328	−0.135367
$554$	−42.2134	−1.79348
$555$	4.19684	0.178146
$556$	−105.688	−4.48218
$557$	39.3710	1.66820	0.834102	−	0.551610i	$-0.185986\pi$
$557$	39.3710	1.66820	0.834102	+	0.551610i	$0.185986\pi$
$558$	1.22161	0.0517149
$559$	5.91480	0.250169
$560$	7.49739	0.316823
$561$	−19.6706	−0.830492
$562$	−53.2179	−2.24486
$563$	−22.9380	−0.966722	−0.483361	−	0.875421i	$-0.660584\pi$
$563$	−22.9380	−0.966722	−0.483361	+	0.875421i	$0.660584\pi$
$564$	−99.9610	−4.20912
$565$	6.25797	0.263275
$566$	−4.96825	−0.208831
$567$	−11.3813	−0.477968
$568$	107.293	4.50193
$569$	−8.76379	−0.367397	−0.183699	−	0.982983i	$-0.558807\pi$
$569$	−8.76379	−0.367397	−0.183699	+	0.982983i	$0.558807\pi$
$570$	6.74745	0.282620
$571$	32.5623	1.36269	0.681344	−	0.731963i	$-0.261396\pi$
$571$	32.5623	1.36269	0.681344	+	0.731963i	$0.261396\pi$
$572$	6.39751	0.267493
$573$	−12.3427	−0.515625
$574$	−20.2386	−0.844741
$575$	2.56592	0.107006
$576$	1.01565	0.0423187
$577$	40.2664	1.67631	0.838156	−	0.545431i	$-0.183634\pi$
$577$	40.2664	1.67631	0.838156	+	0.545431i	$0.183634\pi$
$578$	−15.4282	−0.641729
$579$	28.5533	1.18664
$580$	7.73375	0.321126
$581$	−5.86160	−0.243180
$582$	71.8515	2.97834
$583$	−10.7140	−0.443728
$584$	−88.6854	−3.66983
$585$	0.0450143	0.00186111
$586$	−68.6225	−2.83477
$587$	20.2642	0.836395	0.418197	−	0.908356i	$-0.362662\pi$
$587$	20.2642	0.836395	0.418197	+	0.908356i	$0.362662\pi$
$588$	47.1760	1.94550
$589$	−8.62642	−0.355445
$590$	−2.95635	−0.121711
$591$	32.2852	1.32804
$592$	−36.2496	−1.48985
$593$	8.40970	0.345345	0.172673	−	0.984979i	$-0.444760\pi$
$593$	8.40970	0.345345	0.172673	+	0.984979i	$0.444760\pi$
$594$	−30.8991	−1.26780
$595$	3.69836	0.151618
$596$	106.869	4.37751
$597$	−33.4071	−1.36726
$598$	0.830565	0.0339643
$599$	15.8064	0.645834	0.322917	−	0.946427i	$-0.395337\pi$
$599$	15.8064	0.645834	0.322917	+	0.946427i	$0.395337\pi$
$600$	−60.1967	−2.45752
$601$	−32.9172	−1.34272	−0.671359	−	0.741132i	$-0.734289\pi$
$601$	−32.9172	−1.34272	−0.671359	+	0.741132i	$0.734289\pi$
$602$	−33.0452	−1.34682
$603$	0.779225	0.0317325
$604$	−87.7177	−3.56918
$605$	−3.55367	−0.144477
$606$	64.5317	2.62142
$607$	−16.5774	−0.672857	−0.336428	−	0.941709i	$-0.609219\pi$
$607$	−16.5774	−0.672857	−0.336428	+	0.941709i	$0.609219\pi$
$608$	−24.2273	−0.982548
$609$	−5.41247	−0.219324
$610$	4.91800	0.199124
$611$	6.65734	0.269327
$612$	2.87978	0.116408
$613$	41.4845	1.67554	0.837772	−	0.546020i	$-0.183858\pi$
$613$	41.4845	1.67554	0.837772	+	0.546020i	$0.183858\pi$
$614$	39.5570	1.59639
$615$	−7.14954	−0.288297
$616$	−20.9550	−0.844301
$617$	−43.5895	−1.75485	−0.877423	−	0.479718i	$-0.840739\pi$
$617$	−43.5895	−1.75485	−0.877423	+	0.479718i	$0.840739\pi$
$618$	47.2937	1.90243
$619$	13.5986	0.546572	0.273286	−	0.961933i	$-0.411889\pi$
$619$	13.5986	0.546572	0.273286	+	0.961933i	$0.411889\pi$
$620$	−11.5314	−0.463114
$621$	−2.83757	−0.113868
$622$	−56.3373	−2.25892
$623$	9.70442	0.388799
$624$	−9.75913	−0.390678
$625$	19.1065	0.764261
$626$	−58.2012	−2.32619
$627$	−9.44549	−0.377217
$628$	−52.8369	−2.10842
$629$	−17.8814	−0.712978
$630$	−0.251489	−0.0100196
$631$	1.43274	0.0570365	0.0285183	−	0.999593i	$-0.490921\pi$
$631$	1.43274	0.0570365	0.0285183	+	0.999593i	$0.490921\pi$
$632$	19.3931	0.771416
$633$	5.75502	0.228741
$634$	62.7872	2.49360
$635$	−7.22858	−0.286857
$636$	39.3707	1.56115
$637$	−3.14189	−0.124486
$638$	−15.3051	−0.605936
$639$	−1.80256	−0.0713081
$640$	−0.155007	−0.00612721
$641$	−9.10492	−0.359623	−0.179811	−	0.983701i	$-0.557549\pi$
$641$	−9.10492	−0.359623	−0.179811	+	0.983701i	$0.557549\pi$
$642$	82.1738	3.24314
$643$	−15.6431	−0.616905	−0.308452	−	0.951240i	$-0.599811\pi$
$643$	−15.6431	−0.616905	−0.308452	+	0.951240i	$0.599811\pi$
$644$	−3.28231	−0.129341
$645$	−11.6736	−0.459649
$646$	−28.7487	−1.13110
$647$	−13.1595	−0.517353	−0.258677	−	0.965964i	$-0.583286\pi$
$647$	−13.1595	−0.517353	−0.258677	+	0.965964i	$0.583286\pi$
$648$	69.3366	2.72380
$649$	4.13848	0.162450
$650$	6.83809	0.268212
$651$	8.07029	0.316299
$652$	−94.8919	−3.71625
$653$	−2.05167	−0.0802882	−0.0401441	−	0.999194i	$-0.512782\pi$
$653$	−2.05167	−0.0802882	−0.0401441	+	0.999194i	$0.512782\pi$
$654$	−11.7313	−0.458728
$655$	−1.69561	−0.0662528
$656$	61.7531	2.41105
$657$	1.48994	0.0581280
$658$	−37.1936	−1.44996
$659$	0.884049	0.0344377	0.0172188	−	0.999852i	$-0.494519\pi$
$659$	0.884049	0.0344377	0.0172188	+	0.999852i	$0.494519\pi$
$660$	−12.6263	−0.491480
$661$	−0.984166	−0.0382796	−0.0191398	−	0.999817i	$-0.506093\pi$
$661$	−0.984166	−0.0382796	−0.0191398	+	0.999817i	$0.506093\pi$
$662$	65.3046	2.53814
$663$	−4.81404	−0.186962
$664$	35.7098	1.38581
$665$	1.77589	0.0688661
$666$	1.21594	0.0471167
$667$	−1.40552	−0.0544220
$668$	87.5082	3.38579
$669$	31.5905	1.22136
$670$	−10.3986	−0.401733
$671$	−6.88451	−0.265774
$672$	22.6654	0.874339
$673$	−28.7172	−1.10697	−0.553484	−	0.832860i	$-0.686702\pi$
$673$	−28.7172	−1.10697	−0.553484	+	0.832860i	$0.686702\pi$
$674$	8.42989	0.324707
$675$	−23.3619	−0.899198
$676$	−61.2795	−2.35690
$677$	20.5258	0.788871	0.394436	−	0.918924i	$-0.370940\pi$
$677$	20.5258	0.788871	0.394436	+	0.918924i	$0.370940\pi$
$678$	45.5095	1.74778
$679$	18.9109	0.725734
$680$	−22.5310	−0.864024
$681$	11.9762	0.458929
$682$	22.8208	0.873852
$683$	14.4392	0.552500	0.276250	−	0.961086i	$-0.410908\pi$
$683$	14.4392	0.552500	0.276250	+	0.961086i	$0.410908\pi$
$684$	1.38282	0.0528736
$685$	−8.22115	−0.314114
$686$	39.8096	1.51994
$687$	−27.6460	−1.05476
$688$	100.829	3.84408
$689$	−2.62207	−0.0998927
$690$	−1.63923	−0.0624045
$691$	−36.2435	−1.37877	−0.689383	−	0.724397i	$-0.742118\pi$
$691$	−36.2435	−1.37877	−0.689383	+	0.724397i	$0.742118\pi$
$692$	−99.0891	−3.76680
$693$	0.352049	0.0133733
$694$	−30.4625	−1.15634
$695$	−13.8920	−0.526953
$696$	32.9737	1.24986
$697$	30.4619	1.15383
$698$	−54.4969	−2.06274
$699$	−41.8676	−1.58358
$700$	−27.0234	−1.02139
$701$	26.8016	1.01228	0.506142	−	0.862450i	$-0.331071\pi$
$701$	26.8016	1.01228	0.506142	+	0.862450i	$0.331071\pi$
$702$	−7.56202	−0.285410
$703$	−8.58636	−0.323841
$704$	18.9732	0.715079
$705$	−13.1392	−0.494849
$706$	−54.7868	−2.06193
$707$	16.9844	0.638763
$708$	−15.2077	−0.571540
$709$	4.03134	0.151400	0.0757001	−	0.997131i	$-0.475881\pi$
$709$	4.03134	0.151400	0.0757001	+	0.997131i	$0.475881\pi$
$710$	24.0548	0.902760
$711$	−0.325809	−0.0122188
$712$	−59.1209	−2.21565
$713$	2.09571	0.0784849
$714$	26.8954	1.00653
$715$	0.840907	0.0314481
$716$	−84.3962	−3.15403
$717$	−30.5390	−1.14050
$718$	75.8082	2.82914
$719$	45.4399	1.69462	0.847310	−	0.531098i	$-0.178220\pi$
$719$	45.4399	1.69462	0.847310	+	0.531098i	$0.178220\pi$
$720$	0.767358	0.0285977
$721$	12.4474	0.463567
$722$	35.8658	1.33479
$723$	−27.3149	−1.01585
$724$	−76.9138	−2.85848
$725$	−11.5717	−0.429764
$726$	−25.8432	−0.959130
$727$	−17.8773	−0.663033	−0.331517	−	0.943449i	$-0.607560\pi$
$727$	−17.8773	−0.663033	−0.331517	+	0.943449i	$0.607560\pi$
$728$	−5.12838	−0.190070
$729$	25.7886	0.955135
$730$	−19.8829	−0.735900
$731$	49.7377	1.83961
$732$	25.2985	0.935061
$733$	19.4001	0.716560	0.358280	−	0.933614i	$-0.383363\pi$
$733$	19.4001	0.716560	0.358280	+	0.933614i	$0.383363\pi$
$734$	−52.3865	−1.93362
$735$	6.20094	0.228725
$736$	5.88581	0.216954
$737$	14.5566	0.536199
$738$	−2.07142	−0.0762499
$739$	−2.25640	−0.0830029	−0.0415014	−	0.999138i	$-0.513214\pi$
$739$	−2.25640	−0.0830029	−0.0415014	+	0.999138i	$0.513214\pi$
$740$	−11.4779	−0.421935
$741$	−2.31162	−0.0849196
$742$	14.6491	0.537786
$743$	8.03870	0.294911	0.147456	−	0.989069i	$-0.452892\pi$
$743$	8.03870	0.294911	0.147456	+	0.989069i	$0.452892\pi$
$744$	−49.1655	−1.80250
$745$	14.0471	0.514646
$746$	−58.1785	−2.13007
$747$	−0.599935	−0.0219505
$748$	53.7968	1.96701
$749$	21.6277	0.790258
$750$	−28.1773	−1.02889
$751$	−25.0536	−0.914219	−0.457110	−	0.889410i	$-0.651115\pi$
$751$	−25.0536	−0.914219	−0.457110	+	0.889410i	$0.651115\pi$
$752$	113.487	4.13846
$753$	42.2717	1.54047
$754$	−3.74567	−0.136409
$755$	−11.5299	−0.419615
$756$	29.8844	1.08688
$757$	31.6208	1.14928	0.574639	−	0.818407i	$-0.305142\pi$
$757$	31.6208	1.14928	0.574639	+	0.818407i	$0.305142\pi$
$758$	2.54836	0.0925605
$759$	2.29470	0.0832922
$760$	−10.8190	−0.392447
$761$	18.3459	0.665037	0.332519	−	0.943097i	$-0.392102\pi$
$761$	18.3459	0.665037	0.332519	+	0.943097i	$0.392102\pi$
$762$	−52.5680	−1.90434
$763$	−3.08760	−0.111779
$764$	33.7560	1.22125
$765$	0.378527	0.0136857
$766$	26.5605	0.959670
$767$	1.01282	0.0365709
$768$	27.7173	1.00016
$769$	−16.2301	−0.585271	−0.292635	−	0.956224i	$-0.594532\pi$
$769$	−16.2301	−0.585271	−0.292635	+	0.956224i	$0.594532\pi$
$770$	−4.69803	−0.169305
$771$	27.3567	0.985226
$772$	−78.0902	−2.81053
$773$	32.6118	1.17296	0.586482	−	0.809962i	$-0.300513\pi$
$773$	32.6118	1.17296	0.586482	+	0.809962i	$0.300513\pi$
$774$	−3.38217	−0.121570
$775$	17.2541	0.619785
$776$	−115.208	−4.13574
$777$	8.03281	0.288175
$778$	−30.2219	−1.08351
$779$	14.6273	0.524078
$780$	−3.09008	−0.110643
$781$	−33.6733	−1.20493
$782$	6.98424	0.249756
$783$	12.7968	0.457321
$784$	−53.5597	−1.91285
$785$	−6.94504	−0.247879
$786$	−12.3309	−0.439828
$787$	8.55390	0.304914	0.152457	−	0.988310i	$-0.451281\pi$
$787$	8.55390	0.304914	0.152457	+	0.988310i	$0.451281\pi$
$788$	−88.2964	−3.14543
$789$	−40.8297	−1.45358
$790$	4.34786	0.154690
$791$	11.9778	0.425883
$792$	−2.14474	−0.0762101
$793$	−1.68487	−0.0598314
$794$	−64.7720	−2.29867
$795$	5.17500	0.183538
$796$	91.3647	3.23833
$797$	−7.27866	−0.257823	−0.128912	−	0.991656i	$-0.541148\pi$
$797$	−7.27866	−0.257823	−0.128912	+	0.991656i	$0.541148\pi$
$798$	12.9147	0.457176
$799$	55.9817	1.98049
$800$	48.4582	1.71326
$801$	0.993247	0.0350947
$802$	4.49330	0.158664
$803$	27.8333	0.982217
$804$	−53.4911	−1.88649
$805$	−0.431436	−0.0152061
$806$	5.58499	0.196723
$807$	−19.3549	−0.681324
$808$	−103.472	−3.64012
$809$	29.7966	1.04759	0.523797	−	0.851843i	$-0.324515\pi$
$809$	29.7966	1.04759	0.523797	+	0.851843i	$0.324515\pi$
$810$	15.5450	0.546196
$811$	−51.0519	−1.79268	−0.896338	−	0.443371i	$-0.853782\pi$
$811$	−51.0519	−1.79268	−0.896338	+	0.443371i	$0.853782\pi$
$812$	14.8025	0.519466
$813$	−49.2263	−1.72644
$814$	22.7148	0.796153
$815$	−12.4729	−0.436905
$816$	−82.0647	−2.87284
$817$	23.8832	0.835568
$818$	−86.3353	−3.01864
$819$	0.0861581	0.00301061
$820$	19.5532	0.682827
$821$	22.6256	0.789640	0.394820	−	0.918758i	$-0.370807\pi$
$821$	22.6256	0.789640	0.394820	+	0.918758i	$0.370807\pi$
$822$	−59.7863	−2.08529
$823$	−19.2625	−0.671450	−0.335725	−	0.941960i	$-0.608981\pi$
$823$	−19.2625	−0.671450	−0.335725	+	0.941960i	$0.608981\pi$
$824$	−75.8319	−2.64173
$825$	18.8924	0.657747
$826$	−5.65850	−0.196884
$827$	−3.69779	−0.128585	−0.0642924	−	0.997931i	$-0.520479\pi$
$827$	−3.69779	−0.128585	−0.0642924	+	0.997931i	$0.520479\pi$
$828$	−0.335945	−0.0116749
$829$	−4.98917	−0.173281	−0.0866404	−	0.996240i	$-0.527613\pi$
$829$	−4.98917	−0.173281	−0.0866404	+	0.996240i	$0.527613\pi$
$830$	8.00601	0.277893
$831$	−28.5427	−0.990134
$832$	4.64337	0.160980
$833$	−26.4202	−0.915407
$834$	−101.026	−3.49824
$835$	11.5023	0.398054
$836$	25.8324	0.893431
$837$	−19.0807	−0.659527
$838$	−44.1013	−1.52346
$839$	11.3927	0.393319	0.196660	−	0.980472i	$-0.436991\pi$
$839$	11.3927	0.393319	0.196660	+	0.980472i	$0.436991\pi$
$840$	10.1215	0.349226
$841$	−22.6614	−0.781428
$842$	−91.2084	−3.14325
$843$	−35.9834	−1.23933
$844$	−15.7393	−0.541770
$845$	−8.05475	−0.277092
$846$	−3.80677	−0.130879
$847$	−6.80178	−0.233712
$848$	−44.6983	−1.53494
$849$	−3.35929	−0.115290
$850$	57.5016	1.97229
$851$	2.08598	0.0715063
$852$	123.739	4.23924
$853$	29.5432	1.01154	0.505769	−	0.862669i	$-0.331209\pi$
$853$	29.5432	1.01154	0.505769	+	0.862669i	$0.331209\pi$
$854$	9.41312	0.322110
$855$	0.181762	0.00621614
$856$	−131.759	−4.50344
$857$	−10.1137	−0.345479	−0.172739	−	0.984968i	$-0.555262\pi$
$857$	−10.1137	−0.345479	−0.172739	+	0.984968i	$0.555262\pi$
$858$	6.11528	0.208772
$859$	1.95689	0.0667683	0.0333842	−	0.999443i	$-0.489372\pi$
$859$	1.95689	0.0667683	0.0333842	+	0.999443i	$0.489372\pi$
$860$	31.9261	1.08867
$861$	−13.6843	−0.466361
$862$	19.7395	0.672330
$863$	40.6127	1.38247	0.691237	−	0.722629i	$-0.257066\pi$
$863$	40.6127	1.38247	0.691237	+	0.722629i	$0.257066\pi$
$864$	−53.5884	−1.82311
$865$	−13.0245	−0.442848
$866$	16.6607	0.566154
$867$	−10.4318	−0.354283
$868$	−22.0713	−0.749150
$869$	−6.08640	−0.206467
$870$	7.39258	0.250632
$871$	3.56248	0.120710
$872$	18.8102	0.636992
$873$	1.93553	0.0655078
$874$	3.35372	0.113441
$875$	−7.41611	−0.250710
$876$	−102.279	−3.45569
$877$	35.3103	1.19235	0.596173	−	0.802856i	$-0.296687\pi$
$877$	35.3103	1.19235	0.596173	+	0.802856i	$0.296687\pi$
$878$	−26.2364	−0.885435
$879$	−46.3992	−1.56501
$880$	14.3349	0.483230
$881$	−32.5461	−1.09651	−0.548253	−	0.836312i	$-0.684707\pi$
$881$	−32.5461	−1.09651	−0.548253	+	0.836312i	$0.684707\pi$
$882$	1.79658	0.0604941
$883$	51.3315	1.72744	0.863721	−	0.503970i	$-0.168128\pi$
$883$	51.3315	1.72744	0.863721	+	0.503970i	$0.168128\pi$
$884$	13.1658	0.442815
$885$	−1.99894	−0.0671937
$886$	99.3021	3.33612
$887$	41.0965	1.37988	0.689942	−	0.723864i	$-0.257636\pi$
$887$	41.0965	1.37988	0.689942	+	0.723864i	$0.257636\pi$
$888$	−48.9372	−1.64222
$889$	−13.8356	−0.464031
$890$	−13.2547	−0.444298
$891$	−21.7608	−0.729015
$892$	−86.3964	−2.89277
$893$	26.8815	0.899556
$894$	102.154	3.41654
$895$	−11.0933	−0.370807
$896$	−0.296687	−0.00991160
$897$	0.561588	0.0187509
$898$	3.06995	0.102445
$899$	−9.45119	−0.315215
$900$	−2.76585	−0.0921950
$901$	−22.0490	−0.734559
$902$	−38.6959	−1.28843
$903$	−22.3435	−0.743546
$904$	−72.9710	−2.42698
$905$	−10.1098	−0.336060
$906$	−83.8480	−2.78566
$907$	18.4224	0.611705	0.305852	−	0.952079i	$-0.401059\pi$
$907$	18.4224	0.611705	0.305852	+	0.952079i	$0.401059\pi$
$908$	−32.7536	−1.08697
$909$	1.73835	0.0576574
$910$	−1.14976	−0.0381143
$911$	17.9016	0.593106	0.296553	−	0.955016i	$-0.404163\pi$
$911$	17.9016	0.593106	0.296553	+	0.955016i	$0.404163\pi$
$912$	−39.4062	−1.30487
$913$	−11.2073	−0.370908
$914$	−24.7408	−0.818352
$915$	3.32531	0.109931
$916$	75.6086	2.49818
$917$	−3.24541	−0.107173
$918$	−63.5892	−2.09876
$919$	−46.5329	−1.53498	−0.767490	−	0.641061i	$-0.778494\pi$
$919$	−46.5329	−1.53498	−0.767490	+	0.641061i	$0.778494\pi$
$920$	2.62838	0.0866551
$921$	26.7465	0.881328
$922$	43.4138	1.42976
$923$	−8.24097	−0.271255
$924$	−24.1670	−0.795036
$925$	17.1740	0.564676
$926$	77.2502	2.53860
$927$	1.27400	0.0418435
$928$	−26.5437	−0.871340
$929$	11.6709	0.382911	0.191455	−	0.981501i	$-0.438679\pi$
$929$	11.6709	0.382911	0.191455	+	0.981501i	$0.438679\pi$
$930$	−11.0227	−0.361449
$931$	−12.6866	−0.415786
$932$	114.503	3.75068
$933$	−38.0925	−1.24709
$934$	−73.8817	−2.41748
$935$	7.07121	0.231253
$936$	−0.524889	−0.0171565
$937$	29.9619	0.978812	0.489406	−	0.872056i	$-0.337214\pi$
$937$	29.9619	0.978812	0.489406	+	0.872056i	$0.337214\pi$
$938$	−19.9031	−0.649859
$939$	−39.3528	−1.28423
$940$	35.9341	1.17204
$941$	47.1043	1.53556	0.767778	−	0.640715i	$-0.221362\pi$
$941$	47.1043	1.53556	0.767778	+	0.640715i	$0.221362\pi$
$942$	−50.5060	−1.64558
$943$	−3.55357	−0.115720
$944$	17.2656	0.561946
$945$	3.92808	0.127781
$946$	−63.1819	−2.05422
$947$	−0.464077	−0.0150805	−0.00754023	−	0.999972i	$-0.502400\pi$
$947$	−0.464077	−0.0150805	−0.00754023	+	0.999972i	$0.502400\pi$
$948$	22.3657	0.726404
$949$	6.81173	0.221118
$950$	27.6114	0.895831
$951$	42.4537	1.37665
$952$	−43.1246	−1.39768
$953$	28.0049	0.907168	0.453584	−	0.891214i	$-0.350145\pi$
$953$	28.0049	0.907168	0.453584	+	0.891214i	$0.350145\pi$
$954$	1.49934	0.0485428
$955$	4.43698	0.143577
$956$	83.5208	2.70126
$957$	−10.3486	−0.334522
$958$	−28.2271	−0.911975
$959$	−15.7354	−0.508123
$960$	−9.16430	−0.295777
$961$	−16.9078	−0.545412
$962$	5.55905	0.179231
$963$	2.21359	0.0713320
$964$	74.7032	2.40603
$965$	−10.2644	−0.330422
$966$	−3.13751	−0.100948
$967$	−39.5032	−1.27034	−0.635169	−	0.772373i	$-0.719069\pi$
$967$	−39.5032	−1.27034	−0.635169	+	0.772373i	$0.719069\pi$
$968$	41.4375	1.33185
$969$	−19.4385	−0.624454
$970$	−25.8293	−0.829328
$971$	−30.0799	−0.965311	−0.482656	−	0.875810i	$-0.660328\pi$
$971$	−30.0799	−0.965311	−0.482656	+	0.875810i	$0.660328\pi$
$972$	6.24974	0.200460
$973$	−26.5894	−0.852418
$974$	48.3719	1.54993
$975$	4.62358	0.148073
$976$	−28.7219	−0.919365
$977$	−41.0253	−1.31252	−0.656258	−	0.754536i	$-0.727862\pi$
$977$	−41.0253	−1.31252	−0.656258	+	0.754536i	$0.727862\pi$
$978$	−90.7057	−2.90045
$979$	18.5547	0.593011
$980$	−16.9589	−0.541732
$981$	−0.316016	−0.0100896
$982$	−23.1753	−0.739552
$983$	−34.4956	−1.10024	−0.550119	−	0.835086i	$-0.685418\pi$
$983$	−34.4956	−1.10024	−0.550119	+	0.835086i	$0.685418\pi$
$984$	83.3672	2.65765
$985$	−11.6059	−0.369795
$986$	−31.4974	−1.00308
$987$	−25.1485	−0.800486
$988$	6.32203	0.201131
$989$	−5.80221	−0.184500
$990$	−0.480843	−0.0152822
$991$	−5.26566	−0.167269	−0.0836345	−	0.996496i	$-0.526653\pi$
$991$	−5.26566	−0.167269	−0.0836345	+	0.996496i	$0.526653\pi$
$992$	39.5781	1.25661
$993$	44.1558	1.40124
$994$	46.0412	1.46034
$995$	12.0092	0.380718
$996$	41.1835	1.30495
$997$	34.1314	1.08095	0.540476	−	0.841359i	$-0.318244\pi$
$997$	34.1314	1.08095	0.540476	+	0.841359i	$0.318244\pi$
$998$	−80.6614	−2.55329
$999$	−18.9921	−0.600884

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4003.2.a.c.1.8	✓	179

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4003.2.a.c.1.8	✓	179	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:	\( N \)	\(=\)	\( 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61424 q^{2} -1.76762 q^{3} +4.83424 q^{4} +0.635426 q^{5} +4.62098 q^{6} +1.21621 q^{7} -7.40938 q^{8} +0.124480 q^{9} +O(q^{10})\)	\(q-2.61424 q^{2} -1.76762 q^{3} +4.83424 q^{4} +0.635426 q^{5} +4.62098 q^{6} +1.21621 q^{7} -7.40938 q^{8} +0.124480 q^{9} -1.66116 q^{10} +2.32539 q^{11} -8.54510 q^{12} +0.569099 q^{13} -3.17948 q^{14} -1.12319 q^{15} +9.70141 q^{16} +4.78556 q^{17} -0.325419 q^{18} +2.29795 q^{19} +3.07180 q^{20} -2.14981 q^{21} -6.07911 q^{22} -0.558266 q^{23} +13.0970 q^{24} -4.59623 q^{25} -1.48776 q^{26} +5.08283 q^{27} +5.87948 q^{28} +2.51766 q^{29} +2.93629 q^{30} -3.75396 q^{31} -10.5430 q^{32} -4.11040 q^{33} -12.5106 q^{34} +0.772815 q^{35} +0.601764 q^{36} -3.73653 q^{37} -6.00739 q^{38} -1.00595 q^{39} -4.70812 q^{40} +6.36538 q^{41} +5.62010 q^{42} +10.3933 q^{43} +11.2415 q^{44} +0.0790976 q^{45} +1.45944 q^{46} +11.6980 q^{47} -17.1484 q^{48} -5.52082 q^{49} +12.0156 q^{50} -8.45906 q^{51} +2.75116 q^{52} -4.60740 q^{53} -13.2877 q^{54} +1.47761 q^{55} -9.01140 q^{56} -4.06190 q^{57} -6.58176 q^{58} +1.77970 q^{59} -5.42978 q^{60} -2.96059 q^{61} +9.81375 q^{62} +0.151394 q^{63} +8.15916 q^{64} +0.361620 q^{65} +10.7456 q^{66} +6.25986 q^{67} +23.1346 q^{68} +0.986802 q^{69} -2.02032 q^{70} -14.4807 q^{71} -0.922317 q^{72} +11.9693 q^{73} +9.76817 q^{74} +8.12439 q^{75} +11.1088 q^{76} +2.82817 q^{77} +2.62979 q^{78} -2.61737 q^{79} +6.16453 q^{80} -9.35794 q^{81} -16.6406 q^{82} -4.81954 q^{83} -10.3927 q^{84} +3.04087 q^{85} -27.1705 q^{86} -4.45026 q^{87} -17.2297 q^{88} +7.97919 q^{89} -0.206780 q^{90} +0.692146 q^{91} -2.69879 q^{92} +6.63558 q^{93} -30.5815 q^{94} +1.46018 q^{95} +18.6361 q^{96} +15.5490 q^{97} +14.4327 q^{98} +0.289463 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 179 q + 22 q^{2} + 16 q^{3} + 196 q^{4} + 61 q^{5} + 7 q^{6} + 21 q^{7} + 60 q^{8} + 221 q^{9}+O(q^{10}) \) 179 * q + 22 * q^2 + 16 * q^3 + 196 * q^4 + 61 * q^5 + 7 * q^6 + 21 * q^7 + 60 * q^8 + 221 * q^9	\( 179 q + 22 q^{2} + 16 q^{3} + 196 q^{4} + 61 q^{5} + 7 q^{6} + 21 q^{7} + 60 q^{8} + 221 q^{9} + 9 q^{10} + 46 q^{11} + 33 q^{12} + 47 q^{13} + 22 q^{14} + 36 q^{15} + 222 q^{16} + 103 q^{17} + 43 q^{18} + 12 q^{19} + 102 q^{20} + 50 q^{21} + 39 q^{22} + 121 q^{23} - 3 q^{24} + 246 q^{25} + 52 q^{26} + 49 q^{27} + 41 q^{28} + 138 q^{29} + 28 q^{30} + 5 q^{31} + 137 q^{32} + 63 q^{33} + 2 q^{34} + 72 q^{35} + 279 q^{36} + 118 q^{37} + 123 q^{38} + q^{39} + 9 q^{40} + 50 q^{41} + 48 q^{42} + 48 q^{43} + 108 q^{44} + 158 q^{45} + 13 q^{46} + 85 q^{47} + 50 q^{48} + 230 q^{49} + 78 q^{50} + 15 q^{51} + 41 q^{52} + 399 q^{53} - 5 q^{54} + 24 q^{55} + 53 q^{56} + 45 q^{57} + 27 q^{58} + 48 q^{59} + 66 q^{60} + 46 q^{61} + 81 q^{62} + 78 q^{63} + 252 q^{64} + 153 q^{65} + 6 q^{66} + 70 q^{67} + 240 q^{68} + 120 q^{69} - 31 q^{70} + 86 q^{71} + 89 q^{72} + 45 q^{73} + 68 q^{74} + 17 q^{75} - 13 q^{76} + 362 q^{77} + 69 q^{78} + 31 q^{79} + 169 q^{80} + 303 q^{81} + 25 q^{82} + 106 q^{83} + 13 q^{84} + 115 q^{85} + 95 q^{86} + 32 q^{87} + 83 q^{88} + 105 q^{89} - 38 q^{90} + 3 q^{91} + 310 q^{92} + 298 q^{93} - 17 q^{94} + 102 q^{95} - 82 q^{96} + 34 q^{97} + 81 q^{98} + 58 q^{99}+O(q^{100}) \) 179 * q + 22 * q^2 + 16 * q^3 + 196 * q^4 + 61 * q^5 + 7 * q^6 + 21 * q^7 + 60 * q^8 + 221 * q^9 + 9 * q^10 + 46 * q^11 + 33 * q^12 + 47 * q^13 + 22 * q^14 + 36 * q^15 + 222 * q^16 + 103 * q^17 + 43 * q^18 + 12 * q^19 + 102 * q^20 + 50 * q^21 + 39 * q^22 + 121 * q^23 - 3 * q^24 + 246 * q^25 + 52 * q^26 + 49 * q^27 + 41 * q^28 + 138 * q^29 + 28 * q^30 + 5 * q^31 + 137 * q^32 + 63 * q^33 + 2 * q^34 + 72 * q^35 + 279 * q^36 + 118 * q^37 + 123 * q^38 + q^39 + 9 * q^40 + 50 * q^41 + 48 * q^42 + 48 * q^43 + 108 * q^44 + 158 * q^45 + 13 * q^46 + 85 * q^47 + 50 * q^48 + 230 * q^49 + 78 * q^50 + 15 * q^51 + 41 * q^52 + 399 * q^53 - 5 * q^54 + 24 * q^55 + 53 * q^56 + 45 * q^57 + 27 * q^58 + 48 * q^59 + 66 * q^60 + 46 * q^61 + 81 * q^62 + 78 * q^63 + 252 * q^64 + 153 * q^65 + 6 * q^66 + 70 * q^67 + 240 * q^68 + 120 * q^69 - 31 * q^70 + 86 * q^71 + 89 * q^72 + 45 * q^73 + 68 * q^74 + 17 * q^75 - 13 * q^76 + 362 * q^77 + 69 * q^78 + 31 * q^79 + 169 * q^80 + 303 * q^81 + 25 * q^82 + 106 * q^83 + 13 * q^84 + 115 * q^85 + 95 * q^86 + 32 * q^87 + 83 * q^88 + 105 * q^89 - 38 * q^90 + 3 * q^91 + 310 * q^92 + 298 * q^93 - 17 * q^94 + 102 * q^95 - 82 * q^96 + 34 * q^97 + 81 * q^98 + 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more