LMFDB - Embedded newform 4033.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4033.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4033 = 37 \cdot 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4033.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.2036671352$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	4033.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58335 q^{2} +3.13276 q^{3} +4.67371 q^{4} +4.37342 q^{5} -8.09303 q^{6} +0.189689 q^{7} -6.90713 q^{8} +6.81421 q^{9} +O(q^{10})$	\(q-2.58335 q^{2} +3.13276 q^{3} +4.67371 q^{4} +4.37342 q^{5} -8.09303 q^{6} +0.189689 q^{7} -6.90713 q^{8} +6.81421 q^{9} -11.2981 q^{10} +0.892119 q^{11} +14.6416 q^{12} +0.153333 q^{13} -0.490034 q^{14} +13.7009 q^{15} +8.49612 q^{16} -0.0610933 q^{17} -17.6035 q^{18} -2.64738 q^{19} +20.4401 q^{20} +0.594252 q^{21} -2.30466 q^{22} +1.72299 q^{23} -21.6384 q^{24} +14.1268 q^{25} -0.396113 q^{26} +11.9490 q^{27} +0.886552 q^{28} -1.29554 q^{29} -35.3942 q^{30} +3.31704 q^{31} -8.13422 q^{32} +2.79480 q^{33} +0.157825 q^{34} +0.829590 q^{35} +31.8476 q^{36} -1.00000 q^{37} +6.83911 q^{38} +0.480356 q^{39} -30.2077 q^{40} -1.36451 q^{41} -1.53516 q^{42} +9.87356 q^{43} +4.16951 q^{44} +29.8014 q^{45} -4.45108 q^{46} -6.27304 q^{47} +26.6164 q^{48} -6.96402 q^{49} -36.4944 q^{50} -0.191391 q^{51} +0.716634 q^{52} -2.05851 q^{53} -30.8686 q^{54} +3.90161 q^{55} -1.31021 q^{56} -8.29362 q^{57} +3.34685 q^{58} +7.04721 q^{59} +64.0339 q^{60} -4.89431 q^{61} -8.56908 q^{62} +1.29258 q^{63} +4.02132 q^{64} +0.670589 q^{65} -7.21995 q^{66} -14.4770 q^{67} -0.285532 q^{68} +5.39771 q^{69} -2.14312 q^{70} -10.3069 q^{71} -47.0666 q^{72} -10.1179 q^{73} +2.58335 q^{74} +44.2559 q^{75} -12.3731 q^{76} +0.169226 q^{77} -1.24093 q^{78} -1.79498 q^{79} +37.1571 q^{80} +16.9909 q^{81} +3.52500 q^{82} -8.06202 q^{83} +2.77736 q^{84} -0.267186 q^{85} -25.5069 q^{86} -4.05864 q^{87} -6.16198 q^{88} +11.5255 q^{89} -76.9875 q^{90} +0.0290856 q^{91} +8.05273 q^{92} +10.3915 q^{93} +16.2055 q^{94} -11.5781 q^{95} -25.4826 q^{96} +9.96625 q^{97} +17.9905 q^{98} +6.07909 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 10 q^{2} + 17 q^{3} + 88 q^{4} + 22 q^{5} + 4 q^{6} + 15 q^{7} + 33 q^{8} + 91 q^{9}+O(q^{10}) $ 82 * q + 10 * q^2 + 17 * q^3 + 88 * q^4 + 22 * q^5 + 4 * q^6 + 15 * q^7 + 33 * q^8 + 91 * q^9	\( 82 q + 10 q^{2} + 17 q^{3} + 88 q^{4} + 22 q^{5} + 4 q^{6} + 15 q^{7} + 33 q^{8} + 91 q^{9} + 13 q^{10} + 2 q^{11} + 36 q^{12} + 23 q^{13} + 24 q^{14} + 35 q^{15} + 104 q^{16} + 43 q^{17} + 42 q^{18} + 15 q^{19} + 59 q^{20} + 9 q^{21} + 6 q^{22} + 70 q^{23} + 15 q^{24} + 98 q^{25} + 10 q^{26} + 65 q^{27} + 37 q^{28} + 27 q^{29} + 17 q^{30} + 59 q^{31} + 46 q^{32} + 16 q^{33} - 16 q^{34} + 80 q^{35} + 88 q^{36} - 82 q^{37} + 82 q^{38} + 13 q^{39} + 14 q^{40} + 3 q^{41} + 62 q^{42} + 7 q^{43} - 11 q^{44} + 42 q^{45} - 11 q^{46} + 123 q^{47} + 45 q^{48} + 105 q^{49} + 27 q^{50} + 3 q^{51} - 30 q^{52} + 82 q^{53} - 27 q^{54} + 37 q^{55} + 66 q^{56} + 29 q^{57} - 34 q^{58} + 60 q^{59} + 94 q^{60} + 9 q^{61} - 2 q^{62} + 106 q^{63} + 93 q^{64} + 9 q^{65} + 63 q^{66} + 113 q^{68} + 48 q^{69} + 47 q^{70} + 59 q^{71} + 63 q^{72} + 21 q^{73} - 10 q^{74} + 77 q^{75} + 22 q^{76} + 30 q^{77} + 29 q^{78} + 55 q^{79} + 88 q^{80} + 42 q^{81} + 4 q^{82} + 92 q^{83} + 43 q^{84} - 7 q^{85} + 17 q^{86} + 147 q^{87} - 13 q^{88} + 100 q^{89} + 91 q^{90} + 28 q^{91} + 127 q^{92} + 3 q^{93} + 30 q^{94} + 48 q^{95} - 31 q^{96} + 53 q^{97} + 101 q^{98} - 20 q^{99}+O(q^{100}) \) 82 * q + 10 * q^2 + 17 * q^3 + 88 * q^4 + 22 * q^5 + 4 * q^6 + 15 * q^7 + 33 * q^8 + 91 * q^9 + 13 * q^10 + 2 * q^11 + 36 * q^12 + 23 * q^13 + 24 * q^14 + 35 * q^15 + 104 * q^16 + 43 * q^17 + 42 * q^18 + 15 * q^19 + 59 * q^20 + 9 * q^21 + 6 * q^22 + 70 * q^23 + 15 * q^24 + 98 * q^25 + 10 * q^26 + 65 * q^27 + 37 * q^28 + 27 * q^29 + 17 * q^30 + 59 * q^31 + 46 * q^32 + 16 * q^33 - 16 * q^34 + 80 * q^35 + 88 * q^36 - 82 * q^37 + 82 * q^38 + 13 * q^39 + 14 * q^40 + 3 * q^41 + 62 * q^42 + 7 * q^43 - 11 * q^44 + 42 * q^45 - 11 * q^46 + 123 * q^47 + 45 * q^48 + 105 * q^49 + 27 * q^50 + 3 * q^51 - 30 * q^52 + 82 * q^53 - 27 * q^54 + 37 * q^55 + 66 * q^56 + 29 * q^57 - 34 * q^58 + 60 * q^59 + 94 * q^60 + 9 * q^61 - 2 * q^62 + 106 * q^63 + 93 * q^64 + 9 * q^65 + 63 * q^66 + 113 * q^68 + 48 * q^69 + 47 * q^70 + 59 * q^71 + 63 * q^72 + 21 * q^73 - 10 * q^74 + 77 * q^75 + 22 * q^76 + 30 * q^77 + 29 * q^78 + 55 * q^79 + 88 * q^80 + 42 * q^81 + 4 * q^82 + 92 * q^83 + 43 * q^84 - 7 * q^85 + 17 * q^86 + 147 * q^87 - 13 * q^88 + 100 * q^89 + 91 * q^90 + 28 * q^91 + 127 * q^92 + 3 * q^93 + 30 * q^94 + 48 * q^95 - 31 * q^96 + 53 * q^97 + 101 * q^98 - 20 * 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 $

$2$	−2.58335	−1.82671	−0.913353	−	0.407169i	$-0.866516\pi$
$2$	−2.58335	−1.82671	−0.913353	+	0.407169i	$0.866516\pi$
$3$	3.13276	1.80870	0.904351	−	0.426789i	$-0.140355\pi$
$3$	3.13276	1.80870	0.904351	+	0.426789i	$0.140355\pi$
$4$	4.67371	2.33685
$5$	4.37342	1.95585	0.977926	−	0.208953i	$-0.0670055\pi$
$5$	4.37342	1.95585	0.977926	+	0.208953i	$0.0670055\pi$
$6$	−8.09303	−3.30397
$7$	0.189689	0.0716958	0.0358479	−	0.999357i	$-0.488587\pi$
$7$	0.189689	0.0716958	0.0358479	+	0.999357i	$0.488587\pi$
$8$	−6.90713	−2.44204
$9$	6.81421	2.27140
$10$	−11.2981	−3.57276
$11$	0.892119	0.268984	0.134492	−	0.990915i	$-0.457060\pi$
$11$	0.892119	0.268984	0.134492	+	0.990915i	$0.457060\pi$
$12$	14.6416	4.22667
$13$	0.153333	0.0425269	0.0212635	−	0.999774i	$-0.493231\pi$
$13$	0.153333	0.0425269	0.0212635	+	0.999774i	$0.493231\pi$
$14$	−0.490034	−0.130967
$15$	13.7009	3.53755
$16$	8.49612	2.12403
$17$	−0.0610933	−0.0148173	−0.00740865	−	0.999973i	$-0.502358\pi$
$17$	−0.0610933	−0.0148173	−0.00740865	+	0.999973i	$0.502358\pi$
$18$	−17.6035	−4.14919
$19$	−2.64738	−0.607351	−0.303675	−	0.952776i	$-0.598214\pi$
$19$	−2.64738	−0.607351	−0.303675	+	0.952776i	$0.598214\pi$
$20$	20.4401	4.57054
$21$	0.594252	0.129676
$22$	−2.30466	−0.491355
$23$	1.72299	0.359268	0.179634	−	0.983734i	$-0.442509\pi$
$23$	1.72299	0.359268	0.179634	+	0.983734i	$0.442509\pi$
$24$	−21.6384	−4.41692
$25$	14.1268	2.82536
$26$	−0.396113	−0.0776842
$27$	11.9490	2.29959
$28$	0.886552	0.167543
$29$	−1.29554	−0.240577	−0.120288	−	0.992739i	$-0.538382\pi$
$29$	−1.29554	−0.240577	−0.120288	+	0.992739i	$0.538382\pi$
$30$	−35.3942	−6.46207
$31$	3.31704	0.595758	0.297879	−	0.954604i	$-0.403721\pi$
$31$	3.31704	0.595758	0.297879	+	0.954604i	$0.403721\pi$
$32$	−8.13422	−1.43794
$33$	2.79480	0.486512
$34$	0.157825	0.0270668
$35$	0.829590	0.140226
$36$	31.8476	5.30794
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	6.83911	1.10945
$39$	0.480356	0.0769186
$40$	−30.2077	−4.77626
$41$	−1.36451	−0.213100	−0.106550	−	0.994307i	$-0.533980\pi$
$41$	−1.36451	−0.213100	−0.106550	+	0.994307i	$0.533980\pi$
$42$	−1.53516	−0.236881
$43$	9.87356	1.50570	0.752852	−	0.658190i	$-0.228677\pi$
$43$	9.87356	1.50570	0.752852	+	0.658190i	$0.228677\pi$
$44$	4.16951	0.628577
$45$	29.8014	4.44253
$46$	−4.45108	−0.656276
$47$	−6.27304	−0.915017	−0.457509	−	0.889205i	$-0.651258\pi$
$47$	−6.27304	−0.915017	−0.457509	+	0.889205i	$0.651258\pi$
$48$	26.6164	3.84174
$49$	−6.96402	−0.994860
$50$	−36.4944	−5.16109
$51$	−0.191391	−0.0268001
$52$	0.716634	0.0993792
$53$	−2.05851	−0.282758	−0.141379	−	0.989956i	$-0.545154\pi$
$53$	−2.05851	−0.282758	−0.141379	+	0.989956i	$0.545154\pi$
$54$	−30.8686	−4.20068
$55$	3.90161	0.526093
$56$	−1.31021	−0.175084
$57$	−8.29362	−1.09852
$58$	3.34685	0.439463
$59$	7.04721	0.917468	0.458734	−	0.888574i	$-0.348303\pi$
$59$	7.04721	0.917468	0.458734	+	0.888574i	$0.348303\pi$
$60$	64.0339	8.26674
$61$	−4.89431	−0.626652	−0.313326	−	0.949646i	$-0.601443\pi$
$61$	−4.89431	−0.626652	−0.313326	+	0.949646i	$0.601443\pi$
$62$	−8.56908	−1.08827
$63$	1.29258	0.162850
$64$	4.02132	0.502665
$65$	0.670589	0.0831763
$66$	−7.21995	−0.888715
$67$	−14.4770	−1.76865	−0.884323	−	0.466876i	$-0.845379\pi$
$67$	−14.4770	−1.76865	−0.884323	+	0.466876i	$0.845379\pi$
$68$	−0.285532	−0.0346258
$69$	5.39771	0.649808
$70$	−2.14312	−0.256152
$71$	−10.3069	−1.22321	−0.611603	−	0.791165i	$-0.709475\pi$
$71$	−10.3069	−1.22321	−0.611603	+	0.791165i	$0.709475\pi$
$72$	−47.0666	−5.54686
$73$	−10.1179	−1.18421	−0.592104	−	0.805862i	$-0.701702\pi$
$73$	−10.1179	−1.18421	−0.592104	+	0.805862i	$0.701702\pi$
$74$	2.58335	0.300309
$75$	44.2559	5.11023
$76$	−12.3731	−1.41929
$77$	0.169226	0.0192850
$78$	−1.24093	−0.140508
$79$	−1.79498	−0.201951	−0.100975	−	0.994889i	$-0.532196\pi$
$79$	−1.79498	−0.201951	−0.100975	+	0.994889i	$0.532196\pi$
$80$	37.1571	4.15429
$81$	16.9909	1.88788
$82$	3.52500	0.389272
$83$	−8.06202	−0.884921	−0.442461	−	0.896788i	$-0.645894\pi$
$83$	−8.06202	−0.884921	−0.442461	+	0.896788i	$0.645894\pi$
$84$	2.77736	0.303035
$85$	−0.267186	−0.0289804
$86$	−25.5069	−2.75048
$87$	−4.05864	−0.435132
$88$	−6.16198	−0.656869
$89$	11.5255	1.22170	0.610849	−	0.791747i	$-0.290828\pi$
$89$	11.5255	1.22170	0.610849	+	0.791747i	$0.290828\pi$
$90$	−76.9875	−8.11520
$91$	0.0290856	0.00304900
$92$	8.05273	0.839556
$93$	10.3915	1.07755
$94$	16.2055	1.67147
$95$	−11.5781	−1.18789
$96$	−25.4826	−2.60081
$97$	9.96625	1.01192	0.505960	−	0.862557i	$-0.331139\pi$
$97$	9.96625	1.01192	0.505960	+	0.862557i	$0.331139\pi$
$98$	17.9905	1.81732
$99$	6.07909	0.610972
$100$	66.0244	6.60244
$101$	−1.90108	−0.189165	−0.0945824	−	0.995517i	$-0.530152\pi$
$101$	−1.90108	−0.189165	−0.0945824	+	0.995517i	$0.530152\pi$
$102$	0.494430	0.0489559
$103$	10.0842	0.993630	0.496815	−	0.867856i	$-0.334503\pi$
$103$	10.0842	0.993630	0.496815	+	0.867856i	$0.334503\pi$
$104$	−1.05909	−0.103852
$105$	2.59891	0.253628
$106$	5.31785	0.516515
$107$	15.3101	1.48008	0.740041	−	0.672562i	$-0.234806\pi$
$107$	15.3101	1.48008	0.740041	+	0.672562i	$0.234806\pi$
$108$	55.8463	5.37381
$109$	1.00000	0.0957826
$109$	1.00000	0.0957826
$110$	−10.0792	−0.961017
$111$	−3.13276	−0.297349
$112$	1.61162	0.152284
$113$	−18.0582	−1.69878	−0.849388	−	0.527770i	$-0.823028\pi$
$113$	−18.0582	−1.69878	−0.849388	+	0.527770i	$0.823028\pi$
$114$	21.4253	2.00667
$115$	7.53534	0.702674
$116$	−6.05500	−0.562192
$117$	1.04484	0.0965959
$118$	−18.2054	−1.67594
$119$	−0.0115887	−0.00106234
$120$	−94.6337	−8.63884
$121$	−10.2041	−0.927648
$122$	12.6437	1.14471
$123$	−4.27468	−0.385435
$124$	15.5029	1.39220
$125$	39.9152	3.57012
$126$	−3.33920	−0.297479
$127$	−1.20747	−0.107146	−0.0535728	−	0.998564i	$-0.517061\pi$
$127$	−1.20747	−0.107146	−0.0535728	+	0.998564i	$0.517061\pi$
$128$	5.87997	0.519721
$129$	30.9316	2.72337
$130$	−1.73237	−0.151939
$131$	−1.84259	−0.160988	−0.0804940	−	0.996755i	$-0.525650\pi$
$131$	−1.84259	−0.160988	−0.0804940	+	0.996755i	$0.525650\pi$
$132$	13.0621	1.13691
$133$	−0.502180	−0.0435445
$134$	37.3991	3.23079
$135$	52.2581	4.49766
$136$	0.421979	0.0361844
$137$	−5.11125	−0.436684	−0.218342	−	0.975872i	$-0.570065\pi$
$137$	−5.11125	−0.436684	−0.218342	+	0.975872i	$0.570065\pi$
$138$	−13.9442	−1.18701
$139$	−9.31026	−0.789686	−0.394843	−	0.918749i	$-0.629201\pi$
$139$	−9.31026	−0.789686	−0.394843	+	0.918749i	$0.629201\pi$
$140$	3.87726	0.327689
$141$	−19.6520	−1.65499
$142$	26.6264	2.23444
$143$	0.136791	0.0114391
$144$	57.8944	4.82453
$145$	−5.66596	−0.470532
$146$	26.1380	2.16320
$147$	−21.8166	−1.79941
$148$	−4.67371	−0.384176
$149$	0.907748	0.0743657	0.0371828	−	0.999308i	$-0.488162\pi$
$149$	0.907748	0.0743657	0.0371828	+	0.999308i	$0.488162\pi$
$150$	−114.328	−9.33488
$151$	9.60452	0.781605	0.390802	−	0.920475i	$-0.372198\pi$
$151$	9.60452	0.781605	0.390802	+	0.920475i	$0.372198\pi$
$152$	18.2858	1.48317
$153$	−0.416303	−0.0336561
$154$	−0.437169	−0.0352281
$155$	14.5068	1.16521
$156$	2.24504	0.179747
$157$	8.67783	0.692566	0.346283	−	0.938130i	$-0.387444\pi$
$157$	8.67783	0.692566	0.346283	+	0.938130i	$0.387444\pi$
$158$	4.63706	0.368904
$159$	−6.44882	−0.511425
$160$	−35.5744	−2.81240
$161$	0.326832	0.0257580
$162$	−43.8934	−3.44859
$163$	2.49709	0.195587	0.0977936	−	0.995207i	$-0.468821\pi$
$163$	2.49709	0.195587	0.0977936	+	0.995207i	$0.468821\pi$
$164$	−6.37731	−0.497984
$165$	12.2228	0.951546
$166$	20.8270	1.61649
$167$	−9.79796	−0.758189	−0.379094	−	0.925358i	$-0.623764\pi$
$167$	−9.79796	−0.758189	−0.379094	+	0.925358i	$0.623764\pi$
$168$	−4.10457	−0.316675
$169$	−12.9765	−0.998191
$170$	0.690236	0.0529387
$171$	−18.0398	−1.37954
$172$	46.1461	3.51861
$173$	9.41232	0.715606	0.357803	−	0.933797i	$-0.383526\pi$
$173$	9.41232	0.715606	0.357803	+	0.933797i	$0.383526\pi$
$174$	10.4849	0.794857
$175$	2.67970	0.202566
$176$	7.57956	0.571331
$177$	22.0772	1.65943
$178$	−29.7744	−2.23168
$179$	3.06185	0.228853	0.114427	−	0.993432i	$-0.463497\pi$
$179$	3.06185	0.228853	0.114427	+	0.993432i	$0.463497\pi$
$180$	139.283	10.3815
$181$	14.7409	1.09569	0.547843	−	0.836581i	$-0.315449\pi$
$181$	14.7409	1.09569	0.547843	+	0.836581i	$0.315449\pi$
$182$	−0.0751384	−0.00556963
$183$	−15.3327	−1.13343
$184$	−11.9009	−0.877345
$185$	−4.37342	−0.321540
$186$	−26.8449	−1.96836
$187$	−0.0545025	−0.00398562
$188$	−29.3184	−2.13826
$189$	2.26660	0.164871
$190$	29.9103	2.16992
$191$	25.6317	1.85465	0.927323	−	0.374263i	$-0.122104\pi$
$191$	25.6317	1.85465	0.927323	+	0.374263i	$0.122104\pi$
$192$	12.5978	0.909171
$193$	−19.1038	−1.37512	−0.687561	−	0.726127i	$-0.741318\pi$
$193$	−19.1038	−1.37512	−0.687561	+	0.726127i	$0.741318\pi$
$194$	−25.7463	−1.84848
$195$	2.10080	0.150441
$196$	−32.5478	−2.32484
$197$	18.2896	1.30308	0.651541	−	0.758613i	$-0.274123\pi$
$197$	18.2896	1.30308	0.651541	+	0.758613i	$0.274123\pi$
$198$	−15.7044	−1.11607
$199$	−17.8337	−1.26420	−0.632099	−	0.774888i	$-0.717806\pi$
$199$	−17.8337	−1.26420	−0.632099	+	0.774888i	$0.717806\pi$
$200$	−97.5754	−6.89962
$201$	−45.3530	−3.19895
$202$	4.91117	0.345548
$203$	−0.245751	−0.0172483
$204$	−0.894505	−0.0626279
$205$	−5.96756	−0.416793
$206$	−26.0512	−1.81507
$207$	11.7408	0.816042
$208$	1.30274	0.0903285
$209$	−2.36178	−0.163368
$210$	−6.71390	−0.463303
$211$	26.5086	1.82493	0.912464	−	0.409156i	$-0.134177\pi$
$211$	26.5086	1.82493	0.912464	+	0.409156i	$0.134177\pi$
$212$	−9.62086	−0.660764
$213$	−32.2891	−2.21241
$214$	−39.5513	−2.70367
$215$	43.1812	2.94493
$216$	−82.5335	−5.61569
$217$	0.629207	0.0427133
$218$	−2.58335	−0.174967
$219$	−31.6969	−2.14188
$220$	18.2350	1.22940
$221$	−0.00936762	−0.000630134 0
$222$	8.09303	0.543169
$223$	−25.4869	−1.70673	−0.853366	−	0.521313i	$-0.825442\pi$
$223$	−25.4869	−1.70673	−0.853366	+	0.521313i	$0.825442\pi$
$224$	−1.54298	−0.103094
$225$	96.2629	6.41753
$226$	46.6508	3.10316
$227$	2.51199	0.166727	0.0833633	−	0.996519i	$-0.473434\pi$
$227$	2.51199	0.166727	0.0833633	+	0.996519i	$0.473434\pi$
$228$	−38.7619	−2.56707
$229$	−8.77425	−0.579819	−0.289909	−	0.957054i	$-0.593625\pi$
$229$	−8.77425	−0.579819	−0.289909	+	0.957054i	$0.593625\pi$
$230$	−19.4664	−1.28358
$231$	0.530144	0.0348809
$232$	8.94849	0.587497
$233$	−18.2359	−1.19467	−0.597336	−	0.801991i	$-0.703774\pi$
$233$	−18.2359	−1.19467	−0.597336	+	0.801991i	$0.703774\pi$
$234$	−2.69920	−0.176452
$235$	−27.4346	−1.78964
$236$	32.9366	2.14399
$237$	−5.62324	−0.365269
$238$	0.0299378	0.00194058
$239$	−26.7441	−1.72993	−0.864965	−	0.501832i	$-0.832660\pi$
$239$	−26.7441	−1.72993	−0.864965	+	0.501832i	$0.832660\pi$
$240$	116.404	7.51387
$241$	−26.6736	−1.71820	−0.859099	−	0.511810i	$-0.828975\pi$
$241$	−26.6736	−1.71820	−0.859099	+	0.511810i	$0.828975\pi$
$242$	26.3608	1.69454
$243$	17.3813	1.11501
$244$	−22.8746	−1.46439
$245$	−30.4566	−1.94580
$246$	11.0430	0.704077
$247$	−0.405931	−0.0258288
$248$	−22.9112	−1.45486
$249$	−25.2564	−1.60056
$250$	−103.115	−6.52156
$251$	−15.3535	−0.969105	−0.484553	−	0.874762i	$-0.661018\pi$
$251$	−15.3535	−0.969105	−0.484553	+	0.874762i	$0.661018\pi$
$252$	6.04116	0.380557
$253$	1.53711	0.0966373
$254$	3.11932	0.195723
$255$	−0.837032	−0.0524170
$256$	−23.2327	−1.45204
$257$	10.9370	0.682229	0.341114	−	0.940022i	$-0.389196\pi$
$257$	10.9370	0.682229	0.341114	+	0.940022i	$0.389196\pi$
$258$	−79.9071	−4.97480
$259$	−0.189689	−0.0117867
$260$	3.13414	0.194371
$261$	−8.82812	−0.546447
$262$	4.76006	0.294078
$263$	−1.03677	−0.0639303	−0.0319651	−	0.999489i	$-0.510177\pi$
$263$	−1.03677	−0.0639303	−0.0319651	+	0.999489i	$0.510177\pi$
$264$	−19.3040	−1.18808
$265$	−9.00271	−0.553032
$266$	1.29731	0.0795430
$267$	36.1066	2.20969
$268$	−67.6612	−4.13306
$269$	17.4339	1.06296	0.531482	−	0.847069i	$-0.321635\pi$
$269$	17.4339	1.06296	0.531482	+	0.847069i	$0.321635\pi$
$270$	−135.001	−8.21591
$271$	−31.7025	−1.92579	−0.962895	−	0.269878i	$-0.913017\pi$
$271$	−31.7025	−1.92579	−0.962895	+	0.269878i	$0.913017\pi$
$272$	−0.519056	−0.0314724
$273$	0.0911184	0.00551474
$274$	13.2042	0.797693
$275$	12.6028	0.759976
$276$	25.2273	1.51851
$277$	3.68650	0.221500	0.110750	−	0.993848i	$-0.464675\pi$
$277$	3.68650	0.221500	0.110750	+	0.993848i	$0.464675\pi$
$278$	24.0517	1.44252
$279$	22.6030	1.35321
$280$	−5.73009	−0.342438
$281$	24.6755	1.47202	0.736010	−	0.676971i	$-0.236708\pi$
$281$	24.6755	1.47202	0.736010	+	0.676971i	$0.236708\pi$
$282$	50.7679	3.02319
$283$	27.6103	1.64126	0.820632	−	0.571457i	$-0.193622\pi$
$283$	27.6103	1.64126	0.820632	+	0.571457i	$0.193622\pi$
$284$	−48.1715	−2.85845
$285$	−36.2714	−2.14854
$286$	−0.353380	−0.0208958
$287$	−0.258833	−0.0152784
$288$	−55.4284	−3.26615
$289$	−16.9963	−0.999780
$290$	14.6372	0.859524
$291$	31.2219	1.83026
$292$	−47.2880	−2.76732
$293$	11.2978	0.660023	0.330012	−	0.943977i	$-0.392947\pi$
$293$	11.2978	0.660023	0.330012	+	0.943977i	$0.392947\pi$
$294$	56.3600	3.28698
$295$	30.8204	1.79443
$296$	6.90713	0.401469
$297$	10.6600	0.618554
$298$	−2.34503	−0.135844
$299$	0.264191	0.0152785
$300$	206.839	11.9419
$301$	1.87291	0.107953
$302$	−24.8118	−1.42776
$303$	−5.95565	−0.342143
$304$	−22.4925	−1.29003
$305$	−21.4049	−1.22564
$306$	1.07546	0.0614797
$307$	13.2752	0.757655	0.378827	−	0.925467i	$-0.376327\pi$
$307$	13.2752	0.757655	0.378827	+	0.925467i	$0.376327\pi$
$308$	0.790911	0.0450663
$309$	31.5916	1.79718
$310$	−37.4761	−2.12850
$311$	9.54570	0.541287	0.270644	−	0.962680i	$-0.412764\pi$
$311$	9.54570	0.541287	0.270644	+	0.962680i	$0.412764\pi$
$312$	−3.31788	−0.187838
$313$	−23.9593	−1.35426	−0.677129	−	0.735864i	$-0.736776\pi$
$313$	−23.9593	−1.35426	−0.677129	+	0.735864i	$0.736776\pi$
$314$	−22.4179	−1.26511
$315$	5.65301	0.318511
$316$	−8.38919	−0.471929
$317$	14.1714	0.795943	0.397971	−	0.917398i	$-0.369714\pi$
$317$	14.1714	0.795943	0.397971	+	0.917398i	$0.369714\pi$
$318$	16.6596	0.934222
$319$	−1.15578	−0.0647113
$320$	17.5869	0.983137
$321$	47.9629	2.67703
$322$	−0.844322	−0.0470522
$323$	0.161737	0.00899929
$324$	79.4104	4.41169
$325$	2.16610	0.120154
$326$	−6.45086	−0.357280
$327$	3.13276	0.173242
$328$	9.42483	0.520399
$329$	−1.18993	−0.0656029
$330$	−31.5759	−1.73819
$331$	−5.43825	−0.298913	−0.149457	−	0.988768i	$-0.547752\pi$
$331$	−5.43825	−0.298913	−0.149457	+	0.988768i	$0.547752\pi$
$332$	−37.6795	−2.06793
$333$	−6.81421	−0.373417
$334$	25.3116	1.38499
$335$	−63.3139	−3.45921
$336$	5.04884	0.275437
$337$	−18.6365	−1.01520	−0.507598	−	0.861594i	$-0.669466\pi$
$337$	−18.6365	−1.01520	−0.507598	+	0.861594i	$0.669466\pi$
$338$	33.5228	1.82340
$339$	−56.5722	−3.07258
$340$	−1.24875	−0.0677230
$341$	2.95919	0.160249
$342$	46.6032	2.52001
$343$	−2.64883	−0.143023
$344$	−68.1980	−3.67699
$345$	23.6064	1.27093
$346$	−24.3153	−1.30720
$347$	−3.45387	−0.185413	−0.0927067	−	0.995693i	$-0.529552\pi$
$347$	−3.45387	−0.185413	−0.0927067	+	0.995693i	$0.529552\pi$
$348$	−18.9689	−1.01684
$349$	12.6209	0.675582	0.337791	−	0.941221i	$-0.390320\pi$
$349$	12.6209	0.675582	0.337791	+	0.941221i	$0.390320\pi$
$350$	−6.92260	−0.370029
$351$	1.83218	0.0977946
$352$	−7.25670	−0.386783
$353$	25.3393	1.34868	0.674338	−	0.738423i	$-0.264429\pi$
$353$	25.3393	1.34868	0.674338	+	0.738423i	$0.264429\pi$
$354$	−57.0333	−3.03128
$355$	−45.0764	−2.39241
$356$	53.8667	2.85493
$357$	−0.0363048	−0.00192145
$358$	−7.90984	−0.418048
$359$	−12.7518	−0.673013	−0.336507	−	0.941681i	$-0.609245\pi$
$359$	−12.7518	−0.673013	−0.336507	+	0.941681i	$0.609245\pi$
$360$	−205.842	−10.8488
$361$	−11.9914	−0.631125
$362$	−38.0810	−2.00149
$363$	−31.9671	−1.67784
$364$	0.135938	0.00712507
$365$	−44.2497	−2.31613
$366$	39.6098	2.07044
$367$	2.88227	0.150453	0.0752265	−	0.997166i	$-0.476032\pi$
$367$	2.88227	0.150453	0.0752265	+	0.997166i	$0.476032\pi$
$368$	14.6387	0.763095
$369$	−9.29805	−0.484037
$370$	11.2981	0.587359
$371$	−0.390477	−0.0202726
$372$	48.5668	2.51807
$373$	2.92673	0.151541	0.0757703	−	0.997125i	$-0.475858\pi$
$373$	2.92673	0.151541	0.0757703	+	0.997125i	$0.475858\pi$
$374$	0.140799	0.00728055
$375$	125.045	6.45729
$376$	43.3287	2.23451
$377$	−0.198650	−0.0102310
$378$	−5.85544	−0.301171
$379$	36.9755	1.89931	0.949653	−	0.313303i	$-0.101436\pi$
$379$	36.9755	1.89931	0.949653	+	0.313303i	$0.101436\pi$
$380$	−54.1126	−2.77592
$381$	−3.78272	−0.193794
$382$	−66.2157	−3.38789
$383$	11.0314	0.563679	0.281839	−	0.959462i	$-0.409055\pi$
$383$	11.0314	0.563679	0.281839	+	0.959462i	$0.409055\pi$
$384$	18.4206	0.940021
$385$	0.740094	0.0377187
$386$	49.3518	2.51194
$387$	67.2806	3.42006
$388$	46.5793	2.36471
$389$	−22.9962	−1.16595	−0.582977	−	0.812488i	$-0.698112\pi$
$389$	−22.9962	−1.16595	−0.582977	+	0.812488i	$0.698112\pi$
$390$	−5.42710	−0.274812
$391$	−0.105263	−0.00532337
$392$	48.1014	2.42949
$393$	−5.77241	−0.291179
$394$	−47.2486	−2.38035
$395$	−7.85018	−0.394985
$396$	28.4119	1.42775
$397$	11.7766	0.591053	0.295526	−	0.955335i	$-0.404505\pi$
$397$	11.7766	0.591053	0.295526	+	0.955335i	$0.404505\pi$
$398$	46.0707	2.30932
$399$	−1.57321	−0.0787591
$400$	120.023	6.00114
$401$	3.08976	0.154295	0.0771476	−	0.997020i	$-0.475419\pi$
$401$	3.08976	0.154295	0.0771476	+	0.997020i	$0.475419\pi$
$402$	117.163	5.84355
$403$	0.508611	0.0253357
$404$	−8.88511	−0.442051
$405$	74.3082	3.69240
$406$	0.634861	0.0315076
$407$	−0.892119	−0.0442207
$408$	1.32196	0.0654468
$409$	−19.2846	−0.953560	−0.476780	−	0.879023i	$-0.658196\pi$
$409$	−19.2846	−0.953560	−0.476780	+	0.879023i	$0.658196\pi$
$410$	15.4163	0.761358
$411$	−16.0124	−0.789831
$412$	47.1308	2.32197
$413$	1.33678	0.0657786
$414$	−30.3306	−1.49067
$415$	−35.2586	−1.73077
$416$	−1.24725	−0.0611512
$417$	−29.1669	−1.42831
$418$	6.10131	0.298425
$419$	16.4121	0.801784	0.400892	−	0.916125i	$-0.368700\pi$
$419$	16.4121	0.801784	0.400892	+	0.916125i	$0.368700\pi$
$420$	12.1466	0.592691
$421$	−17.6981	−0.862552	−0.431276	−	0.902220i	$-0.641937\pi$
$421$	−17.6981	−0.862552	−0.431276	+	0.902220i	$0.641937\pi$
$422$	−68.4811	−3.33361
$423$	−42.7458	−2.07837
$424$	14.2184	0.690505
$425$	−0.863051	−0.0418641
$426$	83.4142	4.04143
$427$	−0.928398	−0.0449283
$428$	71.5548	3.45873
$429$	0.428535	0.0206899
$430$	−111.552	−5.37953
$431$	0.241742	0.0116443	0.00582215	−	0.999983i	$-0.498147\pi$
$431$	0.241742	0.0116443	0.00582215	+	0.999983i	$0.498147\pi$
$432$	101.520	4.88441
$433$	−21.8411	−1.04962	−0.524809	−	0.851220i	$-0.675863\pi$
$433$	−21.8411	−1.04962	−0.524809	+	0.851220i	$0.675863\pi$
$434$	−1.62546	−0.0780247
$435$	−17.7501	−0.851053
$436$	4.67371	0.223830
$437$	−4.56140	−0.218201
$438$	81.8843	3.91258
$439$	−35.4838	−1.69355	−0.846774	−	0.531952i	$-0.821459\pi$
$439$	−35.4838	−1.69355	−0.846774	+	0.531952i	$0.821459\pi$
$440$	−26.9489	−1.28474
$441$	−47.4543	−2.25973
$442$	0.0241998	0.00115107
$443$	7.44825	0.353877	0.176938	−	0.984222i	$-0.443381\pi$
$443$	7.44825	0.353877	0.176938	+	0.984222i	$0.443381\pi$
$444$	−14.6416	−0.694861
$445$	50.4057	2.38946
$446$	65.8417	3.11770
$447$	2.84376	0.134505
$448$	0.762801	0.0360390
$449$	12.8817	0.607924	0.303962	−	0.952684i	$-0.401690\pi$
$449$	12.8817	0.607924	0.303962	+	0.952684i	$0.401690\pi$
$450$	−248.681	−11.7229
$451$	−1.21730	−0.0573206
$452$	−84.3989	−3.96979
$453$	30.0887	1.41369
$454$	−6.48935	−0.304560
$455$	0.127204	0.00596340
$456$	57.2851	2.68262
$457$	−2.66622	−0.124721	−0.0623603	−	0.998054i	$-0.519863\pi$
$457$	−2.66622	−0.124721	−0.0623603	+	0.998054i	$0.519863\pi$
$458$	22.6670	1.05916
$459$	−0.730006	−0.0340738
$460$	35.2180	1.64205
$461$	−4.48258	−0.208775	−0.104387	−	0.994537i	$-0.533288\pi$
$461$	−4.48258	−0.208775	−0.104387	+	0.994537i	$0.533288\pi$
$462$	−1.36955	−0.0637171
$463$	28.0885	1.30538	0.652691	−	0.757624i	$-0.273640\pi$
$463$	28.0885	1.30538	0.652691	+	0.757624i	$0.273640\pi$
$464$	−11.0071	−0.510992
$465$	45.4464	2.10752
$466$	47.1097	2.18232
$467$	26.1489	1.21003	0.605013	−	0.796215i	$-0.293168\pi$
$467$	26.1489	1.21003	0.605013	+	0.796215i	$0.293168\pi$
$468$	4.88329	0.225730
$469$	−2.74613	−0.126804
$470$	70.8733	3.26914
$471$	27.1856	1.25265
$472$	−48.6759	−2.24049
$473$	8.80840	0.405011
$474$	14.5268	0.667238
$475$	−37.3989	−1.71598
$476$	−0.0541624	−0.00248253
$477$	−14.0271	−0.642257
$478$	69.0893	3.16007
$479$	4.58561	0.209522	0.104761	−	0.994497i	$-0.466592\pi$
$479$	4.58561	0.209522	0.104761	+	0.994497i	$0.466592\pi$
$480$	−111.446	−5.08679
$481$	−0.153333	−0.00699138
$482$	68.9073	3.13864
$483$	1.02389	0.0465885
$484$	−47.6911	−2.16778
$485$	43.5866	1.97916
$486$	−44.9020	−2.03680
$487$	8.25225	0.373945	0.186973	−	0.982365i	$-0.440132\pi$
$487$	8.25225	0.373945	0.186973	+	0.982365i	$0.440132\pi$
$488$	33.8056	1.53031
$489$	7.82280	0.353759
$490$	78.6800	3.55440
$491$	6.77923	0.305942	0.152971	−	0.988231i	$-0.451116\pi$
$491$	6.77923	0.305942	0.152971	+	0.988231i	$0.451116\pi$
$492$	−19.9786	−0.900706
$493$	0.0791491	0.00356469
$494$	1.04866	0.0471815
$495$	26.5864	1.19497
$496$	28.1820	1.26541
$497$	−1.95511	−0.0876987
$498$	65.2462	2.92375
$499$	4.95058	0.221618	0.110809	−	0.993842i	$-0.464656\pi$
$499$	4.95058	0.221618	0.110809	+	0.993842i	$0.464656\pi$
$500$	186.552	8.34286
$501$	−30.6947	−1.37134
$502$	39.6635	1.77027
$503$	9.55662	0.426109	0.213054	−	0.977040i	$-0.431659\pi$
$503$	9.55662	0.426109	0.213054	+	0.977040i	$0.431659\pi$
$504$	−8.92804	−0.397686
$505$	−8.31423	−0.369978
$506$	−3.97090	−0.176528
$507$	−40.6523	−1.80543
$508$	−5.64336	−0.250383
$509$	−21.1116	−0.935753	−0.467877	−	0.883794i	$-0.654981\pi$
$509$	−21.1116	−0.935753	−0.467877	+	0.883794i	$0.654981\pi$
$510$	2.16235	0.0957504
$511$	−1.91925	−0.0849027
$512$	48.2582	2.13273
$513$	−31.6336	−1.39666
$514$	−28.2540	−1.24623
$515$	44.1026	1.94339
$516$	144.565	6.36412
$517$	−5.59630	−0.246125
$518$	0.490034	0.0215309
$519$	29.4866	1.29432
$520$	−4.63184	−0.203120
$521$	23.2800	1.01991	0.509957	−	0.860200i	$-0.329661\pi$
$521$	23.2800	1.01991	0.509957	+	0.860200i	$0.329661\pi$
$522$	22.8061	0.998198
$523$	−40.2167	−1.75855	−0.879276	−	0.476312i	$-0.841973\pi$
$523$	−40.2167	−1.75855	−0.879276	+	0.476312i	$0.841973\pi$
$524$	−8.61174	−0.376205
$525$	8.39486	0.366382
$526$	2.67835	0.116782
$527$	−0.202649	−0.00882752
$528$	23.7450	1.03337
$529$	−20.0313	−0.870927
$530$	23.2572	1.01023
$531$	48.0212	2.08394
$532$	−2.34704	−0.101757
$533$	−0.209224	−0.00906250
$534$	−93.2761	−4.03645
$535$	66.9574	2.89482
$536$	99.9944	4.31910
$537$	9.59206	0.413928
$538$	−45.0379	−1.94172
$539$	−6.21274	−0.267601
$540$	244.239	10.5104
$541$	34.2146	1.47100	0.735501	−	0.677524i	$-0.236947\pi$
$541$	34.2146	1.47100	0.735501	+	0.677524i	$0.236947\pi$
$542$	81.8987	3.51785
$543$	46.1799	1.98177
$544$	0.496946	0.0213064
$545$	4.37342	0.187337
$546$	−0.235391	−0.0100738
$547$	32.4262	1.38645	0.693223	−	0.720724i	$-0.256190\pi$
$547$	32.4262	1.38645	0.693223	+	0.720724i	$0.256190\pi$
$548$	−23.8885	−1.02047
$549$	−33.3509	−1.42338
$550$	−32.5574	−1.38825
$551$	3.42980	0.146114
$552$	−37.2827	−1.58686
$553$	−0.340488	−0.0144790
$554$	−9.52353	−0.404616
$555$	−13.7009	−0.581570
$556$	−43.5134	−1.84538
$557$	−26.8440	−1.13742	−0.568709	−	0.822539i	$-0.692557\pi$
$557$	−26.8440	−1.13742	−0.568709	+	0.822539i	$0.692557\pi$
$558$	−58.3915	−2.47191
$559$	1.51394	0.0640330
$560$	7.04830	0.297845
$561$	−0.170744	−0.00720880
$562$	−63.7456	−2.68895
$563$	10.3318	0.435434	0.217717	−	0.976012i	$-0.430139\pi$
$563$	10.3318	0.435434	0.217717	+	0.976012i	$0.430139\pi$
$564$	−91.8475	−3.86748
$565$	−78.9762	−3.32255
$566$	−71.3272	−2.99811
$567$	3.22299	0.135353
$568$	71.1911	2.98711
$569$	3.46863	0.145413	0.0727064	−	0.997353i	$-0.476836\pi$
$569$	3.46863	0.145413	0.0727064	+	0.997353i	$0.476836\pi$
$570$	93.7019	3.92474
$571$	−10.2712	−0.429836	−0.214918	−	0.976632i	$-0.568948\pi$
$571$	−10.2712	−0.429836	−0.214918	+	0.976632i	$0.568948\pi$
$572$	0.639323	0.0267314
$573$	80.2981	3.35450
$574$	0.668656	0.0279092
$575$	24.3402	1.01506
$576$	27.4021	1.14175
$577$	43.2035	1.79859	0.899293	−	0.437346i	$-0.144082\pi$
$577$	43.2035	1.79859	0.899293	+	0.437346i	$0.144082\pi$
$578$	43.9073	1.82630
$579$	−59.8477	−2.48718
$580$	−26.4810	−1.09956
$581$	−1.52928	−0.0634452
$582$	−80.6572	−3.34335
$583$	−1.83643	−0.0760574
$584$	69.8854	2.89188
$585$	4.56954	0.188927
$586$	−29.1861	−1.20567
$587$	5.28061	0.217954	0.108977	−	0.994044i	$-0.465242\pi$
$587$	5.28061	0.217954	0.108977	+	0.994044i	$0.465242\pi$
$588$	−101.965	−4.20495
$589$	−8.78146	−0.361834
$590$	−79.6199	−3.27790
$591$	57.2971	2.35689
$592$	−8.49612	−0.349189
$593$	−1.31290	−0.0539144	−0.0269572	−	0.999637i	$-0.508582\pi$
$593$	−1.31290	−0.0539144	−0.0269572	+	0.999637i	$0.508582\pi$
$594$	−27.5385	−1.12992
$595$	−0.0506824	−0.00207778
$596$	4.24255	0.173782
$597$	−55.8688	−2.28656
$598$	−0.682498	−0.0279094
$599$	23.6308	0.965530	0.482765	−	0.875750i	$-0.339633\pi$
$599$	23.6308	0.965530	0.482765	+	0.875750i	$0.339633\pi$
$600$	−305.681	−12.4794
$601$	−28.4939	−1.16229	−0.581146	−	0.813799i	$-0.697396\pi$
$601$	−28.4939	−1.16229	−0.581146	+	0.813799i	$0.697396\pi$
$602$	−4.83839	−0.197198
$603$	−98.6493	−4.01731
$604$	44.8887	1.82650
$605$	−44.6269	−1.81434
$606$	15.3855	0.624994
$607$	18.7211	0.759864	0.379932	−	0.925014i	$-0.375947\pi$
$607$	18.7211	0.759864	0.379932	+	0.925014i	$0.375947\pi$
$608$	21.5344	0.873335
$609$	−0.769880	−0.0311971
$610$	55.2963	2.23888
$611$	−0.961864	−0.0389129
$612$	−1.94568	−0.0786493
$613$	−41.6730	−1.68315	−0.841577	−	0.540136i	$-0.818373\pi$
$613$	−41.6730	−1.68315	−0.841577	+	0.540136i	$0.818373\pi$
$614$	−34.2945	−1.38401
$615$	−18.6950	−0.753854
$616$	−1.16886	−0.0470948
$617$	15.5323	0.625306	0.312653	−	0.949867i	$-0.398782\pi$
$617$	15.5323	0.625306	0.312653	+	0.949867i	$0.398782\pi$
$618$	−81.6122	−3.28292
$619$	−37.3241	−1.50018	−0.750092	−	0.661334i	$-0.769991\pi$
$619$	−37.3241	−1.50018	−0.750092	+	0.661334i	$0.769991\pi$
$620$	67.8005	2.72293
$621$	20.5880	0.826169
$622$	−24.6599	−0.988772
$623$	2.18626	0.0875907
$624$	4.08117	0.163377
$625$	103.932	4.15728
$626$	61.8952	2.47383
$627$	−7.39890	−0.295484
$628$	40.5576	1.61843
$629$	0.0610933	0.00243595
$630$	−14.6037	−0.581826
$631$	−41.2271	−1.64122	−0.820612	−	0.571486i	$-0.806367\pi$
$631$	−41.2271	−1.64122	−0.820612	+	0.571486i	$0.806367\pi$
$632$	12.3981	0.493171
$633$	83.0453	3.30075
$634$	−36.6096	−1.45395
$635$	−5.28076	−0.209561
$636$	−30.1399	−1.19512
$637$	−1.06781	−0.0423083
$638$	2.98579	0.118208
$639$	−70.2335	−2.77839
$640$	25.7156	1.01650
$641$	−30.1671	−1.19153	−0.595764	−	0.803160i	$-0.703151\pi$
$641$	−30.1671	−1.19153	−0.595764	+	0.803160i	$0.703151\pi$
$642$	−123.905	−4.89014
$643$	−29.8258	−1.17622	−0.588108	−	0.808783i	$-0.700127\pi$
$643$	−29.8258	−1.17622	−0.588108	+	0.808783i	$0.700127\pi$
$644$	1.52752	0.0601926
$645$	135.277	5.32651
$646$	−0.417824	−0.0164391
$647$	35.7560	1.40571	0.702856	−	0.711332i	$-0.251908\pi$
$647$	35.7560	1.40571	0.702856	+	0.711332i	$0.251908\pi$
$648$	−117.358	−4.61026
$649$	6.28695	0.246784
$650$	−5.59580	−0.219485
$651$	1.97116	0.0772557
$652$	11.6707	0.457059
$653$	−38.2410	−1.49649	−0.748243	−	0.663425i	$-0.769102\pi$
$653$	−38.2410	−1.49649	−0.748243	+	0.663425i	$0.769102\pi$
$654$	−8.09303	−0.316463
$655$	−8.05842	−0.314869
$656$	−11.5930	−0.452632
$657$	−68.9453	−2.68981
$658$	3.07401	0.119837
$659$	37.6525	1.46673	0.733367	−	0.679833i	$-0.237948\pi$
$659$	37.6525	1.46673	0.733367	+	0.679833i	$0.237948\pi$
$660$	57.1259	2.22362
$661$	−31.5721	−1.22801	−0.614006	−	0.789301i	$-0.710443\pi$
$661$	−31.5721	−1.22801	−0.614006	+	0.789301i	$0.710443\pi$
$662$	14.0489	0.546027
$663$	−0.0293465	−0.00113973
$664$	55.6854	2.16101
$665$	−2.19624	−0.0851666
$666$	17.6035	0.682122
$667$	−2.23221	−0.0864314
$668$	−45.7928	−1.77178
$669$	−79.8446	−3.08697
$670$	163.562	6.31895
$671$	−4.36631	−0.168559
$672$	−4.83378	−0.186467
$673$	23.4429	0.903657	0.451829	−	0.892105i	$-0.350772\pi$
$673$	23.4429	0.903657	0.451829	+	0.892105i	$0.350772\pi$
$674$	48.1447	1.85446
$675$	168.801	6.49717
$676$	−60.6483	−2.33263
$677$	−1.19181	−0.0458051	−0.0229026	−	0.999738i	$-0.507291\pi$
$677$	−1.19181	−0.0458051	−0.0229026	+	0.999738i	$0.507291\pi$
$678$	146.146	5.61270
$679$	1.89049	0.0725504
$680$	1.84549	0.0707713
$681$	7.86947	0.301559
$682$	−7.64464	−0.292728
$683$	−10.3088	−0.394456	−0.197228	−	0.980358i	$-0.563194\pi$
$683$	−10.3088	−0.394456	−0.197228	+	0.980358i	$0.563194\pi$
$684$	−84.3128	−3.22378
$685$	−22.3536	−0.854089
$686$	6.84285	0.261261
$687$	−27.4876	−1.04872
$688$	83.8870	3.19816
$689$	−0.315637	−0.0120248
$690$	−60.9837	−2.32161
$691$	27.7868	1.05706	0.528531	−	0.848914i	$-0.322743\pi$
$691$	27.7868	1.05706	0.528531	+	0.848914i	$0.322743\pi$
$692$	43.9904	1.67227
$693$	1.15314	0.0438041
$694$	8.92256	0.338696
$695$	−40.7177	−1.54451
$696$	28.0335	1.06261
$697$	0.0833623	0.00315757
$698$	−32.6043	−1.23409
$699$	−57.1287	−2.16081
$700$	12.5241	0.473367
$701$	28.2951	1.06869	0.534346	−	0.845266i	$-0.320558\pi$
$701$	28.2951	1.06869	0.534346	+	0.845266i	$0.320558\pi$
$702$	−4.73317	−0.178642
$703$	2.64738	0.0998478
$704$	3.58750	0.135209
$705$	−85.9462	−3.23692
$706$	−65.4604	−2.46363
$707$	−0.360615	−0.0135623
$708$	103.183	3.87784
$709$	36.0508	1.35391	0.676957	−	0.736022i	$-0.263298\pi$
$709$	36.0508	1.35391	0.676957	+	0.736022i	$0.263298\pi$
$710$	116.448	4.37023
$711$	−12.2314	−0.458712
$712$	−79.6080	−2.98343
$713$	5.71521	0.214036
$714$	0.0937881	0.00350993
$715$	0.598246	0.0223731
$716$	14.3102	0.534797
$717$	−83.7829	−3.12893
$718$	32.9423	1.22940
$719$	−11.0875	−0.413494	−0.206747	−	0.978394i	$-0.566288\pi$
$719$	−11.0875	−0.413494	−0.206747	+	0.978394i	$0.566288\pi$
$720$	253.196	9.43607
$721$	1.91287	0.0712392
$722$	30.9780	1.15288
$723$	−83.5621	−3.10771
$724$	68.8948	2.56046
$725$	−18.3019	−0.679714
$726$	82.5823	3.06492
$727$	5.54880	0.205794	0.102897	−	0.994692i	$-0.467189\pi$
$727$	5.54880	0.205794	0.102897	+	0.994692i	$0.467189\pi$
$728$	−0.200898	−0.00744578
$729$	3.47893	0.128849
$730$	114.312	4.23089
$731$	−0.603208	−0.0223105
$732$	−71.6606	−2.64865
$733$	−10.1115	−0.373476	−0.186738	−	0.982410i	$-0.559792\pi$
$733$	−10.1115	−0.373476	−0.186738	+	0.982410i	$0.559792\pi$
$734$	−7.44591	−0.274833
$735$	−95.4132	−3.51937
$736$	−14.0152	−0.516606
$737$	−12.9152	−0.475738
$738$	24.0201	0.884194
$739$	19.8481	0.730126	0.365063	−	0.930983i	$-0.381047\pi$
$739$	19.8481	0.730126	0.365063	+	0.930983i	$0.381047\pi$
$740$	−20.4401	−0.751392
$741$	−1.27169	−0.0467165
$742$	1.00874	0.0370320
$743$	1.78821	0.0656030	0.0328015	−	0.999462i	$-0.489557\pi$
$743$	1.78821	0.0656030	0.0328015	+	0.999462i	$0.489557\pi$
$744$	−71.7754	−2.63141
$745$	3.96996	0.145448
$746$	−7.56078	−0.276820
$747$	−54.9363	−2.01001
$748$	−0.254729	−0.00931380
$749$	2.90416	0.106116
$750$	−323.035	−11.7956
$751$	30.3433	1.10724	0.553621	−	0.832769i	$-0.313246\pi$
$751$	30.3433	1.10724	0.553621	+	0.832769i	$0.313246\pi$
$752$	−53.2965	−1.94352
$753$	−48.0989	−1.75282
$754$	0.513182	0.0186890
$755$	42.0046	1.52870
$756$	10.5934	0.385280
$757$	−4.37313	−0.158944	−0.0794721	−	0.996837i	$-0.525323\pi$
$757$	−4.37313	−0.158944	−0.0794721	+	0.996837i	$0.525323\pi$
$758$	−95.5209	−3.46947
$759$	4.81540	0.174788
$760$	79.9714	2.90087
$761$	−32.9035	−1.19275	−0.596375	−	0.802706i	$-0.703393\pi$
$761$	−32.9035	−1.19275	−0.596375	+	0.802706i	$0.703393\pi$
$762$	9.77209	0.354005
$763$	0.189689	0.00686721
$764$	119.795	4.33403
$765$	−1.82067	−0.0658263
$766$	−28.4980	−1.02968
$767$	1.08057	0.0390171
$768$	−72.7825	−2.62631
$769$	−31.1333	−1.12270	−0.561348	−	0.827580i	$-0.689717\pi$
$769$	−31.1333	−1.12270	−0.561348	+	0.827580i	$0.689717\pi$
$770$	−1.91192	−0.0689009
$771$	34.2629	1.23395
$772$	−89.2855	−3.21346
$773$	−28.4150	−1.02202	−0.511008	−	0.859576i	$-0.670728\pi$
$773$	−28.4150	−1.02202	−0.511008	+	0.859576i	$0.670728\pi$
$774$	−173.809	−6.24745
$775$	46.8591	1.68323
$776$	−68.8381	−2.47114
$777$	−0.594252	−0.0213187
$778$	59.4073	2.12986
$779$	3.61237	0.129427
$780$	9.81851	0.351559
$781$	−9.19500	−0.329023
$782$	0.271931	0.00972424
$783$	−15.4805	−0.553228
$784$	−59.1672	−2.11311
$785$	37.9518	1.35456
$786$	14.9122	0.531899
$787$	36.3618	1.29616	0.648079	−	0.761573i	$-0.275572\pi$
$787$	36.3618	1.29616	0.648079	+	0.761573i	$0.275572\pi$
$788$	85.4804	3.04511
$789$	−3.24797	−0.115631
$790$	20.2798	0.721522
$791$	−3.42545	−0.121795
$792$	−41.9891	−1.49202
$793$	−0.750459	−0.0266496
$794$	−30.4232	−1.07968
$795$	−28.2034	−1.00027
$796$	−83.3495	−2.95424
$797$	−26.2603	−0.930189	−0.465094	−	0.885261i	$-0.653980\pi$
$797$	−26.2603	−0.930189	−0.465094	+	0.885261i	$0.653980\pi$
$798$	4.06416	0.143870
$799$	0.383241	0.0135581
$800$	−114.910	−4.06269
$801$	78.5371	2.77497
$802$	−7.98193	−0.281852
$803$	−9.02635	−0.318533
$804$	−211.967	−7.47548
$805$	1.42937	0.0503788
$806$	−1.31392	−0.0462809
$807$	54.6164	1.92259
$808$	13.1310	0.461948
$809$	45.9544	1.61567	0.807836	−	0.589408i	$-0.200639\pi$
$809$	45.9544	1.61567	0.807836	+	0.589408i	$0.200639\pi$
$810$	−191.964	−6.74494
$811$	46.0495	1.61702	0.808509	−	0.588484i	$-0.200275\pi$
$811$	46.0495	1.61702	0.808509	+	0.588484i	$0.200275\pi$
$812$	−1.14857	−0.0403068
$813$	−99.3164	−3.48318
$814$	2.30466	0.0807782
$815$	10.9208	0.382540
$816$	−1.62608	−0.0569242
$817$	−26.1391	−0.914491
$818$	49.8188	1.74187
$819$	0.198196	0.00692552
$820$	−27.8906	−0.973984
$821$	9.61637	0.335614	0.167807	−	0.985820i	$-0.446332\pi$
$821$	9.61637	0.335614	0.167807	+	0.985820i	$0.446332\pi$
$822$	41.3656	1.44279
$823$	14.2738	0.497552	0.248776	−	0.968561i	$-0.419972\pi$
$823$	14.2738	0.497552	0.248776	+	0.968561i	$0.419972\pi$
$824$	−69.6532	−2.42648
$825$	39.4815	1.37457
$826$	−3.45337	−0.120158
$827$	−30.7249	−1.06841	−0.534204	−	0.845355i	$-0.679389\pi$
$827$	−30.7249	−1.06841	−0.534204	+	0.845355i	$0.679389\pi$
$828$	54.8731	1.90697
$829$	−2.95832	−0.102747	−0.0513734	−	0.998680i	$-0.516360\pi$
$829$	−2.95832	−0.102747	−0.0513734	+	0.998680i	$0.516360\pi$
$830$	91.0853	3.16162
$831$	11.5489	0.400628
$832$	0.616601	0.0213768
$833$	0.425455	0.0147411
$834$	75.3483	2.60910
$835$	−42.8506	−1.48290
$836$	−11.0383	−0.381766
$837$	39.6354	1.37000
$838$	−42.3983	−1.46462
$839$	−26.4222	−0.912195	−0.456098	−	0.889930i	$-0.650753\pi$
$839$	−26.4222	−0.912195	−0.456098	+	0.889930i	$0.650753\pi$
$840$	−17.9510	−0.619369
$841$	−27.3216	−0.942123
$842$	45.7204	1.57563
$843$	77.3027	2.66244
$844$	123.894	4.26459
$845$	−56.7516	−1.95231
$846$	110.428	3.79658
$847$	−1.93561	−0.0665085
$848$	−17.4893	−0.600586
$849$	86.4967	2.96856
$850$	2.22956	0.0764734
$851$	−1.72299	−0.0590632
$852$	−150.910	−5.17009
$853$	−51.6281	−1.76771	−0.883857	−	0.467758i	$-0.845062\pi$
$853$	−51.6281	−1.76771	−0.883857	+	0.467758i	$0.845062\pi$
$854$	2.39838	0.0820708
$855$	−78.8956	−2.69817
$856$	−105.749	−3.61442
$857$	30.8514	1.05386	0.526932	−	0.849907i	$-0.323342\pi$
$857$	30.8514	1.05386	0.526932	+	0.849907i	$0.323342\pi$
$858$	−1.10706	−0.0377943
$859$	−31.2555	−1.06643	−0.533213	−	0.845981i	$-0.679015\pi$
$859$	−31.2555	−1.06643	−0.533213	+	0.845981i	$0.679015\pi$
$860$	201.816	6.88188
$861$	−0.810862	−0.0276341
$862$	−0.624505	−0.0212707
$863$	50.1723	1.70789	0.853943	−	0.520366i	$-0.174205\pi$
$863$	50.1723	1.70789	0.853943	+	0.520366i	$0.174205\pi$
$864$	−97.1962	−3.30668
$865$	41.1640	1.39962
$866$	56.4233	1.91734
$867$	−53.2453	−1.80831
$868$	2.94073	0.0998148
$869$	−1.60133	−0.0543215
$870$	45.8548	1.55462
$871$	−2.21980	−0.0752150
$872$	−6.90713	−0.233905
$873$	67.9121	2.29848
$874$	11.7837	0.398590
$875$	7.57149	0.255963
$876$	−148.142	−5.00526
$877$	−31.2632	−1.05568	−0.527841	−	0.849343i	$-0.676998\pi$
$877$	−31.2632	−1.05568	−0.527841	+	0.849343i	$0.676998\pi$
$878$	91.6671	3.09361
$879$	35.3933	1.19379
$880$	33.1486	1.11744
$881$	11.0811	0.373332	0.186666	−	0.982423i	$-0.440232\pi$
$881$	11.0811	0.373332	0.186666	+	0.982423i	$0.440232\pi$
$882$	122.591	4.12786
$883$	36.5579	1.23027	0.615136	−	0.788421i	$-0.289101\pi$
$883$	36.5579	1.23027	0.615136	+	0.788421i	$0.289101\pi$
$884$	−0.0437815	−0.00147253
$885$	96.5530	3.24559
$886$	−19.2414	−0.646429
$887$	25.2542	0.847952	0.423976	−	0.905673i	$-0.360634\pi$
$887$	25.2542	0.847952	0.423976	+	0.905673i	$0.360634\pi$
$888$	21.6384	0.726137
$889$	−0.229044	−0.00768189
$890$	−130.216	−4.36484
$891$	15.1579	0.507809
$892$	−119.118	−3.98838
$893$	16.6071	0.555736
$894$	−7.34644	−0.245702
$895$	13.3907	0.447603
$896$	1.11537	0.0372618
$897$	0.827647	0.0276343
$898$	−33.2779	−1.11050
$899$	−4.29737	−0.143325
$900$	449.905	14.9968
$901$	0.125761	0.00418971
$902$	3.14473	0.104708
$903$	5.86739	0.195254
$904$	124.730	4.14847
$905$	64.4683	2.14300
$906$	−77.7297	−2.58240
$907$	34.0279	1.12988	0.564939	−	0.825133i	$-0.308900\pi$
$907$	34.0279	1.12988	0.564939	+	0.825133i	$0.308900\pi$
$908$	11.7403	0.389616
$909$	−12.9544	−0.429670
$910$	−0.328612	−0.0108934
$911$	−9.20705	−0.305043	−0.152522	−	0.988300i	$-0.548739\pi$
$911$	−9.20705	−0.305043	−0.152522	+	0.988300i	$0.548739\pi$
$912$	−70.4636	−2.33328
$913$	−7.19228	−0.238030
$914$	6.88779	0.227828
$915$	−67.0564	−2.21682
$916$	−41.0083	−1.35495
$917$	−0.349520	−0.0115422
$918$	1.88586	0.0622427
$919$	−40.4927	−1.33573	−0.667865	−	0.744283i	$-0.732792\pi$
$919$	−40.4927	−1.33573	−0.667865	+	0.744283i	$0.732792\pi$
$920$	−52.0475	−1.71596
$921$	41.5880	1.37037
$922$	11.5801	0.381370
$923$	−1.58039	−0.0520192
$924$	2.47774	0.0815116
$925$	−14.1268	−0.464486
$926$	−72.5624	−2.38455
$927$	68.7162	2.25694
$928$	10.5383	0.345935
$929$	−53.0794	−1.74148	−0.870740	−	0.491744i	$-0.836360\pi$
$929$	−53.0794	−1.74148	−0.870740	+	0.491744i	$0.836360\pi$
$930$	−117.404	−3.84983
$931$	18.4364	0.604229
$932$	−85.2292	−2.79177
$933$	29.9044	0.979027
$934$	−67.5518	−2.21036
$935$	−0.238362	−0.00779528
$936$	−7.21687	−0.235891
$937$	5.54739	0.181225	0.0906127	−	0.995886i	$-0.471117\pi$
$937$	5.54739	0.181225	0.0906127	+	0.995886i	$0.471117\pi$
$938$	7.09422	0.231634
$939$	−75.0587	−2.44945
$940$	−128.221	−4.18212
$941$	33.8031	1.10195	0.550975	−	0.834522i	$-0.314256\pi$
$941$	33.8031	1.10195	0.550975	+	0.834522i	$0.314256\pi$
$942$	−70.2300	−2.28822
$943$	−2.35103	−0.0765600
$944$	59.8739	1.94873
$945$	9.91281	0.322464
$946$	−22.7552	−0.739835
$947$	−33.4059	−1.08555	−0.542774	−	0.839879i	$-0.682626\pi$
$947$	−33.4059	−1.08555	−0.542774	+	0.839879i	$0.682626\pi$
$948$	−26.2814	−0.853579
$949$	−1.55140	−0.0503607
$950$	96.6146	3.13459
$951$	44.3955	1.43962
$952$	0.0800449	0.00259427
$953$	12.0997	0.391949	0.195975	−	0.980609i	$-0.437213\pi$
$953$	12.0997	0.391949	0.195975	+	0.980609i	$0.437213\pi$
$954$	36.2370	1.17322
$955$	112.098	3.62741
$956$	−124.994	−4.04259
$957$	−3.62079	−0.117043
$958$	−11.8462	−0.382735
$959$	−0.969550	−0.0313084
$960$	55.0956	1.77820
$961$	−19.9973	−0.645073
$962$	0.396113	0.0127712
$963$	104.326	3.36186
$964$	−124.665	−4.01518
$965$	−83.5488	−2.68953
$966$	−2.64506	−0.0851035
$967$	36.8736	1.18578	0.592888	−	0.805285i	$-0.297988\pi$
$967$	36.8736	1.18578	0.592888	+	0.805285i	$0.297988\pi$
$968$	70.4812	2.26535
$969$	0.506684	0.0162770
$970$	−112.599	−3.61535
$971$	14.5034	0.465436	0.232718	−	0.972544i	$-0.425238\pi$
$971$	14.5034	0.465436	0.232718	+	0.972544i	$0.425238\pi$
$972$	81.2352	2.60562
$973$	−1.76606	−0.0566172
$974$	−21.3185	−0.683088
$975$	6.78588	0.217322
$976$	−41.5827	−1.33103
$977$	−39.7000	−1.27011	−0.635057	−	0.772465i	$-0.719023\pi$
$977$	−39.7000	−1.27011	−0.635057	+	0.772465i	$0.719023\pi$
$978$	−20.2090	−0.646214
$979$	10.2821	0.328618
$980$	−142.345	−4.54704
$981$	6.81421	0.217561
$982$	−17.5131	−0.558867
$983$	32.3076	1.03045	0.515227	−	0.857054i	$-0.327708\pi$
$983$	32.3076	1.03045	0.515227	+	0.857054i	$0.327708\pi$
$984$	29.5258	0.941247
$985$	79.9882	2.54864
$986$	−0.204470	−0.00651165
$987$	−3.72777	−0.118656
$988$	−1.89720	−0.0603580
$989$	17.0120	0.540951
$990$	−68.6821	−2.18286
$991$	−51.4203	−1.63342	−0.816710	−	0.577048i	$-0.804205\pi$
$991$	−51.4203	−1.63342	−0.816710	+	0.577048i	$0.804205\pi$
$992$	−26.9815	−0.856665
$993$	−17.0368	−0.540645
$994$	5.05074	0.160200
$995$	−77.9942	−2.47258
$996$	−118.041	−3.74027
$997$	55.1677	1.74718	0.873589	−	0.486665i	$-0.161787\pi$
$997$	55.1677	1.74718	0.873589	+	0.486665i	$0.161787\pi$
$998$	−12.7891	−0.404832
$999$	−11.9490	−0.378051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4033.2.a.e.1.4	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4033.2.a.e.1.4	✓	82	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4033 = 37 \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4033.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58335 q^{2} +3.13276 q^{3} +4.67371 q^{4} +4.37342 q^{5} -8.09303 q^{6} +0.189689 q^{7} -6.90713 q^{8} +6.81421 q^{9} +O(q^{10})\)	\(q-2.58335 q^{2} +3.13276 q^{3} +4.67371 q^{4} +4.37342 q^{5} -8.09303 q^{6} +0.189689 q^{7} -6.90713 q^{8} +6.81421 q^{9} -11.2981 q^{10} +0.892119 q^{11} +14.6416 q^{12} +0.153333 q^{13} -0.490034 q^{14} +13.7009 q^{15} +8.49612 q^{16} -0.0610933 q^{17} -17.6035 q^{18} -2.64738 q^{19} +20.4401 q^{20} +0.594252 q^{21} -2.30466 q^{22} +1.72299 q^{23} -21.6384 q^{24} +14.1268 q^{25} -0.396113 q^{26} +11.9490 q^{27} +0.886552 q^{28} -1.29554 q^{29} -35.3942 q^{30} +3.31704 q^{31} -8.13422 q^{32} +2.79480 q^{33} +0.157825 q^{34} +0.829590 q^{35} +31.8476 q^{36} -1.00000 q^{37} +6.83911 q^{38} +0.480356 q^{39} -30.2077 q^{40} -1.36451 q^{41} -1.53516 q^{42} +9.87356 q^{43} +4.16951 q^{44} +29.8014 q^{45} -4.45108 q^{46} -6.27304 q^{47} +26.6164 q^{48} -6.96402 q^{49} -36.4944 q^{50} -0.191391 q^{51} +0.716634 q^{52} -2.05851 q^{53} -30.8686 q^{54} +3.90161 q^{55} -1.31021 q^{56} -8.29362 q^{57} +3.34685 q^{58} +7.04721 q^{59} +64.0339 q^{60} -4.89431 q^{61} -8.56908 q^{62} +1.29258 q^{63} +4.02132 q^{64} +0.670589 q^{65} -7.21995 q^{66} -14.4770 q^{67} -0.285532 q^{68} +5.39771 q^{69} -2.14312 q^{70} -10.3069 q^{71} -47.0666 q^{72} -10.1179 q^{73} +2.58335 q^{74} +44.2559 q^{75} -12.3731 q^{76} +0.169226 q^{77} -1.24093 q^{78} -1.79498 q^{79} +37.1571 q^{80} +16.9909 q^{81} +3.52500 q^{82} -8.06202 q^{83} +2.77736 q^{84} -0.267186 q^{85} -25.5069 q^{86} -4.05864 q^{87} -6.16198 q^{88} +11.5255 q^{89} -76.9875 q^{90} +0.0290856 q^{91} +8.05273 q^{92} +10.3915 q^{93} +16.2055 q^{94} -11.5781 q^{95} -25.4826 q^{96} +9.96625 q^{97} +17.9905 q^{98} +6.07909 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 10 q^{2} + 17 q^{3} + 88 q^{4} + 22 q^{5} + 4 q^{6} + 15 q^{7} + 33 q^{8} + 91 q^{9}+O(q^{10}) \) 82 * q + 10 * q^2 + 17 * q^3 + 88 * q^4 + 22 * q^5 + 4 * q^6 + 15 * q^7 + 33 * q^8 + 91 * q^9	\( 82 q + 10 q^{2} + 17 q^{3} + 88 q^{4} + 22 q^{5} + 4 q^{6} + 15 q^{7} + 33 q^{8} + 91 q^{9} + 13 q^{10} + 2 q^{11} + 36 q^{12} + 23 q^{13} + 24 q^{14} + 35 q^{15} + 104 q^{16} + 43 q^{17} + 42 q^{18} + 15 q^{19} + 59 q^{20} + 9 q^{21} + 6 q^{22} + 70 q^{23} + 15 q^{24} + 98 q^{25} + 10 q^{26} + 65 q^{27} + 37 q^{28} + 27 q^{29} + 17 q^{30} + 59 q^{31} + 46 q^{32} + 16 q^{33} - 16 q^{34} + 80 q^{35} + 88 q^{36} - 82 q^{37} + 82 q^{38} + 13 q^{39} + 14 q^{40} + 3 q^{41} + 62 q^{42} + 7 q^{43} - 11 q^{44} + 42 q^{45} - 11 q^{46} + 123 q^{47} + 45 q^{48} + 105 q^{49} + 27 q^{50} + 3 q^{51} - 30 q^{52} + 82 q^{53} - 27 q^{54} + 37 q^{55} + 66 q^{56} + 29 q^{57} - 34 q^{58} + 60 q^{59} + 94 q^{60} + 9 q^{61} - 2 q^{62} + 106 q^{63} + 93 q^{64} + 9 q^{65} + 63 q^{66} + 113 q^{68} + 48 q^{69} + 47 q^{70} + 59 q^{71} + 63 q^{72} + 21 q^{73} - 10 q^{74} + 77 q^{75} + 22 q^{76} + 30 q^{77} + 29 q^{78} + 55 q^{79} + 88 q^{80} + 42 q^{81} + 4 q^{82} + 92 q^{83} + 43 q^{84} - 7 q^{85} + 17 q^{86} + 147 q^{87} - 13 q^{88} + 100 q^{89} + 91 q^{90} + 28 q^{91} + 127 q^{92} + 3 q^{93} + 30 q^{94} + 48 q^{95} - 31 q^{96} + 53 q^{97} + 101 q^{98} - 20 q^{99}+O(q^{100}) \) 82 * q + 10 * q^2 + 17 * q^3 + 88 * q^4 + 22 * q^5 + 4 * q^6 + 15 * q^7 + 33 * q^8 + 91 * q^9 + 13 * q^10 + 2 * q^11 + 36 * q^12 + 23 * q^13 + 24 * q^14 + 35 * q^15 + 104 * q^16 + 43 * q^17 + 42 * q^18 + 15 * q^19 + 59 * q^20 + 9 * q^21 + 6 * q^22 + 70 * q^23 + 15 * q^24 + 98 * q^25 + 10 * q^26 + 65 * q^27 + 37 * q^28 + 27 * q^29 + 17 * q^30 + 59 * q^31 + 46 * q^32 + 16 * q^33 - 16 * q^34 + 80 * q^35 + 88 * q^36 - 82 * q^37 + 82 * q^38 + 13 * q^39 + 14 * q^40 + 3 * q^41 + 62 * q^42 + 7 * q^43 - 11 * q^44 + 42 * q^45 - 11 * q^46 + 123 * q^47 + 45 * q^48 + 105 * q^49 + 27 * q^50 + 3 * q^51 - 30 * q^52 + 82 * q^53 - 27 * q^54 + 37 * q^55 + 66 * q^56 + 29 * q^57 - 34 * q^58 + 60 * q^59 + 94 * q^60 + 9 * q^61 - 2 * q^62 + 106 * q^63 + 93 * q^64 + 9 * q^65 + 63 * q^66 + 113 * q^68 + 48 * q^69 + 47 * q^70 + 59 * q^71 + 63 * q^72 + 21 * q^73 - 10 * q^74 + 77 * q^75 + 22 * q^76 + 30 * q^77 + 29 * q^78 + 55 * q^79 + 88 * q^80 + 42 * q^81 + 4 * q^82 + 92 * q^83 + 43 * q^84 - 7 * q^85 + 17 * q^86 + 147 * q^87 - 13 * q^88 + 100 * q^89 + 91 * q^90 + 28 * q^91 + 127 * q^92 + 3 * q^93 + 30 * q^94 + 48 * q^95 - 31 * q^96 + 53 * q^97 + 101 * q^98 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more