LMFDB - Embedded newform 7381.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7381,2,Mod(1,7381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7381, 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("7381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7381 = 11^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7381.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:	$58.9375817319$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.69542$ of defining polynomial
Character	$\chi$	$=$	7381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69542 q^{2} -1.41452 q^{3} +5.26531 q^{4} +1.92020 q^{5} +3.81272 q^{6} -2.04756 q^{7} -8.80138 q^{8} -0.999141 q^{9} +O(q^{10})$	\(q-2.69542 q^{2} -1.41452 q^{3} +5.26531 q^{4} +1.92020 q^{5} +3.81272 q^{6} -2.04756 q^{7} -8.80138 q^{8} -0.999141 q^{9} -5.17575 q^{10} -7.44787 q^{12} +5.05790 q^{13} +5.51904 q^{14} -2.71615 q^{15} +13.1928 q^{16} +1.14759 q^{17} +2.69311 q^{18} +8.12286 q^{19} +10.1104 q^{20} +2.89631 q^{21} +3.53584 q^{23} +12.4497 q^{24} -1.31284 q^{25} -13.6332 q^{26} +5.65685 q^{27} -10.7810 q^{28} -6.96857 q^{29} +7.32118 q^{30} +3.42923 q^{31} -17.9575 q^{32} -3.09324 q^{34} -3.93172 q^{35} -5.26078 q^{36} -7.94224 q^{37} -21.8945 q^{38} -7.15449 q^{39} -16.9004 q^{40} -8.02845 q^{41} -7.80678 q^{42} +0.771854 q^{43} -1.91855 q^{45} -9.53060 q^{46} -5.08985 q^{47} -18.6615 q^{48} -2.80750 q^{49} +3.53866 q^{50} -1.62328 q^{51} +26.6314 q^{52} +11.4851 q^{53} -15.2476 q^{54} +18.0214 q^{56} -11.4899 q^{57} +18.7832 q^{58} -13.6413 q^{59} -14.3014 q^{60} +1.00000 q^{61} -9.24323 q^{62} +2.04580 q^{63} +22.0174 q^{64} +9.71217 q^{65} -11.4195 q^{67} +6.04241 q^{68} -5.00151 q^{69} +10.5976 q^{70} -10.0671 q^{71} +8.79382 q^{72} +5.70524 q^{73} +21.4077 q^{74} +1.85704 q^{75} +42.7693 q^{76} +19.2844 q^{78} -3.76482 q^{79} +25.3329 q^{80} -5.00429 q^{81} +21.6401 q^{82} +8.31723 q^{83} +15.2499 q^{84} +2.20360 q^{85} -2.08047 q^{86} +9.85716 q^{87} +1.94450 q^{89} +5.17130 q^{90} -10.3564 q^{91} +18.6173 q^{92} -4.85071 q^{93} +13.7193 q^{94} +15.5975 q^{95} +25.4012 q^{96} +15.3762 q^{97} +7.56740 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9	$ 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9} - 7 q^{10} - 4 q^{12} - 8 q^{13} - 11 q^{14} - 5 q^{15} + 31 q^{16} - 9 q^{17} - 10 q^{18} - 17 q^{19} - 6 q^{20} - 18 q^{21} - 10 q^{23} - 26 q^{24} + 45 q^{25} + 5 q^{26} - 33 q^{27} - 36 q^{28} - 27 q^{29} + 30 q^{30} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 38 q^{36} + 20 q^{37} - 37 q^{38} - 24 q^{39} - 10 q^{40} - 19 q^{41} + 21 q^{42} - 20 q^{43} - 32 q^{45} - 41 q^{46} - 19 q^{47} + 5 q^{48} + 42 q^{49} - 36 q^{50} - 47 q^{51} + 28 q^{52} + 3 q^{53} + 33 q^{54} - 44 q^{56} - 11 q^{57} + 23 q^{58} - 28 q^{59} - 96 q^{60} + 19 q^{61} + 11 q^{62} + 32 q^{63} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} - 19 q^{71} - 34 q^{72} - 20 q^{73} + 22 q^{74} - 50 q^{75} + 25 q^{76} - 94 q^{78} - 69 q^{79} - 36 q^{80} + 47 q^{81} - 61 q^{82} - q^{83} + 28 q^{84} - 24 q^{85} - 27 q^{86} + 58 q^{87} - 36 q^{90} + 24 q^{91} - 67 q^{92} - 14 q^{93} - 64 q^{94} + 3 q^{95} + 26 q^{96} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9 - 7 * q^10 - 4 * q^12 - 8 * q^13 - 11 * q^14 - 5 * q^15 + 31 * q^16 - 9 * q^17 - 10 * q^18 - 17 * q^19 - 6 * q^20 - 18 * q^21 - 10 * q^23 - 26 * q^24 + 45 * q^25 + 5 * q^26 - 33 * q^27 - 36 * q^28 - 27 * q^29 + 30 * q^30 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 38 * q^36 + 20 * q^37 - 37 * q^38 - 24 * q^39 - 10 * q^40 - 19 * q^41 + 21 * q^42 - 20 * q^43 - 32 * q^45 - 41 * q^46 - 19 * q^47 + 5 * q^48 + 42 * q^49 - 36 * q^50 - 47 * q^51 + 28 * q^52 + 3 * q^53 + 33 * q^54 - 44 * q^56 - 11 * q^57 + 23 * q^58 - 28 * q^59 - 96 * q^60 + 19 * q^61 + 11 * q^62 + 32 * q^63 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 - 19 * q^71 - 34 * q^72 - 20 * q^73 + 22 * q^74 - 50 * q^75 + 25 * q^76 - 94 * q^78 - 69 * q^79 - 36 * q^80 + 47 * q^81 - 61 * q^82 - q^83 + 28 * q^84 - 24 * q^85 - 27 * q^86 + 58 * q^87 - 36 * q^90 + 24 * q^91 - 67 * q^92 - 14 * q^93 - 64 * q^94 + 3 * q^95 + 26 * q^96 + 21 * q^97 + 87 * q^98

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 $

$2$	−2.69542	−1.90595	−0.952976	−	0.303046i	$-0.901997\pi$
$2$	−2.69542	−1.90595	−0.952976	+	0.303046i	$0.901997\pi$
$3$	−1.41452	−0.816672	−0.408336	−	0.912832i	$-0.633891\pi$
$3$	−1.41452	−0.816672	−0.408336	+	0.912832i	$0.633891\pi$
$4$	5.26531	2.63265
$5$	1.92020	0.858739	0.429369	−	0.903129i	$-0.358736\pi$
$5$	1.92020	0.858739	0.429369	+	0.903129i	$0.358736\pi$
$6$	3.81272	1.55654
$7$	−2.04756	−0.773905	−0.386952	−	0.922100i	$-0.626472\pi$
$7$	−2.04756	−0.773905	−0.386952	+	0.922100i	$0.626472\pi$
$8$	−8.80138	−3.11176
$9$	−0.999141	−0.333047
$10$	−5.17575	−1.63671
$11$	0	0
$11$	0	0
$12$	−7.44787	−2.15001
$13$	5.05790	1.40281	0.701405	−	0.712763i	$-0.252556\pi$
$13$	5.05790	1.40281	0.701405	+	0.712763i	$0.252556\pi$
$14$	5.51904	1.47503
$15$	−2.71615	−0.701308
$16$	13.1928	3.29821
$17$	1.14759	0.278331	0.139166	−	0.990269i	$-0.455558\pi$
$17$	1.14759	0.278331	0.139166	+	0.990269i	$0.455558\pi$
$18$	2.69311	0.634772
$19$	8.12286	1.86351	0.931756	−	0.363086i	$-0.118277\pi$
$19$	8.12286	1.86351	0.931756	+	0.363086i	$0.118277\pi$
$20$	10.1104	2.26076
$21$	2.89631	0.632026
$22$	0	0
$23$	3.53584	0.737275	0.368637	−	0.929573i	$-0.379824\pi$
$23$	3.53584	0.737275	0.368637	+	0.929573i	$0.379824\pi$
$24$	12.4497	2.54129
$25$	−1.31284	−0.262568
$26$	−13.6332	−2.67369
$27$	5.65685	1.08866
$28$	−10.7810	−2.03742
$29$	−6.96857	−1.29403	−0.647015	−	0.762477i	$-0.723983\pi$
$29$	−6.96857	−1.29403	−0.647015	+	0.762477i	$0.723983\pi$
$30$	7.32118	1.33666
$31$	3.42923	0.615908	0.307954	−	0.951401i	$-0.400356\pi$
$31$	3.42923	0.615908	0.307954	+	0.951401i	$0.400356\pi$
$32$	−17.9575	−3.17447
$33$	0	0
$34$	−3.09324	−0.530486
$35$	−3.93172	−0.664582
$36$	−5.26078	−0.876797
$37$	−7.94224	−1.30570	−0.652848	−	0.757489i	$-0.726426\pi$
$37$	−7.94224	−1.30570	−0.652848	+	0.757489i	$0.726426\pi$
$38$	−21.8945	−3.55176
$39$	−7.15449	−1.14564
$40$	−16.9004	−2.67219
$41$	−8.02845	−1.25383	−0.626917	−	0.779086i	$-0.715683\pi$
$41$	−8.02845	−1.25383	−0.626917	+	0.779086i	$0.715683\pi$
$42$	−7.80678	−1.20461
$43$	0.771854	0.117707	0.0588533	−	0.998267i	$-0.481256\pi$
$43$	0.771854	0.117707	0.0588533	+	0.998267i	$0.481256\pi$
$44$	0	0
$45$	−1.91855	−0.286000
$46$	−9.53060	−1.40521
$47$	−5.08985	−0.742430	−0.371215	−	0.928547i	$-0.621059\pi$
$47$	−5.08985	−0.742430	−0.371215	+	0.928547i	$0.621059\pi$
$48$	−18.6615	−2.69355
$49$	−2.80750	−0.401071
$50$	3.53866	0.500442
$51$	−1.62328	−0.227305
$52$	26.6314	3.69311
$53$	11.4851	1.57760	0.788801	−	0.614648i	$-0.210702\pi$
$53$	11.4851	1.57760	0.788801	+	0.614648i	$0.210702\pi$
$54$	−15.2476	−2.07494
$55$	0	0
$56$	18.0214	2.40820
$57$	−11.4899	−1.52188
$58$	18.7832	2.46636
$59$	−13.6413	−1.77594	−0.887972	−	0.459898i	$-0.847886\pi$
$59$	−13.6413	−1.77594	−0.887972	+	0.459898i	$0.847886\pi$
$60$	−14.3014	−1.84630
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−9.24323	−1.17389
$63$	2.04580	0.257747
$64$	22.0174	2.75218
$65$	9.71217	1.20465
$66$	0	0
$67$	−11.4195	−1.39512	−0.697559	−	0.716528i	$-0.745730\pi$
$67$	−11.4195	−1.39512	−0.697559	+	0.716528i	$0.745730\pi$
$68$	6.04241	0.732749
$69$	−5.00151	−0.602111
$70$	10.5976	1.26666
$71$	−10.0671	−1.19474	−0.597370	−	0.801966i	$-0.703788\pi$
$71$	−10.0671	−1.19474	−0.597370	+	0.801966i	$0.703788\pi$
$72$	8.79382	1.03636
$73$	5.70524	0.667748	0.333874	−	0.942618i	$-0.391644\pi$
$73$	5.70524	0.667748	0.333874	+	0.942618i	$0.391644\pi$
$74$	21.4077	2.48859
$75$	1.85704	0.214432
$76$	42.7693	4.90598
$77$	0	0
$78$	19.2844	2.18353
$79$	−3.76482	−0.423575	−0.211788	−	0.977316i	$-0.567928\pi$
$79$	−3.76482	−0.423575	−0.211788	+	0.977316i	$0.567928\pi$
$80$	25.3329	2.83230
$81$	−5.00429	−0.556033
$82$	21.6401	2.38975
$83$	8.31723	0.912935	0.456468	−	0.889740i	$-0.349114\pi$
$83$	8.31723	0.912935	0.456468	+	0.889740i	$0.349114\pi$
$84$	15.2499	1.66391
$85$	2.20360	0.239014
$86$	−2.08047	−0.224343
$87$	9.85716	1.05680
$88$	0	0
$89$	1.94450	0.206117	0.103058	−	0.994675i	$-0.467137\pi$
$89$	1.94450	0.206117	0.103058	+	0.994675i	$0.467137\pi$
$90$	5.17130	0.545103
$91$	−10.3564	−1.08564
$92$	18.6173	1.94099
$93$	−4.85071	−0.502995
$94$	13.7193	1.41504
$95$	15.5975	1.60027
$96$	25.4012	2.59250
$97$	15.3762	1.56122	0.780611	−	0.625018i	$-0.214908\pi$
$97$	15.3762	1.56122	0.780611	+	0.625018i	$0.214908\pi$
$98$	7.56740	0.764423
$99$	0	0
$100$	−6.91251	−0.691251
$101$	−17.3436	−1.72576	−0.862878	−	0.505412i	$-0.831341\pi$
$101$	−17.3436	−1.72576	−0.862878	+	0.505412i	$0.831341\pi$
$102$	4.37544	0.433233
$103$	3.27640	0.322833	0.161417	−	0.986886i	$-0.448394\pi$
$103$	3.27640	0.322833	0.161417	+	0.986886i	$0.448394\pi$
$104$	−44.5165	−4.36521
$105$	5.56148	0.542745
$106$	−30.9573	−3.00683
$107$	−4.54584	−0.439463	−0.219732	−	0.975560i	$-0.570518\pi$
$107$	−4.54584	−0.439463	−0.219732	+	0.975560i	$0.570518\pi$
$108$	29.7851	2.86607
$109$	−7.99283	−0.765574	−0.382787	−	0.923837i	$-0.625036\pi$
$109$	−7.99283	−0.765574	−0.382787	+	0.923837i	$0.625036\pi$
$110$	0	0
$111$	11.2344	1.06632
$112$	−27.0131	−2.55250
$113$	−5.30740	−0.499278	−0.249639	−	0.968339i	$-0.580312\pi$
$113$	−5.30740	−0.499278	−0.249639	+	0.968339i	$0.580312\pi$
$114$	30.9702	2.90062
$115$	6.78952	0.633126
$116$	−36.6916	−3.40673
$117$	−5.05356	−0.467202
$118$	36.7690	3.38486
$119$	−2.34976	−0.215402
$120$	23.9059	2.18230
$121$	0	0
$122$	−2.69542	−0.244032
$123$	11.3564	1.02397
$124$	18.0559	1.62147
$125$	−12.1219	−1.08422
$126$	−5.51430	−0.491253
$127$	−11.9457	−1.06001	−0.530006	−	0.847994i	$-0.677810\pi$
$127$	−11.9457	−1.06001	−0.530006	+	0.847994i	$0.677810\pi$
$128$	−23.4313	−2.07105
$129$	−1.09180	−0.0961277
$130$	−26.1784	−2.29600
$131$	1.68158	0.146920	0.0734600	−	0.997298i	$-0.476596\pi$
$131$	1.68158	0.146920	0.0734600	+	0.997298i	$0.476596\pi$
$132$	0	0
$133$	−16.6320	−1.44218
$134$	30.7805	2.65903
$135$	10.8623	0.934876
$136$	−10.1004	−0.866099
$137$	18.2640	1.56040	0.780199	−	0.625532i	$-0.215118\pi$
$137$	18.2640	1.56040	0.780199	+	0.625532i	$0.215118\pi$
$138$	13.4812	1.14760
$139$	0.862740	0.0731767	0.0365883	−	0.999330i	$-0.488351\pi$
$139$	0.862740	0.0731767	0.0365883	+	0.999330i	$0.488351\pi$
$140$	−20.7017	−1.74961
$141$	7.19968	0.606322
$142$	27.1350	2.27712
$143$	0	0
$144$	−13.1815	−1.09846
$145$	−13.3810	−1.11123
$146$	−15.3780	−1.27270
$147$	3.97126	0.327544
$148$	−41.8183	−3.43744
$149$	6.60379	0.541004	0.270502	−	0.962719i	$-0.412810\pi$
$149$	6.60379	0.541004	0.270502	+	0.962719i	$0.412810\pi$
$150$	−5.00550	−0.408697
$151$	−11.0977	−0.903119	−0.451559	−	0.892241i	$-0.649132\pi$
$151$	−11.0977	−0.903119	−0.451559	+	0.892241i	$0.649132\pi$
$152$	−71.4924	−5.79880
$153$	−1.14660	−0.0926974
$154$	0	0
$155$	6.58480	0.528904
$156$	−37.6706	−3.01606
$157$	5.27512	0.421000	0.210500	−	0.977594i	$-0.432491\pi$
$157$	5.27512	0.421000	0.210500	+	0.977594i	$0.432491\pi$
$158$	10.1478	0.807314
$159$	−16.2459	−1.28838
$160$	−34.4820	−2.72604
$161$	−7.23985	−0.570580
$162$	13.4887	1.05977
$163$	−3.79414	−0.297180	−0.148590	−	0.988899i	$-0.547473\pi$
$163$	−3.79414	−0.297180	−0.148590	+	0.988899i	$0.547473\pi$
$164$	−42.2723	−3.30091
$165$	0	0
$166$	−22.4185	−1.74001
$167$	4.98070	0.385418	0.192709	−	0.981256i	$-0.438273\pi$
$167$	4.98070	0.385418	0.192709	+	0.981256i	$0.438273\pi$
$168$	−25.4915	−1.96671
$169$	12.5824	0.967876
$170$	−5.93963	−0.455549
$171$	−8.11588	−0.620637
$172$	4.06405	0.309881
$173$	13.7955	1.04885	0.524427	−	0.851455i	$-0.324279\pi$
$173$	13.7955	1.04885	0.524427	+	0.851455i	$0.324279\pi$
$174$	−26.5692	−2.01421
$175$	2.68812	0.203203
$176$	0	0
$177$	19.2958	1.45036
$178$	−5.24126	−0.392849
$179$	−21.8696	−1.63461	−0.817306	−	0.576204i	$-0.804533\pi$
$179$	−21.8696	−1.63461	−0.817306	+	0.576204i	$0.804533\pi$
$180$	−10.1017	−0.752940
$181$	11.4327	0.849787	0.424893	−	0.905243i	$-0.360312\pi$
$181$	11.4327	0.849787	0.424893	+	0.905243i	$0.360312\pi$
$182$	27.9148	2.06918
$183$	−1.41452	−0.104564
$184$	−31.1203	−2.29422
$185$	−15.2507	−1.12125
$186$	13.0747	0.958684
$187$	0	0
$188$	−26.7996	−1.95456
$189$	−11.5827	−0.842521
$190$	−42.0418	−3.05004
$191$	8.87583	0.642233	0.321116	−	0.947040i	$-0.395942\pi$
$191$	8.87583	0.642233	0.321116	+	0.947040i	$0.395942\pi$
$192$	−31.1440	−2.24763
$193$	15.3781	1.10694	0.553471	−	0.832868i	$-0.313303\pi$
$193$	15.3781	1.10694	0.553471	+	0.832868i	$0.313303\pi$
$194$	−41.4455	−2.97561
$195$	−13.7380	−0.983801
$196$	−14.7823	−1.05588
$197$	−10.4220	−0.742540	−0.371270	−	0.928525i	$-0.621078\pi$
$197$	−10.4220	−0.742540	−0.371270	+	0.928525i	$0.621078\pi$
$198$	0	0
$199$	0.694713	0.0492469	0.0246235	−	0.999697i	$-0.492161\pi$
$199$	0.694713	0.0492469	0.0246235	+	0.999697i	$0.492161\pi$
$200$	11.5548	0.817049
$201$	16.1531	1.13935
$202$	46.7484	3.28921
$203$	14.2686	1.00146
$204$	−8.54709	−0.598416
$205$	−15.4162	−1.07672
$206$	−8.83128	−0.615304
$207$	−3.53281	−0.245547
$208$	66.7281	4.62676
$209$	0	0
$210$	−14.9906	−1.03445
$211$	1.04837	0.0721727	0.0360863	−	0.999349i	$-0.488511\pi$
$211$	1.04837	0.0721727	0.0360863	+	0.999349i	$0.488511\pi$
$212$	60.4727	4.15328
$213$	14.2400	0.975711
$214$	12.2530	0.837596
$215$	1.48211	0.101079
$216$	−49.7881	−3.38765
$217$	−7.02155	−0.476654
$218$	21.5441	1.45915
$219$	−8.07017	−0.545331
$220$	0	0
$221$	5.80439	0.390446
$222$	−30.2815	−2.03236
$223$	1.06818	0.0715306	0.0357653	−	0.999360i	$-0.488613\pi$
$223$	1.06818	0.0715306	0.0357653	+	0.999360i	$0.488613\pi$
$224$	36.7691	2.45674
$225$	1.31171	0.0874476
$226$	14.3057	0.951601
$227$	−7.21212	−0.478685	−0.239343	−	0.970935i	$-0.576932\pi$
$227$	−7.21212	−0.478685	−0.239343	+	0.970935i	$0.576932\pi$
$228$	−60.4979	−4.00657
$229$	17.7437	1.17253	0.586267	−	0.810118i	$-0.300597\pi$
$229$	17.7437	1.17253	0.586267	+	0.810118i	$0.300597\pi$
$230$	−18.3006	−1.20671
$231$	0	0
$232$	61.3330	4.02671
$233$	−21.4789	−1.40713	−0.703566	−	0.710630i	$-0.748410\pi$
$233$	−21.4789	−1.40713	−0.703566	+	0.710630i	$0.748410\pi$
$234$	13.6215	0.890464
$235$	−9.77351	−0.637554
$236$	−71.8255	−4.67544
$237$	5.32540	0.345922
$238$	6.33359	0.410546
$239$	2.44108	0.157900	0.0789502	−	0.996879i	$-0.474843\pi$
$239$	2.44108	0.157900	0.0789502	+	0.996879i	$0.474843\pi$
$240$	−35.8338	−2.31306
$241$	8.61322	0.554826	0.277413	−	0.960751i	$-0.410523\pi$
$241$	8.61322	0.554826	0.277413	+	0.960751i	$0.410523\pi$
$242$	0	0
$243$	−9.89190	−0.634566
$244$	5.26531	0.337077
$245$	−5.39095	−0.344415
$246$	−30.6103	−1.95164
$247$	41.0846	2.61415
$248$	−30.1820	−1.91656
$249$	−11.7649	−0.745568
$250$	32.6737	2.06646
$251$	−8.32074	−0.525200	−0.262600	−	0.964905i	$-0.584580\pi$
$251$	−8.32074	−0.525200	−0.262600	+	0.964905i	$0.584580\pi$
$252$	10.7718	0.678558
$253$	0	0
$254$	32.1988	2.02033
$255$	−3.11703	−0.195196
$256$	19.1223	1.19514
$257$	−20.6208	−1.28629	−0.643144	−	0.765745i	$-0.722370\pi$
$257$	−20.6208	−1.28629	−0.643144	+	0.765745i	$0.722370\pi$
$258$	2.94286	0.183215
$259$	16.2622	1.01048
$260$	51.1376	3.17142
$261$	6.96258	0.430973
$262$	−4.53256	−0.280023
$263$	−0.976262	−0.0601989	−0.0300995	−	0.999547i	$-0.509582\pi$
$263$	−0.976262	−0.0601989	−0.0300995	+	0.999547i	$0.509582\pi$
$264$	0	0
$265$	22.0537	1.35475
$266$	44.8304	2.74873
$267$	−2.75053	−0.168330
$268$	−60.1273	−3.67286
$269$	30.0972	1.83506	0.917530	−	0.397666i	$-0.130180\pi$
$269$	30.0972	1.83506	0.917530	+	0.397666i	$0.130180\pi$
$270$	−29.2784	−1.78183
$271$	−22.2148	−1.34945	−0.674726	−	0.738069i	$-0.735738\pi$
$271$	−22.2148	−1.34945	−0.674726	+	0.738069i	$0.735738\pi$
$272$	15.1400	0.917994
$273$	14.6492	0.886613
$274$	−49.2292	−2.97404
$275$	0	0
$276$	−26.3345	−1.58515
$277$	−17.6386	−1.05980	−0.529900	−	0.848060i	$-0.677770\pi$
$277$	−17.6386	−1.05980	−0.529900	+	0.848060i	$0.677770\pi$
$278$	−2.32545	−0.139471
$279$	−3.42629	−0.205126
$280$	34.6046	2.06802
$281$	−2.53996	−0.151521	−0.0757607	−	0.997126i	$-0.524139\pi$
$281$	−2.53996	−0.151521	−0.0757607	+	0.997126i	$0.524139\pi$
$282$	−19.4062	−1.15562
$283$	10.2899	0.611669	0.305834	−	0.952085i	$-0.401065\pi$
$283$	10.2899	0.611669	0.305834	+	0.952085i	$0.401065\pi$
$284$	−53.0062	−3.14534
$285$	−22.0629	−1.30689
$286$	0	0
$287$	16.4387	0.970348
$288$	17.9421	1.05725
$289$	−15.6830	−0.922532
$290$	36.0675	2.11796
$291$	−21.7500	−1.27501
$292$	30.0399	1.75795
$293$	−4.28066	−0.250079	−0.125039	−	0.992152i	$-0.539906\pi$
$293$	−4.28066	−0.250079	−0.125039	+	0.992152i	$0.539906\pi$
$294$	−10.7042	−0.624283
$295$	−26.1940	−1.52507
$296$	69.9027	4.06301
$297$	0	0
$298$	−17.8000	−1.03113
$299$	17.8840	1.03426
$300$	9.77786	0.564525
$301$	−1.58042	−0.0910937
$302$	29.9130	1.72130
$303$	24.5329	1.40938
$304$	107.164	6.14625
$305$	1.92020	0.109950
$306$	3.09058	0.176677
$307$	−6.19669	−0.353664	−0.176832	−	0.984241i	$-0.556585\pi$
$307$	−6.19669	−0.353664	−0.176832	+	0.984241i	$0.556585\pi$
$308$	0	0
$309$	−4.63452	−0.263649
$310$	−17.7488	−1.00807
$311$	2.33194	0.132232	0.0661160	−	0.997812i	$-0.478939\pi$
$311$	2.33194	0.132232	0.0661160	+	0.997812i	$0.478939\pi$
$312$	62.9694	3.56494
$313$	2.29322	0.129620	0.0648102	−	0.997898i	$-0.479356\pi$
$313$	2.29322	0.129620	0.0648102	+	0.997898i	$0.479356\pi$
$314$	−14.2187	−0.802407
$315$	3.92834	0.221337
$316$	−19.8229	−1.11513
$317$	25.8628	1.45260	0.726299	−	0.687379i	$-0.241239\pi$
$317$	25.8628	1.45260	0.726299	+	0.687379i	$0.241239\pi$
$318$	43.7896	2.45560
$319$	0	0
$320$	42.2778	2.36340
$321$	6.43018	0.358897
$322$	19.5145	1.08750
$323$	9.32170	0.518673
$324$	−26.3491	−1.46384
$325$	−6.64022	−0.368333
$326$	10.2268	0.566411
$327$	11.3060	0.625223
$328$	70.6615	3.90163
$329$	10.4218	0.574571
$330$	0	0
$331$	−2.52575	−0.138828	−0.0694139	−	0.997588i	$-0.522113\pi$
$331$	−2.52575	−0.138828	−0.0694139	+	0.997588i	$0.522113\pi$
$332$	43.7928	2.40344
$333$	7.93541	0.434858
$334$	−13.4251	−0.734588
$335$	−21.9277	−1.19804
$336$	38.2105	2.08456
$337$	−11.2141	−0.610871	−0.305436	−	0.952213i	$-0.598802\pi$
$337$	−11.2141	−0.610871	−0.305436	+	0.952213i	$0.598802\pi$
$338$	−33.9148	−1.84472
$339$	7.50741	0.407747
$340$	11.6026	0.629240
$341$	0	0
$342$	21.8757	1.18290
$343$	20.0814	1.08430
$344$	−6.79338	−0.366275
$345$	−9.60389	−0.517056
$346$	−37.1848	−1.99907
$347$	−13.7325	−0.737197	−0.368599	−	0.929589i	$-0.620162\pi$
$347$	−13.7325	−0.737197	−0.368599	+	0.929589i	$0.620162\pi$
$348$	51.9010	2.78218
$349$	24.4459	1.30856	0.654280	−	0.756252i	$-0.272972\pi$
$349$	24.4459	1.30856	0.654280	+	0.756252i	$0.272972\pi$
$350$	−7.24562	−0.387295
$351$	28.6118	1.52719
$352$	0	0
$353$	21.3254	1.13504	0.567519	−	0.823361i	$-0.307903\pi$
$353$	21.3254	1.13504	0.567519	+	0.823361i	$0.307903\pi$
$354$	−52.0104	−2.76432
$355$	−19.3307	−1.02597
$356$	10.2384	0.542634
$357$	3.32377	0.175913
$358$	58.9479	3.11549
$359$	0.928492	0.0490039	0.0245020	−	0.999700i	$-0.492200\pi$
$359$	0.928492	0.0490039	0.0245020	+	0.999700i	$0.492200\pi$
$360$	16.8859	0.889964
$361$	46.9808	2.47267
$362$	−30.8160	−1.61965
$363$	0	0
$364$	−54.5294	−2.85812
$365$	10.9552	0.573421
$366$	3.81272	0.199294
$367$	−28.6651	−1.49630	−0.748152	−	0.663527i	$-0.769059\pi$
$367$	−28.6651	−1.49630	−0.748152	+	0.663527i	$0.769059\pi$
$368$	46.6478	2.43169
$369$	8.02156	0.417586
$370$	41.1070	2.13705
$371$	−23.5165	−1.22091
$372$	−25.5405	−1.32421
$373$	17.8961	0.926627	0.463314	−	0.886194i	$-0.346660\pi$
$373$	17.8961	0.926627	0.463314	+	0.886194i	$0.346660\pi$
$374$	0	0
$375$	17.1466	0.885449
$376$	44.7977	2.31026
$377$	−35.2463	−1.81528
$378$	31.2204	1.60580
$379$	12.1289	0.623021	0.311511	−	0.950243i	$-0.399165\pi$
$379$	12.1289	0.623021	0.311511	+	0.950243i	$0.399165\pi$
$380$	82.1256	4.21295
$381$	16.8974	0.865682
$382$	−23.9241	−1.22406
$383$	−8.54411	−0.436584	−0.218292	−	0.975884i	$-0.570048\pi$
$383$	−8.54411	−0.436584	−0.218292	+	0.975884i	$0.570048\pi$
$384$	33.1439	1.69137
$385$	0	0
$386$	−41.4506	−2.10978
$387$	−0.771191	−0.0392018
$388$	80.9607	4.11015
$389$	13.9503	0.707310	0.353655	−	0.935376i	$-0.384939\pi$
$389$	13.9503	0.707310	0.353655	+	0.935376i	$0.384939\pi$
$390$	37.0298	1.87508
$391$	4.05770	0.205207
$392$	24.7099	1.24804
$393$	−2.37862	−0.119985
$394$	28.0918	1.41525
$395$	−7.22919	−0.363740
$396$	0	0
$397$	−38.5097	−1.93275	−0.966373	−	0.257143i	$-0.917219\pi$
$397$	−38.5097	−1.93275	−0.966373	+	0.257143i	$0.917219\pi$
$398$	−1.87255	−0.0938623
$399$	23.5263	1.17779
$400$	−17.3201	−0.866005
$401$	−24.0504	−1.20102	−0.600510	−	0.799617i	$-0.705036\pi$
$401$	−24.0504	−1.20102	−0.600510	+	0.799617i	$0.705036\pi$
$402$	−43.5395	−2.17155
$403$	17.3447	0.864002
$404$	−91.3196	−4.54332
$405$	−9.60923	−0.477487
$406$	−38.4598	−1.90873
$407$	0	0
$408$	14.2871	0.707319
$409$	−19.8989	−0.983938	−0.491969	−	0.870613i	$-0.663723\pi$
$409$	−19.8989	−0.983938	−0.491969	+	0.870613i	$0.663723\pi$
$410$	41.5532	2.05217
$411$	−25.8347	−1.27433
$412$	17.2512	0.849907
$413$	27.9313	1.37441
$414$	9.52241	0.468001
$415$	15.9707	0.783973
$416$	−90.8274	−4.45318
$417$	−1.22036	−0.0597613
$418$	0	0
$419$	−12.9815	−0.634189	−0.317094	−	0.948394i	$-0.602707\pi$
$419$	−12.9815	−0.634189	−0.317094	+	0.948394i	$0.602707\pi$
$420$	29.2829	1.42886
$421$	−3.38577	−0.165012	−0.0825062	−	0.996591i	$-0.526292\pi$
$421$	−3.38577	−0.165012	−0.0825062	+	0.996591i	$0.526292\pi$
$422$	−2.82580	−0.137558
$423$	5.08548	0.247264
$424$	−101.085	−4.90912
$425$	−1.50660	−0.0730809
$426$	−38.3829	−1.85966
$427$	−2.04756	−0.0990884
$428$	−23.9353	−1.15695
$429$	0	0
$430$	−3.99492	−0.192652
$431$	−39.1250	−1.88459	−0.942293	−	0.334790i	$-0.891335\pi$
$431$	−39.1250	−1.88459	−0.942293	+	0.334790i	$0.891335\pi$
$432$	74.6300	3.59064
$433$	−33.7007	−1.61955	−0.809776	−	0.586740i	$-0.800411\pi$
$433$	−33.7007	−1.61955	−0.809776	+	0.586740i	$0.800411\pi$
$434$	18.9261	0.908480
$435$	18.9277	0.907513
$436$	−42.0847	−2.01549
$437$	28.7212	1.37392
$438$	21.7525	1.03938
$439$	−36.3857	−1.73660	−0.868298	−	0.496043i	$-0.834786\pi$
$439$	−36.3857	−1.73660	−0.868298	+	0.496043i	$0.834786\pi$
$440$	0	0
$441$	2.80509	0.133576
$442$	−15.6453	−0.744171
$443$	27.5870	1.31070	0.655349	−	0.755327i	$-0.272522\pi$
$443$	27.5870	1.31070	0.655349	+	0.755327i	$0.272522\pi$
$444$	59.1527	2.80726
$445$	3.73383	0.177000
$446$	−2.87920	−0.136334
$447$	−9.34118	−0.441822
$448$	−45.0820	−2.12992
$449$	−33.3765	−1.57513	−0.787567	−	0.616229i	$-0.788660\pi$
$449$	−33.3765	−1.57513	−0.787567	+	0.616229i	$0.788660\pi$
$450$	−3.53562	−0.166671
$451$	0	0
$452$	−27.9451	−1.31443
$453$	15.6979	0.737552
$454$	19.4397	0.912351
$455$	−19.8863	−0.932282
$456$	101.127	4.73571
$457$	10.6901	0.500064	0.250032	−	0.968238i	$-0.419559\pi$
$457$	10.6901	0.500064	0.250032	+	0.968238i	$0.419559\pi$
$458$	−47.8267	−2.23479
$459$	6.49174	0.303009
$460$	35.7489	1.66680
$461$	−32.5513	−1.51607	−0.758034	−	0.652215i	$-0.773840\pi$
$461$	−32.5513	−1.51607	−0.758034	+	0.652215i	$0.773840\pi$
$462$	0	0
$463$	26.4666	1.23001	0.615004	−	0.788524i	$-0.289154\pi$
$463$	26.4666	1.23001	0.615004	+	0.788524i	$0.289154\pi$
$464$	−91.9352	−4.26798
$465$	−9.31431	−0.431941
$466$	57.8948	2.68193
$467$	−31.2536	−1.44624	−0.723122	−	0.690720i	$-0.757294\pi$
$467$	−31.2536	−1.44624	−0.723122	+	0.690720i	$0.757294\pi$
$468$	−26.6085	−1.22998
$469$	23.3822	1.07969
$470$	26.3438	1.21515
$471$	−7.46175	−0.343819
$472$	120.062	5.52631
$473$	0	0
$474$	−14.3542	−0.659310
$475$	−10.6640	−0.489299
$476$	−12.3722	−0.567078
$477$	−11.4753	−0.525416
$478$	−6.57975	−0.300951
$479$	−38.2909	−1.74956	−0.874778	−	0.484524i	$-0.838993\pi$
$479$	−38.2909	−1.74956	−0.874778	+	0.484524i	$0.838993\pi$
$480$	48.7754	2.22628
$481$	−40.1711	−1.83164
$482$	−23.2163	−1.05747
$483$	10.2409	0.465977
$484$	0	0
$485$	29.5254	1.34068
$486$	26.6629	1.20945
$487$	20.3803	0.923517	0.461759	−	0.887006i	$-0.347219\pi$
$487$	20.3803	0.923517	0.461759	+	0.887006i	$0.347219\pi$
$488$	−8.80138	−0.398420
$489$	5.36688	0.242699
$490$	14.5309	0.656439
$491$	40.6301	1.83361	0.916805	−	0.399335i	$-0.130759\pi$
$491$	40.6301	1.83361	0.916805	+	0.399335i	$0.130759\pi$
$492$	59.7949	2.69576
$493$	−7.99705	−0.360169
$494$	−110.740	−4.98245
$495$	0	0
$496$	45.2413	2.03139
$497$	20.6129	0.924615
$498$	31.7113	1.42102
$499$	29.5333	1.32209	0.661046	−	0.750345i	$-0.270113\pi$
$499$	29.5333	1.32209	0.661046	+	0.750345i	$0.270113\pi$
$500$	−63.8255	−2.85436
$501$	−7.04528	−0.314760
$502$	22.4279	1.00101
$503$	−8.57603	−0.382386	−0.191193	−	0.981552i	$-0.561236\pi$
$503$	−8.57603	−0.382386	−0.191193	+	0.981552i	$0.561236\pi$
$504$	−18.0059	−0.802046
$505$	−33.3032	−1.48197
$506$	0	0
$507$	−17.7980	−0.790437
$508$	−62.8980	−2.79065
$509$	25.9327	1.14945	0.574724	−	0.818348i	$-0.305110\pi$
$509$	25.9327	1.14945	0.574724	+	0.818348i	$0.305110\pi$
$510$	8.40170	0.372034
$511$	−11.6818	−0.516774
$512$	−4.68019	−0.206837
$513$	45.9498	2.02873
$514$	55.5817	2.45160
$515$	6.29133	0.277229
$516$	−5.74866	−0.253071
$517$	0	0
$518$	−43.8335	−1.92593
$519$	−19.5140	−0.856570
$520$	−85.4806	−3.74857
$521$	24.3706	1.06770	0.533848	−	0.845580i	$-0.320745\pi$
$521$	24.3706	1.06770	0.533848	+	0.845580i	$0.320745\pi$
$522$	−18.7671	−0.821414
$523$	33.0409	1.44478	0.722390	−	0.691486i	$-0.243044\pi$
$523$	33.0409	1.44478	0.722390	+	0.691486i	$0.243044\pi$
$524$	8.85402	0.386790
$525$	−3.80239	−0.165950
$526$	2.63144	0.114736
$527$	3.93535	0.171426
$528$	0	0
$529$	−10.4978	−0.456426
$530$	−59.4441	−2.58208
$531$	13.6296	0.591473
$532$	−87.5727	−3.79676
$533$	−40.6071	−1.75889
$534$	7.41385	0.320828
$535$	−8.72892	−0.377384
$536$	100.508	4.34127
$537$	30.9349	1.33494
$538$	−81.1248	−3.49754
$539$	0	0
$540$	57.1932	2.46120
$541$	3.17301	0.136418	0.0682091	−	0.997671i	$-0.478271\pi$
$541$	3.17301	0.136418	0.0682091	+	0.997671i	$0.478271\pi$
$542$	59.8782	2.57199
$543$	−16.1718	−0.693997
$544$	−20.6078	−0.883554
$545$	−15.3478	−0.657428
$546$	−39.4859	−1.68984
$547$	−1.72864	−0.0739115	−0.0369558	−	0.999317i	$-0.511766\pi$
$547$	−1.72864	−0.0739115	−0.0369558	+	0.999317i	$0.511766\pi$
$548$	96.1655	4.10799
$549$	−0.999141	−0.0426423
$550$	0	0
$551$	−56.6047	−2.41144
$552$	44.0202	1.87363
$553$	7.70869	0.327807
$554$	47.5434	2.01993
$555$	21.5723	0.915694
$556$	4.54259	0.192649
$557$	−0.868836	−0.0368138	−0.0184069	−	0.999831i	$-0.505859\pi$
$557$	−0.868836	−0.0368138	−0.0184069	+	0.999831i	$0.505859\pi$
$558$	9.23529	0.390961
$559$	3.90396	0.165120
$560$	−51.8705	−2.19193
$561$	0	0
$562$	6.84627	0.288792
$563$	−5.84546	−0.246357	−0.123178	−	0.992385i	$-0.539309\pi$
$563$	−5.84546	−0.246357	−0.123178	+	0.992385i	$0.539309\pi$
$564$	37.9085	1.59624
$565$	−10.1913	−0.428750
$566$	−27.7355	−1.16581
$567$	10.2466	0.430316
$568$	88.6041	3.71774
$569$	16.6614	0.698482	0.349241	−	0.937033i	$-0.386439\pi$
$569$	16.6614	0.698482	0.349241	+	0.937033i	$0.386439\pi$
$570$	59.4689	2.49088
$571$	17.3260	0.725072	0.362536	−	0.931970i	$-0.381911\pi$
$571$	17.3260	0.725072	0.362536	+	0.931970i	$0.381911\pi$
$572$	0	0
$573$	−12.5550	−0.524493
$574$	−44.3094	−1.84944
$575$	−4.64200	−0.193585
$576$	−21.9985	−0.916605
$577$	4.55759	0.189735	0.0948674	−	0.995490i	$-0.469757\pi$
$577$	4.55759	0.189735	0.0948674	+	0.995490i	$0.469757\pi$
$578$	42.2724	1.75830
$579$	−21.7526	−0.904009
$580$	−70.4552	−2.92549
$581$	−17.0300	−0.706525
$582$	58.6254	2.43010
$583$	0	0
$584$	−50.2140	−2.07787
$585$	−9.70383	−0.401204
$586$	11.5382	0.476638
$587$	−26.8384	−1.10774	−0.553869	−	0.832604i	$-0.686849\pi$
$587$	−26.8384	−1.10774	−0.553869	+	0.832604i	$0.686849\pi$
$588$	20.9099	0.862309
$589$	27.8551	1.14775
$590$	70.6038	2.90671
$591$	14.7422	0.606411
$592$	−104.781	−4.30646
$593$	7.92520	0.325449	0.162725	−	0.986672i	$-0.447972\pi$
$593$	7.92520	0.325449	0.162725	+	0.986672i	$0.447972\pi$
$594$	0	0
$595$	−4.51200	−0.184974
$596$	34.7710	1.42427
$597$	−0.982684	−0.0402186
$598$	−48.2048	−1.97124
$599$	−42.0884	−1.71968	−0.859842	−	0.510560i	$-0.829438\pi$
$599$	−42.0884	−1.71968	−0.859842	+	0.510560i	$0.829438\pi$
$600$	−16.3445	−0.667261
$601$	−22.6153	−0.922497	−0.461249	−	0.887271i	$-0.652598\pi$
$601$	−22.6153	−0.922497	−0.461249	+	0.887271i	$0.652598\pi$
$602$	4.25989	0.173620
$603$	11.4097	0.464640
$604$	−58.4328	−2.37760
$605$	0	0
$606$	−66.1265	−2.68620
$607$	15.4013	0.625121	0.312560	−	0.949898i	$-0.398813\pi$
$607$	15.4013	0.625121	0.312560	+	0.949898i	$0.398813\pi$
$608$	−145.866	−5.91566
$609$	−20.1831	−0.817861
$610$	−5.17575	−0.209560
$611$	−25.7440	−1.04149
$612$	−6.03722	−0.244040
$613$	8.83937	0.357019	0.178509	−	0.983938i	$-0.442873\pi$
$613$	8.83937	0.357019	0.178509	+	0.983938i	$0.442873\pi$
$614$	16.7027	0.674067
$615$	21.8065	0.879323
$616$	0	0
$617$	−44.2618	−1.78191	−0.890957	−	0.454087i	$-0.849966\pi$
$617$	−44.2618	−1.78191	−0.890957	+	0.454087i	$0.849966\pi$
$618$	12.4920	0.502502
$619$	−10.4503	−0.420033	−0.210017	−	0.977698i	$-0.567352\pi$
$619$	−10.4503	−0.420033	−0.210017	+	0.977698i	$0.567352\pi$
$620$	34.6710	1.39242
$621$	20.0018	0.802643
$622$	−6.28556	−0.252028
$623$	−3.98148	−0.159515
$624$	−94.3880	−3.77855
$625$	−16.7122	−0.668490
$626$	−6.18119	−0.247050
$627$	0	0
$628$	27.7751	1.10835
$629$	−9.11442	−0.363416
$630$	−10.5885	−0.421858
$631$	17.1012	0.680787	0.340394	−	0.940283i	$-0.389440\pi$
$631$	17.1012	0.680787	0.340394	+	0.940283i	$0.389440\pi$
$632$	33.1356	1.31806
$633$	−1.48294	−0.0589414
$634$	−69.7111	−2.76858
$635$	−22.9382	−0.910274
$636$	−85.5396	−3.39187
$637$	−14.2001	−0.562627
$638$	0	0
$639$	10.0584	0.397905
$640$	−44.9927	−1.77849
$641$	−1.95931	−0.0773880	−0.0386940	−	0.999251i	$-0.512320\pi$
$641$	−1.95931	−0.0773880	−0.0386940	+	0.999251i	$0.512320\pi$
$642$	−17.3320	−0.684041
$643$	21.9759	0.866644	0.433322	−	0.901239i	$-0.357341\pi$
$643$	21.9759	0.866644	0.433322	+	0.901239i	$0.357341\pi$
$644$	−38.1200	−1.50214
$645$	−2.09647	−0.0825485
$646$	−25.1259	−0.988566
$647$	−4.74294	−0.186464	−0.0932321	−	0.995644i	$-0.529720\pi$
$647$	−4.74294	−0.186464	−0.0932321	+	0.995644i	$0.529720\pi$
$648$	44.0447	1.73024
$649$	0	0
$650$	17.8982	0.702026
$651$	9.93211	0.389270
$652$	−19.9773	−0.782372
$653$	−22.2797	−0.871871	−0.435935	−	0.899978i	$-0.643582\pi$
$653$	−22.2797	−0.871871	−0.435935	+	0.899978i	$0.643582\pi$
$654$	−30.4744	−1.19164
$655$	3.22896	0.126166
$656$	−105.918	−4.13541
$657$	−5.70034	−0.222392
$658$	−28.0911	−1.09510
$659$	28.0012	1.09077	0.545386	−	0.838185i	$-0.316383\pi$
$659$	28.0012	1.09077	0.545386	+	0.838185i	$0.316383\pi$
$660$	0	0
$661$	41.7104	1.62235	0.811173	−	0.584806i	$-0.198830\pi$
$661$	41.7104	1.62235	0.811173	+	0.584806i	$0.198830\pi$
$662$	6.80797	0.264599
$663$	−8.21041	−0.318866
$664$	−73.2032	−2.84083
$665$	−31.9368	−1.23846
$666$	−21.3893	−0.828819
$667$	−24.6398	−0.954056
$668$	26.2249	1.01467
$669$	−1.51096	−0.0584170
$670$	59.1046	2.28341
$671$	0	0
$672$	−52.0105	−2.00635
$673$	10.0629	0.387898	0.193949	−	0.981012i	$-0.437870\pi$
$673$	10.0629	0.387898	0.193949	+	0.981012i	$0.437870\pi$
$674$	30.2268	1.16429
$675$	−7.42655	−0.285848
$676$	66.2501	2.54808
$677$	−23.1706	−0.890517	−0.445258	−	0.895402i	$-0.646888\pi$
$677$	−23.1706	−0.890517	−0.445258	+	0.895402i	$0.646888\pi$
$678$	−20.2357	−0.777146
$679$	−31.4838	−1.20824
$680$	−19.3947	−0.743753
$681$	10.2017	0.390929
$682$	0	0
$683$	−44.9098	−1.71842	−0.859212	−	0.511619i	$-0.829046\pi$
$683$	−44.9098	−1.71842	−0.859212	+	0.511619i	$0.829046\pi$
$684$	−42.7326	−1.63392
$685$	35.0705	1.33997
$686$	−54.1280	−2.06662
$687$	−25.0987	−0.957575
$688$	10.1829	0.388221
$689$	58.0906	2.21308
$690$	25.8866	0.985484
$691$	22.4821	0.855258	0.427629	−	0.903954i	$-0.359349\pi$
$691$	22.4821	0.855258	0.427629	+	0.903954i	$0.359349\pi$
$692$	72.6377	2.76127
$693$	0	0
$694$	37.0148	1.40506
$695$	1.65663	0.0628396
$696$	−86.7566	−3.28850
$697$	−9.21336	−0.348981
$698$	−65.8921	−2.49405
$699$	30.3823	1.14916
$700$	14.1538	0.534962
$701$	10.4901	0.396206	0.198103	−	0.980181i	$-0.436522\pi$
$701$	10.4901	0.396206	0.198103	+	0.980181i	$0.436522\pi$
$702$	−77.1210	−2.91074
$703$	−64.5136	−2.43318
$704$	0	0
$705$	13.8248	0.520672
$706$	−57.4810	−2.16333
$707$	35.5121	1.33557
$708$	101.598	3.81830
$709$	49.1227	1.84484	0.922421	−	0.386186i	$-0.126208\pi$
$709$	49.1227	1.84484	0.922421	+	0.386186i	$0.126208\pi$
$710$	52.1045	1.95545
$711$	3.76158	0.141070
$712$	−17.1143	−0.641386
$713$	12.1252	0.454093
$714$	−8.95897	−0.335281
$715$	0	0
$716$	−115.150	−4.30337
$717$	−3.45295	−0.128953
$718$	−2.50268	−0.0933991
$719$	−11.6568	−0.434726	−0.217363	−	0.976091i	$-0.569746\pi$
$719$	−11.6568	−0.434726	−0.217363	+	0.976091i	$0.569746\pi$
$720$	−25.3111	−0.943289
$721$	−6.70862	−0.249842
$722$	−126.633	−4.71280
$723$	−12.1835	−0.453111
$724$	60.1967	2.23719
$725$	9.14862	0.339771
$726$	0	0
$727$	21.6498	0.802948	0.401474	−	0.915870i	$-0.368498\pi$
$727$	21.6498	0.802948	0.401474	+	0.915870i	$0.368498\pi$
$728$	91.1503	3.37825
$729$	29.0051	1.07426
$730$	−29.5289	−1.09291
$731$	0.885771	0.0327614
$732$	−7.44787	−0.275281
$733$	−6.09963	−0.225295	−0.112647	−	0.993635i	$-0.535933\pi$
$733$	−6.09963	−0.225295	−0.112647	+	0.993635i	$0.535933\pi$
$734$	77.2645	2.85188
$735$	7.62560	0.281274
$736$	−63.4950	−2.34046
$737$	0	0
$738$	−21.6215	−0.795898
$739$	0.976933	0.0359370	0.0179685	−	0.999839i	$-0.494280\pi$
$739$	0.976933	0.0359370	0.0179685	+	0.999839i	$0.494280\pi$
$740$	−80.2994	−2.95186
$741$	−58.1149	−2.13490
$742$	63.3868	2.32700
$743$	−0.821800	−0.0301489	−0.0150745	−	0.999886i	$-0.504799\pi$
$743$	−0.821800	−0.0301489	−0.0150745	+	0.999886i	$0.504799\pi$
$744$	42.6929	1.56520
$745$	12.6806	0.464581
$746$	−48.2377	−1.76611
$747$	−8.31009	−0.304050
$748$	0	0
$749$	9.30789	0.340103
$750$	−46.2174	−1.68762
$751$	−49.0864	−1.79119	−0.895594	−	0.444873i	$-0.853249\pi$
$751$	−49.0864	−1.79119	−0.895594	+	0.444873i	$0.853249\pi$
$752$	−67.1495	−2.44869
$753$	11.7698	0.428916
$754$	95.0038	3.45983
$755$	−21.3098	−0.775543
$756$	−60.9867	−2.21806
$757$	−32.3794	−1.17685	−0.588425	−	0.808552i	$-0.700252\pi$
$757$	−32.3794	−1.17685	−0.588425	+	0.808552i	$0.700252\pi$
$758$	−32.6926	−1.18745
$759$	0	0
$760$	−137.279	−4.97965
$761$	13.8551	0.502247	0.251124	−	0.967955i	$-0.419200\pi$
$761$	13.8551	0.502247	0.251124	+	0.967955i	$0.419200\pi$
$762$	−45.5458	−1.64995
$763$	16.3658	0.592481
$764$	46.7340	1.69078
$765$	−2.20170	−0.0796028
$766$	23.0300	0.832107
$767$	−68.9963	−2.49131
$768$	−27.0488	−0.976041
$769$	−18.2227	−0.657128	−0.328564	−	0.944482i	$-0.606565\pi$
$769$	−18.2227	−0.657128	−0.328564	+	0.944482i	$0.606565\pi$
$770$	0	0
$771$	29.1684	1.05048
$772$	80.9706	2.91420
$773$	−48.6444	−1.74962	−0.874809	−	0.484467i	$-0.839013\pi$
$773$	−48.6444	−1.74962	−0.874809	+	0.484467i	$0.839013\pi$
$774$	2.07869	0.0747168
$775$	−4.50203	−0.161718
$776$	−135.332	−4.85814
$777$	−23.0032	−0.825234
$778$	−37.6020	−1.34810
$779$	−65.2140	−2.33653
$780$	−72.3350	−2.59001
$781$	0	0
$782$	−10.9372	−0.391114
$783$	−39.4202	−1.40876
$784$	−37.0389	−1.32282
$785$	10.1293	0.361529
$786$	6.41139	0.228687
$787$	−26.7698	−0.954240	−0.477120	−	0.878838i	$-0.658319\pi$
$787$	−26.7698	−0.954240	−0.477120	+	0.878838i	$0.658319\pi$
$788$	−54.8752	−1.95485
$789$	1.38094	0.0491628
$790$	19.4857	0.693271
$791$	10.8672	0.386394
$792$	0	0
$793$	5.05790	0.179611
$794$	103.800	3.68372
$795$	−31.1953	−1.10638
$796$	3.65788	0.129650
$797$	−17.9706	−0.636551	−0.318276	−	0.947998i	$-0.603104\pi$
$797$	−17.9706	−0.636551	−0.318276	+	0.947998i	$0.603104\pi$
$798$	−63.4133	−2.24481
$799$	−5.84105	−0.206642
$800$	23.5754	0.833515
$801$	−1.94283	−0.0686466
$802$	64.8261	2.28909
$803$	0	0
$804$	85.0511	2.99952
$805$	−13.9019	−0.489979
$806$	−46.7513	−1.64675
$807$	−42.5730	−1.49864
$808$	152.648	5.37014
$809$	−39.0710	−1.37366	−0.686832	−	0.726816i	$-0.740999\pi$
$809$	−39.0710	−1.37366	−0.686832	+	0.726816i	$0.740999\pi$
$810$	25.9009	0.910067
$811$	13.3866	0.470068	0.235034	−	0.971987i	$-0.424480\pi$
$811$	13.3866	0.470068	0.235034	+	0.971987i	$0.424480\pi$
$812$	75.1283	2.63649
$813$	31.4232	1.10206
$814$	0	0
$815$	−7.28550	−0.255200
$816$	−21.4157	−0.749700
$817$	6.26966	0.219348
$818$	53.6360	1.87534
$819$	10.3475	0.361570
$820$	−81.1711	−2.83462
$821$	0.907897	0.0316858	0.0158429	−	0.999874i	$-0.494957\pi$
$821$	0.907897	0.0316858	0.0158429	+	0.999874i	$0.494957\pi$
$822$	69.6355	2.42882
$823$	−49.9622	−1.74157	−0.870787	−	0.491661i	$-0.836390\pi$
$823$	−49.9622	−1.74157	−0.870787	+	0.491661i	$0.836390\pi$
$824$	−28.8368	−1.00458
$825$	0	0
$826$	−75.2868	−2.61956
$827$	29.4460	1.02394	0.511968	−	0.859004i	$-0.328917\pi$
$827$	29.4460	1.02394	0.511968	+	0.859004i	$0.328917\pi$
$828$	−18.6013	−0.646440
$829$	7.03763	0.244427	0.122214	−	0.992504i	$-0.461001\pi$
$829$	7.03763	0.244427	0.122214	+	0.992504i	$0.461001\pi$
$830$	−43.0479	−1.49421
$831$	24.9501	0.865508
$832$	111.362	3.86078
$833$	−3.22186	−0.111631
$834$	3.28939	0.113902
$835$	9.56392	0.330973
$836$	0	0
$837$	19.3987	0.670516
$838$	34.9907	1.20873
$839$	−7.08555	−0.244620	−0.122310	−	0.992492i	$-0.539030\pi$
$839$	−7.08555	−0.244620	−0.122310	+	0.992492i	$0.539030\pi$
$840$	−48.9488	−1.68889
$841$	19.5609	0.674515
$842$	9.12609	0.314506
$843$	3.59282	0.123743
$844$	5.51998	0.190006
$845$	24.1607	0.831152
$846$	−13.7075	−0.471274
$847$	0	0
$848$	151.521	5.20326
$849$	−14.5552	−0.499533
$850$	4.06093	0.139289
$851$	−28.0825	−0.962656
$852$	74.9781	2.56871
$853$	14.9177	0.510771	0.255386	−	0.966839i	$-0.417798\pi$
$853$	14.9177	0.510771	0.255386	+	0.966839i	$0.417798\pi$
$854$	5.51904	0.188858
$855$	−15.5841	−0.532965
$856$	40.0097	1.36750
$857$	−2.69868	−0.0921850	−0.0460925	−	0.998937i	$-0.514677\pi$
$857$	−2.69868	−0.0921850	−0.0460925	+	0.998937i	$0.514677\pi$
$858$	0	0
$859$	−53.7843	−1.83510	−0.917549	−	0.397623i	$-0.869835\pi$
$859$	−53.7843	−1.83510	−0.917549	+	0.397623i	$0.869835\pi$
$860$	7.80377	0.266106
$861$	−23.2529	−0.792456
$862$	105.458	3.59193
$863$	26.3132	0.895711	0.447855	−	0.894106i	$-0.352188\pi$
$863$	26.3132	0.895711	0.447855	+	0.894106i	$0.352188\pi$
$864$	−101.583	−3.45593
$865$	26.4901	0.900692
$866$	90.8376	3.08679
$867$	22.1839	0.753406
$868$	−36.9706	−1.25487
$869$	0	0
$870$	−51.0181	−1.72968
$871$	−57.7589	−1.95708
$872$	70.3479	2.38228
$873$	−15.3630	−0.519960
$874$	−77.4157	−2.61862
$875$	24.8203	0.839080
$876$	−42.4919	−1.43567
$877$	17.5620	0.593027	0.296514	−	0.955029i	$-0.404176\pi$
$877$	17.5620	0.593027	0.296514	+	0.955029i	$0.404176\pi$
$878$	98.0749	3.30987
$879$	6.05507	0.204232
$880$	0	0
$881$	42.3553	1.42699	0.713493	−	0.700662i	$-0.247112\pi$
$881$	42.3553	1.42699	0.713493	+	0.700662i	$0.247112\pi$
$882$	−7.56090	−0.254589
$883$	−47.8079	−1.60886	−0.804432	−	0.594045i	$-0.797530\pi$
$883$	−47.8079	−1.60886	−0.804432	+	0.594045i	$0.797530\pi$
$884$	30.5619	1.02791
$885$	37.0518	1.24548
$886$	−74.3586	−2.49813
$887$	19.8353	0.666004	0.333002	−	0.942926i	$-0.391938\pi$
$887$	19.8353	0.666004	0.333002	+	0.942926i	$0.391938\pi$
$888$	−98.8785	−3.31815
$889$	24.4596	0.820349
$890$	−10.0642	−0.337354
$891$	0	0
$892$	5.62429	0.188315
$893$	−41.3441	−1.38353
$894$	25.1784	0.842092
$895$	−41.9940	−1.40370
$896$	47.9769	1.60280
$897$	−25.2972	−0.844648
$898$	89.9638	3.00213
$899$	−23.8968	−0.797004
$900$	6.90657	0.230219
$901$	13.1802	0.439096
$902$	0	0
$903$	2.23553	0.0743937
$904$	46.7125	1.55363
$905$	21.9531	0.729745
$906$	−42.3125	−1.40574
$907$	−3.15204	−0.104662	−0.0523308	−	0.998630i	$-0.516665\pi$
$907$	−3.15204	−0.104662	−0.0523308	+	0.998630i	$0.516665\pi$
$908$	−37.9740	−1.26021
$909$	17.3287	0.574758
$910$	53.6019	1.77688
$911$	−13.2153	−0.437843	−0.218921	−	0.975742i	$-0.570254\pi$
$911$	−13.2153	−0.437843	−0.218921	+	0.975742i	$0.570254\pi$
$912$	−151.585	−5.01947
$913$	0	0
$914$	−28.8145	−0.953098
$915$	−2.71615	−0.0897932
$916$	93.4258	3.08688
$917$	−3.44313	−0.113702
$918$	−17.4980	−0.577520
$919$	−52.6183	−1.73572	−0.867859	−	0.496810i	$-0.834505\pi$
$919$	−52.6183	−1.73572	−0.867859	+	0.496810i	$0.834505\pi$
$920$	−59.7572	−1.97014
$921$	8.76533	0.288827
$922$	87.7397	2.88955
$923$	−50.9182	−1.67599
$924$	0	0
$925$	10.4269	0.342834
$926$	−71.3388	−2.34434
$927$	−3.27358	−0.107519
$928$	125.138	4.10786
$929$	−39.1701	−1.28513	−0.642565	−	0.766231i	$-0.722130\pi$
$929$	−39.1701	−1.28513	−0.642565	+	0.766231i	$0.722130\pi$
$930$	25.1060	0.823259
$931$	−22.8049	−0.747401
$932$	−113.093	−3.70449
$933$	−3.29857	−0.107990
$934$	84.2417	2.75647
$935$	0	0
$936$	44.4783	1.45382
$937$	50.8259	1.66041	0.830205	−	0.557459i	$-0.188224\pi$
$937$	50.8259	1.66041	0.830205	+	0.557459i	$0.188224\pi$
$938$	−63.0248	−2.05783
$939$	−3.24380	−0.105857
$940$	−51.4605	−1.67846
$941$	0.673372	0.0219513	0.0109757	−	0.999940i	$-0.496506\pi$
$941$	0.673372	0.0219513	0.0109757	+	0.999940i	$0.496506\pi$
$942$	20.1126	0.655303
$943$	−28.3874	−0.924420
$944$	−179.967	−5.85743
$945$	−22.2412	−0.723505
$946$	0	0
$947$	−39.9715	−1.29890	−0.649450	−	0.760405i	$-0.725001\pi$
$947$	−39.9715	−1.29890	−0.649450	+	0.760405i	$0.725001\pi$
$948$	28.0399	0.910692
$949$	28.8566	0.936724
$950$	28.7440	0.932580
$951$	−36.5833	−1.18630
$952$	20.6811	0.670278
$953$	−5.14686	−0.166723	−0.0833616	−	0.996519i	$-0.526566\pi$
$953$	−5.14686	−0.166723	−0.0833616	+	0.996519i	$0.526566\pi$
$954$	30.9307	1.00142
$955$	17.0433	0.551510
$956$	12.8530	0.415697
$957$	0	0
$958$	103.210	3.33457
$959$	−37.3966	−1.20760
$960$	−59.8027	−1.93012
$961$	−19.2404	−0.620657
$962$	108.278	3.49102
$963$	4.54194	0.146362
$964$	45.3512	1.46067
$965$	29.5291	0.950574
$966$	−27.6036	−0.888130
$967$	−50.0761	−1.61034	−0.805170	−	0.593044i	$-0.797926\pi$
$967$	−50.0761	−1.61034	−0.805170	+	0.593044i	$0.797926\pi$
$968$	0	0
$969$	−13.1857	−0.423586
$970$	−79.5835	−2.55527
$971$	−19.8968	−0.638519	−0.319259	−	0.947667i	$-0.603434\pi$
$971$	−19.8968	−0.638519	−0.319259	+	0.947667i	$0.603434\pi$
$972$	−52.0839	−1.67059
$973$	−1.76651	−0.0566318
$974$	−54.9334	−1.76018
$975$	9.39271	0.300807
$976$	13.1928	0.422292
$977$	38.3757	1.22775	0.613874	−	0.789404i	$-0.289610\pi$
$977$	38.3757	1.22775	0.613874	+	0.789404i	$0.289610\pi$
$978$	−14.4660	−0.462572
$979$	0	0
$980$	−28.3850	−0.906726
$981$	7.98596	0.254972
$982$	−109.515	−3.49477
$983$	−5.85548	−0.186761	−0.0933804	−	0.995631i	$-0.529767\pi$
$983$	−5.85548	−0.186761	−0.0933804	+	0.995631i	$0.529767\pi$
$984$	−99.9519	−3.18635
$985$	−20.0124	−0.637647
$986$	21.5554	0.686465
$987$	−14.7418	−0.469236
$988$	216.323	6.88215
$989$	2.72916	0.0867821
$990$	0	0
$991$	−35.9901	−1.14326	−0.571631	−	0.820510i	$-0.693689\pi$
$991$	−35.9901	−1.14326	−0.571631	+	0.820510i	$0.693689\pi$
$992$	−61.5805	−1.95518
$993$	3.57272	0.113377
$994$	−55.5605	−1.76227
$995$	1.33399	0.0422902
$996$	−61.9456	−1.96282
$997$	41.2696	1.30702	0.653510	−	0.756918i	$-0.273296\pi$
$997$	41.2696	1.30702	0.653510	+	0.756918i	$0.273296\pi$
$998$	−79.6048	−2.51984
$999$	−44.9281	−1.42146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7381.2.a.i.1.2		19
11.10	odd	2		671.2.a.c.1.18	✓	19
33.32	even	2		6039.2.a.k.1.2		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.18	✓	19	11.10	odd	2
6039.2.a.k.1.2		19	33.32	even	2
7381.2.a.i.1.2		19	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:	\( N \)	\(=\)	\( 7381 = 11^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7381.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69542 q^{2} -1.41452 q^{3} +5.26531 q^{4} +1.92020 q^{5} +3.81272 q^{6} -2.04756 q^{7} -8.80138 q^{8} -0.999141 q^{9} +O(q^{10})\)	\(q-2.69542 q^{2} -1.41452 q^{3} +5.26531 q^{4} +1.92020 q^{5} +3.81272 q^{6} -2.04756 q^{7} -8.80138 q^{8} -0.999141 q^{9} -5.17575 q^{10} -7.44787 q^{12} +5.05790 q^{13} +5.51904 q^{14} -2.71615 q^{15} +13.1928 q^{16} +1.14759 q^{17} +2.69311 q^{18} +8.12286 q^{19} +10.1104 q^{20} +2.89631 q^{21} +3.53584 q^{23} +12.4497 q^{24} -1.31284 q^{25} -13.6332 q^{26} +5.65685 q^{27} -10.7810 q^{28} -6.96857 q^{29} +7.32118 q^{30} +3.42923 q^{31} -17.9575 q^{32} -3.09324 q^{34} -3.93172 q^{35} -5.26078 q^{36} -7.94224 q^{37} -21.8945 q^{38} -7.15449 q^{39} -16.9004 q^{40} -8.02845 q^{41} -7.80678 q^{42} +0.771854 q^{43} -1.91855 q^{45} -9.53060 q^{46} -5.08985 q^{47} -18.6615 q^{48} -2.80750 q^{49} +3.53866 q^{50} -1.62328 q^{51} +26.6314 q^{52} +11.4851 q^{53} -15.2476 q^{54} +18.0214 q^{56} -11.4899 q^{57} +18.7832 q^{58} -13.6413 q^{59} -14.3014 q^{60} +1.00000 q^{61} -9.24323 q^{62} +2.04580 q^{63} +22.0174 q^{64} +9.71217 q^{65} -11.4195 q^{67} +6.04241 q^{68} -5.00151 q^{69} +10.5976 q^{70} -10.0671 q^{71} +8.79382 q^{72} +5.70524 q^{73} +21.4077 q^{74} +1.85704 q^{75} +42.7693 q^{76} +19.2844 q^{78} -3.76482 q^{79} +25.3329 q^{80} -5.00429 q^{81} +21.6401 q^{82} +8.31723 q^{83} +15.2499 q^{84} +2.20360 q^{85} -2.08047 q^{86} +9.85716 q^{87} +1.94450 q^{89} +5.17130 q^{90} -10.3564 q^{91} +18.6173 q^{92} -4.85071 q^{93} +13.7193 q^{94} +15.5975 q^{95} +25.4012 q^{96} +15.3762 q^{97} +7.56740 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9	\( 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9} - 7 q^{10} - 4 q^{12} - 8 q^{13} - 11 q^{14} - 5 q^{15} + 31 q^{16} - 9 q^{17} - 10 q^{18} - 17 q^{19} - 6 q^{20} - 18 q^{21} - 10 q^{23} - 26 q^{24} + 45 q^{25} + 5 q^{26} - 33 q^{27} - 36 q^{28} - 27 q^{29} + 30 q^{30} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 38 q^{36} + 20 q^{37} - 37 q^{38} - 24 q^{39} - 10 q^{40} - 19 q^{41} + 21 q^{42} - 20 q^{43} - 32 q^{45} - 41 q^{46} - 19 q^{47} + 5 q^{48} + 42 q^{49} - 36 q^{50} - 47 q^{51} + 28 q^{52} + 3 q^{53} + 33 q^{54} - 44 q^{56} - 11 q^{57} + 23 q^{58} - 28 q^{59} - 96 q^{60} + 19 q^{61} + 11 q^{62} + 32 q^{63} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} - 19 q^{71} - 34 q^{72} - 20 q^{73} + 22 q^{74} - 50 q^{75} + 25 q^{76} - 94 q^{78} - 69 q^{79} - 36 q^{80} + 47 q^{81} - 61 q^{82} - q^{83} + 28 q^{84} - 24 q^{85} - 27 q^{86} + 58 q^{87} - 36 q^{90} + 24 q^{91} - 67 q^{92} - 14 q^{93} - 64 q^{94} + 3 q^{95} + 26 q^{96} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9 - 7 * q^10 - 4 * q^12 - 8 * q^13 - 11 * q^14 - 5 * q^15 + 31 * q^16 - 9 * q^17 - 10 * q^18 - 17 * q^19 - 6 * q^20 - 18 * q^21 - 10 * q^23 - 26 * q^24 + 45 * q^25 + 5 * q^26 - 33 * q^27 - 36 * q^28 - 27 * q^29 + 30 * q^30 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 38 * q^36 + 20 * q^37 - 37 * q^38 - 24 * q^39 - 10 * q^40 - 19 * q^41 + 21 * q^42 - 20 * q^43 - 32 * q^45 - 41 * q^46 - 19 * q^47 + 5 * q^48 + 42 * q^49 - 36 * q^50 - 47 * q^51 + 28 * q^52 + 3 * q^53 + 33 * q^54 - 44 * q^56 - 11 * q^57 + 23 * q^58 - 28 * q^59 - 96 * q^60 + 19 * q^61 + 11 * q^62 + 32 * q^63 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 - 19 * q^71 - 34 * q^72 - 20 * q^73 + 22 * q^74 - 50 * q^75 + 25 * q^76 - 94 * q^78 - 69 * q^79 - 36 * q^80 + 47 * q^81 - 61 * q^82 - q^83 + 28 * q^84 - 24 * q^85 - 27 * q^86 + 58 * q^87 - 36 * q^90 + 24 * q^91 - 67 * q^92 - 14 * q^93 - 64 * q^94 + 3 * q^95 + 26 * q^96 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more