LMFDB - Embedded newform 7381.2.a.i.1.3

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			1.3
Root			$2.34443$ 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)

$f(q)$	$=$	$q-2.34443 q^{2} -3.17986 q^{3} +3.49635 q^{4} +3.40234 q^{5} +7.45497 q^{6} +2.95885 q^{7} -3.50809 q^{8} +7.11153 q^{9} +O(q^{10})$	\(q-2.34443 q^{2} -3.17986 q^{3} +3.49635 q^{4} +3.40234 q^{5} +7.45497 q^{6} +2.95885 q^{7} -3.50809 q^{8} +7.11153 q^{9} -7.97654 q^{10} -11.1179 q^{12} -5.60121 q^{13} -6.93681 q^{14} -10.8190 q^{15} +1.23176 q^{16} +3.44142 q^{17} -16.6725 q^{18} -6.93678 q^{19} +11.8958 q^{20} -9.40873 q^{21} +0.836947 q^{23} +11.1552 q^{24} +6.57591 q^{25} +13.1316 q^{26} -13.0741 q^{27} +10.3452 q^{28} -6.68910 q^{29} +25.3643 q^{30} +0.117341 q^{31} +4.12839 q^{32} -8.06817 q^{34} +10.0670 q^{35} +24.8644 q^{36} -0.152675 q^{37} +16.2628 q^{38} +17.8111 q^{39} -11.9357 q^{40} +10.6206 q^{41} +22.0581 q^{42} -5.87242 q^{43} +24.1958 q^{45} -1.96216 q^{46} +0.889406 q^{47} -3.91684 q^{48} +1.75477 q^{49} -15.4168 q^{50} -10.9433 q^{51} -19.5838 q^{52} +3.79401 q^{53} +30.6513 q^{54} -10.3799 q^{56} +22.0580 q^{57} +15.6821 q^{58} -9.37456 q^{59} -37.8269 q^{60} +1.00000 q^{61} -0.275097 q^{62} +21.0419 q^{63} -12.1423 q^{64} -19.0572 q^{65} +2.66764 q^{67} +12.0324 q^{68} -2.66138 q^{69} -23.6014 q^{70} +1.13373 q^{71} -24.9479 q^{72} +15.2793 q^{73} +0.357937 q^{74} -20.9105 q^{75} -24.2534 q^{76} -41.7568 q^{78} -5.23851 q^{79} +4.19087 q^{80} +20.2393 q^{81} -24.8993 q^{82} +17.7327 q^{83} -32.8962 q^{84} +11.7089 q^{85} +13.7675 q^{86} +21.2704 q^{87} +2.71742 q^{89} -56.7255 q^{90} -16.5731 q^{91} +2.92626 q^{92} -0.373127 q^{93} -2.08515 q^{94} -23.6013 q^{95} -13.1277 q^{96} -16.6565 q^{97} -4.11394 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.34443	−1.65776	−0.828881	−	0.559425i	$-0.811022\pi$
$2$	−2.34443	−1.65776	−0.828881	+	0.559425i	$0.811022\pi$
$3$	−3.17986	−1.83590	−0.917948	−	0.396702i	$-0.870155\pi$
$3$	−3.17986	−1.83590	−0.917948	+	0.396702i	$0.870155\pi$
$4$	3.49635	1.74817
$5$	3.40234	1.52157	0.760786	−	0.649003i	$-0.224814\pi$
$5$	3.40234	1.52157	0.760786	+	0.649003i	$0.224814\pi$
$6$	7.45497	3.04348
$7$	2.95885	1.11834	0.559169	−	0.829053i	$-0.311120\pi$
$7$	2.95885	1.11834	0.559169	+	0.829053i	$0.311120\pi$
$8$	−3.50809	−1.24030
$9$	7.11153	2.37051
$10$	−7.97654	−2.52240
$11$	0	0
$11$	0	0
$12$	−11.1179	−3.20947
$13$	−5.60121	−1.55349	−0.776747	−	0.629812i	$-0.783132\pi$
$13$	−5.60121	−1.55349	−0.776747	+	0.629812i	$0.783132\pi$
$14$	−6.93681	−1.85394
$15$	−10.8190	−2.79345
$16$	1.23176	0.307941
$17$	3.44142	0.834667	0.417334	−	0.908753i	$-0.362965\pi$
$17$	3.44142	0.834667	0.417334	+	0.908753i	$0.362965\pi$
$18$	−16.6725	−3.92974
$19$	−6.93678	−1.59141	−0.795703	−	0.605687i	$-0.792898\pi$
$19$	−6.93678	−1.59141	−0.795703	+	0.605687i	$0.792898\pi$
$20$	11.8958	2.65997
$21$	−9.40873	−2.05315
$22$	0	0
$23$	0.836947	0.174516	0.0872578	−	0.996186i	$-0.472190\pi$
$23$	0.836947	0.174516	0.0872578	+	0.996186i	$0.472190\pi$
$24$	11.1552	2.27705
$25$	6.57591	1.31518
$26$	13.1316	2.57532
$27$	−13.0741	−2.51611
$28$	10.3452	1.95505
$29$	−6.68910	−1.24213	−0.621067	−	0.783757i	$-0.713301\pi$
$29$	−6.68910	−1.24213	−0.621067	+	0.783757i	$0.713301\pi$
$30$	25.3643	4.63087
$31$	0.117341	0.0210750	0.0105375	−	0.999944i	$-0.496646\pi$
$31$	0.117341	0.0210750	0.0105375	+	0.999944i	$0.496646\pi$
$32$	4.12839	0.729804
$33$	0	0
$34$	−8.06817	−1.38368
$35$	10.0670	1.70163
$36$	24.8644	4.14407
$37$	−0.152675	−0.0250997	−0.0125498	−	0.999921i	$-0.503995\pi$
$37$	−0.152675	−0.0250997	−0.0125498	+	0.999921i	$0.503995\pi$
$38$	16.2628	2.63817
$39$	17.8111	2.85205
$40$	−11.9357	−1.88720
$41$	10.6206	1.65866	0.829330	−	0.558759i	$-0.188722\pi$
$41$	10.6206	1.65866	0.829330	+	0.558759i	$0.188722\pi$
$42$	22.0581	3.40364
$43$	−5.87242	−0.895536	−0.447768	−	0.894150i	$-0.647781\pi$
$43$	−5.87242	−0.895536	−0.447768	+	0.894150i	$0.647781\pi$
$44$	0	0
$45$	24.1958	3.60690
$46$	−1.96216	−0.289305
$47$	0.889406	0.129733	0.0648666	−	0.997894i	$-0.479338\pi$
$47$	0.889406	0.129733	0.0648666	+	0.997894i	$0.479338\pi$
$48$	−3.91684	−0.565347
$49$	1.75477	0.250682
$50$	−15.4168	−2.18026
$51$	−10.9433	−1.53236
$52$	−19.5838	−2.71578
$53$	3.79401	0.521148	0.260574	−	0.965454i	$-0.416088\pi$
$53$	3.79401	0.521148	0.260574	+	0.965454i	$0.416088\pi$
$54$	30.6513	4.17112
$55$	0	0
$56$	−10.3799	−1.38707
$57$	22.0580	2.92165
$58$	15.6821	2.05916
$59$	−9.37456	−1.22046	−0.610232	−	0.792223i	$-0.708924\pi$
$59$	−9.37456	−1.22046	−0.610232	+	0.792223i	$0.708924\pi$
$60$	−37.8269	−4.88343
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−0.275097	−0.0349373
$63$	21.0419	2.65103
$64$	−12.1423	−1.51778
$65$	−19.0572	−2.36375
$66$	0	0
$67$	2.66764	0.325904	0.162952	−	0.986634i	$-0.447898\pi$
$67$	2.66764	0.325904	0.162952	+	0.986634i	$0.447898\pi$
$68$	12.0324	1.45914
$69$	−2.66138	−0.320392
$70$	−23.6014	−2.82090
$71$	1.13373	0.134549	0.0672745	−	0.997735i	$-0.478570\pi$
$71$	1.13373	0.134549	0.0672745	+	0.997735i	$0.478570\pi$
$72$	−24.9479	−2.94014
$73$	15.2793	1.78830	0.894151	−	0.447766i	$-0.147780\pi$
$73$	15.2793	1.78830	0.894151	+	0.447766i	$0.147780\pi$
$74$	0.357937	0.0416093
$75$	−20.9105	−2.41454
$76$	−24.2534	−2.78206
$77$	0	0
$78$	−41.7568	−4.72803
$79$	−5.23851	−0.589378	−0.294689	−	0.955593i	$-0.595216\pi$
$79$	−5.23851	−0.589378	−0.294689	+	0.955593i	$0.595216\pi$
$80$	4.19087	0.468554
$81$	20.2393	2.24881
$82$	−24.8993	−2.74966
$83$	17.7327	1.94641	0.973207	−	0.229929i	$-0.0738493\pi$
$83$	17.7327	1.94641	0.973207	+	0.229929i	$0.0738493\pi$
$84$	−32.8962	−3.58927
$85$	11.7089	1.27001
$86$	13.7675	1.48458
$87$	21.2704	2.28043
$88$	0	0
$89$	2.71742	0.288046	0.144023	−	0.989574i	$-0.453996\pi$
$89$	2.71742	0.288046	0.144023	+	0.989574i	$0.453996\pi$
$90$	−56.7255	−5.97939
$91$	−16.5731	−1.73733
$92$	2.92626	0.305084
$93$	−0.373127	−0.0386915
$94$	−2.08515	−0.215067
$95$	−23.6013	−2.42144
$96$	−13.1277	−1.33984
$97$	−16.6565	−1.69121	−0.845604	−	0.533811i	$-0.820759\pi$
$97$	−16.6565	−1.69121	−0.845604	+	0.533811i	$0.820759\pi$
$98$	−4.11394	−0.415571
$99$	0	0
$100$	22.9917	2.29917
$101$	−5.98398	−0.595429	−0.297714	−	0.954655i	$-0.596224\pi$
$101$	−5.98398	−0.595429	−0.297714	+	0.954655i	$0.596224\pi$
$102$	25.6557	2.54029
$103$	−6.59746	−0.650067	−0.325033	−	0.945703i	$-0.605376\pi$
$103$	−6.59746	−0.650067	−0.325033	+	0.945703i	$0.605376\pi$
$104$	19.6495	1.92679
$105$	−32.0117	−3.12402
$106$	−8.89480	−0.863939
$107$	−9.19690	−0.889098	−0.444549	−	0.895754i	$-0.646636\pi$
$107$	−9.19690	−0.889098	−0.444549	+	0.895754i	$0.646636\pi$
$108$	−45.7117	−4.39861
$109$	8.38506	0.803143	0.401571	−	0.915828i	$-0.368464\pi$
$109$	8.38506	0.803143	0.401571	+	0.915828i	$0.368464\pi$
$110$	0	0
$111$	0.485487	0.0460804
$112$	3.64460	0.344382
$113$	−17.3735	−1.63437	−0.817183	−	0.576379i	$-0.804465\pi$
$113$	−17.3735	−1.63437	−0.817183	+	0.576379i	$0.804465\pi$
$114$	−51.7134	−4.84341
$115$	2.84758	0.265538
$116$	−23.3874	−2.17147
$117$	−39.8332	−3.68258
$118$	21.9780	2.02324
$119$	10.1826	0.933441
$120$	37.9539	3.46470
$121$	0	0
$122$	−2.34443	−0.212255
$123$	−33.7721	−3.04513
$124$	0.410264	0.0368428
$125$	5.36178	0.479572
$126$	−49.3313	−4.39478
$127$	11.1692	0.991102	0.495551	−	0.868579i	$-0.334966\pi$
$127$	11.1692	0.991102	0.495551	+	0.868579i	$0.334966\pi$
$128$	20.2099	1.78632
$129$	18.6735	1.64411
$130$	44.6783	3.91854
$131$	0.0869930	0.00760061	0.00380031	−	0.999993i	$-0.498790\pi$
$131$	0.0869930	0.00760061	0.00380031	+	0.999993i	$0.498790\pi$
$132$	0	0
$133$	−20.5249	−1.77973
$134$	−6.25409	−0.540271
$135$	−44.4826	−3.82845
$136$	−12.0728	−1.03523
$137$	10.4873	0.895989	0.447994	−	0.894036i	$-0.352138\pi$
$137$	10.4873	0.895989	0.447994	+	0.894036i	$0.352138\pi$
$138$	6.23941	0.531134
$139$	−7.79620	−0.661265	−0.330633	−	0.943760i	$-0.607262\pi$
$139$	−7.79620	−0.661265	−0.330633	+	0.943760i	$0.607262\pi$
$140$	35.1978	2.97475
$141$	−2.82819	−0.238177
$142$	−2.65795	−0.223050
$143$	0	0
$144$	8.75972	0.729977
$145$	−22.7586	−1.89000
$146$	−35.8211	−2.96458
$147$	−5.57994	−0.460226
$148$	−0.533806	−0.0438786
$149$	−10.3850	−0.850770	−0.425385	−	0.905012i	$-0.639861\pi$
$149$	−10.3850	−0.850770	−0.425385	+	0.905012i	$0.639861\pi$
$150$	49.0232	4.00273
$151$	3.38814	0.275723	0.137862	−	0.990452i	$-0.455977\pi$
$151$	3.38814	0.275723	0.137862	+	0.990452i	$0.455977\pi$
$152$	24.3348	1.97381
$153$	24.4738	1.97859
$154$	0	0
$155$	0.399233	0.0320671
$156$	62.2737	4.98589
$157$	6.15931	0.491567	0.245783	−	0.969325i	$-0.420955\pi$
$157$	6.15931	0.491567	0.245783	+	0.969325i	$0.420955\pi$
$158$	12.2813	0.977049
$159$	−12.0644	−0.956773
$160$	14.0462	1.11045
$161$	2.47640	0.195168
$162$	−47.4496	−3.72800
$163$	−0.500868	−0.0392310	−0.0196155	−	0.999808i	$-0.506244\pi$
$163$	−0.500868	−0.0392310	−0.0196155	+	0.999808i	$0.506244\pi$
$164$	37.1333	2.89963
$165$	0	0
$166$	−41.5730	−3.22669
$167$	−4.82285	−0.373203	−0.186602	−	0.982436i	$-0.559747\pi$
$167$	−4.82285	−0.373203	−0.186602	+	0.982436i	$0.559747\pi$
$168$	33.0066	2.54652
$169$	18.3735	1.41335
$170$	−27.4507	−2.10537
$171$	−49.3311	−3.77245
$172$	−20.5320	−1.56555
$173$	−5.75856	−0.437816	−0.218908	−	0.975746i	$-0.570249\pi$
$173$	−5.75856	−0.437816	−0.218908	+	0.975746i	$0.570249\pi$
$174$	−49.8670	−3.78041
$175$	19.4571	1.47082
$176$	0	0
$177$	29.8098	2.24064
$178$	−6.37079	−0.477511
$179$	12.1227	0.906090	0.453045	−	0.891488i	$-0.350338\pi$
$179$	12.1227	0.906090	0.453045	+	0.891488i	$0.350338\pi$
$180$	84.5971	6.30550
$181$	26.0278	1.93463	0.967317	−	0.253571i	$-0.0816051\pi$
$181$	26.0278	1.93463	0.967317	+	0.253571i	$0.0816051\pi$
$182$	38.8545	2.88009
$183$	−3.17986	−0.235062
$184$	−2.93608	−0.216451
$185$	−0.519453	−0.0381910
$186$	0.874771	0.0641413
$187$	0	0
$188$	3.10967	0.226796
$189$	−38.6843	−2.81387
$190$	55.3315	4.01417
$191$	−7.55414	−0.546599	−0.273299	−	0.961929i	$-0.588115\pi$
$191$	−7.55414	−0.546599	−0.273299	+	0.961929i	$0.588115\pi$
$192$	38.6107	2.78649
$193$	−7.78558	−0.560418	−0.280209	−	0.959939i	$-0.590404\pi$
$193$	−7.78558	−0.560418	−0.280209	+	0.959939i	$0.590404\pi$
$194$	39.0499	2.80362
$195$	60.5993	4.33961
$196$	6.13530	0.438236
$197$	−10.6392	−0.758011	−0.379006	−	0.925394i	$-0.623734\pi$
$197$	−10.6392	−0.758011	−0.379006	+	0.925394i	$0.623734\pi$
$198$	0	0
$199$	−14.2510	−1.01023	−0.505113	−	0.863053i	$-0.668549\pi$
$199$	−14.2510	−1.01023	−0.505113	+	0.863053i	$0.668549\pi$
$200$	−23.0689	−1.63122
$201$	−8.48273	−0.598325
$202$	14.0290	0.987079
$203$	−19.7920	−1.38913
$204$	−38.2614	−2.67884
$205$	36.1349	2.52377
$206$	15.4673	1.07766
$207$	5.95198	0.413691
$208$	−6.89935	−0.478384
$209$	0	0
$210$	75.0491	5.17888
$211$	−12.0462	−0.829293	−0.414646	−	0.909983i	$-0.636095\pi$
$211$	−12.0462	−0.829293	−0.414646	+	0.909983i	$0.636095\pi$
$212$	13.2652	0.911058
$213$	−3.60511	−0.247018
$214$	21.5615	1.47391
$215$	−19.9800	−1.36262
$216$	45.8651	3.12073
$217$	0.347193	0.0235690
$218$	−19.6582	−1.33142
$219$	−48.5860	−3.28313
$220$	0	0
$221$	−19.2761	−1.29665
$222$	−1.13819	−0.0763903
$223$	13.9356	0.933199	0.466600	−	0.884469i	$-0.345479\pi$
$223$	13.9356	0.933199	0.466600	+	0.884469i	$0.345479\pi$
$224$	12.2153	0.816168
$225$	46.7648	3.11765
$226$	40.7311	2.70939
$227$	6.52993	0.433407	0.216703	−	0.976237i	$-0.430470\pi$
$227$	6.52993	0.433407	0.216703	+	0.976237i	$0.430470\pi$
$228$	77.1225	5.10756
$229$	−11.8163	−0.780841	−0.390420	−	0.920637i	$-0.627670\pi$
$229$	−11.8163	−0.780841	−0.390420	+	0.920637i	$0.627670\pi$
$230$	−6.67595	−0.440199
$231$	0	0
$232$	23.4659	1.54061
$233$	24.9037	1.63149	0.815747	−	0.578409i	$-0.196326\pi$
$233$	24.9037	1.63149	0.815747	+	0.578409i	$0.196326\pi$
$234$	93.3860	6.10484
$235$	3.02606	0.197398
$236$	−32.7768	−2.13358
$237$	16.6577	1.08204
$238$	−23.8725	−1.54742
$239$	−9.06407	−0.586306	−0.293153	−	0.956066i	$-0.594704\pi$
$239$	−9.06407	−0.586306	−0.293153	+	0.956066i	$0.594704\pi$
$240$	−13.3264	−0.860216
$241$	−18.2687	−1.17679	−0.588396	−	0.808573i	$-0.700240\pi$
$241$	−18.2687	−1.17679	−0.588396	+	0.808573i	$0.700240\pi$
$242$	0	0
$243$	−25.1359	−1.61247
$244$	3.49635	0.223831
$245$	5.97033	0.381431
$246$	79.1762	5.04809
$247$	38.8543	2.47224
$248$	−0.411641	−0.0261392
$249$	−56.3875	−3.57341
$250$	−12.5703	−0.795017
$251$	−21.8990	−1.38225	−0.691126	−	0.722734i	$-0.742885\pi$
$251$	−21.8990	−1.38225	−0.691126	+	0.722734i	$0.742885\pi$
$252$	73.5700	4.63447
$253$	0	0
$254$	−26.1853	−1.64301
$255$	−37.2327	−2.33160
$256$	−23.0961	−1.44351
$257$	3.24672	0.202525	0.101263	−	0.994860i	$-0.467712\pi$
$257$	3.24672	0.202525	0.101263	+	0.994860i	$0.467712\pi$
$258$	−43.7787	−2.72554
$259$	−0.451743	−0.0280699
$260$	−66.6306	−4.13226
$261$	−47.5698	−2.94449
$262$	−0.203949	−0.0126000
$263$	−26.4462	−1.63074	−0.815372	−	0.578938i	$-0.803467\pi$
$263$	−26.4462	−1.63074	−0.815372	+	0.578938i	$0.803467\pi$
$264$	0	0
$265$	12.9085	0.792964
$266$	48.1191	2.95037
$267$	−8.64102	−0.528822
$268$	9.32700	0.569737
$269$	−6.95561	−0.424091	−0.212046	−	0.977260i	$-0.568013\pi$
$269$	−6.95561	−0.424091	−0.212046	+	0.977260i	$0.568013\pi$
$270$	104.286	6.34666
$271$	−11.3494	−0.689428	−0.344714	−	0.938708i	$-0.612024\pi$
$271$	−11.3494	−0.689428	−0.344714	+	0.938708i	$0.612024\pi$
$272$	4.23901	0.257028
$273$	52.7002	3.18956
$274$	−24.5867	−1.48534
$275$	0	0
$276$	−9.30511	−0.560102
$277$	18.8926	1.13515	0.567574	−	0.823322i	$-0.307882\pi$
$277$	18.8926	1.13515	0.567574	+	0.823322i	$0.307882\pi$
$278$	18.2776	1.09622
$279$	0.834472	0.0499585
$280$	−35.3159	−2.11053
$281$	15.2462	0.909514	0.454757	−	0.890615i	$-0.349726\pi$
$281$	15.2462	0.909514	0.454757	+	0.890615i	$0.349726\pi$
$282$	6.63049	0.394840
$283$	10.3061	0.612631	0.306316	−	0.951930i	$-0.400904\pi$
$283$	10.3061	0.612631	0.306316	+	0.951930i	$0.400904\pi$
$284$	3.96391	0.235215
$285$	75.0488	4.44551
$286$	0	0
$287$	31.4247	1.85494
$288$	29.3592	1.73001
$289$	−5.15662	−0.303330
$290$	53.3559	3.13317
$291$	52.9653	3.10488
$292$	53.4216	3.12626
$293$	17.6417	1.03064	0.515320	−	0.856998i	$-0.327673\pi$
$293$	17.6417	1.03064	0.515320	+	0.856998i	$0.327673\pi$
$294$	13.0818	0.762945
$295$	−31.8954	−1.85702
$296$	0.535598	0.0311310
$297$	0	0
$298$	24.3468	1.41037
$299$	−4.68791	−0.271109
$300$	−73.1104	−4.22103
$301$	−17.3756	−1.00151
$302$	−7.94326	−0.457083
$303$	19.0283	1.09314
$304$	−8.54446	−0.490058
$305$	3.40234	0.194817
$306$	−57.3771	−3.28003
$307$	8.18683	0.467247	0.233624	−	0.972327i	$-0.424942\pi$
$307$	8.18683	0.467247	0.233624	+	0.972327i	$0.424942\pi$
$308$	0	0
$309$	20.9790	1.19345
$310$	−0.935973	−0.0531597
$311$	−9.35473	−0.530458	−0.265229	−	0.964185i	$-0.585448\pi$
$311$	−9.35473	−0.530458	−0.265229	+	0.964185i	$0.585448\pi$
$312$	−62.4828	−3.53739
$313$	−8.48617	−0.479667	−0.239833	−	0.970814i	$-0.577093\pi$
$313$	−8.48617	−0.479667	−0.239833	+	0.970814i	$0.577093\pi$
$314$	−14.4401	−0.814900
$315$	71.5918	4.03374
$316$	−18.3157	−1.03034
$317$	−16.8444	−0.946074	−0.473037	−	0.881043i	$-0.656842\pi$
$317$	−16.8444	−0.946074	−0.473037	+	0.881043i	$0.656842\pi$
$318$	28.2842	1.58610
$319$	0	0
$320$	−41.3121	−2.30941
$321$	29.2449	1.63229
$322$	−5.80574	−0.323541
$323$	−23.8724	−1.32829
$324$	70.7637	3.93132
$325$	−36.8330	−2.04313
$326$	1.17425	0.0650357
$327$	−26.6633	−1.47449
$328$	−37.2580	−2.05723
$329$	2.63162	0.145086
$330$	0	0
$331$	−3.26480	−0.179450	−0.0897249	−	0.995967i	$-0.528599\pi$
$331$	−3.26480	−0.179450	−0.0897249	+	0.995967i	$0.528599\pi$
$332$	61.9997	3.40267
$333$	−1.08576	−0.0594991
$334$	11.3068	0.618683
$335$	9.07621	0.495886
$336$	−11.5893	−0.632249
$337$	10.7004	0.582888	0.291444	−	0.956588i	$-0.405864\pi$
$337$	10.7004	0.582888	0.291444	+	0.956588i	$0.405864\pi$
$338$	−43.0754	−2.34299
$339$	55.2455	3.00052
$340$	40.9384	2.22019
$341$	0	0
$342$	115.653	6.25382
$343$	−15.5198	−0.837992
$344$	20.6010	1.11073
$345$	−9.05491	−0.487500
$346$	13.5005	0.725794
$347$	−26.7520	−1.43612	−0.718061	−	0.695981i	$-0.754970\pi$
$347$	−26.7520	−1.43612	−0.718061	+	0.695981i	$0.754970\pi$
$348$	74.3689	3.98659
$349$	−36.1274	−1.93386	−0.966928	−	0.255048i	$-0.917909\pi$
$349$	−36.1274	−1.93386	−0.966928	+	0.255048i	$0.917909\pi$
$350$	−45.6158	−2.43827
$351$	73.2308	3.90877
$352$	0	0
$353$	−11.3578	−0.604514	−0.302257	−	0.953227i	$-0.597740\pi$
$353$	−11.3578	−0.604514	−0.302257	+	0.953227i	$0.597740\pi$
$354$	−69.8871	−3.71446
$355$	3.85733	0.204726
$356$	9.50104	0.503554
$357$	−32.3794	−1.71370
$358$	−28.4207	−1.50208
$359$	−21.9983	−1.16102	−0.580512	−	0.814252i	$-0.697148\pi$
$359$	−21.9983	−1.16102	−0.580512	+	0.814252i	$0.697148\pi$
$360$	−84.8811	−4.47363
$361$	29.1189	1.53257
$362$	−61.0204	−3.20716
$363$	0	0
$364$	−57.9454	−3.03716
$365$	51.9852	2.72103
$366$	7.45497	0.389677
$367$	−22.0610	−1.15158	−0.575788	−	0.817599i	$-0.695305\pi$
$367$	−22.0610	−1.15158	−0.575788	+	0.817599i	$0.695305\pi$
$368$	1.03092	0.0537404
$369$	75.5288	3.93187
$370$	1.21782	0.0633115
$371$	11.2259	0.582820
$372$	−1.30458	−0.0676395
$373$	17.4877	0.905477	0.452738	−	0.891643i	$-0.350447\pi$
$373$	17.4877	0.905477	0.452738	+	0.891643i	$0.350447\pi$
$374$	0	0
$375$	−17.0497	−0.880445
$376$	−3.12011	−0.160908
$377$	37.4670	1.92965
$378$	90.6926	4.66473
$379$	−21.5859	−1.10879	−0.554396	−	0.832253i	$-0.687051\pi$
$379$	−21.5859	−1.10879	−0.554396	+	0.832253i	$0.687051\pi$
$380$	−82.5183	−4.23310
$381$	−35.5164	−1.81956
$382$	17.7102	0.906130
$383$	−18.2673	−0.933414	−0.466707	−	0.884412i	$-0.654560\pi$
$383$	−18.2673	−0.933414	−0.466707	+	0.884412i	$0.654560\pi$
$384$	−64.2646	−3.27949
$385$	0	0
$386$	18.2527	0.929040
$387$	−41.7619	−2.12288
$388$	−58.2368	−2.95653
$389$	−33.0652	−1.67647	−0.838236	−	0.545307i	$-0.816413\pi$
$389$	−33.0652	−1.67647	−0.838236	+	0.545307i	$0.816413\pi$
$390$	−142.071	−7.19403
$391$	2.88029	0.145662
$392$	−6.15590	−0.310920
$393$	−0.276626	−0.0139539
$394$	24.9428	1.25660
$395$	−17.8232	−0.896781
$396$	0	0
$397$	30.1806	1.51472	0.757359	−	0.652998i	$-0.226489\pi$
$397$	30.1806	1.51472	0.757359	+	0.652998i	$0.226489\pi$
$398$	33.4104	1.67471
$399$	65.2663	3.26740
$400$	8.09996	0.404998
$401$	19.5558	0.976569	0.488284	−	0.872685i	$-0.337623\pi$
$401$	19.5558	0.976569	0.488284	+	0.872685i	$0.337623\pi$
$402$	19.8872	0.991881
$403$	−0.657249	−0.0327399
$404$	−20.9221	−1.04091
$405$	68.8610	3.42173
$406$	46.4010	2.30284
$407$	0	0
$408$	38.3899	1.90058
$409$	−23.6297	−1.16841	−0.584206	−	0.811605i	$-0.698594\pi$
$409$	−23.6297	−1.16841	−0.584206	+	0.811605i	$0.698594\pi$
$410$	−84.7157	−4.18381
$411$	−33.3481	−1.64494
$412$	−23.0670	−1.13643
$413$	−27.7379	−1.36489
$414$	−13.9540	−0.685801
$415$	60.3326	2.96161
$416$	−23.1240	−1.13375
$417$	24.7909	1.21401
$418$	0	0
$419$	2.18707	0.106845	0.0534227	−	0.998572i	$-0.482987\pi$
$419$	2.18707	0.106845	0.0534227	+	0.998572i	$0.482987\pi$
$420$	−111.924	−5.46133
$421$	−18.1274	−0.883476	−0.441738	−	0.897144i	$-0.645638\pi$
$421$	−18.1274	−0.883476	−0.441738	+	0.897144i	$0.645638\pi$
$422$	28.2414	1.37477
$423$	6.32504	0.307534
$424$	−13.3097	−0.646378
$425$	22.6305	1.09774
$426$	8.45191	0.409497
$427$	2.95885	0.143189
$428$	−32.1556	−1.55430
$429$	0	0
$430$	46.8416	2.25890
$431$	−6.17100	−0.297247	−0.148623	−	0.988894i	$-0.547484\pi$
$431$	−6.17100	−0.297247	−0.148623	+	0.988894i	$0.547484\pi$
$432$	−16.1042	−0.774814
$433$	39.8873	1.91686	0.958431	−	0.285326i	$-0.0921018\pi$
$433$	39.8873	1.91686	0.958431	+	0.285326i	$0.0921018\pi$
$434$	−0.813970	−0.0390718
$435$	72.3692	3.46984
$436$	29.3171	1.40403
$437$	−5.80572	−0.277725
$438$	113.906	5.44266
$439$	−1.12159	−0.0535304	−0.0267652	−	0.999642i	$-0.508521\pi$
$439$	−1.12159	−0.0535304	−0.0267652	+	0.999642i	$0.508521\pi$
$440$	0	0
$441$	12.4791	0.594244
$442$	45.1915	2.14954
$443$	19.2213	0.913231	0.456616	−	0.889664i	$-0.349061\pi$
$443$	19.2213	0.913231	0.456616	+	0.889664i	$0.349061\pi$
$444$	1.69743	0.0805565
$445$	9.24557	0.438282
$446$	−32.6711	−1.54702
$447$	33.0228	1.56193
$448$	−35.9271	−1.69739
$449$	−29.0769	−1.37222	−0.686111	−	0.727497i	$-0.740684\pi$
$449$	−29.0769	−1.37222	−0.686111	+	0.727497i	$0.740684\pi$
$450$	−109.637	−5.16833
$451$	0	0
$452$	−60.7440	−2.85716
$453$	−10.7738	−0.506199
$454$	−15.3090	−0.718485
$455$	−56.3873	−2.64348
$456$	−77.3814	−3.62372
$457$	19.8618	0.929096	0.464548	−	0.885548i	$-0.346217\pi$
$457$	19.8618	0.929096	0.464548	+	0.885548i	$0.346217\pi$
$458$	27.7024	1.29445
$459$	−44.9935	−2.10012
$460$	9.95613	0.464207
$461$	7.77290	0.362020	0.181010	−	0.983481i	$-0.442063\pi$
$461$	7.77290	0.362020	0.181010	+	0.983481i	$0.442063\pi$
$462$	0	0
$463$	−6.57813	−0.305712	−0.152856	−	0.988248i	$-0.548847\pi$
$463$	−6.57813	−0.305712	−0.152856	+	0.988248i	$0.548847\pi$
$464$	−8.23938	−0.382504
$465$	−1.26951	−0.0588719
$466$	−58.3849	−2.70463
$467$	−26.7754	−1.23902	−0.619509	−	0.784990i	$-0.712668\pi$
$467$	−26.7754	−1.23902	−0.619509	+	0.784990i	$0.712668\pi$
$468$	−139.271	−6.43779
$469$	7.89313	0.364471
$470$	−7.09439	−0.327240
$471$	−19.5858	−0.902465
$472$	32.8868	1.51374
$473$	0	0
$474$	−39.0529	−1.79376
$475$	−45.6156	−2.09299
$476$	35.6021	1.63182
$477$	26.9813	1.23539
$478$	21.2501	0.971955
$479$	18.7776	0.857970	0.428985	−	0.903312i	$-0.358871\pi$
$479$	18.7776	0.857970	0.428985	+	0.903312i	$0.358871\pi$
$480$	−44.6650	−2.03867
$481$	0.855166	0.0389922
$482$	42.8297	1.95084
$483$	−7.87461	−0.358307
$484$	0	0
$485$	−56.6709	−2.57329
$486$	58.9293	2.67309
$487$	33.6902	1.52665	0.763324	−	0.646016i	$-0.223566\pi$
$487$	33.6902	1.52665	0.763324	+	0.646016i	$0.223566\pi$
$488$	−3.50809	−0.158804
$489$	1.59269	0.0720240
$490$	−13.9970	−0.632321
$491$	5.96513	0.269203	0.134601	−	0.990900i	$-0.457025\pi$
$491$	5.96513	0.269203	0.134601	+	0.990900i	$0.457025\pi$
$492$	−118.079	−5.32341
$493$	−23.0200	−1.03677
$494$	−91.0912	−4.09839
$495$	0	0
$496$	0.144536	0.00648985
$497$	3.35453	0.150471
$498$	132.197	5.92387
$499$	−29.5064	−1.32089	−0.660445	−	0.750874i	$-0.729632\pi$
$499$	−29.5064	−1.32089	−0.660445	+	0.750874i	$0.729632\pi$
$500$	18.7467	0.838377
$501$	15.3360	0.685162
$502$	51.3407	2.29145
$503$	−22.2297	−0.991173	−0.495586	−	0.868559i	$-0.665047\pi$
$503$	−22.2297	−0.991173	−0.495586	+	0.868559i	$0.665047\pi$
$504$	−73.8169	−3.28807
$505$	−20.3595	−0.905988
$506$	0	0
$507$	−58.4252	−2.59476
$508$	39.0513	1.73262
$509$	−22.8655	−1.01350	−0.506749	−	0.862094i	$-0.669153\pi$
$509$	−22.8655	−1.01350	−0.506749	+	0.862094i	$0.669153\pi$
$510$	87.2893	3.86524
$511$	45.2090	1.99993
$512$	13.7275	0.606674
$513$	90.6922	4.00416
$514$	−7.61171	−0.335738
$515$	−22.4468	−0.989124
$516$	65.2891	2.87419
$517$	0	0
$518$	1.05908	0.0465333
$519$	18.3115	0.803784
$520$	66.8543	2.93176
$521$	10.0702	0.441182	0.220591	−	0.975366i	$-0.429201\pi$
$521$	10.0702	0.441182	0.220591	+	0.975366i	$0.429201\pi$
$522$	111.524	4.88127
$523$	−28.0628	−1.22710	−0.613551	−	0.789655i	$-0.710260\pi$
$523$	−28.0628	−1.22710	−0.613551	+	0.789655i	$0.710260\pi$
$524$	0.304158	0.0132872
$525$	−61.8710	−2.70027
$526$	62.0013	2.70338
$527$	0.403819	0.0175906
$528$	0	0
$529$	−22.2995	−0.969544
$530$	−30.2631	−1.31455
$531$	−66.6675	−2.89312
$532$	−71.7621	−3.11128
$533$	−59.4882	−2.57672
$534$	20.2583	0.876660
$535$	−31.2910	−1.35283
$536$	−9.35831	−0.404217
$537$	−38.5484	−1.66349
$538$	16.3069	0.703042
$539$	0	0
$540$	−155.527	−6.69280
$541$	−28.2703	−1.21544	−0.607718	−	0.794153i	$-0.707915\pi$
$541$	−28.2703	−1.21544	−0.607718	+	0.794153i	$0.707915\pi$
$542$	26.6079	1.14291
$543$	−82.7650	−3.55178
$544$	14.2075	0.609143
$545$	28.5288	1.22204
$546$	−123.552	−5.28754
$547$	7.09345	0.303294	0.151647	−	0.988435i	$-0.451542\pi$
$547$	7.09345	0.303294	0.151647	+	0.988435i	$0.451542\pi$
$548$	36.6672	1.56635
$549$	7.11153	0.303513
$550$	0	0
$551$	46.4008	1.97674
$552$	9.33634	0.397381
$553$	−15.4999	−0.659124
$554$	−44.2924	−1.88180
$555$	1.65179	0.0701146
$556$	−27.2582	−1.15601
$557$	−26.9056	−1.14002	−0.570012	−	0.821636i	$-0.693062\pi$
$557$	−26.9056	−1.14002	−0.570012	+	0.821636i	$0.693062\pi$
$558$	−1.95636	−0.0828194
$559$	32.8926	1.39121
$560$	12.4001	0.524002
$561$	0	0
$562$	−35.7437	−1.50776
$563$	16.1624	0.681164	0.340582	−	0.940215i	$-0.389376\pi$
$563$	16.1624	0.681164	0.340582	+	0.940215i	$0.389376\pi$
$564$	−9.88834	−0.416374
$565$	−59.1107	−2.48681
$566$	−24.1618	−1.01560
$567$	59.8850	2.51493
$568$	−3.97722	−0.166880
$569$	−6.60974	−0.277095	−0.138547	−	0.990356i	$-0.544243\pi$
$569$	−6.60974	−0.277095	−0.138547	+	0.990356i	$0.544243\pi$
$570$	−175.947	−7.36959
$571$	15.9453	0.667291	0.333646	−	0.942699i	$-0.391721\pi$
$571$	15.9453	0.667291	0.333646	+	0.942699i	$0.391721\pi$
$572$	0	0
$573$	24.0211	1.00350
$574$	−73.6731	−3.07506
$575$	5.50369	0.229520
$576$	−86.3500	−3.59792
$577$	−3.98449	−0.165876	−0.0829382	−	0.996555i	$-0.526430\pi$
$577$	−3.98449	−0.165876	−0.0829382	+	0.996555i	$0.526430\pi$
$578$	12.0893	0.502850
$579$	24.7571	1.02887
$580$	−79.5720	−3.30405
$581$	52.4683	2.17675
$582$	−124.173	−5.14715
$583$	0	0
$584$	−53.6010	−2.21802
$585$	−135.526	−5.60331
$586$	−41.3597	−1.70856
$587$	−11.1945	−0.462046	−0.231023	−	0.972948i	$-0.574207\pi$
$587$	−11.1945	−0.462046	−0.231023	+	0.972948i	$0.574207\pi$
$588$	−19.5094	−0.804555
$589$	−0.813966	−0.0335389
$590$	74.7766	3.07850
$591$	33.8312	1.39163
$592$	−0.188060	−0.00772921
$593$	1.90798	0.0783514	0.0391757	−	0.999232i	$-0.487527\pi$
$593$	1.90798	0.0783514	0.0391757	+	0.999232i	$0.487527\pi$
$594$	0	0
$595$	34.6448	1.42030
$596$	−36.3095	−1.48730
$597$	45.3162	1.85467
$598$	10.9905	0.449434
$599$	−0.764630	−0.0312419	−0.0156210	−	0.999878i	$-0.504973\pi$
$599$	−0.764630	−0.0312419	−0.0156210	+	0.999878i	$0.504973\pi$
$600$	73.3558	2.99474
$601$	−31.5123	−1.28541	−0.642706	−	0.766113i	$-0.722188\pi$
$601$	−31.5123	−1.28541	−0.642706	+	0.766113i	$0.722188\pi$
$602$	40.7358	1.66027
$603$	18.9710	0.772559
$604$	11.8461	0.482012
$605$	0	0
$606$	−44.6104	−1.81217
$607$	21.1448	0.858242	0.429121	−	0.903247i	$-0.358823\pi$
$607$	21.1448	0.858242	0.429121	+	0.903247i	$0.358823\pi$
$608$	−28.6377	−1.16141
$609$	62.9359	2.55029
$610$	−7.97654	−0.322961
$611$	−4.98175	−0.201540
$612$	85.5689	3.45892
$613$	−3.15314	−0.127354	−0.0636771	−	0.997971i	$-0.520283\pi$
$613$	−3.15314	−0.127354	−0.0636771	+	0.997971i	$0.520283\pi$
$614$	−19.1934	−0.774584
$615$	−114.904	−4.63338
$616$	0	0
$617$	−7.47829	−0.301065	−0.150532	−	0.988605i	$-0.548099\pi$
$617$	−7.47829	−0.301065	−0.150532	+	0.988605i	$0.548099\pi$
$618$	−49.1838	−1.97846
$619$	12.4737	0.501359	0.250679	−	0.968070i	$-0.419346\pi$
$619$	12.4737	0.501359	0.250679	+	0.968070i	$0.419346\pi$
$620$	1.39586	0.0560590
$621$	−10.9423	−0.439101
$622$	21.9315	0.879373
$623$	8.04042	0.322133
$624$	21.9390	0.878263
$625$	−14.6370	−0.585478
$626$	19.8952	0.795173
$627$	0	0
$628$	21.5351	0.859344
$629$	−0.525420	−0.0209499
$630$	−167.842	−6.68698
$631$	31.4546	1.25219	0.626095	−	0.779747i	$-0.284652\pi$
$631$	31.4546	1.25219	0.626095	+	0.779747i	$0.284652\pi$
$632$	18.3771	0.731003
$633$	38.3052	1.52249
$634$	39.4904	1.56837
$635$	38.0013	1.50803
$636$	−42.1815	−1.67261
$637$	−9.82885	−0.389433
$638$	0	0
$639$	8.06255	0.318950
$640$	68.7608	2.71801
$641$	20.2670	0.800500	0.400250	−	0.916406i	$-0.368923\pi$
$641$	20.2670	0.800500	0.400250	+	0.916406i	$0.368923\pi$
$642$	−68.5626	−2.70595
$643$	−30.4684	−1.20156	−0.600779	−	0.799415i	$-0.705143\pi$
$643$	−30.4684	−1.20156	−0.600779	+	0.799415i	$0.705143\pi$
$644$	8.65835	0.341187
$645$	63.5336	2.50163
$646$	55.9671	2.20200
$647$	14.1881	0.557792	0.278896	−	0.960321i	$-0.410032\pi$
$647$	14.1881	0.557792	0.278896	+	0.960321i	$0.410032\pi$
$648$	−71.0012	−2.78919
$649$	0	0
$650$	86.3524	3.38702
$651$	−1.10403	−0.0432702
$652$	−1.75121	−0.0685826
$653$	2.19721	0.0859833	0.0429917	−	0.999075i	$-0.486311\pi$
$653$	2.19721	0.0859833	0.0429917	+	0.999075i	$0.486311\pi$
$654$	62.5103	2.44435
$655$	0.295980	0.0115649
$656$	13.0821	0.510769
$657$	108.659	4.23919
$658$	−6.16964	−0.240518
$659$	17.7527	0.691547	0.345774	−	0.938318i	$-0.387616\pi$
$659$	17.7527	0.691547	0.345774	+	0.938318i	$0.387616\pi$
$660$	0	0
$661$	30.9265	1.20290	0.601451	−	0.798910i	$-0.294590\pi$
$661$	30.9265	1.20290	0.601451	+	0.798910i	$0.294590\pi$
$662$	7.65410	0.297485
$663$	61.2954	2.38052
$664$	−62.2078	−2.41413
$665$	−69.8325	−2.70799
$666$	2.54548	0.0986353
$667$	−5.59842	−0.216772
$668$	−16.8624	−0.652425
$669$	−44.3134	−1.71326
$670$	−21.2785	−0.822062
$671$	0	0
$672$	−38.8429	−1.49840
$673$	40.4599	1.55961	0.779806	−	0.626021i	$-0.215318\pi$
$673$	40.4599	1.55961	0.779806	+	0.626021i	$0.215318\pi$
$674$	−25.0863	−0.966289
$675$	−85.9742	−3.30915
$676$	64.2402	2.47078
$677$	−39.8009	−1.52967	−0.764837	−	0.644224i	$-0.777180\pi$
$677$	−39.8009	−1.52967	−0.764837	+	0.644224i	$0.777180\pi$
$678$	−129.519	−4.97415
$679$	−49.2839	−1.89134
$680$	−41.0758	−1.57518
$681$	−20.7643	−0.795690
$682$	0	0
$683$	−6.44784	−0.246720	−0.123360	−	0.992362i	$-0.539367\pi$
$683$	−6.44784	−0.246720	−0.123360	+	0.992362i	$0.539367\pi$
$684$	−172.479	−6.59489
$685$	35.6813	1.36331
$686$	36.3851	1.38919
$687$	37.5741	1.43354
$688$	−7.23342	−0.275772
$689$	−21.2511	−0.809601
$690$	21.2286	0.808159
$691$	15.7322	0.598481	0.299241	−	0.954178i	$-0.403267\pi$
$691$	15.7322	0.598481	0.299241	+	0.954178i	$0.403267\pi$
$692$	−20.1340	−0.765378
$693$	0	0
$694$	62.7181	2.38075
$695$	−26.5253	−1.00616
$696$	−74.6185	−2.82841
$697$	36.5500	1.38443
$698$	84.6982	3.20587
$699$	−79.1903	−2.99525
$700$	68.0289	2.57125
$701$	2.82672	0.106764	0.0533818	−	0.998574i	$-0.483000\pi$
$701$	2.82672	0.106764	0.0533818	+	0.998574i	$0.483000\pi$
$702$	−171.684	−6.47981
$703$	1.05907	0.0399438
$704$	0	0
$705$	−9.62246	−0.362403
$706$	26.6275	1.00214
$707$	−17.7057	−0.665891
$708$	104.226	3.91704
$709$	46.2111	1.73549	0.867747	−	0.497007i	$-0.165568\pi$
$709$	46.2111	1.73549	0.867747	+	0.497007i	$0.165568\pi$
$710$	−9.04324	−0.339387
$711$	−37.2538	−1.39713
$712$	−9.53293	−0.357262
$713$	0.0982080	0.00367792
$714$	75.9112	2.84091
$715$	0	0
$716$	42.3851	1.58400
$717$	28.8225	1.07640
$718$	51.5734	1.92470
$719$	−26.3866	−0.984054	−0.492027	−	0.870580i	$-0.663744\pi$
$719$	−26.3866	−0.984054	−0.492027	+	0.870580i	$0.663744\pi$
$720$	29.8035	1.11071
$721$	−19.5209	−0.726995
$722$	−68.2672	−2.54064
$723$	58.0920	2.16047
$724$	91.0024	3.38208
$725$	−43.9869	−1.63363
$726$	0	0
$727$	17.8295	0.661261	0.330631	−	0.943760i	$-0.392739\pi$
$727$	17.8295	0.661261	0.330631	+	0.943760i	$0.392739\pi$
$728$	58.1399	2.15481
$729$	19.2108	0.711511
$730$	−121.876	−4.51082
$731$	−20.2095	−0.747474
$732$	−11.1179	−0.410930
$733$	4.62017	0.170650	0.0853250	−	0.996353i	$-0.472807\pi$
$733$	4.62017	0.170650	0.0853250	+	0.996353i	$0.472807\pi$
$734$	51.7205	1.90904
$735$	−18.9848	−0.700267
$736$	3.45525	0.127362
$737$	0	0
$738$	−177.072	−6.51811
$739$	0.782472	0.0287837	0.0143919	−	0.999896i	$-0.495419\pi$
$739$	0.782472	0.0287837	0.0143919	+	0.999896i	$0.495419\pi$
$740$	−1.81619	−0.0667645
$741$	−123.551	−4.53878
$742$	−26.3183	−0.966177
$743$	−18.1688	−0.666547	−0.333274	−	0.942830i	$-0.608153\pi$
$743$	−18.1688	−0.666547	−0.333274	+	0.942830i	$0.608153\pi$
$744$	1.30896	0.0479889
$745$	−35.3332	−1.29451
$746$	−40.9986	−1.50106
$747$	126.107	4.61400
$748$	0	0
$749$	−27.2122	−0.994313
$750$	39.9719	1.45957
$751$	−21.7126	−0.792305	−0.396152	−	0.918185i	$-0.629655\pi$
$751$	−21.7126	−0.792305	−0.396152	+	0.918185i	$0.629655\pi$
$752$	1.09554	0.0399501
$753$	69.6359	2.53767
$754$	−87.8388	−3.19890
$755$	11.5276	0.419533
$756$	−135.254	−4.91914
$757$	21.2185	0.771201	0.385600	−	0.922666i	$-0.373994\pi$
$757$	21.2185	0.771201	0.385600	+	0.922666i	$0.373994\pi$
$758$	50.6066	1.83811
$759$	0	0
$760$	82.7953	3.00330
$761$	49.6485	1.79976	0.899878	−	0.436142i	$-0.143656\pi$
$761$	49.6485	1.79976	0.899878	+	0.436142i	$0.143656\pi$
$762$	83.2657	3.01640
$763$	24.8101	0.898186
$764$	−26.4119	−0.955550
$765$	83.2681	3.01057
$766$	42.8264	1.54738
$767$	52.5089	1.89599
$768$	73.4425	2.65013
$769$	−38.0830	−1.37331	−0.686654	−	0.726984i	$-0.740921\pi$
$769$	−38.0830	−1.37331	−0.686654	+	0.726984i	$0.740921\pi$
$770$	0	0
$771$	−10.3241	−0.371815
$772$	−27.2211	−0.979709
$773$	14.0384	0.504925	0.252462	−	0.967607i	$-0.418760\pi$
$773$	14.0384	0.504925	0.252462	+	0.967607i	$0.418760\pi$
$774$	97.9079	3.51923
$775$	0.771622	0.0277175
$776$	58.4323	2.09760
$777$	1.43648	0.0515335
$778$	77.5190	2.77919
$779$	−73.6728	−2.63960
$780$	211.876	7.58639
$781$	0	0
$782$	−6.75263	−0.241474
$783$	87.4541	3.12535
$784$	2.16146	0.0771951
$785$	20.9561	0.747954
$786$	0.648530	0.0231323
$787$	−35.9870	−1.28280	−0.641398	−	0.767208i	$-0.721645\pi$
$787$	−35.9870	−1.28280	−0.641398	+	0.767208i	$0.721645\pi$
$788$	−37.1983	−1.32514
$789$	84.0954	2.99387
$790$	41.7852	1.48665
$791$	−51.4057	−1.82777
$792$	0	0
$793$	−5.60121	−0.198905
$794$	−70.7562	−2.51104
$795$	−41.0473	−1.45580
$796$	−49.8264	−1.76605
$797$	−0.425077	−0.0150570	−0.00752850	−	0.999972i	$-0.502396\pi$
$797$	−0.425077	−0.0150570	−0.00752850	+	0.999972i	$0.502396\pi$
$798$	−153.012	−5.41657
$799$	3.06082	0.108284
$800$	27.1479	0.959825
$801$	19.3250	0.682815
$802$	−45.8471	−1.61892
$803$	0	0
$804$	−29.6586	−1.04598
$805$	8.42555	0.296962
$806$	1.54087	0.0542750
$807$	22.1179	0.778587
$808$	20.9923	0.738508
$809$	2.50013	0.0879000	0.0439500	−	0.999034i	$-0.486006\pi$
$809$	2.50013	0.0879000	0.0439500	+	0.999034i	$0.486006\pi$
$810$	−161.440	−5.67241
$811$	4.72280	0.165840	0.0829199	−	0.996556i	$-0.473575\pi$
$811$	4.72280	0.165840	0.0829199	+	0.996556i	$0.473575\pi$
$812$	−69.1998	−2.42844
$813$	36.0896	1.26572
$814$	0	0
$815$	−1.70412	−0.0596928
$816$	−13.4795	−0.471876
$817$	40.7357	1.42516
$818$	55.3981	1.93695
$819$	−117.860	−4.11837
$820$	126.340	4.41199
$821$	47.3691	1.65319	0.826596	−	0.562796i	$-0.190274\pi$
$821$	47.3691	1.65319	0.826596	+	0.562796i	$0.190274\pi$
$822$	78.1823	2.72692
$823$	39.5678	1.37925	0.689623	−	0.724169i	$-0.257776\pi$
$823$	39.5678	1.37925	0.689623	+	0.724169i	$0.257776\pi$
$824$	23.1445	0.806275
$825$	0	0
$826$	65.0295	2.26267
$827$	−47.3942	−1.64806	−0.824029	−	0.566547i	$-0.808279\pi$
$827$	−47.3942	−1.64806	−0.824029	+	0.566547i	$0.808279\pi$
$828$	20.8102	0.723204
$829$	−9.95290	−0.345679	−0.172839	−	0.984950i	$-0.555294\pi$
$829$	−9.95290	−0.345679	−0.172839	+	0.984950i	$0.555294\pi$
$830$	−141.446	−4.90965
$831$	−60.0760	−2.08401
$832$	68.0113	2.35787
$833$	6.03891	0.209236
$834$	−58.1204	−2.01255
$835$	−16.4090	−0.567856
$836$	0	0
$837$	−1.53413	−0.0530271
$838$	−5.12743	−0.177124
$839$	−28.5042	−0.984074	−0.492037	−	0.870574i	$-0.663748\pi$
$839$	−28.5042	−0.984074	−0.492037	+	0.870574i	$0.663748\pi$
$840$	112.300	3.87471
$841$	15.7441	0.542899
$842$	42.4984	1.46459
$843$	−48.4810	−1.66977
$844$	−42.1176	−1.44975
$845$	62.5129	2.15051
$846$	−14.8286	−0.509818
$847$	0	0
$848$	4.67332	0.160483
$849$	−32.7718	−1.12473
$850$	−53.0556	−1.81979
$851$	−0.127781	−0.00438028
$852$	−12.6047	−0.431830
$853$	−15.0590	−0.515610	−0.257805	−	0.966197i	$-0.582999\pi$
$853$	−15.0590	−0.515610	−0.257805	+	0.966197i	$0.582999\pi$
$854$	−6.93681	−0.237373
$855$	−167.841	−5.74005
$856$	32.2635	1.10274
$857$	16.3111	0.557176	0.278588	−	0.960411i	$-0.410134\pi$
$857$	16.3111	0.557176	0.278588	+	0.960411i	$0.410134\pi$
$858$	0	0
$859$	18.4124	0.628223	0.314111	−	0.949386i	$-0.398293\pi$
$859$	18.4124	0.628223	0.314111	+	0.949386i	$0.398293\pi$
$860$	−69.8569	−2.38210
$861$	−99.9264	−3.40548
$862$	14.4675	0.492764
$863$	12.4595	0.424127	0.212064	−	0.977256i	$-0.431982\pi$
$863$	12.4595	0.424127	0.212064	+	0.977256i	$0.431982\pi$
$864$	−53.9751	−1.83627
$865$	−19.5926	−0.666168
$866$	−93.5130	−3.17770
$867$	16.3973	0.556883
$868$	1.21391	0.0412027
$869$	0	0
$870$	−169.664	−5.75217
$871$	−14.9420	−0.506290
$872$	−29.4155	−0.996135
$873$	−118.453	−4.00903
$874$	13.6111	0.460402
$875$	15.8647	0.536325
$876$	−169.873	−5.73949
$877$	−53.9500	−1.82176	−0.910880	−	0.412671i	$-0.864596\pi$
$877$	−53.9500	−1.82176	−0.910880	+	0.412671i	$0.864596\pi$
$878$	2.62948	0.0887407
$879$	−56.0982	−1.89215
$880$	0	0
$881$	−43.0352	−1.44989	−0.724946	−	0.688805i	$-0.758135\pi$
$881$	−43.0352	−1.44989	−0.724946	+	0.688805i	$0.758135\pi$
$882$	−29.2564	−0.985115
$883$	9.74848	0.328063	0.164031	−	0.986455i	$-0.447550\pi$
$883$	9.74848	0.328063	0.164031	+	0.986455i	$0.447550\pi$
$884$	−67.3960	−2.26677
$885$	101.423	3.40930
$886$	−45.0630	−1.51392
$887$	−38.3592	−1.28797	−0.643987	−	0.765036i	$-0.722721\pi$
$887$	−38.3592	−1.28797	−0.643987	+	0.765036i	$0.722721\pi$
$888$	−1.70313	−0.0571533
$889$	33.0478	1.10839
$890$	−21.6756	−0.726568
$891$	0	0
$892$	48.7238	1.63140
$893$	−6.16961	−0.206458
$894$	−77.4196	−2.58930
$895$	41.2454	1.37868
$896$	59.7979	1.99771
$897$	14.9069	0.497728
$898$	68.1687	2.27482
$899$	−0.784904	−0.0261780
$900$	163.506	5.45020
$901$	13.0568	0.434985
$902$	0	0
$903$	55.2520	1.83867
$904$	60.9479	2.02710
$905$	88.5555	2.94368
$906$	25.2585	0.839157
$907$	−49.6502	−1.64861	−0.824304	−	0.566148i	$-0.808433\pi$
$907$	−49.6502	−1.64861	−0.824304	+	0.566148i	$0.808433\pi$
$908$	22.8309	0.757671
$909$	−42.5553	−1.41147
$910$	132.196	4.38226
$911$	43.8140	1.45162	0.725812	−	0.687893i	$-0.241464\pi$
$911$	43.8140	1.45162	0.725812	+	0.687893i	$0.241464\pi$
$912$	27.1702	0.899696
$913$	0	0
$914$	−46.5646	−1.54022
$915$	−10.8190	−0.357664
$916$	−41.3138	−1.36505
$917$	0.257399	0.00850006
$918$	105.484	3.48150
$919$	−5.04128	−0.166296	−0.0831482	−	0.996537i	$-0.526497\pi$
$919$	−5.04128	−0.166296	−0.0831482	+	0.996537i	$0.526497\pi$
$920$	−9.98955	−0.329346
$921$	−26.0330	−0.857817
$922$	−18.2230	−0.600143
$923$	−6.35025	−0.209021
$924$	0	0
$925$	−1.00398	−0.0330106
$926$	15.4220	0.506797
$927$	−46.9180	−1.54099
$928$	−27.6152	−0.906515
$929$	−17.0037	−0.557873	−0.278936	−	0.960310i	$-0.589982\pi$
$929$	−17.0037	−0.557873	−0.278936	+	0.960310i	$0.589982\pi$
$930$	2.97627	0.0975956
$931$	−12.1725	−0.398937
$932$	87.0719	2.85214
$933$	29.7468	0.973865
$934$	62.7730	2.05400
$935$	0	0
$936$	139.738	4.56749
$937$	−19.4635	−0.635846	−0.317923	−	0.948117i	$-0.602985\pi$
$937$	−19.4635	−0.635846	−0.317923	+	0.948117i	$0.602985\pi$
$938$	−18.5049	−0.604206
$939$	26.9849	0.880618
$940$	10.5802	0.345087
$941$	−19.4710	−0.634736	−0.317368	−	0.948302i	$-0.602799\pi$
$941$	−19.4710	−0.634736	−0.317368	+	0.948302i	$0.602799\pi$
$942$	45.9175	1.49607
$943$	8.88888	0.289462
$944$	−11.5472	−0.375830
$945$	−131.617	−4.28151
$946$	0	0
$947$	30.1329	0.979187	0.489593	−	0.871951i	$-0.337145\pi$
$947$	30.1329	0.979187	0.489593	+	0.871951i	$0.337145\pi$
$948$	58.2413	1.89159
$949$	−85.5823	−2.77812
$950$	106.943	3.46968
$951$	53.5628	1.73689
$952$	−35.7216	−1.15774
$953$	35.5897	1.15286	0.576432	−	0.817145i	$-0.304445\pi$
$953$	35.5897	1.15286	0.576432	+	0.817145i	$0.304445\pi$
$954$	−63.2557	−2.04798
$955$	−25.7018	−0.831689
$956$	−31.6911	−1.02496
$957$	0	0
$958$	−44.0227	−1.42231
$959$	31.0303	1.00202
$960$	131.367	4.23984
$961$	−30.9862	−0.999556
$962$	−2.00488	−0.0646398
$963$	−65.4041	−2.10762
$964$	−63.8738	−2.05724
$965$	−26.4892	−0.852717
$966$	18.4615	0.593988
$967$	0.0995826	0.00320236	0.00160118	−	0.999999i	$-0.499490\pi$
$967$	0.0995826	0.00320236	0.00160118	+	0.999999i	$0.499490\pi$
$968$	0	0
$969$	75.9109	2.43861
$970$	132.861	4.26591
$971$	−15.9640	−0.512310	−0.256155	−	0.966636i	$-0.582456\pi$
$971$	−15.9640	−0.512310	−0.256155	+	0.966636i	$0.582456\pi$
$972$	−87.8839	−2.81888
$973$	−23.0678	−0.739519
$974$	−78.9842	−2.53082
$975$	117.124	3.75097
$976$	1.23176	0.0394277
$977$	23.2681	0.744412	0.372206	−	0.928150i	$-0.378601\pi$
$977$	23.2681	0.744412	0.372206	+	0.928150i	$0.378601\pi$
$978$	−3.73395	−0.119399
$979$	0	0
$980$	20.8744	0.666807
$981$	59.6306	1.90386
$982$	−13.9848	−0.446274
$983$	−38.0596	−1.21391	−0.606956	−	0.794735i	$-0.707610\pi$
$983$	−38.0596	−1.21391	−0.606956	+	0.794735i	$0.707610\pi$
$984$	118.475	3.77686
$985$	−36.1981	−1.15337
$986$	53.9688	1.71872
$987$	−8.36818	−0.266362
$988$	135.848	4.32191
$989$	−4.91491	−0.156285
$990$	0	0
$991$	−13.6967	−0.435090	−0.217545	−	0.976050i	$-0.569805\pi$
$991$	−13.6967	−0.435090	−0.217545	+	0.976050i	$0.569805\pi$
$992$	0.484428	0.0153806
$993$	10.3816	0.329451
$994$	−7.86446	−0.249446
$995$	−48.4867	−1.53713
$996$	−197.150	−6.24695
$997$	18.8111	0.595753	0.297877	−	0.954604i	$-0.403722\pi$
$997$	18.8111	0.595753	0.297877	+	0.954604i	$0.403722\pi$
$998$	69.1758	2.18972
$999$	1.99610	0.0631537

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7381.2.a.i.1.3		19
11.10	odd	2		671.2.a.c.1.17	✓	19
33.32	even	2		6039.2.a.k.1.3		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.17	✓	19	11.10	odd	2
6039.2.a.k.1.3		19	33.32	even	2
7381.2.a.i.1.3		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)

\(f(q)\)	\(=\)	\(q-2.34443 q^{2} -3.17986 q^{3} +3.49635 q^{4} +3.40234 q^{5} +7.45497 q^{6} +2.95885 q^{7} -3.50809 q^{8} +7.11153 q^{9} +O(q^{10})\)	\(q-2.34443 q^{2} -3.17986 q^{3} +3.49635 q^{4} +3.40234 q^{5} +7.45497 q^{6} +2.95885 q^{7} -3.50809 q^{8} +7.11153 q^{9} -7.97654 q^{10} -11.1179 q^{12} -5.60121 q^{13} -6.93681 q^{14} -10.8190 q^{15} +1.23176 q^{16} +3.44142 q^{17} -16.6725 q^{18} -6.93678 q^{19} +11.8958 q^{20} -9.40873 q^{21} +0.836947 q^{23} +11.1552 q^{24} +6.57591 q^{25} +13.1316 q^{26} -13.0741 q^{27} +10.3452 q^{28} -6.68910 q^{29} +25.3643 q^{30} +0.117341 q^{31} +4.12839 q^{32} -8.06817 q^{34} +10.0670 q^{35} +24.8644 q^{36} -0.152675 q^{37} +16.2628 q^{38} +17.8111 q^{39} -11.9357 q^{40} +10.6206 q^{41} +22.0581 q^{42} -5.87242 q^{43} +24.1958 q^{45} -1.96216 q^{46} +0.889406 q^{47} -3.91684 q^{48} +1.75477 q^{49} -15.4168 q^{50} -10.9433 q^{51} -19.5838 q^{52} +3.79401 q^{53} +30.6513 q^{54} -10.3799 q^{56} +22.0580 q^{57} +15.6821 q^{58} -9.37456 q^{59} -37.8269 q^{60} +1.00000 q^{61} -0.275097 q^{62} +21.0419 q^{63} -12.1423 q^{64} -19.0572 q^{65} +2.66764 q^{67} +12.0324 q^{68} -2.66138 q^{69} -23.6014 q^{70} +1.13373 q^{71} -24.9479 q^{72} +15.2793 q^{73} +0.357937 q^{74} -20.9105 q^{75} -24.2534 q^{76} -41.7568 q^{78} -5.23851 q^{79} +4.19087 q^{80} +20.2393 q^{81} -24.8993 q^{82} +17.7327 q^{83} -32.8962 q^{84} +11.7089 q^{85} +13.7675 q^{86} +21.2704 q^{87} +2.71742 q^{89} -56.7255 q^{90} -16.5731 q^{91} +2.92626 q^{92} -0.373127 q^{93} -2.08515 q^{94} -23.6013 q^{95} -13.1277 q^{96} -16.6565 q^{97} -4.11394 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9}+O(q^{10}) \) 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

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