LMFDB - Embedded newform 1075.6.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,6,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1075.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1075.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:	$172.412606299$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 $ x^10 - 2x^9 - 256x^8 + 266x^7 + 21986x^6 - 10450x^5 - 719484x^4 + 384582x^3 + 8437093x^2 - 5752252*x - 22734604
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.48720$ of defining polynomial
Character	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.48720 q^{2} +27.5943 q^{3} -25.8138 q^{4} -68.6327 q^{6} -15.7005 q^{7} +143.795 q^{8} +518.448 q^{9} +O(q^{10})$	\(q-2.48720 q^{2} +27.5943 q^{3} -25.8138 q^{4} -68.6327 q^{6} -15.7005 q^{7} +143.795 q^{8} +518.448 q^{9} +394.739 q^{11} -712.316 q^{12} -666.939 q^{13} +39.0503 q^{14} +468.396 q^{16} -172.038 q^{17} -1289.48 q^{18} -1280.86 q^{19} -433.245 q^{21} -981.796 q^{22} -569.882 q^{23} +3967.92 q^{24} +1658.81 q^{26} +7600.80 q^{27} +405.290 q^{28} -6328.41 q^{29} +7795.01 q^{31} -5766.42 q^{32} +10892.6 q^{33} +427.892 q^{34} -13383.1 q^{36} -16252.3 q^{37} +3185.75 q^{38} -18403.7 q^{39} +7454.95 q^{41} +1077.57 q^{42} -1849.00 q^{43} -10189.7 q^{44} +1417.41 q^{46} +5628.68 q^{47} +12925.1 q^{48} -16560.5 q^{49} -4747.27 q^{51} +17216.2 q^{52} -22460.1 q^{53} -18904.7 q^{54} -2257.65 q^{56} -35344.4 q^{57} +15740.0 q^{58} -9061.48 q^{59} -18280.8 q^{61} -19387.8 q^{62} -8139.89 q^{63} -646.417 q^{64} -27092.0 q^{66} +27428.5 q^{67} +4440.95 q^{68} -15725.5 q^{69} -12860.9 q^{71} +74550.0 q^{72} +63446.5 q^{73} +40422.7 q^{74} +33063.8 q^{76} -6197.60 q^{77} +45773.8 q^{78} -1911.26 q^{79} +83756.3 q^{81} -18542.0 q^{82} +52124.9 q^{83} +11183.7 q^{84} +4598.84 q^{86} -174628. q^{87} +56761.4 q^{88} +66185.2 q^{89} +10471.3 q^{91} +14710.8 q^{92} +215098. q^{93} -13999.7 q^{94} -159121. q^{96} -37497.8 q^{97} +41189.3 q^{98} +204652. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9}+O(q^{10}) $ 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9} + 745 q^{11} - 4627 q^{12} - 1917 q^{13} + 1936 q^{14} + 5354 q^{16} - 4017 q^{17} + 2725 q^{18} - 2404 q^{19} - 228 q^{21} + 5836 q^{22} - 1733 q^{23} - 10711 q^{24} - 1484 q^{26} + 2324 q^{27} + 15028 q^{28} + 6996 q^{29} - 4899 q^{31} + 7554 q^{32} + 15734 q^{33} - 27033 q^{34} + 4433 q^{36} - 1466 q^{37} - 13905 q^{38} - 26542 q^{39} + 10297 q^{41} + 37642 q^{42} - 18490 q^{43} - 36140 q^{44} + 17991 q^{46} - 48592 q^{47} - 83607 q^{48} + 29458 q^{49} + 92972 q^{51} - 14232 q^{52} - 127165 q^{53} - 92002 q^{54} - 7780 q^{56} - 34060 q^{57} + 10305 q^{58} + 99372 q^{59} + 17408 q^{61} - 28265 q^{62} - 2244 q^{63} + 47202 q^{64} - 150292 q^{66} + 2021 q^{67} - 192151 q^{68} + 1654 q^{69} + 11286 q^{71} + 298365 q^{72} - 49892 q^{73} - 125431 q^{74} - 249803 q^{76} - 98144 q^{77} + 28494 q^{78} - 91524 q^{79} - 26450 q^{81} + 158909 q^{82} + 105203 q^{83} - 357682 q^{84} + 14792 q^{86} - 181200 q^{87} + 461824 q^{88} - 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 238430 q^{93} + 7259 q^{94} - 162399 q^{96} - 108383 q^{97} - 354656 q^{98} - 270499 q^{99}+O(q^{100}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9 + 745 * q^11 - 4627 * q^12 - 1917 * q^13 + 1936 * q^14 + 5354 * q^16 - 4017 * q^17 + 2725 * q^18 - 2404 * q^19 - 228 * q^21 + 5836 * q^22 - 1733 * q^23 - 10711 * q^24 - 1484 * q^26 + 2324 * q^27 + 15028 * q^28 + 6996 * q^29 - 4899 * q^31 + 7554 * q^32 + 15734 * q^33 - 27033 * q^34 + 4433 * q^36 - 1466 * q^37 - 13905 * q^38 - 26542 * q^39 + 10297 * q^41 + 37642 * q^42 - 18490 * q^43 - 36140 * q^44 + 17991 * q^46 - 48592 * q^47 - 83607 * q^48 + 29458 * q^49 + 92972 * q^51 - 14232 * q^52 - 127165 * q^53 - 92002 * q^54 - 7780 * q^56 - 34060 * q^57 + 10305 * q^58 + 99372 * q^59 + 17408 * q^61 - 28265 * q^62 - 2244 * q^63 + 47202 * q^64 - 150292 * q^66 + 2021 * q^67 - 192151 * q^68 + 1654 * q^69 + 11286 * q^71 + 298365 * q^72 - 49892 * q^73 - 125431 * q^74 - 249803 * q^76 - 98144 * q^77 + 28494 * q^78 - 91524 * q^79 - 26450 * q^81 + 158909 * q^82 + 105203 * q^83 - 357682 * q^84 + 14792 * q^86 - 181200 * q^87 + 461824 * q^88 - 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 238430 * q^93 + 7259 * q^94 - 162399 * q^96 - 108383 * q^97 - 354656 * q^98 - 270499 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.48720	−0.439679	−0.219840	−	0.975536i	$-0.570553\pi$
$2$	−2.48720	−0.439679	−0.219840	+	0.975536i	$0.570553\pi$
$3$	27.5943	1.77018	0.885089	−	0.465422i	$-0.154097\pi$
$3$	27.5943	1.77018	0.885089	+	0.465422i	$0.154097\pi$
$4$	−25.8138	−0.806682
$5$	0	0
$5$	0	0
$6$	−68.6327	−0.778311
$7$	−15.7005	−0.121107	−0.0605534	−	0.998165i	$-0.519287\pi$
$7$	−15.7005	−0.121107	−0.0605534	+	0.998165i	$0.519287\pi$
$8$	143.795	0.794361
$9$	518.448	2.13353
$10$	0	0
$11$	394.739	0.983623	0.491811	−	0.870702i	$-0.336335\pi$
$11$	394.739	0.983623	0.491811	+	0.870702i	$0.336335\pi$
$12$	−712.316	−1.42797
$13$	−666.939	−1.09453	−0.547265	−	0.836959i	$-0.684331\pi$
$13$	−666.939	−1.09453	−0.547265	+	0.836959i	$0.684331\pi$
$14$	39.0503	0.0532481
$15$	0	0
$16$	468.396	0.457418
$17$	−172.038	−0.144378	−0.0721890	−	0.997391i	$-0.522998\pi$
$17$	−172.038	−0.144378	−0.0721890	+	0.997391i	$0.522998\pi$
$18$	−1289.48	−0.938069
$19$	−1280.86	−0.813985	−0.406992	−	0.913432i	$-0.633422\pi$
$19$	−1280.86	−0.813985	−0.406992	+	0.913432i	$0.633422\pi$
$20$	0	0
$21$	−433.245	−0.214380
$22$	−981.796	−0.432479
$23$	−569.882	−0.224629	−0.112314	−	0.993673i	$-0.535826\pi$
$23$	−569.882	−0.224629	−0.112314	+	0.993673i	$0.535826\pi$
$24$	3967.92	1.40616
$25$	0	0
$26$	1658.81	0.481242
$27$	7600.80	2.00655
$28$	405.290	0.0976946
$29$	−6328.41	−1.39733	−0.698666	−	0.715448i	$-0.746223\pi$
$29$	−6328.41	−1.39733	−0.698666	+	0.715448i	$0.746223\pi$
$30$	0	0
$31$	7795.01	1.45684	0.728421	−	0.685129i	$-0.240254\pi$
$31$	7795.01	1.45684	0.728421	+	0.685129i	$0.240254\pi$
$32$	−5766.42	−0.995478
$33$	10892.6	1.74119
$34$	427.892	0.0634800
$35$	0	0
$36$	−13383.1	−1.72108
$37$	−16252.3	−1.95169	−0.975844	−	0.218469i	$-0.929894\pi$
$37$	−16252.3	−1.95169	−0.975844	+	0.218469i	$0.929894\pi$
$38$	3185.75	0.357892
$39$	−18403.7	−1.93751
$40$	0	0
$41$	7454.95	0.692604	0.346302	−	0.938123i	$-0.387437\pi$
$41$	7454.95	0.692604	0.346302	+	0.938123i	$0.387437\pi$
$42$	1077.57	0.0942587
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	−10189.7	−0.793471
$45$	0	0
$46$	1417.41	0.0987646
$47$	5628.68	0.371673	0.185837	−	0.982581i	$-0.440500\pi$
$47$	5628.68	0.371673	0.185837	+	0.982581i	$0.440500\pi$
$48$	12925.1	0.809712
$49$	−16560.5	−0.985333
$50$	0	0
$51$	−4747.27	−0.255575
$52$	17216.2	0.882938
$53$	−22460.1	−1.09830	−0.549151	−	0.835723i	$-0.685049\pi$
$53$	−22460.1	−1.09830	−0.549151	+	0.835723i	$0.685049\pi$
$54$	−18904.7	−0.882238
$55$	0	0
$56$	−2257.65	−0.0962024
$57$	−35344.4	−1.44090
$58$	15740.0	0.614378
$59$	−9061.48	−0.338898	−0.169449	−	0.985539i	$-0.554199\pi$
$59$	−9061.48	−0.338898	−0.169449	+	0.985539i	$0.554199\pi$
$60$	0	0
$61$	−18280.8	−0.629029	−0.314514	−	0.949253i	$-0.601842\pi$
$61$	−18280.8	−0.629029	−0.314514	+	0.949253i	$0.601842\pi$
$62$	−19387.8	−0.640544
$63$	−8139.89	−0.258385
$64$	−646.417	−0.0197271
$65$	0	0
$66$	−27092.0	−0.765564
$67$	27428.5	0.746474	0.373237	−	0.927736i	$-0.378248\pi$
$67$	27428.5	0.746474	0.373237	+	0.927736i	$0.378248\pi$
$68$	4440.95	0.116467
$69$	−15725.5	−0.397633
$70$	0	0
$71$	−12860.9	−0.302780	−0.151390	−	0.988474i	$-0.548375\pi$
$71$	−12860.9	−0.302780	−0.151390	+	0.988474i	$0.548375\pi$
$72$	74550.0	1.69479
$73$	63446.5	1.39348	0.696740	−	0.717324i	$-0.254633\pi$
$73$	63446.5	1.39348	0.696740	+	0.717324i	$0.254633\pi$
$74$	40422.7	0.858117
$75$	0	0
$76$	33063.8	0.656627
$77$	−6197.60	−0.119123
$78$	45773.8	0.851884
$79$	−1911.26	−0.0344549	−0.0172275	−	0.999852i	$-0.505484\pi$
$79$	−1911.26	−0.0344549	−0.0172275	+	0.999852i	$0.505484\pi$
$80$	0	0
$81$	83756.3	1.41842
$82$	−18542.0	−0.304524
$83$	52124.9	0.830520	0.415260	−	0.909703i	$-0.363691\pi$
$83$	52124.9	0.830520	0.415260	+	0.909703i	$0.363691\pi$
$84$	11183.7	0.172937
$85$	0	0
$86$	4598.84	0.0670505
$87$	−174628.	−2.47353
$88$	56761.4	0.781351
$89$	66185.2	0.885698	0.442849	−	0.896596i	$-0.353968\pi$
$89$	66185.2	0.885698	0.442849	+	0.896596i	$0.353968\pi$
$90$	0	0
$91$	10471.3	0.132555
$92$	14710.8	0.181204
$93$	215098.	2.57887
$94$	−13999.7	−0.163417
$95$	0	0
$96$	−159121.	−1.76217
$97$	−37497.8	−0.404648	−0.202324	−	0.979319i	$-0.564849\pi$
$97$	−37497.8	−0.404648	−0.202324	+	0.979319i	$0.564849\pi$
$98$	41189.3	0.433231
$99$	204652.	2.09859
$100$	0	0
$101$	−35537.7	−0.346646	−0.173323	−	0.984865i	$-0.555450\pi$
$101$	−35537.7	−0.346646	−0.173323	+	0.984865i	$0.555450\pi$
$102$	11807.4	0.112371
$103$	−177955.	−1.65279	−0.826395	−	0.563091i	$-0.809612\pi$
$103$	−177955.	−1.65279	−0.826395	+	0.563091i	$0.809612\pi$
$104$	−95902.2	−0.869451
$105$	0	0
$106$	55862.8	0.482901
$107$	−43589.8	−0.368065	−0.184033	−	0.982920i	$-0.558915\pi$
$107$	−43589.8	−0.368065	−0.184033	+	0.982920i	$0.558915\pi$
$108$	−196206.	−1.61865
$109$	−49084.5	−0.395711	−0.197856	−	0.980231i	$-0.563398\pi$
$109$	−49084.5	−0.395711	−0.197856	+	0.980231i	$0.563398\pi$
$110$	0	0
$111$	−448471.	−3.45484
$112$	−7354.05	−0.0553964
$113$	34530.0	0.254390	0.127195	−	0.991878i	$-0.459403\pi$
$113$	34530.0	0.254390	0.127195	+	0.991878i	$0.459403\pi$
$114$	87908.6	0.633533
$115$	0	0
$116$	163360.	1.12720
$117$	−345773.	−2.33521
$118$	22537.7	0.149006
$119$	2701.08	0.0174852
$120$	0	0
$121$	−5231.89	−0.0324859
$122$	45468.0	0.276571
$123$	205715.	1.22603
$124$	−201219.	−1.17521
$125$	0	0
$126$	20245.5	0.113606
$127$	10008.6	0.0550637	0.0275318	−	0.999621i	$-0.491235\pi$
$127$	10008.6	0.0550637	0.0275318	+	0.999621i	$0.491235\pi$
$128$	186133.	1.00415
$129$	−51021.9	−0.269950
$130$	0	0
$131$	336036.	1.71083	0.855415	−	0.517943i	$-0.173302\pi$
$131$	336036.	1.71083	0.855415	+	0.517943i	$0.173302\pi$
$132$	−281179.	−1.40458
$133$	20110.1	0.0985791
$134$	−68220.2	−0.328209
$135$	0	0
$136$	−24738.1	−0.114688
$137$	1217.37	0.00554141	0.00277071	−	0.999996i	$-0.499118\pi$
$137$	1217.37	0.00554141	0.00277071	+	0.999996i	$0.499118\pi$
$138$	39112.5	0.174831
$139$	−157629.	−0.691989	−0.345995	−	0.938237i	$-0.612458\pi$
$139$	−157629.	−0.691989	−0.345995	+	0.938237i	$0.612458\pi$
$140$	0	0
$141$	155320.	0.657928
$142$	31987.8	0.133126
$143$	−263267.	−1.07660
$144$	242839.	0.975915
$145$	0	0
$146$	−157804.	−0.612684
$147$	−456976.	−1.74422
$148$	419534.	1.57439
$149$	−501641.	−1.85109	−0.925545	−	0.378639i	$-0.876392\pi$
$149$	−501641.	−1.85109	−0.925545	+	0.378639i	$0.876392\pi$
$150$	0	0
$151$	−430865.	−1.53780	−0.768899	−	0.639370i	$-0.779195\pi$
$151$	−430865.	−1.53780	−0.768899	+	0.639370i	$0.779195\pi$
$152$	−184180.	−0.646598
$153$	−89192.5	−0.308035
$154$	15414.7	0.0523761
$155$	0	0
$156$	475071.	1.56296
$157$	−157952.	−0.511418	−0.255709	−	0.966754i	$-0.582309\pi$
$157$	−157952.	−0.511418	−0.255709	+	0.966754i	$0.582309\pi$
$158$	4753.68	0.0151491
$159$	−619772.	−1.94419
$160$	0	0
$161$	8947.43	0.0272041
$162$	−208319.	−0.623650
$163$	−405846.	−1.19645	−0.598223	−	0.801330i	$-0.704126\pi$
$163$	−405846.	−1.19645	−0.598223	+	0.801330i	$0.704126\pi$
$164$	−192441.	−0.558712
$165$	0	0
$166$	−129645.	−0.365162
$167$	−166792.	−0.462789	−0.231394	−	0.972860i	$-0.574329\pi$
$167$	−166792.	−0.462789	−0.231394	+	0.972860i	$0.574329\pi$
$168$	−62298.3	−0.170295
$169$	73514.3	0.197995
$170$	0	0
$171$	−664057.	−1.73666
$172$	47729.8	0.123018
$173$	287751.	0.730973	0.365487	−	0.930817i	$-0.380903\pi$
$173$	287751.	0.730973	0.365487	+	0.930817i	$0.380903\pi$
$174$	434336.	1.08756
$175$	0	0
$176$	184894.	0.449927
$177$	−250045.	−0.599910
$178$	−164616.	−0.389423
$179$	522579.	1.21904	0.609522	−	0.792769i	$-0.291361\pi$
$179$	522579.	1.21904	0.609522	+	0.792769i	$0.291361\pi$
$180$	0	0
$181$	81268.9	0.184386	0.0921930	−	0.995741i	$-0.470612\pi$
$181$	81268.9	0.184386	0.0921930	+	0.995741i	$0.470612\pi$
$182$	−26044.2	−0.0582817
$183$	−504447.	−1.11349
$184$	−81946.0	−0.178436
$185$	0	0
$186$	−534993.	−1.13388
$187$	−67910.0	−0.142014
$188$	−145298.	−0.299822
$189$	−119336.	−0.243007
$190$	0	0
$191$	−608799.	−1.20751	−0.603755	−	0.797170i	$-0.706329\pi$
$191$	−608799.	−1.20751	−0.603755	+	0.797170i	$0.706329\pi$
$192$	−17837.5	−0.0349205
$193$	−184959.	−0.357423	−0.178712	−	0.983902i	$-0.557193\pi$
$193$	−184959.	−0.357423	−0.178712	+	0.983902i	$0.557193\pi$
$194$	93264.7	0.177915
$195$	0	0
$196$	427490.	0.794851
$197$	−200949.	−0.368910	−0.184455	−	0.982841i	$-0.559052\pi$
$197$	−200949.	−0.368910	−0.184455	+	0.982841i	$0.559052\pi$
$198$	−509010.	−0.922706
$199$	−233653.	−0.418253	−0.209127	−	0.977889i	$-0.567062\pi$
$199$	−233653.	−0.418253	−0.209127	+	0.977889i	$0.567062\pi$
$200$	0	0
$201$	756871.	1.32139
$202$	88389.5	0.152413
$203$	99359.2	0.169226
$204$	122545.	0.206168
$205$	0	0
$206$	442610.	0.726697
$207$	−295454.	−0.479252
$208$	−312392.	−0.500658
$209$	−505604.	−0.800654
$210$	0	0
$211$	200513.	0.310054	0.155027	−	0.987910i	$-0.450454\pi$
$211$	200513.	0.310054	0.155027	+	0.987910i	$0.450454\pi$
$212$	579781.	0.885981
$213$	−354889.	−0.535974
$214$	108417.	0.161831
$215$	0	0
$216$	1.09295e6	1.59392
$217$	−122386.	−0.176433
$218$	122083.	0.173986
$219$	1.75076e6	2.46671
$220$	0	0
$221$	114739.	0.158026
$222$	1.11544e6	1.51902
$223$	−347051.	−0.467338	−0.233669	−	0.972316i	$-0.575073\pi$
$223$	−347051.	−0.467338	−0.233669	+	0.972316i	$0.575073\pi$
$224$	90535.7	0.120559
$225$	0	0
$226$	−85883.2	−0.111850
$227$	212111.	0.273212	0.136606	−	0.990625i	$-0.456381\pi$
$227$	212111.	0.273212	0.136606	+	0.990625i	$0.456381\pi$
$228$	912374.	1.16235
$229$	−470629.	−0.593049	−0.296524	−	0.955025i	$-0.595828\pi$
$229$	−470629.	−0.593049	−0.296524	+	0.955025i	$0.595828\pi$
$230$	0	0
$231$	−171019.	−0.210870
$232$	−909991.	−1.10999
$233$	952248.	1.14911	0.574553	−	0.818467i	$-0.305176\pi$
$233$	952248.	1.14911	0.574553	+	0.818467i	$0.305176\pi$
$234$	860007.	1.02674
$235$	0	0
$236$	233911.	0.273383
$237$	−52739.9	−0.0609914
$238$	−6718.12	−0.00768786
$239$	−1.63690e6	−1.85365	−0.926824	−	0.375497i	$-0.877472\pi$
$239$	−1.63690e6	−1.85365	−0.926824	+	0.375497i	$0.877472\pi$
$240$	0	0
$241$	−695944.	−0.771848	−0.385924	−	0.922531i	$-0.626117\pi$
$241$	−695944.	−0.771848	−0.385924	+	0.922531i	$0.626117\pi$
$242$	13012.8	0.0142834
$243$	464206.	0.504307
$244$	471897.	0.507426
$245$	0	0
$246$	−511654.	−0.539061
$247$	854253.	0.890931
$248$	1.12088e6	1.15726
$249$	1.43835e6	1.47017
$250$	0	0
$251$	657886.	0.659123	0.329561	−	0.944134i	$-0.393099\pi$
$251$	657886.	0.659123	0.329561	+	0.944134i	$0.393099\pi$
$252$	210122.	0.208434
$253$	−224955.	−0.220950
$254$	−24893.5	−0.0242104
$255$	0	0
$256$	−442266.	−0.421778
$257$	599821.	0.566486	0.283243	−	0.959048i	$-0.408590\pi$
$257$	599821.	0.566486	0.283243	+	0.959048i	$0.408590\pi$
$258$	126902.	0.118691
$259$	255169.	0.236363
$260$	0	0
$261$	−3.28095e6	−2.98125
$262$	−835788.	−0.752217
$263$	1.62868e6	1.45194	0.725968	−	0.687728i	$-0.241392\pi$
$263$	1.62868e6	1.45194	0.725968	+	0.687728i	$0.241392\pi$
$264$	1.56629e6	1.38313
$265$	0	0
$266$	−50017.8	−0.0433432
$267$	1.82634e6	1.56784
$268$	−708034.	−0.602167
$269$	1.99405e6	1.68017	0.840087	−	0.542451i	$-0.182504\pi$
$269$	1.99405e6	1.68017	0.840087	+	0.542451i	$0.182504\pi$
$270$	0	0
$271$	−1.89207e6	−1.56500	−0.782501	−	0.622650i	$-0.786056\pi$
$271$	−1.89207e6	−1.56500	−0.782501	+	0.622650i	$0.786056\pi$
$272$	−80581.8	−0.0660411
$273$	288948.	0.234646
$274$	−3027.84	−0.00243644
$275$	0	0
$276$	405936.	0.320763
$277$	2.11477e6	1.65601	0.828007	−	0.560717i	$-0.189475\pi$
$277$	2.11477e6	1.65601	0.828007	+	0.560717i	$0.189475\pi$
$278$	392055.	0.304253
$279$	4.04131e6	3.10822
$280$	0	0
$281$	−80948.7	−0.0611567	−0.0305784	−	0.999532i	$-0.509735\pi$
$281$	−80948.7	−0.0611567	−0.0305784	+	0.999532i	$0.509735\pi$
$282$	−386311.	−0.289277
$283$	−1.01158e6	−0.750814	−0.375407	−	0.926860i	$-0.622497\pi$
$283$	−1.01158e6	−0.750814	−0.375407	+	0.926860i	$0.622497\pi$
$284$	331990.	0.244247
$285$	0	0
$286$	654798.	0.473361
$287$	−117046.	−0.0838791
$288$	−2.98959e6	−2.12388
$289$	−1.39026e6	−0.979155
$290$	0	0
$291$	−1.03473e6	−0.716298
$292$	−1.63780e6	−1.12409
$293$	−786392.	−0.535143	−0.267572	−	0.963538i	$-0.586221\pi$
$293$	−786392.	−0.535143	−0.267572	+	0.963538i	$0.586221\pi$
$294$	1.13659e6	0.766895
$295$	0	0
$296$	−2.33699e6	−1.55034
$297$	3.00034e6	1.97369
$298$	1.24768e6	0.813886
$299$	380076.	0.245863
$300$	0	0
$301$	29030.2	0.0184686
$302$	1.07165e6	0.676138
$303$	−980641.	−0.613625
$304$	−599948.	−0.372332
$305$	0	0
$306$	221840.	0.135437
$307$	−106306.	−0.0643744	−0.0321872	−	0.999482i	$-0.510247\pi$
$307$	−106306.	−0.0643744	−0.0321872	+	0.999482i	$0.510247\pi$
$308$	159984.	0.0960947
$309$	−4.91056e6	−2.92573
$310$	0	0
$311$	−82617.8	−0.0484365	−0.0242183	−	0.999707i	$-0.507710\pi$
$311$	−82617.8	−0.0484365	−0.0242183	+	0.999707i	$0.507710\pi$
$312$	−2.64636e6	−1.53908
$313$	−1.38431e6	−0.798681	−0.399341	−	0.916803i	$-0.630761\pi$
$313$	−1.38431e6	−0.798681	−0.399341	+	0.916803i	$0.630761\pi$
$314$	392859.	0.224860
$315$	0	0
$316$	49336.9	0.0277942
$317$	−964944.	−0.539329	−0.269665	−	0.962954i	$-0.586913\pi$
$317$	−964944.	−0.539329	−0.269665	+	0.962954i	$0.586913\pi$
$318$	1.54150e6	0.854820
$319$	−2.49807e6	−1.37445
$320$	0	0
$321$	−1.20283e6	−0.651541
$322$	−22254.1	−0.0119611
$323$	220356.	0.117522
$324$	−2.16207e6	−1.14421
$325$	0	0
$326$	1.00942e6	0.526052
$327$	−1.35446e6	−0.700479
$328$	1.07198e6	0.550178
$329$	−88373.0	−0.0450122
$330$	0	0
$331$	−3.21355e6	−1.61218	−0.806092	−	0.591790i	$-0.798422\pi$
$331$	−3.21355e6	−1.61218	−0.806092	+	0.591790i	$0.798422\pi$
$332$	−1.34554e6	−0.669966
$333$	−8.42597e6	−4.16398
$334$	414844.	0.203479
$335$	0	0
$336$	−202930.	−0.0980615
$337$	−835398.	−0.400699	−0.200350	−	0.979724i	$-0.564208\pi$
$337$	−835398.	−0.400699	−0.200350	+	0.979724i	$0.564208\pi$
$338$	−182845.	−0.0870544
$339$	952834.	0.450316
$340$	0	0
$341$	3.07700e6	1.43298
$342$	1.65164e6	0.763574
$343$	523886.	0.240437
$344$	−265876.	−0.121139
$345$	0	0
$346$	−715695.	−0.321394
$347$	−2.19943e6	−0.980589	−0.490295	−	0.871557i	$-0.663111\pi$
$347$	−2.19943e6	−0.980589	−0.490295	+	0.871557i	$0.663111\pi$
$348$	4.50782e6	1.99535
$349$	2.90995e6	1.27886	0.639428	−	0.768851i	$-0.279171\pi$
$349$	2.90995e6	1.27886	0.639428	+	0.768851i	$0.279171\pi$
$350$	0	0
$351$	−5.06927e6	−2.19623
$352$	−2.27623e6	−0.979175
$353$	−2.63861e6	−1.12704	−0.563518	−	0.826104i	$-0.690552\pi$
$353$	−2.63861e6	−1.12704	−0.563518	+	0.826104i	$0.690552\pi$
$354$	621914.	0.263768
$355$	0	0
$356$	−1.70849e6	−0.714477
$357$	74534.4	0.0309518
$358$	−1.29976e6	−0.535988
$359$	−37197.0	−0.0152325	−0.00761626	−	0.999971i	$-0.502424\pi$
$359$	−37197.0	−0.0152325	−0.00761626	+	0.999971i	$0.502424\pi$
$360$	0	0
$361$	−835506.	−0.337428
$362$	−202132.	−0.0810707
$363$	−144371.	−0.0575059
$364$	−270304.	−0.106930
$365$	0	0
$366$	1.25466e6	0.489580
$367$	−2.27179e6	−0.880447	−0.440224	−	0.897888i	$-0.645101\pi$
$367$	−2.27179e6	−0.880447	−0.440224	+	0.897888i	$0.645101\pi$
$368$	−266930.	−0.102749
$369$	3.86500e6	1.47769
$370$	0	0
$371$	352635.	0.133012
$372$	−5.55251e6	−2.08033
$373$	−2.42341e6	−0.901892	−0.450946	−	0.892551i	$-0.648913\pi$
$373$	−2.42341e6	−0.901892	−0.450946	+	0.892551i	$0.648913\pi$
$374$	168906.	0.0624404
$375$	0	0
$376$	809373.	0.295243
$377$	4.22066e6	1.52942
$378$	296814.	0.106845
$379$	−2.23180e6	−0.798101	−0.399050	−	0.916929i	$-0.630660\pi$
$379$	−2.23180e6	−0.798101	−0.399050	+	0.916929i	$0.630660\pi$
$380$	0	0
$381$	276182.	0.0974725
$382$	1.51421e6	0.530917
$383$	1.13758e6	0.396263	0.198131	−	0.980175i	$-0.436513\pi$
$383$	1.13758e6	0.396263	0.198131	+	0.980175i	$0.436513\pi$
$384$	5.13623e6	1.77753
$385$	0	0
$386$	460031.	0.157152
$387$	−958610.	−0.325360
$388$	967963.	0.326422
$389$	1.97034e6	0.660188	0.330094	−	0.943948i	$-0.392920\pi$
$389$	1.97034e6	0.660188	0.330094	+	0.943948i	$0.392920\pi$
$390$	0	0
$391$	98041.1	0.0324315
$392$	−2.38131e6	−0.782710
$393$	9.27268e6	3.02847
$394$	499800.	0.162202
$395$	0	0
$396$	−5.28284e6	−1.69289
$397$	5.13128e6	1.63399	0.816996	−	0.576644i	$-0.195638\pi$
$397$	5.13128e6	1.63399	0.816996	+	0.576644i	$0.195638\pi$
$398$	581143.	0.183897
$399$	554925.	0.174503
$400$	0	0
$401$	−2.92332e6	−0.907852	−0.453926	−	0.891039i	$-0.649977\pi$
$401$	−2.92332e6	−0.907852	−0.453926	+	0.891039i	$0.649977\pi$
$402$	−1.88249e6	−0.580989
$403$	−5.19880e6	−1.59456
$404$	917365.	0.279633
$405$	0	0
$406$	−247126.	−0.0744053
$407$	−6.41542e6	−1.91973
$408$	−682631.	−0.203019
$409$	−2.10623e6	−0.622582	−0.311291	−	0.950315i	$-0.600761\pi$
$409$	−2.10623e6	−0.622582	−0.311291	+	0.950315i	$0.600761\pi$
$410$	0	0
$411$	33592.5	0.00980929
$412$	4.59370e6	1.33328
$413$	142270.	0.0410428
$414$	734854.	0.210717
$415$	0	0
$416$	3.84585e6	1.08958
$417$	−4.34967e6	−1.22494
$418$	1.25754e6	0.352031
$419$	−787697.	−0.219192	−0.109596	−	0.993976i	$-0.534956\pi$
$419$	−787697.	−0.219192	−0.109596	+	0.993976i	$0.534956\pi$
$420$	0	0
$421$	−2.73861e6	−0.753051	−0.376526	−	0.926406i	$-0.622881\pi$
$421$	−2.73861e6	−0.753051	−0.376526	+	0.926406i	$0.622881\pi$
$422$	−498717.	−0.136324
$423$	2.91817e6	0.792977
$424$	−3.22964e6	−0.872448
$425$	0	0
$426$	882681.	0.235657
$427$	287018.	0.0761796
$428$	1.12522e6	0.296912
$429$	−7.26468e6	−1.90578
$430$	0	0
$431$	2.23642e6	0.579911	0.289955	−	0.957040i	$-0.406360\pi$
$431$	2.23642e6	0.579911	0.289955	+	0.957040i	$0.406360\pi$
$432$	3.56019e6	0.917832
$433$	7.32309e6	1.87704	0.938522	−	0.345219i	$-0.112196\pi$
$433$	7.32309e6	1.87704	0.938522	+	0.345219i	$0.112196\pi$
$434$	304398.	0.0775741
$435$	0	0
$436$	1.26706e6	0.319213
$437$	729937.	0.182844
$438$	−4.35450e6	−1.08456
$439$	4.05522e6	1.00428	0.502138	−	0.864788i	$-0.332547\pi$
$439$	4.05522e6	1.00428	0.502138	+	0.864788i	$0.332547\pi$
$440$	0	0
$441$	−8.58575e6	−2.10224
$442$	−285378.	−0.0694808
$443$	−2.23385e6	−0.540810	−0.270405	−	0.962747i	$-0.587158\pi$
$443$	−2.23385e6	−0.540810	−0.270405	+	0.962747i	$0.587158\pi$
$444$	1.15768e7	2.78695
$445$	0	0
$446$	863185.	0.205479
$447$	−1.38425e7	−3.27676
$448$	10149.1	0.00238908
$449$	−3.90186e6	−0.913389	−0.456695	−	0.889623i	$-0.650967\pi$
$449$	−3.90186e6	−0.913389	−0.456695	+	0.889623i	$0.650967\pi$
$450$	0	0
$451$	2.94276e6	0.681262
$452$	−891352.	−0.205212
$453$	−1.18894e7	−2.72218
$454$	−527564.	−0.120126
$455$	0	0
$456$	−5.08233e6	−1.14459
$457$	184098.	0.0412343	0.0206172	−	0.999787i	$-0.493437\pi$
$457$	184098.	0.0412343	0.0206172	+	0.999787i	$0.493437\pi$
$458$	1.17055e6	0.260751
$459$	−1.30762e6	−0.289702
$460$	0	0
$461$	−4.89953e6	−1.07375	−0.536873	−	0.843663i	$-0.680395\pi$
$461$	−4.89953e6	−1.07375	−0.536873	+	0.843663i	$0.680395\pi$
$462$	425358.	0.0927150
$463$	2.72830e6	0.591479	0.295740	−	0.955269i	$-0.404434\pi$
$463$	2.72830e6	0.591479	0.295740	+	0.955269i	$0.404434\pi$
$464$	−2.96420e6	−0.639165
$465$	0	0
$466$	−2.36843e6	−0.505238
$467$	−3.18682e6	−0.676185	−0.338093	−	0.941113i	$-0.609782\pi$
$467$	−3.18682e6	−0.676185	−0.338093	+	0.941113i	$0.609782\pi$
$468$	8.92572e6	1.88377
$469$	−430641.	−0.0904031
$470$	0	0
$471$	−4.35859e6	−0.905302
$472$	−1.30299e6	−0.269207
$473$	−729873.	−0.150001
$474$	131175.	0.0268167
$475$	0	0
$476$	−69725.1	−0.0141050
$477$	−1.16444e7	−2.34326
$478$	4.07130e6	0.815010
$479$	4.93825e6	0.983410	0.491705	−	0.870762i	$-0.336374\pi$
$479$	4.93825e6	0.983410	0.491705	+	0.870762i	$0.336374\pi$
$480$	0	0
$481$	1.08393e7	2.13618
$482$	1.73095e6	0.339365
$483$	246898.	0.0481560
$484$	135055.	0.0262058
$485$	0	0
$486$	−1.15457e6	−0.221733
$487$	−534531.	−0.102129	−0.0510646	−	0.998695i	$-0.516261\pi$
$487$	−534531.	−0.102129	−0.0510646	+	0.998695i	$0.516261\pi$
$488$	−2.62868e6	−0.499676
$489$	−1.11991e7	−2.11792
$490$	0	0
$491$	4.65073e6	0.870597	0.435299	−	0.900286i	$-0.356643\pi$
$491$	4.65073e6	0.870597	0.435299	+	0.900286i	$0.356643\pi$
$492$	−5.31028e6	−0.989019
$493$	1.08872e6	0.201744
$494$	−2.12470e6	−0.391724
$495$	0	0
$496$	3.65115e6	0.666386
$497$	201923.	0.0366687
$498$	−3.57747e6	−0.646403
$499$	−215791.	−0.0387955	−0.0193977	−	0.999812i	$-0.506175\pi$
$499$	−215791.	−0.0387955	−0.0193977	+	0.999812i	$0.506175\pi$
$500$	0	0
$501$	−4.60250e6	−0.819219
$502$	−1.63629e6	−0.289803
$503$	6.45815e6	1.13812	0.569060	−	0.822296i	$-0.307307\pi$
$503$	6.45815e6	1.13812	0.569060	+	0.822296i	$0.307307\pi$
$504$	−1.17047e6	−0.205251
$505$	0	0
$506$	559508.	0.0971471
$507$	2.02858e6	0.350487
$508$	−258361.	−0.0444189
$509$	3.88028e6	0.663847	0.331924	−	0.943306i	$-0.392302\pi$
$509$	3.88028e6	0.663847	0.331924	+	0.943306i	$0.392302\pi$
$510$	0	0
$511$	−996142.	−0.168760
$512$	−4.85626e6	−0.818705
$513$	−9.73553e6	−1.63330
$514$	−1.49188e6	−0.249072
$515$	0	0
$516$	1.31707e6	0.217764
$517$	2.22186e6	0.365587
$518$	−634657.	−0.103924
$519$	7.94030e6	1.29395
$520$	0	0
$521$	2.45313e6	0.395937	0.197969	−	0.980208i	$-0.436566\pi$
$521$	2.45313e6	0.395937	0.197969	+	0.980208i	$0.436566\pi$
$522$	8.16038e6	1.31079
$523$	1.91833e6	0.306668	0.153334	−	0.988174i	$-0.450999\pi$
$523$	1.91833e6	0.306668	0.153334	+	0.988174i	$0.450999\pi$
$524$	−8.67436e6	−1.38010
$525$	0	0
$526$	−4.05087e6	−0.638386
$527$	−1.34104e6	−0.210336
$528$	5.10204e6	0.796451
$529$	−6.11158e6	−0.949542
$530$	0	0
$531$	−4.69790e6	−0.723049
$532$	−519118.	−0.0795220
$533$	−4.97200e6	−0.758076
$534$	−4.54247e6	−0.689348
$535$	0	0
$536$	3.94407e6	0.592970
$537$	1.44202e7	2.15792
$538$	−4.95959e6	−0.738738
$539$	−6.53708e6	−0.969196
$540$	0	0
$541$	−6.89363e6	−1.01264	−0.506320	−	0.862346i	$-0.668995\pi$
$541$	−6.89363e6	−1.01264	−0.506320	+	0.862346i	$0.668995\pi$
$542$	4.70597e6	0.688099
$543$	2.24256e6	0.326396
$544$	992042.	0.143725
$545$	0	0
$546$	−718672.	−0.103169
$547$	−4.78180e6	−0.683319	−0.341659	−	0.939824i	$-0.610989\pi$
$547$	−4.78180e6	−0.683319	−0.341659	+	0.939824i	$0.610989\pi$
$548$	−31424.9	−0.00447016
$549$	−9.47764e6	−1.34205
$550$	0	0
$551$	8.10578e6	1.13741
$552$	−2.26125e6	−0.315864
$553$	30007.7	0.00417273
$554$	−5.25986e6	−0.728115
$555$	0	0
$556$	4.06901e6	0.558215
$557$	1.16813e7	1.59534	0.797672	−	0.603091i	$-0.206065\pi$
$557$	1.16813e7	1.59534	0.797672	+	0.603091i	$0.206065\pi$
$558$	−1.00515e7	−1.36662
$559$	1.23317e6	0.166914
$560$	0	0
$561$	−1.87393e6	−0.251389
$562$	201336.	0.0268893
$563$	−9.15272e6	−1.21697	−0.608484	−	0.793566i	$-0.708222\pi$
$563$	−9.15272e6	−1.21697	−0.608484	+	0.793566i	$0.708222\pi$
$564$	−4.00939e6	−0.530739
$565$	0	0
$566$	2.51599e6	0.330117
$567$	−1.31502e6	−0.171780
$568$	−1.84933e6	−0.240516
$569$	−4.46016e6	−0.577523	−0.288762	−	0.957401i	$-0.593243\pi$
$569$	−4.46016e6	−0.577523	−0.288762	+	0.957401i	$0.593243\pi$
$570$	0	0
$571$	3.82289e6	0.490684	0.245342	−	0.969437i	$-0.421100\pi$
$571$	3.82289e6	0.490684	0.245342	+	0.969437i	$0.421100\pi$
$572$	6.79593e6	0.868478
$573$	−1.67994e7	−2.13751
$574$	291118.	0.0368799
$575$	0	0
$576$	−335134.	−0.0420883
$577$	−1.15777e7	−1.44772	−0.723858	−	0.689949i	$-0.757633\pi$
$577$	−1.15777e7	−1.44772	−0.723858	+	0.689949i	$0.757633\pi$
$578$	3.45786e6	0.430514
$579$	−5.10383e6	−0.632702
$580$	0	0
$581$	−818387.	−0.100582
$582$	2.57358e6	0.314941
$583$	−8.86588e6	−1.08032
$584$	9.12327e6	1.10693
$585$	0	0
$586$	1.95592e6	0.235291
$587$	−2.16236e6	−0.259020	−0.129510	−	0.991578i	$-0.541340\pi$
$587$	−2.16236e6	−0.259020	−0.129510	+	0.991578i	$0.541340\pi$
$588$	1.17963e7	1.40703
$589$	−9.98429e6	−1.18585
$590$	0	0
$591$	−5.54505e6	−0.653036
$592$	−7.61251e6	−0.892738
$593$	−1.42792e7	−1.66751	−0.833753	−	0.552137i	$-0.813812\pi$
$593$	−1.42792e7	−1.66751	−0.833753	+	0.552137i	$0.813812\pi$
$594$	−7.46244e6	−0.867790
$595$	0	0
$596$	1.29493e7	1.49324
$597$	−6.44751e6	−0.740382
$598$	−945326.	−0.108101
$599$	1.13983e7	1.29799	0.648997	−	0.760791i	$-0.275189\pi$
$599$	1.13983e7	1.29799	0.648997	+	0.760791i	$0.275189\pi$
$600$	0	0
$601$	−665939.	−0.0752053	−0.0376026	−	0.999293i	$-0.511972\pi$
$601$	−665939.	−0.0752053	−0.0376026	+	0.999293i	$0.511972\pi$
$602$	−72204.0	−0.00812026
$603$	1.42202e7	1.59263
$604$	1.11223e7	1.24051
$605$	0	0
$606$	2.43905e6	0.269798
$607$	1.35927e7	1.49738	0.748692	−	0.662918i	$-0.230682\pi$
$607$	1.35927e7	1.49738	0.748692	+	0.662918i	$0.230682\pi$
$608$	7.38596e6	0.810304
$609$	2.74175e6	0.299561
$610$	0	0
$611$	−3.75398e6	−0.406808
$612$	2.30240e6	0.248486
$613$	1.08635e7	1.16767	0.583833	−	0.811874i	$-0.301552\pi$
$613$	1.08635e7	1.16767	0.583833	+	0.811874i	$0.301552\pi$
$614$	264405.	0.0283041
$615$	0	0
$616$	−891182.	−0.0946269
$617$	1.62403e6	0.171744	0.0858720	−	0.996306i	$-0.472632\pi$
$617$	1.62403e6	0.171744	0.0858720	+	0.996306i	$0.472632\pi$
$618$	1.22135e7	1.28638
$619$	3.82440e6	0.401177	0.200589	−	0.979676i	$-0.435715\pi$
$619$	3.82440e6	0.401177	0.200589	+	0.979676i	$0.435715\pi$
$620$	0	0
$621$	−4.33156e6	−0.450729
$622$	205487.	0.0212965
$623$	−1.03914e6	−0.107264
$624$	−8.62024e6	−0.886253
$625$	0	0
$626$	3.44307e6	0.351164
$627$	−1.39518e7	−1.41730
$628$	4.07735e6	0.412552
$629$	2.79601e6	0.281781
$630$	0	0
$631$	−4.03571e6	−0.403503	−0.201751	−	0.979437i	$-0.564663\pi$
$631$	−4.03571e6	−0.403503	−0.201751	+	0.979437i	$0.564663\pi$
$632$	−274829.	−0.0273697
$633$	5.53303e6	0.548850
$634$	2.40001e6	0.237132
$635$	0	0
$636$	1.59987e7	1.56834
$637$	1.10448e7	1.07848
$638$	6.21321e6	0.604316
$639$	−6.66773e6	−0.645990
$640$	0	0
$641$	1.03975e7	0.999505	0.499753	−	0.866168i	$-0.333424\pi$
$641$	1.03975e7	0.999505	0.499753	+	0.866168i	$0.333424\pi$
$642$	2.99168e6	0.286469
$643$	703463.	0.0670986	0.0335493	−	0.999437i	$-0.489319\pi$
$643$	703463.	0.0670986	0.0335493	+	0.999437i	$0.489319\pi$
$644$	−230967.	−0.0219450
$645$	0	0
$646$	−548069.	−0.0516718
$647$	1.39550e7	1.31060	0.655300	−	0.755369i	$-0.272542\pi$
$647$	1.39550e7	1.31060	0.655300	+	0.755369i	$0.272542\pi$
$648$	1.20437e7	1.12674
$649$	−3.57692e6	−0.333348
$650$	0	0
$651$	−3.37715e6	−0.312319
$652$	1.04765e7	0.965151
$653$	−4.10881e6	−0.377079	−0.188540	−	0.982066i	$-0.560375\pi$
$653$	−4.10881e6	−0.377079	−0.188540	+	0.982066i	$0.560375\pi$
$654$	3.36880e6	0.307986
$655$	0	0
$656$	3.49187e6	0.316810
$657$	3.28937e7	2.97303
$658$	219802.	0.0197909
$659$	4.28147e6	0.384042	0.192021	−	0.981391i	$-0.438496\pi$
$659$	4.28147e6	0.384042	0.192021	+	0.981391i	$0.438496\pi$
$660$	0	0
$661$	−6.69686e6	−0.596166	−0.298083	−	0.954540i	$-0.596347\pi$
$661$	−6.69686e6	−0.596166	−0.298083	+	0.954540i	$0.596347\pi$
$662$	7.99274e6	0.708844
$663$	3.16614e6	0.279734
$664$	7.49528e6	0.659732
$665$	0	0
$666$	2.09571e7	1.83082
$667$	3.60645e6	0.313881
$668$	4.30553e6	0.373323
$669$	−9.57663e6	−0.827271
$670$	0	0
$671$	−7.21615e6	−0.618727
$672$	2.49827e6	0.213411
$673$	1.29636e7	1.10329	0.551644	−	0.834080i	$-0.314001\pi$
$673$	1.29636e7	1.10329	0.551644	+	0.834080i	$0.314001\pi$
$674$	2.07780e6	0.176179
$675$	0	0
$676$	−1.89768e6	−0.159719
$677$	−1.00392e7	−0.841833	−0.420916	−	0.907099i	$-0.638291\pi$
$677$	−1.00392e7	−0.841833	−0.420916	+	0.907099i	$0.638291\pi$
$678$	−2.36989e6	−0.197995
$679$	588735.	0.0490055
$680$	0	0
$681$	5.85308e6	0.483633
$682$	−7.65312e6	−0.630053
$683$	−1.37778e7	−1.13013	−0.565063	−	0.825048i	$-0.691148\pi$
$683$	−1.37778e7	−1.13013	−0.565063	+	0.825048i	$0.691148\pi$
$684$	1.71419e7	1.40093
$685$	0	0
$686$	−1.30301e6	−0.105715
$687$	−1.29867e7	−1.04980
$688$	−866065.	−0.0697556
$689$	1.49795e7	1.20212
$690$	0	0
$691$	−2.12787e7	−1.69532	−0.847658	−	0.530544i	$-0.821988\pi$
$691$	−2.12787e7	−1.69532	−0.847658	+	0.530544i	$0.821988\pi$
$692$	−7.42795e6	−0.589663
$693$	−3.21313e6	−0.254153
$694$	5.47044e6	0.431145
$695$	0	0
$696$	−2.51106e7	−1.96487
$697$	−1.28253e6	−0.0999969
$698$	−7.23763e6	−0.562287
$699$	2.62767e7	2.03412
$700$	0	0
$701$	2.07790e7	1.59709	0.798547	−	0.601933i	$-0.205602\pi$
$701$	2.07790e7	1.59709	0.798547	+	0.601933i	$0.205602\pi$
$702$	1.26083e7	0.965636
$703$	2.08169e7	1.58864
$704$	−255166.	−0.0194040
$705$	0	0
$706$	6.56274e6	0.495534
$707$	557960.	0.0419812
$708$	6.45463e6	0.483936
$709$	4.16247e6	0.310982	0.155491	−	0.987837i	$-0.450304\pi$
$709$	4.16247e6	0.310982	0.155491	+	0.987837i	$0.450304\pi$
$710$	0	0
$711$	−990888.	−0.0735107
$712$	9.51708e6	0.703564
$713$	−4.44224e6	−0.327249
$714$	−185382.	−0.0136089
$715$	0	0
$716$	−1.34898e7	−0.983381
$717$	−4.51692e7	−3.28129
$718$	92516.4	0.00669742
$719$	−1.69124e7	−1.22007	−0.610033	−	0.792376i	$-0.708844\pi$
$719$	−1.69124e7	−1.22007	−0.610033	+	0.792376i	$0.708844\pi$
$720$	0	0
$721$	2.79399e6	0.200164
$722$	2.07807e6	0.148360
$723$	−1.92041e7	−1.36631
$724$	−2.09786e6	−0.148741
$725$	0	0
$726$	359079.	0.0252841
$727$	−1.27084e7	−0.891776	−0.445888	−	0.895089i	$-0.647112\pi$
$727$	−1.27084e7	−0.891776	−0.445888	+	0.895089i	$0.647112\pi$
$728$	1.50571e6	0.105296
$729$	−7.54333e6	−0.525708
$730$	0	0
$731$	318098.	0.0220174
$732$	1.30217e7	0.898235
$733$	−1.32015e6	−0.0907535	−0.0453767	−	0.998970i	$-0.514449\pi$
$733$	−1.32015e6	−0.0907535	−0.0453767	+	0.998970i	$0.514449\pi$
$734$	5.65040e6	0.387114
$735$	0	0
$736$	3.28618e6	0.223613
$737$	1.08271e7	0.734249
$738$	−9.61304e6	−0.649711
$739$	2.42511e7	1.63351	0.816753	−	0.576988i	$-0.195772\pi$
$739$	2.42511e7	1.63351	0.816753	+	0.576988i	$0.195772\pi$
$740$	0	0
$741$	2.35725e7	1.57711
$742$	−877073.	−0.0584825
$743$	1.31970e7	0.877009	0.438504	−	0.898729i	$-0.355508\pi$
$743$	1.31970e7	0.877009	0.438504	+	0.898729i	$0.355508\pi$
$744$	3.09300e7	2.04855
$745$	0	0
$746$	6.02750e6	0.396543
$747$	2.70240e7	1.77194
$748$	1.75302e6	0.114560
$749$	684381.	0.0445752
$750$	0	0
$751$	−1.06321e7	−0.687891	−0.343945	−	0.938990i	$-0.611763\pi$
$751$	−1.06321e7	−0.687891	−0.343945	+	0.938990i	$0.611763\pi$
$752$	2.63645e6	0.170010
$753$	1.81539e7	1.16676
$754$	−1.04976e7	−0.672455
$755$	0	0
$756$	3.08053e6	0.196029
$757$	−2.44667e7	−1.55180	−0.775900	−	0.630856i	$-0.782704\pi$
$757$	−2.44667e7	−1.55180	−0.775900	+	0.630856i	$0.782704\pi$
$758$	5.55094e6	0.350908
$759$	−6.20748e6	−0.391121
$760$	0	0
$761$	2.20641e7	1.38110	0.690550	−	0.723285i	$-0.257369\pi$
$761$	2.20641e7	1.38110	0.690550	+	0.723285i	$0.257369\pi$
$762$	−686919.	−0.0428566
$763$	770652.	0.0479233
$764$	1.57154e7	0.974076
$765$	0	0
$766$	−2.82938e6	−0.174228
$767$	6.04345e6	0.370934
$768$	−1.22040e7	−0.746621
$769$	2.58590e7	1.57687	0.788434	−	0.615119i	$-0.210892\pi$
$769$	2.58590e7	1.57687	0.788434	+	0.615119i	$0.210892\pi$
$770$	0	0
$771$	1.65517e7	1.00278
$772$	4.77450e6	0.288327
$773$	1.57927e7	0.950620	0.475310	−	0.879818i	$-0.342336\pi$
$773$	1.57927e7	0.950620	0.475310	+	0.879818i	$0.342336\pi$
$774$	2.38426e6	0.143054
$775$	0	0
$776$	−5.39199e6	−0.321436
$777$	7.04122e6	0.418404
$778$	−4.90064e6	−0.290271
$779$	−9.54872e6	−0.563770
$780$	0	0
$781$	−5.07672e6	−0.297821
$782$	−243848.	−0.0142594
$783$	−4.81010e7	−2.80382
$784$	−7.75687e6	−0.450709
$785$	0	0
$786$	−2.30630e7	−1.33156
$787$	2.14880e7	1.23669	0.618344	−	0.785908i	$-0.287804\pi$
$787$	2.14880e7	1.23669	0.618344	+	0.785908i	$0.287804\pi$
$788$	5.18726e6	0.297593
$789$	4.49425e7	2.57019
$790$	0	0
$791$	−542139.	−0.0308084
$792$	2.94278e7	1.66704
$793$	1.21922e7	0.688491
$794$	−1.27625e7	−0.718432
$795$	0	0
$796$	6.03148e6	0.337397
$797$	1.54431e7	0.861170	0.430585	−	0.902550i	$-0.358307\pi$
$797$	1.54431e7	0.861170	0.430585	+	0.902550i	$0.358307\pi$
$798$	−1.38021e6	−0.0767251
$799$	−968344.	−0.0536615
$800$	0	0
$801$	3.43136e7	1.88966
$802$	7.27088e6	0.399164
$803$	2.50448e7	1.37066
$804$	−1.95377e7	−1.06594
$805$	0	0
$806$	1.29305e7	0.701094
$807$	5.50244e7	2.97421
$808$	−5.11014e6	−0.275362
$809$	−5.05884e6	−0.271756	−0.135878	−	0.990726i	$-0.543386\pi$
$809$	−5.05884e6	−0.271756	−0.135878	+	0.990726i	$0.543386\pi$
$810$	0	0
$811$	2.95840e7	1.57944	0.789722	−	0.613465i	$-0.210225\pi$
$811$	2.95840e7	1.57944	0.789722	+	0.613465i	$0.210225\pi$
$812$	−2.56484e6	−0.136512
$813$	−5.22105e7	−2.77033
$814$	1.59564e7	0.844063
$815$	0	0
$816$	−2.22360e6	−0.116905
$817$	2.36830e6	0.124132
$818$	5.23861e6	0.273737
$819$	5.42881e6	0.282810
$820$	0	0
$821$	4.56831e6	0.236536	0.118268	−	0.992982i	$-0.462266\pi$
$821$	4.56831e6	0.236536	0.118268	+	0.992982i	$0.462266\pi$
$822$	−83551.3	−0.00431294
$823$	−3.99029e6	−0.205355	−0.102677	−	0.994715i	$-0.532741\pi$
$823$	−3.99029e6	−0.205355	−0.102677	+	0.994715i	$0.532741\pi$
$824$	−2.55890e7	−1.31291
$825$	0	0
$826$	−353853.	−0.0180457
$827$	−7.46365e6	−0.379479	−0.189739	−	0.981835i	$-0.560764\pi$
$827$	−7.46365e6	−0.379479	−0.189739	+	0.981835i	$0.560764\pi$
$828$	7.62680e6	0.386604
$829$	−1.22180e7	−0.617467	−0.308734	−	0.951149i	$-0.599905\pi$
$829$	−1.22180e7	−0.617467	−0.308734	+	0.951149i	$0.599905\pi$
$830$	0	0
$831$	5.83557e7	2.93144
$832$	431121.	0.0215919
$833$	2.84903e6	0.142260
$834$	1.08185e7	0.538582
$835$	0	0
$836$	1.30516e7	0.645874
$837$	5.92484e7	2.92323
$838$	1.95916e6	0.0963740
$839$	−1.83694e7	−0.900929	−0.450464	−	0.892794i	$-0.648742\pi$
$839$	−1.83694e7	−0.900929	−0.450464	+	0.892794i	$0.648742\pi$
$840$	0	0
$841$	1.95376e7	0.952536
$842$	6.81147e6	0.331101
$843$	−2.23373e6	−0.108258
$844$	−5.17601e6	−0.250115
$845$	0	0
$846$	−7.25809e6	−0.348655
$847$	82143.3	0.00393426
$848$	−1.05202e7	−0.502383
$849$	−2.79138e7	−1.32907
$850$	0	0
$851$	9.26189e6	0.438405
$852$	9.16105e6	0.432361
$853$	3.51699e7	1.65500	0.827501	−	0.561465i	$-0.189762\pi$
$853$	3.51699e7	1.65500	0.827501	+	0.561465i	$0.189762\pi$
$854$	−713871.	−0.0334946
$855$	0	0
$856$	−6.26798e6	−0.292377
$857$	1.23215e7	0.573076	0.286538	−	0.958069i	$-0.407496\pi$
$857$	1.23215e7	0.573076	0.286538	+	0.958069i	$0.407496\pi$
$858$	1.80687e7	0.837933
$859$	2.32476e6	0.107497	0.0537484	−	0.998555i	$-0.482883\pi$
$859$	2.32476e6	0.107497	0.0537484	+	0.998555i	$0.482883\pi$
$860$	0	0
$861$	−3.22982e6	−0.148481
$862$	−5.56244e6	−0.254975
$863$	−1.53407e7	−0.701163	−0.350582	−	0.936532i	$-0.614016\pi$
$863$	−1.53407e7	−0.701163	−0.350582	+	0.936532i	$0.614016\pi$
$864$	−4.38294e7	−1.99748
$865$	0	0
$866$	−1.82140e7	−0.825297
$867$	−3.83633e7	−1.73328
$868$	3.15924e6	0.142326
$869$	−754449.	−0.0338907
$870$	0	0
$871$	−1.82931e7	−0.817038
$872$	−7.05809e6	−0.314337
$873$	−1.94407e7	−0.863328
$874$	−1.81550e6	−0.0803929
$875$	0	0
$876$	−4.51939e7	−1.98985
$877$	−1.21207e7	−0.532142	−0.266071	−	0.963953i	$-0.585726\pi$
$877$	−1.21207e7	−0.532142	−0.266071	+	0.963953i	$0.585726\pi$
$878$	−1.00861e7	−0.441559
$879$	−2.17000e7	−0.947299
$880$	0	0
$881$	208397.	0.00904589	0.00452294	−	0.999990i	$-0.498560\pi$
$881$	208397.	0.00904589	0.00452294	+	0.999990i	$0.498560\pi$
$882$	2.13545e7	0.924310
$883$	3.57371e7	1.54247	0.771237	−	0.636548i	$-0.219638\pi$
$883$	3.57371e7	1.54247	0.771237	+	0.636548i	$0.219638\pi$
$884$	−2.96184e6	−0.127477
$885$	0	0
$886$	5.55604e6	0.237783
$887$	−4.64203e7	−1.98107	−0.990533	−	0.137276i	$-0.956165\pi$
$887$	−4.64203e7	−1.98107	−0.990533	+	0.137276i	$0.956165\pi$
$888$	−6.44878e7	−2.74439
$889$	−157140.	−0.00666858
$890$	0	0
$891$	3.30619e7	1.39519
$892$	8.95871e6	0.376993
$893$	−7.20952e6	−0.302537
$894$	3.44290e7	1.44072
$895$	0	0
$896$	−2.92239e6	−0.121610
$897$	1.04880e7	0.435221
$898$	9.70472e6	0.401598
$899$	−4.93300e7	−2.03569
$900$	0	0
$901$	3.86398e6	0.158571
$902$	−7.31924e6	−0.299537
$903$	801070.	0.0326927
$904$	4.96523e6	0.202078
$905$	0	0
$906$	2.95715e7	1.19688
$907$	2.81640e7	1.13678	0.568390	−	0.822759i	$-0.307566\pi$
$907$	2.81640e7	1.13678	0.568390	+	0.822759i	$0.307566\pi$
$908$	−5.47541e6	−0.220395
$909$	−1.84245e7	−0.739580
$910$	0	0
$911$	8.25253e6	0.329451	0.164726	−	0.986339i	$-0.447326\pi$
$911$	8.25253e6	0.329451	0.164726	+	0.986339i	$0.447326\pi$
$912$	−1.65552e7	−0.659093
$913$	2.05758e7	0.816919
$914$	−457889.	−0.0181299
$915$	0	0
$916$	1.21487e7	0.478402
$917$	−5.27593e6	−0.207193
$918$	3.25232e6	0.127376
$919$	−1.14244e6	−0.0446217	−0.0223108	−	0.999751i	$-0.507102\pi$
$919$	−1.14244e6	−0.0446217	−0.0223108	+	0.999751i	$0.507102\pi$
$920$	0	0
$921$	−2.93346e6	−0.113954
$922$	1.21861e7	0.472104
$923$	8.57746e6	0.331401
$924$	4.41465e6	0.170105
$925$	0	0
$926$	−6.78583e6	−0.260061
$927$	−9.22605e7	−3.52628
$928$	3.64923e7	1.39101
$929$	−3.37164e7	−1.28175	−0.640873	−	0.767647i	$-0.721428\pi$
$929$	−3.37164e7	−1.28175	−0.640873	+	0.767647i	$0.721428\pi$
$930$	0	0
$931$	2.12116e7	0.802046
$932$	−2.45812e7	−0.926963
$933$	−2.27979e6	−0.0857413
$934$	7.92627e6	0.297305
$935$	0	0
$936$	−4.97203e7	−1.85500
$937$	4.06327e7	1.51191	0.755955	−	0.654623i	$-0.227173\pi$
$937$	4.06327e7	1.51191	0.755955	+	0.654623i	$0.227173\pi$
$938$	1.07109e6	0.0397484
$939$	−3.81992e7	−1.41381
$940$	0	0
$941$	4.51496e7	1.66219	0.831093	−	0.556134i	$-0.187716\pi$
$941$	4.51496e7	1.66219	0.831093	+	0.556134i	$0.187716\pi$
$942$	1.08407e7	0.398042
$943$	−4.24844e6	−0.155579
$944$	−4.24436e6	−0.155018
$945$	0	0
$946$	1.81534e6	0.0659524
$947$	1.85178e7	0.670986	0.335493	−	0.942043i	$-0.391097\pi$
$947$	1.85178e7	0.670986	0.335493	+	0.942043i	$0.391097\pi$
$948$	1.36142e6	0.0492007
$949$	−4.23149e7	−1.52520
$950$	0	0
$951$	−2.66270e7	−0.954709
$952$	388400.	0.0138895
$953$	3.17478e7	1.13235	0.566176	−	0.824284i	$-0.308422\pi$
$953$	3.17478e7	1.13235	0.566176	+	0.824284i	$0.308422\pi$
$954$	2.89619e7	1.03028
$955$	0	0
$956$	4.22546e7	1.49530
$957$	−6.89326e7	−2.43302
$958$	−1.22824e7	−0.432385
$959$	−19113.3	−0.000671103 0
$960$	0	0
$961$	3.21331e7	1.12239
$962$	−2.69595e7	−0.939234
$963$	−2.25990e7	−0.785279
$964$	1.79650e7	0.622636
$965$	0	0
$966$	−614086.	−0.0211732
$967$	−4.45707e7	−1.53279	−0.766396	−	0.642368i	$-0.777952\pi$
$967$	−4.45707e7	−1.53279	−0.766396	+	0.642368i	$0.777952\pi$
$968$	−752318.	−0.0258055
$969$	6.08057e6	0.208034
$970$	0	0
$971$	2.22017e7	0.755679	0.377840	−	0.925871i	$-0.376667\pi$
$971$	2.22017e7	0.755679	0.377840	+	0.925871i	$0.376667\pi$
$972$	−1.19829e7	−0.406815
$973$	2.47485e6	0.0838045
$974$	1.32949e6	0.0449041
$975$	0	0
$976$	−8.56265e6	−0.287729
$977$	3.20865e6	0.107544	0.0537719	−	0.998553i	$-0.482876\pi$
$977$	3.20865e6	0.107544	0.0537719	+	0.998553i	$0.482876\pi$
$978$	2.78543e7	0.931206
$979$	2.61259e7	0.871193
$980$	0	0
$981$	−2.54478e7	−0.844262
$982$	−1.15673e7	−0.382783
$983$	4.20453e7	1.38782	0.693910	−	0.720061i	$-0.255886\pi$
$983$	4.20453e7	1.38782	0.693910	+	0.720061i	$0.255886\pi$
$984$	2.95806e7	0.973913
$985$	0	0
$986$	−2.70788e6	−0.0887027
$987$	−2.43860e6	−0.0796795
$988$	−2.20515e7	−0.718698
$989$	1.05371e6	0.0342556
$990$	0	0
$991$	1.08440e7	0.350755	0.175377	−	0.984501i	$-0.443885\pi$
$991$	1.08440e7	0.350755	0.175377	+	0.984501i	$0.443885\pi$
$992$	−4.49494e7	−1.45025
$993$	−8.86757e7	−2.85385
$994$	−502224.	−0.0161225
$995$	0	0
$996$	−3.71294e7	−1.18596
$997$	−3.29287e7	−1.04915	−0.524574	−	0.851365i	$-0.675775\pi$
$997$	−3.29287e7	−1.04915	−0.524574	+	0.851365i	$0.675775\pi$
$998$	536715.	0.0170576
$999$	−1.23530e8	−3.91616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.b.1.5		10
5.4	even	2		43.6.a.b.1.6	✓	10
15.14	odd	2		387.6.a.e.1.5		10
20.19	odd	2		688.6.a.h.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.6	✓	10	5.4	even	2
387.6.a.e.1.5		10	15.14	odd	2
688.6.a.h.1.10		10	20.19	odd	2
1075.6.a.b.1.5		10	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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q-2.48720 q^{2} +27.5943 q^{3} -25.8138 q^{4} -68.6327 q^{6} -15.7005 q^{7} +143.795 q^{8} +518.448 q^{9} +O(q^{10})\)	\(q-2.48720 q^{2} +27.5943 q^{3} -25.8138 q^{4} -68.6327 q^{6} -15.7005 q^{7} +143.795 q^{8} +518.448 q^{9} +394.739 q^{11} -712.316 q^{12} -666.939 q^{13} +39.0503 q^{14} +468.396 q^{16} -172.038 q^{17} -1289.48 q^{18} -1280.86 q^{19} -433.245 q^{21} -981.796 q^{22} -569.882 q^{23} +3967.92 q^{24} +1658.81 q^{26} +7600.80 q^{27} +405.290 q^{28} -6328.41 q^{29} +7795.01 q^{31} -5766.42 q^{32} +10892.6 q^{33} +427.892 q^{34} -13383.1 q^{36} -16252.3 q^{37} +3185.75 q^{38} -18403.7 q^{39} +7454.95 q^{41} +1077.57 q^{42} -1849.00 q^{43} -10189.7 q^{44} +1417.41 q^{46} +5628.68 q^{47} +12925.1 q^{48} -16560.5 q^{49} -4747.27 q^{51} +17216.2 q^{52} -22460.1 q^{53} -18904.7 q^{54} -2257.65 q^{56} -35344.4 q^{57} +15740.0 q^{58} -9061.48 q^{59} -18280.8 q^{61} -19387.8 q^{62} -8139.89 q^{63} -646.417 q^{64} -27092.0 q^{66} +27428.5 q^{67} +4440.95 q^{68} -15725.5 q^{69} -12860.9 q^{71} +74550.0 q^{72} +63446.5 q^{73} +40422.7 q^{74} +33063.8 q^{76} -6197.60 q^{77} +45773.8 q^{78} -1911.26 q^{79} +83756.3 q^{81} -18542.0 q^{82} +52124.9 q^{83} +11183.7 q^{84} +4598.84 q^{86} -174628. q^{87} +56761.4 q^{88} +66185.2 q^{89} +10471.3 q^{91} +14710.8 q^{92} +215098. q^{93} -13999.7 q^{94} -159121. q^{96} -37497.8 q^{97} +41189.3 q^{98} +204652. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9}+O(q^{10}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9} + 745 q^{11} - 4627 q^{12} - 1917 q^{13} + 1936 q^{14} + 5354 q^{16} - 4017 q^{17} + 2725 q^{18} - 2404 q^{19} - 228 q^{21} + 5836 q^{22} - 1733 q^{23} - 10711 q^{24} - 1484 q^{26} + 2324 q^{27} + 15028 q^{28} + 6996 q^{29} - 4899 q^{31} + 7554 q^{32} + 15734 q^{33} - 27033 q^{34} + 4433 q^{36} - 1466 q^{37} - 13905 q^{38} - 26542 q^{39} + 10297 q^{41} + 37642 q^{42} - 18490 q^{43} - 36140 q^{44} + 17991 q^{46} - 48592 q^{47} - 83607 q^{48} + 29458 q^{49} + 92972 q^{51} - 14232 q^{52} - 127165 q^{53} - 92002 q^{54} - 7780 q^{56} - 34060 q^{57} + 10305 q^{58} + 99372 q^{59} + 17408 q^{61} - 28265 q^{62} - 2244 q^{63} + 47202 q^{64} - 150292 q^{66} + 2021 q^{67} - 192151 q^{68} + 1654 q^{69} + 11286 q^{71} + 298365 q^{72} - 49892 q^{73} - 125431 q^{74} - 249803 q^{76} - 98144 q^{77} + 28494 q^{78} - 91524 q^{79} - 26450 q^{81} + 158909 q^{82} + 105203 q^{83} - 357682 q^{84} + 14792 q^{86} - 181200 q^{87} + 461824 q^{88} - 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 238430 q^{93} + 7259 q^{94} - 162399 q^{96} - 108383 q^{97} - 354656 q^{98} - 270499 q^{99}+O(q^{100}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9 + 745 * q^11 - 4627 * q^12 - 1917 * q^13 + 1936 * q^14 + 5354 * q^16 - 4017 * q^17 + 2725 * q^18 - 2404 * q^19 - 228 * q^21 + 5836 * q^22 - 1733 * q^23 - 10711 * q^24 - 1484 * q^26 + 2324 * q^27 + 15028 * q^28 + 6996 * q^29 - 4899 * q^31 + 7554 * q^32 + 15734 * q^33 - 27033 * q^34 + 4433 * q^36 - 1466 * q^37 - 13905 * q^38 - 26542 * q^39 + 10297 * q^41 + 37642 * q^42 - 18490 * q^43 - 36140 * q^44 + 17991 * q^46 - 48592 * q^47 - 83607 * q^48 + 29458 * q^49 + 92972 * q^51 - 14232 * q^52 - 127165 * q^53 - 92002 * q^54 - 7780 * q^56 - 34060 * q^57 + 10305 * q^58 + 99372 * q^59 + 17408 * q^61 - 28265 * q^62 - 2244 * q^63 + 47202 * q^64 - 150292 * q^66 + 2021 * q^67 - 192151 * q^68 + 1654 * q^69 + 11286 * q^71 + 298365 * q^72 - 49892 * q^73 - 125431 * q^74 - 249803 * q^76 - 98144 * q^77 + 28494 * q^78 - 91524 * q^79 - 26450 * q^81 + 158909 * q^82 + 105203 * q^83 - 357682 * q^84 + 14792 * q^86 - 181200 * q^87 + 461824 * q^88 - 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 238430 * q^93 + 7259 * q^94 - 162399 * q^96 - 108383 * q^97 - 354656 * q^98 - 270499 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more