LMFDB - Embedded newform 387.10.a.e.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,10,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("387.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	387.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:	$199.318868595$
Analytic rank:	$1$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 3 x^{16} - 6541 x^{15} + 10299 x^{14} + 17445509 x^{13} - 2347983 x^{12} + \cdots - 37\!\cdots\!40 $ x^17 - 3x^16 - 6541x^15 + 10299x^14 + 17445509x^13 - 2347983x^12 - 24275147889x^11 - 32817396993x^10 + 18668870018715x^9 + 54284997199479x^8 - 7688031373215127x^7 - 34414384567646799x^6 + 1472871456600457271x^5 + 8285298256630038219x^4 - 82173286944147552299x^3 - 301942019130667260459x^2 + 866271940633182378600x - 378136193010971707440
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$40.5953$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+37.5953 q^{2} +901.404 q^{4} +205.302 q^{5} +7435.13 q^{7} +14639.7 q^{8} +O(q^{10})$	\(q+37.5953 q^{2} +901.404 q^{4} +205.302 q^{5} +7435.13 q^{7} +14639.7 q^{8} +7718.39 q^{10} -30246.9 q^{11} -147839. q^{13} +279526. q^{14} +88865.7 q^{16} +141391. q^{17} -297987. q^{19} +185060. q^{20} -1.13714e6 q^{22} -1.01694e6 q^{23} -1.91098e6 q^{25} -5.55805e6 q^{26} +6.70205e6 q^{28} -3.61847e6 q^{29} +287505. q^{31} -4.15461e6 q^{32} +5.31565e6 q^{34} +1.52645e6 q^{35} -7.45525e6 q^{37} -1.12029e7 q^{38} +3.00557e6 q^{40} +1.37565e7 q^{41} +3.41880e6 q^{43} -2.72646e7 q^{44} -3.82323e7 q^{46} -2.60924e7 q^{47} +1.49275e7 q^{49} -7.18436e7 q^{50} -1.33263e8 q^{52} +4.82445e6 q^{53} -6.20975e6 q^{55} +1.08848e8 q^{56} -1.36037e8 q^{58} -1.03849e8 q^{59} -1.20615e8 q^{61} +1.08088e7 q^{62} -2.01693e8 q^{64} -3.03517e7 q^{65} +2.55811e8 q^{67} +1.27451e8 q^{68} +5.73873e7 q^{70} -3.21080e7 q^{71} +2.71959e8 q^{73} -2.80282e8 q^{74} -2.68607e8 q^{76} -2.24889e8 q^{77} +4.52387e8 q^{79} +1.82443e7 q^{80} +5.17179e8 q^{82} -2.25247e8 q^{83} +2.90280e7 q^{85} +1.28531e8 q^{86} -4.42806e8 q^{88} +7.81736e7 q^{89} -1.09920e9 q^{91} -9.16677e8 q^{92} -9.80951e8 q^{94} -6.11775e7 q^{95} +1.42865e9 q^{97} +5.61205e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 48 q^{2} + 4522 q^{4} - 4033 q^{5} - 76 q^{7} - 41046 q^{8}+O(q^{10}) $ 17 * q - 48 * q^2 + 4522 * q^4 - 4033 * q^5 - 76 * q^7 - 41046 * q^8	\( 17 q - 48 q^{2} + 4522 q^{4} - 4033 q^{5} - 76 q^{7} - 41046 q^{8} + 23763 q^{10} - 78370 q^{11} + 114452 q^{13} + 376208 q^{14} + 412586 q^{16} - 726937 q^{17} + 544263 q^{19} - 3642183 q^{20} + 5269148 q^{22} - 5575241 q^{23} + 10874708 q^{25} - 8009180 q^{26} + 12534764 q^{28} - 8223345 q^{29} + 13054147 q^{31} - 37111710 q^{32} + 27991291 q^{34} - 17826330 q^{35} + 46733879 q^{37} - 15733789 q^{38} + 52241669 q^{40} - 53667013 q^{41} + 58119617 q^{43} - 81727236 q^{44} + 146859355 q^{46} - 122945511 q^{47} + 111396073 q^{49} + 96642133 q^{50} - 54447944 q^{52} + 993146 q^{53} - 248155792 q^{55} + 141048116 q^{56} - 466599837 q^{58} + 95519644 q^{59} - 311752038 q^{61} + 212471691 q^{62} - 829842590 q^{64} + 107969830 q^{65} - 292438130 q^{67} + 88281129 q^{68} - 1650972530 q^{70} + 13576908 q^{71} - 501490738 q^{73} + 494831691 q^{74} - 1248630771 q^{76} - 787365348 q^{77} + 740350275 q^{79} + 27802861 q^{80} - 1600400057 q^{82} - 754109940 q^{83} + 1071609956 q^{85} - 164102448 q^{86} + 1863375104 q^{88} - 1470581868 q^{89} + 2895349644 q^{91} - 1041082071 q^{92} - 706582361 q^{94} - 3297255729 q^{95} + 1949310583 q^{97} - 6695989160 q^{98}+O(q^{100}) \) 17 * q - 48 * q^2 + 4522 * q^4 - 4033 * q^5 - 76 * q^7 - 41046 * q^8 + 23763 * q^10 - 78370 * q^11 + 114452 * q^13 + 376208 * q^14 + 412586 * q^16 - 726937 * q^17 + 544263 * q^19 - 3642183 * q^20 + 5269148 * q^22 - 5575241 * q^23 + 10874708 * q^25 - 8009180 * q^26 + 12534764 * q^28 - 8223345 * q^29 + 13054147 * q^31 - 37111710 * q^32 + 27991291 * q^34 - 17826330 * q^35 + 46733879 * q^37 - 15733789 * q^38 + 52241669 * q^40 - 53667013 * q^41 + 58119617 * q^43 - 81727236 * q^44 + 146859355 * q^46 - 122945511 * q^47 + 111396073 * q^49 + 96642133 * q^50 - 54447944 * q^52 + 993146 * q^53 - 248155792 * q^55 + 141048116 * q^56 - 466599837 * q^58 + 95519644 * q^59 - 311752038 * q^61 + 212471691 * q^62 - 829842590 * q^64 + 107969830 * q^65 - 292438130 * q^67 + 88281129 * q^68 - 1650972530 * q^70 + 13576908 * q^71 - 501490738 * q^73 + 494831691 * q^74 - 1248630771 * q^76 - 787365348 * q^77 + 740350275 * q^79 + 27802861 * q^80 - 1600400057 * q^82 - 754109940 * q^83 + 1071609956 * q^85 - 164102448 * q^86 + 1863375104 * q^88 - 1470581868 * q^89 + 2895349644 * q^91 - 1041082071 * q^92 - 706582361 * q^94 - 3297255729 * q^95 + 1949310583 * q^97 - 6695989160 * 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 $

$2$	37.5953	1.66149	0.830746	−	0.556652i	$-0.187914\pi$
$2$	37.5953	1.66149	0.830746	+	0.556652i	$0.187914\pi$
$3$	0	0
$3$	0	0
$4$	901.404	1.76055
$5$	205.302	0.146902	0.0734512	−	0.997299i	$-0.476599\pi$
$5$	205.302	0.146902	0.0734512	+	0.997299i	$0.476599\pi$
$6$	0	0
$7$	7435.13	1.17044	0.585218	−	0.810876i	$-0.301009\pi$
$7$	7435.13	1.17044	0.585218	+	0.810876i	$0.301009\pi$
$8$	14639.7	1.26365
$9$	0	0
$10$	7718.39	0.244077
$11$	−30246.9	−0.622893	−0.311446	−	0.950264i	$-0.600813\pi$
$11$	−30246.9	−0.622893	−0.311446	+	0.950264i	$0.600813\pi$
$12$	0	0
$13$	−147839.	−1.43564	−0.717818	−	0.696231i	$-0.754859\pi$
$13$	−147839.	−1.43564	−0.717818	+	0.696231i	$0.754859\pi$
$14$	279526.	1.94467
$15$	0	0
$16$	88865.7	0.338996
$17$	141391.	0.410585	0.205292	−	0.978701i	$-0.434185\pi$
$17$	141391.	0.410585	0.205292	+	0.978701i	$0.434185\pi$
$18$	0	0
$19$	−297987.	−0.524574	−0.262287	−	0.964990i	$-0.584477\pi$
$19$	−297987.	−0.524574	−0.262287	+	0.964990i	$0.584477\pi$
$20$	185060.	0.258630
$21$	0	0
$22$	−1.13714e6	−1.03493
$23$	−1.01694e6	−0.757743	−0.378872	−	0.925449i	$-0.623688\pi$
$23$	−1.01694e6	−0.757743	−0.378872	+	0.925449i	$0.623688\pi$
$24$	0	0
$25$	−1.91098e6	−0.978420
$26$	−5.55805e6	−2.38530
$27$	0	0
$28$	6.70205e6	2.06061
$29$	−3.61847e6	−0.950023	−0.475011	−	0.879980i	$-0.657556\pi$
$29$	−3.61847e6	−0.950023	−0.475011	+	0.879980i	$0.657556\pi$
$30$	0	0
$31$	287505.	0.0559136	0.0279568	−	0.999609i	$-0.491100\pi$
$31$	287505.	0.0559136	0.0279568	+	0.999609i	$0.491100\pi$
$32$	−4.15461e6	−0.700415
$33$	0	0
$34$	5.31565e6	0.682183
$35$	1.52645e6	0.171940
$36$	0	0
$37$	−7.45525e6	−0.653965	−0.326982	−	0.945030i	$-0.606032\pi$
$37$	−7.45525e6	−0.653965	−0.326982	+	0.945030i	$0.606032\pi$
$38$	−1.12029e7	−0.871575
$39$	0	0
$40$	3.00557e6	0.185634
$41$	1.37565e7	0.760292	0.380146	−	0.924927i	$-0.375874\pi$
$41$	1.37565e7	0.760292	0.380146	+	0.924927i	$0.375874\pi$
$42$	0	0
$43$	3.41880e6	0.152499
$43$	3.41880e6	0.152499
$44$	−2.72646e7	−1.09664
$45$	0	0
$46$	−3.82323e7	−1.25898
$47$	−2.60924e7	−0.779963	−0.389981	−	0.920823i	$-0.627519\pi$
$47$	−2.60924e7	−0.779963	−0.389981	+	0.920823i	$0.627519\pi$
$48$	0	0
$49$	1.49275e7	0.369918
$50$	−7.18436e7	−1.62564
$51$	0	0
$52$	−1.33263e8	−2.52751
$53$	4.82445e6	0.0839859	0.0419930	−	0.999118i	$-0.486629\pi$
$53$	4.82445e6	0.0839859	0.0419930	+	0.999118i	$0.486629\pi$
$54$	0	0
$55$	−6.20975e6	−0.0915044
$56$	1.08848e8	1.47902
$57$	0	0
$58$	−1.36037e8	−1.57845
$59$	−1.03849e8	−1.11576	−0.557878	−	0.829923i	$-0.688384\pi$
$59$	−1.03849e8	−1.11576	−0.557878	+	0.829923i	$0.688384\pi$
$60$	0	0
$61$	−1.20615e8	−1.11536	−0.557681	−	0.830055i	$-0.688309\pi$
$61$	−1.20615e8	−1.11536	−0.557681	+	0.830055i	$0.688309\pi$
$62$	1.08088e7	0.0928999
$63$	0	0
$64$	−2.01693e8	−1.50273
$65$	−3.03517e7	−0.210898
$66$	0	0
$67$	2.55811e8	1.55090	0.775449	−	0.631410i	$-0.217523\pi$
$67$	2.55811e8	1.55090	0.775449	+	0.631410i	$0.217523\pi$
$68$	1.27451e8	0.722857
$69$	0	0
$70$	5.73873e7	0.285676
$71$	−3.21080e7	−0.149951	−0.0749757	−	0.997185i	$-0.523888\pi$
$71$	−3.21080e7	−0.149951	−0.0749757	+	0.997185i	$0.523888\pi$
$72$	0	0
$73$	2.71959e8	1.12086	0.560430	−	0.828202i	$-0.310636\pi$
$73$	2.71959e8	1.12086	0.560430	+	0.828202i	$0.310636\pi$
$74$	−2.80282e8	−1.08656
$75$	0	0
$76$	−2.68607e8	−0.923540
$77$	−2.24889e8	−0.729055
$78$	0	0
$79$	4.52387e8	1.30674	0.653368	−	0.757041i	$-0.273356\pi$
$79$	4.52387e8	1.30674	0.653368	+	0.757041i	$0.273356\pi$
$80$	1.82443e7	0.0497993
$81$	0	0
$82$	5.17179e8	1.26322
$83$	−2.25247e8	−0.520963	−0.260482	−	0.965479i	$-0.583881\pi$
$83$	−2.25247e8	−0.520963	−0.260482	+	0.965479i	$0.583881\pi$
$84$	0	0
$85$	2.90280e7	0.0603159
$86$	1.28531e8	0.253375
$87$	0	0
$88$	−4.42806e8	−0.787120
$89$	7.81736e7	0.132070	0.0660351	−	0.997817i	$-0.478965\pi$
$89$	7.81736e7	0.132070	0.0660351	+	0.997817i	$0.478965\pi$
$90$	0	0
$91$	−1.09920e9	−1.68032
$92$	−9.16677e8	−1.33405
$93$	0	0
$94$	−9.80951e8	−1.29590
$95$	−6.11775e7	−0.0770611
$96$	0	0
$97$	1.42865e9	1.63853	0.819264	−	0.573416i	$-0.194382\pi$
$97$	1.42865e9	1.63853	0.819264	+	0.573416i	$0.194382\pi$
$98$	5.61205e8	0.614616
$99$	0	0
$100$	−1.72256e9	−1.72256
$101$	−1.49225e9	−1.42690	−0.713452	−	0.700704i	$-0.752869\pi$
$101$	−1.49225e9	−1.42690	−0.713452	+	0.700704i	$0.752869\pi$
$102$	0	0
$103$	1.55520e9	1.36151	0.680754	−	0.732513i	$-0.261652\pi$
$103$	1.55520e9	1.36151	0.680754	+	0.732513i	$0.261652\pi$
$104$	−2.16432e9	−1.81415
$105$	0	0
$106$	1.81377e8	0.139542
$107$	−1.02284e9	−0.754361	−0.377181	−	0.926140i	$-0.623106\pi$
$107$	−1.02284e9	−0.754361	−0.377181	+	0.926140i	$0.623106\pi$
$108$	0	0
$109$	−2.01983e8	−0.137055	−0.0685275	−	0.997649i	$-0.521830\pi$
$109$	−2.01983e8	−0.137055	−0.0685275	+	0.997649i	$0.521830\pi$
$110$	−2.33457e8	−0.152034
$111$	0	0
$112$	6.60728e8	0.396773
$113$	2.14053e9	1.23501	0.617503	−	0.786569i	$-0.288144\pi$
$113$	2.14053e9	1.23501	0.617503	+	0.786569i	$0.288144\pi$
$114$	0	0
$115$	−2.08781e8	−0.111314
$116$	−3.26170e9	−1.67257
$117$	0	0
$118$	−3.90424e9	−1.85382
$119$	1.05126e9	0.480563
$120$	0	0
$121$	−1.44308e9	−0.612005
$122$	−4.53454e9	−1.85316
$123$	0	0
$124$	2.59158e8	0.0984388
$125$	−7.93309e8	−0.290635
$126$	0	0
$127$	−3.05275e9	−1.04130	−0.520649	−	0.853771i	$-0.674310\pi$
$127$	−3.05275e9	−1.04130	−0.520649	+	0.853771i	$0.674310\pi$
$128$	−5.45554e9	−1.79636
$129$	0	0
$130$	−1.14108e9	−0.350406
$131$	3.33156e8	0.0988386	0.0494193	−	0.998778i	$-0.484263\pi$
$131$	3.33156e8	0.0988386	0.0494193	+	0.998778i	$0.484263\pi$
$132$	0	0
$133$	−2.21557e9	−0.613980
$134$	9.61730e9	2.57680
$135$	0	0
$136$	2.06993e9	0.518837
$137$	1.70389e9	0.413238	0.206619	−	0.978422i	$-0.433754\pi$
$137$	1.70389e9	0.413238	0.206619	+	0.978422i	$0.433754\pi$
$138$	0	0
$139$	−2.83716e8	−0.0644641	−0.0322320	−	0.999480i	$-0.510262\pi$
$139$	−2.83716e8	−0.0644641	−0.0322320	+	0.999480i	$0.510262\pi$
$140$	1.37595e9	0.302709
$141$	0	0
$142$	−1.20711e9	−0.249143
$143$	4.47167e9	0.894247
$144$	0	0
$145$	−7.42881e8	−0.139561
$146$	1.02244e10	1.86230
$147$	0	0
$148$	−6.72019e9	−1.15134
$149$	−9.39868e9	−1.56217	−0.781086	−	0.624424i	$-0.785334\pi$
$149$	−9.39868e9	−1.56217	−0.781086	+	0.624424i	$0.785334\pi$
$150$	0	0
$151$	−4.46677e9	−0.699193	−0.349597	−	0.936900i	$-0.613681\pi$
$151$	−4.46677e9	−0.699193	−0.349597	+	0.936900i	$0.613681\pi$
$152$	−4.36245e9	−0.662880
$153$	0	0
$154$	−8.45477e9	−1.21132
$155$	5.90254e7	0.00821384
$156$	0	0
$157$	9.18696e9	1.20677	0.603383	−	0.797451i	$-0.293819\pi$
$157$	9.18696e9	1.20677	0.603383	+	0.797451i	$0.293819\pi$
$158$	1.70076e10	2.17113
$159$	0	0
$160$	−8.52951e8	−0.102893
$161$	−7.56111e9	−0.886889
$162$	0	0
$163$	1.88421e8	0.0209067	0.0104533	−	0.999945i	$-0.496673\pi$
$163$	1.88421e8	0.0209067	0.0104533	+	0.999945i	$0.496673\pi$
$164$	1.24001e10	1.33853
$165$	0	0
$166$	−8.46821e9	−0.865576
$167$	−3.71215e9	−0.369319	−0.184659	−	0.982803i	$-0.559118\pi$
$167$	−3.71215e9	−0.369319	−0.184659	+	0.982803i	$0.559118\pi$
$168$	0	0
$169$	1.12519e10	1.06105
$170$	1.09131e9	0.100214
$171$	0	0
$172$	3.08172e9	0.268482
$173$	−1.36778e10	−1.16094	−0.580468	−	0.814283i	$-0.697130\pi$
$173$	−1.36778e10	−1.16094	−0.580468	+	0.814283i	$0.697130\pi$
$174$	0	0
$175$	−1.42084e10	−1.14518
$176$	−2.68791e9	−0.211158
$177$	0	0
$178$	2.93896e9	0.219434
$179$	747875.	5.44490e−5 0	2.72245e−5	−	1.00000i	$-0.499991\pi$
$179$	747875.	5.44490e−5 0	2.72245e−5		1.00000i	$0.499991\pi$
$180$	0	0
$181$	−8.19784e9	−0.567736	−0.283868	−	0.958863i	$-0.591618\pi$
$181$	−8.19784e9	−0.567736	−0.283868	+	0.958863i	$0.591618\pi$
$182$	−4.13248e10	−2.79183
$183$	0	0
$184$	−1.48878e10	−0.957525
$185$	−1.53058e9	−0.0960690
$186$	0	0
$187$	−4.27665e9	−0.255750
$188$	−2.35198e10	−1.37317
$189$	0	0
$190$	−2.29998e9	−0.128036
$191$	−2.11368e10	−1.14918	−0.574592	−	0.818440i	$-0.694839\pi$
$191$	−2.11368e10	−1.14918	−0.574592	+	0.818440i	$0.694839\pi$
$192$	0	0
$193$	2.16539e10	1.12339	0.561693	−	0.827346i	$-0.310150\pi$
$193$	2.16539e10	1.12339	0.561693	+	0.827346i	$0.310150\pi$
$194$	5.37106e10	2.72240
$195$	0	0
$196$	1.34557e10	0.651261
$197$	−1.03360e10	−0.488937	−0.244469	−	0.969657i	$-0.578614\pi$
$197$	−1.03360e10	−0.488937	−0.244469	+	0.969657i	$0.578614\pi$
$198$	0	0
$199$	−1.13167e10	−0.511541	−0.255770	−	0.966738i	$-0.582329\pi$
$199$	−1.13167e10	−0.511541	−0.255770	+	0.966738i	$0.582329\pi$
$200$	−2.79762e10	−1.23638
$201$	0	0
$202$	−5.61014e10	−2.37079
$203$	−2.69038e10	−1.11194
$204$	0	0
$205$	2.82424e9	0.111689
$206$	5.84683e10	2.26213
$207$	0	0
$208$	−1.31378e10	−0.486675
$209$	9.01318e9	0.326753
$210$	0	0
$211$	−5.27721e10	−1.83288	−0.916438	−	0.400177i	$-0.868949\pi$
$211$	−5.27721e10	−1.83288	−0.916438	+	0.400177i	$0.868949\pi$
$212$	4.34878e9	0.147862
$213$	0	0
$214$	−3.84538e10	−1.25336
$215$	7.01888e8	0.0224024
$216$	0	0
$217$	2.13763e9	0.0654432
$218$	−7.59360e9	−0.227716
$219$	0	0
$220$	−5.59749e9	−0.161098
$221$	−2.09032e10	−0.589450
$222$	0	0
$223$	6.42441e10	1.73965	0.869824	−	0.493362i	$-0.164232\pi$
$223$	6.42441e10	1.73965	0.869824	+	0.493362i	$0.164232\pi$
$224$	−3.08901e10	−0.819790
$225$	0	0
$226$	8.04739e10	2.05195
$227$	−5.86771e10	−1.46674	−0.733369	−	0.679831i	$-0.762053\pi$
$227$	−5.86771e10	−1.46674	−0.733369	+	0.679831i	$0.762053\pi$
$228$	0	0
$229$	−7.08361e10	−1.70214	−0.851070	−	0.525053i	$-0.824046\pi$
$229$	−7.08361e10	−1.70214	−0.851070	+	0.525053i	$0.824046\pi$
$230$	−7.84918e9	−0.184948
$231$	0	0
$232$	−5.29734e10	−1.20050
$233$	−5.80001e10	−1.28922	−0.644610	−	0.764512i	$-0.722980\pi$
$233$	−5.80001e10	−1.28922	−0.644610	+	0.764512i	$0.722980\pi$
$234$	0	0
$235$	−5.35683e9	−0.114578
$236$	−9.36101e10	−1.96435
$237$	0	0
$238$	3.95225e10	0.798451
$239$	6.10507e10	1.21032	0.605159	−	0.796104i	$-0.293109\pi$
$239$	6.10507e10	1.21032	0.605159	+	0.796104i	$0.293109\pi$
$240$	0	0
$241$	6.36098e10	1.21464	0.607320	−	0.794458i	$-0.292245\pi$
$241$	6.36098e10	1.21464	0.607320	+	0.794458i	$0.292245\pi$
$242$	−5.42528e10	−1.01684
$243$	0	0
$244$	−1.08722e11	−1.96365
$245$	3.06466e9	0.0543419
$246$	0	0
$247$	4.40542e10	0.753097
$248$	4.20899e9	0.0706554
$249$	0	0
$250$	−2.98247e10	−0.482887
$251$	−6.19985e10	−0.985938	−0.492969	−	0.870047i	$-0.664088\pi$
$251$	−6.19985e10	−0.985938	−0.492969	+	0.870047i	$0.664088\pi$
$252$	0	0
$253$	3.07594e10	0.471993
$254$	−1.14769e11	−1.73011
$255$	0	0
$256$	−1.01836e11	−1.48190
$257$	1.30416e11	1.86480	0.932402	−	0.361423i	$-0.117709\pi$
$257$	1.30416e11	1.86480	0.932402	+	0.361423i	$0.117709\pi$
$258$	0	0
$259$	−5.54308e10	−0.765424
$260$	−2.73591e10	−0.371298
$261$	0	0
$262$	1.25251e10	0.164219
$263$	9.79046e10	1.26183	0.630917	−	0.775850i	$-0.282679\pi$
$263$	9.79046e10	1.26183	0.630917	+	0.775850i	$0.282679\pi$
$264$	0	0
$265$	9.90471e8	0.0123377
$266$	−8.32951e10	−1.02012
$267$	0	0
$268$	2.30589e11	2.73044
$269$	−5.69781e9	−0.0663472	−0.0331736	−	0.999450i	$-0.510561\pi$
$269$	−5.69781e9	−0.0663472	−0.0331736	+	0.999450i	$0.510561\pi$
$270$	0	0
$271$	−8.31043e10	−0.935970	−0.467985	−	0.883736i	$-0.655020\pi$
$271$	−8.31043e10	−0.935970	−0.467985	+	0.883736i	$0.655020\pi$
$272$	1.25649e10	0.139187
$273$	0	0
$274$	6.40583e10	0.686591
$275$	5.78010e10	0.609450
$276$	0	0
$277$	−1.00062e11	−1.02120	−0.510599	−	0.859819i	$-0.670577\pi$
$277$	−1.00062e11	−1.02120	−0.510599	+	0.859819i	$0.670577\pi$
$278$	−1.06664e10	−0.107106
$279$	0	0
$280$	2.23468e10	0.217272
$281$	1.58185e11	1.51351	0.756757	−	0.653696i	$-0.226782\pi$
$281$	1.58185e11	1.51351	0.756757	+	0.653696i	$0.226782\pi$
$282$	0	0
$283$	1.60571e11	1.48809	0.744044	−	0.668131i	$-0.232905\pi$
$283$	1.60571e11	1.48809	0.744044	+	0.668131i	$0.232905\pi$
$284$	−2.89423e10	−0.263998
$285$	0	0
$286$	1.68114e11	1.48578
$287$	1.02281e11	0.889872
$288$	0	0
$289$	−9.85963e10	−0.831420
$290$	−2.79288e10	−0.231879
$291$	0	0
$292$	2.45145e11	1.97333
$293$	−4.09516e10	−0.324613	−0.162307	−	0.986740i	$-0.551893\pi$
$293$	−4.09516e10	−0.324613	−0.162307	+	0.986740i	$0.551893\pi$
$294$	0	0
$295$	−2.13205e10	−0.163907
$296$	−1.09143e11	−0.826385
$297$	0	0
$298$	−3.53346e11	−2.59553
$299$	1.50344e11	1.08784
$300$	0	0
$301$	2.54192e10	0.178490
$302$	−1.67929e11	−1.16170
$303$	0	0
$304$	−2.64809e10	−0.177828
$305$	−2.47625e10	−0.163849
$306$	0	0
$307$	1.75403e11	1.12697	0.563487	−	0.826125i	$-0.309460\pi$
$307$	1.75403e11	1.12697	0.563487	+	0.826125i	$0.309460\pi$
$308$	−2.02716e11	−1.28354
$309$	0	0
$310$	2.21907e9	0.0136472
$311$	1.24919e11	0.757196	0.378598	−	0.925561i	$-0.376406\pi$
$311$	1.24919e11	0.757196	0.378598	+	0.925561i	$0.376406\pi$
$312$	0	0
$313$	8.33656e10	0.490950	0.245475	−	0.969403i	$-0.421056\pi$
$313$	8.33656e10	0.490950	0.245475	+	0.969403i	$0.421056\pi$
$314$	3.45386e11	2.00503
$315$	0	0
$316$	4.07783e11	2.30058
$317$	−2.99907e10	−0.166809	−0.0834047	−	0.996516i	$-0.526579\pi$
$317$	−2.99907e10	−0.166809	−0.0834047	+	0.996516i	$0.526579\pi$
$318$	0	0
$319$	1.09447e11	0.591762
$320$	−4.14080e10	−0.220755
$321$	0	0
$322$	−2.84262e11	−1.47356
$323$	−4.21328e10	−0.215382
$324$	0	0
$325$	2.82517e11	1.40465
$326$	7.08373e9	0.0347362
$327$	0	0
$328$	2.01391e11	0.960745
$329$	−1.94000e11	−0.912896
$330$	0	0
$331$	4.17144e11	1.91012	0.955059	−	0.296417i	$-0.0957919\pi$
$331$	4.17144e11	1.91012	0.955059	+	0.296417i	$0.0957919\pi$
$332$	−2.03038e11	−0.917184
$333$	0	0
$334$	−1.39559e11	−0.613620
$335$	5.25187e10	0.227831
$336$	0	0
$337$	3.40845e11	1.43954	0.719768	−	0.694215i	$-0.244248\pi$
$337$	3.40845e11	1.43954	0.719768	+	0.694215i	$0.244248\pi$
$338$	4.23018e11	1.76292
$339$	0	0
$340$	2.61659e10	0.106189
$341$	−8.69611e9	−0.0348281
$342$	0	0
$343$	−1.89046e11	−0.737470
$344$	5.00503e10	0.192705
$345$	0	0
$346$	−5.14220e11	−1.92889
$347$	8.95748e10	0.331668	0.165834	−	0.986154i	$-0.446968\pi$
$347$	8.95748e10	0.331668	0.165834	+	0.986154i	$0.446968\pi$
$348$	0	0
$349$	−8.79678e10	−0.317402	−0.158701	−	0.987327i	$-0.550731\pi$
$349$	−8.79678e10	−0.317402	−0.158701	+	0.987327i	$0.550731\pi$
$350$	−5.34167e11	−1.90270
$351$	0	0
$352$	1.25664e11	0.436283
$353$	2.35915e10	0.0808665	0.0404333	−	0.999182i	$-0.487126\pi$
$353$	2.35915e10	0.0808665	0.0404333	+	0.999182i	$0.487126\pi$
$354$	0	0
$355$	−6.59185e9	−0.0220282
$356$	7.04659e10	0.232517
$357$	0	0
$358$	2.81165e7	9.04666e−5 0
$359$	−4.74628e11	−1.50809	−0.754046	−	0.656821i	$-0.771901\pi$
$359$	−4.74628e11	−1.50809	−0.754046	+	0.656821i	$0.771901\pi$
$360$	0	0
$361$	−2.33891e11	−0.724822
$362$	−3.08200e11	−0.943288
$363$	0	0
$364$	−9.90825e11	−2.95829
$365$	5.58339e10	0.164657
$366$	0	0
$367$	−5.48861e11	−1.57930	−0.789651	−	0.613556i	$-0.789739\pi$
$367$	−5.48861e11	−1.57930	−0.789651	+	0.613556i	$0.789739\pi$
$368$	−9.03715e10	−0.256872
$369$	0	0
$370$	−5.75426e10	−0.159618
$371$	3.58704e10	0.0983001
$372$	0	0
$373$	−1.12720e11	−0.301517	−0.150759	−	0.988571i	$-0.548172\pi$
$373$	−1.12720e11	−0.301517	−0.150759	+	0.988571i	$0.548172\pi$
$374$	−1.60782e11	−0.424927
$375$	0	0
$376$	−3.81986e11	−0.985603
$377$	5.34951e11	1.36389
$378$	0	0
$379$	5.00203e11	1.24529	0.622644	−	0.782505i	$-0.286059\pi$
$379$	5.00203e11	1.24529	0.622644	+	0.782505i	$0.286059\pi$
$380$	−5.51456e10	−0.135670
$381$	0	0
$382$	−7.94644e11	−1.90936
$383$	−4.50346e11	−1.06943	−0.534715	−	0.845033i	$-0.679581\pi$
$383$	−4.50346e11	−1.06943	−0.534715	+	0.845033i	$0.679581\pi$
$384$	0	0
$385$	−4.61703e10	−0.107100
$386$	8.14086e11	1.86650
$387$	0	0
$388$	1.28779e12	2.88472
$389$	5.17694e11	1.14630	0.573152	−	0.819449i	$-0.305721\pi$
$389$	5.17694e11	1.14630	0.573152	+	0.819449i	$0.305721\pi$
$390$	0	0
$391$	−1.43787e11	−0.311118
$392$	2.18535e11	0.467449
$393$	0	0
$394$	−3.88583e11	−0.812365
$395$	9.28760e10	0.191963
$396$	0	0
$397$	4.30362e11	0.869513	0.434757	−	0.900548i	$-0.356834\pi$
$397$	4.30362e11	0.869513	0.434757	+	0.900548i	$0.356834\pi$
$398$	−4.25453e11	−0.849920
$399$	0	0
$400$	−1.69820e11	−0.331680
$401$	−7.47987e11	−1.44459	−0.722294	−	0.691586i	$-0.756912\pi$
$401$	−7.47987e11	−1.44459	−0.722294	+	0.691586i	$0.756912\pi$
$402$	0	0
$403$	−4.25044e10	−0.0802715
$404$	−1.34512e12	−2.51214
$405$	0	0
$406$	−1.01146e12	−1.84748
$407$	2.25498e11	0.407350
$408$	0	0
$409$	1.06852e12	1.88812	0.944058	−	0.329780i	$-0.106975\pi$
$409$	1.06852e12	1.88812	0.944058	+	0.329780i	$0.106975\pi$
$410$	1.06178e11	0.185570
$411$	0	0
$412$	1.40187e12	2.39701
$413$	−7.72133e11	−1.30592
$414$	0	0
$415$	−4.62437e10	−0.0765308
$416$	6.14214e11	1.00554
$417$	0	0
$418$	3.38853e11	0.542897
$419$	−8.06512e11	−1.27834	−0.639172	−	0.769064i	$-0.720723\pi$
$419$	−8.06512e11	−1.27834	−0.639172	+	0.769064i	$0.720723\pi$
$420$	0	0
$421$	−8.57772e11	−1.33077	−0.665384	−	0.746501i	$-0.731732\pi$
$421$	−8.57772e11	−1.33077	−0.665384	+	0.746501i	$0.731732\pi$
$422$	−1.98398e12	−3.04531
$423$	0	0
$424$	7.06287e10	0.106129
$425$	−2.70196e11	−0.401724
$426$	0	0
$427$	−8.96785e11	−1.30546
$428$	−9.21989e11	−1.32809
$429$	0	0
$430$	2.63877e10	0.0372214
$431$	−6.67092e11	−0.931189	−0.465594	−	0.884998i	$-0.654159\pi$
$431$	−6.67092e11	−0.931189	−0.465594	+	0.884998i	$0.654159\pi$
$432$	0	0
$433$	1.60041e11	0.218795	0.109397	−	0.993998i	$-0.465108\pi$
$433$	1.60041e11	0.218795	0.109397	+	0.993998i	$0.465108\pi$
$434$	8.03649e10	0.108733
$435$	0	0
$436$	−1.82068e11	−0.241293
$437$	3.03037e11	0.397492
$438$	0	0
$439$	8.08663e11	1.03915	0.519574	−	0.854426i	$-0.326091\pi$
$439$	8.08663e11	1.03915	0.519574	+	0.854426i	$0.326091\pi$
$440$	−9.09091e10	−0.115630
$441$	0	0
$442$	−7.85860e11	−0.979366
$443$	4.30605e11	0.531205	0.265602	−	0.964083i	$-0.414429\pi$
$443$	4.30605e11	0.531205	0.265602	+	0.964083i	$0.414429\pi$
$444$	0	0
$445$	1.60492e10	0.0194014
$446$	2.41527e12	2.89041
$447$	0	0
$448$	−1.49961e12	−1.75885
$449$	8.16570e11	0.948167	0.474083	−	0.880480i	$-0.342779\pi$
$449$	8.16570e11	0.948167	0.474083	+	0.880480i	$0.342779\pi$
$450$	0	0
$451$	−4.16091e11	−0.473580
$452$	1.92948e12	2.17429
$453$	0	0
$454$	−2.20598e12	−2.43697
$455$	−2.25669e11	−0.246843
$456$	0	0
$457$	−3.51130e11	−0.376569	−0.188285	−	0.982115i	$-0.560293\pi$
$457$	−3.51130e11	−0.376569	−0.188285	+	0.982115i	$0.560293\pi$
$458$	−2.66310e12	−2.82809
$459$	0	0
$460$	−1.88196e11	−0.195975
$461$	5.39021e11	0.555842	0.277921	−	0.960604i	$-0.410355\pi$
$461$	5.39021e11	0.555842	0.277921	+	0.960604i	$0.410355\pi$
$462$	0	0
$463$	−8.43400e11	−0.852941	−0.426470	−	0.904501i	$-0.640243\pi$
$463$	−8.43400e11	−0.852941	−0.426470	+	0.904501i	$0.640243\pi$
$464$	−3.21558e11	−0.322054
$465$	0	0
$466$	−2.18053e12	−2.14203
$467$	−7.46110e11	−0.725900	−0.362950	−	0.931809i	$-0.618231\pi$
$467$	−7.46110e11	−0.725900	−0.362950	+	0.931809i	$0.618231\pi$
$468$	0	0
$469$	1.90199e12	1.81523
$470$	−2.01391e11	−0.190371
$471$	0	0
$472$	−1.52033e12	−1.40993
$473$	−1.03408e11	−0.0949902
$474$	0	0
$475$	5.69447e11	0.513253
$476$	9.47613e11	0.846057
$477$	0	0
$478$	2.29522e12	2.01093
$479$	−4.41994e11	−0.383625	−0.191812	−	0.981432i	$-0.561437\pi$
$479$	−4.41994e11	−0.383625	−0.191812	+	0.981432i	$0.561437\pi$
$480$	0	0
$481$	1.10218e12	0.938855
$482$	2.39143e12	2.01811
$483$	0	0
$484$	−1.30079e12	−1.07747
$485$	2.93306e11	0.240704
$486$	0	0
$487$	−2.17097e12	−1.74893	−0.874465	−	0.485088i	$-0.838788\pi$
$487$	−2.17097e12	−1.74893	−0.874465	+	0.485088i	$0.838788\pi$
$488$	−1.76577e12	−1.40943
$489$	0	0
$490$	1.15217e11	0.0902886
$491$	1.29154e12	1.00286	0.501430	−	0.865198i	$-0.332807\pi$
$491$	1.29154e12	1.00286	0.501430	+	0.865198i	$0.332807\pi$
$492$	0	0
$493$	−5.11621e11	−0.390065
$494$	1.65623e12	1.25126
$495$	0	0
$496$	2.55493e10	0.0189545
$497$	−2.38727e11	−0.175508
$498$	0	0
$499$	−4.40903e11	−0.318340	−0.159170	−	0.987251i	$-0.550882\pi$
$499$	−4.40903e11	−0.318340	−0.159170	+	0.987251i	$0.550882\pi$
$500$	−7.15092e11	−0.511678
$501$	0	0
$502$	−2.33085e12	−1.63813
$503$	2.51027e12	1.74849	0.874246	−	0.485483i	$-0.161356\pi$
$503$	2.51027e12	1.74849	0.874246	+	0.485483i	$0.161356\pi$
$504$	0	0
$505$	−3.06362e11	−0.209616
$506$	1.15641e12	0.784212
$507$	0	0
$508$	−2.75176e12	−1.83326
$509$	−2.25234e12	−1.48732	−0.743658	−	0.668560i	$-0.766911\pi$
$509$	−2.25234e12	−1.48732	−0.743658	+	0.668560i	$0.766911\pi$
$510$	0	0
$511$	2.02205e12	1.31189
$512$	−1.03530e12	−0.665811
$513$	0	0
$514$	4.90304e12	3.09836
$515$	3.19287e11	0.200009
$516$	0	0
$517$	7.89213e11	0.485833
$518$	−2.08393e12	−1.27174
$519$	0	0
$520$	−4.44341e11	−0.266502
$521$	1.96536e12	1.16862	0.584310	−	0.811531i	$-0.301365\pi$
$521$	1.96536e12	1.16862	0.584310	+	0.811531i	$0.301365\pi$
$522$	0	0
$523$	−1.94685e11	−0.113782	−0.0568912	−	0.998380i	$-0.518119\pi$
$523$	−1.94685e11	−0.113782	−0.0568912	+	0.998380i	$0.518119\pi$
$524$	3.00308e11	0.174011
$525$	0	0
$526$	3.68075e12	2.09653
$527$	4.06507e10	0.0229573
$528$	0	0
$529$	−7.66977e11	−0.425825
$530$	3.72370e10	0.0204990
$531$	0	0
$532$	−1.99713e12	−1.08094
$533$	−2.03375e12	−1.09150
$534$	0	0
$535$	−2.09991e11	−0.110817
$536$	3.74501e12	1.95980
$537$	0	0
$538$	−2.14211e11	−0.110235
$539$	−4.51511e11	−0.230419
$540$	0	0
$541$	3.96660e12	1.99081	0.995406	−	0.0957430i	$-0.0305227\pi$
$541$	3.96660e12	1.99081	0.995406	+	0.0957430i	$0.0305227\pi$
$542$	−3.12433e12	−1.55511
$543$	0	0
$544$	−5.87426e11	−0.287580
$545$	−4.14675e10	−0.0201337
$546$	0	0
$547$	6.79936e11	0.324732	0.162366	−	0.986731i	$-0.448088\pi$
$547$	6.79936e11	0.324732	0.162366	+	0.986731i	$0.448088\pi$
$548$	1.53590e12	0.727527
$549$	0	0
$550$	2.17304e12	1.01260
$551$	1.07826e12	0.498357
$552$	0	0
$553$	3.36355e12	1.52945
$554$	−3.76186e12	−1.69671
$555$	0	0
$556$	−2.55743e11	−0.113492
$557$	2.27773e12	1.00266	0.501329	−	0.865256i	$-0.332844\pi$
$557$	2.27773e12	1.00266	0.501329	+	0.865256i	$0.332844\pi$
$558$	0	0
$559$	−5.05432e11	−0.218932
$560$	1.35649e11	0.0582869
$561$	0	0
$562$	5.94700e12	2.51469
$563$	−3.02050e11	−0.126704	−0.0633520	−	0.997991i	$-0.520179\pi$
$563$	−3.02050e11	−0.126704	−0.0633520	+	0.997991i	$0.520179\pi$
$564$	0	0
$565$	4.39457e11	0.181425
$566$	6.03671e12	2.47244
$567$	0	0
$568$	−4.70052e11	−0.189487
$569$	4.19779e12	1.67887	0.839433	−	0.543464i	$-0.182887\pi$
$569$	4.19779e12	1.67887	0.839433	+	0.543464i	$0.182887\pi$
$570$	0	0
$571$	−5.90682e11	−0.232536	−0.116268	−	0.993218i	$-0.537093\pi$
$571$	−5.90682e11	−0.232536	−0.116268	+	0.993218i	$0.537093\pi$
$572$	4.03078e12	1.57437
$573$	0	0
$574$	3.84529e12	1.47851
$575$	1.94336e12	0.741391
$576$	0	0
$577$	1.62218e12	0.609267	0.304634	−	0.952470i	$-0.401466\pi$
$577$	1.62218e12	0.609267	0.304634	+	0.952470i	$0.401466\pi$
$578$	−3.70676e12	−1.38140
$579$	0	0
$580$	−6.69635e11	−0.245704
$581$	−1.67474e12	−0.609754
$582$	0	0
$583$	−1.45924e11	−0.0523142
$584$	3.98141e12	1.41638
$585$	0	0
$586$	−1.53959e12	−0.539342
$587$	8.50011e11	0.295497	0.147749	−	0.989025i	$-0.452797\pi$
$587$	8.50011e11	0.295497	0.147749	+	0.989025i	$0.452797\pi$
$588$	0	0
$589$	−8.56727e10	−0.0293308
$590$	−8.01550e11	−0.272331
$591$	0	0
$592$	−6.62516e11	−0.221691
$593$	−1.57424e12	−0.522787	−0.261394	−	0.965232i	$-0.584182\pi$
$593$	−1.57424e12	−0.522787	−0.261394	+	0.965232i	$0.584182\pi$
$594$	0	0
$595$	2.15827e11	0.0705958
$596$	−8.47201e12	−2.75029
$597$	0	0
$598$	5.65223e12	1.80744
$599$	−3.95743e12	−1.25601	−0.628004	−	0.778210i	$-0.716128\pi$
$599$	−3.95743e12	−1.25601	−0.628004	+	0.778210i	$0.716128\pi$
$600$	0	0
$601$	1.35180e12	0.422646	0.211323	−	0.977416i	$-0.432223\pi$
$601$	1.35180e12	0.422646	0.211323	+	0.977416i	$0.432223\pi$
$602$	9.55642e11	0.296559
$603$	0	0
$604$	−4.02636e12	−1.23097
$605$	−2.96267e11	−0.0899050
$606$	0	0
$607$	−2.55817e12	−0.764856	−0.382428	−	0.923985i	$-0.624912\pi$
$607$	−2.55817e12	−0.764856	−0.382428	+	0.923985i	$0.624912\pi$
$608$	1.23802e12	0.367419
$609$	0	0
$610$	−9.30951e11	−0.272234
$611$	3.85748e12	1.11974
$612$	0	0
$613$	1.25908e12	0.360148	0.180074	−	0.983653i	$-0.442366\pi$
$613$	1.25908e12	0.360148	0.180074	+	0.983653i	$0.442366\pi$
$614$	6.59431e12	1.87246
$615$	0	0
$616$	−3.29232e12	−0.921273
$617$	−1.16942e12	−0.324853	−0.162427	−	0.986721i	$-0.551932\pi$
$617$	−1.16942e12	−0.324853	−0.162427	+	0.986721i	$0.551932\pi$
$618$	0	0
$619$	−5.33978e12	−1.46189	−0.730946	−	0.682435i	$-0.760921\pi$
$619$	−5.33978e12	−1.46189	−0.730946	+	0.682435i	$0.760921\pi$
$620$	5.32057e10	0.0144609
$621$	0	0
$622$	4.69638e12	1.25807
$623$	5.81231e11	0.154580
$624$	0	0
$625$	3.56951e12	0.935725
$626$	3.13415e12	0.815710
$627$	0	0
$628$	8.28115e12	2.12458
$629$	−1.05411e12	−0.268508
$630$	0	0
$631$	−3.83054e11	−0.0961896	−0.0480948	−	0.998843i	$-0.515315\pi$
$631$	−3.83054e11	−0.0961896	−0.0480948	+	0.998843i	$0.515315\pi$
$632$	6.62282e12	1.65126
$633$	0	0
$634$	−1.12751e12	−0.277152
$635$	−6.26737e11	−0.152969
$636$	0	0
$637$	−2.20687e12	−0.531068
$638$	4.11470e12	0.983208
$639$	0	0
$640$	−1.12003e12	−0.263889
$641$	−3.52121e12	−0.823816	−0.411908	−	0.911225i	$-0.635138\pi$
$641$	−3.52121e12	−0.823816	−0.411908	+	0.911225i	$0.635138\pi$
$642$	0	0
$643$	6.26646e12	1.44568	0.722841	−	0.691015i	$-0.242836\pi$
$643$	6.26646e12	1.44568	0.722841	+	0.691015i	$0.242836\pi$
$644$	−6.81562e12	−1.56142
$645$	0	0
$646$	−1.58400e12	−0.357855
$647$	2.33520e12	0.523907	0.261954	−	0.965080i	$-0.415633\pi$
$647$	2.33520e12	0.523907	0.261954	+	0.965080i	$0.415633\pi$
$648$	0	0
$649$	3.14112e12	0.694997
$650$	1.06213e13	2.33382
$651$	0	0
$652$	1.69843e11	0.0368073
$653$	−7.16958e12	−1.54307	−0.771533	−	0.636189i	$-0.780510\pi$
$653$	−7.16958e12	−1.54307	−0.771533	+	0.636189i	$0.780510\pi$
$654$	0	0
$655$	6.83977e10	0.0145196
$656$	1.22248e12	0.257736
$657$	0	0
$658$	−7.29350e12	−1.51677
$659$	−4.49352e12	−0.928115	−0.464058	−	0.885805i	$-0.653607\pi$
$659$	−4.49352e12	−0.928115	−0.464058	+	0.885805i	$0.653607\pi$
$660$	0	0
$661$	−4.35714e12	−0.887759	−0.443879	−	0.896087i	$-0.646398\pi$
$661$	−4.35714e12	−0.887759	−0.443879	+	0.896087i	$0.646398\pi$
$662$	1.56826e13	3.17364
$663$	0	0
$664$	−3.29755e12	−0.658317
$665$	−4.54863e11	−0.0901951
$666$	0	0
$667$	3.67978e12	0.719873
$668$	−3.34614e12	−0.650205
$669$	0	0
$670$	1.97445e12	0.378539
$671$	3.64821e12	0.694751
$672$	0	0
$673$	−3.51923e12	−0.661272	−0.330636	−	0.943758i	$-0.607263\pi$
$673$	−3.51923e12	−0.661272	−0.330636	+	0.943758i	$0.607263\pi$
$674$	1.28142e13	2.39178
$675$	0	0
$676$	1.01425e13	1.86803
$677$	6.52874e12	1.19448	0.597242	−	0.802061i	$-0.296263\pi$
$677$	6.52874e12	1.19448	0.597242	+	0.802061i	$0.296263\pi$
$678$	0	0
$679$	1.06222e13	1.91779
$680$	4.24962e11	0.0762184
$681$	0	0
$682$	−3.26932e11	−0.0578667
$683$	5.20881e12	0.915894	0.457947	−	0.888979i	$-0.348585\pi$
$683$	5.20881e12	0.915894	0.457947	+	0.888979i	$0.348585\pi$
$684$	0	0
$685$	3.49813e11	0.0607056
$686$	−7.10724e12	−1.22530
$687$	0	0
$688$	3.03814e11	0.0516964
$689$	−7.13243e11	−0.120573
$690$	0	0
$691$	−1.52736e12	−0.254853	−0.127426	−	0.991848i	$-0.540672\pi$
$691$	−1.52736e12	−0.254853	−0.127426	+	0.991848i	$0.540672\pi$
$692$	−1.23292e13	−2.04389
$693$	0	0
$694$	3.36759e12	0.551063
$695$	−5.82476e10	−0.00946992
$696$	0	0
$697$	1.94505e12	0.312164
$698$	−3.30717e12	−0.527360
$699$	0	0
$700$	−1.28075e13	−2.01615
$701$	9.66521e12	1.51175	0.755876	−	0.654715i	$-0.227211\pi$
$701$	9.66521e12	1.51175	0.755876	+	0.654715i	$0.227211\pi$
$702$	0	0
$703$	2.22157e12	0.343053
$704$	6.10058e12	0.936039
$705$	0	0
$706$	8.86928e11	0.134359
$707$	−1.10951e13	−1.67010
$708$	0	0
$709$	−2.96426e12	−0.440563	−0.220281	−	0.975436i	$-0.570698\pi$
$709$	−2.96426e12	−0.440563	−0.220281	+	0.975436i	$0.570698\pi$
$710$	−2.47822e11	−0.0365997
$711$	0	0
$712$	1.14444e12	0.166891
$713$	−2.92376e11	−0.0423681
$714$	0	0
$715$	9.18044e11	0.131367
$716$	6.74137e8	9.58605e−5 0
$717$	0	0
$718$	−1.78437e13	−2.50568
$719$	−9.52433e12	−1.32909	−0.664545	−	0.747248i	$-0.731375\pi$
$719$	−9.52433e12	−1.32909	−0.664545	+	0.747248i	$0.731375\pi$
$720$	0	0
$721$	1.15631e13	1.59356
$722$	−8.79320e12	−1.20429
$723$	0	0
$724$	−7.38957e12	−0.999529
$725$	6.91481e12	0.929521
$726$	0	0
$727$	−8.82435e12	−1.17160	−0.585798	−	0.810457i	$-0.699219\pi$
$727$	−8.82435e12	−1.17160	−0.585798	+	0.810457i	$0.699219\pi$
$728$	−1.60920e13	−2.12334
$729$	0	0
$730$	2.09909e12	0.273576
$731$	4.83389e11	0.0626136
$732$	0	0
$733$	−3.54439e12	−0.453497	−0.226748	−	0.973953i	$-0.572810\pi$
$733$	−3.54439e12	−0.453497	−0.226748	+	0.973953i	$0.572810\pi$
$734$	−2.06346e13	−2.62400
$735$	0	0
$736$	4.22501e12	0.530735
$737$	−7.73749e12	−0.966043
$738$	0	0
$739$	1.30193e13	1.60578	0.802890	−	0.596127i	$-0.203295\pi$
$739$	1.30193e13	1.60578	0.802890	+	0.596127i	$0.203295\pi$
$740$	−1.37967e12	−0.169135
$741$	0	0
$742$	1.34856e12	0.163325
$743$	1.15519e13	1.39061	0.695303	−	0.718717i	$-0.255270\pi$
$743$	1.15519e13	1.39061	0.695303	+	0.718717i	$0.255270\pi$
$744$	0	0
$745$	−1.92957e12	−0.229487
$746$	−4.23775e12	−0.500969
$747$	0	0
$748$	−3.85498e12	−0.450262
$749$	−7.60492e12	−0.882931
$750$	0	0
$751$	3.12550e12	0.358542	0.179271	−	0.983800i	$-0.442626\pi$
$751$	3.12550e12	0.358542	0.179271	+	0.983800i	$0.442626\pi$
$752$	−2.31872e12	−0.264404
$753$	0	0
$754$	2.01116e13	2.26609
$755$	−9.17038e11	−0.102713
$756$	0	0
$757$	1.11743e13	1.23677	0.618387	−	0.785874i	$-0.287786\pi$
$757$	1.11743e13	1.23677	0.618387	+	0.785874i	$0.287786\pi$
$758$	1.88053e13	2.06904
$759$	0	0
$760$	−8.95622e11	−0.0973786
$761$	9.37537e12	1.01335	0.506673	−	0.862139i	$-0.330875\pi$
$761$	9.37537e12	1.01335	0.506673	+	0.862139i	$0.330875\pi$
$762$	0	0
$763$	−1.50177e12	−0.160414
$764$	−1.90528e13	−2.02320
$765$	0	0
$766$	−1.69309e13	−1.77685
$767$	1.53530e13	1.60182
$768$	0	0
$769$	−1.27720e13	−1.31701	−0.658506	−	0.752575i	$-0.728811\pi$
$769$	−1.27720e13	−1.31701	−0.658506	+	0.752575i	$0.728811\pi$
$770$	−1.73578e12	−0.177946
$771$	0	0
$772$	1.95189e13	1.97778
$773$	1.71214e13	1.72477	0.862385	−	0.506253i	$-0.168970\pi$
$773$	1.71214e13	1.72477	0.862385	+	0.506253i	$0.168970\pi$
$774$	0	0
$775$	−5.49414e11	−0.0547069
$776$	2.09151e13	2.07053
$777$	0	0
$778$	1.94628e13	1.90457
$779$	−4.09926e12	−0.398829
$780$	0	0
$781$	9.71166e11	0.0934036
$782$	−5.40572e12	−0.516920
$783$	0	0
$784$	1.32655e12	0.125401
$785$	1.88610e12	0.177277
$786$	0	0
$787$	−9.90899e12	−0.920753	−0.460376	−	0.887724i	$-0.652286\pi$
$787$	−9.90899e12	−0.920753	−0.460376	+	0.887724i	$0.652286\pi$
$788$	−9.31688e12	−0.860801
$789$	0	0
$790$	3.49170e12	0.318944
$791$	1.59151e13	1.44549
$792$	0	0
$793$	1.78316e13	1.60125
$794$	1.61796e13	1.44469
$795$	0	0
$796$	−1.02009e13	−0.900595
$797$	−1.22447e13	−1.07494	−0.537470	−	0.843283i	$-0.680620\pi$
$797$	−1.22447e13	−1.07494	−0.537470	+	0.843283i	$0.680620\pi$
$798$	0	0
$799$	−3.68924e12	−0.320241
$800$	7.93936e12	0.685300
$801$	0	0
$802$	−2.81207e13	−2.40017
$803$	−8.22591e12	−0.698175
$804$	0	0
$805$	−1.55231e12	−0.130286
$806$	−1.59796e12	−0.133370
$807$	0	0
$808$	−2.18461e13	−1.80311
$809$	−1.75854e13	−1.44339	−0.721695	−	0.692211i	$-0.756637\pi$
$809$	−1.75854e13	−1.44339	−0.721695	+	0.692211i	$0.756637\pi$
$810$	0	0
$811$	4.40402e12	0.357483	0.178742	−	0.983896i	$-0.442797\pi$
$811$	4.40402e12	0.357483	0.178742	+	0.983896i	$0.442797\pi$
$812$	−2.42512e13	−1.95763
$813$	0	0
$814$	8.47765e12	0.676808
$815$	3.86832e10	0.00307124
$816$	0	0
$817$	−1.01876e12	−0.0799968
$818$	4.01714e13	3.13709
$819$	0	0
$820$	2.54578e12	0.196634
$821$	−3.71434e12	−0.285323	−0.142662	−	0.989772i	$-0.545566\pi$
$821$	−3.71434e12	−0.285323	−0.142662	+	0.989772i	$0.545566\pi$
$822$	0	0
$823$	1.59197e12	0.120958	0.0604790	−	0.998169i	$-0.480737\pi$
$823$	1.59197e12	0.120958	0.0604790	+	0.998169i	$0.480737\pi$
$824$	2.27678e13	1.72047
$825$	0	0
$826$	−2.90285e13	−2.16978
$827$	−6.05682e11	−0.0450267	−0.0225133	−	0.999747i	$-0.507167\pi$
$827$	−6.05682e11	−0.0450267	−0.0225133	+	0.999747i	$0.507167\pi$
$828$	0	0
$829$	−4.55788e12	−0.335172	−0.167586	−	0.985857i	$-0.553597\pi$
$829$	−4.55788e12	−0.335172	−0.167586	+	0.985857i	$0.553597\pi$
$830$	−1.73854e12	−0.127155
$831$	0	0
$832$	2.98181e13	2.15737
$833$	2.11063e12	0.151883
$834$	0	0
$835$	−7.62113e11	−0.0542538
$836$	8.12451e12	0.575266
$837$	0	0
$838$	−3.03210e13	−2.12396
$839$	−1.13152e13	−0.788376	−0.394188	−	0.919030i	$-0.628974\pi$
$839$	−1.13152e13	−0.788376	−0.394188	+	0.919030i	$0.628974\pi$
$840$	0	0
$841$	−1.41381e12	−0.0974564
$842$	−3.22482e13	−2.21106
$843$	0	0
$844$	−4.75689e13	−3.22688
$845$	2.31004e12	0.155871
$846$	0	0
$847$	−1.07295e13	−0.716312
$848$	4.28728e11	0.0284709
$849$	0	0
$850$	−1.01581e13	−0.667461
$851$	7.58158e12	0.495537
$852$	0	0
$853$	−4.71326e12	−0.304825	−0.152413	−	0.988317i	$-0.548704\pi$
$853$	−4.71326e12	−0.304825	−0.152413	+	0.988317i	$0.548704\pi$
$854$	−3.37149e13	−2.16901
$855$	0	0
$856$	−1.49740e13	−0.953251
$857$	2.17755e13	1.37897	0.689486	−	0.724299i	$-0.257837\pi$
$857$	2.17755e13	1.37897	0.689486	+	0.724299i	$0.257837\pi$
$858$	0	0
$859$	9.89222e12	0.619904	0.309952	−	0.950752i	$-0.399687\pi$
$859$	9.89222e12	0.619904	0.309952	+	0.950752i	$0.399687\pi$
$860$	6.32684e11	0.0394406
$861$	0	0
$862$	−2.50795e13	−1.54716
$863$	6.68756e12	0.410411	0.205206	−	0.978719i	$-0.434214\pi$
$863$	6.68756e12	0.410411	0.205206	+	0.978719i	$0.434214\pi$
$864$	0	0
$865$	−2.80808e12	−0.170544
$866$	6.01679e12	0.363525
$867$	0	0
$868$	1.92687e12	0.115216
$869$	−1.36833e13	−0.813956
$870$	0	0
$871$	−3.78189e13	−2.22653
$872$	−2.95697e12	−0.173190
$873$	0	0
$874$	1.13927e13	0.660430
$875$	−5.89836e12	−0.340169
$876$	0	0
$877$	−5.11595e12	−0.292030	−0.146015	−	0.989282i	$-0.546645\pi$
$877$	−5.11595e12	−0.292030	−0.146015	+	0.989282i	$0.546645\pi$
$878$	3.04019e13	1.72654
$879$	0	0
$880$	−5.51834e11	−0.0310196
$881$	−1.95601e13	−1.09391	−0.546953	−	0.837163i	$-0.684212\pi$
$881$	−1.95601e13	−1.09391	−0.546953	+	0.837163i	$0.684212\pi$
$882$	0	0
$883$	2.19947e13	1.21757	0.608786	−	0.793335i	$-0.291657\pi$
$883$	2.19947e13	1.21757	0.608786	+	0.793335i	$0.291657\pi$
$884$	−1.88422e13	−1.03776
$885$	0	0
$886$	1.61887e13	0.882592
$887$	−3.41991e13	−1.85506	−0.927530	−	0.373749i	$-0.878072\pi$
$887$	−3.41991e13	−1.85506	−0.927530	+	0.373749i	$0.878072\pi$
$888$	0	0
$889$	−2.26976e13	−1.21877
$890$	6.03374e11	0.0322353
$891$	0	0
$892$	5.79099e13	3.06274
$893$	7.77521e12	0.409148
$894$	0	0
$895$	1.53540e8	7.99870e−6 0
$896$	−4.05626e13	−2.10252
$897$	0	0
$898$	3.06992e13	1.57537
$899$	−1.04033e12	−0.0531192
$900$	0	0
$901$	6.82136e11	0.0344833
$902$	−1.56430e13	−0.786849
$903$	0	0
$904$	3.13368e13	1.56062
$905$	−1.68304e12	−0.0834017
$906$	0	0
$907$	1.61616e13	0.792963	0.396481	−	0.918043i	$-0.370231\pi$
$907$	1.61616e13	0.792963	0.396481	+	0.918043i	$0.370231\pi$
$908$	−5.28918e13	−2.58227
$909$	0	0
$910$	−8.48408e12	−0.410127
$911$	−1.85018e13	−0.889984	−0.444992	−	0.895534i	$-0.646794\pi$
$911$	−1.85018e13	−0.889984	−0.444992	+	0.895534i	$0.646794\pi$
$912$	0	0
$913$	6.81301e12	0.324504
$914$	−1.32008e13	−0.625667
$915$	0	0
$916$	−6.38519e13	−2.99671
$917$	2.47706e12	0.115684
$918$	0	0
$919$	−1.57446e13	−0.728134	−0.364067	−	0.931373i	$-0.618612\pi$
$919$	−1.57446e13	−0.728134	−0.364067	+	0.931373i	$0.618612\pi$
$920$	−3.05650e12	−0.140663
$921$	0	0
$922$	2.02646e13	0.923527
$923$	4.74682e12	0.215276
$924$	0	0
$925$	1.42468e13	0.639852
$926$	−3.17078e13	−1.41715
$927$	0	0
$928$	1.50333e13	0.665410
$929$	−1.87066e12	−0.0823992	−0.0411996	−	0.999151i	$-0.513118\pi$
$929$	−1.87066e12	−0.0823992	−0.0411996	+	0.999151i	$0.513118\pi$
$930$	0	0
$931$	−4.44822e12	−0.194050
$932$	−5.22815e13	−2.26974
$933$	0	0
$934$	−2.80502e13	−1.20608
$935$	−8.78005e11	−0.0375703
$936$	0	0
$937$	−2.58528e13	−1.09567	−0.547834	−	0.836587i	$-0.684548\pi$
$937$	−2.58528e13	−1.09567	−0.547834	+	0.836587i	$0.684548\pi$
$938$	7.15059e13	3.01598
$939$	0	0
$940$	−4.82867e12	−0.201721
$941$	−2.80468e13	−1.16608	−0.583042	−	0.812442i	$-0.698138\pi$
$941$	−2.80468e13	−1.16608	−0.583042	+	0.812442i	$0.698138\pi$
$942$	0	0
$943$	−1.39896e13	−0.576106
$944$	−9.22865e12	−0.378237
$945$	0	0
$946$	−3.88765e12	−0.157825
$947$	−2.67293e13	−1.07997	−0.539987	−	0.841673i	$-0.681571\pi$
$947$	−2.67293e13	−1.07997	−0.539987	+	0.841673i	$0.681571\pi$
$948$	0	0
$949$	−4.02062e13	−1.60915
$950$	2.14085e13	0.852766
$951$	0	0
$952$	1.53902e13	0.607265
$953$	3.84609e13	1.51043	0.755217	−	0.655475i	$-0.227532\pi$
$953$	3.84609e13	1.51043	0.755217	+	0.655475i	$0.227532\pi$
$954$	0	0
$955$	−4.33944e12	−0.168818
$956$	5.50313e13	2.13083
$957$	0	0
$958$	−1.66169e13	−0.637389
$959$	1.26687e13	0.483668
$960$	0	0
$961$	−2.63570e13	−0.996874
$962$	4.14366e13	1.55990
$963$	0	0
$964$	5.73381e13	2.13844
$965$	4.44561e12	0.165028
$966$	0	0
$967$	−2.53959e13	−0.933994	−0.466997	−	0.884259i	$-0.654664\pi$
$967$	−2.53959e13	−0.933994	−0.466997	+	0.884259i	$0.654664\pi$
$968$	−2.11262e13	−0.773362
$969$	0	0
$970$	1.10269e13	0.399927
$971$	−4.57054e13	−1.64999	−0.824994	−	0.565142i	$-0.808822\pi$
$971$	−4.57054e13	−1.64999	−0.824994	+	0.565142i	$0.808822\pi$
$972$	0	0
$973$	−2.10947e12	−0.0754510
$974$	−8.16180e13	−2.90583
$975$	0	0
$976$	−1.07185e13	−0.378103
$977$	2.08990e13	0.733839	0.366919	−	0.930253i	$-0.380412\pi$
$977$	2.08990e13	0.733839	0.366919	+	0.930253i	$0.380412\pi$
$978$	0	0
$979$	−2.36450e12	−0.0822656
$980$	2.76250e12	0.0956719
$981$	0	0
$982$	4.85557e13	1.66624
$983$	3.50805e12	0.119832	0.0599162	−	0.998203i	$-0.480917\pi$
$983$	3.50805e12	0.119832	0.0599162	+	0.998203i	$0.480917\pi$
$984$	0	0
$985$	−2.12200e12	−0.0718261
$986$	−1.92345e13	−0.648090
$987$	0	0
$988$	3.97106e13	1.32587
$989$	−3.47673e12	−0.115555
$990$	0	0
$991$	5.36093e13	1.76567	0.882833	−	0.469687i	$-0.155633\pi$
$991$	5.36093e13	1.76567	0.882833	+	0.469687i	$0.155633\pi$
$992$	−1.19447e12	−0.0391627
$993$	0	0
$994$	−8.97501e12	−0.291606
$995$	−2.32334e12	−0.0751465
$996$	0	0
$997$	−1.81254e11	−0.00580976	−0.00290488	−	0.999996i	$-0.500925\pi$
$997$	−1.81254e11	−0.00580976	−0.00290488	+	0.999996i	$0.500925\pi$
$998$	−1.65759e13	−0.528919
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.10.a.e.1.17		17
3.2	odd	2		43.10.a.b.1.1	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.a.b.1.1	✓	17	3.2	odd	2
387.10.a.e.1.17		17	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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q+37.5953 q^{2} +901.404 q^{4} +205.302 q^{5} +7435.13 q^{7} +14639.7 q^{8} +O(q^{10})\)	\(q+37.5953 q^{2} +901.404 q^{4} +205.302 q^{5} +7435.13 q^{7} +14639.7 q^{8} +7718.39 q^{10} -30246.9 q^{11} -147839. q^{13} +279526. q^{14} +88865.7 q^{16} +141391. q^{17} -297987. q^{19} +185060. q^{20} -1.13714e6 q^{22} -1.01694e6 q^{23} -1.91098e6 q^{25} -5.55805e6 q^{26} +6.70205e6 q^{28} -3.61847e6 q^{29} +287505. q^{31} -4.15461e6 q^{32} +5.31565e6 q^{34} +1.52645e6 q^{35} -7.45525e6 q^{37} -1.12029e7 q^{38} +3.00557e6 q^{40} +1.37565e7 q^{41} +3.41880e6 q^{43} -2.72646e7 q^{44} -3.82323e7 q^{46} -2.60924e7 q^{47} +1.49275e7 q^{49} -7.18436e7 q^{50} -1.33263e8 q^{52} +4.82445e6 q^{53} -6.20975e6 q^{55} +1.08848e8 q^{56} -1.36037e8 q^{58} -1.03849e8 q^{59} -1.20615e8 q^{61} +1.08088e7 q^{62} -2.01693e8 q^{64} -3.03517e7 q^{65} +2.55811e8 q^{67} +1.27451e8 q^{68} +5.73873e7 q^{70} -3.21080e7 q^{71} +2.71959e8 q^{73} -2.80282e8 q^{74} -2.68607e8 q^{76} -2.24889e8 q^{77} +4.52387e8 q^{79} +1.82443e7 q^{80} +5.17179e8 q^{82} -2.25247e8 q^{83} +2.90280e7 q^{85} +1.28531e8 q^{86} -4.42806e8 q^{88} +7.81736e7 q^{89} -1.09920e9 q^{91} -9.16677e8 q^{92} -9.80951e8 q^{94} -6.11775e7 q^{95} +1.42865e9 q^{97} +5.61205e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 48 q^{2} + 4522 q^{4} - 4033 q^{5} - 76 q^{7} - 41046 q^{8}+O(q^{10}) \) 17 * q - 48 * q^2 + 4522 * q^4 - 4033 * q^5 - 76 * q^7 - 41046 * q^8	\( 17 q - 48 q^{2} + 4522 q^{4} - 4033 q^{5} - 76 q^{7} - 41046 q^{8} + 23763 q^{10} - 78370 q^{11} + 114452 q^{13} + 376208 q^{14} + 412586 q^{16} - 726937 q^{17} + 544263 q^{19} - 3642183 q^{20} + 5269148 q^{22} - 5575241 q^{23} + 10874708 q^{25} - 8009180 q^{26} + 12534764 q^{28} - 8223345 q^{29} + 13054147 q^{31} - 37111710 q^{32} + 27991291 q^{34} - 17826330 q^{35} + 46733879 q^{37} - 15733789 q^{38} + 52241669 q^{40} - 53667013 q^{41} + 58119617 q^{43} - 81727236 q^{44} + 146859355 q^{46} - 122945511 q^{47} + 111396073 q^{49} + 96642133 q^{50} - 54447944 q^{52} + 993146 q^{53} - 248155792 q^{55} + 141048116 q^{56} - 466599837 q^{58} + 95519644 q^{59} - 311752038 q^{61} + 212471691 q^{62} - 829842590 q^{64} + 107969830 q^{65} - 292438130 q^{67} + 88281129 q^{68} - 1650972530 q^{70} + 13576908 q^{71} - 501490738 q^{73} + 494831691 q^{74} - 1248630771 q^{76} - 787365348 q^{77} + 740350275 q^{79} + 27802861 q^{80} - 1600400057 q^{82} - 754109940 q^{83} + 1071609956 q^{85} - 164102448 q^{86} + 1863375104 q^{88} - 1470581868 q^{89} + 2895349644 q^{91} - 1041082071 q^{92} - 706582361 q^{94} - 3297255729 q^{95} + 1949310583 q^{97} - 6695989160 q^{98}+O(q^{100}) \) 17 * q - 48 * q^2 + 4522 * q^4 - 4033 * q^5 - 76 * q^7 - 41046 * q^8 + 23763 * q^10 - 78370 * q^11 + 114452 * q^13 + 376208 * q^14 + 412586 * q^16 - 726937 * q^17 + 544263 * q^19 - 3642183 * q^20 + 5269148 * q^22 - 5575241 * q^23 + 10874708 * q^25 - 8009180 * q^26 + 12534764 * q^28 - 8223345 * q^29 + 13054147 * q^31 - 37111710 * q^32 + 27991291 * q^34 - 17826330 * q^35 + 46733879 * q^37 - 15733789 * q^38 + 52241669 * q^40 - 53667013 * q^41 + 58119617 * q^43 - 81727236 * q^44 + 146859355 * q^46 - 122945511 * q^47 + 111396073 * q^49 + 96642133 * q^50 - 54447944 * q^52 + 993146 * q^53 - 248155792 * q^55 + 141048116 * q^56 - 466599837 * q^58 + 95519644 * q^59 - 311752038 * q^61 + 212471691 * q^62 - 829842590 * q^64 + 107969830 * q^65 - 292438130 * q^67 + 88281129 * q^68 - 1650972530 * q^70 + 13576908 * q^71 - 501490738 * q^73 + 494831691 * q^74 - 1248630771 * q^76 - 787365348 * q^77 + 740350275 * q^79 + 27802861 * q^80 - 1600400057 * q^82 - 754109940 * q^83 + 1071609956 * q^85 - 164102448 * q^86 + 1863375104 * q^88 - 1470581868 * q^89 + 2895349644 * q^91 - 1041082071 * q^92 - 706582361 * q^94 - 3297255729 * q^95 + 1949310583 * q^97 - 6695989160 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more