LMFDB - Embedded newform 315.10.b.a.251.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,10,Mod(251,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("315.251");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	315.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$162.236288392$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.20
Character	$\chi$	$=$	315.251
Dual form			315.10.b.a.251.29

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-9.87231i q^{2} +414.537 q^{4} -625.000 q^{5} +(-1874.58 + 6069.56i) q^{7} -9147.07i q^{8} +6170.19i q^{10} +9091.98i q^{11} +193005. i q^{13} +(59920.6 + 18506.5i) q^{14} +121940. q^{16} +308913. q^{17} +692032. i q^{19} -259086. q^{20} +89758.8 q^{22} -1.05080e6i q^{23} +390625. q^{25} +1.90541e6 q^{26} +(-777085. + 2.51606e6i) q^{28} +1.04695e6i q^{29} -1.84445e6i q^{31} -5.88713e6i q^{32} -3.04969e6i q^{34} +(1.17161e6 - 3.79347e6i) q^{35} -1.21304e7 q^{37} +6.83195e6 q^{38} +5.71692e6i q^{40} -4.58376e6 q^{41} +1.45992e7 q^{43} +3.76897e6i q^{44} -1.03738e7 q^{46} +1.04158e7 q^{47} +(-3.33255e7 - 2.27558e7i) q^{49} -3.85637e6i q^{50} +8.00079e7i q^{52} +7.23056e7i q^{53} -5.68249e6i q^{55} +(5.55187e7 + 1.71469e7i) q^{56} +1.03358e7 q^{58} +2.66164e6 q^{59} -8.43983e7i q^{61} -1.82090e7 q^{62} +4.31393e6 q^{64} -1.20628e8i q^{65} -2.63873e8 q^{67} +1.28056e8 q^{68} +(-3.74504e7 - 1.15665e7i) q^{70} +2.96944e8i q^{71} -2.76294e8i q^{73} +1.19755e8i q^{74} +2.86873e8i q^{76} +(-5.51843e7 - 1.70437e7i) q^{77} +2.86106e7 q^{79} -7.62128e7 q^{80} +4.52523e7i q^{82} -3.83366e8 q^{83} -1.93071e8 q^{85} -1.44128e8i q^{86} +8.31649e7 q^{88} -7.43699e8 q^{89} +(-1.17146e9 - 3.61804e8i) q^{91} -4.35596e8i q^{92} -1.02828e8i q^{94} -4.32520e8i q^{95} +5.61411e8i q^{97} +(-2.24652e8 + 3.29000e8i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 12288 q^{4} - 30000 q^{5} + 1824 q^{7} + 442908 q^{14} + 3258948 q^{16} + 7680000 q^{20} - 2860668 q^{22} + 18750000 q^{25} - 5432976 q^{26} - 3685092 q^{28} - 1140000 q^{35} + 7750344 q^{37} + 17423136 q^{38}+ \cdots - 2546372484 q^{98}+O(q^{100}) $ 48 * q - 12288 * q^4 - 30000 * q^5 + 1824 * q^7 + 442908 * q^14 + 3258948 * q^16 + 7680000 * q^20 - 2860668 * q^22 + 18750000 * q^25 - 5432976 * q^26 - 3685092 * q^28 - 1140000 * q^35 + 7750344 * q^37 + 17423136 * q^38 - 70626600 * q^41 + 16073832 * q^43 - 15083820 * q^46 - 33200856 * q^47 - 39367500 * q^49 - 318486852 * q^56 + 293819940 * q^58 + 342990864 * q^59 + 896801400 * q^62 - 1639789824 * q^64 - 980229936 * q^67 - 438669576 * q^68 - 276817500 * q^70 - 900236484 * q^77 + 325977144 * q^79 - 2036842500 * q^80 + 1415370096 * q^83 + 3319557300 * q^88 - 1132967016 * q^89 - 1162515060 * q^91 - 2546372484 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		−	9.87231i		−	0.436299i	−0.975915	−	0.218149i	$-0.929998\pi$
$2$		−	9.87231i		−	0.436299i	0.975915	−	0.218149i	$-0.0700020\pi$
$3$		0			0
$3$		0			0
$4$	414.537			0.809643
$5$	−625.000			−0.447214
$5$	−625.000			−0.447214
$6$		0			0
$7$	−1874.58	+	6069.56i	−0.295096	+	0.955468i
$7$	−1874.58	+	6069.56i	−0.295096	+	0.955468i
$8$		−	9147.07i		−	0.789545i
$9$		0			0
$10$			6170.19i			0.195119i
$11$			9091.98i			0.187237i	0.995608	+	0.0936184i	$0.0298434\pi$
$11$			9091.98i			0.187237i	−0.995608	+	0.0936184i	$0.970157\pi$
$12$		0			0
$13$			193005.i			1.87423i	0.349015	+	0.937117i	$0.386516\pi$
$13$			193005.i			1.87423i	−0.349015	+	0.937117i	$0.613484\pi$
$14$	59920.6	+	18506.5i	0.416869	+	0.128750i
$15$		0			0
$16$	121940.			0.465166
$17$	308913.			0.897049			0.448525	−	0.893770i	$-0.351950\pi$
$17$	308913.			0.897049			0.448525	+	0.893770i	$0.351950\pi$
$18$		0			0
$19$			692032.i			1.21825i	0.793076	+	0.609123i	$0.208478\pi$
$19$			692032.i			1.21825i	−0.793076	+	0.609123i	$0.791522\pi$
$20$	−259086.			−0.362084
$21$		0			0
$22$	89758.8			0.0816912
$23$		−	1.05080e6i		−	0.782970i	−0.920184	−	0.391485i	$-0.871961\pi$
$23$		−	1.05080e6i		−	0.782970i	0.920184	−	0.391485i	$-0.128039\pi$
$24$		0			0
$25$	390625.			0.200000
$26$	1.90541e6			0.817726
$27$		0			0
$28$	−777085.	+	2.51606e6i	−0.238923	+	0.773588i
$29$			1.04695e6i			0.274874i	0.990511	+	0.137437i	$0.0438865\pi$
$29$			1.04695e6i			0.274874i	−0.990511	+	0.137437i	$0.956114\pi$
$30$		0			0
$31$		−	1.84445e6i		−	0.358706i	−0.983785	−	0.179353i	$-0.942600\pi$
$31$		−	1.84445e6i		−	0.358706i	0.983785	−	0.179353i	$-0.0574004\pi$
$32$		−	5.88713e6i		−	0.992496i
$33$		0			0
$34$		−	3.04969e6i		−	0.391381i
$35$	1.17161e6	−	3.79347e6i	0.131971	−	0.427298i
$36$		0			0
$37$	−1.21304e7			−1.06406			−0.532030	−	0.846725i	$-0.678571\pi$
$37$	−1.21304e7			−1.06406			−0.532030	+	0.846725i	$0.678571\pi$
$38$	6.83195e6			0.531519
$39$		0			0
$40$			5.71692e6i			0.353095i
$41$	−4.58376e6			−0.253335			−0.126667	−	0.991945i	$-0.540428\pi$
$41$	−4.58376e6			−0.253335			−0.126667	+	0.991945i	$0.540428\pi$
$42$		0			0
$43$	1.45992e7			0.651210			0.325605	−	0.945506i	$-0.394432\pi$
$43$	1.45992e7			0.651210			0.325605	+	0.945506i	$0.394432\pi$
$44$			3.76897e6i			0.151595i
$45$		0			0
$46$	−1.03738e7			−0.341609
$47$	1.04158e7			0.311354			0.155677	−	0.987808i	$-0.450244\pi$
$47$	1.04158e7			0.311354			0.155677	+	0.987808i	$0.450244\pi$
$48$		0			0
$49$	−3.33255e7	−	2.27558e7i	−0.825837	−	0.563910i
$50$		−	3.85637e6i		−	0.0872597i
$51$		0			0
$52$			8.00079e7i			1.51746i
$53$			7.23056e7i			1.25872i	0.777112	+	0.629362i	$0.216684\pi$
$53$			7.23056e7i			1.25872i	−0.777112	+	0.629362i	$0.783316\pi$
$54$		0			0
$55$		−	5.68249e6i		−	0.0837349i
$56$	5.55187e7	+	1.71469e7i	0.754385	+	0.232992i
$57$		0			0
$58$	1.03358e7			0.119927
$59$	2.66164e6			0.0285967			0.0142983	−	0.999898i	$-0.495449\pi$
$59$	2.66164e6			0.0285967			0.0142983	+	0.999898i	$0.495449\pi$
$60$		0			0
$61$		−	8.43983e7i		−	0.780458i	−0.920718	−	0.390229i	$-0.872396\pi$
$61$		−	8.43983e7i		−	0.780458i	0.920718	−	0.390229i	$-0.127604\pi$
$62$	−1.82090e7			−0.156503
$63$		0			0
$64$	4.31393e6			0.0321413
$65$		−	1.20628e8i		−	0.838183i
$66$		0			0
$67$	−2.63873e8			−1.59977			−0.799886	−	0.600153i	$-0.795107\pi$
$67$	−2.63873e8			−1.59977			−0.799886	+	0.600153i	$0.795107\pi$
$68$	1.28056e8			0.726290
$69$		0			0
$70$	−3.74504e7	−	1.15665e7i	−0.186430	−	0.0575788i
$71$			2.96944e8i			1.38680i	0.720555	+	0.693398i	$0.243887\pi$
$71$			2.96944e8i			1.38680i	−0.720555	+	0.693398i	$0.756113\pi$
$72$		0			0
$73$		−	2.76294e8i		−	1.13872i	−0.822087	−	0.569362i	$-0.807190\pi$
$73$		−	2.76294e8i		−	1.13872i	0.822087	−	0.569362i	$-0.192810\pi$
$74$			1.19755e8i			0.464248i
$75$		0			0
$76$			2.86873e8i			0.986345i
$77$	−5.51843e7	−	1.70437e7i	−0.178899	−	0.0552529i
$78$		0			0
$79$	2.86106e7			0.0826427			0.0413214	−	0.999146i	$-0.486843\pi$
$79$	2.86106e7			0.0826427			0.0413214	+	0.999146i	$0.486843\pi$
$80$	−7.62128e7			−0.208029
$81$		0			0
$82$			4.52523e7i			0.110530i
$83$	−3.83366e8			−0.886671			−0.443335	−	0.896356i	$-0.646205\pi$
$83$	−3.83366e8			−0.886671			−0.443335	+	0.896356i	$0.646205\pi$
$84$		0			0
$85$	−1.93071e8			−0.401173
$86$		−	1.44128e8i		−	0.284122i
$87$		0			0
$88$	8.31649e7			0.147832
$89$	−7.43699e8			−1.25644			−0.628220	−	0.778036i	$-0.716216\pi$
$89$	−7.43699e8			−1.25644			−0.628220	+	0.778036i	$0.716216\pi$
$90$		0			0
$91$	−1.17146e9	−	3.61804e8i	−1.79077	−	0.553079i
$92$		−	4.35596e8i		−	0.633927i
$93$		0			0
$94$		−	1.02828e8i		−	0.135843i
$95$		−	4.32520e8i		−	0.544816i
$96$		0			0
$97$			5.61411e8i			0.643885i	0.946759	+	0.321943i	$0.104336\pi$
$97$			5.61411e8i			0.643885i	−0.946759	+	0.321943i	$0.895664\pi$
$98$	−2.24652e8	+	3.29000e8i	−0.246033	+	0.360311i
$99$		0			0
$100$	1.61929e8			0.161929
$101$	6.06932e8			0.580356			0.290178	−	0.956973i	$-0.406286\pi$
$101$	6.06932e8			0.580356			0.290178	+	0.956973i	$0.406286\pi$
$102$		0			0
$103$			1.76874e9i			1.54845i	0.632909	+	0.774226i	$0.281861\pi$
$103$			1.76874e9i			1.54845i	−0.632909	+	0.774226i	$0.718139\pi$
$104$	1.76543e9			1.47979
$105$		0			0
$106$	7.13823e8			0.549179
$107$		−	2.54973e9i		−	1.88047i	−0.340524	−	0.940236i	$-0.610605\pi$
$107$		−	2.54973e9i		−	1.88047i	0.340524	−	0.940236i	$-0.389395\pi$
$108$		0			0
$109$	1.74908e9			1.18683			0.593417	−	0.804895i	$-0.297778\pi$
$109$	1.74908e9			1.18683			0.593417	+	0.804895i	$0.297778\pi$
$110$	−5.60993e7			−0.0365334
$111$		0			0
$112$	−2.28588e8	+	7.40125e8i	−0.137269	+	0.444451i
$113$		−	5.46552e8i		−	0.315339i	−0.987492	−	0.157670i	$-0.949602\pi$
$113$		−	5.46552e8i		−	0.315339i	0.987492	−	0.157670i	$-0.0503981\pi$
$114$		0			0
$115$			6.56751e8i			0.350155i
$116$			4.33999e8i			0.222550i
$117$		0			0
$118$		−	2.62765e7i		−	0.0124767i
$119$	−5.79083e8	+	1.87497e9i	−0.264716	+	0.857102i
$120$		0			0
$121$	2.27528e9			0.964942
$122$	−8.33207e8			−0.340513
$123$		0			0
$124$		−	7.64592e8i		−	0.290424i
$125$	−2.44141e8			−0.0894427
$126$		0			0
$127$	−3.32729e9			−1.13494			−0.567472	−	0.823393i	$-0.692079\pi$
$127$	−3.32729e9			−1.13494			−0.567472	+	0.823393i	$0.692079\pi$
$128$		−	3.05680e9i		−	1.00652i
$129$		0			0
$130$	−1.19088e9			−0.365698
$131$	−1.76600e9			−0.523927			−0.261963	−	0.965078i	$-0.584370\pi$
$131$	−1.76600e9			−0.523927			−0.261963	+	0.965078i	$0.584370\pi$
$132$		0			0
$133$	−4.20033e9	−	1.29727e9i	−1.16399	−	0.359500i
$134$			2.60503e9i			0.697978i
$135$		0			0
$136$		−	2.82565e9i		−	0.708261i
$137$			2.56645e9i			0.622429i	0.950340	+	0.311214i	$0.100736\pi$
$137$			2.56645e9i			0.622429i	−0.950340	+	0.311214i	$0.899264\pi$
$138$		0			0
$139$		−	5.31906e9i		−	1.20856i	−0.796772	−	0.604280i	$-0.793461\pi$
$139$		−	5.31906e9i		−	1.20856i	0.796772	−	0.604280i	$-0.206539\pi$
$140$	4.85678e8	−	1.57254e9i	0.106849	−	0.345959i
$141$		0			0
$142$	2.93153e9			0.605057
$143$	−1.75480e9			−0.350926
$144$		0			0
$145$		−	6.54342e8i		−	0.122927i
$146$	−2.72766e9			−0.496823
$147$		0			0
$148$	−5.02849e9			−0.861510
$149$		−	8.43692e8i		−	0.140231i	−0.997539	−	0.0701157i	$-0.977663\pi$
$149$		−	8.43692e8i		−	0.140231i	0.997539	−	0.0701157i	$-0.0223369\pi$
$150$		0			0
$151$	−1.60909e9			−0.251875			−0.125937	−	0.992038i	$-0.540194\pi$
$151$	−1.60909e9			−0.251875			−0.125937	+	0.992038i	$0.540194\pi$
$152$	6.33006e9			0.961860
$153$		0			0
$154$	−1.68260e8	+	5.44797e8i	−0.0241067	+	0.0780533i
$155$			1.15278e9i			0.160418i
$156$		0			0
$157$			7.22801e9i			0.949445i	0.880135	+	0.474723i	$0.157452\pi$
$157$			7.22801e9i			0.949445i	−0.880135	+	0.474723i	$0.842548\pi$
$158$		−	2.82453e8i		−	0.0360569i
$159$		0			0
$160$			3.67946e9i			0.443858i
$161$	6.37790e9	+	1.96981e9i	0.748103	+	0.231052i
$162$		0			0
$163$	−6.22104e9			−0.690270			−0.345135	−	0.938553i	$-0.612167\pi$
$163$	−6.22104e9			−0.690270			−0.345135	+	0.938553i	$0.612167\pi$
$164$	−1.90014e9			−0.205111
$165$		0			0
$166$			3.78471e9i			0.386853i
$167$	1.60345e10			1.59526			0.797632	−	0.603144i	$-0.206086\pi$
$167$	1.60345e10			1.59526			0.797632	+	0.603144i	$0.206086\pi$
$168$		0			0
$169$	−2.66465e10			−2.51275
$170$			1.90605e9i			0.175031i
$171$		0			0
$172$	6.05191e9			0.527248
$173$	−2.08146e10			−1.76669			−0.883346	−	0.468722i	$-0.844715\pi$
$173$	−2.08146e10			−1.76669			−0.883346	+	0.468722i	$0.844715\pi$
$174$		0			0
$175$	−7.32259e8	+	2.37092e9i	−0.0590192	+	0.191094i
$176$			1.10868e9i			0.0870962i
$177$		0			0
$178$			7.34202e9i			0.548183i
$179$		−	1.39947e10i		−	1.01888i	−0.860506	−	0.509441i	$-0.829852\pi$
$179$		−	1.39947e10i		−	1.01888i	0.860506	−	0.509441i	$-0.170148\pi$
$180$		0			0
$181$			5.26759e9i			0.364803i	0.983224	+	0.182401i	$0.0583870\pi$
$181$			5.26759e9i			0.364803i	−0.983224	+	0.182401i	$0.941613\pi$
$182$	−3.57184e9	+	1.15650e10i	−0.241308	+	0.781311i
$183$		0			0
$184$	−9.61175e9			−0.618190
$185$	7.58148e9			0.475862
$186$		0			0
$187$			2.80863e9i			0.167961i
$188$	4.31776e9			0.252085
$189$		0			0
$190$	−4.26997e9			−0.237703
$191$			2.56251e10i			1.39321i	0.717456	+	0.696603i	$0.245306\pi$
$191$			2.56251e10i			1.39321i	−0.717456	+	0.696603i	$0.754694\pi$
$192$		0			0
$193$	−1.95133e10			−1.01233			−0.506165	−	0.862437i	$-0.668937\pi$
$193$	−1.95133e10			−1.01233			−0.506165	+	0.862437i	$0.668937\pi$
$194$	5.54243e9			0.280926
$195$		0			0
$196$	−1.38147e10	−	9.43313e9i	−0.668633	−	0.456566i
$197$			3.83001e10i			1.81177i	0.423527	+	0.905884i	$0.360792\pi$
$197$			3.83001e10i			1.81177i	−0.423527	+	0.905884i	$0.639208\pi$
$198$		0			0
$199$			2.48464e9i			0.112312i	0.998422	+	0.0561559i	$0.0178844\pi$
$199$			2.48464e9i			0.112312i	−0.998422	+	0.0561559i	$0.982116\pi$
$200$		−	3.57307e9i		−	0.157909i
$201$		0			0
$202$		−	5.99183e9i		−	0.253208i
$203$	−6.35451e9	−	1.96259e9i	−0.262633	−	0.0811143i
$204$		0			0
$205$	2.86485e9			0.113295
$206$	1.74616e10			0.675587
$207$		0			0
$208$			2.35351e10i			0.871830i
$209$	−6.29194e9			−0.228100
$210$		0			0
$211$	−6.96172e9			−0.241794			−0.120897	−	0.992665i	$-0.538577\pi$
$211$	−6.96172e9			−0.241794			−0.120897	+	0.992665i	$0.538577\pi$
$212$			2.99734e10i			1.01912i
$213$		0			0
$214$	−2.51717e10			−0.820447
$215$	−9.12450e9			−0.291230
$216$		0			0
$217$	1.11950e10	+	3.45757e9i	0.342732	+	0.105853i
$218$		−	1.72675e10i		−	0.517814i
$219$		0			0
$220$		−	2.35560e9i		−	0.0677954i
$221$			5.96219e10i			1.68128i
$222$		0			0
$223$			1.87379e10i			0.507399i	0.967283	+	0.253700i	$0.0816474\pi$
$223$			1.87379e10i			0.507399i	−0.967283	+	0.253700i	$0.918353\pi$
$224$	3.57323e10	+	1.10359e10i	0.948298	+	0.292882i
$225$		0			0
$226$	−5.39573e9			−0.137582
$227$	9.34142e9			0.233505			0.116753	−	0.993161i	$-0.462752\pi$
$227$	9.34142e9			0.233505			0.116753	+	0.993161i	$0.462752\pi$
$228$		0			0
$229$			2.26203e10i			0.543548i	0.962361	+	0.271774i	$0.0876103\pi$
$229$			2.26203e10i			0.543548i	−0.962361	+	0.271774i	$0.912390\pi$
$230$	6.48365e9			0.152772
$231$		0			0
$232$	9.57649e9			0.217025
$233$			2.06251e10i			0.458453i	0.973373	+	0.229227i	$0.0736197\pi$
$233$			2.06251e10i			0.458453i	−0.973373	+	0.229227i	$0.926380\pi$
$234$		0			0
$235$	−6.50990e9			−0.139242
$236$	1.10335e9			0.0231531
$237$		0			0
$238$	1.85103e10	+	5.71689e9i	0.373952	+	0.115495i
$239$			2.62097e10i			0.519604i	0.965662	+	0.259802i	$0.0836572\pi$
$239$			2.62097e10i			0.519604i	−0.965662	+	0.259802i	$0.916343\pi$
$240$		0			0
$241$			4.08033e10i			0.779145i	0.920996	+	0.389572i	$0.127377\pi$
$241$			4.08033e10i			0.779145i	−0.920996	+	0.389572i	$0.872623\pi$
$242$		−	2.24623e10i		−	0.421003i
$243$		0			0
$244$		−	3.49863e10i		−	0.631893i
$245$	2.08284e10	+	1.42224e10i	0.369325	+	0.252188i
$246$		0			0
$247$	−1.33566e11			−2.28328
$248$	−1.68713e10			−0.283214
$249$		0			0
$250$			2.41023e9i			0.0390237i
$251$	5.39495e9			0.0857938			0.0428969	−	0.999080i	$-0.486341\pi$
$251$	5.39495e9			0.0857938			0.0428969	+	0.999080i	$0.486341\pi$
$252$		0			0
$253$	9.55386e9			0.146601
$254$			3.28481e10i			0.495175i
$255$		0			0
$256$	−2.79690e10			−0.407002
$257$	−9.54415e10			−1.36470			−0.682352	−	0.731024i	$-0.739043\pi$
$257$	−9.54415e10			−1.36470			−0.682352	+	0.731024i	$0.739043\pi$
$258$		0			0
$259$	2.27394e10	−	7.36260e10i	0.314000	−	1.01668i
$260$		−	5.00049e10i		−	0.678629i
$261$		0			0
$262$			1.74345e10i			0.228589i
$263$			1.25967e11i			1.62351i	0.583996	+	0.811757i	$0.301489\pi$
$263$			1.25967e11i			1.62351i	−0.583996	+	0.811757i	$0.698511\pi$
$264$		0			0
$265$		−	4.51910e10i		−	0.562918i
$266$	−1.28071e10	+	4.14669e10i	−0.156849	+	0.507849i
$267$		0			0
$268$	−1.09385e11			−1.29524
$269$	1.12140e11			1.30579			0.652895	−	0.757448i	$-0.273554\pi$
$269$	1.12140e11			1.30579			0.652895	+	0.757448i	$0.273554\pi$
$270$		0			0
$271$		−	4.67479e10i		−	0.526502i	−0.964727	−	0.263251i	$-0.915205\pi$
$271$		−	4.67479e10i		−	0.526502i	0.964727	−	0.263251i	$-0.0847948\pi$
$272$	3.76690e10			0.417277
$273$		0			0
$274$	2.53368e10			0.271565
$275$			3.55155e9i			0.0374474i
$276$		0			0
$277$	−5.38407e10			−0.549480			−0.274740	−	0.961519i	$-0.588592\pi$
$277$	−5.38407e10			−0.549480			−0.274740	+	0.961519i	$0.588592\pi$
$278$	−5.25114e10			−0.527293
$279$		0			0
$280$	−3.46992e10	−	1.07168e10i	−0.337371	−	0.104197i
$281$		−	1.59057e11i		−	1.52186i	−0.648835	−	0.760929i	$-0.724744\pi$
$281$		−	1.59057e11i		−	1.52186i	0.648835	−	0.760929i	$-0.275256\pi$
$282$		0			0
$283$			4.11793e10i			0.381628i	0.981626	+	0.190814i	$0.0611127\pi$
$283$			4.11793e10i			0.381628i	−0.981626	+	0.190814i	$0.938887\pi$
$284$			1.23095e11i			1.12281i
$285$		0			0
$286$			1.73239e10i			0.153108i
$287$	8.59264e9	−	2.78214e10i	0.0747581	−	0.242053i
$288$		0			0
$289$	−2.31605e10			−0.195303
$290$	−6.45987e9			−0.0536331
$291$		0			0
$292$		−	1.14534e11i		−	0.921960i
$293$	6.07795e10			0.481784			0.240892	−	0.970552i	$-0.422560\pi$
$293$	6.07795e10			0.481784			0.240892	+	0.970552i	$0.422560\pi$
$294$		0			0
$295$	−1.66352e9			−0.0127888
$296$			1.10957e11i			0.840124i
$297$		0			0
$298$	−8.32919e9			−0.0611828
$299$	2.02810e11			1.46747
$300$		0			0
$301$	−2.73674e10	+	8.86107e10i	−0.192169	+	0.622210i
$302$			1.58854e10i			0.109893i
$303$		0			0
$304$			8.43867e10i			0.566687i
$305$			5.27490e10i			0.349031i
$306$		0			0
$307$			1.87074e11i			1.20196i	0.799263	+	0.600981i	$0.205223\pi$
$307$			1.87074e11i			1.20196i	−0.799263	+	0.600981i	$0.794777\pi$
$308$	−2.28760e10	−	7.06524e9i	−0.144844	−	0.0447351i
$309$		0			0
$310$	1.13806e10			0.0699902
$311$	−1.10552e11			−0.670109			−0.335055	−	0.942199i	$-0.608755\pi$
$311$	−1.10552e11			−0.670109			−0.335055	+	0.942199i	$0.608755\pi$
$312$		0			0
$313$			2.81748e11i			1.65925i	0.558323	+	0.829623i	$0.311445\pi$
$313$			2.81748e11i			1.65925i	−0.558323	+	0.829623i	$0.688555\pi$
$314$	7.13571e10			0.414242
$315$		0			0
$316$	1.18602e10			0.0669112
$317$		−	1.50058e11i		−	0.834629i	−0.908762	−	0.417314i	$-0.862971\pi$
$317$		−	1.50058e11i		−	0.834629i	0.908762	−	0.417314i	$-0.137029\pi$
$318$		0			0
$319$	−9.51882e9			−0.0514665
$320$	−2.69620e9			−0.0143740
$321$		0			0
$322$	1.94466e10	−	6.29646e10i	0.100807	−	0.326396i
$323$			2.13778e11i			1.09283i
$324$		0			0
$325$			7.53927e10i			0.374847i
$326$			6.14161e10i			0.301164i
$327$		0			0
$328$			4.19280e10i			0.200019i
$329$	−1.95254e10	+	6.32196e10i	−0.0918793	+	0.297488i
$330$		0			0
$331$	−2.19661e11			−1.00583			−0.502917	−	0.864335i	$-0.667740\pi$
$331$	−2.19661e11			−1.00583			−0.502917	+	0.864335i	$0.667740\pi$
$332$	−1.58920e11			−0.717887
$333$		0			0
$334$		−	1.58298e11i		−	0.696012i
$335$	1.64920e11			0.715439
$336$		0			0
$337$	−3.66810e11			−1.54920			−0.774599	−	0.632453i	$-0.782048\pi$
$337$	−3.66810e11			−1.54920			−0.774599	+	0.632453i	$0.782048\pi$
$338$			2.63063e11i			1.09631i
$339$		0			0
$340$	−8.00351e10			−0.324807
$341$	1.67697e10			0.0671629
$342$		0			0
$343$	2.00589e11	−	1.59613e11i	0.782498	−	0.622653i
$344$		−	1.33540e11i		−	0.514159i
$345$		0			0
$346$			2.05488e11i			0.770805i
$347$			1.77869e11i			0.658594i	0.944226	+	0.329297i	$0.106812\pi$
$347$			1.77869e11i			0.658594i	−0.944226	+	0.329297i	$0.893188\pi$
$348$		0			0
$349$			2.07302e11i			0.747977i	0.927433	+	0.373988i	$0.122010\pi$
$349$			2.07302e11i			0.747977i	−0.927433	+	0.373988i	$0.877990\pi$
$350$	2.34065e10	+	7.22909e9i	0.0833738	+	0.0257500i
$351$		0			0
$352$	5.35257e10			0.185832
$353$	4.14378e11			1.42040			0.710200	−	0.704000i	$-0.248604\pi$
$353$	4.14378e11			1.42040			0.710200	+	0.704000i	$0.248604\pi$
$354$		0			0
$355$		−	1.85590e11i		−	0.620194i
$356$	−3.08291e11			−1.01727
$357$		0			0
$358$	−1.38160e11			−0.444537
$359$		−	3.50290e11i		−	1.11302i	−0.830841	−	0.556510i	$-0.812140\pi$
$359$		−	3.50290e11i		−	1.11302i	0.830841	−	0.556510i	$-0.187860\pi$
$360$		0			0
$361$	−1.56220e11			−0.484123
$362$	5.20033e10			0.159163
$363$		0			0
$364$	−4.85613e11	−	1.49981e11i	−1.44989	−	0.447797i
$365$			1.72684e11i			0.509252i
$366$		0			0
$367$		−	1.48569e11i		−	0.427495i	−0.976889	−	0.213747i	$-0.931433\pi$
$367$		−	1.48569e11i		−	0.427495i	0.976889	−	0.213747i	$-0.0685669\pi$
$368$		−	1.28135e11i		−	0.364211i
$369$		0			0
$370$		−	7.48468e10i		−	0.207618i
$371$	−4.38863e11	−	1.35543e11i	−1.20267	−	0.371444i
$372$		0			0
$373$	3.24299e11			0.867474			0.433737	−	0.901040i	$-0.357195\pi$
$373$	3.24299e11			0.867474			0.433737	+	0.901040i	$0.357195\pi$
$374$	2.77277e10			0.0732810
$375$		0			0
$376$		−	9.52744e10i		−	0.245828i
$377$	−2.02066e11			−0.515178
$378$		0			0
$379$	−6.60230e11			−1.64369			−0.821843	−	0.569714i	$-0.807054\pi$
$379$	−6.60230e11			−1.64369			−0.821843	+	0.569714i	$0.807054\pi$
$380$		−	1.79296e11i		−	0.441107i
$381$		0			0
$382$	2.52979e11			0.607854
$383$	−7.42061e11			−1.76216			−0.881079	−	0.472969i	$-0.843182\pi$
$383$	−7.42061e11			−1.76216			−0.881079	+	0.472969i	$0.843182\pi$
$384$		0			0
$385$	3.44902e10	+	1.06523e10i	0.0800059	+	0.0247098i
$386$			1.92641e11i			0.441678i
$387$		0			0
$388$			2.32726e11i			0.521317i
$389$		−	4.88598e11i		−	1.08188i	−0.841062	−	0.540939i	$-0.818069\pi$
$389$		−	4.88598e11i		−	1.08188i	0.841062	−	0.540939i	$-0.181931\pi$
$390$		0			0
$391$		−	3.24606e11i		−	0.702363i
$392$	−2.08149e11	+	3.04830e11i	−0.445232	+	0.652035i
$393$		0			0
$394$	3.78111e11			0.790472
$395$	−1.78816e10			−0.0369590
$396$		0			0
$397$		−	4.92713e11i		−	0.995491i	−0.867323	−	0.497745i	$-0.834161\pi$
$397$		−	4.92713e11i		−	0.995491i	0.867323	−	0.497745i	$-0.165839\pi$
$398$	2.45292e10			0.0490015
$399$		0			0
$400$	4.76330e10			0.0930332
$401$		−	4.50235e11i		−	0.869541i	−0.900541	−	0.434770i	$-0.856830\pi$
$401$		−	4.50235e11i		−	0.869541i	0.900541	−	0.434770i	$-0.143170\pi$
$402$		0			0
$403$	3.55988e11			0.672299
$404$	2.51596e11			0.469881
$405$		0			0
$406$	−1.93753e10	+	6.27337e10i	−0.0353900	+	0.114587i
$407$		−	1.10289e11i		−	0.199231i
$408$		0			0
$409$			2.52843e11i			0.446783i	0.974729	+	0.223392i	$0.0717129\pi$
$409$			2.52843e11i			0.446783i	−0.974729	+	0.223392i	$0.928287\pi$
$410$		−	2.82827e10i		−	0.0494303i
$411$		0			0
$412$			7.33211e11i			1.25369i
$413$	−4.98946e9	+	1.61550e10i	−0.00843876	+	0.0273232i
$414$		0			0
$415$	2.39604e11			0.396531
$416$	1.13625e12			1.86017
$417$		0			0
$418$			6.21160e10i			0.0995199i
$419$	−2.27387e11			−0.360414			−0.180207	−	0.983629i	$-0.557677\pi$
$419$	−2.27387e11			−0.360414			−0.180207	+	0.983629i	$0.557677\pi$
$420$		0			0
$421$	−9.22990e10			−0.143195			−0.0715975	−	0.997434i	$-0.522810\pi$
$421$	−9.22990e10			−0.143195			−0.0715975	+	0.997434i	$0.522810\pi$
$422$			6.87283e10i			0.105494i
$423$		0			0
$424$	6.61384e11			0.993819
$425$	1.20669e11			0.179410
$426$		0			0
$427$	5.12261e11	+	1.58212e11i	0.745702	+	0.230310i
$428$		−	1.05696e12i		−	1.52251i
$429$		0			0
$430$			9.00799e10i			0.127063i
$431$		−	9.57397e11i		−	1.33642i	−0.743971	−	0.668212i	$-0.767060\pi$
$431$		−	9.57397e11i		−	1.33642i	0.743971	−	0.668212i	$-0.232940\pi$
$432$		0			0
$433$		−	1.76256e11i		−	0.240962i	−0.992716	−	0.120481i	$-0.961556\pi$
$433$		−	1.76256e11i		−	0.240962i	0.992716	−	0.120481i	$-0.0384437\pi$
$434$	3.41342e10	−	1.10520e11i	0.0461834	−	0.149533i
$435$		0			0
$436$	7.25059e11			0.960913
$437$	7.27188e11			0.953850
$438$		0			0
$439$		−	5.96656e11i		−	0.766714i	−0.923600	−	0.383357i	$-0.874768\pi$
$439$		−	5.96656e11i		−	0.766714i	0.923600	−	0.383357i	$-0.125232\pi$
$440$	−5.19781e10			−0.0661124
$441$		0			0
$442$	5.88605e11			0.733540
$443$		−	6.18899e11i		−	0.763490i	−0.924268	−	0.381745i	$-0.875323\pi$
$443$		−	6.18899e11i		−	0.763490i	0.924268	−	0.381745i	$-0.124677\pi$
$444$		0			0
$445$	4.64812e11			0.561897
$446$	1.84987e11			0.221378
$447$		0			0
$448$	−8.08682e9	+	2.61836e10i	−0.00948476	+	0.0307099i
$449$		−	2.65606e11i		−	0.308410i	−0.988039	−	0.154205i	$-0.950718\pi$
$449$		−	2.65606e11i		−	0.308410i	0.988039	−	0.154205i	$-0.0492817\pi$
$450$		0			0
$451$		−	4.16755e10i		−	0.0474336i
$452$		−	2.26566e11i		−	0.255313i
$453$		0			0
$454$		−	9.22214e10i		−	0.101878i
$455$	7.32160e11	+	2.26128e11i	0.800857	+	0.247345i
$456$		0			0
$457$	6.13817e11			0.658288			0.329144	−	0.944280i	$-0.393240\pi$
$457$	6.13817e11			0.658288			0.329144	+	0.944280i	$0.393240\pi$
$458$	2.23314e11			0.237149
$459$		0			0
$460$			2.72248e11i			0.283501i
$461$	−1.66916e12			−1.72125			−0.860624	−	0.509241i	$-0.829926\pi$
$461$	−1.66916e12			−1.72125			−0.860624	+	0.509241i	$0.829926\pi$
$462$		0			0
$463$	6.65897e11			0.673430			0.336715	−	0.941607i	$-0.390684\pi$
$463$	6.65897e11			0.673430			0.336715	+	0.941607i	$0.390684\pi$
$464$			1.27665e11i			0.127862i
$465$		0			0
$466$	2.03618e11			0.200023
$467$	1.25130e11			0.121740			0.0608701	−	0.998146i	$-0.480612\pi$
$467$	1.25130e11			0.121740			0.0608701	+	0.998146i	$0.480612\pi$
$468$		0			0
$469$	4.94651e11	−	1.60159e12i	0.472086	−	1.52853i
$470$			6.42678e10i			0.0607509i
$471$		0			0
$472$		−	2.43462e10i		−	0.0225783i
$473$			1.32736e11i			0.121930i
$474$		0			0
$475$			2.70325e11i			0.243649i
$476$	−2.40052e11	+	7.77244e11i	−0.214325	+	0.693947i
$477$		0			0
$478$	2.58751e11			0.226702
$479$	−2.13896e12			−1.85649			−0.928245	−	0.371969i	$-0.878683\pi$
$479$	−2.13896e12			−1.85649			−0.928245	+	0.371969i	$0.878683\pi$
$480$		0			0
$481$		−	2.34122e12i		−	1.99430i
$482$	4.02823e11			0.339940
$483$		0			0
$484$	9.43190e11			0.781259
$485$		−	3.50882e11i		−	0.287954i
$486$		0			0
$487$	3.36595e11			0.271161			0.135581	−	0.990766i	$-0.456710\pi$
$487$	3.36595e11			0.271161			0.135581	+	0.990766i	$0.456710\pi$
$488$	−7.71997e11			−0.616207
$489$		0			0
$490$	1.40408e11	−	2.05625e11i	0.110029	−	0.161136i
$491$			4.00595e9i			0.00311056i	0.999999	+	0.00155528i	$0.000495062\pi$
$491$			4.00595e9i			0.00311056i	−0.999999	+	0.00155528i	$0.999505\pi$
$492$		0			0
$493$			3.23416e11i			0.246576i
$494$			1.31860e12i			0.996191i
$495$		0			0
$496$		−	2.24913e11i		−	0.166858i
$497$	−1.80232e12	−	5.56647e11i	−1.32504	−	0.409238i
$498$		0			0
$499$	1.22179e12			0.882155			0.441078	−	0.897469i	$-0.354596\pi$
$499$	1.22179e12			0.882155			0.441078	+	0.897469i	$0.354596\pi$
$500$	−1.01205e11			−0.0724167
$501$		0			0
$502$		−	5.32606e10i		−	0.0374317i
$503$	1.49348e12			1.04027			0.520133	−	0.854085i	$-0.325882\pi$
$503$	1.49348e12			1.04027			0.520133	+	0.854085i	$0.325882\pi$
$504$		0			0
$505$	−3.79333e11			−0.259543
$506$		−	9.43187e10i		−	0.0639618i
$507$		0			0
$508$	−1.37929e12			−0.918900
$509$	7.49531e11			0.494948			0.247474	−	0.968895i	$-0.420399\pi$
$509$	7.49531e11			0.494948			0.247474	+	0.968895i	$0.420399\pi$
$510$		0			0
$511$	1.67698e12	+	5.17936e11i	1.08801	+	0.336033i
$512$		−	1.28896e12i		−	0.828945i
$513$		0			0
$514$			9.42228e11i			0.595418i
$515$		−	1.10547e12i		−	0.692489i
$516$		0			0
$517$			9.47006e10i			0.0582969i
$518$	−7.26859e11	−	2.24490e11i	−0.443574	−	0.136998i
$519$		0			0
$520$	−1.10339e12			−0.661783
$521$	2.62961e12			1.56358			0.781792	−	0.623540i	$-0.214306\pi$
$521$	2.62961e12			1.56358			0.781792	+	0.623540i	$0.214306\pi$
$522$		0			0
$523$			2.02241e12i			1.18199i	0.806677	+	0.590993i	$0.201264\pi$
$523$			2.02241e12i			1.18199i	−0.806677	+	0.590993i	$0.798736\pi$
$524$	−7.32075e11			−0.424194
$525$		0			0
$526$	1.24359e12			0.708337
$527$		−	5.69774e11i		−	0.321777i
$528$		0			0
$529$	6.96970e11			0.386957
$530$	−4.46140e11			−0.245601
$531$		0			0
$532$	−1.74119e12	−	5.37768e11i	−0.942420	−	0.291066i
$533$		−	8.84690e11i		−	0.474809i
$534$		0			0
$535$			1.59358e12i			0.840972i
$536$			2.41366e12i			1.26309i
$537$		0			0
$538$		−	1.10708e12i		−	0.569715i
$539$	2.06895e11	−	3.02995e11i	0.105585	−	0.154627i
$540$		0			0
$541$	−2.03050e12			−1.01910			−0.509548	−	0.860442i	$-0.670187\pi$
$541$	−2.03050e12			−1.01910			−0.509548	+	0.860442i	$0.670187\pi$
$542$	−4.61510e11			−0.229712
$543$		0			0
$544$		−	1.81861e12i		−	0.890318i
$545$	−1.09317e12			−0.530768
$546$		0			0
$547$	2.87309e12			1.37217			0.686083	−	0.727523i	$-0.259329\pi$
$547$	2.87309e12			1.37217			0.686083	+	0.727523i	$0.259329\pi$
$548$			1.06389e12i			0.503945i
$549$		0			0
$550$	3.50620e10			0.0163382
$551$	−7.24521e11			−0.334864
$552$		0			0
$553$	−5.36329e10	+	1.73654e11i	−0.0243876	+	0.0789625i
$554$			5.31532e11i			0.239737i
$555$		0			0
$556$		−	2.20495e12i		−	0.978503i
$557$		−	6.65951e11i		−	0.293153i	−0.989199	−	0.146576i	$-0.953175\pi$
$557$		−	6.65951e11i		−	0.293153i	0.989199	−	0.146576i	$-0.0468254\pi$
$558$		0			0
$559$			2.81772e12i			1.22052i
$560$	1.42867e11	−	4.62578e11i	0.0613884	−	0.198765i
$561$		0			0
$562$	−1.57026e12			−0.663985
$563$	4.17795e12			1.75257			0.876286	−	0.481791i	$-0.160014\pi$
$563$	4.17795e12			1.75257			0.876286	+	0.481791i	$0.160014\pi$
$564$		0			0
$565$			3.41595e11i			0.141024i
$566$	4.06535e11			0.166504
$567$		0			0
$568$	2.71617e12			1.09494
$569$			2.53247e12i			1.01284i	0.862288	+	0.506419i	$0.169031\pi$
$569$			2.53247e12i			1.01284i	−0.862288	+	0.506419i	$0.830969\pi$
$570$		0			0
$571$	2.78435e12			1.09613			0.548065	−	0.836436i	$-0.315365\pi$
$571$	2.78435e12			1.09613			0.548065	+	0.836436i	$0.315365\pi$
$572$	−7.27430e11			−0.284125
$573$		0			0
$574$	−2.74662e11	−	8.48292e10i	−0.105607	−	0.0326169i
$575$		−	4.10469e11i		−	0.156594i
$576$		0			0
$577$		−	1.57510e12i		−	0.591584i	−0.955252	−	0.295792i	$-0.904416\pi$
$577$		−	1.57510e12i		−	0.591584i	0.955252	−	0.295792i	$-0.0955836\pi$
$578$			2.28648e11i			0.0852102i
$579$		0			0
$580$		−	2.71249e11i		−	0.0995274i
$581$	7.18652e11	−	2.32686e12i	0.261653	−	0.847185i
$582$		0			0
$583$	−6.57401e11			−0.235679
$584$	−2.52728e12			−0.899073
$585$		0			0
$586$		−	6.00034e11i		−	0.210202i
$587$	−5.32861e11			−0.185243			−0.0926217	−	0.995701i	$-0.529525\pi$
$587$	−5.32861e11			−0.185243			−0.0926217	+	0.995701i	$0.529525\pi$
$588$		0			0
$589$	1.27642e12			0.436992
$590$			1.64228e10i			0.00557974i
$591$		0			0
$592$	−1.47918e12			−0.494965
$593$	−4.46126e12			−1.48153			−0.740767	−	0.671762i	$-0.765538\pi$
$593$	−4.46126e12			−1.48153			−0.740767	+	0.671762i	$0.765538\pi$
$594$		0			0
$595$	3.61927e11	−	1.17185e12i	0.118384	−	0.383307i
$596$		−	3.49742e11i		−	0.113538i
$597$		0			0
$598$		−	2.00220e12i		−	0.640255i
$599$		−	2.53748e12i		−	0.805345i	−0.915344	−	0.402672i	$-0.868081\pi$
$599$		−	2.53748e12i		−	0.805345i	0.915344	−	0.402672i	$-0.131919\pi$
$600$		0			0
$601$		−	4.08771e12i		−	1.27804i	−0.769189	−	0.639021i	$-0.779340\pi$
$601$		−	4.08771e12i		−	1.27804i	0.769189	−	0.639021i	$-0.220660\pi$
$602$	8.74792e11	+	2.70180e11i	0.271469	+	0.0838433i
$603$		0			0
$604$	−6.67029e11			−0.203929
$605$	−1.42205e12			−0.431535
$606$		0			0
$607$			1.95721e12i			0.585180i	0.956238	+	0.292590i	$0.0945170\pi$
$607$			1.95721e12i			0.585180i	−0.956238	+	0.292590i	$0.905483\pi$
$608$	4.07408e12			1.20910
$609$		0			0
$610$	5.20754e11			0.152282
$611$			2.01031e12i			0.583550i
$612$		0			0
$613$	3.05561e12			0.874030			0.437015	−	0.899454i	$-0.356036\pi$
$613$	3.05561e12			0.874030			0.437015	+	0.899454i	$0.356036\pi$
$614$	1.84685e12			0.524414
$615$		0			0
$616$	−1.55900e11	+	5.04774e11i	−0.0436246	+	0.141249i
$617$			6.50182e12i			1.80614i	0.429492	+	0.903071i	$0.358693\pi$
$617$			6.50182e12i			1.80614i	−0.429492	+	0.903071i	$0.641307\pi$
$618$		0			0
$619$			6.96244e12i			1.90614i	0.302757	+	0.953068i	$0.402093\pi$
$619$			6.96244e12i			1.90614i	−0.302757	+	0.953068i	$0.597907\pi$
$620$			4.77870e11i			0.129882i
$621$		0			0
$622$			1.09141e12i			0.292368i
$623$	1.39412e12	−	4.51392e12i	0.370771	−	1.20049i
$624$		0			0
$625$	1.52588e11			0.0400000
$626$	2.78150e12			0.723927
$627$		0			0
$628$			2.99628e12i			0.768712i
$629$	−3.74723e12			−0.954515
$630$		0			0
$631$	−4.36566e12			−1.09627			−0.548135	−	0.836390i	$-0.684662\pi$
$631$	−4.36566e12			−1.09627			−0.548135	+	0.836390i	$0.684662\pi$
$632$		−	2.61703e11i		−	0.0652502i
$633$		0			0
$634$	−1.48142e12			−0.364147
$635$	2.07956e12			0.507562
$636$		0			0
$637$	4.39198e12	−	6.43199e12i	1.05690	−	1.54781i
$638$			9.39727e10i			0.0224548i
$639$		0			0
$640$			1.91050e12i			0.450129i
$641$		−	2.77432e12i		−	0.649075i	−0.945873	−	0.324538i	$-0.894791\pi$
$641$		−	2.77432e12i		−	0.649075i	0.945873	−	0.324538i	$-0.105209\pi$
$642$		0			0
$643$			4.14311e12i			0.955822i	0.878408	+	0.477911i	$0.158606\pi$
$643$			4.14311e12i			0.955822i	−0.878408	+	0.477911i	$0.841394\pi$
$644$	2.64388e12	+	8.16562e11i	0.605697	+	0.187069i
$645$		0			0
$646$	2.11048e12			0.476799
$647$	−8.28816e11			−0.185947			−0.0929735	−	0.995669i	$-0.529637\pi$
$647$	−8.28816e11			−0.185947			−0.0929735	+	0.995669i	$0.529637\pi$
$648$		0			0
$649$			2.41996e10i			0.00535435i
$650$	7.44300e11			0.163545
$651$		0			0
$652$	−2.57885e12			−0.558872
$653$		−	5.49378e12i		−	1.18239i	−0.806527	−	0.591197i	$-0.798656\pi$
$653$		−	5.49378e12i		−	1.18239i	0.806527	−	0.591197i	$-0.201344\pi$
$654$		0			0
$655$	1.10375e12			0.234307
$656$	−5.58946e11			−0.117843
$657$		0			0
$658$	6.24123e11	+	1.92760e11i	0.129794	+	0.0400868i
$659$		−	5.86097e12i		−	1.21056i	−0.796014	−	0.605278i	$-0.793062\pi$
$659$		−	5.86097e12i		−	1.21056i	0.796014	−	0.605278i	$-0.206938\pi$
$660$		0			0
$661$		−	4.93386e12i		−	1.00526i	−0.864500	−	0.502632i	$-0.832365\pi$
$661$		−	4.93386e12i		−	1.00526i	0.864500	−	0.502632i	$-0.167635\pi$
$662$			2.16856e12i			0.438844i
$663$		0			0
$664$			3.50668e12i			0.700067i
$665$	2.62521e12	+	8.10794e11i	0.520554	+	0.160773i
$666$		0			0
$667$	1.10013e12			0.215218
$668$	6.64692e12			1.29160
$669$		0			0
$670$		−	1.62815e12i		−	0.312145i
$671$	7.67348e11			0.146130
$672$		0			0
$673$	−8.89144e12			−1.67072			−0.835361	−	0.549701i	$-0.814742\pi$
$673$	−8.89144e12			−1.67072			−0.835361	+	0.549701i	$0.814742\pi$
$674$			3.62127e12i			0.675913i
$675$		0			0
$676$	−1.10460e13			−2.03444
$677$	−2.23287e12			−0.408521			−0.204260	−	0.978917i	$-0.565479\pi$
$677$	−2.23287e12			−0.408521			−0.204260	+	0.978917i	$0.565479\pi$
$678$		0			0
$679$	−3.40752e12	−	1.05241e12i	−0.615211	−	0.190008i
$680$			1.76603e12i			0.316744i
$681$		0			0
$682$		−	1.65555e11i		−	0.0293031i
$683$		−	3.70327e12i		−	0.651166i	−0.945513	−	0.325583i	$-0.894439\pi$
$683$		−	3.70327e12i		−	0.651166i	0.945513	−	0.325583i	$-0.105561\pi$
$684$		0			0
$685$		−	1.60403e12i		−	0.278359i
$686$	−1.57575e12	−	1.98028e12i	−0.271662	−	0.341403i
$687$		0			0
$688$	1.78023e12			0.302921
$689$	−1.39554e13			−2.35914
$690$		0			0
$691$			2.70215e12i			0.450877i	0.974257	+	0.225438i	$0.0723814\pi$
$691$			2.70215e12i			0.450877i	−0.974257	+	0.225438i	$0.927619\pi$
$692$	−8.62843e12			−1.43039
$693$		0			0
$694$	1.75598e12			0.287344
$695$			3.32441e12i			0.540485i
$696$		0			0
$697$	−1.41598e12			−0.227254
$698$	2.04655e12			0.326341
$699$		0			0
$700$	−3.03549e11	+	9.82836e11i	−0.0477845	+	0.154718i
$701$			4.58407e12i			0.717002i	0.933529	+	0.358501i	$0.116712\pi$
$701$			4.58407e12i			0.717002i	−0.933529	+	0.358501i	$0.883288\pi$
$702$		0			0
$703$		−	8.39460e12i		−	1.29629i
$704$			3.92221e10i			0.00601803i
$705$		0			0
$706$		−	4.09087e12i		−	0.619719i
$707$	−1.13775e12	+	3.68381e12i	−0.171261	+	0.554511i
$708$		0			0
$709$	−7.74026e12			−1.15040			−0.575198	−	0.818014i	$-0.695075\pi$
$709$	−7.74026e12			−1.15040			−0.575198	+	0.818014i	$0.695075\pi$
$710$	−1.83220e12			−0.270590
$711$		0			0
$712$			6.80266e12i			0.992016i
$713$	−1.93815e12			−0.280856
$714$		0			0
$715$	1.09675e12			0.156939
$716$		−	5.80131e12i		−	0.824931i
$717$		0			0
$718$	−3.45818e12			−0.485610
$719$	−2.51152e12			−0.350474			−0.175237	−	0.984526i	$-0.556069\pi$
$719$	−2.51152e12			−0.350474			−0.175237	+	0.984526i	$0.556069\pi$
$720$		0			0
$721$	−1.07355e13	−	3.31566e12i	−1.47950	−	0.456942i
$722$			1.54226e12i			0.211222i
$723$		0			0
$724$			2.18361e12i			0.295360i
$725$			4.08964e11i			0.0549748i
$726$		0			0
$727$			7.04621e11i			0.0935515i	0.998905	+	0.0467757i	$0.0148946\pi$
$727$			7.04621e11i			0.0935515i	−0.998905	+	0.0467757i	$0.985105\pi$
$728$	−3.30945e12	+	1.07154e13i	−0.436681	+	1.41389i
$729$		0			0
$730$	1.70479e12			0.222186
$731$	4.50989e12			0.584167
$732$		0			0
$733$			9.82007e12i			1.25645i	0.778030	+	0.628227i	$0.216219\pi$
$733$			9.82007e12i			1.25645i	−0.778030	+	0.628227i	$0.783781\pi$
$734$	−1.46672e12			−0.186515
$735$		0			0
$736$	−6.18621e12			−0.777095
$737$		−	2.39912e12i		−	0.299536i
$738$		0			0
$739$	1.54546e12			0.190616			0.0953080	−	0.995448i	$-0.469616\pi$
$739$	1.54546e12			0.190616			0.0953080	+	0.995448i	$0.469616\pi$
$740$	3.14281e12			0.385279
$741$		0			0
$742$	−1.33812e12	+	4.33259e12i	−0.162061	+	0.524723i
$743$			9.57090e12i			1.15213i	0.817403	+	0.576067i	$0.195413\pi$
$743$			9.57090e12i			1.15213i	−0.817403	+	0.576067i	$0.804587\pi$
$744$		0			0
$745$			5.27307e11i			0.0627134i
$746$		−	3.20158e12i		−	0.378478i
$747$		0			0
$748$			1.16428e12i			0.135988i
$749$	1.54757e13	+	4.77968e12i	1.79673	+	0.554920i
$750$		0			0
$751$	7.91742e12			0.908247			0.454124	−	0.890939i	$-0.349952\pi$
$751$	7.91742e12			0.908247			0.454124	+	0.890939i	$0.349952\pi$
$752$	1.27011e12			0.144831
$753$		0			0
$754$			1.99486e12i			0.224772i
$755$	1.00568e12			0.112642
$756$		0			0
$757$	−7.97671e12			−0.882861			−0.441430	−	0.897296i	$-0.645529\pi$
$757$	−7.97671e12			−0.882861			−0.441430	+	0.897296i	$0.645529\pi$
$758$			6.51800e12i			0.717138i
$759$		0			0
$760$	−3.95629e12			−0.430157
$761$	2.41105e12			0.260601			0.130300	−	0.991475i	$-0.458406\pi$
$761$	2.41105e12			0.260601			0.130300	+	0.991475i	$0.458406\pi$
$762$		0			0
$763$	−3.27879e12	+	1.06161e13i	−0.350230	+	1.13398i
$764$			1.06226e13i			1.12800i
$765$		0			0
$766$			7.32585e12i			0.768827i
$767$			5.13710e11i			0.0535968i
$768$		0			0
$769$			3.47089e12i			0.357908i	0.983857	+	0.178954i	$0.0572714\pi$
$769$			3.47089e12i			0.357908i	−0.983857	+	0.178954i	$0.942729\pi$
$770$	1.05163e11	−	3.40498e11i	0.0107809	−	0.0349065i
$771$		0			0
$772$	−8.08898e12			−0.819626
$773$	1.79684e13			1.81010			0.905048	−	0.425310i	$-0.139835\pi$
$773$	1.79684e13			1.81010			0.905048	+	0.425310i	$0.139835\pi$
$774$		0			0
$775$		−	7.20487e11i		−	0.0717412i
$776$	5.13527e12			0.508376
$777$		0			0
$778$	−4.82359e12			−0.472022
$779$		−	3.17211e12i		−	0.308624i
$780$		0			0
$781$	−2.69981e12			−0.259659
$782$	−3.20462e12			−0.306440
$783$		0			0
$784$	−4.06373e12	−	2.77485e12i	−0.384151	−	0.262312i
$785$		−	4.51750e12i		−	0.424605i
$786$		0			0
$787$		−	2.01598e12i		−	0.187327i	−0.995604	−	0.0936634i	$-0.970142\pi$
$787$		−	2.01598e12i		−	0.187327i	0.995604	−	0.0936634i	$-0.0298578\pi$
$788$			1.58768e13i			1.46689i
$789$		0			0
$790$			1.76533e11i			0.0161251i
$791$	3.31733e12	+	1.02456e12i	0.301297	+	0.0930555i
$792$		0			0
$793$	1.62893e13			1.46276
$794$	−4.86422e12			−0.434331
$795$		0			0
$796$			1.02998e12i			0.0909325i
$797$	−6.10534e12			−0.535979			−0.267989	−	0.963422i	$-0.586359\pi$
$797$	−6.10534e12			−0.535979			−0.267989	+	0.963422i	$0.586359\pi$
$798$		0			0
$799$	3.21759e12			0.279300
$800$		−	2.29966e12i		−	0.198499i
$801$		0			0
$802$	−4.44486e12			−0.379379
$803$	2.51206e12			0.213211
$804$		0			0
$805$	−3.98619e12	−	1.23113e12i	−0.334562	−	0.103329i
$806$		−	3.51442e12i		−	0.293323i
$807$		0			0
$808$		−	5.55165e12i		−	0.458217i
$809$			2.90954e12i			0.238812i	0.992846	+	0.119406i	$0.0380990\pi$
$809$			2.90954e12i			0.238812i	−0.992846	+	0.119406i	$0.961901\pi$
$810$		0			0
$811$		−	6.95479e12i		−	0.564534i	−0.959336	−	0.282267i	$-0.908914\pi$
$811$		−	6.95479e12i		−	0.564534i	0.959336	−	0.282267i	$-0.0910865\pi$
$812$	−2.63418e12	−	8.13567e11i	−0.212639	−	0.0656736i
$813$		0			0
$814$	−1.08881e12			−0.0869244
$815$	3.88815e12			0.308698
$816$		0			0
$817$			1.01031e13i			0.793334i
$818$	2.49615e12			0.194931
$819$		0			0
$820$	1.18759e12			0.0917283
$821$			6.37371e12i			0.489608i	0.969573	+	0.244804i	$0.0787236\pi$
$821$			6.37371e12i			0.489608i	−0.969573	+	0.244804i	$0.921276\pi$
$822$		0			0
$823$	−1.65168e13			−1.25495			−0.627474	−	0.778638i	$-0.715911\pi$
$823$	−1.65168e13			−1.25495			−0.627474	+	0.778638i	$0.715911\pi$
$824$	1.61788e13			1.22257
$825$		0			0
$826$	1.59487e11	+	4.92575e10i	0.0119211	+	0.00368182i
$827$			5.22858e12i			0.388695i	0.980933	+	0.194348i	$0.0622590\pi$
$827$			5.22858e12i			0.388695i	−0.980933	+	0.194348i	$0.937741\pi$
$828$		0			0
$829$			7.30826e9i			0.000537426i	1.00000		0.000268713i	$8.55340e-5\pi$
$829$			7.30826e9i			0.000537426i	−1.00000		0.000268713i	$0.999914\pi$
$830$		−	2.36544e12i		−	0.173006i
$831$		0			0
$832$			8.32610e11i			0.0602403i
$833$	−1.02947e13	−	7.02956e12i	−0.740816	−	0.505855i
$834$		0			0
$835$	−1.00216e13			−0.713424
$836$	−2.60824e12			−0.184680
$837$		0			0
$838$			2.24483e12i			0.157248i
$839$	−2.43555e13			−1.69694			−0.848472	−	0.529240i	$-0.822477\pi$
$839$	−2.43555e13			−1.69694			−0.848472	+	0.529240i	$0.822477\pi$
$840$		0			0
$841$	1.34110e13			0.924444
$842$			9.11205e11i			0.0624757i
$843$		0			0
$844$	−2.88589e12			−0.195767
$845$	1.66541e13			1.12374
$846$		0			0
$847$	−4.26521e12	+	1.38100e13i	−0.284751	+	0.921971i
$848$			8.81698e12i			0.585516i
$849$		0			0
$850$		−	1.19128e12i		−	0.0782763i
$851$			1.27466e13i			0.833128i
$852$		0			0
$853$		−	2.56288e12i		−	0.165752i	−0.996560	−	0.0828758i	$-0.973590\pi$
$853$		−	2.56288e12i		−	0.165752i	0.996560	−	0.0828758i	$-0.0264105\pi$
$854$	1.56191e12	−	5.05720e12i	0.100484	−	0.325349i
$855$		0			0
$856$	−2.33225e13			−1.48472
$857$	−9.94658e12			−0.629883			−0.314942	−	0.949111i	$-0.601985\pi$
$857$	−9.94658e12			−0.629883			−0.314942	+	0.949111i	$0.601985\pi$
$858$		0			0
$859$		−	8.41384e12i		−	0.527260i	−0.964624	−	0.263630i	$-0.915080\pi$
$859$		−	8.41384e12i		−	0.527260i	0.964624	−	0.263630i	$-0.0849198\pi$
$860$	−3.78245e12			−0.235792
$861$		0			0
$862$	−9.45172e12			−0.583080
$863$		−	5.22459e12i		−	0.320630i	−0.987066	−	0.160315i	$-0.948749\pi$
$863$		−	5.22459e12i		−	0.320630i	0.987066	−	0.160315i	$-0.0512510\pi$
$864$		0			0
$865$	1.30091e13			0.790089
$866$	−1.74005e12			−0.105131
$867$		0			0
$868$	4.64074e12	+	1.43329e12i	0.277491	+	0.0857030i
$869$			2.60127e11i			0.0154738i
$870$		0			0
$871$		−	5.09288e13i		−	2.99835i
$872$		−	1.59989e13i		−	0.937059i
$873$		0			0
$874$		−	7.17903e12i		−	0.416164i
$875$	4.57662e11	−	1.48183e12i	0.0263942	−	0.0854596i
$876$		0			0
$877$	1.09318e13			0.624010			0.312005	−	0.950080i	$-0.398999\pi$
$877$	1.09318e13			0.624010			0.312005	+	0.950080i	$0.398999\pi$
$878$	−5.89037e12			−0.334516
$879$		0			0
$880$		−	6.92925e11i		−	0.0389506i
$881$	7.78847e12			0.435573			0.217786	−	0.975996i	$-0.430116\pi$
$881$	7.78847e12			0.435573			0.217786	+	0.975996i	$0.430116\pi$
$882$		0			0
$883$	2.98937e13			1.65484			0.827420	−	0.561584i	$-0.189808\pi$
$883$	2.98937e13			1.65484			0.827420	+	0.561584i	$0.189808\pi$
$884$			2.47155e13i			1.36124i
$885$		0			0
$886$	−6.10996e12			−0.333109
$887$	−6.46135e12			−0.350483			−0.175241	−	0.984525i	$-0.556071\pi$
$887$	−6.46135e12			−0.350483			−0.175241	+	0.984525i	$0.556071\pi$
$888$		0			0
$889$	6.23729e12	−	2.01952e13i	0.334918	−	1.08440i
$890$		−	4.58877e12i		−	0.245155i
$891$		0			0
$892$			7.76757e12i			0.410812i
$893$			7.20809e12i			0.379305i
$894$		0			0
$895$			8.74667e12i			0.455658i
$896$	1.85534e13	+	5.73023e12i	0.961697	+	0.297020i
$897$		0			0
$898$	−2.62214e12			−0.134559
$899$	1.93104e12			0.0985989
$900$		0			0
$901$			2.23361e13i			1.12914i
$902$	−4.11433e11			−0.0206952
$903$		0			0
$904$	−4.99935e12			−0.248975
$905$		−	3.29224e12i		−	0.163145i
$906$		0			0
$907$	2.86046e13			1.40347			0.701734	−	0.712439i	$-0.252409\pi$
$907$	2.86046e13			1.40347			0.701734	+	0.712439i	$0.252409\pi$
$908$	3.87237e12			0.189056
$909$		0			0
$910$	2.23240e12	−	7.22811e12i	0.107916	−	0.349413i
$911$		−	2.18144e13i		−	1.04933i	−0.851309	−	0.524664i	$-0.824191\pi$
$911$		−	2.18144e13i		−	1.04933i	0.851309	−	0.524664i	$-0.175809\pi$
$912$		0			0
$913$		−	3.48556e12i		−	0.166017i
$914$		−	6.05979e12i		−	0.287210i
$915$		0			0
$916$			9.37695e12i			0.440080i
$917$	3.31052e12	−	1.07189e13i	0.154609	−	0.500595i
$918$		0			0
$919$	−1.86500e13			−0.862500			−0.431250	−	0.902232i	$-0.641927\pi$
$919$	−1.86500e13			−0.862500			−0.431250	+	0.902232i	$0.641927\pi$
$920$	6.00734e12			0.276463
$921$		0			0
$922$			1.64785e13i			0.750978i
$923$	−5.73118e13			−2.59918
$924$		0			0
$925$	−4.73843e12			−0.212812
$926$		−	6.57394e12i		−	0.293817i
$927$		0			0
$928$	6.16352e12			0.272811
$929$	1.90907e13			0.840912			0.420456	−	0.907313i	$-0.361870\pi$
$929$	1.90907e13			0.840912			0.420456	+	0.907313i	$0.361870\pi$
$930$		0			0
$931$	1.57477e13	−	2.30623e13i	0.686980	−	1.00607i
$932$			8.54989e12i			0.371184i
$933$		0			0
$934$		−	1.23532e12i		−	0.0531151i
$935$		−	1.75539e12i		−	0.0751143i
$936$		0			0
$937$			5.62619e12i			0.238444i	0.992868	+	0.119222i	$0.0380400\pi$
$937$			5.62619e12i			0.238444i	−0.992868	+	0.119222i	$0.961960\pi$
$938$	−1.58114e13	−	4.88335e12i	−0.666895	−	0.205971i
$939$		0			0
$940$	−2.69860e12			−0.112736
$941$	3.77421e13			1.56918			0.784590	−	0.620015i	$-0.212874\pi$
$941$	3.77421e13			1.56918			0.784590	+	0.620015i	$0.212874\pi$
$942$		0			0
$943$			4.81662e12i			0.198354i
$944$	3.24562e11			0.0133022
$945$		0			0
$946$	1.31041e12			0.0531981
$947$			3.22230e13i			1.30194i	0.759104	+	0.650970i	$0.225638\pi$
$947$			3.22230e13i			1.30194i	−0.759104	+	0.650970i	$0.774362\pi$
$948$		0			0
$949$	5.33261e13			2.13423
$950$	2.66873e12			0.106304
$951$		0			0
$952$	1.71504e13	+	5.29691e12i	0.676720	+	0.209005i
$953$		−	4.85657e13i		−	1.90727i	−0.300970	−	0.953634i	$-0.597310\pi$
$953$		−	4.85657e13i		−	1.90727i	0.300970	−	0.953634i	$-0.402690\pi$
$954$		0			0
$955$		−	1.60157e13i		−	0.623061i
$956$			1.08649e13i			0.420694i
$957$		0			0
$958$			2.11165e13i			0.809984i
$959$	−1.55772e13	−	4.81102e12i	−0.594711	−	0.183676i
$960$		0			0
$961$	2.30376e13			0.871330
$962$	−2.31133e13			−0.870110
$963$		0			0
$964$			1.69145e13i			0.630830i
$965$	1.21958e13			0.452728
$966$		0			0
$967$	4.61088e13			1.69576			0.847880	−	0.530188i	$-0.177879\pi$
$967$	4.61088e13			1.69576			0.847880	+	0.530188i	$0.177879\pi$
$968$		−	2.08122e13i		−	0.761865i
$969$		0			0
$970$	−3.46402e12			−0.125634
$971$	2.36455e13			0.853615			0.426807	−	0.904343i	$-0.359638\pi$
$971$	2.36455e13			0.853615			0.426807	+	0.904343i	$0.359638\pi$
$972$		0			0
$973$	3.22844e13	+	9.97102e12i	1.15474	+	0.356641i
$974$		−	3.32297e12i		−	0.118307i
$975$		0			0
$976$		−	1.02916e13i		−	0.363043i
$977$			3.41278e13i			1.19835i	0.800619	+	0.599173i	$0.204504\pi$
$977$			3.41278e13i			1.19835i	−0.800619	+	0.599173i	$0.795496\pi$
$978$		0			0
$979$		−	6.76169e12i		−	0.235252i
$980$	8.63416e12	+	5.89570e12i	0.299022	+	0.204182i
$981$		0			0
$982$	3.95480e10			0.00135713
$983$	−1.43069e13			−0.488715			−0.244357	−	0.969685i	$-0.578577\pi$
$983$	−1.43069e13			−0.488715			−0.244357	+	0.969685i	$0.578577\pi$
$984$		0			0
$985$		−	2.39376e13i		−	0.810247i
$986$	3.19286e12			0.107581
$987$		0			0
$988$	−5.53680e13			−1.84864
$989$		−	1.53409e13i		−	0.509878i
$990$		0			0
$991$	−3.24032e12			−0.106723			−0.0533614	−	0.998575i	$-0.516994\pi$
$991$	−3.24032e12			−0.106723			−0.0533614	+	0.998575i	$0.516994\pi$
$992$	−1.08585e13			−0.356014
$993$		0			0
$994$	−5.49539e12	+	1.77931e13i	−0.178550	+	0.578113i
$995$		−	1.55290e12i		−	0.0502274i
$996$		0			0
$997$			2.28403e13i			0.732105i	0.930594	+	0.366053i	$0.119291\pi$
$997$			2.28403e13i			0.732105i	−0.930594	+	0.366053i	$0.880709\pi$
$998$		−	1.20619e13i		−	0.384883i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.b.a.251.20	✓	48
3.2	odd	2		315.10.b.b.251.29	yes	48
7.6	odd	2		315.10.b.b.251.20	yes	48
21.20	even	2	inner	315.10.b.a.251.29	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.10.b.a.251.20	✓	48	1.1	even	1	trivial
315.10.b.a.251.29	yes	48	21.20	even	2	inner
315.10.b.b.251.20	yes	48	7.6	odd	2
315.10.b.b.251.29	yes	48	3.2	odd	2

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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	315.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.87231i q^{2} +414.537 q^{4} -625.000 q^{5} +(-1874.58 + 6069.56i) q^{7} -9147.07i q^{8} +6170.19i q^{10} +9091.98i q^{11} +193005. i q^{13} +(59920.6 + 18506.5i) q^{14} +121940. q^{16} +308913. q^{17} +692032. i q^{19} -259086. q^{20} +89758.8 q^{22} -1.05080e6i q^{23} +390625. q^{25} +1.90541e6 q^{26} +(-777085. + 2.51606e6i) q^{28} +1.04695e6i q^{29} -1.84445e6i q^{31} -5.88713e6i q^{32} -3.04969e6i q^{34} +(1.17161e6 - 3.79347e6i) q^{35} -1.21304e7 q^{37} +6.83195e6 q^{38} +5.71692e6i q^{40} -4.58376e6 q^{41} +1.45992e7 q^{43} +3.76897e6i q^{44} -1.03738e7 q^{46} +1.04158e7 q^{47} +(-3.33255e7 - 2.27558e7i) q^{49} -3.85637e6i q^{50} +8.00079e7i q^{52} +7.23056e7i q^{53} -5.68249e6i q^{55} +(5.55187e7 + 1.71469e7i) q^{56} +1.03358e7 q^{58} +2.66164e6 q^{59} -8.43983e7i q^{61} -1.82090e7 q^{62} +4.31393e6 q^{64} -1.20628e8i q^{65} -2.63873e8 q^{67} +1.28056e8 q^{68} +(-3.74504e7 - 1.15665e7i) q^{70} +2.96944e8i q^{71} -2.76294e8i q^{73} +1.19755e8i q^{74} +2.86873e8i q^{76} +(-5.51843e7 - 1.70437e7i) q^{77} +2.86106e7 q^{79} -7.62128e7 q^{80} +4.52523e7i q^{82} -3.83366e8 q^{83} -1.93071e8 q^{85} -1.44128e8i q^{86} +8.31649e7 q^{88} -7.43699e8 q^{89} +(-1.17146e9 - 3.61804e8i) q^{91} -4.35596e8i q^{92} -1.02828e8i q^{94} -4.32520e8i q^{95} +5.61411e8i q^{97} +(-2.24652e8 + 3.29000e8i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12288 q^{4} - 30000 q^{5} + 1824 q^{7} + 442908 q^{14} + 3258948 q^{16} + 7680000 q^{20} - 2860668 q^{22} + 18750000 q^{25} - 5432976 q^{26} - 3685092 q^{28} - 1140000 q^{35} + 7750344 q^{37} + 17423136 q^{38}+ \cdots - 2546372484 q^{98}+O(q^{100}) \) 48 * q - 12288 * q^4 - 30000 * q^5 + 1824 * q^7 + 442908 * q^14 + 3258948 * q^16 + 7680000 * q^20 - 2860668 * q^22 + 18750000 * q^25 - 5432976 * q^26 - 3685092 * q^28 - 1140000 * q^35 + 7750344 * q^37 + 17423136 * q^38 - 70626600 * q^41 + 16073832 * q^43 - 15083820 * q^46 - 33200856 * q^47 - 39367500 * q^49 - 318486852 * q^56 + 293819940 * q^58 + 342990864 * q^59 + 896801400 * q^62 - 1639789824 * q^64 - 980229936 * q^67 - 438669576 * q^68 - 276817500 * q^70 - 900236484 * q^77 + 325977144 * q^79 - 2036842500 * q^80 + 1415370096 * q^83 + 3319557300 * q^88 - 1132967016 * q^89 - 1162515060 * q^91 - 2546372484 * q^98

Introduction

L-functions

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