LMFDB - Embedded newform 315.10.b.a.251.12

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			251.12
Character	$\chi$	$=$	315.251
Dual form			315.10.b.a.251.37

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-27.0056i q^{2} -217.303 q^{4} -625.000 q^{5} +(6343.29 + 341.079i) q^{7} -7958.48i q^{8} +16878.5i q^{10} +55883.2i q^{11} -109769. i q^{13} +(9211.04 - 171304. i) q^{14} -326183. q^{16} +280977. q^{17} +403251. i q^{19} +135814. q^{20} +1.50916e6 q^{22} -644398. i q^{23} +390625. q^{25} -2.96437e6 q^{26} +(-1.37841e6 - 74117.3i) q^{28} +3.85346e6i q^{29} -7.05000e6i q^{31} +4.73401e6i q^{32} -7.58795e6i q^{34} +(-3.96455e6 - 213174. i) q^{35} -1.31664e7 q^{37} +1.08900e7 q^{38} +4.97405e6i q^{40} -1.26870e7 q^{41} +1.72814e7 q^{43} -1.21436e7i q^{44} -1.74024e7 q^{46} +5.70134e7 q^{47} +(4.01209e7 + 4.32712e6i) q^{49} -1.05491e7i q^{50} +2.38530e7i q^{52} +7.58499e7i q^{53} -3.49270e7i q^{55} +(2.71447e6 - 5.04829e7i) q^{56} +1.04065e8 q^{58} +1.60813e8 q^{59} +1.34644e7i q^{61} -1.90390e8 q^{62} -3.91606e7 q^{64} +6.86055e7i q^{65} -8.67690e6 q^{67} -6.10570e7 q^{68} +(-5.75690e6 + 1.07065e8i) q^{70} -4.14842e8i q^{71} +2.93937e8i q^{73} +3.55566e8i q^{74} -8.76274e7i q^{76} +(-1.90606e7 + 3.54483e8i) q^{77} -5.32251e8 q^{79} +2.03864e8 q^{80} +3.42621e8i q^{82} +7.08646e8 q^{83} -1.75610e8 q^{85} -4.66694e8i q^{86} +4.44745e8 q^{88} +9.09623e8 q^{89} +(3.74398e7 - 6.96295e8i) q^{91} +1.40029e8i q^{92} -1.53968e9i q^{94} -2.52032e8i q^{95} -1.19759e9i q^{97} +(1.16857e8 - 1.08349e9i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	27.0056i		−	1.19349i	−0.802431	−	0.596745i	$-0.796460\pi$
$2$		−	27.0056i		−	1.19349i	0.802431	−	0.596745i	$-0.203540\pi$
$3$		0			0
$3$		0			0
$4$	−217.303			−0.424419
$5$	−625.000			−0.447214
$5$	−625.000			−0.447214
$6$		0			0
$7$	6343.29	+	341.079i	0.998558	+	0.0536925i
$7$	6343.29	+	341.079i	0.998558	+	0.0536925i
$8$		−	7958.48i		−	0.686950i
$9$		0			0
$10$			16878.5i			0.533745i
$11$			55883.2i			1.15084i	0.817859	+	0.575419i	$0.195161\pi$
$11$			55883.2i			1.15084i	−0.817859	+	0.575419i	$0.804839\pi$
$12$		0			0
$13$		−	109769.i		−	1.06594i	−0.846133	−	0.532972i	$-0.821075\pi$
$13$		−	109769.i		−	1.06594i	0.846133	−	0.532972i	$-0.178925\pi$
$14$	9211.04	−	171304.i	0.0640815	−	1.19177i
$15$		0			0
$16$	−326183.			−1.24429
$17$	280977.			0.815925			0.407963	−	0.912999i	$-0.366239\pi$
$17$	280977.			0.815925			0.407963	+	0.912999i	$0.366239\pi$
$18$		0			0
$19$			403251.i			0.709879i	0.934889	+	0.354939i	$0.115498\pi$
$19$			403251.i			0.709879i	−0.934889	+	0.354939i	$0.884502\pi$
$20$	135814.			0.189806
$21$		0			0
$22$	1.50916e6			1.37351
$23$		−	644398.i		−	0.480152i	−0.970754	−	0.240076i	$-0.922828\pi$
$23$		−	644398.i		−	0.480152i	0.970754	−	0.240076i	$-0.0771724\pi$
$24$		0			0
$25$	390625.			0.200000
$26$	−2.96437e6			−1.27219
$27$		0			0
$28$	−1.37841e6	−	74117.3i	−0.423807	−	0.0227881i
$29$			3.85346e6i			1.01172i	0.862616	+	0.505860i	$0.168825\pi$
$29$			3.85346e6i			1.01172i	−0.862616	+	0.505860i	$0.831175\pi$
$30$		0			0
$31$		−	7.05000e6i		−	1.37108i	−0.728037	−	0.685538i	$-0.759567\pi$
$31$		−	7.05000e6i		−	1.37108i	0.728037	−	0.685538i	$-0.240433\pi$
$32$			4.73401e6i			0.798095i
$33$		0			0
$34$		−	7.58795e6i		−	0.973799i
$35$	−3.96455e6	−	213174.i	−0.446568	−	0.0240120i
$36$		0			0
$37$	−1.31664e7			−1.15494			−0.577469	−	0.816413i	$-0.695959\pi$
$37$	−1.31664e7			−1.15494			−0.577469	+	0.816413i	$0.695959\pi$
$38$	1.08900e7			0.847233
$39$		0			0
$40$			4.97405e6i			0.307214i
$41$	−1.26870e7			−0.701185			−0.350593	−	0.936528i	$-0.614020\pi$
$41$	−1.26870e7			−0.701185			−0.350593	+	0.936528i	$0.614020\pi$
$42$		0			0
$43$	1.72814e7			0.770851			0.385426	−	0.922739i	$-0.374055\pi$
$43$	1.72814e7			0.770851			0.385426	+	0.922739i	$0.374055\pi$
$44$		−	1.21436e7i		−	0.488437i
$45$		0			0
$46$	−1.74024e7			−0.573057
$47$	5.70134e7			1.70426			0.852131	−	0.523328i	$-0.175310\pi$
$47$	5.70134e7			1.70426			0.852131	+	0.523328i	$0.175310\pi$
$48$		0			0
$49$	4.01209e7	+	4.32712e6i	0.994234	+	0.107230i
$50$		−	1.05491e7i		−	0.238698i
$51$		0			0
$52$			2.38530e7i			0.452406i
$53$			7.58499e7i			1.32043i	0.751079	+	0.660213i	$0.229534\pi$
$53$			7.58499e7i			1.32043i	−0.751079	+	0.660213i	$0.770466\pi$
$54$		0			0
$55$		−	3.49270e7i		−	0.514670i
$56$	2.71447e6	−	5.04829e7i	0.0368841	−	0.685959i
$57$		0			0
$58$	1.04065e8			1.20748
$59$	1.60813e8			1.72778			0.863889	−	0.503683i	$-0.168022\pi$
$59$	1.60813e8			1.72778			0.863889	+	0.503683i	$0.168022\pi$
$60$		0			0
$61$			1.34644e7i			0.124509i	0.998060	+	0.0622546i	$0.0198291\pi$
$61$			1.34644e7i			0.124509i	−0.998060	+	0.0622546i	$0.980171\pi$
$62$	−1.90390e8			−1.63637
$63$		0			0
$64$	−3.91606e7			−0.291769
$65$			6.86055e7i			0.476704i
$66$		0			0
$67$	−8.67690e6			−0.0526051			−0.0263026	−	0.999654i	$-0.508373\pi$
$67$	−8.67690e6			−0.0526051			−0.0263026	+	0.999654i	$0.508373\pi$
$68$	−6.10570e7			−0.346294
$69$		0			0
$70$	−5.75690e6	+	1.07065e8i	−0.0286581	+	0.532975i
$71$		−	4.14842e8i		−	1.93741i	−0.248223	−	0.968703i	$-0.579847\pi$
$71$		−	4.14842e8i		−	1.93741i	0.248223	−	0.968703i	$-0.420153\pi$
$72$		0			0
$73$			2.93937e8i			1.21144i	0.795679	+	0.605718i	$0.207114\pi$
$73$			2.93937e8i			1.21144i	−0.795679	+	0.605718i	$0.792886\pi$
$74$			3.55566e8i			1.37841i
$75$		0			0
$76$		−	8.76274e7i		−	0.301286i
$77$	−1.90606e7	+	3.54483e8i	−0.0617913	+	1.14918i
$78$		0			0
$79$	−5.32251e8			−1.53743			−0.768714	−	0.639593i	$-0.779103\pi$
$79$	−5.32251e8			−1.53743			−0.768714	+	0.639593i	$0.779103\pi$
$80$	2.03864e8			0.556462
$81$		0			0
$82$			3.42621e8i			0.836858i
$83$	7.08646e8			1.63900			0.819498	−	0.573082i	$-0.194252\pi$
$83$	7.08646e8			1.63900			0.819498	+	0.573082i	$0.194252\pi$
$84$		0			0
$85$	−1.75610e8			−0.364893
$86$		−	4.66694e8i		−	0.920003i
$87$		0			0
$88$	4.44745e8			0.790568
$89$	9.09623e8			1.53676			0.768380	−	0.639993i	$-0.221063\pi$
$89$	9.09623e8			1.53676			0.768380	+	0.639993i	$0.221063\pi$
$90$		0			0
$91$	3.74398e7	−	6.96295e8i	0.0572332	−	1.06441i
$92$			1.40029e8i			0.203786i
$93$		0			0
$94$		−	1.53968e9i		−	2.03402i
$95$		−	2.52032e8i		−	0.317467i
$96$		0			0
$97$		−	1.19759e9i		−	1.37352i	−0.726883	−	0.686761i	$-0.759032\pi$
$97$		−	1.19759e9i		−	1.37352i	0.726883	−	0.686761i	$-0.240968\pi$
$98$	1.16857e8	−	1.08349e9i	0.127978	−	1.18661i
$99$		0			0
$100$	−8.48838e7			−0.0848838
$101$	1.52865e9			1.46172			0.730858	−	0.682530i	$-0.239120\pi$
$101$	1.52865e9			1.46172			0.730858	+	0.682530i	$0.239120\pi$
$102$		0			0
$103$		−	1.53181e9i		−	1.34103i	−0.741898	−	0.670513i	$-0.766074\pi$
$103$		−	1.53181e9i		−	1.34103i	0.741898	−	0.670513i	$-0.233926\pi$
$104$	−8.73594e8			−0.732250
$105$		0			0
$106$	2.04837e9			1.57591
$107$			9.16093e8i			0.675636i	0.941211	+	0.337818i	$0.109689\pi$
$107$			9.16093e8i			0.675636i	−0.941211	+	0.337818i	$0.890311\pi$
$108$		0			0
$109$	−6.73302e8			−0.456868			−0.228434	−	0.973559i	$-0.573361\pi$
$109$	−6.73302e8			−0.456868			−0.228434	+	0.973559i	$0.573361\pi$
$110$	−9.43224e8			−0.614254
$111$		0			0
$112$	−2.06907e9	−	1.11254e8i	−1.24249	−	0.0668089i
$113$		−	5.34863e8i		−	0.308595i	−0.988024	−	0.154298i	$-0.950689\pi$
$113$		−	5.34863e8i		−	0.308595i	0.988024	−	0.154298i	$-0.0493115\pi$
$114$		0			0
$115$			4.02749e8i			0.214731i
$116$		−	8.37367e8i		−	0.429393i
$117$		0			0
$118$		−	4.34286e9i		−	2.06209i
$119$	1.78232e9	+	9.58352e7i	0.814748	+	0.0438091i
$120$		0			0
$121$	−7.64981e8			−0.324427
$122$	3.63613e8			0.148601
$123$		0			0
$124$			1.53198e9i			0.581911i
$125$	−2.44141e8			−0.0894427
$126$		0			0
$127$	1.27644e9			0.435396			0.217698	−	0.976016i	$-0.430145\pi$
$127$	1.27644e9			0.435396			0.217698	+	0.976016i	$0.430145\pi$
$128$			3.48137e9i			1.14632i
$129$		0			0
$130$	1.85273e9			0.568942
$131$	−4.63787e9			−1.37594			−0.687968	−	0.725741i	$-0.741497\pi$
$131$	−4.63787e9			−1.37594			−0.687968	+	0.725741i	$0.741497\pi$
$132$		0			0
$133$	−1.37540e8	+	2.55794e9i	−0.0381152	+	0.708855i
$134$			2.34325e8i			0.0627837i
$135$		0			0
$136$		−	2.23615e9i		−	0.560500i
$137$		−	1.30313e9i		−	0.316042i	−0.987436	−	0.158021i	$-0.949489\pi$
$137$		−	1.30313e9i		−	0.316042i	0.987436	−	0.158021i	$-0.0505113\pi$
$138$		0			0
$139$		−	5.56356e9i		−	1.26411i	−0.774922	−	0.632056i	$-0.782211\pi$
$139$		−	5.56356e9i		−	1.26411i	0.774922	−	0.632056i	$-0.217789\pi$
$140$	8.61507e8	+	4.63233e7i	0.189532	+	0.0101912i
$141$		0			0
$142$	−1.12031e10			−2.31227
$143$	6.13423e9			1.22673
$144$		0			0
$145$		−	2.40841e9i		−	0.452455i
$146$	7.93793e9			1.44584
$147$		0			0
$148$	2.86109e9			0.490177
$149$		−	5.34293e9i		−	0.888058i	−0.896012	−	0.444029i	$-0.853549\pi$
$149$		−	5.34293e9i		−	0.888058i	0.896012	−	0.444029i	$-0.146451\pi$
$150$		0			0
$151$	7.95331e8			0.124495			0.0622475	−	0.998061i	$-0.480173\pi$
$151$	7.95331e8			0.124495			0.0622475	+	0.998061i	$0.480173\pi$
$152$	3.20927e9			0.487651
$153$		0			0
$154$	9.57302e9	+	5.14742e8i	1.37153	+	0.0737474i
$155$			4.40625e9i			0.613164i
$156$		0			0
$157$		−	6.81194e9i		−	0.894793i	−0.894336	−	0.447396i	$-0.852351\pi$
$157$		−	6.81194e9i		−	0.894793i	0.894336	−	0.447396i	$-0.147649\pi$
$158$			1.43738e10i			1.83491i
$159$		0			0
$160$		−	2.95876e9i		−	0.356919i
$161$	2.19791e8	−	4.08760e9i	0.0257806	−	0.479459i
$162$		0			0
$163$	6.23408e9			0.691717			0.345858	−	0.938287i	$-0.387588\pi$
$163$	6.23408e9			0.691717			0.345858	+	0.938287i	$0.387588\pi$
$164$	2.75692e9			0.297596
$165$		0			0
$166$		−	1.91374e10i		−	1.95613i
$167$	1.21756e10			1.21134			0.605669	−	0.795717i	$-0.292906\pi$
$167$	1.21756e10			1.21134			0.605669	+	0.795717i	$0.292906\pi$
$168$		0			0
$169$	−1.44470e9			−0.136235
$170$			4.74247e9i			0.435496i
$171$		0			0
$172$	−3.75529e9			−0.327164
$173$	−1.22845e10			−1.04268			−0.521339	−	0.853350i	$-0.674567\pi$
$173$	−1.22845e10			−1.04268			−0.521339	+	0.853350i	$0.674567\pi$
$174$		0			0
$175$	2.47785e9	+	1.33234e8i	0.199712	+	0.0107385i
$176$		−	1.82281e10i		−	1.43197i
$177$		0			0
$178$		−	2.45649e10i		−	1.83411i
$179$		−	2.34710e10i		−	1.70880i	−0.519613	−	0.854402i	$-0.673924\pi$
$179$		−	2.34710e10i		−	1.70880i	0.519613	−	0.854402i	$-0.326076\pi$
$180$		0			0
$181$			1.97718e10i			1.36928i	0.728882	+	0.684639i	$0.240040\pi$
$181$			1.97718e10i			1.36928i	−0.728882	+	0.684639i	$0.759960\pi$
$182$	−1.88039e10	−	1.01109e9i	−1.27036	−	0.0683072i
$183$		0			0
$184$	−5.12843e9			−0.329841
$185$	8.22898e9			0.516504
$186$		0			0
$187$			1.57019e10i			0.938997i
$188$	−1.23892e10			−0.723321
$189$		0			0
$190$	−6.80627e9			−0.378894
$191$			2.98816e10i			1.62463i	0.583221	+	0.812314i	$0.301792\pi$
$191$			2.98816e10i			1.62463i	−0.583221	+	0.812314i	$0.698208\pi$
$192$		0			0
$193$	5.53792e9			0.287302			0.143651	−	0.989628i	$-0.454116\pi$
$193$	5.53792e9			0.287302			0.143651	+	0.989628i	$0.454116\pi$
$194$	−3.23417e10			−1.63928
$195$		0			0
$196$	−8.71838e9	−	9.40294e8i	−0.421972	−	0.0455105i
$197$		−	1.60172e9i		−	0.0757685i	−0.999282	−	0.0378842i	$-0.987938\pi$
$197$		−	1.60172e9i		−	0.0757685i	0.999282	−	0.0378842i	$-0.0120618\pi$
$198$		0			0
$199$		−	1.61669e10i		−	0.730782i	−0.930854	−	0.365391i	$-0.880935\pi$
$199$		−	1.61669e10i		−	0.730782i	0.930854	−	0.365391i	$-0.119065\pi$
$200$		−	3.10878e9i		−	0.137390i
$201$		0			0
$202$		−	4.12822e10i		−	1.74454i
$203$	−1.31433e9	+	2.44436e10i	−0.0543218	+	1.01026i
$204$		0			0
$205$	7.92940e9			0.313580
$206$	−4.13674e10			−1.60050
$207$		0			0
$208$			3.58047e10i			1.32634i
$209$	−2.25349e10			−0.816955
$210$		0			0
$211$	−5.11331e10			−1.77595			−0.887976	−	0.459890i	$-0.847889\pi$
$211$	−5.11331e10			−1.77595			−0.887976	+	0.459890i	$0.847889\pi$
$212$		−	1.64824e10i		−	0.560414i
$213$		0			0
$214$	2.47396e10			0.806365
$215$	−1.08009e10			−0.344735
$216$		0			0
$217$	2.40461e9	−	4.47202e10i	0.0736165	−	1.36910i
$218$			1.81829e10i			0.545268i
$219$		0			0
$220$			7.58972e9i			0.218436i
$221$		−	3.08425e10i		−	0.869730i
$222$		0			0
$223$			1.25396e10i			0.339555i	0.985482	+	0.169777i	$0.0543049\pi$
$223$			1.25396e10i			0.339555i	−0.985482	+	0.169777i	$0.945695\pi$
$224$	−1.61467e9	+	3.00292e10i	−0.0428517	+	0.796943i
$225$		0			0
$226$	−1.44443e10			−0.368306
$227$	−5.35340e10			−1.33818			−0.669088	−	0.743183i	$-0.733315\pi$
$227$	−5.35340e10			−1.33818			−0.669088	+	0.743183i	$0.733315\pi$
$228$		0			0
$229$		−	1.05694e10i		−	0.253975i	−0.991904	−	0.126987i	$-0.959469\pi$
$229$		−	1.05694e10i		−	0.253975i	0.991904	−	0.126987i	$-0.0405308\pi$
$230$	1.08765e10			0.256279
$231$		0			0
$232$	3.06677e10			0.695001
$233$		−	3.72843e10i		−	0.828751i	−0.910106	−	0.414375i	$-0.864000\pi$
$233$		−	3.72843e10i		−	0.828751i	0.910106	−	0.414375i	$-0.136000\pi$
$234$		0			0
$235$	−3.56334e10			−0.762169
$236$	−3.49451e10			−0.733302
$237$		0			0
$238$	2.58809e9	−	4.81325e10i	0.0522857	−	0.972394i
$239$		−	1.92553e10i		−	0.381732i	−0.981616	−	0.190866i	$-0.938870\pi$
$239$		−	1.92553e10i		−	0.381732i	0.981616	−	0.190866i	$-0.0611297\pi$
$240$		0			0
$241$		−	5.70917e10i		−	1.09018i	−0.838379	−	0.545088i	$-0.816496\pi$
$241$		−	5.70917e10i		−	1.09018i	0.838379	−	0.545088i	$-0.183504\pi$
$242$			2.06588e10i			0.387200i
$243$		0			0
$244$		−	2.92584e9i		−	0.0528441i
$245$	−2.50756e10	−	2.70445e9i	−0.444635	−	0.0479548i
$246$		0			0
$247$	4.42644e10			0.756690
$248$	−5.61073e10			−0.941861
$249$		0			0
$250$			6.59316e9i			0.106749i
$251$	−1.01207e11			−1.60945			−0.804724	−	0.593649i	$-0.797687\pi$
$251$	−1.01207e11			−1.60945			−0.804724	+	0.593649i	$0.797687\pi$
$252$		0			0
$253$	3.60110e10			0.552577
$254$		−	3.44711e10i		−	0.519641i
$255$		0			0
$256$	7.39662e10			1.07635
$257$	−5.59267e9			−0.0799687			−0.0399843	−	0.999200i	$-0.512731\pi$
$257$	−5.59267e9			−0.0799687			−0.0399843	+	0.999200i	$0.512731\pi$
$258$		0			0
$259$	−8.35181e10	−	4.49077e9i	−1.15327	−	0.0620115i
$260$		−	1.49082e10i		−	0.202322i
$261$		0			0
$262$			1.25249e11i			1.64217i
$263$			7.91755e10i			1.02045i	0.860042	+	0.510223i	$0.170437\pi$
$263$			7.91755e10i			1.02045i	−0.860042	+	0.510223i	$0.829563\pi$
$264$		0			0
$265$		−	4.74062e10i		−	0.590512i
$266$	6.90786e10	+	3.71436e9i	0.846011	+	0.0454901i
$267$		0			0
$268$	1.88551e9			0.0223266
$269$	−5.73315e10			−0.667587			−0.333793	−	0.942646i	$-0.608329\pi$
$269$	−5.73315e10			−0.667587			−0.333793	+	0.942646i	$0.608329\pi$
$270$		0			0
$271$			1.78660e10i			0.201217i	0.994926	+	0.100609i	$0.0320790\pi$
$271$			1.78660e10i			0.201217i	−0.994926	+	0.100609i	$0.967921\pi$
$272$	−9.16497e10			−1.01525
$273$		0			0
$274$	−3.51918e10			−0.377193
$275$			2.18294e10i			0.230167i
$276$		0			0
$277$	6.48551e10			0.661889			0.330944	−	0.943650i	$-0.392633\pi$
$277$	6.48551e10			0.661889			0.330944	+	0.943650i	$0.392633\pi$
$278$	−1.50247e11			−1.50871
$279$		0			0
$280$	−1.69654e9	+	3.15518e10i	−0.0164951	+	0.306770i
$281$			1.91282e10i			0.183018i	0.995804	+	0.0915092i	$0.0291691\pi$
$281$			1.91282e10i			0.183018i	−0.995804	+	0.0915092i	$0.970831\pi$
$282$		0			0
$283$		−	1.40410e11i		−	1.30124i	−0.759402	−	0.650622i	$-0.774508\pi$
$283$		−	1.40410e11i		−	1.30124i	0.759402	−	0.650622i	$-0.225492\pi$
$284$			9.01463e10i			0.822272i
$285$		0			0
$286$		−	1.65659e11i		−	1.46409i
$287$	−8.04775e10	−	4.32728e9i	−0.700174	−	0.0376484i
$288$		0			0
$289$	−3.96399e10			−0.334266
$290$	−6.50407e10			−0.540000
$291$		0			0
$292$		−	6.38731e10i		−	0.514157i
$293$	1.12972e11			0.895505			0.447753	−	0.894158i	$-0.352225\pi$
$293$	1.12972e11			0.895505			0.447753	+	0.894158i	$0.352225\pi$
$294$		0			0
$295$	−1.00508e11			−0.772686
$296$			1.04784e11i			0.793384i
$297$		0			0
$298$	−1.44289e11			−1.05989
$299$	−7.07348e10			−0.511815
$300$		0			0
$301$	1.09621e11	+	5.89432e9i	0.769739	+	0.0413889i
$302$		−	2.14784e10i		−	0.148583i
$303$		0			0
$304$		−	1.31533e11i		−	0.883293i
$305$		−	8.41523e9i		−	0.0556822i
$306$		0			0
$307$		−	1.25030e11i		−	0.803323i	−0.915788	−	0.401662i	$-0.868433\pi$
$307$		−	1.25030e11i		−	0.803323i	0.915788	−	0.401662i	$-0.131567\pi$
$308$	4.14191e9	−	7.70300e10i	0.0262254	−	0.487733i
$309$		0			0
$310$	1.18993e11			0.731805
$311$	−7.33308e10			−0.444493			−0.222246	−	0.974991i	$-0.571339\pi$
$311$	−7.33308e10			−0.444493			−0.222246	+	0.974991i	$0.571339\pi$
$312$		0			0
$313$		−	1.99834e11i		−	1.17685i	−0.808553	−	0.588423i	$-0.799749\pi$
$313$		−	1.99834e11i		−	1.17685i	0.808553	−	0.588423i	$-0.200251\pi$
$314$	−1.83961e11			−1.06793
$315$		0			0
$316$	1.15660e11			0.652514
$317$		−	1.94471e11i		−	1.08165i	−0.841135	−	0.540825i	$-0.818112\pi$
$317$		−	1.94471e11i		−	1.08165i	0.841135	−	0.540825i	$-0.181888\pi$
$318$		0			0
$319$	−2.15344e11			−1.16432
$320$	2.44754e10			0.130483
$321$		0			0
$322$	−1.10388e11	−	5.93558e9i	−0.572230	−	0.0307689i
$323$			1.13304e11i			0.579208i
$324$		0			0
$325$		−	4.28785e10i		−	0.213189i
$326$		−	1.68355e11i		−	0.825557i
$327$		0			0
$328$			1.00970e11i			0.481679i
$329$	3.61652e11	+	1.94461e10i	1.70180	+	0.0915061i
$330$		0			0
$331$	2.67740e11			1.22599			0.612996	−	0.790086i	$-0.289964\pi$
$331$	2.67740e11			1.22599			0.612996	+	0.790086i	$0.289964\pi$
$332$	−1.53991e11			−0.695621
$333$		0			0
$334$		−	3.28809e11i		−	1.44572i
$335$	5.42306e9			0.0235257
$336$		0			0
$337$	9.86386e8			0.00416593			0.00208297	−	0.999998i	$-0.499337\pi$
$337$	9.86386e8			0.00416593			0.00208297	+	0.999998i	$0.499337\pi$
$338$			3.90150e10i			0.162595i
$339$		0			0
$340$	3.81606e10			0.154867
$341$	3.93977e11			1.57789
$342$		0			0
$343$	2.53023e11	+	4.11326e10i	0.987043	+	0.160458i
$344$		−	1.37534e11i		−	0.529536i
$345$		0			0
$346$			3.31750e11i			1.24443i
$347$			3.34712e11i			1.23934i	0.784864	+	0.619668i	$0.212733\pi$
$347$			3.34712e11i			1.23934i	−0.784864	+	0.619668i	$0.787267\pi$
$348$		0			0
$349$			2.62595e11i			0.947484i	0.880664	+	0.473742i	$0.157097\pi$
$349$			2.62595e11i			0.947484i	−0.880664	+	0.473742i	$0.842903\pi$
$350$	3.59806e9	−	6.69157e10i	0.0128163	−	0.238354i
$351$		0			0
$352$	−2.64552e11			−0.918477
$353$	2.72882e10			0.0935380			0.0467690	−	0.998906i	$-0.485108\pi$
$353$	2.72882e10			0.0935380			0.0467690	+	0.998906i	$0.485108\pi$
$354$		0			0
$355$			2.59277e11i			0.866434i
$356$	−1.97663e11			−0.652230
$357$		0			0
$358$	−6.33847e11			−2.03944
$359$		−	5.18365e11i		−	1.64706i	−0.567270	−	0.823532i	$-0.692001\pi$
$359$		−	5.18365e11i		−	1.64706i	0.567270	−	0.823532i	$-0.307999\pi$
$360$		0			0
$361$	1.60076e11			0.496072
$362$	5.33949e11			1.63422
$363$		0			0
$364$	−8.13577e9	+	1.51307e11i	−0.0242908	+	0.451754i
$365$		−	1.83710e11i		−	0.541771i
$366$		0			0
$367$		−	5.01432e11i		−	1.44283i	−0.692504	−	0.721414i	$-0.743492\pi$
$367$		−	5.01432e11i		−	1.44283i	0.692504	−	0.721414i	$-0.256508\pi$
$368$			2.10191e11i			0.597447i
$369$		0			0
$370$		−	2.22229e11i		−	0.616442i
$371$	−2.58708e10	+	4.81138e11i	−0.0708970	+	1.31852i
$372$		0			0
$373$	−5.11137e11			−1.36725			−0.683625	−	0.729834i	$-0.739597\pi$
$373$	−5.11137e11			−1.36725			−0.683625	+	0.729834i	$0.739597\pi$
$374$	4.24039e11			1.12068
$375$		0			0
$376$		−	4.53740e11i		−	1.17074i
$377$	4.22990e11			1.07844
$378$		0			0
$379$	3.51354e11			0.874720			0.437360	−	0.899287i	$-0.355914\pi$
$379$	3.51354e11			0.874720			0.437360	+	0.899287i	$0.355914\pi$
$380$			5.47671e10i			0.134739i
$381$		0			0
$382$	8.06971e11			1.93898
$383$	2.35978e11			0.560373			0.280187	−	0.959946i	$-0.409604\pi$
$383$	2.35978e11			0.560373			0.280187	+	0.959946i	$0.409604\pi$
$384$		0			0
$385$	1.19129e10	−	2.21552e11i	0.0276339	−	0.513928i
$386$		−	1.49555e11i		−	0.342892i
$387$		0			0
$388$			2.60239e11i			0.582949i
$389$			2.40784e11i			0.533157i	0.963813	+	0.266578i	$0.0858931\pi$
$389$			2.40784e11i			0.533157i	−0.963813	+	0.266578i	$0.914107\pi$
$390$		0			0
$391$		−	1.81061e11i		−	0.391768i
$392$	3.44373e10	−	3.19302e11i	0.0736618	−	0.682990i
$393$		0			0
$394$	−4.32554e10			−0.0904290
$395$	3.32657e11			0.687559
$396$		0			0
$397$			3.22406e11i			0.651398i	0.945474	+	0.325699i	$0.105600\pi$
$397$			3.22406e11i			0.651398i	−0.945474	+	0.325699i	$0.894400\pi$
$398$	−4.36597e11			−0.872181
$399$		0			0
$400$	−1.27415e11			−0.248858
$401$			2.20901e11i			0.426627i	0.976984	+	0.213314i	$0.0684256\pi$
$401$			2.20901e11i			0.426627i	−0.976984	+	0.213314i	$0.931574\pi$
$402$		0			0
$403$	−7.73871e11			−1.46149
$404$	−3.32180e11			−0.620380
$405$		0			0
$406$	6.60114e11	+	3.54944e10i	1.20574	+	0.0648325i
$407$		−	7.35779e11i		−	1.32914i
$408$		0			0
$409$		−	7.50987e11i		−	1.32702i	−0.748167	−	0.663510i	$-0.769066\pi$
$409$		−	7.50987e11i		−	1.32702i	0.748167	−	0.663510i	$-0.230934\pi$
$410$		−	2.14138e11i		−	0.374254i
$411$		0			0
$412$			3.32866e11i			0.569157i
$413$	1.02008e12	+	5.48500e10i	1.72529	+	0.0927687i
$414$		0			0
$415$	−4.42904e11			−0.732981
$416$	5.19647e11			0.850724
$417$		0			0
$418$			6.08570e11i			0.975028i
$419$	6.93260e11			1.09884			0.549419	−	0.835547i	$-0.314849\pi$
$419$	6.93260e11			1.09884			0.549419	+	0.835547i	$0.314849\pi$
$420$		0			0
$421$	3.27013e11			0.507336			0.253668	−	0.967291i	$-0.418363\pi$
$421$	3.27013e11			0.507336			0.253668	+	0.967291i	$0.418363\pi$
$422$			1.38088e12i			2.11958i
$423$		0			0
$424$	6.03651e11			0.907067
$425$	1.09757e11			0.163185
$426$		0			0
$427$	−4.59241e9	+	8.54083e10i	−0.00668521	+	0.124330i
$428$		−	1.99069e11i		−	0.286753i
$429$		0			0
$430$			2.91684e11i			0.411438i
$431$		−	1.04434e12i		−	1.45779i	−0.684624	−	0.728897i	$-0.740034\pi$
$431$		−	1.04434e12i		−	1.45779i	0.684624	−	0.728897i	$-0.259966\pi$
$432$		0			0
$433$		−	3.43746e11i		−	0.469940i	−0.972003	−	0.234970i	$-0.924501\pi$
$433$		−	3.43746e11i		−	0.469940i	0.972003	−	0.234970i	$-0.0754991\pi$
$434$	−1.20770e12	−	6.49379e10i	−1.63401	−	0.0878606i
$435$		0			0
$436$	1.46310e11			0.193903
$437$	2.59854e11			0.340850
$438$		0			0
$439$			9.43539e11i			1.21247i	0.795287	+	0.606233i	$0.207320\pi$
$439$			9.43539e11i			1.21247i	−0.795287	+	0.606233i	$0.792680\pi$
$440$	−2.77966e11			−0.353553
$441$		0			0
$442$	−8.32920e11			−1.03801
$443$			4.98440e11i			0.614888i	0.951566	+	0.307444i	$0.0994737\pi$
$443$			4.98440e11i			0.614888i	−0.951566	+	0.307444i	$0.900526\pi$
$444$		0			0
$445$	−5.68514e11			−0.687260
$446$	3.38638e11			0.405256
$447$		0			0
$448$	−2.48407e11	−	1.33569e10i	−0.291348	−	0.0156658i
$449$			9.76815e11i			1.13424i	0.823636	+	0.567118i	$0.191942\pi$
$449$			9.76815e11i			1.13424i	−0.823636	+	0.567118i	$0.808058\pi$
$450$		0			0
$451$		−	7.08992e11i		−	0.806950i
$452$			1.16227e11i			0.130974i
$453$		0			0
$454$			1.44572e12i			1.59710i
$455$	−2.33999e10	+	4.35184e11i	−0.0255954	+	0.476017i
$456$		0			0
$457$	1.40246e12			1.50407			0.752036	−	0.659122i	$-0.229072\pi$
$457$	1.40246e12			1.50407			0.752036	+	0.659122i	$0.229072\pi$
$458$	−2.85433e11			−0.303117
$459$		0			0
$460$		−	8.75183e10i		−	0.0911357i
$461$	−5.74667e11			−0.592601			−0.296301	−	0.955095i	$-0.595753\pi$
$461$	−5.74667e11			−0.592601			−0.296301	+	0.955095i	$0.595753\pi$
$462$		0			0
$463$	1.67308e12			1.69201			0.846004	−	0.533177i	$-0.179002\pi$
$463$	1.67308e12			1.69201			0.846004	+	0.533177i	$0.179002\pi$
$464$		−	1.25693e12i		−	1.25887i
$465$		0			0
$466$	−1.00688e12			−0.989106
$467$	4.77367e11			0.464436			0.232218	−	0.972664i	$-0.425402\pi$
$467$	4.77367e11			0.464436			0.232218	+	0.972664i	$0.425402\pi$
$468$		0			0
$469$	−5.50401e10	−	2.95951e9i	−0.0525292	−	0.00282450i
$470$			9.62301e11i			0.909642i
$471$		0			0
$472$		−	1.27983e12i		−	1.18690i
$473$			9.65739e11i			0.887124i
$474$		0			0
$475$			1.57520e11i			0.141976i
$476$	−3.87302e11	−	2.08252e10i	−0.345795	−	0.0185934i
$477$		0			0
$478$	−5.20000e11			−0.455594
$479$	1.77523e11			0.154080			0.0770399	−	0.997028i	$-0.475453\pi$
$479$	1.77523e11			0.154080			0.0770399	+	0.997028i	$0.475453\pi$
$480$		0			0
$481$			1.44526e12i			1.23110i
$482$	−1.54180e12			−1.30111
$483$		0			0
$484$	1.66232e11			0.137693
$485$			7.48494e11i			0.614258i
$486$		0			0
$487$	1.84300e11			0.148473			0.0742363	−	0.997241i	$-0.476348\pi$
$487$	1.84300e11			0.148473			0.0742363	+	0.997241i	$0.476348\pi$
$488$	1.07156e11			0.0855317
$489$		0			0
$490$	−7.30353e10	+	6.77181e11i	−0.0572335	+	0.530668i
$491$		−	1.51219e12i		−	1.17419i	−0.809517	−	0.587097i	$-0.800271\pi$
$491$		−	1.51219e12i		−	1.17419i	0.809517	−	0.587097i	$-0.199729\pi$
$492$		0			0
$493$			1.08273e12i			0.825487i
$494$		−	1.19539e12i		−	0.903102i
$495$		0			0
$496$			2.29959e12i			1.70601i
$497$	1.41494e11	−	2.63146e12i	0.104024	−	1.93461i
$498$		0			0
$499$	−1.09105e12			−0.787754			−0.393877	−	0.919163i	$-0.628866\pi$
$499$	−1.09105e12			−0.787754			−0.393877	+	0.919163i	$0.628866\pi$
$500$	5.30524e10			0.0379612
$501$		0			0
$502$			2.73314e12i			1.92086i
$503$	6.98526e11			0.486549			0.243274	−	0.969957i	$-0.421778\pi$
$503$	6.98526e11			0.486549			0.243274	+	0.969957i	$0.421778\pi$
$504$		0			0
$505$	−9.55408e11			−0.653699
$506$		−	9.72499e11i		−	0.659495i
$507$		0			0
$508$	−2.77374e11			−0.184790
$509$	1.13356e12			0.748537			0.374268	−	0.927320i	$-0.377894\pi$
$509$	1.13356e12			0.748537			0.374268	+	0.927320i	$0.377894\pi$
$510$		0			0
$511$	−1.00256e11	+	1.86452e12i	−0.0650451	+	1.20969i
$512$		−	2.15042e11i		−	0.138295i
$513$		0			0
$514$			1.51033e11i			0.0954418i
$515$			9.57381e11i			0.599725i
$516$		0			0
$517$			3.18609e12i			1.96133i
$518$	−1.21276e11	+	2.25546e12i	−0.0740101	+	1.37642i
$519$		0			0
$520$	5.45996e11			0.327472
$521$	2.44217e12			1.45213			0.726067	−	0.687624i	$-0.241346\pi$
$521$	2.44217e12			1.45213			0.726067	+	0.687624i	$0.241346\pi$
$522$		0			0
$523$			1.67491e11i			0.0978888i	0.998802	+	0.0489444i	$0.0155857\pi$
$523$			1.67491e11i			0.0978888i	−0.998802	+	0.0489444i	$0.984414\pi$
$524$	1.00782e12			0.583973
$525$		0			0
$526$	2.13818e12			1.21789
$527$		−	1.98089e12i		−	1.11870i
$528$		0			0
$529$	1.38590e12			0.769454
$530$	−1.28023e12			−0.704771
$531$		0			0
$532$	2.98879e10	−	5.55846e11i	0.0161768	−	0.300851i
$533$			1.39264e12i			0.747423i
$534$		0			0
$535$		−	5.72558e11i		−	0.302154i
$536$			6.90550e10i			0.0361371i
$537$		0			0
$538$			1.54827e12i			0.796758i
$539$	−2.41813e11	+	2.24209e12i	−0.123404	+	1.14420i
$540$		0			0
$541$	2.71744e12			1.36387			0.681934	−	0.731414i	$-0.261139\pi$
$541$	2.71744e12			1.36387			0.681934	+	0.731414i	$0.261139\pi$
$542$	4.82482e11			0.240151
$543$		0			0
$544$			1.33015e12i			0.651185i
$545$	4.20814e11			0.204318
$546$		0			0
$547$	−2.87949e12			−1.37522			−0.687612	−	0.726079i	$-0.741341\pi$
$547$	−2.87949e12			−1.37522			−0.687612	+	0.726079i	$0.741341\pi$
$548$			2.83173e11i			0.134134i
$549$		0			0
$550$	5.89515e11			0.274703
$551$	−1.55391e12			−0.718198
$552$		0			0
$553$	−3.37622e12	−	1.81540e11i	−1.53521	−	0.0825484i
$554$		−	1.75145e12i		−	0.789958i
$555$		0			0
$556$			1.20897e12i			0.536513i
$557$		−	1.03236e12i		−	0.454445i	−0.973843	−	0.227222i	$-0.927036\pi$
$557$		−	1.03236e12i		−	0.454445i	0.973843	−	0.227222i	$-0.0729644\pi$
$558$		0			0
$559$		−	1.89696e12i		−	0.821683i
$560$	1.29317e12	+	6.95337e10i	0.555660	+	0.0298779i
$561$		0			0
$562$	5.16568e11			0.218431
$563$	−2.08559e12			−0.874866			−0.437433	−	0.899251i	$-0.644112\pi$
$563$	−2.08559e12			−0.874866			−0.437433	+	0.899251i	$0.644112\pi$
$564$		0			0
$565$			3.34289e11i			0.138008i
$566$	−3.79185e12			−1.55302
$567$		0			0
$568$	−3.30152e12			−1.33090
$569$		−	4.97701e10i		−	0.0199051i	−0.999950	−	0.00995253i	$-0.996832\pi$
$569$		−	4.97701e10i		−	0.0199051i	0.999950	−	0.00995253i	$-0.00316804\pi$
$570$		0			0
$571$	2.65371e12			1.04470			0.522349	−	0.852732i	$-0.325056\pi$
$571$	2.65371e12			1.04470			0.522349	+	0.852732i	$0.325056\pi$
$572$	−1.33298e12			−0.520646
$573$		0			0
$574$	−1.16861e11	+	2.17334e12i	−0.0449330	+	0.835650i
$575$		−	2.51718e11i		−	0.0960304i
$576$		0			0
$577$			2.78100e12i			1.04450i	0.852792	+	0.522251i	$0.174908\pi$
$577$			2.78100e12i			1.04450i	−0.852792	+	0.522251i	$0.825092\pi$
$578$			1.07050e12i			0.398944i
$579$		0			0
$580$			5.23354e11i			0.192030i
$581$	4.49514e12	+	2.41704e11i	1.63663	+	0.0880018i
$582$		0			0
$583$	−4.23874e12			−1.51959
$584$	2.33929e12			0.832197
$585$		0			0
$586$		−	3.05089e12i		−	1.06878i
$587$	−2.37706e12			−0.826360			−0.413180	−	0.910649i	$-0.635582\pi$
$587$	−2.37706e12			−0.826360			−0.413180	+	0.910649i	$0.635582\pi$
$588$		0			0
$589$	2.84292e12			0.973298
$590$			2.71429e12i			0.922193i
$591$		0			0
$592$	4.29464e12			1.43707
$593$	1.16491e12			0.386853			0.193426	−	0.981115i	$-0.438040\pi$
$593$	1.16491e12			0.386853			0.193426	+	0.981115i	$0.438040\pi$
$594$		0			0
$595$	−1.11395e12	−	5.98970e10i	−0.364366	−	0.0195920i
$596$			1.16103e12i			0.376909i
$597$		0			0
$598$			1.91024e12i			0.610846i
$599$			1.04441e12i			0.331474i	0.986170	+	0.165737i	$0.0530003\pi$
$599$			1.04441e12i			0.331474i	−0.986170	+	0.165737i	$0.947000\pi$
$600$		0			0
$601$		−	4.94098e12i		−	1.54482i	−0.635124	−	0.772410i	$-0.719051\pi$
$601$		−	4.94098e12i		−	1.54482i	0.635124	−	0.772410i	$-0.280949\pi$
$602$	1.59180e11	−	2.96037e12i	0.0493973	−	0.918676i
$603$		0			0
$604$	−1.72827e11			−0.0528380
$605$	4.78113e11			0.145088
$606$		0			0
$607$		−	4.32601e12i		−	1.29342i	−0.762738	−	0.646708i	$-0.776145\pi$
$607$		−	4.32601e12i		−	1.29342i	0.762738	−	0.646708i	$-0.223855\pi$
$608$	−1.90899e12			−0.566550
$609$		0			0
$610$	−2.27258e11			−0.0664562
$611$		−	6.25829e12i		−	1.81665i
$612$		0			0
$613$	−3.08449e12			−0.882290			−0.441145	−	0.897436i	$-0.645428\pi$
$613$	−3.08449e12			−0.882290			−0.441145	+	0.897436i	$0.645428\pi$
$614$	−3.37650e12			−0.958758
$615$		0			0
$616$	2.82115e12	+	1.51693e11i	0.789428	+	0.0424476i
$617$			9.25749e11i			0.257164i	0.991699	+	0.128582i	$0.0410425\pi$
$617$			9.25749e11i			0.257164i	−0.991699	+	0.128582i	$0.958957\pi$
$618$		0			0
$619$		−	3.64418e12i		−	0.997682i	−0.866694	−	0.498841i	$-0.833759\pi$
$619$		−	3.64418e12i		−	0.997682i	0.866694	−	0.498841i	$-0.166241\pi$
$620$		−	9.57490e11i		−	0.260238i
$621$		0			0
$622$			1.98034e12i			0.530498i
$623$	5.77000e12	+	3.10253e11i	1.53454	+	0.0825125i
$624$		0			0
$625$	1.52588e11			0.0400000
$626$	−5.39664e12			−1.40455
$627$		0			0
$628$			1.48025e12i			0.379767i
$629$	−3.69944e12			−0.942342
$630$		0			0
$631$	5.74499e12			1.44264			0.721319	−	0.692603i	$-0.243536\pi$
$631$	5.74499e12			1.44264			0.721319	+	0.692603i	$0.243536\pi$
$632$			4.23591e12i			1.05614i
$633$		0			0
$634$	−5.25179e12			−1.29094
$635$	−7.97777e11			−0.194715
$636$		0			0
$637$	4.74983e11	−	4.40403e12i	0.114301	−	1.05980i
$638$			5.81549e12i			1.38961i
$639$		0			0
$640$		−	2.17586e12i		−	0.512649i
$641$			5.23972e10i			0.0122588i	0.999981	+	0.00612939i	$0.00195106\pi$
$641$			5.23972e10i			0.0122588i	−0.999981	+	0.00612939i	$0.998049\pi$
$642$		0			0
$643$			2.11895e12i			0.488846i	0.969669	+	0.244423i	$0.0785985\pi$
$643$			2.11895e12i			0.488846i	−0.969669	+	0.244423i	$0.921402\pi$
$644$	−4.77610e10	+	8.88246e11i	−0.0109418	+	0.203492i
$645$		0			0
$646$	3.05985e12			0.691279
$647$	7.15695e12			1.60568			0.802840	−	0.596195i	$-0.203321\pi$
$647$	7.15695e12			1.60568			0.802840	+	0.596195i	$0.203321\pi$
$648$		0			0
$649$			8.98676e12i			1.98839i
$650$	−1.15796e12			−0.254439
$651$		0			0
$652$	−1.35468e12			−0.293578
$653$			1.19090e12i			0.256311i	0.991754	+	0.128155i	$0.0409057\pi$
$653$			1.19090e12i			0.256311i	−0.991754	+	0.128155i	$0.959094\pi$
$654$		0			0
$655$	2.89867e12			0.615337
$656$	4.13829e12			0.872476
$657$		0			0
$658$	5.25153e11	−	9.76664e12i	0.109212	−	2.03109i
$659$			1.41037e12i			0.291306i	0.989336	+	0.145653i	$0.0465283\pi$
$659$			1.41037e12i			0.291306i	−0.989336	+	0.145653i	$0.953472\pi$
$660$		0			0
$661$			1.41937e12i			0.289193i	0.989491	+	0.144597i	$0.0461885\pi$
$661$			1.41937e12i			0.289193i	−0.989491	+	0.144597i	$0.953812\pi$
$662$		−	7.23048e12i		−	1.46321i
$663$		0			0
$664$		−	5.63975e12i		−	1.12591i
$665$	8.59627e10	−	1.59871e12i	0.0170456	−	0.317009i
$666$		0			0
$667$	2.48316e12			0.485779
$668$	−2.64578e12			−0.514115
$669$		0			0
$670$		−	1.46453e11i		−	0.0280777i
$671$	−7.52431e11			−0.143290
$672$		0			0
$673$	−8.08652e12			−1.51948			−0.759738	−	0.650229i	$-0.774673\pi$
$673$	−8.08652e12			−1.51948			−0.759738	+	0.650229i	$0.774673\pi$
$674$		−	2.66380e10i		−	0.00497200i
$675$		0			0
$676$	3.13937e11			0.0578206
$677$	−2.89331e12			−0.529353			−0.264677	−	0.964337i	$-0.585265\pi$
$677$	−2.89331e12			−0.529353			−0.264677	+	0.964337i	$0.585265\pi$
$678$		0			0
$679$	4.08473e11	−	7.59666e12i	0.0737478	−	1.37154i
$680$			1.39759e12i			0.250663i
$681$		0			0
$682$		−	1.06396e13i		−	1.88319i
$683$			1.59990e12i			0.281320i	0.990058	+	0.140660i	$0.0449224\pi$
$683$			1.59990e12i			0.281320i	−0.990058	+	0.140660i	$0.955078\pi$
$684$		0			0
$685$			8.14455e11i			0.141338i
$686$	1.11081e12	−	6.83303e12i	0.191505	−	1.17803i
$687$		0			0
$688$	−5.63688e12			−0.959160
$689$	8.32596e12			1.40750
$690$		0			0
$691$		−	4.79904e11i		−	0.0800761i	−0.999198	−	0.0400380i	$-0.987252\pi$
$691$		−	4.79904e11i		−	0.0800761i	0.999198	−	0.0400380i	$-0.0127479\pi$
$692$	2.66945e12			0.442532
$693$		0			0
$694$	9.03911e12			1.47914
$695$			3.47722e12i			0.565328i
$696$		0			0
$697$	−3.56476e12			−0.572114
$698$	7.09153e12			1.13081
$699$		0			0
$700$	−5.38442e11	−	2.89521e10i	−0.0847613	−	0.00455762i
$701$		−	4.40918e12i		−	0.689646i	−0.938668	−	0.344823i	$-0.887939\pi$
$701$		−	4.40918e12i		−	0.689646i	0.938668	−	0.344823i	$-0.112061\pi$
$702$		0			0
$703$		−	5.30935e12i		−	0.819865i
$704$		−	2.18842e12i		−	0.335779i
$705$		0			0
$706$		−	7.36933e11i		−	0.111637i
$707$	9.69669e12	+	5.21391e11i	1.45961	+	0.0784832i
$708$		0			0
$709$	2.60735e12			0.387518			0.193759	−	0.981049i	$-0.437932\pi$
$709$	2.60735e12			0.387518			0.193759	+	0.981049i	$0.437932\pi$
$710$	7.00192e12			1.03408
$711$		0			0
$712$		−	7.23922e12i		−	1.05568i
$713$	−4.54301e12			−0.658325
$714$		0			0
$715$	−3.83389e12			−0.548609
$716$			5.10030e12i			0.725249i
$717$		0			0
$718$	−1.39987e13			−1.96575
$719$	−9.95833e10			−0.0138965			−0.00694827	−	0.999976i	$-0.502212\pi$
$719$	−9.95833e10			−0.0138965			−0.00694827	+	0.999976i	$0.502212\pi$
$720$		0			0
$721$	5.22468e11	−	9.71671e12i	0.0720031	−	1.33909i
$722$		−	4.32296e12i		−	0.592058i
$723$		0			0
$724$		−	4.29645e12i		−	0.581148i
$725$			1.50526e12i			0.202344i
$726$		0			0
$727$			2.07918e12i			0.276049i	0.990429	+	0.138025i	$0.0440753\pi$
$727$			2.07918e12i			0.276049i	−0.990429	+	0.138025i	$0.955925\pi$
$728$	−5.54145e12	−	2.97964e11i	−0.731194	−	0.0393163i
$729$		0			0
$730$	−4.96121e12			−0.646598
$731$	4.85567e12			0.628957
$732$		0			0
$733$			2.60457e12i			0.333249i	0.986020	+	0.166624i	$0.0532867\pi$
$733$			2.60457e12i			0.333249i	−0.986020	+	0.166624i	$0.946713\pi$
$734$	−1.35415e13			−1.72200
$735$		0			0
$736$	3.05059e12			0.383207
$737$		−	4.84893e11i		−	0.0605399i
$738$		0			0
$739$	−1.08275e13			−1.33545			−0.667725	−	0.744408i	$-0.732732\pi$
$739$	−1.08275e13			−1.33545			−0.667725	+	0.744408i	$0.732732\pi$
$740$	−1.78818e12			−0.219214
$741$		0			0
$742$	1.29934e13	+	6.98657e11i	1.57364	+	0.0846148i
$743$			3.39705e12i			0.408933i	0.978874	+	0.204466i	$0.0655459\pi$
$743$			3.39705e12i			0.408933i	−0.978874	+	0.204466i	$0.934454\pi$
$744$		0			0
$745$			3.33933e12i			0.397152i
$746$			1.38036e13i			1.63180i
$747$		0			0
$748$		−	3.41206e12i		−	0.398528i
$749$	−3.12460e11	+	5.81104e12i	−0.0362766	+	0.674661i
$750$		0			0
$751$	1.22671e13			1.40722			0.703608	−	0.710588i	$-0.251571\pi$
$751$	1.22671e13			1.40722			0.703608	+	0.710588i	$0.251571\pi$
$752$	−1.85968e13			−2.12059
$753$		0			0
$754$		−	1.14231e13i		−	1.28710i
$755$	−4.97082e11			−0.0556758
$756$		0			0
$757$	−2.80656e12			−0.310630			−0.155315	−	0.987865i	$-0.549639\pi$
$757$	−2.80656e12			−0.310630			−0.155315	+	0.987865i	$0.549639\pi$
$758$		−	9.48854e12i		−	1.04397i
$759$		0			0
$760$	−2.00579e12			−0.218084
$761$	−4.31887e11			−0.0466809			−0.0233404	−	0.999728i	$-0.507430\pi$
$761$	−4.31887e11			−0.0466809			−0.0233404	+	0.999728i	$0.507430\pi$
$762$		0			0
$763$	−4.27095e12	−	2.29649e11i	−0.456209	−	0.0245304i
$764$		−	6.49335e12i		−	0.689523i
$765$		0			0
$766$		−	6.37273e12i		−	0.668800i
$767$		−	1.76523e13i		−	1.84171i
$768$		0			0
$769$			1.40687e13i			1.45073i	0.688365	+	0.725364i	$0.258329\pi$
$769$			1.40687e13i			1.45073i	−0.688365	+	0.725364i	$0.741671\pi$
$770$	−5.98314e12	−	3.21714e11i	−0.613368	−	0.0329808i
$771$		0			0
$772$	−1.20340e12			−0.121936
$773$	5.29728e12			0.533636			0.266818	−	0.963747i	$-0.414028\pi$
$773$	5.29728e12			0.533636			0.266818	+	0.963747i	$0.414028\pi$
$774$		0			0
$775$		−	2.75391e12i		−	0.274215i
$776$	−9.53100e12			−0.943541
$777$		0			0
$778$	6.50253e12			0.636318
$779$		−	5.11606e12i		−	0.497756i
$780$		0			0
$781$	2.31827e13			2.22964
$782$	−4.88966e12			−0.467571
$783$		0			0
$784$	−1.30867e13	−	1.41143e12i	−1.23711	−	0.133425i
$785$			4.25746e12i			0.400163i
$786$		0			0
$787$			4.64787e12i			0.431884i	0.976406	+	0.215942i	$0.0692822\pi$
$787$			4.64787e12i			0.431884i	−0.976406	+	0.215942i	$0.930718\pi$
$788$			3.48058e11i			0.0321576i
$789$		0			0
$790$		−	8.98360e12i		−	0.820595i
$791$	1.82430e11	−	3.39279e12i	0.0165693	−	0.308150i
$792$		0			0
$793$	1.47797e12			0.132720
$794$	8.70677e12			0.777437
$795$		0			0
$796$			3.51311e12i			0.310158i
$797$	−9.50187e12			−0.834155			−0.417077	−	0.908871i	$-0.636946\pi$
$797$	−9.50187e12			−0.834155			−0.417077	+	0.908871i	$0.636946\pi$
$798$		0			0
$799$	1.60194e13			1.39055
$800$			1.84922e12i			0.159619i
$801$		0			0
$802$	5.96557e12			0.509175
$803$	−1.64261e13			−1.39417
$804$		0			0
$805$	−1.37369e11	+	2.55475e12i	−0.0115294	+	0.214421i
$806$			2.08988e13i			1.74427i
$807$		0			0
$808$		−	1.21658e13i		−	1.00413i
$809$		−	1.02505e12i		−	0.0841354i	−0.999115	−	0.0420677i	$-0.986605\pi$
$809$		−	1.02505e12i		−	0.0841354i	0.999115	−	0.0420677i	$-0.0133945\pi$
$810$		0			0
$811$			1.74059e13i			1.41287i	0.707778	+	0.706435i	$0.249698\pi$
$811$			1.74059e13i			1.41287i	−0.707778	+	0.706435i	$0.750302\pi$
$812$	2.85608e11	−	5.31166e12i	0.0230552	−	0.428773i
$813$		0			0
$814$	−1.98701e13			−1.58632
$815$	−3.89630e12			−0.309345
$816$		0			0
$817$			6.96873e12i			0.547211i
$818$	−2.02809e13			−1.58379
$819$		0			0
$820$	−1.72308e12			−0.133089
$821$			1.96970e13i			1.51306i	0.653958	+	0.756530i	$0.273107\pi$
$821$			1.96970e13i			1.51306i	−0.653958	+	0.756530i	$0.726893\pi$
$822$		0			0
$823$	−1.78396e13			−1.35546			−0.677728	−	0.735313i	$-0.737035\pi$
$823$	−1.78396e13			−1.35546			−0.677728	+	0.735313i	$0.737035\pi$
$824$	−1.21909e13			−0.921218
$825$		0			0
$826$	1.48126e12	−	2.75480e13i	0.110719	−	2.05911i
$827$		−	1.78314e12i		−	0.132559i	−0.997801	−	0.0662796i	$-0.978887\pi$
$827$		−	1.78314e12i		−	0.132559i	0.997801	−	0.0662796i	$-0.0211129\pi$
$828$		0			0
$829$		−	1.39879e13i		−	1.02862i	−0.857603	−	0.514312i	$-0.828047\pi$
$829$		−	1.39879e13i		−	1.02862i	0.857603	−	0.514312i	$-0.171953\pi$
$830$			1.19609e13i			0.874806i
$831$		0			0
$832$			4.29862e12i			0.311010i
$833$	1.12731e13	+	1.21582e12i	0.811221	+	0.0874917i
$834$		0			0
$835$	−7.60973e12			−0.541727
$836$	4.89690e12			0.346731
$837$		0			0
$838$		−	1.87219e13i		−	1.31145i
$839$	5.75101e12			0.400696			0.200348	−	0.979725i	$-0.435793\pi$
$839$	5.75101e12			0.400696			0.200348	+	0.979725i	$0.435793\pi$
$840$		0			0
$841$	−3.42021e11			−0.0235760
$842$		−	8.83119e12i		−	0.605501i
$843$		0			0
$844$	1.11114e13			0.753747
$845$	9.02939e11			0.0609261
$846$		0			0
$847$	−4.85250e12	−	2.60919e11i	−0.323959	−	0.0174193i
$848$		−	2.47409e13i		−	1.64299i
$849$		0			0
$850$		−	2.96404e12i		−	0.194760i
$851$			8.48438e12i			0.554545i
$852$		0			0
$853$			2.24460e13i			1.45167i	0.687868	+	0.725836i	$0.258547\pi$
$853$			2.24460e13i			1.45167i	−0.687868	+	0.725836i	$0.741453\pi$
$854$	2.30650e12	+	1.24021e11i	0.148386	+	0.00797874i
$855$		0			0
$856$	7.29071e12			0.464128
$857$	1.11334e13			0.705039			0.352520	−	0.935804i	$-0.385325\pi$
$857$	1.11334e13			0.705039			0.352520	+	0.935804i	$0.385325\pi$
$858$		0			0
$859$			1.62152e13i			1.01614i	0.861315	+	0.508071i	$0.169641\pi$
$859$			1.62152e13i			1.01614i	−0.861315	+	0.508071i	$0.830359\pi$
$860$	2.34705e12			0.146312
$861$		0			0
$862$	−2.82031e13			−1.73986
$863$			1.61385e13i			0.990411i	0.868776	+	0.495205i	$0.164907\pi$
$863$			1.61385e13i			0.990411i	−0.868776	+	0.495205i	$0.835093\pi$
$864$		0			0
$865$	7.67781e12			0.466300
$866$	−9.28307e12			−0.560868
$867$		0			0
$868$	−5.22527e11	+	9.71781e12i	−0.0312443	+	0.581071i
$869$		−	2.97439e13i		−	1.76933i
$870$		0			0
$871$			9.52453e11i			0.0560741i
$872$			5.35846e12i			0.313846i
$873$		0			0
$874$		−	7.01751e12i		−	0.406801i
$875$	−1.54865e12	−	8.32712e10i	−0.0893137	−	0.00480240i
$876$		0			0
$877$	−1.60083e12			−0.0913793			−0.0456896	−	0.998956i	$-0.514549\pi$
$877$	−1.60083e12			−0.0913793			−0.0456896	+	0.998956i	$0.514549\pi$
$878$	2.54808e13			1.44707
$879$		0			0
$880$			1.13926e13i			0.640398i
$881$	−2.64903e13			−1.48148			−0.740738	−	0.671794i	$-0.765524\pi$
$881$	−2.64903e13			−1.48148			−0.740738	+	0.671794i	$0.765524\pi$
$882$		0			0
$883$	6.32750e12			0.350275			0.175137	−	0.984544i	$-0.443963\pi$
$883$	6.32750e12			0.350275			0.175137	+	0.984544i	$0.443963\pi$
$884$			6.70215e12i			0.369130i
$885$		0			0
$886$	1.34607e13			0.733863
$887$	−3.33242e12			−0.180761			−0.0903803	−	0.995907i	$-0.528808\pi$
$887$	−3.33242e12			−0.180761			−0.0903803	+	0.995907i	$0.528808\pi$
$888$		0			0
$889$	8.09684e12	+	4.35368e11i	0.434768	+	0.0233775i
$890$			1.53531e13i			0.820239i
$891$		0			0
$892$		−	2.72488e12i		−	0.144114i
$893$			2.29907e13i			1.20982i
$894$		0			0
$895$			1.46693e13i			0.764200i
$896$	−1.18742e12	+	2.20833e13i	−0.0615487	+	1.14466i
$897$		0			0
$898$	2.63795e13			1.35370
$899$	2.71669e13			1.38714
$900$		0			0
$901$			2.13121e13i			1.07737i
$902$	−1.91467e13			−0.963087
$903$		0			0
$904$	−4.25670e12			−0.211990
$905$		−	1.23574e13i		−	0.612360i
$906$		0			0
$907$	−1.52610e13			−0.748773			−0.374387	−	0.927273i	$-0.622147\pi$
$907$	−1.52610e13			−0.748773			−0.374387	+	0.927273i	$0.622147\pi$
$908$	1.16331e13			0.567947
$909$		0			0
$910$	1.17524e13	+	6.31928e11i	0.568121	+	0.0305479i
$911$			1.20397e13i			0.579141i	0.957157	+	0.289570i	$0.0935125\pi$
$911$			1.20397e13i			0.579141i	−0.957157	+	0.289570i	$0.906488\pi$
$912$		0			0
$913$			3.96014e13i			1.88622i
$914$		−	3.78744e13i		−	1.79510i
$915$		0			0
$916$			2.29676e12i			0.107792i
$917$	−2.94193e13	−	1.58188e12i	−1.37395	−	0.0738774i
$918$		0			0
$919$	1.36607e11			0.00631762			0.00315881	−	0.999995i	$-0.498995\pi$
$919$	1.36607e11			0.00631762			0.00315881	+	0.999995i	$0.498995\pi$
$920$	3.20527e12			0.147509
$921$		0			0
$922$			1.55192e13i			0.707264i
$923$	−4.55368e13			−2.06516
$924$		0			0
$925$	−5.14311e12			−0.230987
$926$		−	4.51825e13i		−	2.01939i
$927$		0			0
$928$	−1.82423e13			−0.807448
$929$	−3.71234e13			−1.63522			−0.817612	−	0.575769i	$-0.804703\pi$
$929$	−3.71234e13			−1.63522			−0.817612	+	0.575769i	$0.804703\pi$
$930$		0			0
$931$	−1.74492e12	+	1.61788e13i	−0.0761204	+	0.705786i
$932$			8.10196e12i			0.351738i
$933$		0			0
$934$		−	1.28916e13i		−	0.554300i
$935$		−	9.81367e12i		−	0.419932i
$936$		0			0
$937$		−	1.26403e13i		−	0.535708i	−0.963460	−	0.267854i	$-0.913686\pi$
$937$		−	1.26403e13i		−	0.535708i	0.963460	−	0.267854i	$-0.0863144\pi$
$938$	−7.99233e10	+	1.48639e12i	−0.00337101	+	0.0626931i
$939$		0			0
$940$	7.74322e12			0.323479
$941$	1.20085e13			0.499269			0.249634	−	0.968340i	$-0.419690\pi$
$941$	1.20085e13			0.499269			0.249634	+	0.968340i	$0.419690\pi$
$942$		0			0
$943$			8.17550e12i			0.336676i
$944$	−5.24545e13			−2.14985
$945$		0			0
$946$	2.60803e13			1.05877
$947$		−	2.20936e13i		−	0.892672i	−0.894865	−	0.446336i	$-0.852729\pi$
$947$		−	2.20936e13i		−	0.892672i	0.894865	−	0.446336i	$-0.147271\pi$
$948$		0			0
$949$	3.22651e13			1.29132
$950$	4.25392e12			0.169447
$951$		0			0
$952$	7.62703e11	−	1.41845e13i	0.0300947	−	0.559691i
$953$			3.66500e13i			1.43931i	0.694329	+	0.719657i	$0.255701\pi$
$953$			3.66500e13i			1.43931i	−0.694329	+	0.719657i	$0.744299\pi$
$954$		0			0
$955$		−	1.86760e13i		−	0.726555i
$956$			4.18422e12i			0.162014i
$957$		0			0
$958$		−	4.79412e12i		−	0.183893i
$959$	4.44470e11	−	8.26611e12i	0.0169691	−	0.315586i
$960$		0			0
$961$	−2.32629e13			−0.879850
$962$	3.90300e13			1.46930
$963$		0			0
$964$			1.24062e13i			0.462691i
$965$	−3.46120e12			−0.128485
$966$		0			0
$967$	−2.44347e13			−0.898645			−0.449323	−	0.893370i	$-0.648335\pi$
$967$	−2.44347e13			−0.898645			−0.449323	+	0.893370i	$0.648335\pi$
$968$			6.08809e12i			0.222865i
$969$		0			0
$970$	2.02135e13			0.733111
$971$	−1.63686e12			−0.0590913			−0.0295457	−	0.999563i	$-0.509406\pi$
$971$	−1.63686e12			−0.0590913			−0.0295457	+	0.999563i	$0.509406\pi$
$972$		0			0
$973$	1.89761e12	−	3.52912e13i	0.0678734	−	1.26229i
$974$		−	4.97715e12i		−	0.177201i
$975$		0			0
$976$		−	4.39184e12i		−	0.154925i
$977$		−	3.85081e13i		−	1.35215i	−0.736831	−	0.676077i	$-0.763679\pi$
$977$		−	3.85081e13i		−	1.35215i	0.736831	−	0.676077i	$-0.236321\pi$
$978$		0			0
$979$			5.08326e13i			1.76856i
$980$	5.44899e12	+	5.87684e11i	0.188712	+	0.0203529i
$981$		0			0
$982$	−4.08376e13			−1.40139
$983$	2.19840e13			0.750957			0.375479	−	0.926831i	$-0.377478\pi$
$983$	2.19840e13			0.750957			0.375479	+	0.926831i	$0.377478\pi$
$984$		0			0
$985$			1.00108e12i			0.0338847i
$986$	2.92399e13			0.985211
$987$		0			0
$988$	−9.61876e12			−0.321154
$989$		−	1.11361e13i		−	0.370126i
$990$		0			0
$991$	−7.87267e12			−0.259293			−0.129646	−	0.991560i	$-0.541384\pi$
$991$	−7.87267e12			−0.259293			−0.129646	+	0.991560i	$0.541384\pi$
$992$	3.33748e13			1.09425
$993$		0			0
$994$	−7.10643e13	−	3.82113e12i	−2.30894	−	0.124152i
$995$			1.01043e13i			0.326815i
$996$		0			0
$997$			3.28270e13i			1.05221i	0.850419	+	0.526106i	$0.176349\pi$
$997$			3.28270e13i			1.05221i	−0.850419	+	0.526106i	$0.823651\pi$
$998$			2.94644e13i			0.940177i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.b.a.251.12	✓	48
3.2	odd	2		315.10.b.b.251.37	yes	48
7.6	odd	2		315.10.b.b.251.12	yes	48
21.20	even	2	inner	315.10.b.a.251.37	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.10.b.a.251.12	✓	48	1.1	even	1	trivial
315.10.b.a.251.37	yes	48	21.20	even	2	inner
315.10.b.b.251.12	yes	48	7.6	odd	2
315.10.b.b.251.37	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)

\(f(q)\)	\(=\)	\(q-27.0056i q^{2} -217.303 q^{4} -625.000 q^{5} +(6343.29 + 341.079i) q^{7} -7958.48i q^{8} +16878.5i q^{10} +55883.2i q^{11} -109769. i q^{13} +(9211.04 - 171304. i) q^{14} -326183. q^{16} +280977. q^{17} +403251. i q^{19} +135814. q^{20} +1.50916e6 q^{22} -644398. i q^{23} +390625. q^{25} -2.96437e6 q^{26} +(-1.37841e6 - 74117.3i) q^{28} +3.85346e6i q^{29} -7.05000e6i q^{31} +4.73401e6i q^{32} -7.58795e6i q^{34} +(-3.96455e6 - 213174. i) q^{35} -1.31664e7 q^{37} +1.08900e7 q^{38} +4.97405e6i q^{40} -1.26870e7 q^{41} +1.72814e7 q^{43} -1.21436e7i q^{44} -1.74024e7 q^{46} +5.70134e7 q^{47} +(4.01209e7 + 4.32712e6i) q^{49} -1.05491e7i q^{50} +2.38530e7i q^{52} +7.58499e7i q^{53} -3.49270e7i q^{55} +(2.71447e6 - 5.04829e7i) q^{56} +1.04065e8 q^{58} +1.60813e8 q^{59} +1.34644e7i q^{61} -1.90390e8 q^{62} -3.91606e7 q^{64} +6.86055e7i q^{65} -8.67690e6 q^{67} -6.10570e7 q^{68} +(-5.75690e6 + 1.07065e8i) q^{70} -4.14842e8i q^{71} +2.93937e8i q^{73} +3.55566e8i q^{74} -8.76274e7i q^{76} +(-1.90606e7 + 3.54483e8i) q^{77} -5.32251e8 q^{79} +2.03864e8 q^{80} +3.42621e8i q^{82} +7.08646e8 q^{83} -1.75610e8 q^{85} -4.66694e8i q^{86} +4.44745e8 q^{88} +9.09623e8 q^{89} +(3.74398e7 - 6.96295e8i) q^{91} +1.40029e8i q^{92} -1.53968e9i q^{94} -2.52032e8i q^{95} -1.19759e9i q^{97} +(1.16857e8 - 1.08349e9i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12288 q^{4} - 30000 q^{5} + 1824 q^{7} + 442908 q^{14} + 3258948 q^{16} + 7680000 q^{20} - 2860668 q^{22} + 18750000 q^{25} - 5432976 q^{26} - 3685092 q^{28} - 1140000 q^{35} + 7750344 q^{37} + 17423136 q^{38}+ \cdots - 2546372484 q^{98}+O(q^{100}) \) 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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Database

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