LMFDB - Embedded newform 76.7.c.b.37.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,7,Mod(37,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("76.37");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	76.c (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:	$17.4841103551$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5090x^{6} + 8905881x^{4} + 5831691048x^{2} + 827887219200 $ x^8 + 5090x^6 + 8905881x^4 + 5831691048*x^2 + 827887219200
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.7
Root			$40.3415i$ of defining polynomial
Character	$\chi$	$=$	76.37
Dual form			76.7.c.b.37.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+40.3415i q^{3} -186.796 q^{5} +202.293 q^{7} -898.437 q^{9} +O(q^{10})$	$q+40.3415i q^{3} -186.796 q^{5} +202.293 q^{7} -898.437 q^{9} -1591.27 q^{11} +143.277i q^{13} -7535.62i q^{15} +4771.21 q^{17} +(-1010.48 - 6784.16i) q^{19} +8160.80i q^{21} +7737.11 q^{23} +19267.6 q^{25} -6835.36i q^{27} +8820.72i q^{29} -51550.3i q^{31} -64194.3i q^{33} -37787.4 q^{35} -94640.6i q^{37} -5779.99 q^{39} +78747.2i q^{41} -129047. q^{43} +167824. q^{45} -97220.5 q^{47} -76726.6 q^{49} +192478. i q^{51} -46143.8i q^{53} +297243. q^{55} +(273683. - 40764.1i) q^{57} +61729.4i q^{59} -85634.0 q^{61} -181747. q^{63} -26763.5i q^{65} -49044.4i q^{67} +312127. i q^{69} +408517. i q^{71} -130029. q^{73} +777286. i q^{75} -321903. q^{77} -106942. i q^{79} -379212. q^{81} -12258.3 q^{83} -891242. q^{85} -355841. q^{87} -561920. i q^{89} +28983.8i q^{91} +2.07962e6 q^{93} +(188753. + 1.26725e6i) q^{95} -851962. i q^{97} +1.42966e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{5} + 362 q^{7} - 4348 q^{9}+O(q^{10}) $ 8 * q + 2 * q^5 + 362 * q^7 - 4348 * q^9	$ 8 q + 2 q^{5} + 362 q^{7} - 4348 q^{9} + 902 q^{11} + 1550 q^{17} + 6232 q^{19} - 18820 q^{23} - 12158 q^{25} - 101762 q^{35} + 167028 q^{39} - 335042 q^{43} - 57230 q^{45} - 570394 q^{47} + 448182 q^{49} + 1089198 q^{55} + 341316 q^{57} - 632014 q^{61} + 328174 q^{63} - 852938 q^{73} + 1850530 q^{77} - 1819456 q^{81} + 441200 q^{83} - 1828374 q^{85} + 1483380 q^{87} + 2131176 q^{93} + 627950 q^{95} - 865394 q^{99}+O(q^{100}) $ 8 * q + 2 * q^5 + 362 * q^7 - 4348 * q^9 + 902 * q^11 + 1550 * q^17 + 6232 * q^19 - 18820 * q^23 - 12158 * q^25 - 101762 * q^35 + 167028 * q^39 - 335042 * q^43 - 57230 * q^45 - 570394 * q^47 + 448182 * q^49 + 1089198 * q^55 + 341316 * q^57 - 632014 * q^61 + 328174 * q^63 - 852938 * q^73 + 1850530 * q^77 - 1819456 * q^81 + 441200 * q^83 - 1828374 * q^85 + 1483380 * q^87 + 2131176 * q^93 + 627950 * q^95 - 865394 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			40.3415i			1.49413i	0.664751	+	0.747065i	$0.268538\pi$
$3$			40.3415i			1.49413i	−0.664751	+	0.747065i	$0.731462\pi$
$4$		0			0
$5$	−186.796			−1.49437			−0.747183	−	0.664618i	$-0.768594\pi$
$5$	−186.796			−1.49437			−0.747183	+	0.664618i	$0.768594\pi$
$6$		0			0
$7$	202.293			0.589775			0.294888	−	0.955532i	$-0.404718\pi$
$7$	202.293			0.589775			0.294888	+	0.955532i	$0.404718\pi$
$8$		0			0
$9$	−898.437			−1.23242
$10$		0			0
$11$	−1591.27			−1.19555			−0.597773	−	0.801666i	$-0.703947\pi$
$11$	−1591.27			−1.19555			−0.597773	+	0.801666i	$0.703947\pi$
$12$		0			0
$13$			143.277i			0.0652146i	0.999468	+	0.0326073i	$0.0103811\pi$
$13$			143.277i			0.0652146i	−0.999468	+	0.0326073i	$0.989619\pi$
$14$		0			0
$15$		−	7535.62i		−	2.23278i
$16$		0			0
$17$	4771.21			0.971141			0.485570	−	0.874198i	$-0.338612\pi$
$17$	4771.21			0.971141			0.485570	+	0.874198i	$0.338612\pi$
$18$		0			0
$19$	−1010.48	−	6784.16i	−0.147321	−	0.989089i
$19$	−1010.48	−	6784.16i	−0.147321	−	0.989089i
$20$		0			0
$21$			8160.80i			0.881201i
$22$		0			0
$23$	7737.11			0.635910			0.317955	−	0.948106i	$-0.397004\pi$
$23$	7737.11			0.635910			0.317955	+	0.948106i	$0.397004\pi$
$24$		0			0
$25$	19267.6			1.23313
$26$		0			0
$27$		−	6835.36i		−	0.347272i
$28$		0			0
$29$			8820.72i			0.361668i	0.983514	+	0.180834i	$0.0578796\pi$
$29$			8820.72i			0.361668i	−0.983514	+	0.180834i	$0.942120\pi$
$30$		0			0
$31$		−	51550.3i		−	1.73040i	−0.501428	−	0.865199i	$-0.667192\pi$
$31$		−	51550.3i		−	1.73040i	0.501428	−	0.865199i	$-0.332808\pi$
$32$		0			0
$33$		−	64194.3i		−	1.78630i
$34$		0			0
$35$	−37787.4			−0.881340
$36$		0			0
$37$		−	94640.6i		−	1.86841i	−0.356735	−	0.934206i	$-0.616110\pi$
$37$		−	94640.6i		−	1.86841i	0.356735	−	0.934206i	$-0.383890\pi$
$38$		0			0
$39$	−5779.99			−0.0974392
$40$		0			0
$41$			78747.2i			1.14257i	0.820751	+	0.571286i	$0.193555\pi$
$41$			78747.2i			1.14257i	−0.820751	+	0.571286i	$0.806445\pi$
$42$		0			0
$43$	−129047.			−1.62308			−0.811542	−	0.584294i	$-0.801371\pi$
$43$	−129047.			−1.62308			−0.811542	+	0.584294i	$0.801371\pi$
$44$		0			0
$45$	167824.			1.84169
$46$		0			0
$47$	−97220.5			−0.936406			−0.468203	−	0.883621i	$-0.655098\pi$
$47$	−97220.5			−0.936406			−0.468203	+	0.883621i	$0.655098\pi$
$48$		0			0
$49$	−76726.6			−0.652165
$50$		0			0
$51$			192478.i			1.45101i
$52$		0			0
$53$		−	46143.8i		−	0.309946i	−0.987919	−	0.154973i	$-0.950471\pi$
$53$		−	46143.8i		−	0.309946i	0.987919	−	0.154973i	$-0.0495290\pi$
$54$		0			0
$55$	297243.			1.78658
$56$		0			0
$57$	273683.	−	40764.1i	1.47783	−	0.220117i
$58$		0			0
$59$			61729.4i			0.300563i	0.988643	+	0.150282i	$0.0480181\pi$
$59$			61729.4i			0.300563i	−0.988643	+	0.150282i	$0.951982\pi$
$60$		0			0
$61$	−85634.0			−0.377274			−0.188637	−	0.982047i	$-0.560407\pi$
$61$	−85634.0			−0.377274			−0.188637	+	0.982047i	$0.560407\pi$
$62$		0			0
$63$	−181747.			−0.726853
$64$		0			0
$65$		−	26763.5i		−	0.0974545i
$66$		0			0
$67$		−	49044.4i		−	0.163067i	−0.996671	−	0.0815334i	$-0.974018\pi$
$67$		−	49044.4i		−	0.163067i	0.996671	−	0.0815334i	$-0.0259817\pi$
$68$		0			0
$69$			312127.i			0.950132i
$70$		0			0
$71$			408517.i			1.14139i	0.821161	+	0.570696i	$0.193327\pi$
$71$			408517.i			1.14139i	−0.821161	+	0.570696i	$0.806673\pi$
$72$		0			0
$73$	−130029.			−0.334250			−0.167125	−	0.985936i	$-0.553448\pi$
$73$	−130029.			−0.334250			−0.167125	+	0.985936i	$0.553448\pi$
$74$		0			0
$75$			777286.i			1.84246i
$76$		0			0
$77$	−321903.			−0.705103
$78$		0			0
$79$		−	106942.i		−	0.216904i	−0.994102	−	0.108452i	$-0.965411\pi$
$79$		−	106942.i		−	0.216904i	0.994102	−	0.108452i	$-0.0345893\pi$
$80$		0			0
$81$	−379212.			−0.713554
$82$		0			0
$83$	−12258.3			−0.0214386			−0.0107193	−	0.999943i	$-0.503412\pi$
$83$	−12258.3			−0.0214386			−0.0107193	+	0.999943i	$0.503412\pi$
$84$		0			0
$85$	−891242.			−1.45124
$86$		0			0
$87$	−355841.			−0.540379
$88$		0			0
$89$		−	561920.i		−	0.797085i	−0.917150	−	0.398542i	$-0.869516\pi$
$89$		−	561920.i		−	0.797085i	0.917150	−	0.398542i	$-0.130484\pi$
$90$		0			0
$91$			28983.8i			0.0384620i
$92$		0			0
$93$	2.07962e6			2.58544
$94$		0			0
$95$	188753.	+	1.26725e6i	0.220152	+	1.47806i
$96$		0			0
$97$		−	851962.i		−	0.933480i	−0.884395	−	0.466740i	$-0.845428\pi$
$97$		−	851962.i		−	0.933480i	0.884395	−	0.466740i	$-0.154572\pi$
$98$		0			0
$99$	1.42966e6			1.47342
$100$		0			0
$101$	−1.73735e6			−1.68625			−0.843127	−	0.537714i	$-0.819288\pi$
$101$	−1.73735e6			−1.68625			−0.843127	+	0.537714i	$0.819288\pi$
$102$		0			0
$103$		−	1.34950e6i		−	1.23498i	−0.786577	−	0.617492i	$-0.788149\pi$
$103$		−	1.34950e6i		−	1.23498i	0.786577	−	0.617492i	$-0.211851\pi$
$104$		0			0
$105$		−	1.52440e6i		−	1.31684i
$106$		0			0
$107$		−	1.32161e6i		−	1.07883i	−0.842042	−	0.539413i	$-0.818646\pi$
$107$		−	1.32161e6i		−	1.07883i	0.842042	−	0.539413i	$-0.181354\pi$
$108$		0			0
$109$			1.99638e6i			1.54157i	0.637093	+	0.770787i	$0.280137\pi$
$109$			1.99638e6i			1.54157i	−0.637093	+	0.770787i	$0.719863\pi$
$110$		0			0
$111$	3.81795e6			2.79165
$112$		0			0
$113$		−	1.42114e6i		−	0.984920i	−0.870335	−	0.492460i	$-0.836098\pi$
$113$		−	1.42114e6i		−	0.984920i	0.870335	−	0.492460i	$-0.163902\pi$
$114$		0			0
$115$	−1.44526e6			−0.950282
$116$		0			0
$117$		−	128725.i		−	0.0803721i
$118$		0			0
$119$	965182.			0.572754
$120$		0			0
$121$	760581.			0.429328
$122$		0			0
$123$	−3.17678e6			−1.70715
$124$		0			0
$125$	−680431.			−0.348381
$126$		0			0
$127$			3.62081e6i			1.76764i	0.467823	+	0.883822i	$0.345039\pi$
$127$			3.62081e6i			1.76764i	−0.467823	+	0.883822i	$0.654961\pi$
$128$		0			0
$129$		−	5.20593e6i		−	2.42510i
$130$		0			0
$131$	−4.12259e6			−1.83382			−0.916909	−	0.399096i	$-0.869324\pi$
$131$	−4.12259e6			−1.83382			−0.916909	+	0.399096i	$0.869324\pi$
$132$		0			0
$133$	−204412.	−	1.37239e6i	−0.0868864	−	0.583340i
$134$		0			0
$135$			1.27682e6i			0.518952i
$136$		0			0
$137$	3.08355e6			1.19920			0.599598	−	0.800302i	$-0.295327\pi$
$137$	3.08355e6			1.19920			0.599598	+	0.800302i	$0.295327\pi$
$138$		0			0
$139$	−1.59964e6			−0.595631			−0.297816	−	0.954623i	$-0.596258\pi$
$139$	−1.59964e6			−0.595631			−0.297816	+	0.954623i	$0.596258\pi$
$140$		0			0
$141$		−	3.92202e6i		−	1.39911i
$142$		0			0
$143$		−	227992.i		−	0.0779671i
$144$		0			0
$145$		−	1.64767e6i		−	0.540464i
$146$		0			0
$147$		−	3.09527e6i		−	0.974420i
$148$		0			0
$149$	3.38080e6			1.02202			0.511011	−	0.859574i	$-0.329271\pi$
$149$	3.38080e6			1.02202			0.511011	+	0.859574i	$0.329271\pi$
$150$		0			0
$151$			3.08302e6i			0.895458i	0.894169	+	0.447729i	$0.147767\pi$
$151$			3.08302e6i			0.895458i	−0.894169	+	0.447729i	$0.852233\pi$
$152$		0			0
$153$	−4.28664e6			−1.19686
$154$		0			0
$155$			9.62938e6i			2.58585i
$156$		0			0
$157$	3.24712e6			0.839072			0.419536	−	0.907739i	$-0.362193\pi$
$157$	3.24712e6			0.839072			0.419536	+	0.907739i	$0.362193\pi$
$158$		0			0
$159$	1.86151e6			0.463099
$160$		0			0
$161$	1.56516e6			0.375044
$162$		0			0
$163$	−2.32461e6			−0.536769			−0.268385	−	0.963312i	$-0.586490\pi$
$163$	−2.32461e6			−0.536769			−0.268385	+	0.963312i	$0.586490\pi$
$164$		0			0
$165$			1.19912e7i			2.66939i
$166$		0			0
$167$		−	274073.i		−	0.0588460i	−0.999567	−	0.0294230i	$-0.990633\pi$
$167$		−	274073.i		−	0.0588460i	0.999567	−	0.0294230i	$-0.00936698\pi$
$168$		0			0
$169$	4.80628e6			0.995747
$170$		0			0
$171$	907850.	+	6.09514e6i	0.181562	+	1.21898i
$172$		0			0
$173$			2.32755e6i			0.449532i	0.974413	+	0.224766i	$0.0721617\pi$
$173$			2.32755e6i			0.449532i	−0.974413	+	0.224766i	$0.927838\pi$
$174$		0			0
$175$	3.89771e6			0.727269
$176$		0			0
$177$	−2.49026e6			−0.449081
$178$		0			0
$179$		−	6.83222e6i		−	1.19125i	−0.803263	−	0.595625i	$-0.796905\pi$
$179$		−	6.83222e6i		−	1.19125i	0.803263	−	0.595625i	$-0.203095\pi$
$180$		0			0
$181$		−	9.40294e6i		−	1.58573i	−0.609400	−	0.792863i	$-0.708590\pi$
$181$		−	9.40294e6i		−	1.58573i	0.609400	−	0.792863i	$-0.291410\pi$
$182$		0			0
$183$		−	3.45460e6i		−	0.563696i
$184$		0			0
$185$			1.76785e7i			2.79209i
$186$		0			0
$187$	−7.59229e6			−1.16104
$188$		0			0
$189$		−	1.38275e6i		−	0.204813i
$190$		0			0
$191$	−1.12644e7			−1.61662			−0.808309	−	0.588759i	$-0.799617\pi$
$191$	−1.12644e7			−1.61662			−0.808309	+	0.588759i	$0.799617\pi$
$192$		0			0
$193$			1.26456e6i			0.175901i	0.996125	+	0.0879504i	$0.0280317\pi$
$193$			1.26456e6i			0.175901i	−0.996125	+	0.0879504i	$0.971968\pi$
$194$		0			0
$195$	1.07968e6			0.145610
$196$		0			0
$197$	−5.92683e6			−0.775218			−0.387609	−	0.921824i	$-0.626699\pi$
$197$	−5.92683e6			−0.775218			−0.387609	+	0.921824i	$0.626699\pi$
$198$		0			0
$199$	5.49979e6			0.697889			0.348945	−	0.937143i	$-0.386540\pi$
$199$	5.49979e6			0.697889			0.348945	+	0.937143i	$0.386540\pi$
$200$		0			0
$201$	1.97853e6			0.243643
$202$		0			0
$203$			1.78437e6i			0.213303i
$204$		0			0
$205$		−	1.47096e7i		−	1.70742i
$206$		0			0
$207$	−6.95131e6			−0.783711
$208$		0			0
$209$	1.60794e6	+	1.07954e7i	0.176129	+	1.18250i
$210$		0			0
$211$		−	1.20580e7i		−	1.28359i	−0.766876	−	0.641795i	$-0.778190\pi$
$211$		−	1.20580e7i		−	1.28359i	0.766876	−	0.641795i	$-0.221810\pi$
$212$		0			0
$213$	−1.64802e7			−1.70539
$214$		0			0
$215$	2.41053e7			2.42548
$216$		0			0
$217$		−	1.04283e7i		−	1.02055i
$218$		0			0
$219$		−	5.24556e6i		−	0.499412i
$220$		0			0
$221$			683603.i			0.0633326i
$222$		0			0
$223$			1.50611e7i			1.35813i	0.734077	+	0.679066i	$0.237615\pi$
$223$			1.50611e7i			1.35813i	−0.734077	+	0.679066i	$0.762385\pi$
$224$		0			0
$225$	−1.73108e7			−1.51974
$226$		0			0
$227$			1.46485e7i			1.25232i	0.779694	+	0.626161i	$0.215375\pi$
$227$			1.46485e7i			1.25232i	−0.779694	+	0.626161i	$0.784625\pi$
$228$		0			0
$229$	1.69472e7			1.41121			0.705603	−	0.708607i	$-0.250676\pi$
$229$	1.69472e7			1.41121			0.705603	+	0.708607i	$0.250676\pi$
$230$		0			0
$231$		−	1.29860e7i		−	1.05352i
$232$		0			0
$233$	−3.81937e6			−0.301942			−0.150971	−	0.988538i	$-0.548240\pi$
$233$	−3.81937e6			−0.301942			−0.150971	+	0.988538i	$0.548240\pi$
$234$		0			0
$235$	1.81604e7			1.39933
$236$		0			0
$237$	4.31420e6			0.324082
$238$		0			0
$239$	2.51010e7			1.83864			0.919322	−	0.393507i	$-0.128738\pi$
$239$	2.51010e7			1.83864			0.919322	+	0.393507i	$0.128738\pi$
$240$		0			0
$241$		−	7.55161e6i		−	0.539496i	−0.962931	−	0.269748i	$-0.913060\pi$
$241$		−	7.55161e6i		−	0.539496i	0.962931	−	0.269748i	$-0.0869404\pi$
$242$		0			0
$243$		−	2.02810e7i		−	1.41342i
$244$		0			0
$245$	1.43322e7			0.974574
$246$		0			0
$247$	972011.	−	144778.i	0.0645031	−	0.00960750i
$248$		0			0
$249$		−	494519.i		−	0.0320321i
$250$		0			0
$251$	−2.68658e7			−1.69894			−0.849471	−	0.527635i	$-0.823079\pi$
$251$	−2.68658e7			−1.69894			−0.849471	+	0.527635i	$0.823079\pi$
$252$		0			0
$253$	−1.23118e7			−0.760259
$254$		0			0
$255$		−	3.59541e7i		−	2.16834i
$256$		0			0
$257$		−	1.72065e7i		−	1.01366i	−0.862046	−	0.506831i	$-0.830817\pi$
$257$		−	1.72065e7i		−	1.01366i	0.862046	−	0.506831i	$-0.169183\pi$
$258$		0			0
$259$		−	1.91451e7i		−	1.10194i
$260$		0			0
$261$		−	7.92486e6i		−	0.445728i
$262$		0			0
$263$	−7.35861e6			−0.404510			−0.202255	−	0.979333i	$-0.564827\pi$
$263$	−7.35861e6			−0.404510			−0.202255	+	0.979333i	$0.564827\pi$
$264$		0			0
$265$			8.61947e6i			0.463173i
$266$		0			0
$267$	2.26687e7			1.19095
$268$		0			0
$269$			2.68576e7i			1.37978i	0.723913	+	0.689891i	$0.242342\pi$
$269$			2.68576e7i			1.37978i	−0.723913	+	0.689891i	$0.757658\pi$
$270$		0			0
$271$	−1.84648e6			−0.0927762			−0.0463881	−	0.998923i	$-0.514771\pi$
$271$	−1.84648e6			−0.0927762			−0.0463881	+	0.998923i	$0.514771\pi$
$272$		0			0
$273$	−1.16925e6			−0.0574672
$274$		0			0
$275$	−3.06600e7			−1.47426
$276$		0			0
$277$	1.00493e7			0.472819			0.236410	−	0.971653i	$-0.424029\pi$
$277$	1.00493e7			0.472819			0.236410	+	0.971653i	$0.424029\pi$
$278$		0			0
$279$			4.63147e7i			2.13259i
$280$		0			0
$281$			1.79351e7i			0.808321i	0.914688	+	0.404160i	$0.132436\pi$
$281$			1.79351e7i			0.808321i	−0.914688	+	0.404160i	$0.867564\pi$
$282$		0			0
$283$	−3.11035e7			−1.37230			−0.686151	−	0.727460i	$-0.740701\pi$
$283$	−3.11035e7			−1.37230			−0.686151	+	0.727460i	$0.740701\pi$
$284$		0			0
$285$	−5.11229e7	+	7.61457e6i	−2.20841	+	0.328936i
$286$		0			0
$287$			1.59300e7i			0.673860i
$288$		0			0
$289$	−1.37309e6			−0.0568860
$290$		0			0
$291$	3.43694e7			1.39474
$292$		0			0
$293$			2.91396e7i			1.15846i	0.815165	+	0.579229i	$0.196646\pi$
$293$			2.91396e7i			1.15846i	−0.815165	+	0.579229i	$0.803354\pi$
$294$		0			0
$295$		−	1.15308e7i		−	0.449152i
$296$		0			0
$297$			1.08769e7i			0.415180i
$298$		0			0
$299$			1.10855e6i			0.0414706i
$300$		0			0
$301$	−2.61052e7			−0.957254
$302$		0			0
$303$		−	7.00873e7i		−	2.51948i
$304$		0			0
$305$	1.59961e7			0.563785
$306$		0			0
$307$			4.02183e7i			1.38998i	0.719020	+	0.694989i	$0.244591\pi$
$307$			4.02183e7i			1.38998i	−0.719020	+	0.694989i	$0.755409\pi$
$308$		0			0
$309$	5.44409e7			1.84523
$310$		0			0
$311$	−1.73795e7			−0.577771			−0.288885	−	0.957364i	$-0.593285\pi$
$311$	−1.73795e7			−0.577771			−0.288885	+	0.957364i	$0.593285\pi$
$312$		0			0
$313$	3.25313e7			1.06089			0.530443	−	0.847721i	$-0.322026\pi$
$313$	3.25313e7			1.06089			0.530443	+	0.847721i	$0.322026\pi$
$314$		0			0
$315$	3.39497e7			1.08618
$316$		0			0
$317$			2.86504e7i			0.899400i	0.893180	+	0.449700i	$0.148469\pi$
$317$			2.86504e7i			0.899400i	−0.893180	+	0.449700i	$0.851531\pi$
$318$		0			0
$319$		−	1.40361e7i		−	0.432390i
$320$		0			0
$321$	5.33156e7			1.61190
$322$		0			0
$323$	−4.82120e6	−	3.23687e7i	−0.143070	−	0.960544i
$324$		0			0
$325$			2.76060e6i			0.0804181i
$326$		0			0
$327$	−8.05371e7			−2.30331
$328$		0			0
$329$	−1.96670e7			−0.552269
$330$		0			0
$331$			4.74259e7i			1.30777i	0.756594	+	0.653885i	$0.226862\pi$
$331$			4.74259e7i			1.30777i	−0.756594	+	0.653885i	$0.773138\pi$
$332$		0			0
$333$			8.50287e7i			2.30268i
$334$		0			0
$335$			9.16129e6i			0.243681i
$336$		0			0
$337$		−	2.20894e7i		−	0.577157i	−0.957456	−	0.288579i	$-0.906817\pi$
$337$		−	2.20894e7i		−	0.577157i	0.957456	−	0.288579i	$-0.0931827\pi$
$338$		0			0
$339$	5.73309e7			1.47160
$340$		0			0
$341$			8.20305e7i			2.06877i
$342$		0			0
$343$	−3.93208e7			−0.974406
$344$		0			0
$345$		−	5.83039e7i		−	1.41984i
$346$		0			0
$347$	2.73262e7			0.654019			0.327009	−	0.945021i	$-0.393959\pi$
$347$	2.73262e7			0.654019			0.327009	+	0.945021i	$0.393959\pi$
$348$		0			0
$349$	4.12946e7			0.971442			0.485721	−	0.874114i	$-0.338557\pi$
$349$	4.12946e7			0.971442			0.485721	+	0.874114i	$0.338557\pi$
$350$		0			0
$351$	979348.			0.0226473
$352$		0			0
$353$	−5.44872e7			−1.23871			−0.619356	−	0.785110i	$-0.712606\pi$
$353$	−5.44872e7			−1.23871			−0.619356	+	0.785110i	$0.712606\pi$
$354$		0			0
$355$		−	7.63092e7i		−	1.70566i
$356$		0			0
$357$			3.89369e7i			0.855770i
$358$		0			0
$359$	−1.80050e7			−0.389144			−0.194572	−	0.980888i	$-0.562332\pi$
$359$	−1.80050e7			−0.389144			−0.194572	+	0.980888i	$0.562332\pi$
$360$		0			0
$361$	−4.50038e7	+	1.37105e7i	−0.956593	+	0.291428i
$362$		0			0
$363$			3.06830e7i			0.641472i
$364$		0			0
$365$	2.42888e7			0.499491
$366$		0			0
$367$	−8.33600e7			−1.68640			−0.843198	−	0.537603i	$-0.819330\pi$
$367$	−8.33600e7			−1.68640			−0.843198	+	0.537603i	$0.819330\pi$
$368$		0			0
$369$		−	7.07494e7i		−	1.40813i
$370$		0			0
$371$		−	9.33456e6i		−	0.182798i
$372$		0			0
$373$		−	2.46066e7i		−	0.474159i	−0.971490	−	0.237080i	$-0.923810\pi$
$373$		−	2.46066e7i		−	0.474159i	0.971490	−	0.237080i	$-0.0761902\pi$
$374$		0			0
$375$		−	2.74496e7i		−	0.520526i
$376$		0			0
$377$	−1.26380e6			−0.0235860
$378$		0			0
$379$		−	1.80987e7i		−	0.332452i	−0.986088	−	0.166226i	$-0.946842\pi$
$379$		−	1.80987e7i		−	0.332452i	0.986088	−	0.166226i	$-0.0531581\pi$
$380$		0			0
$381$	−1.46069e8			−2.64109
$382$		0			0
$383$		−	4.68217e7i		−	0.833395i	−0.909045	−	0.416697i	$-0.863188\pi$
$383$		−	4.68217e7i		−	0.833395i	0.909045	−	0.416697i	$-0.136812\pi$
$384$		0			0
$385$	6.01300e7			1.05368
$386$		0			0
$387$	1.15940e8			2.00033
$388$		0			0
$389$	−2.49752e7			−0.424288			−0.212144	−	0.977238i	$-0.568045\pi$
$389$	−2.49752e7			−0.424288			−0.212144	+	0.977238i	$0.568045\pi$
$390$		0			0
$391$	3.69154e7			0.617558
$392$		0			0
$393$		−	1.66312e8i		−	2.73996i
$394$		0			0
$395$			1.99763e7i			0.324134i
$396$		0			0
$397$	1.03759e6			0.0165827			0.00829133	−	0.999966i	$-0.497361\pi$
$397$	1.03759e6			0.0165827			0.00829133	+	0.999966i	$0.497361\pi$
$398$		0			0
$399$	5.53642e7	−	8.24630e6i	0.871586	−	0.129820i
$400$		0			0
$401$		−	1.02883e8i		−	1.59555i	−0.602953	−	0.797777i	$-0.706009\pi$
$401$		−	1.02883e8i		−	1.59555i	0.602953	−	0.797777i	$-0.293991\pi$
$402$		0			0
$403$	7.38595e6			0.112847
$404$		0			0
$405$	7.08352e7			1.06631
$406$		0			0
$407$			1.50599e8i			2.23377i
$408$		0			0
$409$		−	7.78751e7i		−	1.13823i	−0.822259	−	0.569113i	$-0.807287\pi$
$409$		−	7.78751e7i		−	1.13823i	0.822259	−	0.569113i	$-0.192713\pi$
$410$		0			0
$411$			1.24395e8i			1.79175i
$412$		0			0
$413$			1.24874e7i			0.177265i
$414$		0			0
$415$	2.28980e6			0.0320371
$416$		0			0
$417$		−	6.45318e7i		−	0.889950i
$418$		0			0
$419$	−1.20628e8			−1.63986			−0.819932	−	0.572461i	$-0.805989\pi$
$419$	−1.20628e8			−1.63986			−0.819932	+	0.572461i	$0.805989\pi$
$420$		0			0
$421$		−	9.94059e7i		−	1.33219i	−0.745867	−	0.666095i	$-0.767965\pi$
$421$		−	9.94059e7i		−	1.33219i	0.745867	−	0.666095i	$-0.232035\pi$
$422$		0			0
$423$	8.73465e7			1.15405
$424$		0			0
$425$	9.19301e7			1.19754
$426$		0			0
$427$	−1.73231e7			−0.222507
$428$		0			0
$429$	9.19753e6			0.116493
$430$		0			0
$431$		−	3.81811e7i		−	0.476888i	−0.971156	−	0.238444i	$-0.923363\pi$
$431$		−	3.81811e7i		−	0.476888i	0.971156	−	0.238444i	$-0.0766373\pi$
$432$		0			0
$433$			1.37273e8i			1.69091i	0.534043	+	0.845457i	$0.320672\pi$
$433$			1.37273e8i			1.69091i	−0.534043	+	0.845457i	$0.679328\pi$
$434$		0			0
$435$	6.64696e7			0.807524
$436$		0			0
$437$	−7.81817e6	−	5.24898e7i	−0.0936830	−	0.628971i
$438$		0			0
$439$			9.17437e6i			0.108438i	0.998529	+	0.0542192i	$0.0172670\pi$
$439$			9.17437e6i			0.108438i	−0.998529	+	0.0542192i	$0.982733\pi$
$440$		0			0
$441$	6.89341e7			0.803745
$442$		0			0
$443$	1.79271e6			0.0206204			0.0103102	−	0.999947i	$-0.496718\pi$
$443$	1.79271e6			0.0206204			0.0103102	+	0.999947i	$0.496718\pi$
$444$		0			0
$445$			1.04964e8i			1.19114i
$446$		0			0
$447$			1.36387e8i			1.52703i
$448$		0			0
$449$			6.43497e6i			0.0710898i	0.999368	+	0.0355449i	$0.0113167\pi$
$449$			6.43497e6i			0.0710898i	−0.999368	+	0.0355449i	$0.988683\pi$
$450$		0			0
$451$		−	1.25308e8i		−	1.36600i
$452$		0			0
$453$	−1.24374e8			−1.33793
$454$		0			0
$455$		−	5.41406e6i		−	0.0574763i
$456$		0			0
$457$	−1.28044e8			−1.34156			−0.670779	−	0.741658i	$-0.734040\pi$
$457$	−1.28044e8			−1.34156			−0.670779	+	0.741658i	$0.734040\pi$
$458$		0			0
$459$		−	3.26130e7i		−	0.337250i
$460$		0			0
$461$	3.10758e7			0.317190			0.158595	−	0.987344i	$-0.449304\pi$
$461$	3.10758e7			0.317190			0.158595	+	0.987344i	$0.449304\pi$
$462$		0			0
$463$	8.91490e7			0.898201			0.449100	−	0.893481i	$-0.351745\pi$
$463$	8.91490e7			0.898201			0.449100	+	0.893481i	$0.351745\pi$
$464$		0			0
$465$	−3.88464e8			−3.86359
$466$		0			0
$467$	−2.26320e7			−0.222214			−0.111107	−	0.993808i	$-0.535440\pi$
$467$	−2.26320e7			−0.222214			−0.111107	+	0.993808i	$0.535440\pi$
$468$		0			0
$469$		−	9.92134e6i		−	0.0961727i
$470$		0			0
$471$			1.30994e8i			1.25368i
$472$		0			0
$473$	2.05348e8			1.94047
$474$		0			0
$475$	−1.94695e7	−	1.30715e8i	−0.181666	−	1.21967i
$476$		0			0
$477$			4.14573e7i			0.381985i
$478$		0			0
$479$	−1.34746e8			−1.22605			−0.613026	−	0.790062i	$-0.710048\pi$
$479$	−1.34746e8			−1.22605			−0.613026	+	0.790062i	$0.710048\pi$
$480$		0			0
$481$	1.35598e7			0.121848
$482$		0			0
$483$			6.31410e7i			0.560364i
$484$		0			0
$485$			1.59143e8i			1.39496i
$486$		0			0
$487$		−	4.94767e7i		−	0.428365i	−0.976794	−	0.214182i	$-0.931291\pi$
$487$		−	4.94767e7i		−	0.428365i	0.976794	−	0.214182i	$-0.0687087\pi$
$488$		0			0
$489$		−	9.37783e7i		−	0.802003i
$490$		0			0
$491$	1.86984e8			1.57965			0.789825	−	0.613333i	$-0.210172\pi$
$491$	1.86984e8			1.57965			0.789825	+	0.613333i	$0.210172\pi$
$492$		0			0
$493$			4.20855e7i			0.351230i
$494$		0			0
$495$	−2.67054e8			−2.20183
$496$		0			0
$497$			8.26400e7i			0.673165i
$498$		0			0
$499$	−6.68560e6			−0.0538070			−0.0269035	−	0.999638i	$-0.508565\pi$
$499$	−6.68560e6			−0.0538070			−0.0269035	+	0.999638i	$0.508565\pi$
$500$		0			0
$501$	1.10565e7			0.0879236
$502$		0			0
$503$	−1.07046e8			−0.841139			−0.420570	−	0.907260i	$-0.638170\pi$
$503$	−1.07046e8			−0.841139			−0.420570	+	0.907260i	$0.638170\pi$
$504$		0			0
$505$	3.24529e8			2.51988
$506$		0			0
$507$			1.93893e8i			1.48778i
$508$		0			0
$509$		−	8.78261e7i		−	0.665994i	−0.942928	−	0.332997i	$-0.891940\pi$
$509$		−	8.78261e7i		−	0.665994i	0.942928	−	0.332997i	$-0.108060\pi$
$510$		0			0
$511$	−2.63039e7			−0.197132
$512$		0			0
$513$	−4.63722e7	+	6.90697e6i	−0.343483	+	0.0511606i
$514$		0			0
$515$			2.52081e8i			1.84552i
$516$		0			0
$517$	1.54704e8			1.11952
$518$		0			0
$519$	−9.38968e7			−0.671659
$520$		0			0
$521$			5.29466e7i			0.374391i	0.982323	+	0.187195i	$0.0599397\pi$
$521$			5.29466e7i			0.374391i	−0.982323	+	0.187195i	$0.940060\pi$
$522$		0			0
$523$			2.04408e8i			1.42887i	0.699701	+	0.714436i	$0.253317\pi$
$523$			2.04408e8i			1.42887i	−0.699701	+	0.714436i	$0.746683\pi$
$524$		0			0
$525$			1.57239e8i			1.08663i
$526$		0			0
$527$		−	2.45958e8i		−	1.68046i
$528$		0			0
$529$	−8.81730e7			−0.595619
$530$		0			0
$531$		−	5.54600e7i		−	0.370422i
$532$		0			0
$533$	−1.12826e7			−0.0745124
$534$		0			0
$535$			2.46871e8i			1.61216i
$536$		0			0
$537$	2.75622e8			1.77988
$538$		0			0
$539$	1.22093e8			0.779693
$540$		0			0
$541$	1.07904e7			0.0681469			0.0340735	−	0.999419i	$-0.489152\pi$
$541$	1.07904e7			0.0681469			0.0340735	+	0.999419i	$0.489152\pi$
$542$		0			0
$543$	3.79329e8			2.36928
$544$		0			0
$545$		−	3.72916e8i		−	2.30368i
$546$		0			0
$547$			5.89052e7i			0.359908i	0.983675	+	0.179954i	$0.0575949\pi$
$547$			5.89052e7i			0.359908i	−0.983675	+	0.179954i	$0.942405\pi$
$548$		0			0
$549$	7.69368e7			0.464961
$550$		0			0
$551$	5.98411e7	−	8.91313e6i	0.357722	−	0.0532814i
$552$		0			0
$553$		−	2.16336e7i		−	0.127924i
$554$		0			0
$555$	−7.13176e8			−4.17175
$556$		0			0
$557$	4.50709e7			0.260814			0.130407	−	0.991461i	$-0.458372\pi$
$557$	4.50709e7			0.260814			0.130407	+	0.991461i	$0.458372\pi$
$558$		0			0
$559$		−	1.84893e7i		−	0.105849i
$560$		0			0
$561$		−	3.06285e8i		−	1.73475i
$562$		0			0
$563$		−	2.83179e8i		−	1.58685i	−0.608670	−	0.793424i	$-0.708297\pi$
$563$		−	2.83179e8i		−	1.58685i	0.608670	−	0.793424i	$-0.291703\pi$
$564$		0			0
$565$			2.65463e8i			1.47183i
$566$		0			0
$567$	−7.67119e7			−0.420837
$568$		0			0
$569$		−	2.97743e8i		−	1.61624i	−0.589021	−	0.808118i	$-0.700486\pi$
$569$		−	2.97743e8i		−	1.61624i	0.589021	−	0.808118i	$-0.299514\pi$
$570$		0			0
$571$	4.51806e7			0.242685			0.121343	−	0.992611i	$-0.461280\pi$
$571$	4.51806e7			0.242685			0.121343	+	0.992611i	$0.461280\pi$
$572$		0			0
$573$		−	4.54422e8i		−	2.41544i
$574$		0			0
$575$	1.49076e8			0.784159
$576$		0			0
$577$	−1.15048e8			−0.598896			−0.299448	−	0.954113i	$-0.596802\pi$
$577$	−1.15048e8			−0.598896			−0.299448	+	0.954113i	$0.596802\pi$
$578$		0			0
$579$	−5.10143e7			−0.262819
$580$		0			0
$581$	−2.47977e6			−0.0126440
$582$		0			0
$583$			7.34273e7i			0.370554i
$584$		0			0
$585$			2.40453e7i			0.120105i
$586$		0			0
$587$	1.59622e8			0.789183			0.394592	−	0.918857i	$-0.370886\pi$
$587$	1.59622e8			0.789183			0.394592	+	0.918857i	$0.370886\pi$
$588$		0			0
$589$	−3.49726e8	+	5.20904e7i	−1.71152	+	0.254925i
$590$		0			0
$591$		−	2.39097e8i		−	1.15828i
$592$		0			0
$593$	−6.60997e7			−0.316983			−0.158491	−	0.987360i	$-0.550663\pi$
$593$	−6.60997e7			−0.316983			−0.158491	+	0.987360i	$0.550663\pi$
$594$		0			0
$595$	−1.80292e8			−0.855905
$596$		0			0
$597$			2.21870e8i			1.04274i
$598$		0			0
$599$		−	1.30989e8i		−	0.609475i	−0.952436	−	0.304738i	$-0.901431\pi$
$599$		−	1.30989e8i		−	0.609475i	0.952436	−	0.304738i	$-0.0985688\pi$
$600$		0			0
$601$			2.96588e8i			1.36625i	0.730303	+	0.683124i	$0.239379\pi$
$601$			2.96588e8i			1.36625i	−0.730303	+	0.683124i	$0.760621\pi$
$602$		0			0
$603$			4.40634e7i			0.200967i
$604$		0			0
$605$	−1.42073e8			−0.641573
$606$		0			0
$607$			7.29083e7i			0.325995i	0.986626	+	0.162997i	$0.0521162\pi$
$607$			7.29083e7i			0.325995i	−0.986626	+	0.162997i	$0.947884\pi$
$608$		0			0
$609$	−7.19841e7			−0.318702
$610$		0			0
$611$		−	1.39294e7i		−	0.0610674i
$612$		0			0
$613$	3.92971e8			1.70600			0.853000	−	0.521910i	$-0.174780\pi$
$613$	3.92971e8			1.70600			0.853000	+	0.521910i	$0.174780\pi$
$614$		0			0
$615$	5.93409e8			2.55111
$616$		0			0
$617$	−2.27576e8			−0.968884			−0.484442	−	0.874823i	$-0.660977\pi$
$617$	−2.27576e8			−0.968884			−0.484442	+	0.874823i	$0.660977\pi$
$618$		0			0
$619$	−1.93252e8			−0.814804			−0.407402	−	0.913249i	$-0.633565\pi$
$619$	−1.93252e8			−0.814804			−0.407402	+	0.913249i	$0.633565\pi$
$620$		0			0
$621$		−	5.28860e7i		−	0.220834i
$622$		0			0
$623$		−	1.13672e8i		−	0.470101i
$624$		0			0
$625$	−1.73955e8			−0.712521
$626$		0			0
$627$	−4.35504e8	+	6.48668e7i	−1.76681	+	0.263160i
$628$		0			0
$629$		−	4.51551e8i		−	1.81449i
$630$		0			0
$631$	1.93900e7			0.0771773			0.0385887	−	0.999255i	$-0.487714\pi$
$631$	1.93900e7			0.0771773			0.0385887	+	0.999255i	$0.487714\pi$
$632$		0			0
$633$	4.86436e8			1.91785
$634$		0			0
$635$		−	6.76353e8i		−	2.64151i
$636$		0			0
$637$		−	1.09931e7i		−	0.0425307i
$638$		0			0
$639$		−	3.67027e8i		−	1.40668i
$640$		0			0
$641$			3.28211e8i			1.24617i	0.782152	+	0.623087i	$0.214122\pi$
$641$			3.28211e8i			1.24617i	−0.782152	+	0.623087i	$0.785878\pi$
$642$		0			0
$643$	2.37495e7			0.0893349			0.0446675	−	0.999002i	$-0.485777\pi$
$643$	2.37495e7			0.0893349			0.0446675	+	0.999002i	$0.485777\pi$
$644$		0			0
$645$			9.72446e8i			3.62398i
$646$		0			0
$647$	1.17583e7			0.0434143			0.0217072	−	0.999764i	$-0.493090\pi$
$647$	1.17583e7			0.0434143			0.0217072	+	0.999764i	$0.493090\pi$
$648$		0			0
$649$		−	9.82282e7i		−	0.359337i
$650$		0			0
$651$	4.20692e8			1.52483
$652$		0			0
$653$	1.27271e8			0.457079			0.228539	−	0.973535i	$-0.426605\pi$
$653$	1.27271e8			0.457079			0.228539	+	0.973535i	$0.426605\pi$
$654$		0			0
$655$	7.70082e8			2.74040
$656$		0			0
$657$	1.16823e8			0.411937
$658$		0			0
$659$			2.32452e8i			0.812227i	0.913823	+	0.406114i	$0.133116\pi$
$659$			2.32452e8i			0.812227i	−0.913823	+	0.406114i	$0.866884\pi$
$660$		0			0
$661$		−	3.14463e8i		−	1.08884i	−0.838812	−	0.544421i	$-0.816750\pi$
$661$		−	3.14463e8i		−	1.08884i	0.838812	−	0.544421i	$-0.183250\pi$
$662$		0			0
$663$	−2.75776e7			−0.0946271
$664$		0			0
$665$	3.81833e7	+	2.56356e8i	0.129840	+	0.871723i
$666$		0			0
$667$			6.82469e7i			0.229988i
$668$		0			0
$669$	−6.07587e8			−2.02923
$670$		0			0
$671$	1.36267e8			0.451048
$672$		0			0
$673$		−	2.76451e8i		−	0.906929i	−0.891274	−	0.453465i	$-0.850188\pi$
$673$		−	2.76451e8i		−	0.906929i	0.891274	−	0.453465i	$-0.149812\pi$
$674$		0			0
$675$		−	1.31701e8i		−	0.428232i
$676$		0			0
$677$			4.08200e8i			1.31555i	0.753215	+	0.657775i	$0.228502\pi$
$677$			4.08200e8i			1.31555i	−0.753215	+	0.657775i	$0.771498\pi$
$678$		0			0
$679$		−	1.72346e8i		−	0.550543i
$680$		0			0
$681$	−5.90943e8			−1.87113
$682$		0			0
$683$		−	3.61070e8i		−	1.13326i	−0.823973	−	0.566630i	$-0.808247\pi$
$683$		−	3.61070e8i		−	1.13326i	0.823973	−	0.566630i	$-0.191753\pi$
$684$		0			0
$685$	−5.75995e8			−1.79204
$686$		0			0
$687$			6.83674e8i			2.10853i
$688$		0			0
$689$	6.61133e6			0.0202130
$690$		0			0
$691$	3.47750e8			1.05398			0.526991	−	0.849871i	$-0.323320\pi$
$691$	3.47750e8			1.05398			0.526991	+	0.849871i	$0.323320\pi$
$692$		0			0
$693$	2.89209e8			0.868986
$694$		0			0
$695$	2.98806e8			0.890091
$696$		0			0
$697$			3.75720e8i			1.10960i
$698$		0			0
$699$		−	1.54079e8i		−	0.451141i
$700$		0			0
$701$	−8.95408e7			−0.259936			−0.129968	−	0.991518i	$-0.541487\pi$
$701$	−8.95408e7			−0.259936			−0.129968	+	0.991518i	$0.541487\pi$
$702$		0			0
$703$	−6.42057e8	+	9.56321e7i	−1.84802	+	0.275257i
$704$		0			0
$705$			7.32617e8i			2.09079i
$706$		0			0
$707$	−3.51453e8			−0.994511
$708$		0			0
$709$	1.93298e8			0.542362			0.271181	−	0.962528i	$-0.412586\pi$
$709$	1.93298e8			0.542362			0.271181	+	0.962528i	$0.412586\pi$
$710$		0			0
$711$			9.60807e7i			0.267317i
$712$		0			0
$713$		−	3.98851e8i		−	1.10038i
$714$		0			0
$715$			4.25879e7i			0.116511i
$716$		0			0
$717$			1.01261e9i			2.74717i
$718$		0			0
$719$	−3.52698e8			−0.948891			−0.474445	−	0.880285i	$-0.657351\pi$
$719$	−3.52698e8			−0.948891			−0.474445	+	0.880285i	$0.657351\pi$
$720$		0			0
$721$		−	2.72994e8i		−	0.728363i
$722$		0			0
$723$	3.04643e8			0.806077
$724$		0			0
$725$			1.69954e8i			0.445983i
$726$		0			0
$727$	−2.02827e8			−0.527865			−0.263932	−	0.964541i	$-0.585020\pi$
$727$	−2.02827e8			−0.527865			−0.263932	+	0.964541i	$0.585020\pi$
$728$		0			0
$729$	5.41719e8			1.39827
$730$		0			0
$731$	−6.15709e8			−1.57624
$732$		0			0
$733$	−4.60785e8			−1.17000			−0.585001	−	0.811033i	$-0.698906\pi$
$733$	−4.60785e8			−1.17000			−0.585001	+	0.811033i	$0.698906\pi$
$734$		0			0
$735$			5.78183e8i			1.45614i
$736$		0			0
$737$			7.80430e7i			0.194954i
$738$		0			0
$739$	−2.61415e8			−0.647734			−0.323867	−	0.946103i	$-0.604983\pi$
$739$	−2.61415e8			−0.647734			−0.323867	+	0.946103i	$0.604983\pi$
$740$		0			0
$741$	5.84055e6	+	3.92124e7i	0.0143549	+	0.0963760i
$742$		0			0
$743$			3.25053e8i			0.792480i	0.918147	+	0.396240i	$0.129685\pi$
$743$			3.25053e8i			0.792480i	−0.918147	+	0.396240i	$0.870315\pi$
$744$		0			0
$745$	−6.31519e8			−1.52728
$746$		0			0
$747$	1.10133e7			0.0264215
$748$		0			0
$749$		−	2.67352e8i		−	0.636264i
$750$		0			0
$751$			2.90899e8i			0.686787i	0.939192	+	0.343393i	$0.111576\pi$
$751$			2.90899e8i			0.686787i	−0.939192	+	0.343393i	$0.888424\pi$
$752$		0			0
$753$		−	1.08381e9i		−	2.53844i
$754$		0			0
$755$		−	5.75895e8i		−	1.33814i
$756$		0			0
$757$	7.60667e8			1.75350			0.876752	−	0.480943i	$-0.159706\pi$
$757$	7.60667e8			1.75350			0.876752	+	0.480943i	$0.159706\pi$
$758$		0			0
$759$		−	4.96678e8i		−	1.13593i
$760$		0			0
$761$	−3.20532e8			−0.727306			−0.363653	−	0.931534i	$-0.618471\pi$
$761$	−3.20532e8			−0.727306			−0.363653	+	0.931534i	$0.618471\pi$
$762$		0			0
$763$			4.03854e8i			0.909182i
$764$		0			0
$765$	8.00726e8			1.78854
$766$		0			0
$767$	−8.84438e6			−0.0196011
$768$		0			0
$769$	4.20733e8			0.925183			0.462592	−	0.886571i	$-0.346920\pi$
$769$	4.20733e8			0.925183			0.462592	+	0.886571i	$0.346920\pi$
$770$		0			0
$771$	6.94136e8			1.51454
$772$		0			0
$773$			6.21759e8i			1.34612i	0.739588	+	0.673060i	$0.235020\pi$
$773$			6.21759e8i			1.34612i	−0.739588	+	0.673060i	$0.764980\pi$
$774$		0			0
$775$		−	9.93253e8i		−	2.13381i
$776$		0			0
$777$	7.72343e8			1.64645
$778$		0			0
$779$	5.34233e8	−	7.95722e7i	1.13010	−	0.168325i
$780$		0			0
$781$		−	6.50061e8i		−	1.36459i
$782$		0			0
$783$	6.02928e7			0.125597
$784$		0			0
$785$	−6.06548e8			−1.25388
$786$		0			0
$787$			1.97860e7i			0.0405913i	0.999794	+	0.0202956i	$0.00646075\pi$
$787$			1.97860e7i			0.0405913i	−0.999794	+	0.0202956i	$0.993539\pi$
$788$		0			0
$789$		−	2.96858e8i		−	0.604390i
$790$		0			0
$791$		−	2.87486e8i		−	0.580881i
$792$		0			0
$793$		−	1.22693e7i		−	0.0246038i
$794$		0			0
$795$	−3.47722e8			−0.692040
$796$		0			0
$797$			5.31780e8i			1.05041i	0.850977	+	0.525203i	$0.176010\pi$
$797$			5.31780e8i			1.05041i	−0.850977	+	0.525203i	$0.823990\pi$
$798$		0			0
$799$	−4.63860e8			−0.909382
$800$		0			0
$801$			5.04850e8i			0.982347i
$802$		0			0
$803$	2.06911e8			0.399610
$804$		0			0
$805$	−2.92366e8			−0.560452
$806$		0			0
$807$	−1.08348e9			−2.06157
$808$		0			0
$809$	8.55853e8			1.61642			0.808209	−	0.588896i	$-0.200437\pi$
$809$	8.55853e8			1.61642			0.808209	+	0.588896i	$0.200437\pi$
$810$		0			0
$811$			6.99910e8i			1.31214i	0.754701	+	0.656069i	$0.227782\pi$
$811$			6.99910e8i			1.31214i	−0.754701	+	0.656069i	$0.772218\pi$
$812$		0			0
$813$		−	7.44898e7i		−	0.138620i
$814$		0			0
$815$	4.34228e8			0.802130
$816$		0			0
$817$	1.30398e8	+	8.75472e8i	0.239115	+	1.60537i
$818$		0			0
$819$		−	2.60402e7i		−	0.0474015i
$820$		0			0
$821$	5.84681e8			1.05655			0.528274	−	0.849074i	$-0.322839\pi$
$821$	5.84681e8			1.05655			0.528274	+	0.849074i	$0.322839\pi$
$822$		0			0
$823$	4.37701e8			0.785196			0.392598	−	0.919710i	$-0.371576\pi$
$823$	4.37701e8			0.785196			0.392598	+	0.919710i	$0.371576\pi$
$824$		0			0
$825$		−	1.23687e9i		−	2.20274i
$826$		0			0
$827$		−	1.36193e8i		−	0.240790i	−0.992726	−	0.120395i	$-0.961584\pi$
$827$		−	1.36193e8i		−	0.240790i	0.992726	−	0.120395i	$-0.0384162\pi$
$828$		0			0
$829$			7.38167e7i			0.129566i	0.997899	+	0.0647830i	$0.0206355\pi$
$829$			7.38167e7i			0.129566i	−0.997899	+	0.0647830i	$0.979364\pi$
$830$		0			0
$831$			4.05403e8i			0.706454i
$832$		0			0
$833$	−3.66079e8			−0.633344
$834$		0			0
$835$			5.11957e7i			0.0879375i
$836$		0			0
$837$	−3.52365e8			−0.600920
$838$		0			0
$839$		−	4.83661e8i		−	0.818946i	−0.912322	−	0.409473i	$-0.865713\pi$
$839$		−	4.83661e8i		−	0.818946i	0.912322	−	0.409473i	$-0.134287\pi$
$840$		0			0
$841$	5.17018e8			0.869196
$842$		0			0
$843$	−7.23527e8			−1.20774
$844$		0			0
$845$	−8.97793e8			−1.48801
$846$		0			0
$847$	1.53860e8			0.253207
$848$		0			0
$849$		−	1.25476e9i		−	2.05040i
$850$		0			0
$851$		−	7.32245e8i		−	1.18814i
$852$		0			0
$853$	−1.52710e8			−0.246048			−0.123024	−	0.992404i	$-0.539259\pi$
$853$	−1.52710e8			−0.246048			−0.123024	+	0.992404i	$0.539259\pi$
$854$		0			0
$855$	−1.69583e8	−	1.13855e9i	−0.271321	−	1.82160i
$856$		0			0
$857$		−	1.05617e9i		−	1.67799i	−0.544138	−	0.838996i	$-0.683143\pi$
$857$		−	1.05617e9i		−	1.67799i	0.544138	−	0.838996i	$-0.316857\pi$
$858$		0			0
$859$	3.53777e8			0.558150			0.279075	−	0.960269i	$-0.409972\pi$
$859$	3.53777e8			0.558150			0.279075	+	0.960269i	$0.409972\pi$
$860$		0			0
$861$	−6.42640e8			−1.00683
$862$		0			0
$863$		−	1.01186e8i		−	0.157431i	−0.996897	−	0.0787153i	$-0.974918\pi$
$863$		−	1.01186e8i		−	0.157431i	0.996897	−	0.0787153i	$-0.0250818\pi$
$864$		0			0
$865$		−	4.34776e8i		−	0.671765i
$866$		0			0
$867$		−	5.53925e7i		−	0.0849951i
$868$		0			0
$869$			1.70174e8i			0.259318i
$870$		0			0
$871$	7.02692e6			0.0106343
$872$		0			0
$873$			7.65434e8i			1.15044i
$874$		0			0
$875$	−1.37646e8			−0.205466
$876$		0			0
$877$		−	1.10266e9i		−	1.63472i	−0.576127	−	0.817360i	$-0.695437\pi$
$877$		−	1.10266e9i		−	1.63472i	0.576127	−	0.817360i	$-0.304563\pi$
$878$		0			0
$879$	−1.17553e9			−1.73089
$880$		0			0
$881$	9.09645e7			0.133028			0.0665142	−	0.997785i	$-0.478812\pi$
$881$	9.09645e7			0.133028			0.0665142	+	0.997785i	$0.478812\pi$
$882$		0			0
$883$	−3.92658e8			−0.570338			−0.285169	−	0.958477i	$-0.592050\pi$
$883$	−3.92658e8			−0.570338			−0.285169	+	0.958477i	$0.592050\pi$
$884$		0			0
$885$	4.65169e8			0.671091
$886$		0			0
$887$			3.46247e8i			0.496152i	0.968741	+	0.248076i	$0.0797983\pi$
$887$			3.46247e8i			0.496152i	−0.968741	+	0.248076i	$0.920202\pi$
$888$		0			0
$889$			7.32465e8i			1.04251i
$890$		0			0
$891$	6.03429e8			0.853086
$892$		0			0
$893$	9.82390e7	+	6.59559e8i	0.137952	+	0.926189i
$894$		0			0
$895$			1.27623e9i			1.78016i
$896$		0			0
$897$	−4.47205e7			−0.0619625
$898$		0			0
$899$	4.54711e8			0.625830
$900$		0			0
$901$		−	2.20162e8i		−	0.301001i
$902$		0			0
$903$		−	1.05312e9i		−	1.43026i
$904$		0			0
$905$			1.75643e9i			2.36965i
$906$		0			0
$907$			4.69987e8i			0.629888i	0.949110	+	0.314944i	$0.101986\pi$
$907$			4.69987e8i			0.629888i	−0.949110	+	0.314944i	$0.898014\pi$
$908$		0			0
$909$	1.56090e9			2.07818
$910$		0			0
$911$		−	2.84138e8i		−	0.375816i	−0.982187	−	0.187908i	$-0.939829\pi$
$911$		−	2.84138e8i		−	0.375816i	0.982187	−	0.187908i	$-0.0601706\pi$
$912$		0			0
$913$	1.95063e7			0.0256308
$914$		0			0
$915$			6.45305e8i			0.842368i
$916$		0			0
$917$	−8.33971e8			−1.08154
$918$		0			0
$919$	1.06131e9			1.36740			0.683700	−	0.729763i	$-0.260370\pi$
$919$	1.06131e9			1.36740			0.683700	+	0.729763i	$0.260370\pi$
$920$		0			0
$921$	−1.62247e9			−2.07681
$922$		0			0
$923$	−5.85309e7			−0.0744355
$924$		0			0
$925$		−	1.82350e9i		−	2.30399i
$926$		0			0
$927$			1.21244e9i			1.52203i
$928$		0			0
$929$	5.01662e8			0.625697			0.312848	−	0.949803i	$-0.398717\pi$
$929$	5.01662e8			0.625697			0.312848	+	0.949803i	$0.398717\pi$
$930$		0			0
$931$	7.75304e7	+	5.20526e8i	0.0960778	+	0.645049i
$932$		0			0
$933$		−	7.01114e8i		−	0.863264i
$934$		0			0
$935$	1.41821e9			1.73502
$936$		0			0
$937$	−6.16919e8			−0.749910			−0.374955	−	0.927043i	$-0.622342\pi$
$937$	−6.16919e8			−0.749910			−0.374955	+	0.927043i	$0.622342\pi$
$938$		0			0
$939$			1.31236e9i			1.58510i
$940$		0			0
$941$			1.00994e9i			1.21206i	0.795441	+	0.606031i	$0.207239\pi$
$941$			1.00994e9i			1.21206i	−0.795441	+	0.606031i	$0.792761\pi$
$942$		0			0
$943$			6.09276e8i			0.726572i
$944$		0			0
$945$			2.58291e8i			0.306065i
$946$		0			0
$947$	6.91940e8			0.814739			0.407369	−	0.913263i	$-0.366446\pi$
$947$	6.91940e8			0.814739			0.407369	+	0.913263i	$0.366446\pi$
$948$		0			0
$949$		−	1.86301e7i		−	0.0217980i
$950$		0			0
$951$	−1.15580e9			−1.34382
$952$		0			0
$953$		−	4.95481e8i		−	0.572464i	−0.958160	−	0.286232i	$-0.907597\pi$
$953$		−	4.95481e8i		−	0.572464i	0.958160	−	0.286232i	$-0.0924028\pi$
$954$		0			0
$955$	2.10414e9			2.41582
$956$		0			0
$957$	5.66239e8			0.646047
$958$		0			0
$959$	6.23781e8			0.707256
$960$		0			0
$961$	−1.76993e9			−1.99428
$962$		0			0
$963$			1.18738e9i			1.32957i
$964$		0			0
$965$		−	2.36215e8i		−	0.262860i
$966$		0			0
$967$	−1.38648e9			−1.53332			−0.766661	−	0.642052i	$-0.778083\pi$
$967$	−1.38648e9			−1.53332			−0.766661	+	0.642052i	$0.778083\pi$
$968$		0			0
$969$	1.30580e9	−	1.94494e8i	1.43518	−	0.213765i
$970$		0			0
$971$			3.47483e8i			0.379556i	0.981827	+	0.189778i	$0.0607769\pi$
$971$			3.47483e8i			0.379556i	−0.981827	+	0.189778i	$0.939223\pi$
$972$		0			0
$973$	−3.23595e8			−0.351288
$974$		0			0
$975$	−1.11367e8			−0.120155
$976$		0			0
$977$			1.77225e9i			1.90038i	0.311664	+	0.950192i	$0.399114\pi$
$977$			1.77225e9i			1.90038i	−0.311664	+	0.950192i	$0.600886\pi$
$978$		0			0
$979$			8.94167e8i			0.952951i
$980$		0			0
$981$		−	1.79363e9i		−	1.89987i
$982$		0			0
$983$		−	2.01843e8i		−	0.212497i	−0.994340	−	0.106248i	$-0.966116\pi$
$983$		−	2.01843e8i		−	0.212497i	0.994340	−	0.106248i	$-0.0338839\pi$
$984$		0			0
$985$	1.10711e9			1.15846
$986$		0			0
$987$		−	7.93397e8i		−	0.825161i
$988$		0			0
$989$	−9.98447e8			−1.03213
$990$		0			0
$991$			7.84564e8i			0.806135i	0.915170	+	0.403067i	$0.132056\pi$
$991$			7.84564e8i			0.806135i	−0.915170	+	0.403067i	$0.867944\pi$
$992$		0			0
$993$	−1.91323e9			−1.95398
$994$		0			0
$995$	−1.02734e9			−1.04290
$996$		0			0
$997$	1.23743e9			1.24863			0.624316	−	0.781172i	$-0.285378\pi$
$997$	1.23743e9			1.24863			0.624316	+	0.781172i	$0.285378\pi$
$998$		0			0
$999$	−6.46903e8			−0.648848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.c.b.37.7	yes	8
3.2	odd	2		684.7.h.c.37.8		8
4.3	odd	2		304.7.e.c.113.2		8
19.18	odd	2	inner	76.7.c.b.37.2	✓	8
57.56	even	2		684.7.h.c.37.7		8
76.75	even	2		304.7.e.c.113.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.c.b.37.2	✓	8	19.18	odd	2	inner
76.7.c.b.37.7	yes	8	1.1	even	1	trivial
304.7.e.c.113.2		8	4.3	odd	2
304.7.e.c.113.7		8	76.75	even	2
684.7.h.c.37.7		8	57.56	even	2
684.7.h.c.37.8		8	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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	76.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+40.3415i q^{3} -186.796 q^{5} +202.293 q^{7} -898.437 q^{9} +O(q^{10})\)	\(q+40.3415i q^{3} -186.796 q^{5} +202.293 q^{7} -898.437 q^{9} -1591.27 q^{11} +143.277i q^{13} -7535.62i q^{15} +4771.21 q^{17} +(-1010.48 - 6784.16i) q^{19} +8160.80i q^{21} +7737.11 q^{23} +19267.6 q^{25} -6835.36i q^{27} +8820.72i q^{29} -51550.3i q^{31} -64194.3i q^{33} -37787.4 q^{35} -94640.6i q^{37} -5779.99 q^{39} +78747.2i q^{41} -129047. q^{43} +167824. q^{45} -97220.5 q^{47} -76726.6 q^{49} +192478. i q^{51} -46143.8i q^{53} +297243. q^{55} +(273683. - 40764.1i) q^{57} +61729.4i q^{59} -85634.0 q^{61} -181747. q^{63} -26763.5i q^{65} -49044.4i q^{67} +312127. i q^{69} +408517. i q^{71} -130029. q^{73} +777286. i q^{75} -321903. q^{77} -106942. i q^{79} -379212. q^{81} -12258.3 q^{83} -891242. q^{85} -355841. q^{87} -561920. i q^{89} +28983.8i q^{91} +2.07962e6 q^{93} +(188753. + 1.26725e6i) q^{95} -851962. i q^{97} +1.42966e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{5} + 362 q^{7} - 4348 q^{9}+O(q^{10}) \) 8 * q + 2 * q^5 + 362 * q^7 - 4348 * q^9	\( 8 q + 2 q^{5} + 362 q^{7} - 4348 q^{9} + 902 q^{11} + 1550 q^{17} + 6232 q^{19} - 18820 q^{23} - 12158 q^{25} - 101762 q^{35} + 167028 q^{39} - 335042 q^{43} - 57230 q^{45} - 570394 q^{47} + 448182 q^{49} + 1089198 q^{55} + 341316 q^{57} - 632014 q^{61} + 328174 q^{63} - 852938 q^{73} + 1850530 q^{77} - 1819456 q^{81} + 441200 q^{83} - 1828374 q^{85} + 1483380 q^{87} + 2131176 q^{93} + 627950 q^{95} - 865394 q^{99}+O(q^{100}) \) 8 * q + 2 * q^5 + 362 * q^7 - 4348 * q^9 + 902 * q^11 + 1550 * q^17 + 6232 * q^19 - 18820 * q^23 - 12158 * q^25 - 101762 * q^35 + 167028 * q^39 - 335042 * q^43 - 57230 * q^45 - 570394 * q^47 + 448182 * q^49 + 1089198 * q^55 + 341316 * q^57 - 632014 * q^61 + 328174 * q^63 - 852938 * q^73 + 1850530 * q^77 - 1819456 * q^81 + 441200 * q^83 - 1828374 * q^85 + 1483380 * q^87 + 2131176 * q^93 + 627950 * q^95 - 865394 * q^99

Introduction

L-functions

Modular forms

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Database

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