LMFDB - Embedded newform 304.9.e.e.113.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,9,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not 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:	$123.843097459$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 $ x^12 + 46118x^10 + 738386961x^8 + 5214446299656x^6 + 17370647163698184x^4 + 24830681474333400768*x^2 + 9218779084612644462864
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.7
Root			$23.4825i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.9.e.e.113.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.7546i q^{3} -919.278 q^{5} +343.629 q^{7} +6445.34 q^{9} +O(q^{10})$	$q+10.7546i q^{3} -919.278 q^{5} +343.629 q^{7} +6445.34 q^{9} -4190.29 q^{11} +16507.2i q^{13} -9886.47i q^{15} +5020.47 q^{17} +(-39747.8 - 124112. i) q^{19} +3695.59i q^{21} -350263. q^{23} +454447. q^{25} +139878. i q^{27} +484397. i q^{29} +263837. i q^{31} -45064.9i q^{33} -315891. q^{35} +2.12516e6i q^{37} -177529. q^{39} +3.03570e6i q^{41} +2.35738e6 q^{43} -5.92506e6 q^{45} -727875. q^{47} -5.64672e6 q^{49} +53993.1i q^{51} +6.39013e6i q^{53} +3.85204e6 q^{55} +(1.33477e6 - 427472. i) q^{57} +1.53982e7i q^{59} +963914. q^{61} +2.21481e6 q^{63} -1.51747e7i q^{65} -3.33818e7i q^{67} -3.76694e6i q^{69} -2.44350e7i q^{71} +2.36687e7 q^{73} +4.88740e6i q^{75} -1.43991e6 q^{77} -5.45025e7i q^{79} +4.07835e7 q^{81} -4.12252e7 q^{83} -4.61521e6 q^{85} -5.20949e6 q^{87} -7.15445e7i q^{89} +5.67237e6i q^{91} -2.83746e6 q^{93} +(3.65393e7 + 1.14093e8i) q^{95} -1.33289e8i q^{97} -2.70078e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 558 q^{5} + 5422 q^{7} - 15592 q^{9}+O(q^{10}) $ 12 * q + 558 * q^5 + 5422 * q^7 - 15592 * q^9	$ 12 q + 558 q^{5} + 5422 q^{7} - 15592 q^{9} + 12546 q^{11} + 270810 q^{17} - 41512 q^{19} + 823956 q^{23} + 865538 q^{25} + 1194378 q^{35} - 5786100 q^{39} - 7586646 q^{43} + 2226046 q^{45} + 20260530 q^{47} - 19498842 q^{49} + 14858554 q^{55} + 14430564 q^{57} - 41363266 q^{61} - 84235798 q^{63} + 87906498 q^{73} - 78817962 q^{77} - 100904812 q^{81} + 55944960 q^{83} + 25440254 q^{85} - 119189604 q^{87} + 105500856 q^{93} - 81396774 q^{95} + 85554938 q^{99}+O(q^{100}) $ 12 * q + 558 * q^5 + 5422 * q^7 - 15592 * q^9 + 12546 * q^11 + 270810 * q^17 - 41512 * q^19 + 823956 * q^23 + 865538 * q^25 + 1194378 * q^35 - 5786100 * q^39 - 7586646 * q^43 + 2226046 * q^45 + 20260530 * q^47 - 19498842 * q^49 + 14858554 * q^55 + 14430564 * q^57 - 41363266 * q^61 - 84235798 * q^63 + 87906498 * q^73 - 78817962 * q^77 - 100904812 * q^81 + 55944960 * q^83 + 25440254 * q^85 - 119189604 * q^87 + 105500856 * q^93 - 81396774 * q^95 + 85554938 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			10.7546i			0.132773i	0.997794	+	0.0663864i	$0.0211470\pi$
$3$			10.7546i			0.132773i	−0.997794	+	0.0663864i	$0.978853\pi$
$4$		0			0
$5$	−919.278			−1.47085			−0.735423	−	0.677609i	$-0.763016\pi$
$5$	−919.278			−1.47085			−0.735423	+	0.677609i	$0.763016\pi$
$6$		0			0
$7$	343.629			0.143119			0.0715596	−	0.997436i	$-0.477202\pi$
$7$	343.629			0.143119			0.0715596	+	0.997436i	$0.477202\pi$
$8$		0			0
$9$	6445.34			0.982371
$10$		0			0
$11$	−4190.29			−0.286202			−0.143101	−	0.989708i	$-0.545707\pi$
$11$	−4190.29			−0.286202			−0.143101	+	0.989708i	$0.545707\pi$
$12$		0			0
$13$			16507.2i			0.577964i	0.957335	+	0.288982i	$0.0933168\pi$
$13$			16507.2i			0.577964i	−0.957335	+	0.288982i	$0.906683\pi$
$14$		0			0
$15$		−	9886.47i		−	0.195288i
$16$		0			0
$17$	5020.47			0.0601103			0.0300551	−	0.999548i	$-0.490432\pi$
$17$	5020.47			0.0601103			0.0300551	+	0.999548i	$0.490432\pi$
$18$		0			0
$19$	−39747.8	−	124112.i	−0.304999	−	0.952353i
$19$	−39747.8	−	124112.i	−0.304999	−	0.952353i
$20$		0			0
$21$			3695.59i			0.0190023i
$22$		0			0
$23$	−350263.			−1.25165			−0.625825	−	0.779963i	$-0.715238\pi$
$23$	−350263.			−1.25165			−0.625825	+	0.779963i	$0.715238\pi$
$24$		0			0
$25$	454447.			1.16339
$26$		0			0
$27$			139878.i			0.263205i
$28$		0			0
$29$			484397.i			0.684872i	0.939541	+	0.342436i	$0.111252\pi$
$29$			484397.i			0.684872i	−0.939541	+	0.342436i	$0.888748\pi$
$30$		0			0
$31$			263837.i			0.285686i	0.989745	+	0.142843i	$0.0456244\pi$
$31$			263837.i			0.285686i	−0.989745	+	0.142843i	$0.954376\pi$
$32$		0			0
$33$		−	45064.9i		−	0.0379999i
$34$		0			0
$35$	−315891.			−0.210506
$36$		0			0
$37$			2.12516e6i			1.13393i	0.823743	+	0.566963i	$0.191882\pi$
$37$			2.12516e6i			1.13393i	−0.823743	+	0.566963i	$0.808118\pi$
$38$		0			0
$39$	−177529.			−0.0767379
$40$		0			0
$41$			3.03570e6i			1.07429i	0.843488	+	0.537147i	$0.180498\pi$
$41$			3.03570e6i			1.07429i	−0.843488	+	0.537147i	$0.819502\pi$
$42$		0			0
$43$	2.35738e6			0.689533			0.344767	−	0.938688i	$-0.387958\pi$
$43$	2.35738e6			0.689533			0.344767	+	0.938688i	$0.387958\pi$
$44$		0			0
$45$	−5.92506e6			−1.44492
$46$		0			0
$47$	−727875.			−0.149164			−0.0745822	−	0.997215i	$-0.523762\pi$
$47$	−727875.			−0.149164			−0.0745822	+	0.997215i	$0.523762\pi$
$48$		0			0
$49$	−5.64672e6			−0.979517
$50$		0			0
$51$			53993.1i			0.00798101i
$52$		0			0
$53$			6.39013e6i			0.809853i	0.914349	+	0.404927i	$0.132703\pi$
$53$			6.39013e6i			0.809853i	−0.914349	+	0.404927i	$0.867297\pi$
$54$		0			0
$55$	3.85204e6			0.420959
$56$		0			0
$57$	1.33477e6	−	427472.i	0.126447	−	0.0404956i
$58$		0			0
$59$			1.53982e7i			1.27075i	0.772203	+	0.635376i	$0.219155\pi$
$59$			1.53982e7i			1.27075i	−0.772203	+	0.635376i	$0.780845\pi$
$60$		0			0
$61$	963914.			0.0696176			0.0348088	−	0.999394i	$-0.488918\pi$
$61$	963914.			0.0696176			0.0348088	+	0.999394i	$0.488918\pi$
$62$		0			0
$63$	2.21481e6			0.140596
$64$		0			0
$65$		−	1.51747e7i		−	0.850095i
$66$		0			0
$67$		−	3.33818e7i		−	1.65657i	−0.560306	−	0.828286i	$-0.689316\pi$
$67$		−	3.33818e7i		−	1.65657i	0.560306	−	0.828286i	$-0.310684\pi$
$68$		0			0
$69$		−	3.76694e6i		−	0.166185i
$70$		0			0
$71$		−	2.44350e7i		−	0.961567i	−0.876839	−	0.480783i	$-0.840352\pi$
$71$		−	2.44350e7i		−	0.961567i	0.876839	−	0.480783i	$-0.159648\pi$
$72$		0			0
$73$	2.36687e7			0.833455			0.416728	−	0.909031i	$-0.363177\pi$
$73$	2.36687e7			0.833455			0.416728	+	0.909031i	$0.363177\pi$
$74$		0			0
$75$			4.88740e6i			0.154466i
$76$		0			0
$77$	−1.43991e6			−0.0409610
$78$		0			0
$79$		−	5.45025e7i		−	1.39929i	−0.714490	−	0.699646i	$-0.753341\pi$
$79$		−	5.45025e7i		−	1.39929i	0.714490	−	0.699646i	$-0.246659\pi$
$80$		0			0
$81$	4.07835e7			0.947425
$82$		0			0
$83$	−4.12252e7			−0.868662			−0.434331	−	0.900753i	$-0.643015\pi$
$83$	−4.12252e7			−0.868662			−0.434331	+	0.900753i	$0.643015\pi$
$84$		0			0
$85$	−4.61521e6			−0.0884129
$86$		0			0
$87$	−5.20949e6			−0.0909324
$88$		0			0
$89$		−	7.15445e7i		−	1.14029i	−0.821543	−	0.570146i	$-0.806887\pi$
$89$		−	7.15445e7i		−	1.14029i	0.821543	−	0.570146i	$-0.193113\pi$
$90$		0			0
$91$			5.67237e6i			0.0827178i
$92$		0			0
$93$	−2.83746e6			−0.0379313
$94$		0			0
$95$	3.65393e7	+	1.14093e8i	0.448607	+	1.40076i
$96$		0			0
$97$		−	1.33289e8i		−	1.50559i	−0.658255	−	0.752795i	$-0.728705\pi$
$97$		−	1.33289e8i		−	1.50559i	0.658255	−	0.752795i	$-0.271295\pi$
$98$		0			0
$99$	−2.70078e7			−0.281157
$100$		0			0
$101$	−2.20883e7			−0.212264			−0.106132	−	0.994352i	$-0.533847\pi$
$101$	−2.20883e7			−0.212264			−0.106132	+	0.994352i	$0.533847\pi$
$102$		0			0
$103$			1.42603e7i			0.126701i	0.997991	+	0.0633503i	$0.0201786\pi$
$103$			1.42603e7i			0.126701i	−0.997991	+	0.0633503i	$0.979821\pi$
$104$		0			0
$105$		−	3.39728e6i		−	0.0279495i
$106$		0			0
$107$		−	5.24572e6i		−	0.0400193i	−0.999800	−	0.0200097i	$-0.993630\pi$
$107$		−	5.24572e6i		−	0.0400193i	0.999800	−	0.0200097i	$-0.00636970\pi$
$108$		0			0
$109$			1.90488e8i			1.34947i	0.738061	+	0.674734i	$0.235742\pi$
$109$			1.90488e8i			1.34947i	−0.738061	+	0.674734i	$0.764258\pi$
$110$		0			0
$111$	−2.28552e7			−0.150555
$112$		0			0
$113$		−	2.04711e8i		−	1.25553i	−0.778403	−	0.627765i	$-0.783970\pi$
$113$		−	2.04711e8i		−	1.25553i	0.778403	−	0.627765i	$-0.216030\pi$
$114$		0			0
$115$	3.21989e8			1.84098
$116$		0			0
$117$			1.06395e8i			0.567775i
$118$		0			0
$119$	1.72518e6			0.00860293
$120$		0			0
$121$	−1.96800e8			−0.918088
$122$		0			0
$123$	−3.26477e7			−0.142637
$124$		0			0
$125$	−5.86705e7			−0.240314
$126$		0			0
$127$			1.24877e7i			0.0480027i	0.999712	+	0.0240014i	$0.00764061\pi$
$127$			1.24877e7i			0.0480027i	−0.999712	+	0.0240014i	$0.992359\pi$
$128$		0			0
$129$			2.53526e7i			0.0915513i
$130$		0			0
$131$	−1.65537e8			−0.562094			−0.281047	−	0.959694i	$-0.590682\pi$
$131$	−1.65537e8			−0.562094			−0.281047	+	0.959694i	$0.590682\pi$
$132$		0			0
$133$	−1.36585e7	−	4.26484e7i	−0.0436512	−	0.136300i
$134$		0			0
$135$		−	1.28587e8i		−	0.387134i
$136$		0			0
$137$	5.08631e8			1.44384			0.721922	−	0.691974i	$-0.243259\pi$
$137$	5.08631e8			1.44384			0.721922	+	0.691974i	$0.243259\pi$
$138$		0			0
$139$	−1.29060e8			−0.345727			−0.172864	−	0.984946i	$-0.555302\pi$
$139$	−1.29060e8			−0.345727			−0.172864	+	0.984946i	$0.555302\pi$
$140$		0			0
$141$		−	7.82800e6i		−	0.0198050i
$142$		0			0
$143$		−	6.91700e7i		−	0.165415i
$144$		0			0
$145$		−	4.45295e8i		−	1.00734i
$146$		0			0
$147$		−	6.07282e7i		−	0.130053i
$148$		0			0
$149$	7.74775e8			1.57192			0.785961	−	0.618277i	$-0.212169\pi$
$149$	7.74775e8			1.57192			0.785961	+	0.618277i	$0.212169\pi$
$150$		0			0
$151$		−	7.13683e8i		−	1.37277i	−0.727238	−	0.686385i	$-0.759197\pi$
$151$		−	7.13683e8i		−	1.37277i	0.727238	−	0.686385i	$-0.240803\pi$
$152$		0			0
$153$	3.23586e7			0.0590506
$154$		0			0
$155$		−	2.42539e8i		−	0.420200i
$156$		0			0
$157$	−7.63715e8			−1.25699			−0.628497	−	0.777812i	$-0.716329\pi$
$157$	−7.63715e8			−1.25699			−0.628497	+	0.777812i	$0.716329\pi$
$158$		0			0
$159$	−6.87233e7			−0.107527
$160$		0			0
$161$	−1.20361e8			−0.179135
$162$		0			0
$163$	5.73887e8			0.812972			0.406486	−	0.913657i	$-0.366754\pi$
$163$	5.73887e8			0.812972			0.406486	+	0.913657i	$0.366754\pi$
$164$		0			0
$165$			4.14271e7i			0.0558919i
$166$		0			0
$167$		−	5.55377e8i		−	0.714040i	−0.934097	−	0.357020i	$-0.883793\pi$
$167$		−	5.55377e8i		−	0.714040i	0.934097	−	0.357020i	$-0.116207\pi$
$168$		0			0
$169$	5.43242e8			0.665958
$170$		0			0
$171$	−2.56188e8	−	7.99941e8i	−0.299622	−	0.935564i
$172$		0			0
$173$			6.56811e8i			0.733256i	0.930368	+	0.366628i	$0.119488\pi$
$173$			6.56811e8i			0.733256i	−0.930368	+	0.366628i	$0.880512\pi$
$174$		0			0
$175$	1.56161e8			0.166503
$176$		0			0
$177$	−1.65601e8			−0.168721
$178$		0			0
$179$		−	1.17573e9i		−	1.14523i	−0.819823	−	0.572617i	$-0.805928\pi$
$179$		−	1.17573e9i		−	1.14523i	0.819823	−	0.572617i	$-0.194072\pi$
$180$		0			0
$181$		−	6.73071e8i		−	0.627114i	−0.949569	−	0.313557i	$-0.898479\pi$
$181$		−	6.73071e8i		−	0.627114i	0.949569	−	0.313557i	$-0.101521\pi$
$182$		0			0
$183$			1.03665e7i			0.00924332i
$184$		0			0
$185$		−	1.95361e9i		−	1.66783i
$186$		0			0
$187$	−2.10372e7			−0.0172037
$188$		0			0
$189$			4.80662e7i			0.0376697i
$190$		0			0
$191$	2.49800e9			1.87698			0.938488	−	0.345311i	$-0.112227\pi$
$191$	2.49800e9			1.87698			0.938488	+	0.345311i	$0.112227\pi$
$192$		0			0
$193$		−	2.38380e9i		−	1.71807i	−0.511921	−	0.859033i	$-0.671066\pi$
$193$		−	2.38380e9i		−	1.71807i	0.511921	−	0.859033i	$-0.328934\pi$
$194$		0			0
$195$	1.63198e8			0.112870
$196$		0			0
$197$	−4.08382e8			−0.271145			−0.135573	−	0.990767i	$-0.543287\pi$
$197$	−4.08382e8			−0.271145			−0.135573	+	0.990767i	$0.543287\pi$
$198$		0			0
$199$	1.99524e9			1.27228			0.636139	−	0.771574i	$-0.280530\pi$
$199$	1.99524e9			1.27228			0.636139	+	0.771574i	$0.280530\pi$
$200$		0			0
$201$	3.59008e8			0.219948
$202$		0			0
$203$			1.66453e8i			0.0980183i
$204$		0			0
$205$		−	2.79065e9i		−	1.58012i
$206$		0			0
$207$	−2.25756e9			−1.22959
$208$		0			0
$209$	1.66555e8	+	5.20063e8i	0.0872915	+	0.272565i
$210$		0			0
$211$		−	2.21085e9i		−	1.11540i	−0.830044	−	0.557698i	$-0.811685\pi$
$211$		−	2.21085e9i		−	1.11540i	0.830044	−	0.557698i	$-0.188315\pi$
$212$		0			0
$213$	2.62789e8			0.127670
$214$		0			0
$215$	−2.16708e9			−1.01420
$216$		0			0
$217$			9.06621e7i			0.0408871i
$218$		0			0
$219$			2.54547e8i			0.110660i
$220$		0			0
$221$			8.28740e7i			0.0347416i
$222$		0			0
$223$		−	1.00945e9i		−	0.408193i	−0.978951	−	0.204096i	$-0.934574\pi$
$223$		−	1.00945e9i		−	0.408193i	0.978951	−	0.204096i	$-0.0654256\pi$
$224$		0			0
$225$	2.92907e9			1.14288
$226$		0			0
$227$			1.07230e9i			0.403844i	0.979402	+	0.201922i	$0.0647187\pi$
$227$			1.07230e9i			0.403844i	−0.979402	+	0.201922i	$0.935281\pi$
$228$		0			0
$229$	−2.18741e9			−0.795406			−0.397703	−	0.917514i	$-0.630193\pi$
$229$	−2.18741e9			−0.795406			−0.397703	+	0.917514i	$0.630193\pi$
$230$		0			0
$231$		−	1.54856e7i		−	0.00543851i
$232$		0			0
$233$	−1.60064e9			−0.543087			−0.271544	−	0.962426i	$-0.587534\pi$
$233$	−1.60064e9			−0.543087			−0.271544	+	0.962426i	$0.587534\pi$
$234$		0			0
$235$	6.69119e8			0.219398
$236$		0			0
$237$	5.86153e8			0.185788
$238$		0			0
$239$	−2.42499e9			−0.743222			−0.371611	−	0.928388i	$-0.621195\pi$
$239$	−2.42499e9			−0.743222			−0.371611	+	0.928388i	$0.621195\pi$
$240$		0			0
$241$		−	1.99574e9i		−	0.591610i	−0.955248	−	0.295805i	$-0.904412\pi$
$241$		−	1.99574e9i		−	0.591610i	0.955248	−	0.295805i	$-0.0955879\pi$
$242$		0			0
$243$			1.35635e9i			0.388997i
$244$		0			0
$245$	5.19091e9			1.44072
$246$		0			0
$247$	2.04874e9	−	6.56126e8i	0.550426	−	0.176279i
$248$		0			0
$249$		−	4.43361e8i		−	0.115335i
$250$		0			0
$251$	−2.54465e9			−0.641110			−0.320555	−	0.947230i	$-0.603869\pi$
$251$	−2.54465e9			−0.641110			−0.320555	+	0.947230i	$0.603869\pi$
$252$		0			0
$253$	1.46770e9			0.358225
$254$		0			0
$255$		−	4.96347e7i		−	0.0117388i
$256$		0			0
$257$			5.40032e9i			1.23790i	0.785429	+	0.618952i	$0.212442\pi$
$257$			5.40032e9i			1.23790i	−0.785429	+	0.618952i	$0.787558\pi$
$258$		0			0
$259$			7.30267e8i			0.162287i
$260$		0			0
$261$			3.12210e9i			0.672798i
$262$		0			0
$263$	−4.20312e9			−0.878514			−0.439257	−	0.898361i	$-0.644758\pi$
$263$	−4.20312e9			−0.878514			−0.439257	+	0.898361i	$0.644758\pi$
$264$		0			0
$265$		−	5.87431e9i		−	1.19117i
$266$		0			0
$267$	7.69432e8			0.151400
$268$		0			0
$269$			1.65507e7i			0.00316087i	0.999999	+	0.00158044i	$0.000503069\pi$
$269$			1.65507e7i			0.00316087i	−0.999999	+	0.00158044i	$0.999497\pi$
$270$		0			0
$271$	8.10231e9			1.50221			0.751107	−	0.660180i	$-0.229520\pi$
$271$	8.10231e9			1.50221			0.751107	+	0.660180i	$0.229520\pi$
$272$		0			0
$273$	−6.10040e7			−0.0109827
$274$		0			0
$275$	−1.90427e9			−0.332964
$276$		0			0
$277$	−4.75322e9			−0.807363			−0.403682	−	0.914900i	$-0.632270\pi$
$277$	−4.75322e9			−0.807363			−0.403682	+	0.914900i	$0.632270\pi$
$278$		0			0
$279$			1.70052e9i			0.280650i
$280$		0			0
$281$			1.26345e9i			0.202643i	0.994854	+	0.101321i	$0.0323070\pi$
$281$			1.26345e9i			0.202643i	−0.994854	+	0.101321i	$0.967693\pi$
$282$		0			0
$283$	6.11988e9			0.954107			0.477054	−	0.878874i	$-0.341705\pi$
$283$	6.11988e9			0.954107			0.477054	+	0.878874i	$0.341705\pi$
$284$		0			0
$285$	−1.22702e9	+	3.92965e8i	−0.185983	+	0.0595628i
$286$		0			0
$287$			1.04316e9i			0.153752i
$288$		0			0
$289$	−6.95055e9			−0.996387
$290$		0			0
$291$	1.43347e9			0.199901
$292$		0			0
$293$		−	8.14824e9i		−	1.10559i	−0.833318	−	0.552794i	$-0.813562\pi$
$293$		−	8.14824e9i		−	1.10559i	0.833318	−	0.552794i	$-0.186438\pi$
$294$		0			0
$295$		−	1.41552e10i		−	1.86908i
$296$		0			0
$297$		−	5.86129e8i		−	0.0753299i
$298$		0			0
$299$		−	5.78187e9i		−	0.723409i
$300$		0			0
$301$	8.10063e8			0.0986854
$302$		0			0
$303$		−	2.37550e8i		−	0.0281829i
$304$		0			0
$305$	−8.86105e8			−0.102397
$306$		0			0
$307$		−	7.97886e9i		−	0.898229i	−0.893474	−	0.449115i	$-0.851739\pi$
$307$		−	7.97886e9i		−	0.898229i	0.893474	−	0.449115i	$-0.148261\pi$
$308$		0			0
$309$	−1.53364e8			−0.0168224
$310$		0			0
$311$	1.02790e10			1.09878			0.549390	−	0.835566i	$-0.314860\pi$
$311$	1.02790e10			1.09878			0.549390	+	0.835566i	$0.314860\pi$
$312$		0			0
$313$	6.84361e9			0.713030			0.356515	−	0.934290i	$-0.383965\pi$
$313$	6.84361e9			0.713030			0.356515	+	0.934290i	$0.383965\pi$
$314$		0			0
$315$	−2.03602e9			−0.206795
$316$		0			0
$317$		−	1.07817e10i		−	1.06770i	−0.845579	−	0.533851i	$-0.820744\pi$
$317$		−	1.07817e10i		−	1.06770i	0.845579	−	0.533851i	$-0.179256\pi$
$318$		0			0
$319$		−	2.02976e9i		−	0.196012i
$320$		0			0
$321$	5.64156e7			0.00531348
$322$		0			0
$323$	−1.99553e8	−	6.23098e8i	−0.0183336	−	0.0572462i
$324$		0			0
$325$			7.50167e9i			0.672395i
$326$		0			0
$327$	−2.04863e9			−0.179173
$328$		0			0
$329$	−2.50119e8			−0.0213483
$330$		0			0
$331$		−	4.83111e9i		−	0.402471i	−0.979543	−	0.201235i	$-0.935504\pi$
$331$		−	4.83111e9i		−	0.402471i	0.979543	−	0.201235i	$-0.0644956\pi$
$332$		0			0
$333$			1.36974e10i			1.11394i
$334$		0			0
$335$			3.06871e10i			2.43656i
$336$		0			0
$337$			1.24834e10i			0.967860i	0.875107	+	0.483930i	$0.160791\pi$
$337$			1.24834e10i			0.967860i	−0.875107	+	0.483930i	$0.839209\pi$
$338$		0			0
$339$	2.20158e9			0.166700
$340$		0			0
$341$		−	1.10555e9i		−	0.0817639i
$342$		0			0
$343$	−3.92133e9			−0.283307
$344$		0			0
$345$			3.46287e9i			0.244433i
$346$		0			0
$347$	−2.24032e10			−1.54522			−0.772612	−	0.634879i	$-0.781050\pi$
$347$	−2.24032e10			−1.54522			−0.772612	+	0.634879i	$0.781050\pi$
$348$		0			0
$349$	2.47555e10			1.66867			0.834335	−	0.551257i	$-0.185852\pi$
$349$	2.47555e10			1.66867			0.834335	+	0.551257i	$0.185852\pi$
$350$		0			0
$351$	−2.30900e9			−0.152123
$352$		0			0
$353$	7.58628e9			0.488573			0.244287	−	0.969703i	$-0.421446\pi$
$353$	7.58628e9			0.488573			0.244287	+	0.969703i	$0.421446\pi$
$354$		0			0
$355$			2.24626e10i			1.41432i
$356$		0			0
$357$			1.85536e7i			0.00114224i
$358$		0			0
$359$	6.91505e9			0.416311			0.208155	−	0.978096i	$-0.433254\pi$
$359$	6.91505e9			0.416311			0.208155	+	0.978096i	$0.433254\pi$
$360$		0			0
$361$	−1.38238e10	+	9.86632e9i	−0.813951	+	0.580934i
$362$		0			0
$363$		−	2.11651e9i		−	0.121897i
$364$		0			0
$365$	−2.17581e10			−1.22588
$366$		0			0
$367$	−3.46443e10			−1.90971			−0.954855	−	0.297072i	$-0.903990\pi$
$367$	−3.46443e10			−1.90971			−0.954855	+	0.297072i	$0.903990\pi$
$368$		0			0
$369$			1.95661e10i			1.05536i
$370$		0			0
$371$			2.19584e9i			0.115906i
$372$		0			0
$373$		−	1.32696e10i		−	0.685521i	−0.939423	−	0.342761i	$-0.888638\pi$
$373$		−	1.32696e10i		−	0.685521i	0.939423	−	0.342761i	$-0.111362\pi$
$374$		0			0
$375$		−	6.30978e8i		−	0.0319072i
$376$		0			0
$377$	−7.99605e9			−0.395831
$378$		0			0
$379$			1.37892e10i			0.668318i	0.942517	+	0.334159i	$0.108452\pi$
$379$			1.37892e10i			0.668318i	−0.942517	+	0.334159i	$0.891548\pi$
$380$		0			0
$381$	−1.34300e8			−0.00637346
$382$		0			0
$383$			2.21901e10i			1.03125i	0.856814	+	0.515626i	$0.172440\pi$
$383$			2.21901e10i			1.03125i	−0.856814	+	0.515626i	$0.827560\pi$
$384$		0			0
$385$	1.32367e9			0.0602474
$386$		0			0
$387$	1.51941e10			0.677378
$388$		0			0
$389$	5.26015e9			0.229721			0.114860	−	0.993382i	$-0.463358\pi$
$389$	5.26015e9			0.229721			0.114860	+	0.993382i	$0.463358\pi$
$390$		0			0
$391$	−1.75848e9			−0.0752370
$392$		0			0
$393$		−	1.78028e9i		−	0.0746308i
$394$		0			0
$395$			5.01030e10i			2.05814i
$396$		0			0
$397$	−8.36911e9			−0.336913			−0.168456	−	0.985709i	$-0.553878\pi$
$397$	−8.36911e9			−0.336913			−0.168456	+	0.985709i	$0.553878\pi$
$398$		0			0
$399$	4.58666e8	−	1.46892e8i	0.0180969	−	0.00579570i
$400$		0			0
$401$			3.79985e10i			1.46957i	0.678302	+	0.734783i	$0.262716\pi$
$401$			3.79985e10i			1.46957i	−0.678302	+	0.734783i	$0.737284\pi$
$402$		0			0
$403$	−4.35522e9			−0.165116
$404$		0			0
$405$	−3.74914e10			−1.39352
$406$		0			0
$407$		−	8.90503e9i		−	0.324532i
$408$		0			0
$409$			4.18057e9i			0.149397i	0.997206	+	0.0746985i	$0.0237994\pi$
$409$			4.18057e9i			0.149397i	−0.997206	+	0.0746985i	$0.976201\pi$
$410$		0			0
$411$			5.47012e9i			0.191703i
$412$		0			0
$413$			5.29126e9i			0.181869i
$414$		0			0
$415$	3.78975e10			1.27767
$416$		0			0
$417$		−	1.38799e9i		−	0.0459032i
$418$		0			0
$419$	−6.81207e9			−0.221016			−0.110508	−	0.993875i	$-0.535248\pi$
$419$	−6.81207e9			−0.221016			−0.110508	+	0.993875i	$0.535248\pi$
$420$		0			0
$421$		−	4.75587e9i		−	0.151392i	−0.997131	−	0.0756958i	$-0.975882\pi$
$421$		−	4.75587e9i		−	0.151392i	0.997131	−	0.0756958i	$-0.0241178\pi$
$422$		0			0
$423$	−4.69140e9			−0.146535
$424$		0			0
$425$	2.28154e9			0.0699314
$426$		0			0
$427$	3.31229e8			0.00996362
$428$		0			0
$429$	7.43896e8			0.0219626
$430$		0			0
$431$			5.49100e10i			1.59126i	0.605780	+	0.795632i	$0.292861\pi$
$431$			5.49100e10i			1.59126i	−0.605780	+	0.795632i	$0.707139\pi$
$432$		0			0
$433$		−	5.67002e10i		−	1.61300i	−0.591237	−	0.806498i	$-0.701360\pi$
$433$		−	5.67002e10i		−	1.61300i	0.591237	−	0.806498i	$-0.298640\pi$
$434$		0			0
$435$	4.78897e9			0.133747
$436$		0			0
$437$	1.39222e10	+	4.34717e10i	0.381752	+	1.19201i
$438$		0			0
$439$		−	6.67486e10i		−	1.79715i	−0.438822	−	0.898574i	$-0.644604\pi$
$439$		−	6.67486e10i		−	1.79715i	0.438822	−	0.898574i	$-0.355396\pi$
$440$		0			0
$441$	−3.63950e10			−0.962249
$442$		0			0
$443$	4.33866e9			0.112653			0.0563263	−	0.998412i	$-0.482061\pi$
$443$	4.33866e9			0.112653			0.0563263	+	0.998412i	$0.482061\pi$
$444$		0			0
$445$			6.57693e10i			1.67719i
$446$		0			0
$447$			8.33240e9i			0.208708i
$448$		0			0
$449$		−	7.17821e10i		−	1.76616i	−0.469219	−	0.883082i	$-0.655464\pi$
$449$		−	7.17821e10i		−	1.76616i	0.469219	−	0.883082i	$-0.344536\pi$
$450$		0			0
$451$		−	1.27205e10i		−	0.307466i
$452$		0			0
$453$	7.67538e9			0.182267
$454$		0			0
$455$		−	5.21448e9i		−	0.121665i
$456$		0			0
$457$	8.06293e10			1.84854			0.924268	−	0.381743i	$-0.124676\pi$
$457$	8.06293e10			1.84854			0.924268	+	0.381743i	$0.124676\pi$
$458$		0			0
$459$			7.02253e8i			0.0158213i
$460$		0			0
$461$	−5.53201e9			−0.122484			−0.0612420	−	0.998123i	$-0.519506\pi$
$461$	−5.53201e9			−0.122484			−0.0612420	+	0.998123i	$0.519506\pi$
$462$		0			0
$463$	−4.32453e10			−0.941054			−0.470527	−	0.882386i	$-0.655936\pi$
$463$	−4.32453e10			−0.941054			−0.470527	+	0.882386i	$0.655936\pi$
$464$		0			0
$465$	2.60841e9			0.0557911
$466$		0			0
$467$	−2.49413e10			−0.524386			−0.262193	−	0.965015i	$-0.584446\pi$
$467$	−2.49413e10			−0.524386			−0.262193	+	0.965015i	$0.584446\pi$
$468$		0			0
$469$		−	1.14710e10i		−	0.237087i
$470$		0			0
$471$		−	8.21345e9i		−	0.166895i
$472$		0			0
$473$	−9.87809e9			−0.197346
$474$		0			0
$475$	−1.80633e10	−	5.64022e10i	−0.354832	−	1.10795i
$476$		0			0
$477$			4.11866e10i			0.795577i
$478$		0			0
$479$	−5.76944e10			−1.09595			−0.547977	−	0.836494i	$-0.684602\pi$
$479$	−5.76944e10			−1.09595			−0.547977	+	0.836494i	$0.684602\pi$
$480$		0			0
$481$	−3.50805e10			−0.655369
$482$		0			0
$483$		−	1.29443e9i		−	0.0237843i
$484$		0			0
$485$			1.22529e11i			2.21449i
$486$		0			0
$487$			3.94004e10i			0.700462i	0.936663	+	0.350231i	$0.113897\pi$
$487$			3.94004e10i			0.700462i	−0.936663	+	0.350231i	$0.886103\pi$
$488$		0			0
$489$			6.17192e9i			0.107941i
$490$		0			0
$491$	1.08173e11			1.86120			0.930599	−	0.366041i	$-0.119287\pi$
$491$	1.08173e11			1.86120			0.930599	+	0.366041i	$0.119287\pi$
$492$		0			0
$493$			2.43190e9i			0.0411678i
$494$		0			0
$495$	2.48277e10			0.413538
$496$		0			0
$497$		−	8.39659e9i		−	0.137619i
$498$		0			0
$499$	−8.97074e10			−1.44686			−0.723430	−	0.690398i	$-0.757435\pi$
$499$	−8.97074e10			−1.44686			−0.723430	+	0.690398i	$0.757435\pi$
$500$		0			0
$501$	5.97286e9			0.0948051
$502$		0			0
$503$	7.42237e10			1.15950			0.579750	−	0.814795i	$-0.303150\pi$
$503$	7.42237e10			1.15950			0.579750	+	0.814795i	$0.303150\pi$
$504$		0			0
$505$	2.03053e10			0.312207
$506$		0			0
$507$			5.84235e9i			0.0884211i
$508$		0			0
$509$			3.30960e10i			0.493066i	0.969134	+	0.246533i	$0.0792913\pi$
$509$			3.30960e10i			0.493066i	−0.969134	+	0.246533i	$0.920709\pi$
$510$		0			0
$511$	8.13324e9			0.119283
$512$		0			0
$513$	1.73605e10	−	5.55984e9i	0.250664	−	0.0802773i
$514$		0			0
$515$		−	1.31092e10i		−	0.186357i
$516$		0			0
$517$	3.05000e9			0.0426912
$518$		0			0
$519$	−7.06373e9			−0.0973565
$520$		0			0
$521$		−	1.15808e11i		−	1.57177i	−0.618372	−	0.785886i	$-0.712207\pi$
$521$		−	1.15808e11i		−	1.57177i	0.618372	−	0.785886i	$-0.287793\pi$
$522$		0			0
$523$		−	1.39644e11i		−	1.86644i	−0.359305	−	0.933220i	$-0.616986\pi$
$523$		−	1.39644e11i		−	1.86644i	0.359305	−	0.933220i	$-0.383014\pi$
$524$		0			0
$525$			1.67945e9i			0.0221070i
$526$		0			0
$527$			1.32458e9i			0.0171726i
$528$		0			0
$529$	4.43733e10			0.566629
$530$		0			0
$531$			9.92463e10i			1.24835i
$532$		0			0
$533$	−5.01110e10			−0.620904
$534$		0			0
$535$			4.82227e9i			0.0588622i
$536$		0			0
$537$	1.26445e10			0.152056
$538$		0			0
$539$	2.36614e10			0.280340
$540$		0			0
$541$	−1.09728e11			−1.28094			−0.640470	−	0.767983i	$-0.721260\pi$
$541$	−1.09728e11			−1.28094			−0.640470	+	0.767983i	$0.721260\pi$
$542$		0			0
$543$	7.23861e9			0.0832637
$544$		0			0
$545$		−	1.75112e11i		−	1.98486i
$546$		0			0
$547$			7.70367e10i			0.860495i	0.902711	+	0.430247i	$0.141574\pi$
$547$			7.70367e10i			0.860495i	−0.902711	+	0.430247i	$0.858426\pi$
$548$		0			0
$549$	6.21275e9			0.0683903
$550$		0			0
$551$	6.01192e10	−	1.92537e10i	0.652239	−	0.208885i
$552$		0			0
$553$		−	1.87287e10i		−	0.200266i
$554$		0			0
$555$	2.10103e10			0.221442
$556$		0			0
$557$	−8.33803e10			−0.866248			−0.433124	−	0.901334i	$-0.642589\pi$
$557$	−8.33803e10			−0.866248			−0.433124	+	0.901334i	$0.642589\pi$
$558$		0			0
$559$			3.89138e10i			0.398525i
$560$		0			0
$561$		−	2.26247e8i		−	0.00228418i
$562$		0			0
$563$			1.15774e11i			1.15233i	0.817334	+	0.576164i	$0.195451\pi$
$563$			1.15774e11i			1.15233i	−0.817334	+	0.576164i	$0.804549\pi$
$564$		0			0
$565$			1.88186e11i			1.84669i
$566$		0			0
$567$	1.40144e10			0.135595
$568$		0			0
$569$		−	7.99151e10i		−	0.762394i	−0.924494	−	0.381197i	$-0.875512\pi$
$569$		−	7.99151e10i		−	0.762394i	0.924494	−	0.381197i	$-0.124488\pi$
$570$		0			0
$571$	1.62039e11			1.52432			0.762159	−	0.647390i	$-0.224139\pi$
$571$	1.62039e11			1.52432			0.762159	+	0.647390i	$0.224139\pi$
$572$		0			0
$573$			2.68650e10i			0.249212i
$574$		0			0
$575$	−1.59176e11			−1.45615
$576$		0			0
$577$	1.78544e11			1.61080			0.805402	−	0.592728i	$-0.201949\pi$
$577$	1.78544e11			1.61080			0.805402	+	0.592728i	$0.201949\pi$
$578$		0			0
$579$	2.56368e10			0.228112
$580$		0			0
$581$	−1.41662e10			−0.124322
$582$		0			0
$583$		−	2.67765e10i		−	0.231782i
$584$		0			0
$585$		−	9.78063e10i		−	0.835109i
$586$		0			0
$587$	−2.06207e11			−1.73681			−0.868404	−	0.495857i	$-0.834854\pi$
$587$	−2.06207e11			−1.73681			−0.868404	+	0.495857i	$0.834854\pi$
$588$		0			0
$589$	3.27452e10	−	1.04869e10i	0.272074	−	0.0871339i
$590$		0			0
$591$		−	4.39199e9i		−	0.0360007i
$592$		0			0
$593$	−8.30747e10			−0.671815			−0.335908	−	0.941895i	$-0.609043\pi$
$593$	−8.30747e10			−0.671815			−0.335908	+	0.941895i	$0.609043\pi$
$594$		0			0
$595$	−1.58592e9			−0.0126536
$596$		0			0
$597$			2.14580e10i			0.168924i
$598$		0			0
$599$		−	2.05057e11i		−	1.59282i	−0.604757	−	0.796410i	$-0.706730\pi$
$599$		−	2.05057e11i		−	1.59282i	0.604757	−	0.796410i	$-0.293270\pi$
$600$		0			0
$601$		−	1.04583e11i		−	0.801611i	−0.916163	−	0.400805i	$-0.868730\pi$
$601$		−	1.04583e11i		−	0.801611i	0.916163	−	0.400805i	$-0.131270\pi$
$602$		0			0
$603$		−	2.15157e11i		−	1.62737i
$604$		0			0
$605$	1.80914e11			1.35037
$606$		0			0
$607$			1.21178e11i			0.892626i	0.894877	+	0.446313i	$0.147263\pi$
$607$			1.21178e11i			0.892626i	−0.894877	+	0.446313i	$0.852737\pi$
$608$		0			0
$609$	−1.79013e9			−0.0130142
$610$		0			0
$611$		−	1.20152e10i		−	0.0862117i
$612$		0			0
$613$	−5.78476e10			−0.409678			−0.204839	−	0.978796i	$-0.565667\pi$
$613$	−5.78476e10			−0.409678			−0.204839	+	0.978796i	$0.565667\pi$
$614$		0			0
$615$	3.00124e10			0.209797
$616$		0			0
$617$	1.37356e11			0.947776			0.473888	−	0.880585i	$-0.342850\pi$
$617$	1.37356e11			0.947776			0.473888	+	0.880585i	$0.342850\pi$
$618$		0			0
$619$	8.69099e10			0.591979			0.295990	−	0.955191i	$-0.404351\pi$
$619$	8.69099e10			0.591979			0.295990	+	0.955191i	$0.404351\pi$
$620$		0			0
$621$		−	4.89941e10i		−	0.329441i
$622$		0			0
$623$		−	2.45848e10i		−	0.163198i
$624$		0			0
$625$	−1.23584e11			−0.809920
$626$		0			0
$627$	−5.59307e9	+	1.79123e9i	−0.0361893	+	0.0115899i
$628$		0			0
$629$			1.06693e10i			0.0681606i
$630$		0			0
$631$	−1.40125e11			−0.883889			−0.441945	−	0.897042i	$-0.645711\pi$
$631$	−1.40125e11			−0.883889			−0.441945	+	0.897042i	$0.645711\pi$
$632$		0			0
$633$	2.37768e10			0.148094
$634$		0			0
$635$		−	1.14796e10i		−	0.0706046i
$636$		0			0
$637$		−	9.32117e10i		−	0.566125i
$638$		0			0
$639$		−	1.57492e11i		−	0.944616i
$640$		0			0
$641$		−	2.18831e11i		−	1.29621i	−0.761549	−	0.648107i	$-0.775561\pi$
$641$		−	2.18831e11i		−	1.29621i	0.761549	−	0.648107i	$-0.224439\pi$
$642$		0			0
$643$	−6.72532e10			−0.393431			−0.196716	−	0.980461i	$-0.563028\pi$
$643$	−6.72532e10			−0.393431			−0.196716	+	0.980461i	$0.563028\pi$
$644$		0			0
$645$		−	2.33061e10i		−	0.134658i
$646$		0			0
$647$	−2.70481e9			−0.0154355			−0.00771773	−	0.999970i	$-0.502457\pi$
$647$	−2.70481e9			−0.0154355			−0.00771773	+	0.999970i	$0.502457\pi$
$648$		0			0
$649$		−	6.45227e10i		−	0.363692i
$650$		0			0
$651$	−9.75034e8			−0.00542870
$652$		0			0
$653$	2.53210e11			1.39260			0.696302	−	0.717749i	$-0.254828\pi$
$653$	2.53210e11			1.39260			0.696302	+	0.717749i	$0.254828\pi$
$654$		0			0
$655$	1.52174e11			0.826753
$656$		0			0
$657$	1.52553e11			0.818762
$658$		0			0
$659$		−	1.31513e9i		−	0.00697310i	−0.999994	−	0.00348655i	$-0.998890\pi$
$659$		−	1.31513e9i		−	0.00697310i	0.999994	−	0.00348655i	$-0.00110981\pi$
$660$		0			0
$661$		−	1.05346e11i		−	0.551836i	−0.961181	−	0.275918i	$-0.911018\pi$
$661$		−	1.05346e11i		−	0.551836i	0.961181	−	0.275918i	$-0.0889819\pi$
$662$		0			0
$663$	−8.91277e8			−0.00461274
$664$		0			0
$665$	1.25560e10	+	3.92057e10i	0.0642042	+	0.200476i
$666$		0			0
$667$		−	1.69666e11i		−	0.857220i
$668$		0			0
$669$	1.08562e10			0.0541969
$670$		0			0
$671$	−4.03908e9			−0.0199247
$672$		0			0
$673$			1.01126e11i			0.492948i	0.969149	+	0.246474i	$0.0792719\pi$
$673$			1.01126e11i			0.492948i	−0.969149	+	0.246474i	$0.920728\pi$
$674$		0			0
$675$			6.35672e10i			0.306209i
$676$		0			0
$677$		−	1.62812e11i		−	0.775053i	−0.921858	−	0.387527i	$-0.873329\pi$
$677$		−	1.62812e11i		−	0.775053i	0.921858	−	0.387527i	$-0.126671\pi$
$678$		0			0
$679$		−	4.58019e10i		−	0.215479i
$680$		0			0
$681$	−1.15322e10			−0.0536195
$682$		0			0
$683$		−	4.24838e11i		−	1.95227i	−0.217157	−	0.976137i	$-0.569678\pi$
$683$		−	4.24838e11i		−	1.95227i	0.217157	−	0.976137i	$-0.430322\pi$
$684$		0			0
$685$	−4.67573e11			−2.12367
$686$		0			0
$687$		−	2.35247e10i		−	0.105608i
$688$		0			0
$689$	−1.05483e11			−0.468066
$690$		0			0
$691$	4.07739e11			1.78842			0.894210	−	0.447647i	$-0.147738\pi$
$691$	4.07739e11			1.78842			0.894210	+	0.447647i	$0.147738\pi$
$692$		0			0
$693$	−9.28068e9			−0.0402390
$694$		0			0
$695$	1.18642e11			0.508511
$696$		0			0
$697$			1.52406e10i			0.0645761i
$698$		0			0
$699$		−	1.72142e10i		−	0.0721072i
$700$		0			0
$701$	−6.60489e10			−0.273523			−0.136761	−	0.990604i	$-0.543669\pi$
$701$	−6.60489e10			−0.273523			−0.136761	+	0.990604i	$0.543669\pi$
$702$		0			0
$703$	2.63757e11	−	8.44704e10i	1.07990	−	0.345847i
$704$		0			0
$705$			7.19611e9i			0.0291301i
$706$		0			0
$707$	−7.59017e9			−0.0303790
$708$		0			0
$709$	−8.16942e10			−0.323300			−0.161650	−	0.986848i	$-0.551682\pi$
$709$	−8.16942e10			−0.323300			−0.161650	+	0.986848i	$0.551682\pi$
$710$		0			0
$711$		−	3.51287e11i		−	1.37462i
$712$		0			0
$713$		−	9.24123e10i		−	0.357579i
$714$		0			0
$715$			6.35865e10i			0.243299i
$716$		0			0
$717$		−	2.60798e10i		−	0.0986797i
$718$		0			0
$719$	2.01728e11			0.754834			0.377417	−	0.926043i	$-0.376812\pi$
$719$	2.01728e11			0.754834			0.377417	+	0.926043i	$0.376812\pi$
$720$		0			0
$721$			4.90025e9i			0.0181333i
$722$		0			0
$723$	2.14634e10			0.0785498
$724$		0			0
$725$			2.20133e11i			0.796770i
$726$		0			0
$727$	−1.68993e11			−0.604966			−0.302483	−	0.953155i	$-0.597816\pi$
$727$	−1.68993e11			−0.604966			−0.302483	+	0.953155i	$0.597816\pi$
$728$		0			0
$729$	2.52994e11			0.895777
$730$		0			0
$731$	1.18351e10			0.0414480
$732$		0			0
$733$	3.42616e11			1.18684			0.593420	−	0.804893i	$-0.297777\pi$
$733$	3.42616e11			1.18684			0.593420	+	0.804893i	$0.297777\pi$
$734$		0			0
$735$			5.58261e10i			0.191288i
$736$		0			0
$737$			1.39879e11i			0.474115i
$738$		0			0
$739$	−5.33253e11			−1.78795			−0.893975	−	0.448117i	$-0.852095\pi$
$739$	−5.33253e11			−1.78795			−0.893975	+	0.448117i	$0.852095\pi$
$740$		0			0
$741$	7.05637e9	+	2.20334e10i	0.0234050	+	0.0730815i
$742$		0			0
$743$			3.52979e10i			0.115823i	0.998322	+	0.0579113i	$0.0184441\pi$
$743$			3.52979e10i			0.115823i	−0.998322	+	0.0579113i	$0.981556\pi$
$744$		0			0
$745$	−7.12234e11			−2.31205
$746$		0			0
$747$	−2.65711e11			−0.853349
$748$		0			0
$749$		−	1.80258e9i		−	0.00572754i
$750$		0			0
$751$			4.77175e11i			1.50009i	0.661386	+	0.750045i	$0.269968\pi$
$751$			4.77175e11i			1.50009i	−0.661386	+	0.750045i	$0.730032\pi$
$752$		0			0
$753$		−	2.73667e10i		−	0.0851220i
$754$		0			0
$755$			6.56073e11i			2.01913i
$756$		0			0
$757$	3.54055e11			1.07817			0.539084	−	0.842252i	$-0.318770\pi$
$757$	3.54055e11			1.07817			0.539084	+	0.842252i	$0.318770\pi$
$758$		0			0
$759$			1.57846e10i			0.0475626i
$760$		0			0
$761$	−3.22849e11			−0.962632			−0.481316	−	0.876547i	$-0.659841\pi$
$761$	−3.22849e11			−0.962632			−0.481316	+	0.876547i	$0.659841\pi$
$762$		0			0
$763$			6.54574e10i			0.193135i
$764$		0			0
$765$	−2.97466e10			−0.0868543
$766$		0			0
$767$	−2.54181e11			−0.734449
$768$		0			0
$769$	8.99891e10			0.257326			0.128663	−	0.991688i	$-0.458931\pi$
$769$	8.99891e10			0.257326			0.128663	+	0.991688i	$0.458931\pi$
$770$		0			0
$771$	−5.80783e10			−0.164360
$772$		0			0
$773$			3.43312e11i			0.961548i	0.876845	+	0.480774i	$0.159644\pi$
$773$			3.43312e11i			0.961548i	−0.876845	+	0.480774i	$0.840356\pi$
$774$		0			0
$775$			1.19900e11i			0.332363i
$776$		0			0
$777$	−7.85373e9			−0.0215473
$778$		0			0
$779$	3.76765e11	−	1.20662e11i	1.02311	−	0.327659i
$780$		0			0
$781$			1.02390e11i			0.275203i
$782$		0			0
$783$	−6.77564e10			−0.180262
$784$		0			0
$785$	7.02067e11			1.84884
$786$		0			0
$787$			2.82012e11i			0.735137i	0.929997	+	0.367568i	$0.119810\pi$
$787$			2.82012e11i			0.735137i	−0.929997	+	0.367568i	$0.880190\pi$
$788$		0			0
$789$		−	4.52029e10i		−	0.116643i
$790$		0			0
$791$		−	7.03446e10i		−	0.179690i
$792$		0			0
$793$			1.59116e10i			0.0402365i
$794$		0			0
$795$	6.31758e10			0.158155
$796$		0			0
$797$			6.48442e11i			1.60708i	0.595250	+	0.803540i	$0.297053\pi$
$797$			6.48442e11i			1.60708i	−0.595250	+	0.803540i	$0.702947\pi$
$798$		0			0
$799$	−3.65427e9			−0.00896631
$800$		0			0
$801$		−	4.61129e11i		−	1.12019i
$802$		0			0
$803$	−9.91785e10			−0.238537
$804$		0			0
$805$	1.10645e11			0.263480
$806$		0			0
$807$	−1.77996e8			−0.000419678
$808$		0			0
$809$	6.90473e11			1.61195			0.805977	−	0.591947i	$-0.201641\pi$
$809$	6.90473e11			1.61195			0.805977	+	0.591947i	$0.201641\pi$
$810$		0			0
$811$		−	1.59384e11i		−	0.368434i	−0.982886	−	0.184217i	$-0.941025\pi$
$811$		−	1.59384e11i		−	0.368434i	0.982886	−	0.184217i	$-0.0589750\pi$
$812$		0			0
$813$			8.71371e10i			0.199453i
$814$		0			0
$815$	−5.27561e11			−1.19576
$816$		0			0
$817$	−9.37005e10	−	2.92578e11i	−0.210307	−	0.656679i
$818$		0			0
$819$			3.65603e10i			0.0812596i
$820$		0			0
$821$	−7.47845e11			−1.64603			−0.823017	−	0.568016i	$-0.807711\pi$
$821$	−7.47845e11			−1.64603			−0.823017	+	0.568016i	$0.807711\pi$
$822$		0			0
$823$	−7.29538e10			−0.159019			−0.0795095	−	0.996834i	$-0.525335\pi$
$823$	−7.29538e10			−0.159019			−0.0795095	+	0.996834i	$0.525335\pi$
$824$		0			0
$825$		−	2.04796e10i		−	0.0442085i
$826$		0			0
$827$			1.71778e11i			0.367236i	0.982998	+	0.183618i	$0.0587809\pi$
$827$			1.71778e11i			0.367236i	−0.982998	+	0.183618i	$0.941219\pi$
$828$		0			0
$829$			8.66395e11i			1.83442i	0.398408	+	0.917208i	$0.369563\pi$
$829$			8.66395e11i			1.83442i	−0.398408	+	0.917208i	$0.630437\pi$
$830$		0			0
$831$		−	5.11190e10i		−	0.107196i
$832$		0			0
$833$	−2.83492e10			−0.0588790
$834$		0			0
$835$			5.10546e11i			1.05024i
$836$		0			0
$837$	−3.69050e10			−0.0751940
$838$		0			0
$839$		−	2.46507e11i		−	0.497487i	−0.968569	−	0.248743i	$-0.919982\pi$
$839$		−	2.46507e11i		−	0.497487i	0.968569	−	0.248743i	$-0.0800176\pi$
$840$		0			0
$841$	2.65606e11			0.530951
$842$		0			0
$843$	−1.35878e10			−0.0269055
$844$		0			0
$845$	−4.99391e11			−0.979521
$846$		0			0
$847$	−6.76264e10			−0.131396
$848$		0			0
$849$			6.58169e10i			0.126680i
$850$		0			0
$851$		−	7.44365e11i		−	1.41928i
$852$		0			0
$853$	−6.05188e11			−1.14313			−0.571563	−	0.820558i	$-0.693663\pi$
$853$	−6.05188e11			−1.14313			−0.571563	+	0.820558i	$0.693663\pi$
$854$		0			0
$855$	2.35508e11	+	7.35368e11i	0.440698	+	1.37607i
$856$		0			0
$857$		−	8.23654e11i		−	1.52694i	−0.645844	−	0.763469i	$-0.723495\pi$
$857$		−	8.23654e11i		−	1.52694i	0.645844	−	0.763469i	$-0.276505\pi$
$858$		0			0
$859$	3.99209e11			0.733209			0.366605	−	0.930377i	$-0.380520\pi$
$859$	3.99209e11			0.733209			0.366605	+	0.930377i	$0.380520\pi$
$860$		0			0
$861$	−1.12187e10			−0.0204141
$862$		0			0
$863$			6.65968e11i			1.20063i	0.799762	+	0.600317i	$0.204959\pi$
$863$			6.65968e11i			1.20063i	−0.799762	+	0.600317i	$0.795041\pi$
$864$		0			0
$865$		−	6.03792e11i		−	1.07851i
$866$		0			0
$867$		−	7.47504e10i		−	0.132293i
$868$		0			0
$869$			2.28381e11i			0.400480i
$870$		0			0
$871$	5.51041e11			0.957439
$872$		0			0
$873$		−	8.59091e11i		−	1.47905i
$874$		0			0
$875$	−2.01609e10			−0.0343936
$876$		0			0
$877$			1.42492e11i			0.240875i	0.992721	+	0.120438i	$0.0384297\pi$
$877$			1.42492e11i			0.240875i	−0.992721	+	0.120438i	$0.961570\pi$
$878$		0			0
$879$	8.76310e10			0.146792
$880$		0			0
$881$	−6.50983e10			−0.108060			−0.0540302	−	0.998539i	$-0.517207\pi$
$881$	−6.50983e10			−0.108060			−0.0540302	+	0.998539i	$0.517207\pi$
$882$		0			0
$883$	−3.09709e11			−0.509462			−0.254731	−	0.967012i	$-0.581987\pi$
$883$	−3.09709e11			−0.509462			−0.254731	+	0.967012i	$0.581987\pi$
$884$		0			0
$885$	1.52233e11			0.248163
$886$		0			0
$887$		−	4.50470e11i		−	0.727732i	−0.931451	−	0.363866i	$-0.881457\pi$
$887$		−	4.50470e11i		−	0.727732i	0.931451	−	0.363866i	$-0.118543\pi$
$888$		0			0
$889$			4.29112e9i			0.00687012i
$890$		0			0
$891$	−1.70895e11			−0.271155
$892$		0			0
$893$	2.89314e10	+	9.03377e10i	0.0454950	+	0.142057i
$894$		0			0
$895$			1.08082e12i			1.68446i
$896$		0			0
$897$	6.21817e10			0.0960490
$898$		0			0
$899$	−1.27802e11			−0.195658
$900$		0			0
$901$			3.20815e10i			0.0486805i
$902$		0			0
$903$			8.71191e9i			0.0131027i
$904$		0			0
$905$			6.18739e11i			0.922387i
$906$		0			0
$907$		−	5.45609e11i		−	0.806218i	−0.915152	−	0.403109i	$-0.867929\pi$
$907$		−	5.45609e11i		−	0.806218i	0.915152	−	0.403109i	$-0.132071\pi$
$908$		0			0
$909$	−1.42366e11			−0.208522
$910$		0			0
$911$		−	5.13209e11i		−	0.745111i	−0.928010	−	0.372556i	$-0.878482\pi$
$911$		−	5.13209e11i		−	0.745111i	0.928010	−	0.372556i	$-0.121518\pi$
$912$		0			0
$913$	1.72746e11			0.248613
$914$		0			0
$915$		−	9.52971e9i		−	0.0135955i
$916$		0			0
$917$	−5.68832e10			−0.0804464
$918$		0			0
$919$	1.31908e10			0.0184930			0.00924652	−	0.999957i	$-0.497057\pi$
$919$	1.31908e10			0.0184930			0.00924652	+	0.999957i	$0.497057\pi$
$920$		0			0
$921$	8.58094e10			0.119260
$922$		0			0
$923$	4.03355e11			0.555751
$924$		0			0
$925$			9.65774e11i			1.31919i
$926$		0			0
$927$			9.19123e10i			0.124467i
$928$		0			0
$929$	−6.52670e11			−0.876256			−0.438128	−	0.898913i	$-0.644358\pi$
$929$	−6.52670e11			−0.876256			−0.438128	+	0.898913i	$0.644358\pi$
$930$		0			0
$931$	2.24445e11	+	7.00823e11i	0.298752	+	0.932845i
$932$		0			0
$933$			1.10547e11i			0.145888i
$934$		0			0
$935$	1.93390e10			0.0253040
$936$		0			0
$937$	−4.27565e11			−0.554681			−0.277341	−	0.960772i	$-0.589453\pi$
$937$	−4.27565e11			−0.554681			−0.277341	+	0.960772i	$0.589453\pi$
$938$		0			0
$939$			7.36003e10i			0.0946711i
$940$		0			0
$941$			1.36175e12i			1.73676i	0.495898	+	0.868381i	$0.334839\pi$
$941$			1.36175e12i			1.73676i	−0.495898	+	0.868381i	$0.665161\pi$
$942$		0			0
$943$		−	1.06329e12i		−	1.34464i
$944$		0			0
$945$		−	4.41862e10i		−	0.0554063i
$946$		0			0
$947$	1.00481e12			1.24935			0.624675	−	0.780885i	$-0.285232\pi$
$947$	1.00481e12			1.24935			0.624675	+	0.780885i	$0.285232\pi$
$948$		0			0
$949$			3.90704e11i			0.481707i
$950$		0			0
$951$	1.15953e11			0.141762
$952$		0			0
$953$			3.27699e11i			0.397286i	0.980072	+	0.198643i	$0.0636534\pi$
$953$			3.27699e11i			0.397286i	−0.980072	+	0.198643i	$0.936347\pi$
$954$		0			0
$955$	−2.29636e12			−2.76074
$956$		0			0
$957$	2.18293e10			0.0260250
$958$		0			0
$959$	1.74780e11			0.206642
$960$		0			0
$961$	7.83281e11			0.918384
$962$		0			0
$963$		−	3.38104e10i		−	0.0393138i
$964$		0			0
$965$			2.19137e12i			2.52701i
$966$		0			0
$967$	−1.10076e12			−1.25888			−0.629442	−	0.777048i	$-0.716716\pi$
$967$	−1.10076e12			−1.25888			−0.629442	+	0.777048i	$0.716716\pi$
$968$		0			0
$969$	6.70117e9	−	2.14611e9i	0.00760073	−	0.00243420i
$970$		0			0
$971$			8.23490e11i			0.926363i	0.886263	+	0.463182i	$0.153292\pi$
$971$			8.23490e11i			0.926363i	−0.886263	+	0.463182i	$0.846708\pi$
$972$		0			0
$973$	−4.43489e10			−0.0494802
$974$		0			0
$975$	−8.06774e10			−0.0892758
$976$		0			0
$977$		−	1.22250e12i		−	1.34174i	−0.741574	−	0.670871i	$-0.765920\pi$
$977$		−	1.22250e12i		−	1.34174i	0.741574	−	0.670871i	$-0.234080\pi$
$978$		0			0
$979$			2.99792e11i			0.326354i
$980$		0			0
$981$			1.22776e12i			1.32568i
$982$		0			0
$983$			1.79480e12i			1.92222i	0.276167	+	0.961110i	$0.410936\pi$
$983$			1.79480e12i			1.92222i	−0.276167	+	0.961110i	$0.589064\pi$
$984$		0			0
$985$	3.75417e11			0.398813
$986$		0			0
$987$		−	2.68993e9i		−	0.00283447i
$988$		0			0
$989$	−8.25702e11			−0.863054
$990$		0			0
$991$		−	1.61330e11i		−	0.167271i	−0.996496	−	0.0836355i	$-0.973347\pi$
$991$		−	1.61330e11i		−	0.167271i	0.996496	−	0.0836355i	$-0.0266532\pi$
$992$		0			0
$993$	5.19566e10			0.0534372
$994$		0			0
$995$	−1.83418e12			−1.87132
$996$		0			0
$997$	6.29339e11			0.636948			0.318474	−	0.947932i	$-0.396830\pi$
$997$	6.29339e11			0.636948			0.318474	+	0.947932i	$0.396830\pi$
$998$		0			0
$999$	−2.97263e11			−0.298455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.9.e.e.113.7		12
4.3	odd	2		38.9.b.a.37.3	✓	12
12.11	even	2		342.9.d.a.37.12		12
19.18	odd	2	inner	304.9.e.e.113.6		12
76.75	even	2		38.9.b.a.37.10	yes	12
228.227	odd	2		342.9.d.a.37.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.b.a.37.3	✓	12	4.3	odd	2
38.9.b.a.37.10	yes	12	76.75	even	2
304.9.e.e.113.6		12	19.18	odd	2	inner
304.9.e.e.113.7		12	1.1	even	1	trivial
342.9.d.a.37.6		12	228.227	odd	2
342.9.d.a.37.12		12	12.11	even	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.7546i q^{3} -919.278 q^{5} +343.629 q^{7} +6445.34 q^{9} +O(q^{10})\)	\(q+10.7546i q^{3} -919.278 q^{5} +343.629 q^{7} +6445.34 q^{9} -4190.29 q^{11} +16507.2i q^{13} -9886.47i q^{15} +5020.47 q^{17} +(-39747.8 - 124112. i) q^{19} +3695.59i q^{21} -350263. q^{23} +454447. q^{25} +139878. i q^{27} +484397. i q^{29} +263837. i q^{31} -45064.9i q^{33} -315891. q^{35} +2.12516e6i q^{37} -177529. q^{39} +3.03570e6i q^{41} +2.35738e6 q^{43} -5.92506e6 q^{45} -727875. q^{47} -5.64672e6 q^{49} +53993.1i q^{51} +6.39013e6i q^{53} +3.85204e6 q^{55} +(1.33477e6 - 427472. i) q^{57} +1.53982e7i q^{59} +963914. q^{61} +2.21481e6 q^{63} -1.51747e7i q^{65} -3.33818e7i q^{67} -3.76694e6i q^{69} -2.44350e7i q^{71} +2.36687e7 q^{73} +4.88740e6i q^{75} -1.43991e6 q^{77} -5.45025e7i q^{79} +4.07835e7 q^{81} -4.12252e7 q^{83} -4.61521e6 q^{85} -5.20949e6 q^{87} -7.15445e7i q^{89} +5.67237e6i q^{91} -2.83746e6 q^{93} +(3.65393e7 + 1.14093e8i) q^{95} -1.33289e8i q^{97} -2.70078e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 558 q^{5} + 5422 q^{7} - 15592 q^{9}+O(q^{10}) \) 12 * q + 558 * q^5 + 5422 * q^7 - 15592 * q^9	\( 12 q + 558 q^{5} + 5422 q^{7} - 15592 q^{9} + 12546 q^{11} + 270810 q^{17} - 41512 q^{19} + 823956 q^{23} + 865538 q^{25} + 1194378 q^{35} - 5786100 q^{39} - 7586646 q^{43} + 2226046 q^{45} + 20260530 q^{47} - 19498842 q^{49} + 14858554 q^{55} + 14430564 q^{57} - 41363266 q^{61} - 84235798 q^{63} + 87906498 q^{73} - 78817962 q^{77} - 100904812 q^{81} + 55944960 q^{83} + 25440254 q^{85} - 119189604 q^{87} + 105500856 q^{93} - 81396774 q^{95} + 85554938 q^{99}+O(q^{100}) \) 12 * q + 558 * q^5 + 5422 * q^7 - 15592 * q^9 + 12546 * q^11 + 270810 * q^17 - 41512 * q^19 + 823956 * q^23 + 865538 * q^25 + 1194378 * q^35 - 5786100 * q^39 - 7586646 * q^43 + 2226046 * q^45 + 20260530 * q^47 - 19498842 * q^49 + 14858554 * q^55 + 14430564 * q^57 - 41363266 * q^61 - 84235798 * q^63 + 87906498 * q^73 - 78817962 * q^77 - 100904812 * q^81 + 55944960 * q^83 + 25440254 * q^85 - 119189604 * q^87 + 105500856 * q^93 - 81396774 * q^95 + 85554938 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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