LMFDB - Embedded newform 1568.2.i.q.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(961,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1568.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.i (of order $3$, degree $2$, 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:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1568.961
Dual form			1568.2.i.q.1537.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 1.22474i) q^{3} +(0.500000 - 0.866025i) q^{9} +O(q^{10})$	$q+(-0.707107 - 1.22474i) q^{3} +(0.500000 - 0.866025i) q^{9} +(-1.00000 - 1.73205i) q^{11} +2.82843 q^{13} +(-2.12132 - 3.67423i) q^{17} +(2.12132 - 3.67423i) q^{19} +(-4.00000 + 6.92820i) q^{23} +(2.50000 + 4.33013i) q^{25} -5.65685 q^{27} +6.00000 q^{29} +(-4.24264 - 7.34847i) q^{31} +(-1.41421 + 2.44949i) q^{33} +(1.00000 - 1.73205i) q^{37} +(-2.00000 - 3.46410i) q^{39} -4.24264 q^{41} +6.00000 q^{43} +(1.41421 - 2.44949i) q^{47} +(-3.00000 + 5.19615i) q^{51} +(-3.00000 - 5.19615i) q^{53} -6.00000 q^{57} +(-6.36396 - 11.0227i) q^{59} +(2.82843 - 4.89898i) q^{61} +(-6.00000 - 10.3923i) q^{67} +11.3137 q^{69} -4.00000 q^{71} +(0.707107 + 1.22474i) q^{73} +(3.53553 - 6.12372i) q^{75} +(-6.00000 + 10.3923i) q^{79} +(2.50000 + 4.33013i) q^{81} -9.89949 q^{83} +(-4.24264 - 7.34847i) q^{87} +(-2.12132 + 3.67423i) q^{89} +(-6.00000 + 10.3923i) q^{93} -18.3848 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^9	$ 4 q + 2 q^{9} - 4 q^{11} - 16 q^{23} + 10 q^{25} + 24 q^{29} + 4 q^{37} - 8 q^{39} + 24 q^{43} - 12 q^{51} - 12 q^{53} - 24 q^{57} - 24 q^{67} - 16 q^{71} - 24 q^{79} + 10 q^{81} - 24 q^{93} - 8 q^{99}+O(q^{100}) $ 4 * q + 2 * q^9 - 4 * q^11 - 16 * q^23 + 10 * q^25 + 24 * q^29 + 4 * q^37 - 8 * q^39 + 24 * q^43 - 12 * q^51 - 12 * q^53 - 24 * q^57 - 24 * q^67 - 16 * q^71 - 24 * q^79 + 10 * q^81 - 24 * q^93 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	−0.707107	−	1.22474i	−0.408248	−	0.707107i	0.586445	−	0.809989i	$-0.300527\pi$
$3$	−0.707107	−	1.22474i	−0.408248	−	0.707107i	−0.994694	+	0.102882i	$0.967194\pi$
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	0.500000	−	0.866025i	0.166667	−	0.288675i
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	2.82843			0.784465			0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843			0.784465			0.392232	+	0.919866i	$0.371703\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.12132	−	3.67423i	−0.514496	−	0.891133i	−0.999859	−	0.0168199i	$-0.994646\pi$
$17$	−2.12132	−	3.67423i	−0.514496	−	0.891133i	0.485363	−	0.874313i	$-0.338688\pi$
$18$		0			0
$19$	2.12132	−	3.67423i	0.486664	−	0.842927i	−0.513218	−	0.858258i	$-0.671547\pi$
$19$	2.12132	−	3.67423i	0.486664	−	0.842927i	0.999882	+	0.0153309i	$0.00488018\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000	+	6.92820i	−0.834058	+	1.44463i	0.0607377	+	0.998154i	$0.480655\pi$
$23$	−4.00000	+	6.92820i	−0.834058	+	1.44463i	−0.894795	+	0.446476i	$0.852679\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$	−5.65685			−1.08866
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	−4.24264	−	7.34847i	−0.762001	−	1.31982i	−0.941818	−	0.336124i	$-0.890884\pi$
$31$	−4.24264	−	7.34847i	−0.762001	−	1.31982i	0.179817	−	0.983700i	$-0.442449\pi$
$32$		0			0
$33$	−1.41421	+	2.44949i	−0.246183	+	0.426401i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.00000	−	1.73205i	0.164399	−	0.284747i	−0.772043	−	0.635571i	$-0.780765\pi$
$37$	1.00000	−	1.73205i	0.164399	−	0.284747i	0.936442	+	0.350823i	$0.114098\pi$
$38$		0			0
$39$	−2.00000	−	3.46410i	−0.320256	−	0.554700i
$40$		0			0
$41$	−4.24264			−0.662589			−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264			−0.662589			−0.331295	+	0.943527i	$0.607485\pi$
$42$		0			0
$43$	6.00000			0.914991			0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000			0.914991			0.457496	+	0.889212i	$0.348747\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.41421	−	2.44949i	0.206284	−	0.357295i	−0.744257	−	0.667893i	$-0.767196\pi$
$47$	1.41421	−	2.44949i	0.206284	−	0.357295i	0.950541	+	0.310599i	$0.100530\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−3.00000	+	5.19615i	−0.420084	+	0.727607i
$52$		0			0
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.00000			−0.794719
$58$		0			0
$59$	−6.36396	−	11.0227i	−0.828517	−	1.43503i	−0.899201	−	0.437535i	$-0.855852\pi$
$59$	−6.36396	−	11.0227i	−0.828517	−	1.43503i	0.0706842	−	0.997499i	$-0.477482\pi$
$60$		0			0
$61$	2.82843	−	4.89898i	0.362143	−	0.627250i	−0.626170	−	0.779686i	$-0.715379\pi$
$61$	2.82843	−	4.89898i	0.362143	−	0.627250i	0.988313	+	0.152436i	$0.0487119\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	−0.955588	−	0.294706i	$-0.904778\pi$
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	0.222571	−	0.974916i	$-0.428555\pi$
$68$		0			0
$69$	11.3137			1.36201
$70$		0			0
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$	0.707107	+	1.22474i	0.0827606	+	0.143346i	0.904435	−	0.426612i	$-0.140293\pi$
$73$	0.707107	+	1.22474i	0.0827606	+	0.143346i	−0.821674	+	0.569958i	$0.806960\pi$
$74$		0			0
$75$	3.53553	−	6.12372i	0.408248	−	0.707107i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.00000	+	10.3923i	−0.675053	+	1.16923i	0.301401	+	0.953498i	$0.402546\pi$
$79$	−6.00000	+	10.3923i	−0.675053	+	1.16923i	−0.976453	+	0.215728i	$0.930788\pi$
$80$		0			0
$81$	2.50000	+	4.33013i	0.277778	+	0.481125i
$82$		0			0
$83$	−9.89949			−1.08661			−0.543305	−	0.839535i	$-0.682827\pi$
$83$	−9.89949			−1.08661			−0.543305	+	0.839535i	$0.682827\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−4.24264	−	7.34847i	−0.454859	−	0.787839i
$88$		0			0
$89$	−2.12132	+	3.67423i	−0.224860	+	0.389468i	−0.956277	−	0.292462i	$-0.905526\pi$
$89$	−2.12132	+	3.67423i	−0.224860	+	0.389468i	0.731418	+	0.681930i	$0.238859\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−6.00000	+	10.3923i	−0.622171	+	1.07763i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−18.3848			−1.86669			−0.933346	−	0.358979i	$-0.883125\pi$
$97$	−18.3848			−1.86669			−0.933346	+	0.358979i	$0.883125\pi$
$98$		0			0
$99$	−2.00000			−0.201008
$100$		0			0
$101$	8.48528	+	14.6969i	0.844317	+	1.46240i	0.886213	+	0.463278i	$0.153327\pi$
$101$	8.48528	+	14.6969i	0.844317	+	1.46240i	−0.0418959	+	0.999122i	$0.513340\pi$
$102$		0			0
$103$	4.24264	−	7.34847i	0.418040	−	0.724066i	−0.577702	−	0.816247i	$-0.696051\pi$
$103$	4.24264	−	7.34847i	0.418040	−	0.724066i	0.995742	+	0.0921813i	$0.0293839\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.00000	+	3.46410i	−0.193347	+	0.334887i	−0.946357	−	0.323122i	$-0.895268\pi$
$107$	−2.00000	+	3.46410i	−0.193347	+	0.334887i	0.753010	+	0.658009i	$0.228601\pi$
$108$		0			0
$109$	−9.00000	−	15.5885i	−0.862044	−	1.49310i	−0.869953	−	0.493135i	$-0.835851\pi$
$109$	−9.00000	−	15.5885i	−0.862044	−	1.49310i	0.00790932	−	0.999969i	$-0.497482\pi$
$110$		0			0
$111$	−2.82843			−0.268462
$112$		0			0
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.41421	−	2.44949i	0.130744	−	0.226455i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$	3.00000	+	5.19615i	0.270501	+	0.468521i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$	−4.24264	−	7.34847i	−0.373544	−	0.646997i
$130$		0			0
$131$	0.707107	−	1.22474i	0.0617802	−	0.107006i	−0.833481	−	0.552548i	$-0.813656\pi$
$131$	0.707107	−	1.22474i	0.0617802	−	0.107006i	0.895261	+	0.445542i	$0.146989\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	0.999893	+	0.0146279i	$0.00465636\pi$
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	−0.487278	+	0.873247i	$0.662010\pi$
$138$		0			0
$139$	12.7279			1.07957			0.539784	−	0.841803i	$-0.318506\pi$
$139$	12.7279			1.07957			0.539784	+	0.841803i	$0.318506\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$		0			0
$143$	−2.82843	−	4.89898i	−0.236525	−	0.409673i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	−0.716578	−	0.697507i	$-0.754293\pi$
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	0.962348	+	0.271821i	$0.0876260\pi$
$150$		0			0
$151$	−6.00000	−	10.3923i	−0.488273	−	0.845714i	0.511636	−	0.859202i	$-0.329040\pi$
$151$	−6.00000	−	10.3923i	−0.488273	−	0.845714i	−0.999909	+	0.0134886i	$0.995706\pi$
$152$		0			0
$153$	−4.24264			−0.342997
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.41421	+	2.44949i	0.112867	+	0.195491i	0.916925	−	0.399060i	$-0.130663\pi$
$157$	1.41421	+	2.44949i	0.112867	+	0.195491i	−0.804058	+	0.594550i	$0.797330\pi$
$158$		0			0
$159$	−4.24264	+	7.34847i	−0.336463	+	0.582772i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3.00000	+	5.19615i	−0.234978	+	0.406994i	−0.959266	−	0.282503i	$-0.908835\pi$
$163$	−3.00000	+	5.19615i	−0.234978	+	0.406994i	0.724288	+	0.689497i	$0.242169\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.1421			−1.09435			−0.547176	−	0.837018i	$-0.684297\pi$
$167$	−14.1421			−1.09435			−0.547176	+	0.837018i	$0.684297\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$	−2.12132	−	3.67423i	−0.162221	−	0.280976i
$172$		0			0
$173$	4.24264	−	7.34847i	0.322562	−	0.558694i	−0.658454	−	0.752621i	$-0.728789\pi$
$173$	4.24264	−	7.34847i	0.322562	−	0.558694i	0.981016	+	0.193927i	$0.0621226\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−9.00000	+	15.5885i	−0.676481	+	1.17170i
$178$		0			0
$179$	2.00000	+	3.46410i	0.149487	+	0.258919i	0.931038	−	0.364922i	$-0.118904\pi$
$179$	2.00000	+	3.46410i	0.149487	+	0.258919i	−0.781551	+	0.623841i	$0.785571\pi$
$180$		0			0
$181$	14.1421			1.05118			0.525588	−	0.850739i	$-0.323845\pi$
$181$	14.1421			1.05118			0.525588	+	0.850739i	$0.323845\pi$
$182$		0			0
$183$	−8.00000			−0.591377
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−4.24264	+	7.34847i	−0.310253	+	0.537373i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.0000	−	17.3205i	0.723575	−	1.25327i	−0.235983	−	0.971757i	$-0.575831\pi$
$191$	10.0000	−	17.3205i	0.723575	−	1.25327i	0.959558	−	0.281511i	$-0.0908356\pi$
$192$		0			0
$193$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$193$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	12.7279	+	22.0454i	0.902258	+	1.56276i	0.824556	+	0.565781i	$0.191425\pi$
$199$	12.7279	+	22.0454i	0.902258	+	1.56276i	0.0777029	+	0.996977i	$0.475241\pi$
$200$		0			0
$201$	−8.48528	+	14.6969i	−0.598506	+	1.03664i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	4.00000	+	6.92820i	0.278019	+	0.481543i
$208$		0			0
$209$	−8.48528			−0.586939
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$	2.82843	+	4.89898i	0.193801	+	0.335673i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	1.00000	−	1.73205i	0.0675737	−	0.117041i
$220$		0			0
$221$	−6.00000	−	10.3923i	−0.403604	−	0.699062i
$222$		0			0
$223$	16.9706			1.13643			0.568216	−	0.822879i	$-0.307634\pi$
$223$	16.9706			1.13643			0.568216	+	0.822879i	$0.307634\pi$
$224$		0			0
$225$	5.00000			0.333333
$226$		0			0
$227$	6.36396	+	11.0227i	0.422391	+	0.731603i	0.996173	−	0.0874056i	$-0.0278576\pi$
$227$	6.36396	+	11.0227i	0.422391	+	0.731603i	−0.573782	+	0.819008i	$0.694524\pi$
$228$		0			0
$229$	−1.41421	+	2.44949i	−0.0934539	+	0.161867i	−0.908962	−	0.416878i	$-0.863124\pi$
$229$	−1.41421	+	2.44949i	−0.0934539	+	0.161867i	0.815508	+	0.578745i	$0.196457\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.0000	−	20.7846i	0.786146	−	1.36165i	−0.142166	−	0.989843i	$-0.545407\pi$
$233$	12.0000	−	20.7846i	0.786146	−	1.36165i	0.928312	−	0.371802i	$-0.121260\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	16.9706			1.10236
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	−9.19239	−	15.9217i	−0.592134	−	1.02561i	−0.993945	−	0.109883i	$-0.964952\pi$
$241$	−9.19239	−	15.9217i	−0.592134	−	1.02561i	0.401811	−	0.915723i	$-0.368381\pi$
$242$		0			0
$243$	−4.94975	+	8.57321i	−0.317526	+	0.549972i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	6.00000	−	10.3923i	0.381771	−	0.661247i
$248$		0			0
$249$	7.00000	+	12.1244i	0.443607	+	0.768350i
$250$		0			0
$251$	12.7279			0.803379			0.401690	−	0.915776i	$-0.368423\pi$
$251$	12.7279			0.803379			0.401690	+	0.915776i	$0.368423\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.6066	−	18.3712i	0.661622	−	1.14596i	−0.318568	−	0.947900i	$-0.603202\pi$
$257$	10.6066	−	18.3712i	0.661622	−	1.14596i	0.980189	−	0.198062i	$-0.0634648\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.00000	−	5.19615i	0.185695	−	0.321634i
$262$		0			0
$263$	10.0000	+	17.3205i	0.616626	+	1.06803i	0.990097	+	0.140386i	$0.0448344\pi$
$263$	10.0000	+	17.3205i	0.616626	+	1.06803i	−0.373470	+	0.927642i	$0.621832\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	6.00000			0.367194
$268$		0			0
$269$	−12.7279	−	22.0454i	−0.776035	−	1.34413i	−0.934211	−	0.356721i	$-0.883895\pi$
$269$	−12.7279	−	22.0454i	−0.776035	−	1.34413i	0.158176	−	0.987411i	$-0.449439\pi$
$270$		0			0
$271$	8.48528	−	14.6969i	0.515444	−	0.892775i	−0.484395	−	0.874849i	$-0.660960\pi$
$271$	8.48528	−	14.6969i	0.515444	−	0.892775i	0.999839	−	0.0179261i	$-0.00570637\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.00000	−	8.66025i	0.301511	−	0.522233i
$276$		0			0
$277$	9.00000	+	15.5885i	0.540758	+	0.936620i	0.998861	+	0.0477206i	$0.0151957\pi$
$277$	9.00000	+	15.5885i	0.540758	+	0.936620i	−0.458103	+	0.888899i	$0.651471\pi$
$278$		0			0
$279$	−8.48528			−0.508001
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	10.6066	+	18.3712i	0.630497	+	1.09205i	0.987450	+	0.157931i	$0.0504822\pi$
$283$	10.6066	+	18.3712i	0.630497	+	1.09205i	−0.356953	+	0.934122i	$0.616184\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.500000	+	0.866025i	−0.0294118	+	0.0509427i
$290$		0			0
$291$	13.0000	+	22.5167i	0.762073	+	1.31995i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	5.65685	+	9.79796i	0.328244	+	0.568535i
$298$		0			0
$299$	−11.3137	+	19.5959i	−0.654289	+	1.13326i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	12.0000	−	20.7846i	0.689382	−	1.19404i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	21.2132			1.21070			0.605351	−	0.795959i	$-0.293033\pi$
$307$	21.2132			1.21070			0.605351	+	0.795959i	$0.293033\pi$
$308$		0			0
$309$	−12.0000			−0.682656
$310$		0			0
$311$	5.65685	+	9.79796i	0.320771	+	0.555591i	0.980647	−	0.195783i	$-0.0627248\pi$
$311$	5.65685	+	9.79796i	0.320771	+	0.555591i	−0.659877	+	0.751374i	$0.729391\pi$
$312$		0			0
$313$	−9.19239	+	15.9217i	−0.519584	+	0.899947i	0.480156	+	0.877183i	$0.340580\pi$
$313$	−9.19239	+	15.9217i	−0.519584	+	0.899947i	−0.999741	+	0.0227638i	$0.992753\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$	−6.00000	−	10.3923i	−0.335936	−	0.581857i
$320$		0			0
$321$	5.65685			0.315735
$322$		0			0
$323$	−18.0000			−1.00155
$324$		0			0
$325$	7.07107	+	12.2474i	0.392232	+	0.679366i
$326$		0			0
$327$	−12.7279	+	22.0454i	−0.703856	+	1.21911i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	9.00000	−	15.5885i	0.494685	−	0.856819i	−0.505296	−	0.862946i	$-0.668617\pi$
$331$	9.00000	−	15.5885i	0.494685	−	0.856819i	0.999981	+	0.00612670i	$0.00195020\pi$
$332$		0			0
$333$	−1.00000	−	1.73205i	−0.0547997	−	0.0949158i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	18.0000			0.980522			0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000			0.980522			0.490261	+	0.871576i	$0.336901\pi$
$338$		0			0
$339$	8.48528	+	14.6969i	0.460857	+	0.798228i
$340$		0			0
$341$	−8.48528	+	14.6969i	−0.459504	+	0.795884i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.00000	+	8.66025i	0.268414	+	0.464907i	0.968452	−	0.249198i	$-0.0801671\pi$
$347$	5.00000	+	8.66025i	0.268414	+	0.464907i	−0.700038	+	0.714105i	$0.746834\pi$
$348$		0			0
$349$	−19.7990			−1.05982			−0.529908	−	0.848055i	$-0.677773\pi$
$349$	−19.7990			−1.05982			−0.529908	+	0.848055i	$0.677773\pi$
$350$		0			0
$351$	−16.0000			−0.854017
$352$		0			0
$353$	−10.6066	−	18.3712i	−0.564532	−	0.977799i	−0.997093	−	0.0761940i	$-0.975723\pi$
$353$	−10.6066	−	18.3712i	−0.564532	−	0.977799i	0.432561	−	0.901605i	$-0.357610\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.00000	−	3.46410i	0.105556	−	0.182828i	−0.808409	−	0.588621i	$-0.799671\pi$
$359$	2.00000	−	3.46410i	0.105556	−	0.182828i	0.913965	+	0.405793i	$0.133004\pi$
$360$		0			0
$361$	0.500000	+	0.866025i	0.0263158	+	0.0455803i
$362$		0			0
$363$	−9.89949			−0.519589
$364$		0			0
$365$		0			0
$366$		0			0
$367$	16.9706	+	29.3939i	0.885856	+	1.53435i	0.844729	+	0.535194i	$0.179761\pi$
$367$	16.9706	+	29.3939i	0.885856	+	1.53435i	0.0411270	+	0.999154i	$0.486905\pi$
$368$		0			0
$369$	−2.12132	+	3.67423i	−0.110432	+	0.191273i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.0000	−	19.0526i	0.569558	−	0.986504i	−0.427051	−	0.904227i	$-0.640448\pi$
$373$	11.0000	−	19.0526i	0.569558	−	0.986504i	0.996610	−	0.0822766i	$-0.0262191\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	16.9706			0.874028
$378$		0			0
$379$	18.0000			0.924598			0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000			0.924598			0.462299	+	0.886724i	$0.347025\pi$
$380$		0			0
$381$	−8.48528	−	14.6969i	−0.434714	−	0.752947i
$382$		0			0
$383$	−12.7279	+	22.0454i	−0.650366	+	1.12647i	0.332668	+	0.943044i	$0.392051\pi$
$383$	−12.7279	+	22.0454i	−0.650366	+	1.12647i	−0.983034	+	0.183424i	$0.941282\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	3.00000	−	5.19615i	0.152499	−	0.264135i
$388$		0			0
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	0.779895	−	0.625910i	$-0.215272\pi$
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	−0.932002	+	0.362454i	$0.881939\pi$
$390$		0			0
$391$	33.9411			1.71648
$392$		0			0
$393$	−2.00000			−0.100887
$394$		0			0
$395$		0			0
$396$		0			0
$397$	18.3848	−	31.8434i	0.922705	−	1.59817i	0.127495	−	0.991839i	$-0.459306\pi$
$397$	18.3848	−	31.8434i	0.922705	−	1.59817i	0.795210	−	0.606334i	$-0.207360\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	0.781345	+	0.624099i	$0.214534\pi$
$402$		0			0
$403$	−12.0000	−	20.7846i	−0.597763	−	1.03536i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.00000			−0.198273
$408$		0			0
$409$	17.6777	+	30.6186i	0.874105	+	1.51399i	0.857714	+	0.514128i	$0.171884\pi$
$409$	17.6777	+	30.6186i	0.874105	+	1.51399i	0.0163909	+	0.999866i	$0.494782\pi$
$410$		0			0
$411$	8.48528	−	14.6969i	0.418548	−	0.724947i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−9.00000	−	15.5885i	−0.440732	−	0.763370i
$418$		0			0
$419$	−12.7279			−0.621800			−0.310900	−	0.950443i	$-0.600630\pi$
$419$	−12.7279			−0.621800			−0.310900	+	0.950443i	$0.600630\pi$
$420$		0			0
$421$	−18.0000			−0.877266			−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000			−0.877266			−0.438633	+	0.898666i	$0.644537\pi$
$422$		0			0
$423$	−1.41421	−	2.44949i	−0.0687614	−	0.119098i
$424$		0			0
$425$	10.6066	−	18.3712i	0.514496	−	0.891133i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−4.00000	+	6.92820i	−0.193122	+	0.334497i
$430$		0			0
$431$	20.0000	+	34.6410i	0.963366	+	1.66860i	0.713942	+	0.700205i	$0.246908\pi$
$431$	20.0000	+	34.6410i	0.963366	+	1.66860i	0.249424	+	0.968394i	$0.419759\pi$
$432$		0			0
$433$	−1.41421			−0.0679628			−0.0339814	−	0.999422i	$-0.510819\pi$
$433$	−1.41421			−0.0679628			−0.0339814	+	0.999422i	$0.510819\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.9706	+	29.3939i	0.811812	+	1.40610i
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.0000	−	17.3205i	0.475114	−	0.822922i	−0.524479	−	0.851423i	$-0.675740\pi$
$443$	10.0000	−	17.3205i	0.475114	−	0.822922i	0.999594	+	0.0285009i	$0.00907336\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−8.48528			−0.401340
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	4.24264	+	7.34847i	0.199778	+	0.346026i
$452$		0			0
$453$	−8.48528	+	14.6969i	−0.398673	+	0.690522i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.00000	+	6.92820i	−0.187112	+	0.324088i	−0.944286	−	0.329125i	$-0.893246\pi$
$457$	−4.00000	+	6.92820i	−0.187112	+	0.324088i	0.757174	+	0.653213i	$0.226579\pi$
$458$		0			0
$459$	12.0000	+	20.7846i	0.560112	+	0.970143i
$460$		0			0
$461$	−25.4558			−1.18560			−0.592798	−	0.805351i	$-0.701977\pi$
$461$	−25.4558			−1.18560			−0.592798	+	0.805351i	$0.701977\pi$
$462$		0			0
$463$	24.0000			1.11537			0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000			1.11537			0.557687	+	0.830051i	$0.311689\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.36396	+	11.0227i	−0.294489	+	0.510070i	−0.974866	−	0.222792i	$-0.928483\pi$
$467$	−6.36396	+	11.0227i	−0.294489	+	0.510070i	0.680377	+	0.732862i	$0.261816\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	2.00000	−	3.46410i	0.0921551	−	0.159617i
$472$		0			0
$473$	−6.00000	−	10.3923i	−0.275880	−	0.477839i
$474$		0			0
$475$	21.2132			0.973329
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	−1.41421	−	2.44949i	−0.0646171	−	0.111920i	0.831907	−	0.554915i	$-0.187249\pi$
$479$	−1.41421	−	2.44949i	−0.0646171	−	0.111920i	−0.896524	+	0.442995i	$0.853916\pi$
$480$		0			0
$481$	2.82843	−	4.89898i	0.128965	−	0.223374i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$487$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$488$		0			0
$489$	8.48528			0.383718
$490$		0			0
$491$	−28.0000			−1.26362			−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000			−1.26362			−0.631811	+	0.775122i	$0.717688\pi$
$492$		0			0
$493$	−12.7279	−	22.0454i	−0.573237	−	0.992875i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6.00000	−	10.3923i	0.268597	−	0.465223i	−0.699903	−	0.714238i	$-0.746773\pi$
$499$	6.00000	−	10.3923i	0.268597	−	0.465223i	0.968500	+	0.249015i	$0.0801067\pi$
$500$		0			0
$501$	10.0000	+	17.3205i	0.446767	+	0.773823i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	3.53553	+	6.12372i	0.157019	+	0.271964i
$508$		0			0
$509$	−21.2132	+	36.7423i	−0.940259	+	1.62858i	−0.175282	+	0.984518i	$0.556084\pi$
$509$	−21.2132	+	36.7423i	−0.940259	+	1.62858i	−0.764977	+	0.644058i	$0.777250\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−12.0000	+	20.7846i	−0.529813	+	0.917663i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−5.65685			−0.248788
$518$		0			0
$519$	−12.0000			−0.526742
$520$		0			0
$521$	14.8492	+	25.7196i	0.650557	+	1.12680i	0.982988	+	0.183670i	$0.0587979\pi$
$521$	14.8492	+	25.7196i	0.650557	+	1.12680i	−0.332431	+	0.943128i	$0.607869\pi$
$522$		0			0
$523$	6.36396	−	11.0227i	0.278277	−	0.481989i	−0.692680	−	0.721245i	$-0.743570\pi$
$523$	6.36396	−	11.0227i	0.278277	−	0.481989i	0.970957	+	0.239256i	$0.0769035\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−18.0000	+	31.1769i	−0.784092	+	1.35809i
$528$		0			0
$529$	−20.5000	−	35.5070i	−0.891304	−	1.54378i
$530$		0			0
$531$	−12.7279			−0.552345
$532$		0			0
$533$	−12.0000			−0.519778
$534$		0			0
$535$		0			0
$536$		0			0
$537$	2.82843	−	4.89898i	0.122056	−	0.211407i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−9.00000	+	15.5885i	−0.386940	+	0.670200i	−0.992036	−	0.125952i	$-0.959801\pi$
$541$	−9.00000	+	15.5885i	−0.386940	+	0.670200i	0.605096	+	0.796152i	$0.293135\pi$
$542$		0			0
$543$	−10.0000	−	17.3205i	−0.429141	−	0.743294i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	42.0000			1.79579			0.897895	−	0.440209i	$-0.145096\pi$
$547$	42.0000			1.79579			0.897895	+	0.440209i	$0.145096\pi$
$548$		0			0
$549$	−2.82843	−	4.89898i	−0.120714	−	0.209083i
$550$		0			0
$551$	12.7279	−	22.0454i	0.542228	−	0.939166i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	0.986394	+	0.164399i	$0.0525683\pi$
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	−0.350824	+	0.936442i	$0.614098\pi$
$558$		0			0
$559$	16.9706			0.717778
$560$		0			0
$561$	12.0000			0.506640
$562$		0			0
$563$	−6.36396	−	11.0227i	−0.268209	−	0.464552i	0.700190	−	0.713956i	$-0.253098\pi$
$563$	−6.36396	−	11.0227i	−0.268209	−	0.464552i	−0.968399	+	0.249405i	$0.919765\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.0000	−	25.9808i	0.628833	−	1.08917i	−0.358954	−	0.933355i	$-0.616866\pi$
$569$	15.0000	−	25.9808i	0.628833	−	1.08917i	0.987786	−	0.155815i	$-0.0498003\pi$
$570$		0			0
$571$	−21.0000	−	36.3731i	−0.878823	−	1.52217i	−0.852634	−	0.522508i	$-0.824996\pi$
$571$	−21.0000	−	36.3731i	−0.878823	−	1.52217i	−0.0261885	−	0.999657i	$-0.508337\pi$
$572$		0			0
$573$	−28.2843			−1.18159
$574$		0			0
$575$	−40.0000			−1.66812
$576$		0			0
$577$	−0.707107	−	1.22474i	−0.0294372	−	0.0509868i	0.850931	−	0.525277i	$-0.176038\pi$
$577$	−0.707107	−	1.22474i	−0.0294372	−	0.0509868i	−0.880369	+	0.474290i	$0.842705\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−6.00000	+	10.3923i	−0.248495	+	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−38.1838			−1.57601			−0.788006	−	0.615667i	$-0.788887\pi$
$587$	−38.1838			−1.57601			−0.788006	+	0.615667i	$0.788887\pi$
$588$		0			0
$589$	−36.0000			−1.48335
$590$		0			0
$591$	−12.7279	−	22.0454i	−0.523557	−	0.906827i
$592$		0			0
$593$	14.8492	−	25.7196i	0.609785	−	1.05618i	−0.381490	−	0.924373i	$-0.624589\pi$
$593$	14.8492	−	25.7196i	0.609785	−	1.05618i	0.991275	−	0.131806i	$-0.0420777\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	18.0000	−	31.1769i	0.736691	−	1.27599i
$598$		0			0
$599$	−4.00000	−	6.92820i	−0.163436	−	0.283079i	0.772663	−	0.634816i	$-0.218924\pi$
$599$	−4.00000	−	6.92820i	−0.163436	−	0.283079i	−0.936099	+	0.351738i	$0.885591\pi$
$600$		0			0
$601$	18.3848			0.749931			0.374965	−	0.927039i	$-0.377655\pi$
$601$	18.3848			0.749931			0.374965	+	0.927039i	$0.377655\pi$
$602$		0			0
$603$	−12.0000			−0.488678
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−8.48528	+	14.6969i	−0.344407	+	0.596530i	−0.985246	−	0.171145i	$-0.945253\pi$
$607$	−8.48528	+	14.6969i	−0.344407	+	0.596530i	0.640839	+	0.767675i	$0.278587\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.00000	−	6.92820i	0.161823	−	0.280285i
$612$		0			0
$613$	9.00000	+	15.5885i	0.363507	+	0.629612i	0.988535	−	0.150990i	$-0.0482461\pi$
$613$	9.00000	+	15.5885i	0.363507	+	0.629612i	−0.625029	+	0.780602i	$0.714913\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$		0			0
$619$	−2.12132	−	3.67423i	−0.0852631	−	0.147680i	0.820240	−	0.572019i	$-0.193840\pi$
$619$	−2.12132	−	3.67423i	−0.0852631	−	0.147680i	−0.905503	+	0.424339i	$0.860506\pi$
$620$		0			0
$621$	22.6274	−	39.1918i	0.908007	−	1.57271i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$	6.00000	+	10.3923i	0.239617	+	0.415029i
$628$		0			0
$629$	−8.48528			−0.338330
$630$		0			0
$631$	12.0000			0.477712			0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000			0.477712			0.238856	+	0.971055i	$0.423228\pi$
$632$		0			0
$633$	8.48528	+	14.6969i	0.337260	+	0.584151i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−2.00000	+	3.46410i	−0.0791188	+	0.137038i
$640$		0			0
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	0.631721	−	0.775196i	$-0.282349\pi$
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	−0.987200	+	0.159489i	$0.949015\pi$
$642$		0			0
$643$	12.7279			0.501940			0.250970	−	0.967995i	$-0.419250\pi$
$643$	12.7279			0.501940			0.250970	+	0.967995i	$0.419250\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.89949	−	17.1464i	−0.389189	−	0.674096i	0.603151	−	0.797627i	$-0.293911\pi$
$647$	−9.89949	−	17.1464i	−0.389189	−	0.674096i	−0.992341	+	0.123531i	$0.960578\pi$
$648$		0			0
$649$	−12.7279	+	22.0454i	−0.499615	+	0.865358i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.00000	+	15.5885i	−0.352197	+	0.610023i	−0.986634	−	0.162951i	$-0.947899\pi$
$653$	−9.00000	+	15.5885i	−0.352197	+	0.610023i	0.634437	+	0.772975i	$0.281232\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.41421			0.0551737
$658$		0			0
$659$	34.0000			1.32445			0.662226	−	0.749304i	$-0.269612\pi$
$659$	34.0000			1.32445			0.662226	+	0.749304i	$0.269612\pi$
$660$		0			0
$661$	−11.3137	−	19.5959i	−0.440052	−	0.762193i	0.557641	−	0.830083i	$-0.311707\pi$
$661$	−11.3137	−	19.5959i	−0.440052	−	0.762193i	−0.997693	+	0.0678896i	$0.978373\pi$
$662$		0			0
$663$	−8.48528	+	14.6969i	−0.329541	+	0.570782i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−24.0000	+	41.5692i	−0.929284	+	1.60957i
$668$		0			0
$669$	−12.0000	−	20.7846i	−0.463947	−	0.803579i
$670$		0			0
$671$	−11.3137			−0.436761
$672$		0			0
$673$	28.0000			1.07932			0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000			1.07932			0.539660	+	0.841883i	$0.318553\pi$
$674$		0			0
$675$	−14.1421	−	24.4949i	−0.544331	−	0.942809i
$676$		0			0
$677$	−21.2132	+	36.7423i	−0.815290	+	1.41212i	0.0938301	+	0.995588i	$0.470089\pi$
$677$	−21.2132	+	36.7423i	−0.815290	+	1.41212i	−0.909120	+	0.416535i	$0.863244\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	9.00000	−	15.5885i	0.344881	−	0.597351i
$682$		0			0
$683$	−2.00000	−	3.46410i	−0.0765279	−	0.132550i	0.825222	−	0.564809i	$-0.191050\pi$
$683$	−2.00000	−	3.46410i	−0.0765279	−	0.132550i	−0.901750	+	0.432259i	$0.857717\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	4.00000			0.152610
$688$		0			0
$689$	−8.48528	−	14.6969i	−0.323263	−	0.559909i
$690$		0			0
$691$	23.3345	−	40.4166i	0.887687	−	1.53752i	0.0450846	−	0.998983i	$-0.485644\pi$
$691$	23.3345	−	40.4166i	0.887687	−	1.53752i	0.842602	−	0.538536i	$-0.181022\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	9.00000	+	15.5885i	0.340899	+	0.590455i
$698$		0			0
$699$	−33.9411			−1.28377
$700$		0			0
$701$	42.0000			1.58632			0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000			1.58632			0.793159	+	0.609015i	$0.208435\pi$
$702$		0			0
$703$	−4.24264	−	7.34847i	−0.160014	−	0.277153i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	9.00000	−	15.5885i	0.338002	−	0.585437i	−0.646055	−	0.763291i	$-0.723582\pi$
$709$	9.00000	−	15.5885i	0.338002	−	0.585437i	0.984057	+	0.177854i	$0.0569156\pi$
$710$		0			0
$711$	6.00000	+	10.3923i	0.225018	+	0.389742i
$712$		0			0
$713$	67.8823			2.54221
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−5.65685	−	9.79796i	−0.211259	−	0.365911i
$718$		0			0
$719$	12.7279	−	22.0454i	0.474671	−	0.822155i	−0.524908	−	0.851159i	$-0.675900\pi$
$719$	12.7279	−	22.0454i	0.474671	−	0.822155i	0.999579	+	0.0290041i	$0.00923357\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−13.0000	+	22.5167i	−0.483475	+	0.837404i
$724$		0			0
$725$	15.0000	+	25.9808i	0.557086	+	0.964901i
$726$		0			0
$727$	8.48528			0.314702			0.157351	−	0.987543i	$-0.449705\pi$
$727$	8.48528			0.314702			0.157351	+	0.987543i	$0.449705\pi$
$728$		0			0
$729$	29.0000			1.07407
$730$		0			0
$731$	−12.7279	−	22.0454i	−0.470759	−	0.815379i
$732$		0			0
$733$	−14.1421	+	24.4949i	−0.522352	+	0.904740i	0.477310	+	0.878735i	$0.341612\pi$
$733$	−14.1421	+	24.4949i	−0.522352	+	0.904740i	−0.999662	+	0.0260048i	$0.991721\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−12.0000	+	20.7846i	−0.442026	+	0.765611i
$738$		0			0
$739$	−15.0000	−	25.9808i	−0.551784	−	0.955718i	−0.998146	−	0.0608653i	$-0.980614\pi$
$739$	−15.0000	−	25.9808i	−0.551784	−	0.955718i	0.446362	−	0.894852i	$-0.352719\pi$
$740$		0			0
$741$	−16.9706			−0.623429
$742$		0			0
$743$	44.0000			1.61420			0.807102	−	0.590412i	$-0.201035\pi$
$743$	44.0000			1.61420			0.807102	+	0.590412i	$0.201035\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−4.94975	+	8.57321i	−0.181102	+	0.313678i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−12.0000	+	20.7846i	−0.437886	+	0.758441i	−0.997526	−	0.0702946i	$-0.977606\pi$
$751$	−12.0000	+	20.7846i	−0.437886	+	0.758441i	0.559640	+	0.828736i	$0.310939\pi$
$752$		0			0
$753$	−9.00000	−	15.5885i	−0.327978	−	0.568075i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−18.0000			−0.654221			−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000			−0.654221			−0.327111	+	0.944986i	$0.606075\pi$
$758$		0			0
$759$	−11.3137	−	19.5959i	−0.410662	−	0.711287i
$760$		0			0
$761$	6.36396	−	11.0227i	0.230693	−	0.399573i	−0.727319	−	0.686300i	$-0.759234\pi$
$761$	6.36396	−	11.0227i	0.230693	−	0.399573i	0.958012	+	0.286727i	$0.0925672\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.0000	−	31.1769i	−0.649942	−	1.12573i
$768$		0			0
$769$	35.3553			1.27495			0.637473	−	0.770473i	$-0.279980\pi$
$769$	35.3553			1.27495			0.637473	+	0.770473i	$0.279980\pi$
$770$		0			0
$771$	−30.0000			−1.08042
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	21.2132	−	36.7423i	0.762001	−	1.31982i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.00000	+	15.5885i	−0.322458	+	0.558514i
$780$		0			0
$781$	4.00000	+	6.92820i	0.143131	+	0.247911i
$782$		0			0
$783$	−33.9411			−1.21296
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−6.36396	−	11.0227i	−0.226851	−	0.392917i	0.730022	−	0.683423i	$-0.239510\pi$
$787$	−6.36396	−	11.0227i	−0.226851	−	0.392917i	−0.956873	+	0.290506i	$0.906176\pi$
$788$		0			0
$789$	14.1421	−	24.4949i	0.503473	−	0.872041i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	8.00000	−	13.8564i	0.284088	−	0.492055i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−42.4264			−1.50282			−0.751410	−	0.659835i	$-0.770626\pi$
$797$	−42.4264			−1.50282			−0.751410	+	0.659835i	$0.770626\pi$
$798$		0			0
$799$	−12.0000			−0.424529
$800$		0			0
$801$	2.12132	+	3.67423i	0.0749532	+	0.129823i
$802$		0			0
$803$	1.41421	−	2.44949i	0.0499065	−	0.0864406i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−18.0000	+	31.1769i	−0.633630	+	1.09748i
$808$		0			0
$809$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$809$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$810$		0			0
$811$	29.6985			1.04285			0.521427	−	0.853296i	$-0.325400\pi$
$811$	29.6985			1.04285			0.521427	+	0.853296i	$0.325400\pi$
$812$		0			0
$813$	−24.0000			−0.841717
$814$		0			0
$815$		0			0
$816$		0			0
$817$	12.7279	−	22.0454i	0.445294	−	0.771271i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.00000	+	5.19615i	−0.104701	+	0.181347i	−0.913616	−	0.406578i	$-0.866722\pi$
$821$	−3.00000	+	5.19615i	−0.104701	+	0.181347i	0.808915	+	0.587925i	$0.200055\pi$
$822$		0			0
$823$	−12.0000	−	20.7846i	−0.418294	−	0.724506i	0.577474	−	0.816409i	$-0.304038\pi$
$823$	−12.0000	−	20.7846i	−0.418294	−	0.724506i	−0.995768	+	0.0919029i	$0.970705\pi$
$824$		0			0
$825$	−14.1421			−0.492366
$826$		0			0
$827$	−4.00000			−0.139094			−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000			−0.139094			−0.0695468	+	0.997579i	$0.522155\pi$
$828$		0			0
$829$	−19.7990	−	34.2929i	−0.687647	−	1.19104i	−0.972597	−	0.232498i	$-0.925310\pi$
$829$	−19.7990	−	34.2929i	−0.687647	−	1.19104i	0.284950	−	0.958542i	$-0.408023\pi$
$830$		0			0
$831$	12.7279	−	22.0454i	0.441527	−	0.764747i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	24.0000	+	41.5692i	0.829561	+	1.43684i
$838$		0			0
$839$	−48.0833			−1.66002			−0.830009	−	0.557750i	$-0.811665\pi$
$839$	−48.0833			−1.66002			−0.830009	+	0.557750i	$0.811665\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	15.0000	−	25.9808i	0.514799	−	0.891657i
$850$		0			0
$851$	8.00000	+	13.8564i	0.274236	+	0.474991i
$852$		0			0
$853$	19.7990			0.677905			0.338952	−	0.940804i	$-0.389927\pi$
$853$	19.7990			0.677905			0.338952	+	0.940804i	$0.389927\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.36396	−	11.0227i	−0.217389	−	0.376528i	0.736620	−	0.676307i	$-0.236421\pi$
$857$	−6.36396	−	11.0227i	−0.217389	−	0.376528i	−0.954009	+	0.299778i	$0.903087\pi$
$858$		0			0
$859$	−2.12132	+	3.67423i	−0.0723785	+	0.125363i	−0.899943	−	0.436007i	$-0.856392\pi$
$859$	−2.12132	+	3.67423i	−0.0723785	+	0.125363i	0.827565	+	0.561370i	$0.189726\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.00000	+	3.46410i	−0.0680808	+	0.117919i	−0.898056	−	0.439880i	$-0.855021\pi$
$863$	−2.00000	+	3.46410i	−0.0680808	+	0.117919i	0.829976	+	0.557800i	$0.188354\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.41421			0.0480292
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$	−16.9706	−	29.3939i	−0.575026	−	0.995974i
$872$		0			0
$873$	−9.19239	+	15.9217i	−0.311115	+	0.538867i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	9.00000	−	15.5885i	0.303908	−	0.526385i	−0.673109	−	0.739543i	$-0.735042\pi$
$877$	9.00000	−	15.5885i	0.303908	−	0.526385i	0.977018	+	0.213158i	$0.0683750\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.24264			−0.142938			−0.0714691	−	0.997443i	$-0.522769\pi$
$881$	−4.24264			−0.142938			−0.0714691	+	0.997443i	$0.522769\pi$
$882$		0			0
$883$	−12.0000			−0.403832			−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000			−0.403832			−0.201916	+	0.979403i	$0.564717\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.3848	+	31.8434i	−0.617300	+	1.06920i	0.372676	+	0.927962i	$0.378440\pi$
$887$	−18.3848	+	31.8434i	−0.617300	+	1.06920i	−0.989976	+	0.141234i	$0.954893\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	5.00000	−	8.66025i	0.167506	−	0.290129i
$892$		0			0
$893$	−6.00000	−	10.3923i	−0.200782	−	0.347765i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	32.0000			1.06845
$898$		0			0
$899$	−25.4558	−	44.0908i	−0.849000	−	1.47051i
$900$		0			0
$901$	−12.7279	+	22.0454i	−0.424029	+	0.734439i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−18.0000	−	31.1769i	−0.597680	−	1.03521i	−0.993163	−	0.116739i	$-0.962756\pi$
$907$	−18.0000	−	31.1769i	−0.597680	−	1.03521i	0.395482	−	0.918474i	$-0.370577\pi$
$908$		0			0
$909$	16.9706			0.562878
$910$		0			0
$911$	−20.0000			−0.662630			−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000			−0.662630			−0.331315	+	0.943520i	$0.607492\pi$
$912$		0			0
$913$	9.89949	+	17.1464i	0.327625	+	0.567464i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$919$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$920$		0			0
$921$	−15.0000	−	25.9808i	−0.494267	−	0.856095i
$922$		0			0
$923$	−11.3137			−0.372395
$924$		0			0
$925$	10.0000			0.328798
$926$		0			0
$927$	−4.24264	−	7.34847i	−0.139347	−	0.241355i
$928$		0			0
$929$	−2.12132	+	3.67423i	−0.0695983	+	0.120548i	−0.898725	−	0.438514i	$-0.855505\pi$
$929$	−2.12132	+	3.67423i	−0.0695983	+	0.120548i	0.829126	+	0.559061i	$0.188838\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	8.00000	−	13.8564i	0.261908	−	0.453638i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−24.0416			−0.785406			−0.392703	−	0.919665i	$-0.628460\pi$
$937$	−24.0416			−0.785406			−0.392703	+	0.919665i	$0.628460\pi$
$938$		0			0
$939$	26.0000			0.848478
$940$		0			0
$941$	−16.9706	−	29.3939i	−0.553225	−	0.958213i	−0.998039	−	0.0625904i	$-0.980064\pi$
$941$	−16.9706	−	29.3939i	−0.553225	−	0.958213i	0.444815	−	0.895623i	$-0.353270\pi$
$942$		0			0
$943$	16.9706	−	29.3939i	0.552638	−	0.957196i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.0000	+	29.4449i	−0.552426	+	0.956830i	0.445673	+	0.895196i	$0.352964\pi$
$947$	−17.0000	+	29.4449i	−0.552426	+	0.956830i	−0.998099	+	0.0616337i	$0.980369\pi$
$948$		0			0
$949$	2.00000	+	3.46410i	0.0649227	+	0.112449i
$950$		0			0
$951$	25.4558			0.825462
$952$		0			0
$953$	−42.0000			−1.36051			−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000			−1.36051			−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−8.48528	+	14.6969i	−0.274290	+	0.475085i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−20.5000	+	35.5070i	−0.661290	+	1.14539i
$962$		0			0
$963$	2.00000	+	3.46410i	0.0644491	+	0.111629i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	48.0000			1.54358			0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000			1.54358			0.771788	+	0.635880i	$0.219363\pi$
$968$		0			0
$969$	12.7279	+	22.0454i	0.408880	+	0.708201i
$970$		0			0
$971$	21.9203	−	37.9671i	0.703456	−	1.21842i	−0.263790	−	0.964580i	$-0.584972\pi$
$971$	21.9203	−	37.9671i	0.703456	−	1.21842i	0.967246	−	0.253842i	$-0.0816942\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	10.0000	−	17.3205i	0.320256	−	0.554700i
$976$		0			0
$977$	6.00000	+	10.3923i	0.191957	+	0.332479i	0.945899	−	0.324462i	$-0.105183\pi$
$977$	6.00000	+	10.3923i	0.191957	+	0.332479i	−0.753942	+	0.656941i	$0.771850\pi$
$978$		0			0
$979$	8.48528			0.271191
$980$		0			0
$981$	−18.0000			−0.574696
$982$		0			0
$983$	−12.7279	−	22.0454i	−0.405958	−	0.703139i	0.588475	−	0.808516i	$-0.299729\pi$
$983$	−12.7279	−	22.0454i	−0.405958	−	0.703139i	−0.994432	+	0.105376i	$0.966395\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−24.0000	+	41.5692i	−0.763156	+	1.32182i
$990$		0			0
$991$	−24.0000	−	41.5692i	−0.762385	−	1.32049i	−0.941618	−	0.336683i	$-0.890695\pi$
$991$	−24.0000	−	41.5692i	−0.762385	−	1.32049i	0.179233	−	0.983807i	$-0.442638\pi$
$992$		0			0
$993$	−25.4558			−0.807817
$994$		0			0
$995$		0			0
$996$		0			0
$997$	28.2843	+	48.9898i	0.895772	+	1.55152i	0.832846	+	0.553504i	$0.186710\pi$
$997$	28.2843	+	48.9898i	0.895772	+	1.55152i	0.0629254	+	0.998018i	$0.479957\pi$
$998$		0			0
$999$	−5.65685	+	9.79796i	−0.178975	+	0.309994i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.i.q.961.1		4
4.3	odd	2		1568.2.i.r.961.2		4
7.2	even	3		1568.2.a.r.1.2	yes	2
7.3	odd	6	inner	1568.2.i.q.1537.2		4
7.4	even	3	inner	1568.2.i.q.1537.1		4
7.5	odd	6		1568.2.a.r.1.1	yes	2
7.6	odd	2	inner	1568.2.i.q.961.2		4
28.3	even	6		1568.2.i.r.1537.1		4
28.11	odd	6		1568.2.i.r.1537.2		4
28.19	even	6		1568.2.a.q.1.2	yes	2
28.23	odd	6		1568.2.a.q.1.1	✓	2
28.27	even	2		1568.2.i.r.961.1		4
56.5	odd	6		3136.2.a.bl.1.2		2
56.19	even	6		3136.2.a.bo.1.1		2
56.37	even	6		3136.2.a.bl.1.1		2
56.51	odd	6		3136.2.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1568.2.a.q.1.1	✓	2	28.23	odd	6
1568.2.a.q.1.2	yes	2	28.19	even	6
1568.2.a.r.1.1	yes	2	7.5	odd	6
1568.2.a.r.1.2	yes	2	7.2	even	3
1568.2.i.q.961.1		4	1.1	even	1	trivial
1568.2.i.q.961.2		4	7.6	odd	2	inner
1568.2.i.q.1537.1		4	7.4	even	3	inner
1568.2.i.q.1537.2		4	7.3	odd	6	inner
1568.2.i.r.961.1		4	28.27	even	2
1568.2.i.r.961.2		4	4.3	odd	2
1568.2.i.r.1537.1		4	28.3	even	6
1568.2.i.r.1537.2		4	28.11	odd	6
3136.2.a.bl.1.1		2	56.37	even	6
3136.2.a.bl.1.2		2	56.5	odd	6
3136.2.a.bo.1.1		2	56.19	even	6
3136.2.a.bo.1.2		2	56.51	odd	6

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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{3} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.707107 - 1.22474i) q^{3} +(0.500000 - 0.866025i) q^{9} +(-1.00000 - 1.73205i) q^{11} +2.82843 q^{13} +(-2.12132 - 3.67423i) q^{17} +(2.12132 - 3.67423i) q^{19} +(-4.00000 + 6.92820i) q^{23} +(2.50000 + 4.33013i) q^{25} -5.65685 q^{27} +6.00000 q^{29} +(-4.24264 - 7.34847i) q^{31} +(-1.41421 + 2.44949i) q^{33} +(1.00000 - 1.73205i) q^{37} +(-2.00000 - 3.46410i) q^{39} -4.24264 q^{41} +6.00000 q^{43} +(1.41421 - 2.44949i) q^{47} +(-3.00000 + 5.19615i) q^{51} +(-3.00000 - 5.19615i) q^{53} -6.00000 q^{57} +(-6.36396 - 11.0227i) q^{59} +(2.82843 - 4.89898i) q^{61} +(-6.00000 - 10.3923i) q^{67} +11.3137 q^{69} -4.00000 q^{71} +(0.707107 + 1.22474i) q^{73} +(3.53553 - 6.12372i) q^{75} +(-6.00000 + 10.3923i) q^{79} +(2.50000 + 4.33013i) q^{81} -9.89949 q^{83} +(-4.24264 - 7.34847i) q^{87} +(-2.12132 + 3.67423i) q^{89} +(-6.00000 + 10.3923i) q^{93} -18.3848 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^9	\( 4 q + 2 q^{9} - 4 q^{11} - 16 q^{23} + 10 q^{25} + 24 q^{29} + 4 q^{37} - 8 q^{39} + 24 q^{43} - 12 q^{51} - 12 q^{53} - 24 q^{57} - 24 q^{67} - 16 q^{71} - 24 q^{79} + 10 q^{81} - 24 q^{93} - 8 q^{99}+O(q^{100}) \) 4 * q + 2 * q^9 - 4 * q^11 - 16 * q^23 + 10 * q^25 + 24 * q^29 + 4 * q^37 - 8 * q^39 + 24 * q^43 - 12 * q^51 - 12 * q^53 - 24 * q^57 - 24 * q^67 - 16 * q^71 - 24 * q^79 + 10 * q^81 - 24 * q^93 - 8 * q^99

Introduction

L-functions

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