LMFDB - Embedded newform 1440.2.bi.d.847.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(847,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 0, 1]))

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

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

chi := DirichletCharacter("1440.847");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.bi (of order $4$, 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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 5x^{12} + 28x^{8} + 80x^{4} + 256 $ x^16 + 5x^12 + 28x^8 + 80*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{49}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			847.3
Root			$0.788026 - 1.17431i$ of defining polynomial
Character	$\chi$	$=$	1440.847
Dual form			1440.2.bi.d.1423.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.386289 - 2.20245i) q^{5} +(-1.51606 + 1.51606i) q^{7} +O(q^{10})$	$q+(-0.386289 - 2.20245i) q^{5} +(-1.51606 + 1.51606i) q^{7} -3.92468 q^{11} +(3.56393 + 3.56393i) q^{13} +(-1.37670 - 1.37670i) q^{17} -4.00000i q^{19} +(5.17748 + 5.17748i) q^{23} +(-4.70156 + 1.70156i) q^{25} -5.95005 q^{29} +7.12785i q^{31} +(3.92468 + 2.75341i) q^{35} +(-3.56393 + 3.56393i) q^{37} +2.75341 q^{41} +(-5.40312 + 5.40312i) q^{43} +(1.54515 - 1.54515i) q^{47} +2.40312i q^{49} +(-1.81616 - 1.81616i) q^{53} +(1.51606 + 8.64391i) q^{55} +3.92468i q^{59} +13.1921i q^{61} +(6.47266 - 9.22607i) q^{65} +(5.40312 + 5.40312i) q^{67} +(5.00000 - 5.00000i) q^{73} +(5.95005 - 5.95005i) q^{77} +7.12785 q^{79} +(6.67809 - 6.67809i) q^{83} +(-2.50031 + 3.56393i) q^{85} +18.4521i q^{89} -10.8062 q^{91} +(-8.80980 + 1.54515i) q^{95} +(-10.4031 - 10.4031i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 24 q^{25} + 16 q^{43} - 16 q^{67} + 80 q^{73} + 32 q^{91} - 64 q^{97}+O(q^{100}) $ 16 * q - 24 * q^25 + 16 * q^43 - 16 * q^67 + 80 * q^73 + 32 * q^91 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.386289	−	2.20245i	−0.172754	−	0.984965i
$5$	−0.386289	−	2.20245i	−0.172754	−	0.984965i
$6$		0			0
$7$	−1.51606	+	1.51606i	−0.573017	+	0.573017i	−0.932970	−	0.359953i	$-0.882793\pi$
$7$	−1.51606	+	1.51606i	−0.573017	+	0.573017i	0.359953	+	0.932970i	$0.382793\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.92468			−1.18334			−0.591668	−	0.806182i	$-0.701530\pi$
$11$	−3.92468			−1.18334			−0.591668	+	0.806182i	$0.701530\pi$
$12$		0			0
$13$	3.56393	+	3.56393i	0.988455	+	0.988455i	0.999934	−	0.0114791i	$-0.00365400\pi$
$13$	3.56393	+	3.56393i	0.988455	+	0.988455i	−0.0114791	+	0.999934i	$0.503654\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.37670	−	1.37670i	−0.333900	−	0.333900i	0.520166	−	0.854065i	$-0.325870\pi$
$17$	−1.37670	−	1.37670i	−0.333900	−	0.333900i	−0.854065	+	0.520166i	$0.825870\pi$
$18$		0			0
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.17748	+	5.17748i	1.07958	+	1.07958i	0.996547	+	0.0830312i	$0.0264601\pi$
$23$	5.17748	+	5.17748i	1.07958	+	1.07958i	0.0830312	+	0.996547i	$0.473540\pi$
$24$		0			0
$25$	−4.70156	+	1.70156i	−0.940312	+	0.340312i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.95005			−1.10490			−0.552449	−	0.833547i	$-0.686306\pi$
$29$	−5.95005			−1.10490			−0.552449	+	0.833547i	$0.686306\pi$
$30$		0			0
$31$			7.12785i			1.28020i	0.768292	+	0.640100i	$0.221107\pi$
$31$			7.12785i			1.28020i	−0.768292	+	0.640100i	$0.778893\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.92468	+	2.75341i	0.663392	+	0.465411i
$36$		0			0
$37$	−3.56393	+	3.56393i	−0.585906	+	0.585906i	−0.936520	−	0.350614i	$-0.885973\pi$
$37$	−3.56393	+	3.56393i	−0.585906	+	0.585906i	0.350614	+	0.936520i	$0.385973\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.75341			0.430010			0.215005	−	0.976613i	$-0.431023\pi$
$41$	2.75341			0.430010			0.215005	+	0.976613i	$0.431023\pi$
$42$		0			0
$43$	−5.40312	+	5.40312i	−0.823969	+	0.823969i	−0.986675	−	0.162706i	$-0.947978\pi$
$43$	−5.40312	+	5.40312i	−0.823969	+	0.823969i	0.162706	+	0.986675i	$0.447978\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.54515	−	1.54515i	0.225384	−	0.225384i	−0.585377	−	0.810761i	$-0.699054\pi$
$47$	1.54515	−	1.54515i	0.225384	−	0.225384i	0.810761	+	0.585377i	$0.199054\pi$
$48$		0			0
$49$			2.40312i			0.343303i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.81616	−	1.81616i	−0.249469	−	0.249469i	0.571284	−	0.820753i	$-0.306446\pi$
$53$	−1.81616	−	1.81616i	−0.249469	−	0.249469i	−0.820753	+	0.571284i	$0.806446\pi$
$54$		0			0
$55$	1.51606	+	8.64391i	0.204425	+	1.16554i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			3.92468i			0.510950i	0.966816	+	0.255475i	$0.0822318\pi$
$59$			3.92468i			0.510950i	−0.966816	+	0.255475i	$0.917768\pi$
$60$		0			0
$61$			13.1921i			1.68907i	0.535497	+	0.844537i	$0.320124\pi$
$61$			13.1921i			1.68907i	−0.535497	+	0.844537i	$0.679876\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.47266	−	9.22607i	0.802835	−	1.14435i
$66$		0			0
$67$	5.40312	+	5.40312i	0.660097	+	0.660097i	0.955403	−	0.295306i	$-0.0954216\pi$
$67$	5.40312	+	5.40312i	0.660097	+	0.660097i	−0.295306	+	0.955403i	$0.595422\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	5.00000	−	5.00000i	0.585206	−	0.585206i	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	5.00000	−	5.00000i	0.585206	−	0.585206i	0.936329	+	0.351123i	$0.114200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.95005	−	5.95005i	0.678071	−	0.678071i
$78$		0			0
$79$	7.12785			0.801946			0.400973	−	0.916090i	$-0.368672\pi$
$79$	7.12785			0.801946			0.400973	+	0.916090i	$0.368672\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.67809	−	6.67809i	0.733016	−	0.733016i	−0.238200	−	0.971216i	$-0.576558\pi$
$83$	6.67809	−	6.67809i	0.733016	−	0.733016i	0.971216	+	0.238200i	$0.0765575\pi$
$84$		0			0
$85$	−2.50031	+	3.56393i	−0.271197	+	0.386562i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			18.4521i			1.95592i	0.208787	+	0.977961i	$0.433048\pi$
$89$			18.4521i			1.95592i	−0.208787	+	0.977961i	$0.566952\pi$
$90$		0			0
$91$	−10.8062			−1.13280
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.80980	+	1.54515i	−0.903866	+	0.158530i
$96$		0			0
$97$	−10.4031	−	10.4031i	−1.05628	−	1.05628i	−0.998319	−	0.0579582i	$-0.981541\pi$
$97$	−10.4031	−	10.4031i	−1.05628	−	1.05628i	−0.0579582	−	0.998319i	$-0.518459\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			8.03722i			0.799733i	0.916573	+	0.399867i	$0.130944\pi$
$101$			8.03722i			0.799733i	−0.916573	+	0.399867i	$0.869056\pi$
$102$		0			0
$103$	5.61179	+	5.61179i	0.552946	+	0.552946i	0.927290	−	0.374344i	$-0.122132\pi$
$103$	5.61179	+	5.61179i	0.552946	+	0.552946i	−0.374344	+	0.927290i	$0.622132\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.92468	−	3.92468i	−0.379413	−	0.379413i	0.491477	−	0.870890i	$-0.336457\pi$
$107$	−3.92468	−	3.92468i	−0.379413	−	0.379413i	−0.870890	+	0.491477i	$0.836457\pi$
$108$		0			0
$109$	−6.06424			−0.580849			−0.290424	−	0.956898i	$-0.593796\pi$
$109$	−6.06424			−0.580849			−0.290424	+	0.956898i	$0.593796\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.37670	+	1.37670i	−0.129509	+	0.129509i	−0.768890	−	0.639381i	$-0.779191\pi$
$113$	−1.37670	+	1.37670i	−0.129509	+	0.129509i	0.639381	+	0.768890i	$0.279191\pi$
$114$		0			0
$115$	9.40312	−	13.4031i	0.876846	−	1.24985i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.17433			0.382660
$120$		0			0
$121$	4.40312			0.400284
$122$		0			0
$123$		0			0
$124$		0			0
$125$	5.56376	+	9.69766i	0.497638	+	0.867385i
$126$		0			0
$127$	−11.6760	+	11.6760i	−1.03608	+	1.03608i	−0.0367559	+	0.999324i	$0.511702\pi$
$127$	−11.6760	+	11.6760i	−1.03608	+	1.03608i	−0.999324	+	0.0367559i	$0.988298\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−17.2809			−1.50984			−0.754918	−	0.655819i	$-0.772323\pi$
$131$	−17.2809			−1.50984			−0.754918	+	0.655819i	$0.772323\pi$
$132$		0			0
$133$	6.06424	+	6.06424i	0.525836	+	0.525836i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.3220	−	14.3220i	−1.22361	−	1.22361i	−0.966338	−	0.257275i	$-0.917176\pi$
$137$	−14.3220	−	14.3220i	−1.22361	−	1.22361i	−0.257275	−	0.966338i	$-0.582824\pi$
$138$		0			0
$139$			4.00000i			0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$			4.00000i			0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−13.9873	−	13.9873i	−1.16967	−	1.16967i
$144$		0			0
$145$	2.29844	+	13.1047i	0.190875	+	1.08828i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.03722			0.658435			0.329217	−	0.944254i	$-0.393215\pi$
$149$	8.03722			0.658435			0.329217	+	0.944254i	$0.393215\pi$
$150$		0			0
$151$		−	10.1600i		−	0.826807i	−0.910548	−	0.413403i	$-0.864340\pi$
$151$		−	10.1600i		−	0.826807i	0.910548	−	0.413403i	$-0.135660\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	15.6987	−	2.75341i	1.26095	−	0.221159i
$156$		0			0
$157$	9.62817	−	9.62817i	0.768411	−	0.768411i	−0.209416	−	0.977827i	$-0.567156\pi$
$157$	9.62817	−	9.62817i	0.768411	−	0.768411i	0.977827	+	0.209416i	$0.0671561\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−15.6987			−1.23723
$162$		0			0
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.54515	+	1.54515i	−0.119568	+	0.119568i	−0.764359	−	0.644791i	$-0.776944\pi$
$167$	−1.54515	+	1.54515i	−0.119568	+	0.119568i	0.644791	+	0.764359i	$0.276944\pi$
$168$		0			0
$169$			12.4031i			0.954086i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.13389	+	4.13389i	0.314294	+	0.314294i	0.846571	−	0.532277i	$-0.178663\pi$
$173$	4.13389	+	4.13389i	0.314294	+	0.314294i	−0.532277	+	0.846571i	$0.678663\pi$
$174$		0			0
$175$	4.54818	−	9.70752i	0.343810	−	0.733820i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			14.1166i			1.05512i	0.849517	+	0.527562i	$0.176894\pi$
$179$			14.1166i			1.05512i	−0.849517	+	0.527562i	$0.823106\pi$
$180$		0			0
$181$		−	6.06424i		−	0.450751i	−0.974272	−	0.225376i	$-0.927639\pi$
$181$		−	6.06424i		−	0.450751i	0.974272	−	0.225376i	$-0.0723610\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.22607	+	6.47266i	0.678314	+	0.475879i
$186$		0			0
$187$	5.40312	+	5.40312i	0.395116	+	0.395116i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	23.8002i		−	1.72212i	−0.508501	−	0.861061i	$-0.669800\pi$
$191$		−	23.8002i		−	1.72212i	0.508501	−	0.861061i	$-0.330200\pi$
$192$		0			0
$193$	0.403124	−	0.403124i	0.0290175	−	0.0290175i	−0.692449	−	0.721467i	$-0.743468\pi$
$193$	0.403124	−	0.403124i	0.0290175	−	0.0290175i	0.721467	+	0.692449i	$0.243468\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.44848	+	5.44848i	−0.388188	+	0.388188i	−0.874041	−	0.485853i	$-0.838509\pi$
$197$	−5.44848	+	5.44848i	−0.388188	+	0.388188i	0.485853	+	0.874041i	$0.338509\pi$
$198$		0			0
$199$	−18.3514			−1.30090			−0.650449	−	0.759550i	$-0.725419\pi$
$199$	−18.3514			−1.30090			−0.650449	+	0.759550i	$0.725419\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.02064	−	9.02064i	0.633125	−	0.633125i
$204$		0			0
$205$	−1.06361	−	6.06424i	−0.0742858	−	0.423545i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			15.6987i			1.08590i
$210$		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
$213$		0			0
$214$		0			0
$215$	13.9873	+	9.81294i	0.953924	+	0.669237i
$216$		0			0
$217$	−10.8062	−	10.8062i	−0.733576	−	0.733576i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	9.81294i		−	0.660090i
$222$		0			0
$223$	−11.6760	−	11.6760i	−0.781885	−	0.781885i	0.198264	−	0.980149i	$-0.436470\pi$
$223$	−11.6760	−	11.6760i	−0.781885	−	0.781885i	−0.980149	+	0.198264i	$0.936470\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.9453	−	12.9453i	−0.859211	−	0.859211i	0.132034	−	0.991245i	$-0.457849\pi$
$227$	−12.9453	−	12.9453i	−0.859211	−	0.859211i	−0.991245	+	0.132034i	$0.957849\pi$
$228$		0			0
$229$	13.1921			0.871758			0.435879	−	0.900005i	$-0.356438\pi$
$229$	13.1921			0.871758			0.435879	+	0.900005i	$0.356438\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.3220	−	14.3220i	0.938267	−	0.938267i	−0.0599354	−	0.998202i	$-0.519089\pi$
$233$	14.3220	−	14.3220i	0.938267	−	0.938267i	0.998202	+	0.0599354i	$0.0190895\pi$
$234$		0			0
$235$	−4.00000	−	2.80625i	−0.260931	−	0.183059i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−27.9745			−1.80952			−0.904761	−	0.425919i	$-0.859951\pi$
$239$	−27.9745			−1.80952			−0.904761	+	0.425919i	$0.859951\pi$
$240$		0			0
$241$	18.8062			1.21142			0.605708	−	0.795687i	$-0.292890\pi$
$241$	18.8062			1.21142			0.605708	+	0.795687i	$0.292890\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.29276	−	0.928300i	0.338142	−	0.0593069i
$246$		0			0
$247$	14.2557	−	14.2557i	0.907068	−	0.907068i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.58213			0.0998634			0.0499317	−	0.998753i	$-0.484100\pi$
$251$	1.58213			0.0998634			0.0499317	+	0.998753i	$0.484100\pi$
$252$		0			0
$253$	−20.3199	−	20.3199i	−1.27750	−	1.27750i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.13011	+	4.13011i	0.257629	+	0.257629i	0.824089	−	0.566460i	$-0.191687\pi$
$257$	4.13011	+	4.13011i	0.257629	+	0.257629i	−0.566460	+	0.824089i	$0.691687\pi$
$258$		0			0
$259$		−	10.8062i		−	0.671468i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	19.1647	+	19.1647i	1.18175	+	1.18175i	0.979291	+	0.202458i	$0.0648929\pi$
$263$	19.1647	+	19.1647i	1.18175	+	1.18175i	0.202458	+	0.979291i	$0.435107\pi$
$264$		0			0
$265$	−3.29844	+	4.70156i	−0.202621	+	0.288815i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.86289			−0.235524			−0.117762	−	0.993042i	$-0.537572\pi$
$269$	−3.86289			−0.235524			−0.117762	+	0.993042i	$0.537572\pi$
$270$		0			0
$271$		−	16.2242i		−	0.985551i	−0.870157	−	0.492775i	$-0.835982\pi$
$271$		−	16.2242i		−	0.985551i	0.870157	−	0.492775i	$-0.164018\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	18.4521	−	6.67809i	1.11271	−	0.402704i
$276$		0			0
$277$	10.6918	−	10.6918i	0.642407	−	0.642407i	−0.308740	−	0.951146i	$-0.599907\pi$
$277$	10.6918	−	10.6918i	0.642407	−	0.642407i	0.951146	+	0.308740i	$0.0999073\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.26022			−0.492764			−0.246382	−	0.969173i	$-0.579242\pi$
$281$	−8.26022			−0.492764			−0.246382	+	0.969173i	$0.579242\pi$
$282$		0			0
$283$	5.40312	−	5.40312i	0.321182	−	0.321182i	−0.528038	−	0.849221i	$-0.677072\pi$
$283$	5.40312	−	5.40312i	0.321182	−	0.321182i	0.849221	+	0.528038i	$0.177072\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.17433	+	4.17433i	−0.246403	+	0.246403i
$288$		0			0
$289$		−	13.2094i		−	0.777022i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.81616	+	1.81616i	0.106101	+	0.106101i	0.758165	−	0.652063i	$-0.226096\pi$
$293$	1.81616	+	1.81616i	0.106101	+	0.106101i	−0.652063	+	0.758165i	$0.726096\pi$
$294$		0			0
$295$	8.64391	−	1.51606i	0.503268	−	0.0882684i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			36.9043i			2.13423i
$300$		0			0
$301$		−	16.3829i		−	0.944296i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	29.0549	−	5.09596i	1.66368	−	0.291794i
$306$		0			0
$307$	20.0000	+	20.0000i	1.14146	+	1.14146i	0.988183	+	0.153277i	$0.0489827\pi$
$307$	20.0000	+	20.0000i	1.14146	+	1.14146i	0.153277	+	0.988183i	$0.451017\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			27.9745i			1.58629i	0.609032	+	0.793145i	$0.291558\pi$
$311$			27.9745i			1.58629i	−0.609032	+	0.793145i	$0.708442\pi$
$312$		0			0
$313$	5.80625	−	5.80625i	0.328189	−	0.328189i	−0.523709	−	0.851897i	$-0.675452\pi$
$313$	5.80625	−	5.80625i	0.328189	−	0.328189i	0.851897	+	0.523709i	$0.175452\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.3985	+	11.3985i	−0.640205	+	0.640205i	−0.950606	−	0.310400i	$-0.899537\pi$
$317$	−11.3985	+	11.3985i	−0.640205	+	0.640205i	0.310400	+	0.950606i	$0.399537\pi$
$318$		0			0
$319$	23.3521			1.30746
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.50682	+	5.50682i	−0.306407	+	0.306407i
$324$		0			0
$325$	−22.8203	−	10.6918i	−1.26584	−	0.593073i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			4.68509i			0.258298i
$330$		0			0
$331$	22.8062			1.25354			0.626772	−	0.779202i	$-0.284376\pi$
$331$	22.8062			1.25354			0.626772	+	0.779202i	$0.284376\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	9.81294	−	13.9873i	0.536138	−	0.764206i
$336$		0			0
$337$	10.4031	+	10.4031i	0.566694	+	0.566694i	0.931201	−	0.364507i	$-0.118762\pi$
$337$	10.4031	+	10.4031i	0.566694	+	0.566694i	−0.364507	+	0.931201i	$0.618762\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	27.9745i		−	1.51491i
$342$		0			0
$343$	−14.2557	−	14.2557i	−0.769735	−	0.769735i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.4521	−	18.4521i	−0.990562	−	0.990562i	0.00939345	−	0.999956i	$-0.497010\pi$
$347$	−18.4521	−	18.4521i	−0.990562	−	0.990562i	−0.999956	+	0.00939345i	$0.997010\pi$
$348$		0			0
$349$	−13.1921			−0.706156			−0.353078	−	0.935594i	$-0.614865\pi$
$349$	−13.1921			−0.706156			−0.353078	+	0.935594i	$0.614865\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.13011	−	4.13011i	0.219824	−	0.219824i	−0.588600	−	0.808424i	$-0.700321\pi$
$353$	4.13011	−	4.13011i	0.219824	−	0.219824i	0.808424	+	0.588600i	$0.200321\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	32.1489			1.69675			0.848376	−	0.529394i	$-0.177581\pi$
$359$	32.1489			1.69675			0.848376	+	0.529394i	$0.177581\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.9437	−	9.08080i	−0.677504	−	0.475311i
$366$		0			0
$367$	7.58030	−	7.58030i	0.395688	−	0.395688i	−0.481021	−	0.876709i	$-0.659734\pi$
$367$	7.58030	−	7.58030i	0.395688	−	0.395688i	0.876709	+	0.481021i	$0.159734\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.50682			0.285900
$372$		0			0
$373$	2.50031	+	2.50031i	0.129461	+	0.129461i	0.768868	−	0.639407i	$-0.220820\pi$
$373$	2.50031	+	2.50031i	0.129461	+	0.129461i	−0.639407	+	0.768868i	$0.720820\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−21.2055	−	21.2055i	−1.09214	−	1.09214i
$378$		0			0
$379$		−	6.80625i		−	0.349614i	−0.984603	−	0.174807i	$-0.944070\pi$
$379$		−	6.80625i		−	0.349614i	0.984603	−	0.174807i	$-0.0559301\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.35181	−	9.35181i	−0.477855	−	0.477855i	0.426590	−	0.904445i	$-0.359715\pi$
$383$	−9.35181	−	9.35181i	−0.477855	−	0.477855i	−0.904445	+	0.426590i	$0.859715\pi$
$384$		0			0
$385$	−15.4031	−	10.8062i	−0.785016	−	0.550737i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−27.6631			−1.40257			−0.701287	−	0.712879i	$-0.747391\pi$
$389$	−27.6631			−1.40257			−0.701287	+	0.712879i	$0.747391\pi$
$390$		0			0
$391$		−	14.2557i		−	0.720942i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.75341	−	15.6987i	−0.138539	−	0.789889i
$396$		0			0
$397$	−4.62754	+	4.62754i	−0.232249	+	0.232249i	−0.813631	−	0.581382i	$-0.802512\pi$
$397$	−4.62754	+	4.62754i	−0.232249	+	0.232249i	0.581382	+	0.813631i	$0.302512\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−31.3975			−1.56791			−0.783957	−	0.620815i	$-0.786802\pi$
$401$	−31.3975			−1.56791			−0.783957	+	0.620815i	$0.786802\pi$
$402$		0			0
$403$	−25.4031	+	25.4031i	−1.26542	+	1.26542i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.9873	−	13.9873i	0.693323	−	0.693323i
$408$		0			0
$409$		−	9.40312i		−	0.464955i	−0.972602	−	0.232477i	$-0.925317\pi$
$409$		−	9.40312i		−	0.464955i	0.972602	−	0.232477i	$-0.0746831\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.95005	−	5.95005i	−0.292783	−	0.292783i
$414$		0			0
$415$	−17.2878	−	12.1285i	−0.848626	−	0.595364i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			22.7877i			1.11325i	0.830764	+	0.556625i	$0.187904\pi$
$419$			22.7877i			1.11325i	−0.830764	+	0.556625i	$0.812096\pi$
$420$		0			0
$421$		−	25.3206i		−	1.23405i	−0.786944	−	0.617025i	$-0.788338\pi$
$421$		−	25.3206i		−	1.23405i	0.786944	−	0.617025i	$-0.211662\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.81521	+	4.13011i	0.427600	+	0.200340i
$426$		0			0
$427$	−20.0000	−	20.0000i	−0.967868	−	0.967868i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			4.17433i			0.201070i	0.994933	+	0.100535i	$0.0320555\pi$
$431$			4.17433i			0.201070i	−0.994933	+	0.100535i	$0.967944\pi$
$432$		0			0
$433$	−0.403124	+	0.403124i	−0.0193729	+	0.0193729i	−0.716727	−	0.697354i	$-0.754361\pi$
$433$	−0.403124	+	0.403124i	−0.0193729	+	0.0193729i	0.697354	+	0.716727i	$0.254361\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	20.7099	−	20.7099i	0.990689	−	0.990689i
$438$		0			0
$439$	4.09573			0.195479			0.0977393	−	0.995212i	$-0.468839\pi$
$439$	4.09573			0.195479			0.0977393	+	0.995212i	$0.468839\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.9453	+	12.9453i	−0.615051	+	0.615051i	−0.944258	−	0.329207i	$-0.893219\pi$
$443$	−12.9453	+	12.9453i	−0.615051	+	0.615051i	0.329207	+	0.944258i	$0.393219\pi$
$444$		0			0
$445$	40.6399	−	7.12785i	1.92652	−	0.337893i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			15.6987i			0.740869i	0.928859	+	0.370434i	$0.120791\pi$
$449$			15.6987i			0.740869i	−0.928859	+	0.370434i	$0.879209\pi$
$450$		0			0
$451$	−10.8062			−0.508846
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.17433	+	23.8002i	0.195696	+	1.11577i
$456$		0			0
$457$	15.8062	+	15.8062i	0.739385	+	0.739385i	0.972459	−	0.233074i	$-0.0748784\pi$
$457$	15.8062	+	15.8062i	0.739385	+	0.739385i	−0.233074	+	0.972459i	$0.574878\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			3.86289i			0.179913i	0.995946	+	0.0899563i	$0.0286727\pi$
$461$			3.86289i			0.179913i	−0.995946	+	0.0899563i	$0.971327\pi$
$462$		0			0
$463$	7.58030	+	7.58030i	0.352286	+	0.352286i	0.860960	−	0.508673i	$-0.169864\pi$
$463$	7.58030	+	7.58030i	0.352286	+	0.352286i	−0.508673	+	0.860960i	$0.669864\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.17127	+	1.17127i	0.0542001	+	0.0542001i	0.733687	−	0.679487i	$-0.237798\pi$
$467$	1.17127	+	1.17127i	0.0542001	+	0.0542001i	−0.679487	+	0.733687i	$0.737798\pi$
$468$		0			0
$469$	−16.3829			−0.756493
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21.2055	−	21.2055i	0.975032	−	0.975032i
$474$		0			0
$475$	6.80625	+	18.8062i	0.312292	+	0.862890i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−27.9745			−1.27819			−0.639095	−	0.769128i	$-0.720691\pi$
$479$	−27.9745			−1.27819			−0.639095	+	0.769128i	$0.720691\pi$
$480$		0			0
$481$	−25.4031			−1.15828
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−18.8937	+	26.9310i	−0.857921	+	1.22287i
$486$		0			0
$487$	−8.64391	+	8.64391i	−0.391693	+	0.391693i	−0.875290	−	0.483598i	$-0.839330\pi$
$487$	−8.64391	+	8.64391i	−0.391693	+	0.391693i	0.483598	+	0.875290i	$0.339330\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−27.4728			−1.23983			−0.619914	−	0.784669i	$-0.712833\pi$
$491$	−27.4728			−1.23983			−0.619914	+	0.784669i	$0.712833\pi$
$492$		0			0
$493$	8.19146	+	8.19146i	0.368925	+	0.368925i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			6.80625i			0.304690i	0.988327	+	0.152345i	$0.0486824\pi$
$499$			6.80625i			0.304690i	−0.988327	+	0.152345i	$0.951318\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	8.80980	+	8.80980i	0.392809	+	0.392809i	0.875688	−	0.482878i	$-0.160409\pi$
$503$	8.80980	+	8.80980i	0.392809	+	0.392809i	−0.482878	+	0.875688i	$0.660409\pi$
$504$		0			0
$505$	17.7016	−	3.10469i	0.787709	−	0.138157i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	22.0245			0.976218			0.488109	−	0.872783i	$-0.337687\pi$
$509$	22.0245			0.976218			0.488109	+	0.872783i	$0.337687\pi$
$510$		0			0
$511$			15.1606i			0.670665i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.1919	−	14.5275i	0.449109	−	0.640156i
$516$		0			0
$517$	−6.06424	+	6.06424i	−0.266705	+	0.266705i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.6987			−0.687774			−0.343887	−	0.939011i	$-0.611744\pi$
$521$	−15.6987			−0.687774			−0.343887	+	0.939011i	$0.611744\pi$
$522$		0			0
$523$	30.8062	−	30.8062i	1.34706	−	1.34706i	0.458229	−	0.888834i	$-0.348484\pi$
$523$	30.8062	−	30.8062i	1.34706	−	1.34706i	0.888834	−	0.458229i	$-0.151516\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.81294	−	9.81294i	0.427458	−	0.427458i
$528$		0			0
$529$			30.6125i			1.33098i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.81294	+	9.81294i	0.425046	+	0.425046i
$534$		0			0
$535$	−7.12785	+	10.1600i	−0.308164	+	0.439254i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	9.43150i		−	0.406243i
$540$		0			0
$541$			34.5756i			1.48652i	0.669001	+	0.743261i	$0.266722\pi$
$541$			34.5756i			1.48652i	−0.669001	+	0.743261i	$0.733278\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.34255	+	13.3562i	0.100344	+	0.572116i
$546$		0			0
$547$	−16.2094	−	16.2094i	−0.693063	−	0.693063i	0.269842	−	0.962905i	$-0.413029\pi$
$547$	−16.2094	−	16.2094i	−0.693063	−	0.693063i	−0.962905	+	0.269842i	$0.913029\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			23.8002i			1.01392i
$552$		0			0
$553$	−10.8062	+	10.8062i	−0.459528	+	0.459528i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.2614	−	15.2614i	0.646647	−	0.646647i	−0.305534	−	0.952181i	$-0.598835\pi$
$557$	15.2614	−	15.2614i	0.646647	−	0.646647i	0.952181	+	0.305534i	$0.0988350\pi$
$558$		0			0
$559$	−38.5127			−1.62891
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.4521	−	18.4521i	0.777665	−	0.777665i	−0.201769	−	0.979433i	$-0.564669\pi$
$563$	18.4521	−	18.4521i	0.777665	−	0.777665i	0.979433	+	0.201769i	$0.0646689\pi$
$564$		0			0
$565$	3.56393	+	2.50031i	0.149935	+	0.105189i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			11.0136i			0.461715i	0.972988	+	0.230858i	$0.0741532\pi$
$569$			11.0136i			0.461715i	−0.972988	+	0.230858i	$0.925847\pi$
$570$		0			0
$571$	33.6125			1.40664			0.703320	−	0.710874i	$-0.251700\pi$
$571$	33.6125			1.40664			0.703320	+	0.710874i	$0.251700\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−33.1520	−	15.5324i	−1.38253	−	0.647747i
$576$		0			0
$577$	−0.403124	−	0.403124i	−0.0167823	−	0.0167823i	0.698666	−	0.715448i	$-0.253777\pi$
$577$	−0.403124	−	0.403124i	−0.0167823	−	0.0167823i	−0.715448	+	0.698666i	$0.753777\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			20.2488i			0.840060i
$582$		0			0
$583$	7.12785	+	7.12785i	0.295205	+	0.295205i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.92468	+	3.92468i	0.161989	+	0.161989i	0.783447	−	0.621458i	$-0.213459\pi$
$587$	3.92468	+	3.92468i	0.161989	+	0.161989i	−0.621458	+	0.783447i	$0.713459\pi$
$588$		0			0
$589$	28.5114			1.17479
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.37670	−	1.37670i	0.0565345	−	0.0565345i	−0.678274	−	0.734809i	$-0.737272\pi$
$593$	1.37670	−	1.37670i	0.0565345	−	0.0565345i	0.734809	+	0.678274i	$0.237272\pi$
$594$		0			0
$595$	−1.61250	−	9.19375i	−0.0661059	−	0.376907i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.17433			0.170559			0.0852793	−	0.996357i	$-0.472822\pi$
$599$	4.17433			0.170559			0.0852793	+	0.996357i	$0.472822\pi$
$600$		0			0
$601$	−18.2094			−0.742776			−0.371388	−	0.928478i	$-0.621118\pi$
$601$	−18.2094			−0.742776			−0.371388	+	0.928478i	$0.621118\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.70088	−	9.69766i	−0.0691505	−	0.394266i
$606$		0			0
$607$	9.70752	−	9.70752i	0.394016	−	0.394016i	−0.482100	−	0.876116i	$-0.660126\pi$
$607$	9.70752	−	9.70752i	0.394016	−	0.394016i	0.876116	+	0.482100i	$0.160126\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11.0136			0.445564
$612$		0			0
$613$	4.62754	+	4.62754i	0.186904	+	0.186904i	0.794356	−	0.607452i	$-0.207808\pi$
$613$	4.62754	+	4.62754i	0.186904	+	0.186904i	−0.607452	+	0.794356i	$0.707808\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	19.8288	+	19.8288i	0.798279	+	0.798279i	0.982824	−	0.184545i	$-0.0590812\pi$
$617$	19.8288	+	19.8288i	0.798279	+	0.798279i	−0.184545	+	0.982824i	$0.559081\pi$
$618$		0			0
$619$			14.8062i			0.595113i	0.954704	+	0.297557i	$0.0961717\pi$
$619$			14.8062i			0.595113i	−0.954704	+	0.297557i	$0.903828\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−27.9745	−	27.9745i	−1.12078	−	1.12078i
$624$		0			0
$625$	19.2094	−	16.0000i	0.768375	−	0.640000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9.81294			0.391268
$630$		0			0
$631$			35.6393i			1.41878i	0.704818	+	0.709388i	$0.251029\pi$
$631$			35.6393i			1.41878i	−0.704818	+	0.709388i	$0.748971\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	30.2262	+	21.2055i	1.19949	+	0.841516i
$636$		0			0
$637$	−8.56455	+	8.56455i	−0.339340	+	0.339340i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.4521			0.728815			0.364408	−	0.931240i	$-0.381271\pi$
$641$	18.4521			0.728815			0.364408	+	0.931240i	$0.381271\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.71949	−	5.71949i	0.224856	−	0.224856i	−0.585684	−	0.810540i	$-0.699174\pi$
$647$	5.71949	−	5.71949i	0.224856	−	0.224856i	0.810540	+	0.585684i	$0.199174\pi$
$648$		0			0
$649$		−	15.4031i		−	0.604626i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.67905	+	5.67905i	0.222238	+	0.222238i	0.809440	−	0.587202i	$-0.199771\pi$
$653$	5.67905	+	5.67905i	0.222238	+	0.222238i	−0.587202	+	0.809440i	$0.699771\pi$
$654$		0			0
$655$	6.67540	+	38.0602i	0.260829	+	1.48714i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			11.7740i			0.458652i	0.973350	+	0.229326i	$0.0736521\pi$
$659$			11.7740i			0.458652i	−0.973350	+	0.229326i	$0.926348\pi$
$660$		0			0
$661$			15.3193i			0.595852i	0.954589	+	0.297926i	$0.0962949\pi$
$661$			15.3193i			0.595852i	−0.954589	+	0.297926i	$0.903705\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	11.0136	−	15.6987i	0.427090	−	0.608770i
$666$		0			0
$667$	−30.8062	−	30.8062i	−1.19282	−	1.19282i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	51.7748i		−	1.99874i
$672$		0			0
$673$	15.0000	−	15.0000i	0.578208	−	0.578208i	−0.356202	−	0.934409i	$-0.615928\pi$
$673$	15.0000	−	15.0000i	0.578208	−	0.578208i	0.934409	+	0.356202i	$0.115928\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.4857	+	13.4857i	−0.518298	+	0.518298i	−0.917056	−	0.398758i	$-0.869441\pi$
$677$	−13.4857	+	13.4857i	−0.518298	+	0.518298i	0.398758	+	0.917056i	$0.369441\pi$
$678$		0			0
$679$	31.5435			1.21053
$680$		0			0
$681$		0			0
$682$		0			0
$683$	22.3768	−	22.3768i	0.856225	−	0.856225i	−0.134666	−	0.990891i	$-0.542996\pi$
$683$	22.3768	−	22.3768i	0.856225	−	0.856225i	0.990891	+	0.134666i	$0.0429962\pi$
$684$		0			0
$685$	−26.0111	+	37.0760i	−0.993833	+	1.41660i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	12.9453i		−	0.493177i
$690$		0			0
$691$	−28.0000			−1.06517			−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000			−1.06517			−0.532585	+	0.846376i	$0.678779\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.80980	−	1.54515i	0.334175	−	0.0586111i
$696$		0			0
$697$	−3.79063	−	3.79063i	−0.143580	−	0.143580i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			33.9246i			1.28131i	0.767827	+	0.640657i	$0.221338\pi$
$701$			33.9246i			1.28131i	−0.767827	+	0.640657i	$0.778662\pi$
$702$		0			0
$703$	14.2557	+	14.2557i	0.537664	+	0.537664i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.1849	−	12.1849i	−0.458261	−	0.458261i
$708$		0			0
$709$	15.3193			0.575329			0.287664	−	0.957731i	$-0.407121\pi$
$709$	15.3193			0.575329			0.287664	+	0.957731i	$0.407121\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−36.9043	+	36.9043i	−1.38208	+	1.38208i
$714$		0			0
$715$	−25.4031	+	36.2094i	−0.950023	+	1.35415i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−27.9745			−1.04327			−0.521637	−	0.853167i	$-0.674679\pi$
$719$	−27.9745			−1.04327			−0.521637	+	0.853167i	$0.674679\pi$
$720$		0			0
$721$	−17.0156			−0.633695
$722$		0			0
$723$		0			0
$724$		0			0
$725$	27.9745	−	10.1244i	1.03895	−	0.376010i
$726$		0			0
$727$	−10.7711	+	10.7711i	−0.399479	+	0.399479i	−0.878049	−	0.478570i	$-0.841155\pi$
$727$	−10.7711	+	10.7711i	−0.399479	+	0.399479i	0.478570	+	0.878049i	$0.341155\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	14.8770			0.550246
$732$		0			0
$733$	3.56393	+	3.56393i	0.131637	+	0.131637i	0.769855	−	0.638219i	$-0.220328\pi$
$733$	3.56393	+	3.56393i	0.131637	+	0.131637i	−0.638219	+	0.769855i	$0.720328\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21.2055	−	21.2055i	−0.781116	−	0.781116i
$738$		0			0
$739$			28.4187i			1.04540i	0.852517	+	0.522700i	$0.175075\pi$
$739$			28.4187i			1.04540i	−0.852517	+	0.522700i	$0.824925\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	22.7971	+	22.7971i	0.836343	+	0.836343i	0.988376	−	0.152032i	$-0.0485818\pi$
$743$	22.7971	+	22.7971i	0.836343	+	0.836343i	−0.152032	+	0.988376i	$0.548582\pi$
$744$		0			0
$745$	−3.10469	−	17.7016i	−0.113747	−	0.648535i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	11.9001			0.434820
$750$		0			0
$751$			21.3836i			0.780297i	0.920752	+	0.390148i	$0.127576\pi$
$751$			21.3836i			0.780297i	−0.920752	+	0.390148i	$0.872424\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−22.3768	+	3.92468i	−0.814376	+	0.142834i
$756$		0			0
$757$	−29.9481	+	29.9481i	−1.08848	+	1.08848i	−0.0927974	+	0.995685i	$0.529581\pi$
$757$	−29.9481	+	29.9481i	−1.08848	+	1.08848i	−0.995685	+	0.0927974i	$0.970419\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.68509			−0.169835			−0.0849173	−	0.996388i	$-0.527063\pi$
$761$	−4.68509			−0.169835			−0.0849173	+	0.996388i	$0.527063\pi$
$762$		0			0
$763$	9.19375	−	9.19375i	0.332836	−	0.332836i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−13.9873	+	13.9873i	−0.505051	+	0.505051i
$768$		0			0
$769$		−	11.4031i		−	0.411207i	−0.978635	−	0.205604i	$-0.934084\pi$
$769$		−	11.4031i		−	0.411207i	0.978635	−	0.205604i	$-0.0659157\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	11.9405	+	11.9405i	0.429472	+	0.429472i	0.888448	−	0.458977i	$-0.151784\pi$
$773$	11.9405	+	11.9405i	0.429472	+	0.429472i	−0.458977	+	0.888448i	$0.651784\pi$
$774$		0			0
$775$	−12.1285	−	33.5120i	−0.435668	−	1.20379i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	11.0136i		−	0.394604i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−24.9248	−	17.4863i	−0.889604	−	0.624112i
$786$		0			0
$787$	20.0000	+	20.0000i	0.712923	+	0.712923i	0.967146	−	0.254223i	$-0.0818196\pi$
$787$	20.0000	+	20.0000i	0.712923	+	0.712923i	−0.254223	+	0.967146i	$0.581820\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	4.17433i		−	0.148422i
$792$		0			0
$793$	−47.0156	+	47.0156i	−1.66957	+	1.66957i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.4761	+	28.4761i	−1.00868	+	1.00868i	−0.00871364	+	0.999962i	$0.502774\pi$
$797$	−28.4761	+	28.4761i	−1.00868	+	1.00868i	−0.999962	+	0.00871364i	$0.997226\pi$
$798$		0			0
$799$	−4.25444			−0.150511
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19.6234	+	19.6234i	−0.692495	+	0.692495i
$804$		0			0
$805$	6.06424	+	34.5756i	0.213736	+	1.21863i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			18.4521i			0.648742i	0.945930	+	0.324371i	$0.105153\pi$
$809$			18.4521i			0.648742i	−0.945930	+	0.324371i	$0.894847\pi$
$810$		0			0
$811$	17.1938			0.603754			0.301877	−	0.953347i	$-0.402387\pi$
$811$	17.1938			0.603754			0.301877	+	0.953347i	$0.402387\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	21.6125	+	21.6125i	0.756126	+	0.756126i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	26.1988i		−	0.914345i	−0.889378	−	0.457173i	$-0.848862\pi$
$821$		−	26.1988i		−	0.914345i	0.889378	−	0.457173i	$-0.151138\pi$
$822$		0			0
$823$	31.9960	+	31.9960i	1.11531	+	1.11531i	0.992420	+	0.122889i	$0.0392159\pi$
$823$	31.9960	+	31.9960i	1.11531	+	1.11531i	0.122889	+	0.992420i	$0.460784\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	11.0136	+	11.0136i	0.382981	+	0.382981i	0.872175	−	0.489194i	$-0.162709\pi$
$827$	11.0136	+	11.0136i	0.382981	+	0.382981i	−0.489194	+	0.872175i	$0.662709\pi$
$828$		0			0
$829$	−34.5756			−1.20086			−0.600431	−	0.799677i	$-0.705004\pi$
$829$	−34.5756			−1.20086			−0.600431	+	0.799677i	$0.705004\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.30839	−	3.30839i	0.114629	−	0.114629i
$834$		0			0
$835$	4.00000	+	2.80625i	0.138426	+	0.0971142i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4.17433			0.144114			0.0720570	−	0.997401i	$-0.477044\pi$
$839$	4.17433			0.144114			0.0720570	+	0.997401i	$0.477044\pi$
$840$		0			0
$841$	6.40312			0.220797
$842$		0			0
$843$		0			0
$844$		0			0
$845$	27.3172	−	4.79119i	0.939742	−	0.164822i
$846$		0			0
$847$	−6.67540	+	6.67540i	−0.229369	+	0.229369i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−36.9043			−1.26506
$852$		0			0
$853$	2.50031	+	2.50031i	0.0856091	+	0.0856091i	0.748615	−	0.663005i	$-0.230719\pi$
$853$	2.50031	+	2.50031i	0.0856091	+	0.0856091i	−0.663005	+	0.748615i	$0.730719\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.13011	+	4.13011i	0.141082	+	0.141082i	0.774120	−	0.633038i	$-0.218193\pi$
$857$	4.13011	+	4.13011i	0.141082	+	0.141082i	−0.633038	+	0.774120i	$0.718193\pi$
$858$		0			0
$859$			25.6125i			0.873887i	0.899489	+	0.436944i	$0.143939\pi$
$859$			25.6125i			0.873887i	−0.899489	+	0.436944i	$0.856061\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.63546	−	4.63546i	−0.157793	−	0.157793i	0.623795	−	0.781588i	$-0.285590\pi$
$863$	−4.63546	−	4.63546i	−0.157793	−	0.157793i	−0.781588	+	0.623795i	$0.785590\pi$
$864$		0			0
$865$	7.50781	−	10.7016i	0.255273	−	0.363864i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−27.9745			−0.948971
$870$		0			0
$871$			38.5127i			1.30495i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−23.1372	−	6.26723i	−0.782181	−	0.211871i
$876$		0			0
$877$	−2.50031	+	2.50031i	−0.0844296	+	0.0844296i	−0.748060	−	0.663631i	$-0.769015\pi$
$877$	−2.50031	+	2.50031i	−0.0844296	+	0.0844296i	0.663631	+	0.748060i	$0.269015\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	38.8360			1.30842			0.654208	−	0.756314i	$-0.273002\pi$
$881$	38.8360			1.30842			0.654208	+	0.756314i	$0.273002\pi$
$882$		0			0
$883$	10.8062	−	10.8062i	0.363659	−	0.363659i	−0.501499	−	0.865158i	$-0.667218\pi$
$883$	10.8062	−	10.8062i	0.363659	−	0.363659i	0.865158	+	0.501499i	$0.167218\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−25.3454	+	25.3454i	−0.851014	+	0.851014i	−0.990258	−	0.139244i	$-0.955533\pi$
$887$	−25.3454	+	25.3454i	−0.851014	+	0.851014i	0.139244	+	0.990258i	$0.455533\pi$
$888$		0			0
$889$		−	35.4031i		−	1.18738i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.18062	−	6.18062i	−0.206827	−	0.206827i
$894$		0			0
$895$	31.0911	−	5.45308i	1.03926	−	0.182276i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	42.4111i		−	1.41449i
$900$		0			0
$901$			5.00063i			0.166595i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−13.3562	+	2.34255i	−0.443974	+	0.0778689i
$906$		0			0
$907$	−25.4031	−	25.4031i	−0.843497	−	0.843497i	0.145815	−	0.989312i	$-0.453420\pi$
$907$	−25.4031	−	25.4031i	−0.843497	−	0.843497i	−0.989312	+	0.145815i	$0.953420\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	4.17433i		−	0.138302i	−0.997606	−	0.0691509i	$-0.977971\pi$
$911$		−	4.17433i		−	0.138302i	0.997606	−	0.0691509i	$-0.0220290\pi$
$912$		0			0
$913$	−26.2094	+	26.2094i	−0.867404	+	0.867404i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	26.1988	−	26.1988i	0.865161	−	0.865161i
$918$		0			0
$919$	−17.1291			−0.565037			−0.282519	−	0.959262i	$-0.591170\pi$
$919$	−17.1291			−0.565037			−0.282519	+	0.959262i	$0.591170\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	10.6918	−	22.8203i	0.351543	−	0.750325i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			4.68509i			0.153713i	0.997042	+	0.0768565i	$0.0244883\pi$
$929$			4.68509i			0.153713i	−0.997042	+	0.0768565i	$0.975512\pi$
$930$		0			0
$931$	9.61250			0.315037
$932$		0			0
$933$		0			0
$934$		0			0
$935$	9.81294	−	13.9873i	0.320917	−	0.457433i
$936$		0			0
$937$	9.59688	+	9.59688i	0.313516	+	0.313516i	0.846270	−	0.532754i	$-0.178843\pi$
$937$	9.59688	+	9.59688i	0.313516	+	0.313516i	−0.532754	+	0.846270i	$0.678843\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	31.8374i		−	1.03787i	−0.854814	−	0.518935i	$-0.826329\pi$
$941$		−	31.8374i		−	1.03787i	0.854814	−	0.518935i	$-0.173671\pi$
$942$		0			0
$943$	14.2557	+	14.2557i	0.464229	+	0.464229i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.50682	+	5.50682i	0.178947	+	0.178947i	0.790897	−	0.611949i	$-0.209614\pi$
$947$	5.50682	+	5.50682i	0.178947	+	0.178947i	−0.611949	+	0.790897i	$0.709614\pi$
$948$		0			0
$949$	35.6393			1.15690
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−17.0754	+	17.0754i	−0.553127	+	0.553127i	−0.927342	−	0.374215i	$-0.877912\pi$
$953$	−17.0754	+	17.0754i	−0.553127	+	0.553127i	0.374215	+	0.927342i	$0.377912\pi$
$954$		0			0
$955$	−52.4187	+	9.19375i	−1.69623	+	0.297503i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	43.4261			1.40230
$960$		0			0
$961$	−19.8062			−0.638911
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.04358	−	0.732138i	−0.0335941	−	0.0235684i
$966$		0			0
$967$	−8.64391	+	8.64391i	−0.277969	+	0.277969i	−0.832298	−	0.554329i	$-0.812975\pi$
$967$	−8.64391	+	8.64391i	−0.277969	+	0.277969i	0.554329	+	0.832298i	$0.312975\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−7.08895			−0.227495			−0.113748	−	0.993510i	$-0.536286\pi$
$971$	−7.08895			−0.227495			−0.113748	+	0.993510i	$0.536286\pi$
$972$		0			0
$973$	−6.06424	−	6.06424i	−0.194411	−	0.194411i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.8288	−	19.8288i	−0.634381	−	0.634381i	0.314783	−	0.949164i	$-0.398068\pi$
$977$	−19.8288	−	19.8288i	−0.634381	−	0.634381i	−0.949164	+	0.314783i	$0.898068\pi$
$978$		0			0
$979$		−	72.4187i		−	2.31451i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	22.7971	+	22.7971i	0.727113	+	0.727113i	0.970044	−	0.242930i	$-0.0781086\pi$
$983$	22.7971	+	22.7971i	0.727113	+	0.727113i	−0.242930	+	0.970044i	$0.578109\pi$
$984$		0			0
$985$	14.1047	+	9.89531i	0.449413	+	0.315291i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−55.9491			−1.77908
$990$		0			0
$991$		−	26.5429i		−	0.843163i	−0.906791	−	0.421581i	$-0.861475\pi$
$991$		−	26.5429i		−	0.843163i	0.906791	−	0.421581i	$-0.138525\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.08895	+	40.4181i	0.224735	+	1.28134i
$996$		0			0
$997$	−3.56393	+	3.56393i	−0.112871	+	0.112871i	−0.761286	−	0.648416i	$-0.775432\pi$
$997$	−3.56393	+	3.56393i	−0.112871	+	0.112871i	0.648416	+	0.761286i	$0.275432\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.bi.d.847.3		16
3.2	odd	2	inner	1440.2.bi.d.847.5		16
4.3	odd	2		360.2.w.d.307.7	yes	16
5.3	odd	4	inner	1440.2.bi.d.1423.6		16
8.3	odd	2	inner	1440.2.bi.d.847.6		16
8.5	even	2		360.2.w.d.307.6	yes	16
12.11	even	2		360.2.w.d.307.2	yes	16
15.8	even	4	inner	1440.2.bi.d.1423.4		16
20.3	even	4		360.2.w.d.163.6	yes	16
24.5	odd	2		360.2.w.d.307.3	yes	16
24.11	even	2	inner	1440.2.bi.d.847.4		16
40.3	even	4	inner	1440.2.bi.d.1423.3		16
40.13	odd	4		360.2.w.d.163.7	yes	16
60.23	odd	4		360.2.w.d.163.3	yes	16
120.53	even	4		360.2.w.d.163.2	✓	16
120.83	odd	4	inner	1440.2.bi.d.1423.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.w.d.163.2	✓	16	120.53	even	4
360.2.w.d.163.3	yes	16	60.23	odd	4
360.2.w.d.163.6	yes	16	20.3	even	4
360.2.w.d.163.7	yes	16	40.13	odd	4
360.2.w.d.307.2	yes	16	12.11	even	2
360.2.w.d.307.3	yes	16	24.5	odd	2
360.2.w.d.307.6	yes	16	8.5	even	2
360.2.w.d.307.7	yes	16	4.3	odd	2
1440.2.bi.d.847.3		16	1.1	even	1	trivial
1440.2.bi.d.847.4		16	24.11	even	2	inner
1440.2.bi.d.847.5		16	3.2	odd	2	inner
1440.2.bi.d.847.6		16	8.3	odd	2	inner
1440.2.bi.d.1423.3		16	40.3	even	4	inner
1440.2.bi.d.1423.4		16	15.8	even	4	inner
1440.2.bi.d.1423.5		16	120.83	odd	4	inner
1440.2.bi.d.1423.6		16	5.3	odd	4	inner

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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.bi (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.386289 - 2.20245i) q^{5} +(-1.51606 + 1.51606i) q^{7} +O(q^{10})\)	\(q+(-0.386289 - 2.20245i) q^{5} +(-1.51606 + 1.51606i) q^{7} -3.92468 q^{11} +(3.56393 + 3.56393i) q^{13} +(-1.37670 - 1.37670i) q^{17} -4.00000i q^{19} +(5.17748 + 5.17748i) q^{23} +(-4.70156 + 1.70156i) q^{25} -5.95005 q^{29} +7.12785i q^{31} +(3.92468 + 2.75341i) q^{35} +(-3.56393 + 3.56393i) q^{37} +2.75341 q^{41} +(-5.40312 + 5.40312i) q^{43} +(1.54515 - 1.54515i) q^{47} +2.40312i q^{49} +(-1.81616 - 1.81616i) q^{53} +(1.51606 + 8.64391i) q^{55} +3.92468i q^{59} +13.1921i q^{61} +(6.47266 - 9.22607i) q^{65} +(5.40312 + 5.40312i) q^{67} +(5.00000 - 5.00000i) q^{73} +(5.95005 - 5.95005i) q^{77} +7.12785 q^{79} +(6.67809 - 6.67809i) q^{83} +(-2.50031 + 3.56393i) q^{85} +18.4521i q^{89} -10.8062 q^{91} +(-8.80980 + 1.54515i) q^{95} +(-10.4031 - 10.4031i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 24 q^{25} + 16 q^{43} - 16 q^{67} + 80 q^{73} + 32 q^{91} - 64 q^{97}+O(q^{100}) \) 16 * q - 24 * q^25 + 16 * q^43 - 16 * q^67 + 80 * q^73 + 32 * q^91 - 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more