LMFDB - Embedded newform 612.2.k.d.217.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [612,2,Mod(217,612)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(612, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 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("612.217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 612 = 2^{2} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	612.k (of order $4$, degree $2$, 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:	$4.88684460370$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{34})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 289 $ x^4 + 289
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			217.1
Root			$-2.91548 - 2.91548i$ of defining polynomial
Character	$\chi$	$=$	612.217
Dual form			612.2.k.d.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.91548 - 2.91548i) q^{5} +(3.00000 - 3.00000i) q^{7} +(-2.91548 + 2.91548i) q^{11} -5.00000 q^{13} +(-2.91548 + 2.91548i) q^{17} +1.00000i q^{19} +(2.91548 - 2.91548i) q^{23} +12.0000i q^{25} +(-4.00000 - 4.00000i) q^{31} -17.4929 q^{35} +(-3.00000 - 3.00000i) q^{37} +(2.91548 - 2.91548i) q^{41} +5.00000i q^{43} -11.6619 q^{47} -11.0000i q^{49} -11.6619i q^{53} +17.0000 q^{55} -5.83095i q^{59} +(3.00000 - 3.00000i) q^{61} +(14.5774 + 14.5774i) q^{65} -6.00000 q^{67} +(5.83095 + 5.83095i) q^{71} +(-4.00000 - 4.00000i) q^{73} +17.4929i q^{77} +(5.00000 - 5.00000i) q^{79} -11.6619i q^{83} +17.0000 q^{85} +17.4929 q^{89} +(-15.0000 + 15.0000i) q^{91} +(2.91548 - 2.91548i) q^{95} +(6.00000 + 6.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7} - 20 q^{13} - 16 q^{31} - 12 q^{37} + 68 q^{55} + 12 q^{61} - 24 q^{67} - 16 q^{73} + 20 q^{79} + 68 q^{85} - 60 q^{91} + 24 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 - 20 * q^13 - 16 * q^31 - 12 * q^37 + 68 * q^55 + 12 * q^61 - 24 * q^67 - 16 * q^73 + 20 * q^79 + 68 * q^85 - 60 * q^91 + 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/612\mathbb{Z}\right)^\times$.

$n$	$37$	$137$	$307$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$5$	−2.91548	−	2.91548i	−1.30384	−	1.30384i	−0.925782	−	0.378059i	$-0.876592\pi$
$5$	−2.91548	−	2.91548i	−1.30384	−	1.30384i	−0.378059	−	0.925782i	$-0.623408\pi$
$6$		0			0
$7$	3.00000	−	3.00000i	1.13389	−	1.13389i	0.144370	−	0.989524i	$-0.453885\pi$
$7$	3.00000	−	3.00000i	1.13389	−	1.13389i	0.989524	−	0.144370i	$-0.0461154\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.91548	+	2.91548i	−0.879049	+	0.879049i	−0.993436	−	0.114387i	$-0.963510\pi$
$11$	−2.91548	+	2.91548i	−0.879049	+	0.879049i	0.114387	+	0.993436i	$0.463510\pi$
$12$		0			0
$13$	−5.00000			−1.38675			−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000			−1.38675			−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.91548	+	2.91548i	−0.707107	+	0.707107i
$17$	−2.91548	+	2.91548i	−0.707107	+	0.707107i
$18$		0			0
$19$			1.00000i			0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$			1.00000i			0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.91548	−	2.91548i	0.607919	−	0.607919i	−0.334483	−	0.942402i	$-0.608562\pi$
$23$	2.91548	−	2.91548i	0.607919	−	0.607919i	0.942402	+	0.334483i	$0.108562\pi$
$24$		0			0
$25$			12.0000i			2.40000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$29$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$30$		0			0
$31$	−4.00000	−	4.00000i	−0.718421	−	0.718421i	0.249861	−	0.968282i	$-0.419615\pi$
$31$	−4.00000	−	4.00000i	−0.718421	−	0.718421i	−0.968282	+	0.249861i	$0.919615\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−17.4929			−2.95683
$36$		0			0
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	0.416115	−	0.909312i	$-0.363391\pi$
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	−0.909312	+	0.416115i	$0.863391\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.91548	−	2.91548i	0.455321	−	0.455321i	−0.441795	−	0.897116i	$-0.645658\pi$
$41$	2.91548	−	2.91548i	0.455321	−	0.455321i	0.897116	+	0.441795i	$0.145658\pi$
$42$		0			0
$43$			5.00000i			0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$			5.00000i			0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−11.6619			−1.70106			−0.850532	−	0.525924i	$-0.823720\pi$
$47$	−11.6619			−1.70106			−0.850532	+	0.525924i	$0.823720\pi$
$48$		0			0
$49$		−	11.0000i		−	1.57143i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	11.6619i		−	1.60189i	−0.598741	−	0.800943i	$-0.704332\pi$
$53$		−	11.6619i		−	1.60189i	0.598741	−	0.800943i	$-0.295668\pi$
$54$		0			0
$55$	17.0000			2.29228
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	5.83095i		−	0.759125i	−0.925166	−	0.379563i	$-0.876074\pi$
$59$		−	5.83095i		−	0.759125i	0.925166	−	0.379563i	$-0.123926\pi$
$60$		0			0
$61$	3.00000	−	3.00000i	0.384111	−	0.384111i	−0.488470	−	0.872581i	$-0.662445\pi$
$61$	3.00000	−	3.00000i	0.384111	−	0.384111i	0.872581	+	0.488470i	$0.162445\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	14.5774	+	14.5774i	1.80810	+	1.80810i
$66$		0			0
$67$	−6.00000			−0.733017			−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000			−0.733017			−0.366508	+	0.930415i	$0.619447\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.83095	+	5.83095i	0.692007	+	0.692007i	0.962673	−	0.270667i	$-0.0872441\pi$
$71$	5.83095	+	5.83095i	0.692007	+	0.692007i	−0.270667	+	0.962673i	$0.587244\pi$
$72$		0			0
$73$	−4.00000	−	4.00000i	−0.468165	−	0.468165i	0.433155	−	0.901319i	$-0.357400\pi$
$73$	−4.00000	−	4.00000i	−0.468165	−	0.468165i	−0.901319	+	0.433155i	$0.857400\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			17.4929i			1.99350i
$78$		0			0
$79$	5.00000	−	5.00000i	0.562544	−	0.562544i	−0.367485	−	0.930029i	$-0.619781\pi$
$79$	5.00000	−	5.00000i	0.562544	−	0.562544i	0.930029	+	0.367485i	$0.119781\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	11.6619i		−	1.28006i	−0.768350	−	0.640030i	$-0.778922\pi$
$83$		−	11.6619i		−	1.28006i	0.768350	−	0.640030i	$-0.221078\pi$
$84$		0			0
$85$	17.0000			1.84391
$86$		0			0
$87$		0			0
$88$		0			0
$89$	17.4929			1.85424			0.927119	−	0.374766i	$-0.122277\pi$
$89$	17.4929			1.85424			0.927119	+	0.374766i	$0.122277\pi$
$90$		0			0
$91$	−15.0000	+	15.0000i	−1.57243	+	1.57243i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.91548	−	2.91548i	0.299122	−	0.299122i
$96$		0			0
$97$	6.00000	+	6.00000i	0.609208	+	0.609208i	0.942739	−	0.333531i	$-0.108240\pi$
$97$	6.00000	+	6.00000i	0.609208	+	0.609208i	−0.333531	+	0.942739i	$0.608240\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−13.0000			−1.28093			−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000			−1.28093			−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.91548	−	2.91548i	−0.281850	−	0.281850i	0.551997	−	0.833846i	$-0.313866\pi$
$107$	−2.91548	−	2.91548i	−0.281850	−	0.281850i	−0.833846	+	0.551997i	$0.813866\pi$
$108$		0			0
$109$	−6.00000	+	6.00000i	−0.574696	+	0.574696i	−0.933437	−	0.358741i	$-0.883206\pi$
$109$	−6.00000	+	6.00000i	−0.574696	+	0.574696i	0.358741	+	0.933437i	$0.383206\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.91548	+	2.91548i	−0.274265	+	0.274265i	−0.830814	−	0.556550i	$-0.812125\pi$
$113$	−2.91548	+	2.91548i	−0.274265	+	0.274265i	0.556550	+	0.830814i	$0.312125\pi$
$114$		0			0
$115$	−17.0000			−1.58526
$116$		0			0
$117$		0			0
$118$		0			0
$119$			17.4929i			1.60357i
$120$		0			0
$121$		−	6.00000i		−	0.545455i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	20.4083	−	20.4083i	1.82538	−	1.82538i
$126$		0			0
$127$		−	1.00000i		−	0.0887357i	−0.999015	−	0.0443678i	$-0.985873\pi$
$127$		−	1.00000i		−	0.0887357i	0.999015	−	0.0443678i	$-0.0141274\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.91548	−	2.91548i	−0.254726	−	0.254726i	0.568179	−	0.822905i	$-0.307648\pi$
$131$	−2.91548	−	2.91548i	−0.254726	−	0.254726i	−0.822905	+	0.568179i	$0.807648\pi$
$132$		0			0
$133$	3.00000	+	3.00000i	0.260133	+	0.260133i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.83095			0.498172			0.249086	−	0.968481i	$-0.419870\pi$
$137$	5.83095			0.498172			0.249086	+	0.968481i	$0.419870\pi$
$138$		0			0
$139$	1.00000	+	1.00000i	0.0848189	+	0.0848189i	0.748243	−	0.663424i	$-0.230898\pi$
$139$	1.00000	+	1.00000i	0.0848189	+	0.0848189i	−0.663424	+	0.748243i	$0.730898\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	14.5774	−	14.5774i	1.21902	−	1.21902i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.83095			0.477690			0.238845	−	0.971058i	$-0.423231\pi$
$149$	5.83095			0.477690			0.238845	+	0.971058i	$0.423231\pi$
$150$		0			0
$151$		−	6.00000i		−	0.488273i	−0.969741	−	0.244137i	$-0.921495\pi$
$151$		−	6.00000i		−	0.488273i	0.969741	−	0.244137i	$-0.0785045\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			23.3238i			1.87341i
$156$		0			0
$157$	−11.0000			−0.877896			−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000			−0.877896			−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	17.4929i		−	1.37863i
$162$		0			0
$163$	10.0000	−	10.0000i	0.783260	−	0.783260i	−0.197119	−	0.980380i	$-0.563159\pi$
$163$	10.0000	−	10.0000i	0.783260	−	0.783260i	0.980380	+	0.197119i	$0.0631586\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.91548	−	2.91548i	−0.225606	−	0.225606i	0.585248	−	0.810854i	$-0.300997\pi$
$167$	−2.91548	−	2.91548i	−0.225606	−	0.225606i	−0.810854	+	0.585248i	$0.800997\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.5774	+	14.5774i	1.10830	+	1.10830i	0.993374	+	0.114923i	$0.0366622\pi$
$173$	14.5774	+	14.5774i	1.10830	+	1.10830i	0.114923	+	0.993374i	$0.463338\pi$
$174$		0			0
$175$	36.0000	+	36.0000i	2.72134	+	2.72134i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			11.6619i			0.871652i	0.900031	+	0.435826i	$0.143544\pi$
$179$			11.6619i			0.871652i	−0.900031	+	0.435826i	$0.856456\pi$
$180$		0			0
$181$	11.0000	−	11.0000i	0.817624	−	0.817624i	−0.168140	−	0.985763i	$-0.553776\pi$
$181$	11.0000	−	11.0000i	0.817624	−	0.817624i	0.985763	+	0.168140i	$0.0537759\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			17.4929i			1.28610i
$186$		0			0
$187$		−	17.0000i		−	1.24316i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.83095			−0.421913			−0.210956	−	0.977495i	$-0.567658\pi$
$191$	−5.83095			−0.421913			−0.210956	+	0.977495i	$0.567658\pi$
$192$		0			0
$193$	10.0000	−	10.0000i	0.719816	−	0.719816i	−0.248752	−	0.968567i	$-0.580020\pi$
$193$	10.0000	−	10.0000i	0.719816	−	0.719816i	0.968567	+	0.248752i	$0.0800203\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.91548	+	2.91548i	−0.207719	+	0.207719i	−0.803297	−	0.595578i	$-0.796923\pi$
$197$	−2.91548	+	2.91548i	−0.207719	+	0.207719i	0.595578	+	0.803297i	$0.296923\pi$
$198$		0			0
$199$	−18.0000	−	18.0000i	−1.27599	−	1.27599i	−0.942894	−	0.333092i	$-0.891908\pi$
$199$	−18.0000	−	18.0000i	−1.27599	−	1.27599i	−0.333092	−	0.942894i	$-0.608092\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−17.0000			−1.18733
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.91548	−	2.91548i	−0.201668	−	0.201668i
$210$		0			0
$211$	−2.00000	+	2.00000i	−0.137686	+	0.137686i	−0.772590	−	0.634905i	$-0.781039\pi$
$211$	−2.00000	+	2.00000i	−0.137686	+	0.137686i	0.634905	+	0.772590i	$0.281039\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	14.5774	−	14.5774i	0.994169	−	0.994169i
$216$		0			0
$217$	−24.0000			−1.62923
$218$		0			0
$219$		0			0
$220$		0			0
$221$	14.5774	−	14.5774i	0.980581	−	0.980581i
$222$		0			0
$223$		−	23.0000i		−	1.54019i	−0.637927	−	0.770097i	$-0.720208\pi$
$223$		−	23.0000i		−	1.54019i	0.637927	−	0.770097i	$-0.279792\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.91548	+	2.91548i	−0.193507	+	0.193507i	−0.797210	−	0.603703i	$-0.793691\pi$
$227$	−2.91548	+	2.91548i	−0.193507	+	0.193507i	0.603703	+	0.797210i	$0.293691\pi$
$228$		0			0
$229$			12.0000i			0.792982i	0.918039	+	0.396491i	$0.129772\pi$
$229$			12.0000i			0.792982i	−0.918039	+	0.396491i	$0.870228\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.5774	−	14.5774i	−0.954996	−	0.954996i	0.0440341	−	0.999030i	$-0.485979\pi$
$233$	−14.5774	−	14.5774i	−0.954996	−	0.954996i	−0.999030	+	0.0440341i	$0.985979\pi$
$234$		0			0
$235$	34.0000	+	34.0000i	2.21792	+	2.21792i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−9.00000	−	9.00000i	−0.579741	−	0.579741i	0.355091	−	0.934832i	$-0.384450\pi$
$241$	−9.00000	−	9.00000i	−0.579741	−	0.579741i	−0.934832	+	0.355091i	$0.884450\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−32.0702	+	32.0702i	−2.04889	+	2.04889i
$246$		0			0
$247$		−	5.00000i		−	0.318142i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−17.4929			−1.10414			−0.552070	−	0.833798i	$-0.686162\pi$
$251$	−17.4929			−1.10414			−0.552070	+	0.833798i	$0.686162\pi$
$252$		0			0
$253$			17.0000i			1.06878i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	17.4929i		−	1.09117i	−0.838054	−	0.545587i	$-0.816307\pi$
$257$		−	17.4929i		−	1.09117i	0.838054	−	0.545587i	$-0.183693\pi$
$258$		0			0
$259$	−18.0000			−1.11847
$260$		0			0
$261$		0			0
$262$		0			0
$263$			17.4929i			1.07866i	0.842096	+	0.539328i	$0.181322\pi$
$263$			17.4929i			1.07866i	−0.842096	+	0.539328i	$0.818678\pi$
$264$		0			0
$265$	−34.0000	+	34.0000i	−2.08860	+	2.08860i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.91548	−	2.91548i	−0.177760	−	0.177760i	0.612619	−	0.790379i	$-0.290116\pi$
$269$	−2.91548	−	2.91548i	−0.177760	−	0.177760i	−0.790379	+	0.612619i	$0.790116\pi$
$270$		0			0
$271$	−5.00000			−0.303728			−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000			−0.303728			−0.151864	+	0.988401i	$0.548528\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−34.9857	−	34.9857i	−2.10972	−	2.10972i
$276$		0			0
$277$	8.00000	+	8.00000i	0.480673	+	0.480673i	0.905347	−	0.424673i	$-0.139611\pi$
$277$	8.00000	+	8.00000i	0.480673	+	0.480673i	−0.424673	+	0.905347i	$0.639611\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	17.4929i		−	1.04354i	−0.853087	−	0.521768i	$-0.825273\pi$
$281$		−	17.4929i		−	1.04354i	0.853087	−	0.521768i	$-0.174727\pi$
$282$		0			0
$283$	13.0000	−	13.0000i	0.772770	−	0.772770i	−0.205820	−	0.978590i	$-0.565986\pi$
$283$	13.0000	−	13.0000i	0.772770	−	0.772770i	0.978590	+	0.205820i	$0.0659862\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	17.4929i		−	1.03257i
$288$		0			0
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−17.4929			−1.02194			−0.510972	−	0.859597i	$-0.670714\pi$
$293$	−17.4929			−1.02194			−0.510972	+	0.859597i	$0.670714\pi$
$294$		0			0
$295$	−17.0000	+	17.0000i	−0.989778	+	0.989778i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−14.5774	+	14.5774i	−0.843032	+	0.843032i
$300$		0			0
$301$	15.0000	+	15.0000i	0.864586	+	0.864586i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−17.4929			−1.00164
$306$		0			0
$307$	18.0000			1.02731			0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000			1.02731			0.513657	+	0.857996i	$0.328290\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	17.4929	+	17.4929i	0.991929	+	0.991929i	0.999968	−	0.00803884i	$-0.00255887\pi$
$311$	17.4929	+	17.4929i	0.991929	+	0.991929i	−0.00803884	+	0.999968i	$0.502559\pi$
$312$		0			0
$313$	−12.0000	+	12.0000i	−0.678280	+	0.678280i	−0.959611	−	0.281331i	$-0.909224\pi$
$313$	−12.0000	+	12.0000i	−0.678280	+	0.678280i	0.281331	+	0.959611i	$0.409224\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.91548	−	2.91548i	−0.162221	−	0.162221i
$324$		0			0
$325$		−	60.0000i		−	3.32820i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−34.9857	+	34.9857i	−1.92882	+	1.92882i
$330$		0			0
$331$		−	31.0000i		−	1.70391i	−0.523612	−	0.851957i	$-0.675416\pi$
$331$		−	31.0000i		−	1.70391i	0.523612	−	0.851957i	$-0.324584\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	17.4929	+	17.4929i	0.955737	+	0.955737i
$336$		0			0
$337$	17.0000	+	17.0000i	0.926049	+	0.926049i	0.997448	−	0.0713988i	$-0.0227463\pi$
$337$	17.0000	+	17.0000i	0.926049	+	0.926049i	−0.0713988	+	0.997448i	$0.522746\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	23.3238			1.26305
$342$		0			0
$343$	−12.0000	−	12.0000i	−0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.4929	−	17.4929i	0.939066	−	0.939066i	−0.0591815	−	0.998247i	$-0.518849\pi$
$347$	17.4929	−	17.4929i	0.939066	−	0.939066i	0.998247	+	0.0591815i	$0.0188491\pi$
$348$		0			0
$349$			31.0000i			1.65939i	0.558216	+	0.829696i	$0.311486\pi$
$349$			31.0000i			1.65939i	−0.558216	+	0.829696i	$0.688514\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5.83095			0.310350			0.155175	−	0.987887i	$-0.450406\pi$
$353$	5.83095			0.310350			0.155175	+	0.987887i	$0.450406\pi$
$354$		0			0
$355$		−	34.0000i		−	1.80453i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			17.4929i			0.923238i	0.887078	+	0.461619i	$0.152731\pi$
$359$			17.4929i			0.923238i	−0.887078	+	0.461619i	$0.847269\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$			23.3238i			1.22082i
$366$		0			0
$367$	−9.00000	+	9.00000i	−0.469796	+	0.469796i	−0.901849	−	0.432052i	$-0.857790\pi$
$367$	−9.00000	+	9.00000i	−0.469796	+	0.469796i	0.432052	+	0.901849i	$0.357790\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−34.9857	−	34.9857i	−1.81637	−	1.81637i
$372$		0			0
$373$	4.00000			0.207112			0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000			0.207112			0.103556	+	0.994624i	$0.466978\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	9.00000	+	9.00000i	0.462299	+	0.462299i	0.899408	−	0.437109i	$-0.143998\pi$
$379$	9.00000	+	9.00000i	0.462299	+	0.462299i	−0.437109	+	0.899408i	$0.643998\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	34.9857i		−	1.78769i	−0.448380	−	0.893843i	$-0.647999\pi$
$383$		−	34.9857i		−	1.78769i	0.448380	−	0.893843i	$-0.352001\pi$
$384$		0			0
$385$	51.0000	−	51.0000i	2.59920	−	2.59920i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			17.4929i			0.886923i	0.896293	+	0.443461i	$0.146250\pi$
$389$			17.4929i			0.886923i	−0.896293	+	0.443461i	$0.853750\pi$
$390$		0			0
$391$			17.0000i			0.859727i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−29.1548			−1.46694
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.5774	−	14.5774i	0.727960	−	0.727960i	−0.242253	−	0.970213i	$-0.577887\pi$
$401$	14.5774	−	14.5774i	0.727960	−	0.727960i	0.970213	+	0.242253i	$0.0778865\pi$
$402$		0			0
$403$	20.0000	+	20.0000i	0.996271	+	0.996271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	17.4929			0.867089
$408$		0			0
$409$	−7.00000			−0.346128			−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000			−0.346128			−0.173064	+	0.984911i	$0.555367\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−17.4929	−	17.4929i	−0.860767	−	0.860767i
$414$		0			0
$415$	−34.0000	+	34.0000i	−1.66899	+	1.66899i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−17.4929	+	17.4929i	−0.854582	+	0.854582i	−0.990694	−	0.136112i	$-0.956539\pi$
$419$	−17.4929	+	17.4929i	−0.854582	+	0.854582i	0.136112	+	0.990694i	$0.456539\pi$
$420$		0			0
$421$	−7.00000			−0.341159			−0.170580	−	0.985344i	$-0.554564\pi$
$421$	−7.00000			−0.341159			−0.170580	+	0.985344i	$0.554564\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−34.9857	−	34.9857i	−1.69706	−	1.69706i
$426$		0			0
$427$		−	18.0000i		−	0.871081i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−17.4929	+	17.4929i	−0.842601	+	0.842601i	−0.989197	−	0.146595i	$-0.953168\pi$
$431$	−17.4929	+	17.4929i	−0.842601	+	0.842601i	0.146595	+	0.989197i	$0.453168\pi$
$432$		0			0
$433$			31.0000i			1.48976i	0.667196	+	0.744882i	$0.267494\pi$
$433$			31.0000i			1.48976i	−0.667196	+	0.744882i	$0.732506\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.91548	+	2.91548i	0.139466	+	0.139466i
$438$		0			0
$439$	12.0000	+	12.0000i	0.572729	+	0.572729i	0.932890	−	0.360161i	$-0.117278\pi$
$439$	12.0000	+	12.0000i	0.572729	+	0.572729i	−0.360161	+	0.932890i	$0.617278\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.6619			0.554074			0.277037	−	0.960859i	$-0.410648\pi$
$443$	11.6619			0.554074			0.277037	+	0.960859i	$0.410648\pi$
$444$		0			0
$445$	−51.0000	−	51.0000i	−2.41763	−	2.41763i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−17.4929	+	17.4929i	−0.825539	+	0.825539i	−0.986896	−	0.161357i	$-0.948413\pi$
$449$	−17.4929	+	17.4929i	−0.825539	+	0.825539i	0.161357	+	0.986896i	$0.448413\pi$
$450$		0			0
$451$			17.0000i			0.800499i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	87.4643			4.10039
$456$		0			0
$457$		−	7.00000i		−	0.327446i	−0.986506	−	0.163723i	$-0.947650\pi$
$457$		−	7.00000i		−	0.327446i	0.986506	−	0.163723i	$-0.0523504\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	5.83095i		−	0.271575i	−0.990738	−	0.135787i	$-0.956644\pi$
$461$		−	5.83095i		−	0.271575i	0.990738	−	0.135787i	$-0.0433564\pi$
$462$		0			0
$463$	−6.00000			−0.278844			−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000			−0.278844			−0.139422	+	0.990233i	$0.544524\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	5.83095i		−	0.269824i	−0.990858	−	0.134912i	$-0.956925\pi$
$467$		−	5.83095i		−	0.269824i	0.990858	−	0.134912i	$-0.0430752\pi$
$468$		0			0
$469$	−18.0000	+	18.0000i	−0.831163	+	0.831163i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14.5774	−	14.5774i	−0.670269	−	0.670269i
$474$		0			0
$475$	−12.0000			−0.550598
$476$		0			0
$477$		0			0
$478$		0			0
$479$	20.4083	+	20.4083i	0.932480	+	0.932480i	0.997860	−	0.0653800i	$-0.0208259\pi$
$479$	20.4083	+	20.4083i	0.932480	+	0.932480i	−0.0653800	+	0.997860i	$0.520826\pi$
$480$		0			0
$481$	15.0000	+	15.0000i	0.683941	+	0.683941i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	34.9857i		−	1.58862i
$486$		0			0
$487$	−7.00000	+	7.00000i	−0.317200	+	0.317200i	−0.847691	−	0.530491i	$-0.822008\pi$
$487$	−7.00000	+	7.00000i	−0.317200	+	0.317200i	0.530491	+	0.847691i	$0.322008\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			23.3238i			1.05259i	0.850302	+	0.526294i	$0.176419\pi$
$491$			23.3238i			1.05259i	−0.850302	+	0.526294i	$0.823581\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	34.9857			1.56932
$498$		0			0
$499$	−10.0000	+	10.0000i	−0.447661	+	0.447661i	−0.894576	−	0.446915i	$-0.852523\pi$
$499$	−10.0000	+	10.0000i	−0.447661	+	0.447661i	0.446915	+	0.894576i	$0.352523\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	20.4083	−	20.4083i	0.909963	−	0.909963i	−0.0863061	−	0.996269i	$-0.527506\pi$
$503$	20.4083	−	20.4083i	0.909963	−	0.909963i	0.996269	+	0.0863061i	$0.0275063\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−24.0000			−1.06170
$512$		0			0
$513$		0			0
$514$		0			0
$515$	37.9012	+	37.9012i	1.67013	+	1.67013i
$516$		0			0
$517$	34.0000	−	34.0000i	1.49532	−	1.49532i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−14.5774	+	14.5774i	−0.638647	+	0.638647i	−0.950222	−	0.311575i	$-0.899143\pi$
$521$	−14.5774	+	14.5774i	−0.638647	+	0.638647i	0.311575	+	0.950222i	$0.399143\pi$
$522$		0			0
$523$	−42.0000			−1.83653			−0.918266	−	0.395964i	$-0.870410\pi$
$523$	−42.0000			−1.83653			−0.918266	+	0.395964i	$0.870410\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	23.3238			1.01600
$528$		0			0
$529$			6.00000i			0.260870i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−14.5774	+	14.5774i	−0.631416	+	0.631416i
$534$		0			0
$535$			17.0000i			0.734974i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	32.0702	+	32.0702i	1.38136	+	1.38136i
$540$		0			0
$541$	15.0000	+	15.0000i	0.644900	+	0.644900i	0.951756	−	0.306856i	$-0.0992769\pi$
$541$	15.0000	+	15.0000i	0.644900	+	0.644900i	−0.306856	+	0.951756i	$0.599277\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	34.9857			1.49862
$546$		0			0
$547$	21.0000	+	21.0000i	0.897895	+	0.897895i	0.995250	−	0.0973546i	$-0.0310381\pi$
$547$	21.0000	+	21.0000i	0.897895	+	0.897895i	−0.0973546	+	0.995250i	$0.531038\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	30.0000i		−	1.27573i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.83095			−0.247065			−0.123533	−	0.992341i	$-0.539422\pi$
$557$	−5.83095			−0.247065			−0.123533	+	0.992341i	$0.539422\pi$
$558$		0			0
$559$		−	25.0000i		−	1.05739i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	11.6619i		−	0.491491i	−0.969334	−	0.245745i	$-0.920967\pi$
$563$		−	11.6619i		−	0.491491i	0.969334	−	0.245745i	$-0.0790327\pi$
$564$		0			0
$565$	17.0000			0.715195
$566$		0			0
$567$		0			0
$568$		0			0
$569$			34.9857i			1.46668i	0.679864	+	0.733339i	$0.262039\pi$
$569$			34.9857i			1.46668i	−0.679864	+	0.733339i	$0.737961\pi$
$570$		0			0
$571$	12.0000	−	12.0000i	0.502184	−	0.502184i	−0.409932	−	0.912116i	$-0.634448\pi$
$571$	12.0000	−	12.0000i	0.502184	−	0.502184i	0.912116	+	0.409932i	$0.134448\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	34.9857	+	34.9857i	1.45901	+	1.45901i
$576$		0			0
$577$	1.00000			0.0416305			0.0208153	−	0.999783i	$-0.493374\pi$
$577$	1.00000			0.0416305			0.0208153	+	0.999783i	$0.493374\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−34.9857	−	34.9857i	−1.45145	−	1.45145i
$582$		0			0
$583$	34.0000	+	34.0000i	1.40814	+	1.40814i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	34.9857i		−	1.44401i	−0.691885	−	0.722007i	$-0.743220\pi$
$587$		−	34.9857i		−	1.44401i	0.691885	−	0.722007i	$-0.256780\pi$
$588$		0			0
$589$	4.00000	−	4.00000i	0.164817	−	0.164817i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	23.3238i		−	0.957794i	−0.877871	−	0.478897i	$-0.841037\pi$
$593$		−	23.3238i		−	0.957794i	0.877871	−	0.478897i	$-0.158963\pi$
$594$		0			0
$595$	51.0000	−	51.0000i	2.09080	−	2.09080i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−40.8167			−1.66772			−0.833862	−	0.551973i	$-0.813875\pi$
$599$	−40.8167			−1.66772			−0.833862	+	0.551973i	$0.813875\pi$
$600$		0			0
$601$	−16.0000	+	16.0000i	−0.652654	+	0.652654i	−0.953631	−	0.300978i	$-0.902687\pi$
$601$	−16.0000	+	16.0000i	−0.652654	+	0.652654i	0.300978	+	0.953631i	$0.402687\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−17.4929	+	17.4929i	−0.711186	+	0.711186i
$606$		0			0
$607$	−23.0000	−	23.0000i	−0.933541	−	0.933541i	0.0643840	−	0.997925i	$-0.479492\pi$
$607$	−23.0000	−	23.0000i	−0.933541	−	0.933541i	−0.997925	+	0.0643840i	$0.979492\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	58.3095			2.35895
$612$		0			0
$613$	31.0000			1.25208			0.626039	−	0.779792i	$-0.284675\pi$
$613$	31.0000			1.25208			0.626039	+	0.779792i	$0.284675\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$	23.0000	−	23.0000i	0.924448	−	0.924448i	−0.0728918	−	0.997340i	$-0.523223\pi$
$619$	23.0000	−	23.0000i	0.924448	−	0.924448i	0.997340	+	0.0728918i	$0.0232228\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	52.4786	−	52.4786i	2.10251	−	2.10251i
$624$		0			0
$625$	−59.0000			−2.36000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	17.4929			0.697486
$630$		0			0
$631$		−	25.0000i		−	0.995234i	−0.867397	−	0.497617i	$-0.834208\pi$
$631$		−	25.0000i		−	0.995234i	0.867397	−	0.497617i	$-0.165792\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.91548	+	2.91548i	−0.115697	+	0.115697i
$636$		0			0
$637$			55.0000i			2.17918i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	14.5774	+	14.5774i	0.575772	+	0.575772i	0.933736	−	0.357964i	$-0.116529\pi$
$641$	14.5774	+	14.5774i	0.575772	+	0.575772i	−0.357964	+	0.933736i	$0.616529\pi$
$642$		0			0
$643$	−12.0000	−	12.0000i	−0.473234	−	0.473234i	0.429726	−	0.902959i	$-0.358610\pi$
$643$	−12.0000	−	12.0000i	−0.473234	−	0.473234i	−0.902959	+	0.429726i	$0.858610\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.83095			−0.229238			−0.114619	−	0.993410i	$-0.536565\pi$
$647$	−5.83095			−0.229238			−0.114619	+	0.993410i	$0.536565\pi$
$648$		0			0
$649$	17.0000	+	17.0000i	0.667308	+	0.667308i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.5774	−	14.5774i	0.570457	−	0.570457i	−0.361799	−	0.932256i	$-0.617837\pi$
$653$	14.5774	−	14.5774i	0.570457	−	0.570457i	0.932256	+	0.361799i	$0.117837\pi$
$654$		0			0
$655$			17.0000i			0.664245i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−40.8167			−1.58999			−0.794996	−	0.606615i	$-0.792527\pi$
$659$	−40.8167			−1.58999			−0.794996	+	0.606615i	$0.792527\pi$
$660$		0			0
$661$		−	11.0000i		−	0.427850i	−0.976850	−	0.213925i	$-0.931375\pi$
$661$		−	11.0000i		−	0.427850i	0.976850	−	0.213925i	$-0.0686249\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	17.4929i		−	0.678344i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$			17.4929i			0.675304i
$672$		0			0
$673$	22.0000	−	22.0000i	0.848038	−	0.848038i	−0.141850	−	0.989888i	$-0.545305\pi$
$673$	22.0000	−	22.0000i	0.848038	−	0.848038i	0.989888	+	0.141850i	$0.0453052\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.5774	−	14.5774i	−0.560254	−	0.560254i	0.369125	−	0.929380i	$-0.379657\pi$
$677$	−14.5774	−	14.5774i	−0.560254	−	0.560254i	−0.929380	+	0.369125i	$0.879657\pi$
$678$		0			0
$679$	36.0000			1.38155
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.91548	−	2.91548i	−0.111558	−	0.111558i	0.649125	−	0.760682i	$-0.275135\pi$
$683$	−2.91548	−	2.91548i	−0.111558	−	0.111558i	−0.760682	+	0.649125i	$0.775135\pi$
$684$		0			0
$685$	−17.0000	−	17.0000i	−0.649537	−	0.649537i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			58.3095i			2.22142i
$690$		0			0
$691$	6.00000	−	6.00000i	0.228251	−	0.228251i	−0.583711	−	0.811962i	$-0.698400\pi$
$691$	6.00000	−	6.00000i	0.228251	−	0.228251i	0.811962	+	0.583711i	$0.198400\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	5.83095i		−	0.221181i
$696$		0			0
$697$			17.0000i			0.643921i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	34.9857			1.32139			0.660696	−	0.750654i	$-0.270261\pi$
$701$	34.9857			1.32139			0.660696	+	0.750654i	$0.270261\pi$
$702$		0			0
$703$	3.00000	−	3.00000i	0.113147	−	0.113147i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−26.0000	−	26.0000i	−0.976450	−	0.976450i	0.0232785	−	0.999729i	$-0.492590\pi$
$709$	−26.0000	−	26.0000i	−0.976450	−	0.976450i	−0.999729	+	0.0232785i	$0.992590\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−23.3238			−0.873483
$714$		0			0
$715$	−85.0000			−3.17882
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.91548	+	2.91548i	0.108729	+	0.108729i	0.759378	−	0.650649i	$-0.225503\pi$
$719$	2.91548	+	2.91548i	0.108729	+	0.108729i	−0.650649	+	0.759378i	$0.725503\pi$
$720$		0			0
$721$	−39.0000	+	39.0000i	−1.45244	+	1.45244i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−36.0000			−1.33517			−0.667583	−	0.744535i	$-0.732671\pi$
$727$	−36.0000			−1.33517			−0.667583	+	0.744535i	$0.732671\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14.5774	−	14.5774i	−0.539164	−	0.539164i
$732$		0			0
$733$			24.0000i			0.886460i	0.896408	+	0.443230i	$0.146168\pi$
$733$			24.0000i			0.886460i	−0.896408	+	0.443230i	$0.853832\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	17.4929	−	17.4929i	0.644358	−	0.644358i
$738$		0			0
$739$			19.0000i			0.698926i	0.936950	+	0.349463i	$0.113636\pi$
$739$			19.0000i			0.698926i	−0.936950	+	0.349463i	$0.886364\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−34.9857	−	34.9857i	−1.28350	−	1.28350i	−0.938662	−	0.344840i	$-0.887933\pi$
$743$	−34.9857	−	34.9857i	−1.28350	−	1.28350i	−0.344840	−	0.938662i	$-0.612067\pi$
$744$		0			0
$745$	−17.0000	−	17.0000i	−0.622832	−	0.622832i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−17.4929			−0.639175
$750$		0			0
$751$	16.0000	+	16.0000i	0.583848	+	0.583848i	0.935959	−	0.352110i	$-0.114536\pi$
$751$	16.0000	+	16.0000i	0.583848	+	0.583848i	−0.352110	+	0.935959i	$0.614536\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−17.4929	+	17.4929i	−0.636630	+	0.636630i
$756$		0			0
$757$			29.0000i			1.05402i	0.849858	+	0.527011i	$0.176688\pi$
$757$			29.0000i			1.05402i	−0.849858	+	0.527011i	$0.823312\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	11.6619			0.422744			0.211372	−	0.977406i	$-0.432207\pi$
$761$	11.6619			0.422744			0.211372	+	0.977406i	$0.432207\pi$
$762$		0			0
$763$			36.0000i			1.30329i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			29.1548i			1.05272i
$768$		0			0
$769$	49.0000			1.76699			0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000			1.76699			0.883493	+	0.468445i	$0.155186\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	48.0000	−	48.0000i	1.72421	−	1.72421i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.91548	+	2.91548i	0.104458	+	0.104458i
$780$		0			0
$781$	−34.0000			−1.21662
$782$		0			0
$783$		0			0
$784$		0			0
$785$	32.0702	+	32.0702i	1.14464	+	1.14464i
$786$		0			0
$787$	24.0000	+	24.0000i	0.855508	+	0.855508i	0.990805	−	0.135297i	$-0.0431990\pi$
$787$	24.0000	+	24.0000i	0.855508	+	0.855508i	−0.135297	+	0.990805i	$0.543199\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			17.4929i			0.621974i
$792$		0			0
$793$	−15.0000	+	15.0000i	−0.532666	+	0.532666i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			34.9857i			1.23926i	0.784895	+	0.619629i	$0.212717\pi$
$797$			34.9857i			1.23926i	−0.784895	+	0.619629i	$0.787283\pi$
$798$		0			0
$799$	34.0000	−	34.0000i	1.20283	−	1.20283i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	23.3238			0.823079
$804$		0			0
$805$	−51.0000	+	51.0000i	−1.79751	+	1.79751i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.91548	+	2.91548i	−0.102503	+	0.102503i	−0.756498	−	0.653996i	$-0.773091\pi$
$809$	−2.91548	+	2.91548i	−0.102503	+	0.102503i	0.653996	+	0.756498i	$0.273091\pi$
$810$		0			0
$811$	26.0000	+	26.0000i	0.912983	+	0.912983i	0.996506	−	0.0835224i	$-0.0266170\pi$
$811$	26.0000	+	26.0000i	0.912983	+	0.912983i	−0.0835224	+	0.996506i	$0.526617\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−58.3095			−2.04249
$816$		0			0
$817$	−5.00000			−0.174928
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−37.9012	−	37.9012i	−1.32276	−	1.32276i	−0.911532	−	0.411228i	$-0.865100\pi$
$821$	−37.9012	−	37.9012i	−1.32276	−	1.32276i	−0.411228	−	0.911532i	$-0.634900\pi$
$822$		0			0
$823$	−15.0000	+	15.0000i	−0.522867	+	0.522867i	−0.918436	−	0.395569i	$-0.870547\pi$
$823$	−15.0000	+	15.0000i	−0.522867	+	0.522867i	0.395569	+	0.918436i	$0.370547\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−32.0702	+	32.0702i	−1.11519	+	1.11519i	−0.122754	+	0.992437i	$0.539173\pi$
$827$	−32.0702	+	32.0702i	−1.11519	+	1.11519i	−0.992437	+	0.122754i	$0.960827\pi$
$828$		0			0
$829$	24.0000			0.833554			0.416777	−	0.909009i	$-0.363160\pi$
$829$	24.0000			0.833554			0.416777	+	0.909009i	$0.363160\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	32.0702	+	32.0702i	1.11117	+	1.11117i
$834$		0			0
$835$			17.0000i			0.588309i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	37.9012	−	37.9012i	1.30849	−	1.30849i	0.385992	−	0.922502i	$-0.373859\pi$
$839$	37.9012	−	37.9012i	1.30849	−	1.30849i	0.922502	−	0.385992i	$-0.126141\pi$
$840$		0			0
$841$		−	29.0000i		−	1.00000i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−34.9857	−	34.9857i	−1.20355	−	1.20355i
$846$		0			0
$847$	−18.0000	−	18.0000i	−0.618487	−	0.618487i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−17.4929			−0.599647
$852$		0			0
$853$	−31.0000	−	31.0000i	−1.06142	−	1.06142i	−0.997986	−	0.0634337i	$-0.979795\pi$
$853$	−31.0000	−	31.0000i	−1.06142	−	1.06142i	−0.0634337	−	0.997986i	$-0.520205\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		−	30.0000i		−	1.02359i	−0.859109	−	0.511793i	$-0.828981\pi$
$859$		−	30.0000i		−	1.02359i	0.859109	−	0.511793i	$-0.171019\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−17.4929			−0.595464			−0.297732	−	0.954650i	$-0.596230\pi$
$863$	−17.4929			−0.595464			−0.297732	+	0.954650i	$0.596230\pi$
$864$		0			0
$865$		−	85.0000i		−	2.89009i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			29.1548i			0.989007i
$870$		0			0
$871$	30.0000			1.01651
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	122.450i		−	4.13957i
$876$		0			0
$877$	17.0000	−	17.0000i	0.574049	−	0.574049i	−0.359208	−	0.933257i	$-0.616953\pi$
$877$	17.0000	−	17.0000i	0.574049	−	0.574049i	0.933257	+	0.359208i	$0.116953\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−17.4929	−	17.4929i	−0.589349	−	0.589349i	0.348106	−	0.937455i	$-0.386825\pi$
$881$	−17.4929	−	17.4929i	−0.589349	−	0.589349i	−0.937455	+	0.348106i	$0.886825\pi$
$882$		0			0
$883$	−43.0000			−1.44707			−0.723533	−	0.690290i	$-0.757483\pi$
$883$	−43.0000			−1.44707			−0.723533	+	0.690290i	$0.757483\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2.91548	+	2.91548i	0.0978921	+	0.0978921i	0.754357	−	0.656465i	$-0.227949\pi$
$887$	2.91548	+	2.91548i	0.0978921	+	0.0978921i	−0.656465	+	0.754357i	$0.727949\pi$
$888$		0			0
$889$	−3.00000	−	3.00000i	−0.100617	−	0.100617i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	11.6619i		−	0.390251i
$894$		0			0
$895$	34.0000	−	34.0000i	1.13649	−	1.13649i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	34.0000	+	34.0000i	1.13270	+	1.13270i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−64.1405			−2.13210
$906$		0			0
$907$	11.0000	−	11.0000i	0.365249	−	0.365249i	−0.500492	−	0.865741i	$-0.666848\pi$
$907$	11.0000	−	11.0000i	0.365249	−	0.365249i	0.865741	+	0.500492i	$0.166848\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−20.4083	+	20.4083i	−0.676158	+	0.676158i	−0.959129	−	0.282970i	$-0.908680\pi$
$911$	−20.4083	+	20.4083i	−0.676158	+	0.676158i	0.282970	+	0.959129i	$0.408680\pi$
$912$		0			0
$913$	34.0000	+	34.0000i	1.12524	+	1.12524i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−17.4929			−0.577665
$918$		0			0
$919$	−35.0000			−1.15454			−0.577272	−	0.816552i	$-0.695883\pi$
$919$	−35.0000			−1.15454			−0.577272	+	0.816552i	$0.695883\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−29.1548	−	29.1548i	−0.959641	−	0.959641i
$924$		0			0
$925$	36.0000	−	36.0000i	1.18367	−	1.18367i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−14.5774	+	14.5774i	−0.478268	+	0.478268i	−0.904578	−	0.426309i	$-0.859814\pi$
$929$	−14.5774	+	14.5774i	−0.478268	+	0.478268i	0.426309	+	0.904578i	$0.359814\pi$
$930$		0			0
$931$	11.0000			0.360510
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−49.5631	+	49.5631i	−1.62089	+	1.62089i
$936$		0			0
$937$		−	18.0000i		−	0.588034i	−0.955800	−	0.294017i	$-0.905008\pi$
$937$		−	18.0000i		−	0.588034i	0.955800	−	0.294017i	$-0.0949923\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.6619	−	11.6619i	0.380167	−	0.380167i	−0.490995	−	0.871162i	$-0.663367\pi$
$941$	11.6619	−	11.6619i	0.380167	−	0.380167i	0.871162	+	0.490995i	$0.163367\pi$
$942$		0			0
$943$		−	17.0000i		−	0.553596i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.4929	−	17.4929i	−0.568441	−	0.568441i	0.363250	−	0.931692i	$-0.381667\pi$
$947$	−17.4929	−	17.4929i	−0.568441	−	0.568441i	−0.931692	+	0.363250i	$0.881667\pi$
$948$		0			0
$949$	20.0000	+	20.0000i	0.649227	+	0.649227i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−34.9857			−1.13330			−0.566649	−	0.823959i	$-0.691761\pi$
$953$	−34.9857			−1.13330			−0.566649	+	0.823959i	$0.691761\pi$
$954$		0			0
$955$	17.0000	+	17.0000i	0.550107	+	0.550107i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17.4929	−	17.4929i	0.564874	−	0.564874i
$960$		0			0
$961$			1.00000i			0.0322581i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−58.3095			−1.87705
$966$		0			0
$967$			31.0000i			0.996893i	0.866921	+	0.498446i	$0.166096\pi$
$967$			31.0000i			0.996893i	−0.866921	+	0.498446i	$0.833904\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			11.6619i			0.374248i	0.982336	+	0.187124i	$0.0599167\pi$
$971$			11.6619i			0.374248i	−0.982336	+	0.187124i	$0.940083\pi$
$972$		0			0
$973$	6.00000			0.192351
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	29.1548i		−	0.932743i	−0.884589	−	0.466372i	$-0.845561\pi$
$977$		−	29.1548i		−	0.932743i	0.884589	−	0.466372i	$-0.154439\pi$
$978$		0			0
$979$	−51.0000	+	51.0000i	−1.62997	+	1.62997i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20.4083	+	20.4083i	0.650925	+	0.650925i	0.953216	−	0.302291i	$-0.0977514\pi$
$983$	20.4083	+	20.4083i	0.650925	+	0.650925i	−0.302291	+	0.953216i	$0.597751\pi$
$984$		0			0
$985$	17.0000			0.541665
$986$		0			0
$987$		0			0
$988$		0			0
$989$	14.5774	+	14.5774i	0.463534	+	0.463534i
$990$		0			0
$991$	−30.0000	−	30.0000i	−0.952981	−	0.952981i	0.0459618	−	0.998943i	$-0.485365\pi$
$991$	−30.0000	−	30.0000i	−0.952981	−	0.952981i	−0.998943	+	0.0459618i	$0.985365\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			104.957i			3.32736i
$996$		0			0
$997$	7.00000	−	7.00000i	0.221692	−	0.221692i	−0.587519	−	0.809211i	$-0.699895\pi$
$997$	7.00000	−	7.00000i	0.221692	−	0.221692i	0.809211	+	0.587519i	$0.199895\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	612.2.k.d.217.1	✓	4
3.2	odd	2	inner	612.2.k.d.217.2	yes	4
4.3	odd	2		2448.2.be.q.1441.1		4
12.11	even	2		2448.2.be.q.1441.2		4
17.4	even	4	inner	612.2.k.d.361.1	yes	4
51.38	odd	4	inner	612.2.k.d.361.2	yes	4
68.55	odd	4		2448.2.be.q.1585.1		4
204.191	even	4		2448.2.be.q.1585.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
612.2.k.d.217.1	✓	4	1.1	even	1	trivial
612.2.k.d.217.2	yes	4	3.2	odd	2	inner
612.2.k.d.361.1	yes	4	17.4	even	4	inner
612.2.k.d.361.2	yes	4	51.38	odd	4	inner
2448.2.be.q.1441.1		4	4.3	odd	2
2448.2.be.q.1441.2		4	12.11	even	2
2448.2.be.q.1585.1		4	68.55	odd	4
2448.2.be.q.1585.2		4	204.191	even	4

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:	\( N \)	\(=\)	\( 612 = 2^{2} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	612.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.91548 - 2.91548i) q^{5} +(3.00000 - 3.00000i) q^{7} +(-2.91548 + 2.91548i) q^{11} -5.00000 q^{13} +(-2.91548 + 2.91548i) q^{17} +1.00000i q^{19} +(2.91548 - 2.91548i) q^{23} +12.0000i q^{25} +(-4.00000 - 4.00000i) q^{31} -17.4929 q^{35} +(-3.00000 - 3.00000i) q^{37} +(2.91548 - 2.91548i) q^{41} +5.00000i q^{43} -11.6619 q^{47} -11.0000i q^{49} -11.6619i q^{53} +17.0000 q^{55} -5.83095i q^{59} +(3.00000 - 3.00000i) q^{61} +(14.5774 + 14.5774i) q^{65} -6.00000 q^{67} +(5.83095 + 5.83095i) q^{71} +(-4.00000 - 4.00000i) q^{73} +17.4929i q^{77} +(5.00000 - 5.00000i) q^{79} -11.6619i q^{83} +17.0000 q^{85} +17.4929 q^{89} +(-15.0000 + 15.0000i) q^{91} +(2.91548 - 2.91548i) q^{95} +(6.00000 + 6.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7} - 20 q^{13} - 16 q^{31} - 12 q^{37} + 68 q^{55} + 12 q^{61} - 24 q^{67} - 16 q^{73} + 20 q^{79} + 68 q^{85} - 60 q^{91} + 24 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 - 20 * q^13 - 16 * q^31 - 12 * q^37 + 68 * q^55 + 12 * q^61 - 24 * q^67 - 16 * q^73 + 20 * q^79 + 68 * q^85 - 60 * q^91 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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