LMFDB - Embedded newform 1530.2.c.g.271.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1530,2,Mod(271,1530)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1530, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1530.271"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1530.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,4,0,4,0,0,0,4,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.2171115093$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 510)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.3
Root			$-2.30278i$ of defining polynomial
Character	$\chi$	$=$	1530.271
Dual form			1530.2.c.g.271.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000i q^{5} -4.60555i q^{7} +1.00000 q^{8} +1.00000i q^{10} -0.605551 q^{13} -4.60555i q^{14} +1.00000 q^{16} +(3.60555 - 2.00000i) q^{17} +1.00000i q^{20} +2.00000i q^{23} -1.00000 q^{25} -0.605551 q^{26} -4.60555i q^{28} -7.21110i q^{29} -2.00000i q^{31} +1.00000 q^{32} +(3.60555 - 2.00000i) q^{34} +4.60555 q^{35} -11.2111i q^{37} +1.00000i q^{40} +1.39445i q^{41} -2.60555 q^{43} +2.00000i q^{46} -4.00000 q^{47} -14.2111 q^{49} -1.00000 q^{50} -0.605551 q^{52} +7.21110 q^{53} -4.60555i q^{56} -7.21110i q^{58} +10.6056 q^{59} -3.21110i q^{61} -2.00000i q^{62} +1.00000 q^{64} -0.605551i q^{65} +14.6056 q^{67} +(3.60555 - 2.00000i) q^{68} +4.60555 q^{70} +13.8167i q^{71} -6.60555i q^{73} -11.2111i q^{74} +7.21110i q^{79} +1.00000i q^{80} +1.39445i q^{82} -5.21110 q^{83} +(2.00000 + 3.60555i) q^{85} -2.60555 q^{86} +0.788897 q^{89} +2.78890i q^{91} +2.00000i q^{92} -4.00000 q^{94} +2.60555i q^{97} -14.2111 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 12 q^{13} + 4 q^{16} - 4 q^{25} + 12 q^{26} + 4 q^{32} + 4 q^{35} + 4 q^{43} - 16 q^{47} - 28 q^{49} - 4 q^{50} + 12 q^{52} + 28 q^{59} + 4 q^{64} + 44 q^{67} + 4 q^{70}+ \cdots - 28 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 12 * q^13 + 4 * q^16 - 4 * q^25 + 12 * q^26 + 4 * q^32 + 4 * q^35 + 4 * q^43 - 16 * q^47 - 28 * q^49 - 4 * q^50 + 12 * q^52 + 28 * q^59 + 4 * q^64 + 44 * q^67 + 4 * q^70 + 8 * q^83 + 8 * q^85 + 4 * q^86 + 32 * q^89 - 16 * q^94 - 28 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times$.

$n$	$307$	$1261$	$1361$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	4.60555i		−	1.74073i	−0.492403	−	0.870367i	$-0.663881\pi$
$7$		−	4.60555i		−	1.74073i	0.492403	−	0.870367i	$-0.336119\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$			1.00000i			0.316228i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−0.605551			−0.167950			−0.0839749	−	0.996468i	$-0.526762\pi$
$13$	−0.605551			−0.167950			−0.0839749	+	0.996468i	$0.526762\pi$
$14$		−	4.60555i		−	1.23089i
$15$		0			0
$16$	1.00000			0.250000
$17$	3.60555	−	2.00000i	0.874475	−	0.485071i
$17$	3.60555	−	2.00000i	0.874475	−	0.485071i
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$			1.00000i			0.223607i
$21$		0			0
$22$		0			0
$23$			2.00000i			0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$			2.00000i			0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	−0.605551			−0.118758
$27$		0			0
$28$		−	4.60555i		−	0.870367i
$29$		−	7.21110i		−	1.33907i	−0.742781	−	0.669534i	$-0.766494\pi$
$29$		−	7.21110i		−	1.33907i	0.742781	−	0.669534i	$-0.233506\pi$
$30$		0			0
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$	3.60555	−	2.00000i	0.618347	−	0.342997i
$35$	4.60555			0.778480
$36$		0			0
$37$		−	11.2111i		−	1.84309i	−0.388267	−	0.921547i	$-0.626926\pi$
$37$		−	11.2111i		−	1.84309i	0.388267	−	0.921547i	$-0.373074\pi$
$38$		0			0
$39$		0			0
$40$			1.00000i			0.158114i
$41$			1.39445i			0.217776i	0.994054	+	0.108888i	$0.0347290\pi$
$41$			1.39445i			0.217776i	−0.994054	+	0.108888i	$0.965271\pi$
$42$		0			0
$43$	−2.60555			−0.397343			−0.198671	−	0.980066i	$-0.563663\pi$
$43$	−2.60555			−0.397343			−0.198671	+	0.980066i	$0.563663\pi$
$44$		0			0
$45$		0			0
$46$			2.00000i			0.294884i
$47$	−4.00000			−0.583460			−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000			−0.583460			−0.291730	+	0.956501i	$0.594231\pi$
$48$		0			0
$49$	−14.2111			−2.03016
$50$	−1.00000			−0.141421
$51$		0			0
$52$	−0.605551			−0.0839749
$53$	7.21110			0.990521			0.495261	−	0.868744i	$-0.335073\pi$
$53$	7.21110			0.990521			0.495261	+	0.868744i	$0.335073\pi$
$54$		0			0
$55$		0			0
$56$		−	4.60555i		−	0.615443i
$57$		0			0
$58$		−	7.21110i		−	0.946864i
$59$	10.6056			1.38073			0.690363	−	0.723464i	$-0.257451\pi$
$59$	10.6056			1.38073			0.690363	+	0.723464i	$0.257451\pi$
$60$		0			0
$61$		−	3.21110i		−	0.411140i	−0.978642	−	0.205570i	$-0.934095\pi$
$61$		−	3.21110i		−	0.411140i	0.978642	−	0.205570i	$-0.0659048\pi$
$62$		−	2.00000i		−	0.254000i
$63$		0			0
$64$	1.00000			0.125000
$65$		−	0.605551i		−	0.0751094i
$66$		0			0
$67$	14.6056			1.78435			0.892176	−	0.451688i	$-0.149178\pi$
$67$	14.6056			1.78435			0.892176	+	0.451688i	$0.149178\pi$
$68$	3.60555	−	2.00000i	0.437237	−	0.242536i
$69$		0			0
$70$	4.60555			0.550469
$71$			13.8167i			1.63974i	0.572553	+	0.819868i	$0.305953\pi$
$71$			13.8167i			1.63974i	−0.572553	+	0.819868i	$0.694047\pi$
$72$		0			0
$73$		−	6.60555i		−	0.773121i	−0.922264	−	0.386561i	$-0.873663\pi$
$73$		−	6.60555i		−	0.773121i	0.922264	−	0.386561i	$-0.126337\pi$
$74$		−	11.2111i		−	1.30326i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			7.21110i			0.811312i	0.914026	+	0.405656i	$0.132957\pi$
$79$			7.21110i			0.811312i	−0.914026	+	0.405656i	$0.867043\pi$
$80$			1.00000i			0.111803i
$81$		0			0
$82$			1.39445i			0.153991i
$83$	−5.21110			−0.571993			−0.285996	−	0.958231i	$-0.592325\pi$
$83$	−5.21110			−0.571993			−0.285996	+	0.958231i	$0.592325\pi$
$84$		0			0
$85$	2.00000	+	3.60555i	0.216930	+	0.391077i
$86$	−2.60555			−0.280964
$87$		0			0
$88$		0			0
$89$	0.788897			0.0836230			0.0418115	−	0.999126i	$-0.486687\pi$
$89$	0.788897			0.0836230			0.0418115	+	0.999126i	$0.486687\pi$
$90$		0			0
$91$			2.78890i			0.292356i
$92$			2.00000i			0.208514i
$93$		0			0
$94$	−4.00000			−0.412568
$95$		0			0
$96$		0			0
$97$			2.60555i			0.264554i	0.991213	+	0.132277i	$0.0422288\pi$
$97$			2.60555i			0.264554i	−0.991213	+	0.132277i	$0.957771\pi$
$98$	−14.2111			−1.43554
$99$		0			0
$100$	−1.00000			−0.100000
$101$	−16.6056			−1.65231			−0.826157	−	0.563440i	$-0.809478\pi$
$101$	−16.6056			−1.65231			−0.826157	+	0.563440i	$0.809478\pi$
$102$		0			0
$103$	14.4222			1.42106			0.710531	−	0.703666i	$-0.248455\pi$
$103$	14.4222			1.42106			0.710531	+	0.703666i	$0.248455\pi$
$104$	−0.605551			−0.0593792
$105$		0			0
$106$	7.21110			0.700404
$107$			1.21110i			0.117082i	0.998285	+	0.0585409i	$0.0186448\pi$
$107$			1.21110i			0.117082i	−0.998285	+	0.0585409i	$0.981355\pi$
$108$		0			0
$109$		−	0.788897i		−	0.0755627i	−0.999286	−	0.0377813i	$-0.987971\pi$
$109$		−	0.788897i		−	0.0755627i	0.999286	−	0.0377813i	$-0.0120290\pi$
$110$		0			0
$111$		0			0
$112$		−	4.60555i		−	0.435184i
$113$			5.21110i			0.490219i	0.969495	+	0.245110i	$0.0788240\pi$
$113$			5.21110i			0.490219i	−0.969495	+	0.245110i	$0.921176\pi$
$114$		0			0
$115$	−2.00000			−0.186501
$116$		−	7.21110i		−	0.669534i
$117$		0			0
$118$	10.6056			0.976320
$119$	−9.21110	−	16.6056i	−0.844380	−	1.52223i
$120$		0			0
$121$	11.0000			1.00000
$122$		−	3.21110i		−	0.290720i
$123$		0			0
$124$		−	2.00000i		−	0.179605i
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	9.21110			0.817353			0.408677	−	0.912679i	$-0.365990\pi$
$127$	9.21110			0.817353			0.408677	+	0.912679i	$0.365990\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$		−	0.605551i		−	0.0531104i
$131$			10.4222i			0.910592i	0.890340	+	0.455296i	$0.150467\pi$
$131$			10.4222i			0.910592i	−0.890340	+	0.455296i	$0.849533\pi$
$132$		0			0
$133$		0			0
$134$	14.6056			1.26173
$135$		0			0
$136$	3.60555	−	2.00000i	0.309173	−	0.171499i
$137$	7.21110			0.616086			0.308043	−	0.951372i	$-0.400326\pi$
$137$	7.21110			0.616086			0.308043	+	0.951372i	$0.400326\pi$
$138$		0			0
$139$			17.2111i			1.45983i	0.683540	+	0.729913i	$0.260440\pi$
$139$			17.2111i			1.45983i	−0.683540	+	0.729913i	$0.739560\pi$
$140$	4.60555			0.389240
$141$		0			0
$142$			13.8167i			1.15947i
$143$		0			0
$144$		0			0
$145$	7.21110			0.598849
$146$		−	6.60555i		−	0.546679i
$147$		0			0
$148$		−	11.2111i		−	0.921547i
$149$	−23.0278			−1.88651			−0.943254	−	0.332073i	$-0.892252\pi$
$149$	−23.0278			−1.88651			−0.943254	+	0.332073i	$0.892252\pi$
$150$		0			0
$151$	6.42221			0.522632			0.261316	−	0.965253i	$-0.415844\pi$
$151$	6.42221			0.522632			0.261316	+	0.965253i	$0.415844\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.00000			0.160644
$156$		0			0
$157$	−20.6056			−1.64450			−0.822251	−	0.569125i	$-0.807282\pi$
$157$	−20.6056			−1.64450			−0.822251	+	0.569125i	$0.807282\pi$
$158$			7.21110i			0.573685i
$159$		0			0
$160$			1.00000i			0.0790569i
$161$	9.21110			0.725937
$162$		0			0
$163$			20.0000i			1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$			20.0000i			1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$			1.39445i			0.108888i
$165$		0			0
$166$	−5.21110			−0.404460
$167$			3.21110i			0.248483i	0.992252	+	0.124241i	$0.0396497\pi$
$167$			3.21110i			0.248483i	−0.992252	+	0.124241i	$0.960350\pi$
$168$		0			0
$169$	−12.6333			−0.971793
$170$	2.00000	+	3.60555i	0.153393	+	0.276533i
$171$		0			0
$172$	−2.60555			−0.198671
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$		0			0
$175$			4.60555i			0.348147i
$176$		0			0
$177$		0			0
$178$	0.788897			0.0591304
$179$	−25.0278			−1.87066			−0.935331	−	0.353773i	$-0.884898\pi$
$179$	−25.0278			−1.87066			−0.935331	+	0.353773i	$0.884898\pi$
$180$		0			0
$181$		−	7.21110i		−	0.535997i	−0.963419	−	0.267999i	$-0.913638\pi$
$181$		−	7.21110i		−	0.535997i	0.963419	−	0.267999i	$-0.0863622\pi$
$182$			2.78890i			0.206727i
$183$		0			0
$184$			2.00000i			0.147442i
$185$	11.2111			0.824257
$186$		0			0
$187$		0			0
$188$	−4.00000			−0.291730
$189$		0			0
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$		−	10.6056i		−	0.763404i	−0.924285	−	0.381702i	$-0.875338\pi$
$193$		−	10.6056i		−	0.763404i	0.924285	−	0.381702i	$-0.124662\pi$
$194$			2.60555i			0.187068i
$195$		0			0
$196$	−14.2111			−1.01508
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$		−	14.0000i		−	0.992434i	−0.868199	−	0.496217i	$-0.834722\pi$
$199$		−	14.0000i		−	0.992434i	0.868199	−	0.496217i	$-0.165278\pi$
$200$	−1.00000			−0.0707107
$201$		0			0
$202$	−16.6056			−1.16836
$203$	−33.2111			−2.33096
$204$		0			0
$205$	−1.39445			−0.0973925
$206$	14.4222			1.00484
$207$		0			0
$208$	−0.605551			−0.0419874
$209$		0			0
$210$		0			0
$211$		−	16.0000i		−	1.10149i	−0.834675	−	0.550743i	$-0.814345\pi$
$211$		−	16.0000i		−	1.10149i	0.834675	−	0.550743i	$-0.185655\pi$
$212$	7.21110			0.495261
$213$		0			0
$214$			1.21110i			0.0827893i
$215$		−	2.60555i		−	0.177697i
$216$		0			0
$217$	−9.21110			−0.625290
$218$		−	0.788897i		−	0.0534309i
$219$		0			0
$220$		0			0
$221$	−2.18335	+	1.21110i	−0.146868	+	0.0814676i
$222$		0			0
$223$	−20.0000			−1.33930			−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000			−1.33930			−0.669650	+	0.742677i	$0.733556\pi$
$224$		−	4.60555i		−	0.307721i
$225$		0			0
$226$			5.21110i			0.346637i
$227$			29.2111i			1.93881i	0.245469	+	0.969404i	$0.421058\pi$
$227$			29.2111i			1.93881i	−0.245469	+	0.969404i	$0.578942\pi$
$228$		0			0
$229$	2.00000			0.132164			0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000			0.132164			0.0660819	+	0.997814i	$0.478950\pi$
$230$	−2.00000			−0.131876
$231$		0			0
$232$		−	7.21110i		−	0.473432i
$233$			8.00000i			0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$			8.00000i			0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$		0			0
$235$		−	4.00000i		−	0.260931i
$236$	10.6056			0.690363
$237$		0			0
$238$	−9.21110	−	16.6056i	−0.597067	−	1.07638i
$239$	−10.7889			−0.697876			−0.348938	−	0.937146i	$-0.613458\pi$
$239$	−10.7889			−0.697876			−0.348938	+	0.937146i	$0.613458\pi$
$240$		0			0
$241$			14.4222i			0.929016i	0.885569	+	0.464508i	$0.153769\pi$
$241$			14.4222i			0.929016i	−0.885569	+	0.464508i	$0.846231\pi$
$242$	11.0000			0.707107
$243$		0			0
$244$		−	3.21110i		−	0.205570i
$245$		−	14.2111i		−	0.907914i
$246$		0			0
$247$		0			0
$248$		−	2.00000i		−	0.127000i
$249$		0			0
$250$		−	1.00000i		−	0.0632456i
$251$	−14.6056			−0.921894			−0.460947	−	0.887428i	$-0.652490\pi$
$251$	−14.6056			−0.921894			−0.460947	+	0.887428i	$0.652490\pi$
$252$		0			0
$253$		0			0
$254$	9.21110			0.577956
$255$		0			0
$256$	1.00000			0.0625000
$257$	16.4222			1.02439			0.512195	−	0.858869i	$-0.328833\pi$
$257$	16.4222			1.02439			0.512195	+	0.858869i	$0.328833\pi$
$258$		0			0
$259$	−51.6333			−3.20834
$260$		−	0.605551i		−	0.0375547i
$261$		0			0
$262$			10.4222i			0.643886i
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$		0			0
$265$			7.21110i			0.442975i
$266$		0			0
$267$		0			0
$268$	14.6056			0.892176
$269$			0.422205i			0.0257423i	0.999917	+	0.0128711i	$0.00409713\pi$
$269$			0.422205i			0.0257423i	−0.999917	+	0.0128711i	$0.995903\pi$
$270$		0			0
$271$	20.0000			1.21491			0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000			1.21491			0.607457	+	0.794353i	$0.292190\pi$
$272$	3.60555	−	2.00000i	0.218619	−	0.121268i
$273$		0			0
$274$	7.21110			0.435639
$275$		0			0
$276$		0			0
$277$			19.2111i			1.15428i	0.816644	+	0.577142i	$0.195832\pi$
$277$			19.2111i			1.15428i	−0.816644	+	0.577142i	$0.804168\pi$
$278$			17.2111i			1.03225i
$279$		0			0
$280$	4.60555			0.275234
$281$	23.2111			1.38466			0.692329	−	0.721582i	$-0.256585\pi$
$281$	23.2111			1.38466			0.692329	+	0.721582i	$0.256585\pi$
$282$		0			0
$283$			23.6333i			1.40485i	0.711756	+	0.702427i	$0.247900\pi$
$283$			23.6333i			1.40485i	−0.711756	+	0.702427i	$0.752100\pi$
$284$			13.8167i			0.819868i
$285$		0			0
$286$		0			0
$287$	6.42221			0.379091
$288$		0			0
$289$	9.00000	−	14.4222i	0.529412	−	0.848365i
$290$	7.21110			0.423451
$291$		0			0
$292$		−	6.60555i		−	0.386561i
$293$	−0.422205			−0.0246655			−0.0123327	−	0.999924i	$-0.503926\pi$
$293$	−0.422205			−0.0246655			−0.0123327	+	0.999924i	$0.503926\pi$
$294$		0			0
$295$			10.6056i			0.617479i
$296$		−	11.2111i		−	0.651632i
$297$		0			0
$298$	−23.0278			−1.33396
$299$		−	1.21110i		−	0.0700399i
$300$		0			0
$301$			12.0000i			0.691669i
$302$	6.42221			0.369556
$303$		0			0
$304$		0			0
$305$	3.21110			0.183867
$306$		0			0
$307$	18.6056			1.06187			0.530937	−	0.847411i	$-0.321840\pi$
$307$	18.6056			1.06187			0.530937	+	0.847411i	$0.321840\pi$
$308$		0			0
$309$		0			0
$310$	2.00000			0.113592
$311$		−	3.39445i		−	0.192482i	−0.995358	−	0.0962408i	$-0.969318\pi$
$311$		−	3.39445i		−	0.192482i	0.995358	−	0.0962408i	$-0.0306819\pi$
$312$		0			0
$313$		−	8.18335i		−	0.462550i	−0.972888	−	0.231275i	$-0.925710\pi$
$313$		−	8.18335i		−	0.462550i	0.972888	−	0.231275i	$-0.0742897\pi$
$314$	−20.6056			−1.16284
$315$		0			0
$316$			7.21110i			0.405656i
$317$		−	16.4222i		−	0.922363i	−0.887306	−	0.461181i	$-0.847426\pi$
$317$		−	16.4222i		−	0.922363i	0.887306	−	0.461181i	$-0.152574\pi$
$318$		0			0
$319$		0			0
$320$			1.00000i			0.0559017i
$321$		0			0
$322$	9.21110			0.513315
$323$		0			0
$324$		0			0
$325$	0.605551			0.0335899
$326$			20.0000i			1.10770i
$327$		0			0
$328$			1.39445i			0.0769956i
$329$			18.4222i			1.01565i
$330$		0			0
$331$	11.6333			0.639424			0.319712	−	0.947515i	$-0.396414\pi$
$331$	11.6333			0.639424			0.319712	+	0.947515i	$0.396414\pi$
$332$	−5.21110			−0.285996
$333$		0			0
$334$			3.21110i			0.175704i
$335$			14.6056i			0.797987i
$336$		0			0
$337$		−	6.60555i		−	0.359827i	−0.983682	−	0.179914i	$-0.942418\pi$
$337$		−	6.60555i		−	0.359827i	0.983682	−	0.179914i	$-0.0575818\pi$
$338$	−12.6333			−0.687161
$339$		0			0
$340$	2.00000	+	3.60555i	0.108465	+	0.195538i
$341$		0			0
$342$		0			0
$343$			33.2111i			1.79323i
$344$	−2.60555			−0.140482
$345$		0			0
$346$		−	6.00000i		−	0.322562i
$347$			15.6333i			0.839240i	0.907700	+	0.419620i	$0.137837\pi$
$347$			15.6333i			0.839240i	−0.907700	+	0.419620i	$0.862163\pi$
$348$		0			0
$349$	11.2111			0.600117			0.300058	−	0.953921i	$-0.402994\pi$
$349$	11.2111			0.600117			0.300058	+	0.953921i	$0.402994\pi$
$350$			4.60555i			0.246177i
$351$		0			0
$352$		0			0
$353$	16.4222			0.874066			0.437033	−	0.899446i	$-0.356029\pi$
$353$	16.4222			0.874066			0.437033	+	0.899446i	$0.356029\pi$
$354$		0			0
$355$	−13.8167			−0.733312
$356$	0.788897			0.0418115
$357$		0			0
$358$	−25.0278			−1.32276
$359$	−7.63331			−0.402871			−0.201435	−	0.979502i	$-0.564561\pi$
$359$	−7.63331			−0.402871			−0.201435	+	0.979502i	$0.564561\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		−	7.21110i		−	0.379007i
$363$		0			0
$364$			2.78890i			0.146178i
$365$	6.60555			0.345750
$366$		0			0
$367$			33.4500i			1.74607i	0.487654	+	0.873037i	$0.337853\pi$
$367$			33.4500i			1.74607i	−0.487654	+	0.873037i	$0.662147\pi$
$368$			2.00000i			0.104257i
$369$		0			0
$370$	11.2111			0.582837
$371$		−	33.2111i		−	1.72423i
$372$		0			0
$373$	23.0278			1.19233			0.596166	−	0.802861i	$-0.296690\pi$
$373$	23.0278			1.19233			0.596166	+	0.802861i	$0.296690\pi$
$374$		0			0
$375$		0			0
$376$	−4.00000			−0.206284
$377$			4.36669i			0.224896i
$378$		0			0
$379$		−	37.2111i		−	1.91141i	−0.294333	−	0.955703i	$-0.595097\pi$
$379$		−	37.2111i		−	1.91141i	0.294333	−	0.955703i	$-0.404903\pi$
$380$		0			0
$381$		0			0
$382$	8.00000			0.409316
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		−	10.6056i		−	0.539808i
$387$		0			0
$388$			2.60555i			0.132277i
$389$	12.6056			0.639127			0.319563	−	0.947565i	$-0.396464\pi$
$389$	12.6056			0.639127			0.319563	+	0.947565i	$0.396464\pi$
$390$		0			0
$391$	4.00000	+	7.21110i	0.202289	+	0.364681i
$392$	−14.2111			−0.717769
$393$		0			0
$394$			18.0000i			0.906827i
$395$	−7.21110			−0.362830
$396$		0			0
$397$			32.4222i			1.62722i	0.581408	+	0.813612i	$0.302502\pi$
$397$			32.4222i			1.62722i	−0.581408	+	0.813612i	$0.697498\pi$
$398$		−	14.0000i		−	0.701757i
$399$		0			0
$400$	−1.00000			−0.0500000
$401$			14.6056i			0.729366i	0.931132	+	0.364683i	$0.118823\pi$
$401$			14.6056i			0.729366i	−0.931132	+	0.364683i	$0.881177\pi$
$402$		0			0
$403$			1.21110i			0.0603293i
$404$	−16.6056			−0.826157
$405$		0			0
$406$	−33.2111			−1.64824
$407$		0			0
$408$		0			0
$409$	−13.6333			−0.674124			−0.337062	−	0.941483i	$-0.609433\pi$
$409$	−13.6333			−0.674124			−0.337062	+	0.941483i	$0.609433\pi$
$410$	−1.39445			−0.0688669
$411$		0			0
$412$	14.4222			0.710531
$413$		−	48.8444i		−	2.40348i
$414$		0			0
$415$		−	5.21110i		−	0.255803i
$416$	−0.605551			−0.0296896
$417$		0			0
$418$		0			0
$419$			27.6333i			1.34998i	0.737829	+	0.674988i	$0.235851\pi$
$419$			27.6333i			1.34998i	−0.737829	+	0.674988i	$0.764149\pi$
$420$		0			0
$421$	28.4222			1.38521			0.692607	−	0.721315i	$-0.256462\pi$
$421$	28.4222			1.38521			0.692607	+	0.721315i	$0.256462\pi$
$422$		−	16.0000i		−	0.778868i
$423$		0			0
$424$	7.21110			0.350202
$425$	−3.60555	+	2.00000i	−0.174895	+	0.0970143i
$426$		0			0
$427$	−14.7889			−0.715685
$428$			1.21110i			0.0585409i
$429$		0			0
$430$		−	2.60555i		−	0.125651i
$431$		−	7.02776i		−	0.338515i	−0.985572	−	0.169258i	$-0.945863\pi$
$431$		−	7.02776i		−	0.338515i	0.985572	−	0.169258i	$-0.0541370\pi$
$432$		0			0
$433$	−16.4222			−0.789201			−0.394600	−	0.918853i	$-0.629117\pi$
$433$	−16.4222			−0.789201			−0.394600	+	0.918853i	$0.629117\pi$
$434$	−9.21110			−0.442147
$435$		0			0
$436$		−	0.788897i		−	0.0377813i
$437$		0			0
$438$		0			0
$439$			17.6333i			0.841592i	0.907155	+	0.420796i	$0.138249\pi$
$439$			17.6333i			0.841592i	−0.907155	+	0.420796i	$0.861751\pi$
$440$		0			0
$441$		0			0
$442$	−2.18335	+	1.21110i	−0.103851	+	0.0576063i
$443$	−27.6333			−1.31290			−0.656449	−	0.754370i	$-0.727942\pi$
$443$	−27.6333			−1.31290			−0.656449	+	0.754370i	$0.727942\pi$
$444$		0			0
$445$			0.788897i			0.0373973i
$446$	−20.0000			−0.947027
$447$		0			0
$448$		−	4.60555i		−	0.217592i
$449$		−	25.3944i		−	1.19844i	−0.800585	−	0.599219i	$-0.795478\pi$
$449$		−	25.3944i		−	1.19844i	0.800585	−	0.599219i	$-0.204522\pi$
$450$		0			0
$451$		0			0
$452$			5.21110i			0.245110i
$453$		0			0
$454$			29.2111i			1.37094i
$455$	−2.78890			−0.130746
$456$		0			0
$457$	30.8444			1.44284			0.721420	−	0.692497i	$-0.243490\pi$
$457$	30.8444			1.44284			0.721420	+	0.692497i	$0.243490\pi$
$458$	2.00000			0.0934539
$459$		0			0
$460$	−2.00000			−0.0932505
$461$	0.972244			0.0452819			0.0226409	−	0.999744i	$-0.492793\pi$
$461$	0.972244			0.0452819			0.0226409	+	0.999744i	$0.492793\pi$
$462$		0			0
$463$	−20.0000			−0.929479			−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000			−0.929479			−0.464739	+	0.885448i	$0.653852\pi$
$464$		−	7.21110i		−	0.334767i
$465$		0			0
$466$			8.00000i			0.370593i
$467$	−1.57779			−0.0730116			−0.0365058	−	0.999333i	$-0.511623\pi$
$467$	−1.57779			−0.0730116			−0.0365058	+	0.999333i	$0.511623\pi$
$468$		0			0
$469$		−	67.2666i		−	3.10608i
$470$		−	4.00000i		−	0.184506i
$471$		0			0
$472$	10.6056			0.488160
$473$		0			0
$474$		0			0
$475$		0			0
$476$	−9.21110	−	16.6056i	−0.422190	−	0.761114i
$477$		0			0
$478$	−10.7889			−0.493473
$479$		−	20.6056i		−	0.941492i	−0.882269	−	0.470746i	$-0.843985\pi$
$479$		−	20.6056i		−	0.941492i	0.882269	−	0.470746i	$-0.156015\pi$
$480$		0			0
$481$			6.78890i			0.309547i
$482$			14.4222i			0.656913i
$483$		0			0
$484$	11.0000			0.500000
$485$	−2.60555			−0.118312
$486$		0			0
$487$		−	21.4500i		−	0.971991i	−0.873961	−	0.485995i	$-0.838457\pi$
$487$		−	21.4500i		−	0.971991i	0.873961	−	0.485995i	$-0.161543\pi$
$488$		−	3.21110i		−	0.145360i
$489$		0			0
$490$		−	14.2111i		−	0.641992i
$491$	−33.0278			−1.49052			−0.745261	−	0.666773i	$-0.767675\pi$
$491$	−33.0278			−1.49052			−0.745261	+	0.666773i	$0.767675\pi$
$492$		0			0
$493$	−14.4222	−	26.0000i	−0.649543	−	1.17098i
$494$		0			0
$495$		0			0
$496$		−	2.00000i		−	0.0898027i
$497$	63.6333			2.85434
$498$		0			0
$499$			16.0000i			0.716258i	0.933672	+	0.358129i	$0.116585\pi$
$499$			16.0000i			0.716258i	−0.933672	+	0.358129i	$0.883415\pi$
$500$		−	1.00000i		−	0.0447214i
$501$		0			0
$502$	−14.6056			−0.651878
$503$		−	22.0000i		−	0.980932i	−0.871460	−	0.490466i	$-0.836827\pi$
$503$		−	22.0000i		−	0.980932i	0.871460	−	0.490466i	$-0.163173\pi$
$504$		0			0
$505$		−	16.6056i		−	0.738937i
$506$		0			0
$507$		0			0
$508$	9.21110			0.408677
$509$	−31.0278			−1.37528			−0.687641	−	0.726051i	$-0.741353\pi$
$509$	−31.0278			−1.37528			−0.687641	+	0.726051i	$0.741353\pi$
$510$		0			0
$511$	−30.4222			−1.34580
$512$	1.00000			0.0441942
$513$		0			0
$514$	16.4222			0.724352
$515$			14.4222i			0.635518i
$516$		0			0
$517$		0			0
$518$	−51.6333			−2.26864
$519$		0			0
$520$		−	0.605551i		−	0.0265552i
$521$			38.6056i			1.69134i	0.533706	+	0.845670i	$0.320799\pi$
$521$			38.6056i			1.69134i	−0.533706	+	0.845670i	$0.679201\pi$
$522$		0			0
$523$	1.39445			0.0609750			0.0304875	−	0.999535i	$-0.490294\pi$
$523$	1.39445			0.0609750			0.0304875	+	0.999535i	$0.490294\pi$
$524$			10.4222i			0.455296i
$525$		0			0
$526$	16.0000			0.697633
$527$	−4.00000	−	7.21110i	−0.174243	−	0.314121i
$528$		0			0
$529$	19.0000			0.826087
$530$			7.21110i			0.313230i
$531$		0			0
$532$		0			0
$533$		−	0.844410i		−	0.0365755i
$534$		0			0
$535$	−1.21110			−0.0523605
$536$	14.6056			0.630864
$537$		0			0
$538$			0.422205i			0.0182026i
$539$		0			0
$540$		0			0
$541$			16.7889i			0.721811i	0.932602	+	0.360906i	$0.117532\pi$
$541$			16.7889i			0.721811i	−0.932602	+	0.360906i	$0.882468\pi$
$542$	20.0000			0.859074
$543$		0			0
$544$	3.60555	−	2.00000i	0.154587	−	0.0857493i
$545$	0.788897			0.0337927
$546$		0			0
$547$		−	23.6333i		−	1.01049i	−0.862977	−	0.505244i	$-0.831403\pi$
$547$		−	23.6333i		−	1.01049i	0.862977	−	0.505244i	$-0.168597\pi$
$548$	7.21110			0.308043
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	33.2111			1.41228
$554$			19.2111i			0.816202i
$555$		0			0
$556$			17.2111i			0.729913i
$557$	−15.2111			−0.644515			−0.322258	−	0.946652i	$-0.604442\pi$
$557$	−15.2111			−0.644515			−0.322258	+	0.946652i	$0.604442\pi$
$558$		0			0
$559$	1.57779			0.0667336
$560$	4.60555			0.194620
$561$		0			0
$562$	23.2111			0.979101
$563$	39.6333			1.67034			0.835172	−	0.549988i	$-0.185368\pi$
$563$	39.6333			1.67034			0.835172	+	0.549988i	$0.185368\pi$
$564$		0			0
$565$	−5.21110			−0.219233
$566$			23.6333i			0.993382i
$567$		0			0
$568$			13.8167i			0.579734i
$569$	−25.6333			−1.07460			−0.537302	−	0.843390i	$-0.680556\pi$
$569$	−25.6333			−1.07460			−0.537302	+	0.843390i	$0.680556\pi$
$570$		0			0
$571$		−	10.7889i		−	0.451501i	−0.974185	−	0.225751i	$-0.927517\pi$
$571$		−	10.7889i		−	0.451501i	0.974185	−	0.225751i	$-0.0724835\pi$
$572$		0			0
$573$		0			0
$574$	6.42221			0.268058
$575$		−	2.00000i		−	0.0834058i
$576$		0			0
$577$	−30.8444			−1.28407			−0.642035	−	0.766675i	$-0.721910\pi$
$577$	−30.8444			−1.28407			−0.642035	+	0.766675i	$0.721910\pi$
$578$	9.00000	−	14.4222i	0.374351	−	0.599885i
$579$		0			0
$580$	7.21110			0.299425
$581$			24.0000i			0.995688i
$582$		0			0
$583$		0			0
$584$		−	6.60555i		−	0.273340i
$585$		0			0
$586$	−0.422205			−0.0174411
$587$	30.0555			1.24052			0.620262	−	0.784395i	$-0.287026\pi$
$587$	30.0555			1.24052			0.620262	+	0.784395i	$0.287026\pi$
$588$		0			0
$589$		0			0
$590$			10.6056i			0.436624i
$591$		0			0
$592$		−	11.2111i		−	0.460773i
$593$	36.4222			1.49568			0.747840	−	0.663879i	$-0.231091\pi$
$593$	36.4222			1.49568			0.747840	+	0.663879i	$0.231091\pi$
$594$		0			0
$595$	16.6056	−	9.21110i	0.680761	−	0.377618i
$596$	−23.0278			−0.943254
$597$		0			0
$598$		−	1.21110i		−	0.0495257i
$599$	13.2111			0.539791			0.269896	−	0.962890i	$-0.413011\pi$
$599$	13.2111			0.539791			0.269896	+	0.962890i	$0.413011\pi$
$600$		0			0
$601$			37.2111i			1.51787i	0.651165	+	0.758936i	$0.274281\pi$
$601$			37.2111i			1.51787i	−0.651165	+	0.758936i	$0.725719\pi$
$602$			12.0000i			0.489083i
$603$		0			0
$604$	6.42221			0.261316
$605$			11.0000i			0.447214i
$606$		0			0
$607$		−	12.2389i		−	0.496760i	−0.968663	−	0.248380i	$-0.920102\pi$
$607$		−	12.2389i		−	0.496760i	0.968663	−	0.248380i	$-0.0798982\pi$
$608$		0			0
$609$		0			0
$610$	3.21110			0.130014
$611$	2.42221			0.0979919
$612$		0			0
$613$	21.4500			0.866356			0.433178	−	0.901308i	$-0.357392\pi$
$613$	21.4500			0.866356			0.433178	+	0.901308i	$0.357392\pi$
$614$	18.6056			0.750859
$615$		0			0
$616$		0			0
$617$		−	34.4222i		−	1.38579i	−0.721041	−	0.692893i	$-0.756336\pi$
$617$		−	34.4222i		−	1.38579i	0.721041	−	0.692893i	$-0.243664\pi$
$618$		0			0
$619$		−	30.4222i		−	1.22277i	−0.791333	−	0.611386i	$-0.790612\pi$
$619$		−	30.4222i		−	1.22277i	0.791333	−	0.611386i	$-0.209388\pi$
$620$	2.00000			0.0803219
$621$		0			0
$622$		−	3.39445i		−	0.136105i
$623$		−	3.63331i		−	0.145565i
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	8.18335i		−	0.327072i
$627$		0			0
$628$	−20.6056			−0.822251
$629$	−22.4222	−	40.4222i	−0.894032	−	1.61174i
$630$		0			0
$631$	−39.2666			−1.56318			−0.781590	−	0.623793i	$-0.785591\pi$
$631$	−39.2666			−1.56318			−0.781590	+	0.623793i	$0.785591\pi$
$632$			7.21110i			0.286842i
$633$		0			0
$634$		−	16.4222i		−	0.652209i
$635$			9.21110i			0.365531i
$636$		0			0
$637$	8.60555			0.340964
$638$		0			0
$639$		0			0
$640$			1.00000i			0.0395285i
$641$		−	35.4500i		−	1.40019i	−0.714050	−	0.700095i	$-0.753141\pi$
$641$		−	35.4500i		−	1.40019i	0.714050	−	0.700095i	$-0.246859\pi$
$642$		0			0
$643$			8.84441i			0.348789i	0.984676	+	0.174395i	$0.0557969\pi$
$643$			8.84441i			0.348789i	−0.984676	+	0.174395i	$0.944203\pi$
$644$	9.21110			0.362968
$645$		0			0
$646$		0			0
$647$	18.4222			0.724252			0.362126	−	0.932129i	$-0.382051\pi$
$647$	18.4222			0.724252			0.362126	+	0.932129i	$0.382051\pi$
$648$		0			0
$649$		0			0
$650$	0.605551			0.0237517
$651$		0			0
$652$			20.0000i			0.783260i
$653$		−	22.0000i		−	0.860927i	−0.902608	−	0.430463i	$-0.858350\pi$
$653$		−	22.0000i		−	0.860927i	0.902608	−	0.430463i	$-0.141650\pi$
$654$		0			0
$655$	−10.4222			−0.407229
$656$			1.39445i			0.0544441i
$657$		0			0
$658$			18.4222i			0.718172i
$659$	29.0278			1.13076			0.565380	−	0.824830i	$-0.308729\pi$
$659$	29.0278			1.13076			0.565380	+	0.824830i	$0.308729\pi$
$660$		0			0
$661$	−4.78890			−0.186267			−0.0931333	−	0.995654i	$-0.529688\pi$
$661$	−4.78890			−0.186267			−0.0931333	+	0.995654i	$0.529688\pi$
$662$	11.6333			0.452141
$663$		0			0
$664$	−5.21110			−0.202230
$665$		0			0
$666$		0			0
$667$	14.4222			0.558430
$668$			3.21110i			0.124241i
$669$		0			0
$670$			14.6056i			0.564262i
$671$		0			0
$672$		0			0
$673$			2.97224i			0.114572i	0.998358	+	0.0572858i	$0.0182446\pi$
$673$			2.97224i			0.114572i	−0.998358	+	0.0572858i	$0.981755\pi$
$674$		−	6.60555i		−	0.254436i
$675$		0			0
$676$	−12.6333			−0.485896
$677$		−	30.8444i		−	1.18545i	−0.805406	−	0.592723i	$-0.798053\pi$
$677$		−	30.8444i		−	1.18545i	0.805406	−	0.592723i	$-0.201947\pi$
$678$		0			0
$679$	12.0000			0.460518
$680$	2.00000	+	3.60555i	0.0766965	+	0.138267i
$681$		0			0
$682$		0			0
$683$		−	4.00000i		−	0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$		−	4.00000i		−	0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$		0			0
$685$			7.21110i			0.275522i
$686$			33.2111i			1.26801i
$687$		0			0
$688$	−2.60555			−0.0993357
$689$	−4.36669			−0.166358
$690$		0			0
$691$		−	5.57779i		−	0.212189i	−0.994356	−	0.106095i	$-0.966165\pi$
$691$		−	5.57779i		−	0.212189i	0.994356	−	0.106095i	$-0.0338347\pi$
$692$		−	6.00000i		−	0.228086i
$693$		0			0
$694$			15.6333i			0.593432i
$695$	−17.2111			−0.652854
$696$		0			0
$697$	2.78890	+	5.02776i	0.105637	+	0.190440i
$698$	11.2111			0.424346
$699$		0			0
$700$			4.60555i			0.174073i
$701$	20.6056			0.778261			0.389130	−	0.921183i	$-0.372776\pi$
$701$	20.6056			0.778261			0.389130	+	0.921183i	$0.372776\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	16.4222			0.618058
$707$			76.4777i			2.87624i
$708$		0			0
$709$		−	3.21110i		−	0.120595i	−0.998180	−	0.0602977i	$-0.980795\pi$
$709$		−	3.21110i		−	0.120595i	0.998180	−	0.0602977i	$-0.0192050\pi$
$710$	−13.8167			−0.518530
$711$		0			0
$712$	0.788897			0.0295652
$713$	4.00000			0.149801
$714$		0			0
$715$		0			0
$716$	−25.0278			−0.935331
$717$		0			0
$718$	−7.63331			−0.284873
$719$			3.39445i			0.126592i	0.997995	+	0.0632958i	$0.0201612\pi$
$719$			3.39445i			0.126592i	−0.997995	+	0.0632958i	$0.979839\pi$
$720$		0			0
$721$		−	66.4222i		−	2.47369i
$722$	−19.0000			−0.707107
$723$		0			0
$724$		−	7.21110i		−	0.267999i
$725$			7.21110i			0.267814i
$726$		0			0
$727$	30.0555			1.11470			0.557349	−	0.830279i	$-0.311819\pi$
$727$	30.0555			1.11470			0.557349	+	0.830279i	$0.311819\pi$
$728$			2.78890i			0.103363i
$729$		0			0
$730$	6.60555			0.244482
$731$	−9.39445	+	5.21110i	−0.347466	+	0.192740i
$732$		0			0
$733$	−18.1833			−0.671617			−0.335809	−	0.941930i	$-0.609010\pi$
$733$	−18.1833			−0.671617			−0.335809	+	0.941930i	$0.609010\pi$
$734$			33.4500i			1.23466i
$735$		0			0
$736$			2.00000i			0.0737210i
$737$		0			0
$738$		0			0
$739$	2.78890			0.102591			0.0512956	−	0.998684i	$-0.483665\pi$
$739$	2.78890			0.102591			0.0512956	+	0.998684i	$0.483665\pi$
$740$	11.2111			0.412128
$741$		0			0
$742$		−	33.2111i		−	1.21922i
$743$			27.2111i			0.998279i	0.866522	+	0.499139i	$0.166350\pi$
$743$			27.2111i			0.998279i	−0.866522	+	0.499139i	$0.833650\pi$
$744$		0			0
$745$		−	23.0278i		−	0.843672i
$746$	23.0278			0.843106
$747$		0			0
$748$		0			0
$749$	5.57779			0.203808
$750$		0			0
$751$		−	37.6333i		−	1.37326i	−0.727008	−	0.686629i	$-0.759090\pi$
$751$		−	37.6333i		−	1.37326i	0.727008	−	0.686629i	$-0.240910\pi$
$752$	−4.00000			−0.145865
$753$		0			0
$754$			4.36669i			0.159026i
$755$			6.42221i			0.233728i
$756$		0			0
$757$	47.0278			1.70925			0.854626	−	0.519243i	$-0.173786\pi$
$757$	47.0278			1.70925			0.854626	+	0.519243i	$0.173786\pi$
$758$		−	37.2111i		−	1.35157i
$759$		0			0
$760$		0			0
$761$	23.2111			0.841402			0.420701	−	0.907199i	$-0.361784\pi$
$761$	23.2111			0.841402			0.420701	+	0.907199i	$0.361784\pi$
$762$		0			0
$763$	−3.63331			−0.131535
$764$	8.00000			0.289430
$765$		0			0
$766$		0			0
$767$	−6.42221			−0.231892
$768$		0			0
$769$	1.63331			0.0588986			0.0294493	−	0.999566i	$-0.490625\pi$
$769$	1.63331			0.0588986			0.0294493	+	0.999566i	$0.490625\pi$
$770$		0			0
$771$		0			0
$772$		−	10.6056i		−	0.381702i
$773$	−15.2111			−0.547105			−0.273553	−	0.961857i	$-0.588199\pi$
$773$	−15.2111			−0.547105			−0.273553	+	0.961857i	$0.588199\pi$
$774$		0			0
$775$			2.00000i			0.0718421i
$776$			2.60555i			0.0935338i
$777$		0			0
$778$	12.6056			0.451931
$779$		0			0
$780$		0			0
$781$		0			0
$782$	4.00000	+	7.21110i	0.143040	+	0.257869i
$783$		0			0
$784$	−14.2111			−0.507539
$785$		−	20.6056i		−	0.735444i
$786$		0			0
$787$		−	2.78890i		−	0.0994135i	−0.998764	−	0.0497067i	$-0.984171\pi$
$787$		−	2.78890i		−	0.0994135i	0.998764	−	0.0497067i	$-0.0158287\pi$
$788$			18.0000i			0.641223i
$789$		0			0
$790$	−7.21110			−0.256560
$791$	24.0000			0.853342
$792$		0			0
$793$			1.94449i			0.0690508i
$794$			32.4222i			1.15062i
$795$		0			0
$796$		−	14.0000i		−	0.496217i
$797$	−48.4222			−1.71520			−0.857601	−	0.514315i	$-0.828046\pi$
$797$	−48.4222			−1.71520			−0.857601	+	0.514315i	$0.828046\pi$
$798$		0			0
$799$	−14.4222	+	8.00000i	−0.510221	+	0.283020i
$800$	−1.00000			−0.0353553
$801$		0			0
$802$			14.6056i			0.515740i
$803$		0			0
$804$		0			0
$805$			9.21110i			0.324649i
$806$			1.21110i			0.0426593i
$807$		0			0
$808$	−16.6056			−0.584181
$809$			6.97224i			0.245131i	0.992460	+	0.122566i	$0.0391122\pi$
$809$			6.97224i			0.245131i	−0.992460	+	0.122566i	$0.960888\pi$
$810$		0			0
$811$			31.6333i			1.11080i	0.831585	+	0.555398i	$0.187434\pi$
$811$			31.6333i			1.11080i	−0.831585	+	0.555398i	$0.812566\pi$
$812$	−33.2111			−1.16548
$813$		0			0
$814$		0			0
$815$	−20.0000			−0.700569
$816$		0			0
$817$		0			0
$818$	−13.6333			−0.476677
$819$		0			0
$820$	−1.39445			−0.0486963
$821$		−	46.0000i		−	1.60541i	−0.596376	−	0.802706i	$-0.703393\pi$
$821$		−	46.0000i		−	1.60541i	0.596376	−	0.802706i	$-0.296607\pi$
$822$		0			0
$823$		−	16.2389i		−	0.566051i	−0.959112	−	0.283026i	$-0.908662\pi$
$823$		−	16.2389i		−	0.566051i	0.959112	−	0.283026i	$-0.0913381\pi$
$824$	14.4222			0.502421
$825$		0			0
$826$		−	48.8444i		−	1.69951i
$827$		−	2.42221i		−	0.0842283i	−0.999113	−	0.0421142i	$-0.986591\pi$
$827$		−	2.42221i		−	0.0842283i	0.999113	−	0.0421142i	$-0.0134093\pi$
$828$		0			0
$829$	−36.0555			−1.25226			−0.626130	−	0.779719i	$-0.715362\pi$
$829$	−36.0555			−1.25226			−0.626130	+	0.779719i	$0.715362\pi$
$830$		−	5.21110i		−	0.180880i
$831$		0			0
$832$	−0.605551			−0.0209937
$833$	−51.2389	+	28.4222i	−1.77532	+	0.984771i
$834$		0			0
$835$	−3.21110			−0.111125
$836$		0			0
$837$		0			0
$838$			27.6333i			0.954577i
$839$		−	3.02776i		−	0.104530i	−0.998633	−	0.0522649i	$-0.983356\pi$
$839$		−	3.02776i		−	0.104530i	0.998633	−	0.0522649i	$-0.0166440\pi$
$840$		0			0
$841$	−23.0000			−0.793103
$842$	28.4222			0.979494
$843$		0			0
$844$		−	16.0000i		−	0.550743i
$845$		−	12.6333i		−	0.434599i
$846$		0			0
$847$		−	50.6611i		−	1.74073i
$848$	7.21110			0.247630
$849$		0			0
$850$	−3.60555	+	2.00000i	−0.123669	+	0.0685994i
$851$	22.4222			0.768623
$852$		0			0
$853$			35.2111i			1.20561i	0.797890	+	0.602803i	$0.205949\pi$
$853$			35.2111i			1.20561i	−0.797890	+	0.602803i	$0.794051\pi$
$854$	−14.7889			−0.506066
$855$		0			0
$856$			1.21110i			0.0413946i
$857$		−	40.8444i		−	1.39522i	−0.716478	−	0.697609i	$-0.754247\pi$
$857$		−	40.8444i		−	1.39522i	0.716478	−	0.697609i	$-0.245753\pi$
$858$		0			0
$859$	−41.2111			−1.40610			−0.703052	−	0.711138i	$-0.748180\pi$
$859$	−41.2111			−1.40610			−0.703052	+	0.711138i	$0.748180\pi$
$860$		−	2.60555i		−	0.0888486i
$861$		0			0
$862$		−	7.02776i		−	0.239366i
$863$	54.4222			1.85255			0.926277	−	0.376844i	$-0.122991\pi$
$863$	54.4222			1.85255			0.926277	+	0.376844i	$0.122991\pi$
$864$		0			0
$865$	6.00000			0.204006
$866$	−16.4222			−0.558049
$867$		0			0
$868$	−9.21110			−0.312645
$869$		0			0
$870$		0			0
$871$	−8.84441			−0.299681
$872$		−	0.788897i		−	0.0267154i
$873$		0			0
$874$		0			0
$875$	−4.60555			−0.155696
$876$		0			0
$877$			54.8444i			1.85196i	0.377567	+	0.925982i	$0.376761\pi$
$877$			54.8444i			1.85196i	−0.377567	+	0.925982i	$0.623239\pi$
$878$			17.6333i			0.595095i
$879$		0			0
$880$		0			0
$881$		−	53.0278i		−	1.78655i	−0.449510	−	0.893275i	$-0.648401\pi$
$881$		−	53.0278i		−	1.78655i	0.449510	−	0.893275i	$-0.351599\pi$
$882$		0			0
$883$	19.8167			0.666883			0.333442	−	0.942771i	$-0.391790\pi$
$883$	19.8167			0.666883			0.333442	+	0.942771i	$0.391790\pi$
$884$	−2.18335	+	1.21110i	−0.0734339	+	0.0407338i
$885$		0			0
$886$	−27.6333			−0.928359
$887$			28.4222i			0.954324i	0.878815	+	0.477162i	$0.158335\pi$
$887$			28.4222i			0.954324i	−0.878815	+	0.477162i	$0.841665\pi$
$888$		0			0
$889$		−	42.4222i		−	1.42280i
$890$			0.788897i			0.0264439i
$891$		0			0
$892$	−20.0000			−0.669650
$893$		0			0
$894$		0			0
$895$		−	25.0278i		−	0.836586i
$896$		−	4.60555i		−	0.153861i
$897$		0			0
$898$		−	25.3944i		−	0.847424i
$899$	−14.4222			−0.481007
$900$		0			0
$901$	26.0000	−	14.4222i	0.866186	−	0.480473i
$902$		0			0
$903$		0			0
$904$			5.21110i			0.173319i
$905$	7.21110			0.239705
$906$		0			0
$907$			34.7889i			1.15515i	0.816339	+	0.577573i	$0.196000\pi$
$907$			34.7889i			1.15515i	−0.816339	+	0.577573i	$0.804000\pi$
$908$			29.2111i			0.969404i
$909$		0			0
$910$	−2.78890			−0.0924511
$911$		−	26.6611i		−	0.883320i	−0.897182	−	0.441660i	$-0.854390\pi$
$911$		−	26.6611i		−	0.883320i	0.897182	−	0.441660i	$-0.145610\pi$
$912$		0			0
$913$		0			0
$914$	30.8444			1.02024
$915$		0			0
$916$	2.00000			0.0660819
$917$	48.0000			1.58510
$918$		0			0
$919$	40.8444			1.34733			0.673666	−	0.739036i	$-0.264718\pi$
$919$	40.8444			1.34733			0.673666	+	0.739036i	$0.264718\pi$
$920$	−2.00000			−0.0659380
$921$		0			0
$922$	0.972244			0.0320191
$923$		−	8.36669i		−	0.275393i
$924$		0			0
$925$			11.2111i			0.368619i
$926$	−20.0000			−0.657241
$927$		0			0
$928$		−	7.21110i		−	0.236716i
$929$			27.8167i			0.912635i	0.889817	+	0.456317i	$0.150832\pi$
$929$			27.8167i			0.912635i	−0.889817	+	0.456317i	$0.849168\pi$
$930$		0			0
$931$		0			0
$932$			8.00000i			0.262049i
$933$		0			0
$934$	−1.57779			−0.0516270
$935$		0			0
$936$		0			0
$937$	−39.2111			−1.28097			−0.640485	−	0.767970i	$-0.721267\pi$
$937$	−39.2111			−1.28097			−0.640485	+	0.767970i	$0.721267\pi$
$938$		−	67.2666i		−	2.19633i
$939$		0			0
$940$		−	4.00000i		−	0.130466i
$941$			8.42221i			0.274556i	0.990533	+	0.137278i	$0.0438354\pi$
$941$			8.42221i			0.274556i	−0.990533	+	0.137278i	$0.956165\pi$
$942$		0			0
$943$	−2.78890			−0.0908190
$944$	10.6056			0.345181
$945$		0			0
$946$		0			0
$947$			9.21110i			0.299321i	0.988737	+	0.149660i	$0.0478180\pi$
$947$			9.21110i			0.299321i	−0.988737	+	0.149660i	$0.952182\pi$
$948$		0			0
$949$			4.00000i			0.129845i
$950$		0			0
$951$		0			0
$952$	−9.21110	−	16.6056i	−0.298534	−	0.538189i
$953$	32.0555			1.03838			0.519190	−	0.854659i	$-0.326234\pi$
$953$	32.0555			1.03838			0.519190	+	0.854659i	$0.326234\pi$
$954$		0			0
$955$			8.00000i			0.258874i
$956$	−10.7889			−0.348938
$957$		0			0
$958$		−	20.6056i		−	0.665735i
$959$		−	33.2111i		−	1.07244i
$960$		0			0
$961$	27.0000			0.870968
$962$			6.78890i			0.218883i
$963$		0			0
$964$			14.4222i			0.464508i
$965$	10.6056			0.341405
$966$		0			0
$967$	22.7889			0.732842			0.366421	−	0.930449i	$-0.380583\pi$
$967$	22.7889			0.732842			0.366421	+	0.930449i	$0.380583\pi$
$968$	11.0000			0.353553
$969$		0			0
$970$	−2.60555			−0.0836592
$971$	−23.4500			−0.752545			−0.376273	−	0.926509i	$-0.622794\pi$
$971$	−23.4500			−0.752545			−0.376273	+	0.926509i	$0.622794\pi$
$972$		0			0
$973$	79.2666			2.54117
$974$		−	21.4500i		−	0.687301i
$975$		0			0
$976$		−	3.21110i		−	0.102785i
$977$	−26.8444			−0.858829			−0.429414	−	0.903108i	$-0.641280\pi$
$977$	−26.8444			−0.858829			−0.429414	+	0.903108i	$0.641280\pi$
$978$		0			0
$979$		0			0
$980$		−	14.2111i		−	0.453957i
$981$		0			0
$982$	−33.0278			−1.05396
$983$			39.2111i			1.25064i	0.780368	+	0.625320i	$0.215032\pi$
$983$			39.2111i			1.25064i	−0.780368	+	0.625320i	$0.784968\pi$
$984$		0			0
$985$	−18.0000			−0.573528
$986$	−14.4222	−	26.0000i	−0.459297	−	0.828009i
$987$		0			0
$988$		0			0
$989$		−	5.21110i		−	0.165703i
$990$		0			0
$991$		−	11.5778i		−	0.367781i	−0.982947	−	0.183890i	$-0.941131\pi$
$991$		−	11.5778i		−	0.367781i	0.982947	−	0.183890i	$-0.0588691\pi$
$992$		−	2.00000i		−	0.0635001i
$993$		0			0
$994$	63.6333			2.01833
$995$	14.0000			0.443830
$996$		0			0
$997$		−	9.63331i		−	0.305090i	−0.988297	−	0.152545i	$-0.951253\pi$
$997$		−	9.63331i		−	0.305090i	0.988297	−	0.152545i	$-0.0487469\pi$
$998$			16.0000i			0.506471i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1530.2.c.g.271.3		4
3.2	odd	2		510.2.c.c.271.1	✓	4
12.11	even	2		4080.2.h.q.3841.4		4
15.2	even	4		2550.2.f.p.1699.2		4
15.8	even	4		2550.2.f.s.1699.3		4
15.14	odd	2		2550.2.c.o.1801.4		4
17.16	even	2	inner	1530.2.c.g.271.2		4
51.38	odd	4		8670.2.a.bh.1.2		2
51.47	odd	4		8670.2.a.bk.1.1		2
51.50	odd	2		510.2.c.c.271.4	yes	4
204.203	even	2		4080.2.h.q.3841.1		4
255.152	even	4		2550.2.f.s.1699.1		4
255.203	even	4		2550.2.f.p.1699.4		4
255.254	odd	2		2550.2.c.o.1801.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.c.c.271.1	✓	4	3.2	odd	2
510.2.c.c.271.4	yes	4	51.50	odd	2
1530.2.c.g.271.2		4	17.16	even	2	inner
1530.2.c.g.271.3		4	1.1	even	1	trivial
2550.2.c.o.1801.1		4	255.254	odd	2
2550.2.c.o.1801.4		4	15.14	odd	2
2550.2.f.p.1699.2		4	15.2	even	4
2550.2.f.p.1699.4		4	255.203	even	4
2550.2.f.s.1699.1		4	255.152	even	4
2550.2.f.s.1699.3		4	15.8	even	4
4080.2.h.q.3841.1		4	204.203	even	2
4080.2.h.q.3841.4		4	12.11	even	2
8670.2.a.bh.1.2		2	51.38	odd	4
8670.2.a.bk.1.1		2	51.47	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1530.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000i q^{5} -4.60555i q^{7} +1.00000 q^{8} +1.00000i q^{10} -0.605551 q^{13} -4.60555i q^{14} +1.00000 q^{16} +(3.60555 - 2.00000i) q^{17} +1.00000i q^{20} +2.00000i q^{23} -1.00000 q^{25} -0.605551 q^{26} -4.60555i q^{28} -7.21110i q^{29} -2.00000i q^{31} +1.00000 q^{32} +(3.60555 - 2.00000i) q^{34} +4.60555 q^{35} -11.2111i q^{37} +1.00000i q^{40} +1.39445i q^{41} -2.60555 q^{43} +2.00000i q^{46} -4.00000 q^{47} -14.2111 q^{49} -1.00000 q^{50} -0.605551 q^{52} +7.21110 q^{53} -4.60555i q^{56} -7.21110i q^{58} +10.6056 q^{59} -3.21110i q^{61} -2.00000i q^{62} +1.00000 q^{64} -0.605551i q^{65} +14.6056 q^{67} +(3.60555 - 2.00000i) q^{68} +4.60555 q^{70} +13.8167i q^{71} -6.60555i q^{73} -11.2111i q^{74} +7.21110i q^{79} +1.00000i q^{80} +1.39445i q^{82} -5.21110 q^{83} +(2.00000 + 3.60555i) q^{85} -2.60555 q^{86} +0.788897 q^{89} +2.78890i q^{91} +2.00000i q^{92} -4.00000 q^{94} +2.60555i q^{97} -14.2111 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 12 q^{13} + 4 q^{16} - 4 q^{25} + 12 q^{26} + 4 q^{32} + 4 q^{35} + 4 q^{43} - 16 q^{47} - 28 q^{49} - 4 q^{50} + 12 q^{52} + 28 q^{59} + 4 q^{64} + 44 q^{67} + 4 q^{70}+ \cdots - 28 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 12 * q^13 + 4 * q^16 - 4 * q^25 + 12 * q^26 + 4 * q^32 + 4 * q^35 + 4 * q^43 - 16 * q^47 - 28 * q^49 - 4 * q^50 + 12 * q^52 + 28 * q^59 + 4 * q^64 + 44 * q^67 + 4 * q^70 + 8 * q^83 + 8 * q^85 + 4 * q^86 + 32 * q^89 - 16 * q^94 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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