LMFDB - Embedded newform 1008.3.dc.a.737.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(305,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1008.305");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.dc (of order $6$, 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:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			737.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1008.737
Dual form			1008.3.dc.a.305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 0.707107i) q^{5} +(-6.50000 - 2.59808i) q^{7} +O(q^{10})$	$q+(-1.22474 - 0.707107i) q^{5} +(-6.50000 - 2.59808i) q^{7} +(-6.12372 + 3.53553i) q^{11} +15.0000 q^{13} +(9.79796 - 5.65685i) q^{17} +(-6.50000 + 11.2583i) q^{19} +(-19.5959 - 11.3137i) q^{23} +(-11.5000 - 19.9186i) q^{25} +22.6274i q^{29} +(1.50000 + 2.59808i) q^{31} +(6.12372 + 7.77817i) q^{35} +(-8.50000 + 14.7224i) q^{37} +80.6102i q^{41} +85.0000 q^{43} +(62.4620 + 36.0624i) q^{47} +(35.5000 + 33.7750i) q^{49} +(-29.3939 + 16.9706i) q^{53} +10.0000 q^{55} +(-78.3837 + 45.2548i) q^{59} +(36.0000 - 62.3538i) q^{61} +(-18.3712 - 10.6066i) q^{65} +(21.5000 + 37.2391i) q^{67} +52.3259i q^{71} +(47.5000 + 82.2724i) q^{73} +(48.9898 - 7.07107i) q^{77} +(34.5000 - 59.7558i) q^{79} +60.8112i q^{83} -16.0000 q^{85} +(-117.576 - 67.8823i) q^{89} +(-97.5000 - 38.9711i) q^{91} +(15.9217 - 9.19239i) q^{95} +16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 26 q^{7}+O(q^{10}) $ 4 * q - 26 * q^7	$ 4 q - 26 q^{7} + 60 q^{13} - 26 q^{19} - 46 q^{25} + 6 q^{31} - 34 q^{37} + 340 q^{43} + 142 q^{49} + 40 q^{55} + 144 q^{61} + 86 q^{67} + 190 q^{73} + 138 q^{79} - 64 q^{85} - 390 q^{91} + 64 q^{97}+O(q^{100}) $ 4 * q - 26 * q^7 + 60 * q^13 - 26 * q^19 - 46 * q^25 + 6 * q^31 - 34 * q^37 + 340 * q^43 + 142 * q^49 + 40 * q^55 + 144 * q^61 + 86 * q^67 + 190 * q^73 + 138 * q^79 - 64 * q^85 - 390 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$-1$

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
$5$	−1.22474	−	0.707107i	−0.244949	−	0.141421i	0.372500	−	0.928032i	$-0.378501\pi$
$5$	−1.22474	−	0.707107i	−0.244949	−	0.141421i	−0.617449	+	0.786611i	$0.711834\pi$
$6$		0			0
$7$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$7$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.12372	+	3.53553i	−0.556702	+	0.321412i	−0.751821	−	0.659367i	$-0.770824\pi$
$11$	−6.12372	+	3.53553i	−0.556702	+	0.321412i	0.195119	+	0.980780i	$0.437491\pi$
$12$		0			0
$13$	15.0000			1.15385			0.576923	−	0.816798i	$-0.304253\pi$
$13$	15.0000			1.15385			0.576923	+	0.816798i	$0.304253\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	9.79796	−	5.65685i	0.576351	−	0.332756i	−0.183331	−	0.983051i	$-0.558688\pi$
$17$	9.79796	−	5.65685i	0.576351	−	0.332756i	0.759682	+	0.650295i	$0.225355\pi$
$18$		0			0
$19$	−6.50000	+	11.2583i	−0.342105	+	0.592544i	−0.984823	−	0.173559i	$-0.944473\pi$
$19$	−6.50000	+	11.2583i	−0.342105	+	0.592544i	0.642718	+	0.766103i	$0.277807\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−19.5959	−	11.3137i	−0.851996	−	0.491900i	0.00932753	−	0.999956i	$-0.497031\pi$
$23$	−19.5959	−	11.3137i	−0.851996	−	0.491900i	−0.861324	+	0.508056i	$0.830364\pi$
$24$		0			0
$25$	−11.5000	−	19.9186i	−0.460000	−	0.796743i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			22.6274i			0.780256i	0.920761	+	0.390128i	$0.127569\pi$
$29$			22.6274i			0.780256i	−0.920761	+	0.390128i	$0.872431\pi$
$30$		0			0
$31$	1.50000	+	2.59808i	0.0483871	+	0.0838089i	0.889205	−	0.457510i	$-0.151259\pi$
$31$	1.50000	+	2.59808i	0.0483871	+	0.0838089i	−0.840817	+	0.541319i	$0.817925\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.12372	+	7.77817i	0.174964	+	0.222234i
$36$		0			0
$37$	−8.50000	+	14.7224i	−0.229730	+	0.397904i	−0.957728	−	0.287675i	$-0.907118\pi$
$37$	−8.50000	+	14.7224i	−0.229730	+	0.397904i	0.727998	+	0.685579i	$0.240451\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			80.6102i			1.96610i	0.183333	+	0.983051i	$0.441311\pi$
$41$			80.6102i			1.96610i	−0.183333	+	0.983051i	$0.558689\pi$
$42$		0			0
$43$	85.0000			1.97674			0.988372	−	0.152055i	$-0.0485890\pi$
$43$	85.0000			1.97674			0.988372	+	0.152055i	$0.0485890\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	62.4620	+	36.0624i	1.32898	+	0.767286i	0.985142	−	0.171743i	$-0.0549400\pi$
$47$	62.4620	+	36.0624i	1.32898	+	0.767286i	0.343837	+	0.939029i	$0.388273\pi$
$48$		0			0
$49$	35.5000	+	33.7750i	0.724490	+	0.689286i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−29.3939	+	16.9706i	−0.554601	+	0.320199i	−0.750976	−	0.660330i	$-0.770417\pi$
$53$	−29.3939	+	16.9706i	−0.554601	+	0.320199i	0.196374	+	0.980529i	$0.437083\pi$
$54$		0			0
$55$	10.0000			0.181818
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−78.3837	+	45.2548i	−1.32854	+	0.767031i	−0.985073	−	0.172135i	$-0.944933\pi$
$59$	−78.3837	+	45.2548i	−1.32854	+	0.767031i	−0.343463	+	0.939166i	$0.611600\pi$
$60$		0			0
$61$	36.0000	−	62.3538i	0.590164	−	1.02219i	−0.404046	−	0.914739i	$-0.632396\pi$
$61$	36.0000	−	62.3538i	0.590164	−	1.02219i	0.994210	−	0.107455i	$-0.0342702\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−18.3712	−	10.6066i	−0.282633	−	0.163178i
$66$		0			0
$67$	21.5000	+	37.2391i	0.320896	+	0.555807i	0.980673	−	0.195654i	$-0.0626828\pi$
$67$	21.5000	+	37.2391i	0.320896	+	0.555807i	−0.659778	+	0.751461i	$0.729349\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			52.3259i			0.736985i	0.929631	+	0.368492i	$0.120126\pi$
$71$			52.3259i			0.736985i	−0.929631	+	0.368492i	$0.879874\pi$
$72$		0			0
$73$	47.5000	+	82.2724i	0.650685	+	1.12702i	0.982957	+	0.183836i	$0.0588515\pi$
$73$	47.5000	+	82.2724i	0.650685	+	1.12702i	−0.332272	+	0.943184i	$0.607815\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	48.9898	−	7.07107i	0.636231	−	0.0918320i
$78$		0			0
$79$	34.5000	−	59.7558i	0.436709	−	0.756402i	−0.560725	−	0.828002i	$-0.689477\pi$
$79$	34.5000	−	59.7558i	0.436709	−	0.756402i	0.997433	+	0.0716005i	$0.0228107\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			60.8112i			0.732665i	0.930484	+	0.366332i	$0.119387\pi$
$83$			60.8112i			0.732665i	−0.930484	+	0.366332i	$0.880613\pi$
$84$		0			0
$85$	−16.0000			−0.188235
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−117.576	−	67.8823i	−1.32107	−	0.762722i	−0.337173	−	0.941443i	$-0.609471\pi$
$89$	−117.576	−	67.8823i	−1.32107	−	0.762722i	−0.983900	+	0.178721i	$0.942804\pi$
$90$		0			0
$91$	−97.5000	−	38.9711i	−1.07143	−	0.428254i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	15.9217	−	9.19239i	0.167597	−	0.0967620i
$96$		0			0
$97$	16.0000			0.164948			0.0824742	−	0.996593i	$-0.473718\pi$
$97$	16.0000			0.164948			0.0824742	+	0.996593i	$0.473718\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	60.0125	−	34.6482i	0.594183	−	0.343052i	−0.172567	−	0.984998i	$-0.555206\pi$
$101$	60.0125	−	34.6482i	0.594183	−	0.343052i	0.766750	+	0.641946i	$0.221873\pi$
$102$		0			0
$103$	−30.5000	+	52.8275i	−0.296117	+	0.512889i	−0.975244	−	0.221131i	$-0.929025\pi$
$103$	−30.5000	+	52.8275i	−0.296117	+	0.512889i	0.679127	+	0.734020i	$0.262358\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	146.969	+	84.8528i	1.37355	+	0.793017i	0.991373	−	0.131074i	$-0.0418425\pi$
$107$	146.969	+	84.8528i	1.37355	+	0.793017i	0.382173	+	0.924091i	$0.375176\pi$
$108$		0			0
$109$	32.5000	+	56.2917i	0.298165	+	0.516437i	0.975716	−	0.219039i	$-0.0702921\pi$
$109$	32.5000	+	56.2917i	0.298165	+	0.516437i	−0.677551	+	0.735476i	$0.736959\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			137.179i			1.21397i	0.794713	+	0.606985i	$0.207621\pi$
$113$			137.179i			1.21397i	−0.794713	+	0.606985i	$0.792379\pi$
$114$		0			0
$115$	16.0000	+	27.7128i	0.139130	+	0.240981i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−78.3837	+	11.3137i	−0.658686	+	0.0950732i
$120$		0			0
$121$	−35.5000	+	61.4878i	−0.293388	+	0.508164i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			67.8823i			0.543058i
$126$		0			0
$127$	171.000			1.34646			0.673228	−	0.739435i	$-0.264907\pi$
$127$	171.000			1.34646			0.673228	+	0.739435i	$0.264907\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−101.654	−	58.6899i	−0.775983	−	0.448014i	0.0590215	−	0.998257i	$-0.481202\pi$
$131$	−101.654	−	58.6899i	−0.775983	−	0.448014i	−0.835005	+	0.550242i	$0.814535\pi$
$132$		0			0
$133$	71.5000	−	56.2917i	0.537594	−	0.423245i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−186.161	+	107.480i	−1.35884	+	0.784527i	−0.989468	−	0.144754i	$-0.953761\pi$
$137$	−186.161	+	107.480i	−1.35884	+	0.784527i	−0.369373	+	0.929281i	$0.620428\pi$
$138$		0			0
$139$	83.0000			0.597122			0.298561	−	0.954391i	$-0.403493\pi$
$139$	83.0000			0.597122			0.298561	+	0.954391i	$0.403493\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−91.8559	+	53.0330i	−0.642349	+	0.370860i
$144$		0			0
$145$	16.0000	−	27.7128i	0.110345	−	0.191123i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	88.1816	+	50.9117i	0.591823	+	0.341689i	0.765818	−	0.643057i	$-0.222334\pi$
$149$	88.1816	+	50.9117i	0.591823	+	0.341689i	−0.173995	+	0.984747i	$0.555668\pi$
$150$		0			0
$151$	20.0000	+	34.6410i	0.132450	+	0.229411i	0.924621	−	0.380889i	$-0.124382\pi$
$151$	20.0000	+	34.6410i	0.132450	+	0.229411i	−0.792170	+	0.610300i	$0.791049\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	4.24264i		−	0.0273719i
$156$		0			0
$157$	−148.000	−	256.344i	−0.942675	−	1.63276i	−0.760340	−	0.649525i	$-0.774968\pi$
$157$	−148.000	−	256.344i	−0.942675	−	1.63276i	−0.182335	−	0.983237i	$-0.558365\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	97.9796	+	124.451i	0.608569	+	0.772986i
$162$		0			0
$163$	64.0000	−	110.851i	0.392638	−	0.680069i	−0.600159	−	0.799881i	$-0.704896\pi$
$163$	64.0000	−	110.851i	0.392638	−	0.680069i	0.992797	+	0.119812i	$0.0382292\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	60.8112i		−	0.364139i	−0.983286	−	0.182069i	$-0.941720\pi$
$167$		−	60.8112i		−	0.364139i	0.983286	−	0.182069i	$-0.0582796\pi$
$168$		0			0
$169$	56.0000			0.331361
$170$		0			0
$171$		0			0
$172$		0			0
$173$	97.9796	+	56.5685i	0.566356	+	0.326986i	0.755693	−	0.654926i	$-0.227300\pi$
$173$	97.9796	+	56.5685i	0.566356	+	0.326986i	−0.189337	+	0.981912i	$0.560634\pi$
$174$		0			0
$175$	23.0000	+	159.349i	0.131429	+	0.910564i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−25.7196	+	14.8492i	−0.143685	+	0.0829567i	−0.570119	−	0.821562i	$-0.693103\pi$
$179$	−25.7196	+	14.8492i	−0.143685	+	0.0829567i	0.426434	+	0.904519i	$0.359770\pi$
$180$		0			0
$181$	81.0000			0.447514			0.223757	−	0.974645i	$-0.428168\pi$
$181$	81.0000			0.447514			0.223757	+	0.974645i	$0.428168\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	20.8207	−	12.0208i	0.112544	−	0.0649774i
$186$		0			0
$187$	−40.0000	+	69.2820i	−0.213904	+	0.370492i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	55.1135	+	31.8198i	0.288552	+	0.166596i	0.637289	−	0.770625i	$-0.280056\pi$
$191$	55.1135	+	31.8198i	0.288552	+	0.166596i	−0.348736	+	0.937221i	$0.613389\pi$
$192$		0			0
$193$	111.500	+	193.124i	0.577720	+	1.00064i	0.995740	+	0.0922029i	$0.0293909\pi$
$193$	111.500	+	193.124i	0.577720	+	1.00064i	−0.418020	+	0.908438i	$0.637276\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	158.392i		−	0.804020i	−0.915635	−	0.402010i	$-0.868312\pi$
$197$		−	158.392i		−	0.804020i	0.915635	−	0.402010i	$-0.131688\pi$
$198$		0			0
$199$	−68.0000	−	117.779i	−0.341709	−	0.591857i	0.643042	−	0.765831i	$-0.277672\pi$
$199$	−68.0000	−	117.779i	−0.341709	−	0.591857i	−0.984750	+	0.173975i	$0.944339\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	58.7878	−	147.078i	0.289595	−	0.724523i
$204$		0			0
$205$	57.0000	−	98.7269i	0.278049	−	0.481595i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	91.9239i		−	0.439827i
$210$		0			0
$211$	−272.000			−1.28910			−0.644550	−	0.764562i	$-0.722955\pi$
$211$	−272.000			−1.28910			−0.644550	+	0.764562i	$0.722955\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−104.103	−	60.1041i	−0.484201	−	0.279554i
$216$		0			0
$217$	−3.00000	−	20.7846i	−0.0138249	−	0.0957816i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	146.969	−	84.8528i	0.665020	−	0.383949i
$222$		0			0
$223$	−248.000			−1.11211			−0.556054	−	0.831146i	$-0.687685\pi$
$223$	−248.000			−1.11211			−0.556054	+	0.831146i	$0.687685\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	143.295	−	82.7315i	0.631256	−	0.364456i	−0.149982	−	0.988689i	$-0.547922\pi$
$227$	143.295	−	82.7315i	0.631256	−	0.364456i	0.781238	+	0.624233i	$0.214588\pi$
$228$		0			0
$229$	−216.500	+	374.989i	−0.945415	+	1.63751i	−0.190496	+	0.981688i	$0.561010\pi$
$229$	−216.500	+	374.989i	−0.945415	+	1.63751i	−0.754918	+	0.655819i	$0.772324\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−197.184	−	113.844i	−0.846283	−	0.488602i	0.0131120	−	0.999914i	$-0.495826\pi$
$233$	−197.184	−	113.844i	−0.846283	−	0.488602i	−0.859395	+	0.511312i	$0.829160\pi$
$234$		0			0
$235$	−51.0000	−	88.3346i	−0.217021	−	0.375892i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			343.654i			1.43788i	0.695071	+	0.718941i	$0.255373\pi$
$239$			343.654i			1.43788i	−0.695071	+	0.718941i	$0.744627\pi$
$240$		0			0
$241$	−79.0000	−	136.832i	−0.327801	−	0.567768i	0.654274	−	0.756257i	$-0.272974\pi$
$241$	−79.0000	−	136.832i	−0.327801	−	0.567768i	−0.982075	+	0.188490i	$0.939641\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−19.5959	−	66.4680i	−0.0799833	−	0.271298i
$246$		0			0
$247$	−97.5000	+	168.875i	−0.394737	+	0.683704i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	237.588i		−	0.946565i	−0.880911	−	0.473283i	$-0.843069\pi$
$251$		−	237.588i		−	0.946565i	0.880911	−	0.473283i	$-0.156931\pi$
$252$		0			0
$253$	160.000			0.632411
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−275.568	−	159.099i	−1.07225	−	0.619062i	−0.143453	−	0.989657i	$-0.545821\pi$
$257$	−275.568	−	159.099i	−1.07225	−	0.619062i	−0.928795	+	0.370595i	$0.879154\pi$
$258$		0			0
$259$	93.5000	−	73.6122i	0.361004	−	0.284217i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−146.969	+	84.8528i	−0.558819	+	0.322634i	−0.752671	−	0.658396i	$-0.771235\pi$
$263$	−146.969	+	84.8528i	−0.558819	+	0.322634i	0.193852	+	0.981031i	$0.437902\pi$
$264$		0			0
$265$	48.0000			0.181132
$266$		0			0
$267$		0			0
$268$		0			0
$269$	420.087	−	242.538i	1.56166	−	0.901627i	0.564574	−	0.825382i	$-0.309040\pi$
$269$	420.087	−	242.538i	1.56166	−	0.901627i	0.997089	−	0.0762447i	$-0.0242930\pi$
$270$		0			0
$271$	108.000	−	187.061i	0.398524	−	0.690264i	−0.595020	−	0.803711i	$-0.702856\pi$
$271$	108.000	−	187.061i	0.398524	−	0.690264i	0.993544	+	0.113447i	$0.0361892\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	140.846	+	81.3173i	0.512166	+	0.295699i
$276$		0			0
$277$	−152.500	−	264.138i	−0.550542	−	0.953566i	−0.998236	−	0.0593790i	$-0.981088\pi$
$277$	−152.500	−	264.138i	−0.550542	−	0.953566i	0.447694	−	0.894187i	$-0.352245\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	22.6274i		−	0.0805246i	−0.999189	−	0.0402623i	$-0.987181\pi$
$281$		−	22.6274i		−	0.0805246i	0.999189	−	0.0402623i	$-0.0128194\pi$
$282$		0			0
$283$	78.5000	+	135.966i	0.277385	+	0.480445i	0.970734	−	0.240157i	$-0.0771989\pi$
$283$	78.5000	+	135.966i	0.277385	+	0.480445i	−0.693349	+	0.720602i	$0.743866\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	209.431	−	523.966i	0.729726	−	1.82567i
$288$		0			0
$289$	−80.5000	+	139.430i	−0.278547	+	0.482457i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	101.823i		−	0.347520i	−0.984788	−	0.173760i	$-0.944408\pi$
$293$		−	101.823i		−	0.347520i	0.984788	−	0.173760i	$-0.0555917\pi$
$294$		0			0
$295$	128.000			0.433898
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−293.939	−	169.706i	−0.983073	−	0.567577i
$300$		0			0
$301$	−552.500	−	220.836i	−1.83555	−	0.733676i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−88.1816	+	50.9117i	−0.289120	+	0.166924i
$306$		0			0
$307$	11.0000			0.0358306			0.0179153	−	0.999840i	$-0.494297\pi$
$307$	11.0000			0.0358306			0.0179153	+	0.999840i	$0.494297\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	454.380	−	262.337i	1.46103	−	0.843526i	0.461971	−	0.886895i	$-0.347142\pi$
$311$	454.380	−	262.337i	1.46103	−	0.843526i	0.999059	+	0.0433690i	$0.0138091\pi$
$312$		0			0
$313$	−276.500	+	478.912i	−0.883387	+	1.53007i	−0.0358348	+	0.999358i	$0.511409\pi$
$313$	−276.500	+	478.912i	−0.883387	+	1.53007i	−0.847552	+	0.530713i	$0.821924\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	117.576	+	67.8823i	0.370901	+	0.214140i	0.673852	−	0.738866i	$-0.264639\pi$
$317$	117.576	+	67.8823i	0.370901	+	0.214140i	−0.302951	+	0.953006i	$0.597972\pi$
$318$		0			0
$319$	−80.0000	−	138.564i	−0.250784	−	0.434370i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			147.078i			0.455350i
$324$		0			0
$325$	−172.500	−	298.779i	−0.530769	−	0.919319i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−312.310	−	396.687i	−0.949270	−	1.20574i
$330$		0			0
$331$	−30.5000	+	52.8275i	−0.0921450	+	0.159600i	−0.908413	−	0.418073i	$-0.862706\pi$
$331$	−30.5000	+	52.8275i	−0.0921450	+	0.159600i	0.816268	+	0.577673i	$0.196039\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	60.8112i		−	0.181526i
$336$		0			0
$337$	135.000			0.400593			0.200297	−	0.979735i	$-0.435809\pi$
$337$	135.000			0.400593			0.200297	+	0.979735i	$0.435809\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−18.3712	−	10.6066i	−0.0538744	−	0.0311044i
$342$		0			0
$343$	−143.000	−	311.769i	−0.416910	−	0.908948i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	88.1816	−	50.9117i	0.254126	−	0.146720i	−0.367526	−	0.930013i	$-0.619795\pi$
$347$	88.1816	−	50.9117i	0.254126	−	0.146720i	0.621652	+	0.783294i	$0.286462\pi$
$348$		0			0
$349$	−152.000			−0.435530			−0.217765	−	0.976001i	$-0.569877\pi$
$349$	−152.000			−0.435530			−0.217765	+	0.976001i	$0.569877\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−334.355	+	193.040i	−0.947182	+	0.546856i	−0.892205	−	0.451632i	$-0.850842\pi$
$353$	−334.355	+	193.040i	−0.947182	+	0.546856i	−0.0549778	+	0.998488i	$0.517509\pi$
$354$		0			0
$355$	37.0000	−	64.0859i	0.104225	−	0.180524i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−39.1918	−	22.6274i	−0.109169	−	0.0630290i	0.444421	−	0.895818i	$-0.353409\pi$
$359$	−39.1918	−	22.6274i	−0.109169	−	0.0630290i	−0.553591	+	0.832789i	$0.686743\pi$
$360$		0			0
$361$	96.0000	+	166.277i	0.265928	+	0.460601i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	134.350i		−	0.368083i
$366$		0			0
$367$	50.5000	+	87.4686i	0.137602	+	0.238334i	0.926588	−	0.376077i	$-0.122727\pi$
$367$	50.5000	+	87.4686i	0.137602	+	0.238334i	−0.788986	+	0.614411i	$0.789394\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	235.151	−	33.9411i	0.633830	−	0.0914855i
$372$		0			0
$373$	155.500	−	269.334i	0.416890	−	0.722075i	−0.578735	−	0.815516i	$-0.696453\pi$
$373$	155.500	−	269.334i	0.416890	−	0.722075i	0.995625	+	0.0934411i	$0.0297867\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			339.411i			0.900295i
$378$		0			0
$379$	−91.0000			−0.240106			−0.120053	−	0.992768i	$-0.538306\pi$
$379$	−91.0000			−0.240106			−0.120053	+	0.992768i	$0.538306\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−284.141	−	164.049i	−0.741882	−	0.428326i	0.0808712	−	0.996725i	$-0.474230\pi$
$383$	−284.141	−	164.049i	−0.741882	−	0.428326i	−0.822753	+	0.568399i	$0.807563\pi$
$384$		0			0
$385$	−65.0000	−	25.9808i	−0.168831	−	0.0674825i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	596.451	−	344.361i	1.53329	−	0.885247i	0.534085	−	0.845431i	$-0.320656\pi$
$389$	596.451	−	344.361i	1.53329	−	0.885247i	0.999207	−	0.0398163i	$-0.0126773\pi$
$390$		0			0
$391$	−256.000			−0.654731
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−84.5074	+	48.7904i	−0.213943	+	0.123520i
$396$		0			0
$397$	139.500	−	241.621i	0.351385	−	0.608617i	−0.635107	−	0.772424i	$-0.719044\pi$
$397$	139.500	−	241.621i	0.351385	−	0.608617i	0.986492	+	0.163807i	$0.0523774\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	22.5000	+	38.9711i	0.0558313	+	0.0967026i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	120.208i		−	0.295352i
$408$		0			0
$409$	111.500	+	193.124i	0.272616	+	0.472185i	0.969531	−	0.244969i	$-0.0787778\pi$
$409$	111.500	+	193.124i	0.272616	+	0.472185i	−0.696915	+	0.717154i	$0.745444\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	627.069	−	90.5097i	1.51833	−	0.219152i
$414$		0			0
$415$	43.0000	−	74.4782i	0.103614	−	0.179466i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			581.242i			1.38721i	0.720355	+	0.693606i	$0.243979\pi$
$419$			581.242i			1.38721i	−0.720355	+	0.693606i	$0.756021\pi$
$420$		0			0
$421$	153.000			0.363420			0.181710	−	0.983352i	$-0.441837\pi$
$421$	153.000			0.363420			0.181710	+	0.983352i	$0.441837\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−225.353	−	130.108i	−0.530242	−	0.306136i
$426$		0			0
$427$	−396.000	+	311.769i	−0.927400	+	0.730139i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−121.250	+	70.0036i	−0.281322	+	0.162421i	−0.634022	−	0.773315i	$-0.718597\pi$
$431$	−121.250	+	70.0036i	−0.281322	+	0.162421i	0.352700	+	0.935737i	$0.385264\pi$
$432$		0			0
$433$	137.000			0.316397			0.158199	−	0.987407i	$-0.449431\pi$
$433$	137.000			0.316397			0.158199	+	0.987407i	$0.449431\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	254.747	−	147.078i	0.582945	−	0.336563i
$438$		0			0
$439$	−300.000	+	519.615i	−0.683371	+	1.18363i	0.290574	+	0.956852i	$0.406154\pi$
$439$	−300.000	+	519.615i	−0.683371	+	1.18363i	−0.973946	+	0.226781i	$0.927180\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−548.686	−	316.784i	−1.23857	−	0.715088i	−0.269767	−	0.962926i	$-0.586947\pi$
$443$	−548.686	−	316.784i	−1.23857	−	0.715088i	−0.968801	+	0.247838i	$0.920280\pi$
$444$		0			0
$445$	96.0000	+	166.277i	0.215730	+	0.373656i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			383.252i			0.853568i	0.904354	+	0.426784i	$0.140353\pi$
$449$			383.252i			0.853568i	−0.904354	+	0.426784i	$0.859647\pi$
$450$		0			0
$451$	−285.000	−	493.634i	−0.631929	−	1.09453i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	91.8559	+	116.673i	0.201881	+	0.256423i
$456$		0			0
$457$	119.500	−	206.980i	0.261488	−	0.452910i	−0.705150	−	0.709059i	$-0.749120\pi$
$457$	119.500	−	206.980i	0.261488	−	0.452910i	0.966638	+	0.256148i	$0.0824535\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	452.548i		−	0.981667i	−0.871253	−	0.490833i	$-0.836692\pi$
$461$		−	452.548i		−	0.981667i	0.871253	−	0.490833i	$-0.163308\pi$
$462$		0			0
$463$	−211.000			−0.455724			−0.227862	−	0.973693i	$-0.573173\pi$
$463$	−211.000			−0.455724			−0.227862	+	0.973693i	$0.573173\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−270.669	−	156.271i	−0.579590	−	0.334627i	0.181380	−	0.983413i	$-0.441944\pi$
$467$	−270.669	−	156.271i	−0.579590	−	0.334627i	−0.760971	+	0.648786i	$0.775277\pi$
$468$		0			0
$469$	−43.0000	−	297.913i	−0.0916844	−	0.635208i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−520.517	+	300.520i	−1.10046	+	0.635350i
$474$		0			0
$475$	299.000			0.629474
$476$		0			0
$477$		0			0
$478$		0			0
$479$	607.473	−	350.725i	1.26821	−	0.732202i	0.293562	−	0.955940i	$-0.405159\pi$
$479$	607.473	−	350.725i	1.26821	−	0.732202i	0.974649	+	0.223737i	$0.0718258\pi$
$480$		0			0
$481$	−127.500	+	220.836i	−0.265073	+	0.459119i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−19.5959	−	11.3137i	−0.0404040	−	0.0233272i
$486$		0			0
$487$	209.500	+	362.865i	0.430185	+	0.745102i	0.996889	−	0.0788195i	$-0.0251151\pi$
$487$	209.500	+	362.865i	0.430185	+	0.745102i	−0.566704	+	0.823921i	$0.691782\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	169.706i		−	0.345633i	−0.984954	−	0.172816i	$-0.944713\pi$
$491$		−	169.706i		−	0.345633i	0.984954	−	0.172816i	$-0.0552867\pi$
$492$		0			0
$493$	128.000	+	221.703i	0.259635	+	0.449701i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	135.947	−	340.118i	0.273535	−	0.684343i
$498$		0			0
$499$	93.5000	−	161.947i	0.187375	−	0.324543i	−0.756999	−	0.653416i	$-0.773335\pi$
$499$	93.5000	−	161.947i	0.187375	−	0.324543i	0.944374	+	0.328873i	$0.106669\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	173.948i		−	0.345822i	−0.984937	−	0.172911i	$-0.944683\pi$
$503$		−	173.948i		−	0.345822i	0.984937	−	0.172911i	$-0.0553172\pi$
$504$		0			0
$505$	−98.0000			−0.194059
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−224.128	−	129.401i	−0.440331	−	0.254225i	0.263407	−	0.964685i	$-0.415154\pi$
$509$	−224.128	−	129.401i	−0.440331	−	0.254225i	−0.703738	+	0.710460i	$0.748487\pi$
$510$		0			0
$511$	−95.0000	−	658.179i	−0.185910	−	1.28802i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	74.7094	−	43.1335i	0.145067	−	0.0837544i
$516$		0			0
$517$	−510.000			−0.986460
$518$		0			0
$519$		0			0
$520$		0			0
$521$	352.727	−	203.647i	0.677018	−	0.390877i	−0.121712	−	0.992565i	$-0.538839\pi$
$521$	352.727	−	203.647i	0.677018	−	0.390877i	0.798731	+	0.601689i	$0.205505\pi$
$522$		0			0
$523$	−405.500	+	702.347i	−0.775335	+	1.34292i	0.159272	+	0.987235i	$0.449085\pi$
$523$	−405.500	+	702.347i	−0.775335	+	1.34292i	−0.934606	+	0.355684i	$0.884248\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	29.3939	+	16.9706i	0.0557759	+	0.0322022i
$528$		0			0
$529$	−8.50000	−	14.7224i	−0.0160681	−	0.0278307i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			1209.15i			2.26858i
$534$		0			0
$535$	−120.000	−	207.846i	−0.224299	−	0.388497i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−336.805	−	81.3173i	−0.624870	−	0.150867i
$540$		0			0
$541$	172.500	−	298.779i	0.318854	−	0.552271i	−0.661395	−	0.750038i	$-0.730035\pi$
$541$	172.500	−	298.779i	0.318854	−	0.552271i	0.980249	+	0.197766i	$0.0633687\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	91.9239i		−	0.168668i
$546$		0			0
$547$	864.000			1.57952			0.789762	−	0.613413i	$-0.210204\pi$
$547$	864.000			1.57952			0.789762	+	0.613413i	$0.210204\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−254.747	−	147.078i	−0.462336	−	0.266930i
$552$		0			0
$553$	−379.500	+	298.779i	−0.686257	+	0.540287i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−821.804	+	474.469i	−1.47541	+	0.851829i	−0.999616	−	0.0277272i	$-0.991173\pi$
$557$	−821.804	+	474.469i	−1.47541	+	0.851829i	−0.475795	+	0.879556i	$0.657840\pi$
$558$		0			0
$559$	1275.00			2.28086
$560$		0			0
$561$		0			0
$562$		0			0
$563$	640.542	−	369.817i	1.13773	−	0.656868i	0.191862	−	0.981422i	$-0.438547\pi$
$563$	640.542	−	369.817i	1.13773	−	0.656868i	0.945867	+	0.324554i	$0.105214\pi$
$564$		0			0
$565$	97.0000	−	168.009i	0.171681	−	0.297361i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	665.036	+	383.959i	1.16878	+	0.674796i	0.953393	−	0.301732i	$-0.0975649\pi$
$569$	665.036	+	383.959i	1.16878	+	0.674796i	0.215388	+	0.976528i	$0.430898\pi$
$570$		0			0
$571$	238.500	+	413.094i	0.417688	+	0.723457i	0.995706	−	0.0925665i	$-0.0295071\pi$
$571$	238.500	+	413.094i	0.417688	+	0.723457i	−0.578018	+	0.816024i	$0.696174\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			520.431i			0.905097i
$576$		0			0
$577$	131.500	+	227.765i	0.227903	+	0.394739i	0.957186	−	0.289472i	$-0.0934798\pi$
$577$	131.500	+	227.765i	0.227903	+	0.394739i	−0.729283	+	0.684212i	$0.760146\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	157.992	−	395.273i	0.271931	−	0.680332i
$582$		0			0
$583$	120.000	−	207.846i	0.205832	−	0.356511i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			531.744i			0.905868i	0.891544	+	0.452934i	$0.149623\pi$
$587$			531.744i			0.905868i	−0.891544	+	0.452934i	$0.850377\pi$
$588$		0			0
$589$	−39.0000			−0.0662139
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−694.430	−	400.930i	−1.17105	−	0.676104i	−0.217120	−	0.976145i	$-0.569666\pi$
$593$	−694.430	−	400.930i	−1.17105	−	0.676104i	−0.953926	+	0.300041i	$0.903000\pi$
$594$		0			0
$595$	104.000	+	41.5692i	0.174790	+	0.0698642i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−676.059	+	390.323i	−1.12865	+	0.651624i	−0.943594	−	0.331104i	$-0.892579\pi$
$599$	−676.059	+	390.323i	−1.12865	+	0.651624i	−0.185052	+	0.982729i	$0.559245\pi$
$600$		0			0
$601$	383.000			0.637271			0.318636	−	0.947877i	$-0.396775\pi$
$601$	383.000			0.637271			0.318636	+	0.947877i	$0.396775\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	86.9569	−	50.2046i	0.143730	−	0.0829828i
$606$		0			0
$607$	−382.500	+	662.509i	−0.630148	+	1.09145i	0.357373	+	0.933962i	$0.383673\pi$
$607$	−382.500	+	662.509i	−0.630148	+	1.09145i	−0.987521	+	0.157487i	$0.949661\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	936.930	+	540.937i	1.53344	+	0.885330i
$612$		0			0
$613$	−204.000	−	353.338i	−0.332790	−	0.576408i	0.650268	−	0.759705i	$-0.274657\pi$
$613$	−204.000	−	353.338i	−0.332790	−	0.576408i	−0.983058	+	0.183296i	$0.941323\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			914.996i			1.48298i	0.670966	+	0.741488i	$0.265880\pi$
$617$			914.996i			1.48298i	−0.670966	+	0.741488i	$0.734120\pi$
$618$		0			0
$619$	−473.500	−	820.126i	−0.764943	−	1.32492i	−0.940276	−	0.340412i	$-0.889434\pi$
$619$	−473.500	−	820.126i	−0.764943	−	1.32492i	0.175333	−	0.984509i	$-0.443900\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	587.878	+	746.705i	0.943624	+	1.19856i
$624$		0			0
$625$	−239.500	+	414.826i	−0.383200	+	0.663722i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			192.333i			0.305776i
$630$		0			0
$631$	760.000			1.20444			0.602219	−	0.798331i	$-0.294284\pi$
$631$	760.000			1.20444			0.602219	+	0.798331i	$0.294284\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−209.431	−	120.915i	−0.329813	−	0.190418i
$636$		0			0
$637$	532.500	+	506.625i	0.835950	+	0.795329i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−401.716	+	231.931i	−0.626703	+	0.361827i	−0.779474	−	0.626435i	$-0.784514\pi$
$641$	−401.716	+	231.931i	−0.626703	+	0.361827i	0.152771	+	0.988262i	$0.451180\pi$
$642$		0			0
$643$	19.0000			0.0295490			0.0147745	−	0.999891i	$-0.495297\pi$
$643$	19.0000			0.0295490			0.0147745	+	0.999891i	$0.495297\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1003.07	+	579.120i	−1.55033	+	0.895086i	−0.552220	+	0.833698i	$0.686219\pi$
$647$	−1003.07	+	579.120i	−1.55033	+	0.895086i	−0.998114	+	0.0613872i	$0.980448\pi$
$648$		0			0
$649$	320.000	−	554.256i	0.493066	−	0.854016i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	892.839	+	515.481i	1.36729	+	0.789404i	0.990581	−	0.136928i	$-0.0437230\pi$
$653$	892.839	+	515.481i	1.36729	+	0.789404i	0.376707	+	0.926332i	$0.377056\pi$
$654$		0			0
$655$	83.0000	+	143.760i	0.126718	+	0.219481i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			972.979i			1.47645i	0.674556	+	0.738224i	$0.264335\pi$
$659$			972.979i			1.47645i	−0.674556	+	0.738224i	$0.735665\pi$
$660$		0			0
$661$	200.500	+	347.276i	0.303328	+	0.525380i	0.976888	−	0.213753i	$-0.0685688\pi$
$661$	200.500	+	347.276i	0.303328	+	0.525380i	−0.673559	+	0.739133i	$0.735235\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−127.373	+	18.3848i	−0.191539	+	0.0276463i
$666$		0			0
$667$	256.000	−	443.405i	0.383808	−	0.664775i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			509.117i			0.758743i
$672$		0			0
$673$	665.000			0.988113			0.494056	−	0.869430i	$-0.335514\pi$
$673$	665.000			0.988113			0.494056	+	0.869430i	$0.335514\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−793.635	−	458.205i	−1.17228	−	0.676817i	−0.218065	−	0.975934i	$-0.569975\pi$
$677$	−793.635	−	458.205i	−1.17228	−	0.676817i	−0.954216	+	0.299117i	$0.903308\pi$
$678$		0			0
$679$	−104.000	−	41.5692i	−0.153166	−	0.0612212i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	117.576	−	67.8823i	0.172146	−	0.0993884i	−0.411452	−	0.911432i	$-0.634978\pi$
$683$	117.576	−	67.8823i	0.172146	−	0.0993884i	0.583597	+	0.812043i	$0.301645\pi$
$684$		0			0
$685$	304.000			0.443796
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−440.908	+	254.558i	−0.639925	+	0.369461i
$690$		0			0
$691$	110.500	−	191.392i	0.159913	−	0.276978i	−0.774924	−	0.632054i	$-0.782212\pi$
$691$	110.500	−	191.392i	0.159913	−	0.276978i	0.934837	+	0.355077i	$0.115545\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−101.654	−	58.6899i	−0.146264	−	0.0844458i
$696$		0			0
$697$	456.000	+	789.815i	0.654232	+	1.13316i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			192.333i			0.274370i	0.990545	+	0.137185i	$0.0438054\pi$
$701$			192.333i			0.274370i	−0.990545	+	0.137185i	$0.956195\pi$
$702$		0			0
$703$	−110.500	−	191.392i	−0.157183	−	0.272250i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−480.100	+	69.2965i	−0.679066	+	0.0980148i
$708$		0			0
$709$	164.000	−	284.056i	0.231312	−	0.400644i	−0.726883	−	0.686762i	$-0.759032\pi$
$709$	164.000	−	284.056i	0.231312	−	0.400644i	0.958194	+	0.286118i	$0.0923649\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	67.8823i		−	0.0952065i
$714$		0			0
$715$	150.000			0.209790
$716$		0			0
$717$		0			0
$718$		0			0
$719$	878.142	+	506.996i	1.22134	+	0.705140i	0.965203	−	0.261501i	$-0.0842175\pi$
$719$	878.142	+	506.996i	1.22134	+	0.705140i	0.256135	+	0.966641i	$0.417551\pi$
$720$		0			0
$721$	335.500	−	264.138i	0.465326	−	0.366349i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	450.706	−	260.215i	0.621664	−	0.358918i
$726$		0			0
$727$	−1069.00			−1.47043			−0.735213	−	0.677836i	$-0.762918\pi$
$727$	−1069.00			−1.47043			−0.735213	+	0.677836i	$0.762918\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	832.827	−	480.833i	1.13930	−	0.657774i
$732$		0			0
$733$	−247.500	+	428.683i	−0.337653	+	0.584833i	−0.983991	−	0.178219i	$-0.942967\pi$
$733$	−247.500	+	428.683i	−0.337653	+	0.584833i	0.646337	+	0.763052i	$0.276300\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−263.320	−	152.028i	−0.357286	−	0.206279i
$738$		0			0
$739$	226.500	+	392.310i	0.306495	+	0.530865i	0.977593	−	0.210503i	$-0.0675103\pi$
$739$	226.500	+	392.310i	0.306495	+	0.530865i	−0.671098	+	0.741369i	$0.734177\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	538.815i		−	0.725189i	−0.931947	−	0.362594i	$-0.881891\pi$
$743$		−	538.815i		−	0.725189i	0.931947	−	0.362594i	$-0.118109\pi$
$744$		0			0
$745$	−72.0000	−	124.708i	−0.0966443	−	0.167393i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−734.847	−	933.381i	−0.981104	−	1.24617i
$750$		0			0
$751$	−169.500	+	293.583i	−0.225699	+	0.390922i	−0.956529	−	0.291637i	$-0.905800\pi$
$751$	−169.500	+	293.583i	−0.225699	+	0.390922i	0.730830	+	0.682560i	$0.239133\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	56.5685i		−	0.0749252i
$756$		0			0
$757$	198.000			0.261559			0.130779	−	0.991411i	$-0.458252\pi$
$757$	198.000			0.261559			0.130779	+	0.991411i	$0.458252\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	313.535	+	181.019i	0.412004	+	0.237870i	0.691650	−	0.722233i	$-0.256884\pi$
$761$	313.535	+	181.019i	0.412004	+	0.237870i	−0.279647	+	0.960103i	$0.590217\pi$
$762$		0			0
$763$	−65.0000	−	450.333i	−0.0851900	−	0.590214i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1175.76	+	678.823i	−1.53293	+	0.885036i
$768$		0			0
$769$	−929.000			−1.20806			−0.604031	−	0.796961i	$-0.706440\pi$
$769$	−929.000			−1.20806			−0.604031	+	0.796961i	$0.706440\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	148.194	−	85.5599i	0.191713	−	0.110686i	−0.401071	−	0.916047i	$-0.631362\pi$
$773$	148.194	−	85.5599i	0.191713	−	0.110686i	0.592784	+	0.805361i	$0.298029\pi$
$774$		0			0
$775$	34.5000	−	59.7558i	0.0445161	−	0.0771042i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−907.536	−	523.966i	−1.16500	−	0.672614i
$780$		0			0
$781$	−185.000	−	320.429i	−0.236876	−	0.410281i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			418.607i			0.533258i
$786$		0			0
$787$	45.0000	+	77.9423i	0.0571792	+	0.0990372i	0.893198	−	0.449663i	$-0.148456\pi$
$787$	45.0000	+	77.9423i	0.0571792	+	0.0990372i	−0.836019	+	0.548701i	$0.815123\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	356.401	−	891.662i	0.450570	−	1.12726i
$792$		0			0
$793$	540.000	−	935.307i	0.680958	−	1.17945i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1154.00i			1.44793i	0.689838	+	0.723964i	$0.257682\pi$
$797$			1154.00i			1.44793i	−0.689838	+	0.723964i	$0.742318\pi$
$798$		0			0
$799$	816.000			1.02128
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−581.754	−	335.876i	−0.724475	−	0.418276i
$804$		0			0
$805$	−32.0000	−	221.703i	−0.0397516	−	0.275407i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	148.194	−	85.5599i	0.183182	−	0.105760i	−0.405605	−	0.914049i	$-0.632939\pi$
$809$	148.194	−	85.5599i	0.183182	−	0.105760i	0.588787	+	0.808288i	$0.299606\pi$
$810$		0			0
$811$	554.000			0.683107			0.341554	−	0.939862i	$-0.389047\pi$
$811$	554.000			0.683107			0.341554	+	0.939862i	$0.389047\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−156.767	+	90.5097i	−0.192353	+	0.111055i
$816$		0			0
$817$	−552.500	+	956.958i	−0.676255	+	1.17131i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	919.783	+	531.037i	1.12032	+	0.646818i	0.941483	−	0.337061i	$-0.109433\pi$
$821$	919.783	+	531.037i	1.12032	+	0.646818i	0.178838	+	0.983879i	$0.442766\pi$
$822$		0			0
$823$	428.000	+	741.318i	0.520049	+	0.900751i	0.999728	+	0.0233070i	$0.00741951\pi$
$823$	428.000	+	741.318i	0.520049	+	0.900751i	−0.479680	+	0.877444i	$0.659247\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1022.48i			1.23637i	0.786033	+	0.618184i	$0.212131\pi$
$827$			1022.48i			1.23637i	−0.786033	+	0.618184i	$0.787869\pi$
$828$		0			0
$829$	−519.500	−	899.800i	−0.626659	−	1.08540i	−0.988218	−	0.153055i	$-0.951089\pi$
$829$	−519.500	−	899.800i	−0.626659	−	1.08540i	0.361559	−	0.932349i	$-0.382245\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	538.888	+	130.108i	0.646924	+	0.156192i
$834$		0			0
$835$	−43.0000	+	74.4782i	−0.0514970	+	0.0891954i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	656.195i		−	0.782116i	−0.920366	−	0.391058i	$-0.872109\pi$
$839$		−	656.195i		−	0.782116i	0.920366	−	0.391058i	$-0.127891\pi$
$840$		0			0
$841$	329.000			0.391201
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−68.5857	−	39.5980i	−0.0811665	−	0.0468615i
$846$		0			0
$847$	390.500	−	307.439i	0.461039	−	0.362974i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	333.131	−	192.333i	0.391458	−	0.226008i
$852$		0			0
$853$	1463.00			1.71512			0.857562	−	0.514381i	$-0.171978\pi$
$853$	1463.00			1.71512			0.857562	+	0.514381i	$0.171978\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	58.7878	−	33.9411i	0.0685971	−	0.0396046i	−0.465309	−	0.885148i	$-0.654057\pi$
$857$	58.7878	−	33.9411i	0.0685971	−	0.0396046i	0.533906	+	0.845544i	$0.320724\pi$
$858$		0			0
$859$	−731.000	+	1266.13i	−0.850990	+	1.47396i	0.0293268	+	0.999570i	$0.490664\pi$
$859$	−731.000	+	1266.13i	−0.850990	+	1.47396i	−0.880316	+	0.474387i	$0.842670\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	699.329	+	403.758i	0.810347	+	0.467854i	0.847076	−	0.531471i	$-0.178361\pi$
$863$	699.329	+	403.758i	0.810347	+	0.467854i	−0.0367295	+	0.999325i	$0.511694\pi$
$864$		0			0
$865$	−80.0000	−	138.564i	−0.0924855	−	0.160190i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			487.904i			0.561454i
$870$		0			0
$871$	322.500	+	558.586i	0.370264	+	0.641316i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	176.363	−	441.235i	0.201558	−	0.504268i
$876$		0			0
$877$	−740.000	+	1281.72i	−0.843786	+	1.46148i	0.0428860	+	0.999080i	$0.486345\pi$
$877$	−740.000	+	1281.72i	−0.843786	+	1.46148i	−0.886672	+	0.462400i	$0.846989\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			712.764i			0.809039i	0.914529	+	0.404520i	$0.132561\pi$
$881$			712.764i			0.809039i	−0.914529	+	0.404520i	$0.867439\pi$
$882$		0			0
$883$	115.000			0.130238			0.0651189	−	0.997878i	$-0.479257\pi$
$883$	115.000			0.130238			0.0651189	+	0.997878i	$0.479257\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1032.46	−	596.091i	−1.16399	−	0.672030i	−0.211734	−	0.977327i	$-0.567911\pi$
$887$	−1032.46	−	596.091i	−1.16399	−	0.672030i	−0.952257	+	0.305297i	$0.901244\pi$
$888$		0			0
$889$	−1111.50	−	444.271i	−1.25028	−	0.499742i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−812.006	+	468.812i	−0.909301	+	0.524985i
$894$		0			0
$895$	42.0000			0.0469274
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−58.7878	+	33.9411i	−0.0653924	+	0.0377543i
$900$		0			0
$901$	−192.000	+	332.554i	−0.213097	+	0.369094i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−99.2043	−	57.2756i	−0.109618	−	0.0632880i
$906$		0			0
$907$	−250.500	−	433.879i	−0.276185	−	0.478367i	0.694248	−	0.719736i	$-0.255737\pi$
$907$	−250.500	−	433.879i	−0.276185	−	0.478367i	−0.970433	+	0.241369i	$0.922404\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			203.647i			0.223542i	0.993734	+	0.111771i	$0.0356523\pi$
$911$			203.647i			0.223542i	−0.993734	+	0.111771i	$0.964348\pi$
$912$		0			0
$913$	−215.000	−	372.391i	−0.235487	−	0.407876i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	508.269	+	645.588i	0.554274	+	0.704022i
$918$		0			0
$919$	−326.500	+	565.515i	−0.355277	+	0.615359i	−0.987165	−	0.159701i	$-0.948947\pi$
$919$	−326.500	+	565.515i	−0.355277	+	0.615359i	0.631888	+	0.775060i	$0.282280\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			784.889i			0.850367i
$924$		0			0
$925$	391.000			0.422703
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11.0227	+	6.36396i	0.0118651	+	0.00685033i	0.505921	−	0.862580i	$-0.331153\pi$
$929$	11.0227	+	6.36396i	0.0118651	+	0.00685033i	−0.494056	+	0.869430i	$0.664486\pi$
$930$		0			0
$931$	−611.000	+	180.133i	−0.656284	+	0.193484i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	97.9796	−	56.5685i	0.104791	−	0.0605011i
$936$		0			0
$937$	761.000			0.812166			0.406083	−	0.913836i	$-0.366894\pi$
$937$	761.000			0.812166			0.406083	+	0.913836i	$0.366894\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−244.949	+	141.421i	−0.260307	+	0.150288i	−0.624475	−	0.781045i	$-0.714687\pi$
$941$	−244.949	+	141.421i	−0.260307	+	0.150288i	0.364168	+	0.931333i	$0.381354\pi$
$942$		0			0
$943$	912.000	−	1579.63i	0.967126	−	1.67511i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	934.480	+	539.522i	0.986780	+	0.569718i	0.904310	−	0.426876i	$-0.140386\pi$
$947$	934.480	+	539.522i	0.986780	+	0.569718i	0.0824695	+	0.996594i	$0.473719\pi$
$948$		0			0
$949$	712.500	+	1234.09i	0.750790	+	1.30041i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1074.80i		−	1.12781i	−0.825840	−	0.563905i	$-0.809299\pi$
$953$		−	1074.80i		−	1.12781i	0.825840	−	0.563905i	$-0.190701\pi$
$954$		0			0
$955$	−45.0000	−	77.9423i	−0.0471204	−	0.0816150i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1489.29	−	214.960i	1.55296	−	0.224151i
$960$		0			0
$961$	476.000	−	824.456i	0.495317	−	0.857915i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	315.370i		−	0.326808i
$966$		0			0
$967$	−1163.00			−1.20269			−0.601344	−	0.798990i	$-0.705368\pi$
$967$	−1163.00			−1.20269			−0.601344	+	0.798990i	$0.705368\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1352.12	−	780.646i	−1.39250	−	0.803961i	−0.398909	−	0.916990i	$-0.630611\pi$
$971$	−1352.12	−	780.646i	−1.39250	−	0.803961i	−0.993592	+	0.113030i	$0.963944\pi$
$972$		0			0
$973$	−539.500	−	215.640i	−0.554471	−	0.221624i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	981.021	−	566.393i	1.00412	−	0.579726i	0.0946520	−	0.995510i	$-0.469826\pi$
$977$	981.021	−	566.393i	1.00412	−	0.579726i	0.909463	+	0.415784i	$0.136493\pi$
$978$		0			0
$979$	960.000			0.980592
$980$		0			0
$981$		0			0
$982$		0			0
$983$	264.545	−	152.735i	0.269120	−	0.155376i	−0.359368	−	0.933196i	$-0.617008\pi$
$983$	264.545	−	152.735i	0.269120	−	0.155376i	0.628488	+	0.777820i	$0.283674\pi$
$984$		0			0
$985$	−112.000	+	193.990i	−0.113706	+	0.196944i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1665.65	−	961.665i	−1.68418	−	0.972361i
$990$		0			0
$991$	−486.500	−	842.643i	−0.490918	−	0.850295i	0.509027	−	0.860751i	$-0.330005\pi$
$991$	−486.500	−	842.643i	−0.490918	−	0.850295i	−0.999945	+	0.0104551i	$0.996672\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			192.333i			0.193300i
$996$		0			0
$997$	39.5000	+	68.4160i	0.0396189	+	0.0686219i	0.885155	−	0.465296i	$-0.154052\pi$
$997$	39.5000	+	68.4160i	0.0396189	+	0.0686219i	−0.845536	+	0.533918i	$0.820719\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.dc.a.737.1		4
3.2	odd	2	inner	1008.3.dc.a.737.2		4
4.3	odd	2		126.3.s.b.107.1	yes	4
7.4	even	3	inner	1008.3.dc.a.305.2		4
12.11	even	2		126.3.s.b.107.2	yes	4
21.11	odd	6	inner	1008.3.dc.a.305.1		4
28.3	even	6		882.3.s.c.557.2		4
28.11	odd	6		126.3.s.b.53.2	yes	4
28.19	even	6		882.3.b.d.197.2		2
28.23	odd	6		882.3.b.c.197.2		2
28.27	even	2		882.3.s.c.863.1		4
84.11	even	6		126.3.s.b.53.1	✓	4
84.23	even	6		882.3.b.c.197.1		2
84.47	odd	6		882.3.b.d.197.1		2
84.59	odd	6		882.3.s.c.557.1		4
84.83	odd	2		882.3.s.c.863.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.s.b.53.1	✓	4	84.11	even	6
126.3.s.b.53.2	yes	4	28.11	odd	6
126.3.s.b.107.1	yes	4	4.3	odd	2
126.3.s.b.107.2	yes	4	12.11	even	2
882.3.b.c.197.1		2	84.23	even	6
882.3.b.c.197.2		2	28.23	odd	6
882.3.b.d.197.1		2	84.47	odd	6
882.3.b.d.197.2		2	28.19	even	6
882.3.s.c.557.1		4	84.59	odd	6
882.3.s.c.557.2		4	28.3	even	6
882.3.s.c.863.1		4	28.27	even	2
882.3.s.c.863.2		4	84.83	odd	2
1008.3.dc.a.305.1		4	21.11	odd	6	inner
1008.3.dc.a.305.2		4	7.4	even	3	inner
1008.3.dc.a.737.1		4	1.1	even	1	trivial
1008.3.dc.a.737.2		4	3.2	odd	2	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{5} +(-6.50000 - 2.59808i) q^{7} +O(q^{10})\)	\(q+(-1.22474 - 0.707107i) q^{5} +(-6.50000 - 2.59808i) q^{7} +(-6.12372 + 3.53553i) q^{11} +15.0000 q^{13} +(9.79796 - 5.65685i) q^{17} +(-6.50000 + 11.2583i) q^{19} +(-19.5959 - 11.3137i) q^{23} +(-11.5000 - 19.9186i) q^{25} +22.6274i q^{29} +(1.50000 + 2.59808i) q^{31} +(6.12372 + 7.77817i) q^{35} +(-8.50000 + 14.7224i) q^{37} +80.6102i q^{41} +85.0000 q^{43} +(62.4620 + 36.0624i) q^{47} +(35.5000 + 33.7750i) q^{49} +(-29.3939 + 16.9706i) q^{53} +10.0000 q^{55} +(-78.3837 + 45.2548i) q^{59} +(36.0000 - 62.3538i) q^{61} +(-18.3712 - 10.6066i) q^{65} +(21.5000 + 37.2391i) q^{67} +52.3259i q^{71} +(47.5000 + 82.2724i) q^{73} +(48.9898 - 7.07107i) q^{77} +(34.5000 - 59.7558i) q^{79} +60.8112i q^{83} -16.0000 q^{85} +(-117.576 - 67.8823i) q^{89} +(-97.5000 - 38.9711i) q^{91} +(15.9217 - 9.19239i) q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 26 q^{7}+O(q^{10}) \) 4 * q - 26 * q^7	\( 4 q - 26 q^{7} + 60 q^{13} - 26 q^{19} - 46 q^{25} + 6 q^{31} - 34 q^{37} + 340 q^{43} + 142 q^{49} + 40 q^{55} + 144 q^{61} + 86 q^{67} + 190 q^{73} + 138 q^{79} - 64 q^{85} - 390 q^{91} + 64 q^{97}+O(q^{100}) \) 4 * q - 26 * q^7 + 60 * q^13 - 26 * q^19 - 46 * q^25 + 6 * q^31 - 34 * q^37 + 340 * q^43 + 142 * q^49 + 40 * q^55 + 144 * q^61 + 86 * q^67 + 190 * q^73 + 138 * q^79 - 64 * q^85 - 390 * q^91 + 64 * q^97

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