LMFDB - Embedded newform 756.3.bk.e.53.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,3,Mod(53,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.53"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	756.bk (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-14,0,0,0,0,0,-88] 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:	$20.5995079856$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	756.53
Dual form			756.3.bk.e.485.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.19615 - 3.00000i) q^{5} +(-3.50000 - 6.06218i) q^{7} +(-2.59808 - 1.50000i) q^{11} -22.0000 q^{13} +(-20.7846 - 12.0000i) q^{17} +(5.00000 + 8.66025i) q^{19} +(-25.9808 + 15.0000i) q^{23} +(5.50000 - 9.52628i) q^{25} +33.0000i q^{29} +(6.50000 - 11.2583i) q^{31} +(-36.3731 - 21.0000i) q^{35} +(5.00000 + 8.66025i) q^{37} +30.0000i q^{41} +56.0000 q^{43} +(-24.5000 + 42.4352i) q^{49} +(-36.3731 - 21.0000i) q^{53} -18.0000 q^{55} +(-85.7365 - 49.5000i) q^{59} +(47.0000 + 81.4064i) q^{61} +(-114.315 + 66.0000i) q^{65} +(38.0000 - 65.8179i) q^{67} -18.0000i q^{71} +(27.5000 - 47.6314i) q^{73} +21.0000i q^{77} +(-41.5000 - 71.8801i) q^{79} -147.000i q^{83} -144.000 q^{85} +(-114.315 + 66.0000i) q^{89} +(77.0000 + 133.368i) q^{91} +(51.9615 + 30.0000i) q^{95} -85.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{7} - 88 q^{13} + 20 q^{19} + 22 q^{25} + 26 q^{31} + 20 q^{37} + 224 q^{43} - 98 q^{49} - 72 q^{55} + 188 q^{61} + 152 q^{67} + 110 q^{73} - 166 q^{79} - 576 q^{85} + 308 q^{91} - 340 q^{97}+O(q^{100}) $ 4 * q - 14 * q^7 - 88 * q^13 + 20 * q^19 + 22 * q^25 + 26 * q^31 + 20 * q^37 + 224 * q^43 - 98 * q^49 - 72 * q^55 + 188 * q^61 + 152 * q^67 + 110 * q^73 - 166 * q^79 - 576 * q^85 + 308 * q^91 - 340 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	5.19615	−	3.00000i	1.03923	−	0.600000i	0.119615	−	0.992820i	$-0.461834\pi$
$5$	5.19615	−	3.00000i	1.03923	−	0.600000i	0.919615	+	0.392820i	$0.128501\pi$
$6$		0			0
$7$	−3.50000	−	6.06218i	−0.500000	−	0.866025i
$7$	−3.50000	−	6.06218i	−0.500000	−	0.866025i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.59808	−	1.50000i	−0.236189	−	0.136364i	0.377235	−	0.926118i	$-0.376875\pi$
$11$	−2.59808	−	1.50000i	−0.236189	−	0.136364i	−0.613424	+	0.789754i	$0.710208\pi$
$12$		0			0
$13$	−22.0000			−1.69231			−0.846154	−	0.532939i	$-0.821088\pi$
$13$	−22.0000			−1.69231			−0.846154	+	0.532939i	$0.821088\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−20.7846	−	12.0000i	−1.22262	−	0.705882i	−0.257148	−	0.966372i	$-0.582783\pi$
$17$	−20.7846	−	12.0000i	−1.22262	−	0.705882i	−0.965477	+	0.260490i	$0.916116\pi$
$18$		0			0
$19$	5.00000	+	8.66025i	0.263158	+	0.455803i	0.967079	−	0.254475i	$-0.0819026\pi$
$19$	5.00000	+	8.66025i	0.263158	+	0.455803i	−0.703921	+	0.710278i	$0.748569\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−25.9808	+	15.0000i	−1.12960	+	0.652174i	−0.943834	−	0.330420i	$-0.892809\pi$
$23$	−25.9808	+	15.0000i	−1.12960	+	0.652174i	−0.185764	+	0.982594i	$0.559476\pi$
$24$		0			0
$25$	5.50000	−	9.52628i	0.220000	−	0.381051i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			33.0000i			1.13793i	0.822361	+	0.568966i	$0.192656\pi$
$29$			33.0000i			1.13793i	−0.822361	+	0.568966i	$0.807344\pi$
$30$		0			0
$31$	6.50000	−	11.2583i	0.209677	−	0.363172i	−0.741935	−	0.670471i	$-0.766092\pi$
$31$	6.50000	−	11.2583i	0.209677	−	0.363172i	0.951613	+	0.307299i	$0.0994253\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−36.3731	−	21.0000i	−1.03923	−	0.600000i
$36$		0			0
$37$	5.00000	+	8.66025i	0.135135	+	0.234061i	0.925649	−	0.378383i	$-0.123520\pi$
$37$	5.00000	+	8.66025i	0.135135	+	0.234061i	−0.790514	+	0.612444i	$0.790186\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			30.0000i			0.731707i	0.930672	+	0.365854i	$0.119223\pi$
$41$			30.0000i			0.731707i	−0.930672	+	0.365854i	$0.880777\pi$
$42$		0			0
$43$	56.0000			1.30233			0.651163	−	0.758938i	$-0.274282\pi$
$43$	56.0000			1.30233			0.651163	+	0.758938i	$0.274282\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	−24.5000	+	42.4352i	−0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−36.3731	−	21.0000i	−0.686284	−	0.396226i	0.115934	−	0.993257i	$-0.463014\pi$
$53$	−36.3731	−	21.0000i	−0.686284	−	0.396226i	−0.802219	+	0.597030i	$0.796347\pi$
$54$		0			0
$55$	−18.0000			−0.327273
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−85.7365	−	49.5000i	−1.45316	−	0.838983i	−0.454502	−	0.890746i	$-0.650183\pi$
$59$	−85.7365	−	49.5000i	−1.45316	−	0.838983i	−0.998659	+	0.0517627i	$0.983516\pi$
$60$		0			0
$61$	47.0000	+	81.4064i	0.770492	+	1.33453i	0.937294	+	0.348541i	$0.113323\pi$
$61$	47.0000	+	81.4064i	0.770492	+	1.33453i	−0.166802	+	0.985990i	$0.553344\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−114.315	+	66.0000i	−1.75870	+	1.01538i
$66$		0			0
$67$	38.0000	−	65.8179i	0.567164	−	0.982357i	−0.429681	−	0.902981i	$-0.641374\pi$
$67$	38.0000	−	65.8179i	0.567164	−	0.982357i	0.996845	−	0.0793762i	$-0.0252928\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	18.0000i		−	0.253521i	−0.991933	−	0.126761i	$-0.959542\pi$
$71$		−	18.0000i		−	0.253521i	0.991933	−	0.126761i	$-0.0404580\pi$
$72$		0			0
$73$	27.5000	−	47.6314i	0.376712	−	0.652485i	−0.613869	−	0.789408i	$-0.710388\pi$
$73$	27.5000	−	47.6314i	0.376712	−	0.652485i	0.990582	+	0.136923i	$0.0437212\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			21.0000i			0.272727i
$78$		0			0
$79$	−41.5000	−	71.8801i	−0.525316	−	0.909875i	−0.999565	−	0.0294839i	$-0.990614\pi$
$79$	−41.5000	−	71.8801i	−0.525316	−	0.909875i	0.474249	−	0.880391i	$-0.342720\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	147.000i		−	1.77108i	−0.464559	−	0.885542i	$-0.653787\pi$
$83$		−	147.000i		−	1.77108i	0.464559	−	0.885542i	$-0.346213\pi$
$84$		0			0
$85$	−144.000			−1.69412
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−114.315	+	66.0000i	−1.28444	+	0.741573i	−0.977657	−	0.210206i	$-0.932587\pi$
$89$	−114.315	+	66.0000i	−1.28444	+	0.741573i	−0.306785	+	0.951779i	$0.599253\pi$
$90$		0			0
$91$	77.0000	+	133.368i	0.846154	+	1.46558i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	51.9615	+	30.0000i	0.546963	+	0.315789i
$96$		0			0
$97$	−85.0000			−0.876289			−0.438144	−	0.898905i	$-0.644364\pi$
$97$	−85.0000			−0.876289			−0.438144	+	0.898905i	$0.644364\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.59808	−	1.50000i	−0.0257235	−	0.0148515i	0.487083	−	0.873356i	$-0.338061\pi$
$101$	−2.59808	−	1.50000i	−0.0257235	−	0.0148515i	−0.512807	+	0.858504i	$0.671394\pi$
$102$		0			0
$103$	−61.0000	−	105.655i	−0.592233	−	1.02578i	−0.993931	−	0.110005i	$-0.964913\pi$
$103$	−61.0000	−	105.655i	−0.592233	−	1.02578i	0.401698	−	0.915772i	$-0.368420\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	25.9808	−	15.0000i	0.242811	−	0.140187i	−0.373657	−	0.927567i	$-0.621896\pi$
$107$	25.9808	−	15.0000i	0.242811	−	0.140187i	0.616468	+	0.787380i	$0.288563\pi$
$108$		0			0
$109$	50.0000	−	86.6025i	0.458716	−	0.794519i	−0.540178	−	0.841551i	$-0.681643\pi$
$109$	50.0000	−	86.6025i	0.458716	−	0.794519i	0.998893	+	0.0470322i	$0.0149763\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	78.0000i		−	0.690265i	−0.938554	−	0.345133i	$-0.887834\pi$
$113$		−	78.0000i		−	0.690265i	0.938554	−	0.345133i	$-0.112166\pi$
$114$		0			0
$115$	−90.0000	+	155.885i	−0.782609	+	1.35552i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			168.000i			1.41176i
$120$		0			0
$121$	−56.0000	−	96.9948i	−0.462810	−	0.801610i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			84.0000i			0.672000i
$126$		0			0
$127$	134.000			1.05512			0.527559	−	0.849518i	$-0.323107\pi$
$127$	134.000			1.05512			0.527559	+	0.849518i	$0.323107\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−96.1288	+	55.5000i	−0.733808	+	0.423664i	−0.819814	−	0.572631i	$-0.805923\pi$
$131$	−96.1288	+	55.5000i	−0.733808	+	0.423664i	0.0860058	+	0.996295i	$0.472590\pi$
$132$		0			0
$133$	35.0000	−	60.6218i	0.263158	−	0.455803i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−150.688	−	87.0000i	−1.09992	−	0.635036i	−0.163717	−	0.986507i	$-0.552348\pi$
$137$	−150.688	−	87.0000i	−1.09992	−	0.635036i	−0.936199	+	0.351471i	$0.885682\pi$
$138$		0			0
$139$	−166.000			−1.19424			−0.597122	−	0.802150i	$-0.703689\pi$
$139$	−166.000			−1.19424			−0.597122	+	0.802150i	$0.703689\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	57.1577	+	33.0000i	0.399704	+	0.230769i
$144$		0			0
$145$	99.0000	+	171.473i	0.682759	+	1.18257i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−142.894	+	82.5000i	−0.959021	+	0.553691i	−0.895872	−	0.444313i	$-0.853448\pi$
$149$	−142.894	+	82.5000i	−0.959021	+	0.553691i	−0.0631497	+	0.998004i	$0.520115\pi$
$150$		0			0
$151$	15.5000	−	26.8468i	0.102649	−	0.177793i	−0.810126	−	0.586255i	$-0.800601\pi$
$151$	15.5000	−	26.8468i	0.102649	−	0.177793i	0.912775	+	0.408462i	$0.133935\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	78.0000i		−	0.503226i
$156$		0			0
$157$	59.0000	−	102.191i	0.375796	−	0.650898i	−0.614650	−	0.788800i	$-0.710703\pi$
$157$	59.0000	−	102.191i	0.375796	−	0.650898i	0.990446	+	0.137902i	$0.0440359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	181.865	+	105.000i	1.12960	+	0.652174i
$162$		0			0
$163$	44.0000	+	76.2102i	0.269939	+	0.467547i	0.968846	−	0.247665i	$-0.0796632\pi$
$163$	44.0000	+	76.2102i	0.269939	+	0.467547i	−0.698907	+	0.715212i	$0.746330\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	258.000i		−	1.54491i	−0.635069	−	0.772455i	$-0.719029\pi$
$167$		−	258.000i		−	1.54491i	0.635069	−	0.772455i	$-0.280971\pi$
$168$		0			0
$169$	315.000			1.86391
$170$		0			0
$171$		0			0
$172$		0			0
$173$	226.033	−	130.500i	1.30655	−	0.754335i	0.325029	−	0.945704i	$-0.394626\pi$
$173$	226.033	−	130.500i	1.30655	−	0.754335i	0.981518	+	0.191369i	$0.0612927\pi$
$174$		0			0
$175$	−77.0000			−0.440000
$176$		0			0
$177$		0			0
$178$		0			0
$179$	220.836	+	127.500i	1.23372	+	0.712291i	0.967804	−	0.251705i	$-0.0809912\pi$
$179$	220.836	+	127.500i	1.23372	+	0.712291i	0.265919	+	0.963995i	$0.414325\pi$
$180$		0			0
$181$	44.0000			0.243094			0.121547	−	0.992586i	$-0.461214\pi$
$181$	44.0000			0.243094			0.121547	+	0.992586i	$0.461214\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	51.9615	+	30.0000i	0.280873	+	0.162162i
$186$		0			0
$187$	36.0000	+	62.3538i	0.192513	+	0.333443i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	155.885	−	90.0000i	0.816150	−	0.471204i	−0.0329373	−	0.999457i	$-0.510486\pi$
$191$	155.885	−	90.0000i	0.816150	−	0.471204i	0.849087	+	0.528253i	$0.177153\pi$
$192$		0			0
$193$	−101.500	+	175.803i	−0.525907	+	0.910897i	0.473638	+	0.880720i	$0.342941\pi$
$193$	−101.500	+	175.803i	−0.525907	+	0.910897i	−0.999545	+	0.0301775i	$0.990393\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	81.0000i		−	0.411168i	−0.978640	−	0.205584i	$-0.934091\pi$
$197$		−	81.0000i		−	0.411168i	0.978640	−	0.205584i	$-0.0659093\pi$
$198$		0			0
$199$	21.5000	−	37.2391i	0.108040	−	0.187131i	−0.806936	−	0.590639i	$-0.798876\pi$
$199$	21.5000	−	37.2391i	0.108040	−	0.187131i	0.914976	+	0.403508i	$0.132209\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	200.052	−	115.500i	0.985477	−	0.568966i
$204$		0			0
$205$	90.0000	+	155.885i	0.439024	+	0.760413i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	30.0000i		−	0.143541i
$210$		0			0
$211$	−316.000			−1.49763			−0.748815	−	0.662779i	$-0.769377\pi$
$211$	−316.000			−1.49763			−0.748815	+	0.662779i	$0.769377\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	290.985	−	168.000i	1.35342	−	0.781395i
$216$		0			0
$217$	−91.0000			−0.419355
$218$		0			0
$219$		0			0
$220$		0			0
$221$	457.261	+	264.000i	2.06906	+	1.19457i
$222$		0			0
$223$	275.000			1.23318			0.616592	−	0.787283i	$-0.288513\pi$
$223$	275.000			1.23318			0.616592	+	0.787283i	$0.288513\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.1865	−	10.5000i	−0.0801169	−	0.0462555i	0.459406	−	0.888226i	$-0.348062\pi$
$227$	−18.1865	−	10.5000i	−0.0801169	−	0.0462555i	−0.539523	+	0.841971i	$0.681396\pi$
$228$		0			0
$229$	−124.000	−	214.774i	−0.541485	−	0.937879i	−0.998819	−	0.0485842i	$-0.984529\pi$
$229$	−124.000	−	214.774i	−0.541485	−	0.937879i	0.457334	−	0.889295i	$-0.348804\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−394.908	+	228.000i	−1.69488	+	0.978541i	−0.744415	+	0.667718i	$0.767271\pi$
$233$	−394.908	+	228.000i	−1.69488	+	0.978541i	−0.950468	+	0.310823i	$0.899395\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	24.0000i		−	0.100418i	−0.998739	−	0.0502092i	$-0.984011\pi$
$239$		−	24.0000i		−	0.100418i	0.998739	−	0.0502092i	$-0.0159888\pi$
$240$		0			0
$241$	168.500	−	291.851i	0.699170	−	1.21100i	−0.269584	−	0.962977i	$-0.586886\pi$
$241$	168.500	−	291.851i	0.699170	−	1.21100i	0.968755	−	0.248021i	$-0.0797803\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			294.000i			1.20000i
$246$		0			0
$247$	−110.000	−	190.526i	−0.445344	−	0.771359i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	405.000i		−	1.61355i	−0.590862	−	0.806773i	$-0.701212\pi$
$251$		−	405.000i		−	1.61355i	0.590862	−	0.806773i	$-0.298788\pi$
$252$		0			0
$253$	90.0000			0.355731
$254$		0			0
$255$		0			0
$256$		0			0
$257$	176.669	−	102.000i	0.687429	−	0.396887i	−0.115219	−	0.993340i	$-0.536757\pi$
$257$	176.669	−	102.000i	0.687429	−	0.396887i	0.802648	+	0.596453i	$0.203424\pi$
$258$		0			0
$259$	35.0000	−	60.6218i	0.135135	−	0.234061i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	114.315	+	66.0000i	0.434659	+	0.250951i	0.701330	−	0.712837i	$-0.252590\pi$
$263$	114.315	+	66.0000i	0.434659	+	0.250951i	−0.266670	+	0.963788i	$0.585924\pi$
$264$		0			0
$265$	−252.000			−0.950943
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−215.640	−	124.500i	−0.801637	−	0.462825i	0.0424063	−	0.999100i	$-0.486498\pi$
$269$	−215.640	−	124.500i	−0.801637	−	0.462825i	−0.844043	+	0.536275i	$0.819831\pi$
$270$		0			0
$271$	17.0000	+	29.4449i	0.0627306	+	0.108653i	0.895685	−	0.444689i	$-0.146686\pi$
$271$	17.0000	+	29.4449i	0.0627306	+	0.108653i	−0.832954	+	0.553342i	$0.813352\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−28.5788	+	16.5000i	−0.103923	+	0.0600000i
$276$		0			0
$277$	−154.000	+	266.736i	−0.555957	+	0.962945i	0.441872	+	0.897078i	$0.354315\pi$
$277$	−154.000	+	266.736i	−0.555957	+	0.962945i	−0.997828	+	0.0658670i	$0.979019\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			528.000i			1.87900i	0.342544	+	0.939502i	$0.388711\pi$
$281$			528.000i			1.87900i	−0.342544	+	0.939502i	$0.611289\pi$
$282$		0			0
$283$	−115.000	+	199.186i	−0.406360	+	0.703837i	−0.994479	−	0.104938i	$-0.966536\pi$
$283$	−115.000	+	199.186i	−0.406360	+	0.703837i	0.588118	+	0.808775i	$0.299869\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	181.865	−	105.000i	0.633677	−	0.365854i
$288$		0			0
$289$	143.500	+	248.549i	0.496540	+	0.860032i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			291.000i			0.993174i	0.867987	+	0.496587i	$0.165414\pi$
$293$			291.000i			0.993174i	−0.867987	+	0.496587i	$0.834586\pi$
$294$		0			0
$295$	−594.000			−2.01356
$296$		0			0
$297$		0			0
$298$		0			0
$299$	571.577	−	330.000i	1.91163	−	1.10368i
$300$		0			0
$301$	−196.000	−	339.482i	−0.651163	−	1.12785i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	488.438	+	282.000i	1.60144	+	0.924590i
$306$		0			0
$307$	296.000			0.964169			0.482085	−	0.876125i	$-0.339880\pi$
$307$	296.000			0.964169			0.482085	+	0.876125i	$0.339880\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−124.708	−	72.0000i	−0.400989	−	0.231511i	0.285921	−	0.958253i	$-0.407700\pi$
$311$	−124.708	−	72.0000i	−0.400989	−	0.231511i	−0.686911	+	0.726742i	$0.741034\pi$
$312$		0			0
$313$	−253.000	−	438.209i	−0.808307	−	1.40003i	−0.914036	−	0.405633i	$-0.867051\pi$
$313$	−253.000	−	438.209i	−0.808307	−	1.40003i	0.105729	−	0.994395i	$-0.466282\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−54.5596	+	31.5000i	−0.172112	+	0.0993691i	−0.583581	−	0.812055i	$-0.698349\pi$
$317$	−54.5596	+	31.5000i	−0.172112	+	0.0993691i	0.411469	+	0.911424i	$0.365016\pi$
$318$		0			0
$319$	49.5000	−	85.7365i	0.155172	−	0.268767i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	240.000i		−	0.743034i
$324$		0			0
$325$	−121.000	+	209.578i	−0.372308	+	0.644856i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	170.000	+	294.449i	0.513595	+	0.889573i	0.999876	+	0.0157701i	$0.00502000\pi$
$331$	170.000	+	294.449i	0.513595	+	0.889573i	−0.486280	+	0.873803i	$0.661647\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	456.000i		−	1.36119i
$336$		0			0
$337$	95.0000			0.281899			0.140950	−	0.990017i	$-0.454984\pi$
$337$	95.0000			0.281899			0.140950	+	0.990017i	$0.454984\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−33.7750	+	19.5000i	−0.0990469	+	0.0571848i
$342$		0			0
$343$	343.000			1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$	194.856	+	112.500i	0.561544	+	0.324207i	0.753765	−	0.657144i	$-0.228236\pi$
$347$	194.856	+	112.500i	0.561544	+	0.324207i	−0.192221	+	0.981352i	$0.561569\pi$
$348$		0			0
$349$	608.000			1.74212			0.871060	−	0.491176i	$-0.163433\pi$
$349$	608.000			1.74212			0.871060	+	0.491176i	$0.163433\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	161.081	+	93.0000i	0.456319	+	0.263456i	0.710495	−	0.703702i	$-0.248471\pi$
$353$	161.081	+	93.0000i	0.456319	+	0.263456i	−0.254176	+	0.967158i	$0.581804\pi$
$354$		0			0
$355$	−54.0000	−	93.5307i	−0.152113	−	0.263467i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−285.788	+	165.000i	−0.796068	+	0.459610i	−0.842094	−	0.539330i	$-0.818677\pi$
$359$	−285.788	+	165.000i	−0.796068	+	0.459610i	0.0460265	+	0.998940i	$0.485344\pi$
$360$		0			0
$361$	130.500	−	226.033i	0.361496	−	0.626129i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	330.000i		−	0.904110i
$366$		0			0
$367$	−163.000	+	282.324i	−0.444142	+	0.769276i	−0.997992	−	0.0633403i	$-0.979825\pi$
$367$	−163.000	+	282.324i	−0.444142	+	0.769276i	0.553850	+	0.832616i	$0.313158\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			294.000i			0.792453i
$372$		0			0
$373$	20.0000	+	34.6410i	0.0536193	+	0.0928714i	0.891589	−	0.452845i	$-0.149591\pi$
$373$	20.0000	+	34.6410i	0.0536193	+	0.0928714i	−0.837970	+	0.545716i	$0.816258\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	726.000i		−	1.92573i
$378$		0			0
$379$	224.000			0.591029			0.295515	−	0.955338i	$-0.404509\pi$
$379$	224.000			0.591029			0.295515	+	0.955338i	$0.404509\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−88.3346	+	51.0000i	−0.230639	+	0.133159i	−0.610867	−	0.791734i	$-0.709179\pi$
$383$	−88.3346	+	51.0000i	−0.230639	+	0.133159i	0.380228	+	0.924893i	$0.375846\pi$
$384$		0			0
$385$	63.0000	+	109.119i	0.163636	+	0.283426i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−127.306	−	73.5000i	−0.327264	−	0.188946i	0.327362	−	0.944899i	$-0.393840\pi$
$389$	−127.306	−	73.5000i	−0.327264	−	0.188946i	−0.654626	+	0.755953i	$0.727174\pi$
$390$		0			0
$391$	720.000			1.84143
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−431.281	−	249.000i	−1.09185	−	0.630380i
$396$		0			0
$397$	248.000	+	429.549i	0.624685	+	1.08199i	0.988602	+	0.150555i	$0.0481060\pi$
$397$	248.000	+	429.549i	0.624685	+	1.08199i	−0.363917	+	0.931432i	$0.618561\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−571.577	+	330.000i	−1.42538	+	0.822943i	−0.996751	−	0.0805389i	$-0.974336\pi$
$401$	−571.577	+	330.000i	−1.42538	+	0.822943i	−0.428627	+	0.903482i	$0.641003\pi$
$402$		0			0
$403$	−143.000	+	247.683i	−0.354839	+	0.614599i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	30.0000i		−	0.0737101i
$408$		0			0
$409$	−205.000	+	355.070i	−0.501222	+	0.868143i	0.498777	+	0.866731i	$0.333783\pi$
$409$	−205.000	+	355.070i	−0.501222	+	0.868143i	−0.999999	+	0.00141219i	$0.999550\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			693.000i			1.67797i
$414$		0			0
$415$	−441.000	−	763.834i	−1.06265	−	1.84056i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	330.000i		−	0.787589i	−0.919198	−	0.393795i	$-0.871162\pi$
$419$		−	330.000i		−	0.787589i	0.919198	−	0.393795i	$-0.128838\pi$
$420$		0			0
$421$	−232.000			−0.551069			−0.275534	−	0.961291i	$-0.588855\pi$
$421$	−232.000			−0.551069			−0.275534	+	0.961291i	$0.588855\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−228.631	+	132.000i	−0.537955	+	0.310588i
$426$		0			0
$427$	329.000	−	569.845i	0.770492	−	1.33453i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	25.9808	+	15.0000i	0.0602802	+	0.0348028i	0.529837	−	0.848099i	$-0.322253\pi$
$431$	25.9808	+	15.0000i	0.0602802	+	0.0348028i	−0.469557	+	0.882902i	$0.655586\pi$
$432$		0			0
$433$	−649.000			−1.49885			−0.749423	−	0.662092i	$-0.769669\pi$
$433$	−649.000			−1.49885			−0.749423	+	0.662092i	$0.769669\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−259.808	−	150.000i	−0.594525	−	0.343249i
$438$		0			0
$439$	−20.5000	−	35.5070i	−0.0466970	−	0.0808816i	0.841732	−	0.539895i	$-0.181536\pi$
$439$	−20.5000	−	35.5070i	−0.0466970	−	0.0808816i	−0.888429	+	0.459014i	$0.848203\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−724.863	+	418.500i	−1.63626	+	0.944695i	−0.654155	+	0.756360i	$0.726976\pi$
$443$	−724.863	+	418.500i	−1.63626	+	0.944695i	−0.982105	+	0.188335i	$0.939691\pi$
$444$		0			0
$445$	−396.000	+	685.892i	−0.889888	+	1.54133i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	108.000i		−	0.240535i	−0.992742	−	0.120267i	$-0.961625\pi$
$449$		−	108.000i		−	0.240535i	0.992742	−	0.120267i	$-0.0383752\pi$
$450$		0			0
$451$	45.0000	−	77.9423i	0.0997783	−	0.172821i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	800.207	+	462.000i	1.75870	+	1.01538i
$456$		0			0
$457$	107.000	+	185.329i	0.234136	+	0.405535i	0.959021	−	0.283335i	$-0.0914407\pi$
$457$	107.000	+	185.329i	0.234136	+	0.405535i	−0.724885	+	0.688869i	$0.758107\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	363.000i		−	0.787419i	−0.919235	−	0.393709i	$-0.871192\pi$
$461$		−	363.000i		−	0.787419i	0.919235	−	0.393709i	$-0.128808\pi$
$462$		0			0
$463$	491.000			1.06048			0.530238	−	0.847849i	$-0.322103\pi$
$463$	491.000			1.06048			0.530238	+	0.847849i	$0.322103\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−262.406	+	151.500i	−0.561897	+	0.324411i	−0.753906	−	0.656982i	$-0.771833\pi$
$467$	−262.406	+	151.500i	−0.561897	+	0.324411i	0.192010	+	0.981393i	$0.438499\pi$
$468$		0			0
$469$	−532.000			−1.13433
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−145.492	−	84.0000i	−0.307595	−	0.177590i
$474$		0			0
$475$	110.000			0.231579
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−41.5692	−	24.0000i	−0.0867833	−	0.0501044i	0.455980	−	0.889990i	$-0.349289\pi$
$479$	−41.5692	−	24.0000i	−0.0867833	−	0.0501044i	−0.542764	+	0.839885i	$0.682622\pi$
$480$		0			0
$481$	−110.000	−	190.526i	−0.228690	−	0.396103i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−441.673	+	255.000i	−0.910666	+	0.525773i
$486$		0			0
$487$	9.50000	−	16.4545i	0.0195072	−	0.0337874i	−0.856107	−	0.516799i	$-0.827124\pi$
$487$	9.50000	−	16.4545i	0.0195072	−	0.0337874i	0.875614	+	0.483011i	$0.160457\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	330.000i		−	0.672098i	−0.941845	−	0.336049i	$-0.890909\pi$
$491$		−	330.000i		−	0.672098i	0.941845	−	0.336049i	$-0.109091\pi$
$492$		0			0
$493$	396.000	−	685.892i	0.803245	−	1.39126i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−109.119	+	63.0000i	−0.219556	+	0.126761i
$498$		0			0
$499$	74.0000	+	128.172i	0.148297	+	0.256857i	0.930598	−	0.366043i	$-0.119288\pi$
$499$	74.0000	+	128.172i	0.148297	+	0.256857i	−0.782301	+	0.622900i	$0.785954\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			828.000i			1.64612i	0.567952	+	0.823062i	$0.307736\pi$
$503$			828.000i			1.64612i	−0.567952	+	0.823062i	$0.692264\pi$
$504$		0			0
$505$	−18.0000			−0.0356436
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−688.490	+	397.500i	−1.35263	+	0.780943i	−0.988618	−	0.150450i	$-0.951928\pi$
$509$	−688.490	+	397.500i	−1.35263	+	0.780943i	−0.364015	+	0.931393i	$0.618594\pi$
$510$		0			0
$511$	−385.000			−0.753425
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−633.931	−	366.000i	−1.23093	−	0.710680i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	57.1577	+	33.0000i	0.109708	+	0.0633397i	0.553850	−	0.832617i	$-0.313158\pi$
$521$	57.1577	+	33.0000i	0.109708	+	0.0633397i	−0.444142	+	0.895956i	$0.646491\pi$
$522$		0			0
$523$	−121.000	−	209.578i	−0.231358	−	0.400723i	0.726850	−	0.686796i	$-0.240983\pi$
$523$	−121.000	−	209.578i	−0.231358	−	0.400723i	−0.958208	+	0.286073i	$0.907650\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−270.200	+	156.000i	−0.512713	+	0.296015i
$528$		0			0
$529$	185.500	−	321.295i	0.350662	−	0.607364i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	660.000i		−	1.23827i
$534$		0			0
$535$	90.0000	−	155.885i	0.168224	−	0.291373i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	127.306	−	73.5000i	0.236189	−	0.136364i
$540$		0			0
$541$	176.000	+	304.841i	0.325323	+	0.563477i	0.981578	−	0.191063i	$-0.0611935\pi$
$541$	176.000	+	304.841i	0.325323	+	0.563477i	−0.656254	+	0.754540i	$0.727860\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	600.000i		−	1.10092i
$546$		0			0
$547$	−778.000			−1.42230			−0.711152	−	0.703039i	$-0.751826\pi$
$547$	−778.000			−1.42230			−0.711152	+	0.703039i	$0.751826\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−285.788	+	165.000i	−0.518672	+	0.299456i
$552$		0			0
$553$	−290.500	+	503.161i	−0.525316	+	0.909875i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	314.367	+	181.500i	0.564394	+	0.325853i	0.754907	−	0.655832i	$-0.227682\pi$
$557$	314.367	+	181.500i	0.564394	+	0.325853i	−0.190513	+	0.981685i	$0.561015\pi$
$558$		0			0
$559$	−1232.00			−2.20394
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−950.896	−	549.000i	−1.68898	−	0.975133i	−0.955300	−	0.295638i	$-0.904468\pi$
$563$	−950.896	−	549.000i	−1.68898	−	0.975133i	−0.733680	−	0.679495i	$-0.762199\pi$
$564$		0			0
$565$	−234.000	−	405.300i	−0.414159	−	0.717345i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−685.892	+	396.000i	−1.20543	+	0.695958i	−0.961758	−	0.273899i	$-0.911687\pi$
$569$	−685.892	+	396.000i	−1.20543	+	0.695958i	−0.243676	+	0.969857i	$0.578353\pi$
$570$		0			0
$571$	200.000	−	346.410i	0.350263	−	0.606673i	−0.636033	−	0.771662i	$-0.719426\pi$
$571$	200.000	−	346.410i	0.350263	−	0.606673i	0.986295	+	0.164989i	$0.0527590\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			330.000i			0.573913i
$576$		0			0
$577$	270.500	−	468.520i	0.468804	−	0.811993i	−0.530560	−	0.847647i	$-0.678018\pi$
$577$	270.500	−	468.520i	0.468804	−	0.811993i	0.999364	+	0.0356548i	$0.0113517\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−891.140	+	514.500i	−1.53380	+	0.885542i
$582$		0			0
$583$	63.0000	+	109.119i	0.108062	+	0.187168i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1002.00i		−	1.70698i	−0.521105	−	0.853492i	$-0.674480\pi$
$587$		−	1002.00i		−	1.70698i	0.521105	−	0.853492i	$-0.325520\pi$
$588$		0			0
$589$	130.000			0.220713
$590$		0			0
$591$		0			0
$592$		0			0
$593$	244.219	−	141.000i	0.411837	−	0.237774i	−0.279742	−	0.960075i	$-0.590249\pi$
$593$	244.219	−	141.000i	0.411837	−	0.237774i	0.691579	+	0.722301i	$0.256916\pi$
$594$		0			0
$595$	504.000	+	872.954i	0.847059	+	1.46715i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	452.065	+	261.000i	0.754700	+	0.435726i	0.827390	−	0.561628i	$-0.189825\pi$
$599$	452.065	+	261.000i	0.754700	+	0.435726i	−0.0726897	+	0.997355i	$0.523158\pi$
$600$		0			0
$601$	110.000			0.183028			0.0915141	−	0.995804i	$-0.470829\pi$
$601$	110.000			0.183028			0.0915141	+	0.995804i	$0.470829\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−581.969	−	336.000i	−0.961932	−	0.555372i
$606$		0			0
$607$	492.500	+	853.035i	0.811367	+	1.40533i	0.911907	+	0.410396i	$0.134610\pi$
$607$	492.500	+	853.035i	0.811367	+	1.40533i	−0.100540	+	0.994933i	$0.532057\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	242.000	−	419.156i	0.394780	−	0.683779i	−0.598293	−	0.801277i	$-0.704154\pi$
$613$	242.000	−	419.156i	0.394780	−	0.683779i	0.993073	+	0.117499i	$0.0374876\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	366.000i		−	0.593193i	−0.955003	−	0.296596i	$-0.904148\pi$
$617$		−	366.000i		−	0.593193i	0.955003	−	0.296596i	$-0.0958516\pi$
$618$		0			0
$619$	−181.000	+	313.501i	−0.292407	+	0.506464i	−0.974378	−	0.224915i	$-0.927790\pi$
$619$	−181.000	+	313.501i	−0.292407	+	0.506464i	0.681971	+	0.731379i	$0.261123\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	800.207	+	462.000i	1.28444	+	0.741573i
$624$		0			0
$625$	389.500	+	674.634i	0.623200	+	1.07941i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	240.000i		−	0.381558i
$630$		0			0
$631$	425.000			0.673534			0.336767	−	0.941588i	$-0.390667\pi$
$631$	425.000			0.673534			0.336767	+	0.941588i	$0.390667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	696.284	−	402.000i	1.09651	−	0.633071i
$636$		0			0
$637$	539.000	−	933.575i	0.846154	−	1.46558i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	940.504	+	543.000i	1.46724	+	0.847114i	0.999328	−	0.0366588i	$-0.0116715\pi$
$641$	940.504	+	543.000i	1.46724	+	0.847114i	0.467916	+	0.883773i	$0.345005\pi$
$642$		0			0
$643$	302.000			0.469673			0.234837	−	0.972035i	$-0.424544\pi$
$643$	302.000			0.469673			0.234837	+	0.972035i	$0.424544\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.19615	−	3.00000i	−0.00803115	−	0.00463679i	0.495979	−	0.868334i	$-0.334809\pi$
$647$	−5.19615	−	3.00000i	−0.00803115	−	0.00463679i	−0.504010	+	0.863698i	$0.668143\pi$
$648$		0			0
$649$	148.500	+	257.210i	0.228814	+	0.396317i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−400.104	+	231.000i	−0.612716	+	0.353752i	−0.774028	−	0.633152i	$-0.781761\pi$
$653$	−400.104	+	231.000i	−0.612716	+	0.353752i	0.161311	+	0.986904i	$0.448428\pi$
$654$		0			0
$655$	−333.000	+	576.773i	−0.508397	+	0.880569i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			645.000i			0.978756i	0.872072	+	0.489378i	$0.162776\pi$
$659$			645.000i			0.978756i	−0.872072	+	0.489378i	$0.837224\pi$
$660$		0			0
$661$	128.000	−	221.703i	0.193646	−	0.335405i	−0.752810	−	0.658238i	$-0.771302\pi$
$661$	128.000	−	221.703i	0.193646	−	0.335405i	0.946456	+	0.322833i	$0.104635\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	420.000i		−	0.631579i
$666$		0			0
$667$	−495.000	−	857.365i	−0.742129	−	1.28541i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	282.000i		−	0.420268i
$672$		0			0
$673$	1010.00			1.50074			0.750371	−	0.661016i	$-0.229875\pi$
$673$	1010.00			1.50074			0.750371	+	0.661016i	$0.229875\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	833.982	−	481.500i	1.23188	−	0.711226i	0.264458	−	0.964397i	$-0.414807\pi$
$677$	833.982	−	481.500i	1.23188	−	0.711226i	0.967421	+	0.253171i	$0.0814736\pi$
$678$		0			0
$679$	297.500	+	515.285i	0.438144	+	0.758888i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−885.944	−	511.500i	−1.29714	−	0.748902i	−0.317228	−	0.948349i	$-0.602752\pi$
$683$	−885.944	−	511.500i	−1.29714	−	0.748902i	−0.979909	+	0.199448i	$0.936085\pi$
$684$		0			0
$685$	−1044.00			−1.52409
$686$		0			0
$687$		0			0
$688$		0			0
$689$	800.207	+	462.000i	1.16140	+	0.670537i
$690$		0			0
$691$	−73.0000	−	126.440i	−0.105644	−	0.182981i	0.808357	−	0.588692i	$-0.200357\pi$
$691$	−73.0000	−	126.440i	−0.105644	−	0.182981i	−0.914001	+	0.405712i	$0.867024\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−862.561	+	498.000i	−1.24110	+	0.716547i
$696$		0			0
$697$	360.000	−	623.538i	0.516499	−	0.894603i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			162.000i			0.231098i	0.993302	+	0.115549i	$0.0368628\pi$
$701$			162.000i			0.231098i	−0.993302	+	0.115549i	$0.963137\pi$
$702$		0			0
$703$	−50.0000	+	86.6025i	−0.0711238	+	0.123190i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			21.0000i			0.0297030i
$708$		0			0
$709$	−196.000	−	339.482i	−0.276446	−	0.478818i	0.694053	−	0.719924i	$-0.255823\pi$
$709$	−196.000	−	339.482i	−0.276446	−	0.478818i	−0.970499	+	0.241106i	$0.922490\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			390.000i			0.546985i
$714$		0			0
$715$	396.000			0.553846
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−98.7269	+	57.0000i	−0.137311	+	0.0792768i	−0.567082	−	0.823661i	$-0.691928\pi$
$719$	−98.7269	+	57.0000i	−0.137311	+	0.0792768i	0.429771	+	0.902938i	$0.358594\pi$
$720$		0			0
$721$	−427.000	+	739.586i	−0.592233	+	1.02578i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	314.367	+	181.500i	0.433610	+	0.250345i
$726$		0			0
$727$	−430.000			−0.591472			−0.295736	−	0.955270i	$-0.595565\pi$
$727$	−430.000			−0.591472			−0.295736	+	0.955270i	$0.595565\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1163.94	−	672.000i	−1.59225	−	0.919289i
$732$		0			0
$733$	578.000	+	1001.13i	0.788540	+	1.36579i	0.926861	+	0.375404i	$0.122496\pi$
$733$	578.000	+	1001.13i	0.788540	+	1.36579i	−0.138321	+	0.990387i	$0.544171\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−197.454	+	114.000i	−0.267916	+	0.154681i
$738$		0			0
$739$	473.000	−	819.260i	0.640054	−	1.10861i	−0.345366	−	0.938468i	$-0.612245\pi$
$739$	473.000	−	819.260i	0.640054	−	1.10861i	0.985420	−	0.170138i	$-0.0544214\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1452.00i			1.95424i	0.212690	+	0.977120i	$0.431778\pi$
$743$			1452.00i			1.95424i	−0.212690	+	0.977120i	$0.568222\pi$
$744$		0			0
$745$	−495.000	+	857.365i	−0.664430	+	1.15083i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−181.865	−	105.000i	−0.242811	−	0.140187i
$750$		0			0
$751$	281.000	+	486.706i	0.374168	+	0.648078i	0.990202	−	0.139642i	$-0.0445951\pi$
$751$	281.000	+	486.706i	0.374168	+	0.648078i	−0.616034	+	0.787719i	$0.711262\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	186.000i		−	0.246358i
$756$		0			0
$757$	−970.000			−1.28137			−0.640687	−	0.767802i	$-0.721350\pi$
$757$	−970.000			−1.28137			−0.640687	+	0.767802i	$0.721350\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	659.911	−	381.000i	0.867163	−	0.500657i	0.000758839	−	1.00000i	$-0.499758\pi$
$761$	659.911	−	381.000i	0.867163	−	0.500657i	0.866405	+	0.499343i	$0.166425\pi$
$762$		0			0
$763$	−700.000			−0.917431
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1886.20	+	1089.00i	2.45920	+	1.41982i
$768$		0			0
$769$	887.000			1.15345			0.576723	−	0.816940i	$-0.304331\pi$
$769$	887.000			1.15345			0.576723	+	0.816940i	$0.304331\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−67.5500	−	39.0000i	−0.0873868	−	0.0504528i	0.455670	−	0.890149i	$-0.349400\pi$
$773$	−67.5500	−	39.0000i	−0.0873868	−	0.0504528i	−0.543057	+	0.839696i	$0.682733\pi$
$774$		0			0
$775$	−71.5000	−	123.842i	−0.0922581	−	0.159796i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−259.808	+	150.000i	−0.333514	+	0.192555i
$780$		0			0
$781$	−27.0000	+	46.7654i	−0.0345711	+	0.0598788i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	708.000i		−	0.901911i
$786$		0			0
$787$	320.000	−	554.256i	0.406607	−	0.704265i	−0.587900	−	0.808934i	$-0.700045\pi$
$787$	320.000	−	554.256i	0.406607	−	0.704265i	0.994507	+	0.104669i	$0.0333784\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−472.850	+	273.000i	−0.597787	+	0.345133i
$792$		0			0
$793$	−1034.00	−	1790.94i	−1.30391	−	2.25844i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	915.000i		−	1.14806i	−0.818836	−	0.574028i	$-0.805380\pi$
$797$		−	915.000i		−	1.14806i	0.818836	−	0.574028i	$-0.194620\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−142.894	+	82.5000i	−0.177950	+	0.102740i
$804$		0			0
$805$	1260.00			1.56522
$806$		0			0
$807$		0			0
$808$		0			0
$809$	275.396	+	159.000i	0.340415	+	0.196539i	0.660456	−	0.750865i	$-0.270363\pi$
$809$	275.396	+	159.000i	0.340415	+	0.196539i	−0.320040	+	0.947404i	$0.603696\pi$
$810$		0			0
$811$	−1078.00			−1.32922			−0.664612	−	0.747189i	$-0.731403\pi$
$811$	−1078.00			−1.32922			−0.664612	+	0.747189i	$0.731403\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	457.261	+	264.000i	0.561057	+	0.323926i
$816$		0			0
$817$	280.000	+	484.974i	0.342717	+	0.593604i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1332.81	−	769.500i	1.62340	−	0.937272i	0.637401	−	0.770532i	$-0.280009\pi$
$821$	1332.81	−	769.500i	1.62340	−	0.937272i	0.986001	−	0.166739i	$-0.0533239\pi$
$822$		0			0
$823$	−188.500	+	326.492i	−0.229040	+	0.396709i	−0.957524	−	0.288354i	$-0.906892\pi$
$823$	−188.500	+	326.492i	−0.229040	+	0.396709i	0.728484	+	0.685063i	$0.240225\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	297.000i		−	0.359129i	−0.983746	−	0.179565i	$-0.942531\pi$
$827$		−	297.000i		−	0.359129i	0.983746	−	0.179565i	$-0.0574689\pi$
$828$		0			0
$829$	−22.0000	+	38.1051i	−0.0265380	+	0.0459652i	−0.878989	−	0.476841i	$-0.841782\pi$
$829$	−22.0000	+	38.1051i	−0.0265380	+	0.0459652i	0.852451	+	0.522806i	$0.175115\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1018.45	−	588.000i	1.22262	−	0.705882i
$834$		0			0
$835$	−774.000	−	1340.61i	−0.926946	−	1.60552i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			978.000i			1.16567i	0.812589	+	0.582837i	$0.198057\pi$
$839$			978.000i			1.16567i	−0.812589	+	0.582837i	$0.801943\pi$
$840$		0			0
$841$	−248.000			−0.294887
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1636.79	−	945.000i	1.93703	−	1.11834i
$846$		0			0
$847$	−392.000	+	678.964i	−0.462810	+	0.801610i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−259.808	−	150.000i	−0.305297	−	0.176263i
$852$		0			0
$853$	−160.000			−0.187573			−0.0937866	−	0.995592i	$-0.529897\pi$
$853$	−160.000			−0.187573			−0.0937866	+	0.995592i	$0.529897\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1018.45	+	588.000i	1.18838	+	0.686114i	0.957939	−	0.286971i	$-0.0926483\pi$
$857$	1018.45	+	588.000i	1.18838	+	0.686114i	0.230446	+	0.973085i	$0.425982\pi$
$858$		0			0
$859$	−598.000	−	1035.77i	−0.696158	−	1.20578i	−0.969789	−	0.243947i	$-0.921558\pi$
$859$	−598.000	−	1035.77i	−0.696158	−	1.20578i	0.273630	−	0.961835i	$-0.411776\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1070.41	−	618.000i	1.24033	−	0.716107i	0.271171	−	0.962531i	$-0.412589\pi$
$863$	1070.41	−	618.000i	1.24033	−	0.716107i	0.969162	+	0.246425i	$0.0792558\pi$
$864$		0			0
$865$	783.000	−	1356.20i	0.905202	−	1.56786i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			249.000i			0.286536i
$870$		0			0
$871$	−836.000	+	1447.99i	−0.959816	+	1.66245i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	509.223	−	294.000i	0.581969	−	0.336000i
$876$		0			0
$877$	−49.0000	−	84.8705i	−0.0558723	−	0.0967736i	0.836736	−	0.547606i	$-0.184461\pi$
$877$	−49.0000	−	84.8705i	−0.0558723	−	0.0967736i	−0.892609	+	0.450832i	$0.851127\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1170.00i		−	1.32804i	−0.747716	−	0.664018i	$-0.768850\pi$
$881$		−	1170.00i		−	1.32804i	0.747716	−	0.664018i	$-0.231150\pi$
$882$		0			0
$883$	242.000			0.274066			0.137033	−	0.990567i	$-0.456243\pi$
$883$	242.000			0.274066			0.137033	+	0.990567i	$0.456243\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	488.438	−	282.000i	0.550663	−	0.317926i	−0.198726	−	0.980055i	$-0.563680\pi$
$887$	488.438	−	282.000i	0.550663	−	0.317926i	0.749389	+	0.662129i	$0.230347\pi$
$888$		0			0
$889$	−469.000	−	812.332i	−0.527559	−	0.913759i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	1530.00			1.70950
$896$		0			0
$897$		0			0
$898$		0			0
$899$	371.525	+	214.500i	0.413265	+	0.238598i
$900$		0			0
$901$	504.000	+	872.954i	0.559378	+	0.968872i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	228.631	−	132.000i	0.252631	−	0.145856i
$906$		0			0
$907$	−244.000	+	422.620i	−0.269019	+	0.465954i	−0.968609	−	0.248590i	$-0.920033\pi$
$907$	−244.000	+	422.620i	−0.269019	+	0.465954i	0.699590	+	0.714545i	$0.253366\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	462.000i		−	0.507135i	−0.967318	−	0.253568i	$-0.918396\pi$
$911$		−	462.000i		−	0.507135i	0.967318	−	0.253568i	$-0.0816040\pi$
$912$		0			0
$913$	−220.500	+	381.917i	−0.241512	+	0.418310i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	672.902	+	388.500i	0.733808	+	0.423664i
$918$		0			0
$919$	783.500	+	1357.06i	0.852557	+	1.47667i	0.878893	+	0.477019i	$0.158283\pi$
$919$	783.500	+	1357.06i	0.852557	+	1.47667i	−0.0263358	+	0.999653i	$0.508384\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			396.000i			0.429036i
$924$		0			0
$925$	110.000			0.118919
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−701.481	+	405.000i	−0.755092	+	0.435953i	−0.827531	−	0.561420i	$-0.810255\pi$
$929$	−701.481	+	405.000i	−0.755092	+	0.435953i	0.0724387	+	0.997373i	$0.476922\pi$
$930$		0			0
$931$	−490.000			−0.526316
$932$		0			0
$933$		0			0
$934$		0			0
$935$	374.123	+	216.000i	0.400132	+	0.231016i
$936$		0			0
$937$	−1630.00			−1.73959			−0.869797	−	0.493409i	$-0.835750\pi$
$937$	−1630.00			−1.73959			−0.869797	+	0.493409i	$0.835750\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	335.152	+	193.500i	0.356166	+	0.205632i	0.667397	−	0.744702i	$-0.267408\pi$
$941$	335.152	+	193.500i	0.356166	+	0.205632i	−0.311232	+	0.950334i	$0.600742\pi$
$942$		0			0
$943$	−450.000	−	779.423i	−0.477200	−	0.826535i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	724.863	−	418.500i	0.765431	−	0.441922i	−0.0658112	−	0.997832i	$-0.520964\pi$
$947$	724.863	−	418.500i	0.765431	−	0.441922i	0.831242	+	0.555910i	$0.187630\pi$
$948$		0			0
$949$	−605.000	+	1047.89i	−0.637513	+	1.10421i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1242.00i		−	1.30325i	−0.758540	−	0.651626i	$-0.774087\pi$
$953$		−	1242.00i		−	1.30325i	0.758540	−	0.651626i	$-0.225913\pi$
$954$		0			0
$955$	540.000	−	935.307i	0.565445	−	0.979380i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1218.00i			1.27007i
$960$		0			0
$961$	396.000	+	685.892i	0.412071	+	0.713727i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1218.00i			1.26218i
$966$		0			0
$967$	122.000			0.126163			0.0630817	−	0.998008i	$-0.479907\pi$
$967$	122.000			0.126163			0.0630817	+	0.998008i	$0.479907\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1408.16	+	813.000i	−1.45021	+	0.837281i	−0.998493	−	0.0548784i	$-0.982523\pi$
$971$	−1408.16	+	813.000i	−1.45021	+	0.837281i	−0.451720	+	0.892160i	$0.649190\pi$
$972$		0			0
$973$	581.000	+	1006.32i	0.597122	+	1.03425i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−919.719	−	531.000i	−0.941371	−	0.543501i	−0.0509808	−	0.998700i	$-0.516235\pi$
$977$	−919.719	−	531.000i	−0.941371	−	0.543501i	−0.890390	+	0.455199i	$0.849568\pi$
$978$		0			0
$979$	396.000			0.404494
$980$		0			0
$981$		0			0
$982$		0			0
$983$	129.904	+	75.0000i	0.132150	+	0.0762970i	0.564618	−	0.825353i	$-0.309024\pi$
$983$	129.904	+	75.0000i	0.132150	+	0.0762970i	−0.432467	+	0.901650i	$0.642357\pi$
$984$		0			0
$985$	−243.000	−	420.888i	−0.246701	−	0.427298i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1454.92	+	840.000i	−1.47110	+	0.849343i
$990$		0			0
$991$	869.000	−	1505.15i	0.876892	−	1.51882i	0.0221585	−	0.999754i	$-0.492946\pi$
$991$	869.000	−	1505.15i	0.876892	−	1.51882i	0.854734	−	0.519067i	$-0.173721\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	258.000i		−	0.259296i
$996$		0			0
$997$	944.000	−	1635.06i	0.946841	−	1.63998i	0.194817	−	0.980840i	$-0.437589\pi$
$997$	944.000	−	1635.06i	0.946841	−	1.63998i	0.752024	−	0.659136i	$-0.229078\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.3.bk.e.53.2	yes	4
3.2	odd	2	inner	756.3.bk.e.53.1	✓	4
7.2	even	3	inner	756.3.bk.e.485.1	yes	4
21.2	odd	6	inner	756.3.bk.e.485.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.3.bk.e.53.1	✓	4	3.2	odd	2	inner
756.3.bk.e.53.2	yes	4	1.1	even	1	trivial
756.3.bk.e.485.1	yes	4	7.2	even	3	inner
756.3.bk.e.485.2	yes	4	21.2	odd	6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	756.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(5.19615 - 3.00000i) q^{5} +(-3.50000 - 6.06218i) q^{7} +(-2.59808 - 1.50000i) q^{11} -22.0000 q^{13} +(-20.7846 - 12.0000i) q^{17} +(5.00000 + 8.66025i) q^{19} +(-25.9808 + 15.0000i) q^{23} +(5.50000 - 9.52628i) q^{25} +33.0000i q^{29} +(6.50000 - 11.2583i) q^{31} +(-36.3731 - 21.0000i) q^{35} +(5.00000 + 8.66025i) q^{37} +30.0000i q^{41} +56.0000 q^{43} +(-24.5000 + 42.4352i) q^{49} +(-36.3731 - 21.0000i) q^{53} -18.0000 q^{55} +(-85.7365 - 49.5000i) q^{59} +(47.0000 + 81.4064i) q^{61} +(-114.315 + 66.0000i) q^{65} +(38.0000 - 65.8179i) q^{67} -18.0000i q^{71} +(27.5000 - 47.6314i) q^{73} +21.0000i q^{77} +(-41.5000 - 71.8801i) q^{79} -147.000i q^{83} -144.000 q^{85} +(-114.315 + 66.0000i) q^{89} +(77.0000 + 133.368i) q^{91} +(51.9615 + 30.0000i) q^{95} -85.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{7} - 88 q^{13} + 20 q^{19} + 22 q^{25} + 26 q^{31} + 20 q^{37} + 224 q^{43} - 98 q^{49} - 72 q^{55} + 188 q^{61} + 152 q^{67} + 110 q^{73} - 166 q^{79} - 576 q^{85} + 308 q^{91} - 340 q^{97}+O(q^{100}) \) 4 * q - 14 * q^7 - 88 * q^13 + 20 * q^19 + 22 * q^25 + 26 * q^31 + 20 * q^37 + 224 * q^43 - 98 * q^49 - 72 * q^55 + 188 * q^61 + 152 * q^67 + 110 * q^73 - 166 * q^79 - 576 * q^85 + 308 * q^91 - 340 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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