LMFDB - Embedded newform 756.4.w.a.521.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,4,Mod(341,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 5]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("756.341");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.w (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:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.2
Character	$\chi$	$=$	756.521
Dual form			756.4.w.a.341.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.18977 + 15.9171i) q^{5} +(18.3240 + 2.68897i) q^{7} +O(q^{10})$	$q+(-9.18977 + 15.9171i) q^{5} +(18.3240 + 2.68897i) q^{7} +(47.9540 - 27.6862i) q^{11} +(68.0763 - 39.3039i) q^{13} +(-9.99021 + 17.3036i) q^{17} +(-16.7611 + 9.67700i) q^{19} +(-107.511 - 62.0716i) q^{23} +(-106.404 - 184.296i) q^{25} +(48.2851 + 27.8774i) q^{29} -317.989i q^{31} +(-211.194 + 266.955i) q^{35} +(20.5303 + 35.5595i) q^{37} +(53.1315 + 92.0264i) q^{41} +(279.238 - 483.655i) q^{43} -147.916 q^{47} +(328.539 + 98.5453i) q^{49} +(570.090 + 329.141i) q^{53} +1017.72i q^{55} +568.509 q^{59} +140.951i q^{61} +1444.77i q^{65} -294.283 q^{67} -602.374i q^{71} +(541.701 + 312.751i) q^{73} +(953.157 - 378.377i) q^{77} -534.044 q^{79} +(352.414 - 610.399i) q^{83} +(-183.615 - 318.031i) q^{85} +(38.6630 + 66.9663i) q^{89} +(1353.12 - 537.150i) q^{91} -355.718i q^{95} +(382.120 + 220.617i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 6 q^{7}+O(q^{10}) $ 48 * q + 6 * q^7	$ 48 q + 6 q^{7} - 12 q^{11} - 36 q^{13} + 72 q^{17} - 30 q^{23} - 600 q^{25} - 42 q^{29} - 390 q^{35} + 84 q^{37} + 618 q^{41} - 42 q^{43} - 396 q^{47} + 318 q^{49} + 1620 q^{53} - 1500 q^{59} - 588 q^{67} + 2472 q^{77} + 1608 q^{79} - 360 q^{85} - 1722 q^{89} + 540 q^{91} - 792 q^{97}+O(q^{100}) $ 48 * q + 6 * q^7 - 12 * q^11 - 36 * q^13 + 72 * q^17 - 30 * q^23 - 600 * q^25 - 42 * q^29 - 390 * q^35 + 84 * q^37 + 618 * q^41 - 42 * q^43 - 396 * q^47 + 318 * q^49 + 1620 * q^53 - 1500 * q^59 - 588 * q^67 + 2472 * q^77 + 1608 * q^79 - 360 * q^85 - 1722 * q^89 + 540 * q^91 - 792 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$	−9.18977	+	15.9171i	−0.821958	+	1.42367i	0.0822649	+	0.996611i	$0.473785\pi$
$5$	−9.18977	+	15.9171i	−0.821958	+	1.42367i	−0.904222	+	0.427062i	$0.859549\pi$
$6$		0			0
$7$	18.3240	+	2.68897i	0.989404	+	0.145191i
$7$	18.3240	+	2.68897i	0.989404	+	0.145191i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	47.9540	−	27.6862i	1.31442	−	0.758883i	0.331599	−	0.943420i	$-0.392412\pi$
$11$	47.9540	−	27.6862i	1.31442	−	0.758883i	0.982826	+	0.184537i	$0.0590785\pi$
$12$		0			0
$13$	68.0763	−	39.3039i	1.45238	−	0.838534i	0.453766	−	0.891121i	$-0.350080\pi$
$13$	68.0763	−	39.3039i	1.45238	−	0.838534i	0.998616	+	0.0525871i	$0.0167467\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−9.99021	+	17.3036i	−0.142528	+	0.246866i	−0.928448	−	0.371462i	$-0.878857\pi$
$17$	−9.99021	+	17.3036i	−0.142528	+	0.246866i	0.785920	+	0.618329i	$0.212190\pi$
$18$		0			0
$19$	−16.7611	+	9.67700i	−0.202382	+	0.116845i	−0.597766	−	0.801671i	$-0.703945\pi$
$19$	−16.7611	+	9.67700i	−0.202382	+	0.116845i	0.395384	+	0.918516i	$0.370611\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−107.511	−	62.0716i	−0.974679	−	0.562731i	−0.0740195	−	0.997257i	$-0.523583\pi$
$23$	−107.511	−	62.0716i	−0.974679	−	0.562731i	−0.900659	+	0.434526i	$0.856916\pi$
$24$		0			0
$25$	−106.404	−	184.296i	−0.851229	−	1.47437i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	48.2851	+	27.8774i	0.309183	+	0.178507i	0.646561	−	0.762862i	$-0.276207\pi$
$29$	48.2851	+	27.8774i	0.309183	+	0.178507i	−0.337378	+	0.941369i	$0.609540\pi$
$30$		0			0
$31$		−	317.989i		−	1.84234i	−0.389160	−	0.921170i	$-0.627235\pi$
$31$		−	317.989i		−	1.84234i	0.389160	−	0.921170i	$-0.372765\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−211.194	+	266.955i	−1.01995	+	1.28925i
$36$		0			0
$37$	20.5303	+	35.5595i	0.0912205	+	0.157999i	0.908025	−	0.418916i	$-0.137590\pi$
$37$	20.5303	+	35.5595i	0.0912205	+	0.157999i	−0.816804	+	0.576915i	$0.804257\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	53.1315	+	92.0264i	0.202384	+	0.350539i	0.949296	−	0.314383i	$-0.101798\pi$
$41$	53.1315	+	92.0264i	0.202384	+	0.350539i	−0.746912	+	0.664923i	$0.768464\pi$
$42$		0			0
$43$	279.238	−	483.655i	0.990312	−	1.71527i	0.374898	−	0.927066i	$-0.377678\pi$
$43$	279.238	−	483.655i	0.990312	−	1.71527i	0.615414	−	0.788204i	$-0.288989\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−147.916			−0.459059			−0.229529	−	0.973302i	$-0.573719\pi$
$47$	−147.916			−0.459059			−0.229529	+	0.973302i	$0.573719\pi$
$48$		0			0
$49$	328.539	+	98.5453i	0.957839	+	0.287304i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	570.090	+	329.141i	1.47751	+	0.853038i	0.999677	−	0.0254144i	$-0.00809053\pi$
$53$	570.090	+	329.141i	1.47751	+	0.853038i	0.477829	+	0.878453i	$0.341424\pi$
$54$		0			0
$55$			1017.72i			2.49508i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	568.509			1.25447			0.627234	−	0.778831i	$-0.284187\pi$
$59$	568.509			1.25447			0.627234	+	0.778831i	$0.284187\pi$
$60$		0			0
$61$			140.951i			0.295852i	0.988998	+	0.147926i	$0.0472598\pi$
$61$			140.951i			0.295852i	−0.988998	+	0.147926i	$0.952740\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			1444.77i			2.75696i
$66$		0			0
$67$	−294.283			−0.536603			−0.268302	−	0.963335i	$-0.586462\pi$
$67$	−294.283			−0.536603			−0.268302	+	0.963335i	$0.586462\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	602.374i		−	1.00688i	−0.864030	−	0.503441i	$-0.832067\pi$
$71$		−	602.374i		−	1.00688i	0.864030	−	0.503441i	$-0.167933\pi$
$72$		0			0
$73$	541.701	+	312.751i	0.868510	+	0.501435i	0.866853	−	0.498564i	$-0.166139\pi$
$73$	541.701	+	312.751i	0.868510	+	0.501435i	0.00165750	+	0.999999i	$0.499472\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	953.157	−	378.377i	1.41068	−	0.560000i
$78$		0			0
$79$	−534.044			−0.760565			−0.380282	−	0.924870i	$-0.624173\pi$
$79$	−534.044			−0.760565			−0.380282	+	0.924870i	$0.624173\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	352.414	−	610.399i	0.466054	−	0.807229i	−0.533195	−	0.845993i	$-0.679009\pi$
$83$	352.414	−	610.399i	0.466054	−	0.807229i	0.999248	+	0.0387640i	$0.0123420\pi$
$84$		0			0
$85$	−183.615	−	318.031i	−0.234305	−	0.405827i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	38.6630	+	66.9663i	0.0460480	+	0.0797574i	0.888131	−	0.459591i	$-0.152004\pi$
$89$	38.6630	+	66.9663i	0.0460480	+	0.0797574i	−0.842083	+	0.539348i	$0.818671\pi$
$90$		0			0
$91$	1353.12	−	537.150i	1.55874	−	0.618776i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	355.718i		−	0.384167i
$96$		0			0
$97$	382.120	+	220.617i	0.399984	+	0.230931i	0.686477	−	0.727151i	$-0.259156\pi$
$97$	382.120	+	220.617i	0.399984	+	0.230931i	−0.286493	+	0.958082i	$0.592490\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	559.177	+	968.522i	0.550893	+	0.954174i	0.998210	+	0.0597989i	$0.0190460\pi$
$101$	559.177	+	968.522i	0.550893	+	0.954174i	−0.447318	+	0.894375i	$0.647621\pi$
$102$		0			0
$103$	705.623	+	407.392i	0.675020	+	0.389723i	0.797976	−	0.602689i	$-0.205904\pi$
$103$	705.623	+	407.392i	0.675020	+	0.389723i	−0.122956	+	0.992412i	$0.539237\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1320.94	+	762.645i	−1.19346	+	0.689044i	−0.959089	−	0.283103i	$-0.908636\pi$
$107$	−1320.94	+	762.645i	−1.19346	+	0.689044i	−0.234370	+	0.972147i	$0.575303\pi$
$108$		0			0
$109$	101.432	−	175.685i	0.0891319	−	0.154381i	−0.818013	−	0.575200i	$-0.804924\pi$
$109$	101.432	−	175.685i	0.0891319	−	0.154381i	0.907144	+	0.420819i	$0.138257\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	75.2101	−	43.4226i	0.0626121	−	0.0361491i	−0.468367	−	0.883534i	$-0.655158\pi$
$113$	75.2101	−	43.4226i	0.0626121	−	0.0361491i	0.530979	+	0.847385i	$0.321824\pi$
$114$		0			0
$115$	1976.00	−	1140.85i	1.60229	−	0.925082i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−229.589	+	290.207i	−0.176861	+	0.223557i
$120$		0			0
$121$	867.557	−	1502.65i	0.651808	−	1.12896i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1613.85			1.15478
$126$		0			0
$127$	−486.709			−0.340066			−0.170033	−	0.985438i	$-0.554388\pi$
$127$	−486.709			−0.340066			−0.170033	+	0.985438i	$0.554388\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−400.978	+	694.514i	−0.267432	+	0.463206i	−0.968198	−	0.250186i	$-0.919508\pi$
$131$	−400.978	+	694.514i	−0.267432	+	0.463206i	0.700766	+	0.713391i	$0.252842\pi$
$132$		0			0
$133$	−333.151	+	132.252i	−0.217202	+	0.0862231i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−795.730	+	459.415i	−0.496232	+	0.286500i	−0.727156	−	0.686472i	$-0.759158\pi$
$137$	−795.730	+	459.415i	−0.496232	+	0.286500i	0.230924	+	0.972972i	$0.425825\pi$
$138$		0			0
$139$	−159.738	+	92.2248i	−0.0974734	+	0.0562763i	−0.547944	−	0.836515i	$-0.684589\pi$
$139$	−159.738	+	92.2248i	−0.0974734	+	0.0562763i	0.450471	+	0.892791i	$0.351256\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2176.35	−	3769.56i	1.27270	−	2.20438i
$144$		0			0
$145$	−887.457	+	512.374i	−0.508271	+	0.293450i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1940.53	−	1120.36i	−1.06694	−	0.615998i	−0.139596	−	0.990209i	$-0.544580\pi$
$149$	−1940.53	−	1120.36i	−1.06694	−	0.615998i	−0.927344	+	0.374211i	$0.877914\pi$
$150$		0			0
$151$	1064.55	+	1843.85i	0.573721	+	0.993714i	0.996179	+	0.0873314i	$0.0278339\pi$
$151$	1064.55	+	1843.85i	0.573721	+	0.993714i	−0.422458	+	0.906382i	$0.638833\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5061.48	+	2922.25i	2.62289	+	1.51433i
$156$		0			0
$157$			2774.77i			1.41051i	0.708952	+	0.705256i	$0.249168\pi$
$157$			2774.77i			1.41051i	−0.708952	+	0.705256i	$0.750832\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1803.13	−	1426.49i	−0.882648	−	0.698282i
$162$		0			0
$163$	−536.860	−	929.870i	−0.257976	−	0.446828i	0.707723	−	0.706490i	$-0.249722\pi$
$163$	−536.860	−	929.870i	−0.257976	−	0.446828i	−0.965700	+	0.259661i	$0.916389\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	22.8325	+	39.5471i	0.0105798	+	0.0183248i	0.871267	−	0.490810i	$-0.163299\pi$
$167$	22.8325	+	39.5471i	0.0105798	+	0.0183248i	−0.860687	+	0.509134i	$0.829966\pi$
$168$		0			0
$169$	1991.09	−	3448.67i	0.906277	−	1.56972i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−909.083			−0.399516			−0.199758	−	0.979845i	$-0.564016\pi$
$173$	−909.083			−0.399516			−0.199758	+	0.979845i	$0.564016\pi$
$174$		0			0
$175$	−1454.17	−	3663.16i	−0.628144	−	1.58234i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3140.15	−	1812.96i	−1.31120	−	0.757024i	−0.328909	−	0.944362i	$-0.606681\pi$
$179$	−3140.15	−	1812.96i	−1.31120	−	0.757024i	−0.982296	+	0.187338i	$0.940014\pi$
$180$		0			0
$181$			1659.16i			0.681351i	0.940181	+	0.340676i	$0.110656\pi$
$181$			1659.16i			0.681351i	−0.940181	+	0.340676i	$0.889344\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−754.674			−0.299918
$186$		0			0
$187$			1106.37i			0.432650i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1540.32i			0.583527i	0.956491	+	0.291763i	$0.0942420\pi$
$191$			1540.32i			0.583527i	−0.956491	+	0.291763i	$0.905758\pi$
$192$		0			0
$193$	−2900.09			−1.08162			−0.540811	−	0.841144i	$-0.681883\pi$
$193$	−2900.09			−1.08162			−0.540811	+	0.841144i	$0.681883\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2096.85i			0.758347i	0.925326	+	0.379173i	$0.123792\pi$
$197$			2096.85i			0.758347i	−0.925326	+	0.379173i	$0.876208\pi$
$198$		0			0
$199$	3937.78	+	2273.48i	1.40272	+	0.809862i	0.994671	−	0.103097i	$-0.0328752\pi$
$199$	3937.78	+	2273.48i	1.40272	+	0.809862i	0.408051	+	0.912959i	$0.366209\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	809.815	+	640.663i	0.279990	+	0.221506i
$204$		0			0
$205$	−1953.06			−0.665404
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−535.840	+	928.102i	−0.177344	+	0.307168i
$210$		0			0
$211$	1506.32	+	2609.03i	0.491467	+	0.851245i	0.999952	−	0.00982544i	$-0.00312758\pi$
$211$	1506.32	+	2609.03i	0.491467	+	0.851245i	−0.508485	+	0.861071i	$0.669794\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5132.26	+	8889.34i	1.62799	+	2.81976i
$216$		0			0
$217$	855.062	−	5826.84i	0.267490	−	1.82282i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			1570.62i			0.478059i
$222$		0			0
$223$	230.173	+	132.890i	0.0691188	+	0.0399058i	0.534161	−	0.845383i	$-0.320628\pi$
$223$	230.173	+	132.890i	0.0691188	+	0.0399058i	−0.465042	+	0.885288i	$0.653961\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	579.422	+	1003.59i	0.169417	+	0.293439i	0.938215	−	0.346053i	$-0.112478\pi$
$227$	579.422	+	1003.59i	0.169417	+	0.293439i	−0.768798	+	0.639492i	$0.779145\pi$
$228$		0			0
$229$	989.740	+	571.427i	0.285606	+	0.164895i	0.635959	−	0.771723i	$-0.280605\pi$
$229$	989.740	+	571.427i	0.285606	+	0.164895i	−0.350352	+	0.936618i	$0.613938\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4943.19	−	2853.95i	1.38987	−	0.802440i	0.396567	−	0.918006i	$-0.370201\pi$
$233$	4943.19	−	2853.95i	1.38987	−	0.802440i	0.993300	+	0.115565i	$0.0368680\pi$
$234$		0			0
$235$	1359.31	−	2354.40i	0.377327	−	0.653549i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−310.843	+	179.466i	−0.0841288	+	0.0485718i	−0.541474	−	0.840717i	$-0.682134\pi$
$239$	−310.843	+	179.466i	−0.0841288	+	0.0485718i	0.457345	+	0.889289i	$0.348800\pi$
$240$		0			0
$241$	3845.00	−	2219.91i	1.02771	−	0.593349i	0.111382	−	0.993778i	$-0.464472\pi$
$241$	3845.00	−	2219.91i	1.02771	−	0.593349i	0.916328	+	0.400429i	$0.131139\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4587.75	+	4323.79i	−1.19633	+	1.12750i
$246$		0			0
$247$	−760.688	+	1317.55i	−0.195957	+	0.339408i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3438.88			0.864782			0.432391	−	0.901686i	$-0.357670\pi$
$251$	3438.88			0.864782			0.432391	+	0.901686i	$0.357670\pi$
$252$		0			0
$253$	−6874.12			−1.70819
$254$		0			0
$255$		0			0
$256$		0			0
$257$	910.835	−	1577.61i	0.221075	−	0.382913i	−0.734060	−	0.679085i	$-0.762377\pi$
$257$	910.835	−	1577.61i	0.221075	−	0.382913i	0.955135	+	0.296172i	$0.0957101\pi$
$258$		0			0
$259$	280.579	+	706.798i	0.0673140	+	0.169569i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3674.68	−	2121.58i	0.861561	−	0.497422i	−0.00297373	−	0.999996i	$-0.500947\pi$
$263$	3674.68	−	2121.58i	0.861561	−	0.497422i	0.864535	+	0.502573i	$0.167613\pi$
$264$		0			0
$265$	−10478.0	+	6049.46i	−2.42889	+	1.40232i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1538.72	−	2665.14i	0.348763	−	0.604075i	−0.637267	−	0.770643i	$-0.719935\pi$
$269$	1538.72	−	2665.14i	0.348763	−	0.604075i	0.986030	+	0.166568i	$0.0532685\pi$
$270$		0			0
$271$	−623.012	+	359.696i	−0.139651	+	0.0806273i	−0.568197	−	0.822892i	$-0.692359\pi$
$271$	−623.012	+	359.696i	−0.139651	+	0.0806273i	0.428547	+	0.903520i	$0.359026\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10205.0	−	5891.83i	−2.23775	−	1.29197i
$276$		0			0
$277$	455.590	+	789.105i	0.0988221	+	0.171165i	0.911197	−	0.411970i	$-0.135159\pi$
$277$	455.590	+	789.105i	0.0988221	+	0.171165i	−0.812375	+	0.583135i	$0.801826\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4142.21	−	2391.50i	−0.879372	−	0.507705i	−0.00892043	−	0.999960i	$-0.502839\pi$
$281$	−4142.21	−	2391.50i	−0.879372	−	0.507705i	−0.870451	+	0.492255i	$0.836173\pi$
$282$		0			0
$283$		−	4008.65i		−	0.842013i	−0.907058	−	0.421006i	$-0.861677\pi$
$283$		−	4008.65i		−	0.842013i	0.907058	−	0.421006i	$-0.138323\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	726.126	+	1829.16i	0.149344	+	0.376209i
$288$		0			0
$289$	2256.89	+	3909.05i	0.459371	+	0.795654i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3758.85	−	6510.51i	−0.749468	−	1.29812i	−0.948078	−	0.318038i	$-0.896976\pi$
$293$	−3758.85	−	6510.51i	−0.749468	−	1.29812i	0.198610	−	0.980079i	$-0.436357\pi$
$294$		0			0
$295$	−5224.47	+	9049.04i	−1.03112	+	1.78595i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9758.62			−1.88748
$300$		0			0
$301$	6417.29	−	8111.63i	1.22886	−	1.55331i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2243.54	−	1295.31i	−0.421196	−	0.243178i
$306$		0			0
$307$		−	6149.80i		−	1.14328i	−0.820504	−	0.571641i	$-0.806307\pi$
$307$		−	6149.80i		−	1.14328i	0.820504	−	0.571641i	$-0.193693\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2119.34			−0.386421			−0.193211	−	0.981157i	$-0.561890\pi$
$311$	−2119.34			−0.386421			−0.193211	+	0.981157i	$0.561890\pi$
$312$		0			0
$313$			145.759i			0.0263220i	0.999913	+	0.0131610i	$0.00418939\pi$
$313$			145.759i			0.0263220i	−0.999913	+	0.0131610i	$0.995811\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2607.15i		−	0.461931i	−0.972962	−	0.230966i	$-0.925811\pi$
$317$		−	2607.15i		−	0.461931i	0.972962	−	0.230966i	$-0.0741885\pi$
$318$		0			0
$319$	3087.28			0.541864
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	386.701i		−	0.0666150i
$324$		0			0
$325$	−14487.1	−	8364.15i	−2.47262	−	1.42757i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2710.41	−	397.741i	−0.454194	−	0.0666510i
$330$		0			0
$331$	−10890.0			−1.80837			−0.904185	−	0.427142i	$-0.859521\pi$
$331$	−10890.0			−1.80837			−0.904185	+	0.427142i	$0.859521\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2704.39	−	4684.15i	0.441065	−	0.763947i
$336$		0			0
$337$	1171.50	+	2029.09i	0.189364	+	0.327987i	0.945038	−	0.326960i	$-0.106024\pi$
$337$	1171.50	+	2029.09i	0.189364	+	0.327987i	−0.755675	+	0.654947i	$0.772691\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8803.93	−	15248.8i	−1.39812	−	2.42162i
$342$		0			0
$343$	5755.17	+	2689.18i	0.905976	+	0.423329i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	3903.65i		−	0.603916i	−0.953321	−	0.301958i	$-0.902360\pi$
$347$		−	3903.65i		−	0.603916i	0.953321	−	0.301958i	$-0.0976403\pi$
$348$		0			0
$349$	1985.23	+	1146.17i	0.304490	+	0.175797i	0.644458	−	0.764640i	$-0.277083\pi$
$349$	1985.23	+	1146.17i	0.304490	+	0.175797i	−0.339968	+	0.940437i	$0.610416\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3872.95	−	6708.15i	−0.583956	−	1.01144i	−0.995005	−	0.0998278i	$-0.968171\pi$
$353$	−3872.95	−	6708.15i	−0.583956	−	1.01144i	0.411049	−	0.911613i	$-0.365163\pi$
$354$		0			0
$355$	9588.07	+	5535.67i	1.43347	+	0.827614i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5235.74	−	3022.86i	0.769727	−	0.444402i	−0.0630502	−	0.998010i	$-0.520083\pi$
$359$	5235.74	−	3022.86i	0.769727	−	0.444402i	0.832777	+	0.553608i	$0.186749\pi$
$360$		0			0
$361$	−3242.21	+	5615.67i	−0.472694	+	0.818731i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9956.20	+	5748.22i	−1.42776	+	0.824316i
$366$		0			0
$367$	−9468.66	+	5466.74i	−1.34676	+	0.777551i	−0.987789	−	0.155798i	$-0.950205\pi$
$367$	−9468.66	+	5466.74i	−1.34676	+	0.777551i	−0.358969	+	0.933349i	$0.616872\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9561.28	+	7564.14i	1.33800	+	1.05852i
$372$		0			0
$373$	−1432.56	+	2481.26i	−0.198861	+	0.344437i	−0.948159	−	0.317795i	$-0.897058\pi$
$373$	−1432.56	+	2481.26i	−0.198861	+	0.344437i	0.749298	+	0.662232i	$0.230391\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4382.76			0.598737
$378$		0			0
$379$	−2071.78			−0.280792			−0.140396	−	0.990095i	$-0.544838\pi$
$379$	−2071.78			−0.280792			−0.140396	+	0.990095i	$0.544838\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3714.00	+	6432.84i	−0.495500	+	0.858232i	−0.999987	−	0.00518791i	$-0.998349\pi$
$383$	−3714.00	+	6432.84i	−0.495500	+	0.858232i	0.504486	+	0.863420i	$0.331682\pi$
$384$		0			0
$385$	−2736.62	+	18648.7i	−0.362262	+	2.46864i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6567.95	+	3792.01i	−0.856062	+	0.494248i	−0.862692	−	0.505730i	$-0.831223\pi$
$389$	−6567.95	+	3792.01i	−0.856062	+	0.494248i	0.00662941	+	0.999978i	$0.497890\pi$
$390$		0			0
$391$	2148.12	−	1240.22i	0.277839	−	0.160410i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4907.74	−	8500.45i	0.625152	−	1.08279i
$396$		0			0
$397$	1031.73	−	595.671i	0.130431	−	0.0753044i	−0.433365	−	0.901219i	$-0.642674\pi$
$397$	1031.73	−	595.671i	0.130431	−	0.0753044i	0.563796	+	0.825914i	$0.309340\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5591.97	+	3228.52i	0.696383	+	0.402057i	0.805999	−	0.591917i	$-0.201629\pi$
$401$	5591.97	+	3228.52i	0.696383	+	0.402057i	−0.109616	+	0.993974i	$0.534962\pi$
$402$		0			0
$403$	−12498.2	−	21647.5i	−1.54486	−	2.67578i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1969.02	+	1136.81i	0.239805	+	0.138451i
$408$		0			0
$409$		−	1896.49i		−	0.229280i	−0.993407	−	0.114640i	$-0.963429\pi$
$409$		−	1896.49i		−	0.229280i	0.993407	−	0.114640i	$-0.0365715\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	10417.4	+	1528.70i	1.24117	+	0.182137i
$414$		0			0
$415$	6477.20	+	11218.8i	0.766153	+	1.32702i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7002.37	−	12128.5i	−0.816440	−	1.41412i	−0.908289	−	0.418342i	$-0.862611\pi$
$419$	−7002.37	−	12128.5i	−0.816440	−	1.41412i	0.0918495	−	0.995773i	$-0.470722\pi$
$420$		0			0
$421$	4485.43	−	7769.00i	0.519256	−	0.899377i	−0.480494	−	0.876998i	$-0.659543\pi$
$421$	4485.43	−	7769.00i	0.519256	−	0.899377i	0.999750	−	0.0223789i	$-0.00712403\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4251.98			0.485297
$426$		0			0
$427$	−379.014	+	2582.80i	−0.0429549	+	0.292717i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8722.70	+	5036.05i	0.974844	+	0.562826i	0.900709	−	0.434422i	$-0.143047\pi$
$431$	8722.70	+	5036.05i	0.974844	+	0.562826i	0.0741343	+	0.997248i	$0.476381\pi$
$432$		0			0
$433$			10574.2i			1.17358i	0.809738	+	0.586791i	$0.199609\pi$
$433$			10574.2i			1.17358i	−0.809738	+	0.586791i	$0.800391\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2402.67			0.263010
$438$		0			0
$439$		−	9055.68i		−	0.984520i	−0.870448	−	0.492260i	$-0.836171\pi$
$439$		−	9055.68i		−	0.984520i	0.870448	−	0.492260i	$-0.163829\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			3546.64i			0.380374i	0.981748	+	0.190187i	$0.0609095\pi$
$443$			3546.64i			0.380374i	−0.981748	+	0.190187i	$0.939091\pi$
$444$		0			0
$445$	−1421.22			−0.151398
$446$		0			0
$447$		0			0
$448$		0			0
$449$			200.727i			0.0210977i	0.999944	+	0.0105489i	$0.00335787\pi$
$449$			200.727i			0.0210977i	−0.999944	+	0.0105489i	$0.996642\pi$
$450$		0			0
$451$	5095.73	+	2942.02i	0.532037	+	0.307172i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3884.95	+	26474.1i	−0.400284	+	2.72774i
$456$		0			0
$457$	−8584.04			−0.878654			−0.439327	−	0.898327i	$-0.644783\pi$
$457$	−8584.04			−0.878654			−0.439327	+	0.898327i	$0.644783\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−5807.63	+	10059.1i	−0.586742	+	1.01627i	0.407913	+	0.913021i	$0.366257\pi$
$461$	−5807.63	+	10059.1i	−0.586742	+	1.01627i	−0.994656	+	0.103247i	$0.967077\pi$
$462$		0			0
$463$	5426.15	+	9398.36i	0.544653	+	0.943367i	0.998629	+	0.0523527i	$0.0166720\pi$
$463$	5426.15	+	9398.36i	0.544653	+	0.943367i	−0.453976	+	0.891014i	$0.649995\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3071.32	−	5319.68i	−0.304333	−	0.527120i	0.672780	−	0.739843i	$-0.265100\pi$
$467$	−3071.32	−	5319.68i	−0.304333	−	0.527120i	−0.977113	+	0.212723i	$0.931767\pi$
$468$		0			0
$469$	−5392.45	−	791.317i	−0.530917	−	0.0779097i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	30924.2i		−	3.00612i
$474$		0			0
$475$	3566.87	+	2059.34i	0.344546	+	0.198924i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7606.81	+	13175.4i	0.725603	+	1.25678i	0.958725	+	0.284334i	$0.0917726\pi$
$479$	7606.81	+	13175.4i	0.725603	+	1.25678i	−0.233122	+	0.972448i	$0.574894\pi$
$480$		0			0
$481$	2795.25	+	1613.84i	0.264974	+	0.152983i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7023.19	+	4054.84i	−0.657540	+	0.379631i
$486$		0			0
$487$	−780.525	+	1351.91i	−0.0726262	+	0.125792i	−0.900052	−	0.435783i	$-0.856471\pi$
$487$	−780.525	+	1351.91i	−0.0726262	+	0.125792i	0.827425	+	0.561576i	$0.189805\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5107.22	+	2948.66i	−0.469421	+	0.271020i	−0.715997	−	0.698103i	$-0.754028\pi$
$491$	−5107.22	+	2948.66i	−0.469421	+	0.271020i	0.246576	+	0.969123i	$0.420694\pi$
$492$		0			0
$493$	−964.757	+	557.002i	−0.0881348	+	0.0508846i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1619.76	−	11037.9i	0.146190	−	0.996213i
$498$		0			0
$499$	5845.81	−	10125.2i	0.524438	−	0.908353i	−0.475158	−	0.879901i	$-0.657609\pi$
$499$	5845.81	−	10125.2i	0.524438	−	0.908353i	0.999595	−	0.0284519i	$-0.00905774\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2180.14			−0.193255			−0.0966277	−	0.995321i	$-0.530806\pi$
$503$	−2180.14			−0.193255			−0.0966277	+	0.995321i	$0.530806\pi$
$504$		0			0
$505$	−20554.8			−1.81124
$506$		0			0
$507$		0			0
$508$		0			0
$509$	797.527	−	1381.36i	0.0694494	−	0.120290i	−0.829210	−	0.558938i	$-0.811209\pi$
$509$	797.527	−	1381.36i	0.0694494	−	0.120290i	0.898659	+	0.438648i	$0.144542\pi$
$510$		0			0
$511$	9085.15	+	7187.47i	0.786504	+	0.622221i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12969.0	+	7487.67i	−1.10968	+	0.640672i
$516$		0			0
$517$	−7093.16	+	4095.24i	−0.603398	+	0.348372i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2974.63	+	5152.20i	−0.250136	+	0.433248i	−0.963563	−	0.267481i	$-0.913809\pi$
$521$	−2974.63	+	5152.20i	−0.250136	+	0.433248i	0.713427	+	0.700729i	$0.247142\pi$
$522$		0			0
$523$	12106.0	−	6989.39i	1.01216	−	0.584369i	0.100334	−	0.994954i	$-0.468009\pi$
$523$	12106.0	−	6989.39i	1.01216	−	0.584369i	0.911822	+	0.410585i	$0.134675\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5502.34	+	3176.78i	0.454812	+	0.262586i
$528$		0			0
$529$	1622.26	+	2809.84i	0.133333	+	0.230939i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7233.99	+	4176.55i	0.587878	+	0.339412i
$534$		0			0
$535$		−	28034.1i		−	2.26546i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18483.1	−	4370.37i	1.47704	−	0.349249i
$540$		0			0
$541$	6174.78	+	10695.0i	0.490710	+	0.849935i	0.999943	−	0.0106936i	$-0.00340395\pi$
$541$	6174.78	+	10695.0i	0.490710	+	0.849935i	−0.509232	+	0.860629i	$0.670071\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1864.26	+	3229.00i	0.146525	+	0.253789i
$546$		0			0
$547$	8771.44	−	15192.6i	0.685630	−	1.18755i	−0.287609	−	0.957748i	$-0.592860\pi$
$547$	8771.44	−	15192.6i	0.685630	−	1.18755i	0.973238	−	0.229798i	$-0.0738064\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1079.08			−0.0834307
$552$		0			0
$553$	−9785.82	−	1436.03i	−0.752506	−	0.110427i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19166.8	−	11065.9i	−1.45803	−	0.841794i	−0.459116	−	0.888376i	$-0.651834\pi$
$557$	−19166.8	−	11065.9i	−1.45803	−	0.841794i	−0.998914	+	0.0465824i	$0.985167\pi$
$558$		0			0
$559$		−	43900.6i		−	3.32164i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9616.20			−0.719848			−0.359924	−	0.932982i	$-0.617197\pi$
$563$	−9616.20			−0.719848			−0.359924	+	0.932982i	$0.617197\pi$
$564$		0			0
$565$			1596.17i			0.118852i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	11413.1i		−	0.840881i	−0.907320	−	0.420441i	$-0.861875\pi$
$569$		−	11413.1i		−	0.840881i	0.907320	−	0.420441i	$-0.138125\pi$
$570$		0			0
$571$	−19183.8			−1.40599			−0.702993	−	0.711197i	$-0.748153\pi$
$571$	−19183.8			−1.40599			−0.702993	+	0.711197i	$0.748153\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			26418.5i			1.91605i
$576$		0			0
$577$	9697.32	+	5598.75i	0.699661	+	0.403950i	0.807221	−	0.590249i	$-0.200970\pi$
$577$	9697.32	+	5598.75i	0.699661	+	0.403950i	−0.107560	+	0.994199i	$0.534304\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8098.98	−	10237.3i	0.578317	−	0.731008i
$582$		0			0
$583$	36450.8			2.58943
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5285.80	−	9155.27i	0.371666	−	0.643745i	−0.618156	−	0.786056i	$-0.712120\pi$
$587$	5285.80	−	9155.27i	0.371666	−	0.643745i	0.989822	+	0.142311i	$0.0454532\pi$
$588$		0			0
$589$	3077.18	+	5329.84i	0.215268	+	0.372856i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7667.36	+	13280.3i	0.530963	+	0.919655i	0.999347	+	0.0361297i	$0.0115029\pi$
$593$	7667.36	+	13280.3i	0.530963	+	0.919655i	−0.468384	+	0.883525i	$0.655164\pi$
$594$		0			0
$595$	−2509.40	−	6321.34i	−0.172900	−	0.435546i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	8922.83i		−	0.608643i	−0.952569	−	0.304321i	$-0.901570\pi$
$599$		−	8922.83i		−	0.608643i	0.952569	−	0.304321i	$-0.0984297\pi$
$600$		0			0
$601$	6221.25	+	3591.84i	0.422246	+	0.243784i	0.696038	−	0.718005i	$-0.254945\pi$
$601$	6221.25	+	3591.84i	0.422246	+	0.243784i	−0.273792	+	0.961789i	$0.588278\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15945.3	+	27618.0i	1.07152	+	1.85592i
$606$		0			0
$607$	−4773.64	−	2756.06i	−0.319203	−	0.184292i	0.331834	−	0.943338i	$-0.392332\pi$
$607$	−4773.64	−	2756.06i	−0.319203	−	0.184292i	−0.651037	+	0.759046i	$0.725666\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10069.6	+	5813.67i	−0.666729	+	0.384936i
$612$		0			0
$613$	−12611.1	+	21843.0i	−0.830925	+	1.43920i	0.0663806	+	0.997794i	$0.478855\pi$
$613$	−12611.1	+	21843.0i	−0.830925	+	1.43920i	−0.897306	+	0.441410i	$0.854478\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22013.2	+	12709.3i	−1.43633	+	0.829268i	−0.997593	−	0.0693457i	$-0.977909\pi$
$617$	−22013.2	+	12709.3i	−1.43633	+	0.829268i	−0.438741	+	0.898613i	$0.644576\pi$
$618$		0			0
$619$	9378.16	−	5414.48i	0.608950	−	0.351577i	−0.163604	−	0.986526i	$-0.552312\pi$
$619$	9378.16	−	5414.48i	0.608950	−	0.351577i	0.772554	+	0.634949i	$0.218979\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	528.391	+	1331.05i	0.0339800	+	0.0855980i
$624$		0			0
$625$	−1530.49	+	2650.89i	−0.0979515	+	0.169657i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−820.408			−0.0520061
$630$		0			0
$631$	25750.8			1.62460			0.812301	−	0.583239i	$-0.198215\pi$
$631$	25750.8			1.62460			0.812301	+	0.583239i	$0.198215\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4472.74	−	7747.02i	0.279520	−	0.484143i
$636$		0			0
$637$	26238.9	−	6204.25i	1.63206	−	0.385905i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6561.40	−	3788.22i	0.404305	−	0.233426i	−0.284035	−	0.958814i	$-0.591673\pi$
$641$	6561.40	−	3788.22i	0.404305	−	0.233426i	0.688340	+	0.725388i	$0.258340\pi$
$642$		0			0
$643$	−15581.8	+	8996.16i	−0.955655	+	0.551748i	−0.894833	−	0.446401i	$-0.852706\pi$
$643$	−15581.8	+	8996.16i	−0.955655	+	0.551748i	−0.0608220	+	0.998149i	$0.519372\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8457.22	−	14648.3i	0.513891	−	0.890086i	−0.485979	−	0.873971i	$-0.661537\pi$
$647$	8457.22	−	14648.3i	0.513891	−	0.890086i	0.999870	−	0.0161153i	$-0.00512987\pi$
$648$		0			0
$649$	27262.3	−	15739.9i	1.64890	−	0.951995i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	15111.6	+	8724.69i	0.905609	+	0.522854i	0.879016	−	0.476792i	$-0.158201\pi$
$653$	15111.6	+	8724.69i	0.905609	+	0.522854i	0.0265935	+	0.999646i	$0.491534\pi$
$654$		0			0
$655$	−7369.78	−	12764.8i	−0.439635	−	0.761471i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1730.83	+	999.298i	0.102312	+	0.0590700i	0.550283	−	0.834978i	$-0.314520\pi$
$659$	1730.83	+	999.298i	0.102312	+	0.0590700i	−0.447971	+	0.894048i	$0.647853\pi$
$660$		0			0
$661$		−	33493.6i		−	1.97088i	−0.170023	−	0.985440i	$-0.554384\pi$
$661$		−	33493.6i		−	1.97088i	0.170023	−	0.985440i	$-0.445616\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	956.513	−	6518.17i	0.0557774	−	0.380096i
$666$		0			0
$667$	−3460.79	−	5994.26i	−0.200903	−	0.347974i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3902.42	+	6759.18i	0.224517	+	0.388875i
$672$		0			0
$673$	−16342.6	+	28306.3i	−0.936051	+	1.62129i	−0.163303	+	0.986576i	$0.552215\pi$
$673$	−16342.6	+	28306.3i	−0.936051	+	1.62129i	−0.772749	+	0.634712i	$0.781119\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7327.73			0.415993			0.207997	−	0.978130i	$-0.433306\pi$
$677$	7327.73			0.415993			0.207997	+	0.978130i	$0.433306\pi$
$678$		0			0
$679$	6408.75	+	5070.10i	0.362217	+	0.286558i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12311.2	+	7107.88i	0.689715	+	0.398207i	0.803505	−	0.595298i	$-0.202966\pi$
$683$	12311.2	+	7107.88i	0.689715	+	0.398207i	−0.113790	+	0.993505i	$0.536299\pi$
$684$		0			0
$685$		−	16887.7i		−	0.941962i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	51746.1			2.86121
$690$		0			0
$691$			14322.3i			0.788492i	0.919005	+	0.394246i	$0.128994\pi$
$691$			14322.3i			0.788492i	−0.919005	+	0.394246i	$0.871006\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	3390.10i		−	0.185027i
$696$		0			0
$697$	−2123.18			−0.115382
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	3371.89i		−	0.181675i	−0.995866	−	0.0908377i	$-0.971046\pi$
$701$		−	3371.89i		−	0.181675i	0.995866	−	0.0908377i	$-0.0289544\pi$
$702$		0			0
$703$	−688.219	−	397.344i	−0.0369227	−	0.0213173i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7642.04	+	19250.8i	0.406518	+	1.02405i
$708$		0			0
$709$	−30716.8			−1.62707			−0.813537	−	0.581514i	$-0.802461\pi$
$709$	−30716.8			−1.62707			−0.813537	+	0.581514i	$0.802461\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−19738.1	+	34187.4i	−1.03674	+	1.79569i
$714$		0			0
$715$	40000.4	+	69282.7i	2.09221	+	3.62381i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5218.17	−	9038.14i	−0.270661	−	0.468798i	0.698371	−	0.715736i	$-0.253909\pi$
$719$	−5218.17	−	9038.14i	−0.270661	−	0.468798i	−0.969031	+	0.246939i	$0.920575\pi$
$720$		0			0
$721$	11834.4	+	9362.45i	0.611284	+	0.483600i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	11865.0i		−	0.607801i
$726$		0			0
$727$	22053.0	+	12732.3i	1.12503	+	0.649538i	0.942681	−	0.333696i	$-0.108296\pi$
$727$	22053.0	+	12732.3i	1.12503	+	0.649538i	0.182352	+	0.983233i	$0.441629\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5579.30	+	9663.62i	0.282295	+	0.488949i
$732$		0			0
$733$	−8899.02	−	5137.85i	−0.448421	−	0.258896i	0.258742	−	0.965947i	$-0.416692\pi$
$733$	−8899.02	−	5137.85i	−0.448421	−	0.258896i	−0.707163	+	0.707050i	$0.750025\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−14112.1	+	8147.60i	−0.705324	+	0.407219i
$738$		0			0
$739$	−8232.96	+	14259.9i	−0.409816	+	0.709823i	−0.994869	−	0.101173i	$-0.967741\pi$
$739$	−8232.96	+	14259.9i	−0.409816	+	0.709823i	0.585053	+	0.810995i	$0.301074\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	34426.3	−	19876.0i	1.69984	−	0.981400i	0.753932	−	0.656953i	$-0.228155\pi$
$743$	34426.3	−	19876.0i	1.69984	−	0.981400i	0.945904	−	0.324448i	$-0.105178\pi$
$744$		0			0
$745$	35665.9	−	20591.7i	1.75396	−	1.01265i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−26255.7	+	10422.8i	−1.28086	+	0.508464i
$750$		0			0
$751$	−9238.85	+	16002.2i	−0.448909	+	0.777533i	−0.998315	−	0.0580222i	$-0.981521\pi$
$751$	−9238.85	+	16002.2i	−0.448909	+	0.777533i	0.549406	+	0.835555i	$0.314854\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−39131.9			−1.88630
$756$		0			0
$757$	14830.5			0.712053			0.356026	−	0.934476i	$-0.384131\pi$
$757$	14830.5			0.712053			0.356026	+	0.934476i	$0.384131\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2987.19	−	5173.96i	0.142294	−	0.246460i	−0.786066	−	0.618142i	$-0.787886\pi$
$761$	2987.19	−	5173.96i	0.142294	−	0.246460i	0.928360	+	0.371682i	$0.121219\pi$
$762$		0			0
$763$	2331.04	−	2946.50i	0.110602	−	0.139804i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	38702.0	−	22344.6i	1.82197	−	1.05191i
$768$		0			0
$769$	−16782.8	+	9689.54i	−0.786999	+	0.454374i	−0.838905	−	0.544278i	$-0.816804\pi$
$769$	−16782.8	+	9689.54i	−0.786999	+	0.454374i	0.0519057	+	0.998652i	$0.483470\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8932.67	−	15471.8i	0.415635	−	0.719900i	−0.579860	−	0.814716i	$-0.696893\pi$
$773$	8932.67	−	15471.8i	0.415635	−	0.719900i	0.995495	+	0.0948156i	$0.0302261\pi$
$774$		0			0
$775$	−58604.3	+	33835.2i	−2.71629	+	1.56825i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1781.08	−	1028.31i	−0.0819176	−	0.0472952i
$780$		0			0
$781$	−16677.5	−	28886.2i	−0.764106	−	1.32347i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−44166.3	−	25499.5i	−2.00811	−	1.15938i
$786$		0			0
$787$		−	12028.3i		−	0.544806i	−0.962183	−	0.272403i	$-0.912182\pi$
$787$		−	12028.3i		−	0.544806i	0.962183	−	0.272403i	$-0.0878184\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1494.91	−	593.438i	0.0671972	−	0.0266754i
$792$		0			0
$793$	5539.94	+	9595.45i	0.248082	+	0.429691i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13117.1	+	22719.5i	0.582976	+	1.00974i	0.995124	+	0.0986270i	$0.0314451\pi$
$797$	13117.1	+	22719.5i	0.582976	+	1.00974i	−0.412149	+	0.911117i	$0.635222\pi$
$798$		0			0
$799$	1477.71	−	2559.47i	0.0654289	−	0.113326i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	34635.6			1.52212
$804$		0			0
$805$	39276.0	−	15591.5i	1.71962	−	0.682643i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−24743.3	−	14285.5i	−1.07531	−	0.620832i	−0.145684	−	0.989331i	$-0.546538\pi$
$809$	−24743.3	−	14285.5i	−1.07531	−	0.620832i	−0.929628	+	0.368499i	$0.879872\pi$
$810$		0			0
$811$			36790.7i			1.59297i	0.604661	+	0.796483i	$0.293308\pi$
$811$			36790.7i			1.59297i	−0.604661	+	0.796483i	$0.706692\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	19734.5			0.848183
$816$		0			0
$817$			10808.8i			0.462852i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	27440.8i		−	1.16649i	−0.812295	−	0.583246i	$-0.801782\pi$
$821$		−	27440.8i		−	1.16649i	0.812295	−	0.583246i	$-0.198218\pi$
$822$		0			0
$823$	−29435.0			−1.24671			−0.623354	−	0.781940i	$-0.714230\pi$
$823$	−29435.0			−1.24671			−0.623354	+	0.781940i	$0.714230\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	9108.48i		−	0.382990i	−0.981494	−	0.191495i	$-0.938666\pi$
$827$		−	9108.48i		−	0.382990i	0.981494	−	0.191495i	$-0.0613336\pi$
$828$		0			0
$829$	745.006	+	430.129i	0.0312124	+	0.0180205i	0.515525	−	0.856875i	$-0.327597\pi$
$829$	745.006	+	430.129i	0.0312124	+	0.0180205i	−0.484313	+	0.874895i	$0.660930\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4987.36	+	4700.40i	−0.207445	+	0.195509i
$834$		0			0
$835$	−839.301			−0.0347847
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13869.3	+	24022.3i	−0.570704	+	0.988488i	0.425790	+	0.904822i	$0.359996\pi$
$839$	−13869.3	+	24022.3i	−0.570704	+	0.988488i	−0.996494	+	0.0836658i	$0.973337\pi$
$840$		0			0
$841$	−10640.2	−	18429.4i	−0.436270	−	0.755643i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	36595.3	+	63384.9i	1.48984	+	2.58048i
$846$		0			0
$847$	19937.7	−	25201.8i	0.808816	−	1.02237i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	5097.39i		−	0.205331i
$852$		0			0
$853$	−13605.1	−	7854.94i	−0.546109	−	0.315296i	0.201442	−	0.979500i	$-0.435437\pi$
$853$	−13605.1	−	7854.94i	−0.546109	−	0.315296i	−0.747551	+	0.664204i	$0.768771\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20532.6	−	35563.5i	−0.818414	−	1.41754i	−0.906850	−	0.421453i	$-0.861520\pi$
$857$	−20532.6	−	35563.5i	−0.818414	−	1.41754i	0.0884360	−	0.996082i	$-0.471813\pi$
$858$		0			0
$859$	−3324.84	−	1919.60i	−0.132063	−	0.0762466i	0.432513	−	0.901628i	$-0.357627\pi$
$859$	−3324.84	−	1919.60i	−0.132063	−	0.0762466i	−0.564576	+	0.825381i	$0.690960\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−24681.4	+	14249.8i	−0.973540	+	0.562074i	−0.900314	−	0.435242i	$-0.856663\pi$
$863$	−24681.4	+	14249.8i	−0.973540	+	0.562074i	−0.0732263	+	0.997315i	$0.523330\pi$
$864$		0			0
$865$	8354.26	−	14470.0i	0.328385	−	0.568780i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−25609.5	+	14785.7i	−0.999705	+	0.577180i
$870$		0			0
$871$	−20033.7	+	11566.5i	−0.779353	+	0.449960i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	29572.3	+	4339.60i	1.14254	+	0.167663i
$876$		0			0
$877$	2004.33	−	3471.60i	0.0771737	−	0.133669i	−0.824856	−	0.565343i	$-0.808744\pi$
$877$	2004.33	−	3471.60i	0.0771737	−	0.133669i	0.902029	+	0.431674i	$0.142077\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	20416.0			0.780742			0.390371	−	0.920658i	$-0.372347\pi$
$881$	20416.0			0.780742			0.390371	+	0.920658i	$0.372347\pi$
$882$		0			0
$883$	16822.4			0.641133			0.320566	−	0.947226i	$-0.396127\pi$
$883$	16822.4			0.641133			0.320566	+	0.947226i	$0.396127\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−15727.6	+	27241.1i	−0.595358	+	1.03119i	0.398138	+	0.917325i	$0.369656\pi$
$887$	−15727.6	+	27241.1i	−0.595358	+	1.03119i	−0.993496	+	0.113865i	$0.963677\pi$
$888$		0			0
$889$	−8918.46	−	1308.74i	−0.336463	−	0.0493744i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2479.23	−	1431.38i	0.0929051	−	0.0536388i
$894$		0			0
$895$	57714.4	−	33321.4i	2.15551	−	1.24448i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8864.71	−	15354.1i	0.328871	−	0.569621i
$900$		0			0
$901$	−11390.6	+	6576.38i	−0.421173	+	0.243164i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−26409.1	−	15247.3i	−0.970021	−	0.560042i
$906$		0			0
$907$	108.041	+	187.133i	0.00395529	+	0.00685076i	0.867996	−	0.496571i	$-0.165408\pi$
$907$	108.041	+	187.133i	0.00395529	+	0.00685076i	−0.864041	+	0.503421i	$0.832074\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9169.28	−	5293.88i	−0.333471	−	0.192529i	0.323910	−	0.946088i	$-0.395002\pi$
$911$	−9169.28	−	5293.88i	−0.333471	−	0.192529i	−0.657381	+	0.753558i	$0.728336\pi$
$912$		0			0
$913$		−	39028.1i		−	1.41472i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9215.04	+	11648.1i	−0.331851	+	0.419469i
$918$		0			0
$919$	−14001.8	−	24251.8i	−0.502586	−	0.870504i	−0.999996	−	0.00298831i	$-0.999049\pi$
$919$	−14001.8	−	24251.8i	−0.502586	−	0.870504i	0.497410	−	0.867516i	$-0.334285\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−23675.6	−	41007.4i	−0.844304	−	1.46238i
$924$		0			0
$925$	4368.99	−	7567.32i	0.155299	−	0.268986i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−27296.5			−0.964013			−0.482007	−	0.876168i	$-0.660092\pi$
$929$	−27296.5			−0.964013			−0.482007	+	0.876168i	$0.660092\pi$
$930$		0			0
$931$	−6460.29	+	1527.55i	−0.227419	+	0.0537738i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−17610.2	−	10167.2i	−0.615951	−	0.355620i
$936$		0			0
$937$		−	6443.03i		−	0.224637i	−0.993672	−	0.112318i	$-0.964172\pi$
$937$		−	6443.03i		−	0.224637i	0.993672	−	0.112318i	$-0.0358277\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	35678.2			1.23600			0.617999	−	0.786179i	$-0.287943\pi$
$941$	35678.2			1.23600			0.617999	+	0.786179i	$0.287943\pi$
$942$		0			0
$943$		−	13191.8i		−	0.455551i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			5139.71i			0.176366i	0.996104	+	0.0881828i	$0.0281060\pi$
$947$			5139.71i			0.176366i	−0.996104	+	0.0881828i	$0.971894\pi$
$948$		0			0
$949$	49169.3			1.68188
$950$		0			0
$951$		0			0
$952$		0			0
$953$			12018.0i			0.408502i	0.978919	+	0.204251i	$0.0654758\pi$
$953$			12018.0i			0.408502i	−0.978919	+	0.204251i	$0.934524\pi$
$954$		0			0
$955$	−24517.5	−	14155.2i	−0.830751	−	0.479634i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15816.3	+	6278.63i	−0.532571	+	0.211416i
$960$		0			0
$961$	−71326.1			−2.39422
$962$		0			0
$963$		0			0
$964$		0			0
$965$	26651.2	−	46161.2i	0.889048	−	1.53988i
$966$		0			0
$967$	7474.18	+	12945.7i	0.248556	+	0.430511i	0.963125	−	0.269053i	$-0.0867108\pi$
$967$	7474.18	+	12945.7i	0.248556	+	0.430511i	−0.714570	+	0.699564i	$0.753377\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21188.3	+	36699.3i	0.700274	+	1.21291i	0.968370	+	0.249518i	$0.0802721\pi$
$971$	21188.3	+	36699.3i	0.700274	+	1.21291i	−0.268096	+	0.963392i	$0.586395\pi$
$972$		0			0
$973$	−3175.03	+	1260.40i	−0.104611	+	0.0415278i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			15836.9i			0.518594i	0.965798	+	0.259297i	$0.0834909\pi$
$977$			15836.9i			0.518594i	−0.965798	+	0.259297i	$0.916509\pi$
$978$		0			0
$979$	3708.09	+	2140.87i	0.121053	+	0.0698901i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7127.39	−	12345.0i	−0.231260	−	0.400554i	0.726919	−	0.686723i	$-0.240951\pi$
$983$	−7127.39	−	12345.0i	−0.231260	−	0.400554i	−0.958179	+	0.286169i	$0.907618\pi$
$984$		0			0
$985$	−33375.8	−	19269.6i	−1.07964	−	0.623329i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−60042.4	+	34665.5i	−1.93047	+	1.11456i
$990$		0			0
$991$	1262.03	−	2185.91i	0.0404539	−	0.0700682i	−0.845090	−	0.534625i	$-0.820453\pi$
$991$	1262.03	−	2185.91i	0.0404539	−	0.0700682i	0.885543	+	0.464557i	$0.153786\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−72374.5	+	41785.5i	−2.30596	+	1.33134i
$996$		0			0
$997$	43322.6	−	25012.3i	1.37617	−	0.794531i	0.384472	−	0.923137i	$-0.374384\pi$
$997$	43322.6	−	25012.3i	1.37617	−	0.794531i	0.991696	+	0.128606i	$0.0410503\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.w.a.521.2		48
3.2	odd	2		252.4.w.a.101.2	yes	48
7.5	odd	6		756.4.bm.a.89.2		48
9.4	even	3		252.4.bm.a.185.7	yes	48
9.5	odd	6		756.4.bm.a.17.2		48
21.5	even	6		252.4.bm.a.173.7	yes	48
63.5	even	6	inner	756.4.w.a.341.2		48
63.40	odd	6		252.4.w.a.5.2	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.w.a.5.2	✓	48	63.40	odd	6
252.4.w.a.101.2	yes	48	3.2	odd	2
252.4.bm.a.173.7	yes	48	21.5	even	6
252.4.bm.a.185.7	yes	48	9.4	even	3
756.4.w.a.341.2		48	63.5	even	6	inner
756.4.w.a.521.2		48	1.1	even	1	trivial
756.4.bm.a.17.2		48	9.5	odd	6
756.4.bm.a.89.2		48	7.5	odd	6

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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-9.18977 + 15.9171i) q^{5} +(18.3240 + 2.68897i) q^{7} +O(q^{10})\)	\(q+(-9.18977 + 15.9171i) q^{5} +(18.3240 + 2.68897i) q^{7} +(47.9540 - 27.6862i) q^{11} +(68.0763 - 39.3039i) q^{13} +(-9.99021 + 17.3036i) q^{17} +(-16.7611 + 9.67700i) q^{19} +(-107.511 - 62.0716i) q^{23} +(-106.404 - 184.296i) q^{25} +(48.2851 + 27.8774i) q^{29} -317.989i q^{31} +(-211.194 + 266.955i) q^{35} +(20.5303 + 35.5595i) q^{37} +(53.1315 + 92.0264i) q^{41} +(279.238 - 483.655i) q^{43} -147.916 q^{47} +(328.539 + 98.5453i) q^{49} +(570.090 + 329.141i) q^{53} +1017.72i q^{55} +568.509 q^{59} +140.951i q^{61} +1444.77i q^{65} -294.283 q^{67} -602.374i q^{71} +(541.701 + 312.751i) q^{73} +(953.157 - 378.377i) q^{77} -534.044 q^{79} +(352.414 - 610.399i) q^{83} +(-183.615 - 318.031i) q^{85} +(38.6630 + 66.9663i) q^{89} +(1353.12 - 537.150i) q^{91} -355.718i q^{95} +(382.120 + 220.617i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 6 q^{7}+O(q^{10}) \) 48 * q + 6 * q^7	\( 48 q + 6 q^{7} - 12 q^{11} - 36 q^{13} + 72 q^{17} - 30 q^{23} - 600 q^{25} - 42 q^{29} - 390 q^{35} + 84 q^{37} + 618 q^{41} - 42 q^{43} - 396 q^{47} + 318 q^{49} + 1620 q^{53} - 1500 q^{59} - 588 q^{67} + 2472 q^{77} + 1608 q^{79} - 360 q^{85} - 1722 q^{89} + 540 q^{91} - 792 q^{97}+O(q^{100}) \) 48 * q + 6 * q^7 - 12 * q^11 - 36 * q^13 + 72 * q^17 - 30 * q^23 - 600 * q^25 - 42 * q^29 - 390 * q^35 + 84 * q^37 + 618 * q^41 - 42 * q^43 - 396 * q^47 + 318 * q^49 + 1620 * q^53 - 1500 * q^59 - 588 * q^67 + 2472 * q^77 + 1608 * q^79 - 360 * q^85 - 1722 * q^89 + 540 * q^91 - 792 * q^97

Introduction

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