LMFDB - Embedded newform 756.4.x.a.125.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,4,Mod(125,756)]

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, 3]))

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

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

chi := DirichletCharacter("756.125");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.x (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			125.6
Character	$\chi$	$=$	756.125
Dual form			756.4.x.a.629.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.49690 + 9.52092i) q^{5} +(6.85688 - 17.2042i) q^{7} +O(q^{10})$	$q+(-5.49690 + 9.52092i) q^{5} +(6.85688 - 17.2042i) q^{7} +(-13.8268 + 7.98293i) q^{11} +(-77.4289 - 44.7036i) q^{13} +106.953 q^{17} -8.31634i q^{19} +(123.893 + 71.5294i) q^{23} +(2.06812 + 3.58208i) q^{25} +(-129.721 + 74.8943i) q^{29} +(-37.1952 - 21.4747i) q^{31} +(126.108 + 159.853i) q^{35} +390.706 q^{37} +(-172.486 + 298.755i) q^{41} +(28.5593 + 49.4661i) q^{43} +(-8.13282 - 14.0864i) q^{47} +(-248.966 - 235.934i) q^{49} -445.230i q^{53} -175.526i q^{55} +(-193.350 + 334.892i) q^{59} +(-420.857 + 242.982i) q^{61} +(851.238 - 491.463i) q^{65} +(-251.821 + 436.167i) q^{67} +751.418i q^{71} +507.533i q^{73} +(42.5306 + 292.617i) q^{77} +(381.570 + 660.899i) q^{79} +(607.241 + 1051.77i) q^{83} +(-587.911 + 1018.29i) q^{85} -425.727 q^{89} +(-1300.01 + 1025.57i) q^{91} +(79.1792 + 45.7141i) q^{95} +(494.582 - 285.547i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 6 q^{7}+O(q^{10}) $ 48 * q + 6 * q^7	$ 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) $ 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−5.49690	+	9.52092i	−0.491658	+	0.851577i	−0.999954	−	0.00960594i	$-0.996942\pi$
$5$	−5.49690	+	9.52092i	−0.491658	+	0.851577i	0.508296	+	0.861182i	$0.330276\pi$
$6$		0			0
$7$	6.85688	−	17.2042i	0.370237	−	0.928937i
$7$	6.85688	−	17.2042i	0.370237	−	0.928937i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.8268	+	7.98293i	−0.378996	+	0.218813i	−0.677381	−	0.735632i	$-0.736885\pi$
$11$	−13.8268	+	7.98293i	−0.378996	+	0.218813i	0.298386	+	0.954445i	$0.403552\pi$
$12$		0			0
$13$	−77.4289	−	44.7036i	−1.65192	−	0.953734i	−0.976283	−	0.216496i	$-0.930537\pi$
$13$	−77.4289	−	44.7036i	−1.65192	−	0.953734i	−0.675633	−	0.737238i	$-0.736130\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	106.953			1.52588			0.762939	−	0.646471i	$-0.223756\pi$
$17$	106.953			1.52588			0.762939	+	0.646471i	$0.223756\pi$
$18$		0			0
$19$		−	8.31634i		−	0.100416i	−0.998739	−	0.0502079i	$-0.984012\pi$
$19$		−	8.31634i		−	0.100416i	0.998739	−	0.0502079i	$-0.0159884\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	123.893	+	71.5294i	1.12319	+	0.648474i	0.942214	−	0.335013i	$-0.108741\pi$
$23$	123.893	+	71.5294i	1.12319	+	0.648474i	0.180977	+	0.983487i	$0.442074\pi$
$24$		0			0
$25$	2.06812	+	3.58208i	0.0165449	+	0.0286567i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−129.721	+	74.8943i	−0.830639	+	0.479570i	−0.854072	−	0.520156i	$-0.825874\pi$
$29$	−129.721	+	74.8943i	−0.830639	+	0.479570i	0.0234322	+	0.999725i	$0.492541\pi$
$30$		0			0
$31$	−37.1952	−	21.4747i	−0.215499	−	0.124418i	0.388366	−	0.921505i	$-0.373040\pi$
$31$	−37.1952	−	21.4747i	−0.215499	−	0.124418i	−0.603864	+	0.797087i	$0.706373\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	126.108	+	159.853i	0.609031	+	0.772004i
$36$		0			0
$37$	390.706			1.73599			0.867997	−	0.496570i	$-0.165407\pi$
$37$	390.706			1.73599			0.867997	+	0.496570i	$0.165407\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−172.486	+	298.755i	−0.657021	+	1.13799i	0.324363	+	0.945933i	$0.394850\pi$
$41$	−172.486	+	298.755i	−0.657021	+	1.13799i	−0.981383	+	0.192060i	$0.938483\pi$
$42$		0			0
$43$	28.5593	+	49.4661i	0.101285	+	0.175430i	0.912214	−	0.409714i	$-0.134371\pi$
$43$	28.5593	+	49.4661i	0.101285	+	0.175430i	−0.810929	+	0.585144i	$0.801038\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.13282	−	14.0864i	−0.0252403	−	0.0437174i	0.853129	−	0.521699i	$-0.174702\pi$
$47$	−8.13282	−	14.0864i	−0.0252403	−	0.0437174i	−0.878370	+	0.477982i	$0.841368\pi$
$48$		0			0
$49$	−248.966	−	235.934i	−0.725849	−	0.687854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	445.230i		−	1.15391i	−0.816777	−	0.576954i	$-0.804241\pi$
$53$		−	445.230i		−	1.15391i	0.816777	−	0.576954i	$-0.195759\pi$
$54$		0			0
$55$		−	175.526i		−	0.430325i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−193.350	+	334.892i	−0.426644	+	0.738969i	−0.996572	−	0.0827250i	$-0.973638\pi$
$59$	−193.350	+	334.892i	−0.426644	+	0.738969i	0.569928	+	0.821695i	$0.306971\pi$
$60$		0			0
$61$	−420.857	+	242.982i	−0.883365	+	0.510011i	−0.871766	−	0.489922i	$-0.837025\pi$
$61$	−420.857	+	242.982i	−0.883365	+	0.510011i	−0.0115982	+	0.999933i	$0.503692\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	851.238	−	491.463i	1.62436	−	0.937822i
$66$		0			0
$67$	−251.821	+	436.167i	−0.459177	+	0.795318i	−0.998918	−	0.0465137i	$-0.985189\pi$
$67$	−251.821	+	436.167i	−0.459177	+	0.795318i	0.539741	+	0.841831i	$0.318522\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			751.418i			1.25601i	0.778208	+	0.628007i	$0.216129\pi$
$71$			751.418i			1.25601i	−0.778208	+	0.628007i	$0.783871\pi$
$72$		0			0
$73$			507.533i			0.813729i	0.913489	+	0.406865i	$0.133378\pi$
$73$			507.533i			0.813729i	−0.913489	+	0.406865i	$0.866622\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	42.5306	+	292.617i	0.0629456	+	0.433076i
$78$		0			0
$79$	381.570	+	660.899i	0.543418	+	0.941228i	0.998705	+	0.0508826i	$0.0162034\pi$
$79$	381.570	+	660.899i	0.543418	+	0.941228i	−0.455287	+	0.890345i	$0.650463\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	607.241	+	1051.77i	0.803052	+	1.39093i	0.917598	+	0.397509i	$0.130125\pi$
$83$	607.241	+	1051.77i	0.803052	+	1.39093i	−0.114546	+	0.993418i	$0.536541\pi$
$84$		0			0
$85$	−587.911	+	1018.29i	−0.750210	+	1.29940i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−425.727			−0.507045			−0.253523	−	0.967329i	$-0.581589\pi$
$89$	−425.727			−0.507045			−0.253523	+	0.967329i	$0.581589\pi$
$90$		0			0
$91$	−1300.01	+	1025.57i	−1.49756	+	1.18142i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	79.1792	+	45.7141i	0.0855117	+	0.0493702i
$96$		0			0
$97$	494.582	−	285.547i	0.517703	−	0.298896i	−0.218291	−	0.975884i	$-0.570048\pi$
$97$	494.582	−	285.547i	0.517703	−	0.298896i	0.735994	+	0.676988i	$0.236715\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−353.385	−	612.081i	−0.348150	−	0.603013i	0.637771	−	0.770226i	$-0.279857\pi$
$101$	−353.385	−	612.081i	−0.348150	−	0.603013i	−0.985921	+	0.167213i	$0.946523\pi$
$102$		0			0
$103$	−217.607	−	125.635i	−0.208169	−	0.120187i	0.392291	−	0.919841i	$-0.371683\pi$
$103$	−217.607	−	125.635i	−0.208169	−	0.120187i	−0.600460	+	0.799655i	$0.705016\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			739.962i			0.668549i	0.942476	+	0.334275i	$0.108491\pi$
$107$			739.962i			0.668549i	−0.942476	+	0.334275i	$0.891509\pi$
$108$		0			0
$109$	1497.38			1.31580			0.657902	−	0.753104i	$-0.271444\pi$
$109$	1497.38			1.31580			0.657902	+	0.753104i	$0.271444\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	861.916	+	497.627i	0.717542	+	0.414273i	0.813847	−	0.581079i	$-0.197369\pi$
$113$	861.916	+	497.627i	0.717542	+	0.414273i	−0.0963056	+	0.995352i	$0.530703\pi$
$114$		0			0
$115$	−1362.05	+	786.380i	−1.10445	+	0.637655i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	733.365	−	1840.04i	0.564936	−	1.41744i
$120$		0			0
$121$	−538.046	+	931.922i	−0.404242	+	0.700167i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1419.70			−1.01585
$126$		0			0
$127$	2469.41			1.72539			0.862696	−	0.505724i	$-0.168774\pi$
$127$	2469.41			1.72539			0.862696	+	0.505724i	$0.168774\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−769.787	+	1333.31i	−0.513409	+	0.889251i	0.486470	+	0.873697i	$0.338284\pi$
$131$	−769.787	+	1333.31i	−0.513409	+	0.889251i	−0.999879	+	0.0155535i	$0.995049\pi$
$132$		0			0
$133$	−143.076	−	57.0242i	−0.0932800	−	0.0371776i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1419.80	−	819.722i	0.885414	−	0.511194i	0.0129744	−	0.999916i	$-0.495870\pi$
$137$	1419.80	−	819.722i	0.885414	−	0.511194i	0.872440	+	0.488722i	$0.162537\pi$
$138$		0			0
$139$	−899.533	−	519.345i	−0.548902	−	0.316909i	0.199777	−	0.979841i	$-0.435978\pi$
$139$	−899.533	−	519.345i	−0.548902	−	0.316909i	−0.748679	+	0.662933i	$0.769312\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1427.46			0.834759
$144$		0			0
$145$		−	1646.75i		−	0.943137i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1523.17	+	879.401i	0.837468	+	0.483512i	0.856403	−	0.516308i	$-0.172694\pi$
$149$	1523.17	+	879.401i	0.837468	+	0.483512i	−0.0189346	+	0.999821i	$0.506027\pi$
$150$		0			0
$151$	352.401	+	610.376i	0.189920	+	0.328952i	0.945223	−	0.326424i	$-0.105844\pi$
$151$	352.401	+	610.376i	0.189920	+	0.328952i	−0.755303	+	0.655376i	$0.772510\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	408.917	−	236.088i	0.211903	−	0.122342i
$156$		0			0
$157$	579.051	+	334.315i	0.294352	+	0.169944i	0.639903	−	0.768456i	$-0.278975\pi$
$157$	579.051	+	334.315i	0.294352	+	0.169944i	−0.345551	+	0.938400i	$0.612308\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2080.12	−	1641.00i	1.01824	−	0.803285i
$162$		0			0
$163$	−3926.19			−1.88665			−0.943323	−	0.331877i	$-0.892318\pi$
$163$	−3926.19			−1.88665			−0.943323	+	0.331877i	$0.892318\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	966.800	−	1674.55i	0.447984	−	0.775930i	−0.550271	−	0.834986i	$-0.685476\pi$
$167$	966.800	−	1674.55i	0.447984	−	0.775930i	0.998255	+	0.0590557i	$0.0188090\pi$
$168$		0			0
$169$	2898.32	+	5020.04i	1.31922	+	2.28495i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	850.018	+	1472.27i	0.373559	+	0.647023i	0.990110	−	0.140292i	$-0.0448041\pi$
$173$	850.018	+	1472.27i	0.373559	+	0.647023i	−0.616551	+	0.787315i	$0.711471\pi$
$174$		0			0
$175$	75.8075	−	11.0183i	0.0327458	−	0.00475945i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			1175.15i			0.490697i	0.969435	+	0.245349i	$0.0789024\pi$
$179$			1175.15i			0.490697i	−0.969435	+	0.245349i	$0.921098\pi$
$180$		0			0
$181$			3506.89i			1.44014i	0.693902	+	0.720070i	$0.255890\pi$
$181$			3506.89i			1.44014i	−0.693902	+	0.720070i	$0.744110\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2147.68	+	3719.88i	−0.853515	+	1.47833i
$186$		0			0
$187$	−1478.82	+	853.799i	−0.578301	+	0.333882i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1228.14	+	709.068i	−0.465263	+	0.268620i	−0.714255	−	0.699886i	$-0.753234\pi$
$191$	−1228.14	+	709.068i	−0.465263	+	0.268620i	0.248992	+	0.968506i	$0.419901\pi$
$192$		0			0
$193$	746.819	−	1293.53i	0.278535	−	0.482436i	−0.692486	−	0.721431i	$-0.743485\pi$
$193$	746.819	−	1293.53i	0.278535	−	0.482436i	0.971021	+	0.238995i	$0.0768179\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2384.42i			0.862351i	0.902268	+	0.431175i	$0.141901\pi$
$197$			2384.42i			0.862351i	−0.902268	+	0.431175i	$0.858099\pi$
$198$		0			0
$199$		−	4960.24i		−	1.76695i	−0.468482	−	0.883473i	$-0.655199\pi$
$199$		−	4960.24i		−	1.76695i	0.468482	−	0.883473i	$-0.344801\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	399.014	+	2745.28i	0.137957	+	0.949166i
$204$		0			0
$205$	−1896.28	−	3284.46i	−0.646059	−	1.11901i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	66.3888	+	114.989i	0.0219723	+	0.0380571i
$210$		0			0
$211$	1973.64	−	3418.45i	0.643939	−	1.11534i	−0.340606	−	0.940206i	$-0.610632\pi$
$211$	1973.64	−	3418.45i	0.643939	−	1.11534i	0.984545	−	0.175129i	$-0.0560343\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−627.950			−0.199190
$216$		0			0
$217$	−624.497	+	492.663i	−0.195362	+	0.154121i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8281.26	−	4781.19i	−2.52062	−	1.45528i
$222$		0			0
$223$	−1397.46	+	806.825i	−0.419646	+	0.242283i	−0.694926	−	0.719082i	$-0.744563\pi$
$223$	−1397.46	+	806.825i	−0.419646	+	0.242283i	0.275280	+	0.961364i	$0.411229\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1749.96	−	3031.01i	−0.511668	−	0.886235i	−0.999909	−	0.0135262i	$-0.995694\pi$
$227$	−1749.96	−	3031.01i	−0.511668	−	0.886235i	0.488240	−	0.872709i	$-0.337639\pi$
$228$		0			0
$229$	−1654.24	−	955.078i	−0.477360	−	0.275604i	0.241956	−	0.970287i	$-0.422211\pi$
$229$	−1654.24	−	955.078i	−0.477360	−	0.275604i	−0.719316	+	0.694683i	$0.755544\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			4408.26i			1.23946i	0.784814	+	0.619731i	$0.212758\pi$
$233$			4408.26i			1.23946i	−0.784814	+	0.619731i	$0.787242\pi$
$234$		0			0
$235$	178.821			0.0496383
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3715.64	−	2145.23i	−1.00563	−	0.580599i	−0.0957187	−	0.995408i	$-0.530515\pi$
$239$	−3715.64	−	2145.23i	−1.00563	−	0.580599i	−0.909908	+	0.414809i	$0.863848\pi$
$240$		0			0
$241$	−1413.43	+	816.047i	−0.377790	+	0.218117i	−0.676856	−	0.736115i	$-0.736658\pi$
$241$	−1413.43	+	816.047i	−0.377790	+	0.218117i	0.299066	+	0.954232i	$0.403325\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3614.85	−	1073.48i	0.942630	−	0.279927i
$246$		0			0
$247$	−371.770	+	643.925i	−0.0957700	+	0.165878i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3912.60			0.983908			0.491954	−	0.870621i	$-0.336283\pi$
$251$	3912.60			0.983908			0.491954	+	0.870621i	$0.336283\pi$
$252$		0			0
$253$	−2284.06			−0.567579
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1054.61	+	1826.64i	−0.255971	+	0.443356i	−0.965159	−	0.261664i	$-0.915729\pi$
$257$	−1054.61	+	1826.64i	−0.255971	+	0.443356i	0.709187	+	0.705020i	$0.249062\pi$
$258$		0			0
$259$	2679.03	−	6721.78i	0.642729	−	1.61263i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3827.74	+	2209.95i	−0.897448	+	0.518142i	−0.876371	−	0.481636i	$-0.840043\pi$
$263$	−3827.74	+	2209.95i	−0.897448	+	0.518142i	−0.0210765	+	0.999778i	$0.506709\pi$
$264$		0			0
$265$	4239.00	+	2447.39i	0.982641	+	0.567328i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4557.94			1.03309			0.516547	−	0.856259i	$-0.327217\pi$
$269$	4557.94			1.03309			0.516547	+	0.856259i	$0.327217\pi$
$270$		0			0
$271$			128.189i			0.0287341i	0.999897	+	0.0143671i	$0.00457334\pi$
$271$			128.189i			0.0287341i	−0.999897	+	0.0143671i	$0.995427\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−57.1910	−	33.0193i	−0.0125409	−	0.00724050i
$276$		0			0
$277$	1020.01	+	1766.71i	0.221251	+	0.383218i	0.955188	−	0.296000i	$-0.0956527\pi$
$277$	1020.01	+	1766.71i	0.221251	+	0.383218i	−0.733937	+	0.679217i	$0.762319\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1506.57	+	869.820i	−0.319838	+	0.184659i	−0.651321	−	0.758803i	$-0.725785\pi$
$281$	−1506.57	+	869.820i	−0.319838	+	0.184659i	0.331482	+	0.943462i	$0.392451\pi$
$282$		0			0
$283$	1080.52	+	623.836i	0.226961	+	0.131036i	0.609169	−	0.793040i	$-0.291503\pi$
$283$	1080.52	+	623.836i	0.226961	+	0.131036i	−0.382208	+	0.924076i	$0.624836\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3957.11	+	5016.01i	0.813871	+	1.03166i
$288$		0			0
$289$	6525.95			1.32830
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2395.59	−	4149.28i	0.477651	−	0.827316i	−0.522021	−	0.852933i	$-0.674822\pi$
$293$	2395.59	−	4149.28i	0.477651	−	0.827316i	0.999672	+	0.0256168i	$0.00815497\pi$
$294$		0			0
$295$	−2125.65	−	3681.74i	−0.419526	−	0.726640i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6395.24	−	11076.9i	−1.23694	−	2.14245i
$300$		0			0
$301$	1046.85	−	152.155i	0.200463	−	0.0291364i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	5342.59i		−	1.00300i
$306$		0			0
$307$		−	4679.41i		−	0.869929i	−0.900448	−	0.434965i	$-0.856761\pi$
$307$		−	4679.41i		−	0.869929i	0.900448	−	0.434965i	$-0.143239\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−919.164	+	1592.04i	−0.167592	+	0.290277i	−0.937573	−	0.347790i	$-0.886932\pi$
$311$	−919.164	+	1592.04i	−0.167592	+	0.290277i	0.769981	+	0.638067i	$0.220266\pi$
$312$		0			0
$313$	5160.13	−	2979.20i	0.931846	−	0.538001i	0.0444510	−	0.999012i	$-0.485846\pi$
$313$	5160.13	−	2979.20i	0.931846	−	0.538001i	0.887395	+	0.461010i	$0.152513\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8986.17	−	5188.17i	1.59216	−	0.919232i	0.599220	−	0.800584i	$-0.295477\pi$
$317$	8986.17	−	5188.17i	1.59216	−	0.919232i	0.992936	−	0.118648i	$-0.0378559\pi$
$318$		0			0
$319$	1195.75	−	2071.10i	0.209872	−	0.363510i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	889.458i		−	0.153222i
$324$		0			0
$325$		−	369.809i		−	0.0631179i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−298.111	+	43.3291i	−0.0499557	+	0.00726083i
$330$		0			0
$331$	−2421.65	−	4194.42i	−0.402132	−	0.696514i	0.591851	−	0.806048i	$-0.298398\pi$
$331$	−2421.65	−	4194.42i	−0.402132	−	0.696514i	−0.993983	+	0.109534i	$0.965064\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2768.47	−	4795.13i	−0.451516	−	0.782048i
$336$		0			0
$337$	−559.013	+	968.240i	−0.0903602	+	0.156509i	−0.907663	−	0.419700i	$-0.862135\pi$
$337$	−559.013	+	968.240i	−0.0903602	+	0.156509i	0.817302	+	0.576209i	$0.195469\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	685.723			0.108897
$342$		0			0
$343$	−5766.18	+	2665.49i	−0.907709	+	0.419600i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3834.85	−	2214.05i	−0.593273	−	0.342526i	0.173118	−	0.984901i	$-0.444616\pi$
$347$	−3834.85	−	2214.05i	−0.593273	−	0.342526i	−0.766391	+	0.642375i	$0.777949\pi$
$348$		0			0
$349$	1980.13	−	1143.23i	0.303707	−	0.175345i	−0.340400	−	0.940281i	$-0.610562\pi$
$349$	1980.13	−	1143.23i	0.303707	−	0.175345i	0.644107	+	0.764935i	$0.277229\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4027.88	−	6976.50i	−0.607316	−	1.05190i	−0.991681	−	0.128721i	$-0.958913\pi$
$353$	−4027.88	−	6976.50i	−0.607316	−	1.05190i	0.384364	−	0.923181i	$-0.374421\pi$
$354$		0			0
$355$	−7154.19	−	4130.47i	−1.06959	−	0.617529i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	5537.25i		−	0.814053i	−0.913416	−	0.407027i	$-0.866566\pi$
$359$		−	5537.25i		−	0.814053i	0.913416	−	0.407027i	$-0.133434\pi$
$360$		0			0
$361$	6789.84			0.989917
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4832.18	−	2789.86i	−0.692953	−	0.400076i
$366$		0			0
$367$	4668.87	−	2695.58i	0.664069	−	0.383400i	−0.129757	−	0.991546i	$-0.541420\pi$
$367$	4668.87	−	2695.58i	0.664069	−	0.383400i	0.793825	+	0.608146i	$0.208086\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7659.81	−	3052.89i	−1.07191	−	0.427219i
$372$		0			0
$373$	1062.08	−	1839.58i	0.147433	−	0.255362i	−0.782845	−	0.622217i	$-0.786232\pi$
$373$	1062.08	−	1839.58i	0.147433	−	0.255362i	0.930278	+	0.366855i	$0.119566\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	13392.2			1.82953
$378$		0			0
$379$	−5358.80			−0.726288			−0.363144	−	0.931733i	$-0.618297\pi$
$379$	−5358.80			−0.726288			−0.363144	+	0.931733i	$0.618297\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2729.23	+	4727.16i	−0.364118	+	0.630670i	−0.988634	−	0.150341i	$-0.951963\pi$
$383$	−2729.23	+	4727.16i	−0.364118	+	0.630670i	0.624517	+	0.781012i	$0.285296\pi$
$384$		0			0
$385$	−3019.77	−	1203.56i	−0.399745	−	0.159322i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1416.25	+	817.670i	−0.184593	+	0.106575i	−0.589449	−	0.807806i	$-0.700655\pi$
$389$	−1416.25	+	817.670i	−0.184593	+	0.106575i	0.404856	+	0.914380i	$0.367322\pi$
$390$		0			0
$391$	13250.7	+	7650.29i	1.71385	+	0.989493i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8389.82			−1.06870
$396$		0			0
$397$			7694.41i			0.972724i	0.873757	+	0.486362i	$0.161676\pi$
$397$			7694.41i			0.972724i	−0.873757	+	0.486362i	$0.838324\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4800.46	−	2771.55i	−0.597815	−	0.345148i	0.170367	−	0.985381i	$-0.445505\pi$
$401$	−4800.46	−	2771.55i	−0.597815	−	0.345148i	−0.768181	+	0.640232i	$0.778838\pi$
$402$		0			0
$403$	1919.99	+	3325.52i	0.237324	+	0.411057i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5402.24	+	3118.98i	−0.657934	+	0.379858i
$408$		0			0
$409$	6361.70	+	3672.93i	0.769109	+	0.444045i	0.832557	−	0.553940i	$-0.186876\pi$
$409$	6361.70	+	3672.93i	0.769109	+	0.444045i	−0.0634474	+	0.997985i	$0.520210\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4435.75	+	5622.74i	0.528497	+	0.669920i
$414$		0			0
$415$	−13351.8			−1.57931
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1717.45	+	2974.72i	−0.200246	+	0.346836i	−0.948608	−	0.316455i	$-0.897508\pi$
$419$	−1717.45	+	2974.72i	−0.200246	+	0.346836i	0.748362	+	0.663291i	$0.230841\pi$
$420$		0			0
$421$	−7241.13	−	12542.0i	−0.838269	−	1.45192i	−0.891341	−	0.453334i	$-0.850235\pi$
$421$	−7241.13	−	12542.0i	−0.838269	−	1.45192i	0.0530719	−	0.998591i	$-0.483099\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	221.191	+	383.115i	0.0252455	+	0.0437266i
$426$		0			0
$427$	1294.53	+	8906.59i	0.146714	+	1.00942i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			13632.5i			1.52356i	0.647839	+	0.761778i	$0.275673\pi$
$431$			13632.5i			1.52356i	−0.647839	+	0.761778i	$0.724327\pi$
$432$		0			0
$433$			6735.48i			0.747544i	0.927521	+	0.373772i	$0.121936\pi$
$433$			6735.48i			0.747544i	−0.927521	+	0.373772i	$0.878064\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	594.863	−	1030.33i	0.0651170	−	0.112786i
$438$		0			0
$439$	9181.17	−	5300.75i	0.998162	−	0.576289i	0.0904581	−	0.995900i	$-0.471167\pi$
$439$	9181.17	−	5300.75i	0.998162	−	0.576289i	0.907704	+	0.419611i	$0.137834\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3111.86	−	1796.64i	0.333745	−	0.192688i	−0.323757	−	0.946140i	$-0.604946\pi$
$443$	3111.86	−	1796.64i	0.333745	−	0.192688i	0.657503	+	0.753452i	$0.271613\pi$
$444$		0			0
$445$	2340.18	−	4053.31i	0.249293	−	0.431788i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	8675.98i		−	0.911904i	−0.890004	−	0.455952i	$-0.849299\pi$
$449$		−	8675.98i		−	0.911904i	0.890004	−	0.455952i	$-0.150701\pi$
$450$		0			0
$451$		−	5507.79i		−	0.575059i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2618.36	−	18014.7i	−0.269782	−	1.85614i
$456$		0			0
$457$	−1466.35	−	2539.79i	−0.150094	−	0.259970i	0.781168	−	0.624321i	$-0.214624\pi$
$457$	−1466.35	−	2539.79i	−0.150094	−	0.259970i	−0.931262	+	0.364351i	$0.881291\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3015.81	−	5223.53i	−0.304686	−	0.527731i	0.672506	−	0.740092i	$-0.265218\pi$
$461$	−3015.81	−	5223.53i	−0.304686	−	0.527731i	−0.977191	+	0.212361i	$0.931885\pi$
$462$		0			0
$463$	295.638	−	512.060i	0.0296748	−	0.0513983i	−0.850807	−	0.525479i	$-0.823886\pi$
$463$	295.638	−	512.060i	0.0296748	−	0.0513983i	0.880481	+	0.474081i	$0.157219\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16674.6			−1.65227			−0.826134	−	0.563473i	$-0.809465\pi$
$467$	−16674.6			−1.65227			−0.826134	+	0.563473i	$0.809465\pi$
$468$		0			0
$469$	5777.18	+	7323.12i	0.568796	+	0.721002i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−789.769	−	455.973i	−0.0767730	−	0.0443249i
$474$		0			0
$475$	29.7898	−	17.1992i	0.00287758	−	0.00166137i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8608.73	+	14910.8i	0.821175	+	1.42232i	0.904808	+	0.425820i	$0.140014\pi$
$479$	8608.73	+	14910.8i	0.821175	+	1.42232i	−0.0836331	+	0.996497i	$0.526652\pi$
$480$		0			0
$481$	−30252.0	−	17466.0i	−2.86772	−	1.65568i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			6278.50i			0.587818i
$486$		0			0
$487$	8745.94			0.813792			0.406896	−	0.913475i	$-0.366611\pi$
$487$	8745.94			0.813792			0.406896	+	0.913475i	$0.366611\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−653.319	−	377.194i	−0.0600486	−	0.0346691i	0.469675	−	0.882839i	$-0.344371\pi$
$491$	−653.319	−	377.194i	−0.0600486	−	0.0346691i	−0.529724	+	0.848170i	$0.677704\pi$
$492$		0			0
$493$	−13874.0	+	8010.17i	−1.26745	+	0.731765i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12927.5	+	5152.39i	1.16676	+	0.465022i
$498$		0			0
$499$	−2480.24	+	4295.90i	−0.222506	+	0.385392i	−0.955568	−	0.294769i	$-0.904757\pi$
$499$	−2480.24	+	4295.90i	−0.222506	+	0.385392i	0.733062	+	0.680162i	$0.238090\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5299.83			0.469797			0.234898	−	0.972020i	$-0.424524\pi$
$503$	5299.83			0.469797			0.234898	+	0.972020i	$0.424524\pi$
$504$		0			0
$505$	7770.09			0.684682
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9344.81	+	16185.7i	−0.813755	+	1.40947i	0.0964627	+	0.995337i	$0.469247\pi$
$509$	−9344.81	+	16185.7i	−0.813755	+	1.40947i	−0.910218	+	0.414129i	$0.864086\pi$
$510$		0			0
$511$	8731.68	+	3480.09i	0.755903	+	0.301273i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2392.33	−	1381.21i	0.204696	−	0.118181i
$516$		0			0
$517$	224.902	+	129.847i	0.0191319	+	0.0110458i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8964.93			0.753859			0.376930	−	0.926242i	$-0.376980\pi$
$521$	8964.93			0.753859			0.376930	+	0.926242i	$0.376980\pi$
$522$		0			0
$523$		−	20614.1i		−	1.72350i	−0.507330	−	0.861752i	$-0.669367\pi$
$523$		−	20614.1i		−	1.72350i	0.507330	−	0.861752i	$-0.330633\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3978.14	−	2296.78i	−0.328825	−	0.189847i
$528$		0			0
$529$	4149.41	+	7186.99i	0.341038	+	0.590695i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	26710.9	−	15421.5i	2.17069	−	1.25325i
$534$		0			0
$535$	−7045.11	−	4067.50i	−0.569321	−	0.328698i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5325.86	+	1274.74i	0.425605	+	0.101868i
$540$		0			0
$541$	9376.65			0.745164			0.372582	−	0.927999i	$-0.378473\pi$
$541$	9376.65			0.745164			0.372582	+	0.927999i	$0.378473\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8230.93	+	14256.4i	−0.646925	+	1.12051i
$546$		0			0
$547$	1172.25	+	2030.40i	0.0916305	+	0.158709i	0.908197	−	0.418542i	$-0.137459\pi$
$547$	1172.25	+	2030.40i	0.0916305	+	0.158709i	−0.816567	+	0.577251i	$0.804125\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	622.847	+	1078.80i	0.0481564	+	0.0834093i
$552$		0			0
$553$	13986.6	−	2032.89i	1.07553	−	0.156324i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			15368.4i			1.16908i	0.811364	+	0.584541i	$0.198725\pi$
$557$			15368.4i			1.16908i	−0.811364	+	0.584541i	$0.801275\pi$
$558$		0			0
$559$		−	5106.81i		−	0.386395i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6447.73	−	11167.8i	0.482664	−	0.835998i	−0.517138	−	0.855902i	$-0.673003\pi$
$563$	6447.73	−	11167.8i	0.482664	−	0.835998i	0.999802	+	0.0199041i	$0.00633608\pi$
$564$		0			0
$565$	−9475.73	+	5470.82i	−0.705570	+	0.407361i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−20989.2	+	12118.1i	−1.54642	+	0.892824i	−0.548005	+	0.836475i	$0.684613\pi$
$569$	−20989.2	+	12118.1i	−1.54642	+	0.892824i	−0.998411	+	0.0563492i	$0.982054\pi$
$570$		0			0
$571$	−3678.38	+	6371.14i	−0.269589	+	0.466942i	−0.968756	−	0.248017i	$-0.920221\pi$
$571$	−3678.38	+	6371.14i	−0.269589	+	0.466942i	0.699167	+	0.714959i	$0.253555\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			591.724i			0.0429158i
$576$		0			0
$577$		−	11928.2i		−	0.860617i	−0.902682	−	0.430308i	$-0.858405\pi$
$577$		−	11928.2i		−	0.860617i	0.902682	−	0.430308i	$-0.141595\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	22258.6	−	3235.19i	1.58940	−	0.231013i
$582$		0			0
$583$	3554.24	+	6156.13i	0.252490	+	0.437326i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2638.36	+	4569.77i	0.185514	+	0.321320i	0.943750	−	0.330661i	$-0.107272\pi$
$587$	2638.36	+	4569.77i	0.185514	+	0.321320i	−0.758236	+	0.651981i	$0.773938\pi$
$588$		0			0
$589$	−178.591	+	309.328i	−0.0124935	+	0.0216395i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5290.53			−0.366367			−0.183184	−	0.983079i	$-0.558640\pi$
$593$	−5290.53			−0.366367			−0.183184	+	0.983079i	$0.558640\pi$
$594$		0			0
$595$	13487.6	+	17096.8i	0.929308	+	1.17798i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8568.78	+	4947.19i	0.584492	+	0.337457i	0.762917	−	0.646497i	$-0.223767\pi$
$599$	8568.78	+	4947.19i	0.584492	+	0.337457i	−0.178424	+	0.983954i	$0.557100\pi$
$600$		0			0
$601$	2305.57	−	1331.12i	0.156483	−	0.0903454i	−0.419714	−	0.907656i	$-0.637870\pi$
$601$	2305.57	−	1331.12i	0.156483	−	0.0903454i	0.576197	+	0.817311i	$0.304536\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5915.17	−	10245.4i	−0.397497	−	0.688485i
$606$		0			0
$607$	−9797.65	−	5656.68i	−0.655147	−	0.378249i	0.135278	−	0.990808i	$-0.456807\pi$
$607$	−9797.65	−	5656.68i	−0.655147	−	0.378249i	−0.790425	+	0.612558i	$0.790141\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1454.26i			0.0962901i
$612$		0			0
$613$	−23984.5			−1.58030			−0.790151	−	0.612912i	$-0.789998\pi$
$613$	−23984.5			−1.58030			−0.790151	+	0.612912i	$0.789998\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14063.1	+	8119.31i	0.917597	+	0.529775i	0.882868	−	0.469622i	$-0.155610\pi$
$617$	14063.1	+	8119.31i	0.917597	+	0.529775i	0.0347291	+	0.999397i	$0.488943\pi$
$618$		0			0
$619$	−14971.6	+	8643.88i	−0.972150	+	0.561271i	−0.899891	−	0.436115i	$-0.856354\pi$
$619$	−14971.6	+	8643.88i	−0.972150	+	0.561271i	−0.0722590	+	0.997386i	$0.523021\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2919.16	+	7324.28i	−0.187727	+	0.471013i
$624$		0			0
$625$	7545.43	−	13069.1i	0.482908	−	0.836421i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	41787.2			2.64891
$630$		0			0
$631$	348.827			0.0220073			0.0110036	−	0.999939i	$-0.496497\pi$
$631$	348.827			0.0220073			0.0110036	+	0.999939i	$0.496497\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13574.1	+	23511.0i	−0.848302	+	1.46930i
$636$		0			0
$637$	8730.10	+	29397.8i	0.543013	+	1.82854i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	14508.8	−	8376.64i	0.894013	−	0.516158i	0.0187597	−	0.999824i	$-0.494028\pi$
$641$	14508.8	−	8376.64i	0.894013	−	0.516158i	0.875253	+	0.483666i	$0.160695\pi$
$642$		0			0
$643$	10660.5	+	6154.87i	0.653827	+	0.377487i	0.789921	−	0.613209i	$-0.210122\pi$
$643$	10660.5	+	6154.87i	0.653827	+	0.377487i	−0.136094	+	0.990696i	$0.543455\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−32197.6			−1.95644			−0.978220	−	0.207570i	$-0.933445\pi$
$647$	−32197.6			−1.95644			−0.978220	+	0.207570i	$0.933445\pi$
$648$		0			0
$649$		−	6174.00i		−	0.373422i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	15436.6	+	8912.32i	0.925085	+	0.534098i	0.885254	−	0.465108i	$-0.153985\pi$
$653$	15436.6	+	8912.32i	0.925085	+	0.534098i	0.0398315	+	0.999206i	$0.487318\pi$
$654$		0			0
$655$	−8462.89	−	14658.2i	−0.504843	−	0.874415i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−17782.9	+	10267.0i	−1.05117	+	0.606895i	−0.922978	−	0.384853i	$-0.874252\pi$
$659$	−17782.9	+	10267.0i	−1.05117	+	0.606895i	−0.128196	+	0.991749i	$0.540919\pi$
$660$		0			0
$661$	8819.60	+	5092.00i	0.518975	+	0.299630i	0.736515	−	0.676421i	$-0.236470\pi$
$661$	8819.60	+	5092.00i	0.518975	+	0.299630i	−0.217540	+	0.976051i	$0.569803\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1329.40	−	1048.75i	0.0775214	−	0.0611563i
$666$		0			0
$667$	−21428.6			−1.24395
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3879.42	−	6719.35i	0.223194	−	0.386584i
$672$		0			0
$673$	8773.58	+	15196.3i	0.502521	+	0.870392i	0.999996	+	0.00291344i	$0.000927377\pi$
$673$	8773.58	+	15196.3i	0.502521	+	0.870392i	−0.497475	+	0.867478i	$0.665739\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4832.83	−	8370.70i	−0.274358	−	0.475203i	0.695615	−	0.718415i	$-0.255132\pi$
$677$	−4832.83	−	8370.70i	−0.274358	−	0.475203i	−0.969973	+	0.243212i	$0.921799\pi$
$678$		0			0
$679$	−1521.31	−	10466.8i	−0.0859829	−	0.591576i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	4376.97i		−	0.245213i	−0.992455	−	0.122606i	$-0.960875\pi$
$683$		−	4376.97i		−	0.245213i	0.992455	−	0.122606i	$-0.0391253\pi$
$684$		0			0
$685$			18023.7i			1.00533i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19903.4	+	34473.7i	−1.10052	+	1.90616i
$690$		0			0
$691$	−17526.8	+	10119.1i	−0.964905	+	0.557088i	−0.897679	−	0.440649i	$-0.854748\pi$
$691$	−17526.8	+	10119.1i	−0.964905	+	0.557088i	−0.0672260	+	0.997738i	$0.521415\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9889.29	−	5709.58i	0.539744	−	0.311621i
$696$		0			0
$697$	−18447.9	+	31952.8i	−1.00253	+	1.73644i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	33130.3i		−	1.78504i	−0.451004	−	0.892522i	$-0.648934\pi$
$701$		−	33130.3i		−	1.78504i	0.451004	−	0.892522i	$-0.351066\pi$
$702$		0			0
$703$		−	3249.25i		−	0.174321i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12953.5	+	1882.73i	−0.689059	+	0.100152i
$708$		0			0
$709$	17606.6	+	30495.5i	0.932623	+	1.61535i	0.778819	+	0.627249i	$0.215819\pi$
$709$	17606.6	+	30495.5i	0.932623	+	1.61535i	0.153804	+	0.988101i	$0.450847\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3072.14	−	5321.10i	−0.161364	−	0.279491i
$714$		0			0
$715$	−7846.63	+	13590.8i	−0.410416	+	0.710861i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21789.0			−1.13017			−0.565086	−	0.825032i	$-0.691157\pi$
$719$	−21789.0			−1.13017			−0.565086	+	0.825032i	$0.691157\pi$
$720$		0			0
$721$	−3653.55	+	2882.28i	−0.188718	+	0.148879i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−536.555	−	309.780i	−0.0274857	−	0.0158689i
$726$		0			0
$727$	685.180	−	395.589i	0.0349545	−	0.0201810i	−0.482421	−	0.875940i	$-0.660242\pi$
$727$	685.180	−	395.589i	0.0349545	−	0.0201810i	0.517375	+	0.855759i	$0.326909\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3054.50	+	5290.55i	0.154548	+	0.267685i
$732$		0			0
$733$	195.132	+	112.659i	0.00983269	+	0.00567691i	0.504908	−	0.863173i	$-0.331526\pi$
$733$	195.132	+	112.659i	0.00983269	+	0.00567691i	−0.495076	+	0.868850i	$0.664860\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	8041.08i		−	0.401896i
$738$		0			0
$739$	−2564.36			−0.127648			−0.0638238	−	0.997961i	$-0.520330\pi$
$739$	−2564.36			−0.127648			−0.0638238	+	0.997961i	$0.520330\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	30419.9	+	17563.0i	1.50202	+	0.867190i	0.999997	+	0.00233421i	$0.000743004\pi$
$743$	30419.9	+	17563.0i	1.50202	+	0.867190i	0.502020	+	0.864856i	$0.332590\pi$
$744$		0			0
$745$	−16745.4	+	9667.97i	−0.823496	+	0.475446i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	12730.4	+	5073.83i	0.621041	+	0.247522i
$750$		0			0
$751$	−5213.77	+	9030.52i	−0.253333	+	0.438786i	−0.964441	−	0.264297i	$-0.914860\pi$
$751$	−5213.77	+	9030.52i	−0.253333	+	0.438786i	0.711108	+	0.703083i	$0.248194\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−7748.45			−0.373503
$756$		0			0
$757$	−14031.2			−0.673677			−0.336838	−	0.941562i	$-0.609358\pi$
$757$	−14031.2			−0.673677			−0.336838	+	0.941562i	$0.609358\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5435.63	+	9414.79i	−0.258924	+	0.448470i	−0.965954	−	0.258714i	$-0.916701\pi$
$761$	−5435.63	+	9414.79i	−0.258924	+	0.448470i	0.707030	+	0.707184i	$0.250035\pi$
$762$		0			0
$763$	10267.3	−	25761.1i	0.487159	−	1.22230i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	29941.7	−	17286.9i	1.40956	−	0.813811i
$768$		0			0
$769$	17962.0	+	10370.3i	0.842295	+	0.486299i	0.858044	−	0.513577i	$-0.171680\pi$
$769$	17962.0	+	10370.3i	0.842295	+	0.486299i	−0.0157486	+	0.999876i	$0.505013\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7445.62			−0.346443			−0.173221	−	0.984883i	$-0.555418\pi$
$773$	−7445.62			−0.346443			−0.173221	+	0.984883i	$0.555418\pi$
$774$		0			0
$775$		−	177.648i		−	0.00823396i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2484.55	+	1434.46i	0.114272	+	0.0659752i
$780$		0			0
$781$	−5998.52	−	10389.7i	−0.274832	−	0.476023i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6365.97	+	3675.40i	−0.289441	+	0.167109i
$786$		0			0
$787$	−14417.6	−	8323.99i	−0.653025	−	0.377024i	0.136589	−	0.990628i	$-0.456386\pi$
$787$	−14417.6	−	8323.99i	−0.653025	−	0.377024i	−0.789614	+	0.613603i	$0.789719\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	14471.3	−	11416.4i	0.650494	−	0.513172i
$792$		0			0
$793$	43448.7			1.94566
$794$		0			0
$795$		0			0
$796$		0			0
$797$	14318.8	−	24800.9i	0.636383	−	1.10225i	−0.349837	−	0.936810i	$-0.613763\pi$
$797$	14318.8	−	24800.9i	0.636383	−	1.10225i	0.986220	−	0.165437i	$-0.0529035\pi$
$798$		0			0
$799$	−869.829	−	1506.59i	−0.0385136	−	0.0667075i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4051.60	−	7017.58i	−0.178055	−	0.308400i
$804$		0			0
$805$	4189.59	+	28825.1i	0.183433	+	1.26205i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			49.4260i			0.00214799i	0.999999	+	0.00107400i	$0.000341863\pi$
$809$			49.4260i			0.00214799i	−0.999999	+	0.00107400i	$0.999658\pi$
$810$		0			0
$811$			37292.7i			1.61470i	0.590072	+	0.807350i	$0.299099\pi$
$811$			37292.7i			1.61470i	−0.590072	+	0.807350i	$0.700901\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	21581.9	−	37380.9i	0.927584	−	1.60662i
$816$		0			0
$817$	411.377	−	237.509i	0.0176160	−	0.0101706i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	699.390	−	403.793i	0.0297307	−	0.0171650i	−0.485061	−	0.874480i	$-0.661203\pi$
$821$	699.390	−	403.793i	0.0297307	−	0.0171650i	0.514792	+	0.857315i	$0.327869\pi$
$822$		0			0
$823$	10340.2	−	17909.8i	0.437957	−	0.758563i	−0.559575	−	0.828780i	$-0.689036\pi$
$823$	10340.2	−	17909.8i	0.437957	−	0.758563i	0.997532	+	0.0702164i	$0.0223690\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			44485.5i			1.87051i	0.353976	+	0.935255i	$0.384830\pi$
$827$			44485.5i			1.87051i	−0.353976	+	0.935255i	$0.615170\pi$
$828$		0			0
$829$		−	7797.19i		−	0.326668i	−0.986571	−	0.163334i	$-0.947775\pi$
$829$		−	7797.19i		−	0.326668i	0.986571	−	0.163334i	$-0.0522248\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−26627.7	−	25233.8i	−1.10756	−	1.04958i
$834$		0			0
$835$	10628.8	+	18409.7i	0.440509	+	0.762985i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	14737.7	+	25526.4i	0.606439	+	1.05038i	0.991822	+	0.127626i	$0.0407357\pi$
$839$	14737.7	+	25526.4i	0.606439	+	1.05038i	−0.385384	+	0.922756i	$0.625931\pi$
$840$		0			0
$841$	−976.184	+	1690.80i	−0.0400256	+	0.0693263i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−63727.2			−2.59442
$846$		0			0
$847$	12343.6	+	15646.7i	0.500746	+	0.634743i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	48405.6	+	27947.0i	1.94985	+	1.12575i
$852$		0			0
$853$	−8459.43	+	4884.05i	−0.339561	+	0.196045i	−0.660078	−	0.751197i	$-0.729477\pi$
$853$	−8459.43	+	4884.05i	−0.339561	+	0.196045i	0.320517	+	0.947243i	$0.396143\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	11406.2	+	19756.0i	0.454641	+	0.787461i	0.998667	−	0.0516072i	$-0.0164344\pi$
$857$	11406.2	+	19756.0i	0.454641	+	0.787461i	−0.544027	+	0.839068i	$0.683101\pi$
$858$		0			0
$859$	−30803.4	−	17784.3i	−1.22351	−	0.706396i	−0.257848	−	0.966186i	$-0.583013\pi$
$859$	−30803.4	−	17784.3i	−1.22351	−	0.706396i	−0.965665	+	0.259790i	$0.916347\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	34594.9i		−	1.36457i	−0.731087	−	0.682285i	$-0.760986\pi$
$863$		−	34594.9i		−	1.36457i	0.731087	−	0.682285i	$-0.239014\pi$
$864$		0			0
$865$	−18689.9			−0.734653
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−10551.8	−	6092.10i	−0.411906	−	0.237814i
$870$		0			0
$871$	38996.5	−	22514.6i	1.51704	−	0.875865i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9734.71	+	24424.7i	−0.376106	+	0.943664i
$876$		0			0
$877$	11651.8	−	20181.6i	0.448637	−	0.777063i	−0.549660	−	0.835388i	$-0.685243\pi$
$877$	11651.8	−	20181.6i	0.448637	−	0.777063i	0.998298	+	0.0583256i	$0.0185762\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−8387.34			−0.320745			−0.160373	−	0.987057i	$-0.551270\pi$
$881$	−8387.34			−0.320745			−0.160373	+	0.987057i	$0.551270\pi$
$882$		0			0
$883$	4287.17			0.163391			0.0816957	−	0.996657i	$-0.473966\pi$
$883$	4287.17			0.163391			0.0816957	+	0.996657i	$0.473966\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6730.67	−	11657.9i	0.254784	−	0.441299i	−0.710053	−	0.704149i	$-0.751329\pi$
$887$	6730.67	−	11657.9i	0.254784	−	0.441299i	0.964837	+	0.262849i	$0.0846622\pi$
$888$		0			0
$889$	16932.5	−	42484.1i	0.638803	−	1.60278i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−117.148	+	67.6353i	−0.00438992	+	0.00253452i
$894$		0			0
$895$	−11188.5	−	6459.68i	−0.417866	−	0.241255i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6433.32			0.238669
$900$		0			0
$901$		−	47618.7i		−	1.76072i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−33388.8	−	19277.0i	−1.22639	−	0.708056i
$906$		0			0
$907$	−6942.32	−	12024.4i	−0.254152	−	0.440204i	0.710513	−	0.703684i	$-0.248463\pi$
$907$	−6942.32	−	12024.4i	−0.254152	−	0.440204i	−0.964665	+	0.263480i	$0.915130\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−45460.5	+	26246.6i	−1.65332	+	0.954543i	−0.677622	+	0.735410i	$0.736989\pi$
$911$	−45460.5	+	26246.6i	−1.65332	+	0.954543i	−0.975695	+	0.219133i	$0.929677\pi$
$912$		0			0
$913$	−16792.5	−	9695.13i	−0.608707	−	0.351437i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	17660.1	+	22385.9i	0.635975	+	0.806158i
$918$		0			0
$919$	29789.5			1.06928			0.534639	−	0.845081i	$-0.320448\pi$
$919$	29789.5			1.06928			0.534639	+	0.845081i	$0.320448\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	33591.1	−	58181.5i	1.19790	−	2.07483i
$924$		0			0
$925$	808.026	+	1399.54i	0.0287219	+	0.0497478i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9157.11	+	15860.6i	0.323396	+	0.560139i	0.981186	−	0.193062i	$-0.0618420\pi$
$929$	9157.11	+	15860.6i	0.323396	+	0.560139i	−0.657790	+	0.753201i	$0.728509\pi$
$930$		0			0
$931$	−1962.11	+	2070.49i	−0.0690714	+	0.0728867i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	18773.0i		−	0.656623i
$936$		0			0
$937$		−	4717.18i		−	0.164465i	−0.996613	−	0.0822325i	$-0.973795\pi$
$937$		−	4717.18i		−	0.164465i	0.996613	−	0.0822325i	$-0.0262050\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−20464.4	+	35445.4i	−0.708950	+	1.22794i	0.256298	+	0.966598i	$0.417497\pi$
$941$	−20464.4	+	35445.4i	−0.708950	+	1.22794i	−0.965247	+	0.261339i	$0.915836\pi$
$942$		0			0
$943$	−42739.5	+	24675.7i	−1.47592	+	0.852122i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21626.2	+	12485.9i	−0.742088	+	0.428445i	−0.822828	−	0.568290i	$-0.807605\pi$
$947$	−21626.2	+	12485.9i	−0.742088	+	0.428445i	0.0807398	+	0.996735i	$0.474272\pi$
$948$		0			0
$949$	22688.5	−	39297.7i	0.776081	−	1.34421i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			14607.3i			0.496512i	0.968694	+	0.248256i	$0.0798574\pi$
$953$			14607.3i			0.496512i	−0.968694	+	0.248256i	$0.920143\pi$
$954$		0			0
$955$		−	15590.7i		−	0.528276i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4367.23	−	30047.2i	−0.147054	−	1.01176i
$960$		0			0
$961$	−13973.2	−	24202.3i	−0.469040	−	0.812402i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8210.38	+	14220.8i	0.273888	+	0.474387i
$966$		0			0
$967$	13828.2	−	23951.1i	0.459859	−	0.796499i	−0.539094	−	0.842245i	$-0.681233\pi$
$967$	13828.2	−	23951.1i	0.459859	−	0.796499i	0.998953	+	0.0457467i	$0.0145667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5235.20			−0.173023			−0.0865117	−	0.996251i	$-0.527572\pi$
$971$	−5235.20			−0.173023			−0.0865117	+	0.996251i	$0.527572\pi$
$972$		0			0
$973$	−15102.9	+	11914.6i	−0.497612	+	0.392564i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3972.26	−	2293.38i	−0.130076	−	0.0750991i	0.433550	−	0.901129i	$-0.357261\pi$
$977$	−3972.26	−	2293.38i	−0.130076	−	0.0750991i	−0.563626	+	0.826030i	$0.690594\pi$
$978$		0			0
$979$	5886.47	−	3398.55i	0.192168	−	0.110948i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1967.14	−	3407.18i	−0.0638270	−	0.110552i	0.832346	−	0.554256i	$-0.186997\pi$
$983$	−1967.14	−	3407.18i	−0.0638270	−	0.110552i	−0.896173	+	0.443705i	$0.853664\pi$
$984$		0			0
$985$	−22701.9	−	13106.9i	−0.734358	−	0.423982i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			8171.31i			0.262722i
$990$		0			0
$991$	−17434.1			−0.558841			−0.279421	−	0.960169i	$-0.590142\pi$
$991$	−17434.1			−0.558841			−0.279421	+	0.960169i	$0.590142\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	47226.0	+	27266.0i	1.50469	+	0.868733i
$996$		0			0
$997$	12254.1	−	7074.88i	0.389258	−	0.224738i	−0.292581	−	0.956241i	$-0.594514\pi$
$997$	12254.1	−	7074.88i	0.389258	−	0.224738i	0.681838	+	0.731503i	$0.261181\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.x.a.125.6		48
3.2	odd	2		252.4.x.a.41.14	yes	48
7.6	odd	2	inner	756.4.x.a.125.19		48
9.2	odd	6	inner	756.4.x.a.629.19		48
9.4	even	3		2268.4.f.a.1133.37		48
9.5	odd	6		2268.4.f.a.1133.12		48
9.7	even	3		252.4.x.a.209.11	yes	48
21.20	even	2		252.4.x.a.41.11	✓	48
63.13	odd	6		2268.4.f.a.1133.11		48
63.20	even	6	inner	756.4.x.a.629.6		48
63.34	odd	6		252.4.x.a.209.14	yes	48
63.41	even	6		2268.4.f.a.1133.38		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.11	✓	48	21.20	even	2
252.4.x.a.41.14	yes	48	3.2	odd	2
252.4.x.a.209.11	yes	48	9.7	even	3
252.4.x.a.209.14	yes	48	63.34	odd	6
756.4.x.a.125.6		48	1.1	even	1	trivial
756.4.x.a.125.19		48	7.6	odd	2	inner
756.4.x.a.629.6		48	63.20	even	6	inner
756.4.x.a.629.19		48	9.2	odd	6	inner
2268.4.f.a.1133.11		48	63.13	odd	6
2268.4.f.a.1133.12		48	9.5	odd	6
2268.4.f.a.1133.37		48	9.4	even	3
2268.4.f.a.1133.38		48	63.41	even	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.x (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.49690 + 9.52092i) q^{5} +(6.85688 - 17.2042i) q^{7} +O(q^{10})\)	\(q+(-5.49690 + 9.52092i) q^{5} +(6.85688 - 17.2042i) q^{7} +(-13.8268 + 7.98293i) q^{11} +(-77.4289 - 44.7036i) q^{13} +106.953 q^{17} -8.31634i q^{19} +(123.893 + 71.5294i) q^{23} +(2.06812 + 3.58208i) q^{25} +(-129.721 + 74.8943i) q^{29} +(-37.1952 - 21.4747i) q^{31} +(126.108 + 159.853i) q^{35} +390.706 q^{37} +(-172.486 + 298.755i) q^{41} +(28.5593 + 49.4661i) q^{43} +(-8.13282 - 14.0864i) q^{47} +(-248.966 - 235.934i) q^{49} -445.230i q^{53} -175.526i q^{55} +(-193.350 + 334.892i) q^{59} +(-420.857 + 242.982i) q^{61} +(851.238 - 491.463i) q^{65} +(-251.821 + 436.167i) q^{67} +751.418i q^{71} +507.533i q^{73} +(42.5306 + 292.617i) q^{77} +(381.570 + 660.899i) q^{79} +(607.241 + 1051.77i) q^{83} +(-587.911 + 1018.29i) q^{85} -425.727 q^{89} +(-1300.01 + 1025.57i) q^{91} +(79.1792 + 45.7141i) q^{95} +(494.582 - 285.547i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 6 q^{7}+O(q^{10}) \) 48 * q + 6 * q^7	\( 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) \) 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more