LMFDB - Embedded newform 1764.4.k.bd.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, 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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 27x^{6} + 10x^{5} + 446x^{4} + 62x^{3} + 3061x^{2} + 2142x + 14161 $ x^8 - 2x^7 + 27x^6 + 10x^5 + 446x^4 + 62x^3 + 3061x^2 + 2142*x + 14161
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$1.44795 - 2.50793i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.bd.1549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.32734 - 14.4234i) q^{5} +O(q^{10})$	$q+(-8.32734 - 14.4234i) q^{5} +(-35.8800 + 62.1460i) q^{11} -65.3878 q^{13} +(45.2158 - 78.3160i) q^{17} +(-81.9092 - 141.871i) q^{19} +(39.6144 + 68.6142i) q^{23} +(-76.1893 + 131.964i) q^{25} +43.2995 q^{29} +(-67.8179 + 117.464i) q^{31} +(-135.370 - 234.468i) q^{37} -152.241 q^{41} -177.641 q^{43} +(-22.8166 - 39.5194i) q^{47} +(-79.2188 + 137.211i) q^{53} +1195.14 q^{55} +(-195.884 + 339.282i) q^{59} +(-275.647 - 477.435i) q^{61} +(544.506 + 943.113i) q^{65} +(-229.315 + 397.185i) q^{67} +486.786 q^{71} +(-287.446 + 497.871i) q^{73} +(334.160 + 578.782i) q^{79} -76.2450 q^{83} -1506.11 q^{85} +(683.401 + 1183.69i) q^{89} +(-1364.17 + 2362.82i) q^{95} +242.655 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 48 q^{17} - 192 q^{19} + 192 q^{23} - 324 q^{25} - 192 q^{29} - 48 q^{31} - 256 q^{37} - 2016 q^{41} - 224 q^{43} + 864 q^{47} - 648 q^{53} + 4704 q^{55} + 336 q^{59} - 960 q^{61} - 360 q^{65} - 720 q^{67} + 2688 q^{71} - 672 q^{73} + 1984 q^{79} - 6240 q^{83} + 1360 q^{85} + 2160 q^{89} - 3744 q^{95} + 4032 q^{97}+O(q^{100}) $ 8 * q + 48 * q^17 - 192 * q^19 + 192 * q^23 - 324 * q^25 - 192 * q^29 - 48 * q^31 - 256 * q^37 - 2016 * q^41 - 224 * q^43 + 864 * q^47 - 648 * q^53 + 4704 * q^55 + 336 * q^59 - 960 * q^61 - 360 * q^65 - 720 * q^67 + 2688 * q^71 - 672 * q^73 + 1984 * q^79 - 6240 * q^83 + 1360 * q^85 + 2160 * q^89 - 3744 * q^95 + 4032 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−8.32734	−	14.4234i	−0.744820	−	1.29007i	−0.950279	−	0.311401i	$-0.899202\pi$
$5$	−8.32734	−	14.4234i	−0.744820	−	1.29007i	0.205459	−	0.978666i	$-0.434131\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−35.8800	+	62.1460i	−0.983475	+	1.70343i	−0.334949	+	0.942236i	$0.608719\pi$
$11$	−35.8800	+	62.1460i	−0.983475	+	1.70343i	−0.648526	+	0.761192i	$0.724614\pi$
$12$		0			0
$13$	−65.3878			−1.39502			−0.697512	−	0.716573i	$-0.745709\pi$
$13$	−65.3878			−1.39502			−0.697512	+	0.716573i	$0.745709\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	45.2158	−	78.3160i	0.645085	−	1.11732i	−0.339197	−	0.940715i	$-0.610155\pi$
$17$	45.2158	−	78.3160i	0.645085	−	1.11732i	0.984282	−	0.176604i	$-0.0565112\pi$
$18$		0			0
$19$	−81.9092	−	141.871i	−0.989014	−	1.71302i	−0.622526	−	0.782599i	$-0.713893\pi$
$19$	−81.9092	−	141.871i	−0.989014	−	1.71302i	−0.366488	−	0.930423i	$-0.619440\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	39.6144	+	68.6142i	0.359138	+	0.622045i	0.987817	−	0.155619i	$-0.0497374\pi$
$23$	39.6144	+	68.6142i	0.359138	+	0.622045i	−0.628679	+	0.777665i	$0.716404\pi$
$24$		0			0
$25$	−76.1893	+	131.964i	−0.609514	+	1.05571i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	43.2995			0.277259			0.138630	−	0.990344i	$-0.455730\pi$
$29$	43.2995			0.277259			0.138630	+	0.990344i	$0.455730\pi$
$30$		0			0
$31$	−67.8179	+	117.464i	−0.392918	+	0.680554i	−0.992833	−	0.119510i	$-0.961868\pi$
$31$	−67.8179	+	117.464i	−0.392918	+	0.680554i	0.599915	+	0.800064i	$0.295201\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−135.370	−	234.468i	−0.601479	−	1.04179i	−0.992597	−	0.121452i	$-0.961245\pi$
$37$	−135.370	−	234.468i	−0.601479	−	1.04179i	0.391118	−	0.920340i	$-0.372088\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−152.241			−0.579902			−0.289951	−	0.957041i	$-0.593639\pi$
$41$	−152.241			−0.579902			−0.289951	+	0.957041i	$0.593639\pi$
$42$		0			0
$43$	−177.641			−0.630001			−0.315001	−	0.949091i	$-0.602005\pi$
$43$	−177.641			−0.630001			−0.315001	+	0.949091i	$0.602005\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−22.8166	−	39.5194i	−0.0708114	−	0.122649i	0.828446	−	0.560069i	$-0.189226\pi$
$47$	−22.8166	−	39.5194i	−0.0708114	−	0.122649i	−0.899257	+	0.437420i	$0.855892\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−79.2188	+	137.211i	−0.205312	+	0.355611i	−0.950232	−	0.311543i	$-0.899154\pi$
$53$	−79.2188	+	137.211i	−0.205312	+	0.355611i	0.744920	+	0.667154i	$0.232488\pi$
$54$		0			0
$55$	1195.14			2.93005
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−195.884	+	339.282i	−0.432237	+	0.748656i	−0.997066	−	0.0765519i	$-0.975609\pi$
$59$	−195.884	+	339.282i	−0.432237	+	0.748656i	0.564829	+	0.825208i	$0.308942\pi$
$60$		0			0
$61$	−275.647	−	477.435i	−0.578574	−	1.00212i	−0.995643	−	0.0932450i	$-0.970276\pi$
$61$	−275.647	−	477.435i	−0.578574	−	1.00212i	0.417069	−	0.908875i	$-0.363057\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	544.506	+	943.113i	1.03904	+	1.79967i
$66$		0			0
$67$	−229.315	+	397.185i	−0.418139	+	0.724237i	−0.995752	−	0.0920729i	$-0.970651\pi$
$67$	−229.315	+	397.185i	−0.418139	+	0.724237i	0.577614	+	0.816310i	$0.303984\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	486.786			0.813674			0.406837	−	0.913501i	$-0.366632\pi$
$71$	486.786			0.813674			0.406837	+	0.913501i	$0.366632\pi$
$72$		0			0
$73$	−287.446	+	497.871i	−0.460863	+	0.798238i	−0.999004	−	0.0446164i	$-0.985793\pi$
$73$	−287.446	+	497.871i	−0.460863	+	0.798238i	0.538141	+	0.842855i	$0.319127\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	334.160	+	578.782i	0.475897	+	0.824279i	0.999619	−	0.0276111i	$-0.00878999\pi$
$79$	334.160	+	578.782i	0.475897	+	0.824279i	−0.523721	+	0.851890i	$0.675457\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−76.2450			−0.100831			−0.0504155	−	0.998728i	$-0.516055\pi$
$83$	−76.2450			−0.100831			−0.0504155	+	0.998728i	$0.516055\pi$
$84$		0			0
$85$	−1506.11			−1.92189
$86$		0			0
$87$		0			0
$88$		0			0
$89$	683.401	+	1183.69i	0.813937	+	1.40978i	0.910089	+	0.414413i	$0.136013\pi$
$89$	683.401	+	1183.69i	0.813937	+	1.40978i	−0.0961518	+	0.995367i	$0.530653\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1364.17	+	2362.82i	−1.47327	+	2.55179i
$96$		0			0
$97$	242.655			0.253999			0.127000	−	0.991903i	$-0.459465\pi$
$97$	242.655			0.253999			0.127000	+	0.991903i	$0.459465\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	347.371	−	601.665i	0.342225	−	0.592751i	−0.642620	−	0.766185i	$-0.722153\pi$
$101$	347.371	−	601.665i	0.342225	−	0.592751i	0.984846	+	0.173433i	$0.0554861\pi$
$102$		0			0
$103$	87.1089	+	150.877i	0.0833310	+	0.144334i	0.904679	−	0.426094i	$-0.140111\pi$
$103$	87.1089	+	150.877i	0.0833310	+	0.144334i	−0.821348	+	0.570428i	$0.806777\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	23.8025	+	41.2272i	0.0215054	+	0.0372485i	0.876578	−	0.481260i	$-0.159821\pi$
$107$	23.8025	+	41.2272i	0.0215054	+	0.0372485i	−0.855072	+	0.518509i	$0.826487\pi$
$108$		0			0
$109$	816.549	−	1414.30i	0.717534	−	1.24281i	−0.244440	−	0.969664i	$-0.578604\pi$
$109$	816.549	−	1414.30i	0.717534	−	1.24281i	0.961974	−	0.273141i	$-0.0880626\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1127.29			0.938464			0.469232	−	0.883075i	$-0.344531\pi$
$113$	1127.29			0.938464			0.469232	+	0.883075i	$0.344531\pi$
$114$		0			0
$115$	659.766	−	1142.75i	0.534987	−	0.926624i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1909.25	−	3306.91i	−1.43445	−	2.48453i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	455.982			0.326274
$126$		0			0
$127$	2398.66			1.67596			0.837979	−	0.545702i	$-0.183737\pi$
$127$	2398.66			1.67596			0.837979	+	0.545702i	$0.183737\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	774.482	+	1341.44i	0.516541	+	0.894675i	0.999816	+	0.0192062i	$0.00611389\pi$
$131$	774.482	+	1341.44i	0.516541	+	0.894675i	−0.483275	+	0.875469i	$0.660553\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	761.634	−	1319.19i	0.474969	−	0.822671i	−0.524620	−	0.851337i	$-0.675793\pi$
$137$	761.634	−	1319.19i	0.474969	−	0.822671i	0.999589	+	0.0286657i	$0.00912583\pi$
$138$		0			0
$139$	551.445			0.336496			0.168248	−	0.985745i	$-0.446189\pi$
$139$	551.445			0.336496			0.168248	+	0.985745i	$0.446189\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2346.11	−	4063.59i	1.37197	−	2.37632i
$144$		0			0
$145$	−360.570	−	624.526i	−0.206508	−	0.357683i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1200.12	−	2078.67i	−0.659851	−	1.14290i	−0.980654	−	0.195749i	$-0.937286\pi$
$149$	−1200.12	−	2078.67i	−0.659851	−	1.14290i	0.320803	−	0.947146i	$-0.396047\pi$
$150$		0			0
$151$	−310.780	+	538.287i	−0.167490	+	0.290100i	−0.937537	−	0.347887i	$-0.886899\pi$
$151$	−310.780	+	538.287i	−0.167490	+	0.290100i	0.770047	+	0.637987i	$0.220233\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2258.97			1.17061
$156$		0			0
$157$	1345.79	−	2330.97i	0.684112	−	1.18492i	−0.289603	−	0.957147i	$-0.593523\pi$
$157$	1345.79	−	2330.97i	0.684112	−	1.18492i	0.973715	−	0.227770i	$-0.0731433\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−528.653	−	915.654i	−0.254032	−	0.439997i	0.710600	−	0.703596i	$-0.248424\pi$
$163$	−528.653	−	915.654i	−0.254032	−	0.439997i	−0.964632	+	0.263599i	$0.915090\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	243.840			0.112987			0.0564937	−	0.998403i	$-0.482008\pi$
$167$	243.840			0.112987			0.0564937	+	0.998403i	$0.482008\pi$
$168$		0			0
$169$	2078.56			0.946090
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−843.632	−	1461.21i	−0.370753	−	0.642162i	0.618929	−	0.785447i	$-0.287567\pi$
$173$	−843.632	−	1461.21i	−0.370753	−	0.642162i	−0.989682	+	0.143285i	$0.954234\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−179.570	+	311.024i	−0.0749814	+	0.129872i	−0.901078	−	0.433657i	$-0.857223\pi$
$179$	−179.570	+	311.024i	−0.0749814	+	0.129872i	0.826097	+	0.563528i	$0.190556\pi$
$180$		0			0
$181$	2982.58			1.22483			0.612413	−	0.790538i	$-0.290199\pi$
$181$	2982.58			1.22483			0.612413	+	0.790538i	$0.290199\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2254.55	+	3904.99i	−0.895988	+	1.55190i
$186$		0			0
$187$	3244.68	+	5619.96i	1.26885	+	2.19771i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1745.39	+	3023.10i	0.661213	+	1.14525i	0.980297	+	0.197528i	$0.0632914\pi$
$191$	1745.39	+	3023.10i	0.661213	+	1.14525i	−0.319084	+	0.947726i	$0.603375\pi$
$192$		0			0
$193$	−802.114	+	1389.30i	−0.299158	+	0.518156i	−0.975943	−	0.218024i	$-0.930039\pi$
$193$	−802.114	+	1389.30i	−0.299158	+	0.518156i	0.676786	+	0.736180i	$0.263372\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	455.935			0.164893			0.0824467	−	0.996595i	$-0.473727\pi$
$197$	455.935			0.164893			0.0824467	+	0.996595i	$0.473727\pi$
$198$		0			0
$199$	−1042.08	+	1804.94i	−0.371212	+	0.642958i	−0.989752	−	0.142795i	$-0.954391\pi$
$199$	−1042.08	+	1804.94i	−0.371212	+	0.642958i	0.618541	+	0.785753i	$0.287724\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1267.76	+	2195.83i	0.431923	+	0.748113i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11755.6			3.89068
$210$		0			0
$211$	2568.06			0.837880			0.418940	−	0.908014i	$-0.362402\pi$
$211$	2568.06			0.837880			0.418940	+	0.908014i	$0.362402\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1479.28	+	2562.19i	0.469238	+	0.812743i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2956.56	+	5120.91i	−0.899908	+	1.55869i
$222$		0			0
$223$	−4659.55			−1.39922			−0.699612	−	0.714523i	$-0.746644\pi$
$223$	−4659.55			−1.39922			−0.699612	+	0.714523i	$0.746644\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2414.39	−	4181.84i	0.705940	−	1.22272i	−0.260411	−	0.965498i	$-0.583858\pi$
$227$	2414.39	−	4181.84i	0.705940	−	1.22272i	0.966351	−	0.257227i	$-0.0828087\pi$
$228$		0			0
$229$	1611.29	+	2790.84i	0.464966	+	0.805345i	0.999200	−	0.0399917i	$-0.0127331\pi$
$229$	1611.29	+	2790.84i	0.464966	+	0.805345i	−0.534234	+	0.845337i	$0.679400\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2851.21	−	4938.44i	−0.801670	−	1.38853i	−0.918517	−	0.395383i	$-0.870612\pi$
$233$	−2851.21	−	4938.44i	−0.801670	−	1.38853i	0.116847	−	0.993150i	$-0.462721\pi$
$234$		0			0
$235$	−380.003	+	658.184i	−0.105484	+	0.182703i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1994.47			−0.539796			−0.269898	−	0.962889i	$-0.586990\pi$
$239$	−1994.47			−0.539796			−0.269898	+	0.962889i	$0.586990\pi$
$240$		0			0
$241$	710.443	−	1230.52i	0.189891	−	0.328900i	−0.755323	−	0.655353i	$-0.772520\pi$
$241$	710.443	−	1230.52i	0.189891	−	0.328900i	0.945214	+	0.326452i	$0.105853\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	5355.86	+	9276.62i	1.37970	+	2.38971i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3855.44			0.969534			0.484767	−	0.874643i	$-0.338904\pi$
$251$	3855.44			0.969534			0.484767	+	0.874643i	$0.338904\pi$
$252$		0			0
$253$	−5685.46			−1.41281
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2026.64	+	3510.24i	0.491900	+	0.851996i	0.999956	−	0.00932793i	$-0.00296922\pi$
$257$	2026.64	+	3510.24i	0.491900	+	0.851996i	−0.508056	+	0.861324i	$0.669636\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−250.393	+	433.694i	−0.0587069	+	0.101683i	−0.893885	−	0.448296i	$-0.852031\pi$
$263$	−250.393	+	433.694i	−0.0587069	+	0.101683i	0.835178	+	0.549979i	$0.185364\pi$
$264$		0			0
$265$	2638.73			0.611683
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1640.87	−	2842.07i	0.371917	−	0.644179i	−0.617943	−	0.786223i	$-0.712034\pi$
$269$	1640.87	−	2842.07i	0.371917	−	0.644179i	0.989860	+	0.142043i	$0.0453672\pi$
$270$		0			0
$271$	−1662.18	−	2878.98i	−0.372584	−	0.645334i	0.617378	−	0.786666i	$-0.288195\pi$
$271$	−1662.18	−	2878.98i	−0.372584	−	0.645334i	−0.989962	+	0.141332i	$0.954861\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5467.34	−	9469.71i	−1.19888	−	2.07653i
$276$		0			0
$277$	−4309.87	+	7464.92i	−0.934856	+	1.61922i	−0.159966	+	0.987122i	$0.551139\pi$
$277$	−4309.87	+	7464.92i	−0.934856	+	1.61922i	−0.774890	+	0.632096i	$0.782195\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7710.80			1.63697			0.818484	−	0.574529i	$-0.194815\pi$
$281$	7710.80			1.63697			0.818484	+	0.574529i	$0.194815\pi$
$282$		0			0
$283$	69.6354	−	120.612i	0.0146268	−	0.0253344i	−0.858619	−	0.512614i	$-0.828677\pi$
$283$	69.6354	−	120.612i	0.0146268	−	0.0253344i	0.873246	+	0.487279i	$0.162011\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1632.43	−	2827.46i	−0.332268	−	0.575506i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4437.09			0.884702			0.442351	−	0.896842i	$-0.354145\pi$
$293$	4437.09			0.884702			0.442351	+	0.896842i	$0.354145\pi$
$294$		0			0
$295$	6524.79			1.28776
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2590.30	−	4486.53i	−0.501006	−	0.867768i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4590.82	+	7951.53i	−0.861867	+	1.49280i
$306$		0			0
$307$	−299.480			−0.0556749			−0.0278375	−	0.999612i	$-0.508862\pi$
$307$	−299.480			−0.0556749			−0.0278375	+	0.999612i	$0.508862\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	456.972	−	791.498i	0.0833199	−	0.144314i	−0.821354	−	0.570419i	$-0.806781\pi$
$311$	456.972	−	791.498i	0.0833199	−	0.144314i	0.904674	+	0.426104i	$0.140114\pi$
$312$		0			0
$313$	−1943.69	−	3366.57i	−0.351002	−	0.607954i	0.635423	−	0.772164i	$-0.280826\pi$
$313$	−1943.69	−	3366.57i	−0.351002	−	0.607954i	−0.986425	+	0.164210i	$0.947492\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1439.65	+	2493.55i	0.255075	+	0.441803i	0.964916	−	0.262559i	$-0.0845664\pi$
$317$	1439.65	+	2493.55i	0.255075	+	0.441803i	−0.709841	+	0.704362i	$0.751233\pi$
$318$		0			0
$319$	−1553.59	+	2690.89i	−0.272678	+	0.472291i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−14814.4			−2.55199
$324$		0			0
$325$	4981.85	−	8628.81i	0.850287	−	1.47274i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1405.39	+	2434.21i	0.233376	+	0.404219i	0.958799	−	0.284084i	$-0.0916894\pi$
$331$	1405.39	+	2434.21i	0.233376	+	0.404219i	−0.725424	+	0.688303i	$0.758356\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7638.34			1.24575
$336$		0			0
$337$	2960.90			0.478607			0.239303	−	0.970945i	$-0.423081\pi$
$337$	2960.90			0.478607			0.239303	+	0.970945i	$0.423081\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4866.61	−	8429.22i	−0.772850	−	1.33862i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2142.26	+	3710.50i	−0.331419	+	0.574034i	−0.982790	−	0.184725i	$-0.940861\pi$
$347$	−2142.26	+	3710.50i	−0.331419	+	0.574034i	0.651371	+	0.758759i	$0.274194\pi$
$348$		0			0
$349$	3295.36			0.505434			0.252717	−	0.967540i	$-0.418676\pi$
$349$	3295.36			0.505434			0.252717	+	0.967540i	$0.418676\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5314.83	+	9205.56i	−0.801360	+	1.38800i	0.117362	+	0.993089i	$0.462556\pi$
$353$	−5314.83	+	9205.56i	−0.801360	+	1.38800i	−0.918721	+	0.394906i	$0.870777\pi$
$354$		0			0
$355$	−4053.63	−	7021.09i	−0.606040	−	1.04969i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1592.51	+	2758.30i	0.234121	+	0.405509i	0.959017	−	0.283349i	$-0.0914456\pi$
$359$	1592.51	+	2758.30i	0.234121	+	0.405509i	−0.724896	+	0.688858i	$0.758112\pi$
$360$		0			0
$361$	−9988.74	+	17301.0i	−1.45630	+	2.52238i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9574.65			1.37304
$366$		0			0
$367$	−3801.52	+	6584.42i	−0.540702	+	0.936523i	0.458162	+	0.888869i	$0.348508\pi$
$367$	−3801.52	+	6584.42i	−0.540702	+	0.936523i	−0.998864	+	0.0476544i	$0.984825\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−755.978	−	1309.39i	−0.104941	−	0.181763i	0.808773	−	0.588121i	$-0.200132\pi$
$373$	−755.978	−	1309.39i	−0.104941	−	0.181763i	−0.913714	+	0.406358i	$0.866799\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2831.26			−0.386783
$378$		0			0
$379$	8848.36			1.19923			0.599617	−	0.800287i	$-0.295320\pi$
$379$	8848.36			1.19923			0.599617	+	0.800287i	$0.295320\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2067.05	+	3580.24i	0.275774	+	0.477654i	0.970330	−	0.241784i	$-0.0777326\pi$
$383$	2067.05	+	3580.24i	0.275774	+	0.477654i	−0.694556	+	0.719438i	$0.744399\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3815.30	−	6608.30i	0.497284	−	0.861322i	−0.502711	−	0.864455i	$-0.667664\pi$
$389$	3815.30	−	6608.30i	0.497284	−	0.861322i	0.999995	+	0.00313295i	$0.000997251\pi$
$390$		0			0
$391$	7164.79			0.926698
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5565.32	−	9639.43i	0.708916	−	1.22788i
$396$		0			0
$397$	5809.25	+	10061.9i	0.734403	+	1.27202i	0.954985	+	0.296654i	$0.0958709\pi$
$397$	5809.25	+	10061.9i	0.734403	+	1.27202i	−0.220582	+	0.975368i	$0.570796\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1523.51	+	2638.79i	0.189726	+	0.328616i	0.945159	−	0.326611i	$-0.105907\pi$
$401$	1523.51	+	2638.79i	0.189726	+	0.328616i	−0.755433	+	0.655226i	$0.772573\pi$
$402$		0			0
$403$	4434.46	−	7680.71i	0.548130	−	0.949388i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	19428.3			2.36616
$408$		0			0
$409$	−5791.42	+	10031.0i	−0.700164	+	1.21272i	0.268244	+	0.963351i	$0.413557\pi$
$409$	−5791.42	+	10031.0i	−0.700164	+	1.21272i	−0.968408	+	0.249369i	$0.919777\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	634.918	+	1099.71i	0.0751010	+	0.130079i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1318.83			0.153768			0.0768841	−	0.997040i	$-0.475503\pi$
$419$	1318.83			0.153768			0.0768841	+	0.997040i	$0.475503\pi$
$420$		0			0
$421$	−9733.56			−1.12680			−0.563402	−	0.826183i	$-0.690508\pi$
$421$	−9733.56			−1.12680			−0.563402	+	0.826183i	$0.690508\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6889.92	+	11933.7i	0.786377	+	1.36204i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7214.47	+	12495.8i	−0.806284	+	1.39653i	0.109136	+	0.994027i	$0.465192\pi$
$431$	−7214.47	+	12495.8i	−0.806284	+	1.39653i	−0.915421	+	0.402499i	$0.868142\pi$
$432$		0			0
$433$	−16620.3			−1.84463			−0.922313	−	0.386444i	$-0.873703\pi$
$433$	−16620.3			−1.84463			−0.922313	+	0.386444i	$0.873703\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6489.57	−	11240.3i	0.710385	−	1.23042i
$438$		0			0
$439$	257.783	+	446.493i	0.0280257	+	0.0485420i	0.879698	−	0.475533i	$-0.157745\pi$
$439$	257.783	+	446.493i	0.0280257	+	0.0485420i	−0.851672	+	0.524075i	$0.824411\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4488.05	+	7773.52i	0.481340	+	0.833705i	0.999771	−	0.0214146i	$-0.00681700\pi$
$443$	4488.05	+	7773.52i	0.481340	+	0.833705i	−0.518431	+	0.855120i	$0.673484\pi$
$444$		0			0
$445$	11381.8	−	19713.9i	1.21247	−	2.10007i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2106.83			−0.221442			−0.110721	−	0.993852i	$-0.535316\pi$
$449$	−2106.83			−0.221442			−0.110721	+	0.993852i	$0.535316\pi$
$450$		0			0
$451$	5462.39	−	9461.14i	0.570319	−	0.987822i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7863.08	−	13619.3i	−0.804857	−	1.39405i	−0.916388	−	0.400292i	$-0.868909\pi$
$457$	−7863.08	−	13619.3i	−0.804857	−	1.39405i	0.111531	−	0.993761i	$-0.464425\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7921.16			0.800272			0.400136	−	0.916456i	$-0.368963\pi$
$461$	7921.16			0.800272			0.400136	+	0.916456i	$0.368963\pi$
$462$		0			0
$463$	−5821.20			−0.584306			−0.292153	−	0.956372i	$-0.594372\pi$
$463$	−5821.20			−0.584306			−0.292153	+	0.956372i	$0.594372\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5893.94	+	10208.6i	0.584024	+	1.01156i	0.994996	+	0.0999112i	$0.0318559\pi$
$467$	5893.94	+	10208.6i	0.584024	+	1.01156i	−0.410973	+	0.911648i	$0.634811\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6373.77	−	11039.7i	0.619590	−	1.07316i
$474$		0			0
$475$	24962.4			2.41127
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2657.21	+	4602.43i	−0.253468	+	0.439020i	−0.964478	−	0.264162i	$-0.914905\pi$
$479$	−2657.21	+	4602.43i	−0.253468	+	0.439020i	0.711010	+	0.703182i	$0.248238\pi$
$480$		0			0
$481$	8851.56	+	15331.3i	0.839078	+	1.45332i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2020.67	−	3499.91i	−0.189184	−	0.327676i
$486$		0			0
$487$	9331.34	−	16162.4i	0.868261	−	1.50387i	0.00448933	−	0.999990i	$-0.498571\pi$
$487$	9331.34	−	16162.4i	0.868261	−	1.50387i	0.863772	−	0.503883i	$-0.168096\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−13519.1			−1.24259			−0.621293	−	0.783578i	$-0.713392\pi$
$491$	−13519.1			−1.24259			−0.621293	+	0.783578i	$0.713392\pi$
$492$		0			0
$493$	1957.82	−	3391.05i	0.178856	−	0.309787i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−6916.17	−	11979.2i	−0.620462	−	1.07467i	−0.989400	−	0.145217i	$-0.953612\pi$
$499$	−6916.17	−	11979.2i	−0.620462	−	1.07467i	0.368938	−	0.929454i	$-0.379721\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−13469.0			−1.19394			−0.596970	−	0.802264i	$-0.703629\pi$
$503$	−13469.0			−1.19394			−0.596970	+	0.802264i	$0.703629\pi$
$504$		0			0
$505$	−11570.7			−1.01958
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4387.62	+	7599.59i	0.382079	+	0.661779i	0.991359	−	0.131175i	$-0.0418750\pi$
$509$	4387.62	+	7599.59i	0.382079	+	0.661779i	−0.609281	+	0.792955i	$0.708542\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1450.77	−	2512.81i	0.124133	−	0.215005i
$516$		0			0
$517$	3274.63			0.278565
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−830.382	+	1438.26i	−0.0698267	+	0.120943i	−0.898825	−	0.438308i	$-0.855578\pi$
$521$	−830.382	+	1438.26i	−0.0698267	+	0.120943i	0.828998	+	0.559251i	$0.188911\pi$
$522$		0			0
$523$	−3162.41	−	5477.45i	−0.264402	−	0.457958i	0.703005	−	0.711185i	$-0.251841\pi$
$523$	−3162.41	−	5477.45i	−0.264402	−	0.457958i	−0.967407	+	0.253227i	$0.918508\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6132.88	+	10622.5i	0.506931	+	0.878030i
$528$		0			0
$529$	2944.90	−	5100.71i	0.242040	−	0.419225i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9954.68			0.808977
$534$		0			0
$535$	396.424	−	686.626i	0.0320353	−	0.0554868i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2233.94	−	3869.30i	−0.177532	−	0.307494i	0.763503	−	0.645804i	$-0.223478\pi$
$541$	−2233.94	−	3869.30i	−0.177532	−	0.307494i	−0.941034	+	0.338311i	$0.890145\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−27198.7			−2.13774
$546$		0			0
$547$	−10939.0			−0.855063			−0.427531	−	0.904000i	$-0.640617\pi$
$547$	−10939.0			−0.855063			−0.427531	+	0.904000i	$0.640617\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3546.63	−	6142.94i	−0.274213	−	0.474951i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7162.57	+	12405.9i	−0.544862	+	0.943728i	0.453754	+	0.891127i	$0.350085\pi$
$557$	−7162.57	+	12405.9i	−0.544862	+	0.943728i	−0.998616	+	0.0526011i	$0.983249\pi$
$558$		0			0
$559$	11615.6			0.878867
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10201.4	−	17669.4i	0.763656	−	1.32269i	−0.177299	−	0.984157i	$-0.556736\pi$
$563$	10201.4	−	17669.4i	0.763656	−	1.32269i	0.940954	−	0.338533i	$-0.109931\pi$
$564$		0			0
$565$	−9387.32	−	16259.3i	−0.698987	−	1.21068i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3625.21	−	6279.05i	−0.267095	−	0.462622i	0.701016	−	0.713146i	$-0.252730\pi$
$569$	−3625.21	−	6279.05i	−0.267095	−	0.462622i	−0.968110	+	0.250524i	$0.919397\pi$
$570$		0			0
$571$	−12447.4	+	21559.5i	−0.912272	+	1.58010i	−0.101424	+	0.994843i	$0.532340\pi$
$571$	−12447.4	+	21559.5i	−0.912272	+	1.58010i	−0.810847	+	0.585258i	$0.800993\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12072.8			−0.875599
$576$		0			0
$577$	−681.723	+	1180.78i	−0.0491863	+	0.0851932i	−0.889570	−	0.456798i	$-0.848996\pi$
$577$	−681.723	+	1180.78i	−0.0491863	+	0.0851932i	0.840384	+	0.541991i	$0.182329\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−5684.74	−	9846.26i	−0.403839	−	0.699469i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−23854.8			−1.67733			−0.838666	−	0.544647i	$-0.816664\pi$
$587$	−23854.8			−1.67733			−0.838666	+	0.544647i	$0.816664\pi$
$588$		0			0
$589$	22219.6			1.55440
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1718.60	+	2976.71i	0.119013	+	0.206136i	0.919377	−	0.393378i	$-0.128694\pi$
$593$	1718.60	+	2976.71i	0.119013	+	0.206136i	−0.800364	+	0.599514i	$0.795360\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8255.87	−	14299.6i	0.563148	−	0.975401i	−0.434071	−	0.900879i	$-0.642923\pi$
$599$	8255.87	−	14299.6i	0.563148	−	0.975401i	0.997219	−	0.0745226i	$-0.0237433\pi$
$600$		0			0
$601$	−11148.3			−0.756656			−0.378328	−	0.925672i	$-0.623501\pi$
$601$	−11148.3			−0.756656			−0.378328	+	0.925672i	$0.623501\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−31797.9	+	55075.6i	−2.13681	+	3.70106i
$606$		0			0
$607$	195.791	+	339.121i	0.0130921	+	0.0226763i	0.872497	−	0.488619i	$-0.162499\pi$
$607$	195.791	+	339.121i	0.0130921	+	0.0226763i	−0.859405	+	0.511295i	$0.829166\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1491.92	+	2584.09i	0.0987836	+	0.171098i
$612$		0			0
$613$	5870.25	−	10167.6i	0.386782	−	0.669925i	−0.605233	−	0.796048i	$-0.706920\pi$
$613$	5870.25	−	10167.6i	0.386782	−	0.669925i	0.992015	+	0.126123i	$0.0402534\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8818.95			0.575426			0.287713	−	0.957717i	$-0.407105\pi$
$617$	8818.95			0.575426			0.287713	+	0.957717i	$0.407105\pi$
$618$		0			0
$619$	7962.35	−	13791.2i	0.517018	−	0.895501i	−0.482787	−	0.875738i	$-0.660375\pi$
$619$	7962.35	−	13791.2i	0.517018	−	0.895501i	0.999805	−	0.0197631i	$-0.00629120\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	5726.55	+	9918.67i	0.366499	+	0.634795i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−24483.5			−1.55202
$630$		0			0
$631$	29206.4			1.84261			0.921307	−	0.388836i	$-0.127123\pi$
$631$	29206.4			1.84261			0.921307	+	0.388836i	$0.127123\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−19974.5	−	34596.8i	−1.24829	−	2.16210i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2806.51	−	4861.01i	0.172933	−	0.299530i	−0.766511	−	0.642232i	$-0.778009\pi$
$641$	2806.51	−	4861.01i	0.172933	−	0.299530i	0.939444	+	0.342702i	$0.111342\pi$
$642$		0			0
$643$	−18995.0			−1.16499			−0.582496	−	0.812834i	$-0.697924\pi$
$643$	−18995.0			−1.16499			−0.582496	+	0.812834i	$0.697924\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11332.5	−	19628.5i	0.688604	−	1.19270i	−0.283686	−	0.958917i	$-0.591557\pi$
$647$	11332.5	−	19628.5i	0.688604	−	1.19270i	0.972290	−	0.233779i	$-0.0751093\pi$
$648$		0			0
$649$	−14056.7	−	24346.9i	−0.850188	−	1.47257i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13263.9	+	22973.7i	0.794879	+	1.37677i	0.922916	+	0.385001i	$0.125799\pi$
$653$	13263.9	+	22973.7i	0.794879	+	1.37677i	−0.128037	+	0.991769i	$0.540868\pi$
$654$		0			0
$655$	12898.8	−	22341.3i	0.769460	−	1.33274i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	17114.8			1.01168			0.505839	−	0.862628i	$-0.331183\pi$
$659$	17114.8			1.01168			0.505839	+	0.862628i	$0.331183\pi$
$660$		0			0
$661$	595.908	−	1032.14i	0.0350653	−	0.0607348i	−0.847960	−	0.530060i	$-0.822169\pi$
$661$	595.908	−	1032.14i	0.0350653	−	0.0607348i	0.883025	+	0.469325i	$0.155503\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1715.29	+	2970.96i	0.0995744	+	0.172468i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	39560.9			2.27605
$672$		0			0
$673$	−9415.20			−0.539271			−0.269636	−	0.962962i	$-0.586903\pi$
$673$	−9415.20			−0.539271			−0.269636	+	0.962962i	$0.586903\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3958.56	+	6856.43i	0.224727	+	0.389238i	0.956237	−	0.292592i	$-0.0945177\pi$
$677$	3958.56	+	6856.43i	0.224727	+	0.389238i	−0.731511	+	0.681830i	$0.761184\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	14617.2	−	25317.7i	0.818904	−	1.41838i	−0.0875859	−	0.996157i	$-0.527915\pi$
$683$	14617.2	−	25317.7i	0.818904	−	1.41838i	0.906490	−	0.422227i	$-0.138751\pi$
$684$		0			0
$685$	−25369.6			−1.41507
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5179.94	−	8971.93i	0.286415	−	0.496086i
$690$		0			0
$691$	611.228	+	1058.68i	0.0336501	+	0.0582836i	0.882360	−	0.470575i	$-0.155953\pi$
$691$	611.228	+	1058.68i	0.0336501	+	0.0582836i	−0.848710	+	0.528859i	$0.822620\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4592.07	−	7953.70i	−0.250629	−	0.434102i
$696$		0			0
$697$	−6883.68	+	11922.9i	−0.374086	+	0.647936i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13054.6			−0.703372			−0.351686	−	0.936118i	$-0.614392\pi$
$701$	−13054.6			−0.703372			−0.351686	+	0.936118i	$0.614392\pi$
$702$		0			0
$703$	−22176.1	+	38410.2i	−1.18974	+	2.06069i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	9987.87	+	17299.5i	0.529058	+	0.916356i	0.999426	+	0.0338854i	$0.0107881\pi$
$709$	9987.87	+	17299.5i	0.529058	+	0.916356i	−0.470367	+	0.882471i	$0.655879\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−10746.3			−0.564447
$714$		0			0
$715$	−78147.5			−4.08749
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3256.13	+	5639.78i	0.168892	+	0.292529i	0.938030	−	0.346553i	$-0.112648\pi$
$719$	3256.13	+	5639.78i	0.168892	+	0.292529i	−0.769139	+	0.639082i	$0.779315\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3298.96	+	5713.97i	−0.168994	+	0.292705i
$726$		0			0
$727$	11437.1			0.583467			0.291733	−	0.956500i	$-0.405768\pi$
$727$	11437.1			0.583467			0.291733	+	0.956500i	$0.405768\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8032.19	+	13912.2i	−0.406404	+	0.703913i
$732$		0			0
$733$	6138.02	+	10631.4i	0.309295	+	0.535714i	0.978208	−	0.207626i	$-0.0665738\pi$
$733$	6138.02	+	10631.4i	0.309295	+	0.535714i	−0.668914	+	0.743340i	$0.733240\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16455.6	−	28502.0i	−0.822458	−	1.42454i
$738$		0			0
$739$	−6046.91	+	10473.5i	−0.301000	+	0.521347i	−0.976363	−	0.216138i	$-0.930654\pi$
$739$	−6046.91	+	10473.5i	−0.301000	+	0.521347i	0.675363	+	0.737486i	$0.263987\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−34331.6			−1.69516			−0.847580	−	0.530668i	$-0.821941\pi$
$743$	−34331.6			−1.69516			−0.847580	+	0.530668i	$0.821941\pi$
$744$		0			0
$745$	−19987.6	+	34619.6i	−0.982940	+	1.70250i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	7278.96	+	12607.5i	0.353679	+	0.612590i	0.986891	−	0.161388i	$-0.0515971\pi$
$751$	7278.96	+	12607.5i	0.353679	+	0.612590i	−0.633212	+	0.773979i	$0.718264\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10351.9			0.498998
$756$		0			0
$757$	27115.5			1.30189			0.650944	−	0.759126i	$-0.274373\pi$
$757$	27115.5			1.30189			0.650944	+	0.759126i	$0.274373\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5481.98	−	9495.06i	−0.261132	−	0.452294i	0.705411	−	0.708799i	$-0.250762\pi$
$761$	−5481.98	−	9495.06i	−0.261132	−	0.452294i	−0.966543	+	0.256504i	$0.917429\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12808.4	−	22184.9i	0.602981	−	1.04439i
$768$		0			0
$769$	−7835.53			−0.367434			−0.183717	−	0.982979i	$-0.558813\pi$
$769$	−7835.53			−0.367434			−0.183717	+	0.982979i	$0.558813\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14383.9	+	24913.6i	−0.669278	+	1.15922i	0.308828	+	0.951118i	$0.400063\pi$
$773$	−14383.9	+	24913.6i	−0.669278	+	1.15922i	−0.978106	+	0.208106i	$0.933270\pi$
$774$		0			0
$775$	−10334.0	−	17899.0i	−0.478978	−	0.829614i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12469.9	+	21598.5i	0.573531	+	0.993386i
$780$		0			0
$781$	−17465.9	+	30251.8i	−0.800228	+	1.38603i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−44827.3			−2.03816
$786$		0			0
$787$	1347.91	−	2334.65i	0.0610520	−	0.105745i	−0.833884	−	0.551940i	$-0.813888\pi$
$787$	1347.91	−	2334.65i	0.0610520	−	0.105745i	0.894936	+	0.446195i	$0.147221\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	18024.0	+	31218.4i	0.807125	+	1.39798i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12592.1			−0.559643			−0.279822	−	0.960052i	$-0.590275\pi$
$797$	−12592.1			−0.559643			−0.279822	+	0.960052i	$0.590275\pi$
$798$		0			0
$799$	−4126.67			−0.182717
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20627.1	−	35727.2i	−0.906495	−	1.57009i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3899.74	+	6754.56i	−0.169478	+	0.293545i	−0.938236	−	0.345995i	$-0.887542\pi$
$809$	−3899.74	+	6754.56i	−0.169478	+	0.293545i	0.768758	+	0.639539i	$0.220875\pi$
$810$		0			0
$811$	−19323.5			−0.836669			−0.418335	−	0.908293i	$-0.637386\pi$
$811$	−19323.5			−0.836669			−0.418335	+	0.908293i	$0.637386\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8804.55	+	15249.9i	−0.378417	+	0.655438i
$816$		0			0
$817$	14550.5	+	25202.1i	0.623080	+	1.07921i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	542.649	+	939.896i	0.0230677	+	0.0399545i	0.877329	−	0.479890i	$-0.159323\pi$
$821$	542.649	+	939.896i	0.0230677	+	0.0399545i	−0.854261	+	0.519844i	$0.825990\pi$
$822$		0			0
$823$	12175.4	−	21088.4i	0.515683	−	0.893190i	−0.484151	−	0.874985i	$-0.660871\pi$
$823$	12175.4	−	21088.4i	0.515683	−	0.893190i	0.999834	−	0.0182053i	$-0.00579524\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31664.9			−1.33143			−0.665716	−	0.746205i	$-0.731874\pi$
$827$	−31664.9			−1.33143			−0.665716	+	0.746205i	$0.731874\pi$
$828$		0			0
$829$	1709.01	−	2960.09i	0.0716000	−	0.124015i	−0.828003	−	0.560724i	$-0.810523\pi$
$829$	1709.01	−	2960.09i	0.0716000	−	0.124015i	0.899603	+	0.436709i	$0.143856\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−2030.54	−	3517.00i	−0.0841554	−	0.145761i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9114.25			−0.375040			−0.187520	−	0.982261i	$-0.560045\pi$
$839$	−9114.25			−0.375040			−0.187520	+	0.982261i	$0.560045\pi$
$840$		0			0
$841$	−22514.2			−0.923127
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−17308.9	−	29979.9i	−0.704667	−	1.22052i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10725.2	−	18576.6i	0.432028	−	0.748295i
$852$		0			0
$853$	−36229.5			−1.45425			−0.727124	−	0.686506i	$-0.759143\pi$
$853$	−36229.5			−1.45425			−0.727124	+	0.686506i	$0.759143\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4322.59	−	7486.95i	0.172295	−	0.298424i	−0.766927	−	0.641735i	$-0.778215\pi$
$857$	4322.59	−	7486.95i	0.172295	−	0.298424i	0.939222	+	0.343311i	$0.111548\pi$
$858$		0			0
$859$	4106.29	+	7112.31i	0.163102	+	0.282501i	0.935980	−	0.352054i	$-0.114517\pi$
$859$	4106.29	+	7112.31i	0.163102	+	0.282501i	−0.772877	+	0.634555i	$0.781183\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1305.80	−	2261.72i	−0.0515065	−	0.0892119i	0.839123	−	0.543942i	$-0.183069\pi$
$863$	−1305.80	−	2261.72i	−0.0515065	−	0.0892119i	−0.890629	+	0.454730i	$0.849736\pi$
$864$		0			0
$865$	−14050.4	+	24336.1i	−0.552288	+	0.956591i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−47958.6			−1.87213
$870$		0			0
$871$	14994.4	−	25971.1i	0.583313	−	1.01033i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3223.29	+	5582.90i	0.124108	+	0.214961i	0.921384	−	0.388654i	$-0.127060\pi$
$877$	3223.29	+	5582.90i	0.124108	+	0.214961i	−0.797276	+	0.603615i	$0.793726\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−37166.3			−1.42130			−0.710649	−	0.703547i	$-0.751599\pi$
$881$	−37166.3			−1.42130			−0.710649	+	0.703547i	$0.751599\pi$
$882$		0			0
$883$	24377.3			0.929060			0.464530	−	0.885557i	$-0.346223\pi$
$883$	24377.3			0.929060			0.464530	+	0.885557i	$0.346223\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18172.2	−	31475.2i	−0.687896	−	1.19147i	−0.972517	−	0.232831i	$-0.925201\pi$
$887$	−18172.2	−	31475.2i	−0.687896	−	1.19147i	0.284621	−	0.958640i	$-0.408132\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3737.77	+	6474.01i	−0.140067	+	0.242603i
$894$		0			0
$895$	5981.35			0.223391
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2936.48	+	5086.14i	−0.108940	+	0.188690i
$900$		0			0
$901$	7163.88	+	12408.2i	0.264887	+	0.458798i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−24837.0	−	43018.9i	−0.912275	−	1.58011i
$906$		0			0
$907$	−163.437	+	283.082i	−0.00598329	+	0.0103634i	−0.869002	−	0.494809i	$-0.835238\pi$
$907$	−163.437	+	283.082i	−0.00598329	+	0.0103634i	0.863018	+	0.505173i	$0.168571\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	38230.7			1.39038			0.695191	−	0.718825i	$-0.255320\pi$
$911$	38230.7			1.39038			0.695191	+	0.718825i	$0.255320\pi$
$912$		0			0
$913$	2735.67	−	4738.32i	0.0991648	−	0.171758i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	10024.7	+	17363.2i	0.359829	+	0.623243i	0.987932	−	0.154888i	$-0.0495016\pi$
$919$	10024.7	+	17363.2i	0.359829	+	0.623243i	−0.628103	+	0.778130i	$0.716168\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−31829.8			−1.13509
$924$		0			0
$925$	41255.0			1.46644
$926$		0			0
$927$		0			0
$928$		0			0
$929$	21670.7	+	37534.8i	0.765332	+	1.32559i	0.940071	+	0.340980i	$0.110759\pi$
$929$	21670.7	+	37534.8i	0.765332	+	1.32559i	−0.174738	+	0.984615i	$0.555908\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	54039.2	−	93598.6i	1.89013	−	3.27380i
$936$		0			0
$937$	17530.0			0.611185			0.305592	−	0.952162i	$-0.401146\pi$
$937$	17530.0			0.611185			0.305592	+	0.952162i	$0.401146\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−16312.5	+	28254.1i	−0.565115	+	0.978807i	0.431924	+	0.901910i	$0.357835\pi$
$941$	−16312.5	+	28254.1i	−0.565115	+	0.978807i	−0.997039	+	0.0768974i	$0.975499\pi$
$942$		0			0
$943$	−6030.93	−	10445.9i	−0.208265	−	0.360726i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9697.09	−	16795.8i	−0.332749	−	0.576338i	0.650301	−	0.759677i	$-0.274643\pi$
$947$	−9697.09	−	16795.8i	−0.332749	−	0.576338i	−0.983050	+	0.183339i	$0.941309\pi$
$948$		0			0
$949$	18795.5	−	32554.7i	0.642915	−	1.11356i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	49618.5			1.68657			0.843285	−	0.537467i	$-0.180619\pi$
$953$	49618.5			1.68657			0.843285	+	0.537467i	$0.180619\pi$
$954$		0			0
$955$	29068.9	−	50348.7i	0.984970	−	1.70602i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	5696.96	+	9867.43i	0.191231	+	0.331222i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	26717.9			0.891274
$966$		0			0
$967$	28432.8			0.945538			0.472769	−	0.881186i	$-0.343254\pi$
$967$	28432.8			0.945538			0.472769	+	0.881186i	$0.343254\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21345.1	+	36970.8i	0.705455	+	1.22188i	0.966527	+	0.256565i	$0.0825907\pi$
$971$	21345.1	+	36970.8i	0.705455	+	1.22188i	−0.261072	+	0.965319i	$0.584076\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11367.5	−	19689.1i	0.372241	−	0.644740i	−0.617669	−	0.786438i	$-0.711923\pi$
$977$	11367.5	−	19689.1i	0.372241	−	0.644740i	0.989910	+	0.141698i	$0.0452562\pi$
$978$		0			0
$979$	−98081.7			−3.20195
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2821.91	−	4887.70i	0.0915616	−	0.158589i	−0.816607	−	0.577194i	$-0.804147\pi$
$983$	2821.91	−	4887.70i	0.0915616	−	0.158589i	0.908168	+	0.418605i	$0.137481\pi$
$984$		0			0
$985$	−3796.73	−	6576.12i	−0.122816	−	0.212724i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7037.16	−	12188.7i	−0.226257	−	0.391889i
$990$		0			0
$991$	27422.3	−	47496.8i	0.879009	−	1.52249i	0.0265782	−	0.999647i	$-0.491539\pi$
$991$	27422.3	−	47496.8i	0.879009	−	1.52249i	0.852430	−	0.522841i	$-0.175128\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	34711.0			1.10594
$996$		0			0
$997$	−22440.6	+	38868.2i	−0.712838	+	1.23467i	0.250949	+	0.968000i	$0.419257\pi$
$997$	−22440.6	+	38868.2i	−0.712838	+	1.23467i	−0.963787	+	0.266672i	$0.914076\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.bd.361.1		8
3.2	odd	2		588.4.i.k.361.4		8
7.2	even	3	inner	1764.4.k.bd.1549.1		8
7.3	odd	6		1764.4.a.bc.1.1		4
7.4	even	3		1764.4.a.ba.1.4		4
7.5	odd	6		1764.4.k.bb.1549.4		8
7.6	odd	2		1764.4.k.bb.361.4		8
21.2	odd	6		588.4.i.k.373.4		8
21.5	even	6		588.4.i.l.373.1		8
21.11	odd	6		588.4.a.k.1.1	yes	4
21.17	even	6		588.4.a.j.1.4	✓	4
21.20	even	2		588.4.i.l.361.1		8
84.11	even	6		2352.4.a.cl.1.1		4
84.59	odd	6		2352.4.a.cq.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.a.j.1.4	✓	4	21.17	even	6
588.4.a.k.1.1	yes	4	21.11	odd	6
588.4.i.k.361.4		8	3.2	odd	2
588.4.i.k.373.4		8	21.2	odd	6
588.4.i.l.361.1		8	21.20	even	2
588.4.i.l.373.1		8	21.5	even	6
1764.4.a.ba.1.4		4	7.4	even	3
1764.4.a.bc.1.1		4	7.3	odd	6
1764.4.k.bb.361.4		8	7.6	odd	2
1764.4.k.bb.1549.4		8	7.5	odd	6
1764.4.k.bd.361.1		8	1.1	even	1	trivial
1764.4.k.bd.1549.1		8	7.2	even	3	inner
2352.4.a.cl.1.1		4	84.11	even	6
2352.4.a.cq.1.4		4	84.59	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-8.32734 - 14.4234i) q^{5} +O(q^{10})\)	\(q+(-8.32734 - 14.4234i) q^{5} +(-35.8800 + 62.1460i) q^{11} -65.3878 q^{13} +(45.2158 - 78.3160i) q^{17} +(-81.9092 - 141.871i) q^{19} +(39.6144 + 68.6142i) q^{23} +(-76.1893 + 131.964i) q^{25} +43.2995 q^{29} +(-67.8179 + 117.464i) q^{31} +(-135.370 - 234.468i) q^{37} -152.241 q^{41} -177.641 q^{43} +(-22.8166 - 39.5194i) q^{47} +(-79.2188 + 137.211i) q^{53} +1195.14 q^{55} +(-195.884 + 339.282i) q^{59} +(-275.647 - 477.435i) q^{61} +(544.506 + 943.113i) q^{65} +(-229.315 + 397.185i) q^{67} +486.786 q^{71} +(-287.446 + 497.871i) q^{73} +(334.160 + 578.782i) q^{79} -76.2450 q^{83} -1506.11 q^{85} +(683.401 + 1183.69i) q^{89} +(-1364.17 + 2362.82i) q^{95} +242.655 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 48 q^{17} - 192 q^{19} + 192 q^{23} - 324 q^{25} - 192 q^{29} - 48 q^{31} - 256 q^{37} - 2016 q^{41} - 224 q^{43} + 864 q^{47} - 648 q^{53} + 4704 q^{55} + 336 q^{59} - 960 q^{61} - 360 q^{65} - 720 q^{67} + 2688 q^{71} - 672 q^{73} + 1984 q^{79} - 6240 q^{83} + 1360 q^{85} + 2160 q^{89} - 3744 q^{95} + 4032 q^{97}+O(q^{100}) \) 8 * q + 48 * q^17 - 192 * q^19 + 192 * q^23 - 324 * q^25 - 192 * q^29 - 48 * q^31 - 256 * q^37 - 2016 * q^41 - 224 * q^43 + 864 * q^47 - 648 * q^53 + 4704 * q^55 + 336 * q^59 - 960 * q^61 - 360 * q^65 - 720 * q^67 + 2688 * q^71 - 672 * q^73 + 1984 * q^79 - 6240 * q^83 + 1360 * q^85 + 2160 * q^89 - 3744 * q^95 + 4032 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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