LMFDB - Embedded newform 792.5.j.a.505.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [792,5,Mod(505,792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("792.505");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 792 = 2^{3} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	792.j (of order $2$, degree $1$, 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:	$81.8690107624$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 378x^{10} + 49709x^{8} + 2770310x^{6} + 62444900x^{4} + 470757120x^{2} + 33918976 $ x^12 + 378x^10 + 49709x^8 + 2770310x^6 + 62444900x^4 + 470757120*x^2 + 33918976
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{32}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			505.5
Root			$-12.8952i$ of defining polynomial
Character	$\chi$	$=$	792.505
Dual form			792.5.j.a.505.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.72383 q^{5} -35.8576i q^{7} +O(q^{10})$	$q-6.72383 q^{5} -35.8576i q^{7} +(-120.912 + 4.60990i) q^{11} -138.170i q^{13} +364.891i q^{17} -570.563i q^{19} -70.8767 q^{23} -579.790 q^{25} +317.406i q^{29} +1159.17 q^{31} +241.100i q^{35} -1361.35 q^{37} -1336.82i q^{41} +2158.22i q^{43} -1416.39 q^{47} +1115.24 q^{49} +3013.55 q^{53} +(812.993 - 30.9962i) q^{55} -1663.07 q^{59} +5396.77i q^{61} +929.033i q^{65} -588.504 q^{67} +4021.12 q^{71} -2259.81i q^{73} +(165.300 + 4335.61i) q^{77} -6315.63i q^{79} +777.535i q^{83} -2453.46i q^{85} -4643.68 q^{89} -4954.44 q^{91} +3836.37i q^{95} +6327.31 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 12 q^{11} - 1608 q^{23} + 1484 q^{25} - 2136 q^{31} - 288 q^{37} + 6024 q^{47} - 9940 q^{49} - 3576 q^{53} - 8344 q^{55} + 7704 q^{59} + 10312 q^{67} - 8904 q^{71} - 9120 q^{77} - 4224 q^{89} + 23904 q^{91} - 21280 q^{97}+O(q^{100}) $ 12 * q - 12 * q^11 - 1608 * q^23 + 1484 * q^25 - 2136 * q^31 - 288 * q^37 + 6024 * q^47 - 9940 * q^49 - 3576 * q^53 - 8344 * q^55 + 7704 * q^59 + 10312 * q^67 - 8904 * q^71 - 9120 * q^77 - 4224 * q^89 + 23904 * q^91 - 21280 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/792\mathbb{Z}\right)^\times$.

$n$	$145$	$199$	$353$	$397$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	−6.72383			−0.268953			−0.134477	−	0.990917i	$-0.542935\pi$
$5$	−6.72383			−0.268953			−0.134477	+	0.990917i	$0.542935\pi$
$6$		0			0
$7$		−	35.8576i		−	0.731787i	−0.930657	−	0.365893i	$-0.880763\pi$
$7$		−	35.8576i		−	0.731787i	0.930657	−	0.365893i	$-0.119237\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−120.912	+	4.60990i	−0.999274	+	0.0380984i
$11$	−120.912	+	4.60990i	−0.999274	+	0.0380984i
$12$		0			0
$13$		−	138.170i		−	0.817575i	−0.912630	−	0.408788i	$-0.865952\pi$
$13$		−	138.170i		−	0.817575i	0.912630	−	0.408788i	$-0.134048\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			364.891i			1.26260i	0.775540	+	0.631299i	$0.217478\pi$
$17$			364.891i			1.26260i	−0.775540	+	0.631299i	$0.782522\pi$
$18$		0			0
$19$		−	570.563i		−	1.58051i	−0.612780	−	0.790253i	$-0.709949\pi$
$19$		−	570.563i		−	1.58051i	0.612780	−	0.790253i	$-0.290051\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−70.8767			−0.133982			−0.0669912	−	0.997754i	$-0.521340\pi$
$23$	−70.8767			−0.133982			−0.0669912	+	0.997754i	$0.521340\pi$
$24$		0			0
$25$	−579.790			−0.927664
$26$		0			0
$27$		0			0
$28$		0			0
$29$			317.406i			0.377416i	0.982033	+	0.188708i	$0.0604299\pi$
$29$			317.406i			0.377416i	−0.982033	+	0.188708i	$0.939570\pi$
$30$		0			0
$31$	1159.17			1.20621			0.603105	−	0.797662i	$-0.293930\pi$
$31$	1159.17			1.20621			0.603105	+	0.797662i	$0.293930\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			241.100i			0.196816i
$36$		0			0
$37$	−1361.35			−0.994412			−0.497206	−	0.867632i	$-0.665641\pi$
$37$	−1361.35			−0.994412			−0.497206	+	0.867632i	$0.665641\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1336.82i		−	0.795251i	−0.917548	−	0.397626i	$-0.869834\pi$
$41$		−	1336.82i		−	0.795251i	0.917548	−	0.397626i	$-0.130166\pi$
$42$		0			0
$43$			2158.22i			1.16724i	0.812028	+	0.583619i	$0.198364\pi$
$43$			2158.22i			1.16724i	−0.812028	+	0.583619i	$0.801636\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1416.39			−0.641189			−0.320595	−	0.947217i	$-0.603883\pi$
$47$	−1416.39			−0.641189			−0.320595	+	0.947217i	$0.603883\pi$
$48$		0			0
$49$	1115.24			0.464488
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3013.55			1.07282			0.536410	−	0.843957i	$-0.319780\pi$
$53$	3013.55			1.07282			0.536410	+	0.843957i	$0.319780\pi$
$54$		0			0
$55$	812.993	−	30.9962i	0.268758	−	0.0102467i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1663.07			−0.477756			−0.238878	−	0.971050i	$-0.576780\pi$
$59$	−1663.07			−0.477756			−0.238878	+	0.971050i	$0.576780\pi$
$60$		0			0
$61$			5396.77i			1.45036i	0.688562	+	0.725178i	$0.258242\pi$
$61$			5396.77i			1.45036i	−0.688562	+	0.725178i	$0.741758\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			929.033i			0.219889i
$66$		0			0
$67$	−588.504			−0.131099			−0.0655496	−	0.997849i	$-0.520880\pi$
$67$	−588.504			−0.131099			−0.0655496	+	0.997849i	$0.520880\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4021.12			0.797682			0.398841	−	0.917020i	$-0.369412\pi$
$71$	4021.12			0.797682			0.398841	+	0.917020i	$0.369412\pi$
$72$		0			0
$73$		−	2259.81i		−	0.424058i	−0.977263	−	0.212029i	$-0.931993\pi$
$73$		−	2259.81i		−	0.424058i	0.977263	−	0.212029i	$-0.0680072\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	165.300	+	4335.61i	0.0278799	+	0.731255i
$78$		0			0
$79$		−	6315.63i		−	1.01196i	−0.862546	−	0.505979i	$-0.831131\pi$
$79$		−	6315.63i		−	1.01196i	0.862546	−	0.505979i	$-0.168869\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			777.535i			0.112866i	0.998406	+	0.0564331i	$0.0179728\pi$
$83$			777.535i			0.112866i	−0.998406	+	0.0564331i	$0.982027\pi$
$84$		0			0
$85$		−	2453.46i		−	0.339580i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4643.68			−0.586249			−0.293125	−	0.956074i	$-0.594695\pi$
$89$	−4643.68			−0.586249			−0.293125	+	0.956074i	$0.594695\pi$
$90$		0			0
$91$	−4954.44			−0.598291
$92$		0			0
$93$		0			0
$94$		0			0
$95$			3836.37i			0.425082i
$96$		0			0
$97$	6327.31			0.672474			0.336237	−	0.941777i	$-0.390846\pi$
$97$	6327.31			0.672474			0.336237	+	0.941777i	$0.390846\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			19278.4i			1.88985i	0.327281	+	0.944927i	$0.393868\pi$
$101$			19278.4i			1.88985i	−0.327281	+	0.944927i	$0.606132\pi$
$102$		0			0
$103$	−3753.94			−0.353845			−0.176922	−	0.984225i	$-0.556614\pi$
$103$	−3753.94			−0.353845			−0.176922	+	0.984225i	$0.556614\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			18411.8i			1.60815i	0.594525	+	0.804077i	$0.297340\pi$
$107$			18411.8i			1.60815i	−0.594525	+	0.804077i	$0.702660\pi$
$108$		0			0
$109$			9835.23i			0.827812i	0.910320	+	0.413906i	$0.135836\pi$
$109$			9835.23i			0.827812i	−0.910320	+	0.413906i	$0.864164\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−13378.6			−1.04774			−0.523870	−	0.851798i	$-0.675512\pi$
$113$	−13378.6			−1.04774			−0.523870	+	0.851798i	$0.675512\pi$
$114$		0			0
$115$	476.563			0.0360350
$116$		0			0
$117$		0			0
$118$		0			0
$119$	13084.1			0.923952
$120$		0			0
$121$	14598.5	−	1114.79i	0.997097	−	0.0761414i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8100.80			0.518451
$126$		0			0
$127$			24171.9i			1.49866i	0.662198	+	0.749329i	$0.269624\pi$
$127$			24171.9i			1.49866i	−0.662198	+	0.749329i	$0.730376\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	10327.1i		−	0.601776i	−0.953660	−	0.300888i	$-0.902717\pi$
$131$		−	10327.1i		−	0.601776i	0.953660	−	0.300888i	$-0.0972830\pi$
$132$		0			0
$133$	−20459.0			−1.15659
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−18041.0			−0.961211			−0.480605	−	0.876937i	$-0.659583\pi$
$137$	−18041.0			−0.961211			−0.480605	+	0.876937i	$0.659583\pi$
$138$		0			0
$139$			18076.1i			0.935567i	0.883843	+	0.467783i	$0.154947\pi$
$139$			18076.1i			0.935567i	−0.883843	+	0.467783i	$0.845053\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	636.951	+	16706.5i	0.0311483	+	0.816981i
$144$		0			0
$145$		−	2134.19i		−	0.101507i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	26839.5i		−	1.20893i	−0.796630	−	0.604467i	$-0.793386\pi$
$149$		−	26839.5i		−	1.20893i	0.796630	−	0.604467i	$-0.206614\pi$
$150$		0			0
$151$			15391.1i			0.675018i	0.941322	+	0.337509i	$0.109584\pi$
$151$			15391.1i			0.675018i	−0.941322	+	0.337509i	$0.890416\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7794.04			−0.324414
$156$		0			0
$157$	−34362.6			−1.39408			−0.697038	−	0.717034i	$-0.745499\pi$
$157$	−34362.6			−1.39408			−0.697038	+	0.717034i	$0.745499\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2541.46i			0.0980465i
$162$		0			0
$163$	34043.9			1.28134			0.640670	−	0.767817i	$-0.278657\pi$
$163$	34043.9			1.28134			0.640670	+	0.767817i	$0.278657\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			34873.6i			1.25044i	0.780447	+	0.625222i	$0.214991\pi$
$167$			34873.6i			1.25044i	−0.780447	+	0.625222i	$0.785009\pi$
$168$		0			0
$169$	9470.00			0.331571
$170$		0			0
$171$		0			0
$172$		0			0
$173$			29337.5i			0.980238i	0.871655	+	0.490119i	$0.163047\pi$
$173$			29337.5i			0.980238i	−0.871655	+	0.490119i	$0.836953\pi$
$174$		0			0
$175$			20789.9i			0.678852i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2401.98			−0.0749660			−0.0374830	−	0.999297i	$-0.511934\pi$
$179$	−2401.98			−0.0749660			−0.0374830	+	0.999297i	$0.511934\pi$
$180$		0			0
$181$	−28001.5			−0.854721			−0.427360	−	0.904081i	$-0.640556\pi$
$181$	−28001.5			−0.854721			−0.427360	+	0.904081i	$0.640556\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9153.49			0.267450
$186$		0			0
$187$	−1682.11	−	44119.7i	−0.0481029	−	1.26168i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	17399.1			0.476935			0.238467	−	0.971151i	$-0.423355\pi$
$191$	17399.1			0.476935			0.238467	+	0.971151i	$0.423355\pi$
$192$		0			0
$193$			37593.4i			1.00925i	0.863340	+	0.504623i	$0.168368\pi$
$193$			37593.4i			1.00925i	−0.863340	+	0.504623i	$0.831632\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			19628.9i			0.505781i	0.967495	+	0.252891i	$0.0813813\pi$
$197$			19628.9i			0.505781i	−0.967495	+	0.252891i	$0.918619\pi$
$198$		0			0
$199$	63712.2			1.60885			0.804427	−	0.594052i	$-0.202472\pi$
$199$	63712.2			1.60885			0.804427	+	0.594052i	$0.202472\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	11381.4			0.276188
$204$		0			0
$205$			8988.53i			0.213885i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2630.24	+	68988.0i	0.0602147	+	1.57936i
$210$		0			0
$211$		−	42840.1i		−	0.962245i	−0.876653	−	0.481123i	$-0.840229\pi$
$211$		−	42840.1i		−	0.962245i	0.876653	−	0.481123i	$-0.159771\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	14511.5i		−	0.313932i
$216$		0			0
$217$		−	41564.9i		−	0.882688i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	50417.0			1.03227
$222$		0			0
$223$	59661.7			1.19974			0.599868	−	0.800099i	$-0.295220\pi$
$223$	59661.7			1.19974			0.599868	+	0.800099i	$0.295220\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			24125.2i			0.468187i	0.972214	+	0.234094i	$0.0752122\pi$
$227$			24125.2i			0.468187i	−0.972214	+	0.234094i	$0.924788\pi$
$228$		0			0
$229$	−71548.3			−1.36436			−0.682179	−	0.731185i	$-0.738967\pi$
$229$	−71548.3			−1.36436			−0.682179	+	0.731185i	$0.738967\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			76741.8i			1.41358i	0.707424	+	0.706790i	$0.249857\pi$
$233$			76741.8i			1.41358i	−0.707424	+	0.706790i	$0.750143\pi$
$234$		0			0
$235$	9523.54			0.172450
$236$		0			0
$237$		0			0
$238$		0			0
$239$			97651.3i			1.70955i	0.518997	+	0.854776i	$0.326305\pi$
$239$			97651.3i			1.70955i	−0.518997	+	0.854776i	$0.673695\pi$
$240$		0			0
$241$			109461.i			1.88463i	0.334722	+	0.942317i	$0.391358\pi$
$241$			109461.i			1.88463i	−0.334722	+	0.942317i	$0.608642\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7498.66			−0.124926
$246$		0			0
$247$	−78834.8			−1.29218
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−54758.2			−0.869164			−0.434582	−	0.900632i	$-0.643104\pi$
$251$	−54758.2			−0.869164			−0.434582	+	0.900632i	$0.643104\pi$
$252$		0			0
$253$	8569.85	−	326.734i	0.133885	−	0.00510451i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−92386.3			−1.39875			−0.699377	−	0.714753i	$-0.746539\pi$
$257$	−92386.3			−1.39875			−0.699377	+	0.714753i	$0.746539\pi$
$258$		0			0
$259$			48814.7i			0.727698i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	77399.6i		−	1.11899i	−0.828833	−	0.559496i	$-0.810995\pi$
$263$		−	77399.6i		−	1.11899i	0.828833	−	0.559496i	$-0.189005\pi$
$264$		0			0
$265$	−20262.6			−0.288538
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−66027.2			−0.912470			−0.456235	−	0.889859i	$-0.650802\pi$
$269$	−66027.2			−0.912470			−0.456235	+	0.889859i	$0.650802\pi$
$270$		0			0
$271$			73926.8i			1.00661i	0.864108	+	0.503307i	$0.167884\pi$
$271$			73926.8i			1.00661i	−0.864108	+	0.503307i	$0.832116\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	70103.7	−	2672.78i	0.926991	−	0.0353425i
$276$		0			0
$277$		−	64374.6i		−	0.838987i	−0.907758	−	0.419493i	$-0.862208\pi$
$277$		−	64374.6i		−	0.838987i	0.907758	−	0.419493i	$-0.137792\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	104033.i		−	1.31752i	−0.752354	−	0.658759i	$-0.771081\pi$
$281$		−	104033.i		−	1.31752i	0.752354	−	0.658759i	$-0.228919\pi$
$282$		0			0
$283$			3909.45i			0.0488139i	0.999702	+	0.0244069i	$0.00776974\pi$
$283$			3909.45i			0.0488139i	−0.999702	+	0.0244069i	$0.992230\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−47935.0			−0.581954
$288$		0			0
$289$	−49624.1			−0.594151
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	64081.9i		−	0.746449i	−0.927741	−	0.373224i	$-0.878252\pi$
$293$		−	64081.9i		−	0.746449i	0.927741	−	0.373224i	$-0.121748\pi$
$294$		0			0
$295$	11182.2			0.128494
$296$		0			0
$297$		0			0
$298$		0			0
$299$			9793.04i			0.109541i
$300$		0			0
$301$	77388.6			0.854169
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	36287.0i		−	0.390078i
$306$		0			0
$307$			52705.6i			0.559216i	0.960114	+	0.279608i	$0.0902045\pi$
$307$			52705.6i			0.559216i	−0.960114	+	0.279608i	$0.909795\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	80464.5			0.831924			0.415962	−	0.909382i	$-0.363445\pi$
$311$	80464.5			0.831924			0.415962	+	0.909382i	$0.363445\pi$
$312$		0			0
$313$	−19122.0			−0.195184			−0.0975920	−	0.995227i	$-0.531114\pi$
$313$	−19122.0			−0.195184			−0.0975920	+	0.995227i	$0.531114\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−50197.6			−0.499533			−0.249766	−	0.968306i	$-0.580354\pi$
$317$	−50197.6			−0.499533			−0.249766	+	0.968306i	$0.580354\pi$
$318$		0			0
$319$	−1463.21	−	38378.3i	−0.0143789	−	0.377142i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	208193.			1.99554
$324$		0			0
$325$			80109.7i			0.758435i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			50788.2i			0.469214i
$330$		0			0
$331$	170531.			1.55649			0.778247	−	0.627958i	$-0.216109\pi$
$331$	170531.			1.55649			0.778247	+	0.627958i	$0.216109\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3957.00			0.0352595
$336$		0			0
$337$		−	131558.i		−	1.15840i	−0.815185	−	0.579200i	$-0.803365\pi$
$337$		−	131558.i		−	1.15840i	0.815185	−	0.579200i	$-0.196635\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−140157.	+	5343.65i	−1.20533	+	0.0459546i
$342$		0			0
$343$		−	126084.i		−	1.07169i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			25233.1i			0.209562i	0.994495	+	0.104781i	$0.0334141\pi$
$347$			25233.1i			0.209562i	−0.994495	+	0.104781i	$0.966586\pi$
$348$		0			0
$349$		−	10590.3i		−	0.0869473i	−0.999055	−	0.0434736i	$-0.986158\pi$
$349$		−	10590.3i		−	0.0869473i	0.999055	−	0.0434736i	$-0.0138425\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	184997.			1.48462			0.742311	−	0.670055i	$-0.233730\pi$
$353$	184997.			1.48462			0.742311	+	0.670055i	$0.233730\pi$
$354$		0			0
$355$	−27037.3			−0.214539
$356$		0			0
$357$		0			0
$358$		0			0
$359$			2401.44i			0.0186330i	0.999957	+	0.00931651i	$0.00296558\pi$
$359$			2401.44i			0.0186330i	−0.999957	+	0.00931651i	$0.997034\pi$
$360$		0			0
$361$	−195221.			−1.49800
$362$		0			0
$363$		0			0
$364$		0			0
$365$			15194.6i			0.114052i
$366$		0			0
$367$	43796.3			0.325166			0.162583	−	0.986695i	$-0.448017\pi$
$367$	43796.3			0.325166			0.162583	+	0.986695i	$0.448017\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	108059.i		−	0.785076i
$372$		0			0
$373$		−	116581.i		−	0.837931i	−0.908002	−	0.418966i	$-0.862393\pi$
$373$		−	116581.i		−	0.837931i	0.908002	−	0.418966i	$-0.137607\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	43856.1			0.308566
$378$		0			0
$379$	145996.			1.01640			0.508199	−	0.861240i	$-0.330312\pi$
$379$	145996.			1.01640			0.508199	+	0.861240i	$0.330312\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−173807.			−1.18486			−0.592432	−	0.805620i	$-0.701832\pi$
$383$	−173807.			−1.18486			−0.592432	+	0.805620i	$0.701832\pi$
$384$		0			0
$385$	−1111.45	−	29151.9i	−0.00749838	−	0.196673i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−172951.			−1.14294			−0.571471	−	0.820622i	$-0.693627\pi$
$389$	−172951.			−1.14294			−0.571471	+	0.820622i	$0.693627\pi$
$390$		0			0
$391$		−	25862.2i		−	0.169166i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			42465.2i			0.272169i
$396$		0			0
$397$	−205854.			−1.30611			−0.653053	−	0.757312i	$-0.726512\pi$
$397$	−205854.			−1.30611			−0.653053	+	0.757312i	$0.726512\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	29545.2			0.183738			0.0918689	−	0.995771i	$-0.470716\pi$
$401$	29545.2			0.183738			0.0918689	+	0.995771i	$0.470716\pi$
$402$		0			0
$403$		−	160162.i		−	0.986166i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	164604.	−	6275.69i	0.993690	−	0.0378855i
$408$		0			0
$409$			103104.i			0.616349i	0.951330	+	0.308175i	$0.0997181\pi$
$409$			103104.i			0.616349i	−0.951330	+	0.308175i	$0.900282\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			59633.6i			0.349616i
$414$		0			0
$415$		−	5228.01i		−	0.0303557i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−123880.			−0.705621			−0.352811	−	0.935695i	$-0.614774\pi$
$419$	−123880.			−0.705621			−0.352811	+	0.935695i	$0.614774\pi$
$420$		0			0
$421$	−72192.8			−0.407314			−0.203657	−	0.979042i	$-0.565283\pi$
$421$	−72192.8			−0.407314			−0.203657	+	0.979042i	$0.565283\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	211560.i		−	1.17127i
$426$		0			0
$427$	193515.			1.06135
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	370816.i		−	1.99620i	−0.0616150	−	0.998100i	$-0.519625\pi$
$431$		−	370816.i		−	1.99620i	0.0616150	−	0.998100i	$-0.480375\pi$
$432$		0			0
$433$	−67611.3			−0.360615			−0.180307	−	0.983610i	$-0.557709\pi$
$433$	−67611.3			−0.360615			−0.180307	+	0.983610i	$0.557709\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			40439.6i			0.211760i
$438$		0			0
$439$		−	61838.3i		−	0.320869i	−0.987046	−	0.160435i	$-0.948710\pi$
$439$		−	61838.3i		−	0.320869i	0.987046	−	0.160435i	$-0.0512896\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−225163.			−1.14733			−0.573665	−	0.819090i	$-0.694479\pi$
$443$	−225163.			−1.14733			−0.573665	+	0.819090i	$0.694479\pi$
$444$		0			0
$445$	31223.3			0.157674
$446$		0			0
$447$		0			0
$448$		0			0
$449$	348050.			1.72643			0.863214	−	0.504838i	$-0.168448\pi$
$449$	348050.			1.72643			0.863214	+	0.504838i	$0.168448\pi$
$450$		0			0
$451$	6162.59	+	161637.i	0.0302978	+	0.794674i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	33312.8			0.160912
$456$		0			0
$457$		−	367325.i		−	1.75881i	−0.476077	−	0.879404i	$-0.657942\pi$
$457$		−	367325.i		−	1.75881i	0.476077	−	0.879404i	$-0.342058\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	373933.i		−	1.75951i	−0.475425	−	0.879756i	$-0.657706\pi$
$461$		−	373933.i		−	1.75951i	0.475425	−	0.879756i	$-0.342294\pi$
$462$		0			0
$463$	381923.			1.78161			0.890807	−	0.454382i	$-0.150140\pi$
$463$	381923.			1.78161			0.890807	+	0.454382i	$0.150140\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	198666.			0.910940			0.455470	−	0.890251i	$-0.349471\pi$
$467$	198666.			0.910940			0.455470	+	0.890251i	$0.349471\pi$
$468$		0			0
$469$			21102.3i			0.0959366i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−9949.20	−	260955.i	−0.0444699	−	1.16639i
$474$		0			0
$475$			330807.i			1.46618i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			94150.7i			0.410348i	0.978725	+	0.205174i	$0.0657761\pi$
$479$			94150.7i			0.410348i	−0.978725	+	0.205174i	$0.934224\pi$
$480$		0			0
$481$			188098.i			0.813007i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−42543.8			−0.180864
$486$		0			0
$487$	−86151.1			−0.363248			−0.181624	−	0.983368i	$-0.558135\pi$
$487$	−86151.1			−0.363248			−0.181624	+	0.983368i	$0.558135\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	257966.i		−	1.07004i	−0.844840	−	0.535019i	$-0.820304\pi$
$491$		−	257966.i		−	1.07004i	0.844840	−	0.535019i	$-0.179696\pi$
$492$		0			0
$493$	−115819.			−0.476524
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	144187.i		−	0.583733i
$498$		0			0
$499$	−310264.			−1.24604			−0.623018	−	0.782208i	$-0.714093\pi$
$499$	−310264.			−1.24604			−0.623018	+	0.782208i	$0.714093\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	423920.i		−	1.67552i	−0.546042	−	0.837758i	$-0.683866\pi$
$503$		−	423920.i		−	1.67552i	0.546042	−	0.837758i	$-0.316134\pi$
$504$		0			0
$505$		−	129625.i		−	0.508282i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−157612.			−0.608350			−0.304175	−	0.952616i	$-0.598381\pi$
$509$	−157612.			−0.608350			−0.304175	+	0.952616i	$0.598381\pi$
$510$		0			0
$511$	−81031.1			−0.310320
$512$		0			0
$513$		0			0
$514$		0			0
$515$	25240.8			0.0951676
$516$		0			0
$517$	171258.	−	6529.40i	0.640724	−	0.0244283i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−282422.			−1.04046			−0.520228	−	0.854028i	$-0.674153\pi$
$521$	−282422.			−1.04046			−0.520228	+	0.854028i	$0.674153\pi$
$522$		0			0
$523$		−	423736.i		−	1.54914i	−0.632486	−	0.774572i	$-0.717965\pi$
$523$		−	423736.i		−	1.54914i	0.632486	−	0.774572i	$-0.282035\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			422969.i			1.52296i
$528$		0			0
$529$	−274817.			−0.982049
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−184708.			−0.650177
$534$		0			0
$535$		−	123798.i		−	0.432518i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−134846.	+	5141.13i	−0.464151	+	0.0176962i
$540$		0			0
$541$			335320.i			1.14568i	0.819666	+	0.572841i	$0.194159\pi$
$541$			335320.i			1.14568i	−0.819666	+	0.572841i	$0.805841\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	66130.4i		−	0.222643i
$546$		0			0
$547$		−	62911.8i		−	0.210260i	−0.994458	−	0.105130i	$-0.966474\pi$
$547$		−	62911.8i		−	0.210260i	0.994458	−	0.105130i	$-0.0335259\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	181100.			0.596508
$552$		0			0
$553$	−226463.			−0.740538
$554$		0			0
$555$		0			0
$556$		0			0
$557$			168355.i			0.542646i	0.962488	+	0.271323i	$0.0874612\pi$
$557$			168355.i			0.542646i	−0.962488	+	0.271323i	$0.912539\pi$
$558$		0			0
$559$	298202.			0.954305
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	199140.i		−	0.628265i	−0.949379	−	0.314132i	$-0.898286\pi$
$563$		−	199140.i		−	0.628265i	0.949379	−	0.314132i	$-0.101714\pi$
$564$		0			0
$565$	89955.3			0.281793
$566$		0			0
$567$		0			0
$568$		0			0
$569$			190357.i			0.587956i	0.955812	+	0.293978i	$0.0949792\pi$
$569$			190357.i			0.587956i	−0.955812	+	0.293978i	$0.905021\pi$
$570$		0			0
$571$		−	141142.i		−	0.432896i	−0.976294	−	0.216448i	$-0.930553\pi$
$571$		−	141142.i		−	0.432896i	0.976294	−	0.216448i	$-0.0694472\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	41093.6			0.124291
$576$		0			0
$577$	629490.			1.89076			0.945381	−	0.325967i	$-0.105690\pi$
$577$	629490.			1.89076			0.945381	+	0.325967i	$0.105690\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	27880.5			0.0825940
$582$		0			0
$583$	−364375.	+	13892.2i	−1.07204	+	0.0408727i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	459745.			1.33426			0.667130	−	0.744941i	$-0.267522\pi$
$587$	459745.			1.33426			0.667130	+	0.744941i	$0.267522\pi$
$588$		0			0
$589$		−	661378.i		−	1.90642i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			511611.i			1.45489i	0.686166	+	0.727445i	$0.259292\pi$
$593$			511611.i			1.45489i	−0.686166	+	0.727445i	$0.740708\pi$
$594$		0			0
$595$	−87975.1			−0.248500
$596$		0			0
$597$		0			0
$598$		0			0
$599$	266789.			0.743556			0.371778	−	0.928322i	$-0.378748\pi$
$599$	266789.			0.743556			0.371778	+	0.928322i	$0.378748\pi$
$600$		0			0
$601$			76615.0i			0.212112i	0.994360	+	0.106056i	$0.0338223\pi$
$601$			76615.0i			0.212112i	−0.994360	+	0.106056i	$0.966178\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−98157.8	+	7495.63i	−0.268172	+	0.0204785i
$606$		0			0
$607$		−	155813.i		−	0.422888i	−0.977390	−	0.211444i	$-0.932183\pi$
$607$		−	155813.i		−	0.422888i	0.977390	−	0.211444i	$-0.0678166\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			195702.i			0.524220i
$612$		0			0
$613$			423746.i			1.12768i	0.825885	+	0.563838i	$0.190676\pi$
$613$			423746.i			1.12768i	−0.825885	+	0.563838i	$0.809324\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−190437.			−0.500243			−0.250122	−	0.968214i	$-0.580471\pi$
$617$	−190437.			−0.500243			−0.250122	+	0.968214i	$0.580471\pi$
$618$		0			0
$619$	287818.			0.751168			0.375584	−	0.926788i	$-0.377442\pi$
$619$	287818.			0.751168			0.375584	+	0.926788i	$0.377442\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			166511.i			0.429009i
$624$		0			0
$625$	307900.			0.788225
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	496744.i		−	1.25554i
$630$		0			0
$631$	−95505.3			−0.239866			−0.119933	−	0.992782i	$-0.538268\pi$
$631$	−95505.3			−0.239866			−0.119933	+	0.992782i	$0.538268\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	162527.i		−	0.403069i
$636$		0			0
$637$		−	154092.i		−	0.379754i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−212740.			−0.517765			−0.258883	−	0.965909i	$-0.583354\pi$
$641$	−212740.			−0.517765			−0.258883	+	0.965909i	$0.583354\pi$
$642$		0			0
$643$	−127119.			−0.307460			−0.153730	−	0.988113i	$-0.549129\pi$
$643$	−127119.			−0.307460			−0.153730	+	0.988113i	$0.549129\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	322480.			0.770361			0.385181	−	0.922841i	$-0.374139\pi$
$647$	322480.			0.770361			0.385181	+	0.922841i	$0.374139\pi$
$648$		0			0
$649$	201085.	−	7666.58i	0.477409	−	0.0182017i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	427102.			1.00163			0.500813	−	0.865555i	$-0.333034\pi$
$653$	427102.			1.00163			0.500813	+	0.865555i	$0.333034\pi$
$654$		0			0
$655$			69437.5i			0.161850i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			339223.i			0.781114i	0.920579	+	0.390557i	$0.127718\pi$
$659$			339223.i			0.781114i	−0.920579	+	0.390557i	$0.872282\pi$
$660$		0			0
$661$	−518565.			−1.18686			−0.593431	−	0.804885i	$-0.702227\pi$
$661$	−518565.			−1.18686			−0.593431	+	0.804885i	$0.702227\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	137563.			0.311070
$666$		0			0
$667$		−	22496.7i		−	0.0505670i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−24878.6	−	652536.i	−0.0552562	−	1.44930i
$672$		0			0
$673$			37229.5i			0.0821972i	0.999155	+	0.0410986i	$0.0130858\pi$
$673$			37229.5i			0.0821972i	−0.999155	+	0.0410986i	$0.986914\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			431005.i			0.940383i	0.882564	+	0.470192i	$0.155815\pi$
$677$			431005.i			0.940383i	−0.882564	+	0.470192i	$0.844185\pi$
$678$		0			0
$679$		−	226882.i		−	0.492108i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−788957.			−1.69127			−0.845633	−	0.533765i	$-0.820777\pi$
$683$	−788957.			−1.69127			−0.845633	+	0.533765i	$0.820777\pi$
$684$		0			0
$685$	121304.			0.258521
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	416383.i		−	0.877111i
$690$		0			0
$691$	−281540.			−0.589637			−0.294818	−	0.955553i	$-0.595259\pi$
$691$	−281540.			−0.589637			−0.294818	+	0.955553i	$0.595259\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	121541.i		−	0.251624i
$696$		0			0
$697$	487792.			1.00408
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	225633.i		−	0.459162i	−0.973290	−	0.229581i	$-0.926264\pi$
$701$		−	225633.i		−	0.459162i	0.973290	−	0.229581i	$-0.0737355\pi$
$702$		0			0
$703$			776736.i			1.57168i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	691276.			1.38297
$708$		0			0
$709$	−750874.			−1.49374			−0.746869	−	0.664971i	$-0.768444\pi$
$709$	−750874.			−1.49374			−0.746869	+	0.664971i	$0.768444\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−82157.9			−0.161611
$714$		0			0
$715$	−4282.75	−	112331.i	−0.00837743	−	0.219730i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−888683.			−1.71905			−0.859526	−	0.511092i	$-0.829241\pi$
$719$	−888683.			−1.71905			−0.859526	+	0.511092i	$0.829241\pi$
$720$		0			0
$721$			134607.i			0.258939i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	184029.i		−	0.350115i
$726$		0			0
$727$	−361266.			−0.683530			−0.341765	−	0.939785i	$-0.611025\pi$
$727$	−361266.			−0.683530			−0.341765	+	0.939785i	$0.611025\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−787515.			−1.47375
$732$		0			0
$733$		−	178651.i		−	0.332504i	−0.986083	−	0.166252i	$-0.946833\pi$
$733$		−	178651.i		−	0.332504i	0.986083	−	0.166252i	$-0.0531666\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	71157.3	−	2712.95i	0.131004	−	0.00499466i
$738$		0			0
$739$			791792.i			1.44985i	0.688829	+	0.724924i	$0.258125\pi$
$739$			791792.i			1.44985i	−0.688829	+	0.724924i	$0.741875\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			774816.i			1.40353i	0.712409	+	0.701764i	$0.247604\pi$
$743$			774816.i			1.40353i	−0.712409	+	0.701764i	$0.752396\pi$
$744$		0			0
$745$			180464.i			0.325146i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	660201.			1.17683
$750$		0			0
$751$	905890.			1.60618			0.803092	−	0.595854i	$-0.203186\pi$
$751$	905890.			1.60618			0.803092	+	0.595854i	$0.203186\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	103487.i		−	0.181548i
$756$		0			0
$757$	−37734.7			−0.0658489			−0.0329245	−	0.999458i	$-0.510482\pi$
$757$	−37734.7			−0.0658489			−0.0329245	+	0.999458i	$0.510482\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			793663.i			1.37046i	0.728326	+	0.685231i	$0.240299\pi$
$761$			793663.i			1.37046i	−0.728326	+	0.685231i	$0.759701\pi$
$762$		0			0
$763$	352667.			0.605782
$764$		0			0
$765$		0			0
$766$		0			0
$767$			229787.i			0.390601i
$768$		0			0
$769$			248952.i			0.420982i	0.977596	+	0.210491i	$0.0675062\pi$
$769$			248952.i			0.420982i	−0.977596	+	0.210491i	$0.932494\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	696432.			1.16552			0.582760	−	0.812644i	$-0.301973\pi$
$773$	696432.			1.16552			0.582760	+	0.812644i	$0.301973\pi$
$774$		0			0
$775$	−672074.			−1.11896
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−762738.			−1.25690
$780$		0			0
$781$	−486202.	+	18537.0i	−0.797103	+	0.0303904i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	231048.			0.374941
$786$		0			0
$787$			107990.i			0.174355i	0.996193	+	0.0871776i	$0.0277848\pi$
$787$			107990.i			0.174355i	−0.996193	+	0.0871776i	$0.972215\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			479723.i			0.766722i
$792$		0			0
$793$	745673.			1.18577
$794$		0			0
$795$		0			0
$796$		0			0
$797$	79943.4			0.125854			0.0629268	−	0.998018i	$-0.479957\pi$
$797$	79943.4			0.125854			0.0629268	+	0.998018i	$0.479957\pi$
$798$		0			0
$799$		−	516826.i		−	0.809563i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10417.5	+	273238.i	0.0161559	+	0.423751i
$804$		0			0
$805$		−	17088.4i		−	0.0263699i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	140244.i		−	0.214282i	−0.994244	−	0.107141i	$-0.965830\pi$
$809$		−	140244.i		−	0.214282i	0.994244	−	0.107141i	$-0.0341697\pi$
$810$		0			0
$811$			796857.i			1.21154i	0.795639	+	0.605771i	$0.207135\pi$
$811$			796857.i			1.21154i	−0.795639	+	0.605771i	$0.792865\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−228905.			−0.344620
$816$		0			0
$817$	1.23140e6			1.84483
$818$		0			0
$819$		0			0
$820$		0			0
$821$			1.03290e6i			1.53240i	0.642601	+	0.766201i	$0.277855\pi$
$821$			1.03290e6i			1.53240i	−0.642601	+	0.766201i	$0.722145\pi$
$822$		0			0
$823$	−640524.			−0.945662			−0.472831	−	0.881153i	$-0.656768\pi$
$823$	−640524.			−0.945662			−0.472831	+	0.881153i	$0.656768\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1.26963e6i			1.85637i	0.372114	+	0.928187i	$0.378633\pi$
$827$			1.26963e6i			1.85637i	−0.372114	+	0.928187i	$0.621367\pi$
$828$		0			0
$829$	−975440.			−1.41936			−0.709678	−	0.704526i	$-0.751160\pi$
$829$	−975440.			−1.41936			−0.709678	+	0.704526i	$0.751160\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			406939.i			0.586461i
$834$		0			0
$835$		−	234484.i		−	0.336311i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−376637.			−0.535055			−0.267528	−	0.963550i	$-0.586207\pi$
$839$	−376637.			−0.535055			−0.267528	+	0.963550i	$0.586207\pi$
$840$		0			0
$841$	606534.			0.857558
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−63674.7			−0.0891771
$846$		0			0
$847$	−39973.5	−	523466.i	−0.0557193	−	0.729662i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	96488.0			0.133234
$852$		0			0
$853$			362817.i			0.498643i	0.968421	+	0.249322i	$0.0802076\pi$
$853$			362817.i			0.498643i	−0.968421	+	0.249322i	$0.919792\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			73874.1i			0.100584i	0.998735	+	0.0502922i	$0.0160153\pi$
$857$			73874.1i			0.100584i	−0.998735	+	0.0502922i	$0.983985\pi$
$858$		0			0
$859$	−706662.			−0.957691			−0.478845	−	0.877899i	$-0.658945\pi$
$859$	−706662.			−0.957691			−0.478845	+	0.877899i	$0.658945\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	327867.			0.440227			0.220113	−	0.975474i	$-0.429357\pi$
$863$	327867.			0.440227			0.220113	+	0.975474i	$0.429357\pi$
$864$		0			0
$865$		−	197261.i		−	0.263638i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	29114.4	+	763636.i	0.0385539	+	1.01122i
$870$		0			0
$871$			81313.7i			0.107183i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	290475.i		−	0.379396i
$876$		0			0
$877$		−	661821.i		−	0.860481i	−0.902714	−	0.430240i	$-0.858429\pi$
$877$		−	661821.i		−	0.860481i	0.902714	−	0.430240i	$-0.141571\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−185765.			−0.239338			−0.119669	−	0.992814i	$-0.538183\pi$
$881$	−185765.			−0.239338			−0.119669	+	0.992814i	$0.538183\pi$
$882$		0			0
$883$	−1.04690e6			−1.34272			−0.671359	−	0.741132i	$-0.734289\pi$
$883$	−1.04690e6			−1.34272			−0.671359	+	0.741132i	$0.734289\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	355686.i		−	0.452085i	−0.974117	−	0.226042i	$-0.927421\pi$
$887$		−	355686.i		−	0.452085i	0.974117	−	0.226042i	$-0.0725788\pi$
$888$		0			0
$889$	866744.			1.09670
$890$		0			0
$891$		0			0
$892$		0			0
$893$			808138.i			1.01340i
$894$		0			0
$895$	16150.5			0.0201623
$896$		0			0
$897$		0			0
$898$		0			0
$899$			367927.i			0.455242i
$900$		0			0
$901$			1.09962e6i			1.35454i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	188277.			0.229880
$906$		0			0
$907$	139907.			0.170069			0.0850347	−	0.996378i	$-0.472900\pi$
$907$	139907.			0.170069			0.0850347	+	0.996378i	$0.472900\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	366506.			0.441616			0.220808	−	0.975317i	$-0.429131\pi$
$911$	366506.			0.441616			0.220808	+	0.975317i	$0.429131\pi$
$912$		0			0
$913$	−3584.36	−	94013.5i	−0.00430002	−	0.112784i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−370304.			−0.440372
$918$		0			0
$919$			752164.i			0.890598i	0.895382	+	0.445299i	$0.146903\pi$
$919$			752164.i			0.890598i	−0.895382	+	0.445299i	$0.853097\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	555598.i		−	0.652165i
$924$		0			0
$925$	789297.			0.922480
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−161987.			−0.187693			−0.0938467	−	0.995587i	$-0.529916\pi$
$929$	−161987.			−0.187693			−0.0938467	+	0.995587i	$0.529916\pi$
$930$		0			0
$931$		−	636312.i		−	0.734127i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11310.2	+	296653.i	0.0129374	+	0.339333i
$936$		0			0
$937$			915780.i			1.04307i	0.853231	+	0.521533i	$0.174640\pi$
$937$			915780.i			1.04307i	−0.853231	+	0.521533i	$0.825360\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	784090.i		−	0.885497i	−0.896646	−	0.442748i	$-0.854004\pi$
$941$		−	784090.i		−	0.885497i	0.896646	−	0.442748i	$-0.145996\pi$
$942$		0			0
$943$			94749.1i			0.106550i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.47216e6			1.64155			0.820776	−	0.571250i	$-0.193541\pi$
$947$	1.47216e6			1.64155			0.820776	+	0.571250i	$0.193541\pi$
$948$		0			0
$949$	−312238.			−0.346700
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	372692.i		−	0.410360i	−0.978724	−	0.205180i	$-0.934222\pi$
$953$		−	372692.i		−	0.410360i	0.978724	−	0.205180i	$-0.0657779\pi$
$954$		0			0
$955$	−116988.			−0.128273
$956$		0			0
$957$		0			0
$958$		0			0
$959$			646905.i			0.703401i
$960$		0			0
$961$	420147.			0.454940
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	252772.i		−	0.271440i
$966$		0			0
$967$			1.33645e6i			1.42922i	0.699523	+	0.714610i	$0.253396\pi$
$967$			1.33645e6i			1.42922i	−0.699523	+	0.714610i	$0.746604\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.43833e6			1.52552			0.762762	−	0.646679i	$-0.223843\pi$
$971$	1.43833e6			1.52552			0.762762	+	0.646679i	$0.223843\pi$
$972$		0			0
$973$	648164.			0.684635
$974$		0			0
$975$		0			0
$976$		0			0
$977$	133419.			0.139774			0.0698872	−	0.997555i	$-0.477736\pi$
$977$	133419.			0.139774			0.0698872	+	0.997555i	$0.477736\pi$
$978$		0			0
$979$	561477.	−	21406.9i	0.585824	−	0.0223351i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.04704e6			−1.08357			−0.541783	−	0.840518i	$-0.682251\pi$
$983$	−1.04704e6			−1.08357			−0.541783	+	0.840518i	$0.682251\pi$
$984$		0			0
$985$		−	131981.i		−	0.136032i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	152968.i		−	0.156389i
$990$		0			0
$991$	−869334.			−0.885196			−0.442598	−	0.896720i	$-0.645943\pi$
$991$	−869334.			−0.885196			−0.442598	+	0.896720i	$0.645943\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−428390.			−0.432706
$996$		0			0
$997$			706445.i			0.710703i	0.934733	+	0.355351i	$0.115639\pi$
$997$			706445.i			0.710703i	−0.934733	+	0.355351i	$0.884361\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	792.5.j.a.505.5		12
3.2	odd	2		88.5.h.a.65.5	✓	12
11.10	odd	2	inner	792.5.j.a.505.6		12
12.11	even	2		176.5.h.f.65.8		12
24.5	odd	2		704.5.h.k.65.7		12
24.11	even	2		704.5.h.l.65.6		12
33.32	even	2		88.5.h.a.65.6	yes	12
132.131	odd	2		176.5.h.f.65.7		12
264.131	odd	2		704.5.h.l.65.5		12
264.197	even	2		704.5.h.k.65.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.5.h.a.65.5	✓	12	3.2	odd	2
88.5.h.a.65.6	yes	12	33.32	even	2
176.5.h.f.65.7		12	132.131	odd	2
176.5.h.f.65.8		12	12.11	even	2
704.5.h.k.65.7		12	24.5	odd	2
704.5.h.k.65.8		12	264.197	even	2
704.5.h.l.65.5		12	264.131	odd	2
704.5.h.l.65.6		12	24.11	even	2
792.5.j.a.505.5		12	1.1	even	1	trivial
792.5.j.a.505.6		12	11.10	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 792 = 2^{3} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	792.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.72383 q^{5} -35.8576i q^{7} +O(q^{10})\)	\(q-6.72383 q^{5} -35.8576i q^{7} +(-120.912 + 4.60990i) q^{11} -138.170i q^{13} +364.891i q^{17} -570.563i q^{19} -70.8767 q^{23} -579.790 q^{25} +317.406i q^{29} +1159.17 q^{31} +241.100i q^{35} -1361.35 q^{37} -1336.82i q^{41} +2158.22i q^{43} -1416.39 q^{47} +1115.24 q^{49} +3013.55 q^{53} +(812.993 - 30.9962i) q^{55} -1663.07 q^{59} +5396.77i q^{61} +929.033i q^{65} -588.504 q^{67} +4021.12 q^{71} -2259.81i q^{73} +(165.300 + 4335.61i) q^{77} -6315.63i q^{79} +777.535i q^{83} -2453.46i q^{85} -4643.68 q^{89} -4954.44 q^{91} +3836.37i q^{95} +6327.31 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 12 q^{11} - 1608 q^{23} + 1484 q^{25} - 2136 q^{31} - 288 q^{37} + 6024 q^{47} - 9940 q^{49} - 3576 q^{53} - 8344 q^{55} + 7704 q^{59} + 10312 q^{67} - 8904 q^{71} - 9120 q^{77} - 4224 q^{89} + 23904 q^{91} - 21280 q^{97}+O(q^{100}) \) 12 * q - 12 * q^11 - 1608 * q^23 + 1484 * q^25 - 2136 * q^31 - 288 * q^37 + 6024 * q^47 - 9940 * q^49 - 3576 * q^53 - 8344 * q^55 + 7704 * q^59 + 10312 * q^67 - 8904 * q^71 - 9120 * q^77 - 4224 * q^89 + 23904 * q^91 - 21280 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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