LMFDB - Embedded newform 464.6.e.c.289.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [464,6,Mod(289,464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("464.289"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	464.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,46,0,-20,0,-1574] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$74.4180923932$
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} + 278x^{10} + 28285x^{8} + 1260472x^{6} + 22944832x^{4} + 140087936x^{2} + 966400 $ x^12 + 278x^10 + 28285x^8 + 1260472x^6 + 22944832x^4 + 140087936*x^2 + 966400
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{24}\cdot 5 $
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.3
Root			$4.44887i$ of defining polynomial
Character	$\chi$	$=$	464.289
Dual form			464.6.e.c.289.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-21.5070i q^{3} +90.0922 q^{5} +211.678 q^{7} -219.553 q^{9} +593.900i q^{11} +445.466 q^{13} -1937.62i q^{15} -368.642i q^{17} -761.926i q^{19} -4552.56i q^{21} -728.987 q^{23} +4991.60 q^{25} -504.276i q^{27} +(2787.76 + 3569.25i) q^{29} +9222.49i q^{31} +12773.0 q^{33} +19070.5 q^{35} -3392.70i q^{37} -9580.65i q^{39} +1763.84i q^{41} -5296.04i q^{43} -19780.0 q^{45} +12459.7i q^{47} +28000.4 q^{49} -7928.41 q^{51} -14753.2 q^{53} +53505.8i q^{55} -16386.8 q^{57} +34598.5 q^{59} -9455.51i q^{61} -46474.5 q^{63} +40133.0 q^{65} -41093.0 q^{67} +15678.4i q^{69} +23404.1 q^{71} +25457.6i q^{73} -107355. i q^{75} +125715. i q^{77} +45401.0i q^{79} -64196.9 q^{81} +53492.2 q^{83} -33211.8i q^{85} +(76763.9 - 59956.6i) q^{87} -112498. i q^{89} +94295.2 q^{91} +198349. q^{93} -68643.6i q^{95} -724.916i q^{97} -130393. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 46 q^{5} - 20 q^{7} - 1574 q^{9} + 1362 q^{13} - 5852 q^{23} + 12678 q^{25} + 11328 q^{29} - 22694 q^{33} - 4532 q^{35} - 52816 q^{45} + 102836 q^{49} - 58540 q^{51} + 25650 q^{53} - 32824 q^{57}+ \cdots + 377966 q^{93}+O(q^{100}) $ 12 * q + 46 * q^5 - 20 * q^7 - 1574 * q^9 + 1362 * q^13 - 5852 * q^23 + 12678 * q^25 + 11328 * q^29 - 22694 * q^33 - 4532 * q^35 - 52816 * q^45 + 102836 * q^49 - 58540 * q^51 + 25650 * q^53 - 32824 * q^57 + 3900 * q^59 + 146616 * q^63 + 169574 * q^65 + 28264 * q^67 + 286832 * q^71 - 24084 * q^81 - 85692 * q^83 + 137716 * q^87 + 182372 * q^91 + 377966 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/464\mathbb{Z}\right)^\times$.

$n$	$117$	$175$	$321$
$\chi(n)$	$1$	$1$	$-1$

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
$3$		−	21.5070i		−	1.37968i	−0.723963	−	0.689839i	$-0.757681\pi$
$3$		−	21.5070i		−	1.37968i	0.723963	−	0.689839i	$-0.242319\pi$
$4$		0			0
$5$	90.0922			1.61162			0.805809	−	0.592175i	$-0.201731\pi$
$5$	90.0922			1.61162			0.805809	+	0.592175i	$0.201731\pi$
$6$		0			0
$7$	211.678			1.63279			0.816394	−	0.577495i	$-0.195970\pi$
$7$	211.678			1.63279			0.816394	+	0.577495i	$0.195970\pi$
$8$		0			0
$9$	−219.553			−0.903510
$10$		0			0
$11$			593.900i			1.47990i	0.672663	+	0.739949i	$0.265150\pi$
$11$			593.900i			1.47990i	−0.672663	+	0.739949i	$0.734850\pi$
$12$		0			0
$13$	445.466			0.731065			0.365532	−	0.930799i	$-0.380887\pi$
$13$	445.466			0.731065			0.365532	+	0.930799i	$0.380887\pi$
$14$		0			0
$15$		−	1937.62i		−	2.22351i
$16$		0			0
$17$		−	368.642i		−	0.309373i	−0.987964	−	0.154687i	$-0.950563\pi$
$17$		−	368.642i		−	0.309373i	0.987964	−	0.154687i	$-0.0494368\pi$
$18$		0			0
$19$		−	761.926i		−	0.484205i	−0.970251	−	0.242102i	$-0.922163\pi$
$19$		−	761.926i		−	0.484205i	0.970251	−	0.242102i	$-0.0778370\pi$
$20$		0			0
$21$		−	4552.56i		−	2.25272i
$22$		0			0
$23$	−728.987			−0.287343			−0.143671	−	0.989625i	$-0.545891\pi$
$23$	−728.987			−0.287343			−0.143671	+	0.989625i	$0.545891\pi$
$24$		0			0
$25$	4991.60			1.59731
$26$		0			0
$27$		−	504.276i		−	0.133125i
$28$		0			0
$29$	2787.76	+	3569.25i	0.615547	+	0.788100i
$29$	2787.76	+	3569.25i	0.615547	+	0.788100i
$30$		0			0
$31$			9222.49i			1.72363i	0.507223	+	0.861815i	$0.330672\pi$
$31$			9222.49i			1.72363i	−0.507223	+	0.861815i	$0.669328\pi$
$32$		0			0
$33$	12773.0			2.04178
$34$		0			0
$35$	19070.5			2.63143
$36$		0			0
$37$		−	3392.70i		−	0.407418i	−0.979031	−	0.203709i	$-0.934700\pi$
$37$		−	3392.70i		−	0.407418i	0.979031	−	0.203709i	$-0.0652997\pi$
$38$		0			0
$39$		−	9580.65i		−	1.00863i
$40$		0			0
$41$			1763.84i			0.163870i	0.996638	+	0.0819351i	$0.0261100\pi$
$41$			1763.84i			0.163870i	−0.996638	+	0.0819351i	$0.973890\pi$
$42$		0			0
$43$		−	5296.04i		−	0.436797i	−0.975860	−	0.218399i	$-0.929917\pi$
$43$		−	5296.04i		−	0.436797i	0.975860	−	0.218399i	$-0.0700833\pi$
$44$		0			0
$45$	−19780.0			−1.45611
$46$		0			0
$47$			12459.7i			0.822741i	0.911468	+	0.411371i	$0.134950\pi$
$47$			12459.7i			0.822741i	−0.911468	+	0.411371i	$0.865050\pi$
$48$		0			0
$49$	28000.4			1.66600
$50$		0			0
$51$	−7928.41			−0.426836
$52$		0			0
$53$	−14753.2			−0.721432			−0.360716	−	0.932676i	$-0.617468\pi$
$53$	−14753.2			−0.721432			−0.360716	+	0.932676i	$0.617468\pi$
$54$		0			0
$55$			53505.8i			2.38503i
$56$		0			0
$57$	−16386.8			−0.668046
$58$		0			0
$59$	34598.5			1.29398			0.646989	−	0.762499i	$-0.276028\pi$
$59$	34598.5			1.29398			0.646989	+	0.762499i	$0.276028\pi$
$60$		0			0
$61$		−	9455.51i		−	0.325357i	−0.986679	−	0.162679i	$-0.947987\pi$
$61$		−	9455.51i		−	0.325357i	0.986679	−	0.162679i	$-0.0520133\pi$
$62$		0			0
$63$	−46474.5			−1.47524
$64$		0			0
$65$	40133.0			1.17820
$66$		0			0
$67$	−41093.0			−1.11836			−0.559179	−	0.829047i	$-0.688884\pi$
$67$	−41093.0			−1.11836			−0.559179	+	0.829047i	$0.688884\pi$
$68$		0			0
$69$			15678.4i			0.396440i
$70$		0			0
$71$	23404.1			0.550992			0.275496	−	0.961302i	$-0.411158\pi$
$71$	23404.1			0.550992			0.275496	+	0.961302i	$0.411158\pi$
$72$		0			0
$73$			25457.6i			0.559127i	0.960127	+	0.279563i	$0.0901897\pi$
$73$			25457.6i			0.559127i	−0.960127	+	0.279563i	$0.909810\pi$
$74$		0			0
$75$		−	107355.i		−	2.20378i
$76$		0			0
$77$			125715.i			2.41636i
$78$		0			0
$79$			45401.0i			0.818460i	0.912431	+	0.409230i	$0.134203\pi$
$79$			45401.0i			0.818460i	−0.912431	+	0.409230i	$0.865797\pi$
$80$		0			0
$81$	−64196.9			−1.08718
$82$		0			0
$83$	53492.2			0.852306			0.426153	−	0.904651i	$-0.359869\pi$
$83$	53492.2			0.852306			0.426153	+	0.904651i	$0.359869\pi$
$84$		0			0
$85$		−	33211.8i		−	0.498592i
$86$		0			0
$87$	76763.9	−	59956.6i	1.08732	−	0.849256i
$88$		0			0
$89$		−	112498.i		−	1.50547i	−0.658326	−	0.752733i	$-0.728735\pi$
$89$		−	112498.i		−	1.50547i	0.658326	−	0.752733i	$-0.271265\pi$
$90$		0			0
$91$	94295.2			1.19367
$92$		0			0
$93$	198349.			2.37805
$94$		0			0
$95$		−	68643.6i		−	0.780353i
$96$		0			0
$97$		−	724.916i		−	0.00782273i	−0.999992	−	0.00391137i	$-0.998755\pi$
$97$		−	724.916i		−	0.00782273i	0.999992	−	0.00391137i	$-0.00124503\pi$
$98$		0			0
$99$		−	130393.i		−	1.33710i
$100$		0			0
$101$			55652.7i			0.542854i	0.962459	+	0.271427i	$0.0874955\pi$
$101$			55652.7i			0.542854i	−0.962459	+	0.271427i	$0.912505\pi$
$102$		0			0
$103$	−86986.8			−0.807905			−0.403953	−	0.914780i	$-0.632364\pi$
$103$	−86986.8			−0.807905			−0.403953	+	0.914780i	$0.632364\pi$
$104$		0			0
$105$		−	410150.i		−	3.63053i
$106$		0			0
$107$	67319.9			0.568439			0.284220	−	0.958759i	$-0.408266\pi$
$107$	67319.9			0.568439			0.284220	+	0.958759i	$0.408266\pi$
$108$		0			0
$109$	−107618.			−0.867595			−0.433797	−	0.901010i	$-0.642827\pi$
$109$	−107618.			−0.867595			−0.433797	+	0.901010i	$0.642827\pi$
$110$		0			0
$111$	−72966.9			−0.562106
$112$		0			0
$113$		−	140630.i		−	1.03605i	−0.855365	−	0.518027i	$-0.826667\pi$
$113$		−	140630.i		−	1.03605i	0.855365	−	0.518027i	$-0.173333\pi$
$114$		0			0
$115$	−65676.0			−0.463087
$116$		0			0
$117$	−97803.3			−0.660525
$118$		0			0
$119$		−	78033.4i		−	0.505141i
$120$		0			0
$121$	−191666.			−1.19010
$122$		0			0
$123$	37935.0			0.226088
$124$		0			0
$125$	168167.			0.962642
$126$		0			0
$127$		−	210012.i		−	1.15541i	−0.816247	−	0.577703i	$-0.803949\pi$
$127$		−	210012.i		−	1.15541i	0.816247	−	0.577703i	$-0.196051\pi$
$128$		0			0
$129$	−113902.			−0.602640
$130$		0			0
$131$		−	92160.7i		−	0.469210i	−0.972091	−	0.234605i	$-0.924620\pi$
$131$		−	92160.7i		−	0.469210i	0.972091	−	0.234605i	$-0.0753797\pi$
$132$		0			0
$133$		−	161283.i		−	0.790604i
$134$		0			0
$135$		−	45431.3i		−	0.214546i
$136$		0			0
$137$		−	102506.i		−	0.466602i	−0.972405	−	0.233301i	$-0.925047\pi$
$137$		−	102506.i		−	0.466602i	0.972405	−	0.233301i	$-0.0749528\pi$
$138$		0			0
$139$	199342.			0.875106			0.437553	−	0.899193i	$-0.355845\pi$
$139$	199342.			0.875106			0.437553	+	0.899193i	$0.355845\pi$
$140$		0			0
$141$	267972.			1.13512
$142$		0			0
$143$			264562.i			1.08190i
$144$		0			0
$145$	251156.	+	321561.i	0.992026	+	1.27012i
$146$		0			0
$147$		−	602207.i		−	2.29854i
$148$		0			0
$149$	−9725.12			−0.0358863			−0.0179432	−	0.999839i	$-0.505712\pi$
$149$	−9725.12			−0.0358863			−0.0179432	+	0.999839i	$0.505712\pi$
$150$		0			0
$151$	−483239.			−1.72472			−0.862362	−	0.506292i	$-0.831016\pi$
$151$	−483239.			−1.72472			−0.862362	+	0.506292i	$0.831016\pi$
$152$		0			0
$153$			80936.5i			0.279522i
$154$		0			0
$155$			830874.i			2.77783i
$156$		0			0
$157$		−	84963.6i		−	0.275096i	−0.990495	−	0.137548i	$-0.956078\pi$
$157$		−	84963.6i		−	0.275096i	0.990495	−	0.137548i	$-0.0439221\pi$
$158$		0			0
$159$			317297.i			0.995344i
$160$		0			0
$161$	−154310.			−0.469170
$162$		0			0
$163$		−	531836.i		−	1.56786i	−0.620846	−	0.783932i	$-0.713211\pi$
$163$		−	531836.i		−	1.56786i	0.620846	−	0.783932i	$-0.286789\pi$
$164$		0			0
$165$	1.15075e6			3.29057
$166$		0			0
$167$	505065.			1.40138			0.700690	−	0.713466i	$-0.252876\pi$
$167$	505065.			1.40138			0.700690	+	0.713466i	$0.252876\pi$
$168$		0			0
$169$	−172853.			−0.465544
$170$		0			0
$171$			167283.i			0.437484i
$172$		0			0
$173$	−187852.			−0.477201			−0.238601	−	0.971118i	$-0.576689\pi$
$173$	−187852.			−0.477201			−0.238601	+	0.971118i	$0.576689\pi$
$174$		0			0
$175$	1.05661e6			2.60808
$176$		0			0
$177$		−	744111.i		−	1.78527i
$178$		0			0
$179$	−834374.			−1.94638			−0.973191	−	0.229999i	$-0.926128\pi$
$179$	−834374.			−1.94638			−0.973191	+	0.229999i	$0.926128\pi$
$180$		0			0
$181$	84063.7			0.190727			0.0953635	−	0.995443i	$-0.469599\pi$
$181$	84063.7			0.190727			0.0953635	+	0.995443i	$0.469599\pi$
$182$		0			0
$183$	−203360.			−0.448888
$184$		0			0
$185$		−	305655.i		−	0.656603i
$186$		0			0
$187$	218937.			0.457841
$188$		0			0
$189$		−	106744.i		−	0.217365i
$190$		0			0
$191$		−	454281.i		−	0.901034i	−0.892768	−	0.450517i	$-0.851240\pi$
$191$		−	454281.i		−	0.901034i	0.892768	−	0.450517i	$-0.148760\pi$
$192$		0			0
$193$		−	461063.i		−	0.890979i	−0.895287	−	0.445489i	$-0.853030\pi$
$193$		−	461063.i		−	0.890979i	0.895287	−	0.445489i	$-0.146970\pi$
$194$		0			0
$195$		−	863142.i		−	1.62553i
$196$		0			0
$197$	−401536.			−0.737154			−0.368577	−	0.929597i	$-0.620155\pi$
$197$	−401536.			−0.737154			−0.368577	+	0.929597i	$0.620155\pi$
$198$		0			0
$199$	−107914.			−0.193173			−0.0965863	−	0.995325i	$-0.530792\pi$
$199$	−107914.			−0.193173			−0.0965863	+	0.995325i	$0.530792\pi$
$200$		0			0
$201$			883789.i			1.54297i
$202$		0			0
$203$	590107.	+	755530.i	1.00506	+	1.28680i
$204$		0			0
$205$			158908.i			0.264096i
$206$		0			0
$207$	160051.			0.259617
$208$		0			0
$209$	452508.			0.716573
$210$		0			0
$211$			296058.i			0.457794i	0.973451	+	0.228897i	$0.0735120\pi$
$211$			296058.i			0.457794i	−0.973451	+	0.228897i	$0.926488\pi$
$212$		0			0
$213$		−	503352.i		−	0.760192i
$214$		0			0
$215$		−	477132.i		−	0.703951i
$216$		0			0
$217$			1.95220e6i			2.81432i
$218$		0			0
$219$	547518.			0.771415
$220$		0			0
$221$		−	164218.i		−	0.226172i
$222$		0			0
$223$	−857377.			−1.15454			−0.577271	−	0.816552i	$-0.695882\pi$
$223$	−857377.			−1.15454			−0.577271	+	0.816552i	$0.695882\pi$
$224$		0			0
$225$	−1.09592e6			−1.44319
$226$		0			0
$227$	265585.			0.342089			0.171045	−	0.985263i	$-0.445286\pi$
$227$	265585.			0.342089			0.171045	+	0.985263i	$0.445286\pi$
$228$		0			0
$229$			1.13602e6i			1.43152i	0.698349	+	0.715758i	$0.253919\pi$
$229$			1.13602e6i			1.43152i	−0.698349	+	0.715758i	$0.746081\pi$
$230$		0			0
$231$	2.70377e6			3.33380
$232$		0			0
$233$	−341111.			−0.411629			−0.205814	−	0.978591i	$-0.565984\pi$
$233$	−341111.			−0.411629			−0.205814	+	0.978591i	$0.565984\pi$
$234$		0			0
$235$			1.12252e6i			1.32594i
$236$		0			0
$237$	976441.			1.12921
$238$		0			0
$239$	−1.46515e6			−1.65916			−0.829578	−	0.558391i	$-0.811419\pi$
$239$	−1.46515e6			−1.65916			−0.829578	+	0.558391i	$0.811419\pi$
$240$		0			0
$241$	31493.9			0.0349288			0.0174644	−	0.999847i	$-0.494441\pi$
$241$	31493.9			0.0349288			0.0174644	+	0.999847i	$0.494441\pi$
$242$		0			0
$243$			1.25815e6i			1.36683i
$244$		0			0
$245$	2.52262e6			2.68495
$246$		0			0
$247$		−	339412.i		−	0.353985i
$248$		0			0
$249$		−	1.15046e6i		−	1.17591i
$250$		0			0
$251$		−	974972.i		−	0.976805i	−0.872618	−	0.488402i	$-0.837580\pi$
$251$		−	974972.i		−	0.976805i	0.872618	−	0.488402i	$-0.162420\pi$
$252$		0			0
$253$		−	432945.i		−	0.425238i
$254$		0			0
$255$	−714288.			−0.687896
$256$		0			0
$257$	−127188.			−0.120119			−0.0600596	−	0.998195i	$-0.519129\pi$
$257$	−127188.			−0.120119			−0.0600596	+	0.998195i	$0.519129\pi$
$258$		0			0
$259$		−	718158.i		−	0.665228i
$260$		0			0
$261$	−612062.	−	783639.i	−0.556153	−	0.712057i
$262$		0			0
$263$		−	644306.i		−	0.574385i	−0.957873	−	0.287192i	$-0.907278\pi$
$263$		−	644306.i		−	0.574385i	0.957873	−	0.287192i	$-0.0927219\pi$
$264$		0			0
$265$	−1.32914e6			−1.16267
$266$		0			0
$267$	−2.41951e6			−2.07706
$268$		0			0
$269$			2.20790e6i			1.86037i	0.367095	+	0.930183i	$0.380352\pi$
$269$			2.20790e6i			1.86037i	−0.367095	+	0.930183i	$0.619648\pi$
$270$		0			0
$271$		−	1.15746e6i		−	0.957374i	−0.877986	−	0.478687i	$-0.841113\pi$
$271$		−	1.15746e6i		−	0.957374i	0.877986	−	0.478687i	$-0.158887\pi$
$272$		0			0
$273$		−	2.02801e6i		−	1.64689i
$274$		0			0
$275$			2.96451e6i			2.36386i
$276$		0			0
$277$	−699513.			−0.547768			−0.273884	−	0.961763i	$-0.588308\pi$
$277$	−699513.			−0.547768			−0.273884	+	0.961763i	$0.588308\pi$
$278$		0			0
$279$		−	2.02483e6i		−	1.55732i
$280$		0			0
$281$	−2.48281e6			−1.87576			−0.937881	−	0.346956i	$-0.887215\pi$
$281$	−2.48281e6			−1.87576			−0.937881	+	0.346956i	$0.887215\pi$
$282$		0			0
$283$	−169883.			−0.126091			−0.0630456	−	0.998011i	$-0.520081\pi$
$283$	−169883.			−0.126091			−0.0630456	+	0.998011i	$0.520081\pi$
$284$		0			0
$285$	−1.47632e6			−1.07664
$286$		0			0
$287$			373366.i			0.267565i
$288$		0			0
$289$	1.28396e6			0.904288
$290$		0			0
$291$	−15590.8			−0.0107929
$292$		0			0
$293$			1.09182e6i			0.742985i	0.928436	+	0.371493i	$0.121154\pi$
$293$			1.09182e6i			0.742985i	−0.928436	+	0.371493i	$0.878846\pi$
$294$		0			0
$295$	3.11705e6			2.08540
$296$		0			0
$297$	299490.			0.197011
$298$		0			0
$299$	−324739.			−0.210066
$300$		0			0
$301$		−	1.12105e6i		−	0.713198i
$302$		0			0
$303$	1.19692e6			0.748963
$304$		0			0
$305$		−	851867.i		−	0.524351i
$306$		0			0
$307$		−	2.17551e6i		−	1.31739i	−0.752410	−	0.658696i	$-0.771109\pi$
$307$		−	2.17551e6i		−	1.31739i	0.752410	−	0.658696i	$-0.228891\pi$
$308$		0			0
$309$			1.87083e6i			1.11465i
$310$		0			0
$311$		−	891275.i		−	0.522529i	−0.965267	−	0.261265i	$-0.915860\pi$
$311$		−	891275.i		−	0.522529i	0.965267	−	0.261265i	$-0.0841395\pi$
$312$		0			0
$313$	1.33851e6			0.772252			0.386126	−	0.922446i	$-0.373813\pi$
$313$	1.33851e6			0.772252			0.386126	+	0.922446i	$0.373813\pi$
$314$		0			0
$315$	−4.18699e6			−2.37753
$316$		0			0
$317$		−	1.37267e6i		−	0.767219i	−0.923495	−	0.383610i	$-0.874681\pi$
$317$		−	1.37267e6i		−	0.767219i	0.923495	−	0.383610i	$-0.125319\pi$
$318$		0			0
$319$	−2.11978e6	+	1.65565e6i	−1.16631	+	0.910946i
$320$		0			0
$321$		−	1.44785e6i		−	0.784263i
$322$		0			0
$323$	−280878.			−0.149800
$324$		0			0
$325$	2.22359e6			1.16774
$326$		0			0
$327$			2.31454e6i			1.19700i
$328$		0			0
$329$			2.63744e6i			1.34336i
$330$		0			0
$331$			3.39178e6i			1.70160i	0.525488	+	0.850801i	$0.323883\pi$
$331$			3.39178e6i			1.70160i	−0.525488	+	0.850801i	$0.676117\pi$
$332$		0			0
$333$			744876.i			0.368107i
$334$		0			0
$335$	−3.70216e6			−1.80237
$336$		0			0
$337$		−	3.55335e6i		−	1.70437i	−0.523243	−	0.852184i	$-0.675278\pi$
$337$		−	3.55335e6i		−	1.70437i	0.523243	−	0.852184i	$-0.324722\pi$
$338$		0			0
$339$	−3.02454e6			−1.42942
$340$		0			0
$341$	−5.47724e6			−2.55080
$342$		0			0
$343$	2.36940e6			1.08743
$344$		0			0
$345$			1.41250e6i			0.638910i
$346$		0			0
$347$	−657758.			−0.293253			−0.146626	−	0.989192i	$-0.546842\pi$
$347$	−657758.			−0.293253			−0.146626	+	0.989192i	$0.546842\pi$
$348$		0			0
$349$	−2.16941e6			−0.953408			−0.476704	−	0.879064i	$-0.658169\pi$
$349$	−2.16941e6			−0.953408			−0.476704	+	0.879064i	$0.658169\pi$
$350$		0			0
$351$		−	224638.i		−	0.0973228i
$352$		0			0
$353$	420369.			0.179554			0.0897768	−	0.995962i	$-0.471385\pi$
$353$	420369.			0.179554			0.0897768	+	0.995962i	$0.471385\pi$
$354$		0			0
$355$	2.10852e6			0.887989
$356$		0			0
$357$	−1.67827e6			−0.696932
$358$		0			0
$359$			4.02963e6i			1.65017i	0.565006	+	0.825087i	$0.308874\pi$
$359$			4.02963e6i			1.65017i	−0.565006	+	0.825087i	$0.691126\pi$
$360$		0			0
$361$	1.89557e6			0.765546
$362$		0			0
$363$			4.12218e6i			1.64195i
$364$		0			0
$365$			2.29353e6i			0.901099i
$366$		0			0
$367$		−	4.42973e6i		−	1.71677i	−0.513006	−	0.858385i	$-0.671468\pi$
$367$		−	4.42973e6i		−	1.71677i	0.513006	−	0.858385i	$-0.328532\pi$
$368$		0			0
$369$		−	387257.i		−	0.148058i
$370$		0			0
$371$	−3.12292e6			−1.17795
$372$		0			0
$373$	−185657.			−0.0690939			−0.0345470	−	0.999403i	$-0.510999\pi$
$373$	−185657.			−0.0690939			−0.0345470	+	0.999403i	$0.510999\pi$
$374$		0			0
$375$		−	3.61677e6i		−	1.32813i
$376$		0			0
$377$	1.24185e6	+	1.58998e6i	0.450005	+	0.576153i
$378$		0			0
$379$			2.45032e6i			0.876245i	0.898915	+	0.438123i	$0.144356\pi$
$379$			2.45032e6i			0.876245i	−0.898915	+	0.438123i	$0.855644\pi$
$380$		0			0
$381$	−4.51674e6			−1.59409
$382$		0			0
$383$	3.91750e6			1.36462			0.682310	−	0.731063i	$-0.260975\pi$
$383$	3.91750e6			1.36462			0.682310	+	0.731063i	$0.260975\pi$
$384$		0			0
$385$			1.13260e7i			3.89425i
$386$		0			0
$387$			1.16276e6i			0.394651i
$388$		0			0
$389$		−	571898.i		−	0.191622i	−0.995400	−	0.0958109i	$-0.969456\pi$
$389$		−	571898.i		−	0.191622i	0.995400	−	0.0958109i	$-0.0305444\pi$
$390$		0			0
$391$			268735.i			0.0888962i
$392$		0			0
$393$	−1.98210e6			−0.647359
$394$		0			0
$395$			4.09028e6i			1.31905i
$396$		0			0
$397$	1.49161e6			0.474984			0.237492	−	0.971389i	$-0.423675\pi$
$397$	1.49161e6			0.474984			0.237492	+	0.971389i	$0.423675\pi$
$398$		0			0
$399$	−3.46872e6			−1.09078
$400$		0			0
$401$	1.92589e6			0.598096			0.299048	−	0.954238i	$-0.403331\pi$
$401$	1.92589e6			0.598096			0.299048	+	0.954238i	$0.403331\pi$
$402$		0			0
$403$			4.10830e6i			1.26009i
$404$		0			0
$405$	−5.78364e6			−1.75212
$406$		0			0
$407$	2.01492e6			0.602938
$408$		0			0
$409$			2.15032e6i			0.635615i	0.948155	+	0.317808i	$0.102947\pi$
$409$			2.15032e6i			0.635615i	−0.948155	+	0.317808i	$0.897053\pi$
$410$		0			0
$411$	−2.20460e6			−0.643761
$412$		0			0
$413$	7.32373e6			2.11279
$414$		0			0
$415$	4.81923e6			1.37359
$416$		0			0
$417$		−	4.28725e6i		−	1.20736i
$418$		0			0
$419$	4.31814e6			1.20161			0.600803	−	0.799397i	$-0.294848\pi$
$419$	4.31814e6			1.20161			0.600803	+	0.799397i	$0.294848\pi$
$420$		0			0
$421$			3.57350e6i			0.982627i	0.870983	+	0.491314i	$0.163483\pi$
$421$			3.57350e6i			0.982627i	−0.870983	+	0.491314i	$0.836517\pi$
$422$		0			0
$423$		−	2.73557e6i		−	0.743355i
$424$		0			0
$425$		−	1.84012e6i		−	0.494166i
$426$		0			0
$427$		−	2.00152e6i		−	0.531239i
$428$		0			0
$429$	5.68995e6			1.49268
$430$		0			0
$431$	−3.79128e6			−0.983088			−0.491544	−	0.870853i	$-0.663567\pi$
$431$	−3.79128e6			−0.983088			−0.491544	+	0.870853i	$0.663567\pi$
$432$		0			0
$433$		−	2.38236e6i		−	0.610644i	−0.952249	−	0.305322i	$-0.901236\pi$
$433$		−	2.38236e6i		−	0.610644i	0.952249	−	0.305322i	$-0.0987641\pi$
$434$		0			0
$435$	6.91583e6	−	5.40162e6i	1.75235	−	1.36868i
$436$		0			0
$437$			555434.i			0.139133i
$438$		0			0
$439$	−7.23282e6			−1.79121			−0.895605	−	0.444851i	$-0.853257\pi$
$439$	−7.23282e6			−1.79121			−0.895605	+	0.444851i	$0.853257\pi$
$440$		0			0
$441$	−6.14758e6			−1.50525
$442$		0			0
$443$		−	2.05198e6i		−	0.496780i	−0.968660	−	0.248390i	$-0.920099\pi$
$443$		−	2.05198e6i		−	0.496780i	0.968660	−	0.248390i	$-0.0799015\pi$
$444$		0			0
$445$		−	1.01352e7i		−	2.42624i
$446$		0			0
$447$			209159.i			0.0495116i
$448$		0			0
$449$			3.05219e6i			0.714489i	0.934011	+	0.357245i	$0.116284\pi$
$449$			3.05219e6i			0.714489i	−0.934011	+	0.357245i	$0.883716\pi$
$450$		0			0
$451$	−1.04755e6			−0.242511
$452$		0			0
$453$			1.03930e7i			2.37956i
$454$		0			0
$455$	8.49526e6			1.92375
$456$		0			0
$457$	2.06974e6			0.463581			0.231790	−	0.972766i	$-0.425542\pi$
$457$	2.06974e6			0.463581			0.231790	+	0.972766i	$0.425542\pi$
$458$		0			0
$459$	−185897.			−0.0411853
$460$		0			0
$461$		−	3.36651e6i		−	0.737781i	−0.929473	−	0.368890i	$-0.879738\pi$
$461$		−	3.36651e6i		−	0.737781i	0.929473	−	0.368890i	$-0.120262\pi$
$462$		0			0
$463$	81095.6			0.0175811			0.00879053	−	0.999961i	$-0.497202\pi$
$463$	81095.6			0.0175811			0.00879053	+	0.999961i	$0.497202\pi$
$464$		0			0
$465$	1.78697e7			3.83251
$466$		0			0
$467$		−	1.95508e6i		−	0.414833i	−0.978253	−	0.207416i	$-0.933495\pi$
$467$		−	1.95508e6i		−	0.414833i	0.978253	−	0.207416i	$-0.0665055\pi$
$468$		0			0
$469$	−8.69847e6			−1.82604
$470$		0			0
$471$	−1.82732e6			−0.379543
$472$		0			0
$473$	3.14532e6			0.646416
$474$		0			0
$475$		−	3.80323e6i		−	0.773427i
$476$		0			0
$477$	3.23910e6			0.651821
$478$		0			0
$479$			5.02666e6i			1.00102i	0.865732	+	0.500508i	$0.166853\pi$
$479$			5.02666e6i			1.00102i	−0.865732	+	0.500508i	$0.833147\pi$
$480$		0			0
$481$		−	1.51133e6i		−	0.297849i
$482$		0			0
$483$			3.31876e6i			0.647303i
$484$		0			0
$485$		−	65309.3i		−	0.0126073i
$486$		0			0
$487$	7.01534e6			1.34038			0.670188	−	0.742192i	$-0.266214\pi$
$487$	7.01534e6			1.34038			0.670188	+	0.742192i	$0.266214\pi$
$488$		0			0
$489$	−1.14382e7			−2.16315
$490$		0			0
$491$		−	6.04666e6i		−	1.13191i	−0.824436	−	0.565954i	$-0.808508\pi$
$491$		−	6.04666e6i		−	1.13191i	0.824436	−	0.565954i	$-0.191492\pi$
$492$		0			0
$493$	1.31578e6	−	1.02769e6i	0.243817	−	0.190434i
$494$		0			0
$495$		−	1.17474e7i		−	2.15490i
$496$		0			0
$497$	4.95412e6			0.899654
$498$		0			0
$499$	5.31655e6			0.955825			0.477912	−	0.878408i	$-0.341394\pi$
$499$	5.31655e6			0.955825			0.477912	+	0.878408i	$0.341394\pi$
$500$		0			0
$501$		−	1.08625e7i		−	1.93345i
$502$		0			0
$503$			667123.i			0.117567i	0.998271	+	0.0587835i	$0.0187222\pi$
$503$			667123.i			0.117567i	−0.998271	+	0.0587835i	$0.981278\pi$
$504$		0			0
$505$			5.01387e6i			0.874873i
$506$		0			0
$507$			3.71756e6i			0.642301i
$508$		0			0
$509$	852016.			0.145765			0.0728825	−	0.997341i	$-0.476780\pi$
$509$	852016.			0.145765			0.0728825	+	0.997341i	$0.476780\pi$
$510$		0			0
$511$			5.38881e6i			0.912936i
$512$		0			0
$513$	−384221.			−0.0644596
$514$		0			0
$515$	−7.83684e6			−1.30204
$516$		0			0
$517$	−7.39982e6			−1.21757
$518$		0			0
$519$			4.04015e6i			0.658384i
$520$		0			0
$521$	8.30897e6			1.34107			0.670537	−	0.741876i	$-0.266064\pi$
$521$	8.30897e6			1.34107			0.670537	+	0.741876i	$0.266064\pi$
$522$		0			0
$523$	5.37728e6			0.859625			0.429812	−	0.902918i	$-0.358580\pi$
$523$	5.37728e6			0.859625			0.429812	+	0.902918i	$0.358580\pi$
$524$		0			0
$525$		−	2.27246e7i		−	3.59830i
$526$		0			0
$527$	3.39980e6			0.533245
$528$		0			0
$529$	−5.90492e6			−0.917434
$530$		0			0
$531$	−7.59620e6			−1.16912
$532$		0			0
$533$			785731.i			0.119800i
$534$		0			0
$535$	6.06500e6			0.916107
$536$		0			0
$537$			1.79449e7i			2.68538i
$538$		0			0
$539$			1.66295e7i			2.46551i
$540$		0			0
$541$			7.17375e6i			1.05379i	0.849931	+	0.526894i	$0.176643\pi$
$541$			7.17375e6i			1.05379i	−0.849931	+	0.526894i	$0.823357\pi$
$542$		0			0
$543$		−	1.80796e6i		−	0.263142i
$544$		0			0
$545$	−9.69550e6			−1.39823
$546$		0			0
$547$	−8.04876e6			−1.15017			−0.575083	−	0.818095i	$-0.695030\pi$
$547$	−8.04876e6			−1.15017			−0.575083	+	0.818095i	$0.695030\pi$
$548$		0			0
$549$			2.07598e6i			0.293963i
$550$		0			0
$551$	2.71950e6	−	2.12407e6i	0.381602	−	0.298051i
$552$		0			0
$553$			9.61038e6i			1.33637i
$554$		0			0
$555$	−6.57374e6			−0.905900
$556$		0			0
$557$	1.01144e6			0.138134			0.0690669	−	0.997612i	$-0.477998\pi$
$557$	1.01144e6			0.138134			0.0690669	+	0.997612i	$0.477998\pi$
$558$		0			0
$559$		−	2.35920e6i		−	0.319327i
$560$		0			0
$561$		−	4.70868e6i		−	0.631673i
$562$		0			0
$563$		−	1.26692e6i		−	0.168453i	−0.996447	−	0.0842266i	$-0.973158\pi$
$563$		−	1.26692e6i		−	0.168453i	0.996447	−	0.0842266i	$-0.0268419\pi$
$564$		0			0
$565$		−	1.26697e7i		−	1.66972i
$566$		0			0
$567$	−1.35890e7			−1.77513
$568$		0			0
$569$		−	8.25094e6i		−	1.06837i	−0.845367	−	0.534187i	$-0.820618\pi$
$569$		−	8.25094e6i		−	1.06837i	0.845367	−	0.534187i	$-0.179382\pi$
$570$		0			0
$571$	−1.12215e7			−1.44032			−0.720160	−	0.693808i	$-0.755932\pi$
$571$	−1.12215e7			−1.44032			−0.720160	+	0.693808i	$0.755932\pi$
$572$		0			0
$573$	−9.77025e6			−1.24314
$574$		0			0
$575$	−3.63881e6			−0.458976
$576$		0			0
$577$			7.06718e6i			0.883703i	0.897088	+	0.441852i	$0.145678\pi$
$577$			7.06718e6i			0.883703i	−0.897088	+	0.441852i	$0.854322\pi$
$578$		0			0
$579$	−9.91611e6			−1.22926
$580$		0			0
$581$	1.13231e7			1.39164
$582$		0			0
$583$		−	8.76191e6i		−	1.06765i
$584$		0			0
$585$	−8.81132e6			−1.06451
$586$		0			0
$587$	9.89439e6			1.18521			0.592603	−	0.805495i	$-0.298100\pi$
$587$	9.89439e6			1.18521			0.592603	+	0.805495i	$0.298100\pi$
$588$		0			0
$589$	7.02686e6			0.834589
$590$		0			0
$591$			8.63584e6i			1.01704i
$592$		0			0
$593$	3.83383e6			0.447709			0.223854	−	0.974623i	$-0.428136\pi$
$593$	3.83383e6			0.447709			0.223854	+	0.974623i	$0.428136\pi$
$594$		0			0
$595$		−	7.03020e6i		−	0.814095i
$596$		0			0
$597$			2.32091e6i			0.266516i
$598$		0			0
$599$		−	1.47715e7i		−	1.68212i	−0.540938	−	0.841062i	$-0.681931\pi$
$599$		−	1.47715e7i		−	1.68212i	0.540938	−	0.841062i	$-0.318069\pi$
$600$		0			0
$601$		−	1.70212e7i		−	1.92223i	−0.276155	−	0.961113i	$-0.589060\pi$
$601$		−	1.70212e7i		−	1.92223i	0.276155	−	0.961113i	$-0.410940\pi$
$602$		0			0
$603$	9.02209e6			1.01045
$604$		0			0
$605$	−1.72676e7			−1.91798
$606$		0			0
$607$			1.99520e6i			0.219794i	0.993943	+	0.109897i	$0.0350520\pi$
$607$			1.99520e6i			0.219794i	−0.993943	+	0.109897i	$0.964948\pi$
$608$		0			0
$609$	1.62492e7	−	1.26915e7i	1.77537	−	1.38666i
$610$		0			0
$611$			5.55037e6i			0.601477i
$612$		0			0
$613$	8.31000e6			0.893202			0.446601	−	0.894733i	$-0.352634\pi$
$613$	8.31000e6			0.893202			0.446601	+	0.894733i	$0.352634\pi$
$614$		0			0
$615$	3.41765e6			0.364368
$616$		0			0
$617$		−	3.70169e6i		−	0.391460i	−0.980658	−	0.195730i	$-0.937292\pi$
$617$		−	3.70169e6i		−	0.391460i	0.980658	−	0.195730i	$-0.0627076\pi$
$618$		0			0
$619$			1.29198e7i			1.35528i	0.735395	+	0.677639i	$0.236997\pi$
$619$			1.29198e7i			1.35528i	−0.735395	+	0.677639i	$0.763003\pi$
$620$		0			0
$621$			367611.i			0.0382524i
$622$		0			0
$623$		−	2.38134e7i		−	2.45811i
$624$		0			0
$625$	−448271.			−0.0459029
$626$		0			0
$627$		−	9.73211e6i		−	0.988640i
$628$		0			0
$629$	−1.25069e6			−0.126044
$630$		0			0
$631$	−1.52718e6			−0.152692			−0.0763459	−	0.997081i	$-0.524325\pi$
$631$	−1.52718e6			−0.152692			−0.0763459	+	0.997081i	$0.524325\pi$
$632$		0			0
$633$	6.36733e6			0.631609
$634$		0			0
$635$		−	1.89204e7i		−	1.86207i
$636$		0			0
$637$	1.24732e7			1.21795
$638$		0			0
$639$	−5.13843e6			−0.497827
$640$		0			0
$641$		−	9.43372e6i		−	0.906855i	−0.891293	−	0.453427i	$-0.850201\pi$
$641$		−	9.43372e6i		−	0.906855i	0.891293	−	0.453427i	$-0.149799\pi$
$642$		0			0
$643$	−1.37293e7			−1.30954			−0.654772	−	0.755826i	$-0.727235\pi$
$643$	−1.37293e7			−1.30954			−0.654772	+	0.755826i	$0.727235\pi$
$644$		0			0
$645$	−1.02617e7			−0.971225
$646$		0			0
$647$	1.73477e7			1.62922			0.814611	−	0.580007i	$-0.196950\pi$
$647$	1.73477e7			1.62922			0.814611	+	0.580007i	$0.196950\pi$
$648$		0			0
$649$			2.05480e7i			1.91496i
$650$		0			0
$651$	4.19860e7			3.88286
$652$		0			0
$653$		−	6.34963e6i		−	0.582728i	−0.956612	−	0.291364i	$-0.905891\pi$
$653$		−	6.34963e6i		−	0.582728i	0.956612	−	0.291364i	$-0.0941090\pi$
$654$		0			0
$655$		−	8.30296e6i		−	0.756188i
$656$		0			0
$657$		−	5.58929e6i		−	0.505177i
$658$		0			0
$659$			4.42658e6i			0.397058i	0.980095	+	0.198529i	$0.0636165\pi$
$659$			4.42658e6i			0.397058i	−0.980095	+	0.198529i	$0.936384\pi$
$660$		0			0
$661$	1.51522e7			1.34887			0.674437	−	0.738332i	$-0.264386\pi$
$661$	1.51522e7			1.34887			0.674437	+	0.738332i	$0.264386\pi$
$662$		0			0
$663$	−3.53183e6			−0.312044
$664$		0			0
$665$		−	1.45303e7i		−	1.27415i
$666$		0			0
$667$	−2.03224e6	−	2.60193e6i	−0.176873	−	0.226455i
$668$		0			0
$669$			1.84397e7i			1.59290i
$670$		0			0
$671$	5.61563e6			0.481495
$672$		0			0
$673$	−1.21002e7			−1.02980			−0.514901	−	0.857249i	$-0.672171\pi$
$673$	−1.21002e7			−1.02980			−0.514901	+	0.857249i	$0.672171\pi$
$674$		0			0
$675$		−	2.51715e6i		−	0.212642i
$676$		0			0
$677$			1.94913e7i			1.63444i	0.576326	+	0.817220i	$0.304486\pi$
$677$			1.94913e7i			1.63444i	−0.576326	+	0.817220i	$0.695514\pi$
$678$		0			0
$679$		−	153449.i		−	0.0127729i
$680$		0			0
$681$		−	5.71196e6i		−	0.471973i
$682$		0			0
$683$	1.58984e6			0.130407			0.0652034	−	0.997872i	$-0.479230\pi$
$683$	1.58984e6			0.130407			0.0652034	+	0.997872i	$0.479230\pi$
$684$		0			0
$685$		−	9.23497e6i		−	0.751985i
$686$		0			0
$687$	2.44324e7			1.97503
$688$		0			0
$689$	−6.57203e6			−0.527414
$690$		0			0
$691$	478964.			0.0381599			0.0190800	−	0.999818i	$-0.493926\pi$
$691$	478964.			0.0381599			0.0190800	+	0.999818i	$0.493926\pi$
$692$		0			0
$693$		−	2.76012e7i		−	2.18321i
$694$		0			0
$695$	1.79591e7			1.41034
$696$		0			0
$697$	650227.			0.0506971
$698$		0			0
$699$			7.33629e6i			0.567915i
$700$		0			0
$701$	4.60638e6			0.354050			0.177025	−	0.984206i	$-0.443353\pi$
$701$	4.60638e6			0.354050			0.177025	+	0.984206i	$0.443353\pi$
$702$		0			0
$703$	−2.58498e6			−0.197274
$704$		0			0
$705$	2.41421e7			1.82938
$706$		0			0
$707$			1.17804e7i			0.886365i
$708$		0			0
$709$	9.47025e6			0.707531			0.353766	−	0.935334i	$-0.384901\pi$
$709$	9.47025e6			0.707531			0.353766	+	0.935334i	$0.384901\pi$
$710$		0			0
$711$		−	9.96792e6i		−	0.739487i
$712$		0			0
$713$		−	6.72307e6i		−	0.495272i
$714$		0			0
$715$			2.38350e7i			1.74361i
$716$		0			0
$717$			3.15110e7i			2.28910i
$718$		0			0
$719$	1.63490e7			1.17942			0.589711	−	0.807614i	$-0.299241\pi$
$719$	1.63490e7			1.17942			0.589711	+	0.807614i	$0.299241\pi$
$720$		0			0
$721$	−1.84132e7			−1.31914
$722$		0			0
$723$		−	677341.i		−	0.0481905i
$724$		0			0
$725$	1.39154e7	+	1.78163e7i	0.983221	+	1.25884i
$726$		0			0
$727$			1.79177e7i			1.25732i	0.777679	+	0.628661i	$0.216397\pi$
$727$			1.79177e7i			1.25732i	−0.777679	+	0.628661i	$0.783603\pi$
$728$		0			0
$729$	1.14592e7			0.798609
$730$		0			0
$731$	−1.95234e6			−0.135133
$732$		0			0
$733$		−	1.78468e7i		−	1.22687i	−0.789744	−	0.613436i	$-0.789787\pi$
$733$		−	1.78468e7i		−	1.22687i	0.789744	−	0.613436i	$-0.210213\pi$
$734$		0			0
$735$		−	5.42541e7i		−	3.70437i
$736$		0			0
$737$		−	2.44051e7i		−	1.65506i
$738$		0			0
$739$			2.55426e7i			1.72050i	0.509876	+	0.860248i	$0.329691\pi$
$739$			2.55426e7i			1.72050i	−0.509876	+	0.860248i	$0.670309\pi$
$740$		0			0
$741$	−7.29975e6			−0.488385
$742$		0			0
$743$		−	1.50256e7i		−	0.998528i	−0.866450	−	0.499264i	$-0.833604\pi$
$743$		−	1.50256e7i		−	0.998528i	0.866450	−	0.499264i	$-0.166396\pi$
$744$		0			0
$745$	−876157.			−0.0578351
$746$		0			0
$747$	−1.17444e7			−0.770067
$748$		0			0
$749$	1.42501e7			0.928141
$750$		0			0
$751$			2.09579e6i			0.135597i	0.997699	+	0.0677983i	$0.0215974\pi$
$751$			2.09579e6i			0.135597i	−0.997699	+	0.0677983i	$0.978403\pi$
$752$		0			0
$753$	−2.09688e7			−1.34768
$754$		0			0
$755$	−4.35361e7			−2.77960
$756$		0			0
$757$			2.93991e7i			1.86464i	0.361639	+	0.932318i	$0.382217\pi$
$757$			2.93991e7i			1.86464i	−0.361639	+	0.932318i	$0.617783\pi$
$758$		0			0
$759$	−9.31138e6			−0.586691
$760$		0			0
$761$	−9.58603e6			−0.600036			−0.300018	−	0.953934i	$-0.596993\pi$
$761$	−9.58603e6			−0.600036			−0.300018	+	0.953934i	$0.596993\pi$
$762$		0			0
$763$	−2.27802e7			−1.41660
$764$		0			0
$765$			7.29175e6i			0.450483i
$766$		0			0
$767$	1.54124e7			0.945982
$768$		0			0
$769$			1.57832e7i			0.962453i	0.876596	+	0.481226i	$0.159808\pi$
$769$			1.57832e7i			0.962453i	−0.876596	+	0.481226i	$0.840192\pi$
$770$		0			0
$771$			2.73543e6i			0.165726i
$772$		0			0
$773$			1.49255e7i			0.898423i	0.893425	+	0.449212i	$0.148295\pi$
$773$			1.49255e7i			0.898423i	−0.893425	+	0.449212i	$0.851705\pi$
$774$		0			0
$775$			4.60350e7i			2.75318i
$776$		0			0
$777$	−1.54455e7			−0.917800
$778$		0			0
$779$	1.34392e6			0.0793467
$780$		0			0
$781$			1.38997e7i			0.815412i
$782$		0			0
$783$	1.79988e6	−	1.40580e6i	0.104916	−	0.0819445i
$784$		0			0
$785$		−	7.65456e6i		−	0.443349i
$786$		0			0
$787$	−3.13562e7			−1.80462			−0.902312	−	0.431083i	$-0.858132\pi$
$787$	−3.13562e7			−1.80462			−0.902312	+	0.431083i	$0.858132\pi$
$788$		0			0
$789$	−1.38571e7			−0.792466
$790$		0			0
$791$		−	2.97682e7i		−	1.69166i
$792$		0			0
$793$		−	4.21210e6i		−	0.237857i
$794$		0			0
$795$			2.85860e7i			1.60411i
$796$		0			0
$797$		−	2.64771e6i		−	0.147647i	−0.997271	−	0.0738235i	$-0.976480\pi$
$797$		−	2.64771e6i		−	0.147647i	0.997271	−	0.0738235i	$-0.0235202\pi$
$798$		0			0
$799$	4.59318e6			0.254534
$800$		0			0
$801$			2.46993e7i			1.36020i
$802$		0			0
$803$	−1.51193e7			−0.827451
$804$		0			0
$805$	−1.39021e7			−0.756123
$806$		0			0
$807$	4.74854e7			2.56671
$808$		0			0
$809$		−	9.80290e6i		−	0.526603i	−0.964714	−	0.263301i	$-0.915189\pi$
$809$		−	9.80290e6i		−	0.526603i	0.964714	−	0.263301i	$-0.0848114\pi$
$810$		0			0
$811$	6.02035e6			0.321417			0.160709	−	0.987002i	$-0.448622\pi$
$811$	6.02035e6			0.321417			0.160709	+	0.987002i	$0.448622\pi$
$812$		0			0
$813$	−2.48935e7			−1.32087
$814$		0			0
$815$		−	4.79143e7i		−	2.52680i
$816$		0			0
$817$	−4.03519e6			−0.211499
$818$		0			0
$819$	−2.07028e7			−1.07850
$820$		0			0
$821$	3.70550e7			1.91862			0.959309	−	0.282359i	$-0.0911171\pi$
$821$	3.70550e7			1.91862			0.959309	+	0.282359i	$0.0911171\pi$
$822$		0			0
$823$			2.15600e7i			1.10956i	0.831998	+	0.554779i	$0.187197\pi$
$823$			2.15600e7i			1.10956i	−0.831998	+	0.554779i	$0.812803\pi$
$824$		0			0
$825$	6.37580e7			3.26137
$826$		0			0
$827$			1.21823e7i			0.619391i	0.950836	+	0.309696i	$0.100227\pi$
$827$			1.21823e7i			0.619391i	−0.950836	+	0.309696i	$0.899773\pi$
$828$		0			0
$829$		−	1.49741e6i		−	0.0756756i	−0.999284	−	0.0378378i	$-0.987953\pi$
$829$		−	1.49741e6i		−	0.0756756i	0.999284	−	0.0378378i	$-0.0120470\pi$
$830$		0			0
$831$			1.50445e7i			0.755743i
$832$		0			0
$833$		−	1.03221e7i		−	0.515416i
$834$		0			0
$835$	4.55024e7			2.25849
$836$		0			0
$837$	4.65068e6			0.229458
$838$		0			0
$839$			2.46412e7i			1.20853i	0.796785	+	0.604263i	$0.206532\pi$
$839$			2.46412e7i			1.20853i	−0.796785	+	0.604263i	$0.793468\pi$
$840$		0			0
$841$	−4.96789e6	+	1.99004e7i	−0.242204	+	0.970225i
$842$		0			0
$843$			5.33979e7i			2.58795i
$844$		0			0
$845$	−1.55727e7			−0.750279
$846$		0			0
$847$	−4.05715e7			−1.94318
$848$		0			0
$849$			3.65369e6i			0.173965i
$850$		0			0
$851$			2.47323e6i			0.117069i
$852$		0			0
$853$			2.54111e7i			1.19578i	0.801578	+	0.597890i	$0.203994\pi$
$853$			2.54111e7i			1.19578i	−0.801578	+	0.597890i	$0.796006\pi$
$854$		0			0
$855$			1.50709e7i			0.705057i
$856$		0			0
$857$	−1.32471e7			−0.616127			−0.308063	−	0.951366i	$-0.599681\pi$
$857$	−1.32471e7			−0.616127			−0.308063	+	0.951366i	$0.599681\pi$
$858$		0			0
$859$			8.05724e6i			0.372566i	0.982496	+	0.186283i	$0.0596441\pi$
$859$			8.05724e6i			0.372566i	−0.982496	+	0.186283i	$0.940356\pi$
$860$		0			0
$861$	8.03000e6			0.369154
$862$		0			0
$863$	−2.46723e7			−1.12767			−0.563836	−	0.825887i	$-0.690675\pi$
$863$	−2.46723e7			−1.12767			−0.563836	+	0.825887i	$0.690675\pi$
$864$		0			0
$865$	−1.69240e7			−0.769066
$866$		0			0
$867$		−	2.76142e7i		−	1.24763i
$868$		0			0
$869$	−2.69637e7			−1.21124
$870$		0			0
$871$	−1.83055e7			−0.817592
$872$		0			0
$873$			159158.i			0.00706792i
$874$		0			0
$875$	3.55971e7			1.57179
$876$		0			0
$877$	−4.30471e6			−0.188993			−0.0944963	−	0.995525i	$-0.530124\pi$
$877$	−4.30471e6			−0.188993			−0.0944963	+	0.995525i	$0.530124\pi$
$878$		0			0
$879$	2.34817e7			1.02508
$880$		0			0
$881$			7.17894e6i			0.311617i	0.987787	+	0.155808i	$0.0497982\pi$
$881$			7.17894e6i			0.311617i	−0.987787	+	0.155808i	$0.950202\pi$
$882$		0			0
$883$	2.27258e6			0.0980884			0.0490442	−	0.998797i	$-0.484382\pi$
$883$	2.27258e6			0.0980884			0.0490442	+	0.998797i	$0.484382\pi$
$884$		0			0
$885$		−	6.70386e7i		−	2.87718i
$886$		0			0
$887$			3.19149e7i			1.36202i	0.732273	+	0.681012i	$0.238460\pi$
$887$			3.19149e7i			1.36202i	−0.732273	+	0.681012i	$0.761540\pi$
$888$		0			0
$889$		−	4.44548e7i		−	1.88653i
$890$		0			0
$891$		−	3.81265e7i		−	1.60891i
$892$		0			0
$893$	9.49338e6			0.398375
$894$		0			0
$895$	−7.51706e7			−3.13682
$896$		0			0
$897$			6.98417e6i			0.289824i
$898$		0			0
$899$	−3.29173e7	+	2.57101e7i	−1.35839	+	1.06097i
$900$		0			0
$901$			5.43864e6i			0.223192i
$902$		0			0
$903$	−2.41105e7			−0.983983
$904$		0			0
$905$	7.57348e6			0.307379
$906$		0			0
$907$		−	2.84496e7i		−	1.14831i	−0.818748	−	0.574153i	$-0.805331\pi$
$907$		−	2.84496e7i		−	1.14831i	0.818748	−	0.574153i	$-0.194669\pi$
$908$		0			0
$909$		−	1.22187e7i		−	0.490474i
$910$		0			0
$911$		−	1.64758e7i		−	0.657734i	−0.944376	−	0.328867i	$-0.893333\pi$
$911$		−	1.64758e7i		−	0.657734i	0.944376	−	0.328867i	$-0.106667\pi$
$912$		0			0
$913$			3.17690e7i			1.26133i
$914$		0			0
$915$	−1.83211e7			−0.723436
$916$		0			0
$917$		−	1.95084e7i		−	0.766121i
$918$		0			0
$919$	2.11310e7			0.825336			0.412668	−	0.910881i	$-0.364597\pi$
$919$	2.11310e7			0.825336			0.412668	+	0.910881i	$0.364597\pi$
$920$		0			0
$921$	−4.67887e7			−1.81757
$922$		0			0
$923$	1.04257e7			0.402811
$924$		0			0
$925$		−	1.69350e7i		−	0.650775i
$926$		0			0
$927$	1.90982e7			0.729951
$928$		0			0
$929$	−1.53237e7			−0.582538			−0.291269	−	0.956641i	$-0.594077\pi$
$929$	−1.53237e7			−0.582538			−0.291269	+	0.956641i	$0.594077\pi$
$930$		0			0
$931$		−	2.13343e7i		−	0.806684i
$932$		0			0
$933$	−1.91687e7			−0.720922
$934$		0			0
$935$	1.97245e7			0.737865
$936$		0			0
$937$	−2.82423e7			−1.05087			−0.525437	−	0.850832i	$-0.676098\pi$
$937$	−2.82423e7			−1.05087			−0.525437	+	0.850832i	$0.676098\pi$
$938$		0			0
$939$		−	2.87873e7i		−	1.06546i
$940$		0			0
$941$	−1.91232e7			−0.704022			−0.352011	−	0.935996i	$-0.614502\pi$
$941$	−1.91232e7			−0.704022			−0.352011	+	0.935996i	$0.614502\pi$
$942$		0			0
$943$		−	1.28582e6i		−	0.0470869i
$944$		0			0
$945$		−	9.61680e6i		−	0.350309i
$946$		0			0
$947$		−	916050.i		−	0.0331928i	−0.999862	−	0.0165964i	$-0.994717\pi$
$947$		−	916050.i		−	0.0331928i	0.999862	−	0.0165964i	$-0.00528305\pi$
$948$		0			0
$949$			1.13405e7i			0.408758i
$950$		0			0
$951$	−2.95222e7			−1.05852
$952$		0			0
$953$	7.31338e6			0.260847			0.130423	−	0.991458i	$-0.458366\pi$
$953$	7.31338e6			0.260847			0.130423	+	0.991458i	$0.458366\pi$
$954$		0			0
$955$		−	4.09272e7i		−	1.45212i
$956$		0			0
$957$	3.56082e7	+	4.55901e7i	1.25681	+	1.60913i
$958$		0			0
$959$		−	2.16982e7i		−	0.761863i
$960$		0			0
$961$	−5.64252e7			−1.97090
$962$		0			0
$963$	−1.47803e7			−0.513591
$964$		0			0
$965$		−	4.15382e7i		−	1.43592i
$966$		0			0
$967$			5.71496e6i			0.196538i	0.995160	+	0.0982692i	$0.0313306\pi$
$967$			5.71496e6i			0.196538i	−0.995160	+	0.0982692i	$0.968669\pi$
$968$		0			0
$969$			6.04086e6i			0.206676i
$970$		0			0
$971$			3.76567e7i			1.28172i	0.767657	+	0.640861i	$0.221422\pi$
$971$			3.76567e7i			1.28172i	−0.767657	+	0.640861i	$0.778578\pi$
$972$		0			0
$973$	4.21962e7			1.42886
$974$		0			0
$975$		−	4.78228e7i		−	1.61110i
$976$		0			0
$977$	−1.50840e7			−0.505567			−0.252784	−	0.967523i	$-0.581346\pi$
$977$	−1.50840e7			−0.505567			−0.252784	+	0.967523i	$0.581346\pi$
$978$		0			0
$979$	6.68127e7			2.22793
$980$		0			0
$981$	2.36278e7			0.783881
$982$		0			0
$983$		−	4.82887e7i		−	1.59390i	−0.604043	−	0.796951i	$-0.706445\pi$
$983$		−	4.82887e7i		−	1.59390i	0.604043	−	0.796951i	$-0.293555\pi$
$984$		0			0
$985$	−3.61752e7			−1.18801
$986$		0			0
$987$	5.67236e7			1.85341
$988$		0			0
$989$			3.86074e6i			0.125511i
$990$		0			0
$991$	−4.54332e7			−1.46957			−0.734783	−	0.678303i	$-0.762716\pi$
$991$	−4.54332e7			−1.46957			−0.734783	+	0.678303i	$0.762716\pi$
$992$		0			0
$993$	7.29472e7			2.34766
$994$		0			0
$995$	−9.72222e6			−0.311321
$996$		0			0
$997$			6.29767e6i			0.200651i	0.994955	+	0.100326i	$0.0319884\pi$
$997$			6.29767e6i			0.200651i	−0.994955	+	0.100326i	$0.968012\pi$
$998$		0			0
$999$	−1.71085e6			−0.0542375

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.6.e.c.289.3		12
4.3	odd	2		29.6.b.a.28.9	yes	12
12.11	even	2		261.6.c.b.28.4		12
29.28	even	2	inner	464.6.e.c.289.10		12
116.75	even	4		841.6.a.d.1.9		12
116.99	even	4		841.6.a.d.1.4		12
116.115	odd	2		29.6.b.a.28.4	✓	12
348.347	even	2		261.6.c.b.28.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.b.a.28.4	✓	12	116.115	odd	2
29.6.b.a.28.9	yes	12	4.3	odd	2
261.6.c.b.28.4		12	12.11	even	2
261.6.c.b.28.9		12	348.347	even	2
464.6.e.c.289.3		12	1.1	even	1	trivial
464.6.e.c.289.10		12	29.28	even	2	inner
841.6.a.d.1.4		12	116.99	even	4
841.6.a.d.1.9		12	116.75	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	464.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-21.5070i q^{3} +90.0922 q^{5} +211.678 q^{7} -219.553 q^{9} +593.900i q^{11} +445.466 q^{13} -1937.62i q^{15} -368.642i q^{17} -761.926i q^{19} -4552.56i q^{21} -728.987 q^{23} +4991.60 q^{25} -504.276i q^{27} +(2787.76 + 3569.25i) q^{29} +9222.49i q^{31} +12773.0 q^{33} +19070.5 q^{35} -3392.70i q^{37} -9580.65i q^{39} +1763.84i q^{41} -5296.04i q^{43} -19780.0 q^{45} +12459.7i q^{47} +28000.4 q^{49} -7928.41 q^{51} -14753.2 q^{53} +53505.8i q^{55} -16386.8 q^{57} +34598.5 q^{59} -9455.51i q^{61} -46474.5 q^{63} +40133.0 q^{65} -41093.0 q^{67} +15678.4i q^{69} +23404.1 q^{71} +25457.6i q^{73} -107355. i q^{75} +125715. i q^{77} +45401.0i q^{79} -64196.9 q^{81} +53492.2 q^{83} -33211.8i q^{85} +(76763.9 - 59956.6i) q^{87} -112498. i q^{89} +94295.2 q^{91} +198349. q^{93} -68643.6i q^{95} -724.916i q^{97} -130393. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 46 q^{5} - 20 q^{7} - 1574 q^{9} + 1362 q^{13} - 5852 q^{23} + 12678 q^{25} + 11328 q^{29} - 22694 q^{33} - 4532 q^{35} - 52816 q^{45} + 102836 q^{49} - 58540 q^{51} + 25650 q^{53} - 32824 q^{57}+ \cdots + 377966 q^{93}+O(q^{100}) \) 12 * q + 46 * q^5 - 20 * q^7 - 1574 * q^9 + 1362 * q^13 - 5852 * q^23 + 12678 * q^25 + 11328 * q^29 - 22694 * q^33 - 4532 * q^35 - 52816 * q^45 + 102836 * q^49 - 58540 * q^51 + 25650 * q^53 - 32824 * q^57 + 3900 * q^59 + 146616 * q^63 + 169574 * q^65 + 28264 * q^67 + 286832 * q^71 - 24084 * q^81 - 85692 * q^83 + 137716 * q^87 + 182372 * q^91 + 377966 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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