LMFDB - Embedded newform 3264.2.c.m.577.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3264,2,Mod(577,3264)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3264, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3264.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3264 = 2^{6} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3264.c (of order $2$, degree $1$, 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:	$26.0631712197$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1632)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.2
Root			$-2.30278i$ of defining polynomial
Character	$\chi$	$=$	3264.577
Dual form			3264.2.c.m.577.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.60555i q^{5} -4.00000i q^{7} -1.00000 q^{9} +3.00000i q^{11} +1.00000 q^{13} +3.60555 q^{15} +(2.00000 + 3.60555i) q^{17} +3.60555 q^{19} -4.00000 q^{21} -7.00000i q^{23} -8.00000 q^{25} +1.00000i q^{27} -7.21110i q^{29} +2.00000i q^{31} +3.00000 q^{33} +14.4222 q^{35} +7.21110i q^{37} -1.00000i q^{39} +10.8167i q^{41} -3.60555 q^{43} -3.60555i q^{45} -9.00000 q^{49} +(3.60555 - 2.00000i) q^{51} +4.00000 q^{53} -10.8167 q^{55} -3.60555i q^{57} +14.4222 q^{59} +4.00000i q^{63} +3.60555i q^{65} -7.00000 q^{69} +8.00000i q^{71} +8.00000i q^{75} +12.0000 q^{77} -4.00000i q^{79} +1.00000 q^{81} +(-13.0000 + 7.21110i) q^{85} -7.21110 q^{87} +12.0000 q^{89} -4.00000i q^{91} +2.00000 q^{93} +13.0000i q^{95} -7.21110i q^{97} -3.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} + 4 q^{13} + 8 q^{17} - 16 q^{21} - 32 q^{25} + 12 q^{33} - 36 q^{49} + 16 q^{53} - 28 q^{69} + 48 q^{77} + 4 q^{81} - 52 q^{85} + 48 q^{89} + 8 q^{93}+O(q^{100}) $ 4 * q - 4 * q^9 + 4 * q^13 + 8 * q^17 - 16 * q^21 - 32 * q^25 + 12 * q^33 - 36 * q^49 + 16 * q^53 - 28 * q^69 + 48 * q^77 + 4 * q^81 - 52 * q^85 + 48 * q^89 + 8 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3264\mathbb{Z}\right)^\times$.

$n$	$511$	$2177$	$2245$	$2689$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$			3.60555i			1.61245i	0.591608	+	0.806226i	$0.298493\pi$
$5$			3.60555i			1.61245i	−0.591608	+	0.806226i	$0.701507\pi$
$6$		0			0
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			3.00000i			0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$			3.00000i			0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$		0			0
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$		0			0
$15$	3.60555			0.930949
$16$		0			0
$17$	2.00000	+	3.60555i	0.485071	+	0.874475i
$17$	2.00000	+	3.60555i	0.485071	+	0.874475i
$18$		0			0
$19$	3.60555			0.827170			0.413585	−	0.910465i	$-0.364276\pi$
$19$	3.60555			0.827170			0.413585	+	0.910465i	$0.364276\pi$
$20$		0			0
$21$	−4.00000			−0.872872
$22$		0			0
$23$		−	7.00000i		−	1.45960i	−0.683660	−	0.729800i	$-0.739613\pi$
$23$		−	7.00000i		−	1.45960i	0.683660	−	0.729800i	$-0.260387\pi$
$24$		0			0
$25$	−8.00000			−1.60000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	7.21110i		−	1.33907i	−0.742781	−	0.669534i	$-0.766494\pi$
$29$		−	7.21110i		−	1.33907i	0.742781	−	0.669534i	$-0.233506\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		0			0
$33$	3.00000			0.522233
$34$		0			0
$35$	14.4222			2.43780
$36$		0			0
$37$			7.21110i			1.18550i	0.805387	+	0.592749i	$0.201957\pi$
$37$			7.21110i			1.18550i	−0.805387	+	0.592749i	$0.798043\pi$
$38$		0			0
$39$		−	1.00000i		−	0.160128i
$40$		0			0
$41$			10.8167i			1.68928i	0.535337	+	0.844639i	$0.320185\pi$
$41$			10.8167i			1.68928i	−0.535337	+	0.844639i	$0.679815\pi$
$42$		0			0
$43$	−3.60555			−0.549841			−0.274921	−	0.961467i	$-0.588652\pi$
$43$	−3.60555			−0.549841			−0.274921	+	0.961467i	$0.588652\pi$
$44$		0			0
$45$		−	3.60555i		−	0.537484i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$	3.60555	−	2.00000i	0.504878	−	0.280056i
$52$		0			0
$53$	4.00000			0.549442			0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000			0.549442			0.274721	+	0.961524i	$0.411414\pi$
$54$		0			0
$55$	−10.8167			−1.45852
$56$		0			0
$57$		−	3.60555i		−	0.477567i
$58$		0			0
$59$	14.4222			1.87761			0.938806	−	0.344447i	$-0.111934\pi$
$59$	14.4222			1.87761			0.938806	+	0.344447i	$0.111934\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$			4.00000i			0.503953i
$64$		0			0
$65$			3.60555i			0.447214i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	−7.00000			−0.842701
$70$		0			0
$71$			8.00000i			0.949425i	0.880141	+	0.474713i	$0.157448\pi$
$71$			8.00000i			0.949425i	−0.880141	+	0.474713i	$0.842552\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$			8.00000i			0.923760i
$76$		0			0
$77$	12.0000			1.36753
$78$		0			0
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−13.0000	+	7.21110i	−1.41005	+	0.782154i
$86$		0			0
$87$	−7.21110			−0.773111
$88$		0			0
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		0			0
$91$		−	4.00000i		−	0.419314i
$92$		0			0
$93$	2.00000			0.207390
$94$		0			0
$95$			13.0000i			1.33377i
$96$		0			0
$97$		−	7.21110i		−	0.732177i	−0.930580	−	0.366088i	$-0.880697\pi$
$97$		−	7.21110i		−	0.732177i	0.930580	−	0.366088i	$-0.119303\pi$
$98$		0			0
$99$		−	3.00000i		−	0.301511i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	10.8167			1.06580			0.532898	−	0.846179i	$-0.321103\pi$
$103$	10.8167			1.06580			0.532898	+	0.846179i	$0.321103\pi$
$104$		0			0
$105$		−	14.4222i		−	1.40746i
$106$		0			0
$107$			13.0000i			1.25676i	0.777908	+	0.628379i	$0.216281\pi$
$107$			13.0000i			1.25676i	−0.777908	+	0.628379i	$0.783719\pi$
$108$		0			0
$109$			14.4222i			1.38140i	0.723143	+	0.690698i	$0.242697\pi$
$109$			14.4222i			1.38140i	−0.723143	+	0.690698i	$0.757303\pi$
$110$		0			0
$111$	7.21110			0.684448
$112$		0			0
$113$		−	10.8167i		−	1.01755i	−0.860901	−	0.508773i	$-0.830099\pi$
$113$		−	10.8167i		−	1.01755i	0.860901	−	0.508773i	$-0.169901\pi$
$114$		0			0
$115$	25.2389			2.35354
$116$		0			0
$117$	−1.00000			−0.0924500
$118$		0			0
$119$	14.4222	−	8.00000i	1.32208	−	0.733359i
$120$		0			0
$121$	2.00000			0.181818
$122$		0			0
$123$	10.8167			0.975305
$124$		0			0
$125$		−	10.8167i		−	0.967471i
$126$		0			0
$127$	−18.0278			−1.59970			−0.799852	−	0.600197i	$-0.795089\pi$
$127$	−18.0278			−1.59970			−0.799852	+	0.600197i	$0.795089\pi$
$128$		0			0
$129$			3.60555i			0.317451i
$130$		0			0
$131$			15.0000i			1.31056i	0.755388	+	0.655278i	$0.227449\pi$
$131$			15.0000i			1.31056i	−0.755388	+	0.655278i	$0.772551\pi$
$132$		0			0
$133$		−	14.4222i		−	1.25056i
$134$		0			0
$135$	−3.60555			−0.310316
$136$		0			0
$137$	22.0000			1.87959			0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000			1.87959			0.939793	+	0.341743i	$0.111017\pi$
$138$		0			0
$139$			14.0000i			1.18746i	0.804663	+	0.593732i	$0.202346\pi$
$139$			14.0000i			1.18746i	−0.804663	+	0.593732i	$0.797654\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			3.00000i			0.250873i
$144$		0			0
$145$	26.0000			2.15918
$146$		0			0
$147$			9.00000i			0.742307i
$148$		0			0
$149$	18.0000			1.47462			0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000			1.47462			0.737309	+	0.675556i	$0.236096\pi$
$150$		0			0
$151$	14.4222			1.17366			0.586831	−	0.809709i	$-0.300375\pi$
$151$	14.4222			1.17366			0.586831	+	0.809709i	$0.300375\pi$
$152$		0			0
$153$	−2.00000	−	3.60555i	−0.161690	−	0.291492i
$154$		0			0
$155$	−7.21110			−0.579210
$156$		0			0
$157$	1.00000			0.0798087			0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000			0.0798087			0.0399043	+	0.999204i	$0.487295\pi$
$158$		0			0
$159$		−	4.00000i		−	0.317221i
$160$		0			0
$161$	−28.0000			−2.20671
$162$		0			0
$163$			14.0000i			1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$			14.0000i			1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$		0			0
$165$			10.8167i			0.842075i
$166$		0			0
$167$			11.0000i			0.851206i	0.904910	+	0.425603i	$0.139938\pi$
$167$			11.0000i			0.851206i	−0.904910	+	0.425603i	$0.860062\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$	−3.60555			−0.275723
$172$		0			0
$173$		−	25.2389i		−	1.91887i	−0.281922	−	0.959437i	$-0.590972\pi$
$173$		−	25.2389i		−	1.91887i	0.281922	−	0.959437i	$-0.409028\pi$
$174$		0			0
$175$			32.0000i			2.41897i
$176$		0			0
$177$		−	14.4222i		−	1.08404i
$178$		0			0
$179$	−7.21110			−0.538983			−0.269492	−	0.963003i	$-0.586856\pi$
$179$	−7.21110			−0.538983			−0.269492	+	0.963003i	$0.586856\pi$
$180$		0			0
$181$			7.21110i			0.535997i	0.963419	+	0.267999i	$0.0863622\pi$
$181$			7.21110i			0.535997i	−0.963419	+	0.267999i	$0.913638\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−26.0000			−1.91156
$186$		0			0
$187$	−10.8167	+	6.00000i	−0.790992	+	0.438763i
$188$		0			0
$189$	4.00000			0.290957
$190$		0			0
$191$	−7.21110			−0.521777			−0.260889	−	0.965369i	$-0.584016\pi$
$191$	−7.21110			−0.521777			−0.260889	+	0.965369i	$0.584016\pi$
$192$		0			0
$193$			7.21110i			0.519067i	0.965734	+	0.259533i	$0.0835687\pi$
$193$			7.21110i			0.519067i	−0.965734	+	0.259533i	$0.916431\pi$
$194$		0			0
$195$	3.60555			0.258199
$196$		0			0
$197$			10.8167i			0.770655i	0.922780	+	0.385327i	$0.125911\pi$
$197$			10.8167i			0.770655i	−0.922780	+	0.385327i	$0.874089\pi$
$198$		0			0
$199$		−	26.0000i		−	1.84309i	−0.388270	−	0.921546i	$-0.626927\pi$
$199$		−	26.0000i		−	1.84309i	0.388270	−	0.921546i	$-0.373073\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−28.8444			−2.02448
$204$		0			0
$205$	−39.0000			−2.72388
$206$		0			0
$207$			7.00000i			0.486534i
$208$		0			0
$209$			10.8167i			0.748204i
$210$		0			0
$211$			10.0000i			0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$			10.0000i			0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$		0			0
$213$	8.00000			0.548151
$214$		0			0
$215$		−	13.0000i		−	0.886593i
$216$		0			0
$217$	8.00000			0.543075
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.00000	+	3.60555i	0.134535	+	0.242536i
$222$		0			0
$223$	25.2389			1.69012			0.845060	−	0.534672i	$-0.179565\pi$
$223$	25.2389			1.69012			0.845060	+	0.534672i	$0.179565\pi$
$224$		0			0
$225$	8.00000			0.533333
$226$		0			0
$227$		−	27.0000i		−	1.79205i	−0.444001	−	0.896026i	$-0.646441\pi$
$227$		−	27.0000i		−	1.79205i	0.444001	−	0.896026i	$-0.353559\pi$
$228$		0			0
$229$	−10.0000			−0.660819			−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000			−0.660819			−0.330409	+	0.943838i	$0.607187\pi$
$230$		0			0
$231$		−	12.0000i		−	0.789542i
$232$		0			0
$233$			18.0278i			1.18104i	0.807024	+	0.590519i	$0.201077\pi$
$233$			18.0278i			1.18104i	−0.807024	+	0.590519i	$0.798923\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−4.00000			−0.259828
$238$		0			0
$239$	7.21110			0.466447			0.233224	−	0.972423i	$-0.425073\pi$
$239$	7.21110			0.466447			0.233224	+	0.972423i	$0.425073\pi$
$240$		0			0
$241$			14.4222i			0.929016i	0.885569	+	0.464508i	$0.153769\pi$
$241$			14.4222i			0.929016i	−0.885569	+	0.464508i	$0.846231\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$		−	32.4500i		−	2.07315i
$246$		0			0
$247$	3.60555			0.229416
$248$		0			0
$249$		0			0
$250$		0			0
$251$	21.6333			1.36548			0.682741	−	0.730660i	$-0.260788\pi$
$251$	21.6333			1.36548			0.682741	+	0.730660i	$0.260788\pi$
$252$		0			0
$253$	21.0000			1.32026
$254$		0			0
$255$	7.21110	+	13.0000i	0.451577	+	0.814092i
$256$		0			0
$257$	−12.0000			−0.748539			−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000			−0.748539			−0.374270	+	0.927320i	$0.622107\pi$
$258$		0			0
$259$	28.8444			1.79230
$260$		0			0
$261$			7.21110i			0.446356i
$262$		0			0
$263$	−21.6333			−1.33397			−0.666983	−	0.745073i	$-0.732415\pi$
$263$	−21.6333			−1.33397			−0.666983	+	0.745073i	$0.732415\pi$
$264$		0			0
$265$			14.4222i			0.885949i
$266$		0			0
$267$		−	12.0000i		−	0.734388i
$268$		0			0
$269$			10.8167i			0.659503i	0.944068	+	0.329752i	$0.106965\pi$
$269$			10.8167i			0.659503i	−0.944068	+	0.329752i	$0.893035\pi$
$270$		0			0
$271$	3.60555			0.219022			0.109511	−	0.993986i	$-0.465072\pi$
$271$	3.60555			0.219022			0.109511	+	0.993986i	$0.465072\pi$
$272$		0			0
$273$	−4.00000			−0.242091
$274$		0			0
$275$		−	24.0000i		−	1.44725i
$276$		0			0
$277$			14.4222i			0.866546i	0.901263	+	0.433273i	$0.142641\pi$
$277$			14.4222i			0.866546i	−0.901263	+	0.433273i	$0.857359\pi$
$278$		0			0
$279$		−	2.00000i		−	0.119737i
$280$		0			0
$281$	20.0000			1.19310			0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000			1.19310			0.596550	+	0.802576i	$0.296538\pi$
$282$		0			0
$283$		−	32.0000i		−	1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$		−	32.0000i		−	1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$		0			0
$285$	13.0000			0.770054
$286$		0			0
$287$	43.2666			2.55395
$288$		0			0
$289$	−9.00000	+	14.4222i	−0.529412	+	0.848365i
$290$		0			0
$291$	−7.21110			−0.422722
$292$		0			0
$293$	22.0000			1.28525			0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000			1.28525			0.642627	+	0.766179i	$0.277845\pi$
$294$		0			0
$295$			52.0000i			3.02756i
$296$		0			0
$297$	−3.00000			−0.174078
$298$		0			0
$299$		−	7.00000i		−	0.404820i
$300$		0			0
$301$			14.4222i			0.831282i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	28.8444			1.64624			0.823119	−	0.567869i	$-0.192232\pi$
$307$	28.8444			1.64624			0.823119	+	0.567869i	$0.192232\pi$
$308$		0			0
$309$		−	10.8167i		−	0.615338i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		−	7.21110i		−	0.407596i	−0.979013	−	0.203798i	$-0.934671\pi$
$313$		−	7.21110i		−	0.407596i	0.979013	−	0.203798i	$-0.0653285\pi$
$314$		0			0
$315$	−14.4222			−0.812599
$316$		0			0
$317$		−	7.21110i		−	0.405016i	−0.979281	−	0.202508i	$-0.935091\pi$
$317$		−	7.21110i		−	0.405016i	0.979281	−	0.202508i	$-0.0649092\pi$
$318$		0			0
$319$	21.6333			1.21123
$320$		0			0
$321$	13.0000			0.725589
$322$		0			0
$323$	7.21110	+	13.0000i	0.401236	+	0.723339i
$324$		0			0
$325$	−8.00000			−0.443760
$326$		0			0
$327$	14.4222			0.797550
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−25.2389			−1.38725			−0.693627	−	0.720335i	$-0.743988\pi$
$331$	−25.2389			−1.38725			−0.693627	+	0.720335i	$0.743988\pi$
$332$		0			0
$333$		−	7.21110i		−	0.395166i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	36.0555i		−	1.96407i	−0.188702	−	0.982034i	$-0.560428\pi$
$337$		−	36.0555i		−	1.96407i	0.188702	−	0.982034i	$-0.439572\pi$
$338$		0			0
$339$	−10.8167			−0.587480
$340$		0			0
$341$	−6.00000			−0.324918
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$		−	25.2389i		−	1.35881i
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	1.00000			0.0535288			0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000			0.0535288			0.0267644	+	0.999642i	$0.491480\pi$
$350$		0			0
$351$			1.00000i			0.0533761i
$352$		0			0
$353$	−8.00000			−0.425797			−0.212899	−	0.977074i	$-0.568290\pi$
$353$	−8.00000			−0.425797			−0.212899	+	0.977074i	$0.568290\pi$
$354$		0			0
$355$	−28.8444			−1.53090
$356$		0			0
$357$	−8.00000	−	14.4222i	−0.423405	−	0.763304i
$358$		0			0
$359$	−21.6333			−1.14176			−0.570881	−	0.821033i	$-0.693398\pi$
$359$	−21.6333			−1.14176			−0.570881	+	0.821033i	$0.693398\pi$
$360$		0			0
$361$	−6.00000			−0.315789
$362$		0			0
$363$		−	2.00000i		−	0.104973i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	30.0000i		−	1.56599i	−0.622030	−	0.782994i	$-0.713692\pi$
$367$		−	30.0000i		−	1.56599i	0.622030	−	0.782994i	$-0.286308\pi$
$368$		0			0
$369$		−	10.8167i		−	0.563093i
$370$		0			0
$371$		−	16.0000i		−	0.830679i
$372$		0			0
$373$	−14.0000			−0.724893			−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000			−0.724893			−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$	−10.8167			−0.558570
$376$		0			0
$377$		−	7.21110i		−	0.371391i
$378$		0			0
$379$		−	28.0000i		−	1.43826i	−0.694874	−	0.719132i	$-0.744540\pi$
$379$		−	28.0000i		−	1.43826i	0.694874	−	0.719132i	$-0.255460\pi$
$380$		0			0
$381$			18.0278i			0.923590i
$382$		0			0
$383$	21.6333			1.10541			0.552705	−	0.833377i	$-0.313596\pi$
$383$	21.6333			1.10541			0.552705	+	0.833377i	$0.313596\pi$
$384$		0			0
$385$			43.2666i			2.20507i
$386$		0			0
$387$	3.60555			0.183280
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$	25.2389	−	14.0000i	1.27638	−	0.708010i
$392$		0			0
$393$	15.0000			0.756650
$394$		0			0
$395$	14.4222			0.725660
$396$		0			0
$397$			36.0555i			1.80957i	0.425864	+	0.904787i	$0.359970\pi$
$397$			36.0555i			1.80957i	−0.425864	+	0.904787i	$0.640030\pi$
$398$		0			0
$399$	−14.4222			−0.722013
$400$		0			0
$401$		−	18.0278i		−	0.900263i	−0.892962	−	0.450132i	$-0.851377\pi$
$401$		−	18.0278i		−	0.900263i	0.892962	−	0.450132i	$-0.148623\pi$
$402$		0			0
$403$			2.00000i			0.0996271i
$404$		0			0
$405$			3.60555i			0.179161i
$406$		0			0
$407$	−21.6333			−1.07232
$408$		0			0
$409$	1.00000			0.0494468			0.0247234	−	0.999694i	$-0.492129\pi$
$409$	1.00000			0.0494468			0.0247234	+	0.999694i	$0.492129\pi$
$410$		0			0
$411$		−	22.0000i		−	1.08518i
$412$		0			0
$413$		−	57.6888i		−	2.83868i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	14.0000			0.685583
$418$		0			0
$419$			36.0000i			1.75872i	0.476162	+	0.879358i	$0.342028\pi$
$419$			36.0000i			1.75872i	−0.476162	+	0.879358i	$0.657972\pi$
$420$		0			0
$421$	17.0000			0.828529			0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000			0.828529			0.414265	+	0.910156i	$0.364039\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−16.0000	−	28.8444i	−0.776114	−	1.39916i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	3.00000			0.144841
$430$		0			0
$431$		−	16.0000i		−	0.770693i	−0.922772	−	0.385346i	$-0.874082\pi$
$431$		−	16.0000i		−	0.770693i	0.922772	−	0.385346i	$-0.125918\pi$
$432$		0			0
$433$	5.00000			0.240285			0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000			0.240285			0.120142	+	0.992757i	$0.461665\pi$
$434$		0			0
$435$		−	26.0000i		−	1.24660i
$436$		0			0
$437$		−	25.2389i		−	1.20734i
$438$		0			0
$439$			12.0000i			0.572729i	0.958121	+	0.286364i	$0.0924468\pi$
$439$			12.0000i			0.572729i	−0.958121	+	0.286364i	$0.907553\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$	−14.4222			−0.685220			−0.342610	−	0.939478i	$-0.611311\pi$
$443$	−14.4222			−0.685220			−0.342610	+	0.939478i	$0.611311\pi$
$444$		0			0
$445$			43.2666i			2.05103i
$446$		0			0
$447$		−	18.0000i		−	0.851371i
$448$		0			0
$449$			21.6333i			1.02094i	0.859896	+	0.510469i	$0.170528\pi$
$449$			21.6333i			1.02094i	−0.859896	+	0.510469i	$0.829472\pi$
$450$		0			0
$451$	−32.4500			−1.52801
$452$		0			0
$453$		−	14.4222i		−	0.677614i
$454$		0			0
$455$	14.4222			0.676123
$456$		0			0
$457$	−5.00000			−0.233890			−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000			−0.233890			−0.116945	+	0.993138i	$0.537310\pi$
$458$		0			0
$459$	−3.60555	+	2.00000i	−0.168293	+	0.0933520i
$460$		0			0
$461$	−32.0000			−1.49039			−0.745194	−	0.666847i	$-0.767643\pi$
$461$	−32.0000			−1.49039			−0.745194	+	0.666847i	$0.767643\pi$
$462$		0			0
$463$	14.4222			0.670257			0.335128	−	0.942172i	$-0.391220\pi$
$463$	14.4222			0.670257			0.335128	+	0.942172i	$0.391220\pi$
$464$		0			0
$465$			7.21110i			0.334407i
$466$		0			0
$467$	−21.6333			−1.00107			−0.500535	−	0.865716i	$-0.666863\pi$
$467$	−21.6333			−1.00107			−0.500535	+	0.865716i	$0.666863\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		−	1.00000i		−	0.0460776i
$472$		0			0
$473$		−	10.8167i		−	0.497350i
$474$		0			0
$475$	−28.8444			−1.32347
$476$		0			0
$477$	−4.00000			−0.183147
$478$		0			0
$479$			15.0000i			0.685367i	0.939451	+	0.342684i	$0.111336\pi$
$479$			15.0000i			0.685367i	−0.939451	+	0.342684i	$0.888664\pi$
$480$		0			0
$481$			7.21110i			0.328798i
$482$		0			0
$483$			28.0000i			1.27404i
$484$		0			0
$485$	26.0000			1.18060
$486$		0			0
$487$		−	20.0000i		−	0.906287i	−0.891438	−	0.453143i	$-0.850303\pi$
$487$		−	20.0000i		−	0.906287i	0.891438	−	0.453143i	$-0.149697\pi$
$488$		0			0
$489$	14.0000			0.633102
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	26.0000	−	14.4222i	1.17098	−	0.649543i
$494$		0			0
$495$	10.8167			0.486172
$496$		0			0
$497$	32.0000			1.43540
$498$		0			0
$499$		−	4.00000i		−	0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$		−	4.00000i		−	0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$		0			0
$501$	11.0000			0.491444
$502$		0			0
$503$			3.00000i			0.133763i	0.997761	+	0.0668817i	$0.0213050\pi$
$503$			3.00000i			0.133763i	−0.997761	+	0.0668817i	$0.978695\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			12.0000i			0.532939i
$508$		0			0
$509$	−36.0000			−1.59567			−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000			−1.59567			−0.797836	+	0.602875i	$0.794022\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			3.60555i			0.159189i
$514$		0			0
$515$			39.0000i			1.71855i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−25.2389			−1.10786
$520$		0			0
$521$		−	32.4500i		−	1.42166i	−0.703365	−	0.710829i	$-0.748320\pi$
$521$		−	32.4500i		−	1.42166i	0.703365	−	0.710829i	$-0.251680\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$	32.0000			1.39659
$526$		0			0
$527$	−7.21110	+	4.00000i	−0.314121	+	0.174243i
$528$		0			0
$529$	−26.0000			−1.13043
$530$		0			0
$531$	−14.4222			−0.625870
$532$		0			0
$533$			10.8167i			0.468521i
$534$		0			0
$535$	−46.8722			−2.02646
$536$		0			0
$537$			7.21110i			0.311182i
$538$		0			0
$539$		−	27.0000i		−	1.16297i
$540$		0			0
$541$		−	7.21110i		−	0.310030i	−0.987912	−	0.155015i	$-0.950457\pi$
$541$		−	7.21110i		−	0.310030i	0.987912	−	0.155015i	$-0.0495425\pi$
$542$		0			0
$543$	7.21110			0.309458
$544$		0			0
$545$	−52.0000			−2.22744
$546$		0			0
$547$		−	12.0000i		−	0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$		−	12.0000i		−	0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	26.0000i		−	1.10764i
$552$		0			0
$553$	−16.0000			−0.680389
$554$		0			0
$555$			26.0000i			1.10364i
$556$		0			0
$557$	−28.0000			−1.18640			−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000			−1.18640			−0.593199	+	0.805056i	$0.702135\pi$
$558$		0			0
$559$	−3.60555			−0.152499
$560$		0			0
$561$	6.00000	+	10.8167i	0.253320	+	0.456679i
$562$		0			0
$563$	−7.21110			−0.303912			−0.151956	−	0.988387i	$-0.548557\pi$
$563$	−7.21110			−0.303912			−0.151956	+	0.988387i	$0.548557\pi$
$564$		0			0
$565$	39.0000			1.64074
$566$		0			0
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$		−	26.0000i		−	1.08807i	−0.839064	−	0.544033i	$-0.816897\pi$
$571$		−	26.0000i		−	1.08807i	0.839064	−	0.544033i	$-0.183103\pi$
$572$		0			0
$573$			7.21110i			0.301248i
$574$		0			0
$575$			56.0000i			2.33536i
$576$		0			0
$577$	9.00000			0.374675			0.187337	−	0.982296i	$-0.440014\pi$
$577$	9.00000			0.374675			0.187337	+	0.982296i	$0.440014\pi$
$578$		0			0
$579$	7.21110			0.299683
$580$		0			0
$581$		0			0
$582$		0			0
$583$			12.0000i			0.496989i
$584$		0			0
$585$		−	3.60555i		−	0.149071i
$586$		0			0
$587$	−43.2666			−1.78580			−0.892902	−	0.450251i	$-0.851335\pi$
$587$	−43.2666			−1.78580			−0.892902	+	0.450251i	$0.851335\pi$
$588$		0			0
$589$			7.21110i			0.297128i
$590$		0			0
$591$	10.8167			0.444938
$592$		0			0
$593$	4.00000			0.164260			0.0821302	−	0.996622i	$-0.473828\pi$
$593$	4.00000			0.164260			0.0821302	+	0.996622i	$0.473828\pi$
$594$		0			0
$595$	28.8444	+	52.0000i	1.18251	+	2.13179i
$596$		0			0
$597$	−26.0000			−1.06411
$598$		0			0
$599$	−21.6333			−0.883913			−0.441956	−	0.897036i	$-0.645715\pi$
$599$	−21.6333			−0.883913			−0.441956	+	0.897036i	$0.645715\pi$
$600$		0			0
$601$		−	7.21110i		−	0.294147i	−0.989126	−	0.147074i	$-0.953015\pi$
$601$		−	7.21110i		−	0.294147i	0.989126	−	0.147074i	$-0.0469854\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			7.21110i			0.293173i
$606$		0			0
$607$		−	38.0000i		−	1.54237i	−0.636610	−	0.771186i	$-0.719664\pi$
$607$		−	38.0000i		−	1.54237i	0.636610	−	0.771186i	$-0.280336\pi$
$608$		0			0
$609$			28.8444i			1.16883i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−11.0000			−0.444286			−0.222143	−	0.975014i	$-0.571305\pi$
$613$	−11.0000			−0.444286			−0.222143	+	0.975014i	$0.571305\pi$
$614$		0			0
$615$			39.0000i			1.57263i
$616$		0			0
$617$			21.6333i			0.870924i	0.900207	+	0.435462i	$0.143415\pi$
$617$			21.6333i			0.870924i	−0.900207	+	0.435462i	$0.856585\pi$
$618$		0			0
$619$			36.0000i			1.44696i	0.690344	+	0.723481i	$0.257459\pi$
$619$			36.0000i			1.44696i	−0.690344	+	0.723481i	$0.742541\pi$
$620$		0			0
$621$	7.00000			0.280900
$622$		0			0
$623$		−	48.0000i		−	1.92308i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$	10.8167			0.431976
$628$		0			0
$629$	−26.0000	+	14.4222i	−1.03669	+	0.575051i
$630$		0			0
$631$	−46.8722			−1.86595			−0.932976	−	0.359939i	$-0.882797\pi$
$631$	−46.8722			−1.86595			−0.932976	+	0.359939i	$0.882797\pi$
$632$		0			0
$633$	10.0000			0.397464
$634$		0			0
$635$		−	65.0000i		−	2.57945i
$636$		0			0
$637$	−9.00000			−0.356593
$638$		0			0
$639$		−	8.00000i		−	0.316475i
$640$		0			0
$641$			10.8167i			0.427232i	0.976918	+	0.213616i	$0.0685242\pi$
$641$			10.8167i			0.427232i	−0.976918	+	0.213616i	$0.931476\pi$
$642$		0			0
$643$		−	6.00000i		−	0.236617i	−0.992977	−	0.118308i	$-0.962253\pi$
$643$		−	6.00000i		−	0.236617i	0.992977	−	0.118308i	$-0.0377472\pi$
$644$		0			0
$645$	−13.0000			−0.511875
$646$		0			0
$647$	36.0555			1.41749			0.708744	−	0.705466i	$-0.249262\pi$
$647$	36.0555			1.41749			0.708744	+	0.705466i	$0.249262\pi$
$648$		0			0
$649$			43.2666i			1.69836i
$650$		0			0
$651$		−	8.00000i		−	0.313545i
$652$		0			0
$653$			10.8167i			0.423288i	0.977347	+	0.211644i	$0.0678818\pi$
$653$			10.8167i			0.423288i	−0.977347	+	0.211644i	$0.932118\pi$
$654$		0			0
$655$	−54.0833			−2.11321
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.21110			0.280905			0.140452	−	0.990087i	$-0.455144\pi$
$659$	7.21110			0.280905			0.140452	+	0.990087i	$0.455144\pi$
$660$		0			0
$661$	−27.0000			−1.05018			−0.525089	−	0.851047i	$-0.675968\pi$
$661$	−27.0000			−1.05018			−0.525089	+	0.851047i	$0.675968\pi$
$662$		0			0
$663$	3.60555	−	2.00000i	0.140028	−	0.0776736i
$664$		0			0
$665$	52.0000			2.01647
$666$		0			0
$667$	−50.4777			−1.95451
$668$		0			0
$669$		−	25.2389i		−	0.975791i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	14.4222i		−	0.555935i	−0.960590	−	0.277968i	$-0.910339\pi$
$673$		−	14.4222i		−	0.555935i	0.960590	−	0.277968i	$-0.0896608\pi$
$674$		0			0
$675$		−	8.00000i		−	0.307920i
$676$		0			0
$677$			25.2389i			0.970008i	0.874512	+	0.485004i	$0.161182\pi$
$677$			25.2389i			0.970008i	−0.874512	+	0.485004i	$0.838818\pi$
$678$		0			0
$679$	−28.8444			−1.10695
$680$		0			0
$681$	−27.0000			−1.03464
$682$		0			0
$683$		−	21.0000i		−	0.803543i	−0.915740	−	0.401771i	$-0.868395\pi$
$683$		−	21.0000i		−	0.803543i	0.915740	−	0.401771i	$-0.131605\pi$
$684$		0			0
$685$			79.3221i			3.03074i
$686$		0			0
$687$			10.0000i			0.381524i
$688$		0			0
$689$	4.00000			0.152388
$690$		0			0
$691$		−	16.0000i		−	0.608669i	−0.952565	−	0.304334i	$-0.901566\pi$
$691$		−	16.0000i		−	0.608669i	0.952565	−	0.304334i	$-0.0984340\pi$
$692$		0			0
$693$	−12.0000			−0.455842
$694$		0			0
$695$	−50.4777			−1.91473
$696$		0			0
$697$	−39.0000	+	21.6333i	−1.47723	+	0.819420i
$698$		0			0
$699$	18.0278			0.681872
$700$		0			0
$701$	16.0000			0.604312			0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000			0.604312			0.302156	+	0.953259i	$0.402294\pi$
$702$		0			0
$703$			26.0000i			0.980609i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			14.4222i			0.541637i	0.962630	+	0.270819i	$0.0872944\pi$
$709$			14.4222i			0.541637i	−0.962630	+	0.270819i	$0.912706\pi$
$710$		0			0
$711$			4.00000i			0.150012i
$712$		0			0
$713$	14.0000			0.524304
$714$		0			0
$715$	−10.8167			−0.404520
$716$		0			0
$717$		−	7.21110i		−	0.269304i
$718$		0			0
$719$			17.0000i			0.633993i	0.948427	+	0.316997i	$0.102674\pi$
$719$			17.0000i			0.633993i	−0.948427	+	0.316997i	$0.897326\pi$
$720$		0			0
$721$		−	43.2666i		−	1.61133i
$722$		0			0
$723$	14.4222			0.536368
$724$		0			0
$725$			57.6888i			2.14251i
$726$		0			0
$727$	−43.2666			−1.60467			−0.802335	−	0.596874i	$-0.796409\pi$
$727$	−43.2666			−1.60467			−0.802335	+	0.596874i	$0.796409\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−7.21110	−	13.0000i	−0.266712	−	0.480822i
$732$		0			0
$733$	−22.0000			−0.812589			−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000			−0.812589			−0.406294	+	0.913742i	$0.633179\pi$
$734$		0			0
$735$	−32.4500			−1.19693
$736$		0			0
$737$		0			0
$738$		0			0
$739$	18.0278			0.663162			0.331581	−	0.943427i	$-0.392418\pi$
$739$	18.0278			0.663162			0.331581	+	0.943427i	$0.392418\pi$
$740$		0			0
$741$		−	3.60555i		−	0.132453i
$742$		0			0
$743$			24.0000i			0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$			24.0000i			0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$		0			0
$745$			64.8999i			2.37775i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	52.0000			1.90004
$750$		0			0
$751$			2.00000i			0.0729810i	0.999334	+	0.0364905i	$0.0116179\pi$
$751$			2.00000i			0.0729810i	−0.999334	+	0.0364905i	$0.988382\pi$
$752$		0			0
$753$		−	21.6333i		−	0.788362i
$754$		0			0
$755$			52.0000i			1.89247i
$756$		0			0
$757$	9.00000			0.327111			0.163555	−	0.986534i	$-0.447704\pi$
$757$	9.00000			0.327111			0.163555	+	0.986534i	$0.447704\pi$
$758$		0			0
$759$		−	21.0000i		−	0.762252i
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$	57.6888			2.08848
$764$		0			0
$765$	13.0000	−	7.21110i	0.470016	−	0.260718i
$766$		0			0
$767$	14.4222			0.520756
$768$		0			0
$769$	−11.0000			−0.396670			−0.198335	−	0.980134i	$-0.563553\pi$
$769$	−11.0000			−0.396670			−0.198335	+	0.980134i	$0.563553\pi$
$770$		0			0
$771$			12.0000i			0.432169i
$772$		0			0
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$		−	16.0000i		−	0.574737i
$776$		0			0
$777$		−	28.8444i		−	1.03479i
$778$		0			0
$779$			39.0000i			1.39732i
$780$		0			0
$781$	−24.0000			−0.858788
$782$		0			0
$783$	7.21110			0.257704
$784$		0			0
$785$			3.60555i			0.128688i
$786$		0			0
$787$		−	22.0000i		−	0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$		−	22.0000i		−	0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$		0			0
$789$			21.6333i			0.770166i
$790$		0			0
$791$	−43.2666			−1.53838
$792$		0			0
$793$		0			0
$794$		0			0
$795$	14.4222			0.511503
$796$		0			0
$797$	24.0000			0.850124			0.425062	−	0.905164i	$-0.360252\pi$
$797$	24.0000			0.850124			0.425062	+	0.905164i	$0.360252\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−12.0000			−0.423999
$802$		0			0
$803$		0			0
$804$		0			0
$805$		−	100.955i		−	3.55821i
$806$		0			0
$807$	10.8167			0.380764
$808$		0			0
$809$		−	32.4500i		−	1.14088i	−0.821339	−	0.570440i	$-0.806773\pi$
$809$		−	32.4500i		−	1.14088i	0.821339	−	0.570440i	$-0.193227\pi$
$810$		0			0
$811$			4.00000i			0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$			4.00000i			0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$		0			0
$813$		−	3.60555i		−	0.126452i
$814$		0			0
$815$	−50.4777			−1.76816
$816$		0			0
$817$	−13.0000			−0.454812
$818$		0			0
$819$			4.00000i			0.139771i
$820$		0			0
$821$		−	10.8167i		−	0.377504i	−0.982025	−	0.188752i	$-0.939556\pi$
$821$		−	10.8167i		−	0.377504i	0.982025	−	0.188752i	$-0.0604442\pi$
$822$		0			0
$823$		−	34.0000i		−	1.18517i	−0.805510	−	0.592583i	$-0.798108\pi$
$823$		−	34.0000i		−	1.18517i	0.805510	−	0.592583i	$-0.201892\pi$
$824$		0			0
$825$	−24.0000			−0.835573
$826$		0			0
$827$		−	7.00000i		−	0.243414i	−0.992566	−	0.121707i	$-0.961163\pi$
$827$		−	7.00000i		−	0.243414i	0.992566	−	0.121707i	$-0.0388368\pi$
$828$		0			0
$829$	−38.0000			−1.31979			−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000			−1.31979			−0.659897	+	0.751356i	$0.729400\pi$
$830$		0			0
$831$	14.4222			0.500301
$832$		0			0
$833$	−18.0000	−	32.4500i	−0.623663	−	1.12432i
$834$		0			0
$835$	−39.6611			−1.37253
$836$		0			0
$837$	−2.00000			−0.0691301
$838$		0			0
$839$		−	21.0000i		−	0.725001i	−0.931984	−	0.362500i	$-0.881923\pi$
$839$		−	21.0000i		−	0.725001i	0.931984	−	0.362500i	$-0.118077\pi$
$840$		0			0
$841$	−23.0000			−0.793103
$842$		0			0
$843$		−	20.0000i		−	0.688837i
$844$		0			0
$845$		−	43.2666i		−	1.48842i
$846$		0			0
$847$		−	8.00000i		−	0.274883i
$848$		0			0
$849$	−32.0000			−1.09824
$850$		0			0
$851$	50.4777			1.73035
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		−	13.0000i		−	0.444591i
$856$		0			0
$857$			21.6333i			0.738980i	0.929235	+	0.369490i	$0.120468\pi$
$857$			21.6333i			0.738980i	−0.929235	+	0.369490i	$0.879532\pi$
$858$		0			0
$859$	14.4222			0.492079			0.246040	−	0.969260i	$-0.420871\pi$
$859$	14.4222			0.492079			0.246040	+	0.969260i	$0.420871\pi$
$860$		0			0
$861$		−	43.2666i		−	1.47452i
$862$		0			0
$863$	28.8444			0.981875			0.490938	−	0.871195i	$-0.336654\pi$
$863$	28.8444			0.981875			0.490938	+	0.871195i	$0.336654\pi$
$864$		0			0
$865$	91.0000			3.09409
$866$		0			0
$867$	14.4222	+	9.00000i	0.489804	+	0.305656i
$868$		0			0
$869$	12.0000			0.407072
$870$		0			0
$871$		0			0
$872$		0			0
$873$			7.21110i			0.244059i
$874$		0			0
$875$	−43.2666			−1.46268
$876$		0			0
$877$		−	50.4777i		−	1.70451i	−0.523125	−	0.852256i	$-0.675234\pi$
$877$		−	50.4777i		−	1.70451i	0.523125	−	0.852256i	$-0.324766\pi$
$878$		0			0
$879$		−	22.0000i		−	0.742042i
$880$		0			0
$881$			21.6333i			0.728845i	0.931234	+	0.364422i	$0.118734\pi$
$881$			21.6333i			0.728845i	−0.931234	+	0.364422i	$0.881266\pi$
$882$		0			0
$883$	−18.0278			−0.606682			−0.303341	−	0.952882i	$-0.598102\pi$
$883$	−18.0278			−0.606682			−0.303341	+	0.952882i	$0.598102\pi$
$884$		0			0
$885$	52.0000			1.74796
$886$		0			0
$887$		−	31.0000i		−	1.04088i	−0.853899	−	0.520439i	$-0.825768\pi$
$887$		−	31.0000i		−	1.04088i	0.853899	−	0.520439i	$-0.174232\pi$
$888$		0			0
$889$			72.1110i			2.41853i
$890$		0			0
$891$			3.00000i			0.100504i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	26.0000i		−	0.869084i
$896$		0			0
$897$	−7.00000			−0.233723
$898$		0			0
$899$	14.4222			0.481007
$900$		0			0
$901$	8.00000	+	14.4222i	0.266519	+	0.480473i
$902$		0			0
$903$	14.4222			0.479941
$904$		0			0
$905$	−26.0000			−0.864269
$906$		0			0
$907$		−	58.0000i		−	1.92586i	−0.269754	−	0.962929i	$-0.586942\pi$
$907$		−	58.0000i		−	1.92586i	0.269754	−	0.962929i	$-0.413058\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1.00000i			0.0331315i	0.999863	+	0.0165657i	$0.00527328\pi$
$911$			1.00000i			0.0331315i	−0.999863	+	0.0165657i	$0.994727\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	60.0000			1.98137
$918$		0			0
$919$	3.60555			0.118936			0.0594681	−	0.998230i	$-0.481060\pi$
$919$	3.60555			0.118936			0.0594681	+	0.998230i	$0.481060\pi$
$920$		0			0
$921$		−	28.8444i		−	0.950456i
$922$		0			0
$923$			8.00000i			0.263323i
$924$		0			0
$925$		−	57.6888i		−	1.89680i
$926$		0			0
$927$	−10.8167			−0.355266
$928$		0			0
$929$		−	54.0833i		−	1.77441i	−0.461371	−	0.887207i	$-0.652642\pi$
$929$		−	54.0833i		−	1.77441i	0.461371	−	0.887207i	$-0.347358\pi$
$930$		0			0
$931$	−32.4500			−1.06350
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−21.6333	−	39.0000i	−0.707485	−	1.27544i
$936$		0			0
$937$	14.0000			0.457360			0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000			0.457360			0.228680	+	0.973502i	$0.426559\pi$
$938$		0			0
$939$	−7.21110			−0.235325
$940$		0			0
$941$			36.0555i			1.17538i	0.809088	+	0.587688i	$0.199962\pi$
$941$			36.0555i			1.17538i	−0.809088	+	0.587688i	$0.800038\pi$
$942$		0			0
$943$	75.7166			2.46567
$944$		0			0
$945$			14.4222i			0.469154i
$946$		0			0
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−7.21110			−0.233836
$952$		0			0
$953$	60.0000			1.94359			0.971795	−	0.235826i	$-0.0757795\pi$
$953$	60.0000			1.94359			0.971795	+	0.235826i	$0.0757795\pi$
$954$		0			0
$955$		−	26.0000i		−	0.841340i
$956$		0			0
$957$		−	21.6333i		−	0.699306i
$958$		0			0
$959$		−	88.0000i		−	2.84167i
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$		−	13.0000i		−	0.418919i
$964$		0			0
$965$	−26.0000			−0.836970
$966$		0			0
$967$	−25.2389			−0.811627			−0.405813	−	0.913956i	$-0.633012\pi$
$967$	−25.2389			−0.811627			−0.405813	+	0.913956i	$0.633012\pi$
$968$		0			0
$969$	13.0000	−	7.21110i	0.417620	−	0.231654i
$970$		0			0
$971$	−7.21110			−0.231415			−0.115708	−	0.993283i	$-0.536914\pi$
$971$	−7.21110			−0.231415			−0.115708	+	0.993283i	$0.536914\pi$
$972$		0			0
$973$	56.0000			1.79528
$974$		0			0
$975$			8.00000i			0.256205i
$976$		0			0
$977$	24.0000			0.767828			0.383914	−	0.923369i	$-0.374576\pi$
$977$	24.0000			0.767828			0.383914	+	0.923369i	$0.374576\pi$
$978$		0			0
$979$			36.0000i			1.15056i
$980$		0			0
$981$		−	14.4222i		−	0.460466i
$982$		0			0
$983$			49.0000i			1.56286i	0.623995	+	0.781429i	$0.285509\pi$
$983$			49.0000i			1.56286i	−0.623995	+	0.781429i	$0.714491\pi$
$984$		0			0
$985$	−39.0000			−1.24264
$986$		0			0
$987$		0			0
$988$		0			0
$989$			25.2389i			0.802549i
$990$		0			0
$991$			42.0000i			1.33417i	0.744980	+	0.667087i	$0.232459\pi$
$991$			42.0000i			1.33417i	−0.744980	+	0.667087i	$0.767541\pi$
$992$		0			0
$993$			25.2389i			0.800931i
$994$		0			0
$995$	93.7443			2.97190
$996$		0			0
$997$		−	43.2666i		−	1.37027i	−0.728417	−	0.685134i	$-0.759744\pi$
$997$		−	43.2666i		−	1.37027i	0.728417	−	0.685134i	$-0.240256\pi$
$998$		0			0
$999$	−7.21110			−0.228149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3264.2.c.m.577.2		4
4.3	odd	2	inner	3264.2.c.m.577.4		4
8.3	odd	2		1632.2.c.c.577.1	✓	4
8.5	even	2		1632.2.c.c.577.3	yes	4
17.16	even	2	inner	3264.2.c.m.577.3		4
24.5	odd	2		4896.2.c.g.577.3		4
24.11	even	2		4896.2.c.g.577.4		4
68.67	odd	2	inner	3264.2.c.m.577.1		4
136.67	odd	2		1632.2.c.c.577.4	yes	4
136.101	even	2		1632.2.c.c.577.2	yes	4
408.101	odd	2		4896.2.c.g.577.2		4
408.203	even	2		4896.2.c.g.577.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1632.2.c.c.577.1	✓	4	8.3	odd	2
1632.2.c.c.577.2	yes	4	136.101	even	2
1632.2.c.c.577.3	yes	4	8.5	even	2
1632.2.c.c.577.4	yes	4	136.67	odd	2
3264.2.c.m.577.1		4	68.67	odd	2	inner
3264.2.c.m.577.2		4	1.1	even	1	trivial
3264.2.c.m.577.3		4	17.16	even	2	inner
3264.2.c.m.577.4		4	4.3	odd	2	inner
4896.2.c.g.577.1		4	408.203	even	2
4896.2.c.g.577.2		4	408.101	odd	2
4896.2.c.g.577.3		4	24.5	odd	2
4896.2.c.g.577.4		4	24.11	even	2

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:	\( N \)	\(=\)	\( 3264 = 2^{6} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3264.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.60555i q^{5} -4.00000i q^{7} -1.00000 q^{9} +3.00000i q^{11} +1.00000 q^{13} +3.60555 q^{15} +(2.00000 + 3.60555i) q^{17} +3.60555 q^{19} -4.00000 q^{21} -7.00000i q^{23} -8.00000 q^{25} +1.00000i q^{27} -7.21110i q^{29} +2.00000i q^{31} +3.00000 q^{33} +14.4222 q^{35} +7.21110i q^{37} -1.00000i q^{39} +10.8167i q^{41} -3.60555 q^{43} -3.60555i q^{45} -9.00000 q^{49} +(3.60555 - 2.00000i) q^{51} +4.00000 q^{53} -10.8167 q^{55} -3.60555i q^{57} +14.4222 q^{59} +4.00000i q^{63} +3.60555i q^{65} -7.00000 q^{69} +8.00000i q^{71} +8.00000i q^{75} +12.0000 q^{77} -4.00000i q^{79} +1.00000 q^{81} +(-13.0000 + 7.21110i) q^{85} -7.21110 q^{87} +12.0000 q^{89} -4.00000i q^{91} +2.00000 q^{93} +13.0000i q^{95} -7.21110i q^{97} -3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} + 4 q^{13} + 8 q^{17} - 16 q^{21} - 32 q^{25} + 12 q^{33} - 36 q^{49} + 16 q^{53} - 28 q^{69} + 48 q^{77} + 4 q^{81} - 52 q^{85} + 48 q^{89} + 8 q^{93}+O(q^{100}) \) 4 * q - 4 * q^9 + 4 * q^13 + 8 * q^17 - 16 * q^21 - 32 * q^25 + 12 * q^33 - 36 * q^49 + 16 * q^53 - 28 * q^69 + 48 * q^77 + 4 * q^81 - 52 * q^85 + 48 * q^89 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more