LMFDB - Embedded newform 3264.2.c.a.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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 204)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3264.577
Dual form			3264.2.c.a.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} -2.00000i q^{7} -1.00000 q^{9} +3.00000i q^{11} -3.00000 q^{13} -1.00000 q^{15} +(-4.00000 + 1.00000i) q^{17} -3.00000 q^{19} +2.00000 q^{21} -3.00000i q^{23} +4.00000 q^{25} -1.00000i q^{27} -6.00000i q^{29} +2.00000i q^{31} -3.00000 q^{33} +2.00000 q^{35} -10.0000i q^{37} -3.00000i q^{39} +3.00000i q^{41} -1.00000 q^{43} -1.00000i q^{45} +2.00000 q^{47} +3.00000 q^{49} +(-1.00000 - 4.00000i) q^{51} +6.00000 q^{53} -3.00000 q^{55} -3.00000i q^{57} -10.0000i q^{61} +2.00000i q^{63} -3.00000i q^{65} +4.00000 q^{67} +3.00000 q^{69} +8.00000i q^{71} -14.0000i q^{73} +4.00000i q^{75} +6.00000 q^{77} -12.0000i q^{79} +1.00000 q^{81} -14.0000 q^{83} +(-1.00000 - 4.00000i) q^{85} +6.00000 q^{87} -2.00000 q^{89} +6.00000i q^{91} -2.00000 q^{93} -3.00000i q^{95} -3.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 6 q^{13} - 2 q^{15} - 8 q^{17} - 6 q^{19} + 4 q^{21} + 8 q^{25} - 6 q^{33} + 4 q^{35} - 2 q^{43} + 4 q^{47} + 6 q^{49} - 2 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{67} + 6 q^{69} + 12 q^{77} + 2 q^{81} - 28 q^{83} - 2 q^{85} + 12 q^{87} - 4 q^{89} - 4 q^{93}+O(q^{100}) $ 2 * q - 2 * q^9 - 6 * q^13 - 2 * q^15 - 8 * q^17 - 6 * q^19 + 4 * q^21 + 8 * q^25 - 6 * q^33 + 4 * q^35 - 2 * q^43 + 4 * q^47 + 6 * q^49 - 2 * q^51 + 12 * q^53 - 6 * q^55 + 8 * q^67 + 6 * q^69 + 12 * q^77 + 2 * q^81 - 28 * q^83 - 2 * q^85 + 12 * q^87 - 4 * q^89 - 4 * q^93

Display 100 coefficients 10 coefficients

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
$5$			1.00000i			0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$			1.00000i			0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		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
$13$	−3.00000			−0.832050			−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000			−0.832050			−0.416025	+	0.909353i	$0.636577\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	−4.00000	+	1.00000i	−0.970143	+	0.242536i
$17$	−4.00000	+	1.00000i	−0.970143	+	0.242536i
$18$		0			0
$19$	−3.00000			−0.688247			−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000			−0.688247			−0.344124	+	0.938924i	$0.611824\pi$
$20$		0			0
$21$	2.00000			0.436436
$22$		0			0
$23$		−	3.00000i		−	0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$		−	3.00000i		−	0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		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
$35$	2.00000			0.338062
$36$		0			0
$37$		−	10.0000i		−	1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$		−	10.0000i		−	1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$		0			0
$39$		−	3.00000i		−	0.480384i
$40$		0			0
$41$			3.00000i			0.468521i	0.972174	+	0.234261i	$0.0752669\pi$
$41$			3.00000i			0.468521i	−0.972174	+	0.234261i	$0.924733\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$	2.00000			0.291730			0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000			0.291730			0.145865	+	0.989305i	$0.453403\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	−1.00000	−	4.00000i	−0.140028	−	0.560112i
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$		0			0
$57$		−	3.00000i		−	0.397360i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		−	10.0000i		−	1.28037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$		−	10.0000i		−	1.28037i	0.768221	−	0.640184i	$-0.221142\pi$
$62$		0			0
$63$			2.00000i			0.251976i
$64$		0			0
$65$		−	3.00000i		−	0.372104i
$66$		0			0
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$		0			0
$69$	3.00000			0.361158
$70$		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
$73$		−	14.0000i		−	1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$		−	14.0000i		−	1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$		0			0
$75$			4.00000i			0.461880i
$76$		0			0
$77$	6.00000			0.683763
$78$		0			0
$79$		−	12.0000i		−	1.35011i	−0.737769	−	0.675053i	$-0.764121\pi$
$79$		−	12.0000i		−	1.35011i	0.737769	−	0.675053i	$-0.235879\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−14.0000			−1.53670			−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000			−1.53670			−0.768350	+	0.640030i	$0.778922\pi$
$84$		0			0
$85$	−1.00000	−	4.00000i	−0.108465	−	0.433861i
$86$		0			0
$87$	6.00000			0.643268
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$			6.00000i			0.628971i
$92$		0			0
$93$	−2.00000			−0.207390
$94$		0			0
$95$		−	3.00000i		−	0.307794i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		−	3.00000i		−	0.301511i
$100$		0			0
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$	−13.0000			−1.28093			−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000			−1.28093			−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$			2.00000i			0.195180i
$106$		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
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$	10.0000			0.949158
$112$		0			0
$113$		−	19.0000i		−	1.78737i	−0.448695	−	0.893685i	$-0.648111\pi$
$113$		−	19.0000i		−	1.78737i	0.448695	−	0.893685i	$-0.351889\pi$
$114$		0			0
$115$	3.00000			0.279751
$116$		0			0
$117$	3.00000			0.277350
$118$		0			0
$119$	2.00000	+	8.00000i	0.183340	+	0.733359i
$120$		0			0
$121$	2.00000			0.181818
$122$		0			0
$123$	−3.00000			−0.270501
$124$		0			0
$125$			9.00000i			0.804984i
$126$		0			0
$127$	−13.0000			−1.15356			−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000			−1.15356			−0.576782	+	0.816898i	$0.695692\pi$
$128$		0			0
$129$		−	1.00000i		−	0.0880451i
$130$		0			0
$131$		−	21.0000i		−	1.83478i	−0.397991	−	0.917389i	$-0.630293\pi$
$131$		−	21.0000i		−	1.83478i	0.397991	−	0.917389i	$-0.369707\pi$
$132$		0			0
$133$			6.00000i			0.520266i
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	16.0000			1.36697			0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000			1.36697			0.683486	+	0.729964i	$0.260463\pi$
$138$		0			0
$139$		−	8.00000i		−	0.678551i	−0.940687	−	0.339276i	$-0.889818\pi$
$139$		−	8.00000i		−	0.678551i	0.940687	−	0.339276i	$-0.110182\pi$
$140$		0			0
$141$			2.00000i			0.168430i
$142$		0			0
$143$		−	9.00000i		−	0.752618i
$144$		0			0
$145$	6.00000			0.498273
$146$		0			0
$147$			3.00000i			0.247436i
$148$		0			0
$149$	8.00000			0.655386			0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000			0.655386			0.327693	+	0.944784i	$0.393729\pi$
$150$		0			0
$151$	−20.0000			−1.62758			−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000			−1.62758			−0.813788	+	0.581161i	$0.802599\pi$
$152$		0			0
$153$	4.00000	−	1.00000i	0.323381	−	0.0808452i
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	5.00000			0.399043			0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000			0.399043			0.199522	+	0.979893i	$0.436061\pi$
$158$		0			0
$159$			6.00000i			0.475831i
$160$		0			0
$161$	−6.00000			−0.472866
$162$		0			0
$163$		−	2.00000i		−	0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$		−	2.00000i		−	0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$		0			0
$165$		−	3.00000i		−	0.233550i
$166$		0			0
$167$			15.0000i			1.16073i	0.814355	+	0.580367i	$0.197091\pi$
$167$			15.0000i			1.16073i	−0.814355	+	0.580367i	$0.802909\pi$
$168$		0			0
$169$	−4.00000			−0.307692
$170$		0			0
$171$	3.00000			0.229416
$172$		0			0
$173$		−	19.0000i		−	1.44454i	−0.691609	−	0.722272i	$-0.743098\pi$
$173$		−	19.0000i		−	1.44454i	0.691609	−	0.722272i	$-0.256902\pi$
$174$		0			0
$175$		−	8.00000i		−	0.604743i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	26.0000			1.94333			0.971666	−	0.236360i	$-0.0759544\pi$
$179$	26.0000			1.94333			0.971666	+	0.236360i	$0.0759544\pi$
$180$		0			0
$181$		−	4.00000i		−	0.297318i	−0.988889	−	0.148659i	$-0.952504\pi$
$181$		−	4.00000i		−	0.297318i	0.988889	−	0.148659i	$-0.0474956\pi$
$182$		0			0
$183$	10.0000			0.739221
$184$		0			0
$185$	10.0000			0.735215
$186$		0			0
$187$	−3.00000	−	12.0000i	−0.219382	−	0.877527i
$188$		0			0
$189$	−2.00000			−0.145479
$190$		0			0
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$		0			0
$193$		−	22.0000i		−	1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$		−	22.0000i		−	1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$		0			0
$195$	3.00000			0.214834
$196$		0			0
$197$		−	21.0000i		−	1.49619i	−0.663593	−	0.748094i	$-0.730969\pi$
$197$		−	21.0000i		−	1.49619i	0.663593	−	0.748094i	$-0.269031\pi$
$198$		0			0
$199$			24.0000i			1.70131i	0.525720	+	0.850657i	$0.323796\pi$
$199$			24.0000i			1.70131i	−0.525720	+	0.850657i	$0.676204\pi$
$200$		0			0
$201$			4.00000i			0.282138i
$202$		0			0
$203$	−12.0000			−0.842235
$204$		0			0
$205$	−3.00000			−0.209529
$206$		0			0
$207$			3.00000i			0.208514i
$208$		0			0
$209$		−	9.00000i		−	0.622543i
$210$		0			0
$211$		−	6.00000i		−	0.413057i	−0.978441	−	0.206529i	$-0.933783\pi$
$211$		−	6.00000i		−	0.413057i	0.978441	−	0.206529i	$-0.0662166\pi$
$212$		0			0
$213$	−8.00000			−0.548151
$214$		0			0
$215$		−	1.00000i		−	0.0681994i
$216$		0			0
$217$	4.00000			0.271538
$218$		0			0
$219$	14.0000			0.946032
$220$		0			0
$221$	12.0000	−	3.00000i	0.807207	−	0.201802i
$222$		0			0
$223$	−1.00000			−0.0669650			−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000			−0.0669650			−0.0334825	+	0.999439i	$0.510660\pi$
$224$		0			0
$225$	−4.00000			−0.266667
$226$		0			0
$227$		−	3.00000i		−	0.199117i	−0.995032	−	0.0995585i	$-0.968257\pi$
$227$		−	3.00000i		−	0.199117i	0.995032	−	0.0995585i	$-0.0317430\pi$
$228$		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
$231$			6.00000i			0.394771i
$232$		0			0
$233$		−	19.0000i		−	1.24473i	−0.782727	−	0.622366i	$-0.786172\pi$
$233$		−	19.0000i		−	1.24473i	0.782727	−	0.622366i	$-0.213828\pi$
$234$		0			0
$235$			2.00000i			0.130466i
$236$		0			0
$237$	12.0000			0.779484
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$			14.0000i			0.901819i	0.892570	+	0.450910i	$0.148900\pi$
$241$			14.0000i			0.901819i	−0.892570	+	0.450910i	$0.851100\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$			3.00000i			0.191663i
$246$		0			0
$247$	9.00000			0.572656
$248$		0			0
$249$		−	14.0000i		−	0.887214i
$250$		0			0
$251$	30.0000			1.89358			0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000			1.89358			0.946792	+	0.321847i	$0.104304\pi$
$252$		0			0
$253$	9.00000			0.565825
$254$		0			0
$255$	4.00000	−	1.00000i	0.250490	−	0.0626224i
$256$		0			0
$257$	−26.0000			−1.62184			−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000			−1.62184			−0.810918	+	0.585160i	$0.801032\pi$
$258$		0			0
$259$	−20.0000			−1.24274
$260$		0			0
$261$			6.00000i			0.371391i
$262$		0			0
$263$	14.0000			0.863277			0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000			0.863277			0.431638	+	0.902047i	$0.357936\pi$
$264$		0			0
$265$			6.00000i			0.368577i
$266$		0			0
$267$		−	2.00000i		−	0.122398i
$268$		0			0
$269$			15.0000i			0.914566i	0.889321	+	0.457283i	$0.151177\pi$
$269$			15.0000i			0.914566i	−0.889321	+	0.457283i	$0.848823\pi$
$270$		0			0
$271$	−7.00000			−0.425220			−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000			−0.425220			−0.212610	+	0.977137i	$0.568196\pi$
$272$		0			0
$273$	−6.00000			−0.363137
$274$		0			0
$275$			12.0000i			0.723627i
$276$		0			0
$277$			10.0000i			0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$			10.0000i			0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$		0			0
$279$		−	2.00000i		−	0.119737i
$280$		0			0
$281$	−2.00000			−0.119310			−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000			−0.119310			−0.0596550	+	0.998219i	$0.519000\pi$
$282$		0			0
$283$			20.0000i			1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$			20.0000i			1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$		0			0
$285$	3.00000			0.177705
$286$		0			0
$287$	6.00000			0.354169
$288$		0			0
$289$	15.0000	−	8.00000i	0.882353	−	0.470588i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	3.00000			0.174078
$298$		0			0
$299$			9.00000i			0.520483i
$300$		0			0
$301$			2.00000i			0.115278i
$302$		0			0
$303$		−	12.0000i		−	0.689382i
$304$		0			0
$305$	10.0000			0.572598
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$		−	13.0000i		−	0.739544i
$310$		0			0
$311$		−	8.00000i		−	0.453638i	−0.973937	−	0.226819i	$-0.927167\pi$
$311$		−	8.00000i		−	0.453638i	0.973937	−	0.226819i	$-0.0728326\pi$
$312$		0			0
$313$			8.00000i			0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$			8.00000i			0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$		0			0
$315$	−2.00000			−0.112687
$316$		0			0
$317$		−	6.00000i		−	0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$		−	6.00000i		−	0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$		0			0
$319$	18.0000			1.00781
$320$		0			0
$321$	−13.0000			−0.725589
$322$		0			0
$323$	12.0000	−	3.00000i	0.667698	−	0.166924i
$324$		0			0
$325$	−12.0000			−0.665640
$326$		0			0
$327$	−4.00000			−0.221201
$328$		0			0
$329$		−	4.00000i		−	0.220527i
$330$		0			0
$331$	−15.0000			−0.824475			−0.412237	−	0.911077i	$-0.635253\pi$
$331$	−15.0000			−0.824475			−0.412237	+	0.911077i	$0.635253\pi$
$332$		0			0
$333$			10.0000i			0.547997i
$334$		0			0
$335$			4.00000i			0.218543i
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$		0			0
$339$	19.0000			1.03194
$340$		0			0
$341$	−6.00000			−0.324918
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$			3.00000i			0.161515i
$346$		0			0
$347$			24.0000i			1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$			24.0000i			1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$		0			0
$349$	−23.0000			−1.23116			−0.615581	−	0.788074i	$-0.711079\pi$
$349$	−23.0000			−1.23116			−0.615581	+	0.788074i	$0.711079\pi$
$350$		0			0
$351$			3.00000i			0.160128i
$352$		0			0
$353$	8.00000			0.425797			0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000			0.425797			0.212899	+	0.977074i	$0.431710\pi$
$354$		0			0
$355$	−8.00000			−0.424596
$356$		0			0
$357$	−8.00000	+	2.00000i	−0.423405	+	0.105851i
$358$		0			0
$359$	−30.0000			−1.58334			−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000			−1.58334			−0.791670	+	0.610949i	$0.790788\pi$
$360$		0			0
$361$	−10.0000			−0.526316
$362$		0			0
$363$			2.00000i			0.104973i
$364$		0			0
$365$	14.0000			0.732793
$366$		0			0
$367$		−	22.0000i		−	1.14839i	−0.818718	−	0.574195i	$-0.805315\pi$
$367$		−	22.0000i		−	1.14839i	0.818718	−	0.574195i	$-0.194685\pi$
$368$		0			0
$369$		−	3.00000i		−	0.156174i
$370$		0			0
$371$		−	12.0000i		−	0.623009i
$372$		0			0
$373$	18.0000			0.932005			0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000			0.932005			0.466002	+	0.884783i	$0.345694\pi$
$374$		0			0
$375$	−9.00000			−0.464758
$376$		0			0
$377$			18.0000i			0.927047i
$378$		0			0
$379$		−	10.0000i		−	0.513665i	−0.966456	−	0.256833i	$-0.917321\pi$
$379$		−	10.0000i		−	0.513665i	0.966456	−	0.256833i	$-0.0826790\pi$
$380$		0			0
$381$		−	13.0000i		−	0.666010i
$382$		0			0
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		0			0
$385$			6.00000i			0.305788i
$386$		0			0
$387$	1.00000			0.0508329
$388$		0			0
$389$	−26.0000			−1.31825			−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000			−1.31825			−0.659126	+	0.752032i	$0.729074\pi$
$390$		0			0
$391$	3.00000	+	12.0000i	0.151717	+	0.606866i
$392$		0			0
$393$	21.0000			1.05931
$394$		0			0
$395$	12.0000			0.603786
$396$		0			0
$397$		−	32.0000i		−	1.60603i	−0.595956	−	0.803017i	$-0.703227\pi$
$397$		−	32.0000i		−	1.60603i	0.595956	−	0.803017i	$-0.296773\pi$
$398$		0			0
$399$	−6.00000			−0.300376
$400$		0			0
$401$			15.0000i			0.749064i	0.927214	+	0.374532i	$0.122197\pi$
$401$			15.0000i			0.749064i	−0.927214	+	0.374532i	$0.877803\pi$
$402$		0			0
$403$		−	6.00000i		−	0.298881i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$	30.0000			1.48704
$408$		0			0
$409$	−7.00000			−0.346128			−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000			−0.346128			−0.173064	+	0.984911i	$0.555367\pi$
$410$		0			0
$411$			16.0000i			0.789222i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		−	14.0000i		−	0.687233i
$416$		0			0
$417$	8.00000			0.391762
$418$		0			0
$419$		−	16.0000i		−	0.781651i	−0.920465	−	0.390826i	$-0.872190\pi$
$419$		−	16.0000i		−	0.781651i	0.920465	−	0.390826i	$-0.127810\pi$
$420$		0			0
$421$	25.0000			1.21843			0.609213	−	0.793007i	$-0.291486\pi$
$421$	25.0000			1.21843			0.609213	+	0.793007i	$0.291486\pi$
$422$		0			0
$423$	−2.00000			−0.0972433
$424$		0			0
$425$	−16.0000	+	4.00000i	−0.776114	+	0.194029i
$426$		0			0
$427$	−20.0000			−0.967868
$428$		0			0
$429$	9.00000			0.434524
$430$		0			0
$431$		−	4.00000i		−	0.192673i	−0.995349	−	0.0963366i	$-0.969287\pi$
$431$		−	4.00000i		−	0.192673i	0.995349	−	0.0963366i	$-0.0307125\pi$
$432$		0			0
$433$	−11.0000			−0.528626			−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000			−0.528626			−0.264313	+	0.964437i	$0.585145\pi$
$434$		0			0
$435$			6.00000i			0.287678i
$436$		0			0
$437$			9.00000i			0.430528i
$438$		0			0
$439$			24.0000i			1.14546i	0.819745	+	0.572729i	$0.194115\pi$
$439$			24.0000i			1.14546i	−0.819745	+	0.572729i	$0.805885\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$	−22.0000			−1.04525			−0.522626	−	0.852562i	$-0.675047\pi$
$443$	−22.0000			−1.04525			−0.522626	+	0.852562i	$0.675047\pi$
$444$		0			0
$445$		−	2.00000i		−	0.0948091i
$446$		0			0
$447$			8.00000i			0.378387i
$448$		0			0
$449$			30.0000i			1.41579i	0.706319	+	0.707894i	$0.250354\pi$
$449$			30.0000i			1.41579i	−0.706319	+	0.707894i	$0.749646\pi$
$450$		0			0
$451$	−9.00000			−0.423793
$452$		0			0
$453$		−	20.0000i		−	0.939682i
$454$		0			0
$455$	−6.00000			−0.281284
$456$		0			0
$457$	−25.0000			−1.16945			−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000			−1.16945			−0.584725	+	0.811231i	$0.698798\pi$
$458$		0			0
$459$	1.00000	+	4.00000i	0.0466760	+	0.186704i
$460$		0			0
$461$	16.0000			0.745194			0.372597	−	0.927993i	$-0.378467\pi$
$461$	16.0000			0.745194			0.372597	+	0.927993i	$0.378467\pi$
$462$		0			0
$463$	−24.0000			−1.11537			−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000			−1.11537			−0.557687	+	0.830051i	$0.688311\pi$
$464$		0			0
$465$		−	2.00000i		−	0.0927478i
$466$		0			0
$467$	20.0000			0.925490			0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000			0.925490			0.462745	+	0.886492i	$0.346865\pi$
$468$		0			0
$469$		−	8.00000i		−	0.369406i
$470$		0			0
$471$			5.00000i			0.230388i
$472$		0			0
$473$		−	3.00000i		−	0.137940i
$474$		0			0
$475$	−12.0000			−0.550598
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$		−	25.0000i		−	1.14228i	−0.820853	−	0.571140i	$-0.806501\pi$
$479$		−	25.0000i		−	1.14228i	0.820853	−	0.571140i	$-0.193499\pi$
$480$		0			0
$481$			30.0000i			1.36788i
$482$		0			0
$483$		−	6.00000i		−	0.273009i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	2.00000			0.0904431
$490$		0			0
$491$	−30.0000			−1.35388			−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000			−1.35388			−0.676941	+	0.736038i	$0.736695\pi$
$492$		0			0
$493$	6.00000	+	24.0000i	0.270226	+	1.08091i
$494$		0			0
$495$	3.00000			0.134840
$496$		0			0
$497$	16.0000			0.717698
$498$		0			0
$499$		−	18.0000i		−	0.805791i	−0.915246	−	0.402895i	$-0.868004\pi$
$499$		−	18.0000i		−	0.805791i	0.915246	−	0.402895i	$-0.131996\pi$
$500$		0			0
$501$	−15.0000			−0.670151
$502$		0			0
$503$		−	13.0000i		−	0.579641i	−0.957081	−	0.289821i	$-0.906404\pi$
$503$		−	13.0000i		−	0.579641i	0.957081	−	0.289821i	$-0.0935957\pi$
$504$		0			0
$505$		−	12.0000i		−	0.533993i
$506$		0			0
$507$		−	4.00000i		−	0.177646i
$508$		0			0
$509$	−16.0000			−0.709188			−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000			−0.709188			−0.354594	+	0.935020i	$0.615381\pi$
$510$		0			0
$511$	−28.0000			−1.23865
$512$		0			0
$513$			3.00000i			0.132453i
$514$		0			0
$515$		−	13.0000i		−	0.572848i
$516$		0			0
$517$			6.00000i			0.263880i
$518$		0			0
$519$	19.0000			0.834007
$520$		0			0
$521$			15.0000i			0.657162i	0.944476	+	0.328581i	$0.106570\pi$
$521$			15.0000i			0.657162i	−0.944476	+	0.328581i	$0.893430\pi$
$522$		0			0
$523$	−20.0000			−0.874539			−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000			−0.874539			−0.437269	+	0.899331i	$0.644054\pi$
$524$		0			0
$525$	8.00000			0.349149
$526$		0			0
$527$	−2.00000	−	8.00000i	−0.0871214	−	0.348485i
$528$		0			0
$529$	14.0000			0.608696
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	9.00000i		−	0.389833i
$534$		0			0
$535$	−13.0000			−0.562039
$536$		0			0
$537$			26.0000i			1.12198i
$538$		0			0
$539$			9.00000i			0.387657i
$540$		0			0
$541$			46.0000i			1.97769i	0.148933	+	0.988847i	$0.452416\pi$
$541$			46.0000i			1.97769i	−0.148933	+	0.988847i	$0.547584\pi$
$542$		0			0
$543$	4.00000			0.171656
$544$		0			0
$545$	−4.00000			−0.171341
$546$		0			0
$547$			22.0000i			0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$			22.0000i			0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$		0			0
$549$			10.0000i			0.426790i
$550$		0			0
$551$			18.0000i			0.766826i
$552$		0			0
$553$	−24.0000			−1.02058
$554$		0			0
$555$			10.0000i			0.424476i
$556$		0			0
$557$	−20.0000			−0.847427			−0.423714	−	0.905796i	$-0.639274\pi$
$557$	−20.0000			−0.847427			−0.423714	+	0.905796i	$0.639274\pi$
$558$		0			0
$559$	3.00000			0.126886
$560$		0			0
$561$	12.0000	−	3.00000i	0.506640	−	0.126660i
$562$		0			0
$563$	−18.0000			−0.758610			−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000			−0.758610			−0.379305	+	0.925272i	$0.623837\pi$
$564$		0			0
$565$	19.0000			0.799336
$566$		0			0
$567$		−	2.00000i		−	0.0839921i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$			28.0000i			1.17176i	0.810397	+	0.585882i	$0.199252\pi$
$571$			28.0000i			1.17176i	−0.810397	+	0.585882i	$0.800748\pi$
$572$		0			0
$573$			16.0000i			0.668410i
$574$		0			0
$575$		−	12.0000i		−	0.500435i
$576$		0			0
$577$	−35.0000			−1.45707			−0.728535	−	0.685009i	$-0.759798\pi$
$577$	−35.0000			−1.45707			−0.728535	+	0.685009i	$0.759798\pi$
$578$		0			0
$579$	22.0000			0.914289
$580$		0			0
$581$			28.0000i			1.16164i
$582$		0			0
$583$			18.0000i			0.745484i
$584$		0			0
$585$			3.00000i			0.124035i
$586$		0			0
$587$	36.0000			1.48588			0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000			1.48588			0.742940	+	0.669359i	$0.233431\pi$
$588$		0			0
$589$		−	6.00000i		−	0.247226i
$590$		0			0
$591$	21.0000			0.863825
$592$		0			0
$593$	38.0000			1.56047			0.780236	−	0.625485i	$-0.215099\pi$
$593$	38.0000			1.56047			0.780236	+	0.625485i	$0.215099\pi$
$594$		0			0
$595$	−8.00000	+	2.00000i	−0.327968	+	0.0819920i
$596$		0			0
$597$	−24.0000			−0.982255
$598$		0			0
$599$	−4.00000			−0.163436			−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000			−0.163436			−0.0817178	+	0.996656i	$0.526041\pi$
$600$		0			0
$601$		−	22.0000i		−	0.897399i	−0.893683	−	0.448699i	$-0.851887\pi$
$601$		−	22.0000i		−	0.897399i	0.893683	−	0.448699i	$-0.148113\pi$
$602$		0			0
$603$	−4.00000			−0.162893
$604$		0			0
$605$			2.00000i			0.0813116i
$606$		0			0
$607$		−	28.0000i		−	1.13648i	−0.822861	−	0.568242i	$-0.807624\pi$
$607$		−	28.0000i		−	1.13648i	0.822861	−	0.568242i	$-0.192376\pi$
$608$		0			0
$609$		−	12.0000i		−	0.486265i
$610$		0			0
$611$	−6.00000			−0.242734
$612$		0			0
$613$	25.0000			1.00974			0.504870	−	0.863195i	$-0.331540\pi$
$613$	25.0000			1.00974			0.504870	+	0.863195i	$0.331540\pi$
$614$		0			0
$615$		−	3.00000i		−	0.120972i
$616$		0			0
$617$			42.0000i			1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$			42.0000i			1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$		0			0
$619$			20.0000i			0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$			20.0000i			0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$		0			0
$621$	−3.00000			−0.120386
$622$		0			0
$623$			4.00000i			0.160257i
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$	9.00000			0.359425
$628$		0			0
$629$	10.0000	+	40.0000i	0.398726	+	1.59490i
$630$		0			0
$631$	−29.0000			−1.15447			−0.577236	−	0.816577i	$-0.695869\pi$
$631$	−29.0000			−1.15447			−0.577236	+	0.816577i	$0.695869\pi$
$632$		0			0
$633$	6.00000			0.238479
$634$		0			0
$635$		−	13.0000i		−	0.515889i
$636$		0			0
$637$	−9.00000			−0.356593
$638$		0			0
$639$		−	8.00000i		−	0.316475i
$640$		0			0
$641$		−	37.0000i		−	1.46141i	−0.682692	−	0.730706i	$-0.739191\pi$
$641$		−	37.0000i		−	1.46141i	0.682692	−	0.730706i	$-0.260809\pi$
$642$		0			0
$643$		−	36.0000i		−	1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$643$		−	36.0000i		−	1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$644$		0			0
$645$	1.00000			0.0393750
$646$		0			0
$647$	16.0000			0.629025			0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000			0.629025			0.314512	+	0.949253i	$0.398159\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$			4.00000i			0.156772i
$652$		0			0
$653$			7.00000i			0.273931i	0.990576	+	0.136966i	$0.0437350\pi$
$653$			7.00000i			0.273931i	−0.990576	+	0.136966i	$0.956265\pi$
$654$		0			0
$655$	21.0000			0.820538
$656$		0			0
$657$			14.0000i			0.546192i
$658$		0			0
$659$	28.0000			1.09073			0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000			1.09073			0.545363	+	0.838200i	$0.316392\pi$
$660$		0			0
$661$	17.0000			0.661223			0.330612	−	0.943767i	$-0.392745\pi$
$661$	17.0000			0.661223			0.330612	+	0.943767i	$0.392745\pi$
$662$		0			0
$663$	3.00000	+	12.0000i	0.116510	+	0.466041i
$664$		0			0
$665$	−6.00000			−0.232670
$666$		0			0
$667$	−18.0000			−0.696963
$668$		0			0
$669$		−	1.00000i		−	0.0386622i
$670$		0			0
$671$	30.0000			1.15814
$672$		0			0
$673$		−	6.00000i		−	0.231283i	−0.993291	−	0.115642i	$-0.963108\pi$
$673$		−	6.00000i		−	0.231283i	0.993291	−	0.115642i	$-0.0368924\pi$
$674$		0			0
$675$		−	4.00000i		−	0.153960i
$676$		0			0
$677$			15.0000i			0.576497i	0.957556	+	0.288248i	$0.0930729\pi$
$677$			15.0000i			0.576497i	−0.957556	+	0.288248i	$0.906927\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	3.00000			0.114960
$682$		0			0
$683$			11.0000i			0.420903i	0.977604	+	0.210452i	$0.0674935\pi$
$683$			11.0000i			0.420903i	−0.977604	+	0.210452i	$0.932507\pi$
$684$		0			0
$685$			16.0000i			0.611329i
$686$		0			0
$687$		−	10.0000i		−	0.381524i
$688$		0			0
$689$	−18.0000			−0.685745
$690$		0			0
$691$		−	44.0000i		−	1.67384i	−0.547326	−	0.836919i	$-0.684354\pi$
$691$		−	44.0000i		−	1.67384i	0.547326	−	0.836919i	$-0.315646\pi$
$692$		0			0
$693$	−6.00000			−0.227921
$694$		0			0
$695$	8.00000			0.303457
$696$		0			0
$697$	−3.00000	−	12.0000i	−0.113633	−	0.454532i
$698$		0			0
$699$	19.0000			0.718646
$700$		0			0
$701$	−40.0000			−1.51078			−0.755390	−	0.655276i	$-0.772552\pi$
$701$	−40.0000			−1.51078			−0.755390	+	0.655276i	$0.772552\pi$
$702$		0			0
$703$			30.0000i			1.13147i
$704$		0			0
$705$	−2.00000			−0.0753244
$706$		0			0
$707$			24.0000i			0.902613i
$708$		0			0
$709$			18.0000i			0.676004i	0.941145	+	0.338002i	$0.109751\pi$
$709$			18.0000i			0.676004i	−0.941145	+	0.338002i	$0.890249\pi$
$710$		0			0
$711$			12.0000i			0.450035i
$712$		0			0
$713$	6.00000			0.224702
$714$		0			0
$715$	9.00000			0.336581
$716$		0			0
$717$			12.0000i			0.448148i
$718$		0			0
$719$			37.0000i			1.37987i	0.723873	+	0.689934i	$0.242360\pi$
$719$			37.0000i			1.37987i	−0.723873	+	0.689934i	$0.757640\pi$
$720$		0			0
$721$			26.0000i			0.968291i
$722$		0			0
$723$	−14.0000			−0.520666
$724$		0			0
$725$		−	24.0000i		−	0.891338i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	4.00000	−	1.00000i	0.147945	−	0.0369863i
$732$		0			0
$733$	−14.0000			−0.517102			−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000			−0.517102			−0.258551	+	0.965998i	$0.583245\pi$
$734$		0			0
$735$	−3.00000			−0.110657
$736$		0			0
$737$			12.0000i			0.442026i
$738$		0			0
$739$	−11.0000			−0.404642			−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000			−0.404642			−0.202321	+	0.979319i	$0.564848\pi$
$740$		0			0
$741$			9.00000i			0.330623i
$742$		0			0
$743$			32.0000i			1.17397i	0.809599	+	0.586983i	$0.199684\pi$
$743$			32.0000i			1.17397i	−0.809599	+	0.586983i	$0.800316\pi$
$744$		0			0
$745$			8.00000i			0.293097i
$746$		0			0
$747$	14.0000			0.512233
$748$		0			0
$749$	26.0000			0.950019
$750$		0			0
$751$		−	50.0000i		−	1.82453i	−0.409605	−	0.912263i	$-0.634333\pi$
$751$		−	50.0000i		−	1.82453i	0.409605	−	0.912263i	$-0.365667\pi$
$752$		0			0
$753$			30.0000i			1.09326i
$754$		0			0
$755$		−	20.0000i		−	0.727875i
$756$		0			0
$757$	25.0000			0.908640			0.454320	−	0.890838i	$-0.349882\pi$
$757$	25.0000			0.908640			0.454320	+	0.890838i	$0.349882\pi$
$758$		0			0
$759$			9.00000i			0.326679i
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$	8.00000			0.289619
$764$		0			0
$765$	1.00000	+	4.00000i	0.0361551	+	0.144620i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	21.0000			0.757279			0.378640	−	0.925544i	$-0.376392\pi$
$769$	21.0000			0.757279			0.378640	+	0.925544i	$0.376392\pi$
$770$		0			0
$771$		−	26.0000i		−	0.936367i
$772$		0			0
$773$	−20.0000			−0.719350			−0.359675	−	0.933078i	$-0.617112\pi$
$773$	−20.0000			−0.719350			−0.359675	+	0.933078i	$0.617112\pi$
$774$		0			0
$775$			8.00000i			0.287368i
$776$		0			0
$777$		−	20.0000i		−	0.717496i
$778$		0			0
$779$		−	9.00000i		−	0.322458i
$780$		0			0
$781$	−24.0000			−0.858788
$782$		0			0
$783$	−6.00000			−0.214423
$784$		0			0
$785$			5.00000i			0.178458i
$786$		0			0
$787$		−	28.0000i		−	0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$		−	28.0000i		−	0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$		0			0
$789$			14.0000i			0.498413i
$790$		0			0
$791$	−38.0000			−1.35112
$792$		0			0
$793$			30.0000i			1.06533i
$794$		0			0
$795$	−6.00000			−0.212798
$796$		0			0
$797$	12.0000			0.425062			0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000			0.425062			0.212531	+	0.977154i	$0.431829\pi$
$798$		0			0
$799$	−8.00000	+	2.00000i	−0.283020	+	0.0707549i
$800$		0			0
$801$	2.00000			0.0706665
$802$		0			0
$803$	42.0000			1.48215
$804$		0			0
$805$		−	6.00000i		−	0.211472i
$806$		0			0
$807$	−15.0000			−0.528025
$808$		0			0
$809$		−	21.0000i		−	0.738321i	−0.929366	−	0.369160i	$-0.879645\pi$
$809$		−	21.0000i		−	0.738321i	0.929366	−	0.369160i	$-0.120355\pi$
$810$		0			0
$811$		−	10.0000i		−	0.351147i	−0.984466	−	0.175574i	$-0.943822\pi$
$811$		−	10.0000i		−	0.351147i	0.984466	−	0.175574i	$-0.0561780\pi$
$812$		0			0
$813$		−	7.00000i		−	0.245501i
$814$		0			0
$815$	2.00000			0.0700569
$816$		0			0
$817$	3.00000			0.104957
$818$		0			0
$819$		−	6.00000i		−	0.209657i
$820$		0			0
$821$			13.0000i			0.453703i	0.973929	+	0.226852i	$0.0728432\pi$
$821$			13.0000i			0.453703i	−0.973929	+	0.226852i	$0.927157\pi$
$822$		0			0
$823$			22.0000i			0.766872i	0.923567	+	0.383436i	$0.125259\pi$
$823$			22.0000i			0.766872i	−0.923567	+	0.383436i	$0.874741\pi$
$824$		0			0
$825$	−12.0000			−0.417786
$826$		0			0
$827$			29.0000i			1.00843i	0.863579	+	0.504214i	$0.168218\pi$
$827$			29.0000i			1.00843i	−0.863579	+	0.504214i	$0.831782\pi$
$828$		0			0
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$	−10.0000			−0.346896
$832$		0			0
$833$	−12.0000	+	3.00000i	−0.415775	+	0.103944i
$834$		0			0
$835$	−15.0000			−0.519096
$836$		0			0
$837$	2.00000			0.0691301
$838$		0			0
$839$			43.0000i			1.48452i	0.670109	+	0.742262i	$0.266247\pi$
$839$			43.0000i			1.48452i	−0.670109	+	0.742262i	$0.733753\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		−	2.00000i		−	0.0688837i
$844$		0			0
$845$		−	4.00000i		−	0.137604i
$846$		0			0
$847$		−	4.00000i		−	0.137442i
$848$		0			0
$849$	−20.0000			−0.686398
$850$		0			0
$851$	−30.0000			−1.02839
$852$		0			0
$853$			20.0000i			0.684787i	0.939557	+	0.342393i	$0.111238\pi$
$853$			20.0000i			0.684787i	−0.939557	+	0.342393i	$0.888762\pi$
$854$		0			0
$855$			3.00000i			0.102598i
$856$		0			0
$857$			18.0000i			0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$			18.0000i			0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$		0			0
$859$	−44.0000			−1.50126			−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000			−1.50126			−0.750630	+	0.660722i	$0.770250\pi$
$860$		0			0
$861$			6.00000i			0.204479i
$862$		0			0
$863$	−10.0000			−0.340404			−0.170202	−	0.985409i	$-0.554442\pi$
$863$	−10.0000			−0.340404			−0.170202	+	0.985409i	$0.554442\pi$
$864$		0			0
$865$	19.0000			0.646019
$866$		0			0
$867$	8.00000	+	15.0000i	0.271694	+	0.509427i
$868$		0			0
$869$	36.0000			1.22122
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$		0			0
$874$		0			0
$875$	18.0000			0.608511
$876$		0			0
$877$		−	8.00000i		−	0.270141i	−0.990836	−	0.135070i	$-0.956874\pi$
$877$		−	8.00000i		−	0.270141i	0.990836	−	0.135070i	$-0.0431261\pi$
$878$		0			0
$879$		−	6.00000i		−	0.202375i
$880$		0			0
$881$			2.00000i			0.0673817i	0.999432	+	0.0336909i	$0.0107262\pi$
$881$			2.00000i			0.0673817i	−0.999432	+	0.0336909i	$0.989274\pi$
$882$		0			0
$883$	43.0000			1.44707			0.723533	−	0.690290i	$-0.242517\pi$
$883$	43.0000			1.44707			0.723533	+	0.690290i	$0.242517\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	15.0000i		−	0.503651i	−0.967773	−	0.251825i	$-0.918969\pi$
$887$		−	15.0000i		−	0.503651i	0.967773	−	0.251825i	$-0.0810309\pi$
$888$		0			0
$889$			26.0000i			0.872012i
$890$		0			0
$891$			3.00000i			0.100504i
$892$		0			0
$893$	−6.00000			−0.200782
$894$		0			0
$895$			26.0000i			0.869084i
$896$		0			0
$897$	−9.00000			−0.300501
$898$		0			0
$899$	12.0000			0.400222
$900$		0			0
$901$	−24.0000	+	6.00000i	−0.799556	+	0.199889i
$902$		0			0
$903$	−2.00000			−0.0665558
$904$		0			0
$905$	4.00000			0.132964
$906$		0			0
$907$		−	8.00000i		−	0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$		−	8.00000i		−	0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$		0			0
$909$	12.0000			0.398015
$910$		0			0
$911$		−	31.0000i		−	1.02708i	−0.858067	−	0.513538i	$-0.828335\pi$
$911$		−	31.0000i		−	1.02708i	0.858067	−	0.513538i	$-0.171665\pi$
$912$		0			0
$913$		−	42.0000i		−	1.39000i
$914$		0			0
$915$			10.0000i			0.330590i
$916$		0			0
$917$	−42.0000			−1.38696
$918$		0			0
$919$	45.0000			1.48441			0.742207	−	0.670171i	$-0.233779\pi$
$919$	45.0000			1.48441			0.742207	+	0.670171i	$0.233779\pi$
$920$		0			0
$921$		−	12.0000i		−	0.395413i
$922$		0			0
$923$		−	24.0000i		−	0.789970i
$924$		0			0
$925$		−	40.0000i		−	1.31519i
$926$		0			0
$927$	13.0000			0.426976
$928$		0			0
$929$			5.00000i			0.164045i	0.996630	+	0.0820223i	$0.0261379\pi$
$929$			5.00000i			0.164045i	−0.996630	+	0.0820223i	$0.973862\pi$
$930$		0			0
$931$	−9.00000			−0.294963
$932$		0			0
$933$	8.00000			0.261908
$934$		0			0
$935$	12.0000	−	3.00000i	0.392442	−	0.0981105i
$936$		0			0
$937$	22.0000			0.718709			0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000			0.718709			0.359354	+	0.933201i	$0.382997\pi$
$938$		0			0
$939$	−8.00000			−0.261070
$940$		0			0
$941$			30.0000i			0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$			30.0000i			0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$		0			0
$943$	9.00000			0.293080
$944$		0			0
$945$		−	2.00000i		−	0.0650600i
$946$		0			0
$947$			16.0000i			0.519930i	0.965618	+	0.259965i	$0.0837111\pi$
$947$			16.0000i			0.519930i	−0.965618	+	0.259965i	$0.916289\pi$
$948$		0			0
$949$			42.0000i			1.36338i
$950$		0			0
$951$	6.00000			0.194563
$952$		0			0
$953$	12.0000			0.388718			0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000			0.388718			0.194359	+	0.980930i	$0.437737\pi$
$954$		0			0
$955$			16.0000i			0.517748i
$956$		0			0
$957$			18.0000i			0.581857i
$958$		0			0
$959$		−	32.0000i		−	1.03333i
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$		−	13.0000i		−	0.418919i
$964$		0			0
$965$	22.0000			0.708205
$966$		0			0
$967$	−19.0000			−0.610999			−0.305499	−	0.952192i	$-0.598823\pi$
$967$	−19.0000			−0.610999			−0.305499	+	0.952192i	$0.598823\pi$
$968$		0			0
$969$	3.00000	+	12.0000i	0.0963739	+	0.385496i
$970$		0			0
$971$	−38.0000			−1.21948			−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000			−1.21948			−0.609739	+	0.792602i	$0.708726\pi$
$972$		0			0
$973$	−16.0000			−0.512936
$974$		0			0
$975$		−	12.0000i		−	0.384308i
$976$		0			0
$977$	−40.0000			−1.27971			−0.639857	−	0.768494i	$-0.721006\pi$
$977$	−40.0000			−1.27971			−0.639857	+	0.768494i	$0.721006\pi$
$978$		0			0
$979$		−	6.00000i		−	0.191761i
$980$		0			0
$981$		−	4.00000i		−	0.127710i
$982$		0			0
$983$			1.00000i			0.0318950i	0.999873	+	0.0159475i	$0.00507647\pi$
$983$			1.00000i			0.0318950i	−0.999873	+	0.0159475i	$0.994924\pi$
$984$		0			0
$985$	21.0000			0.669116
$986$		0			0
$987$	4.00000			0.127321
$988$		0			0
$989$			3.00000i			0.0953945i
$990$		0			0
$991$		−	8.00000i		−	0.254128i	−0.991894	−	0.127064i	$-0.959445\pi$
$991$		−	8.00000i		−	0.254128i	0.991894	−	0.127064i	$-0.0405554\pi$
$992$		0			0
$993$		−	15.0000i		−	0.476011i
$994$		0			0
$995$	−24.0000			−0.760851
$996$		0			0
$997$			12.0000i			0.380044i	0.981780	+	0.190022i	$0.0608559\pi$
$997$			12.0000i			0.380044i	−0.981780	+	0.190022i	$0.939144\pi$
$998$		0			0
$999$	−10.0000			−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3264.2.c.a.577.2		2
4.3	odd	2		3264.2.c.b.577.1		2
8.3	odd	2		816.2.c.d.577.2		2
8.5	even	2		204.2.b.a.169.1	✓	2
17.16	even	2	inner	3264.2.c.a.577.1		2
24.5	odd	2		612.2.b.b.577.2		2
24.11	even	2		2448.2.c.o.577.2		2
40.13	odd	4		5100.2.k.d.4249.1		2
40.29	even	2		5100.2.e.b.1801.2		2
40.37	odd	4		5100.2.k.c.4249.1		2
68.67	odd	2		3264.2.c.b.577.2		2
136.13	even	4		3468.2.a.h.1.1		1
136.21	even	4		3468.2.a.b.1.1		1
136.53	even	8		3468.2.j.b.829.1		4
136.67	odd	2		816.2.c.d.577.1		2
136.77	even	8		3468.2.j.b.3217.2		4
136.93	even	8		3468.2.j.b.3217.1		4
136.101	even	2		204.2.b.a.169.2	yes	2
136.117	even	8		3468.2.j.b.829.2		4
408.101	odd	2		612.2.b.b.577.1		2
408.203	even	2		2448.2.c.o.577.1		2
680.237	odd	4		5100.2.k.d.4249.2		2
680.373	odd	4		5100.2.k.c.4249.2		2
680.509	even	2		5100.2.e.b.1801.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
204.2.b.a.169.1	✓	2	8.5	even	2
204.2.b.a.169.2	yes	2	136.101	even	2
612.2.b.b.577.1		2	408.101	odd	2
612.2.b.b.577.2		2	24.5	odd	2
816.2.c.d.577.1		2	136.67	odd	2
816.2.c.d.577.2		2	8.3	odd	2
2448.2.c.o.577.1		2	408.203	even	2
2448.2.c.o.577.2		2	24.11	even	2
3264.2.c.a.577.1		2	17.16	even	2	inner
3264.2.c.a.577.2		2	1.1	even	1	trivial
3264.2.c.b.577.1		2	4.3	odd	2
3264.2.c.b.577.2		2	68.67	odd	2
3468.2.a.b.1.1		1	136.21	even	4
3468.2.a.h.1.1		1	136.13	even	4
3468.2.j.b.829.1		4	136.53	even	8
3468.2.j.b.829.2		4	136.117	even	8
3468.2.j.b.3217.1		4	136.93	even	8
3468.2.j.b.3217.2		4	136.77	even	8
5100.2.e.b.1801.1		2	680.509	even	2
5100.2.e.b.1801.2		2	40.29	even	2
5100.2.k.c.4249.1		2	40.37	odd	4
5100.2.k.c.4249.2		2	680.373	odd	4
5100.2.k.d.4249.1		2	40.13	odd	4
5100.2.k.d.4249.2		2	680.237	odd	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} -2.00000i q^{7} -1.00000 q^{9} +3.00000i q^{11} -3.00000 q^{13} -1.00000 q^{15} +(-4.00000 + 1.00000i) q^{17} -3.00000 q^{19} +2.00000 q^{21} -3.00000i q^{23} +4.00000 q^{25} -1.00000i q^{27} -6.00000i q^{29} +2.00000i q^{31} -3.00000 q^{33} +2.00000 q^{35} -10.0000i q^{37} -3.00000i q^{39} +3.00000i q^{41} -1.00000 q^{43} -1.00000i q^{45} +2.00000 q^{47} +3.00000 q^{49} +(-1.00000 - 4.00000i) q^{51} +6.00000 q^{53} -3.00000 q^{55} -3.00000i q^{57} -10.0000i q^{61} +2.00000i q^{63} -3.00000i q^{65} +4.00000 q^{67} +3.00000 q^{69} +8.00000i q^{71} -14.0000i q^{73} +4.00000i q^{75} +6.00000 q^{77} -12.0000i q^{79} +1.00000 q^{81} -14.0000 q^{83} +(-1.00000 - 4.00000i) q^{85} +6.00000 q^{87} -2.00000 q^{89} +6.00000i q^{91} -2.00000 q^{93} -3.00000i q^{95} -3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 6 q^{13} - 2 q^{15} - 8 q^{17} - 6 q^{19} + 4 q^{21} + 8 q^{25} - 6 q^{33} + 4 q^{35} - 2 q^{43} + 4 q^{47} + 6 q^{49} - 2 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{67} + 6 q^{69} + 12 q^{77} + 2 q^{81} - 28 q^{83} - 2 q^{85} + 12 q^{87} - 4 q^{89} - 4 q^{93}+O(q^{100}) \) 2 * q - 2 * q^9 - 6 * q^13 - 2 * q^15 - 8 * q^17 - 6 * q^19 + 4 * q^21 + 8 * q^25 - 6 * q^33 + 4 * q^35 - 2 * q^43 + 4 * q^47 + 6 * q^49 - 2 * q^51 + 12 * q^53 - 6 * q^55 + 8 * q^67 + 6 * q^69 + 12 * q^77 + 2 * q^81 - 28 * q^83 - 2 * q^85 + 12 * q^87 - 4 * q^89 - 4 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more