LMFDB - Embedded newform 2720.2.l.b.2481.25

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2720,2,Mod(2481,2720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2720.2481"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2720 = 2^{5} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2720.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [36,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$21.7193093498$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 680)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2481.25
Character	$\chi$	$=$	2720.2481
Dual form			2720.2.l.b.2481.26

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46409 q^{3} +1.00000 q^{5} -1.02762i q^{7} -0.856436 q^{9} +0.737326 q^{11} -4.98026i q^{13} +1.46409 q^{15} +(3.59849 + 2.01268i) q^{17} -2.82434i q^{19} -1.50452i q^{21} -4.95563i q^{23} +1.00000 q^{25} -5.64618 q^{27} -4.86957 q^{29} -7.67421i q^{31} +1.07951 q^{33} -1.02762i q^{35} +3.22579 q^{37} -7.29156i q^{39} +4.45258i q^{41} +5.96478i q^{43} -0.856436 q^{45} -11.3771 q^{47} +5.94401 q^{49} +(5.26852 + 2.94675i) q^{51} -0.0517813i q^{53} +0.737326 q^{55} -4.13510i q^{57} -6.48292i q^{59} +14.4629 q^{61} +0.880087i q^{63} -4.98026i q^{65} -4.56989i q^{67} -7.25549i q^{69} -12.9933i q^{71} -0.800933i q^{73} +1.46409 q^{75} -0.757688i q^{77} +7.83068i q^{79} -5.69721 q^{81} -10.7517i q^{83} +(3.59849 + 2.01268i) q^{85} -7.12949 q^{87} -6.70231 q^{89} -5.11780 q^{91} -11.2357i q^{93} -2.82434i q^{95} +7.53723i q^{97} -0.631473 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 4 q^{3} + 36 q^{5} + 36 q^{9} + 8 q^{11} + 4 q^{15} + 36 q^{25} + 16 q^{27} - 8 q^{33} + 36 q^{45} - 20 q^{47} - 36 q^{49} + 8 q^{55} + 4 q^{75} + 44 q^{81} - 24 q^{87} + 56 q^{91} + 40 q^{99}+O(q^{100}) $ 36 * q + 4 * q^3 + 36 * q^5 + 36 * q^9 + 8 * q^11 + 4 * q^15 + 36 * q^25 + 16 * q^27 - 8 * q^33 + 36 * q^45 - 20 * q^47 - 36 * q^49 + 8 * q^55 + 4 * q^75 + 44 * q^81 - 24 * q^87 + 56 * q^91 + 40 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2720\mathbb{Z}\right)^\times$.

$n$	$511$	$1601$	$1701$	$2177$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	1.46409			0.845294			0.422647	−	0.906294i	$-0.361101\pi$
$3$	1.46409			0.845294			0.422647	+	0.906294i	$0.361101\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	1.02762i		−	0.388402i	−0.980962	−	0.194201i	$-0.937789\pi$
$7$		−	1.02762i		−	0.388402i	0.980962	−	0.194201i	$-0.0622114\pi$
$8$		0			0
$9$	−0.856436			−0.285479
$10$		0			0
$11$	0.737326			0.222312			0.111156	−	0.993803i	$-0.464545\pi$
$11$	0.737326			0.222312			0.111156	+	0.993803i	$0.464545\pi$
$12$		0			0
$13$		−	4.98026i		−	1.38128i	−0.723200	−	0.690638i	$-0.757330\pi$
$13$		−	4.98026i		−	1.38128i	0.723200	−	0.690638i	$-0.242670\pi$
$14$		0			0
$15$	1.46409			0.378027
$16$		0			0
$17$	3.59849	+	2.01268i	0.872762	+	0.488146i
$17$	3.59849	+	2.01268i	0.872762	+	0.488146i
$18$		0			0
$19$		−	2.82434i		−	0.647949i	−0.946066	−	0.323974i	$-0.894981\pi$
$19$		−	2.82434i		−	0.647949i	0.946066	−	0.323974i	$-0.105019\pi$
$20$		0			0
$21$		−	1.50452i		−	0.328314i
$22$		0			0
$23$		−	4.95563i		−	1.03332i	−0.856191	−	0.516660i	$-0.827175\pi$
$23$		−	4.95563i		−	1.03332i	0.856191	−	0.516660i	$-0.172825\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−5.64618			−1.08661
$28$		0			0
$29$	−4.86957			−0.904256			−0.452128	−	0.891953i	$-0.649335\pi$
$29$	−4.86957			−0.904256			−0.452128	+	0.891953i	$0.649335\pi$
$30$		0			0
$31$		−	7.67421i		−	1.37833i	−0.724605	−	0.689164i	$-0.757978\pi$
$31$		−	7.67421i		−	1.37833i	0.724605	−	0.689164i	$-0.242022\pi$
$32$		0			0
$33$	1.07951			0.187919
$34$		0			0
$35$		−	1.02762i		−	0.173699i
$36$		0			0
$37$	3.22579			0.530316			0.265158	−	0.964205i	$-0.414576\pi$
$37$	3.22579			0.530316			0.265158	+	0.964205i	$0.414576\pi$
$38$		0			0
$39$		−	7.29156i		−	1.16758i
$40$		0			0
$41$			4.45258i			0.695376i	0.937610	+	0.347688i	$0.113033\pi$
$41$			4.45258i			0.695376i	−0.937610	+	0.347688i	$0.886967\pi$
$42$		0			0
$43$			5.96478i			0.909620i	0.890588	+	0.454810i	$0.150293\pi$
$43$			5.96478i			0.909620i	−0.890588	+	0.454810i	$0.849707\pi$
$44$		0			0
$45$	−0.856436			−0.127670
$46$		0			0
$47$	−11.3771			−1.65952			−0.829761	−	0.558119i	$-0.811523\pi$
$47$	−11.3771			−1.65952			−0.829761	+	0.558119i	$0.811523\pi$
$48$		0			0
$49$	5.94401			0.849144
$50$		0			0
$51$	5.26852	+	2.94675i	0.737740	+	0.412627i
$52$		0			0
$53$		−	0.0517813i		−	0.00711271i	−0.999994	−	0.00355636i	$-0.998868\pi$
$53$		−	0.0517813i		−	0.00711271i	0.999994	−	0.00355636i	$-0.00113203\pi$
$54$		0			0
$55$	0.737326			0.0994211
$56$		0			0
$57$		−	4.13510i		−	0.547707i
$58$		0			0
$59$		−	6.48292i		−	0.844005i	−0.906595	−	0.422002i	$-0.861327\pi$
$59$		−	6.48292i		−	0.844005i	0.906595	−	0.422002i	$-0.138673\pi$
$60$		0			0
$61$	14.4629			1.85178			0.925890	−	0.377792i	$-0.123317\pi$
$61$	14.4629			1.85178			0.925890	+	0.377792i	$0.123317\pi$
$62$		0			0
$63$			0.880087i			0.110881i
$64$		0			0
$65$		−	4.98026i		−	0.617726i
$66$		0			0
$67$		−	4.56989i		−	0.558301i	−0.960247	−	0.279150i	$-0.909947\pi$
$67$		−	4.56989i		−	0.558301i	0.960247	−	0.279150i	$-0.0900528\pi$
$68$		0			0
$69$		−	7.25549i		−	0.873459i
$70$		0			0
$71$		−	12.9933i		−	1.54202i	−0.636822	−	0.771010i	$-0.719752\pi$
$71$		−	12.9933i		−	1.54202i	0.636822	−	0.771010i	$-0.280248\pi$
$72$		0			0
$73$		−	0.800933i		−	0.0937421i	−0.998901	−	0.0468710i	$-0.985075\pi$
$73$		−	0.800933i		−	0.0937421i	0.998901	−	0.0468710i	$-0.0149250\pi$
$74$		0			0
$75$	1.46409			0.169059
$76$		0			0
$77$		−	0.757688i		−	0.0863466i
$78$		0			0
$79$			7.83068i			0.881020i	0.897748	+	0.440510i	$0.145202\pi$
$79$			7.83068i			0.881020i	−0.897748	+	0.440510i	$0.854798\pi$
$80$		0			0
$81$	−5.69721			−0.633023
$82$		0			0
$83$		−	10.7517i		−	1.18016i	−0.807346	−	0.590078i	$-0.799097\pi$
$83$		−	10.7517i		−	1.18016i	0.807346	−	0.590078i	$-0.200903\pi$
$84$		0			0
$85$	3.59849	+	2.01268i	0.390311	+	0.218306i
$86$		0			0
$87$	−7.12949			−0.764362
$88$		0			0
$89$	−6.70231			−0.710443			−0.355222	−	0.934782i	$-0.615595\pi$
$89$	−6.70231			−0.710443			−0.355222	+	0.934782i	$0.615595\pi$
$90$		0			0
$91$	−5.11780			−0.536491
$92$		0			0
$93$		−	11.2357i		−	1.16509i
$94$		0			0
$95$		−	2.82434i		−	0.289771i
$96$		0			0
$97$			7.53723i			0.765289i	0.923896	+	0.382645i	$0.124987\pi$
$97$			7.53723i			0.765289i	−0.923896	+	0.382645i	$0.875013\pi$
$98$		0			0
$99$	−0.631473			−0.0634654
$100$		0			0
$101$			1.99652i			0.198662i	0.995054	+	0.0993308i	$0.0316702\pi$
$101$			1.99652i			0.198662i	−0.995054	+	0.0993308i	$0.968330\pi$
$102$		0			0
$103$	−1.44810			−0.142685			−0.0713426	−	0.997452i	$-0.522728\pi$
$103$	−1.44810			−0.142685			−0.0713426	+	0.997452i	$0.522728\pi$
$104$		0			0
$105$		−	1.50452i		−	0.146826i
$106$		0			0
$107$	4.74210			0.458436			0.229218	−	0.973375i	$-0.426383\pi$
$107$	4.74210			0.458436			0.229218	+	0.973375i	$0.426383\pi$
$108$		0			0
$109$	18.6027			1.78182			0.890910	−	0.454180i	$-0.150068\pi$
$109$	18.6027			1.78182			0.890910	+	0.454180i	$0.150068\pi$
$110$		0			0
$111$	4.72285			0.448273
$112$		0			0
$113$		−	13.6404i		−	1.28319i	−0.767046	−	0.641593i	$-0.778274\pi$
$113$		−	13.6404i		−	1.28319i	0.767046	−	0.641593i	$-0.221726\pi$
$114$		0			0
$115$		−	4.95563i		−	0.462115i
$116$		0			0
$117$			4.26528i			0.394325i
$118$		0			0
$119$	2.06826	−	3.69786i	0.189597	−	0.338983i
$120$		0			0
$121$	−10.4563			−0.950577
$122$		0			0
$123$			6.51898i			0.587797i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	3.25854			0.289149			0.144574	−	0.989494i	$-0.453819\pi$
$127$	3.25854			0.289149			0.144574	+	0.989494i	$0.453819\pi$
$128$		0			0
$129$			8.73298i			0.768896i
$130$		0			0
$131$	19.1586			1.67390			0.836949	−	0.547282i	$-0.184337\pi$
$131$	19.1586			1.67390			0.836949	+	0.547282i	$0.184337\pi$
$132$		0			0
$133$	−2.90234			−0.251665
$134$		0			0
$135$	−5.64618			−0.485945
$136$		0			0
$137$	12.0129			1.02633			0.513164	−	0.858291i	$-0.328473\pi$
$137$	12.0129			1.02633			0.513164	+	0.858291i	$0.328473\pi$
$138$		0			0
$139$	8.85801			0.751326			0.375663	−	0.926756i	$-0.377415\pi$
$139$	8.85801			0.751326			0.375663	+	0.926756i	$0.377415\pi$
$140$		0			0
$141$	−16.6571			−1.40278
$142$		0			0
$143$		−	3.67208i		−	0.307075i
$144$		0			0
$145$	−4.86957			−0.404396
$146$		0			0
$147$	8.70257			0.717776
$148$		0			0
$149$			22.3957i			1.83472i	0.398055	+	0.917362i	$0.369685\pi$
$149$			22.3957i			1.83472i	−0.398055	+	0.917362i	$0.630315\pi$
$150$		0			0
$151$	−15.4701			−1.25894			−0.629469	−	0.777025i	$-0.716728\pi$
$151$	−15.4701			−1.25894			−0.629469	+	0.777025i	$0.716728\pi$
$152$		0			0
$153$	−3.08188	−	1.72373i	−0.249155	−	0.139355i
$154$		0			0
$155$		−	7.67421i		−	0.616407i
$156$		0			0
$157$			11.0348i			0.880670i	0.897833	+	0.440335i	$0.145140\pi$
$157$			11.0348i			0.880670i	−0.897833	+	0.440335i	$0.854860\pi$
$158$		0			0
$159$		−	0.0758126i		−	0.00601233i
$160$		0			0
$161$	−5.09248			−0.401344
$162$		0			0
$163$	0.320808			0.0251276			0.0125638	−	0.999921i	$-0.496001\pi$
$163$	0.320808			0.0251276			0.0125638	+	0.999921i	$0.496001\pi$
$164$		0			0
$165$	1.07951			0.0840400
$166$		0			0
$167$			14.3206i			1.10816i	0.832463	+	0.554080i	$0.186930\pi$
$167$			14.3206i			1.10816i	−0.832463	+	0.554080i	$0.813070\pi$
$168$		0			0
$169$	−11.8030			−0.907925
$170$		0			0
$171$			2.41887i			0.184976i
$172$		0			0
$173$	−17.4844			−1.32932			−0.664658	−	0.747148i	$-0.731423\pi$
$173$	−17.4844			−1.32932			−0.664658	+	0.747148i	$0.731423\pi$
$174$		0			0
$175$		−	1.02762i		−	0.0776805i
$176$		0			0
$177$		−	9.49159i		−	0.713432i
$178$		0			0
$179$			10.0384i			0.750308i	0.926962	+	0.375154i	$0.122410\pi$
$179$			10.0384i			0.750308i	−0.926962	+	0.375154i	$0.877590\pi$
$180$		0			0
$181$	7.21688			0.536427			0.268213	−	0.963360i	$-0.413567\pi$
$181$	7.21688			0.536427			0.268213	+	0.963360i	$0.413567\pi$
$182$		0			0
$183$	21.1750			1.56530
$184$		0			0
$185$	3.22579			0.237165
$186$		0			0
$187$	2.65326	+	1.48400i	0.194026	+	0.108521i
$188$		0			0
$189$			5.80210i			0.422041i
$190$		0			0
$191$	−16.2281			−1.17423			−0.587113	−	0.809505i	$-0.699736\pi$
$191$	−16.2281			−1.17423			−0.587113	+	0.809505i	$0.699736\pi$
$192$		0			0
$193$		−	18.4087i		−	1.32509i	−0.749022	−	0.662545i	$-0.769476\pi$
$193$		−	18.4087i		−	1.32509i	0.749022	−	0.662545i	$-0.230524\pi$
$194$		0			0
$195$		−	7.29156i		−	0.522160i
$196$		0			0
$197$	1.17821			0.0839438			0.0419719	−	0.999119i	$-0.486636\pi$
$197$	1.17821			0.0839438			0.0419719	+	0.999119i	$0.486636\pi$
$198$		0			0
$199$		−	4.78744i		−	0.339373i	−0.985498	−	0.169686i	$-0.945725\pi$
$199$		−	4.78744i		−	0.339373i	0.985498	−	0.169686i	$-0.0542755\pi$
$200$		0			0
$201$		−	6.69073i		−	0.471928i
$202$		0			0
$203$			5.00405i			0.351215i
$204$		0			0
$205$			4.45258i			0.310982i
$206$		0			0
$207$			4.24418i			0.294991i
$208$		0			0
$209$		−	2.08246i		−	0.144047i
$210$		0			0
$211$	15.1007			1.03958			0.519788	−	0.854295i	$-0.326011\pi$
$211$	15.1007			1.03958			0.519788	+	0.854295i	$0.326011\pi$
$212$		0			0
$213$		−	19.0234i		−	1.30346i
$214$		0			0
$215$			5.96478i			0.406794i
$216$		0			0
$217$	−7.88614			−0.535346
$218$		0			0
$219$		−	1.17264i		−	0.0792396i
$220$		0			0
$221$	10.0237	−	17.9214i	0.674265	−	1.20553i
$222$		0			0
$223$	9.12247			0.610886			0.305443	−	0.952210i	$-0.401195\pi$
$223$	9.12247			0.610886			0.305443	+	0.952210i	$0.401195\pi$
$224$		0			0
$225$	−0.856436			−0.0570957
$226$		0			0
$227$	−5.42890			−0.360328			−0.180164	−	0.983637i	$-0.557663\pi$
$227$	−5.42890			−0.360328			−0.180164	+	0.983637i	$0.557663\pi$
$228$		0			0
$229$		−	5.57938i		−	0.368696i	−0.982861	−	0.184348i	$-0.940983\pi$
$229$		−	5.57938i		−	0.368696i	0.982861	−	0.184348i	$-0.0590173\pi$
$230$		0			0
$231$		−	1.10932i		−	0.0729882i
$232$		0			0
$233$			19.6266i			1.28578i	0.765959	+	0.642889i	$0.222264\pi$
$233$			19.6266i			1.28578i	−0.765959	+	0.642889i	$0.777736\pi$
$234$		0			0
$235$	−11.3771			−0.742161
$236$		0			0
$237$			11.4648i			0.744721i
$238$		0			0
$239$	1.63097			0.105499			0.0527494	−	0.998608i	$-0.483202\pi$
$239$	1.63097			0.105499			0.0527494	+	0.998608i	$0.483202\pi$
$240$		0			0
$241$		−	20.9878i		−	1.35194i	−0.736927	−	0.675972i	$-0.763724\pi$
$241$		−	20.9878i		−	1.35194i	0.736927	−	0.675972i	$-0.236276\pi$
$242$		0			0
$243$	8.59729			0.551516
$244$		0			0
$245$	5.94401			0.379749
$246$		0			0
$247$	−14.0660			−0.894996
$248$		0			0
$249$		−	15.7415i		−	0.997579i
$250$		0			0
$251$			7.63860i			0.482144i	0.970507	+	0.241072i	$0.0774990\pi$
$251$			7.63860i			0.482144i	−0.970507	+	0.241072i	$0.922501\pi$
$252$		0			0
$253$		−	3.65392i		−	0.229720i
$254$		0			0
$255$	5.26852	+	2.94675i	0.329927	+	0.184532i
$256$		0			0
$257$	1.38861			0.0866189			0.0433095	−	0.999062i	$-0.486210\pi$
$257$	1.38861			0.0866189			0.0433095	+	0.999062i	$0.486210\pi$
$258$		0			0
$259$		−	3.31487i		−	0.205976i
$260$		0			0
$261$	4.17048			0.258146
$262$		0			0
$263$	12.1867			0.751464			0.375732	−	0.926728i	$-0.377391\pi$
$263$	12.1867			0.751464			0.375732	+	0.926728i	$0.377391\pi$
$264$		0			0
$265$		−	0.0517813i		−	0.00318090i
$266$		0			0
$267$	−9.81279			−0.600533
$268$		0			0
$269$	8.53409			0.520333			0.260166	−	0.965564i	$-0.416223\pi$
$269$	8.53409			0.520333			0.260166	+	0.965564i	$0.416223\pi$
$270$		0			0
$271$	−19.0473			−1.15704			−0.578522	−	0.815667i	$-0.696370\pi$
$271$	−19.0473			−1.15704			−0.578522	+	0.815667i	$0.696370\pi$
$272$		0			0
$273$	−7.49292			−0.453492
$274$		0			0
$275$	0.737326			0.0444625
$276$		0			0
$277$	3.11221			0.186994			0.0934972	−	0.995620i	$-0.470195\pi$
$277$	3.11221			0.186994			0.0934972	+	0.995620i	$0.470195\pi$
$278$		0			0
$279$			6.57247i			0.393483i
$280$		0			0
$281$	−28.7935			−1.71767			−0.858837	−	0.512249i	$-0.828813\pi$
$281$	−28.7935			−1.71767			−0.858837	+	0.512249i	$0.828813\pi$
$282$		0			0
$283$	19.7867			1.17620			0.588100	−	0.808788i	$-0.299876\pi$
$283$	19.7867			1.17620			0.588100	+	0.808788i	$0.299876\pi$
$284$		0			0
$285$		−	4.13510i		−	0.244942i
$286$		0			0
$287$	4.57554			0.270086
$288$		0			0
$289$	8.89825	+	14.4852i	0.523427	+	0.852071i
$290$		0			0
$291$			11.0352i			0.646894i
$292$		0			0
$293$		−	11.3185i		−	0.661232i	−0.943765	−	0.330616i	$-0.892744\pi$
$293$		−	11.3185i		−	0.661232i	0.943765	−	0.330616i	$-0.107256\pi$
$294$		0			0
$295$		−	6.48292i		−	0.377450i
$296$		0			0
$297$	−4.16307			−0.241566
$298$		0			0
$299$	−24.6803			−1.42730
$300$		0			0
$301$	6.12950			0.353298
$302$		0			0
$303$			2.92309i			0.167927i
$304$		0			0
$305$	14.4629			0.828142
$306$		0			0
$307$			34.8808i			1.99075i	0.0960443	+	0.995377i	$0.469381\pi$
$307$			34.8808i			1.99075i	−0.0960443	+	0.995377i	$0.530619\pi$
$308$		0			0
$309$	−2.12015			−0.120611
$310$		0			0
$311$			28.9283i			1.64037i	0.572096	+	0.820186i	$0.306130\pi$
$311$			28.9283i			1.64037i	−0.572096	+	0.820186i	$0.693870\pi$
$312$		0			0
$313$		−	5.33643i		−	0.301633i	−0.988562	−	0.150816i	$-0.951810\pi$
$313$		−	5.33643i		−	0.301633i	0.988562	−	0.150816i	$-0.0481902\pi$
$314$		0			0
$315$			0.880087i			0.0495873i
$316$		0			0
$317$	−15.5468			−0.873195			−0.436598	−	0.899657i	$-0.643817\pi$
$317$	−15.5468			−0.873195			−0.436598	+	0.899657i	$0.643817\pi$
$318$		0			0
$319$	−3.59046			−0.201027
$320$		0			0
$321$	6.94286			0.387513
$322$		0			0
$323$	5.68449	−	10.1634i	0.316294	−	0.565505i
$324$		0			0
$325$		−	4.98026i		−	0.276255i
$326$		0			0
$327$	27.2361			1.50616
$328$		0			0
$329$			11.6913i			0.644562i
$330$		0			0
$331$		−	0.824032i		−	0.0452929i	−0.999744	−	0.0226465i	$-0.992791\pi$
$331$		−	0.824032i		−	0.0452929i	0.999744	−	0.0226465i	$-0.00720921\pi$
$332$		0			0
$333$	−2.76268			−0.151394
$334$		0			0
$335$		−	4.56989i		−	0.249680i
$336$		0			0
$337$		−	10.3903i		−	0.565998i	−0.959120	−	0.282999i	$-0.908671\pi$
$337$		−	10.3903i		−	0.565998i	0.959120	−	0.282999i	$-0.0913294\pi$
$338$		0			0
$339$		−	19.9709i		−	1.08467i
$340$		0			0
$341$		−	5.65840i		−	0.306419i
$342$		0			0
$343$		−	13.3015i		−	0.718212i
$344$		0			0
$345$		−	7.25549i		−	0.390623i
$346$		0			0
$347$	−8.82570			−0.473789			−0.236894	−	0.971535i	$-0.576129\pi$
$347$	−8.82570			−0.473789			−0.236894	+	0.971535i	$0.576129\pi$
$348$		0			0
$349$			7.74760i			0.414720i	0.978265	+	0.207360i	$0.0664871\pi$
$349$			7.74760i			0.414720i	−0.978265	+	0.207360i	$0.933513\pi$
$350$		0			0
$351$			28.1194i			1.50090i
$352$		0			0
$353$	7.97452			0.424441			0.212221	−	0.977222i	$-0.431930\pi$
$353$	7.97452			0.424441			0.212221	+	0.977222i	$0.431930\pi$
$354$		0			0
$355$		−	12.9933i		−	0.689613i
$356$		0			0
$357$	3.02812	−	5.41401i	0.160265	−	0.286540i
$358$		0			0
$359$	19.2883			1.01800			0.508998	−	0.860767i	$-0.330016\pi$
$359$	19.2883			1.01800			0.508998	+	0.860767i	$0.330016\pi$
$360$		0			0
$361$	11.0231			0.580163
$362$		0			0
$363$	−15.3091			−0.803517
$364$		0			0
$365$		−	0.800933i		−	0.0419227i
$366$		0			0
$367$		−	15.8068i		−	0.825109i	−0.910933	−	0.412555i	$-0.864637\pi$
$367$		−	15.8068i		−	0.825109i	0.910933	−	0.412555i	$-0.135363\pi$
$368$		0			0
$369$		−	3.81335i		−	0.198515i
$370$		0			0
$371$	−0.0532113			−0.00276259
$372$		0			0
$373$		−	23.2547i		−	1.20408i	−0.798466	−	0.602040i	$-0.794355\pi$
$373$		−	23.2547i		−	1.20408i	0.798466	−	0.602040i	$-0.205645\pi$
$374$		0			0
$375$	1.46409			0.0756054
$376$		0			0
$377$			24.2517i			1.24903i
$378$		0			0
$379$	−26.7603			−1.37458			−0.687292	−	0.726381i	$-0.741201\pi$
$379$	−26.7603			−1.37458			−0.687292	+	0.726381i	$0.741201\pi$
$380$		0			0
$381$	4.77080			0.244415
$382$		0			0
$383$	−8.46688			−0.432638			−0.216319	−	0.976323i	$-0.569405\pi$
$383$	−8.46688			−0.432638			−0.216319	+	0.976323i	$0.569405\pi$
$384$		0			0
$385$		−	0.757688i		−	0.0386154i
$386$		0			0
$387$		−	5.10845i		−	0.259677i
$388$		0			0
$389$			25.9652i			1.31649i	0.752804	+	0.658245i	$0.228701\pi$
$389$			25.9652i			1.31649i	−0.752804	+	0.658245i	$0.771299\pi$
$390$		0			0
$391$	9.97408	−	17.8328i	0.504411	−	0.901842i
$392$		0			0
$393$	28.0500			1.41493
$394$		0			0
$395$			7.83068i			0.394004i
$396$		0			0
$397$	22.6930			1.13893			0.569466	−	0.822015i	$-0.307150\pi$
$397$	22.6930			1.13893			0.569466	+	0.822015i	$0.307150\pi$
$398$		0			0
$399$	−4.24929			−0.212731
$400$		0			0
$401$			32.7100i			1.63346i	0.577019	+	0.816731i	$0.304216\pi$
$401$			32.7100i			1.63346i	−0.577019	+	0.816731i	$0.695784\pi$
$402$		0			0
$403$	−38.2196			−1.90385
$404$		0			0
$405$	−5.69721			−0.283097
$406$		0			0
$407$	2.37846			0.117896
$408$		0			0
$409$	−25.3481			−1.25338			−0.626691	−	0.779268i	$-0.715591\pi$
$409$	−25.3481			−1.25338			−0.626691	+	0.779268i	$0.715591\pi$
$410$		0			0
$411$	17.5879			0.867549
$412$		0			0
$413$	−6.66195			−0.327813
$414$		0			0
$415$		−	10.7517i		−	0.527782i
$416$		0			0
$417$	12.9689			0.635091
$418$		0			0
$419$	2.11612			0.103379			0.0516897	−	0.998663i	$-0.483539\pi$
$419$	2.11612			0.103379			0.0516897	+	0.998663i	$0.483539\pi$
$420$		0			0
$421$			33.1164i			1.61399i	0.590556	+	0.806996i	$0.298908\pi$
$421$			33.1164i			1.61399i	−0.590556	+	0.806996i	$0.701092\pi$
$422$		0			0
$423$	9.74377			0.473758
$424$		0			0
$425$	3.59849	+	2.01268i	0.174552	+	0.0976292i
$426$		0			0
$427$		−	14.8623i		−	0.719236i
$428$		0			0
$429$		−	5.37626i		−	0.259568i
$430$		0			0
$431$		−	9.16433i		−	0.441430i	−0.975338	−	0.220715i	$-0.929161\pi$
$431$		−	9.16433i		−	0.441430i	0.975338	−	0.220715i	$-0.0708391\pi$
$432$		0			0
$433$	28.6734			1.37796			0.688978	−	0.724783i	$-0.258060\pi$
$433$	28.6734			1.37796			0.688978	+	0.724783i	$0.258060\pi$
$434$		0			0
$435$	−7.12949			−0.341833
$436$		0			0
$437$	−13.9964			−0.669538
$438$		0			0
$439$			25.2376i			1.20452i	0.798299	+	0.602261i	$0.205734\pi$
$439$			25.2376i			1.20452i	−0.798299	+	0.602261i	$0.794266\pi$
$440$		0			0
$441$	−5.09066			−0.242412
$442$		0			0
$443$			7.07363i			0.336078i	0.985780	+	0.168039i	$0.0537435\pi$
$443$			7.07363i			0.336078i	−0.985780	+	0.168039i	$0.946257\pi$
$444$		0			0
$445$	−6.70231			−0.317720
$446$		0			0
$447$			32.7893i			1.55088i
$448$		0			0
$449$		−	14.5442i		−	0.686382i	−0.939266	−	0.343191i	$-0.888492\pi$
$449$		−	14.5442i		−	0.686382i	0.939266	−	0.343191i	$-0.111508\pi$
$450$		0			0
$451$			3.28300i			0.154591i
$452$		0			0
$453$	−22.6496			−1.06417
$454$		0			0
$455$	−5.11780			−0.239926
$456$		0			0
$457$	−36.1784			−1.69236			−0.846178	−	0.532901i	$-0.821102\pi$
$457$	−36.1784			−1.69236			−0.846178	+	0.532901i	$0.821102\pi$
$458$		0			0
$459$	−20.3177	−	11.3639i	−0.948349	−	0.530423i
$460$		0			0
$461$		−	2.76009i		−	0.128550i	−0.997932	−	0.0642751i	$-0.979526\pi$
$461$		−	2.76009i		−	0.128550i	0.997932	−	0.0642751i	$-0.0204735\pi$
$462$		0			0
$463$	−8.26733			−0.384216			−0.192108	−	0.981374i	$-0.561532\pi$
$463$	−8.26733			−0.384216			−0.192108	+	0.981374i	$0.561532\pi$
$464$		0			0
$465$		−	11.2357i		−	0.521045i
$466$		0			0
$467$		−	14.7543i		−	0.682748i	−0.939928	−	0.341374i	$-0.889108\pi$
$467$		−	14.7543i		−	0.682748i	0.939928	−	0.341374i	$-0.110892\pi$
$468$		0			0
$469$	−4.69609			−0.216845
$470$		0			0
$471$			16.1559i			0.744425i
$472$		0			0
$473$			4.39799i			0.202220i
$474$		0			0
$475$		−	2.82434i		−	0.129590i
$476$		0			0
$477$			0.0443474i			0.00203053i
$478$		0			0
$479$		−	22.2732i		−	1.01769i	−0.860859	−	0.508843i	$-0.830073\pi$
$479$		−	22.2732i		−	1.01769i	0.860859	−	0.508843i	$-0.169927\pi$
$480$		0			0
$481$		−	16.0653i		−	0.732514i
$482$		0			0
$483$	−7.45586			−0.339253
$484$		0			0
$485$			7.53723i			0.342248i
$486$		0			0
$487$			16.9274i			0.767055i	0.923529	+	0.383528i	$0.125291\pi$
$487$			16.9274i			0.767055i	−0.923529	+	0.383528i	$0.874709\pi$
$488$		0			0
$489$	0.469692			0.0212402
$490$		0			0
$491$			10.4958i			0.473668i	0.971550	+	0.236834i	$0.0761098\pi$
$491$			10.4958i			0.473668i	−0.971550	+	0.236834i	$0.923890\pi$
$492$		0			0
$493$	−17.5231	−	9.80088i	−0.789200	−	0.441409i
$494$		0			0
$495$	−0.631473			−0.0283826
$496$		0			0
$497$	−13.3521			−0.598924
$498$		0			0
$499$	23.9095			1.07034			0.535168	−	0.844746i	$-0.320248\pi$
$499$	23.9095			1.07034			0.535168	+	0.844746i	$0.320248\pi$
$500$		0			0
$501$			20.9666i			0.936721i
$502$		0			0
$503$			1.84621i			0.0823184i	0.999153	+	0.0411592i	$0.0131051\pi$
$503$			1.84621i			0.0823184i	−0.999153	+	0.0411592i	$0.986895\pi$
$504$		0			0
$505$			1.99652i			0.0888441i
$506$		0			0
$507$	−17.2807			−0.767463
$508$		0			0
$509$			21.9087i			0.971086i	0.874213	+	0.485543i	$0.161378\pi$
$509$			21.9087i			0.971086i	−0.874213	+	0.485543i	$0.838622\pi$
$510$		0			0
$511$	−0.823051			−0.0364096
$512$		0			0
$513$			15.9467i			0.704065i
$514$		0			0
$515$	−1.44810			−0.0638108
$516$		0			0
$517$	−8.38865			−0.368932
$518$		0			0
$519$	−25.5988			−1.12366
$520$		0			0
$521$		−	1.41157i		−	0.0618421i	−0.999522	−	0.0309211i	$-0.990156\pi$
$521$		−	1.41157i		−	0.0618421i	0.999522	−	0.0309211i	$-0.00984405\pi$
$522$		0			0
$523$			20.7749i			0.908421i	0.890894	+	0.454211i	$0.150079\pi$
$523$			20.7749i			0.908421i	−0.890894	+	0.454211i	$0.849921\pi$
$524$		0			0
$525$		−	1.50452i		−	0.0656628i
$526$		0			0
$527$	15.4457	−	27.6156i	0.672826	−	1.20295i
$528$		0			0
$529$	−1.55824			−0.0677498
$530$		0			0
$531$			5.55221i			0.240945i
$532$		0			0
$533$	22.1750			0.960506
$534$		0			0
$535$	4.74210			0.205019
$536$		0			0
$537$			14.6972i			0.634231i
$538$		0			0
$539$	4.38267			0.188775
$540$		0			0
$541$	−4.03830			−0.173620			−0.0868101	−	0.996225i	$-0.527667\pi$
$541$	−4.03830			−0.173620			−0.0868101	+	0.996225i	$0.527667\pi$
$542$		0			0
$543$	10.5662			0.453438
$544$		0			0
$545$	18.6027			0.796854
$546$		0			0
$547$	41.2053			1.76181			0.880906	−	0.473290i	$-0.156934\pi$
$547$	41.2053			1.76181			0.880906	+	0.473290i	$0.156934\pi$
$548$		0			0
$549$	−12.3865			−0.528644
$550$		0			0
$551$			13.7533i			0.585912i
$552$		0			0
$553$	8.04693			0.342190
$554$		0			0
$555$	4.72285			0.200474
$556$		0			0
$557$			13.9646i			0.591699i	0.955235	+	0.295849i	$0.0956027\pi$
$557$			13.9646i			0.591699i	−0.955235	+	0.295849i	$0.904397\pi$
$558$		0			0
$559$	29.7062			1.25644
$560$		0			0
$561$	3.88462	+	2.17271i	0.164009	+	0.0917320i
$562$		0			0
$563$		−	39.0610i		−	1.64622i	−0.567880	−	0.823111i	$-0.692236\pi$
$563$		−	39.0610i		−	1.64622i	0.567880	−	0.823111i	$-0.307764\pi$
$564$		0			0
$565$		−	13.6404i		−	0.573858i
$566$		0			0
$567$			5.85454i			0.245868i
$568$		0			0
$569$	40.3237			1.69046			0.845228	−	0.534405i	$-0.179464\pi$
$569$	40.3237			1.69046			0.845228	+	0.534405i	$0.179464\pi$
$570$		0			0
$571$	−27.1177			−1.13484			−0.567421	−	0.823428i	$-0.692059\pi$
$571$	−27.1177			−1.13484			−0.567421	+	0.823428i	$0.692059\pi$
$572$		0			0
$573$	−23.7595			−0.992566
$574$		0			0
$575$		−	4.95563i		−	0.206664i
$576$		0			0
$577$	−7.77551			−0.323699			−0.161849	−	0.986815i	$-0.551746\pi$
$577$	−7.77551			−0.323699			−0.161849	+	0.986815i	$0.551746\pi$
$578$		0			0
$579$		−	26.9521i		−	1.12009i
$580$		0			0
$581$	−11.0487			−0.458375
$582$		0			0
$583$		−	0.0381798i		−	0.00158124i
$584$		0			0
$585$			4.26528i			0.176348i
$586$		0			0
$587$			35.6192i			1.47016i	0.677980	+	0.735080i	$0.262855\pi$
$587$			35.6192i			1.47016i	−0.677980	+	0.735080i	$0.737145\pi$
$588$		0			0
$589$	−21.6746			−0.893086
$590$		0			0
$591$	1.72500			0.0709572
$592$		0			0
$593$	35.3602			1.45207			0.726035	−	0.687658i	$-0.241361\pi$
$593$	35.3602			1.45207			0.726035	+	0.687658i	$0.241361\pi$
$594$		0			0
$595$	2.06826	−	3.69786i	0.0847904	−	0.151598i
$596$		0			0
$597$		−	7.00925i		−	0.286870i
$598$		0			0
$599$	−25.3800			−1.03700			−0.518500	−	0.855078i	$-0.673509\pi$
$599$	−25.3800			−1.03700			−0.518500	+	0.855078i	$0.673509\pi$
$600$		0			0
$601$			0.940741i			0.0383736i	0.999816	+	0.0191868i	$0.00610773\pi$
$601$			0.940741i			0.0383736i	−0.999816	+	0.0191868i	$0.993892\pi$
$602$		0			0
$603$			3.91382i			0.159383i
$604$		0			0
$605$	−10.4563			−0.425111
$606$		0			0
$607$			9.70858i			0.394059i	0.980398	+	0.197030i	$0.0631295\pi$
$607$			9.70858i			0.394059i	−0.980398	+	0.197030i	$0.936871\pi$
$608$		0			0
$609$			7.32638i			0.296880i
$610$		0			0
$611$			56.6610i			2.29226i
$612$		0			0
$613$		−	30.1125i		−	1.21623i	−0.793848	−	0.608116i	$-0.791925\pi$
$613$		−	30.1125i		−	1.21623i	0.793848	−	0.608116i	$-0.208075\pi$
$614$		0			0
$615$			6.51898i			0.262871i
$616$		0			0
$617$			33.4173i			1.34533i	0.739947	+	0.672665i	$0.234850\pi$
$617$			33.4173i			1.34533i	−0.739947	+	0.672665i	$0.765150\pi$
$618$		0			0
$619$	−5.48862			−0.220606			−0.110303	−	0.993898i	$-0.535182\pi$
$619$	−5.48862			−0.220606			−0.110303	+	0.993898i	$0.535182\pi$
$620$		0			0
$621$			27.9803i			1.12281i
$622$		0			0
$623$			6.88740i			0.275938i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	3.04892i		−	0.121762i
$628$		0			0
$629$	11.6080	+	6.49247i	0.462840	+	0.258872i
$630$		0			0
$631$	−0.444706			−0.0177035			−0.00885173	−	0.999961i	$-0.502818\pi$
$631$	−0.444706			−0.0177035			−0.00885173	+	0.999961i	$0.502818\pi$
$632$		0			0
$633$	22.1088			0.878747
$634$		0			0
$635$	3.25854			0.129311
$636$		0			0
$637$		−	29.6027i		−	1.17290i
$638$		0			0
$639$			11.1279i			0.440214i
$640$		0			0
$641$			24.5740i			0.970616i	0.874343	+	0.485308i	$0.161292\pi$
$641$			24.5740i			0.970616i	−0.874343	+	0.485308i	$0.838708\pi$
$642$		0			0
$643$	37.3639			1.47349			0.736743	−	0.676173i	$-0.236363\pi$
$643$	37.3639			1.47349			0.736743	+	0.676173i	$0.236363\pi$
$644$		0			0
$645$			8.73298i			0.343861i
$646$		0			0
$647$	1.14820			0.0451405			0.0225703	−	0.999745i	$-0.492815\pi$
$647$	1.14820			0.0451405			0.0225703	+	0.999745i	$0.492815\pi$
$648$		0			0
$649$		−	4.78003i		−	0.187633i
$650$		0			0
$651$	−11.5460			−0.452524
$652$		0			0
$653$	4.48681			0.175582			0.0877912	−	0.996139i	$-0.472019\pi$
$653$	4.48681			0.175582			0.0877912	+	0.996139i	$0.472019\pi$
$654$		0			0
$655$	19.1586			0.748590
$656$		0			0
$657$			0.685948i			0.0267614i
$658$		0			0
$659$		−	0.209907i		−	0.00817683i	−0.999992	−	0.00408841i	$-0.998699\pi$
$659$		−	0.209907i		−	0.00817683i	0.999992	−	0.00408841i	$-0.00130139\pi$
$660$		0			0
$661$			0.915250i			0.0355991i	0.999842	+	0.0177996i	$0.00566607\pi$
$661$			0.915250i			0.0355991i	−0.999842	+	0.0177996i	$0.994334\pi$
$662$		0			0
$663$	14.6756	−	26.2386i	0.569952	−	1.01902i
$664$		0			0
$665$	−2.90234			−0.112548
$666$		0			0
$667$			24.1318i			0.934386i
$668$		0			0
$669$	13.3561			0.516378
$670$		0			0
$671$	10.6639			0.411674
$672$		0			0
$673$			28.6506i			1.10440i	0.833712	+	0.552199i	$0.186211\pi$
$673$			28.6506i			1.10440i	−0.833712	+	0.552199i	$0.813789\pi$
$674$		0			0
$675$	−5.64618			−0.217321
$676$		0			0
$677$	−27.5070			−1.05718			−0.528590	−	0.848877i	$-0.677279\pi$
$677$	−27.5070			−1.05718			−0.528590	+	0.848877i	$0.677279\pi$
$678$		0			0
$679$	7.74537			0.297240
$680$		0			0
$681$	−7.94840			−0.304583
$682$		0			0
$683$	−28.8188			−1.10272			−0.551361	−	0.834267i	$-0.685891\pi$
$683$	−28.8188			−1.10272			−0.551361	+	0.834267i	$0.685891\pi$
$684$		0			0
$685$	12.0129			0.458988
$686$		0			0
$687$		−	8.16872i		−	0.311656i
$688$		0			0
$689$	−0.257885			−0.00982463
$690$		0			0
$691$	30.2174			1.14952			0.574761	−	0.818321i	$-0.305095\pi$
$691$	30.2174			1.14952			0.574761	+	0.818321i	$0.305095\pi$
$692$		0			0
$693$			0.648912i			0.0246501i
$694$		0			0
$695$	8.85801			0.336003
$696$		0			0
$697$	−8.96161	+	16.0226i	−0.339445	+	0.606898i
$698$		0			0
$699$			28.7351i			1.08686i
$700$		0			0
$701$			15.1682i			0.572893i	0.958096	+	0.286447i	$0.0924741\pi$
$701$			15.1682i			0.572893i	−0.958096	+	0.286447i	$0.907526\pi$
$702$		0			0
$703$		−	9.11073i		−	0.343618i
$704$		0			0
$705$	−16.6571			−0.627344
$706$		0			0
$707$	2.05166			0.0771606
$708$		0			0
$709$	−1.02190			−0.0383782			−0.0191891	−	0.999816i	$-0.506108\pi$
$709$	−1.02190			−0.0383782			−0.0191891	+	0.999816i	$0.506108\pi$
$710$		0			0
$711$		−	6.70648i		−	0.251513i
$712$		0			0
$713$	−38.0305			−1.42425
$714$		0			0
$715$		−	3.67208i		−	0.137328i
$716$		0			0
$717$	2.38789			0.0891775
$718$		0			0
$719$			46.2539i			1.72498i	0.506073	+	0.862491i	$0.331097\pi$
$719$			46.2539i			1.72498i	−0.506073	+	0.862491i	$0.668903\pi$
$720$		0			0
$721$			1.48809i			0.0554193i
$722$		0			0
$723$		−	30.7281i		−	1.14279i
$724$		0			0
$725$	−4.86957			−0.180851
$726$		0			0
$727$	−43.5602			−1.61556			−0.807779	−	0.589485i	$-0.799331\pi$
$727$	−43.5602			−1.61556			−0.807779	+	0.589485i	$0.799331\pi$
$728$		0			0
$729$	29.6788			1.09922
$730$		0			0
$731$	−12.0052	+	21.4642i	−0.444028	+	0.793882i
$732$		0			0
$733$			5.12531i			0.189308i	0.995510	+	0.0946538i	$0.0301744\pi$
$733$			5.12531i			0.189308i	−0.995510	+	0.0946538i	$0.969826\pi$
$734$		0			0
$735$	8.70257			0.320999
$736$		0			0
$737$		−	3.36950i		−	0.124117i
$738$		0			0
$739$		−	15.8626i		−	0.583514i	−0.956493	−	0.291757i	$-0.905760\pi$
$739$		−	15.8626i		−	0.583514i	0.956493	−	0.291757i	$-0.0942398\pi$
$740$		0			0
$741$	−20.5939			−0.756535
$742$		0			0
$743$		−	28.3848i		−	1.04134i	−0.853759	−	0.520668i	$-0.825683\pi$
$743$		−	28.3848i		−	1.04134i	0.853759	−	0.520668i	$-0.174317\pi$
$744$		0			0
$745$			22.3957i			0.820513i
$746$		0			0
$747$			9.20817i			0.336909i
$748$		0			0
$749$		−	4.87305i		−	0.178058i
$750$		0			0
$751$			31.5903i			1.15275i	0.817187	+	0.576373i	$0.195533\pi$
$751$			31.5903i			1.15275i	−0.817187	+	0.576373i	$0.804467\pi$
$752$		0			0
$753$			11.1836i			0.407553i
$754$		0			0
$755$	−15.4701			−0.563014
$756$		0			0
$757$			15.9749i			0.580619i	0.956933	+	0.290309i	$0.0937582\pi$
$757$			15.9749i			0.580619i	−0.956933	+	0.290309i	$0.906242\pi$
$758$		0			0
$759$		−	5.34967i		−	0.194181i
$760$		0			0
$761$	−0.0742013			−0.00268979			−0.00134490	−	0.999999i	$-0.500428\pi$
$761$	−0.0742013			−0.00268979			−0.00134490	+	0.999999i	$0.500428\pi$
$762$		0			0
$763$		−	19.1165i		−	0.692063i
$764$		0			0
$765$	−3.08188	−	1.72373i	−0.111425	−	0.0623216i
$766$		0			0
$767$	−32.2867			−1.16580
$768$		0			0
$769$	−37.0122			−1.33470			−0.667348	−	0.744746i	$-0.732571\pi$
$769$	−37.0122			−1.33470			−0.667348	+	0.744746i	$0.732571\pi$
$770$		0			0
$771$	2.03305			0.0732184
$772$		0			0
$773$		−	22.3692i		−	0.804563i	−0.915516	−	0.402282i	$-0.868217\pi$
$773$		−	22.3692i		−	0.804563i	0.915516	−	0.402282i	$-0.131783\pi$
$774$		0			0
$775$		−	7.67421i		−	0.275666i
$776$		0			0
$777$		−	4.85327i		−	0.174110i
$778$		0			0
$779$	12.5756			0.450568
$780$		0			0
$781$		−	9.58030i		−	0.342810i
$782$		0			0
$783$	27.4944			0.982571
$784$		0			0
$785$			11.0348i			0.393848i
$786$		0			0
$787$	−10.9625			−0.390769			−0.195385	−	0.980727i	$-0.562596\pi$
$787$	−10.9625			−0.390769			−0.195385	+	0.980727i	$0.562596\pi$
$788$		0			0
$789$	17.8424			0.635208
$790$		0			0
$791$	−14.0171			−0.498392
$792$		0			0
$793$		−	72.0289i		−	2.55782i
$794$		0			0
$795$		−	0.0758126i		−	0.00268880i
$796$		0			0
$797$		−	39.0009i		−	1.38148i	−0.723102	−	0.690741i	$-0.757284\pi$
$797$		−	39.0009i		−	1.38148i	0.723102	−	0.690741i	$-0.242716\pi$
$798$		0			0
$799$	−40.9404	−	22.8985i	−1.44837	−	0.810090i
$800$		0			0
$801$	5.74010			0.202816
$802$		0			0
$803$		−	0.590549i		−	0.0208400i
$804$		0			0
$805$	−5.09248			−0.179486
$806$		0			0
$807$	12.4947			0.439834
$808$		0			0
$809$			39.1418i			1.37615i	0.725638	+	0.688077i	$0.241545\pi$
$809$			39.1418i			1.37615i	−0.725638	+	0.688077i	$0.758455\pi$
$810$		0			0
$811$	−18.0496			−0.633809			−0.316904	−	0.948458i	$-0.602643\pi$
$811$	−18.0496			−0.633809			−0.316904	+	0.948458i	$0.602643\pi$
$812$		0			0
$813$	−27.8871			−0.978042
$814$		0			0
$815$	0.320808			0.0112374
$816$		0			0
$817$	16.8466			0.589387
$818$		0			0
$819$	4.38307			0.153157
$820$		0			0
$821$	−4.04733			−0.141253			−0.0706263	−	0.997503i	$-0.522500\pi$
$821$	−4.04733			−0.141253			−0.0706263	+	0.997503i	$0.522500\pi$
$822$		0			0
$823$		−	10.4958i		−	0.365862i	−0.983126	−	0.182931i	$-0.941441\pi$
$823$		−	10.4958i		−	0.365862i	0.983126	−	0.182931i	$-0.0585585\pi$
$824$		0			0
$825$	1.07951			0.0375838
$826$		0			0
$827$	−35.2909			−1.22719			−0.613593	−	0.789623i	$-0.710276\pi$
$827$	−35.2909			−1.22719			−0.613593	+	0.789623i	$0.710276\pi$
$828$		0			0
$829$			30.4113i			1.05623i	0.849173	+	0.528114i	$0.177101\pi$
$829$			30.4113i			1.05623i	−0.849173	+	0.528114i	$0.822899\pi$
$830$		0			0
$831$	4.55656			0.158065
$832$		0			0
$833$	21.3894	+	11.9634i	0.741100	+	0.414506i
$834$		0			0
$835$			14.3206i			0.495584i
$836$		0			0
$837$			43.3299i			1.49770i
$838$		0			0
$839$			28.3278i			0.977985i	0.872288	+	0.488992i	$0.162635\pi$
$839$			28.3278i			0.977985i	−0.872288	+	0.488992i	$0.837365\pi$
$840$		0			0
$841$	−5.28730			−0.182321
$842$		0			0
$843$	−42.1563			−1.45194
$844$		0			0
$845$	−11.8030			−0.406036
$846$		0			0
$847$			10.7451i			0.369206i
$848$		0			0
$849$	28.9696			0.994234
$850$		0			0
$851$		−	15.9858i		−	0.547986i
$852$		0			0
$853$	49.3618			1.69011			0.845057	−	0.534676i	$-0.179566\pi$
$853$	49.3618			1.69011			0.845057	+	0.534676i	$0.179566\pi$
$854$		0			0
$855$			2.41887i			0.0827236i
$856$		0			0
$857$		−	56.3068i		−	1.92340i	−0.274099	−	0.961701i	$-0.588380\pi$
$857$		−	56.3068i		−	1.92340i	0.274099	−	0.961701i	$-0.411620\pi$
$858$		0			0
$859$			4.79059i			0.163453i	0.996655	+	0.0817264i	$0.0260434\pi$
$859$			4.79059i			0.163453i	−0.996655	+	0.0817264i	$0.973957\pi$
$860$		0			0
$861$	6.69901			0.228302
$862$		0			0
$863$	−21.3707			−0.727469			−0.363734	−	0.931503i	$-0.618498\pi$
$863$	−21.3707			−0.727469			−0.363734	+	0.931503i	$0.618498\pi$
$864$		0			0
$865$	−17.4844			−0.594488
$866$		0			0
$867$	13.0279	+	21.2077i	0.442449	+	0.720250i
$868$		0			0
$869$			5.77377i			0.195862i
$870$		0			0
$871$	−22.7592			−0.771168
$872$		0			0
$873$		−	6.45515i		−	0.218474i
$874$		0			0
$875$		−	1.02762i		−	0.0347398i
$876$		0			0
$877$	13.8064			0.466208			0.233104	−	0.972452i	$-0.425112\pi$
$877$	13.8064			0.466208			0.233104	+	0.972452i	$0.425112\pi$
$878$		0			0
$879$		−	16.5713i		−	0.558935i
$880$		0			0
$881$			53.3373i			1.79698i	0.438996	+	0.898489i	$0.355334\pi$
$881$			53.3373i			1.79698i	−0.438996	+	0.898489i	$0.644666\pi$
$882$		0			0
$883$		−	17.6643i		−	0.594452i	−0.954807	−	0.297226i	$-0.903939\pi$
$883$		−	17.6643i		−	0.594452i	0.954807	−	0.297226i	$-0.0960614\pi$
$884$		0			0
$885$		−	9.49159i		−	0.319056i
$886$		0			0
$887$		−	54.2478i		−	1.82146i	−0.413001	−	0.910731i	$-0.635519\pi$
$887$		−	54.2478i		−	1.82146i	0.413001	−	0.910731i	$-0.364481\pi$
$888$		0			0
$889$		−	3.34853i		−	0.112306i
$890$		0			0
$891$	−4.20070			−0.140729
$892$		0			0
$893$			32.1329i			1.07529i
$894$		0			0
$895$			10.0384i			0.335548i
$896$		0			0
$897$	−36.1343			−1.20649
$898$		0			0
$899$			37.3701i			1.24636i
$900$		0			0
$901$	0.104219	−	0.186335i	0.00347204	−	0.00620771i
$902$		0			0
$903$	8.97415			0.298641
$904$		0			0
$905$	7.21688			0.239897
$906$		0			0
$907$	53.1016			1.76321			0.881605	−	0.471989i	$-0.156464\pi$
$907$	53.1016			1.76321			0.881605	+	0.471989i	$0.156464\pi$
$908$		0			0
$909$		−	1.70989i		−	0.0567136i
$910$		0			0
$911$		−	22.4664i		−	0.744344i	−0.928164	−	0.372172i	$-0.878613\pi$
$911$		−	22.4664i		−	0.744344i	0.928164	−	0.372172i	$-0.121387\pi$
$912$		0			0
$913$		−	7.92754i		−	0.262363i
$914$		0			0
$915$	21.1750			0.700023
$916$		0			0
$917$		−	19.6877i		−	0.650145i
$918$		0			0
$919$	−17.5340			−0.578393			−0.289197	−	0.957270i	$-0.593388\pi$
$919$	−17.5340			−0.578393			−0.289197	+	0.957270i	$0.593388\pi$
$920$		0			0
$921$			51.0687i			1.68277i
$922$		0			0
$923$	−64.7101			−2.12996
$924$		0			0
$925$	3.22579			0.106063
$926$		0			0
$927$	1.24020			0.0407336
$928$		0			0
$929$		−	11.8068i		−	0.387369i	−0.981064	−	0.193684i	$-0.937956\pi$
$929$		−	11.8068i		−	0.387369i	0.981064	−	0.193684i	$-0.0620438\pi$
$930$		0			0
$931$		−	16.7879i		−	0.550201i
$932$		0			0
$933$			42.3537i			1.38660i
$934$		0			0
$935$	2.65326	+	1.48400i	0.0867709	+	0.0485320i
$936$		0			0
$937$	42.2046			1.37876			0.689381	−	0.724398i	$-0.257882\pi$
$937$	42.2046			1.37876			0.689381	+	0.724398i	$0.257882\pi$
$938$		0			0
$939$		−	7.81302i		−	0.254968i
$940$		0			0
$941$	42.9773			1.40102			0.700510	−	0.713643i	$-0.252956\pi$
$941$	42.9773			1.40102			0.700510	+	0.713643i	$0.252956\pi$
$942$		0			0
$943$	22.0653			0.718546
$944$		0			0
$945$			5.80210i			0.188742i
$946$		0			0
$947$	23.7141			0.770605			0.385302	−	0.922790i	$-0.374097\pi$
$947$	23.7141			0.770605			0.385302	+	0.922790i	$0.374097\pi$
$948$		0			0
$949$	−3.98886			−0.129484
$950$		0			0
$951$	−22.7619			−0.738106
$952$		0			0
$953$	25.3815			0.822186			0.411093	−	0.911593i	$-0.365147\pi$
$953$	25.3815			0.822186			0.411093	+	0.911593i	$0.365147\pi$
$954$		0			0
$955$	−16.2281			−0.525130
$956$		0			0
$957$	−5.25676			−0.169927
$958$		0			0
$959$		−	12.3446i		−	0.398628i
$960$		0			0
$961$	−27.8935			−0.899789
$962$		0			0
$963$	−4.06130			−0.130874
$964$		0			0
$965$		−	18.4087i		−	0.592598i
$966$		0			0
$967$	9.84890			0.316719			0.158360	−	0.987382i	$-0.449379\pi$
$967$	9.84890			0.316719			0.158360	+	0.987382i	$0.449379\pi$
$968$		0			0
$969$	8.32262	−	14.8801i	0.267361	−	0.478018i
$970$		0			0
$971$		−	15.5359i		−	0.498572i	−0.968430	−	0.249286i	$-0.919804\pi$
$971$		−	15.5359i		−	0.498572i	0.968430	−	0.249286i	$-0.0801959\pi$
$972$		0			0
$973$		−	9.10263i		−	0.291817i
$974$		0			0
$975$		−	7.29156i		−	0.233517i
$976$		0			0
$977$	−11.3011			−0.361556			−0.180778	−	0.983524i	$-0.557861\pi$
$977$	−11.3011			−0.361556			−0.180778	+	0.983524i	$0.557861\pi$
$978$		0			0
$979$	−4.94179			−0.157940
$980$		0			0
$981$	−15.9321			−0.508672
$982$		0			0
$983$			44.2735i			1.41210i	0.708160	+	0.706052i	$0.249526\pi$
$983$			44.2735i			1.41210i	−0.708160	+	0.706052i	$0.750474\pi$
$984$		0			0
$985$	1.17821			0.0375408
$986$		0			0
$987$			17.1171i			0.544844i
$988$		0			0
$989$	29.5592			0.939928
$990$		0			0
$991$			52.0174i			1.65239i	0.563386	+	0.826194i	$0.309498\pi$
$991$			52.0174i			1.65239i	−0.563386	+	0.826194i	$0.690502\pi$
$992$		0			0
$993$		−	1.20646i		−	0.0382858i
$994$		0			0
$995$		−	4.78744i		−	0.151772i
$996$		0			0
$997$	5.37281			0.170159			0.0850793	−	0.996374i	$-0.472886\pi$
$997$	5.37281			0.170159			0.0850793	+	0.996374i	$0.472886\pi$
$998$		0			0
$999$	−18.2134			−0.576245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2720.2.l.b.2481.25		36
4.3	odd	2		680.2.l.a.101.9	✓	36
8.3	odd	2		680.2.l.b.101.10	yes	36
8.5	even	2		2720.2.l.a.2481.11		36
17.16	even	2		2720.2.l.a.2481.12		36
68.67	odd	2		680.2.l.b.101.9	yes	36
136.67	odd	2		680.2.l.a.101.10	yes	36
136.101	even	2	inner	2720.2.l.b.2481.26		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.l.a.101.9	✓	36	4.3	odd	2
680.2.l.a.101.10	yes	36	136.67	odd	2
680.2.l.b.101.9	yes	36	68.67	odd	2
680.2.l.b.101.10	yes	36	8.3	odd	2
2720.2.l.a.2481.11		36	8.5	even	2
2720.2.l.a.2481.12		36	17.16	even	2
2720.2.l.b.2481.25		36	1.1	even	1	trivial
2720.2.l.b.2481.26		36	136.101	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2720 = 2^{5} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2720.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.46409 q^{3} +1.00000 q^{5} -1.02762i q^{7} -0.856436 q^{9} +0.737326 q^{11} -4.98026i q^{13} +1.46409 q^{15} +(3.59849 + 2.01268i) q^{17} -2.82434i q^{19} -1.50452i q^{21} -4.95563i q^{23} +1.00000 q^{25} -5.64618 q^{27} -4.86957 q^{29} -7.67421i q^{31} +1.07951 q^{33} -1.02762i q^{35} +3.22579 q^{37} -7.29156i q^{39} +4.45258i q^{41} +5.96478i q^{43} -0.856436 q^{45} -11.3771 q^{47} +5.94401 q^{49} +(5.26852 + 2.94675i) q^{51} -0.0517813i q^{53} +0.737326 q^{55} -4.13510i q^{57} -6.48292i q^{59} +14.4629 q^{61} +0.880087i q^{63} -4.98026i q^{65} -4.56989i q^{67} -7.25549i q^{69} -12.9933i q^{71} -0.800933i q^{73} +1.46409 q^{75} -0.757688i q^{77} +7.83068i q^{79} -5.69721 q^{81} -10.7517i q^{83} +(3.59849 + 2.01268i) q^{85} -7.12949 q^{87} -6.70231 q^{89} -5.11780 q^{91} -11.2357i q^{93} -2.82434i q^{95} +7.53723i q^{97} -0.631473 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 4 q^{3} + 36 q^{5} + 36 q^{9} + 8 q^{11} + 4 q^{15} + 36 q^{25} + 16 q^{27} - 8 q^{33} + 36 q^{45} - 20 q^{47} - 36 q^{49} + 8 q^{55} + 4 q^{75} + 44 q^{81} - 24 q^{87} + 56 q^{91} + 40 q^{99}+O(q^{100}) \) 36 * q + 4 * q^3 + 36 * q^5 + 36 * q^9 + 8 * q^11 + 4 * q^15 + 36 * q^25 + 16 * q^27 - 8 * q^33 + 36 * q^45 - 20 * q^47 - 36 * q^49 + 8 * q^55 + 4 * q^75 + 44 * q^81 - 24 * q^87 + 56 * q^91 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more