LMFDB - Embedded newform 720.4.f.i.289.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,4,Mod(289,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("720.289"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$720 = 2^{4} \cdot 3^{2} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	720.f (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-22,0,0,0,0,0,148,0,0,0,0,0,0,0,-160] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$42.4813752041$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{129})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 65x^{2} + 1024$ x^4 + 65*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.3
Root			$-6.17891i$ of defining polynomial
Character	$\chi$	$=$	720.289
Dual form			720.4.f.i.289.4

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.178908 - 11.1789i) q^{5} +35.0735i q^{7} +25.6422 q^{11} -37.6422i q^{13} +95.7891i q^{17} +50.8625 q^{19} -110.863i q^{23} +(-124.936 - 4.00000i) q^{25} -54.5047 q^{29} -198.441 q^{31} +(392.083 + 6.27493i) q^{35} +266.945i q^{37} -103.853 q^{41} +108.000i q^{43} +597.009i q^{47} -887.147 q^{49} -305.642i q^{53} +(4.58760 - 286.652i) q^{55} +223.533 q^{59} +485.450 q^{61} +(-420.799 - 6.73450i) q^{65} +876.166i q^{67} +585.597 q^{71} +1137.60i q^{73} +899.360i q^{77} +685.009 q^{79} -305.725i q^{83} +(1070.82 + 17.1375i) q^{85} +887.175 q^{89} +1320.24 q^{91} +(9.09973 - 568.588i) q^{95} -556.550i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 22 q^{5} + 148 q^{11} - 160 q^{19} + 100 q^{29} + 24 q^{31} + 796 q^{35} - 688 q^{41} - 3276 q^{49} - 1072 q^{55} - 1332 q^{59} + 488 q^{61} - 820 q^{65} + 616 q^{71} + 2104 q^{79} + 2148 q^{85} + 1368 q^{89}+ \cdots + 2944 q^{95}+O(q^{100})$ 4 * q - 22 * q^5 + 148 * q^11 - 160 * q^19 + 100 * q^29 + 24 * q^31 + 796 * q^35 - 688 * q^41 - 3276 * q^49 - 1072 * q^55 - 1332 * q^59 + 488 * q^61 - 820 * q^65 + 616 * q^71 + 2104 * q^79 + 2148 * q^85 + 1368 * q^89 - 1352 * q^91 + 2944 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$ .

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.178908	−	11.1789i	0.0160020	−	0.999872i
$5$	0.178908	−	11.1789i	0.0160020	−	0.999872i
$6$		0			0
$7$			35.0735i			1.89379i	0.321545	+	0.946894i	$0.395798\pi$
$7$			35.0735i			1.89379i	−0.321545	+	0.946894i	$0.604202\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	25.6422			0.702855			0.351428	−	0.936215i	$-0.385696\pi$
$11$	25.6422			0.702855			0.351428	+	0.936215i	$0.385696\pi$
$12$		0			0
$13$		−	37.6422i		−	0.803082i	−0.915841	−	0.401541i	$-0.868475\pi$
$13$		−	37.6422i		−	0.803082i	0.915841	−	0.401541i	$-0.131525\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			95.7891i			1.36660i	0.730136	+	0.683302i	$0.239457\pi$
$17$			95.7891i			1.36660i	−0.730136	+	0.683302i	$0.760543\pi$
$18$		0			0
$19$	50.8625			0.614140			0.307070	−	0.951687i	$-0.400651\pi$
$19$	50.8625			0.614140			0.307070	+	0.951687i	$0.400651\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	110.863i		−	1.00506i	−0.864559	−	0.502531i	$-0.832402\pi$
$23$		−	110.863i		−	1.00506i	0.864559	−	0.502531i	$-0.167598\pi$
$24$		0			0
$25$	−124.936	−	4.00000i	−0.999488	−	0.0320000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−54.5047			−0.349009			−0.174505	−	0.984656i	$-0.555832\pi$
$29$	−54.5047			−0.349009			−0.174505	+	0.984656i	$0.555832\pi$
$30$		0			0
$31$	−198.441			−1.14971			−0.574855	−	0.818255i	$-0.694941\pi$
$31$	−198.441			−1.14971			−0.574855	+	0.818255i	$0.694941\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	392.083	+	6.27493i	1.89355	+	0.0303045i
$36$		0			0
$37$			266.945i			1.18610i	0.805167	+	0.593048i	$0.202076\pi$
$37$			266.945i			1.18610i	−0.805167	+	0.593048i	$0.797924\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−103.853			−0.395589			−0.197794	−	0.980244i	$-0.563378\pi$
$41$	−103.853			−0.395589			−0.197794	+	0.980244i	$0.563378\pi$
$42$		0			0
$43$			108.000i			0.383020i	0.981491	+	0.191510i	$0.0613384\pi$
$43$			108.000i			0.383020i	−0.981491	+	0.191510i	$0.938662\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			597.009i			1.85283i	0.376510	+	0.926413i	$0.377124\pi$
$47$			597.009i			1.85283i	−0.376510	+	0.926413i	$0.622876\pi$
$48$		0			0
$49$	−887.147			−2.58643
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	305.642i		−	0.792136i	−0.918221	−	0.396068i	$-0.870375\pi$
$53$		−	305.642i		−	0.792136i	0.918221	−	0.396068i	$-0.129625\pi$
$54$		0			0
$55$	4.58760	−	286.652i	0.0112471	−	0.702765i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	223.533			0.493246			0.246623	−	0.969111i	$-0.420679\pi$
$59$	223.533			0.493246			0.246623	+	0.969111i	$0.420679\pi$
$60$		0			0
$61$	485.450			1.01894			0.509471	−	0.860488i	$-0.329841\pi$
$61$	485.450			1.01894			0.509471	+	0.860488i	$0.329841\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−420.799	−	6.73450i	−0.802979	−	0.0128510i
$66$		0			0
$67$			876.166i			1.59762i	0.601582	+	0.798811i	$0.294537\pi$
$67$			876.166i			1.59762i	−0.601582	+	0.798811i	$0.705463\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	585.597			0.978839			0.489420	−	0.872048i	$-0.337209\pi$
$71$	585.597			0.978839			0.489420	+	0.872048i	$0.337209\pi$
$72$		0			0
$73$			1137.60i			1.82391i	0.410287	+	0.911957i	$0.365429\pi$
$73$			1137.60i			1.82391i	−0.410287	+	0.911957i	$0.634571\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			899.360i			1.33106i
$78$		0			0
$79$	685.009			0.975564			0.487782	−	0.872965i	$-0.337806\pi$
$79$	685.009			0.975564			0.487782	+	0.872965i	$0.337806\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	305.725i		−	0.404309i	−0.979354	−	0.202155i	$-0.935206\pi$
$83$		−	305.725i		−	0.404309i	0.979354	−	0.202155i	$-0.0647944\pi$
$84$		0			0
$85$	1070.82	+	17.1375i	1.36643	+	0.0218685i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	887.175			1.05663			0.528317	−	0.849047i	$-0.322823\pi$
$89$	887.175			1.05663			0.528317	+	0.849047i	$0.322823\pi$
$90$		0			0
$91$	1320.24			1.52087
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9.09973	−	568.588i	0.00982750	−	0.614062i
$96$		0			0
$97$		−	556.550i		−	0.582568i	−0.956637	−	0.291284i	$-0.905918\pi$
$97$		−	556.550i		−	0.582568i	0.956637	−	0.291284i	$-0.0940824\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1591.09			−1.56752			−0.783760	−	0.621063i	$-0.786701\pi$
$101$	−1591.09			−1.56752			−0.783760	+	0.621063i	$0.786701\pi$
$102$		0			0
$103$			1350.95i			1.29236i	0.763187	+	0.646178i	$0.223634\pi$
$103$			1350.95i			1.29236i	−0.763187	+	0.646178i	$0.776366\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1333.51i			1.20481i	0.798190	+	0.602406i	$0.205791\pi$
$107$			1333.51i			1.20481i	−0.798190	+	0.602406i	$0.794209\pi$
$108$		0			0
$109$	609.910			0.535952			0.267976	−	0.963426i	$-0.413645\pi$
$109$	609.910			0.535952			0.267976	+	0.963426i	$0.413645\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			241.808i			0.201304i	0.994922	+	0.100652i	$0.0320929\pi$
$113$			241.808i			0.201304i	−0.994922	+	0.100652i	$0.967907\pi$
$114$		0			0
$115$	−1239.32	−	19.8342i	−1.00493	−	0.0160831i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3359.65			−2.58806
$120$		0			0
$121$	−673.478			−0.505994
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−67.0677	+	1395.93i	−0.0479898	+	0.998848i
$126$		0			0
$127$			1045.94i			0.730802i	0.930850	+	0.365401i	$0.119068\pi$
$127$			1045.94i			0.730802i	−0.930850	+	0.365401i	$0.880932\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−886.524			−0.591267			−0.295633	−	0.955301i	$-0.595531\pi$
$131$	−886.524			−0.591267			−0.295633	+	0.955301i	$0.595531\pi$
$132$		0			0
$133$			1783.92i			1.16305i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			160.723i			0.100230i	0.998743	+	0.0501150i	$0.0159588\pi$
$137$			160.723i			0.100230i	−0.998743	+	0.0501150i	$0.984041\pi$
$138$		0			0
$139$	−57.2655			−0.0349439			−0.0174719	−	0.999847i	$-0.505562\pi$
$139$	−57.2655			−0.0349439			−0.0174719	+	0.999847i	$0.505562\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	965.228i		−	0.564450i
$144$		0			0
$145$	−9.75135	+	609.303i	−0.00558487	+	0.348965i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1105.15			−0.607633			−0.303817	−	0.952731i	$-0.598261\pi$
$149$	−1105.15			−0.607633			−0.303817	+	0.952731i	$0.598261\pi$
$150$		0			0
$151$	−2289.63			−1.23396			−0.616980	−	0.786979i	$-0.711644\pi$
$151$	−2289.63			−1.23396			−0.616980	+	0.786979i	$0.711644\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−35.5027	+	2218.35i	−0.0183977	+	1.14956i
$156$		0			0
$157$		−	161.514i		−	0.0821034i	−0.999157	−	0.0410517i	$-0.986929\pi$
$157$		−	161.514i		−	0.0821034i	0.999157	−	0.0410517i	$-0.0130708\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3888.33			1.90338
$162$		0			0
$163$			1594.15i			0.766032i	0.923742	+	0.383016i	$0.125115\pi$
$163$			1594.15i			0.766032i	−0.923742	+	0.383016i	$0.874885\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1017.39i			0.471427i	0.971823	+	0.235713i	$0.0757427\pi$
$167$			1017.39i			0.471427i	−0.971823	+	0.235713i	$0.924257\pi$
$168$		0			0
$169$	780.066			0.355060
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	2444.49i		−	1.07428i	−0.843492	−	0.537142i	$-0.819504\pi$
$173$		−	2444.49i		−	1.07428i	0.843492	−	0.537142i	$-0.180496\pi$
$174$		0			0
$175$	140.294	−	4381.94i	0.0606012	−	1.89282i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−222.780			−0.0930242			−0.0465121	−	0.998918i	$-0.514811\pi$
$179$	−222.780			−0.0930242			−0.0465121	+	0.998918i	$0.514811\pi$
$180$		0			0
$181$	−100.034			−0.0410798			−0.0205399	−	0.999789i	$-0.506539\pi$
$181$	−100.034			−0.0410798			−0.0205399	+	0.999789i	$0.506539\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2984.16	+	47.7588i	1.18594	+	0.0189800i
$186$		0			0
$187$			2456.24i			0.960525i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	702.403			0.266095			0.133047	−	0.991110i	$-0.457524\pi$
$191$	702.403			0.266095			0.133047	+	0.991110i	$0.457524\pi$
$192$		0			0
$193$			4126.63i			1.53907i	0.638602	+	0.769537i	$0.279513\pi$
$193$			4126.63i			1.53907i	−0.638602	+	0.769537i	$0.720487\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3104.45i		−	1.12276i	−0.827559	−	0.561378i	$-0.810271\pi$
$197$		−	3104.45i		−	1.12276i	0.827559	−	0.561378i	$-0.189729\pi$
$198$		0			0
$199$	−367.616			−0.130953			−0.0654764	−	0.997854i	$-0.520857\pi$
$199$	−367.616			−0.130953			−0.0654764	+	0.997854i	$0.520857\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1911.67i		−	0.660950i
$204$		0			0
$205$	−18.5802	+	1160.96i	−0.00633023	+	0.395538i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1304.23			0.431652
$210$		0			0
$211$	−2594.85			−0.846619			−0.423310	−	0.905985i	$-0.639132\pi$
$211$	−2594.85			−0.846619			−0.423310	+	0.905985i	$0.639132\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1207.32	+	19.3221i	0.382971	+	0.00612910i
$216$		0			0
$217$		−	6960.00i		−	2.17731i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3605.71			1.09749
$222$		0			0
$223$		−	1834.41i		−	0.550857i	−0.961321	−	0.275429i	$-0.911180\pi$
$223$		−	1834.41i		−	0.550857i	0.961321	−	0.275429i	$-0.0888198\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			175.811i			0.0514053i	0.999670	+	0.0257027i	$0.00818231\pi$
$227$			175.811i			0.0514053i	−0.999670	+	0.0257027i	$0.991818\pi$
$228$		0			0
$229$	−1622.35			−0.468158			−0.234079	−	0.972218i	$-0.575207\pi$
$229$	−1622.35			−0.468158			−0.234079	+	0.972218i	$0.575207\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3965.65i			1.11501i	0.830172	+	0.557507i	$0.188242\pi$
$233$			3965.65i			1.11501i	−0.830172	+	0.557507i	$0.811758\pi$
$234$		0			0
$235$	6673.91	+	106.810i	1.85259	+	0.0296490i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6323.25			1.71137			0.855685	−	0.517497i	$-0.173136\pi$
$239$	6323.25			1.71137			0.855685	+	0.517497i	$0.173136\pi$
$240$		0			0
$241$	3407.51			0.910775			0.455388	−	0.890293i	$-0.349501\pi$
$241$	3407.51			0.910775			0.455388	+	0.890293i	$0.349501\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−158.718	+	9917.33i	−0.0413882	+	2.58610i
$246$		0			0
$247$		−	1914.58i		−	0.493205i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1345.81			0.338433			0.169216	−	0.985579i	$-0.445876\pi$
$251$	1345.81			0.338433			0.169216	+	0.985579i	$0.445876\pi$
$252$		0			0
$253$		−	2842.76i		−	0.706414i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	4697.19i		−	1.14009i	−0.821614	−	0.570044i	$-0.806926\pi$
$257$		−	4697.19i		−	1.14009i	0.821614	−	0.570044i	$-0.193074\pi$
$258$		0			0
$259$	−9362.70			−2.24621
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	4700.66i		−	1.10211i	−0.834468	−	0.551056i	$-0.814225\pi$
$263$		−	4700.66i		−	1.10211i	0.834468	−	0.551056i	$-0.185775\pi$
$264$		0			0
$265$	−3416.75	−	54.6819i	−0.792034	−	0.0126758i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7962.67			−1.80481			−0.902403	−	0.430894i	$-0.858198\pi$
$269$	−7962.67			−1.80481			−0.902403	+	0.430894i	$0.858198\pi$
$270$		0			0
$271$	6122.73			1.37243			0.686217	−	0.727397i	$-0.259270\pi$
$271$	6122.73			1.37243			0.686217	+	0.727397i	$0.259270\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3203.63	−	102.569i	−0.702495	−	0.0224914i
$276$		0			0
$277$		−	8417.57i		−	1.82586i	−0.408117	−	0.912930i	$-0.633814\pi$
$277$		−	8417.57i		−	1.82586i	0.408117	−	0.912930i	$-0.366186\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3030.99			−0.643466			−0.321733	−	0.946830i	$-0.604265\pi$
$281$	−3030.99			−0.643466			−0.321733	+	0.946830i	$0.604265\pi$
$282$		0			0
$283$			2890.81i			0.607211i	0.952798	+	0.303606i	$0.0981905\pi$
$283$			2890.81i			0.607211i	−0.952798	+	0.303606i	$0.901809\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	3642.49i		−	0.749161i
$288$		0			0
$289$	−4262.55			−0.867606
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	8966.75i		−	1.78786i	−0.448206	−	0.893930i	$-0.647937\pi$
$293$		−	8966.75i		−	1.78786i	0.448206	−	0.893930i	$-0.352063\pi$
$294$		0			0
$295$	39.9919	−	2498.86i	0.00789295	−	0.493183i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4173.11			−0.807147
$300$		0			0
$301$	−3787.93			−0.725358
$302$		0			0
$303$		0			0
$304$		0			0
$305$	86.8511	−	5426.80i	0.0163052	−	1.01881i
$306$		0			0
$307$			3298.42i			0.613194i	0.951839	+	0.306597i	$0.0991904\pi$
$307$			3298.42i			0.613194i	−0.951839	+	0.306597i	$0.900810\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3394.92			−0.618998			−0.309499	−	0.950900i	$-0.600161\pi$
$311$	−3394.92			−0.618998			−0.309499	+	0.950900i	$0.600161\pi$
$312$		0			0
$313$			5946.95i			1.07393i	0.843603	+	0.536967i	$0.180430\pi$
$313$			5946.95i			1.07393i	−0.843603	+	0.536967i	$0.819570\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2520.83i		−	0.446637i	−0.974746	−	0.223318i	$-0.928311\pi$
$317$		−	2520.83i		−	0.446637i	0.974746	−	0.223318i	$-0.0716889\pi$
$318$		0			0
$319$	−1397.62			−0.245303
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4872.08i			0.839286i
$324$		0			0
$325$	−150.569	+	4702.86i	−0.0256986	+	0.802671i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−20939.2			−3.50886
$330$		0			0
$331$	4586.86			0.761682			0.380841	−	0.924641i	$-0.375635\pi$
$331$	4586.86			0.761682			0.380841	+	0.924641i	$0.375635\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	9794.58	+	156.753i	1.59742	+	0.0255652i
$336$		0			0
$337$		−	8582.12i		−	1.38723i	−0.720344	−	0.693617i	$-0.756016\pi$
$337$		−	8582.12i		−	1.38723i	0.720344	−	0.693617i	$-0.243984\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5088.45			−0.808080
$342$		0			0
$343$		−	19085.1i		−	3.00437i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			2539.38i			0.392857i	0.980518	+	0.196428i	$0.0629343\pi$
$347$			2539.38i			0.392857i	−0.980518	+	0.196428i	$0.937066\pi$
$348$		0			0
$349$	9002.82			1.38083			0.690415	−	0.723413i	$-0.257428\pi$
$349$	9002.82			1.38083			0.690415	+	0.723413i	$0.257428\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			3928.08i			0.592267i	0.955146	+	0.296134i	$0.0956974\pi$
$353$			3928.08i			0.592267i	−0.955146	+	0.296134i	$0.904303\pi$
$354$		0			0
$355$	104.768	−	6546.34i	0.0156634	−	0.978714i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10001.7			−1.47039			−0.735197	−	0.677854i	$-0.762910\pi$
$359$	−10001.7			−1.47039			−0.735197	+	0.677854i	$0.762910\pi$
$360$		0			0
$361$	−4272.00			−0.622832
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12717.1	+	203.526i	1.82368	+	0.0291863i
$366$		0			0
$367$		−	5967.79i		−	0.848818i	−0.905471	−	0.424409i	$-0.860482\pi$
$367$		−	5967.79i		−	0.848818i	0.905471	−	0.424409i	$-0.139518\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10719.9			1.50014
$372$		0			0
$373$		−	6931.55i		−	0.962204i	−0.876665	−	0.481102i	$-0.840237\pi$
$373$		−	6931.55i		−	0.962204i	0.876665	−	0.481102i	$-0.159763\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2051.68i			0.280283i
$378$		0			0
$379$	9711.70			1.31624			0.658122	−	0.752911i	$-0.271351\pi$
$379$	9711.70			1.31624			0.658122	+	0.752911i	$0.271351\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	5664.84i		−	0.755769i	−0.925853	−	0.377885i	$-0.876652\pi$
$383$		−	5664.84i		−	0.755769i	0.925853	−	0.377885i	$-0.123348\pi$
$384$		0			0
$385$	10053.9	+	160.903i	1.33089	+	0.0212997i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8918.17			1.16239			0.581195	−	0.813765i	$-0.302586\pi$
$389$	8918.17			1.16239			0.581195	+	0.813765i	$0.302586\pi$
$390$		0			0
$391$	10619.4			1.37352
$392$		0			0
$393$		0			0
$394$		0			0
$395$	122.554	−	7657.66i	0.0156110	−	0.975439i
$396$		0			0
$397$			12849.9i			1.62448i	0.583321	+	0.812242i	$0.301753\pi$
$397$			12849.9i			1.62448i	−0.583321	+	0.812242i	$0.698247\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3563.08			−0.443721			−0.221860	−	0.975078i	$-0.571213\pi$
$401$	−3563.08			−0.443721			−0.221860	+	0.975078i	$0.571213\pi$
$402$		0			0
$403$			7469.74i			0.923311i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6845.06i			0.833654i
$408$		0			0
$409$	2026.69			0.245020			0.122510	−	0.992467i	$-0.460906\pi$
$409$	2026.69			0.245020			0.122510	+	0.992467i	$0.460906\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			7840.07i			0.934104i
$414$		0			0
$415$	−3417.67	−	54.6968i	−0.404258	−	0.00646978i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1670.39			0.194759			0.0973793	−	0.995247i	$-0.468954\pi$
$419$	1670.39			0.194759			0.0973793	+	0.995247i	$0.468954\pi$
$420$		0			0
$421$	6079.66			0.703812			0.351906	−	0.936035i	$-0.385534\pi$
$421$	6079.66			0.703812			0.351906	+	0.936035i	$0.385534\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	383.156	−	11967.5i	0.0437313	−	1.36590i
$426$		0			0
$427$			17026.4i			1.92966i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1719.37			0.192155			0.0960777	−	0.995374i	$-0.469370\pi$
$431$	1719.37			0.192155			0.0960777	+	0.995374i	$0.469370\pi$
$432$		0			0
$433$		−	12024.9i		−	1.33459i	−0.744792	−	0.667296i	$-0.767451\pi$
$433$		−	12024.9i		−	1.33459i	0.744792	−	0.667296i	$-0.232549\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	5638.75i		−	0.617249i
$438$		0			0
$439$	−7542.97			−0.820060			−0.410030	−	0.912072i	$-0.634482\pi$
$439$	−7542.97			−0.820060			−0.410030	+	0.912072i	$0.634482\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			4578.13i			0.491001i	0.969396	+	0.245501i	$0.0789524\pi$
$443$			4578.13i			0.491001i	−0.969396	+	0.245501i	$0.921048\pi$
$444$		0			0
$445$	158.723	−	9917.65i	0.0169083	−	1.05650i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6875.21			0.722630			0.361315	−	0.932444i	$-0.382328\pi$
$449$	6875.21			0.722630			0.361315	+	0.932444i	$0.382328\pi$
$450$		0			0
$451$	−2663.02			−0.278042
$452$		0			0
$453$		0			0
$454$		0			0
$455$	236.202	−	14758.9i	0.0243370	−	1.52067i
$456$		0			0
$457$		−	7351.97i		−	0.752540i	−0.926510	−	0.376270i	$-0.877207\pi$
$457$		−	7351.97i		−	0.752540i	0.926510	−	0.376270i	$-0.122793\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15614.5			1.57752			0.788762	−	0.614698i	$-0.210722\pi$
$461$	15614.5			1.57752			0.788762	+	0.614698i	$0.210722\pi$
$462$		0			0
$463$		−	1684.73i		−	0.169106i	−0.996419	−	0.0845530i	$-0.973054\pi$
$463$		−	1684.73i		−	0.169106i	0.996419	−	0.0845530i	$-0.0269462\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			10235.2i			1.01420i	0.861888	+	0.507099i	$0.169282\pi$
$467$			10235.2i			1.01420i	−0.861888	+	0.507099i	$0.830718\pi$
$468$		0			0
$469$	−30730.2			−3.02556
$470$		0			0
$471$		0			0
$472$		0			0
$473$			2769.36i			0.269207i
$474$		0			0
$475$	−6354.56	−	203.450i	−0.613826	−	0.0196525i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12247.2			−1.16825			−0.584123	−	0.811665i	$-0.698561\pi$
$479$	−12247.2			−1.16825			−0.584123	+	0.811665i	$0.698561\pi$
$480$		0			0
$481$	10048.4			0.952532
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6221.62	−	99.5714i	−0.582493	−	0.00932228i
$486$		0			0
$487$			7895.91i			0.734698i	0.930083	+	0.367349i	$0.119735\pi$
$487$			7895.91i			0.734698i	−0.930083	+	0.367349i	$0.880265\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7625.48			−0.700882			−0.350441	−	0.936585i	$-0.613968\pi$
$491$	−7625.48			−0.700882			−0.350441	+	0.936585i	$0.613968\pi$
$492$		0			0
$493$		−	5220.96i		−	0.476958i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			20538.9i			1.85371i
$498$		0			0
$499$	−8655.23			−0.776476			−0.388238	−	0.921559i	$-0.626916\pi$
$499$	−8655.23			−0.776476			−0.388238	+	0.921559i	$0.626916\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	118.441i		−	0.0104990i	−0.999986	−	0.00524951i	$-0.998329\pi$
$503$		−	118.441i		−	0.0104990i	0.999986	−	0.00524951i	$-0.00167098\pi$
$504$		0			0
$505$	−284.660	+	17786.7i	−0.0250835	+	1.56732i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5359.43			0.466704			0.233352	−	0.972392i	$-0.425030\pi$
$509$	5359.43			0.466704			0.233352	+	0.972392i	$0.425030\pi$
$510$		0			0
$511$	−39899.5			−3.45411
$512$		0			0
$513$		0			0
$514$		0			0
$515$	15102.1	+	241.695i	1.29219	+	0.0206803i
$516$		0			0
$517$			15308.6i			1.30227i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10862.4			0.913414			0.456707	−	0.889617i	$-0.349029\pi$
$521$	10862.4			0.913414			0.456707	+	0.889617i	$0.349029\pi$
$522$		0			0
$523$		−	9553.39i		−	0.798740i	−0.916790	−	0.399370i	$-0.869229\pi$
$523$		−	9553.39i		−	0.798740i	0.916790	−	0.399370i	$-0.130771\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	19008.5i		−	1.57120i
$528$		0			0
$529$	−123.501			−0.0101505
$530$		0			0
$531$		0			0
$532$		0			0
$533$			3909.26i			0.317690i
$534$		0			0
$535$	14907.1	+	238.575i	1.20466	+	0.0192795i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−22748.4			−1.81789
$540$		0			0
$541$	−6132.47			−0.487348			−0.243674	−	0.969857i	$-0.578353\pi$
$541$	−6132.47			−0.487348			−0.243674	+	0.969857i	$0.578353\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	109.118	−	6818.12i	0.00857633	−	0.535883i
$546$		0			0
$547$			2853.85i			0.223075i	0.993760	+	0.111537i	$0.0355775\pi$
$547$			2853.85i			0.223075i	−0.993760	+	0.111537i	$0.964422\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2772.25			−0.214341
$552$		0			0
$553$			24025.6i			1.84751i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	5192.60i		−	0.395005i	−0.980302	−	0.197502i	$-0.936717\pi$
$557$		−	5192.60i		−	0.395005i	0.980302	−	0.197502i	$-0.0632830\pi$
$558$		0			0
$559$	4065.36			0.307596
$560$		0			0
$561$		0			0
$562$		0			0
$563$			10907.2i			0.816492i	0.912872	+	0.408246i	$0.133859\pi$
$563$			10907.2i			0.816492i	−0.912872	+	0.408246i	$0.866141\pi$
$564$		0			0
$565$	2703.15	+	43.2615i	0.201278	+	0.00322128i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−155.257			−0.0114389			−0.00571945	−	0.999984i	$-0.501821\pi$
$569$	−155.257			−0.0114389			−0.00571945	+	0.999984i	$0.501821\pi$
$570$		0			0
$571$	4925.15			0.360965			0.180483	−	0.983578i	$-0.442234\pi$
$571$	4925.15			0.360965			0.180483	+	0.983578i	$0.442234\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−443.450	+	13850.7i	−0.0321620	+	1.00455i
$576$		0			0
$577$			5292.05i			0.381822i	0.981607	+	0.190911i	$0.0611441\pi$
$577$			5292.05i			0.381822i	−0.981607	+	0.190911i	$0.938856\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10722.8			0.765677
$582$		0			0
$583$		−	7837.33i		−	0.556757i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			19658.5i			1.38227i	0.722727	+	0.691134i	$0.242888\pi$
$587$			19658.5i			1.38227i	−0.722727	+	0.691134i	$0.757112\pi$
$588$		0			0
$589$	−10093.2			−0.706083
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6578.08i		−	0.455530i	−0.973716	−	0.227765i	$-0.926858\pi$
$593$		−	6578.08i		−	0.455530i	0.973716	−	0.227765i	$-0.0731418\pi$
$594$		0			0
$595$	−601.070	+	37557.3i	−0.0414142	+	2.58773i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16915.9			−1.15386			−0.576931	−	0.816793i	$-0.695750\pi$
$599$	−16915.9			−1.15386			−0.576931	+	0.816793i	$0.695750\pi$
$600$		0			0
$601$	19801.4			1.34395			0.671977	−	0.740572i	$-0.265445\pi$
$601$	19801.4			1.34395			0.671977	+	0.740572i	$0.265445\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−120.491	+	7528.75i	−0.00809695	+	0.505930i
$606$		0			0
$607$		−	22498.4i		−	1.50442i	−0.658926	−	0.752208i	$-0.728989\pi$
$607$		−	22498.4i		−	1.50442i	0.658926	−	0.752208i	$-0.271011\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	22472.7			1.48797
$612$		0			0
$613$		−	23829.3i		−	1.57008i	−0.619446	−	0.785039i	$-0.712643\pi$
$613$		−	23829.3i		−	1.57008i	0.619446	−	0.785039i	$-0.287357\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	4277.52i		−	0.279103i	−0.990215	−	0.139551i	$-0.955434\pi$
$617$		−	4277.52i		−	0.279103i	0.990215	−	0.139551i	$-0.0445660\pi$
$618$		0			0
$619$	4995.99			0.324404			0.162202	−	0.986758i	$-0.448140\pi$
$619$	4995.99			0.324404			0.162202	+	0.986758i	$0.448140\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			31116.3i			2.00104i
$624$		0			0
$625$	15593.0	+	999.488i	0.997952	+	0.0639672i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−25570.5			−1.62092
$630$		0			0
$631$	11328.0			0.714675			0.357337	−	0.933975i	$-0.383685\pi$
$631$	11328.0			0.714675			0.357337	+	0.933975i	$0.383685\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11692.4	+	187.127i	0.730708	+	0.0116943i
$636$		0			0
$637$			33394.1i			2.07712i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5955.35			−0.366961			−0.183481	−	0.983023i	$-0.558736\pi$
$641$	−5955.35			−0.366961			−0.183481	+	0.983023i	$0.558736\pi$
$642$		0			0
$643$			15727.7i			0.964605i	0.876005	+	0.482302i	$0.160199\pi$
$643$			15727.7i			0.964605i	−0.876005	+	0.482302i	$0.839801\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	9582.24i		−	0.582252i	−0.956685	−	0.291126i	$-0.905970\pi$
$647$		−	9582.24i		−	0.582252i	0.956685	−	0.291126i	$-0.0940298\pi$
$648$		0			0
$649$	5731.87			0.346681
$650$		0			0
$651$		0			0
$652$		0			0
$653$			8313.70i			0.498224i	0.968475	+	0.249112i	$0.0801388\pi$
$653$			8313.70i			0.498224i	−0.968475	+	0.249112i	$0.919861\pi$
$654$		0			0
$655$	−158.606	+	9910.37i	−0.00946148	+	0.591191i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15095.5			−0.892317			−0.446158	−	0.894954i	$-0.647208\pi$
$659$	−15095.5			−0.892317			−0.446158	+	0.894954i	$0.647208\pi$
$660$		0			0
$661$	8266.98			0.486457			0.243229	−	0.969969i	$-0.421793\pi$
$661$	8266.98			0.486457			0.243229	+	0.969969i	$0.421793\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	19942.3	+	319.159i	1.16290	+	0.0186112i
$666$		0			0
$667$			6042.53i			0.350776i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12448.0			0.716170
$672$		0			0
$673$		−	11186.7i		−	0.640735i	−0.947293	−	0.320367i	$-0.896194\pi$
$673$		−	11186.7i		−	0.640735i	0.947293	−	0.320367i	$-0.103806\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	7675.84i		−	0.435756i	−0.975976	−	0.217878i	$-0.930087\pi$
$677$		−	7675.84i		−	0.435756i	0.975976	−	0.217878i	$-0.0699134\pi$
$678$		0			0
$679$	19520.1			1.10326
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	10550.9i		−	0.591099i	−0.955327	−	0.295549i	$-0.904497\pi$
$683$		−	10550.9i		−	0.591099i	0.955327	−	0.295549i	$-0.0955027\pi$
$684$		0			0
$685$	1796.71	+	28.7547i	0.100217	+	0.00160388i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−11505.0			−0.636150
$690$		0			0
$691$	26950.9			1.48373			0.741867	−	0.670547i	$-0.233940\pi$
$691$	26950.9			1.48373			0.741867	+	0.670547i	$0.233940\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.2453	+	640.166i	−0.000559173	+	0.0349394i
$696$		0			0
$697$		−	9947.99i		−	0.540613i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−12791.6			−0.689204			−0.344602	−	0.938749i	$-0.611986\pi$
$701$	−12791.6			−0.689204			−0.344602	+	0.938749i	$0.611986\pi$
$702$		0			0
$703$			13577.5i			0.728429i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	55805.1i		−	2.96855i
$708$		0			0
$709$	16238.3			0.860142			0.430071	−	0.902795i	$-0.358489\pi$
$709$	16238.3			0.860142			0.430071	+	0.902795i	$0.358489\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			21999.6i			1.15553i
$714$		0			0
$715$	−10790.2	−	172.687i	−0.564378	−	0.00903236i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24285.8			1.25968			0.629839	−	0.776726i	$-0.283121\pi$
$719$	24285.8			1.25968			0.629839	+	0.776726i	$0.283121\pi$
$720$		0			0
$721$	−47382.3			−2.44745
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6809.60	+	218.019i	0.348831	+	0.0111683i
$726$		0			0
$727$		−	4466.82i		−	0.227875i	−0.993488	−	0.113937i	$-0.963654\pi$
$727$		−	4466.82i		−	0.227875i	0.993488	−	0.113937i	$-0.0363464\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−10345.2			−0.523436
$732$		0			0
$733$			30802.1i			1.55212i	0.630661	+	0.776059i	$0.282784\pi$
$733$			30802.1i			1.55212i	−0.630661	+	0.776059i	$0.717216\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			22466.8i			1.12290i
$738$		0			0
$739$	−12920.5			−0.643151			−0.321576	−	0.946884i	$-0.604212\pi$
$739$	−12920.5			−0.643151			−0.321576	+	0.946884i	$0.604212\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	2571.29i		−	0.126960i	−0.997983	−	0.0634802i	$-0.979780\pi$
$743$		−	2571.29i		−	0.126960i	0.997983	−	0.0634802i	$-0.0202200\pi$
$744$		0			0
$745$	−197.720	+	12354.4i	−0.00972337	+	0.607555i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−46770.7			−2.28166
$750$		0			0
$751$	13427.4			0.652426			0.326213	−	0.945296i	$-0.394227\pi$
$751$	13427.4			0.652426			0.326213	+	0.945296i	$0.394227\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−409.635	+	25595.6i	−0.0197459	+	1.23380i
$756$		0			0
$757$			7103.66i			0.341066i	0.985352	+	0.170533i	$0.0545489\pi$
$757$			7103.66i			0.341066i	−0.985352	+	0.170533i	$0.945451\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19205.2			−0.914831			−0.457416	−	0.889253i	$-0.651225\pi$
$761$	−19205.2			−0.914831			−0.457416	+	0.889253i	$0.651225\pi$
$762$		0			0
$763$			21391.6i			1.01498i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	8414.27i		−	0.396117i
$768$		0			0
$769$	−19508.1			−0.914799			−0.457399	−	0.889261i	$-0.651219\pi$
$769$	−19508.1			−0.914799			−0.457399	+	0.889261i	$0.651219\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			38852.5i			1.80780i	0.427746	+	0.903899i	$0.359308\pi$
$773$			38852.5i			1.80780i	−0.427746	+	0.903899i	$0.640692\pi$
$774$		0			0
$775$	24792.4	+	793.763i	1.14912	+	0.0367907i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5282.23			−0.242947
$780$		0			0
$781$	15016.0			0.687982
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1805.55	−	28.8962i	−0.0820929	−	0.00131382i
$786$		0			0
$787$		−	24851.1i		−	1.12560i	−0.826594	−	0.562799i	$-0.809725\pi$
$787$		−	24851.1i		−	1.12560i	0.826594	−	0.562799i	$-0.190275\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−8481.04			−0.381228
$792$		0			0
$793$		−	18273.4i		−	0.818295i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	33040.1i		−	1.46843i	−0.678916	−	0.734216i	$-0.737550\pi$
$797$		−	33040.1i		−	1.46843i	0.678916	−	0.734216i	$-0.262450\pi$
$798$		0			0
$799$	−57187.0			−2.53208
$800$		0			0
$801$		0			0
$802$		0			0
$803$			29170.5i			1.28195i
$804$		0			0
$805$	695.655	−	43467.3i	0.0304579	−	1.90313i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	10756.6			0.467467			0.233734	−	0.972301i	$-0.424906\pi$
$809$	10756.6			0.467467			0.233734	+	0.972301i	$0.424906\pi$
$810$		0			0
$811$	5985.48			0.259160			0.129580	−	0.991569i	$-0.458637\pi$
$811$	5985.48			0.259160			0.129580	+	0.991569i	$0.458637\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	17820.8	+	285.206i	0.765934	+	0.0122581i
$816$		0			0
$817$			5493.15i			0.235228i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16426.4			−0.698276			−0.349138	−	0.937071i	$-0.613526\pi$
$821$	−16426.4			−0.698276			−0.349138	+	0.937071i	$0.613526\pi$
$822$		0			0
$823$			27124.3i			1.14884i	0.818561	+	0.574419i	$0.194772\pi$
$823$			27124.3i			1.14884i	−0.818561	+	0.574419i	$0.805228\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			32759.9i			1.37748i	0.725011	+	0.688738i	$0.241835\pi$
$827$			32759.9i			1.37748i	−0.725011	+	0.688738i	$0.758165\pi$
$828$		0			0
$829$	18211.8			0.762995			0.381498	−	0.924370i	$-0.375408\pi$
$829$	18211.8			0.762995			0.381498	+	0.924370i	$0.375408\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	84979.0i		−	3.53463i
$834$		0			0
$835$	11373.3	+	182.020i	0.471366	+	0.00754379i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	37500.6			1.54310			0.771552	−	0.636166i	$-0.219481\pi$
$839$	37500.6			1.54310			0.771552	+	0.636166i	$0.219481\pi$
$840$		0			0
$841$	−21418.2			−0.878192
$842$		0			0
$843$		0			0
$844$		0			0
$845$	139.560	−	8720.29i	0.00568168	−	0.355014i
$846$		0			0
$847$		−	23621.2i		−	0.958246i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	29594.2			1.19210
$852$		0			0
$853$			2421.81i			0.0972112i	0.998818	+	0.0486056i	$0.0154777\pi$
$853$			2421.81i			0.0972112i	−0.998818	+	0.0486056i	$0.984522\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			37700.0i			1.50269i	0.659908	+	0.751346i	$0.270595\pi$
$857$			37700.0i			1.50269i	−0.659908	+	0.751346i	$0.729405\pi$
$858$		0			0
$859$	−14709.0			−0.584242			−0.292121	−	0.956381i	$-0.594361\pi$
$859$	−14709.0			−0.584242			−0.292121	+	0.956381i	$0.594361\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	28950.4i		−	1.14193i	−0.820975	−	0.570963i	$-0.806570\pi$
$863$		−	28950.4i		−	1.14193i	0.820975	−	0.570963i	$-0.193430\pi$
$864$		0			0
$865$	−27326.7	−	437.340i	−1.07415	−	0.0171907i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	17565.1			0.685681
$870$		0			0
$871$	32980.8			1.28302
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−48960.2	−	2352.30i	−1.89161	−	0.0908824i
$876$		0			0
$877$			17501.3i			0.673860i	0.941530	+	0.336930i	$0.109389\pi$
$877$			17501.3i			0.673860i	−0.941530	+	0.336930i	$0.890611\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	45136.0			1.72607			0.863037	−	0.505140i	$-0.168559\pi$
$881$	45136.0			1.72607			0.863037	+	0.505140i	$0.168559\pi$
$882$		0			0
$883$		−	51564.8i		−	1.96522i	−0.185673	−	0.982612i	$-0.559446\pi$
$883$		−	51564.8i		−	1.96522i	0.185673	−	0.982612i	$-0.440554\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			19226.4i			0.727800i	0.931438	+	0.363900i	$0.118555\pi$
$887$			19226.4i			0.727800i	−0.931438	+	0.363900i	$0.881445\pi$
$888$		0			0
$889$	−36684.6			−1.38398
$890$		0			0
$891$		0			0
$892$		0			0
$893$			30365.4i			1.13789i
$894$		0			0
$895$	−39.8571	+	2490.43i	−0.00148858	+	0.0930123i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10816.0			0.401259
$900$		0			0
$901$	29277.2			1.08254
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−17.8969	+	1118.27i	−0.000657361	+	0.0410746i
$906$		0			0
$907$		−	32050.1i		−	1.17333i	−0.809831	−	0.586663i	$-0.800441\pi$
$907$		−	32050.1i		−	1.17333i	0.809831	−	0.586663i	$-0.199559\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−29674.6			−1.07921			−0.539606	−	0.841918i	$-0.681427\pi$
$911$	−29674.6			−1.07921			−0.539606	+	0.841918i	$0.681427\pi$
$912$		0			0
$913$		−	7839.46i		−	0.284171i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	31093.4i		−	1.11973i
$918$		0			0
$919$	−6029.99			−0.216443			−0.108221	−	0.994127i	$-0.534516\pi$
$919$	−6029.99			−0.216443			−0.108221	+	0.994127i	$0.534516\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	22043.2i		−	0.786088i
$924$		0			0
$925$	1067.78	−	33351.1i	0.0379551	−	1.18549i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−35365.2			−1.24897			−0.624485	−	0.781037i	$-0.714691\pi$
$929$	−35365.2			−1.24897			−0.624485	+	0.781037i	$0.714691\pi$
$930$		0			0
$931$	−45122.5			−1.58843
$932$		0			0
$933$		0			0
$934$		0			0
$935$	27458.1	+	439.442i	0.960402	+	0.0153704i
$936$		0			0
$937$			35637.0i			1.24249i	0.783618	+	0.621243i	$0.213372\pi$
$937$			35637.0i			1.24249i	−0.783618	+	0.621243i	$0.786628\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6609.70			0.228980			0.114490	−	0.993424i	$-0.463477\pi$
$941$	6609.70			0.228980			0.114490	+	0.993424i	$0.463477\pi$
$942$		0			0
$943$			11513.4i			0.397591i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	42650.9i		−	1.46354i	−0.681553	−	0.731768i	$-0.738695\pi$
$947$		−	42650.9i		−	1.46354i	0.681553	−	0.731768i	$-0.261305\pi$
$948$		0			0
$949$	42821.6			1.46475
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	38382.4i		−	1.30465i	−0.757941	−	0.652323i	$-0.773795\pi$
$953$		−	38382.4i		−	1.30465i	0.757941	−	0.652323i	$-0.226205\pi$
$954$		0			0
$955$	125.666	−	7852.10i	0.00425806	−	0.266061i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5637.11			−0.189814
$960$		0			0
$961$	9587.71			0.321832
$962$		0			0
$963$		0			0
$964$		0			0
$965$	46131.2	+	738.288i	1.53888	+	0.0246283i
$966$		0			0
$967$		−	2611.76i		−	0.0868549i	−0.999057	−	0.0434275i	$-0.986172\pi$
$967$		−	2611.76i		−	0.0868549i	0.999057	−	0.0434275i	$-0.0138277\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	7846.09			0.259313			0.129657	−	0.991559i	$-0.458613\pi$
$971$	7846.09			0.259313			0.129657	+	0.991559i	$0.458613\pi$
$972$		0			0
$973$		−	2008.50i		−	0.0661763i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			35667.2i			1.16796i	0.811769	+	0.583979i	$0.198505\pi$
$977$			35667.2i			1.16796i	−0.811769	+	0.583979i	$0.801495\pi$
$978$		0			0
$979$	22749.1			0.742661
$980$		0			0
$981$		0			0
$982$		0			0
$983$			13421.6i			0.435485i	0.976006	+	0.217742i	$0.0698693\pi$
$983$			13421.6i			0.435485i	−0.976006	+	0.217742i	$0.930131\pi$
$984$		0			0
$985$	−34704.4	−	555.412i	−1.12261	−	0.0179664i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	11973.2			0.384959
$990$		0			0
$991$	−40673.7			−1.30378			−0.651888	−	0.758315i	$-0.726023\pi$
$991$	−40673.7			−1.30378			−0.651888	+	0.758315i	$0.726023\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−65.7696	+	4109.54i	−0.00209551	+	0.130936i
$996$		0			0
$997$		−	29558.8i		−	0.938954i	−0.882945	−	0.469477i	$-0.844442\pi$
$997$		−	29558.8i		−	0.938954i	0.882945	−	0.469477i	$-0.155558\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.4.f.i.289.3		4
3.2	odd	2		240.4.f.g.49.3		4
4.3	odd	2		360.4.f.d.289.3		4
5.4	even	2	inner	720.4.f.i.289.4		4
12.11	even	2		120.4.f.d.49.1	✓	4
15.2	even	4		1200.4.a.bq.1.1		2
15.8	even	4		1200.4.a.bo.1.2		2
15.14	odd	2		240.4.f.g.49.1		4
20.3	even	4		1800.4.a.bl.1.1		2
20.7	even	4		1800.4.a.bn.1.2		2
20.19	odd	2		360.4.f.d.289.4		4
24.5	odd	2		960.4.f.o.769.2		4
24.11	even	2		960.4.f.n.769.4		4
60.23	odd	4		600.4.a.v.1.1		2
60.47	odd	4		600.4.a.t.1.2		2
60.59	even	2		120.4.f.d.49.3	yes	4
120.29	odd	2		960.4.f.o.769.4		4
120.59	even	2		960.4.f.n.769.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.f.d.49.1	✓	4	12.11	even	2
120.4.f.d.49.3	yes	4	60.59	even	2
240.4.f.g.49.1		4	15.14	odd	2
240.4.f.g.49.3		4	3.2	odd	2
360.4.f.d.289.3		4	4.3	odd	2
360.4.f.d.289.4		4	20.19	odd	2
600.4.a.t.1.2		2	60.47	odd	4
600.4.a.v.1.1		2	60.23	odd	4
720.4.f.i.289.3		4	1.1	even	1	trivial
720.4.f.i.289.4		4	5.4	even	2	inner
960.4.f.n.769.2		4	120.59	even	2
960.4.f.n.769.4		4	24.11	even	2
960.4.f.o.769.2		4	24.5	odd	2
960.4.f.o.769.4		4	120.29	odd	2
1200.4.a.bo.1.2		2	15.8	even	4
1200.4.a.bq.1.1		2	15.2	even	4
1800.4.a.bl.1.1		2	20.3	even	4
1800.4.a.bn.1.2		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	720.4.f.i.289.3

Level	$720$
Weight	$4$
Character	720.289
Analytic conductor	$42.481$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$