LMFDB - Embedded newform 2400.2.k.e.1201.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,2,Mod(1201,2400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2400.1201"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.214798336.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 $ x^8 - 2x^7 - 2x^5 + 9x^4 - 4x^3 - 16*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1201.7
Root			$-1.08003 + 0.912978i$ of defining polynomial
Character	$\chi$	$=$	2400.1201
Dual form			2400.2.k.e.1201.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.33411 q^{7} -1.00000 q^{9} -2.94418i q^{11} +2.04184i q^{13} -3.61241 q^{17} -5.35964i q^{19} +1.33411i q^{21} +8.59609 q^{23} -1.00000i q^{27} +5.26432i q^{29} +2.08134 q^{31} +2.94418 q^{33} -6.55659i q^{37} -2.04184 q^{39} +7.02786 q^{41} -8.50078i q^{43} +9.97204 q^{47} -5.22015 q^{49} -3.61241i q^{51} +6.12318i q^{53} +5.35964 q^{57} -4.75190i q^{59} -8.51476i q^{61} -1.33411 q^{63} +10.6961i q^{67} +8.59609i q^{69} +2.62405 q^{71} +15.3875 q^{73} -3.92787i q^{77} -10.4450 q^{79} +1.00000 q^{81} +1.52708i q^{83} -5.26432 q^{87} -12.7193 q^{89} +2.72404i q^{91} +2.08134i q^{93} +13.4450 q^{97} +2.94418i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7} - 8 q^{9} + 8 q^{23} - 8 q^{31} - 8 q^{57} - 8 q^{63} + 40 q^{71} + 16 q^{73} + 16 q^{79} + 8 q^{81} + 24 q^{87} + 8 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 - 8 * q^9 + 8 * q^23 - 8 * q^31 - 8 * q^57 - 8 * q^63 + 40 * q^71 + 16 * q^73 + 16 * q^79 + 8 * q^81 + 24 * q^87 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.33411	0.504247	0.252123	−	0.967695i	$-0.418871\pi$
$7$	1.33411	0.504247	0.252123	+	0.967695i	$0.418871\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 2.94418i	− 0.887705i	−0.896100	−	0.443853i	$-0.853611\pi$
$11$	− 2.94418i	− 0.887705i	0.896100	−	0.443853i	$-0.146389\pi$
$12$	0	0
$13$	2.04184i	0.566304i	0.959075	+	0.283152i	$0.0913800\pi$
$13$	2.04184i	0.566304i	−0.959075	+	0.283152i	$0.908620\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.61241	−0.876138	−0.438069	−	0.898941i	$-0.644337\pi$
$17$	−3.61241	−0.876138	−0.438069	+	0.898941i	$0.644337\pi$
$18$	0	0
$19$	− 5.35964i	− 1.22958i	−0.788689	−	0.614792i	$-0.789240\pi$
$19$	− 5.35964i	− 1.22958i	0.788689	−	0.614792i	$-0.210760\pi$
$20$	0	0
$21$	1.33411i	0.291127i
$22$	0	0
$23$	8.59609	1.79241	0.896205	−	0.443641i	$-0.146313\pi$
$23$	8.59609	1.79241	0.896205	+	0.443641i	$0.146313\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	5.26432i	0.977559i	0.872407	+	0.488780i	$0.162558\pi$
$29$	5.26432i	0.977559i	−0.872407	+	0.488780i	$0.837442\pi$
$30$	0	0
$31$	2.08134	0.373820	0.186910	−	0.982377i	$-0.440153\pi$
$31$	2.08134	0.373820	0.186910	+	0.982377i	$0.440153\pi$
$32$	0	0
$33$	2.94418	0.512517
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.55659i	− 1.07790i	−0.842339	−	0.538949i	$-0.818822\pi$
$37$	− 6.55659i	− 1.07790i	0.842339	−	0.538949i	$-0.181178\pi$
$38$	0	0
$39$	−2.04184	−0.326956
$40$	0	0
$41$	7.02786	1.09757	0.548784	−	0.835964i	$-0.315091\pi$
$41$	7.02786	1.09757	0.548784	+	0.835964i	$0.315091\pi$
$42$	0	0
$43$	− 8.50078i	− 1.29636i	−0.761489	−	0.648178i	$-0.775531\pi$
$43$	− 8.50078i	− 1.29636i	0.761489	−	0.648178i	$-0.224469\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.97204	1.45457	0.727286	−	0.686334i	$-0.240781\pi$
$47$	9.97204	1.45457	0.727286	+	0.686334i	$0.240781\pi$
$48$	0	0
$49$	−5.22015	−0.745735
$50$	0	0
$51$	− 3.61241i	− 0.505838i
$52$	0	0
$53$	6.12318i	0.841083i	0.907273	+	0.420541i	$0.138160\pi$
$53$	6.12318i	0.841083i	−0.907273	+	0.420541i	$0.861840\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.35964	0.709901
$58$	0	0
$59$	− 4.75190i	− 0.618644i	−0.950957	−	0.309322i	$-0.899898\pi$
$59$	− 4.75190i	− 0.618644i	0.950957	−	0.309322i	$-0.100102\pi$
$60$	0	0
$61$	− 8.51476i	− 1.09020i	−0.838370	−	0.545101i	$-0.816491\pi$
$61$	− 8.51476i	− 1.09020i	0.838370	−	0.545101i	$-0.183509\pi$
$62$	0	0
$63$	−1.33411	−0.168082
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.6961i	1.30673i	0.757041	+	0.653367i	$0.226644\pi$
$67$	10.6961i	1.30673i	−0.757041	+	0.653367i	$0.773356\pi$
$68$	0	0
$69$	8.59609i	1.03485i
$70$	0	0
$71$	2.62405	0.311418	0.155709	−	0.987803i	$-0.450234\pi$
$71$	2.62405	0.311418	0.155709	+	0.987803i	$0.450234\pi$
$72$	0	0
$73$	15.3875	1.80097	0.900485	−	0.434887i	$-0.143212\pi$
$73$	15.3875	1.80097	0.900485	+	0.434887i	$0.143212\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.92787i	− 0.447622i
$78$	0	0
$79$	−10.4450	−1.17515	−0.587575	−	0.809170i	$-0.699917\pi$
$79$	−10.4450	−1.17515	−0.587575	+	0.809170i	$0.699917\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.52708i	0.167619i	0.996482	+	0.0838095i	$0.0267087\pi$
$83$	1.52708i	0.167619i	−0.996482	+	0.0838095i	$0.973291\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.26432	−0.564394
$88$	0	0
$89$	−12.7193	−1.34824	−0.674120	−	0.738622i	$-0.735477\pi$
$89$	−12.7193	−1.34824	−0.674120	+	0.738622i	$0.735477\pi$
$90$	0	0
$91$	2.72404i	0.285557i
$92$	0	0
$93$	2.08134i	0.215825i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	13.4450	1.36513	0.682565	−	0.730825i	$-0.260865\pi$
$97$	13.4450	1.36513	0.682565	+	0.730825i	$0.260865\pi$
$98$	0	0
$99$	2.94418i	0.295902i
$100$	0	0
$101$	10.1232i	1.00729i	0.863910	+	0.503647i	$0.168009\pi$
$101$	10.1232i	1.00729i	−0.863910	+	0.503647i	$0.831991\pi$
$102$	0	0
$103$	10.7472	1.05896	0.529478	−	0.848324i	$-0.322388\pi$
$103$	10.7472	1.05896	0.529478	+	0.848324i	$0.322388\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.86518i	− 0.470335i	−0.971955	−	0.235167i	$-0.924436\pi$
$107$	− 4.86518i	− 0.470335i	0.971955	−	0.235167i	$-0.0755638\pi$
$108$	0	0
$109$	− 15.4573i	− 1.48054i	−0.672310	−	0.740270i	$-0.734698\pi$
$109$	− 15.4573i	− 1.48054i	0.672310	−	0.740270i	$-0.265302\pi$
$110$	0	0
$111$	6.55659	0.622324
$112$	0	0
$113$	9.88837	0.930220	0.465110	−	0.885253i	$-0.346015\pi$
$113$	9.88837	0.930220	0.465110	+	0.885253i	$0.346015\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.04184i	− 0.188768i
$118$	0	0
$119$	−4.81936	−0.441790
$120$	0	0
$121$	2.33178	0.211980
$122$	0	0
$123$	7.02786i	0.633681i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.75190	−0.776605	−0.388303	−	0.921532i	$-0.626938\pi$
$127$	−8.75190	−0.776605	−0.388303	+	0.921532i	$0.626938\pi$
$128$	0	0
$129$	8.50078	0.748452
$130$	0	0
$131$	0.471266i	0.0411747i	0.999788	+	0.0205874i	$0.00655362\pi$
$131$	0.471266i	0.0411747i	−0.999788	+	0.0205874i	$0.993446\pi$
$132$	0	0
$133$	− 7.15035i	− 0.620014i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.30382	−0.111393	−0.0556964	−	0.998448i	$-0.517738\pi$
$137$	−1.30382	−0.111393	−0.0556964	+	0.998448i	$0.517738\pi$
$138$	0	0
$139$	− 8.74723i	− 0.741930i	−0.928647	−	0.370965i	$-0.879027\pi$
$139$	− 8.74723i	− 0.741930i	0.928647	−	0.370965i	$-0.120973\pi$
$140$	0	0
$141$	9.97204i	0.839798i
$142$	0	0
$143$	6.01155	0.502711
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 5.22015i	− 0.430550i
$148$	0	0
$149$	15.1411i	1.24041i	0.784439	+	0.620205i	$0.212951\pi$
$149$	15.1411i	1.24041i	−0.784439	+	0.620205i	$0.787049\pi$
$150$	0	0
$151$	23.2782	1.89435	0.947176	−	0.320713i	$-0.103922\pi$
$151$	23.2782	1.89435	0.947176	+	0.320713i	$0.103922\pi$
$152$	0	0
$153$	3.61241	0.292046
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.8976i	1.74762i	0.486270	+	0.873809i	$0.338357\pi$
$157$	21.8976i	1.74762i	−0.486270	+	0.873809i	$0.661643\pi$
$158$	0	0
$159$	−6.12318	−0.485599
$160$	0	0
$161$	11.4682	0.903817
$162$	0	0
$163$	− 11.1643i	− 0.874458i	−0.899350	−	0.437229i	$-0.855960\pi$
$163$	− 11.1643i	− 0.874458i	0.899350	−	0.437229i	$-0.144040\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.0952	0.781192	0.390596	−	0.920562i	$-0.372269\pi$
$167$	10.0952	0.781192	0.390596	+	0.920562i	$0.372269\pi$
$168$	0	0
$169$	8.83090	0.679300
$170$	0	0
$171$	5.35964i	0.409862i
$172$	0	0
$173$	− 13.8162i	− 1.05043i	−0.850970	−	0.525215i	$-0.823985\pi$
$173$	− 13.8162i	− 1.05043i	0.850970	−	0.525215i	$-0.176015\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.75190	0.357174
$178$	0	0
$179$	21.9441i	1.64018i	0.572236	+	0.820089i	$0.306076\pi$
$179$	21.9441i	1.64018i	−0.572236	+	0.820089i	$0.693924\pi$
$180$	0	0
$181$	− 1.93021i	− 0.143471i	−0.997424	−	0.0717356i	$-0.977146\pi$
$181$	− 1.93021i	− 0.143471i	0.997424	−	0.0717356i	$-0.0228538\pi$
$182$	0	0
$183$	8.51476	0.629429
$184$	0	0
$185$	0	0
$186$	0	0
$187$	10.6356i	0.777752i
$188$	0	0
$189$	− 1.33411i	− 0.0970423i
$190$	0	0
$191$	−12.1232	−0.877202	−0.438601	−	0.898682i	$-0.644526\pi$
$191$	−12.1232	−0.877202	−0.438601	+	0.898682i	$0.644526\pi$
$192$	0	0
$193$	1.27431	0.0917267	0.0458634	−	0.998948i	$-0.485396\pi$
$193$	1.27431	0.0917267	0.0458634	+	0.998948i	$0.485396\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.30849i	0.235720i	0.993030	+	0.117860i	$0.0376034\pi$
$197$	3.30849i	0.235720i	−0.993030	+	0.117860i	$0.962397\pi$
$198$	0	0
$199$	−9.02718	−0.639920	−0.319960	−	0.947431i	$-0.603669\pi$
$199$	−9.02718	−0.639920	−0.319960	+	0.947431i	$0.603669\pi$
$200$	0	0
$201$	−10.6961	−0.754443
$202$	0	0
$203$	7.02319i	0.492931i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.59609	−0.597470
$208$	0	0
$209$	−15.7798	−1.09151
$210$	0	0
$211$	− 6.61241i	− 0.455217i	−0.973753	−	0.227608i	$-0.926909\pi$
$211$	− 6.61241i	− 0.455217i	0.973753	−	0.227608i	$-0.0730906\pi$
$212$	0	0
$213$	2.62405i	0.179797i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.77674	0.188497
$218$	0	0
$219$	15.3875i	1.03979i
$220$	0	0
$221$	− 7.37595i	− 0.496160i
$222$	0	0
$223$	0.833237	0.0557976	0.0278988	−	0.999611i	$-0.491118\pi$
$223$	0.833237	0.0557976	0.0278988	+	0.999611i	$0.491118\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.9999i	− 0.730089i	−0.930990	−	0.365045i	$-0.881054\pi$
$227$	− 10.9999i	− 0.730089i	0.930990	−	0.365045i	$-0.118946\pi$
$228$	0	0
$229$	− 15.2061i	− 1.00485i	−0.864622	−	0.502423i	$-0.832442\pi$
$229$	− 15.2061i	− 1.00485i	0.864622	−	0.502423i	$-0.167558\pi$
$230$	0	0
$231$	3.92787	0.258435
$232$	0	0
$233$	−2.47594	−0.162204	−0.0811020	−	0.996706i	$-0.525844\pi$
$233$	−2.47594	−0.162204	−0.0811020	+	0.996706i	$0.525844\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 10.4450i	− 0.678473i
$238$	0	0
$239$	21.0737	1.36314	0.681572	−	0.731751i	$-0.261297\pi$
$239$	21.0737	1.36314	0.681572	+	0.731751i	$0.261297\pi$
$240$	0	0
$241$	−6.10852	−0.393484	−0.196742	−	0.980455i	$-0.563036\pi$
$241$	−6.10852	−0.393484	−0.196742	+	0.980455i	$0.563036\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.9435	0.696318
$248$	0	0
$249$	−1.52708	−0.0967748
$250$	0	0
$251$	22.5286i	1.42199i	0.703195	+	0.710997i	$0.251756\pi$
$251$	22.5286i	1.42199i	−0.703195	+	0.710997i	$0.748244\pi$
$252$	0	0
$253$	− 25.3085i	− 1.59113i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.5286	−0.906271	−0.453136	−	0.891442i	$-0.649695\pi$
$257$	−14.5286	−0.906271	−0.453136	+	0.891442i	$0.649695\pi$
$258$	0	0
$259$	− 8.74723i	− 0.543526i
$260$	0	0
$261$	− 5.26432i	− 0.325853i
$262$	0	0
$263$	5.29694	0.326624	0.163312	−	0.986575i	$-0.447782\pi$
$263$	5.29694	0.326624	0.163312	+	0.986575i	$0.447782\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 12.7193i	− 0.778407i
$268$	0	0
$269$	− 27.0737i	− 1.65071i	−0.564613	−	0.825356i	$-0.690975\pi$
$269$	− 27.0737i	− 1.65071i	0.564613	−	0.825356i	$-0.309025\pi$
$270$	0	0
$271$	−15.8604	−0.963451	−0.481726	−	0.876322i	$-0.659990\pi$
$271$	−15.8604	−0.963451	−0.481726	+	0.876322i	$0.659990\pi$
$272$	0	0
$273$	−2.72404	−0.164866
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.98592i	0.599996i	0.953940	+	0.299998i	$0.0969860\pi$
$277$	9.98592i	0.599996i	−0.953940	+	0.299998i	$0.903014\pi$
$278$	0	0
$279$	−2.08134	−0.124607
$280$	0	0
$281$	13.4218	0.800676	0.400338	−	0.916368i	$-0.368893\pi$
$281$	13.4218	0.800676	0.400338	+	0.916368i	$0.368893\pi$
$282$	0	0
$283$	3.83722i	0.228099i	0.993475	+	0.114050i	$0.0363823\pi$
$283$	3.83722i	0.228099i	−0.993475	+	0.114050i	$0.963618\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.37595	0.553445
$288$	0	0
$289$	−3.95051	−0.232383
$290$	0	0
$291$	13.4450i	0.788158i
$292$	0	0
$293$	26.4450i	1.54493i	0.635057	+	0.772466i	$0.280977\pi$
$293$	26.4450i	1.54493i	−0.635057	+	0.772466i	$0.719023\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.94418	−0.170839
$298$	0	0
$299$	17.5518i	1.01505i
$300$	0	0
$301$	− 11.3410i	− 0.653684i
$302$	0	0
$303$	−10.1232	−0.581561
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.27596i	0.0728230i	0.999337	+	0.0364115i	$0.0115927\pi$
$307$	1.27596i	0.0728230i	−0.999337	+	0.0364115i	$0.988407\pi$
$308$	0	0
$309$	10.7472i	0.611388i
$310$	0	0
$311$	−2.44496	−0.138641	−0.0693205	−	0.997594i	$-0.522083\pi$
$311$	−2.44496	−0.138641	−0.0693205	+	0.997594i	$0.522083\pi$
$312$	0	0
$313$	−22.8325	−1.29057	−0.645283	−	0.763943i	$-0.723261\pi$
$313$	−22.8325	−1.29057	−0.645283	+	0.763943i	$0.723261\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.11163i	0.118601i	0.998240	+	0.0593005i	$0.0188870\pi$
$317$	2.11163i	0.118601i	−0.998240	+	0.0593005i	$0.981113\pi$
$318$	0	0
$319$	15.4991	0.867784
$320$	0	0
$321$	4.86518	0.271548
$322$	0	0
$323$	19.3612i	1.07729i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.4573	0.854790
$328$	0	0
$329$	13.3038	0.733463
$330$	0	0
$331$	− 23.2248i	− 1.27655i	−0.769808	−	0.638276i	$-0.779648\pi$
$331$	− 23.2248i	− 1.27655i	0.769808	−	0.638276i	$-0.220352\pi$
$332$	0	0
$333$	6.55659i	0.359299i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.8884	−0.702074	−0.351037	−	0.936362i	$-0.614171\pi$
$337$	−12.8884	−0.702074	−0.351037	+	0.936362i	$0.614171\pi$
$338$	0	0
$339$	9.88837i	0.537063i
$340$	0	0
$341$	− 6.12785i	− 0.331841i
$342$	0	0
$343$	−16.3030	−0.880281
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.79827i	− 0.364951i	−0.983210	−	0.182475i	$-0.941589\pi$
$347$	− 6.79827i	− 0.364951i	0.983210	−	0.182475i	$-0.0584109\pi$
$348$	0	0
$349$	− 34.6076i	− 1.85250i	−0.376904	−	0.926252i	$-0.623011\pi$
$349$	− 34.6076i	− 1.85250i	0.376904	−	0.926252i	$-0.376989\pi$
$350$	0	0
$351$	2.04184	0.108985
$352$	0	0
$353$	−12.2433	−0.651647	−0.325823	−	0.945431i	$-0.605642\pi$
$353$	−12.2433	−0.651647	−0.325823	+	0.945431i	$0.605642\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.81936i	− 0.255067i
$358$	0	0
$359$	2.01622	0.106412	0.0532059	−	0.998584i	$-0.483056\pi$
$359$	2.01622	0.106412	0.0532059	+	0.998584i	$0.483056\pi$
$360$	0	0
$361$	−9.72569	−0.511878
$362$	0	0
$363$	2.33178i	0.122387i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.4131	−0.700159	−0.350079	−	0.936720i	$-0.613845\pi$
$367$	−13.4131	−0.700159	−0.350079	+	0.936720i	$0.613845\pi$
$368$	0	0
$369$	−7.02786	−0.365856
$370$	0	0
$371$	8.16900i	0.424113i
$372$	0	0
$373$	− 10.0976i	− 0.522832i	−0.965226	−	0.261416i	$-0.915811\pi$
$373$	− 10.0976i	− 0.522832i	0.965226	−	0.261416i	$-0.0841894\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.7489	−0.553595
$378$	0	0
$379$	− 18.2775i	− 0.938853i	−0.882972	−	0.469426i	$-0.844461\pi$
$379$	− 18.2775i	− 0.938853i	0.882972	−	0.469426i	$-0.155539\pi$
$380$	0	0
$381$	− 8.75190i	− 0.448373i
$382$	0	0
$383$	−11.7734	−0.601594	−0.300797	−	0.953688i	$-0.597253\pi$
$383$	−11.7734	−0.601594	−0.300797	+	0.953688i	$0.597253\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.50078i	0.432119i
$388$	0	0
$389$	33.4270i	1.69482i	0.530942	+	0.847408i	$0.321838\pi$
$389$	33.4270i	1.69482i	−0.530942	+	0.847408i	$0.678162\pi$
$390$	0	0
$391$	−31.0526	−1.57040
$392$	0	0
$393$	−0.471266	−0.0237722
$394$	0	0
$395$	0	0
$396$	0	0
$397$	39.0434i	1.95953i	0.200147	+	0.979766i	$0.435858\pi$
$397$	39.0434i	1.95953i	−0.200147	+	0.979766i	$0.564142\pi$
$398$	0	0
$399$	7.15035	0.357965
$400$	0	0
$401$	−24.6140	−1.22916	−0.614581	−	0.788853i	$-0.710675\pi$
$401$	−24.6140	−1.22916	−0.614581	+	0.788853i	$0.710675\pi$
$402$	0	0
$403$	4.24976i	0.211695i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−19.3038	−0.956855
$408$	0	0
$409$	−14.5024	−0.717099	−0.358550	−	0.933511i	$-0.616729\pi$
$409$	−14.5024	−0.717099	−0.358550	+	0.933511i	$0.616729\pi$
$410$	0	0
$411$	− 1.30382i	− 0.0643127i
$412$	0	0
$413$	− 6.33956i	− 0.311949i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.74723	0.428354
$418$	0	0
$419$	− 12.6419i	− 0.617598i	−0.951127	−	0.308799i	$-0.900073\pi$
$419$	− 12.6419i	− 0.617598i	0.951127	−	0.308799i	$-0.0999271\pi$
$420$	0	0
$421$	16.8389i	0.820677i	0.911933	+	0.410338i	$0.134589\pi$
$421$	16.8389i	0.820677i	−0.911933	+	0.410338i	$0.865411\pi$
$422$	0	0
$423$	−9.97204	−0.484857
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 11.3596i	− 0.549731i
$428$	0	0
$429$	6.01155i	0.290240i
$430$	0	0
$431$	−5.98845	−0.288454	−0.144227	−	0.989545i	$-0.546070\pi$
$431$	−5.98845	−0.288454	−0.144227	+	0.989545i	$0.546070\pi$
$432$	0	0
$433$	2.22482	0.106918	0.0534589	−	0.998570i	$-0.482975\pi$
$433$	2.22482	0.106918	0.0534589	+	0.998570i	$0.482975\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 46.0719i	− 2.20392i
$438$	0	0
$439$	2.30460	0.109993	0.0549963	−	0.998487i	$-0.482485\pi$
$439$	2.30460	0.109993	0.0549963	+	0.998487i	$0.482485\pi$
$440$	0	0
$441$	5.22015	0.248578
$442$	0	0
$443$	− 22.1347i	− 1.05165i	−0.850592	−	0.525826i	$-0.823756\pi$
$443$	− 22.1347i	− 1.05165i	0.850592	−	0.525826i	$-0.176244\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−15.1411	−0.716151
$448$	0	0
$449$	21.5861	1.01871	0.509356	−	0.860556i	$-0.329884\pi$
$449$	21.5861	1.01871	0.509356	+	0.860556i	$0.329884\pi$
$450$	0	0
$451$	− 20.6913i	− 0.974316i
$452$	0	0
$453$	23.2782i	1.09371i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.50088	0.116986	0.0584930	−	0.998288i	$-0.481370\pi$
$457$	2.50088	0.116986	0.0584930	+	0.998288i	$0.481370\pi$
$458$	0	0
$459$	3.61241i	0.168613i
$460$	0	0
$461$	2.59609i	0.120912i	0.998171	+	0.0604561i	$0.0192555\pi$
$461$	2.59609i	0.120912i	−0.998171	+	0.0604561i	$0.980744\pi$
$462$	0	0
$463$	−27.8604	−1.29478	−0.647392	−	0.762158i	$-0.724140\pi$
$463$	−27.8604	−1.29478	−0.647392	+	0.762158i	$0.724140\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.75200i	− 0.266171i	−0.991105	−	0.133085i	$-0.957512\pi$
$467$	− 5.75200i	− 0.266171i	0.991105	−	0.133085i	$-0.0424884\pi$
$468$	0	0
$469$	14.2698i	0.658917i
$470$	0	0
$471$	−21.8976	−1.00899
$472$	0	0
$473$	−25.0279	−1.15078
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6.12318i	− 0.280361i
$478$	0	0
$479$	12.5473	0.573299	0.286649	−	0.958036i	$-0.407459\pi$
$479$	12.5473	0.573299	0.286649	+	0.958036i	$0.407459\pi$
$480$	0	0
$481$	13.3875	0.610417
$482$	0	0
$483$	11.4682i	0.521819i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8.60530	−0.389944	−0.194972	−	0.980809i	$-0.562462\pi$
$487$	−8.60530	−0.389944	−0.194972	+	0.980809i	$0.562462\pi$
$488$	0	0
$489$	11.1643	0.504868
$490$	0	0
$491$	− 36.8866i	− 1.66467i	−0.554273	−	0.832335i	$-0.687004\pi$
$491$	− 36.8866i	− 1.66467i	0.554273	−	0.832335i	$-0.312996\pi$
$492$	0	0
$493$	− 19.0169i	− 0.856477i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.50078	0.157031
$498$	0	0
$499$	− 36.2496i	− 1.62275i	−0.584524	−	0.811377i	$-0.698719\pi$
$499$	− 36.2496i	− 1.62275i	0.584524	−	0.811377i	$-0.301281\pi$
$500$	0	0
$501$	10.0952i	0.451021i
$502$	0	0
$503$	−23.3527	−1.04124	−0.520622	−	0.853787i	$-0.674300\pi$
$503$	−23.3527	−1.04124	−0.520622	+	0.853787i	$0.674300\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.83090i	0.392194i
$508$	0	0
$509$	− 3.35506i	− 0.148711i	−0.997232	−	0.0743553i	$-0.976310\pi$
$509$	− 3.35506i	− 0.148711i	0.997232	−	0.0743553i	$-0.0236899\pi$
$510$	0	0
$511$	20.5286	0.908133
$512$	0	0
$513$	−5.35964	−0.236634
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 29.3595i	− 1.29123i
$518$	0	0
$519$	13.8162	0.606466
$520$	0	0
$521$	−33.6029	−1.47217	−0.736084	−	0.676890i	$-0.763327\pi$
$521$	−33.6029	−1.47217	−0.736084	+	0.676890i	$0.763327\pi$
$522$	0	0
$523$	− 0.965721i	− 0.0422280i	−0.999777	−	0.0211140i	$-0.993279\pi$
$523$	− 0.965721i	− 0.0422280i	0.999777	−	0.0211140i	$-0.00672130\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.51865	−0.327517
$528$	0	0
$529$	50.8928	2.21273
$530$	0	0
$531$	4.75190i	0.206215i
$532$	0	0
$533$	14.3497i	0.621556i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−21.9441	−0.946957
$538$	0	0
$539$	15.3691i	0.661993i
$540$	0	0
$541$	− 6.34877i	− 0.272955i	−0.990643	−	0.136478i	$-0.956422\pi$
$541$	− 6.34877i	− 0.272955i	0.990643	−	0.136478i	$-0.0435781\pi$
$542$	0	0
$543$	1.93021	0.0828331
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.07433i	0.0886921i	0.999016	+	0.0443460i	$0.0141204\pi$
$547$	2.07433i	0.0886921i	−0.999016	+	0.0443460i	$0.985880\pi$
$548$	0	0
$549$	8.51476i	0.363401i
$550$	0	0
$551$	28.2148	1.20199
$552$	0	0
$553$	−13.9347	−0.592566
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.6931i	1.17339i	0.809807	+	0.586696i	$0.199572\pi$
$557$	27.6931i	1.17339i	−0.809807	+	0.586696i	$0.800428\pi$
$558$	0	0
$559$	17.3572	0.734131
$560$	0	0
$561$	−10.6356	−0.449035
$562$	0	0
$563$	− 3.80771i	− 0.160476i	−0.996776	−	0.0802380i	$-0.974432\pi$
$563$	− 3.80771i	− 0.160476i	0.996776	−	0.0802380i	$-0.0255680\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.33411	0.0560274
$568$	0	0
$569$	38.6371	1.61975	0.809875	−	0.586603i	$-0.199535\pi$
$569$	38.6371	1.61975	0.809875	+	0.586603i	$0.199535\pi$
$570$	0	0
$571$	6.24976i	0.261544i	0.991412	+	0.130772i	$0.0417456\pi$
$571$	6.24976i	0.261544i	−0.991412	+	0.130772i	$0.958254\pi$
$572$	0	0
$573$	− 12.1232i	− 0.506453i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.17377	0.0904952	0.0452476	−	0.998976i	$-0.485592\pi$
$577$	2.17377	0.0904952	0.0452476	+	0.998976i	$0.485592\pi$
$578$	0	0
$579$	1.27431i	0.0529585i
$580$	0	0
$581$	2.03730i	0.0845213i
$582$	0	0
$583$	18.0278	0.746634
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.1688i	1.41030i	0.709059	+	0.705149i	$0.249120\pi$
$587$	34.1688i	1.41030i	−0.709059	+	0.705149i	$0.750880\pi$
$588$	0	0
$589$	− 11.1552i	− 0.459643i
$590$	0	0
$591$	−3.30849	−0.136093
$592$	0	0
$593$	−12.9952	−0.533650	−0.266825	−	0.963745i	$-0.585975\pi$
$593$	−12.9952	−0.533650	−0.266825	+	0.963745i	$0.585975\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 9.02718i	− 0.369458i
$598$	0	0
$599$	−47.2572	−1.93088	−0.965439	−	0.260628i	$-0.916071\pi$
$599$	−47.2572	−1.93088	−0.965439	+	0.260628i	$0.916071\pi$
$600$	0	0
$601$	−23.5007	−0.958613	−0.479306	−	0.877648i	$-0.659112\pi$
$601$	−23.5007	−0.958613	−0.479306	+	0.877648i	$0.659112\pi$
$602$	0	0
$603$	− 10.6961i	− 0.435578i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−0.218591	−0.00887233	−0.00443617	−	0.999990i	$-0.501412\pi$
$607$	−0.218591	−0.00887233	−0.00443617	+	0.999990i	$0.501412\pi$
$608$	0	0
$609$	−7.02319	−0.284594
$610$	0	0
$611$	20.3613i	0.823730i
$612$	0	0
$613$	35.7488i	1.44388i	0.691956	+	0.721940i	$0.256749\pi$
$613$	35.7488i	1.44388i	−0.691956	+	0.721940i	$0.743251\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.0836	1.33189	0.665947	−	0.745999i	$-0.268028\pi$
$617$	33.0836	1.33189	0.665947	+	0.745999i	$0.268028\pi$
$618$	0	0
$619$	− 25.1084i	− 1.00919i	−0.863355	−	0.504596i	$-0.831641\pi$
$619$	− 25.1084i	− 1.00919i	0.863355	−	0.504596i	$-0.168359\pi$
$620$	0	0
$621$	− 8.59609i	− 0.344949i
$622$	0	0
$623$	−16.9689	−0.679846
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 15.7798i	− 0.630183i
$628$	0	0
$629$	23.6851i	0.944386i
$630$	0	0
$631$	−23.2829	−0.926876	−0.463438	−	0.886129i	$-0.653384\pi$
$631$	−23.2829	−0.926876	−0.463438	+	0.886129i	$0.653384\pi$
$632$	0	0
$633$	6.61241	0.262820
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 10.6587i	− 0.422313i
$638$	0	0
$639$	−2.62405	−0.103806
$640$	0	0
$641$	−38.3021	−1.51284	−0.756420	−	0.654086i	$-0.773054\pi$
$641$	−38.3021	−1.51284	−0.756420	+	0.654086i	$0.773054\pi$
$642$	0	0
$643$	− 45.8045i	− 1.80635i	−0.429269	−	0.903177i	$-0.641229\pi$
$643$	− 45.8045i	− 1.80635i	0.429269	−	0.903177i	$-0.358771\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−48.1114	−1.89146	−0.945728	−	0.324960i	$-0.894649\pi$
$647$	−48.1114	−1.89146	−0.945728	+	0.324960i	$0.894649\pi$
$648$	0	0
$649$	−13.9905	−0.549174
$650$	0	0
$651$	2.77674i	0.108829i
$652$	0	0
$653$	38.3331i	1.50009i	0.661386	+	0.750046i	$0.269969\pi$
$653$	38.3331i	1.50009i	−0.661386	+	0.750046i	$0.730031\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−15.3875	−0.600323
$658$	0	0
$659$	− 5.03253i	− 0.196040i	−0.995184	−	0.0980198i	$-0.968749\pi$
$659$	− 5.03253i	− 0.196040i	0.995184	−	0.0980198i	$-0.0312508\pi$
$660$	0	0
$661$	17.4665i	0.679368i	0.940540	+	0.339684i	$0.110320\pi$
$661$	17.4665i	0.679368i	−0.940540	+	0.339684i	$0.889680\pi$
$662$	0	0
$663$	7.37595	0.286458
$664$	0	0
$665$	0	0
$666$	0	0
$667$	45.2526i	1.75219i
$668$	0	0
$669$	0.833237i	0.0322148i
$670$	0	0
$671$	−25.0690	−0.967779
$672$	0	0
$673$	−32.4448	−1.25065	−0.625327	−	0.780363i	$-0.715034\pi$
$673$	−32.4448	−1.25065	−0.625327	+	0.780363i	$0.715034\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.07213i	− 0.310237i	−0.987896	−	0.155119i	$-0.950424\pi$
$677$	− 8.07213i	− 0.310237i	0.987896	−	0.155119i	$-0.0495760\pi$
$678$	0	0
$679$	17.9371	0.688362
$680$	0	0
$681$	10.9999	0.421517
$682$	0	0
$683$	36.3380i	1.39043i	0.718799	+	0.695217i	$0.244692\pi$
$683$	36.3380i	1.39043i	−0.718799	+	0.695217i	$0.755308\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15.2061	0.580148
$688$	0	0
$689$	−12.5025	−0.476308
$690$	0	0
$691$	15.0016i	0.570686i	0.958425	+	0.285343i	$0.0921075\pi$
$691$	15.0016i	0.570686i	−0.958425	+	0.285343i	$0.907892\pi$
$692$	0	0
$693$	3.92787i	0.149207i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−25.3875	−0.961620
$698$	0	0
$699$	− 2.47594i	− 0.0936485i
$700$	0	0
$701$	13.2874i	0.501859i	0.968005	+	0.250929i	$0.0807362\pi$
$701$	13.2874i	0.501859i	−0.968005	+	0.250929i	$0.919264\pi$
$702$	0	0
$703$	−35.1409	−1.32537
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.5054i	0.507925i
$708$	0	0
$709$	37.8976i	1.42327i	0.702548	+	0.711637i	$0.252046\pi$
$709$	37.8976i	1.42327i	−0.702548	+	0.711637i	$0.747954\pi$
$710$	0	0
$711$	10.4450	0.391717
$712$	0	0
$713$	17.8914	0.670038
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.0737i	0.787011i
$718$	0	0
$719$	4.17909	0.155854	0.0779269	−	0.996959i	$-0.475170\pi$
$719$	4.17909	0.155854	0.0779269	+	0.996959i	$0.475170\pi$
$720$	0	0
$721$	14.3380	0.533975
$722$	0	0
$723$	− 6.10852i	− 0.227178i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.7727	0.992943	0.496471	−	0.868053i	$-0.334629\pi$
$727$	26.7727	0.992943	0.496471	+	0.868053i	$0.334629\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	30.7083i	1.13579i
$732$	0	0
$733$	− 21.3364i	− 0.788080i	−0.919093	−	0.394040i	$-0.871077\pi$
$733$	− 21.3364i	− 0.788080i	0.919093	−	0.394040i	$-0.128923\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31.4912	1.15999
$738$	0	0
$739$	13.1038i	0.482033i	0.970521	+	0.241016i	$0.0774807\pi$
$739$	13.1038i	0.482033i	−0.970521	+	0.241016i	$0.922519\pi$
$740$	0	0
$741$	10.9435i	0.402020i
$742$	0	0
$743$	−33.3595	−1.22384	−0.611921	−	0.790919i	$-0.709603\pi$
$743$	−33.3595	−1.22384	−0.611921	+	0.790919i	$0.709603\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.52708i	− 0.0558730i
$748$	0	0
$749$	− 6.49069i	− 0.237165i
$750$	0	0
$751$	1.92100	0.0700981	0.0350491	−	0.999386i	$-0.488841\pi$
$751$	1.92100	0.0700981	0.0350491	+	0.999386i	$0.488841\pi$
$752$	0	0
$753$	−22.5286	−0.820989
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 13.9908i	− 0.508504i	−0.967138	−	0.254252i	$-0.918171\pi$
$757$	− 13.9908i	− 0.508504i	0.967138	−	0.254252i	$-0.0818292\pi$
$758$	0	0
$759$	25.3085	0.918640
$760$	0	0
$761$	25.6618	0.930240	0.465120	−	0.885248i	$-0.346011\pi$
$761$	25.6618	0.930240	0.465120	+	0.885248i	$0.346011\pi$
$762$	0	0
$763$	− 20.6217i	− 0.746557i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.70260	0.350341
$768$	0	0
$769$	−12.3922	−0.446873	−0.223436	−	0.974719i	$-0.571728\pi$
$769$	−12.3922	−0.446873	−0.223436	+	0.974719i	$0.571728\pi$
$770$	0	0
$771$	− 14.5286i	− 0.523236i
$772$	0	0
$773$	− 38.4843i	− 1.38418i	−0.721810	−	0.692091i	$-0.756689\pi$
$773$	− 38.4843i	− 1.38418i	0.721810	−	0.692091i	$-0.243311\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.74723	0.313805
$778$	0	0
$779$	− 37.6668i	− 1.34955i
$780$	0	0
$781$	− 7.72569i	− 0.276447i
$782$	0	0
$783$	5.26432	0.188131
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.389147i	0.0138716i	0.999976	+	0.00693579i	$0.00220775\pi$
$787$	0.389147i	0.0138716i	−0.999976	+	0.00693579i	$0.997792\pi$
$788$	0	0
$789$	5.29694i	0.188576i
$790$	0	0
$791$	13.1922	0.469060
$792$	0	0
$793$	17.3857	0.617386
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0.854188i	0.0302569i	0.999886	+	0.0151284i	$0.00481572\pi$
$797$	0.854188i	0.0302569i	−0.999886	+	0.0151284i	$0.995184\pi$
$798$	0	0
$799$	−36.0231	−1.27441
$800$	0	0
$801$	12.7193	0.449413
$802$	0	0
$803$	− 45.3036i	− 1.59873i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	27.0737	0.953039
$808$	0	0
$809$	−10.4107	−0.366020	−0.183010	−	0.983111i	$-0.558584\pi$
$809$	−10.4107	−0.366020	−0.183010	+	0.983111i	$0.558584\pi$
$810$	0	0
$811$	6.08825i	0.213787i	0.994270	+	0.106894i	$0.0340904\pi$
$811$	6.08825i	0.213787i	−0.994270	+	0.106894i	$0.965910\pi$
$812$	0	0
$813$	− 15.8604i	− 0.556249i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−45.5611	−1.59398
$818$	0	0
$819$	− 2.72404i	− 0.0951856i
$820$	0	0
$821$	35.3908i	1.23515i	0.786513	+	0.617574i	$0.211884\pi$
$821$	35.3908i	1.23515i	−0.786513	+	0.617574i	$0.788116\pi$
$822$	0	0
$823$	16.2846	0.567646	0.283823	−	0.958877i	$-0.408397\pi$
$823$	16.2846	0.567646	0.283823	+	0.958877i	$0.408397\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 32.1362i	− 1.11748i	−0.829341	−	0.558742i	$-0.811284\pi$
$827$	− 32.1362i	− 1.11748i	0.829341	−	0.558742i	$-0.188716\pi$
$828$	0	0
$829$	− 22.4682i	− 0.780355i	−0.920740	−	0.390177i	$-0.872414\pi$
$829$	− 22.4682i	− 0.780355i	0.920740	−	0.390177i	$-0.127586\pi$
$830$	0	0
$831$	−9.98592	−0.346408
$832$	0	0
$833$	18.8573	0.653367
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.08134i	− 0.0719416i
$838$	0	0
$839$	16.1358	0.557070	0.278535	−	0.960426i	$-0.410151\pi$
$839$	16.1358	0.557070	0.278535	+	0.960426i	$0.410151\pi$
$840$	0	0
$841$	1.28695	0.0443777
$842$	0	0
$843$	13.4218i	0.462270i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.11085	0.106890
$848$	0	0
$849$	−3.83722	−0.131693
$850$	0	0
$851$	− 56.3611i	− 1.93203i
$852$	0	0
$853$	44.6262i	1.52797i	0.645233	+	0.763986i	$0.276760\pi$
$853$	44.6262i	1.52797i	−0.645233	+	0.763986i	$0.723240\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.52553	−0.154589	−0.0772945	−	0.997008i	$-0.524628\pi$
$857$	−4.52553	−0.154589	−0.0772945	+	0.997008i	$0.524628\pi$
$858$	0	0
$859$	42.7783i	1.45958i	0.683673	+	0.729788i	$0.260381\pi$
$859$	42.7783i	1.45958i	−0.683673	+	0.729788i	$0.739619\pi$
$860$	0	0
$861$	9.37595i	0.319531i
$862$	0	0
$863$	23.7734	0.809257	0.404629	−	0.914481i	$-0.367401\pi$
$863$	23.7734	0.809257	0.404629	+	0.914481i	$0.367401\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 3.95051i	− 0.134166i
$868$	0	0
$869$	30.7519i	1.04319i
$870$	0	0
$871$	−21.8397	−0.740009
$872$	0	0
$873$	−13.4450	−0.455043
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.1470i	0.443944i	0.975053	+	0.221972i	$0.0712494\pi$
$877$	13.1470i	0.443944i	−0.975053	+	0.221972i	$0.928751\pi$
$878$	0	0
$879$	−26.4450	−0.891966
$880$	0	0
$881$	38.9132	1.31102	0.655510	−	0.755187i	$-0.272454\pi$
$881$	38.9132	1.31102	0.655510	+	0.755187i	$0.272454\pi$
$882$	0	0
$883$	44.5843i	1.50038i	0.661223	+	0.750190i	$0.270038\pi$
$883$	44.5843i	1.50038i	−0.661223	+	0.750190i	$0.729962\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.3240	−1.08533	−0.542667	−	0.839948i	$-0.682585\pi$
$887$	−32.3240	−1.08533	−0.542667	+	0.839948i	$0.682585\pi$
$888$	0	0
$889$	−11.6760	−0.391601
$890$	0	0
$891$	− 2.94418i	− 0.0986339i
$892$	0	0
$893$	− 53.4465i	− 1.78852i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−17.5518	−0.586038
$898$	0	0
$899$	10.9568i	0.365431i
$900$	0	0
$901$	− 22.1194i	− 0.736904i
$902$	0	0
$903$	11.3410	0.377404
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 14.8309i	− 0.492452i	−0.969212	−	0.246226i	$-0.920809\pi$
$907$	− 14.8309i	− 0.492452i	0.969212	−	0.246226i	$-0.0791905\pi$
$908$	0	0
$909$	− 10.1232i	− 0.335765i
$910$	0	0
$911$	−11.6108	−0.384681	−0.192341	−	0.981328i	$-0.561608\pi$
$911$	−11.6108	−0.384681	−0.192341	+	0.981328i	$0.561608\pi$
$912$	0	0
$913$	4.49601	0.148796
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.628722i	0.0207622i
$918$	0	0
$919$	58.2518	1.92155	0.960775	−	0.277330i	$-0.0894495\pi$
$919$	58.2518	1.92155	0.960775	+	0.277330i	$0.0894495\pi$
$920$	0	0
$921$	−1.27596	−0.0420444
$922$	0	0
$923$	5.35789i	0.176357i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10.7472	−0.352985
$928$	0	0
$929$	18.4433	0.605105	0.302553	−	0.953133i	$-0.402161\pi$
$929$	18.4433	0.605105	0.302553	+	0.953133i	$0.402161\pi$
$930$	0	0
$931$	27.9781i	0.916944i
$932$	0	0
$933$	− 2.44496i	− 0.0800445i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.1005	0.525982	0.262991	−	0.964798i	$-0.415291\pi$
$937$	16.1005	0.525982	0.262991	+	0.964798i	$0.415291\pi$
$938$	0	0
$939$	− 22.8325i	− 0.745109i
$940$	0	0
$941$	− 32.0974i	− 1.04635i	−0.852226	−	0.523173i	$-0.824748\pi$
$941$	− 32.0974i	− 1.04635i	0.852226	−	0.523173i	$-0.175252\pi$
$942$	0	0
$943$	60.4121	1.96729
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.10998i	0.133556i	0.997768	+	0.0667782i	$0.0212720\pi$
$947$	4.10998i	0.133556i	−0.997768	+	0.0667782i	$0.978728\pi$
$948$	0	0
$949$	31.4188i	1.01990i
$950$	0	0
$951$	−2.11163	−0.0684743
$952$	0	0
$953$	31.7208	1.02754	0.513769	−	0.857928i	$-0.328249\pi$
$953$	31.7208	1.02754	0.513769	+	0.857928i	$0.328249\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	15.4991i	0.501016i
$958$	0	0
$959$	−1.73944	−0.0561695
$960$	0	0
$961$	−26.6680	−0.860259
$962$	0	0
$963$	4.86518i	0.156778i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	26.9936	0.868055	0.434027	−	0.900900i	$-0.357092\pi$
$967$	26.9936	0.868055	0.434027	+	0.900900i	$0.357092\pi$
$968$	0	0
$969$	−19.3612	−0.621971
$970$	0	0
$971$	− 14.0559i	− 0.451076i	−0.974234	−	0.225538i	$-0.927586\pi$
$971$	− 14.0559i	− 0.451076i	0.974234	−	0.225538i	$-0.0724139\pi$
$972$	0	0
$973$	− 11.6698i	− 0.374116i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.14251	0.0365520	0.0182760	−	0.999833i	$-0.494182\pi$
$977$	1.14251	0.0365520	0.0182760	+	0.999833i	$0.494182\pi$
$978$	0	0
$979$	37.4479i	1.19684i
$980$	0	0
$981$	15.4573i	0.493513i
$982$	0	0
$983$	−6.41720	−0.204677	−0.102338	−	0.994750i	$-0.532632\pi$
$983$	−6.41720	−0.204677	−0.102338	+	0.994750i	$0.532632\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13.3038i	0.423465i
$988$	0	0
$989$	− 73.0735i	− 2.32360i
$990$	0	0
$991$	7.39470	0.234900	0.117450	−	0.993079i	$-0.462528\pi$
$991$	7.39470	0.234900	0.117450	+	0.993079i	$0.462528\pi$
$992$	0	0
$993$	23.2248	0.737017
$994$	0	0
$995$	0	0
$996$	0	0
$997$	31.6649i	1.00284i	0.865205	+	0.501419i	$0.167188\pi$
$997$	31.6649i	1.00284i	−0.865205	+	0.501419i	$0.832812\pi$
$998$	0	0
$999$	−6.55659	−0.207441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.k.e.1201.7		8
3.2	odd	2		7200.2.k.s.3601.6		8
4.3	odd	2		600.2.k.e.301.4	yes	8
5.2	odd	4		2400.2.d.h.49.6		8
5.3	odd	4		2400.2.d.g.49.3		8
5.4	even	2		2400.2.k.d.1201.2		8
8.3	odd	2		600.2.k.e.301.3	yes	8
8.5	even	2	inner	2400.2.k.e.1201.3		8
12.11	even	2		1800.2.k.q.901.5		8
15.2	even	4		7200.2.d.t.2449.6		8
15.8	even	4		7200.2.d.s.2449.3		8
15.14	odd	2		7200.2.k.r.3601.4		8
20.3	even	4		600.2.d.h.349.8		8
20.7	even	4		600.2.d.g.349.1		8
20.19	odd	2		600.2.k.d.301.5	✓	8
24.5	odd	2		7200.2.k.s.3601.5		8
24.11	even	2		1800.2.k.q.901.6		8
40.3	even	4		600.2.d.g.349.2		8
40.13	odd	4		2400.2.d.h.49.3		8
40.19	odd	2		600.2.k.d.301.6	yes	8
40.27	even	4		600.2.d.h.349.7		8
40.29	even	2		2400.2.k.d.1201.6		8
40.37	odd	4		2400.2.d.g.49.6		8
60.23	odd	4		1800.2.d.s.1549.1		8
60.47	odd	4		1800.2.d.t.1549.8		8
60.59	even	2		1800.2.k.t.901.4		8
120.29	odd	2		7200.2.k.r.3601.3		8
120.53	even	4		7200.2.d.t.2449.3		8
120.59	even	2		1800.2.k.t.901.3		8
120.77	even	4		7200.2.d.s.2449.6		8
120.83	odd	4		1800.2.d.t.1549.7		8
120.107	odd	4		1800.2.d.s.1549.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.d.g.349.1		8	20.7	even	4
600.2.d.g.349.2		8	40.3	even	4
600.2.d.h.349.7		8	40.27	even	4
600.2.d.h.349.8		8	20.3	even	4
600.2.k.d.301.5	✓	8	20.19	odd	2
600.2.k.d.301.6	yes	8	40.19	odd	2
600.2.k.e.301.3	yes	8	8.3	odd	2
600.2.k.e.301.4	yes	8	4.3	odd	2
1800.2.d.s.1549.1		8	60.23	odd	4
1800.2.d.s.1549.2		8	120.107	odd	4
1800.2.d.t.1549.7		8	120.83	odd	4
1800.2.d.t.1549.8		8	60.47	odd	4
1800.2.k.q.901.5		8	12.11	even	2
1800.2.k.q.901.6		8	24.11	even	2
1800.2.k.t.901.3		8	120.59	even	2
1800.2.k.t.901.4		8	60.59	even	2
2400.2.d.g.49.3		8	5.3	odd	4
2400.2.d.g.49.6		8	40.37	odd	4
2400.2.d.h.49.3		8	40.13	odd	4
2400.2.d.h.49.6		8	5.2	odd	4
2400.2.k.d.1201.2		8	5.4	even	2
2400.2.k.d.1201.6		8	40.29	even	2
2400.2.k.e.1201.3		8	8.5	even	2	inner
2400.2.k.e.1201.7		8	1.1	even	1	trivial
7200.2.d.s.2449.3		8	15.8	even	4
7200.2.d.s.2449.6		8	120.77	even	4
7200.2.d.t.2449.3		8	120.53	even	4
7200.2.d.t.2449.6		8	15.2	even	4
7200.2.k.r.3601.3		8	120.29	odd	2
7200.2.k.r.3601.4		8	15.14	odd	2
7200.2.k.s.3601.5		8	24.5	odd	2
7200.2.k.s.3601.6		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.33411 q^{7} -1.00000 q^{9} -2.94418i q^{11} +2.04184i q^{13} -3.61241 q^{17} -5.35964i q^{19} +1.33411i q^{21} +8.59609 q^{23} -1.00000i q^{27} +5.26432i q^{29} +2.08134 q^{31} +2.94418 q^{33} -6.55659i q^{37} -2.04184 q^{39} +7.02786 q^{41} -8.50078i q^{43} +9.97204 q^{47} -5.22015 q^{49} -3.61241i q^{51} +6.12318i q^{53} +5.35964 q^{57} -4.75190i q^{59} -8.51476i q^{61} -1.33411 q^{63} +10.6961i q^{67} +8.59609i q^{69} +2.62405 q^{71} +15.3875 q^{73} -3.92787i q^{77} -10.4450 q^{79} +1.00000 q^{81} +1.52708i q^{83} -5.26432 q^{87} -12.7193 q^{89} +2.72404i q^{91} +2.08134i q^{93} +13.4450 q^{97} +2.94418i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7} - 8 q^{9} + 8 q^{23} - 8 q^{31} - 8 q^{57} - 8 q^{63} + 40 q^{71} + 16 q^{73} + 16 q^{79} + 8 q^{81} + 24 q^{87} + 8 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 - 8 * q^9 + 8 * q^23 - 8 * q^31 - 8 * q^57 - 8 * q^63 + 40 * q^71 + 16 * q^73 + 16 * q^79 + 8 * q^81 + 24 * q^87 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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