LMFDB - Embedded newform 2880.3.e.j.2431.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,3,Mod(2431,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2880.2431");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2880.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$78.4743161358$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.85100625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} + x^{5} + 3x^{4} + 2x^{3} - 8x^{2} - 8x + 16 $ x^8 - x^7 - 2x^6 + x^5 + 3x^4 + 2x^3 - 8x^2 - 8*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.5
Root			$1.40906 - 0.120653i$ of defining polynomial
Character	$\chi$	$=$	2880.2431
Dual form			2880.3.e.j.2431.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{5} -6.33166i q^{7} +O(q^{10})$	$q+2.23607 q^{5} -6.33166i q^{7} +9.27963i q^{11} -18.5674 q^{13} -13.9110 q^{17} +17.2468i q^{19} -33.7148i q^{23} +5.00000 q^{25} -28.6177 q^{29} +23.4939i q^{31} -14.1580i q^{35} +67.3338 q^{37} +44.0791 q^{41} +50.2937i q^{43} -31.1594i q^{47} +8.91003 q^{49} +81.6070 q^{53} +20.7499i q^{55} +19.2751i q^{59} +53.1563 q^{61} -41.5180 q^{65} -4.49911i q^{67} -13.3360i q^{71} +40.8904 q^{73} +58.7555 q^{77} +141.309i q^{79} -69.8503i q^{83} -31.1060 q^{85} +46.3079 q^{89} +117.563i q^{91} +38.5651i q^{95} +68.5543 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{13} + 40 q^{25} + 64 q^{29} + 112 q^{37} + 16 q^{41} - 56 q^{49} + 352 q^{53} + 176 q^{61} + 80 q^{65} - 240 q^{73} - 288 q^{77} - 160 q^{85} - 80 q^{89} + 432 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 + 40 * q^25 + 64 * q^29 + 112 * q^37 + 16 * q^41 - 56 * q^49 + 352 * q^53 + 176 * q^61 + 80 * q^65 - 240 * q^73 - 288 * q^77 - 160 * q^85 - 80 * q^89 + 432 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	− 6.33166i	− 0.904523i	−0.891885	−	0.452262i	$-0.850617\pi$
$7$	− 6.33166i	− 0.904523i	0.891885	−	0.452262i	$-0.149383\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9.27963i	0.843602i	0.906688	+	0.421801i	$0.138602\pi$
$11$	9.27963i	0.843602i	−0.906688	+	0.421801i	$0.861398\pi$
$12$	0	0
$13$	−18.5674	−1.42826	−0.714131	−	0.700012i	$-0.753178\pi$
$13$	−18.5674	−1.42826	−0.714131	+	0.700012i	$0.753178\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−13.9110	−0.818296	−0.409148	−	0.912468i	$-0.634174\pi$
$17$	−13.9110	−0.818296	−0.409148	+	0.912468i	$0.634174\pi$
$18$	0	0
$19$	17.2468i	0.907727i	0.891071	+	0.453864i	$0.149955\pi$
$19$	17.2468i	0.907727i	−0.891071	+	0.453864i	$0.850045\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 33.7148i	− 1.46586i	−0.680303	−	0.732931i	$-0.738152\pi$
$23$	− 33.7148i	− 1.46586i	0.680303	−	0.732931i	$-0.261848\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−28.6177	−0.986817	−0.493409	−	0.869798i	$-0.664249\pi$
$29$	−28.6177	−0.986817	−0.493409	+	0.869798i	$0.664249\pi$
$30$	0	0
$31$	23.4939i	0.757866i	0.925424	+	0.378933i	$0.123709\pi$
$31$	23.4939i	0.757866i	−0.925424	+	0.378933i	$0.876291\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 14.1580i	− 0.404515i
$36$	0	0
$37$	67.3338	1.81983	0.909916	−	0.414793i	$-0.136146\pi$
$37$	67.3338	1.81983	0.909916	+	0.414793i	$0.136146\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	44.0791	1.07510	0.537550	−	0.843232i	$-0.319350\pi$
$41$	44.0791	1.07510	0.537550	+	0.843232i	$0.319350\pi$
$42$	0	0
$43$	50.2937i	1.16962i	0.811170	+	0.584811i	$0.198831\pi$
$43$	50.2937i	1.16962i	−0.811170	+	0.584811i	$0.801169\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 31.1594i	− 0.662967i	−0.943461	−	0.331483i	$-0.892451\pi$
$47$	− 31.1594i	− 0.662967i	0.943461	−	0.331483i	$-0.107549\pi$
$48$	0	0
$49$	8.91003	0.181837
$50$	0	0
$51$	0	0
$52$	0	0
$53$	81.6070	1.53975	0.769877	−	0.638192i	$-0.220318\pi$
$53$	81.6070	1.53975	0.769877	+	0.638192i	$0.220318\pi$
$54$	0	0
$55$	20.7499i	0.377270i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	19.2751i	0.326697i	0.986568	+	0.163349i	$0.0522296\pi$
$59$	19.2751i	0.326697i	−0.986568	+	0.163349i	$0.947770\pi$
$60$	0	0
$61$	53.1563	0.871415	0.435707	−	0.900088i	$-0.356498\pi$
$61$	53.1563	0.871415	0.435707	+	0.900088i	$0.356498\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−41.5180	−0.638738
$66$	0	0
$67$	− 4.49911i	− 0.0671509i	−0.999436	−	0.0335754i	$-0.989311\pi$
$67$	− 4.49911i	− 0.0671509i	0.999436	−	0.0335754i	$-0.0106894\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 13.3360i	− 0.187832i	−0.995580	−	0.0939158i	$-0.970062\pi$
$71$	− 13.3360i	− 0.187832i	0.995580	−	0.0939158i	$-0.0299385\pi$
$72$	0	0
$73$	40.8904	0.560143	0.280071	−	0.959979i	$-0.409642\pi$
$73$	40.8904	0.560143	0.280071	+	0.959979i	$0.409642\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	58.7555	0.763058
$78$	0	0
$79$	141.309i	1.78872i	0.447352	+	0.894358i	$0.352367\pi$
$79$	141.309i	1.78872i	−0.447352	+	0.894358i	$0.647633\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 69.8503i	− 0.841570i	−0.907160	−	0.420785i	$-0.861755\pi$
$83$	− 69.8503i	− 0.841570i	0.907160	−	0.420785i	$-0.138245\pi$
$84$	0	0
$85$	−31.1060	−0.365953
$86$	0	0
$87$	0	0
$88$	0	0
$89$	46.3079	0.520313	0.260157	−	0.965566i	$-0.416226\pi$
$89$	46.3079	0.520313	0.260157	+	0.965566i	$0.416226\pi$
$90$	0	0
$91$	117.563i	1.29190i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	38.5651i	0.405948i
$96$	0	0
$97$	68.5543	0.706745	0.353373	−	0.935483i	$-0.385035\pi$
$97$	68.5543	0.706745	0.353373	+	0.935483i	$0.385035\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−43.3949	−0.429653	−0.214826	−	0.976652i	$-0.568919\pi$
$101$	−43.3949	−0.429653	−0.214826	+	0.976652i	$0.568919\pi$
$102$	0	0
$103$	− 85.7919i	− 0.832931i	−0.909152	−	0.416465i	$-0.863269\pi$
$103$	− 85.7919i	− 0.832931i	0.909152	−	0.416465i	$-0.136731\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 183.075i	− 1.71098i	−0.517818	−	0.855491i	$-0.673255\pi$
$107$	− 183.075i	− 1.71098i	0.517818	−	0.855491i	$-0.326745\pi$
$108$	0	0
$109$	−81.4798	−0.747521	−0.373761	−	0.927525i	$-0.621932\pi$
$109$	−81.4798	−0.747521	−0.373761	+	0.927525i	$0.621932\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	172.814	1.52933	0.764664	−	0.644429i	$-0.222905\pi$
$113$	172.814	1.52933	0.764664	+	0.644429i	$0.222905\pi$
$114$	0	0
$115$	− 75.3886i	− 0.655553i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	88.0800i	0.740168i
$120$	0	0
$121$	34.8885	0.288335
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	22.3785i	0.176208i	0.996111	+	0.0881041i	$0.0280808\pi$
$127$	22.3785i	0.176208i	−0.996111	+	0.0881041i	$0.971919\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.75315i	− 0.0133828i	−0.999978	−	0.00669141i	$-0.997870\pi$
$131$	− 1.75315i	− 0.0133828i	0.999978	−	0.00669141i	$-0.00212996\pi$
$132$	0	0
$133$	109.201	0.821060
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.5084	0.142397	0.0711987	−	0.997462i	$-0.477318\pi$
$137$	19.5084	0.142397	0.0711987	+	0.997462i	$0.477318\pi$
$138$	0	0
$139$	− 257.370i	− 1.85158i	−0.378038	−	0.925790i	$-0.623401\pi$
$139$	− 257.370i	− 1.85158i	0.378038	−	0.925790i	$-0.376599\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 172.299i	− 1.20489i
$144$	0	0
$145$	−63.9911	−0.441318
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−111.673	−0.749486	−0.374743	−	0.927129i	$-0.622269\pi$
$149$	−111.673	−0.749486	−0.374743	+	0.927129i	$0.622269\pi$
$150$	0	0
$151$	6.45275i	0.0427335i	0.999772	+	0.0213667i	$0.00680176\pi$
$151$	6.45275i	0.0427335i	−0.999772	+	0.0213667i	$0.993198\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	52.5339i	0.338928i
$156$	0	0
$157$	75.9075	0.483488	0.241744	−	0.970340i	$-0.422281\pi$
$157$	75.9075	0.483488	0.241744	+	0.970340i	$0.422281\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−213.471	−1.32591
$162$	0	0
$163$	249.298i	1.52944i	0.644364	+	0.764719i	$0.277122\pi$
$163$	249.298i	1.52944i	−0.644364	+	0.764719i	$0.722878\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 79.1883i	− 0.474182i	−0.971487	−	0.237091i	$-0.923806\pi$
$167$	− 79.1883i	− 0.474182i	0.971487	−	0.237091i	$-0.0761939\pi$
$168$	0	0
$169$	175.749	1.03993
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−27.7204	−0.160234	−0.0801168	−	0.996785i	$-0.525529\pi$
$173$	−27.7204	−0.160234	−0.0801168	+	0.996785i	$0.525529\pi$
$174$	0	0
$175$	− 31.6583i	− 0.180905i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 204.324i	− 1.14147i	−0.821133	−	0.570737i	$-0.806658\pi$
$179$	− 204.324i	− 1.14147i	0.821133	−	0.570737i	$-0.193342\pi$
$180$	0	0
$181$	49.8262	0.275283	0.137641	−	0.990482i	$-0.456048\pi$
$181$	49.8262	0.275283	0.137641	+	0.990482i	$0.456048\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	150.563	0.813853
$186$	0	0
$187$	− 129.089i	− 0.690317i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.13703i	− 0.00595301i	−0.999996	−	0.00297651i	$-0.999053\pi$
$191$	− 1.13703i	− 0.00595301i	0.999996	−	0.00297651i	$-0.000947453\pi$
$192$	0	0
$193$	−76.6452	−0.397126	−0.198563	−	0.980088i	$-0.563627\pi$
$193$	−76.6452	−0.397126	−0.198563	+	0.980088i	$0.563627\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	134.496	0.682719	0.341359	−	0.939933i	$-0.389113\pi$
$197$	134.496	0.682719	0.341359	+	0.939933i	$0.389113\pi$
$198$	0	0
$199$	− 176.014i	− 0.884491i	−0.896894	−	0.442245i	$-0.854182\pi$
$199$	− 176.014i	− 0.884491i	0.896894	−	0.442245i	$-0.145818\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	181.198i	0.892599i
$204$	0	0
$205$	98.5638	0.480799
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−160.044	−0.765761
$210$	0	0
$211$	218.087i	1.03359i	0.856110	+	0.516793i	$0.172874\pi$
$211$	218.087i	1.03359i	−0.856110	+	0.516793i	$0.827126\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	112.460i	0.523071i
$216$	0	0
$217$	148.755	0.685508
$218$	0	0
$219$	0	0
$220$	0	0
$221$	258.292	1.16874
$222$	0	0
$223$	328.579i	1.47345i	0.676193	+	0.736724i	$0.263628\pi$
$223$	328.579i	1.47345i	−0.676193	+	0.736724i	$0.736372\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	157.649i	0.694491i	0.937774	+	0.347245i	$0.112883\pi$
$227$	157.649i	0.694491i	−0.937774	+	0.347245i	$0.887117\pi$
$228$	0	0
$229$	273.148	1.19279	0.596393	−	0.802692i	$-0.296600\pi$
$229$	273.148	1.19279	0.596393	+	0.802692i	$0.296600\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−108.746	−0.466720	−0.233360	−	0.972390i	$-0.574972\pi$
$233$	−108.746	−0.466720	−0.233360	+	0.972390i	$0.574972\pi$
$234$	0	0
$235$	− 69.6746i	− 0.296488i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	178.994i	0.748927i	0.927242	+	0.374464i	$0.122173\pi$
$239$	178.994i	0.748927i	−0.927242	+	0.374464i	$0.877827\pi$
$240$	0	0
$241$	358.623	1.48806	0.744032	−	0.668144i	$-0.232911\pi$
$241$	358.623	1.48806	0.744032	+	0.668144i	$0.232911\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	19.9234	0.0813202
$246$	0	0
$247$	− 320.229i	− 1.29647i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	306.220i	1.22000i	0.792401	+	0.610000i	$0.208831\pi$
$251$	306.220i	1.22000i	−0.792401	+	0.610000i	$0.791169\pi$
$252$	0	0
$253$	312.861	1.23660
$254$	0	0
$255$	0	0
$256$	0	0
$257$	251.062	0.976895	0.488447	−	0.872593i	$-0.337563\pi$
$257$	251.062	0.976895	0.488447	+	0.872593i	$0.337563\pi$
$258$	0	0
$259$	− 426.335i	− 1.64608i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 48.7645i	− 0.185416i	−0.995693	−	0.0927082i	$-0.970448\pi$
$263$	− 48.7645i	− 0.185416i	0.995693	−	0.0927082i	$-0.0295524\pi$
$264$	0	0
$265$	182.479	0.688599
$266$	0	0
$267$	0	0
$268$	0	0
$269$	148.696	0.552772	0.276386	−	0.961047i	$-0.410863\pi$
$269$	148.696	0.552772	0.276386	+	0.961047i	$0.410863\pi$
$270$	0	0
$271$	− 83.3415i	− 0.307533i	−0.988107	−	0.153767i	$-0.950860\pi$
$271$	− 83.3415i	− 0.307533i	0.988107	−	0.153767i	$-0.0491404\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	46.3981i	0.168720i
$276$	0	0
$277$	−144.080	−0.520146	−0.260073	−	0.965589i	$-0.583747\pi$
$277$	−144.080	−0.520146	−0.260073	+	0.965589i	$0.583747\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	343.671	1.22303	0.611514	−	0.791233i	$-0.290561\pi$
$281$	343.671	1.22303	0.611514	+	0.791233i	$0.290561\pi$
$282$	0	0
$283$	314.955i	1.11292i	0.830876	+	0.556458i	$0.187840\pi$
$283$	314.955i	1.11292i	−0.830876	+	0.556458i	$0.812160\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 279.094i	− 0.972453i
$288$	0	0
$289$	−95.4831	−0.330391
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.55421	−0.0223693	−0.0111847	−	0.999937i	$-0.503560\pi$
$293$	−6.55421	−0.0223693	−0.0111847	+	0.999937i	$0.503560\pi$
$294$	0	0
$295$	43.1005i	0.146104i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	625.997i	2.09364i
$300$	0	0
$301$	318.443	1.05795
$302$	0	0
$303$	0	0
$304$	0	0
$305$	118.861	0.389709
$306$	0	0
$307$	354.559i	1.15492i	0.816420	+	0.577458i	$0.195955\pi$
$307$	354.559i	1.15492i	−0.816420	+	0.577458i	$0.804045\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	193.387i	0.621823i	0.950439	+	0.310912i	$0.100634\pi$
$311$	193.387i	0.621823i	−0.950439	+	0.310912i	$0.899366\pi$
$312$	0	0
$313$	−23.5224	−0.0751514	−0.0375757	−	0.999294i	$-0.511964\pi$
$313$	−23.5224	−0.0751514	−0.0375757	+	0.999294i	$0.511964\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−214.004	−0.675092	−0.337546	−	0.941309i	$-0.609597\pi$
$317$	−214.004	−0.675092	−0.337546	+	0.941309i	$0.609597\pi$
$318$	0	0
$319$	− 265.562i	− 0.832481i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 239.921i	− 0.742790i
$324$	0	0
$325$	−92.8371	−0.285652
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−197.291	−0.599669
$330$	0	0
$331$	− 412.454i	− 1.24609i	−0.782188	−	0.623043i	$-0.785896\pi$
$331$	− 412.454i	− 1.24609i	0.782188	−	0.623043i	$-0.214104\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 10.0603i	− 0.0300308i
$336$	0	0
$337$	103.268	0.306433	0.153216	−	0.988193i	$-0.451037\pi$
$337$	103.268	0.306433	0.153216	+	0.988193i	$0.451037\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−218.014	−0.639338
$342$	0	0
$343$	− 366.667i	− 1.06900i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	153.211i	0.441531i	0.975327	+	0.220766i	$0.0708556\pi$
$347$	153.211i	0.441531i	−0.975327	+	0.220766i	$0.929144\pi$
$348$	0	0
$349$	84.7317	0.242784	0.121392	−	0.992605i	$-0.461264\pi$
$349$	84.7317	0.242784	0.121392	+	0.992605i	$0.461264\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−256.065	−0.725396	−0.362698	−	0.931907i	$-0.618144\pi$
$353$	−256.065	−0.725396	−0.362698	+	0.931907i	$0.618144\pi$
$354$	0	0
$355$	− 29.8203i	− 0.0840009i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	667.258i	1.85866i	0.369253	+	0.929329i	$0.379614\pi$
$359$	667.258i	1.85866i	−0.369253	+	0.929329i	$0.620386\pi$
$360$	0	0
$361$	63.5473	0.176031
$362$	0	0
$363$	0	0
$364$	0	0
$365$	91.4338	0.250503
$366$	0	0
$367$	245.301i	0.668396i	0.942503	+	0.334198i	$0.108465\pi$
$367$	245.301i	0.668396i	−0.942503	+	0.334198i	$0.891535\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 516.708i	− 1.39274i
$372$	0	0
$373$	−698.787	−1.87342	−0.936712	−	0.350101i	$-0.886147\pi$
$373$	−698.787	−1.87342	−0.936712	+	0.350101i	$0.886147\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	531.357	1.40943
$378$	0	0
$379$	− 208.691i	− 0.550636i	−0.961353	−	0.275318i	$-0.911217\pi$
$379$	− 208.691i	− 0.550636i	0.961353	−	0.275318i	$-0.0887831\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	156.524i	0.408680i	0.978900	+	0.204340i	$0.0655048\pi$
$383$	156.524i	0.408680i	−0.978900	+	0.204340i	$0.934495\pi$
$384$	0	0
$385$	131.381	0.341250
$386$	0	0
$387$	0	0
$388$	0	0
$389$	386.588	0.993801	0.496900	−	0.867808i	$-0.334471\pi$
$389$	386.588	0.993801	0.496900	+	0.867808i	$0.334471\pi$
$390$	0	0
$391$	469.008i	1.19951i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	315.976i	0.799938i
$396$	0	0
$397$	561.155	1.41349	0.706744	−	0.707470i	$-0.250163\pi$
$397$	561.155	1.41349	0.706744	+	0.707470i	$0.250163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.9333	−0.0422276	−0.0211138	−	0.999777i	$-0.506721\pi$
$401$	−16.9333	−0.0422276	−0.0211138	+	0.999777i	$0.506721\pi$
$402$	0	0
$403$	− 436.220i	− 1.08243i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	624.832i	1.53521i
$408$	0	0
$409$	258.490	0.632006	0.316003	−	0.948758i	$-0.397659\pi$
$409$	258.490	0.632006	0.316003	+	0.948758i	$0.397659\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	122.044	0.295505
$414$	0	0
$415$	− 156.190i	− 0.376362i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	258.917i	0.617941i	0.951072	+	0.308970i	$0.0999844\pi$
$419$	258.917i	0.617941i	−0.951072	+	0.308970i	$0.900016\pi$
$420$	0	0
$421$	−97.4654	−0.231509	−0.115755	−	0.993278i	$-0.536929\pi$
$421$	−97.4654	−0.231509	−0.115755	+	0.993278i	$0.536929\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−69.5552	−0.163659
$426$	0	0
$427$	− 336.568i	− 0.788215i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 389.968i	− 0.904799i	−0.891815	−	0.452399i	$-0.850568\pi$
$431$	− 389.968i	− 0.904799i	0.891815	−	0.452399i	$-0.149432\pi$
$432$	0	0
$433$	275.893	0.637166	0.318583	−	0.947895i	$-0.396793\pi$
$433$	275.893	0.637166	0.318583	+	0.947895i	$0.396793\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	581.473	1.33060
$438$	0	0
$439$	− 446.143i	− 1.01627i	−0.861277	−	0.508136i	$-0.830335\pi$
$439$	− 446.143i	− 1.01627i	0.861277	−	0.508136i	$-0.169665\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 794.679i	− 1.79386i	−0.442174	−	0.896929i	$-0.645793\pi$
$443$	− 794.679i	− 1.79386i	0.442174	−	0.896929i	$-0.354207\pi$
$444$	0	0
$445$	103.548	0.232691
$446$	0	0
$447$	0	0
$448$	0	0
$449$	750.226	1.67088	0.835441	−	0.549581i	$-0.185212\pi$
$449$	750.226	1.67088	0.835441	+	0.549581i	$0.185212\pi$
$450$	0	0
$451$	409.037i	0.906957i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	262.878i	0.577754i
$456$	0	0
$457$	101.092	0.221209	0.110604	−	0.993865i	$-0.464721\pi$
$457$	101.092	0.221209	0.110604	+	0.993865i	$0.464721\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.48690	−0.00973297	−0.00486648	−	0.999988i	$-0.501549\pi$
$461$	−4.48690	−0.00973297	−0.00486648	+	0.999988i	$0.501549\pi$
$462$	0	0
$463$	515.108i	1.11254i	0.831000	+	0.556272i	$0.187769\pi$
$463$	515.108i	1.11254i	−0.831000	+	0.556272i	$0.812231\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 295.498i	− 0.632758i	−0.948633	−	0.316379i	$-0.897533\pi$
$467$	− 295.498i	− 0.632758i	0.948633	−	0.316379i	$-0.102467\pi$
$468$	0	0
$469$	−28.4869	−0.0607396
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−466.707	−0.986696
$474$	0	0
$475$	86.2341i	0.181545i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 273.155i	− 0.570260i	−0.958489	−	0.285130i	$-0.907963\pi$
$479$	− 273.155i	− 0.570260i	0.958489	−	0.285130i	$-0.0920368\pi$
$480$	0	0
$481$	−1250.21	−2.59920
$482$	0	0
$483$	0	0
$484$	0	0
$485$	153.292	0.316066
$486$	0	0
$487$	357.751i	0.734601i	0.930102	+	0.367301i	$0.119718\pi$
$487$	357.751i	0.734601i	−0.930102	+	0.367301i	$0.880282\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 422.379i	− 0.860242i	−0.902771	−	0.430121i	$-0.858471\pi$
$491$	− 422.379i	− 0.860242i	0.902771	−	0.430121i	$-0.141529\pi$
$492$	0	0
$493$	398.102	0.807509
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−84.4394	−0.169898
$498$	0	0
$499$	207.096i	0.415021i	0.978233	+	0.207511i	$0.0665362\pi$
$499$	207.096i	0.415021i	−0.978233	+	0.207511i	$0.933464\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 702.853i	− 1.39732i	−0.715452	−	0.698661i	$-0.753779\pi$
$503$	− 702.853i	− 1.39732i	0.715452	−	0.698661i	$-0.246221\pi$
$504$	0	0
$505$	−97.0340	−0.192147
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−389.029	−0.764300	−0.382150	−	0.924100i	$-0.624816\pi$
$509$	−389.029	−0.764300	−0.382150	+	0.924100i	$0.624816\pi$
$510$	0	0
$511$	− 258.904i	− 0.506662i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 191.836i	− 0.372498i
$516$	0	0
$517$	289.148	0.559280
$518$	0	0
$519$	0	0
$520$	0	0
$521$	151.753	0.291273	0.145637	−	0.989338i	$-0.453477\pi$
$521$	151.753	0.291273	0.145637	+	0.989338i	$0.453477\pi$
$522$	0	0
$523$	− 557.762i	− 1.06647i	−0.845968	−	0.533234i	$-0.820977\pi$
$523$	− 557.762i	− 1.06647i	0.845968	−	0.533234i	$-0.179023\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 326.824i	− 0.620159i
$528$	0	0
$529$	−607.689	−1.14875
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−818.435	−1.53552
$534$	0	0
$535$	− 409.368i	− 0.765174i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	82.6818i	0.153398i
$540$	0	0
$541$	−340.979	−0.630275	−0.315137	−	0.949046i	$-0.602051\pi$
$541$	−340.979	−0.630275	−0.315137	+	0.949046i	$0.602051\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−182.194	−0.334302
$546$	0	0
$547$	113.651i	0.207771i	0.994589	+	0.103885i	$0.0331275\pi$
$547$	113.651i	0.207771i	−0.994589	+	0.103885i	$0.966872\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 493.564i	− 0.895761i
$552$	0	0
$553$	894.718	1.61794
$554$	0	0
$555$	0	0
$556$	0	0
$557$	233.232	0.418728	0.209364	−	0.977838i	$-0.432861\pi$
$557$	233.232	0.418728	0.209364	+	0.977838i	$0.432861\pi$
$558$	0	0
$559$	− 933.825i	− 1.67053i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 167.786i	− 0.298021i	−0.988836	−	0.149011i	$-0.952391\pi$
$563$	− 167.786i	− 0.298021i	0.988836	−	0.149011i	$-0.0476088\pi$
$564$	0	0
$565$	386.424	0.683936
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−381.089	−0.669752	−0.334876	−	0.942262i	$-0.608694\pi$
$569$	−381.089	−0.669752	−0.334876	+	0.942262i	$0.608694\pi$
$570$	0	0
$571$	− 453.871i	− 0.794870i	−0.917630	−	0.397435i	$-0.869900\pi$
$571$	− 453.871i	− 0.794870i	0.917630	−	0.397435i	$-0.130100\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 168.574i	− 0.293172i
$576$	0	0
$577$	688.294	1.19288	0.596442	−	0.802656i	$-0.296581\pi$
$577$	688.294	1.19288	0.596442	+	0.802656i	$0.296581\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−442.269	−0.761220
$582$	0	0
$583$	757.282i	1.29894i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	249.163i	0.424468i	0.977219	+	0.212234i	$0.0680739\pi$
$587$	249.163i	0.424468i	−0.977219	+	0.212234i	$0.931926\pi$
$588$	0	0
$589$	−405.194	−0.687936
$590$	0	0
$591$	0	0
$592$	0	0
$593$	163.937	0.276454	0.138227	−	0.990401i	$-0.455860\pi$
$593$	163.937	0.276454	0.138227	+	0.990401i	$0.455860\pi$
$594$	0	0
$595$	196.953i	0.331013i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 170.412i	− 0.284494i	−0.989831	−	0.142247i	$-0.954567\pi$
$599$	− 170.412i	− 0.284494i	0.989831	−	0.142247i	$-0.0454327\pi$
$600$	0	0
$601$	1119.87	1.86335	0.931674	−	0.363295i	$-0.118348\pi$
$601$	1119.87	1.86335	0.931674	+	0.363295i	$0.118348\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	78.0132	0.128947
$606$	0	0
$607$	660.957i	1.08889i	0.838796	+	0.544445i	$0.183260\pi$
$607$	660.957i	1.08889i	−0.838796	+	0.544445i	$0.816740\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	578.550i	0.946890i
$612$	0	0
$613$	179.315	0.292520	0.146260	−	0.989246i	$-0.453276\pi$
$613$	179.315	0.292520	0.146260	+	0.989246i	$0.453276\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	63.6752	0.103201	0.0516007	−	0.998668i	$-0.483568\pi$
$617$	63.6752	0.103201	0.0516007	+	0.998668i	$0.483568\pi$
$618$	0	0
$619$	872.350i	1.40929i	0.709561	+	0.704644i	$0.248893\pi$
$619$	872.350i	1.40929i	−0.709561	+	0.704644i	$0.751107\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 293.206i	− 0.470636i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−936.682	−1.48916
$630$	0	0
$631$	340.783i	0.540068i	0.962851	+	0.270034i	$0.0870349\pi$
$631$	340.783i	0.540068i	−0.962851	+	0.270034i	$0.912965\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	50.0397i	0.0788027i
$636$	0	0
$637$	−165.436	−0.259712
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−766.210	−1.19534	−0.597668	−	0.801744i	$-0.703906\pi$
$641$	−766.210	−1.19534	−0.597668	+	0.801744i	$0.703906\pi$
$642$	0	0
$643$	1163.47i	1.80943i	0.426014	+	0.904717i	$0.359917\pi$
$643$	1163.47i	1.80943i	−0.426014	+	0.904717i	$0.640083\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	740.530i	1.14456i	0.820059	+	0.572279i	$0.193941\pi$
$647$	740.530i	1.14456i	−0.820059	+	0.572279i	$0.806059\pi$
$648$	0	0
$649$	−178.866	−0.275603
$650$	0	0
$651$	0	0
$652$	0	0
$653$	109.569	0.167793	0.0838967	−	0.996474i	$-0.473263\pi$
$653$	109.569	0.167793	0.0838967	+	0.996474i	$0.473263\pi$
$654$	0	0
$655$	− 3.92016i	− 0.00598498i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	723.214i	1.09744i	0.836006	+	0.548721i	$0.184885\pi$
$659$	723.214i	1.09744i	−0.836006	+	0.548721i	$0.815115\pi$
$660$	0	0
$661$	−700.333	−1.05951	−0.529753	−	0.848152i	$-0.677715\pi$
$661$	−700.333	−1.05951	−0.529753	+	0.848152i	$0.677715\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	244.181	0.367189
$666$	0	0
$667$	964.841i	1.44654i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	493.271i	0.735128i
$672$	0	0
$673$	−1221.18	−1.81454	−0.907269	−	0.420552i	$-0.861837\pi$
$673$	−1221.18	−1.81454	−0.907269	+	0.420552i	$0.861837\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	989.373	1.46141	0.730704	−	0.682695i	$-0.239192\pi$
$677$	989.373	1.46141	0.730704	+	0.682695i	$0.239192\pi$
$678$	0	0
$679$	− 434.063i	− 0.639268i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 307.312i	− 0.449945i	−0.974365	−	0.224972i	$-0.927771\pi$
$683$	− 307.312i	− 0.449945i	0.974365	−	0.224972i	$-0.0722292\pi$
$684$	0	0
$685$	43.6222	0.0636821
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1515.23	−2.19917
$690$	0	0
$691$	893.378i	1.29288i	0.762966	+	0.646438i	$0.223742\pi$
$691$	893.378i	1.29288i	−0.762966	+	0.646438i	$0.776258\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 575.496i	− 0.828052i
$696$	0	0
$697$	−613.186	−0.879750
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1127.42	1.60830	0.804149	−	0.594428i	$-0.202622\pi$
$701$	1127.42	1.60830	0.804149	+	0.594428i	$0.202622\pi$
$702$	0	0
$703$	1161.29i	1.65191i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	274.762i	0.388631i
$708$	0	0
$709$	−1093.27	−1.54199	−0.770997	−	0.636839i	$-0.780242\pi$
$709$	−1093.27	−1.54199	−0.770997	+	0.636839i	$0.780242\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	792.091	1.11093
$714$	0	0
$715$	− 385.271i	− 0.538841i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 769.690i	− 1.07050i	−0.844693	−	0.535251i	$-0.820217\pi$
$719$	− 769.690i	− 1.07050i	0.844693	−	0.535251i	$-0.179783\pi$
$720$	0	0
$721$	−543.205	−0.753405
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−143.089	−0.197363
$726$	0	0
$727$	− 295.050i	− 0.405846i	−0.979195	−	0.202923i	$-0.934956\pi$
$727$	− 295.050i	− 0.405846i	0.979195	−	0.202923i	$-0.0650441\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 699.638i	− 0.957097i
$732$	0	0
$733$	−261.200	−0.356344	−0.178172	−	0.983999i	$-0.557018\pi$
$733$	−261.200	−0.356344	−0.178172	+	0.983999i	$0.557018\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	41.7501	0.0566487
$738$	0	0
$739$	− 482.679i	− 0.653151i	−0.945171	−	0.326576i	$-0.894105\pi$
$739$	− 482.679i	− 0.653151i	0.945171	−	0.326576i	$-0.105895\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 23.7067i	− 0.0319067i	−0.999873	−	0.0159534i	$-0.994922\pi$
$743$	− 23.7067i	− 0.0319067i	0.999873	−	0.0159534i	$-0.00507833\pi$
$744$	0	0
$745$	−249.709	−0.335180
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1159.17	−1.54762
$750$	0	0
$751$	395.508i	0.526642i	0.964708	+	0.263321i	$0.0848179\pi$
$751$	395.508i	0.526642i	−0.964708	+	0.263321i	$0.915182\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.4288i	0.0191110i
$756$	0	0
$757$	−393.940	−0.520396	−0.260198	−	0.965555i	$-0.583788\pi$
$757$	−393.940	−0.520396	−0.260198	+	0.965555i	$0.583788\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	369.354	0.485354	0.242677	−	0.970107i	$-0.421975\pi$
$761$	369.354	0.485354	0.242677	+	0.970107i	$0.421975\pi$
$762$	0	0
$763$	515.903i	0.676150i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 357.890i	− 0.466610i
$768$	0	0
$769$	−873.491	−1.13588	−0.567940	−	0.823070i	$-0.692259\pi$
$769$	−873.491	−1.13588	−0.567940	+	0.823070i	$0.692259\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1176.93	−1.52254	−0.761272	−	0.648432i	$-0.775425\pi$
$773$	−1176.93	−1.52254	−0.761272	+	0.648432i	$0.775425\pi$
$774$	0	0
$775$	117.469i	0.151573i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	760.224i	0.975897i
$780$	0	0
$781$	123.754	0.158455
$782$	0	0
$783$	0	0
$784$	0	0
$785$	169.734	0.216222
$786$	0	0
$787$	− 603.482i	− 0.766814i	−0.923580	−	0.383407i	$-0.874751\pi$
$787$	− 603.482i	− 0.766814i	0.923580	−	0.383407i	$-0.125249\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1094.20i	− 1.38331i
$792$	0	0
$793$	−986.975	−1.24461
$794$	0	0
$795$	0	0
$796$	0	0
$797$	860.121	1.07920	0.539599	−	0.841922i	$-0.318576\pi$
$797$	860.121	1.07920	0.539599	+	0.841922i	$0.318576\pi$
$798$	0	0
$799$	433.460i	0.542503i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	379.448i	0.472538i
$804$	0	0
$805$	−477.336	−0.592963
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−941.012	−1.16318	−0.581589	−	0.813483i	$-0.697569\pi$
$809$	−941.012	−1.16318	−0.581589	+	0.813483i	$0.697569\pi$
$810$	0	0
$811$	− 1105.29i	− 1.36287i	−0.731878	−	0.681436i	$-0.761356\pi$
$811$	− 1105.29i	− 1.36287i	0.731878	−	0.681436i	$-0.238644\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	557.448i	0.683985i
$816$	0	0
$817$	−867.407	−1.06170
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−193.170	−0.235286	−0.117643	−	0.993056i	$-0.537534\pi$
$821$	−193.170	−0.235286	−0.117643	+	0.993056i	$0.537534\pi$
$822$	0	0
$823$	178.778i	0.217227i	0.994084	+	0.108614i	$0.0346411\pi$
$823$	178.778i	0.217227i	−0.994084	+	0.108614i	$0.965359\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1558.61i	− 1.88465i	−0.334697	−	0.942326i	$-0.608634\pi$
$827$	− 1558.61i	− 1.88465i	0.334697	−	0.942326i	$-0.391366\pi$
$828$	0	0
$829$	565.477	0.682119	0.341059	−	0.940042i	$-0.389214\pi$
$829$	565.477	0.682119	0.341059	+	0.940042i	$0.389214\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−123.948	−0.148797
$834$	0	0
$835$	− 177.071i	− 0.212061i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1280.25i	1.52592i	0.646443	+	0.762962i	$0.276256\pi$
$839$	1280.25i	1.52592i	−0.646443	+	0.762962i	$0.723744\pi$
$840$	0	0
$841$	−22.0271	−0.0261915
$842$	0	0
$843$	0	0
$844$	0	0
$845$	392.986	0.465072
$846$	0	0
$847$	− 220.903i	− 0.260806i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 2270.15i	− 2.66762i
$852$	0	0
$853$	−120.366	−0.141109	−0.0705546	−	0.997508i	$-0.522477\pi$
$853$	−120.366	−0.141109	−0.0705546	+	0.997508i	$0.522477\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	717.784	0.837554	0.418777	−	0.908089i	$-0.362459\pi$
$857$	717.784	0.837554	0.418777	+	0.908089i	$0.362459\pi$
$858$	0	0
$859$	252.894i	0.294405i	0.989106	+	0.147203i	$0.0470269\pi$
$859$	252.894i	0.294405i	−0.989106	+	0.147203i	$0.952973\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1234.73i	1.43075i	0.698743	+	0.715373i	$0.253743\pi$
$863$	1234.73i	1.43075i	−0.698743	+	0.715373i	$0.746257\pi$
$864$	0	0
$865$	−61.9847	−0.0716586
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1311.29	−1.50897
$870$	0	0
$871$	83.5368i	0.0959091i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 70.7902i	− 0.0809030i
$876$	0	0
$877$	685.723	0.781896	0.390948	−	0.920413i	$-0.372147\pi$
$877$	685.723	0.781896	0.390948	+	0.920413i	$0.372147\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	458.454	0.520379	0.260189	−	0.965558i	$-0.416215\pi$
$881$	458.454	0.520379	0.260189	+	0.965558i	$0.416215\pi$
$882$	0	0
$883$	− 771.505i	− 0.873732i	−0.899527	−	0.436866i	$-0.856088\pi$
$883$	− 771.505i	− 0.873732i	0.899527	−	0.436866i	$-0.143912\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1161.05i	− 1.30896i	−0.756080	−	0.654480i	$-0.772888\pi$
$887$	− 1161.05i	− 1.30896i	0.756080	−	0.654480i	$-0.227112\pi$
$888$	0	0
$889$	141.693	0.159385
$890$	0	0
$891$	0	0
$892$	0	0
$893$	537.401	0.601793
$894$	0	0
$895$	− 456.882i	− 0.510482i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 672.340i	− 0.747876i
$900$	0	0
$901$	−1135.24	−1.25997
$902$	0	0
$903$	0	0
$904$	0	0
$905$	111.415	0.123110
$906$	0	0
$907$	392.544i	0.432793i	0.976306	+	0.216397i	$0.0694304\pi$
$907$	392.544i	0.432793i	−0.976306	+	0.216397i	$0.930570\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1013.40i	1.11240i	0.831048	+	0.556201i	$0.187741\pi$
$911$	1013.40i	1.11240i	−0.831048	+	0.556201i	$0.812259\pi$
$912$	0	0
$913$	648.185	0.709950
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11.1004	−0.0121051
$918$	0	0
$919$	970.018i	1.05551i	0.849395	+	0.527757i	$0.176967\pi$
$919$	970.018i	1.05551i	−0.849395	+	0.527757i	$0.823033\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	247.616i	0.268273i
$924$	0	0
$925$	336.669	0.363966
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−980.857	−1.05582	−0.527910	−	0.849300i	$-0.677024\pi$
$929$	−980.857	−1.05582	−0.527910	+	0.849300i	$0.677024\pi$
$930$	0	0
$931$	153.670i	0.165059i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 288.652i	− 0.308719i
$936$	0	0
$937$	964.666	1.02953	0.514763	−	0.857333i	$-0.327880\pi$
$937$	964.666	1.02953	0.514763	+	0.857333i	$0.327880\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1581.10	−1.68023	−0.840117	−	0.542405i	$-0.817514\pi$
$941$	−1581.10	−1.68023	−0.840117	+	0.542405i	$0.817514\pi$
$942$	0	0
$943$	− 1486.12i	− 1.57595i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1245.27i	1.31497i	0.753469	+	0.657483i	$0.228379\pi$
$947$	1245.27i	1.31497i	−0.753469	+	0.657483i	$0.771621\pi$
$948$	0	0
$949$	−759.229	−0.800031
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1106.52	−1.16109	−0.580546	−	0.814228i	$-0.697161\pi$
$953$	−1106.52	−1.16109	−0.580546	+	0.814228i	$0.697161\pi$
$954$	0	0
$955$	− 2.54247i	− 0.00266227i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 123.521i	− 0.128802i
$960$	0	0
$961$	409.039	0.425638
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−171.384	−0.177600
$966$	0	0
$967$	− 406.453i	− 0.420324i	−0.977667	−	0.210162i	$-0.932601\pi$
$967$	− 406.453i	− 0.420324i	0.977667	−	0.210162i	$-0.0673992\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1815.22i	1.86943i	0.355393	+	0.934717i	$0.384347\pi$
$971$	1815.22i	1.86943i	−0.355393	+	0.934717i	$0.615653\pi$
$972$	0	0
$973$	−1629.58	−1.67480
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1457.74	1.49205	0.746027	−	0.665916i	$-0.231959\pi$
$977$	1457.74	1.49205	0.746027	+	0.665916i	$0.231959\pi$
$978$	0	0
$979$	429.720i	0.438938i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	19.9496i	0.0202946i	0.999949	+	0.0101473i	$0.00323004\pi$
$983$	19.9496i	0.0202946i	−0.999949	+	0.0101473i	$0.996770\pi$
$984$	0	0
$985$	300.741	0.305321
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1695.65	1.71450
$990$	0	0
$991$	− 605.720i	− 0.611221i	−0.952157	−	0.305611i	$-0.901139\pi$
$991$	− 605.720i	− 0.611221i	0.952157	−	0.305611i	$-0.0988605\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 393.578i	− 0.395556i
$996$	0	0
$997$	−1238.47	−1.24220	−0.621099	−	0.783732i	$-0.713314\pi$
$997$	−1238.47	−1.24220	−0.621099	+	0.783732i	$0.713314\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.3.e.j.2431.5		8
3.2	odd	2		960.3.e.c.511.5		8
4.3	odd	2	inner	2880.3.e.j.2431.8		8
8.3	odd	2		180.3.c.b.91.5		8
8.5	even	2		180.3.c.b.91.6		8
12.11	even	2		960.3.e.c.511.2		8
24.5	odd	2		60.3.c.a.31.3	✓	8
24.11	even	2		60.3.c.a.31.4	yes	8
40.3	even	4		900.3.f.f.199.2		16
40.13	odd	4		900.3.f.f.199.16		16
40.19	odd	2		900.3.c.u.451.4		8
40.27	even	4		900.3.f.f.199.15		16
40.29	even	2		900.3.c.u.451.3		8
40.37	odd	4		900.3.f.f.199.1		16
120.29	odd	2		300.3.c.d.151.6		8
120.53	even	4		300.3.f.b.199.1		16
120.59	even	2		300.3.c.d.151.5		8
120.77	even	4		300.3.f.b.199.16		16
120.83	odd	4		300.3.f.b.199.15		16
120.107	odd	4		300.3.f.b.199.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.c.a.31.3	✓	8	24.5	odd	2
60.3.c.a.31.4	yes	8	24.11	even	2
180.3.c.b.91.5		8	8.3	odd	2
180.3.c.b.91.6		8	8.5	even	2
300.3.c.d.151.5		8	120.59	even	2
300.3.c.d.151.6		8	120.29	odd	2
300.3.f.b.199.1		16	120.53	even	4
300.3.f.b.199.2		16	120.107	odd	4
300.3.f.b.199.15		16	120.83	odd	4
300.3.f.b.199.16		16	120.77	even	4
900.3.c.u.451.3		8	40.29	even	2
900.3.c.u.451.4		8	40.19	odd	2
900.3.f.f.199.1		16	40.37	odd	4
900.3.f.f.199.2		16	40.3	even	4
900.3.f.f.199.15		16	40.27	even	4
900.3.f.f.199.16		16	40.13	odd	4
960.3.e.c.511.2		8	12.11	even	2
960.3.e.c.511.5		8	3.2	odd	2
2880.3.e.j.2431.5		8	1.1	even	1	trivial
2880.3.e.j.2431.8		8	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2880.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.23607 q^{5} -6.33166i q^{7} +O(q^{10})\)	\(q+2.23607 q^{5} -6.33166i q^{7} +9.27963i q^{11} -18.5674 q^{13} -13.9110 q^{17} +17.2468i q^{19} -33.7148i q^{23} +5.00000 q^{25} -28.6177 q^{29} +23.4939i q^{31} -14.1580i q^{35} +67.3338 q^{37} +44.0791 q^{41} +50.2937i q^{43} -31.1594i q^{47} +8.91003 q^{49} +81.6070 q^{53} +20.7499i q^{55} +19.2751i q^{59} +53.1563 q^{61} -41.5180 q^{65} -4.49911i q^{67} -13.3360i q^{71} +40.8904 q^{73} +58.7555 q^{77} +141.309i q^{79} -69.8503i q^{83} -31.1060 q^{85} +46.3079 q^{89} +117.563i q^{91} +38.5651i q^{95} +68.5543 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{13} + 40 q^{25} + 64 q^{29} + 112 q^{37} + 16 q^{41} - 56 q^{49} + 352 q^{53} + 176 q^{61} + 80 q^{65} - 240 q^{73} - 288 q^{77} - 160 q^{85} - 80 q^{89} + 432 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 + 40 * q^25 + 64 * q^29 + 112 * q^37 + 16 * q^41 - 56 * q^49 + 352 * q^53 + 176 * q^61 + 80 * q^65 - 240 * q^73 - 288 * q^77 - 160 * q^85 - 80 * q^89 + 432 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more