LMFDB - Embedded newform 2025.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,4,Mod(1,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2025.a (trivial)

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:	yes
Analytic conductor:	$119.478867762$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.37228 q^{2} -6.11684 q^{4} +5.11684 q^{7} -19.3723 q^{8} +O(q^{10})$	$q+1.37228 q^{2} -6.11684 q^{4} +5.11684 q^{7} -19.3723 q^{8} +55.9783 q^{11} -37.5842 q^{13} +7.02175 q^{14} +22.3505 q^{16} -23.6495 q^{17} +39.0516 q^{19} +76.8179 q^{22} -71.0733 q^{23} -51.5761 q^{26} -31.2989 q^{28} -28.3723 q^{29} +12.8832 q^{31} +185.649 q^{32} -32.4537 q^{34} +180.103 q^{37} +53.5898 q^{38} -215.484 q^{41} -61.2337 q^{43} -342.410 q^{44} -97.5326 q^{46} +61.8776 q^{47} -316.818 q^{49} +229.897 q^{52} -492.310 q^{53} -99.1249 q^{56} -38.9348 q^{58} +789.630 q^{59} +521.090 q^{61} +17.6793 q^{62} +75.9590 q^{64} -304.429 q^{67} +144.660 q^{68} +270.391 q^{71} +925.464 q^{73} +247.152 q^{74} -238.873 q^{76} +286.432 q^{77} -1289.03 q^{79} -295.704 q^{82} -713.834 q^{83} -84.0298 q^{86} -1084.43 q^{88} -404.804 q^{89} -192.313 q^{91} +434.745 q^{92} +84.9135 q^{94} -75.0273 q^{97} -434.763 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 5 q^{4} - 7 q^{7} - 33 q^{8}+O(q^{10}) $ 2 * q - 3 * q^2 + 5 * q^4 - 7 * q^7 - 33 * q^8	$ 2 q - 3 q^{2} + 5 q^{4} - 7 q^{7} - 33 q^{8} + 66 q^{11} + 11 q^{13} + 60 q^{14} - 7 q^{16} - 99 q^{17} - 77 q^{19} + 33 q^{22} - 33 q^{23} - 264 q^{26} - 166 q^{28} - 51 q^{29} + 43 q^{31} + 423 q^{32} + 297 q^{34} + 50 q^{37} + 561 q^{38} + 132 q^{41} - 88 q^{43} - 231 q^{44} - 264 q^{46} - 399 q^{47} - 513 q^{49} + 770 q^{52} - 54 q^{53} + 66 q^{56} + 60 q^{58} + 798 q^{59} + 439 q^{61} - 114 q^{62} - 727 q^{64} - 988 q^{67} - 693 q^{68} + 1368 q^{71} + 455 q^{73} + 816 q^{74} - 1529 q^{76} + 165 q^{77} - 803 q^{79} - 1815 q^{82} - 813 q^{83} + 33 q^{86} - 1221 q^{88} - 396 q^{89} - 781 q^{91} + 858 q^{92} + 2100 q^{94} - 736 q^{97} + 423 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 + 5 * q^4 - 7 * q^7 - 33 * q^8 + 66 * q^11 + 11 * q^13 + 60 * q^14 - 7 * q^16 - 99 * q^17 - 77 * q^19 + 33 * q^22 - 33 * q^23 - 264 * q^26 - 166 * q^28 - 51 * q^29 + 43 * q^31 + 423 * q^32 + 297 * q^34 + 50 * q^37 + 561 * q^38 + 132 * q^41 - 88 * q^43 - 231 * q^44 - 264 * q^46 - 399 * q^47 - 513 * q^49 + 770 * q^52 - 54 * q^53 + 66 * q^56 + 60 * q^58 + 798 * q^59 + 439 * q^61 - 114 * q^62 - 727 * q^64 - 988 * q^67 - 693 * q^68 + 1368 * q^71 + 455 * q^73 + 816 * q^74 - 1529 * q^76 + 165 * q^77 - 803 * q^79 - 1815 * q^82 - 813 * q^83 + 33 * q^86 - 1221 * q^88 - 396 * q^89 - 781 * q^91 + 858 * q^92 + 2100 * q^94 - 736 * q^97 + 423 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.37228	0.485175	0.242587	−	0.970130i	$-0.422004\pi$
$2$	1.37228	0.485175	0.242587	+	0.970130i	$0.422004\pi$
$3$	0	0
$3$	0	0
$4$	−6.11684	−0.764605
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.11684	0.276284	0.138142	−	0.990412i	$-0.455887\pi$
$7$	5.11684	0.276284	0.138142	+	0.990412i	$0.455887\pi$
$8$	−19.3723	−0.856142
$9$	0	0
$10$	0	0
$11$	55.9783	1.53437	0.767185	−	0.641425i	$-0.221657\pi$
$11$	55.9783	1.53437	0.767185	+	0.641425i	$0.221657\pi$
$12$	0	0
$13$	−37.5842	−0.801845	−0.400923	−	0.916112i	$-0.631310\pi$
$13$	−37.5842	−0.801845	−0.400923	+	0.916112i	$0.631310\pi$
$14$	7.02175	0.134046
$15$	0	0
$16$	22.3505	0.349227
$17$	−23.6495	−0.337402	−0.168701	−	0.985667i	$-0.553957\pi$
$17$	−23.6495	−0.337402	−0.168701	+	0.985667i	$0.553957\pi$
$18$	0	0
$19$	39.0516	0.471529	0.235764	−	0.971810i	$-0.424241\pi$
$19$	39.0516	0.471529	0.235764	+	0.971810i	$0.424241\pi$
$20$	0	0
$21$	0	0
$22$	76.8179	0.744438
$23$	−71.0733	−0.644340	−0.322170	−	0.946682i	$-0.604412\pi$
$23$	−71.0733	−0.644340	−0.322170	+	0.946682i	$0.604412\pi$
$24$	0	0
$25$	0	0
$26$	−51.5761	−0.389035
$27$	0	0
$28$	−31.2989	−0.211248
$29$	−28.3723	−0.181676	−0.0908379	−	0.995866i	$-0.528955\pi$
$29$	−28.3723	−0.181676	−0.0908379	+	0.995866i	$0.528955\pi$
$30$	0	0
$31$	12.8832	0.0746414	0.0373207	−	0.999303i	$-0.488118\pi$
$31$	12.8832	0.0746414	0.0373207	+	0.999303i	$0.488118\pi$
$32$	185.649	1.02558
$33$	0	0
$34$	−32.4537	−0.163699
$35$	0	0
$36$	0	0
$37$	180.103	0.800237	0.400119	−	0.916463i	$-0.368969\pi$
$37$	180.103	0.800237	0.400119	+	0.916463i	$0.368969\pi$
$38$	53.5898	0.228774
$39$	0	0
$40$	0	0
$41$	−215.484	−0.820802	−0.410401	−	0.911905i	$-0.634611\pi$
$41$	−215.484	−0.820802	−0.410401	+	0.911905i	$0.634611\pi$
$42$	0	0
$43$	−61.2337	−0.217164	−0.108582	−	0.994087i	$-0.534631\pi$
$43$	−61.2337	−0.217164	−0.108582	+	0.994087i	$0.534631\pi$
$44$	−342.410	−1.17319
$45$	0	0
$46$	−97.5326	−0.312617
$47$	61.8776	0.192038	0.0960189	−	0.995380i	$-0.469389\pi$
$47$	61.8776	0.192038	0.0960189	+	0.995380i	$0.469389\pi$
$48$	0	0
$49$	−316.818	−0.923667
$50$	0	0
$51$	0	0
$52$	229.897	0.613095
$53$	−492.310	−1.27592	−0.637962	−	0.770068i	$-0.720222\pi$
$53$	−492.310	−1.27592	−0.637962	+	0.770068i	$0.720222\pi$
$54$	0	0
$55$	0	0
$56$	−99.1249	−0.236538
$57$	0	0
$58$	−38.9348	−0.0881445
$59$	789.630	1.74239	0.871196	−	0.490936i	$-0.163345\pi$
$59$	789.630	1.74239	0.871196	+	0.490936i	$0.163345\pi$
$60$	0	0
$61$	521.090	1.09375	0.546874	−	0.837215i	$-0.315818\pi$
$61$	521.090	1.09375	0.546874	+	0.837215i	$0.315818\pi$
$62$	17.6793	0.0362141
$63$	0	0
$64$	75.9590	0.148358
$65$	0	0
$66$	0	0
$67$	−304.429	−0.555104	−0.277552	−	0.960711i	$-0.589523\pi$
$67$	−304.429	−0.555104	−0.277552	+	0.960711i	$0.589523\pi$
$68$	144.660	0.257980
$69$	0	0
$70$	0	0
$71$	270.391	0.451966	0.225983	−	0.974131i	$-0.427441\pi$
$71$	270.391	0.451966	0.225983	+	0.974131i	$0.427441\pi$
$72$	0	0
$73$	925.464	1.48380	0.741900	−	0.670510i	$-0.233925\pi$
$73$	925.464	1.48380	0.741900	+	0.670510i	$0.233925\pi$
$74$	247.152	0.388255
$75$	0	0
$76$	−238.873	−0.360534
$77$	286.432	0.423921
$78$	0	0
$79$	−1289.03	−1.83579	−0.917897	−	0.396818i	$-0.870114\pi$
$79$	−1289.03	−1.83579	−0.917897	+	0.396818i	$0.870114\pi$
$80$	0	0
$81$	0	0
$82$	−295.704	−0.398232
$83$	−713.834	−0.944018	−0.472009	−	0.881594i	$-0.656471\pi$
$83$	−713.834	−0.944018	−0.472009	+	0.881594i	$0.656471\pi$
$84$	0	0
$85$	0	0
$86$	−84.0298	−0.105362
$87$	0	0
$88$	−1084.43	−1.31364
$89$	−404.804	−0.482125	−0.241063	−	0.970510i	$-0.577496\pi$
$89$	−404.804	−0.482125	−0.241063	+	0.970510i	$0.577496\pi$
$90$	0	0
$91$	−192.313	−0.221537
$92$	434.745	0.492666
$93$	0	0
$94$	84.9135	0.0931719
$95$	0	0
$96$	0	0
$97$	−75.0273	−0.0785347	−0.0392674	−	0.999229i	$-0.512502\pi$
$97$	−75.0273	−0.0785347	−0.0392674	+	0.999229i	$0.512502\pi$
$98$	−434.763	−0.448140
$99$	0	0
$100$	0	0
$101$	−1087.88	−1.07176	−0.535881	−	0.844294i	$-0.680020\pi$
$101$	−1087.88	−1.07176	−0.535881	+	0.844294i	$0.680020\pi$
$102$	0	0
$103$	1091.82	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$103$	1091.82	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$104$	728.092	0.686493
$105$	0	0
$106$	−675.587	−0.619046
$107$	1029.15	0.929833	0.464917	−	0.885354i	$-0.346084\pi$
$107$	1029.15	0.929833	0.464917	+	0.885354i	$0.346084\pi$
$108$	0	0
$109$	1776.52	1.56110	0.780548	−	0.625096i	$-0.214940\pi$
$109$	1776.52	1.56110	0.780548	+	0.625096i	$0.214940\pi$
$110$	0	0
$111$	0	0
$112$	114.364	0.0964857
$113$	1615.94	1.34526	0.672631	−	0.739978i	$-0.265164\pi$
$113$	1615.94	1.34526	0.672631	+	0.739978i	$0.265164\pi$
$114$	0	0
$115$	0	0
$116$	173.549	0.138910
$117$	0	0
$118$	1083.59	0.845364
$119$	−121.011	−0.0932187
$120$	0	0
$121$	1802.56	1.35429
$122$	715.081	0.530659
$123$	0	0
$124$	−78.8043	−0.0570712
$125$	0	0
$126$	0	0
$127$	1206.10	0.842711	0.421356	−	0.906895i	$-0.361554\pi$
$127$	1206.10	0.842711	0.421356	+	0.906895i	$0.361554\pi$
$128$	−1380.96	−0.953599
$129$	0	0
$130$	0	0
$131$	−1027.86	−0.685528	−0.342764	−	0.939422i	$-0.611363\pi$
$131$	−1027.86	−0.685528	−0.342764	+	0.939422i	$0.611363\pi$
$132$	0	0
$133$	199.821	0.130276
$134$	−417.763	−0.269322
$135$	0	0
$136$	458.144	0.288864
$137$	1260.91	0.786326	0.393163	−	0.919469i	$-0.371381\pi$
$137$	1260.91	0.786326	0.393163	+	0.919469i	$0.371381\pi$
$138$	0	0
$139$	461.832	0.281813	0.140907	−	0.990023i	$-0.454998\pi$
$139$	461.832	0.281813	0.140907	+	0.990023i	$0.454998\pi$
$140$	0	0
$141$	0	0
$142$	371.053	0.219282
$143$	−2103.90	−1.23033
$144$	0	0
$145$	0	0
$146$	1270.00	0.719902
$147$	0	0
$148$	−1101.66	−0.611866
$149$	1459.32	0.802365	0.401182	−	0.915998i	$-0.368599\pi$
$149$	1459.32	0.802365	0.401182	+	0.915998i	$0.368599\pi$
$150$	0	0
$151$	1541.32	0.830666	0.415333	−	0.909669i	$-0.363665\pi$
$151$	1541.32	0.830666	0.415333	+	0.909669i	$0.363665\pi$
$152$	−756.518	−0.403696
$153$	0	0
$154$	393.065	0.205676
$155$	0	0
$156$	0	0
$157$	3215.57	1.63459	0.817295	−	0.576220i	$-0.195473\pi$
$157$	3215.57	1.63459	0.817295	+	0.576220i	$0.195473\pi$
$158$	−1768.92	−0.890681
$159$	0	0
$160$	0	0
$161$	−363.671	−0.178021
$162$	0	0
$163$	−947.587	−0.455342	−0.227671	−	0.973738i	$-0.573111\pi$
$163$	−947.587	−0.455342	−0.227671	+	0.973738i	$0.573111\pi$
$164$	1318.08	0.627590
$165$	0	0
$166$	−979.581	−0.458014
$167$	−685.960	−0.317851	−0.158926	−	0.987291i	$-0.550803\pi$
$167$	−685.960	−0.317851	−0.158926	+	0.987291i	$0.550803\pi$
$168$	0	0
$169$	−784.426	−0.357044
$170$	0	0
$171$	0	0
$172$	374.557	0.166045
$173$	2212.83	0.972475	0.486237	−	0.873827i	$-0.338369\pi$
$173$	2212.83	0.972475	0.486237	+	0.873827i	$0.338369\pi$
$174$	0	0
$175$	0	0
$176$	1251.14	0.535844
$177$	0	0
$178$	−555.505	−0.233915
$179$	3023.22	1.26238	0.631190	−	0.775629i	$-0.282567\pi$
$179$	3023.22	1.26238	0.631190	+	0.775629i	$0.282567\pi$
$180$	0	0
$181$	391.445	0.160751	0.0803753	−	0.996765i	$-0.474388\pi$
$181$	391.445	0.160751	0.0803753	+	0.996765i	$0.474388\pi$
$182$	−263.907	−0.107484
$183$	0	0
$184$	1376.85	0.551646
$185$	0	0
$186$	0	0
$187$	−1323.86	−0.517700
$188$	−378.496	−0.146833
$189$	0	0
$190$	0	0
$191$	3485.59	1.32046	0.660231	−	0.751062i	$-0.270458\pi$
$191$	3485.59	1.32046	0.660231	+	0.751062i	$0.270458\pi$
$192$	0	0
$193$	−2215.07	−0.826136	−0.413068	−	0.910700i	$-0.635543\pi$
$193$	−2215.07	−0.826136	−0.413068	+	0.910700i	$0.635543\pi$
$194$	−102.959	−0.0381031
$195$	0	0
$196$	1937.93	0.706241
$197$	3975.11	1.43764	0.718820	−	0.695196i	$-0.244682\pi$
$197$	3975.11	1.43764	0.718820	+	0.695196i	$0.244682\pi$
$198$	0	0
$199$	−1555.34	−0.554046	−0.277023	−	0.960863i	$-0.589348\pi$
$199$	−1555.34	−0.554046	−0.277023	+	0.960863i	$0.589348\pi$
$200$	0	0
$201$	0	0
$202$	−1492.87	−0.519991
$203$	−145.177	−0.0501941
$204$	0	0
$205$	0	0
$206$	1498.28	0.506749
$207$	0	0
$208$	−840.027	−0.280026
$209$	2186.04	0.723500
$210$	0	0
$211$	1747.73	0.570231	0.285115	−	0.958493i	$-0.407968\pi$
$211$	1747.73	0.570231	0.285115	+	0.958493i	$0.407968\pi$
$212$	3011.38	0.975578
$213$	0	0
$214$	1412.29	0.451132
$215$	0	0
$216$	0	0
$217$	65.9211	0.0206222
$218$	2437.88	0.757404
$219$	0	0
$220$	0	0
$221$	888.847	0.270544
$222$	0	0
$223$	2541.94	0.763323	0.381662	−	0.924302i	$-0.375352\pi$
$223$	2541.94	0.763323	0.381662	+	0.924302i	$0.375352\pi$
$224$	949.939	0.283350
$225$	0	0
$226$	2217.52	0.652687
$227$	2993.26	0.875197	0.437598	−	0.899171i	$-0.355829\pi$
$227$	2993.26	0.875197	0.437598	+	0.899171i	$0.355829\pi$
$228$	0	0
$229$	−4305.31	−1.24237	−0.621185	−	0.783664i	$-0.713348\pi$
$229$	−4305.31	−1.24237	−0.621185	+	0.783664i	$0.713348\pi$
$230$	0	0
$231$	0	0
$232$	549.636	0.155540
$233$	5581.34	1.56930	0.784648	−	0.619942i	$-0.212844\pi$
$233$	5581.34	1.56930	0.784648	+	0.619942i	$0.212844\pi$
$234$	0	0
$235$	0	0
$236$	−4830.05	−1.33224
$237$	0	0
$238$	−166.061	−0.0452274
$239$	1409.63	0.381512	0.190756	−	0.981638i	$-0.438906\pi$
$239$	1409.63	0.381512	0.190756	+	0.981638i	$0.438906\pi$
$240$	0	0
$241$	626.572	0.167473	0.0837366	−	0.996488i	$-0.473315\pi$
$241$	626.572	0.167473	0.0837366	+	0.996488i	$0.473315\pi$
$242$	2473.63	0.657069
$243$	0	0
$244$	−3187.42	−0.836286
$245$	0	0
$246$	0	0
$247$	−1467.72	−0.378093
$248$	−249.576	−0.0639036
$249$	0	0
$250$	0	0
$251$	1705.53	0.428892	0.214446	−	0.976736i	$-0.431205\pi$
$251$	1705.53	0.428892	0.214446	+	0.976736i	$0.431205\pi$
$252$	0	0
$253$	−3978.56	−0.988656
$254$	1655.11	0.408862
$255$	0	0
$256$	−2502.74	−0.611020
$257$	3597.38	0.873146	0.436573	−	0.899669i	$-0.356192\pi$
$257$	3597.38	0.873146	0.436573	+	0.899669i	$0.356192\pi$
$258$	0	0
$259$	921.560	0.221092
$260$	0	0
$261$	0	0
$262$	−1410.51	−0.332601
$263$	−4137.50	−0.970074	−0.485037	−	0.874494i	$-0.661194\pi$
$263$	−4137.50	−0.970074	−0.485037	+	0.874494i	$0.661194\pi$
$264$	0	0
$265$	0	0
$266$	274.211	0.0632065
$267$	0	0
$268$	1862.15	0.424436
$269$	6090.99	1.38057	0.690287	−	0.723536i	$-0.257484\pi$
$269$	6090.99	1.38057	0.690287	+	0.723536i	$0.257484\pi$
$270$	0	0
$271$	−3196.62	−0.716534	−0.358267	−	0.933619i	$-0.616632\pi$
$271$	−3196.62	−0.716534	−0.358267	+	0.933619i	$0.616632\pi$
$272$	−528.578	−0.117830
$273$	0	0
$274$	1730.32	0.381505
$275$	0	0
$276$	0	0
$277$	3119.36	0.676622	0.338311	−	0.941034i	$-0.390144\pi$
$277$	3119.36	0.676622	0.338311	+	0.941034i	$0.390144\pi$
$278$	633.763	0.136729
$279$	0	0
$280$	0	0
$281$	4948.33	1.05051	0.525254	−	0.850946i	$-0.323970\pi$
$281$	4948.33	1.05051	0.525254	+	0.850946i	$0.323970\pi$
$282$	0	0
$283$	4544.93	0.954658	0.477329	−	0.878725i	$-0.341605\pi$
$283$	4544.93	0.954658	0.477329	+	0.878725i	$0.341605\pi$
$284$	−1653.94	−0.345575
$285$	0	0
$286$	−2887.14	−0.596924
$287$	−1102.60	−0.226774
$288$	0	0
$289$	−4353.70	−0.886160
$290$	0	0
$291$	0	0
$292$	−5660.92	−1.13452
$293$	−6860.11	−1.36782	−0.683911	−	0.729566i	$-0.739722\pi$
$293$	−6860.11	−1.36782	−0.683911	+	0.729566i	$0.739722\pi$
$294$	0	0
$295$	0	0
$296$	−3489.01	−0.685117
$297$	0	0
$298$	2002.60	0.389287
$299$	2671.24	0.516661
$300$	0	0
$301$	−313.323	−0.0599988
$302$	2115.12	0.403018
$303$	0	0
$304$	872.824	0.164671
$305$	0	0
$306$	0	0
$307$	−6332.25	−1.17720	−0.588600	−	0.808424i	$-0.700321\pi$
$307$	−6332.25	−1.17720	−0.588600	+	0.808424i	$0.700321\pi$
$308$	−1752.06	−0.324133
$309$	0	0
$310$	0	0
$311$	7077.67	1.29048	0.645238	−	0.763982i	$-0.276758\pi$
$311$	7077.67	1.29048	0.645238	+	0.763982i	$0.276758\pi$
$312$	0	0
$313$	−1381.30	−0.249443	−0.124721	−	0.992192i	$-0.539804\pi$
$313$	−1381.30	−0.249443	−0.124721	+	0.992192i	$0.539804\pi$
$314$	4412.67	0.793062
$315$	0	0
$316$	7884.83	1.40366
$317$	8174.93	1.44842	0.724211	−	0.689578i	$-0.242204\pi$
$317$	8174.93	1.44842	0.724211	+	0.689578i	$0.242204\pi$
$318$	0	0
$319$	−1588.23	−0.278758
$320$	0	0
$321$	0	0
$322$	−499.059	−0.0863711
$323$	−923.549	−0.159095
$324$	0	0
$325$	0	0
$326$	−1300.36	−0.220920
$327$	0	0
$328$	4174.41	0.702723
$329$	316.618	0.0530569
$330$	0	0
$331$	−9661.28	−1.60433	−0.802163	−	0.597105i	$-0.796318\pi$
$331$	−9661.28	−1.60433	−0.802163	+	0.597105i	$0.796318\pi$
$332$	4366.41	0.721801
$333$	0	0
$334$	−941.329	−0.154213
$335$	0	0
$336$	0	0
$337$	4956.02	0.801103	0.400552	−	0.916274i	$-0.368819\pi$
$337$	4956.02	0.801103	0.400552	+	0.916274i	$0.368819\pi$
$338$	−1076.45	−0.173229
$339$	0	0
$340$	0	0
$341$	721.177	0.114528
$342$	0	0
$343$	−3376.19	−0.531478
$344$	1186.24	0.185923
$345$	0	0
$346$	3036.62	0.471820
$347$	1015.60	0.157120	0.0785598	−	0.996909i	$-0.474968\pi$
$347$	1015.60	0.157120	0.0785598	+	0.996909i	$0.474968\pi$
$348$	0	0
$349$	12158.6	1.86485	0.932426	−	0.361360i	$-0.117687\pi$
$349$	12158.6	1.86485	0.932426	+	0.361360i	$0.117687\pi$
$350$	0	0
$351$	0	0
$352$	10392.3	1.57362
$353$	−4236.08	−0.638708	−0.319354	−	0.947635i	$-0.603466\pi$
$353$	−4236.08	−0.638708	−0.319354	+	0.947635i	$0.603466\pi$
$354$	0	0
$355$	0	0
$356$	2476.12	0.368636
$357$	0	0
$358$	4148.70	0.612474
$359$	−517.939	−0.0761443	−0.0380721	−	0.999275i	$-0.512122\pi$
$359$	−517.939	−0.0761443	−0.0380721	+	0.999275i	$0.512122\pi$
$360$	0	0
$361$	−5333.97	−0.777660
$362$	537.172	0.0779921
$363$	0	0
$364$	1176.35	0.169388
$365$	0	0
$366$	0	0
$367$	4616.29	0.656590	0.328295	−	0.944575i	$-0.393526\pi$
$367$	4616.29	0.656590	0.328295	+	0.944575i	$0.393526\pi$
$368$	−1588.53	−0.225021
$369$	0	0
$370$	0	0
$371$	−2519.07	−0.352517
$372$	0	0
$373$	−4765.42	−0.661512	−0.330756	−	0.943716i	$-0.607304\pi$
$373$	−4765.42	−0.661512	−0.330756	+	0.943716i	$0.607304\pi$
$374$	−1816.70	−0.251175
$375$	0	0
$376$	−1198.71	−0.164412
$377$	1066.35	0.145676
$378$	0	0
$379$	−2000.33	−0.271108	−0.135554	−	0.990770i	$-0.543281\pi$
$379$	−2000.33	−0.271108	−0.135554	+	0.990770i	$0.543281\pi$
$380$	0	0
$381$	0	0
$382$	4783.21	0.640655
$383$	−990.294	−0.132119	−0.0660596	−	0.997816i	$-0.521043\pi$
$383$	−990.294	−0.132119	−0.0660596	+	0.997816i	$0.521043\pi$
$384$	0	0
$385$	0	0
$386$	−3039.70	−0.400820
$387$	0	0
$388$	458.930	0.0600481
$389$	404.411	0.0527106	0.0263553	−	0.999653i	$-0.491610\pi$
$389$	404.411	0.0527106	0.0263553	+	0.999653i	$0.491610\pi$
$390$	0	0
$391$	1680.85	0.217402
$392$	6137.49	0.790790
$393$	0	0
$394$	5454.97	0.697507
$395$	0	0
$396$	0	0
$397$	−2919.61	−0.369096	−0.184548	−	0.982824i	$-0.559082\pi$
$397$	−2919.61	−0.369096	−0.184548	+	0.982824i	$0.559082\pi$
$398$	−2134.37	−0.268809
$399$	0	0
$400$	0	0
$401$	−10186.2	−1.26852	−0.634258	−	0.773121i	$-0.718694\pi$
$401$	−10186.2	−1.26852	−0.634258	+	0.773121i	$0.718694\pi$
$402$	0	0
$403$	−484.203	−0.0598508
$404$	6654.38	0.819474
$405$	0	0
$406$	−199.223	−0.0243529
$407$	10081.9	1.22786
$408$	0	0
$409$	6914.24	0.835910	0.417955	−	0.908468i	$-0.362747\pi$
$409$	6914.24	0.835910	0.417955	+	0.908468i	$0.362747\pi$
$410$	0	0
$411$	0	0
$412$	−6678.48	−0.798605
$413$	4040.41	0.481394
$414$	0	0
$415$	0	0
$416$	−6977.49	−0.822355
$417$	0	0
$418$	2999.86	0.351024
$419$	5120.31	0.597002	0.298501	−	0.954409i	$-0.403513\pi$
$419$	5120.31	0.597002	0.298501	+	0.954409i	$0.403513\pi$
$420$	0	0
$421$	1866.49	0.216074	0.108037	−	0.994147i	$-0.465543\pi$
$421$	1866.49	0.216074	0.108037	+	0.994147i	$0.465543\pi$
$422$	2398.38	0.276662
$423$	0	0
$424$	9537.16	1.09237
$425$	0	0
$426$	0	0
$427$	2666.33	0.302185
$428$	−6295.18	−0.710956
$429$	0	0
$430$	0	0
$431$	−4090.64	−0.457168	−0.228584	−	0.973524i	$-0.573410\pi$
$431$	−4090.64	−0.457168	−0.228584	+	0.973524i	$0.573410\pi$
$432$	0	0
$433$	−633.052	−0.0702599	−0.0351299	−	0.999383i	$-0.511185\pi$
$433$	−633.052	−0.0702599	−0.0351299	+	0.999383i	$0.511185\pi$
$434$	90.4623	0.0100054
$435$	0	0
$436$	−10866.7	−1.19362
$437$	−2775.53	−0.303825
$438$	0	0
$439$	−11306.5	−1.22923	−0.614614	−	0.788828i	$-0.710688\pi$
$439$	−11306.5	−1.22923	−0.614614	+	0.788828i	$0.710688\pi$
$440$	0	0
$441$	0	0
$442$	1219.75	0.131261
$443$	8281.30	0.888163	0.444082	−	0.895986i	$-0.353530\pi$
$443$	8281.30	0.888163	0.444082	+	0.895986i	$0.353530\pi$
$444$	0	0
$445$	0	0
$446$	3488.26	0.370345
$447$	0	0
$448$	388.671	0.0409887
$449$	6888.40	0.724017	0.362008	−	0.932175i	$-0.382091\pi$
$449$	6888.40	0.724017	0.362008	+	0.932175i	$0.382091\pi$
$450$	0	0
$451$	−12062.4	−1.25941
$452$	−9884.44	−1.02859
$453$	0	0
$454$	4107.59	0.424623
$455$	0	0
$456$	0	0
$457$	−4283.60	−0.438465	−0.219233	−	0.975673i	$-0.570355\pi$
$457$	−4283.60	−0.438465	−0.219233	+	0.975673i	$0.570355\pi$
$458$	−5908.09	−0.602766
$459$	0	0
$460$	0	0
$461$	13778.3	1.39202	0.696009	−	0.718033i	$-0.254958\pi$
$461$	13778.3	1.39202	0.696009	+	0.718033i	$0.254958\pi$
$462$	0	0
$463$	−5734.53	−0.575608	−0.287804	−	0.957689i	$-0.592925\pi$
$463$	−5734.53	−0.575608	−0.287804	+	0.957689i	$0.592925\pi$
$464$	−634.136	−0.0634461
$465$	0	0
$466$	7659.17	0.761383
$467$	8950.97	0.886941	0.443470	−	0.896289i	$-0.353747\pi$
$467$	8950.97	0.886941	0.443470	+	0.896289i	$0.353747\pi$
$468$	0	0
$469$	−1557.72	−0.153366
$470$	0	0
$471$	0	0
$472$	−15296.9	−1.49173
$473$	−3427.75	−0.333210
$474$	0	0
$475$	0	0
$476$	740.203	0.0712755
$477$	0	0
$478$	1934.41	0.185100
$479$	9681.01	0.923459	0.461729	−	0.887021i	$-0.347229\pi$
$479$	9681.01	0.923459	0.461729	+	0.887021i	$0.347229\pi$
$480$	0	0
$481$	−6769.04	−0.641666
$482$	859.833	0.0812538
$483$	0	0
$484$	−11026.0	−1.03550
$485$	0	0
$486$	0	0
$487$	−8704.66	−0.809950	−0.404975	−	0.914328i	$-0.632720\pi$
$487$	−8704.66	−0.809950	−0.404975	+	0.914328i	$0.632720\pi$
$488$	−10094.7	−0.936404
$489$	0	0
$490$	0	0
$491$	−15595.7	−1.43345	−0.716725	−	0.697356i	$-0.754360\pi$
$491$	−15595.7	−1.43345	−0.716725	+	0.697356i	$0.754360\pi$
$492$	0	0
$493$	670.989	0.0612979
$494$	−2014.13	−0.183441
$495$	0	0
$496$	287.945	0.0260668
$497$	1383.55	0.124871
$498$	0	0
$499$	−9696.28	−0.869870	−0.434935	−	0.900462i	$-0.643229\pi$
$499$	−9696.28	−0.869870	−0.434935	+	0.900462i	$0.643229\pi$
$500$	0	0
$501$	0	0
$502$	2340.46	0.208087
$503$	−20949.7	−1.85706	−0.928532	−	0.371253i	$-0.878928\pi$
$503$	−20949.7	−1.85706	−0.928532	+	0.371253i	$0.878928\pi$
$504$	0	0
$505$	0	0
$506$	−5459.71	−0.479671
$507$	0	0
$508$	−7377.55	−0.644342
$509$	11274.7	0.981816	0.490908	−	0.871211i	$-0.336665\pi$
$509$	11274.7	0.981816	0.490908	+	0.871211i	$0.336665\pi$
$510$	0	0
$511$	4735.46	0.409950
$512$	7613.21	0.657148
$513$	0	0
$514$	4936.62	0.423628
$515$	0	0
$516$	0	0
$517$	3463.80	0.294657
$518$	1264.64	0.107268
$519$	0	0
$520$	0	0
$521$	8675.49	0.729520	0.364760	−	0.931102i	$-0.381151\pi$
$521$	8675.49	0.729520	0.364760	+	0.931102i	$0.381151\pi$
$522$	0	0
$523$	4226.14	0.353339	0.176670	−	0.984270i	$-0.443468\pi$
$523$	4226.14	0.353339	0.176670	+	0.984270i	$0.443468\pi$
$524$	6287.23	0.524158
$525$	0	0
$526$	−5677.82	−0.470655
$527$	−304.680	−0.0251842
$528$	0	0
$529$	−7115.58	−0.584826
$530$	0	0
$531$	0	0
$532$	−1222.27	−0.0996095
$533$	8098.78	0.658156
$534$	0	0
$535$	0	0
$536$	5897.49	0.475248
$537$	0	0
$538$	8358.56	0.669820
$539$	−17734.9	−1.41725
$540$	0	0
$541$	13357.8	1.06154	0.530771	−	0.847515i	$-0.321902\pi$
$541$	13357.8	1.06154	0.530771	+	0.847515i	$0.321902\pi$
$542$	−4386.66	−0.347644
$543$	0	0
$544$	−4390.51	−0.346032
$545$	0	0
$546$	0	0
$547$	−21671.1	−1.69395	−0.846974	−	0.531634i	$-0.821578\pi$
$547$	−21671.1	−1.69395	−0.846974	+	0.531634i	$0.821578\pi$
$548$	−7712.78	−0.601229
$549$	0	0
$550$	0	0
$551$	−1107.98	−0.0856654
$552$	0	0
$553$	−6595.79	−0.507200
$554$	4280.64	0.328280
$555$	0	0
$556$	−2824.95	−0.215476
$557$	−7477.63	−0.568828	−0.284414	−	0.958702i	$-0.591799\pi$
$557$	−7477.63	−0.568828	−0.284414	+	0.958702i	$0.591799\pi$
$558$	0	0
$559$	2301.42	0.174132
$560$	0	0
$561$	0	0
$562$	6790.50	0.509680
$563$	23304.7	1.74454	0.872269	−	0.489026i	$-0.162648\pi$
$563$	23304.7	1.74454	0.872269	+	0.489026i	$0.162648\pi$
$564$	0	0
$565$	0	0
$566$	6236.92	0.463176
$567$	0	0
$568$	−5238.10	−0.386947
$569$	−14649.1	−1.07930	−0.539650	−	0.841890i	$-0.681443\pi$
$569$	−14649.1	−1.07930	−0.539650	+	0.841890i	$0.681443\pi$
$570$	0	0
$571$	23164.0	1.69769	0.848846	−	0.528640i	$-0.177298\pi$
$571$	23164.0	1.69769	0.848846	+	0.528640i	$0.177298\pi$
$572$	12869.2	0.940715
$573$	0	0
$574$	−1513.07	−0.110025
$575$	0	0
$576$	0	0
$577$	−7865.97	−0.567529	−0.283765	−	0.958894i	$-0.591583\pi$
$577$	−7865.97	−0.567529	−0.283765	+	0.958894i	$0.591583\pi$
$578$	−5974.50	−0.429942
$579$	0	0
$580$	0	0
$581$	−3652.58	−0.260817
$582$	0	0
$583$	−27558.6	−1.95774
$584$	−17928.4	−1.27034
$585$	0	0
$586$	−9413.99	−0.663632
$587$	956.182	0.0672332	0.0336166	−	0.999435i	$-0.489297\pi$
$587$	956.182	0.0672332	0.0336166	+	0.999435i	$0.489297\pi$
$588$	0	0
$589$	503.108	0.0351956
$590$	0	0
$591$	0	0
$592$	4025.40	0.279465
$593$	16966.0	1.17489	0.587444	−	0.809265i	$-0.300134\pi$
$593$	16966.0	1.17489	0.587444	+	0.809265i	$0.300134\pi$
$594$	0	0
$595$	0	0
$596$	−8926.45	−0.613492
$597$	0	0
$598$	3665.69	0.250671
$599$	−6191.41	−0.422327	−0.211164	−	0.977451i	$-0.567725\pi$
$599$	−6191.41	−0.422327	−0.211164	+	0.977451i	$0.567725\pi$
$600$	0	0
$601$	−2718.54	−0.184512	−0.0922559	−	0.995735i	$-0.529408\pi$
$601$	−2718.54	−0.184512	−0.0922559	+	0.995735i	$0.529408\pi$
$602$	−429.968	−0.0291099
$603$	0	0
$604$	−9428.00	−0.635132
$605$	0	0
$606$	0	0
$607$	−16825.0	−1.12505	−0.562524	−	0.826781i	$-0.690170\pi$
$607$	−16825.0	−1.12505	−0.562524	+	0.826781i	$0.690170\pi$
$608$	7249.91	0.483590
$609$	0	0
$610$	0	0
$611$	−2325.62	−0.153985
$612$	0	0
$613$	20175.1	1.32930	0.664652	−	0.747153i	$-0.268580\pi$
$613$	20175.1	1.32930	0.664652	+	0.747153i	$0.268580\pi$
$614$	−8689.63	−0.571148
$615$	0	0
$616$	−5548.84	−0.362937
$617$	−11310.6	−0.738004	−0.369002	−	0.929429i	$-0.620301\pi$
$617$	−11310.6	−0.738004	−0.369002	+	0.929429i	$0.620301\pi$
$618$	0	0
$619$	−17059.9	−1.10775	−0.553873	−	0.832601i	$-0.686851\pi$
$619$	−17059.9	−1.10775	−0.553873	+	0.832601i	$0.686851\pi$
$620$	0	0
$621$	0	0
$622$	9712.56	0.626106
$623$	−2071.32	−0.133203
$624$	0	0
$625$	0	0
$626$	−1895.53	−0.121023
$627$	0	0
$628$	−19669.2	−1.24982
$629$	−4259.34	−0.270002
$630$	0	0
$631$	−13186.3	−0.831916	−0.415958	−	0.909384i	$-0.636554\pi$
$631$	−13186.3	−0.831916	−0.415958	+	0.909384i	$0.636554\pi$
$632$	24971.5	1.57170
$633$	0	0
$634$	11218.3	0.702738
$635$	0	0
$636$	0	0
$637$	11907.4	0.740638
$638$	−2179.50	−0.135246
$639$	0	0
$640$	0	0
$641$	16362.0	1.00820	0.504102	−	0.863644i	$-0.331823\pi$
$641$	16362.0	1.00820	0.504102	+	0.863644i	$0.331823\pi$
$642$	0	0
$643$	28044.9	1.72004	0.860019	−	0.510262i	$-0.170452\pi$
$643$	28044.9	1.72004	0.860019	+	0.510262i	$0.170452\pi$
$644$	2224.52	0.136115
$645$	0	0
$646$	−1267.37	−0.0771888
$647$	−21247.7	−1.29109	−0.645543	−	0.763724i	$-0.723369\pi$
$647$	−21247.7	−1.29109	−0.645543	+	0.763724i	$0.723369\pi$
$648$	0	0
$649$	44202.1	2.67347
$650$	0	0
$651$	0	0
$652$	5796.24	0.348157
$653$	−1259.86	−0.0755007	−0.0377504	−	0.999287i	$-0.512019\pi$
$653$	−1259.86	−0.0755007	−0.0377504	+	0.999287i	$0.512019\pi$
$654$	0	0
$655$	0	0
$656$	−4816.17	−0.286646
$657$	0	0
$658$	434.489	0.0257419
$659$	−12046.7	−0.712098	−0.356049	−	0.934467i	$-0.615876\pi$
$659$	−12046.7	−0.712098	−0.356049	+	0.934467i	$0.615876\pi$
$660$	0	0
$661$	13108.1	0.771324	0.385662	−	0.922640i	$-0.373973\pi$
$661$	13108.1	0.771324	0.385662	+	0.922640i	$0.373973\pi$
$662$	−13258.0	−0.778379
$663$	0	0
$664$	13828.6	0.808213
$665$	0	0
$666$	0	0
$667$	2016.51	0.117061
$668$	4195.91	0.243031
$669$	0	0
$670$	0	0
$671$	29169.7	1.67822
$672$	0	0
$673$	−2743.65	−0.157147	−0.0785734	−	0.996908i	$-0.525037\pi$
$673$	−2743.65	−0.157147	−0.0785734	+	0.996908i	$0.525037\pi$
$674$	6801.06	0.388675
$675$	0	0
$676$	4798.21	0.272998
$677$	−25004.0	−1.41947	−0.709735	−	0.704468i	$-0.751186\pi$
$677$	−25004.0	−1.41947	−0.709735	+	0.704468i	$0.751186\pi$
$678$	0	0
$679$	−383.903	−0.0216979
$680$	0	0
$681$	0	0
$682$	989.657	0.0555659
$683$	4846.23	0.271502	0.135751	−	0.990743i	$-0.456655\pi$
$683$	4846.23	0.271502	0.135751	+	0.990743i	$0.456655\pi$
$684$	0	0
$685$	0	0
$686$	−4633.08	−0.257860
$687$	0	0
$688$	−1368.61	−0.0758395
$689$	18503.1	1.02309
$690$	0	0
$691$	3484.58	0.191837	0.0959187	−	0.995389i	$-0.469421\pi$
$691$	3484.58	0.191837	0.0959187	+	0.995389i	$0.469421\pi$
$692$	−13535.5	−0.743560
$693$	0	0
$694$	1393.70	0.0762305
$695$	0	0
$696$	0	0
$697$	5096.07	0.276940
$698$	16685.0	0.904780
$699$	0	0
$700$	0	0
$701$	15701.4	0.845981	0.422991	−	0.906134i	$-0.360980\pi$
$701$	15701.4	0.845981	0.422991	+	0.906134i	$0.360980\pi$
$702$	0	0
$703$	7033.32	0.377335
$704$	4252.05	0.227635
$705$	0	0
$706$	−5813.10	−0.309885
$707$	−5566.50	−0.296110
$708$	0	0
$709$	15643.4	0.828634	0.414317	−	0.910133i	$-0.364020\pi$
$709$	15643.4	0.828634	0.414317	+	0.910133i	$0.364020\pi$
$710$	0	0
$711$	0	0
$712$	7841.98	0.412768
$713$	−915.649	−0.0480944
$714$	0	0
$715$	0	0
$716$	−18492.5	−0.965222
$717$	0	0
$718$	−710.759	−0.0369433
$719$	−6964.13	−0.361222	−0.180611	−	0.983555i	$-0.557807\pi$
$719$	−6964.13	−0.361222	−0.180611	+	0.983555i	$0.557807\pi$
$720$	0	0
$721$	5586.66	0.288569
$722$	−7319.71	−0.377301
$723$	0	0
$724$	−2394.41	−0.122911
$725$	0	0
$726$	0	0
$727$	14207.2	0.724782	0.362391	−	0.932026i	$-0.381961\pi$
$727$	14207.2	0.724782	0.362391	+	0.932026i	$0.381961\pi$
$728$	3725.53	0.189667
$729$	0	0
$730$	0	0
$731$	1448.14	0.0732716
$732$	0	0
$733$	−26530.5	−1.33687	−0.668437	−	0.743769i	$-0.733036\pi$
$733$	−26530.5	−1.33687	−0.668437	+	0.743769i	$0.733036\pi$
$734$	6334.85	0.318561
$735$	0	0
$736$	−13194.7	−0.660821
$737$	−17041.4	−0.851735
$738$	0	0
$739$	−5683.47	−0.282909	−0.141455	−	0.989945i	$-0.545178\pi$
$739$	−5683.47	−0.282909	−0.141455	+	0.989945i	$0.545178\pi$
$740$	0	0
$741$	0	0
$742$	−3456.87	−0.171032
$743$	15568.6	0.768715	0.384358	−	0.923184i	$-0.374423\pi$
$743$	15568.6	0.768715	0.384358	+	0.923184i	$0.374423\pi$
$744$	0	0
$745$	0	0
$746$	−6539.50	−0.320949
$747$	0	0
$748$	8097.82	0.395836
$749$	5266.02	0.256898
$750$	0	0
$751$	−8261.64	−0.401427	−0.200713	−	0.979650i	$-0.564326\pi$
$751$	−8261.64	−0.401427	−0.200713	+	0.979650i	$0.564326\pi$
$752$	1383.00	0.0670648
$753$	0	0
$754$	1463.33	0.0706783
$755$	0	0
$756$	0	0
$757$	13381.5	0.642481	0.321240	−	0.946998i	$-0.395900\pi$
$757$	13381.5	0.642481	0.321240	+	0.946998i	$0.395900\pi$
$758$	−2745.01	−0.131535
$759$	0	0
$760$	0	0
$761$	5449.84	0.259601	0.129801	−	0.991540i	$-0.458566\pi$
$761$	5449.84	0.259601	0.129801	+	0.991540i	$0.458566\pi$
$762$	0	0
$763$	9090.15	0.431305
$764$	−21320.8	−1.00963
$765$	0	0
$766$	−1358.96	−0.0641009
$767$	−29677.6	−1.39713
$768$	0	0
$769$	−19364.0	−0.908039	−0.454020	−	0.890992i	$-0.650010\pi$
$769$	−19364.0	−0.908039	−0.454020	+	0.890992i	$0.650010\pi$
$770$	0	0
$771$	0	0
$772$	13549.2	0.631668
$773$	1865.54	0.0868033	0.0434017	−	0.999058i	$-0.486180\pi$
$773$	1865.54	0.0868033	0.0434017	+	0.999058i	$0.486180\pi$
$774$	0	0
$775$	0	0
$776$	1453.45	0.0672369
$777$	0	0
$778$	554.965	0.0255739
$779$	−8414.98	−0.387032
$780$	0	0
$781$	15136.0	0.693483
$782$	2306.59	0.105478
$783$	0	0
$784$	−7081.05	−0.322570
$785$	0	0
$786$	0	0
$787$	19207.3	0.869970	0.434985	−	0.900438i	$-0.356754\pi$
$787$	19207.3	0.869970	0.434985	+	0.900438i	$0.356754\pi$
$788$	−24315.2	−1.09923
$789$	0	0
$790$	0	0
$791$	8268.50	0.371674
$792$	0	0
$793$	−19584.7	−0.877017
$794$	−4006.53	−0.179076
$795$	0	0
$796$	9513.78	0.423627
$797$	−186.074	−0.00826988	−0.00413494	−	0.999991i	$-0.501316\pi$
$797$	−186.074	−0.00826988	−0.00413494	+	0.999991i	$0.501316\pi$
$798$	0	0
$799$	−1463.37	−0.0647940
$800$	0	0
$801$	0	0
$802$	−13978.3	−0.615452
$803$	51805.9	2.27670
$804$	0	0
$805$	0	0
$806$	−664.463	−0.0290381
$807$	0	0
$808$	21074.7	0.917580
$809$	5903.09	0.256541	0.128270	−	0.991739i	$-0.459057\pi$
$809$	5903.09	0.256541	0.128270	+	0.991739i	$0.459057\pi$
$810$	0	0
$811$	23111.0	1.00066	0.500331	−	0.865834i	$-0.333212\pi$
$811$	23111.0	1.00066	0.500331	+	0.865834i	$0.333212\pi$
$812$	888.022	0.0383787
$813$	0	0
$814$	13835.2	0.595727
$815$	0	0
$816$	0	0
$817$	−2391.27	−0.102399
$818$	9488.29	0.405563
$819$	0	0
$820$	0	0
$821$	−9644.29	−0.409973	−0.204987	−	0.978765i	$-0.565715\pi$
$821$	−9644.29	−0.409973	−0.204987	+	0.978765i	$0.565715\pi$
$822$	0	0
$823$	33573.4	1.42199	0.710994	−	0.703198i	$-0.248245\pi$
$823$	33573.4	1.42199	0.710994	+	0.703198i	$0.248245\pi$
$824$	−21151.0	−0.894211
$825$	0	0
$826$	5544.59	0.233560
$827$	25916.1	1.08971	0.544855	−	0.838530i	$-0.316585\pi$
$827$	25916.1	1.08971	0.544855	+	0.838530i	$0.316585\pi$
$828$	0	0
$829$	−28650.6	−1.20033	−0.600166	−	0.799876i	$-0.704899\pi$
$829$	−28650.6	−1.20033	−0.600166	+	0.799876i	$0.704899\pi$
$830$	0	0
$831$	0	0
$832$	−2854.86	−0.118960
$833$	7492.58	0.311647
$834$	0	0
$835$	0	0
$836$	−13371.7	−0.553192
$837$	0	0
$838$	7026.51	0.289650
$839$	712.960	0.0293374	0.0146687	−	0.999892i	$-0.495331\pi$
$839$	712.960	0.0293374	0.0146687	+	0.999892i	$0.495331\pi$
$840$	0	0
$841$	−23584.0	−0.966994
$842$	2561.35	0.104834
$843$	0	0
$844$	−10690.6	−0.436002
$845$	0	0
$846$	0	0
$847$	9223.44	0.374169
$848$	−11003.4	−0.445587
$849$	0	0
$850$	0	0
$851$	−12800.5	−0.515625
$852$	0	0
$853$	30367.2	1.21894	0.609469	−	0.792810i	$-0.291383\pi$
$853$	30367.2	1.21894	0.609469	+	0.792810i	$0.291383\pi$
$854$	3658.96	0.146612
$855$	0	0
$856$	−19937.1	−0.796069
$857$	−9080.70	−0.361950	−0.180975	−	0.983488i	$-0.557925\pi$
$857$	−9080.70	−0.361950	−0.180975	+	0.983488i	$0.557925\pi$
$858$	0	0
$859$	26160.2	1.03909	0.519543	−	0.854444i	$-0.326102\pi$
$859$	26160.2	1.03909	0.519543	+	0.854444i	$0.326102\pi$
$860$	0	0
$861$	0	0
$862$	−5613.51	−0.221806
$863$	40102.0	1.58180	0.790898	−	0.611949i	$-0.209614\pi$
$863$	40102.0	1.58180	0.790898	+	0.611949i	$0.209614\pi$
$864$	0	0
$865$	0	0
$866$	−868.725	−0.0340883
$867$	0	0
$868$	−403.229	−0.0157678
$869$	−72157.9	−2.81679
$870$	0	0
$871$	11441.7	0.445108
$872$	−34415.2	−1.33652
$873$	0	0
$874$	−3808.80	−0.147408
$875$	0	0
$876$	0	0
$877$	−25252.2	−0.972299	−0.486149	−	0.873876i	$-0.661599\pi$
$877$	−25252.2	−0.972299	−0.486149	+	0.873876i	$0.661599\pi$
$878$	−15515.7	−0.596390
$879$	0	0
$880$	0	0
$881$	−2049.26	−0.0783670	−0.0391835	−	0.999232i	$-0.512476\pi$
$881$	−2049.26	−0.0783670	−0.0391835	+	0.999232i	$0.512476\pi$
$882$	0	0
$883$	−39413.4	−1.50211	−0.751057	−	0.660237i	$-0.770456\pi$
$883$	−39413.4	−1.50211	−0.751057	+	0.660237i	$0.770456\pi$
$884$	−5436.94	−0.206860
$885$	0	0
$886$	11364.3	0.430914
$887$	−36968.5	−1.39941	−0.699707	−	0.714430i	$-0.746686\pi$
$887$	−36968.5	−1.39941	−0.699707	+	0.714430i	$0.746686\pi$
$888$	0	0
$889$	6171.44	0.232827
$890$	0	0
$891$	0	0
$892$	−15548.7	−0.583641
$893$	2416.42	0.0905514
$894$	0	0
$895$	0	0
$896$	−7066.15	−0.263464
$897$	0	0
$898$	9452.82	0.351275
$899$	−365.525	−0.0135605
$900$	0	0
$901$	11642.9	0.430499
$902$	−16553.0	−0.611036
$903$	0	0
$904$	−31304.4	−1.15174
$905$	0	0
$906$	0	0
$907$	−2710.62	−0.0992334	−0.0496167	−	0.998768i	$-0.515800\pi$
$907$	−2710.62	−0.0992334	−0.0496167	+	0.998768i	$0.515800\pi$
$908$	−18309.3	−0.669180
$909$	0	0
$910$	0	0
$911$	22996.6	0.836345	0.418172	−	0.908368i	$-0.362671\pi$
$911$	22996.6	0.836345	0.418172	+	0.908368i	$0.362671\pi$
$912$	0	0
$913$	−39959.2	−1.44847
$914$	−5878.31	−0.212732
$915$	0	0
$916$	26334.9	0.949922
$917$	−5259.38	−0.189400
$918$	0	0
$919$	−39103.8	−1.40361	−0.701804	−	0.712370i	$-0.747622\pi$
$919$	−39103.8	−1.40361	−0.701804	+	0.712370i	$0.747622\pi$
$920$	0	0
$921$	0	0
$922$	18907.7	0.675372
$923$	−10162.5	−0.362407
$924$	0	0
$925$	0	0
$926$	−7869.39	−0.279270
$927$	0	0
$928$	−5267.30	−0.186323
$929$	−35954.6	−1.26979	−0.634894	−	0.772600i	$-0.718956\pi$
$929$	−35954.6	−1.26979	−0.634894	+	0.772600i	$0.718956\pi$
$930$	0	0
$931$	−12372.2	−0.435536
$932$	−34140.2	−1.19989
$933$	0	0
$934$	12283.2	0.430321
$935$	0	0
$936$	0	0
$937$	7263.94	0.253258	0.126629	−	0.991950i	$-0.459584\pi$
$937$	7263.94	0.253258	0.126629	+	0.991950i	$0.459584\pi$
$938$	−2137.63	−0.0744094
$939$	0	0
$940$	0	0
$941$	7478.91	0.259092	0.129546	−	0.991573i	$-0.458648\pi$
$941$	7478.91	0.259092	0.129546	+	0.991573i	$0.458648\pi$
$942$	0	0
$943$	15315.1	0.528875
$944$	17648.7	0.608490
$945$	0	0
$946$	−4703.84	−0.161665
$947$	−13491.4	−0.462947	−0.231473	−	0.972841i	$-0.574355\pi$
$947$	−13491.4	−0.462947	−0.231473	+	0.972841i	$0.574355\pi$
$948$	0	0
$949$	−34782.9	−1.18978
$950$	0	0
$951$	0	0
$952$	2344.25	0.0798085
$953$	13981.6	0.475246	0.237623	−	0.971357i	$-0.423632\pi$
$953$	13981.6	0.475246	0.237623	+	0.971357i	$0.423632\pi$
$954$	0	0
$955$	0	0
$956$	−8622.48	−0.291706
$957$	0	0
$958$	13285.1	0.448039
$959$	6451.87	0.217249
$960$	0	0
$961$	−29625.0	−0.994429
$962$	−9289.02	−0.311320
$963$	0	0
$964$	−3832.64	−0.128051
$965$	0	0
$966$	0	0
$967$	9081.47	0.302007	0.151003	−	0.988533i	$-0.451750\pi$
$967$	9081.47	0.302007	0.151003	+	0.988533i	$0.451750\pi$
$968$	−34919.8	−1.15947
$969$	0	0
$970$	0	0
$971$	−9709.13	−0.320887	−0.160443	−	0.987045i	$-0.551292\pi$
$971$	−9709.13	−0.320887	−0.160443	+	0.987045i	$0.551292\pi$
$972$	0	0
$973$	2363.12	0.0778604
$974$	−11945.2	−0.392967
$975$	0	0
$976$	11646.6	0.381967
$977$	−10854.9	−0.355455	−0.177727	−	0.984080i	$-0.556875\pi$
$977$	−10854.9	−0.355455	−0.177727	+	0.984080i	$0.556875\pi$
$978$	0	0
$979$	−22660.2	−0.739759
$980$	0	0
$981$	0	0
$982$	−21401.7	−0.695474
$983$	7510.10	0.243678	0.121839	−	0.992550i	$-0.461121\pi$
$983$	7510.10	0.243678	0.121839	+	0.992550i	$0.461121\pi$
$984$	0	0
$985$	0	0
$986$	920.786	0.0297402
$987$	0	0
$988$	8977.84	0.289092
$989$	4352.08	0.139927
$990$	0	0
$991$	46125.6	1.47854	0.739268	−	0.673412i	$-0.235172\pi$
$991$	46125.6	1.47854	0.739268	+	0.673412i	$0.235172\pi$
$992$	2391.75	0.0765506
$993$	0	0
$994$	1898.62	0.0605841
$995$	0	0
$996$	0	0
$997$	−45350.1	−1.44057	−0.720287	−	0.693677i	$-0.755990\pi$
$997$	−45350.1	−1.44057	−0.720287	+	0.693677i	$0.755990\pi$
$998$	−13306.0	−0.422039
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.g.1.2		2
3.2	odd	2		2025.4.a.n.1.1		2
5.4	even	2		81.4.a.d.1.1		2
9.4	even	3		225.4.e.b.151.1		4
9.7	even	3		225.4.e.b.76.1		4
15.14	odd	2		81.4.a.a.1.2		2
20.19	odd	2		1296.4.a.u.1.2		2
45.4	even	6		9.4.c.a.7.2	yes	4
45.7	odd	12		225.4.k.b.49.3		8
45.13	odd	12		225.4.k.b.124.3		8
45.14	odd	6		27.4.c.a.19.1		4
45.22	odd	12		225.4.k.b.124.2		8
45.29	odd	6		27.4.c.a.10.1		4
45.34	even	6		9.4.c.a.4.2	✓	4
45.43	odd	12		225.4.k.b.49.2		8
60.59	even	2		1296.4.a.i.1.1		2
180.59	even	6		432.4.i.c.289.2		4
180.79	odd	6		144.4.i.c.49.2		4
180.119	even	6		432.4.i.c.145.2		4
180.139	odd	6		144.4.i.c.97.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.4.c.a.4.2	✓	4	45.34	even	6
9.4.c.a.7.2	yes	4	45.4	even	6
27.4.c.a.10.1		4	45.29	odd	6
27.4.c.a.19.1		4	45.14	odd	6
81.4.a.a.1.2		2	15.14	odd	2
81.4.a.d.1.1		2	5.4	even	2
144.4.i.c.49.2		4	180.79	odd	6
144.4.i.c.97.2		4	180.139	odd	6
225.4.e.b.76.1		4	9.7	even	3
225.4.e.b.151.1		4	9.4	even	3
225.4.k.b.49.2		8	45.43	odd	12
225.4.k.b.49.3		8	45.7	odd	12
225.4.k.b.124.2		8	45.22	odd	12
225.4.k.b.124.3		8	45.13	odd	12
432.4.i.c.145.2		4	180.119	even	6
432.4.i.c.289.2		4	180.59	even	6
1296.4.a.i.1.1		2	60.59	even	2
1296.4.a.u.1.2		2	20.19	odd	2
2025.4.a.g.1.2		2	1.1	even	1	trivial
2025.4.a.n.1.1		2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.37228 q^{2} -6.11684 q^{4} +5.11684 q^{7} -19.3723 q^{8} +O(q^{10})\)	\(q+1.37228 q^{2} -6.11684 q^{4} +5.11684 q^{7} -19.3723 q^{8} +55.9783 q^{11} -37.5842 q^{13} +7.02175 q^{14} +22.3505 q^{16} -23.6495 q^{17} +39.0516 q^{19} +76.8179 q^{22} -71.0733 q^{23} -51.5761 q^{26} -31.2989 q^{28} -28.3723 q^{29} +12.8832 q^{31} +185.649 q^{32} -32.4537 q^{34} +180.103 q^{37} +53.5898 q^{38} -215.484 q^{41} -61.2337 q^{43} -342.410 q^{44} -97.5326 q^{46} +61.8776 q^{47} -316.818 q^{49} +229.897 q^{52} -492.310 q^{53} -99.1249 q^{56} -38.9348 q^{58} +789.630 q^{59} +521.090 q^{61} +17.6793 q^{62} +75.9590 q^{64} -304.429 q^{67} +144.660 q^{68} +270.391 q^{71} +925.464 q^{73} +247.152 q^{74} -238.873 q^{76} +286.432 q^{77} -1289.03 q^{79} -295.704 q^{82} -713.834 q^{83} -84.0298 q^{86} -1084.43 q^{88} -404.804 q^{89} -192.313 q^{91} +434.745 q^{92} +84.9135 q^{94} -75.0273 q^{97} -434.763 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 5 q^{4} - 7 q^{7} - 33 q^{8}+O(q^{10}) \) 2 * q - 3 * q^2 + 5 * q^4 - 7 * q^7 - 33 * q^8	\( 2 q - 3 q^{2} + 5 q^{4} - 7 q^{7} - 33 q^{8} + 66 q^{11} + 11 q^{13} + 60 q^{14} - 7 q^{16} - 99 q^{17} - 77 q^{19} + 33 q^{22} - 33 q^{23} - 264 q^{26} - 166 q^{28} - 51 q^{29} + 43 q^{31} + 423 q^{32} + 297 q^{34} + 50 q^{37} + 561 q^{38} + 132 q^{41} - 88 q^{43} - 231 q^{44} - 264 q^{46} - 399 q^{47} - 513 q^{49} + 770 q^{52} - 54 q^{53} + 66 q^{56} + 60 q^{58} + 798 q^{59} + 439 q^{61} - 114 q^{62} - 727 q^{64} - 988 q^{67} - 693 q^{68} + 1368 q^{71} + 455 q^{73} + 816 q^{74} - 1529 q^{76} + 165 q^{77} - 803 q^{79} - 1815 q^{82} - 813 q^{83} + 33 q^{86} - 1221 q^{88} - 396 q^{89} - 781 q^{91} + 858 q^{92} + 2100 q^{94} - 736 q^{97} + 423 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 + 5 * q^4 - 7 * q^7 - 33 * q^8 + 66 * q^11 + 11 * q^13 + 60 * q^14 - 7 * q^16 - 99 * q^17 - 77 * q^19 + 33 * q^22 - 33 * q^23 - 264 * q^26 - 166 * q^28 - 51 * q^29 + 43 * q^31 + 423 * q^32 + 297 * q^34 + 50 * q^37 + 561 * q^38 + 132 * q^41 - 88 * q^43 - 231 * q^44 - 264 * q^46 - 399 * q^47 - 513 * q^49 + 770 * q^52 - 54 * q^53 + 66 * q^56 + 60 * q^58 + 798 * q^59 + 439 * q^61 - 114 * q^62 - 727 * q^64 - 988 * q^67 - 693 * q^68 + 1368 * q^71 + 455 * q^73 + 816 * q^74 - 1529 * q^76 + 165 * q^77 - 803 * q^79 - 1815 * q^82 - 813 * q^83 + 33 * q^86 - 1221 * q^88 - 396 * q^89 - 781 * q^91 + 858 * q^92 + 2100 * q^94 - 736 * q^97 + 423 * q^98

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