LMFDB - Embedded newform 2025.4.a.bi.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2025,4,Mod(1,2025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [12,4,0,48,0,0,-6,45,0,0,29,0,-24,-69] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	yes
Analytic conductor:	$119.478867762$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 64 x^{10} + 241 x^{9} + 1452 x^{8} - 5130 x^{7} - 13834 x^{6} + 45247 x^{5} + \cdots + 73008 $ x^12 - 4x^11 - 64x^10 + 241x^9 + 1452x^8 - 5130x^7 - 13834x^6 + 45247x^5 + 52669x^4 - 144610x^3 - 96816x^2 + 143136*x + 73008
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$5.54419$ 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)

$f(q)$	$=$	$q+5.54419 q^{2} +22.7380 q^{4} -25.7127 q^{7} +81.7101 q^{8} +6.25584 q^{11} +15.2600 q^{13} -142.556 q^{14} +271.112 q^{16} +36.0074 q^{17} -52.7512 q^{19} +34.6835 q^{22} +83.7341 q^{23} +84.6040 q^{26} -584.654 q^{28} +119.091 q^{29} +276.661 q^{31} +849.415 q^{32} +199.631 q^{34} +117.553 q^{37} -292.462 q^{38} -159.322 q^{41} +294.963 q^{43} +142.245 q^{44} +464.237 q^{46} -83.2917 q^{47} +318.141 q^{49} +346.981 q^{52} -149.018 q^{53} -2100.98 q^{56} +660.265 q^{58} +635.461 q^{59} +597.151 q^{61} +1533.86 q^{62} +2540.42 q^{64} +330.015 q^{67} +818.735 q^{68} -1149.79 q^{71} +130.284 q^{73} +651.738 q^{74} -1199.46 q^{76} -160.854 q^{77} -737.835 q^{79} -883.309 q^{82} +369.902 q^{83} +1635.33 q^{86} +511.165 q^{88} -225.638 q^{89} -392.374 q^{91} +1903.94 q^{92} -461.784 q^{94} +11.9134 q^{97} +1763.83 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 48 q^{4} - 6 q^{7} + 45 q^{8} + 29 q^{11} - 24 q^{13} - 69 q^{14} + 192 q^{16} + 79 q^{17} - 75 q^{19} + 18 q^{22} + 318 q^{23} - 154 q^{26} - 96 q^{28} + 106 q^{29} + 60 q^{31} + 914 q^{32}+ \cdots + 7511 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 48 * q^4 - 6 * q^7 + 45 * q^8 + 29 * q^11 - 24 * q^13 - 69 * q^14 + 192 * q^16 + 79 * q^17 - 75 * q^19 + 18 * q^22 + 318 * q^23 - 154 * q^26 - 96 * q^28 + 106 * q^29 + 60 * q^31 + 914 * q^32 - 108 * q^34 + 84 * q^37 + 640 * q^38 - 353 * q^41 + 426 * q^43 + 571 * q^44 + 270 * q^46 + 1210 * q^47 + 666 * q^49 + 75 * q^52 + 448 * q^53 - 570 * q^56 - 594 * q^58 + 482 * q^59 + 402 * q^61 + 2544 * q^62 + 975 * q^64 + 201 * q^67 + 3437 * q^68 - 944 * q^71 + 453 * q^73 + 10 * q^74 - 462 * q^76 + 2652 * q^77 + 258 * q^79 - 873 * q^82 + 3012 * q^83 + 1952 * q^86 + 216 * q^88 - 738 * q^89 - 618 * q^91 + 5232 * q^92 + 63 * q^94 + 318 * q^97 + 7511 * q^98

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 $

$2$	5.54419	1.96017	0.980083	−	0.198590i	$-0.0636361\pi$
$2$	5.54419	1.96017	0.980083	+	0.198590i	$0.0636361\pi$
$3$	0	0
$3$	0	0
$4$	22.7380	2.84225
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−25.7127	−1.38835	−0.694177	−	0.719805i	$-0.744231\pi$
$7$	−25.7127	−1.38835	−0.694177	+	0.719805i	$0.744231\pi$
$8$	81.7101	3.61111
$9$	0	0
$10$	0	0
$11$	6.25584	0.171473	0.0857366	−	0.996318i	$-0.472676\pi$
$11$	6.25584	0.171473	0.0857366	+	0.996318i	$0.472676\pi$
$12$	0	0
$13$	15.2600	0.325565	0.162783	−	0.986662i	$-0.447953\pi$
$13$	15.2600	0.325565	0.162783	+	0.986662i	$0.447953\pi$
$14$	−142.556	−2.72140
$15$	0	0
$16$	271.112	4.23613
$17$	36.0074	0.513710	0.256855	−	0.966450i	$-0.417314\pi$
$17$	36.0074	0.513710	0.256855	+	0.966450i	$0.417314\pi$
$18$	0	0
$19$	−52.7512	−0.636945	−0.318472	−	0.947932i	$-0.603170\pi$
$19$	−52.7512	−0.636945	−0.318472	+	0.947932i	$0.603170\pi$
$20$	0	0
$21$	0	0
$22$	34.6835	0.336116
$23$	83.7341	0.759120	0.379560	−	0.925167i	$-0.376075\pi$
$23$	83.7341	0.759120	0.379560	+	0.925167i	$0.376075\pi$
$24$	0	0
$25$	0	0
$26$	84.6040	0.638162
$27$	0	0
$28$	−584.654	−3.94604
$29$	119.091	0.762576	0.381288	−	0.924456i	$-0.375481\pi$
$29$	119.091	0.762576	0.381288	+	0.924456i	$0.375481\pi$
$30$	0	0
$31$	276.661	1.60290	0.801449	−	0.598064i	$-0.204063\pi$
$31$	276.661	1.60290	0.801449	+	0.598064i	$0.204063\pi$
$32$	849.415	4.69240
$33$	0	0
$34$	199.631	1.00696
$35$	0	0
$36$	0	0
$37$	117.553	0.522315	0.261158	−	0.965296i	$-0.415896\pi$
$37$	117.553	0.522315	0.261158	+	0.965296i	$0.415896\pi$
$38$	−292.462	−1.24852
$39$	0	0
$40$	0	0
$41$	−159.322	−0.606875	−0.303437	−	0.952851i	$-0.598134\pi$
$41$	−159.322	−0.606875	−0.303437	+	0.952851i	$0.598134\pi$
$42$	0	0
$43$	294.963	1.04608	0.523040	−	0.852308i	$-0.324798\pi$
$43$	294.963	1.04608	0.523040	+	0.852308i	$0.324798\pi$
$44$	142.245	0.487370
$45$	0	0
$46$	464.237	1.48800
$47$	−83.2917	−0.258497	−0.129248	−	0.991612i	$-0.541256\pi$
$47$	−83.2917	−0.258497	−0.129248	+	0.991612i	$0.541256\pi$
$48$	0	0
$49$	318.141	0.927524
$50$	0	0
$51$	0	0
$52$	346.981	0.925338
$53$	−149.018	−0.386212	−0.193106	−	0.981178i	$-0.561856\pi$
$53$	−149.018	−0.386212	−0.193106	+	0.981178i	$0.561856\pi$
$54$	0	0
$55$	0	0
$56$	−2100.98	−5.01350
$57$	0	0
$58$	660.265	1.49478
$59$	635.461	1.40220	0.701102	−	0.713061i	$-0.252692\pi$
$59$	635.461	1.40220	0.701102	+	0.713061i	$0.252692\pi$
$60$	0	0
$61$	597.151	1.25340	0.626699	−	0.779261i	$-0.284405\pi$
$61$	597.151	1.25340	0.626699	+	0.779261i	$0.284405\pi$
$62$	1533.86	3.14194
$63$	0	0
$64$	2540.42	4.96175
$65$	0	0
$66$	0	0
$67$	330.015	0.601758	0.300879	−	0.953662i	$-0.402720\pi$
$67$	330.015	0.601758	0.300879	+	0.953662i	$0.402720\pi$
$68$	818.735	1.46009
$69$	0	0
$70$	0	0
$71$	−1149.79	−1.92189	−0.960947	−	0.276732i	$-0.910749\pi$
$71$	−1149.79	−1.92189	−0.960947	+	0.276732i	$0.910749\pi$
$72$	0	0
$73$	130.284	0.208885	0.104442	−	0.994531i	$-0.466694\pi$
$73$	130.284	0.208885	0.104442	+	0.994531i	$0.466694\pi$
$74$	651.738	1.02382
$75$	0	0
$76$	−1199.46	−1.81035
$77$	−160.854	−0.238065
$78$	0	0
$79$	−737.835	−1.05080	−0.525398	−	0.850856i	$-0.676084\pi$
$79$	−737.835	−1.05080	−0.525398	+	0.850856i	$0.676084\pi$
$80$	0	0
$81$	0	0
$82$	−883.309	−1.18957
$83$	369.902	0.489182	0.244591	−	0.969626i	$-0.421346\pi$
$83$	369.902	0.489182	0.244591	+	0.969626i	$0.421346\pi$
$84$	0	0
$85$	0	0
$86$	1635.33	2.05049
$87$	0	0
$88$	511.165	0.619209
$89$	−225.638	−0.268737	−0.134369	−	0.990931i	$-0.542901\pi$
$89$	−225.638	−0.268737	−0.134369	+	0.990931i	$0.542901\pi$
$90$	0	0
$91$	−392.374	−0.452000
$92$	1903.94	2.15761
$93$	0	0
$94$	−461.784	−0.506696
$95$	0	0
$96$	0	0
$97$	11.9134	0.0124704	0.00623518	−	0.999981i	$-0.498015\pi$
$97$	11.9134	0.0124704	0.00623518	+	0.999981i	$0.498015\pi$
$98$	1763.83	1.81810
$99$	0	0
$100$	0	0
$101$	766.534	0.755178	0.377589	−	0.925973i	$-0.376753\pi$
$101$	766.534	0.755178	0.377589	+	0.925973i	$0.376753\pi$
$102$	0	0
$103$	1549.47	1.48227	0.741137	−	0.671354i	$-0.234287\pi$
$103$	1549.47	1.48227	0.741137	+	0.671354i	$0.234287\pi$
$104$	1246.89	1.17565
$105$	0	0
$106$	−826.185	−0.757040
$107$	1218.43	1.10084	0.550421	−	0.834887i	$-0.314467\pi$
$107$	1218.43	1.10084	0.550421	+	0.834887i	$0.314467\pi$
$108$	0	0
$109$	−1327.27	−1.16633	−0.583163	−	0.812355i	$-0.698185\pi$
$109$	−1327.27	−1.16633	−0.583163	+	0.812355i	$0.698185\pi$
$110$	0	0
$111$	0	0
$112$	−6971.02	−5.88124
$113$	433.656	0.361017	0.180509	−	0.983573i	$-0.442226\pi$
$113$	433.656	0.361017	0.180509	+	0.983573i	$0.442226\pi$
$114$	0	0
$115$	0	0
$116$	2707.90	2.16743
$117$	0	0
$118$	3523.11	2.74855
$119$	−925.845	−0.713211
$120$	0	0
$121$	−1291.86	−0.970597
$122$	3310.71	2.45687
$123$	0	0
$124$	6290.72	4.55583
$125$	0	0
$126$	0	0
$127$	−1850.18	−1.29273	−0.646365	−	0.763028i	$-0.723712\pi$
$127$	−1850.18	−1.29273	−0.646365	+	0.763028i	$0.723712\pi$
$128$	7289.22	5.03346
$129$	0	0
$130$	0	0
$131$	−1277.43	−0.851983	−0.425991	−	0.904727i	$-0.640075\pi$
$131$	−1277.43	−0.851983	−0.425991	+	0.904727i	$0.640075\pi$
$132$	0	0
$133$	1356.37	0.884304
$134$	1829.67	1.17954
$135$	0	0
$136$	2942.17	1.85506
$137$	−1533.53	−0.956339	−0.478170	−	0.878268i	$-0.658700\pi$
$137$	−1533.53	−0.956339	−0.478170	+	0.878268i	$0.658700\pi$
$138$	0	0
$139$	1533.34	0.935655	0.467828	−	0.883820i	$-0.345037\pi$
$139$	1533.34	0.935655	0.467828	+	0.883820i	$0.345037\pi$
$140$	0	0
$141$	0	0
$142$	−6374.63	−3.76723
$143$	95.4638	0.0558258
$144$	0	0
$145$	0	0
$146$	722.318	0.409448
$147$	0	0
$148$	2672.93	1.48455
$149$	−2503.07	−1.37624	−0.688120	−	0.725597i	$-0.741564\pi$
$149$	−2503.07	−1.37624	−0.688120	+	0.725597i	$0.741564\pi$
$150$	0	0
$151$	−1520.64	−0.819524	−0.409762	−	0.912192i	$-0.634388\pi$
$151$	−1520.64	−0.819524	−0.409762	+	0.912192i	$0.634388\pi$
$152$	−4310.30	−2.30008
$153$	0	0
$154$	−891.805	−0.466648
$155$	0	0
$156$	0	0
$157$	2592.88	1.31805	0.659026	−	0.752120i	$-0.270969\pi$
$157$	2592.88	1.31805	0.659026	+	0.752120i	$0.270969\pi$
$158$	−4090.69	−2.05974
$159$	0	0
$160$	0	0
$161$	−2153.03	−1.05393
$162$	0	0
$163$	−675.580	−0.324635	−0.162318	−	0.986739i	$-0.551897\pi$
$163$	−675.580	−0.324635	−0.162318	+	0.986739i	$0.551897\pi$
$164$	−3622.65	−1.72489
$165$	0	0
$166$	2050.81	0.958877
$167$	−3895.93	−1.80525	−0.902623	−	0.430432i	$-0.858361\pi$
$167$	−3895.93	−1.80525	−0.902623	+	0.430432i	$0.858361\pi$
$168$	0	0
$169$	−1964.13	−0.894007
$170$	0	0
$171$	0	0
$172$	6706.87	2.97322
$173$	−51.6609	−0.0227035	−0.0113517	−	0.999936i	$-0.503613\pi$
$173$	−51.6609	−0.0227035	−0.0113517	+	0.999936i	$0.503613\pi$
$174$	0	0
$175$	0	0
$176$	1696.03	0.726383
$177$	0	0
$178$	−1250.98	−0.526769
$179$	4553.37	1.90131	0.950656	−	0.310247i	$-0.100412\pi$
$179$	4553.37	1.90131	0.950656	+	0.310247i	$0.100412\pi$
$180$	0	0
$181$	2441.01	1.00243	0.501213	−	0.865324i	$-0.332887\pi$
$181$	2441.01	1.00243	0.501213	+	0.865324i	$0.332887\pi$
$182$	−2175.39	−0.885994
$183$	0	0
$184$	6841.92	2.74127
$185$	0	0
$186$	0	0
$187$	225.256	0.0880875
$188$	−1893.88	−0.734711
$189$	0	0
$190$	0	0
$191$	1326.11	0.502375	0.251188	−	0.967938i	$-0.419179\pi$
$191$	1326.11	0.502375	0.251188	+	0.967938i	$0.419179\pi$
$192$	0	0
$193$	−1448.20	−0.540122	−0.270061	−	0.962843i	$-0.587044\pi$
$193$	−1448.20	−0.540122	−0.270061	+	0.962843i	$0.587044\pi$
$194$	66.0502	0.0244440
$195$	0	0
$196$	7233.88	2.63625
$197$	2398.84	0.867564	0.433782	−	0.901018i	$-0.357179\pi$
$197$	2398.84	0.867564	0.433782	+	0.901018i	$0.357179\pi$
$198$	0	0
$199$	3109.15	1.10755	0.553773	−	0.832667i	$-0.313187\pi$
$199$	3109.15	1.10755	0.553773	+	0.832667i	$0.313187\pi$
$200$	0	0
$201$	0	0
$202$	4249.80	1.48027
$203$	−3062.16	−1.05873
$204$	0	0
$205$	0	0
$206$	8590.57	2.90550
$207$	0	0
$208$	4137.16	1.37914
$209$	−330.003	−0.109219
$210$	0	0
$211$	−3309.59	−1.07982	−0.539910	−	0.841723i	$-0.681542\pi$
$211$	−3309.59	−1.07982	−0.539910	+	0.841723i	$0.681542\pi$
$212$	−3388.38	−1.09771
$213$	0	0
$214$	6755.21	2.15783
$215$	0	0
$216$	0	0
$217$	−7113.69	−2.22539
$218$	−7358.63	−2.28619
$219$	0	0
$220$	0	0
$221$	549.471	0.167246
$222$	0	0
$223$	−2798.84	−0.840467	−0.420233	−	0.907416i	$-0.638052\pi$
$223$	−2798.84	−0.840467	−0.420233	+	0.907416i	$0.638052\pi$
$224$	−21840.7	−6.51471
$225$	0	0
$226$	2404.27	0.707653
$227$	−3129.17	−0.914936	−0.457468	−	0.889226i	$-0.651244\pi$
$227$	−3129.17	−0.914936	−0.457468	+	0.889226i	$0.651244\pi$
$228$	0	0
$229$	2975.58	0.858654	0.429327	−	0.903149i	$-0.358751\pi$
$229$	2975.58	0.858654	0.429327	+	0.903149i	$0.358751\pi$
$230$	0	0
$231$	0	0
$232$	9730.97	2.75375
$233$	925.895	0.260332	0.130166	−	0.991492i	$-0.458449\pi$
$233$	925.895	0.260332	0.130166	+	0.991492i	$0.458449\pi$
$234$	0	0
$235$	0	0
$236$	14449.1	3.98541
$237$	0	0
$238$	−5133.06	−1.39801
$239$	2610.82	0.706610	0.353305	−	0.935508i	$-0.385058\pi$
$239$	2610.82	0.706610	0.353305	+	0.935508i	$0.385058\pi$
$240$	0	0
$241$	−669.473	−0.178940	−0.0894700	−	0.995990i	$-0.528517\pi$
$241$	−669.473	−0.178940	−0.0894700	+	0.995990i	$0.528517\pi$
$242$	−7162.34	−1.90253
$243$	0	0
$244$	13578.0	3.56247
$245$	0	0
$246$	0	0
$247$	−804.981	−0.207367
$248$	22606.0	5.78824
$249$	0	0
$250$	0	0
$251$	−6463.00	−1.62526	−0.812631	−	0.582778i	$-0.801966\pi$
$251$	−6463.00	−1.62526	−0.812631	+	0.582778i	$0.801966\pi$
$252$	0	0
$253$	523.827	0.130169
$254$	−10257.7	−2.53397
$255$	0	0
$256$	20089.5	4.90465
$257$	2835.33	0.688183	0.344091	−	0.938936i	$-0.388187\pi$
$257$	2835.33	0.688183	0.344091	+	0.938936i	$0.388187\pi$
$258$	0	0
$259$	−3022.61	−0.725158
$260$	0	0
$261$	0	0
$262$	−7082.32	−1.67003
$263$	1633.67	0.383029	0.191515	−	0.981490i	$-0.438660\pi$
$263$	1633.67	0.383029	0.191515	+	0.981490i	$0.438660\pi$
$264$	0	0
$265$	0	0
$266$	7519.98	1.73338
$267$	0	0
$268$	7503.88	1.71035
$269$	−4870.30	−1.10389	−0.551947	−	0.833880i	$-0.686115\pi$
$269$	−4870.30	−1.10389	−0.551947	+	0.833880i	$0.686115\pi$
$270$	0	0
$271$	−3913.17	−0.877153	−0.438576	−	0.898694i	$-0.644517\pi$
$271$	−3913.17	−0.877153	−0.438576	+	0.898694i	$0.644517\pi$
$272$	9762.03	2.17614
$273$	0	0
$274$	−8502.18	−1.87458
$275$	0	0
$276$	0	0
$277$	−818.145	−0.177464	−0.0887320	−	0.996056i	$-0.528281\pi$
$277$	−818.145	−0.177464	−0.0887320	+	0.996056i	$0.528281\pi$
$278$	8501.11	1.83404
$279$	0	0
$280$	0	0
$281$	−5162.95	−1.09607	−0.548035	−	0.836455i	$-0.684624\pi$
$281$	−5162.95	−1.09607	−0.548035	+	0.836455i	$0.684624\pi$
$282$	0	0
$283$	7146.57	1.50113	0.750565	−	0.660797i	$-0.229782\pi$
$283$	7146.57	1.50113	0.750565	+	0.660797i	$0.229782\pi$
$284$	−26143.8	−5.46250
$285$	0	0
$286$	529.269	0.109428
$287$	4096.58	0.842556
$288$	0	0
$289$	−3616.47	−0.736102
$290$	0	0
$291$	0	0
$292$	2962.39	0.593702
$293$	1465.10	0.292122	0.146061	−	0.989276i	$-0.453340\pi$
$293$	1465.10	0.292122	0.146061	+	0.989276i	$0.453340\pi$
$294$	0	0
$295$	0	0
$296$	9605.31	1.88614
$297$	0	0
$298$	−13877.5	−2.69766
$299$	1277.78	0.247143
$300$	0	0
$301$	−7584.29	−1.45233
$302$	−8430.73	−1.60640
$303$	0	0
$304$	−14301.5	−2.69818
$305$	0	0
$306$	0	0
$307$	790.554	0.146968	0.0734842	−	0.997296i	$-0.476588\pi$
$307$	790.554	0.146968	0.0734842	+	0.997296i	$0.476588\pi$
$308$	−3657.50	−0.676641
$309$	0	0
$310$	0	0
$311$	3925.31	0.715703	0.357852	−	0.933778i	$-0.383509\pi$
$311$	3925.31	0.715703	0.357852	+	0.933778i	$0.383509\pi$
$312$	0	0
$313$	−5742.37	−1.03699	−0.518495	−	0.855080i	$-0.673508\pi$
$313$	−5742.37	−1.03699	−0.518495	+	0.855080i	$0.673508\pi$
$314$	14375.4	2.58360
$315$	0	0
$316$	−16776.9	−2.98663
$317$	−8136.70	−1.44165	−0.720824	−	0.693118i	$-0.756236\pi$
$317$	−8136.70	−1.44165	−0.720824	+	0.693118i	$0.756236\pi$
$318$	0	0
$319$	745.016	0.130761
$320$	0	0
$321$	0	0
$322$	−11936.8	−2.06587
$323$	−1899.43	−0.327205
$324$	0	0
$325$	0	0
$326$	−3745.54	−0.636338
$327$	0	0
$328$	−13018.2	−2.19149
$329$	2141.65	0.358884
$330$	0	0
$331$	−6326.94	−1.05063	−0.525317	−	0.850906i	$-0.676053\pi$
$331$	−6326.94	−1.05063	−0.525317	+	0.850906i	$0.676053\pi$
$332$	8410.84	1.39038
$333$	0	0
$334$	−21599.8	−3.53858
$335$	0	0
$336$	0	0
$337$	−4774.02	−0.771683	−0.385842	−	0.922565i	$-0.626089\pi$
$337$	−4774.02	−0.771683	−0.385842	+	0.922565i	$0.626089\pi$
$338$	−10889.5	−1.75240
$339$	0	0
$340$	0	0
$341$	1730.75	0.274854
$342$	0	0
$343$	639.195	0.100622
$344$	24101.5	3.77751
$345$	0	0
$346$	−286.417	−0.0445026
$347$	1819.43	0.281477	0.140738	−	0.990047i	$-0.455052\pi$
$347$	1819.43	0.281477	0.140738	+	0.990047i	$0.455052\pi$
$348$	0	0
$349$	−2977.37	−0.456663	−0.228331	−	0.973584i	$-0.573327\pi$
$349$	−2977.37	−0.456663	−0.228331	+	0.973584i	$0.573327\pi$
$350$	0	0
$351$	0	0
$352$	5313.80	0.804621
$353$	−6947.77	−1.04757	−0.523786	−	0.851850i	$-0.675481\pi$
$353$	−6947.77	−1.04757	−0.523786	+	0.851850i	$0.675481\pi$
$354$	0	0
$355$	0	0
$356$	−5130.56	−0.763817
$357$	0	0
$358$	25244.7	3.72689
$359$	−3665.56	−0.538887	−0.269444	−	0.963016i	$-0.586840\pi$
$359$	−3665.56	−0.538887	−0.269444	+	0.963016i	$0.586840\pi$
$360$	0	0
$361$	−4076.31	−0.594302
$362$	13533.4	1.96492
$363$	0	0
$364$	−8921.80	−1.28470
$365$	0	0
$366$	0	0
$367$	−9663.09	−1.37441	−0.687206	−	0.726463i	$-0.741163\pi$
$367$	−9663.09	−1.37441	−0.687206	+	0.726463i	$0.741163\pi$
$368$	22701.3	3.21573
$369$	0	0
$370$	0	0
$371$	3831.66	0.536199
$372$	0	0
$373$	6023.73	0.836185	0.418092	−	0.908405i	$-0.362699\pi$
$373$	6023.73	0.836185	0.418092	+	0.908405i	$0.362699\pi$
$374$	1248.86	0.172666
$375$	0	0
$376$	−6805.77	−0.933460
$377$	1817.33	0.248269
$378$	0	0
$379$	−1229.32	−0.166613	−0.0833063	−	0.996524i	$-0.526548\pi$
$379$	−1229.32	−0.166613	−0.0833063	+	0.996524i	$0.526548\pi$
$380$	0	0
$381$	0	0
$382$	7352.18	0.984739
$383$	2133.59	0.284651	0.142326	−	0.989820i	$-0.454542\pi$
$383$	2133.59	0.284651	0.142326	+	0.989820i	$0.454542\pi$
$384$	0	0
$385$	0	0
$386$	−8029.07	−1.05873
$387$	0	0
$388$	270.887	0.0354439
$389$	9841.52	1.28274	0.641369	−	0.767233i	$-0.278367\pi$
$389$	9841.52	1.28274	0.641369	+	0.767233i	$0.278367\pi$
$390$	0	0
$391$	3015.04	0.389967
$392$	25995.3	3.34939
$393$	0	0
$394$	13299.6	1.70057
$395$	0	0
$396$	0	0
$397$	2548.71	0.322207	0.161104	−	0.986938i	$-0.448495\pi$
$397$	2548.71	0.322207	0.161104	+	0.986938i	$0.448495\pi$
$398$	17237.7	2.17098
$399$	0	0
$400$	0	0
$401$	−13105.0	−1.63200	−0.816001	−	0.578051i	$-0.803813\pi$
$401$	−13105.0	−1.63200	−0.816001	+	0.578051i	$0.803813\pi$
$402$	0	0
$403$	4221.84	0.521848
$404$	17429.4	2.14640
$405$	0	0
$406$	−16977.2	−2.07528
$407$	735.395	0.0895631
$408$	0	0
$409$	6217.24	0.751645	0.375823	−	0.926692i	$-0.377360\pi$
$409$	6217.24	0.751645	0.375823	+	0.926692i	$0.377360\pi$
$410$	0	0
$411$	0	0
$412$	35231.9	4.21299
$413$	−16339.4	−1.94675
$414$	0	0
$415$	0	0
$416$	12962.0	1.52768
$417$	0	0
$418$	−1829.60	−0.214087
$419$	8428.06	0.982667	0.491334	−	0.870971i	$-0.336510\pi$
$419$	8428.06	0.982667	0.491334	+	0.870971i	$0.336510\pi$
$420$	0	0
$421$	−12779.4	−1.47940	−0.739702	−	0.672935i	$-0.765033\pi$
$421$	−12779.4	−1.47940	−0.739702	+	0.672935i	$0.765033\pi$
$422$	−18349.0	−2.11662
$423$	0	0
$424$	−12176.3	−1.39465
$425$	0	0
$426$	0	0
$427$	−15354.3	−1.74016
$428$	27704.7	3.12887
$429$	0	0
$430$	0	0
$431$	3149.70	0.352009	0.176005	−	0.984389i	$-0.443683\pi$
$431$	3149.70	0.352009	0.176005	+	0.984389i	$0.443683\pi$
$432$	0	0
$433$	6999.89	0.776890	0.388445	−	0.921472i	$-0.373012\pi$
$433$	6999.89	0.776890	0.388445	+	0.921472i	$0.373012\pi$
$434$	−39439.6	−4.36213
$435$	0	0
$436$	−30179.5	−3.31499
$437$	−4417.07	−0.483517
$438$	0	0
$439$	−7722.30	−0.839557	−0.419778	−	0.907627i	$-0.637892\pi$
$439$	−7722.30	−0.839557	−0.419778	+	0.907627i	$0.637892\pi$
$440$	0	0
$441$	0	0
$442$	3046.37	0.327830
$443$	6648.73	0.713072	0.356536	−	0.934282i	$-0.383958\pi$
$443$	6648.73	0.713072	0.356536	+	0.934282i	$0.383958\pi$
$444$	0	0
$445$	0	0
$446$	−15517.3	−1.64745
$447$	0	0
$448$	−65320.9	−6.88866
$449$	−11416.8	−1.19998	−0.599989	−	0.800008i	$-0.704829\pi$
$449$	−11416.8	−1.19998	−0.599989	+	0.800008i	$0.704829\pi$
$450$	0	0
$451$	−996.690	−0.104063
$452$	9860.47	1.02610
$453$	0	0
$454$	−17348.7	−1.79343
$455$	0	0
$456$	0	0
$457$	11191.5	1.14555	0.572773	−	0.819714i	$-0.305868\pi$
$457$	11191.5	1.14555	0.572773	+	0.819714i	$0.305868\pi$
$458$	16497.2	1.68310
$459$	0	0
$460$	0	0
$461$	−4901.86	−0.495233	−0.247617	−	0.968858i	$-0.579647\pi$
$461$	−4901.86	−0.495233	−0.247617	+	0.968858i	$0.579647\pi$
$462$	0	0
$463$	2131.64	0.213965	0.106982	−	0.994261i	$-0.465881\pi$
$463$	2131.64	0.213965	0.106982	+	0.994261i	$0.465881\pi$
$464$	32287.1	3.23037
$465$	0	0
$466$	5133.33	0.510294
$467$	17877.0	1.77141	0.885707	−	0.464246i	$-0.153675\pi$
$467$	17877.0	1.77141	0.885707	+	0.464246i	$0.153675\pi$
$468$	0	0
$469$	−8485.57	−0.835452
$470$	0	0
$471$	0	0
$472$	51923.6	5.06351
$473$	1845.24	0.179375
$474$	0	0
$475$	0	0
$476$	−21051.8	−2.02712
$477$	0	0
$478$	14474.9	1.38507
$479$	20417.8	1.94762	0.973811	−	0.227360i	$-0.0730093\pi$
$479$	20417.8	1.94762	0.973811	+	0.227360i	$0.0730093\pi$
$480$	0	0
$481$	1793.86	0.170048
$482$	−3711.68	−0.350752
$483$	0	0
$484$	−29374.4	−2.75868
$485$	0	0
$486$	0	0
$487$	8337.59	0.775795	0.387898	−	0.921702i	$-0.373201\pi$
$487$	8337.59	0.775795	0.387898	+	0.921702i	$0.373201\pi$
$488$	48793.3	4.52616
$489$	0	0
$490$	0	0
$491$	16264.7	1.49494	0.747469	−	0.664296i	$-0.231269\pi$
$491$	16264.7	1.49494	0.747469	+	0.664296i	$0.231269\pi$
$492$	0	0
$493$	4288.17	0.391743
$494$	−4462.96	−0.406474
$495$	0	0
$496$	75006.2	6.79008
$497$	29564.1	2.66827
$498$	0	0
$499$	−16111.7	−1.44540	−0.722702	−	0.691159i	$-0.757100\pi$
$499$	−16111.7	−1.44540	−0.722702	+	0.691159i	$0.757100\pi$
$500$	0	0
$501$	0	0
$502$	−35832.1	−3.18578
$503$	9219.89	0.817285	0.408643	−	0.912695i	$-0.366002\pi$
$503$	9219.89	0.817285	0.408643	+	0.912695i	$0.366002\pi$
$504$	0	0
$505$	0	0
$506$	2904.19	0.255152
$507$	0	0
$508$	−42069.3	−3.67426
$509$	8566.85	0.746009	0.373005	−	0.927829i	$-0.378328\pi$
$509$	8566.85	0.746009	0.373005	+	0.927829i	$0.378328\pi$
$510$	0	0
$511$	−3349.94	−0.290005
$512$	53065.9	4.58048
$513$	0	0
$514$	15719.6	1.34895
$515$	0	0
$516$	0	0
$517$	−521.059	−0.0443252
$518$	−16757.9	−1.42143
$519$	0	0
$520$	0	0
$521$	−2450.94	−0.206099	−0.103050	−	0.994676i	$-0.532860\pi$
$521$	−2450.94	−0.206099	−0.103050	+	0.994676i	$0.532860\pi$
$522$	0	0
$523$	20897.8	1.74722	0.873609	−	0.486628i	$-0.161773\pi$
$523$	20897.8	1.74722	0.873609	+	0.486628i	$0.161773\pi$
$524$	−29046.2	−2.42155
$525$	0	0
$526$	9057.40	0.750801
$527$	9961.84	0.823424
$528$	0	0
$529$	−5155.61	−0.423737
$530$	0	0
$531$	0	0
$532$	30841.2	2.51341
$533$	−2431.24	−0.197577
$534$	0	0
$535$	0	0
$536$	26965.6	2.17301
$537$	0	0
$538$	−27001.8	−2.16381
$539$	1990.24	0.159046
$540$	0	0
$541$	−9960.84	−0.791590	−0.395795	−	0.918339i	$-0.629531\pi$
$541$	−9960.84	−0.791590	−0.395795	+	0.918339i	$0.629531\pi$
$542$	−21695.4	−1.71936
$543$	0	0
$544$	30585.2	2.41053
$545$	0	0
$546$	0	0
$547$	8594.35	0.671788	0.335894	−	0.941900i	$-0.390962\pi$
$547$	8594.35	0.671788	0.335894	+	0.941900i	$0.390962\pi$
$548$	−34869.4	−2.71815
$549$	0	0
$550$	0	0
$551$	−6282.21	−0.485719
$552$	0	0
$553$	18971.7	1.45888
$554$	−4535.94	−0.347859
$555$	0	0
$556$	34865.0	2.65936
$557$	−9469.85	−0.720378	−0.360189	−	0.932879i	$-0.617288\pi$
$557$	−9469.85	−0.720378	−0.360189	+	0.932879i	$0.617288\pi$
$558$	0	0
$559$	4501.13	0.340568
$560$	0	0
$561$	0	0
$562$	−28624.4	−2.14848
$563$	−8.77465	−0.000656852 0	−0.000328426	−	1.00000i	$-0.500105\pi$
$563$	−8.77465	−0.000656852 0	−0.000328426		1.00000i	$0.500105\pi$
$564$	0	0
$565$	0	0
$566$	39621.9	2.94246
$567$	0	0
$568$	−93949.2	−6.94018
$569$	−246.934	−0.0181933	−0.00909667	−	0.999959i	$-0.502896\pi$
$569$	−246.934	−0.0181933	−0.00909667	+	0.999959i	$0.502896\pi$
$570$	0	0
$571$	3447.19	0.252645	0.126323	−	0.991989i	$-0.459683\pi$
$571$	3447.19	0.252645	0.126323	+	0.991989i	$0.459683\pi$
$572$	2170.66	0.158671
$573$	0	0
$574$	22712.2	1.65155
$575$	0	0
$576$	0	0
$577$	−21137.2	−1.52505	−0.762525	−	0.646959i	$-0.776040\pi$
$577$	−21137.2	−1.52505	−0.762525	+	0.646959i	$0.776040\pi$
$578$	−20050.4	−1.44288
$579$	0	0
$580$	0	0
$581$	−9511.18	−0.679157
$582$	0	0
$583$	−932.234	−0.0662250
$584$	10645.5	0.754305
$585$	0	0
$586$	8122.77	0.572608
$587$	−5311.01	−0.373439	−0.186720	−	0.982413i	$-0.559786\pi$
$587$	−5311.01	−0.373439	−0.186720	+	0.982413i	$0.559786\pi$
$588$	0	0
$589$	−14594.2	−1.02096
$590$	0	0
$591$	0	0
$592$	31870.2	2.21259
$593$	−11575.8	−0.801624	−0.400812	−	0.916160i	$-0.631272\pi$
$593$	−11575.8	−0.801624	−0.400812	+	0.916160i	$0.631272\pi$
$594$	0	0
$595$	0	0
$596$	−56914.9	−3.91162
$597$	0	0
$598$	7084.24	0.484442
$599$	−7126.64	−0.486121	−0.243061	−	0.970011i	$-0.578151\pi$
$599$	−7126.64	−0.486121	−0.243061	+	0.970011i	$0.578151\pi$
$600$	0	0
$601$	−7721.85	−0.524095	−0.262047	−	0.965055i	$-0.584398\pi$
$601$	−7721.85	−0.524095	−0.262047	+	0.965055i	$0.584398\pi$
$602$	−42048.7	−2.84681
$603$	0	0
$604$	−34576.4	−2.32929
$605$	0	0
$606$	0	0
$607$	−14361.4	−0.960317	−0.480159	−	0.877182i	$-0.659421\pi$
$607$	−14361.4	−0.960317	−0.480159	+	0.877182i	$0.659421\pi$
$608$	−44807.6	−2.98880
$609$	0	0
$610$	0	0
$611$	−1271.03	−0.0841575
$612$	0	0
$613$	12086.0	0.796330	0.398165	−	0.917314i	$-0.369647\pi$
$613$	12086.0	0.796330	0.398165	+	0.917314i	$0.369647\pi$
$614$	4382.98	0.288082
$615$	0	0
$616$	−13143.4	−0.859681
$617$	25235.8	1.64660	0.823301	−	0.567604i	$-0.192130\pi$
$617$	25235.8	1.64660	0.823301	+	0.567604i	$0.192130\pi$
$618$	0	0
$619$	−4905.44	−0.318524	−0.159262	−	0.987236i	$-0.550911\pi$
$619$	−4905.44	−0.318524	−0.159262	+	0.987236i	$0.550911\pi$
$620$	0	0
$621$	0	0
$622$	21762.6	1.40290
$623$	5801.76	0.373102
$624$	0	0
$625$	0	0
$626$	−31836.8	−2.03267
$627$	0	0
$628$	58956.8	3.74623
$629$	4232.79	0.268318
$630$	0	0
$631$	−11240.9	−0.709181	−0.354590	−	0.935022i	$-0.615380\pi$
$631$	−11240.9	−0.709181	−0.354590	+	0.935022i	$0.615380\pi$
$632$	−60288.6	−3.79454
$633$	0	0
$634$	−45111.4	−2.82587
$635$	0	0
$636$	0	0
$637$	4854.82	0.301970
$638$	4130.51	0.256314
$639$	0	0
$640$	0	0
$641$	11159.0	0.687601	0.343800	−	0.939043i	$-0.388286\pi$
$641$	11159.0	0.687601	0.343800	+	0.939043i	$0.388286\pi$
$642$	0	0
$643$	−27244.8	−1.67097	−0.835483	−	0.549516i	$-0.814812\pi$
$643$	−27244.8	−1.67097	−0.835483	+	0.549516i	$0.814812\pi$
$644$	−48955.5	−2.99552
$645$	0	0
$646$	−10530.8	−0.641375
$647$	−31360.6	−1.90559	−0.952793	−	0.303620i	$-0.901805\pi$
$647$	−31360.6	−1.90559	−0.952793	+	0.303620i	$0.901805\pi$
$648$	0	0
$649$	3975.34	0.240440
$650$	0	0
$651$	0	0
$652$	−15361.3	−0.922693
$653$	−7074.47	−0.423959	−0.211980	−	0.977274i	$-0.567991\pi$
$653$	−7074.47	−0.423959	−0.211980	+	0.977274i	$0.567991\pi$
$654$	0	0
$655$	0	0
$656$	−43194.0	−2.57080
$657$	0	0
$658$	11873.7	0.703473
$659$	18498.6	1.09348	0.546740	−	0.837302i	$-0.315868\pi$
$659$	18498.6	1.09348	0.546740	+	0.837302i	$0.315868\pi$
$660$	0	0
$661$	6400.77	0.376643	0.188321	−	0.982107i	$-0.439695\pi$
$661$	6400.77	0.376643	0.188321	+	0.982107i	$0.439695\pi$
$662$	−35077.7	−2.05942
$663$	0	0
$664$	30224.8	1.76649
$665$	0	0
$666$	0	0
$667$	9972.01	0.578887
$668$	−88585.6	−5.13096
$669$	0	0
$670$	0	0
$671$	3735.68	0.214924
$672$	0	0
$673$	14231.4	0.815124	0.407562	−	0.913177i	$-0.366379\pi$
$673$	14231.4	0.815124	0.407562	+	0.913177i	$0.366379\pi$
$674$	−26468.0	−1.51263
$675$	0	0
$676$	−44660.4	−2.54099
$677$	19908.0	1.13017	0.565085	−	0.825032i	$-0.308843\pi$
$677$	19908.0	1.13017	0.565085	+	0.825032i	$0.308843\pi$
$678$	0	0
$679$	−306.326	−0.0173133
$680$	0	0
$681$	0	0
$682$	9595.58	0.538759
$683$	26204.1	1.46804	0.734021	−	0.679126i	$-0.237641\pi$
$683$	26204.1	1.46804	0.734021	+	0.679126i	$0.237641\pi$
$684$	0	0
$685$	0	0
$686$	3543.81	0.197235
$687$	0	0
$688$	79968.1	4.43133
$689$	−2274.01	−0.125737
$690$	0	0
$691$	−26562.5	−1.46235	−0.731176	−	0.682189i	$-0.761028\pi$
$691$	−26562.5	−1.46235	−0.731176	+	0.682189i	$0.761028\pi$
$692$	−1174.66	−0.0645289
$693$	0	0
$694$	10087.3	0.551741
$695$	0	0
$696$	0	0
$697$	−5736.75	−0.311757
$698$	−16507.1	−0.895134
$699$	0	0
$700$	0	0
$701$	35417.5	1.90827	0.954137	−	0.299372i	$-0.0967771\pi$
$701$	35417.5	1.90827	0.954137	+	0.299372i	$0.0967771\pi$
$702$	0	0
$703$	−6201.08	−0.332686
$704$	15892.4	0.850808
$705$	0	0
$706$	−38519.7	−2.05341
$707$	−19709.6	−1.04845
$708$	0	0
$709$	−16678.4	−0.883456	−0.441728	−	0.897149i	$-0.645634\pi$
$709$	−16678.4	−0.883456	−0.441728	+	0.897149i	$0.645634\pi$
$710$	0	0
$711$	0	0
$712$	−18436.9	−0.970439
$713$	23166.0	1.21679
$714$	0	0
$715$	0	0
$716$	103534.	5.40400
$717$	0	0
$718$	−20322.5	−1.05631
$719$	3273.36	0.169786	0.0848928	−	0.996390i	$-0.472945\pi$
$719$	3273.36	0.169786	0.0848928	+	0.996390i	$0.472945\pi$
$720$	0	0
$721$	−39841.1	−2.05792
$722$	−22599.8	−1.16493
$723$	0	0
$724$	55503.7	2.84914
$725$	0	0
$726$	0	0
$727$	−33940.5	−1.73148	−0.865738	−	0.500497i	$-0.833151\pi$
$727$	−33940.5	−1.73148	−0.865738	+	0.500497i	$0.833151\pi$
$728$	−32060.9	−1.63222
$729$	0	0
$730$	0	0
$731$	10620.8	0.537382
$732$	0	0
$733$	1904.70	0.0959779	0.0479890	−	0.998848i	$-0.484719\pi$
$733$	1904.70	0.0959779	0.0479890	+	0.998848i	$0.484719\pi$
$734$	−53573.9	−2.69407
$735$	0	0
$736$	71125.0	3.56209
$737$	2064.52	0.103185
$738$	0	0
$739$	23159.4	1.15282	0.576408	−	0.817162i	$-0.304454\pi$
$739$	23159.4	1.15282	0.576408	+	0.817162i	$0.304454\pi$
$740$	0	0
$741$	0	0
$742$	21243.4	1.05104
$743$	21886.8	1.08068	0.540342	−	0.841445i	$-0.318295\pi$
$743$	21886.8	1.08068	0.540342	+	0.841445i	$0.318295\pi$
$744$	0	0
$745$	0	0
$746$	33396.7	1.63906
$747$	0	0
$748$	5121.87	0.250367
$749$	−31329.1	−1.52836
$750$	0	0
$751$	−31661.3	−1.53840	−0.769200	−	0.639008i	$-0.779345\pi$
$751$	−31661.3	−1.53840	−0.769200	+	0.639008i	$0.779345\pi$
$752$	−22581.4	−1.09502
$753$	0	0
$754$	10075.6	0.486647
$755$	0	0
$756$	0	0
$757$	−29482.4	−1.41553	−0.707764	−	0.706449i	$-0.750296\pi$
$757$	−29482.4	−1.41553	−0.707764	+	0.706449i	$0.750296\pi$
$758$	−6815.60	−0.326588
$759$	0	0
$760$	0	0
$761$	−26024.0	−1.23964	−0.619822	−	0.784743i	$-0.712795\pi$
$761$	−26024.0	−1.23964	−0.619822	+	0.784743i	$0.712795\pi$
$762$	0	0
$763$	34127.7	1.61927
$764$	30153.0	1.42788
$765$	0	0
$766$	11829.0	0.557964
$767$	9697.11	0.456509
$768$	0	0
$769$	1678.40	0.0787054	0.0393527	−	0.999225i	$-0.487470\pi$
$769$	1678.40	0.0787054	0.0393527	+	0.999225i	$0.487470\pi$
$770$	0	0
$771$	0	0
$772$	−32929.1	−1.53516
$773$	−25777.0	−1.19940	−0.599698	−	0.800226i	$-0.704713\pi$
$773$	−25777.0	−1.19940	−0.599698	+	0.800226i	$0.704713\pi$
$774$	0	0
$775$	0	0
$776$	973.447	0.0450319
$777$	0	0
$778$	54563.2	2.51438
$779$	8404.40	0.386546
$780$	0	0
$781$	−7192.87	−0.329553
$782$	16716.0	0.764401
$783$	0	0
$784$	86251.9	3.92911
$785$	0	0
$786$	0	0
$787$	−40816.6	−1.84874	−0.924368	−	0.381502i	$-0.875407\pi$
$787$	−40816.6	−1.84874	−0.924368	+	0.381502i	$0.875407\pi$
$788$	54544.7	2.46583
$789$	0	0
$790$	0	0
$791$	−11150.4	−0.501219
$792$	0	0
$793$	9112.50	0.408063
$794$	14130.5	0.631579
$795$	0	0
$796$	70695.8	3.14792
$797$	9949.56	0.442198	0.221099	−	0.975251i	$-0.429036\pi$
$797$	9949.56	0.442198	0.221099	+	0.975251i	$0.429036\pi$
$798$	0	0
$799$	−2999.11	−0.132792
$800$	0	0
$801$	0	0
$802$	−72656.6	−3.19899
$803$	815.034	0.0358181
$804$	0	0
$805$	0	0
$806$	23406.7	1.02291
$807$	0	0
$808$	62633.6	2.72703
$809$	−32596.2	−1.41659	−0.708295	−	0.705917i	$-0.750535\pi$
$809$	−32596.2	−1.41659	−0.708295	+	0.705917i	$0.750535\pi$
$810$	0	0
$811$	−7399.23	−0.320373	−0.160186	−	0.987087i	$-0.551210\pi$
$811$	−7399.23	−0.320373	−0.160186	+	0.987087i	$0.551210\pi$
$812$	−69627.3	−3.00916
$813$	0	0
$814$	4077.17	0.175558
$815$	0	0
$816$	0	0
$817$	−15559.7	−0.666295
$818$	34469.6	1.47335
$819$	0	0
$820$	0	0
$821$	−17398.1	−0.739583	−0.369791	−	0.929115i	$-0.620571\pi$
$821$	−17398.1	−0.739583	−0.369791	+	0.929115i	$0.620571\pi$
$822$	0	0
$823$	5269.64	0.223193	0.111597	−	0.993754i	$-0.464404\pi$
$823$	5269.64	0.223193	0.111597	+	0.993754i	$0.464404\pi$
$824$	126608.	5.35266
$825$	0	0
$826$	−90588.6	−3.81596
$827$	9005.90	0.378677	0.189338	−	0.981912i	$-0.439366\pi$
$827$	9005.90	0.378677	0.189338	+	0.981912i	$0.439366\pi$
$828$	0	0
$829$	5545.90	0.232349	0.116174	−	0.993229i	$-0.462937\pi$
$829$	5545.90	0.232349	0.116174	+	0.993229i	$0.462937\pi$
$830$	0	0
$831$	0	0
$832$	38766.7	1.61538
$833$	11455.4	0.476478
$834$	0	0
$835$	0	0
$836$	−7503.60	−0.310427
$837$	0	0
$838$	46726.7	1.92619
$839$	−4082.93	−0.168008	−0.0840038	−	0.996465i	$-0.526771\pi$
$839$	−4082.93	−0.168008	−0.0840038	+	0.996465i	$0.526771\pi$
$840$	0	0
$841$	−10206.2	−0.418477
$842$	−70851.3	−2.89988
$843$	0	0
$844$	−75253.5	−3.06911
$845$	0	0
$846$	0	0
$847$	33217.3	1.34753
$848$	−40400.7	−1.63604
$849$	0	0
$850$	0	0
$851$	9843.23	0.396500
$852$	0	0
$853$	−23676.6	−0.950375	−0.475188	−	0.879885i	$-0.657620\pi$
$853$	−23676.6	−0.950375	−0.475188	+	0.879885i	$0.657620\pi$
$854$	−85127.3	−3.41100
$855$	0	0
$856$	99558.1	3.97527
$857$	16551.7	0.659739	0.329869	−	0.944027i	$-0.392995\pi$
$857$	16551.7	0.659739	0.329869	+	0.944027i	$0.392995\pi$
$858$	0	0
$859$	−100.576	−0.00399487	−0.00199744	−	0.999998i	$-0.500636\pi$
$859$	−100.576	−0.00399487	−0.00199744	+	0.999998i	$0.500636\pi$
$860$	0	0
$861$	0	0
$862$	17462.5	0.689996
$863$	−21496.8	−0.847924	−0.423962	−	0.905680i	$-0.639361\pi$
$863$	−21496.8	−0.847924	−0.423962	+	0.905680i	$0.639361\pi$
$864$	0	0
$865$	0	0
$866$	38808.7	1.52283
$867$	0	0
$868$	−161751.	−6.32510
$869$	−4615.78	−0.180184
$870$	0	0
$871$	5036.02	0.195912
$872$	−108451.	−4.21173
$873$	0	0
$874$	−24489.1	−0.947774
$875$	0	0
$876$	0	0
$877$	−15218.6	−0.585969	−0.292984	−	0.956117i	$-0.594648\pi$
$877$	−15218.6	−0.585969	−0.292984	+	0.956117i	$0.594648\pi$
$878$	−42813.9	−1.64567
$879$	0	0
$880$	0	0
$881$	−29944.2	−1.14511	−0.572557	−	0.819865i	$-0.694048\pi$
$881$	−29944.2	−1.14511	−0.572557	+	0.819865i	$0.694048\pi$
$882$	0	0
$883$	−22598.4	−0.861265	−0.430633	−	0.902527i	$-0.641710\pi$
$883$	−22598.4	−0.861265	−0.430633	+	0.902527i	$0.641710\pi$
$884$	12493.9	0.475355
$885$	0	0
$886$	36861.8	1.39774
$887$	36206.4	1.37057	0.685283	−	0.728277i	$-0.259679\pi$
$887$	36206.4	1.37057	0.685283	+	0.728277i	$0.259679\pi$
$888$	0	0
$889$	47573.0	1.79477
$890$	0	0
$891$	0	0
$892$	−63639.9	−2.38881
$893$	4393.73	0.164648
$894$	0	0
$895$	0	0
$896$	−187425.	−6.98821
$897$	0	0
$898$	−63296.7	−2.35216
$899$	32948.0	1.22233
$900$	0	0
$901$	−5365.76	−0.198401
$902$	−5525.84	−0.203980
$903$	0	0
$904$	35434.1	1.30367
$905$	0	0
$906$	0	0
$907$	29439.8	1.07777	0.538883	−	0.842381i	$-0.318847\pi$
$907$	29439.8	1.07777	0.538883	+	0.842381i	$0.318847\pi$
$908$	−71151.1	−2.60047
$909$	0	0
$910$	0	0
$911$	−36128.0	−1.31391	−0.656956	−	0.753929i	$-0.728156\pi$
$911$	−36128.0	−1.31391	−0.656956	+	0.753929i	$0.728156\pi$
$912$	0	0
$913$	2314.05	0.0838815
$914$	62047.5	2.24546
$915$	0	0
$916$	67658.6	2.44051
$917$	32846.2	1.18285
$918$	0	0
$919$	−9547.59	−0.342705	−0.171353	−	0.985210i	$-0.554814\pi$
$919$	−9547.59	−0.342705	−0.171353	+	0.985210i	$0.554814\pi$
$920$	0	0
$921$	0	0
$922$	−27176.8	−0.970739
$923$	−17545.7	−0.625702
$924$	0	0
$925$	0	0
$926$	11818.2	0.419407
$927$	0	0
$928$	101158.	3.57831
$929$	18491.8	0.653064	0.326532	−	0.945186i	$-0.394120\pi$
$929$	18491.8	0.653064	0.326532	+	0.945186i	$0.394120\pi$
$930$	0	0
$931$	−16782.3	−0.590782
$932$	21053.0	0.739929
$933$	0	0
$934$	99113.5	3.47226
$935$	0	0
$936$	0	0
$937$	−268.075	−0.00934644	−0.00467322	−	0.999989i	$-0.501488\pi$
$937$	−268.075	−0.00934644	−0.00467322	+	0.999989i	$0.501488\pi$
$938$	−47045.6	−1.63762
$939$	0	0
$940$	0	0
$941$	−30432.5	−1.05427	−0.527137	−	0.849781i	$-0.676734\pi$
$941$	−30432.5	−1.05427	−0.527137	+	0.849781i	$0.676734\pi$
$942$	0	0
$943$	−13340.7	−0.460691
$944$	172281.	5.93991
$945$	0	0
$946$	10230.4	0.351604
$947$	−29821.1	−1.02329	−0.511645	−	0.859197i	$-0.670964\pi$
$947$	−29821.1	−1.02329	−0.511645	+	0.859197i	$0.670964\pi$
$948$	0	0
$949$	1988.13	0.0680056
$950$	0	0
$951$	0	0
$952$	−75650.9	−2.57548
$953$	41109.5	1.39734	0.698671	−	0.715443i	$-0.253775\pi$
$953$	41109.5	1.39734	0.698671	+	0.715443i	$0.253775\pi$
$954$	0	0
$955$	0	0
$956$	59364.8	2.00836
$957$	0	0
$958$	113200.	3.81766
$959$	39431.2	1.32774
$960$	0	0
$961$	46750.4	1.56928
$962$	9945.49	0.333322
$963$	0	0
$964$	−15222.5	−0.508592
$965$	0	0
$966$	0	0
$967$	42606.0	1.41687	0.708437	−	0.705774i	$-0.249400\pi$
$967$	42606.0	1.41687	0.708437	+	0.705774i	$0.249400\pi$
$968$	−105558.	−3.50493
$969$	0	0
$970$	0	0
$971$	−33410.2	−1.10421	−0.552103	−	0.833776i	$-0.686175\pi$
$971$	−33410.2	−1.10421	−0.552103	+	0.833776i	$0.686175\pi$
$972$	0	0
$973$	−39426.2	−1.29902
$974$	46225.1	1.52069
$975$	0	0
$976$	161895.	5.30956
$977$	−8384.74	−0.274567	−0.137283	−	0.990532i	$-0.543837\pi$
$977$	−8384.74	−0.274567	−0.137283	+	0.990532i	$0.543837\pi$
$978$	0	0
$979$	−1411.56	−0.0460812
$980$	0	0
$981$	0	0
$982$	90174.4	2.93033
$983$	19554.2	0.634467	0.317234	−	0.948347i	$-0.397246\pi$
$983$	19554.2	0.634467	0.317234	+	0.948347i	$0.397246\pi$
$984$	0	0
$985$	0	0
$986$	23774.4	0.767881
$987$	0	0
$988$	−18303.6	−0.589389
$989$	24698.5	0.794101
$990$	0	0
$991$	32644.3	1.04640	0.523199	−	0.852210i	$-0.324738\pi$
$991$	32644.3	1.04640	0.523199	+	0.852210i	$0.324738\pi$
$992$	235000.	7.52144
$993$	0	0
$994$	163909.	5.23025
$995$	0	0
$996$	0	0
$997$	−21391.2	−0.679505	−0.339752	−	0.940515i	$-0.610343\pi$
$997$	−21391.2	−0.679505	−0.339752	+	0.940515i	$0.610343\pi$
$998$	−89326.0	−2.83323
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bi.1.12		12
3.2	odd	2		2025.4.a.be.1.1		12
5.4	even	2		2025.4.a.bf.1.1		12
9.4	even	3		225.4.e.e.151.1	yes	24
9.7	even	3		225.4.e.e.76.1	✓	24
15.14	odd	2		2025.4.a.bj.1.12		12
45.4	even	6		225.4.e.f.151.12	yes	24
45.7	odd	12		225.4.k.e.49.24		48
45.13	odd	12		225.4.k.e.124.24		48
45.22	odd	12		225.4.k.e.124.1		48
45.34	even	6		225.4.e.f.76.12	yes	24
45.43	odd	12		225.4.k.e.49.1		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.e.e.76.1	✓	24	9.7	even	3
225.4.e.e.151.1	yes	24	9.4	even	3
225.4.e.f.76.12	yes	24	45.34	even	6
225.4.e.f.151.12	yes	24	45.4	even	6
225.4.k.e.49.1		48	45.43	odd	12
225.4.k.e.49.24		48	45.7	odd	12
225.4.k.e.124.1		48	45.22	odd	12
225.4.k.e.124.24		48	45.13	odd	12
2025.4.a.be.1.1		12	3.2	odd	2
2025.4.a.bf.1.1		12	5.4	even	2
2025.4.a.bi.1.12		12	1.1	even	1	trivial
2025.4.a.bj.1.12		12	15.14	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:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q+5.54419 q^{2} +22.7380 q^{4} -25.7127 q^{7} +81.7101 q^{8} +6.25584 q^{11} +15.2600 q^{13} -142.556 q^{14} +271.112 q^{16} +36.0074 q^{17} -52.7512 q^{19} +34.6835 q^{22} +83.7341 q^{23} +84.6040 q^{26} -584.654 q^{28} +119.091 q^{29} +276.661 q^{31} +849.415 q^{32} +199.631 q^{34} +117.553 q^{37} -292.462 q^{38} -159.322 q^{41} +294.963 q^{43} +142.245 q^{44} +464.237 q^{46} -83.2917 q^{47} +318.141 q^{49} +346.981 q^{52} -149.018 q^{53} -2100.98 q^{56} +660.265 q^{58} +635.461 q^{59} +597.151 q^{61} +1533.86 q^{62} +2540.42 q^{64} +330.015 q^{67} +818.735 q^{68} -1149.79 q^{71} +130.284 q^{73} +651.738 q^{74} -1199.46 q^{76} -160.854 q^{77} -737.835 q^{79} -883.309 q^{82} +369.902 q^{83} +1635.33 q^{86} +511.165 q^{88} -225.638 q^{89} -392.374 q^{91} +1903.94 q^{92} -461.784 q^{94} +11.9134 q^{97} +1763.83 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 48 q^{4} - 6 q^{7} + 45 q^{8} + 29 q^{11} - 24 q^{13} - 69 q^{14} + 192 q^{16} + 79 q^{17} - 75 q^{19} + 18 q^{22} + 318 q^{23} - 154 q^{26} - 96 q^{28} + 106 q^{29} + 60 q^{31} + 914 q^{32}+ \cdots + 7511 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 48 * q^4 - 6 * q^7 + 45 * q^8 + 29 * q^11 - 24 * q^13 - 69 * q^14 + 192 * q^16 + 79 * q^17 - 75 * q^19 + 18 * q^22 + 318 * q^23 - 154 * q^26 - 96 * q^28 + 106 * q^29 + 60 * q^31 + 914 * q^32 - 108 * q^34 + 84 * q^37 + 640 * q^38 - 353 * q^41 + 426 * q^43 + 571 * q^44 + 270 * q^46 + 1210 * q^47 + 666 * q^49 + 75 * q^52 + 448 * q^53 - 570 * q^56 - 594 * q^58 + 482 * q^59 + 402 * q^61 + 2544 * q^62 + 975 * q^64 + 201 * q^67 + 3437 * q^68 - 944 * q^71 + 453 * q^73 + 10 * q^74 - 462 * q^76 + 2652 * q^77 + 258 * q^79 - 873 * q^82 + 3012 * q^83 + 1952 * q^86 + 216 * q^88 - 738 * q^89 - 618 * q^91 + 5232 * q^92 + 63 * q^94 + 318 * q^97 + 7511 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more