LMFDB - Embedded newform 2475.4.a.bm.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2475,4,Mod(1,2475)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2475.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,4,0,46,0,0,38,72,0,0,-55] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$146.029727264$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 41x^{3} + 3x^{2} + 294x - 200 $ x^5 - x^4 - 41x^3 + 3x^2 + 294*x - 200
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.729174$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.270826 q^{2} -7.92665 q^{4} +33.4133 q^{7} -4.31336 q^{8} -11.0000 q^{11} +33.5174 q^{13} +9.04920 q^{14} +62.2451 q^{16} -71.1540 q^{17} -48.9667 q^{19} -2.97909 q^{22} +66.7490 q^{23} +9.07738 q^{26} -264.856 q^{28} +66.8864 q^{29} +145.759 q^{31} +51.3644 q^{32} -19.2704 q^{34} +37.8491 q^{37} -13.2615 q^{38} +344.036 q^{41} -34.4830 q^{43} +87.1932 q^{44} +18.0774 q^{46} -270.887 q^{47} +773.448 q^{49} -265.681 q^{52} +666.088 q^{53} -144.123 q^{56} +18.1146 q^{58} -876.971 q^{59} +783.523 q^{61} +39.4753 q^{62} -484.050 q^{64} -876.724 q^{67} +564.013 q^{68} +523.428 q^{71} -91.0973 q^{73} +10.2505 q^{74} +388.142 q^{76} -367.546 q^{77} -96.9287 q^{79} +93.1741 q^{82} -1395.68 q^{83} -9.33889 q^{86} +47.4469 q^{88} -508.117 q^{89} +1119.93 q^{91} -529.096 q^{92} -73.3634 q^{94} +644.082 q^{97} +209.470 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 46 q^{4} + 38 q^{7} + 72 q^{8} - 55 q^{11} - 43 q^{13} - 55 q^{14} + 634 q^{16} + 150 q^{17} - 337 q^{19} - 44 q^{22} - 201 q^{23} - 359 q^{26} + 441 q^{28} + 153 q^{29} + 263 q^{31} + 280 q^{32}+ \cdots + 9417 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 46 * q^4 + 38 * q^7 + 72 * q^8 - 55 * q^11 - 43 * q^13 - 55 * q^14 + 634 * q^16 + 150 * q^17 - 337 * q^19 - 44 * q^22 - 201 * q^23 - 359 * q^26 + 441 * q^28 + 153 * q^29 + 263 * q^31 + 280 * q^32 + 517 * q^34 + 350 * q^37 + 786 * q^38 + 88 * q^41 - 161 * q^43 - 506 * q^44 + 436 * q^46 + 490 * q^47 + 2381 * q^49 - 289 * q^52 + 558 * q^53 - 211 * q^56 + 969 * q^58 - 410 * q^59 - 1130 * q^61 + 463 * q^62 + 3170 * q^64 - 1130 * q^67 + 3333 * q^68 - 1249 * q^71 + 44 * q^73 + 2985 * q^74 - 2556 * q^76 - 418 * q^77 + 2258 * q^79 + 3737 * q^82 - 957 * q^83 + 1435 * q^86 - 792 * q^88 - 347 * q^89 - 612 * q^91 - 6006 * q^92 + 907 * q^94 + 2355 * q^97 + 9417 * 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 $

$2$	0.270826	0.0957515	0.0478758	−	0.998853i	$-0.484755\pi$
$2$	0.270826	0.0957515	0.0478758	+	0.998853i	$0.484755\pi$
$3$	0	0
$3$	0	0
$4$	−7.92665	−0.990832
$5$	0	0
$5$	0	0
$6$	0	0
$7$	33.4133	1.80415	0.902074	−	0.431581i	$-0.142044\pi$
$7$	33.4133	1.80415	0.902074	+	0.431581i	$0.142044\pi$
$8$	−4.31336	−0.190625
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	33.5174	0.715081	0.357540	−	0.933898i	$-0.383615\pi$
$13$	33.5174	0.715081	0.357540	+	0.933898i	$0.383615\pi$
$14$	9.04920	0.172750
$15$	0	0
$16$	62.2451	0.972579
$17$	−71.1540	−1.01514	−0.507570	−	0.861610i	$-0.669456\pi$
$17$	−71.1540	−1.01514	−0.507570	+	0.861610i	$0.669456\pi$
$18$	0	0
$19$	−48.9667	−0.591249	−0.295625	−	0.955304i	$-0.595528\pi$
$19$	−48.9667	−0.591249	−0.295625	+	0.955304i	$0.595528\pi$
$20$	0	0
$21$	0	0
$22$	−2.97909	−0.0288702
$23$	66.7490	0.605136	0.302568	−	0.953128i	$-0.402156\pi$
$23$	66.7490	0.605136	0.302568	+	0.953128i	$0.402156\pi$
$24$	0	0
$25$	0	0
$26$	9.07738	0.0684701
$27$	0	0
$28$	−264.856	−1.78761
$29$	66.8864	0.428293	0.214146	−	0.976802i	$-0.431303\pi$
$29$	66.8864	0.428293	0.214146	+	0.976802i	$0.431303\pi$
$30$	0	0
$31$	145.759	0.844486	0.422243	−	0.906483i	$-0.361243\pi$
$31$	145.759	0.844486	0.422243	+	0.906483i	$0.361243\pi$
$32$	51.3644	0.283751
$33$	0	0
$34$	−19.2704	−0.0972013
$35$	0	0
$36$	0	0
$37$	37.8491	0.168172	0.0840859	−	0.996459i	$-0.473203\pi$
$37$	37.8491	0.168172	0.0840859	+	0.996459i	$0.473203\pi$
$38$	−13.2615	−0.0566130
$39$	0	0
$40$	0	0
$41$	344.036	1.31047	0.655237	−	0.755423i	$-0.272569\pi$
$41$	344.036	1.31047	0.655237	+	0.755423i	$0.272569\pi$
$42$	0	0
$43$	−34.4830	−0.122293	−0.0611465	−	0.998129i	$-0.519476\pi$
$43$	−34.4830	−0.122293	−0.0611465	+	0.998129i	$0.519476\pi$
$44$	87.1932	0.298747
$45$	0	0
$46$	18.0774	0.0579427
$47$	−270.887	−0.840701	−0.420351	−	0.907362i	$-0.638093\pi$
$47$	−270.887	−0.840701	−0.420351	+	0.907362i	$0.638093\pi$
$48$	0	0
$49$	773.448	2.25495
$50$	0	0
$51$	0	0
$52$	−265.681	−0.708524
$53$	666.088	1.72631	0.863154	−	0.504941i	$-0.168486\pi$
$53$	666.088	1.72631	0.863154	+	0.504941i	$0.168486\pi$
$54$	0	0
$55$	0	0
$56$	−144.123	−0.343916
$57$	0	0
$58$	18.1146	0.0410097
$59$	−876.971	−1.93512	−0.967559	−	0.252645i	$-0.918699\pi$
$59$	−876.971	−1.93512	−0.967559	+	0.252645i	$0.918699\pi$
$60$	0	0
$61$	783.523	1.64459	0.822294	−	0.569063i	$-0.192694\pi$
$61$	783.523	1.64459	0.822294	+	0.569063i	$0.192694\pi$
$62$	39.4753	0.0808608
$63$	0	0
$64$	−484.050	−0.945409
$65$	0	0
$66$	0	0
$67$	−876.724	−1.59864	−0.799320	−	0.600906i	$-0.794807\pi$
$67$	−876.724	−1.59864	−0.799320	+	0.600906i	$0.794807\pi$
$68$	564.013	1.00583
$69$	0	0
$70$	0	0
$71$	523.428	0.874922	0.437461	−	0.899237i	$-0.355878\pi$
$71$	523.428	0.874922	0.437461	+	0.899237i	$0.355878\pi$
$72$	0	0
$73$	−91.0973	−0.146057	−0.0730283	−	0.997330i	$-0.523266\pi$
$73$	−91.0973	−0.146057	−0.0730283	+	0.997330i	$0.523266\pi$
$74$	10.2505	0.0161027
$75$	0	0
$76$	388.142	0.585829
$77$	−367.546	−0.543971
$78$	0	0
$79$	−96.9287	−0.138042	−0.0690211	−	0.997615i	$-0.521988\pi$
$79$	−96.9287	−0.138042	−0.0690211	+	0.997615i	$0.521988\pi$
$80$	0	0
$81$	0	0
$82$	93.1741	0.125480
$83$	−1395.68	−1.84574	−0.922868	−	0.385116i	$-0.874161\pi$
$83$	−1395.68	−1.84574	−0.922868	+	0.385116i	$0.874161\pi$
$84$	0	0
$85$	0	0
$86$	−9.33889	−0.0117097
$87$	0	0
$88$	47.4469	0.0574757
$89$	−508.117	−0.605172	−0.302586	−	0.953122i	$-0.597850\pi$
$89$	−508.117	−0.605172	−0.302586	+	0.953122i	$0.597850\pi$
$90$	0	0
$91$	1119.93	1.29011
$92$	−529.096	−0.599588
$93$	0	0
$94$	−73.3634	−0.0804984
$95$	0	0
$96$	0	0
$97$	644.082	0.674192	0.337096	−	0.941470i	$-0.390555\pi$
$97$	644.082	0.674192	0.337096	+	0.941470i	$0.390555\pi$
$98$	209.470	0.215915
$99$	0	0
$100$	0	0
$101$	−1485.24	−1.46324	−0.731618	−	0.681715i	$-0.761234\pi$
$101$	−1485.24	−1.46324	−0.731618	+	0.681715i	$0.761234\pi$
$102$	0	0
$103$	−605.952	−0.579672	−0.289836	−	0.957076i	$-0.593601\pi$
$103$	−605.952	−0.579672	−0.289836	+	0.957076i	$0.593601\pi$
$104$	−144.572	−0.136312
$105$	0	0
$106$	180.394	0.165297
$107$	−352.908	−0.318850	−0.159425	−	0.987210i	$-0.550964\pi$
$107$	−352.908	−0.318850	−0.159425	+	0.987210i	$0.550964\pi$
$108$	0	0
$109$	1413.29	1.24191	0.620957	−	0.783844i	$-0.286744\pi$
$109$	1413.29	1.24191	0.620957	+	0.783844i	$0.286744\pi$
$110$	0	0
$111$	0	0
$112$	2079.81	1.75468
$113$	103.753	0.0863741	0.0431870	−	0.999067i	$-0.486249\pi$
$113$	103.753	0.0863741	0.0431870	+	0.999067i	$0.486249\pi$
$114$	0	0
$115$	0	0
$116$	−530.185	−0.424366
$117$	0	0
$118$	−237.507	−0.185291
$119$	−2377.49	−1.83146
$120$	0	0
$121$	121.000	0.0909091
$122$	212.199	0.157472
$123$	0	0
$124$	−1155.38	−0.836743
$125$	0	0
$126$	0	0
$127$	1576.66	1.10162	0.550809	−	0.834631i	$-0.314319\pi$
$127$	1576.66	1.10162	0.550809	+	0.834631i	$0.314319\pi$
$128$	−542.009	−0.374276
$129$	0	0
$130$	0	0
$131$	1085.16	0.723747	0.361873	−	0.932227i	$-0.382137\pi$
$131$	1085.16	0.723747	0.361873	+	0.932227i	$0.382137\pi$
$132$	0	0
$133$	−1636.14	−1.06670
$134$	−237.440	−0.153072
$135$	0	0
$136$	306.913	0.193511
$137$	1389.46	0.866496	0.433248	−	0.901275i	$-0.357367\pi$
$137$	1389.46	0.866496	0.433248	+	0.901275i	$0.357367\pi$
$138$	0	0
$139$	500.066	0.305144	0.152572	−	0.988292i	$-0.451244\pi$
$139$	500.066	0.305144	0.152572	+	0.988292i	$0.451244\pi$
$140$	0	0
$141$	0	0
$142$	141.758	0.0837751
$143$	−368.691	−0.215605
$144$	0	0
$145$	0	0
$146$	−24.6715	−0.0139851
$147$	0	0
$148$	−300.017	−0.166630
$149$	−685.689	−0.377005	−0.188503	−	0.982073i	$-0.560363\pi$
$149$	−685.689	−0.377005	−0.188503	+	0.982073i	$0.560363\pi$
$150$	0	0
$151$	−1208.96	−0.651550	−0.325775	−	0.945447i	$-0.605625\pi$
$151$	−1208.96	−0.651550	−0.325775	+	0.945447i	$0.605625\pi$
$152$	211.211	0.112707
$153$	0	0
$154$	−99.5412	−0.0520861
$155$	0	0
$156$	0	0
$157$	3322.47	1.68893	0.844465	−	0.535610i	$-0.179918\pi$
$157$	3322.47	1.68893	0.844465	+	0.535610i	$0.179918\pi$
$158$	−26.2509	−0.0132178
$159$	0	0
$160$	0	0
$161$	2230.30	1.09176
$162$	0	0
$163$	−2448.31	−1.17648	−0.588240	−	0.808686i	$-0.700179\pi$
$163$	−2448.31	−1.17648	−0.588240	+	0.808686i	$0.700179\pi$
$164$	−2727.06	−1.29846
$165$	0	0
$166$	−377.988	−0.176732
$167$	−1518.71	−0.703720	−0.351860	−	0.936053i	$-0.614451\pi$
$167$	−1518.71	−0.703720	−0.351860	+	0.936053i	$0.614451\pi$
$168$	0	0
$169$	−1073.59	−0.488660
$170$	0	0
$171$	0	0
$172$	273.334	0.121172
$173$	2636.80	1.15880	0.579399	−	0.815044i	$-0.303287\pi$
$173$	2636.80	1.15880	0.579399	+	0.815044i	$0.303287\pi$
$174$	0	0
$175$	0	0
$176$	−684.696	−0.293244
$177$	0	0
$178$	−137.611	−0.0579461
$179$	1773.31	0.740466	0.370233	−	0.928939i	$-0.379278\pi$
$179$	1773.31	0.740466	0.370233	+	0.928939i	$0.379278\pi$
$180$	0	0
$181$	529.751	0.217547	0.108774	−	0.994067i	$-0.465308\pi$
$181$	529.751	0.217547	0.108774	+	0.994067i	$0.465308\pi$
$182$	303.305	0.123530
$183$	0	0
$184$	−287.912	−0.115354
$185$	0	0
$186$	0	0
$187$	782.694	0.306076
$188$	2147.23	0.832993
$189$	0	0
$190$	0	0
$191$	−2061.93	−0.781130	−0.390565	−	0.920575i	$-0.627720\pi$
$191$	−2061.93	−0.781130	−0.390565	+	0.920575i	$0.627720\pi$
$192$	0	0
$193$	1637.24	0.610627	0.305314	−	0.952252i	$-0.401239\pi$
$193$	1637.24	0.610627	0.305314	+	0.952252i	$0.401239\pi$
$194$	174.434	0.0645550
$195$	0	0
$196$	−6130.86	−2.23428
$197$	3414.86	1.23502	0.617510	−	0.786563i	$-0.288142\pi$
$197$	3414.86	1.23502	0.617510	+	0.786563i	$0.288142\pi$
$198$	0	0
$199$	3938.93	1.40313	0.701566	−	0.712605i	$-0.252485\pi$
$199$	3938.93	1.40313	0.701566	+	0.712605i	$0.252485\pi$
$200$	0	0
$201$	0	0
$202$	−402.242	−0.140107
$203$	2234.90	0.772704
$204$	0	0
$205$	0	0
$206$	−164.108	−0.0555045
$207$	0	0
$208$	2086.29	0.695472
$209$	538.634	0.178268
$210$	0	0
$211$	3543.14	1.15602	0.578009	−	0.816031i	$-0.303830\pi$
$211$	3543.14	1.15602	0.578009	+	0.816031i	$0.303830\pi$
$212$	−5279.85	−1.71048
$213$	0	0
$214$	−95.5768	−0.0305304
$215$	0	0
$216$	0	0
$217$	4870.28	1.52358
$218$	382.756	0.118915
$219$	0	0
$220$	0	0
$221$	−2384.90	−0.725907
$222$	0	0
$223$	850.930	0.255527	0.127763	−	0.991805i	$-0.459220\pi$
$223$	850.930	0.255527	0.127763	+	0.991805i	$0.459220\pi$
$224$	1716.26	0.511929
$225$	0	0
$226$	28.0991	0.00827045
$227$	5040.56	1.47381	0.736903	−	0.675999i	$-0.236288\pi$
$227$	5040.56	1.47381	0.736903	+	0.675999i	$0.236288\pi$
$228$	0	0
$229$	−3624.34	−1.04587	−0.522933	−	0.852374i	$-0.675162\pi$
$229$	−3624.34	−1.04587	−0.522933	+	0.852374i	$0.675162\pi$
$230$	0	0
$231$	0	0
$232$	−288.505	−0.0816434
$233$	−2984.56	−0.839164	−0.419582	−	0.907718i	$-0.637823\pi$
$233$	−2984.56	−0.839164	−0.419582	+	0.907718i	$0.637823\pi$
$234$	0	0
$235$	0	0
$236$	6951.45	1.91738
$237$	0	0
$238$	−643.887	−0.175366
$239$	−1298.11	−0.351331	−0.175665	−	0.984450i	$-0.556208\pi$
$239$	−1298.11	−0.351331	−0.175665	+	0.984450i	$0.556208\pi$
$240$	0	0
$241$	3978.71	1.06345	0.531725	−	0.846917i	$-0.321544\pi$
$241$	3978.71	1.06345	0.531725	+	0.846917i	$0.321544\pi$
$242$	32.7700	0.00870469
$243$	0	0
$244$	−6210.71	−1.62951
$245$	0	0
$246$	0	0
$247$	−1641.24	−0.422791
$248$	−628.710	−0.160980
$249$	0	0
$250$	0	0
$251$	1884.05	0.473784	0.236892	−	0.971536i	$-0.423871\pi$
$251$	1884.05	0.473784	0.236892	+	0.971536i	$0.423871\pi$
$252$	0	0
$253$	−734.239	−0.182455
$254$	427.000	0.105482
$255$	0	0
$256$	3725.61	0.909572
$257$	−4012.23	−0.973836	−0.486918	−	0.873448i	$-0.661879\pi$
$257$	−4012.23	−0.973836	−0.486918	+	0.873448i	$0.661879\pi$
$258$	0	0
$259$	1264.66	0.303407
$260$	0	0
$261$	0	0
$262$	293.890	0.0692999
$263$	290.429	0.0680935	0.0340468	−	0.999420i	$-0.489160\pi$
$263$	290.429	0.0680935	0.0340468	+	0.999420i	$0.489160\pi$
$264$	0	0
$265$	0	0
$266$	−443.110	−0.102138
$267$	0	0
$268$	6949.48	1.58398
$269$	6743.15	1.52839	0.764196	−	0.644985i	$-0.223136\pi$
$269$	6743.15	1.52839	0.764196	+	0.644985i	$0.223136\pi$
$270$	0	0
$271$	7182.11	1.60990	0.804949	−	0.593344i	$-0.202193\pi$
$271$	7182.11	1.60990	0.804949	+	0.593344i	$0.202193\pi$
$272$	−4428.99	−0.987304
$273$	0	0
$274$	376.304	0.0829683
$275$	0	0
$276$	0	0
$277$	5708.35	1.23820	0.619100	−	0.785312i	$-0.287498\pi$
$277$	5708.35	1.23820	0.619100	+	0.785312i	$0.287498\pi$
$278$	135.431	0.0292180
$279$	0	0
$280$	0	0
$281$	−8740.94	−1.85566	−0.927831	−	0.373002i	$-0.878328\pi$
$281$	−8740.94	−1.85566	−0.927831	+	0.373002i	$0.878328\pi$
$282$	0	0
$283$	7883.57	1.65593	0.827967	−	0.560777i	$-0.189497\pi$
$283$	7883.57	1.65593	0.827967	+	0.560777i	$0.189497\pi$
$284$	−4149.03	−0.866900
$285$	0	0
$286$	−99.8512	−0.0206445
$287$	11495.4	2.36429
$288$	0	0
$289$	149.897	0.0305102
$290$	0	0
$291$	0	0
$292$	722.097	0.144718
$293$	2328.54	0.464282	0.232141	−	0.972682i	$-0.425427\pi$
$293$	2328.54	0.464282	0.232141	+	0.972682i	$0.425427\pi$
$294$	0	0
$295$	0	0
$296$	−163.257	−0.0320578
$297$	0	0
$298$	−185.703	−0.0360989
$299$	2237.25	0.432721
$300$	0	0
$301$	−1152.19	−0.220635
$302$	−327.419	−0.0623869
$303$	0	0
$304$	−3047.94	−0.575037
$305$	0	0
$306$	0	0
$307$	−2830.31	−0.526170	−0.263085	−	0.964773i	$-0.584740\pi$
$307$	−2830.31	−0.526170	−0.263085	+	0.964773i	$0.584740\pi$
$308$	2913.41	0.538984
$309$	0	0
$310$	0	0
$311$	6580.67	1.19986	0.599929	−	0.800053i	$-0.295196\pi$
$311$	6580.67	1.19986	0.599929	+	0.800053i	$0.295196\pi$
$312$	0	0
$313$	7837.96	1.41542	0.707712	−	0.706501i	$-0.249728\pi$
$313$	7837.96	1.41542	0.707712	+	0.706501i	$0.249728\pi$
$314$	899.813	0.161718
$315$	0	0
$316$	768.321	0.136777
$317$	5853.61	1.03713	0.518567	−	0.855037i	$-0.326466\pi$
$317$	5853.61	1.03713	0.518567	+	0.855037i	$0.326466\pi$
$318$	0	0
$319$	−735.751	−0.129135
$320$	0	0
$321$	0	0
$322$	604.025	0.104537
$323$	3484.18	0.600201
$324$	0	0
$325$	0	0
$326$	−663.066	−0.112650
$327$	0	0
$328$	−1483.95	−0.249809
$329$	−9051.23	−1.51675
$330$	0	0
$331$	5347.61	0.888010	0.444005	−	0.896024i	$-0.353557\pi$
$331$	5347.61	0.888010	0.444005	+	0.896024i	$0.353557\pi$
$332$	11063.1	1.82881
$333$	0	0
$334$	−411.306	−0.0673823
$335$	0	0
$336$	0	0
$337$	−2763.99	−0.446779	−0.223389	−	0.974729i	$-0.571712\pi$
$337$	−2763.99	−0.446779	−0.223389	+	0.974729i	$0.571712\pi$
$338$	−290.755	−0.0467899
$339$	0	0
$340$	0	0
$341$	−1603.35	−0.254622
$342$	0	0
$343$	14382.7	2.26412
$344$	148.737	0.0233121
$345$	0	0
$346$	714.114	0.110957
$347$	5038.66	0.779509	0.389754	−	0.920919i	$-0.372560\pi$
$347$	5038.66	0.779509	0.389754	+	0.920919i	$0.372560\pi$
$348$	0	0
$349$	−7429.49	−1.13952	−0.569759	−	0.821812i	$-0.692963\pi$
$349$	−7429.49	−1.13952	−0.569759	+	0.821812i	$0.692963\pi$
$350$	0	0
$351$	0	0
$352$	−565.009	−0.0855542
$353$	−3151.84	−0.475228	−0.237614	−	0.971360i	$-0.576365\pi$
$353$	−3151.84	−0.475228	−0.237614	+	0.971360i	$0.576365\pi$
$354$	0	0
$355$	0	0
$356$	4027.67	0.599623
$357$	0	0
$358$	480.259	0.0709008
$359$	−903.954	−0.132894	−0.0664469	−	0.997790i	$-0.521166\pi$
$359$	−903.954	−0.132894	−0.0664469	+	0.997790i	$0.521166\pi$
$360$	0	0
$361$	−4461.26	−0.650424
$362$	143.470	0.0208305
$363$	0	0
$364$	−8877.26	−1.27828
$365$	0	0
$366$	0	0
$367$	−3978.31	−0.565848	−0.282924	−	0.959142i	$-0.591304\pi$
$367$	−3978.31	−0.565848	−0.282924	+	0.959142i	$0.591304\pi$
$368$	4154.79	0.588543
$369$	0	0
$370$	0	0
$371$	22256.2	3.11451
$372$	0	0
$373$	5493.67	0.762605	0.381302	−	0.924450i	$-0.375476\pi$
$373$	5493.67	0.762605	0.381302	+	0.924450i	$0.375476\pi$
$374$	211.974	0.0293073
$375$	0	0
$376$	1168.43	0.160259
$377$	2241.86	0.306264
$378$	0	0
$379$	3035.93	0.411465	0.205733	−	0.978608i	$-0.434042\pi$
$379$	3035.93	0.411465	0.205733	+	0.978608i	$0.434042\pi$
$380$	0	0
$381$	0	0
$382$	−558.424	−0.0747944
$383$	−3556.77	−0.474524	−0.237262	−	0.971446i	$-0.576250\pi$
$383$	−3556.77	−0.474524	−0.237262	+	0.971446i	$0.576250\pi$
$384$	0	0
$385$	0	0
$386$	443.408	0.0584685
$387$	0	0
$388$	−5105.42	−0.668011
$389$	5211.56	0.679272	0.339636	−	0.940557i	$-0.389696\pi$
$389$	5211.56	0.679272	0.339636	+	0.940557i	$0.389696\pi$
$390$	0	0
$391$	−4749.46	−0.614298
$392$	−3336.16	−0.429851
$393$	0	0
$394$	924.835	0.118255
$395$	0	0
$396$	0	0
$397$	6547.28	0.827704	0.413852	−	0.910344i	$-0.364183\pi$
$397$	6547.28	0.827704	0.413852	+	0.910344i	$0.364183\pi$
$398$	1066.76	0.134352
$399$	0	0
$400$	0	0
$401$	1661.59	0.206922	0.103461	−	0.994634i	$-0.467008\pi$
$401$	1661.59	0.206922	0.103461	+	0.994634i	$0.467008\pi$
$402$	0	0
$403$	4885.45	0.603875
$404$	11773.0	1.44982
$405$	0	0
$406$	605.268	0.0739876
$407$	−416.340	−0.0507057
$408$	0	0
$409$	7278.96	0.880004	0.440002	−	0.897997i	$-0.354978\pi$
$409$	7278.96	0.880004	0.440002	+	0.897997i	$0.354978\pi$
$410$	0	0
$411$	0	0
$412$	4803.17	0.574357
$413$	−29302.5	−3.49124
$414$	0	0
$415$	0	0
$416$	1721.60	0.202905
$417$	0	0
$418$	145.876	0.0170695
$419$	5892.15	0.686994	0.343497	−	0.939154i	$-0.388389\pi$
$419$	5892.15	0.686994	0.343497	+	0.939154i	$0.388389\pi$
$420$	0	0
$421$	11086.3	1.28340	0.641700	−	0.766955i	$-0.278229\pi$
$421$	11086.3	1.28340	0.641700	+	0.766955i	$0.278229\pi$
$422$	959.575	0.110690
$423$	0	0
$424$	−2873.08	−0.329078
$425$	0	0
$426$	0	0
$427$	26180.1	2.96708
$428$	2797.38	0.315926
$429$	0	0
$430$	0	0
$431$	−5658.06	−0.632341	−0.316170	−	0.948702i	$-0.602397\pi$
$431$	−5658.06	−0.632341	−0.316170	+	0.948702i	$0.602397\pi$
$432$	0	0
$433$	−3065.03	−0.340176	−0.170088	−	0.985429i	$-0.554405\pi$
$433$	−3065.03	−0.340176	−0.170088	+	0.985429i	$0.554405\pi$
$434$	1319.00	0.145885
$435$	0	0
$436$	−11202.7	−1.23053
$437$	−3268.48	−0.357786
$438$	0	0
$439$	−3017.50	−0.328058	−0.164029	−	0.986456i	$-0.552449\pi$
$439$	−3017.50	−0.328058	−0.164029	+	0.986456i	$0.552449\pi$
$440$	0	0
$441$	0	0
$442$	−645.893	−0.0695067
$443$	17098.3	1.83378	0.916888	−	0.399144i	$-0.130693\pi$
$443$	17098.3	1.83378	0.916888	+	0.399144i	$0.130693\pi$
$444$	0	0
$445$	0	0
$446$	230.454	0.0244671
$447$	0	0
$448$	−16173.7	−1.70566
$449$	10823.4	1.13762	0.568808	−	0.822470i	$-0.307405\pi$
$449$	10823.4	1.13762	0.568808	+	0.822470i	$0.307405\pi$
$450$	0	0
$451$	−3784.40	−0.395123
$452$	−822.415	−0.0855821
$453$	0	0
$454$	1365.12	0.141119
$455$	0	0
$456$	0	0
$457$	6859.01	0.702081	0.351040	−	0.936360i	$-0.385828\pi$
$457$	6859.01	0.702081	0.351040	+	0.936360i	$0.385828\pi$
$458$	−981.567	−0.100143
$459$	0	0
$460$	0	0
$461$	−9651.72	−0.975109	−0.487555	−	0.873093i	$-0.662111\pi$
$461$	−9651.72	−0.975109	−0.487555	+	0.873093i	$0.662111\pi$
$462$	0	0
$463$	4744.49	0.476232	0.238116	−	0.971237i	$-0.423470\pi$
$463$	4744.49	0.476232	0.238116	+	0.971237i	$0.423470\pi$
$464$	4163.35	0.416549
$465$	0	0
$466$	−808.298	−0.0803512
$467$	−11038.1	−1.09376	−0.546878	−	0.837213i	$-0.684184\pi$
$467$	−11038.1	−1.09376	−0.546878	+	0.837213i	$0.684184\pi$
$468$	0	0
$469$	−29294.2	−2.88418
$470$	0	0
$471$	0	0
$472$	3782.69	0.368882
$473$	379.313	0.0368727
$474$	0	0
$475$	0	0
$476$	18845.5	1.81467
$477$	0	0
$478$	−351.563	−0.0336404
$479$	−17476.9	−1.66710	−0.833550	−	0.552444i	$-0.813695\pi$
$479$	−17476.9	−1.66710	−0.833550	+	0.552444i	$0.813695\pi$
$480$	0	0
$481$	1268.60	0.120256
$482$	1077.54	0.101827
$483$	0	0
$484$	−959.125	−0.0900756
$485$	0	0
$486$	0	0
$487$	−17014.6	−1.58317	−0.791586	−	0.611058i	$-0.790744\pi$
$487$	−17014.6	−1.58317	−0.791586	+	0.611058i	$0.790744\pi$
$488$	−3379.61	−0.313500
$489$	0	0
$490$	0	0
$491$	−19673.8	−1.80828	−0.904142	−	0.427233i	$-0.859488\pi$
$491$	−19673.8	−1.80828	−0.904142	+	0.427233i	$0.859488\pi$
$492$	0	0
$493$	−4759.24	−0.434778
$494$	−444.490	−0.0404829
$495$	0	0
$496$	9072.76	0.821329
$497$	17489.4	1.57849
$498$	0	0
$499$	−15754.6	−1.41337	−0.706687	−	0.707526i	$-0.749811\pi$
$499$	−15754.6	−1.41337	−0.706687	+	0.707526i	$0.749811\pi$
$500$	0	0
$501$	0	0
$502$	510.249	0.0453656
$503$	−16457.2	−1.45883	−0.729414	−	0.684073i	$-0.760207\pi$
$503$	−16457.2	−1.45883	−0.729414	+	0.684073i	$0.760207\pi$
$504$	0	0
$505$	0	0
$506$	−198.851	−0.0174704
$507$	0	0
$508$	−12497.6	−1.09152
$509$	−13852.0	−1.20625	−0.603124	−	0.797647i	$-0.706078\pi$
$509$	−13852.0	−1.20625	−0.603124	+	0.797647i	$0.706078\pi$
$510$	0	0
$511$	−3043.86	−0.263508
$512$	5345.06	0.461368
$513$	0	0
$514$	−1086.62	−0.0932463
$515$	0	0
$516$	0	0
$517$	2979.76	0.253481
$518$	342.504	0.0290517
$519$	0	0
$520$	0	0
$521$	12295.8	1.03395	0.516976	−	0.856000i	$-0.327058\pi$
$521$	12295.8	1.03395	0.516976	+	0.856000i	$0.327058\pi$
$522$	0	0
$523$	−1971.83	−0.164861	−0.0824305	−	0.996597i	$-0.526268\pi$
$523$	−1971.83	−0.164861	−0.0824305	+	0.996597i	$0.526268\pi$
$524$	−8601.68	−0.717111
$525$	0	0
$526$	78.6557	0.00652006
$527$	−10371.3	−0.857271
$528$	0	0
$529$	−7711.57	−0.633810
$530$	0	0
$531$	0	0
$532$	12969.1	1.05692
$533$	11531.2	0.937095
$534$	0	0
$535$	0	0
$536$	3781.62	0.304741
$537$	0	0
$538$	1826.22	0.146346
$539$	−8507.93	−0.679893
$540$	0	0
$541$	−8849.45	−0.703267	−0.351634	−	0.936138i	$-0.614374\pi$
$541$	−8849.45	−0.703267	−0.351634	+	0.936138i	$0.614374\pi$
$542$	1945.10	0.154150
$543$	0	0
$544$	−3654.79	−0.288047
$545$	0	0
$546$	0	0
$547$	−10687.9	−0.835435	−0.417717	−	0.908577i	$-0.637170\pi$
$547$	−10687.9	−0.835435	−0.417717	+	0.908577i	$0.637170\pi$
$548$	−11013.8	−0.858552
$549$	0	0
$550$	0	0
$551$	−3275.21	−0.253228
$552$	0	0
$553$	−3238.71	−0.249049
$554$	1545.97	0.118560
$555$	0	0
$556$	−3963.85	−0.302346
$557$	23864.2	1.81537	0.907684	−	0.419655i	$-0.137849\pi$
$557$	23864.2	1.81537	0.907684	+	0.419655i	$0.137849\pi$
$558$	0	0
$559$	−1155.78	−0.0874494
$560$	0	0
$561$	0	0
$562$	−2367.28	−0.177682
$563$	13610.5	1.01886	0.509428	−	0.860513i	$-0.329857\pi$
$563$	13610.5	1.01886	0.509428	+	0.860513i	$0.329857\pi$
$564$	0	0
$565$	0	0
$566$	2135.08	0.158558
$567$	0	0
$568$	−2257.73	−0.166782
$569$	−8564.56	−0.631011	−0.315505	−	0.948924i	$-0.602174\pi$
$569$	−8564.56	−0.631011	−0.315505	+	0.948924i	$0.602174\pi$
$570$	0	0
$571$	15329.9	1.12353	0.561764	−	0.827298i	$-0.310123\pi$
$571$	15329.9	1.12353	0.561764	+	0.827298i	$0.310123\pi$
$572$	2922.49	0.213628
$573$	0	0
$574$	3113.25	0.226384
$575$	0	0
$576$	0	0
$577$	9594.47	0.692241	0.346120	−	0.938190i	$-0.387499\pi$
$577$	9594.47	0.692241	0.346120	+	0.938190i	$0.387499\pi$
$578$	40.5959	0.00292140
$579$	0	0
$580$	0	0
$581$	−46634.4	−3.32998
$582$	0	0
$583$	−7326.97	−0.520501
$584$	392.935	0.0278421
$585$	0	0
$586$	630.629	0.0444557
$587$	16659.5	1.17140	0.585698	−	0.810530i	$-0.300821\pi$
$587$	16659.5	1.17140	0.585698	+	0.810530i	$0.300821\pi$
$588$	0	0
$589$	−7137.33	−0.499302
$590$	0	0
$591$	0	0
$592$	2355.92	0.163560
$593$	459.030	0.0317877	0.0158939	−	0.999874i	$-0.494941\pi$
$593$	459.030	0.0317877	0.0158939	+	0.999874i	$0.494941\pi$
$594$	0	0
$595$	0	0
$596$	5435.22	0.373549
$597$	0	0
$598$	605.906	0.0414337
$599$	−869.305	−0.0592969	−0.0296485	−	0.999560i	$-0.509439\pi$
$599$	−869.305	−0.0592969	−0.0296485	+	0.999560i	$0.509439\pi$
$600$	0	0
$601$	−5455.29	−0.370259	−0.185130	−	0.982714i	$-0.559271\pi$
$601$	−5455.29	−0.370259	−0.185130	+	0.982714i	$0.559271\pi$
$602$	−312.043	−0.0211261
$603$	0	0
$604$	9583.03	0.645576
$605$	0	0
$606$	0	0
$607$	−27152.9	−1.81565	−0.907826	−	0.419347i	$-0.862259\pi$
$607$	−27152.9	−1.81565	−0.907826	+	0.419347i	$0.862259\pi$
$608$	−2515.15	−0.167768
$609$	0	0
$610$	0	0
$611$	−9079.43	−0.601169
$612$	0	0
$613$	10313.6	0.679546	0.339773	−	0.940507i	$-0.389650\pi$
$613$	10313.6	0.679546	0.339773	+	0.940507i	$0.389650\pi$
$614$	−766.521	−0.0503816
$615$	0	0
$616$	1585.36	0.103695
$617$	−27148.4	−1.77140	−0.885699	−	0.464259i	$-0.846321\pi$
$617$	−27148.4	−1.77140	−0.885699	+	0.464259i	$0.846321\pi$
$618$	0	0
$619$	13110.4	0.851294	0.425647	−	0.904889i	$-0.360047\pi$
$619$	13110.4	0.851294	0.425647	+	0.904889i	$0.360047\pi$
$620$	0	0
$621$	0	0
$622$	1782.22	0.114888
$623$	−16977.9	−1.09182
$624$	0	0
$625$	0	0
$626$	2122.72	0.135529
$627$	0	0
$628$	−26336.1	−1.67345
$629$	−2693.12	−0.170718
$630$	0	0
$631$	30199.9	1.90529	0.952647	−	0.304078i	$-0.0983482\pi$
$631$	30199.9	1.90529	0.952647	+	0.304078i	$0.0983482\pi$
$632$	418.088	0.0263143
$633$	0	0
$634$	1585.31	0.0993072
$635$	0	0
$636$	0	0
$637$	25924.0	1.61247
$638$	−199.261	−0.0123649
$639$	0	0
$640$	0	0
$641$	8889.43	0.547756	0.273878	−	0.961764i	$-0.411694\pi$
$641$	8889.43	0.547756	0.273878	+	0.961764i	$0.411694\pi$
$642$	0	0
$643$	16570.2	1.01627	0.508137	−	0.861276i	$-0.330334\pi$
$643$	16570.2	1.01627	0.508137	+	0.861276i	$0.330334\pi$
$644$	−17678.8	−1.08175
$645$	0	0
$646$	943.608	0.0574702
$647$	−13356.3	−0.811576	−0.405788	−	0.913967i	$-0.633003\pi$
$647$	−13356.3	−0.811576	−0.405788	+	0.913967i	$0.633003\pi$
$648$	0	0
$649$	9646.69	0.583460
$650$	0	0
$651$	0	0
$652$	19406.9	1.16569
$653$	32689.5	1.95902	0.979510	−	0.201394i	$-0.0645473\pi$
$653$	32689.5	1.95902	0.979510	+	0.201394i	$0.0645473\pi$
$654$	0	0
$655$	0	0
$656$	21414.6	1.27454
$657$	0	0
$658$	−2451.31	−0.145231
$659$	−16123.3	−0.953071	−0.476536	−	0.879155i	$-0.658108\pi$
$659$	−16123.3	−0.953071	−0.476536	+	0.879155i	$0.658108\pi$
$660$	0	0
$661$	22340.9	1.31461	0.657306	−	0.753624i	$-0.271696\pi$
$661$	22340.9	1.31461	0.657306	+	0.753624i	$0.271696\pi$
$662$	1448.27	0.0850283
$663$	0	0
$664$	6020.08	0.351844
$665$	0	0
$666$	0	0
$667$	4464.60	0.259175
$668$	12038.3	0.697268
$669$	0	0
$670$	0	0
$671$	−8618.75	−0.495862
$672$	0	0
$673$	3415.65	0.195637	0.0978184	−	0.995204i	$-0.468814\pi$
$673$	3415.65	0.195637	0.0978184	+	0.995204i	$0.468814\pi$
$674$	−748.562	−0.0427797
$675$	0	0
$676$	8509.94	0.484180
$677$	32841.9	1.86442	0.932212	−	0.361912i	$-0.117876\pi$
$677$	32841.9	1.86442	0.932212	+	0.361912i	$0.117876\pi$
$678$	0	0
$679$	21520.9	1.21634
$680$	0	0
$681$	0	0
$682$	−434.228	−0.0243804
$683$	−7227.43	−0.404905	−0.202452	−	0.979292i	$-0.564891\pi$
$683$	−7227.43	−0.404905	−0.202452	+	0.979292i	$0.564891\pi$
$684$	0	0
$685$	0	0
$686$	3895.21	0.216793
$687$	0	0
$688$	−2146.39	−0.118940
$689$	22325.5	1.23445
$690$	0	0
$691$	15397.5	0.847682	0.423841	−	0.905737i	$-0.360681\pi$
$691$	15397.5	0.847682	0.423841	+	0.905737i	$0.360681\pi$
$692$	−20901.0	−1.14817
$693$	0	0
$694$	1364.60	0.0746392
$695$	0	0
$696$	0	0
$697$	−24479.6	−1.33032
$698$	−2012.10	−0.109111
$699$	0	0
$700$	0	0
$701$	22130.1	1.19236	0.596180	−	0.802851i	$-0.296685\pi$
$701$	22130.1	1.19236	0.596180	+	0.802851i	$0.296685\pi$
$702$	0	0
$703$	−1853.35	−0.0994315
$704$	5324.55	0.285052
$705$	0	0
$706$	−853.601	−0.0455038
$707$	−49626.7	−2.63989
$708$	0	0
$709$	10062.6	0.533018	0.266509	−	0.963832i	$-0.414130\pi$
$709$	10062.6	0.533018	0.266509	+	0.963832i	$0.414130\pi$
$710$	0	0
$711$	0	0
$712$	2191.69	0.115361
$713$	9729.25	0.511029
$714$	0	0
$715$	0	0
$716$	−14056.4	−0.733677
$717$	0	0
$718$	−244.814	−0.0127248
$719$	18629.7	0.966303	0.483151	−	0.875537i	$-0.339492\pi$
$719$	18629.7	0.966303	0.483151	+	0.875537i	$0.339492\pi$
$720$	0	0
$721$	−20246.9	−1.04581
$722$	−1208.23	−0.0622791
$723$	0	0
$724$	−4199.15	−0.215553
$725$	0	0
$726$	0	0
$727$	21605.8	1.10222	0.551112	−	0.834431i	$-0.314204\pi$
$727$	21605.8	1.10222	0.551112	+	0.834431i	$0.314204\pi$
$728$	−4830.64	−0.245928
$729$	0	0
$730$	0	0
$731$	2453.60	0.124145
$732$	0	0
$733$	−23668.1	−1.19264	−0.596318	−	0.802749i	$-0.703370\pi$
$733$	−23668.1	−1.19264	−0.596318	+	0.802749i	$0.703370\pi$
$734$	−1077.43	−0.0541808
$735$	0	0
$736$	3428.52	0.171708
$737$	9643.96	0.482008
$738$	0	0
$739$	−8475.82	−0.421906	−0.210953	−	0.977496i	$-0.567657\pi$
$739$	−8475.82	−0.421906	−0.210953	+	0.977496i	$0.567657\pi$
$740$	0	0
$741$	0	0
$742$	6027.57	0.298220
$743$	14491.2	0.715520	0.357760	−	0.933813i	$-0.383541\pi$
$743$	14491.2	0.715520	0.357760	+	0.933813i	$0.383541\pi$
$744$	0	0
$745$	0	0
$746$	1487.83	0.0730206
$747$	0	0
$748$	−6204.15	−0.303270
$749$	−11791.8	−0.575252
$750$	0	0
$751$	11780.5	0.572408	0.286204	−	0.958169i	$-0.407607\pi$
$751$	11780.5	0.572408	0.286204	+	0.958169i	$0.407607\pi$
$752$	−16861.4	−0.817648
$753$	0	0
$754$	607.154	0.0293252
$755$	0	0
$756$	0	0
$757$	−22616.7	−1.08589	−0.542945	−	0.839768i	$-0.682691\pi$
$757$	−22616.7	−1.08589	−0.542945	+	0.839768i	$0.682691\pi$
$758$	822.210	0.0393984
$759$	0	0
$760$	0	0
$761$	−2037.48	−0.0970547	−0.0485273	−	0.998822i	$-0.515453\pi$
$761$	−2037.48	−0.0970547	−0.0485273	+	0.998822i	$0.515453\pi$
$762$	0	0
$763$	47222.7	2.24060
$764$	16344.2	0.773968
$765$	0	0
$766$	−963.268	−0.0454364
$767$	−29393.8	−1.38377
$768$	0	0
$769$	16192.1	0.759298	0.379649	−	0.925131i	$-0.376045\pi$
$769$	16192.1	0.759298	0.379649	+	0.925131i	$0.376045\pi$
$770$	0	0
$771$	0	0
$772$	−12977.8	−0.605029
$773$	39726.3	1.84845	0.924227	−	0.381844i	$-0.124711\pi$
$773$	39726.3	1.84845	0.924227	+	0.381844i	$0.124711\pi$
$774$	0	0
$775$	0	0
$776$	−2778.16	−0.128518
$777$	0	0
$778$	1411.43	0.0650413
$779$	−16846.3	−0.774817
$780$	0	0
$781$	−5757.71	−0.263799
$782$	−1286.28	−0.0588200
$783$	0	0
$784$	48143.3	2.19312
$785$	0	0
$786$	0	0
$787$	5160.53	0.233739	0.116870	−	0.993147i	$-0.462714\pi$
$787$	5160.53	0.233739	0.116870	+	0.993147i	$0.462714\pi$
$788$	−27068.4	−1.22370
$789$	0	0
$790$	0	0
$791$	3466.73	0.155832
$792$	0	0
$793$	26261.6	1.17601
$794$	1773.17	0.0792539
$795$	0	0
$796$	−31222.5	−1.39027
$797$	9247.76	0.411007	0.205504	−	0.978656i	$-0.434117\pi$
$797$	9247.76	0.411007	0.205504	+	0.978656i	$0.434117\pi$
$798$	0	0
$799$	19274.7	0.853430
$800$	0	0
$801$	0	0
$802$	450.002	0.0198131
$803$	1002.07	0.0440377
$804$	0	0
$805$	0	0
$806$	1323.11	0.0578220
$807$	0	0
$808$	6406.36	0.278930
$809$	−29359.7	−1.27594	−0.637968	−	0.770063i	$-0.720225\pi$
$809$	−29359.7	−1.27594	−0.637968	+	0.770063i	$0.720225\pi$
$810$	0	0
$811$	13935.0	0.603358	0.301679	−	0.953410i	$-0.402453\pi$
$811$	13935.0	0.603358	0.301679	+	0.953410i	$0.402453\pi$
$812$	−17715.2	−0.765620
$813$	0	0
$814$	−112.756	−0.00485515
$815$	0	0
$816$	0	0
$817$	1688.52	0.0723057
$818$	1971.33	0.0842617
$819$	0	0
$820$	0	0
$821$	−9965.28	−0.423618	−0.211809	−	0.977311i	$-0.567936\pi$
$821$	−9965.28	−0.423618	−0.211809	+	0.977311i	$0.567936\pi$
$822$	0	0
$823$	2339.58	0.0990920	0.0495460	−	0.998772i	$-0.484223\pi$
$823$	2339.58	0.0990920	0.0495460	+	0.998772i	$0.484223\pi$
$824$	2613.69	0.110500
$825$	0	0
$826$	−7935.89	−0.334292
$827$	−15883.4	−0.667861	−0.333931	−	0.942598i	$-0.608375\pi$
$827$	−15883.4	−0.667861	−0.333931	+	0.942598i	$0.608375\pi$
$828$	0	0
$829$	4388.04	0.183839	0.0919196	−	0.995766i	$-0.470700\pi$
$829$	4388.04	0.183839	0.0919196	+	0.995766i	$0.470700\pi$
$830$	0	0
$831$	0	0
$832$	−16224.1	−0.676044
$833$	−55034.0	−2.28909
$834$	0	0
$835$	0	0
$836$	−4269.57	−0.176634
$837$	0	0
$838$	1595.75	0.0657807
$839$	−33830.2	−1.39207	−0.696036	−	0.718007i	$-0.745055\pi$
$839$	−33830.2	−1.39207	−0.696036	+	0.718007i	$0.745055\pi$
$840$	0	0
$841$	−19915.2	−0.816565
$842$	3002.45	0.122888
$843$	0	0
$844$	−28085.2	−1.14542
$845$	0	0
$846$	0	0
$847$	4043.01	0.164013
$848$	41460.7	1.67897
$849$	0	0
$850$	0	0
$851$	2526.39	0.101767
$852$	0	0
$853$	−26047.8	−1.04556	−0.522778	−	0.852469i	$-0.675104\pi$
$853$	−26047.8	−1.04556	−0.522778	+	0.852469i	$0.675104\pi$
$854$	7090.25	0.284102
$855$	0	0
$856$	1522.22	0.0607808
$857$	17210.7	0.686006	0.343003	−	0.939334i	$-0.388556\pi$
$857$	17210.7	0.686006	0.343003	+	0.939334i	$0.388556\pi$
$858$	0	0
$859$	27367.9	1.08706	0.543528	−	0.839391i	$-0.317088\pi$
$859$	27367.9	1.08706	0.543528	+	0.839391i	$0.317088\pi$
$860$	0	0
$861$	0	0
$862$	−1532.35	−0.0605476
$863$	34684.4	1.36810	0.684049	−	0.729436i	$-0.260217\pi$
$863$	34684.4	1.36810	0.684049	+	0.729436i	$0.260217\pi$
$864$	0	0
$865$	0	0
$866$	−830.092	−0.0325724
$867$	0	0
$868$	−38605.0	−1.50961
$869$	1066.22	0.0416213
$870$	0	0
$871$	−29385.5	−1.14316
$872$	−6096.03	−0.236740
$873$	0	0
$874$	−885.190	−0.0342586
$875$	0	0
$876$	0	0
$877$	−45816.6	−1.76410	−0.882051	−	0.471153i	$-0.843838\pi$
$877$	−45816.6	−1.76410	−0.882051	+	0.471153i	$0.843838\pi$
$878$	−817.217	−0.0314120
$879$	0	0
$880$	0	0
$881$	26404.8	1.00976	0.504881	−	0.863189i	$-0.331536\pi$
$881$	26404.8	1.00976	0.504881	+	0.863189i	$0.331536\pi$
$882$	0	0
$883$	−33977.1	−1.29493	−0.647463	−	0.762097i	$-0.724170\pi$
$883$	−33977.1	−1.29493	−0.647463	+	0.762097i	$0.724170\pi$
$884$	18904.2	0.719252
$885$	0	0
$886$	4630.66	0.175587
$887$	−3023.60	−0.114456	−0.0572280	−	0.998361i	$-0.518226\pi$
$887$	−3023.60	−0.114456	−0.0572280	+	0.998361i	$0.518226\pi$
$888$	0	0
$889$	52681.2	1.98748
$890$	0	0
$891$	0	0
$892$	−6745.03	−0.253184
$893$	13264.5	0.497064
$894$	0	0
$895$	0	0
$896$	−18110.3	−0.675249
$897$	0	0
$898$	2931.27	0.108929
$899$	9749.28	0.361687
$900$	0	0
$901$	−47394.9	−1.75244
$902$	−1024.92	−0.0378336
$903$	0	0
$904$	−447.524	−0.0164651
$905$	0	0
$906$	0	0
$907$	−23951.0	−0.876825	−0.438412	−	0.898774i	$-0.644459\pi$
$907$	−23951.0	−0.876825	−0.438412	+	0.898774i	$0.644459\pi$
$908$	−39954.8	−1.46029
$909$	0	0
$910$	0	0
$911$	−24198.9	−0.880071	−0.440035	−	0.897980i	$-0.645034\pi$
$911$	−24198.9	−0.880071	−0.440035	+	0.897980i	$0.645034\pi$
$912$	0	0
$913$	15352.5	0.556510
$914$	1857.60	0.0672253
$915$	0	0
$916$	28728.9	1.03628
$917$	36258.8	1.30575
$918$	0	0
$919$	11056.9	0.396880	0.198440	−	0.980113i	$-0.436413\pi$
$919$	11056.9	0.396880	0.198440	+	0.980113i	$0.436413\pi$
$920$	0	0
$921$	0	0
$922$	−2613.94	−0.0933682
$923$	17543.9	0.625639
$924$	0	0
$925$	0	0
$926$	1284.93	0.0455999
$927$	0	0
$928$	3435.58	0.121529
$929$	48242.5	1.70375	0.851876	−	0.523744i	$-0.175465\pi$
$929$	48242.5	1.70375	0.851876	+	0.523744i	$0.175465\pi$
$930$	0	0
$931$	−37873.2	−1.33324
$932$	23657.6	0.831470
$933$	0	0
$934$	−2989.42	−0.104729
$935$	0	0
$936$	0	0
$937$	48721.9	1.69869	0.849346	−	0.527836i	$-0.176996\pi$
$937$	48721.9	1.69869	0.849346	+	0.527836i	$0.176996\pi$
$938$	−7933.65	−0.276165
$939$	0	0
$940$	0	0
$941$	4978.66	0.172476	0.0862379	−	0.996275i	$-0.472515\pi$
$941$	4978.66	0.172476	0.0862379	+	0.996275i	$0.472515\pi$
$942$	0	0
$943$	22964.1	0.793015
$944$	−54587.1	−1.88205
$945$	0	0
$946$	102.728	0.00353062
$947$	27711.4	0.950896	0.475448	−	0.879744i	$-0.342286\pi$
$947$	27711.4	0.950896	0.475448	+	0.879744i	$0.342286\pi$
$948$	0	0
$949$	−3053.34	−0.104442
$950$	0	0
$951$	0	0
$952$	10255.0	0.349123
$953$	−49175.6	−1.67152	−0.835758	−	0.549098i	$-0.814971\pi$
$953$	−49175.6	−1.67152	−0.835758	+	0.549098i	$0.814971\pi$
$954$	0	0
$955$	0	0
$956$	10289.7	0.348109
$957$	0	0
$958$	−4733.21	−0.159627
$959$	46426.6	1.56329
$960$	0	0
$961$	−8545.37	−0.286844
$962$	343.571	0.0115147
$963$	0	0
$964$	−31537.9	−1.05370
$965$	0	0
$966$	0	0
$967$	52678.5	1.75184	0.875918	−	0.482461i	$-0.160257\pi$
$967$	52678.5	1.75184	0.875918	+	0.482461i	$0.160257\pi$
$968$	−521.916	−0.0173296
$969$	0	0
$970$	0	0
$971$	−46877.5	−1.54930	−0.774651	−	0.632390i	$-0.782074\pi$
$971$	−46877.5	−1.54930	−0.774651	+	0.632390i	$0.782074\pi$
$972$	0	0
$973$	16708.8	0.550525
$974$	−4608.00	−0.151591
$975$	0	0
$976$	48770.4	1.59949
$977$	1335.33	0.0437267	0.0218633	−	0.999761i	$-0.493040\pi$
$977$	1335.33	0.0437267	0.0218633	+	0.999761i	$0.493040\pi$
$978$	0	0
$979$	5589.29	0.182466
$980$	0	0
$981$	0	0
$982$	−5328.19	−0.173146
$983$	15102.0	0.490011	0.245005	−	0.969522i	$-0.421210\pi$
$983$	15102.0	0.490011	0.245005	+	0.969522i	$0.421210\pi$
$984$	0	0
$985$	0	0
$986$	−1288.93	−0.0416306
$987$	0	0
$988$	13009.5	0.418915
$989$	−2301.70	−0.0740039
$990$	0	0
$991$	−45849.8	−1.46969	−0.734847	−	0.678233i	$-0.762746\pi$
$991$	−45849.8	−1.46969	−0.734847	+	0.678233i	$0.762746\pi$
$992$	7486.82	0.239624
$993$	0	0
$994$	4736.60	0.151143
$995$	0	0
$996$	0	0
$997$	−49662.4	−1.57756	−0.788779	−	0.614678i	$-0.789286\pi$
$997$	−49662.4	−1.57756	−0.788779	+	0.614678i	$0.789286\pi$
$998$	−4266.77	−0.135333
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bm.1.3		5
3.2	odd	2		825.4.a.w.1.3	✓	5
5.4	even	2		2475.4.a.bf.1.3		5
15.2	even	4		825.4.c.r.199.5		10
15.8	even	4		825.4.c.r.199.6		10
15.14	odd	2		825.4.a.z.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.w.1.3	✓	5	3.2	odd	2
825.4.a.z.1.3	yes	5	15.14	odd	2
825.4.c.r.199.5		10	15.2	even	4
825.4.c.r.199.6		10	15.8	even	4
2475.4.a.bf.1.3		5	5.4	even	2
2475.4.a.bm.1.3		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+0.270826 q^{2} -7.92665 q^{4} +33.4133 q^{7} -4.31336 q^{8} -11.0000 q^{11} +33.5174 q^{13} +9.04920 q^{14} +62.2451 q^{16} -71.1540 q^{17} -48.9667 q^{19} -2.97909 q^{22} +66.7490 q^{23} +9.07738 q^{26} -264.856 q^{28} +66.8864 q^{29} +145.759 q^{31} +51.3644 q^{32} -19.2704 q^{34} +37.8491 q^{37} -13.2615 q^{38} +344.036 q^{41} -34.4830 q^{43} +87.1932 q^{44} +18.0774 q^{46} -270.887 q^{47} +773.448 q^{49} -265.681 q^{52} +666.088 q^{53} -144.123 q^{56} +18.1146 q^{58} -876.971 q^{59} +783.523 q^{61} +39.4753 q^{62} -484.050 q^{64} -876.724 q^{67} +564.013 q^{68} +523.428 q^{71} -91.0973 q^{73} +10.2505 q^{74} +388.142 q^{76} -367.546 q^{77} -96.9287 q^{79} +93.1741 q^{82} -1395.68 q^{83} -9.33889 q^{86} +47.4469 q^{88} -508.117 q^{89} +1119.93 q^{91} -529.096 q^{92} -73.3634 q^{94} +644.082 q^{97} +209.470 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 46 q^{4} + 38 q^{7} + 72 q^{8} - 55 q^{11} - 43 q^{13} - 55 q^{14} + 634 q^{16} + 150 q^{17} - 337 q^{19} - 44 q^{22} - 201 q^{23} - 359 q^{26} + 441 q^{28} + 153 q^{29} + 263 q^{31} + 280 q^{32}+ \cdots + 9417 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 46 * q^4 + 38 * q^7 + 72 * q^8 - 55 * q^11 - 43 * q^13 - 55 * q^14 + 634 * q^16 + 150 * q^17 - 337 * q^19 - 44 * q^22 - 201 * q^23 - 359 * q^26 + 441 * q^28 + 153 * q^29 + 263 * q^31 + 280 * q^32 + 517 * q^34 + 350 * q^37 + 786 * q^38 + 88 * q^41 - 161 * q^43 - 506 * q^44 + 436 * q^46 + 490 * q^47 + 2381 * q^49 - 289 * q^52 + 558 * q^53 - 211 * q^56 + 969 * q^58 - 410 * q^59 - 1130 * q^61 + 463 * q^62 + 3170 * q^64 - 1130 * q^67 + 3333 * q^68 - 1249 * q^71 + 44 * q^73 + 2985 * q^74 - 2556 * q^76 - 418 * q^77 + 2258 * q^79 + 3737 * q^82 - 957 * q^83 + 1435 * q^86 - 792 * q^88 - 347 * q^89 - 612 * q^91 - 6006 * q^92 + 907 * q^94 + 2355 * q^97 + 9417 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more