LMFDB - Embedded newform 2475.4.a.ca.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$146.029727264$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 103 x^{14} + 4248 x^{12} - 89496 x^{10} + 1015487 x^{8} - 5956953 x^{6} + 15313728 x^{4} + \cdots + 861184 $ x^16 - 103x^14 + 4248x^12 - 89496x^10 + 1015487x^8 - 5956953x^6 + 15313728x^4 - 10273600*x^2 + 861184
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.312176$ 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)

$f(q)$	$=$	$q-0.312176 q^{2} -7.90255 q^{4} +15.1298 q^{7} +4.96439 q^{8} +O(q^{10})$	$q-0.312176 q^{2} -7.90255 q^{4} +15.1298 q^{7} +4.96439 q^{8} +11.0000 q^{11} +80.5049 q^{13} -4.72315 q^{14} +61.6706 q^{16} +84.3893 q^{17} -65.0714 q^{19} -3.43393 q^{22} +150.712 q^{23} -25.1317 q^{26} -119.564 q^{28} +67.6503 q^{29} +149.447 q^{31} -58.9671 q^{32} -26.3443 q^{34} +328.806 q^{37} +20.3137 q^{38} -326.946 q^{41} +108.813 q^{43} -86.9280 q^{44} -47.0484 q^{46} +573.831 q^{47} -114.090 q^{49} -636.194 q^{52} +412.498 q^{53} +75.1100 q^{56} -21.1188 q^{58} +143.336 q^{59} +243.110 q^{61} -46.6536 q^{62} -474.957 q^{64} -423.697 q^{67} -666.891 q^{68} -396.989 q^{71} -363.616 q^{73} -102.645 q^{74} +514.230 q^{76} +166.428 q^{77} -228.250 q^{79} +102.065 q^{82} -825.486 q^{83} -33.9688 q^{86} +54.6082 q^{88} -1066.26 q^{89} +1218.02 q^{91} -1191.00 q^{92} -179.136 q^{94} +576.156 q^{97} +35.6161 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 78 q^{4}+O(q^{10}) $ 16 * q + 78 * q^4	$ 16 q + 78 q^{4} + 176 q^{11} + 124 q^{14} + 306 q^{16} + 84 q^{19} + 532 q^{26} + 652 q^{29} + 80 q^{31} + 360 q^{34} + 1204 q^{41} + 858 q^{44} - 672 q^{46} + 1016 q^{49} + 3332 q^{56} + 712 q^{59} + 880 q^{61} - 962 q^{64} + 1968 q^{71} + 4152 q^{74} - 1048 q^{76} - 1636 q^{79} + 7284 q^{86} + 2776 q^{89} + 144 q^{91} + 1400 q^{94}+O(q^{100}) $ 16 * q + 78 * q^4 + 176 * q^11 + 124 * q^14 + 306 * q^16 + 84 * q^19 + 532 * q^26 + 652 * q^29 + 80 * q^31 + 360 * q^34 + 1204 * q^41 + 858 * q^44 - 672 * q^46 + 1016 * q^49 + 3332 * q^56 + 712 * q^59 + 880 * q^61 - 962 * q^64 + 1968 * q^71 + 4152 * q^74 - 1048 * q^76 - 1636 * q^79 + 7284 * q^86 + 2776 * q^89 + 144 * q^91 + 1400 * q^94

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 $

$2$	−0.312176	−0.110371	−0.0551854	−	0.998476i	$-0.517575\pi$
$2$	−0.312176	−0.110371	−0.0551854	+	0.998476i	$0.517575\pi$
$3$	0	0
$3$	0	0
$4$	−7.90255	−0.987818
$5$	0	0
$5$	0	0
$6$	0	0
$7$	15.1298	0.816931	0.408466	−	0.912774i	$-0.366064\pi$
$7$	15.1298	0.816931	0.408466	+	0.912774i	$0.366064\pi$
$8$	4.96439	0.219397
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	80.5049	1.71754	0.858771	−	0.512360i	$-0.171229\pi$
$13$	80.5049	1.71754	0.858771	+	0.512360i	$0.171229\pi$
$14$	−4.72315	−0.0901653
$15$	0	0
$16$	61.6706	0.963603
$17$	84.3893	1.20397	0.601983	−	0.798509i	$-0.294377\pi$
$17$	84.3893	1.20397	0.601983	+	0.798509i	$0.294377\pi$
$18$	0	0
$19$	−65.0714	−0.785706	−0.392853	−	0.919601i	$-0.628512\pi$
$19$	−65.0714	−0.785706	−0.392853	+	0.919601i	$0.628512\pi$
$20$	0	0
$21$	0	0
$22$	−3.43393	−0.0332780
$23$	150.712	1.36633	0.683164	−	0.730265i	$-0.260604\pi$
$23$	150.712	1.36633	0.683164	+	0.730265i	$0.260604\pi$
$24$	0	0
$25$	0	0
$26$	−25.1317	−0.189566
$27$	0	0
$28$	−119.564	−0.806979
$29$	67.6503	0.433184	0.216592	−	0.976262i	$-0.430506\pi$
$29$	67.6503	0.433184	0.216592	+	0.976262i	$0.430506\pi$
$30$	0	0
$31$	149.447	0.865852	0.432926	−	0.901430i	$-0.357481\pi$
$31$	149.447	0.865852	0.432926	+	0.901430i	$0.357481\pi$
$32$	−58.9671	−0.325751
$33$	0	0
$34$	−26.3443	−0.132883
$35$	0	0
$36$	0	0
$37$	328.806	1.46096	0.730478	−	0.682936i	$-0.239297\pi$
$37$	328.806	1.46096	0.730478	+	0.682936i	$0.239297\pi$
$38$	20.3137	0.0867189
$39$	0	0
$40$	0	0
$41$	−326.946	−1.24538	−0.622688	−	0.782470i	$-0.713959\pi$
$41$	−326.946	−1.24538	−0.622688	+	0.782470i	$0.713959\pi$
$42$	0	0
$43$	108.813	0.385903	0.192952	−	0.981208i	$-0.438194\pi$
$43$	108.813	0.385903	0.192952	+	0.981208i	$0.438194\pi$
$44$	−86.9280	−0.297838
$45$	0	0
$46$	−47.0484	−0.150803
$47$	573.831	1.78089	0.890444	−	0.455092i	$-0.150394\pi$
$47$	573.831	1.78089	0.890444	+	0.455092i	$0.150394\pi$
$48$	0	0
$49$	−114.090	−0.332624
$50$	0	0
$51$	0	0
$52$	−636.194	−1.69662
$53$	412.498	1.06907	0.534537	−	0.845145i	$-0.320486\pi$
$53$	412.498	1.06907	0.534537	+	0.845145i	$0.320486\pi$
$54$	0	0
$55$	0	0
$56$	75.1100	0.179232
$57$	0	0
$58$	−21.1188	−0.0478109
$59$	143.336	0.316283	0.158142	−	0.987416i	$-0.449450\pi$
$59$	143.336	0.316283	0.158142	+	0.987416i	$0.449450\pi$
$60$	0	0
$61$	243.110	0.510279	0.255140	−	0.966904i	$-0.417879\pi$
$61$	243.110	0.510279	0.255140	+	0.966904i	$0.417879\pi$
$62$	−46.6536	−0.0955647
$63$	0	0
$64$	−474.957	−0.927650
$65$	0	0
$66$	0	0
$67$	−423.697	−0.772579	−0.386290	−	0.922378i	$-0.626243\pi$
$67$	−423.697	−0.772579	−0.386290	+	0.922378i	$0.626243\pi$
$68$	−666.891	−1.18930
$69$	0	0
$70$	0	0
$71$	−396.989	−0.663576	−0.331788	−	0.943354i	$-0.607652\pi$
$71$	−396.989	−0.663576	−0.331788	+	0.943354i	$0.607652\pi$
$72$	0	0
$73$	−363.616	−0.582987	−0.291493	−	0.956573i	$-0.594152\pi$
$73$	−363.616	−0.582987	−0.291493	+	0.956573i	$0.594152\pi$
$74$	−102.645	−0.161247
$75$	0	0
$76$	514.230	0.776135
$77$	166.428	0.246314
$78$	0	0
$79$	−228.250	−0.325065	−0.162532	−	0.986703i	$-0.551966\pi$
$79$	−228.250	−0.325065	−0.162532	+	0.986703i	$0.551966\pi$
$80$	0	0
$81$	0	0
$82$	102.065	0.137453
$83$	−825.486	−1.09167	−0.545837	−	0.837891i	$-0.683788\pi$
$83$	−825.486	−1.09167	−0.545837	+	0.837891i	$0.683788\pi$
$84$	0	0
$85$	0	0
$86$	−33.9688	−0.0425924
$87$	0	0
$88$	54.6082	0.0661507
$89$	−1066.26	−1.26993	−0.634963	−	0.772542i	$-0.718985\pi$
$89$	−1066.26	−1.26993	−0.634963	+	0.772542i	$0.718985\pi$
$90$	0	0
$91$	1218.02	1.40311
$92$	−1191.00	−1.34968
$93$	0	0
$94$	−179.136	−0.196558
$95$	0	0
$96$	0	0
$97$	576.156	0.603091	0.301545	−	0.953452i	$-0.402498\pi$
$97$	576.156	0.603091	0.301545	+	0.953452i	$0.402498\pi$
$98$	35.6161	0.0367119
$99$	0	0
$100$	0	0
$101$	−393.344	−0.387517	−0.193758	−	0.981049i	$-0.562068\pi$
$101$	−393.344	−0.387517	−0.193758	+	0.981049i	$0.562068\pi$
$102$	0	0
$103$	1389.87	1.32959	0.664797	−	0.747024i	$-0.268518\pi$
$103$	1389.87	1.32959	0.664797	+	0.747024i	$0.268518\pi$
$104$	399.657	0.376823
$105$	0	0
$106$	−128.772	−0.117995
$107$	−573.001	−0.517702	−0.258851	−	0.965917i	$-0.583344\pi$
$107$	−573.001	−0.517702	−0.258851	+	0.965917i	$0.583344\pi$
$108$	0	0
$109$	−1824.79	−1.60352	−0.801760	−	0.597647i	$-0.796103\pi$
$109$	−1824.79	−1.60352	−0.801760	+	0.597647i	$0.796103\pi$
$110$	0	0
$111$	0	0
$112$	933.062	0.787197
$113$	3.92407	0.00326677	0.00163339	−	0.999999i	$-0.499480\pi$
$113$	3.92407	0.00326677	0.00163339	+	0.999999i	$0.499480\pi$
$114$	0	0
$115$	0	0
$116$	−534.609	−0.427907
$117$	0	0
$118$	−44.7459	−0.0349084
$119$	1276.79	0.983557
$120$	0	0
$121$	121.000	0.0909091
$122$	−75.8930	−0.0563199
$123$	0	0
$124$	−1181.01	−0.855304
$125$	0	0
$126$	0	0
$127$	−359.679	−0.251310	−0.125655	−	0.992074i	$-0.540103\pi$
$127$	−359.679	−0.251310	−0.125655	+	0.992074i	$0.540103\pi$
$128$	620.007	0.428136
$129$	0	0
$130$	0	0
$131$	−1624.89	−1.08372	−0.541861	−	0.840468i	$-0.682280\pi$
$131$	−1624.89	−1.08372	−0.541861	+	0.840468i	$0.682280\pi$
$132$	0	0
$133$	−984.516	−0.641868
$134$	132.268	0.0852701
$135$	0	0
$136$	418.941	0.264146
$137$	2606.13	1.62523	0.812616	−	0.582799i	$-0.198042\pi$
$137$	2606.13	1.62523	0.812616	+	0.582799i	$0.198042\pi$
$138$	0	0
$139$	1904.14	1.16192	0.580959	−	0.813933i	$-0.302678\pi$
$139$	1904.14	1.16192	0.580959	+	0.813933i	$0.302678\pi$
$140$	0	0
$141$	0	0
$142$	123.930	0.0732393
$143$	885.554	0.517858
$144$	0	0
$145$	0	0
$146$	113.512	0.0643447
$147$	0	0
$148$	−2598.40	−1.44316
$149$	2424.03	1.33278	0.666390	−	0.745603i	$-0.267839\pi$
$149$	2424.03	1.33278	0.666390	+	0.745603i	$0.267839\pi$
$150$	0	0
$151$	−2331.84	−1.25670	−0.628351	−	0.777930i	$-0.716270\pi$
$151$	−2331.84	−1.25670	−0.628351	+	0.777930i	$0.716270\pi$
$152$	−323.040	−0.172381
$153$	0	0
$154$	−51.9546	−0.0271859
$155$	0	0
$156$	0	0
$157$	−2345.68	−1.19239	−0.596196	−	0.802839i	$-0.703322\pi$
$157$	−2345.68	−1.19239	−0.596196	+	0.802839i	$0.703322\pi$
$158$	71.2540	0.0358776
$159$	0	0
$160$	0	0
$161$	2280.23	1.11620
$162$	0	0
$163$	1794.19	0.862160	0.431080	−	0.902314i	$-0.358133\pi$
$163$	1794.19	0.862160	0.431080	+	0.902314i	$0.358133\pi$
$164$	2583.71	1.23021
$165$	0	0
$166$	257.697	0.120489
$167$	−3249.22	−1.50558	−0.752792	−	0.658259i	$-0.771293\pi$
$167$	−3249.22	−1.50558	−0.752792	+	0.658259i	$0.771293\pi$
$168$	0	0
$169$	4284.04	1.94995
$170$	0	0
$171$	0	0
$172$	−859.901	−0.381202
$173$	3012.62	1.32396	0.661980	−	0.749521i	$-0.269716\pi$
$173$	3012.62	1.32396	0.661980	+	0.749521i	$0.269716\pi$
$174$	0	0
$175$	0	0
$176$	678.377	0.290537
$177$	0	0
$178$	332.861	0.140163
$179$	729.385	0.304563	0.152282	−	0.988337i	$-0.451338\pi$
$179$	729.385	0.304563	0.152282	+	0.988337i	$0.451338\pi$
$180$	0	0
$181$	−2049.59	−0.841684	−0.420842	−	0.907134i	$-0.638265\pi$
$181$	−2049.59	−0.841684	−0.420842	+	0.907134i	$0.638265\pi$
$182$	−380.236	−0.154863
$183$	0	0
$184$	748.190	0.299768
$185$	0	0
$186$	0	0
$187$	928.283	0.363009
$188$	−4534.72	−1.75919
$189$	0	0
$190$	0	0
$191$	2465.72	0.934100	0.467050	−	0.884231i	$-0.345317\pi$
$191$	2465.72	0.934100	0.467050	+	0.884231i	$0.345317\pi$
$192$	0	0
$193$	−2573.38	−0.959772	−0.479886	−	0.877331i	$-0.659322\pi$
$193$	−2573.38	−0.959772	−0.479886	+	0.877331i	$0.659322\pi$
$194$	−179.862	−0.0665635
$195$	0	0
$196$	901.601	0.328572
$197$	366.221	0.132448	0.0662238	−	0.997805i	$-0.478905\pi$
$197$	366.221	0.132448	0.0662238	+	0.997805i	$0.478905\pi$
$198$	0	0
$199$	2824.12	1.00601	0.503006	−	0.864283i	$-0.332227\pi$
$199$	2824.12	1.00601	0.503006	+	0.864283i	$0.332227\pi$
$200$	0	0
$201$	0	0
$202$	122.792	0.0427705
$203$	1023.53	0.353882
$204$	0	0
$205$	0	0
$206$	−433.884	−0.146748
$207$	0	0
$208$	4964.79	1.65503
$209$	−715.786	−0.236899
$210$	0	0
$211$	−5679.26	−1.85297	−0.926484	−	0.376335i	$-0.877184\pi$
$211$	−5679.26	−1.85297	−0.926484	+	0.376335i	$0.877184\pi$
$212$	−3259.79	−1.05605
$213$	0	0
$214$	178.877	0.0571391
$215$	0	0
$216$	0	0
$217$	2261.09	0.707341
$218$	569.656	0.176982
$219$	0	0
$220$	0	0
$221$	6793.76	2.06786
$222$	0	0
$223$	−4872.88	−1.46328	−0.731642	−	0.681689i	$-0.761246\pi$
$223$	−4872.88	−1.46328	−0.731642	+	0.681689i	$0.761246\pi$
$224$	−892.160	−0.266116
$225$	0	0
$226$	−1.22500	−0.000360556 0
$227$	5788.74	1.69256	0.846282	−	0.532735i	$-0.178836\pi$
$227$	5788.74	1.69256	0.846282	+	0.532735i	$0.178836\pi$
$228$	0	0
$229$	−1040.95	−0.300384	−0.150192	−	0.988657i	$-0.547989\pi$
$229$	−1040.95	−0.300384	−0.150192	+	0.988657i	$0.547989\pi$
$230$	0	0
$231$	0	0
$232$	335.842	0.0950393
$233$	−620.560	−0.174482	−0.0872409	−	0.996187i	$-0.527805\pi$
$233$	−620.560	−0.174482	−0.0872409	+	0.996187i	$0.527805\pi$
$234$	0	0
$235$	0	0
$236$	−1132.72	−0.312430
$237$	0	0
$238$	−398.583	−0.108556
$239$	6482.51	1.75447	0.877236	−	0.480059i	$-0.159385\pi$
$239$	6482.51	1.75447	0.877236	+	0.480059i	$0.159385\pi$
$240$	0	0
$241$	−1395.66	−0.373039	−0.186520	−	0.982451i	$-0.559721\pi$
$241$	−1395.66	−0.373039	−0.186520	+	0.982451i	$0.559721\pi$
$242$	−37.7732	−0.0100337
$243$	0	0
$244$	−1921.19	−0.504063
$245$	0	0
$246$	0	0
$247$	−5238.57	−1.34948
$248$	741.911	0.189965
$249$	0	0
$250$	0	0
$251$	−590.055	−0.148382	−0.0741911	−	0.997244i	$-0.523637\pi$
$251$	−590.055	−0.148382	−0.0741911	+	0.997244i	$0.523637\pi$
$252$	0	0
$253$	1657.83	0.411963
$254$	112.283	0.0277373
$255$	0	0
$256$	3606.10	0.880396
$257$	−502.037	−0.121853	−0.0609264	−	0.998142i	$-0.519406\pi$
$257$	−502.037	−0.121853	−0.0609264	+	0.998142i	$0.519406\pi$
$258$	0	0
$259$	4974.76	1.19350
$260$	0	0
$261$	0	0
$262$	507.252	0.119611
$263$	−3810.66	−0.893443	−0.446722	−	0.894673i	$-0.647409\pi$
$263$	−3810.66	−0.893443	−0.446722	+	0.894673i	$0.647409\pi$
$264$	0	0
$265$	0	0
$266$	307.342	0.0708434
$267$	0	0
$268$	3348.28	0.763168
$269$	5507.02	1.24821	0.624106	−	0.781340i	$-0.285463\pi$
$269$	5507.02	1.24821	0.624106	+	0.781340i	$0.285463\pi$
$270$	0	0
$271$	−1832.70	−0.410806	−0.205403	−	0.978677i	$-0.565851\pi$
$271$	−1832.70	−0.410806	−0.205403	+	0.978677i	$0.565851\pi$
$272$	5204.34	1.16015
$273$	0	0
$274$	−813.571	−0.179378
$275$	0	0
$276$	0	0
$277$	−1663.04	−0.360730	−0.180365	−	0.983600i	$-0.557728\pi$
$277$	−1663.04	−0.360730	−0.180365	+	0.983600i	$0.557728\pi$
$278$	−594.425	−0.128242
$279$	0	0
$280$	0	0
$281$	4445.93	0.943851	0.471926	−	0.881638i	$-0.343559\pi$
$281$	4445.93	0.943851	0.471926	+	0.881638i	$0.343559\pi$
$282$	0	0
$283$	5982.95	1.25671	0.628356	−	0.777926i	$-0.283728\pi$
$283$	5982.95	1.25671	0.628356	+	0.777926i	$0.283728\pi$
$284$	3137.22	0.655492
$285$	0	0
$286$	−276.448	−0.0571564
$287$	−4946.62	−1.01739
$288$	0	0
$289$	2208.56	0.449534
$290$	0	0
$291$	0	0
$292$	2873.49	0.575885
$293$	511.269	0.101941	0.0509704	−	0.998700i	$-0.483769\pi$
$293$	511.269	0.101941	0.0509704	+	0.998700i	$0.483769\pi$
$294$	0	0
$295$	0	0
$296$	1632.32	0.320529
$297$	0	0
$298$	−756.723	−0.147100
$299$	12133.0	2.34672
$300$	0	0
$301$	1646.32	0.315256
$302$	727.942	0.138703
$303$	0	0
$304$	−4013.00	−0.757109
$305$	0	0
$306$	0	0
$307$	3765.47	0.700023	0.350011	−	0.936745i	$-0.386178\pi$
$307$	3765.47	0.700023	0.350011	+	0.936745i	$0.386178\pi$
$308$	−1315.20	−0.243313
$309$	0	0
$310$	0	0
$311$	4908.80	0.895025	0.447513	−	0.894278i	$-0.352310\pi$
$311$	4908.80	0.895025	0.447513	+	0.894278i	$0.352310\pi$
$312$	0	0
$313$	−8505.85	−1.53603	−0.768017	−	0.640429i	$-0.778757\pi$
$313$	−8505.85	−1.53603	−0.768017	+	0.640429i	$0.778757\pi$
$314$	732.263	0.131605
$315$	0	0
$316$	1803.75	0.321105
$317$	−6019.12	−1.06646	−0.533230	−	0.845971i	$-0.679022\pi$
$317$	−6019.12	−1.06646	−0.533230	+	0.845971i	$0.679022\pi$
$318$	0	0
$319$	744.153	0.130610
$320$	0	0
$321$	0	0
$322$	−711.832	−0.123195
$323$	−5491.34	−0.945963
$324$	0	0
$325$	0	0
$326$	−560.103	−0.0951572
$327$	0	0
$328$	−1623.09	−0.273232
$329$	8681.93	1.45486
$330$	0	0
$331$	−4596.46	−0.763276	−0.381638	−	0.924312i	$-0.624640\pi$
$331$	−4596.46	−0.763276	−0.381638	+	0.924312i	$0.624640\pi$
$332$	6523.45	1.07838
$333$	0	0
$334$	1014.33	0.166172
$335$	0	0
$336$	0	0
$337$	7331.49	1.18508	0.592539	−	0.805542i	$-0.298125\pi$
$337$	7331.49	1.18508	0.592539	+	0.805542i	$0.298125\pi$
$338$	−1337.37	−0.215217
$339$	0	0
$340$	0	0
$341$	1643.91	0.261064
$342$	0	0
$343$	−6915.67	−1.08866
$344$	540.190	0.0846660
$345$	0	0
$346$	−940.466	−0.146126
$347$	1820.25	0.281603	0.140801	−	0.990038i	$-0.455032\pi$
$347$	1820.25	0.281603	0.140801	+	0.990038i	$0.455032\pi$
$348$	0	0
$349$	2193.93	0.336500	0.168250	−	0.985744i	$-0.446188\pi$
$349$	2193.93	0.336500	0.168250	+	0.985744i	$0.446188\pi$
$350$	0	0
$351$	0	0
$352$	−648.639	−0.0982175
$353$	12628.5	1.90410	0.952052	−	0.305937i	$-0.0989697\pi$
$353$	12628.5	1.90410	0.952052	+	0.305937i	$0.0989697\pi$
$354$	0	0
$355$	0	0
$356$	8426.18	1.25446
$357$	0	0
$358$	−227.696	−0.0336149
$359$	8716.26	1.28141	0.640706	−	0.767786i	$-0.278642\pi$
$359$	8716.26	1.28141	0.640706	+	0.767786i	$0.278642\pi$
$360$	0	0
$361$	−2624.71	−0.382666
$362$	639.832	0.0928973
$363$	0	0
$364$	−9625.47	−1.38602
$365$	0	0
$366$	0	0
$367$	−3185.29	−0.453054	−0.226527	−	0.974005i	$-0.572737\pi$
$367$	−3185.29	−0.453054	−0.226527	+	0.974005i	$0.572737\pi$
$368$	9294.47	1.31660
$369$	0	0
$370$	0	0
$371$	6241.00	0.873360
$372$	0	0
$373$	11301.9	1.56888	0.784438	−	0.620207i	$-0.212952\pi$
$373$	11301.9	1.56888	0.784438	+	0.620207i	$0.212952\pi$
$374$	−289.787	−0.0400656
$375$	0	0
$376$	2848.72	0.390722
$377$	5446.18	0.744012
$378$	0	0
$379$	82.2637	0.0111493	0.00557467	−	0.999984i	$-0.498226\pi$
$379$	82.2637	0.0111493	0.00557467	+	0.999984i	$0.498226\pi$
$380$	0	0
$381$	0	0
$382$	−769.737	−0.103097
$383$	−2403.91	−0.320716	−0.160358	−	0.987059i	$-0.551265\pi$
$383$	−2403.91	−0.320716	−0.160358	+	0.987059i	$0.551265\pi$
$384$	0	0
$385$	0	0
$386$	803.346	0.105931
$387$	0	0
$388$	−4553.10	−0.595744
$389$	−7208.27	−0.939521	−0.469761	−	0.882794i	$-0.655660\pi$
$389$	−7208.27	−0.939521	−0.469761	+	0.882794i	$0.655660\pi$
$390$	0	0
$391$	12718.4	1.64501
$392$	−566.386	−0.0729766
$393$	0	0
$394$	−114.325	−0.0146183
$395$	0	0
$396$	0	0
$397$	−5897.60	−0.745572	−0.372786	−	0.927917i	$-0.621597\pi$
$397$	−5897.60	−0.745572	−0.372786	+	0.927917i	$0.621597\pi$
$398$	−881.620	−0.111034
$399$	0	0
$400$	0	0
$401$	10059.6	1.25274	0.626372	−	0.779524i	$-0.284539\pi$
$401$	10059.6	1.25274	0.626372	+	0.779524i	$0.284539\pi$
$402$	0	0
$403$	12031.2	1.48714
$404$	3108.42	0.382796
$405$	0	0
$406$	−319.522	−0.0390582
$407$	3616.87	0.440495
$408$	0	0
$409$	−10973.8	−1.32669	−0.663347	−	0.748312i	$-0.730865\pi$
$409$	−10973.8	−1.32669	−0.663347	+	0.748312i	$0.730865\pi$
$410$	0	0
$411$	0	0
$412$	−10983.5	−1.31340
$413$	2168.64	0.258382
$414$	0	0
$415$	0	0
$416$	−4747.14	−0.559490
$417$	0	0
$418$	223.451	0.0261467
$419$	3777.56	0.440443	0.220222	−	0.975450i	$-0.429322\pi$
$419$	3777.56	0.440443	0.220222	+	0.975450i	$0.429322\pi$
$420$	0	0
$421$	−5492.70	−0.635861	−0.317931	−	0.948114i	$-0.602988\pi$
$421$	−5492.70	−0.635861	−0.317931	+	0.948114i	$0.602988\pi$
$422$	1772.92	0.204513
$423$	0	0
$424$	2047.80	0.234552
$425$	0	0
$426$	0	0
$427$	3678.20	0.416863
$428$	4528.17	0.511395
$429$	0	0
$430$	0	0
$431$	−8542.18	−0.954669	−0.477335	−	0.878722i	$-0.658397\pi$
$431$	−8542.18	−0.954669	−0.477335	+	0.878722i	$0.658397\pi$
$432$	0	0
$433$	5254.24	0.583147	0.291573	−	0.956548i	$-0.405821\pi$
$433$	5254.24	0.583147	0.291573	+	0.956548i	$0.405821\pi$
$434$	−705.858	−0.0780697
$435$	0	0
$436$	14420.5	1.58399
$437$	−9807.02	−1.07353
$438$	0	0
$439$	313.904	0.0341271	0.0170636	−	0.999854i	$-0.494568\pi$
$439$	313.904	0.0341271	0.0170636	+	0.999854i	$0.494568\pi$
$440$	0	0
$441$	0	0
$442$	−2120.84	−0.228231
$443$	17.9362	0.00192364	0.000961821	−	1.00000i	$-0.499694\pi$
$443$	17.9362	0.00192364	0.000961821		1.00000i	$0.499694\pi$
$444$	0	0
$445$	0	0
$446$	1521.19	0.161504
$447$	0	0
$448$	−7185.99	−0.757826
$449$	15646.2	1.64452	0.822261	−	0.569110i	$-0.192712\pi$
$449$	15646.2	1.64452	0.822261	+	0.569110i	$0.192712\pi$
$450$	0	0
$451$	−3596.41	−0.375495
$452$	−31.0102	−0.00322698
$453$	0	0
$454$	−1807.10	−0.186810
$455$	0	0
$456$	0	0
$457$	12500.9	1.27958	0.639791	−	0.768549i	$-0.279021\pi$
$457$	12500.9	1.27958	0.639791	+	0.768549i	$0.279021\pi$
$458$	324.959	0.0331536
$459$	0	0
$460$	0	0
$461$	−9848.51	−0.994991	−0.497495	−	0.867467i	$-0.665747\pi$
$461$	−9848.51	−0.994991	−0.497495	+	0.867467i	$0.665747\pi$
$462$	0	0
$463$	−15172.1	−1.52291	−0.761453	−	0.648220i	$-0.775514\pi$
$463$	−15172.1	−1.52291	−0.761453	+	0.648220i	$0.775514\pi$
$464$	4172.03	0.417418
$465$	0	0
$466$	193.724	0.0192577
$467$	−6628.86	−0.656846	−0.328423	−	0.944531i	$-0.606517\pi$
$467$	−6628.86	−0.656846	−0.328423	+	0.944531i	$0.606517\pi$
$468$	0	0
$469$	−6410.44	−0.631144
$470$	0	0
$471$	0	0
$472$	711.574	0.0693916
$473$	1196.94	0.116354
$474$	0	0
$475$	0	0
$476$	−10089.9	−0.971576
$477$	0	0
$478$	−2023.68	−0.193642
$479$	−15841.3	−1.51108	−0.755541	−	0.655102i	$-0.772626\pi$
$479$	−15841.3	−1.51108	−0.755541	+	0.655102i	$0.772626\pi$
$480$	0	0
$481$	26470.5	2.50925
$482$	435.691	0.0411726
$483$	0	0
$484$	−956.208	−0.0898017
$485$	0	0
$486$	0	0
$487$	−7415.14	−0.689963	−0.344982	−	0.938609i	$-0.612115\pi$
$487$	−7415.14	−0.689963	−0.344982	+	0.938609i	$0.612115\pi$
$488$	1206.89	0.111954
$489$	0	0
$490$	0	0
$491$	−14354.5	−1.31936	−0.659682	−	0.751545i	$-0.729309\pi$
$491$	−14354.5	−1.31936	−0.659682	+	0.751545i	$0.729309\pi$
$492$	0	0
$493$	5708.96	0.521539
$494$	1635.35	0.148943
$495$	0	0
$496$	9216.46	0.834338
$497$	−6006.35	−0.542096
$498$	0	0
$499$	−16510.5	−1.48118	−0.740591	−	0.671956i	$-0.765454\pi$
$499$	−16510.5	−1.48118	−0.740591	+	0.671956i	$0.765454\pi$
$500$	0	0
$501$	0	0
$502$	184.201	0.0163770
$503$	−13504.1	−1.19705	−0.598526	−	0.801104i	$-0.704247\pi$
$503$	−13504.1	−1.19705	−0.598526	+	0.801104i	$0.704247\pi$
$504$	0	0
$505$	0	0
$506$	−517.533	−0.0454687
$507$	0	0
$508$	2842.38	0.248249
$509$	19029.1	1.65707	0.828537	−	0.559934i	$-0.189174\pi$
$509$	19029.1	1.65707	0.828537	+	0.559934i	$0.189174\pi$
$510$	0	0
$511$	−5501.43	−0.476260
$512$	−6085.79	−0.525306
$513$	0	0
$514$	156.724	0.0134490
$515$	0	0
$516$	0	0
$517$	6312.14	0.536958
$518$	−1553.00	−0.131727
$519$	0	0
$520$	0	0
$521$	20133.3	1.69301	0.846504	−	0.532383i	$-0.178703\pi$
$521$	20133.3	1.69301	0.846504	+	0.532383i	$0.178703\pi$
$522$	0	0
$523$	332.778	0.0278228	0.0139114	−	0.999903i	$-0.495572\pi$
$523$	332.778	0.0278228	0.0139114	+	0.999903i	$0.495572\pi$
$524$	12840.8	1.07052
$525$	0	0
$526$	1189.60	0.0986099
$527$	12611.7	1.04246
$528$	0	0
$529$	10547.0	0.866850
$530$	0	0
$531$	0	0
$532$	7780.19	0.634049
$533$	−26320.8	−2.13899
$534$	0	0
$535$	0	0
$536$	−2103.39	−0.169501
$537$	0	0
$538$	−1719.16	−0.137766
$539$	−1254.99	−0.100290
$540$	0	0
$541$	−9028.23	−0.717475	−0.358738	−	0.933438i	$-0.616793\pi$
$541$	−9028.23	−0.717475	−0.358738	+	0.933438i	$0.616793\pi$
$542$	572.124	0.0453410
$543$	0	0
$544$	−4976.20	−0.392193
$545$	0	0
$546$	0	0
$547$	−16908.3	−1.32166	−0.660828	−	0.750537i	$-0.729795\pi$
$547$	−16908.3	−1.32166	−0.660828	+	0.750537i	$0.729795\pi$
$548$	−20595.1	−1.60543
$549$	0	0
$550$	0	0
$551$	−4402.10	−0.340355
$552$	0	0
$553$	−3453.37	−0.265555
$554$	519.159	0.0398140
$555$	0	0
$556$	−15047.5	−1.14776
$557$	−17188.3	−1.30752	−0.653762	−	0.756700i	$-0.726810\pi$
$557$	−17188.3	−1.30752	−0.653762	+	0.756700i	$0.726810\pi$
$558$	0	0
$559$	8759.99	0.662805
$560$	0	0
$561$	0	0
$562$	−1387.91	−0.104174
$563$	13958.5	1.04490	0.522450	−	0.852670i	$-0.325018\pi$
$563$	13958.5	1.04490	0.522450	+	0.852670i	$0.325018\pi$
$564$	0	0
$565$	0	0
$566$	−1867.73	−0.138704
$567$	0	0
$568$	−1970.80	−0.145586
$569$	−18194.1	−1.34049	−0.670244	−	0.742141i	$-0.733811\pi$
$569$	−18194.1	−1.34049	−0.670244	+	0.742141i	$0.733811\pi$
$570$	0	0
$571$	−7175.48	−0.525892	−0.262946	−	0.964811i	$-0.584694\pi$
$571$	−7175.48	−0.525892	−0.262946	+	0.964811i	$0.584694\pi$
$572$	−6998.13	−0.511550
$573$	0	0
$574$	1544.21	0.112290
$575$	0	0
$576$	0	0
$577$	11227.0	0.810030	0.405015	−	0.914310i	$-0.367266\pi$
$577$	11227.0	0.810030	0.405015	+	0.914310i	$0.367266\pi$
$578$	−689.459	−0.0496154
$579$	0	0
$580$	0	0
$581$	−12489.4	−0.891822
$582$	0	0
$583$	4537.48	0.322338
$584$	−1805.13	−0.127905
$585$	0	0
$586$	−159.606	−0.0112513
$587$	−16249.6	−1.14257	−0.571287	−	0.820750i	$-0.693556\pi$
$587$	−16249.6	−1.14257	−0.571287	+	0.820750i	$0.693556\pi$
$588$	0	0
$589$	−9724.71	−0.680305
$590$	0	0
$591$	0	0
$592$	20277.7	1.40778
$593$	13965.4	0.967099	0.483549	−	0.875317i	$-0.339347\pi$
$593$	13965.4	0.967099	0.483549	+	0.875317i	$0.339347\pi$
$594$	0	0
$595$	0	0
$596$	−19156.0	−1.31654
$597$	0	0
$598$	−3787.63	−0.259010
$599$	−4571.97	−0.311863	−0.155931	−	0.987768i	$-0.549838\pi$
$599$	−4571.97	−0.311863	−0.155931	+	0.987768i	$0.549838\pi$
$600$	0	0
$601$	2422.42	0.164414	0.0822069	−	0.996615i	$-0.473803\pi$
$601$	2422.42	0.164414	0.0822069	+	0.996615i	$0.473803\pi$
$602$	−513.940	−0.0347951
$603$	0	0
$604$	18427.4	1.24139
$605$	0	0
$606$	0	0
$607$	12195.3	0.815469	0.407735	−	0.913100i	$-0.366319\pi$
$607$	12195.3	0.815469	0.407735	+	0.913100i	$0.366319\pi$
$608$	3837.08	0.255944
$609$	0	0
$610$	0	0
$611$	46196.2	3.05875
$612$	0	0
$613$	334.261	0.0220239	0.0110120	−	0.999939i	$-0.496495\pi$
$613$	334.261	0.0220239	0.0110120	+	0.999939i	$0.496495\pi$
$614$	−1175.49	−0.0772620
$615$	0	0
$616$	826.210	0.0540405
$617$	29206.8	1.90570	0.952852	−	0.303435i	$-0.0981334\pi$
$617$	29206.8	1.90570	0.952852	+	0.303435i	$0.0981334\pi$
$618$	0	0
$619$	25960.6	1.68569	0.842846	−	0.538154i	$-0.180878\pi$
$619$	25960.6	1.68569	0.842846	+	0.538154i	$0.180878\pi$
$620$	0	0
$621$	0	0
$622$	−1532.41	−0.0987846
$623$	−16132.3	−1.03744
$624$	0	0
$625$	0	0
$626$	2655.32	0.169533
$627$	0	0
$628$	18536.8	1.17787
$629$	27747.7	1.75894
$630$	0	0
$631$	−22723.0	−1.43358	−0.716791	−	0.697289i	$-0.754390\pi$
$631$	−22723.0	−1.43358	−0.716791	+	0.697289i	$0.754390\pi$
$632$	−1133.12	−0.0713182
$633$	0	0
$634$	1879.02	0.117706
$635$	0	0
$636$	0	0
$637$	−9184.80	−0.571295
$638$	−232.306	−0.0144155
$639$	0	0
$640$	0	0
$641$	−1102.58	−0.0679395	−0.0339698	−	0.999423i	$-0.510815\pi$
$641$	−1102.58	−0.0679395	−0.0339698	+	0.999423i	$0.510815\pi$
$642$	0	0
$643$	15347.2	0.941266	0.470633	−	0.882329i	$-0.344026\pi$
$643$	15347.2	0.941266	0.470633	+	0.882329i	$0.344026\pi$
$644$	−18019.6	−1.10260
$645$	0	0
$646$	1714.26	0.104407
$647$	22613.8	1.37410	0.687049	−	0.726611i	$-0.258906\pi$
$647$	22613.8	1.37410	0.687049	+	0.726611i	$0.258906\pi$
$648$	0	0
$649$	1576.69	0.0953630
$650$	0	0
$651$	0	0
$652$	−14178.7	−0.851657
$653$	13721.4	0.822299	0.411150	−	0.911568i	$-0.365127\pi$
$653$	13721.4	0.822299	0.411150	+	0.911568i	$0.365127\pi$
$654$	0	0
$655$	0	0
$656$	−20163.0	−1.20005
$657$	0	0
$658$	−2710.29	−0.160574
$659$	5460.89	0.322801	0.161401	−	0.986889i	$-0.448399\pi$
$659$	5460.89	0.322801	0.161401	+	0.986889i	$0.448399\pi$
$660$	0	0
$661$	3381.01	0.198950	0.0994750	−	0.995040i	$-0.468284\pi$
$661$	3381.01	0.198950	0.0994750	+	0.995040i	$0.468284\pi$
$662$	1434.90	0.0842433
$663$	0	0
$664$	−4098.03	−0.239510
$665$	0	0
$666$	0	0
$667$	10195.7	0.591871
$668$	25677.1	1.48724
$669$	0	0
$670$	0	0
$671$	2674.21	0.153855
$672$	0	0
$673$	9722.42	0.556868	0.278434	−	0.960455i	$-0.410185\pi$
$673$	9722.42	0.556868	0.278434	+	0.960455i	$0.410185\pi$
$674$	−2288.71	−0.130798
$675$	0	0
$676$	−33854.8	−1.92620
$677$	10153.9	0.576435	0.288217	−	0.957565i	$-0.406937\pi$
$677$	10153.9	0.576435	0.288217	+	0.957565i	$0.406937\pi$
$678$	0	0
$679$	8717.11	0.492683
$680$	0	0
$681$	0	0
$682$	−513.189	−0.0288138
$683$	−7134.92	−0.399722	−0.199861	−	0.979824i	$-0.564049\pi$
$683$	−7134.92	−0.399722	−0.199861	+	0.979824i	$0.564049\pi$
$684$	0	0
$685$	0	0
$686$	2158.90	0.120156
$687$	0	0
$688$	6710.57	0.371858
$689$	33208.1	1.83618
$690$	0	0
$691$	−13035.2	−0.717629	−0.358815	−	0.933409i	$-0.616819\pi$
$691$	−13035.2	−0.717629	−0.358815	+	0.933409i	$0.616819\pi$
$692$	−23807.4	−1.30783
$693$	0	0
$694$	−568.237	−0.0310807
$695$	0	0
$696$	0	0
$697$	−27590.8	−1.49939
$698$	−684.892	−0.0371397
$699$	0	0
$700$	0	0
$701$	3057.29	0.164725	0.0823624	−	0.996602i	$-0.473753\pi$
$701$	3057.29	0.164725	0.0823624	+	0.996602i	$0.473753\pi$
$702$	0	0
$703$	−21395.9	−1.14788
$704$	−5224.52	−0.279697
$705$	0	0
$706$	−3942.32	−0.210157
$707$	−5951.20	−0.316574
$708$	0	0
$709$	−13033.4	−0.690380	−0.345190	−	0.938533i	$-0.612185\pi$
$709$	−13033.4	−0.690380	−0.345190	+	0.938533i	$0.612185\pi$
$710$	0	0
$711$	0	0
$712$	−5293.33	−0.278618
$713$	22523.3	1.18304
$714$	0	0
$715$	0	0
$716$	−5764.00	−0.300853
$717$	0	0
$718$	−2721.00	−0.141430
$719$	−31186.1	−1.61759	−0.808793	−	0.588093i	$-0.799879\pi$
$719$	−31186.1	−1.61759	−0.808793	+	0.588093i	$0.799879\pi$
$720$	0	0
$721$	21028.5	1.08619
$722$	819.370	0.0422351
$723$	0	0
$724$	16197.0	0.831431
$725$	0	0
$726$	0	0
$727$	6077.40	0.310039	0.155019	−	0.987911i	$-0.450456\pi$
$727$	6077.40	0.310039	0.155019	+	0.987911i	$0.450456\pi$
$728$	6046.73	0.307839
$729$	0	0
$730$	0	0
$731$	9182.67	0.464614
$732$	0	0
$733$	−17539.8	−0.883829	−0.441914	−	0.897057i	$-0.645701\pi$
$733$	−17539.8	−0.883829	−0.441914	+	0.897057i	$0.645701\pi$
$734$	994.369	0.0500039
$735$	0	0
$736$	−8887.03	−0.445082
$737$	−4660.66	−0.232941
$738$	0	0
$739$	13832.5	0.688548	0.344274	−	0.938869i	$-0.388125\pi$
$739$	13832.5	0.688548	0.344274	+	0.938869i	$0.388125\pi$
$740$	0	0
$741$	0	0
$742$	−1948.29	−0.0963934
$743$	19660.0	0.970736	0.485368	−	0.874310i	$-0.338686\pi$
$743$	19660.0	0.970736	0.485368	+	0.874310i	$0.338686\pi$
$744$	0	0
$745$	0	0
$746$	−3528.18	−0.173158
$747$	0	0
$748$	−7335.80	−0.358587
$749$	−8669.38	−0.422927
$750$	0	0
$751$	22509.1	1.09370	0.546849	−	0.837231i	$-0.315827\pi$
$751$	22509.1	1.09370	0.546849	+	0.837231i	$0.315827\pi$
$752$	35388.5	1.71607
$753$	0	0
$754$	−1700.16	−0.0821171
$755$	0	0
$756$	0	0
$757$	−6701.91	−0.321777	−0.160888	−	0.986973i	$-0.551436\pi$
$757$	−6701.91	−0.321777	−0.160888	+	0.986973i	$0.551436\pi$
$758$	−25.6807	−0.00123056
$759$	0	0
$760$	0	0
$761$	−26492.0	−1.26194	−0.630968	−	0.775809i	$-0.717342\pi$
$761$	−26492.0	−1.26194	−0.630968	+	0.775809i	$0.717342\pi$
$762$	0	0
$763$	−27608.7	−1.30996
$764$	−19485.5	−0.922721
$765$	0	0
$766$	750.443	0.0353977
$767$	11539.2	0.543230
$768$	0	0
$769$	−36995.2	−1.73483	−0.867414	−	0.497588i	$-0.834219\pi$
$769$	−36995.2	−1.73483	−0.867414	+	0.497588i	$0.834219\pi$
$770$	0	0
$771$	0	0
$772$	20336.3	0.948080
$773$	−25221.8	−1.17357	−0.586783	−	0.809744i	$-0.699606\pi$
$773$	−25221.8	−1.17357	−0.586783	+	0.809744i	$0.699606\pi$
$774$	0	0
$775$	0	0
$776$	2860.26	0.132316
$777$	0	0
$778$	2250.25	0.103696
$779$	21274.9	0.978499
$780$	0	0
$781$	−4366.87	−0.200076
$782$	−3970.39	−0.181561
$783$	0	0
$784$	−7035.99	−0.320517
$785$	0	0
$786$	0	0
$787$	24811.2	1.12379	0.561895	−	0.827209i	$-0.310073\pi$
$787$	24811.2	1.12379	0.561895	+	0.827209i	$0.310073\pi$
$788$	−2894.08	−0.130834
$789$	0	0
$790$	0	0
$791$	59.3703	0.00266873
$792$	0	0
$793$	19571.5	0.876426
$794$	1841.09	0.0822893
$795$	0	0
$796$	−22317.7	−0.993757
$797$	2979.23	0.132409	0.0662044	−	0.997806i	$-0.478911\pi$
$797$	2979.23	0.132409	0.0662044	+	0.997806i	$0.478911\pi$
$798$	0	0
$799$	48425.2	2.14413
$800$	0	0
$801$	0	0
$802$	−3140.35	−0.138266
$803$	−3999.78	−0.175777
$804$	0	0
$805$	0	0
$806$	−3755.84	−0.164136
$807$	0	0
$808$	−1952.71	−0.0850200
$809$	−39142.7	−1.70109	−0.850547	−	0.525900i	$-0.823729\pi$
$809$	−39142.7	−1.70109	−0.850547	+	0.525900i	$0.823729\pi$
$810$	0	0
$811$	30496.1	1.32042	0.660212	−	0.751079i	$-0.270466\pi$
$811$	30496.1	1.32042	0.660212	+	0.751079i	$0.270466\pi$
$812$	−8088.52	−0.349571
$813$	0	0
$814$	−1129.10	−0.0486177
$815$	0	0
$816$	0	0
$817$	−7080.63	−0.303207
$818$	3425.74	0.146428
$819$	0	0
$820$	0	0
$821$	−14241.1	−0.605382	−0.302691	−	0.953089i	$-0.597885\pi$
$821$	−14241.1	−0.605382	−0.302691	+	0.953089i	$0.597885\pi$
$822$	0	0
$823$	3835.01	0.162430	0.0812151	−	0.996697i	$-0.474120\pi$
$823$	3835.01	0.162430	0.0812151	+	0.996697i	$0.474120\pi$
$824$	6899.86	0.291709
$825$	0	0
$826$	−676.995	−0.0285178
$827$	44562.9	1.87376	0.936882	−	0.349645i	$-0.113698\pi$
$827$	44562.9	1.87376	0.936882	+	0.349645i	$0.113698\pi$
$828$	0	0
$829$	−13121.5	−0.549734	−0.274867	−	0.961482i	$-0.588634\pi$
$829$	−13121.5	−0.549734	−0.274867	+	0.961482i	$0.588634\pi$
$830$	0	0
$831$	0	0
$832$	−38236.3	−1.59328
$833$	−9627.97	−0.400468
$834$	0	0
$835$	0	0
$836$	5656.53	0.234013
$837$	0	0
$838$	−1179.26	−0.0486120
$839$	−12675.2	−0.521571	−0.260786	−	0.965397i	$-0.583982\pi$
$839$	−12675.2	−0.521571	−0.260786	+	0.965397i	$0.583982\pi$
$840$	0	0
$841$	−19812.4	−0.812351
$842$	1714.69	0.0701805
$843$	0	0
$844$	44880.6	1.83039
$845$	0	0
$846$	0	0
$847$	1830.70	0.0742665
$848$	25439.0	1.03016
$849$	0	0
$850$	0	0
$851$	49554.8	1.99614
$852$	0	0
$853$	29232.6	1.17339	0.586697	−	0.809806i	$-0.300428\pi$
$853$	29232.6	1.17339	0.586697	+	0.809806i	$0.300428\pi$
$854$	−1148.24	−0.0460095
$855$	0	0
$856$	−2844.60	−0.113582
$857$	−27054.1	−1.07835	−0.539177	−	0.842193i	$-0.681265\pi$
$857$	−27054.1	−1.07835	−0.539177	+	0.842193i	$0.681265\pi$
$858$	0	0
$859$	−7018.44	−0.278773	−0.139387	−	0.990238i	$-0.544513\pi$
$859$	−7018.44	−0.278773	−0.139387	+	0.990238i	$0.544513\pi$
$860$	0	0
$861$	0	0
$862$	2666.66	0.105368
$863$	31451.3	1.24057	0.620287	−	0.784375i	$-0.287016\pi$
$863$	31451.3	1.24057	0.620287	+	0.784375i	$0.287016\pi$
$864$	0	0
$865$	0	0
$866$	−1640.24	−0.0643623
$867$	0	0
$868$	−17868.4	−0.698724
$869$	−2510.75	−0.0980107
$870$	0	0
$871$	−34109.7	−1.32694
$872$	−9058.98	−0.351807
$873$	0	0
$874$	3061.51	0.118486
$875$	0	0
$876$	0	0
$877$	12759.4	0.491280	0.245640	−	0.969361i	$-0.421002\pi$
$877$	12759.4	0.491280	0.245640	+	0.969361i	$0.421002\pi$
$878$	−97.9931	−0.00376664
$879$	0	0
$880$	0	0
$881$	41181.1	1.57483	0.787416	−	0.616422i	$-0.211418\pi$
$881$	41181.1	1.57483	0.787416	+	0.616422i	$0.211418\pi$
$882$	0	0
$883$	30058.1	1.14557	0.572784	−	0.819706i	$-0.305863\pi$
$883$	30058.1	1.14557	0.572784	+	0.819706i	$0.305863\pi$
$884$	−53688.0	−2.04267
$885$	0	0
$886$	−5.59923	−0.000212314 0
$887$	−15313.2	−0.579669	−0.289834	−	0.957077i	$-0.593600\pi$
$887$	−15313.2	−0.579669	−0.289834	+	0.957077i	$0.593600\pi$
$888$	0	0
$889$	−5441.87	−0.205303
$890$	0	0
$891$	0	0
$892$	38508.2	1.44546
$893$	−37340.0	−1.39925
$894$	0	0
$895$	0	0
$896$	9380.57	0.349758
$897$	0	0
$898$	−4884.37	−0.181507
$899$	10110.1	0.375073
$900$	0	0
$901$	34810.4	1.28713
$902$	1122.71	0.0414437
$903$	0	0
$904$	19.4806	0.000716720 0
$905$	0	0
$906$	0	0
$907$	615.600	0.0225366	0.0112683	−	0.999937i	$-0.496413\pi$
$907$	615.600	0.0225366	0.0112683	+	0.999937i	$0.496413\pi$
$908$	−45745.8	−1.67195
$909$	0	0
$910$	0	0
$911$	−5975.04	−0.217302	−0.108651	−	0.994080i	$-0.534653\pi$
$911$	−5975.04	−0.217302	−0.108651	+	0.994080i	$0.534653\pi$
$912$	0	0
$913$	−9080.35	−0.329152
$914$	−3902.49	−0.141228
$915$	0	0
$916$	8226.15	0.296725
$917$	−24584.3	−0.885326
$918$	0	0
$919$	−29526.7	−1.05984	−0.529921	−	0.848047i	$-0.677778\pi$
$919$	−29526.7	−1.05984	−0.529921	+	0.848047i	$0.677778\pi$
$920$	0	0
$921$	0	0
$922$	3074.46	0.109818
$923$	−31959.5	−1.13972
$924$	0	0
$925$	0	0
$926$	4736.35	0.168084
$927$	0	0
$928$	−3989.14	−0.141110
$929$	−7859.83	−0.277581	−0.138791	−	0.990322i	$-0.544321\pi$
$929$	−7859.83	−0.277581	−0.138791	+	0.990322i	$0.544321\pi$
$930$	0	0
$931$	7423.99	0.261344
$932$	4904.01	0.172356
$933$	0	0
$934$	2069.37	0.0724965
$935$	0	0
$936$	0	0
$937$	−38052.0	−1.32668	−0.663342	−	0.748316i	$-0.730863\pi$
$937$	−38052.0	−1.32668	−0.663342	+	0.748316i	$0.730863\pi$
$938$	2001.18	0.0696598
$939$	0	0
$940$	0	0
$941$	25518.2	0.884026	0.442013	−	0.897009i	$-0.354265\pi$
$941$	25518.2	0.884026	0.442013	+	0.897009i	$0.354265\pi$
$942$	0	0
$943$	−49274.6	−1.70159
$944$	8839.60	0.304772
$945$	0	0
$946$	−373.657	−0.0128421
$947$	−29393.8	−1.00863	−0.504314	−	0.863520i	$-0.668255\pi$
$947$	−29393.8	−1.00863	−0.504314	+	0.863520i	$0.668255\pi$
$948$	0	0
$949$	−29272.9	−1.00130
$950$	0	0
$951$	0	0
$952$	6338.49	0.215789
$953$	19274.5	0.655156	0.327578	−	0.944824i	$-0.393768\pi$
$953$	19274.5	0.655156	0.327578	+	0.944824i	$0.393768\pi$
$954$	0	0
$955$	0	0
$956$	−51228.4	−1.73310
$957$	0	0
$958$	4945.27	0.166779
$959$	39430.2	1.32770
$960$	0	0
$961$	−7456.72	−0.250301
$962$	−8263.44	−0.276948
$963$	0	0
$964$	11029.3	0.368495
$965$	0	0
$966$	0	0
$967$	−7441.47	−0.247468	−0.123734	−	0.992315i	$-0.539487\pi$
$967$	−7441.47	−0.247468	−0.123734	+	0.992315i	$0.539487\pi$
$968$	600.691	0.0199452
$969$	0	0
$970$	0	0
$971$	−3914.52	−0.129375	−0.0646873	−	0.997906i	$-0.520605\pi$
$971$	−3914.52	−0.129375	−0.0646873	+	0.997906i	$0.520605\pi$
$972$	0	0
$973$	28809.1	0.949208
$974$	2314.83	0.0761518
$975$	0	0
$976$	14992.7	0.491707
$977$	8627.24	0.282507	0.141254	−	0.989973i	$-0.454887\pi$
$977$	8627.24	0.282507	0.141254	+	0.989973i	$0.454887\pi$
$978$	0	0
$979$	−11728.9	−0.382897
$980$	0	0
$981$	0	0
$982$	4481.11	0.145619
$983$	−39589.8	−1.28456	−0.642278	−	0.766471i	$-0.722011\pi$
$983$	−39589.8	−1.28456	−0.642278	+	0.766471i	$0.722011\pi$
$984$	0	0
$985$	0	0
$986$	−1782.20	−0.0575626
$987$	0	0
$988$	41398.0	1.33304
$989$	16399.4	0.527270
$990$	0	0
$991$	6882.43	0.220613	0.110307	−	0.993898i	$-0.464817\pi$
$991$	6882.43	0.220613	0.110307	+	0.993898i	$0.464817\pi$
$992$	−8812.44	−0.282052
$993$	0	0
$994$	1875.03	0.0598315
$995$	0	0
$996$	0	0
$997$	50583.1	1.60680	0.803402	−	0.595437i	$-0.203021\pi$
$997$	50583.1	1.60680	0.803402	+	0.595437i	$0.203021\pi$
$998$	5154.16	0.163479
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.ca.1.8		16
3.2	odd	2		2475.4.a.bz.1.9		16
5.2	odd	4		495.4.c.f.199.8	yes	16
5.3	odd	4		495.4.c.f.199.9	yes	16
5.4	even	2	inner	2475.4.a.ca.1.9		16
15.2	even	4		495.4.c.e.199.9	yes	16
15.8	even	4		495.4.c.e.199.8	✓	16
15.14	odd	2		2475.4.a.bz.1.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.c.e.199.8	✓	16	15.8	even	4
495.4.c.e.199.9	yes	16	15.2	even	4
495.4.c.f.199.8	yes	16	5.2	odd	4
495.4.c.f.199.9	yes	16	5.3	odd	4
2475.4.a.bz.1.8		16	15.14	odd	2
2475.4.a.bz.1.9		16	3.2	odd	2
2475.4.a.ca.1.8		16	1.1	even	1	trivial
2475.4.a.ca.1.9		16	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.312176 q^{2} -7.90255 q^{4} +15.1298 q^{7} +4.96439 q^{8} +O(q^{10})\)	\(q-0.312176 q^{2} -7.90255 q^{4} +15.1298 q^{7} +4.96439 q^{8} +11.0000 q^{11} +80.5049 q^{13} -4.72315 q^{14} +61.6706 q^{16} +84.3893 q^{17} -65.0714 q^{19} -3.43393 q^{22} +150.712 q^{23} -25.1317 q^{26} -119.564 q^{28} +67.6503 q^{29} +149.447 q^{31} -58.9671 q^{32} -26.3443 q^{34} +328.806 q^{37} +20.3137 q^{38} -326.946 q^{41} +108.813 q^{43} -86.9280 q^{44} -47.0484 q^{46} +573.831 q^{47} -114.090 q^{49} -636.194 q^{52} +412.498 q^{53} +75.1100 q^{56} -21.1188 q^{58} +143.336 q^{59} +243.110 q^{61} -46.6536 q^{62} -474.957 q^{64} -423.697 q^{67} -666.891 q^{68} -396.989 q^{71} -363.616 q^{73} -102.645 q^{74} +514.230 q^{76} +166.428 q^{77} -228.250 q^{79} +102.065 q^{82} -825.486 q^{83} -33.9688 q^{86} +54.6082 q^{88} -1066.26 q^{89} +1218.02 q^{91} -1191.00 q^{92} -179.136 q^{94} +576.156 q^{97} +35.6161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 78 q^{4}+O(q^{10}) \) 16 * q + 78 * q^4	\( 16 q + 78 q^{4} + 176 q^{11} + 124 q^{14} + 306 q^{16} + 84 q^{19} + 532 q^{26} + 652 q^{29} + 80 q^{31} + 360 q^{34} + 1204 q^{41} + 858 q^{44} - 672 q^{46} + 1016 q^{49} + 3332 q^{56} + 712 q^{59} + 880 q^{61} - 962 q^{64} + 1968 q^{71} + 4152 q^{74} - 1048 q^{76} - 1636 q^{79} + 7284 q^{86} + 2776 q^{89} + 144 q^{91} + 1400 q^{94}+O(q^{100}) \) 16 * q + 78 * q^4 + 176 * q^11 + 124 * q^14 + 306 * q^16 + 84 * q^19 + 532 * q^26 + 652 * q^29 + 80 * q^31 + 360 * q^34 + 1204 * q^41 + 858 * q^44 - 672 * q^46 + 1016 * q^49 + 3332 * q^56 + 712 * q^59 + 880 * q^61 - 962 * q^64 + 1968 * q^71 + 4152 * q^74 - 1048 * q^76 - 1636 * q^79 + 7284 * q^86 + 2776 * q^89 + 144 * q^91 + 1400 * q^94

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