LMFDB - Embedded newform 2475.4.a.bn.1.2

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.2301792529.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 22x^{4} + 101x^{2} - 16 $ x^6 - 22x^4 + 101x^2 - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.50006$ 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)

$f(q)$	$=$	$q-2.50006 q^{2} -1.74972 q^{4} -4.41889 q^{7} +24.3748 q^{8} +O(q^{10})$	$q-2.50006 q^{2} -1.74972 q^{4} -4.41889 q^{7} +24.3748 q^{8} +11.0000 q^{11} -43.9127 q^{13} +11.0475 q^{14} -46.9407 q^{16} +61.6801 q^{17} +58.5188 q^{19} -27.5006 q^{22} -40.2086 q^{23} +109.784 q^{26} +7.73184 q^{28} -87.8973 q^{29} -252.799 q^{31} -77.6446 q^{32} -154.204 q^{34} +388.139 q^{37} -146.300 q^{38} +349.035 q^{41} -299.174 q^{43} -19.2470 q^{44} +100.524 q^{46} +104.433 q^{47} -323.473 q^{49} +76.8352 q^{52} -556.493 q^{53} -107.710 q^{56} +219.748 q^{58} +73.5838 q^{59} +250.061 q^{61} +632.012 q^{62} +569.641 q^{64} +406.943 q^{67} -107.923 q^{68} +585.387 q^{71} +85.4133 q^{73} -970.369 q^{74} -102.392 q^{76} -48.6078 q^{77} -525.476 q^{79} -872.607 q^{82} -700.809 q^{83} +747.953 q^{86} +268.123 q^{88} +377.530 q^{89} +194.045 q^{91} +70.3540 q^{92} -261.089 q^{94} -70.7213 q^{97} +808.701 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{4}+O(q^{10}) $ 6 * q - 4 * q^4	$ 6 q - 4 q^{4} + 66 q^{11} + 18 q^{14} - 108 q^{16} - 258 q^{19} + 356 q^{26} + 494 q^{29} - 514 q^{31} - 6 q^{34} + 824 q^{41} - 44 q^{44} - 940 q^{46} - 496 q^{49} + 130 q^{56} + 200 q^{59} - 2210 q^{61} - 676 q^{64} + 270 q^{71} - 266 q^{74} + 422 q^{76} - 824 q^{79} - 2476 q^{86} - 186 q^{89} + 2000 q^{91} + 660 q^{94}+O(q^{100}) $ 6 * q - 4 * q^4 + 66 * q^11 + 18 * q^14 - 108 * q^16 - 258 * q^19 + 356 * q^26 + 494 * q^29 - 514 * q^31 - 6 * q^34 + 824 * q^41 - 44 * q^44 - 940 * q^46 - 496 * q^49 + 130 * q^56 + 200 * q^59 - 2210 * q^61 - 676 * q^64 + 270 * q^71 - 266 * q^74 + 422 * q^76 - 824 * q^79 - 2476 * q^86 - 186 * q^89 + 2000 * q^91 + 660 * 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 $

$2$	−2.50006	−0.883903	−0.441951	−	0.897039i	$-0.645714\pi$
$2$	−2.50006	−0.883903	−0.441951	+	0.897039i	$0.645714\pi$
$3$	0	0
$3$	0	0
$4$	−1.74972	−0.218716
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.41889	−0.238597	−0.119299	−	0.992858i	$-0.538065\pi$
$7$	−4.41889	−0.238597	−0.119299	+	0.992858i	$0.538065\pi$
$8$	24.3748	1.07723
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−43.9127	−0.936861	−0.468431	−	0.883500i	$-0.655180\pi$
$13$	−43.9127	−0.936861	−0.468431	+	0.883500i	$0.655180\pi$
$14$	11.0475	0.210897
$15$	0	0
$16$	−46.9407	−0.733448
$17$	61.6801	0.879978	0.439989	−	0.898003i	$-0.354982\pi$
$17$	61.6801	0.879978	0.439989	+	0.898003i	$0.354982\pi$
$18$	0	0
$19$	58.5188	0.706585	0.353293	−	0.935513i	$-0.385062\pi$
$19$	58.5188	0.706585	0.353293	+	0.935513i	$0.385062\pi$
$20$	0	0
$21$	0	0
$22$	−27.5006	−0.266507
$23$	−40.2086	−0.364525	−0.182263	−	0.983250i	$-0.558342\pi$
$23$	−40.2086	−0.364525	−0.182263	+	0.983250i	$0.558342\pi$
$24$	0	0
$25$	0	0
$26$	109.784	0.828094
$27$	0	0
$28$	7.73184	0.0521850
$29$	−87.8973	−0.562832	−0.281416	−	0.959586i	$-0.590804\pi$
$29$	−87.8973	−0.562832	−0.281416	+	0.959586i	$0.590804\pi$
$30$	0	0
$31$	−252.799	−1.46465	−0.732324	−	0.680957i	$-0.761564\pi$
$31$	−252.799	−1.46465	−0.732324	+	0.680957i	$0.761564\pi$
$32$	−77.6446	−0.428930
$33$	0	0
$34$	−154.204	−0.777815
$35$	0	0
$36$	0	0
$37$	388.139	1.72458	0.862292	−	0.506411i	$-0.169028\pi$
$37$	388.139	1.72458	0.862292	+	0.506411i	$0.169028\pi$
$38$	−146.300	−0.624553
$39$	0	0
$40$	0	0
$41$	349.035	1.32952	0.664758	−	0.747059i	$-0.268535\pi$
$41$	349.035	1.32952	0.664758	+	0.747059i	$0.268535\pi$
$42$	0	0
$43$	−299.174	−1.06102	−0.530508	−	0.847680i	$-0.677999\pi$
$43$	−299.174	−1.06102	−0.530508	+	0.847680i	$0.677999\pi$
$44$	−19.2470	−0.0659452
$45$	0	0
$46$	100.524	0.322205
$47$	104.433	0.324110	0.162055	−	0.986782i	$-0.448188\pi$
$47$	104.433	0.324110	0.162055	+	0.986782i	$0.448188\pi$
$48$	0	0
$49$	−323.473	−0.943071
$50$	0	0
$51$	0	0
$52$	76.8352	0.204906
$53$	−556.493	−1.44227	−0.721134	−	0.692796i	$-0.756379\pi$
$53$	−556.493	−1.44227	−0.721134	+	0.692796i	$0.756379\pi$
$54$	0	0
$55$	0	0
$56$	−107.710	−0.257023
$57$	0	0
$58$	219.748	0.497489
$59$	73.5838	0.162369	0.0811847	−	0.996699i	$-0.474130\pi$
$59$	73.5838	0.162369	0.0811847	+	0.996699i	$0.474130\pi$
$60$	0	0
$61$	250.061	0.524869	0.262434	−	0.964950i	$-0.415475\pi$
$61$	250.061	0.524869	0.262434	+	0.964950i	$0.415475\pi$
$62$	632.012	1.29461
$63$	0	0
$64$	569.641	1.11258
$65$	0	0
$66$	0	0
$67$	406.943	0.742029	0.371015	−	0.928627i	$-0.379010\pi$
$67$	406.943	0.742029	0.371015	+	0.928627i	$0.379010\pi$
$68$	−107.923	−0.192465
$69$	0	0
$70$	0	0
$71$	585.387	0.978489	0.489244	−	0.872147i	$-0.337273\pi$
$71$	585.387	0.978489	0.489244	+	0.872147i	$0.337273\pi$
$72$	0	0
$73$	85.4133	0.136943	0.0684717	−	0.997653i	$-0.478188\pi$
$73$	85.4133	0.136943	0.0684717	+	0.997653i	$0.478188\pi$
$74$	−970.369	−1.52437
$75$	0	0
$76$	−102.392	−0.154541
$77$	−48.6078	−0.0719398
$78$	0	0
$79$	−525.476	−0.748362	−0.374181	−	0.927356i	$-0.622076\pi$
$79$	−525.476	−0.748362	−0.374181	+	0.927356i	$0.622076\pi$
$80$	0	0
$81$	0	0
$82$	−872.607	−1.17516
$83$	−700.809	−0.926792	−0.463396	−	0.886151i	$-0.653369\pi$
$83$	−700.809	−0.926792	−0.463396	+	0.886151i	$0.653369\pi$
$84$	0	0
$85$	0	0
$86$	747.953	0.937835
$87$	0	0
$88$	268.123	0.324796
$89$	377.530	0.449641	0.224821	−	0.974400i	$-0.427820\pi$
$89$	377.530	0.449641	0.224821	+	0.974400i	$0.427820\pi$
$90$	0	0
$91$	194.045	0.223533
$92$	70.3540	0.0797273
$93$	0	0
$94$	−261.089	−0.286481
$95$	0	0
$96$	0	0
$97$	−70.7213	−0.0740274	−0.0370137	−	0.999315i	$-0.511785\pi$
$97$	−70.7213	−0.0740274	−0.0370137	+	0.999315i	$0.511785\pi$
$98$	808.701	0.833583
$99$	0	0
$100$	0	0
$101$	−1243.02	−1.22461	−0.612303	−	0.790623i	$-0.709757\pi$
$101$	−1243.02	−1.22461	−0.612303	+	0.790623i	$0.709757\pi$
$102$	0	0
$103$	444.511	0.425233	0.212616	−	0.977136i	$-0.431802\pi$
$103$	444.511	0.425233	0.212616	+	0.977136i	$0.431802\pi$
$104$	−1070.37	−1.00921
$105$	0	0
$106$	1391.26	1.27482
$107$	1680.66	1.51846	0.759232	−	0.650820i	$-0.225575\pi$
$107$	1680.66	1.51846	0.759232	+	0.650820i	$0.225575\pi$
$108$	0	0
$109$	−1546.89	−1.35931	−0.679657	−	0.733530i	$-0.737871\pi$
$109$	−1546.89	−1.35931	−0.679657	+	0.733530i	$0.737871\pi$
$110$	0	0
$111$	0	0
$112$	207.425	0.174999
$113$	2189.45	1.82271	0.911353	−	0.411626i	$-0.135039\pi$
$113$	2189.45	1.82271	0.911353	+	0.411626i	$0.135039\pi$
$114$	0	0
$115$	0	0
$116$	153.796	0.123100
$117$	0	0
$118$	−183.964	−0.143519
$119$	−272.557	−0.209960
$120$	0	0
$121$	121.000	0.0909091
$122$	−625.165	−0.463933
$123$	0	0
$124$	442.329	0.320341
$125$	0	0
$126$	0	0
$127$	2269.08	1.58542	0.792709	−	0.609600i	$-0.208670\pi$
$127$	2269.08	1.58542	0.792709	+	0.609600i	$0.208670\pi$
$128$	−802.977	−0.554483
$129$	0	0
$130$	0	0
$131$	−1696.12	−1.13123	−0.565613	−	0.824671i	$-0.691360\pi$
$131$	−1696.12	−1.13123	−0.565613	+	0.824671i	$0.691360\pi$
$132$	0	0
$133$	−258.588	−0.168589
$134$	−1017.38	−0.655882
$135$	0	0
$136$	1503.44	0.947935
$137$	−1610.24	−1.00418	−0.502089	−	0.864816i	$-0.667435\pi$
$137$	−1610.24	−1.00418	−0.502089	+	0.864816i	$0.667435\pi$
$138$	0	0
$139$	895.911	0.546692	0.273346	−	0.961916i	$-0.411870\pi$
$139$	895.911	0.546692	0.273346	+	0.961916i	$0.411870\pi$
$140$	0	0
$141$	0	0
$142$	−1463.50	−0.864889
$143$	−483.040	−0.282474
$144$	0	0
$145$	0	0
$146$	−213.538	−0.121045
$147$	0	0
$148$	−679.136	−0.377194
$149$	902.577	0.496255	0.248127	−	0.968727i	$-0.420185\pi$
$149$	902.577	0.496255	0.248127	+	0.968727i	$0.420185\pi$
$150$	0	0
$151$	−2274.02	−1.22554	−0.612771	−	0.790261i	$-0.709945\pi$
$151$	−2274.02	−1.22554	−0.612771	+	0.790261i	$0.709945\pi$
$152$	1426.39	0.761152
$153$	0	0
$154$	121.522	0.0635878
$155$	0	0
$156$	0	0
$157$	−250.150	−0.127160	−0.0635801	−	0.997977i	$-0.520252\pi$
$157$	−250.150	−0.127160	−0.0635801	+	0.997977i	$0.520252\pi$
$158$	1313.72	0.661480
$159$	0	0
$160$	0	0
$161$	177.677	0.0869747
$162$	0	0
$163$	3480.09	1.67228	0.836140	−	0.548516i	$-0.184807\pi$
$163$	3480.09	1.67228	0.836140	+	0.548516i	$0.184807\pi$
$164$	−610.716	−0.290786
$165$	0	0
$166$	1752.06	0.819194
$167$	3616.03	1.67555	0.837774	−	0.546017i	$-0.183857\pi$
$167$	3616.03	1.67555	0.837774	+	0.546017i	$0.183857\pi$
$168$	0	0
$169$	−268.673	−0.122291
$170$	0	0
$171$	0	0
$172$	523.473	0.232061
$173$	1299.32	0.571016	0.285508	−	0.958376i	$-0.407838\pi$
$173$	1299.32	0.571016	0.285508	+	0.958376i	$0.407838\pi$
$174$	0	0
$175$	0	0
$176$	−516.347	−0.221143
$177$	0	0
$178$	−943.845	−0.397439
$179$	1960.54	0.818645	0.409323	−	0.912390i	$-0.365765\pi$
$179$	1960.54	0.818645	0.409323	+	0.912390i	$0.365765\pi$
$180$	0	0
$181$	−1979.92	−0.813072	−0.406536	−	0.913635i	$-0.633263\pi$
$181$	−1979.92	−0.813072	−0.406536	+	0.913635i	$0.633263\pi$
$182$	−485.124	−0.197581
$183$	0	0
$184$	−980.079	−0.392676
$185$	0	0
$186$	0	0
$187$	678.481	0.265323
$188$	−182.729	−0.0708879
$189$	0	0
$190$	0	0
$191$	1038.31	0.393347	0.196673	−	0.980469i	$-0.436986\pi$
$191$	1038.31	0.393347	0.196673	+	0.980469i	$0.436986\pi$
$192$	0	0
$193$	3326.12	1.24051	0.620257	−	0.784399i	$-0.287028\pi$
$193$	3326.12	1.24051	0.620257	+	0.784399i	$0.287028\pi$
$194$	176.807	0.0654330
$195$	0	0
$196$	565.990	0.206264
$197$	−640.800	−0.231752	−0.115876	−	0.993264i	$-0.536968\pi$
$197$	−640.800	−0.231752	−0.115876	+	0.993264i	$0.536968\pi$
$198$	0	0
$199$	47.0981	0.0167774	0.00838868	−	0.999965i	$-0.497330\pi$
$199$	47.0981	0.0167774	0.00838868	+	0.999965i	$0.497330\pi$
$200$	0	0
$201$	0	0
$202$	3107.62	1.08243
$203$	388.408	0.134290
$204$	0	0
$205$	0	0
$206$	−1111.30	−0.375864
$207$	0	0
$208$	2061.29	0.687139
$209$	643.706	0.213044
$210$	0	0
$211$	−3676.88	−1.19965	−0.599826	−	0.800131i	$-0.704764\pi$
$211$	−3676.88	−1.19965	−0.599826	+	0.800131i	$0.704764\pi$
$212$	973.709	0.315446
$213$	0	0
$214$	−4201.74	−1.34217
$215$	0	0
$216$	0	0
$217$	1117.09	0.349461
$218$	3867.31	1.20150
$219$	0	0
$220$	0	0
$221$	−2708.54	−0.824417
$222$	0	0
$223$	−4684.67	−1.40676	−0.703382	−	0.710812i	$-0.748328\pi$
$223$	−4684.67	−1.40676	−0.703382	+	0.710812i	$0.748328\pi$
$224$	343.102	0.102342
$225$	0	0
$226$	−5473.74	−1.61110
$227$	−120.790	−0.0353177	−0.0176588	−	0.999844i	$-0.505621\pi$
$227$	−120.790	−0.0353177	−0.0176588	+	0.999844i	$0.505621\pi$
$228$	0	0
$229$	−4914.00	−1.41802	−0.709009	−	0.705200i	$-0.750857\pi$
$229$	−4914.00	−1.41802	−0.709009	+	0.705200i	$0.750857\pi$
$230$	0	0
$231$	0	0
$232$	−2142.48	−0.606297
$233$	−4983.91	−1.40132	−0.700659	−	0.713496i	$-0.747110\pi$
$233$	−4983.91	−1.40132	−0.700659	+	0.713496i	$0.747110\pi$
$234$	0	0
$235$	0	0
$236$	−128.751	−0.0355127
$237$	0	0
$238$	681.408	0.185585
$239$	1692.95	0.458191	0.229095	−	0.973404i	$-0.426423\pi$
$239$	1692.95	0.458191	0.229095	+	0.973404i	$0.426423\pi$
$240$	0	0
$241$	−6690.46	−1.78826	−0.894130	−	0.447808i	$-0.852205\pi$
$241$	−6690.46	−1.78826	−0.894130	+	0.447808i	$0.852205\pi$
$242$	−302.507	−0.0803548
$243$	0	0
$244$	−437.537	−0.114797
$245$	0	0
$246$	0	0
$247$	−2569.72	−0.661972
$248$	−6161.94	−1.57776
$249$	0	0
$250$	0	0
$251$	−2968.03	−0.746375	−0.373187	−	0.927756i	$-0.621735\pi$
$251$	−2968.03	−0.746375	−0.373187	+	0.927756i	$0.621735\pi$
$252$	0	0
$253$	−442.295	−0.109908
$254$	−5672.82	−1.40136
$255$	0	0
$256$	−2549.64	−0.622471
$257$	4467.19	1.08426	0.542132	−	0.840293i	$-0.317617\pi$
$257$	4467.19	1.08426	0.542132	+	0.840293i	$0.317617\pi$
$258$	0	0
$259$	−1715.14	−0.411482
$260$	0	0
$261$	0	0
$262$	4240.39	0.999893
$263$	7752.09	1.81754	0.908772	−	0.417293i	$-0.137021\pi$
$263$	7752.09	1.81754	0.908772	+	0.417293i	$0.137021\pi$
$264$	0	0
$265$	0	0
$266$	646.484	0.149017
$267$	0	0
$268$	−712.038	−0.162293
$269$	4341.29	0.983990	0.491995	−	0.870598i	$-0.336268\pi$
$269$	4341.29	0.983990	0.491995	+	0.870598i	$0.336268\pi$
$270$	0	0
$271$	850.478	0.190638	0.0953190	−	0.995447i	$-0.469613\pi$
$271$	850.478	0.190638	0.0953190	+	0.995447i	$0.469613\pi$
$272$	−2895.30	−0.645418
$273$	0	0
$274$	4025.70	0.887596
$275$	0	0
$276$	0	0
$277$	−1256.61	−0.272571	−0.136286	−	0.990670i	$-0.543516\pi$
$277$	−1256.61	−0.272571	−0.136286	+	0.990670i	$0.543516\pi$
$278$	−2239.83	−0.483222
$279$	0	0
$280$	0	0
$281$	7598.61	1.61315	0.806575	−	0.591132i	$-0.201319\pi$
$281$	7598.61	1.61315	0.806575	+	0.591132i	$0.201319\pi$
$282$	0	0
$283$	−7732.86	−1.62428	−0.812140	−	0.583463i	$-0.801697\pi$
$283$	−7732.86	−1.62428	−0.812140	+	0.583463i	$0.801697\pi$
$284$	−1024.27	−0.214011
$285$	0	0
$286$	1207.63	0.249680
$287$	−1542.35	−0.317219
$288$	0	0
$289$	−1108.57	−0.225639
$290$	0	0
$291$	0	0
$292$	−149.450	−0.0299517
$293$	198.966	0.0396715	0.0198357	−	0.999803i	$-0.493686\pi$
$293$	198.966	0.0396715	0.0198357	+	0.999803i	$0.493686\pi$
$294$	0	0
$295$	0	0
$296$	9460.83	1.85777
$297$	0	0
$298$	−2256.49	−0.438641
$299$	1765.67	0.341509
$300$	0	0
$301$	1322.02	0.253156
$302$	5685.16	1.08326
$303$	0	0
$304$	−2746.91	−0.518244
$305$	0	0
$306$	0	0
$307$	−878.792	−0.163372	−0.0816862	−	0.996658i	$-0.526031\pi$
$307$	−878.792	−0.163372	−0.0816862	+	0.996658i	$0.526031\pi$
$308$	85.0502	0.0157344
$309$	0	0
$310$	0	0
$311$	−9808.71	−1.78843	−0.894213	−	0.447641i	$-0.852264\pi$
$311$	−9808.71	−1.78843	−0.894213	+	0.447641i	$0.852264\pi$
$312$	0	0
$313$	−3121.78	−0.563749	−0.281874	−	0.959451i	$-0.590956\pi$
$313$	−3121.78	−0.563749	−0.281874	+	0.959451i	$0.590956\pi$
$314$	625.389	0.112397
$315$	0	0
$316$	919.438	0.163679
$317$	−9893.08	−1.75284	−0.876420	−	0.481547i	$-0.840075\pi$
$317$	−9893.08	−1.75284	−0.876420	+	0.481547i	$0.840075\pi$
$318$	0	0
$319$	−966.871	−0.169700
$320$	0	0
$321$	0	0
$322$	−444.203	−0.0768772
$323$	3609.44	0.621779
$324$	0	0
$325$	0	0
$326$	−8700.42	−1.47813
$327$	0	0
$328$	8507.68	1.43219
$329$	−461.479	−0.0773317
$330$	0	0
$331$	9925.29	1.64817	0.824083	−	0.566469i	$-0.191691\pi$
$331$	9925.29	1.64817	0.824083	+	0.566469i	$0.191691\pi$
$332$	1226.22	0.202704
$333$	0	0
$334$	−9040.26	−1.48102
$335$	0	0
$336$	0	0
$337$	−4746.82	−0.767287	−0.383644	−	0.923481i	$-0.625331\pi$
$337$	−4746.82	−0.767287	−0.383644	+	0.923481i	$0.625331\pi$
$338$	671.698	0.108093
$339$	0	0
$340$	0	0
$341$	−2780.79	−0.441608
$342$	0	0
$343$	2945.07	0.463612
$344$	−7292.33	−1.14295
$345$	0	0
$346$	−3248.38	−0.504722
$347$	−1618.22	−0.250347	−0.125174	−	0.992135i	$-0.539949\pi$
$347$	−1618.22	−0.250347	−0.125174	+	0.992135i	$0.539949\pi$
$348$	0	0
$349$	−2740.92	−0.420396	−0.210198	−	0.977659i	$-0.567411\pi$
$349$	−2740.92	−0.420396	−0.210198	+	0.977659i	$0.567411\pi$
$350$	0	0
$351$	0	0
$352$	−854.090	−0.129327
$353$	−692.622	−0.104432	−0.0522161	−	0.998636i	$-0.516628\pi$
$353$	−692.622	−0.104432	−0.0522161	+	0.998636i	$0.516628\pi$
$354$	0	0
$355$	0	0
$356$	−660.573	−0.0983436
$357$	0	0
$358$	−4901.45	−0.723603
$359$	−10494.1	−1.54278	−0.771392	−	0.636361i	$-0.780439\pi$
$359$	−10494.1	−1.54278	−0.771392	+	0.636361i	$0.780439\pi$
$360$	0	0
$361$	−3434.56	−0.500737
$362$	4949.90	0.718677
$363$	0	0
$364$	−339.526	−0.0488901
$365$	0	0
$366$	0	0
$367$	2757.79	0.392249	0.196125	−	0.980579i	$-0.437164\pi$
$367$	2757.79	0.392249	0.196125	+	0.980579i	$0.437164\pi$
$368$	1887.42	0.267360
$369$	0	0
$370$	0	0
$371$	2459.08	0.344121
$372$	0	0
$373$	5207.71	0.722909	0.361454	−	0.932390i	$-0.382280\pi$
$373$	5207.71	0.722909	0.361454	+	0.932390i	$0.382280\pi$
$374$	−1696.24	−0.234520
$375$	0	0
$376$	2545.54	0.349139
$377$	3859.81	0.527295
$378$	0	0
$379$	−1984.19	−0.268920	−0.134460	−	0.990919i	$-0.542930\pi$
$379$	−1984.19	−0.268920	−0.134460	+	0.990919i	$0.542930\pi$
$380$	0	0
$381$	0	0
$382$	−2595.82	−0.347680
$383$	0.357579	4.77061e−5 0	2.38530e−5	−	1.00000i	$-0.499992\pi$
$383$	0.357579	4.77061e−5 0	2.38530e−5		1.00000i	$0.499992\pi$
$384$	0	0
$385$	0	0
$386$	−8315.48	−1.09649
$387$	0	0
$388$	123.743	0.0161909
$389$	−5797.78	−0.755679	−0.377839	−	0.925871i	$-0.623333\pi$
$389$	−5797.78	−0.755679	−0.377839	+	0.925871i	$0.623333\pi$
$390$	0	0
$391$	−2480.07	−0.320774
$392$	−7884.62	−1.01590
$393$	0	0
$394$	1602.04	0.204846
$395$	0	0
$396$	0	0
$397$	2736.60	0.345960	0.172980	−	0.984925i	$-0.444660\pi$
$397$	2736.60	0.345960	0.172980	+	0.984925i	$0.444660\pi$
$398$	−117.748	−0.0148296
$399$	0	0
$400$	0	0
$401$	11238.3	1.39953	0.699765	−	0.714373i	$-0.253288\pi$
$401$	11238.3	1.39953	0.699765	+	0.714373i	$0.253288\pi$
$402$	0	0
$403$	11101.1	1.37217
$404$	2174.94	0.267840
$405$	0	0
$406$	−971.042	−0.118700
$407$	4269.53	0.519982
$408$	0	0
$409$	8931.67	1.07981	0.539905	−	0.841726i	$-0.318460\pi$
$409$	8931.67	1.07981	0.539905	+	0.841726i	$0.318460\pi$
$410$	0	0
$411$	0	0
$412$	−777.772	−0.0930050
$413$	−325.159	−0.0387409
$414$	0	0
$415$	0	0
$416$	3409.58	0.401848
$417$	0	0
$418$	−1609.30	−0.188310
$419$	−12685.9	−1.47911	−0.739556	−	0.673095i	$-0.764964\pi$
$419$	−12685.9	−1.47911	−0.739556	+	0.673095i	$0.764964\pi$
$420$	0	0
$421$	7155.89	0.828401	0.414201	−	0.910186i	$-0.364061\pi$
$421$	7155.89	0.828401	0.414201	+	0.910186i	$0.364061\pi$
$422$	9192.39	1.06038
$423$	0	0
$424$	−13564.4	−1.55365
$425$	0	0
$426$	0	0
$427$	−1104.99	−0.125232
$428$	−2940.69	−0.332112
$429$	0	0
$430$	0	0
$431$	−14555.4	−1.62670	−0.813350	−	0.581775i	$-0.802358\pi$
$431$	−14555.4	−1.62670	−0.813350	+	0.581775i	$0.802358\pi$
$432$	0	0
$433$	10349.5	1.14865	0.574323	−	0.818629i	$-0.305265\pi$
$433$	10349.5	1.14865	0.574323	+	0.818629i	$0.305265\pi$
$434$	−2792.79	−0.308890
$435$	0	0
$436$	2706.63	0.297303
$437$	−2352.96	−0.257568
$438$	0	0
$439$	−17123.9	−1.86169	−0.930843	−	0.365420i	$-0.880925\pi$
$439$	−17123.9	−1.86169	−0.930843	+	0.365420i	$0.880925\pi$
$440$	0	0
$441$	0	0
$442$	6771.50	0.728705
$443$	−8556.52	−0.917681	−0.458841	−	0.888519i	$-0.651735\pi$
$443$	−8556.52	−0.917681	−0.458841	+	0.888519i	$0.651735\pi$
$444$	0	0
$445$	0	0
$446$	11711.9	1.24344
$447$	0	0
$448$	−2517.18	−0.265459
$449$	−10998.2	−1.15598	−0.577990	−	0.816044i	$-0.696163\pi$
$449$	−10998.2	−1.15598	−0.577990	+	0.816044i	$0.696163\pi$
$450$	0	0
$451$	3839.39	0.400864
$452$	−3830.93	−0.398654
$453$	0	0
$454$	301.982	0.0312174
$455$	0	0
$456$	0	0
$457$	−5994.78	−0.613619	−0.306809	−	0.951771i	$-0.599261\pi$
$457$	−5994.78	−0.613619	−0.306809	+	0.951771i	$0.599261\pi$
$458$	12285.3	1.25339
$459$	0	0
$460$	0	0
$461$	1783.57	0.180193	0.0900965	−	0.995933i	$-0.471282\pi$
$461$	1783.57	0.180193	0.0900965	+	0.995933i	$0.471282\pi$
$462$	0	0
$463$	−12053.9	−1.20992	−0.604959	−	0.796257i	$-0.706811\pi$
$463$	−12053.9	−1.20992	−0.604959	+	0.796257i	$0.706811\pi$
$464$	4125.96	0.412808
$465$	0	0
$466$	12460.1	1.23863
$467$	−15933.2	−1.57880	−0.789401	−	0.613879i	$-0.789608\pi$
$467$	−15933.2	−1.57880	−0.789401	+	0.613879i	$0.789608\pi$
$468$	0	0
$469$	−1798.23	−0.177046
$470$	0	0
$471$	0	0
$472$	1793.59	0.174909
$473$	−3290.92	−0.319908
$474$	0	0
$475$	0	0
$476$	476.900	0.0459216
$477$	0	0
$478$	−4232.46	−0.404996
$479$	6009.27	0.573217	0.286608	−	0.958048i	$-0.407472\pi$
$479$	6009.27	0.573217	0.286608	+	0.958048i	$0.407472\pi$
$480$	0	0
$481$	−17044.2	−1.61570
$482$	16726.5	1.58065
$483$	0	0
$484$	−211.717	−0.0198832
$485$	0	0
$486$	0	0
$487$	5954.77	0.554079	0.277039	−	0.960859i	$-0.410647\pi$
$487$	5954.77	0.554079	0.277039	+	0.960859i	$0.410647\pi$
$488$	6095.19	0.565402
$489$	0	0
$490$	0	0
$491$	−12108.0	−1.11289	−0.556444	−	0.830885i	$-0.687835\pi$
$491$	−12108.0	−1.11289	−0.556444	+	0.830885i	$0.687835\pi$
$492$	0	0
$493$	−5421.52	−0.495280
$494$	6424.43	0.585119
$495$	0	0
$496$	11866.6	1.07424
$497$	−2586.76	−0.233465
$498$	0	0
$499$	5340.56	0.479111	0.239555	−	0.970883i	$-0.422998\pi$
$499$	5340.56	0.479111	0.239555	+	0.970883i	$0.422998\pi$
$500$	0	0
$501$	0	0
$502$	7420.23	0.659723
$503$	−11306.5	−1.00225	−0.501123	−	0.865376i	$-0.667080\pi$
$503$	−11306.5	−1.00225	−0.501123	+	0.865376i	$0.667080\pi$
$504$	0	0
$505$	0	0
$506$	1105.76	0.0971484
$507$	0	0
$508$	−3970.26	−0.346756
$509$	−3620.25	−0.315255	−0.157627	−	0.987499i	$-0.550385\pi$
$509$	−3620.25	−0.315255	−0.157627	+	0.987499i	$0.550385\pi$
$510$	0	0
$511$	−377.432	−0.0326744
$512$	12798.1	1.10469
$513$	0	0
$514$	−11168.2	−0.958384
$515$	0	0
$516$	0	0
$517$	1148.77	0.0977227
$518$	4287.95	0.363710
$519$	0	0
$520$	0	0
$521$	8788.87	0.739054	0.369527	−	0.929220i	$-0.379520\pi$
$521$	8788.87	0.739054	0.369527	+	0.929220i	$0.379520\pi$
$522$	0	0
$523$	−22016.3	−1.84074	−0.920368	−	0.391053i	$-0.872111\pi$
$523$	−22016.3	−1.84074	−0.920368	+	0.391053i	$0.872111\pi$
$524$	2967.74	0.247417
$525$	0	0
$526$	−19380.6	−1.60653
$527$	−15592.7	−1.28886
$528$	0	0
$529$	−10550.3	−0.867121
$530$	0	0
$531$	0	0
$532$	452.457	0.0368732
$533$	−15327.1	−1.24557
$534$	0	0
$535$	0	0
$536$	9919.17	0.799333
$537$	0	0
$538$	−10853.5	−0.869752
$539$	−3558.21	−0.284347
$540$	0	0
$541$	−18514.0	−1.47131	−0.735656	−	0.677355i	$-0.763126\pi$
$541$	−18514.0	−1.47131	−0.735656	+	0.677355i	$0.763126\pi$
$542$	−2126.24	−0.168505
$543$	0	0
$544$	−4789.12	−0.377448
$545$	0	0
$546$	0	0
$547$	−7881.90	−0.616098	−0.308049	−	0.951370i	$-0.599676\pi$
$547$	−7881.90	−0.616098	−0.308049	+	0.951370i	$0.599676\pi$
$548$	2817.48	0.219629
$549$	0	0
$550$	0	0
$551$	−5143.64	−0.397689
$552$	0	0
$553$	2322.02	0.178557
$554$	3141.59	0.240926
$555$	0	0
$556$	−1567.60	−0.119570
$557$	−975.367	−0.0741968	−0.0370984	−	0.999312i	$-0.511811\pi$
$557$	−975.367	−0.0741968	−0.0370984	+	0.999312i	$0.511811\pi$
$558$	0	0
$559$	13137.6	0.994025
$560$	0	0
$561$	0	0
$562$	−18996.9	−1.42587
$563$	−6807.77	−0.509615	−0.254807	−	0.966992i	$-0.582012\pi$
$563$	−6807.77	−0.509615	−0.254807	+	0.966992i	$0.582012\pi$
$564$	0	0
$565$	0	0
$566$	19332.6	1.43571
$567$	0	0
$568$	14268.7	1.05405
$569$	−10068.6	−0.741821	−0.370910	−	0.928669i	$-0.620954\pi$
$569$	−10068.6	−0.741821	−0.370910	+	0.928669i	$0.620954\pi$
$570$	0	0
$571$	−15119.9	−1.10814	−0.554070	−	0.832470i	$-0.686926\pi$
$571$	−15119.9	−1.10814	−0.554070	+	0.832470i	$0.686926\pi$
$572$	845.187	0.0617815
$573$	0	0
$574$	3855.95	0.280391
$575$	0	0
$576$	0	0
$577$	−9861.79	−0.711528	−0.355764	−	0.934576i	$-0.615779\pi$
$577$	−9861.79	−0.711528	−0.355764	+	0.934576i	$0.615779\pi$
$578$	2771.47	0.199443
$579$	0	0
$580$	0	0
$581$	3096.79	0.221130
$582$	0	0
$583$	−6121.42	−0.434860
$584$	2081.94	0.147519
$585$	0	0
$586$	−497.427	−0.0350657
$587$	−15025.2	−1.05648	−0.528242	−	0.849094i	$-0.677149\pi$
$587$	−15025.2	−1.05648	−0.528242	+	0.849094i	$0.677149\pi$
$588$	0	0
$589$	−14793.5	−1.03490
$590$	0	0
$591$	0	0
$592$	−18219.5	−1.26489
$593$	5223.84	0.361749	0.180875	−	0.983506i	$-0.442107\pi$
$593$	5223.84	0.361749	0.180875	+	0.983506i	$0.442107\pi$
$594$	0	0
$595$	0	0
$596$	−1579.26	−0.108539
$597$	0	0
$598$	−4414.27	−0.301861
$599$	961.857	0.0656100	0.0328050	−	0.999462i	$-0.489556\pi$
$599$	961.857	0.0656100	0.0328050	+	0.999462i	$0.489556\pi$
$600$	0	0
$601$	−2933.80	−0.199122	−0.0995608	−	0.995031i	$-0.531744\pi$
$601$	−2933.80	−0.199122	−0.0995608	+	0.995031i	$0.531744\pi$
$602$	−3305.12	−0.223765
$603$	0	0
$604$	3978.90	0.268045
$605$	0	0
$606$	0	0
$607$	−21178.3	−1.41615	−0.708074	−	0.706138i	$-0.750436\pi$
$607$	−21178.3	−1.41615	−0.708074	+	0.706138i	$0.750436\pi$
$608$	−4543.66	−0.303075
$609$	0	0
$610$	0	0
$611$	−4585.95	−0.303646
$612$	0	0
$613$	−81.9184	−0.00539748	−0.00269874	−	0.999996i	$-0.500859\pi$
$613$	−81.9184	−0.00539748	−0.00269874	+	0.999996i	$0.500859\pi$
$614$	2197.03	0.144405
$615$	0	0
$616$	−1184.81	−0.0774955
$617$	−15721.0	−1.02578	−0.512889	−	0.858455i	$-0.671425\pi$
$617$	−15721.0	−1.02578	−0.512889	+	0.858455i	$0.671425\pi$
$618$	0	0
$619$	−1366.75	−0.0887470	−0.0443735	−	0.999015i	$-0.514129\pi$
$619$	−1366.75	−0.0887470	−0.0443735	+	0.999015i	$0.514129\pi$
$620$	0	0
$621$	0	0
$622$	24522.3	1.58080
$623$	−1668.26	−0.107283
$624$	0	0
$625$	0	0
$626$	7804.62	0.498299
$627$	0	0
$628$	437.694	0.0278119
$629$	23940.4	1.51760
$630$	0	0
$631$	7942.37	0.501079	0.250539	−	0.968106i	$-0.419392\pi$
$631$	7942.37	0.501079	0.250539	+	0.968106i	$0.419392\pi$
$632$	−12808.4	−0.806156
$633$	0	0
$634$	24733.2	1.54934
$635$	0	0
$636$	0	0
$637$	14204.6	0.883527
$638$	2417.23	0.149998
$639$	0	0
$640$	0	0
$641$	−6386.87	−0.393551	−0.196775	−	0.980449i	$-0.563047\pi$
$641$	−6386.87	−0.393551	−0.196775	+	0.980449i	$0.563047\pi$
$642$	0	0
$643$	27728.0	1.70060	0.850300	−	0.526298i	$-0.176420\pi$
$643$	27728.0	1.70060	0.850300	+	0.526298i	$0.176420\pi$
$644$	−310.886	−0.0190227
$645$	0	0
$646$	−9023.80	−0.549593
$647$	−17964.8	−1.09161	−0.545803	−	0.837914i	$-0.683775\pi$
$647$	−17964.8	−1.09161	−0.545803	+	0.837914i	$0.683775\pi$
$648$	0	0
$649$	809.422	0.0489562
$650$	0	0
$651$	0	0
$652$	−6089.20	−0.365754
$653$	−14272.5	−0.855322	−0.427661	−	0.903939i	$-0.640662\pi$
$653$	−14272.5	−0.855322	−0.427661	+	0.903939i	$0.640662\pi$
$654$	0	0
$655$	0	0
$656$	−16383.9	−0.975131
$657$	0	0
$658$	1153.72	0.0683538
$659$	8372.12	0.494889	0.247444	−	0.968902i	$-0.420409\pi$
$659$	8372.12	0.494889	0.247444	+	0.968902i	$0.420409\pi$
$660$	0	0
$661$	24993.6	1.47071	0.735353	−	0.677684i	$-0.237016\pi$
$661$	24993.6	1.47071	0.735353	+	0.677684i	$0.237016\pi$
$662$	−24813.8	−1.45682
$663$	0	0
$664$	−17082.1	−0.998365
$665$	0	0
$666$	0	0
$667$	3534.23	0.205166
$668$	−6327.05	−0.366469
$669$	0	0
$670$	0	0
$671$	2750.67	0.158254
$672$	0	0
$673$	−10480.6	−0.600294	−0.300147	−	0.953893i	$-0.597036\pi$
$673$	−10480.6	−0.600294	−0.300147	+	0.953893i	$0.597036\pi$
$674$	11867.3	0.678208
$675$	0	0
$676$	470.104	0.0267470
$677$	23963.1	1.36038	0.680188	−	0.733037i	$-0.261898\pi$
$677$	23963.1	1.36038	0.680188	+	0.733037i	$0.261898\pi$
$678$	0	0
$679$	312.509	0.0176627
$680$	0	0
$681$	0	0
$682$	6952.13	0.390338
$683$	−20279.5	−1.13612	−0.568062	−	0.822986i	$-0.692307\pi$
$683$	−20279.5	−1.13612	−0.568062	+	0.822986i	$0.692307\pi$
$684$	0	0
$685$	0	0
$686$	−7362.84	−0.409788
$687$	0	0
$688$	14043.4	0.778200
$689$	24437.1	1.35120
$690$	0	0
$691$	1415.05	0.0779033	0.0389516	−	0.999241i	$-0.487598\pi$
$691$	1415.05	0.0779033	0.0389516	+	0.999241i	$0.487598\pi$
$692$	−2273.46	−0.124890
$693$	0	0
$694$	4045.63	0.221282
$695$	0	0
$696$	0	0
$697$	21528.5	1.16994
$698$	6852.45	0.371589
$699$	0	0
$700$	0	0
$701$	6834.63	0.368246	0.184123	−	0.982903i	$-0.441056\pi$
$701$	6834.63	0.368246	0.184123	+	0.982903i	$0.441056\pi$
$702$	0	0
$703$	22713.4	1.21857
$704$	6266.05	0.335455
$705$	0	0
$706$	1731.59	0.0923079
$707$	5492.77	0.292188
$708$	0	0
$709$	−11827.3	−0.626494	−0.313247	−	0.949672i	$-0.601417\pi$
$709$	−11827.3	−0.626494	−0.313247	+	0.949672i	$0.601417\pi$
$710$	0	0
$711$	0	0
$712$	9202.23	0.484366
$713$	10164.7	0.533901
$714$	0	0
$715$	0	0
$716$	−3430.40	−0.179051
$717$	0	0
$718$	26235.9	1.36367
$719$	−34763.6	−1.80315	−0.901574	−	0.432624i	$-0.857588\pi$
$719$	−34763.6	−1.80315	−0.901574	+	0.432624i	$0.857588\pi$
$720$	0	0
$721$	−1964.24	−0.101459
$722$	8586.58	0.442603
$723$	0	0
$724$	3464.31	0.177832
$725$	0	0
$726$	0	0
$727$	−3241.41	−0.165361	−0.0826803	−	0.996576i	$-0.526348\pi$
$727$	−3241.41	−0.165361	−0.0826803	+	0.996576i	$0.526348\pi$
$728$	4729.83	0.240795
$729$	0	0
$730$	0	0
$731$	−18453.1	−0.933670
$732$	0	0
$733$	−18872.4	−0.950981	−0.475491	−	0.879721i	$-0.657729\pi$
$733$	−18872.4	−0.950981	−0.475491	+	0.879721i	$0.657729\pi$
$734$	−6894.63	−0.346710
$735$	0	0
$736$	3121.98	0.156356
$737$	4476.37	0.223730
$738$	0	0
$739$	19123.9	0.951939	0.475970	−	0.879462i	$-0.342097\pi$
$739$	19123.9	0.951939	0.475970	+	0.879462i	$0.342097\pi$
$740$	0	0
$741$	0	0
$742$	−6147.83	−0.304170
$743$	18855.4	0.931008	0.465504	−	0.885046i	$-0.345873\pi$
$743$	18855.4	0.931008	0.465504	+	0.885046i	$0.345873\pi$
$744$	0	0
$745$	0	0
$746$	−13019.6	−0.638981
$747$	0	0
$748$	−1187.16	−0.0580303
$749$	−7426.65	−0.362301
$750$	0	0
$751$	−10938.5	−0.531495	−0.265748	−	0.964043i	$-0.585619\pi$
$751$	−10938.5	−0.531495	−0.265748	+	0.964043i	$0.585619\pi$
$752$	−4902.17	−0.237718
$753$	0	0
$754$	−9649.74	−0.466078
$755$	0	0
$756$	0	0
$757$	−5153.49	−0.247433	−0.123717	−	0.992318i	$-0.539481\pi$
$757$	−5153.49	−0.247433	−0.123717	+	0.992318i	$0.539481\pi$
$758$	4960.58	0.237700
$759$	0	0
$760$	0	0
$761$	10563.7	0.503196	0.251598	−	0.967832i	$-0.419044\pi$
$761$	10563.7	0.503196	0.251598	+	0.967832i	$0.419044\pi$
$762$	0	0
$763$	6835.54	0.324329
$764$	−1816.75	−0.0860311
$765$	0	0
$766$	−0.893967	−4.21675e−5 0
$767$	−3231.27	−0.152118
$768$	0	0
$769$	25317.3	1.18721	0.593606	−	0.804756i	$-0.297704\pi$
$769$	25317.3	1.18721	0.593606	+	0.804756i	$0.297704\pi$
$770$	0	0
$771$	0	0
$772$	−5819.79	−0.271320
$773$	9900.99	0.460691	0.230345	−	0.973109i	$-0.426014\pi$
$773$	9900.99	0.460691	0.230345	+	0.973109i	$0.426014\pi$
$774$	0	0
$775$	0	0
$776$	−1723.82	−0.0797443
$777$	0	0
$778$	14494.8	0.667947
$779$	20425.1	0.939417
$780$	0	0
$781$	6439.26	0.295025
$782$	6200.32	0.283533
$783$	0	0
$784$	15184.1	0.691694
$785$	0	0
$786$	0	0
$787$	−13135.6	−0.594961	−0.297481	−	0.954728i	$-0.596146\pi$
$787$	−13135.6	−0.594961	−0.297481	+	0.954728i	$0.596146\pi$
$788$	1121.22	0.0506878
$789$	0	0
$790$	0	0
$791$	−9674.91	−0.434893
$792$	0	0
$793$	−10980.8	−0.491729
$794$	−6841.65	−0.305795
$795$	0	0
$796$	−82.4087	−0.00366947
$797$	−7470.23	−0.332006	−0.166003	−	0.986125i	$-0.553086\pi$
$797$	−7470.23	−0.332006	−0.166003	+	0.986125i	$0.553086\pi$
$798$	0	0
$799$	6441.45	0.285209
$800$	0	0
$801$	0	0
$802$	−28096.3	−1.23705
$803$	939.546	0.0412900
$804$	0	0
$805$	0	0
$806$	−27753.3	−1.21287
$807$	0	0
$808$	−30298.4	−1.31918
$809$	−31942.0	−1.38816	−0.694080	−	0.719897i	$-0.744189\pi$
$809$	−31942.0	−1.38816	−0.694080	+	0.719897i	$0.744189\pi$
$810$	0	0
$811$	40573.0	1.75673	0.878367	−	0.477987i	$-0.158633\pi$
$811$	40573.0	1.75673	0.878367	+	0.477987i	$0.158633\pi$
$812$	−679.608	−0.0293714
$813$	0	0
$814$	−10674.1	−0.459614
$815$	0	0
$816$	0	0
$817$	−17507.3	−0.749698
$818$	−22329.7	−0.954448
$819$	0	0
$820$	0	0
$821$	2510.30	0.106711	0.0533557	−	0.998576i	$-0.483008\pi$
$821$	2510.30	0.106711	0.0533557	+	0.998576i	$0.483008\pi$
$822$	0	0
$823$	−12401.7	−0.525269	−0.262634	−	0.964895i	$-0.584591\pi$
$823$	−12401.7	−0.525269	−0.262634	+	0.964895i	$0.584591\pi$
$824$	10834.9	0.458072
$825$	0	0
$826$	812.914	0.0342432
$827$	13888.6	0.583982	0.291991	−	0.956421i	$-0.405682\pi$
$827$	13888.6	0.583982	0.291991	+	0.956421i	$0.405682\pi$
$828$	0	0
$829$	−34929.4	−1.46339	−0.731694	−	0.681633i	$-0.761270\pi$
$829$	−34929.4	−1.46339	−0.731694	+	0.681633i	$0.761270\pi$
$830$	0	0
$831$	0	0
$832$	−25014.5	−1.04233
$833$	−19951.9	−0.829882
$834$	0	0
$835$	0	0
$836$	−1126.31	−0.0465959
$837$	0	0
$838$	31715.5	1.30739
$839$	7063.43	0.290651	0.145326	−	0.989384i	$-0.453577\pi$
$839$	7063.43	0.290651	0.145326	+	0.989384i	$0.453577\pi$
$840$	0	0
$841$	−16663.1	−0.683220
$842$	−17890.1	−0.732226
$843$	0	0
$844$	6433.52	0.262383
$845$	0	0
$846$	0	0
$847$	−534.685	−0.0216907
$848$	26122.1	1.05783
$849$	0	0
$850$	0	0
$851$	−15606.5	−0.628654
$852$	0	0
$853$	−28435.0	−1.14138	−0.570688	−	0.821167i	$-0.693324\pi$
$853$	−28435.0	−1.14138	−0.570688	+	0.821167i	$0.693324\pi$
$854$	2762.53	0.110693
$855$	0	0
$856$	40965.8	1.63573
$857$	39390.5	1.57007	0.785037	−	0.619449i	$-0.212644\pi$
$857$	39390.5	1.57007	0.785037	+	0.619449i	$0.212644\pi$
$858$	0	0
$859$	33148.4	1.31666	0.658328	−	0.752731i	$-0.271264\pi$
$859$	33148.4	1.31666	0.658328	+	0.752731i	$0.271264\pi$
$860$	0	0
$861$	0	0
$862$	36389.2	1.43784
$863$	13814.9	0.544919	0.272460	−	0.962167i	$-0.412163\pi$
$863$	13814.9	0.544919	0.272460	+	0.962167i	$0.412163\pi$
$864$	0	0
$865$	0	0
$866$	−25874.2	−1.01529
$867$	0	0
$868$	−1954.60	−0.0764326
$869$	−5780.23	−0.225640
$870$	0	0
$871$	−17870.0	−0.695178
$872$	−37705.2	−1.46429
$873$	0	0
$874$	5882.52	0.227665
$875$	0	0
$876$	0	0
$877$	20657.8	0.795399	0.397699	−	0.917516i	$-0.369809\pi$
$877$	20657.8	0.795399	0.397699	+	0.917516i	$0.369809\pi$
$878$	42810.7	1.64555
$879$	0	0
$880$	0	0
$881$	10003.5	0.382552	0.191276	−	0.981536i	$-0.438738\pi$
$881$	10003.5	0.382552	0.191276	+	0.981536i	$0.438738\pi$
$882$	0	0
$883$	3999.71	0.152436	0.0762179	−	0.997091i	$-0.475716\pi$
$883$	3999.71	0.152436	0.0762179	+	0.997091i	$0.475716\pi$
$884$	4739.20	0.180313
$885$	0	0
$886$	21391.8	0.811141
$887$	28472.1	1.07779	0.538895	−	0.842373i	$-0.318842\pi$
$887$	28472.1	1.07779	0.538895	+	0.842373i	$0.318842\pi$
$888$	0	0
$889$	−10026.8	−0.378277
$890$	0	0
$891$	0	0
$892$	8196.88	0.307681
$893$	6111.30	0.229011
$894$	0	0
$895$	0	0
$896$	3548.27	0.132298
$897$	0	0
$898$	27496.0	1.02177
$899$	22220.4	0.824350
$900$	0	0
$901$	−34324.5	−1.26916
$902$	−9598.68	−0.354325
$903$	0	0
$904$	53367.4	1.96347
$905$	0	0
$906$	0	0
$907$	48116.6	1.76151	0.880753	−	0.473576i	$-0.157037\pi$
$907$	48116.6	1.76151	0.880753	+	0.473576i	$0.157037\pi$
$908$	211.349	0.00772453
$909$	0	0
$910$	0	0
$911$	−16066.7	−0.584319	−0.292159	−	0.956370i	$-0.594374\pi$
$911$	−16066.7	−0.584319	−0.292159	+	0.956370i	$0.594374\pi$
$912$	0	0
$913$	−7708.90	−0.279438
$914$	14987.3	0.542379
$915$	0	0
$916$	8598.14	0.310143
$917$	7494.95	0.269907
$918$	0	0
$919$	−2292.28	−0.0822799	−0.0411400	−	0.999153i	$-0.513099\pi$
$919$	−2292.28	−0.0822799	−0.0411400	+	0.999153i	$0.513099\pi$
$920$	0	0
$921$	0	0
$922$	−4459.01	−0.159273
$923$	−25706.0	−0.916708
$924$	0	0
$925$	0	0
$926$	30135.4	1.06945
$927$	0	0
$928$	6824.75	0.241415
$929$	−13431.3	−0.474346	−0.237173	−	0.971467i	$-0.576221\pi$
$929$	−13431.3	−0.474346	−0.237173	+	0.971467i	$0.576221\pi$
$930$	0	0
$931$	−18929.3	−0.666360
$932$	8720.48	0.306490
$933$	0	0
$934$	39833.8	1.39551
$935$	0	0
$936$	0	0
$937$	−30992.8	−1.08057	−0.540283	−	0.841483i	$-0.681683\pi$
$937$	−30992.8	−1.08057	−0.540283	+	0.841483i	$0.681683\pi$
$938$	4495.68	0.156492
$939$	0	0
$940$	0	0
$941$	36352.0	1.25934	0.629671	−	0.776862i	$-0.283190\pi$
$941$	36352.0	1.25934	0.629671	+	0.776862i	$0.283190\pi$
$942$	0	0
$943$	−14034.2	−0.484642
$944$	−3454.07	−0.119090
$945$	0	0
$946$	8227.48	0.282768
$947$	18246.3	0.626109	0.313055	−	0.949735i	$-0.398648\pi$
$947$	18246.3	0.626109	0.313055	+	0.949735i	$0.398648\pi$
$948$	0	0
$949$	−3750.73	−0.128297
$950$	0	0
$951$	0	0
$952$	−6643.54	−0.226175
$953$	2491.25	0.0846794	0.0423397	−	0.999103i	$-0.486519\pi$
$953$	2491.25	0.0846794	0.0423397	+	0.999103i	$0.486519\pi$
$954$	0	0
$955$	0	0
$956$	−2962.19	−0.100213
$957$	0	0
$958$	−15023.5	−0.506668
$959$	7115.48	0.239594
$960$	0	0
$961$	34116.4	1.14519
$962$	42611.5	1.42812
$963$	0	0
$964$	11706.5	0.391120
$965$	0	0
$966$	0	0
$967$	31197.9	1.03750	0.518748	−	0.854927i	$-0.326398\pi$
$967$	31197.9	1.03750	0.518748	+	0.854927i	$0.326398\pi$
$968$	2949.36	0.0979297
$969$	0	0
$970$	0	0
$971$	49222.7	1.62681	0.813404	−	0.581699i	$-0.197612\pi$
$971$	49222.7	1.62681	0.813404	+	0.581699i	$0.197612\pi$
$972$	0	0
$973$	−3958.93	−0.130439
$974$	−14887.3	−0.489752
$975$	0	0
$976$	−11738.0	−0.384964
$977$	3942.22	0.129092	0.0645460	−	0.997915i	$-0.479440\pi$
$977$	3942.22	0.129092	0.0645460	+	0.997915i	$0.479440\pi$
$978$	0	0
$979$	4152.83	0.135572
$980$	0	0
$981$	0	0
$982$	30270.8	0.983685
$983$	−27385.6	−0.888571	−0.444286	−	0.895885i	$-0.646542\pi$
$983$	−27385.6	−0.888571	−0.444286	+	0.895885i	$0.646542\pi$
$984$	0	0
$985$	0	0
$986$	13554.1	0.437779
$987$	0	0
$988$	4496.30	0.144784
$989$	12029.4	0.386767
$990$	0	0
$991$	−43996.6	−1.41029	−0.705145	−	0.709063i	$-0.749118\pi$
$991$	−43996.6	−1.41029	−0.705145	+	0.709063i	$0.749118\pi$
$992$	19628.5	0.628230
$993$	0	0
$994$	6467.04	0.206360
$995$	0	0
$996$	0	0
$997$	33920.8	1.07752	0.538758	−	0.842460i	$-0.318894\pi$
$997$	33920.8	1.07752	0.538758	+	0.842460i	$0.318894\pi$
$998$	−13351.7	−0.423488
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bn.1.2		6
3.2	odd	2		275.4.a.j.1.5		6
5.2	odd	4		495.4.c.a.199.2		6
5.3	odd	4		495.4.c.a.199.5		6
5.4	even	2	inner	2475.4.a.bn.1.5		6
15.2	even	4		55.4.b.a.34.5	yes	6
15.8	even	4		55.4.b.a.34.2	✓	6
15.14	odd	2		275.4.a.j.1.2		6
60.23	odd	4		880.4.b.f.529.3		6
60.47	odd	4		880.4.b.f.529.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.b.a.34.2	✓	6	15.8	even	4
55.4.b.a.34.5	yes	6	15.2	even	4
275.4.a.j.1.2		6	15.14	odd	2
275.4.a.j.1.5		6	3.2	odd	2
495.4.c.a.199.2		6	5.2	odd	4
495.4.c.a.199.5		6	5.3	odd	4
880.4.b.f.529.3		6	60.23	odd	4
880.4.b.f.529.4		6	60.47	odd	4
2475.4.a.bn.1.2		6	1.1	even	1	trivial
2475.4.a.bn.1.5		6	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)

\(f(q)\)	\(=\)	\(q-2.50006 q^{2} -1.74972 q^{4} -4.41889 q^{7} +24.3748 q^{8} +O(q^{10})\)	\(q-2.50006 q^{2} -1.74972 q^{4} -4.41889 q^{7} +24.3748 q^{8} +11.0000 q^{11} -43.9127 q^{13} +11.0475 q^{14} -46.9407 q^{16} +61.6801 q^{17} +58.5188 q^{19} -27.5006 q^{22} -40.2086 q^{23} +109.784 q^{26} +7.73184 q^{28} -87.8973 q^{29} -252.799 q^{31} -77.6446 q^{32} -154.204 q^{34} +388.139 q^{37} -146.300 q^{38} +349.035 q^{41} -299.174 q^{43} -19.2470 q^{44} +100.524 q^{46} +104.433 q^{47} -323.473 q^{49} +76.8352 q^{52} -556.493 q^{53} -107.710 q^{56} +219.748 q^{58} +73.5838 q^{59} +250.061 q^{61} +632.012 q^{62} +569.641 q^{64} +406.943 q^{67} -107.923 q^{68} +585.387 q^{71} +85.4133 q^{73} -970.369 q^{74} -102.392 q^{76} -48.6078 q^{77} -525.476 q^{79} -872.607 q^{82} -700.809 q^{83} +747.953 q^{86} +268.123 q^{88} +377.530 q^{89} +194.045 q^{91} +70.3540 q^{92} -261.089 q^{94} -70.7213 q^{97} +808.701 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{4}+O(q^{10}) \) 6 * q - 4 * q^4	\( 6 q - 4 q^{4} + 66 q^{11} + 18 q^{14} - 108 q^{16} - 258 q^{19} + 356 q^{26} + 494 q^{29} - 514 q^{31} - 6 q^{34} + 824 q^{41} - 44 q^{44} - 940 q^{46} - 496 q^{49} + 130 q^{56} + 200 q^{59} - 2210 q^{61} - 676 q^{64} + 270 q^{71} - 266 q^{74} + 422 q^{76} - 824 q^{79} - 2476 q^{86} - 186 q^{89} + 2000 q^{91} + 660 q^{94}+O(q^{100}) \) 6 * q - 4 * q^4 + 66 * q^11 + 18 * q^14 - 108 * q^16 - 258 * q^19 + 356 * q^26 + 494 * q^29 - 514 * q^31 - 6 * q^34 + 824 * q^41 - 44 * q^44 - 940 * q^46 - 496 * q^49 + 130 * q^56 + 200 * q^59 - 2210 * q^61 - 676 * q^64 + 270 * q^71 - 266 * q^74 + 422 * q^76 - 824 * q^79 - 2476 * q^86 - 186 * q^89 + 2000 * q^91 + 660 * q^94

Introduction

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