LMFDB - Embedded newform 1575.4.a.bd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 1 $ x^3 - x^2 - 13*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.10645$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.10645 q^{2} +8.86293 q^{4} +7.00000 q^{7} +3.54358 q^{8} +8.17432 q^{11} -19.2242 q^{13} +28.7451 q^{14} -56.3519 q^{16} +18.8982 q^{17} -76.5016 q^{19} +33.5675 q^{22} -142.692 q^{23} -78.9433 q^{26} +62.0405 q^{28} +96.1582 q^{29} -270.708 q^{31} -259.755 q^{32} +77.6047 q^{34} -335.614 q^{37} -314.150 q^{38} +122.965 q^{41} +492.574 q^{43} +72.4485 q^{44} -585.957 q^{46} +96.9049 q^{47} +49.0000 q^{49} -170.383 q^{52} -388.602 q^{53} +24.8051 q^{56} +394.869 q^{58} -112.896 q^{59} +347.160 q^{61} -1111.65 q^{62} -615.855 q^{64} +101.653 q^{67} +167.494 q^{68} -304.685 q^{71} -753.987 q^{73} -1378.18 q^{74} -678.029 q^{76} +57.2203 q^{77} -1164.38 q^{79} +504.949 q^{82} +889.188 q^{83} +2022.73 q^{86} +28.9664 q^{88} +938.829 q^{89} -134.570 q^{91} -1264.67 q^{92} +397.935 q^{94} -1206.06 q^{97} +201.216 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} + 21 q^{7} + 21 q^{8} + 66 q^{11} - 102 q^{13} + 7 q^{14} - 69 q^{16} + 152 q^{17} - 138 q^{19} - 186 q^{22} - 180 q^{23} + 98 q^{26} + 21 q^{28} - 170 q^{29} - 366 q^{31} - 151 q^{32}+ \cdots + 49 q^{98}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 + 21 * q^7 + 21 * q^8 + 66 * q^11 - 102 * q^13 + 7 * q^14 - 69 * q^16 + 152 * q^17 - 138 * q^19 - 186 * q^22 - 180 * q^23 + 98 * q^26 + 21 * q^28 - 170 * q^29 - 366 * q^31 - 151 * q^32 - 36 * q^34 - 252 * q^37 - 234 * q^38 + 206 * q^41 + 108 * q^43 + 306 * q^44 - 672 * q^46 - 24 * q^47 + 147 * q^49 - 78 * q^52 + 354 * q^53 + 147 * q^56 + 858 * q^58 + 880 * q^59 - 870 * q^61 - 1366 * q^62 - 813 * q^64 + 96 * q^67 - 512 * q^68 + 1018 * q^71 - 1554 * q^73 - 980 * q^74 - 450 * q^76 + 462 * q^77 - 1620 * q^79 + 1638 * q^82 - 872 * q^83 + 2932 * q^86 + 1326 * q^88 + 1938 * q^89 - 714 * q^91 - 708 * q^92 + 2112 * q^94 - 1878 * q^97 + 49 * 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 $

$2$	4.10645	1.45185	0.725925	−	0.687774i	$-0.241412\pi$
$2$	4.10645	1.45185	0.725925	+	0.687774i	$0.241412\pi$
$3$	0	0
$3$	0	0
$4$	8.86293	1.10787
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	3.54358	0.156606
$9$	0	0
$10$	0	0
$11$	8.17432	0.224059	0.112030	−	0.993705i	$-0.464265\pi$
$11$	8.17432	0.224059	0.112030	+	0.993705i	$0.464265\pi$
$12$	0	0
$13$	−19.2242	−0.410142	−0.205071	−	0.978747i	$-0.565742\pi$
$13$	−19.2242	−0.410142	−0.205071	+	0.978747i	$0.565742\pi$
$14$	28.7451	0.548747
$15$	0	0
$16$	−56.3519	−0.880499
$17$	18.8982	0.269618	0.134809	−	0.990872i	$-0.456958\pi$
$17$	18.8982	0.269618	0.134809	+	0.990872i	$0.456958\pi$
$18$	0	0
$19$	−76.5016	−0.923720	−0.461860	−	0.886953i	$-0.652818\pi$
$19$	−76.5016	−0.923720	−0.461860	+	0.886953i	$0.652818\pi$
$20$	0	0
$21$	0	0
$22$	33.5675	0.325300
$23$	−142.692	−1.29362	−0.646811	−	0.762650i	$-0.723898\pi$
$23$	−142.692	−1.29362	−0.646811	+	0.762650i	$0.723898\pi$
$24$	0	0
$25$	0	0
$26$	−78.9433	−0.595464
$27$	0	0
$28$	62.0405	0.418734
$29$	96.1582	0.615728	0.307864	−	0.951430i	$-0.400386\pi$
$29$	96.1582	0.615728	0.307864	+	0.951430i	$0.400386\pi$
$30$	0	0
$31$	−270.708	−1.56840	−0.784202	−	0.620505i	$-0.786928\pi$
$31$	−270.708	−1.56840	−0.784202	+	0.620505i	$0.786928\pi$
$32$	−259.755	−1.43496
$33$	0	0
$34$	77.6047	0.391444
$35$	0	0
$36$	0	0
$37$	−335.614	−1.49120	−0.745602	−	0.666391i	$-0.767838\pi$
$37$	−335.614	−1.49120	−0.745602	+	0.666391i	$0.767838\pi$
$38$	−314.150	−1.34110
$39$	0	0
$40$	0	0
$41$	122.965	0.468387	0.234194	−	0.972190i	$-0.424755\pi$
$41$	122.965	0.468387	0.234194	+	0.972190i	$0.424755\pi$
$42$	0	0
$43$	492.574	1.74690	0.873452	−	0.486910i	$-0.161876\pi$
$43$	492.574	1.74690	0.873452	+	0.486910i	$0.161876\pi$
$44$	72.4485	0.248228
$45$	0	0
$46$	−585.957	−1.87814
$47$	96.9049	0.300745	0.150373	−	0.988629i	$-0.451953\pi$
$47$	96.9049	0.300745	0.150373	+	0.988629i	$0.451953\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−170.383	−0.454382
$53$	−388.602	−1.00714	−0.503572	−	0.863953i	$-0.667981\pi$
$53$	−388.602	−1.00714	−0.503572	+	0.863953i	$0.667981\pi$
$54$	0	0
$55$	0	0
$56$	24.8051	0.0591914
$57$	0	0
$58$	394.869	0.893945
$59$	−112.896	−0.249116	−0.124558	−	0.992212i	$-0.539751\pi$
$59$	−112.896	−0.249116	−0.124558	+	0.992212i	$0.539751\pi$
$60$	0	0
$61$	347.160	0.728676	0.364338	−	0.931267i	$-0.381295\pi$
$61$	347.160	0.728676	0.364338	+	0.931267i	$0.381295\pi$
$62$	−1111.65	−2.27709
$63$	0	0
$64$	−615.855	−1.20284
$65$	0	0
$66$	0	0
$67$	101.653	0.185356	0.0926781	−	0.995696i	$-0.470457\pi$
$67$	101.653	0.185356	0.0926781	+	0.995696i	$0.470457\pi$
$68$	167.494	0.298700
$69$	0	0
$70$	0	0
$71$	−304.685	−0.509287	−0.254644	−	0.967035i	$-0.581958\pi$
$71$	−304.685	−0.509287	−0.254644	+	0.967035i	$0.581958\pi$
$72$	0	0
$73$	−753.987	−1.20887	−0.604435	−	0.796655i	$-0.706601\pi$
$73$	−753.987	−1.20887	−0.604435	+	0.796655i	$0.706601\pi$
$74$	−1378.18	−2.16500
$75$	0	0
$76$	−678.029	−1.02336
$77$	57.2203	0.0846864
$78$	0	0
$79$	−1164.38	−1.65827	−0.829135	−	0.559049i	$-0.811166\pi$
$79$	−1164.38	−1.65827	−0.829135	+	0.559049i	$0.811166\pi$
$80$	0	0
$81$	0	0
$82$	504.949	0.680027
$83$	889.188	1.17592	0.587958	−	0.808891i	$-0.299932\pi$
$83$	889.188	1.17592	0.587958	+	0.808891i	$0.299932\pi$
$84$	0	0
$85$	0	0
$86$	2022.73	2.53624
$87$	0	0
$88$	28.9664	0.0350889
$89$	938.829	1.11815	0.559077	−	0.829116i	$-0.311156\pi$
$89$	938.829	1.11815	0.559077	+	0.829116i	$0.311156\pi$
$90$	0	0
$91$	−134.570	−0.155019
$92$	−1264.67	−1.43316
$93$	0	0
$94$	397.935	0.436637
$95$	0	0
$96$	0	0
$97$	−1206.06	−1.26244	−0.631218	−	0.775605i	$-0.717445\pi$
$97$	−1206.06	−1.26244	−0.631218	+	0.775605i	$0.717445\pi$
$98$	201.216	0.207407
$99$	0	0
$100$	0	0
$101$	−436.407	−0.429942	−0.214971	−	0.976620i	$-0.568966\pi$
$101$	−436.407	−0.429942	−0.214971	+	0.976620i	$0.568966\pi$
$102$	0	0
$103$	−1765.48	−1.68891	−0.844454	−	0.535629i	$-0.820075\pi$
$103$	−1765.48	−1.68891	−0.844454	+	0.535629i	$0.820075\pi$
$104$	−68.1226	−0.0642305
$105$	0	0
$106$	−1595.78	−1.46222
$107$	304.599	0.275203	0.137601	−	0.990488i	$-0.456061\pi$
$107$	304.599	0.275203	0.137601	+	0.990488i	$0.456061\pi$
$108$	0	0
$109$	−1037.76	−0.911918	−0.455959	−	0.890001i	$-0.650704\pi$
$109$	−1037.76	−0.911918	−0.455959	+	0.890001i	$0.650704\pi$
$110$	0	0
$111$	0	0
$112$	−394.463	−0.332797
$113$	−1479.63	−1.23178	−0.615892	−	0.787830i	$-0.711204\pi$
$113$	−1479.63	−1.23178	−0.615892	+	0.787830i	$0.711204\pi$
$114$	0	0
$115$	0	0
$116$	852.243	0.682145
$117$	0	0
$118$	−463.604	−0.361679
$119$	132.288	0.101906
$120$	0	0
$121$	−1264.18	−0.949797
$122$	1425.59	1.05793
$123$	0	0
$124$	−2399.26	−1.73758
$125$	0	0
$126$	0	0
$127$	1633.52	1.14135	0.570676	−	0.821175i	$-0.306681\pi$
$127$	1633.52	1.14135	0.570676	+	0.821175i	$0.306681\pi$
$128$	−450.940	−0.311389
$129$	0	0
$130$	0	0
$131$	359.287	0.239627	0.119813	−	0.992796i	$-0.461770\pi$
$131$	359.287	0.239627	0.119813	+	0.992796i	$0.461770\pi$
$132$	0	0
$133$	−535.511	−0.349133
$134$	417.432	0.269109
$135$	0	0
$136$	66.9675	0.0422236
$137$	2573.71	1.60501	0.802507	−	0.596643i	$-0.203499\pi$
$137$	2573.71	1.60501	0.802507	+	0.596643i	$0.203499\pi$
$138$	0	0
$139$	−2267.12	−1.38341	−0.691706	−	0.722179i	$-0.743141\pi$
$139$	−2267.12	−1.38341	−0.691706	+	0.722179i	$0.743141\pi$
$140$	0	0
$141$	0	0
$142$	−1251.17	−0.739409
$143$	−157.145	−0.0918960
$144$	0	0
$145$	0	0
$146$	−3096.21	−1.75510
$147$	0	0
$148$	−2974.52	−1.65206
$149$	−457.166	−0.251359	−0.125680	−	0.992071i	$-0.540111\pi$
$149$	−457.166	−0.251359	−0.125680	+	0.992071i	$0.540111\pi$
$150$	0	0
$151$	2260.38	1.21819	0.609096	−	0.793097i	$-0.291532\pi$
$151$	2260.38	1.21819	0.609096	+	0.793097i	$0.291532\pi$
$152$	−271.090	−0.144660
$153$	0	0
$154$	234.972	0.122952
$155$	0	0
$156$	0	0
$157$	−643.877	−0.327305	−0.163653	−	0.986518i	$-0.552328\pi$
$157$	−643.877	−0.327305	−0.163653	+	0.986518i	$0.552328\pi$
$158$	−4781.48	−2.40756
$159$	0	0
$160$	0	0
$161$	−998.843	−0.488943
$162$	0	0
$163$	−3089.66	−1.48467	−0.742334	−	0.670030i	$-0.766281\pi$
$163$	−3089.66	−1.48467	−0.742334	+	0.670030i	$0.766281\pi$
$164$	1089.83	0.518910
$165$	0	0
$166$	3651.41	1.70725
$167$	2726.84	1.26353	0.631764	−	0.775161i	$-0.282331\pi$
$167$	2726.84	1.26353	0.631764	+	0.775161i	$0.282331\pi$
$168$	0	0
$169$	−1827.43	−0.831784
$170$	0	0
$171$	0	0
$172$	4365.65	1.93534
$173$	901.915	0.396366	0.198183	−	0.980165i	$-0.436496\pi$
$173$	901.915	0.396366	0.198183	+	0.980165i	$0.436496\pi$
$174$	0	0
$175$	0	0
$176$	−460.639	−0.197284
$177$	0	0
$178$	3855.25	1.62339
$179$	−4547.62	−1.89891	−0.949455	−	0.313903i	$-0.898363\pi$
$179$	−4547.62	−1.89891	−0.949455	+	0.313903i	$0.898363\pi$
$180$	0	0
$181$	−402.003	−0.165086	−0.0825432	−	0.996587i	$-0.526304\pi$
$181$	−402.003	−0.165086	−0.0825432	+	0.996587i	$0.526304\pi$
$182$	−552.603	−0.225064
$183$	0	0
$184$	−505.640	−0.202589
$185$	0	0
$186$	0	0
$187$	154.480	0.0604103
$188$	858.861	0.333186
$189$	0	0
$190$	0	0
$191$	4399.62	1.66673	0.833365	−	0.552723i	$-0.186411\pi$
$191$	4399.62	1.66673	0.833365	+	0.552723i	$0.186411\pi$
$192$	0	0
$193$	301.958	0.112619	0.0563094	−	0.998413i	$-0.482067\pi$
$193$	301.958	0.112619	0.0563094	+	0.998413i	$0.482067\pi$
$194$	−4952.61	−1.83287
$195$	0	0
$196$	434.284	0.158267
$197$	952.950	0.344644	0.172322	−	0.985041i	$-0.444873\pi$
$197$	952.950	0.344644	0.172322	+	0.985041i	$0.444873\pi$
$198$	0	0
$199$	4124.90	1.46938	0.734690	−	0.678403i	$-0.237328\pi$
$199$	4124.90	1.46938	0.734690	+	0.678403i	$0.237328\pi$
$200$	0	0
$201$	0	0
$202$	−1792.08	−0.624210
$203$	673.107	0.232723
$204$	0	0
$205$	0	0
$206$	−7249.83	−2.45204
$207$	0	0
$208$	1083.32	0.361129
$209$	−625.349	−0.206968
$210$	0	0
$211$	−1014.74	−0.331079	−0.165540	−	0.986203i	$-0.552937\pi$
$211$	−1014.74	−0.331079	−0.165540	+	0.986203i	$0.552937\pi$
$212$	−3444.15	−1.11578
$213$	0	0
$214$	1250.82	0.399553
$215$	0	0
$216$	0	0
$217$	−1894.95	−0.592801
$218$	−4261.50	−1.32397
$219$	0	0
$220$	0	0
$221$	−363.304	−0.110581
$222$	0	0
$223$	5042.95	1.51435	0.757177	−	0.653210i	$-0.226578\pi$
$223$	5042.95	1.51435	0.757177	+	0.653210i	$0.226578\pi$
$224$	−1818.28	−0.542363
$225$	0	0
$226$	−6076.02	−1.78837
$227$	−5831.62	−1.70510	−0.852552	−	0.522643i	$-0.824946\pi$
$227$	−5831.62	−1.70510	−0.852552	+	0.522643i	$0.824946\pi$
$228$	0	0
$229$	18.5657	0.00535744	0.00267872	−	0.999996i	$-0.499147\pi$
$229$	18.5657	0.00535744	0.00267872	+	0.999996i	$0.499147\pi$
$230$	0	0
$231$	0	0
$232$	340.744	0.0964265
$233$	−1732.62	−0.487158	−0.243579	−	0.969881i	$-0.578322\pi$
$233$	−1732.62	−0.487158	−0.243579	+	0.969881i	$0.578322\pi$
$234$	0	0
$235$	0	0
$236$	−1000.59	−0.275988
$237$	0	0
$238$	543.233	0.147952
$239$	1937.54	0.524389	0.262195	−	0.965015i	$-0.415554\pi$
$239$	1937.54	0.524389	0.262195	+	0.965015i	$0.415554\pi$
$240$	0	0
$241$	5613.02	1.50027	0.750137	−	0.661282i	$-0.229987\pi$
$241$	5613.02	1.50027	0.750137	+	0.661282i	$0.229987\pi$
$242$	−5191.29	−1.37896
$243$	0	0
$244$	3076.85	0.807275
$245$	0	0
$246$	0	0
$247$	1470.69	0.378856
$248$	−959.275	−0.245621
$249$	0	0
$250$	0	0
$251$	5132.49	1.29068	0.645338	−	0.763897i	$-0.276716\pi$
$251$	5132.49	1.29068	0.645338	+	0.763897i	$0.276716\pi$
$252$	0	0
$253$	−1166.41	−0.289848
$254$	6707.98	1.65707
$255$	0	0
$256$	3075.08	0.750752
$257$	−4689.21	−1.13815	−0.569075	−	0.822285i	$-0.692699\pi$
$257$	−4689.21	−1.13815	−0.569075	+	0.822285i	$0.692699\pi$
$258$	0	0
$259$	−2349.30	−0.563622
$260$	0	0
$261$	0	0
$262$	1475.40	0.347902
$263$	−1767.86	−0.414491	−0.207246	−	0.978289i	$-0.566450\pi$
$263$	−1767.86	−0.414491	−0.207246	+	0.978289i	$0.566450\pi$
$264$	0	0
$265$	0	0
$266$	−2199.05	−0.506889
$267$	0	0
$268$	900.941	0.205350
$269$	7869.14	1.78361	0.891803	−	0.452425i	$-0.149441\pi$
$269$	7869.14	1.78361	0.891803	+	0.452425i	$0.149441\pi$
$270$	0	0
$271$	4820.52	1.08054	0.540268	−	0.841493i	$-0.318323\pi$
$271$	4820.52	1.08054	0.540268	+	0.841493i	$0.318323\pi$
$272$	−1064.95	−0.237398
$273$	0	0
$274$	10568.8	2.33024
$275$	0	0
$276$	0	0
$277$	−1783.66	−0.386895	−0.193448	−	0.981111i	$-0.561967\pi$
$277$	−1783.66	−0.386895	−0.193448	+	0.981111i	$0.561967\pi$
$278$	−9309.80	−2.00851
$279$	0	0
$280$	0	0
$281$	−1779.27	−0.377730	−0.188865	−	0.982003i	$-0.560481\pi$
$281$	−1779.27	−0.377730	−0.188865	+	0.982003i	$0.560481\pi$
$282$	0	0
$283$	−7374.55	−1.54902	−0.774508	−	0.632564i	$-0.782002\pi$
$283$	−7374.55	−1.54902	−0.774508	+	0.632564i	$0.782002\pi$
$284$	−2700.40	−0.564222
$285$	0	0
$286$	−645.308	−0.133419
$287$	860.753	0.177034
$288$	0	0
$289$	−4555.86	−0.927306
$290$	0	0
$291$	0	0
$292$	−6682.53	−1.33927
$293$	−934.020	−0.186232	−0.0931161	−	0.995655i	$-0.529683\pi$
$293$	−934.020	−0.186232	−0.0931161	+	0.995655i	$0.529683\pi$
$294$	0	0
$295$	0	0
$296$	−1189.27	−0.233531
$297$	0	0
$298$	−1877.33	−0.364936
$299$	2743.14	0.530568
$300$	0	0
$301$	3448.02	0.660268
$302$	9282.13	1.76863
$303$	0	0
$304$	4311.01	0.813334
$305$	0	0
$306$	0	0
$307$	1929.10	0.358630	0.179315	−	0.983792i	$-0.442612\pi$
$307$	1929.10	0.358630	0.179315	+	0.983792i	$0.442612\pi$
$308$	507.139	0.0938212
$309$	0	0
$310$	0	0
$311$	4465.18	0.814138	0.407069	−	0.913397i	$-0.366551\pi$
$311$	4465.18	0.814138	0.407069	+	0.913397i	$0.366551\pi$
$312$	0	0
$313$	7457.31	1.34668	0.673342	−	0.739332i	$-0.264858\pi$
$313$	7457.31	1.34668	0.673342	+	0.739332i	$0.264858\pi$
$314$	−2644.05	−0.475198
$315$	0	0
$316$	−10319.8	−1.83714
$317$	−2024.84	−0.358758	−0.179379	−	0.983780i	$-0.557409\pi$
$317$	−2024.84	−0.358758	−0.179379	+	0.983780i	$0.557409\pi$
$318$	0	0
$319$	786.028	0.137960
$320$	0	0
$321$	0	0
$322$	−4101.70	−0.709872
$323$	−1445.75	−0.249051
$324$	0	0
$325$	0	0
$326$	−12687.5	−2.15551
$327$	0	0
$328$	435.736	0.0733521
$329$	678.334	0.113671
$330$	0	0
$331$	3153.56	0.523671	0.261836	−	0.965112i	$-0.415672\pi$
$331$	3153.56	0.523671	0.261836	+	0.965112i	$0.415672\pi$
$332$	7880.81	1.30276
$333$	0	0
$334$	11197.6	1.83445
$335$	0	0
$336$	0	0
$337$	−5058.35	−0.817644	−0.408822	−	0.912614i	$-0.634060\pi$
$337$	−5058.35	−0.817644	−0.408822	+	0.912614i	$0.634060\pi$
$338$	−7504.25	−1.20762
$339$	0	0
$340$	0	0
$341$	−2212.85	−0.351416
$342$	0	0
$343$	343.000	0.0539949
$344$	1745.48	0.273575
$345$	0	0
$346$	3703.67	0.575464
$347$	−7950.25	−1.22995	−0.614973	−	0.788548i	$-0.710833\pi$
$347$	−7950.25	−1.22995	−0.614973	+	0.788548i	$0.710833\pi$
$348$	0	0
$349$	4711.92	0.722702	0.361351	−	0.932430i	$-0.382316\pi$
$349$	4711.92	0.722702	0.361351	+	0.932430i	$0.382316\pi$
$350$	0	0
$351$	0	0
$352$	−2123.32	−0.321515
$353$	308.497	0.0465146	0.0232573	−	0.999730i	$-0.492596\pi$
$353$	308.497	0.0465146	0.0232573	+	0.999730i	$0.492596\pi$
$354$	0	0
$355$	0	0
$356$	8320.77	1.23876
$357$	0	0
$358$	−18674.6	−2.75693
$359$	7598.91	1.11715	0.558573	−	0.829456i	$-0.311349\pi$
$359$	7598.91	1.11715	0.558573	+	0.829456i	$0.311349\pi$
$360$	0	0
$361$	−1006.50	−0.146742
$362$	−1650.80	−0.239681
$363$	0	0
$364$	−1192.68	−0.171740
$365$	0	0
$366$	0	0
$367$	−2057.48	−0.292642	−0.146321	−	0.989237i	$-0.546743\pi$
$367$	−2057.48	−0.292642	−0.146321	+	0.989237i	$0.546743\pi$
$368$	8040.96	1.13903
$369$	0	0
$370$	0	0
$371$	−2720.22	−0.380665
$372$	0	0
$373$	−5661.44	−0.785893	−0.392947	−	0.919561i	$-0.628544\pi$
$373$	−5661.44	−0.785893	−0.392947	+	0.919561i	$0.628544\pi$
$374$	634.366	0.0877066
$375$	0	0
$376$	343.390	0.0470984
$377$	−1848.57	−0.252536
$378$	0	0
$379$	−1822.92	−0.247063	−0.123532	−	0.992341i	$-0.539422\pi$
$379$	−1822.92	−0.247063	−0.123532	+	0.992341i	$0.539422\pi$
$380$	0	0
$381$	0	0
$382$	18066.8	2.41984
$383$	−1886.34	−0.251664	−0.125832	−	0.992052i	$-0.540160\pi$
$383$	−1886.34	−0.251664	−0.125832	+	0.992052i	$0.540160\pi$
$384$	0	0
$385$	0	0
$386$	1239.98	0.163505
$387$	0	0
$388$	−10689.2	−1.39861
$389$	9079.55	1.18342	0.591711	−	0.806150i	$-0.298453\pi$
$389$	9079.55	1.18342	0.591711	+	0.806150i	$0.298453\pi$
$390$	0	0
$391$	−2696.63	−0.348783
$392$	173.635	0.0223722
$393$	0	0
$394$	3913.24	0.500371
$395$	0	0
$396$	0	0
$397$	−2620.95	−0.331339	−0.165669	−	0.986181i	$-0.552978\pi$
$397$	−2620.95	−0.331339	−0.165669	+	0.986181i	$0.552978\pi$
$398$	16938.7	2.13332
$399$	0	0
$400$	0	0
$401$	10752.1	1.33899	0.669493	−	0.742818i	$-0.266511\pi$
$401$	10752.1	1.33899	0.669493	+	0.742818i	$0.266511\pi$
$402$	0	0
$403$	5204.15	0.643268
$404$	−3867.84	−0.476318
$405$	0	0
$406$	2764.08	0.337879
$407$	−2743.42	−0.334118
$408$	0	0
$409$	13382.0	1.61785	0.808923	−	0.587914i	$-0.200051\pi$
$409$	13382.0	1.61785	0.808923	+	0.587914i	$0.200051\pi$
$410$	0	0
$411$	0	0
$412$	−15647.3	−1.87108
$413$	−790.275	−0.0941571
$414$	0	0
$415$	0	0
$416$	4993.59	0.588536
$417$	0	0
$418$	−2567.96	−0.300486
$419$	−2335.91	−0.272354	−0.136177	−	0.990684i	$-0.543482\pi$
$419$	−2335.91	−0.272354	−0.136177	+	0.990684i	$0.543482\pi$
$420$	0	0
$421$	4262.71	0.493472	0.246736	−	0.969083i	$-0.420642\pi$
$421$	4262.71	0.493472	0.246736	+	0.969083i	$0.420642\pi$
$422$	−4166.99	−0.480677
$423$	0	0
$424$	−1377.04	−0.157724
$425$	0	0
$426$	0	0
$427$	2430.12	0.275414
$428$	2699.64	0.304888
$429$	0	0
$430$	0	0
$431$	−5233.78	−0.584924	−0.292462	−	0.956277i	$-0.594474\pi$
$431$	−5233.78	−0.584924	−0.292462	+	0.956277i	$0.594474\pi$
$432$	0	0
$433$	14627.5	1.62345	0.811726	−	0.584039i	$-0.198528\pi$
$433$	14627.5	1.62345	0.811726	+	0.584039i	$0.198528\pi$
$434$	−7781.54	−0.860658
$435$	0	0
$436$	−9197.57	−1.01028
$437$	10916.2	1.19494
$438$	0	0
$439$	35.2964	0.00383737	0.00191868	−	0.999998i	$-0.499389\pi$
$439$	35.2964	0.00383737	0.00191868	+	0.999998i	$0.499389\pi$
$440$	0	0
$441$	0	0
$442$	−1491.89	−0.160548
$443$	−2805.40	−0.300877	−0.150439	−	0.988619i	$-0.548069\pi$
$443$	−2805.40	−0.300877	−0.150439	+	0.988619i	$0.548069\pi$
$444$	0	0
$445$	0	0
$446$	20708.6	2.19861
$447$	0	0
$448$	−4310.99	−0.454632
$449$	−14903.0	−1.56641	−0.783203	−	0.621766i	$-0.786415\pi$
$449$	−14903.0	−1.56641	−0.783203	+	0.621766i	$0.786415\pi$
$450$	0	0
$451$	1005.15	0.104946
$452$	−13113.8	−1.36465
$453$	0	0
$454$	−23947.3	−2.47555
$455$	0	0
$456$	0	0
$457$	2633.85	0.269598	0.134799	−	0.990873i	$-0.456961\pi$
$457$	2633.85	0.269598	0.134799	+	0.990873i	$0.456961\pi$
$458$	76.2390	0.00777820
$459$	0	0
$460$	0	0
$461$	18415.2	1.86048	0.930239	−	0.366955i	$-0.119600\pi$
$461$	18415.2	1.86048	0.930239	+	0.366955i	$0.119600\pi$
$462$	0	0
$463$	−14811.0	−1.48666	−0.743332	−	0.668923i	$-0.766756\pi$
$463$	−14811.0	−1.48666	−0.743332	+	0.668923i	$0.766756\pi$
$464$	−5418.70	−0.542148
$465$	0	0
$466$	−7114.93	−0.707280
$467$	−14407.1	−1.42758	−0.713790	−	0.700360i	$-0.753023\pi$
$467$	−14407.1	−1.42758	−0.713790	+	0.700360i	$0.753023\pi$
$468$	0	0
$469$	711.569	0.0700580
$470$	0	0
$471$	0	0
$472$	−400.058	−0.0390130
$473$	4026.46	0.391410
$474$	0	0
$475$	0	0
$476$	1172.46	0.112898
$477$	0	0
$478$	7956.41	0.761334
$479$	6629.16	0.632347	0.316174	−	0.948701i	$-0.397602\pi$
$479$	6629.16	0.632347	0.316174	+	0.948701i	$0.397602\pi$
$480$	0	0
$481$	6451.92	0.611605
$482$	23049.6	2.17817
$483$	0	0
$484$	−11204.3	−1.05225
$485$	0	0
$486$	0	0
$487$	3890.46	0.361999	0.180999	−	0.983483i	$-0.442067\pi$
$487$	3890.46	0.361999	0.180999	+	0.983483i	$0.442067\pi$
$488$	1230.19	0.114115
$489$	0	0
$490$	0	0
$491$	12804.1	1.17687	0.588433	−	0.808546i	$-0.299745\pi$
$491$	12804.1	1.17687	0.588433	+	0.808546i	$0.299745\pi$
$492$	0	0
$493$	1817.22	0.166011
$494$	6039.29	0.550042
$495$	0	0
$496$	15254.9	1.38098
$497$	−2132.79	−0.192493
$498$	0	0
$499$	−5842.56	−0.524146	−0.262073	−	0.965048i	$-0.584406\pi$
$499$	−5842.56	−0.524146	−0.262073	+	0.965048i	$0.584406\pi$
$500$	0	0
$501$	0	0
$502$	21076.3	1.87387
$503$	−18045.9	−1.59966	−0.799830	−	0.600227i	$-0.795077\pi$
$503$	−18045.9	−1.59966	−0.799830	+	0.600227i	$0.795077\pi$
$504$	0	0
$505$	0	0
$506$	−4789.80	−0.420816
$507$	0	0
$508$	14477.8	1.26447
$509$	21723.0	1.89166	0.945831	−	0.324659i	$-0.105250\pi$
$509$	21723.0	1.89166	0.945831	+	0.324659i	$0.105250\pi$
$510$	0	0
$511$	−5277.91	−0.456910
$512$	16235.2	1.40137
$513$	0	0
$514$	−19256.0	−1.65242
$515$	0	0
$516$	0	0
$517$	792.132	0.0673847
$518$	−9647.27	−0.818295
$519$	0	0
$520$	0	0
$521$	−17101.2	−1.43804	−0.719018	−	0.694992i	$-0.755408\pi$
$521$	−17101.2	−1.43804	−0.719018	+	0.694992i	$0.755408\pi$
$522$	0	0
$523$	4938.15	0.412869	0.206434	−	0.978460i	$-0.433814\pi$
$523$	4938.15	0.412869	0.206434	+	0.978460i	$0.433814\pi$
$524$	3184.34	0.265474
$525$	0	0
$526$	−7259.64	−0.601779
$527$	−5115.90	−0.422869
$528$	0	0
$529$	8193.97	0.673458
$530$	0	0
$531$	0	0
$532$	−4746.20	−0.386793
$533$	−2363.90	−0.192105
$534$	0	0
$535$	0	0
$536$	360.215	0.0290278
$537$	0	0
$538$	32314.2	2.58953
$539$	400.542	0.0320085
$540$	0	0
$541$	11481.7	0.912451	0.456226	−	0.889864i	$-0.349201\pi$
$541$	11481.7	0.912451	0.456226	+	0.889864i	$0.349201\pi$
$542$	19795.2	1.56878
$543$	0	0
$544$	−4908.91	−0.386890
$545$	0	0
$546$	0	0
$547$	18561.2	1.45086	0.725428	−	0.688298i	$-0.241642\pi$
$547$	18561.2	1.45086	0.725428	+	0.688298i	$0.241642\pi$
$548$	22810.6	1.77814
$549$	0	0
$550$	0	0
$551$	−7356.26	−0.568761
$552$	0	0
$553$	−8150.68	−0.626767
$554$	−7324.53	−0.561714
$555$	0	0
$556$	−20093.3	−1.53264
$557$	−17604.9	−1.33922	−0.669608	−	0.742714i	$-0.733538\pi$
$557$	−17604.9	−1.33922	−0.669608	+	0.742714i	$0.733538\pi$
$558$	0	0
$559$	−9469.36	−0.716478
$560$	0	0
$561$	0	0
$562$	−7306.47	−0.548407
$563$	3586.87	0.268505	0.134253	−	0.990947i	$-0.457137\pi$
$563$	3586.87	0.268505	0.134253	+	0.990947i	$0.457137\pi$
$564$	0	0
$565$	0	0
$566$	−30283.2	−2.24894
$567$	0	0
$568$	−1079.67	−0.0797573
$569$	3213.58	0.236767	0.118383	−	0.992968i	$-0.462229\pi$
$569$	3213.58	0.236767	0.118383	+	0.992968i	$0.462229\pi$
$570$	0	0
$571$	−14706.3	−1.07783	−0.538913	−	0.842361i	$-0.681165\pi$
$571$	−14706.3	−1.07783	−0.538913	+	0.842361i	$0.681165\pi$
$572$	−1392.77	−0.101809
$573$	0	0
$574$	3534.64	0.257026
$575$	0	0
$576$	0	0
$577$	−3632.05	−0.262052	−0.131026	−	0.991379i	$-0.541827\pi$
$577$	−3632.05	−0.262052	−0.131026	+	0.991379i	$0.541827\pi$
$578$	−18708.4	−1.34631
$579$	0	0
$580$	0	0
$581$	6224.32	0.444455
$582$	0	0
$583$	−3176.56	−0.225660
$584$	−2671.81	−0.189316
$585$	0	0
$586$	−3835.51	−0.270381
$587$	15671.9	1.10196	0.550978	−	0.834520i	$-0.314255\pi$
$587$	15671.9	1.10196	0.550978	+	0.834520i	$0.314255\pi$
$588$	0	0
$589$	20709.6	1.44877
$590$	0	0
$591$	0	0
$592$	18912.5	1.31300
$593$	6007.59	0.416024	0.208012	−	0.978126i	$-0.433301\pi$
$593$	6007.59	0.416024	0.208012	+	0.978126i	$0.433301\pi$
$594$	0	0
$595$	0	0
$596$	−4051.83	−0.278472
$597$	0	0
$598$	11264.6	0.770305
$599$	−10787.4	−0.735832	−0.367916	−	0.929859i	$-0.619929\pi$
$599$	−10787.4	−0.735832	−0.367916	+	0.929859i	$0.619929\pi$
$600$	0	0
$601$	−9839.14	−0.667798	−0.333899	−	0.942609i	$-0.608364\pi$
$601$	−9839.14	−0.667798	−0.333899	+	0.942609i	$0.608364\pi$
$602$	14159.1	0.958609
$603$	0	0
$604$	20033.6	1.34959
$605$	0	0
$606$	0	0
$607$	−5572.23	−0.372603	−0.186301	−	0.982493i	$-0.559650\pi$
$607$	−5572.23	−0.372603	−0.186301	+	0.982493i	$0.559650\pi$
$608$	19871.7	1.32550
$609$	0	0
$610$	0	0
$611$	−1862.92	−0.123348
$612$	0	0
$613$	27431.5	1.80742	0.903709	−	0.428147i	$-0.140833\pi$
$613$	27431.5	1.80742	0.903709	+	0.428147i	$0.140833\pi$
$614$	7921.75	0.520677
$615$	0	0
$616$	202.765	0.0132624
$617$	−6390.91	−0.416999	−0.208499	−	0.978022i	$-0.566858\pi$
$617$	−6390.91	−0.416999	−0.208499	+	0.978022i	$0.566858\pi$
$618$	0	0
$619$	5725.64	0.371782	0.185891	−	0.982570i	$-0.440483\pi$
$619$	5725.64	0.371782	0.185891	+	0.982570i	$0.440483\pi$
$620$	0	0
$621$	0	0
$622$	18336.0	1.18201
$623$	6571.80	0.422622
$624$	0	0
$625$	0	0
$626$	30623.1	1.95518
$627$	0	0
$628$	−5706.63	−0.362611
$629$	−6342.51	−0.402055
$630$	0	0
$631$	16835.6	1.06215	0.531074	−	0.847326i	$-0.321789\pi$
$631$	16835.6	1.06215	0.531074	+	0.847326i	$0.321789\pi$
$632$	−4126.08	−0.259694
$633$	0	0
$634$	−8314.91	−0.520863
$635$	0	0
$636$	0	0
$637$	−941.987	−0.0585917
$638$	3227.78	0.200297
$639$	0	0
$640$	0	0
$641$	−12957.8	−0.798446	−0.399223	−	0.916854i	$-0.630720\pi$
$641$	−12957.8	−0.798446	−0.399223	+	0.916854i	$0.630720\pi$
$642$	0	0
$643$	−5665.07	−0.347447	−0.173724	−	0.984794i	$-0.555580\pi$
$643$	−5665.07	−0.347447	−0.173724	+	0.984794i	$0.555580\pi$
$644$	−8852.68	−0.541684
$645$	0	0
$646$	−5936.89	−0.361585
$647$	11556.3	0.702201	0.351101	−	0.936338i	$-0.385808\pi$
$647$	11556.3	0.702201	0.351101	+	0.936338i	$0.385808\pi$
$648$	0	0
$649$	−922.852	−0.0558168
$650$	0	0
$651$	0	0
$652$	−27383.4	−1.64481
$653$	−21891.6	−1.31192	−0.655961	−	0.754795i	$-0.727736\pi$
$653$	−21891.6	−1.31192	−0.655961	+	0.754795i	$0.727736\pi$
$654$	0	0
$655$	0	0
$656$	−6929.30	−0.412414
$657$	0	0
$658$	2785.54	0.165033
$659$	−12118.3	−0.716332	−0.358166	−	0.933658i	$-0.616598\pi$
$659$	−12118.3	−0.716332	−0.358166	+	0.933658i	$0.616598\pi$
$660$	0	0
$661$	739.495	0.0435144	0.0217572	−	0.999763i	$-0.493074\pi$
$661$	739.495	0.0435144	0.0217572	+	0.999763i	$0.493074\pi$
$662$	12949.9	0.760292
$663$	0	0
$664$	3150.91	0.184155
$665$	0	0
$666$	0	0
$667$	−13721.0	−0.796520
$668$	24167.8	1.39982
$669$	0	0
$670$	0	0
$671$	2837.79	0.163267
$672$	0	0
$673$	−21080.0	−1.20739	−0.603695	−	0.797215i	$-0.706306\pi$
$673$	−21080.0	−1.20739	−0.603695	+	0.797215i	$0.706306\pi$
$674$	−20771.9	−1.18710
$675$	0	0
$676$	−16196.4	−0.921505
$677$	−9588.70	−0.544348	−0.272174	−	0.962248i	$-0.587743\pi$
$677$	−9588.70	−0.544348	−0.272174	+	0.962248i	$0.587743\pi$
$678$	0	0
$679$	−8442.39	−0.477156
$680$	0	0
$681$	0	0
$682$	−9086.97	−0.510202
$683$	33084.2	1.85348	0.926742	−	0.375698i	$-0.122597\pi$
$683$	33084.2	1.85348	0.926742	+	0.375698i	$0.122597\pi$
$684$	0	0
$685$	0	0
$686$	1408.51	0.0783925
$687$	0	0
$688$	−27757.5	−1.53815
$689$	7470.58	0.413072
$690$	0	0
$691$	−6292.90	−0.346445	−0.173222	−	0.984883i	$-0.555418\pi$
$691$	−6292.90	−0.346445	−0.173222	+	0.984883i	$0.555418\pi$
$692$	7993.61	0.439120
$693$	0	0
$694$	−32647.3	−1.78570
$695$	0	0
$696$	0	0
$697$	2323.82	0.126285
$698$	19349.2	1.04925
$699$	0	0
$700$	0	0
$701$	−23822.6	−1.28355	−0.641775	−	0.766893i	$-0.721802\pi$
$701$	−23822.6	−1.28355	−0.641775	+	0.766893i	$0.721802\pi$
$702$	0	0
$703$	25675.0	1.37746
$704$	−5034.20	−0.269508
$705$	0	0
$706$	1266.83	0.0675322
$707$	−3054.85	−0.162503
$708$	0	0
$709$	10562.0	0.559471	0.279736	−	0.960077i	$-0.409753\pi$
$709$	10562.0	0.559471	0.279736	+	0.960077i	$0.409753\pi$
$710$	0	0
$711$	0	0
$712$	3326.82	0.175109
$713$	38627.8	2.02892
$714$	0	0
$715$	0	0
$716$	−40305.2	−2.10374
$717$	0	0
$718$	31204.5	1.62193
$719$	3538.42	0.183534	0.0917670	−	0.995781i	$-0.470749\pi$
$719$	3538.42	0.183534	0.0917670	+	0.995781i	$0.470749\pi$
$720$	0	0
$721$	−12358.3	−0.638347
$722$	−4133.14	−0.213047
$723$	0	0
$724$	−3562.92	−0.182894
$725$	0	0
$726$	0	0
$727$	14066.6	0.717608	0.358804	−	0.933413i	$-0.383185\pi$
$727$	14066.6	0.717608	0.358804	+	0.933413i	$0.383185\pi$
$728$	−476.858	−0.0242768
$729$	0	0
$730$	0	0
$731$	9308.79	0.470996
$732$	0	0
$733$	−34204.1	−1.72354	−0.861771	−	0.507298i	$-0.830644\pi$
$733$	−34204.1	−1.72354	−0.861771	+	0.507298i	$0.830644\pi$
$734$	−8448.94	−0.424872
$735$	0	0
$736$	37064.9	1.85629
$737$	830.943	0.0415308
$738$	0	0
$739$	−24949.9	−1.24195	−0.620973	−	0.783832i	$-0.713262\pi$
$739$	−24949.9	−1.24195	−0.620973	+	0.783832i	$0.713262\pi$
$740$	0	0
$741$	0	0
$742$	−11170.4	−0.552668
$743$	−16171.5	−0.798487	−0.399244	−	0.916845i	$-0.630727\pi$
$743$	−16171.5	−0.798487	−0.399244	+	0.916845i	$0.630727\pi$
$744$	0	0
$745$	0	0
$746$	−23248.4	−1.14100
$747$	0	0
$748$	1369.15	0.0669265
$749$	2132.19	0.104017
$750$	0	0
$751$	18661.4	0.906744	0.453372	−	0.891321i	$-0.350221\pi$
$751$	18661.4	0.906744	0.453372	+	0.891321i	$0.350221\pi$
$752$	−5460.77	−0.264806
$753$	0	0
$754$	−7591.05	−0.366644
$755$	0	0
$756$	0	0
$757$	−23051.3	−1.10675	−0.553377	−	0.832931i	$-0.686661\pi$
$757$	−23051.3	−1.10675	−0.553377	+	0.832931i	$0.686661\pi$
$758$	−7485.71	−0.358698
$759$	0	0
$760$	0	0
$761$	18693.9	0.890478	0.445239	−	0.895412i	$-0.353119\pi$
$761$	18693.9	0.890478	0.445239	+	0.895412i	$0.353119\pi$
$762$	0	0
$763$	−7264.30	−0.344673
$764$	38993.5	1.84651
$765$	0	0
$766$	−7746.14	−0.365378
$767$	2170.35	0.102173
$768$	0	0
$769$	6553.93	0.307335	0.153668	−	0.988123i	$-0.450891\pi$
$769$	6553.93	0.307335	0.153668	+	0.988123i	$0.450891\pi$
$770$	0	0
$771$	0	0
$772$	2676.23	0.124767
$773$	10818.2	0.503368	0.251684	−	0.967809i	$-0.419016\pi$
$773$	10818.2	0.503368	0.251684	+	0.967809i	$0.419016\pi$
$774$	0	0
$775$	0	0
$776$	−4273.75	−0.197705
$777$	0	0
$778$	37284.7	1.71815
$779$	−9407.01	−0.432658
$780$	0	0
$781$	−2490.59	−0.114111
$782$	−11073.6	−0.506381
$783$	0	0
$784$	−2761.24	−0.125786
$785$	0	0
$786$	0	0
$787$	−28571.9	−1.29413	−0.647064	−	0.762436i	$-0.724003\pi$
$787$	−28571.9	−1.29413	−0.647064	+	0.762436i	$0.724003\pi$
$788$	8445.93	0.381819
$789$	0	0
$790$	0	0
$791$	−10357.4	−0.465571
$792$	0	0
$793$	−6673.88	−0.298860
$794$	−10762.8	−0.481054
$795$	0	0
$796$	36558.7	1.62788
$797$	531.831	0.0236367	0.0118183	−	0.999930i	$-0.496238\pi$
$797$	531.831	0.0236367	0.0118183	+	0.999930i	$0.496238\pi$
$798$	0	0
$799$	1831.33	0.0810862
$800$	0	0
$801$	0	0
$802$	44152.9	1.94401
$803$	−6163.33	−0.270858
$804$	0	0
$805$	0	0
$806$	21370.6	0.933929
$807$	0	0
$808$	−1546.44	−0.0673313
$809$	−44326.9	−1.92639	−0.963196	−	0.268801i	$-0.913373\pi$
$809$	−44326.9	−1.92639	−0.963196	+	0.268801i	$0.913373\pi$
$810$	0	0
$811$	9509.56	0.411746	0.205873	−	0.978579i	$-0.433997\pi$
$811$	9509.56	0.411746	0.205873	+	0.978579i	$0.433997\pi$
$812$	5965.70	0.257827
$813$	0	0
$814$	−11265.7	−0.485089
$815$	0	0
$816$	0	0
$817$	−37682.7	−1.61365
$818$	54952.7	2.34887
$819$	0	0
$820$	0	0
$821$	−45600.8	−1.93846	−0.969232	−	0.246148i	$-0.920835\pi$
$821$	−45600.8	−1.93846	−0.969232	+	0.246148i	$0.920835\pi$
$822$	0	0
$823$	14111.4	0.597681	0.298841	−	0.954303i	$-0.403400\pi$
$823$	14111.4	0.597681	0.298841	+	0.954303i	$0.403400\pi$
$824$	−6256.10	−0.264492
$825$	0	0
$826$	−3245.23	−0.136702
$827$	−10907.3	−0.458628	−0.229314	−	0.973352i	$-0.573648\pi$
$827$	−10907.3	−0.458628	−0.229314	+	0.973352i	$0.573648\pi$
$828$	0	0
$829$	−6653.10	−0.278736	−0.139368	−	0.990241i	$-0.544507\pi$
$829$	−6653.10	−0.278736	−0.139368	+	0.990241i	$0.544507\pi$
$830$	0	0
$831$	0	0
$832$	11839.3	0.493336
$833$	926.014	0.0385168
$834$	0	0
$835$	0	0
$836$	−5542.43	−0.229293
$837$	0	0
$838$	−9592.28	−0.395418
$839$	4960.81	0.204131	0.102066	−	0.994778i	$-0.467455\pi$
$839$	4960.81	0.204131	0.102066	+	0.994778i	$0.467455\pi$
$840$	0	0
$841$	−15142.6	−0.620878
$842$	17504.6	0.716447
$843$	0	0
$844$	−8993.59	−0.366791
$845$	0	0
$846$	0	0
$847$	−8849.26	−0.358990
$848$	21898.5	0.886789
$849$	0	0
$850$	0	0
$851$	47889.4	1.92906
$852$	0	0
$853$	26053.2	1.04577	0.522887	−	0.852402i	$-0.324855\pi$
$853$	26053.2	1.04577	0.522887	+	0.852402i	$0.324855\pi$
$854$	9979.15	0.399859
$855$	0	0
$856$	1079.37	0.0430983
$857$	−42983.0	−1.71327	−0.856635	−	0.515924i	$-0.827449\pi$
$857$	−42983.0	−1.71327	−0.856635	+	0.515924i	$0.827449\pi$
$858$	0	0
$859$	−31824.0	−1.26405	−0.632027	−	0.774946i	$-0.717777\pi$
$859$	−31824.0	−1.26405	−0.632027	+	0.774946i	$0.717777\pi$
$860$	0	0
$861$	0	0
$862$	−21492.2	−0.849221
$863$	−20388.4	−0.804207	−0.402103	−	0.915594i	$-0.631721\pi$
$863$	−20388.4	−0.804207	−0.402103	+	0.915594i	$0.631721\pi$
$864$	0	0
$865$	0	0
$866$	60067.2	2.35701
$867$	0	0
$868$	−16794.8	−0.656745
$869$	−9518.04	−0.371551
$870$	0	0
$871$	−1954.20	−0.0760223
$872$	−3677.38	−0.142812
$873$	0	0
$874$	44826.7	1.73488
$875$	0	0
$876$	0	0
$877$	15298.2	0.589034	0.294517	−	0.955646i	$-0.404841\pi$
$877$	15298.2	0.589034	0.294517	+	0.955646i	$0.404841\pi$
$878$	144.943	0.00557128
$879$	0	0
$880$	0	0
$881$	−22994.9	−0.879361	−0.439681	−	0.898154i	$-0.644908\pi$
$881$	−22994.9	−0.879361	−0.439681	+	0.898154i	$0.644908\pi$
$882$	0	0
$883$	−799.661	−0.0304765	−0.0152382	−	0.999884i	$-0.504851\pi$
$883$	−799.661	−0.0304765	−0.0152382	+	0.999884i	$0.504851\pi$
$884$	−3219.94	−0.122509
$885$	0	0
$886$	−11520.2	−0.436828
$887$	6611.21	0.250262	0.125131	−	0.992140i	$-0.460065\pi$
$887$	6611.21	0.250262	0.125131	+	0.992140i	$0.460065\pi$
$888$	0	0
$889$	11434.7	0.431391
$890$	0	0
$891$	0	0
$892$	44695.3	1.67770
$893$	−7413.38	−0.277804
$894$	0	0
$895$	0	0
$896$	−3156.58	−0.117694
$897$	0	0
$898$	−61198.4	−2.27418
$899$	−26030.8	−0.965712
$900$	0	0
$901$	−7343.90	−0.271544
$902$	4127.61	0.152366
$903$	0	0
$904$	−5243.18	−0.192904
$905$	0	0
$906$	0	0
$907$	−11367.9	−0.416169	−0.208084	−	0.978111i	$-0.566723\pi$
$907$	−11367.9	−0.416169	−0.208084	+	0.978111i	$0.566723\pi$
$908$	−51685.3	−1.88903
$909$	0	0
$910$	0	0
$911$	18431.4	0.670318	0.335159	−	0.942162i	$-0.391210\pi$
$911$	18431.4	0.670318	0.335159	+	0.942162i	$0.391210\pi$
$912$	0	0
$913$	7268.51	0.263475
$914$	10815.8	0.391415
$915$	0	0
$916$	164.546	0.00593533
$917$	2515.01	0.0905703
$918$	0	0
$919$	−16263.1	−0.583755	−0.291877	−	0.956456i	$-0.594280\pi$
$919$	−16263.1	−0.583755	−0.291877	+	0.956456i	$0.594280\pi$
$920$	0	0
$921$	0	0
$922$	75621.0	2.70113
$923$	5857.33	0.208880
$924$	0	0
$925$	0	0
$926$	−60820.6	−2.15841
$927$	0	0
$928$	−24977.6	−0.883544
$929$	3972.84	0.140306	0.0701532	−	0.997536i	$-0.477651\pi$
$929$	3972.84	0.140306	0.0701532	+	0.997536i	$0.477651\pi$
$930$	0	0
$931$	−3748.58	−0.131960
$932$	−15356.1	−0.539706
$933$	0	0
$934$	−59161.9	−2.07263
$935$	0	0
$936$	0	0
$937$	−144.199	−0.00502750	−0.00251375	−	0.999997i	$-0.500800\pi$
$937$	−144.199	−0.00502750	−0.00251375	+	0.999997i	$0.500800\pi$
$938$	2922.02	0.101714
$939$	0	0
$940$	0	0
$941$	−48589.1	−1.68327	−0.841637	−	0.540044i	$-0.818407\pi$
$941$	−48589.1	−1.68327	−0.841637	+	0.540044i	$0.818407\pi$
$942$	0	0
$943$	−17546.1	−0.605916
$944$	6361.93	0.219347
$945$	0	0
$946$	16534.5	0.568268
$947$	−11977.2	−0.410989	−0.205494	−	0.978658i	$-0.565880\pi$
$947$	−11977.2	−0.410989	−0.205494	+	0.978658i	$0.565880\pi$
$948$	0	0
$949$	14494.8	0.495808
$950$	0	0
$951$	0	0
$952$	468.772	0.0159590
$953$	−45863.7	−1.55894	−0.779471	−	0.626439i	$-0.784512\pi$
$953$	−45863.7	−1.55894	−0.779471	+	0.626439i	$0.784512\pi$
$954$	0	0
$955$	0	0
$956$	17172.3	0.580953
$957$	0	0
$958$	27222.3	0.918073
$959$	18016.0	0.606638
$960$	0	0
$961$	43491.7	1.45989
$962$	26494.5	0.887959
$963$	0	0
$964$	49747.8	1.66210
$965$	0	0
$966$	0	0
$967$	−19843.3	−0.659894	−0.329947	−	0.944000i	$-0.607031\pi$
$967$	−19843.3	−0.659894	−0.329947	+	0.944000i	$0.607031\pi$
$968$	−4479.73	−0.148744
$969$	0	0
$970$	0	0
$971$	45035.7	1.48843	0.744214	−	0.667941i	$-0.232824\pi$
$971$	45035.7	1.48843	0.744214	+	0.667941i	$0.232824\pi$
$972$	0	0
$973$	−15869.8	−0.522881
$974$	15976.0	0.525568
$975$	0	0
$976$	−19563.1	−0.641598
$977$	−25098.2	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$977$	−25098.2	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$978$	0	0
$979$	7674.29	0.250533
$980$	0	0
$981$	0	0
$982$	52579.4	1.70863
$983$	−388.045	−0.0125908	−0.00629538	−	0.999980i	$-0.502004\pi$
$983$	−388.045	−0.0125908	−0.00629538	+	0.999980i	$0.502004\pi$
$984$	0	0
$985$	0	0
$986$	7462.33	0.241023
$987$	0	0
$988$	13034.6	0.419722
$989$	−70286.4	−2.25983
$990$	0	0
$991$	53520.8	1.71558	0.857792	−	0.513997i	$-0.171836\pi$
$991$	53520.8	1.71558	0.857792	+	0.513997i	$0.171836\pi$
$992$	70317.7	2.25059
$993$	0	0
$994$	−8758.20	−0.279470
$995$	0	0
$996$	0	0
$997$	5968.90	0.189606	0.0948028	−	0.995496i	$-0.469778\pi$
$997$	5968.90	0.189606	0.0948028	+	0.995496i	$0.469778\pi$
$998$	−23992.2	−0.760981
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bd.1.3		3
3.2	odd	2		525.4.a.q.1.1		3
5.2	odd	4		315.4.d.a.64.6		6
5.3	odd	4		315.4.d.a.64.1		6
5.4	even	2		1575.4.a.bc.1.1		3
15.2	even	4		105.4.d.a.64.1	✓	6
15.8	even	4		105.4.d.a.64.6	yes	6
15.14	odd	2		525.4.a.r.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.d.a.64.1	✓	6	15.2	even	4
105.4.d.a.64.6	yes	6	15.8	even	4
315.4.d.a.64.1		6	5.3	odd	4
315.4.d.a.64.6		6	5.2	odd	4
525.4.a.q.1.1		3	3.2	odd	2
525.4.a.r.1.3		3	15.14	odd	2
1575.4.a.bc.1.1		3	5.4	even	2
1575.4.a.bd.1.3		3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+4.10645 q^{2} +8.86293 q^{4} +7.00000 q^{7} +3.54358 q^{8} +8.17432 q^{11} -19.2242 q^{13} +28.7451 q^{14} -56.3519 q^{16} +18.8982 q^{17} -76.5016 q^{19} +33.5675 q^{22} -142.692 q^{23} -78.9433 q^{26} +62.0405 q^{28} +96.1582 q^{29} -270.708 q^{31} -259.755 q^{32} +77.6047 q^{34} -335.614 q^{37} -314.150 q^{38} +122.965 q^{41} +492.574 q^{43} +72.4485 q^{44} -585.957 q^{46} +96.9049 q^{47} +49.0000 q^{49} -170.383 q^{52} -388.602 q^{53} +24.8051 q^{56} +394.869 q^{58} -112.896 q^{59} +347.160 q^{61} -1111.65 q^{62} -615.855 q^{64} +101.653 q^{67} +167.494 q^{68} -304.685 q^{71} -753.987 q^{73} -1378.18 q^{74} -678.029 q^{76} +57.2203 q^{77} -1164.38 q^{79} +504.949 q^{82} +889.188 q^{83} +2022.73 q^{86} +28.9664 q^{88} +938.829 q^{89} -134.570 q^{91} -1264.67 q^{92} +397.935 q^{94} -1206.06 q^{97} +201.216 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} + 21 q^{7} + 21 q^{8} + 66 q^{11} - 102 q^{13} + 7 q^{14} - 69 q^{16} + 152 q^{17} - 138 q^{19} - 186 q^{22} - 180 q^{23} + 98 q^{26} + 21 q^{28} - 170 q^{29} - 366 q^{31} - 151 q^{32}+ \cdots + 49 q^{98}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 + 21 * q^7 + 21 * q^8 + 66 * q^11 - 102 * q^13 + 7 * q^14 - 69 * q^16 + 152 * q^17 - 138 * q^19 - 186 * q^22 - 180 * q^23 + 98 * q^26 + 21 * q^28 - 170 * q^29 - 366 * q^31 - 151 * q^32 - 36 * q^34 - 252 * q^37 - 234 * q^38 + 206 * q^41 + 108 * q^43 + 306 * q^44 - 672 * q^46 - 24 * q^47 + 147 * q^49 - 78 * q^52 + 354 * q^53 + 147 * q^56 + 858 * q^58 + 880 * q^59 - 870 * q^61 - 1366 * q^62 - 813 * q^64 + 96 * q^67 - 512 * q^68 + 1018 * q^71 - 1554 * q^73 - 980 * q^74 - 450 * q^76 + 462 * q^77 - 1620 * q^79 + 1638 * q^82 - 872 * q^83 + 2932 * q^86 + 1326 * q^88 + 1938 * q^89 - 714 * q^91 - 708 * q^92 + 2112 * q^94 - 1878 * q^97 + 49 * q^98

Introduction

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