LMFDB - Embedded newform 1575.4.a.bs.1.2

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 50x^{6} + 698x^{4} - 2653x^{2} + 2268 $ x^8 - 50x^6 + 698x^4 - 2653*x^2 + 2268
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.95827$ 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)

$f(q)$	$=$	$q-3.95827 q^{2} +7.66789 q^{4} +7.00000 q^{7} +1.31458 q^{8} -18.4481 q^{11} +48.8596 q^{13} -27.7079 q^{14} -66.5466 q^{16} -124.724 q^{17} -122.717 q^{19} +73.0227 q^{22} -140.628 q^{23} -193.400 q^{26} +53.6752 q^{28} +196.100 q^{29} +289.591 q^{31} +252.893 q^{32} +493.690 q^{34} -401.649 q^{37} +485.747 q^{38} +303.093 q^{41} +202.027 q^{43} -141.458 q^{44} +556.643 q^{46} -308.944 q^{47} +49.0000 q^{49} +374.650 q^{52} -461.082 q^{53} +9.20203 q^{56} -776.217 q^{58} -111.900 q^{59} +416.658 q^{61} -1146.28 q^{62} -468.644 q^{64} +452.322 q^{67} -956.367 q^{68} -1008.59 q^{71} +669.043 q^{73} +1589.84 q^{74} -940.980 q^{76} -129.137 q^{77} +517.398 q^{79} -1199.72 q^{82} +35.3950 q^{83} -799.677 q^{86} -24.2515 q^{88} +1082.09 q^{89} +342.018 q^{91} -1078.32 q^{92} +1222.88 q^{94} +203.138 q^{97} -193.955 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 36 q^{4} + 56 q^{7} - 38 q^{13} + 320 q^{16} + 68 q^{19} + 110 q^{22} + 252 q^{28} + 534 q^{31} + 118 q^{34} + 14 q^{37} + 816 q^{43} + 1588 q^{46} + 392 q^{49} - 266 q^{52} + 716 q^{58} + 874 q^{61}+ \cdots + 428 q^{97}+O(q^{100}) $ 8 * q + 36 * q^4 + 56 * q^7 - 38 * q^13 + 320 * q^16 + 68 * q^19 + 110 * q^22 + 252 * q^28 + 534 * q^31 + 118 * q^34 + 14 * q^37 + 816 * q^43 + 1588 * q^46 + 392 * q^49 - 266 * q^52 + 716 * q^58 + 874 * q^61 + 2502 * q^64 + 254 * q^67 - 1532 * q^73 + 4304 * q^76 + 5190 * q^79 + 1158 * q^82 - 4300 * q^88 - 266 * q^91 + 6236 * q^94 + 428 * q^97

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 $

$2$	−3.95827	−1.39946	−0.699730	−	0.714408i	$-0.746696\pi$
$2$	−3.95827	−1.39946	−0.699730	+	0.714408i	$0.746696\pi$
$3$	0	0
$3$	0	0
$4$	7.66789	0.958486
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	1.31458	0.0580966
$9$	0	0
$10$	0	0
$11$	−18.4481	−0.505666	−0.252833	−	0.967510i	$-0.581362\pi$
$11$	−18.4481	−0.505666	−0.252833	+	0.967510i	$0.581362\pi$
$12$	0	0
$13$	48.8596	1.04240	0.521201	−	0.853434i	$-0.325484\pi$
$13$	48.8596	1.04240	0.521201	+	0.853434i	$0.325484\pi$
$14$	−27.7079	−0.528946
$15$	0	0
$16$	−66.5466	−1.03979
$17$	−124.724	−1.77941	−0.889703	−	0.456539i	$-0.849089\pi$
$17$	−124.724	−1.77941	−0.889703	+	0.456539i	$0.849089\pi$
$18$	0	0
$19$	−122.717	−1.48175	−0.740874	−	0.671644i	$-0.765588\pi$
$19$	−122.717	−1.48175	−0.740874	+	0.671644i	$0.765588\pi$
$20$	0	0
$21$	0	0
$22$	73.0227	0.707659
$23$	−140.628	−1.27491	−0.637456	−	0.770487i	$-0.720013\pi$
$23$	−140.628	−1.27491	−0.637456	+	0.770487i	$0.720013\pi$
$24$	0	0
$25$	0	0
$26$	−193.400	−1.45880
$27$	0	0
$28$	53.6752	0.362274
$29$	196.100	1.25569	0.627843	−	0.778340i	$-0.283938\pi$
$29$	196.100	1.25569	0.627843	+	0.778340i	$0.283938\pi$
$30$	0	0
$31$	289.591	1.67781	0.838905	−	0.544277i	$-0.183196\pi$
$31$	289.591	1.67781	0.838905	+	0.544277i	$0.183196\pi$
$32$	252.893	1.39705
$33$	0	0
$34$	493.690	2.49021
$35$	0	0
$36$	0	0
$37$	−401.649	−1.78461	−0.892307	−	0.451429i	$-0.850914\pi$
$37$	−401.649	−1.78461	−0.892307	+	0.451429i	$0.850914\pi$
$38$	485.747	2.07365
$39$	0	0
$40$	0	0
$41$	303.093	1.15452	0.577258	−	0.816562i	$-0.304123\pi$
$41$	303.093	1.15452	0.577258	+	0.816562i	$0.304123\pi$
$42$	0	0
$43$	202.027	0.716485	0.358242	−	0.933629i	$-0.383376\pi$
$43$	202.027	0.716485	0.358242	+	0.933629i	$0.383376\pi$
$44$	−141.458	−0.484674
$45$	0	0
$46$	556.643	1.78419
$47$	−308.944	−0.958811	−0.479405	−	0.877594i	$-0.659148\pi$
$47$	−308.944	−0.958811	−0.479405	+	0.877594i	$0.659148\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	374.650	0.999128
$53$	−461.082	−1.19499	−0.597495	−	0.801873i	$-0.703837\pi$
$53$	−461.082	−1.19499	−0.597495	+	0.801873i	$0.703837\pi$
$54$	0	0
$55$	0	0
$56$	9.20203	0.0219584
$57$	0	0
$58$	−776.217	−1.75728
$59$	−111.900	−0.246918	−0.123459	−	0.992350i	$-0.539399\pi$
$59$	−111.900	−0.246918	−0.123459	+	0.992350i	$0.539399\pi$
$60$	0	0
$61$	416.658	0.874551	0.437276	−	0.899327i	$-0.355943\pi$
$61$	416.658	0.874551	0.437276	+	0.899327i	$0.355943\pi$
$62$	−1146.28	−2.34803
$63$	0	0
$64$	−468.644	−0.915321
$65$	0	0
$66$	0	0
$67$	452.322	0.824775	0.412388	−	0.911009i	$-0.364695\pi$
$67$	452.322	0.824775	0.412388	+	0.911009i	$0.364695\pi$
$68$	−956.367	−1.70554
$69$	0	0
$70$	0	0
$71$	−1008.59	−1.68589	−0.842944	−	0.538001i	$-0.819180\pi$
$71$	−1008.59	−1.68589	−0.842944	+	0.538001i	$0.819180\pi$
$72$	0	0
$73$	669.043	1.07268	0.536339	−	0.844002i	$-0.319807\pi$
$73$	669.043	1.07268	0.536339	+	0.844002i	$0.319807\pi$
$74$	1589.84	2.49749
$75$	0	0
$76$	−940.980	−1.42024
$77$	−129.137	−0.191124
$78$	0	0
$79$	517.398	0.736859	0.368429	−	0.929656i	$-0.379896\pi$
$79$	517.398	0.736859	0.368429	+	0.929656i	$0.379896\pi$
$80$	0	0
$81$	0	0
$82$	−1199.72	−1.61570
$83$	35.3950	0.0468085	0.0234042	−	0.999726i	$-0.492550\pi$
$83$	35.3950	0.0468085	0.0234042	+	0.999726i	$0.492550\pi$
$84$	0	0
$85$	0	0
$86$	−799.677	−1.00269
$87$	0	0
$88$	−24.2515	−0.0293775
$89$	1082.09	1.28878	0.644389	−	0.764698i	$-0.277112\pi$
$89$	1082.09	1.28878	0.644389	+	0.764698i	$0.277112\pi$
$90$	0	0
$91$	342.018	0.393991
$92$	−1078.32	−1.22199
$93$	0	0
$94$	1222.88	1.34182
$95$	0	0
$96$	0	0
$97$	203.138	0.212634	0.106317	−	0.994332i	$-0.466094\pi$
$97$	203.138	0.212634	0.106317	+	0.994332i	$0.466094\pi$
$98$	−193.955	−0.199923
$99$	0	0
$100$	0	0
$101$	−1567.11	−1.54390	−0.771949	−	0.635684i	$-0.780718\pi$
$101$	−1567.11	−1.54390	−0.771949	+	0.635684i	$0.780718\pi$
$102$	0	0
$103$	−473.964	−0.453409	−0.226704	−	0.973964i	$-0.572795\pi$
$103$	−473.964	−0.453409	−0.226704	+	0.973964i	$0.572795\pi$
$104$	64.2297	0.0605600
$105$	0	0
$106$	1825.09	1.67234
$107$	−496.898	−0.448943	−0.224472	−	0.974481i	$-0.572066\pi$
$107$	−496.898	−0.448943	−0.224472	+	0.974481i	$0.572066\pi$
$108$	0	0
$109$	1125.46	0.988983	0.494492	−	0.869182i	$-0.335354\pi$
$109$	1125.46	0.988983	0.494492	+	0.869182i	$0.335354\pi$
$110$	0	0
$111$	0	0
$112$	−465.826	−0.393004
$113$	184.508	0.153603	0.0768013	−	0.997046i	$-0.475529\pi$
$113$	184.508	0.153603	0.0768013	+	0.997046i	$0.475529\pi$
$114$	0	0
$115$	0	0
$116$	1503.67	1.20356
$117$	0	0
$118$	442.931	0.345552
$119$	−873.065	−0.672553
$120$	0	0
$121$	−990.666	−0.744302
$122$	−1649.25	−1.22390
$123$	0	0
$124$	2220.55	1.60816
$125$	0	0
$126$	0	0
$127$	2835.92	1.98148	0.990738	−	0.135784i	$-0.0433553\pi$
$127$	2835.92	1.98148	0.990738	+	0.135784i	$0.0433553\pi$
$128$	−168.121	−0.116093
$129$	0	0
$130$	0	0
$131$	−189.998	−0.126719	−0.0633595	−	0.997991i	$-0.520181\pi$
$131$	−189.998	−0.126719	−0.0633595	+	0.997991i	$0.520181\pi$
$132$	0	0
$133$	−859.019	−0.560048
$134$	−1790.41	−1.15424
$135$	0	0
$136$	−163.959	−0.103377
$137$	−2230.24	−1.39082	−0.695408	−	0.718615i	$-0.744777\pi$
$137$	−2230.24	−1.39082	−0.695408	+	0.718615i	$0.744777\pi$
$138$	0	0
$139$	−398.286	−0.243037	−0.121519	−	0.992589i	$-0.538776\pi$
$139$	−398.286	−0.243037	−0.121519	+	0.992589i	$0.538776\pi$
$140$	0	0
$141$	0	0
$142$	3992.29	2.35933
$143$	−901.370	−0.527107
$144$	0	0
$145$	0	0
$146$	−2648.25	−1.50117
$147$	0	0
$148$	−3079.80	−1.71053
$149$	3556.30	1.95532	0.977662	−	0.210183i	$-0.0674060\pi$
$149$	3556.30	1.95532	0.977662	+	0.210183i	$0.0674060\pi$
$150$	0	0
$151$	−955.955	−0.515196	−0.257598	−	0.966252i	$-0.582931\pi$
$151$	−955.955	−0.515196	−0.257598	+	0.966252i	$0.582931\pi$
$152$	−161.321	−0.0860845
$153$	0	0
$154$	511.159	0.267470
$155$	0	0
$156$	0	0
$157$	−3095.78	−1.57370	−0.786848	−	0.617147i	$-0.788288\pi$
$157$	−3095.78	−1.57370	−0.786848	+	0.617147i	$0.788288\pi$
$158$	−2048.00	−1.03120
$159$	0	0
$160$	0	0
$161$	−984.396	−0.481871
$162$	0	0
$163$	−2665.44	−1.28082	−0.640409	−	0.768034i	$-0.721235\pi$
$163$	−2665.44	−1.28082	−0.640409	+	0.768034i	$0.721235\pi$
$164$	2324.08	1.10659
$165$	0	0
$166$	−140.103	−0.0655065
$167$	967.998	0.448538	0.224269	−	0.974527i	$-0.428000\pi$
$167$	967.998	0.448538	0.224269	+	0.974527i	$0.428000\pi$
$168$	0	0
$169$	190.265	0.0866023
$170$	0	0
$171$	0	0
$172$	1549.12	0.686741
$173$	2793.34	1.22759	0.613796	−	0.789465i	$-0.289642\pi$
$173$	2793.34	1.22759	0.613796	+	0.789465i	$0.289642\pi$
$174$	0	0
$175$	0	0
$176$	1227.66	0.525786
$177$	0	0
$178$	−4283.20	−1.80359
$179$	449.319	0.187618	0.0938091	−	0.995590i	$-0.470096\pi$
$179$	449.319	0.187618	0.0938091	+	0.995590i	$0.470096\pi$
$180$	0	0
$181$	2054.72	0.843790	0.421895	−	0.906645i	$-0.361365\pi$
$181$	2054.72	0.843790	0.421895	+	0.906645i	$0.361365\pi$
$182$	−1353.80	−0.551374
$183$	0	0
$184$	−184.866	−0.0740680
$185$	0	0
$186$	0	0
$187$	2300.92	0.899785
$188$	−2368.95	−0.919007
$189$	0	0
$190$	0	0
$191$	−42.1490	−0.0159675	−0.00798376	−	0.999968i	$-0.502541\pi$
$191$	−42.1490	−0.0159675	−0.00798376	+	0.999968i	$0.502541\pi$
$192$	0	0
$193$	3613.73	1.34778	0.673890	−	0.738831i	$-0.264622\pi$
$193$	3613.73	1.34778	0.673890	+	0.738831i	$0.264622\pi$
$194$	−804.074	−0.297573
$195$	0	0
$196$	375.727	0.136927
$197$	−2429.72	−0.878733	−0.439366	−	0.898308i	$-0.644797\pi$
$197$	−2429.72	−0.878733	−0.439366	+	0.898308i	$0.644797\pi$
$198$	0	0
$199$	3540.39	1.26116	0.630582	−	0.776123i	$-0.282816\pi$
$199$	3540.39	1.26116	0.630582	+	0.776123i	$0.282816\pi$
$200$	0	0
$201$	0	0
$202$	6203.06	2.16062
$203$	1372.70	0.474604
$204$	0	0
$205$	0	0
$206$	1876.08	0.634527
$207$	0	0
$208$	−3251.44	−1.08388
$209$	2263.90	0.749269
$210$	0	0
$211$	−2183.66	−0.712461	−0.356230	−	0.934398i	$-0.615938\pi$
$211$	−2183.66	−0.712461	−0.356230	+	0.934398i	$0.615938\pi$
$212$	−3535.53	−1.14538
$213$	0	0
$214$	1966.85	0.628278
$215$	0	0
$216$	0	0
$217$	2027.14	0.634153
$218$	−4454.86	−1.38404
$219$	0	0
$220$	0	0
$221$	−6093.95	−1.85486
$222$	0	0
$223$	−5276.91	−1.58461	−0.792305	−	0.610126i	$-0.791119\pi$
$223$	−5276.91	−1.58461	−0.792305	+	0.610126i	$0.791119\pi$
$224$	1770.25	0.528034
$225$	0	0
$226$	−730.334	−0.214960
$227$	4472.50	1.30771	0.653855	−	0.756620i	$-0.273151\pi$
$227$	4472.50	1.30771	0.653855	+	0.756620i	$0.273151\pi$
$228$	0	0
$229$	1698.75	0.490202	0.245101	−	0.969498i	$-0.421179\pi$
$229$	1698.75	0.490202	0.245101	+	0.969498i	$0.421179\pi$
$230$	0	0
$231$	0	0
$232$	257.788	0.0729510
$233$	4857.72	1.36583	0.682917	−	0.730496i	$-0.260711\pi$
$233$	4857.72	1.36583	0.682917	+	0.730496i	$0.260711\pi$
$234$	0	0
$235$	0	0
$236$	−858.039	−0.236668
$237$	0	0
$238$	3455.83	0.941210
$239$	−375.571	−0.101647	−0.0508235	−	0.998708i	$-0.516185\pi$
$239$	−375.571	−0.101647	−0.0508235	+	0.998708i	$0.516185\pi$
$240$	0	0
$241$	−1005.92	−0.268868	−0.134434	−	0.990923i	$-0.542922\pi$
$241$	−1005.92	−0.268868	−0.134434	+	0.990923i	$0.542922\pi$
$242$	3921.32	1.04162
$243$	0	0
$244$	3194.89	0.838246
$245$	0	0
$246$	0	0
$247$	−5995.91	−1.54458
$248$	380.690	0.0974751
$249$	0	0
$250$	0	0
$251$	5462.83	1.37375	0.686874	−	0.726776i	$-0.258982\pi$
$251$	5462.83	1.37375	0.686874	+	0.726776i	$0.258982\pi$
$252$	0	0
$253$	2594.33	0.644679
$254$	−11225.3	−2.77300
$255$	0	0
$256$	4414.62	1.07779
$257$	427.791	0.103832	0.0519161	−	0.998651i	$-0.483467\pi$
$257$	427.791	0.103832	0.0519161	+	0.998651i	$0.483467\pi$
$258$	0	0
$259$	−2811.54	−0.674521
$260$	0	0
$261$	0	0
$262$	752.062	0.177338
$263$	520.645	0.122070	0.0610349	−	0.998136i	$-0.480560\pi$
$263$	520.645	0.122070	0.0610349	+	0.998136i	$0.480560\pi$
$264$	0	0
$265$	0	0
$266$	3400.23	0.783764
$267$	0	0
$268$	3468.36	0.790536
$269$	1995.64	0.452328	0.226164	−	0.974089i	$-0.427381\pi$
$269$	1995.64	0.452328	0.226164	+	0.974089i	$0.427381\pi$
$270$	0	0
$271$	5212.99	1.16851	0.584256	−	0.811569i	$-0.301386\pi$
$271$	5212.99	1.16851	0.584256	+	0.811569i	$0.301386\pi$
$272$	8299.93	1.85021
$273$	0	0
$274$	8827.87	1.94639
$275$	0	0
$276$	0	0
$277$	6338.81	1.37495	0.687477	−	0.726206i	$-0.258718\pi$
$277$	6338.81	1.37495	0.687477	+	0.726206i	$0.258718\pi$
$278$	1576.52	0.340121
$279$	0	0
$280$	0	0
$281$	4861.28	1.03203	0.516014	−	0.856580i	$-0.327415\pi$
$281$	4861.28	1.03203	0.516014	+	0.856580i	$0.327415\pi$
$282$	0	0
$283$	−1040.23	−0.218499	−0.109250	−	0.994014i	$-0.534845\pi$
$283$	−1040.23	−0.218499	−0.109250	+	0.994014i	$0.534845\pi$
$284$	−7733.79	−1.61590
$285$	0	0
$286$	3567.86	0.737665
$287$	2121.65	0.436366
$288$	0	0
$289$	10643.0	2.16629
$290$	0	0
$291$	0	0
$292$	5130.15	1.02815
$293$	−3807.01	−0.759072	−0.379536	−	0.925177i	$-0.623916\pi$
$293$	−3807.01	−0.759072	−0.379536	+	0.925177i	$0.623916\pi$
$294$	0	0
$295$	0	0
$296$	−527.998	−0.103680
$297$	0	0
$298$	−14076.8	−2.73640
$299$	−6871.03	−1.32897
$300$	0	0
$301$	1414.19	0.270806
$302$	3783.93	0.720995
$303$	0	0
$304$	8166.39	1.54071
$305$	0	0
$306$	0	0
$307$	−1313.26	−0.244142	−0.122071	−	0.992521i	$-0.538954\pi$
$307$	−1313.26	−0.244142	−0.122071	+	0.992521i	$0.538954\pi$
$308$	−990.208	−0.183189
$309$	0	0
$310$	0	0
$311$	566.183	0.103232	0.0516162	−	0.998667i	$-0.483563\pi$
$311$	566.183	0.103232	0.0516162	+	0.998667i	$0.483563\pi$
$312$	0	0
$313$	−1146.47	−0.207036	−0.103518	−	0.994628i	$-0.533010\pi$
$313$	−1146.47	−0.207036	−0.103518	+	0.994628i	$0.533010\pi$
$314$	12253.9	2.20232
$315$	0	0
$316$	3967.35	0.706269
$317$	2562.45	0.454012	0.227006	−	0.973893i	$-0.427106\pi$
$317$	2562.45	0.454012	0.227006	+	0.973893i	$0.427106\pi$
$318$	0	0
$319$	−3617.68	−0.634957
$320$	0	0
$321$	0	0
$322$	3896.50	0.674359
$323$	15305.7	2.63663
$324$	0	0
$325$	0	0
$326$	10550.5	1.79245
$327$	0	0
$328$	398.438	0.0670734
$329$	−2162.61	−0.362396
$330$	0	0
$331$	−917.092	−0.152290	−0.0761450	−	0.997097i	$-0.524261\pi$
$331$	−917.092	−0.152290	−0.0761450	+	0.997097i	$0.524261\pi$
$332$	271.405	0.0448653
$333$	0	0
$334$	−3831.60	−0.627711
$335$	0	0
$336$	0	0
$337$	−2446.17	−0.395405	−0.197702	−	0.980262i	$-0.563348\pi$
$337$	−2446.17	−0.395405	−0.197702	+	0.980262i	$0.563348\pi$
$338$	−753.121	−0.121196
$339$	0	0
$340$	0	0
$341$	−5342.42	−0.848412
$342$	0	0
$343$	343.000	0.0539949
$344$	265.580	0.0416253
$345$	0	0
$346$	−11056.8	−1.71796
$347$	3598.36	0.556687	0.278343	−	0.960482i	$-0.410215\pi$
$347$	3598.36	0.556687	0.278343	+	0.960482i	$0.410215\pi$
$348$	0	0
$349$	−247.567	−0.0379712	−0.0189856	−	0.999820i	$-0.506044\pi$
$349$	−247.567	−0.0379712	−0.0189856	+	0.999820i	$0.506044\pi$
$350$	0	0
$351$	0	0
$352$	−4665.40	−0.706439
$353$	−81.3189	−0.0122611	−0.00613055	−	0.999981i	$-0.501951\pi$
$353$	−81.3189	−0.0122611	−0.00613055	+	0.999981i	$0.501951\pi$
$354$	0	0
$355$	0	0
$356$	8297.35	1.23528
$357$	0	0
$358$	−1778.52	−0.262564
$359$	−4547.08	−0.668483	−0.334242	−	0.942487i	$-0.608480\pi$
$359$	−4547.08	−0.668483	−0.334242	+	0.942487i	$0.608480\pi$
$360$	0	0
$361$	8200.46	1.19558
$362$	−8133.12	−1.18085
$363$	0	0
$364$	2622.55	0.377635
$365$	0	0
$366$	0	0
$367$	−11297.4	−1.60686	−0.803429	−	0.595401i	$-0.796993\pi$
$367$	−11297.4	−1.60686	−0.803429	+	0.595401i	$0.796993\pi$
$368$	9358.31	1.32564
$369$	0	0
$370$	0	0
$371$	−3227.57	−0.451664
$372$	0	0
$373$	2880.15	0.399808	0.199904	−	0.979816i	$-0.435937\pi$
$373$	2880.15	0.399808	0.199904	+	0.979816i	$0.435937\pi$
$374$	−9107.65	−1.25921
$375$	0	0
$376$	−406.130	−0.0557036
$377$	9581.38	1.30893
$378$	0	0
$379$	−10505.6	−1.42384	−0.711922	−	0.702259i	$-0.752175\pi$
$379$	−10505.6	−1.42384	−0.711922	+	0.702259i	$0.752175\pi$
$380$	0	0
$381$	0	0
$382$	166.837	0.0223459
$383$	−522.066	−0.0696510	−0.0348255	−	0.999393i	$-0.511088\pi$
$383$	−522.066	−0.0696510	−0.0348255	+	0.999393i	$0.511088\pi$
$384$	0	0
$385$	0	0
$386$	−14304.1	−1.88616
$387$	0	0
$388$	1557.64	0.203807
$389$	6513.24	0.848932	0.424466	−	0.905444i	$-0.360462\pi$
$389$	6513.24	0.848932	0.424466	+	0.905444i	$0.360462\pi$
$390$	0	0
$391$	17539.6	2.26859
$392$	64.4142	0.00829951
$393$	0	0
$394$	9617.48	1.22975
$395$	0	0
$396$	0	0
$397$	−9106.98	−1.15130	−0.575650	−	0.817696i	$-0.695251\pi$
$397$	−9106.98	−1.15130	−0.575650	+	0.817696i	$0.695251\pi$
$398$	−14013.8	−1.76495
$399$	0	0
$400$	0	0
$401$	3204.43	0.399056	0.199528	−	0.979892i	$-0.436059\pi$
$401$	3204.43	0.399056	0.199528	+	0.979892i	$0.436059\pi$
$402$	0	0
$403$	14149.3	1.74895
$404$	−12016.5	−1.47981
$405$	0	0
$406$	−5433.52	−0.664190
$407$	7409.68	0.902418
$408$	0	0
$409$	6396.33	0.773296	0.386648	−	0.922227i	$-0.373633\pi$
$409$	6396.33	0.773296	0.386648	+	0.922227i	$0.373633\pi$
$410$	0	0
$411$	0	0
$412$	−3634.31	−0.434586
$413$	−783.302	−0.0933263
$414$	0	0
$415$	0	0
$416$	12356.2	1.45629
$417$	0	0
$418$	−8961.13	−1.04857
$419$	11046.1	1.28791	0.643956	−	0.765062i	$-0.277292\pi$
$419$	11046.1	1.28791	0.643956	+	0.765062i	$0.277292\pi$
$420$	0	0
$421$	15873.7	1.83762	0.918811	−	0.394698i	$-0.129151\pi$
$421$	15873.7	1.83762	0.918811	+	0.394698i	$0.129151\pi$
$422$	8643.51	0.997060
$423$	0	0
$424$	−606.127	−0.0694248
$425$	0	0
$426$	0	0
$427$	2916.61	0.330549
$428$	−3810.16	−0.430306
$429$	0	0
$430$	0	0
$431$	4878.52	0.545221	0.272611	−	0.962124i	$-0.412113\pi$
$431$	4878.52	0.545221	0.272611	+	0.962124i	$0.412113\pi$
$432$	0	0
$433$	8450.44	0.937880	0.468940	−	0.883230i	$-0.344636\pi$
$433$	8450.44	0.937880	0.468940	+	0.883230i	$0.344636\pi$
$434$	−8023.96	−0.887471
$435$	0	0
$436$	8629.88	0.947927
$437$	17257.4	1.88910
$438$	0	0
$439$	10996.0	1.19546	0.597732	−	0.801696i	$-0.296069\pi$
$439$	10996.0	1.19546	0.597732	+	0.801696i	$0.296069\pi$
$440$	0	0
$441$	0	0
$442$	24121.5	2.59580
$443$	13494.3	1.44725	0.723626	−	0.690193i	$-0.242474\pi$
$443$	13494.3	1.44725	0.723626	+	0.690193i	$0.242474\pi$
$444$	0	0
$445$	0	0
$446$	20887.4	2.21760
$447$	0	0
$448$	−3280.51	−0.345959
$449$	−15411.4	−1.61984	−0.809920	−	0.586540i	$-0.800490\pi$
$449$	−15411.4	−1.61984	−0.809920	+	0.586540i	$0.800490\pi$
$450$	0	0
$451$	−5591.50	−0.583799
$452$	1414.79	0.147226
$453$	0	0
$454$	−17703.3	−1.83009
$455$	0	0
$456$	0	0
$457$	11175.7	1.14393	0.571967	−	0.820277i	$-0.306181\pi$
$457$	11175.7	1.14393	0.571967	+	0.820277i	$0.306181\pi$
$458$	−6724.10	−0.686018
$459$	0	0
$460$	0	0
$461$	6343.45	0.640876	0.320438	−	0.947270i	$-0.396170\pi$
$461$	6343.45	0.640876	0.320438	+	0.947270i	$0.396170\pi$
$462$	0	0
$463$	5947.99	0.597034	0.298517	−	0.954404i	$-0.403508\pi$
$463$	5947.99	0.597034	0.298517	+	0.954404i	$0.403508\pi$
$464$	−13049.8	−1.30565
$465$	0	0
$466$	−19228.1	−1.91143
$467$	11775.8	1.16685	0.583424	−	0.812168i	$-0.301713\pi$
$467$	11775.8	1.16685	0.583424	+	0.812168i	$0.301713\pi$
$468$	0	0
$469$	3166.25	0.311736
$470$	0	0
$471$	0	0
$472$	−147.101	−0.0143451
$473$	−3727.02	−0.362302
$474$	0	0
$475$	0	0
$476$	−6694.57	−0.644632
$477$	0	0
$478$	1486.61	0.142251
$479$	18005.6	1.71753	0.858765	−	0.512370i	$-0.171232\pi$
$479$	18005.6	1.71753	0.858765	+	0.512370i	$0.171232\pi$
$480$	0	0
$481$	−19624.4	−1.86029
$482$	3981.71	0.376269
$483$	0	0
$484$	−7596.32	−0.713403
$485$	0	0
$486$	0	0
$487$	11098.4	1.03268	0.516340	−	0.856383i	$-0.327294\pi$
$487$	11098.4	1.03268	0.516340	+	0.856383i	$0.327294\pi$
$488$	547.729	0.0508085
$489$	0	0
$490$	0	0
$491$	−11253.9	−1.03438	−0.517191	−	0.855870i	$-0.673022\pi$
$491$	−11253.9	−1.03438	−0.517191	+	0.855870i	$0.673022\pi$
$492$	0	0
$493$	−24458.3	−2.23437
$494$	23733.4	2.16157
$495$	0	0
$496$	−19271.3	−1.74457
$497$	−7060.16	−0.637206
$498$	0	0
$499$	−6099.94	−0.547236	−0.273618	−	0.961838i	$-0.588220\pi$
$499$	−6099.94	−0.547236	−0.273618	+	0.961838i	$0.588220\pi$
$500$	0	0
$501$	0	0
$502$	−21623.3	−1.92250
$503$	19384.4	1.71830	0.859152	−	0.511721i	$-0.170992\pi$
$503$	19384.4	1.71830	0.859152	+	0.511721i	$0.170992\pi$
$504$	0	0
$505$	0	0
$506$	−10269.0	−0.902202
$507$	0	0
$508$	21745.6	1.89922
$509$	5257.46	0.457825	0.228912	−	0.973447i	$-0.426483\pi$
$509$	5257.46	0.457825	0.228912	+	0.973447i	$0.426483\pi$
$510$	0	0
$511$	4683.30	0.405434
$512$	−16129.3	−1.39223
$513$	0	0
$514$	−1693.31	−0.145309
$515$	0	0
$516$	0	0
$517$	5699.44	0.484838
$518$	11128.8	0.943964
$519$	0	0
$520$	0	0
$521$	−7762.65	−0.652760	−0.326380	−	0.945239i	$-0.605829\pi$
$521$	−7762.65	−0.652760	−0.326380	+	0.945239i	$0.605829\pi$
$522$	0	0
$523$	−9640.89	−0.806055	−0.403027	−	0.915188i	$-0.632042\pi$
$523$	−9640.89	−0.806055	−0.403027	+	0.915188i	$0.632042\pi$
$524$	−1456.88	−0.121458
$525$	0	0
$526$	−2060.85	−0.170832
$527$	−36118.9	−2.98551
$528$	0	0
$529$	7609.24	0.625399
$530$	0	0
$531$	0	0
$532$	−6586.86	−0.536798
$533$	14809.0	1.20347
$534$	0	0
$535$	0	0
$536$	594.612	0.0479166
$537$	0	0
$538$	−7899.28	−0.633015
$539$	−903.959	−0.0722380
$540$	0	0
$541$	−2039.89	−0.162110	−0.0810551	−	0.996710i	$-0.525829\pi$
$541$	−2039.89	−0.162110	−0.0810551	+	0.996710i	$0.525829\pi$
$542$	−20634.4	−1.63528
$543$	0	0
$544$	−31541.7	−2.48592
$545$	0	0
$546$	0	0
$547$	1274.24	0.0996025	0.0498013	−	0.998759i	$-0.484141\pi$
$547$	1274.24	0.0996025	0.0498013	+	0.998759i	$0.484141\pi$
$548$	−17101.2	−1.33308
$549$	0	0
$550$	0	0
$551$	−24064.8	−1.86061
$552$	0	0
$553$	3621.79	0.278506
$554$	−25090.7	−1.92419
$555$	0	0
$556$	−3054.01	−0.232948
$557$	19965.7	1.51880	0.759402	−	0.650621i	$-0.225492\pi$
$557$	19965.7	1.51880	0.759402	+	0.650621i	$0.225492\pi$
$558$	0	0
$559$	9870.97	0.746865
$560$	0	0
$561$	0	0
$562$	−19242.3	−1.44428
$563$	−13467.9	−1.00818	−0.504090	−	0.863651i	$-0.668172\pi$
$563$	−13467.9	−1.00818	−0.504090	+	0.863651i	$0.668172\pi$
$564$	0	0
$565$	0	0
$566$	4117.51	0.305781
$567$	0	0
$568$	−1325.87	−0.0979444
$569$	−8579.29	−0.632096	−0.316048	−	0.948743i	$-0.602356\pi$
$569$	−8579.29	−0.632096	−0.316048	+	0.948743i	$0.602356\pi$
$570$	0	0
$571$	−2146.06	−0.157285	−0.0786426	−	0.996903i	$-0.525059\pi$
$571$	−2146.06	−0.157285	−0.0786426	+	0.996903i	$0.525059\pi$
$572$	−6911.61	−0.505225
$573$	0	0
$574$	−8398.06	−0.610676
$575$	0	0
$576$	0	0
$577$	−22451.1	−1.61985	−0.809925	−	0.586534i	$-0.800492\pi$
$577$	−22451.1	−1.61985	−0.809925	+	0.586534i	$0.800492\pi$
$578$	−42127.8	−3.03163
$579$	0	0
$580$	0	0
$581$	247.765	0.0176919
$582$	0	0
$583$	8506.10	0.604266
$584$	879.508	0.0623190
$585$	0	0
$586$	15069.2	1.06229
$587$	6015.40	0.422968	0.211484	−	0.977381i	$-0.432170\pi$
$587$	6015.40	0.422968	0.211484	+	0.977381i	$0.432170\pi$
$588$	0	0
$589$	−35537.8	−2.48609
$590$	0	0
$591$	0	0
$592$	26728.4	1.85562
$593$	15737.3	1.08981	0.544903	−	0.838499i	$-0.316566\pi$
$593$	15737.3	1.08981	0.544903	+	0.838499i	$0.316566\pi$
$594$	0	0
$595$	0	0
$596$	27269.3	1.87415
$597$	0	0
$598$	27197.4	1.85984
$599$	25634.5	1.74858	0.874288	−	0.485407i	$-0.161328\pi$
$599$	25634.5	1.74858	0.874288	+	0.485407i	$0.161328\pi$
$600$	0	0
$601$	−6955.91	−0.472109	−0.236054	−	0.971740i	$-0.575854\pi$
$601$	−6955.91	−0.472109	−0.236054	+	0.971740i	$0.575854\pi$
$602$	−5597.74	−0.378982
$603$	0	0
$604$	−7330.16	−0.493808
$605$	0	0
$606$	0	0
$607$	−952.222	−0.0636730	−0.0318365	−	0.999493i	$-0.510136\pi$
$607$	−952.222	−0.0636730	−0.0318365	+	0.999493i	$0.510136\pi$
$608$	−31034.2	−2.07007
$609$	0	0
$610$	0	0
$611$	−15094.9	−0.999466
$612$	0	0
$613$	2962.98	0.195226	0.0976132	−	0.995224i	$-0.468879\pi$
$613$	2962.98	0.195226	0.0976132	+	0.995224i	$0.468879\pi$
$614$	5198.23	0.341667
$615$	0	0
$616$	−169.760	−0.0111036
$617$	−28257.7	−1.84378	−0.921891	−	0.387449i	$-0.873356\pi$
$617$	−28257.7	−1.84378	−0.921891	+	0.387449i	$0.873356\pi$
$618$	0	0
$619$	−10708.5	−0.695336	−0.347668	−	0.937618i	$-0.613026\pi$
$619$	−10708.5	−0.695336	−0.347668	+	0.937618i	$0.613026\pi$
$620$	0	0
$621$	0	0
$622$	−2241.10	−0.144470
$623$	7574.63	0.487113
$624$	0	0
$625$	0	0
$626$	4538.03	0.289738
$627$	0	0
$628$	−23738.1	−1.50837
$629$	50095.1	3.17555
$630$	0	0
$631$	17383.2	1.09669	0.548346	−	0.836251i	$-0.315258\pi$
$631$	17383.2	1.09669	0.548346	+	0.836251i	$0.315258\pi$
$632$	680.159	0.0428090
$633$	0	0
$634$	−10142.9	−0.635371
$635$	0	0
$636$	0	0
$637$	2394.12	0.148915
$638$	14319.8	0.888597
$639$	0	0
$640$	0	0
$641$	−8268.85	−0.509516	−0.254758	−	0.967005i	$-0.581996\pi$
$641$	−8268.85	−0.509516	−0.254758	+	0.967005i	$0.581996\pi$
$642$	0	0
$643$	1676.14	0.102800	0.0514000	−	0.998678i	$-0.483632\pi$
$643$	1676.14	0.102800	0.0514000	+	0.998678i	$0.483632\pi$
$644$	−7548.24	−0.461867
$645$	0	0
$646$	−60584.1	−3.68986
$647$	−17319.8	−1.05242	−0.526208	−	0.850356i	$-0.676387\pi$
$647$	−17319.8	−1.05242	−0.526208	+	0.850356i	$0.676387\pi$
$648$	0	0
$649$	2064.35	0.124858
$650$	0	0
$651$	0	0
$652$	−20438.3	−1.22765
$653$	17191.7	1.03026	0.515132	−	0.857111i	$-0.327743\pi$
$653$	17191.7	1.03026	0.515132	+	0.857111i	$0.327743\pi$
$654$	0	0
$655$	0	0
$656$	−20169.8	−1.20045
$657$	0	0
$658$	8560.18	0.507159
$659$	−1559.35	−0.0921753	−0.0460877	−	0.998937i	$-0.514675\pi$
$659$	−1559.35	−0.0921753	−0.0460877	+	0.998937i	$0.514675\pi$
$660$	0	0
$661$	−17710.2	−1.04213	−0.521065	−	0.853517i	$-0.674465\pi$
$661$	−17710.2	−1.04213	−0.521065	+	0.853517i	$0.674465\pi$
$662$	3630.10	0.213124
$663$	0	0
$664$	46.5294	0.00271941
$665$	0	0
$666$	0	0
$667$	−27577.2	−1.60089
$668$	7422.50	0.429918
$669$	0	0
$670$	0	0
$671$	−7686.57	−0.442231
$672$	0	0
$673$	−15614.1	−0.894322	−0.447161	−	0.894453i	$-0.647565\pi$
$673$	−15614.1	−0.894322	−0.447161	+	0.894453i	$0.647565\pi$
$674$	9682.60	0.553353
$675$	0	0
$676$	1458.93	0.0830071
$677$	−1964.69	−0.111535	−0.0557675	−	0.998444i	$-0.517761\pi$
$677$	−1964.69	−0.111535	−0.0557675	+	0.998444i	$0.517761\pi$
$678$	0	0
$679$	1421.96	0.0803682
$680$	0	0
$681$	0	0
$682$	21146.7	1.18732
$683$	−5230.81	−0.293047	−0.146524	−	0.989207i	$-0.546808\pi$
$683$	−5230.81	−0.293047	−0.146524	+	0.989207i	$0.546808\pi$
$684$	0	0
$685$	0	0
$686$	−1357.69	−0.0755637
$687$	0	0
$688$	−13444.2	−0.744994
$689$	−22528.3	−1.24566
$690$	0	0
$691$	28743.3	1.58241	0.791205	−	0.611551i	$-0.209454\pi$
$691$	28743.3	1.58241	0.791205	+	0.611551i	$0.209454\pi$
$692$	21419.0	1.17663
$693$	0	0
$694$	−14243.3	−0.779061
$695$	0	0
$696$	0	0
$697$	−37802.8	−2.05435
$698$	979.937	0.0531392
$699$	0	0
$700$	0	0
$701$	−15018.5	−0.809188	−0.404594	−	0.914497i	$-0.632587\pi$
$701$	−15018.5	−0.809188	−0.404594	+	0.914497i	$0.632587\pi$
$702$	0	0
$703$	49289.2	2.64435
$704$	8645.62	0.462847
$705$	0	0
$706$	321.882	0.0171589
$707$	−10969.8	−0.583539
$708$	0	0
$709$	2164.88	0.114674	0.0573368	−	0.998355i	$-0.481739\pi$
$709$	2164.88	0.114674	0.0573368	+	0.998355i	$0.481739\pi$
$710$	0	0
$711$	0	0
$712$	1422.49	0.0748737
$713$	−40724.6	−2.13906
$714$	0	0
$715$	0	0
$716$	3445.33	0.179830
$717$	0	0
$718$	17998.5	0.935515
$719$	−16570.9	−0.859516	−0.429758	−	0.902944i	$-0.641401\pi$
$719$	−16570.9	−0.859516	−0.429758	+	0.902944i	$0.641401\pi$
$720$	0	0
$721$	−3317.75	−0.171372
$722$	−32459.6	−1.67316
$723$	0	0
$724$	15755.3	0.808761
$725$	0	0
$726$	0	0
$727$	4179.50	0.213217	0.106609	−	0.994301i	$-0.466001\pi$
$727$	4179.50	0.213217	0.106609	+	0.994301i	$0.466001\pi$
$728$	449.608	0.0228895
$729$	0	0
$730$	0	0
$731$	−25197.5	−1.27492
$732$	0	0
$733$	24312.7	1.22511	0.612557	−	0.790426i	$-0.290141\pi$
$733$	24312.7	1.22511	0.612557	+	0.790426i	$0.290141\pi$
$734$	44717.9	2.24873
$735$	0	0
$736$	−35563.8	−1.78111
$737$	−8344.50	−0.417061
$738$	0	0
$739$	−8812.62	−0.438671	−0.219335	−	0.975650i	$-0.570389\pi$
$739$	−8812.62	−0.438671	−0.219335	+	0.975650i	$0.570389\pi$
$740$	0	0
$741$	0	0
$742$	12775.6	0.632085
$743$	−30132.1	−1.48780	−0.743902	−	0.668288i	$-0.767027\pi$
$743$	−30132.1	−1.48780	−0.743902	+	0.668288i	$0.767027\pi$
$744$	0	0
$745$	0	0
$746$	−11400.4	−0.559515
$747$	0	0
$748$	17643.2	0.862432
$749$	−3478.28	−0.169685
$750$	0	0
$751$	1000.93	0.0486343	0.0243172	−	0.999704i	$-0.492259\pi$
$751$	1000.93	0.0486343	0.0243172	+	0.999704i	$0.492259\pi$
$752$	20559.2	0.996962
$753$	0	0
$754$	−37925.7	−1.83179
$755$	0	0
$756$	0	0
$757$	−21708.6	−1.04229	−0.521144	−	0.853469i	$-0.674495\pi$
$757$	−21708.6	−1.04229	−0.521144	+	0.853469i	$0.674495\pi$
$758$	41584.0	1.99261
$759$	0	0
$760$	0	0
$761$	33904.7	1.61504	0.807520	−	0.589840i	$-0.200809\pi$
$761$	33904.7	1.61504	0.807520	+	0.589840i	$0.200809\pi$
$762$	0	0
$763$	7878.20	0.373801
$764$	−323.194	−0.0153047
$765$	0	0
$766$	2066.48	0.0974738
$767$	−5467.41	−0.257388
$768$	0	0
$769$	−6114.14	−0.286712	−0.143356	−	0.989671i	$-0.545789\pi$
$769$	−6114.14	−0.286712	−0.143356	+	0.989671i	$0.545789\pi$
$770$	0	0
$771$	0	0
$772$	27709.7	1.29183
$773$	12350.0	0.574644	0.287322	−	0.957834i	$-0.407235\pi$
$773$	12350.0	0.574644	0.287322	+	0.957834i	$0.407235\pi$
$774$	0	0
$775$	0	0
$776$	267.040	0.0123533
$777$	0	0
$778$	−25781.2	−1.18805
$779$	−37194.6	−1.71070
$780$	0	0
$781$	18606.7	0.852496
$782$	−69426.6	−3.17479
$783$	0	0
$784$	−3260.78	−0.148541
$785$	0	0
$786$	0	0
$787$	−13597.6	−0.615888	−0.307944	−	0.951405i	$-0.599641\pi$
$787$	−13597.6	−0.615888	−0.307944	+	0.951405i	$0.599641\pi$
$788$	−18630.8	−0.842254
$789$	0	0
$790$	0	0
$791$	1291.56	0.0580563
$792$	0	0
$793$	20357.8	0.911634
$794$	36047.9	1.61120
$795$	0	0
$796$	27147.3	1.20881
$797$	18951.1	0.842262	0.421131	−	0.907000i	$-0.361633\pi$
$797$	18951.1	0.842262	0.421131	+	0.907000i	$0.361633\pi$
$798$	0	0
$799$	38532.6	1.70611
$800$	0	0
$801$	0	0
$802$	−12684.0	−0.558463
$803$	−12342.6	−0.542417
$804$	0	0
$805$	0	0
$806$	−56006.9	−2.44759
$807$	0	0
$808$	−2060.09	−0.0896952
$809$	27695.1	1.20359	0.601797	−	0.798649i	$-0.294451\pi$
$809$	27695.1	1.20359	0.601797	+	0.798649i	$0.294451\pi$
$810$	0	0
$811$	1439.61	0.0623325	0.0311663	−	0.999514i	$-0.490078\pi$
$811$	1439.61	0.0623325	0.0311663	+	0.999514i	$0.490078\pi$
$812$	10525.7	0.454902
$813$	0	0
$814$	−29329.5	−1.26290
$815$	0	0
$816$	0	0
$817$	−24792.2	−1.06165
$818$	−25318.4	−1.08220
$819$	0	0
$820$	0	0
$821$	−11212.2	−0.476626	−0.238313	−	0.971188i	$-0.576594\pi$
$821$	−11212.2	−0.476626	−0.238313	+	0.971188i	$0.576594\pi$
$822$	0	0
$823$	14736.8	0.624170	0.312085	−	0.950054i	$-0.398973\pi$
$823$	14736.8	0.624170	0.312085	+	0.950054i	$0.398973\pi$
$824$	−623.062	−0.0263415
$825$	0	0
$826$	3100.52	0.130606
$827$	9129.69	0.383882	0.191941	−	0.981406i	$-0.438522\pi$
$827$	9129.69	0.383882	0.191941	+	0.981406i	$0.438522\pi$
$828$	0	0
$829$	13177.4	0.552074	0.276037	−	0.961147i	$-0.410979\pi$
$829$	13177.4	0.552074	0.276037	+	0.961147i	$0.410979\pi$
$830$	0	0
$831$	0	0
$832$	−22897.8	−0.954133
$833$	−6111.46	−0.254201
$834$	0	0
$835$	0	0
$836$	17359.3	0.718164
$837$	0	0
$838$	−43723.3	−1.80238
$839$	40633.0	1.67200	0.835999	−	0.548731i	$-0.184889\pi$
$839$	40633.0	1.67200	0.835999	+	0.548731i	$0.184889\pi$
$840$	0	0
$841$	14066.2	0.576746
$842$	−62832.5	−2.57168
$843$	0	0
$844$	−16744.1	−0.682884
$845$	0	0
$846$	0	0
$847$	−6934.66	−0.281320
$848$	30683.4	1.24254
$849$	0	0
$850$	0	0
$851$	56483.1	2.27522
$852$	0	0
$853$	16268.8	0.653027	0.326514	−	0.945192i	$-0.394126\pi$
$853$	16268.8	0.653027	0.326514	+	0.945192i	$0.394126\pi$
$854$	−11544.7	−0.462590
$855$	0	0
$856$	−653.210	−0.0260821
$857$	2376.68	0.0947325	0.0473663	−	0.998878i	$-0.484917\pi$
$857$	2376.68	0.0947325	0.0473663	+	0.998878i	$0.484917\pi$
$858$	0	0
$859$	−18821.4	−0.747589	−0.373794	−	0.927512i	$-0.621943\pi$
$859$	−18821.4	−0.747589	−0.373794	+	0.927512i	$0.621943\pi$
$860$	0	0
$861$	0	0
$862$	−19310.5	−0.763015
$863$	20478.1	0.807743	0.403871	−	0.914816i	$-0.367664\pi$
$863$	20478.1	0.807743	0.403871	+	0.914816i	$0.367664\pi$
$864$	0	0
$865$	0	0
$866$	−33449.1	−1.31252
$867$	0	0
$868$	15543.9	0.607827
$869$	−9545.03	−0.372604
$870$	0	0
$871$	22100.3	0.859747
$872$	1479.50	0.0574566
$873$	0	0
$874$	−68309.6	−2.64371
$875$	0	0
$876$	0	0
$877$	46886.5	1.80530	0.902648	−	0.430379i	$-0.141620\pi$
$877$	46886.5	1.80530	0.902648	+	0.430379i	$0.141620\pi$
$878$	−43525.0	−1.67300
$879$	0	0
$880$	0	0
$881$	16858.4	0.644691	0.322346	−	0.946622i	$-0.395529\pi$
$881$	16858.4	0.644691	0.322346	+	0.946622i	$0.395529\pi$
$882$	0	0
$883$	38016.2	1.44886	0.724432	−	0.689346i	$-0.242102\pi$
$883$	38016.2	1.44886	0.724432	+	0.689346i	$0.242102\pi$
$884$	−46727.8	−1.77786
$885$	0	0
$886$	−53414.0	−2.02537
$887$	−25246.2	−0.955675	−0.477838	−	0.878448i	$-0.658579\pi$
$887$	−25246.2	−0.955675	−0.477838	+	0.878448i	$0.658579\pi$
$888$	0	0
$889$	19851.5	0.748928
$890$	0	0
$891$	0	0
$892$	−40462.7	−1.51883
$893$	37912.7	1.42072
$894$	0	0
$895$	0	0
$896$	−1176.85	−0.0438791
$897$	0	0
$898$	61002.4	2.26690
$899$	56788.9	2.10680
$900$	0	0
$901$	57507.8	2.12637
$902$	22132.7	0.817003
$903$	0	0
$904$	242.550	0.00892378
$905$	0	0
$906$	0	0
$907$	−11679.5	−0.427576	−0.213788	−	0.976880i	$-0.568580\pi$
$907$	−11679.5	−0.427576	−0.213788	+	0.976880i	$0.568580\pi$
$908$	34294.6	1.25342
$909$	0	0
$910$	0	0
$911$	23655.6	0.860312	0.430156	−	0.902755i	$-0.358459\pi$
$911$	23655.6	0.860312	0.430156	+	0.902755i	$0.358459\pi$
$912$	0	0
$913$	−652.971	−0.0236694
$914$	−44236.4	−1.60089
$915$	0	0
$916$	13025.8	0.469852
$917$	−1329.98	−0.0478953
$918$	0	0
$919$	19902.0	0.714371	0.357185	−	0.934033i	$-0.383736\pi$
$919$	19902.0	0.714371	0.357185	+	0.934033i	$0.383736\pi$
$920$	0	0
$921$	0	0
$922$	−25109.1	−0.896880
$923$	−49279.5	−1.75737
$924$	0	0
$925$	0	0
$926$	−23543.8	−0.835525
$927$	0	0
$928$	49592.3	1.75425
$929$	−48903.6	−1.72710	−0.863550	−	0.504262i	$-0.831764\pi$
$929$	−48903.6	−1.72710	−0.863550	+	0.504262i	$0.831764\pi$
$930$	0	0
$931$	−6013.13	−0.211678
$932$	37248.4	1.30913
$933$	0	0
$934$	−46611.7	−1.63296
$935$	0	0
$936$	0	0
$937$	−1179.03	−0.0411068	−0.0205534	−	0.999789i	$-0.506543\pi$
$937$	−1179.03	−0.0411068	−0.0205534	+	0.999789i	$0.506543\pi$
$938$	−12532.9	−0.436261
$939$	0	0
$940$	0	0
$941$	−6253.60	−0.216643	−0.108322	−	0.994116i	$-0.534548\pi$
$941$	−6253.60	−0.216643	−0.108322	+	0.994116i	$0.534548\pi$
$942$	0	0
$943$	−42623.3	−1.47191
$944$	7446.58	0.256743
$945$	0	0
$946$	14752.6	0.507027
$947$	29027.3	0.996050	0.498025	−	0.867163i	$-0.334059\pi$
$947$	29027.3	0.996050	0.498025	+	0.867163i	$0.334059\pi$
$948$	0	0
$949$	32689.2	1.11816
$950$	0	0
$951$	0	0
$952$	−1147.71	−0.0390730
$953$	47968.6	1.63049	0.815245	−	0.579116i	$-0.196602\pi$
$953$	47968.6	1.63049	0.815245	+	0.579116i	$0.196602\pi$
$954$	0	0
$955$	0	0
$956$	−2879.83	−0.0974273
$957$	0	0
$958$	−71271.0	−2.40361
$959$	−15611.7	−0.525679
$960$	0	0
$961$	54072.1	1.81505
$962$	77678.8	2.60339
$963$	0	0
$964$	−7713.29	−0.257706
$965$	0	0
$966$	0	0
$967$	−6940.06	−0.230793	−0.115397	−	0.993319i	$-0.536814\pi$
$967$	−6940.06	−0.230793	−0.115397	+	0.993319i	$0.536814\pi$
$968$	−1302.31	−0.0432414
$969$	0	0
$970$	0	0
$971$	−25930.9	−0.857015	−0.428507	−	0.903538i	$-0.640960\pi$
$971$	−25930.9	−0.857015	−0.428507	+	0.903538i	$0.640960\pi$
$972$	0	0
$973$	−2788.00	−0.0918594
$974$	−43930.4	−1.44519
$975$	0	0
$976$	−27727.2	−0.909350
$977$	−9361.03	−0.306536	−0.153268	−	0.988185i	$-0.548980\pi$
$977$	−9361.03	−0.306536	−0.153268	+	0.988185i	$0.548980\pi$
$978$	0	0
$979$	−19962.6	−0.651691
$980$	0	0
$981$	0	0
$982$	44546.0	1.44757
$983$	5317.34	0.172530	0.0862649	−	0.996272i	$-0.472507\pi$
$983$	5317.34	0.172530	0.0862649	+	0.996272i	$0.472507\pi$
$984$	0	0
$985$	0	0
$986$	96812.6	3.12692
$987$	0	0
$988$	−45976.0	−1.48046
$989$	−28410.7	−0.913455
$990$	0	0
$991$	−38425.3	−1.23170	−0.615852	−	0.787862i	$-0.711188\pi$
$991$	−38425.3	−1.23170	−0.615852	+	0.787862i	$0.711188\pi$
$992$	73235.5	2.34398
$993$	0	0
$994$	27946.0	0.891744
$995$	0	0
$996$	0	0
$997$	29596.3	0.940145	0.470073	−	0.882628i	$-0.344228\pi$
$997$	29596.3	0.940145	0.470073	+	0.882628i	$0.344228\pi$
$998$	24145.2	0.765835
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bs.1.2	yes	8
3.2	odd	2	inner	1575.4.a.bs.1.7	yes	8
5.4	even	2		1575.4.a.br.1.7	yes	8
15.14	odd	2		1575.4.a.br.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.4.a.br.1.2	✓	8	15.14	odd	2
1575.4.a.br.1.7	yes	8	5.4	even	2
1575.4.a.bs.1.2	yes	8	1.1	even	1	trivial
1575.4.a.bs.1.7	yes	8	3.2	odd	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-3.95827 q^{2} +7.66789 q^{4} +7.00000 q^{7} +1.31458 q^{8} -18.4481 q^{11} +48.8596 q^{13} -27.7079 q^{14} -66.5466 q^{16} -124.724 q^{17} -122.717 q^{19} +73.0227 q^{22} -140.628 q^{23} -193.400 q^{26} +53.6752 q^{28} +196.100 q^{29} +289.591 q^{31} +252.893 q^{32} +493.690 q^{34} -401.649 q^{37} +485.747 q^{38} +303.093 q^{41} +202.027 q^{43} -141.458 q^{44} +556.643 q^{46} -308.944 q^{47} +49.0000 q^{49} +374.650 q^{52} -461.082 q^{53} +9.20203 q^{56} -776.217 q^{58} -111.900 q^{59} +416.658 q^{61} -1146.28 q^{62} -468.644 q^{64} +452.322 q^{67} -956.367 q^{68} -1008.59 q^{71} +669.043 q^{73} +1589.84 q^{74} -940.980 q^{76} -129.137 q^{77} +517.398 q^{79} -1199.72 q^{82} +35.3950 q^{83} -799.677 q^{86} -24.2515 q^{88} +1082.09 q^{89} +342.018 q^{91} -1078.32 q^{92} +1222.88 q^{94} +203.138 q^{97} -193.955 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{4} + 56 q^{7} - 38 q^{13} + 320 q^{16} + 68 q^{19} + 110 q^{22} + 252 q^{28} + 534 q^{31} + 118 q^{34} + 14 q^{37} + 816 q^{43} + 1588 q^{46} + 392 q^{49} - 266 q^{52} + 716 q^{58} + 874 q^{61}+ \cdots + 428 q^{97}+O(q^{100}) \) 8 * q + 36 * q^4 + 56 * q^7 - 38 * q^13 + 320 * q^16 + 68 * q^19 + 110 * q^22 + 252 * q^28 + 534 * q^31 + 118 * q^34 + 14 * q^37 + 816 * q^43 + 1588 * q^46 + 392 * q^49 - 266 * q^52 + 716 * q^58 + 874 * q^61 + 2502 * q^64 + 254 * q^67 - 1532 * q^73 + 4304 * q^76 + 5190 * q^79 + 1158 * q^82 - 4300 * q^88 - 266 * q^91 + 6236 * q^94 + 428 * q^97

Introduction

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Database

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