LMFDB - Embedded newform 1575.4.a.br.1.7

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			1.7
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	$-0.253304\pi$
$2$	3.95827	1.39946	0.699730	+	0.714408i	$0.253304\pi$
$3$	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	$-0.674516\pi$
$13$	−48.8596	−1.04240	−0.521201	+	0.853434i	$0.674516\pi$
$14$	−27.7079	−0.528946
$15$	0	0
$16$	−66.5466	−1.03979
$17$	124.724	1.77941	0.889703	−	0.456539i	$-0.150911\pi$
$17$	124.724	1.77941	0.889703	+	0.456539i	$0.150911\pi$
$18$	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	$-0.279987\pi$
$23$	140.628	1.27491	0.637456	+	0.770487i	$0.279987\pi$
$24$	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	$-0.149086\pi$
$37$	401.649	1.78461	0.892307	+	0.451429i	$0.149086\pi$
$38$	−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	$-0.616624\pi$
$43$	−202.027	−0.716485	−0.358242	+	0.933629i	$0.616624\pi$
$44$	−141.458	−0.484674
$45$	0	0
$46$	556.643	1.78419
$47$	308.944	0.958811	0.479405	−	0.877594i	$-0.340852\pi$
$47$	308.944	0.958811	0.479405	+	0.877594i	$0.340852\pi$
$48$	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	$-0.296163\pi$
$53$	461.082	1.19499	0.597495	+	0.801873i	$0.296163\pi$
$54$	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	$-0.635305\pi$
$67$	−452.322	−0.824775	−0.412388	+	0.911009i	$0.635305\pi$
$68$	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	$-0.680193\pi$
$73$	−669.043	−1.07268	−0.536339	+	0.844002i	$0.680193\pi$
$74$	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	$-0.507450\pi$
$83$	−35.3950	−0.0468085	−0.0234042	+	0.999726i	$0.507450\pi$
$84$	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	$-0.533906\pi$
$97$	−203.138	−0.212634	−0.106317	+	0.994332i	$0.533906\pi$
$98$	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	$-0.427205\pi$
$103$	473.964	0.453409	0.226704	+	0.973964i	$0.427205\pi$
$104$	64.2297	0.0605600
$105$	0	0
$106$	1825.09	1.67234
$107$	496.898	0.448943	0.224472	−	0.974481i	$-0.427934\pi$
$107$	496.898	0.448943	0.224472	+	0.974481i	$0.427934\pi$
$108$	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	$-0.524471\pi$
$113$	−184.508	−0.153603	−0.0768013	+	0.997046i	$0.524471\pi$
$114$	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	$-0.956645\pi$
$127$	−2835.92	−1.98148	−0.990738	+	0.135784i	$0.956645\pi$
$128$	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	$-0.255223\pi$
$137$	2230.24	1.39082	0.695408	+	0.718615i	$0.255223\pi$
$138$	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	$-0.211712\pi$
$157$	3095.78	1.57370	0.786848	+	0.617147i	$0.211712\pi$
$158$	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	$-0.278765\pi$
$163$	2665.44	1.28082	0.640409	+	0.768034i	$0.278765\pi$
$164$	2324.08	1.10659
$165$	0	0
$166$	−140.103	−0.0655065
$167$	−967.998	−0.448538	−0.224269	−	0.974527i	$-0.572000\pi$
$167$	−967.998	−0.448538	−0.224269	+	0.974527i	$0.572000\pi$
$168$	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	$-0.710358\pi$
$173$	−2793.34	−1.22759	−0.613796	+	0.789465i	$0.710358\pi$
$174$	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	$-0.735378\pi$
$193$	−3613.73	−1.34778	−0.673890	+	0.738831i	$0.735378\pi$
$194$	−804.074	−0.297573
$195$	0	0
$196$	375.727	0.136927
$197$	2429.72	0.878733	0.439366	−	0.898308i	$-0.355203\pi$
$197$	2429.72	0.878733	0.439366	+	0.898308i	$0.355203\pi$
$198$	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	$-0.208881\pi$
$223$	5276.91	1.58461	0.792305	+	0.610126i	$0.208881\pi$
$224$	1770.25	0.528034
$225$	0	0
$226$	−730.334	−0.214960
$227$	−4472.50	−1.30771	−0.653855	−	0.756620i	$-0.726849\pi$
$227$	−4472.50	−1.30771	−0.653855	+	0.756620i	$0.726849\pi$
$228$	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	$-0.739289\pi$
$233$	−4857.72	−1.36583	−0.682917	+	0.730496i	$0.739289\pi$
$234$	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	$-0.516533\pi$
$257$	−427.791	−0.103832	−0.0519161	+	0.998651i	$0.516533\pi$
$258$	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	$-0.519440\pi$
$263$	−520.645	−0.122070	−0.0610349	+	0.998136i	$0.519440\pi$
$264$	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	$-0.741282\pi$
$277$	−6338.81	−1.37495	−0.687477	+	0.726206i	$0.741282\pi$
$278$	−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	$-0.465155\pi$
$283$	1040.23	0.218499	0.109250	+	0.994014i	$0.465155\pi$
$284$	−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	$-0.376084\pi$
$293$	3807.01	0.759072	0.379536	+	0.925177i	$0.376084\pi$
$294$	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	$-0.461046\pi$
$307$	1313.26	0.244142	0.122071	+	0.992521i	$0.461046\pi$
$308$	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	$-0.466990\pi$
$313$	1146.47	0.207036	0.103518	+	0.994628i	$0.466990\pi$
$314$	12253.9	2.20232
$315$	0	0
$316$	3967.35	0.706269
$317$	−2562.45	−0.454012	−0.227006	−	0.973893i	$-0.572894\pi$
$317$	−2562.45	−0.454012	−0.227006	+	0.973893i	$0.572894\pi$
$318$	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	$-0.436652\pi$
$337$	2446.17	0.395405	0.197702	+	0.980262i	$0.436652\pi$
$338$	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	$-0.589785\pi$
$347$	−3598.36	−0.556687	−0.278343	+	0.960482i	$0.589785\pi$
$348$	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	$-0.498049\pi$
$353$	81.3189	0.0122611	0.00613055	+	0.999981i	$0.498049\pi$
$354$	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	$-0.203007\pi$
$367$	11297.4	1.60686	0.803429	+	0.595401i	$0.203007\pi$
$368$	−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	$-0.564063\pi$
$373$	−2880.15	−0.399808	−0.199904	+	0.979816i	$0.564063\pi$
$374$	−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	$-0.488912\pi$
$383$	522.066	0.0696510	0.0348255	+	0.999393i	$0.488912\pi$
$384$	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	$-0.304749\pi$
$397$	9106.98	1.15130	0.575650	+	0.817696i	$0.304749\pi$
$398$	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	$-0.655364\pi$
$433$	−8450.44	−0.937880	−0.468940	+	0.883230i	$0.655364\pi$
$434$	−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	$-0.757526\pi$
$443$	−13494.3	−1.44725	−0.723626	+	0.690193i	$0.757526\pi$
$444$	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	$-0.693819\pi$
$457$	−11175.7	−1.14393	−0.571967	+	0.820277i	$0.693819\pi$
$458$	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	$-0.596492\pi$
$463$	−5947.99	−0.597034	−0.298517	+	0.954404i	$0.596492\pi$
$464$	−13049.8	−1.30565
$465$	0	0
$466$	−19228.1	−1.91143
$467$	−11775.8	−1.16685	−0.583424	−	0.812168i	$-0.698287\pi$
$467$	−11775.8	−1.16685	−0.583424	+	0.812168i	$0.698287\pi$
$468$	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	$-0.672706\pi$
$487$	−11098.4	−1.03268	−0.516340	+	0.856383i	$0.672706\pi$
$488$	−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	$-0.829008\pi$
$503$	−19384.4	−1.71830	−0.859152	+	0.511721i	$0.829008\pi$
$504$	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	$-0.367958\pi$
$523$	9640.89	0.806055	0.403027	+	0.915188i	$0.367958\pi$
$524$	−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	$-0.515859\pi$
$547$	−1274.24	−0.0996025	−0.0498013	+	0.998759i	$0.515859\pi$
$548$	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	$-0.774508\pi$
$557$	−19965.7	−1.51880	−0.759402	+	0.650621i	$0.774508\pi$
$558$	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	$-0.331828\pi$
$563$	13467.9	1.00818	0.504090	+	0.863651i	$0.331828\pi$
$564$	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	$-0.199508\pi$
$577$	22451.1	1.61985	0.809925	+	0.586534i	$0.199508\pi$
$578$	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	$-0.567830\pi$
$587$	−6015.40	−0.422968	−0.211484	+	0.977381i	$0.567830\pi$
$588$	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	$-0.683434\pi$
$593$	−15737.3	−1.08981	−0.544903	+	0.838499i	$0.683434\pi$
$594$	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	$-0.489864\pi$
$607$	952.222	0.0636730	0.0318365	+	0.999493i	$0.489864\pi$
$608$	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	$-0.531121\pi$
$613$	−2962.98	−0.195226	−0.0976132	+	0.995224i	$0.531121\pi$
$614$	5198.23	0.341667
$615$	0	0
$616$	−169.760	−0.0111036
$617$	28257.7	1.84378	0.921891	−	0.387449i	$-0.126644\pi$
$617$	28257.7	1.84378	0.921891	+	0.387449i	$0.126644\pi$
$618$	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	$-0.516368\pi$
$643$	−1676.14	−0.102800	−0.0514000	+	0.998678i	$0.516368\pi$
$644$	−7548.24	−0.461867
$645$	0	0
$646$	−60584.1	−3.68986
$647$	17319.8	1.05242	0.526208	−	0.850356i	$-0.323613\pi$
$647$	17319.8	1.05242	0.526208	+	0.850356i	$0.323613\pi$
$648$	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	$-0.672257\pi$
$653$	−17191.7	−1.03026	−0.515132	+	0.857111i	$0.672257\pi$
$654$	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	$-0.352435\pi$
$673$	15614.1	0.894322	0.447161	+	0.894453i	$0.352435\pi$
$674$	9682.60	0.553353
$675$	0	0
$676$	1458.93	0.0830071
$677$	1964.69	0.111535	0.0557675	−	0.998444i	$-0.482239\pi$
$677$	1964.69	0.111535	0.0557675	+	0.998444i	$0.482239\pi$
$678$	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	$-0.453192\pi$
$683$	5230.81	0.293047	0.146524	+	0.989207i	$0.453192\pi$
$684$	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	$-0.533999\pi$
$727$	−4179.50	−0.213217	−0.106609	+	0.994301i	$0.533999\pi$
$728$	−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	$-0.709859\pi$
$733$	−24312.7	−1.22511	−0.612557	+	0.790426i	$0.709859\pi$
$734$	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	$-0.232973\pi$
$743$	30132.1	1.48780	0.743902	+	0.668288i	$0.232973\pi$
$744$	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	$-0.325505\pi$
$757$	21708.6	1.04229	0.521144	+	0.853469i	$0.325505\pi$
$758$	−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	$-0.592765\pi$
$773$	−12350.0	−0.574644	−0.287322	+	0.957834i	$0.592765\pi$
$774$	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	$-0.400359\pi$
$787$	13597.6	0.615888	0.307944	+	0.951405i	$0.400359\pi$
$788$	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	$-0.638367\pi$
$797$	−18951.1	−0.842262	−0.421131	+	0.907000i	$0.638367\pi$
$798$	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	$-0.601027\pi$
$823$	−14736.8	−0.624170	−0.312085	+	0.950054i	$0.601027\pi$
$824$	−623.062	−0.0263415
$825$	0	0
$826$	3100.52	0.130606
$827$	−9129.69	−0.383882	−0.191941	−	0.981406i	$-0.561478\pi$
$827$	−9129.69	−0.383882	−0.191941	+	0.981406i	$0.561478\pi$
$828$	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	$-0.605874\pi$
$853$	−16268.8	−0.653027	−0.326514	+	0.945192i	$0.605874\pi$
$854$	−11544.7	−0.462590
$855$	0	0
$856$	−653.210	−0.0260821
$857$	−2376.68	−0.0947325	−0.0473663	−	0.998878i	$-0.515083\pi$
$857$	−2376.68	−0.0947325	−0.0473663	+	0.998878i	$0.515083\pi$
$858$	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	$-0.632336\pi$
$863$	−20478.1	−0.807743	−0.403871	+	0.914816i	$0.632336\pi$
$864$	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	$-0.858380\pi$
$877$	−46886.5	−1.80530	−0.902648	+	0.430379i	$0.858380\pi$
$878$	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	$-0.757898\pi$
$883$	−38016.2	−1.44886	−0.724432	+	0.689346i	$0.757898\pi$
$884$	−46727.8	−1.77786
$885$	0	0
$886$	−53414.0	−2.02537
$887$	25246.2	0.955675	0.477838	−	0.878448i	$-0.341421\pi$
$887$	25246.2	0.955675	0.477838	+	0.878448i	$0.341421\pi$
$888$	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	$-0.431420\pi$
$907$	11679.5	0.427576	0.213788	+	0.976880i	$0.431420\pi$
$908$	−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	$-0.493457\pi$
$937$	1179.03	0.0411068	0.0205534	+	0.999789i	$0.493457\pi$
$938$	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	$-0.665941\pi$
$947$	−29027.3	−0.996050	−0.498025	+	0.867163i	$0.665941\pi$
$948$	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	$-0.803398\pi$
$953$	−47968.6	−1.63049	−0.815245	+	0.579116i	$0.803398\pi$
$954$	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	$-0.463186\pi$
$967$	6940.06	0.230793	0.115397	+	0.993319i	$0.463186\pi$
$968$	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	$-0.451020\pi$
$977$	9361.03	0.306536	0.153268	+	0.988185i	$0.451020\pi$
$978$	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	$-0.527493\pi$
$983$	−5317.34	−0.172530	−0.0862649	+	0.996272i	$0.527493\pi$
$984$	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	$-0.655772\pi$
$997$	−29596.3	−0.940145	−0.470073	+	0.882628i	$0.655772\pi$
$998$	−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.br.1.7	yes	8
3.2	odd	2	inner	1575.4.a.br.1.2	✓	8
5.4	even	2		1575.4.a.bs.1.2	yes	8
15.14	odd	2		1575.4.a.bs.1.7	yes	8

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

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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