LMFDB - Embedded newform 1575.4.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,-7,0,17,0,0,14,-63,0,0,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q-5.56155 q^{2} +22.9309 q^{4} +7.00000 q^{7} -83.0388 q^{8} +33.6155 q^{11} +38.3542 q^{13} -38.9309 q^{14} +278.378 q^{16} +65.7235 q^{17} +33.3996 q^{19} -186.955 q^{22} +207.447 q^{23} -213.309 q^{26} +160.516 q^{28} +189.170 q^{29} +202.108 q^{31} -883.902 q^{32} -365.525 q^{34} +16.5227 q^{37} -185.754 q^{38} -388.617 q^{41} -41.8144 q^{43} +770.833 q^{44} -1153.73 q^{46} +368.648 q^{47} +49.0000 q^{49} +879.494 q^{52} +458.172 q^{53} -581.272 q^{56} -1052.08 q^{58} -256.216 q^{59} -123.511 q^{61} -1124.03 q^{62} +2688.85 q^{64} +336.277 q^{67} +1507.10 q^{68} +453.312 q^{71} -22.0436 q^{73} -91.8920 q^{74} +765.882 q^{76} +235.309 q^{77} +385.417 q^{79} +2161.32 q^{82} +23.7501 q^{83} +232.553 q^{86} -2791.39 q^{88} +1482.81 q^{89} +268.479 q^{91} +4756.94 q^{92} -2050.25 q^{94} -51.9867 q^{97} -272.516 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8} + 26 q^{11} - 14 q^{13} - 49 q^{14} + 297 q^{16} + 16 q^{17} + 174 q^{19} - 176 q^{22} + 184 q^{23} - 138 q^{26} + 119 q^{28} + 32 q^{29} + 330 q^{31} - 1071 q^{32}+ \cdots - 343 q^{98}+O(q^{100}) $ 2 * q - 7 * q^2 + 17 * q^4 + 14 * q^7 - 63 * q^8 + 26 * q^11 - 14 * q^13 - 49 * q^14 + 297 * q^16 + 16 * q^17 + 174 * q^19 - 176 * q^22 + 184 * q^23 - 138 * q^26 + 119 * q^28 + 32 * q^29 + 330 * q^31 - 1071 * q^32 - 294 * q^34 + 132 * q^37 - 388 * q^38 - 200 * q^41 - 364 * q^43 + 816 * q^44 - 1120 * q^46 + 292 * q^47 + 98 * q^49 + 1190 * q^52 + 34 * q^53 - 441 * q^56 - 826 * q^58 - 364 * q^59 + 792 * q^61 - 1308 * q^62 + 2809 * q^64 + 788 * q^67 + 1802 * q^68 - 454 * q^71 - 778 * q^73 - 258 * q^74 - 68 * q^76 + 182 * q^77 + 408 * q^79 + 1890 * q^82 + 1136 * q^83 + 696 * q^86 - 2944 * q^88 - 36 * q^89 - 98 * q^91 + 4896 * q^92 - 1940 * q^94 + 498 * q^97 - 343 * 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 $

$2$	−5.56155	−1.96631	−0.983153	−	0.182785i	$-0.941489\pi$
$2$	−5.56155	−1.96631	−0.983153	+	0.182785i	$0.941489\pi$
$3$	0	0
$3$	0	0
$4$	22.9309	2.86636
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−83.0388	−3.66983
$9$	0	0
$10$	0	0
$11$	33.6155	0.921406	0.460703	−	0.887554i	$-0.347597\pi$
$11$	33.6155	0.921406	0.460703	+	0.887554i	$0.347597\pi$
$12$	0	0
$13$	38.3542	0.818272	0.409136	−	0.912474i	$-0.365830\pi$
$13$	38.3542	0.818272	0.409136	+	0.912474i	$0.365830\pi$
$14$	−38.9309	−0.743194
$15$	0	0
$16$	278.378	4.34965
$17$	65.7235	0.937664	0.468832	−	0.883287i	$-0.344675\pi$
$17$	65.7235	0.937664	0.468832	+	0.883287i	$0.344675\pi$
$18$	0	0
$19$	33.3996	0.403284	0.201642	−	0.979459i	$-0.435372\pi$
$19$	33.3996	0.403284	0.201642	+	0.979459i	$0.435372\pi$
$20$	0	0
$21$	0	0
$22$	−186.955	−1.81177
$23$	207.447	1.88068	0.940341	−	0.340234i	$-0.110506\pi$
$23$	207.447	1.88068	0.940341	+	0.340234i	$0.110506\pi$
$24$	0	0
$25$	0	0
$26$	−213.309	−1.60897
$27$	0	0
$28$	160.516	1.08338
$29$	189.170	1.21131	0.605656	−	0.795726i	$-0.292911\pi$
$29$	189.170	1.21131	0.605656	+	0.795726i	$0.292911\pi$
$30$	0	0
$31$	202.108	1.17096	0.585478	−	0.810688i	$-0.300907\pi$
$31$	202.108	1.17096	0.585478	+	0.810688i	$0.300907\pi$
$32$	−883.902	−4.88292
$33$	0	0
$34$	−365.525	−1.84373
$35$	0	0
$36$	0	0
$37$	16.5227	0.0734141	0.0367070	−	0.999326i	$-0.488313\pi$
$37$	16.5227	0.0734141	0.0367070	+	0.999326i	$0.488313\pi$
$38$	−185.754	−0.792980
$39$	0	0
$40$	0	0
$41$	−388.617	−1.48029	−0.740144	−	0.672448i	$-0.765243\pi$
$41$	−388.617	−1.48029	−0.740144	+	0.672448i	$0.765243\pi$
$42$	0	0
$43$	−41.8144	−0.148294	−0.0741469	−	0.997247i	$-0.523623\pi$
$43$	−41.8144	−0.148294	−0.0741469	+	0.997247i	$0.523623\pi$
$44$	770.833	2.64108
$45$	0	0
$46$	−1153.73	−3.69800
$47$	368.648	1.14410	0.572051	−	0.820218i	$-0.306148\pi$
$47$	368.648	1.14410	0.572051	+	0.820218i	$0.306148\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	879.494	2.34546
$53$	458.172	1.18745	0.593725	−	0.804668i	$-0.297657\pi$
$53$	458.172	1.18745	0.593725	+	0.804668i	$0.297657\pi$
$54$	0	0
$55$	0	0
$56$	−581.272	−1.38707
$57$	0	0
$58$	−1052.08	−2.38181
$59$	−256.216	−0.565364	−0.282682	−	0.959214i	$-0.591224\pi$
$59$	−256.216	−0.565364	−0.282682	+	0.959214i	$0.591224\pi$
$60$	0	0
$61$	−123.511	−0.259246	−0.129623	−	0.991563i	$-0.541377\pi$
$61$	−123.511	−0.259246	−0.129623	+	0.991563i	$0.541377\pi$
$62$	−1124.03	−2.30246
$63$	0	0
$64$	2688.85	5.25166
$65$	0	0
$66$	0	0
$67$	336.277	0.613175	0.306587	−	0.951843i	$-0.400813\pi$
$67$	336.277	0.613175	0.306587	+	0.951843i	$0.400813\pi$
$68$	1507.10	2.68768
$69$	0	0
$70$	0	0
$71$	453.312	0.757722	0.378861	−	0.925454i	$-0.376316\pi$
$71$	453.312	0.757722	0.378861	+	0.925454i	$0.376316\pi$
$72$	0	0
$73$	−22.0436	−0.0353426	−0.0176713	−	0.999844i	$-0.505625\pi$
$73$	−22.0436	−0.0353426	−0.0176713	+	0.999844i	$0.505625\pi$
$74$	−91.8920	−0.144355
$75$	0	0
$76$	765.882	1.15596
$77$	235.309	0.348259
$78$	0	0
$79$	385.417	0.548896	0.274448	−	0.961602i	$-0.411505\pi$
$79$	385.417	0.548896	0.274448	+	0.961602i	$0.411505\pi$
$80$	0	0
$81$	0	0
$82$	2161.32	2.91070
$83$	23.7501	0.0314085	0.0157043	−	0.999877i	$-0.495001\pi$
$83$	23.7501	0.0314085	0.0157043	+	0.999877i	$0.495001\pi$
$84$	0	0
$85$	0	0
$86$	232.553	0.291591
$87$	0	0
$88$	−2791.39	−3.38140
$89$	1482.81	1.76604	0.883020	−	0.469335i	$-0.155506\pi$
$89$	1482.81	1.76604	0.883020	+	0.469335i	$0.155506\pi$
$90$	0	0
$91$	268.479	0.309278
$92$	4756.94	5.39071
$93$	0	0
$94$	−2050.25	−2.24965
$95$	0	0
$96$	0	0
$97$	−51.9867	−0.0544170	−0.0272085	−	0.999630i	$-0.508662\pi$
$97$	−51.9867	−0.0544170	−0.0272085	+	0.999630i	$0.508662\pi$
$98$	−272.516	−0.280901
$99$	0	0
$100$	0	0
$101$	−1429.30	−1.40812	−0.704062	−	0.710138i	$-0.748632\pi$
$101$	−1429.30	−1.40812	−0.704062	+	0.710138i	$0.748632\pi$
$102$	0	0
$103$	−434.212	−0.415381	−0.207690	−	0.978195i	$-0.566595\pi$
$103$	−434.212	−0.415381	−0.207690	+	0.978195i	$0.566595\pi$
$104$	−3184.88	−3.00292
$105$	0	0
$106$	−2548.15	−2.33489
$107$	666.307	0.602003	0.301001	−	0.953624i	$-0.402679\pi$
$107$	666.307	0.602003	0.301001	+	0.953624i	$0.402679\pi$
$108$	0	0
$109$	−1199.51	−1.05406	−0.527029	−	0.849847i	$-0.676694\pi$
$109$	−1199.51	−1.05406	−0.527029	+	0.849847i	$0.676694\pi$
$110$	0	0
$111$	0	0
$112$	1948.64	1.64401
$113$	−81.5171	−0.0678627	−0.0339314	−	0.999424i	$-0.510803\pi$
$113$	−81.5171	−0.0678627	−0.0339314	+	0.999424i	$0.510803\pi$
$114$	0	0
$115$	0	0
$116$	4337.84	3.47206
$117$	0	0
$118$	1424.96	1.11168
$119$	460.064	0.354404
$120$	0	0
$121$	−200.996	−0.151011
$122$	686.915	0.509757
$123$	0	0
$124$	4634.51	3.35638
$125$	0	0
$126$	0	0
$127$	336.985	0.235453	0.117727	−	0.993046i	$-0.462439\pi$
$127$	336.985	0.235453	0.117727	+	0.993046i	$0.462439\pi$
$128$	−7882.95	−5.44344
$129$	0	0
$130$	0	0
$131$	−2931.15	−1.95493	−0.977465	−	0.211097i	$-0.932297\pi$
$131$	−2931.15	−1.95493	−0.977465	+	0.211097i	$0.932297\pi$
$132$	0	0
$133$	233.797	0.152427
$134$	−1870.22	−1.20569
$135$	0	0
$136$	−5457.60	−3.44107
$137$	−1585.07	−0.988477	−0.494238	−	0.869326i	$-0.664553\pi$
$137$	−1585.07	−0.988477	−0.494238	+	0.869326i	$0.664553\pi$
$138$	0	0
$139$	−1298.85	−0.792569	−0.396284	−	0.918128i	$-0.629701\pi$
$139$	−1298.85	−0.792569	−0.396284	+	0.918128i	$0.629701\pi$
$140$	0	0
$141$	0	0
$142$	−2521.12	−1.48991
$143$	1289.30	0.753960
$144$	0	0
$145$	0	0
$146$	122.597	0.0694943
$147$	0	0
$148$	378.881	0.210431
$149$	2003.29	1.10145	0.550724	−	0.834687i	$-0.314352\pi$
$149$	2003.29	1.10145	0.550724	+	0.834687i	$0.314352\pi$
$150$	0	0
$151$	2740.96	1.47719	0.738596	−	0.674148i	$-0.235489\pi$
$151$	2740.96	1.47719	0.738596	+	0.674148i	$0.235489\pi$
$152$	−2773.47	−1.47999
$153$	0	0
$154$	−1308.68	−0.684783
$155$	0	0
$156$	0	0
$157$	−3644.22	−1.85249	−0.926243	−	0.376928i	$-0.876981\pi$
$157$	−3644.22	−1.85249	−0.926243	+	0.376928i	$0.876981\pi$
$158$	−2143.52	−1.07930
$159$	0	0
$160$	0	0
$161$	1452.13	0.710831
$162$	0	0
$163$	−2774.27	−1.33311	−0.666557	−	0.745454i	$-0.732233\pi$
$163$	−2774.27	−1.33311	−0.666557	+	0.745454i	$0.732233\pi$
$164$	−8911.33	−4.24304
$165$	0	0
$166$	−132.087	−0.0617588
$167$	1154.91	0.535149	0.267574	−	0.963537i	$-0.413778\pi$
$167$	1154.91	0.535149	0.267574	+	0.963537i	$0.413778\pi$
$168$	0	0
$169$	−725.958	−0.330432
$170$	0	0
$171$	0	0
$172$	−958.841	−0.425064
$173$	−3387.46	−1.48869	−0.744346	−	0.667794i	$-0.767239\pi$
$173$	−3387.46	−1.48869	−0.744346	+	0.667794i	$0.767239\pi$
$174$	0	0
$175$	0	0
$176$	9357.82	4.00780
$177$	0	0
$178$	−8246.73	−3.47258
$179$	−1603.32	−0.669486	−0.334743	−	0.942309i	$-0.608650\pi$
$179$	−1603.32	−0.669486	−0.334743	+	0.942309i	$0.608650\pi$
$180$	0	0
$181$	544.220	0.223489	0.111745	−	0.993737i	$-0.464356\pi$
$181$	544.220	0.223489	0.111745	+	0.993737i	$0.464356\pi$
$182$	−1493.16	−0.608134
$183$	0	0
$184$	−17226.2	−6.90179
$185$	0	0
$186$	0	0
$187$	2209.33	0.863969
$188$	8453.41	3.27941
$189$	0	0
$190$	0	0
$191$	−2993.44	−1.13402	−0.567010	−	0.823711i	$-0.691900\pi$
$191$	−2993.44	−1.13402	−0.567010	+	0.823711i	$0.691900\pi$
$192$	0	0
$193$	−1309.32	−0.488325	−0.244163	−	0.969734i	$-0.578513\pi$
$193$	−1309.32	−0.488325	−0.244163	+	0.969734i	$0.578513\pi$
$194$	289.127	0.107001
$195$	0	0
$196$	1123.61	0.409480
$197$	−1141.38	−0.412790	−0.206395	−	0.978469i	$-0.566173\pi$
$197$	−1141.38	−0.412790	−0.206395	+	0.978469i	$0.566173\pi$
$198$	0	0
$199$	2370.23	0.844327	0.422164	−	0.906520i	$-0.361271\pi$
$199$	2370.23	0.844327	0.422164	+	0.906520i	$0.361271\pi$
$200$	0	0
$201$	0	0
$202$	7949.12	2.76880
$203$	1324.19	0.457833
$204$	0	0
$205$	0	0
$206$	2414.89	0.816765
$207$	0	0
$208$	10676.9	3.55920
$209$	1122.75	0.371588
$210$	0	0
$211$	−687.159	−0.224199	−0.112099	−	0.993697i	$-0.535758\pi$
$211$	−687.159	−0.224199	−0.112099	+	0.993697i	$0.535758\pi$
$212$	10506.3	3.40366
$213$	0	0
$214$	−3705.70	−1.18372
$215$	0	0
$216$	0	0
$217$	1414.76	0.442580
$218$	6671.15	2.07260
$219$	0	0
$220$	0	0
$221$	2520.77	0.767264
$222$	0	0
$223$	−990.496	−0.297437	−0.148719	−	0.988880i	$-0.547515\pi$
$223$	−990.496	−0.297437	−0.148719	+	0.988880i	$0.547515\pi$
$224$	−6187.32	−1.84557
$225$	0	0
$226$	453.362	0.133439
$227$	1479.25	0.432517	0.216258	−	0.976336i	$-0.430615\pi$
$227$	1479.25	0.432517	0.216258	+	0.976336i	$0.430615\pi$
$228$	0	0
$229$	6704.47	1.93469	0.967345	−	0.253463i	$-0.0815696\pi$
$229$	6704.47	1.93469	0.967345	+	0.253463i	$0.0815696\pi$
$230$	0	0
$231$	0	0
$232$	−15708.5	−4.44531
$233$	−1749.09	−0.491789	−0.245895	−	0.969297i	$-0.579082\pi$
$233$	−1749.09	−0.491789	−0.245895	+	0.969297i	$0.579082\pi$
$234$	0	0
$235$	0	0
$236$	−5875.25	−1.62054
$237$	0	0
$238$	−2558.67	−0.696866
$239$	6320.89	1.71073	0.855365	−	0.518027i	$-0.173333\pi$
$239$	6320.89	1.71073	0.855365	+	0.518027i	$0.173333\pi$
$240$	0	0
$241$	3359.62	0.897975	0.448988	−	0.893538i	$-0.351785\pi$
$241$	3359.62	0.897975	0.448988	+	0.893538i	$0.351785\pi$
$242$	1117.85	0.296935
$243$	0	0
$244$	−2832.22	−0.743092
$245$	0	0
$246$	0	0
$247$	1281.01	0.329996
$248$	−16782.8	−4.29721
$249$	0	0
$250$	0	0
$251$	1330.50	0.334582	0.167291	−	0.985908i	$-0.446498\pi$
$251$	1330.50	0.334582	0.167291	+	0.985908i	$0.446498\pi$
$252$	0	0
$253$	6973.44	1.73287
$254$	−1874.16	−0.462973
$255$	0	0
$256$	22330.6	5.45182
$257$	−2476.95	−0.601197	−0.300599	−	0.953751i	$-0.597186\pi$
$257$	−2476.95	−0.601197	−0.300599	+	0.953751i	$0.597186\pi$
$258$	0	0
$259$	115.659	0.0277479
$260$	0	0
$261$	0	0
$262$	16301.8	3.84399
$263$	−5152.56	−1.20806	−0.604032	−	0.796960i	$-0.706440\pi$
$263$	−5152.56	−1.20806	−0.604032	+	0.796960i	$0.706440\pi$
$264$	0	0
$265$	0	0
$266$	−1300.28	−0.299718
$267$	0	0
$268$	7711.11	1.75758
$269$	1150.97	0.260876	0.130438	−	0.991456i	$-0.458362\pi$
$269$	1150.97	0.260876	0.130438	+	0.991456i	$0.458362\pi$
$270$	0	0
$271$	1838.32	0.412067	0.206034	−	0.978545i	$-0.433944\pi$
$271$	1838.32	0.412067	0.206034	+	0.978545i	$0.433944\pi$
$272$	18296.0	4.07851
$273$	0	0
$274$	8815.43	1.94365
$275$	0	0
$276$	0	0
$277$	−568.447	−0.123302	−0.0616510	−	0.998098i	$-0.519637\pi$
$277$	−568.447	−0.123302	−0.0616510	+	0.998098i	$0.519637\pi$
$278$	7223.62	1.55843
$279$	0	0
$280$	0	0
$281$	−6015.00	−1.27696	−0.638479	−	0.769640i	$-0.720436\pi$
$281$	−6015.00	−1.27696	−0.638479	+	0.769640i	$0.720436\pi$
$282$	0	0
$283$	−3985.75	−0.837202	−0.418601	−	0.908170i	$-0.637479\pi$
$283$	−3985.75	−0.837202	−0.418601	+	0.908170i	$0.637479\pi$
$284$	10394.8	2.17190
$285$	0	0
$286$	−7170.48	−1.48252
$287$	−2720.32	−0.559497
$288$	0	0
$289$	−593.424	−0.120787
$290$	0	0
$291$	0	0
$292$	−505.479	−0.101305
$293$	−2490.01	−0.496478	−0.248239	−	0.968699i	$-0.579852\pi$
$293$	−2490.01	−0.496478	−0.248239	+	0.968699i	$0.579852\pi$
$294$	0	0
$295$	0	0
$296$	−1372.03	−0.269417
$297$	0	0
$298$	−11141.4	−2.16578
$299$	7956.45	1.53891
$300$	0	0
$301$	−292.701	−0.0560498
$302$	−15244.0	−2.90461
$303$	0	0
$304$	9297.72	1.75415
$305$	0	0
$306$	0	0
$307$	141.853	0.0263712	0.0131856	−	0.999913i	$-0.495803\pi$
$307$	141.853	0.0263712	0.0131856	+	0.999913i	$0.495803\pi$
$308$	5395.83	0.998234
$309$	0	0
$310$	0	0
$311$	−2091.92	−0.381420	−0.190710	−	0.981646i	$-0.561079\pi$
$311$	−2091.92	−0.381420	−0.190710	+	0.981646i	$0.561079\pi$
$312$	0	0
$313$	−5521.44	−0.997094	−0.498547	−	0.866863i	$-0.666133\pi$
$313$	−5521.44	−0.997094	−0.498547	+	0.866863i	$0.666133\pi$
$314$	20267.5	3.64255
$315$	0	0
$316$	8837.94	1.57333
$317$	5351.63	0.948195	0.474097	−	0.880472i	$-0.342774\pi$
$317$	5351.63	0.948195	0.474097	+	0.880472i	$0.342774\pi$
$318$	0	0
$319$	6359.06	1.11611
$320$	0	0
$321$	0	0
$322$	−8076.09	−1.39771
$323$	2195.14	0.378145
$324$	0	0
$325$	0	0
$326$	15429.2	2.62131
$327$	0	0
$328$	32270.3	5.43241
$329$	2580.53	0.432430
$330$	0	0
$331$	−4383.52	−0.727916	−0.363958	−	0.931415i	$-0.618575\pi$
$331$	−4383.52	−0.727916	−0.363958	+	0.931415i	$0.618575\pi$
$332$	544.609	0.0900281
$333$	0	0
$334$	−6423.11	−1.05227
$335$	0	0
$336$	0	0
$337$	7124.57	1.15163	0.575817	−	0.817579i	$-0.304684\pi$
$337$	7124.57	1.15163	0.575817	+	0.817579i	$0.304684\pi$
$338$	4037.46	0.649730
$339$	0	0
$340$	0	0
$341$	6793.97	1.07893
$342$	0	0
$343$	343.000	0.0539949
$344$	3472.22	0.544214
$345$	0	0
$346$	18839.5	2.92722
$347$	507.743	0.0785506	0.0392753	−	0.999228i	$-0.487495\pi$
$347$	507.743	0.0785506	0.0392753	+	0.999228i	$0.487495\pi$
$348$	0	0
$349$	6155.14	0.944060	0.472030	−	0.881582i	$-0.343521\pi$
$349$	6155.14	0.944060	0.472030	+	0.881582i	$0.343521\pi$
$350$	0	0
$351$	0	0
$352$	−29712.8	−4.49915
$353$	−6429.56	−0.969437	−0.484718	−	0.874670i	$-0.661078\pi$
$353$	−6429.56	−0.969437	−0.484718	+	0.874670i	$0.661078\pi$
$354$	0	0
$355$	0	0
$356$	34002.1	5.06211
$357$	0	0
$358$	8916.97	1.31641
$359$	−10075.4	−1.48123	−0.740614	−	0.671931i	$-0.765465\pi$
$359$	−10075.4	−1.48123	−0.740614	+	0.671931i	$0.765465\pi$
$360$	0	0
$361$	−5743.46	−0.837362
$362$	−3026.71	−0.439448
$363$	0	0
$364$	6156.46	0.886501
$365$	0	0
$366$	0	0
$367$	−816.898	−0.116190	−0.0580950	−	0.998311i	$-0.518503\pi$
$367$	−816.898	−0.116190	−0.0580950	+	0.998311i	$0.518503\pi$
$368$	57748.6	8.18031
$369$	0	0
$370$	0	0
$371$	3207.21	0.448814
$372$	0	0
$373$	3737.85	0.518870	0.259435	−	0.965761i	$-0.416464\pi$
$373$	3737.85	0.518870	0.259435	+	0.965761i	$0.416464\pi$
$374$	−12287.3	−1.69883
$375$	0	0
$376$	−30612.1	−4.19866
$377$	7255.47	0.991183
$378$	0	0
$379$	1950.47	0.264351	0.132176	−	0.991226i	$-0.457804\pi$
$379$	1950.47	0.264351	0.132176	+	0.991226i	$0.457804\pi$
$380$	0	0
$381$	0	0
$382$	16648.2	2.22983
$383$	6762.06	0.902155	0.451077	−	0.892485i	$-0.351040\pi$
$383$	6762.06	0.902155	0.451077	+	0.892485i	$0.351040\pi$
$384$	0	0
$385$	0	0
$386$	7281.84	0.960197
$387$	0	0
$388$	−1192.10	−0.155979
$389$	2551.98	0.332624	0.166312	−	0.986073i	$-0.446814\pi$
$389$	2551.98	0.332624	0.166312	+	0.986073i	$0.446814\pi$
$390$	0	0
$391$	13634.1	1.76345
$392$	−4068.90	−0.524262
$393$	0	0
$394$	6347.83	0.811672
$395$	0	0
$396$	0	0
$397$	4097.93	0.518058	0.259029	−	0.965869i	$-0.416597\pi$
$397$	4097.93	0.518058	0.259029	+	0.965869i	$0.416597\pi$
$398$	−13182.2	−1.66021
$399$	0	0
$400$	0	0
$401$	1046.81	0.130362	0.0651811	−	0.997873i	$-0.479238\pi$
$401$	1046.81	0.130362	0.0651811	+	0.997873i	$0.479238\pi$
$402$	0	0
$403$	7751.68	0.958161
$404$	−32775.1	−4.03619
$405$	0	0
$406$	−7364.57	−0.900240
$407$	555.420	0.0676441
$408$	0	0
$409$	6516.92	0.787876	0.393938	−	0.919137i	$-0.371113\pi$
$409$	6516.92	0.787876	0.393938	+	0.919137i	$0.371113\pi$
$410$	0	0
$411$	0	0
$412$	−9956.86	−1.19063
$413$	−1793.51	−0.213687
$414$	0	0
$415$	0	0
$416$	−33901.3	−3.99555
$417$	0	0
$418$	−6244.21	−0.730656
$419$	−12279.1	−1.43168	−0.715838	−	0.698267i	$-0.753955\pi$
$419$	−12279.1	−1.43168	−0.715838	+	0.698267i	$0.753955\pi$
$420$	0	0
$421$	10146.9	1.17465	0.587325	−	0.809351i	$-0.300181\pi$
$421$	10146.9	1.17465	0.587325	+	0.809351i	$0.300181\pi$
$422$	3821.67	0.440844
$423$	0	0
$424$	−38046.1	−4.35774
$425$	0	0
$426$	0	0
$427$	−864.579	−0.0979858
$428$	15279.0	1.72556
$429$	0	0
$430$	0	0
$431$	−7059.04	−0.788914	−0.394457	−	0.918914i	$-0.629067\pi$
$431$	−7059.04	−0.788914	−0.394457	+	0.918914i	$0.629067\pi$
$432$	0	0
$433$	−6468.98	−0.717966	−0.358983	−	0.933344i	$-0.616876\pi$
$433$	−6468.98	−0.717966	−0.358983	+	0.933344i	$0.616876\pi$
$434$	−7868.24	−0.870248
$435$	0	0
$436$	−27505.8	−3.02131
$437$	6928.65	0.758449
$438$	0	0
$439$	4767.13	0.518275	0.259137	−	0.965840i	$-0.416562\pi$
$439$	4767.13	0.518275	0.259137	+	0.965840i	$0.416562\pi$
$440$	0	0
$441$	0	0
$442$	−14019.4	−1.50868
$443$	2366.55	0.253810	0.126905	−	0.991915i	$-0.459496\pi$
$443$	2366.55	0.253810	0.126905	+	0.991915i	$0.459496\pi$
$444$	0	0
$445$	0	0
$446$	5508.70	0.584853
$447$	0	0
$448$	18821.9	1.98494
$449$	−1814.17	−0.190681	−0.0953406	−	0.995445i	$-0.530394\pi$
$449$	−1814.17	−0.190681	−0.0953406	+	0.995445i	$0.530394\pi$
$450$	0	0
$451$	−13063.6	−1.36395
$452$	−1869.26	−0.194519
$453$	0	0
$454$	−8226.93	−0.850460
$455$	0	0
$456$	0	0
$457$	−8284.13	−0.847955	−0.423977	−	0.905673i	$-0.639366\pi$
$457$	−8284.13	−0.847955	−0.423977	+	0.905673i	$0.639366\pi$
$458$	−37287.3	−3.80419
$459$	0	0
$460$	0	0
$461$	−1384.62	−0.139888	−0.0699439	−	0.997551i	$-0.522282\pi$
$461$	−1384.62	−0.139888	−0.0699439	+	0.997551i	$0.522282\pi$
$462$	0	0
$463$	13210.3	1.32599	0.662994	−	0.748624i	$-0.269285\pi$
$463$	13210.3	1.32599	0.662994	+	0.748624i	$0.269285\pi$
$464$	52660.9	5.26879
$465$	0	0
$466$	9727.67	0.967008
$467$	−4574.24	−0.453256	−0.226628	−	0.973981i	$-0.572770\pi$
$467$	−4574.24	−0.453256	−0.226628	+	0.973981i	$0.572770\pi$
$468$	0	0
$469$	2353.94	0.231758
$470$	0	0
$471$	0	0
$472$	21275.9	2.07479
$473$	−1405.61	−0.136639
$474$	0	0
$475$	0	0
$476$	10549.7	1.01585
$477$	0	0
$478$	−35154.0	−3.36382
$479$	11031.8	1.05231	0.526154	−	0.850389i	$-0.323634\pi$
$479$	11031.8	1.05231	0.526154	+	0.850389i	$0.323634\pi$
$480$	0	0
$481$	633.716	0.0600726
$482$	−18684.7	−1.76569
$483$	0	0
$484$	−4609.02	−0.432853
$485$	0	0
$486$	0	0
$487$	5194.06	0.483296	0.241648	−	0.970364i	$-0.422312\pi$
$487$	5194.06	0.483296	0.241648	+	0.970364i	$0.422312\pi$
$488$	10256.2	0.951389
$489$	0	0
$490$	0	0
$491$	−11954.7	−1.09880	−0.549398	−	0.835561i	$-0.685143\pi$
$491$	−11954.7	−1.09880	−0.549398	+	0.835561i	$0.685143\pi$
$492$	0	0
$493$	12432.9	1.13580
$494$	−7124.43	−0.648873
$495$	0	0
$496$	56262.4	5.09326
$497$	3173.19	0.286392
$498$	0	0
$499$	2566.05	0.230205	0.115102	−	0.993354i	$-0.463280\pi$
$499$	2566.05	0.230205	0.115102	+	0.993354i	$0.463280\pi$
$500$	0	0
$501$	0	0
$502$	−7399.62	−0.657891
$503$	21103.5	1.87069	0.935347	−	0.353731i	$-0.115087\pi$
$503$	21103.5	1.87069	0.935347	+	0.353731i	$0.115087\pi$
$504$	0	0
$505$	0	0
$506$	−38783.1	−3.40735
$507$	0	0
$508$	7727.36	0.674894
$509$	781.732	0.0680740	0.0340370	−	0.999421i	$-0.489164\pi$
$509$	781.732	0.0680740	0.0340370	+	0.999421i	$0.489164\pi$
$510$	0	0
$511$	−154.305	−0.0133582
$512$	−61129.5	−5.27650
$513$	0	0
$514$	13775.7	1.18214
$515$	0	0
$516$	0	0
$517$	12392.3	1.05418
$518$	−643.244	−0.0545609
$519$	0	0
$520$	0	0
$521$	−14013.0	−1.17835	−0.589176	−	0.808005i	$-0.700547\pi$
$521$	−14013.0	−1.17835	−0.589176	+	0.808005i	$0.700547\pi$
$522$	0	0
$523$	10310.7	0.862052	0.431026	−	0.902339i	$-0.358152\pi$
$523$	10310.7	0.862052	0.431026	+	0.902339i	$0.358152\pi$
$524$	−67213.8	−5.60353
$525$	0	0
$526$	28656.3	2.37542
$527$	13283.2	1.09796
$528$	0	0
$529$	30867.2	2.53696
$530$	0	0
$531$	0	0
$532$	5361.18	0.436911
$533$	−14905.1	−1.21128
$534$	0	0
$535$	0	0
$536$	−27924.0	−2.25025
$537$	0	0
$538$	−6401.16	−0.512962
$539$	1647.16	0.131629
$540$	0	0
$541$	−17562.9	−1.39572	−0.697862	−	0.716232i	$-0.745865\pi$
$541$	−17562.9	−1.39572	−0.697862	+	0.716232i	$0.745865\pi$
$542$	−10223.9	−0.810250
$543$	0	0
$544$	−58093.1	−4.57853
$545$	0	0
$546$	0	0
$547$	19889.6	1.55469	0.777347	−	0.629072i	$-0.216565\pi$
$547$	19889.6	1.55469	0.777347	+	0.629072i	$0.216565\pi$
$548$	−36346.9	−2.83333
$549$	0	0
$550$	0	0
$551$	6318.22	0.488503
$552$	0	0
$553$	2697.92	0.207463
$554$	3161.45	0.242450
$555$	0	0
$556$	−29783.8	−2.27179
$557$	5579.54	0.424439	0.212219	−	0.977222i	$-0.431931\pi$
$557$	5579.54	0.424439	0.212219	+	0.977222i	$0.431931\pi$
$558$	0	0
$559$	−1603.76	−0.121345
$560$	0	0
$561$	0	0
$562$	33452.8	2.51089
$563$	−24463.2	−1.83126	−0.915630	−	0.402022i	$-0.868307\pi$
$563$	−24463.2	−1.83126	−0.915630	+	0.402022i	$0.868307\pi$
$564$	0	0
$565$	0	0
$566$	22167.0	1.64620
$567$	0	0
$568$	−37642.5	−2.78071
$569$	8582.14	0.632306	0.316153	−	0.948708i	$-0.397609\pi$
$569$	8582.14	0.632306	0.316153	+	0.948708i	$0.397609\pi$
$570$	0	0
$571$	17580.8	1.28850	0.644248	−	0.764816i	$-0.277170\pi$
$571$	17580.8	1.28850	0.644248	+	0.764816i	$0.277170\pi$
$572$	29564.7	2.16112
$573$	0	0
$574$	15129.2	1.10014
$575$	0	0
$576$	0	0
$577$	8692.57	0.627169	0.313585	−	0.949560i	$-0.398470\pi$
$577$	8692.57	0.627169	0.313585	+	0.949560i	$0.398470\pi$
$578$	3300.36	0.237503
$579$	0	0
$580$	0	0
$581$	166.250	0.0118713
$582$	0	0
$583$	15401.7	1.09412
$584$	1830.47	0.129701
$585$	0	0
$586$	13848.3	0.976227
$587$	3584.61	0.252049	0.126024	−	0.992027i	$-0.459778\pi$
$587$	3584.61	0.252049	0.126024	+	0.992027i	$0.459778\pi$
$588$	0	0
$589$	6750.33	0.472228
$590$	0	0
$591$	0	0
$592$	4599.56	0.319326
$593$	−21853.6	−1.51335	−0.756676	−	0.653790i	$-0.773178\pi$
$593$	−21853.6	−1.51335	−0.756676	+	0.653790i	$0.773178\pi$
$594$	0	0
$595$	0	0
$596$	45937.1	3.15714
$597$	0	0
$598$	−44250.2	−3.02596
$599$	−9090.48	−0.620078	−0.310039	−	0.950724i	$-0.600342\pi$
$599$	−9090.48	−0.620078	−0.310039	+	0.950724i	$0.600342\pi$
$600$	0	0
$601$	−19546.1	−1.32663	−0.663314	−	0.748341i	$-0.730851\pi$
$601$	−19546.1	−1.32663	−0.663314	+	0.748341i	$0.730851\pi$
$602$	1627.87	0.110211
$603$	0	0
$604$	62852.6	4.23416
$605$	0	0
$606$	0	0
$607$	15726.0	1.05157	0.525783	−	0.850619i	$-0.323773\pi$
$607$	15726.0	1.05157	0.525783	+	0.850619i	$0.323773\pi$
$608$	−29522.0	−1.96920
$609$	0	0
$610$	0	0
$611$	14139.2	0.936186
$612$	0	0
$613$	13572.5	0.894269	0.447135	−	0.894467i	$-0.352444\pi$
$613$	13572.5	0.894269	0.447135	+	0.894467i	$0.352444\pi$
$614$	−788.921	−0.0518538
$615$	0	0
$616$	−19539.8	−1.27805
$617$	17378.5	1.13393	0.566964	−	0.823743i	$-0.308118\pi$
$617$	17378.5	1.13393	0.566964	+	0.823743i	$0.308118\pi$
$618$	0	0
$619$	−25113.3	−1.63068	−0.815338	−	0.578985i	$-0.803449\pi$
$619$	−25113.3	−1.63068	−0.815338	+	0.578985i	$0.803449\pi$
$620$	0	0
$621$	0	0
$622$	11634.3	0.749989
$623$	10379.7	0.667501
$624$	0	0
$625$	0	0
$626$	30707.8	1.96059
$627$	0	0
$628$	−83565.1	−5.30989
$629$	1085.93	0.0688377
$630$	0	0
$631$	−10814.4	−0.682276	−0.341138	−	0.940013i	$-0.610812\pi$
$631$	−10814.4	−0.682276	−0.341138	+	0.940013i	$0.610812\pi$
$632$	−32004.5	−2.01436
$633$	0	0
$634$	−29763.4	−1.86444
$635$	0	0
$636$	0	0
$637$	1879.35	0.116896
$638$	−35366.3	−2.19461
$639$	0	0
$640$	0	0
$641$	16359.0	1.00802	0.504010	−	0.863698i	$-0.331857\pi$
$641$	16359.0	1.00802	0.504010	+	0.863698i	$0.331857\pi$
$642$	0	0
$643$	−8819.47	−0.540911	−0.270456	−	0.962732i	$-0.587174\pi$
$643$	−8819.47	−0.540911	−0.270456	+	0.962732i	$0.587174\pi$
$644$	33298.6	2.03750
$645$	0	0
$646$	−12208.4	−0.743549
$647$	−13828.8	−0.840290	−0.420145	−	0.907457i	$-0.638021\pi$
$647$	−13828.8	−0.840290	−0.420145	+	0.907457i	$0.638021\pi$
$648$	0	0
$649$	−8612.83	−0.520930
$650$	0	0
$651$	0	0
$652$	−63616.4	−3.82118
$653$	23988.7	1.43760	0.718798	−	0.695219i	$-0.244693\pi$
$653$	23988.7	1.43760	0.718798	+	0.695219i	$0.244693\pi$
$654$	0	0
$655$	0	0
$656$	−108182.	−6.43874
$657$	0	0
$658$	−14351.8	−0.850289
$659$	−3109.28	−0.183794	−0.0918972	−	0.995769i	$-0.529293\pi$
$659$	−3109.28	−0.183794	−0.0918972	+	0.995769i	$0.529293\pi$
$660$	0	0
$661$	22695.0	1.33545	0.667726	−	0.744407i	$-0.267268\pi$
$661$	22695.0	1.33545	0.667726	+	0.744407i	$0.267268\pi$
$662$	24379.2	1.43131
$663$	0	0
$664$	−1972.18	−0.115264
$665$	0	0
$666$	0	0
$667$	39242.8	2.27809
$668$	26483.2	1.53393
$669$	0	0
$670$	0	0
$671$	−4151.90	−0.238871
$672$	0	0
$673$	−22073.4	−1.26429	−0.632145	−	0.774850i	$-0.717825\pi$
$673$	−22073.4	−1.26429	−0.632145	+	0.774850i	$0.717825\pi$
$674$	−39623.7	−2.26446
$675$	0	0
$676$	−16646.9	−0.947136
$677$	−2489.50	−0.141328	−0.0706642	−	0.997500i	$-0.522512\pi$
$677$	−2489.50	−0.141328	−0.0706642	+	0.997500i	$0.522512\pi$
$678$	0	0
$679$	−363.907	−0.0205677
$680$	0	0
$681$	0	0
$682$	−37785.0	−2.12150
$683$	7970.98	0.446561	0.223280	−	0.974754i	$-0.428323\pi$
$683$	7970.98	0.446561	0.223280	+	0.974754i	$0.428323\pi$
$684$	0	0
$685$	0	0
$686$	−1907.61	−0.106171
$687$	0	0
$688$	−11640.2	−0.645027
$689$	17572.8	0.971656
$690$	0	0
$691$	−23892.7	−1.31537	−0.657687	−	0.753292i	$-0.728465\pi$
$691$	−23892.7	−1.31537	−0.657687	+	0.753292i	$0.728465\pi$
$692$	−77677.4	−4.26712
$693$	0	0
$694$	−2823.84	−0.154454
$695$	0	0
$696$	0	0
$697$	−25541.3	−1.38801
$698$	−34232.1	−1.85631
$699$	0	0
$700$	0	0
$701$	−12197.0	−0.657170	−0.328585	−	0.944474i	$-0.606572\pi$
$701$	−12197.0	−0.657170	−0.328585	+	0.944474i	$0.606572\pi$
$702$	0	0
$703$	551.853	0.0296067
$704$	90387.0	4.83891
$705$	0	0
$706$	35758.4	1.90621
$707$	−10005.1	−0.532221
$708$	0	0
$709$	−8982.28	−0.475792	−0.237896	−	0.971291i	$-0.576458\pi$
$709$	−8982.28	−0.475792	−0.237896	+	0.971291i	$0.576458\pi$
$710$	0	0
$711$	0	0
$712$	−123131.	−6.48107
$713$	41926.7	2.20220
$714$	0	0
$715$	0	0
$716$	−36765.6	−1.91899
$717$	0	0
$718$	56035.0	2.91255
$719$	6501.61	0.337231	0.168616	−	0.985682i	$-0.446070\pi$
$719$	6501.61	0.337231	0.168616	+	0.985682i	$0.446070\pi$
$720$	0	0
$721$	−3039.49	−0.156999
$722$	31942.6	1.64651
$723$	0	0
$724$	12479.4	0.640600
$725$	0	0
$726$	0	0
$727$	24228.7	1.23603	0.618013	−	0.786168i	$-0.287938\pi$
$727$	24228.7	1.23603	0.618013	+	0.786168i	$0.287938\pi$
$728$	−22294.2	−1.13500
$729$	0	0
$730$	0	0
$731$	−2748.19	−0.139050
$732$	0	0
$733$	−36719.1	−1.85027	−0.925136	−	0.379636i	$-0.876049\pi$
$733$	−36719.1	−1.85027	−0.925136	+	0.379636i	$0.876049\pi$
$734$	4543.22	0.228465
$735$	0	0
$736$	−183363.	−9.18321
$737$	11304.1	0.564983
$738$	0	0
$739$	23304.5	1.16004	0.580021	−	0.814602i	$-0.303044\pi$
$739$	23304.5	1.16004	0.580021	+	0.814602i	$0.303044\pi$
$740$	0	0
$741$	0	0
$742$	−17837.0	−0.882505
$743$	6875.35	0.339478	0.169739	−	0.985489i	$-0.445708\pi$
$743$	6875.35	0.339478	0.169739	+	0.985489i	$0.445708\pi$
$744$	0	0
$745$	0	0
$746$	−20788.3	−1.02026
$747$	0	0
$748$	50661.8	2.47644
$749$	4664.15	0.227536
$750$	0	0
$751$	1182.65	0.0574640	0.0287320	−	0.999587i	$-0.490853\pi$
$751$	1182.65	0.0574640	0.0287320	+	0.999587i	$0.490853\pi$
$752$	102623.	4.97645
$753$	0	0
$754$	−40351.7	−1.94897
$755$	0	0
$756$	0	0
$757$	−25226.8	−1.21121	−0.605604	−	0.795766i	$-0.707068\pi$
$757$	−25226.8	−1.21121	−0.605604	+	0.795766i	$0.707068\pi$
$758$	−10847.7	−0.519795
$759$	0	0
$760$	0	0
$761$	10909.2	0.519655	0.259827	−	0.965655i	$-0.416334\pi$
$761$	10909.2	0.519655	0.259827	+	0.965655i	$0.416334\pi$
$762$	0	0
$763$	−8396.58	−0.398397
$764$	−68642.2	−3.25051
$765$	0	0
$766$	−37607.6	−1.77391
$767$	−9826.95	−0.462621
$768$	0	0
$769$	−12771.6	−0.598903	−0.299451	−	0.954112i	$-0.596804\pi$
$769$	−12771.6	−0.598903	−0.299451	+	0.954112i	$0.596804\pi$
$770$	0	0
$771$	0	0
$772$	−30023.8	−1.39972
$773$	2199.06	0.102322	0.0511610	−	0.998690i	$-0.483708\pi$
$773$	2199.06	0.102322	0.0511610	+	0.998690i	$0.483708\pi$
$774$	0	0
$775$	0	0
$776$	4316.92	0.199701
$777$	0	0
$778$	−14193.0	−0.654041
$779$	−12979.7	−0.596977
$780$	0	0
$781$	15238.3	0.698170
$782$	−75827.0	−3.46748
$783$	0	0
$784$	13640.5	0.621379
$785$	0	0
$786$	0	0
$787$	19587.7	0.887201	0.443601	−	0.896225i	$-0.353701\pi$
$787$	19587.7	0.887201	0.443601	+	0.896225i	$0.353701\pi$
$788$	−26172.8	−1.18321
$789$	0	0
$790$	0	0
$791$	−570.620	−0.0256497
$792$	0	0
$793$	−4737.17	−0.212134
$794$	−22790.9	−1.01866
$795$	0	0
$796$	54351.4	2.42014
$797$	21699.3	0.964401	0.482200	−	0.876061i	$-0.339838\pi$
$797$	21699.3	0.964401	0.482200	+	0.876061i	$0.339838\pi$
$798$	0	0
$799$	24228.8	1.07278
$800$	0	0
$801$	0	0
$802$	−5821.89	−0.256332
$803$	−741.007	−0.0325649
$804$	0	0
$805$	0	0
$806$	−43111.4	−1.88404
$807$	0	0
$808$	118687.	5.16758
$809$	15649.4	0.680103	0.340051	−	0.940407i	$-0.389556\pi$
$809$	15649.4	0.680103	0.340051	+	0.940407i	$0.389556\pi$
$810$	0	0
$811$	−33267.3	−1.44041	−0.720206	−	0.693760i	$-0.755953\pi$
$811$	−33267.3	−1.44041	−0.720206	+	0.693760i	$0.755953\pi$
$812$	30364.9	1.31231
$813$	0	0
$814$	−3089.00	−0.133009
$815$	0	0
$816$	0	0
$817$	−1396.59	−0.0598046
$818$	−36244.2	−1.54920
$819$	0	0
$820$	0	0
$821$	−5158.58	−0.219288	−0.109644	−	0.993971i	$-0.534971\pi$
$821$	−5158.58	−0.219288	−0.109644	+	0.993971i	$0.534971\pi$
$822$	0	0
$823$	−26333.6	−1.11535	−0.557674	−	0.830060i	$-0.688306\pi$
$823$	−26333.6	−1.11535	−0.557674	+	0.830060i	$0.688306\pi$
$824$	36056.5	1.52438
$825$	0	0
$826$	9974.71	0.420175
$827$	19572.7	0.822988	0.411494	−	0.911413i	$-0.365007\pi$
$827$	19572.7	0.822988	0.411494	+	0.911413i	$0.365007\pi$
$828$	0	0
$829$	9642.26	0.403968	0.201984	−	0.979389i	$-0.435261\pi$
$829$	9642.26	0.403968	0.201984	+	0.979389i	$0.435261\pi$
$830$	0	0
$831$	0	0
$832$	103128.	4.29728
$833$	3220.45	0.133952
$834$	0	0
$835$	0	0
$836$	25745.5	1.06511
$837$	0	0
$838$	68290.7	2.81511
$839$	31081.1	1.27895	0.639475	−	0.768812i	$-0.279152\pi$
$839$	31081.1	1.27895	0.639475	+	0.768812i	$0.279152\pi$
$840$	0	0
$841$	11396.5	0.467278
$842$	−56432.3	−2.30972
$843$	0	0
$844$	−15757.1	−0.642634
$845$	0	0
$846$	0	0
$847$	−1406.97	−0.0570770
$848$	127545.	5.16499
$849$	0	0
$850$	0	0
$851$	3427.59	0.138068
$852$	0	0
$853$	−25780.9	−1.03484	−0.517421	−	0.855731i	$-0.673108\pi$
$853$	−25780.9	−1.03484	−0.517421	+	0.855731i	$0.673108\pi$
$854$	4808.40	0.192670
$855$	0	0
$856$	−55329.3	−2.20925
$857$	14452.6	0.576069	0.288035	−	0.957620i	$-0.406998\pi$
$857$	14452.6	0.576069	0.288035	+	0.957620i	$0.406998\pi$
$858$	0	0
$859$	889.366	0.0353257	0.0176628	−	0.999844i	$-0.494377\pi$
$859$	889.366	0.0353257	0.0176628	+	0.999844i	$0.494377\pi$
$860$	0	0
$861$	0	0
$862$	39259.2	1.55125
$863$	41460.3	1.63537	0.817685	−	0.575665i	$-0.195257\pi$
$863$	41460.3	1.63537	0.817685	+	0.575665i	$0.195257\pi$
$864$	0	0
$865$	0	0
$866$	35977.6	1.41174
$867$	0	0
$868$	32441.6	1.26859
$869$	12956.0	0.505756
$870$	0	0
$871$	12897.6	0.501744
$872$	99606.0	3.86822
$873$	0	0
$874$	−38534.1	−1.49134
$875$	0	0
$876$	0	0
$877$	−21173.0	−0.815236	−0.407618	−	0.913153i	$-0.633640\pi$
$877$	−21173.0	−0.815236	−0.407618	+	0.913153i	$0.633640\pi$
$878$	−26512.6	−1.01909
$879$	0	0
$880$	0	0
$881$	−9883.39	−0.377957	−0.188978	−	0.981981i	$-0.560518\pi$
$881$	−9883.39	−0.377957	−0.188978	+	0.981981i	$0.560518\pi$
$882$	0	0
$883$	45273.9	1.72547	0.862734	−	0.505658i	$-0.168750\pi$
$883$	45273.9	1.72547	0.862734	+	0.505658i	$0.168750\pi$
$884$	57803.4	2.19925
$885$	0	0
$886$	−13161.7	−0.499069
$887$	644.388	0.0243928	0.0121964	−	0.999926i	$-0.496118\pi$
$887$	644.388	0.0243928	0.0121964	+	0.999926i	$0.496118\pi$
$888$	0	0
$889$	2358.89	0.0889930
$890$	0	0
$891$	0	0
$892$	−22712.9	−0.852562
$893$	12312.7	0.461398
$894$	0	0
$895$	0	0
$896$	−55180.6	−2.05743
$897$	0	0
$898$	10089.6	0.374937
$899$	38232.8	1.41839
$900$	0	0
$901$	30112.7	1.11343
$902$	72653.8	2.68194
$903$	0	0
$904$	6769.09	0.249045
$905$	0	0
$906$	0	0
$907$	15065.2	0.551522	0.275761	−	0.961226i	$-0.411070\pi$
$907$	15065.2	0.551522	0.275761	+	0.961226i	$0.411070\pi$
$908$	33920.5	1.23975
$909$	0	0
$910$	0	0
$911$	−28789.9	−1.04704	−0.523520	−	0.852014i	$-0.675381\pi$
$911$	−28789.9	−1.04704	−0.523520	+	0.852014i	$0.675381\pi$
$912$	0	0
$913$	798.371	0.0289400
$914$	46072.6	1.66734
$915$	0	0
$916$	153739.	5.54552
$917$	−20518.1	−0.738894
$918$	0	0
$919$	−24163.8	−0.867345	−0.433673	−	0.901070i	$-0.642783\pi$
$919$	−24163.8	−0.867345	−0.433673	+	0.901070i	$0.642783\pi$
$920$	0	0
$921$	0	0
$922$	7700.65	0.275062
$923$	17386.4	0.620023
$924$	0	0
$925$	0	0
$926$	−73469.5	−2.60730
$927$	0	0
$928$	−167208.	−5.91474
$929$	35115.4	1.24015	0.620075	−	0.784542i	$-0.287102\pi$
$929$	35115.4	1.24015	0.620075	+	0.784542i	$0.287102\pi$
$930$	0	0
$931$	1636.58	0.0576120
$932$	−40108.2	−1.40964
$933$	0	0
$934$	25439.9	0.891239
$935$	0	0
$936$	0	0
$937$	15512.6	0.540849	0.270424	−	0.962741i	$-0.412836\pi$
$937$	15512.6	0.540849	0.270424	+	0.962741i	$0.412836\pi$
$938$	−13091.5	−0.455708
$939$	0	0
$940$	0	0
$941$	53283.8	1.84591	0.922956	−	0.384905i	$-0.125766\pi$
$941$	53283.8	1.84591	0.922956	+	0.384905i	$0.125766\pi$
$942$	0	0
$943$	−80617.5	−2.78395
$944$	−71324.8	−2.45914
$945$	0	0
$946$	7817.39	0.268674
$947$	−55509.7	−1.90478	−0.952388	−	0.304890i	$-0.901380\pi$
$947$	−55509.7	−1.90478	−0.952388	+	0.304890i	$0.901380\pi$
$948$	0	0
$949$	−845.464	−0.0289198
$950$	0	0
$951$	0	0
$952$	−38203.2	−1.30060
$953$	−28080.6	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$953$	−28080.6	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$954$	0	0
$955$	0	0
$956$	144943.	4.90356
$957$	0	0
$958$	−61353.9	−2.06916
$959$	−11095.5	−0.373609
$960$	0	0
$961$	11056.6	0.371140
$962$	−3524.44	−0.118121
$963$	0	0
$964$	77039.0	2.57392
$965$	0	0
$966$	0	0
$967$	−56609.3	−1.88256	−0.941278	−	0.337634i	$-0.890374\pi$
$967$	−56609.3	−1.88256	−0.941278	+	0.337634i	$0.890374\pi$
$968$	16690.5	0.554187
$969$	0	0
$970$	0	0
$971$	6782.17	0.224151	0.112075	−	0.993700i	$-0.464250\pi$
$971$	6782.17	0.224151	0.112075	+	0.993700i	$0.464250\pi$
$972$	0	0
$973$	−9091.95	−0.299563
$974$	−28887.0	−0.950309
$975$	0	0
$976$	−34382.8	−1.12763
$977$	−45655.0	−1.49502	−0.747509	−	0.664252i	$-0.768750\pi$
$977$	−45655.0	−1.49502	−0.747509	+	0.664252i	$0.768750\pi$
$978$	0	0
$979$	49845.5	1.62724
$980$	0	0
$981$	0	0
$982$	66486.8	2.16057
$983$	−10102.3	−0.327785	−0.163893	−	0.986478i	$-0.552405\pi$
$983$	−10102.3	−0.327785	−0.163893	+	0.986478i	$0.552405\pi$
$984$	0	0
$985$	0	0
$986$	−69146.4	−2.23334
$987$	0	0
$988$	29374.8	0.945887
$989$	−8674.27	−0.278894
$990$	0	0
$991$	25416.2	0.814705	0.407353	−	0.913271i	$-0.366452\pi$
$991$	25416.2	0.814705	0.407353	+	0.913271i	$0.366452\pi$
$992$	−178644.	−5.71768
$993$	0	0
$994$	−17647.8	−0.563135
$995$	0	0
$996$	0	0
$997$	−48152.5	−1.52959	−0.764797	−	0.644271i	$-0.777161\pi$
$997$	−48152.5	−1.52959	−0.764797	+	0.644271i	$0.777161\pi$
$998$	−14271.2	−0.452653
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.m.1.1		2
3.2	odd	2		525.4.a.p.1.2		2
5.4	even	2		315.4.a.m.1.2		2
15.2	even	4		525.4.d.i.274.4		4
15.8	even	4		525.4.d.i.274.1		4
15.14	odd	2		105.4.a.c.1.1	✓	2
35.34	odd	2		2205.4.a.bh.1.2		2
60.59	even	2		1680.4.a.bk.1.2		2
105.104	even	2		735.4.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.c.1.1	✓	2	15.14	odd	2
315.4.a.m.1.2		2	5.4	even	2
525.4.a.p.1.2		2	3.2	odd	2
525.4.d.i.274.1		4	15.8	even	4
525.4.d.i.274.4		4	15.2	even	4
735.4.a.k.1.1		2	105.104	even	2
1575.4.a.m.1.1		2	1.1	even	1	trivial
1680.4.a.bk.1.2		2	60.59	even	2
2205.4.a.bh.1.2		2	35.34	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-5.56155 q^{2} +22.9309 q^{4} +7.00000 q^{7} -83.0388 q^{8} +33.6155 q^{11} +38.3542 q^{13} -38.9309 q^{14} +278.378 q^{16} +65.7235 q^{17} +33.3996 q^{19} -186.955 q^{22} +207.447 q^{23} -213.309 q^{26} +160.516 q^{28} +189.170 q^{29} +202.108 q^{31} -883.902 q^{32} -365.525 q^{34} +16.5227 q^{37} -185.754 q^{38} -388.617 q^{41} -41.8144 q^{43} +770.833 q^{44} -1153.73 q^{46} +368.648 q^{47} +49.0000 q^{49} +879.494 q^{52} +458.172 q^{53} -581.272 q^{56} -1052.08 q^{58} -256.216 q^{59} -123.511 q^{61} -1124.03 q^{62} +2688.85 q^{64} +336.277 q^{67} +1507.10 q^{68} +453.312 q^{71} -22.0436 q^{73} -91.8920 q^{74} +765.882 q^{76} +235.309 q^{77} +385.417 q^{79} +2161.32 q^{82} +23.7501 q^{83} +232.553 q^{86} -2791.39 q^{88} +1482.81 q^{89} +268.479 q^{91} +4756.94 q^{92} -2050.25 q^{94} -51.9867 q^{97} -272.516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8} + 26 q^{11} - 14 q^{13} - 49 q^{14} + 297 q^{16} + 16 q^{17} + 174 q^{19} - 176 q^{22} + 184 q^{23} - 138 q^{26} + 119 q^{28} + 32 q^{29} + 330 q^{31} - 1071 q^{32}+ \cdots - 343 q^{98}+O(q^{100}) \) 2 * q - 7 * q^2 + 17 * q^4 + 14 * q^7 - 63 * q^8 + 26 * q^11 - 14 * q^13 - 49 * q^14 + 297 * q^16 + 16 * q^17 + 174 * q^19 - 176 * q^22 + 184 * q^23 - 138 * q^26 + 119 * q^28 + 32 * q^29 + 330 * q^31 - 1071 * q^32 - 294 * q^34 + 132 * q^37 - 388 * q^38 - 200 * q^41 - 364 * q^43 + 816 * q^44 - 1120 * q^46 + 292 * q^47 + 98 * q^49 + 1190 * q^52 + 34 * q^53 - 441 * q^56 - 826 * q^58 - 364 * q^59 + 792 * q^61 - 1308 * q^62 + 2809 * q^64 + 788 * q^67 + 1802 * q^68 - 454 * q^71 - 778 * q^73 - 258 * q^74 - 68 * q^76 + 182 * q^77 + 408 * q^79 + 1890 * q^82 + 1136 * q^83 + 696 * q^86 - 2944 * q^88 - 36 * q^89 - 98 * q^91 + 4896 * q^92 - 1940 * q^94 + 498 * q^97 - 343 * q^98

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