LMFDB - Embedded newform 1575.4.a.bh.1.4

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 = [4,0,0,10,0,0,-28,0,0,0,0] 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:	$1$
Dimension:	$4$
Coefficient field:	4.4.3030748.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 21x^{2} + 28 $ x^4 - 21*x^2 + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.42371$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.42371 q^{2} +11.5692 q^{4} -7.00000 q^{7} +15.7890 q^{8} +23.3775 q^{11} -3.56918 q^{13} -30.9659 q^{14} -22.7075 q^{16} -57.5082 q^{17} -51.1384 q^{19} +103.415 q^{22} -65.6390 q^{23} -15.7890 q^{26} -80.9843 q^{28} -41.6147 q^{29} -167.538 q^{31} -226.763 q^{32} -254.399 q^{34} +224.538 q^{37} -226.221 q^{38} -196.688 q^{41} -58.9685 q^{43} +270.458 q^{44} -290.368 q^{46} -41.9282 q^{47} +49.0000 q^{49} -41.2925 q^{52} +33.3445 q^{53} -110.523 q^{56} -184.091 q^{58} +229.212 q^{59} -700.613 q^{61} -741.138 q^{62} -821.475 q^{64} +453.890 q^{67} -665.322 q^{68} -930.571 q^{71} -370.182 q^{73} +993.289 q^{74} -591.629 q^{76} -163.642 q^{77} -54.6007 q^{79} -870.091 q^{82} +430.045 q^{83} -260.859 q^{86} +369.107 q^{88} +737.202 q^{89} +24.9843 q^{91} -759.390 q^{92} -185.478 q^{94} -150.371 q^{97} +216.762 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{4} - 28 q^{7} + 22 q^{13} + 18 q^{16} - 132 q^{19} + 196 q^{22} - 70 q^{28} - 126 q^{31} - 546 q^{34} + 354 q^{37} + 272 q^{43} - 182 q^{46} + 196 q^{49} - 274 q^{52} + 98 q^{58} - 1170 q^{61}+ \cdots - 1980 q^{97}+O(q^{100}) $ 4 * q + 10 * q^4 - 28 * q^7 + 22 * q^13 + 18 * q^16 - 132 * q^19 + 196 * q^22 - 70 * q^28 - 126 * q^31 - 546 * q^34 + 354 * q^37 + 272 * q^43 - 182 * q^46 + 196 * q^49 - 274 * q^52 + 98 * q^58 - 1170 * q^61 - 1726 * q^64 + 38 * q^67 + 188 * q^73 - 988 * q^76 - 690 * q^79 - 2646 * q^82 + 896 * q^88 - 154 * q^91 - 1540 * q^94 - 1980 * 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 $

$2$	4.42371	1.56402	0.782008	−	0.623268i	$-0.214195\pi$
$2$	4.42371	1.56402	0.782008	+	0.623268i	$0.214195\pi$
$3$	0	0
$3$	0	0
$4$	11.5692	1.44615
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	15.7890	0.697782
$9$	0	0
$10$	0	0
$11$	23.3775	0.640779	0.320390	−	0.947286i	$-0.396186\pi$
$11$	23.3775	0.640779	0.320390	+	0.947286i	$0.396186\pi$
$12$	0	0
$13$	−3.56918	−0.0761471	−0.0380735	−	0.999275i	$-0.512122\pi$
$13$	−3.56918	−0.0761471	−0.0380735	+	0.999275i	$0.512122\pi$
$14$	−30.9659	−0.591143
$15$	0	0
$16$	−22.7075	−0.354805
$17$	−57.5082	−0.820458	−0.410229	−	0.911983i	$-0.634551\pi$
$17$	−57.5082	−0.820458	−0.410229	+	0.911983i	$0.634551\pi$
$18$	0	0
$19$	−51.1384	−0.617471	−0.308735	−	0.951148i	$-0.599906\pi$
$19$	−51.1384	−0.617471	−0.308735	+	0.951148i	$0.599906\pi$
$20$	0	0
$21$	0	0
$22$	103.415	1.00219
$23$	−65.6390	−0.595073	−0.297537	−	0.954710i	$-0.596165\pi$
$23$	−65.6390	−0.595073	−0.297537	+	0.954710i	$0.596165\pi$
$24$	0	0
$25$	0	0
$26$	−15.7890	−0.119095
$27$	0	0
$28$	−80.9843	−0.546592
$29$	−41.6147	−0.266471	−0.133235	−	0.991084i	$-0.542537\pi$
$29$	−41.6147	−0.266471	−0.133235	+	0.991084i	$0.542537\pi$
$30$	0	0
$31$	−167.538	−0.970666	−0.485333	−	0.874329i	$-0.661302\pi$
$31$	−167.538	−0.970666	−0.485333	+	0.874329i	$0.661302\pi$
$32$	−226.763	−1.25270
$33$	0	0
$34$	−254.399	−1.28321
$35$	0	0
$36$	0	0
$37$	224.538	0.997669	0.498835	−	0.866697i	$-0.333761\pi$
$37$	224.538	0.997669	0.498835	+	0.866697i	$0.333761\pi$
$38$	−226.221	−0.965734
$39$	0	0
$40$	0	0
$41$	−196.688	−0.749208	−0.374604	−	0.927185i	$-0.622221\pi$
$41$	−196.688	−0.749208	−0.374604	+	0.927185i	$0.622221\pi$
$42$	0	0
$43$	−58.9685	−0.209131	−0.104565	−	0.994518i	$-0.533345\pi$
$43$	−58.9685	−0.209131	−0.104565	+	0.994518i	$0.533345\pi$
$44$	270.458	0.926661
$45$	0	0
$46$	−290.368	−0.930704
$47$	−41.9282	−0.130125	−0.0650623	−	0.997881i	$-0.520725\pi$
$47$	−41.9282	−0.130125	−0.0650623	+	0.997881i	$0.520725\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−41.2925	−0.110120
$53$	33.3445	0.0864192	0.0432096	−	0.999066i	$-0.486242\pi$
$53$	33.3445	0.0864192	0.0432096	+	0.999066i	$0.486242\pi$
$54$	0	0
$55$	0	0
$56$	−110.523	−0.263737
$57$	0	0
$58$	−184.091	−0.416765
$59$	229.212	0.505777	0.252888	−	0.967495i	$-0.418619\pi$
$59$	229.212	0.505777	0.252888	+	0.967495i	$0.418619\pi$
$60$	0	0
$61$	−700.613	−1.47056	−0.735281	−	0.677762i	$-0.762950\pi$
$61$	−700.613	−1.47056	−0.735281	+	0.677762i	$0.762950\pi$
$62$	−741.138	−1.51814
$63$	0	0
$64$	−821.475	−1.60444
$65$	0	0
$66$	0	0
$67$	453.890	0.827634	0.413817	−	0.910360i	$-0.364195\pi$
$67$	453.890	0.827634	0.413817	+	0.910360i	$0.364195\pi$
$68$	−665.322	−1.18650
$69$	0	0
$70$	0	0
$71$	−930.571	−1.55547	−0.777735	−	0.628592i	$-0.783632\pi$
$71$	−930.571	−1.55547	−0.777735	+	0.628592i	$0.783632\pi$
$72$	0	0
$73$	−370.182	−0.593514	−0.296757	−	0.954953i	$-0.595905\pi$
$73$	−370.182	−0.593514	−0.296757	+	0.954953i	$0.595905\pi$
$74$	993.289	1.56037
$75$	0	0
$76$	−591.629	−0.892954
$77$	−163.642	−0.242192
$78$	0	0
$79$	−54.6007	−0.0777602	−0.0388801	−	0.999244i	$-0.512379\pi$
$79$	−54.6007	−0.0777602	−0.0388801	+	0.999244i	$0.512379\pi$
$80$	0	0
$81$	0	0
$82$	−870.091	−1.17177
$83$	430.045	0.568718	0.284359	−	0.958718i	$-0.408219\pi$
$83$	430.045	0.568718	0.284359	+	0.958718i	$0.408219\pi$
$84$	0	0
$85$	0	0
$86$	−260.859	−0.327084
$87$	0	0
$88$	369.107	0.447124
$89$	737.202	0.878014	0.439007	−	0.898484i	$-0.355330\pi$
$89$	737.202	0.878014	0.439007	+	0.898484i	$0.355330\pi$
$90$	0	0
$91$	24.9843	0.0287809
$92$	−759.390	−0.860564
$93$	0	0
$94$	−185.478	−0.203517
$95$	0	0
$96$	0	0
$97$	−150.371	−0.157401	−0.0787004	−	0.996898i	$-0.525077\pi$
$97$	−150.371	−0.157401	−0.0787004	+	0.996898i	$0.525077\pi$
$98$	216.762	0.223431
$99$	0	0
$100$	0	0
$101$	−134.423	−0.132432	−0.0662158	−	0.997805i	$-0.521093\pi$
$101$	−134.423	−0.132432	−0.0662158	+	0.997805i	$0.521093\pi$
$102$	0	0
$103$	232.494	0.222411	0.111205	−	0.993797i	$-0.464529\pi$
$103$	232.494	0.222411	0.111205	+	0.993797i	$0.464529\pi$
$104$	−56.3538	−0.0531340
$105$	0	0
$106$	147.506	0.135161
$107$	−1539.38	−1.39082	−0.695409	−	0.718614i	$-0.744777\pi$
$107$	−1539.38	−1.39082	−0.695409	+	0.718614i	$0.744777\pi$
$108$	0	0
$109$	−503.462	−0.442412	−0.221206	−	0.975227i	$-0.570999\pi$
$109$	−503.462	−0.442412	−0.221206	+	0.975227i	$0.570999\pi$
$110$	0	0
$111$	0	0
$112$	158.953	0.134104
$113$	−267.084	−0.222347	−0.111173	−	0.993801i	$-0.535461\pi$
$113$	−267.084	−0.222347	−0.111173	+	0.993801i	$0.535461\pi$
$114$	0	0
$115$	0	0
$116$	−481.448	−0.385356
$117$	0	0
$118$	1013.97	0.791043
$119$	402.557	0.310104
$120$	0	0
$121$	−784.494	−0.589402
$122$	−3099.31	−2.29998
$123$	0	0
$124$	−1938.27	−1.40373
$125$	0	0
$126$	0	0
$127$	296.412	0.207105	0.103552	−	0.994624i	$-0.466979\pi$
$127$	296.412	0.207105	0.103552	+	0.994624i	$0.466979\pi$
$128$	−1819.86	−1.25667
$129$	0	0
$130$	0	0
$131$	−2768.20	−1.84625	−0.923123	−	0.384504i	$-0.874373\pi$
$131$	−2768.20	−1.84625	−0.923123	+	0.384504i	$0.874373\pi$
$132$	0	0
$133$	357.969	0.233382
$134$	2007.88	1.29443
$135$	0	0
$136$	−907.997	−0.572500
$137$	−813.962	−0.507602	−0.253801	−	0.967256i	$-0.581681\pi$
$137$	−813.962	−0.507602	−0.253801	+	0.967256i	$0.581681\pi$
$138$	0	0
$139$	99.1631	0.0605101	0.0302550	−	0.999542i	$-0.490368\pi$
$139$	99.1631	0.0605101	0.0302550	+	0.999542i	$0.490368\pi$
$140$	0	0
$141$	0	0
$142$	−4116.57	−2.43278
$143$	−83.4384	−0.0487935
$144$	0	0
$145$	0	0
$146$	−1637.58	−0.928266
$147$	0	0
$148$	2597.72	1.44278
$149$	1427.06	0.784624	0.392312	−	0.919832i	$-0.371675\pi$
$149$	1427.06	0.784624	0.392312	+	0.919832i	$0.371675\pi$
$150$	0	0
$151$	1719.54	0.926718	0.463359	−	0.886171i	$-0.346644\pi$
$151$	1719.54	0.926718	0.463359	+	0.886171i	$0.346644\pi$
$152$	−807.423	−0.430860
$153$	0	0
$154$	−723.906	−0.378792
$155$	0	0
$156$	0	0
$157$	−3270.08	−1.66230	−0.831149	−	0.556049i	$-0.812317\pi$
$157$	−3270.08	−1.66230	−0.831149	+	0.556049i	$0.812317\pi$
$158$	−241.537	−0.121618
$159$	0	0
$160$	0	0
$161$	459.473	0.224917
$162$	0	0
$163$	891.324	0.428306	0.214153	−	0.976800i	$-0.431301\pi$
$163$	891.324	0.428306	0.214153	+	0.976800i	$0.431301\pi$
$164$	−2275.52	−1.08347
$165$	0	0
$166$	1902.39	0.889484
$167$	−1598.41	−0.740649	−0.370324	−	0.928902i	$-0.620754\pi$
$167$	−1598.41	−0.740649	−0.370324	+	0.928902i	$0.620754\pi$
$168$	0	0
$169$	−2184.26	−0.994202
$170$	0	0
$171$	0	0
$172$	−682.217	−0.302434
$173$	−2492.49	−1.09538	−0.547690	−	0.836681i	$-0.684493\pi$
$173$	−2492.49	−1.09538	−0.547690	+	0.836681i	$0.684493\pi$
$174$	0	0
$175$	0	0
$176$	−530.845	−0.227352
$177$	0	0
$178$	3261.16	1.37323
$179$	1410.34	0.588904	0.294452	−	0.955666i	$-0.404863\pi$
$179$	1410.34	0.588904	0.294452	+	0.955666i	$0.404863\pi$
$180$	0	0
$181$	2039.16	0.837399	0.418700	−	0.908125i	$-0.362486\pi$
$181$	2039.16	0.837399	0.418700	+	0.908125i	$0.362486\pi$
$182$	110.523	0.0450138
$183$	0	0
$184$	−1036.37	−0.415231
$185$	0	0
$186$	0	0
$187$	−1344.40	−0.525732
$188$	−485.075	−0.188179
$189$	0	0
$190$	0	0
$191$	2576.79	0.976178	0.488089	−	0.872794i	$-0.337694\pi$
$191$	2576.79	0.976178	0.488089	+	0.872794i	$0.337694\pi$
$192$	0	0
$193$	2873.61	1.07175	0.535874	−	0.844298i	$-0.319982\pi$
$193$	2873.61	1.07175	0.535874	+	0.844298i	$0.319982\pi$
$194$	−665.198	−0.246178
$195$	0	0
$196$	566.890	0.206592
$197$	1489.82	0.538810	0.269405	−	0.963027i	$-0.413173\pi$
$197$	1489.82	0.538810	0.269405	+	0.963027i	$0.413173\pi$
$198$	0	0
$199$	1311.18	0.467071	0.233536	−	0.972348i	$-0.424970\pi$
$199$	1311.18	0.467071	0.233536	+	0.972348i	$0.424970\pi$
$200$	0	0
$201$	0	0
$202$	−594.648	−0.207125
$203$	291.303	0.100716
$204$	0	0
$205$	0	0
$206$	1028.48	0.347854
$207$	0	0
$208$	81.0472	0.0270174
$209$	−1195.49	−0.395662
$210$	0	0
$211$	2671.37	0.871585	0.435793	−	0.900047i	$-0.356468\pi$
$211$	2671.37	0.871585	0.435793	+	0.900047i	$0.356468\pi$
$212$	385.768	0.124975
$213$	0	0
$214$	−6809.77	−2.17526
$215$	0	0
$216$	0	0
$217$	1172.76	0.366877
$218$	−2227.17	−0.691940
$219$	0	0
$220$	0	0
$221$	205.257	0.0624755
$222$	0	0
$223$	3321.87	0.997529	0.498764	−	0.866738i	$-0.333787\pi$
$223$	3321.87	0.997529	0.498764	+	0.866738i	$0.333787\pi$
$224$	1587.34	0.473477
$225$	0	0
$226$	−1181.50	−0.347754
$227$	1756.95	0.513712	0.256856	−	0.966450i	$-0.417313\pi$
$227$	1756.95	0.513712	0.256856	+	0.966450i	$0.417313\pi$
$228$	0	0
$229$	2654.65	0.766045	0.383023	−	0.923739i	$-0.374883\pi$
$229$	2654.65	0.766045	0.383023	+	0.923739i	$0.374883\pi$
$230$	0	0
$231$	0	0
$232$	−657.054	−0.185938
$233$	4379.70	1.23143	0.615716	−	0.787968i	$-0.288867\pi$
$233$	4379.70	1.23143	0.615716	+	0.787968i	$0.288867\pi$
$234$	0	0
$235$	0	0
$236$	2651.79	0.731427
$237$	0	0
$238$	1780.80	0.485008
$239$	932.890	0.252484	0.126242	−	0.991999i	$-0.459708\pi$
$239$	932.890	0.252484	0.126242	+	0.991999i	$0.459708\pi$
$240$	0	0
$241$	−6451.43	−1.72437	−0.862184	−	0.506595i	$-0.830904\pi$
$241$	−6451.43	−1.72437	−0.862184	+	0.506595i	$0.830904\pi$
$242$	−3470.37	−0.921834
$243$	0	0
$244$	−8105.52	−2.12665
$245$	0	0
$246$	0	0
$247$	182.522	0.0470186
$248$	−2645.25	−0.677313
$249$	0	0
$250$	0	0
$251$	−5085.23	−1.27879	−0.639396	−	0.768878i	$-0.720815\pi$
$251$	−5085.23	−1.27879	−0.639396	+	0.768878i	$0.720815\pi$
$252$	0	0
$253$	−1534.47	−0.381311
$254$	1311.24	0.323915
$255$	0	0
$256$	−1478.71	−0.361013
$257$	4817.11	1.16919	0.584597	−	0.811324i	$-0.301253\pi$
$257$	4817.11	1.16919	0.584597	+	0.811324i	$0.301253\pi$
$258$	0	0
$259$	−1571.76	−0.377084
$260$	0	0
$261$	0	0
$262$	−12245.7	−2.88756
$263$	8001.67	1.87606	0.938030	−	0.346553i	$-0.112648\pi$
$263$	8001.67	1.87606	0.938030	+	0.346553i	$0.112648\pi$
$264$	0	0
$265$	0	0
$266$	1583.55	0.365013
$267$	0	0
$268$	5251.13	1.19688
$269$	3477.43	0.788188	0.394094	−	0.919070i	$-0.371058\pi$
$269$	3477.43	0.788188	0.394094	+	0.919070i	$0.371058\pi$
$270$	0	0
$271$	−4589.09	−1.02866	−0.514331	−	0.857592i	$-0.671960\pi$
$271$	−4589.09	−1.02866	−0.514331	+	0.857592i	$0.671960\pi$
$272$	1305.87	0.291103
$273$	0	0
$274$	−3600.73	−0.793898
$275$	0	0
$276$	0	0
$277$	1920.87	0.416657	0.208329	−	0.978059i	$-0.433198\pi$
$277$	1920.87	0.416657	0.208329	+	0.978059i	$0.433198\pi$
$278$	438.668	0.0946388
$279$	0	0
$280$	0	0
$281$	392.063	0.0832331	0.0416166	−	0.999134i	$-0.486749\pi$
$281$	392.063	0.0832331	0.0416166	+	0.999134i	$0.486749\pi$
$282$	0	0
$283$	−7701.96	−1.61779	−0.808894	−	0.587955i	$-0.799933\pi$
$283$	−7701.96	−1.61779	−0.808894	+	0.587955i	$0.799933\pi$
$284$	−10765.9	−2.24944
$285$	0	0
$286$	−369.107	−0.0763138
$287$	1376.82	0.283174
$288$	0	0
$289$	−1605.81	−0.326849
$290$	0	0
$291$	0	0
$292$	−4282.70	−0.858309
$293$	6657.73	1.32747	0.663735	−	0.747968i	$-0.268970\pi$
$293$	6657.73	1.32747	0.663735	+	0.747968i	$0.268970\pi$
$294$	0	0
$295$	0	0
$296$	3545.23	0.696155
$297$	0	0
$298$	6312.88	1.22716
$299$	234.277	0.0453131
$300$	0	0
$301$	412.780	0.0790439
$302$	7606.75	1.44940
$303$	0	0
$304$	1161.23	0.219082
$305$	0	0
$306$	0	0
$307$	5082.53	0.944870	0.472435	−	0.881366i	$-0.343375\pi$
$307$	5082.53	0.944870	0.472435	+	0.881366i	$0.343375\pi$
$308$	−1893.21	−0.350245
$309$	0	0
$310$	0	0
$311$	2627.30	0.479038	0.239519	−	0.970892i	$-0.423010\pi$
$311$	2627.30	0.479038	0.239519	+	0.970892i	$0.423010\pi$
$312$	0	0
$313$	−2378.77	−0.429571	−0.214786	−	0.976661i	$-0.568905\pi$
$313$	−2378.77	−0.429571	−0.214786	+	0.976661i	$0.568905\pi$
$314$	−14465.9	−2.59986
$315$	0	0
$316$	−631.685	−0.112453
$317$	7116.78	1.26094	0.630470	−	0.776213i	$-0.282862\pi$
$317$	7116.78	1.26094	0.630470	+	0.776213i	$0.282862\pi$
$318$	0	0
$319$	−972.846	−0.170749
$320$	0	0
$321$	0	0
$322$	2032.57	0.351773
$323$	2940.87	0.506609
$324$	0	0
$325$	0	0
$326$	3942.96	0.669878
$327$	0	0
$328$	−3105.51	−0.522784
$329$	293.497	0.0491825
$330$	0	0
$331$	−9616.07	−1.59682	−0.798410	−	0.602115i	$-0.794325\pi$
$331$	−9616.07	−1.59682	−0.798410	+	0.602115i	$0.794325\pi$
$332$	4975.27	0.822449
$333$	0	0
$334$	−7070.88	−1.15839
$335$	0	0
$336$	0	0
$337$	11052.0	1.78648	0.893238	−	0.449584i	$-0.148428\pi$
$337$	11052.0	1.78648	0.893238	+	0.449584i	$0.148428\pi$
$338$	−9662.53	−1.55495
$339$	0	0
$340$	0	0
$341$	−3916.61	−0.621983
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−931.054	−0.145927
$345$	0	0
$346$	−11026.1	−1.71319
$347$	2320.86	0.359050	0.179525	−	0.983753i	$-0.442544\pi$
$347$	2320.86	0.359050	0.179525	+	0.983753i	$0.442544\pi$
$348$	0	0
$349$	−7462.68	−1.14461	−0.572304	−	0.820042i	$-0.693950\pi$
$349$	−7462.68	−1.14461	−0.572304	+	0.820042i	$0.693950\pi$
$350$	0	0
$351$	0	0
$352$	−5301.16	−0.802706
$353$	9430.78	1.42195	0.710977	−	0.703216i	$-0.248253\pi$
$353$	9430.78	1.42195	0.710977	+	0.703216i	$0.248253\pi$
$354$	0	0
$355$	0	0
$356$	8528.82	1.26974
$357$	0	0
$358$	6238.94	0.921056
$359$	7449.39	1.09516	0.547582	−	0.836752i	$-0.315549\pi$
$359$	7449.39	1.09516	0.547582	+	0.836752i	$0.315549\pi$
$360$	0	0
$361$	−4243.87	−0.618730
$362$	9020.63	1.30971
$363$	0	0
$364$	289.047	0.0416214
$365$	0	0
$366$	0	0
$367$	−12464.6	−1.77287	−0.886436	−	0.462850i	$-0.846827\pi$
$367$	−12464.6	−1.77287	−0.886436	+	0.462850i	$0.846827\pi$
$368$	1490.50	0.211135
$369$	0	0
$370$	0	0
$371$	−233.411	−0.0326634
$372$	0	0
$373$	−8335.42	−1.15708	−0.578541	−	0.815653i	$-0.696378\pi$
$373$	−8335.42	−1.15708	−0.578541	+	0.815653i	$0.696378\pi$
$374$	−5947.21	−0.822254
$375$	0	0
$376$	−662.004	−0.0907986
$377$	148.530	0.0202910
$378$	0	0
$379$	−11976.6	−1.62321	−0.811603	−	0.584209i	$-0.801405\pi$
$379$	−11976.6	−1.62321	−0.811603	+	0.584209i	$0.801405\pi$
$380$	0	0
$381$	0	0
$382$	11399.0	1.52676
$383$	1607.28	0.214434	0.107217	−	0.994236i	$-0.465806\pi$
$383$	1607.28	0.214434	0.107217	+	0.994236i	$0.465806\pi$
$384$	0	0
$385$	0	0
$386$	12712.0	1.67623
$387$	0	0
$388$	−1739.67	−0.227625
$389$	−10021.6	−1.30621	−0.653103	−	0.757269i	$-0.726533\pi$
$389$	−10021.6	−1.30621	−0.653103	+	0.757269i	$0.726533\pi$
$390$	0	0
$391$	3774.78	0.488233
$392$	773.661	0.0996831
$393$	0	0
$394$	6590.54	0.842708
$395$	0	0
$396$	0	0
$397$	−1247.17	−0.157666	−0.0788331	−	0.996888i	$-0.525119\pi$
$397$	−1247.17	−0.157666	−0.0788331	+	0.996888i	$0.525119\pi$
$398$	5800.28	0.730507
$399$	0	0
$400$	0	0
$401$	5115.73	0.637075	0.318538	−	0.947910i	$-0.396808\pi$
$401$	5115.73	0.637075	0.318538	+	0.947910i	$0.396808\pi$
$402$	0	0
$403$	597.972	0.0739134
$404$	−1555.16	−0.191516
$405$	0	0
$406$	1288.64	0.157522
$407$	5249.12	0.639286
$408$	0	0
$409$	11412.9	1.37979	0.689894	−	0.723911i	$-0.257657\pi$
$409$	11412.9	1.37979	0.689894	+	0.723911i	$0.257657\pi$
$410$	0	0
$411$	0	0
$412$	2689.76	0.321639
$413$	−1604.48	−0.191166
$414$	0	0
$415$	0	0
$416$	809.359	0.0953897
$417$	0	0
$418$	−5288.48	−0.618823
$419$	−10144.2	−1.18276	−0.591381	−	0.806393i	$-0.701417\pi$
$419$	−10144.2	−1.18276	−0.591381	+	0.806393i	$0.701417\pi$
$420$	0	0
$421$	−195.217	−0.0225993	−0.0112996	−	0.999936i	$-0.503597\pi$
$421$	−195.217	−0.0225993	−0.0112996	+	0.999936i	$0.503597\pi$
$422$	11817.3	1.36317
$423$	0	0
$424$	526.476	0.0603017
$425$	0	0
$426$	0	0
$427$	4904.29	0.555820
$428$	−17809.4	−2.01133
$429$	0	0
$430$	0	0
$431$	−8432.54	−0.942415	−0.471208	−	0.882022i	$-0.656182\pi$
$431$	−8432.54	−0.942415	−0.471208	+	0.882022i	$0.656182\pi$
$432$	0	0
$433$	−3039.89	−0.337386	−0.168693	−	0.985669i	$-0.553955\pi$
$433$	−3039.89	−0.337386	−0.168693	+	0.985669i	$0.553955\pi$
$434$	5187.96	0.573802
$435$	0	0
$436$	−5824.65	−0.639793
$437$	3356.67	0.367440
$438$	0	0
$439$	−8748.73	−0.951148	−0.475574	−	0.879676i	$-0.657760\pi$
$439$	−8748.73	−0.951148	−0.475574	+	0.879676i	$0.657760\pi$
$440$	0	0
$441$	0	0
$442$	907.997	0.0977127
$443$	2184.85	0.234324	0.117162	−	0.993113i	$-0.462620\pi$
$443$	2184.85	0.234324	0.117162	+	0.993113i	$0.462620\pi$
$444$	0	0
$445$	0	0
$446$	14695.0	1.56015
$447$	0	0
$448$	5750.32	0.606422
$449$	12511.6	1.31505	0.657525	−	0.753432i	$-0.271603\pi$
$449$	12511.6	1.31505	0.657525	+	0.753432i	$0.271603\pi$
$450$	0	0
$451$	−4598.07	−0.480077
$452$	−3089.95	−0.321546
$453$	0	0
$454$	7772.22	0.803455
$455$	0	0
$456$	0	0
$457$	−10709.8	−1.09625	−0.548123	−	0.836397i	$-0.684658\pi$
$457$	−10709.8	−1.09625	−0.548123	+	0.836397i	$0.684658\pi$
$458$	11743.4	1.19811
$459$	0	0
$460$	0	0
$461$	−73.0989	−0.00738515	−0.00369258	−	0.999993i	$-0.501175\pi$
$461$	−73.0989	−0.00738515	−0.00369258	+	0.999993i	$0.501175\pi$
$462$	0	0
$463$	2307.72	0.231639	0.115820	−	0.993270i	$-0.463051\pi$
$463$	2307.72	0.231639	0.115820	+	0.993270i	$0.463051\pi$
$464$	944.967	0.0945452
$465$	0	0
$466$	19374.5	1.92598
$467$	977.511	0.0968604	0.0484302	−	0.998827i	$-0.484578\pi$
$467$	977.511	0.0968604	0.0484302	+	0.998827i	$0.484578\pi$
$468$	0	0
$469$	−3177.23	−0.312816
$470$	0	0
$471$	0	0
$472$	3619.02	0.352922
$473$	−1378.53	−0.134007
$474$	0	0
$475$	0	0
$476$	4657.26	0.448456
$477$	0	0
$478$	4126.83	0.394889
$479$	−8973.08	−0.855930	−0.427965	−	0.903795i	$-0.640769\pi$
$479$	−8973.08	−0.855930	−0.427965	+	0.903795i	$0.640769\pi$
$480$	0	0
$481$	−801.415	−0.0759696
$482$	−28539.2	−2.69694
$483$	0	0
$484$	−9075.95	−0.852362
$485$	0	0
$486$	0	0
$487$	17259.1	1.60592	0.802961	−	0.596031i	$-0.203257\pi$
$487$	17259.1	1.60592	0.802961	+	0.596031i	$0.203257\pi$
$488$	−11062.0	−1.02613
$489$	0	0
$490$	0	0
$491$	−2612.25	−0.240100	−0.120050	−	0.992768i	$-0.538305\pi$
$491$	−2612.25	−0.240100	−0.120050	+	0.992768i	$0.538305\pi$
$492$	0	0
$493$	2393.18	0.218628
$494$	807.423	0.0735378
$495$	0	0
$496$	3804.37	0.344397
$497$	6513.99	0.587913
$498$	0	0
$499$	16056.2	1.44043	0.720215	−	0.693751i	$-0.244043\pi$
$499$	16056.2	1.44043	0.720215	+	0.693751i	$0.244043\pi$
$500$	0	0
$501$	0	0
$502$	−22495.6	−2.00005
$503$	20352.4	1.80411	0.902054	−	0.431623i	$-0.142059\pi$
$503$	20352.4	1.80411	0.902054	+	0.431623i	$0.142059\pi$
$504$	0	0
$505$	0	0
$506$	−6788.07	−0.596376
$507$	0	0
$508$	3429.24	0.299504
$509$	19740.3	1.71900	0.859502	−	0.511133i	$-0.170774\pi$
$509$	19740.3	1.71900	0.859502	+	0.511133i	$0.170774\pi$
$510$	0	0
$511$	2591.28	0.224327
$512$	8017.47	0.692042
$513$	0	0
$514$	21309.5	1.82864
$515$	0	0
$516$	0	0
$517$	−980.175	−0.0833812
$518$	−6953.02	−0.589765
$519$	0	0
$520$	0	0
$521$	−19115.6	−1.60742	−0.803712	−	0.595019i	$-0.797145\pi$
$521$	−19115.6	−1.60742	−0.803712	+	0.595019i	$0.797145\pi$
$522$	0	0
$523$	−4696.30	−0.392648	−0.196324	−	0.980539i	$-0.562900\pi$
$523$	−4696.30	−0.392648	−0.196324	+	0.980539i	$0.562900\pi$
$524$	−32025.7	−2.66994
$525$	0	0
$526$	35397.0	2.93419
$527$	9634.79	0.796391
$528$	0	0
$529$	−7858.52	−0.645888
$530$	0	0
$531$	0	0
$532$	4141.40	0.337505
$533$	702.016	0.0570500
$534$	0	0
$535$	0	0
$536$	7166.46	0.577508
$537$	0	0
$538$	15383.1	1.23274
$539$	1145.50	0.0915399
$540$	0	0
$541$	−1302.32	−0.103496	−0.0517478	−	0.998660i	$-0.516479\pi$
$541$	−1302.32	−0.103496	−0.0517478	+	0.998660i	$0.516479\pi$
$542$	−20300.8	−1.60885
$543$	0	0
$544$	13040.8	1.02779
$545$	0	0
$546$	0	0
$547$	13145.9	1.02757	0.513784	−	0.857920i	$-0.328243\pi$
$547$	13145.9	1.02757	0.513784	+	0.857920i	$0.328243\pi$
$548$	−9416.87	−0.734067
$549$	0	0
$550$	0	0
$551$	2128.11	0.164538
$552$	0	0
$553$	382.205	0.0293906
$554$	8497.38	0.651659
$555$	0	0
$556$	1147.24	0.0875065
$557$	5167.96	0.393130	0.196565	−	0.980491i	$-0.437021\pi$
$557$	5167.96	0.393130	0.196565	+	0.980491i	$0.437021\pi$
$558$	0	0
$559$	210.469	0.0159247
$560$	0	0
$561$	0	0
$562$	1734.37	0.130178
$563$	1444.18	0.108108	0.0540542	−	0.998538i	$-0.482786\pi$
$563$	1444.18	0.108108	0.0540542	+	0.998538i	$0.482786\pi$
$564$	0	0
$565$	0	0
$566$	−34071.2	−2.53025
$567$	0	0
$568$	−14692.8	−1.08538
$569$	22883.1	1.68596	0.842980	−	0.537945i	$-0.180799\pi$
$569$	22883.1	1.68596	0.842980	+	0.537945i	$0.180799\pi$
$570$	0	0
$571$	6611.79	0.484580	0.242290	−	0.970204i	$-0.422102\pi$
$571$	6611.79	0.484580	0.242290	+	0.970204i	$0.422102\pi$
$572$	−965.313	−0.0705626
$573$	0	0
$574$	6090.64	0.442889
$575$	0	0
$576$	0	0
$577$	−16695.5	−1.20458	−0.602291	−	0.798277i	$-0.705745\pi$
$577$	−16695.5	−1.20458	−0.602291	+	0.798277i	$0.705745\pi$
$578$	−7103.63	−0.511197
$579$	0	0
$580$	0	0
$581$	−3010.31	−0.214955
$582$	0	0
$583$	779.510	0.0553756
$584$	−5844.81	−0.414144
$585$	0	0
$586$	29451.9	2.07619
$587$	−16504.8	−1.16052	−0.580261	−	0.814431i	$-0.697049\pi$
$587$	−16504.8	−1.16052	−0.580261	+	0.814431i	$0.697049\pi$
$588$	0	0
$589$	8567.60	0.599358
$590$	0	0
$591$	0	0
$592$	−5098.70	−0.353978
$593$	16410.8	1.13644	0.568222	−	0.822875i	$-0.307631\pi$
$593$	16410.8	1.13644	0.568222	+	0.822875i	$0.307631\pi$
$594$	0	0
$595$	0	0
$596$	16509.9	1.13468
$597$	0	0
$598$	1036.37	0.0708704
$599$	−6879.60	−0.469270	−0.234635	−	0.972084i	$-0.575390\pi$
$599$	−6879.60	−0.469270	−0.234635	+	0.972084i	$0.575390\pi$
$600$	0	0
$601$	−979.314	−0.0664676	−0.0332338	−	0.999448i	$-0.510581\pi$
$601$	−979.314	−0.0664676	−0.0332338	+	0.999448i	$0.510581\pi$
$602$	1826.02	0.123626
$603$	0	0
$604$	19893.7	1.34017
$605$	0	0
$606$	0	0
$607$	5654.83	0.378126	0.189063	−	0.981965i	$-0.439455\pi$
$607$	5654.83	0.378126	0.189063	+	0.981965i	$0.439455\pi$
$608$	11596.3	0.773507
$609$	0	0
$610$	0	0
$611$	149.649	0.00990861
$612$	0	0
$613$	8619.72	0.567940	0.283970	−	0.958833i	$-0.408348\pi$
$613$	8619.72	0.567940	0.283970	+	0.958833i	$0.408348\pi$
$614$	22483.6	1.47779
$615$	0	0
$616$	−2583.75	−0.168997
$617$	−7061.35	−0.460744	−0.230372	−	0.973103i	$-0.573994\pi$
$617$	−7061.35	−0.460744	−0.230372	+	0.973103i	$0.573994\pi$
$618$	0	0
$619$	5725.08	0.371745	0.185873	−	0.982574i	$-0.440489\pi$
$619$	5725.08	0.371745	0.185873	+	0.982574i	$0.440489\pi$
$620$	0	0
$621$	0	0
$622$	11622.4	0.749223
$623$	−5160.41	−0.331858
$624$	0	0
$625$	0	0
$626$	−10523.0	−0.671857
$627$	0	0
$628$	−37832.2	−2.40393
$629$	−12912.8	−0.818546
$630$	0	0
$631$	−3621.56	−0.228482	−0.114241	−	0.993453i	$-0.536444\pi$
$631$	−3621.56	−0.228482	−0.114241	+	0.993453i	$0.536444\pi$
$632$	−862.090	−0.0542597
$633$	0	0
$634$	31482.5	1.97213
$635$	0	0
$636$	0	0
$637$	−174.890	−0.0108782
$638$	−4303.58	−0.267054
$639$	0	0
$640$	0	0
$641$	−20656.0	−1.27280	−0.636398	−	0.771361i	$-0.719576\pi$
$641$	−20656.0	−1.27280	−0.636398	+	0.771361i	$0.719576\pi$
$642$	0	0
$643$	−15856.5	−0.972504	−0.486252	−	0.873819i	$-0.661636\pi$
$643$	−15856.5	−0.972504	−0.486252	+	0.873819i	$0.661636\pi$
$644$	5315.73	0.325262
$645$	0	0
$646$	13009.6	0.792344
$647$	−5476.66	−0.332782	−0.166391	−	0.986060i	$-0.553211\pi$
$647$	−5476.66	−0.332782	−0.166391	+	0.986060i	$0.553211\pi$
$648$	0	0
$649$	5358.39	0.324091
$650$	0	0
$651$	0	0
$652$	10311.9	0.619394
$653$	−18293.1	−1.09627	−0.548135	−	0.836390i	$-0.684662\pi$
$653$	−18293.1	−1.09627	−0.548135	+	0.836390i	$0.684662\pi$
$654$	0	0
$655$	0	0
$656$	4466.31	0.265823
$657$	0	0
$658$	1298.35	0.0769222
$659$	32480.3	1.91996	0.959979	−	0.280073i	$-0.0903587\pi$
$659$	32480.3	1.91996	0.959979	+	0.280073i	$0.0903587\pi$
$660$	0	0
$661$	−19403.4	−1.14176	−0.570880	−	0.821034i	$-0.693398\pi$
$661$	−19403.4	−1.14176	−0.570880	+	0.821034i	$0.693398\pi$
$662$	−42538.7	−2.49745
$663$	0	0
$664$	6789.98	0.396841
$665$	0	0
$666$	0	0
$667$	2731.55	0.158570
$668$	−18492.2	−1.07109
$669$	0	0
$670$	0	0
$671$	−16378.6	−0.942306
$672$	0	0
$673$	3449.20	0.197559	0.0987793	−	0.995109i	$-0.468506\pi$
$673$	3449.20	0.197559	0.0987793	+	0.995109i	$0.468506\pi$
$674$	48890.9	2.79408
$675$	0	0
$676$	−25270.1	−1.43776
$677$	−23683.0	−1.34448	−0.672239	−	0.740334i	$-0.734667\pi$
$677$	−23683.0	−1.34448	−0.672239	+	0.740334i	$0.734667\pi$
$678$	0	0
$679$	1052.60	0.0594919
$680$	0	0
$681$	0	0
$682$	−17325.9	−0.972792
$683$	14101.5	0.790012	0.395006	−	0.918679i	$-0.370742\pi$
$683$	14101.5	0.790012	0.395006	+	0.918679i	$0.370742\pi$
$684$	0	0
$685$	0	0
$686$	−1517.33	−0.0844489
$687$	0	0
$688$	1339.03	0.0742006
$689$	−119.012	−0.00658057
$690$	0	0
$691$	1327.04	0.0730577	0.0365289	−	0.999333i	$-0.488370\pi$
$691$	1327.04	0.0730577	0.0365289	+	0.999333i	$0.488370\pi$
$692$	−28836.1	−1.58408
$693$	0	0
$694$	10266.8	0.561560
$695$	0	0
$696$	0	0
$697$	11311.2	0.614694
$698$	−33012.7	−1.79019
$699$	0	0
$700$	0	0
$701$	−9073.19	−0.488858	−0.244429	−	0.969667i	$-0.578601\pi$
$701$	−9073.19	−0.488858	−0.244429	+	0.969667i	$0.578601\pi$
$702$	0	0
$703$	−11482.5	−0.616032
$704$	−19204.0	−1.02809
$705$	0	0
$706$	41719.0	2.22396
$707$	940.961	0.0500544
$708$	0	0
$709$	−13479.5	−0.714010	−0.357005	−	0.934102i	$-0.616202\pi$
$709$	−13479.5	−0.714010	−0.357005	+	0.934102i	$0.616202\pi$
$710$	0	0
$711$	0	0
$712$	11639.7	0.612662
$713$	10997.0	0.577618
$714$	0	0
$715$	0	0
$716$	16316.5	0.851642
$717$	0	0
$718$	32953.9	1.71285
$719$	26389.8	1.36881	0.684404	−	0.729103i	$-0.260062\pi$
$719$	26389.8	1.36881	0.684404	+	0.729103i	$0.260062\pi$
$720$	0	0
$721$	−1627.46	−0.0840633
$722$	−18773.6	−0.967704
$723$	0	0
$724$	23591.4	1.21100
$725$	0	0
$726$	0	0
$727$	7983.43	0.407275	0.203638	−	0.979046i	$-0.434724\pi$
$727$	7983.43	0.407275	0.203638	+	0.979046i	$0.434724\pi$
$728$	394.476	0.0200828
$729$	0	0
$730$	0	0
$731$	3391.17	0.171583
$732$	0	0
$733$	−24350.2	−1.22700	−0.613502	−	0.789693i	$-0.710240\pi$
$733$	−24350.2	−1.22700	−0.613502	+	0.789693i	$0.710240\pi$
$734$	−55139.5	−2.77280
$735$	0	0
$736$	14884.5	0.745450
$737$	10610.8	0.530331
$738$	0	0
$739$	1936.37	0.0963880	0.0481940	−	0.998838i	$-0.484653\pi$
$739$	1936.37	0.0963880	0.0481940	+	0.998838i	$0.484653\pi$
$740$	0	0
$741$	0	0
$742$	−1032.54	−0.0510861
$743$	−27212.0	−1.34362	−0.671811	−	0.740722i	$-0.734483\pi$
$743$	−27212.0	−1.34362	−0.671811	+	0.740722i	$0.734483\pi$
$744$	0	0
$745$	0	0
$746$	−36873.5	−1.80970
$747$	0	0
$748$	−15553.6	−0.760287
$749$	10775.7	0.525680
$750$	0	0
$751$	−13542.6	−0.658026	−0.329013	−	0.944325i	$-0.606716\pi$
$751$	−13542.6	−0.658026	−0.329013	+	0.944325i	$0.606716\pi$
$752$	952.086	0.0461689
$753$	0	0
$754$	657.054	0.0317354
$755$	0	0
$756$	0	0
$757$	6328.67	0.303857	0.151928	−	0.988392i	$-0.451452\pi$
$757$	6328.67	0.303857	0.151928	+	0.988392i	$0.451452\pi$
$758$	−52980.8	−2.53872
$759$	0	0
$760$	0	0
$761$	26036.0	1.24021	0.620107	−	0.784517i	$-0.287089\pi$
$761$	26036.0	1.24021	0.620107	+	0.784517i	$0.287089\pi$
$762$	0	0
$763$	3524.24	0.167216
$764$	29811.4	1.41170
$765$	0	0
$766$	7110.15	0.335379
$767$	−818.097	−0.0385134
$768$	0	0
$769$	−6092.16	−0.285681	−0.142841	−	0.989746i	$-0.545624\pi$
$769$	−6092.16	−0.285681	−0.142841	+	0.989746i	$0.545624\pi$
$770$	0	0
$771$	0	0
$772$	33245.3	1.54990
$773$	16076.6	0.748040	0.374020	−	0.927421i	$-0.377979\pi$
$773$	16076.6	0.748040	0.374020	+	0.927421i	$0.377979\pi$
$774$	0	0
$775$	0	0
$776$	−2374.21	−0.109831
$777$	0	0
$778$	−44332.5	−2.04293
$779$	10058.3	0.462614
$780$	0	0
$781$	−21754.4	−0.996713
$782$	16698.5	0.763604
$783$	0	0
$784$	−1112.67	−0.0506865
$785$	0	0
$786$	0	0
$787$	41683.3	1.88799	0.943995	−	0.329961i	$-0.107036\pi$
$787$	41683.3	1.88799	0.943995	+	0.329961i	$0.107036\pi$
$788$	17236.0	0.779199
$789$	0	0
$790$	0	0
$791$	1869.59	0.0840392
$792$	0	0
$793$	2500.61	0.111979
$794$	−5517.10	−0.246593
$795$	0	0
$796$	15169.3	0.675454
$797$	−37247.7	−1.65543	−0.827717	−	0.561145i	$-0.810361\pi$
$797$	−37247.7	−1.65543	−0.827717	+	0.561145i	$0.810361\pi$
$798$	0	0
$799$	2411.21	0.106762
$800$	0	0
$801$	0	0
$802$	22630.5	0.996396
$803$	−8653.92	−0.380312
$804$	0	0
$805$	0	0
$806$	2645.25	0.115602
$807$	0	0
$808$	−2122.40	−0.0924083
$809$	−12084.0	−0.525156	−0.262578	−	0.964911i	$-0.584573\pi$
$809$	−12084.0	−0.525156	−0.262578	+	0.964911i	$0.584573\pi$
$810$	0	0
$811$	−43963.8	−1.90355	−0.951775	−	0.306796i	$-0.900743\pi$
$811$	−43963.8	−1.90355	−0.951775	+	0.306796i	$0.900743\pi$
$812$	3370.13	0.145651
$813$	0	0
$814$	23220.6	0.999854
$815$	0	0
$816$	0	0
$817$	3015.55	0.129132
$818$	50487.4	2.15801
$819$	0	0
$820$	0	0
$821$	−6761.26	−0.287417	−0.143709	−	0.989620i	$-0.545903\pi$
$821$	−6761.26	−0.287417	−0.143709	+	0.989620i	$0.545903\pi$
$822$	0	0
$823$	−21729.5	−0.920343	−0.460172	−	0.887830i	$-0.652212\pi$
$823$	−21729.5	−0.920343	−0.460172	+	0.887830i	$0.652212\pi$
$824$	3670.84	0.155194
$825$	0	0
$826$	−7097.76	−0.298986
$827$	27297.7	1.14780	0.573902	−	0.818924i	$-0.305429\pi$
$827$	27297.7	1.14780	0.573902	+	0.818924i	$0.305429\pi$
$828$	0	0
$829$	−14194.9	−0.594703	−0.297352	−	0.954768i	$-0.596103\pi$
$829$	−14194.9	−0.594703	−0.297352	+	0.954768i	$0.596103\pi$
$830$	0	0
$831$	0	0
$832$	2931.99	0.122174
$833$	−2817.90	−0.117208
$834$	0	0
$835$	0	0
$836$	−13830.8	−0.572186
$837$	0	0
$838$	−44875.0	−1.84986
$839$	−30955.4	−1.27378	−0.636889	−	0.770955i	$-0.719779\pi$
$839$	−30955.4	−1.27378	−0.636889	+	0.770955i	$0.719779\pi$
$840$	0	0
$841$	−22657.2	−0.928993
$842$	−863.583	−0.0353457
$843$	0	0
$844$	30905.5	1.26044
$845$	0	0
$846$	0	0
$847$	5491.46	0.222773
$848$	−757.171	−0.0306620
$849$	0	0
$850$	0	0
$851$	−14738.4	−0.593686
$852$	0	0
$853$	−21925.6	−0.880094	−0.440047	−	0.897975i	$-0.645038\pi$
$853$	−21925.6	−0.880094	−0.440047	+	0.897975i	$0.645038\pi$
$854$	21695.1	0.869312
$855$	0	0
$856$	−24305.3	−0.970487
$857$	−28288.5	−1.12756	−0.563780	−	0.825925i	$-0.690653\pi$
$857$	−28288.5	−1.12756	−0.563780	+	0.825925i	$0.690653\pi$
$858$	0	0
$859$	−12483.8	−0.495858	−0.247929	−	0.968778i	$-0.579750\pi$
$859$	−12483.8	−0.495858	−0.247929	+	0.968778i	$0.579750\pi$
$860$	0	0
$861$	0	0
$862$	−37303.1	−1.47395
$863$	13344.0	0.526346	0.263173	−	0.964749i	$-0.415231\pi$
$863$	13344.0	0.526346	0.263173	+	0.964749i	$0.415231\pi$
$864$	0	0
$865$	0	0
$866$	−13447.6	−0.527677
$867$	0	0
$868$	13567.9	0.530559
$869$	−1276.43	−0.0498271
$870$	0	0
$871$	−1620.01	−0.0630219
$872$	−7949.17	−0.308707
$873$	0	0
$874$	14848.9	0.574683
$875$	0	0
$876$	0	0
$877$	19198.7	0.739219	0.369609	−	0.929187i	$-0.379492\pi$
$877$	19198.7	0.739219	0.369609	+	0.929187i	$0.379492\pi$
$878$	−38701.8	−1.48761
$879$	0	0
$880$	0	0
$881$	−31063.7	−1.18793	−0.593963	−	0.804492i	$-0.702438\pi$
$881$	−31063.7	−1.18793	−0.593963	+	0.804492i	$0.702438\pi$
$882$	0	0
$883$	−37100.6	−1.41397	−0.706985	−	0.707229i	$-0.749945\pi$
$883$	−37100.6	−1.41397	−0.706985	+	0.707229i	$0.749945\pi$
$884$	2374.65	0.0903487
$885$	0	0
$886$	9665.15	0.366487
$887$	4568.24	0.172927	0.0864636	−	0.996255i	$-0.472443\pi$
$887$	4568.24	0.172927	0.0864636	+	0.996255i	$0.472443\pi$
$888$	0	0
$889$	−2074.88	−0.0782782
$890$	0	0
$891$	0	0
$892$	38431.3	1.44257
$893$	2144.14	0.0803481
$894$	0	0
$895$	0	0
$896$	12739.0	0.474977
$897$	0	0
$898$	55347.5	2.05676
$899$	6972.03	0.258654
$900$	0	0
$901$	−1917.58	−0.0709033
$902$	−20340.5	−0.750849
$903$	0	0
$904$	−4216.99	−0.155150
$905$	0	0
$906$	0	0
$907$	776.485	0.0284264	0.0142132	−	0.999899i	$-0.495476\pi$
$907$	776.485	0.0284264	0.0142132	+	0.999899i	$0.495476\pi$
$908$	20326.4	0.742904
$909$	0	0
$910$	0	0
$911$	24605.9	0.894872	0.447436	−	0.894316i	$-0.352337\pi$
$911$	24605.9	0.894872	0.447436	+	0.894316i	$0.352337\pi$
$912$	0	0
$913$	10053.4	0.364423
$914$	−47377.2	−1.71455
$915$	0	0
$916$	30712.1	1.10781
$917$	19377.4	0.697816
$918$	0	0
$919$	−156.897	−0.00563173	−0.00281587	−	0.999996i	$-0.500896\pi$
$919$	−156.897	−0.00563173	−0.00281587	+	0.999996i	$0.500896\pi$
$920$	0	0
$921$	0	0
$922$	−323.368	−0.0115505
$923$	3321.37	0.118445
$924$	0	0
$925$	0	0
$926$	10208.7	0.362287
$927$	0	0
$928$	9436.69	0.333809
$929$	−2740.94	−0.0968002	−0.0484001	−	0.998828i	$-0.515412\pi$
$929$	−2740.94	−0.0968002	−0.0484001	+	0.998828i	$0.515412\pi$
$930$	0	0
$931$	−2505.78	−0.0882101
$932$	50669.5	1.78083
$933$	0	0
$934$	4324.22	0.151491
$935$	0	0
$936$	0	0
$937$	15341.1	0.534870	0.267435	−	0.963576i	$-0.413824\pi$
$937$	15341.1	0.534870	0.267435	+	0.963576i	$0.413824\pi$
$938$	−14055.1	−0.489250
$939$	0	0
$940$	0	0
$941$	40269.2	1.39505	0.697523	−	0.716563i	$-0.254286\pi$
$941$	40269.2	1.39505	0.697523	+	0.716563i	$0.254286\pi$
$942$	0	0
$943$	12910.4	0.445834
$944$	−5204.83	−0.179452
$945$	0	0
$946$	−6098.23	−0.209588
$947$	10173.6	0.349100	0.174550	−	0.984648i	$-0.444153\pi$
$947$	10173.6	0.349100	0.174550	+	0.984648i	$0.444153\pi$
$948$	0	0
$949$	1321.25	0.0451944
$950$	0	0
$951$	0	0
$952$	6355.98	0.216385
$953$	15239.9	0.518015	0.259007	−	0.965875i	$-0.416605\pi$
$953$	15239.9	0.518015	0.259007	+	0.965875i	$0.416605\pi$
$954$	0	0
$955$	0	0
$956$	10792.8	0.365129
$957$	0	0
$958$	−39694.3	−1.33869
$959$	5697.73	0.191855
$960$	0	0
$961$	−1722.13	−0.0578069
$962$	−3545.23	−0.118818
$963$	0	0
$964$	−74637.7	−2.49369
$965$	0	0
$966$	0	0
$967$	−35689.8	−1.18687	−0.593437	−	0.804881i	$-0.702229\pi$
$967$	−35689.8	−1.18687	−0.593437	+	0.804881i	$0.702229\pi$
$968$	−12386.4	−0.411274
$969$	0	0
$970$	0	0
$971$	−22841.5	−0.754913	−0.377456	−	0.926027i	$-0.623201\pi$
$971$	−22841.5	−0.754913	−0.377456	+	0.926027i	$0.623201\pi$
$972$	0	0
$973$	−694.142	−0.0228707
$974$	76349.2	2.51169
$975$	0	0
$976$	15909.2	0.521763
$977$	−14487.1	−0.474394	−0.237197	−	0.971462i	$-0.576229\pi$
$977$	−14487.1	−0.474394	−0.237197	+	0.971462i	$0.576229\pi$
$978$	0	0
$979$	17233.9	0.562613
$980$	0	0
$981$	0	0
$982$	−11555.8	−0.375520
$983$	35291.6	1.14509	0.572547	−	0.819872i	$-0.305955\pi$
$983$	35291.6	1.14509	0.572547	+	0.819872i	$0.305955\pi$
$984$	0	0
$985$	0	0
$986$	10586.7	0.341938
$987$	0	0
$988$	2111.63	0.0679958
$989$	3870.64	0.124448
$990$	0	0
$991$	−29615.5	−0.949310	−0.474655	−	0.880172i	$-0.657427\pi$
$991$	−29615.5	−0.949310	−0.474655	+	0.880172i	$0.657427\pi$
$992$	37991.4	1.21596
$993$	0	0
$994$	28816.0	0.919505
$995$	0	0
$996$	0	0
$997$	45868.5	1.45704	0.728520	−	0.685025i	$-0.240209\pi$
$997$	45868.5	1.45704	0.728520	+	0.685025i	$0.240209\pi$
$998$	71027.9	2.25286
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bh.1.4	yes	4
3.2	odd	2	inner	1575.4.a.bh.1.1	✓	4
5.4	even	2		1575.4.a.bi.1.1	yes	4
15.14	odd	2		1575.4.a.bi.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.4.a.bh.1.1	✓	4	3.2	odd	2	inner
1575.4.a.bh.1.4	yes	4	1.1	even	1	trivial
1575.4.a.bi.1.1	yes	4	5.4	even	2
1575.4.a.bi.1.4	yes	4	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+4.42371 q^{2} +11.5692 q^{4} -7.00000 q^{7} +15.7890 q^{8} +23.3775 q^{11} -3.56918 q^{13} -30.9659 q^{14} -22.7075 q^{16} -57.5082 q^{17} -51.1384 q^{19} +103.415 q^{22} -65.6390 q^{23} -15.7890 q^{26} -80.9843 q^{28} -41.6147 q^{29} -167.538 q^{31} -226.763 q^{32} -254.399 q^{34} +224.538 q^{37} -226.221 q^{38} -196.688 q^{41} -58.9685 q^{43} +270.458 q^{44} -290.368 q^{46} -41.9282 q^{47} +49.0000 q^{49} -41.2925 q^{52} +33.3445 q^{53} -110.523 q^{56} -184.091 q^{58} +229.212 q^{59} -700.613 q^{61} -741.138 q^{62} -821.475 q^{64} +453.890 q^{67} -665.322 q^{68} -930.571 q^{71} -370.182 q^{73} +993.289 q^{74} -591.629 q^{76} -163.642 q^{77} -54.6007 q^{79} -870.091 q^{82} +430.045 q^{83} -260.859 q^{86} +369.107 q^{88} +737.202 q^{89} +24.9843 q^{91} -759.390 q^{92} -185.478 q^{94} -150.371 q^{97} +216.762 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{4} - 28 q^{7} + 22 q^{13} + 18 q^{16} - 132 q^{19} + 196 q^{22} - 70 q^{28} - 126 q^{31} - 546 q^{34} + 354 q^{37} + 272 q^{43} - 182 q^{46} + 196 q^{49} - 274 q^{52} + 98 q^{58} - 1170 q^{61}+ \cdots - 1980 q^{97}+O(q^{100}) \) 4 * q + 10 * q^4 - 28 * q^7 + 22 * q^13 + 18 * q^16 - 132 * q^19 + 196 * q^22 - 70 * q^28 - 126 * q^31 - 546 * q^34 + 354 * q^37 + 272 * q^43 - 182 * q^46 + 196 * q^49 - 274 * q^52 + 98 * q^58 - 1170 * q^61 - 1726 * q^64 + 38 * q^67 + 188 * q^73 - 988 * q^76 - 690 * q^79 - 2646 * q^82 + 896 * q^88 - 154 * q^91 - 1540 * q^94 - 1980 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more