LMFDB - Embedded newform 1875.4.a.f.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1875,4,Mod(1,1875)] 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("1875.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1875, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1875.a (trivial)

Newform invariants

comment:select newform

sage:traces = [14,0,-42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$110.628581261$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + \cdots + 42064 $ x^14 - 81x^12 - 7x^11 + 2512x^10 + 517x^9 - 36970x^8 - 12987x^7 + 257291x^6 + 125779x^5 - 718713x^4 - 371750x^3 + 579848x^2 + 394896x + 42064
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.134967$ of defining polynomial
Character	$\chi$	$=$	1875.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.134967 q^{2} -3.00000 q^{3} -7.98178 q^{4} -0.404902 q^{6} +17.3099 q^{7} -2.15702 q^{8} +9.00000 q^{9} +42.1544 q^{11} +23.9454 q^{12} +84.5240 q^{13} +2.33627 q^{14} +63.5631 q^{16} +45.1024 q^{17} +1.21471 q^{18} +83.6634 q^{19} -51.9297 q^{21} +5.68947 q^{22} -20.9307 q^{23} +6.47105 q^{24} +11.4080 q^{26} -27.0000 q^{27} -138.164 q^{28} +220.065 q^{29} +160.220 q^{31} +25.8351 q^{32} -126.463 q^{33} +6.08735 q^{34} -71.8361 q^{36} +263.227 q^{37} +11.2918 q^{38} -253.572 q^{39} +410.142 q^{41} -7.00881 q^{42} -290.699 q^{43} -336.468 q^{44} -2.82495 q^{46} +460.997 q^{47} -190.689 q^{48} -43.3675 q^{49} -135.307 q^{51} -674.652 q^{52} +142.443 q^{53} -3.64412 q^{54} -37.3377 q^{56} -250.990 q^{57} +29.7016 q^{58} -472.475 q^{59} +54.0384 q^{61} +21.6244 q^{62} +155.789 q^{63} -505.018 q^{64} -17.0684 q^{66} -449.961 q^{67} -359.998 q^{68} +62.7920 q^{69} -626.058 q^{71} -19.4132 q^{72} -48.7415 q^{73} +35.5271 q^{74} -667.783 q^{76} +729.689 q^{77} -34.2239 q^{78} -647.571 q^{79} +81.0000 q^{81} +55.3557 q^{82} -669.851 q^{83} +414.492 q^{84} -39.2349 q^{86} -660.196 q^{87} -90.9278 q^{88} -866.036 q^{89} +1463.10 q^{91} +167.064 q^{92} -480.659 q^{93} +62.2195 q^{94} -77.5052 q^{96} +177.047 q^{97} -5.85319 q^{98} +379.390 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 42 q^{3} + 50 q^{4} + 27 q^{7} - 21 q^{8} + 126 q^{9} - 33 q^{11} - 150 q^{12} + 188 q^{13} - 287 q^{14} + 82 q^{16} + 146 q^{17} - 184 q^{19} - 81 q^{21} + 277 q^{22} + 164 q^{23} + 63 q^{24} + 103 q^{26}+ \cdots - 297 q^{99}+O(q^{100}) $ 14 * q - 42 * q^3 + 50 * q^4 + 27 * q^7 - 21 * q^8 + 126 * q^9 - 33 * q^11 - 150 * q^12 + 188 * q^13 - 287 * q^14 + 82 * q^16 + 146 * q^17 - 184 * q^19 - 81 * q^21 + 277 * q^22 + 164 * q^23 + 63 * q^24 + 103 * q^26 - 378 * q^27 - 224 * q^28 + 252 * q^29 - 889 * q^31 + 422 * q^32 + 99 * q^33 - 230 * q^34 + 450 * q^36 + 642 * q^37 + 1833 * q^38 - 564 * q^39 - 164 * q^41 + 861 * q^42 + 696 * q^43 - 6 * q^44 - 419 * q^46 - 92 * q^47 - 246 * q^48 + 81 * q^49 - 438 * q^51 + 1956 * q^52 + 949 * q^53 - 2380 * q^56 + 552 * q^57 + 1041 * q^58 - 81 * q^59 - 496 * q^61 + 1454 * q^62 + 243 * q^63 - 3903 * q^64 - 831 * q^66 + 1926 * q^67 + 685 * q^68 - 492 * q^69 + 2498 * q^71 - 189 * q^72 - 1026 * q^73 - 707 * q^74 - 5704 * q^76 + 5434 * q^77 - 309 * q^78 - 695 * q^79 + 1134 * q^81 - 886 * q^82 + 5315 * q^83 + 672 * q^84 + 3997 * q^86 - 756 * q^87 + 2969 * q^88 - 1424 * q^89 - 1194 * q^91 + 1607 * q^92 + 2667 * q^93 - 629 * q^94 - 1266 * q^96 + 291 * q^97 - 1018 * q^98 - 297 * q^99

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 $

$2$	0.134967	0.0477181	0.0238591	−	0.999715i	$-0.492405\pi$
$2$	0.134967	0.0477181	0.0238591	+	0.999715i	$0.492405\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.98178	−0.997723
$5$	0	0
$5$	0	0
$6$	−0.404902	−0.0275501
$7$	17.3099	0.934647	0.467323	−	0.884087i	$-0.345218\pi$
$7$	17.3099	0.934647	0.467323	+	0.884087i	$0.345218\pi$
$8$	−2.15702	−0.0953276
$9$	9.00000	0.333333
$10$	0	0
$11$	42.1544	1.15546	0.577729	−	0.816229i	$-0.303939\pi$
$11$	42.1544	1.15546	0.577729	+	0.816229i	$0.303939\pi$
$12$	23.9454	0.576036
$13$	84.5240	1.80329	0.901643	−	0.432480i	$-0.142361\pi$
$13$	84.5240	1.80329	0.901643	+	0.432480i	$0.142361\pi$
$14$	2.33627	0.0445996
$15$	0	0
$16$	63.5631	0.993174
$17$	45.1024	0.643467	0.321734	−	0.946830i	$-0.395734\pi$
$17$	45.1024	0.643467	0.321734	+	0.946830i	$0.395734\pi$
$18$	1.21471	0.0159060
$19$	83.6634	1.01019	0.505097	−	0.863062i	$-0.331457\pi$
$19$	83.6634	1.01019	0.505097	+	0.863062i	$0.331457\pi$
$20$	0	0
$21$	−51.9297	−0.539618
$22$	5.68947	0.0551363
$23$	−20.9307	−0.189754	−0.0948771	−	0.995489i	$-0.530246\pi$
$23$	−20.9307	−0.189754	−0.0948771	+	0.995489i	$0.530246\pi$
$24$	6.47105	0.0550374
$25$	0	0
$26$	11.4080	0.0860495
$27$	−27.0000	−0.192450
$28$	−138.164	−0.932518
$29$	220.065	1.40914	0.704571	−	0.709634i	$-0.251139\pi$
$29$	220.065	1.40914	0.704571	+	0.709634i	$0.251139\pi$
$30$	0	0
$31$	160.220	0.928267	0.464134	−	0.885765i	$-0.346366\pi$
$31$	160.220	0.928267	0.464134	+	0.885765i	$0.346366\pi$
$32$	25.8351	0.142720
$33$	−126.463	−0.667104
$34$	6.08735	0.0307050
$35$	0	0
$36$	−71.8361	−0.332574
$37$	263.227	1.16958	0.584788	−	0.811186i	$-0.301178\pi$
$37$	263.227	1.16958	0.584788	+	0.811186i	$0.301178\pi$
$38$	11.2918	0.0482046
$39$	−253.572	−1.04113
$40$	0	0
$41$	410.142	1.56228	0.781139	−	0.624357i	$-0.214639\pi$
$41$	410.142	1.56228	0.781139	+	0.624357i	$0.214639\pi$
$42$	−7.00881	−0.0257496
$43$	−290.699	−1.03096	−0.515479	−	0.856902i	$-0.672386\pi$
$43$	−290.699	−1.03096	−0.515479	+	0.856902i	$0.672386\pi$
$44$	−336.468	−1.15283
$45$	0	0
$46$	−2.82495	−0.00905471
$47$	460.997	1.43071	0.715355	−	0.698761i	$-0.246265\pi$
$47$	460.997	1.43071	0.715355	+	0.698761i	$0.246265\pi$
$48$	−190.689	−0.573409
$49$	−43.3675	−0.126436
$50$	0	0
$51$	−135.307	−0.371506
$52$	−674.652	−1.79918
$53$	142.443	0.369170	0.184585	−	0.982817i	$-0.440906\pi$
$53$	142.443	0.369170	0.184585	+	0.982817i	$0.440906\pi$
$54$	−3.64412	−0.00918336
$55$	0	0
$56$	−37.3377	−0.0890976
$57$	−250.990	−0.583236
$58$	29.7016	0.0672416
$59$	−472.475	−1.04256	−0.521280	−	0.853386i	$-0.674545\pi$
$59$	−472.475	−1.04256	−0.521280	+	0.853386i	$0.674545\pi$
$60$	0	0
$61$	54.0384	0.113425	0.0567124	−	0.998391i	$-0.481938\pi$
$61$	54.0384	0.113425	0.0567124	+	0.998391i	$0.481938\pi$
$62$	21.6244	0.0442952
$63$	155.789	0.311549
$64$	−505.018	−0.986364
$65$	0	0
$66$	−17.0684	−0.0318330
$67$	−449.961	−0.820470	−0.410235	−	0.911980i	$-0.634553\pi$
$67$	−449.961	−0.820470	−0.410235	+	0.911980i	$0.634553\pi$
$68$	−359.998	−0.642002
$69$	62.7920	0.109555
$70$	0	0
$71$	−626.058	−1.04647	−0.523235	−	0.852188i	$-0.675275\pi$
$71$	−626.058	−1.04647	−0.523235	+	0.852188i	$0.675275\pi$
$72$	−19.4132	−0.0317759
$73$	−48.7415	−0.0781474	−0.0390737	−	0.999236i	$-0.512441\pi$
$73$	−48.7415	−0.0781474	−0.0390737	+	0.999236i	$0.512441\pi$
$74$	35.5271	0.0558099
$75$	0	0
$76$	−667.783	−1.00789
$77$	729.689	1.07995
$78$	−34.2239	−0.0496807
$79$	−647.571	−0.922246	−0.461123	−	0.887336i	$-0.652553\pi$
$79$	−647.571	−0.922246	−0.461123	+	0.887336i	$0.652553\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	55.3557	0.0745490
$83$	−669.851	−0.885852	−0.442926	−	0.896558i	$-0.646060\pi$
$83$	−669.851	−0.885852	−0.442926	+	0.896558i	$0.646060\pi$
$84$	414.492	0.538390
$85$	0	0
$86$	−39.2349	−0.0491954
$87$	−660.196	−0.813569
$88$	−90.9278	−0.110147
$89$	−866.036	−1.03146	−0.515728	−	0.856752i	$-0.672479\pi$
$89$	−866.036	−1.03146	−0.515728	+	0.856752i	$0.672479\pi$
$90$	0	0
$91$	1463.10	1.68544
$92$	167.064	0.189322
$93$	−480.659	−0.535935
$94$	62.2195	0.0682708
$95$	0	0
$96$	−77.5052	−0.0823994
$97$	177.047	0.185324	0.0926621	−	0.995698i	$-0.470462\pi$
$97$	177.047	0.185324	0.0926621	+	0.995698i	$0.470462\pi$
$98$	−5.85319	−0.00603328
$99$	379.390	0.385153
$100$	0	0
$101$	810.637	0.798628	0.399314	−	0.916814i	$-0.369248\pi$
$101$	810.637	0.798628	0.399314	+	0.916814i	$0.369248\pi$
$102$	−18.2620	−0.0177276
$103$	−1364.33	−1.30516	−0.652580	−	0.757720i	$-0.726313\pi$
$103$	−1364.33	−1.30516	−0.652580	+	0.757720i	$0.726313\pi$
$104$	−182.320	−0.171903
$105$	0	0
$106$	19.2251	0.0176161
$107$	1109.53	1.00245	0.501227	−	0.865316i	$-0.332882\pi$
$107$	1109.53	1.00245	0.501227	+	0.865316i	$0.332882\pi$
$108$	215.508	0.192012
$109$	1263.96	1.11069	0.555345	−	0.831620i	$-0.312586\pi$
$109$	1263.96	1.11069	0.555345	+	0.831620i	$0.312586\pi$
$110$	0	0
$111$	−789.682	−0.675255
$112$	1100.27	0.928267
$113$	1144.53	0.952816	0.476408	−	0.879224i	$-0.341938\pi$
$113$	1144.53	0.952816	0.476408	+	0.879224i	$0.341938\pi$
$114$	−33.8755	−0.0278309
$115$	0	0
$116$	−1756.51	−1.40593
$117$	760.716	0.601096
$118$	−63.7686	−0.0497490
$119$	780.718	0.601414
$120$	0	0
$121$	445.997	0.335084
$122$	7.29341	0.00541242
$123$	−1230.43	−0.901982
$124$	−1278.84	−0.926153
$125$	0	0
$126$	21.0264	0.0148665
$127$	2283.93	1.59579	0.797897	−	0.602793i	$-0.205946\pi$
$127$	2283.93	1.59579	0.797897	+	0.602793i	$0.205946\pi$
$128$	−274.842	−0.189787
$129$	872.097	0.595224
$130$	0	0
$131$	−203.205	−0.135527	−0.0677637	−	0.997701i	$-0.521586\pi$
$131$	−203.205	−0.135527	−0.0677637	+	0.997701i	$0.521586\pi$
$132$	1009.40	0.665585
$133$	1448.21	0.944175
$134$	−60.7300	−0.0391513
$135$	0	0
$136$	−97.2867	−0.0613402
$137$	−63.6065	−0.0396662	−0.0198331	−	0.999803i	$-0.506313\pi$
$137$	−63.6065	−0.0396662	−0.0198331	+	0.999803i	$0.506313\pi$
$138$	8.47486	0.00522774
$139$	1603.91	0.978722	0.489361	−	0.872081i	$-0.337230\pi$
$139$	1603.91	0.978722	0.489361	+	0.872081i	$0.337230\pi$
$140$	0	0
$141$	−1382.99	−0.826021
$142$	−84.4973	−0.0499356
$143$	3563.06	2.08362
$144$	572.068	0.331058
$145$	0	0
$146$	−6.57851	−0.00372905
$147$	130.102	0.0729977
$148$	−2101.02	−1.16691
$149$	−2731.00	−1.50156	−0.750780	−	0.660552i	$-0.770322\pi$
$149$	−2731.00	−1.50156	−0.750780	+	0.660552i	$0.770322\pi$
$150$	0	0
$151$	−879.795	−0.474150	−0.237075	−	0.971491i	$-0.576189\pi$
$151$	−879.795	−0.474150	−0.237075	+	0.971491i	$0.576189\pi$
$152$	−180.463	−0.0962994
$153$	405.922	0.214489
$154$	98.4841	0.0515330
$155$	0	0
$156$	2023.96	1.03876
$157$	−366.317	−0.186212	−0.0931059	−	0.995656i	$-0.529680\pi$
$157$	−366.317	−0.186212	−0.0931059	+	0.995656i	$0.529680\pi$
$158$	−87.4009	−0.0440078
$159$	−427.328	−0.213140
$160$	0	0
$161$	−362.308	−0.177353
$162$	10.9323	0.00530201
$163$	−843.200	−0.405181	−0.202591	−	0.979264i	$-0.564936\pi$
$163$	−843.200	−0.405181	−0.202591	+	0.979264i	$0.564936\pi$
$164$	−3273.66	−1.55872
$165$	0	0
$166$	−90.4079	−0.0422712
$167$	2230.09	1.03335	0.516675	−	0.856182i	$-0.327170\pi$
$167$	2230.09	1.03335	0.516675	+	0.856182i	$0.327170\pi$
$168$	112.013	0.0514405
$169$	4947.30	2.25184
$170$	0	0
$171$	752.971	0.336732
$172$	2320.30	1.02861
$173$	2893.55	1.27163	0.635816	−	0.771841i	$-0.280664\pi$
$173$	2893.55	1.27163	0.635816	+	0.771841i	$0.280664\pi$
$174$	−89.1048	−0.0388220
$175$	0	0
$176$	2679.47	1.14757
$177$	1417.42	0.601922
$178$	−116.886	−0.0492192
$179$	−2705.86	−1.12986	−0.564931	−	0.825138i	$-0.691097\pi$
$179$	−2705.86	−1.12986	−0.564931	+	0.825138i	$0.691097\pi$
$180$	0	0
$181$	−3554.78	−1.45980	−0.729902	−	0.683552i	$-0.760434\pi$
$181$	−3554.78	−1.45980	−0.729902	+	0.683552i	$0.760434\pi$
$182$	197.471	0.0804258
$183$	−162.115	−0.0654858
$184$	45.1478	0.0180888
$185$	0	0
$186$	−64.8732	−0.0255738
$187$	1901.27	0.743500
$188$	−3679.58	−1.42745
$189$	−467.367	−0.179873
$190$	0	0
$191$	172.435	0.0653244	0.0326622	−	0.999466i	$-0.489601\pi$
$191$	172.435	0.0653244	0.0326622	+	0.999466i	$0.489601\pi$
$192$	1515.05	0.569477
$193$	−1203.86	−0.448993	−0.224496	−	0.974475i	$-0.572074\pi$
$193$	−1203.86	−0.448993	−0.224496	+	0.974475i	$0.572074\pi$
$194$	23.8956	0.00884332
$195$	0	0
$196$	346.150	0.126148
$197$	−4195.96	−1.51751	−0.758756	−	0.651375i	$-0.774192\pi$
$197$	−4195.96	−1.51751	−0.758756	+	0.651375i	$0.774192\pi$
$198$	51.2052	0.0183788
$199$	−2413.09	−0.859594	−0.429797	−	0.902926i	$-0.641415\pi$
$199$	−2413.09	−0.859594	−0.429797	+	0.902926i	$0.641415\pi$
$200$	0	0
$201$	1349.88	0.473699
$202$	109.409	0.0381090
$203$	3809.31	1.31705
$204$	1079.99	0.370660
$205$	0	0
$206$	−184.140	−0.0622798
$207$	−188.376	−0.0632514
$208$	5372.61	1.79098
$209$	3526.78	1.16724
$210$	0	0
$211$	675.120	0.220271	0.110136	−	0.993917i	$-0.464872\pi$
$211$	675.120	0.220271	0.110136	+	0.993917i	$0.464872\pi$
$212$	−1136.95	−0.368329
$213$	1878.17	0.604180
$214$	149.750	0.0478352
$215$	0	0
$216$	58.2395	0.0183458
$217$	2773.38	0.867602
$218$	170.593	0.0530000
$219$	146.225	0.0451184
$220$	0	0
$221$	3812.23	1.16036
$222$	−106.581	−0.0322219
$223$	5739.58	1.72355	0.861773	−	0.507294i	$-0.169354\pi$
$223$	5739.58	1.72355	0.861773	+	0.507294i	$0.169354\pi$
$224$	447.203	0.133393
$225$	0	0
$226$	154.474	0.0454666
$227$	2649.74	0.774756	0.387378	−	0.921921i	$-0.373381\pi$
$227$	2649.74	0.774756	0.387378	+	0.921921i	$0.373381\pi$
$228$	2003.35	0.581908
$229$	−3685.69	−1.06357	−0.531785	−	0.846879i	$-0.678479\pi$
$229$	−3685.69	−1.06357	−0.531785	+	0.846879i	$0.678479\pi$
$230$	0	0
$231$	−2189.07	−0.623507
$232$	−474.685	−0.134330
$233$	−2551.40	−0.717373	−0.358687	−	0.933458i	$-0.616775\pi$
$233$	−2551.40	−0.717373	−0.358687	+	0.933458i	$0.616775\pi$
$234$	102.672	0.0286832
$235$	0	0
$236$	3771.19	1.04019
$237$	1942.71	0.532459
$238$	105.371	0.0286984
$239$	−6229.73	−1.68606	−0.843028	−	0.537869i	$-0.819230\pi$
$239$	−6229.73	−1.68606	−0.843028	+	0.537869i	$0.819230\pi$
$240$	0	0
$241$	−4084.06	−1.09161	−0.545804	−	0.837913i	$-0.683776\pi$
$241$	−4084.06	−1.09161	−0.545804	+	0.837913i	$0.683776\pi$
$242$	60.1949	0.0159896
$243$	−243.000	−0.0641500
$244$	−431.323	−0.113166
$245$	0	0
$246$	−166.067	−0.0430409
$247$	7071.56	1.82167
$248$	−345.596	−0.0884895
$249$	2009.55	0.511447
$250$	0	0
$251$	−1158.16	−0.291245	−0.145622	−	0.989340i	$-0.546518\pi$
$251$	−1158.16	−0.291245	−0.145622	+	0.989340i	$0.546518\pi$
$252$	−1243.47	−0.310839
$253$	−882.321	−0.219253
$254$	308.256	0.0761483
$255$	0	0
$256$	4003.05	0.977308
$257$	4275.75	1.03780	0.518899	−	0.854836i	$-0.326342\pi$
$257$	4275.75	1.03780	0.518899	+	0.854836i	$0.326342\pi$
$258$	117.705	0.0284030
$259$	4556.44	1.09314
$260$	0	0
$261$	1980.59	0.469714
$262$	−27.4260	−0.00646711
$263$	267.835	0.0627964	0.0313982	−	0.999507i	$-0.490004\pi$
$263$	267.835	0.0627964	0.0313982	+	0.999507i	$0.490004\pi$
$264$	272.784	0.0635934
$265$	0	0
$266$	195.460	0.0450543
$267$	2598.11	0.595512
$268$	3591.49	0.818602
$269$	6221.73	1.41021	0.705103	−	0.709105i	$-0.250901\pi$
$269$	6221.73	1.41021	0.705103	+	0.709105i	$0.250901\pi$
$270$	0	0
$271$	−1485.86	−0.333061	−0.166531	−	0.986036i	$-0.553256\pi$
$271$	−1485.86	−0.333061	−0.166531	+	0.986036i	$0.553256\pi$
$272$	2866.85	0.639075
$273$	−4389.30	−0.973087
$274$	−8.58479	−0.00189280
$275$	0	0
$276$	−501.192	−0.109305
$277$	−1125.01	−0.244025	−0.122013	−	0.992529i	$-0.538935\pi$
$277$	−1125.01	−0.244025	−0.122013	+	0.992529i	$0.538935\pi$
$278$	216.476	0.0467028
$279$	1441.98	0.309422
$280$	0	0
$281$	3631.61	0.770974	0.385487	−	0.922713i	$-0.374033\pi$
$281$	3631.61	0.770974	0.385487	+	0.922713i	$0.374033\pi$
$282$	−186.659	−0.0394162
$283$	−5262.72	−1.10543	−0.552715	−	0.833371i	$-0.686408\pi$
$283$	−5262.72	−1.10543	−0.552715	+	0.833371i	$0.686408\pi$
$284$	4997.06	1.04409
$285$	0	0
$286$	480.896	0.0994266
$287$	7099.51	1.46018
$288$	232.516	0.0475733
$289$	−2878.77	−0.585950
$290$	0	0
$291$	−531.142	−0.106997
$292$	389.044	0.0779695
$293$	−3619.55	−0.721693	−0.360847	−	0.932625i	$-0.617512\pi$
$293$	−3619.55	−0.721693	−0.360847	+	0.932625i	$0.617512\pi$
$294$	17.5596	0.00348331
$295$	0	0
$296$	−567.786	−0.111493
$297$	−1138.17	−0.222368
$298$	−368.596	−0.0716516
$299$	−1769.14	−0.342181
$300$	0	0
$301$	−5031.97	−0.963581
$302$	−118.743	−0.0226256
$303$	−2431.91	−0.461088
$304$	5317.91	1.00330
$305$	0	0
$306$	54.7861	0.0102350
$307$	1819.08	0.338176	0.169088	−	0.985601i	$-0.445918\pi$
$307$	1819.08	0.338176	0.169088	+	0.985601i	$0.445918\pi$
$308$	−5824.22	−1.07749
$309$	4092.99	0.753534
$310$	0	0
$311$	341.860	0.0623315	0.0311657	−	0.999514i	$-0.490078\pi$
$311$	341.860	0.0623315	0.0311657	+	0.999514i	$0.490078\pi$
$312$	546.959	0.0992482
$313$	3913.30	0.706686	0.353343	−	0.935494i	$-0.385045\pi$
$313$	3913.30	0.706686	0.353343	+	0.935494i	$0.385045\pi$
$314$	−49.4408	−0.00888568
$315$	0	0
$316$	5168.77	0.920146
$317$	−6577.26	−1.16535	−0.582675	−	0.812705i	$-0.697994\pi$
$317$	−6577.26	−1.16535	−0.582675	+	0.812705i	$0.697994\pi$
$318$	−57.6752	−0.0101706
$319$	9276.73	1.62820
$320$	0	0
$321$	−3328.60	−0.578767
$322$	−48.8997	−0.00846296
$323$	3773.42	0.650027
$324$	−646.524	−0.110858
$325$	0	0
$326$	−113.804	−0.0193345
$327$	−3791.87	−0.641257
$328$	−884.683	−0.148928
$329$	7979.81	1.33721
$330$	0	0
$331$	−9580.26	−1.59087	−0.795436	−	0.606038i	$-0.792758\pi$
$331$	−9580.26	−1.59087	−0.795436	+	0.606038i	$0.792758\pi$
$332$	5346.61	0.883835
$333$	2369.05	0.389859
$334$	300.989	0.0493095
$335$	0	0
$336$	−3300.81	−0.535935
$337$	1790.88	0.289482	0.144741	−	0.989470i	$-0.453765\pi$
$337$	1790.88	0.289482	0.144741	+	0.989470i	$0.453765\pi$
$338$	667.723	0.107454
$339$	−3433.59	−0.550109
$340$	0	0
$341$	6753.97	1.07257
$342$	101.626	0.0160682
$343$	−6687.98	−1.05282
$344$	627.043	0.0982787
$345$	0	0
$346$	390.534	0.0606799
$347$	9080.61	1.40482	0.702411	−	0.711772i	$-0.252107\pi$
$347$	9080.61	1.40482	0.702411	+	0.711772i	$0.252107\pi$
$348$	5269.54	0.811716
$349$	1036.94	0.159043	0.0795216	−	0.996833i	$-0.474661\pi$
$349$	1036.94	0.159043	0.0795216	+	0.996833i	$0.474661\pi$
$350$	0	0
$351$	−2282.15	−0.347043
$352$	1089.06	0.164907
$353$	−3307.28	−0.498665	−0.249332	−	0.968418i	$-0.580211\pi$
$353$	−3307.28	−0.498665	−0.249332	+	0.968418i	$0.580211\pi$
$354$	191.306	0.0287226
$355$	0	0
$356$	6912.51	1.02911
$357$	−2342.15	−0.347227
$358$	−365.202	−0.0539149
$359$	−12468.5	−1.83304	−0.916521	−	0.399987i	$-0.869015\pi$
$359$	−12468.5	−1.83304	−0.916521	+	0.399987i	$0.869015\pi$
$360$	0	0
$361$	140.567	0.0204939
$362$	−479.779	−0.0696591
$363$	−1337.99	−0.193461
$364$	−11678.2	−1.68160
$365$	0	0
$366$	−21.8802	−0.00312486
$367$	4956.10	0.704923	0.352461	−	0.935826i	$-0.385345\pi$
$367$	4956.10	0.704923	0.352461	+	0.935826i	$0.385345\pi$
$368$	−1330.42	−0.188459
$369$	3691.28	0.520759
$370$	0	0
$371$	2465.67	0.345043
$372$	3836.51	0.534715
$373$	−2903.60	−0.403063	−0.201532	−	0.979482i	$-0.564592\pi$
$373$	−2903.60	−0.403063	−0.201532	+	0.979482i	$0.564592\pi$
$374$	256.609	0.0354784
$375$	0	0
$376$	−994.379	−0.136386
$377$	18600.8	2.54109
$378$	−63.0793	−0.00858319
$379$	4784.12	0.648400	0.324200	−	0.945989i	$-0.394905\pi$
$379$	4784.12	0.648400	0.324200	+	0.945989i	$0.394905\pi$
$380$	0	0
$381$	−6851.79	−0.921333
$382$	23.2731	0.00311716
$383$	−1537.67	−0.205146	−0.102573	−	0.994725i	$-0.532708\pi$
$383$	−1537.67	−0.205146	−0.102573	+	0.994725i	$0.532708\pi$
$384$	824.525	0.109574
$385$	0	0
$386$	−162.481	−0.0214251
$387$	−2616.29	−0.343653
$388$	−1413.15	−0.184902
$389$	−3208.85	−0.418239	−0.209119	−	0.977890i	$-0.567060\pi$
$389$	−3208.85	−0.418239	−0.209119	+	0.977890i	$0.567060\pi$
$390$	0	0
$391$	−944.024	−0.122101
$392$	93.5444	0.0120528
$393$	609.614	0.0782468
$394$	−566.317	−0.0724128
$395$	0	0
$396$	−3028.21	−0.384276
$397$	−6227.13	−0.787231	−0.393616	−	0.919275i	$-0.628776\pi$
$397$	−6227.13	−0.787231	−0.393616	+	0.919275i	$0.628776\pi$
$398$	−325.687	−0.0410182
$399$	−4344.62	−0.545120
$400$	0	0
$401$	−3174.14	−0.395284	−0.197642	−	0.980274i	$-0.563328\pi$
$401$	−3174.14	−0.395284	−0.197642	+	0.980274i	$0.563328\pi$
$402$	182.190	0.0226040
$403$	13542.4	1.67393
$404$	−6470.33	−0.796810
$405$	0	0
$406$	514.132	0.0628471
$407$	11096.2	1.35140
$408$	291.860	0.0354148
$409$	10096.0	1.22057	0.610286	−	0.792182i	$-0.291055\pi$
$409$	10096.0	1.22057	0.610286	+	0.792182i	$0.291055\pi$
$410$	0	0
$411$	190.819	0.0229013
$412$	10889.8	1.30219
$413$	−8178.49	−0.974424
$414$	−25.4246	−0.00301824
$415$	0	0
$416$	2183.68	0.257365
$417$	−4811.74	−0.565065
$418$	476.000	0.0556984
$419$	−4337.34	−0.505711	−0.252855	−	0.967504i	$-0.581370\pi$
$419$	−4337.34	−0.505711	−0.252855	+	0.967504i	$0.581370\pi$
$420$	0	0
$421$	−16427.6	−1.90173	−0.950867	−	0.309598i	$-0.899805\pi$
$421$	−16427.6	−1.90173	−0.950867	+	0.309598i	$0.899805\pi$
$422$	91.1191	0.0105109
$423$	4148.98	0.476903
$424$	−307.251	−0.0351920
$425$	0	0
$426$	253.492	0.0288303
$427$	935.399	0.106012
$428$	−8856.04	−1.00017
$429$	−10689.2	−1.20298
$430$	0	0
$431$	1395.49	0.155959	0.0779797	−	0.996955i	$-0.475153\pi$
$431$	1395.49	0.155959	0.0779797	+	0.996955i	$0.475153\pi$
$432$	−1716.20	−0.191136
$433$	1905.84	0.211522	0.105761	−	0.994392i	$-0.466272\pi$
$433$	1905.84	0.211522	0.105761	+	0.994392i	$0.466272\pi$
$434$	374.316	0.0414003
$435$	0	0
$436$	−10088.6	−1.10816
$437$	−1751.13	−0.191689
$438$	19.7355	0.00215297
$439$	−7973.28	−0.866843	−0.433421	−	0.901191i	$-0.642694\pi$
$439$	−7973.28	−0.866843	−0.433421	+	0.901191i	$0.642694\pi$
$440$	0	0
$441$	−390.307	−0.0421453
$442$	514.527	0.0553700
$443$	−2866.93	−0.307476	−0.153738	−	0.988112i	$-0.549131\pi$
$443$	−2866.93	−0.307476	−0.153738	+	0.988112i	$0.549131\pi$
$444$	6303.07	0.673717
$445$	0	0
$446$	774.655	0.0822444
$447$	8193.01	0.866926
$448$	−8741.81	−0.921902
$449$	−10711.2	−1.12582	−0.562908	−	0.826520i	$-0.690317\pi$
$449$	−10711.2	−1.12582	−0.562908	+	0.826520i	$0.690317\pi$
$450$	0	0
$451$	17289.3	1.80515
$452$	−9135.38	−0.950647
$453$	2639.38	0.273751
$454$	357.628	0.0369699
$455$	0	0
$456$	541.390	0.0555985
$457$	5306.67	0.543185	0.271593	−	0.962412i	$-0.412450\pi$
$457$	5306.67	0.543185	0.271593	+	0.962412i	$0.412450\pi$
$458$	−497.448	−0.0507515
$459$	−1217.77	−0.123835
$460$	0	0
$461$	−8288.61	−0.837395	−0.418697	−	0.908126i	$-0.637513\pi$
$461$	−8288.61	−0.837395	−0.418697	+	0.908126i	$0.637513\pi$
$462$	−295.452	−0.0297526
$463$	−1162.48	−0.116685	−0.0583424	−	0.998297i	$-0.518582\pi$
$463$	−1162.48	−0.116685	−0.0583424	+	0.998297i	$0.518582\pi$
$464$	13988.0	1.39952
$465$	0	0
$466$	−344.356	−0.0342317
$467$	−7585.71	−0.751659	−0.375829	−	0.926689i	$-0.622642\pi$
$467$	−7585.71	−0.751659	−0.375829	+	0.926689i	$0.622642\pi$
$468$	−6071.87	−0.599727
$469$	−7788.78	−0.766849
$470$	0	0
$471$	1098.95	0.107509
$472$	1019.14	0.0993846
$473$	−12254.3	−1.19123
$474$	262.203	0.0254079
$475$	0	0
$476$	−6231.52	−0.600045
$477$	1281.98	0.123057
$478$	−840.809	−0.0804555
$479$	−7436.68	−0.709375	−0.354687	−	0.934985i	$-0.615413\pi$
$479$	−7436.68	−0.709375	−0.354687	+	0.934985i	$0.615413\pi$
$480$	0	0
$481$	22249.0	2.10908
$482$	−551.214	−0.0520895
$483$	1086.92	0.102395
$484$	−3559.85	−0.334321
$485$	0	0
$486$	−32.7970	−0.00306112
$487$	−21171.7	−1.96998	−0.984991	−	0.172604i	$-0.944782\pi$
$487$	−21171.7	−1.96998	−0.984991	+	0.172604i	$0.944782\pi$
$488$	−116.562	−0.0108125
$489$	2529.60	0.233931
$490$	0	0
$491$	9419.29	0.865756	0.432878	−	0.901452i	$-0.357498\pi$
$491$	9419.29	0.865756	0.432878	+	0.901452i	$0.357498\pi$
$492$	9820.99	0.899928
$493$	9925.48	0.906737
$494$	954.429	0.0869267
$495$	0	0
$496$	10184.1	0.921931
$497$	−10837.0	−0.978080
$498$	271.224	0.0244053
$499$	3295.44	0.295640	0.147820	−	0.989014i	$-0.452774\pi$
$499$	3295.44	0.295640	0.147820	+	0.989014i	$0.452774\pi$
$500$	0	0
$501$	−6690.27	−0.596605
$502$	−156.314	−0.0138976
$503$	−5611.35	−0.497411	−0.248706	−	0.968579i	$-0.580005\pi$
$503$	−5611.35	−0.497411	−0.248706	+	0.968579i	$0.580005\pi$
$504$	−336.040	−0.0296992
$505$	0	0
$506$	−119.084	−0.0104623
$507$	−14841.9	−1.30010
$508$	−18229.8	−1.59216
$509$	20552.6	1.78974	0.894872	−	0.446323i	$-0.147267\pi$
$509$	20552.6	1.78974	0.894872	+	0.446323i	$0.147267\pi$
$510$	0	0
$511$	−843.711	−0.0730402
$512$	2739.01	0.236423
$513$	−2258.91	−0.194412
$514$	577.086	0.0495218
$515$	0	0
$516$	−6960.89	−0.593869
$517$	19433.1	1.65313
$518$	614.970	0.0521626
$519$	−8680.64	−0.734177
$520$	0	0
$521$	541.011	0.0454936	0.0227468	−	0.999741i	$-0.492759\pi$
$521$	541.011	0.0454936	0.0227468	+	0.999741i	$0.492759\pi$
$522$	267.315	0.0224139
$523$	−2248.74	−0.188012	−0.0940061	−	0.995572i	$-0.529967\pi$
$523$	−2248.74	−0.188012	−0.0940061	+	0.995572i	$0.529967\pi$
$524$	1621.94	0.135219
$525$	0	0
$526$	36.1490	0.00299652
$527$	7226.29	0.597309
$528$	−8038.41	−0.662551
$529$	−11728.9	−0.963993
$530$	0	0
$531$	−4252.27	−0.347520
$532$	−11559.3	−0.942025
$533$	34666.8	2.81724
$534$	350.659	0.0284167
$535$	0	0
$536$	970.574	0.0782134
$537$	8117.57	0.652326
$538$	839.730	0.0672924
$539$	−1828.13	−0.146091
$540$	0	0
$541$	−18663.3	−1.48318	−0.741590	−	0.670854i	$-0.765928\pi$
$541$	−18663.3	−1.48318	−0.741590	+	0.670854i	$0.765928\pi$
$542$	−200.543	−0.0158931
$543$	10664.3	0.842819
$544$	1165.22	0.0918356
$545$	0	0
$546$	−592.412	−0.0464339
$547$	−10867.7	−0.849484	−0.424742	−	0.905314i	$-0.639635\pi$
$547$	−10867.7	−0.849484	−0.424742	+	0.905314i	$0.639635\pi$
$548$	507.693	0.0395759
$549$	486.346	0.0378082
$550$	0	0
$551$	18411.4	1.42351
$552$	−135.443	−0.0104436
$553$	−11209.4	−0.861974
$554$	−151.839	−0.0116444
$555$	0	0
$556$	−12802.1	−0.976493
$557$	2740.22	0.208450	0.104225	−	0.994554i	$-0.466764\pi$
$557$	2740.22	0.208450	0.104225	+	0.994554i	$0.466764\pi$
$558$	194.619	0.0147651
$559$	−24571.0	−1.85911
$560$	0	0
$561$	−5703.80	−0.429260
$562$	490.148	0.0367894
$563$	19190.9	1.43659	0.718294	−	0.695740i	$-0.244923\pi$
$563$	19190.9	1.43659	0.718294	+	0.695740i	$0.244923\pi$
$564$	11038.7	0.824140
$565$	0	0
$566$	−710.295	−0.0527490
$567$	1402.10	0.103850
$568$	1350.42	0.0997575
$569$	20217.4	1.48956	0.744779	−	0.667311i	$-0.232555\pi$
$569$	20217.4	1.48956	0.744779	+	0.667311i	$0.232555\pi$
$570$	0	0
$571$	5129.67	0.375954	0.187977	−	0.982173i	$-0.439807\pi$
$571$	5129.67	0.375954	0.187977	+	0.982173i	$0.439807\pi$
$572$	−28439.6	−2.07888
$573$	−517.305	−0.0377151
$574$	958.202	0.0696769
$575$	0	0
$576$	−4545.16	−0.328788
$577$	−14063.5	−1.01468	−0.507342	−	0.861745i	$-0.669372\pi$
$577$	−14063.5	−1.01468	−0.507342	+	0.861745i	$0.669372\pi$
$578$	−388.540	−0.0279604
$579$	3611.58	0.259226
$580$	0	0
$581$	−11595.1	−0.827958
$582$	−71.6868	−0.00510569
$583$	6004.58	0.426560
$584$	105.136	0.00744961
$585$	0	0
$586$	−488.520	−0.0344379
$587$	3057.11	0.214958	0.107479	−	0.994207i	$-0.465722\pi$
$587$	3057.11	0.214958	0.107479	+	0.994207i	$0.465722\pi$
$588$	−1038.45	−0.0728315
$589$	13404.5	0.937731
$590$	0	0
$591$	12587.9	0.876136
$592$	16731.6	1.16159
$593$	13287.5	0.920153	0.460077	−	0.887879i	$-0.347822\pi$
$593$	13287.5	0.920153	0.460077	+	0.887879i	$0.347822\pi$
$594$	−153.616	−0.0106110
$595$	0	0
$596$	21798.3	1.49814
$597$	7239.26	0.496287
$598$	−238.776	−0.0163282
$599$	12749.7	0.869682	0.434841	−	0.900507i	$-0.356805\pi$
$599$	12749.7	0.869682	0.434841	+	0.900507i	$0.356805\pi$
$600$	0	0
$601$	21184.3	1.43782	0.718908	−	0.695106i	$-0.244642\pi$
$601$	21184.3	1.43782	0.718908	+	0.695106i	$0.244642\pi$
$602$	−679.151	−0.0459803
$603$	−4049.65	−0.273490
$604$	7022.33	0.473071
$605$	0	0
$606$	−328.228	−0.0220023
$607$	−2387.17	−0.159625	−0.0798124	−	0.996810i	$-0.525432\pi$
$607$	−2387.17	−0.159625	−0.0798124	+	0.996810i	$0.525432\pi$
$608$	2161.45	0.144175
$609$	−11427.9	−0.760399
$610$	0	0
$611$	38965.3	2.57998
$612$	−3239.98	−0.214001
$613$	9470.85	0.624020	0.312010	−	0.950079i	$-0.398998\pi$
$613$	9470.85	0.624020	0.312010	+	0.950079i	$0.398998\pi$
$614$	245.516	0.0161371
$615$	0	0
$616$	−1573.95	−0.102949
$617$	−27465.6	−1.79210	−0.896049	−	0.443955i	$-0.853575\pi$
$617$	−27465.6	−1.79210	−0.896049	+	0.443955i	$0.853575\pi$
$618$	552.420	0.0359572
$619$	−19231.1	−1.24873	−0.624365	−	0.781133i	$-0.714642\pi$
$619$	−19231.1	−1.24873	−0.624365	+	0.781133i	$0.714642\pi$
$620$	0	0
$621$	565.128	0.0365182
$622$	46.1399	0.00297434
$623$	−14991.0	−0.964047
$624$	−16117.8	−1.03402
$625$	0	0
$626$	528.167	0.0337217
$627$	−10580.4	−0.673905
$628$	2923.86	0.185788
$629$	11872.2	0.752583
$630$	0	0
$631$	−894.541	−0.0564360	−0.0282180	−	0.999602i	$-0.508983\pi$
$631$	−894.541	−0.0564360	−0.0282180	+	0.999602i	$0.508983\pi$
$632$	1396.82	0.0879155
$633$	−2025.36	−0.127174
$634$	−887.715	−0.0556083
$635$	0	0
$636$	3410.84	0.212655
$637$	−3665.59	−0.228000
$638$	1252.05	0.0776949
$639$	−5634.52	−0.348823
$640$	0	0
$641$	−8943.89	−0.551111	−0.275556	−	0.961285i	$-0.588862\pi$
$641$	−8943.89	−0.551111	−0.275556	+	0.961285i	$0.588862\pi$
$642$	−449.251	−0.0276177
$643$	4002.17	0.245459	0.122730	−	0.992440i	$-0.460835\pi$
$643$	4002.17	0.245459	0.122730	+	0.992440i	$0.460835\pi$
$644$	2891.86	0.176949
$645$	0	0
$646$	509.288	0.0310181
$647$	13315.6	0.809104	0.404552	−	0.914515i	$-0.367427\pi$
$647$	13315.6	0.809104	0.404552	+	0.914515i	$0.367427\pi$
$648$	−174.718	−0.0105920
$649$	−19916.9	−1.20463
$650$	0	0
$651$	−8320.15	−0.500910
$652$	6730.24	0.404258
$653$	22994.6	1.37802	0.689011	−	0.724750i	$-0.258045\pi$
$653$	22994.6	1.37802	0.689011	+	0.724750i	$0.258045\pi$
$654$	−511.779	−0.0305996
$655$	0	0
$656$	26069.9	1.55161
$657$	−438.674	−0.0260491
$658$	1077.01	0.0638090
$659$	−4002.57	−0.236598	−0.118299	−	0.992978i	$-0.537744\pi$
$659$	−4002.57	−0.236598	−0.118299	+	0.992978i	$0.537744\pi$
$660$	0	0
$661$	−13629.6	−0.802009	−0.401005	−	0.916076i	$-0.631339\pi$
$661$	−13629.6	−0.802009	−0.401005	+	0.916076i	$0.631339\pi$
$662$	−1293.02	−0.0759134
$663$	−11436.7	−0.669932
$664$	1444.88	0.0844461
$665$	0	0
$666$	319.743	0.0186033
$667$	−4606.12	−0.267391
$668$	−17800.1	−1.03100
$669$	−17218.7	−0.995090
$670$	0	0
$671$	2277.96	0.131058
$672$	−1341.61	−0.0770143
$673$	25216.1	1.44429	0.722146	−	0.691741i	$-0.243156\pi$
$673$	25216.1	1.44429	0.722146	+	0.691741i	$0.243156\pi$
$674$	241.710	0.0138135
$675$	0	0
$676$	−39488.3	−2.24672
$677$	−529.842	−0.0300790	−0.0150395	−	0.999887i	$-0.504787\pi$
$677$	−529.842	−0.0300790	−0.0150395	+	0.999887i	$0.504787\pi$
$678$	−463.422	−0.0262502
$679$	3064.67	0.173213
$680$	0	0
$681$	−7949.22	−0.447305
$682$	911.564	0.0511812
$683$	2509.91	0.140614	0.0703068	−	0.997525i	$-0.477602\pi$
$683$	2509.91	0.140614	0.0703068	+	0.997525i	$0.477602\pi$
$684$	−6010.05	−0.335965
$685$	0	0
$686$	−902.658	−0.0502386
$687$	11057.1	0.614052
$688$	−18477.7	−1.02392
$689$	12039.8	0.665719
$690$	0	0
$691$	−25676.8	−1.41359	−0.706796	−	0.707417i	$-0.749860\pi$
$691$	−25676.8	−1.41359	−0.706796	+	0.707417i	$0.749860\pi$
$692$	−23095.7	−1.26874
$693$	6567.20	0.359982
$694$	1225.59	0.0670354
$695$	0	0
$696$	1424.05	0.0775555
$697$	18498.4	1.00527
$698$	139.953	0.00758924
$699$	7654.21	0.414176
$700$	0	0
$701$	34424.0	1.85475	0.927373	−	0.374139i	$-0.122062\pi$
$701$	34424.0	1.85475	0.927373	+	0.374139i	$0.122062\pi$
$702$	−308.015	−0.0165602
$703$	22022.5	1.18150
$704$	−21288.8	−1.13970
$705$	0	0
$706$	−446.374	−0.0237953
$707$	14032.0	0.746435
$708$	−11313.6	−0.600551
$709$	−6915.53	−0.366316	−0.183158	−	0.983083i	$-0.558632\pi$
$709$	−6915.53	−0.366316	−0.183158	+	0.983083i	$0.558632\pi$
$710$	0	0
$711$	−5828.14	−0.307415
$712$	1868.05	0.0983263
$713$	−3353.50	−0.176143
$714$	−316.114	−0.0165690
$715$	0	0
$716$	21597.6	1.12729
$717$	18689.2	0.973445
$718$	−1682.84	−0.0874693
$719$	−29938.5	−1.55288	−0.776438	−	0.630193i	$-0.782976\pi$
$719$	−29938.5	−1.55288	−0.776438	+	0.630193i	$0.782976\pi$
$720$	0	0
$721$	−23616.4	−1.21986
$722$	18.9720	0.000977929 0
$723$	12252.2	0.630240
$724$	28373.5	1.45648
$725$	0	0
$726$	−180.585	−0.00923158
$727$	−17305.3	−0.882832	−0.441416	−	0.897303i	$-0.645524\pi$
$727$	−17305.3	−0.882832	−0.441416	+	0.897303i	$0.645524\pi$
$728$	−3155.93	−0.160669
$729$	729.000	0.0370370
$730$	0	0
$731$	−13111.2	−0.663388
$732$	1293.97	0.0653367
$733$	−4581.82	−0.230878	−0.115439	−	0.993315i	$-0.536827\pi$
$733$	−4581.82	−0.230878	−0.115439	+	0.993315i	$0.536827\pi$
$734$	668.912	0.0336376
$735$	0	0
$736$	−540.746	−0.0270817
$737$	−18967.9	−0.948019
$738$	498.201	0.0248497
$739$	−1715.91	−0.0854139	−0.0427069	−	0.999088i	$-0.513598\pi$
$739$	−1715.91	−0.0854139	−0.0427069	+	0.999088i	$0.513598\pi$
$740$	0	0
$741$	−21214.7	−1.05174
$742$	332.784	0.0164648
$743$	−12824.6	−0.633228	−0.316614	−	0.948554i	$-0.602546\pi$
$743$	−12824.6	−0.633228	−0.316614	+	0.948554i	$0.602546\pi$
$744$	1036.79	0.0510894
$745$	0	0
$746$	−391.890	−0.0192334
$747$	−6028.66	−0.295284
$748$	−15175.5	−0.741807
$749$	19205.9	0.936939
$750$	0	0
$751$	−38918.8	−1.89103	−0.945516	−	0.325574i	$-0.894442\pi$
$751$	−38918.8	−1.89103	−0.945516	+	0.325574i	$0.894442\pi$
$752$	29302.4	1.42094
$753$	3474.48	0.168150
$754$	2510.50	0.121256
$755$	0	0
$756$	3730.42	0.179463
$757$	−22740.4	−1.09183	−0.545915	−	0.837841i	$-0.683818\pi$
$757$	−22740.4	−1.09183	−0.545915	+	0.837841i	$0.683818\pi$
$758$	645.699	0.0309404
$759$	2646.96	0.126586
$760$	0	0
$761$	40270.0	1.91825	0.959123	−	0.282988i	$-0.0913257\pi$
$761$	40270.0	1.91825	0.959123	+	0.282988i	$0.0913257\pi$
$762$	−924.767	−0.0439643
$763$	21879.0	1.03810
$764$	−1376.34	−0.0651757
$765$	0	0
$766$	−207.534	−0.00978919
$767$	−39935.5	−1.88003
$768$	−12009.2	−0.564249
$769$	−30661.9	−1.43784	−0.718919	−	0.695094i	$-0.755363\pi$
$769$	−30661.9	−1.43784	−0.718919	+	0.695094i	$0.755363\pi$
$770$	0	0
$771$	−12827.3	−0.599173
$772$	9608.94	0.447971
$773$	30438.6	1.41630	0.708151	−	0.706061i	$-0.249529\pi$
$773$	30438.6	1.41630	0.708151	+	0.706061i	$0.249529\pi$
$774$	−353.114	−0.0163985
$775$	0	0
$776$	−381.894	−0.0176665
$777$	−13669.3	−0.631125
$778$	−433.089	−0.0199576
$779$	34313.9	1.57821
$780$	0	0
$781$	−26391.1	−1.20915
$782$	−127.412	−0.00582641
$783$	−5941.77	−0.271190
$784$	−2756.57	−0.125573
$785$	0	0
$786$	82.2780	0.00373379
$787$	−24386.3	−1.10455	−0.552274	−	0.833663i	$-0.686240\pi$
$787$	−24386.3	−1.10455	−0.552274	+	0.833663i	$0.686240\pi$
$788$	33491.2	1.51406
$789$	−803.506	−0.0362555
$790$	0	0
$791$	19811.7	0.890547
$792$	−818.351	−0.0367157
$793$	4567.54	0.204537
$794$	−840.459	−0.0375652
$795$	0	0
$796$	19260.7	0.857636
$797$	−41143.2	−1.82857	−0.914283	−	0.405076i	$-0.867245\pi$
$797$	−41143.2	−1.82857	−0.914283	+	0.405076i	$0.867245\pi$
$798$	−586.381	−0.0260121
$799$	20792.1	0.920615
$800$	0	0
$801$	−7794.32	−0.343819
$802$	−428.404	−0.0188622
$803$	−2054.67	−0.0902961
$804$	−10774.5	−0.472620
$805$	0	0
$806$	1827.78	0.0798769
$807$	−18665.2	−0.814183
$808$	−1748.56	−0.0761313
$809$	−16879.6	−0.733566	−0.366783	−	0.930307i	$-0.619541\pi$
$809$	−16879.6	−0.733566	−0.366783	+	0.930307i	$0.619541\pi$
$810$	0	0
$811$	11423.5	0.494616	0.247308	−	0.968937i	$-0.420454\pi$
$811$	11423.5	0.494616	0.247308	+	0.968937i	$0.420454\pi$
$812$	−30405.1	−1.31405
$813$	4457.58	0.192293
$814$	1497.62	0.0644861
$815$	0	0
$816$	−8600.55	−0.368970
$817$	−24320.9	−1.04147
$818$	1362.62	0.0582434
$819$	13167.9	0.561812
$820$	0	0
$821$	−3507.29	−0.149093	−0.0745466	−	0.997218i	$-0.523751\pi$
$821$	−3507.29	−0.149093	−0.0745466	+	0.997218i	$0.523751\pi$
$822$	25.7544	0.00109281
$823$	10475.2	0.443674	0.221837	−	0.975084i	$-0.428795\pi$
$823$	10475.2	0.443674	0.221837	+	0.975084i	$0.428795\pi$
$824$	2942.88	0.124418
$825$	0	0
$826$	−1103.83	−0.0464977
$827$	−26560.2	−1.11679	−0.558397	−	0.829574i	$-0.688583\pi$
$827$	−26560.2	−1.11679	−0.558397	+	0.829574i	$0.688583\pi$
$828$	1503.58	0.0631074
$829$	7296.73	0.305701	0.152850	−	0.988249i	$-0.451155\pi$
$829$	7296.73	0.305701	0.152850	+	0.988249i	$0.451155\pi$
$830$	0	0
$831$	3375.02	0.140888
$832$	−42686.1	−1.77870
$833$	−1955.98	−0.0813573
$834$	−649.428	−0.0269638
$835$	0	0
$836$	−28150.0	−1.16458
$837$	−4325.93	−0.178645
$838$	−585.398	−0.0241316
$839$	46997.4	1.93389	0.966944	−	0.254989i	$-0.0820718\pi$
$839$	46997.4	1.93389	0.966944	+	0.254989i	$0.0820718\pi$
$840$	0	0
$841$	24039.8	0.985681
$842$	−2217.18	−0.0907472
$843$	−10894.8	−0.445122
$844$	−5388.66	−0.219770
$845$	0	0
$846$	559.976	0.0227569
$847$	7720.16	0.313185
$848$	9054.10	0.366650
$849$	15788.2	0.638220
$850$	0	0
$851$	−5509.52	−0.221932
$852$	−14991.2	−0.602804
$853$	−30588.9	−1.22784	−0.613918	−	0.789370i	$-0.710408\pi$
$853$	−30588.9	−1.22784	−0.613918	+	0.789370i	$0.710408\pi$
$854$	126.248	0.00505870
$855$	0	0
$856$	−2393.28	−0.0955614
$857$	39552.7	1.57654	0.788271	−	0.615329i	$-0.210977\pi$
$857$	39552.7	1.57654	0.788271	+	0.615329i	$0.210977\pi$
$858$	−1442.69	−0.0574040
$859$	14881.4	0.591089	0.295545	−	0.955329i	$-0.404499\pi$
$859$	14881.4	0.591089	0.295545	+	0.955329i	$0.404499\pi$
$860$	0	0
$861$	−21298.5	−0.843034
$862$	188.346	0.00744209
$863$	−6318.56	−0.249231	−0.124616	−	0.992205i	$-0.539770\pi$
$863$	−6318.56	−0.249231	−0.124616	+	0.992205i	$0.539770\pi$
$864$	−697.547	−0.0274665
$865$	0	0
$866$	257.226	0.0100934
$867$	8636.32	0.338298
$868$	−22136.6	−0.865626
$869$	−27298.0	−1.06562
$870$	0	0
$871$	−38032.5	−1.47954
$872$	−2726.38	−0.105879
$873$	1593.43	0.0617747
$874$	−236.345	−0.00914703
$875$	0	0
$876$	−1167.13	−0.0450157
$877$	−25188.2	−0.969835	−0.484918	−	0.874560i	$-0.661150\pi$
$877$	−25188.2	−0.969835	−0.484918	+	0.874560i	$0.661150\pi$
$878$	−1076.13	−0.0413641
$879$	10858.6	0.416670
$880$	0	0
$881$	−1209.46	−0.0462518	−0.0231259	−	0.999733i	$-0.507362\pi$
$881$	−1209.46	−0.0462518	−0.0231259	+	0.999733i	$0.507362\pi$
$882$	−52.6787	−0.00201109
$883$	−6253.63	−0.238337	−0.119168	−	0.992874i	$-0.538023\pi$
$883$	−6253.63	−0.238337	−0.119168	+	0.992874i	$0.538023\pi$
$884$	−30428.4	−1.15771
$885$	0	0
$886$	−386.941	−0.0146722
$887$	20271.5	0.767360	0.383680	−	0.923466i	$-0.374656\pi$
$887$	20271.5	0.767360	0.383680	+	0.923466i	$0.374656\pi$
$888$	1703.36	0.0643704
$889$	39534.6	1.49150
$890$	0	0
$891$	3414.51	0.128384
$892$	−45812.1	−1.71962
$893$	38568.6	1.44530
$894$	1105.79	0.0413681
$895$	0	0
$896$	−4757.48	−0.177384
$897$	5307.43	0.197558
$898$	−1445.66	−0.0537218
$899$	35258.8	1.30806
$900$	0	0
$901$	6424.50	0.237548
$902$	2333.49	0.0861382
$903$	15095.9	0.556324
$904$	−2468.77	−0.0908297
$905$	0	0
$906$	356.230	0.0130629
$907$	−20161.0	−0.738077	−0.369039	−	0.929414i	$-0.620313\pi$
$907$	−20161.0	−0.738077	−0.369039	+	0.929414i	$0.620313\pi$
$908$	−21149.7	−0.772991
$909$	7295.74	0.266209
$910$	0	0
$911$	12771.2	0.464468	0.232234	−	0.972660i	$-0.425397\pi$
$911$	12771.2	0.464468	0.232234	+	0.972660i	$0.425397\pi$
$912$	−15953.7	−0.579255
$913$	−28237.2	−1.02356
$914$	716.227	0.0259198
$915$	0	0
$916$	29418.4	1.06115
$917$	−3517.45	−0.126670
$918$	−164.358	−0.00590919
$919$	22979.7	0.824844	0.412422	−	0.910993i	$-0.364683\pi$
$919$	22979.7	0.824844	0.412422	+	0.910993i	$0.364683\pi$
$920$	0	0
$921$	−5457.23	−0.195246
$922$	−1118.69	−0.0399589
$923$	−52916.9	−1.88709
$924$	17472.7	0.622087
$925$	0	0
$926$	−156.897	−0.00556798
$927$	−12279.0	−0.435053
$928$	5685.41	0.201113
$929$	31009.9	1.09516	0.547579	−	0.836754i	$-0.315549\pi$
$929$	31009.9	1.09516	0.547579	+	0.836754i	$0.315549\pi$
$930$	0	0
$931$	−3628.27	−0.127725
$932$	20364.7	0.715740
$933$	−1025.58	−0.0359871
$934$	−1023.82	−0.0358677
$935$	0	0
$936$	−1640.88	−0.0573010
$937$	37193.1	1.29674	0.648371	−	0.761325i	$-0.275451\pi$
$937$	37193.1	1.29674	0.648371	+	0.761325i	$0.275451\pi$
$938$	−1051.23	−0.0365926
$939$	−11739.9	−0.408005
$940$	0	0
$941$	−2159.48	−0.0748109	−0.0374055	−	0.999300i	$-0.511909\pi$
$941$	−2159.48	−0.0748109	−0.0374055	+	0.999300i	$0.511909\pi$
$942$	148.322	0.00513015
$943$	−8584.55	−0.296449
$944$	−30032.0	−1.03544
$945$	0	0
$946$	−1653.92	−0.0568432
$947$	−9462.47	−0.324698	−0.162349	−	0.986733i	$-0.551907\pi$
$947$	−9462.47	−0.324698	−0.162349	+	0.986733i	$0.551907\pi$
$948$	−15506.3	−0.531247
$949$	−4119.83	−0.140922
$950$	0	0
$951$	19731.8	0.672815
$952$	−1684.02	−0.0573314
$953$	11718.0	0.398304	0.199152	−	0.979969i	$-0.436181\pi$
$953$	11718.0	0.398304	0.199152	+	0.979969i	$0.436181\pi$
$954$	173.026	0.00587203
$955$	0	0
$956$	49724.3	1.68222
$957$	−27830.2	−0.940044
$958$	−1003.71	−0.0338500
$959$	−1101.02	−0.0370739
$960$	0	0
$961$	−4120.70	−0.138320
$962$	3002.89	0.100641
$963$	9985.79	0.334151
$964$	32598.1	1.08912
$965$	0	0
$966$	146.699	0.00488609
$967$	40593.1	1.34993	0.674967	−	0.737848i	$-0.264158\pi$
$967$	40593.1	1.34993	0.674967	+	0.737848i	$0.264158\pi$
$968$	−962.022	−0.0319427
$969$	−11320.3	−0.375293
$970$	0	0
$971$	51359.1	1.69742	0.848708	−	0.528862i	$-0.177381\pi$
$971$	51359.1	1.69742	0.848708	+	0.528862i	$0.177381\pi$
$972$	1939.57	0.0640040
$973$	27763.6	0.914759
$974$	−2857.49	−0.0940039
$975$	0	0
$976$	3434.85	0.112651
$977$	45722.4	1.49723	0.748613	−	0.663008i	$-0.230720\pi$
$977$	45722.4	1.49723	0.748613	+	0.663008i	$0.230720\pi$
$978$	341.413	0.0111628
$979$	−36507.3	−1.19181
$980$	0	0
$981$	11375.6	0.370230
$982$	1271.29	0.0413123
$983$	−12933.8	−0.419657	−0.209828	−	0.977738i	$-0.567291\pi$
$983$	−12933.8	−0.419657	−0.209828	+	0.977738i	$0.567291\pi$
$984$	2654.05	0.0859837
$985$	0	0
$986$	1339.61	0.0432678
$987$	−23939.4	−0.772037
$988$	−56443.7	−1.81752
$989$	6084.53	0.195629
$990$	0	0
$991$	13580.2	0.435308	0.217654	−	0.976026i	$-0.430159\pi$
$991$	13580.2	0.435308	0.217654	+	0.976026i	$0.430159\pi$
$992$	4139.28	0.132482
$993$	28740.8	0.918490
$994$	−1462.64	−0.0466721
$995$	0	0
$996$	−16039.8	−0.510282
$997$	−19249.8	−0.611480	−0.305740	−	0.952115i	$-0.598904\pi$
$997$	−19249.8	−0.611480	−0.305740	+	0.952115i	$0.598904\pi$
$998$	444.777	0.0141074
$999$	−7107.14	−0.225085

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.4.a.f.1.7		14
5.4	even	2		1875.4.a.g.1.8		14
25.4	even	10		75.4.g.b.16.4	✓	28
25.19	even	10		75.4.g.b.61.4	yes	28
75.29	odd	10		225.4.h.a.91.4		28
75.44	odd	10		225.4.h.a.136.4		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.16.4	✓	28	25.4	even	10
75.4.g.b.61.4	yes	28	25.19	even	10
225.4.h.a.91.4		28	75.29	odd	10
225.4.h.a.136.4		28	75.44	odd	10
1875.4.a.f.1.7		14	1.1	even	1	trivial
1875.4.a.g.1.8		14	5.4	even	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q+0.134967 q^{2} -3.00000 q^{3} -7.98178 q^{4} -0.404902 q^{6} +17.3099 q^{7} -2.15702 q^{8} +9.00000 q^{9} +42.1544 q^{11} +23.9454 q^{12} +84.5240 q^{13} +2.33627 q^{14} +63.5631 q^{16} +45.1024 q^{17} +1.21471 q^{18} +83.6634 q^{19} -51.9297 q^{21} +5.68947 q^{22} -20.9307 q^{23} +6.47105 q^{24} +11.4080 q^{26} -27.0000 q^{27} -138.164 q^{28} +220.065 q^{29} +160.220 q^{31} +25.8351 q^{32} -126.463 q^{33} +6.08735 q^{34} -71.8361 q^{36} +263.227 q^{37} +11.2918 q^{38} -253.572 q^{39} +410.142 q^{41} -7.00881 q^{42} -290.699 q^{43} -336.468 q^{44} -2.82495 q^{46} +460.997 q^{47} -190.689 q^{48} -43.3675 q^{49} -135.307 q^{51} -674.652 q^{52} +142.443 q^{53} -3.64412 q^{54} -37.3377 q^{56} -250.990 q^{57} +29.7016 q^{58} -472.475 q^{59} +54.0384 q^{61} +21.6244 q^{62} +155.789 q^{63} -505.018 q^{64} -17.0684 q^{66} -449.961 q^{67} -359.998 q^{68} +62.7920 q^{69} -626.058 q^{71} -19.4132 q^{72} -48.7415 q^{73} +35.5271 q^{74} -667.783 q^{76} +729.689 q^{77} -34.2239 q^{78} -647.571 q^{79} +81.0000 q^{81} +55.3557 q^{82} -669.851 q^{83} +414.492 q^{84} -39.2349 q^{86} -660.196 q^{87} -90.9278 q^{88} -866.036 q^{89} +1463.10 q^{91} +167.064 q^{92} -480.659 q^{93} +62.2195 q^{94} -77.5052 q^{96} +177.047 q^{97} -5.85319 q^{98} +379.390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 42 q^{3} + 50 q^{4} + 27 q^{7} - 21 q^{8} + 126 q^{9} - 33 q^{11} - 150 q^{12} + 188 q^{13} - 287 q^{14} + 82 q^{16} + 146 q^{17} - 184 q^{19} - 81 q^{21} + 277 q^{22} + 164 q^{23} + 63 q^{24} + 103 q^{26}+ \cdots - 297 q^{99}+O(q^{100}) \) 14 * q - 42 * q^3 + 50 * q^4 + 27 * q^7 - 21 * q^8 + 126 * q^9 - 33 * q^11 - 150 * q^12 + 188 * q^13 - 287 * q^14 + 82 * q^16 + 146 * q^17 - 184 * q^19 - 81 * q^21 + 277 * q^22 + 164 * q^23 + 63 * q^24 + 103 * q^26 - 378 * q^27 - 224 * q^28 + 252 * q^29 - 889 * q^31 + 422 * q^32 + 99 * q^33 - 230 * q^34 + 450 * q^36 + 642 * q^37 + 1833 * q^38 - 564 * q^39 - 164 * q^41 + 861 * q^42 + 696 * q^43 - 6 * q^44 - 419 * q^46 - 92 * q^47 - 246 * q^48 + 81 * q^49 - 438 * q^51 + 1956 * q^52 + 949 * q^53 - 2380 * q^56 + 552 * q^57 + 1041 * q^58 - 81 * q^59 - 496 * q^61 + 1454 * q^62 + 243 * q^63 - 3903 * q^64 - 831 * q^66 + 1926 * q^67 + 685 * q^68 - 492 * q^69 + 2498 * q^71 - 189 * q^72 - 1026 * q^73 - 707 * q^74 - 5704 * q^76 + 5434 * q^77 - 309 * q^78 - 695 * q^79 + 1134 * q^81 - 886 * q^82 + 5315 * q^83 + 672 * q^84 + 3997 * q^86 - 756 * q^87 + 2969 * q^88 - 1424 * q^89 - 1194 * q^91 + 1607 * q^92 + 2667 * q^93 - 629 * q^94 - 1266 * q^96 + 291 * q^97 - 1018 * q^98 - 297 * q^99

Introduction

L-functions

Modular forms

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