LMFDB - Embedded newform 1875.4.a.g.1.8

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	$110.628581261$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
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} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \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			1.8
Root			$-0.134967$ of defining polynomial
Character	$\chi$	$=$	1875.1

$q$-expansion

$f(q)$	$=$	$q-0.134967 q^{2} +3.00000 q^{3} -7.98178 q^{4} -0.404902 q^{6} -17.3099 q^{7} +2.15702 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.134967 q^{2} +3.00000 q^{3} -7.98178 q^{4} -0.404902 q^{6} -17.3099 q^{7} +2.15702 q^{8} +9.00000 q^{9} +42.1544 q^{11} -23.9454 q^{12} -84.5240 q^{13} +2.33627 q^{14} +63.5631 q^{16} -45.1024 q^{17} -1.21471 q^{18} +83.6634 q^{19} -51.9297 q^{21} -5.68947 q^{22} +20.9307 q^{23} +6.47105 q^{24} +11.4080 q^{26} +27.0000 q^{27} +138.164 q^{28} +220.065 q^{29} +160.220 q^{31} -25.8351 q^{32} +126.463 q^{33} +6.08735 q^{34} -71.8361 q^{36} -263.227 q^{37} -11.2918 q^{38} -253.572 q^{39} +410.142 q^{41} +7.00881 q^{42} +290.699 q^{43} -336.468 q^{44} -2.82495 q^{46} -460.997 q^{47} +190.689 q^{48} -43.3675 q^{49} -135.307 q^{51} +674.652 q^{52} -142.443 q^{53} -3.64412 q^{54} -37.3377 q^{56} +250.990 q^{57} -29.7016 q^{58} -472.475 q^{59} +54.0384 q^{61} -21.6244 q^{62} -155.789 q^{63} -505.018 q^{64} -17.0684 q^{66} +449.961 q^{67} +359.998 q^{68} +62.7920 q^{69} -626.058 q^{71} +19.4132 q^{72} +48.7415 q^{73} +35.5271 q^{74} -667.783 q^{76} -729.689 q^{77} +34.2239 q^{78} -647.571 q^{79} +81.0000 q^{81} -55.3557 q^{82} +669.851 q^{83} +414.492 q^{84} -39.2349 q^{86} +660.196 q^{87} +90.9278 q^{88} -866.036 q^{89} +1463.10 q^{91} -167.064 q^{92} +480.659 q^{93} +62.2195 q^{94} -77.5052 q^{96} -177.047 q^{97} +5.85319 q^{98} +379.390 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9}+O(q^{10}) $ 14 * q + 42 * q^3 + 50 * q^4 - 27 * q^7 + 21 * q^8 + 126 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

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	$-0.507595\pi$
$2$	−0.134967	−0.0477181	−0.0238591	+	0.999715i	$0.507595\pi$
$3$	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	$-0.654782\pi$
$7$	−17.3099	−0.934647	−0.467323	+	0.884087i	$0.654782\pi$
$8$	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	$-0.857639\pi$
$13$	−84.5240	−1.80329	−0.901643	+	0.432480i	$0.857639\pi$
$14$	2.33627	0.0445996
$15$	0	0
$16$	63.5631	0.993174
$17$	−45.1024	−0.643467	−0.321734	−	0.946830i	$-0.604266\pi$
$17$	−45.1024	−0.643467	−0.321734	+	0.946830i	$0.604266\pi$
$18$	−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	$-0.469754\pi$
$23$	20.9307	0.189754	0.0948771	+	0.995489i	$0.469754\pi$
$24$	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	$-0.698822\pi$
$37$	−263.227	−1.16958	−0.584788	+	0.811186i	$0.698822\pi$
$38$	−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	$-0.327614\pi$
$43$	290.699	1.03096	0.515479	+	0.856902i	$0.327614\pi$
$44$	−336.468	−1.15283
$45$	0	0
$46$	−2.82495	−0.00905471
$47$	−460.997	−1.43071	−0.715355	−	0.698761i	$-0.753735\pi$
$47$	−460.997	−1.43071	−0.715355	+	0.698761i	$0.753735\pi$
$48$	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	$-0.559094\pi$
$53$	−142.443	−0.369170	−0.184585	+	0.982817i	$0.559094\pi$
$54$	−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	$-0.365447\pi$
$67$	449.961	0.820470	0.410235	+	0.911980i	$0.365447\pi$
$68$	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	$-0.487559\pi$
$73$	48.7415	0.0781474	0.0390737	+	0.999236i	$0.487559\pi$
$74$	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	$-0.353940\pi$
$83$	669.851	0.885852	0.442926	+	0.896558i	$0.353940\pi$
$84$	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	$-0.529538\pi$
$97$	−177.047	−0.185324	−0.0926621	+	0.995698i	$0.529538\pi$
$98$	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	$-0.273687\pi$
$103$	1364.33	1.30516	0.652580	+	0.757720i	$0.273687\pi$
$104$	−182.320	−0.171903
$105$	0	0
$106$	19.2251	0.0176161
$107$	−1109.53	−1.00245	−0.501227	−	0.865316i	$-0.667118\pi$
$107$	−1109.53	−1.00245	−0.501227	+	0.865316i	$0.667118\pi$
$108$	−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	$-0.658062\pi$
$113$	−1144.53	−0.952816	−0.476408	+	0.879224i	$0.658062\pi$
$114$	−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	$-0.794054\pi$
$127$	−2283.93	−1.59579	−0.797897	+	0.602793i	$0.794054\pi$
$128$	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	$-0.493687\pi$
$137$	63.6065	0.0396662	0.0198331	+	0.999803i	$0.493687\pi$
$138$	−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	$-0.470320\pi$
$157$	366.317	0.186212	0.0931059	+	0.995656i	$0.470320\pi$
$158$	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	$-0.435064\pi$
$163$	843.200	0.405181	0.202591	+	0.979264i	$0.435064\pi$
$164$	−3273.66	−1.55872
$165$	0	0
$166$	−90.4079	−0.0422712
$167$	−2230.09	−1.03335	−0.516675	−	0.856182i	$-0.672830\pi$
$167$	−2230.09	−1.03335	−0.516675	+	0.856182i	$0.672830\pi$
$168$	−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	$-0.719336\pi$
$173$	−2893.55	−1.27163	−0.635816	+	0.771841i	$0.719336\pi$
$174$	−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	$-0.427926\pi$
$193$	1203.86	0.448993	0.224496	+	0.974475i	$0.427926\pi$
$194$	23.8956	0.00884332
$195$	0	0
$196$	346.150	0.126148
$197$	4195.96	1.51751	0.758756	−	0.651375i	$-0.225808\pi$
$197$	4195.96	1.51751	0.758756	+	0.651375i	$0.225808\pi$
$198$	−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	$-0.830646\pi$
$223$	−5739.58	−1.72355	−0.861773	+	0.507294i	$0.830646\pi$
$224$	447.203	0.133393
$225$	0	0
$226$	154.474	0.0454666
$227$	−2649.74	−0.774756	−0.387378	−	0.921921i	$-0.626619\pi$
$227$	−2649.74	−0.774756	−0.387378	+	0.921921i	$0.626619\pi$
$228$	−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	$-0.383225\pi$
$233$	2551.40	0.717373	0.358687	+	0.933458i	$0.383225\pi$
$234$	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	$-0.673658\pi$
$257$	−4275.75	−1.03780	−0.518899	+	0.854836i	$0.673658\pi$
$258$	−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	$-0.509996\pi$
$263$	−267.835	−0.0627964	−0.0313982	+	0.999507i	$0.509996\pi$
$264$	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	$-0.461065\pi$
$277$	1125.01	0.244025	0.122013	+	0.992529i	$0.461065\pi$
$278$	−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	$-0.313592\pi$
$283$	5262.72	1.10543	0.552715	+	0.833371i	$0.313592\pi$
$284$	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	$-0.382488\pi$
$293$	3619.55	0.721693	0.360847	+	0.932625i	$0.382488\pi$
$294$	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	$-0.554082\pi$
$307$	−1819.08	−0.338176	−0.169088	+	0.985601i	$0.554082\pi$
$308$	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	$-0.614955\pi$
$313$	−3913.30	−0.706686	−0.353343	+	0.935494i	$0.614955\pi$
$314$	−49.4408	−0.00888568
$315$	0	0
$316$	5168.77	0.920146
$317$	6577.26	1.16535	0.582675	−	0.812705i	$-0.302006\pi$
$317$	6577.26	1.16535	0.582675	+	0.812705i	$0.302006\pi$
$318$	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	$-0.546235\pi$
$337$	−1790.88	−0.289482	−0.144741	+	0.989470i	$0.546235\pi$
$338$	−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	$-0.747893\pi$
$347$	−9080.61	−1.40482	−0.702411	+	0.711772i	$0.747893\pi$
$348$	−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	$-0.419789\pi$
$353$	3307.28	0.498665	0.249332	+	0.968418i	$0.419789\pi$
$354$	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	$-0.614655\pi$
$367$	−4956.10	−0.704923	−0.352461	+	0.935826i	$0.614655\pi$
$368$	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	$-0.435408\pi$
$373$	2903.60	0.403063	0.201532	+	0.979482i	$0.435408\pi$
$374$	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	$-0.467292\pi$
$383$	1537.67	0.205146	0.102573	+	0.994725i	$0.467292\pi$
$384$	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	$-0.371224\pi$
$397$	6227.13	0.787231	0.393616	+	0.919275i	$0.371224\pi$
$398$	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	$-0.533728\pi$
$433$	−1905.84	−0.211522	−0.105761	+	0.994392i	$0.533728\pi$
$434$	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	$-0.450869\pi$
$443$	2866.93	0.307476	0.153738	+	0.988112i	$0.450869\pi$
$444$	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	$-0.587550\pi$
$457$	−5306.67	−0.543185	−0.271593	+	0.962412i	$0.587550\pi$
$458$	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	$-0.481418\pi$
$463$	1162.48	0.116685	0.0583424	+	0.998297i	$0.481418\pi$
$464$	13988.0	1.39952
$465$	0	0
$466$	−344.356	−0.0342317
$467$	7585.71	0.751659	0.375829	−	0.926689i	$-0.377358\pi$
$467$	7585.71	0.751659	0.375829	+	0.926689i	$0.377358\pi$
$468$	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	$-0.0552182\pi$
$487$	21171.7	1.96998	0.984991	+	0.172604i	$0.0552182\pi$
$488$	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	$-0.419995\pi$
$503$	5611.35	0.497411	0.248706	+	0.968579i	$0.419995\pi$
$504$	−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	$-0.470033\pi$
$523$	2248.74	0.188012	0.0940061	+	0.995572i	$0.470033\pi$
$524$	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	$-0.360365\pi$
$547$	10867.7	0.849484	0.424742	+	0.905314i	$0.360365\pi$
$548$	−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	$-0.533236\pi$
$557$	−2740.22	−0.208450	−0.104225	+	0.994554i	$0.533236\pi$
$558$	−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	$-0.755077\pi$
$563$	−19190.9	−1.43659	−0.718294	+	0.695740i	$0.755077\pi$
$564$	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	$-0.330628\pi$
$577$	14063.5	1.01468	0.507342	+	0.861745i	$0.330628\pi$
$578$	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	$-0.534278\pi$
$587$	−3057.11	−0.214958	−0.107479	+	0.994207i	$0.534278\pi$
$588$	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	$-0.652178\pi$
$593$	−13287.5	−0.920153	−0.460077	+	0.887879i	$0.652178\pi$
$594$	−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	$-0.474568\pi$
$607$	2387.17	0.159625	0.0798124	+	0.996810i	$0.474568\pi$
$608$	−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	$-0.601002\pi$
$613$	−9470.85	−0.624020	−0.312010	+	0.950079i	$0.601002\pi$
$614$	245.516	0.0161371
$615$	0	0
$616$	−1573.95	−0.102949
$617$	27465.6	1.79210	0.896049	−	0.443955i	$-0.146425\pi$
$617$	27465.6	1.79210	0.896049	+	0.443955i	$0.146425\pi$
$618$	−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	$-0.539165\pi$
$643$	−4002.17	−0.245459	−0.122730	+	0.992440i	$0.539165\pi$
$644$	2891.86	0.176949
$645$	0	0
$646$	509.288	0.0310181
$647$	−13315.6	−0.809104	−0.404552	−	0.914515i	$-0.632573\pi$
$647$	−13315.6	−0.809104	−0.404552	+	0.914515i	$0.632573\pi$
$648$	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	$-0.741955\pi$
$653$	−22994.6	−1.37802	−0.689011	+	0.724750i	$0.741955\pi$
$654$	−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	$-0.756844\pi$
$673$	−25216.1	−1.44429	−0.722146	+	0.691741i	$0.756844\pi$
$674$	241.710	0.0138135
$675$	0	0
$676$	−39488.3	−2.24672
$677$	529.842	0.0300790	0.0150395	−	0.999887i	$-0.495213\pi$
$677$	529.842	0.0300790	0.0150395	+	0.999887i	$0.495213\pi$
$678$	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	$-0.522398\pi$
$683$	−2509.91	−0.140614	−0.0703068	+	0.997525i	$0.522398\pi$
$684$	−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	$-0.354476\pi$
$727$	17305.3	0.882832	0.441416	+	0.897303i	$0.354476\pi$
$728$	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	$-0.463173\pi$
$733$	4581.82	0.230878	0.115439	+	0.993315i	$0.463173\pi$
$734$	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	$-0.397454\pi$
$743$	12824.6	0.633228	0.316614	+	0.948554i	$0.397454\pi$
$744$	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	$-0.316182\pi$
$757$	22740.4	1.09183	0.545915	+	0.837841i	$0.316182\pi$
$758$	−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	$-0.750471\pi$
$773$	−30438.6	−1.41630	−0.708151	+	0.706061i	$0.750471\pi$
$774$	−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	$-0.313760\pi$
$787$	24386.3	1.10455	0.552274	+	0.833663i	$0.313760\pi$
$788$	−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	$-0.132755\pi$
$797$	41143.2	1.82857	0.914283	+	0.405076i	$0.132755\pi$
$798$	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	$-0.571205\pi$
$823$	−10475.2	−0.443674	−0.221837	+	0.975084i	$0.571205\pi$
$824$	2942.88	0.124418
$825$	0	0
$826$	−1103.83	−0.0464977
$827$	26560.2	1.11679	0.558397	−	0.829574i	$-0.311417\pi$
$827$	26560.2	1.11679	0.558397	+	0.829574i	$0.311417\pi$
$828$	−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	$-0.289592\pi$
$853$	30588.9	1.22784	0.613918	+	0.789370i	$0.289592\pi$
$854$	126.248	0.00505870
$855$	0	0
$856$	−2393.28	−0.0955614
$857$	−39552.7	−1.57654	−0.788271	−	0.615329i	$-0.789023\pi$
$857$	−39552.7	−1.57654	−0.788271	+	0.615329i	$0.789023\pi$
$858$	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	$-0.460230\pi$
$863$	6318.56	0.249231	0.124616	+	0.992205i	$0.460230\pi$
$864$	−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	$-0.338850\pi$
$877$	25188.2	0.969835	0.484918	+	0.874560i	$0.338850\pi$
$878$	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	$-0.461977\pi$
$883$	6253.63	0.238337	0.119168	+	0.992874i	$0.461977\pi$
$884$	−30428.4	−1.15771
$885$	0	0
$886$	−386.941	−0.0146722
$887$	−20271.5	−0.767360	−0.383680	−	0.923466i	$-0.625344\pi$
$887$	−20271.5	−0.767360	−0.383680	+	0.923466i	$0.625344\pi$
$888$	−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	$-0.379687\pi$
$907$	20161.0	0.738077	0.369039	+	0.929414i	$0.379687\pi$
$908$	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	$-0.724549\pi$
$937$	−37193.1	−1.29674	−0.648371	+	0.761325i	$0.724549\pi$
$938$	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	$-0.448093\pi$
$947$	9462.47	0.324698	0.162349	+	0.986733i	$0.448093\pi$
$948$	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	$-0.563819\pi$
$953$	−11718.0	−0.398304	−0.199152	+	0.979969i	$0.563819\pi$
$954$	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	$-0.735842\pi$
$967$	−40593.1	−1.34993	−0.674967	+	0.737848i	$0.735842\pi$
$968$	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	$-0.769280\pi$
$977$	−45722.4	−1.49723	−0.748613	+	0.663008i	$0.769280\pi$
$978$	−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	$-0.432709\pi$
$983$	12933.8	0.419657	0.209828	+	0.977738i	$0.432709\pi$
$984$	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	$-0.401096\pi$
$997$	19249.8	0.611480	0.305740	+	0.952115i	$0.401096\pi$
$998$	−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.g.1.8		14
5.4	even	2		1875.4.a.f.1.7		14
25.6	even	5		75.4.g.b.61.4	yes	28
25.21	even	5		75.4.g.b.16.4	✓	28
75.56	odd	10		225.4.h.a.136.4		28
75.71	odd	10		225.4.h.a.91.4		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.16.4	✓	28	25.21	even	5
75.4.g.b.61.4	yes	28	25.6	even	5
225.4.h.a.91.4		28	75.71	odd	10
225.4.h.a.136.4		28	75.56	odd	10
1875.4.a.f.1.7		14	5.4	even	2
1875.4.a.g.1.8		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.134967 q^{2} +3.00000 q^{3} -7.98178 q^{4} -0.404902 q^{6} -17.3099 q^{7} +2.15702 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.134967 q^{2} +3.00000 q^{3} -7.98178 q^{4} -0.404902 q^{6} -17.3099 q^{7} +2.15702 q^{8} +9.00000 q^{9} +42.1544 q^{11} -23.9454 q^{12} -84.5240 q^{13} +2.33627 q^{14} +63.5631 q^{16} -45.1024 q^{17} -1.21471 q^{18} +83.6634 q^{19} -51.9297 q^{21} -5.68947 q^{22} +20.9307 q^{23} +6.47105 q^{24} +11.4080 q^{26} +27.0000 q^{27} +138.164 q^{28} +220.065 q^{29} +160.220 q^{31} -25.8351 q^{32} +126.463 q^{33} +6.08735 q^{34} -71.8361 q^{36} -263.227 q^{37} -11.2918 q^{38} -253.572 q^{39} +410.142 q^{41} +7.00881 q^{42} +290.699 q^{43} -336.468 q^{44} -2.82495 q^{46} -460.997 q^{47} +190.689 q^{48} -43.3675 q^{49} -135.307 q^{51} +674.652 q^{52} -142.443 q^{53} -3.64412 q^{54} -37.3377 q^{56} +250.990 q^{57} -29.7016 q^{58} -472.475 q^{59} +54.0384 q^{61} -21.6244 q^{62} -155.789 q^{63} -505.018 q^{64} -17.0684 q^{66} +449.961 q^{67} +359.998 q^{68} +62.7920 q^{69} -626.058 q^{71} +19.4132 q^{72} +48.7415 q^{73} +35.5271 q^{74} -667.783 q^{76} -729.689 q^{77} +34.2239 q^{78} -647.571 q^{79} +81.0000 q^{81} -55.3557 q^{82} +669.851 q^{83} +414.492 q^{84} -39.2349 q^{86} +660.196 q^{87} +90.9278 q^{88} -866.036 q^{89} +1463.10 q^{91} -167.064 q^{92} +480.659 q^{93} +62.2195 q^{94} -77.5052 q^{96} -177.047 q^{97} +5.85319 q^{98} +379.390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9}+O(q^{10}) \) 14 * q + 42 * q^3 + 50 * q^4 - 27 * q^7 + 21 * q^8 + 126 * q^9	\( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9} - 33 q^{11} + 150 q^{12} - 188 q^{13} - 287 q^{14} + 82 q^{16} - 146 q^{17} - 184 q^{19} - 81 q^{21} - 277 q^{22} - 164 q^{23} + 63 q^{24} + 103 q^{26} + 378 q^{27} + 224 q^{28} + 252 q^{29} - 889 q^{31} - 422 q^{32} - 99 q^{33} - 230 q^{34} + 450 q^{36} - 642 q^{37} - 1833 q^{38} - 564 q^{39} - 164 q^{41} - 861 q^{42} - 696 q^{43} - 6 q^{44} - 419 q^{46} + 92 q^{47} + 246 q^{48} + 81 q^{49} - 438 q^{51} - 1956 q^{52} - 949 q^{53} - 2380 q^{56} - 552 q^{57} - 1041 q^{58} - 81 q^{59} - 496 q^{61} - 1454 q^{62} - 243 q^{63} - 3903 q^{64} - 831 q^{66} - 1926 q^{67} - 685 q^{68} - 492 q^{69} + 2498 q^{71} + 189 q^{72} + 1026 q^{73} - 707 q^{74} - 5704 q^{76} - 5434 q^{77} + 309 q^{78} - 695 q^{79} + 1134 q^{81} + 886 q^{82} - 5315 q^{83} + 672 q^{84} + 3997 q^{86} + 756 q^{87} - 2969 q^{88} - 1424 q^{89} - 1194 q^{91} - 1607 q^{92} - 2667 q^{93} - 629 q^{94} - 1266 q^{96} - 291 q^{97} + 1018 q^{98} - 297 q^{99}+O(q^{100}) \) 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

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