LMFDB - Embedded newform 2175.2.a.z.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(1,2175)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.51908$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.51908 q^{2} +1.00000 q^{3} +4.34577 q^{4} +2.51908 q^{6} +0.173311 q^{7} +5.90919 q^{8} +1.00000 q^{9} +5.08250 q^{11} +4.34577 q^{12} +1.82669 q^{13} +0.436584 q^{14} +6.19418 q^{16} -4.24598 q^{17} +2.51908 q^{18} -8.62093 q^{19} +0.173311 q^{21} +12.8032 q^{22} +3.16348 q^{23} +5.90919 q^{24} +4.60158 q^{26} +1.00000 q^{27} +0.753170 q^{28} +1.00000 q^{29} +3.22185 q^{31} +3.78527 q^{32} +5.08250 q^{33} -10.6960 q^{34} +4.34577 q^{36} +1.97828 q^{37} -21.7168 q^{38} +1.82669 q^{39} -9.96541 q^{41} +0.436584 q^{42} +7.91070 q^{43} +22.0874 q^{44} +7.96907 q^{46} -8.66893 q^{47} +6.19418 q^{48} -6.96996 q^{49} -4.24598 q^{51} +7.93837 q^{52} +5.40285 q^{53} +2.51908 q^{54} +1.02413 q^{56} -8.62093 q^{57} +2.51908 q^{58} +7.66286 q^{59} +2.76215 q^{61} +8.11611 q^{62} +0.173311 q^{63} -2.85296 q^{64} +12.8032 q^{66} +8.13872 q^{67} -18.4520 q^{68} +3.16348 q^{69} -6.25938 q^{71} +5.90919 q^{72} -6.74356 q^{73} +4.98345 q^{74} -37.4646 q^{76} +0.880853 q^{77} +4.60158 q^{78} +4.54763 q^{79} +1.00000 q^{81} -25.1037 q^{82} -11.6224 q^{83} +0.753170 q^{84} +19.9277 q^{86} +1.00000 q^{87} +30.0334 q^{88} -16.4911 q^{89} +0.316585 q^{91} +13.7478 q^{92} +3.22185 q^{93} -21.8377 q^{94} +3.78527 q^{96} +2.74419 q^{97} -17.5579 q^{98} +5.08250 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 5 q^{3} + 5 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 5 q^{9} + 12 q^{11} + 5 q^{12} + 2 q^{13} + 6 q^{14} + q^{16} + 3 q^{18} - 2 q^{19} + 8 q^{21} + 14 q^{22} + 8 q^{23} + 9 q^{24} + 5 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) $ 5 * q + 3 * q^2 + 5 * q^3 + 5 * q^4 + 3 * q^6 + 8 * q^7 + 9 * q^8 + 5 * q^9 + 12 * q^11 + 5 * q^12 + 2 * q^13 + 6 * q^14 + q^16 + 3 * q^18 - 2 * q^19 + 8 * q^21 + 14 * q^22 + 8 * q^23 + 9 * q^24 + 5 * q^27 - 6 * q^28 + 5 * q^29 + 2 * q^31 + q^32 + 12 * q^33 + 4 * q^34 + 5 * q^36 + 16 * q^37 - 14 * q^38 + 2 * q^39 - 14 * q^41 + 6 * q^42 + 20 * q^44 - 6 * q^46 + 2 * q^47 + q^48 - 7 * q^49 + 16 * q^52 + 26 * q^53 + 3 * q^54 - 2 * q^56 - 2 * q^57 + 3 * q^58 + 4 * q^59 - 12 * q^61 + 8 * q^63 - 9 * q^64 + 14 * q^66 + 12 * q^67 - 20 * q^68 + 8 * q^69 + 30 * q^71 + 9 * q^72 - 12 * q^73 + 2 * q^74 - 44 * q^76 + 18 * q^77 - 18 * q^79 + 5 * q^81 + 10 * q^82 + 2 * q^83 - 6 * q^84 + 30 * q^86 + 5 * q^87 + 42 * q^88 - 22 * q^89 - 12 * q^91 + 20 * q^92 + 2 * q^93 - 50 * q^94 + q^96 + 20 * q^97 - 9 * q^98 + 12 * 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 $

$2$	2.51908	1.78126	0.890630	−	0.454729i	$-0.150264\pi$
$2$	2.51908	1.78126	0.890630	+	0.454729i	$0.150264\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.34577	2.17289
$5$	0	0
$5$	0	0
$6$	2.51908	1.02841
$7$	0.173311	0.0655054	0.0327527	−	0.999463i	$-0.489573\pi$
$7$	0.173311	0.0655054	0.0327527	+	0.999463i	$0.489573\pi$
$8$	5.90919	2.08921
$9$	1.00000	0.333333
$10$	0	0
$11$	5.08250	1.53243	0.766215	−	0.642584i	$-0.222138\pi$
$11$	5.08250	1.53243	0.766215	+	0.642584i	$0.222138\pi$
$12$	4.34577	1.25452
$13$	1.82669	0.506632	0.253316	−	0.967384i	$-0.418479\pi$
$13$	1.82669	0.506632	0.253316	+	0.967384i	$0.418479\pi$
$14$	0.436584	0.116682
$15$	0	0
$16$	6.19418	1.54854
$17$	−4.24598	−1.02980	−0.514901	−	0.857250i	$-0.672171\pi$
$17$	−4.24598	−1.02980	−0.514901	+	0.857250i	$0.672171\pi$
$18$	2.51908	0.593753
$19$	−8.62093	−1.97778	−0.988889	−	0.148656i	$-0.952505\pi$
$19$	−8.62093	−1.97778	−0.988889	+	0.148656i	$0.952505\pi$
$20$	0	0
$21$	0.173311	0.0378196
$22$	12.8032	2.72966
$23$	3.16348	0.659632	0.329816	−	0.944045i	$-0.393013\pi$
$23$	3.16348	0.659632	0.329816	+	0.944045i	$0.393013\pi$
$24$	5.90919	1.20621
$25$	0	0
$26$	4.60158	0.902444
$27$	1.00000	0.192450
$28$	0.753170	0.142336
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	3.22185	0.578662	0.289331	−	0.957229i	$-0.406567\pi$
$31$	3.22185	0.578662	0.289331	+	0.957229i	$0.406567\pi$
$32$	3.78527	0.669147
$33$	5.08250	0.884749
$34$	−10.6960	−1.83434
$35$	0	0
$36$	4.34577	0.724295
$37$	1.97828	0.325227	0.162614	−	0.986690i	$-0.448008\pi$
$37$	1.97828	0.325227	0.162614	+	0.986690i	$0.448008\pi$
$38$	−21.7168	−3.52294
$39$	1.82669	0.292504
$40$	0	0
$41$	−9.96541	−1.55634	−0.778168	−	0.628056i	$-0.783851\pi$
$41$	−9.96541	−1.55634	−0.778168	+	0.628056i	$0.783851\pi$
$42$	0.436584	0.0673664
$43$	7.91070	1.20637	0.603185	−	0.797601i	$-0.293898\pi$
$43$	7.91070	1.20637	0.603185	+	0.797601i	$0.293898\pi$
$44$	22.0874	3.32980
$45$	0	0
$46$	7.96907	1.17497
$47$	−8.66893	−1.26449	−0.632246	−	0.774767i	$-0.717867\pi$
$47$	−8.66893	−1.26449	−0.632246	+	0.774767i	$0.717867\pi$
$48$	6.19418	0.894053
$49$	−6.96996	−0.995709
$50$	0	0
$51$	−4.24598	−0.594556
$52$	7.93837	1.10085
$53$	5.40285	0.742138	0.371069	−	0.928605i	$-0.378991\pi$
$53$	5.40285	0.742138	0.371069	+	0.928605i	$0.378991\pi$
$54$	2.51908	0.342804
$55$	0	0
$56$	1.02413	0.136855
$57$	−8.62093	−1.14187
$58$	2.51908	0.330772
$59$	7.66286	0.997619	0.498810	−	0.866712i	$-0.333771\pi$
$59$	7.66286	0.997619	0.498810	+	0.866712i	$0.333771\pi$
$60$	0	0
$61$	2.76215	0.353657	0.176828	−	0.984242i	$-0.443416\pi$
$61$	2.76215	0.353657	0.176828	+	0.984242i	$0.443416\pi$
$62$	8.11611	1.03075
$63$	0.173311	0.0218351
$64$	−2.85296	−0.356620
$65$	0	0
$66$	12.8032	1.57597
$67$	8.13872	0.994303	0.497152	−	0.867664i	$-0.334379\pi$
$67$	8.13872	0.994303	0.497152	+	0.867664i	$0.334379\pi$
$68$	−18.4520	−2.23764
$69$	3.16348	0.380838
$70$	0	0
$71$	−6.25938	−0.742852	−0.371426	−	0.928463i	$-0.621131\pi$
$71$	−6.25938	−0.742852	−0.371426	+	0.928463i	$0.621131\pi$
$72$	5.90919	0.696404
$73$	−6.74356	−0.789274	−0.394637	−	0.918837i	$-0.629130\pi$
$73$	−6.74356	−0.789274	−0.394637	+	0.918837i	$0.629130\pi$
$74$	4.98345	0.579314
$75$	0	0
$76$	−37.4646	−4.29748
$77$	0.880853	0.100382
$78$	4.60158	0.521026
$79$	4.54763	0.511649	0.255824	−	0.966723i	$-0.417653\pi$
$79$	4.54763	0.511649	0.255824	+	0.966723i	$0.417653\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−25.1037	−2.77224
$83$	−11.6224	−1.27573	−0.637865	−	0.770149i	$-0.720182\pi$
$83$	−11.6224	−1.27573	−0.637865	+	0.770149i	$0.720182\pi$
$84$	0.753170	0.0821776
$85$	0	0
$86$	19.9277	2.14886
$87$	1.00000	0.107211
$88$	30.0334	3.20157
$89$	−16.4911	−1.74805	−0.874024	−	0.485882i	$-0.838499\pi$
$89$	−16.4911	−1.74805	−0.874024	+	0.485882i	$0.838499\pi$
$90$	0	0
$91$	0.316585	0.0331872
$92$	13.7478	1.43330
$93$	3.22185	0.334090
$94$	−21.8377	−2.25239
$95$	0	0
$96$	3.78527	0.386332
$97$	2.74419	0.278631	0.139315	−	0.990248i	$-0.455510\pi$
$97$	2.74419	0.278631	0.139315	+	0.990248i	$0.455510\pi$
$98$	−17.5579	−1.77362
$99$	5.08250	0.510810
$100$	0	0
$101$	6.41474	0.638290	0.319145	−	0.947706i	$-0.396604\pi$
$101$	6.41474	0.638290	0.319145	+	0.947706i	$0.396604\pi$
$102$	−10.6960	−1.05906
$103$	−7.08642	−0.698245	−0.349123	−	0.937077i	$-0.613520\pi$
$103$	−7.08642	−0.698245	−0.349123	+	0.937077i	$0.613520\pi$
$104$	10.7942	1.05846
$105$	0	0
$106$	13.6102	1.32194
$107$	0.925187	0.0894412	0.0447206	−	0.999000i	$-0.485760\pi$
$107$	0.925187	0.0894412	0.0447206	+	0.999000i	$0.485760\pi$
$108$	4.34577	0.418172
$109$	4.11700	0.394337	0.197169	−	0.980370i	$-0.436825\pi$
$109$	4.11700	0.394337	0.197169	+	0.980370i	$0.436825\pi$
$110$	0	0
$111$	1.97828	0.187770
$112$	1.07352	0.101438
$113$	6.84626	0.644042	0.322021	−	0.946732i	$-0.395638\pi$
$113$	6.84626	0.644042	0.322021	+	0.946732i	$0.395638\pi$
$114$	−21.7168	−2.03397
$115$	0	0
$116$	4.34577	0.403495
$117$	1.82669	0.168877
$118$	19.3034	1.77702
$119$	−0.735875	−0.0674575
$120$	0	0
$121$	14.8318	1.34834
$122$	6.95807	0.629954
$123$	−9.96541	−0.898551
$124$	14.0014	1.25737
$125$	0	0
$126$	0.436584	0.0388940
$127$	−5.25850	−0.466617	−0.233308	−	0.972403i	$-0.574955\pi$
$127$	−5.25850	−0.466617	−0.233308	+	0.972403i	$0.574955\pi$
$128$	−14.7574	−1.30438
$129$	7.91070	0.696498
$130$	0	0
$131$	17.7157	1.54783	0.773913	−	0.633293i	$-0.218297\pi$
$131$	17.7157	1.54783	0.773913	+	0.633293i	$0.218297\pi$
$132$	22.0874	1.92246
$133$	−1.49410	−0.129555
$134$	20.5021	1.77111
$135$	0	0
$136$	−25.0903	−2.15147
$137$	−1.14479	−0.0978057	−0.0489028	−	0.998804i	$-0.515572\pi$
$137$	−1.14479	−0.0978057	−0.0489028	+	0.998804i	$0.515572\pi$
$138$	7.96907	0.678372
$139$	4.08670	0.346630	0.173315	−	0.984866i	$-0.444552\pi$
$139$	4.08670	0.346630	0.173315	+	0.984866i	$0.444552\pi$
$140$	0	0
$141$	−8.66893	−0.730055
$142$	−15.7679	−1.32321
$143$	9.28414	0.776379
$144$	6.19418	0.516182
$145$	0	0
$146$	−16.9876	−1.40590
$147$	−6.96996	−0.574873
$148$	8.59715	0.706682
$149$	1.50268	0.123105	0.0615523	−	0.998104i	$-0.480395\pi$
$149$	1.50268	0.123105	0.0615523	+	0.998104i	$0.480395\pi$
$150$	0	0
$151$	−21.4897	−1.74881	−0.874404	−	0.485199i	$-0.838747\pi$
$151$	−21.4897	−1.74881	−0.874404	+	0.485199i	$0.838747\pi$
$152$	−50.9427	−4.13200
$153$	−4.24598	−0.343267
$154$	2.21894	0.178807
$155$	0	0
$156$	7.93837	0.635578
$157$	11.1167	0.887206	0.443603	−	0.896223i	$-0.353700\pi$
$157$	11.1167	0.887206	0.443603	+	0.896223i	$0.353700\pi$
$158$	11.4559	0.911379
$159$	5.40285	0.428474
$160$	0	0
$161$	0.548266	0.0432094
$162$	2.51908	0.197918
$163$	−12.2180	−0.956988	−0.478494	−	0.878091i	$-0.658817\pi$
$163$	−12.2180	−0.956988	−0.478494	+	0.878091i	$0.658817\pi$
$164$	−43.3074	−3.38174
$165$	0	0
$166$	−29.2779	−2.27240
$167$	21.3634	1.65315	0.826576	−	0.562826i	$-0.190286\pi$
$167$	21.3634	1.65315	0.826576	+	0.562826i	$0.190286\pi$
$168$	1.02413	0.0790131
$169$	−9.66321	−0.743324
$170$	0	0
$171$	−8.62093	−0.659259
$172$	34.3781	2.62130
$173$	13.1686	1.00119	0.500594	−	0.865682i	$-0.333115\pi$
$173$	13.1686	1.00119	0.500594	+	0.865682i	$0.333115\pi$
$174$	2.51908	0.190971
$175$	0	0
$176$	31.4819	2.37304
$177$	7.66286	0.575976
$178$	−41.5423	−3.11373
$179$	10.6947	0.799357	0.399679	−	0.916655i	$-0.369122\pi$
$179$	10.6947	0.799357	0.399679	+	0.916655i	$0.369122\pi$
$180$	0	0
$181$	−15.2511	−1.13361	−0.566804	−	0.823852i	$-0.691820\pi$
$181$	−15.2511	−1.13361	−0.566804	+	0.823852i	$0.691820\pi$
$182$	0.797504	0.0591149
$183$	2.76215	0.204184
$184$	18.6936	1.37811
$185$	0	0
$186$	8.11611	0.595102
$187$	−21.5802	−1.57810
$188$	−37.6732	−2.74760
$189$	0.173311	0.0126065
$190$	0	0
$191$	−25.7131	−1.86053	−0.930266	−	0.366885i	$-0.880424\pi$
$191$	−25.7131	−1.86053	−0.930266	+	0.366885i	$0.880424\pi$
$192$	−2.85296	−0.205895
$193$	−22.8935	−1.64791	−0.823953	−	0.566658i	$-0.808236\pi$
$193$	−22.8935	−1.64791	−0.823953	+	0.566658i	$0.808236\pi$
$194$	6.91284	0.496313
$195$	0	0
$196$	−30.2899	−2.16356
$197$	0.840810	0.0599052	0.0299526	−	0.999551i	$-0.490464\pi$
$197$	0.840810	0.0599052	0.0299526	+	0.999551i	$0.490464\pi$
$198$	12.8032	0.909885
$199$	−26.5996	−1.88560	−0.942799	−	0.333361i	$-0.891817\pi$
$199$	−26.5996	−1.88560	−0.942799	+	0.333361i	$0.891817\pi$
$200$	0	0
$201$	8.13872	0.574061
$202$	16.1592	1.13696
$203$	0.173311	0.0121640
$204$	−18.4520	−1.29190
$205$	0	0
$206$	−17.8513	−1.24376
$207$	3.16348	0.219877
$208$	11.3148	0.784543
$209$	−43.8159	−3.03081
$210$	0	0
$211$	1.86360	0.128296	0.0641478	−	0.997940i	$-0.479567\pi$
$211$	1.86360	0.128296	0.0641478	+	0.997940i	$0.479567\pi$
$212$	23.4795	1.61258
$213$	−6.25938	−0.428886
$214$	2.33062	0.159318
$215$	0	0
$216$	5.90919	0.402069
$217$	0.558382	0.0379055
$218$	10.3711	0.702417
$219$	−6.74356	−0.455687
$220$	0	0
$221$	−7.75608	−0.521731
$222$	4.98345	0.334467
$223$	25.0759	1.67920	0.839602	−	0.543202i	$-0.182788\pi$
$223$	25.0759	1.67920	0.839602	+	0.543202i	$0.182788\pi$
$224$	0.656028	0.0438327
$225$	0	0
$226$	17.2463	1.14721
$227$	20.3877	1.35318	0.676590	−	0.736360i	$-0.263457\pi$
$227$	20.3877	1.35318	0.676590	+	0.736360i	$0.263457\pi$
$228$	−37.4646	−2.48115
$229$	−5.84179	−0.386037	−0.193018	−	0.981195i	$-0.561828\pi$
$229$	−5.84179	−0.386037	−0.193018	+	0.981195i	$0.561828\pi$
$230$	0	0
$231$	0.880853	0.0579558
$232$	5.90919	0.387957
$233$	14.1216	0.925134	0.462567	−	0.886584i	$-0.346928\pi$
$233$	14.1216	0.925134	0.462567	+	0.886584i	$0.346928\pi$
$234$	4.60158	0.300815
$235$	0	0
$236$	33.3010	2.16771
$237$	4.54763	0.295400
$238$	−1.85373	−0.120159
$239$	−6.39059	−0.413373	−0.206686	−	0.978407i	$-0.566268\pi$
$239$	−6.39059	−0.413373	−0.206686	+	0.978407i	$0.566268\pi$
$240$	0	0
$241$	−10.1713	−0.655188	−0.327594	−	0.944819i	$-0.606238\pi$
$241$	−10.1713	−0.655188	−0.327594	+	0.944819i	$0.606238\pi$
$242$	37.3624	2.40175
$243$	1.00000	0.0641500
$244$	12.0037	0.768455
$245$	0	0
$246$	−25.1037	−1.60055
$247$	−15.7478	−1.00201
$248$	19.0385	1.20895
$249$	−11.6224	−0.736543
$250$	0	0
$251$	−15.7587	−0.994678	−0.497339	−	0.867556i	$-0.665690\pi$
$251$	−15.7587	−0.994678	−0.497339	+	0.867556i	$0.665690\pi$
$252$	0.753170	0.0474452
$253$	16.0784	1.01084
$254$	−13.2466	−0.831165
$255$	0	0
$256$	−31.4691	−1.96682
$257$	12.3486	0.770283	0.385142	−	0.922858i	$-0.374153\pi$
$257$	12.3486	0.770283	0.385142	+	0.922858i	$0.374153\pi$
$258$	19.9277	1.24064
$259$	0.342858	0.0213041
$260$	0	0
$261$	1.00000	0.0618984
$262$	44.6272	2.75708
$263$	−4.16928	−0.257089	−0.128545	−	0.991704i	$-0.541031\pi$
$263$	−4.16928	−0.257089	−0.128545	+	0.991704i	$0.541031\pi$
$264$	30.0334	1.84843
$265$	0	0
$266$	−3.76377	−0.230771
$267$	−16.4911	−1.00924
$268$	35.3690	2.16051
$269$	−1.36011	−0.0829273	−0.0414636	−	0.999140i	$-0.513202\pi$
$269$	−1.36011	−0.0829273	−0.0414636	+	0.999140i	$0.513202\pi$
$270$	0	0
$271$	11.1448	0.676998	0.338499	−	0.940967i	$-0.390081\pi$
$271$	11.1448	0.676998	0.338499	+	0.940967i	$0.390081\pi$
$272$	−26.3003	−1.59469
$273$	0.316585	0.0191606
$274$	−2.88381	−0.174217
$275$	0	0
$276$	13.7478	0.827518
$277$	0.704673	0.0423397	0.0211699	−	0.999776i	$-0.493261\pi$
$277$	0.704673	0.0423397	0.0211699	+	0.999776i	$0.493261\pi$
$278$	10.2947	0.617437
$279$	3.22185	0.192887
$280$	0	0
$281$	−13.8119	−0.823950	−0.411975	−	0.911195i	$-0.635161\pi$
$281$	−13.8119	−0.823950	−0.411975	+	0.911195i	$0.635161\pi$
$282$	−21.8377	−1.30042
$283$	8.92142	0.530324	0.265162	−	0.964204i	$-0.414575\pi$
$283$	8.92142	0.530324	0.265162	+	0.964204i	$0.414575\pi$
$284$	−27.2018	−1.61413
$285$	0	0
$286$	23.3875	1.38293
$287$	−1.72712	−0.101948
$288$	3.78527	0.223049
$289$	1.02833	0.0604902
$290$	0	0
$291$	2.74419	0.160867
$292$	−29.3060	−1.71500
$293$	−31.2817	−1.82750	−0.913749	−	0.406278i	$-0.866826\pi$
$293$	−31.2817	−1.82750	−0.913749	+	0.406278i	$0.866826\pi$
$294$	−17.5579	−1.02400
$295$	0	0
$296$	11.6900	0.679469
$297$	5.08250	0.294916
$298$	3.78538	0.219281
$299$	5.77870	0.334191
$300$	0	0
$301$	1.37101	0.0790238
$302$	−54.1343	−3.11508
$303$	6.41474	0.368517
$304$	−53.3996	−3.06268
$305$	0	0
$306$	−10.6960	−0.611448
$307$	−21.8533	−1.24723	−0.623617	−	0.781730i	$-0.714338\pi$
$307$	−21.8533	−1.24723	−0.623617	+	0.781730i	$0.714338\pi$
$308$	3.82798	0.218120
$309$	−7.08642	−0.403132
$310$	0	0
$311$	28.8838	1.63785	0.818925	−	0.573901i	$-0.194570\pi$
$311$	28.8838	1.63785	0.818925	+	0.573901i	$0.194570\pi$
$312$	10.7942	0.611104
$313$	8.24232	0.465884	0.232942	−	0.972491i	$-0.425165\pi$
$313$	8.24232	0.465884	0.232942	+	0.972491i	$0.425165\pi$
$314$	28.0038	1.58034
$315$	0	0
$316$	19.7630	1.11175
$317$	−7.15829	−0.402050	−0.201025	−	0.979586i	$-0.564427\pi$
$317$	−7.15829	−0.402050	−0.201025	+	0.979586i	$0.564427\pi$
$318$	13.6102	0.763223
$319$	5.08250	0.284565
$320$	0	0
$321$	0.925187	0.0516389
$322$	1.38113	0.0769672
$323$	36.6043	2.03672
$324$	4.34577	0.241432
$325$	0	0
$326$	−30.7781	−1.70464
$327$	4.11700	0.227671
$328$	−58.8875	−3.25152
$329$	−1.50242	−0.0828311
$330$	0	0
$331$	−24.6936	−1.35728	−0.678642	−	0.734470i	$-0.737431\pi$
$331$	−24.6936	−1.35728	−0.678642	+	0.734470i	$0.737431\pi$
$332$	−50.5085	−2.77201
$333$	1.97828	0.108409
$334$	53.8162	2.94469
$335$	0	0
$336$	1.07352	0.0585653
$337$	11.7142	0.638116	0.319058	−	0.947735i	$-0.396634\pi$
$337$	11.7142	0.638116	0.319058	+	0.947735i	$0.396634\pi$
$338$	−24.3424	−1.32405
$339$	6.84626	0.371838
$340$	0	0
$341$	16.3751	0.886759
$342$	−21.7168	−1.17431
$343$	−2.42115	−0.130730
$344$	46.7458	2.52036
$345$	0	0
$346$	33.1727	1.78337
$347$	0.828990	0.0445025	0.0222513	−	0.999752i	$-0.492917\pi$
$347$	0.828990	0.0445025	0.0222513	+	0.999752i	$0.492917\pi$
$348$	4.34577	0.232958
$349$	−8.44370	−0.451981	−0.225991	−	0.974129i	$-0.572562\pi$
$349$	−8.44370	−0.451981	−0.225991	+	0.974129i	$0.572562\pi$
$350$	0	0
$351$	1.82669	0.0975014
$352$	19.2386	1.02542
$353$	−4.90553	−0.261095	−0.130547	−	0.991442i	$-0.541673\pi$
$353$	−4.90553	−0.261095	−0.130547	+	0.991442i	$0.541673\pi$
$354$	19.3034	1.02596
$355$	0	0
$356$	−71.6664	−3.79831
$357$	−0.735875	−0.0389466
$358$	26.9407	1.42386
$359$	4.69044	0.247552	0.123776	−	0.992310i	$-0.460500\pi$
$359$	4.69044	0.247552	0.123776	+	0.992310i	$0.460500\pi$
$360$	0	0
$361$	55.3205	2.91161
$362$	−38.4189	−2.01925
$363$	14.8318	0.778466
$364$	1.37581	0.0721119
$365$	0	0
$366$	6.95807	0.363704
$367$	29.2460	1.52663	0.763314	−	0.646028i	$-0.223571\pi$
$367$	29.2460	1.52663	0.763314	+	0.646028i	$0.223571\pi$
$368$	19.5952	1.02147
$369$	−9.96541	−0.518779
$370$	0	0
$371$	0.936373	0.0486140
$372$	14.0014	0.725940
$373$	34.5171	1.78723	0.893613	−	0.448839i	$-0.148162\pi$
$373$	34.5171	1.78723	0.893613	+	0.448839i	$0.148162\pi$
$374$	−54.3622	−2.81100
$375$	0	0
$376$	−51.2263	−2.64179
$377$	1.82669	0.0940793
$378$	0.436584	0.0224555
$379$	−3.20131	−0.164440	−0.0822201	−	0.996614i	$-0.526201\pi$
$379$	−3.20131	−0.164440	−0.0822201	+	0.996614i	$0.526201\pi$
$380$	0	0
$381$	−5.25850	−0.269401
$382$	−64.7733	−3.31409
$383$	19.4424	0.993458	0.496729	−	0.867906i	$-0.334534\pi$
$383$	19.4424	0.993458	0.496729	+	0.867906i	$0.334534\pi$
$384$	−14.7574	−0.753084
$385$	0	0
$386$	−57.6705	−2.93535
$387$	7.91070	0.402123
$388$	11.9256	0.605432
$389$	2.55875	0.129734	0.0648669	−	0.997894i	$-0.479338\pi$
$389$	2.55875	0.129734	0.0648669	+	0.997894i	$0.479338\pi$
$390$	0	0
$391$	−13.4321	−0.679289
$392$	−41.1868	−2.08025
$393$	17.7157	0.893637
$394$	2.11807	0.106707
$395$	0	0
$396$	22.0874	1.10993
$397$	30.6583	1.53869	0.769347	−	0.638831i	$-0.220582\pi$
$397$	30.6583	1.53869	0.769347	+	0.638831i	$0.220582\pi$
$398$	−67.0067	−3.35874
$399$	−1.49410	−0.0747987
$400$	0	0
$401$	11.1628	0.557446	0.278723	−	0.960372i	$-0.410089\pi$
$401$	11.1628	0.557446	0.278723	+	0.960372i	$0.410089\pi$
$402$	20.5021	1.02255
$403$	5.88532	0.293169
$404$	27.8770	1.38693
$405$	0	0
$406$	0.436584	0.0216673
$407$	10.0546	0.498388
$408$	−25.0903	−1.24215
$409$	13.3880	0.661994	0.330997	−	0.943632i	$-0.392615\pi$
$409$	13.3880	0.661994	0.330997	+	0.943632i	$0.392615\pi$
$410$	0	0
$411$	−1.14479	−0.0564681
$412$	−30.7959	−1.51721
$413$	1.32806	0.0653495
$414$	7.96907	0.391658
$415$	0	0
$416$	6.91451	0.339012
$417$	4.08670	0.200127
$418$	−110.376	−5.39865
$419$	29.2982	1.43131	0.715655	−	0.698454i	$-0.246128\pi$
$419$	29.2982	1.43131	0.715655	+	0.698454i	$0.246128\pi$
$420$	0	0
$421$	10.1098	0.492724	0.246362	−	0.969178i	$-0.420765\pi$
$421$	10.1098	0.492724	0.246362	+	0.969178i	$0.420765\pi$
$422$	4.69456	0.228528
$423$	−8.66893	−0.421498
$424$	31.9264	1.55048
$425$	0	0
$426$	−15.7679	−0.763957
$427$	0.478710	0.0231664
$428$	4.02065	0.194345
$429$	9.28414	0.448243
$430$	0	0
$431$	30.3330	1.46109	0.730545	−	0.682865i	$-0.239266\pi$
$431$	30.3330	1.46109	0.730545	+	0.682865i	$0.239266\pi$
$432$	6.19418	0.298018
$433$	−6.50520	−0.312620	−0.156310	−	0.987708i	$-0.549960\pi$
$433$	−6.50520	−0.312620	−0.156310	+	0.987708i	$0.549960\pi$
$434$	1.40661	0.0675195
$435$	0	0
$436$	17.8915	0.856850
$437$	−27.2722	−1.30460
$438$	−16.9876	−0.811698
$439$	−9.40248	−0.448756	−0.224378	−	0.974502i	$-0.572035\pi$
$439$	−9.40248	−0.448756	−0.224378	+	0.974502i	$0.572035\pi$
$440$	0	0
$441$	−6.96996	−0.331903
$442$	−19.5382	−0.929337
$443$	28.1657	1.33819	0.669097	−	0.743175i	$-0.266681\pi$
$443$	28.1657	1.33819	0.669097	+	0.743175i	$0.266681\pi$
$444$	8.59715	0.408003
$445$	0	0
$446$	63.1682	2.99110
$447$	1.50268	0.0710744
$448$	−0.494450	−0.0233605
$449$	−32.2625	−1.52256	−0.761281	−	0.648422i	$-0.775429\pi$
$449$	−32.2625	−1.52256	−0.761281	+	0.648422i	$0.775429\pi$
$450$	0	0
$451$	−50.6492	−2.38498
$452$	29.7523	1.39943
$453$	−21.4897	−1.00967
$454$	51.3583	2.41037
$455$	0	0
$456$	−50.9427	−2.38561
$457$	27.1466	1.26986	0.634932	−	0.772568i	$-0.281028\pi$
$457$	27.1466	1.26986	0.634932	+	0.772568i	$0.281028\pi$
$458$	−14.7160	−0.687631
$459$	−4.24598	−0.198185
$460$	0	0
$461$	7.00940	0.326460	0.163230	−	0.986588i	$-0.447809\pi$
$461$	7.00940	0.326460	0.163230	+	0.986588i	$0.447809\pi$
$462$	2.21894	0.103234
$463$	−21.7590	−1.01123	−0.505613	−	0.862760i	$-0.668734\pi$
$463$	−21.7590	−1.01123	−0.505613	+	0.862760i	$0.668734\pi$
$464$	6.19418	0.287558
$465$	0	0
$466$	35.5733	1.64790
$467$	30.7825	1.42445	0.712223	−	0.701953i	$-0.247688\pi$
$467$	30.7825	1.42445	0.712223	+	0.701953i	$0.247688\pi$
$468$	7.93837	0.366951
$469$	1.41053	0.0651322
$470$	0	0
$471$	11.1167	0.512228
$472$	45.2813	2.08424
$473$	40.2061	1.84868
$474$	11.4559	0.526185
$475$	0	0
$476$	−3.19794	−0.146577
$477$	5.40285	0.247379
$478$	−16.0984	−0.736324
$479$	−6.29237	−0.287506	−0.143753	−	0.989614i	$-0.545917\pi$
$479$	−6.29237	−0.287506	−0.143753	+	0.989614i	$0.545917\pi$
$480$	0	0
$481$	3.61370	0.164771
$482$	−25.6222	−1.16706
$483$	0.548266	0.0249470
$484$	64.4555	2.92979
$485$	0	0
$486$	2.51908	0.114268
$487$	−30.3406	−1.37487	−0.687433	−	0.726248i	$-0.741262\pi$
$487$	−30.3406	−1.37487	−0.687433	+	0.726248i	$0.741262\pi$
$488$	16.3220	0.738864
$489$	−12.2180	−0.552517
$490$	0	0
$491$	17.7822	0.802502	0.401251	−	0.915968i	$-0.368576\pi$
$491$	17.7822	0.802502	0.401251	+	0.915968i	$0.368576\pi$
$492$	−43.3074	−1.95245
$493$	−4.24598	−0.191229
$494$	−39.6699	−1.78483
$495$	0	0
$496$	19.9567	0.896083
$497$	−1.08482	−0.0486608
$498$	−29.2779	−1.31197
$499$	0.262248	0.0117399	0.00586993	−	0.999983i	$-0.498132\pi$
$499$	0.262248	0.0117399	0.00586993	+	0.999983i	$0.498132\pi$
$500$	0	0
$501$	21.3634	0.954447
$502$	−39.6974	−1.77178
$503$	−15.0486	−0.670986	−0.335493	−	0.942043i	$-0.608903\pi$
$503$	−15.0486	−0.670986	−0.335493	+	0.942043i	$0.608903\pi$
$504$	1.02413	0.0456182
$505$	0	0
$506$	40.5028	1.80057
$507$	−9.66321	−0.429158
$508$	−22.8522	−1.01390
$509$	39.8752	1.76744	0.883719	−	0.468019i	$-0.155032\pi$
$509$	39.8752	1.76744	0.883719	+	0.468019i	$0.155032\pi$
$510$	0	0
$511$	−1.16873	−0.0517017
$512$	−49.7585	−2.19904
$513$	−8.62093	−0.380624
$514$	31.1071	1.37207
$515$	0	0
$516$	34.3781	1.51341
$517$	−44.0598	−1.93775
$518$	0.863687	0.0379482
$519$	13.1686	0.578036
$520$	0	0
$521$	18.3187	0.802557	0.401279	−	0.915956i	$-0.368566\pi$
$521$	18.3187	0.802557	0.401279	+	0.915956i	$0.368566\pi$
$522$	2.51908	0.110257
$523$	−6.04576	−0.264363	−0.132181	−	0.991226i	$-0.542198\pi$
$523$	−6.04576	−0.264363	−0.132181	+	0.991226i	$0.542198\pi$
$524$	76.9882	3.36325
$525$	0	0
$526$	−10.5028	−0.457942
$527$	−13.6799	−0.595906
$528$	31.4819	1.37007
$529$	−12.9924	−0.564886
$530$	0	0
$531$	7.66286	0.332540
$532$	−6.49303	−0.281508
$533$	−18.2037	−0.788490
$534$	−41.5423	−1.79771
$535$	0	0
$536$	48.0932	2.07731
$537$	10.6947	0.461509
$538$	−3.42622	−0.147715
$539$	−35.4248	−1.52585
$540$	0	0
$541$	−35.6643	−1.53333	−0.766664	−	0.642049i	$-0.778085\pi$
$541$	−35.6643	−1.53333	−0.766664	+	0.642049i	$0.778085\pi$
$542$	28.0746	1.20591
$543$	−15.2511	−0.654489
$544$	−16.0722	−0.689088
$545$	0	0
$546$	0.797504	0.0341300
$547$	−5.23087	−0.223656	−0.111828	−	0.993728i	$-0.535671\pi$
$547$	−5.23087	−0.223656	−0.111828	+	0.993728i	$0.535671\pi$
$548$	−4.97498	−0.212520
$549$	2.76215	0.117886
$550$	0	0
$551$	−8.62093	−0.367264
$552$	18.6936	0.795653
$553$	0.788155	0.0335157
$554$	1.77513	0.0754180
$555$	0	0
$556$	17.7599	0.753186
$557$	32.9893	1.39780	0.698900	−	0.715219i	$-0.253673\pi$
$557$	32.9893	1.39780	0.698900	+	0.715219i	$0.253673\pi$
$558$	8.11611	0.343582
$559$	14.4504	0.611186
$560$	0	0
$561$	−21.5802	−0.911116
$562$	−34.7933	−1.46767
$563$	5.94931	0.250734	0.125367	−	0.992110i	$-0.459989\pi$
$563$	5.94931	0.250734	0.125367	+	0.992110i	$0.459989\pi$
$564$	−37.6732	−1.58633
$565$	0	0
$566$	22.4738	0.944644
$567$	0.173311	0.00727838
$568$	−36.9878	−1.55198
$569$	−14.6940	−0.616003	−0.308002	−	0.951386i	$-0.599660\pi$
$569$	−14.6940	−0.616003	−0.308002	+	0.951386i	$0.599660\pi$
$570$	0	0
$571$	7.86509	0.329144	0.164572	−	0.986365i	$-0.447376\pi$
$571$	7.86509	0.329144	0.164572	+	0.986365i	$0.447376\pi$
$572$	40.3467	1.68698
$573$	−25.7131	−1.07418
$574$	−4.35074	−0.181597
$575$	0	0
$576$	−2.85296	−0.118873
$577$	29.7454	1.23832	0.619159	−	0.785265i	$-0.287474\pi$
$577$	29.7454	1.23832	0.619159	+	0.785265i	$0.287474\pi$
$578$	2.59045	0.107749
$579$	−22.8935	−0.951419
$580$	0	0
$581$	−2.01430	−0.0835671
$582$	6.91284	0.286547
$583$	27.4599	1.13727
$584$	−39.8489	−1.64896
$585$	0	0
$586$	−78.8013	−3.25525
$587$	−46.1639	−1.90539	−0.952695	−	0.303929i	$-0.901701\pi$
$587$	−46.1639	−1.90539	−0.952695	+	0.303929i	$0.901701\pi$
$588$	−30.2899	−1.24913
$589$	−27.7754	−1.14446
$590$	0	0
$591$	0.840810	0.0345863
$592$	12.2538	0.503629
$593$	−22.6806	−0.931382	−0.465691	−	0.884948i	$-0.654194\pi$
$593$	−22.6806	−0.931382	−0.465691	+	0.884948i	$0.654194\pi$
$594$	12.8032	0.525323
$595$	0	0
$596$	6.53031	0.267492
$597$	−26.5996	−1.08865
$598$	14.5570	0.595280
$599$	22.8764	0.934704	0.467352	−	0.884071i	$-0.345208\pi$
$599$	22.8764	0.934704	0.467352	+	0.884071i	$0.345208\pi$
$600$	0	0
$601$	14.6918	0.599291	0.299646	−	0.954051i	$-0.403132\pi$
$601$	14.6918	0.599291	0.299646	+	0.954051i	$0.403132\pi$
$602$	3.45369	0.140762
$603$	8.13872	0.331434
$604$	−93.3893	−3.79996
$605$	0	0
$606$	16.1592	0.656424
$607$	−20.9484	−0.850270	−0.425135	−	0.905130i	$-0.639773\pi$
$607$	−20.9484	−0.850270	−0.425135	+	0.905130i	$0.639773\pi$
$608$	−32.6325	−1.32342
$609$	0.173311	0.00702292
$610$	0	0
$611$	−15.8354	−0.640633
$612$	−18.4520	−0.745880
$613$	−42.0373	−1.69787	−0.848936	−	0.528496i	$-0.822756\pi$
$613$	−42.0373	−1.69787	−0.848936	+	0.528496i	$0.822756\pi$
$614$	−55.0503	−2.22165
$615$	0	0
$616$	5.20512	0.209720
$617$	44.6853	1.79896	0.899482	−	0.436958i	$-0.143944\pi$
$617$	44.6853	1.79896	0.899482	+	0.436958i	$0.143944\pi$
$618$	−17.8513	−0.718083
$619$	−22.1966	−0.892157	−0.446079	−	0.894994i	$-0.647180\pi$
$619$	−22.1966	−0.892157	−0.446079	+	0.894994i	$0.647180\pi$
$620$	0	0
$621$	3.16348	0.126946
$622$	72.7606	2.91744
$623$	−2.85808	−0.114507
$624$	11.3148	0.452956
$625$	0	0
$626$	20.7631	0.829859
$627$	−43.8159	−1.74984
$628$	48.3104	1.92780
$629$	−8.39974	−0.334919
$630$	0	0
$631$	−5.29765	−0.210896	−0.105448	−	0.994425i	$-0.533628\pi$
$631$	−5.29765	−0.210896	−0.105448	+	0.994425i	$0.533628\pi$
$632$	26.8728	1.06894
$633$	1.86360	0.0740715
$634$	−18.0323	−0.716155
$635$	0	0
$636$	23.4795	0.931024
$637$	−12.7320	−0.504458
$638$	12.8032	0.506884
$639$	−6.25938	−0.247617
$640$	0	0
$641$	3.21592	0.127021	0.0635107	−	0.997981i	$-0.479770\pi$
$641$	3.21592	0.127021	0.0635107	+	0.997981i	$0.479770\pi$
$642$	2.33062	0.0919823
$643$	32.0630	1.26444	0.632221	−	0.774788i	$-0.282143\pi$
$643$	32.0630	1.26444	0.632221	+	0.774788i	$0.282143\pi$
$644$	2.38264	0.0938891
$645$	0	0
$646$	92.2092	3.62792
$647$	9.26571	0.364273	0.182136	−	0.983273i	$-0.441699\pi$
$647$	9.26571	0.364273	0.182136	+	0.983273i	$0.441699\pi$
$648$	5.90919	0.232135
$649$	38.9465	1.52878
$650$	0	0
$651$	0.558382	0.0218847
$652$	−53.0966	−2.07942
$653$	22.2362	0.870171	0.435085	−	0.900389i	$-0.356718\pi$
$653$	22.2362	0.870171	0.435085	+	0.900389i	$0.356718\pi$
$654$	10.3711	0.405541
$655$	0	0
$656$	−61.7275	−2.41006
$657$	−6.74356	−0.263091
$658$	−3.78472	−0.147544
$659$	34.6708	1.35058	0.675291	−	0.737552i	$-0.264018\pi$
$659$	34.6708	1.35058	0.675291	+	0.737552i	$0.264018\pi$
$660$	0	0
$661$	−9.67217	−0.376204	−0.188102	−	0.982150i	$-0.560234\pi$
$661$	−9.67217	−0.376204	−0.188102	+	0.982150i	$0.560234\pi$
$662$	−62.2052	−2.41767
$663$	−7.75608	−0.301221
$664$	−68.6792	−2.66527
$665$	0	0
$666$	4.98345	0.193105
$667$	3.16348	0.122491
$668$	92.8405	3.59211
$669$	25.0759	0.969489
$670$	0	0
$671$	14.0386	0.541954
$672$	0.656028	0.0253068
$673$	24.7689	0.954770	0.477385	−	0.878694i	$-0.341585\pi$
$673$	24.7689	0.954770	0.477385	+	0.878694i	$0.341585\pi$
$674$	29.5091	1.13665
$675$	0	0
$676$	−41.9941	−1.61516
$677$	33.3636	1.28227	0.641133	−	0.767430i	$-0.278465\pi$
$677$	33.3636	1.28227	0.641133	+	0.767430i	$0.278465\pi$
$678$	17.2463	0.662340
$679$	0.475599	0.0182518
$680$	0	0
$681$	20.3877	0.781259
$682$	41.2501	1.57955
$683$	−13.9516	−0.533842	−0.266921	−	0.963718i	$-0.586006\pi$
$683$	−13.9516	−0.533842	−0.266921	+	0.963718i	$0.586006\pi$
$684$	−37.4646	−1.43249
$685$	0	0
$686$	−6.09907	−0.232864
$687$	−5.84179	−0.222878
$688$	49.0003	1.86812
$689$	9.86932	0.375991
$690$	0	0
$691$	50.5611	1.92344	0.961718	−	0.274042i	$-0.0883607\pi$
$691$	50.5611	1.92344	0.961718	+	0.274042i	$0.0883607\pi$
$692$	57.2276	2.17547
$693$	0.880853	0.0334608
$694$	2.08829	0.0792705
$695$	0	0
$696$	5.90919	0.223987
$697$	42.3129	1.60272
$698$	−21.2704	−0.805096
$699$	14.1216	0.534126
$700$	0	0
$701$	−24.0803	−0.909499	−0.454749	−	0.890620i	$-0.650271\pi$
$701$	−24.0803	−0.909499	−0.454749	+	0.890620i	$0.650271\pi$
$702$	4.60158	0.173675
$703$	−17.0546	−0.643227
$704$	−14.5002	−0.546496
$705$	0	0
$706$	−12.3574	−0.465078
$707$	1.11174	0.0418114
$708$	33.3010	1.25153
$709$	0.769569	0.0289018	0.0144509	−	0.999896i	$-0.495400\pi$
$709$	0.769569	0.0289018	0.0144509	+	0.999896i	$0.495400\pi$
$710$	0	0
$711$	4.54763	0.170550
$712$	−97.4487	−3.65205
$713$	10.1923	0.381703
$714$	−1.85373	−0.0693740
$715$	0	0
$716$	46.4766	1.73691
$717$	−6.39059	−0.238661
$718$	11.8156	0.440955
$719$	4.92456	0.183655	0.0918275	−	0.995775i	$-0.470729\pi$
$719$	4.92456	0.183655	0.0918275	+	0.995775i	$0.470729\pi$
$720$	0	0
$721$	−1.22815	−0.0457388
$722$	139.357	5.18632
$723$	−10.1713	−0.378273
$724$	−66.2780	−2.46320
$725$	0	0
$726$	37.3624	1.38665
$727$	−41.8838	−1.55339	−0.776693	−	0.629880i	$-0.783104\pi$
$727$	−41.8838	−1.55339	−0.776693	+	0.629880i	$0.783104\pi$
$728$	1.87076	0.0693350
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.5887	−1.24232
$732$	12.0037	0.443668
$733$	27.1372	1.00234	0.501168	−	0.865350i	$-0.332904\pi$
$733$	27.1372	1.00234	0.501168	+	0.865350i	$0.332904\pi$
$734$	73.6730	2.71932
$735$	0	0
$736$	11.9746	0.441390
$737$	41.3650	1.52370
$738$	−25.1037	−0.924079
$739$	−3.41445	−0.125603	−0.0628013	−	0.998026i	$-0.520003\pi$
$739$	−3.41445	−0.125603	−0.0628013	+	0.998026i	$0.520003\pi$
$740$	0	0
$741$	−15.7478	−0.578509
$742$	2.35880	0.0865942
$743$	−3.73961	−0.137193	−0.0685966	−	0.997644i	$-0.521852\pi$
$743$	−3.73961	−0.137193	−0.0685966	+	0.997644i	$0.521852\pi$
$744$	19.0385	0.697986
$745$	0	0
$746$	86.9513	3.18351
$747$	−11.6224	−0.425243
$748$	−93.7825	−3.42903
$749$	0.160345	0.00585888
$750$	0	0
$751$	−24.1900	−0.882707	−0.441354	−	0.897333i	$-0.645502\pi$
$751$	−24.1900	−0.882707	−0.441354	+	0.897333i	$0.645502\pi$
$752$	−53.6969	−1.95812
$753$	−15.7587	−0.574278
$754$	4.60158	0.167580
$755$	0	0
$756$	0.753170	0.0273925
$757$	3.55451	0.129191	0.0645953	−	0.997912i	$-0.479424\pi$
$757$	3.55451	0.129191	0.0645953	+	0.997912i	$0.479424\pi$
$758$	−8.06436	−0.292911
$759$	16.0784	0.583608
$760$	0	0
$761$	−44.2390	−1.60366	−0.801832	−	0.597550i	$-0.796141\pi$
$761$	−44.2390	−1.60366	−0.801832	+	0.597550i	$0.796141\pi$
$762$	−13.2466	−0.479873
$763$	0.713522	0.0258312
$764$	−111.743	−4.04272
$765$	0	0
$766$	48.9769	1.76961
$767$	13.9977	0.505426
$768$	−31.4691	−1.13554
$769$	6.30558	0.227385	0.113692	−	0.993516i	$-0.463732\pi$
$769$	6.30558	0.227385	0.113692	+	0.993516i	$0.463732\pi$
$770$	0	0
$771$	12.3486	0.444723
$772$	−99.4897	−3.58071
$773$	24.2649	0.872747	0.436374	−	0.899766i	$-0.356263\pi$
$773$	24.2649	0.872747	0.436374	+	0.899766i	$0.356263\pi$
$774$	19.9277	0.716286
$775$	0	0
$776$	16.2159	0.582118
$777$	0.342858	0.0123000
$778$	6.44570	0.231090
$779$	85.9111	3.07809
$780$	0	0
$781$	−31.8133	−1.13837
$782$	−33.8365	−1.20999
$783$	1.00000	0.0357371
$784$	−43.1732	−1.54190
$785$	0	0
$786$	44.6272	1.59180
$787$	−39.1338	−1.39497	−0.697484	−	0.716600i	$-0.745697\pi$
$787$	−39.1338	−1.39497	−0.697484	+	0.716600i	$0.745697\pi$
$788$	3.65397	0.130167
$789$	−4.16928	−0.148430
$790$	0	0
$791$	1.18653	0.0421882
$792$	30.0334	1.06719
$793$	5.04558	0.179174
$794$	77.2307	2.74081
$795$	0	0
$796$	−115.596	−4.09719
$797$	29.3166	1.03845	0.519224	−	0.854638i	$-0.326221\pi$
$797$	29.3166	1.03845	0.519224	+	0.854638i	$0.326221\pi$
$798$	−3.76377	−0.133236
$799$	36.8081	1.30218
$800$	0	0
$801$	−16.4911	−0.582683
$802$	28.1201	0.992956
$803$	−34.2741	−1.20951
$804$	35.3690	1.24737
$805$	0	0
$806$	14.8256	0.522210
$807$	−1.36011	−0.0478781
$808$	37.9059	1.33352
$809$	−28.7412	−1.01049	−0.505244	−	0.862977i	$-0.668597\pi$
$809$	−28.7412	−1.01049	−0.505244	+	0.862977i	$0.668597\pi$
$810$	0	0
$811$	4.04326	0.141978	0.0709891	−	0.997477i	$-0.477384\pi$
$811$	4.04326	0.141978	0.0709891	+	0.997477i	$0.477384\pi$
$812$	0.753170	0.0264311
$813$	11.1448	0.390865
$814$	25.3284	0.887759
$815$	0	0
$816$	−26.3003	−0.920696
$817$	−68.1976	−2.38593
$818$	33.7255	1.17918
$819$	0.316585	0.0110624
$820$	0	0
$821$	1.38325	0.0482758	0.0241379	−	0.999709i	$-0.492316\pi$
$821$	1.38325	0.0482758	0.0241379	+	0.999709i	$0.492316\pi$
$822$	−2.88381	−0.100584
$823$	−11.7734	−0.410395	−0.205197	−	0.978721i	$-0.565784\pi$
$823$	−11.7734	−0.410395	−0.205197	+	0.978721i	$0.565784\pi$
$824$	−41.8749	−1.45878
$825$	0	0
$826$	3.34549	0.116404
$827$	19.4700	0.677040	0.338520	−	0.940959i	$-0.390074\pi$
$827$	19.4700	0.677040	0.338520	+	0.940959i	$0.390074\pi$
$828$	13.7478	0.477768
$829$	16.9194	0.587634	0.293817	−	0.955862i	$-0.405074\pi$
$829$	16.9194	0.587634	0.293817	+	0.955862i	$0.405074\pi$
$830$	0	0
$831$	0.704673	0.0244448
$832$	−5.21147	−0.180675
$833$	29.5943	1.02538
$834$	10.2947	0.356478
$835$	0	0
$836$	−190.414	−6.58560
$837$	3.22185	0.111363
$838$	73.8045	2.54953
$839$	17.9764	0.620615	0.310308	−	0.950636i	$-0.399568\pi$
$839$	17.9764	0.620615	0.310308	+	0.950636i	$0.399568\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	25.4675	0.877669
$843$	−13.8119	−0.475708
$844$	8.09878	0.278772
$845$	0	0
$846$	−21.8377	−0.750797
$847$	2.57051	0.0883237
$848$	33.4662	1.14923
$849$	8.92142	0.306182
$850$	0	0
$851$	6.25826	0.214530
$852$	−27.2018	−0.931919
$853$	−25.1608	−0.861489	−0.430744	−	0.902474i	$-0.641749\pi$
$853$	−25.1608	−0.861489	−0.430744	+	0.902474i	$0.641749\pi$
$854$	1.20591	0.0412654
$855$	0	0
$856$	5.46710	0.186862
$857$	27.8748	0.952184	0.476092	−	0.879395i	$-0.342053\pi$
$857$	27.8748	0.952184	0.476092	+	0.879395i	$0.342053\pi$
$858$	23.3875	0.798436
$859$	21.0421	0.717947	0.358973	−	0.933348i	$-0.383127\pi$
$859$	21.0421	0.717947	0.358973	+	0.933348i	$0.383127\pi$
$860$	0	0
$861$	−1.72712	−0.0588599
$862$	76.4113	2.60258
$863$	−29.1647	−0.992779	−0.496390	−	0.868100i	$-0.665341\pi$
$863$	−29.1647	−0.992779	−0.496390	+	0.868100i	$0.665341\pi$
$864$	3.78527	0.128777
$865$	0	0
$866$	−16.3871	−0.556858
$867$	1.02833	0.0349240
$868$	2.42660	0.0823642
$869$	23.1133	0.784066
$870$	0	0
$871$	14.8669	0.503746
$872$	24.3281	0.823854
$873$	2.74419	0.0928768
$874$	−68.7008	−2.32384
$875$	0	0
$876$	−29.3060	−0.990156
$877$	11.5743	0.390836	0.195418	−	0.980720i	$-0.437394\pi$
$877$	11.5743	0.390836	0.195418	+	0.980720i	$0.437394\pi$
$878$	−23.6856	−0.799350
$879$	−31.2817	−1.05511
$880$	0	0
$881$	21.8843	0.737299	0.368650	−	0.929568i	$-0.379820\pi$
$881$	21.8843	0.737299	0.368650	+	0.929568i	$0.379820\pi$
$882$	−17.5579	−0.591205
$883$	−20.2591	−0.681773	−0.340887	−	0.940104i	$-0.610727\pi$
$883$	−20.2591	−0.681773	−0.340887	+	0.940104i	$0.610727\pi$
$884$	−33.7062	−1.13366
$885$	0	0
$886$	70.9517	2.38367
$887$	−37.5330	−1.26023	−0.630117	−	0.776500i	$-0.716993\pi$
$887$	−37.5330	−1.26023	−0.630117	+	0.776500i	$0.716993\pi$
$888$	11.6900	0.392292
$889$	−0.911356	−0.0305659
$890$	0	0
$891$	5.08250	0.170270
$892$	108.974	3.64872
$893$	74.7342	2.50089
$894$	3.78538	0.126602
$895$	0	0
$896$	−2.55762	−0.0854439
$897$	5.77870	0.192945
$898$	−81.2719	−2.71208
$899$	3.22185	0.107455
$900$	0	0
$901$	−22.9404	−0.764254
$902$	−127.589	−4.24826
$903$	1.37101	0.0456244
$904$	40.4558	1.34554
$905$	0	0
$906$	−54.1343	−1.79849
$907$	17.7096	0.588037	0.294018	−	0.955800i	$-0.405007\pi$
$907$	17.7096	0.588037	0.294018	+	0.955800i	$0.405007\pi$
$908$	88.6004	2.94031
$909$	6.41474	0.212763
$910$	0	0
$911$	−6.63407	−0.219796	−0.109898	−	0.993943i	$-0.535052\pi$
$911$	−6.63407	−0.219796	−0.109898	+	0.993943i	$0.535052\pi$
$912$	−53.3996	−1.76824
$913$	−59.0710	−1.95497
$914$	68.3845	2.26196
$915$	0	0
$916$	−25.3871	−0.838813
$917$	3.07032	0.101391
$918$	−10.6960	−0.353019
$919$	4.71782	0.155626	0.0778132	−	0.996968i	$-0.475206\pi$
$919$	4.71782	0.155626	0.0778132	+	0.996968i	$0.475206\pi$
$920$	0	0
$921$	−21.8533	−0.720091
$922$	17.6572	0.581510
$923$	−11.4339	−0.376353
$924$	3.82798	0.125931
$925$	0	0
$926$	−54.8127	−1.80126
$927$	−7.08642	−0.232748
$928$	3.78527	0.124257
$929$	43.4668	1.42610	0.713049	−	0.701114i	$-0.247314\pi$
$929$	43.4668	1.42610	0.713049	+	0.701114i	$0.247314\pi$
$930$	0	0
$931$	60.0876	1.96929
$932$	61.3690	2.01021
$933$	28.8838	0.945613
$934$	77.5437	2.53731
$935$	0	0
$936$	10.7942	0.352821
$937$	50.9680	1.66505	0.832526	−	0.553986i	$-0.186894\pi$
$937$	50.9680	1.66505	0.832526	+	0.553986i	$0.186894\pi$
$938$	3.55324	0.116017
$939$	8.24232	0.268978
$940$	0	0
$941$	11.0790	0.361165	0.180583	−	0.983560i	$-0.442202\pi$
$941$	11.0790	0.361165	0.180583	+	0.983560i	$0.442202\pi$
$942$	28.0038	0.912412
$943$	−31.5254	−1.02661
$944$	47.4651	1.54486
$945$	0	0
$946$	101.282	3.29298
$947$	58.5923	1.90399	0.951997	−	0.306107i	$-0.0990264\pi$
$947$	58.5923	1.90399	0.951997	+	0.306107i	$0.0990264\pi$
$948$	19.7630	0.641871
$949$	−12.3184	−0.399872
$950$	0	0
$951$	−7.15829	−0.232124
$952$	−4.34842	−0.140933
$953$	22.9017	0.741859	0.370929	−	0.928661i	$-0.379039\pi$
$953$	22.9017	0.741859	0.370929	+	0.928661i	$0.379039\pi$
$954$	13.6102	0.440647
$955$	0	0
$956$	−27.7720	−0.898211
$957$	5.08250	0.164294
$958$	−15.8510	−0.512123
$959$	−0.198404	−0.00640680
$960$	0	0
$961$	−20.6197	−0.665151
$962$	9.10321	0.293499
$963$	0.925187	0.0298137
$964$	−44.2019	−1.42365
$965$	0	0
$966$	1.38113	0.0444370
$967$	−49.6623	−1.59703	−0.798516	−	0.601973i	$-0.794381\pi$
$967$	−49.6623	−1.59703	−0.798516	+	0.601973i	$0.794381\pi$
$968$	87.6437	2.81698
$969$	36.6043	1.17590
$970$	0	0
$971$	−6.90899	−0.221720	−0.110860	−	0.993836i	$-0.535361\pi$
$971$	−6.90899	−0.221720	−0.110860	+	0.993836i	$0.535361\pi$
$972$	4.34577	0.139391
$973$	0.708271	0.0227061
$974$	−76.4305	−2.44899
$975$	0	0
$976$	17.1092	0.547653
$977$	−8.17310	−0.261481	−0.130740	−	0.991417i	$-0.541735\pi$
$977$	−8.17310	−0.261481	−0.130740	+	0.991417i	$0.541735\pi$
$978$	−30.7781	−0.984176
$979$	−83.8158	−2.67876
$980$	0	0
$981$	4.11700	0.131446
$982$	44.7949	1.42946
$983$	−24.1451	−0.770108	−0.385054	−	0.922894i	$-0.625817\pi$
$983$	−24.1451	−0.770108	−0.385054	+	0.922894i	$0.625817\pi$
$984$	−58.8875	−1.87726
$985$	0	0
$986$	−10.6960	−0.340629
$987$	−1.50242	−0.0478226
$988$	−68.4362	−2.17724
$989$	25.0253	0.795760
$990$	0	0
$991$	−51.7396	−1.64356	−0.821781	−	0.569803i	$-0.807020\pi$
$991$	−51.7396	−1.64356	−0.821781	+	0.569803i	$0.807020\pi$
$992$	12.1956	0.387210
$993$	−24.6936	−0.783628
$994$	−2.73275	−0.0866775
$995$	0	0
$996$	−50.5085	−1.60042
$997$	−9.07880	−0.287528	−0.143764	−	0.989612i	$-0.545921\pi$
$997$	−9.07880	−0.287528	−0.143764	+	0.989612i	$0.545921\pi$
$998$	0.660625	0.0209117
$999$	1.97828	0.0625900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.z.1.5		5
3.2	odd	2		6525.2.a.bl.1.1		5
5.2	odd	4		435.2.c.e.349.10	yes	10
5.3	odd	4		435.2.c.e.349.1	✓	10
5.4	even	2		2175.2.a.w.1.1		5
15.2	even	4		1305.2.c.j.784.1		10
15.8	even	4		1305.2.c.j.784.10		10
15.14	odd	2		6525.2.a.bs.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.c.e.349.1	✓	10	5.3	odd	4
435.2.c.e.349.10	yes	10	5.2	odd	4
1305.2.c.j.784.1		10	15.2	even	4
1305.2.c.j.784.10		10	15.8	even	4
2175.2.a.w.1.1		5	5.4	even	2
2175.2.a.z.1.5		5	1.1	even	1	trivial
6525.2.a.bl.1.1		5	3.2	odd	2
6525.2.a.bs.1.5		5	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q+2.51908 q^{2} +1.00000 q^{3} +4.34577 q^{4} +2.51908 q^{6} +0.173311 q^{7} +5.90919 q^{8} +1.00000 q^{9} +5.08250 q^{11} +4.34577 q^{12} +1.82669 q^{13} +0.436584 q^{14} +6.19418 q^{16} -4.24598 q^{17} +2.51908 q^{18} -8.62093 q^{19} +0.173311 q^{21} +12.8032 q^{22} +3.16348 q^{23} +5.90919 q^{24} +4.60158 q^{26} +1.00000 q^{27} +0.753170 q^{28} +1.00000 q^{29} +3.22185 q^{31} +3.78527 q^{32} +5.08250 q^{33} -10.6960 q^{34} +4.34577 q^{36} +1.97828 q^{37} -21.7168 q^{38} +1.82669 q^{39} -9.96541 q^{41} +0.436584 q^{42} +7.91070 q^{43} +22.0874 q^{44} +7.96907 q^{46} -8.66893 q^{47} +6.19418 q^{48} -6.96996 q^{49} -4.24598 q^{51} +7.93837 q^{52} +5.40285 q^{53} +2.51908 q^{54} +1.02413 q^{56} -8.62093 q^{57} +2.51908 q^{58} +7.66286 q^{59} +2.76215 q^{61} +8.11611 q^{62} +0.173311 q^{63} -2.85296 q^{64} +12.8032 q^{66} +8.13872 q^{67} -18.4520 q^{68} +3.16348 q^{69} -6.25938 q^{71} +5.90919 q^{72} -6.74356 q^{73} +4.98345 q^{74} -37.4646 q^{76} +0.880853 q^{77} +4.60158 q^{78} +4.54763 q^{79} +1.00000 q^{81} -25.1037 q^{82} -11.6224 q^{83} +0.753170 q^{84} +19.9277 q^{86} +1.00000 q^{87} +30.0334 q^{88} -16.4911 q^{89} +0.316585 q^{91} +13.7478 q^{92} +3.22185 q^{93} -21.8377 q^{94} +3.78527 q^{96} +2.74419 q^{97} -17.5579 q^{98} +5.08250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 5 q^{3} + 5 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 5 q^{9} + 12 q^{11} + 5 q^{12} + 2 q^{13} + 6 q^{14} + q^{16} + 3 q^{18} - 2 q^{19} + 8 q^{21} + 14 q^{22} + 8 q^{23} + 9 q^{24} + 5 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) \) 5 * q + 3 * q^2 + 5 * q^3 + 5 * q^4 + 3 * q^6 + 8 * q^7 + 9 * q^8 + 5 * q^9 + 12 * q^11 + 5 * q^12 + 2 * q^13 + 6 * q^14 + q^16 + 3 * q^18 - 2 * q^19 + 8 * q^21 + 14 * q^22 + 8 * q^23 + 9 * q^24 + 5 * q^27 - 6 * q^28 + 5 * q^29 + 2 * q^31 + q^32 + 12 * q^33 + 4 * q^34 + 5 * q^36 + 16 * q^37 - 14 * q^38 + 2 * q^39 - 14 * q^41 + 6 * q^42 + 20 * q^44 - 6 * q^46 + 2 * q^47 + q^48 - 7 * q^49 + 16 * q^52 + 26 * q^53 + 3 * q^54 - 2 * q^56 - 2 * q^57 + 3 * q^58 + 4 * q^59 - 12 * q^61 + 8 * q^63 - 9 * q^64 + 14 * q^66 + 12 * q^67 - 20 * q^68 + 8 * q^69 + 30 * q^71 + 9 * q^72 - 12 * q^73 + 2 * q^74 - 44 * q^76 + 18 * q^77 - 18 * q^79 + 5 * q^81 + 10 * q^82 + 2 * q^83 - 6 * q^84 + 30 * q^86 + 5 * q^87 + 42 * q^88 - 22 * q^89 - 12 * q^91 + 20 * q^92 + 2 * q^93 - 50 * q^94 + q^96 + 20 * q^97 - 9 * q^98 + 12 * q^99

Introduction

L-functions

Modular forms

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