LMFDB - Embedded newform 2175.4.a.n.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,4,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, 4, names="a")

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

chi := DirichletCharacter("2175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
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:	$128.329154262$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 37x^{5} + 55x^{4} + 336x^{3} - 227x^{2} - 824x - 166 $ x^7 - 2x^6 - 37x^5 + 55x^4 + 336x^3 - 227x^2 - 824x - 166
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$5.24052$ 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)

$f(q)$	$=$	\(q+5.24052 q^{2} +3.00000 q^{3} +19.4631 q^{4} +15.7216 q^{6} +18.4475 q^{7} +60.0724 q^{8} +9.00000 q^{9} +11.5384 q^{11} +58.3892 q^{12} +34.4618 q^{13} +96.6746 q^{14} +159.106 q^{16} -70.7681 q^{17} +47.1647 q^{18} +3.52127 q^{19} +55.3426 q^{21} +60.4671 q^{22} +18.3117 q^{23} +180.217 q^{24} +180.598 q^{26} +27.0000 q^{27} +359.045 q^{28} +29.0000 q^{29} -120.558 q^{31} +353.221 q^{32} +34.6151 q^{33} -370.862 q^{34} +175.168 q^{36} +182.505 q^{37} +18.4533 q^{38} +103.385 q^{39} +74.4691 q^{41} +290.024 q^{42} +405.899 q^{43} +224.572 q^{44} +95.9626 q^{46} +327.143 q^{47} +477.319 q^{48} -2.68899 q^{49} -212.304 q^{51} +670.732 q^{52} -409.067 q^{53} +141.494 q^{54} +1108.19 q^{56} +10.5638 q^{57} +151.975 q^{58} +120.722 q^{59} +34.6082 q^{61} -631.788 q^{62} +166.028 q^{63} +578.210 q^{64} +181.401 q^{66} -671.739 q^{67} -1377.36 q^{68} +54.9350 q^{69} +873.524 q^{71} +540.652 q^{72} -432.426 q^{73} +956.419 q^{74} +68.5347 q^{76} +212.854 q^{77} +541.793 q^{78} -548.355 q^{79} +81.0000 q^{81} +390.257 q^{82} -511.077 q^{83} +1077.14 q^{84} +2127.12 q^{86} +87.0000 q^{87} +693.138 q^{88} -788.135 q^{89} +635.734 q^{91} +356.401 q^{92} -361.675 q^{93} +1714.40 q^{94} +1059.66 q^{96} -967.070 q^{97} -14.0917 q^{98} +103.845 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 21 q^{3} + 22 q^{4} + 6 q^{6} + 50 q^{7} + 33 q^{8} + 63 q^{9} + 76 q^{11} + 66 q^{12} - 30 q^{13} + 89 q^{14} + 138 q^{16} + 140 q^{17} + 18 q^{18} + 90 q^{19} + 150 q^{21} - 61 q^{22}+ \cdots + 684 q^{99}+O(q^{100}) $ 7 * q + 2 * q^2 + 21 * q^3 + 22 * q^4 + 6 * q^6 + 50 * q^7 + 33 * q^8 + 63 * q^9 + 76 * q^11 + 66 * q^12 - 30 * q^13 + 89 * q^14 + 138 * q^16 + 140 * q^17 + 18 * q^18 + 90 * q^19 + 150 * q^21 - 61 * q^22 - 34 * q^23 + 99 * q^24 - 241 * q^26 + 189 * q^27 + 57 * q^28 + 203 * q^29 + 524 * q^31 + 6 * q^32 + 228 * q^33 + 255 * q^34 + 198 * q^36 + 28 * q^37 - 222 * q^38 - 90 * q^39 + 1532 * q^41 + 267 * q^42 + 464 * q^43 + 1475 * q^44 + 72 * q^46 + 360 * q^47 + 414 * q^48 + 569 * q^49 + 420 * q^51 + 205 * q^52 - 282 * q^53 + 54 * q^54 + 1102 * q^56 + 270 * q^57 + 58 * q^58 + 766 * q^59 + 1200 * q^61 - 2856 * q^62 + 450 * q^63 + 701 * q^64 - 183 * q^66 - 1546 * q^67 - 1801 * q^68 - 102 * q^69 + 1802 * q^71 + 297 * q^72 + 220 * q^73 + 1594 * q^74 + 1960 * q^76 - 3222 * q^77 - 723 * q^78 + 1298 * q^79 + 567 * q^81 - 856 * q^82 - 1652 * q^83 + 171 * q^84 + 7628 * q^86 + 609 * q^87 - 550 * q^88 + 2846 * q^89 - 816 * q^91 - 472 * q^92 + 1572 * q^93 + 745 * q^94 + 18 * q^96 - 1110 * q^97 + 761 * q^98 + 684 * 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 $

$2$	5.24052	1.85280	0.926402	−	0.376536i	$-0.122885\pi$
$2$	5.24052	1.85280	0.926402	+	0.376536i	$0.122885\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	19.4631	2.43288
$5$	0	0
$5$	0	0
$6$	15.7216	1.06972
$7$	18.4475	0.996072	0.498036	−	0.867156i	$-0.334055\pi$
$7$	18.4475	0.996072	0.498036	+	0.867156i	$0.334055\pi$
$8$	60.0724	2.65485
$9$	9.00000	0.333333
$10$	0	0
$11$	11.5384	0.316268	0.158134	−	0.987418i	$-0.449452\pi$
$11$	11.5384	0.316268	0.158134	+	0.987418i	$0.449452\pi$
$12$	58.3892	1.40463
$13$	34.4618	0.735229	0.367614	−	0.929978i	$-0.380175\pi$
$13$	34.4618	0.735229	0.367614	+	0.929978i	$0.380175\pi$
$14$	96.6746	1.84553
$15$	0	0
$16$	159.106	2.48604
$17$	−70.7681	−1.00963	−0.504817	−	0.863226i	$-0.668440\pi$
$17$	−70.7681	−1.00963	−0.504817	+	0.863226i	$0.668440\pi$
$18$	47.1647	0.617601
$19$	3.52127	0.0425176	0.0212588	−	0.999774i	$-0.493233\pi$
$19$	3.52127	0.0425176	0.0212588	+	0.999774i	$0.493233\pi$
$20$	0	0
$21$	55.3426	0.575083
$22$	60.4671	0.585983
$23$	18.3117	0.166011	0.0830053	−	0.996549i	$-0.473548\pi$
$23$	18.3117	0.166011	0.0830053	+	0.996549i	$0.473548\pi$
$24$	180.217	1.53278
$25$	0	0
$26$	180.598	1.36224
$27$	27.0000	0.192450
$28$	359.045	2.42333
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−120.558	−0.698481	−0.349240	−	0.937033i	$-0.613560\pi$
$31$	−120.558	−0.698481	−0.349240	+	0.937033i	$0.613560\pi$
$32$	353.221	1.95129
$33$	34.6151	0.182598
$34$	−370.862	−1.87065
$35$	0	0
$36$	175.168	0.810961
$37$	182.505	0.810907	0.405454	−	0.914116i	$-0.367114\pi$
$37$	182.505	0.810907	0.405454	+	0.914116i	$0.367114\pi$
$38$	18.4533	0.0787768
$39$	103.385	0.424485
$40$	0	0
$41$	74.4691	0.283661	0.141831	−	0.989891i	$-0.454701\pi$
$41$	74.4691	0.283661	0.141831	+	0.989891i	$0.454701\pi$
$42$	290.024	1.06552
$43$	405.899	1.43951	0.719756	−	0.694227i	$-0.244254\pi$
$43$	405.899	1.43951	0.719756	+	0.694227i	$0.244254\pi$
$44$	224.572	0.769444
$45$	0	0
$46$	95.9626	0.307585
$47$	327.143	1.01529	0.507646	−	0.861566i	$-0.330516\pi$
$47$	327.143	1.01529	0.507646	+	0.861566i	$0.330516\pi$
$48$	477.319	1.43531
$49$	−2.68899	−0.00783963
$50$	0	0
$51$	−212.304	−0.582913
$52$	670.732	1.78873
$53$	−409.067	−1.06018	−0.530092	−	0.847940i	$-0.677843\pi$
$53$	−409.067	−1.06018	−0.530092	+	0.847940i	$0.677843\pi$
$54$	141.494	0.356572
$55$	0	0
$56$	1108.19	2.64442
$57$	10.5638	0.0245475
$58$	151.975	0.344057
$59$	120.722	0.266384	0.133192	−	0.991090i	$-0.457477\pi$
$59$	120.722	0.266384	0.133192	+	0.991090i	$0.457477\pi$
$60$	0	0
$61$	34.6082	0.0726415	0.0363207	−	0.999340i	$-0.488436\pi$
$61$	34.6082	0.0726415	0.0363207	+	0.999340i	$0.488436\pi$
$62$	−631.788	−1.29415
$63$	166.028	0.332024
$64$	578.210	1.12932
$65$	0	0
$66$	181.401	0.338318
$67$	−671.739	−1.22487	−0.612433	−	0.790523i	$-0.709809\pi$
$67$	−671.739	−1.22487	−0.612433	+	0.790523i	$0.709809\pi$
$68$	−1377.36	−2.45632
$69$	54.9350	0.0958463
$70$	0	0
$71$	873.524	1.46012	0.730058	−	0.683386i	$-0.239493\pi$
$71$	873.524	1.46012	0.730058	+	0.683386i	$0.239493\pi$
$72$	540.652	0.884951
$73$	−432.426	−0.693311	−0.346655	−	0.937993i	$-0.612683\pi$
$73$	−432.426	−0.693311	−0.346655	+	0.937993i	$0.612683\pi$
$74$	956.419	1.50245
$75$	0	0
$76$	68.5347	0.103440
$77$	212.854	0.315026
$78$	541.793	0.786487
$79$	−548.355	−0.780946	−0.390473	−	0.920614i	$-0.627689\pi$
$79$	−548.355	−0.780946	−0.390473	+	0.920614i	$0.627689\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	390.257	0.525569
$83$	−511.077	−0.675879	−0.337940	−	0.941168i	$-0.609730\pi$
$83$	−511.077	−0.675879	−0.337940	+	0.941168i	$0.609730\pi$
$84$	1077.14	1.39911
$85$	0	0
$86$	2127.12	2.66713
$87$	87.0000	0.107211
$88$	693.138	0.839646
$89$	−788.135	−0.938676	−0.469338	−	0.883019i	$-0.655507\pi$
$89$	−788.135	−0.938676	−0.469338	+	0.883019i	$0.655507\pi$
$90$	0	0
$91$	635.734	0.732341
$92$	356.401	0.403884
$93$	−361.675	−0.403268
$94$	1714.40	1.88114
$95$	0	0
$96$	1059.66	1.12658
$97$	−967.070	−1.01228	−0.506140	−	0.862452i	$-0.668928\pi$
$97$	−967.070	−1.01228	−0.506140	+	0.862452i	$0.668928\pi$
$98$	−14.0917	−0.0145253
$99$	103.845	0.105423
$100$	0	0
$101$	510.016	0.502460	0.251230	−	0.967927i	$-0.419165\pi$
$101$	510.016	0.502460	0.251230	+	0.967927i	$0.419165\pi$
$102$	−1112.59	−1.08002
$103$	−1645.72	−1.57435	−0.787175	−	0.616730i	$-0.788457\pi$
$103$	−1645.72	−1.57435	−0.787175	+	0.616730i	$0.788457\pi$
$104$	2070.20	1.95192
$105$	0	0
$106$	−2143.73	−1.96431
$107$	1305.35	1.17937	0.589687	−	0.807632i	$-0.299251\pi$
$107$	1305.35	1.17937	0.589687	+	0.807632i	$0.299251\pi$
$108$	525.503	0.468209
$109$	932.125	0.819095	0.409548	−	0.912289i	$-0.365687\pi$
$109$	932.125	0.819095	0.409548	+	0.912289i	$0.365687\pi$
$110$	0	0
$111$	547.514	0.468178
$112$	2935.12	2.47627
$113$	−1719.89	−1.43180	−0.715901	−	0.698202i	$-0.753984\pi$
$113$	−1719.89	−1.43180	−0.715901	+	0.698202i	$0.753984\pi$
$114$	55.3598	0.0454818
$115$	0	0
$116$	564.429	0.451775
$117$	310.156	0.245076
$118$	632.646	0.493558
$119$	−1305.50	−1.00567
$120$	0	0
$121$	−1197.87	−0.899974
$122$	181.365	0.134590
$123$	223.407	0.163772
$124$	−2346.43	−1.69932
$125$	0	0
$126$	870.072	0.615176
$127$	883.792	0.617511	0.308756	−	0.951141i	$-0.400088\pi$
$127$	883.792	0.617511	0.308756	+	0.951141i	$0.400088\pi$
$128$	204.357	0.141116
$129$	1217.70	0.831102
$130$	0	0
$131$	−937.386	−0.625190	−0.312595	−	0.949887i	$-0.601198\pi$
$131$	−937.386	−0.625190	−0.312595	+	0.949887i	$0.601198\pi$
$132$	673.717	0.444239
$133$	64.9587	0.0423506
$134$	−3520.26	−2.26944
$135$	0	0
$136$	−4251.21	−2.68043
$137$	−237.650	−0.148203	−0.0741014	−	0.997251i	$-0.523609\pi$
$137$	−237.650	−0.148203	−0.0741014	+	0.997251i	$0.523609\pi$
$138$	287.888	0.177584
$139$	1706.72	1.04146	0.520728	−	0.853723i	$-0.325661\pi$
$139$	1706.72	1.04146	0.520728	+	0.853723i	$0.325661\pi$
$140$	0	0
$141$	981.430	0.586179
$142$	4577.72	2.70531
$143$	397.633	0.232530
$144$	1431.96	0.828679
$145$	0	0
$146$	−2266.14	−1.28457
$147$	−8.06698	−0.00452621
$148$	3552.10	1.97284
$149$	3546.59	1.94999	0.974994	−	0.222230i	$-0.0713334\pi$
$149$	3546.59	1.94999	0.974994	+	0.222230i	$0.0713334\pi$
$150$	0	0
$151$	1818.23	0.979901	0.489951	−	0.871750i	$-0.337015\pi$
$151$	1818.23	0.979901	0.489951	+	0.871750i	$0.337015\pi$
$152$	211.531	0.112878
$153$	−636.913	−0.336545
$154$	1115.47	0.583682
$155$	0	0
$156$	2012.20	1.03272
$157$	37.7540	0.0191917	0.00959585	−	0.999954i	$-0.496945\pi$
$157$	37.7540	0.0191917	0.00959585	+	0.999954i	$0.496945\pi$
$158$	−2873.66	−1.44694
$159$	−1227.20	−0.612097
$160$	0	0
$161$	337.805	0.165359
$162$	424.482	0.205867
$163$	−1541.87	−0.740913	−0.370457	−	0.928850i	$-0.620799\pi$
$163$	−1541.87	−0.740913	−0.370457	+	0.928850i	$0.620799\pi$
$164$	1449.40	0.690115
$165$	0	0
$166$	−2678.31	−1.25227
$167$	−1669.49	−0.773586	−0.386793	−	0.922167i	$-0.626417\pi$
$167$	−1669.49	−0.773586	−0.386793	+	0.922167i	$0.626417\pi$
$168$	3324.56	1.52676
$169$	−1009.39	−0.459438
$170$	0	0
$171$	31.6914	0.0141725
$172$	7900.04	3.50216
$173$	728.623	0.320209	0.160105	−	0.987100i	$-0.448817\pi$
$173$	728.623	0.320209	0.160105	+	0.987100i	$0.448817\pi$
$174$	455.925	0.198641
$175$	0	0
$176$	1835.83	0.786255
$177$	362.166	0.153797
$178$	−4130.24	−1.73918
$179$	1926.46	0.804416	0.402208	−	0.915548i	$-0.368243\pi$
$179$	1926.46	0.804416	0.402208	+	0.915548i	$0.368243\pi$
$180$	0	0
$181$	3695.62	1.51764	0.758821	−	0.651299i	$-0.225776\pi$
$181$	3695.62	1.51764	0.758821	+	0.651299i	$0.225776\pi$
$182$	3331.58	1.35689
$183$	103.825	0.0419396
$184$	1100.03	0.440733
$185$	0	0
$186$	−1895.36	−0.747177
$187$	−816.549	−0.319315
$188$	6367.21	2.47009
$189$	498.083	0.191694
$190$	0	0
$191$	2490.19	0.943370	0.471685	−	0.881767i	$-0.343646\pi$
$191$	2490.19	0.943370	0.471685	+	0.881767i	$0.343646\pi$
$192$	1734.63	0.652012
$193$	−4493.27	−1.67582	−0.837909	−	0.545809i	$-0.816222\pi$
$193$	−4493.27	−1.67582	−0.837909	+	0.545809i	$0.816222\pi$
$194$	−5067.95	−1.87555
$195$	0	0
$196$	−52.3361	−0.0190729
$197$	−1990.23	−0.719787	−0.359894	−	0.932993i	$-0.617187\pi$
$197$	−1990.23	−0.719787	−0.359894	+	0.932993i	$0.617187\pi$
$198$	544.204	0.195328
$199$	−83.8021	−0.0298521	−0.0149261	−	0.999889i	$-0.504751\pi$
$199$	−83.8021	−0.0298521	−0.0149261	+	0.999889i	$0.504751\pi$
$200$	0	0
$201$	−2015.22	−0.707176
$202$	2672.75	0.930960
$203$	534.978	0.184966
$204$	−4132.09	−1.41816
$205$	0	0
$206$	−8624.45	−2.91696
$207$	164.805	0.0553369
$208$	5483.09	1.82781
$209$	40.6297	0.0134470
$210$	0	0
$211$	−4369.95	−1.42578	−0.712891	−	0.701275i	$-0.752615\pi$
$211$	−4369.95	−1.42578	−0.712891	+	0.701275i	$0.752615\pi$
$212$	−7961.71	−2.57930
$213$	2620.57	0.842998
$214$	6840.72	2.18515
$215$	0	0
$216$	1621.96	0.510926
$217$	−2224.00	−0.695738
$218$	4884.82	1.51762
$219$	−1297.28	−0.400283
$220$	0	0
$221$	−2438.79	−0.742313
$222$	2869.26	0.867441
$223$	−3633.76	−1.09119	−0.545594	−	0.838050i	$-0.683696\pi$
$223$	−3633.76	−1.09119	−0.545594	+	0.838050i	$0.683696\pi$
$224$	6516.05	1.94362
$225$	0	0
$226$	−9013.12	−2.65285
$227$	3692.30	1.07959	0.539794	−	0.841797i	$-0.318502\pi$
$227$	3692.30	1.07959	0.539794	+	0.841797i	$0.318502\pi$
$228$	205.604	0.0597213
$229$	4833.73	1.39486	0.697428	−	0.716655i	$-0.254328\pi$
$229$	4833.73	1.39486	0.697428	+	0.716655i	$0.254328\pi$
$230$	0	0
$231$	638.563	0.181880
$232$	1742.10	0.492994
$233$	4834.45	1.35929	0.679647	−	0.733539i	$-0.262133\pi$
$233$	4834.45	1.35929	0.679647	+	0.733539i	$0.262133\pi$
$234$	1625.38	0.454078
$235$	0	0
$236$	2349.62	0.648081
$237$	−1645.06	−0.450879
$238$	−6841.48	−1.86331
$239$	4019.80	1.08795	0.543974	−	0.839102i	$-0.316919\pi$
$239$	4019.80	1.08795	0.543974	+	0.839102i	$0.316919\pi$
$240$	0	0
$241$	1076.81	0.287816	0.143908	−	0.989591i	$-0.454033\pi$
$241$	1076.81	0.287816	0.143908	+	0.989591i	$0.454033\pi$
$242$	−6277.44	−1.66748
$243$	243.000	0.0641500
$244$	673.582	0.176728
$245$	0	0
$246$	1170.77	0.303437
$247$	121.349	0.0312602
$248$	−7242.23	−1.85436
$249$	−1533.23	−0.390219
$250$	0	0
$251$	−3179.99	−0.799677	−0.399839	−	0.916586i	$-0.630934\pi$
$251$	−3179.99	−0.799677	−0.399839	+	0.916586i	$0.630934\pi$
$252$	3231.41	0.807776
$253$	211.287	0.0525039
$254$	4631.53	1.14413
$255$	0	0
$256$	−3554.74	−0.867858
$257$	−3125.29	−0.758561	−0.379280	−	0.925282i	$-0.623828\pi$
$257$	−3125.29	−0.758561	−0.379280	+	0.925282i	$0.623828\pi$
$258$	6381.37	1.53987
$259$	3366.76	0.807722
$260$	0	0
$261$	261.000	0.0618984
$262$	−4912.39	−1.15835
$263$	4474.30	1.04904	0.524519	−	0.851399i	$-0.324245\pi$
$263$	4474.30	1.04904	0.524519	+	0.851399i	$0.324245\pi$
$264$	2079.42	0.484770
$265$	0	0
$266$	340.417	0.0784674
$267$	−2364.41	−0.541945
$268$	−13074.1	−2.97995
$269$	4053.04	0.918655	0.459327	−	0.888267i	$-0.348091\pi$
$269$	4053.04	0.918655	0.459327	+	0.888267i	$0.348091\pi$
$270$	0	0
$271$	−8045.96	−1.80353	−0.901766	−	0.432225i	$-0.857729\pi$
$271$	−8045.96	−1.80353	−0.901766	+	0.432225i	$0.857729\pi$
$272$	−11259.7	−2.50999
$273$	1907.20	0.422817
$274$	−1245.41	−0.274591
$275$	0	0
$276$	1069.20	0.233183
$277$	−4625.74	−1.00337	−0.501685	−	0.865050i	$-0.667286\pi$
$277$	−4625.74	−1.00337	−0.501685	+	0.865050i	$0.667286\pi$
$278$	8944.11	1.92961
$279$	−1085.02	−0.232827
$280$	0	0
$281$	−69.7964	−0.0148174	−0.00740872	−	0.999973i	$-0.502358\pi$
$281$	−69.7964	−0.0148174	−0.00740872	+	0.999973i	$0.502358\pi$
$282$	5143.20	1.08608
$283$	−1903.58	−0.399845	−0.199922	−	0.979812i	$-0.564069\pi$
$283$	−1903.58	−0.399845	−0.199922	+	0.979812i	$0.564069\pi$
$284$	17001.4	3.55229
$285$	0	0
$286$	2083.80	0.430832
$287$	1373.77	0.282547
$288$	3178.99	0.650429
$289$	95.1246	0.0193618
$290$	0	0
$291$	−2901.21	−0.584440
$292$	−8416.34	−1.68674
$293$	−623.198	−0.124258	−0.0621290	−	0.998068i	$-0.519789\pi$
$293$	−623.198	−0.124258	−0.0621290	+	0.998068i	$0.519789\pi$
$294$	−42.2752	−0.00838619
$295$	0	0
$296$	10963.5	2.15284
$297$	311.536	0.0608659
$298$	18586.0	3.61295
$299$	631.052	0.122056
$300$	0	0
$301$	7487.83	1.43386
$302$	9528.45	1.81557
$303$	1530.05	0.290095
$304$	560.256	0.105700
$305$	0	0
$306$	−3337.76	−0.623552
$307$	6899.90	1.28273	0.641365	−	0.767236i	$-0.278368\pi$
$307$	6899.90	1.28273	0.641365	+	0.767236i	$0.278368\pi$
$308$	4142.80	0.766422
$309$	−4937.17	−0.908951
$310$	0	0
$311$	−181.369	−0.0330692	−0.0165346	−	0.999863i	$-0.505263\pi$
$311$	−181.369	−0.0330692	−0.0165346	+	0.999863i	$0.505263\pi$
$312$	6210.61	1.12694
$313$	3839.22	0.693308	0.346654	−	0.937993i	$-0.387318\pi$
$313$	3839.22	0.693308	0.346654	+	0.937993i	$0.387318\pi$
$314$	197.851	0.0355585
$315$	0	0
$316$	−10672.7	−1.89995
$317$	−7000.08	−1.24026	−0.620132	−	0.784497i	$-0.712921\pi$
$317$	−7000.08	−1.24026	−0.620132	+	0.784497i	$0.712921\pi$
$318$	−6431.18	−1.13410
$319$	334.613	0.0587296
$320$	0	0
$321$	3916.05	0.680912
$322$	1770.27	0.306377
$323$	−249.193	−0.0429272
$324$	1576.51	0.270320
$325$	0	0
$326$	−8080.22	−1.37277
$327$	2796.38	0.472905
$328$	4473.54	0.753079
$329$	6034.98	1.01130
$330$	0	0
$331$	−1176.84	−0.195423	−0.0977115	−	0.995215i	$-0.531152\pi$
$331$	−1176.84	−0.195423	−0.0977115	+	0.995215i	$0.531152\pi$
$332$	−9947.12	−1.64433
$333$	1642.54	0.270302
$334$	−8748.99	−1.43330
$335$	0	0
$336$	8805.35	1.42968
$337$	174.049	0.0281336	0.0140668	−	0.999901i	$-0.495522\pi$
$337$	174.049	0.0281336	0.0140668	+	0.999901i	$0.495522\pi$
$338$	−5289.71	−0.851249
$339$	−5159.67	−0.826652
$340$	0	0
$341$	−1391.05	−0.220907
$342$	166.080	0.0262589
$343$	−6377.10	−1.00388
$344$	24383.3	3.82169
$345$	0	0
$346$	3818.37	0.593285
$347$	10895.5	1.68560	0.842799	−	0.538228i	$-0.180906\pi$
$347$	10895.5	1.68560	0.842799	+	0.538228i	$0.180906\pi$
$348$	1693.29	0.260832
$349$	−11820.3	−1.81298	−0.906488	−	0.422232i	$-0.861247\pi$
$349$	−11820.3	−1.81298	−0.906488	+	0.422232i	$0.861247\pi$
$350$	0	0
$351$	930.468	0.141495
$352$	4075.60	0.617131
$353$	−9795.13	−1.47689	−0.738445	−	0.674314i	$-0.764439\pi$
$353$	−9795.13	−1.47689	−0.738445	+	0.674314i	$0.764439\pi$
$354$	1897.94	0.284956
$355$	0	0
$356$	−15339.5	−2.28369
$357$	−3916.49	−0.580623
$358$	10095.7	1.49042
$359$	8894.06	1.30755	0.653775	−	0.756689i	$-0.273184\pi$
$359$	8894.06	1.30755	0.653775	+	0.756689i	$0.273184\pi$
$360$	0	0
$361$	−6846.60	−0.998192
$362$	19367.0	2.81189
$363$	−3593.60	−0.519600
$364$	12373.3	1.78170
$365$	0	0
$366$	544.096	0.0777058
$367$	−11113.1	−1.58066	−0.790329	−	0.612683i	$-0.790090\pi$
$367$	−11113.1	−1.58066	−0.790329	+	0.612683i	$0.790090\pi$
$368$	2913.50	0.412708
$369$	670.222	0.0945538
$370$	0	0
$371$	−7546.28	−1.05602
$372$	−7039.30	−0.981104
$373$	−13469.8	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$373$	−13469.8	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$374$	−4279.14	−0.591629
$375$	0	0
$376$	19652.3	2.69545
$377$	999.391	0.136529
$378$	2610.21	0.355172
$379$	−3750.71	−0.508341	−0.254170	−	0.967159i	$-0.581802\pi$
$379$	−3750.71	−0.508341	−0.254170	+	0.967159i	$0.581802\pi$
$380$	0	0
$381$	2651.38	0.356520
$382$	13049.9	1.74788
$383$	−12404.3	−1.65490	−0.827451	−	0.561537i	$-0.810210\pi$
$383$	−12404.3	−1.65490	−0.827451	+	0.561537i	$0.810210\pi$
$384$	613.072	0.0814732
$385$	0	0
$386$	−23547.1	−3.10496
$387$	3653.09	0.479837
$388$	−18822.1	−2.46276
$389$	−7718.67	−1.00605	−0.503023	−	0.864273i	$-0.667779\pi$
$389$	−7718.67	−1.00605	−0.503023	+	0.864273i	$0.667779\pi$
$390$	0	0
$391$	−1295.88	−0.167610
$392$	−161.534	−0.0208131
$393$	−2812.16	−0.360953
$394$	−10429.8	−1.33362
$395$	0	0
$396$	2021.15	0.256481
$397$	6747.40	0.853004	0.426502	−	0.904487i	$-0.359746\pi$
$397$	6747.40	0.853004	0.426502	+	0.904487i	$0.359746\pi$
$398$	−439.167	−0.0553101
$399$	194.876	0.0244511
$400$	0	0
$401$	−5047.40	−0.628567	−0.314283	−	0.949329i	$-0.601764\pi$
$401$	−5047.40	−0.628567	−0.314283	+	0.949329i	$0.601764\pi$
$402$	−10560.8	−1.31026
$403$	−4154.65	−0.513543
$404$	9926.47	1.22243
$405$	0	0
$406$	2803.56	0.342706
$407$	2105.81	0.256464
$408$	−12753.6	−1.54755
$409$	5317.14	0.642826	0.321413	−	0.946939i	$-0.395842\pi$
$409$	5317.14	0.642826	0.321413	+	0.946939i	$0.395842\pi$
$410$	0	0
$411$	−712.949	−0.0855649
$412$	−32030.8	−3.83021
$413$	2227.02	0.265338
$414$	863.663	0.102528
$415$	0	0
$416$	12172.6	1.43464
$417$	5120.17	0.601284
$418$	212.921	0.0249146
$419$	2953.43	0.344355	0.172177	−	0.985066i	$-0.444920\pi$
$419$	2953.43	0.344355	0.172177	+	0.985066i	$0.444920\pi$
$420$	0	0
$421$	7427.88	0.859888	0.429944	−	0.902855i	$-0.358533\pi$
$421$	7427.88	0.859888	0.429944	+	0.902855i	$0.358533\pi$
$422$	−22900.8	−2.64170
$423$	2944.29	0.338431
$424$	−24573.7	−2.81463
$425$	0	0
$426$	13733.2	1.56191
$427$	638.436	0.0723562
$428$	25406.1	2.86928
$429$	1192.90	0.134251
$430$	0	0
$431$	3302.08	0.369038	0.184519	−	0.982829i	$-0.440927\pi$
$431$	3302.08	0.369038	0.184519	+	0.982829i	$0.440927\pi$
$432$	4295.87	0.478438
$433$	−9352.89	−1.03804	−0.519020	−	0.854762i	$-0.673703\pi$
$433$	−9352.89	−1.03804	−0.519020	+	0.854762i	$0.673703\pi$
$434$	−11654.9	−1.28907
$435$	0	0
$436$	18142.0	1.99276
$437$	64.4802	0.00705837
$438$	−6798.42	−0.741646
$439$	−6109.74	−0.664241	−0.332120	−	0.943237i	$-0.607764\pi$
$439$	−6109.74	−0.664241	−0.332120	+	0.943237i	$0.607764\pi$
$440$	0	0
$441$	−24.2009	−0.00261321
$442$	−12780.6	−1.37536
$443$	−14282.0	−1.53173	−0.765865	−	0.643002i	$-0.777689\pi$
$443$	−14282.0	−1.53173	−0.765865	+	0.643002i	$0.777689\pi$
$444$	10656.3	1.13902
$445$	0	0
$446$	−19042.8	−2.02176
$447$	10639.8	1.12583
$448$	10666.5	1.12488
$449$	−16406.0	−1.72439	−0.862193	−	0.506581i	$-0.830909\pi$
$449$	−16406.0	−1.72439	−0.862193	+	0.506581i	$0.830909\pi$
$450$	0	0
$451$	859.252	0.0897131
$452$	−33474.3	−3.48341
$453$	5454.68	0.565746
$454$	19349.6	2.00026
$455$	0	0
$456$	634.593	0.0651701
$457$	15726.2	1.60971	0.804856	−	0.593470i	$-0.202243\pi$
$457$	15726.2	1.60971	0.804856	+	0.593470i	$0.202243\pi$
$458$	25331.3	2.58440
$459$	−1910.74	−0.194304
$460$	0	0
$461$	−14091.2	−1.42363	−0.711816	−	0.702366i	$-0.752127\pi$
$461$	−14091.2	−1.42363	−0.711816	+	0.702366i	$0.752127\pi$
$462$	3346.41	0.336989
$463$	8794.40	0.882743	0.441372	−	0.897324i	$-0.354492\pi$
$463$	8794.40	0.882743	0.441372	+	0.897324i	$0.354492\pi$
$464$	4614.08	0.461645
$465$	0	0
$466$	25335.0	2.51850
$467$	8746.05	0.866636	0.433318	−	0.901241i	$-0.357343\pi$
$467$	8746.05	0.866636	0.433318	+	0.901241i	$0.357343\pi$
$468$	6036.59	0.596242
$469$	−12391.9	−1.22005
$470$	0	0
$471$	113.262	0.0110803
$472$	7252.06	0.707210
$473$	4683.41	0.455272
$474$	−8620.99	−0.835391
$475$	0	0
$476$	−25409.0	−2.44668
$477$	−3681.61	−0.353395
$478$	21065.9	2.01575
$479$	−11452.6	−1.09245	−0.546225	−	0.837638i	$-0.683936\pi$
$479$	−11452.6	−1.09245	−0.546225	+	0.837638i	$0.683936\pi$
$480$	0	0
$481$	6289.43	0.596203
$482$	5643.06	0.533266
$483$	1013.41	0.0954698
$484$	−23314.1	−2.18953
$485$	0	0
$486$	1273.45	0.118857
$487$	−1635.31	−0.152163	−0.0760813	−	0.997102i	$-0.524241\pi$
$487$	−1635.31	−0.152163	−0.0760813	+	0.997102i	$0.524241\pi$
$488$	2079.00	0.192852
$489$	−4625.62	−0.427766
$490$	0	0
$491$	2106.92	0.193654	0.0968270	−	0.995301i	$-0.469131\pi$
$491$	2106.92	0.193654	0.0968270	+	0.995301i	$0.469131\pi$
$492$	4348.19	0.398438
$493$	−2052.28	−0.187484
$494$	635.933	0.0579190
$495$	0	0
$496$	−19181.6	−1.73645
$497$	16114.3	1.45438
$498$	−8034.93	−0.722999
$499$	−8865.74	−0.795361	−0.397681	−	0.917524i	$-0.630185\pi$
$499$	−8865.74	−0.795361	−0.397681	+	0.917524i	$0.630185\pi$
$500$	0	0
$501$	−5008.46	−0.446630
$502$	−16664.8	−1.48165
$503$	14591.9	1.29348	0.646738	−	0.762712i	$-0.276133\pi$
$503$	14591.9	1.29348	0.646738	+	0.762712i	$0.276133\pi$
$504$	9973.69	0.881475
$505$	0	0
$506$	1107.25	0.0972794
$507$	−3028.16	−0.265257
$508$	17201.3	1.50233
$509$	13600.3	1.18432	0.592162	−	0.805819i	$-0.298274\pi$
$509$	13600.3	1.18432	0.592162	+	0.805819i	$0.298274\pi$
$510$	0	0
$511$	−7977.19	−0.690588
$512$	−20263.6	−1.74909
$513$	95.0742	0.00818251
$514$	−16378.1	−1.40546
$515$	0	0
$516$	23700.1	2.02197
$517$	3774.70	0.321105
$518$	17643.6	1.49655
$519$	2185.87	0.184873
$520$	0	0
$521$	8355.92	0.702647	0.351324	−	0.936254i	$-0.385732\pi$
$521$	8355.92	0.702647	0.351324	+	0.936254i	$0.385732\pi$
$522$	1367.78	0.114686
$523$	−926.372	−0.0774520	−0.0387260	−	0.999250i	$-0.512330\pi$
$523$	−926.372	−0.0774520	−0.0387260	+	0.999250i	$0.512330\pi$
$524$	−18244.4	−1.52101
$525$	0	0
$526$	23447.7	1.94366
$527$	8531.68	0.705210
$528$	5507.49	0.453944
$529$	−11831.7	−0.972440
$530$	0	0
$531$	1086.50	0.0887947
$532$	1264.29	0.103034
$533$	2566.34	0.208556
$534$	−12390.7	−1.00412
$535$	0	0
$536$	−40353.0	−3.25184
$537$	5779.38	0.464430
$538$	21240.0	1.70209
$539$	−31.0266	−0.00247943
$540$	0	0
$541$	13503.6	1.07313	0.536565	−	0.843859i	$-0.319722\pi$
$541$	13503.6	1.07313	0.536565	+	0.843859i	$0.319722\pi$
$542$	−42165.0	−3.34159
$543$	11086.9	0.876211
$544$	−24996.8	−1.97009
$545$	0	0
$546$	9994.74	0.783398
$547$	20425.4	1.59658	0.798288	−	0.602275i	$-0.205739\pi$
$547$	20425.4	1.59658	0.798288	+	0.602275i	$0.205739\pi$
$548$	−4625.39	−0.360560
$549$	311.474	0.0242138
$550$	0	0
$551$	102.117	0.00789532
$552$	3300.08	0.254458
$553$	−10115.8	−0.777879
$554$	−24241.3	−1.85905
$555$	0	0
$556$	33218.0	2.53374
$557$	3929.12	0.298891	0.149446	−	0.988770i	$-0.452251\pi$
$557$	3929.12	0.298891	0.149446	+	0.988770i	$0.452251\pi$
$558$	−5686.09	−0.431383
$559$	13988.0	1.05837
$560$	0	0
$561$	−2449.65	−0.184357
$562$	−365.769	−0.0274538
$563$	−4452.67	−0.333318	−0.166659	−	0.986015i	$-0.553298\pi$
$563$	−4452.67	−0.333318	−0.166659	+	0.986015i	$0.553298\pi$
$564$	19101.6	1.42611
$565$	0	0
$566$	−9975.75	−0.740834
$567$	1494.25	0.110675
$568$	52474.7	3.87639
$569$	−5462.20	−0.402438	−0.201219	−	0.979546i	$-0.564490\pi$
$569$	−5462.20	−0.402438	−0.201219	+	0.979546i	$0.564490\pi$
$570$	0	0
$571$	−7228.87	−0.529805	−0.264902	−	0.964275i	$-0.585340\pi$
$571$	−7228.87	−0.529805	−0.264902	+	0.964275i	$0.585340\pi$
$572$	7739.16	0.565717
$573$	7470.57	0.544655
$574$	7199.27	0.523505
$575$	0	0
$576$	5203.89	0.376439
$577$	−12771.6	−0.921468	−0.460734	−	0.887538i	$-0.652414\pi$
$577$	−12771.6	−0.921468	−0.460734	+	0.887538i	$0.652414\pi$
$578$	498.502	0.0358736
$579$	−13479.8	−0.967534
$580$	0	0
$581$	−9428.10	−0.673224
$582$	−15203.9	−1.08285
$583$	−4719.97	−0.335303
$584$	−25976.9	−1.84064
$585$	0	0
$586$	−3265.88	−0.230226
$587$	−6986.93	−0.491280	−0.245640	−	0.969361i	$-0.578998\pi$
$587$	−6986.93	−0.491280	−0.245640	+	0.969361i	$0.578998\pi$
$588$	−157.008	−0.0110117
$589$	−424.518	−0.0296977
$590$	0	0
$591$	−5970.69	−0.415569
$592$	29037.6	2.01595
$593$	−4323.42	−0.299396	−0.149698	−	0.988732i	$-0.547830\pi$
$593$	−4323.42	−0.299396	−0.149698	+	0.988732i	$0.547830\pi$
$594$	1632.61	0.112773
$595$	0	0
$596$	69027.6	4.74409
$597$	−251.406	−0.0172351
$598$	3307.04	0.226145
$599$	13443.7	0.917019	0.458510	−	0.888689i	$-0.348383\pi$
$599$	13443.7	0.917019	0.458510	+	0.888689i	$0.348383\pi$
$600$	0	0
$601$	−7179.72	−0.487299	−0.243650	−	0.969863i	$-0.578345\pi$
$601$	−7179.72	−0.487299	−0.243650	+	0.969863i	$0.578345\pi$
$602$	39240.1	2.65666
$603$	−6045.65	−0.408288
$604$	35388.2	2.38399
$605$	0	0
$606$	8018.24	0.537490
$607$	8560.81	0.572442	0.286221	−	0.958164i	$-0.407601\pi$
$607$	8560.81	0.572442	0.286221	+	0.958164i	$0.407601\pi$
$608$	1243.79	0.0829641
$609$	1604.93	0.106790
$610$	0	0
$611$	11273.9	0.746472
$612$	−12396.3	−0.818774
$613$	−19634.3	−1.29367	−0.646837	−	0.762628i	$-0.723909\pi$
$613$	−19634.3	−1.29367	−0.646837	+	0.762628i	$0.723909\pi$
$614$	36159.1	2.37665
$615$	0	0
$616$	12786.7	0.836348
$617$	−14268.5	−0.931003	−0.465502	−	0.885047i	$-0.654126\pi$
$617$	−14268.5	−0.931003	−0.465502	+	0.885047i	$0.654126\pi$
$618$	−25873.3	−1.68411
$619$	1663.48	0.108014	0.0540071	−	0.998541i	$-0.482801\pi$
$619$	1663.48	0.108014	0.0540071	+	0.998541i	$0.482801\pi$
$620$	0	0
$621$	494.415	0.0319488
$622$	−950.470	−0.0612707
$623$	−14539.1	−0.934990
$624$	16449.3	1.05528
$625$	0	0
$626$	20119.5	1.28456
$627$	121.889	0.00776361
$628$	734.808	0.0466912
$629$	−12915.5	−0.818720
$630$	0	0
$631$	13448.3	0.848444	0.424222	−	0.905558i	$-0.360548\pi$
$631$	13448.3	0.848444	0.424222	+	0.905558i	$0.360548\pi$
$632$	−32941.0	−2.07330
$633$	−13109.9	−0.823176
$634$	−36684.1	−2.29797
$635$	0	0
$636$	−23885.1	−1.48916
$637$	−92.6675	−0.00576392
$638$	1753.55	0.108814
$639$	7861.71	0.486705
$640$	0	0
$641$	7644.92	0.471070	0.235535	−	0.971866i	$-0.424316\pi$
$641$	7644.92	0.471070	0.235535	+	0.971866i	$0.424316\pi$
$642$	20522.1	1.26160
$643$	−1814.60	−0.111292	−0.0556462	−	0.998451i	$-0.517722\pi$
$643$	−1814.60	−0.111292	−0.0556462	+	0.998451i	$0.517722\pi$
$644$	6574.71	0.402298
$645$	0	0
$646$	−1305.90	−0.0795357
$647$	−18746.6	−1.13911	−0.569556	−	0.821952i	$-0.692885\pi$
$647$	−18746.6	−1.13911	−0.569556	+	0.821952i	$0.692885\pi$
$648$	4865.87	0.294984
$649$	1392.94	0.0842489
$650$	0	0
$651$	−6672.00	−0.401684
$652$	−30009.6	−1.80256
$653$	−8299.18	−0.497354	−0.248677	−	0.968586i	$-0.579996\pi$
$653$	−8299.18	−0.497354	−0.248677	+	0.968586i	$0.579996\pi$
$654$	14654.5	0.876200
$655$	0	0
$656$	11848.5	0.705192
$657$	−3891.84	−0.231104
$658$	31626.4	1.87375
$659$	12624.7	0.746262	0.373131	−	0.927779i	$-0.378284\pi$
$659$	12624.7	0.746262	0.373131	+	0.927779i	$0.378284\pi$
$660$	0	0
$661$	−10912.8	−0.642144	−0.321072	−	0.947055i	$-0.604043\pi$
$661$	−10912.8	−0.642144	−0.321072	+	0.947055i	$0.604043\pi$
$662$	−6167.26	−0.362081
$663$	−7316.38	−0.428574
$664$	−30701.6	−1.79436
$665$	0	0
$666$	8607.77	0.500817
$667$	531.038	0.0308274
$668$	−32493.4	−1.88204
$669$	−10901.3	−0.629998
$670$	0	0
$671$	399.323	0.0229742
$672$	19548.1	1.12215
$673$	−6292.15	−0.360393	−0.180197	−	0.983631i	$-0.557673\pi$
$673$	−6292.15	−0.360393	−0.180197	+	0.983631i	$0.557673\pi$
$674$	912.105	0.0521261
$675$	0	0
$676$	−19645.7	−1.11776
$677$	−16630.8	−0.944125	−0.472062	−	0.881565i	$-0.656490\pi$
$677$	−16630.8	−0.944125	−0.472062	+	0.881565i	$0.656490\pi$
$678$	−27039.4	−1.53162
$679$	−17840.0	−1.00830
$680$	0	0
$681$	11076.9	0.623300
$682$	−7289.81	−0.409298
$683$	−11843.7	−0.663521	−0.331761	−	0.943364i	$-0.607643\pi$
$683$	−11843.7	−0.663521	−0.331761	+	0.943364i	$0.607643\pi$
$684$	616.812	0.0344801
$685$	0	0
$686$	−33419.4	−1.86000
$687$	14501.2	0.805321
$688$	64581.1	3.57868
$689$	−14097.2	−0.779478
$690$	0	0
$691$	20458.3	1.12630	0.563149	−	0.826356i	$-0.309590\pi$
$691$	20458.3	1.12630	0.563149	+	0.826356i	$0.309590\pi$
$692$	14181.2	0.779032
$693$	1915.69	0.105009
$694$	57098.2	3.12308
$695$	0	0
$696$	5226.30	0.284630
$697$	−5270.03	−0.286394
$698$	−61944.8	−3.35909
$699$	14503.4	0.784789
$700$	0	0
$701$	23987.6	1.29244	0.646220	−	0.763151i	$-0.276349\pi$
$701$	23987.6	1.29244	0.646220	+	0.763151i	$0.276349\pi$
$702$	4876.14	0.262162
$703$	642.648	0.0344778
$704$	6671.61	0.357167
$705$	0	0
$706$	−51331.6	−2.73639
$707$	9408.52	0.500487
$708$	7048.86	0.374170
$709$	6275.84	0.332432	0.166216	−	0.986089i	$-0.446845\pi$
$709$	6275.84	0.332432	0.166216	+	0.986089i	$0.446845\pi$
$710$	0	0
$711$	−4935.19	−0.260315
$712$	−47345.2	−2.49205
$713$	−2207.62	−0.115955
$714$	−20524.4	−1.07578
$715$	0	0
$716$	37494.8	1.95705
$717$	12059.4	0.628127
$718$	46609.5	2.42263
$719$	−15544.4	−0.806273	−0.403136	−	0.915140i	$-0.632080\pi$
$719$	−15544.4	−0.806273	−0.403136	+	0.915140i	$0.632080\pi$
$720$	0	0
$721$	−30359.5	−1.56817
$722$	−35879.8	−1.84945
$723$	3230.44	0.166170
$724$	71928.1	3.69225
$725$	0	0
$726$	−18832.3	−0.962718
$727$	25002.9	1.27552	0.637762	−	0.770233i	$-0.279860\pi$
$727$	25002.9	1.27552	0.637762	+	0.770233i	$0.279860\pi$
$728$	38190.1	1.94426
$729$	729.000	0.0370370
$730$	0	0
$731$	−28724.7	−1.45338
$732$	2020.75	0.102034
$733$	16981.1	0.855678	0.427839	−	0.903855i	$-0.359275\pi$
$733$	16981.1	0.855678	0.427839	+	0.903855i	$0.359275\pi$
$734$	−58238.7	−2.92865
$735$	0	0
$736$	6468.06	0.323934
$737$	−7750.78	−0.387386
$738$	3512.31	0.175190
$739$	920.528	0.0458216	0.0229108	−	0.999738i	$-0.492707\pi$
$739$	920.528	0.0458216	0.0229108	+	0.999738i	$0.492707\pi$
$740$	0	0
$741$	364.047	0.0180481
$742$	−39546.4	−1.95660
$743$	−6708.21	−0.331225	−0.165613	−	0.986191i	$-0.552960\pi$
$743$	−6708.21	−0.331225	−0.165613	+	0.986191i	$0.552960\pi$
$744$	−21726.7	−1.07062
$745$	0	0
$746$	−70588.9	−3.46440
$747$	−4599.69	−0.225293
$748$	−15892.5	−0.776857
$749$	24080.5	1.17474
$750$	0	0
$751$	34311.4	1.66717	0.833583	−	0.552394i	$-0.186286\pi$
$751$	34311.4	1.66717	0.833583	+	0.552394i	$0.186286\pi$
$752$	52050.6	2.52405
$753$	−9539.96	−0.461694
$754$	5237.33	0.252961
$755$	0	0
$756$	9694.22	0.466370
$757$	12405.6	0.595628	0.297814	−	0.954624i	$-0.403743\pi$
$757$	12405.6	0.595628	0.297814	+	0.954624i	$0.403743\pi$
$758$	−19655.7	−0.941856
$759$	633.860	0.0303131
$760$	0	0
$761$	−18869.8	−0.898857	−0.449428	−	0.893316i	$-0.648372\pi$
$761$	−18869.8	−0.898857	−0.449428	+	0.893316i	$0.648372\pi$
$762$	13894.6	0.660562
$763$	17195.4	0.815878
$764$	48466.7	2.29511
$765$	0	0
$766$	−65004.8	−3.06621
$767$	4160.29	0.195853
$768$	−10664.2	−0.501058
$769$	38269.7	1.79459	0.897295	−	0.441432i	$-0.145529\pi$
$769$	38269.7	1.79459	0.897295	+	0.441432i	$0.145529\pi$
$770$	0	0
$771$	−9375.86	−0.437955
$772$	−87452.9	−4.07707
$773$	−37398.4	−1.74014	−0.870069	−	0.492930i	$-0.835926\pi$
$773$	−37398.4	−1.74014	−0.870069	+	0.492930i	$0.835926\pi$
$774$	19144.1	0.889044
$775$	0	0
$776$	−58094.2	−2.68745
$777$	10100.3	0.466339
$778$	−40449.9	−1.86401
$779$	262.226	0.0120606
$780$	0	0
$781$	10079.0	0.461788
$782$	−6791.09	−0.310549
$783$	783.000	0.0357371
$784$	−427.836	−0.0194896
$785$	0	0
$786$	−14737.2	−0.668776
$787$	22352.2	1.01241	0.506206	−	0.862413i	$-0.331048\pi$
$787$	22352.2	1.01241	0.506206	+	0.862413i	$0.331048\pi$
$788$	−38736.0	−1.75116
$789$	13422.9	0.605663
$790$	0	0
$791$	−31727.7	−1.42618
$792$	6238.25	0.279882
$793$	1192.66	0.0534081
$794$	35359.9	1.58045
$795$	0	0
$796$	−1631.05	−0.0726267
$797$	29993.8	1.33304	0.666521	−	0.745487i	$-0.267783\pi$
$797$	29993.8	1.33304	0.666521	+	0.745487i	$0.267783\pi$
$798$	1021.25	0.0453032
$799$	−23151.3	−1.02507
$800$	0	0
$801$	−7093.22	−0.312892
$802$	−26451.0	−1.16461
$803$	−4989.50	−0.219272
$804$	−39222.3	−1.72048
$805$	0	0
$806$	−21772.5	−0.951495
$807$	12159.1	0.530385
$808$	30637.9	1.33396
$809$	28328.9	1.23114	0.615569	−	0.788083i	$-0.288926\pi$
$809$	28328.9	1.23114	0.615569	+	0.788083i	$0.288926\pi$
$810$	0	0
$811$	4087.14	0.176965	0.0884826	−	0.996078i	$-0.471798\pi$
$811$	4087.14	0.176965	0.0884826	+	0.996078i	$0.471798\pi$
$812$	10412.3	0.450001
$813$	−24137.9	−1.04127
$814$	11035.5	0.475178
$815$	0	0
$816$	−33779.0	−1.44914
$817$	1429.28	0.0612046
$818$	27864.6	1.19103
$819$	5721.61	0.244114
$820$	0	0
$821$	−9552.90	−0.406088	−0.203044	−	0.979170i	$-0.565084\pi$
$821$	−9552.90	−0.406088	−0.203044	+	0.979170i	$0.565084\pi$
$822$	−3736.23	−0.158535
$823$	−26863.5	−1.13779	−0.568895	−	0.822410i	$-0.692629\pi$
$823$	−26863.5	−1.13779	−0.568895	+	0.822410i	$0.692629\pi$
$824$	−98862.6	−4.17966
$825$	0	0
$826$	11670.8	0.491619
$827$	43901.7	1.84596	0.922981	−	0.384846i	$-0.125746\pi$
$827$	43901.7	1.84596	0.922981	+	0.384846i	$0.125746\pi$
$828$	3207.61	0.134628
$829$	−9622.68	−0.403148	−0.201574	−	0.979473i	$-0.564606\pi$
$829$	−9622.68	−0.403148	−0.201574	+	0.979473i	$0.564606\pi$
$830$	0	0
$831$	−13877.2	−0.579296
$832$	19926.2	0.830307
$833$	190.295	0.00791516
$834$	26832.3	1.11406
$835$	0	0
$836$	790.779	0.0327149
$837$	−3255.07	−0.134423
$838$	15477.5	0.638022
$839$	26742.6	1.10043	0.550214	−	0.835024i	$-0.314546\pi$
$839$	26742.6	1.10043	0.550214	+	0.835024i	$0.314546\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	38926.0	1.59320
$843$	−209.389	−0.00855486
$844$	−85052.7	−3.46876
$845$	0	0
$846$	15429.6	0.627046
$847$	−22097.7	−0.896440
$848$	−65085.2	−2.63566
$849$	−5710.74	−0.230851
$850$	0	0
$851$	3341.96	0.134619
$852$	51004.3	2.05092
$853$	16600.5	0.666341	0.333171	−	0.942867i	$-0.391881\pi$
$853$	16600.5	0.666341	0.333171	+	0.942867i	$0.391881\pi$
$854$	3345.74	0.134062
$855$	0	0
$856$	78415.6	3.13106
$857$	−15752.2	−0.627869	−0.313935	−	0.949445i	$-0.601647\pi$
$857$	−15752.2	−0.627869	−0.313935	+	0.949445i	$0.601647\pi$
$858$	6251.41	0.248741
$859$	−35249.8	−1.40012	−0.700062	−	0.714082i	$-0.746844\pi$
$859$	−35249.8	−1.40012	−0.700062	+	0.714082i	$0.746844\pi$
$860$	0	0
$861$	4121.31	0.163129
$862$	17304.6	0.683755
$863$	36020.7	1.42081	0.710404	−	0.703794i	$-0.248512\pi$
$863$	36020.7	1.42081	0.710404	+	0.703794i	$0.248512\pi$
$864$	9536.96	0.375526
$865$	0	0
$866$	−49014.0	−1.92328
$867$	285.374	0.0111785
$868$	−43285.9	−1.69265
$869$	−6327.12	−0.246988
$870$	0	0
$871$	−23149.3	−0.900556
$872$	55995.0	2.17458
$873$	−8703.63	−0.337426
$874$	337.910	0.0130778
$875$	0	0
$876$	−25249.0	−0.973842
$877$	−1353.74	−0.0521239	−0.0260620	−	0.999660i	$-0.508297\pi$
$877$	−1353.74	−0.0521239	−0.0260620	+	0.999660i	$0.508297\pi$
$878$	−32018.2	−1.23071
$879$	−1869.59	−0.0717404
$880$	0	0
$881$	−1774.07	−0.0678435	−0.0339217	−	0.999424i	$-0.510800\pi$
$881$	−1774.07	−0.0678435	−0.0339217	+	0.999424i	$0.510800\pi$
$882$	−126.826	−0.00484177
$883$	−1483.09	−0.0565233	−0.0282616	−	0.999601i	$-0.508997\pi$
$883$	−1483.09	−0.0565233	−0.0282616	+	0.999601i	$0.508997\pi$
$884$	−47466.4	−1.80596
$885$	0	0
$886$	−74844.9	−2.83799
$887$	−18524.8	−0.701244	−0.350622	−	0.936517i	$-0.614030\pi$
$887$	−18524.8	−0.701244	−0.350622	+	0.936517i	$0.614030\pi$
$888$	32890.5	1.24294
$889$	16303.8	0.615086
$890$	0	0
$891$	934.609	0.0351409
$892$	−70724.2	−2.65473
$893$	1151.96	0.0431678
$894$	55758.0	2.08594
$895$	0	0
$896$	3769.89	0.140561
$897$	1893.16	0.0704689
$898$	−85976.2	−3.19495
$899$	−3496.19	−0.129705
$900$	0	0
$901$	28948.9	1.07040
$902$	4502.93	0.166221
$903$	22463.5	0.827838
$904$	−103318.	−3.80122
$905$	0	0
$906$	28585.4	1.04822
$907$	26687.1	0.976991	0.488495	−	0.872567i	$-0.337546\pi$
$907$	26687.1	0.976991	0.488495	+	0.872567i	$0.337546\pi$
$908$	71863.4	2.62651
$909$	4590.14	0.167487
$910$	0	0
$911$	−4443.10	−0.161588	−0.0807939	−	0.996731i	$-0.525746\pi$
$911$	−4443.10	−0.161588	−0.0807939	+	0.996731i	$0.525746\pi$
$912$	1680.77	0.0610261
$913$	−5897.00	−0.213759
$914$	82413.2	2.98248
$915$	0	0
$916$	94079.3	3.39352
$917$	−17292.5	−0.622734
$918$	−10013.3	−0.360008
$919$	17674.0	0.634398	0.317199	−	0.948359i	$-0.397258\pi$
$919$	17674.0	0.634398	0.317199	+	0.948359i	$0.397258\pi$
$920$	0	0
$921$	20699.7	0.740585
$922$	−73845.4	−2.63771
$923$	30103.2	1.07352
$924$	12428.4	0.442494
$925$	0	0
$926$	46087.2	1.63555
$927$	−14811.5	−0.524783
$928$	10243.4	0.362345
$929$	10760.5	0.380021	0.190010	−	0.981782i	$-0.439148\pi$
$929$	10760.5	0.380021	0.190010	+	0.981782i	$0.439148\pi$
$930$	0	0
$931$	−9.46867	−0.000333322 0
$932$	94093.2	3.30700
$933$	−544.108	−0.0190925
$934$	45833.9	1.60571
$935$	0	0
$936$	18631.8	0.650641
$937$	−22210.3	−0.774365	−0.387182	−	0.922003i	$-0.626552\pi$
$937$	−22210.3	−0.774365	−0.387182	+	0.922003i	$0.626552\pi$
$938$	−64940.1	−2.26052
$939$	11517.6	0.400281
$940$	0	0
$941$	34977.1	1.21171	0.605855	−	0.795575i	$-0.292831\pi$
$941$	34977.1	1.21171	0.605855	+	0.795575i	$0.292831\pi$
$942$	593.552	0.0205297
$943$	1363.65	0.0470908
$944$	19207.6	0.662241
$945$	0	0
$946$	24543.5	0.843530
$947$	−42798.4	−1.46860	−0.734298	−	0.678827i	$-0.762489\pi$
$947$	−42798.4	−1.46860	−0.734298	+	0.678827i	$0.762489\pi$
$948$	−32018.0	−1.09694
$949$	−14902.2	−0.509742
$950$	0	0
$951$	−21000.2	−0.716067
$952$	−78424.3	−2.66990
$953$	15363.5	0.522217	0.261108	−	0.965309i	$-0.415912\pi$
$953$	15363.5	0.522217	0.261108	+	0.965309i	$0.415912\pi$
$954$	−19293.5	−0.654771
$955$	0	0
$956$	78237.7	2.64685
$957$	1003.84	0.0339075
$958$	−60017.7	−2.02410
$959$	−4384.05	−0.147621
$960$	0	0
$961$	−15256.7	−0.512125
$962$	32959.9	1.10465
$963$	11748.2	0.393125
$964$	20958.1	0.700222
$965$	0	0
$966$	5310.82	0.176887
$967$	44310.2	1.47355	0.736774	−	0.676139i	$-0.236348\pi$
$967$	44310.2	1.47355	0.736774	+	0.676139i	$0.236348\pi$
$968$	−71958.7	−2.38930
$969$	−747.580	−0.0247840
$970$	0	0
$971$	50163.2	1.65789	0.828946	−	0.559328i	$-0.188941\pi$
$971$	50163.2	1.65789	0.828946	+	0.559328i	$0.188941\pi$
$972$	4729.52	0.156070
$973$	31484.8	1.03736
$974$	−8569.90	−0.281927
$975$	0	0
$976$	5506.39	0.180589
$977$	2549.65	0.0834908	0.0417454	−	0.999128i	$-0.486708\pi$
$977$	2549.65	0.0834908	0.0417454	+	0.999128i	$0.486708\pi$
$978$	−24240.7	−0.792568
$979$	−9093.80	−0.296874
$980$	0	0
$981$	8389.13	0.273032
$982$	11041.4	0.358803
$983$	−25382.0	−0.823561	−0.411781	−	0.911283i	$-0.635093\pi$
$983$	−25382.0	−0.823561	−0.411781	+	0.911283i	$0.635093\pi$
$984$	13420.6	0.434790
$985$	0	0
$986$	−10755.0	−0.347372
$987$	18104.9	0.583877
$988$	2361.83	0.0760523
$989$	7432.68	0.238974
$990$	0	0
$991$	−44583.2	−1.42909	−0.714547	−	0.699588i	$-0.753367\pi$
$991$	−44583.2	−1.42909	−0.714547	+	0.699588i	$0.753367\pi$
$992$	−42583.7	−1.36294
$993$	−3530.52	−0.112828
$994$	84447.6	2.69468
$995$	0	0
$996$	−29841.4	−0.949357
$997$	39668.0	1.26008	0.630039	−	0.776563i	$-0.283039\pi$
$997$	39668.0	1.26008	0.630039	+	0.776563i	$0.283039\pi$
$998$	−46461.1	−1.47365
$999$	4927.62	0.156059

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.4.a.n.1.7		7
5.4	even	2		435.4.a.i.1.1	✓	7
15.14	odd	2		1305.4.a.n.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.i.1.1	✓	7	5.4	even	2
1305.4.a.n.1.7		7	15.14	odd	2
2175.4.a.n.1.7		7	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q+5.24052 q^{2} +3.00000 q^{3} +19.4631 q^{4} +15.7216 q^{6} +18.4475 q^{7} +60.0724 q^{8} +9.00000 q^{9} +11.5384 q^{11} +58.3892 q^{12} +34.4618 q^{13} +96.6746 q^{14} +159.106 q^{16} -70.7681 q^{17} +47.1647 q^{18} +3.52127 q^{19} +55.3426 q^{21} +60.4671 q^{22} +18.3117 q^{23} +180.217 q^{24} +180.598 q^{26} +27.0000 q^{27} +359.045 q^{28} +29.0000 q^{29} -120.558 q^{31} +353.221 q^{32} +34.6151 q^{33} -370.862 q^{34} +175.168 q^{36} +182.505 q^{37} +18.4533 q^{38} +103.385 q^{39} +74.4691 q^{41} +290.024 q^{42} +405.899 q^{43} +224.572 q^{44} +95.9626 q^{46} +327.143 q^{47} +477.319 q^{48} -2.68899 q^{49} -212.304 q^{51} +670.732 q^{52} -409.067 q^{53} +141.494 q^{54} +1108.19 q^{56} +10.5638 q^{57} +151.975 q^{58} +120.722 q^{59} +34.6082 q^{61} -631.788 q^{62} +166.028 q^{63} +578.210 q^{64} +181.401 q^{66} -671.739 q^{67} -1377.36 q^{68} +54.9350 q^{69} +873.524 q^{71} +540.652 q^{72} -432.426 q^{73} +956.419 q^{74} +68.5347 q^{76} +212.854 q^{77} +541.793 q^{78} -548.355 q^{79} +81.0000 q^{81} +390.257 q^{82} -511.077 q^{83} +1077.14 q^{84} +2127.12 q^{86} +87.0000 q^{87} +693.138 q^{88} -788.135 q^{89} +635.734 q^{91} +356.401 q^{92} -361.675 q^{93} +1714.40 q^{94} +1059.66 q^{96} -967.070 q^{97} -14.0917 q^{98} +103.845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 21 q^{3} + 22 q^{4} + 6 q^{6} + 50 q^{7} + 33 q^{8} + 63 q^{9} + 76 q^{11} + 66 q^{12} - 30 q^{13} + 89 q^{14} + 138 q^{16} + 140 q^{17} + 18 q^{18} + 90 q^{19} + 150 q^{21} - 61 q^{22}+ \cdots + 684 q^{99}+O(q^{100}) \) 7 * q + 2 * q^2 + 21 * q^3 + 22 * q^4 + 6 * q^6 + 50 * q^7 + 33 * q^8 + 63 * q^9 + 76 * q^11 + 66 * q^12 - 30 * q^13 + 89 * q^14 + 138 * q^16 + 140 * q^17 + 18 * q^18 + 90 * q^19 + 150 * q^21 - 61 * q^22 - 34 * q^23 + 99 * q^24 - 241 * q^26 + 189 * q^27 + 57 * q^28 + 203 * q^29 + 524 * q^31 + 6 * q^32 + 228 * q^33 + 255 * q^34 + 198 * q^36 + 28 * q^37 - 222 * q^38 - 90 * q^39 + 1532 * q^41 + 267 * q^42 + 464 * q^43 + 1475 * q^44 + 72 * q^46 + 360 * q^47 + 414 * q^48 + 569 * q^49 + 420 * q^51 + 205 * q^52 - 282 * q^53 + 54 * q^54 + 1102 * q^56 + 270 * q^57 + 58 * q^58 + 766 * q^59 + 1200 * q^61 - 2856 * q^62 + 450 * q^63 + 701 * q^64 - 183 * q^66 - 1546 * q^67 - 1801 * q^68 - 102 * q^69 + 1802 * q^71 + 297 * q^72 + 220 * q^73 + 1594 * q^74 + 1960 * q^76 - 3222 * q^77 - 723 * q^78 + 1298 * q^79 + 567 * q^81 - 856 * q^82 - 1652 * q^83 + 171 * q^84 + 7628 * q^86 + 609 * q^87 - 550 * q^88 + 2846 * q^89 - 816 * q^91 - 472 * q^92 + 1572 * q^93 + 745 * q^94 + 18 * q^96 - 1110 * q^97 + 761 * q^98 + 684 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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