LMFDB - Embedded newform 1575.4.a.bm.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,6,0,16,0,0,28,93,0,0,-57] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.26729725.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 18x^{2} + 19x + 44 $ x^4 - 2x^3 - 18x^2 + 19*x + 44
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.21734$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.217342 q^{2} -7.95276 q^{4} +7.00000 q^{7} +3.46721 q^{8} +30.6085 q^{11} +25.3178 q^{13} -1.52140 q^{14} +62.8685 q^{16} -72.8676 q^{17} +122.711 q^{19} -6.65253 q^{22} +194.258 q^{23} -5.50264 q^{26} -55.6693 q^{28} -48.6103 q^{29} -288.907 q^{31} -41.4017 q^{32} +15.8372 q^{34} +15.8251 q^{37} -26.6702 q^{38} -452.905 q^{41} -152.574 q^{43} -243.422 q^{44} -42.2205 q^{46} +164.435 q^{47} +49.0000 q^{49} -201.347 q^{52} +591.600 q^{53} +24.2705 q^{56} +10.5651 q^{58} +180.823 q^{59} +115.773 q^{61} +62.7918 q^{62} -493.950 q^{64} -605.264 q^{67} +579.499 q^{68} +990.917 q^{71} -863.756 q^{73} -3.43947 q^{74} -975.889 q^{76} +214.260 q^{77} +965.930 q^{79} +98.4355 q^{82} +160.924 q^{83} +33.1608 q^{86} +106.126 q^{88} +51.6227 q^{89} +177.225 q^{91} -1544.89 q^{92} -35.7387 q^{94} -1497.31 q^{97} -10.6498 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{2} + 16 q^{4} + 28 q^{7} + 93 q^{8} - 57 q^{11} - 43 q^{13} + 42 q^{14} + 216 q^{16} + 99 q^{17} - 12 q^{19} + 41 q^{22} + 156 q^{23} + 81 q^{26} + 112 q^{28} - 378 q^{29} - 93 q^{31} + 690 q^{32}+ \cdots + 294 q^{98}+O(q^{100}) $ 4 * q + 6 * q^2 + 16 * q^4 + 28 * q^7 + 93 * q^8 - 57 * q^11 - 43 * q^13 + 42 * q^14 + 216 * q^16 + 99 * q^17 - 12 * q^19 + 41 * q^22 + 156 * q^23 + 81 * q^26 + 112 * q^28 - 378 * q^29 - 93 * q^31 + 690 * q^32 + 783 * q^34 - 81 * q^37 + 216 * q^38 + 465 * q^41 + 64 * q^43 - 681 * q^44 + 310 * q^46 + 744 * q^47 + 196 * q^49 + 727 * q^52 + 729 * q^53 + 651 * q^56 - 1172 * q^58 - 231 * q^59 - 1353 * q^61 - 165 * q^62 + 3107 * q^64 - 1487 * q^67 + 2577 * q^68 + 1725 * q^71 - 512 * q^73 + 1953 * q^74 - 3046 * q^76 - 399 * q^77 + 1629 * q^79 + 693 * q^82 + 321 * q^83 + 4542 * q^86 - 3482 * q^88 + 978 * q^89 - 301 * q^91 + 852 * q^92 + 2480 * q^94 - 2616 * q^97 + 294 * q^98

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 $

$2$	−0.217342	−0.0768421	−0.0384211	−	0.999262i	$-0.512233\pi$
$2$	−0.217342	−0.0768421	−0.0384211	+	0.999262i	$0.512233\pi$
$3$	0	0
$3$	0	0
$4$	−7.95276	−0.994095
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	3.46721	0.153231
$9$	0	0
$10$	0	0
$11$	30.6085	0.838983	0.419492	−	0.907759i	$-0.362208\pi$
$11$	30.6085	0.838983	0.419492	+	0.907759i	$0.362208\pi$
$12$	0	0
$13$	25.3178	0.540146	0.270073	−	0.962840i	$-0.412952\pi$
$13$	25.3178	0.540146	0.270073	+	0.962840i	$0.412952\pi$
$14$	−1.52140	−0.0290436
$15$	0	0
$16$	62.8685	0.982321
$17$	−72.8676	−1.03959	−0.519794	−	0.854292i	$-0.673991\pi$
$17$	−72.8676	−1.03959	−0.519794	+	0.854292i	$0.673991\pi$
$18$	0	0
$19$	122.711	1.48167	0.740836	−	0.671686i	$-0.234430\pi$
$19$	122.711	1.48167	0.740836	+	0.671686i	$0.234430\pi$
$20$	0	0
$21$	0	0
$22$	−6.65253	−0.0644692
$23$	194.258	1.76111	0.880556	−	0.473943i	$-0.157170\pi$
$23$	194.258	1.76111	0.880556	+	0.473943i	$0.157170\pi$
$24$	0	0
$25$	0	0
$26$	−5.50264	−0.0415060
$27$	0	0
$28$	−55.6693	−0.375733
$29$	−48.6103	−0.311266	−0.155633	−	0.987815i	$-0.549742\pi$
$29$	−48.6103	−0.311266	−0.155633	+	0.987815i	$0.549742\pi$
$30$	0	0
$31$	−288.907	−1.67385	−0.836924	−	0.547320i	$-0.815648\pi$
$31$	−288.907	−1.67385	−0.836924	+	0.547320i	$0.815648\pi$
$32$	−41.4017	−0.228714
$33$	0	0
$34$	15.8372	0.0798841
$35$	0	0
$36$	0	0
$37$	15.8251	0.0703144	0.0351572	−	0.999382i	$-0.488807\pi$
$37$	15.8251	0.0703144	0.0351572	+	0.999382i	$0.488807\pi$
$38$	−26.6702	−0.113855
$39$	0	0
$40$	0	0
$41$	−452.905	−1.72517	−0.862584	−	0.505914i	$-0.831155\pi$
$41$	−452.905	−1.72517	−0.862584	+	0.505914i	$0.831155\pi$
$42$	0	0
$43$	−152.574	−0.541101	−0.270550	−	0.962706i	$-0.587206\pi$
$43$	−152.574	−0.541101	−0.270550	+	0.962706i	$0.587206\pi$
$44$	−243.422	−0.834029
$45$	0	0
$46$	−42.2205	−0.135328
$47$	164.435	0.510325	0.255163	−	0.966898i	$-0.417871\pi$
$47$	164.435	0.510325	0.255163	+	0.966898i	$0.417871\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−201.347	−0.536957
$53$	591.600	1.53326	0.766628	−	0.642092i	$-0.221933\pi$
$53$	591.600	1.53326	0.766628	+	0.642092i	$0.221933\pi$
$54$	0	0
$55$	0	0
$56$	24.2705	0.0579157
$57$	0	0
$58$	10.5651	0.0239183
$59$	180.823	0.399003	0.199501	−	0.979898i	$-0.436068\pi$
$59$	180.823	0.399003	0.199501	+	0.979898i	$0.436068\pi$
$60$	0	0
$61$	115.773	0.243004	0.121502	−	0.992591i	$-0.461229\pi$
$61$	115.773	0.243004	0.121502	+	0.992591i	$0.461229\pi$
$62$	62.7918	0.128622
$63$	0	0
$64$	−493.950	−0.964746
$65$	0	0
$66$	0	0
$67$	−605.264	−1.10365	−0.551827	−	0.833959i	$-0.686069\pi$
$67$	−605.264	−1.10365	−0.551827	+	0.833959i	$0.686069\pi$
$68$	579.499	1.03345
$69$	0	0
$70$	0	0
$71$	990.917	1.65634	0.828170	−	0.560477i	$-0.189382\pi$
$71$	990.917	1.65634	0.828170	+	0.560477i	$0.189382\pi$
$72$	0	0
$73$	−863.756	−1.38486	−0.692431	−	0.721484i	$-0.743460\pi$
$73$	−863.756	−1.38486	−0.692431	+	0.721484i	$0.743460\pi$
$74$	−3.43947	−0.00540311
$75$	0	0
$76$	−975.889	−1.47292
$77$	214.260	0.317106
$78$	0	0
$79$	965.930	1.37564	0.687821	−	0.725881i	$-0.258568\pi$
$79$	965.930	1.37564	0.687821	+	0.725881i	$0.258568\pi$
$80$	0	0
$81$	0	0
$82$	98.4355	0.132566
$83$	160.924	0.212816	0.106408	−	0.994323i	$-0.466065\pi$
$83$	160.924	0.212816	0.106408	+	0.994323i	$0.466065\pi$
$84$	0	0
$85$	0	0
$86$	33.1608	0.0415793
$87$	0	0
$88$	106.126	0.128558
$89$	51.6227	0.0614831	0.0307415	−	0.999527i	$-0.490213\pi$
$89$	51.6227	0.0614831	0.0307415	+	0.999527i	$0.490213\pi$
$90$	0	0
$91$	177.225	0.204156
$92$	−1544.89	−1.75071
$93$	0	0
$94$	−35.7387	−0.0392145
$95$	0	0
$96$	0	0
$97$	−1497.31	−1.56731	−0.783656	−	0.621195i	$-0.786647\pi$
$97$	−1497.31	−1.56731	−0.783656	+	0.621195i	$0.786647\pi$
$98$	−10.6498	−0.0109774
$99$	0	0
$100$	0	0
$101$	485.333	0.478143	0.239071	−	0.971002i	$-0.423157\pi$
$101$	485.333	0.478143	0.239071	+	0.971002i	$0.423157\pi$
$102$	0	0
$103$	110.629	0.105831	0.0529156	−	0.998599i	$-0.483149\pi$
$103$	110.629	0.105831	0.0529156	+	0.998599i	$0.483149\pi$
$104$	87.7822	0.0827669
$105$	0	0
$106$	−128.580	−0.117819
$107$	361.440	0.326559	0.163279	−	0.986580i	$-0.447793\pi$
$107$	361.440	0.326559	0.163279	+	0.986580i	$0.447793\pi$
$108$	0	0
$109$	571.242	0.501973	0.250986	−	0.967991i	$-0.419245\pi$
$109$	571.242	0.501973	0.250986	+	0.967991i	$0.419245\pi$
$110$	0	0
$111$	0	0
$112$	440.080	0.371282
$113$	1389.82	1.15702	0.578508	−	0.815677i	$-0.303635\pi$
$113$	1389.82	1.15702	0.578508	+	0.815677i	$0.303635\pi$
$114$	0	0
$115$	0	0
$116$	386.586	0.309428
$117$	0	0
$118$	−39.3005	−0.0306602
$119$	−510.073	−0.392927
$120$	0	0
$121$	−394.119	−0.296107
$122$	−25.1624	−0.0186729
$123$	0	0
$124$	2297.61	1.66396
$125$	0	0
$126$	0	0
$127$	777.868	0.543501	0.271751	−	0.962368i	$-0.412397\pi$
$127$	777.868	0.543501	0.271751	+	0.962368i	$0.412397\pi$
$128$	438.570	0.302847
$129$	0	0
$130$	0	0
$131$	172.164	0.114825	0.0574125	−	0.998351i	$-0.481715\pi$
$131$	172.164	0.114825	0.0574125	+	0.998351i	$0.481715\pi$
$132$	0	0
$133$	858.975	0.560019
$134$	131.549	0.0848070
$135$	0	0
$136$	−252.647	−0.159297
$137$	−2010.10	−1.25354	−0.626769	−	0.779205i	$-0.715623\pi$
$137$	−2010.10	−1.25354	−0.626769	+	0.779205i	$0.715623\pi$
$138$	0	0
$139$	−14.4226	−0.00880080	−0.00440040	−	0.999990i	$-0.501401\pi$
$139$	−14.4226	−0.00880080	−0.00440040	+	0.999990i	$0.501401\pi$
$140$	0	0
$141$	0	0
$142$	−215.368	−0.127277
$143$	774.941	0.453174
$144$	0	0
$145$	0	0
$146$	187.731	0.106416
$147$	0	0
$148$	−125.853	−0.0698992
$149$	1286.13	0.707140	0.353570	−	0.935408i	$-0.384968\pi$
$149$	1286.13	0.707140	0.353570	+	0.935408i	$0.384968\pi$
$150$	0	0
$151$	3048.17	1.64276	0.821381	−	0.570380i	$-0.193204\pi$
$151$	3048.17	1.64276	0.821381	+	0.570380i	$0.193204\pi$
$152$	425.464	0.227037
$153$	0	0
$154$	−46.5677	−0.0243671
$155$	0	0
$156$	0	0
$157$	318.632	0.161972	0.0809859	−	0.996715i	$-0.474193\pi$
$157$	318.632	0.161972	0.0809859	+	0.996715i	$0.474193\pi$
$158$	−209.938	−0.105707
$159$	0	0
$160$	0	0
$161$	1359.80	0.665638
$162$	0	0
$163$	499.091	0.239827	0.119914	−	0.992784i	$-0.461738\pi$
$163$	499.091	0.239827	0.119914	+	0.992784i	$0.461738\pi$
$164$	3601.85	1.71498
$165$	0	0
$166$	−34.9756	−0.0163532
$167$	1908.27	0.884231	0.442116	−	0.896958i	$-0.354228\pi$
$167$	1908.27	0.884231	0.442116	+	0.896958i	$0.354228\pi$
$168$	0	0
$169$	−1556.01	−0.708242
$170$	0	0
$171$	0	0
$172$	1213.39	0.537906
$173$	−1401.96	−0.616121	−0.308061	−	0.951367i	$-0.599680\pi$
$173$	−1401.96	−0.616121	−0.308061	+	0.951367i	$0.599680\pi$
$174$	0	0
$175$	0	0
$176$	1924.31	0.824150
$177$	0	0
$178$	−11.2198	−0.00472449
$179$	3529.16	1.47364	0.736821	−	0.676088i	$-0.236326\pi$
$179$	3529.16	1.47364	0.736821	+	0.676088i	$0.236326\pi$
$180$	0	0
$181$	4356.04	1.78885	0.894425	−	0.447217i	$-0.147585\pi$
$181$	4356.04	1.78885	0.894425	+	0.447217i	$0.147585\pi$
$182$	−38.5185	−0.0156878
$183$	0	0
$184$	673.533	0.269856
$185$	0	0
$186$	0	0
$187$	−2230.37	−0.872197
$188$	−1307.71	−0.507312
$189$	0	0
$190$	0	0
$191$	−1203.33	−0.455864	−0.227932	−	0.973677i	$-0.573196\pi$
$191$	−1203.33	−0.455864	−0.227932	+	0.973677i	$0.573196\pi$
$192$	0	0
$193$	−1553.31	−0.579323	−0.289662	−	0.957129i	$-0.593543\pi$
$193$	−1553.31	−0.579323	−0.289662	+	0.957129i	$0.593543\pi$
$194$	325.430	0.120436
$195$	0	0
$196$	−389.685	−0.142014
$197$	−1826.02	−0.660397	−0.330199	−	0.943912i	$-0.607116\pi$
$197$	−1826.02	−0.660397	−0.330199	+	0.943912i	$0.607116\pi$
$198$	0	0
$199$	596.394	0.212448	0.106224	−	0.994342i	$-0.466124\pi$
$199$	596.394	0.212448	0.106224	+	0.994342i	$0.466124\pi$
$200$	0	0
$201$	0	0
$202$	−105.483	−0.0367415
$203$	−340.272	−0.117647
$204$	0	0
$205$	0	0
$206$	−24.0444	−0.00813229
$207$	0	0
$208$	1591.69	0.530597
$209$	3755.99	1.24310
$210$	0	0
$211$	122.070	0.0398276	0.0199138	−	0.999802i	$-0.493661\pi$
$211$	122.070	0.0398276	0.0199138	+	0.999802i	$0.493661\pi$
$212$	−4704.86	−1.52420
$213$	0	0
$214$	−78.5563	−0.0250935
$215$	0	0
$216$	0	0
$217$	−2022.35	−0.632655
$218$	−124.155	−0.0385726
$219$	0	0
$220$	0	0
$221$	−1844.85	−0.561530
$222$	0	0
$223$	3619.02	1.08676	0.543380	−	0.839487i	$-0.317144\pi$
$223$	3619.02	1.08676	0.543380	+	0.839487i	$0.317144\pi$
$224$	−289.812	−0.0864458
$225$	0	0
$226$	−302.066	−0.0889076
$227$	5648.79	1.65165	0.825823	−	0.563929i	$-0.190711\pi$
$227$	5648.79	1.65165	0.825823	+	0.563929i	$0.190711\pi$
$228$	0	0
$229$	−619.352	−0.178725	−0.0893623	−	0.995999i	$-0.528483\pi$
$229$	−619.352	−0.178725	−0.0893623	+	0.995999i	$0.528483\pi$
$230$	0	0
$231$	0	0
$232$	−168.542	−0.0476954
$233$	675.465	0.189919	0.0949597	−	0.995481i	$-0.469728\pi$
$233$	675.465	0.189919	0.0949597	+	0.995481i	$0.469728\pi$
$234$	0	0
$235$	0	0
$236$	−1438.04	−0.396647
$237$	0	0
$238$	110.861	0.0301934
$239$	5799.50	1.56962	0.784809	−	0.619738i	$-0.212761\pi$
$239$	5799.50	1.56962	0.784809	+	0.619738i	$0.212761\pi$
$240$	0	0
$241$	−2855.14	−0.763136	−0.381568	−	0.924341i	$-0.624616\pi$
$241$	−2855.14	−0.763136	−0.381568	+	0.924341i	$0.624616\pi$
$242$	85.6587	0.0227535
$243$	0	0
$244$	−920.716	−0.241569
$245$	0	0
$246$	0	0
$247$	3106.77	0.800319
$248$	−1001.70	−0.256484
$249$	0	0
$250$	0	0
$251$	6378.07	1.60390	0.801952	−	0.597388i	$-0.203795\pi$
$251$	6378.07	1.60390	0.801952	+	0.597388i	$0.203795\pi$
$252$	0	0
$253$	5945.94	1.47754
$254$	−169.064	−0.0417638
$255$	0	0
$256$	3856.28	0.941474
$257$	−7570.27	−1.83743	−0.918716	−	0.394918i	$-0.870773\pi$
$257$	−7570.27	−1.83743	−0.918716	+	0.394918i	$0.870773\pi$
$258$	0	0
$259$	110.776	0.0265763
$260$	0	0
$261$	0	0
$262$	−37.4186	−0.00882339
$263$	4258.56	0.998456	0.499228	−	0.866471i	$-0.333617\pi$
$263$	4258.56	0.998456	0.499228	+	0.866471i	$0.333617\pi$
$264$	0	0
$265$	0	0
$266$	−186.692	−0.0430331
$267$	0	0
$268$	4813.52	1.09714
$269$	6518.96	1.47758	0.738788	−	0.673938i	$-0.235399\pi$
$269$	6518.96	1.47758	0.738788	+	0.673938i	$0.235399\pi$
$270$	0	0
$271$	1565.51	0.350914	0.175457	−	0.984487i	$-0.443860\pi$
$271$	1565.51	0.350914	0.175457	+	0.984487i	$0.443860\pi$
$272$	−4581.08	−1.02121
$273$	0	0
$274$	436.881	0.0963245
$275$	0	0
$276$	0	0
$277$	2396.83	0.519897	0.259948	−	0.965623i	$-0.416294\pi$
$277$	2396.83	0.519897	0.259948	+	0.965623i	$0.416294\pi$
$278$	3.13465	0.000676272 0
$279$	0	0
$280$	0	0
$281$	−774.094	−0.164336	−0.0821682	−	0.996618i	$-0.526184\pi$
$281$	−774.094	−0.164336	−0.0821682	+	0.996618i	$0.526184\pi$
$282$	0	0
$283$	4711.07	0.989554	0.494777	−	0.869020i	$-0.335250\pi$
$283$	4711.07	0.989554	0.494777	+	0.869020i	$0.335250\pi$
$284$	−7880.52	−1.64656
$285$	0	0
$286$	−168.428	−0.0348228
$287$	−3170.34	−0.652052
$288$	0	0
$289$	396.690	0.0807428
$290$	0	0
$291$	0	0
$292$	6869.24	1.37668
$293$	1141.66	0.227634	0.113817	−	0.993502i	$-0.463692\pi$
$293$	1141.66	0.227634	0.113817	+	0.993502i	$0.463692\pi$
$294$	0	0
$295$	0	0
$296$	54.8690	0.0107743
$297$	0	0
$298$	−279.531	−0.0543382
$299$	4918.19	0.951258
$300$	0	0
$301$	−1068.02	−0.204517
$302$	−662.498	−0.126233
$303$	0	0
$304$	7714.64	1.45548
$305$	0	0
$306$	0	0
$307$	−8177.38	−1.52022	−0.760110	−	0.649795i	$-0.774855\pi$
$307$	−8177.38	−1.52022	−0.760110	+	0.649795i	$0.774855\pi$
$308$	−1703.96	−0.315233
$309$	0	0
$310$	0	0
$311$	−728.496	−0.132827	−0.0664136	−	0.997792i	$-0.521156\pi$
$311$	−728.496	−0.132827	−0.0664136	+	0.997792i	$0.521156\pi$
$312$	0	0
$313$	−5505.37	−0.994191	−0.497096	−	0.867696i	$-0.665600\pi$
$313$	−5505.37	−0.994191	−0.497096	+	0.867696i	$0.665600\pi$
$314$	−69.2521	−0.0124463
$315$	0	0
$316$	−7681.81	−1.36752
$317$	4947.15	0.876529	0.438264	−	0.898846i	$-0.355593\pi$
$317$	4947.15	0.876529	0.438264	+	0.898846i	$0.355593\pi$
$318$	0	0
$319$	−1487.89	−0.261146
$320$	0	0
$321$	0	0
$322$	−295.543	−0.0511490
$323$	−8941.63	−1.54033
$324$	0	0
$325$	0	0
$326$	−108.474	−0.0184288
$327$	0	0
$328$	−1570.32	−0.264348
$329$	1151.04	0.192885
$330$	0	0
$331$	5893.79	0.978707	0.489353	−	0.872086i	$-0.337233\pi$
$331$	5893.79	0.978707	0.489353	+	0.872086i	$0.337233\pi$
$332$	−1279.79	−0.211559
$333$	0	0
$334$	−414.749	−0.0679462
$335$	0	0
$336$	0	0
$337$	−10148.7	−1.64046	−0.820231	−	0.572033i	$-0.806155\pi$
$337$	−10148.7	−1.64046	−0.820231	+	0.572033i	$0.806155\pi$
$338$	338.186	0.0544228
$339$	0	0
$340$	0	0
$341$	−8843.02	−1.40433
$342$	0	0
$343$	343.000	0.0539949
$344$	−529.007	−0.0829131
$345$	0	0
$346$	304.705	0.0473441
$347$	−9389.81	−1.45266	−0.726328	−	0.687349i	$-0.758774\pi$
$347$	−9389.81	−1.45266	−0.726328	+	0.687349i	$0.758774\pi$
$348$	0	0
$349$	−1302.83	−0.199825	−0.0999127	−	0.994996i	$-0.531856\pi$
$349$	−1302.83	−0.199825	−0.0999127	+	0.994996i	$0.531856\pi$
$350$	0	0
$351$	0	0
$352$	−1267.24	−0.191887
$353$	9448.64	1.42465	0.712323	−	0.701851i	$-0.247643\pi$
$353$	9448.64	1.42465	0.712323	+	0.701851i	$0.247643\pi$
$354$	0	0
$355$	0	0
$356$	−410.543	−0.0611201
$357$	0	0
$358$	−767.036	−0.113238
$359$	7937.83	1.16697	0.583486	−	0.812124i	$-0.301688\pi$
$359$	7937.83	1.16697	0.583486	+	0.812124i	$0.301688\pi$
$360$	0	0
$361$	8198.90	1.19535
$362$	−946.752	−0.137459
$363$	0	0
$364$	−1409.43	−0.202951
$365$	0	0
$366$	0	0
$367$	−2030.61	−0.288821	−0.144410	−	0.989518i	$-0.546129\pi$
$367$	−2030.61	−0.288821	−0.144410	+	0.989518i	$0.546129\pi$
$368$	12212.7	1.72998
$369$	0	0
$370$	0	0
$371$	4141.20	0.579516
$372$	0	0
$373$	−10507.7	−1.45863	−0.729314	−	0.684179i	$-0.760161\pi$
$373$	−10507.7	−1.45863	−0.729314	+	0.684179i	$0.760161\pi$
$374$	484.754	0.0670214
$375$	0	0
$376$	570.130	0.0781974
$377$	−1230.71	−0.168129
$378$	0	0
$379$	10524.4	1.42639	0.713196	−	0.700965i	$-0.247247\pi$
$379$	10524.4	1.42639	0.713196	+	0.700965i	$0.247247\pi$
$380$	0	0
$381$	0	0
$382$	261.535	0.0350296
$383$	10464.3	1.39609	0.698045	−	0.716053i	$-0.254053\pi$
$383$	10464.3	1.39609	0.698045	+	0.716053i	$0.254053\pi$
$384$	0	0
$385$	0	0
$386$	337.599	0.0445164
$387$	0	0
$388$	11907.8	1.55806
$389$	−7674.91	−1.00034	−0.500172	−	0.865926i	$-0.666730\pi$
$389$	−7674.91	−1.00034	−0.500172	+	0.865926i	$0.666730\pi$
$390$	0	0
$391$	−14155.1	−1.83083
$392$	169.893	0.0218901
$393$	0	0
$394$	396.870	0.0507463
$395$	0	0
$396$	0	0
$397$	7.14346	0.000903072 0	0.000451536	−	1.00000i	$-0.499856\pi$
$397$	7.14346	0.000903072 0	0.000451536		1.00000i	$0.499856\pi$
$398$	−129.622	−0.0163250
$399$	0	0
$400$	0	0
$401$	−1756.24	−0.218709	−0.109355	−	0.994003i	$-0.534878\pi$
$401$	−1756.24	−0.218709	−0.109355	+	0.994003i	$0.534878\pi$
$402$	0	0
$403$	−7314.50	−0.904122
$404$	−3859.74	−0.475320
$405$	0	0
$406$	73.9555	0.00904027
$407$	484.383	0.0589926
$408$	0	0
$409$	−11484.0	−1.38838	−0.694192	−	0.719790i	$-0.744238\pi$
$409$	−11484.0	−1.38838	−0.694192	+	0.719790i	$0.744238\pi$
$410$	0	0
$411$	0	0
$412$	−879.807	−0.105206
$413$	1265.76	0.150809
$414$	0	0
$415$	0	0
$416$	−1048.20	−0.123539
$417$	0	0
$418$	−816.336	−0.0955222
$419$	3229.63	0.376558	0.188279	−	0.982116i	$-0.439709\pi$
$419$	3229.63	0.376558	0.188279	+	0.982116i	$0.439709\pi$
$420$	0	0
$421$	11091.5	1.28400	0.642001	−	0.766703i	$-0.278104\pi$
$421$	11091.5	1.28400	0.642001	+	0.766703i	$0.278104\pi$
$422$	−26.5309	−0.00306044
$423$	0	0
$424$	2051.20	0.234942
$425$	0	0
$426$	0	0
$427$	810.411	0.0918467
$428$	−2874.45	−0.324630
$429$	0	0
$430$	0	0
$431$	−12807.8	−1.43139	−0.715695	−	0.698413i	$-0.753890\pi$
$431$	−12807.8	−1.43139	−0.715695	+	0.698413i	$0.753890\pi$
$432$	0	0
$433$	7189.27	0.797908	0.398954	−	0.916971i	$-0.369373\pi$
$433$	7189.27	0.797908	0.398954	+	0.916971i	$0.369373\pi$
$434$	439.542	0.0486145
$435$	0	0
$436$	−4542.95	−0.499009
$437$	23837.5	2.60939
$438$	0	0
$439$	4112.88	0.447146	0.223573	−	0.974687i	$-0.428228\pi$
$439$	4112.88	0.447146	0.223573	+	0.974687i	$0.428228\pi$
$440$	0	0
$441$	0	0
$442$	400.964	0.0431491
$443$	8575.57	0.919724	0.459862	−	0.887991i	$-0.347899\pi$
$443$	8575.57	0.919724	0.459862	+	0.887991i	$0.347899\pi$
$444$	0	0
$445$	0	0
$446$	−786.566	−0.0835089
$447$	0	0
$448$	−3457.65	−0.364640
$449$	−17872.4	−1.87850	−0.939252	−	0.343227i	$-0.888480\pi$
$449$	−17872.4	−1.87850	−0.939252	+	0.343227i	$0.888480\pi$
$450$	0	0
$451$	−13862.8	−1.44739
$452$	−11052.9	−1.15018
$453$	0	0
$454$	−1227.72	−0.126916
$455$	0	0
$456$	0	0
$457$	12602.8	1.29000	0.645002	−	0.764181i	$-0.276856\pi$
$457$	12602.8	1.29000	0.645002	+	0.764181i	$0.276856\pi$
$458$	134.611	0.0137336
$459$	0	0
$460$	0	0
$461$	18008.7	1.81941	0.909703	−	0.415259i	$-0.136309\pi$
$461$	18008.7	1.81941	0.909703	+	0.415259i	$0.136309\pi$
$462$	0	0
$463$	−15173.3	−1.52303	−0.761513	−	0.648150i	$-0.775543\pi$
$463$	−15173.3	−1.52303	−0.761513	+	0.648150i	$0.775543\pi$
$464$	−3056.06	−0.305763
$465$	0	0
$466$	−146.807	−0.0145938
$467$	8759.79	0.867998	0.433999	−	0.900913i	$-0.357102\pi$
$467$	8759.79	0.867998	0.433999	+	0.900913i	$0.357102\pi$
$468$	0	0
$469$	−4236.85	−0.417142
$470$	0	0
$471$	0	0
$472$	626.952	0.0611394
$473$	−4670.07	−0.453974
$474$	0	0
$475$	0	0
$476$	4056.49	0.390607
$477$	0	0
$478$	−1260.48	−0.120613
$479$	−9047.02	−0.862983	−0.431491	−	0.902117i	$-0.642012\pi$
$479$	−9047.02	−0.862983	−0.431491	+	0.902117i	$0.642012\pi$
$480$	0	0
$481$	400.657	0.0379800
$482$	620.543	0.0586410
$483$	0	0
$484$	3134.33	0.294359
$485$	0	0
$486$	0	0
$487$	1832.63	0.170523	0.0852614	−	0.996359i	$-0.472827\pi$
$487$	1832.63	0.170523	0.0852614	+	0.996359i	$0.472827\pi$
$488$	401.410	0.0372356
$489$	0	0
$490$	0	0
$491$	20593.6	1.89282	0.946411	−	0.322964i	$-0.104679\pi$
$491$	20593.6	1.89282	0.946411	+	0.322964i	$0.104679\pi$
$492$	0	0
$493$	3542.11	0.323588
$494$	−675.232	−0.0614982
$495$	0	0
$496$	−18163.2	−1.64425
$497$	6936.42	0.626038
$498$	0	0
$499$	4863.99	0.436357	0.218178	−	0.975909i	$-0.429989\pi$
$499$	4863.99	0.436357	0.218178	+	0.975909i	$0.429989\pi$
$500$	0	0
$501$	0	0
$502$	−1386.22	−0.123247
$503$	−11688.3	−1.03609	−0.518047	−	0.855352i	$-0.673341\pi$
$503$	−11688.3	−1.03609	−0.518047	+	0.855352i	$0.673341\pi$
$504$	0	0
$505$	0	0
$506$	−1292.31	−0.113538
$507$	0	0
$508$	−6186.20	−0.540292
$509$	16917.0	1.47315	0.736573	−	0.676358i	$-0.236443\pi$
$509$	16917.0	1.47315	0.736573	+	0.676358i	$0.236443\pi$
$510$	0	0
$511$	−6046.29	−0.523429
$512$	−4346.69	−0.375192
$513$	0	0
$514$	1645.34	0.141192
$515$	0	0
$516$	0	0
$517$	5033.11	0.428154
$518$	−24.0763	−0.00204218
$519$	0	0
$520$	0	0
$521$	13797.3	1.16022	0.580108	−	0.814540i	$-0.303010\pi$
$521$	13797.3	1.16022	0.580108	+	0.814540i	$0.303010\pi$
$522$	0	0
$523$	−19819.3	−1.65705	−0.828524	−	0.559953i	$-0.810819\pi$
$523$	−19819.3	−1.65705	−0.828524	+	0.559953i	$0.810819\pi$
$524$	−1369.18	−0.114147
$525$	0	0
$526$	−925.565	−0.0767234
$527$	21052.0	1.74011
$528$	0	0
$529$	25569.1	2.10151
$530$	0	0
$531$	0	0
$532$	−6831.22	−0.556712
$533$	−11466.6	−0.931843
$534$	0	0
$535$	0	0
$536$	−2098.58	−0.169113
$537$	0	0
$538$	−1416.85	−0.113540
$539$	1499.82	0.119855
$540$	0	0
$541$	14896.9	1.18386	0.591930	−	0.805989i	$-0.298366\pi$
$541$	14896.9	1.18386	0.591930	+	0.805989i	$0.298366\pi$
$542$	−340.251	−0.0269650
$543$	0	0
$544$	3016.84	0.237768
$545$	0	0
$546$	0	0
$547$	−17673.2	−1.38145	−0.690724	−	0.723119i	$-0.742708\pi$
$547$	−17673.2	−1.38145	−0.690724	+	0.723119i	$0.742708\pi$
$548$	15985.9	1.24614
$549$	0	0
$550$	0	0
$551$	−5965.00	−0.461193
$552$	0	0
$553$	6761.51	0.519944
$554$	−520.932	−0.0399500
$555$	0	0
$556$	114.700	0.00874883
$557$	−10008.9	−0.761382	−0.380691	−	0.924702i	$-0.624314\pi$
$557$	−10008.9	−0.761382	−0.380691	+	0.924702i	$0.624314\pi$
$558$	0	0
$559$	−3862.84	−0.292274
$560$	0	0
$561$	0	0
$562$	168.243	0.0126280
$563$	−5504.86	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$563$	−5504.86	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$564$	0	0
$565$	0	0
$566$	−1023.91	−0.0760394
$567$	0	0
$568$	3435.72	0.253802
$569$	−9097.13	−0.670248	−0.335124	−	0.942174i	$-0.608778\pi$
$569$	−9097.13	−0.670248	−0.335124	+	0.942174i	$0.608778\pi$
$570$	0	0
$571$	21105.4	1.54682	0.773408	−	0.633908i	$-0.218550\pi$
$571$	21105.4	1.54682	0.773408	+	0.633908i	$0.218550\pi$
$572$	−6162.92	−0.450498
$573$	0	0
$574$	689.048	0.0501051
$575$	0	0
$576$	0	0
$577$	7916.35	0.571165	0.285582	−	0.958354i	$-0.407813\pi$
$577$	7916.35	0.571165	0.285582	+	0.958354i	$0.407813\pi$
$578$	−86.2174	−0.00620445
$579$	0	0
$580$	0	0
$581$	1126.47	0.0804369
$582$	0	0
$583$	18108.0	1.28638
$584$	−2994.82	−0.212203
$585$	0	0
$586$	−248.132	−0.0174919
$587$	4843.75	0.340584	0.170292	−	0.985394i	$-0.445529\pi$
$587$	4843.75	0.340584	0.170292	+	0.985394i	$0.445529\pi$
$588$	0	0
$589$	−35452.0	−2.48009
$590$	0	0
$591$	0	0
$592$	994.901	0.0690713
$593$	−24436.8	−1.69224	−0.846122	−	0.532989i	$-0.821069\pi$
$593$	−24436.8	−1.69224	−0.846122	+	0.532989i	$0.821069\pi$
$594$	0	0
$595$	0	0
$596$	−10228.3	−0.702965
$597$	0	0
$598$	−1068.93	−0.0730967
$599$	−19665.1	−1.34139	−0.670695	−	0.741733i	$-0.734004\pi$
$599$	−19665.1	−1.34139	−0.670695	+	0.741733i	$0.734004\pi$
$600$	0	0
$601$	5290.41	0.359069	0.179534	−	0.983752i	$-0.442541\pi$
$601$	5290.41	0.359069	0.179534	+	0.983752i	$0.442541\pi$
$602$	232.126	0.0157155
$603$	0	0
$604$	−24241.4	−1.63306
$605$	0	0
$606$	0	0
$607$	13324.8	0.890997	0.445499	−	0.895283i	$-0.353026\pi$
$607$	13324.8	0.890997	0.445499	+	0.895283i	$0.353026\pi$
$608$	−5080.43	−0.338879
$609$	0	0
$610$	0	0
$611$	4163.13	0.275650
$612$	0	0
$613$	−9606.30	−0.632944	−0.316472	−	0.948602i	$-0.602498\pi$
$613$	−9606.30	−0.632944	−0.316472	+	0.948602i	$0.602498\pi$
$614$	1777.29	0.116817
$615$	0	0
$616$	742.883	0.0485903
$617$	7675.38	0.500809	0.250404	−	0.968141i	$-0.419436\pi$
$617$	7675.38	0.500809	0.250404	+	0.968141i	$0.419436\pi$
$618$	0	0
$619$	8301.91	0.539066	0.269533	−	0.962991i	$-0.413131\pi$
$619$	8301.91	0.539066	0.269533	+	0.962991i	$0.413131\pi$
$620$	0	0
$621$	0	0
$622$	158.333	0.0102067
$623$	361.359	0.0232384
$624$	0	0
$625$	0	0
$626$	1196.55	0.0763958
$627$	0	0
$628$	−2534.00	−0.161015
$629$	−1153.14	−0.0730980
$630$	0	0
$631$	−18189.1	−1.14754	−0.573768	−	0.819018i	$-0.694519\pi$
$631$	−18189.1	−1.14754	−0.573768	+	0.819018i	$0.694519\pi$
$632$	3349.08	0.210790
$633$	0	0
$634$	−1075.23	−0.0673543
$635$	0	0
$636$	0	0
$637$	1240.57	0.0771638
$638$	323.381	0.0200671
$639$	0	0
$640$	0	0
$641$	−7924.18	−0.488278	−0.244139	−	0.969740i	$-0.578505\pi$
$641$	−7924.18	−0.488278	−0.244139	+	0.969740i	$0.578505\pi$
$642$	0	0
$643$	−18621.9	−1.14211	−0.571054	−	0.820913i	$-0.693465\pi$
$643$	−18621.9	−1.14211	−0.571054	+	0.820913i	$0.693465\pi$
$644$	−10814.2	−0.661707
$645$	0	0
$646$	1943.40	0.118362
$647$	13479.1	0.819039	0.409520	−	0.912301i	$-0.365696\pi$
$647$	13479.1	0.819039	0.409520	+	0.912301i	$0.365696\pi$
$648$	0	0
$649$	5534.73	0.334757
$650$	0	0
$651$	0	0
$652$	−3969.15	−0.238411
$653$	1979.08	0.118602	0.0593011	−	0.998240i	$-0.481113\pi$
$653$	1979.08	0.118602	0.0593011	+	0.998240i	$0.481113\pi$
$654$	0	0
$655$	0	0
$656$	−28473.5	−1.69467
$657$	0	0
$658$	−250.171	−0.0148217
$659$	11366.4	0.671882	0.335941	−	0.941883i	$-0.390946\pi$
$659$	11366.4	0.671882	0.335941	+	0.941883i	$0.390946\pi$
$660$	0	0
$661$	−7806.27	−0.459348	−0.229674	−	0.973268i	$-0.573766\pi$
$661$	−7806.27	−0.459348	−0.229674	+	0.973268i	$0.573766\pi$
$662$	−1280.97	−0.0752059
$663$	0	0
$664$	557.958	0.0326099
$665$	0	0
$666$	0	0
$667$	−9442.93	−0.548173
$668$	−15176.0	−0.879010
$669$	0	0
$670$	0	0
$671$	3543.64	0.203876
$672$	0	0
$673$	8629.80	0.494286	0.247143	−	0.968979i	$-0.420508\pi$
$673$	8629.80	0.494286	0.247143	+	0.968979i	$0.420508\pi$
$674$	2205.74	0.126057
$675$	0	0
$676$	12374.6	0.704060
$677$	−2482.54	−0.140933	−0.0704666	−	0.997514i	$-0.522449\pi$
$677$	−2482.54	−0.140933	−0.0704666	+	0.997514i	$0.522449\pi$
$678$	0	0
$679$	−10481.2	−0.592388
$680$	0	0
$681$	0	0
$682$	1921.96	0.107912
$683$	21007.2	1.17689	0.588446	−	0.808536i	$-0.299740\pi$
$683$	21007.2	1.17689	0.588446	+	0.808536i	$0.299740\pi$
$684$	0	0
$685$	0	0
$686$	−74.5484	−0.00414908
$687$	0	0
$688$	−9592.11	−0.531534
$689$	14978.0	0.828182
$690$	0	0
$691$	28126.3	1.54844	0.774222	−	0.632914i	$-0.218142\pi$
$691$	28126.3	1.54844	0.774222	+	0.632914i	$0.218142\pi$
$692$	11149.4	0.612483
$693$	0	0
$694$	2040.80	0.111625
$695$	0	0
$696$	0	0
$697$	33002.1	1.79346
$698$	283.161	0.0153550
$699$	0	0
$700$	0	0
$701$	−6685.41	−0.360206	−0.180103	−	0.983648i	$-0.557643\pi$
$701$	−6685.41	−0.360206	−0.180103	+	0.983648i	$0.557643\pi$
$702$	0	0
$703$	1941.91	0.104183
$704$	−15119.1	−0.809405
$705$	0	0
$706$	−2053.59	−0.109473
$707$	3397.33	0.180721
$708$	0	0
$709$	6066.14	0.321324	0.160662	−	0.987009i	$-0.448637\pi$
$709$	6066.14	0.321324	0.160662	+	0.987009i	$0.448637\pi$
$710$	0	0
$711$	0	0
$712$	178.987	0.00942109
$713$	−56122.5	−2.94783
$714$	0	0
$715$	0	0
$716$	−28066.6	−1.46494
$717$	0	0
$718$	−1725.23	−0.0896725
$719$	−33959.8	−1.76146	−0.880728	−	0.473623i	$-0.842946\pi$
$719$	−33959.8	−1.76146	−0.880728	+	0.473623i	$0.842946\pi$
$720$	0	0
$721$	774.404	0.0400004
$722$	−1781.97	−0.0918532
$723$	0	0
$724$	−34642.6	−1.77829
$725$	0	0
$726$	0	0
$727$	−1200.78	−0.0612578	−0.0306289	−	0.999531i	$-0.509751\pi$
$727$	−1200.78	−0.0612578	−0.0306289	+	0.999531i	$0.509751\pi$
$728$	614.476	0.0312829
$729$	0	0
$730$	0	0
$731$	11117.7	0.562522
$732$	0	0
$733$	14536.9	0.732513	0.366257	−	0.930514i	$-0.380639\pi$
$733$	14536.9	0.732513	0.366257	+	0.930514i	$0.380639\pi$
$734$	441.338	0.0221936
$735$	0	0
$736$	−8042.60	−0.402791
$737$	−18526.2	−0.925946
$738$	0	0
$739$	−33712.0	−1.67810	−0.839049	−	0.544056i	$-0.816888\pi$
$739$	−33712.0	−1.67810	−0.839049	+	0.544056i	$0.816888\pi$
$740$	0	0
$741$	0	0
$742$	−900.059	−0.0445313
$743$	−34986.0	−1.72747	−0.863737	−	0.503944i	$-0.831882\pi$
$743$	−34986.0	−1.72747	−0.863737	+	0.503944i	$0.831882\pi$
$744$	0	0
$745$	0	0
$746$	2283.77	0.112084
$747$	0	0
$748$	17737.6	0.867047
$749$	2530.08	0.123428
$750$	0	0
$751$	−23260.7	−1.13022	−0.565109	−	0.825016i	$-0.691166\pi$
$751$	−23260.7	−1.13022	−0.565109	+	0.825016i	$0.691166\pi$
$752$	10337.8	0.501303
$753$	0	0
$754$	267.485	0.0129194
$755$	0	0
$756$	0	0
$757$	27313.4	1.31139	0.655694	−	0.755026i	$-0.272376\pi$
$757$	27313.4	1.31139	0.655694	+	0.755026i	$0.272376\pi$
$758$	−2287.40	−0.109607
$759$	0	0
$760$	0	0
$761$	−24853.4	−1.18389	−0.591943	−	0.805980i	$-0.701639\pi$
$761$	−24853.4	−1.18389	−0.591943	+	0.805980i	$0.701639\pi$
$762$	0	0
$763$	3998.69	0.189728
$764$	9569.81	0.453172
$765$	0	0
$766$	−2274.34	−0.107279
$767$	4578.05	0.215520
$768$	0	0
$769$	8507.18	0.398929	0.199465	−	0.979905i	$-0.436080\pi$
$769$	8507.18	0.398929	0.199465	+	0.979905i	$0.436080\pi$
$770$	0	0
$771$	0	0
$772$	12353.1	0.575903
$773$	−1265.93	−0.0589032	−0.0294516	−	0.999566i	$-0.509376\pi$
$773$	−1265.93	−0.0589032	−0.0294516	+	0.999566i	$0.509376\pi$
$774$	0	0
$775$	0	0
$776$	−5191.50	−0.240160
$777$	0	0
$778$	1668.08	0.0768685
$779$	−55576.3	−2.55613
$780$	0	0
$781$	30330.5	1.38964
$782$	3076.50	0.140685
$783$	0	0
$784$	3080.56	0.140332
$785$	0	0
$786$	0	0
$787$	1435.78	0.0650316	0.0325158	−	0.999471i	$-0.489648\pi$
$787$	1435.78	0.0650316	0.0325158	+	0.999471i	$0.489648\pi$
$788$	14521.9	0.656498
$789$	0	0
$790$	0	0
$791$	9728.71	0.437311
$792$	0	0
$793$	2931.12	0.131258
$794$	−1.55258	−6.93940e−5 0
$795$	0	0
$796$	−4742.98	−0.211194
$797$	−23340.9	−1.03736	−0.518680	−	0.854968i	$-0.673576\pi$
$797$	−23340.9	−1.03736	−0.518680	+	0.854968i	$0.673576\pi$
$798$	0	0
$799$	−11982.0	−0.530528
$800$	0	0
$801$	0	0
$802$	381.705	0.0168061
$803$	−26438.3	−1.16188
$804$	0	0
$805$	0	0
$806$	1589.75	0.0694747
$807$	0	0
$808$	1682.75	0.0732661
$809$	−1425.33	−0.0619430	−0.0309715	−	0.999520i	$-0.509860\pi$
$809$	−1425.33	−0.0619430	−0.0309715	+	0.999520i	$0.509860\pi$
$810$	0	0
$811$	5914.80	0.256099	0.128050	−	0.991768i	$-0.459128\pi$
$811$	5914.80	0.256099	0.128050	+	0.991768i	$0.459128\pi$
$812$	2706.10	0.116953
$813$	0	0
$814$	−105.277	−0.00453311
$815$	0	0
$816$	0	0
$817$	−18722.5	−0.801733
$818$	2495.97	0.106686
$819$	0	0
$820$	0	0
$821$	−561.660	−0.0238759	−0.0119379	−	0.999929i	$-0.503800\pi$
$821$	−561.660	−0.0238759	−0.0119379	+	0.999929i	$0.503800\pi$
$822$	0	0
$823$	28178.0	1.19347	0.596734	−	0.802439i	$-0.296465\pi$
$823$	28178.0	1.19347	0.596734	+	0.802439i	$0.296465\pi$
$824$	383.574	0.0162166
$825$	0	0
$826$	−275.104	−0.0115885
$827$	12836.7	0.539754	0.269877	−	0.962895i	$-0.413017\pi$
$827$	12836.7	0.539754	0.269877	+	0.962895i	$0.413017\pi$
$828$	0	0
$829$	1411.97	0.0591552	0.0295776	−	0.999562i	$-0.490584\pi$
$829$	1411.97	0.0591552	0.0295776	+	0.999562i	$0.490584\pi$
$830$	0	0
$831$	0	0
$832$	−12505.7	−0.521104
$833$	−3570.51	−0.148513
$834$	0	0
$835$	0	0
$836$	−29870.5	−1.23576
$837$	0	0
$838$	−701.935	−0.0289355
$839$	−23633.2	−0.972476	−0.486238	−	0.873826i	$-0.661631\pi$
$839$	−23633.2	−0.972476	−0.486238	+	0.873826i	$0.661631\pi$
$840$	0	0
$841$	−22026.0	−0.903114
$842$	−2410.65	−0.0986655
$843$	0	0
$844$	−970.791	−0.0395924
$845$	0	0
$846$	0	0
$847$	−2758.83	−0.111918
$848$	37193.0	1.50615
$849$	0	0
$850$	0	0
$851$	3074.15	0.123831
$852$	0	0
$853$	22987.9	0.922734	0.461367	−	0.887210i	$-0.347359\pi$
$853$	22987.9	0.922734	0.461367	+	0.887210i	$0.347359\pi$
$854$	−176.137	−0.00705770
$855$	0	0
$856$	1253.19	0.0500387
$857$	−10505.6	−0.418745	−0.209372	−	0.977836i	$-0.567142\pi$
$857$	−10505.6	−0.418745	−0.209372	+	0.977836i	$0.567142\pi$
$858$	0	0
$859$	−4159.29	−0.165207	−0.0826037	−	0.996582i	$-0.526324\pi$
$859$	−4159.29	−0.165207	−0.0826037	+	0.996582i	$0.526324\pi$
$860$	0	0
$861$	0	0
$862$	2783.67	0.109991
$863$	1464.49	0.0577657	0.0288828	−	0.999583i	$-0.490805\pi$
$863$	1464.49	0.0577657	0.0288828	+	0.999583i	$0.490805\pi$
$864$	0	0
$865$	0	0
$866$	−1562.53	−0.0613129
$867$	0	0
$868$	16083.3	0.628919
$869$	29565.7	1.15414
$870$	0	0
$871$	−15324.0	−0.596134
$872$	1980.62	0.0769175
$873$	0	0
$874$	−5180.90	−0.200511
$875$	0	0
$876$	0	0
$877$	−16909.1	−0.651062	−0.325531	−	0.945531i	$-0.605543\pi$
$877$	−16909.1	−0.651062	−0.325531	+	0.945531i	$0.605543\pi$
$878$	−893.904	−0.0343597
$879$	0	0
$880$	0	0
$881$	−16144.3	−0.617384	−0.308692	−	0.951162i	$-0.599891\pi$
$881$	−16144.3	−0.617384	−0.308692	+	0.951162i	$0.599891\pi$
$882$	0	0
$883$	−12346.9	−0.470562	−0.235281	−	0.971927i	$-0.575601\pi$
$883$	−12346.9	−0.470562	−0.235281	+	0.971927i	$0.575601\pi$
$884$	14671.7	0.558214
$885$	0	0
$886$	−1863.83	−0.0706735
$887$	−37140.9	−1.40594	−0.702971	−	0.711219i	$-0.748144\pi$
$887$	−37140.9	−1.40594	−0.702971	+	0.711219i	$0.748144\pi$
$888$	0	0
$889$	5445.08	0.205424
$890$	0	0
$891$	0	0
$892$	−28781.2	−1.08034
$893$	20177.9	0.756134
$894$	0	0
$895$	0	0
$896$	3069.99	0.114466
$897$	0	0
$898$	3884.42	0.144348
$899$	14043.9	0.521011
$900$	0	0
$901$	−43108.5	−1.59395
$902$	3012.96	0.111220
$903$	0	0
$904$	4818.78	0.177290
$905$	0	0
$906$	0	0
$907$	44511.9	1.62954	0.814770	−	0.579784i	$-0.196863\pi$
$907$	44511.9	1.62954	0.814770	+	0.579784i	$0.196863\pi$
$908$	−44923.5	−1.64189
$909$	0	0
$910$	0	0
$911$	−33943.4	−1.23446	−0.617232	−	0.786781i	$-0.711746\pi$
$911$	−33943.4	−1.23446	−0.617232	+	0.786781i	$0.711746\pi$
$912$	0	0
$913$	4925.65	0.178549
$914$	−2739.11	−0.0991267
$915$	0	0
$916$	4925.56	0.177669
$917$	1205.15	0.0433998
$918$	0	0
$919$	14553.6	0.522392	0.261196	−	0.965286i	$-0.415883\pi$
$919$	14553.6	0.522392	0.261196	+	0.965286i	$0.415883\pi$
$920$	0	0
$921$	0	0
$922$	−3914.04	−0.139807
$923$	25087.9	0.894666
$924$	0	0
$925$	0	0
$926$	3297.79	0.117033
$927$	0	0
$928$	2012.55	0.0711908
$929$	38414.8	1.35667	0.678335	−	0.734752i	$-0.262702\pi$
$929$	38414.8	1.35667	0.678335	+	0.734752i	$0.262702\pi$
$930$	0	0
$931$	6012.82	0.211667
$932$	−5371.82	−0.188798
$933$	0	0
$934$	−1903.87	−0.0666988
$935$	0	0
$936$	0	0
$937$	−31745.6	−1.10681	−0.553407	−	0.832911i	$-0.686672\pi$
$937$	−31745.6	−1.10681	−0.553407	+	0.832911i	$0.686672\pi$
$938$	920.846	0.0320541
$939$	0	0
$940$	0	0
$941$	45611.1	1.58011	0.790053	−	0.613039i	$-0.210053\pi$
$941$	45611.1	1.58011	0.790053	+	0.613039i	$0.210053\pi$
$942$	0	0
$943$	−87980.4	−3.03821
$944$	11368.1	0.391949
$945$	0	0
$946$	1015.00	0.0348844
$947$	29473.9	1.01138	0.505689	−	0.862716i	$-0.331238\pi$
$947$	29473.9	1.01138	0.505689	+	0.862716i	$0.331238\pi$
$948$	0	0
$949$	−21868.4	−0.748028
$950$	0	0
$951$	0	0
$952$	−1768.53	−0.0602084
$953$	−15280.3	−0.519390	−0.259695	−	0.965691i	$-0.583622\pi$
$953$	−15280.3	−0.519390	−0.259695	+	0.965691i	$0.583622\pi$
$954$	0	0
$955$	0	0
$956$	−46122.1	−1.56035
$957$	0	0
$958$	1966.30	0.0663134
$959$	−14070.7	−0.473793
$960$	0	0
$961$	53676.4	1.80176
$962$	−87.0798	−0.00291847
$963$	0	0
$964$	22706.3	0.758630
$965$	0	0
$966$	0	0
$967$	35221.4	1.17130	0.585648	−	0.810565i	$-0.300840\pi$
$967$	35221.4	1.17130	0.585648	+	0.810565i	$0.300840\pi$
$968$	−1366.49	−0.0453727
$969$	0	0
$970$	0	0
$971$	20190.6	0.667298	0.333649	−	0.942697i	$-0.391720\pi$
$971$	20190.6	0.667298	0.333649	+	0.942697i	$0.391720\pi$
$972$	0	0
$973$	−100.958	−0.00332639
$974$	−398.309	−0.0131033
$975$	0	0
$976$	7278.48	0.238707
$977$	54320.0	1.77876	0.889382	−	0.457165i	$-0.151135\pi$
$977$	54320.0	1.77876	0.889382	+	0.457165i	$0.151135\pi$
$978$	0	0
$979$	1580.09	0.0515833
$980$	0	0
$981$	0	0
$982$	−4475.86	−0.145449
$983$	22357.2	0.725415	0.362707	−	0.931903i	$-0.381852\pi$
$983$	22357.2	0.725415	0.362707	+	0.931903i	$0.381852\pi$
$984$	0	0
$985$	0	0
$986$	−769.851	−0.0248652
$987$	0	0
$988$	−24707.4	−0.795594
$989$	−29638.7	−0.952939
$990$	0	0
$991$	−35261.4	−1.13029	−0.565144	−	0.824993i	$-0.691179\pi$
$991$	−35261.4	−1.13029	−0.565144	+	0.824993i	$0.691179\pi$
$992$	11961.2	0.382833
$993$	0	0
$994$	−1507.58	−0.0481061
$995$	0	0
$996$	0	0
$997$	15862.5	0.503881	0.251940	−	0.967743i	$-0.418931\pi$
$997$	15862.5	0.503881	0.251940	+	0.967743i	$0.418931\pi$
$998$	−1057.15	−0.0335306
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bm.1.2		4
3.2	odd	2		525.4.a.s.1.3	✓	4
5.4	even	2		1575.4.a.bf.1.3		4
15.2	even	4		525.4.d.o.274.5		8
15.8	even	4		525.4.d.o.274.4		8
15.14	odd	2		525.4.a.v.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.s.1.3	✓	4	3.2	odd	2
525.4.a.v.1.2	yes	4	15.14	odd	2
525.4.d.o.274.4		8	15.8	even	4
525.4.d.o.274.5		8	15.2	even	4
1575.4.a.bf.1.3		4	5.4	even	2
1575.4.a.bm.1.2		4	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-0.217342 q^{2} -7.95276 q^{4} +7.00000 q^{7} +3.46721 q^{8} +30.6085 q^{11} +25.3178 q^{13} -1.52140 q^{14} +62.8685 q^{16} -72.8676 q^{17} +122.711 q^{19} -6.65253 q^{22} +194.258 q^{23} -5.50264 q^{26} -55.6693 q^{28} -48.6103 q^{29} -288.907 q^{31} -41.4017 q^{32} +15.8372 q^{34} +15.8251 q^{37} -26.6702 q^{38} -452.905 q^{41} -152.574 q^{43} -243.422 q^{44} -42.2205 q^{46} +164.435 q^{47} +49.0000 q^{49} -201.347 q^{52} +591.600 q^{53} +24.2705 q^{56} +10.5651 q^{58} +180.823 q^{59} +115.773 q^{61} +62.7918 q^{62} -493.950 q^{64} -605.264 q^{67} +579.499 q^{68} +990.917 q^{71} -863.756 q^{73} -3.43947 q^{74} -975.889 q^{76} +214.260 q^{77} +965.930 q^{79} +98.4355 q^{82} +160.924 q^{83} +33.1608 q^{86} +106.126 q^{88} +51.6227 q^{89} +177.225 q^{91} -1544.89 q^{92} -35.7387 q^{94} -1497.31 q^{97} -10.6498 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{2} + 16 q^{4} + 28 q^{7} + 93 q^{8} - 57 q^{11} - 43 q^{13} + 42 q^{14} + 216 q^{16} + 99 q^{17} - 12 q^{19} + 41 q^{22} + 156 q^{23} + 81 q^{26} + 112 q^{28} - 378 q^{29} - 93 q^{31} + 690 q^{32}+ \cdots + 294 q^{98}+O(q^{100}) \) 4 * q + 6 * q^2 + 16 * q^4 + 28 * q^7 + 93 * q^8 - 57 * q^11 - 43 * q^13 + 42 * q^14 + 216 * q^16 + 99 * q^17 - 12 * q^19 + 41 * q^22 + 156 * q^23 + 81 * q^26 + 112 * q^28 - 378 * q^29 - 93 * q^31 + 690 * q^32 + 783 * q^34 - 81 * q^37 + 216 * q^38 + 465 * q^41 + 64 * q^43 - 681 * q^44 + 310 * q^46 + 744 * q^47 + 196 * q^49 + 727 * q^52 + 729 * q^53 + 651 * q^56 - 1172 * q^58 - 231 * q^59 - 1353 * q^61 - 165 * q^62 + 3107 * q^64 - 1487 * q^67 + 2577 * q^68 + 1725 * q^71 - 512 * q^73 + 1953 * q^74 - 3046 * q^76 - 399 * q^77 + 1629 * q^79 + 693 * q^82 + 321 * q^83 + 4542 * q^86 - 3482 * q^88 + 978 * q^89 - 301 * q^91 + 852 * q^92 + 2480 * q^94 - 2616 * q^97 + 294 * q^98

Introduction

L-functions

Modular forms

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