LMFDB - Embedded newform 1225.4.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 32x^{2} - 35x + 120 $ x^4 - 32x^2 - 35x + 120
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.84167$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.84167 q^{2} -9.58084 q^{3} +15.4418 q^{4} +46.3873 q^{6} -36.0308 q^{8} +64.7926 q^{9} +62.1962 q^{11} -147.945 q^{12} +14.0934 q^{13} +50.9148 q^{16} +63.5104 q^{17} -313.704 q^{18} -48.7094 q^{19} -301.134 q^{22} +99.3784 q^{23} +345.205 q^{24} -68.2354 q^{26} -362.085 q^{27} -69.0571 q^{29} +9.68658 q^{31} +41.7334 q^{32} -595.892 q^{33} -307.497 q^{34} +1000.51 q^{36} -240.290 q^{37} +235.835 q^{38} -135.026 q^{39} -335.306 q^{41} -51.2582 q^{43} +960.421 q^{44} -481.158 q^{46} -451.564 q^{47} -487.806 q^{48} -608.484 q^{51} +217.627 q^{52} +180.014 q^{53} +1753.10 q^{54} +466.677 q^{57} +334.352 q^{58} -268.600 q^{59} +323.925 q^{61} -46.8992 q^{62} -609.378 q^{64} +2885.12 q^{66} +541.910 q^{67} +980.715 q^{68} -952.129 q^{69} -161.433 q^{71} -2334.53 q^{72} +305.751 q^{73} +1163.40 q^{74} -752.161 q^{76} +653.753 q^{78} -504.722 q^{79} +1719.68 q^{81} +1623.44 q^{82} -513.838 q^{83} +248.176 q^{86} +661.625 q^{87} -2240.98 q^{88} -543.158 q^{89} +1534.58 q^{92} -92.8056 q^{93} +2186.32 q^{94} -399.842 q^{96} +1863.06 q^{97} +4029.85 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9} + 100 q^{11} - 165 q^{12} + 44 q^{13} + 160 q^{16} - 53 q^{17} - 433 q^{18} + 29 q^{19} + 152 q^{22} - 295 q^{23} + 21 q^{24} - 700 q^{26}+ \cdots + 2383 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 - 3 * q^3 + 36 * q^4 - q^6 - 27 * q^8 + 61 * q^9 + 100 * q^11 - 165 * q^12 + 44 * q^13 + 160 * q^16 - 53 * q^17 - 433 * q^18 + 29 * q^19 + 152 * q^22 - 295 * q^23 + 21 * q^24 - 700 * q^26 - 441 * q^27 + 129 * q^29 - 114 * q^31 + 310 * q^32 - 865 * q^33 - 203 * q^34 + 1101 * q^36 - 403 * q^37 + 555 * q^38 + 674 * q^39 - 671 * q^41 + 411 * q^43 + 438 * q^44 - 997 * q^46 - 8 * q^47 - 523 * q^48 - 885 * q^51 + 74 * q^52 - 90 * q^53 + 2777 * q^54 - 233 * q^57 - 673 * q^58 - 1018 * q^59 - 50 * q^61 - 1626 * q^62 - 2421 * q^64 + 3841 * q^66 - 424 * q^67 - 617 * q^68 - 1080 * q^69 + 215 * q^71 - 2940 * q^72 - 1207 * q^73 + 623 * q^74 + 3257 * q^76 - 278 * q^78 - 951 * q^79 + 28 * q^81 + 1695 * q^82 - 3035 * q^83 - 99 * q^86 + 2210 * q^87 - 163 * q^88 - 2819 * q^89 - 3073 * q^92 + 852 * q^93 + 3056 * q^94 + 1345 * q^96 - 1100 * q^97 + 2383 * 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 $

$2$	−4.84167	−1.71179	−0.855895	−	0.517150i	$-0.826993\pi$
$2$	−4.84167	−1.71179	−0.855895	+	0.517150i	$0.826993\pi$
$3$	−9.58084	−1.84383	−0.921917	−	0.387387i	$-0.873378\pi$
$3$	−9.58084	−1.84383	−0.921917	+	0.387387i	$0.873378\pi$
$4$	15.4418	1.93022
$5$	0	0
$5$	0	0
$6$	46.3873	3.15626
$7$	0	0
$7$	0	0
$8$	−36.0308	−1.59235
$9$	64.7926	2.39972
$10$	0	0
$11$	62.1962	1.70481	0.852403	−	0.522886i	$-0.175145\pi$
$11$	62.1962	1.70481	0.852403	+	0.522886i	$0.175145\pi$
$12$	−147.945	−3.55901
$13$	14.0934	0.300676	0.150338	−	0.988635i	$-0.451964\pi$
$13$	14.0934	0.300676	0.150338	+	0.988635i	$0.451964\pi$
$14$	0	0
$15$	0	0
$16$	50.9148	0.795543
$17$	63.5104	0.906091	0.453045	−	0.891487i	$-0.350338\pi$
$17$	63.5104	0.906091	0.453045	+	0.891487i	$0.350338\pi$
$18$	−313.704	−4.10782
$19$	−48.7094	−0.588143	−0.294071	−	0.955783i	$-0.595010\pi$
$19$	−48.7094	−0.588143	−0.294071	+	0.955783i	$0.595010\pi$
$20$	0	0
$21$	0	0
$22$	−301.134	−2.91827
$23$	99.3784	0.900949	0.450475	−	0.892789i	$-0.351255\pi$
$23$	99.3784	0.900949	0.450475	+	0.892789i	$0.351255\pi$
$24$	345.205	2.93603
$25$	0	0
$26$	−68.2354	−0.514695
$27$	−362.085	−2.58086
$28$	0	0
$29$	−69.0571	−0.442193	−0.221096	−	0.975252i	$-0.570964\pi$
$29$	−69.0571	−0.442193	−0.221096	+	0.975252i	$0.570964\pi$
$30$	0	0
$31$	9.68658	0.0561213	0.0280607	−	0.999606i	$-0.491067\pi$
$31$	9.68658	0.0561213	0.0280607	+	0.999606i	$0.491067\pi$
$32$	41.7334	0.230547
$33$	−595.892	−3.14338
$34$	−307.497	−1.55104
$35$	0	0
$36$	1000.51	4.63201
$37$	−240.290	−1.06766	−0.533829	−	0.845592i	$-0.679248\pi$
$37$	−240.290	−1.06766	−0.533829	+	0.845592i	$0.679248\pi$
$38$	235.835	1.00678
$39$	−135.026	−0.554398
$40$	0	0
$41$	−335.306	−1.27722	−0.638610	−	0.769531i	$-0.720490\pi$
$41$	−335.306	−1.27722	−0.638610	+	0.769531i	$0.720490\pi$
$42$	0	0
$43$	−51.2582	−0.181786	−0.0908931	−	0.995861i	$-0.528972\pi$
$43$	−51.2582	−0.181786	−0.0908931	+	0.995861i	$0.528972\pi$
$44$	960.421	3.29066
$45$	0	0
$46$	−481.158	−1.54224
$47$	−451.564	−1.40143	−0.700716	−	0.713440i	$-0.747136\pi$
$47$	−451.564	−1.40143	−0.700716	+	0.713440i	$0.747136\pi$
$48$	−487.806	−1.46685
$49$	0	0
$50$	0	0
$51$	−608.484	−1.67068
$52$	217.627	0.580373
$53$	180.014	0.466545	0.233273	−	0.972411i	$-0.425057\pi$
$53$	180.014	0.466545	0.233273	+	0.972411i	$0.425057\pi$
$54$	1753.10	4.41789
$55$	0	0
$56$	0	0
$57$	466.677	1.08444
$58$	334.352	0.756941
$59$	−268.600	−0.592691	−0.296345	−	0.955081i	$-0.595768\pi$
$59$	−268.600	−0.592691	−0.296345	+	0.955081i	$0.595768\pi$
$60$	0	0
$61$	323.925	0.679906	0.339953	−	0.940442i	$-0.389589\pi$
$61$	323.925	0.679906	0.339953	+	0.940442i	$0.389589\pi$
$62$	−46.8992	−0.0960679
$63$	0	0
$64$	−609.378	−1.19019
$65$	0	0
$66$	2885.12	5.38080
$67$	541.910	0.988132	0.494066	−	0.869424i	$-0.335510\pi$
$67$	541.910	0.988132	0.494066	+	0.869424i	$0.335510\pi$
$68$	980.715	1.74896
$69$	−952.129	−1.66120
$70$	0	0
$71$	−161.433	−0.269839	−0.134919	−	0.990857i	$-0.543078\pi$
$71$	−161.433	−0.269839	−0.134919	+	0.990857i	$0.543078\pi$
$72$	−2334.53	−3.82120
$73$	305.751	0.490212	0.245106	−	0.969496i	$-0.421177\pi$
$73$	305.751	0.490212	0.245106	+	0.969496i	$0.421177\pi$
$74$	1163.40	1.82761
$75$	0	0
$76$	−752.161	−1.13525
$77$	0	0
$78$	653.753	0.949012
$79$	−504.722	−0.718805	−0.359403	−	0.933183i	$-0.617020\pi$
$79$	−504.722	−0.718805	−0.359403	+	0.933183i	$0.617020\pi$
$80$	0	0
$81$	1719.68	2.35895
$82$	1623.44	2.18633
$83$	−513.838	−0.679531	−0.339766	−	0.940510i	$-0.610348\pi$
$83$	−513.838	−0.679531	−0.339766	+	0.940510i	$0.610348\pi$
$84$	0	0
$85$	0	0
$86$	248.176	0.311180
$87$	661.625	0.815330
$88$	−2240.98	−2.71465
$89$	−543.158	−0.646907	−0.323453	−	0.946244i	$-0.604844\pi$
$89$	−543.158	−0.646907	−0.323453	+	0.946244i	$0.604844\pi$
$90$	0	0
$91$	0	0
$92$	1534.58	1.73903
$93$	−92.8056	−0.103478
$94$	2186.32	2.39896
$95$	0	0
$96$	−399.842	−0.425090
$97$	1863.06	1.95016	0.975079	−	0.221857i	$-0.0712118\pi$
$97$	1863.06	1.95016	0.975079	+	0.221857i	$0.0712118\pi$
$98$	0	0
$99$	4029.85	4.09106
$100$	0	0
$101$	1685.70	1.66073	0.830365	−	0.557221i	$-0.188132\pi$
$101$	1685.70	1.66073	0.830365	+	0.557221i	$0.188132\pi$
$102$	2946.08	2.85986
$103$	−1014.19	−0.970203	−0.485102	−	0.874458i	$-0.661217\pi$
$103$	−1014.19	−0.970203	−0.485102	+	0.874458i	$0.661217\pi$
$104$	−507.794	−0.478782
$105$	0	0
$106$	−871.571	−0.798627
$107$	−913.161	−0.825033	−0.412517	−	0.910950i	$-0.635350\pi$
$107$	−913.161	−0.825033	−0.412517	+	0.910950i	$0.635350\pi$
$108$	−5591.24	−4.98164
$109$	−1397.13	−1.22772	−0.613859	−	0.789416i	$-0.710383\pi$
$109$	−1397.13	−1.22772	−0.613859	+	0.789416i	$0.710383\pi$
$110$	0	0
$111$	2302.18	1.96859
$112$	0	0
$113$	−2082.49	−1.73366	−0.866831	−	0.498603i	$-0.833847\pi$
$113$	−2082.49	−1.73366	−0.866831	+	0.498603i	$0.833847\pi$
$114$	−2259.50	−1.85633
$115$	0	0
$116$	−1066.37	−0.853531
$117$	913.145	0.721541
$118$	1300.47	1.01456
$119$	0	0
$120$	0	0
$121$	2537.37	1.90636
$122$	−1568.34	−1.16386
$123$	3212.51	2.35498
$124$	149.578	0.108327
$125$	0	0
$126$	0	0
$127$	230.691	0.161185	0.0805925	−	0.996747i	$-0.474319\pi$
$127$	230.691	0.161185	0.0805925	+	0.996747i	$0.474319\pi$
$128$	2616.54	1.80681
$129$	491.097	0.335184
$130$	0	0
$131$	973.877	0.649527	0.324764	−	0.945795i	$-0.394715\pi$
$131$	973.877	0.649527	0.324764	+	0.945795i	$0.394715\pi$
$132$	−9201.65	−6.06743
$133$	0	0
$134$	−2623.75	−1.69148
$135$	0	0
$136$	−2288.33	−1.44281
$137$	−695.734	−0.433873	−0.216936	−	0.976186i	$-0.569606\pi$
$137$	−695.734	−0.433873	−0.216936	+	0.976186i	$0.569606\pi$
$138$	4609.90	2.84363
$139$	298.530	0.182165	0.0910826	−	0.995843i	$-0.470967\pi$
$139$	298.530	0.182165	0.0910826	+	0.995843i	$0.470967\pi$
$140$	0	0
$141$	4326.36	2.58401
$142$	781.605	0.461908
$143$	876.553	0.512595
$144$	3298.90	1.90908
$145$	0	0
$146$	−1480.35	−0.839140
$147$	0	0
$148$	−3710.50	−2.06082
$149$	−1792.02	−0.985290	−0.492645	−	0.870230i	$-0.663970\pi$
$149$	−1792.02	−0.985290	−0.492645	+	0.870230i	$0.663970\pi$
$150$	0	0
$151$	1201.27	0.647403	0.323701	−	0.946159i	$-0.395073\pi$
$151$	1201.27	0.647403	0.323701	+	0.946159i	$0.395073\pi$
$152$	1755.04	0.936529
$153$	4115.00	2.17437
$154$	0	0
$155$	0	0
$156$	−2085.05	−1.07011
$157$	409.798	0.208315	0.104158	−	0.994561i	$-0.466785\pi$
$157$	409.798	0.208315	0.104158	+	0.994561i	$0.466785\pi$
$158$	2443.70	1.23044
$159$	−1724.69	−0.860232
$160$	0	0
$161$	0	0
$162$	−8326.12	−4.03803
$163$	2030.71	0.975811	0.487905	−	0.872896i	$-0.337761\pi$
$163$	2030.71	0.975811	0.487905	+	0.872896i	$0.337761\pi$
$164$	−5177.73	−2.46532
$165$	0	0
$166$	2487.84	1.16321
$167$	520.014	0.240958	0.120479	−	0.992716i	$-0.461557\pi$
$167$	520.014	0.240958	0.120479	+	0.992716i	$0.461557\pi$
$168$	0	0
$169$	−1998.38	−0.909594
$170$	0	0
$171$	−3156.01	−1.41138
$172$	−791.520	−0.350888
$173$	−2561.90	−1.12588	−0.562942	−	0.826497i	$-0.690331\pi$
$173$	−2561.90	−1.12588	−0.562942	+	0.826497i	$0.690331\pi$
$174$	−3203.37	−1.39567
$175$	0	0
$176$	3166.71	1.35625
$177$	2573.42	1.09282
$178$	2629.80	1.10737
$179$	−3042.08	−1.27025	−0.635127	−	0.772408i	$-0.719052\pi$
$179$	−3042.08	−1.27025	−0.635127	+	0.772408i	$0.719052\pi$
$180$	0	0
$181$	3648.02	1.49809	0.749047	−	0.662516i	$-0.230512\pi$
$181$	3648.02	1.49809	0.749047	+	0.662516i	$0.230512\pi$
$182$	0	0
$183$	−3103.47	−1.25363
$184$	−3580.68	−1.43463
$185$	0	0
$186$	449.334	0.177133
$187$	3950.11	1.54471
$188$	−6972.95	−2.70508
$189$	0	0
$190$	0	0
$191$	−4091.12	−1.54986	−0.774929	−	0.632049i	$-0.782214\pi$
$191$	−4091.12	−1.54986	−0.774929	+	0.632049i	$0.782214\pi$
$192$	5838.35	2.19451
$193$	−3051.70	−1.13817	−0.569084	−	0.822280i	$-0.692702\pi$
$193$	−3051.70	−1.13817	−0.569084	+	0.822280i	$0.692702\pi$
$194$	−9020.34	−3.33826
$195$	0	0
$196$	0	0
$197$	−3011.44	−1.08912	−0.544560	−	0.838722i	$-0.683303\pi$
$197$	−3011.44	−1.08912	−0.544560	+	0.838722i	$0.683303\pi$
$198$	−19511.2	−7.00304
$199$	−199.943	−0.0712239	−0.0356119	−	0.999366i	$-0.511338\pi$
$199$	−199.943	−0.0712239	−0.0356119	+	0.999366i	$0.511338\pi$
$200$	0	0
$201$	−5191.96	−1.82195
$202$	−8161.62	−2.84282
$203$	0	0
$204$	−9396.08	−3.22479
$205$	0	0
$206$	4910.37	1.66078
$207$	6438.98	2.16203
$208$	717.560	0.239201
$209$	−3029.54	−1.00267
$210$	0	0
$211$	−297.442	−0.0970461	−0.0485231	−	0.998822i	$-0.515451\pi$
$211$	−297.442	−0.0970461	−0.0485231	+	0.998822i	$0.515451\pi$
$212$	2779.75	0.900537
$213$	1546.66	0.497538
$214$	4421.23	1.41228
$215$	0	0
$216$	13046.2	4.10963
$217$	0	0
$218$	6764.47	2.10159
$219$	−2929.36	−0.903870
$220$	0	0
$221$	895.075	0.272440
$222$	−11146.4	−3.36981
$223$	−6282.68	−1.88663	−0.943317	−	0.331894i	$-0.892312\pi$
$223$	−6282.68	−1.88663	−0.943317	+	0.331894i	$0.892312\pi$
$224$	0	0
$225$	0	0
$226$	10082.7	2.96766
$227$	−2357.28	−0.689243	−0.344621	−	0.938742i	$-0.611993\pi$
$227$	−2357.28	−0.689243	−0.344621	+	0.938742i	$0.611993\pi$
$228$	7206.34	2.09321
$229$	2476.81	0.714727	0.357363	−	0.933965i	$-0.383676\pi$
$229$	2476.81	0.714727	0.357363	+	0.933965i	$0.383676\pi$
$230$	0	0
$231$	0	0
$232$	2488.18	0.704125
$233$	5142.47	1.44590	0.722950	−	0.690900i	$-0.242786\pi$
$233$	5142.47	1.44590	0.722950	+	0.690900i	$0.242786\pi$
$234$	−4421.15	−1.23513
$235$	0	0
$236$	−4147.67	−1.14403
$237$	4835.66	1.32536
$238$	0	0
$239$	−831.250	−0.224975	−0.112488	−	0.993653i	$-0.535882\pi$
$239$	−831.250	−0.224975	−0.112488	+	0.993653i	$0.535882\pi$
$240$	0	0
$241$	−4538.85	−1.21317	−0.606584	−	0.795020i	$-0.707460\pi$
$241$	−4538.85	−1.21317	−0.606584	+	0.795020i	$0.707460\pi$
$242$	−12285.1	−3.26329
$243$	−6699.68	−1.76866
$244$	5001.98	1.31237
$245$	0	0
$246$	−15553.9	−4.03123
$247$	−686.479	−0.176841
$248$	−349.015	−0.0893648
$249$	4923.01	1.25294
$250$	0	0
$251$	−7232.00	−1.81865	−0.909323	−	0.416091i	$-0.863400\pi$
$251$	−7232.00	−1.81865	−0.909323	+	0.416091i	$0.863400\pi$
$252$	0	0
$253$	6180.96	1.53594
$254$	−1116.93	−0.275915
$255$	0	0
$256$	−7793.41	−1.90269
$257$	4242.66	1.02977	0.514883	−	0.857260i	$-0.327835\pi$
$257$	4242.66	1.02977	0.514883	+	0.857260i	$0.327835\pi$
$258$	−2377.73	−0.573764
$259$	0	0
$260$	0	0
$261$	−4474.39	−1.06114
$262$	−4715.20	−1.11185
$263$	−6604.75	−1.54854	−0.774271	−	0.632855i	$-0.781883\pi$
$263$	−6604.75	−1.54854	−0.774271	+	0.632855i	$0.781883\pi$
$264$	21470.4	5.00536
$265$	0	0
$266$	0	0
$267$	5203.92	1.19279
$268$	8368.07	1.90732
$269$	−4612.29	−1.04541	−0.522707	−	0.852512i	$-0.675078\pi$
$269$	−4612.29	−1.04541	−0.522707	+	0.852512i	$0.675078\pi$
$270$	0	0
$271$	3542.29	0.794018	0.397009	−	0.917815i	$-0.370048\pi$
$271$	3542.29	0.794018	0.397009	+	0.917815i	$0.370048\pi$
$272$	3233.62	0.720834
$273$	0	0
$274$	3368.52	0.742699
$275$	0	0
$276$	−14702.6	−3.20649
$277$	−19.5351	−0.00423737	−0.00211868	−	0.999998i	$-0.500674\pi$
$277$	−19.5351	−0.00423737	−0.00211868	+	0.999998i	$0.500674\pi$
$278$	−1445.38	−0.311829
$279$	627.618	0.134676
$280$	0	0
$281$	6769.20	1.43707	0.718535	−	0.695491i	$-0.244813\pi$
$281$	6769.20	1.43707	0.718535	+	0.695491i	$0.244813\pi$
$282$	−20946.8	−4.42328
$283$	1269.89	0.266740	0.133370	−	0.991066i	$-0.457420\pi$
$283$	1269.89	0.266740	0.133370	+	0.991066i	$0.457420\pi$
$284$	−2492.81	−0.520850
$285$	0	0
$286$	−4243.98	−0.877455
$287$	0	0
$288$	2704.02	0.553249
$289$	−879.424	−0.178999
$290$	0	0
$291$	−17849.7	−3.59577
$292$	4721.35	0.946220
$293$	6349.74	1.26606	0.633030	−	0.774127i	$-0.281811\pi$
$293$	6349.74	1.26606	0.633030	+	0.774127i	$0.281811\pi$
$294$	0	0
$295$	0	0
$296$	8657.82	1.70009
$297$	−22520.3	−4.39987
$298$	8676.39	1.68661
$299$	1400.58	0.270894
$300$	0	0
$301$	0	0
$302$	−5816.15	−1.10822
$303$	−16150.4	−3.06211
$304$	−2480.03	−0.467893
$305$	0	0
$306$	−19923.5	−3.72206
$307$	−4772.37	−0.887211	−0.443605	−	0.896222i	$-0.646301\pi$
$307$	−4772.37	−0.887211	−0.443605	+	0.896222i	$0.646301\pi$
$308$	0	0
$309$	9716.78	1.78889
$310$	0	0
$311$	−740.703	−0.135053	−0.0675264	−	0.997717i	$-0.521511\pi$
$311$	−740.703	−0.135053	−0.0675264	+	0.997717i	$0.521511\pi$
$312$	4865.10	0.882795
$313$	−2279.68	−0.411678	−0.205839	−	0.978586i	$-0.565992\pi$
$313$	−2279.68	−0.411678	−0.205839	+	0.978586i	$0.565992\pi$
$314$	−1984.11	−0.356592
$315$	0	0
$316$	−7793.81	−1.38746
$317$	10198.1	1.80689	0.903446	−	0.428702i	$-0.141029\pi$
$317$	10198.1	1.80689	0.903446	+	0.428702i	$0.141029\pi$
$318$	8350.39	1.47254
$319$	−4295.09	−0.753852
$320$	0	0
$321$	8748.85	1.52122
$322$	0	0
$323$	−3093.56	−0.532911
$324$	26554.9	4.55331
$325$	0	0
$326$	−9832.01	−1.67038
$327$	13385.7	2.26371
$328$	12081.3	2.03378
$329$	0	0
$330$	0	0
$331$	−78.0380	−0.0129588	−0.00647939	−	0.999979i	$-0.502062\pi$
$331$	−78.0380	−0.0129588	−0.00647939	+	0.999979i	$0.502062\pi$
$332$	−7934.59	−1.31165
$333$	−15569.0	−2.56209
$334$	−2517.74	−0.412469
$335$	0	0
$336$	0	0
$337$	4164.73	0.673196	0.336598	−	0.941648i	$-0.390724\pi$
$337$	4164.73	0.673196	0.336598	+	0.941648i	$0.390724\pi$
$338$	9675.49	1.55703
$339$	19952.0	3.19658
$340$	0	0
$341$	602.468	0.0956759
$342$	15280.4	2.41599
$343$	0	0
$344$	1846.87	0.289467
$345$	0	0
$346$	12403.9	1.92728
$347$	−1795.19	−0.277726	−0.138863	−	0.990312i	$-0.544345\pi$
$347$	−1795.19	−0.277726	−0.138863	+	0.990312i	$0.544345\pi$
$348$	10216.7	1.57377
$349$	11751.4	1.80241	0.901203	−	0.433398i	$-0.142686\pi$
$349$	11751.4	1.80241	0.901203	+	0.433398i	$0.142686\pi$
$350$	0	0
$351$	−5102.99	−0.776004
$352$	2595.66	0.393038
$353$	2882.32	0.434590	0.217295	−	0.976106i	$-0.430277\pi$
$353$	2882.32	0.434590	0.217295	+	0.976106i	$0.430277\pi$
$354$	−12459.6	−1.87069
$355$	0	0
$356$	−8387.34	−1.24868
$357$	0	0
$358$	14728.7	2.17441
$359$	−1193.78	−0.175503	−0.0877513	−	0.996142i	$-0.527968\pi$
$359$	−1193.78	−0.175503	−0.0877513	+	0.996142i	$0.527968\pi$
$360$	0	0
$361$	−4486.39	−0.654088
$362$	−17662.5	−2.56442
$363$	−24310.1	−3.51502
$364$	0	0
$365$	0	0
$366$	15026.0	2.14596
$367$	8858.80	1.26001	0.630007	−	0.776589i	$-0.283052\pi$
$367$	8858.80	1.26001	0.630007	+	0.776589i	$0.283052\pi$
$368$	5059.83	0.716744
$369$	−21725.3	−3.06497
$370$	0	0
$371$	0	0
$372$	−1433.09	−0.199737
$373$	2287.78	0.317578	0.158789	−	0.987313i	$-0.449241\pi$
$373$	2287.78	0.317578	0.158789	+	0.987313i	$0.449241\pi$
$374$	−19125.1	−2.64422
$375$	0	0
$376$	16270.2	2.23157
$377$	−973.246	−0.132957
$378$	0	0
$379$	8870.18	1.20219	0.601095	−	0.799177i	$-0.294731\pi$
$379$	8870.18	1.20219	0.601095	+	0.799177i	$0.294731\pi$
$380$	0	0
$381$	−2210.21	−0.297198
$382$	19807.8	2.65303
$383$	13359.2	1.78231	0.891155	−	0.453700i	$-0.149896\pi$
$383$	13359.2	1.78231	0.891155	+	0.453700i	$0.149896\pi$
$384$	−25068.7	−3.33146
$385$	0	0
$386$	14775.3	1.94830
$387$	−3321.15	−0.436237
$388$	28769.0	3.76424
$389$	11324.7	1.47605	0.738025	−	0.674773i	$-0.235759\pi$
$389$	11324.7	1.47605	0.738025	+	0.674773i	$0.235759\pi$
$390$	0	0
$391$	6311.57	0.816342
$392$	0	0
$393$	−9330.57	−1.19762
$394$	14580.4	1.86434
$395$	0	0
$396$	62228.2	7.89667
$397$	−13242.8	−1.67415	−0.837076	−	0.547087i	$-0.815737\pi$
$397$	−13242.8	−1.67415	−0.837076	+	0.547087i	$0.815737\pi$
$398$	968.056	0.121920
$399$	0	0
$400$	0	0
$401$	1952.96	0.243207	0.121604	−	0.992579i	$-0.461196\pi$
$401$	1952.96	0.243207	0.121604	+	0.992579i	$0.461196\pi$
$402$	25137.8	3.11880
$403$	136.516	0.0168744
$404$	26030.3	3.20558
$405$	0	0
$406$	0	0
$407$	−14945.1	−1.82015
$408$	21924.1	2.66031
$409$	10597.8	1.28124	0.640618	−	0.767860i	$-0.278678\pi$
$409$	10597.8	1.28124	0.640618	+	0.767860i	$0.278678\pi$
$410$	0	0
$411$	6665.72	0.799989
$412$	−15660.9	−1.87271
$413$	0	0
$414$	−31175.5	−3.70094
$415$	0	0
$416$	588.164	0.0693200
$417$	−2860.17	−0.335882
$418$	14668.1	1.71636
$419$	1631.18	0.190187	0.0950935	−	0.995468i	$-0.469685\pi$
$419$	1631.18	0.190187	0.0950935	+	0.995468i	$0.469685\pi$
$420$	0	0
$421$	13181.2	1.52592	0.762961	−	0.646445i	$-0.223745\pi$
$421$	13181.2	1.52592	0.762961	+	0.646445i	$0.223745\pi$
$422$	1440.12	0.166123
$423$	−29258.0	−3.36305
$424$	−6486.06	−0.742903
$425$	0	0
$426$	−7488.44	−0.851681
$427$	0	0
$428$	−14100.8	−1.59250
$429$	−8398.12	−0.945140
$430$	0	0
$431$	9159.40	1.02365	0.511825	−	0.859090i	$-0.328970\pi$
$431$	9159.40	1.02365	0.511825	+	0.859090i	$0.328970\pi$
$432$	−18435.5	−2.05319
$433$	270.519	0.0300238	0.0150119	−	0.999887i	$-0.495221\pi$
$433$	270.519	0.0300238	0.0150119	+	0.999887i	$0.495221\pi$
$434$	0	0
$435$	0	0
$436$	−21574.3	−2.36977
$437$	−4840.67	−0.529887
$438$	14183.0	1.54724
$439$	−17008.0	−1.84908	−0.924541	−	0.381082i	$-0.875552\pi$
$439$	−17008.0	−1.84908	−0.924541	+	0.381082i	$0.875552\pi$
$440$	0	0
$441$	0	0
$442$	−4333.66	−0.466360
$443$	−3968.12	−0.425578	−0.212789	−	0.977098i	$-0.568255\pi$
$443$	−3968.12	−0.425578	−0.212789	+	0.977098i	$0.568255\pi$
$444$	35549.8	3.79981
$445$	0	0
$446$	30418.7	3.22952
$447$	17169.1	1.81671
$448$	0	0
$449$	−12480.3	−1.31177	−0.655884	−	0.754862i	$-0.727704\pi$
$449$	−12480.3	−1.31177	−0.655884	+	0.754862i	$0.727704\pi$
$450$	0	0
$451$	−20854.8	−2.17741
$452$	−32157.3	−3.34636
$453$	−11509.2	−1.19370
$454$	11413.2	1.17984
$455$	0	0
$456$	−16814.7	−1.72680
$457$	−5782.90	−0.591932	−0.295966	−	0.955199i	$-0.595641\pi$
$457$	−5782.90	−0.591932	−0.295966	+	0.955199i	$0.595641\pi$
$458$	−11991.9	−1.22346
$459$	−22996.2	−2.33849
$460$	0	0
$461$	−10049.6	−1.01531	−0.507653	−	0.861562i	$-0.669487\pi$
$461$	−10049.6	−1.01531	−0.507653	+	0.861562i	$0.669487\pi$
$462$	0	0
$463$	659.842	0.0662321	0.0331161	−	0.999452i	$-0.489457\pi$
$463$	659.842	0.0662321	0.0331161	+	0.999452i	$0.489457\pi$
$464$	−3516.03	−0.351783
$465$	0	0
$466$	−24898.2	−2.47508
$467$	−4467.60	−0.442690	−0.221345	−	0.975196i	$-0.571045\pi$
$467$	−4467.60	−0.442690	−0.221345	+	0.975196i	$0.571045\pi$
$468$	14100.6	1.39274
$469$	0	0
$470$	0	0
$471$	−3926.21	−0.384099
$472$	9677.87	0.943771
$473$	−3188.07	−0.309910
$474$	−23412.7	−2.26873
$475$	0	0
$476$	0	0
$477$	11663.6	1.11958
$478$	4024.64	0.385110
$479$	−7633.87	−0.728185	−0.364092	−	0.931363i	$-0.618621\pi$
$479$	−7633.87	−0.728185	−0.364092	+	0.931363i	$0.618621\pi$
$480$	0	0
$481$	−3386.49	−0.321020
$482$	21975.6	2.07669
$483$	0	0
$484$	39181.5	3.67971
$485$	0	0
$486$	32437.6	3.02758
$487$	−6289.84	−0.585256	−0.292628	−	0.956226i	$-0.594530\pi$
$487$	−6289.84	−0.585256	−0.292628	+	0.956226i	$0.594530\pi$
$488$	−11671.2	−1.08265
$489$	−19455.9	−1.79923
$490$	0	0
$491$	3562.54	0.327445	0.163722	−	0.986506i	$-0.447650\pi$
$491$	3562.54	0.327445	0.163722	+	0.986506i	$0.447650\pi$
$492$	49607.0	4.54564
$493$	−4385.85	−0.400667
$494$	3323.71	0.302714
$495$	0	0
$496$	493.190	0.0446469
$497$	0	0
$498$	−23835.6	−2.14478
$499$	−18916.9	−1.69706	−0.848532	−	0.529144i	$-0.822513\pi$
$499$	−18916.9	−1.69706	−0.848532	+	0.529144i	$0.822513\pi$
$500$	0	0
$501$	−4982.17	−0.444286
$502$	35015.0	3.11314
$503$	2565.91	0.227452	0.113726	−	0.993512i	$-0.463721\pi$
$503$	2565.91	0.227452	0.113726	+	0.993512i	$0.463721\pi$
$504$	0	0
$505$	0	0
$506$	−29926.2	−2.62921
$507$	19146.1	1.67714
$508$	3562.28	0.311123
$509$	−5447.84	−0.474403	−0.237202	−	0.971460i	$-0.576230\pi$
$509$	−5447.84	−0.474403	−0.237202	+	0.971460i	$0.576230\pi$
$510$	0	0
$511$	0	0
$512$	16800.8	1.45019
$513$	17636.9	1.51791
$514$	−20541.6	−1.76274
$515$	0	0
$516$	7583.42	0.646980
$517$	−28085.5	−2.38917
$518$	0	0
$519$	24545.2	2.07594
$520$	0	0
$521$	4732.95	0.397993	0.198997	−	0.980000i	$-0.436232\pi$
$521$	4732.95	0.397993	0.198997	+	0.980000i	$0.436232\pi$
$522$	21663.5	1.81645
$523$	−9182.96	−0.767769	−0.383884	−	0.923381i	$-0.625414\pi$
$523$	−9182.96	−0.767769	−0.383884	+	0.923381i	$0.625414\pi$
$524$	15038.4	1.25373
$525$	0	0
$526$	31978.0	2.65078
$527$	615.199	0.0508510
$528$	−30339.7	−2.50069
$529$	−2290.93	−0.188290
$530$	0	0
$531$	−17403.3	−1.42230
$532$	0	0
$533$	−4725.59	−0.384030
$534$	−25195.7	−2.04180
$535$	0	0
$536$	−19525.4	−1.57345
$537$	29145.7	2.34214
$538$	22331.2	1.78953
$539$	0	0
$540$	0	0
$541$	14572.9	1.15811	0.579055	−	0.815288i	$-0.303422\pi$
$541$	14572.9	1.15811	0.579055	+	0.815288i	$0.303422\pi$
$542$	−17150.6	−1.35919
$543$	−34951.1	−2.76224
$544$	2650.51	0.208896
$545$	0	0
$546$	0	0
$547$	−11290.3	−0.882519	−0.441260	−	0.897380i	$-0.645468\pi$
$547$	−11290.3	−0.882519	−0.441260	+	0.897380i	$0.645468\pi$
$548$	−10743.4	−0.837472
$549$	20987.9	1.63159
$550$	0	0
$551$	3363.73	0.260072
$552$	34305.9	2.64521
$553$	0	0
$554$	94.5826	0.00725349
$555$	0	0
$556$	4609.84	0.351620
$557$	−3919.93	−0.298191	−0.149096	−	0.988823i	$-0.547636\pi$
$557$	−3919.93	−0.298191	−0.149096	+	0.988823i	$0.547636\pi$
$558$	−3038.72	−0.230537
$559$	−722.401	−0.0546588
$560$	0	0
$561$	−37845.4	−2.84819
$562$	−32774.2	−2.45996
$563$	−7444.71	−0.557295	−0.278647	−	0.960393i	$-0.589886\pi$
$563$	−7444.71	−0.557295	−0.278647	+	0.960393i	$0.589886\pi$
$564$	66806.8	4.98772
$565$	0	0
$566$	−6148.41	−0.456602
$567$	0	0
$568$	5816.55	0.429678
$569$	8529.24	0.628408	0.314204	−	0.949355i	$-0.398262\pi$
$569$	8529.24	0.628408	0.314204	+	0.949355i	$0.398262\pi$
$570$	0	0
$571$	10324.8	0.756706	0.378353	−	0.925661i	$-0.376491\pi$
$571$	10324.8	0.756706	0.378353	+	0.925661i	$0.376491\pi$
$572$	13535.6	0.989423
$573$	39196.3	2.85768
$574$	0	0
$575$	0	0
$576$	−39483.1	−2.85613
$577$	6467.10	0.466601	0.233301	−	0.972405i	$-0.425047\pi$
$577$	6467.10	0.466601	0.233301	+	0.972405i	$0.425047\pi$
$578$	4257.88	0.306409
$579$	29237.9	2.09859
$580$	0	0
$581$	0	0
$582$	86422.5	6.15520
$583$	11196.2	0.795369
$584$	−11016.5	−0.780589
$585$	0	0
$586$	−30743.4	−2.16723
$587$	7114.68	0.500263	0.250131	−	0.968212i	$-0.419526\pi$
$587$	7114.68	0.500263	0.250131	+	0.968212i	$0.419526\pi$
$588$	0	0
$589$	−471.828	−0.0330073
$590$	0	0
$591$	28852.2	2.00816
$592$	−12234.3	−0.849369
$593$	−21270.8	−1.47299	−0.736497	−	0.676441i	$-0.763522\pi$
$593$	−21270.8	−1.47299	−0.736497	+	0.676441i	$0.763522\pi$
$594$	109036.	7.53165
$595$	0	0
$596$	−27672.1	−1.90183
$597$	1915.62	0.131325
$598$	−6781.13	−0.463714
$599$	−4697.43	−0.320420	−0.160210	−	0.987083i	$-0.551217\pi$
$599$	−4697.43	−0.320420	−0.160210	+	0.987083i	$0.551217\pi$
$600$	0	0
$601$	−8719.81	−0.591828	−0.295914	−	0.955215i	$-0.595624\pi$
$601$	−8719.81	−0.591828	−0.295914	+	0.955215i	$0.595624\pi$
$602$	0	0
$603$	35111.8	2.37125
$604$	18549.7	1.24963
$605$	0	0
$606$	78195.2	5.24169
$607$	−23097.1	−1.54445	−0.772227	−	0.635347i	$-0.780857\pi$
$607$	−23097.1	−1.54445	−0.772227	+	0.635347i	$0.780857\pi$
$608$	−2032.81	−0.135594
$609$	0	0
$610$	0	0
$611$	−6364.05	−0.421378
$612$	63543.1	4.19702
$613$	16846.2	1.10997	0.554986	−	0.831860i	$-0.312724\pi$
$613$	16846.2	1.10997	0.554986	+	0.831860i	$0.312724\pi$
$614$	23106.3	1.51872
$615$	0	0
$616$	0	0
$617$	−3476.46	−0.226834	−0.113417	−	0.993547i	$-0.536180\pi$
$617$	−3476.46	−0.226834	−0.113417	+	0.993547i	$0.536180\pi$
$618$	−47045.5	−3.06221
$619$	5407.16	0.351102	0.175551	−	0.984470i	$-0.443829\pi$
$619$	5407.16	0.351102	0.175551	+	0.984470i	$0.443829\pi$
$620$	0	0
$621$	−35983.4	−2.32522
$622$	3586.24	0.231182
$623$	0	0
$624$	−6874.83	−0.441047
$625$	0	0
$626$	11037.5	0.704706
$627$	29025.6	1.84876
$628$	6328.02	0.402095
$629$	−15260.9	−0.967396
$630$	0	0
$631$	−10498.7	−0.662358	−0.331179	−	0.943568i	$-0.607446\pi$
$631$	−10498.7	−0.662358	−0.331179	+	0.943568i	$0.607446\pi$
$632$	18185.5	1.14459
$633$	2849.74	0.178937
$634$	−49376.1	−3.09302
$635$	0	0
$636$	−26632.3	−1.66044
$637$	0	0
$638$	20795.4	1.29044
$639$	−10459.7	−0.647539
$640$	0	0
$641$	14363.3	0.885050	0.442525	−	0.896756i	$-0.354083\pi$
$641$	14363.3	0.885050	0.442525	+	0.896756i	$0.354083\pi$
$642$	−42359.1	−2.60402
$643$	16883.3	1.03548	0.517739	−	0.855539i	$-0.326774\pi$
$643$	16883.3	1.03548	0.517739	+	0.855539i	$0.326774\pi$
$644$	0	0
$645$	0	0
$646$	14978.0	0.912231
$647$	−27365.2	−1.66281	−0.831404	−	0.555668i	$-0.812462\pi$
$647$	−27365.2	−1.66281	−0.831404	+	0.555668i	$0.812462\pi$
$648$	−61961.3	−3.75628
$649$	−16705.9	−1.01042
$650$	0	0
$651$	0	0
$652$	31357.8	1.88353
$653$	−28643.3	−1.71654	−0.858269	−	0.513200i	$-0.828460\pi$
$653$	−28643.3	−1.71654	−0.858269	+	0.513200i	$0.828460\pi$
$654$	−64809.3	−3.87499
$655$	0	0
$656$	−17072.0	−1.01608
$657$	19810.4	1.17637
$658$	0	0
$659$	−4102.07	−0.242480	−0.121240	−	0.992623i	$-0.538687\pi$
$659$	−4102.07	−0.242480	−0.121240	+	0.992623i	$0.538687\pi$
$660$	0	0
$661$	13784.0	0.811098	0.405549	−	0.914073i	$-0.367080\pi$
$661$	13784.0	0.811098	0.405549	+	0.914073i	$0.367080\pi$
$662$	377.834	0.0221827
$663$	−8575.58	−0.502334
$664$	18514.0	1.08205
$665$	0	0
$666$	75379.9	4.38576
$667$	−6862.79	−0.398393
$668$	8029.95	0.465102
$669$	60193.4	3.47864
$670$	0	0
$671$	20146.9	1.15911
$672$	0	0
$673$	−4676.98	−0.267882	−0.133941	−	0.990989i	$-0.542763\pi$
$673$	−4676.98	−0.267882	−0.133941	+	0.990989i	$0.542763\pi$
$674$	−20164.2	−1.15237
$675$	0	0
$676$	−30858.5	−1.75572
$677$	3849.54	0.218537	0.109269	−	0.994012i	$-0.465149\pi$
$677$	3849.54	0.218537	0.109269	+	0.994012i	$0.465149\pi$
$678$	−96600.9	−5.47188
$679$	0	0
$680$	0	0
$681$	22584.7	1.27085
$682$	−2916.96	−0.163777
$683$	10647.5	0.596508	0.298254	−	0.954487i	$-0.403596\pi$
$683$	10647.5	0.596508	0.298254	+	0.954487i	$0.403596\pi$
$684$	−48734.5	−2.72428
$685$	0	0
$686$	0	0
$687$	−23730.0	−1.31784
$688$	−2609.80	−0.144619
$689$	2537.01	0.140279
$690$	0	0
$691$	−22410.8	−1.23379	−0.616894	−	0.787046i	$-0.711609\pi$
$691$	−22410.8	−1.23379	−0.616894	+	0.787046i	$0.711609\pi$
$692$	−39560.4	−2.17321
$693$	0	0
$694$	8691.72	0.475408
$695$	0	0
$696$	−23838.9	−1.29829
$697$	−21295.4	−1.15728
$698$	−56896.6	−3.08534
$699$	−49269.2	−2.66600
$700$	0	0
$701$	6040.55	0.325461	0.162731	−	0.986671i	$-0.447970\pi$
$701$	6040.55	0.325461	0.162731	+	0.986671i	$0.447970\pi$
$702$	24707.0	1.32836
$703$	11704.4	0.627936
$704$	−37901.0	−2.02904
$705$	0	0
$706$	−13955.2	−0.743926
$707$	0	0
$708$	39738.2	2.10940
$709$	19582.2	1.03727	0.518634	−	0.854996i	$-0.326441\pi$
$709$	19582.2	1.03727	0.518634	+	0.854996i	$0.326441\pi$
$710$	0	0
$711$	−32702.2	−1.72493
$712$	19570.4	1.03010
$713$	962.637	0.0505625
$714$	0	0
$715$	0	0
$716$	−46975.2	−2.45188
$717$	7964.07	0.414817
$718$	5779.91	0.300424
$719$	−9257.31	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$719$	−9257.31	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$720$	0	0
$721$	0	0
$722$	21721.6	1.11966
$723$	43486.1	2.23688
$724$	56332.0	2.89166
$725$	0	0
$726$	117702.	6.01697
$727$	11915.5	0.607870	0.303935	−	0.952693i	$-0.401699\pi$
$727$	11915.5	0.607870	0.303935	+	0.952693i	$0.401699\pi$
$728$	0	0
$729$	17757.3	0.902162
$730$	0	0
$731$	−3255.43	−0.164715
$732$	−47923.2	−2.41980
$733$	−15608.0	−0.786486	−0.393243	−	0.919435i	$-0.628647\pi$
$733$	−15608.0	−0.786486	−0.393243	+	0.919435i	$0.628647\pi$
$734$	−42891.4	−2.15688
$735$	0	0
$736$	4147.40	0.207711
$737$	33704.8	1.68457
$738$	105187.	5.24659
$739$	−26189.9	−1.30367	−0.651835	−	0.758361i	$-0.726000\pi$
$739$	−26189.9	−1.30367	−0.651835	+	0.758361i	$0.726000\pi$
$740$	0	0
$741$	6577.05	0.326065
$742$	0	0
$743$	−37914.9	−1.87209	−0.936044	−	0.351882i	$-0.885542\pi$
$743$	−37914.9	−1.87209	−0.936044	+	0.351882i	$0.885542\pi$
$744$	3343.86	0.164774
$745$	0	0
$746$	−11076.7	−0.543627
$747$	−33292.9	−1.63069
$748$	60996.8	2.98164
$749$	0	0
$750$	0	0
$751$	−21401.6	−1.03989	−0.519944	−	0.854200i	$-0.674047\pi$
$751$	−21401.6	−1.03989	−0.519944	+	0.854200i	$0.674047\pi$
$752$	−22991.3	−1.11490
$753$	69288.7	3.35328
$754$	4712.14	0.227594
$755$	0	0
$756$	0	0
$757$	−24094.8	−1.15686	−0.578428	−	0.815734i	$-0.696333\pi$
$757$	−24094.8	−1.15686	−0.578428	+	0.815734i	$0.696333\pi$
$758$	−42946.5	−2.05790
$759$	−59218.8	−2.83203
$760$	0	0
$761$	−19101.1	−0.909872	−0.454936	−	0.890524i	$-0.650338\pi$
$761$	−19101.1	−0.909872	−0.454936	+	0.890524i	$0.650338\pi$
$762$	10701.1	0.508741
$763$	0	0
$764$	−63174.2	−2.99157
$765$	0	0
$766$	−64681.0	−3.05094
$767$	−3785.48	−0.178208
$768$	74667.5	3.50824
$769$	32718.1	1.53426	0.767129	−	0.641493i	$-0.221685\pi$
$769$	32718.1	1.53426	0.767129	+	0.641493i	$0.221685\pi$
$770$	0	0
$771$	−40648.3	−1.89872
$772$	−47123.8	−2.19692
$773$	−4707.97	−0.219061	−0.109530	−	0.993983i	$-0.534935\pi$
$773$	−4707.97	−0.219061	−0.109530	+	0.993983i	$0.534935\pi$
$774$	16079.9	0.746746
$775$	0	0
$776$	−67127.6	−3.10533
$777$	0	0
$778$	−54830.3	−2.52669
$779$	16332.6	0.751187
$780$	0	0
$781$	−10040.5	−0.460023
$782$	−30558.5	−1.39741
$783$	25004.5	1.14124
$784$	0	0
$785$	0	0
$786$	45175.6	2.05007
$787$	3218.47	0.145776	0.0728882	−	0.997340i	$-0.476778\pi$
$787$	3218.47	0.145776	0.0728882	+	0.997340i	$0.476778\pi$
$788$	−46502.1	−2.10224
$789$	63279.1	2.85525
$790$	0	0
$791$	0	0
$792$	−145199.	−6.51440
$793$	4565.18	0.204432
$794$	64117.4	2.86580
$795$	0	0
$796$	−3087.47	−0.137478
$797$	−15548.9	−0.691054	−0.345527	−	0.938409i	$-0.612300\pi$
$797$	−15548.9	−0.691054	−0.345527	+	0.938409i	$0.612300\pi$
$798$	0	0
$799$	−28679.0	−1.26982
$800$	0	0
$801$	−35192.6	−1.55240
$802$	−9455.58	−0.416319
$803$	19016.6	0.835717
$804$	−80173.2	−3.51678
$805$	0	0
$806$	−660.968	−0.0288854
$807$	44189.6	1.92757
$808$	−60737.1	−2.64446
$809$	−3106.83	−0.135019	−0.0675095	−	0.997719i	$-0.521505\pi$
$809$	−3106.83	−0.135019	−0.0675095	+	0.997719i	$0.521505\pi$
$810$	0	0
$811$	44061.1	1.90776	0.953882	−	0.300183i	$-0.0970477\pi$
$811$	44061.1	1.90776	0.953882	+	0.300183i	$0.0970477\pi$
$812$	0	0
$813$	−33938.1	−1.46404
$814$	72359.3	3.11572
$815$	0	0
$816$	−30980.8	−1.32910
$817$	2496.76	0.106916
$818$	−51310.9	−2.19321
$819$	0	0
$820$	0	0
$821$	13977.3	0.594169	0.297084	−	0.954851i	$-0.403986\pi$
$821$	13977.3	0.594169	0.297084	+	0.954851i	$0.403986\pi$
$822$	−32273.2	−1.36941
$823$	−3287.14	−0.139225	−0.0696127	−	0.997574i	$-0.522176\pi$
$823$	−3287.14	−0.139225	−0.0696127	+	0.997574i	$0.522176\pi$
$824$	36542.0	1.54490
$825$	0	0
$826$	0	0
$827$	2454.69	0.103214	0.0516069	−	0.998667i	$-0.483566\pi$
$827$	2454.69	0.103214	0.0516069	+	0.998667i	$0.483566\pi$
$828$	99429.5	4.17320
$829$	38510.6	1.61342	0.806711	−	0.590946i	$-0.201245\pi$
$829$	38510.6	1.61342	0.806711	+	0.590946i	$0.201245\pi$
$830$	0	0
$831$	187.163	0.00781301
$832$	−8588.18	−0.357862
$833$	0	0
$834$	13848.0	0.574960
$835$	0	0
$836$	−46781.6	−1.93538
$837$	−3507.36	−0.144841
$838$	−7897.64	−0.325560
$839$	−32983.5	−1.35723	−0.678616	−	0.734493i	$-0.737420\pi$
$839$	−32983.5	−1.35723	−0.678616	+	0.734493i	$0.737420\pi$
$840$	0	0
$841$	−19620.1	−0.804466
$842$	−63819.1	−2.61206
$843$	−64854.6	−2.64972
$844$	−4593.04	−0.187321
$845$	0	0
$846$	141658.	5.75684
$847$	0	0
$848$	9165.39	0.371157
$849$	−12166.7	−0.491824
$850$	0	0
$851$	−23879.6	−0.961906
$852$	23883.3	0.960361
$853$	−13620.1	−0.546710	−0.273355	−	0.961913i	$-0.588133\pi$
$853$	−13620.1	−0.546710	−0.273355	+	0.961913i	$0.588133\pi$
$854$	0	0
$855$	0	0
$856$	32901.9	1.31374
$857$	24493.2	0.976279	0.488139	−	0.872766i	$-0.337676\pi$
$857$	24493.2	0.976279	0.488139	+	0.872766i	$0.337676\pi$
$858$	40661.0	1.61788
$859$	−3742.34	−0.148646	−0.0743231	−	0.997234i	$-0.523680\pi$
$859$	−3742.34	−0.148646	−0.0743231	+	0.997234i	$0.523680\pi$
$860$	0	0
$861$	0	0
$862$	−44346.8	−1.75227
$863$	−165.238	−0.00651770	−0.00325885	−	0.999995i	$-0.501037\pi$
$863$	−165.238	−0.00651770	−0.00325885	+	0.999995i	$0.501037\pi$
$864$	−15111.0	−0.595009
$865$	0	0
$866$	−1309.77	−0.0513945
$867$	8425.63	0.330045
$868$	0	0
$869$	−31391.8	−1.22542
$870$	0	0
$871$	7637.33	0.297108
$872$	50339.8	1.95495
$873$	120713.	4.67984
$874$	23436.9	0.907055
$875$	0	0
$876$	−45234.5	−1.74467
$877$	−7230.06	−0.278383	−0.139192	−	0.990265i	$-0.544450\pi$
$877$	−7230.06	−0.278383	−0.139192	+	0.990265i	$0.544450\pi$
$878$	82347.2	3.16524
$879$	−60835.9	−2.33441
$880$	0	0
$881$	18707.9	0.715422	0.357711	−	0.933832i	$-0.383557\pi$
$881$	18707.9	0.715422	0.357711	+	0.933832i	$0.383557\pi$
$882$	0	0
$883$	28766.9	1.09636	0.548179	−	0.836361i	$-0.315321\pi$
$883$	28766.9	1.09636	0.548179	+	0.836361i	$0.315321\pi$
$884$	13821.6	0.525871
$885$	0	0
$886$	19212.3	0.728500
$887$	−33980.4	−1.28630	−0.643152	−	0.765738i	$-0.722374\pi$
$887$	−33980.4	−1.28630	−0.643152	+	0.765738i	$0.722374\pi$
$888$	−82949.2	−3.13468
$889$	0	0
$890$	0	0
$891$	106957.	4.02156
$892$	−97015.8	−3.64163
$893$	21995.4	0.824242
$894$	−83127.1	−3.10983
$895$	0	0
$896$	0	0
$897$	−13418.7	−0.499484
$898$	60425.7	2.24547
$899$	−668.927	−0.0248164
$900$	0	0
$901$	11432.8	0.422732
$902$	100972.	3.72727
$903$	0	0
$904$	75033.5	2.76059
$905$	0	0
$906$	55723.6	2.04337
$907$	35753.2	1.30889	0.654446	−	0.756109i	$-0.272902\pi$
$907$	35753.2	1.30889	0.654446	+	0.756109i	$0.272902\pi$
$908$	−36400.6	−1.33039
$909$	109221.	3.98529
$910$	0	0
$911$	16039.7	0.583334	0.291667	−	0.956520i	$-0.405790\pi$
$911$	16039.7	0.583334	0.291667	+	0.956520i	$0.405790\pi$
$912$	23760.8	0.862717
$913$	−31958.8	−1.15847
$914$	27998.9	1.01326
$915$	0	0
$916$	38246.5	1.37958
$917$	0	0
$918$	111340.	4.00301
$919$	22104.9	0.793441	0.396720	−	0.917940i	$-0.370148\pi$
$919$	22104.9	0.793441	0.396720	+	0.917940i	$0.370148\pi$
$920$	0	0
$921$	45723.3	1.63587
$922$	48656.8	1.73799
$923$	−2275.13	−0.0811342
$924$	0	0
$925$	0	0
$926$	−3194.74	−0.113376
$927$	−65711.9	−2.32822
$928$	−2881.99	−0.101946
$929$	−20234.8	−0.714621	−0.357310	−	0.933986i	$-0.616306\pi$
$929$	−20234.8	−0.714621	−0.357310	+	0.933986i	$0.616306\pi$
$930$	0	0
$931$	0	0
$932$	79409.1	2.79091
$933$	7096.56	0.249015
$934$	21630.7	0.757792
$935$	0	0
$936$	−32901.3	−1.14894
$937$	−37551.2	−1.30923	−0.654614	−	0.755964i	$-0.727169\pi$
$937$	−37551.2	−1.30923	−0.654614	+	0.755964i	$0.727169\pi$
$938$	0	0
$939$	21841.3	0.759066
$940$	0	0
$941$	−19093.3	−0.661449	−0.330725	−	0.943727i	$-0.607293\pi$
$941$	−19093.3	−0.661449	−0.330725	+	0.943727i	$0.607293\pi$
$942$	19009.4	0.657496
$943$	−33322.2	−1.15071
$944$	−13675.7	−0.471511
$945$	0	0
$946$	15435.6	0.530501
$947$	10117.0	0.347158	0.173579	−	0.984820i	$-0.444467\pi$
$947$	10117.0	0.347158	0.173579	+	0.984820i	$0.444467\pi$
$948$	74671.3	2.55824
$949$	4309.06	0.147395
$950$	0	0
$951$	−97706.8	−3.33161
$952$	0	0
$953$	−6272.48	−0.213206	−0.106603	−	0.994302i	$-0.533997\pi$
$953$	−6272.48	−0.213206	−0.106603	+	0.994302i	$0.533997\pi$
$954$	−56471.3	−1.91649
$955$	0	0
$956$	−12836.0	−0.434253
$957$	41150.6	1.38998
$958$	36960.7	1.24650
$959$	0	0
$960$	0	0
$961$	−29697.2	−0.996850
$962$	16396.3	0.549519
$963$	−59166.0	−1.97985
$964$	−70088.1	−2.34169
$965$	0	0
$966$	0	0
$967$	−26660.8	−0.886613	−0.443307	−	0.896370i	$-0.646195\pi$
$967$	−26660.8	−0.886613	−0.443307	+	0.896370i	$0.646195\pi$
$968$	−91423.3	−3.03560
$969$	29638.9	0.982599
$970$	0	0
$971$	461.190	0.0152423	0.00762116	−	0.999971i	$-0.497574\pi$
$971$	461.190	0.0152423	0.00762116	+	0.999971i	$0.497574\pi$
$972$	−103455.	−3.41391
$973$	0	0
$974$	30453.3	1.00184
$975$	0	0
$976$	16492.5	0.540895
$977$	4058.08	0.132886	0.0664429	−	0.997790i	$-0.478835\pi$
$977$	4058.08	0.132886	0.0664429	+	0.997790i	$0.478835\pi$
$978$	94199.0	3.07991
$979$	−33782.4	−1.10285
$980$	0	0
$981$	−90523.9	−2.94618
$982$	−17248.7	−0.560517
$983$	3259.92	0.105773	0.0528867	−	0.998601i	$-0.483158\pi$
$983$	3259.92	0.105773	0.0528867	+	0.998601i	$0.483158\pi$
$984$	−115749.	−3.74995
$985$	0	0
$986$	21234.8	0.685857
$987$	0	0
$988$	−10600.5	−0.341342
$989$	−5093.96	−0.163780
$990$	0	0
$991$	−21398.9	−0.685933	−0.342967	−	0.939348i	$-0.611432\pi$
$991$	−21398.9	−0.685933	−0.342967	+	0.939348i	$0.611432\pi$
$992$	404.254	0.0129386
$993$	747.669	0.0238938
$994$	0	0
$995$	0	0
$996$	76020.1	2.41846
$997$	−28609.5	−0.908798	−0.454399	−	0.890798i	$-0.650146\pi$
$997$	−28609.5	−0.908798	−0.454399	+	0.890798i	$0.650146\pi$
$998$	91589.2	2.90502
$999$	87005.2	2.75548

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.z.1.1		4
5.4	even	2		1225.4.a.bd.1.4		4
7.6	odd	2		175.4.a.g.1.1	✓	4
21.20	even	2		1575.4.a.bl.1.4		4
35.13	even	4		175.4.b.f.99.7		8
35.27	even	4		175.4.b.f.99.2		8
35.34	odd	2		175.4.a.h.1.4	yes	4
105.104	even	2		1575.4.a.bg.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.g.1.1	✓	4	7.6	odd	2
175.4.a.h.1.4	yes	4	35.34	odd	2
175.4.b.f.99.2		8	35.27	even	4
175.4.b.f.99.7		8	35.13	even	4
1225.4.a.z.1.1		4	1.1	even	1	trivial
1225.4.a.bd.1.4		4	5.4	even	2
1575.4.a.bg.1.1		4	105.104	even	2
1575.4.a.bl.1.4		4	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-4.84167 q^{2} -9.58084 q^{3} +15.4418 q^{4} +46.3873 q^{6} -36.0308 q^{8} +64.7926 q^{9} +62.1962 q^{11} -147.945 q^{12} +14.0934 q^{13} +50.9148 q^{16} +63.5104 q^{17} -313.704 q^{18} -48.7094 q^{19} -301.134 q^{22} +99.3784 q^{23} +345.205 q^{24} -68.2354 q^{26} -362.085 q^{27} -69.0571 q^{29} +9.68658 q^{31} +41.7334 q^{32} -595.892 q^{33} -307.497 q^{34} +1000.51 q^{36} -240.290 q^{37} +235.835 q^{38} -135.026 q^{39} -335.306 q^{41} -51.2582 q^{43} +960.421 q^{44} -481.158 q^{46} -451.564 q^{47} -487.806 q^{48} -608.484 q^{51} +217.627 q^{52} +180.014 q^{53} +1753.10 q^{54} +466.677 q^{57} +334.352 q^{58} -268.600 q^{59} +323.925 q^{61} -46.8992 q^{62} -609.378 q^{64} +2885.12 q^{66} +541.910 q^{67} +980.715 q^{68} -952.129 q^{69} -161.433 q^{71} -2334.53 q^{72} +305.751 q^{73} +1163.40 q^{74} -752.161 q^{76} +653.753 q^{78} -504.722 q^{79} +1719.68 q^{81} +1623.44 q^{82} -513.838 q^{83} +248.176 q^{86} +661.625 q^{87} -2240.98 q^{88} -543.158 q^{89} +1534.58 q^{92} -92.8056 q^{93} +2186.32 q^{94} -399.842 q^{96} +1863.06 q^{97} +4029.85 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9} + 100 q^{11} - 165 q^{12} + 44 q^{13} + 160 q^{16} - 53 q^{17} - 433 q^{18} + 29 q^{19} + 152 q^{22} - 295 q^{23} + 21 q^{24} - 700 q^{26}+ \cdots + 2383 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 3 * q^3 + 36 * q^4 - q^6 - 27 * q^8 + 61 * q^9 + 100 * q^11 - 165 * q^12 + 44 * q^13 + 160 * q^16 - 53 * q^17 - 433 * q^18 + 29 * q^19 + 152 * q^22 - 295 * q^23 + 21 * q^24 - 700 * q^26 - 441 * q^27 + 129 * q^29 - 114 * q^31 + 310 * q^32 - 865 * q^33 - 203 * q^34 + 1101 * q^36 - 403 * q^37 + 555 * q^38 + 674 * q^39 - 671 * q^41 + 411 * q^43 + 438 * q^44 - 997 * q^46 - 8 * q^47 - 523 * q^48 - 885 * q^51 + 74 * q^52 - 90 * q^53 + 2777 * q^54 - 233 * q^57 - 673 * q^58 - 1018 * q^59 - 50 * q^61 - 1626 * q^62 - 2421 * q^64 + 3841 * q^66 - 424 * q^67 - 617 * q^68 - 1080 * q^69 + 215 * q^71 - 2940 * q^72 - 1207 * q^73 + 623 * q^74 + 3257 * q^76 - 278 * q^78 - 951 * q^79 + 28 * q^81 + 1695 * q^82 - 3035 * q^83 - 99 * q^86 + 2210 * q^87 - 163 * q^88 - 2819 * q^89 - 3073 * q^92 + 852 * q^93 + 3056 * q^94 + 1345 * q^96 - 1100 * q^97 + 2383 * q^99

Introduction

L-functions

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