LMFDB - Embedded newform 1225.4.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1225,4,Mod(1,1225)] 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("1225.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [2,-8,2,20,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.2773397570$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	\(q-2.58579 q^{2} +6.65685 q^{3} -1.31371 q^{4} -17.2132 q^{6} +24.0833 q^{8} +17.3137 q^{9} +38.2548 q^{11} -8.74517 q^{12} +19.3431 q^{13} -51.7645 q^{16} -87.2254 q^{17} -44.7696 q^{18} +44.2254 q^{19} -98.9188 q^{22} -218.167 q^{23} +160.319 q^{24} -50.0172 q^{26} -64.4802 q^{27} -46.9411 q^{29} -194.558 q^{31} -58.8141 q^{32} +254.657 q^{33} +225.546 q^{34} -22.7452 q^{36} -366.853 q^{37} -114.357 q^{38} +128.765 q^{39} +339.362 q^{41} +226.167 q^{43} -50.2557 q^{44} +564.132 q^{46} +11.6762 q^{47} -344.589 q^{48} -580.647 q^{51} -25.4113 q^{52} +209.019 q^{53} +166.732 q^{54} +294.402 q^{57} +121.380 q^{58} +616.000 q^{59} -320.735 q^{61} +503.087 q^{62} +566.197 q^{64} -658.488 q^{66} -14.5097 q^{67} +114.589 q^{68} -1452.30 q^{69} -952.000 q^{71} +416.971 q^{72} +824.489 q^{73} +948.603 q^{74} -58.0993 q^{76} -332.958 q^{78} +156.275 q^{79} -896.706 q^{81} -877.519 q^{82} -1036.53 q^{83} -584.818 q^{86} -312.480 q^{87} +921.301 q^{88} +170.225 q^{89} +286.607 q^{92} -1295.15 q^{93} -30.1921 q^{94} -391.517 q^{96} +1059.87 q^{97} +662.333 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9} - 14 q^{11} - 108 q^{12} + 50 q^{13} + 168 q^{16} - 50 q^{17} - 16 q^{18} - 36 q^{19} + 184 q^{22} - 244 q^{23} + 496 q^{24} - 216 q^{26}+ \cdots + 940 q^{99}+O(q^{100}) $ 2 * q - 8 * q^2 + 2 * q^3 + 20 * q^4 + 8 * q^6 - 48 * q^8 + 12 * q^9 - 14 * q^11 - 108 * q^12 + 50 * q^13 + 168 * q^16 - 50 * q^17 - 16 * q^18 - 36 * q^19 + 184 * q^22 - 244 * q^23 + 496 * q^24 - 216 * q^26 + 86 * q^27 - 26 * q^29 + 120 * q^31 - 672 * q^32 + 498 * q^33 + 24 * q^34 - 136 * q^36 - 564 * q^37 + 320 * q^38 - 14 * q^39 + 328 * q^41 + 260 * q^43 - 1164 * q^44 + 704 * q^46 - 350 * q^47 - 1368 * q^48 - 754 * q^51 + 628 * q^52 + 56 * q^53 - 648 * q^54 + 668 * q^57 + 8 * q^58 + 1232 * q^59 - 336 * q^61 - 1200 * q^62 + 2128 * q^64 - 1976 * q^66 + 152 * q^67 + 908 * q^68 - 1332 * q^69 - 1904 * q^71 + 800 * q^72 + 676 * q^73 + 2016 * q^74 - 1768 * q^76 + 440 * q^78 + 1014 * q^79 - 1454 * q^81 - 816 * q^82 - 376 * q^83 - 768 * q^86 - 410 * q^87 + 4688 * q^88 + 216 * q^89 - 264 * q^92 - 2760 * q^93 + 1928 * q^94 + 2464 * q^96 + 2742 * q^97 + 940 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.58579	−0.914214	−0.457107	−	0.889412i	$-0.651114\pi$
$2$	−2.58579	−0.914214	−0.457107	+	0.889412i	$0.651114\pi$
$3$	6.65685	1.28111	0.640556	−	0.767911i	$-0.278704\pi$
$3$	6.65685	1.28111	0.640556	+	0.767911i	$0.278704\pi$
$4$	−1.31371	−0.164214
$5$	0	0
$5$	0	0
$6$	−17.2132	−1.17121
$7$	0	0
$7$	0	0
$8$	24.0833	1.06434
$9$	17.3137	0.641248
$10$	0	0
$11$	38.2548	1.04857	0.524285	−	0.851543i	$-0.324333\pi$
$11$	38.2548	1.04857	0.524285	+	0.851543i	$0.324333\pi$
$12$	−8.74517	−0.210376
$13$	19.3431	0.412679	0.206339	−	0.978480i	$-0.433845\pi$
$13$	19.3431	0.412679	0.206339	+	0.978480i	$0.433845\pi$
$14$	0	0
$15$	0	0
$16$	−51.7645	−0.808820
$17$	−87.2254	−1.24443	−0.622214	−	0.782847i	$-0.713767\pi$
$17$	−87.2254	−1.24443	−0.622214	+	0.782847i	$0.713767\pi$
$18$	−44.7696	−0.586238
$19$	44.2254	0.534000	0.267000	−	0.963697i	$-0.413968\pi$
$19$	44.2254	0.534000	0.267000	+	0.963697i	$0.413968\pi$
$20$	0	0
$21$	0	0
$22$	−98.9188	−0.958617
$23$	−218.167	−1.97786	−0.988932	−	0.148371i	$-0.952597\pi$
$23$	−218.167	−1.97786	−0.988932	+	0.148371i	$0.952597\pi$
$24$	160.319	1.36354
$25$	0	0
$26$	−50.0172	−0.377276
$27$	−64.4802	−0.459601
$28$	0	0
$29$	−46.9411	−0.300578	−0.150289	−	0.988642i	$-0.548020\pi$
$29$	−46.9411	−0.300578	−0.150289	+	0.988642i	$0.548020\pi$
$30$	0	0
$31$	−194.558	−1.12722	−0.563609	−	0.826042i	$-0.690587\pi$
$31$	−194.558	−1.12722	−0.563609	+	0.826042i	$0.690587\pi$
$32$	−58.8141	−0.324905
$33$	254.657	1.34334
$34$	225.546	1.13767
$35$	0	0
$36$	−22.7452	−0.105302
$37$	−366.853	−1.63001	−0.815003	−	0.579457i	$-0.803265\pi$
$37$	−366.853	−1.63001	−0.815003	+	0.579457i	$0.803265\pi$
$38$	−114.357	−0.488190
$39$	128.765	0.528688
$40$	0	0
$41$	339.362	1.29267	0.646336	−	0.763053i	$-0.276301\pi$
$41$	339.362	1.29267	0.646336	+	0.763053i	$0.276301\pi$
$42$	0	0
$43$	226.167	0.802095	0.401047	−	0.916057i	$-0.368646\pi$
$43$	226.167	0.802095	0.401047	+	0.916057i	$0.368646\pi$
$44$	−50.2557	−0.172189
$45$	0	0
$46$	564.132	1.80819
$47$	11.6762	0.0362372	0.0181186	−	0.999836i	$-0.494232\pi$
$47$	11.6762	0.0362372	0.0181186	+	0.999836i	$0.494232\pi$
$48$	−344.589	−1.03619
$49$	0	0
$50$	0	0
$51$	−580.647	−1.59425
$52$	−25.4113	−0.0677674
$53$	209.019	0.541717	0.270859	−	0.962619i	$-0.412692\pi$
$53$	209.019	0.541717	0.270859	+	0.962619i	$0.412692\pi$
$54$	166.732	0.420173
$55$	0	0
$56$	0	0
$57$	294.402	0.684114
$58$	121.380	0.274792
$59$	616.000	1.35926	0.679630	−	0.733555i	$-0.262140\pi$
$59$	616.000	1.35926	0.679630	+	0.733555i	$0.262140\pi$
$60$	0	0
$61$	−320.735	−0.673212	−0.336606	−	0.941646i	$-0.609279\pi$
$61$	−320.735	−0.673212	−0.336606	+	0.941646i	$0.609279\pi$
$62$	503.087	1.03052
$63$	0	0
$64$	566.197	1.10585
$65$	0	0
$66$	−658.488	−1.22810
$67$	−14.5097	−0.0264573	−0.0132286	−	0.999912i	$-0.504211\pi$
$67$	−14.5097	−0.0264573	−0.0132286	+	0.999912i	$0.504211\pi$
$68$	114.589	0.204352
$69$	−1452.30	−2.53387
$70$	0	0
$71$	−952.000	−1.59129	−0.795645	−	0.605763i	$-0.792868\pi$
$71$	−952.000	−1.59129	−0.795645	+	0.605763i	$0.792868\pi$
$72$	416.971	0.682506
$73$	824.489	1.32191	0.660953	−	0.750427i	$-0.270152\pi$
$73$	824.489	1.32191	0.660953	+	0.750427i	$0.270152\pi$
$74$	948.603	1.49017
$75$	0	0
$76$	−58.0993	−0.0876901
$77$	0	0
$78$	−332.958	−0.483334
$79$	156.275	0.222561	0.111280	−	0.993789i	$-0.464505\pi$
$79$	156.275	0.222561	0.111280	+	0.993789i	$0.464505\pi$
$80$	0	0
$81$	−896.706	−1.23005
$82$	−877.519	−1.18178
$83$	−1036.53	−1.37077	−0.685384	−	0.728182i	$-0.740366\pi$
$83$	−1036.53	−1.37077	−0.685384	+	0.728182i	$0.740366\pi$
$84$	0	0
$85$	0	0
$86$	−584.818	−0.733286
$87$	−312.480	−0.385074
$88$	921.301	1.11603
$89$	170.225	0.202740	0.101370	−	0.994849i	$-0.467677\pi$
$89$	170.225	0.202740	0.101370	+	0.994849i	$0.467677\pi$
$90$	0	0
$91$	0	0
$92$	286.607	0.324792
$93$	−1295.15	−1.44409
$94$	−30.1921	−0.0331285
$95$	0	0
$96$	−391.517	−0.416240
$97$	1059.87	1.10942	0.554710	−	0.832044i	$-0.312829\pi$
$97$	1059.87	1.10942	0.554710	+	0.832044i	$0.312829\pi$
$98$	0	0
$99$	662.333	0.672394
$100$	0	0
$101$	241.833	0.238251	0.119125	−	0.992879i	$-0.461991\pi$
$101$	241.833	0.238251	0.119125	+	0.992879i	$0.461991\pi$
$102$	1501.43	1.45749
$103$	−1679.58	−1.60673	−0.803367	−	0.595484i	$-0.796960\pi$
$103$	−1679.58	−1.60673	−0.803367	+	0.595484i	$0.796960\pi$
$104$	465.846	0.439230
$105$	0	0
$106$	−540.479	−0.495245
$107$	−1506.88	−1.36146	−0.680728	−	0.732537i	$-0.738336\pi$
$107$	−1506.88	−1.36146	−0.680728	+	0.732537i	$0.738336\pi$
$108$	84.7082	0.0754727
$109$	−1252.41	−1.10054	−0.550271	−	0.834986i	$-0.685476\pi$
$109$	−1252.41	−1.10054	−0.550271	+	0.834986i	$0.685476\pi$
$110$	0	0
$111$	−2442.09	−2.08822
$112$	0	0
$113$	−1370.20	−1.14069	−0.570345	−	0.821405i	$-0.693190\pi$
$113$	−1370.20	−1.14069	−0.570345	+	0.821405i	$0.693190\pi$
$114$	−761.261	−0.625426
$115$	0	0
$116$	61.6670	0.0493589
$117$	334.902	0.264630
$118$	−1592.84	−1.24265
$119$	0	0
$120$	0	0
$121$	132.432	0.0994984
$122$	829.352	0.615459
$123$	2259.09	1.65606
$124$	255.593	0.185104
$125$	0	0
$126$	0	0
$127$	−1213.49	−0.847873	−0.423936	−	0.905692i	$-0.639352\pi$
$127$	−1213.49	−0.847873	−0.423936	+	0.905692i	$0.639352\pi$
$128$	−993.551	−0.686081
$129$	1505.56	1.02757
$130$	0	0
$131$	1982.42	1.32217	0.661087	−	0.750309i	$-0.270096\pi$
$131$	1982.42	1.32217	0.661087	+	0.750309i	$0.270096\pi$
$132$	−334.545	−0.220594
$133$	0	0
$134$	37.5189	0.0241876
$135$	0	0
$136$	−2100.67	−1.32449
$137$	−2210.95	−1.37879	−0.689394	−	0.724386i	$-0.742123\pi$
$137$	−2210.95	−1.37879	−0.689394	+	0.724386i	$0.742123\pi$
$138$	3755.34	2.31649
$139$	−528.039	−0.322213	−0.161107	−	0.986937i	$-0.551506\pi$
$139$	−528.039	−0.322213	−0.161107	+	0.986937i	$0.551506\pi$
$140$	0	0
$141$	77.7267	0.0464239
$142$	2461.67	1.45478
$143$	739.969	0.432722
$144$	−896.235	−0.518655
$145$	0	0
$146$	−2131.95	−1.20851
$147$	0	0
$148$	481.938	0.267669
$149$	−328.372	−0.180545	−0.0902727	−	0.995917i	$-0.528774\pi$
$149$	−328.372	−0.180545	−0.0902727	+	0.995917i	$0.528774\pi$
$150$	0	0
$151$	1029.43	0.554793	0.277396	−	0.960756i	$-0.410528\pi$
$151$	1029.43	0.554793	0.277396	+	0.960756i	$0.410528\pi$
$152$	1065.09	0.568358
$153$	−1510.20	−0.797987
$154$	0	0
$155$	0	0
$156$	−169.159	−0.0868177
$157$	525.098	0.266926	0.133463	−	0.991054i	$-0.457390\pi$
$157$	525.098	0.266926	0.133463	+	0.991054i	$0.457390\pi$
$158$	−404.094	−0.203468
$159$	1391.41	0.694001
$160$	0	0
$161$	0	0
$162$	2318.69	1.12453
$163$	−1002.63	−0.481790	−0.240895	−	0.970551i	$-0.577441\pi$
$163$	−1002.63	−0.481790	−0.240895	+	0.970551i	$0.577441\pi$
$164$	−445.823	−0.212274
$165$	0	0
$166$	2680.24	1.25317
$167$	−359.422	−0.166544	−0.0832722	−	0.996527i	$-0.526537\pi$
$167$	−359.422	−0.166544	−0.0832722	+	0.996527i	$0.526537\pi$
$168$	0	0
$169$	−1822.84	−0.829696
$170$	0	0
$171$	765.706	0.342427
$172$	−297.117	−0.131715
$173$	3293.65	1.44747	0.723733	−	0.690080i	$-0.242425\pi$
$173$	3293.65	1.44747	0.723733	+	0.690080i	$0.242425\pi$
$174$	808.007	0.352039
$175$	0	0
$176$	−1980.24	−0.848104
$177$	4100.62	1.74137
$178$	−440.167	−0.185348
$179$	2978.82	1.24384	0.621921	−	0.783080i	$-0.286353\pi$
$179$	2978.82	1.24384	0.621921	+	0.783080i	$0.286353\pi$
$180$	0	0
$181$	−1462.31	−0.600514	−0.300257	−	0.953858i	$-0.597072\pi$
$181$	−1462.31	−0.600514	−0.300257	+	0.953858i	$0.597072\pi$
$182$	0	0
$183$	−2135.09	−0.862460
$184$	−5254.16	−2.10512
$185$	0	0
$186$	3348.97	1.32021
$187$	−3336.79	−1.30487
$188$	−15.3391	−0.00595064
$189$	0	0
$190$	0	0
$191$	−374.923	−0.142034	−0.0710169	−	0.997475i	$-0.522624\pi$
$191$	−374.923	−0.142034	−0.0710169	+	0.997475i	$0.522624\pi$
$192$	3769.09	1.41672
$193$	−733.028	−0.273391	−0.136696	−	0.990613i	$-0.543648\pi$
$193$	−733.028	−0.273391	−0.136696	+	0.990613i	$0.543648\pi$
$194$	−2740.61	−1.01425
$195$	0	0
$196$	0	0
$197$	2093.24	0.757043	0.378521	−	0.925593i	$-0.376433\pi$
$197$	2093.24	0.757043	0.378521	+	0.925593i	$0.376433\pi$
$198$	−1712.65	−0.614711
$199$	−2865.04	−1.02059	−0.510295	−	0.860000i	$-0.670464\pi$
$199$	−2865.04	−1.02059	−0.510295	+	0.860000i	$0.670464\pi$
$200$	0	0
$201$	−96.5887	−0.0338948
$202$	−625.330	−0.217812
$203$	0	0
$204$	762.801	0.261798
$205$	0	0
$206$	4343.03	1.46890
$207$	−3777.27	−1.26830
$208$	−1001.29	−0.333783
$209$	1691.84	0.559936
$210$	0	0
$211$	5643.65	1.84135	0.920674	−	0.390331i	$-0.127640\pi$
$211$	5643.65	1.84135	0.920674	+	0.390331i	$0.127640\pi$
$212$	−274.590	−0.0889573
$213$	−6337.33	−2.03862
$214$	3896.47	1.24466
$215$	0	0
$216$	−1552.89	−0.489172
$217$	0	0
$218$	3238.46	1.00613
$219$	5488.51	1.69351
$220$	0	0
$221$	−1687.21	−0.513549
$222$	6314.71	1.90908
$223$	−6369.16	−1.91260	−0.956302	−	0.292381i	$-0.905552\pi$
$223$	−6369.16	−1.91260	−0.956302	+	0.292381i	$0.905552\pi$
$224$	0	0
$225$	0	0
$226$	3543.05	1.04283
$227$	−1015.67	−0.296972	−0.148486	−	0.988914i	$-0.547440\pi$
$227$	−1015.67	−0.296972	−0.148486	+	0.988914i	$0.547440\pi$
$228$	−386.758	−0.112341
$229$	−4108.35	−1.18554	−0.592768	−	0.805373i	$-0.701965\pi$
$229$	−4108.35	−1.18554	−0.592768	+	0.805373i	$0.701965\pi$
$230$	0	0
$231$	0	0
$232$	−1130.50	−0.319917
$233$	−608.431	−0.171071	−0.0855357	−	0.996335i	$-0.527260\pi$
$233$	−608.431	−0.171071	−0.0855357	+	0.996335i	$0.527260\pi$
$234$	−865.984	−0.241928
$235$	0	0
$236$	−809.244	−0.223209
$237$	1040.30	0.285126
$238$	0	0
$239$	−5054.44	−1.36797	−0.683985	−	0.729496i	$-0.739755\pi$
$239$	−5054.44	−1.36797	−0.683985	+	0.729496i	$0.739755\pi$
$240$	0	0
$241$	−4.86782	−0.00130109	−0.000650547	−	1.00000i	$-0.500207\pi$
$241$	−4.86782	−0.00130109	−0.000650547		1.00000i	$0.500207\pi$
$242$	−342.442	−0.0909628
$243$	−4228.27	−1.11623
$244$	421.352	0.110551
$245$	0	0
$246$	−5841.52	−1.51399
$247$	855.458	0.220370
$248$	−4685.60	−1.19974
$249$	−6900.02	−1.75611
$250$	0	0
$251$	547.921	0.137787	0.0688934	−	0.997624i	$-0.478053\pi$
$251$	547.921	0.137787	0.0688934	+	0.997624i	$0.478053\pi$
$252$	0	0
$253$	−8345.92	−2.07393
$254$	3137.83	0.775137
$255$	0	0
$256$	−1960.46	−0.478629
$257$	−1774.61	−0.430729	−0.215364	−	0.976534i	$-0.569094\pi$
$257$	−1774.61	−0.430729	−0.215364	+	0.976534i	$0.569094\pi$
$258$	−3893.05	−0.939421
$259$	0	0
$260$	0	0
$261$	−812.725	−0.192745
$262$	−5126.11	−1.20875
$263$	1199.09	0.281138	0.140569	−	0.990071i	$-0.455107\pi$
$263$	1199.09	0.281138	0.140569	+	0.990071i	$0.455107\pi$
$264$	6132.97	1.42977
$265$	0	0
$266$	0	0
$267$	1133.17	0.259733
$268$	19.0615	0.00434464
$269$	−3250.29	−0.736706	−0.368353	−	0.929686i	$-0.620078\pi$
$269$	−3250.29	−0.736706	−0.368353	+	0.929686i	$0.620078\pi$
$270$	0	0
$271$	896.143	0.200874	0.100437	−	0.994943i	$-0.467976\pi$
$271$	896.143	0.200874	0.100437	+	0.994943i	$0.467976\pi$
$272$	4515.18	1.00652
$273$	0	0
$274$	5717.04	1.26051
$275$	0	0
$276$	1907.90	0.416095
$277$	386.562	0.0838492	0.0419246	−	0.999121i	$-0.486651\pi$
$277$	386.562	0.0838492	0.0419246	+	0.999121i	$0.486651\pi$
$278$	1365.40	0.294572
$279$	−3368.53	−0.722826
$280$	0	0
$281$	−3335.10	−0.708025	−0.354013	−	0.935241i	$-0.615183\pi$
$281$	−3335.10	−0.708025	−0.354013	+	0.935241i	$0.615183\pi$
$282$	−200.985	−0.0424414
$283$	5412.26	1.13684	0.568419	−	0.822739i	$-0.307555\pi$
$283$	5412.26	1.13684	0.568419	+	0.822739i	$0.307555\pi$
$284$	1250.65	0.261311
$285$	0	0
$286$	−1913.40	−0.395601
$287$	0	0
$288$	−1018.29	−0.208345
$289$	2695.27	0.548600
$290$	0	0
$291$	7055.42	1.42129
$292$	−1083.14	−0.217075
$293$	282.211	0.0562695	0.0281347	−	0.999604i	$-0.491043\pi$
$293$	282.211	0.0562695	0.0281347	+	0.999604i	$0.491043\pi$
$294$	0	0
$295$	0	0
$296$	−8835.01	−1.73488
$297$	−2466.68	−0.481924
$298$	849.099	0.165057
$299$	−4220.03	−0.816222
$300$	0	0
$301$	0	0
$302$	−2661.88	−0.507199
$303$	1609.85	0.305226
$304$	−2289.31	−0.431910
$305$	0	0
$306$	3905.04	0.729531
$307$	1919.67	0.356878	0.178439	−	0.983951i	$-0.442895\pi$
$307$	1919.67	0.356878	0.178439	+	0.983951i	$0.442895\pi$
$308$	0	0
$309$	−11180.7	−2.05841
$310$	0	0
$311$	−1213.31	−0.221223	−0.110612	−	0.993864i	$-0.535281\pi$
$311$	−1213.31	−0.221223	−0.110612	+	0.993864i	$0.535281\pi$
$312$	3101.07	0.562703
$313$	−1434.00	−0.258960	−0.129480	−	0.991582i	$-0.541331\pi$
$313$	−1434.00	−0.258960	−0.129480	+	0.991582i	$0.541331\pi$
$314$	−1357.79	−0.244028
$315$	0	0
$316$	−205.300	−0.0365475
$317$	−6496.95	−1.15112	−0.575560	−	0.817760i	$-0.695216\pi$
$317$	−6496.95	−1.15112	−0.575560	+	0.817760i	$0.695216\pi$
$318$	−3597.89	−0.634465
$319$	−1795.72	−0.315176
$320$	0	0
$321$	−10031.1	−1.74418
$322$	0	0
$323$	−3857.58	−0.664524
$324$	1178.01	0.201991
$325$	0	0
$326$	2592.58	0.440459
$327$	−8337.11	−1.40992
$328$	8172.96	1.37584
$329$	0	0
$330$	0	0
$331$	−9683.88	−1.60808	−0.804039	−	0.594576i	$-0.797320\pi$
$331$	−9683.88	−1.60808	−0.804039	+	0.594576i	$0.797320\pi$
$332$	1361.70	0.225099
$333$	−6351.58	−1.04524
$334$	929.389	0.152257
$335$	0	0
$336$	0	0
$337$	−29.1319	−0.00470895	−0.00235447	−	0.999997i	$-0.500749\pi$
$337$	−29.1319	−0.00470895	−0.00235447	+	0.999997i	$0.500749\pi$
$338$	4713.48	0.758520
$339$	−9121.24	−1.46135
$340$	0	0
$341$	−7442.80	−1.18197
$342$	−1979.95	−0.313051
$343$	0	0
$344$	5446.83	0.853701
$345$	0	0
$346$	−8516.68	−1.32329
$347$	7848.58	1.21422	0.607110	−	0.794618i	$-0.292329\pi$
$347$	7848.58	1.21422	0.607110	+	0.794618i	$0.292329\pi$
$348$	410.508	0.0632343
$349$	10269.6	1.57513	0.787567	−	0.616229i	$-0.211341\pi$
$349$	10269.6	1.57513	0.787567	+	0.616229i	$0.211341\pi$
$350$	0	0
$351$	−1247.25	−0.189668
$352$	−2249.93	−0.340686
$353$	2799.93	0.422168	0.211084	−	0.977468i	$-0.432301\pi$
$353$	2799.93	0.422168	0.211084	+	0.977468i	$0.432301\pi$
$354$	−10603.3	−1.59198
$355$	0	0
$356$	−223.627	−0.0332927
$357$	0	0
$358$	−7702.60	−1.13714
$359$	−3163.29	−0.465048	−0.232524	−	0.972591i	$-0.574698\pi$
$359$	−3163.29	−0.465048	−0.232524	+	0.972591i	$0.574698\pi$
$360$	0	0
$361$	−4903.11	−0.714844
$362$	3781.23	0.548998
$363$	881.583	0.127469
$364$	0	0
$365$	0	0
$366$	5520.88	0.788472
$367$	3182.85	0.452706	0.226353	−	0.974045i	$-0.427320\pi$
$367$	3182.85	0.452706	0.226353	+	0.974045i	$0.427320\pi$
$368$	11293.3	1.59974
$369$	5875.62	0.828923
$370$	0	0
$371$	0	0
$372$	1701.45	0.237139
$373$	2615.14	0.363021	0.181510	−	0.983389i	$-0.441901\pi$
$373$	2615.14	0.363021	0.181510	+	0.983389i	$0.441901\pi$
$374$	8628.23	1.19293
$375$	0	0
$376$	281.201	0.0385687
$377$	−907.989	−0.124042
$378$	0	0
$379$	−672.434	−0.0911362	−0.0455681	−	0.998961i	$-0.514510\pi$
$379$	−672.434	−0.0911362	−0.0455681	+	0.998961i	$0.514510\pi$
$380$	0	0
$381$	−8078.03	−1.08622
$382$	969.470	0.129849
$383$	1169.86	0.156075	0.0780377	−	0.996950i	$-0.475135\pi$
$383$	1169.86	0.156075	0.0780377	+	0.996950i	$0.475135\pi$
$384$	−6613.92	−0.878946
$385$	0	0
$386$	1895.45	0.249938
$387$	3915.78	0.514342
$388$	−1392.36	−0.182182
$389$	−1122.22	−0.146269	−0.0731347	−	0.997322i	$-0.523300\pi$
$389$	−1122.22	−0.146269	−0.0731347	+	0.997322i	$0.523300\pi$
$390$	0	0
$391$	19029.7	2.46131
$392$	0	0
$393$	13196.7	1.69385
$394$	−5412.68	−0.692099
$395$	0	0
$396$	−870.113	−0.110416
$397$	−1985.93	−0.251060	−0.125530	−	0.992090i	$-0.540063\pi$
$397$	−1985.93	−0.251060	−0.125530	+	0.992090i	$0.540063\pi$
$398$	7408.38	0.933037
$399$	0	0
$400$	0	0
$401$	−4172.38	−0.519597	−0.259799	−	0.965663i	$-0.583656\pi$
$401$	−4172.38	−0.519597	−0.259799	+	0.965663i	$0.583656\pi$
$402$	249.758	0.0309870
$403$	−3763.37	−0.465178
$404$	−317.699	−0.0391240
$405$	0	0
$406$	0	0
$407$	−14033.9	−1.70918
$408$	−13983.9	−1.69682
$409$	11700.8	1.41459	0.707295	−	0.706919i	$-0.249915\pi$
$409$	11700.8	1.41459	0.707295	+	0.706919i	$0.249915\pi$
$410$	0	0
$411$	−14718.0	−1.76638
$412$	2206.47	0.263848
$413$	0	0
$414$	9767.22	1.15950
$415$	0	0
$416$	−1137.65	−0.134082
$417$	−3515.08	−0.412791
$418$	−4374.72	−0.511901
$419$	−2733.20	−0.318677	−0.159339	−	0.987224i	$-0.550936\pi$
$419$	−2733.20	−0.318677	−0.159339	+	0.987224i	$0.550936\pi$
$420$	0	0
$421$	13549.4	1.56854	0.784272	−	0.620417i	$-0.213037\pi$
$421$	13549.4	1.56854	0.784272	+	0.620417i	$0.213037\pi$
$422$	−14593.3	−1.68339
$423$	202.158	0.0232370
$424$	5033.87	0.576571
$425$	0	0
$426$	16387.0	1.86374
$427$	0	0
$428$	1979.60	0.223569
$429$	4925.86	0.554366
$430$	0	0
$431$	−6429.25	−0.718530	−0.359265	−	0.933236i	$-0.616973\pi$
$431$	−6429.25	−0.718530	−0.359265	+	0.933236i	$0.616973\pi$
$432$	3337.79	0.371735
$433$	8022.03	0.890333	0.445166	−	0.895448i	$-0.353145\pi$
$433$	8022.03	0.890333	0.445166	+	0.895448i	$0.353145\pi$
$434$	0	0
$435$	0	0
$436$	1645.30	0.180724
$437$	−9648.50	−1.05618
$438$	−14192.1	−1.54823
$439$	5569.88	0.605549	0.302774	−	0.953062i	$-0.402087\pi$
$439$	5569.88	0.605549	0.302774	+	0.953062i	$0.402087\pi$
$440$	0	0
$441$	0	0
$442$	4362.77	0.469493
$443$	5486.21	0.588392	0.294196	−	0.955745i	$-0.404948\pi$
$443$	5486.21	0.588392	0.294196	+	0.955745i	$0.404948\pi$
$444$	3208.19	0.342914
$445$	0	0
$446$	16469.3	1.74853
$447$	−2185.92	−0.231299
$448$	0	0
$449$	−7232.67	−0.760203	−0.380101	−	0.924945i	$-0.624111\pi$
$449$	−7232.67	−0.760203	−0.380101	+	0.924945i	$0.624111\pi$
$450$	0	0
$451$	12982.3	1.35546
$452$	1800.05	0.187317
$453$	6852.76	0.710752
$454$	2626.32	0.271496
$455$	0	0
$456$	7090.16	0.728130
$457$	2900.51	0.296893	0.148446	−	0.988920i	$-0.452573\pi$
$457$	2900.51	0.296893	0.148446	+	0.988920i	$0.452573\pi$
$458$	10623.3	1.08383
$459$	5624.31	0.571940
$460$	0	0
$461$	−6073.57	−0.613611	−0.306805	−	0.951772i	$-0.599260\pi$
$461$	−6073.57	−0.613611	−0.306805	+	0.951772i	$0.599260\pi$
$462$	0	0
$463$	18922.8	1.89939	0.949693	−	0.313183i	$-0.101395\pi$
$463$	18922.8	1.89939	0.949693	+	0.313183i	$0.101395\pi$
$464$	2429.88	0.243113
$465$	0	0
$466$	1573.27	0.156396
$467$	−6776.71	−0.671496	−0.335748	−	0.941952i	$-0.608989\pi$
$467$	−6776.71	−0.671496	−0.335748	+	0.941952i	$0.608989\pi$
$468$	−439.963	−0.0434558
$469$	0	0
$470$	0	0
$471$	3495.50	0.341962
$472$	14835.3	1.44672
$473$	8651.96	0.841052
$474$	−2689.99	−0.260666
$475$	0	0
$476$	0	0
$477$	3618.90	0.347375
$478$	13069.7	1.25062
$479$	−2397.32	−0.228677	−0.114338	−	0.993442i	$-0.536475\pi$
$479$	−2397.32	−0.228677	−0.114338	+	0.993442i	$0.536475\pi$
$480$	0	0
$481$	−7096.09	−0.672669
$482$	12.5871	0.00118948
$483$	0	0
$484$	−173.977	−0.0163390
$485$	0	0
$486$	10933.4	1.02047
$487$	−5586.17	−0.519781	−0.259890	−	0.965638i	$-0.583686\pi$
$487$	−5586.17	−0.519781	−0.259890	+	0.965638i	$0.583686\pi$
$488$	−7724.35	−0.716526
$489$	−6674.33	−0.617227
$490$	0	0
$491$	537.392	0.0493934	0.0246967	−	0.999695i	$-0.492138\pi$
$491$	537.392	0.0493934	0.0246967	+	0.999695i	$0.492138\pi$
$492$	−2967.78	−0.271947
$493$	4094.46	0.374047
$494$	−2212.03	−0.201466
$495$	0	0
$496$	10071.2	0.911716
$497$	0	0
$498$	17842.0	1.60546
$499$	598.965	0.0537342	0.0268671	−	0.999639i	$-0.491447\pi$
$499$	598.965	0.0537342	0.0268671	+	0.999639i	$0.491447\pi$
$500$	0	0
$501$	−2392.62	−0.213362
$502$	−1416.81	−0.125966
$503$	4426.76	0.392405	0.196202	−	0.980563i	$-0.437139\pi$
$503$	4426.76	0.392405	0.196202	+	0.980563i	$0.437139\pi$
$504$	0	0
$505$	0	0
$506$	21580.8	1.89601
$507$	−12134.4	−1.06293
$508$	1594.17	0.139232
$509$	−17727.7	−1.54374	−0.771872	−	0.635779i	$-0.780679\pi$
$509$	−17727.7	−1.54374	−0.771872	+	0.635779i	$0.780679\pi$
$510$	0	0
$511$	0	0
$512$	13017.7	1.12365
$513$	−2851.66	−0.245427
$514$	4588.77	0.393778
$515$	0	0
$516$	−1977.86	−0.168741
$517$	446.671	0.0379972
$518$	0	0
$519$	21925.4	1.85437
$520$	0	0
$521$	−8662.79	−0.728453	−0.364226	−	0.931310i	$-0.618667\pi$
$521$	−8662.79	−0.728453	−0.364226	+	0.931310i	$0.618667\pi$
$522$	2101.53	0.176210
$523$	−7770.40	−0.649667	−0.324833	−	0.945771i	$-0.605308\pi$
$523$	−7770.40	−0.649667	−0.324833	+	0.945771i	$0.605308\pi$
$524$	−2604.32	−0.217119
$525$	0	0
$526$	−3100.60	−0.257020
$527$	16970.4	1.40274
$528$	−13182.2	−1.08652
$529$	35429.6	2.91194
$530$	0	0
$531$	10665.2	0.871624
$532$	0	0
$533$	6564.34	0.533458
$534$	−2930.12	−0.237451
$535$	0	0
$536$	−349.440	−0.0281595
$537$	19829.6	1.59350
$538$	8404.56	0.673506
$539$	0	0
$540$	0	0
$541$	21641.0	1.71981	0.859906	−	0.510453i	$-0.170522\pi$
$541$	21641.0	1.71981	0.859906	+	0.510453i	$0.170522\pi$
$542$	−2317.23	−0.183642
$543$	−9734.41	−0.769325
$544$	5130.09	0.404321
$545$	0	0
$546$	0	0
$547$	−7489.29	−0.585409	−0.292705	−	0.956203i	$-0.594555\pi$
$547$	−7489.29	−0.585409	−0.292705	+	0.956203i	$0.594555\pi$
$548$	2904.54	0.226416
$549$	−5553.11	−0.431696
$550$	0	0
$551$	−2075.99	−0.160508
$552$	−34976.2	−2.69689
$553$	0	0
$554$	−999.566	−0.0766561
$555$	0	0
$556$	693.689	0.0529118
$557$	−25297.9	−1.92443	−0.962214	−	0.272295i	$-0.912217\pi$
$557$	−25297.9	−1.92443	−0.962214	+	0.272295i	$0.912217\pi$
$558$	8710.29	0.660817
$559$	4374.77	0.331007
$560$	0	0
$561$	−22212.5	−1.67168
$562$	8623.85	0.647286
$563$	15661.3	1.17237	0.586186	−	0.810177i	$-0.300629\pi$
$563$	15661.3	1.17237	0.586186	+	0.810177i	$0.300629\pi$
$564$	−102.110	−0.00762343
$565$	0	0
$566$	−13994.9	−1.03931
$567$	0	0
$568$	−22927.3	−1.69367
$569$	−9982.75	−0.735498	−0.367749	−	0.929925i	$-0.619871\pi$
$569$	−9982.75	−0.735498	−0.367749	+	0.929925i	$0.619871\pi$
$570$	0	0
$571$	−11583.6	−0.848966	−0.424483	−	0.905436i	$-0.639544\pi$
$571$	−11583.6	−0.848966	−0.424483	+	0.905436i	$0.639544\pi$
$572$	−972.103	−0.0710589
$573$	−2495.81	−0.181961
$574$	0	0
$575$	0	0
$576$	9802.97	0.709127
$577$	−595.378	−0.0429565	−0.0214783	−	0.999769i	$-0.506837\pi$
$577$	−595.378	−0.0429565	−0.0214783	+	0.999769i	$0.506837\pi$
$578$	−6969.39	−0.501537
$579$	−4879.66	−0.350245
$580$	0	0
$581$	0	0
$582$	−18243.8	−1.29936
$583$	7996.00	0.568028
$584$	19856.4	1.40696
$585$	0	0
$586$	−729.738	−0.0514423
$587$	−15750.3	−1.10747	−0.553736	−	0.832693i	$-0.686798\pi$
$587$	−15750.3	−1.10747	−0.553736	+	0.832693i	$0.686798\pi$
$588$	0	0
$589$	−8604.42	−0.601934
$590$	0	0
$591$	13934.4	0.969856
$592$	18990.0	1.31838
$593$	−417.878	−0.0289379	−0.0144690	−	0.999895i	$-0.504606\pi$
$593$	−417.878	−0.0289379	−0.0144690	+	0.999895i	$0.504606\pi$
$594$	6378.31	0.440581
$595$	0	0
$596$	431.385	0.0296480
$597$	−19072.2	−1.30749
$598$	10912.1	0.746201
$599$	−19997.3	−1.36406	−0.682028	−	0.731326i	$-0.738902\pi$
$599$	−19997.3	−1.36406	−0.682028	+	0.731326i	$0.738902\pi$
$600$	0	0
$601$	15992.6	1.08545	0.542723	−	0.839912i	$-0.317393\pi$
$601$	15992.6	1.08545	0.542723	+	0.839912i	$0.317393\pi$
$602$	0	0
$603$	−251.216	−0.0169657
$604$	−1352.37	−0.0911045
$605$	0	0
$606$	−4162.73	−0.279042
$607$	14159.2	0.946793	0.473396	−	0.880850i	$-0.343028\pi$
$607$	14159.2	0.946793	0.473396	+	0.880850i	$0.343028\pi$
$608$	−2601.08	−0.173499
$609$	0	0
$610$	0	0
$611$	225.854	0.0149543
$612$	1983.96	0.131040
$613$	4629.41	0.305025	0.152512	−	0.988302i	$-0.451264\pi$
$613$	4629.41	0.305025	0.152512	+	0.988302i	$0.451264\pi$
$614$	−4963.87	−0.326263
$615$	0	0
$616$	0	0
$617$	23165.3	1.51151	0.755753	−	0.654857i	$-0.227271\pi$
$617$	23165.3	1.51151	0.755753	+	0.654857i	$0.227271\pi$
$618$	28910.9	1.88182
$619$	−12370.6	−0.803258	−0.401629	−	0.915803i	$-0.631556\pi$
$619$	−12370.6	−0.803258	−0.401629	+	0.915803i	$0.631556\pi$
$620$	0	0
$621$	14067.4	0.909028
$622$	3137.36	0.202245
$623$	0	0
$624$	−6665.43	−0.427613
$625$	0	0
$626$	3708.02	0.236745
$627$	11262.3	0.717341
$628$	−689.826	−0.0438329
$629$	31998.9	2.02842
$630$	0	0
$631$	13980.2	0.882002	0.441001	−	0.897507i	$-0.354623\pi$
$631$	13980.2	0.882002	0.441001	+	0.897507i	$0.354623\pi$
$632$	3763.61	0.236880
$633$	37568.9	2.35897
$634$	16799.7	1.05237
$635$	0	0
$636$	−1827.91	−0.113964
$637$	0	0
$638$	4643.36	0.288139
$639$	−16482.7	−1.02041
$640$	0	0
$641$	−16060.9	−0.989655	−0.494828	−	0.868991i	$-0.664769\pi$
$641$	−16060.9	−0.989655	−0.494828	+	0.868991i	$0.664769\pi$
$642$	25938.3	1.59455
$643$	−4502.17	−0.276125	−0.138063	−	0.990424i	$-0.544087\pi$
$643$	−4502.17	−0.276125	−0.138063	+	0.990424i	$0.544087\pi$
$644$	0	0
$645$	0	0
$646$	9974.87	0.607517
$647$	29414.8	1.78735	0.893675	−	0.448715i	$-0.148118\pi$
$647$	29414.8	1.78735	0.893675	+	0.448715i	$0.148118\pi$
$648$	−21595.6	−1.30919
$649$	23565.0	1.42528
$650$	0	0
$651$	0	0
$652$	1317.16	0.0791164
$653$	−13013.6	−0.779882	−0.389941	−	0.920840i	$-0.627505\pi$
$653$	−13013.6	−0.779882	−0.389941	+	0.920840i	$0.627505\pi$
$654$	21558.0	1.28897
$655$	0	0
$656$	−17566.9	−1.04554
$657$	14275.0	0.847671
$658$	0	0
$659$	23474.2	1.38759	0.693797	−	0.720171i	$-0.255937\pi$
$659$	23474.2	1.38759	0.693797	+	0.720171i	$0.255937\pi$
$660$	0	0
$661$	9266.36	0.545264	0.272632	−	0.962118i	$-0.412106\pi$
$661$	9266.36	0.545264	0.272632	+	0.962118i	$0.412106\pi$
$662$	25040.4	1.47013
$663$	−11231.5	−0.657914
$664$	−24963.0	−1.45896
$665$	0	0
$666$	16423.8	0.955572
$667$	10241.0	0.594501
$668$	472.176	0.0273489
$669$	−42398.6	−2.45026
$670$	0	0
$671$	−12269.7	−0.705909
$672$	0	0
$673$	25067.2	1.43576	0.717882	−	0.696164i	$-0.245112\pi$
$673$	25067.2	1.43576	0.717882	+	0.696164i	$0.245112\pi$
$674$	75.3288	0.00430498
$675$	0	0
$676$	2394.68	0.136247
$677$	−22409.6	−1.27219	−0.636093	−	0.771613i	$-0.719450\pi$
$677$	−22409.6	−1.27219	−0.636093	+	0.771613i	$0.719450\pi$
$678$	23585.6	1.33599
$679$	0	0
$680$	0	0
$681$	−6761.20	−0.380455
$682$	19245.5	1.08057
$683$	8757.53	0.490626	0.245313	−	0.969444i	$-0.421109\pi$
$683$	8757.53	0.490626	0.245313	+	0.969444i	$0.421109\pi$
$684$	−1005.91	−0.0562311
$685$	0	0
$686$	0	0
$687$	−27348.7	−1.51880
$688$	−11707.4	−0.648750
$689$	4043.09	0.223555
$690$	0	0
$691$	−8468.42	−0.466214	−0.233107	−	0.972451i	$-0.574889\pi$
$691$	−8468.42	−0.466214	−0.233107	+	0.972451i	$0.574889\pi$
$692$	−4326.90	−0.237694
$693$	0	0
$694$	−20294.8	−1.11006
$695$	0	0
$696$	−7525.54	−0.409849
$697$	−29601.0	−1.60864
$698$	−26555.1	−1.44001
$699$	−4050.23	−0.219162
$700$	0	0
$701$	15996.9	0.861906	0.430953	−	0.902374i	$-0.358177\pi$
$701$	15996.9	0.861906	0.430953	+	0.902374i	$0.358177\pi$
$702$	3225.12	0.173397
$703$	−16224.2	−0.870423
$704$	21659.8	1.15956
$705$	0	0
$706$	−7240.03	−0.385952
$707$	0	0
$708$	−5387.02	−0.285956
$709$	19903.0	1.05426	0.527131	−	0.849784i	$-0.323268\pi$
$709$	19903.0	1.05426	0.527131	+	0.849784i	$0.323268\pi$
$710$	0	0
$711$	2705.70	0.142717
$712$	4099.58	0.215784
$713$	42446.1	2.22948
$714$	0	0
$715$	0	0
$716$	−3913.30	−0.204256
$717$	−33646.7	−1.75252
$718$	8179.59	0.425153
$719$	11073.1	0.574347	0.287174	−	0.957879i	$-0.407284\pi$
$719$	11073.1	0.574347	0.287174	+	0.957879i	$0.407284\pi$
$720$	0	0
$721$	0	0
$722$	12678.4	0.653520
$723$	−32.4043	−0.00166685
$724$	1921.06	0.0986125
$725$	0	0
$726$	−2279.58	−0.116533
$727$	31652.7	1.61476	0.807382	−	0.590029i	$-0.200884\pi$
$727$	31652.7	1.61476	0.807382	+	0.590029i	$0.200884\pi$
$728$	0	0
$729$	−3935.94	−0.199967
$730$	0	0
$731$	−19727.5	−0.998149
$732$	2804.88	0.141628
$733$	16958.3	0.854528	0.427264	−	0.904127i	$-0.359477\pi$
$733$	16958.3	0.854528	0.427264	+	0.904127i	$0.359477\pi$
$734$	−8230.16	−0.413870
$735$	0	0
$736$	12831.3	0.642618
$737$	−555.065	−0.0277423
$738$	−15193.1	−0.757813
$739$	−11616.6	−0.578245	−0.289123	−	0.957292i	$-0.593364\pi$
$739$	−11616.6	−0.578245	−0.289123	+	0.957292i	$0.593364\pi$
$740$	0	0
$741$	5694.66	0.282319
$742$	0	0
$743$	−15928.0	−0.786464	−0.393232	−	0.919439i	$-0.628643\pi$
$743$	−15928.0	−0.786464	−0.393232	+	0.919439i	$0.628643\pi$
$744$	−31191.4	−1.53700
$745$	0	0
$746$	−6762.19	−0.331879
$747$	−17946.1	−0.879003
$748$	4383.57	0.214277
$749$	0	0
$750$	0	0
$751$	25571.9	1.24252	0.621260	−	0.783604i	$-0.286621\pi$
$751$	25571.9	1.24252	0.621260	+	0.783604i	$0.286621\pi$
$752$	−604.412	−0.0293094
$753$	3647.43	0.176520
$754$	2347.87	0.113401
$755$	0	0
$756$	0	0
$757$	−6202.41	−0.297794	−0.148897	−	0.988853i	$-0.547572\pi$
$757$	−6202.41	−0.297794	−0.148897	+	0.988853i	$0.547572\pi$
$758$	1738.77	0.0833179
$759$	−55557.6	−2.65693
$760$	0	0
$761$	29199.1	1.39089	0.695444	−	0.718580i	$-0.255208\pi$
$761$	29199.1	1.39089	0.695444	+	0.718580i	$0.255208\pi$
$762$	20888.1	0.993037
$763$	0	0
$764$	492.539	0.0233239
$765$	0	0
$766$	−3025.00	−0.142686
$767$	11915.4	0.560938
$768$	−13050.5	−0.613177
$769$	−21838.2	−1.02407	−0.512033	−	0.858966i	$-0.671108\pi$
$769$	−21838.2	−1.02407	−0.512033	+	0.858966i	$0.671108\pi$
$770$	0	0
$771$	−11813.3	−0.551812
$772$	962.985	0.0448945
$773$	−25544.8	−1.18859	−0.594296	−	0.804246i	$-0.702569\pi$
$773$	−25544.8	−1.18859	−0.594296	+	0.804246i	$0.702569\pi$
$774$	−10125.4	−0.470218
$775$	0	0
$776$	25525.2	1.18080
$777$	0	0
$778$	2901.82	0.133721
$779$	15008.4	0.690286
$780$	0	0
$781$	−36418.6	−1.66858
$782$	−49206.6	−2.25016
$783$	3026.77	0.138146
$784$	0	0
$785$	0	0
$786$	−34123.8	−1.54854
$787$	−37223.2	−1.68598	−0.842989	−	0.537931i	$-0.819206\pi$
$787$	−37223.2	−1.68598	−0.842989	+	0.537931i	$0.819206\pi$
$788$	−2749.91	−0.124317
$789$	7982.19	0.360169
$790$	0	0
$791$	0	0
$792$	15951.1	0.715655
$793$	−6204.03	−0.277820
$794$	5135.18	0.229522
$795$	0	0
$796$	3763.83	0.167595
$797$	−40384.6	−1.79485	−0.897425	−	0.441168i	$-0.854564\pi$
$797$	−40384.6	−1.79485	−0.897425	+	0.441168i	$0.854564\pi$
$798$	0	0
$799$	−1018.46	−0.0450945
$800$	0	0
$801$	2947.23	0.130007
$802$	10788.9	0.475023
$803$	31540.7	1.38611
$804$	126.889	0.00556598
$805$	0	0
$806$	9731.28	0.425272
$807$	−21636.7	−0.943803
$808$	5824.14	0.253580
$809$	−1955.76	−0.0849948	−0.0424974	−	0.999097i	$-0.513531\pi$
$809$	−1955.76	−0.0849948	−0.0424974	+	0.999097i	$0.513531\pi$
$810$	0	0
$811$	34301.8	1.48520	0.742600	−	0.669735i	$-0.233592\pi$
$811$	34301.8	1.48520	0.742600	+	0.669735i	$0.233592\pi$
$812$	0	0
$813$	5965.49	0.257342
$814$	36288.7	1.56255
$815$	0	0
$816$	30056.9	1.28946
$817$	10002.3	0.428319
$818$	−30255.8	−1.29324
$819$	0	0
$820$	0	0
$821$	−13665.6	−0.580918	−0.290459	−	0.956887i	$-0.593808\pi$
$821$	−13665.6	−0.580918	−0.290459	+	0.956887i	$0.593808\pi$
$822$	38057.5	1.61485
$823$	21519.5	0.911449	0.455724	−	0.890121i	$-0.349380\pi$
$823$	21519.5	0.911449	0.455724	+	0.890121i	$0.349380\pi$
$824$	−40449.7	−1.71011
$825$	0	0
$826$	0	0
$827$	−35220.6	−1.48094	−0.740471	−	0.672088i	$-0.765398\pi$
$827$	−35220.6	−1.48094	−0.740471	+	0.672088i	$0.765398\pi$
$828$	4962.23	0.208272
$829$	−31365.5	−1.31408	−0.657039	−	0.753857i	$-0.728191\pi$
$829$	−31365.5	−1.31408	−0.657039	+	0.753857i	$0.728191\pi$
$830$	0	0
$831$	2573.28	0.107420
$832$	10952.0	0.456362
$833$	0	0
$834$	9089.24	0.377380
$835$	0	0
$836$	−2222.58	−0.0919491
$837$	12545.2	0.518070
$838$	7067.48	0.291339
$839$	28287.1	1.16398	0.581990	−	0.813196i	$-0.302274\pi$
$839$	28287.1	1.16398	0.581990	+	0.813196i	$0.302274\pi$
$840$	0	0
$841$	−22185.5	−0.909653
$842$	−35035.8	−1.43398
$843$	−22201.2	−0.907060
$844$	−7414.11	−0.302374
$845$	0	0
$846$	−522.738	−0.0212436
$847$	0	0
$848$	−10819.8	−0.438152
$849$	36028.6	1.45642
$850$	0	0
$851$	80035.0	3.22393
$852$	8325.40	0.334769
$853$	9405.41	0.377533	0.188766	−	0.982022i	$-0.439551\pi$
$853$	9405.41	0.377533	0.188766	+	0.982022i	$0.439551\pi$
$854$	0	0
$855$	0	0
$856$	−36290.6	−1.44905
$857$	−27966.9	−1.11474	−0.557369	−	0.830265i	$-0.688189\pi$
$857$	−27966.9	−1.11474	−0.557369	+	0.830265i	$0.688189\pi$
$858$	−12737.2	−0.506809
$859$	−6281.11	−0.249486	−0.124743	−	0.992189i	$-0.539811\pi$
$859$	−6281.11	−0.249486	−0.124743	+	0.992189i	$0.539811\pi$
$860$	0	0
$861$	0	0
$862$	16624.7	0.656890
$863$	4757.13	0.187642	0.0938208	−	0.995589i	$-0.470092\pi$
$863$	4757.13	0.187642	0.0938208	+	0.995589i	$0.470092\pi$
$864$	3792.35	0.149327
$865$	0	0
$866$	−20743.3	−0.813954
$867$	17942.0	0.702818
$868$	0	0
$869$	5978.28	0.233371
$870$	0	0
$871$	−280.663	−0.0109184
$872$	−30162.1	−1.17135
$873$	18350.3	0.711414
$874$	24949.0	0.965574
$875$	0	0
$876$	−7210.30	−0.278097
$877$	30240.5	1.16437	0.582184	−	0.813057i	$-0.302198\pi$
$877$	30240.5	1.16437	0.582184	+	0.813057i	$0.302198\pi$
$878$	−14402.5	−0.553601
$879$	1878.64	0.0720875
$880$	0	0
$881$	−44875.5	−1.71611	−0.858056	−	0.513556i	$-0.828328\pi$
$881$	−44875.5	−1.71611	−0.858056	+	0.513556i	$0.828328\pi$
$882$	0	0
$883$	−4892.13	−0.186448	−0.0932238	−	0.995645i	$-0.529717\pi$
$883$	−4892.13	−0.186448	−0.0932238	+	0.995645i	$0.529717\pi$
$884$	2216.51	0.0843317
$885$	0	0
$886$	−14186.2	−0.537916
$887$	1761.40	0.0666765	0.0333382	−	0.999444i	$-0.489386\pi$
$887$	1761.40	0.0666765	0.0333382	+	0.999444i	$0.489386\pi$
$888$	−58813.4	−2.22258
$889$	0	0
$890$	0	0
$891$	−34303.3	−1.28979
$892$	8367.22	0.314075
$893$	516.384	0.0193507
$894$	5652.33	0.211457
$895$	0	0
$896$	0	0
$897$	−28092.1	−1.04567
$898$	18702.2	0.694988
$899$	9132.79	0.338816
$900$	0	0
$901$	−18231.8	−0.674128
$902$	−33569.3	−1.23918
$903$	0	0
$904$	−32999.0	−1.21408
$905$	0	0
$906$	−17719.8	−0.649779
$907$	−23689.1	−0.867238	−0.433619	−	0.901096i	$-0.642764\pi$
$907$	−23689.1	−0.867238	−0.433619	+	0.901096i	$0.642764\pi$
$908$	1334.30	0.0487669
$909$	4187.03	0.152778
$910$	0	0
$911$	−13877.3	−0.504692	−0.252346	−	0.967637i	$-0.581202\pi$
$911$	−13877.3	−0.504692	−0.252346	+	0.967637i	$0.581202\pi$
$912$	−15239.6	−0.553325
$913$	−39652.2	−1.43735
$914$	−7500.09	−0.271423
$915$	0	0
$916$	5397.18	0.194681
$917$	0	0
$918$	−14543.3	−0.522875
$919$	14331.6	0.514426	0.257213	−	0.966355i	$-0.417196\pi$
$919$	14331.6	0.514426	0.257213	+	0.966355i	$0.417196\pi$
$920$	0	0
$921$	12779.0	0.457201
$922$	15705.0	0.560971
$923$	−18414.7	−0.656692
$924$	0	0
$925$	0	0
$926$	−48930.2	−1.73644
$927$	−29079.7	−1.03032
$928$	2760.80	0.0976592
$929$	−16668.4	−0.588668	−0.294334	−	0.955703i	$-0.595098\pi$
$929$	−16668.4	−0.588668	−0.294334	+	0.955703i	$0.595098\pi$
$930$	0	0
$931$	0	0
$932$	799.300	0.0280922
$933$	−8076.82	−0.283412
$934$	17523.1	0.613891
$935$	0	0
$936$	8065.52	0.281656
$937$	30384.9	1.05937	0.529685	−	0.848194i	$-0.322310\pi$
$937$	30384.9	1.05937	0.529685	+	0.848194i	$0.322310\pi$
$938$	0	0
$939$	−9545.94	−0.331757
$940$	0	0
$941$	1196.35	0.0414452	0.0207226	−	0.999785i	$-0.493403\pi$
$941$	1196.35	0.0414452	0.0207226	+	0.999785i	$0.493403\pi$
$942$	−9038.63	−0.312627
$943$	−74037.5	−2.55673
$944$	−31886.9	−1.09940
$945$	0	0
$946$	−22372.1	−0.768901
$947$	−1788.41	−0.0613681	−0.0306840	−	0.999529i	$-0.509769\pi$
$947$	−1788.41	−0.0613681	−0.0306840	+	0.999529i	$0.509769\pi$
$948$	−1366.65	−0.0468215
$949$	15948.2	0.545523
$950$	0	0
$951$	−43249.2	−1.47471
$952$	0	0
$953$	−8578.60	−0.291593	−0.145796	−	0.989315i	$-0.546574\pi$
$953$	−8578.60	−0.291593	−0.145796	+	0.989315i	$0.546574\pi$
$954$	−9357.70	−0.317575
$955$	0	0
$956$	6640.06	0.224639
$957$	−11953.9	−0.403776
$958$	6198.95	0.209059
$959$	0	0
$960$	0	0
$961$	8061.99	0.270618
$962$	18349.0	0.614963
$963$	−26089.7	−0.873031
$964$	6.39489	0.000213657 0
$965$	0	0
$966$	0	0
$967$	55459.3	1.84431	0.922156	−	0.386818i	$-0.126426\pi$
$967$	55459.3	1.84431	0.922156	+	0.386818i	$0.126426\pi$
$968$	3189.40	0.105900
$969$	−25679.3	−0.851330
$970$	0	0
$971$	−22047.3	−0.728662	−0.364331	−	0.931270i	$-0.618702\pi$
$971$	−22047.3	−0.728662	−0.364331	+	0.931270i	$0.618702\pi$
$972$	5554.72	0.183300
$973$	0	0
$974$	14444.6	0.475191
$975$	0	0
$976$	16602.7	0.544507
$977$	−14402.3	−0.471617	−0.235809	−	0.971800i	$-0.575774\pi$
$977$	−14402.3	−0.471617	−0.235809	+	0.971800i	$0.575774\pi$
$978$	17258.4	0.564277
$979$	6511.94	0.212587
$980$	0	0
$981$	−21683.9	−0.705721
$982$	−1389.58	−0.0451561
$983$	7817.11	0.253639	0.126819	−	0.991926i	$-0.459523\pi$
$983$	7817.11	0.253639	0.126819	+	0.991926i	$0.459523\pi$
$984$	54406.2	1.76261
$985$	0	0
$986$	−10587.4	−0.341959
$987$	0	0
$988$	−1123.82	−0.0361878
$989$	−49342.0	−1.58643
$990$	0	0
$991$	24501.6	0.785386	0.392693	−	0.919670i	$-0.371543\pi$
$991$	24501.6	0.785386	0.392693	+	0.919670i	$0.371543\pi$
$992$	11442.8	0.366239
$993$	−64464.2	−2.06013
$994$	0	0
$995$	0	0
$996$	9064.61	0.288377
$997$	−50696.0	−1.61039	−0.805195	−	0.593010i	$-0.797940\pi$
$997$	−50696.0	−1.61039	−0.805195	+	0.593010i	$0.797940\pi$
$998$	−1548.80	−0.0491245
$999$	23654.8	0.749152

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.m.1.2		2
5.4	even	2		245.4.a.k.1.1		2
7.6	odd	2		175.4.a.c.1.2		2
15.14	odd	2		2205.4.a.u.1.2		2
21.20	even	2		1575.4.a.z.1.1		2
35.4	even	6		245.4.e.i.226.2		4
35.9	even	6		245.4.e.i.116.2		4
35.13	even	4		175.4.b.c.99.3		4
35.19	odd	6		245.4.e.h.116.2		4
35.24	odd	6		245.4.e.h.226.2		4
35.27	even	4		175.4.b.c.99.2		4
35.34	odd	2		35.4.a.b.1.1	✓	2
105.104	even	2		315.4.a.f.1.2		2
140.139	even	2		560.4.a.r.1.1		2
280.69	odd	2		2240.4.a.bn.1.1		2
280.139	even	2		2240.4.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.b.1.1	✓	2	35.34	odd	2
175.4.a.c.1.2		2	7.6	odd	2
175.4.b.c.99.2		4	35.27	even	4
175.4.b.c.99.3		4	35.13	even	4
245.4.a.k.1.1		2	5.4	even	2
245.4.e.h.116.2		4	35.19	odd	6
245.4.e.h.226.2		4	35.24	odd	6
245.4.e.i.116.2		4	35.9	even	6
245.4.e.i.226.2		4	35.4	even	6
315.4.a.f.1.2		2	105.104	even	2
560.4.a.r.1.1		2	140.139	even	2
1225.4.a.m.1.2		2	1.1	even	1	trivial
1575.4.a.z.1.1		2	21.20	even	2
2205.4.a.u.1.2		2	15.14	odd	2
2240.4.a.bn.1.1		2	280.69	odd	2
2240.4.a.bo.1.2		2	280.139	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.58579 q^{2} +6.65685 q^{3} -1.31371 q^{4} -17.2132 q^{6} +24.0833 q^{8} +17.3137 q^{9} +38.2548 q^{11} -8.74517 q^{12} +19.3431 q^{13} -51.7645 q^{16} -87.2254 q^{17} -44.7696 q^{18} +44.2254 q^{19} -98.9188 q^{22} -218.167 q^{23} +160.319 q^{24} -50.0172 q^{26} -64.4802 q^{27} -46.9411 q^{29} -194.558 q^{31} -58.8141 q^{32} +254.657 q^{33} +225.546 q^{34} -22.7452 q^{36} -366.853 q^{37} -114.357 q^{38} +128.765 q^{39} +339.362 q^{41} +226.167 q^{43} -50.2557 q^{44} +564.132 q^{46} +11.6762 q^{47} -344.589 q^{48} -580.647 q^{51} -25.4113 q^{52} +209.019 q^{53} +166.732 q^{54} +294.402 q^{57} +121.380 q^{58} +616.000 q^{59} -320.735 q^{61} +503.087 q^{62} +566.197 q^{64} -658.488 q^{66} -14.5097 q^{67} +114.589 q^{68} -1452.30 q^{69} -952.000 q^{71} +416.971 q^{72} +824.489 q^{73} +948.603 q^{74} -58.0993 q^{76} -332.958 q^{78} +156.275 q^{79} -896.706 q^{81} -877.519 q^{82} -1036.53 q^{83} -584.818 q^{86} -312.480 q^{87} +921.301 q^{88} +170.225 q^{89} +286.607 q^{92} -1295.15 q^{93} -30.1921 q^{94} -391.517 q^{96} +1059.87 q^{97} +662.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9} - 14 q^{11} - 108 q^{12} + 50 q^{13} + 168 q^{16} - 50 q^{17} - 16 q^{18} - 36 q^{19} + 184 q^{22} - 244 q^{23} + 496 q^{24} - 216 q^{26}+ \cdots + 940 q^{99}+O(q^{100}) \) 2 * q - 8 * q^2 + 2 * q^3 + 20 * q^4 + 8 * q^6 - 48 * q^8 + 12 * q^9 - 14 * q^11 - 108 * q^12 + 50 * q^13 + 168 * q^16 - 50 * q^17 - 16 * q^18 - 36 * q^19 + 184 * q^22 - 244 * q^23 + 496 * q^24 - 216 * q^26 + 86 * q^27 - 26 * q^29 + 120 * q^31 - 672 * q^32 + 498 * q^33 + 24 * q^34 - 136 * q^36 - 564 * q^37 + 320 * q^38 - 14 * q^39 + 328 * q^41 + 260 * q^43 - 1164 * q^44 + 704 * q^46 - 350 * q^47 - 1368 * q^48 - 754 * q^51 + 628 * q^52 + 56 * q^53 - 648 * q^54 + 668 * q^57 + 8 * q^58 + 1232 * q^59 - 336 * q^61 - 1200 * q^62 + 2128 * q^64 - 1976 * q^66 + 152 * q^67 + 908 * q^68 - 1332 * q^69 - 1904 * q^71 + 800 * q^72 + 676 * q^73 + 2016 * q^74 - 1768 * q^76 + 440 * q^78 + 1014 * q^79 - 1454 * q^81 - 816 * q^82 - 376 * q^83 - 768 * q^86 - 410 * q^87 + 4688 * q^88 + 216 * q^89 - 264 * q^92 - 2760 * q^93 + 1928 * q^94 + 2464 * q^96 + 2742 * q^97 + 940 * q^99

Introduction

L-functions

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