LMFDB - Embedded newform 1225.4.a.bd.1.3

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:	$0$
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			1.3
Root			$-3.53510$ 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)

$f(q)$	$=$	\(q+4.53510 q^{2} -6.46622 q^{3} +12.5671 q^{4} -29.3249 q^{6} +20.7124 q^{8} +14.8119 q^{9} -54.0684 q^{11} -81.2617 q^{12} -75.2159 q^{13} -6.60441 q^{16} +71.2538 q^{17} +67.1736 q^{18} +65.5100 q^{19} -245.206 q^{22} +125.688 q^{23} -133.931 q^{24} -341.111 q^{26} +78.8106 q^{27} +190.405 q^{29} +193.105 q^{31} -195.650 q^{32} +349.618 q^{33} +323.143 q^{34} +186.144 q^{36} +114.673 q^{37} +297.094 q^{38} +486.362 q^{39} -216.896 q^{41} -413.032 q^{43} -679.484 q^{44} +570.007 q^{46} +113.555 q^{47} +42.7055 q^{48} -460.743 q^{51} -945.247 q^{52} +584.366 q^{53} +357.414 q^{54} -423.602 q^{57} +863.504 q^{58} -203.748 q^{59} +162.539 q^{61} +875.749 q^{62} -834.459 q^{64} +1585.55 q^{66} +477.534 q^{67} +895.456 q^{68} -812.725 q^{69} +822.294 q^{71} +306.790 q^{72} +798.993 q^{73} +520.052 q^{74} +823.273 q^{76} +2205.70 q^{78} -468.087 q^{79} -909.529 q^{81} -983.643 q^{82} +310.333 q^{83} -1873.14 q^{86} -1231.20 q^{87} -1119.88 q^{88} -1314.90 q^{89} +1579.53 q^{92} -1248.66 q^{93} +514.985 q^{94} +1265.12 q^{96} +1314.66 q^{97} -800.858 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.53510	1.60340	0.801700	−	0.597727i	$-0.203929\pi$
$2$	4.53510	1.60340	0.801700	+	0.597727i	$0.203929\pi$
$3$	−6.46622	−1.24442	−0.622212	−	0.782849i	$-0.713766\pi$
$3$	−6.46622	−1.24442	−0.622212	+	0.782849i	$0.713766\pi$
$4$	12.5671	1.57089
$5$	0	0
$5$	0	0
$6$	−29.3249	−1.99531
$7$	0	0
$7$	0	0
$8$	20.7124	0.915365
$9$	14.8119	0.548591
$10$	0	0
$11$	−54.0684	−1.48202	−0.741011	−	0.671493i	$-0.765653\pi$
$11$	−54.0684	−1.48202	−0.741011	+	0.671493i	$0.765653\pi$
$12$	−81.2617	−1.95485
$13$	−75.2159	−1.60470	−0.802351	−	0.596853i	$-0.796418\pi$
$13$	−75.2159	−1.60470	−0.802351	+	0.596853i	$0.796418\pi$
$14$	0	0
$15$	0	0
$16$	−6.60441	−0.103194
$17$	71.2538	1.01656	0.508282	−	0.861191i	$-0.330281\pi$
$17$	71.2538	1.01656	0.508282	+	0.861191i	$0.330281\pi$
$18$	67.1736	0.879610
$19$	65.5100	0.791002	0.395501	−	0.918466i	$-0.370571\pi$
$19$	65.5100	0.791002	0.395501	+	0.918466i	$0.370571\pi$
$20$	0	0
$21$	0	0
$22$	−245.206	−2.37627
$23$	125.688	1.13947	0.569733	−	0.821830i	$-0.307047\pi$
$23$	125.688	1.13947	0.569733	+	0.821830i	$0.307047\pi$
$24$	−133.931	−1.13910
$25$	0	0
$26$	−341.111	−2.57298
$27$	78.8106	0.561745
$28$	0	0
$29$	190.405	1.21922	0.609608	−	0.792703i	$-0.291327\pi$
$29$	190.405	1.21922	0.609608	+	0.792703i	$0.291327\pi$
$30$	0	0
$31$	193.105	1.11879	0.559397	−	0.828900i	$-0.311033\pi$
$31$	193.105	1.11879	0.559397	+	0.828900i	$0.311033\pi$
$32$	−195.650	−1.08083
$33$	349.618	1.84426
$34$	323.143	1.62996
$35$	0	0
$36$	186.144	0.861776
$37$	114.673	0.509516	0.254758	−	0.967005i	$-0.418004\pi$
$37$	114.673	0.509516	0.254758	+	0.967005i	$0.418004\pi$
$38$	297.094	1.26829
$39$	486.362	1.99693
$40$	0	0
$41$	−216.896	−0.826180	−0.413090	−	0.910690i	$-0.635551\pi$
$41$	−216.896	−0.826180	−0.413090	+	0.910690i	$0.635551\pi$
$42$	0	0
$43$	−413.032	−1.46481	−0.732405	−	0.680870i	$-0.761602\pi$
$43$	−413.032	−1.46481	−0.732405	+	0.680870i	$0.761602\pi$
$44$	−679.484	−2.32809
$45$	0	0
$46$	570.007	1.82702
$47$	113.555	0.352421	0.176210	−	0.984353i	$-0.443616\pi$
$47$	113.555	0.352421	0.176210	+	0.984353i	$0.443616\pi$
$48$	42.7055	0.128417
$49$	0	0
$50$	0	0
$51$	−460.743	−1.26504
$52$	−945.247	−2.52081
$53$	584.366	1.51451	0.757253	−	0.653122i	$-0.226541\pi$
$53$	584.366	1.51451	0.757253	+	0.653122i	$0.226541\pi$
$54$	357.414	0.900701
$55$	0	0
$56$	0	0
$57$	−423.602	−0.984341
$58$	863.504	1.95489
$59$	−203.748	−0.449589	−0.224795	−	0.974406i	$-0.572171\pi$
$59$	−203.748	−0.449589	−0.224795	+	0.974406i	$0.572171\pi$
$60$	0	0
$61$	162.539	0.341163	0.170581	−	0.985344i	$-0.445435\pi$
$61$	162.539	0.341163	0.170581	+	0.985344i	$0.445435\pi$
$62$	875.749	1.79387
$63$	0	0
$64$	−834.459	−1.62980
$65$	0	0
$66$	1585.55	2.95709
$67$	477.534	0.870747	0.435374	−	0.900250i	$-0.356616\pi$
$67$	477.534	0.870747	0.435374	+	0.900250i	$0.356616\pi$
$68$	895.456	1.59691
$69$	−812.725	−1.41798
$70$	0	0
$71$	822.294	1.37448	0.687242	−	0.726429i	$-0.258821\pi$
$71$	822.294	1.37448	0.687242	+	0.726429i	$0.258821\pi$
$72$	306.790	0.502161
$73$	798.993	1.28103	0.640514	−	0.767947i	$-0.278721\pi$
$73$	798.993	1.28103	0.640514	+	0.767947i	$0.278721\pi$
$74$	520.052	0.816957
$75$	0	0
$76$	823.273	1.24258
$77$	0	0
$78$	2205.70	3.20188
$79$	−468.087	−0.666632	−0.333316	−	0.942815i	$-0.608168\pi$
$79$	−468.087	−0.666632	−0.333316	+	0.942815i	$0.608168\pi$
$80$	0	0
$81$	−909.529	−1.24764
$82$	−983.643	−1.32470
$83$	310.333	0.410403	0.205202	−	0.978720i	$-0.434215\pi$
$83$	310.333	0.410403	0.205202	+	0.978720i	$0.434215\pi$
$84$	0	0
$85$	0	0
$86$	−1873.14	−2.34867
$87$	−1231.20	−1.51722
$88$	−1119.88	−1.35659
$89$	−1314.90	−1.56606	−0.783029	−	0.621985i	$-0.786326\pi$
$89$	−1314.90	−1.56606	−0.783029	+	0.621985i	$0.786326\pi$
$90$	0	0
$91$	0	0
$92$	1579.53	1.78998
$93$	−1248.66	−1.39225
$94$	514.985	0.565071
$95$	0	0
$96$	1265.12	1.34501
$97$	1314.66	1.37612	0.688058	−	0.725656i	$-0.258463\pi$
$97$	1314.66	1.37612	0.688058	+	0.725656i	$0.258463\pi$
$98$	0	0
$99$	−800.858	−0.813023
$100$	0	0
$101$	401.240	0.395296	0.197648	−	0.980273i	$-0.436670\pi$
$101$	401.240	0.395296	0.197648	+	0.980273i	$0.436670\pi$
$102$	−2089.51	−2.02836
$103$	−291.844	−0.279187	−0.139594	−	0.990209i	$-0.544580\pi$
$103$	−291.844	−0.279187	−0.139594	+	0.990209i	$0.544580\pi$
$104$	−1557.90	−1.46889
$105$	0	0
$106$	2650.16	2.42836
$107$	−21.9594	−0.0198402	−0.00992009	−	0.999951i	$-0.503158\pi$
$107$	−21.9594	−0.0198402	−0.00992009	+	0.999951i	$0.503158\pi$
$108$	990.422	0.882439
$109$	−17.3019	−0.0152039	−0.00760194	−	0.999971i	$-0.502420\pi$
$109$	−17.3019	−0.0152039	−0.00760194	+	0.999971i	$0.502420\pi$
$110$	0	0
$111$	−741.498	−0.634053
$112$	0	0
$113$	58.8182	0.0489660	0.0244830	−	0.999700i	$-0.492206\pi$
$113$	58.8182	0.0489660	0.0244830	+	0.999700i	$0.492206\pi$
$114$	−1921.08	−1.57829
$115$	0	0
$116$	2392.84	1.91526
$117$	−1114.09	−0.880324
$118$	−924.018	−0.720871
$119$	0	0
$120$	0	0
$121$	1592.39	1.19639
$122$	737.129	0.547020
$123$	1402.49	1.02812
$124$	2426.77	1.75750
$125$	0	0
$126$	0	0
$127$	1862.03	1.30101	0.650507	−	0.759501i	$-0.274557\pi$
$127$	1862.03	1.30101	0.650507	+	0.759501i	$0.274557\pi$
$128$	−2219.15	−1.53240
$129$	2670.75	1.82284
$130$	0	0
$131$	−775.161	−0.516994	−0.258497	−	0.966012i	$-0.583227\pi$
$131$	−775.161	−0.516994	−0.258497	+	0.966012i	$0.583227\pi$
$132$	4393.69	2.89714
$133$	0	0
$134$	2165.66	1.39616
$135$	0	0
$136$	1475.83	0.930528
$137$	−1027.95	−0.641051	−0.320525	−	0.947240i	$-0.603859\pi$
$137$	−1027.95	−0.641051	−0.320525	+	0.947240i	$0.603859\pi$
$138$	−3685.79	−2.27359
$139$	1029.66	0.628309	0.314154	−	0.949372i	$-0.398279\pi$
$139$	1029.66	0.628309	0.314154	+	0.949372i	$0.398279\pi$
$140$	0	0
$141$	−734.274	−0.438561
$142$	3729.18	2.20385
$143$	4066.80	2.37820
$144$	−97.8241	−0.0566112
$145$	0	0
$146$	3623.51	2.05400
$147$	0	0
$148$	1441.11	0.800393
$149$	1414.67	0.777814	0.388907	−	0.921277i	$-0.372853\pi$
$149$	1414.67	0.777814	0.388907	+	0.921277i	$0.372853\pi$
$150$	0	0
$151$	2094.96	1.12904	0.564522	−	0.825418i	$-0.309060\pi$
$151$	2094.96	1.12904	0.564522	+	0.825418i	$0.309060\pi$
$152$	1356.87	0.724056
$153$	1055.41	0.557678
$154$	0	0
$155$	0	0
$156$	6112.17	3.13696
$157$	709.495	0.360662	0.180331	−	0.983606i	$-0.442283\pi$
$157$	709.495	0.360662	0.180331	+	0.983606i	$0.442283\pi$
$158$	−2122.82	−1.06888
$159$	−3778.64	−1.88469
$160$	0	0
$161$	0	0
$162$	−4124.80	−2.00046
$163$	3276.23	1.57432	0.787160	−	0.616749i	$-0.211551\pi$
$163$	3276.23	1.57432	0.787160	+	0.616749i	$0.211551\pi$
$164$	−2725.75	−1.29784
$165$	0	0
$166$	1407.39	0.658040
$167$	−2860.69	−1.32555	−0.662775	−	0.748819i	$-0.730621\pi$
$167$	−2860.69	−1.32555	−0.662775	+	0.748819i	$0.730621\pi$
$168$	0	0
$169$	3460.42	1.57507
$170$	0	0
$171$	970.331	0.433936
$172$	−5190.62	−2.30105
$173$	−2803.44	−1.23203	−0.616015	−	0.787734i	$-0.711254\pi$
$173$	−2803.44	−1.23203	−0.616015	+	0.787734i	$0.711254\pi$
$174$	−5583.61	−2.43271
$175$	0	0
$176$	357.090	0.152936
$177$	1317.48	0.559479
$178$	−5963.20	−2.51102
$179$	248.617	0.103813	0.0519064	−	0.998652i	$-0.483470\pi$
$179$	248.617	0.103813	0.0519064	+	0.998652i	$0.483470\pi$
$180$	0	0
$181$	−2075.56	−0.852347	−0.426174	−	0.904641i	$-0.640139\pi$
$181$	−2075.56	−0.852347	−0.426174	+	0.904641i	$0.640139\pi$
$182$	0	0
$183$	−1051.01	−0.424551
$184$	2603.29	1.04303
$185$	0	0
$186$	−5662.78	−2.23234
$187$	−3852.58	−1.50657
$188$	1427.07	0.553614
$189$	0	0
$190$	0	0
$191$	−1252.95	−0.474663	−0.237331	−	0.971429i	$-0.576273\pi$
$191$	−1252.95	−0.474663	−0.237331	+	0.971429i	$0.576273\pi$
$192$	5395.79	2.02817
$193$	1602.12	0.597527	0.298764	−	0.954327i	$-0.403426\pi$
$193$	1602.12	0.597527	0.298764	+	0.954327i	$0.403426\pi$
$194$	5962.10	2.20646
$195$	0	0
$196$	0	0
$197$	2346.38	0.848593	0.424296	−	0.905523i	$-0.360521\pi$
$197$	2346.38	0.848593	0.424296	+	0.905523i	$0.360521\pi$
$198$	−3631.97	−1.30360
$199$	1993.53	0.710140	0.355070	−	0.934840i	$-0.384457\pi$
$199$	1993.53	0.710140	0.355070	+	0.934840i	$0.384457\pi$
$200$	0	0
$201$	−3087.84	−1.08358
$202$	1819.67	0.633818
$203$	0	0
$204$	−5790.21	−1.98723
$205$	0	0
$206$	−1323.54	−0.447649
$207$	1861.68	0.625101
$208$	496.756	0.165595
$209$	−3542.02	−1.17228
$210$	0	0
$211$	−5852.55	−1.90951	−0.954754	−	0.297396i	$-0.903882\pi$
$211$	−5852.55	−1.90951	−0.954754	+	0.297396i	$0.903882\pi$
$212$	7343.80	2.37912
$213$	−5317.13	−1.71044
$214$	−99.5883	−0.0318117
$215$	0	0
$216$	1632.35	0.514202
$217$	0	0
$218$	−78.4659	−0.0243779
$219$	−5166.46	−1.59414
$220$	0	0
$221$	−5359.42	−1.63128
$222$	−3362.77	−1.01664
$223$	1719.58	0.516374	0.258187	−	0.966095i	$-0.416875\pi$
$223$	1719.58	0.516374	0.258187	+	0.966095i	$0.416875\pi$
$224$	0	0
$225$	0	0
$226$	266.747	0.0785120
$227$	5686.00	1.66252	0.831262	−	0.555881i	$-0.187619\pi$
$227$	5686.00	1.66252	0.831262	+	0.555881i	$0.187619\pi$
$228$	−5323.46	−1.54629
$229$	3087.40	0.890923	0.445461	−	0.895301i	$-0.353040\pi$
$229$	3087.40	0.890923	0.445461	+	0.895301i	$0.353040\pi$
$230$	0	0
$231$	0	0
$232$	3943.73	1.11603
$233$	3997.55	1.12398	0.561991	−	0.827143i	$-0.310035\pi$
$233$	3997.55	1.12398	0.561991	+	0.827143i	$0.310035\pi$
$234$	−5052.52	−1.41151
$235$	0	0
$236$	−2560.53	−0.706255
$237$	3026.75	0.829573
$238$	0	0
$239$	−4499.79	−1.21786	−0.608928	−	0.793226i	$-0.708400\pi$
$239$	−4499.79	−1.21786	−0.608928	+	0.793226i	$0.708400\pi$
$240$	0	0
$241$	−3633.26	−0.971115	−0.485557	−	0.874205i	$-0.661383\pi$
$241$	−3633.26	−0.971115	−0.485557	+	0.874205i	$0.661383\pi$
$242$	7221.66	1.91829
$243$	3753.32	0.990847
$244$	2042.64	0.535929
$245$	0	0
$246$	6360.45	1.64849
$247$	−4927.39	−1.26932
$248$	3999.65	1.02411
$249$	−2006.68	−0.510715
$250$	0	0
$251$	1211.21	0.304585	0.152292	−	0.988335i	$-0.451334\pi$
$251$	1211.21	0.304585	0.152292	+	0.988335i	$0.451334\pi$
$252$	0	0
$253$	−6795.74	−1.68871
$254$	8444.50	2.08604
$255$	0	0
$256$	−3388.39	−0.827245
$257$	6225.81	1.51111	0.755556	−	0.655085i	$-0.227367\pi$
$257$	6225.81	1.51111	0.755556	+	0.655085i	$0.227367\pi$
$258$	12112.1	2.92275
$259$	0	0
$260$	0	0
$261$	2820.27	0.668851
$262$	−3515.43	−0.828947
$263$	−757.377	−0.177574	−0.0887869	−	0.996051i	$-0.528299\pi$
$263$	−757.377	−0.177574	−0.0887869	+	0.996051i	$0.528299\pi$
$264$	7241.41	1.68817
$265$	0	0
$266$	0	0
$267$	8502.43	1.94884
$268$	6001.23	1.36785
$269$	−1750.59	−0.396786	−0.198393	−	0.980123i	$-0.563572\pi$
$269$	−1750.59	−0.396786	−0.198393	+	0.980123i	$0.563572\pi$
$270$	0	0
$271$	−4451.86	−0.997902	−0.498951	−	0.866630i	$-0.666281\pi$
$271$	−4451.86	−0.997902	−0.498951	+	0.866630i	$0.666281\pi$
$272$	−470.589	−0.104903
$273$	0	0
$274$	−4661.87	−1.02786
$275$	0	0
$276$	−10213.6	−2.22749
$277$	−7691.69	−1.66841	−0.834204	−	0.551456i	$-0.814072\pi$
$277$	−7691.69	−1.66841	−0.834204	+	0.551456i	$0.814072\pi$
$278$	4669.62	1.00743
$279$	2860.26	0.613760
$280$	0	0
$281$	−199.034	−0.0422540	−0.0211270	−	0.999777i	$-0.506725\pi$
$281$	−199.034	−0.0422540	−0.0211270	+	0.999777i	$0.506725\pi$
$282$	−3330.01	−0.703188
$283$	799.307	0.167893	0.0839467	−	0.996470i	$-0.473247\pi$
$283$	799.307	0.167893	0.0839467	+	0.996470i	$0.473247\pi$
$284$	10333.9	2.15916
$285$	0	0
$286$	18443.3	3.81321
$287$	0	0
$288$	−2897.96	−0.592931
$289$	164.110	0.0334031
$290$	0	0
$291$	−8500.86	−1.71247
$292$	10041.0	2.01235
$293$	−3302.75	−0.658528	−0.329264	−	0.944238i	$-0.606801\pi$
$293$	−3302.75	−0.658528	−0.329264	+	0.944238i	$0.606801\pi$
$294$	0	0
$295$	0	0
$296$	2375.14	0.466393
$297$	−4261.16	−0.832518
$298$	6415.67	1.24715
$299$	−9453.72	−1.82850
$300$	0	0
$301$	0	0
$302$	9500.87	1.81031
$303$	−2594.51	−0.491916
$304$	−432.655	−0.0816265
$305$	0	0
$306$	4786.38	0.894180
$307$	8628.17	1.60402	0.802012	−	0.597308i	$-0.203763\pi$
$307$	8628.17	1.60402	0.802012	+	0.597308i	$0.203763\pi$
$308$	0	0
$309$	1887.13	0.347427
$310$	0	0
$311$	4900.60	0.893530	0.446765	−	0.894651i	$-0.352576\pi$
$311$	4900.60	0.893530	0.446765	+	0.894651i	$0.352576\pi$
$312$	10073.7	1.82792
$313$	9114.26	1.64590	0.822952	−	0.568110i	$-0.192325\pi$
$313$	9114.26	1.64590	0.822952	+	0.568110i	$0.192325\pi$
$314$	3217.63	0.578285
$315$	0	0
$316$	−5882.51	−1.04721
$317$	−8022.82	−1.42147	−0.710736	−	0.703459i	$-0.751638\pi$
$317$	−8022.82	−1.42147	−0.710736	+	0.703459i	$0.751638\pi$
$318$	−17136.5	−3.02191
$319$	−10294.9	−1.80691
$320$	0	0
$321$	141.995	0.0246896
$322$	0	0
$323$	4667.84	0.804104
$324$	−11430.2	−1.95990
$325$	0	0
$326$	14858.0	2.52426
$327$	111.878	0.0189201
$328$	−4492.42	−0.756257
$329$	0	0
$330$	0	0
$331$	−1333.58	−0.221451	−0.110725	−	0.993851i	$-0.535317\pi$
$331$	−1333.58	−0.221451	−0.110725	+	0.993851i	$0.535317\pi$
$332$	3899.99	0.644698
$333$	1698.53	0.279515
$334$	−12973.5	−2.12539
$335$	0	0
$336$	0	0
$337$	−1687.33	−0.272744	−0.136372	−	0.990658i	$-0.543544\pi$
$337$	−1687.33	−0.272744	−0.136372	+	0.990658i	$0.543544\pi$
$338$	15693.4	2.52546
$339$	−380.331	−0.0609344
$340$	0	0
$341$	−10440.9	−1.65808
$342$	4400.55	0.695773
$343$	0	0
$344$	−8554.87	−1.34084
$345$	0	0
$346$	−12713.9	−1.97544
$347$	5955.18	0.921299	0.460649	−	0.887582i	$-0.347617\pi$
$347$	5955.18	0.921299	0.460649	+	0.887582i	$0.347617\pi$
$348$	−15472.6	−2.38339
$349$	−6876.59	−1.05471	−0.527357	−	0.849644i	$-0.676817\pi$
$349$	−6876.59	−1.05471	−0.527357	+	0.849644i	$0.676817\pi$
$350$	0	0
$351$	−5927.81	−0.901433
$352$	10578.5	1.60181
$353$	−1395.95	−0.210479	−0.105239	−	0.994447i	$-0.533561\pi$
$353$	−1395.95	−0.210479	−0.105239	+	0.994447i	$0.533561\pi$
$354$	5974.90	0.897069
$355$	0	0
$356$	−16524.5	−2.46011
$357$	0	0
$358$	1127.50	0.166453
$359$	−382.988	−0.0563046	−0.0281523	−	0.999604i	$-0.508962\pi$
$359$	−382.988	−0.0563046	−0.0281523	+	0.999604i	$0.508962\pi$
$360$	0	0
$361$	−2567.44	−0.374316
$362$	−9412.85	−1.36665
$363$	−10296.8	−1.48881
$364$	0	0
$365$	0	0
$366$	−4766.43	−0.680725
$367$	10178.4	1.44771	0.723856	−	0.689951i	$-0.242368\pi$
$367$	10178.4	1.44771	0.723856	+	0.689951i	$0.242368\pi$
$368$	−830.094	−0.117586
$369$	−3212.65	−0.453235
$370$	0	0
$371$	0	0
$372$	−15692.0	−2.18708
$373$	−2991.00	−0.415196	−0.207598	−	0.978214i	$-0.566565\pi$
$373$	−2991.00	−0.415196	−0.207598	+	0.978214i	$0.566565\pi$
$374$	−17471.8	−2.41563
$375$	0	0
$376$	2352.00	0.322594
$377$	−14321.5	−1.95648
$378$	0	0
$379$	7400.66	1.00302	0.501512	−	0.865150i	$-0.332777\pi$
$379$	7400.66	1.00302	0.501512	+	0.865150i	$0.332777\pi$
$380$	0	0
$381$	−12040.3	−1.61901
$382$	−5682.27	−0.761074
$383$	10769.6	1.43682	0.718408	−	0.695622i	$-0.244871\pi$
$383$	10769.6	1.43682	0.718408	+	0.695622i	$0.244871\pi$
$384$	14349.5	1.90695
$385$	0	0
$386$	7265.75	0.958075
$387$	−6117.81	−0.803580
$388$	16521.5	2.16173
$389$	−11193.3	−1.45893	−0.729466	−	0.684017i	$-0.760231\pi$
$389$	−11193.3	−1.45893	−0.729466	+	0.684017i	$0.760231\pi$
$390$	0	0
$391$	8955.74	1.15834
$392$	0	0
$393$	5012.36	0.643359
$394$	10641.1	1.36063
$395$	0	0
$396$	−10064.5	−1.27717
$397$	−10371.9	−1.31121	−0.655605	−	0.755104i	$-0.727586\pi$
$397$	−10371.9	−1.31121	−0.655605	+	0.755104i	$0.727586\pi$
$398$	9040.88	1.13864
$399$	0	0
$400$	0	0
$401$	5402.20	0.672751	0.336375	−	0.941728i	$-0.390799\pi$
$401$	5402.20	0.672751	0.336375	+	0.941728i	$0.390799\pi$
$402$	−14003.7	−1.73741
$403$	−14524.5	−1.79533
$404$	5042.44	0.620967
$405$	0	0
$406$	0	0
$407$	−6200.17	−0.755113
$408$	−9543.07	−1.15797
$409$	12829.9	1.55110	0.775549	−	0.631288i	$-0.217473\pi$
$409$	12829.9	1.55110	0.775549	+	0.631288i	$0.217473\pi$
$410$	0	0
$411$	6646.97	0.797739
$412$	−3667.64	−0.438572
$413$	0	0
$414$	8442.91	1.00229
$415$	0	0
$416$	14716.0	1.73440
$417$	−6658.02	−0.781882
$418$	−16063.4	−1.87964
$419$	−5620.85	−0.655362	−0.327681	−	0.944788i	$-0.606267\pi$
$419$	−5620.85	−0.655362	−0.327681	+	0.944788i	$0.606267\pi$
$420$	0	0
$421$	−2177.11	−0.252033	−0.126017	−	0.992028i	$-0.540219\pi$
$421$	−2177.11	−0.252033	−0.126017	+	0.992028i	$0.540219\pi$
$422$	−26541.9	−3.06171
$423$	1681.98	0.193335
$424$	12103.6	1.38633
$425$	0	0
$426$	−24113.7	−2.74252
$427$	0	0
$428$	−275.967	−0.0311668
$429$	−26296.8	−2.95949
$430$	0	0
$431$	10396.4	1.16190	0.580950	−	0.813939i	$-0.302681\pi$
$431$	10396.4	1.16190	0.580950	+	0.813939i	$0.302681\pi$
$432$	−520.497	−0.0579686
$433$	−11875.6	−1.31803	−0.659015	−	0.752129i	$-0.729027\pi$
$433$	−11875.6	−1.31803	−0.659015	+	0.752129i	$0.729027\pi$
$434$	0	0
$435$	0	0
$436$	−217.435	−0.0238836
$437$	8233.81	0.901320
$438$	−23430.4	−2.55605
$439$	−640.369	−0.0696200	−0.0348100	−	0.999394i	$-0.511083\pi$
$439$	−640.369	−0.0696200	−0.0348100	+	0.999394i	$0.511083\pi$
$440$	0	0
$441$	0	0
$442$	−24305.5	−2.61560
$443$	−854.852	−0.0916823	−0.0458411	−	0.998949i	$-0.514597\pi$
$443$	−854.852	−0.0916823	−0.0458411	+	0.998949i	$0.514597\pi$
$444$	−9318.50	−0.996028
$445$	0	0
$446$	7798.46	0.827954
$447$	−9147.56	−0.967930
$448$	0	0
$449$	3427.44	0.360247	0.180123	−	0.983644i	$-0.442350\pi$
$449$	3427.44	0.360247	0.180123	+	0.983644i	$0.442350\pi$
$450$	0	0
$451$	11727.2	1.22442
$452$	739.176	0.0769202
$453$	−13546.5	−1.40501
$454$	25786.6	2.66569
$455$	0	0
$456$	−8773.79	−0.901032
$457$	1709.31	0.174963	0.0874814	−	0.996166i	$-0.472118\pi$
$457$	1709.31	0.174963	0.0874814	+	0.996166i	$0.472118\pi$
$458$	14001.7	1.42851
$459$	5615.56	0.571050
$460$	0	0
$461$	−2251.18	−0.227436	−0.113718	−	0.993513i	$-0.536276\pi$
$461$	−2251.18	−0.227436	−0.113718	+	0.993513i	$0.536276\pi$
$462$	0	0
$463$	−3200.80	−0.321282	−0.160641	−	0.987013i	$-0.551356\pi$
$463$	−3200.80	−0.321282	−0.160641	+	0.987013i	$0.551356\pi$
$464$	−1257.51	−0.125816
$465$	0	0
$466$	18129.3	1.80219
$467$	−5477.39	−0.542748	−0.271374	−	0.962474i	$-0.587478\pi$
$467$	−5477.39	−0.542748	−0.271374	+	0.962474i	$0.587478\pi$
$468$	−14000.9	−1.38289
$469$	0	0
$470$	0	0
$471$	−4587.75	−0.448816
$472$	−4220.11	−0.411538
$473$	22332.0	2.17088
$474$	13726.6	1.33014
$475$	0	0
$476$	0	0
$477$	8655.60	0.830844
$478$	−20407.0	−1.95271
$479$	14182.0	1.35280	0.676399	−	0.736535i	$-0.263539\pi$
$479$	14182.0	1.35280	0.676399	+	0.736535i	$0.263539\pi$
$480$	0	0
$481$	−8625.20	−0.817620
$482$	−16477.2	−1.55709
$483$	0	0
$484$	20011.8	1.87940
$485$	0	0
$486$	17021.7	1.58872
$487$	−5320.83	−0.495092	−0.247546	−	0.968876i	$-0.579624\pi$
$487$	−5320.83	−0.495092	−0.247546	+	0.968876i	$0.579624\pi$
$488$	3366.56	0.312289
$489$	−21184.8	−1.95912
$490$	0	0
$491$	−6574.80	−0.604311	−0.302155	−	0.953259i	$-0.597706\pi$
$491$	−6574.80	−0.604311	−0.302155	+	0.953259i	$0.597706\pi$
$492$	17625.3	1.61506
$493$	13567.1	1.23941
$494$	−22346.2	−2.03523
$495$	0	0
$496$	−1275.34	−0.115453
$497$	0	0
$498$	−9100.49	−0.818881
$499$	−1507.68	−0.135256	−0.0676282	−	0.997711i	$-0.521543\pi$
$499$	−1507.68	−0.135256	−0.0676282	+	0.997711i	$0.521543\pi$
$500$	0	0
$501$	18497.8	1.64955
$502$	5492.95	0.488371
$503$	8782.24	0.778491	0.389245	−	0.921134i	$-0.372736\pi$
$503$	8782.24	0.778491	0.389245	+	0.921134i	$0.372736\pi$
$504$	0	0
$505$	0	0
$506$	−30819.4	−2.70768
$507$	−22375.9	−1.96005
$508$	23400.4	2.04375
$509$	−12696.6	−1.10563	−0.552815	−	0.833304i	$-0.686447\pi$
$509$	−12696.6	−1.10563	−0.552815	+	0.833304i	$0.686447\pi$
$510$	0	0
$511$	0	0
$512$	2386.50	0.205995
$513$	5162.88	0.444341
$514$	28234.7	2.42291
$515$	0	0
$516$	33563.7	2.86349
$517$	−6139.76	−0.522295
$518$	0	0
$519$	18127.6	1.53317
$520$	0	0
$521$	10434.7	0.877455	0.438727	−	0.898620i	$-0.355429\pi$
$521$	10434.7	0.877455	0.438727	+	0.898620i	$0.355429\pi$
$522$	12790.2	1.07243
$523$	8403.03	0.702560	0.351280	−	0.936270i	$-0.385746\pi$
$523$	8403.03	0.702560	0.351280	+	0.936270i	$0.385746\pi$
$524$	−9741.55	−0.812140
$525$	0	0
$526$	−3434.78	−0.284722
$527$	13759.4	1.13733
$528$	−2309.02	−0.190317
$529$	3630.43	0.298384
$530$	0	0
$531$	−3017.91	−0.246640
$532$	0	0
$533$	16314.0	1.32577
$534$	38559.4	3.12477
$535$	0	0
$536$	9890.85	0.797052
$537$	−1607.61	−0.129187
$538$	−7939.11	−0.636207
$539$	0	0
$540$	0	0
$541$	8349.66	0.663549	0.331775	−	0.943359i	$-0.392353\pi$
$541$	8349.66	0.663549	0.331775	+	0.943359i	$0.392353\pi$
$542$	−20189.6	−1.60004
$543$	13421.0	1.06068
$544$	−13940.8	−1.09873
$545$	0	0
$546$	0	0
$547$	12185.7	0.952509	0.476254	−	0.879308i	$-0.341994\pi$
$547$	12185.7	0.952509	0.476254	+	0.879308i	$0.341994\pi$
$548$	−12918.4	−1.00702
$549$	2407.51	0.187159
$550$	0	0
$551$	12473.4	0.964402
$552$	−16833.4	−1.29797
$553$	0	0
$554$	−34882.6	−2.67512
$555$	0	0
$556$	12939.9	0.987004
$557$	−540.209	−0.0410940	−0.0205470	−	0.999789i	$-0.506541\pi$
$557$	−540.209	−0.0410940	−0.0205470	+	0.999789i	$0.506541\pi$
$558$	12971.5	0.984103
$559$	31066.6	2.35058
$560$	0	0
$561$	24911.6	1.87481
$562$	−902.639	−0.0677501
$563$	1949.26	0.145917	0.0729587	−	0.997335i	$-0.476756\pi$
$563$	1949.26	0.145917	0.0729587	+	0.997335i	$0.476756\pi$
$564$	−9227.71	−0.688930
$565$	0	0
$566$	3624.94	0.269200
$567$	0	0
$568$	17031.6	1.25815
$569$	9487.28	0.698994	0.349497	−	0.936938i	$-0.386353\pi$
$569$	9487.28	0.698994	0.349497	+	0.936938i	$0.386353\pi$
$570$	0	0
$571$	−20172.3	−1.47843	−0.739215	−	0.673470i	$-0.764803\pi$
$571$	−20172.3	−1.47843	−0.739215	+	0.673470i	$0.764803\pi$
$572$	51108.0	3.73590
$573$	8101.87	0.590681
$574$	0	0
$575$	0	0
$576$	−12360.0	−0.894094
$577$	−12937.3	−0.933423	−0.466712	−	0.884410i	$-0.654561\pi$
$577$	−12937.3	−0.933423	−0.466712	+	0.884410i	$0.654561\pi$
$578$	744.254	0.0535586
$579$	−10359.6	−0.743577
$580$	0	0
$581$	0	0
$582$	−38552.2	−2.74578
$583$	−31595.7	−2.24453
$584$	16549.0	1.17261
$585$	0	0
$586$	−14978.3	−1.05588
$587$	6489.43	0.456299	0.228149	−	0.973626i	$-0.426733\pi$
$587$	6489.43	0.456299	0.228149	+	0.973626i	$0.426733\pi$
$588$	0	0
$589$	12650.3	0.884968
$590$	0	0
$591$	−15172.2	−1.05601
$592$	−757.345	−0.0525789
$593$	16803.6	1.16364	0.581821	−	0.813317i	$-0.302340\pi$
$593$	16803.6	1.16364	0.581821	+	0.813317i	$0.302340\pi$
$594$	−19324.8	−1.33486
$595$	0	0
$596$	17778.3	1.22186
$597$	−12890.6	−0.883716
$598$	−42873.5	−2.93182
$599$	19139.6	1.30554	0.652772	−	0.757554i	$-0.273606\pi$
$599$	19139.6	1.30554	0.652772	+	0.757554i	$0.273606\pi$
$600$	0	0
$601$	−14304.0	−0.970838	−0.485419	−	0.874282i	$-0.661333\pi$
$601$	−14304.0	−0.970838	−0.485419	+	0.874282i	$0.661333\pi$
$602$	0	0
$603$	7073.21	0.477684
$604$	26327.7	1.77361
$605$	0	0
$606$	−11766.3	−0.788738
$607$	2018.10	0.134946	0.0674730	−	0.997721i	$-0.478506\pi$
$607$	2018.10	0.134946	0.0674730	+	0.997721i	$0.478506\pi$
$608$	−12817.1	−0.854935
$609$	0	0
$610$	0	0
$611$	−8541.17	−0.565530
$612$	13263.4	0.876050
$613$	21479.4	1.41524	0.707621	−	0.706592i	$-0.249768\pi$
$613$	21479.4	1.41524	0.707621	+	0.706592i	$0.249768\pi$
$614$	39129.6	2.57189
$615$	0	0
$616$	0	0
$617$	−5856.83	−0.382151	−0.191075	−	0.981575i	$-0.561197\pi$
$617$	−5856.83	−0.382151	−0.191075	+	0.981575i	$0.561197\pi$
$618$	8558.31	0.557065
$619$	17619.3	1.14407	0.572035	−	0.820229i	$-0.306154\pi$
$619$	17619.3	1.14407	0.572035	+	0.820229i	$0.306154\pi$
$620$	0	0
$621$	9905.53	0.640089
$622$	22224.7	1.43269
$623$	0	0
$624$	−3212.13	−0.206071
$625$	0	0
$626$	41334.1	2.63904
$627$	22903.5	1.45882
$628$	8916.31	0.566560
$629$	8170.87	0.517955
$630$	0	0
$631$	26395.7	1.66529	0.832643	−	0.553810i	$-0.186827\pi$
$631$	26395.7	1.66529	0.832643	+	0.553810i	$0.186827\pi$
$632$	−9695.19	−0.610212
$633$	37843.9	2.37624
$634$	−36384.3	−2.27919
$635$	0	0
$636$	−47486.6	−2.96064
$637$	0	0
$638$	−46688.3	−2.89719
$639$	12179.8	0.754029
$640$	0	0
$641$	3209.30	0.197753	0.0988764	−	0.995100i	$-0.468475\pi$
$641$	3209.30	0.197753	0.0988764	+	0.995100i	$0.468475\pi$
$642$	643.959	0.0395873
$643$	1762.48	0.108096	0.0540478	−	0.998538i	$-0.482788\pi$
$643$	1762.48	0.108096	0.0540478	+	0.998538i	$0.482788\pi$
$644$	0	0
$645$	0	0
$646$	21169.1	1.28930
$647$	24372.7	1.48098	0.740488	−	0.672070i	$-0.234595\pi$
$647$	24372.7	1.48098	0.740488	+	0.672070i	$0.234595\pi$
$648$	−18838.5	−1.14205
$649$	11016.3	0.666301
$650$	0	0
$651$	0	0
$652$	41172.8	2.47308
$653$	6389.97	0.382938	0.191469	−	0.981499i	$-0.438675\pi$
$653$	6389.97	0.382938	0.191469	+	0.981499i	$0.438675\pi$
$654$	507.377	0.0303364
$655$	0	0
$656$	1432.47	0.0852567
$657$	11834.6	0.702760
$658$	0	0
$659$	−6111.68	−0.361270	−0.180635	−	0.983550i	$-0.557815\pi$
$659$	−6111.68	−0.361270	−0.180635	+	0.983550i	$0.557815\pi$
$660$	0	0
$661$	−16391.1	−0.964510	−0.482255	−	0.876031i	$-0.660182\pi$
$661$	−16391.1	−0.964510	−0.482255	+	0.876031i	$0.660182\pi$
$662$	−6047.92	−0.355074
$663$	34655.2	2.03001
$664$	6427.72	0.375669
$665$	0	0
$666$	7702.98	0.448175
$667$	23931.6	1.38926
$668$	−35950.6	−2.08229
$669$	−11119.2	−0.642588
$670$	0	0
$671$	−8788.20	−0.505611
$672$	0	0
$673$	−13055.9	−0.747799	−0.373900	−	0.927469i	$-0.621980\pi$
$673$	−13055.9	−0.747799	−0.373900	+	0.927469i	$0.621980\pi$
$674$	−7652.20	−0.437317
$675$	0	0
$676$	43487.6	2.47426
$677$	3154.53	0.179082	0.0895410	−	0.995983i	$-0.471460\pi$
$677$	3154.53	0.179082	0.0895410	+	0.995983i	$0.471460\pi$
$678$	−1724.84	−0.0977022
$679$	0	0
$680$	0	0
$681$	−36766.9	−2.06888
$682$	−47350.4	−2.65856
$683$	17282.7	0.968233	0.484117	−	0.875004i	$-0.339141\pi$
$683$	17282.7	0.968233	0.484117	+	0.875004i	$0.339141\pi$
$684$	12194.3	0.681666
$685$	0	0
$686$	0	0
$687$	−19963.8	−1.10869
$688$	2727.83	0.151159
$689$	−43953.6	−2.43033
$690$	0	0
$691$	−26838.3	−1.47753	−0.738766	−	0.673961i	$-0.764591\pi$
$691$	−26838.3	−1.47753	−0.738766	+	0.673961i	$0.764591\pi$
$692$	−35231.1	−1.93538
$693$	0	0
$694$	27007.3	1.47721
$695$	0	0
$696$	−25501.0	−1.38881
$697$	−15454.6	−0.839866
$698$	−31186.0	−1.69113
$699$	−25849.0	−1.39871
$700$	0	0
$701$	−8374.52	−0.451214	−0.225607	−	0.974218i	$-0.572437\pi$
$701$	−8374.52	−0.451214	−0.225607	+	0.974218i	$0.572437\pi$
$702$	−26883.2	−1.44536
$703$	7512.21	0.403028
$704$	45117.9	2.41540
$705$	0	0
$706$	−6330.79	−0.337482
$707$	0	0
$708$	16556.9	0.878881
$709$	11929.4	0.631902	0.315951	−	0.948775i	$-0.397676\pi$
$709$	11929.4	0.631902	0.315951	+	0.948775i	$0.397676\pi$
$710$	0	0
$711$	−6933.28	−0.365708
$712$	−27234.7	−1.43352
$713$	24270.9	1.27483
$714$	0	0
$715$	0	0
$716$	3124.40	0.163078
$717$	29096.6	1.51553
$718$	−1736.89	−0.0902788
$719$	23392.3	1.21333	0.606666	−	0.794957i	$-0.292507\pi$
$719$	23392.3	1.21333	0.606666	+	0.794957i	$0.292507\pi$
$720$	0	0
$721$	0	0
$722$	−11643.6	−0.600179
$723$	23493.4	1.20848
$724$	−26083.8	−1.33894
$725$	0	0
$726$	−46696.8	−2.38717
$727$	−10659.8	−0.543810	−0.271905	−	0.962324i	$-0.587654\pi$
$727$	−10659.8	−0.543810	−0.271905	+	0.962324i	$0.587654\pi$
$728$	0	0
$729$	287.476	0.0146053
$730$	0	0
$731$	−29430.1	−1.48907
$732$	−13208.2	−0.666923
$733$	−23919.1	−1.20528	−0.602642	−	0.798011i	$-0.705885\pi$
$733$	−23919.1	−1.20528	−0.602642	+	0.798011i	$0.705885\pi$
$734$	46160.2	2.32126
$735$	0	0
$736$	−24590.9	−1.23157
$737$	−25819.5	−1.29047
$738$	−14569.7	−0.726717
$739$	34535.1	1.71907	0.859536	−	0.511074i	$-0.170752\pi$
$739$	34535.1	1.71907	0.859536	+	0.511074i	$0.170752\pi$
$740$	0	0
$741$	31861.6	1.57957
$742$	0	0
$743$	17526.8	0.865403	0.432702	−	0.901537i	$-0.357560\pi$
$743$	17526.8	0.865403	0.432702	+	0.901537i	$0.357560\pi$
$744$	−25862.6	−1.27442
$745$	0	0
$746$	−13564.5	−0.665725
$747$	4596.63	0.225143
$748$	−48415.9	−2.36666
$749$	0	0
$750$	0	0
$751$	18534.3	0.900565	0.450283	−	0.892886i	$-0.351323\pi$
$751$	18534.3	0.900565	0.450283	+	0.892886i	$0.351323\pi$
$752$	−749.966	−0.0363676
$753$	−7831.94	−0.379033
$754$	−64949.2	−3.13702
$755$	0	0
$756$	0	0
$757$	−9309.64	−0.446981	−0.223491	−	0.974706i	$-0.571745\pi$
$757$	−9309.64	−0.446981	−0.223491	+	0.974706i	$0.571745\pi$
$758$	33562.7	1.60825
$759$	43942.7	2.10148
$760$	0	0
$761$	2968.41	0.141399	0.0706996	−	0.997498i	$-0.477477\pi$
$761$	2968.41	0.141399	0.0706996	+	0.997498i	$0.477477\pi$
$762$	−54604.0	−2.59592
$763$	0	0
$764$	−15746.0	−0.745643
$765$	0	0
$766$	48841.1	2.30379
$767$	15325.1	0.721457
$768$	21910.1	1.02944
$769$	34932.9	1.63812	0.819060	−	0.573708i	$-0.194495\pi$
$769$	34932.9	1.63812	0.819060	+	0.573708i	$0.194495\pi$
$770$	0	0
$771$	−40257.5	−1.88046
$772$	20134.0	0.938650
$773$	13595.4	0.632591	0.316295	−	0.948661i	$-0.397561\pi$
$773$	13595.4	0.632591	0.316295	+	0.948661i	$0.397561\pi$
$774$	−27744.9	−1.28846
$775$	0	0
$776$	27229.7	1.25965
$777$	0	0
$778$	−50762.9	−2.33925
$779$	−14208.8	−0.653510
$780$	0	0
$781$	−44460.1	−2.03701
$782$	40615.2	1.85728
$783$	15005.9	0.684888
$784$	0	0
$785$	0	0
$786$	22731.6	1.03156
$787$	14228.1	0.644444	0.322222	−	0.946664i	$-0.395570\pi$
$787$	14228.1	0.644444	0.322222	+	0.946664i	$0.395570\pi$
$788$	29487.3	1.33305
$789$	4897.36	0.220977
$790$	0	0
$791$	0	0
$792$	−16587.7	−0.744213
$793$	−12225.5	−0.547465
$794$	−47037.5	−2.10239
$795$	0	0
$796$	25053.0	1.11555
$797$	−11442.3	−0.508539	−0.254270	−	0.967133i	$-0.581835\pi$
$797$	−11442.3	−0.508539	−0.254270	+	0.967133i	$0.581835\pi$
$798$	0	0
$799$	8091.26	0.358258
$800$	0	0
$801$	−19476.2	−0.859125
$802$	24499.5	1.07869
$803$	−43200.3	−1.89851
$804$	−38805.2	−1.70218
$805$	0	0
$806$	−65870.2	−2.87863
$807$	11319.7	0.493770
$808$	8310.63	0.361840
$809$	22732.9	0.987944	0.493972	−	0.869478i	$-0.335545\pi$
$809$	22732.9	0.987944	0.493972	+	0.869478i	$0.335545\pi$
$810$	0	0
$811$	29768.2	1.28890	0.644452	−	0.764644i	$-0.277085\pi$
$811$	29768.2	1.28890	0.644452	+	0.764644i	$0.277085\pi$
$812$	0	0
$813$	28786.7	1.24181
$814$	−28118.4	−1.21075
$815$	0	0
$816$	3042.93	0.130544
$817$	−27057.7	−1.15867
$818$	58185.0	2.48703
$819$	0	0
$820$	0	0
$821$	6408.35	0.272416	0.136208	−	0.990680i	$-0.456509\pi$
$821$	6408.35	0.272416	0.136208	+	0.990680i	$0.456509\pi$
$822$	30144.6	1.27909
$823$	−20876.0	−0.884194	−0.442097	−	0.896967i	$-0.645765\pi$
$823$	−20876.0	−0.884194	−0.442097	+	0.896967i	$0.645765\pi$
$824$	−6044.78	−0.255558
$825$	0	0
$826$	0	0
$827$	−41650.1	−1.75129	−0.875645	−	0.482955i	$-0.839563\pi$
$827$	−41650.1	−1.75129	−0.875645	+	0.482955i	$0.839563\pi$
$828$	23396.0	0.981964
$829$	−17194.7	−0.720380	−0.360190	−	0.932879i	$-0.617288\pi$
$829$	−17194.7	−0.720380	−0.360190	+	0.932879i	$0.617288\pi$
$830$	0	0
$831$	49736.1	2.07621
$832$	62764.5	2.61535
$833$	0	0
$834$	−30194.8	−1.25367
$835$	0	0
$836$	−44513.0	−1.84153
$837$	15218.7	0.628477
$838$	−25491.1	−1.05081
$839$	−40583.7	−1.66997	−0.834985	−	0.550273i	$-0.814524\pi$
$839$	−40583.7	−1.66997	−0.834985	+	0.550273i	$0.814524\pi$
$840$	0	0
$841$	11865.0	0.486489
$842$	−9873.42	−0.404110
$843$	1287.00	0.0525819
$844$	−73549.7	−2.99963
$845$	0	0
$846$	7627.93	0.309993
$847$	0	0
$848$	−3859.39	−0.156288
$849$	−5168.49	−0.208931
$850$	0	0
$851$	14413.0	0.580576
$852$	−66821.0	−2.68691
$853$	−23020.1	−0.924023	−0.462012	−	0.886874i	$-0.652872\pi$
$853$	−23020.1	−0.924023	−0.462012	+	0.886874i	$0.652872\pi$
$854$	0	0
$855$	0	0
$856$	−454.832	−0.0181610
$857$	18075.5	0.720476	0.360238	−	0.932861i	$-0.382696\pi$
$857$	18075.5	0.720476	0.360238	+	0.932861i	$0.382696\pi$
$858$	−119259.	−4.74525
$859$	−20308.2	−0.806643	−0.403321	−	0.915058i	$-0.632144\pi$
$859$	−20308.2	−0.806643	−0.403321	+	0.915058i	$0.632144\pi$
$860$	0	0
$861$	0	0
$862$	47148.9	1.86299
$863$	−31023.5	−1.22370	−0.611849	−	0.790975i	$-0.709574\pi$
$863$	−31023.5	−1.22370	−0.611849	+	0.790975i	$0.709574\pi$
$864$	−15419.3	−0.607148
$865$	0	0
$866$	−53857.2	−2.11333
$867$	−1061.17	−0.0415677
$868$	0	0
$869$	25308.7	0.987963
$870$	0	0
$871$	−35918.1	−1.39729
$872$	−358.363	−0.0139171
$873$	19472.6	0.754924
$874$	37341.2	1.44518
$875$	0	0
$876$	−64927.5	−2.50422
$877$	−28344.1	−1.09135	−0.545674	−	0.837997i	$-0.683726\pi$
$877$	−28344.1	−1.09135	−0.545674	+	0.837997i	$0.683726\pi$
$878$	−2904.14	−0.111629
$879$	21356.3	0.819488
$880$	0	0
$881$	−41264.5	−1.57802	−0.789010	−	0.614380i	$-0.789406\pi$
$881$	−41264.5	−1.57802	−0.789010	+	0.614380i	$0.789406\pi$
$882$	0	0
$883$	−7995.59	−0.304726	−0.152363	−	0.988325i	$-0.548688\pi$
$883$	−7995.59	−0.304726	−0.152363	+	0.988325i	$0.548688\pi$
$884$	−67352.5	−2.56257
$885$	0	0
$886$	−3876.84	−0.147003
$887$	−24438.3	−0.925093	−0.462547	−	0.886595i	$-0.653064\pi$
$887$	−24438.3	−0.925093	−0.462547	+	0.886595i	$0.653064\pi$
$888$	−15358.2	−0.580390
$889$	0	0
$890$	0	0
$891$	49176.8	1.84903
$892$	21610.1	0.811167
$893$	7439.02	0.278765
$894$	−41485.1	−1.55198
$895$	0	0
$896$	0	0
$897$	61129.8	2.27543
$898$	15543.8	0.577620
$899$	36768.0	1.36405
$900$	0	0
$901$	41638.3	1.53959
$902$	53184.0	1.96323
$903$	0	0
$904$	1218.26	0.0448218
$905$	0	0
$906$	−61434.7	−2.25279
$907$	17241.6	0.631201	0.315601	−	0.948892i	$-0.397794\pi$
$907$	17241.6	0.631201	0.315601	+	0.948892i	$0.397794\pi$
$908$	71456.6	2.61164
$909$	5943.15	0.216856
$910$	0	0
$911$	17407.8	0.633092	0.316546	−	0.948577i	$-0.397477\pi$
$911$	17407.8	0.633092	0.316546	+	0.948577i	$0.397477\pi$
$912$	2797.64	0.101578
$913$	−16779.2	−0.608226
$914$	7751.87	0.280535
$915$	0	0
$916$	38799.8	1.39954
$917$	0	0
$918$	25467.1	0.915621
$919$	7704.64	0.276553	0.138277	−	0.990394i	$-0.455844\pi$
$919$	7704.64	0.276553	0.138277	+	0.990394i	$0.455844\pi$
$920$	0	0
$921$	−55791.6	−1.99609
$922$	−10209.3	−0.364671
$923$	−61849.5	−2.20564
$924$	0	0
$925$	0	0
$926$	−14515.9	−0.515144
$927$	−4322.78	−0.153159
$928$	−37252.8	−1.31776
$929$	48184.3	1.70170	0.850849	−	0.525411i	$-0.176088\pi$
$929$	48184.3	1.70170	0.850849	+	0.525411i	$0.176088\pi$
$930$	0	0
$931$	0	0
$932$	50237.7	1.76565
$933$	−31688.4	−1.11193
$934$	−24840.5	−0.870242
$935$	0	0
$936$	−23075.5	−0.805818
$937$	−23371.6	−0.814853	−0.407427	−	0.913238i	$-0.633574\pi$
$937$	−23371.6	−0.814853	−0.407427	+	0.913238i	$0.633574\pi$
$938$	0	0
$939$	−58934.7	−2.04820
$940$	0	0
$941$	−18048.6	−0.625258	−0.312629	−	0.949875i	$-0.601210\pi$
$941$	−18048.6	−0.625258	−0.312629	+	0.949875i	$0.601210\pi$
$942$	−20805.9	−0.719631
$943$	−27261.1	−0.941405
$944$	1345.64	0.0463948
$945$	0	0
$946$	101278.	3.48079
$947$	10346.2	0.355022	0.177511	−	0.984119i	$-0.443195\pi$
$947$	10346.2	0.355022	0.177511	+	0.984119i	$0.443195\pi$
$948$	38037.6	1.30317
$949$	−60096.9	−2.05567
$950$	0	0
$951$	51877.3	1.76891
$952$	0	0
$953$	8691.77	0.295440	0.147720	−	0.989029i	$-0.452807\pi$
$953$	8691.77	0.295440	0.147720	+	0.989029i	$0.452807\pi$
$954$	39254.0	1.33217
$955$	0	0
$956$	−56549.5	−1.91312
$957$	66568.9	2.24856
$958$	64316.6	2.16908
$959$	0	0
$960$	0	0
$961$	7498.41	0.251701
$962$	−39116.2	−1.31097
$963$	−325.262	−0.0108841
$964$	−45659.6	−1.52552
$965$	0	0
$966$	0	0
$967$	8971.58	0.298352	0.149176	−	0.988811i	$-0.452338\pi$
$967$	8971.58	0.298352	0.149176	+	0.988811i	$0.452338\pi$
$968$	32982.2	1.09513
$969$	−30183.3	−1.00065
$970$	0	0
$971$	−29275.1	−0.967541	−0.483771	−	0.875195i	$-0.660733\pi$
$971$	−29275.1	−0.967541	−0.483771	+	0.875195i	$0.660733\pi$
$972$	47168.5	1.55651
$973$	0	0
$974$	−24130.5	−0.793831
$975$	0	0
$976$	−1073.47	−0.0352059
$977$	−16477.5	−0.539572	−0.269786	−	0.962920i	$-0.586953\pi$
$977$	−16477.5	−0.539572	−0.269786	+	0.962920i	$0.586953\pi$
$978$	−96075.2	−3.14125
$979$	71094.6	2.32093
$980$	0	0
$981$	−256.275	−0.00834071
$982$	−29817.4	−0.968952
$983$	−45912.9	−1.48972	−0.744859	−	0.667222i	$-0.767483\pi$
$983$	−45912.9	−1.48972	−0.744859	+	0.667222i	$0.767483\pi$
$984$	29048.9	0.941104
$985$	0	0
$986$	61528.0	1.98727
$987$	0	0
$988$	−61923.2	−1.99397
$989$	−51913.1	−1.66910
$990$	0	0
$991$	46124.8	1.47851	0.739255	−	0.673426i	$-0.235178\pi$
$991$	46124.8	1.47851	0.739255	+	0.673426i	$0.235178\pi$
$992$	−37781.0	−1.20922
$993$	8623.22	0.275579
$994$	0	0
$995$	0	0
$996$	−25218.2	−0.802278
$997$	30984.5	0.984240	0.492120	−	0.870527i	$-0.336222\pi$
$997$	30984.5	0.984240	0.492120	+	0.870527i	$0.336222\pi$
$998$	−6837.47	−0.216870
$999$	9037.42	0.286218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.bd.1.3		4
5.4	even	2		1225.4.a.z.1.2		4
7.6	odd	2		175.4.a.h.1.3	yes	4
21.20	even	2		1575.4.a.bg.1.2		4
35.13	even	4		175.4.b.f.99.3		8
35.27	even	4		175.4.b.f.99.6		8
35.34	odd	2		175.4.a.g.1.2	✓	4
105.104	even	2		1575.4.a.bl.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.g.1.2	✓	4	35.34	odd	2
175.4.a.h.1.3	yes	4	7.6	odd	2
175.4.b.f.99.3		8	35.13	even	4
175.4.b.f.99.6		8	35.27	even	4
1225.4.a.z.1.2		4	5.4	even	2
1225.4.a.bd.1.3		4	1.1	even	1	trivial
1575.4.a.bg.1.2		4	21.20	even	2
1575.4.a.bl.1.3		4	105.104	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)

\(f(q)\)	\(=\)	\(q+4.53510 q^{2} -6.46622 q^{3} +12.5671 q^{4} -29.3249 q^{6} +20.7124 q^{8} +14.8119 q^{9} -54.0684 q^{11} -81.2617 q^{12} -75.2159 q^{13} -6.60441 q^{16} +71.2538 q^{17} +67.1736 q^{18} +65.5100 q^{19} -245.206 q^{22} +125.688 q^{23} -133.931 q^{24} -341.111 q^{26} +78.8106 q^{27} +190.405 q^{29} +193.105 q^{31} -195.650 q^{32} +349.618 q^{33} +323.143 q^{34} +186.144 q^{36} +114.673 q^{37} +297.094 q^{38} +486.362 q^{39} -216.896 q^{41} -413.032 q^{43} -679.484 q^{44} +570.007 q^{46} +113.555 q^{47} +42.7055 q^{48} -460.743 q^{51} -945.247 q^{52} +584.366 q^{53} +357.414 q^{54} -423.602 q^{57} +863.504 q^{58} -203.748 q^{59} +162.539 q^{61} +875.749 q^{62} -834.459 q^{64} +1585.55 q^{66} +477.534 q^{67} +895.456 q^{68} -812.725 q^{69} +822.294 q^{71} +306.790 q^{72} +798.993 q^{73} +520.052 q^{74} +823.273 q^{76} +2205.70 q^{78} -468.087 q^{79} -909.529 q^{81} -983.643 q^{82} +310.333 q^{83} -1873.14 q^{86} -1231.20 q^{87} -1119.88 q^{88} -1314.90 q^{89} +1579.53 q^{92} -1248.66 q^{93} +514.985 q^{94} +1265.12 q^{96} +1314.66 q^{97} -800.858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 3 q^{3} + 36 q^{4} - q^{6} + 27 q^{8} + 61 q^{9} + 100 q^{11} + 165 q^{12} - 44 q^{13} + 160 q^{16} + 53 q^{17} + 433 q^{18} + 29 q^{19} - 152 q^{22} + 295 q^{23} + 21 q^{24} - 700 q^{26}+ \cdots + 2383 q^{99}+O(q^{100}) \) 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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