LMFDB - Embedded newform 245.4.a.k.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [245,4,Mod(1,245)] 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("245.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	245.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,8,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.4554679514$
Analytic rank:	$0$
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			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	245.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.58579 q^{2} -6.65685 q^{3} -1.31371 q^{4} +5.00000 q^{5} -17.2132 q^{6} -24.0833 q^{8} +17.3137 q^{9} +12.9289 q^{10} +38.2548 q^{11} +8.74517 q^{12} -19.3431 q^{13} -33.2843 q^{15} -51.7645 q^{16} +87.2254 q^{17} +44.7696 q^{18} +44.2254 q^{19} -6.56854 q^{20} +98.9188 q^{22} +218.167 q^{23} +160.319 q^{24} +25.0000 q^{25} -50.0172 q^{26} +64.4802 q^{27} -46.9411 q^{29} -86.0660 q^{30} -194.558 q^{31} +58.8141 q^{32} -254.657 q^{33} +225.546 q^{34} -22.7452 q^{36} +366.853 q^{37} +114.357 q^{38} +128.765 q^{39} -120.416 q^{40} +339.362 q^{41} -226.167 q^{43} -50.2557 q^{44} +86.5685 q^{45} +564.132 q^{46} -11.6762 q^{47} +344.589 q^{48} +64.6447 q^{50} -580.647 q^{51} +25.4113 q^{52} -209.019 q^{53} +166.732 q^{54} +191.274 q^{55} -294.402 q^{57} -121.380 q^{58} +616.000 q^{59} +43.7258 q^{60} -320.735 q^{61} -503.087 q^{62} +566.197 q^{64} -96.7157 q^{65} -658.488 q^{66} +14.5097 q^{67} -114.589 q^{68} -1452.30 q^{69} -952.000 q^{71} -416.971 q^{72} -824.489 q^{73} +948.603 q^{74} -166.421 q^{75} -58.0993 q^{76} +332.958 q^{78} +156.275 q^{79} -258.823 q^{80} -896.706 q^{81} +877.519 q^{82} +1036.53 q^{83} +436.127 q^{85} -584.818 q^{86} +312.480 q^{87} -921.301 q^{88} +170.225 q^{89} +223.848 q^{90} -286.607 q^{92} +1295.15 q^{93} -30.1921 q^{94} +221.127 q^{95} -391.517 q^{96} -1059.87 q^{97} +662.333 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} - 2 q^{3} + 20 q^{4} + 10 q^{5} + 8 q^{6} + 48 q^{8} + 12 q^{9} + 40 q^{10} - 14 q^{11} + 108 q^{12} - 50 q^{13} - 10 q^{15} + 168 q^{16} + 50 q^{17} + 16 q^{18} - 36 q^{19} + 100 q^{20} - 184 q^{22}+ \cdots + 940 q^{99}+O(q^{100}) $ 2 * q + 8 * q^2 - 2 * q^3 + 20 * q^4 + 10 * q^5 + 8 * q^6 + 48 * q^8 + 12 * q^9 + 40 * q^10 - 14 * q^11 + 108 * q^12 - 50 * q^13 - 10 * q^15 + 168 * q^16 + 50 * q^17 + 16 * q^18 - 36 * q^19 + 100 * q^20 - 184 * q^22 + 244 * q^23 + 496 * q^24 + 50 * q^25 - 216 * q^26 - 86 * q^27 - 26 * q^29 + 40 * q^30 + 120 * q^31 + 672 * q^32 - 498 * q^33 + 24 * q^34 - 136 * q^36 + 564 * q^37 - 320 * q^38 - 14 * q^39 + 240 * q^40 + 328 * q^41 - 260 * q^43 - 1164 * q^44 + 60 * q^45 + 704 * q^46 + 350 * q^47 + 1368 * q^48 + 200 * q^50 - 754 * q^51 - 628 * q^52 - 56 * q^53 - 648 * q^54 - 70 * q^55 - 668 * q^57 - 8 * q^58 + 1232 * q^59 + 540 * q^60 - 336 * q^61 + 1200 * q^62 + 2128 * q^64 - 250 * q^65 - 1976 * q^66 - 152 * q^67 - 908 * q^68 - 1332 * q^69 - 1904 * q^71 - 800 * q^72 - 676 * q^73 + 2016 * q^74 - 50 * q^75 - 1768 * q^76 - 440 * q^78 + 1014 * q^79 + 840 * q^80 - 1454 * q^81 + 816 * q^82 + 376 * q^83 + 250 * q^85 - 768 * q^86 + 410 * q^87 - 4688 * q^88 + 216 * q^89 + 80 * q^90 + 264 * q^92 + 2760 * q^93 + 1928 * q^94 - 180 * q^95 + 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	$-0.348886\pi$
$2$	2.58579	0.914214	0.457107	+	0.889412i	$0.348886\pi$
$3$	−6.65685	−1.28111	−0.640556	−	0.767911i	$-0.721296\pi$
$3$	−6.65685	−1.28111	−0.640556	+	0.767911i	$0.721296\pi$
$4$	−1.31371	−0.164214
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−17.2132	−1.17121
$7$	0	0
$7$	0	0
$8$	−24.0833	−1.06434
$9$	17.3137	0.641248
$10$	12.9289	0.408849
$11$	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	$-0.566155\pi$
$13$	−19.3431	−0.412679	−0.206339	+	0.978480i	$0.566155\pi$
$14$	0	0
$15$	−33.2843	−0.572931
$16$	−51.7645	−0.808820
$17$	87.2254	1.24443	0.622214	−	0.782847i	$-0.286233\pi$
$17$	87.2254	1.24443	0.622214	+	0.782847i	$0.286233\pi$
$18$	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	$0.413968\pi$
$20$	−6.56854	−0.0734385
$21$	0	0
$22$	98.9188	0.958617
$23$	218.167	1.97786	0.988932	−	0.148371i	$-0.0474028\pi$
$23$	218.167	1.97786	0.988932	+	0.148371i	$0.0474028\pi$
$24$	160.319	1.36354
$25$	25.0000	0.200000
$26$	−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	$0.548020\pi$
$30$	−86.0660	−0.523781
$31$	−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	$-0.196735\pi$
$37$	366.853	1.63001	0.815003	+	0.579457i	$0.196735\pi$
$38$	114.357	0.488190
$39$	128.765	0.528688
$40$	−120.416	−0.475987
$41$	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	$-0.631354\pi$
$43$	−226.167	−0.802095	−0.401047	+	0.916057i	$0.631354\pi$
$44$	−50.2557	−0.172189
$45$	86.5685	0.286775
$46$	564.132	1.80819
$47$	−11.6762	−0.0362372	−0.0181186	−	0.999836i	$-0.505768\pi$
$47$	−11.6762	−0.0362372	−0.0181186	+	0.999836i	$0.505768\pi$
$48$	344.589	1.03619
$49$	0	0
$50$	64.6447	0.182843
$51$	−580.647	−1.59425
$52$	25.4113	0.0677674
$53$	−209.019	−0.541717	−0.270859	−	0.962619i	$-0.587308\pi$
$53$	−209.019	−0.541717	−0.270859	+	0.962619i	$0.587308\pi$
$54$	166.732	0.420173
$55$	191.274	0.468935
$56$	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	$0.262140\pi$
$60$	43.7258	0.0940830
$61$	−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
$65$	−96.7157	−0.184556
$66$	−658.488	−1.22810
$67$	14.5097	0.0264573	0.0132286	−	0.999912i	$-0.495789\pi$
$67$	14.5097	0.0264573	0.0132286	+	0.999912i	$0.495789\pi$
$68$	−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	$-0.729848\pi$
$73$	−824.489	−1.32191	−0.660953	+	0.750427i	$0.729848\pi$
$74$	948.603	1.49017
$75$	−166.421	−0.256222
$76$	−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	$0.464505\pi$
$80$	−258.823	−0.361715
$81$	−896.706	−1.23005
$82$	877.519	1.18178
$83$	1036.53	1.37077	0.685384	−	0.728182i	$-0.259634\pi$
$83$	1036.53	1.37077	0.685384	+	0.728182i	$0.259634\pi$
$84$	0	0
$85$	436.127	0.556525
$86$	−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	$0.467677\pi$
$90$	223.848	0.262174
$91$	0	0
$92$	−286.607	−0.324792
$93$	1295.15	1.44409
$94$	−30.1921	−0.0331285
$95$	221.127	0.238812
$96$	−391.517	−0.416240
$97$	−1059.87	−1.10942	−0.554710	−	0.832044i	$-0.687171\pi$
$97$	−1059.87	−1.10942	−0.554710	+	0.832044i	$0.687171\pi$
$98$	0	0
$99$	662.333	0.672394
$100$	−32.8427	−0.0328427
$101$	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	$-0.203040\pi$
$103$	1679.58	1.60673	0.803367	+	0.595484i	$0.203040\pi$
$104$	465.846	0.439230
$105$	0	0
$106$	−540.479	−0.495245
$107$	1506.88	1.36146	0.680728	−	0.732537i	$-0.261664\pi$
$107$	1506.88	1.36146	0.680728	+	0.732537i	$0.261664\pi$
$108$	−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	$0.685476\pi$
$110$	494.594	0.428706
$111$	−2442.09	−2.08822
$112$	0	0
$113$	1370.20	1.14069	0.570345	−	0.821405i	$-0.306810\pi$
$113$	1370.20	1.14069	0.570345	+	0.821405i	$0.306810\pi$
$114$	−761.261	−0.625426
$115$	1090.83	0.884528
$116$	61.6670	0.0493589
$117$	−334.902	−0.264630
$118$	1592.84	1.24265
$119$	0	0
$120$	801.594	0.609793
$121$	132.432	0.0994984
$122$	−829.352	−0.615459
$123$	−2259.09	−1.65606
$124$	255.593	0.185104
$125$	125.000	0.0894427
$126$	0	0
$127$	1213.49	0.847873	0.423936	−	0.905692i	$-0.360648\pi$
$127$	1213.49	0.847873	0.423936	+	0.905692i	$0.360648\pi$
$128$	993.551	0.686081
$129$	1505.56	1.02757
$130$	−250.086	−0.168723
$131$	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
$135$	322.401	0.205540
$136$	−2100.67	−1.32449
$137$	2210.95	1.37879	0.689394	−	0.724386i	$-0.257877\pi$
$137$	2210.95	1.37879	0.689394	+	0.724386i	$0.257877\pi$
$138$	−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
$145$	−234.706	−0.134422
$146$	−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	$0.528774\pi$
$150$	−430.330	−0.234242
$151$	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
$155$	−972.792	−0.504107
$156$	−169.159	−0.0868177
$157$	−525.098	−0.266926	−0.133463	−	0.991054i	$-0.542610\pi$
$157$	−525.098	−0.266926	−0.133463	+	0.991054i	$0.542610\pi$
$158$	404.094	0.203468
$159$	1391.41	0.694001
$160$	294.071	0.145302
$161$	0	0
$162$	−2318.69	−1.12453
$163$	1002.63	0.481790	0.240895	−	0.970551i	$-0.422559\pi$
$163$	1002.63	0.481790	0.240895	+	0.970551i	$0.422559\pi$
$164$	−445.823	−0.212274
$165$	−1273.28	−0.600758
$166$	2680.24	1.25317
$167$	359.422	0.166544	0.0832722	−	0.996527i	$-0.473463\pi$
$167$	359.422	0.166544	0.0832722	+	0.996527i	$0.473463\pi$
$168$	0	0
$169$	−1822.84	−0.829696
$170$	1127.73	0.508783
$171$	765.706	0.342427
$172$	297.117	0.131715
$173$	−3293.65	−1.44747	−0.723733	−	0.690080i	$-0.757575\pi$
$173$	−3293.65	−1.44747	−0.723733	+	0.690080i	$0.757575\pi$
$174$	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	$0.286353\pi$
$180$	−113.726	−0.0470923
$181$	−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
$185$	1834.26	0.728961
$186$	3348.97	1.32021
$187$	3336.79	1.30487
$188$	15.3391	0.00595064
$189$	0	0
$190$	571.787	0.218325
$191$	−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	$-0.456352\pi$
$193$	733.028	0.273391	0.136696	+	0.990613i	$0.456352\pi$
$194$	−2740.61	−1.01425
$195$	643.823	0.236436
$196$	0	0
$197$	−2093.24	−0.757043	−0.378521	−	0.925593i	$-0.623567\pi$
$197$	−2093.24	−0.757043	−0.378521	+	0.925593i	$0.623567\pi$
$198$	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	$0.670464\pi$
$200$	−602.082	−0.212868
$201$	−96.5887	−0.0338948
$202$	625.330	0.217812
$203$	0	0
$204$	762.801	0.261798
$205$	1696.81	0.578100
$206$	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
$215$	−1130.83	−0.358708
$216$	−1552.89	−0.489172
$217$	0	0
$218$	−3238.46	−1.00613
$219$	5488.51	1.69351
$220$	−251.279	−0.0770054
$221$	−1687.21	−0.513549
$222$	−6314.71	−1.90908
$223$	6369.16	1.91260	0.956302	−	0.292381i	$-0.0944477\pi$
$223$	6369.16	1.91260	0.956302	+	0.292381i	$0.0944477\pi$
$224$	0	0
$225$	432.843	0.128250
$226$	3543.05	1.04283
$227$	1015.67	0.296972	0.148486	−	0.988914i	$-0.452560\pi$
$227$	1015.67	0.296972	0.148486	+	0.988914i	$0.452560\pi$
$228$	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	$0.701965\pi$
$230$	2820.66	0.808647
$231$	0	0
$232$	1130.50	0.319917
$233$	608.431	0.171071	0.0855357	−	0.996335i	$-0.472740\pi$
$233$	608.431	0.171071	0.0855357	+	0.996335i	$0.472740\pi$
$234$	−865.984	−0.241928
$235$	−58.3810	−0.0162058
$236$	−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	$0.739755\pi$
$240$	1722.94	0.463398
$241$	−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
$250$	323.223	0.0817697
$251$	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
$255$	−2903.23	−0.712971
$256$	−1960.46	−0.478629
$257$	1774.61	0.430729	0.215364	−	0.976534i	$-0.430906\pi$
$257$	1774.61	0.430729	0.215364	+	0.976534i	$0.430906\pi$
$258$	3893.05	0.939421
$259$	0	0
$260$	127.056	0.0303065
$261$	−812.725	−0.192745
$262$	5126.11	1.20875
$263$	−1199.09	−0.281138	−0.140569	−	0.990071i	$-0.544893\pi$
$263$	−1199.09	−0.281138	−0.140569	+	0.990071i	$0.544893\pi$
$264$	6132.97	1.42977
$265$	−1045.10	−0.242263
$266$	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	$0.620078\pi$
$270$	833.661	0.187907
$271$	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
$275$	956.371	0.209714
$276$	1907.90	0.416095
$277$	−386.562	−0.0838492	−0.0419246	−	0.999121i	$-0.513349\pi$
$277$	−386.562	−0.0838492	−0.0419246	+	0.999121i	$0.513349\pi$
$278$	−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	$-0.692445\pi$
$283$	−5412.26	−1.13684	−0.568419	+	0.822739i	$0.692445\pi$
$284$	1250.65	0.261311
$285$	−1472.01	−0.305945
$286$	−1913.40	−0.395601
$287$	0	0
$288$	1018.29	0.208345
$289$	2695.27	0.548600
$290$	−606.899	−0.122891
$291$	7055.42	1.42129
$292$	1083.14	0.217075
$293$	−282.211	−0.0562695	−0.0281347	−	0.999604i	$-0.508957\pi$
$293$	−282.211	−0.0562695	−0.0281347	+	0.999604i	$0.508957\pi$
$294$	0	0
$295$	3080.00	0.607880
$296$	−8835.01	−1.73488
$297$	2466.68	0.481924
$298$	−849.099	−0.165057
$299$	−4220.03	−0.816222
$300$	218.629	0.0420752
$301$	0	0
$302$	2661.88	0.507199
$303$	−1609.85	−0.305226
$304$	−2289.31	−0.431910
$305$	−1603.68	−0.301069
$306$	3905.04	0.729531
$307$	−1919.67	−0.356878	−0.178439	−	0.983951i	$-0.557105\pi$
$307$	−1919.67	−0.356878	−0.178439	+	0.983951i	$0.557105\pi$
$308$	0	0
$309$	−11180.7	−2.05841
$310$	−2515.43	−0.460861
$311$	−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	$-0.458669\pi$
$313$	1434.00	0.258960	0.129480	+	0.991582i	$0.458669\pi$
$314$	−1357.79	−0.244028
$315$	0	0
$316$	−205.300	−0.0365475
$317$	6496.95	1.15112	0.575560	−	0.817760i	$-0.304784\pi$
$317$	6496.95	1.15112	0.575560	+	0.817760i	$0.304784\pi$
$318$	3597.89	0.634465
$319$	−1795.72	−0.315176
$320$	2830.98	0.494553
$321$	−10031.1	−1.74418
$322$	0	0
$323$	3857.58	0.664524
$324$	1178.01	0.201991
$325$	−483.579	−0.0825357
$326$	2592.58	0.440459
$327$	8337.11	1.40992
$328$	−8172.96	−1.37584
$329$	0	0
$330$	−3292.44	−0.549221
$331$	−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
$335$	72.5483	0.0118321
$336$	0	0
$337$	29.1319	0.00470895	0.00235447	−	0.999997i	$-0.499251\pi$
$337$	29.1319	0.00470895	0.00235447	+	0.999997i	$0.499251\pi$
$338$	−4713.48	−0.758520
$339$	−9121.24	−1.46135
$340$	−572.944	−0.0913889
$341$	−7442.80	−1.18197
$342$	1979.95	0.313051
$343$	0	0
$344$	5446.83	0.853701
$345$	−7261.51	−1.13318
$346$	−8516.68	−1.32329
$347$	−7848.58	−1.21422	−0.607110	−	0.794618i	$-0.707671\pi$
$347$	−7848.58	−1.21422	−0.607110	+	0.794618i	$0.707671\pi$
$348$	−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	$-0.567699\pi$
$353$	−2799.93	−0.422168	−0.211084	+	0.977468i	$0.567699\pi$
$354$	−10603.3	−1.59198
$355$	−4760.00	−0.711647
$356$	−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	$0.574698\pi$
$360$	−2084.85	−0.305226
$361$	−4903.11	−0.714844
$362$	−3781.23	−0.548998
$363$	−881.583	−0.127469
$364$	0	0
$365$	−4122.45	−0.591175
$366$	5520.88	0.788472
$367$	−3182.85	−0.452706	−0.226353	−	0.974045i	$-0.572680\pi$
$367$	−3182.85	−0.452706	−0.226353	+	0.974045i	$0.572680\pi$
$368$	−11293.3	−1.59974
$369$	5875.62	0.828923
$370$	4743.02	0.666426
$371$	0	0
$372$	−1701.45	−0.237139
$373$	−2615.14	−0.363021	−0.181510	−	0.983389i	$-0.558099\pi$
$373$	−2615.14	−0.363021	−0.181510	+	0.983389i	$0.558099\pi$
$374$	8628.23	1.19293
$375$	−832.107	−0.114586
$376$	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	$0.514510\pi$
$380$	−290.496	−0.0392162
$381$	−8078.03	−1.08622
$382$	−969.470	−0.129849
$383$	−1169.86	−0.156075	−0.0780377	−	0.996950i	$-0.524865\pi$
$383$	−1169.86	−0.156075	−0.0780377	+	0.996950i	$0.524865\pi$
$384$	−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	$0.523300\pi$
$390$	1664.79	0.216153
$391$	19029.7	2.46131
$392$	0	0
$393$	−13196.7	−1.69385
$394$	−5412.68	−0.692099
$395$	781.375	0.0995323
$396$	−870.113	−0.110416
$397$	1985.93	0.251060	0.125530	−	0.992090i	$-0.459937\pi$
$397$	1985.93	0.251060	0.125530	+	0.992090i	$0.459937\pi$
$398$	−7408.38	−0.933037
$399$	0	0
$400$	−1294.11	−0.161764
$401$	−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
$405$	−4483.53	−0.550095
$406$	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	$0.249915\pi$
$410$	4387.59	0.528507
$411$	−14718.0	−1.76638
$412$	−2206.47	−0.263848
$413$	0	0
$414$	9767.22	1.15950
$415$	5182.64	0.613026
$416$	−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
$425$	2180.63	0.248885
$426$	16387.0	1.86374
$427$	0	0
$428$	−1979.60	−0.223569
$429$	4925.86	0.554366
$430$	−2924.09	−0.327935
$431$	−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	$-0.646855\pi$
$433$	−8022.03	−0.890333	−0.445166	+	0.895448i	$0.646855\pi$
$434$	0	0
$435$	1562.40	0.172210
$436$	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	$0.402087\pi$
$440$	−4606.51	−0.499106
$441$	0	0
$442$	−4362.77	−0.469493
$443$	−5486.21	−0.588392	−0.294196	−	0.955745i	$-0.595052\pi$
$443$	−5486.21	−0.588392	−0.294196	+	0.955745i	$0.595052\pi$
$444$	3208.19	0.342914
$445$	851.127	0.0906681
$446$	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	$0.624111\pi$
$450$	1119.24	0.117248
$451$	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	$-0.547427\pi$
$457$	−2900.51	−0.296893	−0.148446	+	0.988920i	$0.547427\pi$
$458$	−10623.3	−1.08383
$459$	5624.31	0.571940
$460$	−1433.04	−0.145251
$461$	−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	$-0.898605\pi$
$463$	−18922.8	−1.89939	−0.949693	+	0.313183i	$0.898605\pi$
$464$	2429.88	0.243113
$465$	6475.74	0.645817
$466$	1573.27	0.156396
$467$	6776.71	0.671496	0.335748	−	0.941952i	$-0.391011\pi$
$467$	6776.71	0.671496	0.335748	+	0.941952i	$0.391011\pi$
$468$	439.963	0.0434558
$469$	0	0
$470$	−150.961	−0.0148155
$471$	3495.50	0.341962
$472$	−14835.3	−1.44672
$473$	−8651.96	−0.841052
$474$	−2689.99	−0.260666
$475$	1105.63	0.106800
$476$	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	$0.536475\pi$
$480$	−1957.59	−0.186148
$481$	−7096.09	−0.672669
$482$	−12.5871	−0.00118948
$483$	0	0
$484$	−173.977	−0.0163390
$485$	−5299.37	−0.496148
$486$	10933.4	1.02047
$487$	5586.17	0.519781	0.259890	−	0.965638i	$-0.416314\pi$
$487$	5586.17	0.519781	0.259890	+	0.965638i	$0.416314\pi$
$488$	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
$495$	3311.67	0.300704
$496$	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	$0.491447\pi$
$500$	−164.214	−0.0146877
$501$	−2392.62	−0.213362
$502$	1416.81	0.125966
$503$	−4426.76	−0.392405	−0.196202	−	0.980563i	$-0.562861\pi$
$503$	−4426.76	−0.392405	−0.196202	+	0.980563i	$0.562861\pi$
$504$	0	0
$505$	1209.17	0.106549
$506$	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	$0.780679\pi$
$510$	−7507.14	−0.651808
$511$	0	0
$512$	−13017.7	−1.12365
$513$	2851.66	0.245427
$514$	4588.77	0.393778
$515$	8397.88	0.718553
$516$	−1977.86	−0.168741
$517$	−446.671	−0.0379972
$518$	0	0
$519$	21925.4	1.85437
$520$	2329.23	0.196430
$521$	−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	$-0.394692\pi$
$523$	7770.40	0.649667	0.324833	+	0.945771i	$0.394692\pi$
$524$	−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
$530$	−2702.40	−0.221480
$531$	10665.2	0.871624
$532$	0	0
$533$	−6564.34	−0.533458
$534$	−2930.12	−0.237451
$535$	7534.41	0.608861
$536$	−349.440	−0.0281595
$537$	−19829.6	−1.59350
$538$	−8404.56	−0.673506
$539$	0	0
$540$	−423.541	−0.0337524
$541$	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
$545$	−6262.05	−0.492177
$546$	0	0
$547$	7489.29	0.585409	0.292705	−	0.956203i	$-0.405445\pi$
$547$	7489.29	0.585409	0.292705	+	0.956203i	$0.405445\pi$
$548$	−2904.54	−0.226416
$549$	−5553.11	−0.431696
$550$	2472.97	0.191723
$551$	−2075.99	−0.160508
$552$	34976.2	2.69689
$553$	0	0
$554$	−999.566	−0.0766561
$555$	−12210.4	−0.933881
$556$	693.689	0.0529118
$557$	25297.9	1.92443	0.962214	−	0.272295i	$-0.0877826\pi$
$557$	25297.9	1.92443	0.962214	+	0.272295i	$0.0877826\pi$
$558$	−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	$-0.699371\pi$
$563$	−15661.3	−1.17237	−0.586186	+	0.810177i	$0.699371\pi$
$564$	−102.110	−0.00762343
$565$	6851.02	0.510132
$566$	−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	$0.619871\pi$
$570$	−3806.30	−0.279699
$571$	−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
$575$	5454.16	0.395573
$576$	9802.97	0.709127
$577$	595.378	0.0429565	0.0214783	−	0.999769i	$-0.493163\pi$
$577$	595.378	0.0429565	0.0214783	+	0.999769i	$0.493163\pi$
$578$	6969.39	0.501537
$579$	−4879.66	−0.350245
$580$	308.335	0.0220740
$581$	0	0
$582$	18243.8	1.29936
$583$	−7996.00	−0.568028
$584$	19856.4	1.40696
$585$	−1674.51	−0.118346
$586$	−729.738	−0.0514423
$587$	15750.3	1.10747	0.553736	−	0.832693i	$-0.313202\pi$
$587$	15750.3	1.10747	0.553736	+	0.832693i	$0.313202\pi$
$588$	0	0
$589$	−8604.42	−0.601934
$590$	7964.22	0.555732
$591$	13934.4	0.969856
$592$	−18990.0	−1.31838
$593$	417.878	0.0289379	0.0144690	−	0.999895i	$-0.495394\pi$
$593$	417.878	0.0289379	0.0144690	+	0.999895i	$0.495394\pi$
$594$	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	$0.738902\pi$
$600$	4007.97	0.272708
$601$	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
$605$	662.162	0.0444970
$606$	−4162.73	−0.279042
$607$	−14159.2	−0.946793	−0.473396	−	0.880850i	$-0.656972\pi$
$607$	−14159.2	−0.946793	−0.473396	+	0.880850i	$0.656972\pi$
$608$	2601.08	0.173499
$609$	0	0
$610$	−4146.76	−0.275242
$611$	225.854	0.0149543
$612$	−1983.96	−0.131040
$613$	−4629.41	−0.305025	−0.152512	−	0.988302i	$-0.548736\pi$
$613$	−4629.41	−0.305025	−0.152512	+	0.988302i	$0.548736\pi$
$614$	−4963.87	−0.326263
$615$	−11295.4	−0.740611
$616$	0	0
$617$	−23165.3	−1.51151	−0.755753	−	0.654857i	$-0.772729\pi$
$617$	−23165.3	−1.51151	−0.755753	+	0.654857i	$0.772729\pi$
$618$	−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	$0.631556\pi$
$620$	1277.97	0.0827812
$621$	14067.4	0.909028
$622$	−3137.36	−0.202245
$623$	0	0
$624$	−6665.43	−0.427613
$625$	625.000	0.0400000
$626$	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
$635$	6067.45	0.379180
$636$	−1827.91	−0.113964
$637$	0	0
$638$	−4643.36	−0.288139
$639$	−16482.7	−1.02041
$640$	4967.75	0.306825
$641$	−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	$-0.455913\pi$
$643$	4502.17	0.276125	0.138063	+	0.990424i	$0.455913\pi$
$644$	0	0
$645$	7527.79	0.459545
$646$	9974.87	0.607517
$647$	−29414.8	−1.78735	−0.893675	−	0.448715i	$-0.851882\pi$
$647$	−29414.8	−1.78735	−0.893675	+	0.448715i	$0.851882\pi$
$648$	21595.6	1.30919
$649$	23565.0	1.42528
$650$	−1250.43	−0.0754553
$651$	0	0
$652$	−1317.16	−0.0791164
$653$	13013.6	0.779882	0.389941	−	0.920840i	$-0.372495\pi$
$653$	13013.6	0.779882	0.389941	+	0.920840i	$0.372495\pi$
$654$	21558.0	1.28897
$655$	9912.09	0.591294
$656$	−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	$0.255937\pi$
$660$	1672.72	0.0986526
$661$	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
$670$	187.595	0.0108170
$671$	−12269.7	−0.705909
$672$	0	0
$673$	−25067.2	−1.43576	−0.717882	−	0.696164i	$-0.754888\pi$
$673$	−25067.2	−1.43576	−0.717882	+	0.696164i	$0.754888\pi$
$674$	75.3288	0.00430498
$675$	1612.01	0.0919202
$676$	2394.68	0.136247
$677$	22409.6	1.27219	0.636093	−	0.771613i	$-0.280550\pi$
$677$	22409.6	1.27219	0.636093	+	0.771613i	$0.280550\pi$
$678$	−23585.6	−1.33599
$679$	0	0
$680$	−10503.4	−0.592332
$681$	−6761.20	−0.380455
$682$	−19245.5	−1.08057
$683$	−8757.53	−0.490626	−0.245313	−	0.969444i	$-0.578891\pi$
$683$	−8757.53	−0.490626	−0.245313	+	0.969444i	$0.578891\pi$
$684$	−1005.91	−0.0562311
$685$	11054.7	0.616613
$686$	0	0
$687$	27348.7	1.51880
$688$	11707.4	0.648750
$689$	4043.09	0.223555
$690$	−18776.7	−1.03597
$691$	−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
$695$	−2640.19	−0.144098
$696$	−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
$705$	388.633	0.0207614
$706$	−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	$0.323268\pi$
$710$	−12308.3	−0.650597
$711$	2705.70	0.142717
$712$	−4099.58	−0.215784
$713$	−42446.1	−2.22948
$714$	0	0
$715$	−3699.84	−0.193519
$716$	−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	$0.407284\pi$
$720$	−4481.18	−0.231949
$721$	0	0
$722$	−12678.4	−0.653520
$723$	32.4043	0.00166685
$724$	1921.06	0.0986125
$725$	−1173.53	−0.0601155
$726$	−2279.58	−0.116533
$727$	−31652.7	−1.61476	−0.807382	−	0.590029i	$-0.799116\pi$
$727$	−31652.7	−1.61476	−0.807382	+	0.590029i	$0.799116\pi$
$728$	0	0
$729$	−3935.94	−0.199967
$730$	−10659.8	−0.540460
$731$	−19727.5	−0.998149
$732$	−2804.88	−0.141628
$733$	−16958.3	−0.854528	−0.427264	−	0.904127i	$-0.640523\pi$
$733$	−16958.3	−0.854528	−0.427264	+	0.904127i	$0.640523\pi$
$734$	−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	$0.593364\pi$
$740$	−2409.69	−0.119705
$741$	5694.66	0.282319
$742$	0	0
$743$	15928.0	0.786464	0.393232	−	0.919439i	$-0.371357\pi$
$743$	15928.0	0.786464	0.393232	+	0.919439i	$0.371357\pi$
$744$	−31191.4	−1.53700
$745$	−1641.86	−0.0807423
$746$	−6762.19	−0.331879
$747$	17946.1	0.879003
$748$	−4383.57	−0.214277
$749$	0	0
$750$	−2151.65	−0.104756
$751$	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
$755$	5147.14	0.248111
$756$	0	0
$757$	6202.41	0.297794	0.148897	−	0.988853i	$-0.452428\pi$
$757$	6202.41	0.297794	0.148897	+	0.988853i	$0.452428\pi$
$758$	−1738.77	−0.0833179
$759$	−55557.6	−2.65693
$760$	−5325.46	−0.254177
$761$	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
$765$	7550.98	0.356871
$766$	−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	$-0.297431\pi$
$773$	25544.8	1.18859	0.594296	+	0.804246i	$0.297431\pi$
$774$	−10125.4	−0.470218
$775$	−4863.96	−0.225443
$776$	25525.2	1.18080
$777$	0	0
$778$	−2901.82	−0.133721
$779$	15008.4	0.690286
$780$	−845.795	−0.0388261
$781$	−36418.6	−1.66858
$782$	49206.6	2.25016
$783$	−3026.77	−0.138146
$784$	0	0
$785$	−2625.49	−0.119373
$786$	−34123.8	−1.54854
$787$	37223.2	1.68598	0.842989	−	0.537931i	$-0.180794\pi$
$787$	37223.2	1.68598	0.842989	+	0.537931i	$0.180794\pi$
$788$	2749.91	0.124317
$789$	7982.19	0.360169
$790$	2020.47	0.0909938
$791$	0	0
$792$	−15951.1	−0.715655
$793$	6204.03	0.277820
$794$	5135.18	0.229522
$795$	6957.06	0.310366
$796$	3763.83	0.167595
$797$	40384.6	1.79485	0.897425	−	0.441168i	$-0.145436\pi$
$797$	40384.6	1.79485	0.897425	+	0.441168i	$0.145436\pi$
$798$	0	0
$799$	−1018.46	−0.0450945
$800$	1470.35	0.0649811
$801$	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	$0.513531\pi$
$810$	−11593.4	−0.502904
$811$	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
$815$	5013.13	0.215463
$816$	30056.9	1.28946
$817$	−10002.3	−0.428319
$818$	30255.8	1.29324
$819$	0	0
$820$	−2229.12	−0.0949319
$821$	−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	$-0.650620\pi$
$823$	−21519.5	−0.911449	−0.455724	+	0.890121i	$0.650620\pi$
$824$	−40449.7	−1.71011
$825$	−6366.42	−0.268667
$826$	0	0
$827$	35220.6	1.48094	0.740471	−	0.672088i	$-0.234602\pi$
$827$	35220.6	1.48094	0.740471	+	0.672088i	$0.234602\pi$
$828$	−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	$0.728191\pi$
$830$	13401.2	0.560437
$831$	2573.28	0.107420
$832$	−10952.0	−0.456362
$833$	0	0
$834$	9089.24	0.377380
$835$	1797.11	0.0744810
$836$	−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
$845$	−9114.21	−0.371051
$846$	−522.738	−0.0212436
$847$	0	0
$848$	10819.8	0.438152
$849$	36028.6	1.45642
$850$	5638.66	0.227534
$851$	80035.0	3.22393
$852$	−8325.40	−0.334769
$853$	−9405.41	−0.377533	−0.188766	−	0.982022i	$-0.560449\pi$
$853$	−9405.41	−0.377533	−0.188766	+	0.982022i	$0.560449\pi$
$854$	0	0
$855$	3828.53	0.153138
$856$	−36290.6	−1.44905
$857$	27966.9	1.11474	0.557369	−	0.830265i	$-0.311811\pi$
$857$	27966.9	1.11474	0.557369	+	0.830265i	$0.311811\pi$
$858$	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	$0.539811\pi$
$860$	1485.58	0.0589047
$861$	0	0
$862$	−16624.7	−0.656890
$863$	−4757.13	−0.187642	−0.0938208	−	0.995589i	$-0.529908\pi$
$863$	−4757.13	−0.187642	−0.0938208	+	0.995589i	$0.529908\pi$
$864$	3792.35	0.149327
$865$	−16468.3	−0.647327
$866$	−20743.3	−0.813954
$867$	−17942.0	−0.702818
$868$	0	0
$869$	5978.28	0.233371
$870$	4040.04	0.157437
$871$	−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	$-0.697802\pi$
$877$	−30240.5	−1.16437	−0.582184	+	0.813057i	$0.697802\pi$
$878$	14402.5	0.553601
$879$	1878.64	0.0720875
$880$	−9901.21	−0.379284
$881$	−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	$-0.470283\pi$
$883$	4892.13	0.186448	0.0932238	+	0.995645i	$0.470283\pi$
$884$	2216.51	0.0843317
$885$	−20503.1	−0.778762
$886$	−14186.2	−0.537916
$887$	−1761.40	−0.0666765	−0.0333382	−	0.999444i	$-0.510614\pi$
$887$	−1761.40	−0.0666765	−0.0333382	+	0.999444i	$0.510614\pi$
$888$	58813.4	2.22258
$889$	0	0
$890$	2200.83	0.0828900
$891$	−34303.3	−1.28979
$892$	−8367.22	−0.314075
$893$	−516.384	−0.0193507
$894$	5652.33	0.211457
$895$	14894.1	0.556263
$896$	0	0
$897$	28092.1	1.04567
$898$	−18702.2	−0.694988
$899$	9132.79	0.338816
$900$	−568.629	−0.0210603
$901$	−18231.8	−0.674128
$902$	33569.3	1.23918
$903$	0	0
$904$	−32999.0	−1.21408
$905$	−7311.57	−0.268558
$906$	−17719.8	−0.649779
$907$	23689.1	0.867238	0.433619	−	0.901096i	$-0.357236\pi$
$907$	23689.1	0.867238	0.433619	+	0.901096i	$0.357236\pi$
$908$	−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
$915$	10675.4	0.385704
$916$	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	$0.417196\pi$
$920$	−26270.8	−0.941438
$921$	12779.0	0.457201
$922$	−15705.0	−0.560971
$923$	18414.7	0.656692
$924$	0	0
$925$	9171.32	0.326001
$926$	−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	$0.595098\pi$
$930$	16744.9	0.590415
$931$	0	0
$932$	−799.300	−0.0280922
$933$	8076.82	0.283412
$934$	17523.1	0.613891
$935$	16684.0	0.583555
$936$	8065.52	0.281656
$937$	−30384.9	−1.05937	−0.529685	−	0.848194i	$-0.677690\pi$
$937$	−30384.9	−1.05937	−0.529685	+	0.848194i	$0.677690\pi$
$938$	0	0
$939$	−9545.94	−0.331757
$940$	76.6956	0.00266121
$941$	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	$-0.490231\pi$
$947$	1788.41	0.0613681	0.0306840	+	0.999529i	$0.490231\pi$
$948$	1366.65	0.0468215
$949$	15948.2	0.545523
$950$	2858.94	0.0976380
$951$	−43249.2	−1.47471
$952$	0	0
$953$	8578.60	0.291593	0.145796	−	0.989315i	$-0.453426\pi$
$953$	8578.60	0.291593	0.145796	+	0.989315i	$0.453426\pi$
$954$	−9357.70	−0.317575
$955$	−1874.61	−0.0635194
$956$	6640.06	0.224639
$957$	11953.9	0.403776
$958$	−6198.95	−0.209059
$959$	0	0
$960$	−18845.4	−0.633577
$961$	8061.99	0.270618
$962$	−18349.0	−0.614963
$963$	26089.7	0.873031
$964$	6.39489	0.000213657 0
$965$	3665.14	0.122264
$966$	0	0
$967$	−55459.3	−1.84431	−0.922156	−	0.386818i	$-0.873574\pi$
$967$	−55459.3	−1.84431	−0.922156	+	0.386818i	$0.873574\pi$
$968$	−3189.40	−0.105900
$969$	−25679.3	−0.851330
$970$	−13703.0	−0.453585
$971$	−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
$975$	3219.11	0.105738
$976$	16602.7	0.544507
$977$	14402.3	0.471617	0.235809	−	0.971800i	$-0.424226\pi$
$977$	14402.3	0.471617	0.235809	+	0.971800i	$0.424226\pi$
$978$	−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	$-0.540477\pi$
$983$	−7817.11	−0.253639	−0.126819	+	0.991926i	$0.540477\pi$
$984$	54406.2	1.76261
$985$	−10466.2	−0.338560
$986$	−10587.4	−0.341959
$987$	0	0
$988$	1123.82	0.0361878
$989$	−49342.0	−1.58643
$990$	8563.26	0.274907
$991$	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
$995$	−14325.2	−0.456421
$996$	9064.61	0.288377
$997$	50696.0	1.61039	0.805195	−	0.593010i	$-0.202060\pi$
$997$	50696.0	1.61039	0.805195	+	0.593010i	$0.202060\pi$
$998$	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	245.4.a.k.1.1		2
3.2	odd	2		2205.4.a.u.1.2		2
5.4	even	2		1225.4.a.m.1.2		2
7.2	even	3		245.4.e.i.116.2		4
7.3	odd	6		245.4.e.h.226.2		4
7.4	even	3		245.4.e.i.226.2		4
7.5	odd	6		245.4.e.h.116.2		4
7.6	odd	2		35.4.a.b.1.1	✓	2
21.20	even	2		315.4.a.f.1.2		2
28.27	even	2		560.4.a.r.1.1		2
35.13	even	4		175.4.b.c.99.2		4
35.27	even	4		175.4.b.c.99.3		4
35.34	odd	2		175.4.a.c.1.2		2
56.13	odd	2		2240.4.a.bn.1.1		2
56.27	even	2		2240.4.a.bo.1.2		2
105.104	even	2		1575.4.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.b.1.1	✓	2	7.6	odd	2
175.4.a.c.1.2		2	35.34	odd	2
175.4.b.c.99.2		4	35.13	even	4
175.4.b.c.99.3		4	35.27	even	4
245.4.a.k.1.1		2	1.1	even	1	trivial
245.4.e.h.116.2		4	7.5	odd	6
245.4.e.h.226.2		4	7.3	odd	6
245.4.e.i.116.2		4	7.2	even	3
245.4.e.i.226.2		4	7.4	even	3
315.4.a.f.1.2		2	21.20	even	2
560.4.a.r.1.1		2	28.27	even	2
1225.4.a.m.1.2		2	5.4	even	2
1575.4.a.z.1.1		2	105.104	even	2
2205.4.a.u.1.2		2	3.2	odd	2
2240.4.a.bn.1.1		2	56.13	odd	2
2240.4.a.bo.1.2		2	56.27	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:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

\(f(q)\)	\(=\)	\(q+2.58579 q^{2} -6.65685 q^{3} -1.31371 q^{4} +5.00000 q^{5} -17.2132 q^{6} -24.0833 q^{8} +17.3137 q^{9} +12.9289 q^{10} +38.2548 q^{11} +8.74517 q^{12} -19.3431 q^{13} -33.2843 q^{15} -51.7645 q^{16} +87.2254 q^{17} +44.7696 q^{18} +44.2254 q^{19} -6.56854 q^{20} +98.9188 q^{22} +218.167 q^{23} +160.319 q^{24} +25.0000 q^{25} -50.0172 q^{26} +64.4802 q^{27} -46.9411 q^{29} -86.0660 q^{30} -194.558 q^{31} +58.8141 q^{32} -254.657 q^{33} +225.546 q^{34} -22.7452 q^{36} +366.853 q^{37} +114.357 q^{38} +128.765 q^{39} -120.416 q^{40} +339.362 q^{41} -226.167 q^{43} -50.2557 q^{44} +86.5685 q^{45} +564.132 q^{46} -11.6762 q^{47} +344.589 q^{48} +64.6447 q^{50} -580.647 q^{51} +25.4113 q^{52} -209.019 q^{53} +166.732 q^{54} +191.274 q^{55} -294.402 q^{57} -121.380 q^{58} +616.000 q^{59} +43.7258 q^{60} -320.735 q^{61} -503.087 q^{62} +566.197 q^{64} -96.7157 q^{65} -658.488 q^{66} +14.5097 q^{67} -114.589 q^{68} -1452.30 q^{69} -952.000 q^{71} -416.971 q^{72} -824.489 q^{73} +948.603 q^{74} -166.421 q^{75} -58.0993 q^{76} +332.958 q^{78} +156.275 q^{79} -258.823 q^{80} -896.706 q^{81} +877.519 q^{82} +1036.53 q^{83} +436.127 q^{85} -584.818 q^{86} +312.480 q^{87} -921.301 q^{88} +170.225 q^{89} +223.848 q^{90} -286.607 q^{92} +1295.15 q^{93} -30.1921 q^{94} +221.127 q^{95} -391.517 q^{96} -1059.87 q^{97} +662.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} - 2 q^{3} + 20 q^{4} + 10 q^{5} + 8 q^{6} + 48 q^{8} + 12 q^{9} + 40 q^{10} - 14 q^{11} + 108 q^{12} - 50 q^{13} - 10 q^{15} + 168 q^{16} + 50 q^{17} + 16 q^{18} - 36 q^{19} + 100 q^{20} - 184 q^{22}+ \cdots + 940 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 - 2 * q^3 + 20 * q^4 + 10 * q^5 + 8 * q^6 + 48 * q^8 + 12 * q^9 + 40 * q^10 - 14 * q^11 + 108 * q^12 - 50 * q^13 - 10 * q^15 + 168 * q^16 + 50 * q^17 + 16 * q^18 - 36 * q^19 + 100 * q^20 - 184 * q^22 + 244 * q^23 + 496 * q^24 + 50 * q^25 - 216 * q^26 - 86 * q^27 - 26 * q^29 + 40 * q^30 + 120 * q^31 + 672 * q^32 - 498 * q^33 + 24 * q^34 - 136 * q^36 + 564 * q^37 - 320 * q^38 - 14 * q^39 + 240 * q^40 + 328 * q^41 - 260 * q^43 - 1164 * q^44 + 60 * q^45 + 704 * q^46 + 350 * q^47 + 1368 * q^48 + 200 * q^50 - 754 * q^51 - 628 * q^52 - 56 * q^53 - 648 * q^54 - 70 * q^55 - 668 * q^57 - 8 * q^58 + 1232 * q^59 + 540 * q^60 - 336 * q^61 + 1200 * q^62 + 2128 * q^64 - 250 * q^65 - 1976 * q^66 - 152 * q^67 - 908 * q^68 - 1332 * q^69 - 1904 * q^71 - 800 * q^72 - 676 * q^73 + 2016 * q^74 - 50 * q^75 - 1768 * q^76 - 440 * q^78 + 1014 * q^79 + 840 * q^80 - 1454 * q^81 + 816 * q^82 + 376 * q^83 + 250 * q^85 - 768 * q^86 + 410 * q^87 - 4688 * q^88 + 216 * q^89 + 80 * q^90 + 264 * q^92 + 2760 * q^93 + 1928 * q^94 - 180 * q^95 + 2464 * q^96 - 2742 * q^97 + 940 * q^99

Introduction

L-functions

Modular forms

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Database

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