LMFDB - Embedded newform 225.8.a.z.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,8,Mod(1,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-9,0,333,0,0,1188] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$70.2866307339$
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} - 81x^{2} - 150x + 400 $ x^4 - 81x^2 - 150x + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-7.26440$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.02250 q^{2} -118.865 q^{4} -1505.28 q^{7} -746.147 q^{8} -1596.37 q^{11} -956.100 q^{13} -4549.69 q^{14} +12959.4 q^{16} -32470.6 q^{17} -39168.1 q^{19} -4825.04 q^{22} -59361.5 q^{23} -2889.81 q^{26} +178924. q^{28} -66150.5 q^{29} -19664.1 q^{31} +134677. q^{32} -98142.2 q^{34} +376045. q^{37} -118386. q^{38} -385003. q^{41} +466410. q^{43} +189752. q^{44} -179420. q^{46} -468903. q^{47} +1.44232e6 q^{49} +113646. q^{52} -1.60516e6 q^{53} +1.12316e6 q^{56} -199940. q^{58} +2.04044e6 q^{59} -378667. q^{61} -59434.6 q^{62} -1.25175e6 q^{64} -4644.40 q^{67} +3.85960e6 q^{68} +2.79333e6 q^{71} -2.01174e6 q^{73} +1.13660e6 q^{74} +4.65570e6 q^{76} +2.40299e6 q^{77} +1.76767e6 q^{79} -1.16367e6 q^{82} +3.06625e6 q^{83} +1.40972e6 q^{86} +1.19113e6 q^{88} +6.14397e6 q^{89} +1.43920e6 q^{91} +7.05598e6 q^{92} -1.41726e6 q^{94} +3.02969e6 q^{97} +4.35940e6 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 9 q^{2} + 333 q^{4} + 1188 q^{7} - 2043 q^{8} - 5376 q^{11} + 8424 q^{13} - 6762 q^{14} + 43265 q^{16} - 4896 q^{17} + 15232 q^{19} - 44118 q^{22} - 110016 q^{23} + 396762 q^{26} + 674514 q^{28} + 120036 q^{29}+ \cdots + 20819979 q^{98}+O(q^{100}) $ 4 * q - 9 * q^2 + 333 * q^4 + 1188 * q^7 - 2043 * q^8 - 5376 * q^11 + 8424 * q^13 - 6762 * q^14 + 43265 * q^16 - 4896 * q^17 + 15232 * q^19 - 44118 * q^22 - 110016 * q^23 + 396762 * q^26 + 674514 * q^28 + 120036 * q^29 + 116864 * q^31 - 621963 * q^32 - 92374 * q^34 + 663768 * q^37 - 1888776 * q^38 - 253824 * q^41 + 1092960 * q^43 + 1289286 * q^44 + 331204 * q^46 - 132840 * q^47 + 1633580 * q^49 - 779274 * q^52 - 1994616 * q^53 - 352830 * q^56 - 742608 * q^58 + 545712 * q^59 - 3216760 * q^61 - 476856 * q^62 + 284297 * q^64 + 2013336 * q^67 + 7540038 * q^68 + 690912 * q^71 - 5498064 * q^73 + 11980638 * q^74 + 6557832 * q^76 + 795744 * q^77 + 7190080 * q^79 - 26456814 * q^82 + 13158432 * q^83 + 9918804 * q^86 - 22119642 * q^88 + 22889448 * q^89 + 12037824 * q^91 + 17368812 * q^92 + 22864448 * q^94 + 17873136 * q^97 + 20819979 * q^98

Coefficient data

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

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

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

$2$	3.02250	0.267153	0.133577	−	0.991038i	$-0.457354\pi$
$2$	3.02250	0.267153	0.133577	+	0.991038i	$0.457354\pi$
$3$	0	0
$3$	0	0
$4$	−118.865	−0.928629
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1505.28	−1.65872	−0.829361	−	0.558714i	$-0.811295\pi$
$7$	−1505.28	−1.65872	−0.829361	+	0.558714i	$0.811295\pi$
$8$	−746.147	−0.515240
$9$	0	0
$10$	0	0
$11$	−1596.37	−0.361627	−0.180813	−	0.983517i	$-0.557873\pi$
$11$	−1596.37	−0.361627	−0.180813	+	0.983517i	$0.557873\pi$
$12$	0	0
$13$	−956.100	−0.120698	−0.0603492	−	0.998177i	$-0.519221\pi$
$13$	−956.100	−0.120698	−0.0603492	+	0.998177i	$0.519221\pi$
$14$	−4549.69	−0.443133
$15$	0	0
$16$	12959.4	0.790981
$17$	−32470.6	−1.60295	−0.801473	−	0.598031i	$-0.795950\pi$
$17$	−32470.6	−1.60295	−0.801473	+	0.598031i	$0.795950\pi$
$18$	0	0
$19$	−39168.1	−1.31007	−0.655036	−	0.755598i	$-0.727347\pi$
$19$	−39168.1	−1.31007	−0.655036	+	0.755598i	$0.727347\pi$
$20$	0	0
$21$	0	0
$22$	−4825.04	−0.0966098
$23$	−59361.5	−1.01732	−0.508660	−	0.860968i	$-0.669859\pi$
$23$	−59361.5	−1.01732	−0.508660	+	0.860968i	$0.669859\pi$
$24$	0	0
$25$	0	0
$26$	−2889.81	−0.0322450
$27$	0	0
$28$	178924.	1.54034
$29$	−66150.5	−0.503663	−0.251831	−	0.967771i	$-0.581033\pi$
$29$	−66150.5	−0.503663	−0.251831	+	0.967771i	$0.581033\pi$
$30$	0	0
$31$	−19664.1	−0.118552	−0.0592758	−	0.998242i	$-0.518879\pi$
$31$	−19664.1	−0.118552	−0.0592758	+	0.998242i	$0.518879\pi$
$32$	134677.	0.726553
$33$	0	0
$34$	−98142.2	−0.428232
$35$	0	0
$36$	0	0
$37$	376045.	1.22049	0.610245	−	0.792213i	$-0.291071\pi$
$37$	376045.	1.22049	0.610245	+	0.792213i	$0.291071\pi$
$38$	−118386.	−0.349990
$39$	0	0
$40$	0	0
$41$	−385003.	−0.872409	−0.436205	−	0.899848i	$-0.643678\pi$
$41$	−385003.	−0.872409	−0.436205	+	0.899848i	$0.643678\pi$
$42$	0	0
$43$	466410.	0.894600	0.447300	−	0.894384i	$-0.352386\pi$
$43$	466410.	0.894600	0.447300	+	0.894384i	$0.352386\pi$
$44$	189752.	0.335817
$45$	0	0
$46$	−179420.	−0.271780
$47$	−468903.	−0.658779	−0.329390	−	0.944194i	$-0.606843\pi$
$47$	−468903.	−0.658779	−0.329390	+	0.944194i	$0.606843\pi$
$48$	0	0
$49$	1.44232e6	1.75136
$50$	0	0
$51$	0	0
$52$	113646.	0.112084
$53$	−1.60516e6	−1.48100	−0.740498	−	0.672058i	$-0.765410\pi$
$53$	−1.60516e6	−1.48100	−0.740498	+	0.672058i	$0.765410\pi$
$54$	0	0
$55$	0	0
$56$	1.12316e6	0.854639
$57$	0	0
$58$	−199940.	−0.134555
$59$	2.04044e6	1.29342	0.646712	−	0.762734i	$-0.276144\pi$
$59$	2.04044e6	1.29342	0.646712	+	0.762734i	$0.276144\pi$
$60$	0	0
$61$	−378667.	−0.213601	−0.106800	−	0.994280i	$-0.534061\pi$
$61$	−378667.	−0.213601	−0.106800	+	0.994280i	$0.534061\pi$
$62$	−59434.6	−0.0316715
$63$	0	0
$64$	−1.25175e6	−0.596880
$65$	0	0
$66$	0	0
$67$	−4644.40	−0.00188655	−0.000943274	−	1.00000i	$-0.500300\pi$
$67$	−4644.40	−0.00188655	−0.000943274		1.00000i	$0.500300\pi$
$68$	3.85960e6	1.48854
$69$	0	0
$70$	0	0
$71$	2.79333e6	0.926229	0.463115	−	0.886298i	$-0.346732\pi$
$71$	2.79333e6	0.926229	0.463115	+	0.886298i	$0.346732\pi$
$72$	0	0
$73$	−2.01174e6	−0.605259	−0.302630	−	0.953108i	$-0.597864\pi$
$73$	−2.01174e6	−0.605259	−0.302630	+	0.953108i	$0.597864\pi$
$74$	1.13660e6	0.326058
$75$	0	0
$76$	4.65570e6	1.21657
$77$	2.40299e6	0.599838
$78$	0	0
$79$	1.76767e6	0.403372	0.201686	−	0.979450i	$-0.435358\pi$
$79$	1.76767e6	0.403372	0.201686	+	0.979450i	$0.435358\pi$
$80$	0	0
$81$	0	0
$82$	−1.16367e6	−0.233067
$83$	3.06625e6	0.588619	0.294310	−	0.955710i	$-0.404910\pi$
$83$	3.06625e6	0.588619	0.294310	+	0.955710i	$0.404910\pi$
$84$	0	0
$85$	0	0
$86$	1.40972e6	0.238995
$87$	0	0
$88$	1.19113e6	0.186324
$89$	6.14397e6	0.923813	0.461907	−	0.886929i	$-0.347166\pi$
$89$	6.14397e6	0.923813	0.461907	+	0.886929i	$0.347166\pi$
$90$	0	0
$91$	1.43920e6	0.200205
$92$	7.05598e6	0.944713
$93$	0	0
$94$	−1.41726e6	−0.175995
$95$	0	0
$96$	0	0
$97$	3.02969e6	0.337053	0.168526	−	0.985697i	$-0.446099\pi$
$97$	3.02969e6	0.337053	0.168526	+	0.985697i	$0.446099\pi$
$98$	4.35940e6	0.467881
$99$	0	0
$100$	0	0
$101$	−7.17509e6	−0.692951	−0.346475	−	0.938059i	$-0.612622\pi$
$101$	−7.17509e6	−0.692951	−0.346475	+	0.938059i	$0.612622\pi$
$102$	0	0
$103$	−1.45694e7	−1.31374	−0.656872	−	0.754002i	$-0.728121\pi$
$103$	−1.45694e7	−1.31374	−0.656872	+	0.754002i	$0.728121\pi$
$104$	713391.	0.0621886
$105$	0	0
$106$	−4.85160e6	−0.395653
$107$	−1.64442e7	−1.29769	−0.648843	−	0.760923i	$-0.724747\pi$
$107$	−1.64442e7	−1.29769	−0.648843	+	0.760923i	$0.724747\pi$
$108$	0	0
$109$	2.34832e7	1.73686	0.868430	−	0.495812i	$-0.165129\pi$
$109$	2.34832e7	1.73686	0.868430	+	0.495812i	$0.165129\pi$
$110$	0	0
$111$	0	0
$112$	−1.95075e7	−1.31202
$113$	−1.25747e7	−0.819826	−0.409913	−	0.912125i	$-0.634441\pi$
$113$	−1.25747e7	−0.819826	−0.409913	+	0.912125i	$0.634441\pi$
$114$	0	0
$115$	0	0
$116$	7.86294e6	0.467716
$117$	0	0
$118$	6.16721e6	0.345543
$119$	4.88772e7	2.65884
$120$	0	0
$121$	−1.69388e7	−0.869226
$122$	−1.14452e6	−0.0570642
$123$	0	0
$124$	2.33736e6	0.110091
$125$	0	0
$126$	0	0
$127$	2.41070e7	1.04431	0.522155	−	0.852850i	$-0.325128\pi$
$127$	2.41070e7	1.04431	0.522155	+	0.852850i	$0.325128\pi$
$128$	−2.10220e7	−0.886012
$129$	0	0
$130$	0	0
$131$	2.08040e7	0.808533	0.404266	−	0.914641i	$-0.367527\pi$
$131$	2.08040e7	0.808533	0.404266	+	0.914641i	$0.367527\pi$
$132$	0	0
$133$	5.89589e7	2.17304
$134$	−14037.7	−0.000503998 0
$135$	0	0
$136$	2.42278e7	0.825901
$137$	−9.94308e6	−0.330369	−0.165184	−	0.986263i	$-0.552822\pi$
$137$	−9.94308e6	−0.330369	−0.165184	+	0.986263i	$0.552822\pi$
$138$	0	0
$139$	1.47414e7	0.465571	0.232785	−	0.972528i	$-0.425216\pi$
$139$	1.47414e7	0.465571	0.232785	+	0.972528i	$0.425216\pi$
$140$	0	0
$141$	0	0
$142$	8.44284e6	0.247445
$143$	1.52629e6	0.0436478
$144$	0	0
$145$	0	0
$146$	−6.08047e6	−0.161697
$147$	0	0
$148$	−4.46985e7	−1.13338
$149$	2.21899e7	0.549545	0.274772	−	0.961509i	$-0.411398\pi$
$149$	2.21899e7	0.549545	0.274772	+	0.961509i	$0.411398\pi$
$150$	0	0
$151$	4.90759e7	1.15998	0.579988	−	0.814625i	$-0.303057\pi$
$151$	4.90759e7	1.15998	0.579988	+	0.814625i	$0.303057\pi$
$152$	2.92252e7	0.675001
$153$	0	0
$154$	7.26302e6	0.160249
$155$	0	0
$156$	0	0
$157$	−6.21368e6	−0.128145	−0.0640723	−	0.997945i	$-0.520409\pi$
$157$	−6.21368e6	−0.128145	−0.0640723	+	0.997945i	$0.520409\pi$
$158$	5.34277e6	0.107762
$159$	0	0
$160$	0	0
$161$	8.93555e7	1.68745
$162$	0	0
$163$	1.54205e7	0.278895	0.139448	−	0.990229i	$-0.455467\pi$
$163$	1.54205e7	0.278895	0.139448	+	0.990229i	$0.455467\pi$
$164$	4.57632e7	0.810145
$165$	0	0
$166$	9.26773e6	0.157252
$167$	−7.74611e6	−0.128699	−0.0643496	−	0.997927i	$-0.520497\pi$
$167$	−7.74611e6	−0.128699	−0.0643496	+	0.997927i	$0.520497\pi$
$168$	0	0
$169$	−6.18344e7	−0.985432
$170$	0	0
$171$	0	0
$172$	−5.54397e7	−0.830751
$173$	−1.00086e7	−0.146965	−0.0734824	−	0.997297i	$-0.523411\pi$
$173$	−1.00086e7	−0.146965	−0.0734824	+	0.997297i	$0.523411\pi$
$174$	0	0
$175$	0	0
$176$	−2.06881e7	−0.286040
$177$	0	0
$178$	1.85701e7	0.246800
$179$	−1.09231e8	−1.42350	−0.711751	−	0.702431i	$-0.752098\pi$
$179$	−1.09231e8	−1.42350	−0.711751	+	0.702431i	$0.752098\pi$
$180$	0	0
$181$	−1.27824e7	−0.160228	−0.0801138	−	0.996786i	$-0.525528\pi$
$181$	−1.27824e7	−0.160228	−0.0801138	+	0.996786i	$0.525528\pi$
$182$	4.34996e6	0.0534854
$183$	0	0
$184$	4.42924e7	0.524164
$185$	0	0
$186$	0	0
$187$	5.18352e7	0.579668
$188$	5.57359e7	0.611762
$189$	0	0
$190$	0	0
$191$	−1.54482e8	−1.60421	−0.802103	−	0.597185i	$-0.796286\pi$
$191$	−1.54482e8	−1.60421	−0.802103	+	0.597185i	$0.796286\pi$
$192$	0	0
$193$	−1.51365e8	−1.51556	−0.757782	−	0.652507i	$-0.773717\pi$
$193$	−1.51365e8	−1.51556	−0.757782	+	0.652507i	$0.773717\pi$
$194$	9.15724e6	0.0900448
$195$	0	0
$196$	−1.71440e8	−1.62636
$197$	−1.23829e8	−1.15396	−0.576981	−	0.816758i	$-0.695769\pi$
$197$	−1.23829e8	−1.15396	−0.576981	+	0.816758i	$0.695769\pi$
$198$	0	0
$199$	4.77920e7	0.429902	0.214951	−	0.976625i	$-0.431041\pi$
$199$	4.77920e7	0.429902	0.214951	+	0.976625i	$0.431041\pi$
$200$	0	0
$201$	0	0
$202$	−2.16867e7	−0.185124
$203$	9.95748e7	0.835436
$204$	0	0
$205$	0	0
$206$	−4.40359e7	−0.350971
$207$	0	0
$208$	−1.23905e7	−0.0954701
$209$	6.25270e7	0.473757
$210$	0	0
$211$	6.28256e7	0.460413	0.230207	−	0.973142i	$-0.426060\pi$
$211$	6.28256e7	0.460413	0.230207	+	0.973142i	$0.426060\pi$
$212$	1.90797e8	1.37530
$213$	0	0
$214$	−4.97025e7	−0.346681
$215$	0	0
$216$	0	0
$217$	2.95999e7	0.196644
$218$	7.09779e7	0.464008
$219$	0	0
$220$	0	0
$221$	3.10451e7	0.193473
$222$	0	0
$223$	5.17882e6	0.0312726	0.0156363	−	0.999878i	$-0.495023\pi$
$223$	5.17882e6	0.0312726	0.0156363	+	0.999878i	$0.495023\pi$
$224$	−2.02726e8	−1.20515
$225$	0	0
$226$	−3.80069e7	−0.219019
$227$	2.43478e8	1.38156	0.690778	−	0.723067i	$-0.257268\pi$
$227$	2.43478e8	1.38156	0.690778	+	0.723067i	$0.257268\pi$
$228$	0	0
$229$	−1.09734e8	−0.603832	−0.301916	−	0.953335i	$-0.597626\pi$
$229$	−1.09734e8	−0.603832	−0.301916	+	0.953335i	$0.597626\pi$
$230$	0	0
$231$	0	0
$232$	4.93580e7	0.259507
$233$	−1.85771e8	−0.962129	−0.481064	−	0.876685i	$-0.659750\pi$
$233$	−1.85771e8	−0.962129	−0.481064	+	0.876685i	$0.659750\pi$
$234$	0	0
$235$	0	0
$236$	−2.42535e8	−1.20111
$237$	0	0
$238$	1.47731e8	0.710318
$239$	−3.06316e8	−1.45137	−0.725684	−	0.688028i	$-0.758477\pi$
$239$	−3.06316e8	−1.45137	−0.725684	+	0.688028i	$0.758477\pi$
$240$	0	0
$241$	−5.78177e7	−0.266073	−0.133037	−	0.991111i	$-0.542473\pi$
$241$	−5.78177e7	−0.266073	−0.133037	+	0.991111i	$0.542473\pi$
$242$	−5.11973e7	−0.232217
$243$	0	0
$244$	4.50101e7	0.198356
$245$	0	0
$246$	0	0
$247$	3.74486e7	0.158124
$248$	1.46723e7	0.0610826
$249$	0	0
$250$	0	0
$251$	3.90109e8	1.55714	0.778571	−	0.627557i	$-0.215945\pi$
$251$	3.90109e8	1.55714	0.778571	+	0.627557i	$0.215945\pi$
$252$	0	0
$253$	9.47632e7	0.367890
$254$	7.28632e7	0.278991
$255$	0	0
$256$	9.66848e7	0.360179
$257$	−5.32200e8	−1.95573	−0.977865	−	0.209237i	$-0.932902\pi$
$257$	−5.32200e8	−1.95573	−0.977865	+	0.209237i	$0.932902\pi$
$258$	0	0
$259$	−5.66053e8	−2.02445
$260$	0	0
$261$	0	0
$262$	6.28800e7	0.216002
$263$	−3.12888e8	−1.06058	−0.530291	−	0.847816i	$-0.677917\pi$
$263$	−3.12888e8	−1.06058	−0.530291	+	0.847816i	$0.677917\pi$
$264$	0	0
$265$	0	0
$266$	1.78203e8	0.580536
$267$	0	0
$268$	552054.	0.00175190
$269$	−2.95534e8	−0.925710	−0.462855	−	0.886434i	$-0.653175\pi$
$269$	−2.95534e8	−0.925710	−0.462855	+	0.886434i	$0.653175\pi$
$270$	0	0
$271$	3.96274e8	1.20949	0.604746	−	0.796419i	$-0.293275\pi$
$271$	3.96274e8	1.20949	0.604746	+	0.796419i	$0.293275\pi$
$272$	−4.20800e8	−1.26790
$273$	0	0
$274$	−3.00529e7	−0.0882591
$275$	0	0
$276$	0	0
$277$	−4.00402e8	−1.13192	−0.565961	−	0.824432i	$-0.691495\pi$
$277$	−4.00402e8	−1.13192	−0.565961	+	0.824432i	$0.691495\pi$
$278$	4.45557e7	0.124379
$279$	0	0
$280$	0	0
$281$	−3.58146e8	−0.962916	−0.481458	−	0.876469i	$-0.659893\pi$
$281$	−3.58146e8	−0.962916	−0.481458	+	0.876469i	$0.659893\pi$
$282$	0	0
$283$	3.95688e8	1.03777	0.518884	−	0.854845i	$-0.326348\pi$
$283$	3.95688e8	1.03777	0.518884	+	0.854845i	$0.326348\pi$
$284$	−3.32028e8	−0.860123
$285$	0	0
$286$	4.61322e6	0.0116606
$287$	5.79536e8	1.44708
$288$	0	0
$289$	6.44000e8	1.56943
$290$	0	0
$291$	0	0
$292$	2.39124e8	0.562061
$293$	−1.93280e8	−0.448900	−0.224450	−	0.974486i	$-0.572059\pi$
$293$	−1.93280e8	−0.448900	−0.224450	+	0.974486i	$0.572059\pi$
$294$	0	0
$295$	0	0
$296$	−2.80585e8	−0.628845
$297$	0	0
$298$	6.70688e7	0.146813
$299$	5.67555e7	0.122789
$300$	0	0
$301$	−7.02077e8	−1.48389
$302$	1.48332e8	0.309892
$303$	0	0
$304$	−5.07597e8	−1.03624
$305$	0	0
$306$	0	0
$307$	−2.11267e8	−0.416722	−0.208361	−	0.978052i	$-0.566813\pi$
$307$	−2.11267e8	−0.416722	−0.208361	+	0.978052i	$0.566813\pi$
$308$	−2.85630e8	−0.557027
$309$	0	0
$310$	0	0
$311$	−2.59442e8	−0.489079	−0.244540	−	0.969639i	$-0.578637\pi$
$311$	−2.59442e8	−0.489079	−0.244540	+	0.969639i	$0.578637\pi$
$312$	0	0
$313$	−9.29772e8	−1.71384	−0.856922	−	0.515446i	$-0.827626\pi$
$313$	−9.29772e8	−1.71384	−0.856922	+	0.515446i	$0.827626\pi$
$314$	−1.87808e7	−0.0342343
$315$	0	0
$316$	−2.10113e8	−0.374583
$317$	1.63510e8	0.288295	0.144147	−	0.989556i	$-0.453956\pi$
$317$	1.63510e8	0.288295	0.144147	+	0.989556i	$0.453956\pi$
$318$	0	0
$319$	1.05601e8	0.182138
$320$	0	0
$321$	0	0
$322$	2.70077e8	0.450808
$323$	1.27181e9	2.09997
$324$	0	0
$325$	0	0
$326$	4.66083e7	0.0745079
$327$	0	0
$328$	2.87269e8	0.449500
$329$	7.05828e8	1.09273
$330$	0	0
$331$	−2.57948e8	−0.390962	−0.195481	−	0.980707i	$-0.562627\pi$
$331$	−2.57948e8	−0.390962	−0.195481	+	0.980707i	$0.562627\pi$
$332$	−3.64469e8	−0.546609
$333$	0	0
$334$	−2.34126e7	−0.0343825
$335$	0	0
$336$	0	0
$337$	2.34282e8	0.333452	0.166726	−	0.986003i	$-0.446680\pi$
$337$	2.34282e8	0.333452	0.166726	+	0.986003i	$0.446680\pi$
$338$	−1.86894e8	−0.263261
$339$	0	0
$340$	0	0
$341$	3.13912e7	0.0428715
$342$	0	0
$343$	−9.31426e8	−1.24629
$344$	−3.48011e8	−0.460933
$345$	0	0
$346$	−3.02511e7	−0.0392622
$347$	6.73974e8	0.865944	0.432972	−	0.901407i	$-0.357465\pi$
$347$	6.73974e8	0.865944	0.432972	+	0.901407i	$0.357465\pi$
$348$	0	0
$349$	8.77464e8	1.10494	0.552472	−	0.833531i	$-0.313685\pi$
$349$	8.77464e8	1.10494	0.552472	+	0.833531i	$0.313685\pi$
$350$	0	0
$351$	0	0
$352$	−2.14994e8	−0.262741
$353$	6.75088e8	0.816863	0.408431	−	0.912789i	$-0.366076\pi$
$353$	6.75088e8	0.816863	0.408431	+	0.912789i	$0.366076\pi$
$354$	0	0
$355$	0	0
$356$	−7.30300e8	−0.857880
$357$	0	0
$358$	−3.30149e8	−0.380294
$359$	7.40244e8	0.844393	0.422196	−	0.906504i	$-0.361259\pi$
$359$	7.40244e8	0.844393	0.422196	+	0.906504i	$0.361259\pi$
$360$	0	0
$361$	6.40271e8	0.716290
$362$	−3.86348e7	−0.0428054
$363$	0	0
$364$	−1.71069e8	−0.185916
$365$	0	0
$366$	0	0
$367$	−1.05986e9	−1.11922	−0.559611	−	0.828755i	$-0.689049\pi$
$367$	−1.05986e9	−1.11922	−0.559611	+	0.828755i	$0.689049\pi$
$368$	−7.69291e8	−0.804680
$369$	0	0
$370$	0	0
$371$	2.41622e9	2.45656
$372$	0	0
$373$	1.28239e9	1.27950	0.639748	−	0.768585i	$-0.279039\pi$
$373$	1.28239e9	1.27950	0.639748	+	0.768585i	$0.279039\pi$
$374$	1.56672e8	0.154860
$375$	0	0
$376$	3.49870e8	0.339429
$377$	6.32465e7	0.0607913
$378$	0	0
$379$	−2.97355e8	−0.280568	−0.140284	−	0.990111i	$-0.544802\pi$
$379$	−2.97355e8	−0.280568	−0.140284	+	0.990111i	$0.544802\pi$
$380$	0	0
$381$	0	0
$382$	−4.66920e8	−0.428569
$383$	8.10508e8	0.737160	0.368580	−	0.929596i	$-0.379844\pi$
$383$	8.10508e8	0.737160	0.368580	+	0.929596i	$0.379844\pi$
$384$	0	0
$385$	0	0
$386$	−4.57500e8	−0.404888
$387$	0	0
$388$	−3.60123e8	−0.312997
$389$	3.38899e8	0.291908	0.145954	−	0.989291i	$-0.453375\pi$
$389$	3.38899e8	0.291908	0.145954	+	0.989291i	$0.453375\pi$
$390$	0	0
$391$	1.92750e9	1.63071
$392$	−1.07618e9	−0.902368
$393$	0	0
$394$	−3.74273e8	−0.308285
$395$	0	0
$396$	0	0
$397$	1.00265e9	0.804235	0.402117	−	0.915588i	$-0.368274\pi$
$397$	1.00265e9	0.804235	0.402117	+	0.915588i	$0.368274\pi$
$398$	1.44451e8	0.114850
$399$	0	0
$400$	0	0
$401$	1.09336e9	0.846755	0.423377	−	0.905953i	$-0.360844\pi$
$401$	1.09336e9	0.846755	0.423377	+	0.905953i	$0.360844\pi$
$402$	0	0
$403$	1.88008e7	0.0143090
$404$	8.52863e8	0.643494
$405$	0	0
$406$	3.00964e8	0.223190
$407$	−6.00309e8	−0.441362
$408$	0	0
$409$	1.09511e9	0.791453	0.395726	−	0.918368i	$-0.370493\pi$
$409$	1.09511e9	0.791453	0.395726	+	0.918368i	$0.370493\pi$
$410$	0	0
$411$	0	0
$412$	1.73178e9	1.21998
$413$	−3.07142e9	−2.14543
$414$	0	0
$415$	0	0
$416$	−1.28764e8	−0.0876938
$417$	0	0
$418$	1.88988e8	0.126566
$419$	1.75196e9	1.16352	0.581762	−	0.813359i	$-0.302364\pi$
$419$	1.75196e9	1.16352	0.581762	+	0.813359i	$0.302364\pi$
$420$	0	0
$421$	−4.58026e8	−0.299160	−0.149580	−	0.988750i	$-0.547792\pi$
$421$	−4.58026e8	−0.299160	−0.149580	+	0.988750i	$0.547792\pi$
$422$	1.89890e8	0.123001
$423$	0	0
$424$	1.19769e9	0.763068
$425$	0	0
$426$	0	0
$427$	5.69999e8	0.354304
$428$	1.95463e9	1.20507
$429$	0	0
$430$	0	0
$431$	7.60129e8	0.457317	0.228658	−	0.973507i	$-0.426566\pi$
$431$	7.60129e8	0.457317	0.228658	+	0.973507i	$0.426566\pi$
$432$	0	0
$433$	2.48461e9	1.47079	0.735396	−	0.677638i	$-0.236996\pi$
$433$	2.48461e9	1.47079	0.735396	+	0.677638i	$0.236996\pi$
$434$	8.94656e7	0.0525342
$435$	0	0
$436$	−2.79132e9	−1.61290
$437$	2.32508e9	1.33276
$438$	0	0
$439$	1.69934e8	0.0958637	0.0479318	−	0.998851i	$-0.484737\pi$
$439$	1.69934e8	0.0958637	0.0479318	+	0.998851i	$0.484737\pi$
$440$	0	0
$441$	0	0
$442$	9.38337e7	0.0516870
$443$	−1.53246e9	−0.837481	−0.418741	−	0.908106i	$-0.637528\pi$
$443$	−1.53246e9	−0.837481	−0.418741	+	0.908106i	$0.637528\pi$
$444$	0	0
$445$	0	0
$446$	1.56530e7	0.00835458
$447$	0	0
$448$	1.88423e9	0.990057
$449$	3.45325e9	1.80039	0.900194	−	0.435489i	$-0.143425\pi$
$449$	3.45325e9	1.80039	0.900194	+	0.435489i	$0.143425\pi$
$450$	0	0
$451$	6.14609e8	0.315486
$452$	1.49468e9	0.761315
$453$	0	0
$454$	7.35910e8	0.369087
$455$	0	0
$456$	0	0
$457$	−7.56279e8	−0.370659	−0.185330	−	0.982676i	$-0.559335\pi$
$457$	−7.56279e8	−0.370659	−0.185330	+	0.982676i	$0.559335\pi$
$458$	−3.31670e8	−0.161316
$459$	0	0
$460$	0	0
$461$	8.78272e8	0.417518	0.208759	−	0.977967i	$-0.433058\pi$
$461$	8.78272e8	0.417518	0.208759	+	0.977967i	$0.433058\pi$
$462$	0	0
$463$	−3.54973e9	−1.66212	−0.831058	−	0.556185i	$-0.812264\pi$
$463$	−3.54973e9	−1.66212	−0.831058	+	0.556185i	$0.812264\pi$
$464$	−8.57273e8	−0.398388
$465$	0	0
$466$	−5.61493e8	−0.257036
$467$	−3.03363e9	−1.37833	−0.689165	−	0.724605i	$-0.742022\pi$
$467$	−3.03363e9	−1.37833	−0.689165	+	0.724605i	$0.742022\pi$
$468$	0	0
$469$	6.99111e6	0.00312926
$470$	0	0
$471$	0	0
$472$	−1.52246e9	−0.666423
$473$	−7.44566e8	−0.323511
$474$	0	0
$475$	0	0
$476$	−5.80977e9	−2.46908
$477$	0	0
$478$	−9.25840e8	−0.387738
$479$	−3.00343e9	−1.24866	−0.624328	−	0.781162i	$-0.714627\pi$
$479$	−3.00343e9	−1.24866	−0.624328	+	0.781162i	$0.714627\pi$
$480$	0	0
$481$	−3.59537e8	−0.147311
$482$	−1.74754e8	−0.0710823
$483$	0	0
$484$	2.01342e9	0.807189
$485$	0	0
$486$	0	0
$487$	3.28649e8	0.128938	0.0644690	−	0.997920i	$-0.479465\pi$
$487$	3.28649e8	0.128938	0.0644690	+	0.997920i	$0.479465\pi$
$488$	2.82541e8	0.110056
$489$	0	0
$490$	0	0
$491$	6.97275e8	0.265839	0.132919	−	0.991127i	$-0.457565\pi$
$491$	6.97275e8	0.265839	0.132919	+	0.991127i	$0.457565\pi$
$492$	0	0
$493$	2.14794e9	0.807344
$494$	1.13188e8	0.0422433
$495$	0	0
$496$	−2.54835e8	−0.0937721
$497$	−4.20474e9	−1.53636
$498$	0	0
$499$	−3.18226e9	−1.14652	−0.573262	−	0.819372i	$-0.694322\pi$
$499$	−3.18226e9	−1.14652	−0.573262	+	0.819372i	$0.694322\pi$
$500$	0	0
$501$	0	0
$502$	1.17910e9	0.415996
$503$	4.07550e9	1.42788	0.713942	−	0.700205i	$-0.246908\pi$
$503$	4.07550e9	1.42788	0.713942	+	0.700205i	$0.246908\pi$
$504$	0	0
$505$	0	0
$506$	2.86421e8	0.0982830
$507$	0	0
$508$	−2.86546e9	−0.969777
$509$	−4.18725e8	−0.140740	−0.0703699	−	0.997521i	$-0.522418\pi$
$509$	−4.18725e8	−0.140740	−0.0703699	+	0.997521i	$0.522418\pi$
$510$	0	0
$511$	3.02822e9	1.00396
$512$	2.98305e9	0.982235
$513$	0	0
$514$	−1.60857e9	−0.522480
$515$	0	0
$516$	0	0
$517$	7.48544e8	0.238232
$518$	−1.71089e9	−0.540839
$519$	0	0
$520$	0	0
$521$	2.02201e9	0.626400	0.313200	−	0.949687i	$-0.398599\pi$
$521$	2.02201e9	0.626400	0.313200	+	0.949687i	$0.398599\pi$
$522$	0	0
$523$	−1.87087e8	−0.0571857	−0.0285928	−	0.999591i	$-0.509103\pi$
$523$	−1.87087e8	−0.0571857	−0.0285928	+	0.999591i	$0.509103\pi$
$524$	−2.47286e9	−0.750827
$525$	0	0
$526$	−9.45704e8	−0.283338
$527$	6.38504e8	0.190032
$528$	0	0
$529$	1.18962e8	0.0349392
$530$	0	0
$531$	0	0
$532$	−7.00812e9	−2.01795
$533$	3.68101e8	0.105298
$534$	0	0
$535$	0	0
$536$	3.46541e6	0.000972025 0
$537$	0	0
$538$	−8.93251e8	−0.247307
$539$	−2.30248e9	−0.633337
$540$	0	0
$541$	3.94897e9	1.07224	0.536122	−	0.844140i	$-0.319889\pi$
$541$	3.94897e9	1.07224	0.536122	+	0.844140i	$0.319889\pi$
$542$	1.19774e9	0.323120
$543$	0	0
$544$	−4.37303e9	−1.16463
$545$	0	0
$546$	0	0
$547$	−1.71845e9	−0.448932	−0.224466	−	0.974482i	$-0.572064\pi$
$547$	−1.71845e9	−0.448932	−0.224466	+	0.974482i	$0.572064\pi$
$548$	1.18188e9	0.306790
$549$	0	0
$550$	0	0
$551$	2.59099e9	0.659835
$552$	0	0
$553$	−2.66083e9	−0.669082
$554$	−1.21021e9	−0.302397
$555$	0	0
$556$	−1.75222e9	−0.432342
$557$	3.22406e9	0.790516	0.395258	−	0.918570i	$-0.370655\pi$
$557$	3.22406e9	0.790516	0.395258	+	0.918570i	$0.370655\pi$
$558$	0	0
$559$	−4.45935e8	−0.107977
$560$	0	0
$561$	0	0
$562$	−1.08250e9	−0.257246
$563$	5.69907e9	1.34594	0.672968	−	0.739671i	$-0.265019\pi$
$563$	5.69907e9	1.34594	0.672968	+	0.739671i	$0.265019\pi$
$564$	0	0
$565$	0	0
$566$	1.19596e9	0.277243
$567$	0	0
$568$	−2.08424e9	−0.477230
$569$	−3.70804e9	−0.843823	−0.421912	−	0.906637i	$-0.638641\pi$
$569$	−3.70804e9	−0.843823	−0.421912	+	0.906637i	$0.638641\pi$
$570$	0	0
$571$	−3.34811e9	−0.752615	−0.376308	−	0.926495i	$-0.622806\pi$
$571$	−3.34811e9	−0.752615	−0.376308	+	0.926495i	$0.622806\pi$
$572$	−1.81422e8	−0.0405326
$573$	0	0
$574$	1.75164e9	0.386593
$575$	0	0
$576$	0	0
$577$	8.37348e9	1.81464	0.907320	−	0.420441i	$-0.138124\pi$
$577$	8.37348e9	1.81464	0.907320	+	0.420441i	$0.138124\pi$
$578$	1.94649e9	0.419280
$579$	0	0
$580$	0	0
$581$	−4.61556e9	−0.976355
$582$	0	0
$583$	2.56244e9	0.535568
$584$	1.50105e9	0.311854
$585$	0	0
$586$	−5.84188e8	−0.119925
$587$	9.32230e8	0.190235	0.0951174	−	0.995466i	$-0.469677\pi$
$587$	9.32230e8	0.190235	0.0951174	+	0.995466i	$0.469677\pi$
$588$	0	0
$589$	7.70205e8	0.155311
$590$	0	0
$591$	0	0
$592$	4.87334e9	0.965385
$593$	−4.69858e9	−0.925283	−0.462642	−	0.886545i	$-0.653098\pi$
$593$	−4.69858e9	−0.925283	−0.462642	+	0.886545i	$0.653098\pi$
$594$	0	0
$595$	0	0
$596$	−2.63759e9	−0.510323
$597$	0	0
$598$	1.71543e8	0.0328035
$599$	2.66487e9	0.506620	0.253310	−	0.967385i	$-0.418481\pi$
$599$	2.66487e9	0.506620	0.253310	+	0.967385i	$0.418481\pi$
$600$	0	0
$601$	−1.21738e9	−0.228753	−0.114376	−	0.993437i	$-0.536487\pi$
$601$	−1.21738e9	−0.228753	−0.114376	+	0.993437i	$0.536487\pi$
$602$	−2.12203e9	−0.396427
$603$	0	0
$604$	−5.83339e9	−1.07719
$605$	0	0
$606$	0	0
$607$	−2.74917e9	−0.498932	−0.249466	−	0.968384i	$-0.580255\pi$
$607$	−2.74917e9	−0.498932	−0.249466	+	0.968384i	$0.580255\pi$
$608$	−5.27503e9	−0.951837
$609$	0	0
$610$	0	0
$611$	4.48318e8	0.0795136
$612$	0	0
$613$	7.89654e9	1.38460	0.692301	−	0.721609i	$-0.256597\pi$
$613$	7.89654e9	1.38460	0.692301	+	0.721609i	$0.256597\pi$
$614$	−6.38552e8	−0.111329
$615$	0	0
$616$	−1.79298e9	−0.309060
$617$	−3.64303e9	−0.624403	−0.312202	−	0.950016i	$-0.601066\pi$
$617$	−3.64303e9	−0.624403	−0.312202	+	0.950016i	$0.601066\pi$
$618$	0	0
$619$	7.22253e9	1.22397	0.611987	−	0.790868i	$-0.290370\pi$
$619$	7.22253e9	1.22397	0.611987	+	0.790868i	$0.290370\pi$
$620$	0	0
$621$	0	0
$622$	−7.84163e8	−0.130659
$623$	−9.24838e9	−1.53235
$624$	0	0
$625$	0	0
$626$	−2.81023e9	−0.457859
$627$	0	0
$628$	7.38586e8	0.118999
$629$	−1.22104e10	−1.95638
$630$	0	0
$631$	−1.55879e9	−0.246993	−0.123496	−	0.992345i	$-0.539411\pi$
$631$	−1.55879e9	−0.246993	−0.123496	+	0.992345i	$0.539411\pi$
$632$	−1.31894e9	−0.207833
$633$	0	0
$634$	4.94208e8	0.0770189
$635$	0	0
$636$	0	0
$637$	−1.37900e9	−0.211386
$638$	3.19178e8	0.0486588
$639$	0	0
$640$	0	0
$641$	−3.18649e9	−0.477869	−0.238935	−	0.971036i	$-0.576798\pi$
$641$	−3.18649e9	−0.477869	−0.238935	+	0.971036i	$0.576798\pi$
$642$	0	0
$643$	1.65144e9	0.244976	0.122488	−	0.992470i	$-0.460913\pi$
$643$	1.65144e9	0.244976	0.122488	+	0.992470i	$0.460913\pi$
$644$	−1.06212e10	−1.56701
$645$	0	0
$646$	3.84405e9	0.561015
$647$	−3.95797e9	−0.574523	−0.287262	−	0.957852i	$-0.592745\pi$
$647$	−3.95797e9	−0.574523	−0.287262	+	0.957852i	$0.592745\pi$
$648$	0	0
$649$	−3.25730e9	−0.467736
$650$	0	0
$651$	0	0
$652$	−1.83295e9	−0.258990
$653$	2.09998e9	0.295134	0.147567	−	0.989052i	$-0.452856\pi$
$653$	2.09998e9	0.295134	0.147567	+	0.989052i	$0.452856\pi$
$654$	0	0
$655$	0	0
$656$	−4.98942e9	−0.690059
$657$	0	0
$658$	2.13336e9	0.291927
$659$	1.18536e10	1.61344	0.806718	−	0.590937i	$-0.201242\pi$
$659$	1.18536e10	1.61344	0.806718	+	0.590937i	$0.201242\pi$
$660$	0	0
$661$	−1.14873e10	−1.54709	−0.773543	−	0.633744i	$-0.781517\pi$
$661$	−1.14873e10	−1.54709	−0.773543	+	0.633744i	$0.781517\pi$
$662$	−7.79648e8	−0.104447
$663$	0	0
$664$	−2.28787e9	−0.303280
$665$	0	0
$666$	0	0
$667$	3.92679e9	0.512386
$668$	9.20738e8	0.119514
$669$	0	0
$670$	0	0
$671$	6.04494e8	0.0772437
$672$	0	0
$673$	−1.99478e9	−0.252257	−0.126128	−	0.992014i	$-0.540255\pi$
$673$	−1.99478e9	−0.252257	−0.126128	+	0.992014i	$0.540255\pi$
$674$	7.08115e8	0.0890829
$675$	0	0
$676$	7.34992e9	0.915101
$677$	3.21860e9	0.398664	0.199332	−	0.979932i	$-0.436123\pi$
$677$	3.21860e9	0.398664	0.199332	+	0.979932i	$0.436123\pi$
$678$	0	0
$679$	−4.56053e9	−0.559076
$680$	0	0
$681$	0	0
$682$	9.48799e7	0.0114533
$683$	1.87259e9	0.224890	0.112445	−	0.993658i	$-0.464132\pi$
$683$	1.87259e9	0.224890	0.112445	+	0.993658i	$0.464132\pi$
$684$	0	0
$685$	0	0
$686$	−2.81523e9	−0.332950
$687$	0	0
$688$	6.04442e9	0.707611
$689$	1.53470e9	0.178754
$690$	0	0
$691$	3.08223e8	0.0355380	0.0177690	−	0.999842i	$-0.494344\pi$
$691$	3.08223e8	0.0355380	0.0177690	+	0.999842i	$0.494344\pi$
$692$	1.18967e9	0.136476
$693$	0	0
$694$	2.03708e9	0.231340
$695$	0	0
$696$	0	0
$697$	1.25013e10	1.39842
$698$	2.65213e9	0.295190
$699$	0	0
$700$	0	0
$701$	1.39953e10	1.53450	0.767251	−	0.641346i	$-0.221624\pi$
$701$	1.39953e10	1.53450	0.767251	+	0.641346i	$0.221624\pi$
$702$	0	0
$703$	−1.47290e10	−1.59893
$704$	1.99826e9	0.215848
$705$	0	0
$706$	2.04045e9	0.218228
$707$	1.08005e10	1.14941
$708$	0	0
$709$	−1.02356e10	−1.07858	−0.539290	−	0.842120i	$-0.681307\pi$
$709$	−1.02356e10	−1.07858	−0.539290	+	0.842120i	$0.681307\pi$
$710$	0	0
$711$	0	0
$712$	−4.58430e9	−0.475985
$713$	1.16729e9	0.120605
$714$	0	0
$715$	0	0
$716$	1.29836e10	1.32191
$717$	0	0
$718$	2.23739e9	0.225582
$719$	−9.63078e9	−0.966297	−0.483148	−	0.875538i	$-0.660507\pi$
$719$	−9.63078e9	−0.966297	−0.483148	+	0.875538i	$0.660507\pi$
$720$	0	0
$721$	2.19309e10	2.17913
$722$	1.93522e9	0.191359
$723$	0	0
$724$	1.51937e9	0.148792
$725$	0	0
$726$	0	0
$727$	−1.62740e10	−1.57081	−0.785407	−	0.618980i	$-0.787546\pi$
$727$	−1.62740e10	−1.57081	−0.785407	+	0.618980i	$0.787546\pi$
$728$	−1.07385e9	−0.103154
$729$	0	0
$730$	0	0
$731$	−1.51446e10	−1.43399
$732$	0	0
$733$	1.62212e10	1.52132	0.760659	−	0.649152i	$-0.224876\pi$
$733$	1.62212e10	1.52132	0.760659	+	0.649152i	$0.224876\pi$
$734$	−3.20342e9	−0.299004
$735$	0	0
$736$	−7.99461e9	−0.739137
$737$	7.41420e6	0.000682226 0
$738$	0	0
$739$	1.78796e10	1.62968	0.814841	−	0.579684i	$-0.196824\pi$
$739$	1.78796e10	1.62968	0.814841	+	0.579684i	$0.196824\pi$
$740$	0	0
$741$	0	0
$742$	7.30301e9	0.656278
$743$	1.10072e10	0.984499	0.492249	−	0.870454i	$-0.336175\pi$
$743$	1.10072e10	0.984499	0.492249	+	0.870454i	$0.336175\pi$
$744$	0	0
$745$	0	0
$746$	3.87602e9	0.341822
$747$	0	0
$748$	−6.16137e9	−0.538296
$749$	2.47531e10	2.15250
$750$	0	0
$751$	−8.82495e9	−0.760278	−0.380139	−	0.924929i	$-0.624124\pi$
$751$	−8.82495e9	−0.760278	−0.380139	+	0.924929i	$0.624124\pi$
$752$	−6.07671e9	−0.521082
$753$	0	0
$754$	1.91162e8	0.0162406
$755$	0	0
$756$	0	0
$757$	−1.26747e10	−1.06194	−0.530972	−	0.847389i	$-0.678173\pi$
$757$	−1.26747e10	−1.06194	−0.530972	+	0.847389i	$0.678173\pi$
$758$	−8.98754e8	−0.0749547
$759$	0	0
$760$	0	0
$761$	4.76538e9	0.391969	0.195984	−	0.980607i	$-0.437210\pi$
$761$	4.76538e9	0.391969	0.195984	+	0.980607i	$0.437210\pi$
$762$	0	0
$763$	−3.53487e10	−2.88097
$764$	1.83624e10	1.48971
$765$	0	0
$766$	2.44976e9	0.196935
$767$	−1.95086e9	−0.156114
$768$	0	0
$769$	−3.33620e9	−0.264552	−0.132276	−	0.991213i	$-0.542228\pi$
$769$	−3.33620e9	−0.264552	−0.132276	+	0.991213i	$0.542228\pi$
$770$	0	0
$771$	0	0
$772$	1.79919e10	1.40740
$773$	−1.22441e9	−0.0953453	−0.0476726	−	0.998863i	$-0.515180\pi$
$773$	−1.22441e9	−0.0953453	−0.0476726	+	0.998863i	$0.515180\pi$
$774$	0	0
$775$	0	0
$776$	−2.26060e9	−0.173663
$777$	0	0
$778$	1.02432e9	0.0779843
$779$	1.50798e10	1.14292
$780$	0	0
$781$	−4.45921e9	−0.334949
$782$	5.82587e9	0.435649
$783$	0	0
$784$	1.86916e10	1.38529
$785$	0	0
$786$	0	0
$787$	−1.72256e10	−1.25969	−0.629844	−	0.776721i	$-0.716881\pi$
$787$	−1.72256e10	−1.25969	−0.629844	+	0.776721i	$0.716881\pi$
$788$	1.47189e10	1.07160
$789$	0	0
$790$	0	0
$791$	1.89284e10	1.35986
$792$	0	0
$793$	3.62043e8	0.0257813
$794$	3.03051e9	0.214854
$795$	0	0
$796$	−5.68077e9	−0.399219
$797$	−1.01065e10	−0.707127	−0.353564	−	0.935411i	$-0.615030\pi$
$797$	−1.01065e10	−0.707127	−0.353564	+	0.935411i	$0.615030\pi$
$798$	0	0
$799$	1.52255e10	1.05599
$800$	0	0
$801$	0	0
$802$	3.30467e9	0.226213
$803$	3.21149e9	0.218878
$804$	0	0
$805$	0	0
$806$	5.68254e7	0.00382270
$807$	0	0
$808$	5.35367e9	0.357036
$809$	−9.46505e9	−0.628497	−0.314248	−	0.949341i	$-0.601753\pi$
$809$	−9.46505e9	−0.628497	−0.314248	+	0.949341i	$0.601753\pi$
$810$	0	0
$811$	1.67478e10	1.10251	0.551257	−	0.834335i	$-0.314148\pi$
$811$	1.67478e10	1.10251	0.551257	+	0.834335i	$0.314148\pi$
$812$	−1.18359e10	−0.775810
$813$	0	0
$814$	−1.81443e9	−0.117911
$815$	0	0
$816$	0	0
$817$	−1.82684e10	−1.17199
$818$	3.30996e9	0.211439
$819$	0	0
$820$	0	0
$821$	3.34978e9	0.211259	0.105630	−	0.994406i	$-0.466314\pi$
$821$	3.34978e9	0.211259	0.105630	+	0.994406i	$0.466314\pi$
$822$	0	0
$823$	6.69632e9	0.418732	0.209366	−	0.977837i	$-0.432860\pi$
$823$	6.69632e9	0.418732	0.209366	+	0.977837i	$0.432860\pi$
$824$	1.08709e10	0.676893
$825$	0	0
$826$	−9.28336e9	−0.573159
$827$	5.81000e9	0.357196	0.178598	−	0.983922i	$-0.442844\pi$
$827$	5.81000e9	0.357196	0.178598	+	0.983922i	$0.442844\pi$
$828$	0	0
$829$	6.84048e9	0.417009	0.208505	−	0.978021i	$-0.433140\pi$
$829$	6.84048e9	0.417009	0.208505	+	0.978021i	$0.433140\pi$
$830$	0	0
$831$	0	0
$832$	1.19680e9	0.0720424
$833$	−4.68329e10	−2.80733
$834$	0	0
$835$	0	0
$836$	−7.43224e9	−0.439945
$837$	0	0
$838$	5.29529e9	0.310839
$839$	1.04085e8	0.00608445	0.00304223	−	0.999995i	$-0.499032\pi$
$839$	1.04085e8	0.00608445	0.00304223	+	0.999995i	$0.499032\pi$
$840$	0	0
$841$	−1.28740e10	−0.746324
$842$	−1.38438e9	−0.0799215
$843$	0	0
$844$	−7.46773e9	−0.427553
$845$	0	0
$846$	0	0
$847$	2.54975e10	1.44180
$848$	−2.08020e10	−1.17144
$849$	0	0
$850$	0	0
$851$	−2.23226e10	−1.24163
$852$	0	0
$853$	6.47246e9	0.357066	0.178533	−	0.983934i	$-0.442865\pi$
$853$	6.47246e9	0.357066	0.178533	+	0.983934i	$0.442865\pi$
$854$	1.72282e9	0.0946536
$855$	0	0
$856$	1.22698e10	0.668619
$857$	2.27261e10	1.23337	0.616684	−	0.787211i	$-0.288476\pi$
$857$	2.27261e10	1.23337	0.616684	+	0.787211i	$0.288476\pi$
$858$	0	0
$859$	−1.08363e10	−0.583315	−0.291657	−	0.956523i	$-0.594207\pi$
$859$	−1.08363e10	−0.583315	−0.291657	+	0.956523i	$0.594207\pi$
$860$	0	0
$861$	0	0
$862$	2.29749e9	0.122174
$863$	3.23218e10	1.71182	0.855910	−	0.517125i	$-0.172998\pi$
$863$	3.23218e10	1.71182	0.855910	+	0.517125i	$0.172998\pi$
$864$	0	0
$865$	0	0
$866$	7.50973e9	0.392927
$867$	0	0
$868$	−3.51838e9	−0.182610
$869$	−2.82186e9	−0.145870
$870$	0	0
$871$	4.44051e6	0.000227703 0
$872$	−1.75219e10	−0.894899
$873$	0	0
$874$	7.02754e9	0.356052
$875$	0	0
$876$	0	0
$877$	4.21367e9	0.210941	0.105471	−	0.994422i	$-0.466365\pi$
$877$	4.21367e9	0.210941	0.105471	+	0.994422i	$0.466365\pi$
$878$	5.13624e8	0.0256103
$879$	0	0
$880$	0	0
$881$	−5.03021e9	−0.247840	−0.123920	−	0.992292i	$-0.539547\pi$
$881$	−5.03021e9	−0.247840	−0.123920	+	0.992292i	$0.539547\pi$
$882$	0	0
$883$	2.81606e10	1.37651	0.688255	−	0.725469i	$-0.258377\pi$
$883$	2.81606e10	1.37651	0.688255	+	0.725469i	$0.258377\pi$
$884$	−3.69016e9	−0.179665
$885$	0	0
$886$	−4.63184e9	−0.223736
$887$	1.59422e10	0.767038	0.383519	−	0.923533i	$-0.374712\pi$
$887$	1.59422e10	0.767038	0.383519	+	0.923533i	$0.374712\pi$
$888$	0	0
$889$	−3.62877e10	−1.73222
$890$	0	0
$891$	0	0
$892$	−6.15578e8	−0.0290406
$893$	1.83660e10	0.863049
$894$	0	0
$895$	0	0
$896$	3.16440e10	1.46965
$897$	0	0
$898$	1.04374e10	0.480980
$899$	1.30079e9	0.0597101
$900$	0	0
$901$	5.21206e10	2.37396
$902$	1.85765e9	0.0842833
$903$	0	0
$904$	9.38255e9	0.422407
$905$	0	0
$906$	0	0
$907$	2.20381e10	0.980729	0.490364	−	0.871517i	$-0.336864\pi$
$907$	2.20381e10	0.980729	0.490364	+	0.871517i	$0.336864\pi$
$908$	−2.89408e10	−1.28295
$909$	0	0
$910$	0	0
$911$	−1.50904e10	−0.661283	−0.330641	−	0.943756i	$-0.607265\pi$
$911$	−1.50904e10	−0.661283	−0.330641	+	0.943756i	$0.607265\pi$
$912$	0	0
$913$	−4.89489e9	−0.212860
$914$	−2.28585e9	−0.0990229
$915$	0	0
$916$	1.30434e10	0.560736
$917$	−3.13158e10	−1.34113
$918$	0	0
$919$	9.21105e9	0.391476	0.195738	−	0.980656i	$-0.437290\pi$
$919$	9.21105e9	0.391476	0.195738	+	0.980656i	$0.437290\pi$
$920$	0	0
$921$	0	0
$922$	2.65457e9	0.111541
$923$	−2.67071e9	−0.111794
$924$	0	0
$925$	0	0
$926$	−1.07290e10	−0.444040
$927$	0	0
$928$	−8.90892e9	−0.365938
$929$	−4.08831e10	−1.67297	−0.836487	−	0.547987i	$-0.815394\pi$
$929$	−4.08831e10	−1.67297	−0.836487	+	0.547987i	$0.815394\pi$
$930$	0	0
$931$	−5.64929e10	−2.29440
$932$	2.20816e10	0.893461
$933$	0	0
$934$	−9.16912e9	−0.368225
$935$	0	0
$936$	0	0
$937$	−3.38015e10	−1.34229	−0.671147	−	0.741324i	$-0.734198\pi$
$937$	−3.38015e10	−1.34229	−0.671147	+	0.741324i	$0.734198\pi$
$938$	2.11306e7	0.000835992 0
$939$	0	0
$940$	0	0
$941$	−2.16815e10	−0.848252	−0.424126	−	0.905603i	$-0.639419\pi$
$941$	−2.16815e10	−0.848252	−0.424126	+	0.905603i	$0.639419\pi$
$942$	0	0
$943$	2.28543e10	0.887519
$944$	2.64429e10	1.02307
$945$	0	0
$946$	−2.25045e9	−0.0864271
$947$	−1.47060e10	−0.562692	−0.281346	−	0.959606i	$-0.590781\pi$
$947$	−1.47060e10	−0.562692	−0.281346	+	0.959606i	$0.590781\pi$
$948$	0	0
$949$	1.92342e9	0.0730538
$950$	0	0
$951$	0	0
$952$	−3.64696e10	−1.36994
$953$	−1.26663e10	−0.474050	−0.237025	−	0.971504i	$-0.576172\pi$
$953$	−1.26663e10	−0.474050	−0.237025	+	0.971504i	$0.576172\pi$
$954$	0	0
$955$	0	0
$956$	3.64102e10	1.34778
$957$	0	0
$958$	−9.07785e9	−0.333583
$959$	1.49671e10	0.547989
$960$	0	0
$961$	−2.71259e10	−0.985945
$962$	−1.08670e9	−0.0393547
$963$	0	0
$964$	6.87247e9	0.247083
$965$	0	0
$966$	0	0
$967$	2.68713e10	0.955644	0.477822	−	0.878457i	$-0.341426\pi$
$967$	2.68713e10	0.955644	0.477822	+	0.878457i	$0.341426\pi$
$968$	1.26388e10	0.447860
$969$	0	0
$970$	0	0
$971$	−1.85571e9	−0.0650494	−0.0325247	−	0.999471i	$-0.510355\pi$
$971$	−1.85571e9	−0.0650494	−0.0325247	+	0.999471i	$0.510355\pi$
$972$	0	0
$973$	−2.21898e10	−0.772252
$974$	9.93340e8	0.0344462
$975$	0	0
$976$	−4.90731e9	−0.168954
$977$	−3.13360e10	−1.07501	−0.537506	−	0.843260i	$-0.680633\pi$
$977$	−3.13360e10	−1.07501	−0.537506	+	0.843260i	$0.680633\pi$
$978$	0	0
$979$	−9.80808e9	−0.334075
$980$	0	0
$981$	0	0
$982$	2.10751e9	0.0710198
$983$	2.78355e9	0.0934679	0.0467339	−	0.998907i	$-0.485119\pi$
$983$	2.78355e9	0.0934679	0.0467339	+	0.998907i	$0.485119\pi$
$984$	0	0
$985$	0	0
$986$	6.49215e9	0.215685
$987$	0	0
$988$	−4.45132e9	−0.146838
$989$	−2.76868e10	−0.910094
$990$	0	0
$991$	2.20283e10	0.718991	0.359495	−	0.933147i	$-0.382949\pi$
$991$	2.20283e10	0.718991	0.359495	+	0.933147i	$0.382949\pi$
$992$	−2.64829e9	−0.0861341
$993$	0	0
$994$	−1.27088e10	−0.410443
$995$	0	0
$996$	0	0
$997$	5.02880e10	1.60706	0.803528	−	0.595266i	$-0.202953\pi$
$997$	5.02880e10	1.60706	0.803528	+	0.595266i	$0.202953\pi$
$998$	−9.61836e9	−0.306298
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.8.a.z.1.3		4
3.2	odd	2		75.8.a.j.1.2		4
5.2	odd	4		45.8.b.d.19.5		8
5.3	odd	4		45.8.b.d.19.4		8
5.4	even	2		225.8.a.bb.1.2		4
15.2	even	4		15.8.b.a.4.4	✓	8
15.8	even	4		15.8.b.a.4.5	yes	8
15.14	odd	2		75.8.a.i.1.3		4
60.23	odd	4		240.8.f.e.49.1		8
60.47	odd	4		240.8.f.e.49.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.b.a.4.4	✓	8	15.2	even	4
15.8.b.a.4.5	yes	8	15.8	even	4
45.8.b.d.19.4		8	5.3	odd	4
45.8.b.d.19.5		8	5.2	odd	4
75.8.a.i.1.3		4	15.14	odd	2
75.8.a.j.1.2		4	3.2	odd	2
225.8.a.z.1.3		4	1.1	even	1	trivial
225.8.a.bb.1.2		4	5.4	even	2
240.8.f.e.49.1		8	60.23	odd	4
240.8.f.e.49.5		8	60.47	odd	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+3.02250 q^{2} -118.865 q^{4} -1505.28 q^{7} -746.147 q^{8} -1596.37 q^{11} -956.100 q^{13} -4549.69 q^{14} +12959.4 q^{16} -32470.6 q^{17} -39168.1 q^{19} -4825.04 q^{22} -59361.5 q^{23} -2889.81 q^{26} +178924. q^{28} -66150.5 q^{29} -19664.1 q^{31} +134677. q^{32} -98142.2 q^{34} +376045. q^{37} -118386. q^{38} -385003. q^{41} +466410. q^{43} +189752. q^{44} -179420. q^{46} -468903. q^{47} +1.44232e6 q^{49} +113646. q^{52} -1.60516e6 q^{53} +1.12316e6 q^{56} -199940. q^{58} +2.04044e6 q^{59} -378667. q^{61} -59434.6 q^{62} -1.25175e6 q^{64} -4644.40 q^{67} +3.85960e6 q^{68} +2.79333e6 q^{71} -2.01174e6 q^{73} +1.13660e6 q^{74} +4.65570e6 q^{76} +2.40299e6 q^{77} +1.76767e6 q^{79} -1.16367e6 q^{82} +3.06625e6 q^{83} +1.40972e6 q^{86} +1.19113e6 q^{88} +6.14397e6 q^{89} +1.43920e6 q^{91} +7.05598e6 q^{92} -1.41726e6 q^{94} +3.02969e6 q^{97} +4.35940e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 9 q^{2} + 333 q^{4} + 1188 q^{7} - 2043 q^{8} - 5376 q^{11} + 8424 q^{13} - 6762 q^{14} + 43265 q^{16} - 4896 q^{17} + 15232 q^{19} - 44118 q^{22} - 110016 q^{23} + 396762 q^{26} + 674514 q^{28} + 120036 q^{29}+ \cdots + 20819979 q^{98}+O(q^{100}) \) 4 * q - 9 * q^2 + 333 * q^4 + 1188 * q^7 - 2043 * q^8 - 5376 * q^11 + 8424 * q^13 - 6762 * q^14 + 43265 * q^16 - 4896 * q^17 + 15232 * q^19 - 44118 * q^22 - 110016 * q^23 + 396762 * q^26 + 674514 * q^28 + 120036 * q^29 + 116864 * q^31 - 621963 * q^32 - 92374 * q^34 + 663768 * q^37 - 1888776 * q^38 - 253824 * q^41 + 1092960 * q^43 + 1289286 * q^44 + 331204 * q^46 - 132840 * q^47 + 1633580 * q^49 - 779274 * q^52 - 1994616 * q^53 - 352830 * q^56 - 742608 * q^58 + 545712 * q^59 - 3216760 * q^61 - 476856 * q^62 + 284297 * q^64 + 2013336 * q^67 + 7540038 * q^68 + 690912 * q^71 - 5498064 * q^73 + 11980638 * q^74 + 6557832 * q^76 + 795744 * q^77 + 7190080 * q^79 - 26456814 * q^82 + 13158432 * q^83 + 9918804 * q^86 - 22119642 * q^88 + 22889448 * q^89 + 12037824 * q^91 + 17368812 * q^92 + 22864448 * q^94 + 17873136 * q^97 + 20819979 * q^98

Introduction

L-functions

Modular forms

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