LMFDB - Embedded newform 476.4.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [476,4,Mod(1,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("476.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 476 = 2^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	476.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:	$28.0849091627$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 71x^{4} - 32x^{3} + 547x^{2} + 268x - 720 $ x^6 - 2x^5 - 71x^4 - 32x^3 + 547x^2 + 268*x - 720
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.00920$ of defining polynomial
Character	$\chi$	$=$	476.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00920 q^{3} +15.3947 q^{5} -7.00000 q^{7} -25.9815 q^{9} +25.7348 q^{11} +16.0232 q^{13} +15.5364 q^{15} +17.0000 q^{17} +157.385 q^{19} -7.06441 q^{21} -152.339 q^{23} +111.998 q^{25} -53.4690 q^{27} +293.792 q^{29} +246.190 q^{31} +25.9715 q^{33} -107.763 q^{35} -102.495 q^{37} +16.1707 q^{39} -80.0959 q^{41} -311.232 q^{43} -399.979 q^{45} +449.527 q^{47} +49.0000 q^{49} +17.1564 q^{51} -518.705 q^{53} +396.180 q^{55} +158.833 q^{57} +810.255 q^{59} +101.071 q^{61} +181.871 q^{63} +246.673 q^{65} -233.289 q^{67} -153.740 q^{69} +981.570 q^{71} -274.218 q^{73} +113.029 q^{75} -180.143 q^{77} -218.065 q^{79} +647.540 q^{81} +432.623 q^{83} +261.711 q^{85} +296.495 q^{87} +1019.40 q^{89} -112.163 q^{91} +248.456 q^{93} +2422.91 q^{95} +466.343 q^{97} -668.628 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 14 q^{5} - 42 q^{7} - 16 q^{9} - 2 q^{11} + 8 q^{13} - 16 q^{15} + 102 q^{17} + 170 q^{19} - 14 q^{21} + 102 q^{23} + 140 q^{25} + 422 q^{27} + 278 q^{29} + 380 q^{31} + 98 q^{33} + 98 q^{35}+ \cdots + 450 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 14 * q^5 - 42 * q^7 - 16 * q^9 - 2 * q^11 + 8 * q^13 - 16 * q^15 + 102 * q^17 + 170 * q^19 - 14 * q^21 + 102 * q^23 + 140 * q^25 + 422 * q^27 + 278 * q^29 + 380 * q^31 + 98 * q^33 + 98 * q^35 + 542 * q^37 + 790 * q^39 + 224 * q^41 + 104 * q^43 + 294 * q^45 + 1022 * q^47 + 294 * q^49 + 34 * q^51 + 554 * q^53 + 496 * q^55 + 716 * q^57 + 844 * q^59 + 452 * q^61 + 112 * q^63 + 746 * q^65 + 1376 * q^67 + 664 * q^69 + 1062 * q^71 + 208 * q^73 - 134 * q^75 + 14 * q^77 + 1410 * q^79 + 1850 * q^81 + 232 * q^83 - 238 * q^85 + 1350 * q^87 - 654 * q^89 - 56 * q^91 + 3564 * q^93 + 3186 * q^95 - 1546 * q^97 + 450 * 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 $

$2$	0	0
$2$	0	0
$3$	1.00920	0.194221	0.0971104	−	0.995274i	$-0.469040\pi$
$3$	1.00920	0.194221	0.0971104	+	0.995274i	$0.469040\pi$
$4$	0	0
$5$	15.3947	1.37695	0.688474	−	0.725261i	$-0.258281\pi$
$5$	15.3947	1.37695	0.688474	+	0.725261i	$0.258281\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−25.9815	−0.962278
$10$	0	0
$11$	25.7348	0.705393	0.352696	−	0.935738i	$-0.385265\pi$
$11$	25.7348	0.705393	0.352696	+	0.935738i	$0.385265\pi$
$12$	0	0
$13$	16.0232	0.341849	0.170925	−	0.985284i	$-0.445324\pi$
$13$	16.0232	0.341849	0.170925	+	0.985284i	$0.445324\pi$
$14$	0	0
$15$	15.5364	0.267432
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	157.385	1.90035	0.950175	−	0.311717i	$-0.100904\pi$
$19$	157.385	1.90035	0.950175	+	0.311717i	$0.100904\pi$
$20$	0	0
$21$	−7.06441	−0.0734086
$22$	0	0
$23$	−152.339	−1.38108	−0.690540	−	0.723294i	$-0.742627\pi$
$23$	−152.339	−1.38108	−0.690540	+	0.723294i	$0.742627\pi$
$24$	0	0
$25$	111.998	0.895985
$26$	0	0
$27$	−53.4690	−0.381115
$28$	0	0
$29$	293.792	1.88124	0.940618	−	0.339466i	$-0.110246\pi$
$29$	293.792	1.88124	0.940618	+	0.339466i	$0.110246\pi$
$30$	0	0
$31$	246.190	1.42636	0.713179	−	0.700982i	$-0.247255\pi$
$31$	246.190	1.42636	0.713179	+	0.700982i	$0.247255\pi$
$32$	0	0
$33$	25.9715	0.137002
$34$	0	0
$35$	−107.763	−0.520437
$36$	0	0
$37$	−102.495	−0.455405	−0.227703	−	0.973731i	$-0.573121\pi$
$37$	−102.495	−0.455405	−0.227703	+	0.973731i	$0.573121\pi$
$38$	0	0
$39$	16.1707	0.0663943
$40$	0	0
$41$	−80.0959	−0.305094	−0.152547	−	0.988296i	$-0.548748\pi$
$41$	−80.0959	−0.305094	−0.152547	+	0.988296i	$0.548748\pi$
$42$	0	0
$43$	−311.232	−1.10378	−0.551889	−	0.833918i	$-0.686093\pi$
$43$	−311.232	−1.10378	−0.551889	+	0.833918i	$0.686093\pi$
$44$	0	0
$45$	−399.979	−1.32501
$46$	0	0
$47$	449.527	1.39511	0.697556	−	0.716530i	$-0.254271\pi$
$47$	449.527	1.39511	0.697556	+	0.716530i	$0.254271\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	17.1564	0.0471055
$52$	0	0
$53$	−518.705	−1.34433	−0.672167	−	0.740400i	$-0.734636\pi$
$53$	−518.705	−1.34433	−0.672167	+	0.740400i	$0.734636\pi$
$54$	0	0
$55$	396.180	0.971289
$56$	0	0
$57$	158.833	0.369088
$58$	0	0
$59$	810.255	1.78790	0.893951	−	0.448164i	$-0.147922\pi$
$59$	810.255	1.78790	0.893951	+	0.448164i	$0.147922\pi$
$60$	0	0
$61$	101.071	0.212144	0.106072	−	0.994358i	$-0.466173\pi$
$61$	101.071	0.212144	0.106072	+	0.994358i	$0.466173\pi$
$62$	0	0
$63$	181.871	0.363707
$64$	0	0
$65$	246.673	0.470709
$66$	0	0
$67$	−233.289	−0.425385	−0.212693	−	0.977119i	$-0.568223\pi$
$67$	−233.289	−0.425385	−0.212693	+	0.977119i	$0.568223\pi$
$68$	0	0
$69$	−153.740	−0.268234
$70$	0	0
$71$	981.570	1.64072	0.820358	−	0.571850i	$-0.193774\pi$
$71$	981.570	1.64072	0.820358	+	0.571850i	$0.193774\pi$
$72$	0	0
$73$	−274.218	−0.439654	−0.219827	−	0.975539i	$-0.570549\pi$
$73$	−274.218	−0.439654	−0.219827	+	0.975539i	$0.570549\pi$
$74$	0	0
$75$	113.029	0.174019
$76$	0	0
$77$	−180.143	−0.266613
$78$	0	0
$79$	−218.065	−0.310560	−0.155280	−	0.987870i	$-0.549628\pi$
$79$	−218.065	−0.310560	−0.155280	+	0.987870i	$0.549628\pi$
$80$	0	0
$81$	647.540	0.888258
$82$	0	0
$83$	432.623	0.572127	0.286064	−	0.958211i	$-0.407653\pi$
$83$	432.623	0.572127	0.286064	+	0.958211i	$0.407653\pi$
$84$	0	0
$85$	261.711	0.333959
$86$	0	0
$87$	296.495	0.365375
$88$	0	0
$89$	1019.40	1.21412	0.607058	−	0.794657i	$-0.292349\pi$
$89$	1019.40	1.21412	0.607058	+	0.794657i	$0.292349\pi$
$90$	0	0
$91$	−112.163	−0.129207
$92$	0	0
$93$	248.456	0.277028
$94$	0	0
$95$	2422.91	2.61668
$96$	0	0
$97$	466.343	0.488144	0.244072	−	0.969757i	$-0.421517\pi$
$97$	466.343	0.488144	0.244072	+	0.969757i	$0.421517\pi$
$98$	0	0
$99$	−668.628	−0.678784
$100$	0	0
$101$	317.874	0.313165	0.156583	−	0.987665i	$-0.449952\pi$
$101$	317.874	0.313165	0.156583	+	0.987665i	$0.449952\pi$
$102$	0	0
$103$	659.029	0.630447	0.315224	−	0.949017i	$-0.397920\pi$
$103$	659.029	0.630447	0.315224	+	0.949017i	$0.397920\pi$
$104$	0	0
$105$	−108.755	−0.101080
$106$	0	0
$107$	−471.704	−0.426181	−0.213091	−	0.977032i	$-0.568353\pi$
$107$	−471.704	−0.426181	−0.213091	+	0.977032i	$0.568353\pi$
$108$	0	0
$109$	−500.205	−0.439550	−0.219775	−	0.975551i	$-0.570532\pi$
$109$	−500.205	−0.439550	−0.219775	+	0.975551i	$0.570532\pi$
$110$	0	0
$111$	−103.438	−0.0884492
$112$	0	0
$113$	−123.992	−0.103223	−0.0516115	−	0.998667i	$-0.516436\pi$
$113$	−123.992	−0.103223	−0.0516115	+	0.998667i	$0.516436\pi$
$114$	0	0
$115$	−2345.22	−1.90167
$116$	0	0
$117$	−416.308	−0.328954
$118$	0	0
$119$	−119.000	−0.0916698
$120$	0	0
$121$	−668.722	−0.502421
$122$	0	0
$123$	−80.8328	−0.0592557
$124$	0	0
$125$	−200.161	−0.143224
$126$	0	0
$127$	−1011.83	−0.706974	−0.353487	−	0.935439i	$-0.615004\pi$
$127$	−1011.83	−0.706974	−0.353487	+	0.935439i	$0.615004\pi$
$128$	0	0
$129$	−314.096	−0.214377
$130$	0	0
$131$	686.919	0.458140	0.229070	−	0.973410i	$-0.426431\pi$
$131$	686.919	0.458140	0.229070	+	0.973410i	$0.426431\pi$
$132$	0	0
$133$	−1101.70	−0.718265
$134$	0	0
$135$	−823.141	−0.524776
$136$	0	0
$137$	−2091.92	−1.30456	−0.652279	−	0.757979i	$-0.726187\pi$
$137$	−2091.92	−1.30456	−0.652279	+	0.757979i	$0.726187\pi$
$138$	0	0
$139$	1372.65	0.837605	0.418802	−	0.908077i	$-0.362450\pi$
$139$	1372.65	0.837605	0.418802	+	0.908077i	$0.362450\pi$
$140$	0	0
$141$	453.663	0.270960
$142$	0	0
$143$	412.354	0.241138
$144$	0	0
$145$	4522.86	2.59036
$146$	0	0
$147$	49.4508	0.0277458
$148$	0	0
$149$	−2399.64	−1.31937	−0.659684	−	0.751543i	$-0.729310\pi$
$149$	−2399.64	−1.31937	−0.659684	+	0.751543i	$0.729310\pi$
$150$	0	0
$151$	−3143.05	−1.69389	−0.846946	−	0.531678i	$-0.821562\pi$
$151$	−3143.05	−1.69389	−0.846946	+	0.531678i	$0.821562\pi$
$152$	0	0
$153$	−441.686	−0.233387
$154$	0	0
$155$	3790.04	1.96402
$156$	0	0
$157$	−167.866	−0.0853323	−0.0426662	−	0.999089i	$-0.513585\pi$
$157$	−167.866	−0.0853323	−0.0426662	+	0.999089i	$0.513585\pi$
$158$	0	0
$159$	−523.478	−0.261097
$160$	0	0
$161$	1066.37	0.521999
$162$	0	0
$163$	4127.67	1.98346	0.991729	−	0.128346i	$-0.0409669\pi$
$163$	4127.67	1.98346	0.991729	+	0.128346i	$0.0409669\pi$
$164$	0	0
$165$	399.825	0.188645
$166$	0	0
$167$	−1829.02	−0.847507	−0.423753	−	0.905778i	$-0.639288\pi$
$167$	−1829.02	−0.847507	−0.423753	+	0.905778i	$0.639288\pi$
$168$	0	0
$169$	−1940.26	−0.883139
$170$	0	0
$171$	−4089.11	−1.82867
$172$	0	0
$173$	−1930.68	−0.848481	−0.424240	−	0.905550i	$-0.639459\pi$
$173$	−1930.68	−0.848481	−0.424240	+	0.905550i	$0.639459\pi$
$174$	0	0
$175$	−783.987	−0.338650
$176$	0	0
$177$	817.710	0.347248
$178$	0	0
$179$	−1322.13	−0.552069	−0.276034	−	0.961148i	$-0.589020\pi$
$179$	−1322.13	−0.552069	−0.276034	+	0.961148i	$0.589020\pi$
$180$	0	0
$181$	494.916	0.203242	0.101621	−	0.994823i	$-0.467597\pi$
$181$	494.916	0.203242	0.101621	+	0.994823i	$0.467597\pi$
$182$	0	0
$183$	102.001	0.0412027
$184$	0	0
$185$	−1577.88	−0.627069
$186$	0	0
$187$	437.491	0.171083
$188$	0	0
$189$	374.283	0.144048
$190$	0	0
$191$	124.336	0.0471028	0.0235514	−	0.999723i	$-0.492503\pi$
$191$	124.336	0.0471028	0.0235514	+	0.999723i	$0.492503\pi$
$192$	0	0
$193$	2718.16	1.01377	0.506884	−	0.862014i	$-0.330797\pi$
$193$	2718.16	1.01377	0.506884	+	0.862014i	$0.330797\pi$
$194$	0	0
$195$	248.943	0.0914215
$196$	0	0
$197$	−3498.30	−1.26520	−0.632598	−	0.774481i	$-0.718011\pi$
$197$	−3498.30	−1.26520	−0.632598	+	0.774481i	$0.718011\pi$
$198$	0	0
$199$	1175.36	0.418688	0.209344	−	0.977842i	$-0.432867\pi$
$199$	1175.36	0.418688	0.209344	+	0.977842i	$0.432867\pi$
$200$	0	0
$201$	−235.436	−0.0826187
$202$	0	0
$203$	−2056.55	−0.711041
$204$	0	0
$205$	−1233.06	−0.420099
$206$	0	0
$207$	3957.99	1.32898
$208$	0	0
$209$	4050.27	1.34049
$210$	0	0
$211$	−3063.75	−0.999608	−0.499804	−	0.866138i	$-0.666595\pi$
$211$	−3063.75	−0.999608	−0.499804	+	0.866138i	$0.666595\pi$
$212$	0	0
$213$	990.601	0.318661
$214$	0	0
$215$	−4791.34	−1.51984
$216$	0	0
$217$	−1723.33	−0.539113
$218$	0	0
$219$	−276.741	−0.0853900
$220$	0	0
$221$	272.395	0.0829107
$222$	0	0
$223$	−3458.24	−1.03848	−0.519239	−	0.854629i	$-0.673785\pi$
$223$	−3458.24	−1.03848	−0.519239	+	0.854629i	$0.673785\pi$
$224$	0	0
$225$	−2909.88	−0.862187
$226$	0	0
$227$	−1731.27	−0.506205	−0.253102	−	0.967439i	$-0.581451\pi$
$227$	−1731.27	−0.506205	−0.253102	+	0.967439i	$0.581451\pi$
$228$	0	0
$229$	−4666.12	−1.34649	−0.673245	−	0.739420i	$-0.735100\pi$
$229$	−4666.12	−1.34649	−0.673245	+	0.739420i	$0.735100\pi$
$230$	0	0
$231$	−181.801	−0.0517819
$232$	0	0
$233$	6137.20	1.72559	0.862793	−	0.505558i	$-0.168713\pi$
$233$	6137.20	1.72559	0.862793	+	0.505558i	$0.168713\pi$
$234$	0	0
$235$	6920.35	1.92100
$236$	0	0
$237$	−220.072	−0.0603172
$238$	0	0
$239$	−1399.51	−0.378774	−0.189387	−	0.981903i	$-0.560650\pi$
$239$	−1399.51	−0.378774	−0.189387	+	0.981903i	$0.560650\pi$
$240$	0	0
$241$	1160.08	0.310072	0.155036	−	0.987909i	$-0.450451\pi$
$241$	1160.08	0.310072	0.155036	+	0.987909i	$0.450451\pi$
$242$	0	0
$243$	2097.16	0.553633
$244$	0	0
$245$	754.342	0.196707
$246$	0	0
$247$	2521.82	0.649634
$248$	0	0
$249$	436.604	0.111119
$250$	0	0
$251$	−7093.95	−1.78393	−0.891965	−	0.452105i	$-0.850673\pi$
$251$	−7093.95	−1.78393	−0.891965	+	0.452105i	$0.850673\pi$
$252$	0	0
$253$	−3920.40	−0.974204
$254$	0	0
$255$	264.119	0.0648618
$256$	0	0
$257$	−6646.89	−1.61331	−0.806657	−	0.591019i	$-0.798726\pi$
$257$	−6646.89	−1.61331	−0.806657	+	0.591019i	$0.798726\pi$
$258$	0	0
$259$	717.462	0.172127
$260$	0	0
$261$	−7633.17	−1.81027
$262$	0	0
$263$	7548.59	1.76983	0.884916	−	0.465751i	$-0.154216\pi$
$263$	7548.59	1.76983	0.884916	+	0.465751i	$0.154216\pi$
$264$	0	0
$265$	−7985.33	−1.85108
$266$	0	0
$267$	1028.78	0.235807
$268$	0	0
$269$	−7286.99	−1.65166	−0.825828	−	0.563922i	$-0.809292\pi$
$269$	−7286.99	−1.65166	−0.825828	+	0.563922i	$0.809292\pi$
$270$	0	0
$271$	3457.10	0.774922	0.387461	−	0.921886i	$-0.373352\pi$
$271$	3457.10	0.774922	0.387461	+	0.921886i	$0.373352\pi$
$272$	0	0
$273$	−113.195	−0.0250947
$274$	0	0
$275$	2882.24	0.632021
$276$	0	0
$277$	−2279.17	−0.494377	−0.247188	−	0.968967i	$-0.579507\pi$
$277$	−2279.17	−0.494377	−0.247188	+	0.968967i	$0.579507\pi$
$278$	0	0
$279$	−6396.40	−1.37255
$280$	0	0
$281$	−5833.37	−1.23840	−0.619199	−	0.785234i	$-0.712543\pi$
$281$	−5833.37	−1.23840	−0.619199	+	0.785234i	$0.712543\pi$
$282$	0	0
$283$	5449.45	1.14465	0.572325	−	0.820027i	$-0.306042\pi$
$283$	5449.45	1.14465	0.572325	+	0.820027i	$0.306042\pi$
$284$	0	0
$285$	2445.20	0.508214
$286$	0	0
$287$	560.671	0.115315
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	470.634	0.0948077
$292$	0	0
$293$	2639.75	0.526334	0.263167	−	0.964750i	$-0.415233\pi$
$293$	2639.75	0.526334	0.263167	+	0.964750i	$0.415233\pi$
$294$	0	0
$295$	12473.7	2.46185
$296$	0	0
$297$	−1376.01	−0.268836
$298$	0	0
$299$	−2440.96	−0.472121
$300$	0	0
$301$	2178.63	0.417189
$302$	0	0
$303$	320.799	0.0608232
$304$	0	0
$305$	1555.96	0.292111
$306$	0	0
$307$	−6167.27	−1.14653	−0.573265	−	0.819370i	$-0.694323\pi$
$307$	−6167.27	−1.14653	−0.573265	+	0.819370i	$0.694323\pi$
$308$	0	0
$309$	665.093	0.122446
$310$	0	0
$311$	6519.01	1.18862	0.594308	−	0.804238i	$-0.297426\pi$
$311$	6519.01	1.18862	0.594308	+	0.804238i	$0.297426\pi$
$312$	0	0
$313$	−10196.8	−1.84140	−0.920702	−	0.390267i	$-0.872383\pi$
$313$	−10196.8	−1.84140	−0.920702	+	0.390267i	$0.872383\pi$
$314$	0	0
$315$	2799.85	0.500805
$316$	0	0
$317$	−5456.49	−0.966773	−0.483387	−	0.875407i	$-0.660593\pi$
$317$	−5456.49	−0.966773	−0.483387	+	0.875407i	$0.660593\pi$
$318$	0	0
$319$	7560.67	1.32701
$320$	0	0
$321$	−476.045	−0.0827733
$322$	0	0
$323$	2675.55	0.460903
$324$	0	0
$325$	1794.57	0.306292
$326$	0	0
$327$	−504.807	−0.0853697
$328$	0	0
$329$	−3146.69	−0.527303
$330$	0	0
$331$	42.4952	0.00705663	0.00352832	−	0.999994i	$-0.498877\pi$
$331$	42.4952	0.00705663	0.00352832	+	0.999994i	$0.498877\pi$
$332$	0	0
$333$	2662.96	0.438227
$334$	0	0
$335$	−3591.43	−0.585733
$336$	0	0
$337$	−9804.74	−1.58486	−0.792431	−	0.609962i	$-0.791185\pi$
$337$	−9804.74	−1.58486	−0.792431	+	0.609962i	$0.791185\pi$
$338$	0	0
$339$	−125.133	−0.0200480
$340$	0	0
$341$	6335.65	1.00614
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−2366.79	−0.369345
$346$	0	0
$347$	−6842.22	−1.05853	−0.529265	−	0.848457i	$-0.677532\pi$
$347$	−6842.22	−1.05853	−0.529265	+	0.848457i	$0.677532\pi$
$348$	0	0
$349$	10649.2	1.63335	0.816674	−	0.577099i	$-0.195815\pi$
$349$	10649.2	1.63335	0.816674	+	0.577099i	$0.195815\pi$
$350$	0	0
$351$	−856.746	−0.130284
$352$	0	0
$353$	−1594.80	−0.240461	−0.120231	−	0.992746i	$-0.538363\pi$
$353$	−1594.80	−0.240461	−0.120231	+	0.992746i	$0.538363\pi$
$354$	0	0
$355$	15111.0	2.25918
$356$	0	0
$357$	−120.095	−0.0178042
$358$	0	0
$359$	4241.38	0.623542	0.311771	−	0.950157i	$-0.399078\pi$
$359$	4241.38	0.623542	0.311771	+	0.950157i	$0.399078\pi$
$360$	0	0
$361$	17911.1	2.61133
$362$	0	0
$363$	−674.875	−0.0975806
$364$	0	0
$365$	−4221.51	−0.605381
$366$	0	0
$367$	6814.84	0.969296	0.484648	−	0.874709i	$-0.338948\pi$
$367$	6814.84	0.969296	0.484648	+	0.874709i	$0.338948\pi$
$368$	0	0
$369$	2081.01	0.293586
$370$	0	0
$371$	3630.94	0.508110
$372$	0	0
$373$	1982.82	0.275245	0.137623	−	0.990485i	$-0.456054\pi$
$373$	1982.82	0.275245	0.137623	+	0.990485i	$0.456054\pi$
$374$	0	0
$375$	−202.003	−0.0278170
$376$	0	0
$377$	4707.50	0.643100
$378$	0	0
$379$	−3805.19	−0.515725	−0.257862	−	0.966182i	$-0.583018\pi$
$379$	−3805.19	−0.515725	−0.257862	+	0.966182i	$0.583018\pi$
$380$	0	0
$381$	−1021.14	−0.137309
$382$	0	0
$383$	9577.05	1.27771	0.638857	−	0.769325i	$-0.279408\pi$
$383$	9577.05	1.27771	0.638857	+	0.769325i	$0.279408\pi$
$384$	0	0
$385$	−2773.26	−0.367113
$386$	0	0
$387$	8086.28	1.06214
$388$	0	0
$389$	10499.8	1.36854	0.684269	−	0.729229i	$-0.260121\pi$
$389$	10499.8	1.36854	0.684269	+	0.729229i	$0.260121\pi$
$390$	0	0
$391$	−2589.76	−0.334961
$392$	0	0
$393$	693.239	0.0889804
$394$	0	0
$395$	−3357.06	−0.427625
$396$	0	0
$397$	6497.20	0.821373	0.410687	−	0.911777i	$-0.365289\pi$
$397$	6497.20	0.821373	0.410687	+	0.911777i	$0.365289\pi$
$398$	0	0
$399$	−1111.83	−0.139502
$400$	0	0
$401$	6028.66	0.750766	0.375383	−	0.926870i	$-0.377511\pi$
$401$	6028.66	0.750766	0.375383	+	0.926870i	$0.377511\pi$
$402$	0	0
$403$	3944.76	0.487600
$404$	0	0
$405$	9968.71	1.22308
$406$	0	0
$407$	−2637.67	−0.321240
$408$	0	0
$409$	−4053.69	−0.490079	−0.245039	−	0.969513i	$-0.578801\pi$
$409$	−4053.69	−0.490079	−0.245039	+	0.969513i	$0.578801\pi$
$410$	0	0
$411$	−2111.16	−0.253372
$412$	0	0
$413$	−5671.79	−0.675764
$414$	0	0
$415$	6660.12	0.787789
$416$	0	0
$417$	1385.28	0.162680
$418$	0	0
$419$	−3771.70	−0.439760	−0.219880	−	0.975527i	$-0.570567\pi$
$419$	−3771.70	−0.439760	−0.219880	+	0.975527i	$0.570567\pi$
$420$	0	0
$421$	7683.30	0.889457	0.444728	−	0.895666i	$-0.353300\pi$
$421$	7683.30	0.889457	0.444728	+	0.895666i	$0.353300\pi$
$422$	0	0
$423$	−11679.4	−1.34249
$424$	0	0
$425$	1903.97	0.217308
$426$	0	0
$427$	−707.495	−0.0801828
$428$	0	0
$429$	416.148	0.0468341
$430$	0	0
$431$	−2810.92	−0.314146	−0.157073	−	0.987587i	$-0.550206\pi$
$431$	−2810.92	−0.314146	−0.157073	+	0.987587i	$0.550206\pi$
$432$	0	0
$433$	−12675.1	−1.40676	−0.703380	−	0.710814i	$-0.748327\pi$
$433$	−12675.1	−1.40676	−0.703380	+	0.710814i	$0.748327\pi$
$434$	0	0
$435$	4564.47	0.503103
$436$	0	0
$437$	−23975.9	−2.62453
$438$	0	0
$439$	3256.34	0.354024	0.177012	−	0.984209i	$-0.443357\pi$
$439$	3256.34	0.354024	0.177012	+	0.984209i	$0.443357\pi$
$440$	0	0
$441$	−1273.09	−0.137468
$442$	0	0
$443$	−4751.27	−0.509570	−0.254785	−	0.966998i	$-0.582005\pi$
$443$	−4751.27	−0.509570	−0.254785	+	0.966998i	$0.582005\pi$
$444$	0	0
$445$	15693.4	1.67177
$446$	0	0
$447$	−2421.72	−0.256249
$448$	0	0
$449$	−13214.1	−1.38889	−0.694445	−	0.719546i	$-0.744350\pi$
$449$	−13214.1	−1.38889	−0.694445	+	0.719546i	$0.744350\pi$
$450$	0	0
$451$	−2061.25	−0.215211
$452$	0	0
$453$	−3171.97	−0.328989
$454$	0	0
$455$	−1726.71	−0.177911
$456$	0	0
$457$	8864.12	0.907322	0.453661	−	0.891174i	$-0.350118\pi$
$457$	8864.12	0.907322	0.453661	+	0.891174i	$0.350118\pi$
$458$	0	0
$459$	−908.973	−0.0924340
$460$	0	0
$461$	−17544.6	−1.77252	−0.886261	−	0.463186i	$-0.846706\pi$
$461$	−17544.6	−1.77252	−0.886261	+	0.463186i	$0.846706\pi$
$462$	0	0
$463$	−6190.68	−0.621394	−0.310697	−	0.950509i	$-0.600562\pi$
$463$	−6190.68	−0.621394	−0.310697	+	0.950509i	$0.600562\pi$
$464$	0	0
$465$	3824.91	0.381454
$466$	0	0
$467$	12097.8	1.19876	0.599381	−	0.800464i	$-0.295414\pi$
$467$	12097.8	1.19876	0.599381	+	0.800464i	$0.295414\pi$
$468$	0	0
$469$	1633.02	0.160781
$470$	0	0
$471$	−169.411	−0.0165733
$472$	0	0
$473$	−8009.48	−0.778597
$474$	0	0
$475$	17626.8	1.70268
$476$	0	0
$477$	13476.7	1.29362
$478$	0	0
$479$	−2134.03	−0.203562	−0.101781	−	0.994807i	$-0.532454\pi$
$479$	−2134.03	−0.203562	−0.101781	+	0.994807i	$0.532454\pi$
$480$	0	0
$481$	−1642.29	−0.155680
$482$	0	0
$483$	1076.18	0.101383
$484$	0	0
$485$	7179.23	0.672149
$486$	0	0
$487$	10374.0	0.965278	0.482639	−	0.875820i	$-0.339678\pi$
$487$	10374.0	0.965278	0.482639	+	0.875820i	$0.339678\pi$
$488$	0	0
$489$	4165.64	0.385229
$490$	0	0
$491$	−7307.57	−0.671662	−0.335831	−	0.941922i	$-0.609017\pi$
$491$	−7307.57	−0.671662	−0.335831	+	0.941922i	$0.609017\pi$
$492$	0	0
$493$	4994.47	0.456267
$494$	0	0
$495$	−10293.4	−0.934650
$496$	0	0
$497$	−6870.99	−0.620133
$498$	0	0
$499$	7833.65	0.702770	0.351385	−	0.936231i	$-0.385711\pi$
$499$	7833.65	0.702770	0.351385	+	0.936231i	$0.385711\pi$
$500$	0	0
$501$	−1845.85	−0.164603
$502$	0	0
$503$	−15051.0	−1.33418	−0.667090	−	0.744977i	$-0.732460\pi$
$503$	−15051.0	−1.33418	−0.667090	+	0.744977i	$0.732460\pi$
$504$	0	0
$505$	4893.59	0.431212
$506$	0	0
$507$	−1958.11	−0.171524
$508$	0	0
$509$	−299.106	−0.0260464	−0.0130232	−	0.999915i	$-0.504146\pi$
$509$	−299.106	−0.0260464	−0.0130232	+	0.999915i	$0.504146\pi$
$510$	0	0
$511$	1919.52	0.166174
$512$	0	0
$513$	−8415.23	−0.724253
$514$	0	0
$515$	10145.6	0.868093
$516$	0	0
$517$	11568.5	0.984102
$518$	0	0
$519$	−1948.45	−0.164793
$520$	0	0
$521$	13064.5	1.09859	0.549297	−	0.835627i	$-0.314896\pi$
$521$	13064.5	1.09859	0.549297	+	0.835627i	$0.314896\pi$
$522$	0	0
$523$	−11621.2	−0.971623	−0.485812	−	0.874064i	$-0.661476\pi$
$523$	−11621.2	−0.971623	−0.485812	+	0.874064i	$0.661476\pi$
$524$	0	0
$525$	−791.200	−0.0657730
$526$	0	0
$527$	4185.24	0.345943
$528$	0	0
$529$	11040.1	0.907380
$530$	0	0
$531$	−21051.7	−1.72046
$532$	0	0
$533$	−1283.39	−0.104296
$534$	0	0
$535$	−7261.77	−0.586829
$536$	0	0
$537$	−1334.29	−0.107223
$538$	0	0
$539$	1261.00	0.100770
$540$	0	0
$541$	−552.394	−0.0438988	−0.0219494	−	0.999759i	$-0.506987\pi$
$541$	−552.394	−0.0438988	−0.0219494	+	0.999759i	$0.506987\pi$
$542$	0	0
$543$	499.470	0.0394739
$544$	0	0
$545$	−7700.53	−0.605237
$546$	0	0
$547$	3714.09	0.290316	0.145158	−	0.989408i	$-0.453631\pi$
$547$	3714.09	0.290316	0.145158	+	0.989408i	$0.453631\pi$
$548$	0	0
$549$	−2625.97	−0.204141
$550$	0	0
$551$	46238.6	3.57501
$552$	0	0
$553$	1526.46	0.117381
$554$	0	0
$555$	−1592.39	−0.121790
$556$	0	0
$557$	9082.21	0.690890	0.345445	−	0.938439i	$-0.387728\pi$
$557$	9082.21	0.690890	0.345445	+	0.938439i	$0.387728\pi$
$558$	0	0
$559$	−4986.94	−0.377326
$560$	0	0
$561$	441.516	0.0332279
$562$	0	0
$563$	13260.3	0.992640	0.496320	−	0.868140i	$-0.334684\pi$
$563$	13260.3	0.992640	0.496320	+	0.868140i	$0.334684\pi$
$564$	0	0
$565$	−1908.83	−0.142133
$566$	0	0
$567$	−4532.78	−0.335730
$568$	0	0
$569$	16130.1	1.18842	0.594208	−	0.804312i	$-0.297466\pi$
$569$	16130.1	1.18842	0.594208	+	0.804312i	$0.297466\pi$
$570$	0	0
$571$	−10919.8	−0.800316	−0.400158	−	0.916446i	$-0.631045\pi$
$571$	−10919.8	−0.800316	−0.400158	+	0.916446i	$0.631045\pi$
$572$	0	0
$573$	125.480	0.00914835
$574$	0	0
$575$	−17061.7	−1.23743
$576$	0	0
$577$	−13205.7	−0.952790	−0.476395	−	0.879231i	$-0.658057\pi$
$577$	−13205.7	−0.952790	−0.476395	+	0.879231i	$0.658057\pi$
$578$	0	0
$579$	2743.17	0.196895
$580$	0	0
$581$	−3028.36	−0.216244
$582$	0	0
$583$	−13348.8	−0.948283
$584$	0	0
$585$	−6408.95	−0.452953
$586$	0	0
$587$	11464.8	0.806136	0.403068	−	0.915170i	$-0.367944\pi$
$587$	11464.8	0.806136	0.403068	+	0.915170i	$0.367944\pi$
$588$	0	0
$589$	38746.7	2.71058
$590$	0	0
$591$	−3530.49	−0.245727
$592$	0	0
$593$	10524.3	0.728807	0.364404	−	0.931241i	$-0.381273\pi$
$593$	10524.3	0.728807	0.364404	+	0.931241i	$0.381273\pi$
$594$	0	0
$595$	−1831.97	−0.126225
$596$	0	0
$597$	1186.17	0.0813179
$598$	0	0
$599$	−21190.3	−1.44543	−0.722715	−	0.691146i	$-0.757106\pi$
$599$	−21190.3	−1.44543	−0.722715	+	0.691146i	$0.757106\pi$
$600$	0	0
$601$	−17924.2	−1.21654	−0.608272	−	0.793729i	$-0.708137\pi$
$601$	−17924.2	−1.21654	−0.608272	+	0.793729i	$0.708137\pi$
$602$	0	0
$603$	6061.21	0.409339
$604$	0	0
$605$	−10294.8	−0.691807
$606$	0	0
$607$	11521.8	0.770438	0.385219	−	0.922825i	$-0.374126\pi$
$607$	11521.8	0.770438	0.385219	+	0.922825i	$0.374126\pi$
$608$	0	0
$609$	−2075.47	−0.138099
$610$	0	0
$611$	7202.87	0.476918
$612$	0	0
$613$	8052.51	0.530567	0.265284	−	0.964170i	$-0.414534\pi$
$613$	8052.51	0.530567	0.265284	+	0.964170i	$0.414534\pi$
$614$	0	0
$615$	−1244.40	−0.0815920
$616$	0	0
$617$	−21023.6	−1.37176	−0.685882	−	0.727713i	$-0.740583\pi$
$617$	−21023.6	−1.37176	−0.685882	+	0.727713i	$0.740583\pi$
$618$	0	0
$619$	−29039.3	−1.88560	−0.942802	−	0.333354i	$-0.891819\pi$
$619$	−29039.3	−1.88560	−0.942802	+	0.333354i	$0.891819\pi$
$620$	0	0
$621$	8145.40	0.526350
$622$	0	0
$623$	−7135.81	−0.458893
$624$	0	0
$625$	−17081.2	−1.09320
$626$	0	0
$627$	4087.54	0.260352
$628$	0	0
$629$	−1742.41	−0.110452
$630$	0	0
$631$	−8671.86	−0.547102	−0.273551	−	0.961857i	$-0.588198\pi$
$631$	−8671.86	−0.547102	−0.273551	+	0.961857i	$0.588198\pi$
$632$	0	0
$633$	−3091.94	−0.194145
$634$	0	0
$635$	−15576.9	−0.973467
$636$	0	0
$637$	785.138	0.0488356
$638$	0	0
$639$	−25502.7	−1.57883
$640$	0	0
$641$	−17252.9	−1.06310	−0.531551	−	0.847026i	$-0.678391\pi$
$641$	−17252.9	−1.06310	−0.531551	+	0.847026i	$0.678391\pi$
$642$	0	0
$643$	16000.1	0.981309	0.490655	−	0.871354i	$-0.336758\pi$
$643$	16000.1	0.981309	0.490655	+	0.871354i	$0.336758\pi$
$644$	0	0
$645$	−4835.42	−0.295185
$646$	0	0
$647$	−4349.51	−0.264292	−0.132146	−	0.991230i	$-0.542187\pi$
$647$	−4349.51	−0.264292	−0.132146	+	0.991230i	$0.542187\pi$
$648$	0	0
$649$	20851.7	1.26117
$650$	0	0
$651$	−1739.19	−0.104707
$652$	0	0
$653$	15497.9	0.928761	0.464380	−	0.885636i	$-0.346277\pi$
$653$	15497.9	0.928761	0.464380	+	0.885636i	$0.346277\pi$
$654$	0	0
$655$	10574.9	0.630835
$656$	0	0
$657$	7124.59	0.423070
$658$	0	0
$659$	1040.36	0.0614970	0.0307485	−	0.999527i	$-0.490211\pi$
$659$	1040.36	0.0614970	0.0307485	+	0.999527i	$0.490211\pi$
$660$	0	0
$661$	10737.2	0.631814	0.315907	−	0.948790i	$-0.397691\pi$
$661$	10737.2	0.631814	0.315907	+	0.948790i	$0.397691\pi$
$662$	0	0
$663$	274.901	0.0161030
$664$	0	0
$665$	−16960.3	−0.989013
$666$	0	0
$667$	−44756.0	−2.59814
$668$	0	0
$669$	−3490.05	−0.201694
$670$	0	0
$671$	2601.03	0.149645
$672$	0	0
$673$	21103.1	1.20871	0.604357	−	0.796713i	$-0.293430\pi$
$673$	21103.1	1.20871	0.604357	+	0.796713i	$0.293430\pi$
$674$	0	0
$675$	−5988.43	−0.341473
$676$	0	0
$677$	−8460.36	−0.480292	−0.240146	−	0.970737i	$-0.577195\pi$
$677$	−8460.36	−0.480292	−0.240146	+	0.970737i	$0.577195\pi$
$678$	0	0
$679$	−3264.40	−0.184501
$680$	0	0
$681$	−1747.20	−0.0983155
$682$	0	0
$683$	−8550.65	−0.479036	−0.239518	−	0.970892i	$-0.576989\pi$
$683$	−8550.65	−0.479036	−0.239518	+	0.970892i	$0.576989\pi$
$684$	0	0
$685$	−32204.5	−1.79631
$686$	0	0
$687$	−4709.05	−0.261516
$688$	0	0
$689$	−8311.33	−0.459560
$690$	0	0
$691$	9246.17	0.509032	0.254516	−	0.967069i	$-0.418084\pi$
$691$	9246.17	0.509032	0.254516	+	0.967069i	$0.418084\pi$
$692$	0	0
$693$	4680.40	0.256556
$694$	0	0
$695$	21131.7	1.15334
$696$	0	0
$697$	−1361.63	−0.0739963
$698$	0	0
$699$	6193.67	0.335145
$700$	0	0
$701$	1138.54	0.0613438	0.0306719	−	0.999530i	$-0.490235\pi$
$701$	1138.54	0.0613438	0.0306719	+	0.999530i	$0.490235\pi$
$702$	0	0
$703$	−16131.1	−0.865430
$704$	0	0
$705$	6984.03	0.373097
$706$	0	0
$707$	−2225.12	−0.118365
$708$	0	0
$709$	−7742.45	−0.410118	−0.205059	−	0.978750i	$-0.565739\pi$
$709$	−7742.45	−0.410118	−0.205059	+	0.978750i	$0.565739\pi$
$710$	0	0
$711$	5665.66	0.298845
$712$	0	0
$713$	−37504.3	−1.96991
$714$	0	0
$715$	6348.08	0.332035
$716$	0	0
$717$	−1412.39	−0.0735657
$718$	0	0
$719$	−33796.7	−1.75300	−0.876498	−	0.481406i	$-0.840126\pi$
$719$	−33796.7	−1.75300	−0.876498	+	0.481406i	$0.840126\pi$
$720$	0	0
$721$	−4613.21	−0.238287
$722$	0	0
$723$	1170.75	0.0602224
$724$	0	0
$725$	32904.2	1.68556
$726$	0	0
$727$	27370.1	1.39629	0.698144	−	0.715958i	$-0.254010\pi$
$727$	27370.1	1.39629	0.698144	+	0.715958i	$0.254010\pi$
$728$	0	0
$729$	−15367.1	−0.780731
$730$	0	0
$731$	−5290.95	−0.267706
$732$	0	0
$733$	20572.5	1.03665	0.518324	−	0.855184i	$-0.326556\pi$
$733$	20572.5	1.03665	0.518324	+	0.855184i	$0.326556\pi$
$734$	0	0
$735$	761.283	0.0382046
$736$	0	0
$737$	−6003.64	−0.300064
$738$	0	0
$739$	7596.57	0.378138	0.189069	−	0.981964i	$-0.439453\pi$
$739$	7596.57	0.378138	0.189069	+	0.981964i	$0.439453\pi$
$740$	0	0
$741$	2545.02	0.126172
$742$	0	0
$743$	−10034.0	−0.495441	−0.247721	−	0.968831i	$-0.579682\pi$
$743$	−10034.0	−0.495441	−0.247721	+	0.968831i	$0.579682\pi$
$744$	0	0
$745$	−36941.8	−1.81670
$746$	0	0
$747$	−11240.2	−0.550546
$748$	0	0
$749$	3301.93	0.161081
$750$	0	0
$751$	29250.6	1.42126	0.710632	−	0.703564i	$-0.248409\pi$
$751$	29250.6	1.42126	0.710632	+	0.703564i	$0.248409\pi$
$752$	0	0
$753$	−7159.22	−0.346476
$754$	0	0
$755$	−48386.5	−2.33240
$756$	0	0
$757$	−40135.6	−1.92702	−0.963509	−	0.267676i	$-0.913744\pi$
$757$	−40135.6	−1.92702	−0.963509	+	0.267676i	$0.913744\pi$
$758$	0	0
$759$	−3956.47	−0.189211
$760$	0	0
$761$	5811.28	0.276818	0.138409	−	0.990375i	$-0.455801\pi$
$761$	5811.28	0.276818	0.138409	+	0.990375i	$0.455801\pi$
$762$	0	0
$763$	3501.43	0.166134
$764$	0	0
$765$	−6799.64	−0.321361
$766$	0	0
$767$	12982.9	0.611194
$768$	0	0
$769$	32588.3	1.52817	0.764086	−	0.645114i	$-0.223190\pi$
$769$	32588.3	1.52817	0.764086	+	0.645114i	$0.223190\pi$
$770$	0	0
$771$	−6708.05	−0.313339
$772$	0	0
$773$	34681.7	1.61373	0.806865	−	0.590737i	$-0.201163\pi$
$773$	34681.7	1.61373	0.806865	+	0.590737i	$0.201163\pi$
$774$	0	0
$775$	27572.9	1.27799
$776$	0	0
$777$	724.063	0.0334307
$778$	0	0
$779$	−12605.9	−0.579786
$780$	0	0
$781$	25260.5	1.15735
$782$	0	0
$783$	−15708.8	−0.716968
$784$	0	0
$785$	−2584.26	−0.117498
$786$	0	0
$787$	−14570.5	−0.659953	−0.329976	−	0.943989i	$-0.607041\pi$
$787$	−14570.5	−0.659953	−0.329976	+	0.943989i	$0.607041\pi$
$788$	0	0
$789$	7618.04	0.343738
$790$	0	0
$791$	867.944	0.0390146
$792$	0	0
$793$	1619.48	0.0725212
$794$	0	0
$795$	−8058.81	−0.359518
$796$	0	0
$797$	20881.3	0.928047	0.464023	−	0.885823i	$-0.346405\pi$
$797$	20881.3	0.928047	0.464023	+	0.885823i	$0.346405\pi$
$798$	0	0
$799$	7641.96	0.338364
$800$	0	0
$801$	−26485.6	−1.16832
$802$	0	0
$803$	−7056.93	−0.310129
$804$	0	0
$805$	16416.5	0.718765
$806$	0	0
$807$	−7354.03	−0.320786
$808$	0	0
$809$	−5971.77	−0.259526	−0.129763	−	0.991545i	$-0.541422\pi$
$809$	−5971.77	−0.259526	−0.129763	+	0.991545i	$0.541422\pi$
$810$	0	0
$811$	−41435.5	−1.79408	−0.897039	−	0.441951i	$-0.854286\pi$
$811$	−41435.5	−1.79408	−0.897039	+	0.441951i	$0.854286\pi$
$812$	0	0
$813$	3488.91	0.150506
$814$	0	0
$815$	63544.4	2.73112
$816$	0	0
$817$	−48983.4	−2.09756
$818$	0	0
$819$	2914.15	0.124333
$820$	0	0
$821$	1088.07	0.0462531	0.0231266	−	0.999733i	$-0.492638\pi$
$821$	1088.07	0.0462531	0.0231266	+	0.999733i	$0.492638\pi$
$822$	0	0
$823$	14637.7	0.619973	0.309987	−	0.950741i	$-0.399675\pi$
$823$	14637.7	0.619973	0.309987	+	0.950741i	$0.399675\pi$
$824$	0	0
$825$	2908.76	0.122752
$826$	0	0
$827$	3434.72	0.144422	0.0722110	−	0.997389i	$-0.476994\pi$
$827$	3434.72	0.144422	0.0722110	+	0.997389i	$0.476994\pi$
$828$	0	0
$829$	34138.8	1.43027	0.715133	−	0.698988i	$-0.246366\pi$
$829$	34138.8	1.43027	0.715133	+	0.698988i	$0.246366\pi$
$830$	0	0
$831$	−2300.14	−0.0960182
$832$	0	0
$833$	833.000	0.0346479
$834$	0	0
$835$	−28157.3	−1.16697
$836$	0	0
$837$	−13163.6	−0.543607
$838$	0	0
$839$	−4742.97	−0.195168	−0.0975838	−	0.995227i	$-0.531111\pi$
$839$	−4742.97	−0.195168	−0.0975838	+	0.995227i	$0.531111\pi$
$840$	0	0
$841$	61924.9	2.53905
$842$	0	0
$843$	−5887.05	−0.240523
$844$	0	0
$845$	−29869.7	−1.21604
$846$	0	0
$847$	4681.06	0.189897
$848$	0	0
$849$	5499.59	0.222315
$850$	0	0
$851$	15613.9	0.628951
$852$	0	0
$853$	10776.5	0.432569	0.216284	−	0.976330i	$-0.430606\pi$
$853$	10776.5	0.432569	0.216284	+	0.976330i	$0.430606\pi$
$854$	0	0
$855$	−62950.8	−2.51798
$856$	0	0
$857$	901.774	0.0359440	0.0179720	−	0.999838i	$-0.494279\pi$
$857$	901.774	0.0359440	0.0179720	+	0.999838i	$0.494279\pi$
$858$	0	0
$859$	2052.46	0.0815240	0.0407620	−	0.999169i	$-0.487021\pi$
$859$	2052.46	0.0815240	0.0407620	+	0.999169i	$0.487021\pi$
$860$	0	0
$861$	565.830	0.0223965
$862$	0	0
$863$	7560.61	0.298223	0.149111	−	0.988820i	$-0.452359\pi$
$863$	7560.61	0.298223	0.149111	+	0.988820i	$0.452359\pi$
$864$	0	0
$865$	−29722.4	−1.16831
$866$	0	0
$867$	291.659	0.0114248
$868$	0	0
$869$	−5611.85	−0.219067
$870$	0	0
$871$	−3738.05	−0.145418
$872$	0	0
$873$	−12116.3	−0.469730
$874$	0	0
$875$	1401.13	0.0541334
$876$	0	0
$877$	26675.1	1.02709	0.513543	−	0.858064i	$-0.328333\pi$
$877$	26675.1	1.02709	0.513543	+	0.858064i	$0.328333\pi$
$878$	0	0
$879$	2664.04	0.102225
$880$	0	0
$881$	13711.6	0.524354	0.262177	−	0.965020i	$-0.415560\pi$
$881$	13711.6	0.524354	0.262177	+	0.965020i	$0.415560\pi$
$882$	0	0
$883$	29386.2	1.11996	0.559980	−	0.828506i	$-0.310809\pi$
$883$	29386.2	1.11996	0.559980	+	0.828506i	$0.310809\pi$
$884$	0	0
$885$	12588.4	0.478142
$886$	0	0
$887$	15162.6	0.573969	0.286985	−	0.957935i	$-0.407347\pi$
$887$	15162.6	0.573969	0.286985	+	0.957935i	$0.407347\pi$
$888$	0	0
$889$	7082.84	0.267211
$890$	0	0
$891$	16664.3	0.626571
$892$	0	0
$893$	70748.9	2.65120
$894$	0	0
$895$	−20353.8	−0.760170
$896$	0	0
$897$	−2463.42	−0.0916958
$898$	0	0
$899$	72328.8	2.68332
$900$	0	0
$901$	−8817.99	−0.326049
$902$	0	0
$903$	2198.67	0.0810268
$904$	0	0
$905$	7619.11	0.279854
$906$	0	0
$907$	30643.4	1.12183	0.560914	−	0.827874i	$-0.310450\pi$
$907$	30643.4	1.12183	0.560914	+	0.827874i	$0.310450\pi$
$908$	0	0
$909$	−8258.86	−0.301352
$910$	0	0
$911$	−28467.4	−1.03531	−0.517655	−	0.855589i	$-0.673195\pi$
$911$	−28467.4	−1.03531	−0.517655	+	0.855589i	$0.673195\pi$
$912$	0	0
$913$	11133.5	0.403574
$914$	0	0
$915$	1570.27	0.0567340
$916$	0	0
$917$	−4808.43	−0.173161
$918$	0	0
$919$	15469.7	0.555275	0.277637	−	0.960686i	$-0.410449\pi$
$919$	15469.7	0.555275	0.277637	+	0.960686i	$0.410449\pi$
$920$	0	0
$921$	−6224.01	−0.222680
$922$	0	0
$923$	15727.9	0.560878
$924$	0	0
$925$	−11479.2	−0.408036
$926$	0	0
$927$	−17122.6	−0.606666
$928$	0	0
$929$	24954.0	0.881286	0.440643	−	0.897682i	$-0.354750\pi$
$929$	24954.0	0.881286	0.440643	+	0.897682i	$0.354750\pi$
$930$	0	0
$931$	7711.88	0.271479
$932$	0	0
$933$	6578.99	0.230854
$934$	0	0
$935$	6735.06	0.235572
$936$	0	0
$937$	53097.7	1.85125	0.925627	−	0.378438i	$-0.123539\pi$
$937$	53097.7	1.85125	0.925627	+	0.378438i	$0.123539\pi$
$938$	0	0
$939$	−10290.7	−0.357639
$940$	0	0
$941$	−31108.6	−1.07770	−0.538848	−	0.842403i	$-0.681140\pi$
$941$	−31108.6	−1.07770	−0.538848	+	0.842403i	$0.681140\pi$
$942$	0	0
$943$	12201.7	0.421360
$944$	0	0
$945$	5761.99	0.198347
$946$	0	0
$947$	−37184.9	−1.27597	−0.637987	−	0.770047i	$-0.720233\pi$
$947$	−37184.9	−1.27597	−0.637987	+	0.770047i	$0.720233\pi$
$948$	0	0
$949$	−4393.85	−0.150296
$950$	0	0
$951$	−5506.70	−0.187767
$952$	0	0
$953$	−29917.1	−1.01690	−0.508452	−	0.861090i	$-0.669782\pi$
$953$	−29917.1	−1.01690	−0.508452	+	0.861090i	$0.669782\pi$
$954$	0	0
$955$	1914.12	0.0648581
$956$	0	0
$957$	7630.24	0.257733
$958$	0	0
$959$	14643.4	0.493077
$960$	0	0
$961$	30818.7	1.03450
$962$	0	0
$963$	12255.6	0.410105
$964$	0	0
$965$	41845.3	1.39591
$966$	0	0
$967$	−49683.9	−1.65225	−0.826126	−	0.563485i	$-0.809460\pi$
$967$	−49683.9	−1.65225	−0.826126	+	0.563485i	$0.809460\pi$
$968$	0	0
$969$	2700.17	0.0895169
$970$	0	0
$971$	−38044.4	−1.25737	−0.628684	−	0.777661i	$-0.716406\pi$
$971$	−38044.4	−1.25737	−0.628684	+	0.777661i	$0.716406\pi$
$972$	0	0
$973$	−9608.58	−0.316585
$974$	0	0
$975$	1811.08	0.0594883
$976$	0	0
$977$	−45796.0	−1.49964	−0.749818	−	0.661644i	$-0.769859\pi$
$977$	−45796.0	−1.49964	−0.749818	+	0.661644i	$0.769859\pi$
$978$	0	0
$979$	26234.1	0.856429
$980$	0	0
$981$	12996.1	0.422969
$982$	0	0
$983$	−10.6433	−0.000345340 0	−0.000172670	−	1.00000i	$-0.500055\pi$
$983$	−10.6433	−0.000345340 0	−0.000172670		1.00000i	$0.500055\pi$
$984$	0	0
$985$	−53855.4	−1.74211
$986$	0	0
$987$	−3175.64	−0.102413
$988$	0	0
$989$	47412.7	1.52441
$990$	0	0
$991$	−6252.30	−0.200415	−0.100207	−	0.994967i	$-0.531951\pi$
$991$	−6252.30	−0.200415	−0.100207	+	0.994967i	$0.531951\pi$
$992$	0	0
$993$	42.8862	0.00137055
$994$	0	0
$995$	18094.3	0.576511
$996$	0	0
$997$	−11851.0	−0.376453	−0.188227	−	0.982126i	$-0.560274\pi$
$997$	−11851.0	−0.376453	−0.188227	+	0.982126i	$0.560274\pi$
$998$	0	0
$999$	5480.28	0.173562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	476.4.a.b.1.4	✓	6
4.3	odd	2		1904.4.a.m.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.4.a.b.1.4	✓	6	1.1	even	1	trivial
1904.4.a.m.1.3		6	4.3	odd	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:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	476.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00920 q^{3} +15.3947 q^{5} -7.00000 q^{7} -25.9815 q^{9} +25.7348 q^{11} +16.0232 q^{13} +15.5364 q^{15} +17.0000 q^{17} +157.385 q^{19} -7.06441 q^{21} -152.339 q^{23} +111.998 q^{25} -53.4690 q^{27} +293.792 q^{29} +246.190 q^{31} +25.9715 q^{33} -107.763 q^{35} -102.495 q^{37} +16.1707 q^{39} -80.0959 q^{41} -311.232 q^{43} -399.979 q^{45} +449.527 q^{47} +49.0000 q^{49} +17.1564 q^{51} -518.705 q^{53} +396.180 q^{55} +158.833 q^{57} +810.255 q^{59} +101.071 q^{61} +181.871 q^{63} +246.673 q^{65} -233.289 q^{67} -153.740 q^{69} +981.570 q^{71} -274.218 q^{73} +113.029 q^{75} -180.143 q^{77} -218.065 q^{79} +647.540 q^{81} +432.623 q^{83} +261.711 q^{85} +296.495 q^{87} +1019.40 q^{89} -112.163 q^{91} +248.456 q^{93} +2422.91 q^{95} +466.343 q^{97} -668.628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 14 q^{5} - 42 q^{7} - 16 q^{9} - 2 q^{11} + 8 q^{13} - 16 q^{15} + 102 q^{17} + 170 q^{19} - 14 q^{21} + 102 q^{23} + 140 q^{25} + 422 q^{27} + 278 q^{29} + 380 q^{31} + 98 q^{33} + 98 q^{35}+ \cdots + 450 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 14 * q^5 - 42 * q^7 - 16 * q^9 - 2 * q^11 + 8 * q^13 - 16 * q^15 + 102 * q^17 + 170 * q^19 - 14 * q^21 + 102 * q^23 + 140 * q^25 + 422 * q^27 + 278 * q^29 + 380 * q^31 + 98 * q^33 + 98 * q^35 + 542 * q^37 + 790 * q^39 + 224 * q^41 + 104 * q^43 + 294 * q^45 + 1022 * q^47 + 294 * q^49 + 34 * q^51 + 554 * q^53 + 496 * q^55 + 716 * q^57 + 844 * q^59 + 452 * q^61 + 112 * q^63 + 746 * q^65 + 1376 * q^67 + 664 * q^69 + 1062 * q^71 + 208 * q^73 - 134 * q^75 + 14 * q^77 + 1410 * q^79 + 1850 * q^81 + 232 * q^83 - 238 * q^85 + 1350 * q^87 - 654 * q^89 - 56 * q^91 + 3564 * q^93 + 3186 * q^95 - 1546 * q^97 + 450 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more