LMFDB - Embedded newform 8024.2.a.w.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 29 x^{18} + 51 x^{17} + 341 x^{16} - 514 x^{15} - 2114 x^{14} + 2629 x^{13} + \cdots + 4 $ x^20 - 2x^19 - 29x^18 + 51x^17 + 341x^16 - 514x^15 - 2114x^14 + 2629x^13 + 7532x^12 - 7342x^11 - 15668x^10 + 11260x^9 + 18269x^8 - 9059x^7 - 10680x^6 + 3386x^5 + 2473x^4 - 427x^3 - 193x^2 + 18*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.237078$ of defining polynomial
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.237078 q^{3} +2.74912 q^{5} -2.88085 q^{7} -2.94379 q^{9} +O(q^{10})$	$q-0.237078 q^{3} +2.74912 q^{5} -2.88085 q^{7} -2.94379 q^{9} -2.31199 q^{11} +0.899710 q^{13} -0.651755 q^{15} +1.00000 q^{17} +8.66672 q^{19} +0.682985 q^{21} +2.21439 q^{23} +2.55766 q^{25} +1.40914 q^{27} -4.31561 q^{29} -1.36287 q^{31} +0.548120 q^{33} -7.91980 q^{35} -4.03568 q^{37} -0.213301 q^{39} -8.30878 q^{41} -0.582436 q^{43} -8.09284 q^{45} +3.69851 q^{47} +1.29928 q^{49} -0.237078 q^{51} +1.86324 q^{53} -6.35593 q^{55} -2.05469 q^{57} -1.00000 q^{59} -5.61743 q^{61} +8.48062 q^{63} +2.47341 q^{65} +11.1511 q^{67} -0.524983 q^{69} +0.209332 q^{71} +0.762440 q^{73} -0.606365 q^{75} +6.66048 q^{77} +3.50569 q^{79} +8.49731 q^{81} -7.71241 q^{83} +2.74912 q^{85} +1.02314 q^{87} -3.55723 q^{89} -2.59193 q^{91} +0.323106 q^{93} +23.8259 q^{95} -18.9378 q^{97} +6.80601 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) $ 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9	$ 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + 8 q^{11} - 3 q^{13} - 2 q^{15} + 20 q^{17} - 9 q^{19} - 31 q^{21} - 8 q^{23} - 4 q^{25} - 17 q^{27} - 37 q^{29} - 9 q^{31} - 8 q^{33} - 8 q^{35} - 11 q^{37} - 12 q^{39} - 27 q^{41} + 17 q^{43} - 4 q^{45} + 7 q^{47} - 9 q^{49} - 2 q^{51} - q^{53} - 21 q^{55} - 57 q^{57} - 20 q^{59} - 12 q^{61} - 14 q^{63} - 59 q^{65} - 26 q^{67} + 14 q^{69} - 3 q^{71} - 45 q^{73} + 15 q^{75} + 6 q^{77} + 5 q^{79} + 12 q^{81} - 17 q^{83} - 2 q^{85} - 32 q^{87} - 40 q^{89} - 36 q^{91} - 4 q^{93} + 5 q^{95} - 77 q^{97} + 54 q^{99}+O(q^{100}) $ 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 + 8 * q^11 - 3 * q^13 - 2 * q^15 + 20 * q^17 - 9 * q^19 - 31 * q^21 - 8 * q^23 - 4 * q^25 - 17 * q^27 - 37 * q^29 - 9 * q^31 - 8 * q^33 - 8 * q^35 - 11 * q^37 - 12 * q^39 - 27 * q^41 + 17 * q^43 - 4 * q^45 + 7 * q^47 - 9 * q^49 - 2 * q^51 - q^53 - 21 * q^55 - 57 * q^57 - 20 * q^59 - 12 * q^61 - 14 * q^63 - 59 * q^65 - 26 * q^67 + 14 * q^69 - 3 * q^71 - 45 * q^73 + 15 * q^75 + 6 * q^77 + 5 * q^79 + 12 * q^81 - 17 * q^83 - 2 * q^85 - 32 * q^87 - 40 * q^89 - 36 * q^91 - 4 * q^93 + 5 * q^95 - 77 * q^97 + 54 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−0.237078	−0.136877	−0.0684384	−	0.997655i	$-0.521802\pi$
$3$	−0.237078	−0.136877	−0.0684384	+	0.997655i	$0.521802\pi$
$4$	0	0
$5$	2.74912	1.22944	0.614722	−	0.788744i	$-0.289268\pi$
$5$	2.74912	1.22944	0.614722	+	0.788744i	$0.289268\pi$
$6$	0	0
$7$	−2.88085	−1.08886	−0.544429	−	0.838807i	$-0.683254\pi$
$7$	−2.88085	−1.08886	−0.544429	+	0.838807i	$0.683254\pi$
$8$	0	0
$9$	−2.94379	−0.981265
$10$	0	0
$11$	−2.31199	−0.697090	−0.348545	−	0.937292i	$-0.613324\pi$
$11$	−2.31199	−0.697090	−0.348545	+	0.937292i	$0.613324\pi$
$12$	0	0
$13$	0.899710	0.249535	0.124767	−	0.992186i	$-0.460182\pi$
$13$	0.899710	0.249535	0.124767	+	0.992186i	$0.460182\pi$
$14$	0	0
$15$	−0.651755	−0.168282
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	8.66672	1.98828	0.994141	−	0.108087i	$-0.0344724\pi$
$19$	8.66672	1.98828	0.994141	+	0.108087i	$0.0344724\pi$
$20$	0	0
$21$	0.682985	0.149039
$22$	0	0
$23$	2.21439	0.461733	0.230866	−	0.972985i	$-0.425844\pi$
$23$	2.21439	0.461733	0.230866	+	0.972985i	$0.425844\pi$
$24$	0	0
$25$	2.55766	0.511533
$26$	0	0
$27$	1.40914	0.271189
$28$	0	0
$29$	−4.31561	−0.801389	−0.400695	−	0.916212i	$-0.631231\pi$
$29$	−4.31561	−0.801389	−0.400695	+	0.916212i	$0.631231\pi$
$30$	0	0
$31$	−1.36287	−0.244779	−0.122389	−	0.992482i	$-0.539056\pi$
$31$	−1.36287	−0.244779	−0.122389	+	0.992482i	$0.539056\pi$
$32$	0	0
$33$	0.548120	0.0954155
$34$	0	0
$35$	−7.91980	−1.33869
$36$	0	0
$37$	−4.03568	−0.663462	−0.331731	−	0.943374i	$-0.607633\pi$
$37$	−4.03568	−0.663462	−0.331731	+	0.943374i	$0.607633\pi$
$38$	0	0
$39$	−0.213301	−0.0341555
$40$	0	0
$41$	−8.30878	−1.29761	−0.648807	−	0.760953i	$-0.724732\pi$
$41$	−8.30878	−1.29761	−0.648807	+	0.760953i	$0.724732\pi$
$42$	0	0
$43$	−0.582436	−0.0888207	−0.0444103	−	0.999013i	$-0.514141\pi$
$43$	−0.582436	−0.0888207	−0.0444103	+	0.999013i	$0.514141\pi$
$44$	0	0
$45$	−8.09284	−1.20641
$46$	0	0
$47$	3.69851	0.539484	0.269742	−	0.962933i	$-0.413062\pi$
$47$	3.69851	0.539484	0.269742	+	0.962933i	$0.413062\pi$
$48$	0	0
$49$	1.29928	0.185612
$50$	0	0
$51$	−0.237078	−0.0331975
$52$	0	0
$53$	1.86324	0.255936	0.127968	−	0.991778i	$-0.459154\pi$
$53$	1.86324	0.255936	0.127968	+	0.991778i	$0.459154\pi$
$54$	0	0
$55$	−6.35593	−0.857033
$56$	0	0
$57$	−2.05469	−0.272150
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−5.61743	−0.719239	−0.359619	−	0.933099i	$-0.617093\pi$
$61$	−5.61743	−0.719239	−0.359619	+	0.933099i	$0.617093\pi$
$62$	0	0
$63$	8.48062	1.06846
$64$	0	0
$65$	2.47341	0.306789
$66$	0	0
$67$	11.1511	1.36233	0.681163	−	0.732132i	$-0.261474\pi$
$67$	11.1511	1.36233	0.681163	+	0.732132i	$0.261474\pi$
$68$	0	0
$69$	−0.524983	−0.0632005
$70$	0	0
$71$	0.209332	0.0248432	0.0124216	−	0.999923i	$-0.496046\pi$
$71$	0.209332	0.0248432	0.0124216	+	0.999923i	$0.496046\pi$
$72$	0	0
$73$	0.762440	0.0892369	0.0446184	−	0.999004i	$-0.485793\pi$
$73$	0.762440	0.0892369	0.0446184	+	0.999004i	$0.485793\pi$
$74$	0	0
$75$	−0.606365	−0.0700170
$76$	0	0
$77$	6.66048	0.759032
$78$	0	0
$79$	3.50569	0.394421	0.197210	−	0.980361i	$-0.436812\pi$
$79$	3.50569	0.394421	0.197210	+	0.980361i	$0.436812\pi$
$80$	0	0
$81$	8.49731	0.944145
$82$	0	0
$83$	−7.71241	−0.846547	−0.423273	−	0.906002i	$-0.639119\pi$
$83$	−7.71241	−0.846547	−0.423273	+	0.906002i	$0.639119\pi$
$84$	0	0
$85$	2.74912	0.298184
$86$	0	0
$87$	1.02314	0.109692
$88$	0	0
$89$	−3.55723	−0.377065	−0.188533	−	0.982067i	$-0.560373\pi$
$89$	−3.55723	−0.377065	−0.188533	+	0.982067i	$0.560373\pi$
$90$	0	0
$91$	−2.59193	−0.271708
$92$	0	0
$93$	0.323106	0.0335045
$94$	0	0
$95$	23.8259	2.44448
$96$	0	0
$97$	−18.9378	−1.92284	−0.961422	−	0.275079i	$-0.911296\pi$
$97$	−18.9378	−1.92284	−0.961422	+	0.275079i	$0.911296\pi$
$98$	0	0
$99$	6.80601	0.684030
$100$	0	0
$101$	16.6873	1.66045	0.830224	−	0.557430i	$-0.188213\pi$
$101$	16.6873	1.66045	0.830224	+	0.557430i	$0.188213\pi$
$102$	0	0
$103$	−0.508442	−0.0500982	−0.0250491	−	0.999686i	$-0.507974\pi$
$103$	−0.508442	−0.0500982	−0.0250491	+	0.999686i	$0.507974\pi$
$104$	0	0
$105$	1.87761	0.183236
$106$	0	0
$107$	−0.138777	−0.0134161	−0.00670803	−	0.999978i	$-0.502135\pi$
$107$	−0.138777	−0.0134161	−0.00670803	+	0.999978i	$0.502135\pi$
$108$	0	0
$109$	−13.7242	−1.31454	−0.657268	−	0.753657i	$-0.728288\pi$
$109$	−13.7242	−1.31454	−0.657268	+	0.753657i	$0.728288\pi$
$110$	0	0
$111$	0.956770	0.0908126
$112$	0	0
$113$	−7.30585	−0.687276	−0.343638	−	0.939102i	$-0.611659\pi$
$113$	−7.30585	−0.687276	−0.343638	+	0.939102i	$0.611659\pi$
$114$	0	0
$115$	6.08763	0.567675
$116$	0	0
$117$	−2.64856	−0.244860
$118$	0	0
$119$	−2.88085	−0.264087
$120$	0	0
$121$	−5.65472	−0.514065
$122$	0	0
$123$	1.96983	0.177613
$124$	0	0
$125$	−6.71428	−0.600543
$126$	0	0
$127$	3.28468	0.291468	0.145734	−	0.989324i	$-0.453446\pi$
$127$	3.28468	0.291468	0.145734	+	0.989324i	$0.453446\pi$
$128$	0	0
$129$	0.138083	0.0121575
$130$	0	0
$131$	15.1218	1.32119	0.660597	−	0.750741i	$-0.270303\pi$
$131$	15.1218	1.32119	0.660597	+	0.750741i	$0.270303\pi$
$132$	0	0
$133$	−24.9675	−2.16496
$134$	0	0
$135$	3.87390	0.333412
$136$	0	0
$137$	3.85238	0.329131	0.164565	−	0.986366i	$-0.447378\pi$
$137$	3.85238	0.329131	0.164565	+	0.986366i	$0.447378\pi$
$138$	0	0
$139$	−5.35235	−0.453980	−0.226990	−	0.973897i	$-0.572889\pi$
$139$	−5.35235	−0.453980	−0.226990	+	0.973897i	$0.572889\pi$
$140$	0	0
$141$	−0.876835	−0.0738428
$142$	0	0
$143$	−2.08012	−0.173948
$144$	0	0
$145$	−11.8641	−0.985263
$146$	0	0
$147$	−0.308031	−0.0254060
$148$	0	0
$149$	14.6345	1.19891	0.599454	−	0.800409i	$-0.295384\pi$
$149$	14.6345	1.19891	0.599454	+	0.800409i	$0.295384\pi$
$150$	0	0
$151$	−11.8192	−0.961837	−0.480918	−	0.876765i	$-0.659697\pi$
$151$	−11.8192	−0.961837	−0.480918	+	0.876765i	$0.659697\pi$
$152$	0	0
$153$	−2.94379	−0.237992
$154$	0	0
$155$	−3.74670	−0.300942
$156$	0	0
$157$	−8.52128	−0.680072	−0.340036	−	0.940412i	$-0.610439\pi$
$157$	−8.52128	−0.680072	−0.340036	+	0.940412i	$0.610439\pi$
$158$	0	0
$159$	−0.441734	−0.0350318
$160$	0	0
$161$	−6.37933	−0.502762
$162$	0	0
$163$	−5.50060	−0.430840	−0.215420	−	0.976522i	$-0.569112\pi$
$163$	−5.50060	−0.430840	−0.215420	+	0.976522i	$0.569112\pi$
$164$	0	0
$165$	1.50685	0.117308
$166$	0	0
$167$	−5.60889	−0.434029	−0.217014	−	0.976168i	$-0.569632\pi$
$167$	−5.60889	−0.434029	−0.217014	+	0.976168i	$0.569632\pi$
$168$	0	0
$169$	−12.1905	−0.937732
$170$	0	0
$171$	−25.5131	−1.95103
$172$	0	0
$173$	−24.7553	−1.88211	−0.941055	−	0.338253i	$-0.890164\pi$
$173$	−24.7553	−1.88211	−0.941055	+	0.338253i	$0.890164\pi$
$174$	0	0
$175$	−7.36824	−0.556986
$176$	0	0
$177$	0.237078	0.0178198
$178$	0	0
$179$	−24.0309	−1.79616	−0.898079	−	0.439835i	$-0.855037\pi$
$179$	−24.0309	−1.79616	−0.898079	+	0.439835i	$0.855037\pi$
$180$	0	0
$181$	−5.85017	−0.434839	−0.217420	−	0.976078i	$-0.569764\pi$
$181$	−5.85017	−0.434839	−0.217420	+	0.976078i	$0.569764\pi$
$182$	0	0
$183$	1.33177	0.0984471
$184$	0	0
$185$	−11.0946	−0.815690
$186$	0	0
$187$	−2.31199	−0.169069
$188$	0	0
$189$	−4.05952	−0.295287
$190$	0	0
$191$	25.9534	1.87792	0.938962	−	0.344022i	$-0.111789\pi$
$191$	25.9534	1.87792	0.938962	+	0.344022i	$0.111789\pi$
$192$	0	0
$193$	16.5911	1.19425	0.597127	−	0.802147i	$-0.296309\pi$
$193$	16.5911	1.19425	0.597127	+	0.802147i	$0.296309\pi$
$194$	0	0
$195$	−0.586390	−0.0419923
$196$	0	0
$197$	−15.3874	−1.09630	−0.548152	−	0.836378i	$-0.684669\pi$
$197$	−15.3874	−1.09630	−0.548152	+	0.836378i	$0.684669\pi$
$198$	0	0
$199$	5.12580	0.363359	0.181679	−	0.983358i	$-0.441847\pi$
$199$	5.12580	0.363359	0.181679	+	0.983358i	$0.441847\pi$
$200$	0	0
$201$	−2.64368	−0.186471
$202$	0	0
$203$	12.4326	0.872599
$204$	0	0
$205$	−22.8418	−1.59534
$206$	0	0
$207$	−6.51872	−0.453082
$208$	0	0
$209$	−20.0373	−1.38601
$210$	0	0
$211$	−3.88050	−0.267144	−0.133572	−	0.991039i	$-0.542645\pi$
$211$	−3.88050	−0.267144	−0.133572	+	0.991039i	$0.542645\pi$
$212$	0	0
$213$	−0.0496280	−0.00340045
$214$	0	0
$215$	−1.60119	−0.109200
$216$	0	0
$217$	3.92622	0.266529
$218$	0	0
$219$	−0.180758	−0.0122145
$220$	0	0
$221$	0.899710	0.0605210
$222$	0	0
$223$	−27.7090	−1.85553	−0.927767	−	0.373161i	$-0.878274\pi$
$223$	−27.7090	−1.85553	−0.927767	+	0.373161i	$0.878274\pi$
$224$	0	0
$225$	−7.52923	−0.501949
$226$	0	0
$227$	−24.7822	−1.64485	−0.822425	−	0.568874i	$-0.807379\pi$
$227$	−24.7822	−1.64485	−0.822425	+	0.568874i	$0.807379\pi$
$228$	0	0
$229$	12.6925	0.838742	0.419371	−	0.907815i	$-0.362251\pi$
$229$	12.6925	0.838742	0.419371	+	0.907815i	$0.362251\pi$
$230$	0	0
$231$	−1.57905	−0.103894
$232$	0	0
$233$	−11.3768	−0.745317	−0.372659	−	0.927968i	$-0.621554\pi$
$233$	−11.3768	−0.745317	−0.372659	+	0.927968i	$0.621554\pi$
$234$	0	0
$235$	10.1677	0.663265
$236$	0	0
$237$	−0.831120	−0.0539870
$238$	0	0
$239$	9.79857	0.633817	0.316908	−	0.948456i	$-0.397355\pi$
$239$	9.79857	0.633817	0.316908	+	0.948456i	$0.397355\pi$
$240$	0	0
$241$	−3.01669	−0.194322	−0.0971610	−	0.995269i	$-0.530976\pi$
$241$	−3.01669	−0.194322	−0.0971610	+	0.995269i	$0.530976\pi$
$242$	0	0
$243$	−6.24194	−0.400421
$244$	0	0
$245$	3.57189	0.228200
$246$	0	0
$247$	7.79754	0.496145
$248$	0	0
$249$	1.82844	0.115873
$250$	0	0
$251$	−8.81759	−0.556562	−0.278281	−	0.960500i	$-0.589765\pi$
$251$	−8.81759	−0.556562	−0.278281	+	0.960500i	$0.589765\pi$
$252$	0	0
$253$	−5.11965	−0.321869
$254$	0	0
$255$	−0.651755	−0.0408145
$256$	0	0
$257$	24.6070	1.53495	0.767473	−	0.641082i	$-0.221514\pi$
$257$	24.6070	1.53495	0.767473	+	0.641082i	$0.221514\pi$
$258$	0	0
$259$	11.6262	0.722416
$260$	0	0
$261$	12.7043	0.786375
$262$	0	0
$263$	−16.2021	−0.999063	−0.499531	−	0.866296i	$-0.666494\pi$
$263$	−16.2021	−0.999063	−0.499531	+	0.866296i	$0.666494\pi$
$264$	0	0
$265$	5.12228	0.314659
$266$	0	0
$267$	0.843339	0.0516115
$268$	0	0
$269$	−24.6464	−1.50272	−0.751359	−	0.659893i	$-0.770601\pi$
$269$	−24.6464	−1.50272	−0.751359	+	0.659893i	$0.770601\pi$
$270$	0	0
$271$	2.79208	0.169607	0.0848033	−	0.996398i	$-0.472974\pi$
$271$	2.79208	0.169607	0.0848033	+	0.996398i	$0.472974\pi$
$272$	0	0
$273$	0.614488	0.0371905
$274$	0	0
$275$	−5.91328	−0.356584
$276$	0	0
$277$	−20.3022	−1.21984	−0.609920	−	0.792463i	$-0.708798\pi$
$277$	−20.3022	−1.21984	−0.609920	+	0.792463i	$0.708798\pi$
$278$	0	0
$279$	4.01201	0.240193
$280$	0	0
$281$	0.466352	0.0278202	0.0139101	−	0.999903i	$-0.495572\pi$
$281$	0.466352	0.0278202	0.0139101	+	0.999903i	$0.495572\pi$
$282$	0	0
$283$	17.0677	1.01457	0.507285	−	0.861778i	$-0.330649\pi$
$283$	17.0677	1.01457	0.507285	+	0.861778i	$0.330649\pi$
$284$	0	0
$285$	−5.64858	−0.334593
$286$	0	0
$287$	23.9363	1.41292
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	4.48973	0.263193
$292$	0	0
$293$	−20.5803	−1.20232	−0.601158	−	0.799130i	$-0.705294\pi$
$293$	−20.5803	−1.20232	−0.601158	+	0.799130i	$0.705294\pi$
$294$	0	0
$295$	−2.74912	−0.160060
$296$	0	0
$297$	−3.25791	−0.189043
$298$	0	0
$299$	1.99231	0.115218
$300$	0	0
$301$	1.67791	0.0967131
$302$	0	0
$303$	−3.95618	−0.227277
$304$	0	0
$305$	−15.4430	−0.884264
$306$	0	0
$307$	−10.7985	−0.616305	−0.308152	−	0.951337i	$-0.599711\pi$
$307$	−10.7985	−0.616305	−0.308152	+	0.951337i	$0.599711\pi$
$308$	0	0
$309$	0.120540	0.00685729
$310$	0	0
$311$	10.2457	0.580981	0.290491	−	0.956878i	$-0.406182\pi$
$311$	10.2457	0.580981	0.290491	+	0.956878i	$0.406182\pi$
$312$	0	0
$313$	21.8852	1.23703	0.618513	−	0.785774i	$-0.287735\pi$
$313$	21.8852	1.23703	0.618513	+	0.785774i	$0.287735\pi$
$314$	0	0
$315$	23.3143	1.31361
$316$	0	0
$317$	−17.8305	−1.00146	−0.500729	−	0.865604i	$-0.666935\pi$
$317$	−17.8305	−1.00146	−0.500729	+	0.865604i	$0.666935\pi$
$318$	0	0
$319$	9.97764	0.558640
$320$	0	0
$321$	0.0329009	0.00183635
$322$	0	0
$323$	8.66672	0.482229
$324$	0	0
$325$	2.30115	0.127645
$326$	0	0
$327$	3.25369	0.179930
$328$	0	0
$329$	−10.6549	−0.587421
$330$	0	0
$331$	−3.51021	−0.192939	−0.0964693	−	0.995336i	$-0.530755\pi$
$331$	−3.51021	−0.192939	−0.0964693	+	0.995336i	$0.530755\pi$
$332$	0	0
$333$	11.8802	0.651032
$334$	0	0
$335$	30.6558	1.67490
$336$	0	0
$337$	−16.9342	−0.922466	−0.461233	−	0.887279i	$-0.652593\pi$
$337$	−16.9342	−0.922466	−0.461233	+	0.887279i	$0.652593\pi$
$338$	0	0
$339$	1.73205	0.0940722
$340$	0	0
$341$	3.15094	0.170633
$342$	0	0
$343$	16.4229	0.886753
$344$	0	0
$345$	−1.44324	−0.0777015
$346$	0	0
$347$	−19.3020	−1.03619	−0.518093	−	0.855324i	$-0.673358\pi$
$347$	−19.3020	−1.03619	−0.518093	+	0.855324i	$0.673358\pi$
$348$	0	0
$349$	−4.94096	−0.264484	−0.132242	−	0.991217i	$-0.542218\pi$
$349$	−4.94096	−0.264484	−0.132242	+	0.991217i	$0.542218\pi$
$350$	0	0
$351$	1.26782	0.0676711
$352$	0	0
$353$	22.6572	1.20592	0.602962	−	0.797770i	$-0.293987\pi$
$353$	22.6572	1.20592	0.602962	+	0.797770i	$0.293987\pi$
$354$	0	0
$355$	0.575479	0.0305433
$356$	0	0
$357$	0.682985	0.0361474
$358$	0	0
$359$	−10.1642	−0.536447	−0.268223	−	0.963357i	$-0.586437\pi$
$359$	−10.1642	−0.536447	−0.268223	+	0.963357i	$0.586437\pi$
$360$	0	0
$361$	56.1121	2.95327
$362$	0	0
$363$	1.34061	0.0703636
$364$	0	0
$365$	2.09604	0.109712
$366$	0	0
$367$	2.18620	0.114119	0.0570593	−	0.998371i	$-0.481828\pi$
$367$	2.18620	0.114119	0.0570593	+	0.998371i	$0.481828\pi$
$368$	0	0
$369$	24.4593	1.27330
$370$	0	0
$371$	−5.36772	−0.278678
$372$	0	0
$373$	2.24388	0.116184	0.0580920	−	0.998311i	$-0.481498\pi$
$373$	2.24388	0.116184	0.0580920	+	0.998311i	$0.481498\pi$
$374$	0	0
$375$	1.59181	0.0822005
$376$	0	0
$377$	−3.88280	−0.199974
$378$	0	0
$379$	−1.90160	−0.0976787	−0.0488393	−	0.998807i	$-0.515552\pi$
$379$	−1.90160	−0.0976787	−0.0488393	+	0.998807i	$0.515552\pi$
$380$	0	0
$381$	−0.778724	−0.0398952
$382$	0	0
$383$	−24.8301	−1.26876	−0.634379	−	0.773022i	$-0.718744\pi$
$383$	−24.8301	−1.26876	−0.634379	+	0.773022i	$0.718744\pi$
$384$	0	0
$385$	18.3105	0.933188
$386$	0	0
$387$	1.71457	0.0871566
$388$	0	0
$389$	−28.9665	−1.46866	−0.734329	−	0.678793i	$-0.762503\pi$
$389$	−28.9665	−1.46866	−0.734329	+	0.678793i	$0.762503\pi$
$390$	0	0
$391$	2.21439	0.111987
$392$	0	0
$393$	−3.58503	−0.180841
$394$	0	0
$395$	9.63756	0.484918
$396$	0	0
$397$	24.8207	1.24571	0.622857	−	0.782336i	$-0.285972\pi$
$397$	24.8207	1.24571	0.622857	+	0.782336i	$0.285972\pi$
$398$	0	0
$399$	5.91924	0.296333
$400$	0	0
$401$	8.38186	0.418570	0.209285	−	0.977855i	$-0.432886\pi$
$401$	8.38186	0.418570	0.209285	+	0.977855i	$0.432886\pi$
$402$	0	0
$403$	−1.22619	−0.0610808
$404$	0	0
$405$	23.3601	1.16077
$406$	0	0
$407$	9.33044	0.462493
$408$	0	0
$409$	−18.9230	−0.935683	−0.467841	−	0.883812i	$-0.654968\pi$
$409$	−18.9230	−0.935683	−0.467841	+	0.883812i	$0.654968\pi$
$410$	0	0
$411$	−0.913312	−0.0450504
$412$	0	0
$413$	2.88085	0.141757
$414$	0	0
$415$	−21.2023	−1.04078
$416$	0	0
$417$	1.26892	0.0621394
$418$	0	0
$419$	−28.8403	−1.40894	−0.704470	−	0.709734i	$-0.748815\pi$
$419$	−28.8403	−1.40894	−0.704470	+	0.709734i	$0.748815\pi$
$420$	0	0
$421$	−6.69470	−0.326280	−0.163140	−	0.986603i	$-0.552162\pi$
$421$	−6.69470	−0.326280	−0.163140	+	0.986603i	$0.552162\pi$
$422$	0	0
$423$	−10.8877	−0.529376
$424$	0	0
$425$	2.55766	0.124065
$426$	0	0
$427$	16.1830	0.783149
$428$	0	0
$429$	0.493149	0.0238095
$430$	0	0
$431$	−3.35627	−0.161666	−0.0808330	−	0.996728i	$-0.525758\pi$
$431$	−3.35627	−0.161666	−0.0808330	+	0.996728i	$0.525758\pi$
$432$	0	0
$433$	−22.8497	−1.09809	−0.549043	−	0.835794i	$-0.685008\pi$
$433$	−22.8497	−1.09809	−0.549043	+	0.835794i	$0.685008\pi$
$434$	0	0
$435$	2.81272	0.134860
$436$	0	0
$437$	19.1915	0.918056
$438$	0	0
$439$	−8.41995	−0.401862	−0.200931	−	0.979605i	$-0.564397\pi$
$439$	−8.41995	−0.401862	−0.200931	+	0.979605i	$0.564397\pi$
$440$	0	0
$441$	−3.82483	−0.182135
$442$	0	0
$443$	−41.9985	−1.99541	−0.997705	−	0.0677109i	$-0.978430\pi$
$443$	−41.9985	−1.99541	−0.997705	+	0.0677109i	$0.978430\pi$
$444$	0	0
$445$	−9.77924	−0.463580
$446$	0	0
$447$	−3.46952	−0.164103
$448$	0	0
$449$	8.39799	0.396326	0.198163	−	0.980169i	$-0.436503\pi$
$449$	8.39799	0.396326	0.198163	+	0.980169i	$0.436503\pi$
$450$	0	0
$451$	19.2098	0.904554
$452$	0	0
$453$	2.80208	0.131653
$454$	0	0
$455$	−7.12552	−0.334050
$456$	0	0
$457$	34.1576	1.59782	0.798912	−	0.601447i	$-0.205409\pi$
$457$	34.1576	1.59782	0.798912	+	0.601447i	$0.205409\pi$
$458$	0	0
$459$	1.40914	0.0657730
$460$	0	0
$461$	19.0056	0.885178	0.442589	−	0.896725i	$-0.354060\pi$
$461$	19.0056	0.885178	0.442589	+	0.896725i	$0.354060\pi$
$462$	0	0
$463$	19.6958	0.915341	0.457671	−	0.889122i	$-0.348684\pi$
$463$	19.6958	0.915341	0.457671	+	0.889122i	$0.348684\pi$
$464$	0	0
$465$	0.888258	0.0411920
$466$	0	0
$467$	23.7651	1.09972	0.549859	−	0.835258i	$-0.314682\pi$
$467$	23.7651	1.09972	0.549859	+	0.835258i	$0.314682\pi$
$468$	0	0
$469$	−32.1247	−1.48338
$470$	0	0
$471$	2.02020	0.0930861
$472$	0	0
$473$	1.34658	0.0619160
$474$	0	0
$475$	22.1666	1.01707
$476$	0	0
$477$	−5.48501	−0.251141
$478$	0	0
$479$	2.95171	0.134867	0.0674336	−	0.997724i	$-0.478519\pi$
$479$	2.95171	0.134867	0.0674336	+	0.997724i	$0.478519\pi$
$480$	0	0
$481$	−3.63094	−0.165557
$482$	0	0
$483$	1.51240	0.0688164
$484$	0	0
$485$	−52.0623	−2.36403
$486$	0	0
$487$	−30.8429	−1.39762	−0.698812	−	0.715305i	$-0.746288\pi$
$487$	−30.8429	−1.39762	−0.698812	+	0.715305i	$0.746288\pi$
$488$	0	0
$489$	1.30407	0.0589720
$490$	0	0
$491$	17.7386	0.800531	0.400266	−	0.916399i	$-0.368918\pi$
$491$	17.7386	0.800531	0.400266	+	0.916399i	$0.368918\pi$
$492$	0	0
$493$	−4.31561	−0.194365
$494$	0	0
$495$	18.7105	0.840976
$496$	0	0
$497$	−0.603054	−0.0270507
$498$	0	0
$499$	7.22002	0.323212	0.161606	−	0.986855i	$-0.448333\pi$
$499$	7.22002	0.323212	0.161606	+	0.986855i	$0.448333\pi$
$500$	0	0
$501$	1.32974	0.0594085
$502$	0	0
$503$	20.9367	0.933519	0.466760	−	0.884384i	$-0.345421\pi$
$503$	20.9367	0.933519	0.466760	+	0.884384i	$0.345421\pi$
$504$	0	0
$505$	45.8754	2.04143
$506$	0	0
$507$	2.89010	0.128354
$508$	0	0
$509$	24.6436	1.09231	0.546155	−	0.837684i	$-0.316091\pi$
$509$	24.6436	1.09231	0.546155	+	0.837684i	$0.316091\pi$
$510$	0	0
$511$	−2.19647	−0.0971663
$512$	0	0
$513$	12.2126	0.539201
$514$	0	0
$515$	−1.39777	−0.0615930
$516$	0	0
$517$	−8.55091	−0.376069
$518$	0	0
$519$	5.86893	0.257617
$520$	0	0
$521$	−11.5657	−0.506701	−0.253350	−	0.967375i	$-0.581533\pi$
$521$	−11.5657	−0.506701	−0.253350	+	0.967375i	$0.581533\pi$
$522$	0	0
$523$	−8.59249	−0.375723	−0.187862	−	0.982195i	$-0.560156\pi$
$523$	−8.59249	−0.375723	−0.187862	+	0.982195i	$0.560156\pi$
$524$	0	0
$525$	1.74684	0.0762385
$526$	0	0
$527$	−1.36287	−0.0593676
$528$	0	0
$529$	−18.0965	−0.786803
$530$	0	0
$531$	2.94379	0.127750
$532$	0	0
$533$	−7.47549	−0.323800
$534$	0	0
$535$	−0.381514	−0.0164943
$536$	0	0
$537$	5.69720	0.245852
$538$	0	0
$539$	−3.00393	−0.129388
$540$	0	0
$541$	−26.6430	−1.14547	−0.572736	−	0.819740i	$-0.694118\pi$
$541$	−26.6430	−1.14547	−0.572736	+	0.819740i	$0.694118\pi$
$542$	0	0
$543$	1.38694	0.0595194
$544$	0	0
$545$	−37.7294	−1.61615
$546$	0	0
$547$	−25.6682	−1.09749	−0.548746	−	0.835989i	$-0.684895\pi$
$547$	−25.6682	−1.09749	−0.548746	+	0.835989i	$0.684895\pi$
$548$	0	0
$549$	16.5366	0.705764
$550$	0	0
$551$	−37.4022	−1.59339
$552$	0	0
$553$	−10.0994	−0.429468
$554$	0	0
$555$	2.63028	0.111649
$556$	0	0
$557$	25.4817	1.07969	0.539846	−	0.841764i	$-0.318482\pi$
$557$	25.4817	1.07969	0.539846	+	0.841764i	$0.318482\pi$
$558$	0	0
$559$	−0.524024	−0.0221638
$560$	0	0
$561$	0.548120	0.0231417
$562$	0	0
$563$	−20.7494	−0.874482	−0.437241	−	0.899344i	$-0.644044\pi$
$563$	−20.7494	−0.874482	−0.437241	+	0.899344i	$0.644044\pi$
$564$	0	0
$565$	−20.0847	−0.844968
$566$	0	0
$567$	−24.4794	−1.02804
$568$	0	0
$569$	−22.5059	−0.943498	−0.471749	−	0.881733i	$-0.656377\pi$
$569$	−22.5059	−0.943498	−0.471749	+	0.881733i	$0.656377\pi$
$570$	0	0
$571$	2.68276	0.112270	0.0561350	−	0.998423i	$-0.482122\pi$
$571$	2.68276	0.112270	0.0561350	+	0.998423i	$0.482122\pi$
$572$	0	0
$573$	−6.15297	−0.257044
$574$	0	0
$575$	5.66367	0.236191
$576$	0	0
$577$	1.71059	0.0712129	0.0356065	−	0.999366i	$-0.488664\pi$
$577$	1.71059	0.0712129	0.0356065	+	0.999366i	$0.488664\pi$
$578$	0	0
$579$	−3.93338	−0.163466
$580$	0	0
$581$	22.2183	0.921769
$582$	0	0
$583$	−4.30780	−0.178411
$584$	0	0
$585$	−7.28121	−0.301041
$586$	0	0
$587$	7.43707	0.306961	0.153480	−	0.988152i	$-0.450952\pi$
$587$	7.43707	0.306961	0.153480	+	0.988152i	$0.450952\pi$
$588$	0	0
$589$	−11.8116	−0.486690
$590$	0	0
$591$	3.64800	0.150059
$592$	0	0
$593$	26.0155	1.06833	0.534164	−	0.845381i	$-0.320627\pi$
$593$	26.0155	1.06833	0.534164	+	0.845381i	$0.320627\pi$
$594$	0	0
$595$	−7.91980	−0.324680
$596$	0	0
$597$	−1.21521	−0.0497354
$598$	0	0
$599$	−32.2216	−1.31654	−0.658269	−	0.752783i	$-0.728711\pi$
$599$	−32.2216	−1.31654	−0.658269	+	0.752783i	$0.728711\pi$
$600$	0	0
$601$	−27.2334	−1.11088	−0.555438	−	0.831558i	$-0.687449\pi$
$601$	−27.2334	−1.11088	−0.555438	+	0.831558i	$0.687449\pi$
$602$	0	0
$603$	−32.8266	−1.33680
$604$	0	0
$605$	−15.5455	−0.632015
$606$	0	0
$607$	6.78953	0.275578	0.137789	−	0.990462i	$-0.456000\pi$
$607$	6.78953	0.275578	0.137789	+	0.990462i	$0.456000\pi$
$608$	0	0
$609$	−2.94750	−0.119439
$610$	0	0
$611$	3.32759	0.134620
$612$	0	0
$613$	26.1462	1.05603	0.528017	−	0.849234i	$-0.322936\pi$
$613$	26.1462	1.05603	0.528017	+	0.849234i	$0.322936\pi$
$614$	0	0
$615$	5.41529	0.218366
$616$	0	0
$617$	4.02522	0.162049	0.0810246	−	0.996712i	$-0.474181\pi$
$617$	4.02522	0.162049	0.0810246	+	0.996712i	$0.474181\pi$
$618$	0	0
$619$	35.2373	1.41631	0.708154	−	0.706058i	$-0.249528\pi$
$619$	35.2373	1.41631	0.708154	+	0.706058i	$0.249528\pi$
$620$	0	0
$621$	3.12039	0.125217
$622$	0	0
$623$	10.2478	0.410570
$624$	0	0
$625$	−31.2467	−1.24987
$626$	0	0
$627$	4.75041	0.189713
$628$	0	0
$629$	−4.03568	−0.160913
$630$	0	0
$631$	−48.5640	−1.93330	−0.966651	−	0.256096i	$-0.917564\pi$
$631$	−48.5640	−1.93330	−0.966651	+	0.256096i	$0.917564\pi$
$632$	0	0
$633$	0.919979	0.0365659
$634$	0	0
$635$	9.02998	0.358344
$636$	0	0
$637$	1.16898	0.0463166
$638$	0	0
$639$	−0.616231	−0.0243777
$640$	0	0
$641$	6.05073	0.238989	0.119495	−	0.992835i	$-0.461873\pi$
$641$	6.05073	0.238989	0.119495	+	0.992835i	$0.461873\pi$
$642$	0	0
$643$	32.8378	1.29499	0.647497	−	0.762068i	$-0.275816\pi$
$643$	32.8378	1.29499	0.647497	+	0.762068i	$0.275816\pi$
$644$	0	0
$645$	0.379606	0.0149470
$646$	0	0
$647$	−3.37044	−0.132506	−0.0662528	−	0.997803i	$-0.521104\pi$
$647$	−3.37044	−0.132506	−0.0662528	+	0.997803i	$0.521104\pi$
$648$	0	0
$649$	2.31199	0.0907534
$650$	0	0
$651$	−0.930820	−0.0364817
$652$	0	0
$653$	6.31124	0.246978	0.123489	−	0.992346i	$-0.460592\pi$
$653$	6.31124	0.246978	0.123489	+	0.992346i	$0.460592\pi$
$654$	0	0
$655$	41.5715	1.62433
$656$	0	0
$657$	−2.24447	−0.0875650
$658$	0	0
$659$	7.34249	0.286023	0.143011	−	0.989721i	$-0.454321\pi$
$659$	7.34249	0.286023	0.143011	+	0.989721i	$0.454321\pi$
$660$	0	0
$661$	23.7236	0.922740	0.461370	−	0.887208i	$-0.347358\pi$
$661$	23.7236	0.922740	0.461370	+	0.887208i	$0.347358\pi$
$662$	0	0
$663$	−0.213301	−0.00828393
$664$	0	0
$665$	−68.6387	−2.66169
$666$	0	0
$667$	−9.55646	−0.370028
$668$	0	0
$669$	6.56919	0.253980
$670$	0	0
$671$	12.9874	0.501374
$672$	0	0
$673$	39.1218	1.50804	0.754018	−	0.656854i	$-0.228113\pi$
$673$	39.1218	1.50804	0.754018	+	0.656854i	$0.228113\pi$
$674$	0	0
$675$	3.60411	0.138722
$676$	0	0
$677$	4.29806	0.165188	0.0825938	−	0.996583i	$-0.473680\pi$
$677$	4.29806	0.165188	0.0825938	+	0.996583i	$0.473680\pi$
$678$	0	0
$679$	54.5569	2.09370
$680$	0	0
$681$	5.87529	0.225142
$682$	0	0
$683$	−21.5704	−0.825368	−0.412684	−	0.910874i	$-0.635409\pi$
$683$	−21.5704	−0.825368	−0.412684	+	0.910874i	$0.635409\pi$
$684$	0	0
$685$	10.5906	0.404648
$686$	0	0
$687$	−3.00910	−0.114804
$688$	0	0
$689$	1.67638	0.0638650
$690$	0	0
$691$	−9.19332	−0.349730	−0.174865	−	0.984592i	$-0.555949\pi$
$691$	−9.19332	−0.349730	−0.174865	+	0.984592i	$0.555949\pi$
$692$	0	0
$693$	−19.6071	−0.744812
$694$	0	0
$695$	−14.7142	−0.558143
$696$	0	0
$697$	−8.30878	−0.314718
$698$	0	0
$699$	2.69718	0.102017
$700$	0	0
$701$	−4.06109	−0.153385	−0.0766926	−	0.997055i	$-0.524436\pi$
$701$	−4.06109	−0.153385	−0.0766926	+	0.997055i	$0.524436\pi$
$702$	0	0
$703$	−34.9762	−1.31915
$704$	0	0
$705$	−2.41052	−0.0907856
$706$	0	0
$707$	−48.0736	−1.80799
$708$	0	0
$709$	−21.9896	−0.825838	−0.412919	−	0.910768i	$-0.635491\pi$
$709$	−21.9896	−0.825838	−0.412919	+	0.910768i	$0.635491\pi$
$710$	0	0
$711$	−10.3200	−0.387031
$712$	0	0
$713$	−3.01793	−0.113022
$714$	0	0
$715$	−5.71849	−0.213859
$716$	0	0
$717$	−2.32302	−0.0867548
$718$	0	0
$719$	44.1996	1.64837	0.824184	−	0.566322i	$-0.191634\pi$
$719$	44.1996	1.64837	0.824184	+	0.566322i	$0.191634\pi$
$720$	0	0
$721$	1.46474	0.0545499
$722$	0	0
$723$	0.715190	0.0265982
$724$	0	0
$725$	−11.0379	−0.409937
$726$	0	0
$727$	6.98704	0.259135	0.129568	−	0.991571i	$-0.458641\pi$
$727$	6.98704	0.259135	0.129568	+	0.991571i	$0.458641\pi$
$728$	0	0
$729$	−24.0121	−0.889337
$730$	0	0
$731$	−0.582436	−0.0215422
$732$	0	0
$733$	−37.5065	−1.38533	−0.692666	−	0.721258i	$-0.743564\pi$
$733$	−37.5065	−1.38533	−0.692666	+	0.721258i	$0.743564\pi$
$734$	0	0
$735$	−0.846815	−0.0312352
$736$	0	0
$737$	−25.7812	−0.949664
$738$	0	0
$739$	10.0463	0.369560	0.184780	−	0.982780i	$-0.440843\pi$
$739$	10.0463	0.369560	0.184780	+	0.982780i	$0.440843\pi$
$740$	0	0
$741$	−1.84862	−0.0679108
$742$	0	0
$743$	23.6249	0.866712	0.433356	−	0.901223i	$-0.357329\pi$
$743$	23.6249	0.866712	0.433356	+	0.901223i	$0.357329\pi$
$744$	0	0
$745$	40.2321	1.47399
$746$	0	0
$747$	22.7037	0.830687
$748$	0	0
$749$	0.399795	0.0146082
$750$	0	0
$751$	−20.9995	−0.766284	−0.383142	−	0.923689i	$-0.625158\pi$
$751$	−20.9995	−0.766284	−0.383142	+	0.923689i	$0.625158\pi$
$752$	0	0
$753$	2.09045	0.0761804
$754$	0	0
$755$	−32.4925	−1.18252
$756$	0	0
$757$	−14.1312	−0.513605	−0.256803	−	0.966464i	$-0.582669\pi$
$757$	−14.1312	−0.513605	−0.256803	+	0.966464i	$0.582669\pi$
$758$	0	0
$759$	1.21375	0.0440565
$760$	0	0
$761$	27.1139	0.982879	0.491440	−	0.870912i	$-0.336471\pi$
$761$	27.1139	0.982879	0.491440	+	0.870912i	$0.336471\pi$
$762$	0	0
$763$	39.5372	1.43134
$764$	0	0
$765$	−8.09284	−0.292597
$766$	0	0
$767$	−0.899710	−0.0324866
$768$	0	0
$769$	−39.9224	−1.43964	−0.719819	−	0.694162i	$-0.755775\pi$
$769$	−39.9224	−1.43964	−0.719819	+	0.694162i	$0.755775\pi$
$770$	0	0
$771$	−5.83378	−0.210098
$772$	0	0
$773$	35.4925	1.27658	0.638288	−	0.769798i	$-0.279643\pi$
$773$	35.4925	1.27658	0.638288	+	0.769798i	$0.279643\pi$
$774$	0	0
$775$	−3.48576	−0.125212
$776$	0	0
$777$	−2.75631	−0.0988820
$778$	0	0
$779$	−72.0099	−2.58002
$780$	0	0
$781$	−0.483973	−0.0173179
$782$	0	0
$783$	−6.08131	−0.217328
$784$	0	0
$785$	−23.4260	−0.836110
$786$	0	0
$787$	19.8537	0.707707	0.353853	−	0.935301i	$-0.384871\pi$
$787$	19.8537	0.707707	0.353853	+	0.935301i	$0.384871\pi$
$788$	0	0
$789$	3.84115	0.136749
$790$	0	0
$791$	21.0470	0.748346
$792$	0	0
$793$	−5.05406	−0.179475
$794$	0	0
$795$	−1.21438	−0.0430696
$796$	0	0
$797$	−7.91451	−0.280346	−0.140173	−	0.990127i	$-0.544766\pi$
$797$	−7.91451	−0.280346	−0.140173	+	0.990127i	$0.544766\pi$
$798$	0	0
$799$	3.69851	0.130844
$800$	0	0
$801$	10.4717	0.370001
$802$	0	0
$803$	−1.76275	−0.0622061
$804$	0	0
$805$	−17.5375	−0.618117
$806$	0	0
$807$	5.84311	0.205687
$808$	0	0
$809$	21.8629	0.768657	0.384329	−	0.923196i	$-0.374433\pi$
$809$	21.8629	0.768657	0.384329	+	0.923196i	$0.374433\pi$
$810$	0	0
$811$	−12.0109	−0.421759	−0.210879	−	0.977512i	$-0.567633\pi$
$811$	−12.0109	−0.421759	−0.210879	+	0.977512i	$0.567633\pi$
$812$	0	0
$813$	−0.661939	−0.0232152
$814$	0	0
$815$	−15.1218	−0.529694
$816$	0	0
$817$	−5.04781	−0.176601
$818$	0	0
$819$	7.63010	0.266617
$820$	0	0
$821$	−34.7226	−1.21183	−0.605913	−	0.795531i	$-0.707192\pi$
$821$	−34.7226	−1.21183	−0.605913	+	0.795531i	$0.707192\pi$
$822$	0	0
$823$	10.1524	0.353892	0.176946	−	0.984221i	$-0.443378\pi$
$823$	10.1524	0.353892	0.176946	+	0.984221i	$0.443378\pi$
$824$	0	0
$825$	1.40191	0.0488081
$826$	0	0
$827$	23.6554	0.822579	0.411290	−	0.911505i	$-0.365078\pi$
$827$	23.6554	0.822579	0.411290	+	0.911505i	$0.365078\pi$
$828$	0	0
$829$	9.95680	0.345814	0.172907	−	0.984938i	$-0.444684\pi$
$829$	9.95680	0.345814	0.172907	+	0.984938i	$0.444684\pi$
$830$	0	0
$831$	4.81319	0.166968
$832$	0	0
$833$	1.29928	0.0450175
$834$	0	0
$835$	−15.4195	−0.533614
$836$	0	0
$837$	−1.92048	−0.0663814
$838$	0	0
$839$	12.3601	0.426719	0.213359	−	0.976974i	$-0.431559\pi$
$839$	12.3601	0.426719	0.213359	+	0.976974i	$0.431559\pi$
$840$	0	0
$841$	−10.3755	−0.357775
$842$	0	0
$843$	−0.110562	−0.00380794
$844$	0	0
$845$	−33.5132	−1.15289
$846$	0	0
$847$	16.2904	0.559744
$848$	0	0
$849$	−4.04637	−0.138871
$850$	0	0
$851$	−8.93659	−0.306342
$852$	0	0
$853$	41.2998	1.41408	0.707040	−	0.707174i	$-0.250030\pi$
$853$	41.2998	1.41408	0.707040	+	0.707174i	$0.250030\pi$
$854$	0	0
$855$	−70.1385	−2.39868
$856$	0	0
$857$	13.5683	0.463483	0.231741	−	0.972777i	$-0.425558\pi$
$857$	13.5683	0.463483	0.231741	+	0.972777i	$0.425558\pi$
$858$	0	0
$859$	−1.87046	−0.0638192	−0.0319096	−	0.999491i	$-0.510159\pi$
$859$	−1.87046	−0.0638192	−0.0319096	+	0.999491i	$0.510159\pi$
$860$	0	0
$861$	−5.67477	−0.193396
$862$	0	0
$863$	−20.7587	−0.706633	−0.353317	−	0.935504i	$-0.614946\pi$
$863$	−20.7587	−0.706633	−0.353317	+	0.935504i	$0.614946\pi$
$864$	0	0
$865$	−68.0553	−2.31395
$866$	0	0
$867$	−0.237078	−0.00805158
$868$	0	0
$869$	−8.10510	−0.274947
$870$	0	0
$871$	10.0328	0.339948
$872$	0	0
$873$	55.7490	1.88682
$874$	0	0
$875$	19.3428	0.653906
$876$	0	0
$877$	48.1541	1.62605	0.813024	−	0.582230i	$-0.197820\pi$
$877$	48.1541	1.62605	0.813024	+	0.582230i	$0.197820\pi$
$878$	0	0
$879$	4.87913	0.164569
$880$	0	0
$881$	−17.8091	−0.600003	−0.300002	−	0.953939i	$-0.596987\pi$
$881$	−17.8091	−0.600003	−0.300002	+	0.953939i	$0.596987\pi$
$882$	0	0
$883$	51.5837	1.73593	0.867965	−	0.496625i	$-0.165428\pi$
$883$	51.5837	1.73593	0.867965	+	0.496625i	$0.165428\pi$
$884$	0	0
$885$	0.651755	0.0219085
$886$	0	0
$887$	−15.8067	−0.530738	−0.265369	−	0.964147i	$-0.585494\pi$
$887$	−15.8067	−0.530738	−0.265369	+	0.964147i	$0.585494\pi$
$888$	0	0
$889$	−9.46266	−0.317367
$890$	0	0
$891$	−19.6457	−0.658154
$892$	0	0
$893$	32.0540	1.07265
$894$	0	0
$895$	−66.0640	−2.20827
$896$	0	0
$897$	−0.472332	−0.0157707
$898$	0	0
$899$	5.88162	0.196163
$900$	0	0
$901$	1.86324	0.0620737
$902$	0	0
$903$	−0.397795	−0.0132378
$904$	0	0
$905$	−16.0828	−0.534611
$906$	0	0
$907$	−37.8422	−1.25653	−0.628265	−	0.777999i	$-0.716235\pi$
$907$	−37.8422	−1.25653	−0.628265	+	0.777999i	$0.716235\pi$
$908$	0	0
$909$	−49.1240	−1.62934
$910$	0	0
$911$	−18.5106	−0.613282	−0.306641	−	0.951825i	$-0.599205\pi$
$911$	−18.5106	−0.613282	−0.306641	+	0.951825i	$0.599205\pi$
$912$	0	0
$913$	17.8310	0.590119
$914$	0	0
$915$	3.66119	0.121035
$916$	0	0
$917$	−43.5635	−1.43859
$918$	0	0
$919$	16.9318	0.558528	0.279264	−	0.960214i	$-0.409910\pi$
$919$	16.9318	0.558528	0.279264	+	0.960214i	$0.409910\pi$
$920$	0	0
$921$	2.56009	0.0843579
$922$	0	0
$923$	0.188338	0.00619923
$924$	0	0
$925$	−10.3219	−0.339383
$926$	0	0
$927$	1.49675	0.0491596
$928$	0	0
$929$	−39.9795	−1.31168	−0.655842	−	0.754898i	$-0.727686\pi$
$929$	−39.9795	−1.31168	−0.655842	+	0.754898i	$0.727686\pi$
$930$	0	0
$931$	11.2605	0.369049
$932$	0	0
$933$	−2.42903	−0.0795228
$934$	0	0
$935$	−6.35593	−0.207861
$936$	0	0
$937$	−3.02537	−0.0988347	−0.0494173	−	0.998778i	$-0.515736\pi$
$937$	−3.02537	−0.0988347	−0.0494173	+	0.998778i	$0.515736\pi$
$938$	0	0
$939$	−5.18850	−0.169320
$940$	0	0
$941$	−38.9721	−1.27045	−0.635226	−	0.772326i	$-0.719093\pi$
$941$	−38.9721	−1.27045	−0.635226	+	0.772326i	$0.719093\pi$
$942$	0	0
$943$	−18.3989	−0.599151
$944$	0	0
$945$	−11.1601	−0.363038
$946$	0	0
$947$	45.6619	1.48381	0.741906	−	0.670504i	$-0.233922\pi$
$947$	45.6619	1.48381	0.741906	+	0.670504i	$0.233922\pi$
$948$	0	0
$949$	0.685975	0.0222677
$950$	0	0
$951$	4.22720	0.137076
$952$	0	0
$953$	49.8061	1.61338	0.806689	−	0.590977i	$-0.201258\pi$
$953$	49.8061	1.61338	0.806689	+	0.590977i	$0.201258\pi$
$954$	0	0
$955$	71.3491	2.30880
$956$	0	0
$957$	−2.36547	−0.0764649
$958$	0	0
$959$	−11.0981	−0.358377
$960$	0	0
$961$	−29.1426	−0.940083
$962$	0	0
$963$	0.408530	0.0131647
$964$	0	0
$965$	45.6109	1.46827
$966$	0	0
$967$	−8.65020	−0.278172	−0.139086	−	0.990280i	$-0.544416\pi$
$967$	−8.65020	−0.278172	−0.139086	+	0.990280i	$0.544416\pi$
$968$	0	0
$969$	−2.05469	−0.0660060
$970$	0	0
$971$	−5.27235	−0.169198	−0.0845988	−	0.996415i	$-0.526961\pi$
$971$	−5.27235	−0.169198	−0.0845988	+	0.996415i	$0.526961\pi$
$972$	0	0
$973$	15.4193	0.494320
$974$	0	0
$975$	−0.545552	−0.0174717
$976$	0	0
$977$	−7.82113	−0.250220	−0.125110	−	0.992143i	$-0.539928\pi$
$977$	−7.82113	−0.250220	−0.125110	+	0.992143i	$0.539928\pi$
$978$	0	0
$979$	8.22426	0.262848
$980$	0	0
$981$	40.4011	1.28991
$982$	0	0
$983$	28.7105	0.915722	0.457861	−	0.889024i	$-0.348616\pi$
$983$	28.7105	0.915722	0.457861	+	0.889024i	$0.348616\pi$
$984$	0	0
$985$	−42.3017	−1.34785
$986$	0	0
$987$	2.52603	0.0804043
$988$	0	0
$989$	−1.28974	−0.0410114
$990$	0	0
$991$	46.7541	1.48519	0.742597	−	0.669739i	$-0.233594\pi$
$991$	46.7541	1.48519	0.742597	+	0.669739i	$0.233594\pi$
$992$	0	0
$993$	0.832192	0.0264088
$994$	0	0
$995$	14.0914	0.446729
$996$	0	0
$997$	32.4232	1.02685	0.513426	−	0.858134i	$-0.328376\pi$
$997$	32.4232	1.02685	0.513426	+	0.858134i	$0.328376\pi$
$998$	0	0
$999$	−5.68685	−0.179924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.w.1.10	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.w.1.10	✓	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.237078 q^{3} +2.74912 q^{5} -2.88085 q^{7} -2.94379 q^{9} +O(q^{10})\)	\(q-0.237078 q^{3} +2.74912 q^{5} -2.88085 q^{7} -2.94379 q^{9} -2.31199 q^{11} +0.899710 q^{13} -0.651755 q^{15} +1.00000 q^{17} +8.66672 q^{19} +0.682985 q^{21} +2.21439 q^{23} +2.55766 q^{25} +1.40914 q^{27} -4.31561 q^{29} -1.36287 q^{31} +0.548120 q^{33} -7.91980 q^{35} -4.03568 q^{37} -0.213301 q^{39} -8.30878 q^{41} -0.582436 q^{43} -8.09284 q^{45} +3.69851 q^{47} +1.29928 q^{49} -0.237078 q^{51} +1.86324 q^{53} -6.35593 q^{55} -2.05469 q^{57} -1.00000 q^{59} -5.61743 q^{61} +8.48062 q^{63} +2.47341 q^{65} +11.1511 q^{67} -0.524983 q^{69} +0.209332 q^{71} +0.762440 q^{73} -0.606365 q^{75} +6.66048 q^{77} +3.50569 q^{79} +8.49731 q^{81} -7.71241 q^{83} +2.74912 q^{85} +1.02314 q^{87} -3.55723 q^{89} -2.59193 q^{91} +0.323106 q^{93} +23.8259 q^{95} -18.9378 q^{97} +6.80601 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) \) 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9	\( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + 8 q^{11} - 3 q^{13} - 2 q^{15} + 20 q^{17} - 9 q^{19} - 31 q^{21} - 8 q^{23} - 4 q^{25} - 17 q^{27} - 37 q^{29} - 9 q^{31} - 8 q^{33} - 8 q^{35} - 11 q^{37} - 12 q^{39} - 27 q^{41} + 17 q^{43} - 4 q^{45} + 7 q^{47} - 9 q^{49} - 2 q^{51} - q^{53} - 21 q^{55} - 57 q^{57} - 20 q^{59} - 12 q^{61} - 14 q^{63} - 59 q^{65} - 26 q^{67} + 14 q^{69} - 3 q^{71} - 45 q^{73} + 15 q^{75} + 6 q^{77} + 5 q^{79} + 12 q^{81} - 17 q^{83} - 2 q^{85} - 32 q^{87} - 40 q^{89} - 36 q^{91} - 4 q^{93} + 5 q^{95} - 77 q^{97} + 54 q^{99}+O(q^{100}) \) 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 + 8 * q^11 - 3 * q^13 - 2 * q^15 + 20 * q^17 - 9 * q^19 - 31 * q^21 - 8 * q^23 - 4 * q^25 - 17 * q^27 - 37 * q^29 - 9 * q^31 - 8 * q^33 - 8 * q^35 - 11 * q^37 - 12 * q^39 - 27 * q^41 + 17 * q^43 - 4 * q^45 + 7 * q^47 - 9 * q^49 - 2 * q^51 - q^53 - 21 * q^55 - 57 * q^57 - 20 * q^59 - 12 * q^61 - 14 * q^63 - 59 * q^65 - 26 * q^67 + 14 * q^69 - 3 * q^71 - 45 * q^73 + 15 * q^75 + 6 * q^77 + 5 * q^79 + 12 * q^81 - 17 * q^83 - 2 * q^85 - 32 * q^87 - 40 * q^89 - 36 * q^91 - 4 * q^93 + 5 * q^95 - 77 * q^97 + 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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