LMFDB - Embedded newform 7248.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7248,2,Mod(1,7248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7248, 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("7248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7248 = 2^{4} \cdot 3 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7248.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:	$57.8755713850$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 27x^{8} + 45x^{7} + 258x^{6} - 289x^{5} - 1133x^{4} + 510x^{3} + 2070x^{2} + 341x - 500 $ x^10 - 2x^9 - 27x^8 + 45x^7 + 258x^6 - 289x^5 - 1133x^4 + 510x^3 + 2070x^2 + 341*x - 500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3624)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.87603$ of defining polynomial
Character	$\chi$	$=$	7248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.87603 q^{5} -0.0631878 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.87603 q^{5} -0.0631878 q^{7} +1.00000 q^{9} -5.45893 q^{11} +5.11055 q^{13} +1.87603 q^{15} -1.55394 q^{17} -1.69819 q^{19} +0.0631878 q^{21} -2.04714 q^{23} -1.48051 q^{25} -1.00000 q^{27} +0.725269 q^{29} +9.80329 q^{31} +5.45893 q^{33} +0.118542 q^{35} +10.7914 q^{37} -5.11055 q^{39} -1.79379 q^{41} +7.31972 q^{43} -1.87603 q^{45} -2.10491 q^{47} -6.99601 q^{49} +1.55394 q^{51} +9.70810 q^{53} +10.2411 q^{55} +1.69819 q^{57} -7.68657 q^{59} +1.95220 q^{61} -0.0631878 q^{63} -9.58755 q^{65} +2.58343 q^{67} +2.04714 q^{69} -5.70100 q^{71} +0.697303 q^{73} +1.48051 q^{75} +0.344938 q^{77} -6.35995 q^{79} +1.00000 q^{81} -0.371395 q^{83} +2.91523 q^{85} -0.725269 q^{87} -6.15488 q^{89} -0.322925 q^{91} -9.80329 q^{93} +3.18587 q^{95} -2.20968 q^{97} -5.45893 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9	$ 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9} - 7 q^{11} + 6 q^{13} - 2 q^{15} + 7 q^{17} + 8 q^{21} - 25 q^{23} + 8 q^{25} - 10 q^{27} + 12 q^{29} - 11 q^{31} + 7 q^{33} - 9 q^{35} - 3 q^{37} - 6 q^{39} + 12 q^{41} + 2 q^{45} - 31 q^{47} + 14 q^{49} - 7 q^{51} + q^{53} - 9 q^{55} - 19 q^{59} + 24 q^{61} - 8 q^{63} + 20 q^{65} + q^{67} + 25 q^{69} - 34 q^{71} - 18 q^{73} - 8 q^{75} + 27 q^{77} - 25 q^{79} + 10 q^{81} - 14 q^{83} - 3 q^{85} - 12 q^{87} + 20 q^{89} + 12 q^{91} + 11 q^{93} - 48 q^{95} - 15 q^{97} - 7 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9 - 7 * q^11 + 6 * q^13 - 2 * q^15 + 7 * q^17 + 8 * q^21 - 25 * q^23 + 8 * q^25 - 10 * q^27 + 12 * q^29 - 11 * q^31 + 7 * q^33 - 9 * q^35 - 3 * q^37 - 6 * q^39 + 12 * q^41 + 2 * q^45 - 31 * q^47 + 14 * q^49 - 7 * q^51 + q^53 - 9 * q^55 - 19 * q^59 + 24 * q^61 - 8 * q^63 + 20 * q^65 + q^67 + 25 * q^69 - 34 * q^71 - 18 * q^73 - 8 * q^75 + 27 * q^77 - 25 * q^79 + 10 * q^81 - 14 * q^83 - 3 * q^85 - 12 * q^87 + 20 * q^89 + 12 * q^91 + 11 * q^93 - 48 * q^95 - 15 * q^97 - 7 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.87603	−0.838986	−0.419493	−	0.907758i	$-0.637792\pi$
$5$	−1.87603	−0.838986	−0.419493	+	0.907758i	$0.637792\pi$
$6$	0	0
$7$	−0.0631878	−0.0238827	−0.0119414	−	0.999929i	$-0.503801\pi$
$7$	−0.0631878	−0.0238827	−0.0119414	+	0.999929i	$0.503801\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.45893	−1.64593	−0.822965	−	0.568092i	$-0.807682\pi$
$11$	−5.45893	−1.64593	−0.822965	+	0.568092i	$0.807682\pi$
$12$	0	0
$13$	5.11055	1.41741	0.708706	−	0.705504i	$-0.249279\pi$
$13$	5.11055	1.41741	0.708706	+	0.705504i	$0.249279\pi$
$14$	0	0
$15$	1.87603	0.484389
$16$	0	0
$17$	−1.55394	−0.376885	−0.188443	−	0.982084i	$-0.560344\pi$
$17$	−1.55394	−0.376885	−0.188443	+	0.982084i	$0.560344\pi$
$18$	0	0
$19$	−1.69819	−0.389593	−0.194796	−	0.980844i	$-0.562405\pi$
$19$	−1.69819	−0.389593	−0.194796	+	0.980844i	$0.562405\pi$
$20$	0	0
$21$	0.0631878	0.0137887
$22$	0	0
$23$	−2.04714	−0.426859	−0.213430	−	0.976958i	$-0.568463\pi$
$23$	−2.04714	−0.426859	−0.213430	+	0.976958i	$0.568463\pi$
$24$	0	0
$25$	−1.48051	−0.296102
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.725269	0.134679	0.0673395	−	0.997730i	$-0.478549\pi$
$29$	0.725269	0.134679	0.0673395	+	0.997730i	$0.478549\pi$
$30$	0	0
$31$	9.80329	1.76072	0.880362	−	0.474303i	$-0.157300\pi$
$31$	9.80329	1.76072	0.880362	+	0.474303i	$0.157300\pi$
$32$	0	0
$33$	5.45893	0.950278
$34$	0	0
$35$	0.118542	0.0200373
$36$	0	0
$37$	10.7914	1.77410	0.887050	−	0.461673i	$-0.152751\pi$
$37$	10.7914	1.77410	0.887050	+	0.461673i	$0.152751\pi$
$38$	0	0
$39$	−5.11055	−0.818343
$40$	0	0
$41$	−1.79379	−0.280142	−0.140071	−	0.990141i	$-0.544733\pi$
$41$	−1.79379	−0.280142	−0.140071	+	0.990141i	$0.544733\pi$
$42$	0	0
$43$	7.31972	1.11625	0.558124	−	0.829758i	$-0.311522\pi$
$43$	7.31972	1.11625	0.558124	+	0.829758i	$0.311522\pi$
$44$	0	0
$45$	−1.87603	−0.279662
$46$	0	0
$47$	−2.10491	−0.307032	−0.153516	−	0.988146i	$-0.549060\pi$
$47$	−2.10491	−0.307032	−0.153516	+	0.988146i	$0.549060\pi$
$48$	0	0
$49$	−6.99601	−0.999430
$50$	0	0
$51$	1.55394	0.217595
$52$	0	0
$53$	9.70810	1.33351	0.666755	−	0.745277i	$-0.267683\pi$
$53$	9.70810	1.33351	0.666755	+	0.745277i	$0.267683\pi$
$54$	0	0
$55$	10.2411	1.38091
$56$	0	0
$57$	1.69819	0.224931
$58$	0	0
$59$	−7.68657	−1.00071	−0.500353	−	0.865822i	$-0.666796\pi$
$59$	−7.68657	−1.00071	−0.500353	+	0.865822i	$0.666796\pi$
$60$	0	0
$61$	1.95220	0.249953	0.124977	−	0.992160i	$-0.460114\pi$
$61$	1.95220	0.249953	0.124977	+	0.992160i	$0.460114\pi$
$62$	0	0
$63$	−0.0631878	−0.00796092
$64$	0	0
$65$	−9.58755	−1.18919
$66$	0	0
$67$	2.58343	0.315617	0.157808	−	0.987470i	$-0.449557\pi$
$67$	2.58343	0.315617	0.157808	+	0.987470i	$0.449557\pi$
$68$	0	0
$69$	2.04714	0.246447
$70$	0	0
$71$	−5.70100	−0.676584	−0.338292	−	0.941041i	$-0.609849\pi$
$71$	−5.70100	−0.676584	−0.338292	+	0.941041i	$0.609849\pi$
$72$	0	0
$73$	0.697303	0.0816132	0.0408066	−	0.999167i	$-0.487007\pi$
$73$	0.697303	0.0816132	0.0408066	+	0.999167i	$0.487007\pi$
$74$	0	0
$75$	1.48051	0.170954
$76$	0	0
$77$	0.344938	0.0393093
$78$	0	0
$79$	−6.35995	−0.715550	−0.357775	−	0.933808i	$-0.616465\pi$
$79$	−6.35995	−0.715550	−0.357775	+	0.933808i	$0.616465\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.371395	−0.0407659	−0.0203829	−	0.999792i	$-0.506489\pi$
$83$	−0.371395	−0.0407659	−0.0203829	+	0.999792i	$0.506489\pi$
$84$	0	0
$85$	2.91523	0.316202
$86$	0	0
$87$	−0.725269	−0.0777570
$88$	0	0
$89$	−6.15488	−0.652416	−0.326208	−	0.945298i	$-0.605771\pi$
$89$	−6.15488	−0.652416	−0.326208	+	0.945298i	$0.605771\pi$
$90$	0	0
$91$	−0.322925	−0.0338517
$92$	0	0
$93$	−9.80329	−1.01655
$94$	0	0
$95$	3.18587	0.326863
$96$	0	0
$97$	−2.20968	−0.224359	−0.112180	−	0.993688i	$-0.535783\pi$
$97$	−2.20968	−0.224359	−0.112180	+	0.993688i	$0.535783\pi$
$98$	0	0
$99$	−5.45893	−0.548643
$100$	0	0
$101$	7.90739	0.786814	0.393407	−	0.919364i	$-0.371296\pi$
$101$	7.90739	0.786814	0.393407	+	0.919364i	$0.371296\pi$
$102$	0	0
$103$	−3.98793	−0.392942	−0.196471	−	0.980510i	$-0.562948\pi$
$103$	−3.98793	−0.392942	−0.196471	+	0.980510i	$0.562948\pi$
$104$	0	0
$105$	−0.118542	−0.0115685
$106$	0	0
$107$	−9.28677	−0.897786	−0.448893	−	0.893586i	$-0.648182\pi$
$107$	−9.28677	−0.897786	−0.448893	+	0.893586i	$0.648182\pi$
$108$	0	0
$109$	−9.69803	−0.928903	−0.464451	−	0.885599i	$-0.653748\pi$
$109$	−9.69803	−0.928903	−0.464451	+	0.885599i	$0.653748\pi$
$110$	0	0
$111$	−10.7914	−1.02428
$112$	0	0
$113$	5.00461	0.470794	0.235397	−	0.971899i	$-0.424361\pi$
$113$	5.00461	0.470794	0.235397	+	0.971899i	$0.424361\pi$
$114$	0	0
$115$	3.84051	0.358129
$116$	0	0
$117$	5.11055	0.472471
$118$	0	0
$119$	0.0981899	0.00900105
$120$	0	0
$121$	18.7999	1.70908
$122$	0	0
$123$	1.79379	0.161740
$124$	0	0
$125$	12.1576	1.08741
$126$	0	0
$127$	−12.9444	−1.14863	−0.574317	−	0.818633i	$-0.694732\pi$
$127$	−12.9444	−1.14863	−0.574317	+	0.818633i	$0.694732\pi$
$128$	0	0
$129$	−7.31972	−0.644466
$130$	0	0
$131$	16.7274	1.46148	0.730741	−	0.682655i	$-0.239175\pi$
$131$	16.7274	1.46148	0.730741	+	0.682655i	$0.239175\pi$
$132$	0	0
$133$	0.107305	0.00930454
$134$	0	0
$135$	1.87603	0.161463
$136$	0	0
$137$	−7.33167	−0.626387	−0.313194	−	0.949689i	$-0.601399\pi$
$137$	−7.33167	−0.626387	−0.313194	+	0.949689i	$0.601399\pi$
$138$	0	0
$139$	−4.88939	−0.414713	−0.207356	−	0.978265i	$-0.566486\pi$
$139$	−4.88939	−0.414713	−0.207356	+	0.978265i	$0.566486\pi$
$140$	0	0
$141$	2.10491	0.177265
$142$	0	0
$143$	−27.8982	−2.33296
$144$	0	0
$145$	−1.36063	−0.112994
$146$	0	0
$147$	6.99601	0.577021
$148$	0	0
$149$	16.5995	1.35989	0.679944	−	0.733264i	$-0.262004\pi$
$149$	16.5995	1.35989	0.679944	+	0.733264i	$0.262004\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	−1.55394	−0.125628
$154$	0	0
$155$	−18.3913	−1.47722
$156$	0	0
$157$	24.4076	1.94794	0.973968	−	0.226685i	$-0.0727887\pi$
$157$	24.4076	1.94794	0.973968	+	0.226685i	$0.0727887\pi$
$158$	0	0
$159$	−9.70810	−0.769902
$160$	0	0
$161$	0.129355	0.0101946
$162$	0	0
$163$	1.90327	0.149075	0.0745376	−	0.997218i	$-0.476252\pi$
$163$	1.90327	0.149075	0.0745376	+	0.997218i	$0.476252\pi$
$164$	0	0
$165$	−10.2411	−0.797270
$166$	0	0
$167$	−9.52817	−0.737312	−0.368656	−	0.929566i	$-0.620182\pi$
$167$	−9.52817	−0.737312	−0.368656	+	0.929566i	$0.620182\pi$
$168$	0	0
$169$	13.1177	1.00906
$170$	0	0
$171$	−1.69819	−0.129864
$172$	0	0
$173$	12.6353	0.960642	0.480321	−	0.877093i	$-0.340520\pi$
$173$	12.6353	0.960642	0.480321	+	0.877093i	$0.340520\pi$
$174$	0	0
$175$	0.0935501	0.00707172
$176$	0	0
$177$	7.68657	0.577758
$178$	0	0
$179$	−23.3258	−1.74345	−0.871727	−	0.489993i	$-0.836999\pi$
$179$	−23.3258	−1.74345	−0.871727	+	0.489993i	$0.836999\pi$
$180$	0	0
$181$	−20.2062	−1.50191	−0.750957	−	0.660351i	$-0.770408\pi$
$181$	−20.2062	−1.50191	−0.750957	+	0.660351i	$0.770408\pi$
$182$	0	0
$183$	−1.95220	−0.144311
$184$	0	0
$185$	−20.2451	−1.48845
$186$	0	0
$187$	8.48284	0.620326
$188$	0	0
$189$	0.0631878	0.00459624
$190$	0	0
$191$	8.07723	0.584448	0.292224	−	0.956350i	$-0.405605\pi$
$191$	8.07723	0.584448	0.292224	+	0.956350i	$0.405605\pi$
$192$	0	0
$193$	5.43095	0.390928	0.195464	−	0.980711i	$-0.437379\pi$
$193$	5.43095	0.390928	0.195464	+	0.980711i	$0.437379\pi$
$194$	0	0
$195$	9.58755	0.686579
$196$	0	0
$197$	3.75564	0.267578	0.133789	−	0.991010i	$-0.457286\pi$
$197$	3.75564	0.267578	0.133789	+	0.991010i	$0.457286\pi$
$198$	0	0
$199$	2.90269	0.205766	0.102883	−	0.994693i	$-0.467193\pi$
$199$	2.90269	0.205766	0.102883	+	0.994693i	$0.467193\pi$
$200$	0	0
$201$	−2.58343	−0.182221
$202$	0	0
$203$	−0.0458282	−0.00321651
$204$	0	0
$205$	3.36520	0.235036
$206$	0	0
$207$	−2.04714	−0.142286
$208$	0	0
$209$	9.27033	0.641242
$210$	0	0
$211$	4.27024	0.293975	0.146988	−	0.989138i	$-0.453042\pi$
$211$	4.27024	0.293975	0.146988	+	0.989138i	$0.453042\pi$
$212$	0	0
$213$	5.70100	0.390626
$214$	0	0
$215$	−13.7320	−0.936516
$216$	0	0
$217$	−0.619449	−0.0420509
$218$	0	0
$219$	−0.697303	−0.0471194
$220$	0	0
$221$	−7.94148	−0.534202
$222$	0	0
$223$	9.61150	0.643634	0.321817	−	0.946802i	$-0.395706\pi$
$223$	9.61150	0.643634	0.321817	+	0.946802i	$0.395706\pi$
$224$	0	0
$225$	−1.48051	−0.0987006
$226$	0	0
$227$	1.03580	0.0687485	0.0343742	−	0.999409i	$-0.489056\pi$
$227$	1.03580	0.0687485	0.0343742	+	0.999409i	$0.489056\pi$
$228$	0	0
$229$	−10.0963	−0.667180	−0.333590	−	0.942718i	$-0.608260\pi$
$229$	−10.0963	−0.667180	−0.333590	+	0.942718i	$0.608260\pi$
$230$	0	0
$231$	−0.344938	−0.0226952
$232$	0	0
$233$	−7.32352	−0.479780	−0.239890	−	0.970800i	$-0.577111\pi$
$233$	−7.32352	−0.479780	−0.239890	+	0.970800i	$0.577111\pi$
$234$	0	0
$235$	3.94887	0.257596
$236$	0	0
$237$	6.35995	0.413123
$238$	0	0
$239$	−24.8200	−1.60547	−0.802735	−	0.596335i	$-0.796623\pi$
$239$	−24.8200	−1.60547	−0.802735	+	0.596335i	$0.796623\pi$
$240$	0	0
$241$	−15.4981	−0.998317	−0.499159	−	0.866511i	$-0.666358\pi$
$241$	−15.4981	−0.998317	−0.499159	+	0.866511i	$0.666358\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	13.1247	0.838508
$246$	0	0
$247$	−8.67871	−0.552213
$248$	0	0
$249$	0.371395	0.0235362
$250$	0	0
$251$	−2.93034	−0.184961	−0.0924806	−	0.995714i	$-0.529480\pi$
$251$	−2.93034	−0.184961	−0.0924806	+	0.995714i	$0.529480\pi$
$252$	0	0
$253$	11.1752	0.702580
$254$	0	0
$255$	−2.91523	−0.182559
$256$	0	0
$257$	−2.78348	−0.173628	−0.0868142	−	0.996225i	$-0.527669\pi$
$257$	−2.78348	−0.173628	−0.0868142	+	0.996225i	$0.527669\pi$
$258$	0	0
$259$	−0.681887	−0.0423704
$260$	0	0
$261$	0.725269	0.0448930
$262$	0	0
$263$	14.8315	0.914549	0.457274	−	0.889326i	$-0.348826\pi$
$263$	14.8315	0.914549	0.457274	+	0.889326i	$0.348826\pi$
$264$	0	0
$265$	−18.2127	−1.11880
$266$	0	0
$267$	6.15488	0.376672
$268$	0	0
$269$	−16.8481	−1.02725	−0.513625	−	0.858015i	$-0.671698\pi$
$269$	−16.8481	−1.02725	−0.513625	+	0.858015i	$0.671698\pi$
$270$	0	0
$271$	−18.9488	−1.15106	−0.575528	−	0.817782i	$-0.695203\pi$
$271$	−18.9488	−1.15106	−0.575528	+	0.817782i	$0.695203\pi$
$272$	0	0
$273$	0.322925	0.0195443
$274$	0	0
$275$	8.08199	0.487363
$276$	0	0
$277$	−24.9035	−1.49631	−0.748154	−	0.663525i	$-0.769060\pi$
$277$	−24.9035	−1.49631	−0.748154	+	0.663525i	$0.769060\pi$
$278$	0	0
$279$	9.80329	0.586908
$280$	0	0
$281$	−3.39550	−0.202558	−0.101279	−	0.994858i	$-0.532294\pi$
$281$	−3.39550	−0.202558	−0.101279	+	0.994858i	$0.532294\pi$
$282$	0	0
$283$	−22.4553	−1.33483	−0.667414	−	0.744687i	$-0.732599\pi$
$283$	−22.4553	−1.33483	−0.667414	+	0.744687i	$0.732599\pi$
$284$	0	0
$285$	−3.18587	−0.188714
$286$	0	0
$287$	0.113345	0.00669057
$288$	0	0
$289$	−14.5853	−0.857958
$290$	0	0
$291$	2.20968	0.129534
$292$	0	0
$293$	−28.8963	−1.68814	−0.844068	−	0.536235i	$-0.819846\pi$
$293$	−28.8963	−1.68814	−0.844068	+	0.536235i	$0.819846\pi$
$294$	0	0
$295$	14.4202	0.839578
$296$	0	0
$297$	5.45893	0.316759
$298$	0	0
$299$	−10.4620	−0.605035
$300$	0	0
$301$	−0.462517	−0.0266590
$302$	0	0
$303$	−7.90739	−0.454267
$304$	0	0
$305$	−3.66238	−0.209707
$306$	0	0
$307$	−1.24516	−0.0710649	−0.0355325	−	0.999369i	$-0.511313\pi$
$307$	−1.24516	−0.0710649	−0.0355325	+	0.999369i	$0.511313\pi$
$308$	0	0
$309$	3.98793	0.226865
$310$	0	0
$311$	−31.5989	−1.79181	−0.895905	−	0.444246i	$-0.853472\pi$
$311$	−31.5989	−1.79181	−0.895905	+	0.444246i	$0.853472\pi$
$312$	0	0
$313$	−11.6871	−0.660593	−0.330297	−	0.943877i	$-0.607149\pi$
$313$	−11.6871	−0.660593	−0.330297	+	0.943877i	$0.607149\pi$
$314$	0	0
$315$	0.118542	0.00667910
$316$	0	0
$317$	6.09051	0.342077	0.171039	−	0.985264i	$-0.445288\pi$
$317$	6.09051	0.342077	0.171039	+	0.985264i	$0.445288\pi$
$318$	0	0
$319$	−3.95919	−0.221672
$320$	0	0
$321$	9.28677	0.518337
$322$	0	0
$323$	2.63889	0.146832
$324$	0	0
$325$	−7.56622	−0.419698
$326$	0	0
$327$	9.69803	0.536302
$328$	0	0
$329$	0.133004	0.00733278
$330$	0	0
$331$	1.77617	0.0976270	0.0488135	−	0.998808i	$-0.484456\pi$
$331$	1.77617	0.0976270	0.0488135	+	0.998808i	$0.484456\pi$
$332$	0	0
$333$	10.7914	0.591367
$334$	0	0
$335$	−4.84660	−0.264798
$336$	0	0
$337$	19.3669	1.05498	0.527492	−	0.849560i	$-0.323132\pi$
$337$	19.3669	1.05498	0.527492	+	0.849560i	$0.323132\pi$
$338$	0	0
$339$	−5.00461	−0.271813
$340$	0	0
$341$	−53.5155	−2.89803
$342$	0	0
$343$	0.884377	0.0477519
$344$	0	0
$345$	−3.84051	−0.206766
$346$	0	0
$347$	18.0675	0.969914	0.484957	−	0.874538i	$-0.338835\pi$
$347$	18.0675	0.969914	0.484957	+	0.874538i	$0.338835\pi$
$348$	0	0
$349$	−28.9385	−1.54904	−0.774521	−	0.632548i	$-0.782009\pi$
$349$	−28.9385	−1.54904	−0.774521	+	0.632548i	$0.782009\pi$
$350$	0	0
$351$	−5.11055	−0.272781
$352$	0	0
$353$	−4.01868	−0.213893	−0.106946	−	0.994265i	$-0.534107\pi$
$353$	−4.01868	−0.213893	−0.106946	+	0.994265i	$0.534107\pi$
$354$	0	0
$355$	10.6953	0.567645
$356$	0	0
$357$	−0.0981899	−0.00519676
$358$	0	0
$359$	7.87164	0.415449	0.207725	−	0.978187i	$-0.433394\pi$
$359$	7.87164	0.415449	0.207725	+	0.978187i	$0.433394\pi$
$360$	0	0
$361$	−16.1161	−0.848218
$362$	0	0
$363$	−18.7999	−0.986740
$364$	0	0
$365$	−1.30816	−0.0684723
$366$	0	0
$367$	13.7868	0.719663	0.359832	−	0.933017i	$-0.382834\pi$
$367$	13.7868	0.719663	0.359832	+	0.933017i	$0.382834\pi$
$368$	0	0
$369$	−1.79379	−0.0933807
$370$	0	0
$371$	−0.613434	−0.0318479
$372$	0	0
$373$	−28.6873	−1.48537	−0.742686	−	0.669640i	$-0.766448\pi$
$373$	−28.6873	−1.48537	−0.742686	+	0.669640i	$0.766448\pi$
$374$	0	0
$375$	−12.1576	−0.627817
$376$	0	0
$377$	3.70652	0.190896
$378$	0	0
$379$	−4.87163	−0.250239	−0.125120	−	0.992142i	$-0.539931\pi$
$379$	−4.87163	−0.250239	−0.125120	+	0.992142i	$0.539931\pi$
$380$	0	0
$381$	12.9444	0.663164
$382$	0	0
$383$	9.14941	0.467513	0.233756	−	0.972295i	$-0.424898\pi$
$383$	9.14941	0.467513	0.233756	+	0.972295i	$0.424898\pi$
$384$	0	0
$385$	−0.647114	−0.0329800
$386$	0	0
$387$	7.31972	0.372082
$388$	0	0
$389$	10.8974	0.552520	0.276260	−	0.961083i	$-0.410905\pi$
$389$	10.8974	0.552520	0.276260	+	0.961083i	$0.410905\pi$
$390$	0	0
$391$	3.18113	0.160877
$392$	0	0
$393$	−16.7274	−0.843787
$394$	0	0
$395$	11.9315	0.600337
$396$	0	0
$397$	−0.242973	−0.0121945	−0.00609724	−	0.999981i	$-0.501941\pi$
$397$	−0.242973	−0.0121945	−0.00609724	+	0.999981i	$0.501941\pi$
$398$	0	0
$399$	−0.107305	−0.00537198
$400$	0	0
$401$	−2.19191	−0.109459	−0.0547293	−	0.998501i	$-0.517430\pi$
$401$	−2.19191	−0.109459	−0.0547293	+	0.998501i	$0.517430\pi$
$402$	0	0
$403$	50.1002	2.49567
$404$	0	0
$405$	−1.87603	−0.0932207
$406$	0	0
$407$	−58.9097	−2.92004
$408$	0	0
$409$	−11.3973	−0.563561	−0.281780	−	0.959479i	$-0.590925\pi$
$409$	−11.3973	−0.563561	−0.281780	+	0.959479i	$0.590925\pi$
$410$	0	0
$411$	7.33167	0.361645
$412$	0	0
$413$	0.485697	0.0238996
$414$	0	0
$415$	0.696748	0.0342020
$416$	0	0
$417$	4.88939	0.239435
$418$	0	0
$419$	−12.1390	−0.593029	−0.296514	−	0.955028i	$-0.595824\pi$
$419$	−12.1390	−0.593029	−0.296514	+	0.955028i	$0.595824\pi$
$420$	0	0
$421$	−22.7611	−1.10931	−0.554654	−	0.832081i	$-0.687149\pi$
$421$	−22.7611	−1.10931	−0.554654	+	0.832081i	$0.687149\pi$
$422$	0	0
$423$	−2.10491	−0.102344
$424$	0	0
$425$	2.30062	0.111596
$426$	0	0
$427$	−0.123355	−0.00596957
$428$	0	0
$429$	27.8982	1.34694
$430$	0	0
$431$	−21.1842	−1.02041	−0.510204	−	0.860053i	$-0.670430\pi$
$431$	−21.1842	−1.02041	−0.510204	+	0.860053i	$0.670430\pi$
$432$	0	0
$433$	−35.5533	−1.70858	−0.854291	−	0.519794i	$-0.826009\pi$
$433$	−35.5533	−1.70858	−0.854291	+	0.519794i	$0.826009\pi$
$434$	0	0
$435$	1.36063	0.0652371
$436$	0	0
$437$	3.47645	0.166301
$438$	0	0
$439$	−14.9775	−0.714838	−0.357419	−	0.933944i	$-0.616343\pi$
$439$	−14.9775	−0.714838	−0.357419	+	0.933944i	$0.616343\pi$
$440$	0	0
$441$	−6.99601	−0.333143
$442$	0	0
$443$	27.5957	1.31111	0.655556	−	0.755146i	$-0.272434\pi$
$443$	27.5957	1.31111	0.655556	+	0.755146i	$0.272434\pi$
$444$	0	0
$445$	11.5467	0.547368
$446$	0	0
$447$	−16.5995	−0.785131
$448$	0	0
$449$	34.9418	1.64901	0.824503	−	0.565858i	$-0.191455\pi$
$449$	34.9418	1.64901	0.824503	+	0.565858i	$0.191455\pi$
$450$	0	0
$451$	9.79215	0.461094
$452$	0	0
$453$	1.00000	0.0469841
$454$	0	0
$455$	0.605817	0.0284011
$456$	0	0
$457$	29.7743	1.39278	0.696391	−	0.717663i	$-0.254788\pi$
$457$	29.7743	1.39278	0.696391	+	0.717663i	$0.254788\pi$
$458$	0	0
$459$	1.55394	0.0725316
$460$	0	0
$461$	23.1444	1.07794	0.538970	−	0.842325i	$-0.318814\pi$
$461$	23.1444	1.07794	0.538970	+	0.842325i	$0.318814\pi$
$462$	0	0
$463$	−9.17930	−0.426598	−0.213299	−	0.976987i	$-0.568421\pi$
$463$	−9.17930	−0.426598	−0.213299	+	0.976987i	$0.568421\pi$
$464$	0	0
$465$	18.3913	0.852875
$466$	0	0
$467$	2.74799	0.127162	0.0635810	−	0.997977i	$-0.479748\pi$
$467$	2.74799	0.127162	0.0635810	+	0.997977i	$0.479748\pi$
$468$	0	0
$469$	−0.163241	−0.00753779
$470$	0	0
$471$	−24.4076	−1.12464
$472$	0	0
$473$	−39.9579	−1.83726
$474$	0	0
$475$	2.51419	0.115359
$476$	0	0
$477$	9.70810	0.444503
$478$	0	0
$479$	−11.9757	−0.547184	−0.273592	−	0.961846i	$-0.588212\pi$
$479$	−11.9757	−0.547184	−0.273592	+	0.961846i	$0.588212\pi$
$480$	0	0
$481$	55.1502	2.51463
$482$	0	0
$483$	−0.129355	−0.00588584
$484$	0	0
$485$	4.14543	0.188234
$486$	0	0
$487$	−23.6493	−1.07165	−0.535825	−	0.844329i	$-0.679999\pi$
$487$	−23.6493	−1.07165	−0.535825	+	0.844329i	$0.679999\pi$
$488$	0	0
$489$	−1.90327	−0.0860687
$490$	0	0
$491$	−13.3044	−0.600419	−0.300210	−	0.953873i	$-0.597057\pi$
$491$	−13.3044	−0.600419	−0.300210	+	0.953873i	$0.597057\pi$
$492$	0	0
$493$	−1.12702	−0.0507585
$494$	0	0
$495$	10.2411	0.460304
$496$	0	0
$497$	0.360234	0.0161587
$498$	0	0
$499$	−5.28398	−0.236543	−0.118272	−	0.992981i	$-0.537735\pi$
$499$	−5.28398	−0.236543	−0.118272	+	0.992981i	$0.537735\pi$
$500$	0	0
$501$	9.52817	0.425687
$502$	0	0
$503$	29.3327	1.30788	0.653940	−	0.756546i	$-0.273115\pi$
$503$	29.3327	1.30788	0.653940	+	0.756546i	$0.273115\pi$
$504$	0	0
$505$	−14.8345	−0.660127
$506$	0	0
$507$	−13.1177	−0.582580
$508$	0	0
$509$	−2.70246	−0.119784	−0.0598922	−	0.998205i	$-0.519076\pi$
$509$	−2.70246	−0.119784	−0.0598922	+	0.998205i	$0.519076\pi$
$510$	0	0
$511$	−0.0440611	−0.00194915
$512$	0	0
$513$	1.69819	0.0749771
$514$	0	0
$515$	7.48148	0.329673
$516$	0	0
$517$	11.4905	0.505354
$518$	0	0
$519$	−12.6353	−0.554627
$520$	0	0
$521$	1.04407	0.0457414	0.0228707	−	0.999738i	$-0.492719\pi$
$521$	1.04407	0.0457414	0.0228707	+	0.999738i	$0.492719\pi$
$522$	0	0
$523$	−12.1324	−0.530514	−0.265257	−	0.964178i	$-0.585457\pi$
$523$	−12.1324	−0.530514	−0.265257	+	0.964178i	$0.585457\pi$
$524$	0	0
$525$	−0.0935501	−0.00408286
$526$	0	0
$527$	−15.2337	−0.663590
$528$	0	0
$529$	−18.8092	−0.817791
$530$	0	0
$531$	−7.68657	−0.333569
$532$	0	0
$533$	−9.16724	−0.397077
$534$	0	0
$535$	17.4223	0.753230
$536$	0	0
$537$	23.3258	1.00658
$538$	0	0
$539$	38.1907	1.64499
$540$	0	0
$541$	22.1487	0.952248	0.476124	−	0.879378i	$-0.342041\pi$
$541$	22.1487	0.952248	0.476124	+	0.879378i	$0.342041\pi$
$542$	0	0
$543$	20.2062	0.867131
$544$	0	0
$545$	18.1938	0.779337
$546$	0	0
$547$	0.525660	0.0224756	0.0112378	−	0.999937i	$-0.496423\pi$
$547$	0.525660	0.0224756	0.0112378	+	0.999937i	$0.496423\pi$
$548$	0	0
$549$	1.95220	0.0833178
$550$	0	0
$551$	−1.23165	−0.0524700
$552$	0	0
$553$	0.401871	0.0170893
$554$	0	0
$555$	20.2451	0.859355
$556$	0	0
$557$	−19.8098	−0.839367	−0.419683	−	0.907671i	$-0.637859\pi$
$557$	−19.8098	−0.839367	−0.419683	+	0.907671i	$0.637859\pi$
$558$	0	0
$559$	37.4078	1.58218
$560$	0	0
$561$	−8.48284	−0.358146
$562$	0	0
$563$	35.0174	1.47581	0.737905	−	0.674905i	$-0.235815\pi$
$563$	35.0174	1.47581	0.737905	+	0.674905i	$0.235815\pi$
$564$	0	0
$565$	−9.38880	−0.394990
$566$	0	0
$567$	−0.0631878	−0.00265364
$568$	0	0
$569$	29.2428	1.22592	0.612961	−	0.790114i	$-0.289978\pi$
$569$	29.2428	1.22592	0.612961	+	0.790114i	$0.289978\pi$
$570$	0	0
$571$	−29.9760	−1.25446	−0.627228	−	0.778836i	$-0.715810\pi$
$571$	−29.9760	−1.25446	−0.627228	+	0.778836i	$0.715810\pi$
$572$	0	0
$573$	−8.07723	−0.337431
$574$	0	0
$575$	3.03081	0.126394
$576$	0	0
$577$	8.97453	0.373615	0.186807	−	0.982397i	$-0.440186\pi$
$577$	8.97453	0.373615	0.186807	+	0.982397i	$0.440186\pi$
$578$	0	0
$579$	−5.43095	−0.225703
$580$	0	0
$581$	0.0234676	0.000973601 0
$582$	0	0
$583$	−52.9958	−2.19486
$584$	0	0
$585$	−9.58755	−0.396397
$586$	0	0
$587$	18.9977	0.784117	0.392059	−	0.919940i	$-0.371763\pi$
$587$	18.9977	0.784117	0.392059	+	0.919940i	$0.371763\pi$
$588$	0	0
$589$	−16.6479	−0.685965
$590$	0	0
$591$	−3.75564	−0.154486
$592$	0	0
$593$	−41.2997	−1.69598	−0.847988	−	0.530015i	$-0.822186\pi$
$593$	−41.2997	−1.69598	−0.847988	+	0.530015i	$0.822186\pi$
$594$	0	0
$595$	−0.184207	−0.00755176
$596$	0	0
$597$	−2.90269	−0.118799
$598$	0	0
$599$	−19.3622	−0.791120	−0.395560	−	0.918440i	$-0.629449\pi$
$599$	−19.3622	−0.791120	−0.395560	+	0.918440i	$0.629449\pi$
$600$	0	0
$601$	−38.1922	−1.55789	−0.778945	−	0.627092i	$-0.784245\pi$
$601$	−38.1922	−1.55789	−0.778945	+	0.627092i	$0.784245\pi$
$602$	0	0
$603$	2.58343	0.105206
$604$	0	0
$605$	−35.2692	−1.43390
$606$	0	0
$607$	38.1463	1.54831	0.774155	−	0.632996i	$-0.218175\pi$
$607$	38.1463	1.54831	0.774155	+	0.632996i	$0.218175\pi$
$608$	0	0
$609$	0.0458282	0.00185705
$610$	0	0
$611$	−10.7572	−0.435191
$612$	0	0
$613$	−22.5590	−0.911148	−0.455574	−	0.890198i	$-0.650566\pi$
$613$	−22.5590	−0.911148	−0.455574	+	0.890198i	$0.650566\pi$
$614$	0	0
$615$	−3.36520	−0.135698
$616$	0	0
$617$	−44.6831	−1.79887	−0.899437	−	0.437051i	$-0.856023\pi$
$617$	−44.6831	−1.79887	−0.899437	+	0.437051i	$0.856023\pi$
$618$	0	0
$619$	−19.9223	−0.800743	−0.400371	−	0.916353i	$-0.631119\pi$
$619$	−19.9223	−0.800743	−0.400371	+	0.916353i	$0.631119\pi$
$620$	0	0
$621$	2.04714	0.0821491
$622$	0	0
$623$	0.388913	0.0155815
$624$	0	0
$625$	−15.4056	−0.616222
$626$	0	0
$627$	−9.27033	−0.370221
$628$	0	0
$629$	−16.7692	−0.668632
$630$	0	0
$631$	9.22963	0.367426	0.183713	−	0.982980i	$-0.441188\pi$
$631$	9.22963	0.367426	0.183713	+	0.982980i	$0.441188\pi$
$632$	0	0
$633$	−4.27024	−0.169727
$634$	0	0
$635$	24.2842	0.963688
$636$	0	0
$637$	−35.7535	−1.41660
$638$	0	0
$639$	−5.70100	−0.225528
$640$	0	0
$641$	4.79357	0.189334	0.0946672	−	0.995509i	$-0.469821\pi$
$641$	4.79357	0.189334	0.0946672	+	0.995509i	$0.469821\pi$
$642$	0	0
$643$	−26.7894	−1.05647	−0.528236	−	0.849098i	$-0.677146\pi$
$643$	−26.7894	−1.05647	−0.528236	+	0.849098i	$0.677146\pi$
$644$	0	0
$645$	13.7320	0.540698
$646$	0	0
$647$	−21.9905	−0.864535	−0.432267	−	0.901745i	$-0.642286\pi$
$647$	−21.9905	−0.864535	−0.432267	+	0.901745i	$0.642286\pi$
$648$	0	0
$649$	41.9604	1.64709
$650$	0	0
$651$	0.619449	0.0242781
$652$	0	0
$653$	11.1016	0.434440	0.217220	−	0.976123i	$-0.430301\pi$
$653$	11.1016	0.434440	0.217220	+	0.976123i	$0.430301\pi$
$654$	0	0
$655$	−31.3812	−1.22616
$656$	0	0
$657$	0.697303	0.0272044
$658$	0	0
$659$	−2.69237	−0.104880	−0.0524400	−	0.998624i	$-0.516700\pi$
$659$	−2.69237	−0.104880	−0.0524400	+	0.998624i	$0.516700\pi$
$660$	0	0
$661$	15.6508	0.608744	0.304372	−	0.952553i	$-0.401553\pi$
$661$	15.6508	0.608744	0.304372	+	0.952553i	$0.401553\pi$
$662$	0	0
$663$	7.94148	0.308421
$664$	0	0
$665$	−0.201308	−0.00780638
$666$	0	0
$667$	−1.48473	−0.0574890
$668$	0	0
$669$	−9.61150	−0.371602
$670$	0	0
$671$	−10.6569	−0.411406
$672$	0	0
$673$	−21.4863	−0.828238	−0.414119	−	0.910223i	$-0.635910\pi$
$673$	−21.4863	−0.828238	−0.414119	+	0.910223i	$0.635910\pi$
$674$	0	0
$675$	1.48051	0.0569848
$676$	0	0
$677$	−34.4639	−1.32456	−0.662278	−	0.749258i	$-0.730410\pi$
$677$	−34.4639	−1.32456	−0.662278	+	0.749258i	$0.730410\pi$
$678$	0	0
$679$	0.139625	0.00535832
$680$	0	0
$681$	−1.03580	−0.0396920
$682$	0	0
$683$	−30.4210	−1.16403	−0.582014	−	0.813179i	$-0.697735\pi$
$683$	−30.4210	−1.16403	−0.582014	+	0.813179i	$0.697735\pi$
$684$	0	0
$685$	13.7544	0.525530
$686$	0	0
$687$	10.0963	0.385196
$688$	0	0
$689$	49.6138	1.89013
$690$	0	0
$691$	21.4478	0.815913	0.407957	−	0.913001i	$-0.366241\pi$
$691$	21.4478	0.815913	0.407957	+	0.913001i	$0.366241\pi$
$692$	0	0
$693$	0.344938	0.0131031
$694$	0	0
$695$	9.17265	0.347938
$696$	0	0
$697$	2.78743	0.105581
$698$	0	0
$699$	7.32352	0.277001
$700$	0	0
$701$	−41.0618	−1.55088	−0.775441	−	0.631420i	$-0.782472\pi$
$701$	−41.0618	−1.55088	−0.775441	+	0.631420i	$0.782472\pi$
$702$	0	0
$703$	−18.3259	−0.691176
$704$	0	0
$705$	−3.94887	−0.148723
$706$	0	0
$707$	−0.499650	−0.0187913
$708$	0	0
$709$	−5.07044	−0.190424	−0.0952122	−	0.995457i	$-0.530353\pi$
$709$	−5.07044	−0.190424	−0.0952122	+	0.995457i	$0.530353\pi$
$710$	0	0
$711$	−6.35995	−0.238517
$712$	0	0
$713$	−20.0688	−0.751581
$714$	0	0
$715$	52.3378	1.95732
$716$	0	0
$717$	24.8200	0.926919
$718$	0	0
$719$	−2.92759	−0.109181	−0.0545903	−	0.998509i	$-0.517385\pi$
$719$	−2.92759	−0.109181	−0.0545903	+	0.998509i	$0.517385\pi$
$720$	0	0
$721$	0.251989	0.00938454
$722$	0	0
$723$	15.4981	0.576379
$724$	0	0
$725$	−1.07377	−0.0398787
$726$	0	0
$727$	12.8586	0.476898	0.238449	−	0.971155i	$-0.423361\pi$
$727$	12.8586	0.476898	0.238449	+	0.971155i	$0.423361\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.3744	−0.420697
$732$	0	0
$733$	16.0969	0.594554	0.297277	−	0.954791i	$-0.403922\pi$
$733$	16.0969	0.594554	0.297277	+	0.954791i	$0.403922\pi$
$734$	0	0
$735$	−13.1247	−0.484113
$736$	0	0
$737$	−14.1028	−0.519483
$738$	0	0
$739$	12.6385	0.464916	0.232458	−	0.972606i	$-0.425323\pi$
$739$	12.6385	0.464916	0.232458	+	0.972606i	$0.425323\pi$
$740$	0	0
$741$	8.67871	0.318820
$742$	0	0
$743$	20.5020	0.752144	0.376072	−	0.926590i	$-0.377275\pi$
$743$	20.5020	0.752144	0.376072	+	0.926590i	$0.377275\pi$
$744$	0	0
$745$	−31.1413	−1.14093
$746$	0	0
$747$	−0.371395	−0.0135886
$748$	0	0
$749$	0.586811	0.0214416
$750$	0	0
$751$	39.7446	1.45030	0.725150	−	0.688591i	$-0.241770\pi$
$751$	39.7446	1.45030	0.725150	+	0.688591i	$0.241770\pi$
$752$	0	0
$753$	2.93034	0.106787
$754$	0	0
$755$	1.87603	0.0682758
$756$	0	0
$757$	−10.5136	−0.382122	−0.191061	−	0.981578i	$-0.561193\pi$
$757$	−10.5136	−0.382122	−0.191061	+	0.981578i	$0.561193\pi$
$758$	0	0
$759$	−11.1752	−0.405635
$760$	0	0
$761$	9.11810	0.330531	0.165265	−	0.986249i	$-0.447152\pi$
$761$	9.11810	0.330531	0.165265	+	0.986249i	$0.447152\pi$
$762$	0	0
$763$	0.612797	0.0221847
$764$	0	0
$765$	2.91523	0.105401
$766$	0	0
$767$	−39.2826	−1.41841
$768$	0	0
$769$	10.3469	0.373120	0.186560	−	0.982444i	$-0.440266\pi$
$769$	10.3469	0.373120	0.186560	+	0.982444i	$0.440266\pi$
$770$	0	0
$771$	2.78348	0.100244
$772$	0	0
$773$	27.6399	0.994138	0.497069	−	0.867711i	$-0.334410\pi$
$773$	27.6399	0.994138	0.497069	+	0.867711i	$0.334410\pi$
$774$	0	0
$775$	−14.5139	−0.521353
$776$	0	0
$777$	0.681887	0.0244626
$778$	0	0
$779$	3.04620	0.109141
$780$	0	0
$781$	31.1214	1.11361
$782$	0	0
$783$	−0.725269	−0.0259190
$784$	0	0
$785$	−45.7894	−1.63429
$786$	0	0
$787$	−43.3507	−1.54529	−0.772643	−	0.634840i	$-0.781066\pi$
$787$	−43.3507	−1.54529	−0.772643	+	0.634840i	$0.781066\pi$
$788$	0	0
$789$	−14.8315	−0.528015
$790$	0	0
$791$	−0.316230	−0.0112439
$792$	0	0
$793$	9.97681	0.354287
$794$	0	0
$795$	18.2127	0.645938
$796$	0	0
$797$	36.7318	1.30111	0.650554	−	0.759460i	$-0.274537\pi$
$797$	36.7318	1.30111	0.650554	+	0.759460i	$0.274537\pi$
$798$	0	0
$799$	3.27089	0.115716
$800$	0	0
$801$	−6.15488	−0.217472
$802$	0	0
$803$	−3.80653	−0.134329
$804$	0	0
$805$	−0.242673	−0.00855311
$806$	0	0
$807$	16.8481	0.593082
$808$	0	0
$809$	5.93288	0.208589	0.104295	−	0.994546i	$-0.466742\pi$
$809$	5.93288	0.208589	0.104295	+	0.994546i	$0.466742\pi$
$810$	0	0
$811$	25.2374	0.886207	0.443103	−	0.896471i	$-0.353877\pi$
$811$	25.2374	0.886207	0.443103	+	0.896471i	$0.353877\pi$
$812$	0	0
$813$	18.9488	0.664563
$814$	0	0
$815$	−3.57059	−0.125072
$816$	0	0
$817$	−12.4303	−0.434882
$818$	0	0
$819$	−0.322925	−0.0112839
$820$	0	0
$821$	20.6806	0.721758	0.360879	−	0.932613i	$-0.382477\pi$
$821$	20.6806	0.721758	0.360879	+	0.932613i	$0.382477\pi$
$822$	0	0
$823$	19.1728	0.668323	0.334161	−	0.942516i	$-0.391547\pi$
$823$	19.1728	0.668323	0.334161	+	0.942516i	$0.391547\pi$
$824$	0	0
$825$	−8.08199	−0.281379
$826$	0	0
$827$	−9.72437	−0.338149	−0.169075	−	0.985603i	$-0.554078\pi$
$827$	−9.72437	−0.338149	−0.169075	+	0.985603i	$0.554078\pi$
$828$	0	0
$829$	15.7353	0.546508	0.273254	−	0.961942i	$-0.411900\pi$
$829$	15.7353	0.546508	0.273254	+	0.961942i	$0.411900\pi$
$830$	0	0
$831$	24.9035	0.863894
$832$	0	0
$833$	10.8714	0.376670
$834$	0	0
$835$	17.8751	0.618595
$836$	0	0
$837$	−9.80329	−0.338851
$838$	0	0
$839$	−19.7067	−0.680351	−0.340176	−	0.940362i	$-0.610487\pi$
$839$	−19.7067	−0.680351	−0.340176	+	0.940362i	$0.610487\pi$
$840$	0	0
$841$	−28.4740	−0.981862
$842$	0	0
$843$	3.39550	0.116947
$844$	0	0
$845$	−24.6093	−0.846586
$846$	0	0
$847$	−1.18793	−0.0408176
$848$	0	0
$849$	22.4553	0.770663
$850$	0	0
$851$	−22.0916	−0.757291
$852$	0	0
$853$	−37.5902	−1.28707	−0.643533	−	0.765419i	$-0.722532\pi$
$853$	−37.5902	−1.28707	−0.643533	+	0.765419i	$0.722532\pi$
$854$	0	0
$855$	3.18587	0.108954
$856$	0	0
$857$	−16.8284	−0.574846	−0.287423	−	0.957804i	$-0.592799\pi$
$857$	−16.8284	−0.574846	−0.287423	+	0.957804i	$0.592799\pi$
$858$	0	0
$859$	34.6530	1.18235	0.591173	−	0.806545i	$-0.298665\pi$
$859$	34.6530	1.18235	0.591173	+	0.806545i	$0.298665\pi$
$860$	0	0
$861$	−0.113345	−0.00386280
$862$	0	0
$863$	−2.11171	−0.0718836	−0.0359418	−	0.999354i	$-0.511443\pi$
$863$	−2.11171	−0.0718836	−0.0359418	+	0.999354i	$0.511443\pi$
$864$	0	0
$865$	−23.7042	−0.805966
$866$	0	0
$867$	14.5853	0.495342
$868$	0	0
$869$	34.7185	1.17774
$870$	0	0
$871$	13.2028	0.447359
$872$	0	0
$873$	−2.20968	−0.0747864
$874$	0	0
$875$	−0.768214	−0.0259704
$876$	0	0
$877$	33.6058	1.13479	0.567394	−	0.823447i	$-0.307952\pi$
$877$	33.6058	1.13479	0.567394	+	0.823447i	$0.307952\pi$
$878$	0	0
$879$	28.8963	0.974646
$880$	0	0
$881$	30.4809	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$881$	30.4809	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$882$	0	0
$883$	45.9825	1.54744	0.773718	−	0.633531i	$-0.218395\pi$
$883$	45.9825	1.54744	0.773718	+	0.633531i	$0.218395\pi$
$884$	0	0
$885$	−14.4202	−0.484731
$886$	0	0
$887$	12.3031	0.413097	0.206548	−	0.978436i	$-0.433777\pi$
$887$	12.3031	0.413097	0.206548	+	0.978436i	$0.433777\pi$
$888$	0	0
$889$	0.817931	0.0274325
$890$	0	0
$891$	−5.45893	−0.182881
$892$	0	0
$893$	3.57454	0.119617
$894$	0	0
$895$	43.7599	1.46273
$896$	0	0
$897$	10.4620	0.349317
$898$	0	0
$899$	7.11002	0.237133
$900$	0	0
$901$	−15.0858	−0.502580
$902$	0	0
$903$	0.462517	0.0153916
$904$	0	0
$905$	37.9074	1.26009
$906$	0	0
$907$	0.0189188	0.000628189 0	0.000314094	−	1.00000i	$-0.499900\pi$
$907$	0.0189188	0.000628189 0	0.000314094		1.00000i	$0.499900\pi$
$908$	0	0
$909$	7.90739	0.262271
$910$	0	0
$911$	−17.4030	−0.576587	−0.288294	−	0.957542i	$-0.593088\pi$
$911$	−17.4030	−0.576587	−0.288294	+	0.957542i	$0.593088\pi$
$912$	0	0
$913$	2.02742	0.0670977
$914$	0	0
$915$	3.66238	0.121075
$916$	0	0
$917$	−1.05697	−0.0349042
$918$	0	0
$919$	59.1626	1.95159	0.975797	−	0.218680i	$-0.0701750\pi$
$919$	59.1626	1.95159	0.975797	+	0.218680i	$0.0701750\pi$
$920$	0	0
$921$	1.24516	0.0410293
$922$	0	0
$923$	−29.1353	−0.958999
$924$	0	0
$925$	−15.9768	−0.525314
$926$	0	0
$927$	−3.98793	−0.130981
$928$	0	0
$929$	−6.45135	−0.211662	−0.105831	−	0.994384i	$-0.533750\pi$
$929$	−6.45135	−0.211662	−0.105831	+	0.994384i	$0.533750\pi$
$930$	0	0
$931$	11.8806	0.389370
$932$	0	0
$933$	31.5989	1.03450
$934$	0	0
$935$	−15.9141	−0.520445
$936$	0	0
$937$	35.4132	1.15690	0.578450	−	0.815718i	$-0.303658\pi$
$937$	35.4132	1.15690	0.578450	+	0.815718i	$0.303658\pi$
$938$	0	0
$939$	11.6871	0.381394
$940$	0	0
$941$	−50.6539	−1.65127	−0.825635	−	0.564205i	$-0.809183\pi$
$941$	−50.6539	−1.65127	−0.825635	+	0.564205i	$0.809183\pi$
$942$	0	0
$943$	3.67214	0.119581
$944$	0	0
$945$	−0.118542	−0.00385618
$946$	0	0
$947$	−30.4749	−0.990303	−0.495151	−	0.868807i	$-0.664887\pi$
$947$	−30.4749	−0.990303	−0.495151	+	0.868807i	$0.664887\pi$
$948$	0	0
$949$	3.56360	0.115679
$950$	0	0
$951$	−6.09051	−0.197498
$952$	0	0
$953$	−3.27468	−0.106077	−0.0530387	−	0.998592i	$-0.516891\pi$
$953$	−3.27468	−0.106077	−0.0530387	+	0.998592i	$0.516891\pi$
$954$	0	0
$955$	−15.1531	−0.490344
$956$	0	0
$957$	3.95919	0.127983
$958$	0	0
$959$	0.463272	0.0149598
$960$	0	0
$961$	65.1045	2.10015
$962$	0	0
$963$	−9.28677	−0.299262
$964$	0	0
$965$	−10.1886	−0.327984
$966$	0	0
$967$	50.5768	1.62644	0.813220	−	0.581956i	$-0.197712\pi$
$967$	50.5768	1.62644	0.813220	+	0.581956i	$0.197712\pi$
$968$	0	0
$969$	−2.63889	−0.0847733
$970$	0	0
$971$	−19.2683	−0.618349	−0.309175	−	0.951005i	$-0.600053\pi$
$971$	−19.2683	−0.618349	−0.309175	+	0.951005i	$0.600053\pi$
$972$	0	0
$973$	0.308950	0.00990448
$974$	0	0
$975$	7.56622	0.242313
$976$	0	0
$977$	−29.0422	−0.929141	−0.464571	−	0.885536i	$-0.653791\pi$
$977$	−29.0422	−0.929141	−0.464571	+	0.885536i	$0.653791\pi$
$978$	0	0
$979$	33.5990	1.07383
$980$	0	0
$981$	−9.69803	−0.309634
$982$	0	0
$983$	−7.53729	−0.240402	−0.120201	−	0.992750i	$-0.538354\pi$
$983$	−7.53729	−0.240402	−0.120201	+	0.992750i	$0.538354\pi$
$984$	0	0
$985$	−7.04570	−0.224495
$986$	0	0
$987$	−0.133004	−0.00423358
$988$	0	0
$989$	−14.9845	−0.476480
$990$	0	0
$991$	−55.4015	−1.75989	−0.879943	−	0.475079i	$-0.842420\pi$
$991$	−55.4015	−1.75989	−0.879943	+	0.475079i	$0.842420\pi$
$992$	0	0
$993$	−1.77617	−0.0563650
$994$	0	0
$995$	−5.44554	−0.172635
$996$	0	0
$997$	7.87502	0.249404	0.124702	−	0.992194i	$-0.460202\pi$
$997$	7.87502	0.249404	0.124702	+	0.992194i	$0.460202\pi$
$998$	0	0
$999$	−10.7914	−0.341426

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7248.2.a.bm.1.3		10
4.3	odd	2		3624.2.a.l.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3624.2.a.l.1.3	✓	10	4.3	odd	2
7248.2.a.bm.1.3		10	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:	\( N \)	\(=\)	\( 7248 = 2^{4} \cdot 3 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7248.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.87603 q^{5} -0.0631878 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.87603 q^{5} -0.0631878 q^{7} +1.00000 q^{9} -5.45893 q^{11} +5.11055 q^{13} +1.87603 q^{15} -1.55394 q^{17} -1.69819 q^{19} +0.0631878 q^{21} -2.04714 q^{23} -1.48051 q^{25} -1.00000 q^{27} +0.725269 q^{29} +9.80329 q^{31} +5.45893 q^{33} +0.118542 q^{35} +10.7914 q^{37} -5.11055 q^{39} -1.79379 q^{41} +7.31972 q^{43} -1.87603 q^{45} -2.10491 q^{47} -6.99601 q^{49} +1.55394 q^{51} +9.70810 q^{53} +10.2411 q^{55} +1.69819 q^{57} -7.68657 q^{59} +1.95220 q^{61} -0.0631878 q^{63} -9.58755 q^{65} +2.58343 q^{67} +2.04714 q^{69} -5.70100 q^{71} +0.697303 q^{73} +1.48051 q^{75} +0.344938 q^{77} -6.35995 q^{79} +1.00000 q^{81} -0.371395 q^{83} +2.91523 q^{85} -0.725269 q^{87} -6.15488 q^{89} -0.322925 q^{91} -9.80329 q^{93} +3.18587 q^{95} -2.20968 q^{97} -5.45893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9	\( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9} - 7 q^{11} + 6 q^{13} - 2 q^{15} + 7 q^{17} + 8 q^{21} - 25 q^{23} + 8 q^{25} - 10 q^{27} + 12 q^{29} - 11 q^{31} + 7 q^{33} - 9 q^{35} - 3 q^{37} - 6 q^{39} + 12 q^{41} + 2 q^{45} - 31 q^{47} + 14 q^{49} - 7 q^{51} + q^{53} - 9 q^{55} - 19 q^{59} + 24 q^{61} - 8 q^{63} + 20 q^{65} + q^{67} + 25 q^{69} - 34 q^{71} - 18 q^{73} - 8 q^{75} + 27 q^{77} - 25 q^{79} + 10 q^{81} - 14 q^{83} - 3 q^{85} - 12 q^{87} + 20 q^{89} + 12 q^{91} + 11 q^{93} - 48 q^{95} - 15 q^{97} - 7 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9 - 7 * q^11 + 6 * q^13 - 2 * q^15 + 7 * q^17 + 8 * q^21 - 25 * q^23 + 8 * q^25 - 10 * q^27 + 12 * q^29 - 11 * q^31 + 7 * q^33 - 9 * q^35 - 3 * q^37 - 6 * q^39 + 12 * q^41 + 2 * q^45 - 31 * q^47 + 14 * q^49 - 7 * q^51 + q^53 - 9 * q^55 - 19 * q^59 + 24 * q^61 - 8 * q^63 + 20 * q^65 + q^67 + 25 * q^69 - 34 * q^71 - 18 * q^73 - 8 * q^75 + 27 * q^77 - 25 * q^79 + 10 * q^81 - 14 * q^83 - 3 * q^85 - 12 * q^87 + 20 * q^89 + 12 * q^91 + 11 * q^93 - 48 * q^95 - 15 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

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