LMFDB - Embedded newform 7248.2.a.bm.1.10

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			1.10
Root			$3.48207$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.48207 q^{5} -0.907352 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.48207 q^{5} -0.907352 q^{7} +1.00000 q^{9} -1.45070 q^{11} +4.85095 q^{13} -3.48207 q^{15} +1.91500 q^{17} +0.837962 q^{19} +0.907352 q^{21} -9.04299 q^{23} +7.12481 q^{25} -1.00000 q^{27} -9.91697 q^{29} -3.55684 q^{31} +1.45070 q^{33} -3.15946 q^{35} -6.58205 q^{37} -4.85095 q^{39} -6.41964 q^{41} +4.33239 q^{43} +3.48207 q^{45} -0.221433 q^{47} -6.17671 q^{49} -1.91500 q^{51} -10.3790 q^{53} -5.05145 q^{55} -0.837962 q^{57} -6.52315 q^{59} -11.5657 q^{61} -0.907352 q^{63} +16.8913 q^{65} +0.0316033 q^{67} +9.04299 q^{69} +0.0698850 q^{71} +12.1679 q^{73} -7.12481 q^{75} +1.31630 q^{77} -7.91478 q^{79} +1.00000 q^{81} +3.01324 q^{83} +6.66816 q^{85} +9.91697 q^{87} -3.72751 q^{89} -4.40152 q^{91} +3.55684 q^{93} +2.91784 q^{95} +11.5723 q^{97} -1.45070 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.48207	1.55723	0.778615	−	0.627502i	$-0.215923\pi$
$5$	3.48207	1.55723	0.778615	+	0.627502i	$0.215923\pi$
$6$	0	0
$7$	−0.907352	−0.342947	−0.171473	−	0.985189i	$-0.554853\pi$
$7$	−0.907352	−0.342947	−0.171473	+	0.985189i	$0.554853\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.45070	−0.437404	−0.218702	−	0.975792i	$-0.570182\pi$
$11$	−1.45070	−0.437404	−0.218702	+	0.975792i	$0.570182\pi$
$12$	0	0
$13$	4.85095	1.34541	0.672705	−	0.739911i	$-0.265132\pi$
$13$	4.85095	1.34541	0.672705	+	0.739911i	$0.265132\pi$
$14$	0	0
$15$	−3.48207	−0.899067
$16$	0	0
$17$	1.91500	0.464456	0.232228	−	0.972661i	$-0.425398\pi$
$17$	1.91500	0.464456	0.232228	+	0.972661i	$0.425398\pi$
$18$	0	0
$19$	0.837962	0.192242	0.0961209	−	0.995370i	$-0.469356\pi$
$19$	0.837962	0.192242	0.0961209	+	0.995370i	$0.469356\pi$
$20$	0	0
$21$	0.907352	0.198000
$22$	0	0
$23$	−9.04299	−1.88559	−0.942796	−	0.333369i	$-0.891814\pi$
$23$	−9.04299	−1.88559	−0.942796	+	0.333369i	$0.891814\pi$
$24$	0	0
$25$	7.12481	1.42496
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.91697	−1.84154	−0.920768	−	0.390111i	$-0.872437\pi$
$29$	−9.91697	−1.84154	−0.920768	+	0.390111i	$0.872437\pi$
$30$	0	0
$31$	−3.55684	−0.638828	−0.319414	−	0.947615i	$-0.603486\pi$
$31$	−3.55684	−0.638828	−0.319414	+	0.947615i	$0.603486\pi$
$32$	0	0
$33$	1.45070	0.252535
$34$	0	0
$35$	−3.15946	−0.534047
$36$	0	0
$37$	−6.58205	−1.08208	−0.541041	−	0.840996i	$-0.681970\pi$
$37$	−6.58205	−1.08208	−0.541041	+	0.840996i	$0.681970\pi$
$38$	0	0
$39$	−4.85095	−0.776773
$40$	0	0
$41$	−6.41964	−1.00258	−0.501290	−	0.865280i	$-0.667141\pi$
$41$	−6.41964	−1.00258	−0.501290	+	0.865280i	$0.667141\pi$
$42$	0	0
$43$	4.33239	0.660683	0.330342	−	0.943861i	$-0.392836\pi$
$43$	4.33239	0.660683	0.330342	+	0.943861i	$0.392836\pi$
$44$	0	0
$45$	3.48207	0.519076
$46$	0	0
$47$	−0.221433	−0.0322994	−0.0161497	−	0.999870i	$-0.505141\pi$
$47$	−0.221433	−0.0322994	−0.0161497	+	0.999870i	$0.505141\pi$
$48$	0	0
$49$	−6.17671	−0.882388
$50$	0	0
$51$	−1.91500	−0.268154
$52$	0	0
$53$	−10.3790	−1.42567	−0.712833	−	0.701334i	$-0.752588\pi$
$53$	−10.3790	−1.42567	−0.712833	+	0.701334i	$0.752588\pi$
$54$	0	0
$55$	−5.05145	−0.681138
$56$	0	0
$57$	−0.837962	−0.110991
$58$	0	0
$59$	−6.52315	−0.849242	−0.424621	−	0.905371i	$-0.639593\pi$
$59$	−6.52315	−0.849242	−0.424621	+	0.905371i	$0.639593\pi$
$60$	0	0
$61$	−11.5657	−1.48084	−0.740421	−	0.672144i	$-0.765374\pi$
$61$	−11.5657	−1.48084	−0.740421	+	0.672144i	$0.765374\pi$
$62$	0	0
$63$	−0.907352	−0.114316
$64$	0	0
$65$	16.8913	2.09511
$66$	0	0
$67$	0.0316033	0.00386096	0.00193048	−	0.999998i	$-0.499386\pi$
$67$	0.0316033	0.00386096	0.00193048	+	0.999998i	$0.499386\pi$
$68$	0	0
$69$	9.04299	1.08865
$70$	0	0
$71$	0.0698850	0.00829382	0.00414691	−	0.999991i	$-0.498680\pi$
$71$	0.0698850	0.00829382	0.00414691	+	0.999991i	$0.498680\pi$
$72$	0	0
$73$	12.1679	1.42414	0.712071	−	0.702107i	$-0.247757\pi$
$73$	12.1679	1.42414	0.712071	+	0.702107i	$0.247757\pi$
$74$	0	0
$75$	−7.12481	−0.822702
$76$	0	0
$77$	1.31630	0.150006
$78$	0	0
$79$	−7.91478	−0.890482	−0.445241	−	0.895411i	$-0.646882\pi$
$79$	−7.91478	−0.890482	−0.445241	+	0.895411i	$0.646882\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.01324	0.330746	0.165373	−	0.986231i	$-0.447117\pi$
$83$	3.01324	0.330746	0.165373	+	0.986231i	$0.447117\pi$
$84$	0	0
$85$	6.66816	0.723264
$86$	0	0
$87$	9.91697	1.06321
$88$	0	0
$89$	−3.72751	−0.395115	−0.197558	−	0.980291i	$-0.563301\pi$
$89$	−3.72751	−0.395115	−0.197558	+	0.980291i	$0.563301\pi$
$90$	0	0
$91$	−4.40152	−0.461404
$92$	0	0
$93$	3.55684	0.368827
$94$	0	0
$95$	2.91784	0.299364
$96$	0	0
$97$	11.5723	1.17499	0.587493	−	0.809229i	$-0.300115\pi$
$97$	11.5723	1.17499	0.587493	+	0.809229i	$0.300115\pi$
$98$	0	0
$99$	−1.45070	−0.145801
$100$	0	0
$101$	−9.68576	−0.963769	−0.481884	−	0.876235i	$-0.660047\pi$
$101$	−9.68576	−0.963769	−0.481884	+	0.876235i	$0.660047\pi$
$102$	0	0
$103$	18.9203	1.86427	0.932137	−	0.362106i	$-0.117942\pi$
$103$	18.9203	1.86427	0.932137	+	0.362106i	$0.117942\pi$
$104$	0	0
$105$	3.15946	0.308332
$106$	0	0
$107$	−6.48409	−0.626841	−0.313420	−	0.949614i	$-0.601475\pi$
$107$	−6.48409	−0.626841	−0.313420	+	0.949614i	$0.601475\pi$
$108$	0	0
$109$	−1.02001	−0.0976990	−0.0488495	−	0.998806i	$-0.515555\pi$
$109$	−1.02001	−0.0976990	−0.0488495	+	0.998806i	$0.515555\pi$
$110$	0	0
$111$	6.58205	0.624741
$112$	0	0
$113$	−15.2789	−1.43732	−0.718658	−	0.695363i	$-0.755244\pi$
$113$	−15.2789	−1.43732	−0.718658	+	0.695363i	$0.755244\pi$
$114$	0	0
$115$	−31.4883	−2.93630
$116$	0	0
$117$	4.85095	0.448470
$118$	0	0
$119$	−1.73758	−0.159284
$120$	0	0
$121$	−8.89546	−0.808678
$122$	0	0
$123$	6.41964	0.578839
$124$	0	0
$125$	7.39875	0.661764
$126$	0	0
$127$	3.26192	0.289448	0.144724	−	0.989472i	$-0.453771\pi$
$127$	3.26192	0.289448	0.144724	+	0.989472i	$0.453771\pi$
$128$	0	0
$129$	−4.33239	−0.381446
$130$	0	0
$131$	10.3652	0.905612	0.452806	−	0.891609i	$-0.350423\pi$
$131$	10.3652	0.905612	0.452806	+	0.891609i	$0.350423\pi$
$132$	0	0
$133$	−0.760327	−0.0659287
$134$	0	0
$135$	−3.48207	−0.299689
$136$	0	0
$137$	9.64844	0.824322	0.412161	−	0.911111i	$-0.364774\pi$
$137$	9.64844	0.824322	0.412161	+	0.911111i	$0.364774\pi$
$138$	0	0
$139$	−14.3445	−1.21669	−0.608344	−	0.793674i	$-0.708166\pi$
$139$	−14.3445	−1.21669	−0.608344	+	0.793674i	$0.708166\pi$
$140$	0	0
$141$	0.221433	0.0186481
$142$	0	0
$143$	−7.03729	−0.588487
$144$	0	0
$145$	−34.5316	−2.86769
$146$	0	0
$147$	6.17671	0.509447
$148$	0	0
$149$	8.09211	0.662931	0.331466	−	0.943467i	$-0.392457\pi$
$149$	8.09211	0.662931	0.331466	+	0.943467i	$0.392457\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	1.91500	0.154819
$154$	0	0
$155$	−12.3852	−0.994801
$156$	0	0
$157$	4.35018	0.347182	0.173591	−	0.984818i	$-0.444463\pi$
$157$	4.35018	0.347182	0.173591	+	0.984818i	$0.444463\pi$
$158$	0	0
$159$	10.3790	0.823108
$160$	0	0
$161$	8.20517	0.646658
$162$	0	0
$163$	5.01383	0.392713	0.196357	−	0.980533i	$-0.437089\pi$
$163$	5.01383	0.392713	0.196357	+	0.980533i	$0.437089\pi$
$164$	0	0
$165$	5.05145	0.393255
$166$	0	0
$167$	−23.0883	−1.78662	−0.893312	−	0.449437i	$-0.851625\pi$
$167$	−23.0883	−1.78662	−0.893312	+	0.449437i	$0.851625\pi$
$168$	0	0
$169$	10.5317	0.810129
$170$	0	0
$171$	0.837962	0.0640806
$172$	0	0
$173$	14.0483	1.06807	0.534037	−	0.845461i	$-0.320674\pi$
$173$	14.0483	1.06807	0.534037	+	0.845461i	$0.320674\pi$
$174$	0	0
$175$	−6.46471	−0.488686
$176$	0	0
$177$	6.52315	0.490310
$178$	0	0
$179$	21.8801	1.63539	0.817696	−	0.575650i	$-0.195251\pi$
$179$	21.8801	1.63539	0.817696	+	0.575650i	$0.195251\pi$
$180$	0	0
$181$	−12.8671	−0.956406	−0.478203	−	0.878249i	$-0.658712\pi$
$181$	−12.8671	−0.956406	−0.478203	+	0.878249i	$0.658712\pi$
$182$	0	0
$183$	11.5657	0.854964
$184$	0	0
$185$	−22.9192	−1.68505
$186$	0	0
$187$	−2.77810	−0.203155
$188$	0	0
$189$	0.907352	0.0660001
$190$	0	0
$191$	11.6060	0.839778	0.419889	−	0.907575i	$-0.362069\pi$
$191$	11.6060	0.839778	0.419889	+	0.907575i	$0.362069\pi$
$192$	0	0
$193$	−19.6706	−1.41592	−0.707960	−	0.706252i	$-0.750384\pi$
$193$	−19.6706	−1.41592	−0.707960	+	0.706252i	$0.750384\pi$
$194$	0	0
$195$	−16.8913	−1.20961
$196$	0	0
$197$	18.3464	1.30712	0.653562	−	0.756873i	$-0.273274\pi$
$197$	18.3464	1.30712	0.653562	+	0.756873i	$0.273274\pi$
$198$	0	0
$199$	16.8814	1.19669	0.598346	−	0.801238i	$-0.295825\pi$
$199$	16.8814	1.19669	0.598346	+	0.801238i	$0.295825\pi$
$200$	0	0
$201$	−0.0316033	−0.00222912
$202$	0	0
$203$	8.99818	0.631549
$204$	0	0
$205$	−22.3536	−1.56125
$206$	0	0
$207$	−9.04299	−0.628531
$208$	0	0
$209$	−1.21564	−0.0840873
$210$	0	0
$211$	−6.49552	−0.447170	−0.223585	−	0.974684i	$-0.571776\pi$
$211$	−6.49552	−0.447170	−0.223585	+	0.974684i	$0.571776\pi$
$212$	0	0
$213$	−0.0698850	−0.00478844
$214$	0	0
$215$	15.0857	1.02884
$216$	0	0
$217$	3.22731	0.219084
$218$	0	0
$219$	−12.1679	−0.822229
$220$	0	0
$221$	9.28956	0.624884
$222$	0	0
$223$	−28.2895	−1.89441	−0.947204	−	0.320632i	$-0.896105\pi$
$223$	−28.2895	−1.89441	−0.947204	+	0.320632i	$0.896105\pi$
$224$	0	0
$225$	7.12481	0.474988
$226$	0	0
$227$	−20.4996	−1.36061	−0.680303	−	0.732931i	$-0.738152\pi$
$227$	−20.4996	−1.36061	−0.680303	+	0.732931i	$0.738152\pi$
$228$	0	0
$229$	2.94356	0.194516	0.0972578	−	0.995259i	$-0.468993\pi$
$229$	2.94356	0.194516	0.0972578	+	0.995259i	$0.468993\pi$
$230$	0	0
$231$	−1.31630	−0.0866061
$232$	0	0
$233$	−0.618074	−0.0404914	−0.0202457	−	0.999795i	$-0.506445\pi$
$233$	−0.618074	−0.0404914	−0.0202457	+	0.999795i	$0.506445\pi$
$234$	0	0
$235$	−0.771047	−0.0502975
$236$	0	0
$237$	7.91478	0.514120
$238$	0	0
$239$	0.466443	0.0301717	0.0150858	−	0.999886i	$-0.495198\pi$
$239$	0.466443	0.0301717	0.0150858	+	0.999886i	$0.495198\pi$
$240$	0	0
$241$	1.75734	0.113200	0.0566002	−	0.998397i	$-0.481974\pi$
$241$	1.75734	0.113200	0.0566002	+	0.998397i	$0.481974\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−21.5077	−1.37408
$246$	0	0
$247$	4.06491	0.258644
$248$	0	0
$249$	−3.01324	−0.190956
$250$	0	0
$251$	10.4734	0.661074	0.330537	−	0.943793i	$-0.392770\pi$
$251$	10.4734	0.661074	0.330537	+	0.943793i	$0.392770\pi$
$252$	0	0
$253$	13.1187	0.824765
$254$	0	0
$255$	−6.66816	−0.417577
$256$	0	0
$257$	24.4255	1.52362	0.761809	−	0.647802i	$-0.224312\pi$
$257$	24.4255	1.52362	0.761809	+	0.647802i	$0.224312\pi$
$258$	0	0
$259$	5.97224	0.371097
$260$	0	0
$261$	−9.91697	−0.613845
$262$	0	0
$263$	7.24585	0.446798	0.223399	−	0.974727i	$-0.428285\pi$
$263$	7.24585	0.446798	0.223399	+	0.974727i	$0.428285\pi$
$264$	0	0
$265$	−36.1404	−2.22009
$266$	0	0
$267$	3.72751	0.228120
$268$	0	0
$269$	−16.4898	−1.00540	−0.502701	−	0.864460i	$-0.667660\pi$
$269$	−16.4898	−1.00540	−0.502701	+	0.864460i	$0.667660\pi$
$270$	0	0
$271$	−4.37387	−0.265694	−0.132847	−	0.991137i	$-0.542412\pi$
$271$	−4.37387	−0.265694	−0.132847	+	0.991137i	$0.542412\pi$
$272$	0	0
$273$	4.40152	0.266392
$274$	0	0
$275$	−10.3360	−0.623284
$276$	0	0
$277$	15.2825	0.918236	0.459118	−	0.888375i	$-0.348166\pi$
$277$	15.2825	0.918236	0.459118	+	0.888375i	$0.348166\pi$
$278$	0	0
$279$	−3.55684	−0.212943
$280$	0	0
$281$	29.6109	1.76644	0.883219	−	0.468961i	$-0.155372\pi$
$281$	29.6109	1.76644	0.883219	+	0.468961i	$0.155372\pi$
$282$	0	0
$283$	13.7350	0.816463	0.408231	−	0.912878i	$-0.366146\pi$
$283$	13.7350	0.816463	0.408231	+	0.912878i	$0.366146\pi$
$284$	0	0
$285$	−2.91784	−0.172838
$286$	0	0
$287$	5.82487	0.343831
$288$	0	0
$289$	−13.3328	−0.784281
$290$	0	0
$291$	−11.5723	−0.678379
$292$	0	0
$293$	18.2736	1.06755	0.533777	−	0.845625i	$-0.320772\pi$
$293$	18.2736	1.06755	0.533777	+	0.845625i	$0.320772\pi$
$294$	0	0
$295$	−22.7141	−1.32246
$296$	0	0
$297$	1.45070	0.0841784
$298$	0	0
$299$	−43.8670	−2.53690
$300$	0	0
$301$	−3.93100	−0.226579
$302$	0	0
$303$	9.68576	0.556432
$304$	0	0
$305$	−40.2727	−2.30601
$306$	0	0
$307$	−7.22044	−0.412092	−0.206046	−	0.978542i	$-0.566060\pi$
$307$	−7.22044	−0.412092	−0.206046	+	0.978542i	$0.566060\pi$
$308$	0	0
$309$	−18.9203	−1.07634
$310$	0	0
$311$	18.9913	1.07690	0.538449	−	0.842658i	$-0.319010\pi$
$311$	18.9913	1.07690	0.538449	+	0.842658i	$0.319010\pi$
$312$	0	0
$313$	2.22051	0.125511	0.0627554	−	0.998029i	$-0.480011\pi$
$313$	2.22051	0.125511	0.0627554	+	0.998029i	$0.480011\pi$
$314$	0	0
$315$	−3.15946	−0.178016
$316$	0	0
$317$	−26.4283	−1.48436	−0.742180	−	0.670200i	$-0.766208\pi$
$317$	−26.4283	−1.48436	−0.742180	+	0.670200i	$0.766208\pi$
$318$	0	0
$319$	14.3866	0.805494
$320$	0	0
$321$	6.48409	0.361907
$322$	0	0
$323$	1.60470	0.0892878
$324$	0	0
$325$	34.5621	1.91716
$326$	0	0
$327$	1.02001	0.0564066
$328$	0	0
$329$	0.200918	0.0110770
$330$	0	0
$331$	33.7212	1.85348	0.926741	−	0.375700i	$-0.122598\pi$
$331$	33.7212	1.85348	0.926741	+	0.375700i	$0.122598\pi$
$332$	0	0
$333$	−6.58205	−0.360694
$334$	0	0
$335$	0.110045	0.00601239
$336$	0	0
$337$	3.43252	0.186981	0.0934906	−	0.995620i	$-0.470197\pi$
$337$	3.43252	0.186981	0.0934906	+	0.995620i	$0.470197\pi$
$338$	0	0
$339$	15.2789	0.829835
$340$	0	0
$341$	5.15992	0.279426
$342$	0	0
$343$	11.9559	0.645559
$344$	0	0
$345$	31.4883	1.69527
$346$	0	0
$347$	−30.0834	−1.61496	−0.807481	−	0.589894i	$-0.799170\pi$
$347$	−30.0834	−1.61496	−0.807481	+	0.589894i	$0.799170\pi$
$348$	0	0
$349$	0.905510	0.0484709	0.0242354	−	0.999706i	$-0.492285\pi$
$349$	0.905510	0.0484709	0.0242354	+	0.999706i	$0.492285\pi$
$350$	0	0
$351$	−4.85095	−0.258924
$352$	0	0
$353$	10.0760	0.536289	0.268145	−	0.963379i	$-0.413590\pi$
$353$	10.0760	0.536289	0.268145	+	0.963379i	$0.413590\pi$
$354$	0	0
$355$	0.243344	0.0129154
$356$	0	0
$357$	1.73758	0.0919624
$358$	0	0
$359$	−23.5468	−1.24275	−0.621376	−	0.783512i	$-0.713426\pi$
$359$	−23.5468	−1.24275	−0.621376	+	0.783512i	$0.713426\pi$
$360$	0	0
$361$	−18.2978	−0.963043
$362$	0	0
$363$	8.89546	0.466890
$364$	0	0
$365$	42.3694	2.21772
$366$	0	0
$367$	14.5929	0.761741	0.380871	−	0.924628i	$-0.375624\pi$
$367$	14.5929	0.761741	0.380871	+	0.924628i	$0.375624\pi$
$368$	0	0
$369$	−6.41964	−0.334193
$370$	0	0
$371$	9.41741	0.488927
$372$	0	0
$373$	−14.9765	−0.775453	−0.387727	−	0.921774i	$-0.626740\pi$
$373$	−14.9765	−0.775453	−0.387727	+	0.921774i	$0.626740\pi$
$374$	0	0
$375$	−7.39875	−0.382070
$376$	0	0
$377$	−48.1067	−2.47762
$378$	0	0
$379$	3.51655	0.180633	0.0903165	−	0.995913i	$-0.471212\pi$
$379$	3.51655	0.180633	0.0903165	+	0.995913i	$0.471212\pi$
$380$	0	0
$381$	−3.26192	−0.167113
$382$	0	0
$383$	−23.7162	−1.21184	−0.605920	−	0.795525i	$-0.707195\pi$
$383$	−23.7162	−1.21184	−0.605920	+	0.795525i	$0.707195\pi$
$384$	0	0
$385$	4.58344	0.233594
$386$	0	0
$387$	4.33239	0.220228
$388$	0	0
$389$	14.8989	0.755406	0.377703	−	0.925927i	$-0.376714\pi$
$389$	14.8989	0.755406	0.377703	+	0.925927i	$0.376714\pi$
$390$	0	0
$391$	−17.3173	−0.875774
$392$	0	0
$393$	−10.3652	−0.522855
$394$	0	0
$395$	−27.5598	−1.38668
$396$	0	0
$397$	27.3528	1.37280	0.686400	−	0.727224i	$-0.259190\pi$
$397$	27.3528	1.37280	0.686400	+	0.727224i	$0.259190\pi$
$398$	0	0
$399$	0.760327	0.0380640
$400$	0	0
$401$	11.1532	0.556963	0.278481	−	0.960442i	$-0.410169\pi$
$401$	11.1532	0.556963	0.278481	+	0.960442i	$0.410169\pi$
$402$	0	0
$403$	−17.2540	−0.859485
$404$	0	0
$405$	3.48207	0.173025
$406$	0	0
$407$	9.54861	0.473307
$408$	0	0
$409$	17.6182	0.871164	0.435582	−	0.900149i	$-0.356543\pi$
$409$	17.6182	0.871164	0.435582	+	0.900149i	$0.356543\pi$
$410$	0	0
$411$	−9.64844	−0.475923
$412$	0	0
$413$	5.91880	0.291245
$414$	0	0
$415$	10.4923	0.515047
$416$	0	0
$417$	14.3445	0.702455
$418$	0	0
$419$	7.52047	0.367399	0.183700	−	0.982982i	$-0.441193\pi$
$419$	7.52047	0.367399	0.183700	+	0.982982i	$0.441193\pi$
$420$	0	0
$421$	2.40642	0.117282	0.0586408	−	0.998279i	$-0.481323\pi$
$421$	2.40642	0.117282	0.0586408	+	0.998279i	$0.481323\pi$
$422$	0	0
$423$	−0.221433	−0.0107665
$424$	0	0
$425$	13.6440	0.661832
$426$	0	0
$427$	10.4942	0.507850
$428$	0	0
$429$	7.03729	0.339763
$430$	0	0
$431$	−12.2742	−0.591227	−0.295614	−	0.955308i	$-0.595524\pi$
$431$	−12.2742	−0.591227	−0.295614	+	0.955308i	$0.595524\pi$
$432$	0	0
$433$	−28.8444	−1.38617	−0.693087	−	0.720854i	$-0.743750\pi$
$433$	−28.8444	−1.38617	−0.693087	+	0.720854i	$0.743750\pi$
$434$	0	0
$435$	34.5316	1.65566
$436$	0	0
$437$	−7.57768	−0.362490
$438$	0	0
$439$	5.51501	0.263217	0.131609	−	0.991302i	$-0.457986\pi$
$439$	5.51501	0.263217	0.131609	+	0.991302i	$0.457986\pi$
$440$	0	0
$441$	−6.17671	−0.294129
$442$	0	0
$443$	−12.8643	−0.611202	−0.305601	−	0.952160i	$-0.598857\pi$
$443$	−12.8643	−0.611202	−0.305601	+	0.952160i	$0.598857\pi$
$444$	0	0
$445$	−12.9794	−0.615285
$446$	0	0
$447$	−8.09211	−0.382744
$448$	0	0
$449$	4.11149	0.194033	0.0970166	−	0.995283i	$-0.469070\pi$
$449$	4.11149	0.194033	0.0970166	+	0.995283i	$0.469070\pi$
$450$	0	0
$451$	9.31300	0.438532
$452$	0	0
$453$	1.00000	0.0469841
$454$	0	0
$455$	−15.3264	−0.718512
$456$	0	0
$457$	−3.05162	−0.142749	−0.0713744	−	0.997450i	$-0.522739\pi$
$457$	−3.05162	−0.142749	−0.0713744	+	0.997450i	$0.522739\pi$
$458$	0	0
$459$	−1.91500	−0.0893845
$460$	0	0
$461$	2.18791	0.101901	0.0509506	−	0.998701i	$-0.483775\pi$
$461$	2.18791	0.101901	0.0509506	+	0.998701i	$0.483775\pi$
$462$	0	0
$463$	−13.1783	−0.612448	−0.306224	−	0.951959i	$-0.599066\pi$
$463$	−13.1783	−0.612448	−0.306224	+	0.951959i	$0.599066\pi$
$464$	0	0
$465$	12.3852	0.574349
$466$	0	0
$467$	−42.5920	−1.97092	−0.985460	−	0.169905i	$-0.945654\pi$
$467$	−42.5920	−1.97092	−0.985460	+	0.169905i	$0.945654\pi$
$468$	0	0
$469$	−0.0286753	−0.00132410
$470$	0	0
$471$	−4.35018	−0.200446
$472$	0	0
$473$	−6.28502	−0.288985
$474$	0	0
$475$	5.97033	0.273937
$476$	0	0
$477$	−10.3790	−0.475222
$478$	0	0
$479$	−21.8890	−1.00013	−0.500067	−	0.865987i	$-0.666691\pi$
$479$	−21.8890	−1.00013	−0.500067	+	0.865987i	$0.666691\pi$
$480$	0	0
$481$	−31.9292	−1.45585
$482$	0	0
$483$	−8.20517	−0.373348
$484$	0	0
$485$	40.2955	1.82972
$486$	0	0
$487$	7.47697	0.338814	0.169407	−	0.985546i	$-0.445815\pi$
$487$	7.47697	0.338814	0.169407	+	0.985546i	$0.445815\pi$
$488$	0	0
$489$	−5.01383	−0.226733
$490$	0	0
$491$	7.12012	0.321326	0.160663	−	0.987009i	$-0.448637\pi$
$491$	7.12012	0.321326	0.160663	+	0.987009i	$0.448637\pi$
$492$	0	0
$493$	−18.9910	−0.855312
$494$	0	0
$495$	−5.05145	−0.227046
$496$	0	0
$497$	−0.0634103	−0.00284434
$498$	0	0
$499$	22.6724	1.01496	0.507478	−	0.861664i	$-0.330578\pi$
$499$	22.6724	1.01496	0.507478	+	0.861664i	$0.330578\pi$
$500$	0	0
$501$	23.0883	1.03151
$502$	0	0
$503$	−20.3537	−0.907528	−0.453764	−	0.891122i	$-0.649919\pi$
$503$	−20.3537	−0.907528	−0.453764	+	0.891122i	$0.649919\pi$
$504$	0	0
$505$	−33.7265	−1.50081
$506$	0	0
$507$	−10.5317	−0.467728
$508$	0	0
$509$	−34.2529	−1.51823	−0.759117	−	0.650955i	$-0.774369\pi$
$509$	−34.2529	−1.51823	−0.759117	+	0.650955i	$0.774369\pi$
$510$	0	0
$511$	−11.0405	−0.488405
$512$	0	0
$513$	−0.837962	−0.0369969
$514$	0	0
$515$	65.8819	2.90310
$516$	0	0
$517$	0.321234	0.0141279
$518$	0	0
$519$	−14.0483	−0.616652
$520$	0	0
$521$	3.32169	0.145526	0.0727630	−	0.997349i	$-0.476818\pi$
$521$	3.32169	0.145526	0.0727630	+	0.997349i	$0.476818\pi$
$522$	0	0
$523$	1.27077	0.0555668	0.0277834	−	0.999614i	$-0.491155\pi$
$523$	1.27077	0.0555668	0.0277834	+	0.999614i	$0.491155\pi$
$524$	0	0
$525$	6.46471	0.282143
$526$	0	0
$527$	−6.81135	−0.296707
$528$	0	0
$529$	58.7756	2.55546
$530$	0	0
$531$	−6.52315	−0.283081
$532$	0	0
$533$	−31.1413	−1.34888
$534$	0	0
$535$	−22.5781	−0.976135
$536$	0	0
$537$	−21.8801	−0.944194
$538$	0	0
$539$	8.96058	0.385960
$540$	0	0
$541$	−14.4475	−0.621145	−0.310572	−	0.950550i	$-0.600521\pi$
$541$	−14.4475	−0.621145	−0.310572	+	0.950550i	$0.600521\pi$
$542$	0	0
$543$	12.8671	0.552181
$544$	0	0
$545$	−3.55174	−0.152140
$546$	0	0
$547$	−16.8212	−0.719224	−0.359612	−	0.933102i	$-0.617091\pi$
$547$	−16.8212	−0.719224	−0.359612	+	0.933102i	$0.617091\pi$
$548$	0	0
$549$	−11.5657	−0.493614
$550$	0	0
$551$	−8.31005	−0.354020
$552$	0	0
$553$	7.18149	0.305388
$554$	0	0
$555$	22.9192	0.972864
$556$	0	0
$557$	−28.9032	−1.22467	−0.612335	−	0.790599i	$-0.709770\pi$
$557$	−28.9032	−1.22467	−0.612335	+	0.790599i	$0.709770\pi$
$558$	0	0
$559$	21.0162	0.888890
$560$	0	0
$561$	2.77810	0.117291
$562$	0	0
$563$	−16.3407	−0.688678	−0.344339	−	0.938845i	$-0.611897\pi$
$563$	−16.3407	−0.688678	−0.344339	+	0.938845i	$0.611897\pi$
$564$	0	0
$565$	−53.2022	−2.23823
$566$	0	0
$567$	−0.907352	−0.0381052
$568$	0	0
$569$	19.2693	0.807811	0.403906	−	0.914801i	$-0.367652\pi$
$569$	19.2693	0.807811	0.403906	+	0.914801i	$0.367652\pi$
$570$	0	0
$571$	26.7810	1.12075	0.560376	−	0.828238i	$-0.310657\pi$
$571$	26.7810	1.12075	0.560376	+	0.828238i	$0.310657\pi$
$572$	0	0
$573$	−11.6060	−0.484846
$574$	0	0
$575$	−64.4296	−2.68690
$576$	0	0
$577$	−40.9699	−1.70560	−0.852799	−	0.522240i	$-0.825097\pi$
$577$	−40.9699	−1.70560	−0.852799	+	0.522240i	$0.825097\pi$
$578$	0	0
$579$	19.6706	0.817482
$580$	0	0
$581$	−2.73407	−0.113428
$582$	0	0
$583$	15.0569	0.623591
$584$	0	0
$585$	16.8913	0.698371
$586$	0	0
$587$	9.03614	0.372961	0.186481	−	0.982459i	$-0.440292\pi$
$587$	9.03614	0.372961	0.186481	+	0.982459i	$0.440292\pi$
$588$	0	0
$589$	−2.98050	−0.122809
$590$	0	0
$591$	−18.3464	−0.754668
$592$	0	0
$593$	−9.70347	−0.398474	−0.199237	−	0.979951i	$-0.563846\pi$
$593$	−9.70347	−0.398474	−0.199237	+	0.979951i	$0.563846\pi$
$594$	0	0
$595$	−6.05037	−0.248041
$596$	0	0
$597$	−16.8814	−0.690910
$598$	0	0
$599$	−0.165602	−0.00676630	−0.00338315	−	0.999994i	$-0.501077\pi$
$599$	−0.165602	−0.00676630	−0.00338315	+	0.999994i	$0.501077\pi$
$600$	0	0
$601$	−27.5519	−1.12386	−0.561932	−	0.827184i	$-0.689942\pi$
$601$	−27.5519	−1.12386	−0.561932	+	0.827184i	$0.689942\pi$
$602$	0	0
$603$	0.0316033	0.00128699
$604$	0	0
$605$	−30.9746	−1.25930
$606$	0	0
$607$	27.5789	1.11939	0.559696	−	0.828698i	$-0.310918\pi$
$607$	27.5789	1.11939	0.559696	+	0.828698i	$0.310918\pi$
$608$	0	0
$609$	−8.99818	−0.364625
$610$	0	0
$611$	−1.07416	−0.0434559
$612$	0	0
$613$	23.6977	0.957140	0.478570	−	0.878049i	$-0.341155\pi$
$613$	23.6977	0.957140	0.478570	+	0.878049i	$0.341155\pi$
$614$	0	0
$615$	22.3536	0.901386
$616$	0	0
$617$	33.9562	1.36703	0.683513	−	0.729938i	$-0.260451\pi$
$617$	33.9562	1.36703	0.683513	+	0.729938i	$0.260451\pi$
$618$	0	0
$619$	46.5477	1.87091	0.935456	−	0.353443i	$-0.114989\pi$
$619$	46.5477	1.87091	0.935456	+	0.353443i	$0.114989\pi$
$620$	0	0
$621$	9.04299	0.362882
$622$	0	0
$623$	3.38216	0.135503
$624$	0	0
$625$	−9.86111	−0.394444
$626$	0	0
$627$	1.21564	0.0485478
$628$	0	0
$629$	−12.6046	−0.502579
$630$	0	0
$631$	35.1299	1.39850	0.699250	−	0.714878i	$-0.253518\pi$
$631$	35.1299	1.39850	0.699250	+	0.714878i	$0.253518\pi$
$632$	0	0
$633$	6.49552	0.258174
$634$	0	0
$635$	11.3582	0.450737
$636$	0	0
$637$	−29.9629	−1.18717
$638$	0	0
$639$	0.0698850	0.00276461
$640$	0	0
$641$	−19.4672	−0.768910	−0.384455	−	0.923144i	$-0.625611\pi$
$641$	−19.4672	−0.768910	−0.384455	+	0.923144i	$0.625611\pi$
$642$	0	0
$643$	26.1569	1.03153	0.515764	−	0.856731i	$-0.327508\pi$
$643$	26.1569	1.03153	0.515764	+	0.856731i	$0.327508\pi$
$644$	0	0
$645$	−15.0857	−0.593998
$646$	0	0
$647$	−49.6125	−1.95047	−0.975234	−	0.221175i	$-0.929011\pi$
$647$	−49.6125	−1.95047	−0.975234	+	0.221175i	$0.929011\pi$
$648$	0	0
$649$	9.46316	0.371462
$650$	0	0
$651$	−3.22731	−0.126488
$652$	0	0
$653$	−12.5165	−0.489809	−0.244904	−	0.969547i	$-0.578757\pi$
$653$	−12.5165	−0.489809	−0.244904	+	0.969547i	$0.578757\pi$
$654$	0	0
$655$	36.0924	1.41025
$656$	0	0
$657$	12.1679	0.474714
$658$	0	0
$659$	13.1242	0.511247	0.255624	−	0.966776i	$-0.417719\pi$
$659$	13.1242	0.511247	0.255624	+	0.966776i	$0.417719\pi$
$660$	0	0
$661$	33.3238	1.29615	0.648073	−	0.761579i	$-0.275575\pi$
$661$	33.3238	1.29615	0.648073	+	0.761579i	$0.275575\pi$
$662$	0	0
$663$	−9.28956	−0.360777
$664$	0	0
$665$	−2.64751	−0.102666
$666$	0	0
$667$	89.6790	3.47239
$668$	0	0
$669$	28.2895	1.09374
$670$	0	0
$671$	16.7785	0.647725
$672$	0	0
$673$	−38.3445	−1.47807	−0.739035	−	0.673667i	$-0.764718\pi$
$673$	−38.3445	−1.47807	−0.739035	+	0.673667i	$0.764718\pi$
$674$	0	0
$675$	−7.12481	−0.274234
$676$	0	0
$677$	7.17915	0.275917	0.137959	−	0.990438i	$-0.455946\pi$
$677$	7.17915	0.275917	0.137959	+	0.990438i	$0.455946\pi$
$678$	0	0
$679$	−10.5001	−0.402958
$680$	0	0
$681$	20.4996	0.785546
$682$	0	0
$683$	−7.81897	−0.299184	−0.149592	−	0.988748i	$-0.547796\pi$
$683$	−7.81897	−0.299184	−0.149592	+	0.988748i	$0.547796\pi$
$684$	0	0
$685$	33.5966	1.28366
$686$	0	0
$687$	−2.94356	−0.112304
$688$	0	0
$689$	−50.3480	−1.91811
$690$	0	0
$691$	−7.14698	−0.271884	−0.135942	−	0.990717i	$-0.543406\pi$
$691$	−7.14698	−0.271884	−0.135942	+	0.990717i	$0.543406\pi$
$692$	0	0
$693$	1.31630	0.0500021
$694$	0	0
$695$	−49.9487	−1.89466
$696$	0	0
$697$	−12.2936	−0.465654
$698$	0	0
$699$	0.618074	0.0233777
$700$	0	0
$701$	−4.33917	−0.163888	−0.0819442	−	0.996637i	$-0.526113\pi$
$701$	−4.33917	−0.163888	−0.0819442	+	0.996637i	$0.526113\pi$
$702$	0	0
$703$	−5.51551	−0.208021
$704$	0	0
$705$	0.771047	0.0290393
$706$	0	0
$707$	8.78839	0.330521
$708$	0	0
$709$	45.7083	1.71661	0.858306	−	0.513139i	$-0.171517\pi$
$709$	45.7083	1.71661	0.858306	+	0.513139i	$0.171517\pi$
$710$	0	0
$711$	−7.91478	−0.296827
$712$	0	0
$713$	32.1645	1.20457
$714$	0	0
$715$	−24.5043	−0.916410
$716$	0	0
$717$	−0.466443	−0.0174196
$718$	0	0
$719$	−32.7134	−1.22000	−0.610002	−	0.792400i	$-0.708831\pi$
$719$	−32.7134	−1.22000	−0.610002	+	0.792400i	$0.708831\pi$
$720$	0	0
$721$	−17.1674	−0.639347
$722$	0	0
$723$	−1.75734	−0.0653563
$724$	0	0
$725$	−70.6566	−2.62412
$726$	0	0
$727$	−40.2892	−1.49424	−0.747122	−	0.664687i	$-0.768565\pi$
$727$	−40.2892	−1.49424	−0.747122	+	0.664687i	$0.768565\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.29653	0.306858
$732$	0	0
$733$	40.5716	1.49855	0.749274	−	0.662260i	$-0.230403\pi$
$733$	40.5716	1.49855	0.749274	+	0.662260i	$0.230403\pi$
$734$	0	0
$735$	21.5077	0.793325
$736$	0	0
$737$	−0.0458470	−0.00168880
$738$	0	0
$739$	−22.1341	−0.814217	−0.407109	−	0.913380i	$-0.633463\pi$
$739$	−22.1341	−0.814217	−0.407109	+	0.913380i	$0.633463\pi$
$740$	0	0
$741$	−4.06491	−0.149328
$742$	0	0
$743$	38.7052	1.41996	0.709979	−	0.704223i	$-0.248705\pi$
$743$	38.7052	1.41996	0.709979	+	0.704223i	$0.248705\pi$
$744$	0	0
$745$	28.1773	1.03234
$746$	0	0
$747$	3.01324	0.110249
$748$	0	0
$749$	5.88335	0.214973
$750$	0	0
$751$	−11.8480	−0.432339	−0.216170	−	0.976356i	$-0.569356\pi$
$751$	−11.8480	−0.432339	−0.216170	+	0.976356i	$0.569356\pi$
$752$	0	0
$753$	−10.4734	−0.381672
$754$	0	0
$755$	−3.48207	−0.126726
$756$	0	0
$757$	−41.6762	−1.51475	−0.757374	−	0.652982i	$-0.773518\pi$
$757$	−41.6762	−1.51475	−0.757374	+	0.652982i	$0.773518\pi$
$758$	0	0
$759$	−13.1187	−0.476178
$760$	0	0
$761$	−24.0125	−0.870454	−0.435227	−	0.900321i	$-0.643332\pi$
$761$	−24.0125	−0.870454	−0.435227	+	0.900321i	$0.643332\pi$
$762$	0	0
$763$	0.925506	0.0335056
$764$	0	0
$765$	6.66816	0.241088
$766$	0	0
$767$	−31.6435	−1.14258
$768$	0	0
$769$	−46.6751	−1.68315	−0.841573	−	0.540144i	$-0.818370\pi$
$769$	−46.6751	−1.68315	−0.841573	+	0.540144i	$0.818370\pi$
$770$	0	0
$771$	−24.4255	−0.879661
$772$	0	0
$773$	−40.7399	−1.46531	−0.732656	−	0.680599i	$-0.761720\pi$
$773$	−40.7399	−1.46531	−0.732656	+	0.680599i	$0.761720\pi$
$774$	0	0
$775$	−25.3418	−0.910305
$776$	0	0
$777$	−5.97224	−0.214253
$778$	0	0
$779$	−5.37942	−0.192738
$780$	0	0
$781$	−0.101382	−0.00362775
$782$	0	0
$783$	9.91697	0.354404
$784$	0	0
$785$	15.1476	0.540642
$786$	0	0
$787$	−22.9231	−0.817121	−0.408560	−	0.912731i	$-0.633969\pi$
$787$	−22.9231	−0.817121	−0.408560	+	0.912731i	$0.633969\pi$
$788$	0	0
$789$	−7.24585	−0.257959
$790$	0	0
$791$	13.8633	0.492923
$792$	0	0
$793$	−56.1048	−1.99234
$794$	0	0
$795$	36.1404	1.28177
$796$	0	0
$797$	−13.5724	−0.480759	−0.240379	−	0.970679i	$-0.577272\pi$
$797$	−13.5724	−0.480759	−0.240379	+	0.970679i	$0.577272\pi$
$798$	0	0
$799$	−0.424045	−0.0150016
$800$	0	0
$801$	−3.72751	−0.131705
$802$	0	0
$803$	−17.6520	−0.622925
$804$	0	0
$805$	28.5710	1.00699
$806$	0	0
$807$	16.4898	0.580469
$808$	0	0
$809$	10.1812	0.357953	0.178977	−	0.983853i	$-0.442721\pi$
$809$	10.1812	0.357953	0.178977	+	0.983853i	$0.442721\pi$
$810$	0	0
$811$	−30.1141	−1.05745	−0.528724	−	0.848794i	$-0.677329\pi$
$811$	−30.1141	−1.05745	−0.528724	+	0.848794i	$0.677329\pi$
$812$	0	0
$813$	4.37387	0.153398
$814$	0	0
$815$	17.4585	0.611544
$816$	0	0
$817$	3.63038	0.127011
$818$	0	0
$819$	−4.40152	−0.153801
$820$	0	0
$821$	−35.9778	−1.25563	−0.627817	−	0.778361i	$-0.716051\pi$
$821$	−35.9778	−1.25563	−0.627817	+	0.778361i	$0.716051\pi$
$822$	0	0
$823$	−39.8830	−1.39023	−0.695117	−	0.718897i	$-0.744647\pi$
$823$	−39.8830	−1.39023	−0.695117	+	0.718897i	$0.744647\pi$
$824$	0	0
$825$	10.3360	0.359853
$826$	0	0
$827$	43.3079	1.50596	0.752981	−	0.658042i	$-0.228615\pi$
$827$	43.3079	1.50596	0.752981	+	0.658042i	$0.228615\pi$
$828$	0	0
$829$	−39.7456	−1.38042	−0.690211	−	0.723608i	$-0.742482\pi$
$829$	−39.7456	−1.38042	−0.690211	+	0.723608i	$0.742482\pi$
$830$	0	0
$831$	−15.2825	−0.530144
$832$	0	0
$833$	−11.8284	−0.409830
$834$	0	0
$835$	−80.3950	−2.78218
$836$	0	0
$837$	3.55684	0.122942
$838$	0	0
$839$	−37.0829	−1.28024	−0.640122	−	0.768273i	$-0.721116\pi$
$839$	−37.0829	−1.28024	−0.640122	+	0.768273i	$0.721116\pi$
$840$	0	0
$841$	69.3463	2.39125
$842$	0	0
$843$	−29.6109	−1.01985
$844$	0	0
$845$	36.6721	1.26156
$846$	0	0
$847$	8.07131	0.277334
$848$	0	0
$849$	−13.7350	−0.471385
$850$	0	0
$851$	59.5214	2.04037
$852$	0	0
$853$	10.8405	0.371173	0.185586	−	0.982628i	$-0.440582\pi$
$853$	10.8405	0.371173	0.185586	+	0.982628i	$0.440582\pi$
$854$	0	0
$855$	2.91784	0.0997882
$856$	0	0
$857$	−38.9863	−1.33175	−0.665874	−	0.746064i	$-0.731941\pi$
$857$	−38.9863	−1.33175	−0.665874	+	0.746064i	$0.731941\pi$
$858$	0	0
$859$	−1.80885	−0.0617172	−0.0308586	−	0.999524i	$-0.509824\pi$
$859$	−1.80885	−0.0617172	−0.0308586	+	0.999524i	$0.509824\pi$
$860$	0	0
$861$	−5.82487	−0.198511
$862$	0	0
$863$	−11.9277	−0.406022	−0.203011	−	0.979176i	$-0.565073\pi$
$863$	−11.9277	−0.406022	−0.203011	+	0.979176i	$0.565073\pi$
$864$	0	0
$865$	48.9172	1.66323
$866$	0	0
$867$	13.3328	0.452805
$868$	0	0
$869$	11.4820	0.389500
$870$	0	0
$871$	0.153306	0.00519457
$872$	0	0
$873$	11.5723	0.391662
$874$	0	0
$875$	−6.71327	−0.226950
$876$	0	0
$877$	−7.30034	−0.246515	−0.123257	−	0.992375i	$-0.539334\pi$
$877$	−7.30034	−0.246515	−0.123257	+	0.992375i	$0.539334\pi$
$878$	0	0
$879$	−18.2736	−0.616353
$880$	0	0
$881$	−19.4415	−0.655000	−0.327500	−	0.944851i	$-0.606206\pi$
$881$	−19.4415	−0.655000	−0.327500	+	0.944851i	$0.606206\pi$
$882$	0	0
$883$	55.0025	1.85098	0.925490	−	0.378771i	$-0.123653\pi$
$883$	55.0025	1.85098	0.925490	+	0.378771i	$0.123653\pi$
$884$	0	0
$885$	22.7141	0.763525
$886$	0	0
$887$	43.3996	1.45722	0.728608	−	0.684931i	$-0.240168\pi$
$887$	43.3996	1.45722	0.728608	+	0.684931i	$0.240168\pi$
$888$	0	0
$889$	−2.95971	−0.0992653
$890$	0	0
$891$	−1.45070	−0.0486004
$892$	0	0
$893$	−0.185553	−0.00620929
$894$	0	0
$895$	76.1879	2.54668
$896$	0	0
$897$	43.8670	1.46468
$898$	0	0
$899$	35.2731	1.17642
$900$	0	0
$901$	−19.8758	−0.662158
$902$	0	0
$903$	3.93100	0.130816
$904$	0	0
$905$	−44.8042	−1.48934
$906$	0	0
$907$	40.8621	1.35680	0.678401	−	0.734691i	$-0.262673\pi$
$907$	40.8621	1.35680	0.678401	+	0.734691i	$0.262673\pi$
$908$	0	0
$909$	−9.68576	−0.321256
$910$	0	0
$911$	−4.26197	−0.141206	−0.0706028	−	0.997505i	$-0.522492\pi$
$911$	−4.26197	−0.141206	−0.0706028	+	0.997505i	$0.522492\pi$
$912$	0	0
$913$	−4.37131	−0.144669
$914$	0	0
$915$	40.2727	1.33137
$916$	0	0
$917$	−9.40489	−0.310577
$918$	0	0
$919$	−9.78271	−0.322702	−0.161351	−	0.986897i	$-0.551585\pi$
$919$	−9.78271	−0.322702	−0.161351	+	0.986897i	$0.551585\pi$
$920$	0	0
$921$	7.22044	0.237922
$922$	0	0
$923$	0.339008	0.0111586
$924$	0	0
$925$	−46.8959	−1.54193
$926$	0	0
$927$	18.9203	0.621425
$928$	0	0
$929$	−48.2818	−1.58407	−0.792037	−	0.610473i	$-0.790979\pi$
$929$	−48.2818	−1.58407	−0.792037	+	0.610473i	$0.790979\pi$
$930$	0	0
$931$	−5.17585	−0.169632
$932$	0	0
$933$	−18.9913	−0.621748
$934$	0	0
$935$	−9.67353	−0.316358
$936$	0	0
$937$	7.61741	0.248850	0.124425	−	0.992229i	$-0.460291\pi$
$937$	7.61741	0.248850	0.124425	+	0.992229i	$0.460291\pi$
$938$	0	0
$939$	−2.22051	−0.0724637
$940$	0	0
$941$	58.9593	1.92202	0.961009	−	0.276517i	$-0.0891803\pi$
$941$	58.9593	1.92202	0.961009	+	0.276517i	$0.0891803\pi$
$942$	0	0
$943$	58.0527	1.89046
$944$	0	0
$945$	3.15946	0.102777
$946$	0	0
$947$	−14.3941	−0.467746	−0.233873	−	0.972267i	$-0.575140\pi$
$947$	−14.3941	−0.467746	−0.233873	+	0.972267i	$0.575140\pi$
$948$	0	0
$949$	59.0257	1.91606
$950$	0	0
$951$	26.4283	0.856996
$952$	0	0
$953$	27.4561	0.889389	0.444694	−	0.895682i	$-0.353312\pi$
$953$	27.4561	0.889389	0.444694	+	0.895682i	$0.353312\pi$
$954$	0	0
$955$	40.4128	1.30773
$956$	0	0
$957$	−14.3866	−0.465052
$958$	0	0
$959$	−8.75453	−0.282699
$960$	0	0
$961$	−18.3489	−0.591899
$962$	0	0
$963$	−6.48409	−0.208947
$964$	0	0
$965$	−68.4944	−2.20491
$966$	0	0
$967$	−1.95419	−0.0628425	−0.0314213	−	0.999506i	$-0.510003\pi$
$967$	−1.95419	−0.0628425	−0.0314213	+	0.999506i	$0.510003\pi$
$968$	0	0
$969$	−1.60470	−0.0515503
$970$	0	0
$971$	−37.9035	−1.21638	−0.608191	−	0.793791i	$-0.708104\pi$
$971$	−37.9035	−1.21638	−0.608191	+	0.793791i	$0.708104\pi$
$972$	0	0
$973$	13.0155	0.417259
$974$	0	0
$975$	−34.5621	−1.10687
$976$	0	0
$977$	6.19521	0.198202	0.0991011	−	0.995077i	$-0.468403\pi$
$977$	6.19521	0.198202	0.0991011	+	0.995077i	$0.468403\pi$
$978$	0	0
$979$	5.40751	0.172825
$980$	0	0
$981$	−1.02001	−0.0325663
$982$	0	0
$983$	14.8038	0.472168	0.236084	−	0.971733i	$-0.424136\pi$
$983$	14.8038	0.472168	0.236084	+	0.971733i	$0.424136\pi$
$984$	0	0
$985$	63.8833	2.03549
$986$	0	0
$987$	−0.200918	−0.00639529
$988$	0	0
$989$	−39.1777	−1.24578
$990$	0	0
$991$	0.893732	0.0283903	0.0141952	−	0.999899i	$-0.495481\pi$
$991$	0.893732	0.0283903	0.0141952	+	0.999899i	$0.495481\pi$
$992$	0	0
$993$	−33.7212	−1.07011
$994$	0	0
$995$	58.7823	1.86352
$996$	0	0
$997$	6.02433	0.190792	0.0953962	−	0.995439i	$-0.469588\pi$
$997$	6.02433	0.190792	0.0953962	+	0.995439i	$0.469588\pi$
$998$	0	0
$999$	6.58205	0.208247

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7248.2.a.bm.1.10		10
4.3	odd	2		3624.2.a.l.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3624.2.a.l.1.10	✓	10	4.3	odd	2
7248.2.a.bm.1.10		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.48207 q^{5} -0.907352 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.48207 q^{5} -0.907352 q^{7} +1.00000 q^{9} -1.45070 q^{11} +4.85095 q^{13} -3.48207 q^{15} +1.91500 q^{17} +0.837962 q^{19} +0.907352 q^{21} -9.04299 q^{23} +7.12481 q^{25} -1.00000 q^{27} -9.91697 q^{29} -3.55684 q^{31} +1.45070 q^{33} -3.15946 q^{35} -6.58205 q^{37} -4.85095 q^{39} -6.41964 q^{41} +4.33239 q^{43} +3.48207 q^{45} -0.221433 q^{47} -6.17671 q^{49} -1.91500 q^{51} -10.3790 q^{53} -5.05145 q^{55} -0.837962 q^{57} -6.52315 q^{59} -11.5657 q^{61} -0.907352 q^{63} +16.8913 q^{65} +0.0316033 q^{67} +9.04299 q^{69} +0.0698850 q^{71} +12.1679 q^{73} -7.12481 q^{75} +1.31630 q^{77} -7.91478 q^{79} +1.00000 q^{81} +3.01324 q^{83} +6.66816 q^{85} +9.91697 q^{87} -3.72751 q^{89} -4.40152 q^{91} +3.55684 q^{93} +2.91784 q^{95} +11.5723 q^{97} -1.45070 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 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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