LMFDB - Embedded newform 7448.2.a.bv.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 27 x^{12} + 46 x^{11} + 286 x^{10} - 386 x^{9} - 1525 x^{8} + 1414 x^{7} + \cdots + 392 $ x^14 - 2x^13 - 27x^12 + 46x^11 + 286x^10 - 386x^9 - 1525x^8 + 1414x^7 + 4390x^6 - 1910x^5 - 6603x^4 - 610x^3 + 4031x^2 + 2352*x + 392
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-1.89634$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.89634 q^{3} -2.28827 q^{5} +0.596117 q^{9} +O(q^{10})$	$q+1.89634 q^{3} -2.28827 q^{5} +0.596117 q^{9} -2.00048 q^{11} -0.811145 q^{13} -4.33935 q^{15} +6.47139 q^{17} +1.00000 q^{19} -6.43798 q^{23} +0.236188 q^{25} -4.55859 q^{27} +3.19681 q^{29} +10.4468 q^{31} -3.79361 q^{33} -5.90974 q^{37} -1.53821 q^{39} -5.42367 q^{41} +9.43194 q^{43} -1.36408 q^{45} +11.9368 q^{47} +12.2720 q^{51} +6.30580 q^{53} +4.57765 q^{55} +1.89634 q^{57} -12.3616 q^{59} -3.96425 q^{61} +1.85612 q^{65} -4.00524 q^{67} -12.2086 q^{69} -2.72658 q^{71} -12.7797 q^{73} +0.447893 q^{75} -7.60892 q^{79} -10.4330 q^{81} -10.8514 q^{83} -14.8083 q^{85} +6.06225 q^{87} -10.9161 q^{89} +19.8107 q^{93} -2.28827 q^{95} -18.5203 q^{97} -1.19252 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{3} - 2 q^{5} + 16 q^{9}+O(q^{10}) $ 14 * q - 2 * q^3 - 2 * q^5 + 16 * q^9	$ 14 q - 2 q^{3} - 2 q^{5} + 16 q^{9} - 6 q^{11} - 16 q^{13} - 4 q^{15} - 20 q^{17} + 14 q^{19} + 4 q^{23} + 16 q^{25} - 20 q^{27} + 6 q^{29} - 34 q^{33} - 6 q^{37} + 8 q^{39} - 46 q^{41} - 18 q^{43} + 10 q^{47} - 4 q^{51} - 2 q^{53} - 28 q^{55} - 2 q^{57} + 22 q^{59} - 26 q^{61} + 8 q^{65} - 12 q^{67} - 48 q^{69} + 18 q^{71} - 28 q^{73} + 24 q^{75} - 10 q^{79} - 2 q^{81} - 8 q^{83} - 16 q^{85} - 16 q^{87} - 78 q^{89} - 2 q^{95} - 54 q^{97} + 26 q^{99}+O(q^{100}) $ 14 * q - 2 * q^3 - 2 * q^5 + 16 * q^9 - 6 * q^11 - 16 * q^13 - 4 * q^15 - 20 * q^17 + 14 * q^19 + 4 * q^23 + 16 * q^25 - 20 * q^27 + 6 * q^29 - 34 * q^33 - 6 * q^37 + 8 * q^39 - 46 * q^41 - 18 * q^43 + 10 * q^47 - 4 * q^51 - 2 * q^53 - 28 * q^55 - 2 * q^57 + 22 * q^59 - 26 * q^61 + 8 * q^65 - 12 * q^67 - 48 * q^69 + 18 * q^71 - 28 * q^73 + 24 * q^75 - 10 * q^79 - 2 * q^81 - 8 * q^83 - 16 * q^85 - 16 * q^87 - 78 * q^89 - 2 * q^95 - 54 * q^97 + 26 * 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
$3$	1.89634	1.09485	0.547427	−	0.836853i	$-0.315607\pi$
$3$	1.89634	1.09485	0.547427	+	0.836853i	$0.315607\pi$
$4$	0	0
$5$	−2.28827	−1.02335	−0.511673	−	0.859180i	$-0.670974\pi$
$5$	−2.28827	−1.02335	−0.511673	+	0.859180i	$0.670974\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.596117	0.198706
$10$	0	0
$11$	−2.00048	−0.603169	−0.301584	−	0.953439i	$-0.597516\pi$
$11$	−2.00048	−0.603169	−0.301584	+	0.953439i	$0.597516\pi$
$12$	0	0
$13$	−0.811145	−0.224971	−0.112486	−	0.993653i	$-0.535881\pi$
$13$	−0.811145	−0.224971	−0.112486	+	0.993653i	$0.535881\pi$
$14$	0	0
$15$	−4.33935	−1.12041
$16$	0	0
$17$	6.47139	1.56954	0.784771	−	0.619785i	$-0.212780\pi$
$17$	6.47139	1.56954	0.784771	+	0.619785i	$0.212780\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.43798	−1.34241	−0.671206	−	0.741271i	$-0.734223\pi$
$23$	−6.43798	−1.34241	−0.671206	+	0.741271i	$0.734223\pi$
$24$	0	0
$25$	0.236188	0.0472376
$26$	0	0
$27$	−4.55859	−0.877300
$28$	0	0
$29$	3.19681	0.593633	0.296816	−	0.954935i	$-0.404075\pi$
$29$	3.19681	0.593633	0.296816	+	0.954935i	$0.404075\pi$
$30$	0	0
$31$	10.4468	1.87630	0.938150	−	0.346228i	$-0.112538\pi$
$31$	10.4468	1.87630	0.938150	+	0.346228i	$0.112538\pi$
$32$	0	0
$33$	−3.79361	−0.660382
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.90974	−0.971556	−0.485778	−	0.874082i	$-0.661464\pi$
$37$	−5.90974	−0.971556	−0.485778	+	0.874082i	$0.661464\pi$
$38$	0	0
$39$	−1.53821	−0.246310
$40$	0	0
$41$	−5.42367	−0.847035	−0.423517	−	0.905888i	$-0.639205\pi$
$41$	−5.42367	−0.847035	−0.423517	+	0.905888i	$0.639205\pi$
$42$	0	0
$43$	9.43194	1.43836	0.719179	−	0.694825i	$-0.244518\pi$
$43$	9.43194	1.43836	0.719179	+	0.694825i	$0.244518\pi$
$44$	0	0
$45$	−1.36408	−0.203345
$46$	0	0
$47$	11.9368	1.74117	0.870584	−	0.492020i	$-0.163741\pi$
$47$	11.9368	1.74117	0.870584	+	0.492020i	$0.163741\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	12.2720	1.71842
$52$	0	0
$53$	6.30580	0.866168	0.433084	−	0.901354i	$-0.357425\pi$
$53$	6.30580	0.866168	0.433084	+	0.901354i	$0.357425\pi$
$54$	0	0
$55$	4.57765	0.617251
$56$	0	0
$57$	1.89634	0.251177
$58$	0	0
$59$	−12.3616	−1.60934	−0.804670	−	0.593723i	$-0.797658\pi$
$59$	−12.3616	−1.60934	−0.804670	+	0.593723i	$0.797658\pi$
$60$	0	0
$61$	−3.96425	−0.507570	−0.253785	−	0.967261i	$-0.581676\pi$
$61$	−3.96425	−0.507570	−0.253785	+	0.967261i	$0.581676\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.85612	0.230223
$66$	0	0
$67$	−4.00524	−0.489318	−0.244659	−	0.969609i	$-0.578676\pi$
$67$	−4.00524	−0.489318	−0.244659	+	0.969609i	$0.578676\pi$
$68$	0	0
$69$	−12.2086	−1.46975
$70$	0	0
$71$	−2.72658	−0.323585	−0.161793	−	0.986825i	$-0.551728\pi$
$71$	−2.72658	−0.323585	−0.161793	+	0.986825i	$0.551728\pi$
$72$	0	0
$73$	−12.7797	−1.49575	−0.747875	−	0.663839i	$-0.768926\pi$
$73$	−12.7797	−1.49575	−0.747875	+	0.663839i	$0.768926\pi$
$74$	0	0
$75$	0.447893	0.0517183
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.60892	−0.856070	−0.428035	−	0.903762i	$-0.640794\pi$
$79$	−7.60892	−0.856070	−0.428035	+	0.903762i	$0.640794\pi$
$80$	0	0
$81$	−10.4330	−1.15922
$82$	0	0
$83$	−10.8514	−1.19110	−0.595550	−	0.803318i	$-0.703066\pi$
$83$	−10.8514	−1.19110	−0.595550	+	0.803318i	$0.703066\pi$
$84$	0	0
$85$	−14.8083	−1.60619
$86$	0	0
$87$	6.06225	0.649941
$88$	0	0
$89$	−10.9161	−1.15710	−0.578551	−	0.815646i	$-0.696382\pi$
$89$	−10.9161	−1.15710	−0.578551	+	0.815646i	$0.696382\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	19.8107	2.05428
$94$	0	0
$95$	−2.28827	−0.234772
$96$	0	0
$97$	−18.5203	−1.88045	−0.940224	−	0.340557i	$-0.889384\pi$
$97$	−18.5203	−1.88045	−0.940224	+	0.340557i	$0.889384\pi$
$98$	0	0
$99$	−1.19252	−0.119853
$100$	0	0
$101$	11.5866	1.15291	0.576454	−	0.817130i	$-0.304436\pi$
$101$	11.5866	1.15291	0.576454	+	0.817130i	$0.304436\pi$
$102$	0	0
$103$	−15.1626	−1.49401	−0.747006	−	0.664817i	$-0.768510\pi$
$103$	−15.1626	−1.49401	−0.747006	+	0.664817i	$0.768510\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.59151	−0.250531	−0.125266	−	0.992123i	$-0.539978\pi$
$107$	−2.59151	−0.250531	−0.125266	+	0.992123i	$0.539978\pi$
$108$	0	0
$109$	9.79001	0.937713	0.468857	−	0.883274i	$-0.344666\pi$
$109$	9.79001	0.937713	0.468857	+	0.883274i	$0.344666\pi$
$110$	0	0
$111$	−11.2069	−1.06371
$112$	0	0
$113$	5.65934	0.532386	0.266193	−	0.963920i	$-0.414234\pi$
$113$	5.65934	0.532386	0.266193	+	0.963920i	$0.414234\pi$
$114$	0	0
$115$	14.7319	1.37375
$116$	0	0
$117$	−0.483537	−0.0447030
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−6.99806	−0.636187
$122$	0	0
$123$	−10.2851	−0.927380
$124$	0	0
$125$	10.9009	0.975006
$126$	0	0
$127$	−17.9822	−1.59566	−0.797829	−	0.602884i	$-0.794018\pi$
$127$	−17.9822	−1.59566	−0.797829	+	0.602884i	$0.794018\pi$
$128$	0	0
$129$	17.8862	1.57479
$130$	0	0
$131$	−6.68344	−0.583935	−0.291968	−	0.956428i	$-0.594310\pi$
$131$	−6.68344	−0.583935	−0.291968	+	0.956428i	$0.594310\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	10.4313	0.897782
$136$	0	0
$137$	8.22748	0.702921	0.351461	−	0.936203i	$-0.385685\pi$
$137$	8.22748	0.702921	0.351461	+	0.936203i	$0.385685\pi$
$138$	0	0
$139$	−12.7837	−1.08430	−0.542151	−	0.840281i	$-0.682390\pi$
$139$	−12.7837	−1.08430	−0.542151	+	0.840281i	$0.682390\pi$
$140$	0	0
$141$	22.6363	1.90632
$142$	0	0
$143$	1.62268	0.135696
$144$	0	0
$145$	−7.31517	−0.607492
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.43078	0.690676	0.345338	−	0.938478i	$-0.387764\pi$
$149$	8.43078	0.690676	0.345338	+	0.938478i	$0.387764\pi$
$150$	0	0
$151$	−5.73286	−0.466534	−0.233267	−	0.972413i	$-0.574942\pi$
$151$	−5.73286	−0.466534	−0.233267	+	0.972413i	$0.574942\pi$
$152$	0	0
$153$	3.85771	0.311877
$154$	0	0
$155$	−23.9051	−1.92011
$156$	0	0
$157$	−5.87304	−0.468720	−0.234360	−	0.972150i	$-0.575299\pi$
$157$	−5.87304	−0.468720	−0.234360	+	0.972150i	$0.575299\pi$
$158$	0	0
$159$	11.9580	0.948327
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.5959	0.986591	0.493295	−	0.869862i	$-0.335792\pi$
$163$	12.5959	0.986591	0.493295	+	0.869862i	$0.335792\pi$
$164$	0	0
$165$	8.68080	0.675799
$166$	0	0
$167$	4.48662	0.347185	0.173593	−	0.984818i	$-0.444462\pi$
$167$	4.48662	0.347185	0.173593	+	0.984818i	$0.444462\pi$
$168$	0	0
$169$	−12.3420	−0.949388
$170$	0	0
$171$	0.596117	0.0455862
$172$	0	0
$173$	−2.52653	−0.192089	−0.0960443	−	0.995377i	$-0.530619\pi$
$173$	−2.52653	−0.192089	−0.0960443	+	0.995377i	$0.530619\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−23.4418	−1.76199
$178$	0	0
$179$	−16.8831	−1.26190	−0.630950	−	0.775823i	$-0.717335\pi$
$179$	−16.8831	−1.26190	−0.630950	+	0.775823i	$0.717335\pi$
$180$	0	0
$181$	−20.2364	−1.50416	−0.752081	−	0.659071i	$-0.770950\pi$
$181$	−20.2364	−1.50416	−0.752081	+	0.659071i	$0.770950\pi$
$182$	0	0
$183$	−7.51757	−0.555715
$184$	0	0
$185$	13.5231	0.994238
$186$	0	0
$187$	−12.9459	−0.946699
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.70197	0.123150	0.0615751	−	0.998102i	$-0.480388\pi$
$191$	1.70197	0.123150	0.0615751	+	0.998102i	$0.480388\pi$
$192$	0	0
$193$	7.58611	0.546060	0.273030	−	0.962006i	$-0.411974\pi$
$193$	7.58611	0.546060	0.273030	+	0.962006i	$0.411974\pi$
$194$	0	0
$195$	3.51984	0.252061
$196$	0	0
$197$	−3.47298	−0.247439	−0.123720	−	0.992317i	$-0.539482\pi$
$197$	−3.47298	−0.247439	−0.123720	+	0.992317i	$0.539482\pi$
$198$	0	0
$199$	−24.8626	−1.76246	−0.881231	−	0.472685i	$-0.843285\pi$
$199$	−24.8626	−1.76246	−0.881231	+	0.472685i	$0.843285\pi$
$200$	0	0
$201$	−7.59532	−0.535732
$202$	0	0
$203$	0	0
$204$	0	0
$205$	12.4108	0.866810
$206$	0	0
$207$	−3.83779	−0.266745
$208$	0	0
$209$	−2.00048	−0.138376
$210$	0	0
$211$	−7.22567	−0.497435	−0.248718	−	0.968576i	$-0.580009\pi$
$211$	−7.22567	−0.497435	−0.248718	+	0.968576i	$0.580009\pi$
$212$	0	0
$213$	−5.17053	−0.354279
$214$	0	0
$215$	−21.5828	−1.47194
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−24.2347	−1.63763
$220$	0	0
$221$	−5.24923	−0.353102
$222$	0	0
$223$	9.92832	0.664849	0.332425	−	0.943130i	$-0.392133\pi$
$223$	9.92832	0.664849	0.332425	+	0.943130i	$0.392133\pi$
$224$	0	0
$225$	0.140796	0.00938638
$226$	0	0
$227$	−4.28272	−0.284254	−0.142127	−	0.989848i	$-0.545394\pi$
$227$	−4.28272	−0.284254	−0.142127	+	0.989848i	$0.545394\pi$
$228$	0	0
$229$	17.0098	1.12404	0.562021	−	0.827123i	$-0.310024\pi$
$229$	17.0098	1.12404	0.562021	+	0.827123i	$0.310024\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.60881	0.498470	0.249235	−	0.968443i	$-0.419821\pi$
$233$	7.60881	0.498470	0.249235	+	0.968443i	$0.419821\pi$
$234$	0	0
$235$	−27.3147	−1.78182
$236$	0	0
$237$	−14.4291	−0.937272
$238$	0	0
$239$	4.66914	0.302022	0.151011	−	0.988532i	$-0.451747\pi$
$239$	4.66914	0.302022	0.151011	+	0.988532i	$0.451747\pi$
$240$	0	0
$241$	−0.651496	−0.0419665	−0.0209833	−	0.999780i	$-0.506680\pi$
$241$	−0.651496	−0.0419665	−0.0209833	+	0.999780i	$0.506680\pi$
$242$	0	0
$243$	−6.10878	−0.391878
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.811145	−0.0516119
$248$	0	0
$249$	−20.5780	−1.30408
$250$	0	0
$251$	20.1582	1.27238	0.636188	−	0.771534i	$-0.280510\pi$
$251$	20.1582	1.27238	0.636188	+	0.771534i	$0.280510\pi$
$252$	0	0
$253$	12.8791	0.809701
$254$	0	0
$255$	−28.0816	−1.75854
$256$	0	0
$257$	−23.6524	−1.47540	−0.737698	−	0.675131i	$-0.764087\pi$
$257$	−23.6524	−1.47540	−0.737698	+	0.675131i	$0.764087\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.90567	0.117958
$262$	0	0
$263$	−6.98779	−0.430886	−0.215443	−	0.976516i	$-0.569119\pi$
$263$	−6.98779	−0.430886	−0.215443	+	0.976516i	$0.569119\pi$
$264$	0	0
$265$	−14.4294	−0.886389
$266$	0	0
$267$	−20.7006	−1.26686
$268$	0	0
$269$	20.3785	1.24250	0.621250	−	0.783612i	$-0.286625\pi$
$269$	20.3785	1.24250	0.621250	+	0.783612i	$0.286625\pi$
$270$	0	0
$271$	16.8380	1.02284	0.511418	−	0.859332i	$-0.329120\pi$
$271$	16.8380	1.02284	0.511418	+	0.859332i	$0.329120\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.472490	−0.0284922
$276$	0	0
$277$	26.3196	1.58139	0.790697	−	0.612208i	$-0.209718\pi$
$277$	26.3196	1.58139	0.790697	+	0.612208i	$0.209718\pi$
$278$	0	0
$279$	6.22751	0.372831
$280$	0	0
$281$	−12.2158	−0.728733	−0.364366	−	0.931256i	$-0.618714\pi$
$281$	−12.2158	−0.728733	−0.364366	+	0.931256i	$0.618714\pi$
$282$	0	0
$283$	6.22108	0.369805	0.184902	−	0.982757i	$-0.440803\pi$
$283$	6.22108	0.369805	0.184902	+	0.982757i	$0.440803\pi$
$284$	0	0
$285$	−4.33935	−0.257041
$286$	0	0
$287$	0	0
$288$	0	0
$289$	24.8789	1.46346
$290$	0	0
$291$	−35.1208	−2.05882
$292$	0	0
$293$	3.53386	0.206450	0.103225	−	0.994658i	$-0.467084\pi$
$293$	3.53386	0.206450	0.103225	+	0.994658i	$0.467084\pi$
$294$	0	0
$295$	28.2866	1.64691
$296$	0	0
$297$	9.11938	0.529160
$298$	0	0
$299$	5.22213	0.302004
$300$	0	0
$301$	0	0
$302$	0	0
$303$	21.9721	1.26227
$304$	0	0
$305$	9.07127	0.519419
$306$	0	0
$307$	16.8753	0.963125	0.481563	−	0.876412i	$-0.340069\pi$
$307$	16.8753	0.963125	0.481563	+	0.876412i	$0.340069\pi$
$308$	0	0
$309$	−28.7534	−1.63573
$310$	0	0
$311$	10.7816	0.611369	0.305685	−	0.952133i	$-0.401115\pi$
$311$	10.7816	0.611369	0.305685	+	0.952133i	$0.401115\pi$
$312$	0	0
$313$	−4.66747	−0.263821	−0.131910	−	0.991262i	$-0.542111\pi$
$313$	−4.66747	−0.263821	−0.131910	+	0.991262i	$0.542111\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.5593	−0.986230	−0.493115	−	0.869964i	$-0.664142\pi$
$317$	−17.5593	−0.986230	−0.493115	+	0.869964i	$0.664142\pi$
$318$	0	0
$319$	−6.39517	−0.358061
$320$	0	0
$321$	−4.91440	−0.274295
$322$	0	0
$323$	6.47139	0.360078
$324$	0	0
$325$	−0.191583	−0.0106271
$326$	0	0
$327$	18.5652	1.02666
$328$	0	0
$329$	0	0
$330$	0	0
$331$	11.2158	0.616478	0.308239	−	0.951309i	$-0.400260\pi$
$331$	11.2158	0.616478	0.308239	+	0.951309i	$0.400260\pi$
$332$	0	0
$333$	−3.52290	−0.193054
$334$	0	0
$335$	9.16509	0.500742
$336$	0	0
$337$	−24.9197	−1.35746	−0.678732	−	0.734386i	$-0.737470\pi$
$337$	−24.9197	−1.35746	−0.678732	+	0.734386i	$0.737470\pi$
$338$	0	0
$339$	10.7320	0.582885
$340$	0	0
$341$	−20.8987	−1.13173
$342$	0	0
$343$	0	0
$344$	0	0
$345$	27.9367	1.50406
$346$	0	0
$347$	0.708986	0.0380603	0.0190302	−	0.999819i	$-0.493942\pi$
$347$	0.708986	0.0380603	0.0190302	+	0.999819i	$0.493942\pi$
$348$	0	0
$349$	−27.5425	−1.47432	−0.737159	−	0.675719i	$-0.763833\pi$
$349$	−27.5425	−1.47432	−0.737159	+	0.675719i	$0.763833\pi$
$350$	0	0
$351$	3.69767	0.197367
$352$	0	0
$353$	−30.9629	−1.64799	−0.823995	−	0.566598i	$-0.808259\pi$
$353$	−30.9629	−1.64799	−0.823995	+	0.566598i	$0.808259\pi$
$354$	0	0
$355$	6.23915	0.331140
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.6612	1.93490	0.967452	−	0.253054i	$-0.0814349\pi$
$359$	36.6612	1.93490	0.967452	+	0.253054i	$0.0814349\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−13.2707	−0.696532
$364$	0	0
$365$	29.2434	1.53067
$366$	0	0
$367$	−24.6440	−1.28640	−0.643202	−	0.765697i	$-0.722394\pi$
$367$	−24.6440	−1.28640	−0.643202	+	0.765697i	$0.722394\pi$
$368$	0	0
$369$	−3.23314	−0.168311
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−35.8970	−1.85868	−0.929339	−	0.369228i	$-0.879622\pi$
$373$	−35.8970	−1.85868	−0.929339	+	0.369228i	$0.879622\pi$
$374$	0	0
$375$	20.6718	1.06749
$376$	0	0
$377$	−2.59307	−0.133550
$378$	0	0
$379$	17.0472	0.875656	0.437828	−	0.899059i	$-0.355748\pi$
$379$	17.0472	0.875656	0.437828	+	0.899059i	$0.355748\pi$
$380$	0	0
$381$	−34.1003	−1.74701
$382$	0	0
$383$	−19.2549	−0.983879	−0.491940	−	0.870629i	$-0.663712\pi$
$383$	−19.2549	−0.983879	−0.491940	+	0.870629i	$0.663712\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.62254	0.285810
$388$	0	0
$389$	−1.76203	−0.0893382	−0.0446691	−	0.999002i	$-0.514223\pi$
$389$	−1.76203	−0.0893382	−0.0446691	+	0.999002i	$0.514223\pi$
$390$	0	0
$391$	−41.6627	−2.10697
$392$	0	0
$393$	−12.6741	−0.639324
$394$	0	0
$395$	17.4113	0.876056
$396$	0	0
$397$	−6.77728	−0.340142	−0.170071	−	0.985432i	$-0.554400\pi$
$397$	−6.77728	−0.340142	−0.170071	+	0.985432i	$0.554400\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.7222	0.885006	0.442503	−	0.896767i	$-0.354091\pi$
$401$	17.7222	0.885006	0.442503	+	0.896767i	$0.354091\pi$
$402$	0	0
$403$	−8.47387	−0.422113
$404$	0	0
$405$	23.8735	1.18629
$406$	0	0
$407$	11.8224	0.586012
$408$	0	0
$409$	−33.9897	−1.68068	−0.840341	−	0.542058i	$-0.817645\pi$
$409$	−33.9897	−1.68068	−0.840341	+	0.542058i	$0.817645\pi$
$410$	0	0
$411$	15.6021	0.769596
$412$	0	0
$413$	0	0
$414$	0	0
$415$	24.8310	1.21891
$416$	0	0
$417$	−24.2424	−1.18715
$418$	0	0
$419$	−15.8623	−0.774922	−0.387461	−	0.921886i	$-0.626648\pi$
$419$	−15.8623	−0.774922	−0.387461	+	0.921886i	$0.626648\pi$
$420$	0	0
$421$	−4.84953	−0.236352	−0.118176	−	0.992993i	$-0.537705\pi$
$421$	−4.84953	−0.236352	−0.118176	+	0.992993i	$0.537705\pi$
$422$	0	0
$423$	7.11575	0.345980
$424$	0	0
$425$	1.52846	0.0741414
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3.07716	0.148567
$430$	0	0
$431$	−8.98213	−0.432654	−0.216327	−	0.976321i	$-0.569408\pi$
$431$	−8.98213	−0.432654	−0.216327	+	0.976321i	$0.569408\pi$
$432$	0	0
$433$	6.63971	0.319084	0.159542	−	0.987191i	$-0.448998\pi$
$433$	6.63971	0.319084	0.159542	+	0.987191i	$0.448998\pi$
$434$	0	0
$435$	−13.8721	−0.665115
$436$	0	0
$437$	−6.43798	−0.307970
$438$	0	0
$439$	22.1296	1.05619	0.528094	−	0.849186i	$-0.322907\pi$
$439$	22.1296	1.05619	0.528094	+	0.849186i	$0.322907\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.2781	0.535840	0.267920	−	0.963441i	$-0.413664\pi$
$443$	11.2781	0.535840	0.267920	+	0.963441i	$0.413664\pi$
$444$	0	0
$445$	24.9789	1.18412
$446$	0	0
$447$	15.9877	0.756190
$448$	0	0
$449$	−31.4545	−1.48443	−0.742214	−	0.670163i	$-0.766224\pi$
$449$	−31.4545	−1.48443	−0.742214	+	0.670163i	$0.766224\pi$
$450$	0	0
$451$	10.8500	0.510905
$452$	0	0
$453$	−10.8715	−0.510787
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.2628	0.807520	0.403760	−	0.914865i	$-0.367703\pi$
$457$	17.2628	0.807520	0.403760	+	0.914865i	$0.367703\pi$
$458$	0	0
$459$	−29.5004	−1.37696
$460$	0	0
$461$	25.7461	1.19911	0.599557	−	0.800332i	$-0.295343\pi$
$461$	25.7461	1.19911	0.599557	+	0.800332i	$0.295343\pi$
$462$	0	0
$463$	−15.3593	−0.713808	−0.356904	−	0.934141i	$-0.616168\pi$
$463$	−15.3593	−0.713808	−0.356904	+	0.934141i	$0.616168\pi$
$464$	0	0
$465$	−45.3323	−2.10224
$466$	0	0
$467$	−19.6226	−0.908025	−0.454013	−	0.890995i	$-0.650008\pi$
$467$	−19.6226	−0.908025	−0.454013	+	0.890995i	$0.650008\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−11.1373	−0.513180
$472$	0	0
$473$	−18.8685	−0.867573
$474$	0	0
$475$	0.236188	0.0108370
$476$	0	0
$477$	3.75899	0.172112
$478$	0	0
$479$	−28.7858	−1.31526	−0.657628	−	0.753343i	$-0.728440\pi$
$479$	−28.7858	−1.31526	−0.657628	+	0.753343i	$0.728440\pi$
$480$	0	0
$481$	4.79366	0.218572
$482$	0	0
$483$	0	0
$484$	0	0
$485$	42.3794	1.92435
$486$	0	0
$487$	19.2887	0.874053	0.437027	−	0.899449i	$-0.356032\pi$
$487$	19.2887	0.874053	0.437027	+	0.899449i	$0.356032\pi$
$488$	0	0
$489$	23.8862	1.08017
$490$	0	0
$491$	−6.82855	−0.308168	−0.154084	−	0.988058i	$-0.549243\pi$
$491$	−6.82855	−0.308168	−0.154084	+	0.988058i	$0.549243\pi$
$492$	0	0
$493$	20.6878	0.931732
$494$	0	0
$495$	2.72882	0.122651
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15.4747	0.692743	0.346371	−	0.938097i	$-0.387414\pi$
$499$	15.4747	0.692743	0.346371	+	0.938097i	$0.387414\pi$
$500$	0	0
$501$	8.50817	0.380117
$502$	0	0
$503$	17.0423	0.759879	0.379939	−	0.925011i	$-0.375945\pi$
$503$	17.0423	0.759879	0.379939	+	0.925011i	$0.375945\pi$
$504$	0	0
$505$	−26.5133	−1.17982
$506$	0	0
$507$	−23.4047	−1.03944
$508$	0	0
$509$	10.8621	0.481456	0.240728	−	0.970593i	$-0.422614\pi$
$509$	10.8621	0.481456	0.240728	+	0.970593i	$0.422614\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.55859	−0.201267
$514$	0	0
$515$	34.6961	1.52889
$516$	0	0
$517$	−23.8795	−1.05022
$518$	0	0
$519$	−4.79117	−0.210309
$520$	0	0
$521$	−16.9182	−0.741198	−0.370599	−	0.928793i	$-0.620848\pi$
$521$	−16.9182	−0.741198	−0.370599	+	0.928793i	$0.620848\pi$
$522$	0	0
$523$	−3.43198	−0.150070	−0.0750350	−	0.997181i	$-0.523907\pi$
$523$	−3.43198	−0.150070	−0.0750350	+	0.997181i	$0.523907\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	67.6053	2.94493
$528$	0	0
$529$	18.4476	0.802071
$530$	0	0
$531$	−7.36894	−0.319785
$532$	0	0
$533$	4.39938	0.190558
$534$	0	0
$535$	5.93009	0.256380
$536$	0	0
$537$	−32.0161	−1.38160
$538$	0	0
$539$	0	0
$540$	0	0
$541$	28.3445	1.21863	0.609313	−	0.792930i	$-0.291445\pi$
$541$	28.3445	1.21863	0.609313	+	0.792930i	$0.291445\pi$
$542$	0	0
$543$	−38.3752	−1.64684
$544$	0	0
$545$	−22.4022	−0.959605
$546$	0	0
$547$	36.1606	1.54611	0.773057	−	0.634336i	$-0.218726\pi$
$547$	36.1606	1.54611	0.773057	+	0.634336i	$0.218726\pi$
$548$	0	0
$549$	−2.36315	−0.100857
$550$	0	0
$551$	3.19681	0.136189
$552$	0	0
$553$	0	0
$554$	0	0
$555$	25.6444	1.08855
$556$	0	0
$557$	13.7616	0.583096	0.291548	−	0.956556i	$-0.405830\pi$
$557$	13.7616	0.583096	0.291548	+	0.956556i	$0.405830\pi$
$558$	0	0
$559$	−7.65067	−0.323589
$560$	0	0
$561$	−24.5499	−1.03650
$562$	0	0
$563$	29.2093	1.23102	0.615512	−	0.788127i	$-0.288949\pi$
$563$	29.2093	1.23102	0.615512	+	0.788127i	$0.288949\pi$
$564$	0	0
$565$	−12.9501	−0.544815
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.68082	−0.0704637	−0.0352319	−	0.999379i	$-0.511217\pi$
$569$	−1.68082	−0.0704637	−0.0352319	+	0.999379i	$0.511217\pi$
$570$	0	0
$571$	19.9577	0.835203	0.417602	−	0.908630i	$-0.362871\pi$
$571$	19.9577	0.835203	0.417602	+	0.908630i	$0.362871\pi$
$572$	0	0
$573$	3.22752	0.134831
$574$	0	0
$575$	−1.52057	−0.0634123
$576$	0	0
$577$	−2.80390	−0.116728	−0.0583638	−	0.998295i	$-0.518588\pi$
$577$	−2.80390	−0.116728	−0.0583638	+	0.998295i	$0.518588\pi$
$578$	0	0
$579$	14.3859	0.597856
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12.6146	−0.522445
$584$	0	0
$585$	1.10646	0.0457467
$586$	0	0
$587$	−38.4051	−1.58515	−0.792575	−	0.609775i	$-0.791260\pi$
$587$	−38.4051	−1.58515	−0.792575	+	0.609775i	$0.791260\pi$
$588$	0	0
$589$	10.4468	0.430453
$590$	0	0
$591$	−6.58595	−0.270910
$592$	0	0
$593$	29.3676	1.20598	0.602991	−	0.797748i	$-0.293976\pi$
$593$	29.3676	1.20598	0.602991	+	0.797748i	$0.293976\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−47.1480	−1.92964
$598$	0	0
$599$	−25.5438	−1.04369	−0.521847	−	0.853039i	$-0.674757\pi$
$599$	−25.5438	−1.04369	−0.521847	+	0.853039i	$0.674757\pi$
$600$	0	0
$601$	20.3565	0.830359	0.415179	−	0.909740i	$-0.363719\pi$
$601$	20.3565	0.830359	0.415179	+	0.909740i	$0.363719\pi$
$602$	0	0
$603$	−2.38759	−0.0972303
$604$	0	0
$605$	16.0135	0.651040
$606$	0	0
$607$	−21.5585	−0.875032	−0.437516	−	0.899211i	$-0.644142\pi$
$607$	−21.5585	−0.875032	−0.437516	+	0.899211i	$0.644142\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.68250	−0.391712
$612$	0	0
$613$	46.2723	1.86892	0.934460	−	0.356069i	$-0.115883\pi$
$613$	46.2723	1.86892	0.934460	+	0.356069i	$0.115883\pi$
$614$	0	0
$615$	23.5352	0.949030
$616$	0	0
$617$	49.2011	1.98076	0.990381	−	0.138369i	$-0.0441861\pi$
$617$	49.2011	1.98076	0.990381	+	0.138369i	$0.0441861\pi$
$618$	0	0
$619$	36.7671	1.47779	0.738897	−	0.673818i	$-0.235347\pi$
$619$	36.7671	1.47779	0.738897	+	0.673818i	$0.235347\pi$
$620$	0	0
$621$	29.3481	1.17770
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−26.1252	−1.04501
$626$	0	0
$627$	−3.79361	−0.151502
$628$	0	0
$629$	−38.2443	−1.52490
$630$	0	0
$631$	−26.9353	−1.07228	−0.536140	−	0.844129i	$-0.680118\pi$
$631$	−26.9353	−1.07228	−0.536140	+	0.844129i	$0.680118\pi$
$632$	0	0
$633$	−13.7023	−0.544619
$634$	0	0
$635$	41.1481	1.63291
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.62536	−0.0642982
$640$	0	0
$641$	32.7685	1.29428	0.647138	−	0.762373i	$-0.275966\pi$
$641$	32.7685	1.29428	0.647138	+	0.762373i	$0.275966\pi$
$642$	0	0
$643$	−12.8047	−0.504968	−0.252484	−	0.967601i	$-0.581248\pi$
$643$	−12.8047	−0.504968	−0.252484	+	0.967601i	$0.581248\pi$
$644$	0	0
$645$	−40.9285	−1.61156
$646$	0	0
$647$	2.42585	0.0953701	0.0476850	−	0.998862i	$-0.484816\pi$
$647$	2.42585	0.0953701	0.0476850	+	0.998862i	$0.484816\pi$
$648$	0	0
$649$	24.7291	0.970703
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−34.3350	−1.34363	−0.671817	−	0.740717i	$-0.734486\pi$
$653$	−34.3350	−1.34363	−0.671817	+	0.740717i	$0.734486\pi$
$654$	0	0
$655$	15.2935	0.597568
$656$	0	0
$657$	−7.61819	−0.297214
$658$	0	0
$659$	19.6230	0.764402	0.382201	−	0.924079i	$-0.375166\pi$
$659$	19.6230	0.764402	0.382201	+	0.924079i	$0.375166\pi$
$660$	0	0
$661$	−29.9310	−1.16418	−0.582091	−	0.813124i	$-0.697765\pi$
$661$	−29.9310	−1.16418	−0.582091	+	0.813124i	$0.697765\pi$
$662$	0	0
$663$	−9.95435	−0.386595
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.5810	−0.796900
$668$	0	0
$669$	18.8275	0.727913
$670$	0	0
$671$	7.93041	0.306150
$672$	0	0
$673$	−4.89057	−0.188518	−0.0942588	−	0.995548i	$-0.530048\pi$
$673$	−4.89057	−0.188518	−0.0942588	+	0.995548i	$0.530048\pi$
$674$	0	0
$675$	−1.07668	−0.0414416
$676$	0	0
$677$	32.5652	1.25158	0.625791	−	0.779991i	$-0.284776\pi$
$677$	32.5652	1.25158	0.625791	+	0.779991i	$0.284776\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−8.12151	−0.311217
$682$	0	0
$683$	−29.0095	−1.11002	−0.555008	−	0.831845i	$-0.687285\pi$
$683$	−29.0095	−1.11002	−0.555008	+	0.831845i	$0.687285\pi$
$684$	0	0
$685$	−18.8267	−0.719332
$686$	0	0
$687$	32.2565	1.23066
$688$	0	0
$689$	−5.11491	−0.194863
$690$	0	0
$691$	−38.6835	−1.47159	−0.735794	−	0.677205i	$-0.763191\pi$
$691$	−38.6835	−1.47159	−0.735794	+	0.677205i	$0.763191\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	29.2527	1.10962
$696$	0	0
$697$	−35.0987	−1.32946
$698$	0	0
$699$	14.4289	0.545752
$700$	0	0
$701$	−1.58537	−0.0598787	−0.0299393	−	0.999552i	$-0.509531\pi$
$701$	−1.58537	−0.0598787	−0.0299393	+	0.999552i	$0.509531\pi$
$702$	0	0
$703$	−5.90974	−0.222890
$704$	0	0
$705$	−51.7981	−1.95083
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−21.8240	−0.819618	−0.409809	−	0.912171i	$-0.634405\pi$
$709$	−21.8240	−0.819618	−0.409809	+	0.912171i	$0.634405\pi$
$710$	0	0
$711$	−4.53580	−0.170106
$712$	0	0
$713$	−67.2563	−2.51877
$714$	0	0
$715$	−3.71314	−0.138863
$716$	0	0
$717$	8.85430	0.330670
$718$	0	0
$719$	32.7395	1.22098	0.610490	−	0.792024i	$-0.290973\pi$
$719$	32.7395	1.22098	0.610490	+	0.792024i	$0.290973\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.23546	−0.0459472
$724$	0	0
$725$	0.755048	0.0280418
$726$	0	0
$727$	10.0086	0.371199	0.185599	−	0.982625i	$-0.440577\pi$
$727$	10.0086	0.371199	0.185599	+	0.982625i	$0.440577\pi$
$728$	0	0
$729$	19.7146	0.730172
$730$	0	0
$731$	61.0378	2.25756
$732$	0	0
$733$	−21.4733	−0.793136	−0.396568	−	0.918005i	$-0.629799\pi$
$733$	−21.4733	−0.793136	−0.396568	+	0.918005i	$0.629799\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.01243	0.295142
$738$	0	0
$739$	16.5193	0.607673	0.303836	−	0.952724i	$-0.401732\pi$
$739$	16.5193	0.607673	0.303836	+	0.952724i	$0.401732\pi$
$740$	0	0
$741$	−1.53821	−0.0565075
$742$	0	0
$743$	36.8210	1.35083	0.675415	−	0.737438i	$-0.263964\pi$
$743$	36.8210	1.35083	0.675415	+	0.737438i	$0.263964\pi$
$744$	0	0
$745$	−19.2919	−0.706801
$746$	0	0
$747$	−6.46872	−0.236678
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26.0404	−0.950229	−0.475115	−	0.879924i	$-0.657593\pi$
$751$	−26.0404	−0.950229	−0.475115	+	0.879924i	$0.657593\pi$
$752$	0	0
$753$	38.2269	1.39307
$754$	0	0
$755$	13.1184	0.477426
$756$	0	0
$757$	−36.9091	−1.34148	−0.670742	−	0.741691i	$-0.734024\pi$
$757$	−36.9091	−1.34148	−0.670742	+	0.741691i	$0.734024\pi$
$758$	0	0
$759$	24.4232	0.886505
$760$	0	0
$761$	−53.6184	−1.94366	−0.971832	−	0.235674i	$-0.924270\pi$
$761$	−53.6184	−1.94366	−0.971832	+	0.235674i	$0.924270\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−8.82748	−0.319158
$766$	0	0
$767$	10.0270	0.362055
$768$	0	0
$769$	51.2222	1.84712	0.923560	−	0.383455i	$-0.125266\pi$
$769$	51.2222	1.84712	0.923560	+	0.383455i	$0.125266\pi$
$770$	0	0
$771$	−44.8531	−1.61534
$772$	0	0
$773$	−14.9397	−0.537342	−0.268671	−	0.963232i	$-0.586584\pi$
$773$	−14.9397	−0.537342	−0.268671	+	0.963232i	$0.586584\pi$
$774$	0	0
$775$	2.46741	0.0886319
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.42367	−0.194323
$780$	0	0
$781$	5.45448	0.195177
$782$	0	0
$783$	−14.5729	−0.520794
$784$	0	0
$785$	13.4391	0.479663
$786$	0	0
$787$	−12.0523	−0.429619	−0.214809	−	0.976656i	$-0.568913\pi$
$787$	−12.0523	−0.429619	−0.214809	+	0.976656i	$0.568913\pi$
$788$	0	0
$789$	−13.2512	−0.471757
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.21558	0.114188
$794$	0	0
$795$	−27.3630	−0.970467
$796$	0	0
$797$	−14.0906	−0.499113	−0.249557	−	0.968360i	$-0.580285\pi$
$797$	−14.0906	−0.499113	−0.249557	+	0.968360i	$0.580285\pi$
$798$	0	0
$799$	77.2480	2.73284
$800$	0	0
$801$	−6.50726	−0.229923
$802$	0	0
$803$	25.5656	0.902190
$804$	0	0
$805$	0	0
$806$	0	0
$807$	38.6447	1.36036
$808$	0	0
$809$	−17.1994	−0.604697	−0.302349	−	0.953197i	$-0.597771\pi$
$809$	−17.1994	−0.604697	−0.302349	+	0.953197i	$0.597771\pi$
$810$	0	0
$811$	−36.1232	−1.26846	−0.634229	−	0.773145i	$-0.718682\pi$
$811$	−36.1232	−1.26846	−0.634229	+	0.773145i	$0.718682\pi$
$812$	0	0
$813$	31.9306	1.11986
$814$	0	0
$815$	−28.8229	−1.00962
$816$	0	0
$817$	9.43194	0.329982
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.62285	−0.300940	−0.150470	−	0.988615i	$-0.548079\pi$
$821$	−8.62285	−0.300940	−0.150470	+	0.988615i	$0.548079\pi$
$822$	0	0
$823$	2.74191	0.0955771	0.0477885	−	0.998857i	$-0.484783\pi$
$823$	2.74191	0.0955771	0.0477885	+	0.998857i	$0.484783\pi$
$824$	0	0
$825$	−0.896004	−0.0311949
$826$	0	0
$827$	43.8831	1.52597	0.762983	−	0.646418i	$-0.223734\pi$
$827$	43.8831	1.52597	0.762983	+	0.646418i	$0.223734\pi$
$828$	0	0
$829$	−2.55472	−0.0887290	−0.0443645	−	0.999015i	$-0.514126\pi$
$829$	−2.55472	−0.0887290	−0.0443645	+	0.999015i	$0.514126\pi$
$830$	0	0
$831$	49.9111	1.73140
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−10.2666	−0.355291
$836$	0	0
$837$	−47.6226	−1.64608
$838$	0	0
$839$	1.26628	0.0437168	0.0218584	−	0.999761i	$-0.493042\pi$
$839$	1.26628	0.0437168	0.0218584	+	0.999761i	$0.493042\pi$
$840$	0	0
$841$	−18.7804	−0.647600
$842$	0	0
$843$	−23.1653	−0.797856
$844$	0	0
$845$	28.2420	0.971553
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.7973	0.404883
$850$	0	0
$851$	38.0468	1.30423
$852$	0	0
$853$	44.4111	1.52061	0.760303	−	0.649568i	$-0.225050\pi$
$853$	44.4111	1.52061	0.760303	+	0.649568i	$0.225050\pi$
$854$	0	0
$855$	−1.36408	−0.0466505
$856$	0	0
$857$	11.6714	0.398686	0.199343	−	0.979930i	$-0.436119\pi$
$857$	11.6714	0.398686	0.199343	+	0.979930i	$0.436119\pi$
$858$	0	0
$859$	−32.2514	−1.10040	−0.550202	−	0.835032i	$-0.685449\pi$
$859$	−32.2514	−1.10040	−0.550202	+	0.835032i	$0.685449\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.0855	0.717759	0.358880	−	0.933384i	$-0.383159\pi$
$863$	21.0855	0.717759	0.358880	+	0.933384i	$0.383159\pi$
$864$	0	0
$865$	5.78139	0.196573
$866$	0	0
$867$	47.1789	1.60228
$868$	0	0
$869$	15.2215	0.516355
$870$	0	0
$871$	3.24883	0.110082
$872$	0	0
$873$	−11.0402	−0.373656
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.8426	−1.17655	−0.588275	−	0.808661i	$-0.700193\pi$
$877$	−34.8426	−1.17655	−0.588275	+	0.808661i	$0.700193\pi$
$878$	0	0
$879$	6.70142	0.226033
$880$	0	0
$881$	−23.1285	−0.779217	−0.389609	−	0.920981i	$-0.627390\pi$
$881$	−23.1285	−0.779217	−0.389609	+	0.920981i	$0.627390\pi$
$882$	0	0
$883$	−31.7628	−1.06890	−0.534452	−	0.845199i	$-0.679482\pi$
$883$	−31.7628	−1.06890	−0.534452	+	0.845199i	$0.679482\pi$
$884$	0	0
$885$	53.6412	1.80313
$886$	0	0
$887$	−31.3552	−1.05280	−0.526402	−	0.850236i	$-0.676459\pi$
$887$	−31.3552	−1.05280	−0.526402	+	0.850236i	$0.676459\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	20.8710	0.699206
$892$	0	0
$893$	11.9368	0.399451
$894$	0	0
$895$	38.6331	1.29136
$896$	0	0
$897$	9.90296	0.330650
$898$	0	0
$899$	33.3964	1.11383
$900$	0	0
$901$	40.8073	1.35949
$902$	0	0
$903$	0	0
$904$	0	0
$905$	46.3064	1.53928
$906$	0	0
$907$	18.2904	0.607323	0.303662	−	0.952780i	$-0.401791\pi$
$907$	18.2904	0.607323	0.303662	+	0.952780i	$0.401791\pi$
$908$	0	0
$909$	6.90696	0.229089
$910$	0	0
$911$	−47.0090	−1.55748	−0.778738	−	0.627349i	$-0.784140\pi$
$911$	−47.0090	−1.55748	−0.778738	+	0.627349i	$0.784140\pi$
$912$	0	0
$913$	21.7081	0.718434
$914$	0	0
$915$	17.2022	0.568689
$916$	0	0
$917$	0	0
$918$	0	0
$919$	15.9661	0.526673	0.263336	−	0.964704i	$-0.415177\pi$
$919$	15.9661	0.526673	0.263336	+	0.964704i	$0.415177\pi$
$920$	0	0
$921$	32.0014	1.05448
$922$	0	0
$923$	2.21165	0.0727973
$924$	0	0
$925$	−1.39581	−0.0458940
$926$	0	0
$927$	−9.03867	−0.296869
$928$	0	0
$929$	1.53048	0.0502135	0.0251068	−	0.999685i	$-0.492007\pi$
$929$	1.53048	0.0502135	0.0251068	+	0.999685i	$0.492007\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	20.4456	0.669360
$934$	0	0
$935$	29.6238	0.968801
$936$	0	0
$937$	−9.93360	−0.324516	−0.162258	−	0.986748i	$-0.551878\pi$
$937$	−9.93360	−0.324516	−0.162258	+	0.986748i	$0.551878\pi$
$938$	0	0
$939$	−8.85112	−0.288845
$940$	0	0
$941$	2.41263	0.0786496	0.0393248	−	0.999226i	$-0.487479\pi$
$941$	2.41263	0.0786496	0.0393248	+	0.999226i	$0.487479\pi$
$942$	0	0
$943$	34.9175	1.13707
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.0308	1.26833	0.634166	−	0.773197i	$-0.281344\pi$
$947$	39.0308	1.26833	0.634166	+	0.773197i	$0.281344\pi$
$948$	0	0
$949$	10.3662	0.336501
$950$	0	0
$951$	−33.2985	−1.07978
$952$	0	0
$953$	−24.6841	−0.799595	−0.399798	−	0.916603i	$-0.630920\pi$
$953$	−24.6841	−0.799595	−0.399798	+	0.916603i	$0.630920\pi$
$954$	0	0
$955$	−3.89457	−0.126025
$956$	0	0
$957$	−12.1274	−0.392024
$958$	0	0
$959$	0	0
$960$	0	0
$961$	78.1356	2.52050
$962$	0	0
$963$	−1.54484	−0.0497819
$964$	0	0
$965$	−17.3591	−0.558808
$966$	0	0
$967$	15.9458	0.512782	0.256391	−	0.966573i	$-0.417467\pi$
$967$	15.9458	0.512782	0.256391	+	0.966573i	$0.417467\pi$
$968$	0	0
$969$	12.2720	0.394233
$970$	0	0
$971$	24.5390	0.787494	0.393747	−	0.919219i	$-0.371179\pi$
$971$	24.5390	0.787494	0.393747	+	0.919219i	$0.371179\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−0.363306	−0.0116351
$976$	0	0
$977$	12.8928	0.412477	0.206238	−	0.978502i	$-0.433878\pi$
$977$	12.8928	0.412477	0.206238	+	0.978502i	$0.433878\pi$
$978$	0	0
$979$	21.8374	0.697928
$980$	0	0
$981$	5.83599	0.186329
$982$	0	0
$983$	13.0355	0.415767	0.207883	−	0.978154i	$-0.433343\pi$
$983$	13.0355	0.415767	0.207883	+	0.978154i	$0.433343\pi$
$984$	0	0
$985$	7.94711	0.253216
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−60.7227	−1.93087
$990$	0	0
$991$	29.3547	0.932482	0.466241	−	0.884658i	$-0.345608\pi$
$991$	29.3547	0.932482	0.466241	+	0.884658i	$0.345608\pi$
$992$	0	0
$993$	21.2691	0.674954
$994$	0	0
$995$	56.8924	1.80361
$996$	0	0
$997$	−8.13934	−0.257775	−0.128888	−	0.991659i	$-0.541141\pi$
$997$	−8.13934	−0.257775	−0.128888	+	0.991659i	$0.541141\pi$
$998$	0	0
$999$	26.9401	0.852346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bv.1.12	✓	14
7.6	odd	2		7448.2.a.bw.1.3	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bv.1.12	✓	14	1.1	even	1	trivial
7448.2.a.bw.1.3	yes	14	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+1.89634 q^{3} -2.28827 q^{5} +0.596117 q^{9} +O(q^{10})\)	\(q+1.89634 q^{3} -2.28827 q^{5} +0.596117 q^{9} -2.00048 q^{11} -0.811145 q^{13} -4.33935 q^{15} +6.47139 q^{17} +1.00000 q^{19} -6.43798 q^{23} +0.236188 q^{25} -4.55859 q^{27} +3.19681 q^{29} +10.4468 q^{31} -3.79361 q^{33} -5.90974 q^{37} -1.53821 q^{39} -5.42367 q^{41} +9.43194 q^{43} -1.36408 q^{45} +11.9368 q^{47} +12.2720 q^{51} +6.30580 q^{53} +4.57765 q^{55} +1.89634 q^{57} -12.3616 q^{59} -3.96425 q^{61} +1.85612 q^{65} -4.00524 q^{67} -12.2086 q^{69} -2.72658 q^{71} -12.7797 q^{73} +0.447893 q^{75} -7.60892 q^{79} -10.4330 q^{81} -10.8514 q^{83} -14.8083 q^{85} +6.06225 q^{87} -10.9161 q^{89} +19.8107 q^{93} -2.28827 q^{95} -18.5203 q^{97} -1.19252 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} - 2 q^{5} + 16 q^{9}+O(q^{10}) \) 14 * q - 2 * q^3 - 2 * q^5 + 16 * q^9	\( 14 q - 2 q^{3} - 2 q^{5} + 16 q^{9} - 6 q^{11} - 16 q^{13} - 4 q^{15} - 20 q^{17} + 14 q^{19} + 4 q^{23} + 16 q^{25} - 20 q^{27} + 6 q^{29} - 34 q^{33} - 6 q^{37} + 8 q^{39} - 46 q^{41} - 18 q^{43} + 10 q^{47} - 4 q^{51} - 2 q^{53} - 28 q^{55} - 2 q^{57} + 22 q^{59} - 26 q^{61} + 8 q^{65} - 12 q^{67} - 48 q^{69} + 18 q^{71} - 28 q^{73} + 24 q^{75} - 10 q^{79} - 2 q^{81} - 8 q^{83} - 16 q^{85} - 16 q^{87} - 78 q^{89} - 2 q^{95} - 54 q^{97} + 26 q^{99}+O(q^{100}) \) 14 * q - 2 * q^3 - 2 * q^5 + 16 * q^9 - 6 * q^11 - 16 * q^13 - 4 * q^15 - 20 * q^17 + 14 * q^19 + 4 * q^23 + 16 * q^25 - 20 * q^27 + 6 * q^29 - 34 * q^33 - 6 * q^37 + 8 * q^39 - 46 * q^41 - 18 * q^43 + 10 * q^47 - 4 * q^51 - 2 * q^53 - 28 * q^55 - 2 * q^57 + 22 * q^59 - 26 * q^61 + 8 * q^65 - 12 * q^67 - 48 * q^69 + 18 * q^71 - 28 * q^73 + 24 * q^75 - 10 * q^79 - 2 * q^81 - 8 * q^83 - 16 * q^85 - 16 * q^87 - 78 * q^89 - 2 * q^95 - 54 * q^97 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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