LMFDB - Embedded newform 7448.2.a.br.1.7

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} - x^{5} + 66x^{4} + 4x^{3} - 76x^{2} + 8x + 16 $ x^8 - 15x^6 - x^5 + 66x^4 + 4x^3 - 76x^2 + 8*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.46993$ 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)

$f(q)$	$=$	$q+2.46993 q^{3} +4.03871 q^{5} +3.10055 q^{9} +O(q^{10})$	$q+2.46993 q^{3} +4.03871 q^{5} +3.10055 q^{9} +1.60209 q^{11} -3.15476 q^{13} +9.97534 q^{15} -0.652775 q^{17} -1.00000 q^{19} +2.91726 q^{23} +11.3112 q^{25} +0.248363 q^{27} +2.12163 q^{29} +7.02322 q^{31} +3.95705 q^{33} -5.30985 q^{37} -7.79204 q^{39} +2.00000 q^{41} +7.97365 q^{43} +12.5223 q^{45} +0.605361 q^{47} -1.61231 q^{51} +12.4824 q^{53} +6.47038 q^{55} -2.46993 q^{57} -14.4517 q^{59} -9.34534 q^{61} -12.7412 q^{65} -2.78539 q^{67} +7.20542 q^{69} -9.50073 q^{71} +16.6278 q^{73} +27.9379 q^{75} +4.85650 q^{79} -8.68822 q^{81} +8.73966 q^{83} -2.63637 q^{85} +5.24028 q^{87} +2.27610 q^{89} +17.3469 q^{93} -4.03871 q^{95} -1.24899 q^{97} +4.96736 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{5} + 6 q^{9}+O(q^{10}) $ 8 * q + q^5 + 6 * q^9	$ 8 q + q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} - 4 q^{17} - 8 q^{19} + 25 q^{23} + 15 q^{25} - 3 q^{27} + 6 q^{29} - 8 q^{33} + 13 q^{37} - 11 q^{39} + 16 q^{41} + 17 q^{43} - 17 q^{45} + 24 q^{47} + 5 q^{51} + 2 q^{53} - 5 q^{55} - 2 q^{59} + 13 q^{61} - 26 q^{65} + 2 q^{67} + 11 q^{69} + 10 q^{71} - 5 q^{73} + 20 q^{75} + 16 q^{79} - 12 q^{81} + 43 q^{83} + 24 q^{85} - 20 q^{87} - 8 q^{89} + 2 q^{93} - q^{95} + 12 q^{97} + 37 q^{99}+O(q^{100}) $ 8 * q + q^5 + 6 * q^9 + 9 * q^11 + 8 * q^15 - 4 * q^17 - 8 * q^19 + 25 * q^23 + 15 * q^25 - 3 * q^27 + 6 * q^29 - 8 * q^33 + 13 * q^37 - 11 * q^39 + 16 * q^41 + 17 * q^43 - 17 * q^45 + 24 * q^47 + 5 * q^51 + 2 * q^53 - 5 * q^55 - 2 * q^59 + 13 * q^61 - 26 * q^65 + 2 * q^67 + 11 * q^69 + 10 * q^71 - 5 * q^73 + 20 * q^75 + 16 * q^79 - 12 * q^81 + 43 * q^83 + 24 * q^85 - 20 * q^87 - 8 * q^89 + 2 * q^93 - q^95 + 12 * q^97 + 37 * 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
$3$	2.46993	1.42601	0.713007	−	0.701157i	$-0.247333\pi$
$3$	2.46993	1.42601	0.713007	+	0.701157i	$0.247333\pi$
$4$	0	0
$5$	4.03871	1.80617	0.903084	−	0.429464i	$-0.141297\pi$
$5$	4.03871	1.80617	0.903084	+	0.429464i	$0.141297\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.10055	1.03352
$10$	0	0
$11$	1.60209	0.483048	0.241524	−	0.970395i	$-0.422353\pi$
$11$	1.60209	0.483048	0.241524	+	0.970395i	$0.422353\pi$
$12$	0	0
$13$	−3.15476	−0.874974	−0.437487	−	0.899225i	$-0.644131\pi$
$13$	−3.15476	−0.874974	−0.437487	+	0.899225i	$0.644131\pi$
$14$	0	0
$15$	9.97534	2.57562
$16$	0	0
$17$	−0.652775	−0.158321	−0.0791605	−	0.996862i	$-0.525224\pi$
$17$	−0.652775	−0.158321	−0.0791605	+	0.996862i	$0.525224\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.91726	0.608290	0.304145	−	0.952626i	$-0.401629\pi$
$23$	2.91726	0.608290	0.304145	+	0.952626i	$0.401629\pi$
$24$	0	0
$25$	11.3112	2.26224
$26$	0	0
$27$	0.248363	0.0477975
$28$	0	0
$29$	2.12163	0.393977	0.196988	−	0.980406i	$-0.436884\pi$
$29$	2.12163	0.393977	0.196988	+	0.980406i	$0.436884\pi$
$30$	0	0
$31$	7.02322	1.26141	0.630704	−	0.776024i	$-0.282766\pi$
$31$	7.02322	1.26141	0.630704	+	0.776024i	$0.282766\pi$
$32$	0	0
$33$	3.95705	0.688833
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.30985	−0.872934	−0.436467	−	0.899720i	$-0.643770\pi$
$37$	−5.30985	−0.872934	−0.436467	+	0.899720i	$0.643770\pi$
$38$	0	0
$39$	−7.79204	−1.24773
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	7.97365	1.21597	0.607985	−	0.793948i	$-0.291978\pi$
$43$	7.97365	1.21597	0.607985	+	0.793948i	$0.291978\pi$
$44$	0	0
$45$	12.5223	1.86671
$46$	0	0
$47$	0.605361	0.0883009	0.0441505	−	0.999025i	$-0.485942\pi$
$47$	0.605361	0.0883009	0.0441505	+	0.999025i	$0.485942\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.61231	−0.225768
$52$	0	0
$53$	12.4824	1.71459	0.857296	−	0.514823i	$-0.172142\pi$
$53$	12.4824	1.71459	0.857296	+	0.514823i	$0.172142\pi$
$54$	0	0
$55$	6.47038	0.872465
$56$	0	0
$57$	−2.46993	−0.327150
$58$	0	0
$59$	−14.4517	−1.88146	−0.940728	−	0.339163i	$-0.889856\pi$
$59$	−14.4517	−1.88146	−0.940728	+	0.339163i	$0.889856\pi$
$60$	0	0
$61$	−9.34534	−1.19655	−0.598274	−	0.801292i	$-0.704146\pi$
$61$	−9.34534	−1.19655	−0.598274	+	0.801292i	$0.704146\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.7412	−1.58035
$66$	0	0
$67$	−2.78539	−0.340289	−0.170145	−	0.985419i	$-0.554423\pi$
$67$	−2.78539	−0.340289	−0.170145	+	0.985419i	$0.554423\pi$
$68$	0	0
$69$	7.20542	0.867430
$70$	0	0
$71$	−9.50073	−1.12753	−0.563764	−	0.825936i	$-0.690647\pi$
$71$	−9.50073	−1.12753	−0.563764	+	0.825936i	$0.690647\pi$
$72$	0	0
$73$	16.6278	1.94613	0.973067	−	0.230523i	$-0.0740436\pi$
$73$	16.6278	1.94613	0.973067	+	0.230523i	$0.0740436\pi$
$74$	0	0
$75$	27.9379	3.22599
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.85650	0.546399	0.273199	−	0.961957i	$-0.411918\pi$
$79$	4.85650	0.546399	0.273199	+	0.961957i	$0.411918\pi$
$80$	0	0
$81$	−8.68822	−0.965358
$82$	0	0
$83$	8.73966	0.959302	0.479651	−	0.877459i	$-0.340763\pi$
$83$	8.73966	0.959302	0.479651	+	0.877459i	$0.340763\pi$
$84$	0	0
$85$	−2.63637	−0.285954
$86$	0	0
$87$	5.24028	0.561816
$88$	0	0
$89$	2.27610	0.241266	0.120633	−	0.992697i	$-0.461508\pi$
$89$	2.27610	0.241266	0.120633	+	0.992697i	$0.461508\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	17.3469	1.79879
$94$	0	0
$95$	−4.03871	−0.414363
$96$	0	0
$97$	−1.24899	−0.126815	−0.0634077	−	0.997988i	$-0.520197\pi$
$97$	−1.24899	−0.126815	−0.0634077	+	0.997988i	$0.520197\pi$
$98$	0	0
$99$	4.96736	0.499239
$100$	0	0
$101$	11.2461	1.11903	0.559515	−	0.828820i	$-0.310987\pi$
$101$	11.2461	1.11903	0.559515	+	0.828820i	$0.310987\pi$
$102$	0	0
$103$	−3.59297	−0.354025	−0.177013	−	0.984209i	$-0.556643\pi$
$103$	−3.59297	−0.354025	−0.177013	+	0.984209i	$0.556643\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.74632	−0.168823	−0.0844117	−	0.996431i	$-0.526901\pi$
$107$	−1.74632	−0.168823	−0.0844117	+	0.996431i	$0.526901\pi$
$108$	0	0
$109$	−1.91848	−0.183757	−0.0918786	−	0.995770i	$-0.529287\pi$
$109$	−1.91848	−0.183757	−0.0918786	+	0.995770i	$0.529287\pi$
$110$	0	0
$111$	−13.1150	−1.24482
$112$	0	0
$113$	−11.8303	−1.11290	−0.556451	−	0.830881i	$-0.687837\pi$
$113$	−11.8303	−1.11290	−0.556451	+	0.830881i	$0.687837\pi$
$114$	0	0
$115$	11.7820	1.09867
$116$	0	0
$117$	−9.78152	−0.904301
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.43331	−0.766665
$122$	0	0
$123$	4.93986	0.445412
$124$	0	0
$125$	25.4892	2.27982
$126$	0	0
$127$	−15.3583	−1.36283	−0.681416	−	0.731896i	$-0.738636\pi$
$127$	−15.3583	−1.36283	−0.681416	+	0.731896i	$0.738636\pi$
$128$	0	0
$129$	19.6944	1.73399
$130$	0	0
$131$	13.3828	1.16926	0.584631	−	0.811299i	$-0.301239\pi$
$131$	13.3828	1.16926	0.584631	+	0.811299i	$0.301239\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00307	0.0863303
$136$	0	0
$137$	−3.16704	−0.270579	−0.135289	−	0.990806i	$-0.543196\pi$
$137$	−3.16704	−0.270579	−0.135289	+	0.990806i	$0.543196\pi$
$138$	0	0
$139$	16.3904	1.39021	0.695106	−	0.718907i	$-0.255358\pi$
$139$	16.3904	1.39021	0.695106	+	0.718907i	$0.255358\pi$
$140$	0	0
$141$	1.49520	0.125918
$142$	0	0
$143$	−5.05421	−0.422654
$144$	0	0
$145$	8.56865	0.711588
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.0220	1.14873	0.574366	−	0.818599i	$-0.305249\pi$
$149$	14.0220	1.14873	0.574366	+	0.818599i	$0.305249\pi$
$150$	0	0
$151$	18.5461	1.50926	0.754632	−	0.656149i	$-0.227816\pi$
$151$	18.5461	1.50926	0.754632	+	0.656149i	$0.227816\pi$
$152$	0	0
$153$	−2.02396	−0.163628
$154$	0	0
$155$	28.3648	2.27831
$156$	0	0
$157$	−23.5512	−1.87959	−0.939796	−	0.341735i	$-0.888986\pi$
$157$	−23.5512	−1.87959	−0.939796	+	0.341735i	$0.888986\pi$
$158$	0	0
$159$	30.8307	2.44503
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.07130	0.710519	0.355260	−	0.934768i	$-0.384392\pi$
$163$	9.07130	0.710519	0.355260	+	0.934768i	$0.384392\pi$
$164$	0	0
$165$	15.9814	1.24415
$166$	0	0
$167$	0.191778	0.0148402	0.00742011	−	0.999972i	$-0.497638\pi$
$167$	0.191778	0.0148402	0.00742011	+	0.999972i	$0.497638\pi$
$168$	0	0
$169$	−3.04747	−0.234421
$170$	0	0
$171$	−3.10055	−0.237105
$172$	0	0
$173$	6.98927	0.531384	0.265692	−	0.964058i	$-0.414400\pi$
$173$	6.98927	0.531384	0.265692	+	0.964058i	$0.414400\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−35.6948	−2.68298
$178$	0	0
$179$	−15.9087	−1.18907	−0.594534	−	0.804070i	$-0.702664\pi$
$179$	−15.9087	−1.18907	−0.594534	+	0.804070i	$0.702664\pi$
$180$	0	0
$181$	−16.6258	−1.23579	−0.617894	−	0.786261i	$-0.712014\pi$
$181$	−16.6258	−1.23579	−0.617894	+	0.786261i	$0.712014\pi$
$182$	0	0
$183$	−23.0823	−1.70629
$184$	0	0
$185$	−21.4450	−1.57667
$186$	0	0
$187$	−1.04580	−0.0764766
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.0101	1.73731	0.868654	−	0.495419i	$-0.164985\pi$
$191$	24.0101	1.73731	0.868654	+	0.495419i	$0.164985\pi$
$192$	0	0
$193$	0.109875	0.00790894	0.00395447	−	0.999992i	$-0.498741\pi$
$193$	0.109875	0.00790894	0.00395447	+	0.999992i	$0.498741\pi$
$194$	0	0
$195$	−31.4698	−2.25360
$196$	0	0
$197$	−10.9282	−0.778601	−0.389301	−	0.921111i	$-0.627283\pi$
$197$	−10.9282	−0.778601	−0.389301	+	0.921111i	$0.627283\pi$
$198$	0	0
$199$	−22.5372	−1.59762	−0.798810	−	0.601584i	$-0.794537\pi$
$199$	−22.5372	−1.59762	−0.798810	+	0.601584i	$0.794537\pi$
$200$	0	0
$201$	−6.87971	−0.485258
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.07743	0.564152
$206$	0	0
$207$	9.04511	0.628679
$208$	0	0
$209$	−1.60209	−0.110819
$210$	0	0
$211$	23.7342	1.63393	0.816966	−	0.576685i	$-0.195654\pi$
$211$	23.7342	1.63393	0.816966	+	0.576685i	$0.195654\pi$
$212$	0	0
$213$	−23.4661	−1.60787
$214$	0	0
$215$	32.2033	2.19625
$216$	0	0
$217$	0	0
$218$	0	0
$219$	41.0694	2.77522
$220$	0	0
$221$	2.05935	0.138527
$222$	0	0
$223$	−4.36632	−0.292390	−0.146195	−	0.989256i	$-0.546703\pi$
$223$	−4.36632	−0.292390	−0.146195	+	0.989256i	$0.546703\pi$
$224$	0	0
$225$	35.0710	2.33807
$226$	0	0
$227$	−21.9914	−1.45962	−0.729810	−	0.683650i	$-0.760391\pi$
$227$	−21.9914	−1.45962	−0.729810	+	0.683650i	$0.760391\pi$
$228$	0	0
$229$	−4.40344	−0.290988	−0.145494	−	0.989359i	$-0.546477\pi$
$229$	−4.40344	−0.290988	−0.145494	+	0.989359i	$0.546477\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.0560	−1.31391	−0.656956	−	0.753929i	$-0.728156\pi$
$233$	−20.0560	−1.31391	−0.656956	+	0.753929i	$0.728156\pi$
$234$	0	0
$235$	2.44488	0.159486
$236$	0	0
$237$	11.9952	0.779173
$238$	0	0
$239$	21.7477	1.40674	0.703370	−	0.710824i	$-0.251678\pi$
$239$	21.7477	1.40674	0.703370	+	0.710824i	$0.251678\pi$
$240$	0	0
$241$	−30.9910	−1.99630	−0.998152	−	0.0607661i	$-0.980646\pi$
$241$	−30.9910	−1.99630	−0.998152	+	0.0607661i	$0.980646\pi$
$242$	0	0
$243$	−22.2044	−1.42441
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.15476	0.200733
$248$	0	0
$249$	21.5863	1.36798
$250$	0	0
$251$	−5.24483	−0.331051	−0.165525	−	0.986206i	$-0.552932\pi$
$251$	−5.24483	−0.331051	−0.165525	+	0.986206i	$0.552932\pi$
$252$	0	0
$253$	4.67370	0.293833
$254$	0	0
$255$	−6.51165	−0.407775
$256$	0	0
$257$	28.5942	1.78366	0.891828	−	0.452374i	$-0.149423\pi$
$257$	28.5942	1.78366	0.891828	+	0.452374i	$0.149423\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.57823	0.407182
$262$	0	0
$263$	5.53993	0.341607	0.170803	−	0.985305i	$-0.445364\pi$
$263$	5.53993	0.341607	0.170803	+	0.985305i	$0.445364\pi$
$264$	0	0
$265$	50.4130	3.09684
$266$	0	0
$267$	5.62181	0.344049
$268$	0	0
$269$	4.29154	0.261660	0.130830	−	0.991405i	$-0.458236\pi$
$269$	4.29154	0.261660	0.130830	+	0.991405i	$0.458236\pi$
$270$	0	0
$271$	0.890669	0.0541043	0.0270521	−	0.999634i	$-0.491388\pi$
$271$	0.890669	0.0541043	0.0270521	+	0.999634i	$0.491388\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.1216	1.09277
$276$	0	0
$277$	−12.0106	−0.721648	−0.360824	−	0.932634i	$-0.617505\pi$
$277$	−12.0106	−0.721648	−0.360824	+	0.932634i	$0.617505\pi$
$278$	0	0
$279$	21.7759	1.30369
$280$	0	0
$281$	−17.4819	−1.04288	−0.521442	−	0.853287i	$-0.674606\pi$
$281$	−17.4819	−1.04288	−0.521442	+	0.853287i	$0.674606\pi$
$282$	0	0
$283$	−20.4863	−1.21778	−0.608891	−	0.793254i	$-0.708385\pi$
$283$	−20.4863	−1.21778	−0.608891	+	0.793254i	$0.708385\pi$
$284$	0	0
$285$	−9.97534	−0.590888
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.5739	−0.974934
$290$	0	0
$291$	−3.08491	−0.180841
$292$	0	0
$293$	22.9489	1.34069	0.670346	−	0.742049i	$-0.266146\pi$
$293$	22.9489	1.34069	0.670346	+	0.742049i	$0.266146\pi$
$294$	0	0
$295$	−58.3664	−3.39822
$296$	0	0
$297$	0.397899	0.0230885
$298$	0	0
$299$	−9.20325	−0.532238
$300$	0	0
$301$	0	0
$302$	0	0
$303$	27.7771	1.59575
$304$	0	0
$305$	−37.7431	−2.16117
$306$	0	0
$307$	−14.1287	−0.806365	−0.403183	−	0.915120i	$-0.632096\pi$
$307$	−14.1287	−0.806365	−0.403183	+	0.915120i	$0.632096\pi$
$308$	0	0
$309$	−8.87437	−0.504846
$310$	0	0
$311$	13.5304	0.767240	0.383620	−	0.923491i	$-0.374677\pi$
$311$	13.5304	0.767240	0.383620	+	0.923491i	$0.374677\pi$
$312$	0	0
$313$	−26.4488	−1.49498	−0.747489	−	0.664275i	$-0.768741\pi$
$313$	−26.4488	−1.49498	−0.747489	+	0.664275i	$0.768741\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.31283	−0.186067	−0.0930335	−	0.995663i	$-0.529656\pi$
$317$	−3.31283	−0.186067	−0.0930335	+	0.995663i	$0.529656\pi$
$318$	0	0
$319$	3.39904	0.190310
$320$	0	0
$321$	−4.31329	−0.240745
$322$	0	0
$323$	0.652775	0.0363213
$324$	0	0
$325$	−35.6842	−1.97940
$326$	0	0
$327$	−4.73851	−0.262040
$328$	0	0
$329$	0	0
$330$	0	0
$331$	11.7870	0.647874	0.323937	−	0.946079i	$-0.394993\pi$
$331$	11.7870	0.647874	0.323937	+	0.946079i	$0.394993\pi$
$332$	0	0
$333$	−16.4635	−0.902194
$334$	0	0
$335$	−11.2494	−0.614620
$336$	0	0
$337$	11.3688	0.619298	0.309649	−	0.950851i	$-0.399789\pi$
$337$	11.3688	0.619298	0.309649	+	0.950851i	$0.399789\pi$
$338$	0	0
$339$	−29.2200	−1.58701
$340$	0	0
$341$	11.2518	0.609320
$342$	0	0
$343$	0	0
$344$	0	0
$345$	29.1006	1.56672
$346$	0	0
$347$	−0.0849530	−0.00456052	−0.00228026	−	0.999997i	$-0.500726\pi$
$347$	−0.0849530	−0.00456052	−0.00228026	+	0.999997i	$0.500726\pi$
$348$	0	0
$349$	−10.5549	−0.564989	−0.282494	−	0.959269i	$-0.591162\pi$
$349$	−10.5549	−0.564989	−0.282494	+	0.959269i	$0.591162\pi$
$350$	0	0
$351$	−0.783526	−0.0418215
$352$	0	0
$353$	0.617246	0.0328527	0.0164263	−	0.999865i	$-0.494771\pi$
$353$	0.617246	0.0328527	0.0164263	+	0.999865i	$0.494771\pi$
$354$	0	0
$355$	−38.3707	−2.03651
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.59248	−0.189604	−0.0948020	−	0.995496i	$-0.530222\pi$
$359$	−3.59248	−0.189604	−0.0948020	+	0.995496i	$0.530222\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−20.8297	−1.09328
$364$	0	0
$365$	67.1548	3.51505
$366$	0	0
$367$	6.18758	0.322989	0.161494	−	0.986874i	$-0.448369\pi$
$367$	6.18758	0.322989	0.161494	+	0.986874i	$0.448369\pi$
$368$	0	0
$369$	6.20111	0.322817
$370$	0	0
$371$	0	0
$372$	0	0
$373$	18.2122	0.942992	0.471496	−	0.881868i	$-0.343714\pi$
$373$	18.2122	0.942992	0.471496	+	0.881868i	$0.343714\pi$
$374$	0	0
$375$	62.9566	3.25106
$376$	0	0
$377$	−6.69324	−0.344719
$378$	0	0
$379$	0.847819	0.0435495	0.0217748	−	0.999763i	$-0.493068\pi$
$379$	0.847819	0.0435495	0.0217748	+	0.999763i	$0.493068\pi$
$380$	0	0
$381$	−37.9340	−1.94342
$382$	0	0
$383$	−33.2555	−1.69928	−0.849639	−	0.527365i	$-0.823180\pi$
$383$	−33.2555	−1.69928	−0.849639	+	0.527365i	$0.823180\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	24.7227	1.25673
$388$	0	0
$389$	−17.7449	−0.899703	−0.449852	−	0.893103i	$-0.648523\pi$
$389$	−17.7449	−0.899703	−0.449852	+	0.893103i	$0.648523\pi$
$390$	0	0
$391$	−1.90431	−0.0963051
$392$	0	0
$393$	33.0546	1.66739
$394$	0	0
$395$	19.6140	0.986888
$396$	0	0
$397$	7.61524	0.382198	0.191099	−	0.981571i	$-0.438795\pi$
$397$	7.61524	0.382198	0.191099	+	0.981571i	$0.438795\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−34.8573	−1.74069	−0.870345	−	0.492442i	$-0.836104\pi$
$401$	−34.8573	−1.74069	−0.870345	+	0.492442i	$0.836104\pi$
$402$	0	0
$403$	−22.1566	−1.10370
$404$	0	0
$405$	−35.0893	−1.74360
$406$	0	0
$407$	−8.50685	−0.421669
$408$	0	0
$409$	3.21019	0.158734	0.0793668	−	0.996845i	$-0.474710\pi$
$409$	3.21019	0.158734	0.0793668	+	0.996845i	$0.474710\pi$
$410$	0	0
$411$	−7.82237	−0.385849
$412$	0	0
$413$	0	0
$414$	0	0
$415$	35.2970	1.73266
$416$	0	0
$417$	40.4830	1.98246
$418$	0	0
$419$	16.7962	0.820550	0.410275	−	0.911962i	$-0.365433\pi$
$419$	16.7962	0.820550	0.410275	+	0.911962i	$0.365433\pi$
$420$	0	0
$421$	−20.3496	−0.991780	−0.495890	−	0.868385i	$-0.665158\pi$
$421$	−20.3496	−0.991780	−0.495890	+	0.868385i	$0.665158\pi$
$422$	0	0
$423$	1.87695	0.0912606
$424$	0	0
$425$	−7.38367	−0.358161
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−12.4835	−0.602711
$430$	0	0
$431$	−32.5259	−1.56672	−0.783358	−	0.621571i	$-0.786495\pi$
$431$	−32.5259	−1.56672	−0.783358	+	0.621571i	$0.786495\pi$
$432$	0	0
$433$	−40.2853	−1.93599	−0.967994	−	0.250975i	$-0.919249\pi$
$433$	−40.2853	−1.93599	−0.967994	+	0.250975i	$0.919249\pi$
$434$	0	0
$435$	21.1640	1.01474
$436$	0	0
$437$	−2.91726	−0.139551
$438$	0	0
$439$	−6.28387	−0.299913	−0.149956	−	0.988693i	$-0.547913\pi$
$439$	−6.28387	−0.299913	−0.149956	+	0.988693i	$0.547913\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.6409	1.21824	0.609118	−	0.793079i	$-0.291523\pi$
$443$	25.6409	1.21824	0.609118	+	0.793079i	$0.291523\pi$
$444$	0	0
$445$	9.19253	0.435768
$446$	0	0
$447$	34.6335	1.63811
$448$	0	0
$449$	−28.7456	−1.35659	−0.678294	−	0.734791i	$-0.737280\pi$
$449$	−28.7456	−1.35659	−0.678294	+	0.734791i	$0.737280\pi$
$450$	0	0
$451$	3.20418	0.150879
$452$	0	0
$453$	45.8077	2.15223
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.9390	1.30693	0.653467	−	0.756955i	$-0.273314\pi$
$457$	27.9390	1.30693	0.653467	+	0.756955i	$0.273314\pi$
$458$	0	0
$459$	−0.162125	−0.00756735
$460$	0	0
$461$	−4.95920	−0.230973	−0.115486	−	0.993309i	$-0.536843\pi$
$461$	−4.95920	−0.230973	−0.115486	+	0.993309i	$0.536843\pi$
$462$	0	0
$463$	13.4715	0.626072	0.313036	−	0.949741i	$-0.398654\pi$
$463$	13.4715	0.626072	0.313036	+	0.949741i	$0.398654\pi$
$464$	0	0
$465$	70.0590	3.24891
$466$	0	0
$467$	−30.9267	−1.43112	−0.715558	−	0.698553i	$-0.753827\pi$
$467$	−30.9267	−1.43112	−0.715558	+	0.698553i	$0.753827\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−58.1699	−2.68033
$472$	0	0
$473$	12.7745	0.587372
$474$	0	0
$475$	−11.3112	−0.518994
$476$	0	0
$477$	38.7024	1.77206
$478$	0	0
$479$	−23.9952	−1.09637	−0.548183	−	0.836358i	$-0.684680\pi$
$479$	−23.9952	−1.09637	−0.548183	+	0.836358i	$0.684680\pi$
$480$	0	0
$481$	16.7513	0.763795
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.04430	−0.229050
$486$	0	0
$487$	39.2762	1.77977	0.889887	−	0.456181i	$-0.150783\pi$
$487$	39.2762	1.77977	0.889887	+	0.456181i	$0.150783\pi$
$488$	0	0
$489$	22.4055	1.01321
$490$	0	0
$491$	28.1575	1.27073	0.635365	−	0.772212i	$-0.280849\pi$
$491$	28.1575	1.27073	0.635365	+	0.772212i	$0.280849\pi$
$492$	0	0
$493$	−1.38495	−0.0623748
$494$	0	0
$495$	20.0618	0.901709
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−32.3051	−1.44617	−0.723087	−	0.690757i	$-0.757278\pi$
$499$	−32.3051	−1.44617	−0.723087	+	0.690757i	$0.757278\pi$
$500$	0	0
$501$	0.473678	0.0211624
$502$	0	0
$503$	−22.4333	−1.00025	−0.500125	−	0.865953i	$-0.666713\pi$
$503$	−22.4333	−1.00025	−0.500125	+	0.865953i	$0.666713\pi$
$504$	0	0
$505$	45.4198	2.02116
$506$	0	0
$507$	−7.52704	−0.334288
$508$	0	0
$509$	2.56739	0.113798	0.0568988	−	0.998380i	$-0.481879\pi$
$509$	2.56739	0.113798	0.0568988	+	0.998380i	$0.481879\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−0.248363	−0.0109655
$514$	0	0
$515$	−14.5110	−0.639429
$516$	0	0
$517$	0.969841	0.0426536
$518$	0	0
$519$	17.2630	0.757762
$520$	0	0
$521$	28.2588	1.23804	0.619020	−	0.785376i	$-0.287530\pi$
$521$	28.2588	1.23804	0.619020	+	0.785376i	$0.287530\pi$
$522$	0	0
$523$	12.4023	0.542316	0.271158	−	0.962535i	$-0.412593\pi$
$523$	12.4023	0.542316	0.271158	+	0.962535i	$0.412593\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.58458	−0.199707
$528$	0	0
$529$	−14.4896	−0.629984
$530$	0	0
$531$	−44.8084	−1.94452
$532$	0	0
$533$	−6.30953	−0.273296
$534$	0	0
$535$	−7.05290	−0.304923
$536$	0	0
$537$	−39.2933	−1.69563
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.18990	0.0511576	0.0255788	−	0.999673i	$-0.491857\pi$
$541$	1.18990	0.0511576	0.0255788	+	0.999673i	$0.491857\pi$
$542$	0	0
$543$	−41.0646	−1.76225
$544$	0	0
$545$	−7.74820	−0.331896
$546$	0	0
$547$	−15.1927	−0.649593	−0.324797	−	0.945784i	$-0.605296\pi$
$547$	−15.1927	−0.649593	−0.324797	+	0.945784i	$0.605296\pi$
$548$	0	0
$549$	−28.9757	−1.23665
$550$	0	0
$551$	−2.12163	−0.0903844
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−52.9676	−2.24835
$556$	0	0
$557$	−24.5874	−1.04180	−0.520902	−	0.853617i	$-0.674404\pi$
$557$	−24.5874	−1.04180	−0.520902	+	0.853617i	$0.674404\pi$
$558$	0	0
$559$	−25.1550	−1.06394
$560$	0	0
$561$	−2.58306	−0.109057
$562$	0	0
$563$	33.8283	1.42569	0.712847	−	0.701319i	$-0.247405\pi$
$563$	33.8283	1.42569	0.712847	+	0.701319i	$0.247405\pi$
$564$	0	0
$565$	−47.7792	−2.01009
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.85831	0.371360	0.185680	−	0.982610i	$-0.440551\pi$
$569$	8.85831	0.371360	0.185680	+	0.982610i	$0.440551\pi$
$570$	0	0
$571$	−36.4577	−1.52571	−0.762854	−	0.646571i	$-0.776202\pi$
$571$	−36.4577	−1.52571	−0.762854	+	0.646571i	$0.776202\pi$
$572$	0	0
$573$	59.3032	2.47743
$574$	0	0
$575$	32.9977	1.37610
$576$	0	0
$577$	−42.2188	−1.75759	−0.878795	−	0.477199i	$-0.841652\pi$
$577$	−42.2188	−1.75759	−0.878795	+	0.477199i	$0.841652\pi$
$578$	0	0
$579$	0.271382	0.0112783
$580$	0	0
$581$	0	0
$582$	0	0
$583$	19.9979	0.828230
$584$	0	0
$585$	−39.5048	−1.63332
$586$	0	0
$587$	24.3851	1.00648	0.503240	−	0.864146i	$-0.332141\pi$
$587$	24.3851	1.00648	0.503240	+	0.864146i	$0.332141\pi$
$588$	0	0
$589$	−7.02322	−0.289387
$590$	0	0
$591$	−26.9919	−1.11030
$592$	0	0
$593$	17.2356	0.707783	0.353892	−	0.935286i	$-0.384858\pi$
$593$	17.2356	0.707783	0.353892	+	0.935286i	$0.384858\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−55.6653	−2.27823
$598$	0	0
$599$	−5.88690	−0.240532	−0.120266	−	0.992742i	$-0.538375\pi$
$599$	−5.88690	−0.240532	−0.120266	+	0.992742i	$0.538375\pi$
$600$	0	0
$601$	13.6088	0.555114	0.277557	−	0.960709i	$-0.410475\pi$
$601$	13.6088	0.555114	0.277557	+	0.960709i	$0.410475\pi$
$602$	0	0
$603$	−8.63625	−0.351695
$604$	0	0
$605$	−34.0597	−1.38473
$606$	0	0
$607$	13.6021	0.552091	0.276045	−	0.961145i	$-0.410976\pi$
$607$	13.6021	0.552091	0.276045	+	0.961145i	$0.410976\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.90977	−0.0772610
$612$	0	0
$613$	−32.9386	−1.33038	−0.665188	−	0.746676i	$-0.731649\pi$
$613$	−32.9386	−1.33038	−0.665188	+	0.746676i	$0.731649\pi$
$614$	0	0
$615$	19.9507	0.804489
$616$	0	0
$617$	39.2585	1.58049	0.790244	−	0.612792i	$-0.209954\pi$
$617$	39.2585	1.58049	0.790244	+	0.612792i	$0.209954\pi$
$618$	0	0
$619$	7.81918	0.314279	0.157140	−	0.987576i	$-0.449773\pi$
$619$	7.81918	0.314279	0.157140	+	0.987576i	$0.449773\pi$
$620$	0	0
$621$	0.724538	0.0290747
$622$	0	0
$623$	0	0
$624$	0	0
$625$	46.3876	1.85550
$626$	0	0
$627$	−3.95705	−0.158029
$628$	0	0
$629$	3.46614	0.138204
$630$	0	0
$631$	−15.0926	−0.600828	−0.300414	−	0.953809i	$-0.597125\pi$
$631$	−15.0926	−0.600828	−0.300414	+	0.953809i	$0.597125\pi$
$632$	0	0
$633$	58.6219	2.33001
$634$	0	0
$635$	−62.0279	−2.46150
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−29.4575	−1.16532
$640$	0	0
$641$	−23.0518	−0.910493	−0.455247	−	0.890365i	$-0.650449\pi$
$641$	−23.0518	−0.910493	−0.455247	+	0.890365i	$0.650449\pi$
$642$	0	0
$643$	−31.5297	−1.24341	−0.621706	−	0.783251i	$-0.713560\pi$
$643$	−31.5297	−1.24341	−0.621706	+	0.783251i	$0.713560\pi$
$644$	0	0
$645$	79.5399	3.13188
$646$	0	0
$647$	−6.44280	−0.253293	−0.126646	−	0.991948i	$-0.540421\pi$
$647$	−6.44280	−0.253293	−0.126646	+	0.991948i	$0.540421\pi$
$648$	0	0
$649$	−23.1529	−0.908833
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.1146	0.669746	0.334873	−	0.942263i	$-0.391307\pi$
$653$	17.1146	0.669746	0.334873	+	0.942263i	$0.391307\pi$
$654$	0	0
$655$	54.0494	2.11188
$656$	0	0
$657$	51.5553	2.01136
$658$	0	0
$659$	27.1362	1.05707	0.528537	−	0.848910i	$-0.322741\pi$
$659$	27.1362	1.05707	0.528537	+	0.848910i	$0.322741\pi$
$660$	0	0
$661$	−9.45983	−0.367945	−0.183972	−	0.982931i	$-0.558896\pi$
$661$	−9.45983	−0.367945	−0.183972	+	0.982931i	$0.558896\pi$
$662$	0	0
$663$	5.08645	0.197541
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.18933	0.239652
$668$	0	0
$669$	−10.7845	−0.416953
$670$	0	0
$671$	−14.9721	−0.577990
$672$	0	0
$673$	−13.6374	−0.525682	−0.262841	−	0.964839i	$-0.584659\pi$
$673$	−13.6374	−0.525682	−0.262841	+	0.964839i	$0.584659\pi$
$674$	0	0
$675$	2.80929	0.108130
$676$	0	0
$677$	2.50516	0.0962810	0.0481405	−	0.998841i	$-0.484670\pi$
$677$	2.50516	0.0962810	0.0481405	+	0.998841i	$0.484670\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−54.3172	−2.08144
$682$	0	0
$683$	−24.0553	−0.920449	−0.460224	−	0.887803i	$-0.652231\pi$
$683$	−24.0553	−0.920449	−0.460224	+	0.887803i	$0.652231\pi$
$684$	0	0
$685$	−12.7908	−0.488710
$686$	0	0
$687$	−10.8762	−0.414953
$688$	0	0
$689$	−39.3791	−1.50022
$690$	0	0
$691$	10.5411	0.401003	0.200501	−	0.979693i	$-0.435743\pi$
$691$	10.5411	0.401003	0.200501	+	0.979693i	$0.435743\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	66.1960	2.51096
$696$	0	0
$697$	−1.30555	−0.0494512
$698$	0	0
$699$	−49.5369	−1.87366
$700$	0	0
$701$	−1.75242	−0.0661880	−0.0330940	−	0.999452i	$-0.510536\pi$
$701$	−1.75242	−0.0661880	−0.0330940	+	0.999452i	$0.510536\pi$
$702$	0	0
$703$	5.30985	0.200265
$704$	0	0
$705$	6.03868	0.227430
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.89668	0.0712312	0.0356156	−	0.999366i	$-0.488661\pi$
$709$	1.89668	0.0712312	0.0356156	+	0.999366i	$0.488661\pi$
$710$	0	0
$711$	15.0578	0.564713
$712$	0	0
$713$	20.4885	0.767301
$714$	0	0
$715$	−20.4125	−0.763384
$716$	0	0
$717$	53.7152	2.00603
$718$	0	0
$719$	−14.3276	−0.534329	−0.267164	−	0.963651i	$-0.586087\pi$
$719$	−14.3276	−0.534329	−0.267164	+	0.963651i	$0.586087\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−76.5455	−2.84676
$724$	0	0
$725$	23.9982	0.891271
$726$	0	0
$727$	42.5021	1.57632	0.788158	−	0.615472i	$-0.211035\pi$
$727$	42.5021	1.57632	0.788158	+	0.615472i	$0.211035\pi$
$728$	0	0
$729$	−28.7786	−1.06588
$730$	0	0
$731$	−5.20500	−0.192514
$732$	0	0
$733$	−19.1811	−0.708471	−0.354235	−	0.935156i	$-0.615259\pi$
$733$	−19.1811	−0.708471	−0.354235	+	0.935156i	$0.615259\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.46244	−0.164376
$738$	0	0
$739$	−12.2726	−0.451454	−0.225727	−	0.974191i	$-0.572476\pi$
$739$	−12.2726	−0.451454	−0.225727	+	0.974191i	$0.572476\pi$
$740$	0	0
$741$	7.79204	0.286248
$742$	0	0
$743$	36.7738	1.34910	0.674549	−	0.738230i	$-0.264338\pi$
$743$	36.7738	1.34910	0.674549	+	0.738230i	$0.264338\pi$
$744$	0	0
$745$	56.6311	2.07480
$746$	0	0
$747$	27.0978	0.991456
$748$	0	0
$749$	0	0
$750$	0	0
$751$	37.4842	1.36782	0.683908	−	0.729568i	$-0.260279\pi$
$751$	37.4842	1.36782	0.683908	+	0.729568i	$0.260279\pi$
$752$	0	0
$753$	−12.9544	−0.472083
$754$	0	0
$755$	74.9026	2.72598
$756$	0	0
$757$	11.0641	0.402132	0.201066	−	0.979578i	$-0.435559\pi$
$757$	11.0641	0.402132	0.201066	+	0.979578i	$0.435559\pi$
$758$	0	0
$759$	11.5437	0.419010
$760$	0	0
$761$	50.5192	1.83132	0.915659	−	0.401955i	$-0.131669\pi$
$761$	50.5192	1.83132	0.915659	+	0.401955i	$0.131669\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−8.17421	−0.295539
$766$	0	0
$767$	45.5918	1.64622
$768$	0	0
$769$	10.7883	0.389037	0.194518	−	0.980899i	$-0.437686\pi$
$769$	10.7883	0.389037	0.194518	+	0.980899i	$0.437686\pi$
$770$	0	0
$771$	70.6257	2.54352
$772$	0	0
$773$	−4.67288	−0.168072	−0.0840358	−	0.996463i	$-0.526781\pi$
$773$	−4.67288	−0.168072	−0.0840358	+	0.996463i	$0.526781\pi$
$774$	0	0
$775$	79.4412	2.85361
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	−15.2210	−0.544650
$782$	0	0
$783$	0.526934	0.0188311
$784$	0	0
$785$	−95.1167	−3.39486
$786$	0	0
$787$	−13.5788	−0.484032	−0.242016	−	0.970272i	$-0.577809\pi$
$787$	−13.5788	−0.484032	−0.242016	+	0.970272i	$0.577809\pi$
$788$	0	0
$789$	13.6832	0.487136
$790$	0	0
$791$	0	0
$792$	0	0
$793$	29.4823	1.04695
$794$	0	0
$795$	124.516	4.41614
$796$	0	0
$797$	−25.3577	−0.898214	−0.449107	−	0.893478i	$-0.648258\pi$
$797$	−25.3577	−0.898214	−0.449107	+	0.893478i	$0.648258\pi$
$798$	0	0
$799$	−0.395164	−0.0139799
$800$	0	0
$801$	7.05718	0.249353
$802$	0	0
$803$	26.6392	0.940076
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.5998	0.373131
$808$	0	0
$809$	1.53872	0.0540985	0.0270493	−	0.999634i	$-0.491389\pi$
$809$	1.53872	0.0540985	0.0270493	+	0.999634i	$0.491389\pi$
$810$	0	0
$811$	−37.5288	−1.31781	−0.658907	−	0.752225i	$-0.728981\pi$
$811$	−37.5288	−1.31781	−0.658907	+	0.752225i	$0.728981\pi$
$812$	0	0
$813$	2.19989	0.0771535
$814$	0	0
$815$	36.6364	1.28332
$816$	0	0
$817$	−7.97365	−0.278963
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.83486	−0.308339	−0.154169	−	0.988044i	$-0.549270\pi$
$821$	−8.83486	−0.308339	−0.154169	+	0.988044i	$0.549270\pi$
$822$	0	0
$823$	−30.8210	−1.07435	−0.537177	−	0.843470i	$-0.680509\pi$
$823$	−30.8210	−1.07435	−0.537177	+	0.843470i	$0.680509\pi$
$824$	0	0
$825$	44.7590	1.55831
$826$	0	0
$827$	0.938262	0.0326266	0.0163133	−	0.999867i	$-0.494807\pi$
$827$	0.938262	0.0326266	0.0163133	+	0.999867i	$0.494807\pi$
$828$	0	0
$829$	6.78457	0.235638	0.117819	−	0.993035i	$-0.462410\pi$
$829$	6.78457	0.235638	0.117819	+	0.993035i	$0.462410\pi$
$830$	0	0
$831$	−29.6654	−1.02908
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0.774537	0.0268039
$836$	0	0
$837$	1.74431	0.0602921
$838$	0	0
$839$	11.6858	0.403437	0.201718	−	0.979444i	$-0.435347\pi$
$839$	11.6858	0.403437	0.201718	+	0.979444i	$0.435347\pi$
$840$	0	0
$841$	−24.4987	−0.844782
$842$	0	0
$843$	−43.1791	−1.48717
$844$	0	0
$845$	−12.3079	−0.423403
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−50.5996	−1.73657
$850$	0	0
$851$	−15.4902	−0.530997
$852$	0	0
$853$	−19.5533	−0.669493	−0.334746	−	0.942308i	$-0.608651\pi$
$853$	−19.5533	−0.669493	−0.334746	+	0.942308i	$0.608651\pi$
$854$	0	0
$855$	−12.5223	−0.428252
$856$	0	0
$857$	−51.4154	−1.75632	−0.878159	−	0.478369i	$-0.841228\pi$
$857$	−51.4154	−1.75632	−0.878159	+	0.478369i	$0.841228\pi$
$858$	0	0
$859$	27.8957	0.951789	0.475894	−	0.879502i	$-0.342124\pi$
$859$	27.8957	0.951789	0.475894	+	0.879502i	$0.342124\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.2264	1.53953	0.769763	−	0.638330i	$-0.220374\pi$
$863$	45.2264	1.53953	0.769763	+	0.638330i	$0.220374\pi$
$864$	0	0
$865$	28.2277	0.959769
$866$	0	0
$867$	−40.9363	−1.39027
$868$	0	0
$869$	7.78054	0.263937
$870$	0	0
$871$	8.78724	0.297744
$872$	0	0
$873$	−3.87255	−0.131066
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.91265	0.300959	0.150479	−	0.988613i	$-0.451918\pi$
$877$	8.91265	0.300959	0.150479	+	0.988613i	$0.451918\pi$
$878$	0	0
$879$	56.6823	1.91185
$880$	0	0
$881$	45.9579	1.54836	0.774180	−	0.632965i	$-0.218163\pi$
$881$	45.9579	1.54836	0.774180	+	0.632965i	$0.218163\pi$
$882$	0	0
$883$	28.6951	0.965666	0.482833	−	0.875712i	$-0.339608\pi$
$883$	28.6951	0.965666	0.482833	+	0.875712i	$0.339608\pi$
$884$	0	0
$885$	−144.161	−4.84592
$886$	0	0
$887$	24.1058	0.809394	0.404697	−	0.914451i	$-0.367377\pi$
$887$	24.1058	0.809394	0.404697	+	0.914451i	$0.367377\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−13.9193	−0.466314
$892$	0	0
$893$	−0.605361	−0.0202576
$894$	0	0
$895$	−64.2505	−2.14766
$896$	0	0
$897$	−22.7314	−0.758979
$898$	0	0
$899$	14.9007	0.496965
$900$	0	0
$901$	−8.14821	−0.271456
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−67.1470	−2.23204
$906$	0	0
$907$	34.3249	1.13974	0.569869	−	0.821735i	$-0.306994\pi$
$907$	34.3249	1.13974	0.569869	+	0.821735i	$0.306994\pi$
$908$	0	0
$909$	34.8692	1.15654
$910$	0	0
$911$	25.9008	0.858133	0.429066	−	0.903273i	$-0.358843\pi$
$911$	25.9008	0.858133	0.429066	+	0.903273i	$0.358843\pi$
$912$	0	0
$913$	14.0017	0.463389
$914$	0	0
$915$	−93.2229	−3.08186
$916$	0	0
$917$	0	0
$918$	0	0
$919$	24.2185	0.798894	0.399447	−	0.916756i	$-0.369202\pi$
$919$	24.2185	0.798894	0.399447	+	0.916756i	$0.369202\pi$
$920$	0	0
$921$	−34.8968	−1.14989
$922$	0	0
$923$	29.9725	0.986558
$924$	0	0
$925$	−60.0609	−1.97479
$926$	0	0
$927$	−11.1402	−0.365892
$928$	0	0
$929$	12.6889	0.416310	0.208155	−	0.978096i	$-0.433254\pi$
$929$	12.6889	0.416310	0.208155	+	0.978096i	$0.433254\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	33.4192	1.09410
$934$	0	0
$935$	−4.22370	−0.138130
$936$	0	0
$937$	−44.2503	−1.44559	−0.722797	−	0.691060i	$-0.757144\pi$
$937$	−44.2503	−1.44559	−0.722797	+	0.691060i	$0.757144\pi$
$938$	0	0
$939$	−65.3268	−2.13186
$940$	0	0
$941$	13.7250	0.447421	0.223710	−	0.974656i	$-0.428183\pi$
$941$	13.7250	0.447421	0.223710	+	0.974656i	$0.428183\pi$
$942$	0	0
$943$	5.83451	0.189998
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.6683	−0.671630	−0.335815	−	0.941928i	$-0.609012\pi$
$947$	−20.6683	−0.671630	−0.335815	+	0.941928i	$0.609012\pi$
$948$	0	0
$949$	−52.4567	−1.70282
$950$	0	0
$951$	−8.18246	−0.265334
$952$	0	0
$953$	16.3518	0.529686	0.264843	−	0.964292i	$-0.414680\pi$
$953$	16.3518	0.529686	0.264843	+	0.964292i	$0.414680\pi$
$954$	0	0
$955$	96.9699	3.13787
$956$	0	0
$957$	8.39538	0.271384
$958$	0	0
$959$	0	0
$960$	0	0
$961$	18.3256	0.591150
$962$	0	0
$963$	−5.41457	−0.174482
$964$	0	0
$965$	0.443752	0.0142849
$966$	0	0
$967$	20.6838	0.665145	0.332573	−	0.943078i	$-0.392083\pi$
$967$	20.6838	0.665145	0.332573	+	0.943078i	$0.392083\pi$
$968$	0	0
$969$	1.61231	0.0517948
$970$	0	0
$971$	2.05777	0.0660368	0.0330184	−	0.999455i	$-0.489488\pi$
$971$	2.05777	0.0660368	0.0330184	+	0.999455i	$0.489488\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−88.1375	−2.82266
$976$	0	0
$977$	5.75307	0.184057	0.0920285	−	0.995756i	$-0.470665\pi$
$977$	5.75307	0.184057	0.0920285	+	0.995756i	$0.470665\pi$
$978$	0	0
$979$	3.64652	0.116543
$980$	0	0
$981$	−5.94835	−0.189916
$982$	0	0
$983$	26.9379	0.859186	0.429593	−	0.903023i	$-0.358657\pi$
$983$	26.9379	0.859186	0.429593	+	0.903023i	$0.358657\pi$
$984$	0	0
$985$	−44.1358	−1.40628
$986$	0	0
$987$	0	0
$988$	0	0
$989$	23.2612	0.739662
$990$	0	0
$991$	41.9464	1.33247	0.666235	−	0.745742i	$-0.267905\pi$
$991$	41.9464	1.33247	0.666235	+	0.745742i	$0.267905\pi$
$992$	0	0
$993$	29.1132	0.923878
$994$	0	0
$995$	−91.0213	−2.88557
$996$	0	0
$997$	41.9695	1.32919	0.664593	−	0.747205i	$-0.268605\pi$
$997$	41.9695	1.32919	0.664593	+	0.747205i	$0.268605\pi$
$998$	0	0
$999$	−1.31877	−0.0417240

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.br.1.7		8
7.3	odd	6		1064.2.q.n.457.7	yes	16
7.5	odd	6		1064.2.q.n.305.7	✓	16
7.6	odd	2		7448.2.a.bq.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.n.305.7	✓	16	7.5	odd	6
1064.2.q.n.457.7	yes	16	7.3	odd	6
7448.2.a.bq.1.2		8	7.6	odd	2
7448.2.a.br.1.7		8	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+2.46993 q^{3} +4.03871 q^{5} +3.10055 q^{9} +O(q^{10})\)	\(q+2.46993 q^{3} +4.03871 q^{5} +3.10055 q^{9} +1.60209 q^{11} -3.15476 q^{13} +9.97534 q^{15} -0.652775 q^{17} -1.00000 q^{19} +2.91726 q^{23} +11.3112 q^{25} +0.248363 q^{27} +2.12163 q^{29} +7.02322 q^{31} +3.95705 q^{33} -5.30985 q^{37} -7.79204 q^{39} +2.00000 q^{41} +7.97365 q^{43} +12.5223 q^{45} +0.605361 q^{47} -1.61231 q^{51} +12.4824 q^{53} +6.47038 q^{55} -2.46993 q^{57} -14.4517 q^{59} -9.34534 q^{61} -12.7412 q^{65} -2.78539 q^{67} +7.20542 q^{69} -9.50073 q^{71} +16.6278 q^{73} +27.9379 q^{75} +4.85650 q^{79} -8.68822 q^{81} +8.73966 q^{83} -2.63637 q^{85} +5.24028 q^{87} +2.27610 q^{89} +17.3469 q^{93} -4.03871 q^{95} -1.24899 q^{97} +4.96736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{5} + 6 q^{9}+O(q^{10}) \) 8 * q + q^5 + 6 * q^9	\( 8 q + q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} - 4 q^{17} - 8 q^{19} + 25 q^{23} + 15 q^{25} - 3 q^{27} + 6 q^{29} - 8 q^{33} + 13 q^{37} - 11 q^{39} + 16 q^{41} + 17 q^{43} - 17 q^{45} + 24 q^{47} + 5 q^{51} + 2 q^{53} - 5 q^{55} - 2 q^{59} + 13 q^{61} - 26 q^{65} + 2 q^{67} + 11 q^{69} + 10 q^{71} - 5 q^{73} + 20 q^{75} + 16 q^{79} - 12 q^{81} + 43 q^{83} + 24 q^{85} - 20 q^{87} - 8 q^{89} + 2 q^{93} - q^{95} + 12 q^{97} + 37 q^{99}+O(q^{100}) \) 8 * q + q^5 + 6 * q^9 + 9 * q^11 + 8 * q^15 - 4 * q^17 - 8 * q^19 + 25 * q^23 + 15 * q^25 - 3 * q^27 + 6 * q^29 - 8 * q^33 + 13 * q^37 - 11 * q^39 + 16 * q^41 + 17 * q^43 - 17 * q^45 + 24 * q^47 + 5 * q^51 + 2 * q^53 - 5 * q^55 - 2 * q^59 + 13 * q^61 - 26 * q^65 + 2 * q^67 + 11 * q^69 + 10 * q^71 - 5 * q^73 + 20 * q^75 + 16 * q^79 - 12 * q^81 + 43 * q^83 + 24 * q^85 - 20 * q^87 - 8 * q^89 + 2 * q^93 - q^95 + 12 * q^97 + 37 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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