LMFDB - Embedded newform 7448.2.a.br.1.8

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			1.8
Root			$-2.59528$ 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)

$f(q)$	$=$	$q+2.59528 q^{3} -1.66621 q^{5} +3.73546 q^{9} +O(q^{10})$	$q+2.59528 q^{3} -1.66621 q^{5} +3.73546 q^{9} +0.145289 q^{11} +1.47949 q^{13} -4.32429 q^{15} +5.90894 q^{17} -1.00000 q^{19} +6.22006 q^{23} -2.22373 q^{25} +1.90873 q^{27} -3.13688 q^{29} +0.882526 q^{31} +0.377064 q^{33} -1.96240 q^{37} +3.83969 q^{39} +2.00000 q^{41} -2.05954 q^{43} -6.22408 q^{45} +0.174405 q^{47} +15.3353 q^{51} +8.43843 q^{53} -0.242082 q^{55} -2.59528 q^{57} +1.58188 q^{59} +8.55244 q^{61} -2.46515 q^{65} +1.33931 q^{67} +16.1428 q^{69} +7.37135 q^{71} -4.72425 q^{73} -5.77120 q^{75} +11.4986 q^{79} -6.25271 q^{81} -4.37769 q^{83} -9.84555 q^{85} -8.14108 q^{87} +1.39298 q^{89} +2.29040 q^{93} +1.66621 q^{95} -10.6777 q^{97} +0.542720 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.59528	1.49838	0.749192	−	0.662353i	$-0.230442\pi$
$3$	2.59528	1.49838	0.749192	+	0.662353i	$0.230442\pi$
$4$	0	0
$5$	−1.66621	−0.745153	−0.372577	−	0.928001i	$-0.621526\pi$
$5$	−1.66621	−0.745153	−0.372577	+	0.928001i	$0.621526\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.73546	1.24515
$10$	0	0
$11$	0.145289	0.0438062	0.0219031	−	0.999760i	$-0.493027\pi$
$11$	0.145289	0.0438062	0.0219031	+	0.999760i	$0.493027\pi$
$12$	0	0
$13$	1.47949	0.410337	0.205169	−	0.978727i	$-0.434226\pi$
$13$	1.47949	0.410337	0.205169	+	0.978727i	$0.434226\pi$
$14$	0	0
$15$	−4.32429	−1.11653
$16$	0	0
$17$	5.90894	1.43313	0.716564	−	0.697521i	$-0.245714\pi$
$17$	5.90894	1.43313	0.716564	+	0.697521i	$0.245714\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.22006	1.29697	0.648486	−	0.761227i	$-0.275403\pi$
$23$	6.22006	1.29697	0.648486	+	0.761227i	$0.275403\pi$
$24$	0	0
$25$	−2.22373	−0.444747
$26$	0	0
$27$	1.90873	0.367335
$28$	0	0
$29$	−3.13688	−0.582504	−0.291252	−	0.956646i	$-0.594072\pi$
$29$	−3.13688	−0.582504	−0.291252	+	0.956646i	$0.594072\pi$
$30$	0	0
$31$	0.882526	0.158506	0.0792532	−	0.996855i	$-0.474746\pi$
$31$	0.882526	0.158506	0.0792532	+	0.996855i	$0.474746\pi$
$32$	0	0
$33$	0.377064	0.0656384
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.96240	−0.322616	−0.161308	−	0.986904i	$-0.551571\pi$
$37$	−1.96240	−0.322616	−0.161308	+	0.986904i	$0.551571\pi$
$38$	0	0
$39$	3.83969	0.614842
$40$	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
$43$	−2.05954	−0.314076	−0.157038	−	0.987593i	$-0.550195\pi$
$43$	−2.05954	−0.314076	−0.157038	+	0.987593i	$0.550195\pi$
$44$	0	0
$45$	−6.22408	−0.927831
$46$	0	0
$47$	0.174405	0.0254395	0.0127198	−	0.999919i	$-0.495951\pi$
$47$	0.174405	0.0254395	0.0127198	+	0.999919i	$0.495951\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	15.3353	2.14738
$52$	0	0
$53$	8.43843	1.15911	0.579553	−	0.814934i	$-0.303227\pi$
$53$	8.43843	1.15911	0.579553	+	0.814934i	$0.303227\pi$
$54$	0	0
$55$	−0.242082	−0.0326423
$56$	0	0
$57$	−2.59528	−0.343753
$58$	0	0
$59$	1.58188	0.205943	0.102971	−	0.994684i	$-0.467165\pi$
$59$	1.58188	0.205943	0.102971	+	0.994684i	$0.467165\pi$
$60$	0	0
$61$	8.55244	1.09503	0.547514	−	0.836797i	$-0.315574\pi$
$61$	8.55244	1.09503	0.547514	+	0.836797i	$0.315574\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.46515	−0.305764
$66$	0	0
$67$	1.33931	0.163622	0.0818112	−	0.996648i	$-0.473930\pi$
$67$	1.33931	0.163622	0.0818112	+	0.996648i	$0.473930\pi$
$68$	0	0
$69$	16.1428	1.94336
$70$	0	0
$71$	7.37135	0.874819	0.437409	−	0.899263i	$-0.355896\pi$
$71$	7.37135	0.874819	0.437409	+	0.899263i	$0.355896\pi$
$72$	0	0
$73$	−4.72425	−0.552931	−0.276466	−	0.961024i	$-0.589163\pi$
$73$	−4.72425	−0.552931	−0.276466	+	0.961024i	$0.589163\pi$
$74$	0	0
$75$	−5.77120	−0.666401
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.4986	1.29369	0.646846	−	0.762621i	$-0.276088\pi$
$79$	11.4986	1.29369	0.646846	+	0.762621i	$0.276088\pi$
$80$	0	0
$81$	−6.25271	−0.694746
$82$	0	0
$83$	−4.37769	−0.480514	−0.240257	−	0.970709i	$-0.577232\pi$
$83$	−4.37769	−0.480514	−0.240257	+	0.970709i	$0.577232\pi$
$84$	0	0
$85$	−9.84555	−1.06790
$86$	0	0
$87$	−8.14108	−0.872815
$88$	0	0
$89$	1.39298	0.147655	0.0738277	−	0.997271i	$-0.476479\pi$
$89$	1.39298	0.147655	0.0738277	+	0.997271i	$0.476479\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.29040	0.237503
$94$	0	0
$95$	1.66621	0.170950
$96$	0	0
$97$	−10.6777	−1.08416	−0.542080	−	0.840327i	$-0.682363\pi$
$97$	−10.6777	−1.08416	−0.542080	+	0.840327i	$0.682363\pi$
$98$	0	0
$99$	0.542720	0.0545454
$100$	0	0
$101$	1.20245	0.119649	0.0598244	−	0.998209i	$-0.480946\pi$
$101$	1.20245	0.119649	0.0598244	+	0.998209i	$0.480946\pi$
$102$	0	0
$103$	19.5472	1.92605	0.963024	−	0.269417i	$-0.0868309\pi$
$103$	19.5472	1.92605	0.963024	+	0.269417i	$0.0868309\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.02110	0.775429	0.387714	−	0.921780i	$-0.373265\pi$
$107$	8.02110	0.775429	0.387714	+	0.921780i	$0.373265\pi$
$108$	0	0
$109$	20.3842	1.95245	0.976226	−	0.216755i	$-0.0695472\pi$
$109$	20.3842	1.95245	0.976226	+	0.216755i	$0.0695472\pi$
$110$	0	0
$111$	−5.09297	−0.483403
$112$	0	0
$113$	8.93908	0.840918	0.420459	−	0.907312i	$-0.361869\pi$
$113$	8.93908	0.840918	0.420459	+	0.907312i	$0.361869\pi$
$114$	0	0
$115$	−10.3639	−0.966443
$116$	0	0
$117$	5.52658	0.510933
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9789	−0.998081
$122$	0	0
$123$	5.19055	0.468016
$124$	0	0
$125$	12.0363	1.07656
$126$	0	0
$127$	1.76063	0.156231	0.0781154	−	0.996944i	$-0.475110\pi$
$127$	1.76063	0.156231	0.0781154	+	0.996944i	$0.475110\pi$
$128$	0	0
$129$	−5.34506	−0.470607
$130$	0	0
$131$	−4.04271	−0.353213	−0.176607	−	0.984281i	$-0.556512\pi$
$131$	−4.04271	−0.353213	−0.176607	+	0.984281i	$0.556512\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.18035	−0.273721
$136$	0	0
$137$	−20.4084	−1.74361	−0.871805	−	0.489853i	$-0.837051\pi$
$137$	−20.4084	−1.74361	−0.871805	+	0.489853i	$0.837051\pi$
$138$	0	0
$139$	−11.2435	−0.953662	−0.476831	−	0.878995i	$-0.658215\pi$
$139$	−11.2435	−0.953662	−0.476831	+	0.878995i	$0.658215\pi$
$140$	0	0
$141$	0.452628	0.0381182
$142$	0	0
$143$	0.214953	0.0179753
$144$	0	0
$145$	5.22672	0.434055
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.58697	−0.457702	−0.228851	−	0.973461i	$-0.573497\pi$
$149$	−5.58697	−0.457702	−0.228851	+	0.973461i	$0.573497\pi$
$150$	0	0
$151$	−19.3575	−1.57529	−0.787644	−	0.616131i	$-0.788699\pi$
$151$	−19.3575	−1.57529	−0.787644	+	0.616131i	$0.788699\pi$
$152$	0	0
$153$	22.0726	1.78446
$154$	0	0
$155$	−1.47048	−0.118112
$156$	0	0
$157$	10.9235	0.871792	0.435896	−	0.899997i	$-0.356432\pi$
$157$	10.9235	0.871792	0.435896	+	0.899997i	$0.356432\pi$
$158$	0	0
$159$	21.9000	1.73679
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.2988	1.27662	0.638310	−	0.769779i	$-0.279634\pi$
$163$	16.2988	1.27662	0.638310	+	0.769779i	$0.279634\pi$
$164$	0	0
$165$	−0.628269	−0.0489107
$166$	0	0
$167$	−16.7379	−1.29522	−0.647610	−	0.761972i	$-0.724231\pi$
$167$	−16.7379	−1.29522	−0.647610	+	0.761972i	$0.724231\pi$
$168$	0	0
$169$	−10.8111	−0.831623
$170$	0	0
$171$	−3.73546	−0.285658
$172$	0	0
$173$	18.2402	1.38678	0.693389	−	0.720564i	$-0.256117\pi$
$173$	18.2402	1.38678	0.693389	+	0.720564i	$0.256117\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.10540	0.308581
$178$	0	0
$179$	−2.03827	−0.152347	−0.0761737	−	0.997095i	$-0.524270\pi$
$179$	−2.03827	−0.152347	−0.0761737	+	0.997095i	$0.524270\pi$
$180$	0	0
$181$	8.70643	0.647144	0.323572	−	0.946204i	$-0.395116\pi$
$181$	8.70643	0.647144	0.323572	+	0.946204i	$0.395116\pi$
$182$	0	0
$183$	22.1960	1.64077
$184$	0	0
$185$	3.26977	0.240399
$186$	0	0
$187$	0.858501	0.0627798
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.6580	1.06062	0.530309	−	0.847805i	$-0.322076\pi$
$191$	14.6580	1.06062	0.530309	+	0.847805i	$0.322076\pi$
$192$	0	0
$193$	16.7805	1.20789	0.603943	−	0.797028i	$-0.293596\pi$
$193$	16.7805	1.20789	0.603943	+	0.797028i	$0.293596\pi$
$194$	0	0
$195$	−6.39774	−0.458152
$196$	0	0
$197$	24.0559	1.71391	0.856955	−	0.515391i	$-0.172353\pi$
$197$	24.0559	1.71391	0.856955	+	0.515391i	$0.172353\pi$
$198$	0	0
$199$	−6.26427	−0.444062	−0.222031	−	0.975040i	$-0.571269\pi$
$199$	−6.26427	−0.444062	−0.222031	+	0.975040i	$0.571269\pi$
$200$	0	0
$201$	3.47587	0.245169
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.33243	−0.232747
$206$	0	0
$207$	23.2348	1.61493
$208$	0	0
$209$	−0.145289	−0.0100498
$210$	0	0
$211$	−4.39876	−0.302823	−0.151412	−	0.988471i	$-0.548382\pi$
$211$	−4.39876	−0.302823	−0.151412	+	0.988471i	$0.548382\pi$
$212$	0	0
$213$	19.1307	1.31081
$214$	0	0
$215$	3.43163	0.234035
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−12.2607	−0.828504
$220$	0	0
$221$	8.74222	0.588066
$222$	0	0
$223$	−19.7309	−1.32128	−0.660639	−	0.750703i	$-0.729715\pi$
$223$	−19.7309	−1.32128	−0.660639	+	0.750703i	$0.729715\pi$
$224$	0	0
$225$	−8.30667	−0.553778
$226$	0	0
$227$	8.14720	0.540748	0.270374	−	0.962755i	$-0.412853\pi$
$227$	8.14720	0.540748	0.270374	+	0.962755i	$0.412853\pi$
$228$	0	0
$229$	16.9770	1.12187	0.560937	−	0.827859i	$-0.310441\pi$
$229$	16.9770	1.12187	0.560937	+	0.827859i	$0.310441\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.19191	−0.209109	−0.104555	−	0.994519i	$-0.533342\pi$
$233$	−3.19191	−0.209109	−0.104555	+	0.994519i	$0.533342\pi$
$234$	0	0
$235$	−0.290595	−0.0189563
$236$	0	0
$237$	29.8420	1.93845
$238$	0	0
$239$	−19.9212	−1.28859	−0.644297	−	0.764775i	$-0.722850\pi$
$239$	−19.9212	−1.28859	−0.644297	+	0.764775i	$0.722850\pi$
$240$	0	0
$241$	8.29145	0.534099	0.267050	−	0.963683i	$-0.413951\pi$
$241$	8.29145	0.534099	0.267050	+	0.963683i	$0.413951\pi$
$242$	0	0
$243$	−21.9537	−1.40833
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.47949	−0.0941378
$248$	0	0
$249$	−11.3613	−0.719995
$250$	0	0
$251$	10.2504	0.647002	0.323501	−	0.946228i	$-0.395140\pi$
$251$	10.2504	0.647002	0.323501	+	0.946228i	$0.395140\pi$
$252$	0	0
$253$	0.903703	0.0568153
$254$	0	0
$255$	−25.5519	−1.60012
$256$	0	0
$257$	9.14284	0.570315	0.285157	−	0.958481i	$-0.407954\pi$
$257$	9.14284	0.570315	0.285157	+	0.958481i	$0.407954\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−11.7177	−0.725308
$262$	0	0
$263$	2.67921	0.165207	0.0826036	−	0.996582i	$-0.473676\pi$
$263$	2.67921	0.165207	0.0826036	+	0.996582i	$0.473676\pi$
$264$	0	0
$265$	−14.0602	−0.863712
$266$	0	0
$267$	3.61516	0.221244
$268$	0	0
$269$	−25.2063	−1.53685	−0.768426	−	0.639938i	$-0.778960\pi$
$269$	−25.2063	−1.53685	−0.768426	+	0.639938i	$0.778960\pi$
$270$	0	0
$271$	12.6703	0.769663	0.384831	−	0.922987i	$-0.374260\pi$
$271$	12.6703	0.769663	0.384831	+	0.922987i	$0.374260\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.323083	−0.0194826
$276$	0	0
$277$	22.0486	1.32477	0.662386	−	0.749163i	$-0.269544\pi$
$277$	22.0486	1.32477	0.662386	+	0.749163i	$0.269544\pi$
$278$	0	0
$279$	3.29664	0.197365
$280$	0	0
$281$	10.9385	0.652539	0.326270	−	0.945277i	$-0.394208\pi$
$281$	10.9385	0.652539	0.326270	+	0.945277i	$0.394208\pi$
$282$	0	0
$283$	−20.6217	−1.22583	−0.612916	−	0.790148i	$-0.710004\pi$
$283$	−20.6217	−1.22583	−0.612916	+	0.790148i	$0.710004\pi$
$284$	0	0
$285$	4.32429	0.256149
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.9156	1.05386
$290$	0	0
$291$	−27.7117	−1.62449
$292$	0	0
$293$	−23.8496	−1.39331	−0.696654	−	0.717407i	$-0.745329\pi$
$293$	−23.8496	−1.39331	−0.696654	+	0.717407i	$0.745329\pi$
$294$	0	0
$295$	−2.63574	−0.153459
$296$	0	0
$297$	0.277316	0.0160915
$298$	0	0
$299$	9.20252	0.532196
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3.12070	0.179280
$304$	0	0
$305$	−14.2502	−0.815964
$306$	0	0
$307$	14.6943	0.838647	0.419323	−	0.907837i	$-0.362267\pi$
$307$	14.6943	0.838647	0.419323	+	0.907837i	$0.362267\pi$
$308$	0	0
$309$	50.7305	2.88596
$310$	0	0
$311$	−5.91507	−0.335413	−0.167706	−	0.985837i	$-0.553636\pi$
$311$	−5.91507	−0.335413	−0.167706	+	0.985837i	$0.553636\pi$
$312$	0	0
$313$	−0.910652	−0.0514731	−0.0257365	−	0.999669i	$-0.508193\pi$
$313$	−0.910652	−0.0514731	−0.0257365	+	0.999669i	$0.508193\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.4108	−1.42721	−0.713607	−	0.700546i	$-0.752940\pi$
$317$	−25.4108	−1.42721	−0.713607	+	0.700546i	$0.752940\pi$
$318$	0	0
$319$	−0.455753	−0.0255173
$320$	0	0
$321$	20.8170	1.16189
$322$	0	0
$323$	−5.90894	−0.328782
$324$	0	0
$325$	−3.28999	−0.182496
$326$	0	0
$327$	52.9026	2.92552
$328$	0	0
$329$	0	0
$330$	0	0
$331$	9.79525	0.538396	0.269198	−	0.963085i	$-0.413241\pi$
$331$	9.79525	0.538396	0.269198	+	0.963085i	$0.413241\pi$
$332$	0	0
$333$	−7.33046	−0.401707
$334$	0	0
$335$	−2.23157	−0.121924
$336$	0	0
$337$	23.8358	1.29842	0.649210	−	0.760609i	$-0.275100\pi$
$337$	23.8358	1.29842	0.649210	+	0.760609i	$0.275100\pi$
$338$	0	0
$339$	23.1994	1.26002
$340$	0	0
$341$	0.128221	0.00694356
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−26.8973	−1.44810
$346$	0	0
$347$	26.6393	1.43007	0.715035	−	0.699089i	$-0.246411\pi$
$347$	26.6393	1.43007	0.715035	+	0.699089i	$0.246411\pi$
$348$	0	0
$349$	6.31857	0.338225	0.169113	−	0.985597i	$-0.445910\pi$
$349$	6.31857	0.338225	0.169113	+	0.985597i	$0.445910\pi$
$350$	0	0
$351$	2.82395	0.150731
$352$	0	0
$353$	−3.12471	−0.166311	−0.0831557	−	0.996537i	$-0.526500\pi$
$353$	−3.12471	−0.166311	−0.0831557	+	0.996537i	$0.526500\pi$
$354$	0	0
$355$	−12.2822	−0.651874
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.1244	−0.798234	−0.399117	−	0.916900i	$-0.630683\pi$
$359$	−15.1244	−0.798234	−0.399117	+	0.916900i	$0.630683\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−28.4933	−1.49551
$364$	0	0
$365$	7.87161	0.412019
$366$	0	0
$367$	−4.45173	−0.232378	−0.116189	−	0.993227i	$-0.537068\pi$
$367$	−4.45173	−0.232378	−0.116189	+	0.993227i	$0.537068\pi$
$368$	0	0
$369$	7.47092	0.388921
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.6612	0.603792	0.301896	−	0.953341i	$-0.402380\pi$
$373$	11.6612	0.603792	0.301896	+	0.953341i	$0.402380\pi$
$374$	0	0
$375$	31.2375	1.61310
$376$	0	0
$377$	−4.64099	−0.239023
$378$	0	0
$379$	36.0926	1.85395	0.926976	−	0.375122i	$-0.122399\pi$
$379$	36.0926	1.85395	0.926976	+	0.375122i	$0.122399\pi$
$380$	0	0
$381$	4.56933	0.234094
$382$	0	0
$383$	−14.5992	−0.745984	−0.372992	−	0.927834i	$-0.621668\pi$
$383$	−14.5992	−0.745984	−0.372992	+	0.927834i	$0.621668\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.69332	−0.391073
$388$	0	0
$389$	4.53987	0.230181	0.115090	−	0.993355i	$-0.463284\pi$
$389$	4.53987	0.230181	0.115090	+	0.993355i	$0.463284\pi$
$390$	0	0
$391$	36.7539	1.85873
$392$	0	0
$393$	−10.4920	−0.529249
$394$	0	0
$395$	−19.1591	−0.963998
$396$	0	0
$397$	−21.8844	−1.09835	−0.549175	−	0.835708i	$-0.685058\pi$
$397$	−21.8844	−1.09835	−0.549175	+	0.835708i	$0.685058\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.07346	0.403169	0.201585	−	0.979471i	$-0.435391\pi$
$401$	8.07346	0.403169	0.201585	+	0.979471i	$0.435391\pi$
$402$	0	0
$403$	1.30569	0.0650411
$404$	0	0
$405$	10.4184	0.517692
$406$	0	0
$407$	−0.285114	−0.0141326
$408$	0	0
$409$	−37.5265	−1.85557	−0.927784	−	0.373118i	$-0.878289\pi$
$409$	−37.5265	−1.85557	−0.927784	+	0.373118i	$0.878289\pi$
$410$	0	0
$411$	−52.9655	−2.61260
$412$	0	0
$413$	0	0
$414$	0	0
$415$	7.29417	0.358057
$416$	0	0
$417$	−29.1800	−1.42895
$418$	0	0
$419$	24.9638	1.21956	0.609782	−	0.792569i	$-0.291257\pi$
$419$	24.9638	1.21956	0.609782	+	0.792569i	$0.291257\pi$
$420$	0	0
$421$	17.4710	0.851485	0.425743	−	0.904844i	$-0.360013\pi$
$421$	17.4710	0.851485	0.425743	+	0.904844i	$0.360013\pi$
$422$	0	0
$423$	0.651482	0.0316761
$424$	0	0
$425$	−13.1399	−0.637379
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.557863	0.0269339
$430$	0	0
$431$	25.5759	1.23195	0.615973	−	0.787767i	$-0.288763\pi$
$431$	25.5759	1.23195	0.615973	+	0.787767i	$0.288763\pi$
$432$	0	0
$433$	21.5603	1.03612	0.518060	−	0.855344i	$-0.326654\pi$
$433$	21.5603	1.03612	0.518060	+	0.855344i	$0.326654\pi$
$434$	0	0
$435$	13.5648	0.650381
$436$	0	0
$437$	−6.22006	−0.297546
$438$	0	0
$439$	−25.8430	−1.23342	−0.616710	−	0.787190i	$-0.711535\pi$
$439$	−25.8430	−1.23342	−0.616710	+	0.787190i	$0.711535\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.0570	1.38054	0.690270	−	0.723552i	$-0.257492\pi$
$443$	29.0570	1.38054	0.690270	+	0.723552i	$0.257492\pi$
$444$	0	0
$445$	−2.32100	−0.110026
$446$	0	0
$447$	−14.4997	−0.685814
$448$	0	0
$449$	29.6266	1.39817	0.699083	−	0.715040i	$-0.253592\pi$
$449$	29.6266	1.39817	0.699083	+	0.715040i	$0.253592\pi$
$450$	0	0
$451$	0.290577	0.0136827
$452$	0	0
$453$	−50.2380	−2.36039
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.03231	−0.188624	−0.0943119	−	0.995543i	$-0.530065\pi$
$457$	−4.03231	−0.188624	−0.0943119	+	0.995543i	$0.530065\pi$
$458$	0	0
$459$	11.2785	0.526438
$460$	0	0
$461$	−17.6181	−0.820555	−0.410277	−	0.911961i	$-0.634568\pi$
$461$	−17.6181	−0.820555	−0.410277	+	0.911961i	$0.634568\pi$
$462$	0	0
$463$	−26.2595	−1.22038	−0.610192	−	0.792253i	$-0.708908\pi$
$463$	−26.2595	−1.22038	−0.610192	+	0.792253i	$0.708908\pi$
$464$	0	0
$465$	−3.81630	−0.176976
$466$	0	0
$467$	−34.9518	−1.61738	−0.808688	−	0.588238i	$-0.799822\pi$
$467$	−34.9518	−1.61738	−0.808688	+	0.588238i	$0.799822\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	28.3496	1.30628
$472$	0	0
$473$	−0.299227	−0.0137585
$474$	0	0
$475$	2.22373	0.102032
$476$	0	0
$477$	31.5214	1.44327
$478$	0	0
$479$	27.0479	1.23585	0.617925	−	0.786237i	$-0.287974\pi$
$479$	27.0479	1.23585	0.617925	+	0.786237i	$0.287974\pi$
$480$	0	0
$481$	−2.90335	−0.132381
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.7914	0.807865
$486$	0	0
$487$	19.2167	0.870790	0.435395	−	0.900239i	$-0.356609\pi$
$487$	19.2167	0.870790	0.435395	+	0.900239i	$0.356609\pi$
$488$	0	0
$489$	42.2999	1.91287
$490$	0	0
$491$	31.6972	1.43048	0.715238	−	0.698881i	$-0.246318\pi$
$491$	31.6972	1.43048	0.715238	+	0.698881i	$0.246318\pi$
$492$	0	0
$493$	−18.5356	−0.834803
$494$	0	0
$495$	−0.904287	−0.0406447
$496$	0	0
$497$	0	0
$498$	0	0
$499$	21.5590	0.965112	0.482556	−	0.875865i	$-0.339708\pi$
$499$	21.5590	0.965112	0.482556	+	0.875865i	$0.339708\pi$
$500$	0	0
$501$	−43.4396	−1.94074
$502$	0	0
$503$	7.27173	0.324230	0.162115	−	0.986772i	$-0.448168\pi$
$503$	7.27173	0.324230	0.162115	+	0.986772i	$0.448168\pi$
$504$	0	0
$505$	−2.00355	−0.0891566
$506$	0	0
$507$	−28.0578	−1.24609
$508$	0	0
$509$	−40.6805	−1.80313	−0.901567	−	0.432640i	$-0.857582\pi$
$509$	−40.6805	−1.80313	−0.901567	+	0.432640i	$0.857582\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.90873	−0.0842723
$514$	0	0
$515$	−32.5699	−1.43520
$516$	0	0
$517$	0.0253390	0.00111441
$518$	0	0
$519$	47.3384	2.07792
$520$	0	0
$521$	2.03869	0.0893167	0.0446583	−	0.999002i	$-0.485780\pi$
$521$	2.03869	0.0893167	0.0446583	+	0.999002i	$0.485780\pi$
$522$	0	0
$523$	37.5930	1.64383	0.821914	−	0.569611i	$-0.192906\pi$
$523$	37.5930	1.64383	0.821914	+	0.569611i	$0.192906\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.21479	0.227160
$528$	0	0
$529$	15.6891	0.682135
$530$	0	0
$531$	5.90904	0.256430
$532$	0	0
$533$	2.95898	0.128168
$534$	0	0
$535$	−13.3649	−0.577813
$536$	0	0
$537$	−5.28987	−0.228275
$538$	0	0
$539$	0	0
$540$	0	0
$541$	32.3190	1.38950	0.694751	−	0.719250i	$-0.255514\pi$
$541$	32.3190	1.38950	0.694751	+	0.719250i	$0.255514\pi$
$542$	0	0
$543$	22.5956	0.969670
$544$	0	0
$545$	−33.9644	−1.45488
$546$	0	0
$547$	5.52462	0.236216	0.118108	−	0.993001i	$-0.462317\pi$
$547$	5.52462	0.236216	0.118108	+	0.993001i	$0.462317\pi$
$548$	0	0
$549$	31.9473	1.36348
$550$	0	0
$551$	3.13688	0.133636
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.48597	0.360209
$556$	0	0
$557$	−31.7060	−1.34342	−0.671712	−	0.740812i	$-0.734441\pi$
$557$	−31.7060	−1.34342	−0.671712	+	0.740812i	$0.734441\pi$
$558$	0	0
$559$	−3.04707	−0.128877
$560$	0	0
$561$	2.22805	0.0940683
$562$	0	0
$563$	−34.7554	−1.46477	−0.732383	−	0.680893i	$-0.761592\pi$
$563$	−34.7554	−1.46477	−0.732383	+	0.680893i	$0.761592\pi$
$564$	0	0
$565$	−14.8944	−0.626613
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.8986	−1.50495	−0.752473	−	0.658623i	$-0.771139\pi$
$569$	−35.8986	−1.50495	−0.752473	+	0.658623i	$0.771139\pi$
$570$	0	0
$571$	−43.0882	−1.80318	−0.901592	−	0.432588i	$-0.857600\pi$
$571$	−43.0882	−1.80318	−0.901592	+	0.432588i	$0.857600\pi$
$572$	0	0
$573$	38.0416	1.58921
$574$	0	0
$575$	−13.8317	−0.576824
$576$	0	0
$577$	−46.6165	−1.94067	−0.970335	−	0.241766i	$-0.922273\pi$
$577$	−46.6165	−1.94067	−0.970335	+	0.241766i	$0.922273\pi$
$578$	0	0
$579$	43.5500	1.80988
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.22601	0.0507760
$584$	0	0
$585$	−9.20847	−0.380723
$586$	0	0
$587$	−46.4370	−1.91666	−0.958331	−	0.285661i	$-0.907787\pi$
$587$	−46.4370	−1.91666	−0.958331	+	0.285661i	$0.907787\pi$
$588$	0	0
$589$	−0.882526	−0.0363639
$590$	0	0
$591$	62.4317	2.56810
$592$	0	0
$593$	−40.3003	−1.65494	−0.827468	−	0.561513i	$-0.810219\pi$
$593$	−40.3003	−1.65494	−0.827468	+	0.561513i	$0.810219\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.2575	−0.665376
$598$	0	0
$599$	−19.7924	−0.808695	−0.404348	−	0.914605i	$-0.632501\pi$
$599$	−19.7924	−0.808695	−0.404348	+	0.914605i	$0.632501\pi$
$600$	0	0
$601$	−39.2064	−1.59926	−0.799631	−	0.600492i	$-0.794971\pi$
$601$	−39.2064	−1.59926	−0.799631	+	0.600492i	$0.794971\pi$
$602$	0	0
$603$	5.00293	0.203735
$604$	0	0
$605$	18.2932	0.743723
$606$	0	0
$607$	17.2293	0.699316	0.349658	−	0.936877i	$-0.386298\pi$
$607$	17.2293	0.699316	0.349658	+	0.936877i	$0.386298\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.258030	0.0104388
$612$	0	0
$613$	24.1131	0.973917	0.486959	−	0.873425i	$-0.338106\pi$
$613$	24.1131	0.973917	0.486959	+	0.873425i	$0.338106\pi$
$614$	0	0
$615$	−8.64857	−0.348744
$616$	0	0
$617$	−44.6161	−1.79618	−0.898089	−	0.439814i	$-0.855044\pi$
$617$	−44.6161	−1.79618	−0.898089	+	0.439814i	$0.855044\pi$
$618$	0	0
$619$	20.5549	0.826172	0.413086	−	0.910692i	$-0.364451\pi$
$619$	20.5549	0.826172	0.413086	+	0.910692i	$0.364451\pi$
$620$	0	0
$621$	11.8724	0.476423
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8.93635	−0.357454
$626$	0	0
$627$	−0.377064	−0.0150585
$628$	0	0
$629$	−11.5957	−0.462350
$630$	0	0
$631$	3.35794	0.133678	0.0668388	−	0.997764i	$-0.478709\pi$
$631$	3.35794	0.133678	0.0668388	+	0.997764i	$0.478709\pi$
$632$	0	0
$633$	−11.4160	−0.453745
$634$	0	0
$635$	−2.93359	−0.116416
$636$	0	0
$637$	0	0
$638$	0	0
$639$	27.5354	1.08928
$640$	0	0
$641$	−4.65861	−0.184004	−0.0920020	−	0.995759i	$-0.529327\pi$
$641$	−4.65861	−0.184004	−0.0920020	+	0.995759i	$0.529327\pi$
$642$	0	0
$643$	−9.50546	−0.374859	−0.187429	−	0.982278i	$-0.560016\pi$
$643$	−9.50546	−0.374859	−0.187429	+	0.982278i	$0.560016\pi$
$644$	0	0
$645$	8.90602	0.350674
$646$	0	0
$647$	−13.0308	−0.512294	−0.256147	−	0.966638i	$-0.582453\pi$
$647$	−13.0308	−0.512294	−0.256147	+	0.966638i	$0.582453\pi$
$648$	0	0
$649$	0.229828	0.00902156
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.02455	0.274892	0.137446	−	0.990509i	$-0.456111\pi$
$653$	7.02455	0.274892	0.137446	+	0.990509i	$0.456111\pi$
$654$	0	0
$655$	6.73602	0.263198
$656$	0	0
$657$	−17.6472	−0.688485
$658$	0	0
$659$	−7.92031	−0.308532	−0.154266	−	0.988029i	$-0.549301\pi$
$659$	−7.92031	−0.308532	−0.154266	+	0.988029i	$0.549301\pi$
$660$	0	0
$661$	−10.7536	−0.418265	−0.209133	−	0.977887i	$-0.567064\pi$
$661$	−10.7536	−0.418265	−0.209133	+	0.977887i	$0.567064\pi$
$662$	0	0
$663$	22.6885	0.881148
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−19.5116	−0.755492
$668$	0	0
$669$	−51.2071	−1.97978
$670$	0	0
$671$	1.24257	0.0479690
$672$	0	0
$673$	−32.6941	−1.26027	−0.630133	−	0.776487i	$-0.717000\pi$
$673$	−32.6941	−1.26027	−0.630133	+	0.776487i	$0.717000\pi$
$674$	0	0
$675$	−4.24450	−0.163371
$676$	0	0
$677$	−36.5994	−1.40663	−0.703314	−	0.710879i	$-0.748297\pi$
$677$	−36.5994	−1.40663	−0.703314	+	0.710879i	$0.748297\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	21.1442	0.810249
$682$	0	0
$683$	2.97440	0.113812	0.0569062	−	0.998380i	$-0.481876\pi$
$683$	2.97440	0.113812	0.0569062	+	0.998380i	$0.481876\pi$
$684$	0	0
$685$	34.0048	1.29926
$686$	0	0
$687$	44.0601	1.68100
$688$	0	0
$689$	12.4846	0.475625
$690$	0	0
$691$	1.82312	0.0693546	0.0346773	−	0.999399i	$-0.488960\pi$
$691$	1.82312	0.0693546	0.0346773	+	0.999399i	$0.488960\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.7341	0.710625
$696$	0	0
$697$	11.8179	0.447634
$698$	0	0
$699$	−8.28390	−0.313326
$700$	0	0
$701$	−49.7699	−1.87978	−0.939891	−	0.341474i	$-0.889074\pi$
$701$	−49.7699	−1.87978	−0.939891	+	0.341474i	$0.889074\pi$
$702$	0	0
$703$	1.96240	0.0740133
$704$	0	0
$705$	−0.754175	−0.0284039
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−14.9525	−0.561552	−0.280776	−	0.959773i	$-0.590592\pi$
$709$	−14.9525	−0.561552	−0.280776	+	0.959773i	$0.590592\pi$
$710$	0	0
$711$	42.9525	1.61084
$712$	0	0
$713$	5.48936	0.205578
$714$	0	0
$715$	−0.358158	−0.0133943
$716$	0	0
$717$	−51.7010	−1.93081
$718$	0	0
$719$	−14.2696	−0.532166	−0.266083	−	0.963950i	$-0.585730\pi$
$719$	−14.2696	−0.532166	−0.266083	+	0.963950i	$0.585730\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	21.5186	0.800286
$724$	0	0
$725$	6.97559	0.259067
$726$	0	0
$727$	2.40032	0.0890231	0.0445115	−	0.999009i	$-0.485827\pi$
$727$	2.40032	0.0890231	0.0445115	+	0.999009i	$0.485827\pi$
$728$	0	0
$729$	−38.2178	−1.41547
$730$	0	0
$731$	−12.1697	−0.450111
$732$	0	0
$733$	12.4073	0.458273	0.229137	−	0.973394i	$-0.426410\pi$
$733$	12.4073	0.458273	0.229137	+	0.973394i	$0.426410\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.194586	0.00716767
$738$	0	0
$739$	−0.0480938	−0.00176916	−0.000884580	−	1.00000i	$-0.500282\pi$
$739$	−0.0480938	−0.00176916	−0.000884580		1.00000i	$0.500282\pi$
$740$	0	0
$741$	−3.83969	−0.141055
$742$	0	0
$743$	−28.5507	−1.04742	−0.523712	−	0.851895i	$-0.675453\pi$
$743$	−28.5507	−1.04742	−0.523712	+	0.851895i	$0.675453\pi$
$744$	0	0
$745$	9.30908	0.341058
$746$	0	0
$747$	−16.3527	−0.598314
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−10.8556	−0.396128	−0.198064	−	0.980189i	$-0.563465\pi$
$751$	−10.8556	−0.396128	−0.198064	+	0.980189i	$0.563465\pi$
$752$	0	0
$753$	26.6027	0.969457
$754$	0	0
$755$	32.2537	1.17383
$756$	0	0
$757$	−20.1699	−0.733089	−0.366545	−	0.930400i	$-0.619459\pi$
$757$	−20.1699	−0.733089	−0.366545	+	0.930400i	$0.619459\pi$
$758$	0	0
$759$	2.34536	0.0851312
$760$	0	0
$761$	29.9226	1.08469	0.542347	−	0.840155i	$-0.317536\pi$
$761$	29.9226	1.08469	0.542347	+	0.840155i	$0.317536\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−36.7777	−1.32970
$766$	0	0
$767$	2.34037	0.0845059
$768$	0	0
$769$	15.7701	0.568685	0.284343	−	0.958723i	$-0.408225\pi$
$769$	15.7701	0.568685	0.284343	+	0.958723i	$0.408225\pi$
$770$	0	0
$771$	23.7282	0.854550
$772$	0	0
$773$	−35.2277	−1.26705	−0.633526	−	0.773721i	$-0.718393\pi$
$773$	−35.2277	−1.26705	−0.633526	+	0.773721i	$0.718393\pi$
$774$	0	0
$775$	−1.96250	−0.0704952
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	1.07097	0.0383224
$782$	0	0
$783$	−5.98745	−0.213974
$784$	0	0
$785$	−18.2009	−0.649618
$786$	0	0
$787$	−45.9116	−1.63657	−0.818286	−	0.574811i	$-0.805076\pi$
$787$	−45.9116	−1.63657	−0.818286	+	0.574811i	$0.805076\pi$
$788$	0	0
$789$	6.95329	0.247544
$790$	0	0
$791$	0	0
$792$	0	0
$793$	12.6533	0.449331
$794$	0	0
$795$	−36.4902	−1.29417
$796$	0	0
$797$	2.14407	0.0759468	0.0379734	−	0.999279i	$-0.487910\pi$
$797$	2.14407	0.0759468	0.0379734	+	0.999279i	$0.487910\pi$
$798$	0	0
$799$	1.03055	0.0364581
$800$	0	0
$801$	5.20342	0.183854
$802$	0	0
$803$	−0.686379	−0.0242218
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−65.4172	−2.30279
$808$	0	0
$809$	−23.3363	−0.820461	−0.410230	−	0.911982i	$-0.634552\pi$
$809$	−23.3363	−0.820461	−0.410230	+	0.911982i	$0.634552\pi$
$810$	0	0
$811$	−41.5527	−1.45911	−0.729556	−	0.683921i	$-0.760273\pi$
$811$	−41.5527	−1.45911	−0.729556	+	0.683921i	$0.760273\pi$
$812$	0	0
$813$	32.8828	1.15325
$814$	0	0
$815$	−27.1573	−0.951278
$816$	0	0
$817$	2.05954	0.0720540
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.1695	1.08782	0.543912	−	0.839142i	$-0.316943\pi$
$821$	31.1695	1.08782	0.543912	+	0.839142i	$0.316943\pi$
$822$	0	0
$823$	−17.2573	−0.601551	−0.300775	−	0.953695i	$-0.597245\pi$
$823$	−17.2573	−0.601551	−0.300775	+	0.953695i	$0.597245\pi$
$824$	0	0
$825$	−0.838490	−0.0291925
$826$	0	0
$827$	−35.1385	−1.22188	−0.610942	−	0.791675i	$-0.709209\pi$
$827$	−35.1385	−1.22188	−0.610942	+	0.791675i	$0.709209\pi$
$828$	0	0
$829$	29.2471	1.01579	0.507897	−	0.861418i	$-0.330423\pi$
$829$	29.2471	1.01579	0.507897	+	0.861418i	$0.330423\pi$
$830$	0	0
$831$	57.2222	1.98502
$832$	0	0
$833$	0	0
$834$	0	0
$835$	27.8890	0.965138
$836$	0	0
$837$	1.68450	0.0582249
$838$	0	0
$839$	16.6637	0.575294	0.287647	−	0.957737i	$-0.407127\pi$
$839$	16.6637	0.575294	0.287647	+	0.957737i	$0.407127\pi$
$840$	0	0
$841$	−19.1600	−0.660689
$842$	0	0
$843$	28.3886	0.977754
$844$	0	0
$845$	18.0136	0.619687
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−53.5190	−1.83677
$850$	0	0
$851$	−12.2062	−0.418424
$852$	0	0
$853$	−20.0759	−0.687386	−0.343693	−	0.939082i	$-0.611678\pi$
$853$	−20.0759	−0.687386	−0.343693	+	0.939082i	$0.611678\pi$
$854$	0	0
$855$	6.22408	0.212859
$856$	0	0
$857$	−21.4823	−0.733823	−0.366911	−	0.930256i	$-0.619585\pi$
$857$	−21.4823	−0.733823	−0.366911	+	0.930256i	$0.619585\pi$
$858$	0	0
$859$	−13.9460	−0.475830	−0.237915	−	0.971286i	$-0.576464\pi$
$859$	−13.9460	−0.475830	−0.237915	+	0.971286i	$0.576464\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−10.0298	−0.341419	−0.170709	−	0.985321i	$-0.554606\pi$
$863$	−10.0298	−0.341419	−0.170709	+	0.985321i	$0.554606\pi$
$864$	0	0
$865$	−30.3921	−1.03336
$866$	0	0
$867$	46.4958	1.57908
$868$	0	0
$869$	1.67061	0.0566717
$870$	0	0
$871$	1.98149	0.0671403
$872$	0	0
$873$	−39.8863	−1.34995
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.2658	0.481722	0.240861	−	0.970560i	$-0.422570\pi$
$877$	14.2658	0.481722	0.240861	+	0.970560i	$0.422570\pi$
$878$	0	0
$879$	−61.8963	−2.08771
$880$	0	0
$881$	−33.3253	−1.12276	−0.561379	−	0.827559i	$-0.689729\pi$
$881$	−33.3253	−1.12276	−0.561379	+	0.827559i	$0.689729\pi$
$882$	0	0
$883$	40.1473	1.35107	0.675533	−	0.737330i	$-0.263914\pi$
$883$	40.1473	1.35107	0.675533	+	0.737330i	$0.263914\pi$
$884$	0	0
$885$	−6.84048	−0.229940
$886$	0	0
$887$	17.5383	0.588879	0.294440	−	0.955670i	$-0.404867\pi$
$887$	17.5383	0.588879	0.294440	+	0.955670i	$0.404867\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.908448	−0.0304341
$892$	0	0
$893$	−0.174405	−0.00583623
$894$	0	0
$895$	3.39619	0.113522
$896$	0	0
$897$	23.8831	0.797433
$898$	0	0
$899$	−2.76838	−0.0923307
$900$	0	0
$901$	49.8621	1.66115
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.5068	−0.482221
$906$	0	0
$907$	30.7972	1.02260	0.511301	−	0.859401i	$-0.329164\pi$
$907$	30.7972	1.02260	0.511301	+	0.859401i	$0.329164\pi$
$908$	0	0
$909$	4.49172	0.148981
$910$	0	0
$911$	−26.5837	−0.880758	−0.440379	−	0.897812i	$-0.645156\pi$
$911$	−26.5837	−0.880758	−0.440379	+	0.897812i	$0.645156\pi$
$912$	0	0
$913$	−0.636029	−0.0210495
$914$	0	0
$915$	−36.9832	−1.22263
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−12.1469	−0.400690	−0.200345	−	0.979725i	$-0.564206\pi$
$919$	−12.1469	−0.400690	−0.200345	+	0.979725i	$0.564206\pi$
$920$	0	0
$921$	38.1357	1.25661
$922$	0	0
$923$	10.9059	0.358971
$924$	0	0
$925$	4.36385	0.143482
$926$	0	0
$927$	73.0180	2.39823
$928$	0	0
$929$	26.8183	0.879881	0.439940	−	0.898027i	$-0.355000\pi$
$929$	26.8183	0.879881	0.439940	+	0.898027i	$0.355000\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−15.3512	−0.502577
$934$	0	0
$935$	−1.43045	−0.0467806
$936$	0	0
$937$	−42.6816	−1.39435	−0.697174	−	0.716902i	$-0.745559\pi$
$937$	−42.6816	−1.39435	−0.697174	+	0.716902i	$0.745559\pi$
$938$	0	0
$939$	−2.36339	−0.0771264
$940$	0	0
$941$	−26.4552	−0.862416	−0.431208	−	0.902253i	$-0.641912\pi$
$941$	−26.4552	−0.862416	−0.431208	+	0.902253i	$0.641912\pi$
$942$	0	0
$943$	12.4401	0.405106
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.25108	−0.235628	−0.117814	−	0.993036i	$-0.537589\pi$
$947$	−7.25108	−0.235628	−0.117814	+	0.993036i	$0.537589\pi$
$948$	0	0
$949$	−6.98949	−0.226888
$950$	0	0
$951$	−65.9482	−2.13852
$952$	0	0
$953$	−19.1832	−0.621404	−0.310702	−	0.950507i	$-0.600564\pi$
$953$	−19.1832	−0.621404	−0.310702	+	0.950507i	$0.600564\pi$
$954$	0	0
$955$	−24.4234	−0.790322
$956$	0	0
$957$	−1.18281	−0.0382347
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.2211	−0.974876
$962$	0	0
$963$	29.9625	0.965528
$964$	0	0
$965$	−27.9599	−0.900060
$966$	0	0
$967$	−25.6363	−0.824407	−0.412203	−	0.911092i	$-0.635241\pi$
$967$	−25.6363	−0.824407	−0.412203	+	0.911092i	$0.635241\pi$
$968$	0	0
$969$	−15.3353	−0.492642
$970$	0	0
$971$	−36.2893	−1.16458	−0.582289	−	0.812982i	$-0.697843\pi$
$971$	−36.2893	−1.16458	−0.582289	+	0.812982i	$0.697843\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−8.53844	−0.273449
$976$	0	0
$977$	2.29103	0.0732967	0.0366483	−	0.999328i	$-0.488332\pi$
$977$	2.29103	0.0732967	0.0366483	+	0.999328i	$0.488332\pi$
$978$	0	0
$979$	0.202384	0.00646822
$980$	0	0
$981$	76.1444	2.43110
$982$	0	0
$983$	10.6705	0.340336	0.170168	−	0.985415i	$-0.445569\pi$
$983$	10.6705	0.340336	0.170168	+	0.985415i	$0.445569\pi$
$984$	0	0
$985$	−40.0822	−1.27713
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.8104	−0.407348
$990$	0	0
$991$	18.7242	0.594792	0.297396	−	0.954754i	$-0.403882\pi$
$991$	18.7242	0.594792	0.297396	+	0.954754i	$0.403882\pi$
$992$	0	0
$993$	25.4214	0.806723
$994$	0	0
$995$	10.4376	0.330895
$996$	0	0
$997$	14.4493	0.457614	0.228807	−	0.973472i	$-0.426518\pi$
$997$	14.4493	0.457614	0.228807	+	0.973472i	$0.426518\pi$
$998$	0	0
$999$	−3.74568	−0.118508

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.br.1.8		8
7.3	odd	6		1064.2.q.n.457.8	yes	16
7.5	odd	6		1064.2.q.n.305.8	✓	16
7.6	odd	2		7448.2.a.bq.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.n.305.8	✓	16	7.5	odd	6
1064.2.q.n.457.8	yes	16	7.3	odd	6
7448.2.a.bq.1.1		8	7.6	odd	2
7448.2.a.br.1.8		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)

\(f(q)\)	\(=\)	\(q+2.59528 q^{3} -1.66621 q^{5} +3.73546 q^{9} +O(q^{10})\)	\(q+2.59528 q^{3} -1.66621 q^{5} +3.73546 q^{9} +0.145289 q^{11} +1.47949 q^{13} -4.32429 q^{15} +5.90894 q^{17} -1.00000 q^{19} +6.22006 q^{23} -2.22373 q^{25} +1.90873 q^{27} -3.13688 q^{29} +0.882526 q^{31} +0.377064 q^{33} -1.96240 q^{37} +3.83969 q^{39} +2.00000 q^{41} -2.05954 q^{43} -6.22408 q^{45} +0.174405 q^{47} +15.3353 q^{51} +8.43843 q^{53} -0.242082 q^{55} -2.59528 q^{57} +1.58188 q^{59} +8.55244 q^{61} -2.46515 q^{65} +1.33931 q^{67} +16.1428 q^{69} +7.37135 q^{71} -4.72425 q^{73} -5.77120 q^{75} +11.4986 q^{79} -6.25271 q^{81} -4.37769 q^{83} -9.84555 q^{85} -8.14108 q^{87} +1.39298 q^{89} +2.29040 q^{93} +1.66621 q^{95} -10.6777 q^{97} +0.542720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{5} + 6 q^{9}+O(q^{10}) \) 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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Representations

Groups

Database

Properties

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