LMFDB - Embedded newform 3724.2.a.m.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3724,2,Mod(1,3724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3724, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3724 = 2^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3724.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:	$29.7362897127$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 18x^{5} + 84x^{3} - 4x^{2} - 44x + 8 $ x^7 - 18x^5 + 84x^3 - 4x^2 - 44x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.77904$ of defining polynomial
Character	$\chi$	$=$	3724.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.77904 q^{3} -3.08629 q^{5} +4.72304 q^{9} +5.16388 q^{11} +4.04970 q^{13} -8.57690 q^{15} +1.08292 q^{17} -1.00000 q^{19} +5.72837 q^{23} +4.52516 q^{25} +4.78838 q^{27} -3.16724 q^{29} -8.18192 q^{31} +14.3506 q^{33} -4.39755 q^{37} +11.2543 q^{39} -8.68032 q^{41} +7.99159 q^{43} -14.5766 q^{45} -0.190404 q^{47} +3.00948 q^{51} +7.44586 q^{53} -15.9372 q^{55} -2.77904 q^{57} +13.2439 q^{59} +9.35092 q^{61} -12.4985 q^{65} +8.16585 q^{67} +15.9193 q^{69} -4.71652 q^{71} +6.50522 q^{73} +12.5756 q^{75} -2.31841 q^{79} -0.862030 q^{81} +14.7229 q^{83} -3.34221 q^{85} -8.80188 q^{87} -3.67499 q^{89} -22.7378 q^{93} +3.08629 q^{95} +16.2607 q^{97} +24.3892 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{5} + 15 q^{9} + 14 q^{11} + 12 q^{15} - 10 q^{17} - 7 q^{19} + 7 q^{23} + 21 q^{25} + 2 q^{29} - 4 q^{31} + 24 q^{33} - 12 q^{37} + 18 q^{39} - 4 q^{41} - 6 q^{45} + 16 q^{47} + 2 q^{51} - 6 q^{53}+ \cdots + 40 q^{99}+O(q^{100}) $ 7 * q - 2 * q^5 + 15 * q^9 + 14 * q^11 + 12 * q^15 - 10 * q^17 - 7 * q^19 + 7 * q^23 + 21 * q^25 + 2 * q^29 - 4 * q^31 + 24 * q^33 - 12 * q^37 + 18 * q^39 - 4 * q^41 - 6 * q^45 + 16 * q^47 + 2 * q^51 - 6 * q^53 - 9 * q^55 + 4 * q^59 + 28 * q^61 - 8 * q^65 + 22 * q^67 - 34 * q^69 + 34 * q^71 - 16 * q^73 + 32 * q^75 + 14 * q^79 + 51 * q^81 - 13 * q^83 - 9 * q^85 - 10 * q^87 + 16 * q^89 - 36 * q^93 + 2 * q^95 + 4 * q^97 + 40 * q^99

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
$3$	2.77904	1.60448	0.802238	−	0.597004i	$-0.203642\pi$
$3$	2.77904	1.60448	0.802238	+	0.597004i	$0.203642\pi$
$4$	0	0
$5$	−3.08629	−1.38023	−0.690115	−	0.723700i	$-0.742440\pi$
$5$	−3.08629	−1.38023	−0.690115	+	0.723700i	$0.742440\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	4.72304	1.57435
$10$	0	0
$11$	5.16388	1.55697	0.778484	−	0.627664i	$-0.215989\pi$
$11$	5.16388	1.55697	0.778484	+	0.627664i	$0.215989\pi$
$12$	0	0
$13$	4.04970	1.12318	0.561592	−	0.827414i	$-0.310189\pi$
$13$	4.04970	1.12318	0.561592	+	0.827414i	$0.310189\pi$
$14$	0	0
$15$	−8.57690	−2.21455
$16$	0	0
$17$	1.08292	0.262647	0.131324	−	0.991340i	$-0.458077\pi$
$17$	1.08292	0.262647	0.131324	+	0.991340i	$0.458077\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.72837	1.19445	0.597224	−	0.802075i	$-0.296271\pi$
$23$	5.72837	1.19445	0.597224	+	0.802075i	$0.296271\pi$
$24$	0	0
$25$	4.52516	0.905032
$26$	0	0
$27$	4.78838	0.921525
$28$	0	0
$29$	−3.16724	−0.588142	−0.294071	−	0.955784i	$-0.595010\pi$
$29$	−3.16724	−0.588142	−0.294071	+	0.955784i	$0.595010\pi$
$30$	0	0
$31$	−8.18192	−1.46952	−0.734758	−	0.678329i	$-0.762704\pi$
$31$	−8.18192	−1.46952	−0.734758	+	0.678329i	$0.762704\pi$
$32$	0	0
$33$	14.3506	2.49812
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.39755	−0.722953	−0.361477	−	0.932381i	$-0.617727\pi$
$37$	−4.39755	−0.722953	−0.361477	+	0.932381i	$0.617727\pi$
$38$	0	0
$39$	11.2543	1.80212
$40$	0	0
$41$	−8.68032	−1.35564	−0.677819	−	0.735229i	$-0.737075\pi$
$41$	−8.68032	−1.35564	−0.677819	+	0.735229i	$0.737075\pi$
$42$	0	0
$43$	7.99159	1.21871	0.609353	−	0.792899i	$-0.291429\pi$
$43$	7.99159	1.21871	0.609353	+	0.792899i	$0.291429\pi$
$44$	0	0
$45$	−14.5766	−2.17296
$46$	0	0
$47$	−0.190404	−0.0277733	−0.0138866	−	0.999904i	$-0.504420\pi$
$47$	−0.190404	−0.0277733	−0.0138866	+	0.999904i	$0.504420\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.00948	0.421412
$52$	0	0
$53$	7.44586	1.02277	0.511383	−	0.859353i	$-0.329133\pi$
$53$	7.44586	1.02277	0.511383	+	0.859353i	$0.329133\pi$
$54$	0	0
$55$	−15.9372	−2.14897
$56$	0	0
$57$	−2.77904	−0.368092
$58$	0	0
$59$	13.2439	1.72421	0.862107	−	0.506726i	$-0.169145\pi$
$59$	13.2439	1.72421	0.862107	+	0.506726i	$0.169145\pi$
$60$	0	0
$61$	9.35092	1.19726	0.598631	−	0.801025i	$-0.295711\pi$
$61$	9.35092	1.19726	0.598631	+	0.801025i	$0.295711\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.4985	−1.55025
$66$	0	0
$67$	8.16585	0.997617	0.498808	−	0.866712i	$-0.333771\pi$
$67$	8.16585	0.997617	0.498808	+	0.866712i	$0.333771\pi$
$68$	0	0
$69$	15.9193	1.91646
$70$	0	0
$71$	−4.71652	−0.559748	−0.279874	−	0.960037i	$-0.590293\pi$
$71$	−4.71652	−0.559748	−0.279874	+	0.960037i	$0.590293\pi$
$72$	0	0
$73$	6.50522	0.761378	0.380689	−	0.924703i	$-0.375687\pi$
$73$	6.50522	0.761378	0.380689	+	0.924703i	$0.375687\pi$
$74$	0	0
$75$	12.5756	1.45210
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.31841	−0.260842	−0.130421	−	0.991459i	$-0.541633\pi$
$79$	−2.31841	−0.260842	−0.130421	+	0.991459i	$0.541633\pi$
$80$	0	0
$81$	−0.862030	−0.0957811
$82$	0	0
$83$	14.7229	1.61605	0.808027	−	0.589146i	$-0.200536\pi$
$83$	14.7229	1.61605	0.808027	+	0.589146i	$0.200536\pi$
$84$	0	0
$85$	−3.34221	−0.362514
$86$	0	0
$87$	−8.80188	−0.943661
$88$	0	0
$89$	−3.67499	−0.389548	−0.194774	−	0.980848i	$-0.562397\pi$
$89$	−3.67499	−0.389548	−0.194774	+	0.980848i	$0.562397\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−22.7378	−2.35780
$94$	0	0
$95$	3.08629	0.316646
$96$	0	0
$97$	16.2607	1.65102	0.825511	−	0.564386i	$-0.190887\pi$
$97$	16.2607	1.65102	0.825511	+	0.564386i	$0.190887\pi$
$98$	0	0
$99$	24.3892	2.45121
$100$	0	0
$101$	−7.28741	−0.725124	−0.362562	−	0.931960i	$-0.618098\pi$
$101$	−7.28741	−0.725124	−0.362562	+	0.931960i	$0.618098\pi$
$102$	0	0
$103$	−16.2875	−1.60486	−0.802429	−	0.596747i	$-0.796460\pi$
$103$	−16.2875	−1.60486	−0.802429	+	0.596747i	$0.796460\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.56624	−0.441435	−0.220718	−	0.975338i	$-0.570840\pi$
$107$	−4.56624	−0.441435	−0.220718	+	0.975338i	$0.570840\pi$
$108$	0	0
$109$	13.5687	1.29965	0.649824	−	0.760084i	$-0.274842\pi$
$109$	13.5687	1.29965	0.649824	+	0.760084i	$0.274842\pi$
$110$	0	0
$111$	−12.2210	−1.15996
$112$	0	0
$113$	−2.16842	−0.203988	−0.101994	−	0.994785i	$-0.532522\pi$
$113$	−2.16842	−0.203988	−0.101994	+	0.994785i	$0.532522\pi$
$114$	0	0
$115$	−17.6794	−1.64861
$116$	0	0
$117$	19.1269	1.76828
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.6657	1.42415
$122$	0	0
$123$	−24.1229	−2.17509
$124$	0	0
$125$	1.46548	0.131077
$126$	0	0
$127$	4.27599	0.379433	0.189716	−	0.981839i	$-0.439243\pi$
$127$	4.27599	0.379433	0.189716	+	0.981839i	$0.439243\pi$
$128$	0	0
$129$	22.2089	1.95538
$130$	0	0
$131$	−10.9881	−0.960039	−0.480019	−	0.877258i	$-0.659370\pi$
$131$	−10.9881	−0.960039	−0.480019	+	0.877258i	$0.659370\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−14.7783	−1.27191
$136$	0	0
$137$	15.1226	1.29201	0.646004	−	0.763334i	$-0.276439\pi$
$137$	15.1226	1.29201	0.646004	+	0.763334i	$0.276439\pi$
$138$	0	0
$139$	−20.9836	−1.77981	−0.889904	−	0.456149i	$-0.849229\pi$
$139$	−20.9836	−1.77981	−0.889904	+	0.456149i	$0.849229\pi$
$140$	0	0
$141$	−0.529140	−0.0445616
$142$	0	0
$143$	20.9122	1.74876
$144$	0	0
$145$	9.77502	0.811771
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.419466	−0.0343640	−0.0171820	−	0.999852i	$-0.505469\pi$
$149$	−0.419466	−0.0343640	−0.0171820	+	0.999852i	$0.505469\pi$
$150$	0	0
$151$	3.47957	0.283164	0.141582	−	0.989927i	$-0.454781\pi$
$151$	3.47957	0.283164	0.141582	+	0.989927i	$0.454781\pi$
$152$	0	0
$153$	5.11469	0.413498
$154$	0	0
$155$	25.2517	2.02827
$156$	0	0
$157$	6.77536	0.540732	0.270366	−	0.962758i	$-0.412855\pi$
$157$	6.77536	0.540732	0.270366	+	0.962758i	$0.412855\pi$
$158$	0	0
$159$	20.6923	1.64101
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.5176	0.902131	0.451066	−	0.892491i	$-0.351044\pi$
$163$	11.5176	0.902131	0.451066	+	0.892491i	$0.351044\pi$
$164$	0	0
$165$	−44.2901	−3.44798
$166$	0	0
$167$	14.3708	1.11205	0.556024	−	0.831166i	$-0.312326\pi$
$167$	14.3708	1.11205	0.556024	+	0.831166i	$0.312326\pi$
$168$	0	0
$169$	3.40006	0.261543
$170$	0	0
$171$	−4.72304	−0.361180
$172$	0	0
$173$	10.1302	0.770181	0.385091	−	0.922879i	$-0.374170\pi$
$173$	10.1302	0.770181	0.385091	+	0.922879i	$0.374170\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	36.8054	2.76646
$178$	0	0
$179$	17.2266	1.28757	0.643786	−	0.765205i	$-0.277362\pi$
$179$	17.2266	1.28757	0.643786	+	0.765205i	$0.277362\pi$
$180$	0	0
$181$	−7.55315	−0.561421	−0.280711	−	0.959792i	$-0.590570\pi$
$181$	−7.55315	−0.561421	−0.280711	+	0.959792i	$0.590570\pi$
$182$	0	0
$183$	25.9865	1.92098
$184$	0	0
$185$	13.5721	0.997841
$186$	0	0
$187$	5.59208	0.408934
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.4102	−1.25976	−0.629878	−	0.776694i	$-0.716895\pi$
$191$	−17.4102	−1.25976	−0.629878	+	0.776694i	$0.716895\pi$
$192$	0	0
$193$	−4.31688	−0.310736	−0.155368	−	0.987857i	$-0.549656\pi$
$193$	−4.31688	−0.310736	−0.155368	+	0.987857i	$0.549656\pi$
$194$	0	0
$195$	−34.7339	−2.48734
$196$	0	0
$197$	3.63036	0.258652	0.129326	−	0.991602i	$-0.458719\pi$
$197$	3.63036	0.258652	0.129326	+	0.991602i	$0.458719\pi$
$198$	0	0
$199$	−18.4689	−1.30922	−0.654611	−	0.755966i	$-0.727168\pi$
$199$	−18.4689	−1.30922	−0.654611	+	0.755966i	$0.727168\pi$
$200$	0	0
$201$	22.6932	1.60065
$202$	0	0
$203$	0	0
$204$	0	0
$205$	26.7899	1.87109
$206$	0	0
$207$	27.0553	1.88047
$208$	0	0
$209$	−5.16388	−0.357193
$210$	0	0
$211$	−14.4742	−0.996448	−0.498224	−	0.867048i	$-0.666014\pi$
$211$	−14.4742	−0.996448	−0.498224	+	0.867048i	$0.666014\pi$
$212$	0	0
$213$	−13.1074	−0.898102
$214$	0	0
$215$	−24.6643	−1.68209
$216$	0	0
$217$	0	0
$218$	0	0
$219$	18.0782	1.22161
$220$	0	0
$221$	4.38551	0.295001
$222$	0	0
$223$	19.0914	1.27845	0.639226	−	0.769019i	$-0.279255\pi$
$223$	19.0914	1.27845	0.639226	+	0.769019i	$0.279255\pi$
$224$	0	0
$225$	21.3725	1.42483
$226$	0	0
$227$	−0.119758	−0.00794860	−0.00397430	−	0.999992i	$-0.501265\pi$
$227$	−0.119758	−0.00794860	−0.00397430	+	0.999992i	$0.501265\pi$
$228$	0	0
$229$	−13.8077	−0.912438	−0.456219	−	0.889868i	$-0.650797\pi$
$229$	−13.8077	−0.912438	−0.456219	+	0.889868i	$0.650797\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.4113	1.33719	0.668593	−	0.743629i	$-0.266897\pi$
$233$	20.4113	1.33719	0.668593	+	0.743629i	$0.266897\pi$
$234$	0	0
$235$	0.587642	0.0383335
$236$	0	0
$237$	−6.44295	−0.418515
$238$	0	0
$239$	3.06182	0.198053	0.0990263	−	0.995085i	$-0.468427\pi$
$239$	3.06182	0.198053	0.0990263	+	0.995085i	$0.468427\pi$
$240$	0	0
$241$	−26.1291	−1.68313	−0.841563	−	0.540159i	$-0.818364\pi$
$241$	−26.1291	−1.68313	−0.841563	+	0.540159i	$0.818364\pi$
$242$	0	0
$243$	−16.7608	−1.07520
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.04970	−0.257676
$248$	0	0
$249$	40.9156	2.59292
$250$	0	0
$251$	−17.5062	−1.10498	−0.552490	−	0.833520i	$-0.686322\pi$
$251$	−17.5062	−1.10498	−0.552490	+	0.833520i	$0.686322\pi$
$252$	0	0
$253$	29.5806	1.85972
$254$	0	0
$255$	−9.28812	−0.581645
$256$	0	0
$257$	6.93487	0.432585	0.216293	−	0.976329i	$-0.430603\pi$
$257$	6.93487	0.432585	0.216293	+	0.976329i	$0.430603\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−14.9590	−0.925939
$262$	0	0
$263$	−10.3126	−0.635902	−0.317951	−	0.948107i	$-0.602995\pi$
$263$	−10.3126	−0.635902	−0.317951	+	0.948107i	$0.602995\pi$
$264$	0	0
$265$	−22.9800	−1.41165
$266$	0	0
$267$	−10.2129	−0.625021
$268$	0	0
$269$	20.0185	1.22055	0.610276	−	0.792189i	$-0.291059\pi$
$269$	20.0185	1.22055	0.610276	+	0.792189i	$0.291059\pi$
$270$	0	0
$271$	−5.60895	−0.340719	−0.170360	−	0.985382i	$-0.554493\pi$
$271$	−5.60895	−0.340719	−0.170360	+	0.985382i	$0.554493\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	23.3674	1.40911
$276$	0	0
$277$	−1.10303	−0.0662747	−0.0331374	−	0.999451i	$-0.510550\pi$
$277$	−1.10303	−0.0662747	−0.0331374	+	0.999451i	$0.510550\pi$
$278$	0	0
$279$	−38.6435	−2.31353
$280$	0	0
$281$	12.5761	0.750229	0.375115	−	0.926978i	$-0.377603\pi$
$281$	12.5761	0.750229	0.375115	+	0.926978i	$0.377603\pi$
$282$	0	0
$283$	−31.9245	−1.89771	−0.948857	−	0.315706i	$-0.897759\pi$
$283$	−31.9245	−1.89771	−0.948857	+	0.315706i	$0.897759\pi$
$284$	0	0
$285$	8.57690	0.508052
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.8273	−0.931016
$290$	0	0
$291$	45.1890	2.64903
$292$	0	0
$293$	−0.574159	−0.0335428	−0.0167714	−	0.999859i	$-0.505339\pi$
$293$	−0.574159	−0.0335428	−0.0167714	+	0.999859i	$0.505339\pi$
$294$	0	0
$295$	−40.8746	−2.37981
$296$	0	0
$297$	24.7266	1.43478
$298$	0	0
$299$	23.1982	1.34158
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−20.2520	−1.16344
$304$	0	0
$305$	−28.8596	−1.65250
$306$	0	0
$307$	7.42536	0.423788	0.211894	−	0.977293i	$-0.432037\pi$
$307$	7.42536	0.423788	0.211894	+	0.977293i	$0.432037\pi$
$308$	0	0
$309$	−45.2636	−2.57496
$310$	0	0
$311$	−32.8935	−1.86522	−0.932610	−	0.360887i	$-0.882474\pi$
$311$	−32.8935	−1.86522	−0.932610	+	0.360887i	$0.882474\pi$
$312$	0	0
$313$	−19.4604	−1.09997	−0.549985	−	0.835175i	$-0.685366\pi$
$313$	−19.4604	−1.09997	−0.549985	+	0.835175i	$0.685366\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.4794	−0.644749	−0.322375	−	0.946612i	$-0.604481\pi$
$317$	−11.4794	−0.644749	−0.322375	+	0.946612i	$0.604481\pi$
$318$	0	0
$319$	−16.3553	−0.915719
$320$	0	0
$321$	−12.6897	−0.708272
$322$	0	0
$323$	−1.08292	−0.0602554
$324$	0	0
$325$	18.3255	1.01652
$326$	0	0
$327$	37.7080	2.08526
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.5371	0.908963	0.454481	−	0.890756i	$-0.349825\pi$
$331$	16.5371	0.908963	0.454481	+	0.890756i	$0.349825\pi$
$332$	0	0
$333$	−20.7698	−1.13818
$334$	0	0
$335$	−25.2021	−1.37694
$336$	0	0
$337$	−11.7147	−0.638139	−0.319069	−	0.947731i	$-0.603370\pi$
$337$	−11.7147	−0.638139	−0.319069	+	0.947731i	$0.603370\pi$
$338$	0	0
$339$	−6.02612	−0.327294
$340$	0	0
$341$	−42.2504	−2.28799
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−49.1316	−2.64516
$346$	0	0
$347$	−32.6556	−1.75304	−0.876522	−	0.481361i	$-0.840143\pi$
$347$	−32.6556	−1.75304	−0.876522	+	0.481361i	$0.840143\pi$
$348$	0	0
$349$	−12.9112	−0.691123	−0.345561	−	0.938396i	$-0.612312\pi$
$349$	−12.9112	−0.691123	−0.345561	+	0.938396i	$0.612312\pi$
$350$	0	0
$351$	19.3915	1.03504
$352$	0	0
$353$	8.76458	0.466492	0.233246	−	0.972418i	$-0.425065\pi$
$353$	8.76458	0.466492	0.233246	+	0.972418i	$0.425065\pi$
$354$	0	0
$355$	14.5565	0.772580
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.0423	1.16335	0.581673	−	0.813423i	$-0.302398\pi$
$359$	22.0423	1.16335	0.581673	+	0.813423i	$0.302398\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	43.5354	2.28502
$364$	0	0
$365$	−20.0770	−1.05088
$366$	0	0
$367$	−20.4585	−1.06792	−0.533961	−	0.845509i	$-0.679297\pi$
$367$	−20.4585	−1.06792	−0.533961	+	0.845509i	$0.679297\pi$
$368$	0	0
$369$	−40.9975	−2.13424
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−15.9329	−0.824975	−0.412488	−	0.910963i	$-0.635340\pi$
$373$	−15.9329	−0.824975	−0.412488	+	0.910963i	$0.635340\pi$
$374$	0	0
$375$	4.07263	0.210310
$376$	0	0
$377$	−12.8264	−0.660592
$378$	0	0
$379$	−1.32037	−0.0678227	−0.0339113	−	0.999425i	$-0.510796\pi$
$379$	−1.32037	−0.0678227	−0.0339113	+	0.999425i	$0.510796\pi$
$380$	0	0
$381$	11.8831	0.608791
$382$	0	0
$383$	−25.3496	−1.29530	−0.647651	−	0.761937i	$-0.724248\pi$
$383$	−25.3496	−1.29530	−0.647651	+	0.761937i	$0.724248\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	37.7446	1.91866
$388$	0	0
$389$	−26.9070	−1.36424	−0.682119	−	0.731241i	$-0.738941\pi$
$389$	−26.9070	−1.36424	−0.682119	+	0.731241i	$0.738941\pi$
$390$	0	0
$391$	6.20338	0.313718
$392$	0	0
$393$	−30.5364	−1.54036
$394$	0	0
$395$	7.15529	0.360022
$396$	0	0
$397$	−5.96986	−0.299619	−0.149809	−	0.988715i	$-0.547866\pi$
$397$	−5.96986	−0.299619	−0.149809	+	0.988715i	$0.547866\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	34.9373	1.74469	0.872344	−	0.488893i	$-0.162599\pi$
$401$	34.9373	1.74469	0.872344	+	0.488893i	$0.162599\pi$
$402$	0	0
$403$	−33.1343	−1.65054
$404$	0	0
$405$	2.66047	0.132200
$406$	0	0
$407$	−22.7084	−1.12562
$408$	0	0
$409$	−4.48194	−0.221618	−0.110809	−	0.993842i	$-0.535344\pi$
$409$	−4.48194	−0.221618	−0.110809	+	0.993842i	$0.535344\pi$
$410$	0	0
$411$	42.0261	2.07300
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−45.4392	−2.23052
$416$	0	0
$417$	−58.3142	−2.85566
$418$	0	0
$419$	−0.577870	−0.0282308	−0.0141154	−	0.999900i	$-0.504493\pi$
$419$	−0.577870	−0.0282308	−0.0141154	+	0.999900i	$0.504493\pi$
$420$	0	0
$421$	6.21914	0.303103	0.151551	−	0.988449i	$-0.451573\pi$
$421$	6.21914	0.303103	0.151551	+	0.988449i	$0.451573\pi$
$422$	0	0
$423$	−0.899286	−0.0437248
$424$	0	0
$425$	4.90040	0.237704
$426$	0	0
$427$	0	0
$428$	0	0
$429$	58.1156	2.80585
$430$	0	0
$431$	33.0876	1.59377	0.796887	−	0.604128i	$-0.206478\pi$
$431$	33.0876	1.59377	0.796887	+	0.604128i	$0.206478\pi$
$432$	0	0
$433$	−7.81123	−0.375384	−0.187692	−	0.982228i	$-0.560101\pi$
$433$	−7.81123	−0.375384	−0.187692	+	0.982228i	$0.560101\pi$
$434$	0	0
$435$	27.1651	1.30247
$436$	0	0
$437$	−5.72837	−0.274025
$438$	0	0
$439$	−29.5411	−1.40992	−0.704960	−	0.709248i	$-0.749035\pi$
$439$	−29.5411	−1.40992	−0.704960	+	0.709248i	$0.749035\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.89611	−0.0900871	−0.0450436	−	0.998985i	$-0.514343\pi$
$443$	−1.89611	−0.0900871	−0.0450436	+	0.998985i	$0.514343\pi$
$444$	0	0
$445$	11.3421	0.537666
$446$	0	0
$447$	−1.16571	−0.0551363
$448$	0	0
$449$	10.1049	0.476881	0.238441	−	0.971157i	$-0.423364\pi$
$449$	10.1049	0.476881	0.238441	+	0.971157i	$0.423364\pi$
$450$	0	0
$451$	−44.8241	−2.11069
$452$	0	0
$453$	9.66985	0.454329
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.0503	−0.563688	−0.281844	−	0.959460i	$-0.590946\pi$
$457$	−12.0503	−0.563688	−0.281844	+	0.959460i	$0.590946\pi$
$458$	0	0
$459$	5.18545	0.242036
$460$	0	0
$461$	23.7612	1.10667	0.553335	−	0.832959i	$-0.313355\pi$
$461$	23.7612	1.10667	0.553335	+	0.832959i	$0.313355\pi$
$462$	0	0
$463$	−21.2804	−0.988985	−0.494493	−	0.869182i	$-0.664646\pi$
$463$	−21.2804	−0.988985	−0.494493	+	0.869182i	$0.664646\pi$
$464$	0	0
$465$	70.1755	3.25431
$466$	0	0
$467$	−11.5986	−0.536719	−0.268360	−	0.963319i	$-0.586482\pi$
$467$	−11.5986	−0.536719	−0.268360	+	0.963319i	$0.586482\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	18.8290	0.867592
$472$	0	0
$473$	41.2676	1.89749
$474$	0	0
$475$	−4.52516	−0.207629
$476$	0	0
$477$	35.1671	1.61019
$478$	0	0
$479$	11.7538	0.537046	0.268523	−	0.963273i	$-0.413464\pi$
$479$	11.7538	0.537046	0.268523	+	0.963273i	$0.413464\pi$
$480$	0	0
$481$	−17.8088	−0.812010
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−50.1851	−2.27879
$486$	0	0
$487$	23.5092	1.06530	0.532651	−	0.846335i	$-0.321196\pi$
$487$	23.5092	1.06530	0.532651	+	0.846335i	$0.321196\pi$
$488$	0	0
$489$	32.0079	1.44745
$490$	0	0
$491$	29.3052	1.32252	0.661262	−	0.750155i	$-0.270021\pi$
$491$	29.3052	1.32252	0.661262	+	0.750155i	$0.270021\pi$
$492$	0	0
$493$	−3.42988	−0.154474
$494$	0	0
$495$	−75.2720	−3.38323
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−37.1336	−1.66233	−0.831164	−	0.556027i	$-0.812325\pi$
$499$	−37.1336	−1.66233	−0.831164	+	0.556027i	$0.812325\pi$
$500$	0	0
$501$	39.9370	1.78426
$502$	0	0
$503$	31.5636	1.40735	0.703675	−	0.710522i	$-0.251541\pi$
$503$	31.5636	1.40735	0.703675	+	0.710522i	$0.251541\pi$
$504$	0	0
$505$	22.4910	1.00084
$506$	0	0
$507$	9.44888	0.419639
$508$	0	0
$509$	3.81890	0.169270	0.0846350	−	0.996412i	$-0.473028\pi$
$509$	3.81890	0.169270	0.0846350	+	0.996412i	$0.473028\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.78838	−0.211412
$514$	0	0
$515$	50.2680	2.21507
$516$	0	0
$517$	−0.983224	−0.0432421
$518$	0	0
$519$	28.1520	1.23574
$520$	0	0
$521$	−16.3956	−0.718302	−0.359151	−	0.933279i	$-0.616934\pi$
$521$	−16.3956	−0.718302	−0.359151	+	0.933279i	$0.616934\pi$
$522$	0	0
$523$	28.0981	1.22864	0.614322	−	0.789055i	$-0.289429\pi$
$523$	28.0981	1.22864	0.614322	+	0.789055i	$0.289429\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.86039	−0.385965
$528$	0	0
$529$	9.81419	0.426704
$530$	0	0
$531$	62.5516	2.71451
$532$	0	0
$533$	−35.1527	−1.52263
$534$	0	0
$535$	14.0927	0.609282
$536$	0	0
$537$	47.8732	2.06588
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−32.5542	−1.39961	−0.699807	−	0.714332i	$-0.746731\pi$
$541$	−32.5542	−1.39961	−0.699807	+	0.714332i	$0.746731\pi$
$542$	0	0
$543$	−20.9905	−0.900787
$544$	0	0
$545$	−41.8770	−1.79381
$546$	0	0
$547$	−34.6056	−1.47963	−0.739814	−	0.672812i	$-0.765086\pi$
$547$	−34.6056	−1.47963	−0.739814	+	0.672812i	$0.765086\pi$
$548$	0	0
$549$	44.1647	1.88491
$550$	0	0
$551$	3.16724	0.134929
$552$	0	0
$553$	0	0
$554$	0	0
$555$	37.7174	1.60101
$556$	0	0
$557$	18.1093	0.767314	0.383657	−	0.923476i	$-0.374665\pi$
$557$	18.1093	0.767314	0.383657	+	0.923476i	$0.374665\pi$
$558$	0	0
$559$	32.3635	1.36883
$560$	0	0
$561$	15.5406	0.656125
$562$	0	0
$563$	17.0718	0.719489	0.359745	−	0.933051i	$-0.382864\pi$
$563$	17.0718	0.719489	0.359745	+	0.933051i	$0.382864\pi$
$564$	0	0
$565$	6.69237	0.281550
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.33801	0.391470	0.195735	−	0.980657i	$-0.437291\pi$
$569$	9.33801	0.391470	0.195735	+	0.980657i	$0.437291\pi$
$570$	0	0
$571$	3.93740	0.164775	0.0823876	−	0.996600i	$-0.473745\pi$
$571$	3.93740	0.164775	0.0823876	+	0.996600i	$0.473745\pi$
$572$	0	0
$573$	−48.3835	−2.02125
$574$	0	0
$575$	25.9218	1.08101
$576$	0	0
$577$	−16.4363	−0.684254	−0.342127	−	0.939654i	$-0.611147\pi$
$577$	−16.4363	−0.684254	−0.342127	+	0.939654i	$0.611147\pi$
$578$	0	0
$579$	−11.9968	−0.498569
$580$	0	0
$581$	0	0
$582$	0	0
$583$	38.4495	1.59242
$584$	0	0
$585$	−59.0310	−2.44063
$586$	0	0
$587$	31.7915	1.31218	0.656089	−	0.754684i	$-0.272210\pi$
$587$	31.7915	1.31218	0.656089	+	0.754684i	$0.272210\pi$
$588$	0	0
$589$	8.18192	0.337130
$590$	0	0
$591$	10.0889	0.415001
$592$	0	0
$593$	−28.8411	−1.18436	−0.592182	−	0.805805i	$-0.701733\pi$
$593$	−28.8411	−1.18436	−0.592182	+	0.805805i	$0.701733\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−51.3256	−2.10062
$598$	0	0
$599$	−44.0348	−1.79921	−0.899606	−	0.436703i	$-0.856146\pi$
$599$	−44.0348	−1.79921	−0.899606	+	0.436703i	$0.856146\pi$
$600$	0	0
$601$	−2.40658	−0.0981666	−0.0490833	−	0.998795i	$-0.515630\pi$
$601$	−2.40658	−0.0981666	−0.0490833	+	0.998795i	$0.515630\pi$
$602$	0	0
$603$	38.5676	1.57059
$604$	0	0
$605$	−48.3487	−1.96565
$606$	0	0
$607$	−3.13678	−0.127318	−0.0636590	−	0.997972i	$-0.520277\pi$
$607$	−3.13678	−0.127318	−0.0636590	+	0.997972i	$0.520277\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.771079	−0.0311945
$612$	0	0
$613$	−9.28527	−0.375028	−0.187514	−	0.982262i	$-0.560043\pi$
$613$	−9.28527	−0.375028	−0.187514	+	0.982262i	$0.560043\pi$
$614$	0	0
$615$	74.4502	3.00212
$616$	0	0
$617$	−16.4951	−0.664067	−0.332034	−	0.943268i	$-0.607735\pi$
$617$	−16.4951	−0.664067	−0.332034	+	0.943268i	$0.607735\pi$
$618$	0	0
$619$	−11.6429	−0.467969	−0.233984	−	0.972240i	$-0.575176\pi$
$619$	−11.6429	−0.467969	−0.233984	+	0.972240i	$0.575176\pi$
$620$	0	0
$621$	27.4296	1.10071
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−27.1487	−1.08595
$626$	0	0
$627$	−14.3506	−0.573108
$628$	0	0
$629$	−4.76221	−0.189882
$630$	0	0
$631$	−28.3865	−1.13005	−0.565024	−	0.825075i	$-0.691133\pi$
$631$	−28.3865	−1.13005	−0.565024	+	0.825075i	$0.691133\pi$
$632$	0	0
$633$	−40.2244	−1.59878
$634$	0	0
$635$	−13.1969	−0.523704
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−22.2763	−0.881237
$640$	0	0
$641$	−5.97591	−0.236034	−0.118017	−	0.993012i	$-0.537654\pi$
$641$	−5.97591	−0.236034	−0.118017	+	0.993012i	$0.537654\pi$
$642$	0	0
$643$	−1.33053	−0.0524711	−0.0262356	−	0.999656i	$-0.508352\pi$
$643$	−1.33053	−0.0524711	−0.0262356	+	0.999656i	$0.508352\pi$
$644$	0	0
$645$	−68.5430	−2.69888
$646$	0	0
$647$	5.81167	0.228480	0.114240	−	0.993453i	$-0.463557\pi$
$647$	5.81167	0.228480	0.114240	+	0.993453i	$0.463557\pi$
$648$	0	0
$649$	68.3901	2.68455
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.2035	1.29935	0.649676	−	0.760211i	$-0.274905\pi$
$653$	33.2035	1.29935	0.649676	+	0.760211i	$0.274905\pi$
$654$	0	0
$655$	33.9126	1.32507
$656$	0	0
$657$	30.7244	1.19867
$658$	0	0
$659$	15.5795	0.606890	0.303445	−	0.952849i	$-0.401863\pi$
$659$	15.5795	0.606890	0.303445	+	0.952849i	$0.401863\pi$
$660$	0	0
$661$	−20.3700	−0.792299	−0.396150	−	0.918186i	$-0.629654\pi$
$661$	−20.3700	−0.792299	−0.396150	+	0.918186i	$0.629654\pi$
$662$	0	0
$663$	12.1875	0.473323
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.1431	−0.702505
$668$	0	0
$669$	53.0556	2.05125
$670$	0	0
$671$	48.2870	1.86410
$672$	0	0
$673$	36.1669	1.39413	0.697065	−	0.717008i	$-0.254489\pi$
$673$	36.1669	1.39413	0.697065	+	0.717008i	$0.254489\pi$
$674$	0	0
$675$	21.6682	0.834010
$676$	0	0
$677$	−18.8558	−0.724686	−0.362343	−	0.932045i	$-0.618023\pi$
$677$	−18.8558	−0.724686	−0.362343	+	0.932045i	$0.618023\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−0.332811	−0.0127534
$682$	0	0
$683$	−22.2251	−0.850420	−0.425210	−	0.905095i	$-0.639800\pi$
$683$	−22.2251	−0.850420	−0.425210	+	0.905095i	$0.639800\pi$
$684$	0	0
$685$	−46.6725	−1.78327
$686$	0	0
$687$	−38.3720	−1.46398
$688$	0	0
$689$	30.1535	1.14876
$690$	0	0
$691$	20.1072	0.764912	0.382456	−	0.923974i	$-0.375078\pi$
$691$	20.1072	0.764912	0.382456	+	0.923974i	$0.375078\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	64.7614	2.45654
$696$	0	0
$697$	−9.40012	−0.356055
$698$	0	0
$699$	56.7236	2.14548
$700$	0	0
$701$	−0.702956	−0.0265503	−0.0132751	−	0.999912i	$-0.504226\pi$
$701$	−0.702956	−0.0265503	−0.0132751	+	0.999912i	$0.504226\pi$
$702$	0	0
$703$	4.39755	0.165857
$704$	0	0
$705$	1.63308	0.0615052
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−13.7387	−0.515969	−0.257985	−	0.966149i	$-0.583058\pi$
$709$	−13.7387	−0.515969	−0.257985	+	0.966149i	$0.583058\pi$
$710$	0	0
$711$	−10.9500	−0.410655
$712$	0	0
$713$	−46.8690	−1.75526
$714$	0	0
$715$	−64.5409	−2.41369
$716$	0	0
$717$	8.50890	0.317771
$718$	0	0
$719$	−3.00236	−0.111969	−0.0559845	−	0.998432i	$-0.517830\pi$
$719$	−3.00236	−0.111969	−0.0559845	+	0.998432i	$0.517830\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−72.6138	−2.70054
$724$	0	0
$725$	−14.3323	−0.532288
$726$	0	0
$727$	31.2768	1.15999	0.579997	−	0.814619i	$-0.303054\pi$
$727$	31.2768	1.15999	0.579997	+	0.814619i	$0.303054\pi$
$728$	0	0
$729$	−43.9926	−1.62936
$730$	0	0
$731$	8.65427	0.320090
$732$	0	0
$733$	−28.4488	−1.05078	−0.525391	−	0.850861i	$-0.676081\pi$
$733$	−28.4488	−1.05078	−0.525391	+	0.850861i	$0.676081\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	42.1674	1.55326
$738$	0	0
$739$	−32.1040	−1.18096	−0.590482	−	0.807051i	$-0.701062\pi$
$739$	−32.1040	−1.18096	−0.590482	+	0.807051i	$0.701062\pi$
$740$	0	0
$741$	−11.2543	−0.413435
$742$	0	0
$743$	−17.2263	−0.631972	−0.315986	−	0.948764i	$-0.602335\pi$
$743$	−17.2263	−0.631972	−0.315986	+	0.948764i	$0.602335\pi$
$744$	0	0
$745$	1.29459	0.0474302
$746$	0	0
$747$	69.5370	2.54423
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−24.1134	−0.879911	−0.439955	−	0.898020i	$-0.645006\pi$
$751$	−24.1134	−0.879911	−0.439955	+	0.898020i	$0.645006\pi$
$752$	0	0
$753$	−48.6503	−1.77291
$754$	0	0
$755$	−10.7390	−0.390831
$756$	0	0
$757$	−15.4641	−0.562054	−0.281027	−	0.959700i	$-0.590675\pi$
$757$	−15.4641	−0.562054	−0.281027	+	0.959700i	$0.590675\pi$
$758$	0	0
$759$	82.2055	2.98387
$760$	0	0
$761$	−12.2374	−0.443604	−0.221802	−	0.975092i	$-0.571194\pi$
$761$	−12.2374	−0.443604	−0.221802	+	0.975092i	$0.571194\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−15.7854	−0.570722
$766$	0	0
$767$	53.6340	1.93661
$768$	0	0
$769$	28.4514	1.02598	0.512992	−	0.858393i	$-0.328537\pi$
$769$	28.4514	1.02598	0.512992	+	0.858393i	$0.328537\pi$
$770$	0	0
$771$	19.2723	0.694073
$772$	0	0
$773$	−1.84646	−0.0664125	−0.0332063	−	0.999449i	$-0.510572\pi$
$773$	−1.84646	−0.0664125	−0.0332063	+	0.999449i	$0.510572\pi$
$774$	0	0
$775$	−37.0245	−1.32996
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.68032	0.311005
$780$	0	0
$781$	−24.3555	−0.871510
$782$	0	0
$783$	−15.1660	−0.541988
$784$	0	0
$785$	−20.9107	−0.746334
$786$	0	0
$787$	37.5576	1.33878	0.669392	−	0.742909i	$-0.266555\pi$
$787$	37.5576	1.33878	0.669392	+	0.742909i	$0.266555\pi$
$788$	0	0
$789$	−28.6591	−1.02029
$790$	0	0
$791$	0	0
$792$	0	0
$793$	37.8684	1.34475
$794$	0	0
$795$	−63.8623	−2.26496
$796$	0	0
$797$	23.6411	0.837411	0.418706	−	0.908122i	$-0.362484\pi$
$797$	23.6411	0.837411	0.418706	+	0.908122i	$0.362484\pi$
$798$	0	0
$799$	−0.206193	−0.00729458
$800$	0	0
$801$	−17.3571	−0.613283
$802$	0	0
$803$	33.5922	1.18544
$804$	0	0
$805$	0	0
$806$	0	0
$807$	55.6322	1.95835
$808$	0	0
$809$	7.98180	0.280625	0.140313	−	0.990107i	$-0.455189\pi$
$809$	7.98180	0.280625	0.140313	+	0.990107i	$0.455189\pi$
$810$	0	0
$811$	21.7435	0.763517	0.381758	−	0.924262i	$-0.375319\pi$
$811$	21.7435	0.763517	0.381758	+	0.924262i	$0.375319\pi$
$812$	0	0
$813$	−15.5875	−0.546676
$814$	0	0
$815$	−35.5467	−1.24515
$816$	0	0
$817$	−7.99159	−0.279590
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29.5062	−1.02977	−0.514886	−	0.857259i	$-0.672166\pi$
$821$	−29.5062	−1.02977	−0.514886	+	0.857259i	$0.672166\pi$
$822$	0	0
$823$	47.1421	1.64327	0.821635	−	0.570013i	$-0.193062\pi$
$823$	47.1421	1.64327	0.821635	+	0.570013i	$0.193062\pi$
$824$	0	0
$825$	64.9388	2.26088
$826$	0	0
$827$	9.47543	0.329493	0.164747	−	0.986336i	$-0.447319\pi$
$827$	9.47543	0.329493	0.164747	+	0.986336i	$0.447319\pi$
$828$	0	0
$829$	20.1639	0.700322	0.350161	−	0.936689i	$-0.386127\pi$
$829$	20.1639	0.700322	0.350161	+	0.936689i	$0.386127\pi$
$830$	0	0
$831$	−3.06536	−0.106336
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−44.3525	−1.53488
$836$	0	0
$837$	−39.1782	−1.35420
$838$	0	0
$839$	−12.0088	−0.414591	−0.207296	−	0.978278i	$-0.566466\pi$
$839$	−12.0088	−0.414591	−0.207296	+	0.978278i	$0.566466\pi$
$840$	0	0
$841$	−18.9686	−0.654089
$842$	0	0
$843$	34.9495	1.20373
$844$	0	0
$845$	−10.4935	−0.360989
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−88.7193	−3.04484
$850$	0	0
$851$	−25.1908	−0.863530
$852$	0	0
$853$	−11.3292	−0.387904	−0.193952	−	0.981011i	$-0.562131\pi$
$853$	−11.3292	−0.387904	−0.193952	+	0.981011i	$0.562131\pi$
$854$	0	0
$855$	14.5766	0.498511
$856$	0	0
$857$	43.5287	1.48691	0.743456	−	0.668785i	$-0.233185\pi$
$857$	43.5287	1.48691	0.743456	+	0.668785i	$0.233185\pi$
$858$	0	0
$859$	46.8152	1.59731	0.798657	−	0.601786i	$-0.205544\pi$
$859$	46.8152	1.59731	0.798657	+	0.601786i	$0.205544\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.4038	0.932835	0.466417	−	0.884565i	$-0.345544\pi$
$863$	27.4038	0.932835	0.466417	+	0.884565i	$0.345544\pi$
$864$	0	0
$865$	−31.2645	−1.06303
$866$	0	0
$867$	−43.9846	−1.49379
$868$	0	0
$869$	−11.9720	−0.406122
$870$	0	0
$871$	33.0692	1.12051
$872$	0	0
$873$	76.7998	2.59928
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.4575	−1.02848	−0.514238	−	0.857648i	$-0.671925\pi$
$877$	−30.4575	−1.02848	−0.514238	+	0.857648i	$0.671925\pi$
$878$	0	0
$879$	−1.59561	−0.0538186
$880$	0	0
$881$	−48.9723	−1.64992	−0.824960	−	0.565191i	$-0.808802\pi$
$881$	−48.9723	−1.64992	−0.824960	+	0.565191i	$0.808802\pi$
$882$	0	0
$883$	−39.2534	−1.32098	−0.660490	−	0.750835i	$-0.729652\pi$
$883$	−39.2534	−1.32098	−0.660490	+	0.750835i	$0.729652\pi$
$884$	0	0
$885$	−113.592	−3.81835
$886$	0	0
$887$	−43.0844	−1.44663	−0.723316	−	0.690517i	$-0.757383\pi$
$887$	−43.0844	−1.44663	−0.723316	+	0.690517i	$0.757383\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.45142	−0.149128
$892$	0	0
$893$	0.190404	0.00637163
$894$	0	0
$895$	−53.1661	−1.77715
$896$	0	0
$897$	64.4685	2.15254
$898$	0	0
$899$	25.9141	0.864284
$900$	0	0
$901$	8.06329	0.268627
$902$	0	0
$903$	0	0
$904$	0	0
$905$	23.3112	0.774890
$906$	0	0
$907$	22.0151	0.731000	0.365500	−	0.930811i	$-0.380898\pi$
$907$	22.0151	0.731000	0.365500	+	0.930811i	$0.380898\pi$
$908$	0	0
$909$	−34.4187	−1.14160
$910$	0	0
$911$	27.6824	0.917158	0.458579	−	0.888654i	$-0.348359\pi$
$911$	27.6824	0.917158	0.458579	+	0.888654i	$0.348359\pi$
$912$	0	0
$913$	76.0275	2.51614
$914$	0	0
$915$	−80.2019	−2.65139
$916$	0	0
$917$	0	0
$918$	0	0
$919$	29.0442	0.958079	0.479040	−	0.877793i	$-0.340985\pi$
$919$	29.0442	0.958079	0.479040	+	0.877793i	$0.340985\pi$
$920$	0	0
$921$	20.6353	0.679957
$922$	0	0
$923$	−19.1005	−0.628700
$924$	0	0
$925$	−19.8996	−0.654296
$926$	0	0
$927$	−76.9266	−2.52660
$928$	0	0
$929$	−6.20578	−0.203605	−0.101803	−	0.994805i	$-0.532461\pi$
$929$	−6.20578	−0.203605	−0.101803	+	0.994805i	$0.532461\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−91.4122	−2.99270
$934$	0	0
$935$	−17.2588	−0.564422
$936$	0	0
$937$	23.4286	0.765380	0.382690	−	0.923877i	$-0.374998\pi$
$937$	23.4286	0.765380	0.382690	+	0.923877i	$0.374998\pi$
$938$	0	0
$939$	−54.0813	−1.76487
$940$	0	0
$941$	−7.80972	−0.254590	−0.127295	−	0.991865i	$-0.540629\pi$
$941$	−7.80972	−0.254590	−0.127295	+	0.991865i	$0.540629\pi$
$942$	0	0
$943$	−49.7240	−1.61924
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.9744	1.13652	0.568258	−	0.822850i	$-0.307618\pi$
$947$	34.9744	1.13652	0.568258	+	0.822850i	$0.307618\pi$
$948$	0	0
$949$	26.3442	0.855168
$950$	0	0
$951$	−31.9018	−1.03449
$952$	0	0
$953$	22.4430	0.727001	0.363501	−	0.931594i	$-0.381581\pi$
$953$	22.4430	0.727001	0.363501	+	0.931594i	$0.381581\pi$
$954$	0	0
$955$	53.7328	1.73875
$956$	0	0
$957$	−45.4519	−1.46925
$958$	0	0
$959$	0	0
$960$	0	0
$961$	35.9438	1.15948
$962$	0	0
$963$	−21.5665	−0.694971
$964$	0	0
$965$	13.3231	0.428887
$966$	0	0
$967$	−15.1114	−0.485949	−0.242975	−	0.970033i	$-0.578123\pi$
$967$	−15.1114	−0.485949	−0.242975	+	0.970033i	$0.578123\pi$
$968$	0	0
$969$	−3.00948	−0.0966785
$970$	0	0
$971$	−6.54519	−0.210045	−0.105023	−	0.994470i	$-0.533491\pi$
$971$	−6.54519	−0.210045	−0.105023	+	0.994470i	$0.533491\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	50.9273	1.63098
$976$	0	0
$977$	−6.11997	−0.195795	−0.0978976	−	0.995196i	$-0.531212\pi$
$977$	−6.11997	−0.195795	−0.0978976	+	0.995196i	$0.531212\pi$
$978$	0	0
$979$	−18.9772	−0.606514
$980$	0	0
$981$	64.0856	2.04610
$982$	0	0
$983$	−20.8829	−0.666061	−0.333031	−	0.942916i	$-0.608071\pi$
$983$	−20.8829	−0.666061	−0.333031	+	0.942916i	$0.608071\pi$
$984$	0	0
$985$	−11.2043	−0.356999
$986$	0	0
$987$	0	0
$988$	0	0
$989$	45.7787	1.45568
$990$	0	0
$991$	32.4748	1.03159	0.515797	−	0.856711i	$-0.327496\pi$
$991$	32.4748	1.03159	0.515797	+	0.856711i	$0.327496\pi$
$992$	0	0
$993$	45.9573	1.45841
$994$	0	0
$995$	57.0002	1.80703
$996$	0	0
$997$	−39.9989	−1.26678	−0.633389	−	0.773833i	$-0.718337\pi$
$997$	−39.9989	−1.26678	−0.633389	+	0.773833i	$0.718337\pi$
$998$	0	0
$999$	−21.0572	−0.666219

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3724.2.a.m.1.6		7
7.2	even	3		532.2.i.c.305.2	✓	14
7.4	even	3		532.2.i.c.457.2	yes	14
7.6	odd	2		3724.2.a.n.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.i.c.305.2	✓	14	7.2	even	3
532.2.i.c.457.2	yes	14	7.4	even	3
3724.2.a.m.1.6		7	1.1	even	1	trivial
3724.2.a.n.1.2		7	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:	\( N \)	\(=\)	\( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3724.a (trivial)

\(f(q)\)	\(=\)	\(q+2.77904 q^{3} -3.08629 q^{5} +4.72304 q^{9} +5.16388 q^{11} +4.04970 q^{13} -8.57690 q^{15} +1.08292 q^{17} -1.00000 q^{19} +5.72837 q^{23} +4.52516 q^{25} +4.78838 q^{27} -3.16724 q^{29} -8.18192 q^{31} +14.3506 q^{33} -4.39755 q^{37} +11.2543 q^{39} -8.68032 q^{41} +7.99159 q^{43} -14.5766 q^{45} -0.190404 q^{47} +3.00948 q^{51} +7.44586 q^{53} -15.9372 q^{55} -2.77904 q^{57} +13.2439 q^{59} +9.35092 q^{61} -12.4985 q^{65} +8.16585 q^{67} +15.9193 q^{69} -4.71652 q^{71} +6.50522 q^{73} +12.5756 q^{75} -2.31841 q^{79} -0.862030 q^{81} +14.7229 q^{83} -3.34221 q^{85} -8.80188 q^{87} -3.67499 q^{89} -22.7378 q^{93} +3.08629 q^{95} +16.2607 q^{97} +24.3892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{5} + 15 q^{9} + 14 q^{11} + 12 q^{15} - 10 q^{17} - 7 q^{19} + 7 q^{23} + 21 q^{25} + 2 q^{29} - 4 q^{31} + 24 q^{33} - 12 q^{37} + 18 q^{39} - 4 q^{41} - 6 q^{45} + 16 q^{47} + 2 q^{51} - 6 q^{53}+ \cdots + 40 q^{99}+O(q^{100}) \) 7 * q - 2 * q^5 + 15 * q^9 + 14 * q^11 + 12 * q^15 - 10 * q^17 - 7 * q^19 + 7 * q^23 + 21 * q^25 + 2 * q^29 - 4 * q^31 + 24 * q^33 - 12 * q^37 + 18 * q^39 - 4 * q^41 - 6 * q^45 + 16 * q^47 + 2 * q^51 - 6 * q^53 - 9 * q^55 + 4 * q^59 + 28 * q^61 - 8 * q^65 + 22 * q^67 - 34 * q^69 + 34 * q^71 - 16 * q^73 + 32 * q^75 + 14 * q^79 + 51 * q^81 - 13 * q^83 - 9 * q^85 - 10 * q^87 + 16 * q^89 - 36 * q^93 + 2 * q^95 + 4 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more