LMFDB - Embedded newform 3332.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, 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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.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:	$26.6061539535$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	3332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79129 q^{3} +2.79129 q^{5} +0.208712 q^{9} +O(q^{10})$	$q-1.79129 q^{3} +2.79129 q^{5} +0.208712 q^{9} +5.58258 q^{11} -4.00000 q^{13} -5.00000 q^{15} -1.00000 q^{17} -2.00000 q^{19} -7.58258 q^{23} +2.79129 q^{25} +5.00000 q^{27} -4.79129 q^{31} -10.0000 q^{33} -3.58258 q^{37} +7.16515 q^{39} -5.79129 q^{41} +1.79129 q^{43} +0.582576 q^{45} -7.58258 q^{47} +1.79129 q^{51} -5.20871 q^{53} +15.5826 q^{55} +3.58258 q^{57} -2.41742 q^{59} -2.20871 q^{61} -11.1652 q^{65} +11.9564 q^{67} +13.5826 q^{69} +13.5826 q^{71} -10.2087 q^{73} -5.00000 q^{75} -10.0000 q^{79} -9.58258 q^{81} +0.417424 q^{83} -2.79129 q^{85} -7.16515 q^{89} +8.58258 q^{93} -5.58258 q^{95} -5.20871 q^{97} +1.16515 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + q^{5} + 5 q^{9}+O(q^{10}) $ 2 * q + q^3 + q^5 + 5 * q^9	$ 2 q + q^{3} + q^{5} + 5 q^{9} + 2 q^{11} - 8 q^{13} - 10 q^{15} - 2 q^{17} - 4 q^{19} - 6 q^{23} + q^{25} + 10 q^{27} - 5 q^{31} - 20 q^{33} + 2 q^{37} - 4 q^{39} - 7 q^{41} - q^{43} - 8 q^{45} - 6 q^{47} - q^{51} - 15 q^{53} + 22 q^{55} - 2 q^{57} - 14 q^{59} - 9 q^{61} - 4 q^{65} + q^{67} + 18 q^{69} + 18 q^{71} - 25 q^{73} - 10 q^{75} - 20 q^{79} - 10 q^{81} + 10 q^{83} - q^{85} + 4 q^{89} + 8 q^{93} - 2 q^{95} - 15 q^{97} - 16 q^{99}+O(q^{100}) $ 2 * q + q^3 + q^5 + 5 * q^9 + 2 * q^11 - 8 * q^13 - 10 * q^15 - 2 * q^17 - 4 * q^19 - 6 * q^23 + q^25 + 10 * q^27 - 5 * q^31 - 20 * q^33 + 2 * q^37 - 4 * q^39 - 7 * q^41 - q^43 - 8 * q^45 - 6 * q^47 - q^51 - 15 * q^53 + 22 * q^55 - 2 * q^57 - 14 * q^59 - 9 * q^61 - 4 * q^65 + q^67 + 18 * q^69 + 18 * q^71 - 25 * q^73 - 10 * q^75 - 20 * q^79 - 10 * q^81 + 10 * q^83 - q^85 + 4 * q^89 + 8 * q^93 - 2 * q^95 - 15 * q^97 - 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.79129	−1.03420	−0.517100	−	0.855925i	$-0.672989\pi$
$3$	−1.79129	−1.03420	−0.517100	+	0.855925i	$0.672989\pi$
$4$	0	0
$5$	2.79129	1.24830	0.624151	−	0.781304i	$-0.285445\pi$
$5$	2.79129	1.24830	0.624151	+	0.781304i	$0.285445\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.208712	0.0695707
$10$	0	0
$11$	5.58258	1.68321	0.841605	−	0.540094i	$-0.181611\pi$
$11$	5.58258	1.68321	0.841605	+	0.540094i	$0.181611\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	−5.00000	−1.29099
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.58258	−1.58108	−0.790538	−	0.612413i	$-0.790199\pi$
$23$	−7.58258	−1.58108	−0.790538	+	0.612413i	$0.790199\pi$
$24$	0	0
$25$	2.79129	0.558258
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.79129	−0.860541	−0.430270	−	0.902700i	$-0.641582\pi$
$31$	−4.79129	−0.860541	−0.430270	+	0.902700i	$0.641582\pi$
$32$	0	0
$33$	−10.0000	−1.74078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.58258	−0.588972	−0.294486	−	0.955656i	$-0.595148\pi$
$37$	−3.58258	−0.588972	−0.294486	+	0.955656i	$0.595148\pi$
$38$	0	0
$39$	7.16515	1.14734
$40$	0	0
$41$	−5.79129	−0.904447	−0.452224	−	0.891905i	$-0.649369\pi$
$41$	−5.79129	−0.904447	−0.452224	+	0.891905i	$0.649369\pi$
$42$	0	0
$43$	1.79129	0.273169	0.136584	−	0.990628i	$-0.456388\pi$
$43$	1.79129	0.273169	0.136584	+	0.990628i	$0.456388\pi$
$44$	0	0
$45$	0.582576	0.0868453
$46$	0	0
$47$	−7.58258	−1.10603	−0.553016	−	0.833171i	$-0.686523\pi$
$47$	−7.58258	−1.10603	−0.553016	+	0.833171i	$0.686523\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.79129	0.250830
$52$	0	0
$53$	−5.20871	−0.715472	−0.357736	−	0.933823i	$-0.616451\pi$
$53$	−5.20871	−0.715472	−0.357736	+	0.933823i	$0.616451\pi$
$54$	0	0
$55$	15.5826	2.10115
$56$	0	0
$57$	3.58258	0.474524
$58$	0	0
$59$	−2.41742	−0.314722	−0.157361	−	0.987541i	$-0.550299\pi$
$59$	−2.41742	−0.314722	−0.157361	+	0.987541i	$0.550299\pi$
$60$	0	0
$61$	−2.20871	−0.282797	−0.141398	−	0.989953i	$-0.545160\pi$
$61$	−2.20871	−0.282797	−0.141398	+	0.989953i	$0.545160\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11.1652	−1.38487
$66$	0	0
$67$	11.9564	1.46071	0.730356	−	0.683067i	$-0.239354\pi$
$67$	11.9564	1.46071	0.730356	+	0.683067i	$0.239354\pi$
$68$	0	0
$69$	13.5826	1.63515
$70$	0	0
$71$	13.5826	1.61196	0.805978	−	0.591946i	$-0.201640\pi$
$71$	13.5826	1.61196	0.805978	+	0.591946i	$0.201640\pi$
$72$	0	0
$73$	−10.2087	−1.19484	−0.597420	−	0.801929i	$-0.703807\pi$
$73$	−10.2087	−1.19484	−0.597420	+	0.801929i	$0.703807\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	−9.58258	−1.06473
$82$	0	0
$83$	0.417424	0.0458183	0.0229091	−	0.999738i	$-0.492707\pi$
$83$	0.417424	0.0458183	0.0229091	+	0.999738i	$0.492707\pi$
$84$	0	0
$85$	−2.79129	−0.302758
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.16515	−0.759505	−0.379752	−	0.925088i	$-0.623991\pi$
$89$	−7.16515	−0.759505	−0.379752	+	0.925088i	$0.623991\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.58258	0.889972
$94$	0	0
$95$	−5.58258	−0.572760
$96$	0	0
$97$	−5.20871	−0.528865	−0.264432	−	0.964404i	$-0.585185\pi$
$97$	−5.20871	−0.528865	−0.264432	+	0.964404i	$0.585185\pi$
$98$	0	0
$99$	1.16515	0.117102
$100$	0	0
$101$	19.1652	1.90700	0.953502	−	0.301387i	$-0.0974496\pi$
$101$	19.1652	1.90700	0.953502	+	0.301387i	$0.0974496\pi$
$102$	0	0
$103$	−1.58258	−0.155936	−0.0779679	−	0.996956i	$-0.524843\pi$
$103$	−1.58258	−0.155936	−0.0779679	+	0.996956i	$0.524843\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.5826	1.50642	0.753212	−	0.657778i	$-0.228503\pi$
$107$	15.5826	1.50642	0.753212	+	0.657778i	$0.228503\pi$
$108$	0	0
$109$	10.4174	0.997808	0.498904	−	0.866657i	$-0.333736\pi$
$109$	10.4174	0.997808	0.498904	+	0.866657i	$0.333736\pi$
$110$	0	0
$111$	6.41742	0.609115
$112$	0	0
$113$	−10.4174	−0.979989	−0.489994	−	0.871726i	$-0.663001\pi$
$113$	−10.4174	−0.979989	−0.489994	+	0.871726i	$0.663001\pi$
$114$	0	0
$115$	−21.1652	−1.97366
$116$	0	0
$117$	−0.834849	−0.0771818
$118$	0	0
$119$	0	0
$120$	0	0
$121$	20.1652	1.83320
$122$	0	0
$123$	10.3739	0.935380
$124$	0	0
$125$	−6.16515	−0.551428
$126$	0	0
$127$	15.2087	1.34955	0.674777	−	0.738021i	$-0.264240\pi$
$127$	15.2087	1.34955	0.674777	+	0.738021i	$0.264240\pi$
$128$	0	0
$129$	−3.20871	−0.282511
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	13.9564	1.20118
$136$	0	0
$137$	−16.5390	−1.41302	−0.706512	−	0.707701i	$-0.749732\pi$
$137$	−16.5390	−1.41302	−0.706512	+	0.707701i	$0.749732\pi$
$138$	0	0
$139$	−12.7913	−1.08494	−0.542471	−	0.840074i	$-0.682511\pi$
$139$	−12.7913	−1.08494	−0.542471	+	0.840074i	$0.682511\pi$
$140$	0	0
$141$	13.5826	1.14386
$142$	0	0
$143$	−22.3303	−1.86735
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.5390	−1.35493	−0.677464	−	0.735556i	$-0.736921\pi$
$149$	−16.5390	−1.35493	−0.677464	+	0.735556i	$0.736921\pi$
$150$	0	0
$151$	16.9564	1.37990	0.689948	−	0.723859i	$-0.257633\pi$
$151$	16.9564	1.37990	0.689948	+	0.723859i	$0.257633\pi$
$152$	0	0
$153$	−0.208712	−0.0168734
$154$	0	0
$155$	−13.3739	−1.07421
$156$	0	0
$157$	−13.1652	−1.05069	−0.525347	−	0.850888i	$-0.676064\pi$
$157$	−13.1652	−1.05069	−0.525347	+	0.850888i	$0.676064\pi$
$158$	0	0
$159$	9.33030	0.739941
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.58258	0.750565	0.375283	−	0.926910i	$-0.377546\pi$
$163$	9.58258	0.750565	0.375283	+	0.926910i	$0.377546\pi$
$164$	0	0
$165$	−27.9129	−2.17301
$166$	0	0
$167$	−9.79129	−0.757673	−0.378836	−	0.925464i	$-0.623676\pi$
$167$	−9.79129	−0.757673	−0.378836	+	0.925464i	$0.623676\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−0.417424	−0.0319212
$172$	0	0
$173$	−25.3739	−1.92914	−0.964570	−	0.263829i	$-0.915015\pi$
$173$	−25.3739	−1.92914	−0.964570	+	0.263829i	$0.915015\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.33030	0.325485
$178$	0	0
$179$	−17.7913	−1.32978	−0.664892	−	0.746940i	$-0.731522\pi$
$179$	−17.7913	−1.32978	−0.664892	+	0.746940i	$0.731522\pi$
$180$	0	0
$181$	17.1652	1.27588	0.637938	−	0.770088i	$-0.279788\pi$
$181$	17.1652	1.27588	0.637938	+	0.770088i	$0.279788\pi$
$182$	0	0
$183$	3.95644	0.292468
$184$	0	0
$185$	−10.0000	−0.735215
$186$	0	0
$187$	−5.58258	−0.408238
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.20871	−0.232174	−0.116087	−	0.993239i	$-0.537035\pi$
$191$	−3.20871	−0.232174	−0.116087	+	0.993239i	$0.537035\pi$
$192$	0	0
$193$	−6.74773	−0.485712	−0.242856	−	0.970062i	$-0.578084\pi$
$193$	−6.74773	−0.485712	−0.242856	+	0.970062i	$0.578084\pi$
$194$	0	0
$195$	20.0000	1.43223
$196$	0	0
$197$	4.83485	0.344469	0.172234	−	0.985056i	$-0.444901\pi$
$197$	4.83485	0.344469	0.172234	+	0.985056i	$0.444901\pi$
$198$	0	0
$199$	16.5390	1.17242	0.586210	−	0.810159i	$-0.300619\pi$
$199$	16.5390	1.17242	0.586210	+	0.810159i	$0.300619\pi$
$200$	0	0
$201$	−21.4174	−1.51067
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−16.1652	−1.12902
$206$	0	0
$207$	−1.58258	−0.109997
$208$	0	0
$209$	−11.1652	−0.772310
$210$	0	0
$211$	−23.9129	−1.64623	−0.823115	−	0.567874i	$-0.807766\pi$
$211$	−23.9129	−1.64623	−0.823115	+	0.567874i	$0.807766\pi$
$212$	0	0
$213$	−24.3303	−1.66708
$214$	0	0
$215$	5.00000	0.340997
$216$	0	0
$217$	0	0
$218$	0	0
$219$	18.2867	1.23570
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	3.16515	0.211954	0.105977	−	0.994369i	$-0.466203\pi$
$223$	3.16515	0.211954	0.105977	+	0.994369i	$0.466203\pi$
$224$	0	0
$225$	0.582576	0.0388384
$226$	0	0
$227$	−13.6261	−0.904398	−0.452199	−	0.891917i	$-0.649360\pi$
$227$	−13.6261	−0.904398	−0.452199	+	0.891917i	$0.649360\pi$
$228$	0	0
$229$	10.7477	0.710230	0.355115	−	0.934823i	$-0.384442\pi$
$229$	10.7477	0.710230	0.355115	+	0.934823i	$0.384442\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.7477	−0.835131	−0.417566	−	0.908647i	$-0.637117\pi$
$233$	−12.7477	−0.835131	−0.417566	+	0.908647i	$0.637117\pi$
$234$	0	0
$235$	−21.1652	−1.38066
$236$	0	0
$237$	17.9129	1.16357
$238$	0	0
$239$	−1.37386	−0.0888678	−0.0444339	−	0.999012i	$-0.514148\pi$
$239$	−1.37386	−0.0888678	−0.0444339	+	0.999012i	$0.514148\pi$
$240$	0	0
$241$	−19.5390	−1.25862	−0.629309	−	0.777155i	$-0.716662\pi$
$241$	−19.5390	−1.25862	−0.629309	+	0.777155i	$0.716662\pi$
$242$	0	0
$243$	2.16515	0.138895
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	−0.747727	−0.0473853
$250$	0	0
$251$	19.5826	1.23604	0.618021	−	0.786162i	$-0.287935\pi$
$251$	19.5826	1.23604	0.618021	+	0.786162i	$0.287935\pi$
$252$	0	0
$253$	−42.3303	−2.66128
$254$	0	0
$255$	5.00000	0.313112
$256$	0	0
$257$	−28.3303	−1.76720	−0.883598	−	0.468247i	$-0.844886\pi$
$257$	−28.3303	−1.76720	−0.883598	+	0.468247i	$0.844886\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.33030	−0.390343	−0.195172	−	0.980769i	$-0.562526\pi$
$263$	−6.33030	−0.390343	−0.195172	+	0.980769i	$0.562526\pi$
$264$	0	0
$265$	−14.5390	−0.893125
$266$	0	0
$267$	12.8348	0.785480
$268$	0	0
$269$	21.1652	1.29046	0.645231	−	0.763988i	$-0.276761\pi$
$269$	21.1652	1.29046	0.645231	+	0.763988i	$0.276761\pi$
$270$	0	0
$271$	−15.1652	−0.921217	−0.460609	−	0.887603i	$-0.652369\pi$
$271$	−15.1652	−0.921217	−0.460609	+	0.887603i	$0.652369\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	15.5826	0.939665
$276$	0	0
$277$	0.747727	0.0449266	0.0224633	−	0.999748i	$-0.492849\pi$
$277$	0.747727	0.0449266	0.0224633	+	0.999748i	$0.492849\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	5.20871	0.310726	0.155363	−	0.987857i	$-0.450345\pi$
$281$	5.20871	0.310726	0.155363	+	0.987857i	$0.450345\pi$
$282$	0	0
$283$	15.1216	0.898885	0.449443	−	0.893309i	$-0.351623\pi$
$283$	15.1216	0.898885	0.449443	+	0.893309i	$0.351623\pi$
$284$	0	0
$285$	10.0000	0.592349
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	9.33030	0.546952
$292$	0	0
$293$	14.7477	0.861571	0.430786	−	0.902454i	$-0.358236\pi$
$293$	14.7477	0.861571	0.430786	+	0.902454i	$0.358236\pi$
$294$	0	0
$295$	−6.74773	−0.392868
$296$	0	0
$297$	27.9129	1.61967
$298$	0	0
$299$	30.3303	1.75405
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−34.3303	−1.97222
$304$	0	0
$305$	−6.16515	−0.353016
$306$	0	0
$307$	1.58258	0.0903224	0.0451612	−	0.998980i	$-0.485620\pi$
$307$	1.58258	0.0903224	0.0451612	+	0.998980i	$0.485620\pi$
$308$	0	0
$309$	2.83485	0.161269
$310$	0	0
$311$	24.7913	1.40578	0.702892	−	0.711296i	$-0.251891\pi$
$311$	24.7913	1.40578	0.702892	+	0.711296i	$0.251891\pi$
$312$	0	0
$313$	−18.7042	−1.05722	−0.528611	−	0.848864i	$-0.677287\pi$
$313$	−18.7042	−1.05722	−0.528611	+	0.848864i	$0.677287\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.3303	−1.47886	−0.739429	−	0.673235i	$-0.764904\pi$
$317$	−26.3303	−1.47886	−0.739429	+	0.673235i	$0.764904\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−27.9129	−1.55794
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	−11.1652	−0.619331
$326$	0	0
$327$	−18.6606	−1.03193
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6.37386	0.350339	0.175170	−	0.984538i	$-0.443953\pi$
$331$	6.37386	0.350339	0.175170	+	0.984538i	$0.443953\pi$
$332$	0	0
$333$	−0.747727	−0.0409752
$334$	0	0
$335$	33.3739	1.82341
$336$	0	0
$337$	1.16515	0.0634698	0.0317349	−	0.999496i	$-0.489897\pi$
$337$	1.16515	0.0634698	0.0317349	+	0.999496i	$0.489897\pi$
$338$	0	0
$339$	18.6606	1.01350
$340$	0	0
$341$	−26.7477	−1.44847
$342$	0	0
$343$	0	0
$344$	0	0
$345$	37.9129	2.04116
$346$	0	0
$347$	19.1652	1.02884	0.514420	−	0.857539i	$-0.328007\pi$
$347$	19.1652	1.02884	0.514420	+	0.857539i	$0.328007\pi$
$348$	0	0
$349$	−9.58258	−0.512944	−0.256472	−	0.966552i	$-0.582560\pi$
$349$	−9.58258	−0.512944	−0.256472	+	0.966552i	$0.582560\pi$
$350$	0	0
$351$	−20.0000	−1.06752
$352$	0	0
$353$	8.83485	0.470232	0.235116	−	0.971967i	$-0.424453\pi$
$353$	8.83485	0.470232	0.235116	+	0.971967i	$0.424453\pi$
$354$	0	0
$355$	37.9129	2.01221
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.95644	−0.208813	−0.104406	−	0.994535i	$-0.533294\pi$
$359$	−3.95644	−0.208813	−0.104406	+	0.994535i	$0.533294\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−36.1216	−1.89589
$364$	0	0
$365$	−28.4955	−1.49152
$366$	0	0
$367$	−14.6261	−0.763478	−0.381739	−	0.924270i	$-0.624675\pi$
$367$	−14.6261	−0.763478	−0.381739	+	0.924270i	$0.624675\pi$
$368$	0	0
$369$	−1.20871	−0.0629230
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−24.1216	−1.24897	−0.624484	−	0.781037i	$-0.714691\pi$
$373$	−24.1216	−1.24897	−0.624484	+	0.781037i	$0.714691\pi$
$374$	0	0
$375$	11.0436	0.570287
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−26.3303	−1.35250	−0.676248	−	0.736674i	$-0.736395\pi$
$379$	−26.3303	−1.35250	−0.676248	+	0.736674i	$0.736395\pi$
$380$	0	0
$381$	−27.2432	−1.39571
$382$	0	0
$383$	17.5826	0.898428	0.449214	−	0.893424i	$-0.351704\pi$
$383$	17.5826	0.898428	0.449214	+	0.893424i	$0.351704\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.373864	0.0190046
$388$	0	0
$389$	21.1216	1.07091	0.535454	−	0.844565i	$-0.320141\pi$
$389$	21.1216	1.07091	0.535454	+	0.844565i	$0.320141\pi$
$390$	0	0
$391$	7.58258	0.383467
$392$	0	0
$393$	21.4955	1.08430
$394$	0	0
$395$	−27.9129	−1.40445
$396$	0	0
$397$	28.2867	1.41967	0.709835	−	0.704368i	$-0.248769\pi$
$397$	28.2867	1.41967	0.709835	+	0.704368i	$0.248769\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.1652	−1.05694	−0.528469	−	0.848953i	$-0.677234\pi$
$401$	−21.1652	−1.05694	−0.528469	+	0.848953i	$0.677234\pi$
$402$	0	0
$403$	19.1652	0.954684
$404$	0	0
$405$	−26.7477	−1.32911
$406$	0	0
$407$	−20.0000	−0.991363
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	29.6261	1.46135
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.16515	0.0571950
$416$	0	0
$417$	22.9129	1.12205
$418$	0	0
$419$	28.7913	1.40655	0.703273	−	0.710920i	$-0.251721\pi$
$419$	28.7913	1.40655	0.703273	+	0.710920i	$0.251721\pi$
$420$	0	0
$421$	−13.7913	−0.672146	−0.336073	−	0.941836i	$-0.609099\pi$
$421$	−13.7913	−0.672146	−0.336073	+	0.941836i	$0.609099\pi$
$422$	0	0
$423$	−1.58258	−0.0769475
$424$	0	0
$425$	−2.79129	−0.135397
$426$	0	0
$427$	0	0
$428$	0	0
$429$	40.0000	1.93122
$430$	0	0
$431$	−9.58258	−0.461576	−0.230788	−	0.973004i	$-0.574130\pi$
$431$	−9.58258	−0.461576	−0.230788	+	0.973004i	$0.574130\pi$
$432$	0	0
$433$	−24.4174	−1.17343	−0.586713	−	0.809795i	$-0.699578\pi$
$433$	−24.4174	−1.17343	−0.586713	+	0.809795i	$0.699578\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15.1652	0.725448
$438$	0	0
$439$	20.9564	1.00020	0.500098	−	0.865969i	$-0.333297\pi$
$439$	20.9564	1.00020	0.500098	+	0.865969i	$0.333297\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−38.3303	−1.82113	−0.910564	−	0.413369i	$-0.864352\pi$
$443$	−38.3303	−1.82113	−0.910564	+	0.413369i	$0.864352\pi$
$444$	0	0
$445$	−20.0000	−0.948091
$446$	0	0
$447$	29.6261	1.40127
$448$	0	0
$449$	7.58258	0.357844	0.178922	−	0.983863i	$-0.442739\pi$
$449$	7.58258	0.357844	0.178922	+	0.983863i	$0.442739\pi$
$450$	0	0
$451$	−32.3303	−1.52237
$452$	0	0
$453$	−30.3739	−1.42709
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.20871	0.243653	0.121827	−	0.992551i	$-0.461125\pi$
$457$	5.20871	0.243653	0.121827	+	0.992551i	$0.461125\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	0	0
$461$	−3.25227	−0.151473	−0.0757367	−	0.997128i	$-0.524131\pi$
$461$	−3.25227	−0.151473	−0.0757367	+	0.997128i	$0.524131\pi$
$462$	0	0
$463$	18.6261	0.865630	0.432815	−	0.901483i	$-0.357520\pi$
$463$	18.6261	0.865630	0.432815	+	0.901483i	$0.357520\pi$
$464$	0	0
$465$	23.9564	1.11095
$466$	0	0
$467$	−12.8348	−0.593926	−0.296963	−	0.954889i	$-0.595974\pi$
$467$	−12.8348	−0.593926	−0.296963	+	0.954889i	$0.595974\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	23.5826	1.08663
$472$	0	0
$473$	10.0000	0.459800
$474$	0	0
$475$	−5.58258	−0.256146
$476$	0	0
$477$	−1.08712	−0.0497759
$478$	0	0
$479$	−34.8693	−1.59322	−0.796610	−	0.604494i	$-0.793375\pi$
$479$	−34.8693	−1.59322	−0.796610	+	0.604494i	$0.793375\pi$
$480$	0	0
$481$	14.3303	0.653406
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.5390	−0.660183
$486$	0	0
$487$	−5.25227	−0.238003	−0.119002	−	0.992894i	$-0.537969\pi$
$487$	−5.25227	−0.238003	−0.119002	+	0.992894i	$0.537969\pi$
$488$	0	0
$489$	−17.1652	−0.776235
$490$	0	0
$491$	23.9564	1.08114	0.540569	−	0.841299i	$-0.318209\pi$
$491$	23.9564	1.08114	0.540569	+	0.841299i	$0.318209\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	3.25227	0.146179
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−9.58258	−0.428975	−0.214488	−	0.976727i	$-0.568808\pi$
$499$	−9.58258	−0.428975	−0.214488	+	0.976727i	$0.568808\pi$
$500$	0	0
$501$	17.5390	0.783585
$502$	0	0
$503$	−0.791288	−0.0352818	−0.0176409	−	0.999844i	$-0.505616\pi$
$503$	−0.791288	−0.0352818	−0.0176409	+	0.999844i	$0.505616\pi$
$504$	0	0
$505$	53.4955	2.38052
$506$	0	0
$507$	−5.37386	−0.238662
$508$	0	0
$509$	28.3303	1.25572	0.627859	−	0.778327i	$-0.283931\pi$
$509$	28.3303	1.25572	0.627859	+	0.778327i	$0.283931\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−10.0000	−0.441511
$514$	0	0
$515$	−4.41742	−0.194655
$516$	0	0
$517$	−42.3303	−1.86168
$518$	0	0
$519$	45.4519	1.99512
$520$	0	0
$521$	28.9564	1.26860	0.634302	−	0.773085i	$-0.281287\pi$
$521$	28.9564	1.26860	0.634302	+	0.773085i	$0.281287\pi$
$522$	0	0
$523$	−31.0780	−1.35895	−0.679474	−	0.733700i	$-0.737792\pi$
$523$	−31.0780	−1.35895	−0.679474	+	0.733700i	$0.737792\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.79129	0.208712
$528$	0	0
$529$	34.4955	1.49980
$530$	0	0
$531$	−0.504546	−0.0218954
$532$	0	0
$533$	23.1652	1.00339
$534$	0	0
$535$	43.4955	1.88047
$536$	0	0
$537$	31.8693	1.37526
$538$	0	0
$539$	0	0
$540$	0	0
$541$	31.1652	1.33989	0.669947	−	0.742409i	$-0.266317\pi$
$541$	31.1652	1.33989	0.669947	+	0.742409i	$0.266317\pi$
$542$	0	0
$543$	−30.7477	−1.31951
$544$	0	0
$545$	29.0780	1.24557
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	−0.460985	−0.0196744
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	17.9129	0.760359
$556$	0	0
$557$	12.3303	0.522452	0.261226	−	0.965278i	$-0.415873\pi$
$557$	12.3303	0.522452	0.261226	+	0.965278i	$0.415873\pi$
$558$	0	0
$559$	−7.16515	−0.303054
$560$	0	0
$561$	10.0000	0.422200
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	−29.0780	−1.22332
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.04356	−0.295281	−0.147641	−	0.989041i	$-0.547168\pi$
$569$	−7.04356	−0.295281	−0.147641	+	0.989041i	$0.547168\pi$
$570$	0	0
$571$	−6.41742	−0.268561	−0.134280	−	0.990943i	$-0.542872\pi$
$571$	−6.41742	−0.268561	−0.134280	+	0.990943i	$0.542872\pi$
$572$	0	0
$573$	5.74773	0.240115
$574$	0	0
$575$	−21.1652	−0.882648
$576$	0	0
$577$	35.9129	1.49507	0.747536	−	0.664221i	$-0.231237\pi$
$577$	35.9129	1.49507	0.747536	+	0.664221i	$0.231237\pi$
$578$	0	0
$579$	12.0871	0.502324
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−29.0780	−1.20429
$584$	0	0
$585$	−2.33030	−0.0963462
$586$	0	0
$587$	−35.5826	−1.46865	−0.734325	−	0.678798i	$-0.762501\pi$
$587$	−35.5826	−1.46865	−0.734325	+	0.678798i	$0.762501\pi$
$588$	0	0
$589$	9.58258	0.394843
$590$	0	0
$591$	−8.66061	−0.356250
$592$	0	0
$593$	39.0780	1.60474	0.802371	−	0.596825i	$-0.203571\pi$
$593$	39.0780	1.60474	0.802371	+	0.596825i	$0.203571\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−29.6261	−1.21252
$598$	0	0
$599$	−17.7913	−0.726932	−0.363466	−	0.931607i	$-0.618407\pi$
$599$	−17.7913	−0.726932	−0.363466	+	0.931607i	$0.618407\pi$
$600$	0	0
$601$	−31.4955	−1.28473	−0.642363	−	0.766400i	$-0.722046\pi$
$601$	−31.4955	−1.28473	−0.642363	+	0.766400i	$0.722046\pi$
$602$	0	0
$603$	2.49545	0.101623
$604$	0	0
$605$	56.2867	2.28838
$606$	0	0
$607$	25.6261	1.04013	0.520066	−	0.854126i	$-0.325907\pi$
$607$	25.6261	1.04013	0.520066	+	0.854126i	$0.325907\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.3303	1.22703
$612$	0	0
$613$	36.2867	1.46561	0.732804	−	0.680440i	$-0.238211\pi$
$613$	36.2867	1.46561	0.732804	+	0.680440i	$0.238211\pi$
$614$	0	0
$615$	28.9564	1.16764
$616$	0	0
$617$	39.4955	1.59003	0.795014	−	0.606592i	$-0.207464\pi$
$617$	39.4955	1.59003	0.795014	+	0.606592i	$0.207464\pi$
$618$	0	0
$619$	−44.6606	−1.79506	−0.897531	−	0.440952i	$-0.854641\pi$
$619$	−44.6606	−1.79506	−0.897531	+	0.440952i	$0.854641\pi$
$620$	0	0
$621$	−37.9129	−1.52139
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.1652	−1.24661
$626$	0	0
$627$	20.0000	0.798723
$628$	0	0
$629$	3.58258	0.142847
$630$	0	0
$631$	3.95644	0.157503	0.0787517	−	0.996894i	$-0.474907\pi$
$631$	3.95644	0.157503	0.0787517	+	0.996894i	$0.474907\pi$
$632$	0	0
$633$	42.8348	1.70253
$634$	0	0
$635$	42.4519	1.68465
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.83485	0.112145
$640$	0	0
$641$	3.91288	0.154549	0.0772747	−	0.997010i	$-0.475378\pi$
$641$	3.91288	0.154549	0.0772747	+	0.997010i	$0.475378\pi$
$642$	0	0
$643$	20.9564	0.826441	0.413221	−	0.910631i	$-0.364404\pi$
$643$	20.9564	0.826441	0.413221	+	0.910631i	$0.364404\pi$
$644$	0	0
$645$	−8.95644	−0.352659
$646$	0	0
$647$	23.1652	0.910716	0.455358	−	0.890308i	$-0.349511\pi$
$647$	23.1652	0.910716	0.455358	+	0.890308i	$0.349511\pi$
$648$	0	0
$649$	−13.4955	−0.529743
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.9129	−1.01405	−0.507025	−	0.861932i	$-0.669255\pi$
$653$	−25.9129	−1.01405	−0.507025	+	0.861932i	$0.669255\pi$
$654$	0	0
$655$	−33.4955	−1.30878
$656$	0	0
$657$	−2.13068	−0.0831258
$658$	0	0
$659$	−4.04356	−0.157515	−0.0787574	−	0.996894i	$-0.525095\pi$
$659$	−4.04356	−0.157515	−0.0787574	+	0.996894i	$0.525095\pi$
$660$	0	0
$661$	11.4955	0.447121	0.223561	−	0.974690i	$-0.428232\pi$
$661$	11.4955	0.447121	0.223561	+	0.974690i	$0.428232\pi$
$662$	0	0
$663$	−7.16515	−0.278271
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−5.66970	−0.219203
$670$	0	0
$671$	−12.3303	−0.476006
$672$	0	0
$673$	51.0780	1.96891	0.984457	−	0.175628i	$-0.0561955\pi$
$673$	51.0780	1.96891	0.984457	+	0.175628i	$0.0561955\pi$
$674$	0	0
$675$	13.9564	0.537184
$676$	0	0
$677$	45.1652	1.73584	0.867919	−	0.496706i	$-0.165457\pi$
$677$	45.1652	1.73584	0.867919	+	0.496706i	$0.165457\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	24.4083	0.935329
$682$	0	0
$683$	5.25227	0.200973	0.100486	−	0.994938i	$-0.467960\pi$
$683$	5.25227	0.200973	0.100486	+	0.994938i	$0.467960\pi$
$684$	0	0
$685$	−46.1652	−1.76388
$686$	0	0
$687$	−19.2523	−0.734520
$688$	0	0
$689$	20.8348	0.793745
$690$	0	0
$691$	33.3739	1.26960	0.634801	−	0.772676i	$-0.281082\pi$
$691$	33.3739	1.26960	0.634801	+	0.772676i	$0.281082\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−35.7042	−1.35434
$696$	0	0
$697$	5.79129	0.219361
$698$	0	0
$699$	22.8348	0.863693
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	7.16515	0.270239
$704$	0	0
$705$	37.9129	1.42788
$706$	0	0
$707$	0	0
$708$	0	0
$709$	49.1652	1.84644	0.923218	−	0.384277i	$-0.125549\pi$
$709$	49.1652	1.84644	0.923218	+	0.384277i	$0.125549\pi$
$710$	0	0
$711$	−2.08712	−0.0782732
$712$	0	0
$713$	36.3303	1.36058
$714$	0	0
$715$	−62.3303	−2.33102
$716$	0	0
$717$	2.46099	0.0919072
$718$	0	0
$719$	50.8693	1.89711	0.948553	−	0.316619i	$-0.102548\pi$
$719$	50.8693	1.89711	0.948553	+	0.316619i	$0.102548\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	35.0000	1.30166
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.33030	0.160602	0.0803010	−	0.996771i	$-0.474412\pi$
$727$	4.33030	0.160602	0.0803010	+	0.996771i	$0.474412\pi$
$728$	0	0
$729$	24.8693	0.921086
$730$	0	0
$731$	−1.79129	−0.0662532
$732$	0	0
$733$	−12.7477	−0.470848	−0.235424	−	0.971893i	$-0.575648\pi$
$733$	−12.7477	−0.470848	−0.235424	+	0.971893i	$0.575648\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	66.7477	2.45868
$738$	0	0
$739$	7.04356	0.259102	0.129551	−	0.991573i	$-0.458646\pi$
$739$	7.04356	0.259102	0.129551	+	0.991573i	$0.458646\pi$
$740$	0	0
$741$	−14.3303	−0.526437
$742$	0	0
$743$	26.8348	0.984475	0.492238	−	0.870461i	$-0.336179\pi$
$743$	26.8348	0.984475	0.492238	+	0.870461i	$0.336179\pi$
$744$	0	0
$745$	−46.1652	−1.69136
$746$	0	0
$747$	0.0871215	0.00318761
$748$	0	0
$749$	0	0
$750$	0	0
$751$	25.1652	0.918289	0.459145	−	0.888361i	$-0.348156\pi$
$751$	25.1652	0.918289	0.459145	+	0.888361i	$0.348156\pi$
$752$	0	0
$753$	−35.0780	−1.27831
$754$	0	0
$755$	47.3303	1.72253
$756$	0	0
$757$	−41.6170	−1.51260	−0.756299	−	0.654227i	$-0.772994\pi$
$757$	−41.6170	−1.51260	−0.756299	+	0.654227i	$0.772994\pi$
$758$	0	0
$759$	75.8258	2.75230
$760$	0	0
$761$	12.8348	0.465263	0.232631	−	0.972565i	$-0.425266\pi$
$761$	12.8348	0.465263	0.232631	+	0.972565i	$0.425266\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.582576	−0.0210631
$766$	0	0
$767$	9.66970	0.349153
$768$	0	0
$769$	17.4955	0.630902	0.315451	−	0.948942i	$-0.397844\pi$
$769$	17.4955	0.630902	0.315451	+	0.948942i	$0.397844\pi$
$770$	0	0
$771$	50.7477	1.82763
$772$	0	0
$773$	13.9129	0.500411	0.250206	−	0.968193i	$-0.419502\pi$
$773$	13.9129	0.500411	0.250206	+	0.968193i	$0.419502\pi$
$774$	0	0
$775$	−13.3739	−0.480403
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.5826	0.414989
$780$	0	0
$781$	75.8258	2.71326
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−36.7477	−1.31158
$786$	0	0
$787$	14.3303	0.510820	0.255410	−	0.966833i	$-0.417790\pi$
$787$	14.3303	0.510820	0.255410	+	0.966833i	$0.417790\pi$
$788$	0	0
$789$	11.3394	0.403693
$790$	0	0
$791$	0	0
$792$	0	0
$793$	8.83485	0.313735
$794$	0	0
$795$	26.0436	0.923670
$796$	0	0
$797$	−41.0780	−1.45506	−0.727529	−	0.686077i	$-0.759331\pi$
$797$	−41.0780	−1.45506	−0.727529	+	0.686077i	$0.759331\pi$
$798$	0	0
$799$	7.58258	0.268252
$800$	0	0
$801$	−1.49545	−0.0528393
$802$	0	0
$803$	−56.9909	−2.01117
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−37.9129	−1.33460
$808$	0	0
$809$	27.4955	0.966689	0.483344	−	0.875430i	$-0.339422\pi$
$809$	27.4955	0.966689	0.483344	+	0.875430i	$0.339422\pi$
$810$	0	0
$811$	−51.7042	−1.81558	−0.907789	−	0.419426i	$-0.862231\pi$
$811$	−51.7042	−1.81558	−0.907789	+	0.419426i	$0.862231\pi$
$812$	0	0
$813$	27.1652	0.952723
$814$	0	0
$815$	26.7477	0.936932
$816$	0	0
$817$	−3.58258	−0.125338
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−53.0780	−1.85018	−0.925092	−	0.379743i	$-0.876012\pi$
$823$	−53.0780	−1.85018	−0.925092	+	0.379743i	$0.876012\pi$
$824$	0	0
$825$	−27.9129	−0.971802
$826$	0	0
$827$	3.66970	0.127608	0.0638039	−	0.997962i	$-0.479677\pi$
$827$	3.66970	0.127608	0.0638039	+	0.997962i	$0.479677\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	−1.33939	−0.0464631
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−27.3303	−0.945804
$836$	0	0
$837$	−23.9564	−0.828056
$838$	0	0
$839$	32.8348	1.13358	0.566792	−	0.823861i	$-0.308184\pi$
$839$	32.8348	1.13358	0.566792	+	0.823861i	$0.308184\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−9.33030	−0.321353
$844$	0	0
$845$	8.37386	0.288070
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−27.0871	−0.929628
$850$	0	0
$851$	27.1652	0.931209
$852$	0	0
$853$	−43.4955	−1.48926	−0.744628	−	0.667480i	$-0.767373\pi$
$853$	−43.4955	−1.48926	−0.744628	+	0.667480i	$0.767373\pi$
$854$	0	0
$855$	−1.16515	−0.0398473
$856$	0	0
$857$	−56.2867	−1.92272	−0.961359	−	0.275297i	$-0.911224\pi$
$857$	−56.2867	−1.92272	−0.961359	+	0.275297i	$0.911224\pi$
$858$	0	0
$859$	37.9129	1.29357	0.646785	−	0.762672i	$-0.276113\pi$
$859$	37.9129	1.29357	0.646785	+	0.762672i	$0.276113\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.37386	0.0808073	0.0404036	−	0.999183i	$-0.487136\pi$
$863$	2.37386	0.0808073	0.0404036	+	0.999183i	$0.487136\pi$
$864$	0	0
$865$	−70.8258	−2.40815
$866$	0	0
$867$	−1.79129	−0.0608353
$868$	0	0
$869$	−55.8258	−1.89376
$870$	0	0
$871$	−47.8258	−1.62051
$872$	0	0
$873$	−1.08712	−0.0367935
$874$	0	0
$875$	0	0
$876$	0	0
$877$	21.9129	0.739945	0.369973	−	0.929043i	$-0.379367\pi$
$877$	21.9129	0.739945	0.369973	+	0.929043i	$0.379367\pi$
$878$	0	0
$879$	−26.4174	−0.891038
$880$	0	0
$881$	−1.12159	−0.0377873	−0.0188937	−	0.999821i	$-0.506014\pi$
$881$	−1.12159	−0.0377873	−0.0188937	+	0.999821i	$0.506014\pi$
$882$	0	0
$883$	−2.46099	−0.0828187	−0.0414094	−	0.999142i	$-0.513185\pi$
$883$	−2.46099	−0.0828187	−0.0414094	+	0.999142i	$0.513185\pi$
$884$	0	0
$885$	12.0871	0.406304
$886$	0	0
$887$	−24.7913	−0.832410	−0.416205	−	0.909271i	$-0.636640\pi$
$887$	−24.7913	−0.832410	−0.416205	+	0.909271i	$0.636640\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−53.4955	−1.79217
$892$	0	0
$893$	15.1652	0.507482
$894$	0	0
$895$	−49.6606	−1.65997
$896$	0	0
$897$	−54.3303	−1.81404
$898$	0	0
$899$	0	0
$900$	0	0
$901$	5.20871	0.173527
$902$	0	0
$903$	0	0
$904$	0	0
$905$	47.9129	1.59268
$906$	0	0
$907$	−4.41742	−0.146678	−0.0733391	−	0.997307i	$-0.523366\pi$
$907$	−4.41742	−0.146678	−0.0733391	+	0.997307i	$0.523366\pi$
$908$	0	0
$909$	4.00000	0.132672
$910$	0	0
$911$	−46.3303	−1.53499	−0.767496	−	0.641054i	$-0.778497\pi$
$911$	−46.3303	−1.53499	−0.767496	+	0.641054i	$0.778497\pi$
$912$	0	0
$913$	2.33030	0.0771218
$914$	0	0
$915$	11.0436	0.365089
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−56.1216	−1.85128	−0.925640	−	0.378405i	$-0.876473\pi$
$919$	−56.1216	−1.85128	−0.925640	+	0.378405i	$0.876473\pi$
$920$	0	0
$921$	−2.83485	−0.0934114
$922$	0	0
$923$	−54.3303	−1.78830
$924$	0	0
$925$	−10.0000	−0.328798
$926$	0	0
$927$	−0.330303	−0.0108486
$928$	0	0
$929$	8.37386	0.274738	0.137369	−	0.990520i	$-0.456135\pi$
$929$	8.37386	0.274738	0.137369	+	0.990520i	$0.456135\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−44.4083	−1.45386
$934$	0	0
$935$	−15.5826	−0.509605
$936$	0	0
$937$	56.6606	1.85102	0.925511	−	0.378722i	$-0.123636\pi$
$937$	56.6606	1.85102	0.925511	+	0.378722i	$0.123636\pi$
$938$	0	0
$939$	33.5045	1.09338
$940$	0	0
$941$	55.2867	1.80230	0.901148	−	0.433511i	$-0.142726\pi$
$941$	55.2867	1.80230	0.901148	+	0.433511i	$0.142726\pi$
$942$	0	0
$943$	43.9129	1.43000
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.9129	1.49197	0.745984	−	0.665964i	$-0.231979\pi$
$947$	45.9129	1.49197	0.745984	+	0.665964i	$0.231979\pi$
$948$	0	0
$949$	40.8348	1.32556
$950$	0	0
$951$	47.1652	1.52943
$952$	0	0
$953$	−58.7042	−1.90161	−0.950807	−	0.309783i	$-0.899744\pi$
$953$	−58.7042	−1.90161	−0.950807	+	0.309783i	$0.899744\pi$
$954$	0	0
$955$	−8.95644	−0.289824
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−8.04356	−0.259470
$962$	0	0
$963$	3.25227	0.104803
$964$	0	0
$965$	−18.8348	−0.606315
$966$	0	0
$967$	7.46099	0.239929	0.119965	−	0.992778i	$-0.461722\pi$
$967$	7.46099	0.239929	0.119965	+	0.992778i	$0.461722\pi$
$968$	0	0
$969$	−3.58258	−0.115089
$970$	0	0
$971$	38.8348	1.24627	0.623135	−	0.782114i	$-0.285859\pi$
$971$	38.8348	1.24627	0.623135	+	0.782114i	$0.285859\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	20.0000	0.640513
$976$	0	0
$977$	−16.4610	−0.526634	−0.263317	−	0.964709i	$-0.584816\pi$
$977$	−16.4610	−0.526634	−0.263317	+	0.964709i	$0.584816\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	0	0
$981$	2.17424	0.0694182
$982$	0	0
$983$	22.3739	0.713615	0.356808	−	0.934178i	$-0.383865\pi$
$983$	22.3739	0.713615	0.356808	+	0.934178i	$0.383865\pi$
$984$	0	0
$985$	13.4955	0.430001
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.5826	−0.431901
$990$	0	0
$991$	−19.0780	−0.606034	−0.303017	−	0.952985i	$-0.597994\pi$
$991$	−19.0780	−0.606034	−0.303017	+	0.952985i	$0.597994\pi$
$992$	0	0
$993$	−11.4174	−0.362321
$994$	0	0
$995$	46.1652	1.46353
$996$	0	0
$997$	−51.7042	−1.63749	−0.818744	−	0.574159i	$-0.805329\pi$
$997$	−51.7042	−1.63749	−0.818744	+	0.574159i	$0.805329\pi$
$998$	0	0
$999$	−17.9129	−0.566738

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.m.1.1	yes	2
7.6	odd	2		3332.2.a.i.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.a.i.1.2	✓	2	7.6	odd	2
3332.2.a.m.1.1	yes	2	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:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79129 q^{3} +2.79129 q^{5} +0.208712 q^{9} +O(q^{10})\)	\(q-1.79129 q^{3} +2.79129 q^{5} +0.208712 q^{9} +5.58258 q^{11} -4.00000 q^{13} -5.00000 q^{15} -1.00000 q^{17} -2.00000 q^{19} -7.58258 q^{23} +2.79129 q^{25} +5.00000 q^{27} -4.79129 q^{31} -10.0000 q^{33} -3.58258 q^{37} +7.16515 q^{39} -5.79129 q^{41} +1.79129 q^{43} +0.582576 q^{45} -7.58258 q^{47} +1.79129 q^{51} -5.20871 q^{53} +15.5826 q^{55} +3.58258 q^{57} -2.41742 q^{59} -2.20871 q^{61} -11.1652 q^{65} +11.9564 q^{67} +13.5826 q^{69} +13.5826 q^{71} -10.2087 q^{73} -5.00000 q^{75} -10.0000 q^{79} -9.58258 q^{81} +0.417424 q^{83} -2.79129 q^{85} -7.16515 q^{89} +8.58258 q^{93} -5.58258 q^{95} -5.20871 q^{97} +1.16515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{5} + 5 q^{9}+O(q^{10}) \) 2 * q + q^3 + q^5 + 5 * q^9	\( 2 q + q^{3} + q^{5} + 5 q^{9} + 2 q^{11} - 8 q^{13} - 10 q^{15} - 2 q^{17} - 4 q^{19} - 6 q^{23} + q^{25} + 10 q^{27} - 5 q^{31} - 20 q^{33} + 2 q^{37} - 4 q^{39} - 7 q^{41} - q^{43} - 8 q^{45} - 6 q^{47} - q^{51} - 15 q^{53} + 22 q^{55} - 2 q^{57} - 14 q^{59} - 9 q^{61} - 4 q^{65} + q^{67} + 18 q^{69} + 18 q^{71} - 25 q^{73} - 10 q^{75} - 20 q^{79} - 10 q^{81} + 10 q^{83} - q^{85} + 4 q^{89} + 8 q^{93} - 2 q^{95} - 15 q^{97} - 16 q^{99}+O(q^{100}) \) 2 * q + q^3 + q^5 + 5 * q^9 + 2 * q^11 - 8 * q^13 - 10 * q^15 - 2 * q^17 - 4 * q^19 - 6 * q^23 + q^25 + 10 * q^27 - 5 * q^31 - 20 * q^33 + 2 * q^37 - 4 * q^39 - 7 * q^41 - q^43 - 8 * q^45 - 6 * q^47 - q^51 - 15 * q^53 + 22 * q^55 - 2 * q^57 - 14 * q^59 - 9 * q^61 - 4 * q^65 + q^67 + 18 * q^69 + 18 * q^71 - 25 * q^73 - 10 * q^75 - 20 * q^79 - 10 * q^81 + 10 * q^83 - q^85 + 4 * q^89 + 8 * q^93 - 2 * q^95 - 15 * q^97 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more