LMFDB - Embedded newform 3300.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3300,2,Mod(1,3300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3300, 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("3300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3300.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:	$26.3506326670$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 660)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	3300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.60555 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.60555 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.60555 q^{13} -6.60555 q^{17} -7.21110 q^{19} -4.60555 q^{21} +1.00000 q^{27} +8.00000 q^{29} +9.21110 q^{31} +1.00000 q^{33} +3.21110 q^{37} +4.60555 q^{39} +8.00000 q^{41} +3.39445 q^{43} +5.21110 q^{47} +14.2111 q^{49} -6.60555 q^{51} -2.00000 q^{53} -7.21110 q^{57} +8.00000 q^{59} +7.21110 q^{61} -4.60555 q^{63} +4.00000 q^{67} -14.4222 q^{71} -0.605551 q^{73} -4.60555 q^{77} -11.2111 q^{79} +1.00000 q^{81} +10.6056 q^{83} +8.00000 q^{87} +6.00000 q^{89} -21.2111 q^{91} +9.21110 q^{93} +12.4222 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 6 q^{17} - 2 q^{21} + 2 q^{27} + 16 q^{29} + 4 q^{31} + 2 q^{33} - 8 q^{37} + 2 q^{39} + 16 q^{41} + 14 q^{43} - 4 q^{47} + 14 q^{49} - 6 q^{51} - 4 q^{53} + 16 q^{59} - 2 q^{63} + 8 q^{67} + 6 q^{73} - 2 q^{77} - 8 q^{79} + 2 q^{81} + 14 q^{83} + 16 q^{87} + 12 q^{89} - 28 q^{91} + 4 q^{93} - 4 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 - 6 * q^17 - 2 * q^21 + 2 * q^27 + 16 * q^29 + 4 * q^31 + 2 * q^33 - 8 * q^37 + 2 * q^39 + 16 * q^41 + 14 * q^43 - 4 * q^47 + 14 * q^49 - 6 * q^51 - 4 * q^53 + 16 * q^59 - 2 * q^63 + 8 * q^67 + 6 * q^73 - 2 * q^77 - 8 * q^79 + 2 * q^81 + 14 * q^83 + 16 * q^87 + 12 * q^89 - 28 * q^91 + 4 * q^93 - 4 * q^97 + 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.60555	−1.74073	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−4.60555	−1.74073	−0.870367	+	0.492403i	$0.836119\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.60555	1.27735	0.638675	−	0.769477i	$-0.279483\pi$
$13$	4.60555	1.27735	0.638675	+	0.769477i	$0.279483\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.60555	−1.60208	−0.801041	−	0.598610i	$-0.795720\pi$
$17$	−6.60555	−1.60208	−0.801041	+	0.598610i	$0.795720\pi$
$18$	0	0
$19$	−7.21110	−1.65434	−0.827170	−	0.561951i	$-0.810051\pi$
$19$	−7.21110	−1.65434	−0.827170	+	0.561951i	$0.810051\pi$
$20$	0	0
$21$	−4.60555	−1.00501
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	9.21110	1.65436	0.827181	−	0.561935i	$-0.189943\pi$
$31$	9.21110	1.65436	0.827181	+	0.561935i	$0.189943\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.21110	0.527902	0.263951	−	0.964536i	$-0.414974\pi$
$37$	3.21110	0.527902	0.263951	+	0.964536i	$0.414974\pi$
$38$	0	0
$39$	4.60555	0.737478
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	3.39445	0.517649	0.258824	−	0.965924i	$-0.416665\pi$
$43$	3.39445	0.517649	0.258824	+	0.965924i	$0.416665\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.21110	0.760117	0.380059	−	0.924962i	$-0.375904\pi$
$47$	5.21110	0.760117	0.380059	+	0.924962i	$0.375904\pi$
$48$	0	0
$49$	14.2111	2.03016
$50$	0	0
$51$	−6.60555	−0.924962
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.21110	−0.955134
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	7.21110	0.923287	0.461644	−	0.887066i	$-0.347260\pi$
$61$	7.21110	0.923287	0.461644	+	0.887066i	$0.347260\pi$
$62$	0	0
$63$	−4.60555	−0.580245
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.4222	−1.71160	−0.855800	−	0.517306i	$-0.826935\pi$
$71$	−14.4222	−1.71160	−0.855800	+	0.517306i	$0.826935\pi$
$72$	0	0
$73$	−0.605551	−0.0708744	−0.0354372	−	0.999372i	$-0.511282\pi$
$73$	−0.605551	−0.0708744	−0.0354372	+	0.999372i	$0.511282\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.60555	−0.524851
$78$	0	0
$79$	−11.2111	−1.26135	−0.630674	−	0.776048i	$-0.717221\pi$
$79$	−11.2111	−1.26135	−0.630674	+	0.776048i	$0.717221\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.6056	1.16411	0.582055	−	0.813149i	$-0.302249\pi$
$83$	10.6056	1.16411	0.582055	+	0.813149i	$0.302249\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−21.2111	−2.22353
$92$	0	0
$93$	9.21110	0.955147
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.4222	1.26128	0.630642	−	0.776074i	$-0.282792\pi$
$97$	12.4222	1.26128	0.630642	+	0.776074i	$0.282792\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	5.21110	0.518524	0.259262	−	0.965807i	$-0.416521\pi$
$101$	5.21110	0.518524	0.259262	+	0.965807i	$0.416521\pi$
$102$	0	0
$103$	−1.21110	−0.119333	−0.0596667	−	0.998218i	$-0.519004\pi$
$103$	−1.21110	−0.119333	−0.0596667	+	0.998218i	$0.519004\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.8167	−1.52905	−0.764527	−	0.644592i	$-0.777027\pi$
$107$	−15.8167	−1.52905	−0.764527	+	0.644592i	$0.777027\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	3.21110	0.304784
$112$	0	0
$113$	0.788897	0.0742132	0.0371066	−	0.999311i	$-0.488186\pi$
$113$	0.788897	0.0742132	0.0371066	+	0.999311i	$0.488186\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.60555	0.425783
$118$	0	0
$119$	30.4222	2.78880
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.00000	0.721336
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.60555	0.763619	0.381810	−	0.924241i	$-0.375301\pi$
$127$	8.60555	0.763619	0.381810	+	0.924241i	$0.375301\pi$
$128$	0	0
$129$	3.39445	0.298865
$130$	0	0
$131$	−6.78890	−0.593149	−0.296574	−	0.955010i	$-0.595844\pi$
$131$	−6.78890	−0.593149	−0.296574	+	0.955010i	$0.595844\pi$
$132$	0	0
$133$	33.2111	2.87977
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	16.4222	1.39291	0.696457	−	0.717599i	$-0.254759\pi$
$139$	16.4222	1.39291	0.696457	+	0.717599i	$0.254759\pi$
$140$	0	0
$141$	5.21110	0.438854
$142$	0	0
$143$	4.60555	0.385136
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.2111	1.17211
$148$	0	0
$149$	−9.21110	−0.754603	−0.377301	−	0.926090i	$-0.623148\pi$
$149$	−9.21110	−0.754603	−0.377301	+	0.926090i	$0.623148\pi$
$150$	0	0
$151$	−19.2111	−1.56338	−0.781689	−	0.623669i	$-0.785641\pi$
$151$	−19.2111	−1.56338	−0.781689	+	0.623669i	$0.785641\pi$
$152$	0	0
$153$	−6.60555	−0.534027
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.21110	−0.256274	−0.128137	−	0.991756i	$-0.540900\pi$
$157$	−3.21110	−0.256274	−0.128137	+	0.991756i	$0.540900\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	5.21110	0.408165	0.204083	−	0.978954i	$-0.434579\pi$
$163$	5.21110	0.408165	0.204083	+	0.978954i	$0.434579\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.8167	1.53346	0.766729	−	0.641970i	$-0.221883\pi$
$167$	19.8167	1.53346	0.766729	+	0.641970i	$0.221883\pi$
$168$	0	0
$169$	8.21110	0.631623
$170$	0	0
$171$	−7.21110	−0.551447
$172$	0	0
$173$	−0.183346	−0.0139396	−0.00696978	−	0.999976i	$-0.502219\pi$
$173$	−0.183346	−0.0139396	−0.00696978	+	0.999976i	$0.502219\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	23.2111	1.72527	0.862634	−	0.505829i	$-0.168813\pi$
$181$	23.2111	1.72527	0.862634	+	0.505829i	$0.168813\pi$
$182$	0	0
$183$	7.21110	0.533060
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.60555	−0.483046
$188$	0	0
$189$	−4.60555	−0.335005
$190$	0	0
$191$	18.4222	1.33298	0.666492	−	0.745512i	$-0.267795\pi$
$191$	18.4222	1.33298	0.666492	+	0.745512i	$0.267795\pi$
$192$	0	0
$193$	21.8167	1.57040	0.785199	−	0.619244i	$-0.212561\pi$
$193$	21.8167	1.57040	0.785199	+	0.619244i	$0.212561\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.3944	0.954315	0.477157	−	0.878818i	$-0.341667\pi$
$197$	13.3944	0.954315	0.477157	+	0.878818i	$0.341667\pi$
$198$	0	0
$199$	−10.4222	−0.738811	−0.369405	−	0.929268i	$-0.620439\pi$
$199$	−10.4222	−0.738811	−0.369405	+	0.929268i	$0.620439\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	−36.8444	−2.58597
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.21110	−0.498802
$210$	0	0
$211$	−23.2111	−1.59792	−0.798959	−	0.601385i	$-0.794616\pi$
$211$	−23.2111	−1.59792	−0.798959	+	0.601385i	$0.794616\pi$
$212$	0	0
$213$	−14.4222	−0.988193
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−42.4222	−2.87981
$218$	0	0
$219$	−0.605551	−0.0409194
$220$	0	0
$221$	−30.4222	−2.04642
$222$	0	0
$223$	−9.21110	−0.616821	−0.308411	−	0.951253i	$-0.599797\pi$
$223$	−9.21110	−0.616821	−0.308411	+	0.951253i	$0.599797\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.8167	1.04979	0.524894	−	0.851168i	$-0.324105\pi$
$227$	15.8167	1.04979	0.524894	+	0.851168i	$0.324105\pi$
$228$	0	0
$229$	−8.42221	−0.556555	−0.278277	−	0.960501i	$-0.589763\pi$
$229$	−8.42221	−0.556555	−0.278277	+	0.960501i	$0.589763\pi$
$230$	0	0
$231$	−4.60555	−0.303023
$232$	0	0
$233$	10.6056	0.694793	0.347396	−	0.937718i	$-0.387066\pi$
$233$	10.6056	0.694793	0.347396	+	0.937718i	$0.387066\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−11.2111	−0.728239
$238$	0	0
$239$	6.78890	0.439137	0.219569	−	0.975597i	$-0.429535\pi$
$239$	6.78890	0.439137	0.219569	+	0.975597i	$0.429535\pi$
$240$	0	0
$241$	−15.2111	−0.979833	−0.489917	−	0.871769i	$-0.662973\pi$
$241$	−15.2111	−0.979833	−0.489917	+	0.871769i	$0.662973\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−33.2111	−2.11317
$248$	0	0
$249$	10.6056	0.672100
$250$	0	0
$251$	−10.4222	−0.657844	−0.328922	−	0.944357i	$-0.606685\pi$
$251$	−10.4222	−0.657844	−0.328922	+	0.944357i	$0.606685\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.2111	1.44787	0.723934	−	0.689869i	$-0.242332\pi$
$257$	23.2111	1.44787	0.723934	+	0.689869i	$0.242332\pi$
$258$	0	0
$259$	−14.7889	−0.918937
$260$	0	0
$261$	8.00000	0.495188
$262$	0	0
$263$	17.0278	1.04998	0.524988	−	0.851109i	$-0.324070\pi$
$263$	17.0278	1.04998	0.524988	+	0.851109i	$0.324070\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	−20.4222	−1.24516	−0.622582	−	0.782555i	$-0.713916\pi$
$269$	−20.4222	−1.24516	−0.622582	+	0.782555i	$0.713916\pi$
$270$	0	0
$271$	−4.78890	−0.290905	−0.145452	−	0.989365i	$-0.546464\pi$
$271$	−4.78890	−0.290905	−0.145452	+	0.989365i	$0.546464\pi$
$272$	0	0
$273$	−21.2111	−1.28375
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−32.2389	−1.93705	−0.968523	−	0.248925i	$-0.919923\pi$
$277$	−32.2389	−1.93705	−0.968523	+	0.248925i	$0.919923\pi$
$278$	0	0
$279$	9.21110	0.551454
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−4.60555	−0.273772	−0.136886	−	0.990587i	$-0.543709\pi$
$283$	−4.60555	−0.273772	−0.136886	+	0.990587i	$0.543709\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−36.8444	−2.17486
$288$	0	0
$289$	26.6333	1.56667
$290$	0	0
$291$	12.4222	0.728203
$292$	0	0
$293$	−19.8167	−1.15770	−0.578851	−	0.815434i	$-0.696499\pi$
$293$	−19.8167	−1.15770	−0.578851	+	0.815434i	$0.696499\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−15.6333	−0.901089
$302$	0	0
$303$	5.21110	0.299370
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−8.60555	−0.491145	−0.245572	−	0.969378i	$-0.578976\pi$
$307$	−8.60555	−0.491145	−0.245572	+	0.969378i	$0.578976\pi$
$308$	0	0
$309$	−1.21110	−0.0688972
$310$	0	0
$311$	30.4222	1.72508	0.862542	−	0.505985i	$-0.168871\pi$
$311$	30.4222	1.72508	0.862542	+	0.505985i	$0.168871\pi$
$312$	0	0
$313$	−8.78890	−0.496778	−0.248389	−	0.968660i	$-0.579901\pi$
$313$	−8.78890	−0.496778	−0.248389	+	0.968660i	$0.579901\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.2111	−0.629678	−0.314839	−	0.949145i	$-0.601951\pi$
$317$	−11.2111	−0.629678	−0.314839	+	0.949145i	$0.601951\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	−15.8167	−0.882800
$322$	0	0
$323$	47.6333	2.65039
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−10.0000	−0.553001
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−6.78890	−0.373152	−0.186576	−	0.982441i	$-0.559739\pi$
$331$	−6.78890	−0.373152	−0.186576	+	0.982441i	$0.559739\pi$
$332$	0	0
$333$	3.21110	0.175967
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.2389	1.53827	0.769134	−	0.639087i	$-0.220688\pi$
$337$	28.2389	1.53827	0.769134	+	0.639087i	$0.220688\pi$
$338$	0	0
$339$	0.788897	0.0428470
$340$	0	0
$341$	9.21110	0.498809
$342$	0	0
$343$	−33.2111	−1.79323
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.0278	1.55829	0.779146	−	0.626843i	$-0.215653\pi$
$347$	29.0278	1.55829	0.779146	+	0.626843i	$0.215653\pi$
$348$	0	0
$349$	−8.78890	−0.470459	−0.235229	−	0.971940i	$-0.575584\pi$
$349$	−8.78890	−0.470459	−0.235229	+	0.971940i	$0.575584\pi$
$350$	0	0
$351$	4.60555	0.245826
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	30.4222	1.61011
$358$	0	0
$359$	18.7889	0.991640	0.495820	−	0.868425i	$-0.334868\pi$
$359$	18.7889	0.991640	0.495820	+	0.868425i	$0.334868\pi$
$360$	0	0
$361$	33.0000	1.73684
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6.78890	−0.354378	−0.177189	−	0.984177i	$-0.556700\pi$
$367$	−6.78890	−0.354378	−0.177189	+	0.984177i	$0.556700\pi$
$368$	0	0
$369$	8.00000	0.416463
$370$	0	0
$371$	9.21110	0.478217
$372$	0	0
$373$	−4.60555	−0.238466	−0.119233	−	0.992866i	$-0.538044\pi$
$373$	−4.60555	−0.238466	−0.119233	+	0.992866i	$0.538044\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	36.8444	1.89758
$378$	0	0
$379$	−18.4222	−0.946285	−0.473143	−	0.880986i	$-0.656880\pi$
$379$	−18.4222	−0.946285	−0.473143	+	0.880986i	$0.656880\pi$
$380$	0	0
$381$	8.60555	0.440876
$382$	0	0
$383$	−18.7889	−0.960068	−0.480034	−	0.877250i	$-0.659376\pi$
$383$	−18.7889	−0.960068	−0.480034	+	0.877250i	$0.659376\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.39445	0.172550
$388$	0	0
$389$	30.8444	1.56387	0.781937	−	0.623358i	$-0.214232\pi$
$389$	30.8444	1.56387	0.781937	+	0.623358i	$0.214232\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−6.78890	−0.342455
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−28.4222	−1.42647	−0.713235	−	0.700925i	$-0.752771\pi$
$397$	−28.4222	−1.42647	−0.713235	+	0.700925i	$0.752771\pi$
$398$	0	0
$399$	33.2111	1.66263
$400$	0	0
$401$	20.4222	1.01984	0.509918	−	0.860223i	$-0.329676\pi$
$401$	20.4222	1.01984	0.509918	+	0.860223i	$0.329676\pi$
$402$	0	0
$403$	42.4222	2.11320
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.21110	0.159168
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	−36.8444	−1.81299
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.4222	0.804199
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−19.2111	−0.936292	−0.468146	−	0.883651i	$-0.655078\pi$
$421$	−19.2111	−0.936292	−0.468146	+	0.883651i	$0.655078\pi$
$422$	0	0
$423$	5.21110	0.253372
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−33.2111	−1.60720
$428$	0	0
$429$	4.60555	0.222358
$430$	0	0
$431$	−1.57779	−0.0759997	−0.0379999	−	0.999278i	$-0.512099\pi$
$431$	−1.57779	−0.0759997	−0.0379999	+	0.999278i	$0.512099\pi$
$432$	0	0
$433$	−8.42221	−0.404745	−0.202373	−	0.979309i	$-0.564865\pi$
$433$	−8.42221	−0.404745	−0.202373	+	0.979309i	$0.564865\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	14.2111	0.676719
$442$	0	0
$443$	−25.2111	−1.19782	−0.598908	−	0.800818i	$-0.704398\pi$
$443$	−25.2111	−1.19782	−0.598908	+	0.800818i	$0.704398\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−9.21110	−0.435670
$448$	0	0
$449$	18.8444	0.889323	0.444661	−	0.895699i	$-0.353324\pi$
$449$	18.8444	0.889323	0.444661	+	0.895699i	$0.353324\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	−19.2111	−0.902616
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.60555	0.215439	0.107719	−	0.994181i	$-0.465645\pi$
$457$	4.60555	0.215439	0.107719	+	0.994181i	$0.465645\pi$
$458$	0	0
$459$	−6.60555	−0.308321
$460$	0	0
$461$	−9.21110	−0.429004	−0.214502	−	0.976724i	$-0.568813\pi$
$461$	−9.21110	−0.429004	−0.214502	+	0.976724i	$0.568813\pi$
$462$	0	0
$463$	−1.21110	−0.0562847	−0.0281424	−	0.999604i	$-0.508959\pi$
$463$	−1.21110	−0.0562847	−0.0281424	+	0.999604i	$0.508959\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	0	0
$469$	−18.4222	−0.850658
$470$	0	0
$471$	−3.21110	−0.147960
$472$	0	0
$473$	3.39445	0.156077
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−14.7889	−0.675722	−0.337861	−	0.941196i	$-0.609703\pi$
$479$	−14.7889	−0.675722	−0.337861	+	0.941196i	$0.609703\pi$
$480$	0	0
$481$	14.7889	0.674316
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.57779	−0.252754	−0.126377	−	0.991982i	$-0.540335\pi$
$487$	−5.57779	−0.252754	−0.126377	+	0.991982i	$0.540335\pi$
$488$	0	0
$489$	5.21110	0.235654
$490$	0	0
$491$	−5.57779	−0.251722	−0.125861	−	0.992048i	$-0.540169\pi$
$491$	−5.57779	−0.251722	−0.125861	+	0.992048i	$0.540169\pi$
$492$	0	0
$493$	−52.8444	−2.37999
$494$	0	0
$495$	0	0
$496$	0	0
$497$	66.4222	2.97944
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	19.8167	0.885343
$502$	0	0
$503$	−4.18335	−0.186526	−0.0932631	−	0.995641i	$-0.529730\pi$
$503$	−4.18335	−0.186526	−0.0932631	+	0.995641i	$0.529730\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.21110	0.364668
$508$	0	0
$509$	−28.4222	−1.25979	−0.629896	−	0.776679i	$-0.716903\pi$
$509$	−28.4222	−1.25979	−0.629896	+	0.776679i	$0.716903\pi$
$510$	0	0
$511$	2.78890	0.123374
$512$	0	0
$513$	−7.21110	−0.318378
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.21110	0.229184
$518$	0	0
$519$	−0.183346	−0.00804800
$520$	0	0
$521$	−8.42221	−0.368984	−0.184492	−	0.982834i	$-0.559064\pi$
$521$	−8.42221	−0.368984	−0.184492	+	0.982834i	$0.559064\pi$
$522$	0	0
$523$	−28.2389	−1.23480	−0.617400	−	0.786650i	$-0.711814\pi$
$523$	−28.2389	−1.23480	−0.617400	+	0.786650i	$0.711814\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−60.8444	−2.65042
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	36.8444	1.59591
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	14.2111	0.612116
$540$	0	0
$541$	3.57779	0.153821	0.0769107	−	0.997038i	$-0.475494\pi$
$541$	3.57779	0.153821	0.0769107	+	0.997038i	$0.475494\pi$
$542$	0	0
$543$	23.2111	0.996084
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.1833	0.606436	0.303218	−	0.952921i	$-0.401939\pi$
$547$	14.1833	0.606436	0.303218	+	0.952921i	$0.401939\pi$
$548$	0	0
$549$	7.21110	0.307762
$550$	0	0
$551$	−57.6888	−2.45763
$552$	0	0
$553$	51.6333	2.19567
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.97224	0.295423	0.147712	−	0.989030i	$-0.452809\pi$
$557$	6.97224	0.295423	0.147712	+	0.989030i	$0.452809\pi$
$558$	0	0
$559$	15.6333	0.661218
$560$	0	0
$561$	−6.60555	−0.278887
$562$	0	0
$563$	19.8167	0.835172	0.417586	−	0.908637i	$-0.362876\pi$
$563$	19.8167	0.835172	0.417586	+	0.908637i	$0.362876\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.60555	−0.193415
$568$	0	0
$569$	−27.6333	−1.15845	−0.579224	−	0.815168i	$-0.696644\pi$
$569$	−27.6333	−1.15845	−0.579224	+	0.815168i	$0.696644\pi$
$570$	0	0
$571$	12.4222	0.519853	0.259927	−	0.965628i	$-0.416302\pi$
$571$	12.4222	0.519853	0.259927	+	0.965628i	$0.416302\pi$
$572$	0	0
$573$	18.4222	0.769599
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.7889	−1.19850	−0.599249	−	0.800563i	$-0.704534\pi$
$577$	−28.7889	−1.19850	−0.599249	+	0.800563i	$0.704534\pi$
$578$	0	0
$579$	21.8167	0.906669
$580$	0	0
$581$	−48.8444	−2.02641
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.42221	−0.265073	−0.132536	−	0.991178i	$-0.542312\pi$
$587$	−6.42221	−0.265073	−0.132536	+	0.991178i	$0.542312\pi$
$588$	0	0
$589$	−66.4222	−2.73688
$590$	0	0
$591$	13.3944	0.550974
$592$	0	0
$593$	−13.0278	−0.534986	−0.267493	−	0.963560i	$-0.586195\pi$
$593$	−13.0278	−0.534986	−0.267493	+	0.963560i	$0.586195\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10.4222	−0.426552
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−8.78890	−0.358507	−0.179253	−	0.983803i	$-0.557368\pi$
$601$	−8.78890	−0.358507	−0.179253	+	0.983803i	$0.557368\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.97224	−0.201817	−0.100909	−	0.994896i	$-0.532175\pi$
$607$	−4.97224	−0.201817	−0.100909	+	0.994896i	$0.532175\pi$
$608$	0	0
$609$	−36.8444	−1.49301
$610$	0	0
$611$	24.0000	0.970936
$612$	0	0
$613$	21.8167	0.881166	0.440583	−	0.897712i	$-0.354772\pi$
$613$	21.8167	0.881166	0.440583	+	0.897712i	$0.354772\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.7889	0.997963	0.498982	−	0.866613i	$-0.333707\pi$
$617$	24.7889	0.997963	0.498982	+	0.866613i	$0.333707\pi$
$618$	0	0
$619$	−19.6333	−0.789129	−0.394565	−	0.918868i	$-0.629105\pi$
$619$	−19.6333	−0.789129	−0.394565	+	0.918868i	$0.629105\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−27.6333	−1.10711
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−7.21110	−0.287984
$628$	0	0
$629$	−21.2111	−0.845742
$630$	0	0
$631$	36.8444	1.46675	0.733376	−	0.679823i	$-0.237943\pi$
$631$	36.8444	1.46675	0.733376	+	0.679823i	$0.237943\pi$
$632$	0	0
$633$	−23.2111	−0.922559
$634$	0	0
$635$	0	0
$636$	0	0
$637$	65.4500	2.59322
$638$	0	0
$639$	−14.4222	−0.570534
$640$	0	0
$641$	18.8444	0.744309	0.372155	−	0.928171i	$-0.378619\pi$
$641$	18.8444	0.744309	0.372155	+	0.928171i	$0.378619\pi$
$642$	0	0
$643$	−0.366692	−0.0144609	−0.00723047	−	0.999974i	$-0.502302\pi$
$643$	−0.366692	−0.0144609	−0.00723047	+	0.999974i	$0.502302\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−45.2111	−1.77743	−0.888716	−	0.458458i	$-0.848402\pi$
$647$	−45.2111	−1.77743	−0.888716	+	0.458458i	$0.848402\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	0	0
$651$	−42.4222	−1.66266
$652$	0	0
$653$	−4.78890	−0.187404	−0.0937020	−	0.995600i	$-0.529870\pi$
$653$	−4.78890	−0.187404	−0.0937020	+	0.995600i	$0.529870\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−0.605551	−0.0236248
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	45.2666	1.76067	0.880334	−	0.474355i	$-0.157319\pi$
$661$	45.2666	1.76067	0.880334	+	0.474355i	$0.157319\pi$
$662$	0	0
$663$	−30.4222	−1.18150
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−9.21110	−0.356122
$670$	0	0
$671$	7.21110	0.278382
$672$	0	0
$673$	11.3944	0.439224	0.219612	−	0.975587i	$-0.429521\pi$
$673$	11.3944	0.439224	0.219612	+	0.975587i	$0.429521\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.39445	−0.207326	−0.103663	−	0.994613i	$-0.533056\pi$
$677$	−5.39445	−0.207326	−0.103663	+	0.994613i	$0.533056\pi$
$678$	0	0
$679$	−57.2111	−2.19556
$680$	0	0
$681$	15.8167	0.606095
$682$	0	0
$683$	30.7889	1.17810	0.589052	−	0.808095i	$-0.299501\pi$
$683$	30.7889	1.17810	0.589052	+	0.808095i	$0.299501\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.42221	−0.321327
$688$	0	0
$689$	−9.21110	−0.350915
$690$	0	0
$691$	−13.5778	−0.516524	−0.258262	−	0.966075i	$-0.583150\pi$
$691$	−13.5778	−0.516524	−0.258262	+	0.966075i	$0.583150\pi$
$692$	0	0
$693$	−4.60555	−0.174950
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−52.8444	−2.00162
$698$	0	0
$699$	10.6056	0.401139
$700$	0	0
$701$	46.4222	1.75334	0.876671	−	0.481090i	$-0.159759\pi$
$701$	46.4222	1.75334	0.876671	+	0.481090i	$0.159759\pi$
$702$	0	0
$703$	−23.1556	−0.873330
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−24.0000	−0.902613
$708$	0	0
$709$	5.63331	0.211563	0.105782	−	0.994389i	$-0.466266\pi$
$709$	5.63331	0.211563	0.105782	+	0.994389i	$0.466266\pi$
$710$	0	0
$711$	−11.2111	−0.420449
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.78890	0.253536
$718$	0	0
$719$	−26.4222	−0.985382	−0.492691	−	0.870204i	$-0.663987\pi$
$719$	−26.4222	−0.985382	−0.492691	+	0.870204i	$0.663987\pi$
$720$	0	0
$721$	5.57779	0.207728
$722$	0	0
$723$	−15.2111	−0.565707
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.8444	1.06978	0.534890	−	0.844922i	$-0.320353\pi$
$727$	28.8444	1.06978	0.534890	+	0.844922i	$0.320353\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−22.4222	−0.829315
$732$	0	0
$733$	7.39445	0.273120	0.136560	−	0.990632i	$-0.456395\pi$
$733$	7.39445	0.273120	0.136560	+	0.990632i	$0.456395\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	41.2666	1.51802	0.759008	−	0.651081i	$-0.225684\pi$
$739$	41.2666	1.51802	0.759008	+	0.651081i	$0.225684\pi$
$740$	0	0
$741$	−33.2111	−1.22004
$742$	0	0
$743$	−37.3944	−1.37187	−0.685935	−	0.727663i	$-0.740606\pi$
$743$	−37.3944	−1.37187	−0.685935	+	0.727663i	$0.740606\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10.6056	0.388037
$748$	0	0
$749$	72.8444	2.66168
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	−10.4222	−0.379806
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.7889	0.464820	0.232410	−	0.972618i	$-0.425339\pi$
$757$	12.7889	0.464820	0.232410	+	0.972618i	$0.425339\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.6333	1.00171	0.500853	−	0.865532i	$-0.333020\pi$
$761$	27.6333	1.00171	0.500853	+	0.865532i	$0.333020\pi$
$762$	0	0
$763$	46.0555	1.66732
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.8444	1.33037
$768$	0	0
$769$	−43.2111	−1.55823	−0.779116	−	0.626880i	$-0.784332\pi$
$769$	−43.2111	−1.55823	−0.779116	+	0.626880i	$0.784332\pi$
$770$	0	0
$771$	23.2111	0.835927
$772$	0	0
$773$	48.0555	1.72844	0.864218	−	0.503117i	$-0.167814\pi$
$773$	48.0555	1.72844	0.864218	+	0.503117i	$0.167814\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−14.7889	−0.530549
$778$	0	0
$779$	−57.6888	−2.06692
$780$	0	0
$781$	−14.4222	−0.516067
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−39.0278	−1.39119	−0.695595	−	0.718434i	$-0.744859\pi$
$787$	−39.0278	−1.39119	−0.695595	+	0.718434i	$0.744859\pi$
$788$	0	0
$789$	17.0278	0.606204
$790$	0	0
$791$	−3.63331	−0.129186
$792$	0	0
$793$	33.2111	1.17936
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−40.0555	−1.41884	−0.709420	−	0.704786i	$-0.751043\pi$
$797$	−40.0555	−1.41884	−0.709420	+	0.704786i	$0.751043\pi$
$798$	0	0
$799$	−34.4222	−1.21777
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	−0.605551	−0.0213694
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−20.4222	−0.718896
$808$	0	0
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	−7.57779	−0.266092	−0.133046	−	0.991110i	$-0.542476\pi$
$811$	−7.57779	−0.266092	−0.133046	+	0.991110i	$0.542476\pi$
$812$	0	0
$813$	−4.78890	−0.167954
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.4777	−0.856367
$818$	0	0
$819$	−21.2111	−0.741176
$820$	0	0
$821$	−12.8444	−0.448273	−0.224137	−	0.974558i	$-0.571956\pi$
$821$	−12.8444	−0.448273	−0.224137	+	0.974558i	$0.571956\pi$
$822$	0	0
$823$	−41.2111	−1.43653	−0.718264	−	0.695770i	$-0.755063\pi$
$823$	−41.2111	−1.43653	−0.718264	+	0.695770i	$0.755063\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.60555	0.0906039	0.0453019	−	0.998973i	$-0.485575\pi$
$827$	2.60555	0.0906039	0.0453019	+	0.998973i	$0.485575\pi$
$828$	0	0
$829$	6.36669	0.221124	0.110562	−	0.993869i	$-0.464735\pi$
$829$	6.36669	0.221124	0.110562	+	0.993869i	$0.464735\pi$
$830$	0	0
$831$	−32.2389	−1.11835
$832$	0	0
$833$	−93.8722	−3.25248
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.21110	0.318382
$838$	0	0
$839$	30.4222	1.05029	0.525146	−	0.851012i	$-0.324011\pi$
$839$	30.4222	1.05029	0.525146	+	0.851012i	$0.324011\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.60555	−0.158249
$848$	0	0
$849$	−4.60555	−0.158062
$850$	0	0
$851$	0	0
$852$	0	0
$853$	40.2389	1.37775	0.688876	−	0.724879i	$-0.258104\pi$
$853$	40.2389	1.37775	0.688876	+	0.724879i	$0.258104\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.0278	−0.854932	−0.427466	−	0.904031i	$-0.640594\pi$
$857$	−25.0278	−0.854932	−0.427466	+	0.904031i	$0.640594\pi$
$858$	0	0
$859$	57.2111	1.95202	0.976009	−	0.217731i	$-0.0698656\pi$
$859$	57.2111	1.95202	0.976009	+	0.217731i	$0.0698656\pi$
$860$	0	0
$861$	−36.8444	−1.25565
$862$	0	0
$863$	−39.6333	−1.34913	−0.674567	−	0.738214i	$-0.735670\pi$
$863$	−39.6333	−1.34913	−0.674567	+	0.738214i	$0.735670\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.6333	0.904515
$868$	0	0
$869$	−11.2111	−0.380311
$870$	0	0
$871$	18.4222	0.624213
$872$	0	0
$873$	12.4222	0.420428
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.2389	0.818488	0.409244	−	0.912425i	$-0.365792\pi$
$877$	24.2389	0.818488	0.409244	+	0.912425i	$0.365792\pi$
$878$	0	0
$879$	−19.8167	−0.668399
$880$	0	0
$881$	26.8444	0.904411	0.452206	−	0.891914i	$-0.350637\pi$
$881$	26.8444	0.904411	0.452206	+	0.891914i	$0.350637\pi$
$882$	0	0
$883$	−6.42221	−0.216124	−0.108062	−	0.994144i	$-0.534465\pi$
$883$	−6.42221	−0.216124	−0.108062	+	0.994144i	$0.534465\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.0278	−0.437429	−0.218715	−	0.975789i	$-0.570186\pi$
$887$	−13.0278	−0.437429	−0.218715	+	0.975789i	$0.570186\pi$
$888$	0	0
$889$	−39.6333	−1.32926
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−37.5778	−1.25749
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	73.6888	2.45766
$900$	0	0
$901$	13.2111	0.440126
$902$	0	0
$903$	−15.6333	−0.520244
$904$	0	0
$905$	0	0
$906$	0	0
$907$	34.0555	1.13079	0.565397	−	0.824819i	$-0.308723\pi$
$907$	34.0555	1.13079	0.565397	+	0.824819i	$0.308723\pi$
$908$	0	0
$909$	5.21110	0.172841
$910$	0	0
$911$	18.4222	0.610355	0.305177	−	0.952296i	$-0.401284\pi$
$911$	18.4222	0.610355	0.305177	+	0.952296i	$0.401284\pi$
$912$	0	0
$913$	10.6056	0.350993
$914$	0	0
$915$	0	0
$916$	0	0
$917$	31.2666	1.03251
$918$	0	0
$919$	−26.0000	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$920$	0	0
$921$	−8.60555	−0.283563
$922$	0	0
$923$	−66.4222	−2.18631
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.21110	−0.0397778
$928$	0	0
$929$	26.8444	0.880737	0.440368	−	0.897817i	$-0.354848\pi$
$929$	26.8444	0.880737	0.440368	+	0.897817i	$0.354848\pi$
$930$	0	0
$931$	−102.478	−3.35857
$932$	0	0
$933$	30.4222	0.995978
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20.9722	−0.685133	−0.342567	−	0.939494i	$-0.611296\pi$
$937$	−20.9722	−0.685133	−0.342567	+	0.939494i	$0.611296\pi$
$938$	0	0
$939$	−8.78890	−0.286815
$940$	0	0
$941$	53.2111	1.73463	0.867316	−	0.497758i	$-0.165843\pi$
$941$	53.2111	1.73463	0.867316	+	0.497758i	$0.165843\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.63331	0.118067	0.0590333	−	0.998256i	$-0.481198\pi$
$947$	3.63331	0.118067	0.0590333	+	0.998256i	$0.481198\pi$
$948$	0	0
$949$	−2.78890	−0.0905314
$950$	0	0
$951$	−11.2111	−0.363545
$952$	0	0
$953$	35.4500	1.14834	0.574168	−	0.818737i	$-0.305325\pi$
$953$	35.4500	1.14834	0.574168	+	0.818737i	$0.305325\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.00000	0.258603
$958$	0	0
$959$	−27.6333	−0.892326
$960$	0	0
$961$	53.8444	1.73692
$962$	0	0
$963$	−15.8167	−0.509685
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−25.8167	−0.830208	−0.415104	−	0.909774i	$-0.636255\pi$
$967$	−25.8167	−0.830208	−0.415104	+	0.909774i	$0.636255\pi$
$968$	0	0
$969$	47.6333	1.53020
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−75.6333	−2.42469
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	−52.4777	−1.67378	−0.836890	−	0.547372i	$-0.815628\pi$
$983$	−52.4777	−1.67378	−0.836890	+	0.547372i	$0.815628\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−24.0000	−0.763928
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−35.6333	−1.13193	−0.565965	−	0.824430i	$-0.691496\pi$
$991$	−35.6333	−1.13193	−0.565965	+	0.824430i	$0.691496\pi$
$992$	0	0
$993$	−6.78890	−0.215439
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−15.0278	−0.475934	−0.237967	−	0.971273i	$-0.576481\pi$
$997$	−15.0278	−0.475934	−0.237967	+	0.971273i	$0.576481\pi$
$998$	0	0
$999$	3.21110	0.101595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3300.2.a.w.1.1		2
3.2	odd	2		9900.2.a.bl.1.1		2
5.2	odd	4		3300.2.c.l.1849.1		4
5.3	odd	4		3300.2.c.l.1849.4		4
5.4	even	2		660.2.a.e.1.2	✓	2
15.2	even	4		9900.2.c.q.5149.1		4
15.8	even	4		9900.2.c.q.5149.4		4
15.14	odd	2		1980.2.a.h.1.2		2
20.19	odd	2		2640.2.a.bc.1.1		2
55.54	odd	2		7260.2.a.w.1.1		2
60.59	even	2		7920.2.a.bo.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
660.2.a.e.1.2	✓	2	5.4	even	2
1980.2.a.h.1.2		2	15.14	odd	2
2640.2.a.bc.1.1		2	20.19	odd	2
3300.2.a.w.1.1		2	1.1	even	1	trivial
3300.2.c.l.1849.1		4	5.2	odd	4
3300.2.c.l.1849.4		4	5.3	odd	4
7260.2.a.w.1.1		2	55.54	odd	2
7920.2.a.bo.1.1		2	60.59	even	2
9900.2.a.bl.1.1		2	3.2	odd	2
9900.2.c.q.5149.1		4	15.2	even	4
9900.2.c.q.5149.4		4	15.8	even	4

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:	\( N \)	\(=\)	\( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3300.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.60555 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.60555 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.60555 q^{13} -6.60555 q^{17} -7.21110 q^{19} -4.60555 q^{21} +1.00000 q^{27} +8.00000 q^{29} +9.21110 q^{31} +1.00000 q^{33} +3.21110 q^{37} +4.60555 q^{39} +8.00000 q^{41} +3.39445 q^{43} +5.21110 q^{47} +14.2111 q^{49} -6.60555 q^{51} -2.00000 q^{53} -7.21110 q^{57} +8.00000 q^{59} +7.21110 q^{61} -4.60555 q^{63} +4.00000 q^{67} -14.4222 q^{71} -0.605551 q^{73} -4.60555 q^{77} -11.2111 q^{79} +1.00000 q^{81} +10.6056 q^{83} +8.00000 q^{87} +6.00000 q^{89} -21.2111 q^{91} +9.21110 q^{93} +12.4222 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 6 q^{17} - 2 q^{21} + 2 q^{27} + 16 q^{29} + 4 q^{31} + 2 q^{33} - 8 q^{37} + 2 q^{39} + 16 q^{41} + 14 q^{43} - 4 q^{47} + 14 q^{49} - 6 q^{51} - 4 q^{53} + 16 q^{59} - 2 q^{63} + 8 q^{67} + 6 q^{73} - 2 q^{77} - 8 q^{79} + 2 q^{81} + 14 q^{83} + 16 q^{87} + 12 q^{89} - 28 q^{91} + 4 q^{93} - 4 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 - 6 * q^17 - 2 * q^21 + 2 * q^27 + 16 * q^29 + 4 * q^31 + 2 * q^33 - 8 * q^37 + 2 * q^39 + 16 * q^41 + 14 * q^43 - 4 * q^47 + 14 * q^49 - 6 * q^51 - 4 * q^53 + 16 * q^59 - 2 * q^63 + 8 * q^67 + 6 * q^73 - 2 * q^77 - 8 * q^79 + 2 * q^81 + 14 * q^83 + 16 * q^87 + 12 * q^89 - 28 * q^91 + 4 * q^93 - 4 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more