LMFDB - Embedded newform 2112.2.f.b.1057.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(1057,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("2112.1057");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.f (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1057.3
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2112.1057
Dual form			2112.2.f.b.1057.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -4.24264i q^{5} +1.41421 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -4.24264i q^{5} +1.41421 q^{7} -1.00000 q^{9} +1.00000i q^{11} -4.24264i q^{13} +4.24264 q^{15} -4.00000 q^{17} -4.00000i q^{19} +1.41421i q^{21} +1.41421 q^{23} -13.0000 q^{25} -1.00000i q^{27} -2.82843i q^{29} -1.00000 q^{33} -6.00000i q^{35} +5.65685i q^{37} +4.24264 q^{39} +10.0000 q^{41} +8.00000i q^{43} +4.24264i q^{45} -9.89949 q^{47} -5.00000 q^{49} -4.00000i q^{51} -7.07107i q^{53} +4.24264 q^{55} +4.00000 q^{57} -8.00000i q^{59} +7.07107i q^{61} -1.41421 q^{63} -18.0000 q^{65} +2.00000i q^{67} +1.41421i q^{69} -1.41421 q^{71} -6.00000 q^{73} -13.0000i q^{75} +1.41421i q^{77} -12.7279 q^{79} +1.00000 q^{81} -6.00000i q^{83} +16.9706i q^{85} +2.82843 q^{87} -14.0000 q^{89} -6.00000i q^{91} -16.9706 q^{95} -4.00000 q^{97} -1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 16 q^{17} - 52 q^{25} - 4 q^{33} + 40 q^{41} - 20 q^{49} + 16 q^{57} - 72 q^{65} - 24 q^{73} + 4 q^{81} - 56 q^{89} - 16 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 - 16 * q^17 - 52 * q^25 - 4 * q^33 + 40 * q^41 - 20 * q^49 + 16 * q^57 - 72 * q^65 - 24 * q^73 + 4 * q^81 - 56 * q^89 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	− 4.24264i	− 1.89737i	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	− 4.24264i	− 1.89737i	0.316228	−	0.948683i	$-0.397584\pi$
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000i	0.301511i
$11$	1.00000i	0.301511i
$12$	0	0
$13$	− 4.24264i	− 1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$	− 4.24264i	− 1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$	0	0
$15$	4.24264	1.09545
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	1.41421i	0.308607i
$22$	0	0
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	0	0
$25$	−13.0000	−2.60000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 2.82843i	− 0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$	− 2.82843i	− 0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	− 6.00000i	− 1.01419i
$36$	0	0
$37$	5.65685i	0.929981i	0.885316	+	0.464991i	$0.153942\pi$
$37$	5.65685i	0.929981i	−0.885316	+	0.464991i	$0.846058\pi$
$38$	0	0
$39$	4.24264	0.679366
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	4.24264i	0.632456i
$46$	0	0
$47$	−9.89949	−1.44399	−0.721995	−	0.691898i	$-0.756775\pi$
$47$	−9.89949	−1.44399	−0.721995	+	0.691898i	$0.756775\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	− 4.00000i	− 0.560112i
$52$	0	0
$53$	− 7.07107i	− 0.971286i	−0.874157	−	0.485643i	$-0.838586\pi$
$53$	− 7.07107i	− 0.971286i	0.874157	−	0.485643i	$-0.161414\pi$
$54$	0	0
$55$	4.24264	0.572078
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	− 8.00000i	− 1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$	− 8.00000i	− 1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$	0	0
$61$	7.07107i	0.905357i	0.891674	+	0.452679i	$0.149532\pi$
$61$	7.07107i	0.905357i	−0.891674	+	0.452679i	$0.850468\pi$
$62$	0	0
$63$	−1.41421	−0.178174
$64$	0	0
$65$	−18.0000	−2.23263
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	0	0
$69$	1.41421i	0.170251i
$70$	0	0
$71$	−1.41421	−0.167836	−0.0839181	−	0.996473i	$-0.526743\pi$
$71$	−1.41421	−0.167836	−0.0839181	+	0.996473i	$0.526743\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	− 13.0000i	− 1.50111i
$76$	0	0
$77$	1.41421i	0.161165i
$78$	0	0
$79$	−12.7279	−1.43200	−0.716002	−	0.698099i	$-0.754030\pi$
$79$	−12.7279	−1.43200	−0.716002	+	0.698099i	$0.754030\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	16.9706i	1.84072i
$86$	0	0
$87$	2.82843	0.303239
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	− 6.00000i	− 0.628971i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.9706	−1.74114
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	− 1.00000i	− 0.100504i
$100$	0	0
$101$	− 19.7990i	− 1.97007i	−0.172345	−	0.985037i	$-0.555135\pi$
$101$	− 19.7990i	− 1.97007i	0.172345	−	0.985037i	$-0.444865\pi$
$102$	0	0
$103$	19.7990	1.95085	0.975426	−	0.220326i	$-0.0707122\pi$
$103$	19.7990	1.95085	0.975426	+	0.220326i	$0.0707122\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	1.41421i	0.135457i	0.997704	+	0.0677285i	$0.0215752\pi$
$109$	1.41421i	0.135457i	−0.997704	+	0.0677285i	$0.978425\pi$
$110$	0	0
$111$	−5.65685	−0.536925
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	− 6.00000i	− 0.559503i
$116$	0	0
$117$	4.24264i	0.392232i
$118$	0	0
$119$	−5.65685	−0.518563
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	10.0000i	0.901670i
$124$	0	0
$125$	33.9411i	3.03579i
$126$	0	0
$127$	4.24264	0.376473	0.188237	−	0.982124i	$-0.439723\pi$
$127$	4.24264	0.376473	0.188237	+	0.982124i	$0.439723\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	− 22.0000i	− 1.92215i	−0.276289	−	0.961074i	$-0.589105\pi$
$131$	− 22.0000i	− 1.92215i	0.276289	−	0.961074i	$-0.410895\pi$
$132$	0	0
$133$	− 5.65685i	− 0.490511i
$134$	0	0
$135$	−4.24264	−0.365148
$136$	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
$139$	12.0000i	1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$	12.0000i	1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$	0	0
$141$	− 9.89949i	− 0.833688i
$142$	0	0
$143$	4.24264	0.354787
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	− 5.00000i	− 0.412393i
$148$	0	0
$149$	5.65685i	0.463428i	0.972784	+	0.231714i	$0.0744333\pi$
$149$	5.65685i	0.463428i	−0.972784	+	0.231714i	$0.925567\pi$
$150$	0	0
$151$	−12.7279	−1.03578	−0.517892	−	0.855446i	$-0.673283\pi$
$151$	−12.7279	−1.03578	−0.517892	+	0.855446i	$0.673283\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.48528i	− 0.677199i	−0.940931	−	0.338600i	$-0.890047\pi$
$157$	− 8.48528i	− 0.677199i	0.940931	−	0.338600i	$-0.109953\pi$
$158$	0	0
$159$	7.07107	0.560772
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	0	0
$165$	4.24264i	0.330289i
$166$	0	0
$167$	−2.82843	−0.218870	−0.109435	−	0.993994i	$-0.534904\pi$
$167$	−2.82843	−0.218870	−0.109435	+	0.993994i	$0.534904\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	− 14.1421i	− 1.07521i	−0.843198	−	0.537603i	$-0.819330\pi$
$173$	− 14.1421i	− 1.07521i	0.843198	−	0.537603i	$-0.180670\pi$
$174$	0	0
$175$	−18.3848	−1.38976
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	− 20.0000i	− 1.49487i	−0.664335	−	0.747435i	$-0.731285\pi$
$179$	− 20.0000i	− 1.49487i	0.664335	−	0.747435i	$-0.268715\pi$
$180$	0	0
$181$	8.48528i	0.630706i	0.948974	+	0.315353i	$0.102123\pi$
$181$	8.48528i	0.630706i	−0.948974	+	0.315353i	$0.897877\pi$
$182$	0	0
$183$	−7.07107	−0.522708
$184$	0	0
$185$	24.0000	1.76452
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	− 1.41421i	− 0.102869i
$190$	0	0
$191$	18.3848	1.33028	0.665138	−	0.746721i	$-0.268373\pi$
$191$	18.3848	1.33028	0.665138	+	0.746721i	$0.268373\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	− 18.0000i	− 1.28901i
$196$	0	0
$197$	− 19.7990i	− 1.41062i	−0.708899	−	0.705310i	$-0.750808\pi$
$197$	− 19.7990i	− 1.41062i	0.708899	−	0.705310i	$-0.249192\pi$
$198$	0	0
$199$	25.4558	1.80452	0.902258	−	0.431196i	$-0.141908\pi$
$199$	25.4558	1.80452	0.902258	+	0.431196i	$0.141908\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	− 4.00000i	− 0.280745i
$204$	0	0
$205$	− 42.4264i	− 2.96319i
$206$	0	0
$207$	−1.41421	−0.0982946
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	24.0000i	1.65223i	0.563503	+	0.826114i	$0.309453\pi$
$211$	24.0000i	1.65223i	−0.563503	+	0.826114i	$0.690547\pi$
$212$	0	0
$213$	− 1.41421i	− 0.0969003i
$214$	0	0
$215$	33.9411	2.31477
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 6.00000i	− 0.405442i
$220$	0	0
$221$	16.9706i	1.14156i
$222$	0	0
$223$	5.65685	0.378811	0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685	0.378811	0.189405	+	0.981899i	$0.439344\pi$
$224$	0	0
$225$	13.0000	0.866667
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	− 25.4558i	− 1.68217i	−0.540903	−	0.841085i	$-0.681918\pi$
$229$	− 25.4558i	− 1.68217i	0.540903	−	0.841085i	$-0.318082\pi$
$230$	0	0
$231$	−1.41421	−0.0930484
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	42.0000i	2.73978i
$236$	0	0
$237$	− 12.7279i	− 0.826767i
$238$	0	0
$239$	25.4558	1.64660	0.823301	−	0.567605i	$-0.192130\pi$
$239$	25.4558	1.64660	0.823301	+	0.567605i	$0.192130\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	21.2132i	1.35526i
$246$	0	0
$247$	−16.9706	−1.07981
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	1.41421i	0.0889108i
$254$	0	0
$255$	−16.9706	−1.06274
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	8.00000i	0.497096i
$260$	0	0
$261$	2.82843i	0.175075i
$262$	0	0
$263$	−5.65685	−0.348817	−0.174408	−	0.984673i	$-0.555801\pi$
$263$	−5.65685	−0.348817	−0.174408	+	0.984673i	$0.555801\pi$
$264$	0	0
$265$	−30.0000	−1.84289
$266$	0	0
$267$	− 14.0000i	− 0.856786i
$268$	0	0
$269$	9.89949i	0.603583i	0.953374	+	0.301791i	$0.0975846\pi$
$269$	9.89949i	0.603583i	−0.953374	+	0.301791i	$0.902415\pi$
$270$	0	0
$271$	26.8701	1.63224	0.816120	−	0.577883i	$-0.196121\pi$
$271$	26.8701	1.63224	0.816120	+	0.577883i	$0.196121\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	− 13.0000i	− 0.783929i
$276$	0	0
$277$	− 24.0416i	− 1.44452i	−0.691621	−	0.722261i	$-0.743103\pi$
$277$	− 24.0416i	− 1.44452i	0.691621	−	0.722261i	$-0.256897\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	− 16.9706i	− 1.00525i
$286$	0	0
$287$	14.1421	0.834784
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	− 4.00000i	− 0.234484i
$292$	0	0
$293$	5.65685i	0.330477i	0.986254	+	0.165238i	$0.0528394\pi$
$293$	5.65685i	0.330477i	−0.986254	+	0.165238i	$0.947161\pi$
$294$	0	0
$295$	−33.9411	−1.97613
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	− 6.00000i	− 0.346989i
$300$	0	0
$301$	11.3137i	0.652111i
$302$	0	0
$303$	19.7990	1.13742
$304$	0	0
$305$	30.0000	1.71780
$306$	0	0
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	0	0
$309$	19.7990i	1.12633i
$310$	0	0
$311$	−1.41421	−0.0801927	−0.0400963	−	0.999196i	$-0.512766\pi$
$311$	−1.41421	−0.0801927	−0.0400963	+	0.999196i	$0.512766\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	6.00000i	0.338062i
$316$	0	0
$317$	18.3848i	1.03259i	0.856410	+	0.516296i	$0.172690\pi$
$317$	18.3848i	1.03259i	−0.856410	+	0.516296i	$0.827310\pi$
$318$	0	0
$319$	2.82843	0.158362
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	16.0000i	0.890264i
$324$	0	0
$325$	55.1543i	3.05941i
$326$	0	0
$327$	−1.41421	−0.0782062
$328$	0	0
$329$	−14.0000	−0.771845
$330$	0	0
$331$	12.0000i	0.659580i	0.944054	+	0.329790i	$0.106978\pi$
$331$	12.0000i	0.659580i	−0.944054	+	0.329790i	$0.893022\pi$
$332$	0	0
$333$	− 5.65685i	− 0.309994i
$334$	0	0
$335$	8.48528	0.463600
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	10.0000i	0.543125i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	0	0
$349$	− 9.89949i	− 0.529908i	−0.964261	−	0.264954i	$-0.914643\pi$
$349$	− 9.89949i	− 0.529908i	0.964261	−	0.264954i	$-0.0853567\pi$
$350$	0	0
$351$	−4.24264	−0.226455
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	6.00000i	0.318447i
$356$	0	0
$357$	− 5.65685i	− 0.299392i
$358$	0	0
$359$	−31.1127	−1.64207	−0.821033	−	0.570881i	$-0.806602\pi$
$359$	−31.1127	−1.64207	−0.821033	+	0.570881i	$0.806602\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	25.4558i	1.33242i
$366$	0	0
$367$	−5.65685	−0.295285	−0.147643	−	0.989041i	$-0.547169\pi$
$367$	−5.65685	−0.295285	−0.147643	+	0.989041i	$0.547169\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	− 10.0000i	− 0.519174i
$372$	0	0
$373$	15.5563i	0.805477i	0.915315	+	0.402739i	$0.131942\pi$
$373$	15.5563i	0.805477i	−0.915315	+	0.402739i	$0.868058\pi$
$374$	0	0
$375$	−33.9411	−1.75271
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	4.00000i	0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$	4.00000i	0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	0	0
$381$	4.24264i	0.217357i
$382$	0	0
$383$	7.07107	0.361315	0.180657	−	0.983546i	$-0.442177\pi$
$383$	7.07107	0.361315	0.180657	+	0.983546i	$0.442177\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	− 8.00000i	− 0.406663i
$388$	0	0
$389$	4.24264i	0.215110i	0.994199	+	0.107555i	$0.0343022\pi$
$389$	4.24264i	0.215110i	−0.994199	+	0.107555i	$0.965698\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	22.0000	1.10975
$394$	0	0
$395$	54.0000i	2.71703i
$396$	0	0
$397$	− 25.4558i	− 1.27759i	−0.769376	−	0.638796i	$-0.779433\pi$
$397$	− 25.4558i	− 1.27759i	0.769376	−	0.638796i	$-0.220567\pi$
$398$	0	0
$399$	5.65685	0.283197
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 4.24264i	− 0.210819i
$406$	0	0
$407$	−5.65685	−0.280400
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	6.00000i	0.295958i
$412$	0	0
$413$	− 11.3137i	− 0.556711i
$414$	0	0
$415$	−25.4558	−1.24958
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	24.0000i	1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$	24.0000i	1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$	0	0
$421$	− 16.9706i	− 0.827095i	−0.910483	−	0.413547i	$-0.864290\pi$
$421$	− 16.9706i	− 0.827095i	0.910483	−	0.413547i	$-0.135710\pi$
$422$	0	0
$423$	9.89949	0.481330
$424$	0	0
$425$	52.0000	2.52237
$426$	0	0
$427$	10.0000i	0.483934i
$428$	0	0
$429$	4.24264i	0.204837i
$430$	0	0
$431$	−22.6274	−1.08992	−0.544962	−	0.838461i	$-0.683456\pi$
$431$	−22.6274	−1.08992	−0.544962	+	0.838461i	$0.683456\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	− 12.0000i	− 0.575356i
$436$	0	0
$437$	− 5.65685i	− 0.270604i
$438$	0	0
$439$	−32.5269	−1.55242	−0.776212	−	0.630471i	$-0.782862\pi$
$439$	−32.5269	−1.55242	−0.776212	+	0.630471i	$0.782862\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	0	0
$443$	32.0000i	1.52037i	0.649709	+	0.760183i	$0.274891\pi$
$443$	32.0000i	1.52037i	−0.649709	+	0.760183i	$0.725109\pi$
$444$	0	0
$445$	59.3970i	2.81569i
$446$	0	0
$447$	−5.65685	−0.267560
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	10.0000i	0.470882i
$452$	0	0
$453$	− 12.7279i	− 0.598010i
$454$	0	0
$455$	−25.4558	−1.19339
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0	0
$459$	4.00000i	0.186704i
$460$	0	0
$461$	− 28.2843i	− 1.31733i	−0.752436	−	0.658665i	$-0.771121\pi$
$461$	− 28.2843i	− 1.31733i	0.752436	−	0.658665i	$-0.228879\pi$
$462$	0	0
$463$	36.7696	1.70883	0.854413	−	0.519594i	$-0.173917\pi$
$463$	36.7696	1.70883	0.854413	+	0.519594i	$0.173917\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 32.0000i	− 1.48078i	−0.672176	−	0.740392i	$-0.734640\pi$
$467$	− 32.0000i	− 1.48078i	0.672176	−	0.740392i	$-0.265360\pi$
$468$	0	0
$469$	2.82843i	0.130605i
$470$	0	0
$471$	8.48528	0.390981
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	52.0000i	2.38592i
$476$	0	0
$477$	7.07107i	0.323762i
$478$	0	0
$479$	11.3137	0.516937	0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137	0.516937	0.258468	+	0.966020i	$0.416782\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	0	0
$483$	2.00000i	0.0910032i
$484$	0	0
$485$	16.9706i	0.770594i
$486$	0	0
$487$	31.1127	1.40985	0.704925	−	0.709281i	$-0.250980\pi$
$487$	31.1127	1.40985	0.704925	+	0.709281i	$0.250980\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	− 12.0000i	− 0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$	− 12.0000i	− 0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$	0	0
$493$	11.3137i	0.509544i
$494$	0	0
$495$	−4.24264	−0.190693
$496$	0	0
$497$	−2.00000	−0.0897123
$498$	0	0
$499$	− 36.0000i	− 1.61158i	−0.592200	−	0.805791i	$-0.701741\pi$
$499$	− 36.0000i	− 1.61158i	0.592200	−	0.805791i	$-0.298259\pi$
$500$	0	0
$501$	− 2.82843i	− 0.126365i
$502$	0	0
$503$	14.1421	0.630567	0.315283	−	0.948998i	$-0.397900\pi$
$503$	14.1421	0.630567	0.315283	+	0.948998i	$0.397900\pi$
$504$	0	0
$505$	−84.0000	−3.73795
$506$	0	0
$507$	− 5.00000i	− 0.222058i
$508$	0	0
$509$	− 15.5563i	− 0.689523i	−0.938690	−	0.344762i	$-0.887960\pi$
$509$	− 15.5563i	− 0.689523i	0.938690	−	0.344762i	$-0.112040\pi$
$510$	0	0
$511$	−8.48528	−0.375367
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	− 84.0000i	− 3.70148i
$516$	0	0
$517$	− 9.89949i	− 0.435379i
$518$	0	0
$519$	14.1421	0.620771
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	40.0000i	1.74908i	0.484955	+	0.874539i	$0.338836\pi$
$523$	40.0000i	1.74908i	−0.484955	+	0.874539i	$0.661164\pi$
$524$	0	0
$525$	− 18.3848i	− 0.802377i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	8.00000i	0.347170i
$532$	0	0
$533$	− 42.4264i	− 1.83769i
$534$	0	0
$535$	16.9706	0.733701
$536$	0	0
$537$	20.0000	0.863064
$538$	0	0
$539$	− 5.00000i	− 0.215365i
$540$	0	0
$541$	29.6985i	1.27684i	0.769689	+	0.638419i	$0.220411\pi$
$541$	29.6985i	1.27684i	−0.769689	+	0.638419i	$0.779589\pi$
$542$	0	0
$543$	−8.48528	−0.364138
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	0	0
$549$	− 7.07107i	− 0.301786i
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	−18.0000	−0.765438
$554$	0	0
$555$	24.0000i	1.01874i
$556$	0	0
$557$	− 5.65685i	− 0.239689i	−0.992793	−	0.119844i	$-0.961760\pi$
$557$	− 5.65685i	− 0.239689i	0.992793	−	0.119844i	$-0.0382395\pi$
$558$	0	0
$559$	33.9411	1.43556
$560$	0	0
$561$	4.00000	0.168880
$562$	0	0
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	0	0
$565$	− 42.4264i	− 1.78489i
$566$	0	0
$567$	1.41421	0.0593914
$568$	0	0
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	0	0
$571$	12.0000i	0.502184i	0.967963	+	0.251092i	$0.0807897\pi$
$571$	12.0000i	0.502184i	−0.967963	+	0.251092i	$0.919210\pi$
$572$	0	0
$573$	18.3848i	0.768035i
$574$	0	0
$575$	−18.3848	−0.766698
$576$	0	0
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	0	0
$579$	− 22.0000i	− 0.914289i
$580$	0	0
$581$	− 8.48528i	− 0.352029i
$582$	0	0
$583$	7.07107	0.292854
$584$	0	0
$585$	18.0000	0.744208
$586$	0	0
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	19.7990	0.814422
$592$	0	0
$593$	16.0000	0.657041	0.328521	−	0.944497i	$-0.393450\pi$
$593$	16.0000	0.657041	0.328521	+	0.944497i	$0.393450\pi$
$594$	0	0
$595$	24.0000i	0.983904i
$596$	0	0
$597$	25.4558i	1.04184i
$598$	0	0
$599$	15.5563	0.635615	0.317808	−	0.948155i	$-0.397053\pi$
$599$	15.5563	0.635615	0.317808	+	0.948155i	$0.397053\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	− 2.00000i	− 0.0814463i
$604$	0	0
$605$	4.24264i	0.172488i
$606$	0	0
$607$	35.3553	1.43503	0.717514	−	0.696544i	$-0.245280\pi$
$607$	35.3553	1.43503	0.717514	+	0.696544i	$0.245280\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	42.0000i	1.69914i
$612$	0	0
$613$	38.1838i	1.54223i	0.636697	+	0.771114i	$0.280300\pi$
$613$	38.1838i	1.54223i	−0.636697	+	0.771114i	$0.719700\pi$
$614$	0	0
$615$	42.4264	1.71080
$616$	0	0
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	0	0
$619$	− 26.0000i	− 1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$	− 26.0000i	− 1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$	0	0
$621$	− 1.41421i	− 0.0567504i
$622$	0	0
$623$	−19.7990	−0.793230
$624$	0	0
$625$	79.0000	3.16000
$626$	0	0
$627$	4.00000i	0.159745i
$628$	0	0
$629$	− 22.6274i	− 0.902214i
$630$	0	0
$631$	28.2843	1.12598	0.562990	−	0.826464i	$-0.309651\pi$
$631$	28.2843	1.12598	0.562990	+	0.826464i	$0.309651\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	0	0
$635$	− 18.0000i	− 0.714308i
$636$	0	0
$637$	21.2132i	0.840498i
$638$	0	0
$639$	1.41421	0.0559454
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	30.0000i	1.18308i	0.806274	+	0.591542i	$0.201481\pi$
$643$	30.0000i	1.18308i	−0.806274	+	0.591542i	$0.798519\pi$
$644$	0	0
$645$	33.9411i	1.33643i
$646$	0	0
$647$	−7.07107	−0.277992	−0.138996	−	0.990293i	$-0.544388\pi$
$647$	−7.07107	−0.277992	−0.138996	+	0.990293i	$0.544388\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 26.8701i	− 1.05151i	−0.850637	−	0.525753i	$-0.823784\pi$
$653$	− 26.8701i	− 1.05151i	0.850637	−	0.525753i	$-0.176216\pi$
$654$	0	0
$655$	−93.3381	−3.64702
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	20.0000i	0.779089i	0.921008	+	0.389545i	$0.127368\pi$
$659$	20.0000i	0.779089i	−0.921008	+	0.389545i	$0.872632\pi$
$660$	0	0
$661$	− 42.4264i	− 1.65020i	−0.564990	−	0.825098i	$-0.691120\pi$
$661$	− 42.4264i	− 1.65020i	0.564990	−	0.825098i	$-0.308880\pi$
$662$	0	0
$663$	−16.9706	−0.659082
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	− 4.00000i	− 0.154881i
$668$	0	0
$669$	5.65685i	0.218707i
$670$	0	0
$671$	−7.07107	−0.272976
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	13.0000i	0.500370i
$676$	0	0
$677$	− 31.1127i	− 1.19576i	−0.801586	−	0.597879i	$-0.796010\pi$
$677$	− 31.1127i	− 1.19576i	0.801586	−	0.597879i	$-0.203990\pi$
$678$	0	0
$679$	−5.65685	−0.217090
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	20.0000i	0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$	20.0000i	0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$	0	0
$685$	− 25.4558i	− 0.972618i
$686$	0	0
$687$	25.4558	0.971201
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	− 14.0000i	− 0.532585i	−0.963892	−	0.266293i	$-0.914201\pi$
$691$	− 14.0000i	− 0.532585i	0.963892	−	0.266293i	$-0.0857987\pi$
$692$	0	0
$693$	− 1.41421i	− 0.0537215i
$694$	0	0
$695$	50.9117	1.93119
$696$	0	0
$697$	−40.0000	−1.51511
$698$	0	0
$699$	− 10.0000i	− 0.378235i
$700$	0	0
$701$	− 39.5980i	− 1.49560i	−0.663927	−	0.747798i	$-0.731111\pi$
$701$	− 39.5980i	− 1.49560i	0.663927	−	0.747798i	$-0.268889\pi$
$702$	0	0
$703$	22.6274	0.853409
$704$	0	0
$705$	−42.0000	−1.58181
$706$	0	0
$707$	− 28.0000i	− 1.05305i
$708$	0	0
$709$	− 31.1127i	− 1.16846i	−0.811587	−	0.584231i	$-0.801396\pi$
$709$	− 31.1127i	− 1.16846i	0.811587	−	0.584231i	$-0.198604\pi$
$710$	0	0
$711$	12.7279	0.477334
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 18.0000i	− 0.673162i
$716$	0	0
$717$	25.4558i	0.950666i
$718$	0	0
$719$	18.3848	0.685636	0.342818	−	0.939402i	$-0.388619\pi$
$719$	18.3848	0.685636	0.342818	+	0.939402i	$0.388619\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	0	0
$723$	− 2.00000i	− 0.0743808i
$724$	0	0
$725$	36.7696i	1.36559i
$726$	0	0
$727$	14.1421	0.524503	0.262251	−	0.965000i	$-0.415535\pi$
$727$	14.1421	0.524503	0.262251	+	0.965000i	$0.415535\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 32.0000i	− 1.18356i
$732$	0	0
$733$	− 15.5563i	− 0.574587i	−0.957843	−	0.287293i	$-0.907245\pi$
$733$	− 15.5563i	− 0.574587i	0.957843	−	0.287293i	$-0.0927555\pi$
$734$	0	0
$735$	−21.2132	−0.782461
$736$	0	0
$737$	−2.00000	−0.0736709
$738$	0	0
$739$	− 52.0000i	− 1.91285i	−0.291977	−	0.956425i	$-0.594313\pi$
$739$	− 52.0000i	− 1.91285i	0.291977	−	0.956425i	$-0.405687\pi$
$740$	0	0
$741$	− 16.9706i	− 0.623429i
$742$	0	0
$743$	−36.7696	−1.34894	−0.674472	−	0.738300i	$-0.735629\pi$
$743$	−36.7696	−1.34894	−0.674472	+	0.738300i	$0.735629\pi$
$744$	0	0
$745$	24.0000	0.879292
$746$	0	0
$747$	6.00000i	0.219529i
$748$	0	0
$749$	5.65685i	0.206697i
$750$	0	0
$751$	11.3137	0.412843	0.206422	−	0.978463i	$-0.433818\pi$
$751$	11.3137	0.412843	0.206422	+	0.978463i	$0.433818\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	54.0000i	1.96526i
$756$	0	0
$757$	39.5980i	1.43921i	0.694382	+	0.719607i	$0.255678\pi$
$757$	39.5980i	1.43921i	−0.694382	+	0.719607i	$0.744322\pi$
$758$	0	0
$759$	−1.41421	−0.0513327
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	2.00000i	0.0724049i
$764$	0	0
$765$	− 16.9706i	− 0.613572i
$766$	0	0
$767$	−33.9411	−1.22554
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	− 6.00000i	− 0.216085i
$772$	0	0
$773$	18.3848i	0.661254i	0.943761	+	0.330627i	$0.107260\pi$
$773$	18.3848i	0.661254i	−0.943761	+	0.330627i	$0.892740\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	− 40.0000i	− 1.43315i
$780$	0	0
$781$	− 1.41421i	− 0.0506045i
$782$	0	0
$783$	−2.82843	−0.101080
$784$	0	0
$785$	−36.0000	−1.28490
$786$	0	0
$787$	− 44.0000i	− 1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$	− 44.0000i	− 1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$	0	0
$789$	− 5.65685i	− 0.201389i
$790$	0	0
$791$	14.1421	0.502836
$792$	0	0
$793$	30.0000	1.06533
$794$	0	0
$795$	− 30.0000i	− 1.06399i
$796$	0	0
$797$	− 26.8701i	− 0.951786i	−0.879503	−	0.475893i	$-0.842125\pi$
$797$	− 26.8701i	− 0.951786i	0.879503	−	0.475893i	$-0.157875\pi$
$798$	0	0
$799$	39.5980	1.40088
$800$	0	0
$801$	14.0000	0.494666
$802$	0	0
$803$	− 6.00000i	− 0.211735i
$804$	0	0
$805$	− 8.48528i	− 0.299067i
$806$	0	0
$807$	−9.89949	−0.348479
$808$	0	0
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	0	0
$811$	− 28.0000i	− 0.983213i	−0.870817	−	0.491606i	$-0.836410\pi$
$811$	− 28.0000i	− 0.983213i	0.870817	−	0.491606i	$-0.163590\pi$
$812$	0	0
$813$	26.8701i	0.942374i
$814$	0	0
$815$	−84.8528	−2.97226
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	6.00000i	0.209657i
$820$	0	0
$821$	48.0833i	1.67812i	0.544041	+	0.839059i	$0.316894\pi$
$821$	48.0833i	1.67812i	−0.544041	+	0.839059i	$0.683106\pi$
$822$	0	0
$823$	5.65685	0.197186	0.0985928	−	0.995128i	$-0.468566\pi$
$823$	5.65685	0.197186	0.0985928	+	0.995128i	$0.468566\pi$
$824$	0	0
$825$	13.0000	0.452602
$826$	0	0
$827$	6.00000i	0.208640i	0.994544	+	0.104320i	$0.0332667\pi$
$827$	6.00000i	0.208640i	−0.994544	+	0.104320i	$0.966733\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	24.0416	0.833995
$832$	0	0
$833$	20.0000	0.692959
$834$	0	0
$835$	12.0000i	0.415277i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21.2132	0.732361	0.366181	−	0.930544i	$-0.380665\pi$
$839$	21.2132	0.732361	0.366181	+	0.930544i	$0.380665\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	16.0000i	0.551069i
$844$	0	0
$845$	21.2132i	0.729756i
$846$	0	0
$847$	−1.41421	−0.0485930
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	8.00000i	0.274236i
$852$	0	0
$853$	21.2132i	0.726326i	0.931726	+	0.363163i	$0.118303\pi$
$853$	21.2132i	0.726326i	−0.931726	+	0.363163i	$0.881697\pi$
$854$	0	0
$855$	16.9706	0.580381
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	− 4.00000i	− 0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$	− 4.00000i	− 0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$	0	0
$861$	14.1421i	0.481963i
$862$	0	0
$863$	−21.2132	−0.722106	−0.361053	−	0.932545i	$-0.617583\pi$
$863$	−21.2132	−0.722106	−0.361053	+	0.932545i	$0.617583\pi$
$864$	0	0
$865$	−60.0000	−2.04006
$866$	0	0
$867$	− 1.00000i	− 0.0339618i
$868$	0	0
$869$	− 12.7279i	− 0.431765i
$870$	0	0
$871$	8.48528	0.287513
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	48.0000i	1.62270i
$876$	0	0
$877$	21.2132i	0.716319i	0.933660	+	0.358159i	$0.116596\pi$
$877$	21.2132i	0.716319i	−0.933660	+	0.358159i	$0.883404\pi$
$878$	0	0
$879$	−5.65685	−0.190801
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	− 22.0000i	− 0.740359i	−0.928960	−	0.370179i	$-0.879296\pi$
$883$	− 22.0000i	− 0.740359i	0.928960	−	0.370179i	$-0.120704\pi$
$884$	0	0
$885$	− 33.9411i	− 1.14092i
$886$	0	0
$887$	36.7696	1.23460	0.617300	−	0.786728i	$-0.288226\pi$
$887$	36.7696	1.23460	0.617300	+	0.786728i	$0.288226\pi$
$888$	0	0
$889$	6.00000	0.201234
$890$	0	0
$891$	1.00000i	0.0335013i
$892$	0	0
$893$	39.5980i	1.32510i
$894$	0	0
$895$	−84.8528	−2.83632
$896$	0	0
$897$	6.00000	0.200334
$898$	0	0
$899$	0	0
$900$	0	0
$901$	28.2843i	0.942286i
$902$	0	0
$903$	−11.3137	−0.376497
$904$	0	0
$905$	36.0000	1.19668
$906$	0	0
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	0	0
$909$	19.7990i	0.656691i
$910$	0	0
$911$	9.89949	0.327985	0.163992	−	0.986462i	$-0.447563\pi$
$911$	9.89949	0.327985	0.163992	+	0.986462i	$0.447563\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	0	0
$915$	30.0000i	0.991769i
$916$	0	0
$917$	− 31.1127i	− 1.02743i
$918$	0	0
$919$	−38.1838	−1.25957	−0.629783	−	0.776771i	$-0.716856\pi$
$919$	−38.1838	−1.25957	−0.629783	+	0.776771i	$0.716856\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	6.00000i	0.197492i
$924$	0	0
$925$	− 73.5391i	− 2.41795i
$926$	0	0
$927$	−19.7990	−0.650284
$928$	0	0
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	0	0
$931$	20.0000i	0.655474i
$932$	0	0
$933$	− 1.41421i	− 0.0462993i
$934$	0	0
$935$	−16.9706	−0.554997
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	6.00000i	0.195803i
$940$	0	0
$941$	− 31.1127i	− 1.01424i	−0.861874	−	0.507122i	$-0.830709\pi$
$941$	− 31.1127i	− 1.01424i	0.861874	−	0.507122i	$-0.169291\pi$
$942$	0	0
$943$	14.1421	0.460531
$944$	0	0
$945$	−6.00000	−0.195180
$946$	0	0
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	0	0
$949$	25.4558i	0.826332i
$950$	0	0
$951$	−18.3848	−0.596167
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	− 78.0000i	− 2.52402i
$956$	0	0
$957$	2.82843i	0.0914301i
$958$	0	0
$959$	8.48528	0.274004
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	0	0
$965$	93.3381i	3.00466i
$966$	0	0
$967$	7.07107	0.227390	0.113695	−	0.993516i	$-0.463731\pi$
$967$	7.07107	0.227390	0.113695	+	0.993516i	$0.463731\pi$
$968$	0	0
$969$	−16.0000	−0.513994
$970$	0	0
$971$	12.0000i	0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$	12.0000i	0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$	0	0
$973$	16.9706i	0.544051i
$974$	0	0
$975$	−55.1543	−1.76635
$976$	0	0
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	− 14.0000i	− 0.447442i
$980$	0	0
$981$	− 1.41421i	− 0.0451524i
$982$	0	0
$983$	−41.0122	−1.30809	−0.654043	−	0.756457i	$-0.726928\pi$
$983$	−41.0122	−1.30809	−0.654043	+	0.756457i	$0.726928\pi$
$984$	0	0
$985$	−84.0000	−2.67646
$986$	0	0
$987$	− 14.0000i	− 0.445625i
$988$	0	0
$989$	11.3137i	0.359755i
$990$	0	0
$991$	5.65685	0.179696	0.0898479	−	0.995955i	$-0.471362\pi$
$991$	5.65685	0.179696	0.0898479	+	0.995955i	$0.471362\pi$
$992$	0	0
$993$	−12.0000	−0.380808
$994$	0	0
$995$	− 108.000i	− 3.42383i
$996$	0	0
$997$	24.0416i	0.761406i	0.924697	+	0.380703i	$0.124318\pi$
$997$	24.0416i	0.761406i	−0.924697	+	0.380703i	$0.875682\pi$
$998$	0	0
$999$	5.65685	0.178975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.f.b.1057.3	yes	4
3.2	odd	2		6336.2.f.g.3169.4		4
4.3	odd	2	inner	2112.2.f.b.1057.1	✓	4
8.3	odd	2	inner	2112.2.f.b.1057.4	yes	4
8.5	even	2	inner	2112.2.f.b.1057.2	yes	4
12.11	even	2		6336.2.f.g.3169.3		4
16.3	odd	4		8448.2.a.bs.1.1		2
16.5	even	4		8448.2.a.bs.1.2		2
16.11	odd	4		8448.2.a.bg.1.2		2
16.13	even	4		8448.2.a.bg.1.1		2
24.5	odd	2		6336.2.f.g.3169.2		4
24.11	even	2		6336.2.f.g.3169.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2112.2.f.b.1057.1	✓	4	4.3	odd	2	inner
2112.2.f.b.1057.2	yes	4	8.5	even	2	inner
2112.2.f.b.1057.3	yes	4	1.1	even	1	trivial
2112.2.f.b.1057.4	yes	4	8.3	odd	2	inner
6336.2.f.g.3169.1		4	24.11	even	2
6336.2.f.g.3169.2		4	24.5	odd	2
6336.2.f.g.3169.3		4	12.11	even	2
6336.2.f.g.3169.4		4	3.2	odd	2
8448.2.a.bg.1.1		2	16.13	even	4
8448.2.a.bg.1.2		2	16.11	odd	4
8448.2.a.bs.1.1		2	16.3	odd	4
8448.2.a.bs.1.2		2	16.5	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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -4.24264i q^{5} +1.41421 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -4.24264i q^{5} +1.41421 q^{7} -1.00000 q^{9} +1.00000i q^{11} -4.24264i q^{13} +4.24264 q^{15} -4.00000 q^{17} -4.00000i q^{19} +1.41421i q^{21} +1.41421 q^{23} -13.0000 q^{25} -1.00000i q^{27} -2.82843i q^{29} -1.00000 q^{33} -6.00000i q^{35} +5.65685i q^{37} +4.24264 q^{39} +10.0000 q^{41} +8.00000i q^{43} +4.24264i q^{45} -9.89949 q^{47} -5.00000 q^{49} -4.00000i q^{51} -7.07107i q^{53} +4.24264 q^{55} +4.00000 q^{57} -8.00000i q^{59} +7.07107i q^{61} -1.41421 q^{63} -18.0000 q^{65} +2.00000i q^{67} +1.41421i q^{69} -1.41421 q^{71} -6.00000 q^{73} -13.0000i q^{75} +1.41421i q^{77} -12.7279 q^{79} +1.00000 q^{81} -6.00000i q^{83} +16.9706i q^{85} +2.82843 q^{87} -14.0000 q^{89} -6.00000i q^{91} -16.9706 q^{95} -4.00000 q^{97} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 16 q^{17} - 52 q^{25} - 4 q^{33} + 40 q^{41} - 20 q^{49} + 16 q^{57} - 72 q^{65} - 24 q^{73} + 4 q^{81} - 56 q^{89} - 16 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 - 16 * q^17 - 52 * q^25 - 4 * q^33 + 40 * q^41 - 20 * q^49 + 16 * q^57 - 72 * q^65 - 24 * q^73 + 4 * q^81 - 56 * q^89 - 16 * q^97

Introduction

L-functions

Modular forms

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Database

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