LMFDB - Embedded newform 2112.2.f.g.1057.7

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:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1057.7
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2112.1057
Dual form			2112.2.f.g.1057.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.82843i q^{5} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.82843i q^{5} -1.00000 q^{9} -1.00000i q^{11} -6.29253i q^{13} -2.82843 q^{15} -6.89898 q^{17} -4.89898i q^{19} +6.29253 q^{23} -3.00000 q^{25} -1.00000i q^{27} -0.635674i q^{29} -9.12096 q^{31} +1.00000 q^{33} -6.92820i q^{37} +6.29253 q^{39} -10.8990 q^{41} -8.89898i q^{43} -2.82843i q^{45} +0.635674 q^{47} -7.00000 q^{49} -6.89898i q^{51} -9.75663i q^{53} +2.82843 q^{55} +4.89898 q^{57} +5.79796i q^{59} +5.02118i q^{61} +17.7980 q^{65} +13.7980i q^{67} +6.29253i q^{69} +11.9494 q^{71} +7.79796 q^{73} -3.00000i q^{75} +6.92820 q^{79} +1.00000 q^{81} +9.79796i q^{83} -19.5133i q^{85} +0.635674 q^{87} -15.7980 q^{89} -9.12096i q^{93} +13.8564 q^{95} -6.00000 q^{97} +1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 16 q^{17} - 24 q^{25} + 8 q^{33} - 48 q^{41} - 56 q^{49} + 64 q^{65} - 16 q^{73} + 8 q^{81} - 48 q^{89} - 48 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 16 * q^17 - 24 * q^25 + 8 * q^33 - 48 * q^41 - 56 * q^49 + 64 * q^65 - 16 * q^73 + 8 * q^81 - 48 * q^89 - 48 * 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
$5$	2.82843i	1.26491i	0.774597	+	0.632456i	$0.217953\pi$
$5$	2.82843i	1.26491i	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	− 6.29253i	− 1.74523i	−0.488406	−	0.872617i	$-0.662421\pi$
$13$	− 6.29253i	− 1.74523i	0.488406	−	0.872617i	$-0.337579\pi$
$14$	0	0
$15$	−2.82843	−0.730297
$16$	0	0
$17$	−6.89898	−1.67325	−0.836624	−	0.547777i	$-0.815474\pi$
$17$	−6.89898	−1.67325	−0.836624	+	0.547777i	$0.815474\pi$
$18$	0	0
$19$	− 4.89898i	− 1.12390i	−0.827170	−	0.561951i	$-0.810051\pi$
$19$	− 4.89898i	− 1.12390i	0.827170	−	0.561951i	$-0.189949\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.29253	1.31208	0.656041	−	0.754725i	$-0.272230\pi$
$23$	6.29253	1.31208	0.656041	+	0.754725i	$0.272230\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 0.635674i	− 0.118042i	−0.998257	−	0.0590209i	$-0.981202\pi$
$29$	− 0.635674i	− 0.118042i	0.998257	−	0.0590209i	$-0.0187979\pi$
$30$	0	0
$31$	−9.12096	−1.63817	−0.819086	−	0.573671i	$-0.805519\pi$
$31$	−9.12096	−1.63817	−0.819086	+	0.573671i	$0.805519\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.92820i	− 1.13899i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	− 6.92820i	− 1.13899i	0.821995	−	0.569495i	$-0.192861\pi$
$38$	0	0
$39$	6.29253	1.00761
$40$	0	0
$41$	−10.8990	−1.70213	−0.851067	−	0.525057i	$-0.824044\pi$
$41$	−10.8990	−1.70213	−0.851067	+	0.525057i	$0.824044\pi$
$42$	0	0
$43$	− 8.89898i	− 1.35708i	−0.734563	−	0.678541i	$-0.762613\pi$
$43$	− 8.89898i	− 1.35708i	0.734563	−	0.678541i	$-0.237387\pi$
$44$	0	0
$45$	− 2.82843i	− 0.421637i
$46$	0	0
$47$	0.635674	0.0927227	0.0463613	−	0.998925i	$-0.485237\pi$
$47$	0.635674	0.0927227	0.0463613	+	0.998925i	$0.485237\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	− 6.89898i	− 0.966050i
$52$	0	0
$53$	− 9.75663i	− 1.34018i	−0.742282	−	0.670088i	$-0.766256\pi$
$53$	− 9.75663i	− 1.34018i	0.742282	−	0.670088i	$-0.233744\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	4.89898	0.648886
$58$	0	0
$59$	5.79796i	0.754830i	0.926044	+	0.377415i	$0.123187\pi$
$59$	5.79796i	0.754830i	−0.926044	+	0.377415i	$0.876813\pi$
$60$	0	0
$61$	5.02118i	0.642896i	0.946927	+	0.321448i	$0.104170\pi$
$61$	5.02118i	0.642896i	−0.946927	+	0.321448i	$0.895830\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	17.7980	2.20757
$66$	0	0
$67$	13.7980i	1.68569i	0.538157	+	0.842844i	$0.319121\pi$
$67$	13.7980i	1.68569i	−0.538157	+	0.842844i	$0.680879\pi$
$68$	0	0
$69$	6.29253i	0.757531i
$70$	0	0
$71$	11.9494	1.41813	0.709065	−	0.705143i	$-0.249117\pi$
$71$	11.9494	1.41813	0.709065	+	0.705143i	$0.249117\pi$
$72$	0	0
$73$	7.79796	0.912682	0.456341	−	0.889805i	$-0.349160\pi$
$73$	7.79796	0.912682	0.456341	+	0.889805i	$0.349160\pi$
$74$	0	0
$75$	− 3.00000i	− 0.346410i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.92820	0.779484	0.389742	−	0.920924i	$-0.372564\pi$
$79$	6.92820	0.779484	0.389742	+	0.920924i	$0.372564\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.79796i	1.07547i	0.843115	+	0.537733i	$0.180719\pi$
$83$	9.79796i	1.07547i	−0.843115	+	0.537733i	$0.819281\pi$
$84$	0	0
$85$	− 19.5133i	− 2.11651i
$86$	0	0
$87$	0.635674	0.0681515
$88$	0	0
$89$	−15.7980	−1.67458	−0.837290	−	0.546759i	$-0.815861\pi$
$89$	−15.7980	−1.67458	−0.837290	+	0.546759i	$0.815861\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 9.12096i	− 0.945799i
$94$	0	0
$95$	13.8564	1.42164
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	1.00000i	0.100504i
$100$	0	0
$101$	− 5.02118i	− 0.499626i	−0.968294	−	0.249813i	$-0.919631\pi$
$101$	− 5.02118i	− 0.499626i	0.968294	−	0.249813i	$-0.0803692\pi$
$102$	0	0
$103$	9.12096	0.898714	0.449357	−	0.893352i	$-0.351653\pi$
$103$	9.12096	0.898714	0.449357	+	0.893352i	$0.351653\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	0	0
$109$	− 11.9494i	− 1.14454i	−0.820064	−	0.572272i	$-0.806062\pi$
$109$	− 11.9494i	− 1.14454i	0.820064	−	0.572272i	$-0.193938\pi$
$110$	0	0
$111$	6.92820	0.657596
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	17.7980i	1.65967i
$116$	0	0
$117$	6.29253i	0.581744i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	− 10.8990i	− 0.982728i
$124$	0	0
$125$	5.65685i	0.505964i
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	8.89898	0.783511
$130$	0	0
$131$	− 13.7980i	− 1.20553i	−0.797918	−	0.602767i	$-0.794065\pi$
$131$	− 13.7980i	− 1.20553i	0.797918	−	0.602767i	$-0.205935\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.82843	0.243432
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	− 16.8990i	− 1.43335i	−0.697406	−	0.716676i	$-0.745662\pi$
$139$	− 16.8990i	− 1.43335i	0.697406	−	0.716676i	$-0.254338\pi$
$140$	0	0
$141$	0.635674i	0.0535334i
$142$	0	0
$143$	−6.29253	−0.526208
$144$	0	0
$145$	1.79796	0.149312
$146$	0	0
$147$	− 7.00000i	− 0.577350i
$148$	0	0
$149$	0.635674i	0.0520765i	0.999661	+	0.0260382i	$0.00828917\pi$
$149$	0.635674i	0.0520765i	−0.999661	+	0.0260382i	$0.991711\pi$
$150$	0	0
$151$	−6.92820	−0.563809	−0.281905	−	0.959442i	$-0.590966\pi$
$151$	−6.92820	−0.563809	−0.281905	+	0.959442i	$0.590966\pi$
$152$	0	0
$153$	6.89898	0.557749
$154$	0	0
$155$	− 25.7980i	− 2.07214i
$156$	0	0
$157$	5.65685i	0.451466i	0.974189	+	0.225733i	$0.0724777\pi$
$157$	5.65685i	0.451466i	−0.974189	+	0.225733i	$0.927522\pi$
$158$	0	0
$159$	9.75663	0.773751
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.20204i	0.172477i	0.996275	+	0.0862386i	$0.0274847\pi$
$163$	2.20204i	0.172477i	−0.996275	+	0.0862386i	$0.972515\pi$
$164$	0	0
$165$	2.82843i	0.220193i
$166$	0	0
$167$	−10.0424	−0.777101	−0.388551	−	0.921427i	$-0.627024\pi$
$167$	−10.0424	−0.777101	−0.388551	+	0.921427i	$0.627024\pi$
$168$	0	0
$169$	−26.5959	−2.04584
$170$	0	0
$171$	4.89898i	0.374634i
$172$	0	0
$173$	− 1.90702i	− 0.144988i	−0.997369	−	0.0724942i	$-0.976904\pi$
$173$	− 1.90702i	− 0.144988i	0.997369	−	0.0724942i	$-0.0230959\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.79796	−0.435801
$178$	0	0
$179$	4.00000i	0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$	4.00000i	0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$	0	0
$181$	10.0424i	0.746443i	0.927742	+	0.373221i	$0.121747\pi$
$181$	10.0424i	0.746443i	−0.927742	+	0.373221i	$0.878253\pi$
$182$	0	0
$183$	−5.02118	−0.371176
$184$	0	0
$185$	19.5959	1.44072
$186$	0	0
$187$	6.89898i	0.504503i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.8776	1.36593	0.682967	−	0.730449i	$-0.260689\pi$
$191$	18.8776	1.36593	0.682967	+	0.730449i	$0.260689\pi$
$192$	0	0
$193$	−21.5959	−1.55451	−0.777254	−	0.629187i	$-0.783388\pi$
$193$	−21.5959	−1.55451	−0.777254	+	0.629187i	$0.783388\pi$
$194$	0	0
$195$	17.7980i	1.27454i
$196$	0	0
$197$	20.1489i	1.43555i	0.696274	+	0.717776i	$0.254840\pi$
$197$	20.1489i	1.43555i	−0.696274	+	0.717776i	$0.745160\pi$
$198$	0	0
$199$	3.46410	0.245564	0.122782	−	0.992434i	$-0.460818\pi$
$199$	3.46410	0.245564	0.122782	+	0.992434i	$0.460818\pi$
$200$	0	0
$201$	−13.7980	−0.973233
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 30.8270i	− 2.15305i
$206$	0	0
$207$	−6.29253	−0.437361
$208$	0	0
$209$	−4.89898	−0.338869
$210$	0	0
$211$	− 14.6969i	− 1.01178i	−0.862598	−	0.505889i	$-0.831164\pi$
$211$	− 14.6969i	− 1.01178i	0.862598	−	0.505889i	$-0.168836\pi$
$212$	0	0
$213$	11.9494i	0.818758i
$214$	0	0
$215$	25.1701	1.71659
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.79796i	0.526937i
$220$	0	0
$221$	43.4120i	2.92021i
$222$	0	0
$223$	−24.2487	−1.62381	−0.811907	−	0.583787i	$-0.801570\pi$
$223$	−24.2487	−1.62381	−0.811907	+	0.583787i	$0.801570\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	− 5.79796i	− 0.384824i	−0.981314	−	0.192412i	$-0.938369\pi$
$227$	− 5.79796i	− 0.384824i	0.981314	−	0.192412i	$-0.0616310\pi$
$228$	0	0
$229$	− 26.4415i	− 1.74730i	−0.486554	−	0.873651i	$-0.661746\pi$
$229$	− 26.4415i	− 1.74730i	0.486554	−	0.873651i	$-0.338254\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.8990	0.714016	0.357008	−	0.934101i	$-0.383797\pi$
$233$	10.8990	0.714016	0.357008	+	0.934101i	$0.383797\pi$
$234$	0	0
$235$	1.79796i	0.117286i
$236$	0	0
$237$	6.92820i	0.450035i
$238$	0	0
$239$	1.27135	0.0822367	0.0411184	−	0.999154i	$-0.486908\pi$
$239$	1.27135	0.0822367	0.0411184	+	0.999154i	$0.486908\pi$
$240$	0	0
$241$	−11.7980	−0.759973	−0.379987	−	0.924992i	$-0.624071\pi$
$241$	−11.7980	−0.759973	−0.379987	+	0.924992i	$0.624071\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 19.7990i	− 1.26491i
$246$	0	0
$247$	−30.8270	−1.96147
$248$	0	0
$249$	−9.79796	−0.620920
$250$	0	0
$251$	2.20204i	0.138992i	0.997582	+	0.0694958i	$0.0221390\pi$
$251$	2.20204i	0.138992i	−0.997582	+	0.0694958i	$0.977861\pi$
$252$	0	0
$253$	− 6.29253i	− 0.395608i
$254$	0	0
$255$	19.5133	1.22197
$256$	0	0
$257$	4.20204	0.262116	0.131058	−	0.991375i	$-0.458163\pi$
$257$	4.20204	0.262116	0.131058	+	0.991375i	$0.458163\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0.635674i	0.0393473i
$262$	0	0
$263$	−11.3137	−0.697633	−0.348817	−	0.937191i	$-0.613416\pi$
$263$	−11.3137	−0.697633	−0.348817	+	0.937191i	$0.613416\pi$
$264$	0	0
$265$	27.5959	1.69520
$266$	0	0
$267$	− 15.7980i	− 0.966819i
$268$	0	0
$269$	− 26.7272i	− 1.62959i	−0.579752	−	0.814793i	$-0.696851\pi$
$269$	− 26.7272i	− 1.62959i	0.579752	−	0.814793i	$-0.303149\pi$
$270$	0	0
$271$	−6.92820	−0.420858	−0.210429	−	0.977609i	$-0.567486\pi$
$271$	−6.92820	−0.420858	−0.210429	+	0.977609i	$0.567486\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.00000i	0.180907i
$276$	0	0
$277$	10.6780i	0.641581i	0.947150	+	0.320790i	$0.103949\pi$
$277$	10.6780i	0.641581i	−0.947150	+	0.320790i	$0.896051\pi$
$278$	0	0
$279$	9.12096	0.546057
$280$	0	0
$281$	10.8990	0.650179	0.325089	−	0.945683i	$-0.394606\pi$
$281$	10.8990	0.650179	0.325089	+	0.945683i	$0.394606\pi$
$282$	0	0
$283$	20.8990i	1.24232i	0.783686	+	0.621158i	$0.213337\pi$
$283$	20.8990i	1.24232i	−0.783686	+	0.621158i	$0.786663\pi$
$284$	0	0
$285$	13.8564i	0.820783i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	30.5959	1.79976
$290$	0	0
$291$	− 6.00000i	− 0.351726i
$292$	0	0
$293$	− 17.6062i	− 1.02857i	−0.857620	−	0.514284i	$-0.828058\pi$
$293$	− 17.6062i	− 1.02857i	0.857620	−	0.514284i	$-0.171942\pi$
$294$	0	0
$295$	−16.3991	−0.954793
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	− 39.5959i	− 2.28989i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.02118	0.288459
$304$	0	0
$305$	−14.2020	−0.813207
$306$	0	0
$307$	− 2.69694i	− 0.153922i	−0.997034	−	0.0769612i	$-0.975478\pi$
$307$	− 2.69694i	− 0.153922i	0.997034	−	0.0769612i	$-0.0245218\pi$
$308$	0	0
$309$	9.12096i	0.518873i
$310$	0	0
$311$	−0.635674	−0.0360458	−0.0180229	−	0.999838i	$-0.505737\pi$
$311$	−0.635674	−0.0360458	−0.0180229	+	0.999838i	$0.505737\pi$
$312$	0	0
$313$	−21.5959	−1.22067	−0.610337	−	0.792142i	$-0.708966\pi$
$313$	−21.5959	−1.22067	−0.610337	+	0.792142i	$0.708966\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.82843i	0.158860i	0.996840	+	0.0794301i	$0.0253101\pi$
$317$	2.82843i	0.158860i	−0.996840	+	0.0794301i	$0.974690\pi$
$318$	0	0
$319$	−0.635674	−0.0355909
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	33.7980i	1.88057i
$324$	0	0
$325$	18.8776i	1.04714i
$326$	0	0
$327$	11.9494	0.660802
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.3939i	1.39577i	0.716208	+	0.697887i	$0.245876\pi$
$331$	25.3939i	1.39577i	−0.716208	+	0.697887i	$0.754124\pi$
$332$	0	0
$333$	6.92820i	0.379663i
$334$	0	0
$335$	−39.0265	−2.13225
$336$	0	0
$337$	−12.2020	−0.664688	−0.332344	−	0.943158i	$-0.607839\pi$
$337$	−12.2020	−0.664688	−0.332344	+	0.943158i	$0.607839\pi$
$338$	0	0
$339$	6.00000i	0.325875i
$340$	0	0
$341$	9.12096i	0.493927i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−17.7980	−0.958210
$346$	0	0
$347$	− 6.20204i	− 0.332943i	−0.986046	−	0.166472i	$-0.946763\pi$
$347$	− 6.20204i	− 0.332943i	0.986046	−	0.166472i	$-0.0532374\pi$
$348$	0	0
$349$	9.40669i	0.503528i	0.967789	+	0.251764i	$0.0810107\pi$
$349$	9.40669i	0.503528i	−0.967789	+	0.251764i	$0.918989\pi$
$350$	0	0
$351$	−6.29253	−0.335870
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	33.7980i	1.79381i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.3137	−0.597115	−0.298557	−	0.954392i	$-0.596505\pi$
$359$	−11.3137	−0.597115	−0.298557	+	0.954392i	$0.596505\pi$
$360$	0	0
$361$	−5.00000	−0.263158
$362$	0	0
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	22.0560i	1.15446i
$366$	0	0
$367$	19.1633	1.00032	0.500158	−	0.865934i	$-0.333275\pi$
$367$	19.1633	1.00032	0.500158	+	0.865934i	$0.333275\pi$
$368$	0	0
$369$	10.8990	0.567378
$370$	0	0
$371$	0	0
$372$	0	0
$373$	27.0771i	1.40200i	0.713161	+	0.701001i	$0.247263\pi$
$373$	27.0771i	1.40200i	−0.713161	+	0.701001i	$0.752737\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	− 33.3939i	− 1.71533i	−0.514210	−	0.857664i	$-0.671915\pi$
$379$	− 33.3939i	− 1.71533i	0.514210	−	0.857664i	$-0.328085\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.7634	−0.805474	−0.402737	−	0.915316i	$-0.631941\pi$
$383$	−15.7634	−0.805474	−0.402737	+	0.915316i	$0.631941\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.89898i	0.452361i
$388$	0	0
$389$	2.82843i	0.143407i	0.997426	+	0.0717035i	$0.0228435\pi$
$389$	2.82843i	0.143407i	−0.997426	+	0.0717035i	$0.977156\pi$
$390$	0	0
$391$	−43.4120	−2.19544
$392$	0	0
$393$	13.7980	0.696015
$394$	0	0
$395$	19.5959i	0.985978i
$396$	0	0
$397$	− 16.9706i	− 0.851728i	−0.904787	−	0.425864i	$-0.859970\pi$
$397$	− 16.9706i	− 0.851728i	0.904787	−	0.425864i	$-0.140030\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.7980	1.18841	0.594207	−	0.804312i	$-0.297466\pi$
$401$	23.7980	1.18841	0.594207	+	0.804312i	$0.297466\pi$
$402$	0	0
$403$	57.3939i	2.85899i
$404$	0	0
$405$	2.82843i	0.140546i
$406$	0	0
$407$	−6.92820	−0.343418
$408$	0	0
$409$	27.7980	1.37452	0.687260	−	0.726411i	$-0.258813\pi$
$409$	27.7980	1.37452	0.687260	+	0.726411i	$0.258813\pi$
$410$	0	0
$411$	− 2.00000i	− 0.0986527i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−27.7128	−1.36037
$416$	0	0
$417$	16.8990	0.827547
$418$	0	0
$419$	2.20204i	0.107577i	0.998552	+	0.0537884i	$0.0171296\pi$
$419$	2.20204i	0.107577i	−0.998552	+	0.0537884i	$0.982870\pi$
$420$	0	0
$421$	2.54270i	0.123924i	0.998079	+	0.0619618i	$0.0197357\pi$
$421$	2.54270i	0.123924i	−0.998079	+	0.0619618i	$0.980264\pi$
$422$	0	0
$423$	−0.635674	−0.0309076
$424$	0	0
$425$	20.6969	1.00395
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 6.29253i	− 0.303806i
$430$	0	0
$431$	−1.27135	−0.0612387	−0.0306194	−	0.999531i	$-0.509748\pi$
$431$	−1.27135	−0.0612387	−0.0306194	+	0.999531i	$0.509748\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	1.79796i	0.0862055i
$436$	0	0
$437$	− 30.8270i	− 1.47465i
$438$	0	0
$439$	13.8564	0.661330	0.330665	−	0.943748i	$-0.392727\pi$
$439$	13.8564	0.661330	0.330665	+	0.943748i	$0.392727\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	− 44.6834i	− 2.11820i
$446$	0	0
$447$	−0.635674	−0.0300664
$448$	0	0
$449$	−7.79796	−0.368008	−0.184004	−	0.982925i	$-0.558906\pi$
$449$	−7.79796	−0.368008	−0.184004	+	0.982925i	$0.558906\pi$
$450$	0	0
$451$	10.8990i	0.513213i
$452$	0	0
$453$	− 6.92820i	− 0.325515i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.20204	0.383675	0.191838	−	0.981427i	$-0.438555\pi$
$457$	8.20204	0.383675	0.191838	+	0.981427i	$0.438555\pi$
$458$	0	0
$459$	6.89898i	0.322017i
$460$	0	0
$461$	35.2767i	1.64300i	0.570209	+	0.821500i	$0.306862\pi$
$461$	35.2767i	1.64300i	−0.570209	+	0.821500i	$0.693138\pi$
$462$	0	0
$463$	24.8202	1.15349	0.576746	−	0.816924i	$-0.304322\pi$
$463$	24.8202	1.15349	0.576746	+	0.816924i	$0.304322\pi$
$464$	0	0
$465$	25.7980	1.19635
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5.65685	−0.260654
$472$	0	0
$473$	−8.89898	−0.409176
$474$	0	0
$475$	14.6969i	0.674342i
$476$	0	0
$477$	9.75663i	0.446725i
$478$	0	0
$479$	26.4415	1.20814	0.604071	−	0.796931i	$-0.293544\pi$
$479$	26.4415	1.20814	0.604071	+	0.796931i	$0.293544\pi$
$480$	0	0
$481$	−43.5959	−1.98780
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 16.9706i	− 0.770594i
$486$	0	0
$487$	−17.3205	−0.784867	−0.392434	−	0.919780i	$-0.628367\pi$
$487$	−17.3205	−0.784867	−0.392434	+	0.919780i	$0.628367\pi$
$488$	0	0
$489$	−2.20204	−0.0995797
$490$	0	0
$491$	− 24.0000i	− 1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$491$	− 24.0000i	− 1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$492$	0	0
$493$	4.38551i	0.197513i
$494$	0	0
$495$	−2.82843	−0.127128
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	− 10.0424i	− 0.448660i
$502$	0	0
$503$	−27.7128	−1.23565	−0.617827	−	0.786314i	$-0.711987\pi$
$503$	−27.7128	−1.23565	−0.617827	+	0.786314i	$0.711987\pi$
$504$	0	0
$505$	14.2020	0.631983
$506$	0	0
$507$	− 26.5959i	− 1.18117i
$508$	0	0
$509$	32.3840i	1.43540i	0.696354	+	0.717699i	$0.254805\pi$
$509$	32.3840i	1.43540i	−0.696354	+	0.717699i	$0.745195\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.89898	−0.216295
$514$	0	0
$515$	25.7980i	1.13679i
$516$	0	0
$517$	− 0.635674i	− 0.0279569i
$518$	0	0
$519$	1.90702	0.0837090
$520$	0	0
$521$	19.7980	0.867364	0.433682	−	0.901066i	$-0.357214\pi$
$521$	19.7980	0.867364	0.433682	+	0.901066i	$0.357214\pi$
$522$	0	0
$523$	− 20.4949i	− 0.896179i	−0.893989	−	0.448090i	$-0.852105\pi$
$523$	− 20.4949i	− 0.896179i	0.893989	−	0.448090i	$-0.147895\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	62.9253	2.74107
$528$	0	0
$529$	16.5959	0.721562
$530$	0	0
$531$	− 5.79796i	− 0.251610i
$532$	0	0
$533$	68.5821i	2.97062i
$534$	0	0
$535$	22.6274	0.978269
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	7.00000i	0.301511i
$540$	0	0
$541$	− 20.7204i	− 0.890839i	−0.895322	−	0.445420i	$-0.853054\pi$
$541$	− 20.7204i	− 0.890839i	0.895322	−	0.445420i	$-0.146946\pi$
$542$	0	0
$543$	−10.0424	−0.430959
$544$	0	0
$545$	33.7980	1.44775
$546$	0	0
$547$	1.30306i	0.0557149i	0.999612	+	0.0278574i	$0.00886845\pi$
$547$	1.30306i	0.0557149i	−0.999612	+	0.0278574i	$0.991132\pi$
$548$	0	0
$549$	− 5.02118i	− 0.214299i
$550$	0	0
$551$	−3.11416	−0.132668
$552$	0	0
$553$	0	0
$554$	0	0
$555$	19.5959i	0.831800i
$556$	0	0
$557$	2.47848i	0.105017i	0.998620	+	0.0525083i	$0.0167216\pi$
$557$	2.47848i	0.105017i	−0.998620	+	0.0525083i	$0.983278\pi$
$558$	0	0
$559$	−55.9971	−2.36842
$560$	0	0
$561$	−6.89898	−0.291275
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	16.9706i	0.713957i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.6969	−0.867661	−0.433830	−	0.900995i	$-0.642838\pi$
$569$	−20.6969	−0.867661	−0.433830	+	0.900995i	$0.642838\pi$
$570$	0	0
$571$	− 18.6969i	− 0.782443i	−0.920297	−	0.391221i	$-0.872053\pi$
$571$	− 18.6969i	− 0.782443i	0.920297	−	0.391221i	$-0.127947\pi$
$572$	0	0
$573$	18.8776i	0.788622i
$574$	0	0
$575$	−18.8776	−0.787250
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	− 21.5959i	− 0.897496i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−9.75663	−0.404078
$584$	0	0
$585$	−17.7980	−0.735855
$586$	0	0
$587$	− 21.7980i	− 0.899698i	−0.893105	−	0.449849i	$-0.851478\pi$
$587$	− 21.7980i	− 0.899698i	0.893105	−	0.449849i	$-0.148522\pi$
$588$	0	0
$589$	44.6834i	1.84115i
$590$	0	0
$591$	−20.1489	−0.828816
$592$	0	0
$593$	13.1010	0.537994	0.268997	−	0.963141i	$-0.413308\pi$
$593$	13.1010	0.537994	0.268997	+	0.963141i	$0.413308\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.46410i	0.141776i
$598$	0	0
$599$	3.74983	0.153214	0.0766070	−	0.997061i	$-0.475591\pi$
$599$	3.74983	0.153214	0.0766070	+	0.997061i	$0.475591\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	− 13.7980i	− 0.561896i
$604$	0	0
$605$	− 2.82843i	− 0.114992i
$606$	0	0
$607$	−8.77101	−0.356004	−0.178002	−	0.984030i	$-0.556963\pi$
$607$	−8.77101	−0.356004	−0.178002	+	0.984030i	$0.556963\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 4.00000i	− 0.161823i
$612$	0	0
$613$	− 30.1913i	− 1.21941i	−0.792627	−	0.609707i	$-0.791287\pi$
$613$	− 30.1913i	− 1.21941i	0.792627	−	0.609707i	$-0.208713\pi$
$614$	0	0
$615$	30.8270	1.24306
$616$	0	0
$617$	−37.1918	−1.49729	−0.748643	−	0.662973i	$-0.769294\pi$
$617$	−37.1918	−1.49729	−0.748643	+	0.662973i	$0.769294\pi$
$618$	0	0
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	0	0
$621$	− 6.29253i	− 0.252510i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	− 4.89898i	− 0.195646i
$628$	0	0
$629$	47.7975i	1.90581i
$630$	0	0
$631$	−36.1339	−1.43847	−0.719233	−	0.694768i	$-0.755507\pi$
$631$	−36.1339	−1.43847	−0.719233	+	0.694768i	$0.755507\pi$
$632$	0	0
$633$	14.6969	0.584151
$634$	0	0
$635$	0	0
$636$	0	0
$637$	44.0477i	1.74523i
$638$	0	0
$639$	−11.9494	−0.472710
$640$	0	0
$641$	−5.59592	−0.221025	−0.110513	−	0.993875i	$-0.535249\pi$
$641$	−5.59592	−0.221025	−0.110513	+	0.993875i	$0.535249\pi$
$642$	0	0
$643$	− 23.5959i	− 0.930532i	−0.885171	−	0.465266i	$-0.845959\pi$
$643$	− 23.5959i	− 0.930532i	0.885171	−	0.465266i	$-0.154041\pi$
$644$	0	0
$645$	25.1701i	0.991072i
$646$	0	0
$647$	−46.5904	−1.83166	−0.915829	−	0.401569i	$-0.868465\pi$
$647$	−46.5904	−1.83166	−0.915829	+	0.401569i	$0.868465\pi$
$648$	0	0
$649$	5.79796	0.227590
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.64247i	0.259940i	0.991518	+	0.129970i	$0.0414881\pi$
$653$	6.64247i	0.259940i	−0.991518	+	0.129970i	$0.958512\pi$
$654$	0	0
$655$	39.0265	1.52489
$656$	0	0
$657$	−7.79796	−0.304227
$658$	0	0
$659$	21.3939i	0.833387i	0.909047	+	0.416694i	$0.136811\pi$
$659$	21.3939i	0.833387i	−0.909047	+	0.416694i	$0.863189\pi$
$660$	0	0
$661$	10.7423i	0.417825i	0.977934	+	0.208913i	$0.0669924\pi$
$661$	10.7423i	0.417825i	−0.977934	+	0.208913i	$0.933008\pi$
$662$	0	0
$663$	−43.4120	−1.68598
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.00000i	− 0.154881i
$668$	0	0
$669$	− 24.2487i	− 0.937509i
$670$	0	0
$671$	5.02118	0.193840
$672$	0	0
$673$	−29.5959	−1.14084	−0.570419	−	0.821354i	$-0.693219\pi$
$673$	−29.5959	−1.14084	−0.570419	+	0.821354i	$0.693219\pi$
$674$	0	0
$675$	3.00000i	0.115470i
$676$	0	0
$677$	− 3.74983i	− 0.144118i	−0.997400	−	0.0720589i	$-0.977043\pi$
$677$	− 3.74983i	− 0.144118i	0.997400	−	0.0720589i	$-0.0229569\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5.79796	0.222178
$682$	0	0
$683$	− 37.7980i	− 1.44630i	−0.690692	−	0.723149i	$-0.742694\pi$
$683$	− 37.7980i	− 1.44630i	0.690692	−	0.723149i	$-0.257306\pi$
$684$	0	0
$685$	− 5.65685i	− 0.216137i
$686$	0	0
$687$	26.4415	1.00880
$688$	0	0
$689$	−61.3939	−2.33892
$690$	0	0
$691$	20.0000i	0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$	20.0000i	0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	47.7975	1.81306
$696$	0	0
$697$	75.1918	2.84809
$698$	0	0
$699$	10.8990i	0.412237i
$700$	0	0
$701$	− 35.2767i	− 1.33238i	−0.745781	−	0.666191i	$-0.767924\pi$
$701$	− 35.2767i	− 1.33238i	0.745781	−	0.666191i	$-0.232076\pi$
$702$	0	0
$703$	−33.9411	−1.28011
$704$	0	0
$705$	−1.79796	−0.0677151
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 3.81405i	− 0.143240i	−0.997432	−	0.0716198i	$-0.977183\pi$
$709$	− 3.81405i	− 0.143240i	0.997432	−	0.0716198i	$-0.0228168\pi$
$710$	0	0
$711$	−6.92820	−0.259828
$712$	0	0
$713$	−57.3939	−2.14942
$714$	0	0
$715$	− 17.7980i	− 0.665606i
$716$	0	0
$717$	1.27135i	0.0474794i
$718$	0	0
$719$	−23.2631	−0.867567	−0.433783	−	0.901017i	$-0.642822\pi$
$719$	−23.2631	−0.867567	−0.433783	+	0.901017i	$0.642822\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 11.7980i	− 0.438771i
$724$	0	0
$725$	1.90702i	0.0708251i
$726$	0	0
$727$	7.84961	0.291126	0.145563	−	0.989349i	$-0.453501\pi$
$727$	7.84961	0.291126	0.145563	+	0.989349i	$0.453501\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	61.3939i	2.27073i
$732$	0	0
$733$	35.8481i	1.32408i	0.749468	+	0.662041i	$0.230309\pi$
$733$	35.8481i	1.32408i	−0.749468	+	0.662041i	$0.769691\pi$
$734$	0	0
$735$	19.7990	0.730297
$736$	0	0
$737$	13.7980	0.508254
$738$	0	0
$739$	− 17.3031i	− 0.636503i	−0.948006	−	0.318252i	$-0.896904\pi$
$739$	− 17.3031i	− 0.636503i	0.948006	−	0.318252i	$-0.103096\pi$
$740$	0	0
$741$	− 30.8270i	− 1.13246i
$742$	0	0
$743$	15.1278	0.554984	0.277492	−	0.960728i	$-0.410497\pi$
$743$	15.1278	0.554984	0.277492	+	0.960728i	$0.410497\pi$
$744$	0	0
$745$	−1.79796	−0.0658721
$746$	0	0
$747$	− 9.79796i	− 0.358489i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	27.3629	0.998485	0.499243	−	0.866462i	$-0.333612\pi$
$751$	27.3629	0.998485	0.499243	+	0.866462i	$0.333612\pi$
$752$	0	0
$753$	−2.20204	−0.0802468
$754$	0	0
$755$	− 19.5959i	− 0.713168i
$756$	0	0
$757$	− 39.0265i	− 1.41844i	−0.704986	−	0.709222i	$-0.749047\pi$
$757$	− 39.0265i	− 1.41844i	0.704986	−	0.709222i	$-0.250953\pi$
$758$	0	0
$759$	6.29253	0.228404
$760$	0	0
$761$	19.3031	0.699735	0.349868	−	0.936799i	$-0.386227\pi$
$761$	19.3031	0.699735	0.349868	+	0.936799i	$0.386227\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	19.5133i	0.705503i
$766$	0	0
$767$	36.4838	1.31735
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	4.20204i	0.151333i
$772$	0	0
$773$	10.3281i	0.371476i	0.982599	+	0.185738i	$0.0594675\pi$
$773$	10.3281i	0.371476i	−0.982599	+	0.185738i	$0.940533\pi$
$774$	0	0
$775$	27.3629	0.982903
$776$	0	0
$777$	0	0
$778$	0	0
$779$	53.3939i	1.91303i
$780$	0	0
$781$	− 11.9494i	− 0.427583i
$782$	0	0
$783$	−0.635674	−0.0227172
$784$	0	0
$785$	−16.0000	−0.571064
$786$	0	0
$787$	− 46.2929i	− 1.65016i	−0.565014	−	0.825081i	$-0.691129\pi$
$787$	− 46.2929i	− 1.65016i	0.565014	−	0.825081i	$-0.308871\pi$
$788$	0	0
$789$	− 11.3137i	− 0.402779i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	31.5959	1.12200
$794$	0	0
$795$	27.5959i	0.978726i
$796$	0	0
$797$	− 22.9131i	− 0.811625i	−0.913956	−	0.405813i	$-0.866989\pi$
$797$	− 22.9131i	− 0.811625i	0.913956	−	0.405813i	$-0.133011\pi$
$798$	0	0
$799$	−4.38551	−0.155148
$800$	0	0
$801$	15.7980	0.558193
$802$	0	0
$803$	− 7.79796i	− 0.275184i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.7272	0.940842
$808$	0	0
$809$	−5.50510	−0.193549	−0.0967745	−	0.995306i	$-0.530853\pi$
$809$	−5.50510	−0.193549	−0.0967745	+	0.995306i	$0.530853\pi$
$810$	0	0
$811$	− 2.69694i	− 0.0947023i	−0.998878	−	0.0473512i	$-0.984922\pi$
$811$	− 2.69694i	− 0.0947023i	0.998878	−	0.0473512i	$-0.0150780\pi$
$812$	0	0
$813$	− 6.92820i	− 0.242983i
$814$	0	0
$815$	−6.22831	−0.218168
$816$	0	0
$817$	−43.5959	−1.52523
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.74983i	0.130870i	0.997857	+	0.0654350i	$0.0208435\pi$
$821$	3.74983i	0.130870i	−0.997857	+	0.0654350i	$0.979157\pi$
$822$	0	0
$823$	0.921404	0.0321181	0.0160591	−	0.999871i	$-0.494888\pi$
$823$	0.921404	0.0321181	0.0160591	+	0.999871i	$0.494888\pi$
$824$	0	0
$825$	−3.00000	−0.104447
$826$	0	0
$827$	51.1918i	1.78011i	0.455849	+	0.890057i	$0.349336\pi$
$827$	51.1918i	1.78011i	−0.455849	+	0.890057i	$0.650664\pi$
$828$	0	0
$829$	− 47.7975i	− 1.66008i	−0.557706	−	0.830038i	$-0.688318\pi$
$829$	− 47.7975i	− 1.66008i	0.557706	−	0.830038i	$-0.311682\pi$
$830$	0	0
$831$	−10.6780	−0.370417
$832$	0	0
$833$	48.2929	1.67325
$834$	0	0
$835$	− 28.4041i	− 0.982964i
$836$	0	0
$837$	9.12096i	0.315266i
$838$	0	0
$839$	44.0477	1.52070	0.760348	−	0.649516i	$-0.225029\pi$
$839$	44.0477	1.52070	0.760348	+	0.649516i	$0.225029\pi$
$840$	0	0
$841$	28.5959	0.986066
$842$	0	0
$843$	10.8990i	0.375381i
$844$	0	0
$845$	− 75.2246i	− 2.58781i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−20.8990	−0.717251
$850$	0	0
$851$	− 43.5959i	− 1.49445i
$852$	0	0
$853$	− 21.9917i	− 0.752983i	−0.926420	−	0.376491i	$-0.877130\pi$
$853$	− 21.9917i	− 0.752983i	0.926420	−	0.376491i	$-0.122870\pi$
$854$	0	0
$855$	−13.8564	−0.473879
$856$	0	0
$857$	6.89898	0.235665	0.117832	−	0.993034i	$-0.462405\pi$
$857$	6.89898	0.235665	0.117832	+	0.993034i	$0.462405\pi$
$858$	0	0
$859$	− 2.20204i	− 0.0751327i	−0.999294	−	0.0375663i	$-0.988039\pi$
$859$	− 2.20204i	− 0.0751327i	0.999294	−	0.0375663i	$-0.0119606\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−42.2049	−1.43667	−0.718336	−	0.695697i	$-0.755096\pi$
$863$	−42.2049	−1.43667	−0.718336	+	0.695697i	$0.755096\pi$
$864$	0	0
$865$	5.39388	0.183397
$866$	0	0
$867$	30.5959i	1.03909i
$868$	0	0
$869$	− 6.92820i	− 0.235023i
$870$	0	0
$871$	86.8241	2.94192
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	40.2337i	1.35859i	0.733863	+	0.679297i	$0.237715\pi$
$877$	40.2337i	1.35859i	−0.733863	+	0.679297i	$0.762285\pi$
$878$	0	0
$879$	17.6062	0.593844
$880$	0	0
$881$	−4.20204	−0.141570	−0.0707852	−	0.997492i	$-0.522550\pi$
$881$	−4.20204	−0.141570	−0.0707852	+	0.997492i	$0.522550\pi$
$882$	0	0
$883$	29.7980i	1.00278i	0.865221	+	0.501391i	$0.167178\pi$
$883$	29.7980i	1.00278i	−0.865221	+	0.501391i	$0.832822\pi$
$884$	0	0
$885$	− 16.3991i	− 0.551250i
$886$	0	0
$887$	40.2979	1.35307	0.676535	−	0.736410i	$-0.263481\pi$
$887$	40.2979	1.35307	0.676535	+	0.736410i	$0.263481\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 1.00000i	− 0.0335013i
$892$	0	0
$893$	− 3.11416i	− 0.104211i
$894$	0	0
$895$	−11.3137	−0.378176
$896$	0	0
$897$	39.5959	1.32207
$898$	0	0
$899$	5.79796i	0.193373i
$900$	0	0
$901$	67.3108i	2.24245i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−28.4041	−0.944184
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	0	0
$909$	5.02118i	0.166542i
$910$	0	0
$911$	−44.0477	−1.45937	−0.729683	−	0.683786i	$-0.760332\pi$
$911$	−44.0477	−1.45937	−0.729683	+	0.683786i	$0.760332\pi$
$912$	0	0
$913$	9.79796	0.324265
$914$	0	0
$915$	− 14.2020i	− 0.469505i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	30.2555	0.998037	0.499019	−	0.866591i	$-0.333694\pi$
$919$	30.2555	0.998037	0.499019	+	0.866591i	$0.333694\pi$
$920$	0	0
$921$	2.69694	0.0888671
$922$	0	0
$923$	− 75.1918i	− 2.47497i
$924$	0	0
$925$	20.7846i	0.683394i
$926$	0	0
$927$	−9.12096	−0.299571
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	34.2929i	1.12390i
$932$	0	0
$933$	− 0.635674i	− 0.0208110i
$934$	0	0
$935$	−19.5133	−0.638152
$936$	0	0
$937$	−5.59592	−0.182811	−0.0914053	−	0.995814i	$-0.529136\pi$
$937$	−5.59592	−0.182811	−0.0914053	+	0.995814i	$0.529136\pi$
$938$	0	0
$939$	− 21.5959i	− 0.704756i
$940$	0	0
$941$	17.6062i	0.573947i	0.957939	+	0.286973i	$0.0926492\pi$
$941$	17.6062i	0.573947i	−0.957939	+	0.286973i	$0.907351\pi$
$942$	0	0
$943$	−68.5821	−2.23334
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.5959i	1.02673i	0.858171	+	0.513365i	$0.171601\pi$
$947$	31.5959i	1.02673i	−0.858171	+	0.513365i	$0.828399\pi$
$948$	0	0
$949$	− 49.0689i	− 1.59284i
$950$	0	0
$951$	−2.82843	−0.0917180
$952$	0	0
$953$	−32.2929	−1.04607	−0.523034	−	0.852312i	$-0.675200\pi$
$953$	−32.2929	−1.04607	−0.523034	+	0.852312i	$0.675200\pi$
$954$	0	0
$955$	53.3939i	1.72779i
$956$	0	0
$957$	− 0.635674i	− 0.0205484i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	52.1918	1.68361
$962$	0	0
$963$	8.00000i	0.257796i
$964$	0	0
$965$	− 61.0825i	− 1.96631i
$966$	0	0
$967$	39.0265	1.25501	0.627504	−	0.778613i	$-0.284077\pi$
$967$	39.0265	1.25501	0.627504	+	0.778613i	$0.284077\pi$
$968$	0	0
$969$	−33.7980	−1.08575
$970$	0	0
$971$	− 31.5959i	− 1.01396i	−0.861957	−	0.506981i	$-0.830762\pi$
$971$	− 31.5959i	− 1.01396i	0.861957	−	0.506981i	$-0.169238\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−18.8776	−0.604567
$976$	0	0
$977$	7.79796	0.249479	0.124739	−	0.992190i	$-0.460191\pi$
$977$	7.79796	0.249479	0.124739	+	0.992190i	$0.460191\pi$
$978$	0	0
$979$	15.7980i	0.504905i
$980$	0	0
$981$	11.9494i	0.381514i
$982$	0	0
$983$	−28.3485	−0.904176	−0.452088	−	0.891973i	$-0.649321\pi$
$983$	−28.3485	−0.904176	−0.452088	+	0.891973i	$0.649321\pi$
$984$	0	0
$985$	−56.9898	−1.81585
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 55.9971i	− 1.78060i
$990$	0	0
$991$	57.6184	1.83031	0.915154	−	0.403104i	$-0.132069\pi$
$991$	57.6184	1.83031	0.915154	+	0.403104i	$0.132069\pi$
$992$	0	0
$993$	−25.3939	−0.805850
$994$	0	0
$995$	9.79796i	0.310616i
$996$	0	0
$997$	− 32.0341i	− 1.01453i	−0.861790	−	0.507265i	$-0.830656\pi$
$997$	− 32.0341i	− 1.01453i	0.861790	−	0.507265i	$-0.169344\pi$
$998$	0	0
$999$	−6.92820	−0.219199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.f.g.1057.7	yes	8
3.2	odd	2		6336.2.f.l.3169.3		8
4.3	odd	2	inner	2112.2.f.g.1057.3	yes	8
8.3	odd	2	inner	2112.2.f.g.1057.6	yes	8
8.5	even	2	inner	2112.2.f.g.1057.2	✓	8
12.11	even	2		6336.2.f.l.3169.1		8
16.3	odd	4		8448.2.a.ct.1.4		4
16.5	even	4		8448.2.a.ct.1.1		4
16.11	odd	4		8448.2.a.cm.1.1		4
16.13	even	4		8448.2.a.cm.1.4		4
24.5	odd	2		6336.2.f.l.3169.6		8
24.11	even	2		6336.2.f.l.3169.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2112.2.f.g.1057.2	✓	8	8.5	even	2	inner
2112.2.f.g.1057.3	yes	8	4.3	odd	2	inner
2112.2.f.g.1057.6	yes	8	8.3	odd	2	inner
2112.2.f.g.1057.7	yes	8	1.1	even	1	trivial
6336.2.f.l.3169.1		8	12.11	even	2
6336.2.f.l.3169.3		8	3.2	odd	2
6336.2.f.l.3169.6		8	24.5	odd	2
6336.2.f.l.3169.8		8	24.11	even	2
8448.2.a.cm.1.1		4	16.11	odd	4
8448.2.a.cm.1.4		4	16.13	even	4
8448.2.a.ct.1.1		4	16.5	even	4
8448.2.a.ct.1.4		4	16.3	odd	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.82843i q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.82843i q^{5} -1.00000 q^{9} -1.00000i q^{11} -6.29253i q^{13} -2.82843 q^{15} -6.89898 q^{17} -4.89898i q^{19} +6.29253 q^{23} -3.00000 q^{25} -1.00000i q^{27} -0.635674i q^{29} -9.12096 q^{31} +1.00000 q^{33} -6.92820i q^{37} +6.29253 q^{39} -10.8990 q^{41} -8.89898i q^{43} -2.82843i q^{45} +0.635674 q^{47} -7.00000 q^{49} -6.89898i q^{51} -9.75663i q^{53} +2.82843 q^{55} +4.89898 q^{57} +5.79796i q^{59} +5.02118i q^{61} +17.7980 q^{65} +13.7980i q^{67} +6.29253i q^{69} +11.9494 q^{71} +7.79796 q^{73} -3.00000i q^{75} +6.92820 q^{79} +1.00000 q^{81} +9.79796i q^{83} -19.5133i q^{85} +0.635674 q^{87} -15.7980 q^{89} -9.12096i q^{93} +13.8564 q^{95} -6.00000 q^{97} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 16 q^{17} - 24 q^{25} + 8 q^{33} - 48 q^{41} - 56 q^{49} + 64 q^{65} - 16 q^{73} + 8 q^{81} - 48 q^{89} - 48 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 16 * q^17 - 24 * q^25 + 8 * q^33 - 48 * q^41 - 56 * q^49 + 64 * q^65 - 16 * q^73 + 8 * q^81 - 48 * q^89 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more