LMFDB - Embedded newform 2268.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(1,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2268.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.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:	$18.1100711784$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.31342$ of defining polynomial
Character	$\chi$	$=$	2268.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.31342 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-1.31342 q^{5} -1.00000 q^{7} -0.418627 q^{11} +2.27492 q^{13} +4.77753 q^{17} -8.54983 q^{19} +8.71780 q^{23} -3.27492 q^{25} -8.24163 q^{29} -6.00000 q^{31} +1.31342 q^{35} +3.54983 q^{37} +6.92820 q^{41} +1.27492 q^{43} -3.46410 q^{47} +1.00000 q^{49} -2.20822 q^{53} +0.549834 q^{55} -8.71780 q^{59} -3.72508 q^{61} -2.98793 q^{65} -7.27492 q^{67} -3.88273 q^{71} -10.2749 q^{73} +0.418627 q^{77} -11.2749 q^{79} -9.55505 q^{83} -6.27492 q^{85} +2.98793 q^{89} -2.27492 q^{91} +11.2296 q^{95} -0.549834 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} - 6 q^{13} - 4 q^{19} + 2 q^{25} - 24 q^{31} - 16 q^{37} - 10 q^{43} + 4 q^{49} - 28 q^{55} - 30 q^{61} - 14 q^{67} - 26 q^{73} - 30 q^{79} - 10 q^{85} + 6 q^{91} + 28 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 6 * q^13 - 4 * q^19 + 2 * q^25 - 24 * q^31 - 16 * q^37 - 10 * q^43 + 4 * q^49 - 28 * q^55 - 30 * q^61 - 14 * q^67 - 26 * q^73 - 30 * q^79 - 10 * q^85 + 6 * q^91 + 28 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.31342	−0.587381	−0.293691	−	0.955901i	$-0.594884\pi$
$5$	−1.31342	−0.587381	−0.293691	+	0.955901i	$0.594884\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.418627	−0.126221	−0.0631104	−	0.998007i	$-0.520102\pi$
$11$	−0.418627	−0.126221	−0.0631104	+	0.998007i	$0.520102\pi$
$12$	0	0
$13$	2.27492	0.630949	0.315474	−	0.948934i	$-0.397836\pi$
$13$	2.27492	0.630949	0.315474	+	0.948934i	$0.397836\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.77753	1.15872	0.579360	−	0.815072i	$-0.303303\pi$
$17$	4.77753	1.15872	0.579360	+	0.815072i	$0.303303\pi$
$18$	0	0
$19$	−8.54983	−1.96147	−0.980733	−	0.195352i	$-0.937415\pi$
$19$	−8.54983	−1.96147	−0.980733	+	0.195352i	$0.937415\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.71780	1.81779	0.908893	−	0.417029i	$-0.136929\pi$
$23$	8.71780	1.81779	0.908893	+	0.417029i	$0.136929\pi$
$24$	0	0
$25$	−3.27492	−0.654983
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.24163	−1.53043	−0.765216	−	0.643774i	$-0.777368\pi$
$29$	−8.24163	−1.53043	−0.765216	+	0.643774i	$0.777368\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.31342	0.222009
$36$	0	0
$37$	3.54983	0.583589	0.291795	−	0.956481i	$-0.405748\pi$
$37$	3.54983	0.583589	0.291795	+	0.956481i	$0.405748\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	1.27492	0.194423	0.0972115	−	0.995264i	$-0.469008\pi$
$43$	1.27492	0.194423	0.0972115	+	0.995264i	$0.469008\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.20822	−0.303323	−0.151661	−	0.988433i	$-0.548462\pi$
$53$	−2.20822	−0.303323	−0.151661	+	0.988433i	$0.548462\pi$
$54$	0	0
$55$	0.549834	0.0741397
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.71780	−1.13496	−0.567480	−	0.823387i	$-0.692082\pi$
$59$	−8.71780	−1.13496	−0.567480	+	0.823387i	$0.692082\pi$
$60$	0	0
$61$	−3.72508	−0.476948	−0.238474	−	0.971149i	$-0.576647\pi$
$61$	−3.72508	−0.476948	−0.238474	+	0.971149i	$0.576647\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.98793	−0.370607
$66$	0	0
$67$	−7.27492	−0.888773	−0.444386	−	0.895835i	$-0.646578\pi$
$67$	−7.27492	−0.888773	−0.444386	+	0.895835i	$0.646578\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.88273	−0.460795	−0.230398	−	0.973097i	$-0.574003\pi$
$71$	−3.88273	−0.460795	−0.230398	+	0.973097i	$0.574003\pi$
$72$	0	0
$73$	−10.2749	−1.20259	−0.601294	−	0.799028i	$-0.705348\pi$
$73$	−10.2749	−1.20259	−0.601294	+	0.799028i	$0.705348\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.418627	0.0477069
$78$	0	0
$79$	−11.2749	−1.26853	−0.634264	−	0.773117i	$-0.718697\pi$
$79$	−11.2749	−1.26853	−0.634264	+	0.773117i	$0.718697\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.55505	−1.04880	−0.524402	−	0.851471i	$-0.675711\pi$
$83$	−9.55505	−1.04880	−0.524402	+	0.851471i	$0.675711\pi$
$84$	0	0
$85$	−6.27492	−0.680610
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.98793	0.316720	0.158360	−	0.987381i	$-0.449379\pi$
$89$	2.98793	0.316720	0.158360	+	0.987381i	$0.449379\pi$
$90$	0	0
$91$	−2.27492	−0.238476
$92$	0	0
$93$	0	0
$94$	0	0
$95$	11.2296	1.15213
$96$	0	0
$97$	−0.549834	−0.0558272	−0.0279136	−	0.999610i	$-0.508886\pi$
$97$	−0.549834	−0.0558272	−0.0279136	+	0.999610i	$0.508886\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.1819	−1.21214	−0.606072	−	0.795410i	$-0.707256\pi$
$101$	−12.1819	−1.21214	−0.606072	+	0.795410i	$0.707256\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.13642	0.883252	0.441626	−	0.897199i	$-0.354402\pi$
$107$	9.13642	0.883252	0.441626	+	0.897199i	$0.354402\pi$
$108$	0	0
$109$	−8.82475	−0.845258	−0.422629	−	0.906303i	$-0.638893\pi$
$109$	−8.82475	−0.845258	−0.422629	+	0.906303i	$0.638893\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.2153	1.71355	0.856776	−	0.515689i	$-0.172464\pi$
$113$	18.2153	1.71355	0.856776	+	0.515689i	$0.172464\pi$
$114$	0	0
$115$	−11.4502	−1.06773
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.77753	−0.437955
$120$	0	0
$121$	−10.8248	−0.984068
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.8685	0.972106
$126$	0	0
$127$	3.82475	0.339392	0.169696	−	0.985496i	$-0.445721\pi$
$127$	3.82475	0.339392	0.169696	+	0.985496i	$0.445721\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.09095	0.532169	0.266084	−	0.963950i	$-0.414270\pi$
$131$	6.09095	0.532169	0.266084	+	0.963950i	$0.414270\pi$
$132$	0	0
$133$	8.54983	0.741365
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.3781	−1.48471	−0.742354	−	0.670008i	$-0.766291\pi$
$137$	−17.3781	−1.48471	−0.742354	+	0.670008i	$0.766291\pi$
$138$	0	0
$139$	−12.5498	−1.06446	−0.532232	−	0.846599i	$-0.678646\pi$
$139$	−12.5498	−1.06446	−0.532232	+	0.846599i	$0.678646\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.952341	−0.0796388
$144$	0	0
$145$	10.8248	0.898947
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−23.4690	−1.92266	−0.961328	−	0.275407i	$-0.911187\pi$
$149$	−23.4690	−1.92266	−0.961328	+	0.275407i	$0.911187\pi$
$150$	0	0
$151$	13.8248	1.12504	0.562521	−	0.826783i	$-0.309831\pi$
$151$	13.8248	1.12504	0.562521	+	0.826783i	$0.309831\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.88054	0.632981
$156$	0	0
$157$	2.82475	0.225440	0.112720	−	0.993627i	$-0.464044\pi$
$157$	2.82475	0.225440	0.112720	+	0.993627i	$0.464044\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.71780	−0.687059
$162$	0	0
$163$	−10.7251	−0.840053	−0.420027	−	0.907512i	$-0.637979\pi$
$163$	−10.7251	−0.840053	−0.420027	+	0.907512i	$0.637979\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.55505	0.739392	0.369696	−	0.929153i	$-0.379462\pi$
$167$	9.55505	0.739392	0.369696	+	0.929153i	$0.379462\pi$
$168$	0	0
$169$	−7.82475	−0.601904
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.7057	−0.889970	−0.444985	−	0.895538i	$-0.646791\pi$
$173$	−11.7057	−0.889970	−0.444985	+	0.895538i	$0.646791\pi$
$174$	0	0
$175$	3.27492	0.247560
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.3923	−0.776757	−0.388379	−	0.921500i	$-0.626965\pi$
$179$	−10.3923	−0.776757	−0.388379	+	0.921500i	$0.626965\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.66244	−0.342789
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.67232	0.410435	0.205217	−	0.978716i	$-0.434210\pi$
$191$	5.67232	0.410435	0.205217	+	0.978716i	$0.434210\pi$
$192$	0	0
$193$	25.3746	1.82650	0.913251	−	0.407397i	$-0.133563\pi$
$193$	25.3746	1.82650	0.913251	+	0.407397i	$0.133563\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.9140	0.991328	0.495664	−	0.868514i	$-0.334925\pi$
$197$	13.9140	0.991328	0.495664	+	0.868514i	$0.334925\pi$
$198$	0	0
$199$	−19.0997	−1.35394	−0.676970	−	0.736011i	$-0.736707\pi$
$199$	−19.0997	−1.35394	−0.676970	+	0.736011i	$0.736707\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.24163	0.578449
$204$	0	0
$205$	−9.09967	−0.635548
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.57919	0.247578
$210$	0	0
$211$	19.2749	1.32694	0.663470	−	0.748203i	$-0.269083\pi$
$211$	19.2749	1.32694	0.663470	+	0.748203i	$0.269083\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.67451	−0.114200
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.8685	0.731093
$222$	0	0
$223$	−14.5498	−0.974329	−0.487164	−	0.873310i	$-0.661969\pi$
$223$	−14.5498	−0.974329	−0.487164	+	0.873310i	$0.661969\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.8564	0.919682	0.459841	−	0.888001i	$-0.347906\pi$
$227$	13.8564	0.919682	0.459841	+	0.888001i	$0.347906\pi$
$228$	0	0
$229$	−5.72508	−0.378324	−0.189162	−	0.981946i	$-0.560577\pi$
$229$	−5.72508	−0.378324	−0.189162	+	0.981946i	$0.560577\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.0504	1.51008	0.755040	−	0.655679i	$-0.227617\pi$
$233$	23.0504	1.51008	0.755040	+	0.655679i	$0.227617\pi$
$234$	0	0
$235$	4.54983	0.296798
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.18408	0.529384	0.264692	−	0.964333i	$-0.414730\pi$
$239$	8.18408	0.529384	0.264692	+	0.964333i	$0.414730\pi$
$240$	0	0
$241$	4.27492	0.275372	0.137686	−	0.990476i	$-0.456034\pi$
$241$	4.27492	0.275372	0.137686	+	0.990476i	$0.456034\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.31342	−0.0839116
$246$	0	0
$247$	−19.4502	−1.23758
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.2487	−1.53057	−0.765283	−	0.643695i	$-0.777401\pi$
$251$	−24.2487	−1.53057	−0.765283	+	0.643695i	$0.777401\pi$
$252$	0	0
$253$	−3.64950	−0.229442
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.8685	−0.677957	−0.338978	−	0.940794i	$-0.610081\pi$
$257$	−10.8685	−0.677957	−0.338978	+	0.940794i	$0.610081\pi$
$258$	0	0
$259$	−3.54983	−0.220576
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.67232	−0.349770	−0.174885	−	0.984589i	$-0.555955\pi$
$263$	−5.67232	−0.349770	−0.174885	+	0.984589i	$0.555955\pi$
$264$	0	0
$265$	2.90033	0.178166
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.3994	1.60960	0.804800	−	0.593547i	$-0.202273\pi$
$269$	26.3994	1.60960	0.804800	+	0.593547i	$0.202273\pi$
$270$	0	0
$271$	−15.0997	−0.917240	−0.458620	−	0.888633i	$-0.651656\pi$
$271$	−15.0997	−0.917240	−0.458620	+	0.888633i	$0.651656\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.37097	0.0826725
$276$	0	0
$277$	5.82475	0.349975	0.174988	−	0.984571i	$-0.444011\pi$
$277$	5.82475	0.349975	0.174988	+	0.984571i	$0.444011\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.7512	−0.879983	−0.439992	−	0.898002i	$-0.645019\pi$
$281$	−14.7512	−0.879983	−0.439992	+	0.898002i	$0.645019\pi$
$282$	0	0
$283$	−18.5498	−1.10267	−0.551337	−	0.834283i	$-0.685882\pi$
$283$	−18.5498	−1.10267	−0.551337	+	0.834283i	$0.685882\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.92820	−0.408959
$288$	0	0
$289$	5.82475	0.342632
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.40437	−0.432568	−0.216284	−	0.976330i	$-0.569394\pi$
$293$	−7.40437	−0.432568	−0.216284	+	0.976330i	$0.569394\pi$
$294$	0	0
$295$	11.4502	0.666654
$296$	0	0
$297$	0	0
$298$	0	0
$299$	19.8323	1.14693
$300$	0	0
$301$	−1.27492	−0.0734850
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.89261	0.280150
$306$	0	0
$307$	5.45017	0.311057	0.155529	−	0.987831i	$-0.450292\pi$
$307$	5.45017	0.311057	0.155529	+	0.987831i	$0.450292\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−29.5024	−1.67293	−0.836464	−	0.548022i	$-0.815381\pi$
$311$	−29.5024	−1.67293	−0.836464	+	0.548022i	$0.815381\pi$
$312$	0	0
$313$	16.8248	0.950991	0.475496	−	0.879718i	$-0.342269\pi$
$313$	16.8248	0.950991	0.475496	+	0.879718i	$0.342269\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.8443	0.946072	0.473036	−	0.881043i	$-0.343158\pi$
$317$	16.8443	0.946072	0.473036	+	0.881043i	$0.343158\pi$
$318$	0	0
$319$	3.45017	0.193172
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−40.8471	−2.27279
$324$	0	0
$325$	−7.45017	−0.413261
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.46410	0.190982
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.55505	0.522048
$336$	0	0
$337$	21.8248	1.18887	0.594435	−	0.804144i	$-0.297376\pi$
$337$	21.8248	1.18887	0.594435	+	0.804144i	$0.297376\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.51176	0.136019
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.71998	0.253382	0.126691	−	0.991942i	$-0.459564\pi$
$347$	4.71998	0.253382	0.126691	+	0.991942i	$0.459564\pi$
$348$	0	0
$349$	28.1993	1.50948	0.754738	−	0.656026i	$-0.227764\pi$
$349$	28.1993	1.50948	0.754738	+	0.656026i	$0.227764\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	36.5457	1.94513	0.972566	−	0.232629i	$-0.0747327\pi$
$353$	36.5457	1.94513	0.972566	+	0.232629i	$0.0747327\pi$
$354$	0	0
$355$	5.09967	0.270662
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.3901	−0.759482	−0.379741	−	0.925093i	$-0.623987\pi$
$359$	−14.3901	−0.759482	−0.379741	+	0.925093i	$0.623987\pi$
$360$	0	0
$361$	54.0997	2.84735
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.4953	0.706378
$366$	0	0
$367$	−20.5498	−1.07269	−0.536346	−	0.843998i	$-0.680196\pi$
$367$	−20.5498	−1.07269	−0.536346	+	0.843998i	$0.680196\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.20822	0.114645
$372$	0	0
$373$	16.7251	0.865992	0.432996	−	0.901396i	$-0.357456\pi$
$373$	16.7251	0.865992	0.432996	+	0.901396i	$0.357456\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.7490	−0.965624
$378$	0	0
$379$	24.3746	1.25204	0.626019	−	0.779808i	$-0.284683\pi$
$379$	24.3746	1.25204	0.626019	+	0.779808i	$0.284683\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−31.2920	−1.59895	−0.799473	−	0.600702i	$-0.794888\pi$
$383$	−31.2920	−1.59895	−0.799473	+	0.600702i	$0.794888\pi$
$384$	0	0
$385$	−0.549834	−0.0280222
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.3638	1.23529	0.617647	−	0.786456i	$-0.288086\pi$
$389$	24.3638	1.23529	0.617647	+	0.786456i	$0.288086\pi$
$390$	0	0
$391$	41.6495	2.10631
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.8087	0.745109
$396$	0	0
$397$	−24.8248	−1.24592	−0.622959	−	0.782254i	$-0.714070\pi$
$397$	−24.8248	−1.24592	−0.622959	+	0.782254i	$0.714070\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.779710	−0.0389368	−0.0194684	−	0.999810i	$-0.506197\pi$
$401$	−0.779710	−0.0389368	−0.0194684	+	0.999810i	$0.506197\pi$
$402$	0	0
$403$	−13.6495	−0.679930
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.48606	−0.0736610
$408$	0	0
$409$	−27.3746	−1.35359	−0.676793	−	0.736173i	$-0.736631\pi$
$409$	−27.3746	−1.35359	−0.676793	+	0.736173i	$0.736631\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.71780	0.428975
$414$	0	0
$415$	12.5498	0.616047
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.6893	1.10844	0.554222	−	0.832369i	$-0.313016\pi$
$419$	22.6893	1.10844	0.554222	+	0.832369i	$0.313016\pi$
$420$	0	0
$421$	36.0997	1.75939	0.879695	−	0.475538i	$-0.157747\pi$
$421$	36.0997	1.75939	0.879695	+	0.475538i	$0.157747\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15.6460	−0.758943
$426$	0	0
$427$	3.72508	0.180269
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9.43996	−0.454707	−0.227354	−	0.973812i	$-0.573007\pi$
$431$	−9.43996	−0.454707	−0.227354	+	0.973812i	$0.573007\pi$
$432$	0	0
$433$	10.2749	0.493781	0.246891	−	0.969043i	$-0.420591\pi$
$433$	10.2749	0.493781	0.246891	+	0.969043i	$0.420591\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−74.5357	−3.56553
$438$	0	0
$439$	−10.5498	−0.503516	−0.251758	−	0.967790i	$-0.581009\pi$
$439$	−10.5498	−0.503516	−0.251758	+	0.967790i	$0.581009\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.115088	−0.00546798	−0.00273399	−	0.999996i	$-0.500870\pi$
$443$	−0.115088	−0.00546798	−0.00273399	+	0.999996i	$0.500870\pi$
$444$	0	0
$445$	−3.92442	−0.186035
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.9019	−0.797649	−0.398825	−	0.917027i	$-0.630582\pi$
$449$	−16.9019	−0.797649	−0.398825	+	0.917027i	$0.630582\pi$
$450$	0	0
$451$	−2.90033	−0.136571
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.98793	0.140076
$456$	0	0
$457$	22.0997	1.03378	0.516889	−	0.856052i	$-0.327090\pi$
$457$	22.0997	1.03378	0.516889	+	0.856052i	$0.327090\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16.5983	−0.773062	−0.386531	−	0.922276i	$-0.626327\pi$
$461$	−16.5983	−0.773062	−0.386531	+	0.922276i	$0.626327\pi$
$462$	0	0
$463$	−37.4743	−1.74158	−0.870788	−	0.491658i	$-0.836391\pi$
$463$	−37.4743	−1.74158	−0.870788	+	0.491658i	$0.836391\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.55505	0.442155	0.221078	−	0.975256i	$-0.429043\pi$
$467$	9.55505	0.442155	0.221078	+	0.975256i	$0.429043\pi$
$468$	0	0
$469$	7.27492	0.335924
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.533714	−0.0245402
$474$	0	0
$475$	28.0000	1.28473
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.81312	0.311299	0.155650	−	0.987812i	$-0.450253\pi$
$479$	6.81312	0.311299	0.155650	+	0.987812i	$0.450253\pi$
$480$	0	0
$481$	8.07558	0.368215
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.722166	0.0327919
$486$	0	0
$487$	−25.8248	−1.17023	−0.585116	−	0.810950i	$-0.698951\pi$
$487$	−25.8248	−1.17023	−0.585116	+	0.810950i	$0.698951\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.43996	0.426020	0.213010	−	0.977050i	$-0.431673\pi$
$491$	9.43996	0.426020	0.213010	+	0.977050i	$0.431673\pi$
$492$	0	0
$493$	−39.3746	−1.77334
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.88273	0.174164
$498$	0	0
$499$	17.6495	0.790100	0.395050	−	0.918660i	$-0.370727\pi$
$499$	17.6495	0.790100	0.395050	+	0.918660i	$0.370727\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0.952341	0.0424628	0.0212314	−	0.999775i	$-0.493241\pi$
$503$	0.952341	0.0424628	0.0212314	+	0.999775i	$0.493241\pi$
$504$	0	0
$505$	16.0000	0.711991
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.43996	−0.418419	−0.209210	−	0.977871i	$-0.567089\pi$
$509$	−9.43996	−0.418419	−0.209210	+	0.977871i	$0.567089\pi$
$510$	0	0
$511$	10.2749	0.454536
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.25370	0.231506
$516$	0	0
$517$	1.45017	0.0637782
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.2728	0.800548	0.400274	−	0.916395i	$-0.368915\pi$
$521$	18.2728	0.800548	0.400274	+	0.916395i	$0.368915\pi$
$522$	0	0
$523$	39.6495	1.73375	0.866876	−	0.498524i	$-0.166124\pi$
$523$	39.6495	1.73375	0.866876	+	0.498524i	$0.166124\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28.6652	−1.24867
$528$	0	0
$529$	53.0000	2.30435
$530$	0	0
$531$	0	0
$532$	0	0
$533$	15.7611	0.682689
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.418627	−0.0180315
$540$	0	0
$541$	−15.1993	−0.653471	−0.326735	−	0.945116i	$-0.605949\pi$
$541$	−15.1993	−0.653471	−0.326735	+	0.945116i	$0.605949\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11.5906	0.496489
$546$	0	0
$547$	20.3746	0.871154	0.435577	−	0.900151i	$-0.356544\pi$
$547$	20.3746	0.871154	0.435577	+	0.900151i	$0.356544\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	70.4645	3.00189
$552$	0	0
$553$	11.2749	0.479458
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−33.1391	−1.40415	−0.702075	−	0.712103i	$-0.747743\pi$
$557$	−33.1391	−1.40415	−0.702075	+	0.712103i	$0.747743\pi$
$558$	0	0
$559$	2.90033	0.122671
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.8038	−1.42466	−0.712329	−	0.701845i	$-0.752360\pi$
$563$	−33.8038	−1.42466	−0.712329	+	0.701845i	$0.752360\pi$
$564$	0	0
$565$	−23.9244	−1.00651
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.19615	0.217834	0.108917	−	0.994051i	$-0.465262\pi$
$569$	5.19615	0.217834	0.108917	+	0.994051i	$0.465262\pi$
$570$	0	0
$571$	4.54983	0.190405	0.0952023	−	0.995458i	$-0.469650\pi$
$571$	4.54983	0.190405	0.0952023	+	0.995458i	$0.469650\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−28.5501	−1.19062
$576$	0	0
$577$	7.72508	0.321599	0.160800	−	0.986987i	$-0.448593\pi$
$577$	7.72508	0.321599	0.160800	+	0.986987i	$0.448593\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.55505	0.396410
$582$	0	0
$583$	0.924421	0.0382856
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.4306	1.50365	0.751826	−	0.659361i	$-0.229173\pi$
$587$	36.4306	1.50365	0.751826	+	0.659361i	$0.229173\pi$
$588$	0	0
$589$	51.2990	2.11374
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.0937	1.23580	0.617899	−	0.786257i	$-0.287984\pi$
$593$	30.0937	1.23580	0.617899	+	0.786257i	$0.287984\pi$
$594$	0	0
$595$	6.27492	0.257247
$596$	0	0
$597$	0	0
$598$	0	0
$599$	28.3616	1.15882	0.579412	−	0.815035i	$-0.303282\pi$
$599$	28.3616	1.15882	0.579412	+	0.815035i	$0.303282\pi$
$600$	0	0
$601$	−10.8248	−0.441551	−0.220775	−	0.975325i	$-0.570859\pi$
$601$	−10.8248	−0.441551	−0.220775	+	0.975325i	$0.570859\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	14.2175	0.578023
$606$	0	0
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.88054	−0.318813
$612$	0	0
$613$	2.17525	0.0878575	0.0439287	−	0.999035i	$-0.486013\pi$
$613$	2.17525	0.0878575	0.0439287	+	0.999035i	$0.486013\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.28929	0.293456	0.146728	−	0.989177i	$-0.453126\pi$
$617$	7.28929	0.293456	0.146728	+	0.989177i	$0.453126\pi$
$618$	0	0
$619$	−9.45017	−0.379834	−0.189917	−	0.981800i	$-0.560822\pi$
$619$	−9.45017	−0.379834	−0.189917	+	0.981800i	$0.560822\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.98793	−0.119709
$624$	0	0
$625$	2.09967	0.0839868
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.9594	0.676217
$630$	0	0
$631$	27.4502	1.09277	0.546387	−	0.837533i	$-0.316003\pi$
$631$	27.4502	1.09277	0.546387	+	0.837533i	$0.316003\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.02352	−0.199352
$636$	0	0
$637$	2.27492	0.0901355
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.77534	−0.346605	−0.173303	−	0.984869i	$-0.555444\pi$
$641$	−8.77534	−0.346605	−0.173303	+	0.984869i	$0.555444\pi$
$642$	0	0
$643$	9.64950	0.380539	0.190270	−	0.981732i	$-0.439064\pi$
$643$	9.64950	0.380539	0.190270	+	0.981732i	$0.439064\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.7846	0.817127	0.408564	−	0.912730i	$-0.366030\pi$
$647$	20.7846	0.817127	0.408564	+	0.912730i	$0.366030\pi$
$648$	0	0
$649$	3.64950	0.143256
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−43.7774	−1.71314	−0.856572	−	0.516028i	$-0.827410\pi$
$653$	−43.7774	−1.71314	−0.856572	+	0.516028i	$0.827410\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8.29917	−0.323290	−0.161645	−	0.986849i	$-0.551680\pi$
$659$	−8.29917	−0.323290	−0.161645	+	0.986849i	$0.551680\pi$
$660$	0	0
$661$	19.9244	0.774970	0.387485	−	0.921876i	$-0.373344\pi$
$661$	19.9244	0.774970	0.387485	+	0.921876i	$0.373344\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−11.2296	−0.435464
$666$	0	0
$667$	−71.8488	−2.78200
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.55942	0.0602007
$672$	0	0
$673$	1.90033	0.0732524	0.0366262	−	0.999329i	$-0.488339\pi$
$673$	1.90033	0.0732524	0.0366262	+	0.999329i	$0.488339\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.51176	−0.0965348	−0.0482674	−	0.998834i	$-0.515370\pi$
$677$	−2.51176	−0.0965348	−0.0482674	+	0.998834i	$0.515370\pi$
$678$	0	0
$679$	0.549834	0.0211007
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−38.7539	−1.48288	−0.741439	−	0.671021i	$-0.765856\pi$
$683$	−38.7539	−1.48288	−0.741439	+	0.671021i	$0.765856\pi$
$684$	0	0
$685$	22.8248	0.872089
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.02352	−0.191381
$690$	0	0
$691$	−11.6495	−0.443168	−0.221584	−	0.975141i	$-0.571123\pi$
$691$	−11.6495	−0.443168	−0.221584	+	0.975141i	$0.571123\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16.4833	0.625245
$696$	0	0
$697$	33.0997	1.25374
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20.1200	0.759921	0.379961	−	0.925003i	$-0.375938\pi$
$701$	20.1200	0.759921	0.379961	+	0.925003i	$0.375938\pi$
$702$	0	0
$703$	−30.3505	−1.14469
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.1819	0.458147
$708$	0	0
$709$	−15.9003	−0.597149	−0.298575	−	0.954386i	$-0.596511\pi$
$709$	−15.9003	−0.597149	−0.298575	+	0.954386i	$0.596511\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−52.3068	−1.95890
$714$	0	0
$715$	1.25083	0.0467783
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32.1293	−1.19822	−0.599110	−	0.800667i	$-0.704479\pi$
$719$	−32.1293	−1.19822	−0.599110	+	0.800667i	$0.704479\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	26.9906	1.00241
$726$	0	0
$727$	5.45017	0.202135	0.101068	−	0.994880i	$-0.467774\pi$
$727$	5.45017	0.202135	0.101068	+	0.994880i	$0.467774\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.09095	0.225282
$732$	0	0
$733$	−52.5498	−1.94097	−0.970486	−	0.241157i	$-0.922473\pi$
$733$	−52.5498	−1.94097	−0.970486	+	0.241157i	$0.922473\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.04547	0.112182
$738$	0	0
$739$	23.2749	0.856182	0.428091	−	0.903736i	$-0.359186\pi$
$739$	23.2749	0.856182	0.428091	+	0.903736i	$0.359186\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.4093	1.00555	0.502774	−	0.864418i	$-0.332313\pi$
$743$	27.4093	1.00555	0.502774	+	0.864418i	$0.332313\pi$
$744$	0	0
$745$	30.8248	1.12933
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.13642	−0.333838
$750$	0	0
$751$	−24.9244	−0.909505	−0.454753	−	0.890618i	$-0.650272\pi$
$751$	−24.9244	−0.909505	−0.454753	+	0.890618i	$0.650272\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−18.1578	−0.660829
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	54.2273	1.96574	0.982869	−	0.184306i	$-0.0590038\pi$
$761$	54.2273	1.96574	0.982869	+	0.184306i	$0.0590038\pi$
$762$	0	0
$763$	8.82475	0.319477
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19.8323	−0.716102
$768$	0	0
$769$	−14.8248	−0.534594	−0.267297	−	0.963614i	$-0.586131\pi$
$769$	−14.8248	−0.534594	−0.267297	+	0.963614i	$0.586131\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29.8635	1.07412	0.537058	−	0.843546i	$-0.319536\pi$
$773$	29.8635	1.07412	0.537058	+	0.843546i	$0.319536\pi$
$774$	0	0
$775$	19.6495	0.705831
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−59.2350	−2.12231
$780$	0	0
$781$	1.62541	0.0581619
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.71010	−0.132419
$786$	0	0
$787$	−48.0000	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$787$	−48.0000	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−18.2153	−0.647662
$792$	0	0
$793$	−8.47425	−0.300930
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.7967	−0.630391	−0.315195	−	0.949027i	$-0.602070\pi$
$797$	−17.7967	−0.630391	−0.315195	+	0.949027i	$0.602070\pi$
$798$	0	0
$799$	−16.5498	−0.585491
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.30136	0.151792
$804$	0	0
$805$	11.4502	0.403565
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16.6559	0.585590	0.292795	−	0.956175i	$-0.405415\pi$
$809$	16.6559	0.585590	0.292795	+	0.956175i	$0.405415\pi$
$810$	0	0
$811$	−45.6495	−1.60297	−0.801485	−	0.598014i	$-0.795957\pi$
$811$	−45.6495	−1.60297	−0.801485	+	0.598014i	$0.795957\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	14.0866	0.493431
$816$	0	0
$817$	−10.9003	−0.381354
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.9616	0.452363	0.226182	−	0.974085i	$-0.427376\pi$
$821$	12.9616	0.452363	0.226182	+	0.974085i	$0.427376\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.9928	0.799539	0.399770	−	0.916616i	$-0.369090\pi$
$827$	22.9928	0.799539	0.399770	+	0.916616i	$0.369090\pi$
$828$	0	0
$829$	−6.35050	−0.220562	−0.110281	−	0.993900i	$-0.535175\pi$
$829$	−6.35050	−0.220562	−0.110281	+	0.993900i	$0.535175\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.77753	0.165531
$834$	0	0
$835$	−12.5498	−0.434305
$836$	0	0
$837$	0	0
$838$	0	0
$839$	27.9430	0.964699	0.482350	−	0.875979i	$-0.339784\pi$
$839$	27.9430	0.964699	0.482350	+	0.875979i	$0.339784\pi$
$840$	0	0
$841$	38.9244	1.34222
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.2772	0.353547
$846$	0	0
$847$	10.8248	0.371943
$848$	0	0
$849$	0	0
$850$	0	0
$851$	30.9467	1.06084
$852$	0	0
$853$	16.9003	0.578656	0.289328	−	0.957230i	$-0.406568\pi$
$853$	16.9003	0.578656	0.289328	+	0.957230i	$0.406568\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.0219	1.26464	0.632321	−	0.774706i	$-0.282102\pi$
$857$	37.0219	1.26464	0.632321	+	0.774706i	$0.282102\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48.3090	1.64446	0.822228	−	0.569158i	$-0.192731\pi$
$863$	48.3090	1.64446	0.822228	+	0.569158i	$0.192731\pi$
$864$	0	0
$865$	15.3746	0.522752
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.71998	0.160114
$870$	0	0
$871$	−16.5498	−0.560770
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10.8685	−0.367422
$876$	0	0
$877$	−44.0997	−1.48914	−0.744570	−	0.667544i	$-0.767345\pi$
$877$	−44.0997	−1.48914	−0.744570	+	0.667544i	$0.767345\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−43.4739	−1.46467	−0.732336	−	0.680943i	$-0.761570\pi$
$881$	−43.4739	−1.46467	−0.732336	+	0.680943i	$0.761570\pi$
$882$	0	0
$883$	−36.3746	−1.22410	−0.612051	−	0.790818i	$-0.709655\pi$
$883$	−36.3746	−1.22410	−0.612051	+	0.790818i	$0.709655\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26.7605	−0.898529	−0.449264	−	0.893399i	$-0.648314\pi$
$887$	−26.7605	−0.898529	−0.449264	+	0.893399i	$0.648314\pi$
$888$	0	0
$889$	−3.82475	−0.128278
$890$	0	0
$891$	0	0
$892$	0	0
$893$	29.6175	0.991112
$894$	0	0
$895$	13.6495	0.456253
$896$	0	0
$897$	0	0
$898$	0	0
$899$	49.4498	1.64924
$900$	0	0
$901$	−10.5498	−0.351466
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.88054	−0.261958
$906$	0	0
$907$	−15.8248	−0.525452	−0.262726	−	0.964870i	$-0.584622\pi$
$907$	−15.8248	−0.525452	−0.262726	+	0.964870i	$0.584622\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−40.9621	−1.35714	−0.678568	−	0.734537i	$-0.737399\pi$
$911$	−40.9621	−1.35714	−0.678568	+	0.734537i	$0.737399\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.09095	−0.201141
$918$	0	0
$919$	1.82475	0.0601930	0.0300965	−	0.999547i	$-0.490419\pi$
$919$	1.82475	0.0601930	0.0300965	+	0.999547i	$0.490419\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.83289	−0.290738
$924$	0	0
$925$	−11.6254	−0.382241
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−57.8065	−1.89657	−0.948285	−	0.317422i	$-0.897183\pi$
$929$	−57.8065	−1.89657	−0.948285	+	0.317422i	$0.897183\pi$
$930$	0	0
$931$	−8.54983	−0.280210
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.62685	0.0859071
$936$	0	0
$937$	10.2749	0.335667	0.167833	−	0.985815i	$-0.446323\pi$
$937$	10.2749	0.335667	0.167833	+	0.985815i	$0.446323\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−45.7397	−1.49107	−0.745535	−	0.666466i	$-0.767806\pi$
$941$	−45.7397	−1.49107	−0.745535	+	0.666466i	$0.767806\pi$
$942$	0	0
$943$	60.3987	1.96685
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35.4783	−1.15289	−0.576444	−	0.817136i	$-0.695560\pi$
$947$	−35.4783	−1.15289	−0.576444	+	0.817136i	$0.695560\pi$
$948$	0	0
$949$	−23.3746	−0.758771
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−50.0410	−1.62099	−0.810494	−	0.585747i	$-0.800801\pi$
$953$	−50.0410	−1.62099	−0.810494	+	0.585747i	$0.800801\pi$
$954$	0	0
$955$	−7.45017	−0.241082
$956$	0	0
$957$	0	0
$958$	0	0
$959$	17.3781	0.561167
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−33.3276	−1.07285
$966$	0	0
$967$	27.4502	0.882738	0.441369	−	0.897326i	$-0.354493\pi$
$967$	27.4502	0.882738	0.441369	+	0.897326i	$0.354493\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.3205	−0.555842	−0.277921	−	0.960604i	$-0.589645\pi$
$971$	−17.3205	−0.555842	−0.277921	+	0.960604i	$0.589645\pi$
$972$	0	0
$973$	12.5498	0.402329
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.92820	0.221653	0.110826	−	0.993840i	$-0.464650\pi$
$977$	6.92820	0.221653	0.110826	+	0.993840i	$0.464650\pi$
$978$	0	0
$979$	−1.25083	−0.0399766
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.2011	−0.803789	−0.401894	−	0.915686i	$-0.631648\pi$
$983$	−25.2011	−0.803789	−0.401894	+	0.915686i	$0.631648\pi$
$984$	0	0
$985$	−18.2749	−0.582287
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11.1145	0.353420
$990$	0	0
$991$	38.9244	1.23647	0.618237	−	0.785991i	$-0.287847\pi$
$991$	38.9244	1.23647	0.618237	+	0.785991i	$0.287847\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.0860	0.795279
$996$	0	0
$997$	−61.9244	−1.96117	−0.980583	−	0.196104i	$-0.937171\pi$
$997$	−61.9244	−1.96117	−0.980583	+	0.196104i	$0.937171\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.a.k.1.2	✓	4
3.2	odd	2	inner	2268.2.a.k.1.3	yes	4
4.3	odd	2		9072.2.a.ch.1.2		4
9.2	odd	6		2268.2.j.r.757.2		8
9.4	even	3		2268.2.j.r.1513.3		8
9.5	odd	6		2268.2.j.r.1513.2		8
9.7	even	3		2268.2.j.r.757.3		8
12.11	even	2		9072.2.a.ch.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.a.k.1.2	✓	4	1.1	even	1	trivial
2268.2.a.k.1.3	yes	4	3.2	odd	2	inner
2268.2.j.r.757.2		8	9.2	odd	6
2268.2.j.r.757.3		8	9.7	even	3
2268.2.j.r.1513.2		8	9.5	odd	6
2268.2.j.r.1513.3		8	9.4	even	3
9072.2.a.ch.1.2		4	4.3	odd	2
9072.2.a.ch.1.3		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.a (trivial)

\(f(q)\)	\(=\)	\(q-1.31342 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-1.31342 q^{5} -1.00000 q^{7} -0.418627 q^{11} +2.27492 q^{13} +4.77753 q^{17} -8.54983 q^{19} +8.71780 q^{23} -3.27492 q^{25} -8.24163 q^{29} -6.00000 q^{31} +1.31342 q^{35} +3.54983 q^{37} +6.92820 q^{41} +1.27492 q^{43} -3.46410 q^{47} +1.00000 q^{49} -2.20822 q^{53} +0.549834 q^{55} -8.71780 q^{59} -3.72508 q^{61} -2.98793 q^{65} -7.27492 q^{67} -3.88273 q^{71} -10.2749 q^{73} +0.418627 q^{77} -11.2749 q^{79} -9.55505 q^{83} -6.27492 q^{85} +2.98793 q^{89} -2.27492 q^{91} +11.2296 q^{95} -0.549834 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} - 6 q^{13} - 4 q^{19} + 2 q^{25} - 24 q^{31} - 16 q^{37} - 10 q^{43} + 4 q^{49} - 28 q^{55} - 30 q^{61} - 14 q^{67} - 26 q^{73} - 30 q^{79} - 10 q^{85} + 6 q^{91} + 28 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 6 * q^13 - 4 * q^19 + 2 * q^25 - 24 * q^31 - 16 * q^37 - 10 * q^43 + 4 * q^49 - 28 * q^55 - 30 * q^61 - 14 * q^67 - 26 * q^73 - 30 * q^79 - 10 * q^85 + 6 * q^91 + 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more