LMFDB - Embedded newform 6760.2.a.bi.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6760,2,Mod(1,6760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6760, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6760 = 2^{3} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6760.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:	$53.9788717664$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.2249737.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 9x^{3} + 14x^{2} - 8x - 8 $ x^6 - 2x^5 - 7x^4 + 9x^3 + 14x^2 - 8*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.682674$ of defining polynomial
Character	$\chi$	$=$	6760.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.12772 q^{3} +1.00000 q^{5} -0.445042 q^{7} -1.72826 q^{9} +O(q^{10})$	$q+1.12772 q^{3} +1.00000 q^{5} -0.445042 q^{7} -1.72826 q^{9} -4.23347 q^{11} +1.12772 q^{15} +3.27905 q^{17} +2.88031 q^{19} -0.501881 q^{21} +3.27436 q^{23} +1.00000 q^{25} -5.33213 q^{27} +1.52646 q^{29} -6.43956 q^{31} -4.77416 q^{33} -0.445042 q^{35} -5.00638 q^{37} -4.97595 q^{41} +2.24322 q^{43} -1.72826 q^{45} +3.97316 q^{47} -6.80194 q^{49} +3.69784 q^{51} -12.7301 q^{53} -4.23347 q^{55} +3.24817 q^{57} +11.9756 q^{59} +2.75729 q^{61} +0.769146 q^{63} -0.107348 q^{67} +3.69255 q^{69} -0.137887 q^{71} -5.01314 q^{73} +1.12772 q^{75} +1.88407 q^{77} +0.424756 q^{79} -0.828364 q^{81} -14.7764 q^{83} +3.27905 q^{85} +1.72141 q^{87} -2.91829 q^{89} -7.26200 q^{93} +2.88031 q^{95} -7.55962 q^{97} +7.31652 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 6 * q + 6 * q^5 - 2 * q^7 + 4 * q^9	$ 6 q + 6 q^{5} - 2 q^{7} + 4 q^{9} - 5 q^{11} - 9 q^{17} + 2 q^{19} - 7 q^{21} + 3 q^{23} + 6 q^{25} - 11 q^{29} - 5 q^{31} - 20 q^{33} - 2 q^{35} - 7 q^{37} - 27 q^{41} - 4 q^{43} + 4 q^{45} - 19 q^{47} - 32 q^{49} + 5 q^{51} - 6 q^{53} - 5 q^{55} + 6 q^{57} + 19 q^{59} - 4 q^{61} - 6 q^{63} + 10 q^{67} - 15 q^{69} - 15 q^{71} - 14 q^{73} + 4 q^{77} + 7 q^{79} + 2 q^{81} + 12 q^{83} - 9 q^{85} + 21 q^{87} - 34 q^{89} - 55 q^{93} + 2 q^{95} + 2 q^{97} - 29 q^{99}+O(q^{100}) $ 6 * q + 6 * q^5 - 2 * q^7 + 4 * q^9 - 5 * q^11 - 9 * q^17 + 2 * q^19 - 7 * q^21 + 3 * q^23 + 6 * q^25 - 11 * q^29 - 5 * q^31 - 20 * q^33 - 2 * q^35 - 7 * q^37 - 27 * q^41 - 4 * q^43 + 4 * q^45 - 19 * q^47 - 32 * q^49 + 5 * q^51 - 6 * q^53 - 5 * q^55 + 6 * q^57 + 19 * q^59 - 4 * q^61 - 6 * q^63 + 10 * q^67 - 15 * q^69 - 15 * q^71 - 14 * q^73 + 4 * q^77 + 7 * q^79 + 2 * q^81 + 12 * q^83 - 9 * q^85 + 21 * q^87 - 34 * q^89 - 55 * q^93 + 2 * q^95 + 2 * q^97 - 29 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.12772	0.651087	0.325544	−	0.945527i	$-0.394453\pi$
$3$	1.12772	0.651087	0.325544	+	0.945527i	$0.394453\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.445042	−0.168210	−0.0841050	−	0.996457i	$-0.526803\pi$
$7$	−0.445042	−0.168210	−0.0841050	+	0.996457i	$0.526803\pi$
$8$	0	0
$9$	−1.72826	−0.576085
$10$	0	0
$11$	−4.23347	−1.27644	−0.638220	−	0.769854i	$-0.720329\pi$
$11$	−4.23347	−1.27644	−0.638220	+	0.769854i	$0.720329\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.12772	0.291175
$16$	0	0
$17$	3.27905	0.795287	0.397644	−	0.917540i	$-0.369828\pi$
$17$	3.27905	0.795287	0.397644	+	0.917540i	$0.369828\pi$
$18$	0	0
$19$	2.88031	0.660789	0.330394	−	0.943843i	$-0.392818\pi$
$19$	2.88031	0.660789	0.330394	+	0.943843i	$0.392818\pi$
$20$	0	0
$21$	−0.501881	−0.109519
$22$	0	0
$23$	3.27436	0.682752	0.341376	−	0.939927i	$-0.389107\pi$
$23$	3.27436	0.682752	0.341376	+	0.939927i	$0.389107\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.33213	−1.02617
$28$	0	0
$29$	1.52646	0.283456	0.141728	−	0.989906i	$-0.454734\pi$
$29$	1.52646	0.283456	0.141728	+	0.989906i	$0.454734\pi$
$30$	0	0
$31$	−6.43956	−1.15658	−0.578290	−	0.815831i	$-0.696280\pi$
$31$	−6.43956	−1.15658	−0.578290	+	0.815831i	$0.696280\pi$
$32$	0	0
$33$	−4.77416	−0.831074
$34$	0	0
$35$	−0.445042	−0.0752258
$36$	0	0
$37$	−5.00638	−0.823044	−0.411522	−	0.911400i	$-0.635003\pi$
$37$	−5.00638	−0.823044	−0.411522	+	0.911400i	$0.635003\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.97595	−0.777113	−0.388557	−	0.921425i	$-0.627026\pi$
$41$	−4.97595	−0.777113	−0.388557	+	0.921425i	$0.627026\pi$
$42$	0	0
$43$	2.24322	0.342087	0.171044	−	0.985263i	$-0.445286\pi$
$43$	2.24322	0.342087	0.171044	+	0.985263i	$0.445286\pi$
$44$	0	0
$45$	−1.72826	−0.257633
$46$	0	0
$47$	3.97316	0.579545	0.289773	−	0.957095i	$-0.406420\pi$
$47$	3.97316	0.579545	0.289773	+	0.957095i	$0.406420\pi$
$48$	0	0
$49$	−6.80194	−0.971705
$50$	0	0
$51$	3.69784	0.517802
$52$	0	0
$53$	−12.7301	−1.74861	−0.874303	−	0.485380i	$-0.838681\pi$
$53$	−12.7301	−1.74861	−0.874303	+	0.485380i	$0.838681\pi$
$54$	0	0
$55$	−4.23347	−0.570841
$56$	0	0
$57$	3.24817	0.430231
$58$	0	0
$59$	11.9756	1.55909	0.779546	−	0.626346i	$-0.215450\pi$
$59$	11.9756	1.55909	0.779546	+	0.626346i	$0.215450\pi$
$60$	0	0
$61$	2.75729	0.353034	0.176517	−	0.984298i	$-0.443517\pi$
$61$	2.75729	0.353034	0.176517	+	0.984298i	$0.443517\pi$
$62$	0	0
$63$	0.769146	0.0969033
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.107348	−0.0131147	−0.00655733	−	0.999979i	$-0.502087\pi$
$67$	−0.107348	−0.0131147	−0.00655733	+	0.999979i	$0.502087\pi$
$68$	0	0
$69$	3.69255	0.444531
$70$	0	0
$71$	−0.137887	−0.0163642	−0.00818210	−	0.999967i	$-0.502604\pi$
$71$	−0.137887	−0.0163642	−0.00818210	+	0.999967i	$0.502604\pi$
$72$	0	0
$73$	−5.01314	−0.586744	−0.293372	−	0.955998i	$-0.594777\pi$
$73$	−5.01314	−0.586744	−0.293372	+	0.955998i	$0.594777\pi$
$74$	0	0
$75$	1.12772	0.130217
$76$	0	0
$77$	1.88407	0.214710
$78$	0	0
$79$	0.424756	0.0477888	0.0238944	−	0.999714i	$-0.492393\pi$
$79$	0.424756	0.0477888	0.0238944	+	0.999714i	$0.492393\pi$
$80$	0	0
$81$	−0.828364	−0.0920404
$82$	0	0
$83$	−14.7764	−1.62192	−0.810960	−	0.585101i	$-0.801055\pi$
$83$	−14.7764	−1.62192	−0.810960	+	0.585101i	$0.801055\pi$
$84$	0	0
$85$	3.27905	0.355663
$86$	0	0
$87$	1.72141	0.184555
$88$	0	0
$89$	−2.91829	−0.309338	−0.154669	−	0.987966i	$-0.549431\pi$
$89$	−2.91829	−0.309338	−0.154669	+	0.987966i	$0.549431\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.26200	−0.753034
$94$	0	0
$95$	2.88031	0.295514
$96$	0	0
$97$	−7.55962	−0.767563	−0.383782	−	0.923424i	$-0.625378\pi$
$97$	−7.55962	−0.767563	−0.383782	+	0.923424i	$0.625378\pi$
$98$	0	0
$99$	7.31652	0.735338
$100$	0	0
$101$	13.1532	1.30879	0.654396	−	0.756152i	$-0.272923\pi$
$101$	13.1532	1.30879	0.654396	+	0.756152i	$0.272923\pi$
$102$	0	0
$103$	−7.74587	−0.763223	−0.381612	−	0.924323i	$-0.624631\pi$
$103$	−7.74587	−0.763223	−0.381612	+	0.924323i	$0.624631\pi$
$104$	0	0
$105$	−0.501881	−0.0489786
$106$	0	0
$107$	−1.06949	−0.103392	−0.0516959	−	0.998663i	$-0.516463\pi$
$107$	−1.06949	−0.103392	−0.0516959	+	0.998663i	$0.516463\pi$
$108$	0	0
$109$	−14.1955	−1.35969	−0.679843	−	0.733358i	$-0.737952\pi$
$109$	−14.1955	−1.35969	−0.679843	+	0.733358i	$0.737952\pi$
$110$	0	0
$111$	−5.64578	−0.535873
$112$	0	0
$113$	9.02624	0.849117	0.424559	−	0.905400i	$-0.360429\pi$
$113$	9.02624	0.849117	0.424559	+	0.905400i	$0.360429\pi$
$114$	0	0
$115$	3.27436	0.305336
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.45932	−0.133775
$120$	0	0
$121$	6.92229	0.629299
$122$	0	0
$123$	−5.61146	−0.505969
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.40584	0.568427	0.284213	−	0.958761i	$-0.408268\pi$
$127$	6.40584	0.568427	0.284213	+	0.958761i	$0.408268\pi$
$128$	0	0
$129$	2.52971	0.222729
$130$	0	0
$131$	−9.76604	−0.853263	−0.426632	−	0.904425i	$-0.640300\pi$
$131$	−9.76604	−0.853263	−0.426632	+	0.904425i	$0.640300\pi$
$132$	0	0
$133$	−1.28186	−0.111151
$134$	0	0
$135$	−5.33213	−0.458917
$136$	0	0
$137$	−9.86366	−0.842709	−0.421355	−	0.906896i	$-0.638445\pi$
$137$	−9.86366	−0.842709	−0.421355	+	0.906896i	$0.638445\pi$
$138$	0	0
$139$	0.276643	0.0234646	0.0117323	−	0.999931i	$-0.496265\pi$
$139$	0.276643	0.0234646	0.0117323	+	0.999931i	$0.496265\pi$
$140$	0	0
$141$	4.48060	0.377335
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.52646	0.126766
$146$	0	0
$147$	−7.67066	−0.632665
$148$	0	0
$149$	−19.2203	−1.57459	−0.787295	−	0.616577i	$-0.788519\pi$
$149$	−19.2203	−1.57459	−0.787295	+	0.616577i	$0.788519\pi$
$150$	0	0
$151$	−11.0473	−0.899018	−0.449509	−	0.893276i	$-0.648401\pi$
$151$	−11.0473	−0.899018	−0.449509	+	0.893276i	$0.648401\pi$
$152$	0	0
$153$	−5.66704	−0.458153
$154$	0	0
$155$	−6.43956	−0.517238
$156$	0	0
$157$	−13.6733	−1.09125	−0.545623	−	0.838030i	$-0.683707\pi$
$157$	−13.6733	−1.09125	−0.545623	+	0.838030i	$0.683707\pi$
$158$	0	0
$159$	−14.3559	−1.13850
$160$	0	0
$161$	−1.45723	−0.114846
$162$	0	0
$163$	−11.9563	−0.936491	−0.468245	−	0.883598i	$-0.655114\pi$
$163$	−11.9563	−0.936491	−0.468245	+	0.883598i	$0.655114\pi$
$164$	0	0
$165$	−4.77416	−0.371668
$166$	0	0
$167$	−7.20151	−0.557269	−0.278635	−	0.960397i	$-0.589882\pi$
$167$	−7.20151	−0.557269	−0.278635	+	0.960397i	$0.589882\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−4.97791	−0.380671
$172$	0	0
$173$	−23.0278	−1.75077	−0.875384	−	0.483428i	$-0.839391\pi$
$173$	−23.0278	−1.75077	−0.875384	+	0.483428i	$0.839391\pi$
$174$	0	0
$175$	−0.445042	−0.0336420
$176$	0	0
$177$	13.5051	1.01510
$178$	0	0
$179$	−14.4960	−1.08348	−0.541739	−	0.840547i	$-0.682234\pi$
$179$	−14.4960	−1.08348	−0.541739	+	0.840547i	$0.682234\pi$
$180$	0	0
$181$	7.06545	0.525171	0.262585	−	0.964909i	$-0.415425\pi$
$181$	7.06545	0.525171	0.262585	+	0.964909i	$0.415425\pi$
$182$	0	0
$183$	3.10944	0.229856
$184$	0	0
$185$	−5.00638	−0.368076
$186$	0	0
$187$	−13.8818	−1.01514
$188$	0	0
$189$	2.37302	0.172612
$190$	0	0
$191$	2.74305	0.198480	0.0992402	−	0.995064i	$-0.468359\pi$
$191$	2.74305	0.198480	0.0992402	+	0.995064i	$0.468359\pi$
$192$	0	0
$193$	0.822416	0.0591988	0.0295994	−	0.999562i	$-0.490577\pi$
$193$	0.822416	0.0591988	0.0295994	+	0.999562i	$0.490577\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.6237	−1.46938	−0.734689	−	0.678405i	$-0.762672\pi$
$197$	−20.6237	−1.46938	−0.734689	+	0.678405i	$0.762672\pi$
$198$	0	0
$199$	24.7909	1.75738	0.878691	−	0.477390i	$-0.158417\pi$
$199$	24.7909	1.75738	0.878691	+	0.477390i	$0.158417\pi$
$200$	0	0
$201$	−0.121058	−0.00853879
$202$	0	0
$203$	−0.679339	−0.0476802
$204$	0	0
$205$	−4.97595	−0.347536
$206$	0	0
$207$	−5.65894	−0.393323
$208$	0	0
$209$	−12.1937	−0.843457
$210$	0	0
$211$	−8.37111	−0.576291	−0.288145	−	0.957587i	$-0.593039\pi$
$211$	−8.37111	−0.576291	−0.288145	+	0.957587i	$0.593039\pi$
$212$	0	0
$213$	−0.155498	−0.0106545
$214$	0	0
$215$	2.24322	0.152986
$216$	0	0
$217$	2.86587	0.194548
$218$	0	0
$219$	−5.65340	−0.382022
$220$	0	0
$221$	0	0
$222$	0	0
$223$	11.8154	0.791217	0.395609	−	0.918419i	$-0.370534\pi$
$223$	11.8154	0.791217	0.395609	+	0.918419i	$0.370534\pi$
$224$	0	0
$225$	−1.72826	−0.115217
$226$	0	0
$227$	9.96864	0.661642	0.330821	−	0.943694i	$-0.392674\pi$
$227$	9.96864	0.661642	0.330821	+	0.943694i	$0.392674\pi$
$228$	0	0
$229$	17.7698	1.17426	0.587130	−	0.809493i	$-0.300258\pi$
$229$	17.7698	1.17426	0.587130	+	0.809493i	$0.300258\pi$
$230$	0	0
$231$	2.12470	0.139795
$232$	0	0
$233$	19.5187	1.27871	0.639355	−	0.768912i	$-0.279202\pi$
$233$	19.5187	1.27871	0.639355	+	0.768912i	$0.279202\pi$
$234$	0	0
$235$	3.97316	0.259181
$236$	0	0
$237$	0.479005	0.0311147
$238$	0	0
$239$	−20.4403	−1.32217	−0.661085	−	0.750311i	$-0.729904\pi$
$239$	−20.4403	−1.32217	−0.661085	+	0.750311i	$0.729904\pi$
$240$	0	0
$241$	−22.2745	−1.43483	−0.717413	−	0.696648i	$-0.754674\pi$
$241$	−22.2745	−1.43483	−0.717413	+	0.696648i	$0.754674\pi$
$242$	0	0
$243$	15.0622	0.966243
$244$	0	0
$245$	−6.80194	−0.434560
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−16.6636	−1.05601
$250$	0	0
$251$	28.8067	1.81826	0.909130	−	0.416512i	$-0.136748\pi$
$251$	28.8067	1.81826	0.909130	+	0.416512i	$0.136748\pi$
$252$	0	0
$253$	−13.8619	−0.871492
$254$	0	0
$255$	3.69784	0.231568
$256$	0	0
$257$	−13.7041	−0.854838	−0.427419	−	0.904054i	$-0.640577\pi$
$257$	−13.7041	−0.854838	−0.427419	+	0.904054i	$0.640577\pi$
$258$	0	0
$259$	2.22805	0.138444
$260$	0	0
$261$	−2.63811	−0.163295
$262$	0	0
$263$	0.469354	0.0289416	0.0144708	−	0.999895i	$-0.495394\pi$
$263$	0.469354	0.0289416	0.0144708	+	0.999895i	$0.495394\pi$
$264$	0	0
$265$	−12.7301	−0.782001
$266$	0	0
$267$	−3.29100	−0.201406
$268$	0	0
$269$	0.494786	0.0301677	0.0150838	−	0.999886i	$-0.495198\pi$
$269$	0.494786	0.0301677	0.0150838	+	0.999886i	$0.495198\pi$
$270$	0	0
$271$	−24.0033	−1.45809	−0.729047	−	0.684463i	$-0.760037\pi$
$271$	−24.0033	−1.45809	−0.729047	+	0.684463i	$0.760037\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.23347	−0.255288
$276$	0	0
$277$	27.6177	1.65939	0.829695	−	0.558217i	$-0.188514\pi$
$277$	27.6177	1.65939	0.829695	+	0.558217i	$0.188514\pi$
$278$	0	0
$279$	11.1292	0.666288
$280$	0	0
$281$	28.0115	1.67103	0.835513	−	0.549470i	$-0.185170\pi$
$281$	28.0115	1.67103	0.835513	+	0.549470i	$0.185170\pi$
$282$	0	0
$283$	24.0868	1.43181	0.715907	−	0.698196i	$-0.246014\pi$
$283$	24.0868	1.43181	0.715907	+	0.698196i	$0.246014\pi$
$284$	0	0
$285$	3.24817	0.192405
$286$	0	0
$287$	2.21451	0.130718
$288$	0	0
$289$	−6.24780	−0.367518
$290$	0	0
$291$	−8.52511	−0.499751
$292$	0	0
$293$	−17.6465	−1.03092	−0.515458	−	0.856915i	$-0.672378\pi$
$293$	−17.6465	−1.03092	−0.515458	+	0.856915i	$0.672378\pi$
$294$	0	0
$295$	11.9756	0.697247
$296$	0	0
$297$	22.5734	1.30984
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−0.998326	−0.0575425
$302$	0	0
$303$	14.8331	0.852138
$304$	0	0
$305$	2.75729	0.157882
$306$	0	0
$307$	−10.1087	−0.576932	−0.288466	−	0.957490i	$-0.593145\pi$
$307$	−10.1087	−0.576932	−0.288466	+	0.957490i	$0.593145\pi$
$308$	0	0
$309$	−8.73514	−0.496925
$310$	0	0
$311$	3.36459	0.190788	0.0953941	−	0.995440i	$-0.469589\pi$
$311$	3.36459	0.190788	0.0953941	+	0.995440i	$0.469589\pi$
$312$	0	0
$313$	−16.5757	−0.936914	−0.468457	−	0.883486i	$-0.655190\pi$
$313$	−16.5757	−0.936914	−0.468457	+	0.883486i	$0.655190\pi$
$314$	0	0
$315$	0.769146	0.0433365
$316$	0	0
$317$	0.733244	0.0411831	0.0205915	−	0.999788i	$-0.493445\pi$
$317$	0.733244	0.0411831	0.0205915	+	0.999788i	$0.493445\pi$
$318$	0	0
$319$	−6.46223	−0.361815
$320$	0	0
$321$	−1.20609	−0.0673171
$322$	0	0
$323$	9.44469	0.525517
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−16.0085	−0.885274
$328$	0	0
$329$	−1.76822	−0.0974853
$330$	0	0
$331$	27.8189	1.52906	0.764532	−	0.644586i	$-0.222970\pi$
$331$	27.8189	1.52906	0.764532	+	0.644586i	$0.222970\pi$
$332$	0	0
$333$	8.65231	0.474144
$334$	0	0
$335$	−0.107348	−0.00586505
$336$	0	0
$337$	−4.44917	−0.242362	−0.121181	−	0.992630i	$-0.538668\pi$
$337$	−4.44917	−0.242362	−0.121181	+	0.992630i	$0.538668\pi$
$338$	0	0
$339$	10.1790	0.552850
$340$	0	0
$341$	27.2617	1.47630
$342$	0	0
$343$	6.14244	0.331661
$344$	0	0
$345$	3.69255	0.198800
$346$	0	0
$347$	−4.64234	−0.249214	−0.124607	−	0.992206i	$-0.539767\pi$
$347$	−4.64234	−0.249214	−0.124607	+	0.992206i	$0.539767\pi$
$348$	0	0
$349$	−1.68938	−0.0904306	−0.0452153	−	0.998977i	$-0.514397\pi$
$349$	−1.68938	−0.0904306	−0.0452153	+	0.998977i	$0.514397\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.3045	0.548452	0.274226	−	0.961665i	$-0.411578\pi$
$353$	10.3045	0.548452	0.274226	+	0.961665i	$0.411578\pi$
$354$	0	0
$355$	−0.137887	−0.00731829
$356$	0	0
$357$	−1.64569	−0.0870994
$358$	0	0
$359$	9.90212	0.522614	0.261307	−	0.965256i	$-0.415846\pi$
$359$	9.90212	0.522614	0.261307	+	0.965256i	$0.415846\pi$
$360$	0	0
$361$	−10.7038	−0.563358
$362$	0	0
$363$	7.80638	0.409729
$364$	0	0
$365$	−5.01314	−0.262400
$366$	0	0
$367$	−3.56821	−0.186259	−0.0931296	−	0.995654i	$-0.529687\pi$
$367$	−3.56821	−0.186259	−0.0931296	+	0.995654i	$0.529687\pi$
$368$	0	0
$369$	8.59972	0.447683
$370$	0	0
$371$	5.66541	0.294133
$372$	0	0
$373$	28.0358	1.45164	0.725819	−	0.687886i	$-0.241461\pi$
$373$	28.0358	1.45164	0.725819	+	0.687886i	$0.241461\pi$
$374$	0	0
$375$	1.12772	0.0582350
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−9.51700	−0.488855	−0.244428	−	0.969668i	$-0.578600\pi$
$379$	−9.51700	−0.488855	−0.244428	+	0.969668i	$0.578600\pi$
$380$	0	0
$381$	7.22397	0.370095
$382$	0	0
$383$	−23.1731	−1.18409	−0.592046	−	0.805904i	$-0.701680\pi$
$383$	−23.1731	−1.18409	−0.592046	+	0.805904i	$0.701680\pi$
$384$	0	0
$385$	1.88407	0.0960212
$386$	0	0
$387$	−3.87685	−0.197072
$388$	0	0
$389$	−28.3216	−1.43596	−0.717981	−	0.696063i	$-0.754933\pi$
$389$	−28.3216	−1.43596	−0.717981	+	0.696063i	$0.754933\pi$
$390$	0	0
$391$	10.7368	0.542984
$392$	0	0
$393$	−11.0133	−0.555549
$394$	0	0
$395$	0.424756	0.0213718
$396$	0	0
$397$	3.92556	0.197018	0.0985090	−	0.995136i	$-0.468593\pi$
$397$	3.92556	0.197018	0.0985090	+	0.995136i	$0.468593\pi$
$398$	0	0
$399$	−1.44557	−0.0723692
$400$	0	0
$401$	11.9282	0.595664	0.297832	−	0.954618i	$-0.403737\pi$
$401$	11.9282	0.595664	0.297832	+	0.954618i	$0.403737\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−0.828364	−0.0411617
$406$	0	0
$407$	21.1944	1.05057
$408$	0	0
$409$	32.5690	1.61043	0.805216	−	0.592982i	$-0.202049\pi$
$409$	32.5690	1.61043	0.805216	+	0.592982i	$0.202049\pi$
$410$	0	0
$411$	−11.1234	−0.548677
$412$	0	0
$413$	−5.32965	−0.262255
$414$	0	0
$415$	−14.7764	−0.725345
$416$	0	0
$417$	0.311975	0.0152775
$418$	0	0
$419$	−13.9798	−0.682958	−0.341479	−	0.939889i	$-0.610928\pi$
$419$	−13.9798	−0.682958	−0.341479	+	0.939889i	$0.610928\pi$
$420$	0	0
$421$	12.1344	0.591394	0.295697	−	0.955282i	$-0.404448\pi$
$421$	12.1344	0.591394	0.295697	+	0.955282i	$0.404448\pi$
$422$	0	0
$423$	−6.86664	−0.333868
$424$	0	0
$425$	3.27905	0.159057
$426$	0	0
$427$	−1.22711	−0.0593839
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.14576	0.344199	0.172100	−	0.985080i	$-0.444945\pi$
$431$	7.14576	0.344199	0.172100	+	0.985080i	$0.444945\pi$
$432$	0	0
$433$	−8.31695	−0.399687	−0.199844	−	0.979828i	$-0.564043\pi$
$433$	−8.31695	−0.399687	−0.199844	+	0.979828i	$0.564043\pi$
$434$	0	0
$435$	1.72141	0.0825355
$436$	0	0
$437$	9.43118	0.451155
$438$	0	0
$439$	7.62324	0.363837	0.181919	−	0.983314i	$-0.441769\pi$
$439$	7.62324	0.363837	0.181919	+	0.983314i	$0.441769\pi$
$440$	0	0
$441$	11.7555	0.559785
$442$	0	0
$443$	−26.0801	−1.23910	−0.619550	−	0.784957i	$-0.712685\pi$
$443$	−26.0801	−1.23910	−0.619550	+	0.784957i	$0.712685\pi$
$444$	0	0
$445$	−2.91829	−0.138340
$446$	0	0
$447$	−21.6751	−1.02520
$448$	0	0
$449$	−20.1054	−0.948833	−0.474417	−	0.880300i	$-0.657341\pi$
$449$	−20.1054	−0.948833	−0.474417	+	0.880300i	$0.657341\pi$
$450$	0	0
$451$	21.0656	0.991938
$452$	0	0
$453$	−12.4582	−0.585339
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.27246	0.199857	0.0999287	−	0.994995i	$-0.468139\pi$
$457$	4.27246	0.199857	0.0999287	+	0.994995i	$0.468139\pi$
$458$	0	0
$459$	−17.4843	−0.816099
$460$	0	0
$461$	−7.04966	−0.328336	−0.164168	−	0.986432i	$-0.552494\pi$
$461$	−7.04966	−0.328336	−0.164168	+	0.986432i	$0.552494\pi$
$462$	0	0
$463$	22.2332	1.03326	0.516631	−	0.856208i	$-0.327186\pi$
$463$	22.2332	1.03326	0.516631	+	0.856208i	$0.327186\pi$
$464$	0	0
$465$	−7.26200	−0.336767
$466$	0	0
$467$	33.5593	1.55294	0.776469	−	0.630156i	$-0.217009\pi$
$467$	33.5593	1.55294	0.776469	+	0.630156i	$0.217009\pi$
$468$	0	0
$469$	0.0477744	0.00220602
$470$	0	0
$471$	−15.4196	−0.710497
$472$	0	0
$473$	−9.49660	−0.436654
$474$	0	0
$475$	2.88031	0.132158
$476$	0	0
$477$	22.0008	1.00735
$478$	0	0
$479$	−9.80115	−0.447826	−0.223913	−	0.974609i	$-0.571883\pi$
$479$	−9.80115	−0.447826	−0.223913	+	0.974609i	$0.571883\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−1.64334	−0.0747746
$484$	0	0
$485$	−7.55962	−0.343265
$486$	0	0
$487$	18.6166	0.843598	0.421799	−	0.906689i	$-0.361399\pi$
$487$	18.6166	0.843598	0.421799	+	0.906689i	$0.361399\pi$
$488$	0	0
$489$	−13.4833	−0.609737
$490$	0	0
$491$	10.8926	0.491578	0.245789	−	0.969323i	$-0.420953\pi$
$491$	10.8926	0.491578	0.245789	+	0.969323i	$0.420953\pi$
$492$	0	0
$493$	5.00534	0.225429
$494$	0	0
$495$	7.31652	0.328853
$496$	0	0
$497$	0.0613656	0.00275262
$498$	0	0
$499$	−37.1486	−1.66300	−0.831501	−	0.555523i	$-0.812518\pi$
$499$	−37.1486	−1.66300	−0.831501	+	0.555523i	$0.812518\pi$
$500$	0	0
$501$	−8.12125	−0.362831
$502$	0	0
$503$	24.5502	1.09464	0.547319	−	0.836924i	$-0.315648\pi$
$503$	24.5502	1.09464	0.547319	+	0.836924i	$0.315648\pi$
$504$	0	0
$505$	13.1532	0.585310
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.3406	−1.12320	−0.561601	−	0.827409i	$-0.689814\pi$
$509$	−25.3406	−1.12320	−0.561601	+	0.827409i	$0.689814\pi$
$510$	0	0
$511$	2.23106	0.0986962
$512$	0	0
$513$	−15.3582	−0.678081
$514$	0	0
$515$	−7.74587	−0.341324
$516$	0	0
$517$	−16.8203	−0.739755
$518$	0	0
$519$	−25.9688	−1.13990
$520$	0	0
$521$	−44.8766	−1.96608	−0.983039	−	0.183395i	$-0.941291\pi$
$521$	−44.8766	−1.96608	−0.983039	+	0.183395i	$0.941291\pi$
$522$	0	0
$523$	−41.1637	−1.79996	−0.899981	−	0.435929i	$-0.856420\pi$
$523$	−41.1637	−1.79996	−0.899981	+	0.435929i	$0.856420\pi$
$524$	0	0
$525$	−0.501881	−0.0219039
$526$	0	0
$527$	−21.1157	−0.919813
$528$	0	0
$529$	−12.2785	−0.533850
$530$	0	0
$531$	−20.6969	−0.898169
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1.06949	−0.0462383
$536$	0	0
$537$	−16.3473	−0.705439
$538$	0	0
$539$	28.7958	1.24032
$540$	0	0
$541$	29.2726	1.25853	0.629264	−	0.777191i	$-0.283356\pi$
$541$	29.2726	1.25853	0.629264	+	0.777191i	$0.283356\pi$
$542$	0	0
$543$	7.96782	0.341932
$544$	0	0
$545$	−14.1955	−0.608070
$546$	0	0
$547$	−1.91307	−0.0817971	−0.0408986	−	0.999163i	$-0.513022\pi$
$547$	−1.91307	−0.0817971	−0.0408986	+	0.999163i	$0.513022\pi$
$548$	0	0
$549$	−4.76529	−0.203378
$550$	0	0
$551$	4.39668	0.187305
$552$	0	0
$553$	−0.189034	−0.00803856
$554$	0	0
$555$	−5.64578	−0.239650
$556$	0	0
$557$	−28.9476	−1.22655	−0.613274	−	0.789870i	$-0.710148\pi$
$557$	−28.9476	−1.22655	−0.613274	+	0.789870i	$0.710148\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−15.6547	−0.660943
$562$	0	0
$563$	5.41245	0.228107	0.114054	−	0.993475i	$-0.463616\pi$
$563$	5.41245	0.228107	0.114054	+	0.993475i	$0.463616\pi$
$564$	0	0
$565$	9.02624	0.379737
$566$	0	0
$567$	0.368657	0.0154821
$568$	0	0
$569$	−22.8672	−0.958643	−0.479322	−	0.877639i	$-0.659117\pi$
$569$	−22.8672	−0.958643	−0.479322	+	0.877639i	$0.659117\pi$
$570$	0	0
$571$	29.0092	1.21400	0.606999	−	0.794702i	$-0.292373\pi$
$571$	29.0092	1.21400	0.606999	+	0.794702i	$0.292373\pi$
$572$	0	0
$573$	3.09339	0.129228
$574$	0	0
$575$	3.27436	0.136550
$576$	0	0
$577$	20.5207	0.854287	0.427144	−	0.904184i	$-0.359520\pi$
$577$	20.5207	0.854287	0.427144	+	0.904184i	$0.359520\pi$
$578$	0	0
$579$	0.927452	0.0385436
$580$	0	0
$581$	6.57612	0.272823
$582$	0	0
$583$	53.8923	2.23199
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.6088	1.42846	0.714230	−	0.699911i	$-0.246777\pi$
$587$	34.6088	1.42846	0.714230	+	0.699911i	$0.246777\pi$
$588$	0	0
$589$	−18.5479	−0.764255
$590$	0	0
$591$	−23.2577	−0.956693
$592$	0	0
$593$	−9.37916	−0.385156	−0.192578	−	0.981282i	$-0.561685\pi$
$593$	−9.37916	−0.385156	−0.192578	+	0.981282i	$0.561685\pi$
$594$	0	0
$595$	−1.45932	−0.0598261
$596$	0	0
$597$	27.9571	1.14421
$598$	0	0
$599$	41.7563	1.70612	0.853059	−	0.521815i	$-0.174745\pi$
$599$	41.7563	1.70612	0.853059	+	0.521815i	$0.174745\pi$
$600$	0	0
$601$	−14.2042	−0.579402	−0.289701	−	0.957117i	$-0.593556\pi$
$601$	−14.2042	−0.579402	−0.289701	+	0.957117i	$0.593556\pi$
$602$	0	0
$603$	0.185525	0.00755516
$604$	0	0
$605$	6.92229	0.281431
$606$	0	0
$607$	13.5378	0.549482	0.274741	−	0.961518i	$-0.411408\pi$
$607$	13.5378	0.549482	0.274741	+	0.961518i	$0.411408\pi$
$608$	0	0
$609$	−0.766101	−0.0310440
$610$	0	0
$611$	0	0
$612$	0	0
$613$	26.9171	1.08717	0.543586	−	0.839354i	$-0.317066\pi$
$613$	26.9171	1.08717	0.543586	+	0.839354i	$0.317066\pi$
$614$	0	0
$615$	−5.61146	−0.226276
$616$	0	0
$617$	−38.5309	−1.55120	−0.775599	−	0.631226i	$-0.782552\pi$
$617$	−38.5309	−1.55120	−0.775599	+	0.631226i	$0.782552\pi$
$618$	0	0
$619$	−7.45920	−0.299810	−0.149905	−	0.988700i	$-0.547897\pi$
$619$	−7.45920	−0.299810	−0.149905	+	0.988700i	$0.547897\pi$
$620$	0	0
$621$	−17.4593	−0.700619
$622$	0	0
$623$	1.29876	0.0520338
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−13.7511	−0.549164
$628$	0	0
$629$	−16.4162	−0.654557
$630$	0	0
$631$	−30.9408	−1.23173	−0.615867	−	0.787850i	$-0.711194\pi$
$631$	−30.9408	−1.23173	−0.615867	+	0.787850i	$0.711194\pi$
$632$	0	0
$633$	−9.44024	−0.375216
$634$	0	0
$635$	6.40584	0.254208
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.238304	0.00942718
$640$	0	0
$641$	26.5790	1.04981	0.524904	−	0.851161i	$-0.324101\pi$
$641$	26.5790	1.04981	0.524904	+	0.851161i	$0.324101\pi$
$642$	0	0
$643$	4.57986	0.180612	0.0903060	−	0.995914i	$-0.471215\pi$
$643$	4.57986	0.180612	0.0903060	+	0.995914i	$0.471215\pi$
$644$	0	0
$645$	2.52971	0.0996074
$646$	0	0
$647$	35.7861	1.40690	0.703448	−	0.710747i	$-0.251643\pi$
$647$	35.7861	1.40690	0.703448	+	0.710747i	$0.251643\pi$
$648$	0	0
$649$	−50.6984	−1.99009
$650$	0	0
$651$	3.23189	0.126668
$652$	0	0
$653$	−14.0388	−0.549380	−0.274690	−	0.961533i	$-0.588575\pi$
$653$	−14.0388	−0.549380	−0.274690	+	0.961533i	$0.588575\pi$
$654$	0	0
$655$	−9.76604	−0.381591
$656$	0	0
$657$	8.66400	0.338015
$658$	0	0
$659$	−18.7603	−0.730798	−0.365399	−	0.930851i	$-0.619067\pi$
$659$	−18.7603	−0.730798	−0.365399	+	0.930851i	$0.619067\pi$
$660$	0	0
$661$	20.0370	0.779348	0.389674	−	0.920953i	$-0.372588\pi$
$661$	20.0370	0.779348	0.389674	+	0.920953i	$0.372588\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.28186	−0.0497084
$666$	0	0
$667$	4.99818	0.193530
$668$	0	0
$669$	13.3244	0.515151
$670$	0	0
$671$	−11.6729	−0.450627
$672$	0	0
$673$	−35.8527	−1.38202	−0.691010	−	0.722845i	$-0.742834\pi$
$673$	−35.8527	−1.38202	−0.691010	+	0.722845i	$0.742834\pi$
$674$	0	0
$675$	−5.33213	−0.205234
$676$	0	0
$677$	5.97403	0.229601	0.114800	−	0.993389i	$-0.463377\pi$
$677$	5.97403	0.229601	0.114800	+	0.993389i	$0.463377\pi$
$678$	0	0
$679$	3.36435	0.129112
$680$	0	0
$681$	11.2418	0.430787
$682$	0	0
$683$	29.9698	1.14676	0.573382	−	0.819288i	$-0.305631\pi$
$683$	29.9698	1.14676	0.573382	+	0.819288i	$0.305631\pi$
$684$	0	0
$685$	−9.86366	−0.376871
$686$	0	0
$687$	20.0393	0.764545
$688$	0	0
$689$	0	0
$690$	0	0
$691$	26.1604	0.995188	0.497594	−	0.867410i	$-0.334217\pi$
$691$	26.1604	0.995188	0.497594	+	0.867410i	$0.334217\pi$
$692$	0	0
$693$	−3.25616	−0.123691
$694$	0	0
$695$	0.276643	0.0104937
$696$	0	0
$697$	−16.3164	−0.618028
$698$	0	0
$699$	22.0115	0.832552
$700$	0	0
$701$	5.74566	0.217010	0.108505	−	0.994096i	$-0.465394\pi$
$701$	5.74566	0.217010	0.108505	+	0.994096i	$0.465394\pi$
$702$	0	0
$703$	−14.4199	−0.543858
$704$	0	0
$705$	4.48060	0.168749
$706$	0	0
$707$	−5.85372	−0.220152
$708$	0	0
$709$	25.4361	0.955271	0.477635	−	0.878558i	$-0.341494\pi$
$709$	25.4361	0.955271	0.477635	+	0.878558i	$0.341494\pi$
$710$	0	0
$711$	−0.734087	−0.0275304
$712$	0	0
$713$	−21.0855	−0.789657
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−23.0508	−0.860849
$718$	0	0
$719$	−20.7742	−0.774748	−0.387374	−	0.921923i	$-0.626618\pi$
$719$	−20.7742	−0.774748	−0.387374	+	0.921923i	$0.626618\pi$
$720$	0	0
$721$	3.44724	0.128382
$722$	0	0
$723$	−25.1193	−0.934197
$724$	0	0
$725$	1.52646	0.0566913
$726$	0	0
$727$	23.4532	0.869830	0.434915	−	0.900472i	$-0.356778\pi$
$727$	23.4532	0.869830	0.434915	+	0.900472i	$0.356778\pi$
$728$	0	0
$729$	19.4710	0.721149
$730$	0	0
$731$	7.35563	0.272058
$732$	0	0
$733$	12.9778	0.479347	0.239673	−	0.970854i	$-0.422960\pi$
$733$	12.9778	0.479347	0.239673	+	0.970854i	$0.422960\pi$
$734$	0	0
$735$	−7.67066	−0.282936
$736$	0	0
$737$	0.454455	0.0167401
$738$	0	0
$739$	−16.4369	−0.604642	−0.302321	−	0.953206i	$-0.597762\pi$
$739$	−16.4369	−0.604642	−0.302321	+	0.953206i	$0.597762\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	30.8464	1.13165	0.565823	−	0.824527i	$-0.308558\pi$
$743$	30.8464	1.13165	0.565823	+	0.824527i	$0.308558\pi$
$744$	0	0
$745$	−19.2203	−0.704178
$746$	0	0
$747$	25.5374	0.934365
$748$	0	0
$749$	0.475970	0.0173915
$750$	0	0
$751$	−27.9011	−1.01812	−0.509062	−	0.860730i	$-0.670008\pi$
$751$	−27.9011	−1.01812	−0.509062	+	0.860730i	$0.670008\pi$
$752$	0	0
$753$	32.4857	1.18385
$754$	0	0
$755$	−11.0473	−0.402053
$756$	0	0
$757$	15.0303	0.546287	0.273143	−	0.961973i	$-0.411937\pi$
$757$	15.0303	0.546287	0.273143	+	0.961973i	$0.411937\pi$
$758$	0	0
$759$	−15.6323	−0.567417
$760$	0	0
$761$	20.6415	0.748255	0.374127	−	0.927377i	$-0.377942\pi$
$761$	20.6415	0.748255	0.374127	+	0.927377i	$0.377942\pi$
$762$	0	0
$763$	6.31761	0.228713
$764$	0	0
$765$	−5.66704	−0.204892
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−51.0207	−1.83985	−0.919927	−	0.392090i	$-0.871752\pi$
$769$	−51.0207	−1.83985	−0.919927	+	0.392090i	$0.871752\pi$
$770$	0	0
$771$	−15.4543	−0.556574
$772$	0	0
$773$	−35.9282	−1.29225	−0.646124	−	0.763233i	$-0.723611\pi$
$773$	−35.9282	−1.29225	−0.646124	+	0.763233i	$0.723611\pi$
$774$	0	0
$775$	−6.43956	−0.231316
$776$	0	0
$777$	2.51261	0.0901393
$778$	0	0
$779$	−14.3323	−0.513508
$780$	0	0
$781$	0.583742	0.0208879
$782$	0	0
$783$	−8.13928	−0.290874
$784$	0	0
$785$	−13.6733	−0.488020
$786$	0	0
$787$	−1.91272	−0.0681812	−0.0340906	−	0.999419i	$-0.510853\pi$
$787$	−1.91272	−0.0681812	−0.0340906	+	0.999419i	$0.510853\pi$
$788$	0	0
$789$	0.529298	0.0188435
$790$	0	0
$791$	−4.01705	−0.142830
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−14.3559	−0.509151
$796$	0	0
$797$	2.65811	0.0941552	0.0470776	−	0.998891i	$-0.485009\pi$
$797$	2.65811	0.0941552	0.0470776	+	0.998891i	$0.485009\pi$
$798$	0	0
$799$	13.0282	0.460905
$800$	0	0
$801$	5.04355	0.178205
$802$	0	0
$803$	21.2230	0.748944
$804$	0	0
$805$	−1.45723	−0.0513606
$806$	0	0
$807$	0.557979	0.0196418
$808$	0	0
$809$	38.8246	1.36500	0.682499	−	0.730886i	$-0.260893\pi$
$809$	38.8246	1.36500	0.682499	+	0.730886i	$0.260893\pi$
$810$	0	0
$811$	36.5502	1.28345	0.641725	−	0.766935i	$-0.278219\pi$
$811$	36.5502	1.28345	0.641725	+	0.766935i	$0.278219\pi$
$812$	0	0
$813$	−27.0689	−0.949347
$814$	0	0
$815$	−11.9563	−0.418811
$816$	0	0
$817$	6.46116	0.226048
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−48.7846	−1.70260	−0.851298	−	0.524683i	$-0.824184\pi$
$821$	−48.7846	−1.70260	−0.851298	+	0.524683i	$0.824184\pi$
$822$	0	0
$823$	14.3284	0.499456	0.249728	−	0.968316i	$-0.419659\pi$
$823$	14.3284	0.499456	0.249728	+	0.968316i	$0.419659\pi$
$824$	0	0
$825$	−4.77416	−0.166215
$826$	0	0
$827$	4.32002	0.150222	0.0751109	−	0.997175i	$-0.476069\pi$
$827$	4.32002	0.150222	0.0751109	+	0.997175i	$0.476069\pi$
$828$	0	0
$829$	−28.1846	−0.978890	−0.489445	−	0.872034i	$-0.662801\pi$
$829$	−28.1846	−0.978890	−0.489445	+	0.872034i	$0.662801\pi$
$830$	0	0
$831$	31.1450	1.08041
$832$	0	0
$833$	−22.3039	−0.772785
$834$	0	0
$835$	−7.20151	−0.249218
$836$	0	0
$837$	34.3366	1.18685
$838$	0	0
$839$	44.0913	1.52220	0.761100	−	0.648635i	$-0.224660\pi$
$839$	44.0913	1.52220	0.761100	+	0.648635i	$0.224660\pi$
$840$	0	0
$841$	−26.6699	−0.919652
$842$	0	0
$843$	31.5890	1.08798
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.08071	−0.105854
$848$	0	0
$849$	27.1631	0.932236
$850$	0	0
$851$	−16.3927	−0.561935
$852$	0	0
$853$	12.0427	0.412333	0.206166	−	0.978517i	$-0.433901\pi$
$853$	12.0427	0.412333	0.206166	+	0.978517i	$0.433901\pi$
$854$	0	0
$855$	−4.97791	−0.170241
$856$	0	0
$857$	−55.6733	−1.90176	−0.950881	−	0.309555i	$-0.899820\pi$
$857$	−55.6733	−1.90176	−0.950881	+	0.309555i	$0.899820\pi$
$858$	0	0
$859$	−33.2261	−1.13366	−0.566830	−	0.823835i	$-0.691830\pi$
$859$	−33.2261	−1.13366	−0.566830	+	0.823835i	$0.691830\pi$
$860$	0	0
$861$	2.49734	0.0851090
$862$	0	0
$863$	31.9260	1.08678	0.543388	−	0.839482i	$-0.317141\pi$
$863$	31.9260	1.08678	0.543388	+	0.839482i	$0.317141\pi$
$864$	0	0
$865$	−23.0278	−0.782967
$866$	0	0
$867$	−7.04575	−0.239286
$868$	0	0
$869$	−1.79819	−0.0609996
$870$	0	0
$871$	0	0
$872$	0	0
$873$	13.0650	0.442182
$874$	0	0
$875$	−0.445042	−0.0150452
$876$	0	0
$877$	45.1379	1.52420	0.762099	−	0.647461i	$-0.224169\pi$
$877$	45.1379	1.52420	0.762099	+	0.647461i	$0.224169\pi$
$878$	0	0
$879$	−19.9002	−0.671217
$880$	0	0
$881$	47.8545	1.61226	0.806129	−	0.591739i	$-0.201558\pi$
$881$	47.8545	1.61226	0.806129	+	0.591739i	$0.201558\pi$
$882$	0	0
$883$	44.9904	1.51405	0.757024	−	0.653387i	$-0.226653\pi$
$883$	44.9904	1.51405	0.757024	+	0.653387i	$0.226653\pi$
$884$	0	0
$885$	13.5051	0.453968
$886$	0	0
$887$	5.25364	0.176400	0.0882000	−	0.996103i	$-0.471889\pi$
$887$	5.25364	0.176400	0.0882000	+	0.996103i	$0.471889\pi$
$888$	0	0
$889$	−2.85087	−0.0956150
$890$	0	0
$891$	3.50686	0.117484
$892$	0	0
$893$	11.4439	0.382957
$894$	0	0
$895$	−14.4960	−0.484546
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9.82973	−0.327840
$900$	0	0
$901$	−41.7425	−1.39065
$902$	0	0
$903$	−1.12583	−0.0374652
$904$	0	0
$905$	7.06545	0.234863
$906$	0	0
$907$	30.7814	1.02208	0.511040	−	0.859557i	$-0.329261\pi$
$907$	30.7814	1.02208	0.511040	+	0.859557i	$0.329261\pi$
$908$	0	0
$909$	−22.7321	−0.753976
$910$	0	0
$911$	48.4686	1.60584	0.802918	−	0.596089i	$-0.203280\pi$
$911$	48.4686	1.60584	0.802918	+	0.596089i	$0.203280\pi$
$912$	0	0
$913$	62.5555	2.07028
$914$	0	0
$915$	3.10944	0.102795
$916$	0	0
$917$	4.34630	0.143527
$918$	0	0
$919$	−26.5546	−0.875957	−0.437978	−	0.898985i	$-0.644305\pi$
$919$	−26.5546	−0.875957	−0.437978	+	0.898985i	$0.644305\pi$
$920$	0	0
$921$	−11.3997	−0.375633
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−5.00638	−0.164609
$926$	0	0
$927$	13.3868	0.439682
$928$	0	0
$929$	42.1749	1.38371	0.691857	−	0.722035i	$-0.256793\pi$
$929$	42.1749	1.38371	0.691857	+	0.722035i	$0.256793\pi$
$930$	0	0
$931$	−19.5917	−0.642092
$932$	0	0
$933$	3.79430	0.124220
$934$	0	0
$935$	−13.8818	−0.453983
$936$	0	0
$937$	−21.2597	−0.694523	−0.347262	−	0.937768i	$-0.612888\pi$
$937$	−21.2597	−0.694523	−0.347262	+	0.937768i	$0.612888\pi$
$938$	0	0
$939$	−18.6927	−0.610013
$940$	0	0
$941$	36.8440	1.20108	0.600540	−	0.799595i	$-0.294952\pi$
$941$	36.8440	1.20108	0.600540	+	0.799595i	$0.294952\pi$
$942$	0	0
$943$	−16.2931	−0.530576
$944$	0	0
$945$	2.37302	0.0771944
$946$	0	0
$947$	−59.0365	−1.91843	−0.959215	−	0.282678i	$-0.908777\pi$
$947$	−59.0365	−1.91843	−0.959215	+	0.282678i	$0.908777\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0.826891	0.0268138
$952$	0	0
$953$	3.57905	0.115937	0.0579685	−	0.998318i	$-0.481538\pi$
$953$	3.57905	0.115937	0.0579685	+	0.998318i	$0.481538\pi$
$954$	0	0
$955$	2.74305	0.0887632
$956$	0	0
$957$	−7.28756	−0.235573
$958$	0	0
$959$	4.38974	0.141752
$960$	0	0
$961$	10.4680	0.337676
$962$	0	0
$963$	1.84836	0.0595625
$964$	0	0
$965$	0.822416	0.0264745
$966$	0	0
$967$	10.4947	0.337488	0.168744	−	0.985660i	$-0.446029\pi$
$967$	10.4947	0.337488	0.168744	+	0.985660i	$0.446029\pi$
$968$	0	0
$969$	10.6509	0.342157
$970$	0	0
$971$	−10.0445	−0.322343	−0.161172	−	0.986926i	$-0.551527\pi$
$971$	−10.0445	−0.322343	−0.161172	+	0.986926i	$0.551527\pi$
$972$	0	0
$973$	−0.123118	−0.00394698
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.2899	1.06504	0.532518	−	0.846418i	$-0.321246\pi$
$977$	33.2899	1.06504	0.532518	+	0.846418i	$0.321246\pi$
$978$	0	0
$979$	12.3545	0.394852
$980$	0	0
$981$	24.5335	0.783295
$982$	0	0
$983$	−30.3432	−0.967798	−0.483899	−	0.875124i	$-0.660780\pi$
$983$	−30.3432	−0.967798	−0.483899	+	0.875124i	$0.660780\pi$
$984$	0	0
$985$	−20.6237	−0.657125
$986$	0	0
$987$	−1.99406	−0.0634715
$988$	0	0
$989$	7.34511	0.233561
$990$	0	0
$991$	40.0448	1.27207	0.636033	−	0.771662i	$-0.280574\pi$
$991$	40.0448	1.27207	0.636033	+	0.771662i	$0.280574\pi$
$992$	0	0
$993$	31.3718	0.995554
$994$	0	0
$995$	24.7909	0.785925
$996$	0	0
$997$	−29.2008	−0.924798	−0.462399	−	0.886672i	$-0.653011\pi$
$997$	−29.2008	−0.924798	−0.462399	+	0.886672i	$0.653011\pi$
$998$	0	0
$999$	26.6947	0.844582

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6760.2.a.bi.1.5	yes	6
13.12	even	2		6760.2.a.bh.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6760.2.a.bh.1.5	✓	6	13.12	even	2
6760.2.a.bi.1.5	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6760 = 2^{3} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6760.a (trivial)

\(f(q)\)	\(=\)	\(q+1.12772 q^{3} +1.00000 q^{5} -0.445042 q^{7} -1.72826 q^{9} +O(q^{10})\)	\(q+1.12772 q^{3} +1.00000 q^{5} -0.445042 q^{7} -1.72826 q^{9} -4.23347 q^{11} +1.12772 q^{15} +3.27905 q^{17} +2.88031 q^{19} -0.501881 q^{21} +3.27436 q^{23} +1.00000 q^{25} -5.33213 q^{27} +1.52646 q^{29} -6.43956 q^{31} -4.77416 q^{33} -0.445042 q^{35} -5.00638 q^{37} -4.97595 q^{41} +2.24322 q^{43} -1.72826 q^{45} +3.97316 q^{47} -6.80194 q^{49} +3.69784 q^{51} -12.7301 q^{53} -4.23347 q^{55} +3.24817 q^{57} +11.9756 q^{59} +2.75729 q^{61} +0.769146 q^{63} -0.107348 q^{67} +3.69255 q^{69} -0.137887 q^{71} -5.01314 q^{73} +1.12772 q^{75} +1.88407 q^{77} +0.424756 q^{79} -0.828364 q^{81} -14.7764 q^{83} +3.27905 q^{85} +1.72141 q^{87} -2.91829 q^{89} -7.26200 q^{93} +2.88031 q^{95} -7.55962 q^{97} +7.31652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 6 * q + 6 * q^5 - 2 * q^7 + 4 * q^9	\( 6 q + 6 q^{5} - 2 q^{7} + 4 q^{9} - 5 q^{11} - 9 q^{17} + 2 q^{19} - 7 q^{21} + 3 q^{23} + 6 q^{25} - 11 q^{29} - 5 q^{31} - 20 q^{33} - 2 q^{35} - 7 q^{37} - 27 q^{41} - 4 q^{43} + 4 q^{45} - 19 q^{47} - 32 q^{49} + 5 q^{51} - 6 q^{53} - 5 q^{55} + 6 q^{57} + 19 q^{59} - 4 q^{61} - 6 q^{63} + 10 q^{67} - 15 q^{69} - 15 q^{71} - 14 q^{73} + 4 q^{77} + 7 q^{79} + 2 q^{81} + 12 q^{83} - 9 q^{85} + 21 q^{87} - 34 q^{89} - 55 q^{93} + 2 q^{95} + 2 q^{97} - 29 q^{99}+O(q^{100}) \) 6 * q + 6 * q^5 - 2 * q^7 + 4 * q^9 - 5 * q^11 - 9 * q^17 + 2 * q^19 - 7 * q^21 + 3 * q^23 + 6 * q^25 - 11 * q^29 - 5 * q^31 - 20 * q^33 - 2 * q^35 - 7 * q^37 - 27 * q^41 - 4 * q^43 + 4 * q^45 - 19 * q^47 - 32 * q^49 + 5 * q^51 - 6 * q^53 - 5 * q^55 + 6 * q^57 + 19 * q^59 - 4 * q^61 - 6 * q^63 + 10 * q^67 - 15 * q^69 - 15 * q^71 - 14 * q^73 + 4 * q^77 + 7 * q^79 + 2 * q^81 + 12 * q^83 - 9 * q^85 + 21 * q^87 - 34 * q^89 - 55 * q^93 + 2 * q^95 + 2 * q^97 - 29 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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