LMFDB - Embedded newform 6776.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6776,2,Mod(1,6776)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6776.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6776 = 2^{3} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6776.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,-3,0,2,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$54.1066324096$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	6776.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{3} +0.561553 q^{5} +1.00000 q^{7} +3.56155 q^{9} -3.12311 q^{13} -1.43845 q^{15} +2.00000 q^{17} +5.12311 q^{19} -2.56155 q^{21} -1.43845 q^{23} -4.68466 q^{25} -1.43845 q^{27} +2.00000 q^{29} -10.5616 q^{31} +0.561553 q^{35} +4.56155 q^{37} +8.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} +2.00000 q^{45} -6.24621 q^{47} +1.00000 q^{49} -5.12311 q^{51} -4.24621 q^{53} -13.1231 q^{57} -5.43845 q^{59} +4.24621 q^{61} +3.56155 q^{63} -1.75379 q^{65} -2.56155 q^{67} +3.68466 q^{69} +3.68466 q^{71} +4.24621 q^{73} +12.0000 q^{75} -5.12311 q^{79} -7.00000 q^{81} +8.00000 q^{83} +1.12311 q^{85} -5.12311 q^{87} -12.5616 q^{89} -3.12311 q^{91} +27.0540 q^{93} +2.87689 q^{95} +8.56155 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} + 2 q^{13} - 7 q^{15} + 4 q^{17} + 2 q^{19} - q^{21} - 7 q^{23} + 3 q^{25} - 7 q^{27} + 4 q^{29} - 17 q^{31} - 3 q^{35} + 5 q^{37} + 16 q^{39} + 20 q^{41} - 8 q^{43}+ \cdots + 13 q^{97}+O(q^{100}) $ 2 * q - q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 + 2 * q^13 - 7 * q^15 + 4 * q^17 + 2 * q^19 - q^21 - 7 * q^23 + 3 * q^25 - 7 * q^27 + 4 * q^29 - 17 * q^31 - 3 * q^35 + 5 * q^37 + 16 * q^39 + 20 * q^41 - 8 * q^43 + 4 * q^45 + 4 * q^47 + 2 * q^49 - 2 * q^51 + 8 * q^53 - 18 * q^57 - 15 * q^59 - 8 * q^61 + 3 * q^63 - 20 * q^65 - q^67 - 5 * q^69 - 5 * q^71 - 8 * q^73 + 24 * q^75 - 2 * q^79 - 14 * q^81 + 16 * q^83 - 6 * q^85 - 2 * q^87 - 21 * q^89 + 2 * q^91 + 17 * q^93 + 14 * q^95 + 13 * q^97

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
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	0.561553	0.251134	0.125567	−	0.992085i	$-0.459925\pi$
$5$	0.561553	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.12311	−0.866194	−0.433097	−	0.901347i	$-0.642579\pi$
$13$	−3.12311	−0.866194	−0.433097	+	0.901347i	$0.642579\pi$
$14$	0	0
$15$	−1.43845	−0.371405
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	5.12311	1.17532	0.587661	−	0.809108i	$-0.300049\pi$
$19$	5.12311	1.17532	0.587661	+	0.809108i	$0.300049\pi$
$20$	0	0
$21$	−2.56155	−0.558977
$22$	0	0
$23$	−1.43845	−0.299937	−0.149968	−	0.988691i	$-0.547917\pi$
$23$	−1.43845	−0.299937	−0.149968	+	0.988691i	$0.547917\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−10.5616	−1.89691	−0.948455	−	0.316911i	$-0.897355\pi$
$31$	−10.5616	−1.89691	−0.948455	+	0.316911i	$0.897355\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.561553	0.0949197
$36$	0	0
$37$	4.56155	0.749915	0.374957	−	0.927042i	$-0.377657\pi$
$37$	4.56155	0.749915	0.374957	+	0.927042i	$0.377657\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−6.24621	−0.911104	−0.455552	−	0.890209i	$-0.650558\pi$
$47$	−6.24621	−0.911104	−0.455552	+	0.890209i	$0.650558\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.12311	−0.717378
$52$	0	0
$53$	−4.24621	−0.583262	−0.291631	−	0.956531i	$-0.594198\pi$
$53$	−4.24621	−0.583262	−0.291631	+	0.956531i	$0.594198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−13.1231	−1.73820
$58$	0	0
$59$	−5.43845	−0.708026	−0.354013	−	0.935241i	$-0.615183\pi$
$59$	−5.43845	−0.708026	−0.354013	+	0.935241i	$0.615183\pi$
$60$	0	0
$61$	4.24621	0.543672	0.271836	−	0.962344i	$-0.412369\pi$
$61$	4.24621	0.543672	0.271836	+	0.962344i	$0.412369\pi$
$62$	0	0
$63$	3.56155	0.448713
$64$	0	0
$65$	−1.75379	−0.217531
$66$	0	0
$67$	−2.56155	−0.312943	−0.156472	−	0.987682i	$-0.550012\pi$
$67$	−2.56155	−0.312943	−0.156472	+	0.987682i	$0.550012\pi$
$68$	0	0
$69$	3.68466	0.443581
$70$	0	0
$71$	3.68466	0.437289	0.218644	−	0.975805i	$-0.429837\pi$
$71$	3.68466	0.437289	0.218644	+	0.975805i	$0.429837\pi$
$72$	0	0
$73$	4.24621	0.496981	0.248491	−	0.968634i	$-0.420065\pi$
$73$	4.24621	0.496981	0.248491	+	0.968634i	$0.420065\pi$
$74$	0	0
$75$	12.0000	1.38564
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.12311	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$79$	−5.12311	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	1.12311	0.121818
$86$	0	0
$87$	−5.12311	−0.549255
$88$	0	0
$89$	−12.5616	−1.33152	−0.665761	−	0.746165i	$-0.731893\pi$
$89$	−12.5616	−1.33152	−0.665761	+	0.746165i	$0.731893\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	0	0
$93$	27.0540	2.80537
$94$	0	0
$95$	2.87689	0.295163
$96$	0	0
$97$	8.56155	0.869294	0.434647	−	0.900601i	$-0.356873\pi$
$97$	8.56155	0.869294	0.434647	+	0.900601i	$0.356873\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.3693	−1.33030	−0.665148	−	0.746711i	$-0.731632\pi$
$101$	−13.3693	−1.33030	−0.665148	+	0.746711i	$0.731632\pi$
$102$	0	0
$103$	16.4924	1.62505	0.812523	−	0.582929i	$-0.198093\pi$
$103$	16.4924	1.62505	0.812523	+	0.582929i	$0.198093\pi$
$104$	0	0
$105$	−1.43845	−0.140378
$106$	0	0
$107$	−17.1231	−1.65535	−0.827677	−	0.561205i	$-0.810338\pi$
$107$	−17.1231	−1.65535	−0.827677	+	0.561205i	$0.810338\pi$
$108$	0	0
$109$	9.36932	0.897418	0.448709	−	0.893678i	$-0.351884\pi$
$109$	9.36932	0.897418	0.448709	+	0.893678i	$0.351884\pi$
$110$	0	0
$111$	−11.6847	−1.10906
$112$	0	0
$113$	5.68466	0.534768	0.267384	−	0.963590i	$-0.413841\pi$
$113$	5.68466	0.534768	0.267384	+	0.963590i	$0.413841\pi$
$114$	0	0
$115$	−0.807764	−0.0753244
$116$	0	0
$117$	−11.1231	−1.02833
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−25.6155	−2.30967
$124$	0	0
$125$	−5.43845	−0.486430
$126$	0	0
$127$	5.12311	0.454602	0.227301	−	0.973825i	$-0.427010\pi$
$127$	5.12311	0.454602	0.227301	+	0.973825i	$0.427010\pi$
$128$	0	0
$129$	10.2462	0.902129
$130$	0	0
$131$	−13.1231	−1.14657	−0.573286	−	0.819356i	$-0.694331\pi$
$131$	−13.1231	−1.14657	−0.573286	+	0.819356i	$0.694331\pi$
$132$	0	0
$133$	5.12311	0.444230
$134$	0	0
$135$	−0.807764	−0.0695213
$136$	0	0
$137$	−2.31534	−0.197813	−0.0989065	−	0.995097i	$-0.531534\pi$
$137$	−2.31534	−0.197813	−0.0989065	+	0.995097i	$0.531534\pi$
$138$	0	0
$139$	5.12311	0.434536	0.217268	−	0.976112i	$-0.430285\pi$
$139$	5.12311	0.434536	0.217268	+	0.976112i	$0.430285\pi$
$140$	0	0
$141$	16.0000	1.34744
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.12311	0.0932688
$146$	0	0
$147$	−2.56155	−0.211273
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−5.12311	−0.416912	−0.208456	−	0.978032i	$-0.566844\pi$
$151$	−5.12311	−0.416912	−0.208456	+	0.978032i	$0.566844\pi$
$152$	0	0
$153$	7.12311	0.575869
$154$	0	0
$155$	−5.93087	−0.476379
$156$	0	0
$157$	16.5616	1.32176	0.660878	−	0.750493i	$-0.270184\pi$
$157$	16.5616	1.32176	0.660878	+	0.750493i	$0.270184\pi$
$158$	0	0
$159$	10.8769	0.862594
$160$	0	0
$161$	−1.43845	−0.113366
$162$	0	0
$163$	16.4924	1.29179	0.645893	−	0.763428i	$-0.276485\pi$
$163$	16.4924	1.29179	0.645893	+	0.763428i	$0.276485\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.36932	0.570255	0.285127	−	0.958490i	$-0.407964\pi$
$167$	7.36932	0.570255	0.285127	+	0.958490i	$0.407964\pi$
$168$	0	0
$169$	−3.24621	−0.249709
$170$	0	0
$171$	18.2462	1.39532
$172$	0	0
$173$	23.1231	1.75802	0.879009	−	0.476806i	$-0.158206\pi$
$173$	23.1231	1.75802	0.879009	+	0.476806i	$0.158206\pi$
$174$	0	0
$175$	−4.68466	−0.354127
$176$	0	0
$177$	13.9309	1.04711
$178$	0	0
$179$	−20.8078	−1.55525	−0.777623	−	0.628731i	$-0.783575\pi$
$179$	−20.8078	−1.55525	−0.777623	+	0.628731i	$0.783575\pi$
$180$	0	0
$181$	7.93087	0.589497	0.294748	−	0.955575i	$-0.404764\pi$
$181$	7.93087	0.589497	0.294748	+	0.955575i	$0.404764\pi$
$182$	0	0
$183$	−10.8769	−0.804043
$184$	0	0
$185$	2.56155	0.188329
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.43845	−0.104632
$190$	0	0
$191$	−27.0540	−1.95756	−0.978778	−	0.204921i	$-0.934306\pi$
$191$	−27.0540	−1.95756	−0.978778	+	0.204921i	$0.934306\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	4.49242	0.321709
$196$	0	0
$197$	20.2462	1.44248	0.721241	−	0.692684i	$-0.243572\pi$
$197$	20.2462	1.44248	0.721241	+	0.692684i	$0.243572\pi$
$198$	0	0
$199$	−14.2462	−1.00989	−0.504944	−	0.863152i	$-0.668487\pi$
$199$	−14.2462	−1.00989	−0.504944	+	0.863152i	$0.668487\pi$
$200$	0	0
$201$	6.56155	0.462816
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	5.61553	0.392205
$206$	0	0
$207$	−5.12311	−0.356080
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−25.1231	−1.72955	−0.864773	−	0.502163i	$-0.832538\pi$
$211$	−25.1231	−1.72955	−0.864773	+	0.502163i	$0.832538\pi$
$212$	0	0
$213$	−9.43845	−0.646712
$214$	0	0
$215$	−2.24621	−0.153190
$216$	0	0
$217$	−10.5616	−0.716965
$218$	0	0
$219$	−10.8769	−0.734992
$220$	0	0
$221$	−6.24621	−0.420166
$222$	0	0
$223$	−26.5616	−1.77869	−0.889347	−	0.457234i	$-0.848840\pi$
$223$	−26.5616	−1.77869	−0.889347	+	0.457234i	$0.848840\pi$
$224$	0	0
$225$	−16.6847	−1.11231
$226$	0	0
$227$	10.2462	0.680065	0.340032	−	0.940414i	$-0.389562\pi$
$227$	10.2462	0.680065	0.340032	+	0.940414i	$0.389562\pi$
$228$	0	0
$229$	−2.31534	−0.153002	−0.0765010	−	0.997070i	$-0.524375\pi$
$229$	−2.31534	−0.153002	−0.0765010	+	0.997070i	$0.524375\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.49242	0.163284	0.0816420	−	0.996662i	$-0.473984\pi$
$233$	2.49242	0.163284	0.0816420	+	0.996662i	$0.473984\pi$
$234$	0	0
$235$	−3.50758	−0.228809
$236$	0	0
$237$	13.1231	0.852437
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−5.36932	−0.345868	−0.172934	−	0.984933i	$-0.555325\pi$
$241$	−5.36932	−0.345868	−0.172934	+	0.984933i	$0.555325\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	0.561553	0.0358763
$246$	0	0
$247$	−16.0000	−1.01806
$248$	0	0
$249$	−20.4924	−1.29865
$250$	0	0
$251$	15.0540	0.950198	0.475099	−	0.879932i	$-0.342412\pi$
$251$	15.0540	0.950198	0.475099	+	0.879932i	$0.342412\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2.87689	−0.180158
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	4.56155	0.283441
$260$	0	0
$261$	7.12311	0.440909
$262$	0	0
$263$	−9.61553	−0.592919	−0.296459	−	0.955045i	$-0.595806\pi$
$263$	−9.61553	−0.592919	−0.296459	+	0.955045i	$0.595806\pi$
$264$	0	0
$265$	−2.38447	−0.146477
$266$	0	0
$267$	32.1771	1.96921
$268$	0	0
$269$	7.75379	0.472757	0.236378	−	0.971661i	$-0.424040\pi$
$269$	7.75379	0.472757	0.236378	+	0.971661i	$0.424040\pi$
$270$	0	0
$271$	−18.2462	−1.10838	−0.554189	−	0.832391i	$-0.686972\pi$
$271$	−18.2462	−1.10838	−0.554189	+	0.832391i	$0.686972\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.87689	−0.533361	−0.266680	−	0.963785i	$-0.585927\pi$
$277$	−8.87689	−0.533361	−0.266680	+	0.963785i	$0.585927\pi$
$278$	0	0
$279$	−37.6155	−2.25198
$280$	0	0
$281$	−29.8617	−1.78140	−0.890701	−	0.454590i	$-0.849786\pi$
$281$	−29.8617	−1.78140	−0.890701	+	0.454590i	$0.849786\pi$
$282$	0	0
$283$	−28.4924	−1.69370	−0.846849	−	0.531833i	$-0.821503\pi$
$283$	−28.4924	−1.69370	−0.846849	+	0.531833i	$0.821503\pi$
$284$	0	0
$285$	−7.36932	−0.436521
$286$	0	0
$287$	10.0000	0.590281
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−21.9309	−1.28561
$292$	0	0
$293$	12.2462	0.715431	0.357716	−	0.933831i	$-0.383556\pi$
$293$	12.2462	0.715431	0.357716	+	0.933831i	$0.383556\pi$
$294$	0	0
$295$	−3.05398	−0.177809
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.49242	0.259804
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	34.2462	1.96739
$304$	0	0
$305$	2.38447	0.136534
$306$	0	0
$307$	−20.4924	−1.16956	−0.584782	−	0.811190i	$-0.698820\pi$
$307$	−20.4924	−1.16956	−0.584782	+	0.811190i	$0.698820\pi$
$308$	0	0
$309$	−42.2462	−2.40330
$310$	0	0
$311$	18.7386	1.06257	0.531285	−	0.847193i	$-0.321709\pi$
$311$	18.7386	1.06257	0.531285	+	0.847193i	$0.321709\pi$
$312$	0	0
$313$	15.9309	0.900466	0.450233	−	0.892911i	$-0.351341\pi$
$313$	15.9309	0.900466	0.450233	+	0.892911i	$0.351341\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	−13.0540	−0.733184	−0.366592	−	0.930382i	$-0.619476\pi$
$317$	−13.0540	−0.733184	−0.366592	+	0.930382i	$0.619476\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	43.8617	2.44812
$322$	0	0
$323$	10.2462	0.570114
$324$	0	0
$325$	14.6307	0.811564
$326$	0	0
$327$	−24.0000	−1.32720
$328$	0	0
$329$	−6.24621	−0.344365
$330$	0	0
$331$	12.8078	0.703978	0.351989	−	0.936004i	$-0.385505\pi$
$331$	12.8078	0.703978	0.351989	+	0.936004i	$0.385505\pi$
$332$	0	0
$333$	16.2462	0.890287
$334$	0	0
$335$	−1.43845	−0.0785908
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	−14.5616	−0.790875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.06913	0.111398
$346$	0	0
$347$	−6.87689	−0.369171	−0.184586	−	0.982816i	$-0.559094\pi$
$347$	−6.87689	−0.369171	−0.184586	+	0.982816i	$0.559094\pi$
$348$	0	0
$349$	12.2462	0.655525	0.327762	−	0.944760i	$-0.393705\pi$
$349$	12.2462	0.655525	0.327762	+	0.944760i	$0.393705\pi$
$350$	0	0
$351$	4.49242	0.239788
$352$	0	0
$353$	−20.5616	−1.09438	−0.547191	−	0.837008i	$-0.684303\pi$
$353$	−20.5616	−1.09438	−0.547191	+	0.837008i	$0.684303\pi$
$354$	0	0
$355$	2.06913	0.109818
$356$	0	0
$357$	−5.12311	−0.271144
$358$	0	0
$359$	−2.24621	−0.118550	−0.0592752	−	0.998242i	$-0.518879\pi$
$359$	−2.24621	−0.118550	−0.0592752	+	0.998242i	$0.518879\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.38447	0.124809
$366$	0	0
$367$	−0.315342	−0.0164607	−0.00823035	−	0.999966i	$-0.502620\pi$
$367$	−0.315342	−0.0164607	−0.00823035	+	0.999966i	$0.502620\pi$
$368$	0	0
$369$	35.6155	1.85407
$370$	0	0
$371$	−4.24621	−0.220452
$372$	0	0
$373$	11.6155	0.601429	0.300715	−	0.953714i	$-0.402775\pi$
$373$	11.6155	0.601429	0.300715	+	0.953714i	$0.402775\pi$
$374$	0	0
$375$	13.9309	0.719387
$376$	0	0
$377$	−6.24621	−0.321696
$378$	0	0
$379$	−25.9309	−1.33198	−0.665990	−	0.745961i	$-0.731991\pi$
$379$	−25.9309	−1.33198	−0.665990	+	0.745961i	$0.731991\pi$
$380$	0	0
$381$	−13.1231	−0.672317
$382$	0	0
$383$	9.93087	0.507444	0.253722	−	0.967277i	$-0.418345\pi$
$383$	9.93087	0.507444	0.253722	+	0.967277i	$0.418345\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−14.2462	−0.724176
$388$	0	0
$389$	−29.0540	−1.47310	−0.736548	−	0.676386i	$-0.763545\pi$
$389$	−29.0540	−1.47310	−0.736548	+	0.676386i	$0.763545\pi$
$390$	0	0
$391$	−2.87689	−0.145491
$392$	0	0
$393$	33.6155	1.69568
$394$	0	0
$395$	−2.87689	−0.144752
$396$	0	0
$397$	−10.4924	−0.526600	−0.263300	−	0.964714i	$-0.584811\pi$
$397$	−10.4924	−0.526600	−0.263300	+	0.964714i	$0.584811\pi$
$398$	0	0
$399$	−13.1231	−0.656977
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	32.9848	1.64309
$404$	0	0
$405$	−3.93087	−0.195326
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−34.4924	−1.70554	−0.852770	−	0.522286i	$-0.825079\pi$
$409$	−34.4924	−1.70554	−0.852770	+	0.522286i	$0.825079\pi$
$410$	0	0
$411$	5.93087	0.292548
$412$	0	0
$413$	−5.43845	−0.267608
$414$	0	0
$415$	4.49242	0.220524
$416$	0	0
$417$	−13.1231	−0.642641
$418$	0	0
$419$	18.7386	0.915442	0.457721	−	0.889096i	$-0.348666\pi$
$419$	18.7386	0.915442	0.457721	+	0.889096i	$0.348666\pi$
$420$	0	0
$421$	30.9848	1.51011	0.755054	−	0.655662i	$-0.227610\pi$
$421$	30.9848	1.51011	0.755054	+	0.655662i	$0.227610\pi$
$422$	0	0
$423$	−22.2462	−1.08165
$424$	0	0
$425$	−9.36932	−0.454479
$426$	0	0
$427$	4.24621	0.205489
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.7386	0.709935	0.354968	−	0.934879i	$-0.384492\pi$
$431$	14.7386	0.709935	0.354968	+	0.934879i	$0.384492\pi$
$432$	0	0
$433$	−1.68466	−0.0809595	−0.0404798	−	0.999180i	$-0.512889\pi$
$433$	−1.68466	−0.0809595	−0.0404798	+	0.999180i	$0.512889\pi$
$434$	0	0
$435$	−2.87689	−0.137937
$436$	0	0
$437$	−7.36932	−0.352522
$438$	0	0
$439$	−17.6155	−0.840743	−0.420372	−	0.907352i	$-0.638100\pi$
$439$	−17.6155	−0.840743	−0.420372	+	0.907352i	$0.638100\pi$
$440$	0	0
$441$	3.56155	0.169598
$442$	0	0
$443$	12.1771	0.578551	0.289275	−	0.957246i	$-0.406586\pi$
$443$	12.1771	0.578551	0.289275	+	0.957246i	$0.406586\pi$
$444$	0	0
$445$	−7.05398	−0.334390
$446$	0	0
$447$	−25.6155	−1.21157
$448$	0	0
$449$	−17.6847	−0.834591	−0.417295	−	0.908771i	$-0.637022\pi$
$449$	−17.6847	−0.834591	−0.417295	+	0.908771i	$0.637022\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	13.1231	0.616577
$454$	0	0
$455$	−1.75379	−0.0822189
$456$	0	0
$457$	−37.8617	−1.77110	−0.885549	−	0.464546i	$-0.846217\pi$
$457$	−37.8617	−1.77110	−0.885549	+	0.464546i	$0.846217\pi$
$458$	0	0
$459$	−2.87689	−0.134282
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−39.5464	−1.83788	−0.918938	−	0.394401i	$-0.870952\pi$
$463$	−39.5464	−1.83788	−0.918938	+	0.394401i	$0.870952\pi$
$464$	0	0
$465$	15.1922	0.704523
$466$	0	0
$467$	−30.4233	−1.40782	−0.703911	−	0.710288i	$-0.748565\pi$
$467$	−30.4233	−1.40782	−0.703911	+	0.710288i	$0.748565\pi$
$468$	0	0
$469$	−2.56155	−0.118282
$470$	0	0
$471$	−42.4233	−1.95476
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−24.0000	−1.10120
$476$	0	0
$477$	−15.1231	−0.692439
$478$	0	0
$479$	−20.4924	−0.936323	−0.468161	−	0.883643i	$-0.655083\pi$
$479$	−20.4924	−0.936323	−0.468161	+	0.883643i	$0.655083\pi$
$480$	0	0
$481$	−14.2462	−0.649571
$482$	0	0
$483$	3.68466	0.167658
$484$	0	0
$485$	4.80776	0.218309
$486$	0	0
$487$	−17.4384	−0.790211	−0.395106	−	0.918636i	$-0.629292\pi$
$487$	−17.4384	−0.790211	−0.395106	+	0.918636i	$0.629292\pi$
$488$	0	0
$489$	−42.2462	−1.91044
$490$	0	0
$491$	17.1231	0.772755	0.386377	−	0.922341i	$-0.373726\pi$
$491$	17.1231	0.772755	0.386377	+	0.922341i	$0.373726\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.68466	0.165280
$498$	0	0
$499$	−32.4924	−1.45456	−0.727280	−	0.686341i	$-0.759216\pi$
$499$	−32.4924	−1.45456	−0.727280	+	0.686341i	$0.759216\pi$
$500$	0	0
$501$	−18.8769	−0.843357
$502$	0	0
$503$	−14.7386	−0.657163	−0.328582	−	0.944476i	$-0.606571\pi$
$503$	−14.7386	−0.657163	−0.328582	+	0.944476i	$0.606571\pi$
$504$	0	0
$505$	−7.50758	−0.334083
$506$	0	0
$507$	8.31534	0.369297
$508$	0	0
$509$	−14.8078	−0.656343	−0.328171	−	0.944618i	$-0.606432\pi$
$509$	−14.8078	−0.656343	−0.328171	+	0.944618i	$0.606432\pi$
$510$	0	0
$511$	4.24621	0.187841
$512$	0	0
$513$	−7.36932	−0.325363
$514$	0	0
$515$	9.26137	0.408105
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−59.2311	−2.59995
$520$	0	0
$521$	22.3153	0.977653	0.488826	−	0.872381i	$-0.337425\pi$
$521$	22.3153	0.977653	0.488826	+	0.872381i	$0.337425\pi$
$522$	0	0
$523$	−2.24621	−0.0982200	−0.0491100	−	0.998793i	$-0.515638\pi$
$523$	−2.24621	−0.0982200	−0.0491100	+	0.998793i	$0.515638\pi$
$524$	0	0
$525$	12.0000	0.523723
$526$	0	0
$527$	−21.1231	−0.920137
$528$	0	0
$529$	−20.9309	−0.910038
$530$	0	0
$531$	−19.3693	−0.840557
$532$	0	0
$533$	−31.2311	−1.35277
$534$	0	0
$535$	−9.61553	−0.415716
$536$	0	0
$537$	53.3002	2.30007
$538$	0	0
$539$	0	0
$540$	0	0
$541$	15.1231	0.650193	0.325097	−	0.945681i	$-0.394603\pi$
$541$	15.1231	0.650193	0.325097	+	0.945681i	$0.394603\pi$
$542$	0	0
$543$	−20.3153	−0.871815
$544$	0	0
$545$	5.26137	0.225372
$546$	0	0
$547$	−34.7386	−1.48532	−0.742658	−	0.669670i	$-0.766435\pi$
$547$	−34.7386	−1.48532	−0.742658	+	0.669670i	$0.766435\pi$
$548$	0	0
$549$	15.1231	0.645438
$550$	0	0
$551$	10.2462	0.436503
$552$	0	0
$553$	−5.12311	−0.217857
$554$	0	0
$555$	−6.56155	−0.278522
$556$	0	0
$557$	−40.2462	−1.70529	−0.852643	−	0.522493i	$-0.825002\pi$
$557$	−40.2462	−1.70529	−0.852643	+	0.522493i	$0.825002\pi$
$558$	0	0
$559$	12.4924	0.528373
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.7386	1.29548	0.647739	−	0.761862i	$-0.275715\pi$
$563$	30.7386	1.29548	0.647739	+	0.761862i	$0.275715\pi$
$564$	0	0
$565$	3.19224	0.134298
$566$	0	0
$567$	−7.00000	−0.293972
$568$	0	0
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	40.4924	1.69456	0.847278	−	0.531150i	$-0.178240\pi$
$571$	40.4924	1.69456	0.847278	+	0.531150i	$0.178240\pi$
$572$	0	0
$573$	69.3002	2.89506
$574$	0	0
$575$	6.73863	0.281020
$576$	0	0
$577$	6.31534	0.262911	0.131456	−	0.991322i	$-0.458035\pi$
$577$	6.31534	0.262911	0.131456	+	0.991322i	$0.458035\pi$
$578$	0	0
$579$	5.12311	0.212909
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−6.24621	−0.258249
$586$	0	0
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	−54.1080	−2.22948
$590$	0	0
$591$	−51.8617	−2.13331
$592$	0	0
$593$	−8.24621	−0.338631	−0.169316	−	0.985562i	$-0.554156\pi$
$593$	−8.24621	−0.338631	−0.169316	+	0.985562i	$0.554156\pi$
$594$	0	0
$595$	1.12311	0.0460428
$596$	0	0
$597$	36.4924	1.49354
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−43.1231	−1.75903	−0.879514	−	0.475873i	$-0.842132\pi$
$601$	−43.1231	−1.75903	−0.879514	+	0.475873i	$0.842132\pi$
$602$	0	0
$603$	−9.12311	−0.371522
$604$	0	0
$605$	0	0
$606$	0	0
$607$	48.3542	1.96263	0.981317	−	0.192396i	$-0.0616257\pi$
$607$	48.3542	1.96263	0.981317	+	0.192396i	$0.0616257\pi$
$608$	0	0
$609$	−5.12311	−0.207599
$610$	0	0
$611$	19.5076	0.789192
$612$	0	0
$613$	−23.6155	−0.953822	−0.476911	−	0.878952i	$-0.658244\pi$
$613$	−23.6155	−0.953822	−0.476911	+	0.878952i	$0.658244\pi$
$614$	0	0
$615$	−14.3845	−0.580038
$616$	0	0
$617$	−26.4924	−1.06654	−0.533272	−	0.845944i	$-0.679038\pi$
$617$	−26.4924	−1.06654	−0.533272	+	0.845944i	$0.679038\pi$
$618$	0	0
$619$	27.1922	1.09295	0.546474	−	0.837476i	$-0.315970\pi$
$619$	27.1922	1.09295	0.546474	+	0.837476i	$0.315970\pi$
$620$	0	0
$621$	2.06913	0.0830313
$622$	0	0
$623$	−12.5616	−0.503268
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.12311	0.363762
$630$	0	0
$631$	−37.9309	−1.51000	−0.755002	−	0.655722i	$-0.772364\pi$
$631$	−37.9309	−1.51000	−0.755002	+	0.655722i	$0.772364\pi$
$632$	0	0
$633$	64.3542	2.55785
$634$	0	0
$635$	2.87689	0.114166
$636$	0	0
$637$	−3.12311	−0.123742
$638$	0	0
$639$	13.1231	0.519142
$640$	0	0
$641$	34.8078	1.37482	0.687412	−	0.726268i	$-0.258747\pi$
$641$	34.8078	1.37482	0.687412	+	0.726268i	$0.258747\pi$
$642$	0	0
$643$	43.5464	1.71730	0.858651	−	0.512560i	$-0.171303\pi$
$643$	43.5464	1.71730	0.858651	+	0.512560i	$0.171303\pi$
$644$	0	0
$645$	5.75379	0.226555
$646$	0	0
$647$	−36.1771	−1.42227	−0.711134	−	0.703057i	$-0.751818\pi$
$647$	−36.1771	−1.42227	−0.711134	+	0.703057i	$0.751818\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	27.0540	1.06033
$652$	0	0
$653$	43.3002	1.69447	0.847234	−	0.531220i	$-0.178266\pi$
$653$	43.3002	1.69447	0.847234	+	0.531220i	$0.178266\pi$
$654$	0	0
$655$	−7.36932	−0.287943
$656$	0	0
$657$	15.1231	0.590009
$658$	0	0
$659$	21.6155	0.842021	0.421011	−	0.907056i	$-0.361675\pi$
$659$	21.6155	0.842021	0.421011	+	0.907056i	$0.361675\pi$
$660$	0	0
$661$	−34.6695	−1.34849	−0.674244	−	0.738509i	$-0.735530\pi$
$661$	−34.6695	−1.34849	−0.674244	+	0.738509i	$0.735530\pi$
$662$	0	0
$663$	16.0000	0.621389
$664$	0	0
$665$	2.87689	0.111561
$666$	0	0
$667$	−2.87689	−0.111394
$668$	0	0
$669$	68.0388	2.63053
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−33.3693	−1.28629	−0.643146	−	0.765743i	$-0.722371\pi$
$673$	−33.3693	−1.28629	−0.643146	+	0.765743i	$0.722371\pi$
$674$	0	0
$675$	6.73863	0.259370
$676$	0	0
$677$	28.2462	1.08559	0.542795	−	0.839865i	$-0.317366\pi$
$677$	28.2462	1.08559	0.542795	+	0.839865i	$0.317366\pi$
$678$	0	0
$679$	8.56155	0.328562
$680$	0	0
$681$	−26.2462	−1.00576
$682$	0	0
$683$	−32.4924	−1.24329	−0.621644	−	0.783300i	$-0.713535\pi$
$683$	−32.4924	−1.24329	−0.621644	+	0.783300i	$0.713535\pi$
$684$	0	0
$685$	−1.30019	−0.0496776
$686$	0	0
$687$	5.93087	0.226277
$688$	0	0
$689$	13.2614	0.505218
$690$	0	0
$691$	−28.1771	−1.07191	−0.535953	−	0.844248i	$-0.680048\pi$
$691$	−28.1771	−1.07191	−0.535953	+	0.844248i	$0.680048\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.87689	0.109127
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	−6.38447	−0.241483
$700$	0	0
$701$	−36.1080	−1.36378	−0.681889	−	0.731455i	$-0.738841\pi$
$701$	−36.1080	−1.36378	−0.681889	+	0.731455i	$0.738841\pi$
$702$	0	0
$703$	23.3693	0.881390
$704$	0	0
$705$	8.98485	0.338389
$706$	0	0
$707$	−13.3693	−0.502805
$708$	0	0
$709$	3.30019	0.123941	0.0619706	−	0.998078i	$-0.480262\pi$
$709$	3.30019	0.123941	0.0619706	+	0.998078i	$0.480262\pi$
$710$	0	0
$711$	−18.2462	−0.684286
$712$	0	0
$713$	15.1922	0.568954
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−20.4924	−0.765304
$718$	0	0
$719$	−41.3002	−1.54024	−0.770119	−	0.637901i	$-0.779803\pi$
$719$	−41.3002	−1.54024	−0.770119	+	0.637901i	$0.779803\pi$
$720$	0	0
$721$	16.4924	0.614210
$722$	0	0
$723$	13.7538	0.511509
$724$	0	0
$725$	−9.36932	−0.347968
$726$	0	0
$727$	3.19224	0.118393	0.0591967	−	0.998246i	$-0.481146\pi$
$727$	3.19224	0.118393	0.0591967	+	0.998246i	$0.481146\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	23.1231	0.854071	0.427036	−	0.904235i	$-0.359558\pi$
$733$	23.1231	0.854071	0.427036	+	0.904235i	$0.359558\pi$
$734$	0	0
$735$	−1.43845	−0.0530579
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.6307	0.758912	0.379456	−	0.925210i	$-0.376111\pi$
$739$	20.6307	0.758912	0.379456	+	0.925210i	$0.376111\pi$
$740$	0	0
$741$	40.9848	1.50562
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	5.61553	0.205737
$746$	0	0
$747$	28.4924	1.04248
$748$	0	0
$749$	−17.1231	−0.625665
$750$	0	0
$751$	−37.3002	−1.36110	−0.680552	−	0.732700i	$-0.738260\pi$
$751$	−37.3002	−1.36110	−0.680552	+	0.732700i	$0.738260\pi$
$752$	0	0
$753$	−38.5616	−1.40526
$754$	0	0
$755$	−2.87689	−0.104701
$756$	0	0
$757$	−26.9848	−0.980781	−0.490390	−	0.871503i	$-0.663146\pi$
$757$	−26.9848	−0.980781	−0.490390	+	0.871503i	$0.663146\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.1080	1.45391	0.726956	−	0.686684i	$-0.240934\pi$
$761$	40.1080	1.45391	0.726956	+	0.686684i	$0.240934\pi$
$762$	0	0
$763$	9.36932	0.339192
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	16.9848	0.613287
$768$	0	0
$769$	18.6307	0.671840	0.335920	−	0.941891i	$-0.390953\pi$
$769$	18.6307	0.671840	0.335920	+	0.941891i	$0.390953\pi$
$770$	0	0
$771$	35.8617	1.29153
$772$	0	0
$773$	13.5076	0.485834	0.242917	−	0.970047i	$-0.421896\pi$
$773$	13.5076	0.485834	0.242917	+	0.970047i	$0.421896\pi$
$774$	0	0
$775$	49.4773	1.77728
$776$	0	0
$777$	−11.6847	−0.419185
$778$	0	0
$779$	51.2311	1.83554
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2.87689	−0.102812
$784$	0	0
$785$	9.30019	0.331938
$786$	0	0
$787$	10.8769	0.387719	0.193860	−	0.981029i	$-0.437899\pi$
$787$	10.8769	0.387719	0.193860	+	0.981029i	$0.437899\pi$
$788$	0	0
$789$	24.6307	0.876876
$790$	0	0
$791$	5.68466	0.202123
$792$	0	0
$793$	−13.2614	−0.470925
$794$	0	0
$795$	6.10795	0.216627
$796$	0	0
$797$	12.0691	0.427511	0.213755	−	0.976887i	$-0.431430\pi$
$797$	12.0691	0.427511	0.213755	+	0.976887i	$0.431430\pi$
$798$	0	0
$799$	−12.4924	−0.441950
$800$	0	0
$801$	−44.7386	−1.58076
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−0.807764	−0.0284699
$806$	0	0
$807$	−19.8617	−0.699166
$808$	0	0
$809$	20.7386	0.729132	0.364566	−	0.931178i	$-0.381217\pi$
$809$	20.7386	0.729132	0.364566	+	0.931178i	$0.381217\pi$
$810$	0	0
$811$	5.12311	0.179897	0.0899483	−	0.995946i	$-0.471330\pi$
$811$	5.12311	0.179897	0.0899483	+	0.995946i	$0.471330\pi$
$812$	0	0
$813$	46.7386	1.63920
$814$	0	0
$815$	9.26137	0.324412
$816$	0	0
$817$	−20.4924	−0.716939
$818$	0	0
$819$	−11.1231	−0.388673
$820$	0	0
$821$	36.2462	1.26500	0.632501	−	0.774560i	$-0.282029\pi$
$821$	36.2462	1.26500	0.632501	+	0.774560i	$0.282029\pi$
$822$	0	0
$823$	−52.6695	−1.83594	−0.917972	−	0.396646i	$-0.870174\pi$
$823$	−52.6695	−1.83594	−0.917972	+	0.396646i	$0.870174\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−47.2311	−1.64238	−0.821192	−	0.570651i	$-0.806691\pi$
$827$	−47.2311	−1.64238	−0.821192	+	0.570651i	$0.806691\pi$
$828$	0	0
$829$	−6.17708	−0.214539	−0.107269	−	0.994230i	$-0.534211\pi$
$829$	−6.17708	−0.214539	−0.107269	+	0.994230i	$0.534211\pi$
$830$	0	0
$831$	22.7386	0.788794
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	4.13826	0.143210
$836$	0	0
$837$	15.1922	0.525120
$838$	0	0
$839$	7.68466	0.265304	0.132652	−	0.991163i	$-0.457651\pi$
$839$	7.68466	0.265304	0.132652	+	0.991163i	$0.457651\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	76.4924	2.63454
$844$	0	0
$845$	−1.82292	−0.0627103
$846$	0	0
$847$	0	0
$848$	0	0
$849$	72.9848	2.50483
$850$	0	0
$851$	−6.56155	−0.224927
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	10.2462	0.350413
$856$	0	0
$857$	−32.8769	−1.12305	−0.561527	−	0.827459i	$-0.689786\pi$
$857$	−32.8769	−1.12305	−0.561527	+	0.827459i	$0.689786\pi$
$858$	0	0
$859$	36.8078	1.25586	0.627932	−	0.778268i	$-0.283901\pi$
$859$	36.8078	1.25586	0.627932	+	0.778268i	$0.283901\pi$
$860$	0	0
$861$	−25.6155	−0.872975
$862$	0	0
$863$	51.2311	1.74393	0.871963	−	0.489572i	$-0.162847\pi$
$863$	51.2311	1.74393	0.871963	+	0.489572i	$0.162847\pi$
$864$	0	0
$865$	12.9848	0.441498
$866$	0	0
$867$	33.3002	1.13093
$868$	0	0
$869$	0	0
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	30.4924	1.03201
$874$	0	0
$875$	−5.43845	−0.183853
$876$	0	0
$877$	44.2462	1.49409	0.747044	−	0.664774i	$-0.231472\pi$
$877$	44.2462	1.49409	0.747044	+	0.664774i	$0.231472\pi$
$878$	0	0
$879$	−31.3693	−1.05806
$880$	0	0
$881$	−19.9309	−0.671488	−0.335744	−	0.941953i	$-0.608988\pi$
$881$	−19.9309	−0.671488	−0.335744	+	0.941953i	$0.608988\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	7.82292	0.262965
$886$	0	0
$887$	31.3693	1.05328	0.526639	−	0.850089i	$-0.323452\pi$
$887$	31.3693	1.05328	0.526639	+	0.850089i	$0.323452\pi$
$888$	0	0
$889$	5.12311	0.171823
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−32.0000	−1.07084
$894$	0	0
$895$	−11.6847	−0.390575
$896$	0	0
$897$	−11.5076	−0.384227
$898$	0	0
$899$	−21.1231	−0.704495
$900$	0	0
$901$	−8.49242	−0.282924
$902$	0	0
$903$	10.2462	0.340973
$904$	0	0
$905$	4.45360	0.148043
$906$	0	0
$907$	1.75379	0.0582336	0.0291168	−	0.999576i	$-0.490731\pi$
$907$	1.75379	0.0582336	0.0291168	+	0.999576i	$0.490731\pi$
$908$	0	0
$909$	−47.6155	−1.57931
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−6.10795	−0.201923
$916$	0	0
$917$	−13.1231	−0.433363
$918$	0	0
$919$	38.7386	1.27787	0.638935	−	0.769261i	$-0.279375\pi$
$919$	38.7386	1.27787	0.638935	+	0.769261i	$0.279375\pi$
$920$	0	0
$921$	52.4924	1.72968
$922$	0	0
$923$	−11.5076	−0.378777
$924$	0	0
$925$	−21.3693	−0.702619
$926$	0	0
$927$	58.7386	1.92923
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	5.12311	0.167903
$932$	0	0
$933$	−48.0000	−1.57145
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.8769	0.420670	0.210335	−	0.977629i	$-0.432545\pi$
$937$	12.8769	0.420670	0.210335	+	0.977629i	$0.432545\pi$
$938$	0	0
$939$	−40.8078	−1.33171
$940$	0	0
$941$	−4.73863	−0.154475	−0.0772375	−	0.997013i	$-0.524610\pi$
$941$	−4.73863	−0.154475	−0.0772375	+	0.997013i	$0.524610\pi$
$942$	0	0
$943$	−14.3845	−0.468423
$944$	0	0
$945$	−0.807764	−0.0262766
$946$	0	0
$947$	17.9309	0.582675	0.291337	−	0.956620i	$-0.405900\pi$
$947$	17.9309	0.582675	0.291337	+	0.956620i	$0.405900\pi$
$948$	0	0
$949$	−13.2614	−0.430482
$950$	0	0
$951$	33.4384	1.08432
$952$	0	0
$953$	54.3542	1.76070	0.880352	−	0.474321i	$-0.157306\pi$
$953$	54.3542	1.76070	0.880352	+	0.474321i	$0.157306\pi$
$954$	0	0
$955$	−15.1922	−0.491609
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.31534	−0.0747663
$960$	0	0
$961$	80.5464	2.59827
$962$	0	0
$963$	−60.9848	−1.96521
$964$	0	0
$965$	−1.12311	−0.0361540
$966$	0	0
$967$	37.1231	1.19380	0.596899	−	0.802316i	$-0.296399\pi$
$967$	37.1231	1.19380	0.596899	+	0.802316i	$0.296399\pi$
$968$	0	0
$969$	−26.2462	−0.843150
$970$	0	0
$971$	−47.0540	−1.51003	−0.755017	−	0.655705i	$-0.772371\pi$
$971$	−47.0540	−1.51003	−0.755017	+	0.655705i	$0.772371\pi$
$972$	0	0
$973$	5.12311	0.164239
$974$	0	0
$975$	−37.4773	−1.20023
$976$	0	0
$977$	4.42329	0.141514	0.0707568	−	0.997494i	$-0.477459\pi$
$977$	4.42329	0.141514	0.0707568	+	0.997494i	$0.477459\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	33.3693	1.06540
$982$	0	0
$983$	47.6847	1.52090	0.760452	−	0.649394i	$-0.224977\pi$
$983$	47.6847	1.52090	0.760452	+	0.649394i	$0.224977\pi$
$984$	0	0
$985$	11.3693	0.362257
$986$	0	0
$987$	16.0000	0.509286
$988$	0	0
$989$	5.75379	0.182960
$990$	0	0
$991$	−3.50758	−0.111422	−0.0557109	−	0.998447i	$-0.517743\pi$
$991$	−3.50758	−0.111422	−0.0557109	+	0.998447i	$0.517743\pi$
$992$	0	0
$993$	−32.8078	−1.04112
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	−13.3693	−0.423411	−0.211705	−	0.977334i	$-0.567902\pi$
$997$	−13.3693	−0.423411	−0.211705	+	0.977334i	$0.567902\pi$
$998$	0	0
$999$	−6.56155	−0.207598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6776.2.a.l.1.1		2
11.10	odd	2		616.2.a.f.1.1	✓	2
33.32	even	2		5544.2.a.bf.1.1		2
44.43	even	2		1232.2.a.o.1.2		2
77.76	even	2		4312.2.a.t.1.2		2
88.21	odd	2		4928.2.a.bs.1.2		2
88.43	even	2		4928.2.a.bo.1.1		2
308.307	odd	2		8624.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.a.f.1.1	✓	2	11.10	odd	2
1232.2.a.o.1.2		2	44.43	even	2
4312.2.a.t.1.2		2	77.76	even	2
4928.2.a.bo.1.1		2	88.43	even	2
4928.2.a.bs.1.2		2	88.21	odd	2
5544.2.a.bf.1.1		2	33.32	even	2
6776.2.a.l.1.1		2	1.1	even	1	trivial
8624.2.a.bi.1.1		2	308.307	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6776 = 2^{3} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6776.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} +0.561553 q^{5} +1.00000 q^{7} +3.56155 q^{9} -3.12311 q^{13} -1.43845 q^{15} +2.00000 q^{17} +5.12311 q^{19} -2.56155 q^{21} -1.43845 q^{23} -4.68466 q^{25} -1.43845 q^{27} +2.00000 q^{29} -10.5616 q^{31} +0.561553 q^{35} +4.56155 q^{37} +8.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} +2.00000 q^{45} -6.24621 q^{47} +1.00000 q^{49} -5.12311 q^{51} -4.24621 q^{53} -13.1231 q^{57} -5.43845 q^{59} +4.24621 q^{61} +3.56155 q^{63} -1.75379 q^{65} -2.56155 q^{67} +3.68466 q^{69} +3.68466 q^{71} +4.24621 q^{73} +12.0000 q^{75} -5.12311 q^{79} -7.00000 q^{81} +8.00000 q^{83} +1.12311 q^{85} -5.12311 q^{87} -12.5616 q^{89} -3.12311 q^{91} +27.0540 q^{93} +2.87689 q^{95} +8.56155 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} + 2 q^{13} - 7 q^{15} + 4 q^{17} + 2 q^{19} - q^{21} - 7 q^{23} + 3 q^{25} - 7 q^{27} + 4 q^{29} - 17 q^{31} - 3 q^{35} + 5 q^{37} + 16 q^{39} + 20 q^{41} - 8 q^{43}+ \cdots + 13 q^{97}+O(q^{100}) \) 2 * q - q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 + 2 * q^13 - 7 * q^15 + 4 * q^17 + 2 * q^19 - q^21 - 7 * q^23 + 3 * q^25 - 7 * q^27 + 4 * q^29 - 17 * q^31 - 3 * q^35 + 5 * q^37 + 16 * q^39 + 20 * q^41 - 8 * q^43 + 4 * q^45 + 4 * q^47 + 2 * q^49 - 2 * q^51 + 8 * q^53 - 18 * q^57 - 15 * q^59 - 8 * q^61 + 3 * q^63 - 20 * q^65 - q^67 - 5 * q^69 - 5 * q^71 - 8 * q^73 + 24 * q^75 - 2 * q^79 - 14 * q^81 + 16 * q^83 - 6 * q^85 - 2 * q^87 - 21 * q^89 + 2 * q^91 + 17 * q^93 + 14 * q^95 + 13 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more