LMFDB - Embedded newform 6776.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6776,2,Mod(1,6776)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$54.1066324096$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.11491$ 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)

$f(q)$	$=$	$q+0.642074 q^{3} +0.642074 q^{5} -1.00000 q^{7} -2.58774 q^{9} -2.22982 q^{13} +0.412259 q^{15} -7.17548 q^{17} -4.94567 q^{19} -0.642074 q^{21} +6.30359 q^{23} -4.58774 q^{25} -3.58774 q^{27} +6.45963 q^{29} -1.58774 q^{31} -0.642074 q^{35} +4.30359 q^{37} -1.43171 q^{39} +9.74378 q^{41} -8.45963 q^{43} -1.66152 q^{45} +4.71585 q^{47} +1.00000 q^{49} -4.60719 q^{51} +6.45963 q^{53} -3.17548 q^{57} +5.92622 q^{59} +10.8370 q^{61} +2.58774 q^{63} -1.43171 q^{65} -14.7632 q^{67} +4.04737 q^{69} +10.1560 q^{71} +9.28415 q^{73} -2.94567 q^{75} +7.17548 q^{79} +5.45963 q^{81} +2.22982 q^{83} -4.60719 q^{85} +4.14756 q^{87} +11.4791 q^{89} +2.22982 q^{91} -1.01945 q^{93} -3.17548 q^{95} -17.5877 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + q^{5} - 3 q^{7} + 4 q^{9} + 6 q^{13} + 13 q^{15} + 2 q^{17} - 4 q^{19} - q^{21} + 9 q^{23} - 2 q^{25} + q^{27} - 6 q^{29} + 7 q^{31} - q^{35} + 3 q^{37} - 8 q^{39} + 2 q^{41} + 4 q^{45}+ \cdots - 41 q^{97}+O(q^{100}) $ 3 * q + q^3 + q^5 - 3 * q^7 + 4 * q^9 + 6 * q^13 + 13 * q^15 + 2 * q^17 - 4 * q^19 - q^21 + 9 * q^23 - 2 * q^25 + q^27 - 6 * q^29 + 7 * q^31 - q^35 + 3 * q^37 - 8 * q^39 + 2 * q^41 + 4 * q^45 + 16 * q^47 + 3 * q^49 + 6 * q^51 - 6 * q^53 + 14 * q^57 + 15 * q^59 - 4 * q^63 - 8 * q^65 - 9 * q^67 - 25 * q^69 + 15 * q^71 + 26 * q^73 + 2 * q^75 - 2 * q^79 - 9 * q^81 - 6 * q^83 + 6 * q^85 + 18 * q^87 + q^89 - 6 * q^91 + 5 * q^93 + 14 * q^95 - 41 * 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
$3$	0.642074	0.370701	0.185351	−	0.982672i	$-0.440658\pi$
$3$	0.642074	0.370701	0.185351	+	0.982672i	$0.440658\pi$
$4$	0	0
$5$	0.642074	0.287144	0.143572	−	0.989640i	$-0.454141\pi$
$5$	0.642074	0.287144	0.143572	+	0.989640i	$0.454141\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.58774	−0.862580
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.22982	−0.618439	−0.309220	−	0.950991i	$-0.600068\pi$
$13$	−2.22982	−0.618439	−0.309220	+	0.950991i	$0.600068\pi$
$14$	0	0
$15$	0.412259	0.106445
$16$	0	0
$17$	−7.17548	−1.74031	−0.870155	−	0.492778i	$-0.835981\pi$
$17$	−7.17548	−1.74031	−0.870155	+	0.492778i	$0.835981\pi$
$18$	0	0
$19$	−4.94567	−1.13461	−0.567307	−	0.823506i	$-0.692015\pi$
$19$	−4.94567	−1.13461	−0.567307	+	0.823506i	$0.692015\pi$
$20$	0	0
$21$	−0.642074	−0.140112
$22$	0	0
$23$	6.30359	1.31439	0.657195	−	0.753720i	$-0.271743\pi$
$23$	6.30359	1.31439	0.657195	+	0.753720i	$0.271743\pi$
$24$	0	0
$25$	−4.58774	−0.917548
$26$	0	0
$27$	−3.58774	−0.690461
$28$	0	0
$29$	6.45963	1.19952	0.599762	−	0.800179i	$-0.295262\pi$
$29$	6.45963	1.19952	0.599762	+	0.800179i	$0.295262\pi$
$30$	0	0
$31$	−1.58774	−0.285167	−0.142583	−	0.989783i	$-0.545541\pi$
$31$	−1.58774	−0.285167	−0.142583	+	0.989783i	$0.545541\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.642074	−0.108530
$36$	0	0
$37$	4.30359	0.707507	0.353753	−	0.935339i	$-0.384905\pi$
$37$	4.30359	0.707507	0.353753	+	0.935339i	$0.384905\pi$
$38$	0	0
$39$	−1.43171	−0.229256
$40$	0	0
$41$	9.74378	1.52172	0.760861	−	0.648915i	$-0.224777\pi$
$41$	9.74378	1.52172	0.760861	+	0.648915i	$0.224777\pi$
$42$	0	0
$43$	−8.45963	−1.29008	−0.645041	−	0.764148i	$-0.723160\pi$
$43$	−8.45963	−1.29008	−0.645041	+	0.764148i	$0.723160\pi$
$44$	0	0
$45$	−1.66152	−0.247685
$46$	0	0
$47$	4.71585	0.687878	0.343939	−	0.938992i	$-0.388239\pi$
$47$	4.71585	0.687878	0.343939	+	0.938992i	$0.388239\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.60719	−0.645135
$52$	0	0
$53$	6.45963	0.887298	0.443649	−	0.896201i	$-0.353684\pi$
$53$	6.45963	0.887298	0.443649	+	0.896201i	$0.353684\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.17548	−0.420603
$58$	0	0
$59$	5.92622	0.771528	0.385764	−	0.922597i	$-0.373938\pi$
$59$	5.92622	0.771528	0.385764	+	0.922597i	$0.373938\pi$
$60$	0	0
$61$	10.8370	1.38754	0.693768	−	0.720199i	$-0.255949\pi$
$61$	10.8370	1.38754	0.693768	+	0.720199i	$0.255949\pi$
$62$	0	0
$63$	2.58774	0.326025
$64$	0	0
$65$	−1.43171	−0.177581
$66$	0	0
$67$	−14.7632	−1.80361	−0.901807	−	0.432138i	$-0.857759\pi$
$67$	−14.7632	−1.80361	−0.901807	+	0.432138i	$0.857759\pi$
$68$	0	0
$69$	4.04737	0.487246
$70$	0	0
$71$	10.1560	1.20530	0.602650	−	0.798006i	$-0.294112\pi$
$71$	10.1560	1.20530	0.602650	+	0.798006i	$0.294112\pi$
$72$	0	0
$73$	9.28415	1.08663	0.543314	−	0.839530i	$-0.317169\pi$
$73$	9.28415	1.08663	0.543314	+	0.839530i	$0.317169\pi$
$74$	0	0
$75$	−2.94567	−0.340136
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.17548	0.807305	0.403652	−	0.914912i	$-0.367740\pi$
$79$	7.17548	0.807305	0.403652	+	0.914912i	$0.367740\pi$
$80$	0	0
$81$	5.45963	0.606626
$82$	0	0
$83$	2.22982	0.244754	0.122377	−	0.992484i	$-0.460948\pi$
$83$	2.22982	0.244754	0.122377	+	0.992484i	$0.460948\pi$
$84$	0	0
$85$	−4.60719	−0.499720
$86$	0	0
$87$	4.14756	0.444665
$88$	0	0
$89$	11.4791	1.21678	0.608390	−	0.793638i	$-0.291816\pi$
$89$	11.4791	1.21678	0.608390	+	0.793638i	$0.291816\pi$
$90$	0	0
$91$	2.22982	0.233748
$92$	0	0
$93$	−1.01945	−0.105712
$94$	0	0
$95$	−3.17548	−0.325798
$96$	0	0
$97$	−17.5877	−1.78576	−0.892882	−	0.450290i	$-0.851321\pi$
$97$	−17.5877	−1.78576	−0.892882	+	0.450290i	$0.851321\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.5808	1.64985	0.824925	−	0.565243i	$-0.191217\pi$
$101$	16.5808	1.64985	0.824925	+	0.565243i	$0.191217\pi$
$102$	0	0
$103$	15.2841	1.50599	0.752996	−	0.658025i	$-0.228608\pi$
$103$	15.2841	1.50599	0.752996	+	0.658025i	$0.228608\pi$
$104$	0	0
$105$	−0.412259	−0.0402323
$106$	0	0
$107$	15.6351	1.51150	0.755752	−	0.654858i	$-0.227272\pi$
$107$	15.6351	1.51150	0.755752	+	0.654858i	$0.227272\pi$
$108$	0	0
$109$	9.17548	0.878852	0.439426	−	0.898279i	$-0.355182\pi$
$109$	9.17548	0.878852	0.439426	+	0.898279i	$0.355182\pi$
$110$	0	0
$111$	2.76322	0.262274
$112$	0	0
$113$	−11.3315	−1.06598	−0.532990	−	0.846122i	$-0.678932\pi$
$113$	−11.3315	−1.06598	−0.532990	+	0.846122i	$0.678932\pi$
$114$	0	0
$115$	4.04737	0.377419
$116$	0	0
$117$	5.77018	0.533454
$118$	0	0
$119$	7.17548	0.657775
$120$	0	0
$121$	0	0
$122$	0	0
$123$	6.25622	0.564105
$124$	0	0
$125$	−6.15604	−0.550613
$126$	0	0
$127$	−16.6072	−1.47365	−0.736825	−	0.676084i	$-0.763676\pi$
$127$	−16.6072	−1.47365	−0.736825	+	0.676084i	$0.763676\pi$
$128$	0	0
$129$	−5.43171	−0.478235
$130$	0	0
$131$	10.8370	0.946833	0.473417	−	0.880839i	$-0.343020\pi$
$131$	10.8370	0.946833	0.473417	+	0.880839i	$0.343020\pi$
$132$	0	0
$133$	4.94567	0.428844
$134$	0	0
$135$	−2.30359	−0.198262
$136$	0	0
$137$	−1.73530	−0.148257	−0.0741283	−	0.997249i	$-0.523617\pi$
$137$	−1.73530	−0.148257	−0.0741283	+	0.997249i	$0.523617\pi$
$138$	0	0
$139$	15.5140	1.31588	0.657939	−	0.753072i	$-0.271429\pi$
$139$	15.5140	1.31588	0.657939	+	0.753072i	$0.271429\pi$
$140$	0	0
$141$	3.02792	0.254997
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.14756	0.344436
$146$	0	0
$147$	0.642074	0.0529573
$148$	0	0
$149$	−19.8913	−1.62956	−0.814781	−	0.579769i	$-0.803143\pi$
$149$	−19.8913	−1.62956	−0.814781	+	0.579769i	$0.803143\pi$
$150$	0	0
$151$	4.82452	0.392614	0.196307	−	0.980543i	$-0.437105\pi$
$151$	4.82452	0.392614	0.196307	+	0.980543i	$0.437105\pi$
$152$	0	0
$153$	18.5683	1.50116
$154$	0	0
$155$	−1.01945	−0.0818839
$156$	0	0
$157$	−0.0348853	−0.00278415	−0.00139207	−	0.999999i	$-0.500443\pi$
$157$	−0.0348853	−0.00278415	−0.00139207	+	0.999999i	$0.500443\pi$
$158$	0	0
$159$	4.14756	0.328923
$160$	0	0
$161$	−6.30359	−0.496793
$162$	0	0
$163$	1.89134	0.148141	0.0740704	−	0.997253i	$-0.476401\pi$
$163$	1.89134	0.148141	0.0740704	+	0.997253i	$0.476401\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.7159	0.829218	0.414609	−	0.910000i	$-0.363918\pi$
$167$	10.7159	0.829218	0.414609	+	0.910000i	$0.363918\pi$
$168$	0	0
$169$	−8.02792	−0.617533
$170$	0	0
$171$	12.7981	0.978696
$172$	0	0
$173$	−16.5808	−1.26061	−0.630307	−	0.776346i	$-0.717071\pi$
$173$	−16.5808	−1.26061	−0.630307	+	0.776346i	$0.717071\pi$
$174$	0	0
$175$	4.58774	0.346801
$176$	0	0
$177$	3.80507	0.286007
$178$	0	0
$179$	−11.4402	−0.855079	−0.427540	−	0.903997i	$-0.640620\pi$
$179$	−11.4402	−0.855079	−0.427540	+	0.903997i	$0.640620\pi$
$180$	0	0
$181$	−6.07378	−0.451460	−0.225730	−	0.974190i	$-0.572477\pi$
$181$	−6.07378	−0.451460	−0.225730	+	0.974190i	$0.572477\pi$
$182$	0	0
$183$	6.95815	0.514362
$184$	0	0
$185$	2.76322	0.203156
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.58774	0.260970
$190$	0	0
$191$	22.1560	1.60315	0.801577	−	0.597891i	$-0.203995\pi$
$191$	22.1560	1.60315	0.801577	+	0.597891i	$0.203995\pi$
$192$	0	0
$193$	−8.56829	−0.616759	−0.308380	−	0.951263i	$-0.599787\pi$
$193$	−8.56829	−0.616759	−0.308380	+	0.951263i	$0.599787\pi$
$194$	0	0
$195$	−0.919260	−0.0658296
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−12.2034	−0.865077	−0.432538	−	0.901616i	$-0.642382\pi$
$199$	−12.2034	−0.865077	−0.432538	+	0.901616i	$0.642382\pi$
$200$	0	0
$201$	−9.47908	−0.668603
$202$	0	0
$203$	−6.45963	−0.453377
$204$	0	0
$205$	6.25622	0.436954
$206$	0	0
$207$	−16.3121	−1.13377
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.7717	1.42998	0.714991	−	0.699133i	$-0.246431\pi$
$211$	20.7717	1.42998	0.714991	+	0.699133i	$0.246431\pi$
$212$	0	0
$213$	6.52092	0.446806
$214$	0	0
$215$	−5.43171	−0.370439
$216$	0	0
$217$	1.58774	0.107783
$218$	0	0
$219$	5.96111	0.402814
$220$	0	0
$221$	16.0000	1.07628
$222$	0	0
$223$	−7.47908	−0.500836	−0.250418	−	0.968138i	$-0.580568\pi$
$223$	−7.47908	−0.500836	−0.250418	+	0.968138i	$0.580568\pi$
$224$	0	0
$225$	11.8719	0.791459
$226$	0	0
$227$	14.2298	0.944466	0.472233	−	0.881474i	$-0.343448\pi$
$227$	14.2298	0.944466	0.472233	+	0.881474i	$0.343448\pi$
$228$	0	0
$229$	−17.5613	−1.16049	−0.580243	−	0.814444i	$-0.697042\pi$
$229$	−17.5613	−1.16049	−0.580243	+	0.814444i	$0.697042\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.4876	−1.66975	−0.834873	−	0.550443i	$-0.814459\pi$
$233$	−25.4876	−1.66975	−0.834873	+	0.550443i	$0.814459\pi$
$234$	0	0
$235$	3.02792	0.197520
$236$	0	0
$237$	4.60719	0.299269
$238$	0	0
$239$	−10.3510	−0.669548	−0.334774	−	0.942298i	$-0.608660\pi$
$239$	−10.3510	−0.669548	−0.334774	+	0.942298i	$0.608660\pi$
$240$	0	0
$241$	−6.35097	−0.409102	−0.204551	−	0.978856i	$-0.565573\pi$
$241$	−6.35097	−0.409102	−0.204551	+	0.978856i	$0.565573\pi$
$242$	0	0
$243$	14.2687	0.915338
$244$	0	0
$245$	0.642074	0.0410206
$246$	0	0
$247$	11.0279	0.701690
$248$	0	0
$249$	1.43171	0.0907306
$250$	0	0
$251$	3.81756	0.240962	0.120481	−	0.992716i	$-0.461556\pi$
$251$	3.81756	0.240962	0.120481	+	0.992716i	$0.461556\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2.95815	−0.185247
$256$	0	0
$257$	−9.78267	−0.610226	−0.305113	−	0.952316i	$-0.598694\pi$
$257$	−9.78267	−0.610226	−0.305113	+	0.952316i	$0.598694\pi$
$258$	0	0
$259$	−4.30359	−0.267412
$260$	0	0
$261$	−16.7159	−1.03469
$262$	0	0
$263$	2.03889	0.125724	0.0628618	−	0.998022i	$-0.479977\pi$
$263$	2.03889	0.125724	0.0628618	+	0.998022i	$0.479977\pi$
$264$	0	0
$265$	4.14756	0.254782
$266$	0	0
$267$	7.37041	0.451062
$268$	0	0
$269$	17.6615	1.07684	0.538421	−	0.842676i	$-0.319021\pi$
$269$	17.6615	1.07684	0.538421	+	0.842676i	$0.319021\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	1.43171	0.0866508
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.85244	0.351639	0.175820	−	0.984422i	$-0.443742\pi$
$277$	5.85244	0.351639	0.175820	+	0.984422i	$0.443742\pi$
$278$	0	0
$279$	4.10866	0.245979
$280$	0	0
$281$	−23.7438	−1.41644	−0.708218	−	0.705994i	$-0.750500\pi$
$281$	−23.7438	−1.41644	−0.708218	+	0.705994i	$0.750500\pi$
$282$	0	0
$283$	15.4442	0.918062	0.459031	−	0.888420i	$-0.348197\pi$
$283$	15.4442	0.918062	0.459031	+	0.888420i	$0.348197\pi$
$284$	0	0
$285$	−2.03889	−0.120774
$286$	0	0
$287$	−9.74378	−0.575157
$288$	0	0
$289$	34.4876	2.02868
$290$	0	0
$291$	−11.2926	−0.661985
$292$	0	0
$293$	−0.0264075	−0.00154274	−0.000771371	−	1.00000i	$-0.500246\pi$
$293$	−0.0264075	−0.00154274	−0.000771371		1.00000i	$0.500246\pi$
$294$	0	0
$295$	3.80507	0.221540
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.0558	−0.812871
$300$	0	0
$301$	8.45963	0.487605
$302$	0	0
$303$	10.6461	0.611601
$304$	0	0
$305$	6.95815	0.398423
$306$	0	0
$307$	8.79811	0.502135	0.251067	−	0.967970i	$-0.419218\pi$
$307$	8.79811	0.502135	0.251067	+	0.967970i	$0.419218\pi$
$308$	0	0
$309$	9.81355	0.558273
$310$	0	0
$311$	12.0389	0.682663	0.341332	−	0.939943i	$-0.389122\pi$
$311$	12.0389	0.682663	0.341332	+	0.939943i	$0.389122\pi$
$312$	0	0
$313$	−0.980553	−0.0554241	−0.0277121	−	0.999616i	$-0.508822\pi$
$313$	−0.980553	−0.0554241	−0.0277121	+	0.999616i	$0.508822\pi$
$314$	0	0
$315$	1.66152	0.0936161
$316$	0	0
$317$	−18.6546	−1.04774	−0.523872	−	0.851797i	$-0.675513\pi$
$317$	−18.6546	−1.04774	−0.523872	+	0.851797i	$0.675513\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	10.0389	0.560316
$322$	0	0
$323$	35.4876	1.97458
$324$	0	0
$325$	10.2298	0.567448
$326$	0	0
$327$	5.89134	0.325792
$328$	0	0
$329$	−4.71585	−0.259993
$330$	0	0
$331$	3.44018	0.189090	0.0945448	−	0.995521i	$-0.469860\pi$
$331$	3.44018	0.189090	0.0945448	+	0.995521i	$0.469860\pi$
$332$	0	0
$333$	−11.1366	−0.610281
$334$	0	0
$335$	−9.47908	−0.517897
$336$	0	0
$337$	−2.91926	−0.159022	−0.0795111	−	0.996834i	$-0.525336\pi$
$337$	−2.91926	−0.159022	−0.0795111	+	0.996834i	$0.525336\pi$
$338$	0	0
$339$	−7.27567	−0.395160
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.59871	0.139910
$346$	0	0
$347$	−23.6351	−1.26880	−0.634400	−	0.773005i	$-0.718753\pi$
$347$	−23.6351	−1.26880	−0.634400	+	0.773005i	$0.718753\pi$
$348$	0	0
$349$	−1.62263	−0.0868572	−0.0434286	−	0.999057i	$-0.513828\pi$
$349$	−1.62263	−0.0868572	−0.0434286	+	0.999057i	$0.513828\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	15.6964	0.835435	0.417718	−	0.908577i	$-0.362830\pi$
$353$	15.6964	0.835435	0.417718	+	0.908577i	$0.362830\pi$
$354$	0	0
$355$	6.52092	0.346095
$356$	0	0
$357$	4.60719	0.243838
$358$	0	0
$359$	−9.13659	−0.482211	−0.241105	−	0.970499i	$-0.577510\pi$
$359$	−9.13659	−0.482211	−0.241105	+	0.970499i	$0.577510\pi$
$360$	0	0
$361$	5.45963	0.287349
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.96111	0.312019
$366$	0	0
$367$	29.5180	1.54083	0.770413	−	0.637545i	$-0.220050\pi$
$367$	29.5180	1.54083	0.770413	+	0.637545i	$0.220050\pi$
$368$	0	0
$369$	−25.2144	−1.31261
$370$	0	0
$371$	−6.45963	−0.335367
$372$	0	0
$373$	14.0947	0.729798	0.364899	−	0.931047i	$-0.381103\pi$
$373$	14.0947	0.729798	0.364899	+	0.931047i	$0.381103\pi$
$374$	0	0
$375$	−3.95263	−0.204113
$376$	0	0
$377$	−14.4038	−0.741832
$378$	0	0
$379$	−21.9387	−1.12692	−0.563458	−	0.826145i	$-0.690529\pi$
$379$	−21.9387	−1.12692	−0.563458	+	0.826145i	$0.690529\pi$
$380$	0	0
$381$	−10.6630	−0.546284
$382$	0	0
$383$	−6.72433	−0.343597	−0.171799	−	0.985132i	$-0.554958\pi$
$383$	−6.72433	−0.343597	−0.171799	+	0.985132i	$0.554958\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	21.8913	1.11280
$388$	0	0
$389$	33.3704	1.69195	0.845974	−	0.533225i	$-0.179020\pi$
$389$	33.3704	1.69195	0.845974	+	0.533225i	$0.179020\pi$
$390$	0	0
$391$	−45.2313	−2.28745
$392$	0	0
$393$	6.95815	0.350992
$394$	0	0
$395$	4.60719	0.231813
$396$	0	0
$397$	−10.3385	−0.518873	−0.259437	−	0.965760i	$-0.583537\pi$
$397$	−10.3385	−0.518873	−0.259437	+	0.965760i	$0.583537\pi$
$398$	0	0
$399$	3.17548	0.158973
$400$	0	0
$401$	20.1087	1.00418	0.502089	−	0.864816i	$-0.332565\pi$
$401$	20.1087	1.00418	0.502089	+	0.864816i	$0.332565\pi$
$402$	0	0
$403$	3.54037	0.176358
$404$	0	0
$405$	3.50548	0.174189
$406$	0	0
$407$	0	0
$408$	0	0
$409$	16.3121	0.806580	0.403290	−	0.915072i	$-0.367867\pi$
$409$	16.3121	0.806580	0.403290	+	0.915072i	$0.367867\pi$
$410$	0	0
$411$	−1.11419	−0.0549589
$412$	0	0
$413$	−5.92622	−0.291610
$414$	0	0
$415$	1.43171	0.0702797
$416$	0	0
$417$	9.96111	0.487797
$418$	0	0
$419$	−7.62263	−0.372390	−0.186195	−	0.982513i	$-0.559616\pi$
$419$	−7.62263	−0.372390	−0.186195	+	0.982513i	$0.559616\pi$
$420$	0	0
$421$	−21.0279	−1.02484	−0.512419	−	0.858735i	$-0.671251\pi$
$421$	−21.0279	−1.02484	−0.512419	+	0.858735i	$0.671251\pi$
$422$	0	0
$423$	−12.2034	−0.593350
$424$	0	0
$425$	32.9193	1.59682
$426$	0	0
$427$	−10.8370	−0.524439
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.8385	1.43727	0.718635	−	0.695387i	$-0.244767\pi$
$431$	29.8385	1.43727	0.718635	+	0.695387i	$0.244767\pi$
$432$	0	0
$433$	−3.23678	−0.155550	−0.0777748	−	0.996971i	$-0.524782\pi$
$433$	−3.23678	−0.155550	−0.0777748	+	0.996971i	$0.524782\pi$
$434$	0	0
$435$	2.66304	0.127683
$436$	0	0
$437$	−31.1755	−1.49133
$438$	0	0
$439$	−28.0947	−1.34089	−0.670444	−	0.741960i	$-0.733897\pi$
$439$	−28.0947	−1.34089	−0.670444	+	0.741960i	$0.733897\pi$
$440$	0	0
$441$	−2.58774	−0.123226
$442$	0	0
$443$	9.69641	0.460690	0.230345	−	0.973109i	$-0.426015\pi$
$443$	9.69641	0.460690	0.230345	+	0.973109i	$0.426015\pi$
$444$	0	0
$445$	7.37041	0.349391
$446$	0	0
$447$	−12.7717	−0.604081
$448$	0	0
$449$	−6.04737	−0.285393	−0.142697	−	0.989766i	$-0.545577\pi$
$449$	−6.04737	−0.285393	−0.142697	+	0.989766i	$0.545577\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.09770	0.145542
$454$	0	0
$455$	1.43171	0.0671194
$456$	0	0
$457$	12.0389	0.563156	0.281578	−	0.959538i	$-0.409142\pi$
$457$	12.0389	0.563156	0.281578	+	0.959538i	$0.409142\pi$
$458$	0	0
$459$	25.7438	1.20162
$460$	0	0
$461$	−19.7563	−0.920141	−0.460070	−	0.887882i	$-0.652176\pi$
$461$	−19.7563	−0.920141	−0.460070	+	0.887882i	$0.652176\pi$
$462$	0	0
$463$	41.4263	1.92524	0.962621	−	0.270853i	$-0.0873056\pi$
$463$	41.4263	1.92524	0.962621	+	0.270853i	$0.0873056\pi$
$464$	0	0
$465$	−0.654560	−0.0303545
$466$	0	0
$467$	−5.77866	−0.267405	−0.133702	−	0.991022i	$-0.542687\pi$
$467$	−5.77866	−0.267405	−0.133702	+	0.991022i	$0.542687\pi$
$468$	0	0
$469$	14.7632	0.681702
$470$	0	0
$471$	−0.0223989	−0.00103209
$472$	0	0
$473$	0	0
$474$	0	0
$475$	22.6894	1.04106
$476$	0	0
$477$	−16.7159	−0.765366
$478$	0	0
$479$	27.9472	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$479$	27.9472	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$480$	0	0
$481$	−9.59622	−0.437550
$482$	0	0
$483$	−4.04737	−0.184162
$484$	0	0
$485$	−11.2926	−0.512772
$486$	0	0
$487$	15.2229	0.689813	0.344907	−	0.938637i	$-0.387911\pi$
$487$	15.2229	0.689813	0.344907	+	0.938637i	$0.387911\pi$
$488$	0	0
$489$	1.21438	0.0549160
$490$	0	0
$491$	−25.8216	−1.16531	−0.582655	−	0.812719i	$-0.697986\pi$
$491$	−25.8216	−1.16531	−0.582655	+	0.812719i	$0.697986\pi$
$492$	0	0
$493$	−46.3510	−2.08754
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.1560	−0.455560
$498$	0	0
$499$	0.972075	0.0435161	0.0217580	−	0.999763i	$-0.493074\pi$
$499$	0.972075	0.0435161	0.0217580	+	0.999763i	$0.493074\pi$
$500$	0	0
$501$	6.88037	0.307392
$502$	0	0
$503$	12.9193	0.576041	0.288021	−	0.957624i	$-0.407003\pi$
$503$	12.9193	0.576041	0.288021	+	0.957624i	$0.407003\pi$
$504$	0	0
$505$	10.6461	0.473744
$506$	0	0
$507$	−5.15452	−0.228920
$508$	0	0
$509$	−19.5055	−0.864565	−0.432283	−	0.901738i	$-0.642292\pi$
$509$	−19.5055	−0.864565	−0.432283	+	0.901738i	$0.642292\pi$
$510$	0	0
$511$	−9.28415	−0.410706
$512$	0	0
$513$	17.7438	0.783407
$514$	0	0
$515$	9.81355	0.432437
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−10.6461	−0.467311
$520$	0	0
$521$	−1.88286	−0.0824895	−0.0412448	−	0.999149i	$-0.513132\pi$
$521$	−1.88286	−0.0824895	−0.0412448	+	0.999149i	$0.513132\pi$
$522$	0	0
$523$	−23.8260	−1.04184	−0.520920	−	0.853606i	$-0.674411\pi$
$523$	−23.8260	−1.04184	−0.520920	+	0.853606i	$0.674411\pi$
$524$	0	0
$525$	2.94567	0.128559
$526$	0	0
$527$	11.3928	0.496279
$528$	0	0
$529$	16.7353	0.727622
$530$	0	0
$531$	−15.3355	−0.665505
$532$	0	0
$533$	−21.7268	−0.941093
$534$	0	0
$535$	10.0389	0.434019
$536$	0	0
$537$	−7.34544	−0.316979
$538$	0	0
$539$	0	0
$540$	0	0
$541$	24.0389	1.03351	0.516756	−	0.856132i	$-0.327139\pi$
$541$	24.0389	1.03351	0.516756	+	0.856132i	$0.327139\pi$
$542$	0	0
$543$	−3.89981	−0.167357
$544$	0	0
$545$	5.89134	0.252357
$546$	0	0
$547$	25.4317	1.08738	0.543691	−	0.839286i	$-0.317027\pi$
$547$	25.4317	1.08738	0.543691	+	0.839286i	$0.317027\pi$
$548$	0	0
$549$	−28.0434	−1.19686
$550$	0	0
$551$	−31.9472	−1.36100
$552$	0	0
$553$	−7.17548	−0.305133
$554$	0	0
$555$	1.77419	0.0753103
$556$	0	0
$557$	6.91926	0.293178	0.146589	−	0.989197i	$-0.453170\pi$
$557$	6.91926	0.293178	0.146589	+	0.989197i	$0.453170\pi$
$558$	0	0
$559$	18.8634	0.797837
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.33848	0.351425	0.175713	−	0.984442i	$-0.443777\pi$
$563$	8.33848	0.351425	0.175713	+	0.984442i	$0.443777\pi$
$564$	0	0
$565$	−7.27567	−0.306090
$566$	0	0
$567$	−5.45963	−0.229283
$568$	0	0
$569$	−38.7019	−1.62247	−0.811235	−	0.584721i	$-0.801204\pi$
$569$	−38.7019	−1.62247	−0.811235	+	0.584721i	$0.801204\pi$
$570$	0	0
$571$	−1.13659	−0.0475648	−0.0237824	−	0.999717i	$-0.507571\pi$
$571$	−1.13659	−0.0475648	−0.0237824	+	0.999717i	$0.507571\pi$
$572$	0	0
$573$	14.2258	0.594292
$574$	0	0
$575$	−28.9193	−1.20602
$576$	0	0
$577$	−29.3176	−1.22051	−0.610254	−	0.792206i	$-0.708933\pi$
$577$	−29.3176	−1.22051	−0.610254	+	0.792206i	$0.708933\pi$
$578$	0	0
$579$	−5.50148	−0.228634
$580$	0	0
$581$	−2.22982	−0.0925083
$582$	0	0
$583$	0	0
$584$	0	0
$585$	3.70488	0.153178
$586$	0	0
$587$	12.3774	0.510869	0.255434	−	0.966826i	$-0.417782\pi$
$587$	12.3774	0.510869	0.255434	+	0.966826i	$0.417782\pi$
$588$	0	0
$589$	7.85244	0.323554
$590$	0	0
$591$	−1.28415	−0.0528228
$592$	0	0
$593$	37.0668	1.52215	0.761076	−	0.648663i	$-0.224671\pi$
$593$	37.0668	1.52215	0.761076	+	0.648663i	$0.224671\pi$
$594$	0	0
$595$	4.60719	0.188876
$596$	0	0
$597$	−7.83549	−0.320685
$598$	0	0
$599$	1.59622	0.0652197	0.0326099	−	0.999468i	$-0.489618\pi$
$599$	1.59622	0.0652197	0.0326099	+	0.999468i	$0.489618\pi$
$600$	0	0
$601$	23.3230	0.951367	0.475683	−	0.879617i	$-0.342201\pi$
$601$	23.3230	0.951367	0.475683	+	0.879617i	$0.342201\pi$
$602$	0	0
$603$	38.2034	1.55576
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9.74378	−0.395488	−0.197744	−	0.980254i	$-0.563361\pi$
$607$	−9.74378	−0.395488	−0.197744	+	0.980254i	$0.563361\pi$
$608$	0	0
$609$	−4.14756	−0.168068
$610$	0	0
$611$	−10.5155	−0.425411
$612$	0	0
$613$	−16.4207	−0.663227	−0.331614	−	0.943415i	$-0.607593\pi$
$613$	−16.4207	−0.663227	−0.331614	+	0.943415i	$0.607593\pi$
$614$	0	0
$615$	4.01696	0.161979
$616$	0	0
$617$	15.3789	0.619131	0.309565	−	0.950878i	$-0.399816\pi$
$617$	15.3789	0.619131	0.309565	+	0.950878i	$0.399816\pi$
$618$	0	0
$619$	35.9651	1.44556	0.722780	−	0.691078i	$-0.242864\pi$
$619$	35.9651	1.44556	0.722780	+	0.691078i	$0.242864\pi$
$620$	0	0
$621$	−22.6157	−0.907535
$622$	0	0
$623$	−11.4791	−0.459900
$624$	0	0
$625$	18.9861	0.759443
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−30.8804	−1.23128
$630$	0	0
$631$	−31.9946	−1.27368	−0.636842	−	0.770995i	$-0.719759\pi$
$631$	−31.9946	−1.27368	−0.636842	+	0.770995i	$0.719759\pi$
$632$	0	0
$633$	13.3370	0.530097
$634$	0	0
$635$	−10.6630	−0.423150
$636$	0	0
$637$	−2.22982	−0.0883485
$638$	0	0
$639$	−26.2812	−1.03967
$640$	0	0
$641$	−2.10019	−0.0829524	−0.0414762	−	0.999139i	$-0.513206\pi$
$641$	−2.10019	−0.0829524	−0.0414762	+	0.999139i	$0.513206\pi$
$642$	0	0
$643$	43.3829	1.71085	0.855427	−	0.517923i	$-0.173295\pi$
$643$	43.3829	1.71085	0.855427	+	0.517923i	$0.173295\pi$
$644$	0	0
$645$	−3.48755	−0.137322
$646$	0	0
$647$	−21.2757	−0.836433	−0.418216	−	0.908347i	$-0.637345\pi$
$647$	−21.2757	−0.836433	−0.418216	+	0.908347i	$0.637345\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1.01945	0.0399553
$652$	0	0
$653$	−17.2926	−0.676713	−0.338356	−	0.941018i	$-0.609871\pi$
$653$	−17.2926	−0.676713	−0.338356	+	0.941018i	$0.609871\pi$
$654$	0	0
$655$	6.95815	0.271878
$656$	0	0
$657$	−24.0250	−0.937303
$658$	0	0
$659$	29.0668	1.13228	0.566141	−	0.824308i	$-0.308436\pi$
$659$	29.0668	1.13228	0.566141	+	0.824308i	$0.308436\pi$
$660$	0	0
$661$	40.8316	1.58816	0.794082	−	0.607811i	$-0.207952\pi$
$661$	40.8316	1.58816	0.794082	+	0.607811i	$0.207952\pi$
$662$	0	0
$663$	10.2732	0.398977
$664$	0	0
$665$	3.17548	0.123140
$666$	0	0
$667$	40.7189	1.57664
$668$	0	0
$669$	−4.80212	−0.185661
$670$	0	0
$671$	0	0
$672$	0	0
$673$	11.4487	0.441313	0.220657	−	0.975352i	$-0.429180\pi$
$673$	11.4487	0.441313	0.220657	+	0.975352i	$0.429180\pi$
$674$	0	0
$675$	16.4596	0.633531
$676$	0	0
$677$	23.8091	0.915057	0.457529	−	0.889195i	$-0.348735\pi$
$677$	23.8091	0.915057	0.457529	+	0.889195i	$0.348735\pi$
$678$	0	0
$679$	17.5877	0.674956
$680$	0	0
$681$	9.13659	0.350115
$682$	0	0
$683$	−2.64608	−0.101250	−0.0506248	−	0.998718i	$-0.516121\pi$
$683$	−2.64608	−0.101250	−0.0506248	+	0.998718i	$0.516121\pi$
$684$	0	0
$685$	−1.11419	−0.0425710
$686$	0	0
$687$	−11.2757	−0.430194
$688$	0	0
$689$	−14.4038	−0.548740
$690$	0	0
$691$	20.2772	0.771381	0.385690	−	0.922628i	$-0.373963\pi$
$691$	20.2772	0.771381	0.385690	+	0.922628i	$0.373963\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.96111	0.377846
$696$	0	0
$697$	−69.9163	−2.64827
$698$	0	0
$699$	−16.3649	−0.618977
$700$	0	0
$701$	39.2313	1.48175	0.740873	−	0.671645i	$-0.234412\pi$
$701$	39.2313	1.48175	0.740873	+	0.671645i	$0.234412\pi$
$702$	0	0
$703$	−21.2841	−0.802747
$704$	0	0
$705$	1.94415	0.0732209
$706$	0	0
$707$	−16.5808	−0.623584
$708$	0	0
$709$	−12.1560	−0.456530	−0.228265	−	0.973599i	$-0.573305\pi$
$709$	−12.1560	−0.456530	−0.228265	+	0.973599i	$0.573305\pi$
$710$	0	0
$711$	−18.5683	−0.696365
$712$	0	0
$713$	−10.0085	−0.374820
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.64608	−0.248202
$718$	0	0
$719$	19.4093	0.723845	0.361922	−	0.932208i	$-0.382120\pi$
$719$	19.4093	0.723845	0.361922	+	0.932208i	$0.382120\pi$
$720$	0	0
$721$	−15.2841	−0.569211
$722$	0	0
$723$	−4.07779	−0.151655
$724$	0	0
$725$	−29.6351	−1.10062
$726$	0	0
$727$	−49.7772	−1.84614	−0.923068	−	0.384638i	$-0.874326\pi$
$727$	−49.7772	−1.84614	−0.923068	+	0.384638i	$0.874326\pi$
$728$	0	0
$729$	−7.21733	−0.267308
$730$	0	0
$731$	60.7019	2.24514
$732$	0	0
$733$	19.3664	0.715314	0.357657	−	0.933853i	$-0.383576\pi$
$733$	19.3664	0.715314	0.357657	+	0.933853i	$0.383576\pi$
$734$	0	0
$735$	0.412259	0.0152064
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−33.2313	−1.22243	−0.611217	−	0.791463i	$-0.709320\pi$
$739$	−33.2313	−1.22243	−0.611217	+	0.791463i	$0.709320\pi$
$740$	0	0
$741$	7.08074	0.260117
$742$	0	0
$743$	21.2144	0.778280	0.389140	−	0.921179i	$-0.372772\pi$
$743$	21.2144	0.778280	0.389140	+	0.921179i	$0.372772\pi$
$744$	0	0
$745$	−12.7717	−0.467919
$746$	0	0
$747$	−5.77018	−0.211120
$748$	0	0
$749$	−15.6351	−0.571295
$750$	0	0
$751$	15.1281	0.552033	0.276016	−	0.961153i	$-0.410986\pi$
$751$	15.1281	0.552033	0.276016	+	0.961153i	$0.410986\pi$
$752$	0	0
$753$	2.45115	0.0893250
$754$	0	0
$755$	3.09770	0.112737
$756$	0	0
$757$	18.4596	0.670927	0.335463	−	0.942053i	$-0.391107\pi$
$757$	18.4596	0.670927	0.335463	+	0.942053i	$0.391107\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.5653	0.999243	0.499621	−	0.866244i	$-0.333472\pi$
$761$	27.5653	0.999243	0.499621	+	0.866244i	$0.333472\pi$
$762$	0	0
$763$	−9.17548	−0.332175
$764$	0	0
$765$	11.9222	0.431048
$766$	0	0
$767$	−13.2144	−0.477143
$768$	0	0
$769$	27.5653	0.994032	0.497016	−	0.867741i	$-0.334429\pi$
$769$	27.5653	0.994032	0.497016	+	0.867741i	$0.334429\pi$
$770$	0	0
$771$	−6.28120	−0.226212
$772$	0	0
$773$	39.3355	1.41480	0.707400	−	0.706813i	$-0.249868\pi$
$773$	39.3355	1.41480	0.707400	+	0.706813i	$0.249868\pi$
$774$	0	0
$775$	7.28415	0.261654
$776$	0	0
$777$	−2.76322	−0.0991301
$778$	0	0
$779$	−48.1895	−1.72657
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−23.1755	−0.828224
$784$	0	0
$785$	−0.0223989	−0.000799451 0
$786$	0	0
$787$	13.6226	0.485594	0.242797	−	0.970077i	$-0.421935\pi$
$787$	13.6226	0.485594	0.242797	+	0.970077i	$0.421935\pi$
$788$	0	0
$789$	1.30912	0.0466059
$790$	0	0
$791$	11.3315	0.402902
$792$	0	0
$793$	−24.1645	−0.858107
$794$	0	0
$795$	2.66304	0.0944482
$796$	0	0
$797$	−24.9541	−0.883921	−0.441961	−	0.897034i	$-0.645717\pi$
$797$	−24.9541	−0.883921	−0.441961	+	0.897034i	$0.645717\pi$
$798$	0	0
$799$	−33.8385	−1.19712
$800$	0	0
$801$	−29.7049	−1.04957
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−4.04737	−0.142651
$806$	0	0
$807$	11.3400	0.399187
$808$	0	0
$809$	−5.08074	−0.178629	−0.0893146	−	0.996003i	$-0.528468\pi$
$809$	−5.08074	−0.178629	−0.0893146	+	0.996003i	$0.528468\pi$
$810$	0	0
$811$	−14.3774	−0.504858	−0.252429	−	0.967615i	$-0.581229\pi$
$811$	−14.3774	−0.504858	−0.252429	+	0.967615i	$0.581229\pi$
$812$	0	0
$813$	−7.70488	−0.270222
$814$	0	0
$815$	1.21438	0.0425378
$816$	0	0
$817$	41.8385	1.46374
$818$	0	0
$819$	−5.77018	−0.201627
$820$	0	0
$821$	−6.45963	−0.225443	−0.112721	−	0.993627i	$-0.535957\pi$
$821$	−6.45963	−0.225443	−0.112721	+	0.993627i	$0.535957\pi$
$822$	0	0
$823$	49.1142	1.71201	0.856007	−	0.516965i	$-0.172938\pi$
$823$	49.1142	1.71201	0.856007	+	0.516965i	$0.172938\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.4876	1.09493	0.547465	−	0.836829i	$-0.315593\pi$
$827$	31.4876	1.09493	0.547465	+	0.836829i	$0.315593\pi$
$828$	0	0
$829$	39.9123	1.38621	0.693106	−	0.720836i	$-0.256242\pi$
$829$	39.9123	1.38621	0.693106	+	0.720836i	$0.256242\pi$
$830$	0	0
$831$	3.75770	0.130353
$832$	0	0
$833$	−7.17548	−0.248616
$834$	0	0
$835$	6.88037	0.238105
$836$	0	0
$837$	5.69641	0.196897
$838$	0	0
$839$	42.3844	1.46327	0.731636	−	0.681695i	$-0.238757\pi$
$839$	42.3844	1.46327	0.731636	+	0.681695i	$0.238757\pi$
$840$	0	0
$841$	12.7268	0.438856
$842$	0	0
$843$	−15.2453	−0.525074
$844$	0	0
$845$	−5.15452	−0.177321
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9.91631	0.340327
$850$	0	0
$851$	27.1281	0.929940
$852$	0	0
$853$	−32.0514	−1.09742	−0.548709	−	0.836013i	$-0.684881\pi$
$853$	−32.0514	−1.09742	−0.548709	+	0.836013i	$0.684881\pi$
$854$	0	0
$855$	8.21733	0.281027
$856$	0	0
$857$	52.4846	1.79284	0.896420	−	0.443206i	$-0.146159\pi$
$857$	52.4846	1.79284	0.896420	+	0.443206i	$0.146159\pi$
$858$	0	0
$859$	−2.53341	−0.0864388	−0.0432194	−	0.999066i	$-0.513761\pi$
$859$	−2.53341	−0.0864388	−0.0432194	+	0.999066i	$0.513761\pi$
$860$	0	0
$861$	−6.25622	−0.213211
$862$	0	0
$863$	18.7328	0.637672	0.318836	−	0.947810i	$-0.396708\pi$
$863$	18.7328	0.637672	0.318836	+	0.947810i	$0.396708\pi$
$864$	0	0
$865$	−10.6461	−0.361978
$866$	0	0
$867$	22.1435	0.752034
$868$	0	0
$869$	0	0
$870$	0	0
$871$	32.9193	1.11543
$872$	0	0
$873$	45.5125	1.54037
$874$	0	0
$875$	6.15604	0.208112
$876$	0	0
$877$	8.35097	0.281992	0.140996	−	0.990010i	$-0.454970\pi$
$877$	8.35097	0.281992	0.140996	+	0.990010i	$0.454970\pi$
$878$	0	0
$879$	−0.0169556	−0.000571897 0
$880$	0	0
$881$	16.5987	0.559225	0.279612	−	0.960113i	$-0.409794\pi$
$881$	16.5987	0.559225	0.279612	+	0.960113i	$0.409794\pi$
$882$	0	0
$883$	26.6461	0.896712	0.448356	−	0.893855i	$-0.352010\pi$
$883$	26.6461	0.896712	0.448356	+	0.893855i	$0.352010\pi$
$884$	0	0
$885$	2.44314	0.0821251
$886$	0	0
$887$	12.1476	0.407875	0.203938	−	0.978984i	$-0.434626\pi$
$887$	12.1476	0.407875	0.203938	+	0.978984i	$0.434626\pi$
$888$	0	0
$889$	16.6072	0.556987
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−23.3230	−0.780476
$894$	0	0
$895$	−7.34544	−0.245531
$896$	0	0
$897$	−9.02489	−0.301332
$898$	0	0
$899$	−10.2562	−0.342064
$900$	0	0
$901$	−46.3510	−1.54417
$902$	0	0
$903$	5.43171	0.180756
$904$	0	0
$905$	−3.89981	−0.129634
$906$	0	0
$907$	−44.9442	−1.49235	−0.746174	−	0.665751i	$-0.768111\pi$
$907$	−44.9442	−1.49235	−0.746174	+	0.665751i	$0.768111\pi$
$908$	0	0
$909$	−42.9068	−1.42313
$910$	0	0
$911$	−14.1336	−0.468268	−0.234134	−	0.972204i	$-0.575225\pi$
$911$	−14.1336	−0.468268	−0.234134	+	0.972204i	$0.575225\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	4.46765	0.147696
$916$	0	0
$917$	−10.8370	−0.357869
$918$	0	0
$919$	25.2144	0.831746	0.415873	−	0.909423i	$-0.363476\pi$
$919$	25.2144	0.831746	0.415873	+	0.909423i	$0.363476\pi$
$920$	0	0
$921$	5.64903	0.186142
$922$	0	0
$923$	−22.6461	−0.745405
$924$	0	0
$925$	−19.7438	−0.649171
$926$	0	0
$927$	−39.5514	−1.29904
$928$	0	0
$929$	29.0279	0.952375	0.476188	−	0.879344i	$-0.342018\pi$
$929$	29.0279	0.952375	0.476188	+	0.879344i	$0.342018\pi$
$930$	0	0
$931$	−4.94567	−0.162088
$932$	0	0
$933$	7.72986	0.253064
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−41.0838	−1.34215	−0.671074	−	0.741390i	$-0.734167\pi$
$937$	−41.0838	−1.34215	−0.671074	+	0.741390i	$0.734167\pi$
$938$	0	0
$939$	−0.629587	−0.0205458
$940$	0	0
$941$	38.6197	1.25897	0.629483	−	0.777014i	$-0.283267\pi$
$941$	38.6197	1.25897	0.629483	+	0.777014i	$0.283267\pi$
$942$	0	0
$943$	61.4208	2.00014
$944$	0	0
$945$	2.30359	0.0749359
$946$	0	0
$947$	28.7493	0.934227	0.467113	−	0.884197i	$-0.345294\pi$
$947$	28.7493	0.934227	0.467113	+	0.884197i	$0.345294\pi$
$948$	0	0
$949$	−20.7019	−0.672013
$950$	0	0
$951$	−11.9776	−0.388400
$952$	0	0
$953$	16.0389	0.519551	0.259775	−	0.965669i	$-0.416351\pi$
$953$	16.0389	0.519551	0.259775	+	0.965669i	$0.416351\pi$
$954$	0	0
$955$	14.2258	0.460336
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.73530	0.0560357
$960$	0	0
$961$	−28.4791	−0.918680
$962$	0	0
$963$	−40.4596	−1.30379
$964$	0	0
$965$	−5.50148	−0.177099
$966$	0	0
$967$	−4.82452	−0.155146	−0.0775730	−	0.996987i	$-0.524717\pi$
$967$	−4.82452	−0.155146	−0.0775730	+	0.996987i	$0.524717\pi$
$968$	0	0
$969$	22.7856	0.731980
$970$	0	0
$971$	18.3161	0.587791	0.293895	−	0.955838i	$-0.405048\pi$
$971$	18.3161	0.587791	0.293895	+	0.955838i	$0.405048\pi$
$972$	0	0
$973$	−15.5140	−0.497355
$974$	0	0
$975$	6.56829	0.210354
$976$	0	0
$977$	38.0474	1.21724	0.608622	−	0.793461i	$-0.291723\pi$
$977$	38.0474	1.21724	0.608622	+	0.793461i	$0.291723\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−23.7438	−0.758080
$982$	0	0
$983$	−17.3704	−0.554030	−0.277015	−	0.960866i	$-0.589345\pi$
$983$	−17.3704	−0.554030	−0.277015	+	0.960866i	$0.589345\pi$
$984$	0	0
$985$	−1.28415	−0.0409163
$986$	0	0
$987$	−3.02792	−0.0963799
$988$	0	0
$989$	−53.3261	−1.69567
$990$	0	0
$991$	25.1616	0.799283	0.399642	−	0.916671i	$-0.369135\pi$
$991$	25.1616	0.799283	0.399642	+	0.916671i	$0.369135\pi$
$992$	0	0
$993$	2.20885	0.0700958
$994$	0	0
$995$	−7.83549	−0.248402
$996$	0	0
$997$	58.8540	1.86392	0.931962	−	0.362557i	$-0.118096\pi$
$997$	58.8540	1.86392	0.931962	+	0.362557i	$0.118096\pi$
$998$	0	0
$999$	−15.4402	−0.488506

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6776.2.a.y.1.2		3
11.10	odd	2		616.2.a.g.1.2	✓	3
33.32	even	2		5544.2.a.bj.1.2		3
44.43	even	2		1232.2.a.q.1.2		3
77.76	even	2		4312.2.a.w.1.2		3
88.21	odd	2		4928.2.a.bw.1.2		3
88.43	even	2		4928.2.a.bz.1.2		3
308.307	odd	2		8624.2.a.cm.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.a.g.1.2	✓	3	11.10	odd	2
1232.2.a.q.1.2		3	44.43	even	2
4312.2.a.w.1.2		3	77.76	even	2
4928.2.a.bw.1.2		3	88.21	odd	2
4928.2.a.bz.1.2		3	88.43	even	2
5544.2.a.bj.1.2		3	33.32	even	2
6776.2.a.y.1.2		3	1.1	even	1	trivial
8624.2.a.cm.1.2		3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.642074 q^{3} +0.642074 q^{5} -1.00000 q^{7} -2.58774 q^{9} -2.22982 q^{13} +0.412259 q^{15} -7.17548 q^{17} -4.94567 q^{19} -0.642074 q^{21} +6.30359 q^{23} -4.58774 q^{25} -3.58774 q^{27} +6.45963 q^{29} -1.58774 q^{31} -0.642074 q^{35} +4.30359 q^{37} -1.43171 q^{39} +9.74378 q^{41} -8.45963 q^{43} -1.66152 q^{45} +4.71585 q^{47} +1.00000 q^{49} -4.60719 q^{51} +6.45963 q^{53} -3.17548 q^{57} +5.92622 q^{59} +10.8370 q^{61} +2.58774 q^{63} -1.43171 q^{65} -14.7632 q^{67} +4.04737 q^{69} +10.1560 q^{71} +9.28415 q^{73} -2.94567 q^{75} +7.17548 q^{79} +5.45963 q^{81} +2.22982 q^{83} -4.60719 q^{85} +4.14756 q^{87} +11.4791 q^{89} +2.22982 q^{91} -1.01945 q^{93} -3.17548 q^{95} -17.5877 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + q^{5} - 3 q^{7} + 4 q^{9} + 6 q^{13} + 13 q^{15} + 2 q^{17} - 4 q^{19} - q^{21} + 9 q^{23} - 2 q^{25} + q^{27} - 6 q^{29} + 7 q^{31} - q^{35} + 3 q^{37} - 8 q^{39} + 2 q^{41} + 4 q^{45}+ \cdots - 41 q^{97}+O(q^{100}) \) 3 * q + q^3 + q^5 - 3 * q^7 + 4 * q^9 + 6 * q^13 + 13 * q^15 + 2 * q^17 - 4 * q^19 - q^21 + 9 * q^23 - 2 * q^25 + q^27 - 6 * q^29 + 7 * q^31 - q^35 + 3 * q^37 - 8 * q^39 + 2 * q^41 + 4 * q^45 + 16 * q^47 + 3 * q^49 + 6 * q^51 - 6 * q^53 + 14 * q^57 + 15 * q^59 - 4 * q^63 - 8 * q^65 - 9 * q^67 - 25 * q^69 + 15 * q^71 + 26 * q^73 + 2 * q^75 - 2 * q^79 - 9 * q^81 - 6 * q^83 + 6 * q^85 + 18 * q^87 + q^89 - 6 * q^91 + 5 * q^93 + 14 * q^95 - 41 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more