LMFDB - Embedded newform 2760.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(1,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2760.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:	$22.0387109579$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.76156$ of defining polynomial
Character	$\chi$	$=$	2760.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.89692 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.89692 q^{7} +1.00000 q^{9} -2.38776 q^{11} -4.38776 q^{13} +1.00000 q^{15} +0.761557 q^{17} +0.864641 q^{19} +1.89692 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.76156 q^{29} +9.35548 q^{31} -2.38776 q^{33} +1.89692 q^{35} -0.103084 q^{37} -4.38776 q^{39} +9.53707 q^{41} +3.25240 q^{43} +1.00000 q^{45} +3.13536 q^{47} -3.40171 q^{49} +0.761557 q^{51} +8.01395 q^{53} -2.38776 q^{55} +0.864641 q^{57} +10.2201 q^{59} +3.64015 q^{61} +1.89692 q^{63} -4.38776 q^{65} +12.6079 q^{67} -1.00000 q^{69} -6.55539 q^{71} -4.92919 q^{73} +1.00000 q^{75} -4.52937 q^{77} -7.04623 q^{79} +1.00000 q^{81} +5.23844 q^{83} +0.761557 q^{85} +8.76156 q^{87} -10.9817 q^{89} -8.32320 q^{91} +9.35548 q^{93} +0.864641 q^{95} -0.0645508 q^{97} -2.38776 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + 8 q^{11} + 2 q^{13} + 3 q^{15} - 4 q^{17} + 2 q^{21} - 3 q^{23} + 3 q^{25} + 3 q^{27} + 20 q^{29} + 14 q^{31} + 8 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{47} + 29 q^{49} - 4 q^{51} + 8 q^{55} + 14 q^{59} - 22 q^{61} + 2 q^{63} + 2 q^{65} + 6 q^{67} - 3 q^{69} - 6 q^{71} - 10 q^{73} + 3 q^{75} - 8 q^{77} + 4 q^{79} + 3 q^{81} + 22 q^{83} - 4 q^{85} + 20 q^{87} - 10 q^{89} - 12 q^{91} + 14 q^{93} + 2 q^{97} + 8 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9 + 8 * q^11 + 2 * q^13 + 3 * q^15 - 4 * q^17 + 2 * q^21 - 3 * q^23 + 3 * q^25 + 3 * q^27 + 20 * q^29 + 14 * q^31 + 8 * q^33 + 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 - 8 * q^43 + 3 * q^45 + 12 * q^47 + 29 * q^49 - 4 * q^51 + 8 * q^55 + 14 * q^59 - 22 * q^61 + 2 * q^63 + 2 * q^65 + 6 * q^67 - 3 * q^69 - 6 * q^71 - 10 * q^73 + 3 * q^75 - 8 * q^77 + 4 * q^79 + 3 * q^81 + 22 * q^83 - 4 * q^85 + 20 * q^87 - 10 * q^89 - 12 * q^91 + 14 * q^93 + 2 * q^97 + 8 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.89692	0.716967	0.358483	−	0.933536i	$-0.383294\pi$
$7$	1.89692	0.716967	0.358483	+	0.933536i	$0.383294\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.38776	−0.719935	−0.359968	−	0.932965i	$-0.617212\pi$
$11$	−2.38776	−0.719935	−0.359968	+	0.932965i	$0.617212\pi$
$12$	0	0
$13$	−4.38776	−1.21694	−0.608472	−	0.793575i	$-0.708217\pi$
$13$	−4.38776	−1.21694	−0.608472	+	0.793575i	$0.708217\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0.761557	0.184705	0.0923524	−	0.995726i	$-0.470561\pi$
$17$	0.761557	0.184705	0.0923524	+	0.995726i	$0.470561\pi$
$18$	0	0
$19$	0.864641	0.198362	0.0991811	−	0.995069i	$-0.468378\pi$
$19$	0.864641	0.198362	0.0991811	+	0.995069i	$0.468378\pi$
$20$	0	0
$21$	1.89692	0.413941
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.76156	1.62698	0.813490	−	0.581579i	$-0.197565\pi$
$29$	8.76156	1.62698	0.813490	+	0.581579i	$0.197565\pi$
$30$	0	0
$31$	9.35548	1.68029	0.840147	−	0.542359i	$-0.182469\pi$
$31$	9.35548	1.68029	0.840147	+	0.542359i	$0.182469\pi$
$32$	0	0
$33$	−2.38776	−0.415655
$34$	0	0
$35$	1.89692	0.320637
$36$	0	0
$37$	−0.103084	−0.0169469	−0.00847343	−	0.999964i	$-0.502697\pi$
$37$	−0.103084	−0.0169469	−0.00847343	+	0.999964i	$0.502697\pi$
$38$	0	0
$39$	−4.38776	−0.702603
$40$	0	0
$41$	9.53707	1.48944	0.744720	−	0.667377i	$-0.232583\pi$
$41$	9.53707	1.48944	0.744720	+	0.667377i	$0.232583\pi$
$42$	0	0
$43$	3.25240	0.495986	0.247993	−	0.968762i	$-0.420229\pi$
$43$	3.25240	0.495986	0.247993	+	0.968762i	$0.420229\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	3.13536	0.457339	0.228670	−	0.973504i	$-0.426562\pi$
$47$	3.13536	0.457339	0.228670	+	0.973504i	$0.426562\pi$
$48$	0	0
$49$	−3.40171	−0.485958
$50$	0	0
$51$	0.761557	0.106639
$52$	0	0
$53$	8.01395	1.10080	0.550401	−	0.834901i	$-0.314475\pi$
$53$	8.01395	1.10080	0.550401	+	0.834901i	$0.314475\pi$
$54$	0	0
$55$	−2.38776	−0.321965
$56$	0	0
$57$	0.864641	0.114524
$58$	0	0
$59$	10.2201	1.33055	0.665273	−	0.746600i	$-0.268315\pi$
$59$	10.2201	1.33055	0.665273	+	0.746600i	$0.268315\pi$
$60$	0	0
$61$	3.64015	0.466074	0.233037	−	0.972468i	$-0.425134\pi$
$61$	3.64015	0.466074	0.233037	+	0.972468i	$0.425134\pi$
$62$	0	0
$63$	1.89692	0.238989
$64$	0	0
$65$	−4.38776	−0.544234
$66$	0	0
$67$	12.6079	1.54030	0.770149	−	0.637865i	$-0.220182\pi$
$67$	12.6079	1.54030	0.770149	+	0.637865i	$0.220182\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−6.55539	−0.777982	−0.388991	−	0.921242i	$-0.627176\pi$
$71$	−6.55539	−0.777982	−0.388991	+	0.921242i	$0.627176\pi$
$72$	0	0
$73$	−4.92919	−0.576918	−0.288459	−	0.957492i	$-0.593143\pi$
$73$	−4.92919	−0.576918	−0.288459	+	0.957492i	$0.593143\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−4.52937	−0.516170
$78$	0	0
$79$	−7.04623	−0.792763	−0.396381	−	0.918086i	$-0.629734\pi$
$79$	−7.04623	−0.792763	−0.396381	+	0.918086i	$0.629734\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.23844	0.574994	0.287497	−	0.957782i	$-0.407177\pi$
$83$	5.23844	0.574994	0.287497	+	0.957782i	$0.407177\pi$
$84$	0	0
$85$	0.761557	0.0826025
$86$	0	0
$87$	8.76156	0.939338
$88$	0	0
$89$	−10.9817	−1.16406	−0.582028	−	0.813169i	$-0.697741\pi$
$89$	−10.9817	−1.16406	−0.582028	+	0.813169i	$0.697741\pi$
$90$	0	0
$91$	−8.32320	−0.872509
$92$	0	0
$93$	9.35548	0.970118
$94$	0	0
$95$	0.864641	0.0887103
$96$	0	0
$97$	−0.0645508	−0.00655414	−0.00327707	−	0.999995i	$-0.501043\pi$
$97$	−0.0645508	−0.00655414	−0.00327707	+	0.999995i	$0.501043\pi$
$98$	0	0
$99$	−2.38776	−0.239978
$100$	0	0
$101$	−1.30299	−0.129653	−0.0648264	−	0.997897i	$-0.520649\pi$
$101$	−1.30299	−0.129653	−0.0648264	+	0.997897i	$0.520649\pi$
$102$	0	0
$103$	−4.98168	−0.490859	−0.245430	−	0.969414i	$-0.578929\pi$
$103$	−4.98168	−0.490859	−0.245430	+	0.969414i	$0.578929\pi$
$104$	0	0
$105$	1.89692	0.185120
$106$	0	0
$107$	−0.697006	−0.0673821	−0.0336911	−	0.999432i	$-0.510726\pi$
$107$	−0.697006	−0.0673821	−0.0336911	+	0.999432i	$0.510726\pi$
$108$	0	0
$109$	−4.92919	−0.472131	−0.236065	−	0.971737i	$-0.575858\pi$
$109$	−4.92919	−0.472131	−0.236065	+	0.971737i	$0.575858\pi$
$110$	0	0
$111$	−0.103084	−0.00978427
$112$	0	0
$113$	−9.53707	−0.897172	−0.448586	−	0.893740i	$-0.648072\pi$
$113$	−9.53707	−0.897172	−0.448586	+	0.893740i	$0.648072\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−4.38776	−0.405648
$118$	0	0
$119$	1.44461	0.132427
$120$	0	0
$121$	−5.29862	−0.481693
$122$	0	0
$123$	9.53707	0.859928
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.38776	0.211879	0.105940	−	0.994373i	$-0.466215\pi$
$127$	2.38776	0.211879	0.105940	+	0.994373i	$0.466215\pi$
$128$	0	0
$129$	3.25240	0.286358
$130$	0	0
$131$	0.953771	0.0833314	0.0416657	−	0.999132i	$-0.486734\pi$
$131$	0.953771	0.0833314	0.0416657	+	0.999132i	$0.486734\pi$
$132$	0	0
$133$	1.64015	0.142219
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	3.93545	0.336228	0.168114	−	0.985768i	$-0.446232\pi$
$137$	3.93545	0.336228	0.168114	+	0.985768i	$0.446232\pi$
$138$	0	0
$139$	−2.87859	−0.244159	−0.122080	−	0.992520i	$-0.538956\pi$
$139$	−2.87859	−0.244159	−0.122080	+	0.992520i	$0.538956\pi$
$140$	0	0
$141$	3.13536	0.264045
$142$	0	0
$143$	10.4769	0.876121
$144$	0	0
$145$	8.76156	0.727608
$146$	0	0
$147$	−3.40171	−0.280568
$148$	0	0
$149$	19.4340	1.59209	0.796047	−	0.605235i	$-0.206921\pi$
$149$	19.4340	1.59209	0.796047	+	0.605235i	$0.206921\pi$
$150$	0	0
$151$	−10.7110	−0.871646	−0.435823	−	0.900033i	$-0.643543\pi$
$151$	−10.7110	−0.871646	−0.435823	+	0.900033i	$0.643543\pi$
$152$	0	0
$153$	0.761557	0.0615682
$154$	0	0
$155$	9.35548	0.751450
$156$	0	0
$157$	−16.3372	−1.30385	−0.651924	−	0.758285i	$-0.726038\pi$
$157$	−16.3372	−1.30385	−0.651924	+	0.758285i	$0.726038\pi$
$158$	0	0
$159$	8.01395	0.635548
$160$	0	0
$161$	−1.89692	−0.149498
$162$	0	0
$163$	11.2524	0.881356	0.440678	−	0.897665i	$-0.354738\pi$
$163$	11.2524	0.881356	0.440678	+	0.897665i	$0.354738\pi$
$164$	0	0
$165$	−2.38776	−0.185886
$166$	0	0
$167$	−6.59392	−0.510253	−0.255127	−	0.966908i	$-0.582117\pi$
$167$	−6.59392	−0.510253	−0.255127	+	0.966908i	$0.582117\pi$
$168$	0	0
$169$	6.25240	0.480954
$170$	0	0
$171$	0.864641	0.0661207
$172$	0	0
$173$	−20.7110	−1.57463	−0.787313	−	0.616554i	$-0.788528\pi$
$173$	−20.7110	−1.57463	−0.787313	+	0.616554i	$0.788528\pi$
$174$	0	0
$175$	1.89692	0.143393
$176$	0	0
$177$	10.2201	0.768191
$178$	0	0
$179$	22.2986	1.66668	0.833339	−	0.552763i	$-0.186426\pi$
$179$	22.2986	1.66668	0.833339	+	0.552763i	$0.186426\pi$
$180$	0	0
$181$	−21.4865	−1.59708	−0.798538	−	0.601944i	$-0.794393\pi$
$181$	−21.4865	−1.59708	−0.798538	+	0.601944i	$0.794393\pi$
$182$	0	0
$183$	3.64015	0.269088
$184$	0	0
$185$	−0.103084	−0.00757886
$186$	0	0
$187$	−1.81841	−0.132975
$188$	0	0
$189$	1.89692	0.137980
$190$	0	0
$191$	−12.8646	−0.930853	−0.465426	−	0.885087i	$-0.654099\pi$
$191$	−12.8646	−0.930853	−0.465426	+	0.885087i	$0.654099\pi$
$192$	0	0
$193$	−18.0279	−1.29768	−0.648839	−	0.760926i	$-0.724745\pi$
$193$	−18.0279	−1.29768	−0.648839	+	0.760926i	$0.724745\pi$
$194$	0	0
$195$	−4.38776	−0.314214
$196$	0	0
$197$	14.2341	1.01414	0.507068	−	0.861906i	$-0.330729\pi$
$197$	14.2341	1.01414	0.507068	+	0.861906i	$0.330729\pi$
$198$	0	0
$199$	12.5693	0.891017	0.445509	−	0.895278i	$-0.353023\pi$
$199$	12.5693	0.891017	0.445509	+	0.895278i	$0.353023\pi$
$200$	0	0
$201$	12.6079	0.889291
$202$	0	0
$203$	16.6199	1.16649
$204$	0	0
$205$	9.53707	0.666098
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−2.06455	−0.142808
$210$	0	0
$211$	−5.12141	−0.352572	−0.176286	−	0.984339i	$-0.556408\pi$
$211$	−5.12141	−0.352572	−0.176286	+	0.984339i	$0.556408\pi$
$212$	0	0
$213$	−6.55539	−0.449168
$214$	0	0
$215$	3.25240	0.221812
$216$	0	0
$217$	17.7466	1.20472
$218$	0	0
$219$	−4.92919	−0.333084
$220$	0	0
$221$	−3.34153	−0.224775
$222$	0	0
$223$	−0.747604	−0.0500633	−0.0250316	−	0.999687i	$-0.507969\pi$
$223$	−0.747604	−0.0500633	−0.0250316	+	0.999687i	$0.507969\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	1.49521	0.0992404	0.0496202	−	0.998768i	$-0.484199\pi$
$227$	1.49521	0.0992404	0.0496202	+	0.998768i	$0.484199\pi$
$228$	0	0
$229$	2.56934	0.169787	0.0848935	−	0.996390i	$-0.472945\pi$
$229$	2.56934	0.169787	0.0848935	+	0.996390i	$0.472945\pi$
$230$	0	0
$231$	−4.52937	−0.298011
$232$	0	0
$233$	9.25240	0.606145	0.303072	−	0.952968i	$-0.401988\pi$
$233$	9.25240	0.606145	0.303072	+	0.952968i	$0.401988\pi$
$234$	0	0
$235$	3.13536	0.204528
$236$	0	0
$237$	−7.04623	−0.457702
$238$	0	0
$239$	21.0602	1.36227	0.681135	−	0.732158i	$-0.261487\pi$
$239$	21.0602	1.36227	0.681135	+	0.732158i	$0.261487\pi$
$240$	0	0
$241$	15.4340	0.994190	0.497095	−	0.867696i	$-0.334400\pi$
$241$	15.4340	0.994190	0.497095	+	0.867696i	$0.334400\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.40171	−0.217327
$246$	0	0
$247$	−3.79383	−0.241396
$248$	0	0
$249$	5.23844	0.331973
$250$	0	0
$251$	0.747604	0.0471883	0.0235942	−	0.999722i	$-0.492489\pi$
$251$	0.747604	0.0471883	0.0235942	+	0.999722i	$0.492489\pi$
$252$	0	0
$253$	2.38776	0.150117
$254$	0	0
$255$	0.761557	0.0476906
$256$	0	0
$257$	−25.9388	−1.61802	−0.809008	−	0.587797i	$-0.799995\pi$
$257$	−25.9388	−1.61802	−0.809008	+	0.587797i	$0.799995\pi$
$258$	0	0
$259$	−0.195541	−0.0121503
$260$	0	0
$261$	8.76156	0.542327
$262$	0	0
$263$	−16.8540	−1.03926	−0.519632	−	0.854390i	$-0.673931\pi$
$263$	−16.8540	−1.03926	−0.519632	+	0.854390i	$0.673931\pi$
$264$	0	0
$265$	8.01395	0.492293
$266$	0	0
$267$	−10.9817	−0.672068
$268$	0	0
$269$	31.6016	1.92678	0.963392	−	0.268095i	$-0.0863943\pi$
$269$	31.6016	1.92678	0.963392	+	0.268095i	$0.0863943\pi$
$270$	0	0
$271$	−3.59829	−0.218581	−0.109290	−	0.994010i	$-0.534858\pi$
$271$	−3.59829	−0.218581	−0.109290	+	0.994010i	$0.534858\pi$
$272$	0	0
$273$	−8.32320	−0.503743
$274$	0	0
$275$	−2.38776	−0.143987
$276$	0	0
$277$	−19.9634	−1.19948	−0.599741	−	0.800194i	$-0.704730\pi$
$277$	−19.9634	−1.19948	−0.599741	+	0.800194i	$0.704730\pi$
$278$	0	0
$279$	9.35548	0.560098
$280$	0	0
$281$	−24.4157	−1.45652	−0.728258	−	0.685303i	$-0.759670\pi$
$281$	−24.4157	−1.45652	−0.728258	+	0.685303i	$0.759670\pi$
$282$	0	0
$283$	29.9248	1.77885	0.889423	−	0.457085i	$-0.151106\pi$
$283$	29.9248	1.77885	0.889423	+	0.457085i	$0.151106\pi$
$284$	0	0
$285$	0.864641	0.0512169
$286$	0	0
$287$	18.0910	1.06788
$288$	0	0
$289$	−16.4200	−0.965884
$290$	0	0
$291$	−0.0645508	−0.00378404
$292$	0	0
$293$	−25.1247	−1.46780	−0.733901	−	0.679256i	$-0.762303\pi$
$293$	−25.1247	−1.46780	−0.733901	+	0.679256i	$0.762303\pi$
$294$	0	0
$295$	10.2201	0.595038
$296$	0	0
$297$	−2.38776	−0.138552
$298$	0	0
$299$	4.38776	0.253750
$300$	0	0
$301$	6.16952	0.355605
$302$	0	0
$303$	−1.30299	−0.0748550
$304$	0	0
$305$	3.64015	0.208434
$306$	0	0
$307$	−5.64015	−0.321900	−0.160950	−	0.986963i	$-0.551456\pi$
$307$	−5.64015	−0.321900	−0.160950	+	0.986963i	$0.551456\pi$
$308$	0	0
$309$	−4.98168	−0.283398
$310$	0	0
$311$	11.0462	0.626374	0.313187	−	0.949691i	$-0.398603\pi$
$311$	11.0462	0.626374	0.313187	+	0.949691i	$0.398603\pi$
$312$	0	0
$313$	−4.30925	−0.243573	−0.121787	−	0.992556i	$-0.538862\pi$
$313$	−4.30925	−0.243573	−0.121787	+	0.992556i	$0.538862\pi$
$314$	0	0
$315$	1.89692	0.106879
$316$	0	0
$317$	8.62183	0.484250	0.242125	−	0.970245i	$-0.422156\pi$
$317$	8.62183	0.484250	0.242125	+	0.970245i	$0.422156\pi$
$318$	0	0
$319$	−20.9205	−1.17132
$320$	0	0
$321$	−0.697006	−0.0389031
$322$	0	0
$323$	0.658473	0.0366384
$324$	0	0
$325$	−4.38776	−0.243389
$326$	0	0
$327$	−4.92919	−0.272585
$328$	0	0
$329$	5.94751	0.327897
$330$	0	0
$331$	−10.8786	−0.597942	−0.298971	−	0.954262i	$-0.596643\pi$
$331$	−10.8786	−0.597942	−0.298971	+	0.954262i	$0.596643\pi$
$332$	0	0
$333$	−0.103084	−0.00564895
$334$	0	0
$335$	12.6079	0.688842
$336$	0	0
$337$	3.55102	0.193436	0.0967182	−	0.995312i	$-0.469165\pi$
$337$	3.55102	0.193436	0.0967182	+	0.995312i	$0.469165\pi$
$338$	0	0
$339$	−9.53707	−0.517982
$340$	0	0
$341$	−22.3386	−1.20970
$342$	0	0
$343$	−19.7312	−1.06538
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	5.31695	0.285429	0.142714	−	0.989764i	$-0.454417\pi$
$347$	5.31695	0.285429	0.142714	+	0.989764i	$0.454417\pi$
$348$	0	0
$349$	−6.57997	−0.352218	−0.176109	−	0.984371i	$-0.556351\pi$
$349$	−6.57997	−0.352218	−0.176109	+	0.984371i	$0.556351\pi$
$350$	0	0
$351$	−4.38776	−0.234201
$352$	0	0
$353$	−9.88296	−0.526017	−0.263009	−	0.964794i	$-0.584715\pi$
$353$	−9.88296	−0.526017	−0.263009	+	0.964794i	$0.584715\pi$
$354$	0	0
$355$	−6.55539	−0.347924
$356$	0	0
$357$	1.44461	0.0764569
$358$	0	0
$359$	15.9109	0.839744	0.419872	−	0.907583i	$-0.362075\pi$
$359$	15.9109	0.839744	0.419872	+	0.907583i	$0.362075\pi$
$360$	0	0
$361$	−18.2524	−0.960652
$362$	0	0
$363$	−5.29862	−0.278106
$364$	0	0
$365$	−4.92919	−0.258006
$366$	0	0
$367$	−21.5125	−1.12294	−0.561471	−	0.827496i	$-0.689765\pi$
$367$	−21.5125	−1.12294	−0.561471	+	0.827496i	$0.689765\pi$
$368$	0	0
$369$	9.53707	0.496480
$370$	0	0
$371$	15.2018	0.789238
$372$	0	0
$373$	−24.2986	−1.25814	−0.629068	−	0.777351i	$-0.716563\pi$
$373$	−24.2986	−1.25814	−0.629068	+	0.777351i	$0.716563\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−38.4436	−1.97994
$378$	0	0
$379$	21.5510	1.10700	0.553501	−	0.832849i	$-0.313292\pi$
$379$	21.5510	1.10700	0.553501	+	0.832849i	$0.313292\pi$
$380$	0	0
$381$	2.38776	0.122328
$382$	0	0
$383$	23.5371	1.20269	0.601344	−	0.798990i	$-0.294632\pi$
$383$	23.5371	1.20269	0.601344	+	0.798990i	$0.294632\pi$
$384$	0	0
$385$	−4.52937	−0.230838
$386$	0	0
$387$	3.25240	0.165329
$388$	0	0
$389$	−21.6926	−1.09986	−0.549930	−	0.835211i	$-0.685346\pi$
$389$	−21.6926	−1.09986	−0.549930	+	0.835211i	$0.685346\pi$
$390$	0	0
$391$	−0.761557	−0.0385136
$392$	0	0
$393$	0.953771	0.0481114
$394$	0	0
$395$	−7.04623	−0.354534
$396$	0	0
$397$	−19.8863	−0.998064	−0.499032	−	0.866583i	$-0.666311\pi$
$397$	−19.8863	−0.998064	−0.499032	+	0.866583i	$0.666311\pi$
$398$	0	0
$399$	1.64015	0.0821103
$400$	0	0
$401$	10.7755	0.538103	0.269052	−	0.963126i	$-0.413290\pi$
$401$	10.7755	0.538103	0.269052	+	0.963126i	$0.413290\pi$
$402$	0	0
$403$	−41.0496	−2.04482
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0.246139	0.0122006
$408$	0	0
$409$	−11.3834	−0.562872	−0.281436	−	0.959580i	$-0.590811\pi$
$409$	−11.3834	−0.562872	−0.281436	+	0.959580i	$0.590811\pi$
$410$	0	0
$411$	3.93545	0.194121
$412$	0	0
$413$	19.3867	0.953958
$414$	0	0
$415$	5.23844	0.257145
$416$	0	0
$417$	−2.87859	−0.140965
$418$	0	0
$419$	−10.6218	−0.518910	−0.259455	−	0.965755i	$-0.583543\pi$
$419$	−10.6218	−0.518910	−0.259455	+	0.965755i	$0.583543\pi$
$420$	0	0
$421$	−3.43398	−0.167362	−0.0836811	−	0.996493i	$-0.526668\pi$
$421$	−3.43398	−0.167362	−0.0836811	+	0.996493i	$0.526668\pi$
$422$	0	0
$423$	3.13536	0.152446
$424$	0	0
$425$	0.761557	0.0369409
$426$	0	0
$427$	6.90506	0.334159
$428$	0	0
$429$	10.4769	0.505829
$430$	0	0
$431$	−23.8217	−1.14745	−0.573726	−	0.819047i	$-0.694503\pi$
$431$	−23.8217	−1.14745	−0.573726	+	0.819047i	$0.694503\pi$
$432$	0	0
$433$	23.7957	1.14355	0.571775	−	0.820411i	$-0.306255\pi$
$433$	23.7957	1.14355	0.571775	+	0.820411i	$0.306255\pi$
$434$	0	0
$435$	8.76156	0.420085
$436$	0	0
$437$	−0.864641	−0.0413614
$438$	0	0
$439$	38.6618	1.84523	0.922614	−	0.385726i	$-0.126049\pi$
$439$	38.6618	1.84523	0.922614	+	0.385726i	$0.126049\pi$
$440$	0	0
$441$	−3.40171	−0.161986
$442$	0	0
$443$	28.3232	1.34568	0.672838	−	0.739790i	$-0.265075\pi$
$443$	28.3232	1.34568	0.672838	+	0.739790i	$0.265075\pi$
$444$	0	0
$445$	−10.9817	−0.520581
$446$	0	0
$447$	19.4340	0.919196
$448$	0	0
$449$	23.8078	1.12356	0.561779	−	0.827287i	$-0.310117\pi$
$449$	23.8078	1.12356	0.561779	+	0.827287i	$0.310117\pi$
$450$	0	0
$451$	−22.7722	−1.07230
$452$	0	0
$453$	−10.7110	−0.503245
$454$	0	0
$455$	−8.32320	−0.390198
$456$	0	0
$457$	32.7003	1.52966	0.764829	−	0.644234i	$-0.222824\pi$
$457$	32.7003	1.52966	0.764829	+	0.644234i	$0.222824\pi$
$458$	0	0
$459$	0.761557	0.0355464
$460$	0	0
$461$	−22.7755	−1.06076	−0.530381	−	0.847760i	$-0.677951\pi$
$461$	−22.7755	−1.06076	−0.530381	+	0.847760i	$0.677951\pi$
$462$	0	0
$463$	−17.4061	−0.808929	−0.404465	−	0.914554i	$-0.632542\pi$
$463$	−17.4061	−0.808929	−0.404465	+	0.914554i	$0.632542\pi$
$464$	0	0
$465$	9.35548	0.433850
$466$	0	0
$467$	24.8540	1.15011	0.575053	−	0.818116i	$-0.304981\pi$
$467$	24.8540	1.15011	0.575053	+	0.818116i	$0.304981\pi$
$468$	0	0
$469$	23.9161	1.10434
$470$	0	0
$471$	−16.3372	−0.752776
$472$	0	0
$473$	−7.76593	−0.357078
$474$	0	0
$475$	0.864641	0.0396724
$476$	0	0
$477$	8.01395	0.366934
$478$	0	0
$479$	17.6402	0.805999	0.403000	−	0.915200i	$-0.367968\pi$
$479$	17.6402	0.805999	0.403000	+	0.915200i	$0.367968\pi$
$480$	0	0
$481$	0.452306	0.0206234
$482$	0	0
$483$	−1.89692	−0.0863127
$484$	0	0
$485$	−0.0645508	−0.00293110
$486$	0	0
$487$	−14.7509	−0.668428	−0.334214	−	0.942497i	$-0.608471\pi$
$487$	−14.7509	−0.668428	−0.334214	+	0.942497i	$0.608471\pi$
$488$	0	0
$489$	11.2524	0.508851
$490$	0	0
$491$	9.80779	0.442619	0.221310	−	0.975204i	$-0.428967\pi$
$491$	9.80779	0.442619	0.221310	+	0.975204i	$0.428967\pi$
$492$	0	0
$493$	6.67243	0.300511
$494$	0	0
$495$	−2.38776	−0.107322
$496$	0	0
$497$	−12.4350	−0.557787
$498$	0	0
$499$	−21.1772	−0.948023	−0.474011	−	0.880519i	$-0.657194\pi$
$499$	−21.1772	−0.948023	−0.474011	+	0.880519i	$0.657194\pi$
$500$	0	0
$501$	−6.59392	−0.294595
$502$	0	0
$503$	22.4542	1.00118	0.500592	−	0.865684i	$-0.333116\pi$
$503$	22.4542	1.00118	0.500592	+	0.865684i	$0.333116\pi$
$504$	0	0
$505$	−1.30299	−0.0579825
$506$	0	0
$507$	6.25240	0.277679
$508$	0	0
$509$	22.9817	1.01864	0.509322	−	0.860576i	$-0.329896\pi$
$509$	22.9817	1.01864	0.509322	+	0.860576i	$0.329896\pi$
$510$	0	0
$511$	−9.35026	−0.413631
$512$	0	0
$513$	0.864641	0.0381748
$514$	0	0
$515$	−4.98168	−0.219519
$516$	0	0
$517$	−7.48647	−0.329255
$518$	0	0
$519$	−20.7110	−0.909110
$520$	0	0
$521$	21.4985	0.941868	0.470934	−	0.882168i	$-0.343917\pi$
$521$	21.4985	0.941868	0.470934	+	0.882168i	$0.343917\pi$
$522$	0	0
$523$	−15.4586	−0.675956	−0.337978	−	0.941154i	$-0.609743\pi$
$523$	−15.4586	−0.675956	−0.337978	+	0.941154i	$0.609743\pi$
$524$	0	0
$525$	1.89692	0.0827882
$526$	0	0
$527$	7.12473	0.310358
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	10.2201	0.443515
$532$	0	0
$533$	−41.8463	−1.81257
$534$	0	0
$535$	−0.697006	−0.0301342
$536$	0	0
$537$	22.2986	0.962257
$538$	0	0
$539$	8.12245	0.349859
$540$	0	0
$541$	−14.0279	−0.603107	−0.301553	−	0.953449i	$-0.597505\pi$
$541$	−14.0279	−0.603107	−0.301553	+	0.953449i	$0.597505\pi$
$542$	0	0
$543$	−21.4865	−0.922073
$544$	0	0
$545$	−4.92919	−0.211143
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	3.64015	0.155358
$550$	0	0
$551$	7.57560	0.322731
$552$	0	0
$553$	−13.3661	−0.568385
$554$	0	0
$555$	−0.103084	−0.00437566
$556$	0	0
$557$	−23.9860	−1.01632	−0.508161	−	0.861262i	$-0.669674\pi$
$557$	−23.9860	−1.01632	−0.508161	+	0.861262i	$0.669674\pi$
$558$	0	0
$559$	−14.2707	−0.603587
$560$	0	0
$561$	−1.81841	−0.0767734
$562$	0	0
$563$	−27.2297	−1.14760	−0.573798	−	0.818997i	$-0.694530\pi$
$563$	−27.2297	−1.14760	−0.573798	+	0.818997i	$0.694530\pi$
$564$	0	0
$565$	−9.53707	−0.401227
$566$	0	0
$567$	1.89692	0.0796630
$568$	0	0
$569$	−10.9046	−0.457145	−0.228573	−	0.973527i	$-0.573406\pi$
$569$	−10.9046	−0.457145	−0.228573	+	0.973527i	$0.573406\pi$
$570$	0	0
$571$	−2.02458	−0.0847260	−0.0423630	−	0.999102i	$-0.513489\pi$
$571$	−2.02458	−0.0847260	−0.0423630	+	0.999102i	$0.513489\pi$
$572$	0	0
$573$	−12.8646	−0.537428
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−22.8034	−0.949319	−0.474659	−	0.880170i	$-0.657429\pi$
$577$	−22.8034	−0.949319	−0.474659	+	0.880170i	$0.657429\pi$
$578$	0	0
$579$	−18.0279	−0.749214
$580$	0	0
$581$	9.93689	0.412252
$582$	0	0
$583$	−19.1354	−0.792506
$584$	0	0
$585$	−4.38776	−0.181411
$586$	0	0
$587$	7.38150	0.304667	0.152334	−	0.988329i	$-0.451321\pi$
$587$	7.38150	0.304667	0.152334	+	0.988329i	$0.451321\pi$
$588$	0	0
$589$	8.08913	0.333307
$590$	0	0
$591$	14.2341	0.585512
$592$	0	0
$593$	5.05829	0.207719	0.103860	−	0.994592i	$-0.466881\pi$
$593$	5.05829	0.207719	0.103860	+	0.994592i	$0.466881\pi$
$594$	0	0
$595$	1.44461	0.0592232
$596$	0	0
$597$	12.5693	0.514429
$598$	0	0
$599$	14.2707	0.583086	0.291543	−	0.956558i	$-0.405831\pi$
$599$	14.2707	0.583086	0.291543	+	0.956558i	$0.405831\pi$
$600$	0	0
$601$	20.4384	0.833698	0.416849	−	0.908976i	$-0.363134\pi$
$601$	20.4384	0.833698	0.416849	+	0.908976i	$0.363134\pi$
$602$	0	0
$603$	12.6079	0.513432
$604$	0	0
$605$	−5.29862	−0.215420
$606$	0	0
$607$	−32.8925	−1.33507	−0.667534	−	0.744580i	$-0.732650\pi$
$607$	−32.8925	−1.33507	−0.667534	+	0.744580i	$0.732650\pi$
$608$	0	0
$609$	16.6199	0.673474
$610$	0	0
$611$	−13.7572	−0.556556
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	0	0
$615$	9.53707	0.384572
$616$	0	0
$617$	1.56497	0.0630035	0.0315017	−	0.999504i	$-0.489971\pi$
$617$	1.56497	0.0630035	0.0315017	+	0.999504i	$0.489971\pi$
$618$	0	0
$619$	34.0925	1.37029	0.685146	−	0.728406i	$-0.259738\pi$
$619$	34.0925	1.37029	0.685146	+	0.728406i	$0.259738\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−20.8313	−0.834589
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.06455	−0.0824502
$628$	0	0
$629$	−0.0785041	−0.00313016
$630$	0	0
$631$	−23.2924	−0.927255	−0.463627	−	0.886030i	$-0.653452\pi$
$631$	−23.2924	−0.927255	−0.463627	+	0.886030i	$0.653452\pi$
$632$	0	0
$633$	−5.12141	−0.203558
$634$	0	0
$635$	2.38776	0.0947552
$636$	0	0
$637$	14.9259	0.591384
$638$	0	0
$639$	−6.55539	−0.259327
$640$	0	0
$641$	41.4252	1.63620	0.818099	−	0.575077i	$-0.195028\pi$
$641$	41.4252	1.63620	0.818099	+	0.575077i	$0.195028\pi$
$642$	0	0
$643$	−35.6820	−1.40716	−0.703581	−	0.710615i	$-0.748417\pi$
$643$	−35.6820	−1.40716	−0.703581	+	0.710615i	$0.748417\pi$
$644$	0	0
$645$	3.25240	0.128063
$646$	0	0
$647$	−26.4436	−1.03960	−0.519802	−	0.854287i	$-0.673994\pi$
$647$	−26.4436	−1.03960	−0.519802	+	0.854287i	$0.673994\pi$
$648$	0	0
$649$	−24.4031	−0.957907
$650$	0	0
$651$	17.7466	0.695543
$652$	0	0
$653$	−3.64015	−0.142450	−0.0712251	−	0.997460i	$-0.522691\pi$
$653$	−3.64015	−0.142450	−0.0712251	+	0.997460i	$0.522691\pi$
$654$	0	0
$655$	0.953771	0.0372669
$656$	0	0
$657$	−4.92919	−0.192306
$658$	0	0
$659$	13.1999	0.514195	0.257098	−	0.966385i	$-0.417234\pi$
$659$	13.1999	0.514195	0.257098	+	0.966385i	$0.417234\pi$
$660$	0	0
$661$	−28.2707	−1.09960	−0.549802	−	0.835295i	$-0.685297\pi$
$661$	−28.2707	−1.09960	−0.549802	+	0.835295i	$0.685297\pi$
$662$	0	0
$663$	−3.34153	−0.129774
$664$	0	0
$665$	1.64015	0.0636023
$666$	0	0
$667$	−8.76156	−0.339249
$668$	0	0
$669$	−0.747604	−0.0289040
$670$	0	0
$671$	−8.69179	−0.335543
$672$	0	0
$673$	−7.22782	−0.278612	−0.139306	−	0.990249i	$-0.544487\pi$
$673$	−7.22782	−0.278612	−0.139306	+	0.990249i	$0.544487\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−26.1064	−1.00335	−0.501675	−	0.865056i	$-0.667283\pi$
$677$	−26.1064	−1.00335	−0.501675	+	0.865056i	$0.667283\pi$
$678$	0	0
$679$	−0.122447	−0.00469910
$680$	0	0
$681$	1.49521	0.0572965
$682$	0	0
$683$	35.3203	1.35149	0.675746	−	0.737134i	$-0.263821\pi$
$683$	35.3203	1.35149	0.675746	+	0.737134i	$0.263821\pi$
$684$	0	0
$685$	3.93545	0.150366
$686$	0	0
$687$	2.56934	0.0980266
$688$	0	0
$689$	−35.1633	−1.33961
$690$	0	0
$691$	39.8496	1.51595	0.757976	−	0.652282i	$-0.226188\pi$
$691$	39.8496	1.51595	0.757976	+	0.652282i	$0.226188\pi$
$692$	0	0
$693$	−4.52937	−0.172057
$694$	0	0
$695$	−2.87859	−0.109191
$696$	0	0
$697$	7.26302	0.275107
$698$	0	0
$699$	9.25240	0.349958
$700$	0	0
$701$	50.1170	1.89289	0.946447	−	0.322859i	$-0.104644\pi$
$701$	50.1170	1.89289	0.946447	+	0.322859i	$0.104644\pi$
$702$	0	0
$703$	−0.0891304	−0.00336162
$704$	0	0
$705$	3.13536	0.118084
$706$	0	0
$707$	−2.47167	−0.0929567
$708$	0	0
$709$	−31.8742	−1.19706	−0.598531	−	0.801100i	$-0.704249\pi$
$709$	−31.8742	−1.19706	−0.598531	+	0.801100i	$0.704249\pi$
$710$	0	0
$711$	−7.04623	−0.264254
$712$	0	0
$713$	−9.35548	−0.350365
$714$	0	0
$715$	10.4769	0.391813
$716$	0	0
$717$	21.0602	0.786507
$718$	0	0
$719$	−19.1526	−0.714273	−0.357136	−	0.934052i	$-0.616247\pi$
$719$	−19.1526	−0.714273	−0.357136	+	0.934052i	$0.616247\pi$
$720$	0	0
$721$	−9.44983	−0.351930
$722$	0	0
$723$	15.4340	0.573996
$724$	0	0
$725$	8.76156	0.325396
$726$	0	0
$727$	−23.7553	−0.881035	−0.440518	−	0.897744i	$-0.645205\pi$
$727$	−23.7553	−0.881035	−0.440518	+	0.897744i	$0.645205\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.47689	0.0916109
$732$	0	0
$733$	−18.9152	−0.698650	−0.349325	−	0.937002i	$-0.613589\pi$
$733$	−18.9152	−0.698650	−0.349325	+	0.937002i	$0.613589\pi$
$734$	0	0
$735$	−3.40171	−0.125474
$736$	0	0
$737$	−30.1045	−1.10891
$738$	0	0
$739$	42.1310	1.54981	0.774907	−	0.632076i	$-0.217797\pi$
$739$	42.1310	1.54981	0.774907	+	0.632076i	$0.217797\pi$
$740$	0	0
$741$	−3.79383	−0.139370
$742$	0	0
$743$	−8.41233	−0.308619	−0.154309	−	0.988023i	$-0.549315\pi$
$743$	−8.41233	−0.308619	−0.154309	+	0.988023i	$0.549315\pi$
$744$	0	0
$745$	19.4340	0.712006
$746$	0	0
$747$	5.23844	0.191665
$748$	0	0
$749$	−1.32216	−0.0483108
$750$	0	0
$751$	−25.5110	−0.930911	−0.465456	−	0.885071i	$-0.654110\pi$
$751$	−25.5110	−0.930911	−0.465456	+	0.885071i	$0.654110\pi$
$752$	0	0
$753$	0.747604	0.0272442
$754$	0	0
$755$	−10.7110	−0.389812
$756$	0	0
$757$	46.0452	1.67354	0.836770	−	0.547554i	$-0.184441\pi$
$757$	46.0452	1.67354	0.836770	+	0.547554i	$0.184441\pi$
$758$	0	0
$759$	2.38776	0.0866700
$760$	0	0
$761$	−37.5929	−1.36274	−0.681370	−	0.731939i	$-0.738616\pi$
$761$	−37.5929	−1.36274	−0.681370	+	0.731939i	$0.738616\pi$
$762$	0	0
$763$	−9.35026	−0.338502
$764$	0	0
$765$	0.761557	0.0275342
$766$	0	0
$767$	−44.8434	−1.61920
$768$	0	0
$769$	−0.230747	−0.00832095	−0.00416047	−	0.999991i	$-0.501324\pi$
$769$	−0.230747	−0.00832095	−0.00416047	+	0.999991i	$0.501324\pi$
$770$	0	0
$771$	−25.9388	−0.934162
$772$	0	0
$773$	−12.3998	−0.445991	−0.222995	−	0.974820i	$-0.571583\pi$
$773$	−12.3998	−0.445991	−0.222995	+	0.974820i	$0.571583\pi$
$774$	0	0
$775$	9.35548	0.336059
$776$	0	0
$777$	−0.195541	−0.00701500
$778$	0	0
$779$	8.24614	0.295449
$780$	0	0
$781$	15.6527	0.560096
$782$	0	0
$783$	8.76156	0.313113
$784$	0	0
$785$	−16.3372	−0.583098
$786$	0	0
$787$	−28.0943	−1.00146	−0.500728	−	0.865605i	$-0.666934\pi$
$787$	−28.0943	−1.00146	−0.500728	+	0.865605i	$0.666934\pi$
$788$	0	0
$789$	−16.8540	−0.600019
$790$	0	0
$791$	−18.0910	−0.643243
$792$	0	0
$793$	−15.9721	−0.567186
$794$	0	0
$795$	8.01395	0.284226
$796$	0	0
$797$	−10.5187	−0.372593	−0.186297	−	0.982494i	$-0.559649\pi$
$797$	−10.5187	−0.372593	−0.186297	+	0.982494i	$0.559649\pi$
$798$	0	0
$799$	2.38776	0.0844727
$800$	0	0
$801$	−10.9817	−0.388019
$802$	0	0
$803$	11.7697	0.415344
$804$	0	0
$805$	−1.89692	−0.0668575
$806$	0	0
$807$	31.6016	1.11243
$808$	0	0
$809$	−0.527483	−0.0185453	−0.00927266	−	0.999957i	$-0.502952\pi$
$809$	−0.527483	−0.0185453	−0.00927266	+	0.999957i	$0.502952\pi$
$810$	0	0
$811$	31.4200	1.10331	0.551653	−	0.834074i	$-0.313997\pi$
$811$	31.4200	1.10331	0.551653	+	0.834074i	$0.313997\pi$
$812$	0	0
$813$	−3.59829	−0.126198
$814$	0	0
$815$	11.2524	0.394154
$816$	0	0
$817$	2.81215	0.0983848
$818$	0	0
$819$	−8.32320	−0.290836
$820$	0	0
$821$	−24.6060	−0.858755	−0.429377	−	0.903125i	$-0.641267\pi$
$821$	−24.6060	−0.858755	−0.429377	+	0.903125i	$0.641267\pi$
$822$	0	0
$823$	−22.0925	−0.770095	−0.385047	−	0.922897i	$-0.625815\pi$
$823$	−22.0925	−0.770095	−0.385047	+	0.922897i	$0.625815\pi$
$824$	0	0
$825$	−2.38776	−0.0831310
$826$	0	0
$827$	−30.0419	−1.04466	−0.522329	−	0.852744i	$-0.674937\pi$
$827$	−30.0419	−1.04466	−0.522329	+	0.852744i	$0.674937\pi$
$828$	0	0
$829$	−30.6358	−1.06402	−0.532012	−	0.846737i	$-0.678564\pi$
$829$	−30.6358	−1.06402	−0.532012	+	0.846737i	$0.678564\pi$
$830$	0	0
$831$	−19.9634	−0.692521
$832$	0	0
$833$	−2.59060	−0.0897588
$834$	0	0
$835$	−6.59392	−0.228192
$836$	0	0
$837$	9.35548	0.323373
$838$	0	0
$839$	−37.3449	−1.28929	−0.644644	−	0.764483i	$-0.722994\pi$
$839$	−37.3449	−1.28929	−0.644644	+	0.764483i	$0.722994\pi$
$840$	0	0
$841$	47.7649	1.64706
$842$	0	0
$843$	−24.4157	−0.840920
$844$	0	0
$845$	6.25240	0.215089
$846$	0	0
$847$	−10.0510	−0.345358
$848$	0	0
$849$	29.9248	1.02702
$850$	0	0
$851$	0.103084	0.00353366
$852$	0	0
$853$	7.78510	0.266557	0.133278	−	0.991079i	$-0.457450\pi$
$853$	7.78510	0.266557	0.133278	+	0.991079i	$0.457450\pi$
$854$	0	0
$855$	0.864641	0.0295701
$856$	0	0
$857$	56.1483	1.91799	0.958994	−	0.283426i	$-0.0914709\pi$
$857$	56.1483	1.91799	0.958994	+	0.283426i	$0.0914709\pi$
$858$	0	0
$859$	−1.25051	−0.0426668	−0.0213334	−	0.999772i	$-0.506791\pi$
$859$	−1.25051	−0.0426668	−0.0213334	+	0.999772i	$0.506791\pi$
$860$	0	0
$861$	18.0910	0.616540
$862$	0	0
$863$	15.4865	0.527166	0.263583	−	0.964637i	$-0.415096\pi$
$863$	15.4865	0.527166	0.263583	+	0.964637i	$0.415096\pi$
$864$	0	0
$865$	−20.7110	−0.704194
$866$	0	0
$867$	−16.4200	−0.557653
$868$	0	0
$869$	16.8247	0.570738
$870$	0	0
$871$	−55.3203	−1.87446
$872$	0	0
$873$	−0.0645508	−0.00218471
$874$	0	0
$875$	1.89692	0.0641275
$876$	0	0
$877$	26.7755	0.904145	0.452072	−	0.891981i	$-0.350685\pi$
$877$	26.7755	0.904145	0.452072	+	0.891981i	$0.350685\pi$
$878$	0	0
$879$	−25.1247	−0.847436
$880$	0	0
$881$	5.75386	0.193853	0.0969263	−	0.995292i	$-0.469099\pi$
$881$	5.75386	0.193853	0.0969263	+	0.995292i	$0.469099\pi$
$882$	0	0
$883$	22.8559	0.769162	0.384581	−	0.923091i	$-0.374346\pi$
$883$	22.8559	0.769162	0.384581	+	0.923091i	$0.374346\pi$
$884$	0	0
$885$	10.2201	0.343546
$886$	0	0
$887$	−39.0741	−1.31198	−0.655991	−	0.754769i	$-0.727749\pi$
$887$	−39.0741	−1.31198	−0.655991	+	0.754769i	$0.727749\pi$
$888$	0	0
$889$	4.52937	0.151910
$890$	0	0
$891$	−2.38776	−0.0799928
$892$	0	0
$893$	2.71096	0.0907188
$894$	0	0
$895$	22.2986	0.745361
$896$	0	0
$897$	4.38776	0.146503
$898$	0	0
$899$	81.9686	2.73380
$900$	0	0
$901$	6.10308	0.203323
$902$	0	0
$903$	6.16952	0.205309
$904$	0	0
$905$	−21.4865	−0.714234
$906$	0	0
$907$	29.2784	0.972174	0.486087	−	0.873910i	$-0.338424\pi$
$907$	29.2784	0.972174	0.486087	+	0.873910i	$0.338424\pi$
$908$	0	0
$909$	−1.30299	−0.0432176
$910$	0	0
$911$	−25.9634	−0.860204	−0.430102	−	0.902780i	$-0.641522\pi$
$911$	−25.9634	−0.860204	−0.430102	+	0.902780i	$0.641522\pi$
$912$	0	0
$913$	−12.5081	−0.413958
$914$	0	0
$915$	3.64015	0.120340
$916$	0	0
$917$	1.80922	0.0597458
$918$	0	0
$919$	−26.9171	−0.887914	−0.443957	−	0.896048i	$-0.646426\pi$
$919$	−26.9171	−0.887914	−0.443957	+	0.896048i	$0.646426\pi$
$920$	0	0
$921$	−5.64015	−0.185849
$922$	0	0
$923$	28.7634	0.946760
$924$	0	0
$925$	−0.103084	−0.00338937
$926$	0	0
$927$	−4.98168	−0.163620
$928$	0	0
$929$	41.9494	1.37632	0.688158	−	0.725561i	$-0.258420\pi$
$929$	41.9494	1.37632	0.688158	+	0.725561i	$0.258420\pi$
$930$	0	0
$931$	−2.94126	−0.0963958
$932$	0	0
$933$	11.0462	0.361637
$934$	0	0
$935$	−1.81841	−0.0594684
$936$	0	0
$937$	34.9817	1.14280	0.571401	−	0.820671i	$-0.306400\pi$
$937$	34.9817	1.14280	0.571401	+	0.820671i	$0.306400\pi$
$938$	0	0
$939$	−4.30925	−0.140627
$940$	0	0
$941$	−4.38776	−0.143037	−0.0715184	−	0.997439i	$-0.522784\pi$
$941$	−4.38776	−0.143037	−0.0715184	+	0.997439i	$0.522784\pi$
$942$	0	0
$943$	−9.53707	−0.310570
$944$	0	0
$945$	1.89692	0.0617067
$946$	0	0
$947$	10.3265	0.335567	0.167784	−	0.985824i	$-0.446339\pi$
$947$	10.3265	0.335567	0.167784	+	0.985824i	$0.446339\pi$
$948$	0	0
$949$	21.6281	0.702077
$950$	0	0
$951$	8.62183	0.279582
$952$	0	0
$953$	−48.5048	−1.57122	−0.785612	−	0.618719i	$-0.787652\pi$
$953$	−48.5048	−1.57122	−0.785612	+	0.618719i	$0.787652\pi$
$954$	0	0
$955$	−12.8646	−0.416290
$956$	0	0
$957$	−20.9205	−0.676262
$958$	0	0
$959$	7.46522	0.241064
$960$	0	0
$961$	56.5250	1.82339
$962$	0	0
$963$	−0.697006	−0.0224607
$964$	0	0
$965$	−18.0279	−0.580339
$966$	0	0
$967$	29.9021	0.961588	0.480794	−	0.876834i	$-0.340349\pi$
$967$	29.9021	0.961588	0.480794	+	0.876834i	$0.340349\pi$
$968$	0	0
$969$	0.658473	0.0211532
$970$	0	0
$971$	−7.15120	−0.229493	−0.114746	−	0.993395i	$-0.536606\pi$
$971$	−7.15120	−0.229493	−0.114746	+	0.993395i	$0.536606\pi$
$972$	0	0
$973$	−5.46045	−0.175054
$974$	0	0
$975$	−4.38776	−0.140521
$976$	0	0
$977$	−30.7249	−0.982977	−0.491489	−	0.870884i	$-0.663547\pi$
$977$	−30.7249	−0.982977	−0.491489	+	0.870884i	$0.663547\pi$
$978$	0	0
$979$	26.2216	0.838045
$980$	0	0
$981$	−4.92919	−0.157377
$982$	0	0
$983$	−31.3588	−1.00019	−0.500095	−	0.865970i	$-0.666702\pi$
$983$	−31.3588	−1.00019	−0.500095	+	0.865970i	$0.666702\pi$
$984$	0	0
$985$	14.2341	0.453535
$986$	0	0
$987$	5.94751	0.189311
$988$	0	0
$989$	−3.25240	−0.103420
$990$	0	0
$991$	−54.1868	−1.72130	−0.860650	−	0.509197i	$-0.829943\pi$
$991$	−54.1868	−1.72130	−0.860650	+	0.509197i	$0.829943\pi$
$992$	0	0
$993$	−10.8786	−0.345222
$994$	0	0
$995$	12.5693	0.398475
$996$	0	0
$997$	27.9354	0.884725	0.442362	−	0.896836i	$-0.354141\pi$
$997$	27.9354	0.884725	0.442362	+	0.896836i	$0.354141\pi$
$998$	0	0
$999$	−0.103084	−0.00326142

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.a.u.1.2	✓	3
3.2	odd	2		8280.2.a.bi.1.2		3
4.3	odd	2		5520.2.a.bz.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.u.1.2	✓	3	1.1	even	1	trivial
5520.2.a.bz.1.2		3	4.3	odd	2
8280.2.a.bi.1.2		3	3.2	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:	\( N \)	\(=\)	\( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2760.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.89692 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.89692 q^{7} +1.00000 q^{9} -2.38776 q^{11} -4.38776 q^{13} +1.00000 q^{15} +0.761557 q^{17} +0.864641 q^{19} +1.89692 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.76156 q^{29} +9.35548 q^{31} -2.38776 q^{33} +1.89692 q^{35} -0.103084 q^{37} -4.38776 q^{39} +9.53707 q^{41} +3.25240 q^{43} +1.00000 q^{45} +3.13536 q^{47} -3.40171 q^{49} +0.761557 q^{51} +8.01395 q^{53} -2.38776 q^{55} +0.864641 q^{57} +10.2201 q^{59} +3.64015 q^{61} +1.89692 q^{63} -4.38776 q^{65} +12.6079 q^{67} -1.00000 q^{69} -6.55539 q^{71} -4.92919 q^{73} +1.00000 q^{75} -4.52937 q^{77} -7.04623 q^{79} +1.00000 q^{81} +5.23844 q^{83} +0.761557 q^{85} +8.76156 q^{87} -10.9817 q^{89} -8.32320 q^{91} +9.35548 q^{93} +0.864641 q^{95} -0.0645508 q^{97} -2.38776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + 8 q^{11} + 2 q^{13} + 3 q^{15} - 4 q^{17} + 2 q^{21} - 3 q^{23} + 3 q^{25} + 3 q^{27} + 20 q^{29} + 14 q^{31} + 8 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{47} + 29 q^{49} - 4 q^{51} + 8 q^{55} + 14 q^{59} - 22 q^{61} + 2 q^{63} + 2 q^{65} + 6 q^{67} - 3 q^{69} - 6 q^{71} - 10 q^{73} + 3 q^{75} - 8 q^{77} + 4 q^{79} + 3 q^{81} + 22 q^{83} - 4 q^{85} + 20 q^{87} - 10 q^{89} - 12 q^{91} + 14 q^{93} + 2 q^{97} + 8 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9 + 8 * q^11 + 2 * q^13 + 3 * q^15 - 4 * q^17 + 2 * q^21 - 3 * q^23 + 3 * q^25 + 3 * q^27 + 20 * q^29 + 14 * q^31 + 8 * q^33 + 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 - 8 * q^43 + 3 * q^45 + 12 * q^47 + 29 * q^49 - 4 * q^51 + 8 * q^55 + 14 * q^59 - 22 * q^61 + 2 * q^63 + 2 * q^65 + 6 * q^67 - 3 * q^69 - 6 * q^71 - 10 * q^73 + 3 * q^75 - 8 * q^77 + 4 * q^79 + 3 * q^81 + 22 * q^83 - 4 * q^85 + 20 * q^87 - 10 * q^89 - 12 * q^91 + 14 * q^93 + 2 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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