LMFDB - Embedded newform 6840.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6840.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:	$54.6176749826$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	6840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +1.19656 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +1.19656 q^{7} -4.97858 q^{11} -6.63565 q^{13} -1.48929 q^{17} +1.00000 q^{19} +0.510711 q^{23} +1.00000 q^{25} +7.88240 q^{29} -2.97858 q^{31} +1.19656 q^{35} -7.14637 q^{37} -1.66442 q^{41} -6.39312 q^{43} +9.95715 q^{47} -5.56825 q^{49} +11.4219 q^{53} -4.97858 q^{55} +11.8396 q^{59} +3.66442 q^{61} -6.63565 q^{65} +7.61423 q^{67} +13.8396 q^{73} -5.95715 q^{77} +12.6858 q^{79} +8.68585 q^{83} -1.48929 q^{85} +4.87819 q^{89} -7.93994 q^{91} +1.00000 q^{95} -6.81079 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - q^7	$ 3 q + 3 q^{5} - q^{7} - 11 q^{13} + 3 q^{17} + 3 q^{19} + 9 q^{23} + 3 q^{25} + 7 q^{29} + 6 q^{31} - q^{35} - 20 q^{37} + 22 q^{41} - 10 q^{43} + 12 q^{49} + 7 q^{53} - 11 q^{59} - 16 q^{61} - 11 q^{65} - q^{67} - 5 q^{73} + 12 q^{77} + 26 q^{79} + 14 q^{83} + 3 q^{85} + 6 q^{89} + 29 q^{91} + 3 q^{95} + 8 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - q^7 - 11 * q^13 + 3 * q^17 + 3 * q^19 + 9 * q^23 + 3 * q^25 + 7 * q^29 + 6 * q^31 - q^35 - 20 * q^37 + 22 * q^41 - 10 * q^43 + 12 * q^49 + 7 * q^53 - 11 * q^59 - 16 * q^61 - 11 * q^65 - q^67 - 5 * q^73 + 12 * q^77 + 26 * q^79 + 14 * q^83 + 3 * q^85 + 6 * q^89 + 29 * q^91 + 3 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.19656	0.452256	0.226128	−	0.974098i	$-0.427393\pi$
$7$	1.19656	0.452256	0.226128	+	0.974098i	$0.427393\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.97858	−1.50110	−0.750549	−	0.660815i	$-0.770211\pi$
$11$	−4.97858	−1.50110	−0.750549	+	0.660815i	$0.770211\pi$
$12$	0	0
$13$	−6.63565	−1.84040	−0.920200	−	0.391449i	$-0.871974\pi$
$13$	−6.63565	−1.84040	−0.920200	+	0.391449i	$0.871974\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.48929	−0.361206	−0.180603	−	0.983556i	$-0.557805\pi$
$17$	−1.48929	−0.361206	−0.180603	+	0.983556i	$0.557805\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.510711	0.106491	0.0532453	−	0.998581i	$-0.483043\pi$
$23$	0.510711	0.106491	0.0532453	+	0.998581i	$0.483043\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.88240	1.46373	0.731863	−	0.681452i	$-0.238651\pi$
$29$	7.88240	1.46373	0.731863	+	0.681452i	$0.238651\pi$
$30$	0	0
$31$	−2.97858	−0.534968	−0.267484	−	0.963562i	$-0.586192\pi$
$31$	−2.97858	−0.534968	−0.267484	+	0.963562i	$0.586192\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.19656	0.202255
$36$	0	0
$37$	−7.14637	−1.17486	−0.587428	−	0.809277i	$-0.699859\pi$
$37$	−7.14637	−1.17486	−0.587428	+	0.809277i	$0.699859\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.66442	−0.259939	−0.129970	−	0.991518i	$-0.541488\pi$
$41$	−1.66442	−0.259939	−0.129970	+	0.991518i	$0.541488\pi$
$42$	0	0
$43$	−6.39312	−0.974941	−0.487470	−	0.873139i	$-0.662080\pi$
$43$	−6.39312	−0.974941	−0.487470	+	0.873139i	$0.662080\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.95715	1.45240	0.726200	−	0.687483i	$-0.241285\pi$
$47$	9.95715	1.45240	0.726200	+	0.687483i	$0.241285\pi$
$48$	0	0
$49$	−5.56825	−0.795464
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.4219	1.56892	0.784458	−	0.620182i	$-0.212941\pi$
$53$	11.4219	1.56892	0.784458	+	0.620182i	$0.212941\pi$
$54$	0	0
$55$	−4.97858	−0.671311
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.8396	1.54138	0.770690	−	0.637211i	$-0.219912\pi$
$59$	11.8396	1.54138	0.770690	+	0.637211i	$0.219912\pi$
$60$	0	0
$61$	3.66442	0.469181	0.234591	−	0.972094i	$-0.424625\pi$
$61$	3.66442	0.469181	0.234591	+	0.972094i	$0.424625\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.63565	−0.823052
$66$	0	0
$67$	7.61423	0.930226	0.465113	−	0.885251i	$-0.346014\pi$
$67$	7.61423	0.930226	0.465113	+	0.885251i	$0.346014\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	13.8396	1.61980	0.809899	−	0.586570i	$-0.199522\pi$
$73$	13.8396	1.61980	0.809899	+	0.586570i	$0.199522\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.95715	−0.678881
$78$	0	0
$79$	12.6858	1.42727	0.713635	−	0.700518i	$-0.247048\pi$
$79$	12.6858	1.42727	0.713635	+	0.700518i	$0.247048\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.68585	0.953395	0.476698	−	0.879067i	$-0.341834\pi$
$83$	8.68585	0.953395	0.476698	+	0.879067i	$0.341834\pi$
$84$	0	0
$85$	−1.48929	−0.161536
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.87819	0.517087	0.258544	−	0.966000i	$-0.416757\pi$
$89$	4.87819	0.517087	0.258544	+	0.966000i	$0.416757\pi$
$90$	0	0
$91$	−7.93994	−0.832332
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−6.81079	−0.691531	−0.345765	−	0.938321i	$-0.612381\pi$
$97$	−6.81079	−0.691531	−0.345765	+	0.938321i	$0.612381\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.29273	0.228135	0.114068	−	0.993473i	$-0.463612\pi$
$101$	2.29273	0.228135	0.114068	+	0.993473i	$0.463612\pi$
$102$	0	0
$103$	−6.51806	−0.642243	−0.321122	−	0.947038i	$-0.604060\pi$
$103$	−6.51806	−0.642243	−0.321122	+	0.947038i	$0.604060\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.71462	−0.745800	−0.372900	−	0.927872i	$-0.621637\pi$
$107$	−7.71462	−0.745800	−0.372900	+	0.927872i	$0.621637\pi$
$108$	0	0
$109$	15.5468	1.48912	0.744558	−	0.667558i	$-0.232660\pi$
$109$	15.5468	1.48912	0.744558	+	0.667558i	$0.232660\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.753250	0.0708598	0.0354299	−	0.999372i	$-0.488720\pi$
$113$	0.753250	0.0708598	0.0354299	+	0.999372i	$0.488720\pi$
$114$	0	0
$115$	0.510711	0.0476241
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.78202	−0.163357
$120$	0	0
$121$	13.7862	1.25329
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.4177	0.924419	0.462210	−	0.886771i	$-0.347057\pi$
$127$	10.4177	0.924419	0.462210	+	0.886771i	$0.347057\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.9572	1.56892	0.784462	−	0.620177i	$-0.212939\pi$
$131$	17.9572	1.56892	0.784462	+	0.620177i	$0.212939\pi$
$132$	0	0
$133$	1.19656	0.103755
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.83956	0.840650	0.420325	−	0.907374i	$-0.361916\pi$
$137$	9.83956	0.840650	0.420325	+	0.907374i	$0.361916\pi$
$138$	0	0
$139$	−0.978577	−0.0830018	−0.0415009	−	0.999138i	$-0.513214\pi$
$139$	−0.978577	−0.0830018	−0.0415009	+	0.999138i	$0.513214\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	33.0361	2.76262
$144$	0	0
$145$	7.88240	0.654598
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.24989	0.675857	0.337928	−	0.941172i	$-0.390274\pi$
$149$	8.24989	0.675857	0.337928	+	0.941172i	$0.390274\pi$
$150$	0	0
$151$	13.8568	1.12765	0.563824	−	0.825895i	$-0.309330\pi$
$151$	13.8568	1.12765	0.563824	+	0.825895i	$0.309330\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.97858	−0.239245
$156$	0	0
$157$	−15.7073	−1.25358	−0.626788	−	0.779190i	$-0.715631\pi$
$157$	−15.7073	−1.25358	−0.626788	+	0.779190i	$0.715631\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.611096	0.0481611
$162$	0	0
$163$	5.80765	0.454891	0.227445	−	0.973791i	$-0.426963\pi$
$163$	5.80765	0.454891	0.227445	+	0.973791i	$0.426963\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.8322	1.07037	0.535184	−	0.844735i	$-0.320242\pi$
$167$	13.8322	1.07037	0.535184	+	0.844735i	$0.320242\pi$
$168$	0	0
$169$	31.0319	2.38707
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.8108	−1.12604	−0.563022	−	0.826442i	$-0.690361\pi$
$173$	−14.8108	−1.12604	−0.563022	+	0.826442i	$0.690361\pi$
$174$	0	0
$175$	1.19656	0.0904513
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.6644	−1.17081	−0.585407	−	0.810740i	$-0.699065\pi$
$179$	−15.6644	−1.17081	−0.585407	+	0.810740i	$0.699065\pi$
$180$	0	0
$181$	−14.7862	−1.09905	−0.549526	−	0.835477i	$-0.685192\pi$
$181$	−14.7862	−1.09905	−0.549526	+	0.835477i	$0.685192\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.14637	−0.525411
$186$	0	0
$187$	7.41454	0.542205
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.36748	0.460735	0.230367	−	0.973104i	$-0.426007\pi$
$191$	6.36748	0.460735	0.230367	+	0.973104i	$0.426007\pi$
$192$	0	0
$193$	−19.5970	−1.41062	−0.705312	−	0.708897i	$-0.749193\pi$
$193$	−19.5970	−1.41062	−0.705312	+	0.708897i	$0.749193\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.7862	−0.768487	−0.384244	−	0.923232i	$-0.625538\pi$
$197$	−10.7862	−0.768487	−0.384244	+	0.923232i	$0.625538\pi$
$198$	0	0
$199$	2.80344	0.198731	0.0993654	−	0.995051i	$-0.468319\pi$
$199$	2.80344	0.198731	0.0993654	+	0.995051i	$0.468319\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.43175	0.661979
$204$	0	0
$205$	−1.66442	−0.116248
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.97858	−0.344375
$210$	0	0
$211$	−11.2541	−0.774764	−0.387382	−	0.921919i	$-0.626621\pi$
$211$	−11.2541	−0.774764	−0.387382	+	0.921919i	$0.626621\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.39312	−0.436007
$216$	0	0
$217$	−3.56404	−0.241943
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.88240	0.664762
$222$	0	0
$223$	17.4966	1.17166	0.585831	−	0.810433i	$-0.300768\pi$
$223$	17.4966	1.17166	0.585831	+	0.810433i	$0.300768\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.84942	0.255495	0.127748	−	0.991807i	$-0.459225\pi$
$227$	3.84942	0.255495	0.127748	+	0.991807i	$0.459225\pi$
$228$	0	0
$229$	−14.6430	−0.967637	−0.483818	−	0.875168i	$-0.660750\pi$
$229$	−14.6430	−0.967637	−0.483818	+	0.875168i	$0.660750\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.9572	−0.783339	−0.391670	−	0.920106i	$-0.628102\pi$
$233$	−11.9572	−0.783339	−0.391670	+	0.920106i	$0.628102\pi$
$234$	0	0
$235$	9.95715	0.649533
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.5897	−1.00841	−0.504206	−	0.863583i	$-0.668215\pi$
$239$	−15.5897	−1.00841	−0.504206	+	0.863583i	$0.668215\pi$
$240$	0	0
$241$	−16.0575	−1.03436	−0.517178	−	0.855878i	$-0.673018\pi$
$241$	−16.0575	−1.03436	−0.517178	+	0.855878i	$0.673018\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.56825	−0.355742
$246$	0	0
$247$	−6.63565	−0.422217
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.9143	−1.50946	−0.754729	−	0.656037i	$-0.772232\pi$
$251$	−23.9143	−1.50946	−0.754729	+	0.656037i	$0.772232\pi$
$252$	0	0
$253$	−2.54262	−0.159853
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.4324	1.52405	0.762025	−	0.647548i	$-0.224206\pi$
$257$	24.4324	1.52405	0.762025	+	0.647548i	$0.224206\pi$
$258$	0	0
$259$	−8.55104	−0.531336
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.4935	0.893707	0.446854	−	0.894607i	$-0.352544\pi$
$263$	14.4935	0.893707	0.446854	+	0.894607i	$0.352544\pi$
$264$	0	0
$265$	11.4219	0.701641
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−6.76060	−0.410677	−0.205339	−	0.978691i	$-0.565830\pi$
$271$	−6.76060	−0.410677	−0.205339	+	0.978691i	$0.565830\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.97858	−0.300219
$276$	0	0
$277$	−25.4292	−1.52789	−0.763947	−	0.645279i	$-0.776741\pi$
$277$	−25.4292	−1.52789	−0.763947	+	0.645279i	$0.776741\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.07896	−0.541605	−0.270803	−	0.962635i	$-0.587289\pi$
$281$	−9.07896	−0.541605	−0.270803	+	0.962635i	$0.587289\pi$
$282$	0	0
$283$	12.0147	0.714199	0.357100	−	0.934066i	$-0.383766\pi$
$283$	12.0147	0.714199	0.357100	+	0.934066i	$0.383766\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.99158	−0.117559
$288$	0	0
$289$	−14.7820	−0.869531
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−23.6718	−1.38292	−0.691460	−	0.722415i	$-0.743032\pi$
$293$	−23.6718	−1.38292	−0.691460	+	0.722415i	$0.743032\pi$
$294$	0	0
$295$	11.8396	0.689326
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.38890	−0.195985
$300$	0	0
$301$	−7.64973	−0.440923
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.66442	0.209824
$306$	0	0
$307$	−16.0246	−0.914570	−0.457285	−	0.889320i	$-0.651178\pi$
$307$	−16.0246	−0.914570	−0.457285	+	0.889320i	$0.651178\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.5468	1.56204	0.781019	−	0.624508i	$-0.214700\pi$
$311$	27.5468	1.56204	0.781019	+	0.624508i	$0.214700\pi$
$312$	0	0
$313$	10.1751	0.575133	0.287566	−	0.957761i	$-0.407154\pi$
$313$	10.1751	0.575133	0.287566	+	0.957761i	$0.407154\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.6289	−1.66413	−0.832063	−	0.554681i	$-0.812840\pi$
$317$	−29.6289	−1.66413	−0.832063	+	0.554681i	$0.812840\pi$
$318$	0	0
$319$	−39.2432	−2.19719
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.48929	−0.0828662
$324$	0	0
$325$	−6.63565	−0.368080
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.9143	0.656857
$330$	0	0
$331$	30.4679	1.67467	0.837333	−	0.546694i	$-0.184114\pi$
$331$	30.4679	1.67467	0.837333	+	0.546694i	$0.184114\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.61423	0.416010
$336$	0	0
$337$	26.1396	1.42392	0.711958	−	0.702222i	$-0.247808\pi$
$337$	26.1396	1.42392	0.711958	+	0.702222i	$0.247808\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.8291	0.803039
$342$	0	0
$343$	−15.0386	−0.812010
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.31415	0.392644	0.196322	−	0.980539i	$-0.437100\pi$
$347$	7.31415	0.392644	0.196322	+	0.980539i	$0.437100\pi$
$348$	0	0
$349$	0.628308	0.0336325	0.0168163	−	0.999859i	$-0.494647\pi$
$349$	0.628308	0.0336325	0.0168163	+	0.999859i	$0.494647\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.23267	0.438181	0.219090	−	0.975705i	$-0.429691\pi$
$353$	8.23267	0.438181	0.219090	+	0.975705i	$0.429691\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.1323	−0.640318	−0.320159	−	0.947364i	$-0.603736\pi$
$359$	−12.1323	−0.640318	−0.320159	+	0.947364i	$0.603736\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.8396	0.724396
$366$	0	0
$367$	17.9572	0.937356	0.468678	−	0.883369i	$-0.344730\pi$
$367$	17.9572	0.937356	0.468678	+	0.883369i	$0.344730\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.6669	0.709552
$372$	0	0
$373$	29.6289	1.53413	0.767064	−	0.641571i	$-0.221717\pi$
$373$	29.6289	1.53413	0.767064	+	0.641571i	$0.221717\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−52.3049	−2.69384
$378$	0	0
$379$	−6.80344	−0.349469	−0.174735	−	0.984616i	$-0.555907\pi$
$379$	−6.80344	−0.349469	−0.174735	+	0.984616i	$0.555907\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.2253	1.03347	0.516733	−	0.856147i	$-0.327148\pi$
$383$	20.2253	1.03347	0.516733	+	0.856147i	$0.327148\pi$
$384$	0	0
$385$	−5.95715	−0.303605
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−0.393115	−0.0199317	−0.00996587	−	0.999950i	$-0.503172\pi$
$389$	−0.393115	−0.0199317	−0.00996587	+	0.999950i	$0.503172\pi$
$390$	0	0
$391$	−0.760597	−0.0384650
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.6858	0.638294
$396$	0	0
$397$	17.4292	0.874748	0.437374	−	0.899280i	$-0.355909\pi$
$397$	17.4292	0.874748	0.437374	+	0.899280i	$0.355909\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13.1281	−0.655585	−0.327792	−	0.944750i	$-0.606305\pi$
$401$	−13.1281	−0.655585	−0.327792	+	0.944750i	$0.606305\pi$
$402$	0	0
$403$	19.7648	0.984555
$404$	0	0
$405$	0	0
$406$	0	0
$407$	35.5787	1.76357
$408$	0	0
$409$	−18.7862	−0.928919	−0.464460	−	0.885594i	$-0.653751\pi$
$409$	−18.7862	−0.928919	−0.464460	+	0.885594i	$0.653751\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	14.1667	0.697098
$414$	0	0
$415$	8.68585	0.426371
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.22219	0.255121	0.127560	−	0.991831i	$-0.459285\pi$
$419$	5.22219	0.255121	0.127560	+	0.991831i	$0.459285\pi$
$420$	0	0
$421$	−7.68164	−0.374380	−0.187190	−	0.982324i	$-0.559938\pi$
$421$	−7.68164	−0.374380	−0.187190	+	0.982324i	$0.559938\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.48929	−0.0722411
$426$	0	0
$427$	4.38469	0.212190
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.7073	0.852929	0.426465	−	0.904504i	$-0.359759\pi$
$431$	17.7073	0.852929	0.426465	+	0.904504i	$0.359759\pi$
$432$	0	0
$433$	17.9901	0.864551	0.432275	−	0.901742i	$-0.357711\pi$
$433$	17.9901	0.864551	0.432275	+	0.901742i	$0.357711\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.510711	0.0244306
$438$	0	0
$439$	14.8438	0.708454	0.354227	−	0.935159i	$-0.384744\pi$
$439$	14.8438	0.708454	0.354227	+	0.935159i	$0.384744\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−22.4078	−1.06463	−0.532314	−	0.846547i	$-0.678677\pi$
$443$	−22.4078	−1.06463	−0.532314	+	0.846547i	$0.678677\pi$
$444$	0	0
$445$	4.87819	0.231249
$446$	0	0
$447$	0	0
$448$	0	0
$449$	40.7434	1.92280	0.961400	−	0.275156i	$-0.0887295\pi$
$449$	40.7434	1.92280	0.961400	+	0.275156i	$0.0887295\pi$
$450$	0	0
$451$	8.28646	0.390194
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.93994	−0.372230
$456$	0	0
$457$	8.01721	0.375029	0.187515	−	0.982262i	$-0.439957\pi$
$457$	8.01721	0.375029	0.187515	+	0.982262i	$0.439957\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	25.1281	1.17033	0.585166	−	0.810914i	$-0.301029\pi$
$461$	25.1281	1.17033	0.585166	+	0.810914i	$0.301029\pi$
$462$	0	0
$463$	−31.7795	−1.47692	−0.738459	−	0.674298i	$-0.764446\pi$
$463$	−31.7795	−1.47692	−0.738459	+	0.674298i	$0.764446\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.9210	−0.968110	−0.484055	−	0.875038i	$-0.660837\pi$
$467$	−20.9210	−0.968110	−0.484055	+	0.875038i	$0.660837\pi$
$468$	0	0
$469$	9.11087	0.420701
$470$	0	0
$471$	0	0
$472$	0	0
$473$	31.8286	1.46348
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−27.7220	−1.26665	−0.633324	−	0.773886i	$-0.718310\pi$
$479$	−27.7220	−1.26665	−0.633324	+	0.773886i	$0.718310\pi$
$480$	0	0
$481$	47.4208	2.16220
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.81079	−0.309262
$486$	0	0
$487$	3.20390	0.145183	0.0725914	−	0.997362i	$-0.476873\pi$
$487$	3.20390	0.145183	0.0725914	+	0.997362i	$0.476873\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.5212	0.971238	0.485619	−	0.874171i	$-0.338594\pi$
$491$	21.5212	0.971238	0.485619	+	0.874171i	$0.338594\pi$
$492$	0	0
$493$	−11.7392	−0.528706
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−23.1709	−1.03727	−0.518637	−	0.854995i	$-0.673560\pi$
$499$	−23.1709	−1.03727	−0.518637	+	0.854995i	$0.673560\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.88240	0.0839322	0.0419661	−	0.999119i	$-0.486638\pi$
$503$	1.88240	0.0839322	0.0419661	+	0.999119i	$0.486638\pi$
$504$	0	0
$505$	2.29273	0.102025
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.46365	−0.242172	−0.121086	−	0.992642i	$-0.538638\pi$
$509$	−5.46365	−0.242172	−0.121086	+	0.992642i	$0.538638\pi$
$510$	0	0
$511$	16.5598	0.732564
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.51806	−0.287220
$516$	0	0
$517$	−49.5725	−2.18019
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11.2222	−0.491653	−0.245827	−	0.969314i	$-0.579059\pi$
$521$	−11.2222	−0.491653	−0.245827	+	0.969314i	$0.579059\pi$
$522$	0	0
$523$	−29.1867	−1.27624	−0.638122	−	0.769935i	$-0.720289\pi$
$523$	−29.1867	−1.27624	−0.638122	+	0.769935i	$0.720289\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.43596	0.193233
$528$	0	0
$529$	−22.7392	−0.988660
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.0445	0.478392
$534$	0	0
$535$	−7.71462	−0.333532
$536$	0	0
$537$	0	0
$538$	0	0
$539$	27.7220	1.19407
$540$	0	0
$541$	20.2499	0.870611	0.435305	−	0.900283i	$-0.356640\pi$
$541$	20.2499	0.870611	0.435305	+	0.900283i	$0.356640\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	15.5468	0.665953
$546$	0	0
$547$	42.8830	1.83355	0.916773	−	0.399409i	$-0.130785\pi$
$547$	42.8830	1.83355	0.916773	+	0.399409i	$0.130785\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.88240	0.335802
$552$	0	0
$553$	15.1793	0.645491
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.2583	−1.19734	−0.598671	−	0.800995i	$-0.704304\pi$
$557$	−28.2583	−1.19734	−0.598671	+	0.800995i	$0.704304\pi$
$558$	0	0
$559$	42.4225	1.79428
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.2253	0.852396	0.426198	−	0.904630i	$-0.359853\pi$
$563$	20.2253	0.852396	0.426198	+	0.904630i	$0.359853\pi$
$564$	0	0
$565$	0.753250	0.0316895
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.1365	1.30531	0.652655	−	0.757655i	$-0.273655\pi$
$569$	31.1365	1.30531	0.652655	+	0.757655i	$0.273655\pi$
$570$	0	0
$571$	27.4868	1.15029	0.575143	−	0.818053i	$-0.304947\pi$
$571$	27.4868	1.15029	0.575143	+	0.818053i	$0.304947\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.510711	0.0212981
$576$	0	0
$577$	23.7820	0.990058	0.495029	−	0.868876i	$-0.335157\pi$
$577$	23.7820	0.990058	0.495029	+	0.868876i	$0.335157\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.3931	0.431179
$582$	0	0
$583$	−56.8647	−2.35510
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.0147	−0.991192	−0.495596	−	0.868553i	$-0.665050\pi$
$587$	−24.0147	−0.991192	−0.495596	+	0.868553i	$0.665050\pi$
$588$	0	0
$589$	−2.97858	−0.122730
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−46.6148	−1.91424	−0.957121	−	0.289688i	$-0.906449\pi$
$593$	−46.6148	−1.91424	−0.957121	+	0.289688i	$0.906449\pi$
$594$	0	0
$595$	−1.78202	−0.0730557
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.6002	1.82231	0.911156	−	0.412061i	$-0.135191\pi$
$599$	44.6002	1.82231	0.911156	+	0.412061i	$0.135191\pi$
$600$	0	0
$601$	−17.5787	−0.717051	−0.358526	−	0.933520i	$-0.616720\pi$
$601$	−17.5787	−0.717051	−0.358526	+	0.933520i	$0.616720\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	13.7862	0.560490
$606$	0	0
$607$	37.2467	1.51180	0.755899	−	0.654688i	$-0.227200\pi$
$607$	37.2467	1.51180	0.755899	+	0.654688i	$0.227200\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−66.0722	−2.67300
$612$	0	0
$613$	−14.8866	−0.601265	−0.300632	−	0.953740i	$-0.597198\pi$
$613$	−14.8866	−0.601265	−0.300632	+	0.953740i	$0.597198\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.70727	−0.149249	−0.0746245	−	0.997212i	$-0.523776\pi$
$617$	−3.70727	−0.149249	−0.0746245	+	0.997212i	$0.523776\pi$
$618$	0	0
$619$	17.7648	0.714028	0.357014	−	0.934099i	$-0.383795\pi$
$619$	17.7648	0.714028	0.357014	+	0.934099i	$0.383795\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.83704	0.233856
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.6430	0.424364
$630$	0	0
$631$	40.3074	1.60461	0.802307	−	0.596912i	$-0.203606\pi$
$631$	40.3074	1.60461	0.802307	+	0.596912i	$0.203606\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.4177	0.413413
$636$	0	0
$637$	36.9490	1.46397
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27.6728	1.09301	0.546506	−	0.837455i	$-0.315958\pi$
$641$	27.6728	1.09301	0.546506	+	0.837455i	$0.315958\pi$
$642$	0	0
$643$	19.0277	0.750379	0.375190	−	0.926948i	$-0.377578\pi$
$643$	19.0277	0.750379	0.375190	+	0.926948i	$0.377578\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.0403	1.25964	0.629818	−	0.776743i	$-0.283130\pi$
$647$	32.0403	1.25964	0.629818	+	0.776743i	$0.283130\pi$
$648$	0	0
$649$	−58.9442	−2.31376
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.3142	−0.990619	−0.495310	−	0.868716i	$-0.664945\pi$
$653$	−25.3142	−0.990619	−0.495310	+	0.868716i	$0.664945\pi$
$654$	0	0
$655$	17.9572	0.701644
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0.860981	0.0335391	0.0167695	−	0.999859i	$-0.494662\pi$
$659$	0.860981	0.0335391	0.0167695	+	0.999859i	$0.494662\pi$
$660$	0	0
$661$	4.55356	0.177113	0.0885564	−	0.996071i	$-0.471775\pi$
$661$	4.55356	0.177113	0.0885564	+	0.996071i	$0.471775\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.19656	0.0464005
$666$	0	0
$667$	4.02563	0.155873
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−18.2436	−0.704287
$672$	0	0
$673$	−11.7465	−0.452795	−0.226398	−	0.974035i	$-0.572695\pi$
$673$	−11.7465	−0.452795	−0.226398	+	0.974035i	$0.572695\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.2854	0.856497	0.428248	−	0.903661i	$-0.359131\pi$
$677$	22.2854	0.856497	0.428248	+	0.903661i	$0.359131\pi$
$678$	0	0
$679$	−8.14950	−0.312749
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7.23833	0.276967	0.138483	−	0.990365i	$-0.455777\pi$
$683$	7.23833	0.276967	0.138483	+	0.990365i	$0.455777\pi$
$684$	0	0
$685$	9.83956	0.375950
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−75.7917	−2.88743
$690$	0	0
$691$	32.7434	1.24562	0.622809	−	0.782374i	$-0.285992\pi$
$691$	32.7434	1.24562	0.622809	+	0.782374i	$0.285992\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.978577	−0.0371195
$696$	0	0
$697$	2.47881	0.0938915
$698$	0	0
$699$	0	0
$700$	0	0
$701$	29.4208	1.11121	0.555604	−	0.831447i	$-0.312487\pi$
$701$	29.4208	1.11121	0.555604	+	0.831447i	$0.312487\pi$
$702$	0	0
$703$	−7.14637	−0.269530
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.74338	0.103176
$708$	0	0
$709$	−5.32885	−0.200129	−0.100065	−	0.994981i	$-0.531905\pi$
$709$	−5.32885	−0.200129	−0.100065	+	0.994981i	$0.531905\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.52119	−0.0569691
$714$	0	0
$715$	33.0361	1.23548
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.45317	−0.315250	−0.157625	−	0.987499i	$-0.550384\pi$
$719$	−8.45317	−0.315250	−0.157625	+	0.987499i	$0.550384\pi$
$720$	0	0
$721$	−7.79923	−0.290459
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.88240	0.292745
$726$	0	0
$727$	2.56825	0.0952511	0.0476256	−	0.998865i	$-0.484835\pi$
$727$	2.56825	0.0952511	0.0476256	+	0.998865i	$0.484835\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.52119	0.352154
$732$	0	0
$733$	35.5212	1.31201	0.656003	−	0.754759i	$-0.272246\pi$
$733$	35.5212	1.31201	0.656003	+	0.754759i	$0.272246\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−37.9080	−1.39636
$738$	0	0
$739$	−37.0508	−1.36294	−0.681468	−	0.731848i	$-0.738658\pi$
$739$	−37.0508	−1.36294	−0.681468	+	0.731848i	$0.738658\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23.2039	−0.851269	−0.425634	−	0.904895i	$-0.639949\pi$
$743$	−23.2039	−0.851269	−0.425634	+	0.904895i	$0.639949\pi$
$744$	0	0
$745$	8.24989	0.302252
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.23098	−0.337293
$750$	0	0
$751$	51.2285	1.86935	0.934677	−	0.355499i	$-0.115689\pi$
$751$	51.2285	1.86935	0.934677	+	0.355499i	$0.115689\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	13.8568	0.504299
$756$	0	0
$757$	−43.3717	−1.57637	−0.788185	−	0.615438i	$-0.788979\pi$
$757$	−43.3717	−1.57637	−0.788185	+	0.615438i	$0.788979\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24.7753	−0.898104	−0.449052	−	0.893506i	$-0.648238\pi$
$761$	−24.7753	−0.898104	−0.449052	+	0.893506i	$0.648238\pi$
$762$	0	0
$763$	18.6027	0.673462
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−78.5632	−2.83675
$768$	0	0
$769$	−11.4893	−0.414314	−0.207157	−	0.978308i	$-0.566421\pi$
$769$	−11.4893	−0.414314	−0.207157	+	0.978308i	$0.566421\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27.0863	−0.974227	−0.487113	−	0.873339i	$-0.661950\pi$
$773$	−27.0863	−0.974227	−0.487113	+	0.873339i	$0.661950\pi$
$774$	0	0
$775$	−2.97858	−0.106994
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.66442	−0.0596342
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−15.7073	−0.560616
$786$	0	0
$787$	13.4563	0.479666	0.239833	−	0.970814i	$-0.422907\pi$
$787$	13.4563	0.479666	0.239833	+	0.970814i	$0.422907\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.901307	0.0320468
$792$	0	0
$793$	−24.3158	−0.863481
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.4496	−1.07858	−0.539290	−	0.842120i	$-0.681307\pi$
$797$	−30.4496	−1.07858	−0.539290	+	0.842120i	$0.681307\pi$
$798$	0	0
$799$	−14.8291	−0.524615
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−68.9013	−2.43147
$804$	0	0
$805$	0.611096	0.0215383
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29.7047	1.04436	0.522182	−	0.852834i	$-0.325118\pi$
$809$	29.7047	1.04436	0.522182	+	0.852834i	$0.325118\pi$
$810$	0	0
$811$	32.2902	1.13386	0.566931	−	0.823765i	$-0.308130\pi$
$811$	32.2902	1.13386	0.566931	+	0.823765i	$0.308130\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.80765	0.203433
$816$	0	0
$817$	−6.39312	−0.223667
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.06427	−0.316345	−0.158173	−	0.987411i	$-0.550560\pi$
$821$	−9.06427	−0.316345	−0.158173	+	0.987411i	$0.550560\pi$
$822$	0	0
$823$	−7.35448	−0.256361	−0.128181	−	0.991751i	$-0.540914\pi$
$823$	−7.35448	−0.256361	−0.128181	+	0.991751i	$0.540914\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.02204	−0.0703132	−0.0351566	−	0.999382i	$-0.511193\pi$
$827$	−2.02204	−0.0703132	−0.0351566	+	0.999382i	$0.511193\pi$
$828$	0	0
$829$	−36.3675	−1.26309	−0.631547	−	0.775337i	$-0.717580\pi$
$829$	−36.3675	−1.26309	−0.631547	+	0.775337i	$0.717580\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.29273	0.287326
$834$	0	0
$835$	13.8322	0.478683
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.8223	−0.891486	−0.445743	−	0.895161i	$-0.647061\pi$
$839$	−25.8223	−0.891486	−0.445743	+	0.895161i	$0.647061\pi$
$840$	0	0
$841$	33.1323	1.14249
$842$	0	0
$843$	0	0
$844$	0	0
$845$	31.0319	1.06753
$846$	0	0
$847$	16.4960	0.566810
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.64973	−0.125111
$852$	0	0
$853$	8.54262	0.292494	0.146247	−	0.989248i	$-0.453281\pi$
$853$	8.54262	0.292494	0.146247	+	0.989248i	$0.453281\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.8322	−0.950730	−0.475365	−	0.879789i	$-0.657684\pi$
$857$	−27.8322	−0.950730	−0.475365	+	0.879789i	$0.657684\pi$
$858$	0	0
$859$	33.4868	1.14255	0.571277	−	0.820757i	$-0.306448\pi$
$859$	33.4868	1.14255	0.571277	+	0.820757i	$0.306448\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29.2614	0.996071	0.498036	−	0.867157i	$-0.334055\pi$
$863$	29.2614	0.996071	0.498036	+	0.867157i	$0.334055\pi$
$864$	0	0
$865$	−14.8108	−0.503582
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−63.1575	−2.14247
$870$	0	0
$871$	−50.5254	−1.71199
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.19656	0.0404510
$876$	0	0
$877$	−46.7852	−1.57982	−0.789911	−	0.613221i	$-0.789873\pi$
$877$	−46.7852	−1.57982	−0.789911	+	0.613221i	$0.789873\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	30.7581	1.03627	0.518133	−	0.855300i	$-0.326627\pi$
$881$	30.7581	1.03627	0.518133	+	0.855300i	$0.326627\pi$
$882$	0	0
$883$	0.435961	0.0146713	0.00733563	−	0.999973i	$-0.497665\pi$
$883$	0.435961	0.0146713	0.00733563	+	0.999973i	$0.497665\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.15310	0.0722939	0.0361469	−	0.999346i	$-0.488492\pi$
$887$	2.15310	0.0722939	0.0361469	+	0.999346i	$0.488492\pi$
$888$	0	0
$889$	12.4653	0.418074
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.95715	0.333203
$894$	0	0
$895$	−15.6644	−0.523604
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−23.4783	−0.783047
$900$	0	0
$901$	−17.0105	−0.566701
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.7862	−0.491511
$906$	0	0
$907$	−36.9002	−1.22525	−0.612626	−	0.790373i	$-0.709887\pi$
$907$	−36.9002	−1.22525	−0.612626	+	0.790373i	$0.709887\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36.9504	1.22422	0.612111	−	0.790772i	$-0.290321\pi$
$911$	36.9504	1.22422	0.612111	+	0.790772i	$0.290321\pi$
$912$	0	0
$913$	−43.2432	−1.43114
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.4868	0.709556
$918$	0	0
$919$	19.1537	0.631823	0.315911	−	0.948789i	$-0.397690\pi$
$919$	19.1537	0.631823	0.315911	+	0.948789i	$0.397690\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−7.14637	−0.234971
$926$	0	0
$927$	0	0
$928$	0	0
$929$	56.9185	1.86744	0.933718	−	0.358009i	$-0.116544\pi$
$929$	56.9185	1.86744	0.933718	+	0.358009i	$0.116544\pi$
$930$	0	0
$931$	−5.56825	−0.182492
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.41454	0.242481
$936$	0	0
$937$	−24.1176	−0.787888	−0.393944	−	0.919135i	$-0.628890\pi$
$937$	−24.1176	−0.787888	−0.393944	+	0.919135i	$0.628890\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.19656	−0.104205	−0.0521024	−	0.998642i	$-0.516592\pi$
$941$	−3.19656	−0.104205	−0.0521024	+	0.998642i	$0.516592\pi$
$942$	0	0
$943$	−0.850040	−0.0276811
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.63504	−0.118123	−0.0590614	−	0.998254i	$-0.518811\pi$
$947$	−3.63504	−0.118123	−0.0590614	+	0.998254i	$0.518811\pi$
$948$	0	0
$949$	−91.8345	−2.98107
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.0821	1.55753	0.778766	−	0.627315i	$-0.215846\pi$
$953$	48.0821	1.55753	0.778766	+	0.627315i	$0.215846\pi$
$954$	0	0
$955$	6.36748	0.206047
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.7736	0.380189
$960$	0	0
$961$	−22.1281	−0.713809
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19.5970	−0.630850
$966$	0	0
$967$	−23.4637	−0.754540	−0.377270	−	0.926103i	$-0.623137\pi$
$967$	−23.4637	−0.754540	−0.377270	+	0.926103i	$0.623137\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.4868	−0.561177	−0.280589	−	0.959828i	$-0.590530\pi$
$971$	−17.4868	−0.561177	−0.280589	+	0.959828i	$0.590530\pi$
$972$	0	0
$973$	−1.17092	−0.0375381
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.2316	−0.647266	−0.323633	−	0.946183i	$-0.604904\pi$
$977$	−20.2316	−0.647266	−0.323633	+	0.946183i	$0.604904\pi$
$978$	0	0
$979$	−24.2865	−0.776199
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.1249	−0.897046	−0.448523	−	0.893771i	$-0.648050\pi$
$983$	−28.1249	−0.897046	−0.448523	+	0.893771i	$0.648050\pi$
$984$	0	0
$985$	−10.7862	−0.343678
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.26504	−0.103822
$990$	0	0
$991$	−33.9718	−1.07915	−0.539576	−	0.841937i	$-0.681415\pi$
$991$	−33.9718	−1.07915	−0.539576	+	0.841937i	$0.681415\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.80344	0.0888751
$996$	0	0
$997$	−23.8223	−0.754461	−0.377231	−	0.926119i	$-0.623124\pi$
$997$	−23.8223	−0.754461	−0.377231	+	0.926119i	$0.623124\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.bm.1.2		3
3.2	odd	2		760.2.a.i.1.1	✓	3
12.11	even	2		1520.2.a.q.1.3		3
15.2	even	4		3800.2.d.n.3649.6		6
15.8	even	4		3800.2.d.n.3649.1		6
15.14	odd	2		3800.2.a.w.1.3		3
24.5	odd	2		6080.2.a.bx.1.3		3
24.11	even	2		6080.2.a.br.1.1		3
60.59	even	2		7600.2.a.bp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.i.1.1	✓	3	3.2	odd	2
1520.2.a.q.1.3		3	12.11	even	2
3800.2.a.w.1.3		3	15.14	odd	2
3800.2.d.n.3649.1		6	15.8	even	4
3800.2.d.n.3649.6		6	15.2	even	4
6080.2.a.br.1.1		3	24.11	even	2
6080.2.a.bx.1.3		3	24.5	odd	2
6840.2.a.bm.1.2		3	1.1	even	1	trivial
7600.2.a.bp.1.1		3	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +1.19656 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +1.19656 q^{7} -4.97858 q^{11} -6.63565 q^{13} -1.48929 q^{17} +1.00000 q^{19} +0.510711 q^{23} +1.00000 q^{25} +7.88240 q^{29} -2.97858 q^{31} +1.19656 q^{35} -7.14637 q^{37} -1.66442 q^{41} -6.39312 q^{43} +9.95715 q^{47} -5.56825 q^{49} +11.4219 q^{53} -4.97858 q^{55} +11.8396 q^{59} +3.66442 q^{61} -6.63565 q^{65} +7.61423 q^{67} +13.8396 q^{73} -5.95715 q^{77} +12.6858 q^{79} +8.68585 q^{83} -1.48929 q^{85} +4.87819 q^{89} -7.93994 q^{91} +1.00000 q^{95} -6.81079 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - q^7	\( 3 q + 3 q^{5} - q^{7} - 11 q^{13} + 3 q^{17} + 3 q^{19} + 9 q^{23} + 3 q^{25} + 7 q^{29} + 6 q^{31} - q^{35} - 20 q^{37} + 22 q^{41} - 10 q^{43} + 12 q^{49} + 7 q^{53} - 11 q^{59} - 16 q^{61} - 11 q^{65} - q^{67} - 5 q^{73} + 12 q^{77} + 26 q^{79} + 14 q^{83} + 3 q^{85} + 6 q^{89} + 29 q^{91} + 3 q^{95} + 8 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - q^7 - 11 * q^13 + 3 * q^17 + 3 * q^19 + 9 * q^23 + 3 * q^25 + 7 * q^29 + 6 * q^31 - q^35 - 20 * q^37 + 22 * q^41 - 10 * q^43 + 12 * q^49 + 7 * q^53 - 11 * q^59 - 16 * q^61 - 11 * q^65 - q^67 - 5 * q^73 + 12 * q^77 + 26 * q^79 + 14 * q^83 + 3 * q^85 + 6 * q^89 + 29 * q^91 + 3 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more