LMFDB - Embedded newform 6840.2.a.bm.1.1

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			1.1
Root			$-1.81361$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} -4.91638 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -4.91638 q^{7} -0.578337 q^{11} -6.39194 q^{13} +0.710831 q^{17} +1.00000 q^{19} +2.71083 q^{23} +1.00000 q^{25} -6.54359 q^{29} +1.42166 q^{31} -4.91638 q^{35} -9.10278 q^{37} +11.0489 q^{41} +5.83276 q^{43} +1.15667 q^{47} +17.1708 q^{49} -13.2736 q^{53} -0.578337 q^{55} -11.3869 q^{59} -9.04888 q^{61} -6.39194 q^{65} +2.97028 q^{67} -9.38692 q^{73} +2.84333 q^{77} +4.37279 q^{79} +0.372787 q^{83} +0.710831 q^{85} +16.6167 q^{89} +31.4252 q^{91} +1.00000 q^{95} +3.94610 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−4.91638	−1.85822	−0.929109	−	0.369807i	$-0.879424\pi$
$7$	−4.91638	−1.85822	−0.929109	+	0.369807i	$0.879424\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.578337	−0.174375	−0.0871876	−	0.996192i	$-0.527788\pi$
$11$	−0.578337	−0.174375	−0.0871876	+	0.996192i	$0.527788\pi$
$12$	0	0
$13$	−6.39194	−1.77281	−0.886403	−	0.462914i	$-0.846804\pi$
$13$	−6.39194	−1.77281	−0.886403	+	0.462914i	$0.846804\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.710831	0.172402	0.0862010	−	0.996278i	$-0.472527\pi$
$17$	0.710831	0.172402	0.0862010	+	0.996278i	$0.472527\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.71083	0.565247	0.282624	−	0.959231i	$-0.408795\pi$
$23$	2.71083	0.565247	0.282624	+	0.959231i	$0.408795\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.54359	−1.21512	−0.607558	−	0.794276i	$-0.707851\pi$
$29$	−6.54359	−1.21512	−0.607558	+	0.794276i	$0.707851\pi$
$30$	0	0
$31$	1.42166	0.255338	0.127669	−	0.991817i	$-0.459250\pi$
$31$	1.42166	0.255338	0.127669	+	0.991817i	$0.459250\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.91638	−0.831020
$36$	0	0
$37$	−9.10278	−1.49649	−0.748243	−	0.663424i	$-0.769103\pi$
$37$	−9.10278	−1.49649	−0.748243	+	0.663424i	$0.769103\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.0489	1.72554	0.862772	−	0.505593i	$-0.168726\pi$
$41$	11.0489	1.72554	0.862772	+	0.505593i	$0.168726\pi$
$42$	0	0
$43$	5.83276	0.889488	0.444744	−	0.895658i	$-0.353295\pi$
$43$	5.83276	0.889488	0.444744	+	0.895658i	$0.353295\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.15667	0.168718	0.0843591	−	0.996435i	$-0.473116\pi$
$47$	1.15667	0.168718	0.0843591	+	0.996435i	$0.473116\pi$
$48$	0	0
$49$	17.1708	2.45297
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13.2736	−1.82327	−0.911633	−	0.411004i	$-0.865178\pi$
$53$	−13.2736	−1.82327	−0.911633	+	0.411004i	$0.865178\pi$
$54$	0	0
$55$	−0.578337	−0.0779830
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.3869	−1.48245	−0.741225	−	0.671256i	$-0.765755\pi$
$59$	−11.3869	−1.48245	−0.741225	+	0.671256i	$0.765755\pi$
$60$	0	0
$61$	−9.04888	−1.15859	−0.579295	−	0.815118i	$-0.696672\pi$
$61$	−9.04888	−1.15859	−0.579295	+	0.815118i	$0.696672\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.39194	−0.792823
$66$	0	0
$67$	2.97028	0.362878	0.181439	−	0.983402i	$-0.441925\pi$
$67$	2.97028	0.362878	0.181439	+	0.983402i	$0.441925\pi$
$68$	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
$73$	−9.38692	−1.09866	−0.549328	−	0.835607i	$-0.685116\pi$
$73$	−9.38692	−1.09866	−0.549328	+	0.835607i	$0.685116\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.84333	0.324027
$78$	0	0
$79$	4.37279	0.491977	0.245988	−	0.969273i	$-0.420887\pi$
$79$	4.37279	0.491977	0.245988	+	0.969273i	$0.420887\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.372787	0.0409187	0.0204593	−	0.999791i	$-0.493487\pi$
$83$	0.372787	0.0409187	0.0204593	+	0.999791i	$0.493487\pi$
$84$	0	0
$85$	0.710831	0.0771005
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.6167	1.76136	0.880681	−	0.473710i	$-0.157086\pi$
$89$	16.6167	1.76136	0.880681	+	0.473710i	$0.157086\pi$
$90$	0	0
$91$	31.4252	3.29426
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	3.94610	0.400666	0.200333	−	0.979728i	$-0.435798\pi$
$97$	3.94610	0.400666	0.200333	+	0.979728i	$0.435798\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.20555	0.617475	0.308738	−	0.951147i	$-0.400093\pi$
$101$	6.20555	0.617475	0.308738	+	0.951147i	$0.400093\pi$
$102$	0	0
$103$	8.15165	0.803206	0.401603	−	0.915814i	$-0.368453\pi$
$103$	8.15165	0.803206	0.401603	+	0.915814i	$0.368453\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.0680	1.26333	0.631667	−	0.775240i	$-0.282371\pi$
$107$	13.0680	1.26333	0.631667	+	0.775240i	$0.282371\pi$
$108$	0	0
$109$	−11.5925	−1.11036	−0.555179	−	0.831731i	$-0.687350\pi$
$109$	−11.5925	−1.11036	−0.555179	+	0.831731i	$0.687350\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.9355	1.40502	0.702509	−	0.711675i	$-0.252063\pi$
$113$	14.9355	1.40502	0.702509	+	0.711675i	$0.252063\pi$
$114$	0	0
$115$	2.71083	0.252786
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.49472	−0.320360
$120$	0	0
$121$	−10.6655	−0.969593
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.8867	1.05477	0.527385	−	0.849626i	$-0.323172\pi$
$127$	11.8867	1.05477	0.527385	+	0.849626i	$0.323172\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.15667	0.800022	0.400011	−	0.916510i	$-0.369006\pi$
$131$	9.15667	0.800022	0.400011	+	0.916510i	$0.369006\pi$
$132$	0	0
$133$	−4.91638	−0.426304
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.3869	−1.14372	−0.571861	−	0.820351i	$-0.693778\pi$
$137$	−13.3869	−1.14372	−0.571861	+	0.820351i	$0.693778\pi$
$138$	0	0
$139$	3.42166	0.290222	0.145111	−	0.989415i	$-0.453646\pi$
$139$	3.42166	0.290222	0.145111	+	0.989415i	$0.453646\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.69670	0.309133
$144$	0	0
$145$	−6.54359	−0.543416
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.36222	0.275444	0.137722	−	0.990471i	$-0.456022\pi$
$149$	3.36222	0.275444	0.137722	+	0.990471i	$0.456022\pi$
$150$	0	0
$151$	21.1950	1.72482	0.862412	−	0.506207i	$-0.168953\pi$
$151$	21.1950	1.72482	0.862412	+	0.506207i	$0.168953\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.42166	0.114191
$156$	0	0
$157$	−11.7944	−0.941300	−0.470650	−	0.882320i	$-0.655980\pi$
$157$	−11.7944	−0.941300	−0.470650	+	0.882320i	$0.655980\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13.3275	−1.05035
$162$	0	0
$163$	−14.2439	−1.11567	−0.557833	−	0.829953i	$-0.688367\pi$
$163$	−14.2439	−1.11567	−0.557833	+	0.829953i	$0.688367\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.47556	0.578476	0.289238	−	0.957257i	$-0.406598\pi$
$167$	7.47556	0.578476	0.289238	+	0.957257i	$0.406598\pi$
$168$	0	0
$169$	27.8569	2.14284
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.05390	−0.308212	−0.154106	−	0.988054i	$-0.549250\pi$
$173$	−4.05390	−0.308212	−0.154106	+	0.988054i	$0.549250\pi$
$174$	0	0
$175$	−4.91638	−0.371644
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.95112	−0.220577	−0.110289	−	0.993900i	$-0.535178\pi$
$179$	−2.95112	−0.220577	−0.110289	+	0.993900i	$0.535178\pi$
$180$	0	0
$181$	9.66553	0.718433	0.359216	−	0.933254i	$-0.383044\pi$
$181$	9.66553	0.718433	0.359216	+	0.933254i	$0.383044\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.10278	−0.669249
$186$	0	0
$187$	−0.411100	−0.0300626
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.9058	1.15090	0.575452	−	0.817835i	$-0.304826\pi$
$191$	15.9058	1.15090	0.575452	+	0.817835i	$0.304826\pi$
$192$	0	0
$193$	15.6116	1.12375	0.561875	−	0.827222i	$-0.310080\pi$
$193$	15.6116	1.12375	0.561875	+	0.827222i	$0.310080\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.6655	0.973628	0.486814	−	0.873506i	$-0.338159\pi$
$197$	13.6655	0.973628	0.486814	+	0.873506i	$0.338159\pi$
$198$	0	0
$199$	8.91638	0.632066	0.316033	−	0.948748i	$-0.397649\pi$
$199$	8.91638	0.632066	0.316033	+	0.948748i	$0.397649\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	32.1708	2.25795
$204$	0	0
$205$	11.0489	0.771687
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.578337	−0.0400044
$210$	0	0
$211$	19.7980	1.36295	0.681476	−	0.731840i	$-0.261338\pi$
$211$	19.7980	1.36295	0.681476	+	0.731840i	$0.261338\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.83276	0.397791
$216$	0	0
$217$	−6.98944	−0.474474
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.54359	−0.305635
$222$	0	0
$223$	−1.57331	−0.105357	−0.0526784	−	0.998612i	$-0.516776\pi$
$223$	−1.57331	−0.105357	−0.0526784	+	0.998612i	$0.516776\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.0575	1.86224	0.931120	−	0.364713i	$-0.118833\pi$
$227$	28.0575	1.86224	0.931120	+	0.364713i	$0.118833\pi$
$228$	0	0
$229$	2.47054	0.163258	0.0816289	−	0.996663i	$-0.473988\pi$
$229$	2.47054	0.163258	0.0816289	+	0.996663i	$0.473988\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.15667	−0.206801	−0.103400	−	0.994640i	$-0.532972\pi$
$233$	−3.15667	−0.206801	−0.103400	+	0.994640i	$0.532972\pi$
$234$	0	0
$235$	1.15667	0.0754531
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.74914	0.177827	0.0889137	−	0.996039i	$-0.471660\pi$
$239$	2.74914	0.177827	0.0889137	+	0.996039i	$0.471660\pi$
$240$	0	0
$241$	8.88164	0.572117	0.286058	−	0.958212i	$-0.407655\pi$
$241$	8.88164	0.572117	0.286058	+	0.958212i	$0.407655\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	17.1708	1.09700
$246$	0	0
$247$	−6.39194	−0.406710
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.31335	−0.398495	−0.199248	−	0.979949i	$-0.563850\pi$
$251$	−6.31335	−0.398495	−0.199248	+	0.979949i	$0.563850\pi$
$252$	0	0
$253$	−1.56777	−0.0985651
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.83830	−0.488940	−0.244470	−	0.969657i	$-0.578614\pi$
$257$	−7.83830	−0.488940	−0.244470	+	0.969657i	$0.578614\pi$
$258$	0	0
$259$	44.7527	2.78080
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.8711	−0.855327	−0.427664	−	0.903938i	$-0.640663\pi$
$263$	−13.8711	−0.855327	−0.427664	+	0.903938i	$0.640663\pi$
$264$	0	0
$265$	−13.2736	−0.815390
$266$	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
$271$	−4.07306	−0.247421	−0.123710	−	0.992318i	$-0.539479\pi$
$271$	−4.07306	−0.247421	−0.123710	+	0.992318i	$0.539479\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.578337	−0.0348750
$276$	0	0
$277$	16.1361	0.969522	0.484761	−	0.874647i	$-0.338907\pi$
$277$	16.1361	0.969522	0.484761	+	0.874647i	$0.338907\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.4600	0.683645	0.341822	−	0.939765i	$-0.388956\pi$
$281$	11.4600	0.683645	0.341822	+	0.939765i	$0.388956\pi$
$282$	0	0
$283$	−21.7250	−1.29142	−0.645708	−	0.763585i	$-0.723437\pi$
$283$	−21.7250	−1.29142	−0.645708	+	0.763585i	$0.723437\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−54.3205	−3.20644
$288$	0	0
$289$	−16.4947	−0.970278
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.91136	0.345345	0.172673	−	0.984979i	$-0.444760\pi$
$293$	5.91136	0.345345	0.172673	+	0.984979i	$0.444760\pi$
$294$	0	0
$295$	−11.3869	−0.662972
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−17.3275	−1.00207
$300$	0	0
$301$	−28.6761	−1.65286
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.04888	−0.518137
$306$	0	0
$307$	−29.7194	−1.69618	−0.848089	−	0.529854i	$-0.822247\pi$
$307$	−29.7194	−1.69618	−0.848089	+	0.529854i	$0.822247\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0.407530	0.0231089	0.0115544	−	0.999933i	$-0.496322\pi$
$311$	0.407530	0.0231089	0.0115544	+	0.999933i	$0.496322\pi$
$312$	0	0
$313$	−0.338044	−0.0191074	−0.00955370	−	0.999954i	$-0.503041\pi$
$313$	−0.338044	−0.0191074	−0.00955370	+	0.999954i	$0.503041\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.75468	0.491712	0.245856	−	0.969306i	$-0.420931\pi$
$317$	8.75468	0.491712	0.245856	+	0.969306i	$0.420931\pi$
$318$	0	0
$319$	3.78440	0.211886
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.710831	0.0395517
$324$	0	0
$325$	−6.39194	−0.354561
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.68665	−0.313515
$330$	0	0
$331$	23.8675	1.31188	0.655938	−	0.754814i	$-0.272273\pi$
$331$	23.8675	1.31188	0.655938	+	0.754814i	$0.272273\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.97028	0.162284
$336$	0	0
$337$	−10.0439	−0.547124	−0.273562	−	0.961854i	$-0.588202\pi$
$337$	−10.0439	−0.547124	−0.273562	+	0.961854i	$0.588202\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.822200	−0.0445246
$342$	0	0
$343$	−50.0036	−2.69994
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.6272	0.838913	0.419456	−	0.907775i	$-0.362221\pi$
$347$	15.6272	0.838913	0.419456	+	0.907775i	$0.362221\pi$
$348$	0	0
$349$	17.2544	0.923608	0.461804	−	0.886982i	$-0.347202\pi$
$349$	17.2544	0.923608	0.461804	+	0.886982i	$0.347202\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−27.2197	−1.44876	−0.724379	−	0.689402i	$-0.757873\pi$
$353$	−27.2197	−1.44876	−0.724379	+	0.689402i	$0.757873\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.18137	0.379018	0.189509	−	0.981879i	$-0.439310\pi$
$359$	7.18137	0.379018	0.189509	+	0.981879i	$0.439310\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.38692	−0.491334
$366$	0	0
$367$	9.15667	0.477975	0.238987	−	0.971023i	$-0.423185\pi$
$367$	9.15667	0.477975	0.238987	+	0.971023i	$0.423185\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	65.2580	3.38803
$372$	0	0
$373$	−8.75468	−0.453300	−0.226650	−	0.973976i	$-0.572777\pi$
$373$	−8.75468	−0.453300	−0.226650	+	0.973976i	$0.572777\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	41.8263	2.15416
$378$	0	0
$379$	−12.9164	−0.663470	−0.331735	−	0.943373i	$-0.607634\pi$
$379$	−12.9164	−0.663470	−0.331735	+	0.943373i	$0.607634\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.64280	0.0839431	0.0419716	−	0.999119i	$-0.486636\pi$
$383$	1.64280	0.0839431	0.0419716	+	0.999119i	$0.486636\pi$
$384$	0	0
$385$	2.84333	0.144909
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11.8328	0.599945	0.299972	−	0.953948i	$-0.403023\pi$
$389$	11.8328	0.599945	0.299972	+	0.953948i	$0.403023\pi$
$390$	0	0
$391$	1.92694	0.0974498
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.37279	0.220019
$396$	0	0
$397$	−24.1361	−1.21135	−0.605677	−	0.795710i	$-0.707098\pi$
$397$	−24.1361	−1.21135	−0.605677	+	0.795710i	$0.707098\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.9789	−0.997697	−0.498849	−	0.866689i	$-0.666244\pi$
$401$	−19.9789	−0.997697	−0.498849	+	0.866689i	$0.666244\pi$
$402$	0	0
$403$	−9.08719	−0.452665
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.26447	0.260950
$408$	0	0
$409$	5.66553	0.280142	0.140071	−	0.990141i	$-0.455267\pi$
$409$	5.66553	0.280142	0.140071	+	0.990141i	$0.455267\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	55.9824	2.75472
$414$	0	0
$415$	0.372787	0.0182994
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−22.6550	−1.10677	−0.553384	−	0.832926i	$-0.686664\pi$
$419$	−22.6550	−1.10677	−0.553384	+	0.832926i	$0.686664\pi$
$420$	0	0
$421$	−25.5330	−1.24440	−0.622202	−	0.782857i	$-0.713762\pi$
$421$	−25.5330	−1.24440	−0.622202	+	0.782857i	$0.713762\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.710831	0.0344804
$426$	0	0
$427$	44.4877	2.15291
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.7944	0.664455	0.332228	−	0.943199i	$-0.392200\pi$
$431$	13.7944	0.664455	0.332228	+	0.943199i	$0.392200\pi$
$432$	0	0
$433$	−29.4444	−1.41501	−0.707504	−	0.706710i	$-0.750179\pi$
$433$	−29.4444	−1.41501	−0.707504	+	0.706710i	$0.750179\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.71083	0.129677
$438$	0	0
$439$	−34.5472	−1.64885	−0.824423	−	0.565974i	$-0.808500\pi$
$439$	−34.5472	−1.64885	−0.824423	+	0.565974i	$0.808500\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.5577	1.11926	0.559631	−	0.828742i	$-0.310943\pi$
$443$	23.5577	1.11926	0.559631	+	0.828742i	$0.310943\pi$
$444$	0	0
$445$	16.6167	0.787705
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.49115	0.353529	0.176765	−	0.984253i	$-0.443437\pi$
$449$	7.49115	0.353529	0.176765	+	0.984253i	$0.443437\pi$
$450$	0	0
$451$	−6.38997	−0.300892
$452$	0	0
$453$	0	0
$454$	0	0
$455$	31.4252	1.47324
$456$	0	0
$457$	38.5819	1.80479	0.902393	−	0.430914i	$-0.141809\pi$
$457$	38.5819	1.80479	0.902393	+	0.430914i	$0.141809\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.9789	1.48940	0.744702	−	0.667397i	$-0.232591\pi$
$461$	31.9789	1.48940	0.744702	+	0.667397i	$0.232591\pi$
$462$	0	0
$463$	30.8122	1.43196	0.715981	−	0.698120i	$-0.245980\pi$
$463$	30.8122	1.43196	0.715981	+	0.698120i	$0.245980\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−41.4600	−1.91854	−0.959269	−	0.282493i	$-0.908839\pi$
$467$	−41.4600	−1.91854	−0.959269	+	0.282493i	$0.908839\pi$
$468$	0	0
$469$	−14.6030	−0.674305
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.37330	−0.155105
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9.93051	0.453737	0.226868	−	0.973925i	$-0.427151\pi$
$479$	9.93051	0.453737	0.226868	+	0.973925i	$0.427151\pi$
$480$	0	0
$481$	58.1844	2.65298
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.94610	0.179183
$486$	0	0
$487$	−19.7789	−0.896266	−0.448133	−	0.893967i	$-0.647911\pi$
$487$	−19.7789	−0.896266	−0.448133	+	0.893967i	$0.647911\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.1461	0.728664	0.364332	−	0.931269i	$-0.381297\pi$
$491$	16.1461	0.728664	0.364332	+	0.931269i	$0.381297\pi$
$492$	0	0
$493$	−4.65139	−0.209488
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−38.8222	−1.73792	−0.868960	−	0.494882i	$-0.835211\pi$
$499$	−38.8222	−1.73792	−0.868960	+	0.494882i	$0.835211\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.5436	−0.559291	−0.279646	−	0.960103i	$-0.590217\pi$
$503$	−12.5436	−0.559291	−0.279646	+	0.960103i	$0.590217\pi$
$504$	0	0
$505$	6.20555	0.276143
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.0278	−1.10934	−0.554668	−	0.832072i	$-0.687155\pi$
$509$	−25.0278	−1.10934	−0.554668	+	0.832072i	$0.687155\pi$
$510$	0	0
$511$	46.1497	2.04154
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.15165	0.359205
$516$	0	0
$517$	−0.668948	−0.0294203
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.6550	0.729667	0.364834	−	0.931073i	$-0.381126\pi$
$521$	16.6550	0.729667	0.364834	+	0.931073i	$0.381126\pi$
$522$	0	0
$523$	24.3608	1.06522	0.532611	−	0.846360i	$-0.321211\pi$
$523$	24.3608	1.06522	0.532611	+	0.846360i	$0.321211\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.01056	0.0440208
$528$	0	0
$529$	−15.6514	−0.680495
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−70.6238	−3.05906
$534$	0	0
$535$	13.0680	0.564980
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9.93051	−0.427738
$540$	0	0
$541$	15.3622	0.660474	0.330237	−	0.943898i	$-0.392871\pi$
$541$	15.3622	0.660474	0.330237	+	0.943898i	$0.392871\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11.5925	−0.496567
$546$	0	0
$547$	−26.5527	−1.13531	−0.567656	−	0.823266i	$-0.692150\pi$
$547$	−26.5527	−1.13531	−0.567656	+	0.823266i	$0.692150\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.54359	−0.278767
$552$	0	0
$553$	−21.4983	−0.914200
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.9583	1.22700	0.613501	−	0.789694i	$-0.289761\pi$
$557$	28.9583	1.22700	0.613501	+	0.789694i	$0.289761\pi$
$558$	0	0
$559$	−37.2827	−1.57689
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.64280	0.0692357	0.0346179	−	0.999401i	$-0.488979\pi$
$563$	1.64280	0.0692357	0.0346179	+	0.999401i	$0.488979\pi$
$564$	0	0
$565$	14.9355	0.628343
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.3416	−0.601232	−0.300616	−	0.953745i	$-0.597192\pi$
$569$	−14.3416	−0.601232	−0.300616	+	0.953745i	$0.597192\pi$
$570$	0	0
$571$	−39.0177	−1.63284	−0.816420	−	0.577458i	$-0.804045\pi$
$571$	−39.0177	−1.63284	−0.816420	+	0.577458i	$0.804045\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.71083	0.113049
$576$	0	0
$577$	25.4947	1.06136	0.530680	−	0.847573i	$-0.321937\pi$
$577$	25.4947	1.06136	0.530680	+	0.847573i	$0.321937\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.83276	−0.0760358
$582$	0	0
$583$	7.67661	0.317933
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.72496	0.401392	0.200696	−	0.979654i	$-0.435680\pi$
$587$	9.72496	0.401392	0.200696	+	0.979654i	$0.435680\pi$
$588$	0	0
$589$	1.42166	0.0585786
$590$	0	0
$591$	0	0
$592$	0	0
$593$	13.0388	0.535441	0.267720	−	0.963497i	$-0.413730\pi$
$593$	13.0388	0.535441	0.267720	+	0.963497i	$0.413730\pi$
$594$	0	0
$595$	−3.49472	−0.143269
$596$	0	0
$597$	0	0
$598$	0	0
$599$	18.6861	0.763495	0.381747	−	0.924267i	$-0.375322\pi$
$599$	18.6861	0.763495	0.381747	+	0.924267i	$0.375322\pi$
$600$	0	0
$601$	12.7355	0.519493	0.259747	−	0.965677i	$-0.416361\pi$
$601$	12.7355	0.519493	0.259747	+	0.965677i	$0.416361\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.6655	−0.433615
$606$	0	0
$607$	23.0645	0.936158	0.468079	−	0.883687i	$-0.344946\pi$
$607$	23.0645	0.936158	0.468079	+	0.883687i	$0.344946\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.39340	−0.299105
$612$	0	0
$613$	25.7038	1.03817	0.519084	−	0.854723i	$-0.326273\pi$
$613$	25.7038	1.03817	0.519084	+	0.854723i	$0.326273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.205550	0.00827514	0.00413757	−	0.999991i	$-0.498683\pi$
$617$	0.205550	0.00827514	0.00413757	+	0.999991i	$0.498683\pi$
$618$	0	0
$619$	−11.0872	−0.445632	−0.222816	−	0.974861i	$-0.571525\pi$
$619$	−11.0872	−0.445632	−0.222816	+	0.974861i	$0.571525\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−81.6938	−3.27299
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.47054	−0.257997
$630$	0	0
$631$	10.4806	0.417226	0.208613	−	0.977998i	$-0.433105\pi$
$631$	10.4806	0.417226	0.208613	+	0.977998i	$0.433105\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.8867	0.471708
$636$	0	0
$637$	−109.755	−4.34864
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−37.3694	−1.47600	−0.738001	−	0.674800i	$-0.764230\pi$
$641$	−37.3694	−1.47600	−0.738001	+	0.674800i	$0.764230\pi$
$642$	0	0
$643$	42.0172	1.65700	0.828498	−	0.559992i	$-0.189196\pi$
$643$	42.0172	1.65700	0.828498	+	0.559992i	$0.189196\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.4635	−0.922447	−0.461224	−	0.887284i	$-0.652589\pi$
$647$	−23.4635	−0.922447	−0.461224	+	0.887284i	$0.652589\pi$
$648$	0	0
$649$	6.58548	0.258503
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−33.6272	−1.31593	−0.657967	−	0.753047i	$-0.728583\pi$
$653$	−33.6272	−1.31593	−0.657967	+	0.753047i	$0.728583\pi$
$654$	0	0
$655$	9.15667	0.357781
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−17.9653	−0.699827	−0.349914	−	0.936782i	$-0.613789\pi$
$659$	−17.9653	−0.699827	−0.349914	+	0.936782i	$0.613789\pi$
$660$	0	0
$661$	15.5542	0.604987	0.302493	−	0.953152i	$-0.402181\pi$
$661$	15.5542	0.604987	0.302493	+	0.953152i	$0.402181\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.91638	−0.190649
$666$	0	0
$667$	−17.7386	−0.686841
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.23330	0.202029
$672$	0	0
$673$	12.2111	0.470703	0.235351	−	0.971910i	$-0.424376\pi$
$673$	12.2111	0.470703	0.235351	+	0.971910i	$0.424376\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.0680	1.65524	0.827619	−	0.561290i	$-0.189695\pi$
$677$	43.0680	1.65524	0.827619	+	0.561290i	$0.189695\pi$
$678$	0	0
$679$	−19.4005	−0.744524
$680$	0	0
$681$	0	0
$682$	0	0
$683$	45.3850	1.73661	0.868303	−	0.496033i	$-0.165211\pi$
$683$	45.3850	1.73661	0.868303	+	0.496033i	$0.165211\pi$
$684$	0	0
$685$	−13.3869	−0.511488
$686$	0	0
$687$	0	0
$688$	0	0
$689$	84.8440	3.23230
$690$	0	0
$691$	−0.508852	−0.0193576	−0.00967882	−	0.999953i	$-0.503081\pi$
$691$	−0.508852	−0.0193576	−0.00967882	+	0.999953i	$0.503081\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.42166	0.129791
$696$	0	0
$697$	7.85389	0.297487
$698$	0	0
$699$	0	0
$700$	0	0
$701$	40.1844	1.51774	0.758872	−	0.651239i	$-0.225751\pi$
$701$	40.1844	1.51774	0.758872	+	0.651239i	$0.225751\pi$
$702$	0	0
$703$	−9.10278	−0.343318
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−30.5089	−1.14740
$708$	0	0
$709$	20.0978	0.754787	0.377393	−	0.926053i	$-0.376820\pi$
$709$	20.0978	0.754787	0.377393	+	0.926053i	$0.376820\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.85389	0.144329
$714$	0	0
$715$	3.69670	0.138249
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−35.5925	−1.32738	−0.663688	−	0.748010i	$-0.731010\pi$
$719$	−35.5925	−1.32738	−0.663688	+	0.748010i	$0.731010\pi$
$720$	0	0
$721$	−40.0766	−1.49253
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.54359	−0.243023
$726$	0	0
$727$	−20.1708	−0.748094	−0.374047	−	0.927410i	$-0.622030\pi$
$727$	−20.1708	−0.748094	−0.374047	+	0.927410i	$0.622030\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.14611	0.153349
$732$	0	0
$733$	30.1461	1.11347	0.556736	−	0.830689i	$-0.312053\pi$
$733$	30.1461	1.11347	0.556736	+	0.830689i	$0.312053\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.71782	−0.0632768
$738$	0	0
$739$	26.0283	0.957465	0.478733	−	0.877961i	$-0.341096\pi$
$739$	26.0283	0.957465	0.478733	+	0.877961i	$0.341096\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−0.221136	−0.00811269	−0.00405635	−	0.999992i	$-0.501291\pi$
$743$	−0.221136	−0.00811269	−0.00405635	+	0.999992i	$0.501291\pi$
$744$	0	0
$745$	3.36222	0.123182
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−64.2474	−2.34755
$750$	0	0
$751$	41.9406	1.53043	0.765216	−	0.643773i	$-0.222632\pi$
$751$	41.9406	1.53043	0.765216	+	0.643773i	$0.222632\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	21.1950	0.771365
$756$	0	0
$757$	−26.7456	−0.972084	−0.486042	−	0.873935i	$-0.661560\pi$
$757$	−26.7456	−0.972084	−0.486042	+	0.873935i	$0.661560\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.6519	0.422381	0.211191	−	0.977445i	$-0.432266\pi$
$761$	11.6519	0.422381	0.211191	+	0.977445i	$0.432266\pi$
$762$	0	0
$763$	56.9930	2.06329
$764$	0	0
$765$	0	0
$766$	0	0
$767$	72.7846	2.62810
$768$	0	0
$769$	−9.28917	−0.334976	−0.167488	−	0.985874i	$-0.553566\pi$
$769$	−9.28917	−0.334976	−0.167488	+	0.985874i	$0.553566\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.3225	0.371273	0.185637	−	0.982618i	$-0.440565\pi$
$773$	10.3225	0.371273	0.185637	+	0.982618i	$0.440565\pi$
$774$	0	0
$775$	1.42166	0.0510676
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.0489	0.395867
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−11.7944	−0.420962
$786$	0	0
$787$	49.8902	1.77839	0.889197	−	0.457524i	$-0.151264\pi$
$787$	49.8902	1.77839	0.889197	+	0.457524i	$0.151264\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−73.4288	−2.61083
$792$	0	0
$793$	57.8399	2.05396
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.7436	−1.01815	−0.509075	−	0.860722i	$-0.670013\pi$
$797$	−28.7436	−1.01815	−0.509075	+	0.860722i	$0.670013\pi$
$798$	0	0
$799$	0.822200	0.0290874
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.42880	0.191578
$804$	0	0
$805$	−13.3275	−0.469732
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.5124	−1.35402	−0.677012	−	0.735972i	$-0.736726\pi$
$809$	−38.5124	−1.35402	−0.677012	+	0.735972i	$0.736726\pi$
$810$	0	0
$811$	−28.1013	−0.986771	−0.493385	−	0.869811i	$-0.664241\pi$
$811$	−28.1013	−0.986771	−0.493385	+	0.869811i	$0.664241\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14.2439	−0.498941
$816$	0	0
$817$	5.83276	0.204063
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.2650	−0.777053	−0.388527	−	0.921437i	$-0.627016\pi$
$821$	−22.2650	−0.777053	−0.388527	+	0.921437i	$0.627016\pi$
$822$	0	0
$823$	39.8363	1.38861	0.694304	−	0.719682i	$-0.255712\pi$
$823$	39.8363	1.38861	0.694304	+	0.719682i	$0.255712\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	48.5874	1.68955	0.844776	−	0.535121i	$-0.179734\pi$
$827$	48.5874	1.68955	0.844776	+	0.535121i	$0.179734\pi$
$828$	0	0
$829$	−45.9058	−1.59437	−0.797187	−	0.603732i	$-0.793680\pi$
$829$	−45.9058	−1.59437	−0.797187	+	0.603732i	$0.793680\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	12.2056	0.422897
$834$	0	0
$835$	7.47556	0.258702
$836$	0	0
$837$	0	0
$838$	0	0
$839$	27.9688	0.965591	0.482796	−	0.875733i	$-0.339621\pi$
$839$	27.9688	0.965591	0.482796	+	0.875733i	$0.339621\pi$
$840$	0	0
$841$	13.8186	0.476504
$842$	0	0
$843$	0	0
$844$	0	0
$845$	27.8569	0.958308
$846$	0	0
$847$	52.4358	1.80172
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24.6761	−0.845885
$852$	0	0
$853$	7.56777	0.259116	0.129558	−	0.991572i	$-0.458644\pi$
$853$	7.56777	0.259116	0.129558	+	0.991572i	$0.458644\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.4756	−0.733591	−0.366796	−	0.930302i	$-0.619545\pi$
$857$	−21.4756	−0.733591	−0.366796	+	0.930302i	$0.619545\pi$
$858$	0	0
$859$	−33.0177	−1.12655	−0.563275	−	0.826270i	$-0.690459\pi$
$859$	−33.0177	−1.12655	−0.563275	+	0.826270i	$0.690459\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.6605	−0.635211	−0.317605	−	0.948223i	$-0.602879\pi$
$863$	−18.6605	−0.635211	−0.317605	+	0.948223i	$0.602879\pi$
$864$	0	0
$865$	−4.05390	−0.137837
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.52894	−0.0857886
$870$	0	0
$871$	−18.9859	−0.643312
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.91638	−0.166204
$876$	0	0
$877$	−57.7925	−1.95151	−0.975757	−	0.218858i	$-0.929767\pi$
$877$	−57.7925	−1.95151	−0.975757	+	0.218858i	$0.929767\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−36.2338	−1.22075	−0.610374	−	0.792113i	$-0.708981\pi$
$881$	−36.2338	−1.22075	−0.610374	+	0.792113i	$0.708981\pi$
$882$	0	0
$883$	−2.98944	−0.100603	−0.0503013	−	0.998734i	$-0.516018\pi$
$883$	−2.98944	−0.100603	−0.0503013	+	0.998734i	$0.516018\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.2494	1.41860	0.709298	−	0.704909i	$-0.249012\pi$
$887$	42.2494	1.41860	0.709298	+	0.704909i	$0.249012\pi$
$888$	0	0
$889$	−58.4394	−1.95999
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.15667	0.0387066
$894$	0	0
$895$	−2.95112	−0.0986452
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9.30279	−0.310265
$900$	0	0
$901$	−9.43528	−0.314335
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.66553	0.321293
$906$	0	0
$907$	1.97080	0.0654392	0.0327196	−	0.999465i	$-0.489583\pi$
$907$	1.97080	0.0654392	0.0327196	+	0.999465i	$0.489583\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.98995	−0.330982	−0.165491	−	0.986211i	$-0.552921\pi$
$911$	−9.98995	−0.330982	−0.165491	+	0.986211i	$0.552921\pi$
$912$	0	0
$913$	−0.215597	−0.00713520
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−45.0177	−1.48662
$918$	0	0
$919$	4.24029	0.139874	0.0699372	−	0.997551i	$-0.477720\pi$
$919$	4.24029	0.139874	0.0699372	+	0.997551i	$0.477720\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−9.10278	−0.299297
$926$	0	0
$927$	0	0
$928$	0	0
$929$	13.1531	0.431539	0.215770	−	0.976444i	$-0.430774\pi$
$929$	13.1531	0.431539	0.215770	+	0.976444i	$0.430774\pi$
$930$	0	0
$931$	17.1708	0.562750
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.411100	−0.0134444
$936$	0	0
$937$	−38.5436	−1.25916	−0.629582	−	0.776934i	$-0.716774\pi$
$937$	−38.5436	−1.25916	−0.629582	+	0.776934i	$0.716774\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.91638	0.0950713	0.0475357	−	0.998870i	$-0.484863\pi$
$941$	2.91638	0.0950713	0.0475357	+	0.998870i	$0.484863\pi$
$942$	0	0
$943$	29.9516	0.975360
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−58.4011	−1.89778	−0.948890	−	0.315608i	$-0.897792\pi$
$947$	−58.4011	−1.89778	−0.948890	+	0.315608i	$0.897792\pi$
$948$	0	0
$949$	60.0007	1.94770
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.8378	1.19329	0.596646	−	0.802504i	$-0.296499\pi$
$953$	36.8378	1.19329	0.596646	+	0.802504i	$0.296499\pi$
$954$	0	0
$955$	15.9058	0.514700
$956$	0	0
$957$	0	0
$958$	0	0
$959$	65.8152	2.12528
$960$	0	0
$961$	−28.9789	−0.934802
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15.6116	0.502556
$966$	0	0
$967$	−43.0278	−1.38368	−0.691840	−	0.722051i	$-0.743199\pi$
$967$	−43.0278	−1.38368	−0.691840	+	0.722051i	$0.743199\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	49.0177	1.57305	0.786526	−	0.617557i	$-0.211877\pi$
$971$	49.0177	1.57305	0.786526	+	0.617557i	$0.211877\pi$
$972$	0	0
$973$	−16.8222	−0.539295
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.2383	−0.647481	−0.323741	−	0.946146i	$-0.604941\pi$
$977$	−20.2383	−0.647481	−0.323741	+	0.946146i	$0.604941\pi$
$978$	0	0
$979$	−9.61003	−0.307138
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.6811	−0.819100	−0.409550	−	0.912288i	$-0.634314\pi$
$983$	−25.6811	−0.819100	−0.409550	+	0.912288i	$0.634314\pi$
$984$	0	0
$985$	13.6655	0.435420
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15.8116	0.502781
$990$	0	0
$991$	8.56829	0.272181	0.136090	−	0.990696i	$-0.456546\pi$
$991$	8.56829	0.272181	0.136090	+	0.990696i	$0.456546\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.91638	0.282668
$996$	0	0
$997$	29.9688	0.949122	0.474561	−	0.880223i	$-0.342607\pi$
$997$	29.9688	0.949122	0.474561	+	0.880223i	$0.342607\pi$
$998$	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.1.1		3
3.2	odd	2		760.2.a.i.1.3	✓	3
12.11	even	2		1520.2.a.q.1.1		3
15.2	even	4		3800.2.d.n.3649.2		6
15.8	even	4		3800.2.d.n.3649.5		6
15.14	odd	2		3800.2.a.w.1.1		3
24.5	odd	2		6080.2.a.bx.1.1		3
24.11	even	2		6080.2.a.br.1.3		3
60.59	even	2		7600.2.a.bp.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.i.1.3	✓	3	3.2	odd	2
1520.2.a.q.1.1		3	12.11	even	2
3800.2.a.w.1.1		3	15.14	odd	2
3800.2.d.n.3649.2		6	15.2	even	4
3800.2.d.n.3649.5		6	15.8	even	4
6080.2.a.br.1.3		3	24.11	even	2
6080.2.a.bx.1.1		3	24.5	odd	2
6840.2.a.bm.1.1		3	1.1	even	1	trivial
7600.2.a.bp.1.3		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -4.91638 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -4.91638 q^{7} -0.578337 q^{11} -6.39194 q^{13} +0.710831 q^{17} +1.00000 q^{19} +2.71083 q^{23} +1.00000 q^{25} -6.54359 q^{29} +1.42166 q^{31} -4.91638 q^{35} -9.10278 q^{37} +11.0489 q^{41} +5.83276 q^{43} +1.15667 q^{47} +17.1708 q^{49} -13.2736 q^{53} -0.578337 q^{55} -11.3869 q^{59} -9.04888 q^{61} -6.39194 q^{65} +2.97028 q^{67} -9.38692 q^{73} +2.84333 q^{77} +4.37279 q^{79} +0.372787 q^{83} +0.710831 q^{85} +16.6167 q^{89} +31.4252 q^{91} +1.00000 q^{95} +3.94610 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - q^{7}+O(q^{10}) \) 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

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