LMFDB - Embedded newform 6840.2.a.x.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} -0.585786 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -0.585786 q^{7} -1.41421 q^{11} -4.24264 q^{13} +2.82843 q^{17} -1.00000 q^{19} -0.828427 q^{23} +1.00000 q^{25} +6.24264 q^{29} -3.17157 q^{31} -0.585786 q^{35} +7.07107 q^{37} +6.24264 q^{41} -10.2426 q^{43} +0.828427 q^{47} -6.65685 q^{49} -4.48528 q^{53} -1.41421 q^{55} -2.82843 q^{59} -8.00000 q^{61} -4.24264 q^{65} +11.3137 q^{67} +10.8284 q^{71} +7.65685 q^{73} +0.828427 q^{77} -13.6569 q^{79} +6.00000 q^{83} +2.82843 q^{85} -17.5563 q^{89} +2.48528 q^{91} -1.00000 q^{95} -9.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 4 * q^7	$ 2 q + 2 q^{5} - 4 q^{7} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 12 q^{31} - 4 q^{35} + 4 q^{41} - 12 q^{43} - 4 q^{47} - 2 q^{49} + 8 q^{53} - 16 q^{61} + 16 q^{71} + 4 q^{73} - 4 q^{77} - 16 q^{79} + 12 q^{83} - 4 q^{89} - 12 q^{91} - 2 q^{95}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^7 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 4 * q^29 - 12 * q^31 - 4 * q^35 + 4 * q^41 - 12 * q^43 - 4 * q^47 - 2 * q^49 + 8 * q^53 - 16 * q^61 + 16 * q^71 + 4 * q^73 - 4 * q^77 - 16 * q^79 + 12 * q^83 - 4 * q^89 - 12 * q^91 - 2 * q^95

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
$7$	−0.585786	−0.221406	−0.110703	−	0.993854i	$-0.535310\pi$
$7$	−0.585786	−0.221406	−0.110703	+	0.993854i	$0.535310\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.828427	−0.172739	−0.0863695	−	0.996263i	$-0.527527\pi$
$23$	−0.828427	−0.172739	−0.0863695	+	0.996263i	$0.527527\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.24264	1.15923	0.579615	−	0.814891i	$-0.303203\pi$
$29$	6.24264	1.15923	0.579615	+	0.814891i	$0.303203\pi$
$30$	0	0
$31$	−3.17157	−0.569631	−0.284816	−	0.958582i	$-0.591932\pi$
$31$	−3.17157	−0.569631	−0.284816	+	0.958582i	$0.591932\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.585786	−0.0990160
$36$	0	0
$37$	7.07107	1.16248	0.581238	−	0.813733i	$-0.302568\pi$
$37$	7.07107	1.16248	0.581238	+	0.813733i	$0.302568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.24264	0.974937	0.487468	−	0.873141i	$-0.337920\pi$
$41$	6.24264	0.974937	0.487468	+	0.873141i	$0.337920\pi$
$42$	0	0
$43$	−10.2426	−1.56199	−0.780994	−	0.624538i	$-0.785287\pi$
$43$	−10.2426	−1.56199	−0.780994	+	0.624538i	$0.785287\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.828427	0.120839	0.0604193	−	0.998173i	$-0.480756\pi$
$47$	0.828427	0.120839	0.0604193	+	0.998173i	$0.480756\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.48528	−0.616101	−0.308050	−	0.951370i	$-0.599677\pi$
$53$	−4.48528	−0.616101	−0.308050	+	0.951370i	$0.599677\pi$
$54$	0	0
$55$	−1.41421	−0.190693
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.82843	−0.368230	−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843	−0.368230	−0.184115	+	0.982905i	$0.558942\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.24264	−0.526235
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.8284	1.28510	0.642549	−	0.766245i	$-0.277877\pi$
$71$	10.8284	1.28510	0.642549	+	0.766245i	$0.277877\pi$
$72$	0	0
$73$	7.65685	0.896167	0.448084	−	0.893992i	$-0.352107\pi$
$73$	7.65685	0.896167	0.448084	+	0.893992i	$0.352107\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.828427	0.0944080
$78$	0	0
$79$	−13.6569	−1.53652	−0.768258	−	0.640140i	$-0.778876\pi$
$79$	−13.6569	−1.53652	−0.768258	+	0.640140i	$0.778876\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	2.82843	0.306786
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17.5563	−1.86097	−0.930485	−	0.366331i	$-0.880614\pi$
$89$	−17.5563	−1.86097	−0.930485	+	0.366331i	$0.880614\pi$
$90$	0	0
$91$	2.48528	0.260528
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−9.89949	−1.00514	−0.502571	−	0.864536i	$-0.667612\pi$
$97$	−9.89949	−1.00514	−0.502571	+	0.864536i	$0.667612\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.17157	0.713598	0.356799	−	0.934181i	$-0.383868\pi$
$101$	7.17157	0.713598	0.356799	+	0.934181i	$0.383868\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.34315	−0.226520	−0.113260	−	0.993565i	$-0.536129\pi$
$107$	−2.34315	−0.226520	−0.113260	+	0.993565i	$0.536129\pi$
$108$	0	0
$109$	−16.1421	−1.54614	−0.773068	−	0.634323i	$-0.781279\pi$
$109$	−16.1421	−1.54614	−0.773068	+	0.634323i	$0.781279\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.65685	0.155864	0.0779319	−	0.996959i	$-0.475168\pi$
$113$	1.65685	0.155864	0.0779319	+	0.996959i	$0.475168\pi$
$114$	0	0
$115$	−0.828427	−0.0772512
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.65685	−0.151884
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.9706	−1.50589	−0.752947	−	0.658081i	$-0.771368\pi$
$127$	−16.9706	−1.50589	−0.752947	+	0.658081i	$0.771368\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.0711	−1.66625	−0.833123	−	0.553087i	$-0.813450\pi$
$131$	−19.0711	−1.66625	−0.833123	+	0.553087i	$0.813450\pi$
$132$	0	0
$133$	0.585786	0.0507941
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.9706	−1.62076	−0.810382	−	0.585901i	$-0.800741\pi$
$137$	−18.9706	−1.62076	−0.810382	+	0.585901i	$0.800741\pi$
$138$	0	0
$139$	4.48528	0.380437	0.190218	−	0.981742i	$-0.439080\pi$
$139$	4.48528	0.380437	0.190218	+	0.981742i	$0.439080\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	6.24264	0.518423
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.343146	−0.0281116	−0.0140558	−	0.999901i	$-0.504474\pi$
$149$	−0.343146	−0.0281116	−0.0140558	+	0.999901i	$0.504474\pi$
$150$	0	0
$151$	−16.8284	−1.36948	−0.684739	−	0.728788i	$-0.740084\pi$
$151$	−16.8284	−1.36948	−0.684739	+	0.728788i	$0.740084\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.17157	−0.254747
$156$	0	0
$157$	20.1421	1.60752	0.803759	−	0.594955i	$-0.202830\pi$
$157$	20.1421	1.60752	0.803759	+	0.594955i	$0.202830\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.485281	0.0382455
$162$	0	0
$163$	−19.8995	−1.55865	−0.779324	−	0.626621i	$-0.784438\pi$
$163$	−19.8995	−1.55865	−0.779324	+	0.626621i	$0.784438\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.34315	−0.336083	−0.168041	−	0.985780i	$-0.553744\pi$
$167$	−4.34315	−0.336083	−0.168041	+	0.985780i	$0.553744\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.34315	−0.178146	−0.0890730	−	0.996025i	$-0.528390\pi$
$173$	−2.34315	−0.178146	−0.0890730	+	0.996025i	$0.528390\pi$
$174$	0	0
$175$	−0.585786	−0.0442813
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.7990	1.18087	0.590436	−	0.807084i	$-0.298956\pi$
$179$	15.7990	1.18087	0.590436	+	0.807084i	$0.298956\pi$
$180$	0	0
$181$	−0.828427	−0.0615765	−0.0307883	−	0.999526i	$-0.509802\pi$
$181$	−0.828427	−0.0615765	−0.0307883	+	0.999526i	$0.509802\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.07107	0.519875
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.5858	−0.765961	−0.382980	−	0.923757i	$-0.625102\pi$
$191$	−10.5858	−0.765961	−0.382980	+	0.923757i	$0.625102\pi$
$192$	0	0
$193$	8.92893	0.642719	0.321359	−	0.946957i	$-0.395860\pi$
$193$	8.92893	0.642719	0.321359	+	0.946957i	$0.395860\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.48528	0.604551	0.302276	−	0.953221i	$-0.402254\pi$
$197$	8.48528	0.604551	0.302276	+	0.953221i	$0.402254\pi$
$198$	0	0
$199$	14.1421	1.00251	0.501255	−	0.865300i	$-0.332872\pi$
$199$	14.1421	1.00251	0.501255	+	0.865300i	$0.332872\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.65685	−0.256661
$204$	0	0
$205$	6.24264	0.436005
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.41421	0.0978232
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.2426	−0.698542
$216$	0	0
$217$	1.85786	0.126120
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−4.68629	−0.313817	−0.156909	−	0.987613i	$-0.550153\pi$
$223$	−4.68629	−0.313817	−0.156909	+	0.987613i	$0.550153\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.3137	−1.14915	−0.574576	−	0.818452i	$-0.694833\pi$
$227$	−17.3137	−1.14915	−0.574576	+	0.818452i	$0.694833\pi$
$228$	0	0
$229$	26.6274	1.75959	0.879795	−	0.475354i	$-0.157680\pi$
$229$	26.6274	1.75959	0.879795	+	0.475354i	$0.157680\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.65685	0.239568	0.119784	−	0.992800i	$-0.461780\pi$
$233$	3.65685	0.239568	0.119784	+	0.992800i	$0.461780\pi$
$234$	0	0
$235$	0.828427	0.0540406
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.75736	0.243043	0.121522	−	0.992589i	$-0.461223\pi$
$239$	3.75736	0.243043	0.121522	+	0.992589i	$0.461223\pi$
$240$	0	0
$241$	18.9706	1.22200	0.611001	−	0.791630i	$-0.290767\pi$
$241$	18.9706	1.22200	0.611001	+	0.791630i	$0.290767\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.65685	−0.425291
$246$	0	0
$247$	4.24264	0.269953
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.727922	0.0459460	0.0229730	−	0.999736i	$-0.492687\pi$
$251$	0.727922	0.0459460	0.0229730	+	0.999736i	$0.492687\pi$
$252$	0	0
$253$	1.17157	0.0736562
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.34315	−0.395675	−0.197837	−	0.980235i	$-0.563392\pi$
$257$	−6.34315	−0.395675	−0.197837	+	0.980235i	$0.563392\pi$
$258$	0	0
$259$	−4.14214	−0.257380
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.34315	0.514460	0.257230	−	0.966350i	$-0.417190\pi$
$263$	8.34315	0.514460	0.257230	+	0.966350i	$0.417190\pi$
$264$	0	0
$265$	−4.48528	−0.275529
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−23.8995	−1.45718	−0.728589	−	0.684951i	$-0.759824\pi$
$269$	−23.8995	−1.45718	−0.728589	+	0.684951i	$0.759824\pi$
$270$	0	0
$271$	−26.1421	−1.58802	−0.794011	−	0.607904i	$-0.792011\pi$
$271$	−26.1421	−1.58802	−0.794011	+	0.607904i	$0.792011\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.41421	−0.0852803
$276$	0	0
$277$	17.3137	1.04028	0.520140	−	0.854081i	$-0.325880\pi$
$277$	17.3137	1.04028	0.520140	+	0.854081i	$0.325880\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.75736	−0.343455	−0.171728	−	0.985144i	$-0.554935\pi$
$281$	−5.75736	−0.343455	−0.171728	+	0.985144i	$0.554935\pi$
$282$	0	0
$283$	−23.8995	−1.42068	−0.710339	−	0.703860i	$-0.751458\pi$
$283$	−23.8995	−1.42068	−0.710339	+	0.703860i	$0.751458\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.65685	−0.215857
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.00000	−0.233682	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000	−0.233682	−0.116841	+	0.993151i	$0.537277\pi$
$294$	0	0
$295$	−2.82843	−0.164677
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.51472	0.203261
$300$	0	0
$301$	6.00000	0.345834
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	6.82843	0.389719	0.194859	−	0.980831i	$-0.437575\pi$
$307$	6.82843	0.389719	0.194859	+	0.980831i	$0.437575\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.58579	0.373446	0.186723	−	0.982413i	$-0.440213\pi$
$311$	6.58579	0.373446	0.186723	+	0.982413i	$0.440213\pi$
$312$	0	0
$313$	−9.79899	−0.553872	−0.276936	−	0.960888i	$-0.589319\pi$
$313$	−9.79899	−0.553872	−0.276936	+	0.960888i	$0.589319\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.48528	0.476581	0.238290	−	0.971194i	$-0.423413\pi$
$317$	8.48528	0.476581	0.238290	+	0.971194i	$0.423413\pi$
$318$	0	0
$319$	−8.82843	−0.494297
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.82843	−0.157378
$324$	0	0
$325$	−4.24264	−0.235339
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.485281	−0.0267544
$330$	0	0
$331$	−7.17157	−0.394185	−0.197093	−	0.980385i	$-0.563150\pi$
$331$	−7.17157	−0.394185	−0.197093	+	0.980385i	$0.563150\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.3137	0.618134
$336$	0	0
$337$	17.8995	0.975048	0.487524	−	0.873110i	$-0.337900\pi$
$337$	17.8995	0.975048	0.487524	+	0.873110i	$0.337900\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.48528	0.242892
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.6569	−0.625773	−0.312886	−	0.949791i	$-0.601296\pi$
$347$	−11.6569	−0.625773	−0.312886	+	0.949791i	$0.601296\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	10.8284	0.574713
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.0416	1.47998	0.739990	−	0.672618i	$-0.234830\pi$
$359$	28.0416	1.47998	0.739990	+	0.672618i	$0.234830\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.65685	0.400778
$366$	0	0
$367$	9.07107	0.473506	0.236753	−	0.971570i	$-0.423917\pi$
$367$	9.07107	0.473506	0.236753	+	0.971570i	$0.423917\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.62742	0.136409
$372$	0	0
$373$	20.0416	1.03772	0.518858	−	0.854860i	$-0.326357\pi$
$373$	20.0416	1.03772	0.518858	+	0.854860i	$0.326357\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−26.4853	−1.36406
$378$	0	0
$379$	6.48528	0.333127	0.166563	−	0.986031i	$-0.446733\pi$
$379$	6.48528	0.333127	0.166563	+	0.986031i	$0.446733\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0.686292	0.0350679	0.0175339	−	0.999846i	$-0.494418\pi$
$383$	0.686292	0.0350679	0.0175339	+	0.999846i	$0.494418\pi$
$384$	0	0
$385$	0.828427	0.0422206
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.3431	0.625822	0.312911	−	0.949782i	$-0.398696\pi$
$389$	12.3431	0.625822	0.312911	+	0.949782i	$0.398696\pi$
$390$	0	0
$391$	−2.34315	−0.118498
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.6569	−0.687151
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−31.2132	−1.55871	−0.779356	−	0.626581i	$-0.784454\pi$
$401$	−31.2132	−1.55871	−0.779356	+	0.626581i	$0.784454\pi$
$402$	0	0
$403$	13.4558	0.670283
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.0000	−0.495682
$408$	0	0
$409$	−7.85786	−0.388546	−0.194273	−	0.980947i	$-0.562235\pi$
$409$	−7.85786	−0.388546	−0.194273	+	0.980947i	$0.562235\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.65685	0.0815285
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−19.7574	−0.965210	−0.482605	−	0.875838i	$-0.660309\pi$
$419$	−19.7574	−0.965210	−0.482605	+	0.875838i	$0.660309\pi$
$420$	0	0
$421$	2.68629	0.130922	0.0654609	−	0.997855i	$-0.479148\pi$
$421$	2.68629	0.130922	0.0654609	+	0.997855i	$0.479148\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.82843	0.137199
$426$	0	0
$427$	4.68629	0.226786
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.85786	0.282163	0.141082	−	0.989998i	$-0.454942\pi$
$431$	5.85786	0.282163	0.141082	+	0.989998i	$0.454942\pi$
$432$	0	0
$433$	20.2426	0.972799	0.486400	−	0.873736i	$-0.338310\pi$
$433$	20.2426	0.972799	0.486400	+	0.873736i	$0.338310\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.828427	0.0396290
$438$	0	0
$439$	−27.3137	−1.30361	−0.651806	−	0.758386i	$-0.725988\pi$
$439$	−27.3137	−1.30361	−0.651806	+	0.758386i	$0.725988\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.1421	1.14703	0.573514	−	0.819196i	$-0.305580\pi$
$443$	24.1421	1.14703	0.573514	+	0.819196i	$0.305580\pi$
$444$	0	0
$445$	−17.5563	−0.832251
$446$	0	0
$447$	0	0
$448$	0	0
$449$	38.5269	1.81820	0.909099	−	0.416581i	$-0.136772\pi$
$449$	38.5269	1.81820	0.909099	+	0.416581i	$0.136772\pi$
$450$	0	0
$451$	−8.82843	−0.415714
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.48528	0.116512
$456$	0	0
$457$	−8.14214	−0.380873	−0.190437	−	0.981700i	$-0.560990\pi$
$457$	−8.14214	−0.380873	−0.190437	+	0.981700i	$0.560990\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−24.6274	−1.14701	−0.573507	−	0.819201i	$-0.694417\pi$
$461$	−24.6274	−1.14701	−0.573507	+	0.819201i	$0.694417\pi$
$462$	0	0
$463$	−34.2426	−1.59139	−0.795695	−	0.605697i	$-0.792894\pi$
$463$	−34.2426	−1.59139	−0.795695	+	0.605697i	$0.792894\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.14214	−0.376773	−0.188387	−	0.982095i	$-0.560326\pi$
$467$	−8.14214	−0.376773	−0.188387	+	0.982095i	$0.560326\pi$
$468$	0	0
$469$	−6.62742	−0.306026
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14.4853	0.666034
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−38.3848	−1.75385	−0.876923	−	0.480632i	$-0.840407\pi$
$479$	−38.3848	−1.75385	−0.876923	+	0.480632i	$0.840407\pi$
$480$	0	0
$481$	−30.0000	−1.36788
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.89949	−0.449513
$486$	0	0
$487$	13.1716	0.596861	0.298430	−	0.954431i	$-0.403537\pi$
$487$	13.1716	0.596861	0.298430	+	0.954431i	$0.403537\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−40.2426	−1.81612	−0.908062	−	0.418835i	$-0.862439\pi$
$491$	−40.2426	−1.81612	−0.908062	+	0.418835i	$0.862439\pi$
$492$	0	0
$493$	17.6569	0.795225
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.34315	−0.284529
$498$	0	0
$499$	9.17157	0.410576	0.205288	−	0.978702i	$-0.434187\pi$
$499$	9.17157	0.410576	0.205288	+	0.978702i	$0.434187\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−30.2843	−1.35031	−0.675154	−	0.737676i	$-0.735923\pi$
$503$	−30.2843	−1.35031	−0.675154	+	0.737676i	$0.735923\pi$
$504$	0	0
$505$	7.17157	0.319131
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.07107	0.0474742	0.0237371	−	0.999718i	$-0.492444\pi$
$509$	1.07107	0.0474742	0.0237371	+	0.999718i	$0.492444\pi$
$510$	0	0
$511$	−4.48528	−0.198417
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−1.17157	−0.0515257
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16.3848	−0.717830	−0.358915	−	0.933370i	$-0.616853\pi$
$521$	−16.3848	−0.717830	−0.358915	+	0.933370i	$0.616853\pi$
$522$	0	0
$523$	9.85786	0.431054	0.215527	−	0.976498i	$-0.430853\pi$
$523$	9.85786	0.431054	0.215527	+	0.976498i	$0.430853\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.97056	−0.390764
$528$	0	0
$529$	−22.3137	−0.970161
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−26.4853	−1.14720
$534$	0	0
$535$	−2.34315	−0.101303
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.41421	0.405499
$540$	0	0
$541$	41.3137	1.77622	0.888108	−	0.459636i	$-0.152020\pi$
$541$	41.3137	1.77622	0.888108	+	0.459636i	$0.152020\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−16.1421	−0.691453
$546$	0	0
$547$	11.5147	0.492334	0.246167	−	0.969227i	$-0.420829\pi$
$547$	11.5147	0.492334	0.246167	+	0.969227i	$0.420829\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.24264	−0.265945
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.9706	0.973294	0.486647	−	0.873599i	$-0.338220\pi$
$557$	22.9706	0.973294	0.486647	+	0.873599i	$0.338220\pi$
$558$	0	0
$559$	43.4558	1.83799
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.6863	−0.450373	−0.225187	−	0.974316i	$-0.572299\pi$
$563$	−10.6863	−0.450373	−0.225187	+	0.974316i	$0.572299\pi$
$564$	0	0
$565$	1.65685	0.0697044
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.8995	1.00192	0.500959	−	0.865471i	$-0.332981\pi$
$569$	23.8995	1.00192	0.500959	+	0.865471i	$0.332981\pi$
$570$	0	0
$571$	−16.4853	−0.689888	−0.344944	−	0.938623i	$-0.612102\pi$
$571$	−16.4853	−0.689888	−0.344944	+	0.938623i	$0.612102\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.828427	−0.0345478
$576$	0	0
$577$	18.4853	0.769552	0.384776	−	0.923010i	$-0.374279\pi$
$577$	18.4853	0.769552	0.384776	+	0.923010i	$0.374279\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.51472	−0.145815
$582$	0	0
$583$	6.34315	0.262706
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.8284	−0.529486	−0.264743	−	0.964319i	$-0.585287\pi$
$587$	−12.8284	−0.529486	−0.264743	+	0.964319i	$0.585287\pi$
$588$	0	0
$589$	3.17157	0.130682
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.31371	−0.0539475	−0.0269738	−	0.999636i	$-0.508587\pi$
$593$	−1.31371	−0.0539475	−0.0269738	+	0.999636i	$0.508587\pi$
$594$	0	0
$595$	−1.65685	−0.0679244
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16.9706	−0.693398	−0.346699	−	0.937976i	$-0.612698\pi$
$599$	−16.9706	−0.693398	−0.346699	+	0.937976i	$0.612698\pi$
$600$	0	0
$601$	−11.8579	−0.483692	−0.241846	−	0.970315i	$-0.577753\pi$
$601$	−11.8579	−0.483692	−0.241846	+	0.970315i	$0.577753\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.00000	−0.365902
$606$	0	0
$607$	5.85786	0.237763	0.118882	−	0.992908i	$-0.462069\pi$
$607$	5.85786	0.237763	0.118882	+	0.992908i	$0.462069\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.51472	−0.142190
$612$	0	0
$613$	−28.1421	−1.13665	−0.568325	−	0.822804i	$-0.692408\pi$
$613$	−28.1421	−1.13665	−0.568325	+	0.822804i	$0.692408\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.1127	−1.25255	−0.626275	−	0.779602i	$-0.715421\pi$
$617$	−31.1127	−1.25255	−0.626275	+	0.779602i	$0.715421\pi$
$618$	0	0
$619$	−10.8284	−0.435231	−0.217616	−	0.976035i	$-0.569828\pi$
$619$	−10.8284	−0.435231	−0.217616	+	0.976035i	$0.569828\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.2843	0.412031
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.0000	0.797452
$630$	0	0
$631$	−24.9706	−0.994062	−0.497031	−	0.867733i	$-0.665577\pi$
$631$	−24.9706	−0.994062	−0.497031	+	0.867733i	$0.665577\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16.9706	−0.673456
$636$	0	0
$637$	28.2426	1.11901
$638$	0	0
$639$	0	0
$640$	0	0
$641$	25.7574	1.01735	0.508677	−	0.860957i	$-0.330135\pi$
$641$	25.7574	1.01735	0.508677	+	0.860957i	$0.330135\pi$
$642$	0	0
$643$	10.2426	0.403930	0.201965	−	0.979393i	$-0.435267\pi$
$643$	10.2426	0.403930	0.201965	+	0.979393i	$0.435267\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−31.6569	−1.24456	−0.622280	−	0.782795i	$-0.713793\pi$
$647$	−31.6569	−1.24456	−0.622280	+	0.782795i	$0.713793\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.48528	−0.175523	−0.0877613	−	0.996142i	$-0.527971\pi$
$653$	−4.48528	−0.175523	−0.0877613	+	0.996142i	$0.527971\pi$
$654$	0	0
$655$	−19.0711	−0.745168
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17.6569	0.687813	0.343907	−	0.939004i	$-0.388250\pi$
$659$	17.6569	0.687813	0.343907	+	0.939004i	$0.388250\pi$
$660$	0	0
$661$	−37.1127	−1.44352	−0.721758	−	0.692145i	$-0.756666\pi$
$661$	−37.1127	−1.44352	−0.721758	+	0.692145i	$0.756666\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.585786	0.0227158
$666$	0	0
$667$	−5.17157	−0.200244
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.3137	0.436761
$672$	0	0
$673$	−10.1005	−0.389346	−0.194673	−	0.980868i	$-0.562365\pi$
$673$	−10.1005	−0.389346	−0.194673	+	0.980868i	$0.562365\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.201010	−0.00772545	−0.00386272	−	0.999993i	$-0.501230\pi$
$677$	−0.201010	−0.00772545	−0.00386272	+	0.999993i	$0.501230\pi$
$678$	0	0
$679$	5.79899	0.222545
$680$	0	0
$681$	0	0
$682$	0	0
$683$	43.3137	1.65735	0.828676	−	0.559728i	$-0.189094\pi$
$683$	43.3137	1.65735	0.828676	+	0.559728i	$0.189094\pi$
$684$	0	0
$685$	−18.9706	−0.724828
$686$	0	0
$687$	0	0
$688$	0	0
$689$	19.0294	0.724964
$690$	0	0
$691$	31.1127	1.18358	0.591791	−	0.806091i	$-0.298421\pi$
$691$	31.1127	1.18358	0.591791	+	0.806091i	$0.298421\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.48528	0.170136
$696$	0	0
$697$	17.6569	0.668801
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.6569	−0.440273	−0.220137	−	0.975469i	$-0.570650\pi$
$701$	−11.6569	−0.440273	−0.220137	+	0.975469i	$0.570650\pi$
$702$	0	0
$703$	−7.07107	−0.266690
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.20101	−0.157995
$708$	0	0
$709$	−29.6569	−1.11379	−0.556893	−	0.830584i	$-0.688007\pi$
$709$	−29.6569	−1.11379	−0.556893	+	0.830584i	$0.688007\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.62742	0.0983975
$714$	0	0
$715$	6.00000	0.224387
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9.61522	−0.358587	−0.179294	−	0.983796i	$-0.557381\pi$
$719$	−9.61522	−0.358587	−0.179294	+	0.983796i	$0.557381\pi$
$720$	0	0
$721$	4.68629	0.174527
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.24264	0.231846
$726$	0	0
$727$	−28.3848	−1.05273	−0.526367	−	0.850258i	$-0.676446\pi$
$727$	−28.3848	−1.05273	−0.526367	+	0.850258i	$0.676446\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−28.9706	−1.07151
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	31.3137	1.15189	0.575947	−	0.817487i	$-0.304634\pi$
$739$	31.3137	1.15189	0.575947	+	0.817487i	$0.304634\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.9411	0.804942	0.402471	−	0.915433i	$-0.368151\pi$
$743$	21.9411	0.804942	0.402471	+	0.915433i	$0.368151\pi$
$744$	0	0
$745$	−0.343146	−0.0125719
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.37258	0.0501531
$750$	0	0
$751$	28.1421	1.02692	0.513461	−	0.858113i	$-0.328363\pi$
$751$	28.1421	1.02692	0.513461	+	0.858113i	$0.328363\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−16.8284	−0.612449
$756$	0	0
$757$	−26.7696	−0.972956	−0.486478	−	0.873693i	$-0.661719\pi$
$757$	−26.7696	−0.972956	−0.486478	+	0.873693i	$0.661719\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.9706	0.687682	0.343841	−	0.939028i	$-0.388272\pi$
$761$	18.9706	0.687682	0.343841	+	0.939028i	$0.388272\pi$
$762$	0	0
$763$	9.45584	0.342325
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	−8.34315	−0.300862	−0.150431	−	0.988621i	$-0.548066\pi$
$769$	−8.34315	−0.300862	−0.150431	+	0.988621i	$0.548066\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7.51472	−0.270286	−0.135143	−	0.990826i	$-0.543149\pi$
$773$	−7.51472	−0.270286	−0.135143	+	0.990826i	$0.543149\pi$
$774$	0	0
$775$	−3.17157	−0.113926
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.24264	−0.223666
$780$	0	0
$781$	−15.3137	−0.547968
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20.1421	0.718904
$786$	0	0
$787$	8.48528	0.302468	0.151234	−	0.988498i	$-0.451675\pi$
$787$	8.48528	0.302468	0.151234	+	0.988498i	$0.451675\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.970563	−0.0345092
$792$	0	0
$793$	33.9411	1.20528
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.02944	0.248995	0.124498	−	0.992220i	$-0.460268\pi$
$797$	7.02944	0.248995	0.124498	+	0.992220i	$0.460268\pi$
$798$	0	0
$799$	2.34315	0.0828945
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.8284	−0.382127
$804$	0	0
$805$	0.485281	0.0171039
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−25.3137	−0.889983	−0.444991	−	0.895535i	$-0.646793\pi$
$809$	−25.3137	−0.889983	−0.444991	+	0.895535i	$0.646793\pi$
$810$	0	0
$811$	8.28427	0.290900	0.145450	−	0.989366i	$-0.453537\pi$
$811$	8.28427	0.290900	0.145450	+	0.989366i	$0.453537\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−19.8995	−0.697049
$816$	0	0
$817$	10.2426	0.358345
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.51472	0.192465	0.0962325	−	0.995359i	$-0.469321\pi$
$821$	5.51472	0.192465	0.0962325	+	0.995359i	$0.469321\pi$
$822$	0	0
$823$	−23.2132	−0.809161	−0.404581	−	0.914502i	$-0.632583\pi$
$823$	−23.2132	−0.809161	−0.404581	+	0.914502i	$0.632583\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.0000	−0.904109	−0.452054	−	0.891990i	$-0.649309\pi$
$827$	−26.0000	−0.904109	−0.452054	+	0.891990i	$0.649309\pi$
$828$	0	0
$829$	37.7990	1.31281	0.656407	−	0.754407i	$-0.272076\pi$
$829$	37.7990	1.31281	0.656407	+	0.754407i	$0.272076\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18.8284	−0.652366
$834$	0	0
$835$	−4.34315	−0.150301
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.4853	1.39771	0.698854	−	0.715265i	$-0.253694\pi$
$839$	40.4853	1.39771	0.698854	+	0.715265i	$0.253694\pi$
$840$	0	0
$841$	9.97056	0.343813
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.00000	0.172005
$846$	0	0
$847$	5.27208	0.181151
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.85786	−0.200805
$852$	0	0
$853$	3.45584	0.118326	0.0591629	−	0.998248i	$-0.481157\pi$
$853$	3.45584	0.118326	0.0591629	+	0.998248i	$0.481157\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	52.0833	1.77913	0.889565	−	0.456808i	$-0.151007\pi$
$857$	52.0833	1.77913	0.889565	+	0.456808i	$0.151007\pi$
$858$	0	0
$859$	−17.9411	−0.612143	−0.306072	−	0.952008i	$-0.599015\pi$
$859$	−17.9411	−0.612143	−0.306072	+	0.952008i	$0.599015\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	−2.34315	−0.0796693
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19.3137	0.655173
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.585786	−0.0198032
$876$	0	0
$877$	34.5858	1.16788	0.583940	−	0.811797i	$-0.301511\pi$
$877$	34.5858	1.16788	0.583940	+	0.811797i	$0.301511\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	29.1127	0.980832	0.490416	−	0.871489i	$-0.336845\pi$
$881$	29.1127	0.980832	0.490416	+	0.871489i	$0.336845\pi$
$882$	0	0
$883$	34.7279	1.16869	0.584344	−	0.811506i	$-0.301352\pi$
$883$	34.7279	1.16869	0.584344	+	0.811506i	$0.301352\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.9706	0.569816	0.284908	−	0.958555i	$-0.408037\pi$
$887$	16.9706	0.569816	0.284908	+	0.958555i	$0.408037\pi$
$888$	0	0
$889$	9.94113	0.333415
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.828427	−0.0277223
$894$	0	0
$895$	15.7990	0.528102
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19.7990	−0.660333
$900$	0	0
$901$	−12.6863	−0.422642
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.828427	−0.0275378
$906$	0	0
$907$	−42.1421	−1.39931	−0.699653	−	0.714482i	$-0.746662\pi$
$907$	−42.1421	−1.39931	−0.699653	+	0.714482i	$0.746662\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	−8.48528	−0.280822
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.1716	0.368918
$918$	0	0
$919$	−6.34315	−0.209241	−0.104621	−	0.994512i	$-0.533363\pi$
$919$	−6.34315	−0.209241	−0.104621	+	0.994512i	$0.533363\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−45.9411	−1.51217
$924$	0	0
$925$	7.07107	0.232495
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.1716	0.628999	0.314499	−	0.949258i	$-0.398163\pi$
$929$	19.1716	0.628999	0.314499	+	0.949258i	$0.398163\pi$
$930$	0	0
$931$	6.65685	0.218170
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	49.1127	1.60444	0.802221	−	0.597027i	$-0.203652\pi$
$937$	49.1127	1.60444	0.802221	+	0.597027i	$0.203652\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26.0416	0.848933	0.424466	−	0.905444i	$-0.360462\pi$
$941$	26.0416	0.848933	0.424466	+	0.905444i	$0.360462\pi$
$942$	0	0
$943$	−5.17157	−0.168410
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.68629	−0.0872927	−0.0436464	−	0.999047i	$-0.513897\pi$
$947$	−2.68629	−0.0872927	−0.0436464	+	0.999047i	$0.513897\pi$
$948$	0	0
$949$	−32.4853	−1.05452
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−45.9411	−1.48818	−0.744090	−	0.668080i	$-0.767116\pi$
$953$	−45.9411	−1.48818	−0.744090	+	0.668080i	$0.767116\pi$
$954$	0	0
$955$	−10.5858	−0.342548
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.1127	0.358848
$960$	0	0
$961$	−20.9411	−0.675520
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.92893	0.287432
$966$	0	0
$967$	11.8995	0.382662	0.191331	−	0.981526i	$-0.438720\pi$
$967$	11.8995	0.382662	0.191331	+	0.981526i	$0.438720\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−23.3137	−0.748173	−0.374086	−	0.927394i	$-0.622044\pi$
$971$	−23.3137	−0.748173	−0.374086	+	0.927394i	$0.622044\pi$
$972$	0	0
$973$	−2.62742	−0.0842311
$974$	0	0
$975$	0	0
$976$	0	0
$977$	59.7990	1.91314	0.956570	−	0.291504i	$-0.0941557\pi$
$977$	59.7990	1.91314	0.956570	+	0.291504i	$0.0941557\pi$
$978$	0	0
$979$	24.8284	0.793520
$980$	0	0
$981$	0	0
$982$	0	0
$983$	57.5980	1.83709	0.918545	−	0.395316i	$-0.129365\pi$
$983$	57.5980	1.83709	0.918545	+	0.395316i	$0.129365\pi$
$984$	0	0
$985$	8.48528	0.270364
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.48528	0.269816
$990$	0	0
$991$	−48.9706	−1.55560	−0.777801	−	0.628511i	$-0.783665\pi$
$991$	−48.9706	−1.55560	−0.777801	+	0.628511i	$0.783665\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.1421	0.448336
$996$	0	0
$997$	10.2010	0.323069	0.161535	−	0.986867i	$-0.448356\pi$
$997$	10.2010	0.323069	0.161535	+	0.986867i	$0.448356\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.x.1.2		2
3.2	odd	2		2280.2.a.o.1.2	✓	2
12.11	even	2		4560.2.a.bg.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.o.1.2	✓	2	3.2	odd	2
4560.2.a.bg.1.1		2	12.11	even	2
6840.2.a.x.1.2		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.585786 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -0.585786 q^{7} -1.41421 q^{11} -4.24264 q^{13} +2.82843 q^{17} -1.00000 q^{19} -0.828427 q^{23} +1.00000 q^{25} +6.24264 q^{29} -3.17157 q^{31} -0.585786 q^{35} +7.07107 q^{37} +6.24264 q^{41} -10.2426 q^{43} +0.828427 q^{47} -6.65685 q^{49} -4.48528 q^{53} -1.41421 q^{55} -2.82843 q^{59} -8.00000 q^{61} -4.24264 q^{65} +11.3137 q^{67} +10.8284 q^{71} +7.65685 q^{73} +0.828427 q^{77} -13.6569 q^{79} +6.00000 q^{83} +2.82843 q^{85} -17.5563 q^{89} +2.48528 q^{91} -1.00000 q^{95} -9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 4 * q^7	\( 2 q + 2 q^{5} - 4 q^{7} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 12 q^{31} - 4 q^{35} + 4 q^{41} - 12 q^{43} - 4 q^{47} - 2 q^{49} + 8 q^{53} - 16 q^{61} + 16 q^{71} + 4 q^{73} - 4 q^{77} - 16 q^{79} + 12 q^{83} - 4 q^{89} - 12 q^{91} - 2 q^{95}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^7 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 4 * q^29 - 12 * q^31 - 4 * q^35 + 4 * q^41 - 12 * q^43 - 4 * q^47 - 2 * q^49 + 8 * q^53 - 16 * q^61 + 16 * q^71 + 4 * q^73 - 4 * q^77 - 16 * q^79 + 12 * q^83 - 4 * q^89 - 12 * q^91 - 2 * q^95

Introduction

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