LMFDB - Embedded newform 3840.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(1,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.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:	$30.6625543762$
Analytic rank:	$0$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 1920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +4.82843 q^{7} +1.00000 q^{9} -0.828427 q^{11} +2.00000 q^{13} -1.00000 q^{15} +5.65685 q^{17} +6.82843 q^{19} -4.82843 q^{21} +1.17157 q^{23} +1.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +4.00000 q^{31} +0.828427 q^{33} +4.82843 q^{35} -7.65685 q^{37} -2.00000 q^{39} -3.65685 q^{41} +9.65685 q^{43} +1.00000 q^{45} +5.17157 q^{47} +16.3137 q^{49} -5.65685 q^{51} +9.31371 q^{53} -0.828427 q^{55} -6.82843 q^{57} -14.4853 q^{59} -4.00000 q^{61} +4.82843 q^{63} +2.00000 q^{65} -4.00000 q^{67} -1.17157 q^{69} -5.65685 q^{71} -13.3137 q^{73} -1.00000 q^{75} -4.00000 q^{77} +4.00000 q^{79} +1.00000 q^{81} -2.34315 q^{83} +5.65685 q^{85} +6.00000 q^{87} +0.343146 q^{89} +9.65685 q^{91} -4.00000 q^{93} +6.82843 q^{95} -14.0000 q^{97} -0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{15} + 8 q^{19} - 4 q^{21} + 8 q^{23} + 2 q^{25} - 2 q^{27} - 12 q^{29} + 8 q^{31} - 4 q^{33} + 4 q^{35} - 4 q^{37} - 4 q^{39} + 4 q^{41}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 + 4 * q^11 + 4 * q^13 - 2 * q^15 + 8 * q^19 - 4 * q^21 + 8 * q^23 + 2 * q^25 - 2 * q^27 - 12 * q^29 + 8 * q^31 - 4 * q^33 + 4 * q^35 - 4 * q^37 - 4 * q^39 + 4 * q^41 + 8 * q^43 + 2 * q^45 + 16 * q^47 + 10 * q^49 - 4 * q^53 + 4 * q^55 - 8 * q^57 - 12 * q^59 - 8 * q^61 + 4 * q^63 + 4 * q^65 - 8 * q^67 - 8 * q^69 - 4 * q^73 - 2 * q^75 - 8 * q^77 + 8 * q^79 + 2 * q^81 - 16 * q^83 + 12 * q^87 + 12 * q^89 + 8 * q^91 - 8 * q^93 + 8 * q^95 - 28 * q^97 + 4 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.65685	1.37199	0.685994	−	0.727607i	$-0.259367\pi$
$17$	5.65685	1.37199	0.685994	+	0.727607i	$0.259367\pi$
$18$	0	0
$19$	6.82843	1.56655	0.783274	−	0.621676i	$-0.213548\pi$
$19$	6.82843	1.56655	0.783274	+	0.621676i	$0.213548\pi$
$20$	0	0
$21$	−4.82843	−1.05365
$22$	0	0
$23$	1.17157	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$23$	1.17157	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	0.828427	0.144211
$34$	0	0
$35$	4.82843	0.816153
$36$	0	0
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	9.65685	1.47266	0.736328	−	0.676625i	$-0.236558\pi$
$43$	9.65685	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	5.17157	0.754351	0.377176	−	0.926142i	$-0.376895\pi$
$47$	5.17157	0.754351	0.377176	+	0.926142i	$0.376895\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	0	0
$51$	−5.65685	−0.792118
$52$	0	0
$53$	9.31371	1.27934	0.639668	−	0.768651i	$-0.279072\pi$
$53$	9.31371	1.27934	0.639668	+	0.768651i	$0.279072\pi$
$54$	0	0
$55$	−0.828427	−0.111705
$56$	0	0
$57$	−6.82843	−0.904447
$58$	0	0
$59$	−14.4853	−1.88582	−0.942912	−	0.333043i	$-0.891924\pi$
$59$	−14.4853	−1.88582	−0.942912	+	0.333043i	$0.891924\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	4.82843	0.608325
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−1.17157	−0.141041
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	−13.3137	−1.55825	−0.779126	−	0.626868i	$-0.784337\pi$
$73$	−13.3137	−1.55825	−0.779126	+	0.626868i	$0.784337\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.34315	−0.257194	−0.128597	−	0.991697i	$-0.541047\pi$
$83$	−2.34315	−0.257194	−0.128597	+	0.991697i	$0.541047\pi$
$84$	0	0
$85$	5.65685	0.613572
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	0.343146	0.0363734	0.0181867	−	0.999835i	$-0.494211\pi$
$89$	0.343146	0.0363734	0.0181867	+	0.999835i	$0.494211\pi$
$90$	0	0
$91$	9.65685	1.01231
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	6.82843	0.700582
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	−17.3137	−1.72278	−0.861389	−	0.507946i	$-0.830405\pi$
$101$	−17.3137	−1.72278	−0.861389	+	0.507946i	$0.830405\pi$
$102$	0	0
$103$	10.4853	1.03315	0.516573	−	0.856243i	$-0.327208\pi$
$103$	10.4853	1.03315	0.516573	+	0.856243i	$0.327208\pi$
$104$	0	0
$105$	−4.82843	−0.471206
$106$	0	0
$107$	−5.65685	−0.546869	−0.273434	−	0.961891i	$-0.588160\pi$
$107$	−5.65685	−0.546869	−0.273434	+	0.961891i	$0.588160\pi$
$108$	0	0
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	7.65685	0.726756
$112$	0	0
$113$	9.65685	0.908440	0.454220	−	0.890889i	$-0.349918\pi$
$113$	9.65685	0.908440	0.454220	+	0.890889i	$0.349918\pi$
$114$	0	0
$115$	1.17157	0.109250
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	27.3137	2.50384
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	3.65685	0.329727
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.828427	−0.0735110	−0.0367555	−	0.999324i	$-0.511702\pi$
$127$	−0.828427	−0.0735110	−0.0367555	+	0.999324i	$0.511702\pi$
$128$	0	0
$129$	−9.65685	−0.850239
$130$	0	0
$131$	8.82843	0.771343	0.385672	−	0.922636i	$-0.373970\pi$
$131$	8.82843	0.771343	0.385672	+	0.922636i	$0.373970\pi$
$132$	0	0
$133$	32.9706	2.85891
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−17.6569	−1.50853	−0.754263	−	0.656572i	$-0.772006\pi$
$137$	−17.6569	−1.50853	−0.754263	+	0.656572i	$0.772006\pi$
$138$	0	0
$139$	8.48528	0.719712	0.359856	−	0.933008i	$-0.382826\pi$
$139$	8.48528	0.719712	0.359856	+	0.933008i	$0.382826\pi$
$140$	0	0
$141$	−5.17157	−0.435525
$142$	0	0
$143$	−1.65685	−0.138553
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	−16.3137	−1.34553
$148$	0	0
$149$	−13.3137	−1.09070	−0.545351	−	0.838208i	$-0.683604\pi$
$149$	−13.3137	−1.09070	−0.545351	+	0.838208i	$0.683604\pi$
$150$	0	0
$151$	10.3431	0.841713	0.420857	−	0.907127i	$-0.361730\pi$
$151$	10.3431	0.841713	0.420857	+	0.907127i	$0.361730\pi$
$152$	0	0
$153$	5.65685	0.457330
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	7.65685	0.611083	0.305542	−	0.952179i	$-0.401162\pi$
$157$	7.65685	0.611083	0.305542	+	0.952179i	$0.401162\pi$
$158$	0	0
$159$	−9.31371	−0.738625
$160$	0	0
$161$	5.65685	0.445823
$162$	0	0
$163$	−12.9706	−1.01593	−0.507966	−	0.861377i	$-0.669603\pi$
$163$	−12.9706	−1.01593	−0.507966	+	0.861377i	$0.669603\pi$
$164$	0	0
$165$	0.828427	0.0644930
$166$	0	0
$167$	6.82843	0.528400	0.264200	−	0.964468i	$-0.414892\pi$
$167$	6.82843	0.528400	0.264200	+	0.964468i	$0.414892\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	6.82843	0.522183
$172$	0	0
$173$	−1.31371	−0.0998794	−0.0499397	−	0.998752i	$-0.515903\pi$
$173$	−1.31371	−0.0998794	−0.0499397	+	0.998752i	$0.515903\pi$
$174$	0	0
$175$	4.82843	0.364995
$176$	0	0
$177$	14.4853	1.08878
$178$	0	0
$179$	2.48528	0.185759	0.0928793	−	0.995677i	$-0.470393\pi$
$179$	2.48528	0.185759	0.0928793	+	0.995677i	$0.470393\pi$
$180$	0	0
$181$	−18.6274	−1.38457	−0.692283	−	0.721627i	$-0.743395\pi$
$181$	−18.6274	−1.38457	−0.692283	+	0.721627i	$0.743395\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	−7.65685	−0.562943
$186$	0	0
$187$	−4.68629	−0.342696
$188$	0	0
$189$	−4.82843	−0.351216
$190$	0	0
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−25.3137	−1.80353	−0.901764	−	0.432230i	$-0.857727\pi$
$197$	−25.3137	−1.80353	−0.901764	+	0.432230i	$0.857727\pi$
$198$	0	0
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	−28.9706	−2.03333
$204$	0	0
$205$	−3.65685	−0.255406
$206$	0	0
$207$	1.17157	0.0814299
$208$	0	0
$209$	−5.65685	−0.391293
$210$	0	0
$211$	16.4853	1.13489	0.567447	−	0.823410i	$-0.307931\pi$
$211$	16.4853	1.13489	0.567447	+	0.823410i	$0.307931\pi$
$212$	0	0
$213$	5.65685	0.387601
$214$	0	0
$215$	9.65685	0.658592
$216$	0	0
$217$	19.3137	1.31110
$218$	0	0
$219$	13.3137	0.899657
$220$	0	0
$221$	11.3137	0.761042
$222$	0	0
$223$	9.51472	0.637153	0.318576	−	0.947897i	$-0.396795\pi$
$223$	9.51472	0.637153	0.318576	+	0.947897i	$0.396795\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	18.6274	1.23635	0.618173	−	0.786042i	$-0.287873\pi$
$227$	18.6274	1.23635	0.618173	+	0.786042i	$0.287873\pi$
$228$	0	0
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	29.6569	1.94289	0.971443	−	0.237275i	$-0.0762542\pi$
$233$	29.6569	1.94289	0.971443	+	0.237275i	$0.0762542\pi$
$234$	0	0
$235$	5.17157	0.337356
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	29.6569	1.91834	0.959171	−	0.282826i	$-0.0912719\pi$
$239$	29.6569	1.91834	0.959171	+	0.282826i	$0.0912719\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	16.3137	1.04224
$246$	0	0
$247$	13.6569	0.868965
$248$	0	0
$249$	2.34315	0.148491
$250$	0	0
$251$	−0.142136	−0.00897152	−0.00448576	−	0.999990i	$-0.501428\pi$
$251$	−0.142136	−0.00897152	−0.00448576	+	0.999990i	$0.501428\pi$
$252$	0	0
$253$	−0.970563	−0.0610188
$254$	0	0
$255$	−5.65685	−0.354246
$256$	0	0
$257$	−9.65685	−0.602378	−0.301189	−	0.953564i	$-0.597384\pi$
$257$	−9.65685	−0.602378	−0.301189	+	0.953564i	$0.597384\pi$
$258$	0	0
$259$	−36.9706	−2.29724
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−26.1421	−1.61199	−0.805997	−	0.591920i	$-0.798370\pi$
$263$	−26.1421	−1.61199	−0.805997	+	0.591920i	$0.798370\pi$
$264$	0	0
$265$	9.31371	0.572137
$266$	0	0
$267$	−0.343146	−0.0210002
$268$	0	0
$269$	5.31371	0.323983	0.161991	−	0.986792i	$-0.448208\pi$
$269$	5.31371	0.323983	0.161991	+	0.986792i	$0.448208\pi$
$270$	0	0
$271$	21.6569	1.31556	0.657780	−	0.753210i	$-0.271496\pi$
$271$	21.6569	1.31556	0.657780	+	0.753210i	$0.271496\pi$
$272$	0	0
$273$	−9.65685	−0.584459
$274$	0	0
$275$	−0.828427	−0.0499560
$276$	0	0
$277$	27.6569	1.66174	0.830870	−	0.556467i	$-0.187843\pi$
$277$	27.6569	1.66174	0.830870	+	0.556467i	$0.187843\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	12.6274	0.753289	0.376644	−	0.926358i	$-0.377078\pi$
$281$	12.6274	0.753289	0.376644	+	0.926358i	$0.377078\pi$
$282$	0	0
$283$	−26.6274	−1.58284	−0.791418	−	0.611276i	$-0.790657\pi$
$283$	−26.6274	−1.58284	−0.791418	+	0.611276i	$0.790657\pi$
$284$	0	0
$285$	−6.82843	−0.404481
$286$	0	0
$287$	−17.6569	−1.04225
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	16.6274	0.971384	0.485692	−	0.874130i	$-0.338568\pi$
$293$	16.6274	0.971384	0.485692	+	0.874130i	$0.338568\pi$
$294$	0	0
$295$	−14.4853	−0.843366
$296$	0	0
$297$	0.828427	0.0480702
$298$	0	0
$299$	2.34315	0.135508
$300$	0	0
$301$	46.6274	2.68756
$302$	0	0
$303$	17.3137	0.994647
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−25.6569	−1.46431	−0.732157	−	0.681136i	$-0.761486\pi$
$307$	−25.6569	−1.46431	−0.732157	+	0.681136i	$0.761486\pi$
$308$	0	0
$309$	−10.4853	−0.596487
$310$	0	0
$311$	2.34315	0.132868	0.0664338	−	0.997791i	$-0.478838\pi$
$311$	2.34315	0.132868	0.0664338	+	0.997791i	$0.478838\pi$
$312$	0	0
$313$	1.31371	0.0742552	0.0371276	−	0.999311i	$-0.488179\pi$
$313$	1.31371	0.0742552	0.0371276	+	0.999311i	$0.488179\pi$
$314$	0	0
$315$	4.82843	0.272051
$316$	0	0
$317$	5.31371	0.298448	0.149224	−	0.988803i	$-0.452323\pi$
$317$	5.31371	0.298448	0.149224	+	0.988803i	$0.452323\pi$
$318$	0	0
$319$	4.97056	0.278298
$320$	0	0
$321$	5.65685	0.315735
$322$	0	0
$323$	38.6274	2.14929
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−12.0000	−0.663602
$328$	0	0
$329$	24.9706	1.37667
$330$	0	0
$331$	18.1421	0.997182	0.498591	−	0.866837i	$-0.333851\pi$
$331$	18.1421	0.997182	0.498591	+	0.866837i	$0.333851\pi$
$332$	0	0
$333$	−7.65685	−0.419593
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−24.6274	−1.34154	−0.670770	−	0.741665i	$-0.734036\pi$
$337$	−24.6274	−1.34154	−0.670770	+	0.741665i	$0.734036\pi$
$338$	0	0
$339$	−9.65685	−0.524488
$340$	0	0
$341$	−3.31371	−0.179447
$342$	0	0
$343$	44.9706	2.42818
$344$	0	0
$345$	−1.17157	−0.0630754
$346$	0	0
$347$	−23.3137	−1.25155	−0.625773	−	0.780005i	$-0.715216\pi$
$347$	−23.3137	−1.25155	−0.625773	+	0.780005i	$0.715216\pi$
$348$	0	0
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	10.3431	0.550510	0.275255	−	0.961371i	$-0.411238\pi$
$353$	10.3431	0.550510	0.275255	+	0.961371i	$0.411238\pi$
$354$	0	0
$355$	−5.65685	−0.300235
$356$	0	0
$357$	−27.3137	−1.44559
$358$	0	0
$359$	20.2843	1.07056	0.535281	−	0.844674i	$-0.320206\pi$
$359$	20.2843	1.07056	0.535281	+	0.844674i	$0.320206\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	0	0
$363$	10.3137	0.541329
$364$	0	0
$365$	−13.3137	−0.696871
$366$	0	0
$367$	7.17157	0.374353	0.187177	−	0.982326i	$-0.440066\pi$
$367$	7.17157	0.374353	0.187177	+	0.982326i	$0.440066\pi$
$368$	0	0
$369$	−3.65685	−0.190368
$370$	0	0
$371$	44.9706	2.33476
$372$	0	0
$373$	10.9706	0.568034	0.284017	−	0.958819i	$-0.408333\pi$
$373$	10.9706	0.568034	0.284017	+	0.958819i	$0.408333\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−10.8284	−0.556219	−0.278109	−	0.960549i	$-0.589708\pi$
$379$	−10.8284	−0.556219	−0.278109	+	0.960549i	$0.589708\pi$
$380$	0	0
$381$	0.828427	0.0424416
$382$	0	0
$383$	−13.1716	−0.673036	−0.336518	−	0.941677i	$-0.609249\pi$
$383$	−13.1716	−0.673036	−0.336518	+	0.941677i	$0.609249\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	9.65685	0.490885
$388$	0	0
$389$	20.6274	1.04585	0.522926	−	0.852378i	$-0.324840\pi$
$389$	20.6274	1.04585	0.522926	+	0.852378i	$0.324840\pi$
$390$	0	0
$391$	6.62742	0.335163
$392$	0	0
$393$	−8.82843	−0.445335
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	0	0
$399$	−32.9706	−1.65059
$400$	0	0
$401$	−34.9706	−1.74635	−0.873173	−	0.487410i	$-0.837942\pi$
$401$	−34.9706	−1.74635	−0.873173	+	0.487410i	$0.837942\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	6.34315	0.314418
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	17.6569	0.870948
$412$	0	0
$413$	−69.9411	−3.44158
$414$	0	0
$415$	−2.34315	−0.115021
$416$	0	0
$417$	−8.48528	−0.415526
$418$	0	0
$419$	−9.51472	−0.464824	−0.232412	−	0.972617i	$-0.574662\pi$
$419$	−9.51472	−0.464824	−0.232412	+	0.972617i	$0.574662\pi$
$420$	0	0
$421$	4.68629	0.228396	0.114198	−	0.993458i	$-0.463570\pi$
$421$	4.68629	0.228396	0.114198	+	0.993458i	$0.463570\pi$
$422$	0	0
$423$	5.17157	0.251450
$424$	0	0
$425$	5.65685	0.274398
$426$	0	0
$427$	−19.3137	−0.934656
$428$	0	0
$429$	1.65685	0.0799937
$430$	0	0
$431$	−20.2843	−0.977059	−0.488529	−	0.872547i	$-0.662467\pi$
$431$	−20.2843	−0.977059	−0.488529	+	0.872547i	$0.662467\pi$
$432$	0	0
$433$	−21.3137	−1.02427	−0.512136	−	0.858905i	$-0.671146\pi$
$433$	−21.3137	−1.02427	−0.512136	+	0.858905i	$0.671146\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	0	0
$437$	8.00000	0.382692
$438$	0	0
$439$	−20.2843	−0.968115	−0.484058	−	0.875036i	$-0.660838\pi$
$439$	−20.2843	−0.968115	−0.484058	+	0.875036i	$0.660838\pi$
$440$	0	0
$441$	16.3137	0.776843
$442$	0	0
$443$	23.3137	1.10767	0.553834	−	0.832627i	$-0.313164\pi$
$443$	23.3137	1.10767	0.553834	+	0.832627i	$0.313164\pi$
$444$	0	0
$445$	0.343146	0.0162667
$446$	0	0
$447$	13.3137	0.629717
$448$	0	0
$449$	−5.31371	−0.250769	−0.125385	−	0.992108i	$-0.540017\pi$
$449$	−5.31371	−0.250769	−0.125385	+	0.992108i	$0.540017\pi$
$450$	0	0
$451$	3.02944	0.142651
$452$	0	0
$453$	−10.3431	−0.485963
$454$	0	0
$455$	9.65685	0.452720
$456$	0	0
$457$	−9.31371	−0.435677	−0.217838	−	0.975985i	$-0.569901\pi$
$457$	−9.31371	−0.435677	−0.217838	+	0.975985i	$0.569901\pi$
$458$	0	0
$459$	−5.65685	−0.264039
$460$	0	0
$461$	12.6274	0.588117	0.294059	−	0.955787i	$-0.404994\pi$
$461$	12.6274	0.588117	0.294059	+	0.955787i	$0.404994\pi$
$462$	0	0
$463$	−18.4853	−0.859084	−0.429542	−	0.903047i	$-0.641325\pi$
$463$	−18.4853	−0.859084	−0.429542	+	0.903047i	$0.641325\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	−21.6569	−1.00216	−0.501080	−	0.865401i	$-0.667064\pi$
$467$	−21.6569	−1.00216	−0.501080	+	0.865401i	$0.667064\pi$
$468$	0	0
$469$	−19.3137	−0.891824
$470$	0	0
$471$	−7.65685	−0.352809
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	6.82843	0.313310
$476$	0	0
$477$	9.31371	0.426445
$478$	0	0
$479$	2.34315	0.107061	0.0535305	−	0.998566i	$-0.482953\pi$
$479$	2.34315	0.107061	0.0535305	+	0.998566i	$0.482953\pi$
$480$	0	0
$481$	−15.3137	−0.698245
$482$	0	0
$483$	−5.65685	−0.257396
$484$	0	0
$485$	−14.0000	−0.635707
$486$	0	0
$487$	−34.4853	−1.56268	−0.781339	−	0.624107i	$-0.785463\pi$
$487$	−34.4853	−1.56268	−0.781339	+	0.624107i	$0.785463\pi$
$488$	0	0
$489$	12.9706	0.586549
$490$	0	0
$491$	7.85786	0.354620	0.177310	−	0.984155i	$-0.443260\pi$
$491$	7.85786	0.354620	0.177310	+	0.984155i	$0.443260\pi$
$492$	0	0
$493$	−33.9411	−1.52863
$494$	0	0
$495$	−0.828427	−0.0372350
$496$	0	0
$497$	−27.3137	−1.22519
$498$	0	0
$499$	6.82843	0.305682	0.152841	−	0.988251i	$-0.451158\pi$
$499$	6.82843	0.305682	0.152841	+	0.988251i	$0.451158\pi$
$500$	0	0
$501$	−6.82843	−0.305072
$502$	0	0
$503$	−25.4558	−1.13502	−0.567510	−	0.823367i	$-0.692093\pi$
$503$	−25.4558	−1.13502	−0.567510	+	0.823367i	$0.692093\pi$
$504$	0	0
$505$	−17.3137	−0.770450
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	6.68629	0.296365	0.148182	−	0.988960i	$-0.452658\pi$
$509$	6.68629	0.296365	0.148182	+	0.988960i	$0.452658\pi$
$510$	0	0
$511$	−64.2843	−2.84377
$512$	0	0
$513$	−6.82843	−0.301482
$514$	0	0
$515$	10.4853	0.462037
$516$	0	0
$517$	−4.28427	−0.188422
$518$	0	0
$519$	1.31371	0.0576654
$520$	0	0
$521$	−37.3137	−1.63474	−0.817372	−	0.576111i	$-0.804570\pi$
$521$	−37.3137	−1.63474	−0.817372	+	0.576111i	$0.804570\pi$
$522$	0	0
$523$	7.31371	0.319806	0.159903	−	0.987133i	$-0.448882\pi$
$523$	7.31371	0.319806	0.159903	+	0.987133i	$0.448882\pi$
$524$	0	0
$525$	−4.82843	−0.210730
$526$	0	0
$527$	22.6274	0.985666
$528$	0	0
$529$	−21.6274	−0.940322
$530$	0	0
$531$	−14.4853	−0.628608
$532$	0	0
$533$	−7.31371	−0.316792
$534$	0	0
$535$	−5.65685	−0.244567
$536$	0	0
$537$	−2.48528	−0.107248
$538$	0	0
$539$	−13.5147	−0.582120
$540$	0	0
$541$	37.9411	1.63122	0.815608	−	0.578605i	$-0.196403\pi$
$541$	37.9411	1.63122	0.815608	+	0.578605i	$0.196403\pi$
$542$	0	0
$543$	18.6274	0.799379
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	−28.9706	−1.23869	−0.619346	−	0.785118i	$-0.712602\pi$
$547$	−28.9706	−1.23869	−0.619346	+	0.785118i	$0.712602\pi$
$548$	0	0
$549$	−4.00000	−0.170716
$550$	0	0
$551$	−40.9706	−1.74540
$552$	0	0
$553$	19.3137	0.821302
$554$	0	0
$555$	7.65685	0.325015
$556$	0	0
$557$	−2.68629	−0.113822	−0.0569109	−	0.998379i	$-0.518125\pi$
$557$	−2.68629	−0.113822	−0.0569109	+	0.998379i	$0.518125\pi$
$558$	0	0
$559$	19.3137	0.816883
$560$	0	0
$561$	4.68629	0.197855
$562$	0	0
$563$	−39.3137	−1.65688	−0.828438	−	0.560081i	$-0.810770\pi$
$563$	−39.3137	−1.65688	−0.828438	+	0.560081i	$0.810770\pi$
$564$	0	0
$565$	9.65685	0.406267
$566$	0	0
$567$	4.82843	0.202775
$568$	0	0
$569$	−15.6569	−0.656369	−0.328185	−	0.944614i	$-0.606437\pi$
$569$	−15.6569	−0.656369	−0.328185	+	0.944614i	$0.606437\pi$
$570$	0	0
$571$	27.5147	1.15146	0.575728	−	0.817642i	$-0.304719\pi$
$571$	27.5147	1.15146	0.575728	+	0.817642i	$0.304719\pi$
$572$	0	0
$573$	−11.3137	−0.472637
$574$	0	0
$575$	1.17157	0.0488580
$576$	0	0
$577$	21.3137	0.887301	0.443651	−	0.896200i	$-0.353683\pi$
$577$	21.3137	0.887301	0.443651	+	0.896200i	$0.353683\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	−11.3137	−0.469372
$582$	0	0
$583$	−7.71573	−0.319553
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−26.3431	−1.08730	−0.543649	−	0.839313i	$-0.682958\pi$
$587$	−26.3431	−1.08730	−0.543649	+	0.839313i	$0.682958\pi$
$588$	0	0
$589$	27.3137	1.12544
$590$	0	0
$591$	25.3137	1.04127
$592$	0	0
$593$	−32.2843	−1.32576	−0.662878	−	0.748727i	$-0.730665\pi$
$593$	−32.2843	−1.32576	−0.662878	+	0.748727i	$0.730665\pi$
$594$	0	0
$595$	27.3137	1.11975
$596$	0	0
$597$	16.9706	0.694559
$598$	0	0
$599$	21.6569	0.884875	0.442438	−	0.896799i	$-0.354114\pi$
$599$	21.6569	0.884875	0.442438	+	0.896799i	$0.354114\pi$
$600$	0	0
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	−10.3137	−0.419312
$606$	0	0
$607$	6.48528	0.263229	0.131615	−	0.991301i	$-0.457984\pi$
$607$	6.48528	0.263229	0.131615	+	0.991301i	$0.457984\pi$
$608$	0	0
$609$	28.9706	1.17395
$610$	0	0
$611$	10.3431	0.418439
$612$	0	0
$613$	25.3137	1.02241	0.511206	−	0.859458i	$-0.329199\pi$
$613$	25.3137	1.02241	0.511206	+	0.859458i	$0.329199\pi$
$614$	0	0
$615$	3.65685	0.147459
$616$	0	0
$617$	28.2843	1.13868	0.569341	−	0.822102i	$-0.307198\pi$
$617$	28.2843	1.13868	0.569341	+	0.822102i	$0.307198\pi$
$618$	0	0
$619$	−15.5147	−0.623589	−0.311795	−	0.950150i	$-0.600930\pi$
$619$	−15.5147	−0.623589	−0.311795	+	0.950150i	$0.600930\pi$
$620$	0	0
$621$	−1.17157	−0.0470136
$622$	0	0
$623$	1.65685	0.0663805
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.65685	0.225913
$628$	0	0
$629$	−43.3137	−1.72703
$630$	0	0
$631$	21.9411	0.873462	0.436731	−	0.899592i	$-0.356136\pi$
$631$	21.9411	0.873462	0.436731	+	0.899592i	$0.356136\pi$
$632$	0	0
$633$	−16.4853	−0.655231
$634$	0	0
$635$	−0.828427	−0.0328751
$636$	0	0
$637$	32.6274	1.29275
$638$	0	0
$639$	−5.65685	−0.223782
$640$	0	0
$641$	−18.6863	−0.738064	−0.369032	−	0.929417i	$-0.620311\pi$
$641$	−18.6863	−0.738064	−0.369032	+	0.929417i	$0.620311\pi$
$642$	0	0
$643$	−3.02944	−0.119469	−0.0597347	−	0.998214i	$-0.519025\pi$
$643$	−3.02944	−0.119469	−0.0597347	+	0.998214i	$0.519025\pi$
$644$	0	0
$645$	−9.65685	−0.380238
$646$	0	0
$647$	−32.4853	−1.27713	−0.638564	−	0.769569i	$-0.720471\pi$
$647$	−32.4853	−1.27713	−0.638564	+	0.769569i	$0.720471\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	−19.3137	−0.756964
$652$	0	0
$653$	−21.3137	−0.834070	−0.417035	−	0.908890i	$-0.636931\pi$
$653$	−21.3137	−0.834070	−0.417035	+	0.908890i	$0.636931\pi$
$654$	0	0
$655$	8.82843	0.344955
$656$	0	0
$657$	−13.3137	−0.519417
$658$	0	0
$659$	27.1716	1.05845	0.529227	−	0.848480i	$-0.322482\pi$
$659$	27.1716	1.05845	0.529227	+	0.848480i	$0.322482\pi$
$660$	0	0
$661$	18.6274	0.724523	0.362261	−	0.932077i	$-0.382005\pi$
$661$	18.6274	0.724523	0.362261	+	0.932077i	$0.382005\pi$
$662$	0	0
$663$	−11.3137	−0.439388
$664$	0	0
$665$	32.9706	1.27854
$666$	0	0
$667$	−7.02944	−0.272181
$668$	0	0
$669$	−9.51472	−0.367860
$670$	0	0
$671$	3.31371	0.127924
$672$	0	0
$673$	1.31371	0.0506397	0.0253199	−	0.999679i	$-0.491940\pi$
$673$	1.31371	0.0506397	0.0253199	+	0.999679i	$0.491940\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−29.3137	−1.12662	−0.563309	−	0.826247i	$-0.690472\pi$
$677$	−29.3137	−1.12662	−0.563309	+	0.826247i	$0.690472\pi$
$678$	0	0
$679$	−67.5980	−2.59417
$680$	0	0
$681$	−18.6274	−0.713804
$682$	0	0
$683$	16.9706	0.649361	0.324680	−	0.945824i	$-0.394743\pi$
$683$	16.9706	0.649361	0.324680	+	0.945824i	$0.394743\pi$
$684$	0	0
$685$	−17.6569	−0.674634
$686$	0	0
$687$	−16.0000	−0.610438
$688$	0	0
$689$	18.6274	0.709648
$690$	0	0
$691$	−13.4558	−0.511884	−0.255942	−	0.966692i	$-0.582386\pi$
$691$	−13.4558	−0.511884	−0.255942	+	0.966692i	$0.582386\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	8.48528	0.321865
$696$	0	0
$697$	−20.6863	−0.783549
$698$	0	0
$699$	−29.6569	−1.12173
$700$	0	0
$701$	16.6274	0.628009	0.314004	−	0.949422i	$-0.398329\pi$
$701$	16.6274	0.628009	0.314004	+	0.949422i	$0.398329\pi$
$702$	0	0
$703$	−52.2843	−1.97194
$704$	0	0
$705$	−5.17157	−0.194773
$706$	0	0
$707$	−83.5980	−3.14403
$708$	0	0
$709$	51.3137	1.92713	0.963563	−	0.267480i	$-0.0861910\pi$
$709$	51.3137	1.92713	0.963563	+	0.267480i	$0.0861910\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	4.68629	0.175503
$714$	0	0
$715$	−1.65685	−0.0619628
$716$	0	0
$717$	−29.6569	−1.10756
$718$	0	0
$719$	−20.6863	−0.771468	−0.385734	−	0.922610i	$-0.626052\pi$
$719$	−20.6863	−0.771468	−0.385734	+	0.922610i	$0.626052\pi$
$720$	0	0
$721$	50.6274	1.88546
$722$	0	0
$723$	−6.00000	−0.223142
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	40.8284	1.51424	0.757121	−	0.653274i	$-0.226605\pi$
$727$	40.8284	1.51424	0.757121	+	0.653274i	$0.226605\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	54.6274	2.02047
$732$	0	0
$733$	−30.2843	−1.11858	−0.559288	−	0.828974i	$-0.688925\pi$
$733$	−30.2843	−1.11858	−0.559288	+	0.828974i	$0.688925\pi$
$734$	0	0
$735$	−16.3137	−0.601740
$736$	0	0
$737$	3.31371	0.122062
$738$	0	0
$739$	5.45584	0.200696	0.100348	−	0.994952i	$-0.468004\pi$
$739$	5.45584	0.200696	0.100348	+	0.994952i	$0.468004\pi$
$740$	0	0
$741$	−13.6569	−0.501697
$742$	0	0
$743$	−46.1421	−1.69279	−0.846395	−	0.532555i	$-0.821232\pi$
$743$	−46.1421	−1.69279	−0.846395	+	0.532555i	$0.821232\pi$
$744$	0	0
$745$	−13.3137	−0.487777
$746$	0	0
$747$	−2.34315	−0.0857312
$748$	0	0
$749$	−27.3137	−0.998021
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	0.142136	0.00517971
$754$	0	0
$755$	10.3431	0.376426
$756$	0	0
$757$	−9.02944	−0.328180	−0.164090	−	0.986445i	$-0.552469\pi$
$757$	−9.02944	−0.328180	−0.164090	+	0.986445i	$0.552469\pi$
$758$	0	0
$759$	0.970563	0.0352292
$760$	0	0
$761$	−16.6274	−0.602743	−0.301372	−	0.953507i	$-0.597444\pi$
$761$	−16.6274	−0.602743	−0.301372	+	0.953507i	$0.597444\pi$
$762$	0	0
$763$	57.9411	2.09761
$764$	0	0
$765$	5.65685	0.204524
$766$	0	0
$767$	−28.9706	−1.04607
$768$	0	0
$769$	−33.3137	−1.20132	−0.600662	−	0.799503i	$-0.705096\pi$
$769$	−33.3137	−1.20132	−0.600662	+	0.799503i	$0.705096\pi$
$770$	0	0
$771$	9.65685	0.347783
$772$	0	0
$773$	−18.6863	−0.672099	−0.336050	−	0.941844i	$-0.609091\pi$
$773$	−18.6863	−0.672099	−0.336050	+	0.941844i	$0.609091\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	36.9706	1.32631
$778$	0	0
$779$	−24.9706	−0.894663
$780$	0	0
$781$	4.68629	0.167689
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	7.65685	0.273285
$786$	0	0
$787$	41.2548	1.47058	0.735288	−	0.677755i	$-0.237047\pi$
$787$	41.2548	1.47058	0.735288	+	0.677755i	$0.237047\pi$
$788$	0	0
$789$	26.1421	0.930685
$790$	0	0
$791$	46.6274	1.65788
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	−9.31371	−0.330323
$796$	0	0
$797$	29.3137	1.03834	0.519172	−	0.854670i	$-0.326240\pi$
$797$	29.3137	1.03834	0.519172	+	0.854670i	$0.326240\pi$
$798$	0	0
$799$	29.2548	1.03496
$800$	0	0
$801$	0.343146	0.0121245
$802$	0	0
$803$	11.0294	0.389220
$804$	0	0
$805$	5.65685	0.199378
$806$	0	0
$807$	−5.31371	−0.187051
$808$	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
$811$	15.5147	0.544795	0.272398	−	0.962185i	$-0.412183\pi$
$811$	15.5147	0.544795	0.272398	+	0.962185i	$0.412183\pi$
$812$	0	0
$813$	−21.6569	−0.759539
$814$	0	0
$815$	−12.9706	−0.454339
$816$	0	0
$817$	65.9411	2.30699
$818$	0	0
$819$	9.65685	0.337438
$820$	0	0
$821$	−33.3137	−1.16266	−0.581328	−	0.813669i	$-0.697467\pi$
$821$	−33.3137	−1.16266	−0.581328	+	0.813669i	$0.697467\pi$
$822$	0	0
$823$	21.1127	0.735942	0.367971	−	0.929837i	$-0.380052\pi$
$823$	21.1127	0.735942	0.367971	+	0.929837i	$0.380052\pi$
$824$	0	0
$825$	0.828427	0.0288421
$826$	0	0
$827$	−42.6274	−1.48230	−0.741150	−	0.671339i	$-0.765719\pi$
$827$	−42.6274	−1.48230	−0.741150	+	0.671339i	$0.765719\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	−27.6569	−0.959406
$832$	0	0
$833$	92.2843	3.19746
$834$	0	0
$835$	6.82843	0.236307
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	−7.02944	−0.242683	−0.121342	−	0.992611i	$-0.538720\pi$
$839$	−7.02944	−0.242683	−0.121342	+	0.992611i	$0.538720\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−12.6274	−0.434911
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−49.7990	−1.71111
$848$	0	0
$849$	26.6274	0.913851
$850$	0	0
$851$	−8.97056	−0.307507
$852$	0	0
$853$	−37.3137	−1.27760	−0.638799	−	0.769374i	$-0.720568\pi$
$853$	−37.3137	−1.27760	−0.638799	+	0.769374i	$0.720568\pi$
$854$	0	0
$855$	6.82843	0.233527
$856$	0	0
$857$	−4.28427	−0.146348	−0.0731740	−	0.997319i	$-0.523313\pi$
$857$	−4.28427	−0.146348	−0.0731740	+	0.997319i	$0.523313\pi$
$858$	0	0
$859$	−44.7696	−1.52752	−0.763759	−	0.645502i	$-0.776648\pi$
$859$	−44.7696	−1.52752	−0.763759	+	0.645502i	$0.776648\pi$
$860$	0	0
$861$	17.6569	0.601744
$862$	0	0
$863$	−9.17157	−0.312204	−0.156102	−	0.987741i	$-0.549893\pi$
$863$	−9.17157	−0.312204	−0.156102	+	0.987741i	$0.549893\pi$
$864$	0	0
$865$	−1.31371	−0.0446674
$866$	0	0
$867$	−15.0000	−0.509427
$868$	0	0
$869$	−3.31371	−0.112410
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	4.82843	0.163231
$876$	0	0
$877$	−16.3431	−0.551869	−0.275934	−	0.961176i	$-0.588987\pi$
$877$	−16.3431	−0.551869	−0.275934	+	0.961176i	$0.588987\pi$
$878$	0	0
$879$	−16.6274	−0.560829
$880$	0	0
$881$	46.9706	1.58248	0.791239	−	0.611507i	$-0.209436\pi$
$881$	46.9706	1.58248	0.791239	+	0.611507i	$0.209436\pi$
$882$	0	0
$883$	−22.3431	−0.751907	−0.375953	−	0.926639i	$-0.622685\pi$
$883$	−22.3431	−0.751907	−0.375953	+	0.926639i	$0.622685\pi$
$884$	0	0
$885$	14.4853	0.486917
$886$	0	0
$887$	−47.7990	−1.60493	−0.802467	−	0.596697i	$-0.796479\pi$
$887$	−47.7990	−1.60493	−0.802467	+	0.596697i	$0.796479\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	−0.828427	−0.0277534
$892$	0	0
$893$	35.3137	1.18173
$894$	0	0
$895$	2.48528	0.0830738
$896$	0	0
$897$	−2.34315	−0.0782354
$898$	0	0
$899$	−24.0000	−0.800445
$900$	0	0
$901$	52.6863	1.75523
$902$	0	0
$903$	−46.6274	−1.55166
$904$	0	0
$905$	−18.6274	−0.619196
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	0	0
$909$	−17.3137	−0.574259
$910$	0	0
$911$	−42.3431	−1.40289	−0.701446	−	0.712723i	$-0.747462\pi$
$911$	−42.3431	−1.40289	−0.701446	+	0.712723i	$0.747462\pi$
$912$	0	0
$913$	1.94113	0.0642419
$914$	0	0
$915$	4.00000	0.132236
$916$	0	0
$917$	42.6274	1.40768
$918$	0	0
$919$	−37.9411	−1.25156	−0.625781	−	0.779999i	$-0.715220\pi$
$919$	−37.9411	−1.25156	−0.625781	+	0.779999i	$0.715220\pi$
$920$	0	0
$921$	25.6569	0.845422
$922$	0	0
$923$	−11.3137	−0.372395
$924$	0	0
$925$	−7.65685	−0.251756
$926$	0	0
$927$	10.4853	0.344382
$928$	0	0
$929$	12.6274	0.414292	0.207146	−	0.978310i	$-0.433582\pi$
$929$	12.6274	0.414292	0.207146	+	0.978310i	$0.433582\pi$
$930$	0	0
$931$	111.397	3.65089
$932$	0	0
$933$	−2.34315	−0.0767111
$934$	0	0
$935$	−4.68629	−0.153258
$936$	0	0
$937$	−31.9411	−1.04347	−0.521736	−	0.853107i	$-0.674715\pi$
$937$	−31.9411	−1.04347	−0.521736	+	0.853107i	$0.674715\pi$
$938$	0	0
$939$	−1.31371	−0.0428713
$940$	0	0
$941$	17.3137	0.564411	0.282205	−	0.959354i	$-0.408934\pi$
$941$	17.3137	0.564411	0.282205	+	0.959354i	$0.408934\pi$
$942$	0	0
$943$	−4.28427	−0.139515
$944$	0	0
$945$	−4.82843	−0.157069
$946$	0	0
$947$	−5.37258	−0.174585	−0.0872927	−	0.996183i	$-0.527822\pi$
$947$	−5.37258	−0.174585	−0.0872927	+	0.996183i	$0.527822\pi$
$948$	0	0
$949$	−26.6274	−0.864363
$950$	0	0
$951$	−5.31371	−0.172309
$952$	0	0
$953$	49.6569	1.60854	0.804272	−	0.594262i	$-0.202556\pi$
$953$	49.6569	1.60854	0.804272	+	0.594262i	$0.202556\pi$
$954$	0	0
$955$	11.3137	0.366103
$956$	0	0
$957$	−4.97056	−0.160675
$958$	0	0
$959$	−85.2548	−2.75302
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−5.65685	−0.182290
$964$	0	0
$965$	6.00000	0.193147
$966$	0	0
$967$	−30.4853	−0.980341	−0.490170	−	0.871627i	$-0.663065\pi$
$967$	−30.4853	−0.980341	−0.490170	+	0.871627i	$0.663065\pi$
$968$	0	0
$969$	−38.6274	−1.24089
$970$	0	0
$971$	48.4264	1.55408	0.777039	−	0.629453i	$-0.216721\pi$
$971$	48.4264	1.55408	0.777039	+	0.629453i	$0.216721\pi$
$972$	0	0
$973$	40.9706	1.31346
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	−7.02944	−0.224892	−0.112446	−	0.993658i	$-0.535868\pi$
$977$	−7.02944	−0.224892	−0.112446	+	0.993658i	$0.535868\pi$
$978$	0	0
$979$	−0.284271	−0.00908535
$980$	0	0
$981$	12.0000	0.383131
$982$	0	0
$983$	32.4853	1.03612	0.518060	−	0.855344i	$-0.326654\pi$
$983$	32.4853	1.03612	0.518060	+	0.855344i	$0.326654\pi$
$984$	0	0
$985$	−25.3137	−0.806562
$986$	0	0
$987$	−24.9706	−0.794822
$988$	0	0
$989$	11.3137	0.359755
$990$	0	0
$991$	7.02944	0.223297	0.111649	−	0.993748i	$-0.464387\pi$
$991$	7.02944	0.223297	0.111649	+	0.993748i	$0.464387\pi$
$992$	0	0
$993$	−18.1421	−0.575723
$994$	0	0
$995$	−16.9706	−0.538003
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	0	0
$999$	7.65685	0.242252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.a.bh.1.2		2
4.3	odd	2		3840.2.a.bl.1.1		2
8.3	odd	2		3840.2.a.bc.1.1		2
8.5	even	2		3840.2.a.bk.1.2		2
16.3	odd	4		1920.2.k.l.961.4	yes	4
16.5	even	4		1920.2.k.i.961.3	yes	4
16.11	odd	4		1920.2.k.l.961.2	yes	4
16.13	even	4		1920.2.k.i.961.1	✓	4
48.5	odd	4		5760.2.k.n.2881.1		4
48.11	even	4		5760.2.k.w.2881.2		4
48.29	odd	4		5760.2.k.n.2881.3		4
48.35	even	4		5760.2.k.w.2881.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.k.i.961.1	✓	4	16.13	even	4
1920.2.k.i.961.3	yes	4	16.5	even	4
1920.2.k.l.961.2	yes	4	16.11	odd	4
1920.2.k.l.961.4	yes	4	16.3	odd	4
3840.2.a.bc.1.1		2	8.3	odd	2
3840.2.a.bh.1.2		2	1.1	even	1	trivial
3840.2.a.bk.1.2		2	8.5	even	2
3840.2.a.bl.1.1		2	4.3	odd	2
5760.2.k.n.2881.1		4	48.5	odd	4
5760.2.k.n.2881.3		4	48.29	odd	4
5760.2.k.w.2881.2		4	48.11	even	4
5760.2.k.w.2881.4		4	48.35	even	4

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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.82843 q^{7} +1.00000 q^{9} -0.828427 q^{11} +2.00000 q^{13} -1.00000 q^{15} +5.65685 q^{17} +6.82843 q^{19} -4.82843 q^{21} +1.17157 q^{23} +1.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +4.00000 q^{31} +0.828427 q^{33} +4.82843 q^{35} -7.65685 q^{37} -2.00000 q^{39} -3.65685 q^{41} +9.65685 q^{43} +1.00000 q^{45} +5.17157 q^{47} +16.3137 q^{49} -5.65685 q^{51} +9.31371 q^{53} -0.828427 q^{55} -6.82843 q^{57} -14.4853 q^{59} -4.00000 q^{61} +4.82843 q^{63} +2.00000 q^{65} -4.00000 q^{67} -1.17157 q^{69} -5.65685 q^{71} -13.3137 q^{73} -1.00000 q^{75} -4.00000 q^{77} +4.00000 q^{79} +1.00000 q^{81} -2.34315 q^{83} +5.65685 q^{85} +6.00000 q^{87} +0.343146 q^{89} +9.65685 q^{91} -4.00000 q^{93} +6.82843 q^{95} -14.0000 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{15} + 8 q^{19} - 4 q^{21} + 8 q^{23} + 2 q^{25} - 2 q^{27} - 12 q^{29} + 8 q^{31} - 4 q^{33} + 4 q^{35} - 4 q^{37} - 4 q^{39} + 4 q^{41}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 + 4 * q^11 + 4 * q^13 - 2 * q^15 + 8 * q^19 - 4 * q^21 + 8 * q^23 + 2 * q^25 - 2 * q^27 - 12 * q^29 + 8 * q^31 - 4 * q^33 + 4 * q^35 - 4 * q^37 - 4 * q^39 + 4 * q^41 + 8 * q^43 + 2 * q^45 + 16 * q^47 + 10 * q^49 - 4 * q^53 + 4 * q^55 - 8 * q^57 - 12 * q^59 - 8 * q^61 + 4 * q^63 + 4 * q^65 - 8 * q^67 - 8 * q^69 - 4 * q^73 - 2 * q^75 - 8 * q^77 + 8 * q^79 + 2 * q^81 - 16 * q^83 + 12 * q^87 + 12 * q^89 + 8 * q^91 - 8 * q^93 + 8 * q^95 - 28 * q^97 + 4 * q^99

Introduction

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