LMFDB - Embedded newform 4160.2.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4160,2,Mod(1,4160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4160, 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("4160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4160 = 2^{6} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4160.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:	$33.2177672409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.765367$ of defining polynomial
Character	$\chi$	$=$	4160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.08239 q^{3} +1.00000 q^{5} -1.53073 q^{7} -1.82843 q^{9} +O(q^{10})$	$q+1.08239 q^{3} +1.00000 q^{5} -1.53073 q^{7} -1.82843 q^{9} -2.61313 q^{11} +1.00000 q^{13} +1.08239 q^{15} +3.65685 q^{17} -2.61313 q^{19} -1.65685 q^{21} +8.47343 q^{23} +1.00000 q^{25} -5.22625 q^{27} +3.17157 q^{29} -5.67459 q^{31} -2.82843 q^{33} -1.53073 q^{35} +6.48528 q^{37} +1.08239 q^{39} -2.00000 q^{41} +9.37011 q^{43} -1.82843 q^{45} +4.59220 q^{47} -4.65685 q^{49} +3.95815 q^{51} +13.3137 q^{53} -2.61313 q^{55} -2.82843 q^{57} +6.94269 q^{59} +3.17157 q^{61} +2.79884 q^{63} +1.00000 q^{65} -9.81845 q^{67} +9.17157 q^{69} +1.71644 q^{71} -4.82843 q^{73} +1.08239 q^{75} +4.00000 q^{77} -9.55582 q^{79} -0.171573 q^{81} -1.53073 q^{83} +3.65685 q^{85} +3.43289 q^{87} -0.343146 q^{89} -1.53073 q^{91} -6.14214 q^{93} -2.61313 q^{95} +17.3137 q^{97} +4.77791 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 4 * q^9	$ 4 q + 4 q^{5} + 4 q^{9} + 4 q^{13} - 8 q^{17} + 16 q^{21} + 4 q^{25} + 24 q^{29} - 8 q^{37} - 8 q^{41} + 4 q^{45} + 4 q^{49} + 8 q^{53} + 24 q^{61} + 4 q^{65} + 48 q^{69} - 8 q^{73} + 16 q^{77} - 12 q^{81} - 8 q^{85} - 24 q^{89} + 32 q^{93} + 24 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^9 + 4 * q^13 - 8 * q^17 + 16 * q^21 + 4 * q^25 + 24 * q^29 - 8 * q^37 - 8 * q^41 + 4 * q^45 + 4 * q^49 + 8 * q^53 + 24 * q^61 + 4 * q^65 + 48 * q^69 - 8 * q^73 + 16 * q^77 - 12 * q^81 - 8 * q^85 - 24 * q^89 + 32 * q^93 + 24 * 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
$3$	1.08239	0.624919	0.312460	−	0.949931i	$-0.398847\pi$
$3$	1.08239	0.624919	0.312460	+	0.949931i	$0.398847\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.53073	−0.578563	−0.289281	−	0.957244i	$-0.593416\pi$
$7$	−1.53073	−0.578563	−0.289281	+	0.957244i	$0.593416\pi$
$8$	0	0
$9$	−1.82843	−0.609476
$10$	0	0
$11$	−2.61313	−0.787887	−0.393944	−	0.919135i	$-0.628889\pi$
$11$	−2.61313	−0.787887	−0.393944	+	0.919135i	$0.628889\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.08239	0.279472
$16$	0	0
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	−2.61313	−0.599492	−0.299746	−	0.954019i	$-0.596902\pi$
$19$	−2.61313	−0.599492	−0.299746	+	0.954019i	$0.596902\pi$
$20$	0	0
$21$	−1.65685	−0.361555
$22$	0	0
$23$	8.47343	1.76683	0.883416	−	0.468590i	$-0.155238\pi$
$23$	8.47343	1.76683	0.883416	+	0.468590i	$0.155238\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.22625	−1.00579
$28$	0	0
$29$	3.17157	0.588946	0.294473	−	0.955660i	$-0.404856\pi$
$29$	3.17157	0.588946	0.294473	+	0.955660i	$0.404856\pi$
$30$	0	0
$31$	−5.67459	−1.01919	−0.509594	−	0.860415i	$-0.670204\pi$
$31$	−5.67459	−1.01919	−0.509594	+	0.860415i	$0.670204\pi$
$32$	0	0
$33$	−2.82843	−0.492366
$34$	0	0
$35$	−1.53073	−0.258741
$36$	0	0
$37$	6.48528	1.06617	0.533087	−	0.846061i	$-0.321032\pi$
$37$	6.48528	1.06617	0.533087	+	0.846061i	$0.321032\pi$
$38$	0	0
$39$	1.08239	0.173321
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	9.37011	1.42893	0.714464	−	0.699672i	$-0.246670\pi$
$43$	9.37011	1.42893	0.714464	+	0.699672i	$0.246670\pi$
$44$	0	0
$45$	−1.82843	−0.272566
$46$	0	0
$47$	4.59220	0.669841	0.334921	−	0.942246i	$-0.391290\pi$
$47$	4.59220	0.669841	0.334921	+	0.942246i	$0.391290\pi$
$48$	0	0
$49$	−4.65685	−0.665265
$50$	0	0
$51$	3.95815	0.554252
$52$	0	0
$53$	13.3137	1.82878	0.914389	−	0.404836i	$-0.132671\pi$
$53$	13.3137	1.82878	0.914389	+	0.404836i	$0.132671\pi$
$54$	0	0
$55$	−2.61313	−0.352354
$56$	0	0
$57$	−2.82843	−0.374634
$58$	0	0
$59$	6.94269	0.903862	0.451931	−	0.892053i	$-0.350735\pi$
$59$	6.94269	0.903862	0.451931	+	0.892053i	$0.350735\pi$
$60$	0	0
$61$	3.17157	0.406078	0.203039	−	0.979171i	$-0.434918\pi$
$61$	3.17157	0.406078	0.203039	+	0.979171i	$0.434918\pi$
$62$	0	0
$63$	2.79884	0.352620
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−9.81845	−1.19951	−0.599757	−	0.800182i	$-0.704736\pi$
$67$	−9.81845	−1.19951	−0.599757	+	0.800182i	$0.704736\pi$
$68$	0	0
$69$	9.17157	1.10413
$70$	0	0
$71$	1.71644	0.203704	0.101852	−	0.994800i	$-0.467523\pi$
$71$	1.71644	0.203704	0.101852	+	0.994800i	$0.467523\pi$
$72$	0	0
$73$	−4.82843	−0.565125	−0.282562	−	0.959249i	$-0.591184\pi$
$73$	−4.82843	−0.565125	−0.282562	+	0.959249i	$0.591184\pi$
$74$	0	0
$75$	1.08239	0.124984
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	−9.55582	−1.07511	−0.537557	−	0.843227i	$-0.680653\pi$
$79$	−9.55582	−1.07511	−0.537557	+	0.843227i	$0.680653\pi$
$80$	0	0
$81$	−0.171573	−0.0190637
$82$	0	0
$83$	−1.53073	−0.168020	−0.0840099	−	0.996465i	$-0.526773\pi$
$83$	−1.53073	−0.168020	−0.0840099	+	0.996465i	$0.526773\pi$
$84$	0	0
$85$	3.65685	0.396642
$86$	0	0
$87$	3.43289	0.368044
$88$	0	0
$89$	−0.343146	−0.0363734	−0.0181867	−	0.999835i	$-0.505789\pi$
$89$	−0.343146	−0.0363734	−0.0181867	+	0.999835i	$0.505789\pi$
$90$	0	0
$91$	−1.53073	−0.160464
$92$	0	0
$93$	−6.14214	−0.636910
$94$	0	0
$95$	−2.61313	−0.268101
$96$	0	0
$97$	17.3137	1.75794	0.878970	−	0.476876i	$-0.158231\pi$
$97$	17.3137	1.75794	0.878970	+	0.476876i	$0.158231\pi$
$98$	0	0
$99$	4.77791	0.480198
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	11.5349	1.13657	0.568284	−	0.822833i	$-0.307608\pi$
$103$	11.5349	1.13657	0.568284	+	0.822833i	$0.307608\pi$
$104$	0	0
$105$	−1.65685	−0.161692
$106$	0	0
$107$	16.7611	1.62036	0.810181	−	0.586180i	$-0.199369\pi$
$107$	16.7611	1.62036	0.810181	+	0.586180i	$0.199369\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	7.01962	0.666273
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	8.47343	0.790151
$116$	0	0
$117$	−1.82843	−0.169038
$118$	0	0
$119$	−5.59767	−0.513138
$120$	0	0
$121$	−4.17157	−0.379234
$122$	0	0
$123$	−2.16478	−0.195192
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.37011	0.831463	0.415731	−	0.909487i	$-0.363526\pi$
$127$	9.37011	0.831463	0.415731	+	0.909487i	$0.363526\pi$
$128$	0	0
$129$	10.1421	0.892965
$130$	0	0
$131$	−1.26810	−0.110795	−0.0553973	−	0.998464i	$-0.517643\pi$
$131$	−1.26810	−0.110795	−0.0553973	+	0.998464i	$0.517643\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	−5.22625	−0.449804
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	20.0083	1.69708	0.848542	−	0.529128i	$-0.177481\pi$
$139$	20.0083	1.69708	0.848542	+	0.529128i	$0.177481\pi$
$140$	0	0
$141$	4.97056	0.418597
$142$	0	0
$143$	−2.61313	−0.218521
$144$	0	0
$145$	3.17157	0.263385
$146$	0	0
$147$	−5.04054	−0.415737
$148$	0	0
$149$	11.6569	0.954967	0.477483	−	0.878641i	$-0.341549\pi$
$149$	11.6569	0.954967	0.477483	+	0.878641i	$0.341549\pi$
$150$	0	0
$151$	−7.83938	−0.637960	−0.318980	−	0.947762i	$-0.603340\pi$
$151$	−7.83938	−0.637960	−0.318980	+	0.947762i	$0.603340\pi$
$152$	0	0
$153$	−6.68629	−0.540555
$154$	0	0
$155$	−5.67459	−0.455794
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	14.4107	1.14284
$160$	0	0
$161$	−12.9706	−1.02222
$162$	0	0
$163$	−11.0866	−0.868366	−0.434183	−	0.900825i	$-0.642963\pi$
$163$	−11.0866	−0.868366	−0.434183	+	0.900825i	$0.642963\pi$
$164$	0	0
$165$	−2.82843	−0.220193
$166$	0	0
$167$	−24.9719	−1.93239	−0.966194	−	0.257818i	$-0.916997\pi$
$167$	−24.9719	−1.93239	−0.966194	+	0.257818i	$0.916997\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.77791	0.365376
$172$	0	0
$173$	18.9706	1.44231	0.721153	−	0.692776i	$-0.243613\pi$
$173$	18.9706	1.44231	0.721153	+	0.692776i	$0.243613\pi$
$174$	0	0
$175$	−1.53073	−0.115713
$176$	0	0
$177$	7.51472	0.564841
$178$	0	0
$179$	−20.9050	−1.56251	−0.781257	−	0.624210i	$-0.785421\pi$
$179$	−20.9050	−1.56251	−0.781257	+	0.624210i	$0.785421\pi$
$180$	0	0
$181$	−8.14214	−0.605200	−0.302600	−	0.953118i	$-0.597855\pi$
$181$	−8.14214	−0.605200	−0.302600	+	0.953118i	$0.597855\pi$
$182$	0	0
$183$	3.43289	0.253766
$184$	0	0
$185$	6.48528	0.476807
$186$	0	0
$187$	−9.55582	−0.698791
$188$	0	0
$189$	8.00000	0.581914
$190$	0	0
$191$	13.5140	0.977837	0.488918	−	0.872330i	$-0.337392\pi$
$191$	13.5140	0.977837	0.488918	+	0.872330i	$0.337392\pi$
$192$	0	0
$193$	17.3137	1.24627	0.623134	−	0.782115i	$-0.285859\pi$
$193$	17.3137	1.24627	0.623134	+	0.782115i	$0.285859\pi$
$194$	0	0
$195$	1.08239	0.0775117
$196$	0	0
$197$	−13.3137	−0.948562	−0.474281	−	0.880373i	$-0.657292\pi$
$197$	−13.3137	−0.948562	−0.474281	+	0.880373i	$0.657292\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−10.6274	−0.749600
$202$	0	0
$203$	−4.85483	−0.340743
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	−15.4930	−1.07684
$208$	0	0
$209$	6.82843	0.472332
$210$	0	0
$211$	−13.5140	−0.930340	−0.465170	−	0.885221i	$-0.654007\pi$
$211$	−13.5140	−0.930340	−0.465170	+	0.885221i	$0.654007\pi$
$212$	0	0
$213$	1.85786	0.127299
$214$	0	0
$215$	9.37011	0.639036
$216$	0	0
$217$	8.68629	0.589664
$218$	0	0
$219$	−5.22625	−0.353157
$220$	0	0
$221$	3.65685	0.245987
$222$	0	0
$223$	−23.7038	−1.58733	−0.793663	−	0.608357i	$-0.791829\pi$
$223$	−23.7038	−1.58733	−0.793663	+	0.608357i	$0.791829\pi$
$224$	0	0
$225$	−1.82843	−0.121895
$226$	0	0
$227$	12.8799	0.854870	0.427435	−	0.904046i	$-0.359417\pi$
$227$	12.8799	0.854870	0.427435	+	0.904046i	$0.359417\pi$
$228$	0	0
$229$	24.6274	1.62743	0.813713	−	0.581267i	$-0.197443\pi$
$229$	24.6274	1.62743	0.813713	+	0.581267i	$0.197443\pi$
$230$	0	0
$231$	4.32957	0.284865
$232$	0	0
$233$	−17.3137	−1.13426	−0.567129	−	0.823629i	$-0.691946\pi$
$233$	−17.3137	−1.13426	−0.567129	+	0.823629i	$0.691946\pi$
$234$	0	0
$235$	4.59220	0.299562
$236$	0	0
$237$	−10.3431	−0.671860
$238$	0	0
$239$	−16.4985	−1.06720	−0.533600	−	0.845737i	$-0.679161\pi$
$239$	−16.4985	−1.06720	−0.533600	+	0.845737i	$0.679161\pi$
$240$	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
$243$	15.4930	0.993879
$244$	0	0
$245$	−4.65685	−0.297516
$246$	0	0
$247$	−2.61313	−0.166269
$248$	0	0
$249$	−1.65685	−0.104999
$250$	0	0
$251$	−2.16478	−0.136640	−0.0683200	−	0.997663i	$-0.521764\pi$
$251$	−2.16478	−0.136640	−0.0683200	+	0.997663i	$0.521764\pi$
$252$	0	0
$253$	−22.1421	−1.39206
$254$	0	0
$255$	3.95815	0.247869
$256$	0	0
$257$	−3.65685	−0.228108	−0.114054	−	0.993475i	$-0.536384\pi$
$257$	−3.65685	−0.228108	−0.114054	+	0.993475i	$0.536384\pi$
$258$	0	0
$259$	−9.92724	−0.616849
$260$	0	0
$261$	−5.79899	−0.358948
$262$	0	0
$263$	6.30864	0.389008	0.194504	−	0.980902i	$-0.437690\pi$
$263$	6.30864	0.389008	0.194504	+	0.980902i	$0.437690\pi$
$264$	0	0
$265$	13.3137	0.817855
$266$	0	0
$267$	−0.371418	−0.0227304
$268$	0	0
$269$	4.34315	0.264806	0.132403	−	0.991196i	$-0.457731\pi$
$269$	4.34315	0.264806	0.132403	+	0.991196i	$0.457731\pi$
$270$	0	0
$271$	−16.4985	−1.00221	−0.501107	−	0.865385i	$-0.667074\pi$
$271$	−16.4985	−1.00221	−0.501107	+	0.865385i	$0.667074\pi$
$272$	0	0
$273$	−1.65685	−0.100277
$274$	0	0
$275$	−2.61313	−0.157577
$276$	0	0
$277$	−14.9706	−0.899494	−0.449747	−	0.893156i	$-0.648486\pi$
$277$	−14.9706	−0.899494	−0.449747	+	0.893156i	$0.648486\pi$
$278$	0	0
$279$	10.3756	0.621170
$280$	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
$283$	3.24718	0.193025	0.0965123	−	0.995332i	$-0.469231\pi$
$283$	3.24718	0.193025	0.0965123	+	0.995332i	$0.469231\pi$
$284$	0	0
$285$	−2.82843	−0.167542
$286$	0	0
$287$	3.06147	0.180713
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	18.7402	1.09857
$292$	0	0
$293$	−2.48528	−0.145192	−0.0725958	−	0.997361i	$-0.523128\pi$
$293$	−2.48528	−0.145192	−0.0725958	+	0.997361i	$0.523128\pi$
$294$	0	0
$295$	6.94269	0.404219
$296$	0	0
$297$	13.6569	0.792451
$298$	0	0
$299$	8.47343	0.490031
$300$	0	0
$301$	−14.3431	−0.826725
$302$	0	0
$303$	10.8239	0.621818
$304$	0	0
$305$	3.17157	0.181604
$306$	0	0
$307$	16.3128	0.931021	0.465511	−	0.885042i	$-0.345871\pi$
$307$	16.3128	0.931021	0.465511	+	0.885042i	$0.345871\pi$
$308$	0	0
$309$	12.4853	0.710263
$310$	0	0
$311$	−12.6173	−0.715461	−0.357730	−	0.933825i	$-0.616449\pi$
$311$	−12.6173	−0.715461	−0.357730	+	0.933825i	$0.616449\pi$
$312$	0	0
$313$	0.343146	0.0193957	0.00969787	−	0.999953i	$-0.496913\pi$
$313$	0.343146	0.0193957	0.00969787	+	0.999953i	$0.496913\pi$
$314$	0	0
$315$	2.79884	0.157696
$316$	0	0
$317$	20.1421	1.13130	0.565648	−	0.824647i	$-0.308626\pi$
$317$	20.1421	1.13130	0.565648	+	0.824647i	$0.308626\pi$
$318$	0	0
$319$	−8.28772	−0.464023
$320$	0	0
$321$	18.1421	1.01260
$322$	0	0
$323$	−9.55582	−0.531700
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	15.1535	0.837990
$328$	0	0
$329$	−7.02944	−0.387545
$330$	0	0
$331$	−33.0740	−1.81791	−0.908954	−	0.416895i	$-0.863118\pi$
$331$	−33.0740	−1.81791	−0.908954	+	0.416895i	$0.863118\pi$
$332$	0	0
$333$	−11.8579	−0.649807
$334$	0	0
$335$	−9.81845	−0.536439
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	−2.16478	−0.117575
$340$	0	0
$341$	14.8284	0.803004
$342$	0	0
$343$	17.8435	0.963461
$344$	0	0
$345$	9.17157	0.493781
$346$	0	0
$347$	−36.7695	−1.97389	−0.986944	−	0.161062i	$-0.948508\pi$
$347$	−36.7695	−1.97389	−0.986944	+	0.161062i	$0.948508\pi$
$348$	0	0
$349$	1.02944	0.0551045	0.0275523	−	0.999620i	$-0.491229\pi$
$349$	1.02944	0.0551045	0.0275523	+	0.999620i	$0.491229\pi$
$350$	0	0
$351$	−5.22625	−0.278957
$352$	0	0
$353$	−10.4853	−0.558075	−0.279038	−	0.960280i	$-0.590015\pi$
$353$	−10.4853	−0.558075	−0.279038	+	0.960280i	$0.590015\pi$
$354$	0	0
$355$	1.71644	0.0910993
$356$	0	0
$357$	−6.05887	−0.320670
$358$	0	0
$359$	−10.0042	−0.527999	−0.264000	−	0.964523i	$-0.585042\pi$
$359$	−10.0042	−0.527999	−0.264000	+	0.964523i	$0.585042\pi$
$360$	0	0
$361$	−12.1716	−0.640609
$362$	0	0
$363$	−4.51528	−0.236991
$364$	0	0
$365$	−4.82843	−0.252731
$366$	0	0
$367$	−14.9678	−0.781312	−0.390656	−	0.920537i	$-0.627752\pi$
$367$	−14.9678	−0.781312	−0.390656	+	0.920537i	$0.627752\pi$
$368$	0	0
$369$	3.65685	0.190368
$370$	0	0
$371$	−20.3797	−1.05806
$372$	0	0
$373$	−6.68629	−0.346203	−0.173102	−	0.984904i	$-0.555379\pi$
$373$	−6.68629	−0.346203	−0.173102	+	0.984904i	$0.555379\pi$
$374$	0	0
$375$	1.08239	0.0558945
$376$	0	0
$377$	3.17157	0.163344
$378$	0	0
$379$	17.7666	0.912610	0.456305	−	0.889823i	$-0.349173\pi$
$379$	17.7666	0.912610	0.456305	+	0.889823i	$0.349173\pi$
$380$	0	0
$381$	10.1421	0.519597
$382$	0	0
$383$	14.5194	0.741909	0.370954	−	0.928651i	$-0.379031\pi$
$383$	14.5194	0.741909	0.370954	+	0.928651i	$0.379031\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	−17.1326	−0.870897
$388$	0	0
$389$	23.6569	1.19945	0.599725	−	0.800206i	$-0.295277\pi$
$389$	23.6569	1.19945	0.599725	+	0.800206i	$0.295277\pi$
$390$	0	0
$391$	30.9861	1.56703
$392$	0	0
$393$	−1.37258	−0.0692377
$394$	0	0
$395$	−9.55582	−0.480806
$396$	0	0
$397$	−27.4558	−1.37797	−0.688985	−	0.724776i	$-0.741943\pi$
$397$	−27.4558	−1.37797	−0.688985	+	0.724776i	$0.741943\pi$
$398$	0	0
$399$	4.32957	0.216750
$400$	0	0
$401$	26.9706	1.34685	0.673423	−	0.739258i	$-0.264823\pi$
$401$	26.9706	1.34685	0.673423	+	0.739258i	$0.264823\pi$
$402$	0	0
$403$	−5.67459	−0.282672
$404$	0	0
$405$	−0.171573	−0.00852552
$406$	0	0
$407$	−16.9469	−0.840025
$408$	0	0
$409$	33.3137	1.64726	0.823628	−	0.567130i	$-0.191946\pi$
$409$	33.3137	1.64726	0.823628	+	0.567130i	$0.191946\pi$
$410$	0	0
$411$	−2.16478	−0.106781
$412$	0	0
$413$	−10.6274	−0.522941
$414$	0	0
$415$	−1.53073	−0.0751408
$416$	0	0
$417$	21.6569	1.06054
$418$	0	0
$419$	21.8017	1.06508	0.532541	−	0.846404i	$-0.321237\pi$
$419$	21.8017	1.06508	0.532541	+	0.846404i	$0.321237\pi$
$420$	0	0
$421$	36.6274	1.78511	0.892556	−	0.450937i	$-0.148910\pi$
$421$	36.6274	1.78511	0.892556	+	0.450937i	$0.148910\pi$
$422$	0	0
$423$	−8.39651	−0.408252
$424$	0	0
$425$	3.65685	0.177383
$426$	0	0
$427$	−4.85483	−0.234942
$428$	0	0
$429$	−2.82843	−0.136558
$430$	0	0
$431$	−22.9929	−1.10753	−0.553764	−	0.832674i	$-0.686809\pi$
$431$	−22.9929	−1.10753	−0.553764	+	0.832674i	$0.686809\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	3.43289	0.164594
$436$	0	0
$437$	−22.1421	−1.05920
$438$	0	0
$439$	−20.9050	−0.997742	−0.498871	−	0.866676i	$-0.666252\pi$
$439$	−20.9050	−0.997742	−0.498871	+	0.866676i	$0.666252\pi$
$440$	0	0
$441$	8.51472	0.405463
$442$	0	0
$443$	21.9874	1.04465	0.522326	−	0.852746i	$-0.325064\pi$
$443$	21.9874	1.04465	0.522326	+	0.852746i	$0.325064\pi$
$444$	0	0
$445$	−0.343146	−0.0162667
$446$	0	0
$447$	12.6173	0.596777
$448$	0	0
$449$	−10.9706	−0.517733	−0.258866	−	0.965913i	$-0.583349\pi$
$449$	−10.9706	−0.517733	−0.258866	+	0.965913i	$0.583349\pi$
$450$	0	0
$451$	5.22625	0.246095
$452$	0	0
$453$	−8.48528	−0.398673
$454$	0	0
$455$	−1.53073	−0.0717619
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	0	0
$459$	−19.1116	−0.892055
$460$	0	0
$461$	−19.6569	−0.915511	−0.457755	−	0.889078i	$-0.651346\pi$
$461$	−19.6569	−0.915511	−0.457755	+	0.889078i	$0.651346\pi$
$462$	0	0
$463$	21.5391	1.00100	0.500502	−	0.865735i	$-0.333149\pi$
$463$	21.5391	1.00100	0.500502	+	0.865735i	$0.333149\pi$
$464$	0	0
$465$	−6.14214	−0.284835
$466$	0	0
$467$	−0.185709	−0.00859359	−0.00429680	−	0.999991i	$-0.501368\pi$
$467$	−0.185709	−0.00859359	−0.00429680	+	0.999991i	$0.501368\pi$
$468$	0	0
$469$	15.0294	0.693995
$470$	0	0
$471$	−2.16478	−0.0997480
$472$	0	0
$473$	−24.4853	−1.12583
$474$	0	0
$475$	−2.61313	−0.119898
$476$	0	0
$477$	−24.3431	−1.11460
$478$	0	0
$479$	34.3421	1.56913	0.784564	−	0.620047i	$-0.212887\pi$
$479$	34.3421	1.56913	0.784564	+	0.620047i	$0.212887\pi$
$480$	0	0
$481$	6.48528	0.295703
$482$	0	0
$483$	−14.0392	−0.638807
$484$	0	0
$485$	17.3137	0.786175
$486$	0	0
$487$	33.2597	1.50714	0.753570	−	0.657368i	$-0.228330\pi$
$487$	33.2597	1.50714	0.753570	+	0.657368i	$0.228330\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	8.28772	0.374020	0.187010	−	0.982358i	$-0.440120\pi$
$491$	8.28772	0.374020	0.187010	+	0.982358i	$0.440120\pi$
$492$	0	0
$493$	11.5980	0.522347
$494$	0	0
$495$	4.77791	0.214751
$496$	0	0
$497$	−2.62742	−0.117856
$498$	0	0
$499$	−6.94269	−0.310798	−0.155399	−	0.987852i	$-0.549666\pi$
$499$	−6.94269	−0.310798	−0.155399	+	0.987852i	$0.549666\pi$
$500$	0	0
$501$	−27.0294	−1.20759
$502$	0	0
$503$	−19.4512	−0.867286	−0.433643	−	0.901085i	$-0.642772\pi$
$503$	−19.4512	−0.867286	−0.433643	+	0.901085i	$0.642772\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	1.08239	0.0480707
$508$	0	0
$509$	7.65685	0.339384	0.169692	−	0.985497i	$-0.445723\pi$
$509$	7.65685	0.339384	0.169692	+	0.985497i	$0.445723\pi$
$510$	0	0
$511$	7.39104	0.326960
$512$	0	0
$513$	13.6569	0.602965
$514$	0	0
$515$	11.5349	0.508288
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	20.5336	0.901325
$520$	0	0
$521$	−8.14214	−0.356713	−0.178357	−	0.983966i	$-0.557078\pi$
$521$	−8.14214	−0.356713	−0.178357	+	0.983966i	$0.557078\pi$
$522$	0	0
$523$	28.8532	1.26166	0.630831	−	0.775921i	$-0.282714\pi$
$523$	28.8532	1.26166	0.630831	+	0.775921i	$0.282714\pi$
$524$	0	0
$525$	−1.65685	−0.0723110
$526$	0	0
$527$	−20.7512	−0.903935
$528$	0	0
$529$	48.7990	2.12170
$530$	0	0
$531$	−12.6942	−0.550882
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	16.7611	0.724648
$536$	0	0
$537$	−22.6274	−0.976445
$538$	0	0
$539$	12.1689	0.524154
$540$	0	0
$541$	−17.3137	−0.744374	−0.372187	−	0.928158i	$-0.621392\pi$
$541$	−17.3137	−0.744374	−0.372187	+	0.928158i	$0.621392\pi$
$542$	0	0
$543$	−8.81298	−0.378201
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	29.0070	1.24025	0.620125	−	0.784503i	$-0.287082\pi$
$547$	29.0070	1.24025	0.620125	+	0.784503i	$0.287082\pi$
$548$	0	0
$549$	−5.79899	−0.247495
$550$	0	0
$551$	−8.28772	−0.353069
$552$	0	0
$553$	14.6274	0.622021
$554$	0	0
$555$	7.01962	0.297966
$556$	0	0
$557$	−40.1421	−1.70088	−0.850438	−	0.526075i	$-0.823663\pi$
$557$	−40.1421	−1.70088	−0.850438	+	0.526075i	$0.823663\pi$
$558$	0	0
$559$	9.37011	0.396313
$560$	0	0
$561$	−10.3431	−0.436688
$562$	0	0
$563$	22.3588	0.942312	0.471156	−	0.882050i	$-0.343837\pi$
$563$	22.3588	0.942312	0.471156	+	0.882050i	$0.343837\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	0.262632	0.0110295
$568$	0	0
$569$	−37.7990	−1.58462	−0.792308	−	0.610121i	$-0.791121\pi$
$569$	−37.7990	−1.58462	−0.792308	+	0.610121i	$0.791121\pi$
$570$	0	0
$571$	−21.2764	−0.890391	−0.445195	−	0.895433i	$-0.646866\pi$
$571$	−21.2764	−0.890391	−0.445195	+	0.895433i	$0.646866\pi$
$572$	0	0
$573$	14.6274	0.611069
$574$	0	0
$575$	8.47343	0.353366
$576$	0	0
$577$	25.7990	1.07403	0.537013	−	0.843574i	$-0.319553\pi$
$577$	25.7990	1.07403	0.537013	+	0.843574i	$0.319553\pi$
$578$	0	0
$579$	18.7402	0.778817
$580$	0	0
$581$	2.34315	0.0972101
$582$	0	0
$583$	−34.7904	−1.44087
$584$	0	0
$585$	−1.82843	−0.0755962
$586$	0	0
$587$	4.22078	0.174210	0.0871052	−	0.996199i	$-0.472238\pi$
$587$	4.22078	0.174210	0.0871052	+	0.996199i	$0.472238\pi$
$588$	0	0
$589$	14.8284	0.610995
$590$	0	0
$591$	−14.4107	−0.592775
$592$	0	0
$593$	−22.6863	−0.931614	−0.465807	−	0.884886i	$-0.654236\pi$
$593$	−22.6863	−0.931614	−0.465807	+	0.884886i	$0.654236\pi$
$594$	0	0
$595$	−5.59767	−0.229482
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11.3492	−0.463715	−0.231858	−	0.972750i	$-0.574480\pi$
$599$	−11.3492	−0.463715	−0.231858	+	0.972750i	$0.574480\pi$
$600$	0	0
$601$	−22.9706	−0.936989	−0.468494	−	0.883466i	$-0.655203\pi$
$601$	−22.9706	−0.936989	−0.468494	+	0.883466i	$0.655203\pi$
$602$	0	0
$603$	17.9523	0.731075
$604$	0	0
$605$	−4.17157	−0.169599
$606$	0	0
$607$	−13.6997	−0.556053	−0.278026	−	0.960573i	$-0.589680\pi$
$607$	−13.6997	−0.556053	−0.278026	+	0.960573i	$0.589680\pi$
$608$	0	0
$609$	−5.25483	−0.212937
$610$	0	0
$611$	4.59220	0.185781
$612$	0	0
$613$	−14.6863	−0.593174	−0.296587	−	0.955006i	$-0.595848\pi$
$613$	−14.6863	−0.593174	−0.296587	+	0.955006i	$0.595848\pi$
$614$	0	0
$615$	−2.16478	−0.0872925
$616$	0	0
$617$	31.9411	1.28590	0.642951	−	0.765908i	$-0.277710\pi$
$617$	31.9411	1.28590	0.642951	+	0.765908i	$0.277710\pi$
$618$	0	0
$619$	−26.9510	−1.08325	−0.541626	−	0.840619i	$-0.682191\pi$
$619$	−26.9510	−1.08325	−0.541626	+	0.840619i	$0.682191\pi$
$620$	0	0
$621$	−44.2843	−1.77707
$622$	0	0
$623$	0.525265	0.0210443
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.39104	0.295170
$628$	0	0
$629$	23.7157	0.945608
$630$	0	0
$631$	−31.8059	−1.26617	−0.633086	−	0.774082i	$-0.718212\pi$
$631$	−31.8059	−1.26617	−0.633086	+	0.774082i	$0.718212\pi$
$632$	0	0
$633$	−14.6274	−0.581388
$634$	0	0
$635$	9.37011	0.371842
$636$	0	0
$637$	−4.65685	−0.184511
$638$	0	0
$639$	−3.13839	−0.124153
$640$	0	0
$641$	−32.3431	−1.27748	−0.638739	−	0.769424i	$-0.720544\pi$
$641$	−32.3431	−1.27748	−0.638739	+	0.769424i	$0.720544\pi$
$642$	0	0
$643$	−11.9832	−0.472573	−0.236286	−	0.971683i	$-0.575930\pi$
$643$	−11.9832	−0.472573	−0.236286	+	0.971683i	$0.575930\pi$
$644$	0	0
$645$	10.1421	0.399346
$646$	0	0
$647$	16.2359	0.638298	0.319149	−	0.947704i	$-0.396603\pi$
$647$	16.2359	0.638298	0.319149	+	0.947704i	$0.396603\pi$
$648$	0	0
$649$	−18.1421	−0.712141
$650$	0	0
$651$	9.40197	0.368492
$652$	0	0
$653$	−38.2843	−1.49818	−0.749090	−	0.662469i	$-0.769509\pi$
$653$	−38.2843	−1.49818	−0.749090	+	0.662469i	$0.769509\pi$
$654$	0	0
$655$	−1.26810	−0.0495488
$656$	0	0
$657$	8.82843	0.344430
$658$	0	0
$659$	−21.8017	−0.849273	−0.424637	−	0.905364i	$-0.639598\pi$
$659$	−21.8017	−0.849273	−0.424637	+	0.905364i	$0.639598\pi$
$660$	0	0
$661$	−6.68629	−0.260067	−0.130033	−	0.991510i	$-0.541508\pi$
$661$	−6.68629	−0.260067	−0.130033	+	0.991510i	$0.541508\pi$
$662$	0	0
$663$	3.95815	0.153722
$664$	0	0
$665$	4.00000	0.155113
$666$	0	0
$667$	26.8741	1.04057
$668$	0	0
$669$	−25.6569	−0.991951
$670$	0	0
$671$	−8.28772	−0.319944
$672$	0	0
$673$	−26.9706	−1.03964	−0.519819	−	0.854276i	$-0.674001\pi$
$673$	−26.9706	−1.03964	−0.519819	+	0.854276i	$0.674001\pi$
$674$	0	0
$675$	−5.22625	−0.201159
$676$	0	0
$677$	−4.62742	−0.177846	−0.0889230	−	0.996038i	$-0.528343\pi$
$677$	−4.62742	−0.177846	−0.0889230	+	0.996038i	$0.528343\pi$
$678$	0	0
$679$	−26.5027	−1.01708
$680$	0	0
$681$	13.9411	0.534225
$682$	0	0
$683$	−9.29319	−0.355594	−0.177797	−	0.984067i	$-0.556897\pi$
$683$	−9.29319	−0.355594	−0.177797	+	0.984067i	$0.556897\pi$
$684$	0	0
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	26.6565	1.01701
$688$	0	0
$689$	13.3137	0.507212
$690$	0	0
$691$	29.6411	1.12760	0.563800	−	0.825912i	$-0.309339\pi$
$691$	29.6411	1.12760	0.563800	+	0.825912i	$0.309339\pi$
$692$	0	0
$693$	−7.31371	−0.277825
$694$	0	0
$695$	20.0083	0.758959
$696$	0	0
$697$	−7.31371	−0.277026
$698$	0	0
$699$	−18.7402	−0.708820
$700$	0	0
$701$	−16.3431	−0.617272	−0.308636	−	0.951180i	$-0.599873\pi$
$701$	−16.3431	−0.617272	−0.308636	+	0.951180i	$0.599873\pi$
$702$	0	0
$703$	−16.9469	−0.639163
$704$	0	0
$705$	4.97056	0.187202
$706$	0	0
$707$	−15.3073	−0.575692
$708$	0	0
$709$	5.31371	0.199561	0.0997803	−	0.995009i	$-0.468186\pi$
$709$	5.31371	0.199561	0.0997803	+	0.995009i	$0.468186\pi$
$710$	0	0
$711$	17.4721	0.655256
$712$	0	0
$713$	−48.0833	−1.80073
$714$	0	0
$715$	−2.61313	−0.0977254
$716$	0	0
$717$	−17.8579	−0.666914
$718$	0	0
$719$	32.1003	1.19714	0.598570	−	0.801070i	$-0.295736\pi$
$719$	32.1003	1.19714	0.598570	+	0.801070i	$0.295736\pi$
$720$	0	0
$721$	−17.6569	−0.657576
$722$	0	0
$723$	6.49435	0.241528
$724$	0	0
$725$	3.17157	0.117789
$726$	0	0
$727$	14.2249	0.527574	0.263787	−	0.964581i	$-0.415028\pi$
$727$	14.2249	0.527574	0.263787	+	0.964581i	$0.415028\pi$
$728$	0	0
$729$	17.2843	0.640158
$730$	0	0
$731$	34.2651	1.26734
$732$	0	0
$733$	25.3137	0.934983	0.467492	−	0.883998i	$-0.345158\pi$
$733$	25.3137	0.934983	0.467492	+	0.883998i	$0.345158\pi$
$734$	0	0
$735$	−5.04054	−0.185923
$736$	0	0
$737$	25.6569	0.945082
$738$	0	0
$739$	−11.6437	−0.428320	−0.214160	−	0.976799i	$-0.568701\pi$
$739$	−11.6437	−0.428320	−0.214160	+	0.976799i	$0.568701\pi$
$740$	0	0
$741$	−2.82843	−0.103905
$742$	0	0
$743$	−17.5809	−0.644981	−0.322490	−	0.946573i	$-0.604520\pi$
$743$	−17.5809	−0.644981	−0.322490	+	0.946573i	$0.604520\pi$
$744$	0	0
$745$	11.6569	0.427074
$746$	0	0
$747$	2.79884	0.102404
$748$	0	0
$749$	−25.6569	−0.937481
$750$	0	0
$751$	−3.95815	−0.144435	−0.0722175	−	0.997389i	$-0.523008\pi$
$751$	−3.95815	−0.144435	−0.0722175	+	0.997389i	$0.523008\pi$
$752$	0	0
$753$	−2.34315	−0.0853890
$754$	0	0
$755$	−7.83938	−0.285304
$756$	0	0
$757$	−0.343146	−0.0124718	−0.00623592	−	0.999981i	$-0.501985\pi$
$757$	−0.343146	−0.0124718	−0.00623592	+	0.999981i	$0.501985\pi$
$758$	0	0
$759$	−23.9665	−0.869928
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	−21.4303	−0.775828
$764$	0	0
$765$	−6.68629	−0.241743
$766$	0	0
$767$	6.94269	0.250686
$768$	0	0
$769$	−2.68629	−0.0968701	−0.0484351	−	0.998826i	$-0.515423\pi$
$769$	−2.68629	−0.0968701	−0.0484351	+	0.998826i	$0.515423\pi$
$770$	0	0
$771$	−3.95815	−0.142549
$772$	0	0
$773$	3.17157	0.114074	0.0570368	−	0.998372i	$-0.481835\pi$
$773$	3.17157	0.114074	0.0570368	+	0.998372i	$0.481835\pi$
$774$	0	0
$775$	−5.67459	−0.203837
$776$	0	0
$777$	−10.7452	−0.385481
$778$	0	0
$779$	5.22625	0.187250
$780$	0	0
$781$	−4.48528	−0.160496
$782$	0	0
$783$	−16.5754	−0.592358
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	7.65367	0.272824	0.136412	−	0.990652i	$-0.456443\pi$
$787$	7.65367	0.272824	0.136412	+	0.990652i	$0.456443\pi$
$788$	0	0
$789$	6.82843	0.243098
$790$	0	0
$791$	3.06147	0.108853
$792$	0	0
$793$	3.17157	0.112626
$794$	0	0
$795$	14.4107	0.511093
$796$	0	0
$797$	9.31371	0.329908	0.164954	−	0.986301i	$-0.447252\pi$
$797$	9.31371	0.329908	0.164954	+	0.986301i	$0.447252\pi$
$798$	0	0
$799$	16.7930	0.594094
$800$	0	0
$801$	0.627417	0.0221687
$802$	0	0
$803$	12.6173	0.445254
$804$	0	0
$805$	−12.9706	−0.457152
$806$	0	0
$807$	4.70099	0.165483
$808$	0	0
$809$	−18.4853	−0.649908	−0.324954	−	0.945730i	$-0.605349\pi$
$809$	−18.4853	−0.649908	−0.324954	+	0.945730i	$0.605349\pi$
$810$	0	0
$811$	−26.9510	−0.946378	−0.473189	−	0.880961i	$-0.656897\pi$
$811$	−26.9510	−0.946378	−0.473189	+	0.880961i	$0.656897\pi$
$812$	0	0
$813$	−17.8579	−0.626303
$814$	0	0
$815$	−11.0866	−0.388345
$816$	0	0
$817$	−24.4853	−0.856632
$818$	0	0
$819$	2.79884	0.0977992
$820$	0	0
$821$	18.2843	0.638125	0.319063	−	0.947734i	$-0.396632\pi$
$821$	18.2843	0.638125	0.319063	+	0.947734i	$0.396632\pi$
$822$	0	0
$823$	49.0153	1.70857	0.854284	−	0.519807i	$-0.173996\pi$
$823$	49.0153	1.70857	0.854284	+	0.519807i	$0.173996\pi$
$824$	0	0
$825$	−2.82843	−0.0984732
$826$	0	0
$827$	31.0949	1.08127	0.540637	−	0.841256i	$-0.318183\pi$
$827$	31.0949	1.08127	0.540637	+	0.841256i	$0.318183\pi$
$828$	0	0
$829$	45.1127	1.56683	0.783414	−	0.621500i	$-0.213476\pi$
$829$	45.1127	1.56683	0.783414	+	0.621500i	$0.213476\pi$
$830$	0	0
$831$	−16.2040	−0.562111
$832$	0	0
$833$	−17.0294	−0.590035
$834$	0	0
$835$	−24.9719	−0.864190
$836$	0	0
$837$	29.6569	1.02509
$838$	0	0
$839$	28.7444	0.992366	0.496183	−	0.868218i	$-0.334734\pi$
$839$	28.7444	0.992366	0.496183	+	0.868218i	$0.334734\pi$
$840$	0	0
$841$	−18.9411	−0.653142
$842$	0	0
$843$	13.6678	0.470745
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	6.38557	0.219411
$848$	0	0
$849$	3.51472	0.120625
$850$	0	0
$851$	54.9526	1.88375
$852$	0	0
$853$	−33.1127	−1.13376	−0.566879	−	0.823801i	$-0.691849\pi$
$853$	−33.1127	−1.13376	−0.566879	+	0.823801i	$0.691849\pi$
$854$	0	0
$855$	4.77791	0.163401
$856$	0	0
$857$	−24.3431	−0.831546	−0.415773	−	0.909468i	$-0.636489\pi$
$857$	−24.3431	−0.831546	−0.415773	+	0.909468i	$0.636489\pi$
$858$	0	0
$859$	−17.4721	−0.596141	−0.298071	−	0.954544i	$-0.596343\pi$
$859$	−17.4721	−0.596141	−0.298071	+	0.954544i	$0.596343\pi$
$860$	0	0
$861$	3.31371	0.112931
$862$	0	0
$863$	22.4357	0.763722	0.381861	−	0.924220i	$-0.375283\pi$
$863$	22.4357	0.763722	0.381861	+	0.924220i	$0.375283\pi$
$864$	0	0
$865$	18.9706	0.645018
$866$	0	0
$867$	−3.92629	−0.133344
$868$	0	0
$869$	24.9706	0.847068
$870$	0	0
$871$	−9.81845	−0.332686
$872$	0	0
$873$	−31.6569	−1.07142
$874$	0	0
$875$	−1.53073	−0.0517482
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	0	0
$879$	−2.69005	−0.0907331
$880$	0	0
$881$	−52.8284	−1.77983	−0.889917	−	0.456122i	$-0.849238\pi$
$881$	−52.8284	−1.77983	−0.889917	+	0.456122i	$0.849238\pi$
$882$	0	0
$883$	−25.5741	−0.860638	−0.430319	−	0.902677i	$-0.641599\pi$
$883$	−25.5741	−0.860638	−0.430319	+	0.902677i	$0.641599\pi$
$884$	0	0
$885$	7.51472	0.252605
$886$	0	0
$887$	48.4901	1.62814	0.814069	−	0.580769i	$-0.197248\pi$
$887$	48.4901	1.62814	0.814069	+	0.580769i	$0.197248\pi$
$888$	0	0
$889$	−14.3431	−0.481054
$890$	0	0
$891$	0.448342	0.0150200
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	−20.9050	−0.698777
$896$	0	0
$897$	9.17157	0.306230
$898$	0	0
$899$	−17.9974	−0.600246
$900$	0	0
$901$	48.6863	1.62198
$902$	0	0
$903$	−15.5249	−0.516637
$904$	0	0
$905$	−8.14214	−0.270654
$906$	0	0
$907$	5.93723	0.197142	0.0985712	−	0.995130i	$-0.468573\pi$
$907$	5.93723	0.197142	0.0985712	+	0.995130i	$0.468573\pi$
$908$	0	0
$909$	−18.2843	−0.606451
$910$	0	0
$911$	26.5027	0.878073	0.439036	−	0.898469i	$-0.355320\pi$
$911$	26.5027	0.878073	0.439036	+	0.898469i	$0.355320\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	0	0
$915$	3.43289	0.113488
$916$	0	0
$917$	1.94113	0.0641016
$918$	0	0
$919$	36.5838	1.20679	0.603393	−	0.797444i	$-0.293815\pi$
$919$	36.5838	1.20679	0.603393	+	0.797444i	$0.293815\pi$
$920$	0	0
$921$	17.6569	0.581813
$922$	0	0
$923$	1.71644	0.0564974
$924$	0	0
$925$	6.48528	0.213235
$926$	0	0
$927$	−21.0907	−0.692710
$928$	0	0
$929$	−26.9706	−0.884875	−0.442438	−	0.896799i	$-0.645886\pi$
$929$	−26.9706	−0.884875	−0.442438	+	0.896799i	$0.645886\pi$
$930$	0	0
$931$	12.1689	0.398821
$932$	0	0
$933$	−13.6569	−0.447105
$934$	0	0
$935$	−9.55582	−0.312509
$936$	0	0
$937$	−43.2548	−1.41307	−0.706537	−	0.707676i	$-0.749744\pi$
$937$	−43.2548	−1.41307	−0.706537	+	0.707676i	$0.749744\pi$
$938$	0	0
$939$	0.371418	0.0121208
$940$	0	0
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	0	0
$943$	−16.9469	−0.551866
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	−10.7151	−0.348195	−0.174098	−	0.984728i	$-0.555701\pi$
$947$	−10.7151	−0.348195	−0.174098	+	0.984728i	$0.555701\pi$
$948$	0	0
$949$	−4.82843	−0.156737
$950$	0	0
$951$	21.8017	0.706968
$952$	0	0
$953$	−59.6569	−1.93248	−0.966238	−	0.257653i	$-0.917051\pi$
$953$	−59.6569	−1.93248	−0.966238	+	0.257653i	$0.917051\pi$
$954$	0	0
$955$	13.5140	0.437302
$956$	0	0
$957$	−8.97056	−0.289977
$958$	0	0
$959$	3.06147	0.0988599
$960$	0	0
$961$	1.20101	0.0387423
$962$	0	0
$963$	−30.6465	−0.987571
$964$	0	0
$965$	17.3137	0.557348
$966$	0	0
$967$	−33.7849	−1.08645	−0.543225	−	0.839587i	$-0.682797\pi$
$967$	−33.7849	−1.08645	−0.543225	+	0.839587i	$0.682797\pi$
$968$	0	0
$969$	−10.3431	−0.332270
$970$	0	0
$971$	25.7598	0.826673	0.413336	−	0.910578i	$-0.364363\pi$
$971$	25.7598	0.826673	0.413336	+	0.910578i	$0.364363\pi$
$972$	0	0
$973$	−30.6274	−0.981870
$974$	0	0
$975$	1.08239	0.0346643
$976$	0	0
$977$	−24.1421	−0.772375	−0.386188	−	0.922420i	$-0.626208\pi$
$977$	−24.1421	−0.772375	−0.386188	+	0.922420i	$0.626208\pi$
$978$	0	0
$979$	0.896683	0.0286581
$980$	0	0
$981$	−25.5980	−0.817281
$982$	0	0
$983$	−1.15932	−0.0369764	−0.0184882	−	0.999829i	$-0.505885\pi$
$983$	−1.15932	−0.0369764	−0.0184882	+	0.999829i	$0.505885\pi$
$984$	0	0
$985$	−13.3137	−0.424210
$986$	0	0
$987$	−7.60861	−0.242185
$988$	0	0
$989$	79.3970	2.52468
$990$	0	0
$991$	43.8210	1.39202	0.696009	−	0.718033i	$-0.254957\pi$
$991$	43.8210	1.39202	0.696009	+	0.718033i	$0.254957\pi$
$992$	0	0
$993$	−35.7990	−1.13605
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	0	0
$999$	−33.8937	−1.07235

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4160.2.a.bu.1.3		4
4.3	odd	2	inner	4160.2.a.bu.1.2		4
8.3	odd	2		2080.2.a.q.1.3	yes	4
8.5	even	2		2080.2.a.q.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2080.2.a.q.1.2	✓	4	8.5	even	2
2080.2.a.q.1.3	yes	4	8.3	odd	2
4160.2.a.bu.1.2		4	4.3	odd	2	inner
4160.2.a.bu.1.3		4	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:	\( N \)	\(=\)	\( 4160 = 2^{6} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4160.a (trivial)

\(f(q)\)	\(=\)	\(q+1.08239 q^{3} +1.00000 q^{5} -1.53073 q^{7} -1.82843 q^{9} +O(q^{10})\)	\(q+1.08239 q^{3} +1.00000 q^{5} -1.53073 q^{7} -1.82843 q^{9} -2.61313 q^{11} +1.00000 q^{13} +1.08239 q^{15} +3.65685 q^{17} -2.61313 q^{19} -1.65685 q^{21} +8.47343 q^{23} +1.00000 q^{25} -5.22625 q^{27} +3.17157 q^{29} -5.67459 q^{31} -2.82843 q^{33} -1.53073 q^{35} +6.48528 q^{37} +1.08239 q^{39} -2.00000 q^{41} +9.37011 q^{43} -1.82843 q^{45} +4.59220 q^{47} -4.65685 q^{49} +3.95815 q^{51} +13.3137 q^{53} -2.61313 q^{55} -2.82843 q^{57} +6.94269 q^{59} +3.17157 q^{61} +2.79884 q^{63} +1.00000 q^{65} -9.81845 q^{67} +9.17157 q^{69} +1.71644 q^{71} -4.82843 q^{73} +1.08239 q^{75} +4.00000 q^{77} -9.55582 q^{79} -0.171573 q^{81} -1.53073 q^{83} +3.65685 q^{85} +3.43289 q^{87} -0.343146 q^{89} -1.53073 q^{91} -6.14214 q^{93} -2.61313 q^{95} +17.3137 q^{97} +4.77791 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 4 * q^9	\( 4 q + 4 q^{5} + 4 q^{9} + 4 q^{13} - 8 q^{17} + 16 q^{21} + 4 q^{25} + 24 q^{29} - 8 q^{37} - 8 q^{41} + 4 q^{45} + 4 q^{49} + 8 q^{53} + 24 q^{61} + 4 q^{65} + 48 q^{69} - 8 q^{73} + 16 q^{77} - 12 q^{81} - 8 q^{85} - 24 q^{89} + 32 q^{93} + 24 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^9 + 4 * q^13 - 8 * q^17 + 16 * q^21 + 4 * q^25 + 24 * q^29 - 8 * q^37 - 8 * q^41 + 4 * q^45 + 4 * q^49 + 8 * q^53 + 24 * q^61 + 4 * q^65 + 48 * q^69 - 8 * q^73 + 16 * q^77 - 12 * q^81 - 8 * q^85 - 24 * q^89 + 32 * q^93 + 24 * q^97

Introduction

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Database

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