LMFDB - Embedded newform 3072.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.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:	$24.5300435009$
Analytic rank:	$1$
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 $
Twist minimal:	no (minimal twist has level 1536)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.765367$ of defining polynomial
Character	$\chi$	$=$	3072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.02734 q^{5} -4.61313 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.02734 q^{5} -4.61313 q^{7} +1.00000 q^{9} +1.53073 q^{11} +5.57900 q^{13} +4.02734 q^{15} -1.29769 q^{17} +4.35916 q^{19} +4.61313 q^{21} -4.00000 q^{23} +11.2195 q^{25} -1.00000 q^{27} +1.86256 q^{29} +2.77791 q^{31} -1.53073 q^{33} +18.5786 q^{35} -3.97682 q^{37} -5.57900 q^{39} -3.03188 q^{41} +3.03188 q^{43} -4.02734 q^{45} -9.65685 q^{47} +14.2809 q^{49} +1.29769 q^{51} +8.58995 q^{53} -6.16478 q^{55} -4.35916 q^{57} +5.73418 q^{59} +7.64725 q^{61} -4.61313 q^{63} -22.4685 q^{65} -8.79565 q^{67} +4.00000 q^{69} +6.88311 q^{71} -3.50114 q^{73} -11.2195 q^{75} -7.06147 q^{77} -8.18252 q^{79} +1.00000 q^{81} +4.12612 q^{83} +5.22625 q^{85} -1.86256 q^{87} +7.98642 q^{89} -25.7366 q^{91} -2.77791 q^{93} -17.5558 q^{95} -1.40461 q^{97} +1.53073 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9} + 8 q^{13} + 8 q^{21} - 16 q^{23} + 4 q^{25} - 4 q^{27} - 8 q^{31} + 16 q^{35} + 8 q^{37} - 8 q^{39} - 16 q^{47} + 4 q^{49} - 16 q^{55} + 16 q^{59} + 24 q^{61} - 8 q^{63} - 8 q^{65} - 16 q^{67} + 16 q^{69} - 16 q^{71} - 8 q^{73} - 4 q^{75} - 16 q^{77} - 24 q^{79} + 4 q^{81} - 8 q^{89} - 16 q^{91} + 8 q^{93} - 32 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9 + 8 * q^13 + 8 * q^21 - 16 * q^23 + 4 * q^25 - 4 * q^27 - 8 * q^31 + 16 * q^35 + 8 * q^37 - 8 * q^39 - 16 * q^47 + 4 * q^49 - 16 * q^55 + 16 * q^59 + 24 * q^61 - 8 * q^63 - 8 * q^65 - 16 * q^67 + 16 * q^69 - 16 * q^71 - 8 * q^73 - 4 * q^75 - 16 * q^77 - 24 * q^79 + 4 * q^81 - 8 * q^89 - 16 * q^91 + 8 * q^93 - 32 * q^95 - 16 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−4.02734	−1.80108	−0.900540	−	0.434772i	$-0.856829\pi$
$5$	−4.02734	−1.80108	−0.900540	+	0.434772i	$0.856829\pi$
$6$	0	0
$7$	−4.61313	−1.74360	−0.871799	−	0.489864i	$-0.837046\pi$
$7$	−4.61313	−1.74360	−0.871799	+	0.489864i	$0.837046\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.53073	0.461534	0.230767	−	0.973009i	$-0.425877\pi$
$11$	1.53073	0.461534	0.230767	+	0.973009i	$0.425877\pi$
$12$	0	0
$13$	5.57900	1.54734	0.773668	−	0.633591i	$-0.218420\pi$
$13$	5.57900	1.54734	0.773668	+	0.633591i	$0.218420\pi$
$14$	0	0
$15$	4.02734	1.03985
$16$	0	0
$17$	−1.29769	−0.314737	−0.157368	−	0.987540i	$-0.550301\pi$
$17$	−1.29769	−0.314737	−0.157368	+	0.987540i	$0.550301\pi$
$18$	0	0
$19$	4.35916	1.00006	0.500030	−	0.866008i	$-0.333322\pi$
$19$	4.35916	1.00006	0.500030	+	0.866008i	$0.333322\pi$
$20$	0	0
$21$	4.61313	1.00667
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	11.2195	2.24389
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.86256	0.345868	0.172934	−	0.984933i	$-0.444675\pi$
$29$	1.86256	0.345868	0.172934	+	0.984933i	$0.444675\pi$
$30$	0	0
$31$	2.77791	0.498927	0.249464	−	0.968384i	$-0.419746\pi$
$31$	2.77791	0.498927	0.249464	+	0.968384i	$0.419746\pi$
$32$	0	0
$33$	−1.53073	−0.266467
$34$	0	0
$35$	18.5786	3.14036
$36$	0	0
$37$	−3.97682	−0.653786	−0.326893	−	0.945061i	$-0.606002\pi$
$37$	−3.97682	−0.653786	−0.326893	+	0.945061i	$0.606002\pi$
$38$	0	0
$39$	−5.57900	−0.893355
$40$	0	0
$41$	−3.03188	−0.473499	−0.236750	−	0.971571i	$-0.576082\pi$
$41$	−3.03188	−0.473499	−0.236750	+	0.971571i	$0.576082\pi$
$42$	0	0
$43$	3.03188	0.462357	0.231178	−	0.972911i	$-0.425742\pi$
$43$	3.03188	0.462357	0.231178	+	0.972911i	$0.425742\pi$
$44$	0	0
$45$	−4.02734	−0.600360
$46$	0	0
$47$	−9.65685	−1.40860	−0.704298	−	0.709904i	$-0.748738\pi$
$47$	−9.65685	−1.40860	−0.704298	+	0.709904i	$0.748738\pi$
$48$	0	0
$49$	14.2809	2.04013
$50$	0	0
$51$	1.29769	0.181713
$52$	0	0
$53$	8.58995	1.17992	0.589960	−	0.807432i	$-0.299143\pi$
$53$	8.58995	1.17992	0.589960	+	0.807432i	$0.299143\pi$
$54$	0	0
$55$	−6.16478	−0.831259
$56$	0	0
$57$	−4.35916	−0.577385
$58$	0	0
$59$	5.73418	0.746527	0.373263	−	0.927725i	$-0.378239\pi$
$59$	5.73418	0.746527	0.373263	+	0.927725i	$0.378239\pi$
$60$	0	0
$61$	7.64725	0.979131	0.489565	−	0.871967i	$-0.337155\pi$
$61$	7.64725	0.979131	0.489565	+	0.871967i	$0.337155\pi$
$62$	0	0
$63$	−4.61313	−0.581199
$64$	0	0
$65$	−22.4685	−2.78688
$66$	0	0
$67$	−8.79565	−1.07456	−0.537280	−	0.843404i	$-0.680548\pi$
$67$	−8.79565	−1.07456	−0.537280	+	0.843404i	$0.680548\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	6.88311	0.816874	0.408437	−	0.912786i	$-0.366074\pi$
$71$	6.88311	0.816874	0.408437	+	0.912786i	$0.366074\pi$
$72$	0	0
$73$	−3.50114	−0.409778	−0.204889	−	0.978785i	$-0.565683\pi$
$73$	−3.50114	−0.409778	−0.204889	+	0.978785i	$0.565683\pi$
$74$	0	0
$75$	−11.2195	−1.29551
$76$	0	0
$77$	−7.06147	−0.804729
$78$	0	0
$79$	−8.18252	−0.920606	−0.460303	−	0.887762i	$-0.652259\pi$
$79$	−8.18252	−0.920606	−0.460303	+	0.887762i	$0.652259\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.12612	0.452901	0.226450	−	0.974023i	$-0.427288\pi$
$83$	4.12612	0.452901	0.226450	+	0.974023i	$0.427288\pi$
$84$	0	0
$85$	5.22625	0.566867
$86$	0	0
$87$	−1.86256	−0.199687
$88$	0	0
$89$	7.98642	0.846559	0.423280	−	0.905999i	$-0.360879\pi$
$89$	7.98642	0.846559	0.423280	+	0.905999i	$0.360879\pi$
$90$	0	0
$91$	−25.7366	−2.69793
$92$	0	0
$93$	−2.77791	−0.288056
$94$	0	0
$95$	−17.5558	−1.80119
$96$	0	0
$97$	−1.40461	−0.142617	−0.0713084	−	0.997454i	$-0.522717\pi$
$97$	−1.40461	−0.142617	−0.0713084	+	0.997454i	$0.522717\pi$
$98$	0	0
$99$	1.53073	0.153845
$100$	0	0
$101$	0.457942	0.0455669	0.0227835	−	0.999740i	$-0.492747\pi$
$101$	0.457942	0.0455669	0.0227835	+	0.999740i	$0.492747\pi$
$102$	0	0
$103$	−7.77563	−0.766155	−0.383078	−	0.923716i	$-0.625136\pi$
$103$	−7.77563	−0.766155	−0.383078	+	0.923716i	$0.625136\pi$
$104$	0	0
$105$	−18.5786	−1.81309
$106$	0	0
$107$	−14.7047	−1.42156	−0.710781	−	0.703414i	$-0.751658\pi$
$107$	−14.7047	−1.42156	−0.710781	+	0.703414i	$0.751658\pi$
$108$	0	0
$109$	1.09372	0.104759	0.0523795	−	0.998627i	$-0.483319\pi$
$109$	1.09372	0.104759	0.0523795	+	0.998627i	$0.483319\pi$
$110$	0	0
$111$	3.97682	0.377463
$112$	0	0
$113$	−7.65685	−0.720296	−0.360148	−	0.932895i	$-0.617274\pi$
$113$	−7.65685	−0.720296	−0.360148	+	0.932895i	$0.617274\pi$
$114$	0	0
$115$	16.1094	1.50221
$116$	0	0
$117$	5.57900	0.515779
$118$	0	0
$119$	5.98642	0.548775
$120$	0	0
$121$	−8.65685	−0.786987
$122$	0	0
$123$	3.03188	0.273375
$124$	0	0
$125$	−25.0479	−2.24035
$126$	0	0
$127$	0.900845	0.0799371	0.0399685	−	0.999201i	$-0.487274\pi$
$127$	0.900845	0.0799371	0.0399685	+	0.999201i	$0.487274\pi$
$128$	0	0
$129$	−3.03188	−0.266942
$130$	0	0
$131$	−15.7206	−1.37352	−0.686758	−	0.726886i	$-0.740967\pi$
$131$	−15.7206	−1.37352	−0.686758	+	0.726886i	$0.740967\pi$
$132$	0	0
$133$	−20.1094	−1.74370
$134$	0	0
$135$	4.02734	0.346618
$136$	0	0
$137$	−1.76377	−0.150689	−0.0753447	−	0.997158i	$-0.524006\pi$
$137$	−1.76377	−0.150689	−0.0753447	+	0.997158i	$0.524006\pi$
$138$	0	0
$139$	−18.3752	−1.55856	−0.779281	−	0.626675i	$-0.784416\pi$
$139$	−18.3752	−1.55856	−0.779281	+	0.626675i	$0.784416\pi$
$140$	0	0
$141$	9.65685	0.813254
$142$	0	0
$143$	8.53996	0.714147
$144$	0	0
$145$	−7.50114	−0.622936
$146$	0	0
$147$	−14.2809	−1.17787
$148$	0	0
$149$	1.68419	0.137975	0.0689873	−	0.997618i	$-0.478023\pi$
$149$	1.68419	0.137975	0.0689873	+	0.997618i	$0.478023\pi$
$150$	0	0
$151$	−10.3118	−0.839165	−0.419582	−	0.907717i	$-0.637823\pi$
$151$	−10.3118	−0.839165	−0.419582	+	0.907717i	$0.637823\pi$
$152$	0	0
$153$	−1.29769	−0.104912
$154$	0	0
$155$	−11.1876	−0.898609
$156$	0	0
$157$	13.6337	1.08809	0.544043	−	0.839057i	$-0.316893\pi$
$157$	13.6337	1.08809	0.544043	+	0.839057i	$0.316893\pi$
$158$	0	0
$159$	−8.58995	−0.681227
$160$	0	0
$161$	18.4525	1.45426
$162$	0	0
$163$	3.17476	0.248666	0.124333	−	0.992241i	$-0.460321\pi$
$163$	3.17476	0.248666	0.124333	+	0.992241i	$0.460321\pi$
$164$	0	0
$165$	6.16478	0.479928
$166$	0	0
$167$	−18.4170	−1.42515	−0.712576	−	0.701595i	$-0.752472\pi$
$167$	−18.4170	−1.42515	−0.712576	+	0.701595i	$0.752472\pi$
$168$	0	0
$169$	18.1252	1.39425
$170$	0	0
$171$	4.35916	0.333353
$172$	0	0
$173$	−11.0341	−0.838909	−0.419455	−	0.907776i	$-0.637779\pi$
$173$	−11.0341	−0.838909	−0.419455	+	0.907776i	$0.637779\pi$
$174$	0	0
$175$	−51.7568	−3.91245
$176$	0	0
$177$	−5.73418	−0.431008
$178$	0	0
$179$	−0.795649	−0.0594696	−0.0297348	−	0.999558i	$-0.509466\pi$
$179$	−0.795649	−0.0594696	−0.0297348	+	0.999558i	$0.509466\pi$
$180$	0	0
$181$	5.24943	0.390187	0.195093	−	0.980785i	$-0.437499\pi$
$181$	5.24943	0.390187	0.195093	+	0.980785i	$0.437499\pi$
$182$	0	0
$183$	−7.64725	−0.565301
$184$	0	0
$185$	16.0160	1.17752
$186$	0	0
$187$	−1.98642	−0.145262
$188$	0	0
$189$	4.61313	0.335556
$190$	0	0
$191$	−25.0060	−1.80937	−0.904687	−	0.426077i	$-0.859895\pi$
$191$	−25.0060	−1.80937	−0.904687	+	0.426077i	$0.859895\pi$
$192$	0	0
$193$	22.9378	1.65110	0.825549	−	0.564331i	$-0.190866\pi$
$193$	22.9378	1.65110	0.825549	+	0.564331i	$0.190866\pi$
$194$	0	0
$195$	22.4685	1.60900
$196$	0	0
$197$	−15.0533	−1.07251	−0.536253	−	0.844058i	$-0.680161\pi$
$197$	−15.0533	−1.07251	−0.536253	+	0.844058i	$0.680161\pi$
$198$	0	0
$199$	19.7485	1.39993	0.699966	−	0.714176i	$-0.253198\pi$
$199$	19.7485	1.39993	0.699966	+	0.714176i	$0.253198\pi$
$200$	0	0
$201$	8.79565	0.620397
$202$	0	0
$203$	−8.59220	−0.603054
$204$	0	0
$205$	12.2104	0.852811
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	6.67271	0.461561
$210$	0	0
$211$	−0.938533	−0.0646112	−0.0323056	−	0.999478i	$-0.510285\pi$
$211$	−0.938533	−0.0646112	−0.0323056	+	0.999478i	$0.510285\pi$
$212$	0	0
$213$	−6.88311	−0.471623
$214$	0	0
$215$	−12.2104	−0.832742
$216$	0	0
$217$	−12.8149	−0.869929
$218$	0	0
$219$	3.50114	0.236585
$220$	0	0
$221$	−7.23983	−0.487004
$222$	0	0
$223$	26.2783	1.75973	0.879863	−	0.475228i	$-0.157634\pi$
$223$	26.2783	1.75973	0.879863	+	0.475228i	$0.157634\pi$
$224$	0	0
$225$	11.2195	0.747964
$226$	0	0
$227$	10.9082	0.724002	0.362001	−	0.932178i	$-0.382094\pi$
$227$	10.9082	0.724002	0.362001	+	0.932178i	$0.382094\pi$
$228$	0	0
$229$	18.8599	1.24630	0.623150	−	0.782103i	$-0.285853\pi$
$229$	18.8599	1.24630	0.623150	+	0.782103i	$0.285853\pi$
$230$	0	0
$231$	7.06147	0.464610
$232$	0	0
$233$	19.7662	1.29493	0.647464	−	0.762096i	$-0.275830\pi$
$233$	19.7662	1.29493	0.647464	+	0.762096i	$0.275830\pi$
$234$	0	0
$235$	38.8914	2.53700
$236$	0	0
$237$	8.18252	0.531512
$238$	0	0
$239$	14.3515	0.928319	0.464160	−	0.885752i	$-0.346356\pi$
$239$	14.3515	0.928319	0.464160	+	0.885752i	$0.346356\pi$
$240$	0	0
$241$	21.7534	1.40126	0.700629	−	0.713525i	$-0.252903\pi$
$241$	21.7534	1.40126	0.700629	+	0.713525i	$0.252903\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−57.5142	−3.67444
$246$	0	0
$247$	24.3197	1.54743
$248$	0	0
$249$	−4.12612	−0.261482
$250$	0	0
$251$	−9.53073	−0.601575	−0.300787	−	0.953691i	$-0.597249\pi$
$251$	−9.53073	−0.601575	−0.300787	+	0.953691i	$0.597249\pi$
$252$	0	0
$253$	−6.12293	−0.384946
$254$	0	0
$255$	−5.22625	−0.327281
$256$	0	0
$257$	−1.20435	−0.0751253	−0.0375627	−	0.999294i	$-0.511959\pi$
$257$	−1.20435	−0.0751253	−0.0375627	+	0.999294i	$0.511959\pi$
$258$	0	0
$259$	18.3456	1.13994
$260$	0	0
$261$	1.86256	0.115289
$262$	0	0
$263$	−27.5777	−1.70052	−0.850258	−	0.526367i	$-0.823554\pi$
$263$	−27.5777	−1.70052	−0.850258	+	0.526367i	$0.823554\pi$
$264$	0	0
$265$	−34.5946	−2.12513
$266$	0	0
$267$	−7.98642	−0.488761
$268$	0	0
$269$	−14.6646	−0.894115	−0.447057	−	0.894505i	$-0.647528\pi$
$269$	−14.6646	−0.894115	−0.447057	+	0.894505i	$0.647528\pi$
$270$	0	0
$271$	20.3574	1.23663	0.618313	−	0.785932i	$-0.287816\pi$
$271$	20.3574	1.23663	0.618313	+	0.785932i	$0.287816\pi$
$272$	0	0
$273$	25.7366	1.55765
$274$	0	0
$275$	17.1740	1.03563
$276$	0	0
$277$	−23.8113	−1.43068	−0.715341	−	0.698776i	$-0.753729\pi$
$277$	−23.8113	−1.43068	−0.715341	+	0.698776i	$0.753729\pi$
$278$	0	0
$279$	2.77791	0.166309
$280$	0	0
$281$	−17.1116	−1.02079	−0.510397	−	0.859939i	$-0.670502\pi$
$281$	−17.1116	−1.02079	−0.510397	+	0.859939i	$0.670502\pi$
$282$	0	0
$283$	2.26582	0.134689	0.0673444	−	0.997730i	$-0.478547\pi$
$283$	2.26582	0.134689	0.0673444	+	0.997730i	$0.478547\pi$
$284$	0	0
$285$	17.5558	1.03992
$286$	0	0
$287$	13.9864	0.825592
$288$	0	0
$289$	−15.3160	−0.900941
$290$	0	0
$291$	1.40461	0.0823399
$292$	0	0
$293$	3.90366	0.228054	0.114027	−	0.993478i	$-0.463625\pi$
$293$	3.90366	0.228054	0.114027	+	0.993478i	$0.463625\pi$
$294$	0	0
$295$	−23.0935	−1.34456
$296$	0	0
$297$	−1.53073	−0.0888222
$298$	0	0
$299$	−22.3160	−1.29057
$300$	0	0
$301$	−13.9864	−0.806164
$302$	0	0
$303$	−0.457942	−0.0263081
$304$	0	0
$305$	−30.7981	−1.76349
$306$	0	0
$307$	26.2323	1.49716	0.748578	−	0.663047i	$-0.230737\pi$
$307$	26.2323	1.49716	0.748578	+	0.663047i	$0.230737\pi$
$308$	0	0
$309$	7.77563	0.442340
$310$	0	0
$311$	−9.12522	−0.517444	−0.258722	−	0.965952i	$-0.583301\pi$
$311$	−9.12522	−0.517444	−0.258722	+	0.965952i	$0.583301\pi$
$312$	0	0
$313$	−9.28093	−0.524589	−0.262295	−	0.964988i	$-0.584479\pi$
$313$	−9.28093	−0.524589	−0.262295	+	0.964988i	$0.584479\pi$
$314$	0	0
$315$	18.5786	1.04679
$316$	0	0
$317$	32.3652	1.81781	0.908906	−	0.417000i	$-0.136919\pi$
$317$	32.3652	1.81781	0.908906	+	0.417000i	$0.136919\pi$
$318$	0	0
$319$	2.85108	0.159630
$320$	0	0
$321$	14.7047	0.820739
$322$	0	0
$323$	−5.65685	−0.314756
$324$	0	0
$325$	62.5934	3.47206
$326$	0	0
$327$	−1.09372	−0.0604827
$328$	0	0
$329$	44.5483	2.45603
$330$	0	0
$331$	21.8435	1.20063	0.600315	−	0.799764i	$-0.295042\pi$
$331$	21.8435	1.20063	0.600315	+	0.799764i	$0.295042\pi$
$332$	0	0
$333$	−3.97682	−0.217929
$334$	0	0
$335$	35.4231	1.93537
$336$	0	0
$337$	−28.8722	−1.57277	−0.786385	−	0.617736i	$-0.788050\pi$
$337$	−28.8722	−1.57277	−0.786385	+	0.617736i	$0.788050\pi$
$338$	0	0
$339$	7.65685	0.415863
$340$	0	0
$341$	4.25224	0.230272
$342$	0	0
$343$	−33.5879	−1.81357
$344$	0	0
$345$	−16.1094	−0.867299
$346$	0	0
$347$	21.9697	1.17939	0.589697	−	0.807625i	$-0.299247\pi$
$347$	21.9697	1.17939	0.589697	+	0.807625i	$0.299247\pi$
$348$	0	0
$349$	−20.0862	−1.07519	−0.537594	−	0.843204i	$-0.680667\pi$
$349$	−20.0862	−1.07519	−0.537594	+	0.843204i	$0.680667\pi$
$350$	0	0
$351$	−5.57900	−0.297785
$352$	0	0
$353$	−36.9570	−1.96702	−0.983511	−	0.180849i	$-0.942116\pi$
$353$	−36.9570	−1.96702	−0.983511	+	0.180849i	$0.942116\pi$
$354$	0	0
$355$	−27.7206	−1.47126
$356$	0	0
$357$	−5.98642	−0.316835
$358$	0	0
$359$	−22.5264	−1.18890	−0.594449	−	0.804134i	$-0.702630\pi$
$359$	−22.5264	−1.18890	−0.594449	+	0.804134i	$0.702630\pi$
$360$	0	0
$361$	0.00228335	0.000120176 0
$362$	0	0
$363$	8.65685	0.454367
$364$	0	0
$365$	14.1003	0.738043
$366$	0	0
$367$	−0.535798	−0.0279684	−0.0139842	−	0.999902i	$-0.504451\pi$
$367$	−0.535798	−0.0279684	−0.0139842	+	0.999902i	$0.504451\pi$
$368$	0	0
$369$	−3.03188	−0.157833
$370$	0	0
$371$	−39.6265	−2.05731
$372$	0	0
$373$	8.83803	0.457616	0.228808	−	0.973472i	$-0.426517\pi$
$373$	8.83803	0.457616	0.228808	+	0.973472i	$0.426517\pi$
$374$	0	0
$375$	25.0479	1.29347
$376$	0	0
$377$	10.3912	0.535174
$378$	0	0
$379$	−24.6887	−1.26817	−0.634087	−	0.773261i	$-0.718624\pi$
$379$	−24.6887	−1.26817	−0.634087	+	0.773261i	$0.718624\pi$
$380$	0	0
$381$	−0.900845	−0.0461517
$382$	0	0
$383$	31.1290	1.59062	0.795308	−	0.606205i	$-0.207309\pi$
$383$	31.1290	1.59062	0.795308	+	0.606205i	$0.207309\pi$
$384$	0	0
$385$	28.4389	1.44938
$386$	0	0
$387$	3.03188	0.154119
$388$	0	0
$389$	−20.3614	−1.03236	−0.516182	−	0.856479i	$-0.672647\pi$
$389$	−20.3614	−1.03236	−0.516182	+	0.856479i	$0.672647\pi$
$390$	0	0
$391$	5.19077	0.262509
$392$	0	0
$393$	15.7206	0.793000
$394$	0	0
$395$	32.9538	1.65809
$396$	0	0
$397$	−3.60090	−0.180724	−0.0903620	−	0.995909i	$-0.528802\pi$
$397$	−3.60090	−0.180724	−0.0903620	+	0.995909i	$0.528802\pi$
$398$	0	0
$399$	20.1094	1.00673
$400$	0	0
$401$	−18.0160	−0.899677	−0.449838	−	0.893110i	$-0.648518\pi$
$401$	−18.0160	−0.899677	−0.449838	+	0.893110i	$0.648518\pi$
$402$	0	0
$403$	15.4980	0.772008
$404$	0	0
$405$	−4.02734	−0.200120
$406$	0	0
$407$	−6.08746	−0.301744
$408$	0	0
$409$	−25.1572	−1.24395	−0.621973	−	0.783039i	$-0.713669\pi$
$409$	−25.1572	−1.24395	−0.621973	+	0.783039i	$0.713669\pi$
$410$	0	0
$411$	1.76377	0.0870006
$412$	0	0
$413$	−26.4525	−1.30164
$414$	0	0
$415$	−16.6173	−0.815711
$416$	0	0
$417$	18.3752	0.899836
$418$	0	0
$419$	6.52845	0.318936	0.159468	−	0.987203i	$-0.449022\pi$
$419$	6.52845	0.318936	0.159468	+	0.987203i	$0.449022\pi$
$420$	0	0
$421$	19.8758	0.968687	0.484343	−	0.874878i	$-0.339059\pi$
$421$	19.8758	0.968687	0.484343	+	0.874878i	$0.339059\pi$
$422$	0	0
$423$	−9.65685	−0.469532
$424$	0	0
$425$	−14.5594	−0.706236
$426$	0	0
$427$	−35.2777	−1.70721
$428$	0	0
$429$	−8.53996	−0.412313
$430$	0	0
$431$	−34.7368	−1.67321	−0.836606	−	0.547805i	$-0.815463\pi$
$431$	−34.7368	−1.67321	−0.836606	+	0.547805i	$0.815463\pi$
$432$	0	0
$433$	−27.0935	−1.30203	−0.651015	−	0.759065i	$-0.725657\pi$
$433$	−27.0935	−1.30203	−0.651015	+	0.759065i	$0.725657\pi$
$434$	0	0
$435$	7.50114	0.359652
$436$	0	0
$437$	−17.4366	−0.834108
$438$	0	0
$439$	3.52976	0.168466	0.0842331	−	0.996446i	$-0.473156\pi$
$439$	3.52976	0.168466	0.0842331	+	0.996446i	$0.473156\pi$
$440$	0	0
$441$	14.2809	0.680044
$442$	0	0
$443$	−5.71742	−0.271643	−0.135821	−	0.990733i	$-0.543367\pi$
$443$	−5.71742	−0.271643	−0.135821	+	0.990733i	$0.543367\pi$
$444$	0	0
$445$	−32.1640	−1.52472
$446$	0	0
$447$	−1.68419	−0.0796596
$448$	0	0
$449$	28.9346	1.36551	0.682754	−	0.730648i	$-0.260782\pi$
$449$	28.9346	1.36551	0.682754	+	0.730648i	$0.260782\pi$
$450$	0	0
$451$	−4.64099	−0.218536
$452$	0	0
$453$	10.3118	0.484492
$454$	0	0
$455$	103.650	4.85919
$456$	0	0
$457$	−33.6233	−1.57283	−0.786416	−	0.617697i	$-0.788066\pi$
$457$	−33.6233	−1.57283	−0.786416	+	0.617697i	$0.788066\pi$
$458$	0	0
$459$	1.29769	0.0605711
$460$	0	0
$461$	−5.27730	−0.245788	−0.122894	−	0.992420i	$-0.539218\pi$
$461$	−5.27730	−0.245788	−0.122894	+	0.992420i	$0.539218\pi$
$462$	0	0
$463$	−11.1430	−0.517857	−0.258929	−	0.965896i	$-0.583369\pi$
$463$	−11.1430	−0.517857	−0.258929	+	0.965896i	$0.583369\pi$
$464$	0	0
$465$	11.1876	0.518812
$466$	0	0
$467$	−26.5013	−1.22633	−0.613167	−	0.789953i	$-0.710105\pi$
$467$	−26.5013	−1.22633	−0.613167	+	0.789953i	$0.710105\pi$
$468$	0	0
$469$	40.5754	1.87360
$470$	0	0
$471$	−13.6337	−0.628207
$472$	0	0
$473$	4.64099	0.213393
$474$	0	0
$475$	48.9074	2.24403
$476$	0	0
$477$	8.58995	0.393307
$478$	0	0
$479$	22.4170	1.02426	0.512130	−	0.858908i	$-0.328857\pi$
$479$	22.4170	1.02426	0.512130	+	0.858908i	$0.328857\pi$
$480$	0	0
$481$	−22.1867	−1.01163
$482$	0	0
$483$	−18.4525	−0.839618
$484$	0	0
$485$	5.65685	0.256865
$486$	0	0
$487$	7.42235	0.336339	0.168169	−	0.985758i	$-0.446214\pi$
$487$	7.42235	0.336339	0.168169	+	0.985758i	$0.446214\pi$
$488$	0	0
$489$	−3.17476	−0.143568
$490$	0	0
$491$	−17.3711	−0.783946	−0.391973	−	0.919977i	$-0.628207\pi$
$491$	−17.3711	−0.783946	−0.391973	+	0.919977i	$0.628207\pi$
$492$	0	0
$493$	−2.41703	−0.108857
$494$	0	0
$495$	−6.16478	−0.277086
$496$	0	0
$497$	−31.7526	−1.42430
$498$	0	0
$499$	−37.2938	−1.66950	−0.834749	−	0.550631i	$-0.814387\pi$
$499$	−37.2938	−1.66950	−0.834749	+	0.550631i	$0.814387\pi$
$500$	0	0
$501$	18.4170	0.822812
$502$	0	0
$503$	13.8672	0.618310	0.309155	−	0.951012i	$-0.399954\pi$
$503$	13.8672	0.618310	0.309155	+	0.951012i	$0.399954\pi$
$504$	0	0
$505$	−1.84429	−0.0820697
$506$	0	0
$507$	−18.1252	−0.804969
$508$	0	0
$509$	−34.6311	−1.53499	−0.767497	−	0.641052i	$-0.778498\pi$
$509$	−34.6311	−1.53499	−0.767497	+	0.641052i	$0.778498\pi$
$510$	0	0
$511$	16.1512	0.714487
$512$	0	0
$513$	−4.35916	−0.192462
$514$	0	0
$515$	31.3151	1.37991
$516$	0	0
$517$	−14.7821	−0.650115
$518$	0	0
$519$	11.0341	0.484344
$520$	0	0
$521$	34.8437	1.52653	0.763265	−	0.646086i	$-0.223595\pi$
$521$	34.8437	1.52653	0.763265	+	0.646086i	$0.223595\pi$
$522$	0	0
$523$	25.5499	1.11722	0.558610	−	0.829430i	$-0.311335\pi$
$523$	25.5499	1.11722	0.558610	+	0.829430i	$0.311335\pi$
$524$	0	0
$525$	51.7568	2.25885
$526$	0	0
$527$	−3.60488	−0.157031
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	5.73418	0.248842
$532$	0	0
$533$	−16.9148	−0.732662
$534$	0	0
$535$	59.2210	2.56035
$536$	0	0
$537$	0.795649	0.0343348
$538$	0	0
$539$	21.8603	0.941590
$540$	0	0
$541$	32.8735	1.41334	0.706671	−	0.707542i	$-0.250196\pi$
$541$	32.8735	1.41334	0.706671	+	0.707542i	$0.250196\pi$
$542$	0	0
$543$	−5.24943	−0.225275
$544$	0	0
$545$	−4.40477	−0.188680
$546$	0	0
$547$	−32.4414	−1.38709	−0.693546	−	0.720412i	$-0.743953\pi$
$547$	−32.4414	−1.38709	−0.693546	+	0.720412i	$0.743953\pi$
$548$	0	0
$549$	7.64725	0.326377
$550$	0	0
$551$	8.11918	0.345889
$552$	0	0
$553$	37.7470	1.60517
$554$	0	0
$555$	−16.0160	−0.679842
$556$	0	0
$557$	−2.25025	−0.0953462	−0.0476731	−	0.998863i	$-0.515181\pi$
$557$	−2.25025	−0.0953462	−0.0476731	+	0.998863i	$0.515181\pi$
$558$	0	0
$559$	16.9148	0.715421
$560$	0	0
$561$	1.98642	0.0838668
$562$	0	0
$563$	−1.37608	−0.0579948	−0.0289974	−	0.999579i	$-0.509231\pi$
$563$	−1.37608	−0.0579948	−0.0289974	+	0.999579i	$0.509231\pi$
$564$	0	0
$565$	30.8368	1.29731
$566$	0	0
$567$	−4.61313	−0.193733
$568$	0	0
$569$	15.5043	0.649975	0.324988	−	0.945718i	$-0.394640\pi$
$569$	15.5043	0.649975	0.324988	+	0.945718i	$0.394640\pi$
$570$	0	0
$571$	7.45659	0.312049	0.156024	−	0.987753i	$-0.450132\pi$
$571$	7.45659	0.312049	0.156024	+	0.987753i	$0.450132\pi$
$572$	0	0
$573$	25.0060	1.04464
$574$	0	0
$575$	−44.8779	−1.87154
$576$	0	0
$577$	−25.9355	−1.07971	−0.539855	−	0.841758i	$-0.681521\pi$
$577$	−25.9355	−1.07971	−0.539855	+	0.841758i	$0.681521\pi$
$578$	0	0
$579$	−22.9378	−0.953262
$580$	0	0
$581$	−19.0343	−0.789676
$582$	0	0
$583$	13.1489	0.544573
$584$	0	0
$585$	−22.4685	−0.928959
$586$	0	0
$587$	33.6233	1.38778	0.693892	−	0.720079i	$-0.255895\pi$
$587$	33.6233	1.38778	0.693892	+	0.720079i	$0.255895\pi$
$588$	0	0
$589$	12.1094	0.498957
$590$	0	0
$591$	15.0533	0.619211
$592$	0	0
$593$	44.0302	1.80810	0.904052	−	0.427422i	$-0.140578\pi$
$593$	44.0302	1.80810	0.904052	+	0.427422i	$0.140578\pi$
$594$	0	0
$595$	−24.1094	−0.988387
$596$	0	0
$597$	−19.7485	−0.808251
$598$	0	0
$599$	−6.09578	−0.249067	−0.124533	−	0.992215i	$-0.539743\pi$
$599$	−6.09578	−0.249067	−0.124533	+	0.992215i	$0.539743\pi$
$600$	0	0
$601$	−11.8126	−0.481845	−0.240922	−	0.970544i	$-0.577450\pi$
$601$	−11.8126	−0.481845	−0.240922	+	0.970544i	$0.577450\pi$
$602$	0	0
$603$	−8.79565	−0.358187
$604$	0	0
$605$	34.8641	1.41743
$606$	0	0
$607$	−4.66330	−0.189277	−0.0946387	−	0.995512i	$-0.530170\pi$
$607$	−4.66330	−0.189277	−0.0946387	+	0.995512i	$0.530170\pi$
$608$	0	0
$609$	8.59220	0.348174
$610$	0	0
$611$	−53.8756	−2.17957
$612$	0	0
$613$	−28.5715	−1.15399	−0.576995	−	0.816748i	$-0.695775\pi$
$613$	−28.5715	−1.15399	−0.576995	+	0.816748i	$0.695775\pi$
$614$	0	0
$615$	−12.2104	−0.492371
$616$	0	0
$617$	−21.2482	−0.855418	−0.427709	−	0.903916i	$-0.640679\pi$
$617$	−21.2482	−0.855418	−0.427709	+	0.903916i	$0.640679\pi$
$618$	0	0
$619$	−11.4412	−0.459861	−0.229931	−	0.973207i	$-0.573850\pi$
$619$	−11.4412	−0.459861	−0.229931	+	0.973207i	$0.573850\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	−36.8424	−1.47606
$624$	0	0
$625$	44.7790	1.79116
$626$	0	0
$627$	−6.67271	−0.266483
$628$	0	0
$629$	5.16070	0.205770
$630$	0	0
$631$	16.9427	0.674478	0.337239	−	0.941419i	$-0.390507\pi$
$631$	16.9427	0.674478	0.337239	+	0.941419i	$0.390507\pi$
$632$	0	0
$633$	0.938533	0.0373033
$634$	0	0
$635$	−3.62801	−0.143973
$636$	0	0
$637$	79.6733	3.15677
$638$	0	0
$639$	6.88311	0.272291
$640$	0	0
$641$	−27.8596	−1.10039	−0.550193	−	0.835037i	$-0.685446\pi$
$641$	−27.8596	−1.10039	−0.550193	+	0.835037i	$0.685446\pi$
$642$	0	0
$643$	−20.6114	−0.812834	−0.406417	−	0.913688i	$-0.633222\pi$
$643$	−20.6114	−0.812834	−0.406417	+	0.913688i	$0.633222\pi$
$644$	0	0
$645$	12.2104	0.480784
$646$	0	0
$647$	−11.7142	−0.460534	−0.230267	−	0.973127i	$-0.573960\pi$
$647$	−11.7142	−0.460534	−0.230267	+	0.973127i	$0.573960\pi$
$648$	0	0
$649$	8.77751	0.344547
$650$	0	0
$651$	12.8149	0.502254
$652$	0	0
$653$	−51.0435	−1.99749	−0.998743	−	0.0501153i	$-0.984041\pi$
$653$	−51.0435	−1.99749	−0.998743	+	0.0501153i	$0.984041\pi$
$654$	0	0
$655$	63.3122	2.47381
$656$	0	0
$657$	−3.50114	−0.136593
$658$	0	0
$659$	5.68401	0.221418	0.110709	−	0.993853i	$-0.464688\pi$
$659$	5.68401	0.221418	0.110709	+	0.993853i	$0.464688\pi$
$660$	0	0
$661$	−36.2611	−1.41039	−0.705197	−	0.709012i	$-0.749141\pi$
$661$	−36.2611	−1.41039	−0.705197	+	0.709012i	$0.749141\pi$
$662$	0	0
$663$	7.23983	0.281172
$664$	0	0
$665$	80.9872	3.14055
$666$	0	0
$667$	−7.45022	−0.288474
$668$	0	0
$669$	−26.2783	−1.01598
$670$	0	0
$671$	11.7059	0.451902
$672$	0	0
$673$	17.6569	0.680622	0.340311	−	0.940313i	$-0.389468\pi$
$673$	17.6569	0.680622	0.340311	+	0.940313i	$0.389468\pi$
$674$	0	0
$675$	−11.2195	−0.431837
$676$	0	0
$677$	−18.0866	−0.695124	−0.347562	−	0.937657i	$-0.612990\pi$
$677$	−18.0866	−0.695124	−0.347562	+	0.937657i	$0.612990\pi$
$678$	0	0
$679$	6.47966	0.248666
$680$	0	0
$681$	−10.9082	−0.418003
$682$	0	0
$683$	20.3448	0.778474	0.389237	−	0.921138i	$-0.372739\pi$
$683$	20.3448	0.778474	0.389237	+	0.921138i	$0.372739\pi$
$684$	0	0
$685$	7.10332	0.271404
$686$	0	0
$687$	−18.8599	−0.719551
$688$	0	0
$689$	47.9233	1.82573
$690$	0	0
$691$	−20.1118	−0.765089	−0.382544	−	0.923937i	$-0.624952\pi$
$691$	−20.1118	−0.765089	−0.382544	+	0.923937i	$0.624952\pi$
$692$	0	0
$693$	−7.06147	−0.268243
$694$	0	0
$695$	74.0031	2.80710
$696$	0	0
$697$	3.93444	0.149028
$698$	0	0
$699$	−19.7662	−0.747627
$700$	0	0
$701$	−19.9263	−0.752606	−0.376303	−	0.926497i	$-0.622805\pi$
$701$	−19.9263	−0.752606	−0.376303	+	0.926497i	$0.622805\pi$
$702$	0	0
$703$	−17.3356	−0.653825
$704$	0	0
$705$	−38.8914	−1.46474
$706$	0	0
$707$	−2.11254	−0.0794504
$708$	0	0
$709$	3.30141	0.123987	0.0619935	−	0.998077i	$-0.480254\pi$
$709$	3.30141	0.123987	0.0619935	+	0.998077i	$0.480254\pi$
$710$	0	0
$711$	−8.18252	−0.306869
$712$	0	0
$713$	−11.1116	−0.416134
$714$	0	0
$715$	−34.3933	−1.28624
$716$	0	0
$717$	−14.3515	−0.535965
$718$	0	0
$719$	40.2104	1.49959	0.749797	−	0.661668i	$-0.230151\pi$
$719$	40.2104	1.49959	0.749797	+	0.661668i	$0.230151\pi$
$720$	0	0
$721$	35.8699	1.33587
$722$	0	0
$723$	−21.7534	−0.809017
$724$	0	0
$725$	20.8969	0.776090
$726$	0	0
$727$	−39.4272	−1.46227	−0.731137	−	0.682230i	$-0.761010\pi$
$727$	−39.4272	−1.46227	−0.731137	+	0.682230i	$0.761010\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.93444	−0.145521
$732$	0	0
$733$	−20.7370	−0.765938	−0.382969	−	0.923761i	$-0.625098\pi$
$733$	−20.7370	−0.765938	−0.382969	+	0.923761i	$0.625098\pi$
$734$	0	0
$735$	57.5142	2.12144
$736$	0	0
$737$	−13.4638	−0.495945
$738$	0	0
$739$	13.9529	0.513266	0.256633	−	0.966509i	$-0.417387\pi$
$739$	13.9529	0.513266	0.256633	+	0.966509i	$0.417387\pi$
$740$	0	0
$741$	−24.3197	−0.893408
$742$	0	0
$743$	5.02418	0.184319	0.0921597	−	0.995744i	$-0.470623\pi$
$743$	5.02418	0.184319	0.0921597	+	0.995744i	$0.470623\pi$
$744$	0	0
$745$	−6.78282	−0.248503
$746$	0	0
$747$	4.12612	0.150967
$748$	0	0
$749$	67.8348	2.47863
$750$	0	0
$751$	5.44425	0.198664	0.0993318	−	0.995054i	$-0.468329\pi$
$751$	5.44425	0.198664	0.0993318	+	0.995054i	$0.468329\pi$
$752$	0	0
$753$	9.53073	0.347319
$754$	0	0
$755$	41.5292	1.51140
$756$	0	0
$757$	14.7698	0.536816	0.268408	−	0.963305i	$-0.413502\pi$
$757$	14.7698	0.536816	0.268408	+	0.963305i	$0.413502\pi$
$758$	0	0
$759$	6.12293	0.222248
$760$	0	0
$761$	−1.96584	−0.0712617	−0.0356308	−	0.999365i	$-0.511344\pi$
$761$	−1.96584	−0.0712617	−0.0356308	+	0.999365i	$0.511344\pi$
$762$	0	0
$763$	−5.04545	−0.182658
$764$	0	0
$765$	5.22625	0.188956
$766$	0	0
$767$	31.9910	1.15513
$768$	0	0
$769$	−7.09244	−0.255760	−0.127880	−	0.991790i	$-0.540817\pi$
$769$	−7.09244	−0.255760	−0.127880	+	0.991790i	$0.540817\pi$
$770$	0	0
$771$	1.20435	0.0433736
$772$	0	0
$773$	27.5696	0.991609	0.495804	−	0.868434i	$-0.334873\pi$
$773$	27.5696	0.991609	0.495804	+	0.868434i	$0.334873\pi$
$774$	0	0
$775$	31.1667	1.11954
$776$	0	0
$777$	−18.3456	−0.658144
$778$	0	0
$779$	−13.2164	−0.473528
$780$	0	0
$781$	10.5362	0.377015
$782$	0	0
$783$	−1.86256	−0.0665623
$784$	0	0
$785$	−54.9074	−1.95973
$786$	0	0
$787$	−29.6529	−1.05701	−0.528506	−	0.848929i	$-0.677248\pi$
$787$	−29.6529	−1.05701	−0.528506	+	0.848929i	$0.677248\pi$
$788$	0	0
$789$	27.5777	0.981793
$790$	0	0
$791$	35.3220	1.25591
$792$	0	0
$793$	42.6640	1.51504
$794$	0	0
$795$	34.5946	1.22695
$796$	0	0
$797$	−5.43832	−0.192635	−0.0963177	−	0.995351i	$-0.530706\pi$
$797$	−5.43832	−0.192635	−0.0963177	+	0.995351i	$0.530706\pi$
$798$	0	0
$799$	12.5316	0.443337
$800$	0	0
$801$	7.98642	0.282186
$802$	0	0
$803$	−5.35932	−0.189126
$804$	0	0
$805$	−74.3145	−2.61924
$806$	0	0
$807$	14.6646	0.516218
$808$	0	0
$809$	17.1548	0.603131	0.301566	−	0.953445i	$-0.402491\pi$
$809$	17.1548	0.603131	0.301566	+	0.953445i	$0.402491\pi$
$810$	0	0
$811$	9.46624	0.332404	0.166202	−	0.986092i	$-0.446850\pi$
$811$	9.46624	0.332404	0.166202	+	0.986092i	$0.446850\pi$
$812$	0	0
$813$	−20.3574	−0.713966
$814$	0	0
$815$	−12.7858	−0.447868
$816$	0	0
$817$	13.2164	0.462384
$818$	0	0
$819$	−25.7366	−0.899310
$820$	0	0
$821$	19.1198	0.667285	0.333642	−	0.942700i	$-0.391722\pi$
$821$	19.1198	0.667285	0.333642	+	0.942700i	$0.391722\pi$
$822$	0	0
$823$	−10.5257	−0.366902	−0.183451	−	0.983029i	$-0.558727\pi$
$823$	−10.5257	−0.366902	−0.183451	+	0.983029i	$0.558727\pi$
$824$	0	0
$825$	−17.1740	−0.597922
$826$	0	0
$827$	26.9841	0.938330	0.469165	−	0.883110i	$-0.344555\pi$
$827$	26.9841	0.938330	0.469165	+	0.883110i	$0.344555\pi$
$828$	0	0
$829$	−12.4704	−0.433116	−0.216558	−	0.976270i	$-0.569483\pi$
$829$	−12.4704	−0.433116	−0.216558	+	0.976270i	$0.569483\pi$
$830$	0	0
$831$	23.8113	0.826005
$832$	0	0
$833$	−18.5323	−0.642105
$834$	0	0
$835$	74.1716	2.56681
$836$	0	0
$837$	−2.77791	−0.0960186
$838$	0	0
$839$	−29.4631	−1.01718	−0.508590	−	0.861009i	$-0.669833\pi$
$839$	−29.4631	−1.01718	−0.508590	+	0.861009i	$0.669833\pi$
$840$	0	0
$841$	−25.5309	−0.880375
$842$	0	0
$843$	17.1116	0.589356
$844$	0	0
$845$	−72.9964	−2.51115
$846$	0	0
$847$	39.9352	1.37219
$848$	0	0
$849$	−2.26582	−0.0777627
$850$	0	0
$851$	15.9073	0.545295
$852$	0	0
$853$	16.1955	0.554525	0.277262	−	0.960794i	$-0.410573\pi$
$853$	16.1955	0.554525	0.277262	+	0.960794i	$0.410573\pi$
$854$	0	0
$855$	−17.5558	−0.600396
$856$	0	0
$857$	−28.3592	−0.968730	−0.484365	−	0.874866i	$-0.660949\pi$
$857$	−28.3592	−0.968730	−0.484365	+	0.874866i	$0.660949\pi$
$858$	0	0
$859$	−2.51562	−0.0858319	−0.0429159	−	0.999079i	$-0.513665\pi$
$859$	−2.51562	−0.0858319	−0.0429159	+	0.999079i	$0.513665\pi$
$860$	0	0
$861$	−13.9864	−0.476656
$862$	0	0
$863$	−13.8944	−0.472971	−0.236485	−	0.971635i	$-0.575996\pi$
$863$	−13.8944	−0.472971	−0.236485	+	0.971635i	$0.575996\pi$
$864$	0	0
$865$	44.4382	1.51094
$866$	0	0
$867$	15.3160	0.520158
$868$	0	0
$869$	−12.5253	−0.424891
$870$	0	0
$871$	−49.0709	−1.66270
$872$	0	0
$873$	−1.40461	−0.0475390
$874$	0	0
$875$	115.549	3.90627
$876$	0	0
$877$	−26.7216	−0.902323	−0.451161	−	0.892442i	$-0.648990\pi$
$877$	−26.7216	−0.902323	−0.451161	+	0.892442i	$0.648990\pi$
$878$	0	0
$879$	−3.90366	−0.131667
$880$	0	0
$881$	11.5460	0.388995	0.194497	−	0.980903i	$-0.437692\pi$
$881$	11.5460	0.388995	0.194497	+	0.980903i	$0.437692\pi$
$882$	0	0
$883$	−53.1868	−1.78988	−0.894940	−	0.446187i	$-0.852782\pi$
$883$	−53.1868	−1.78988	−0.894940	+	0.446187i	$0.852782\pi$
$884$	0	0
$885$	23.0935	0.776279
$886$	0	0
$887$	−44.8123	−1.50465	−0.752325	−	0.658792i	$-0.771068\pi$
$887$	−44.8123	−1.50465	−0.752325	+	0.658792i	$0.771068\pi$
$888$	0	0
$889$	−4.15571	−0.139378
$890$	0	0
$891$	1.53073	0.0512815
$892$	0	0
$893$	−42.0958	−1.40868
$894$	0	0
$895$	3.20435	0.107110
$896$	0	0
$897$	22.3160	0.745109
$898$	0	0
$899$	5.17401	0.172563
$900$	0	0
$901$	−11.1471	−0.371364
$902$	0	0
$903$	13.9864	0.465439
$904$	0	0
$905$	−21.1412	−0.702758
$906$	0	0
$907$	29.3297	0.973877	0.486939	−	0.873436i	$-0.338113\pi$
$907$	29.3297	0.973877	0.486939	+	0.873436i	$0.338113\pi$
$908$	0	0
$909$	0.457942	0.0151890
$910$	0	0
$911$	−4.91483	−0.162835	−0.0814177	−	0.996680i	$-0.525945\pi$
$911$	−4.91483	−0.162835	−0.0814177	+	0.996680i	$0.525945\pi$
$912$	0	0
$913$	6.31599	0.209029
$914$	0	0
$915$	30.7981	1.01815
$916$	0	0
$917$	72.5211	2.39486
$918$	0	0
$919$	−18.9465	−0.624986	−0.312493	−	0.949920i	$-0.601164\pi$
$919$	−18.9465	−0.624986	−0.312493	+	0.949920i	$0.601164\pi$
$920$	0	0
$921$	−26.2323	−0.864383
$922$	0	0
$923$	38.4008	1.26398
$924$	0	0
$925$	−44.6178	−1.46702
$926$	0	0
$927$	−7.77563	−0.255385
$928$	0	0
$929$	−8.40296	−0.275692	−0.137846	−	0.990454i	$-0.544018\pi$
$929$	−8.40296	−0.275692	−0.137846	+	0.990454i	$0.544018\pi$
$930$	0	0
$931$	62.2529	2.04026
$932$	0	0
$933$	9.12522	0.298746
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	−43.8100	−1.43121	−0.715605	−	0.698505i	$-0.753849\pi$
$937$	−43.8100	−1.43121	−0.715605	+	0.698505i	$0.753849\pi$
$938$	0	0
$939$	9.28093	0.302872
$940$	0	0
$941$	16.5363	0.539069	0.269534	−	0.962991i	$-0.413130\pi$
$941$	16.5363	0.539069	0.269534	+	0.962991i	$0.413130\pi$
$942$	0	0
$943$	12.1275	0.394926
$944$	0	0
$945$	−18.5786	−0.604363
$946$	0	0
$947$	46.7639	1.51962	0.759812	−	0.650143i	$-0.225291\pi$
$947$	46.7639	1.51962	0.759812	+	0.650143i	$0.225291\pi$
$948$	0	0
$949$	−19.5329	−0.634064
$950$	0	0
$951$	−32.3652	−1.04951
$952$	0	0
$953$	−43.4436	−1.40728	−0.703639	−	0.710558i	$-0.748443\pi$
$953$	−43.4436	−1.40728	−0.703639	+	0.710558i	$0.748443\pi$
$954$	0	0
$955$	100.708	3.25883
$956$	0	0
$957$	−2.85108	−0.0921622
$958$	0	0
$959$	8.13651	0.262742
$960$	0	0
$961$	−23.2832	−0.751071
$962$	0	0
$963$	−14.7047	−0.473854
$964$	0	0
$965$	−92.3782	−2.97376
$966$	0	0
$967$	41.3718	1.33043	0.665214	−	0.746653i	$-0.268340\pi$
$967$	41.3718	1.33043	0.665214	+	0.746653i	$0.268340\pi$
$968$	0	0
$969$	5.65685	0.181724
$970$	0	0
$971$	−55.1541	−1.76998	−0.884989	−	0.465612i	$-0.845834\pi$
$971$	−55.1541	−1.76998	−0.884989	+	0.465612i	$0.845834\pi$
$972$	0	0
$973$	84.7670	2.71751
$974$	0	0
$975$	−62.5934	−2.00459
$976$	0	0
$977$	−46.9120	−1.50085	−0.750424	−	0.660957i	$-0.770151\pi$
$977$	−46.9120	−1.50085	−0.750424	+	0.660957i	$0.770151\pi$
$978$	0	0
$979$	12.2251	0.390715
$980$	0	0
$981$	1.09372	0.0349197
$982$	0	0
$983$	4.71195	0.150288	0.0751439	−	0.997173i	$-0.476058\pi$
$983$	4.71195	0.150288	0.0751439	+	0.997173i	$0.476058\pi$
$984$	0	0
$985$	60.6249	1.93167
$986$	0	0
$987$	−44.5483	−1.41799
$988$	0	0
$989$	−12.1275	−0.385632
$990$	0	0
$991$	13.3733	0.424817	0.212408	−	0.977181i	$-0.431869\pi$
$991$	13.3733	0.424817	0.212408	+	0.977181i	$0.431869\pi$
$992$	0	0
$993$	−21.8435	−0.693184
$994$	0	0
$995$	−79.5338	−2.52139
$996$	0	0
$997$	24.8445	0.786833	0.393416	−	0.919360i	$-0.371293\pi$
$997$	24.8445	0.786833	0.393416	+	0.919360i	$0.371293\pi$
$998$	0	0
$999$	3.97682	0.125821

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.a.j.1.1		4
3.2	odd	2		9216.2.a.ba.1.4		4
4.3	odd	2		3072.2.a.s.1.1		4
8.3	odd	2		3072.2.a.m.1.4		4
8.5	even	2		3072.2.a.p.1.4		4
12.11	even	2		9216.2.a.bm.1.4		4
16.3	odd	4		3072.2.d.e.1537.8		8
16.5	even	4		3072.2.d.j.1537.5		8
16.11	odd	4		3072.2.d.e.1537.1		8
16.13	even	4		3072.2.d.j.1537.4		8
24.5	odd	2		9216.2.a.z.1.1		4
24.11	even	2		9216.2.a.bl.1.1		4
32.3	odd	8		1536.2.j.j.1153.4	yes	8
32.5	even	8		1536.2.j.f.385.3	yes	8
32.11	odd	8		1536.2.j.j.385.4	yes	8
32.13	even	8		1536.2.j.f.1153.3	yes	8
32.19	odd	8		1536.2.j.e.1153.1	yes	8
32.21	even	8		1536.2.j.i.385.2	yes	8
32.27	odd	8		1536.2.j.e.385.1	✓	8
32.29	even	8		1536.2.j.i.1153.2	yes	8
96.5	odd	8		4608.2.k.bj.3457.4		8
96.11	even	8		4608.2.k.be.3457.1		8
96.29	odd	8		4608.2.k.bc.1153.1		8
96.35	even	8		4608.2.k.be.1153.1		8
96.53	odd	8		4608.2.k.bc.3457.1		8
96.59	even	8		4608.2.k.bh.3457.4		8
96.77	odd	8		4608.2.k.bj.1153.4		8
96.83	even	8		4608.2.k.bh.1153.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.1	✓	8	32.27	odd	8
1536.2.j.e.1153.1	yes	8	32.19	odd	8
1536.2.j.f.385.3	yes	8	32.5	even	8
1536.2.j.f.1153.3	yes	8	32.13	even	8
1536.2.j.i.385.2	yes	8	32.21	even	8
1536.2.j.i.1153.2	yes	8	32.29	even	8
1536.2.j.j.385.4	yes	8	32.11	odd	8
1536.2.j.j.1153.4	yes	8	32.3	odd	8
3072.2.a.j.1.1		4	1.1	even	1	trivial
3072.2.a.m.1.4		4	8.3	odd	2
3072.2.a.p.1.4		4	8.5	even	2
3072.2.a.s.1.1		4	4.3	odd	2
3072.2.d.e.1537.1		8	16.11	odd	4
3072.2.d.e.1537.8		8	16.3	odd	4
3072.2.d.j.1537.4		8	16.13	even	4
3072.2.d.j.1537.5		8	16.5	even	4
4608.2.k.bc.1153.1		8	96.29	odd	8
4608.2.k.bc.3457.1		8	96.53	odd	8
4608.2.k.be.1153.1		8	96.35	even	8
4608.2.k.be.3457.1		8	96.11	even	8
4608.2.k.bh.1153.4		8	96.83	even	8
4608.2.k.bh.3457.4		8	96.59	even	8
4608.2.k.bj.1153.4		8	96.77	odd	8
4608.2.k.bj.3457.4		8	96.5	odd	8
9216.2.a.z.1.1		4	24.5	odd	2
9216.2.a.ba.1.4		4	3.2	odd	2
9216.2.a.bl.1.1		4	24.11	even	2
9216.2.a.bm.1.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.02734 q^{5} -4.61313 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.02734 q^{5} -4.61313 q^{7} +1.00000 q^{9} +1.53073 q^{11} +5.57900 q^{13} +4.02734 q^{15} -1.29769 q^{17} +4.35916 q^{19} +4.61313 q^{21} -4.00000 q^{23} +11.2195 q^{25} -1.00000 q^{27} +1.86256 q^{29} +2.77791 q^{31} -1.53073 q^{33} +18.5786 q^{35} -3.97682 q^{37} -5.57900 q^{39} -3.03188 q^{41} +3.03188 q^{43} -4.02734 q^{45} -9.65685 q^{47} +14.2809 q^{49} +1.29769 q^{51} +8.58995 q^{53} -6.16478 q^{55} -4.35916 q^{57} +5.73418 q^{59} +7.64725 q^{61} -4.61313 q^{63} -22.4685 q^{65} -8.79565 q^{67} +4.00000 q^{69} +6.88311 q^{71} -3.50114 q^{73} -11.2195 q^{75} -7.06147 q^{77} -8.18252 q^{79} +1.00000 q^{81} +4.12612 q^{83} +5.22625 q^{85} -1.86256 q^{87} +7.98642 q^{89} -25.7366 q^{91} -2.77791 q^{93} -17.5558 q^{95} -1.40461 q^{97} +1.53073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9} + 8 q^{13} + 8 q^{21} - 16 q^{23} + 4 q^{25} - 4 q^{27} - 8 q^{31} + 16 q^{35} + 8 q^{37} - 8 q^{39} - 16 q^{47} + 4 q^{49} - 16 q^{55} + 16 q^{59} + 24 q^{61} - 8 q^{63} - 8 q^{65} - 16 q^{67} + 16 q^{69} - 16 q^{71} - 8 q^{73} - 4 q^{75} - 16 q^{77} - 24 q^{79} + 4 q^{81} - 8 q^{89} - 16 q^{91} + 8 q^{93} - 32 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9 + 8 * q^13 + 8 * q^21 - 16 * q^23 + 4 * q^25 - 4 * q^27 - 8 * q^31 + 16 * q^35 + 8 * q^37 - 8 * q^39 - 16 * q^47 + 4 * q^49 - 16 * q^55 + 16 * q^59 + 24 * q^61 - 8 * q^63 - 8 * q^65 - 16 * q^67 + 16 * q^69 - 16 * q^71 - 8 * q^73 - 4 * q^75 - 16 * q^77 - 24 * q^79 + 4 * q^81 - 8 * q^89 - 16 * q^91 + 8 * q^93 - 32 * q^95 - 16 * q^97

Introduction

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