LMFDB - Embedded newform 3072.2.d.e.1537.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1537,3072)]

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, 1, 0]))

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

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

chi := DirichletCharacter("3072.1537");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$24.5300435009$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.8
Root			$-0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.e.1537.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +4.02734i q^{5} -4.61313 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +4.02734i q^{5} -4.61313 q^{7} -1.00000 q^{9} +1.53073i q^{11} +5.57900i q^{13} -4.02734 q^{15} -1.29769 q^{17} -4.35916i q^{19} -4.61313i q^{21} -4.00000 q^{23} -11.2195 q^{25} -1.00000i q^{27} +1.86256i q^{29} -2.77791 q^{31} -1.53073 q^{33} -18.5786i q^{35} +3.97682i q^{37} -5.57900 q^{39} +3.03188 q^{41} +3.03188i q^{43} -4.02734i q^{45} +9.65685 q^{47} +14.2809 q^{49} -1.29769i q^{51} -8.58995i q^{53} -6.16478 q^{55} +4.35916 q^{57} +5.73418i q^{59} +7.64725i q^{61} +4.61313 q^{63} -22.4685 q^{65} +8.79565i q^{67} -4.00000i q^{69} +6.88311 q^{71} +3.50114 q^{73} -11.2195i q^{75} -7.06147i q^{77} +8.18252 q^{79} +1.00000 q^{81} -4.12612i q^{83} -5.22625i q^{85} -1.86256 q^{87} -7.98642 q^{89} -25.7366i q^{91} -2.77791i q^{93} +17.5558 q^{95} -1.40461 q^{97} -1.53073i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{7} - 8 q^{9}+O(q^{10}) $ 8 * q - 16 * q^7 - 8 * q^9	$ 8 q - 16 q^{7} - 8 q^{9} - 32 q^{23} - 8 q^{25} + 16 q^{31} - 16 q^{39} + 32 q^{47} + 8 q^{49} - 32 q^{55} + 16 q^{63} - 16 q^{65} - 32 q^{71} + 16 q^{73} + 48 q^{79} + 8 q^{81} + 16 q^{89} + 64 q^{95} - 32 q^{97}+O(q^{100}) $ 8 * q - 16 * q^7 - 8 * q^9 - 32 * q^23 - 8 * q^25 + 16 * q^31 - 16 * q^39 + 32 * q^47 + 8 * q^49 - 32 * q^55 + 16 * q^63 - 16 * q^65 - 32 * q^71 + 16 * q^73 + 48 * q^79 + 8 * q^81 + 16 * q^89 + 64 * q^95 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$.

$n$	$1025$	$2047$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	4.02734i	1.80108i	0.434772	+	0.900540i	$0.356829\pi$
$5$	4.02734i	1.80108i	−0.434772	+	0.900540i	$0.643171\pi$
$6$	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
$11$	1.53073i	0.461534i	0.973009	+	0.230767i	$0.0741234\pi$
$11$	1.53073i	0.461534i	−0.973009	+	0.230767i	$0.925877\pi$
$12$	0	0
$13$	5.57900i	1.54734i	0.633591	+	0.773668i	$0.281580\pi$
$13$	5.57900i	1.54734i	−0.633591	+	0.773668i	$0.718420\pi$
$14$	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
$19$	− 4.35916i	− 1.00006i	−0.866008	−	0.500030i	$-0.833322\pi$
$19$	− 4.35916i	− 1.00006i	0.866008	−	0.500030i	$-0.166678\pi$
$20$	0	0
$21$	− 4.61313i	− 1.00667i
$22$	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
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	1.86256i	0.345868i	0.984933	+	0.172934i	$0.0553247\pi$
$29$	1.86256i	0.345868i	−0.984933	+	0.172934i	$0.944675\pi$
$30$	0	0
$31$	−2.77791	−0.498927	−0.249464	−	0.968384i	$-0.580254\pi$
$31$	−2.77791	−0.498927	−0.249464	+	0.968384i	$0.580254\pi$
$32$	0	0
$33$	−1.53073	−0.266467
$34$	0	0
$35$	− 18.5786i	− 3.14036i
$36$	0	0
$37$	3.97682i	0.653786i	0.945061	+	0.326893i	$0.106002\pi$
$37$	3.97682i	0.653786i	−0.945061	+	0.326893i	$0.893998\pi$
$38$	0	0
$39$	−5.57900	−0.893355
$40$	0	0
$41$	3.03188	0.473499	0.236750	−	0.971571i	$-0.423918\pi$
$41$	3.03188	0.473499	0.236750	+	0.971571i	$0.423918\pi$
$42$	0	0
$43$	3.03188i	0.462357i	0.972911	+	0.231178i	$0.0742581\pi$
$43$	3.03188i	0.462357i	−0.972911	+	0.231178i	$0.925742\pi$
$44$	0	0
$45$	− 4.02734i	− 0.600360i
$46$	0	0
$47$	9.65685	1.40860	0.704298	−	0.709904i	$-0.251262\pi$
$47$	9.65685	1.40860	0.704298	+	0.709904i	$0.251262\pi$
$48$	0	0
$49$	14.2809	2.04013
$50$	0	0
$51$	− 1.29769i	− 0.181713i
$52$	0	0
$53$	− 8.58995i	− 1.17992i	−0.807432	−	0.589960i	$-0.799143\pi$
$53$	− 8.58995i	− 1.17992i	0.807432	−	0.589960i	$-0.200857\pi$
$54$	0	0
$55$	−6.16478	−0.831259
$56$	0	0
$57$	4.35916	0.577385
$58$	0	0
$59$	5.73418i	0.746527i	0.927725	+	0.373263i	$0.121761\pi$
$59$	5.73418i	0.746527i	−0.927725	+	0.373263i	$0.878239\pi$
$60$	0	0
$61$	7.64725i	0.979131i	0.871967	+	0.489565i	$0.162845\pi$
$61$	7.64725i	0.979131i	−0.871967	+	0.489565i	$0.837155\pi$
$62$	0	0
$63$	4.61313	0.581199
$64$	0	0
$65$	−22.4685	−2.78688
$66$	0	0
$67$	8.79565i	1.07456i	0.843404	+	0.537280i	$0.180548\pi$
$67$	8.79565i	1.07456i	−0.843404	+	0.537280i	$0.819452\pi$
$68$	0	0
$69$	− 4.00000i	− 0.481543i
$70$	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	$-0.434317\pi$
$73$	3.50114	0.409778	0.204889	+	0.978785i	$0.434317\pi$
$74$	0	0
$75$	− 11.2195i	− 1.29551i
$76$	0	0
$77$	− 7.06147i	− 0.804729i
$78$	0	0
$79$	8.18252	0.920606	0.460303	−	0.887762i	$-0.347741\pi$
$79$	8.18252	0.920606	0.460303	+	0.887762i	$0.347741\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.12612i	− 0.452901i	−0.974023	−	0.226450i	$-0.927288\pi$
$83$	− 4.12612i	− 0.452901i	0.974023	−	0.226450i	$-0.0727121\pi$
$84$	0	0
$85$	− 5.22625i	− 0.566867i
$86$	0	0
$87$	−1.86256	−0.199687
$88$	0	0
$89$	−7.98642	−0.846559	−0.423280	−	0.905999i	$-0.639121\pi$
$89$	−7.98642	−0.846559	−0.423280	+	0.905999i	$0.639121\pi$
$90$	0	0
$91$	− 25.7366i	− 2.69793i
$92$	0	0
$93$	− 2.77791i	− 0.288056i
$94$	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
$99$	− 1.53073i	− 0.153845i
$100$	0	0
$101$	− 0.457942i	− 0.0455669i	−0.999740	−	0.0227835i	$-0.992747\pi$
$101$	− 0.457942i	− 0.0455669i	0.999740	−	0.0227835i	$-0.00725283\pi$
$102$	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
$107$	− 14.7047i	− 1.42156i	−0.703414	−	0.710781i	$-0.748342\pi$
$107$	− 14.7047i	− 1.42156i	0.703414	−	0.710781i	$-0.251658\pi$
$108$	0	0
$109$	1.09372i	0.104759i	0.998627	+	0.0523795i	$0.0166806\pi$
$109$	1.09372i	0.104759i	−0.998627	+	0.0523795i	$0.983319\pi$
$110$	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
$115$	− 16.1094i	− 1.50221i
$116$	0	0
$117$	− 5.57900i	− 0.515779i
$118$	0	0
$119$	5.98642	0.548775
$120$	0	0
$121$	8.65685	0.786987
$122$	0	0
$123$	3.03188i	0.273375i
$124$	0	0
$125$	− 25.0479i	− 2.24035i
$126$	0	0
$127$	−0.900845	−0.0799371	−0.0399685	−	0.999201i	$-0.512726\pi$
$127$	−0.900845	−0.0799371	−0.0399685	+	0.999201i	$0.512726\pi$
$128$	0	0
$129$	−3.03188	−0.266942
$130$	0	0
$131$	15.7206i	1.37352i	0.726886	+	0.686758i	$0.240967\pi$
$131$	15.7206i	1.37352i	−0.726886	+	0.686758i	$0.759033\pi$
$132$	0	0
$133$	20.1094i	1.74370i
$134$	0	0
$135$	4.02734	0.346618
$136$	0	0
$137$	1.76377	0.150689	0.0753447	−	0.997158i	$-0.475994\pi$
$137$	1.76377	0.150689	0.0753447	+	0.997158i	$0.475994\pi$
$138$	0	0
$139$	− 18.3752i	− 1.55856i	−0.626675	−	0.779281i	$-0.715584\pi$
$139$	− 18.3752i	− 1.55856i	0.626675	−	0.779281i	$-0.284416\pi$
$140$	0	0
$141$	9.65685i	0.813254i
$142$	0	0
$143$	−8.53996	−0.714147
$144$	0	0
$145$	−7.50114	−0.622936
$146$	0	0
$147$	14.2809i	1.17787i
$148$	0	0
$149$	− 1.68419i	− 0.137975i	−0.997618	−	0.0689873i	$-0.978023\pi$
$149$	− 1.68419i	− 0.137975i	0.997618	−	0.0689873i	$-0.0219768\pi$
$150$	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
$155$	− 11.1876i	− 0.898609i
$156$	0	0
$157$	13.6337i	1.08809i	0.839057	+	0.544043i	$0.183107\pi$
$157$	13.6337i	1.08809i	−0.839057	+	0.544043i	$0.816893\pi$
$158$	0	0
$159$	8.58995	0.681227
$160$	0	0
$161$	18.4525	1.45426
$162$	0	0
$163$	− 3.17476i	− 0.248666i	−0.992241	−	0.124333i	$-0.960321\pi$
$163$	− 3.17476i	− 0.248666i	0.992241	−	0.124333i	$-0.0396791\pi$
$164$	0	0
$165$	− 6.16478i	− 0.479928i
$166$	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
$171$	4.35916i	0.333353i
$172$	0	0
$173$	− 11.0341i	− 0.838909i	−0.907776	−	0.419455i	$-0.862221\pi$
$173$	− 11.0341i	− 0.838909i	0.907776	−	0.419455i	$-0.137779\pi$
$174$	0	0
$175$	51.7568	3.91245
$176$	0	0
$177$	−5.73418	−0.431008
$178$	0	0
$179$	0.795649i	0.0594696i	0.999558	+	0.0297348i	$0.00946628\pi$
$179$	0.795649i	0.0594696i	−0.999558	+	0.0297348i	$0.990534\pi$
$180$	0	0
$181$	− 5.24943i	− 0.390187i	−0.980785	−	0.195093i	$-0.937499\pi$
$181$	− 5.24943i	− 0.390187i	0.980785	−	0.195093i	$-0.0625010\pi$
$182$	0	0
$183$	−7.64725	−0.565301
$184$	0	0
$185$	−16.0160	−1.17752
$186$	0	0
$187$	− 1.98642i	− 0.145262i
$188$	0	0
$189$	4.61313i	0.335556i
$190$	0	0
$191$	25.0060	1.80937	0.904687	−	0.426077i	$-0.140105\pi$
$191$	25.0060	1.80937	0.904687	+	0.426077i	$0.140105\pi$
$192$	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
$195$	− 22.4685i	− 1.60900i
$196$	0	0
$197$	15.0533i	1.07251i	0.844058	+	0.536253i	$0.180161\pi$
$197$	15.0533i	1.07251i	−0.844058	+	0.536253i	$0.819839\pi$
$198$	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
$203$	− 8.59220i	− 0.603054i
$204$	0	0
$205$	12.2104i	0.852811i
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	6.67271	0.461561
$210$	0	0
$211$	0.938533i	0.0646112i	0.999478	+	0.0323056i	$0.0102850\pi$
$211$	0.938533i	0.0646112i	−0.999478	+	0.0323056i	$0.989715\pi$
$212$	0	0
$213$	6.88311i	0.471623i
$214$	0	0
$215$	−12.2104	−0.832742
$216$	0	0
$217$	12.8149	0.869929
$218$	0	0
$219$	3.50114i	0.236585i
$220$	0	0
$221$	− 7.23983i	− 0.487004i
$222$	0	0
$223$	−26.2783	−1.75973	−0.879863	−	0.475228i	$-0.842366\pi$
$223$	−26.2783	−1.75973	−0.879863	+	0.475228i	$0.842366\pi$
$224$	0	0
$225$	11.2195	0.747964
$226$	0	0
$227$	− 10.9082i	− 0.724002i	−0.932178	−	0.362001i	$-0.882094\pi$
$227$	− 10.9082i	− 0.724002i	0.932178	−	0.362001i	$-0.117906\pi$
$228$	0	0
$229$	− 18.8599i	− 1.24630i	−0.782103	−	0.623150i	$-0.785853\pi$
$229$	− 18.8599i	− 1.24630i	0.782103	−	0.623150i	$-0.214147\pi$
$230$	0	0
$231$	7.06147	0.464610
$232$	0	0
$233$	−19.7662	−1.29493	−0.647464	−	0.762096i	$-0.724170\pi$
$233$	−19.7662	−1.29493	−0.647464	+	0.762096i	$0.724170\pi$
$234$	0	0
$235$	38.8914i	2.53700i
$236$	0	0
$237$	8.18252i	0.531512i
$238$	0	0
$239$	−14.3515	−0.928319	−0.464160	−	0.885752i	$-0.653644\pi$
$239$	−14.3515	−0.928319	−0.464160	+	0.885752i	$0.653644\pi$
$240$	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
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	57.5142i	3.67444i
$246$	0	0
$247$	24.3197	1.54743
$248$	0	0
$249$	4.12612	0.261482
$250$	0	0
$251$	− 9.53073i	− 0.601575i	−0.953691	−	0.300787i	$-0.902751\pi$
$251$	− 9.53073i	− 0.601575i	0.953691	−	0.300787i	$-0.0972494\pi$
$252$	0	0
$253$	− 6.12293i	− 0.384946i
$254$	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
$259$	− 18.3456i	− 1.13994i
$260$	0	0
$261$	− 1.86256i	− 0.115289i
$262$	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
$267$	− 7.98642i	− 0.488761i
$268$	0	0
$269$	− 14.6646i	− 0.894115i	−0.894505	−	0.447057i	$-0.852472\pi$
$269$	− 14.6646i	− 0.894115i	0.894505	−	0.447057i	$-0.147528\pi$
$270$	0	0
$271$	−20.3574	−1.23663	−0.618313	−	0.785932i	$-0.712184\pi$
$271$	−20.3574	−1.23663	−0.618313	+	0.785932i	$0.712184\pi$
$272$	0	0
$273$	25.7366	1.55765
$274$	0	0
$275$	− 17.1740i	− 1.03563i
$276$	0	0
$277$	23.8113i	1.43068i	0.698776	+	0.715341i	$0.253729\pi$
$277$	23.8113i	1.43068i	−0.698776	+	0.715341i	$0.746271\pi$
$278$	0	0
$279$	2.77791	0.166309
$280$	0	0
$281$	17.1116	1.02079	0.510397	−	0.859939i	$-0.329498\pi$
$281$	17.1116	1.02079	0.510397	+	0.859939i	$0.329498\pi$
$282$	0	0
$283$	2.26582i	0.134689i	0.997730	+	0.0673444i	$0.0214526\pi$
$283$	2.26582i	0.134689i	−0.997730	+	0.0673444i	$0.978547\pi$
$284$	0	0
$285$	17.5558i	1.03992i
$286$	0	0
$287$	−13.9864	−0.825592
$288$	0	0
$289$	−15.3160	−0.900941
$290$	0	0
$291$	− 1.40461i	− 0.0823399i
$292$	0	0
$293$	− 3.90366i	− 0.228054i	−0.993478	−	0.114027i	$-0.963625\pi$
$293$	− 3.90366i	− 0.228054i	0.993478	−	0.114027i	$-0.0363751\pi$
$294$	0	0
$295$	−23.0935	−1.34456
$296$	0	0
$297$	1.53073	0.0888222
$298$	0	0
$299$	− 22.3160i	− 1.29057i
$300$	0	0
$301$	− 13.9864i	− 0.806164i
$302$	0	0
$303$	0.457942	0.0263081
$304$	0	0
$305$	−30.7981	−1.76349
$306$	0	0
$307$	− 26.2323i	− 1.49716i	−0.663047	−	0.748578i	$-0.730737\pi$
$307$	− 26.2323i	− 1.49716i	0.663047	−	0.748578i	$-0.269263\pi$
$308$	0	0
$309$	− 7.77563i	− 0.442340i
$310$	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	$-0.415521\pi$
$313$	9.28093	0.524589	0.262295	+	0.964988i	$0.415521\pi$
$314$	0	0
$315$	18.5786i	1.04679i
$316$	0	0
$317$	32.3652i	1.81781i	0.417000	+	0.908906i	$0.363081\pi$
$317$	32.3652i	1.81781i	−0.417000	+	0.908906i	$0.636919\pi$
$318$	0	0
$319$	−2.85108	−0.159630
$320$	0	0
$321$	14.7047	0.820739
$322$	0	0
$323$	5.65685i	0.314756i
$324$	0	0
$325$	− 62.5934i	− 3.47206i
$326$	0	0
$327$	−1.09372	−0.0604827
$328$	0	0
$329$	−44.5483	−2.45603
$330$	0	0
$331$	21.8435i	1.20063i	0.799764	+	0.600315i	$0.204958\pi$
$331$	21.8435i	1.20063i	−0.799764	+	0.600315i	$0.795042\pi$
$332$	0	0
$333$	− 3.97682i	− 0.217929i
$334$	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
$339$	− 7.65685i	− 0.415863i
$340$	0	0
$341$	− 4.25224i	− 0.230272i
$342$	0	0
$343$	−33.5879	−1.81357
$344$	0	0
$345$	16.1094	0.867299
$346$	0	0
$347$	21.9697i	1.17939i	0.807625	+	0.589697i	$0.200753\pi$
$347$	21.9697i	1.17939i	−0.807625	+	0.589697i	$0.799247\pi$
$348$	0	0
$349$	− 20.0862i	− 1.07519i	−0.843204	−	0.537594i	$-0.819333\pi$
$349$	− 20.0862i	− 1.07519i	0.843204	−	0.537594i	$-0.180667\pi$
$350$	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
$355$	27.7206i	1.47126i
$356$	0	0
$357$	5.98642i	0.316835i
$358$	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
$363$	8.65685i	0.454367i
$364$	0	0
$365$	14.1003i	0.738043i
$366$	0	0
$367$	0.535798	0.0279684	0.0139842	−	0.999902i	$-0.495549\pi$
$367$	0.535798	0.0279684	0.0139842	+	0.999902i	$0.495549\pi$
$368$	0	0
$369$	−3.03188	−0.157833
$370$	0	0
$371$	39.6265i	2.05731i
$372$	0	0
$373$	− 8.83803i	− 0.457616i	−0.973472	−	0.228808i	$-0.926517\pi$
$373$	− 8.83803i	− 0.457616i	0.973472	−	0.228808i	$-0.0734828\pi$
$374$	0	0
$375$	25.0479	1.29347
$376$	0	0
$377$	−10.3912	−0.535174
$378$	0	0
$379$	− 24.6887i	− 1.26817i	−0.773261	−	0.634087i	$-0.781376\pi$
$379$	− 24.6887i	− 1.26817i	0.773261	−	0.634087i	$-0.218624\pi$
$380$	0	0
$381$	− 0.900845i	− 0.0461517i
$382$	0	0
$383$	−31.1290	−1.59062	−0.795308	−	0.606205i	$-0.792691\pi$
$383$	−31.1290	−1.59062	−0.795308	+	0.606205i	$0.792691\pi$
$384$	0	0
$385$	28.4389	1.44938
$386$	0	0
$387$	− 3.03188i	− 0.154119i
$388$	0	0
$389$	20.3614i	1.03236i	0.856479	+	0.516182i	$0.172647\pi$
$389$	20.3614i	1.03236i	−0.856479	+	0.516182i	$0.827353\pi$
$390$	0	0
$391$	5.19077	0.262509
$392$	0	0
$393$	−15.7206	−0.793000
$394$	0	0
$395$	32.9538i	1.65809i
$396$	0	0
$397$	− 3.60090i	− 0.180724i	−0.995909	−	0.0903620i	$-0.971198\pi$
$397$	− 3.60090i	− 0.180724i	0.995909	−	0.0903620i	$-0.0288024\pi$
$398$	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
$403$	− 15.4980i	− 0.772008i
$404$	0	0
$405$	4.02734i	0.200120i
$406$	0	0
$407$	−6.08746	−0.301744
$408$	0	0
$409$	25.1572	1.24395	0.621973	−	0.783039i	$-0.286331\pi$
$409$	25.1572	1.24395	0.621973	+	0.783039i	$0.286331\pi$
$410$	0	0
$411$	1.76377i	0.0870006i
$412$	0	0
$413$	− 26.4525i	− 1.30164i
$414$	0	0
$415$	16.6173	0.815711
$416$	0	0
$417$	18.3752	0.899836
$418$	0	0
$419$	− 6.52845i	− 0.318936i	−0.987203	−	0.159468i	$-0.949022\pi$
$419$	− 6.52845i	− 0.318936i	0.987203	−	0.159468i	$-0.0509778\pi$
$420$	0	0
$421$	− 19.8758i	− 0.968687i	−0.874878	−	0.484343i	$-0.839059\pi$
$421$	− 19.8758i	− 0.968687i	0.874878	−	0.484343i	$-0.160941\pi$
$422$	0	0
$423$	−9.65685	−0.469532
$424$	0	0
$425$	14.5594	0.706236
$426$	0	0
$427$	− 35.2777i	− 1.70721i
$428$	0	0
$429$	− 8.53996i	− 0.412313i
$430$	0	0
$431$	34.7368	1.67321	0.836606	−	0.547805i	$-0.184537\pi$
$431$	34.7368	1.67321	0.836606	+	0.547805i	$0.184537\pi$
$432$	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
$435$	− 7.50114i	− 0.359652i
$436$	0	0
$437$	17.4366i	0.834108i
$438$	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
$443$	− 5.71742i	− 0.271643i	−0.990733	−	0.135821i	$-0.956633\pi$
$443$	− 5.71742i	− 0.271643i	0.990733	−	0.135821i	$-0.0433673\pi$
$444$	0	0
$445$	− 32.1640i	− 1.52472i
$446$	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
$451$	4.64099i	0.218536i
$452$	0	0
$453$	− 10.3118i	− 0.484492i
$454$	0	0
$455$	103.650	4.85919
$456$	0	0
$457$	33.6233	1.57283	0.786416	−	0.617697i	$-0.211934\pi$
$457$	33.6233	1.57283	0.786416	+	0.617697i	$0.211934\pi$
$458$	0	0
$459$	1.29769i	0.0605711i
$460$	0	0
$461$	− 5.27730i	− 0.245788i	−0.992420	−	0.122894i	$-0.960782\pi$
$461$	− 5.27730i	− 0.245788i	0.992420	−	0.122894i	$-0.0392176\pi$
$462$	0	0
$463$	11.1430	0.517857	0.258929	−	0.965896i	$-0.416631\pi$
$463$	11.1430	0.517857	0.258929	+	0.965896i	$0.416631\pi$
$464$	0	0
$465$	11.1876	0.518812
$466$	0	0
$467$	26.5013i	1.22633i	0.789953	+	0.613167i	$0.210105\pi$
$467$	26.5013i	1.22633i	−0.789953	+	0.613167i	$0.789895\pi$
$468$	0	0
$469$	− 40.5754i	− 1.87360i
$470$	0	0
$471$	−13.6337	−0.628207
$472$	0	0
$473$	−4.64099	−0.213393
$474$	0	0
$475$	48.9074i	2.24403i
$476$	0	0
$477$	8.58995i	0.393307i
$478$	0	0
$479$	−22.4170	−1.02426	−0.512130	−	0.858908i	$-0.671143\pi$
$479$	−22.4170	−1.02426	−0.512130	+	0.858908i	$0.671143\pi$
$480$	0	0
$481$	−22.1867	−1.01163
$482$	0	0
$483$	18.4525i	0.839618i
$484$	0	0
$485$	− 5.65685i	− 0.256865i
$486$	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
$491$	− 17.3711i	− 0.783946i	−0.919977	−	0.391973i	$-0.871793\pi$
$491$	− 17.3711i	− 0.783946i	0.919977	−	0.391973i	$-0.128207\pi$
$492$	0	0
$493$	− 2.41703i	− 0.108857i
$494$	0	0
$495$	6.16478	0.277086
$496$	0	0
$497$	−31.7526	−1.42430
$498$	0	0
$499$	37.2938i	1.66950i	0.550631	+	0.834749i	$0.314387\pi$
$499$	37.2938i	1.66950i	−0.550631	+	0.834749i	$0.685613\pi$
$500$	0	0
$501$	− 18.4170i	− 0.822812i
$502$	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
$507$	− 18.1252i	− 0.804969i
$508$	0	0
$509$	− 34.6311i	− 1.53499i	−0.641052	−	0.767497i	$-0.721502\pi$
$509$	− 34.6311i	− 1.53499i	0.641052	−	0.767497i	$-0.278498\pi$
$510$	0	0
$511$	−16.1512	−0.714487
$512$	0	0
$513$	−4.35916	−0.192462
$514$	0	0
$515$	− 31.3151i	− 1.37991i
$516$	0	0
$517$	14.7821i	0.650115i
$518$	0	0
$519$	11.0341	0.484344
$520$	0	0
$521$	−34.8437	−1.52653	−0.763265	−	0.646086i	$-0.776405\pi$
$521$	−34.8437	−1.52653	−0.763265	+	0.646086i	$0.776405\pi$
$522$	0	0
$523$	25.5499i	1.11722i	0.829430	+	0.558610i	$0.188665\pi$
$523$	25.5499i	1.11722i	−0.829430	+	0.558610i	$0.811335\pi$
$524$	0	0
$525$	51.7568i	2.25885i
$526$	0	0
$527$	3.60488	0.157031
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 5.73418i	− 0.248842i
$532$	0	0
$533$	16.9148i	0.732662i
$534$	0	0
$535$	59.2210	2.56035
$536$	0	0
$537$	−0.795649	−0.0343348
$538$	0	0
$539$	21.8603i	0.941590i
$540$	0	0
$541$	32.8735i	1.41334i	0.707542	+	0.706671i	$0.249804\pi$
$541$	32.8735i	1.41334i	−0.707542	+	0.706671i	$0.750196\pi$
$542$	0	0
$543$	5.24943	0.225275
$544$	0	0
$545$	−4.40477	−0.188680
$546$	0	0
$547$	32.4414i	1.38709i	0.720412	+	0.693546i	$0.243953\pi$
$547$	32.4414i	1.38709i	−0.720412	+	0.693546i	$0.756047\pi$
$548$	0	0
$549$	− 7.64725i	− 0.326377i
$550$	0	0
$551$	8.11918	0.345889
$552$	0	0
$553$	−37.7470	−1.60517
$554$	0	0
$555$	− 16.0160i	− 0.679842i
$556$	0	0
$557$	− 2.25025i	− 0.0953462i	−0.998863	−	0.0476731i	$-0.984819\pi$
$557$	− 2.25025i	− 0.0953462i	0.998863	−	0.0476731i	$-0.0151806\pi$
$558$	0	0
$559$	−16.9148	−0.715421
$560$	0	0
$561$	1.98642	0.0838668
$562$	0	0
$563$	1.37608i	0.0579948i	0.999579	+	0.0289974i	$0.00923145\pi$
$563$	1.37608i	0.0579948i	−0.999579	+	0.0289974i	$0.990769\pi$
$564$	0	0
$565$	− 30.8368i	− 1.29731i
$566$	0	0
$567$	−4.61313	−0.193733
$568$	0	0
$569$	−15.5043	−0.649975	−0.324988	−	0.945718i	$-0.605360\pi$
$569$	−15.5043	−0.649975	−0.324988	+	0.945718i	$0.605360\pi$
$570$	0	0
$571$	7.45659i	0.312049i	0.987753	+	0.156024i	$0.0498678\pi$
$571$	7.45659i	0.312049i	−0.987753	+	0.156024i	$0.950132\pi$
$572$	0	0
$573$	25.0060i	1.04464i
$574$	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
$579$	22.9378i	0.953262i
$580$	0	0
$581$	19.0343i	0.789676i
$582$	0	0
$583$	13.1489	0.544573
$584$	0	0
$585$	22.4685	0.928959
$586$	0	0
$587$	33.6233i	1.38778i	0.720079	+	0.693892i	$0.244105\pi$
$587$	33.6233i	1.38778i	−0.720079	+	0.693892i	$0.755895\pi$
$588$	0	0
$589$	12.1094i	0.498957i
$590$	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
$595$	24.1094i	0.988387i
$596$	0	0
$597$	19.7485i	0.808251i
$598$	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	$-0.422550\pi$
$601$	11.8126	0.481845	0.240922	+	0.970544i	$0.422550\pi$
$602$	0	0
$603$	− 8.79565i	− 0.358187i
$604$	0	0
$605$	34.8641i	1.41743i
$606$	0	0
$607$	4.66330	0.189277	0.0946387	−	0.995512i	$-0.469830\pi$
$607$	4.66330	0.189277	0.0946387	+	0.995512i	$0.469830\pi$
$608$	0	0
$609$	8.59220	0.348174
$610$	0	0
$611$	53.8756i	2.17957i
$612$	0	0
$613$	28.5715i	1.15399i	0.816748	+	0.576995i	$0.195775\pi$
$613$	28.5715i	1.15399i	−0.816748	+	0.576995i	$0.804225\pi$
$614$	0	0
$615$	−12.2104	−0.492371
$616$	0	0
$617$	21.2482	0.855418	0.427709	−	0.903916i	$-0.359321\pi$
$617$	21.2482	0.855418	0.427709	+	0.903916i	$0.359321\pi$
$618$	0	0
$619$	− 11.4412i	− 0.459861i	−0.973207	−	0.229931i	$-0.926150\pi$
$619$	− 11.4412i	− 0.459861i	0.973207	−	0.229931i	$-0.0738499\pi$
$620$	0	0
$621$	4.00000i	0.160514i
$622$	0	0
$623$	36.8424	1.47606
$624$	0	0
$625$	44.7790	1.79116
$626$	0	0
$627$	6.67271i	0.266483i
$628$	0	0
$629$	− 5.16070i	− 0.205770i
$630$	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
$635$	− 3.62801i	− 0.143973i
$636$	0	0
$637$	79.6733i	3.15677i
$638$	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
$643$	20.6114i	0.812834i	0.913688	+	0.406417i	$0.133222\pi$
$643$	20.6114i	0.812834i	−0.913688	+	0.406417i	$0.866778\pi$
$644$	0	0
$645$	− 12.2104i	− 0.480784i
$646$	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
$651$	12.8149i	0.502254i
$652$	0	0
$653$	− 51.0435i	− 1.99749i	−0.0501153	−	0.998743i	$-0.515959\pi$
$653$	− 51.0435i	− 1.99749i	0.0501153	−	0.998743i	$-0.484041\pi$
$654$	0	0
$655$	−63.3122	−2.47381
$656$	0	0
$657$	−3.50114	−0.136593
$658$	0	0
$659$	− 5.68401i	− 0.221418i	−0.993853	−	0.110709i	$-0.964688\pi$
$659$	− 5.68401i	− 0.221418i	0.993853	−	0.110709i	$-0.0353121\pi$
$660$	0	0
$661$	36.2611i	1.41039i	0.709012	+	0.705197i	$0.249141\pi$
$661$	36.2611i	1.41039i	−0.709012	+	0.705197i	$0.750859\pi$
$662$	0	0
$663$	7.23983	0.281172
$664$	0	0
$665$	−80.9872	−3.14055
$666$	0	0
$667$	− 7.45022i	− 0.288474i
$668$	0	0
$669$	− 26.2783i	− 1.01598i
$670$	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
$675$	11.2195i	0.431837i
$676$	0	0
$677$	18.0866i	0.695124i	0.937657	+	0.347562i	$0.112990\pi$
$677$	18.0866i	0.695124i	−0.937657	+	0.347562i	$0.887010\pi$
$678$	0	0
$679$	6.47966	0.248666
$680$	0	0
$681$	10.9082	0.418003
$682$	0	0
$683$	20.3448i	0.778474i	0.921138	+	0.389237i	$0.127261\pi$
$683$	20.3448i	0.778474i	−0.921138	+	0.389237i	$0.872739\pi$
$684$	0	0
$685$	7.10332i	0.271404i
$686$	0	0
$687$	18.8599	0.719551
$688$	0	0
$689$	47.9233	1.82573
$690$	0	0
$691$	20.1118i	0.765089i	0.923937	+	0.382544i	$0.124952\pi$
$691$	20.1118i	0.765089i	−0.923937	+	0.382544i	$0.875048\pi$
$692$	0	0
$693$	7.06147i	0.268243i
$694$	0	0
$695$	74.0031	2.80710
$696$	0	0
$697$	−3.93444	−0.149028
$698$	0	0
$699$	− 19.7662i	− 0.747627i
$700$	0	0
$701$	− 19.9263i	− 0.752606i	−0.926497	−	0.376303i	$-0.877195\pi$
$701$	− 19.9263i	− 0.752606i	0.926497	−	0.376303i	$-0.122805\pi$
$702$	0	0
$703$	17.3356	0.653825
$704$	0	0
$705$	−38.8914	−1.46474
$706$	0	0
$707$	2.11254i	0.0794504i
$708$	0	0
$709$	− 3.30141i	− 0.123987i	−0.998077	−	0.0619935i	$-0.980254\pi$
$709$	− 3.30141i	− 0.123987i	0.998077	−	0.0619935i	$-0.0197458\pi$
$710$	0	0
$711$	−8.18252	−0.306869
$712$	0	0
$713$	11.1116	0.416134
$714$	0	0
$715$	− 34.3933i	− 1.28624i
$716$	0	0
$717$	− 14.3515i	− 0.535965i
$718$	0	0
$719$	−40.2104	−1.49959	−0.749797	−	0.661668i	$-0.769849\pi$
$719$	−40.2104	−1.49959	−0.749797	+	0.661668i	$0.769849\pi$
$720$	0	0
$721$	35.8699	1.33587
$722$	0	0
$723$	21.7534i	0.809017i
$724$	0	0
$725$	− 20.8969i	− 0.776090i
$726$	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
$731$	− 3.93444i	− 0.145521i
$732$	0	0
$733$	− 20.7370i	− 0.765938i	−0.923761	−	0.382969i	$-0.874902\pi$
$733$	− 20.7370i	− 0.765938i	0.923761	−	0.382969i	$-0.125098\pi$
$734$	0	0
$735$	−57.5142	−2.12144
$736$	0	0
$737$	−13.4638	−0.495945
$738$	0	0
$739$	− 13.9529i	− 0.513266i	−0.966509	−	0.256633i	$-0.917387\pi$
$739$	− 13.9529i	− 0.513266i	0.966509	−	0.256633i	$-0.0826131\pi$
$740$	0	0
$741$	24.3197i	0.893408i
$742$	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
$747$	4.12612i	0.150967i
$748$	0	0
$749$	67.8348i	2.47863i
$750$	0	0
$751$	−5.44425	−0.198664	−0.0993318	−	0.995054i	$-0.531671\pi$
$751$	−5.44425	−0.198664	−0.0993318	+	0.995054i	$0.531671\pi$
$752$	0	0
$753$	9.53073	0.347319
$754$	0	0
$755$	− 41.5292i	− 1.51140i
$756$	0	0
$757$	− 14.7698i	− 0.536816i	−0.963305	−	0.268408i	$-0.913502\pi$
$757$	− 14.7698i	− 0.536816i	0.963305	−	0.268408i	$-0.0864976\pi$
$758$	0	0
$759$	6.12293	0.222248
$760$	0	0
$761$	1.96584	0.0712617	0.0356308	−	0.999365i	$-0.488656\pi$
$761$	1.96584	0.0712617	0.0356308	+	0.999365i	$0.488656\pi$
$762$	0	0
$763$	− 5.04545i	− 0.182658i
$764$	0	0
$765$	5.22625i	0.188956i
$766$	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
$771$	− 1.20435i	− 0.0433736i
$772$	0	0
$773$	− 27.5696i	− 0.991609i	−0.868434	−	0.495804i	$-0.834873\pi$
$773$	− 27.5696i	− 0.991609i	0.868434	−	0.495804i	$-0.165127\pi$
$774$	0	0
$775$	31.1667	1.11954
$776$	0	0
$777$	18.3456	0.658144
$778$	0	0
$779$	− 13.2164i	− 0.473528i
$780$	0	0
$781$	10.5362i	0.377015i
$782$	0	0
$783$	1.86256	0.0665623
$784$	0	0
$785$	−54.9074	−1.95973
$786$	0	0
$787$	29.6529i	1.05701i	0.848929	+	0.528506i	$0.177248\pi$
$787$	29.6529i	1.05701i	−0.848929	+	0.528506i	$0.822752\pi$
$788$	0	0
$789$	− 27.5777i	− 0.981793i
$790$	0	0
$791$	35.3220	1.25591
$792$	0	0
$793$	−42.6640	−1.51504
$794$	0	0
$795$	34.5946i	1.22695i
$796$	0	0
$797$	− 5.43832i	− 0.192635i	−0.995351	−	0.0963177i	$-0.969294\pi$
$797$	− 5.43832i	− 0.192635i	0.995351	−	0.0963177i	$-0.0307065\pi$
$798$	0	0
$799$	−12.5316	−0.443337
$800$	0	0
$801$	7.98642	0.282186
$802$	0	0
$803$	5.35932i	0.189126i
$804$	0	0
$805$	74.3145i	2.61924i
$806$	0	0
$807$	14.6646	0.516218
$808$	0	0
$809$	−17.1548	−0.603131	−0.301566	−	0.953445i	$-0.597509\pi$
$809$	−17.1548	−0.603131	−0.301566	+	0.953445i	$0.597509\pi$
$810$	0	0
$811$	9.46624i	0.332404i	0.986092	+	0.166202i	$0.0531505\pi$
$811$	9.46624i	0.332404i	−0.986092	+	0.166202i	$0.946850\pi$
$812$	0	0
$813$	− 20.3574i	− 0.713966i
$814$	0	0
$815$	12.7858	0.447868
$816$	0	0
$817$	13.2164	0.462384
$818$	0	0
$819$	25.7366i	0.899310i
$820$	0	0
$821$	− 19.1198i	− 0.667285i	−0.942700	−	0.333642i	$-0.891722\pi$
$821$	− 19.1198i	− 0.667285i	0.942700	−	0.333642i	$-0.108278\pi$
$822$	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
$827$	26.9841i	0.938330i	0.883110	+	0.469165i	$0.155445\pi$
$827$	26.9841i	0.938330i	−0.883110	+	0.469165i	$0.844555\pi$
$828$	0	0
$829$	− 12.4704i	− 0.433116i	−0.976270	−	0.216558i	$-0.930517\pi$
$829$	− 12.4704i	− 0.433116i	0.976270	−	0.216558i	$-0.0694830\pi$
$830$	0	0
$831$	−23.8113	−0.826005
$832$	0	0
$833$	−18.5323	−0.642105
$834$	0	0
$835$	− 74.1716i	− 2.56681i
$836$	0	0
$837$	2.77791i	0.0960186i
$838$	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
$843$	17.1116i	0.589356i
$844$	0	0
$845$	− 72.9964i	− 2.51115i
$846$	0	0
$847$	−39.9352	−1.37219
$848$	0	0
$849$	−2.26582	−0.0777627
$850$	0	0
$851$	− 15.9073i	− 0.545295i
$852$	0	0
$853$	− 16.1955i	− 0.554525i	−0.960794	−	0.277262i	$-0.910573\pi$
$853$	− 16.1955i	− 0.554525i	0.960794	−	0.277262i	$-0.0894271\pi$
$854$	0	0
$855$	−17.5558	−0.600396
$856$	0	0
$857$	28.3592	0.968730	0.484365	−	0.874866i	$-0.339051\pi$
$857$	28.3592	0.968730	0.484365	+	0.874866i	$0.339051\pi$
$858$	0	0
$859$	− 2.51562i	− 0.0858319i	−0.999079	−	0.0429159i	$-0.986335\pi$
$859$	− 2.51562i	− 0.0858319i	0.999079	−	0.0429159i	$-0.0136648\pi$
$860$	0	0
$861$	− 13.9864i	− 0.476656i
$862$	0	0
$863$	13.8944	0.472971	0.236485	−	0.971635i	$-0.424004\pi$
$863$	13.8944	0.472971	0.236485	+	0.971635i	$0.424004\pi$
$864$	0	0
$865$	44.4382	1.51094
$866$	0	0
$867$	− 15.3160i	− 0.520158i
$868$	0	0
$869$	12.5253i	0.424891i
$870$	0	0
$871$	−49.0709	−1.66270
$872$	0	0
$873$	1.40461	0.0475390
$874$	0	0
$875$	115.549i	3.90627i
$876$	0	0
$877$	− 26.7216i	− 0.902323i	−0.892442	−	0.451161i	$-0.851010\pi$
$877$	− 26.7216i	− 0.902323i	0.892442	−	0.451161i	$-0.148990\pi$
$878$	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
$883$	53.1868i	1.78988i	0.446187	+	0.894940i	$0.352782\pi$
$883$	53.1868i	1.78988i	−0.446187	+	0.894940i	$0.647218\pi$
$884$	0	0
$885$	− 23.0935i	− 0.776279i
$886$	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
$891$	1.53073i	0.0512815i
$892$	0	0
$893$	− 42.0958i	− 1.40868i
$894$	0	0
$895$	−3.20435	−0.107110
$896$	0	0
$897$	22.3160	0.745109
$898$	0	0
$899$	− 5.17401i	− 0.172563i
$900$	0	0
$901$	11.1471i	0.371364i
$902$	0	0
$903$	13.9864	0.465439
$904$	0	0
$905$	21.1412	0.702758
$906$	0	0
$907$	29.3297i	0.973877i	0.873436	+	0.486939i	$0.161887\pi$
$907$	29.3297i	0.973877i	−0.873436	+	0.486939i	$0.838113\pi$
$908$	0	0
$909$	0.457942i	0.0151890i
$910$	0	0
$911$	4.91483	0.162835	0.0814177	−	0.996680i	$-0.474055\pi$
$911$	4.91483	0.162835	0.0814177	+	0.996680i	$0.474055\pi$
$912$	0	0
$913$	6.31599	0.209029
$914$	0	0
$915$	− 30.7981i	− 1.01815i
$916$	0	0
$917$	− 72.5211i	− 2.39486i
$918$	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
$923$	38.4008i	1.26398i
$924$	0	0
$925$	− 44.6178i	− 1.46702i
$926$	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
$931$	− 62.2529i	− 2.04026i
$932$	0	0
$933$	− 9.12522i	− 0.298746i
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	43.8100	1.43121	0.715605	−	0.698505i	$-0.246151\pi$
$937$	43.8100	1.43121	0.715605	+	0.698505i	$0.246151\pi$
$938$	0	0
$939$	9.28093i	0.302872i
$940$	0	0
$941$	16.5363i	0.539069i	0.962991	+	0.269534i	$0.0868698\pi$
$941$	16.5363i	0.539069i	−0.962991	+	0.269534i	$0.913130\pi$
$942$	0	0
$943$	−12.1275	−0.394926
$944$	0	0
$945$	−18.5786	−0.604363
$946$	0	0
$947$	− 46.7639i	− 1.51962i	−0.650143	−	0.759812i	$-0.725291\pi$
$947$	− 46.7639i	− 1.51962i	0.650143	−	0.759812i	$-0.274709\pi$
$948$	0	0
$949$	19.5329i	0.634064i
$950$	0	0
$951$	−32.3652	−1.04951
$952$	0	0
$953$	43.4436	1.40728	0.703639	−	0.710558i	$-0.251557\pi$
$953$	43.4436	1.40728	0.703639	+	0.710558i	$0.251557\pi$
$954$	0	0
$955$	100.708i	3.25883i
$956$	0	0
$957$	− 2.85108i	− 0.0921622i
$958$	0	0
$959$	−8.13651	−0.262742
$960$	0	0
$961$	−23.2832	−0.751071
$962$	0	0
$963$	14.7047i	0.473854i
$964$	0	0
$965$	92.3782i	2.97376i
$966$	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
$971$	− 55.1541i	− 1.76998i	−0.465612	−	0.884989i	$-0.654166\pi$
$971$	− 55.1541i	− 1.76998i	0.465612	−	0.884989i	$-0.345834\pi$
$972$	0	0
$973$	84.7670i	2.71751i
$974$	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
$979$	− 12.2251i	− 0.390715i
$980$	0	0
$981$	− 1.09372i	− 0.0349197i
$982$	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
$987$	− 44.5483i	− 1.41799i
$988$	0	0
$989$	− 12.1275i	− 0.385632i
$990$	0	0
$991$	−13.3733	−0.424817	−0.212408	−	0.977181i	$-0.568131\pi$
$991$	−13.3733	−0.424817	−0.212408	+	0.977181i	$0.568131\pi$
$992$	0	0
$993$	−21.8435	−0.693184
$994$	0	0
$995$	79.5338i	2.52139i
$996$	0	0
$997$	− 24.8445i	− 0.786833i	−0.919360	−	0.393416i	$-0.871293\pi$
$997$	− 24.8445i	− 0.786833i	0.919360	−	0.393416i	$-0.128707\pi$
$998$	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.d.e.1537.8		8
4.3	odd	2		3072.2.d.j.1537.4		8
8.3	odd	2		3072.2.d.j.1537.5		8
8.5	even	2	inner	3072.2.d.e.1537.1		8
16.3	odd	4		3072.2.a.p.1.4		4
16.5	even	4		3072.2.a.s.1.1		4
16.11	odd	4		3072.2.a.j.1.1		4
16.13	even	4		3072.2.a.m.1.4		4
32.3	odd	8		1536.2.j.f.385.3	yes	8
32.5	even	8		1536.2.j.e.1153.1	yes	8
32.11	odd	8		1536.2.j.i.1153.2	yes	8
32.13	even	8		1536.2.j.j.385.4	yes	8
32.19	odd	8		1536.2.j.i.385.2	yes	8
32.21	even	8		1536.2.j.j.1153.4	yes	8
32.27	odd	8		1536.2.j.f.1153.3	yes	8
32.29	even	8		1536.2.j.e.385.1	✓	8
48.5	odd	4		9216.2.a.bm.1.4		4
48.11	even	4		9216.2.a.ba.1.4		4
48.29	odd	4		9216.2.a.bl.1.1		4
48.35	even	4		9216.2.a.z.1.1		4
96.5	odd	8		4608.2.k.bh.1153.4		8
96.11	even	8		4608.2.k.bc.1153.1		8
96.29	odd	8		4608.2.k.bh.3457.4		8
96.35	even	8		4608.2.k.bj.3457.4		8
96.53	odd	8		4608.2.k.be.1153.1		8
96.59	even	8		4608.2.k.bj.1153.4		8
96.77	odd	8		4608.2.k.be.3457.1		8
96.83	even	8		4608.2.k.bc.3457.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.1	✓	8	32.29	even	8
1536.2.j.e.1153.1	yes	8	32.5	even	8
1536.2.j.f.385.3	yes	8	32.3	odd	8
1536.2.j.f.1153.3	yes	8	32.27	odd	8
1536.2.j.i.385.2	yes	8	32.19	odd	8
1536.2.j.i.1153.2	yes	8	32.11	odd	8
1536.2.j.j.385.4	yes	8	32.13	even	8
1536.2.j.j.1153.4	yes	8	32.21	even	8
3072.2.a.j.1.1		4	16.11	odd	4
3072.2.a.m.1.4		4	16.13	even	4
3072.2.a.p.1.4		4	16.3	odd	4
3072.2.a.s.1.1		4	16.5	even	4
3072.2.d.e.1537.1		8	8.5	even	2	inner
3072.2.d.e.1537.8		8	1.1	even	1	trivial
3072.2.d.j.1537.4		8	4.3	odd	2
3072.2.d.j.1537.5		8	8.3	odd	2
4608.2.k.bc.1153.1		8	96.11	even	8
4608.2.k.bc.3457.1		8	96.83	even	8
4608.2.k.be.1153.1		8	96.53	odd	8
4608.2.k.be.3457.1		8	96.77	odd	8
4608.2.k.bh.1153.4		8	96.5	odd	8
4608.2.k.bh.3457.4		8	96.29	odd	8
4608.2.k.bj.1153.4		8	96.59	even	8
4608.2.k.bj.3457.4		8	96.35	even	8
9216.2.a.z.1.1		4	48.35	even	4
9216.2.a.ba.1.4		4	48.11	even	4
9216.2.a.bl.1.1		4	48.29	odd	4
9216.2.a.bm.1.4		4	48.5	odd	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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +4.02734i q^{5} -4.61313 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +4.02734i q^{5} -4.61313 q^{7} -1.00000 q^{9} +1.53073i q^{11} +5.57900i q^{13} -4.02734 q^{15} -1.29769 q^{17} -4.35916i q^{19} -4.61313i q^{21} -4.00000 q^{23} -11.2195 q^{25} -1.00000i q^{27} +1.86256i q^{29} -2.77791 q^{31} -1.53073 q^{33} -18.5786i q^{35} +3.97682i q^{37} -5.57900 q^{39} +3.03188 q^{41} +3.03188i q^{43} -4.02734i q^{45} +9.65685 q^{47} +14.2809 q^{49} -1.29769i q^{51} -8.58995i q^{53} -6.16478 q^{55} +4.35916 q^{57} +5.73418i q^{59} +7.64725i q^{61} +4.61313 q^{63} -22.4685 q^{65} +8.79565i q^{67} -4.00000i q^{69} +6.88311 q^{71} +3.50114 q^{73} -11.2195i q^{75} -7.06147i q^{77} +8.18252 q^{79} +1.00000 q^{81} -4.12612i q^{83} -5.22625i q^{85} -1.86256 q^{87} -7.98642 q^{89} -25.7366i q^{91} -2.77791i q^{93} +17.5558 q^{95} -1.40461 q^{97} -1.53073i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{7} - 8 q^{9}+O(q^{10}) \) 8 * q - 16 * q^7 - 8 * q^9	\( 8 q - 16 q^{7} - 8 q^{9} - 32 q^{23} - 8 q^{25} + 16 q^{31} - 16 q^{39} + 32 q^{47} + 8 q^{49} - 32 q^{55} + 16 q^{63} - 16 q^{65} - 32 q^{71} + 16 q^{73} + 48 q^{79} + 8 q^{81} + 16 q^{89} + 64 q^{95} - 32 q^{97}+O(q^{100}) \) 8 * q - 16 * q^7 - 8 * q^9 - 32 * q^23 - 8 * q^25 + 16 * q^31 - 16 * q^39 + 32 * q^47 + 8 * q^49 - 32 * q^55 + 16 * q^63 - 16 * q^65 - 32 * q^71 + 16 * q^73 + 48 * q^79 + 8 * q^81 + 16 * q^89 + 64 * q^95 - 32 * q^97

Introduction

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