LMFDB - Embedded newform 3072.2.d.h.1537.1

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:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 768)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.1
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.h.1537.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -3.86370i q^{5} -2.44949 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -3.86370i q^{5} -2.44949 q^{7} -1.00000 q^{9} -1.46410i q^{11} +4.24264i q^{13} -3.86370 q^{15} -3.46410 q^{17} -7.46410i q^{19} +2.44949i q^{21} -2.82843 q^{23} -9.92820 q^{25} +1.00000i q^{27} +8.76268i q^{29} +7.34847 q^{31} -1.46410 q^{33} +9.46410i q^{35} -0.656339i q^{37} +4.24264 q^{39} -4.53590 q^{41} -3.46410i q^{43} +3.86370i q^{45} -2.82843 q^{47} -1.00000 q^{49} +3.46410i q^{51} +4.62158i q^{53} -5.65685 q^{55} -7.46410 q^{57} +13.8564i q^{59} -0.656339i q^{61} +2.44949 q^{63} +16.3923 q^{65} -14.9282i q^{67} +2.82843i q^{69} -6.41473 q^{71} -4.00000 q^{73} +9.92820i q^{75} +3.58630i q^{77} -2.44949 q^{79} +1.00000 q^{81} -5.46410i q^{83} +13.3843i q^{85} +8.76268 q^{87} +4.92820 q^{89} -10.3923i q^{91} -7.34847i q^{93} -28.8391 q^{95} +1.07180 q^{97} +1.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 24 q^{25} + 16 q^{33} - 64 q^{41} - 8 q^{49} - 32 q^{57} + 48 q^{65} - 32 q^{73} + 8 q^{81} - 16 q^{89} + 64 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 24 * q^25 + 16 * q^33 - 64 * q^41 - 8 * q^49 - 32 * q^57 + 48 * q^65 - 32 * q^73 + 8 * q^81 - 16 * q^89 + 64 * 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
$5$	− 3.86370i	− 1.72790i	−0.503577	−	0.863950i	$-0.667983\pi$
$5$	− 3.86370i	− 1.72790i	0.503577	−	0.863950i	$-0.332017\pi$
$6$	0	0
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 1.46410i	− 0.441443i	−0.975337	−	0.220722i	$-0.929159\pi$
$11$	− 1.46410i	− 0.441443i	0.975337	−	0.220722i	$-0.0708412\pi$
$12$	0	0
$13$	4.24264i	1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$	4.24264i	1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$	0	0
$15$	−3.86370	−0.997604
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	− 7.46410i	− 1.71238i	−0.516659	−	0.856191i	$-0.672825\pi$
$19$	− 7.46410i	− 1.71238i	0.516659	−	0.856191i	$-0.327175\pi$
$20$	0	0
$21$	2.44949i	0.534522i
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	−9.92820	−1.98564
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	8.76268i	1.62719i	0.581432	+	0.813595i	$0.302492\pi$
$29$	8.76268i	1.62719i	−0.581432	+	0.813595i	$0.697508\pi$
$30$	0	0
$31$	7.34847	1.31982	0.659912	−	0.751343i	$-0.270594\pi$
$31$	7.34847	1.31982	0.659912	+	0.751343i	$0.270594\pi$
$32$	0	0
$33$	−1.46410	−0.254867
$34$	0	0
$35$	9.46410i	1.59973i
$36$	0	0
$37$	− 0.656339i	− 0.107901i	−0.998544	−	0.0539507i	$-0.982819\pi$
$37$	− 0.656339i	− 0.107901i	0.998544	−	0.0539507i	$-0.0171814\pi$
$38$	0	0
$39$	4.24264	0.679366
$40$	0	0
$41$	−4.53590	−0.708388	−0.354194	−	0.935172i	$-0.615245\pi$
$41$	−4.53590	−0.708388	−0.354194	+	0.935172i	$0.615245\pi$
$42$	0	0
$43$	− 3.46410i	− 0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$	− 3.46410i	− 0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$	0	0
$45$	3.86370i	0.575967i
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	3.46410i	0.485071i
$52$	0	0
$53$	4.62158i	0.634823i	0.948288	+	0.317411i	$0.102814\pi$
$53$	4.62158i	0.634823i	−0.948288	+	0.317411i	$0.897186\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	−7.46410	−0.988644
$58$	0	0
$59$	13.8564i	1.80395i	0.431788	+	0.901975i	$0.357883\pi$
$59$	13.8564i	1.80395i	−0.431788	+	0.901975i	$0.642117\pi$
$60$	0	0
$61$	− 0.656339i	− 0.0840356i	−0.999117	−	0.0420178i	$-0.986621\pi$
$61$	− 0.656339i	− 0.0840356i	0.999117	−	0.0420178i	$-0.0133786\pi$
$62$	0	0
$63$	2.44949	0.308607
$64$	0	0
$65$	16.3923	2.03322
$66$	0	0
$67$	− 14.9282i	− 1.82377i	−0.410445	−	0.911885i	$-0.634627\pi$
$67$	− 14.9282i	− 1.82377i	0.410445	−	0.911885i	$-0.365373\pi$
$68$	0	0
$69$	2.82843i	0.340503i
$70$	0	0
$71$	−6.41473	−0.761288	−0.380644	−	0.924722i	$-0.624298\pi$
$71$	−6.41473	−0.761288	−0.380644	+	0.924722i	$0.624298\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	9.92820i	1.14641i
$76$	0	0
$77$	3.58630i	0.408697i
$78$	0	0
$79$	−2.44949	−0.275589	−0.137795	−	0.990461i	$-0.544001\pi$
$79$	−2.44949	−0.275589	−0.137795	+	0.990461i	$0.544001\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 5.46410i	− 0.599763i	−0.953976	−	0.299882i	$-0.903053\pi$
$83$	− 5.46410i	− 0.599763i	0.953976	−	0.299882i	$-0.0969472\pi$
$84$	0	0
$85$	13.3843i	1.45173i
$86$	0	0
$87$	8.76268	0.939458
$88$	0	0
$89$	4.92820	0.522388	0.261194	−	0.965286i	$-0.415884\pi$
$89$	4.92820	0.522388	0.261194	+	0.965286i	$0.415884\pi$
$90$	0	0
$91$	− 10.3923i	− 1.08941i
$92$	0	0
$93$	− 7.34847i	− 0.762001i
$94$	0	0
$95$	−28.8391	−2.95883
$96$	0	0
$97$	1.07180	0.108824	0.0544122	−	0.998519i	$-0.482671\pi$
$97$	1.07180	0.108824	0.0544122	+	0.998519i	$0.482671\pi$
$98$	0	0
$99$	1.46410i	0.147148i
$100$	0	0
$101$	14.4195i	1.43480i	0.696663	+	0.717399i	$0.254667\pi$
$101$	14.4195i	1.43480i	−0.696663	+	0.717399i	$0.745333\pi$
$102$	0	0
$103$	3.20736	0.316031	0.158016	−	0.987437i	$-0.449490\pi$
$103$	3.20736	0.316031	0.158016	+	0.987437i	$0.449490\pi$
$104$	0	0
$105$	9.46410	0.923602
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	1.41421i	0.135457i	0.997704	+	0.0677285i	$0.0215752\pi$
$109$	1.41421i	0.135457i	−0.997704	+	0.0677285i	$0.978425\pi$
$110$	0	0
$111$	−0.656339	−0.0622969
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	10.9282i	1.01906i
$116$	0	0
$117$	− 4.24264i	− 0.392232i
$118$	0	0
$119$	8.48528	0.777844
$120$	0	0
$121$	8.85641	0.805128
$122$	0	0
$123$	4.53590i	0.408988i
$124$	0	0
$125$	19.0411i	1.70309i
$126$	0	0
$127$	14.5211	1.28854	0.644268	−	0.764799i	$-0.277162\pi$
$127$	14.5211	1.28854	0.644268	+	0.764799i	$0.277162\pi$
$128$	0	0
$129$	−3.46410	−0.304997
$130$	0	0
$131$	14.9282i	1.30428i	0.758097	+	0.652142i	$0.226129\pi$
$131$	14.9282i	1.30428i	−0.758097	+	0.652142i	$0.773871\pi$
$132$	0	0
$133$	18.2832i	1.58536i
$134$	0	0
$135$	3.86370	0.332535
$136$	0	0
$137$	−14.3923	−1.22962	−0.614809	−	0.788676i	$-0.710767\pi$
$137$	−14.3923	−1.22962	−0.614809	+	0.788676i	$0.710767\pi$
$138$	0	0
$139$	6.92820i	0.587643i	0.955860	+	0.293821i	$0.0949270\pi$
$139$	6.92820i	0.587643i	−0.955860	+	0.293821i	$0.905073\pi$
$140$	0	0
$141$	2.82843i	0.238197i
$142$	0	0
$143$	6.21166	0.519445
$144$	0	0
$145$	33.8564	2.81162
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	5.37945i	0.440702i	0.975421	+	0.220351i	$0.0707203\pi$
$149$	5.37945i	0.440702i	−0.975421	+	0.220351i	$0.929280\pi$
$150$	0	0
$151$	−6.59059	−0.536335	−0.268167	−	0.963372i	$-0.586418\pi$
$151$	−6.59059	−0.536335	−0.268167	+	0.963372i	$0.586418\pi$
$152$	0	0
$153$	3.46410	0.280056
$154$	0	0
$155$	− 28.3923i	− 2.28052i
$156$	0	0
$157$	2.17209i	0.173352i	0.996237	+	0.0866758i	$0.0276244\pi$
$157$	2.17209i	0.173352i	−0.996237	+	0.0866758i	$0.972376\pi$
$158$	0	0
$159$	4.62158	0.366515
$160$	0	0
$161$	6.92820	0.546019
$162$	0	0
$163$	− 4.53590i	− 0.355279i	−0.984096	−	0.177639i	$-0.943154\pi$
$163$	− 4.53590i	− 0.355279i	0.984096	−	0.177639i	$-0.0568461\pi$
$164$	0	0
$165$	5.65685i	0.440386i
$166$	0	0
$167$	7.72741	0.597965	0.298982	−	0.954259i	$-0.403353\pi$
$167$	7.72741	0.597965	0.298982	+	0.954259i	$0.403353\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	7.46410i	0.570794i
$172$	0	0
$173$	− 17.2480i	− 1.31134i	−0.755048	−	0.655669i	$-0.772387\pi$
$173$	− 17.2480i	− 1.31134i	0.755048	−	0.655669i	$-0.227613\pi$
$174$	0	0
$175$	24.3190	1.83835
$176$	0	0
$177$	13.8564	1.04151
$178$	0	0
$179$	18.9282i	1.41476i	0.706833	+	0.707380i	$0.250123\pi$
$179$	18.9282i	1.41476i	−0.706833	+	0.707380i	$0.749877\pi$
$180$	0	0
$181$	− 7.07107i	− 0.525588i	−0.964852	−	0.262794i	$-0.915356\pi$
$181$	− 7.07107i	− 0.525588i	0.964852	−	0.262794i	$-0.0846440\pi$
$182$	0	0
$183$	−0.656339	−0.0485180
$184$	0	0
$185$	−2.53590	−0.186443
$186$	0	0
$187$	5.07180i	0.370887i
$188$	0	0
$189$	− 2.44949i	− 0.178174i
$190$	0	0
$191$	−7.72741	−0.559136	−0.279568	−	0.960126i	$-0.590191\pi$
$191$	−7.72741	−0.559136	−0.279568	+	0.960126i	$0.590191\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	− 16.3923i	− 1.17388i
$196$	0	0
$197$	8.76268i	0.624315i	0.950030	+	0.312158i	$0.101052\pi$
$197$	8.76268i	0.624315i	−0.950030	+	0.312158i	$0.898948\pi$
$198$	0	0
$199$	−18.6622	−1.32293	−0.661463	−	0.749977i	$-0.730064\pi$
$199$	−18.6622	−1.32293	−0.661463	+	0.749977i	$0.730064\pi$
$200$	0	0
$201$	−14.9282	−1.05295
$202$	0	0
$203$	− 21.4641i	− 1.50648i
$204$	0	0
$205$	17.5254i	1.22402i
$206$	0	0
$207$	2.82843	0.196589
$208$	0	0
$209$	−10.9282	−0.755920
$210$	0	0
$211$	− 6.92820i	− 0.476957i	−0.971148	−	0.238479i	$-0.923351\pi$
$211$	− 6.92820i	− 0.476957i	0.971148	−	0.238479i	$-0.0766487\pi$
$212$	0	0
$213$	6.41473i	0.439530i
$214$	0	0
$215$	−13.3843	−0.912799
$216$	0	0
$217$	−18.0000	−1.22192
$218$	0	0
$219$	4.00000i	0.270295i
$220$	0	0
$221$	− 14.6969i	− 0.988623i
$222$	0	0
$223$	−14.5211	−0.972403	−0.486201	−	0.873847i	$-0.661618\pi$
$223$	−14.5211	−0.972403	−0.486201	+	0.873847i	$0.661618\pi$
$224$	0	0
$225$	9.92820	0.661880
$226$	0	0
$227$	− 11.3205i	− 0.751369i	−0.926748	−	0.375684i	$-0.877408\pi$
$227$	− 11.3205i	− 0.751369i	0.926748	−	0.375684i	$-0.122592\pi$
$228$	0	0
$229$	16.8690i	1.11474i	0.830265	+	0.557368i	$0.188189\pi$
$229$	16.8690i	1.11474i	−0.830265	+	0.557368i	$0.811811\pi$
$230$	0	0
$231$	3.58630	0.235961
$232$	0	0
$233$	−12.9282	−0.846955	−0.423477	−	0.905907i	$-0.639191\pi$
$233$	−12.9282	−0.846955	−0.423477	+	0.905907i	$0.639191\pi$
$234$	0	0
$235$	10.9282i	0.712877i
$236$	0	0
$237$	2.44949i	0.159111i
$238$	0	0
$239$	7.72741	0.499844	0.249922	−	0.968266i	$-0.419595\pi$
$239$	7.72741	0.499844	0.249922	+	0.968266i	$0.419595\pi$
$240$	0	0
$241$	−26.7846	−1.72535	−0.862674	−	0.505760i	$-0.831212\pi$
$241$	−26.7846	−1.72535	−0.862674	+	0.505760i	$0.831212\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	3.86370i	0.246843i
$246$	0	0
$247$	31.6675	2.01495
$248$	0	0
$249$	−5.46410	−0.346273
$250$	0	0
$251$	− 6.53590i	− 0.412542i	−0.978495	−	0.206271i	$-0.933867\pi$
$251$	− 6.53590i	− 0.412542i	0.978495	−	0.206271i	$-0.0661329\pi$
$252$	0	0
$253$	4.14110i	0.260349i
$254$	0	0
$255$	13.3843	0.838155
$256$	0	0
$257$	−22.7846	−1.42126	−0.710632	−	0.703563i	$-0.751591\pi$
$257$	−22.7846	−1.42126	−0.710632	+	0.703563i	$0.751591\pi$
$258$	0	0
$259$	1.60770i	0.0998973i
$260$	0	0
$261$	− 8.76268i	− 0.542396i
$262$	0	0
$263$	−22.6274	−1.39527	−0.697633	−	0.716455i	$-0.745763\pi$
$263$	−22.6274	−1.39527	−0.697633	+	0.716455i	$0.745763\pi$
$264$	0	0
$265$	17.8564	1.09691
$266$	0	0
$267$	− 4.92820i	− 0.301601i
$268$	0	0
$269$	10.2784i	0.626687i	0.949640	+	0.313344i	$0.101449\pi$
$269$	10.2784i	0.626687i	−0.949640	+	0.313344i	$0.898551\pi$
$270$	0	0
$271$	−3.20736	−0.194834	−0.0974168	−	0.995244i	$-0.531058\pi$
$271$	−3.20736	−0.194834	−0.0974168	+	0.995244i	$0.531058\pi$
$272$	0	0
$273$	−10.3923	−0.628971
$274$	0	0
$275$	14.5359i	0.876548i
$276$	0	0
$277$	− 12.5249i	− 0.752545i	−0.926509	−	0.376273i	$-0.877206\pi$
$277$	− 12.5249i	− 0.752545i	0.926509	−	0.376273i	$-0.122794\pi$
$278$	0	0
$279$	−7.34847	−0.439941
$280$	0	0
$281$	−0.928203	−0.0553720	−0.0276860	−	0.999617i	$-0.508814\pi$
$281$	−0.928203	−0.0553720	−0.0276860	+	0.999617i	$0.508814\pi$
$282$	0	0
$283$	9.85641i	0.585903i	0.956127	+	0.292951i	$0.0946374\pi$
$283$	9.85641i	0.585903i	−0.956127	+	0.292951i	$0.905363\pi$
$284$	0	0
$285$	28.8391i	1.70828i
$286$	0	0
$287$	11.1106	0.655840
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	− 1.07180i	− 0.0628298i
$292$	0	0
$293$	− 10.8332i	− 0.632884i	−0.948612	−	0.316442i	$-0.897512\pi$
$293$	− 10.8332i	− 0.632884i	0.948612	−	0.316442i	$-0.102488\pi$
$294$	0	0
$295$	53.5370	3.11705
$296$	0	0
$297$	1.46410	0.0849558
$298$	0	0
$299$	− 12.0000i	− 0.693978i
$300$	0	0
$301$	8.48528i	0.489083i
$302$	0	0
$303$	14.4195	0.828381
$304$	0	0
$305$	−2.53590	−0.145205
$306$	0	0
$307$	− 22.9282i	− 1.30858i	−0.756243	−	0.654291i	$-0.772967\pi$
$307$	− 22.9282i	− 1.30858i	0.756243	−	0.654291i	$-0.227033\pi$
$308$	0	0
$309$	− 3.20736i	− 0.182461i
$310$	0	0
$311$	−32.4254	−1.83867	−0.919337	−	0.393471i	$-0.871274\pi$
$311$	−32.4254	−1.83867	−0.919337	+	0.393471i	$0.871274\pi$
$312$	0	0
$313$	−3.85641	−0.217977	−0.108988	−	0.994043i	$-0.534761\pi$
$313$	−3.85641	−0.217977	−0.108988	+	0.994043i	$0.534761\pi$
$314$	0	0
$315$	− 9.46410i	− 0.533242i
$316$	0	0
$317$	6.69213i	0.375867i	0.982182	+	0.187934i	$0.0601790\pi$
$317$	6.69213i	0.375867i	−0.982182	+	0.187934i	$0.939821\pi$
$318$	0	0
$319$	12.8295	0.718312
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	25.8564i	1.43869i
$324$	0	0
$325$	− 42.1218i	− 2.33650i
$326$	0	0
$327$	1.41421	0.0782062
$328$	0	0
$329$	6.92820	0.381964
$330$	0	0
$331$	1.07180i	0.0589113i	0.999566	+	0.0294556i	$0.00937738\pi$
$331$	1.07180i	0.0589113i	−0.999566	+	0.0294556i	$0.990623\pi$
$332$	0	0
$333$	0.656339i	0.0359671i
$334$	0	0
$335$	−57.6781	−3.15129
$336$	0	0
$337$	9.85641	0.536913	0.268456	−	0.963292i	$-0.413486\pi$
$337$	9.85641	0.536913	0.268456	+	0.963292i	$0.413486\pi$
$338$	0	0
$339$	− 18.0000i	− 0.977626i
$340$	0	0
$341$	− 10.7589i	− 0.582627i
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	10.9282	0.588355
$346$	0	0
$347$	2.53590i	0.136134i	0.997681	+	0.0680671i	$0.0216832\pi$
$347$	2.53590i	0.136134i	−0.997681	+	0.0680671i	$0.978317\pi$
$348$	0	0
$349$	27.4249i	1.46802i	0.679139	+	0.734010i	$0.262353\pi$
$349$	27.4249i	1.46802i	−0.679139	+	0.734010i	$0.737647\pi$
$350$	0	0
$351$	−4.24264	−0.226455
$352$	0	0
$353$	−0.928203	−0.0494033	−0.0247016	−	0.999695i	$-0.507864\pi$
$353$	−0.928203	−0.0494033	−0.0247016	+	0.999695i	$0.507864\pi$
$354$	0	0
$355$	24.7846i	1.31543i
$356$	0	0
$357$	− 8.48528i	− 0.449089i
$358$	0	0
$359$	−18.8380	−0.994234	−0.497117	−	0.867684i	$-0.665608\pi$
$359$	−18.8380	−0.994234	−0.497117	+	0.867684i	$0.665608\pi$
$360$	0	0
$361$	−36.7128	−1.93225
$362$	0	0
$363$	− 8.85641i	− 0.464841i
$364$	0	0
$365$	15.4548i	0.808942i
$366$	0	0
$367$	−16.3886	−0.855476	−0.427738	−	0.903903i	$-0.640689\pi$
$367$	−16.3886	−0.855476	−0.427738	+	0.903903i	$0.640689\pi$
$368$	0	0
$369$	4.53590	0.236129
$370$	0	0
$371$	− 11.3205i	− 0.587731i
$372$	0	0
$373$	13.2827i	0.687753i	0.939015	+	0.343877i	$0.111740\pi$
$373$	13.2827i	0.687753i	−0.939015	+	0.343877i	$0.888260\pi$
$374$	0	0
$375$	19.0411	0.983279
$376$	0	0
$377$	−37.1769	−1.91471
$378$	0	0
$379$	13.3205i	0.684229i	0.939658	+	0.342114i	$0.111143\pi$
$379$	13.3205i	0.684229i	−0.939658	+	0.342114i	$0.888857\pi$
$380$	0	0
$381$	− 14.5211i	− 0.743937i
$382$	0	0
$383$	28.8391	1.47361	0.736804	−	0.676106i	$-0.236334\pi$
$383$	28.8391	1.47361	0.736804	+	0.676106i	$0.236334\pi$
$384$	0	0
$385$	13.8564	0.706188
$386$	0	0
$387$	3.46410i	0.176090i
$388$	0	0
$389$	− 9.52056i	− 0.482711i	−0.970437	−	0.241356i	$-0.922408\pi$
$389$	− 9.52056i	− 0.482711i	0.970437	−	0.241356i	$-0.0775921\pi$
$390$	0	0
$391$	9.79796	0.495504
$392$	0	0
$393$	14.9282	0.753028
$394$	0	0
$395$	9.46410i	0.476191i
$396$	0	0
$397$	− 23.0807i	− 1.15839i	−0.815190	−	0.579193i	$-0.803368\pi$
$397$	− 23.0807i	− 1.15839i	0.815190	−	0.579193i	$-0.196632\pi$
$398$	0	0
$399$	18.2832	0.915307
$400$	0	0
$401$	−18.3923	−0.918468	−0.459234	−	0.888315i	$-0.651876\pi$
$401$	−18.3923	−0.918468	−0.459234	+	0.888315i	$0.651876\pi$
$402$	0	0
$403$	31.1769i	1.55303i
$404$	0	0
$405$	− 3.86370i	− 0.191989i
$406$	0	0
$407$	−0.960947	−0.0476324
$408$	0	0
$409$	9.07180	0.448571	0.224286	−	0.974523i	$-0.427995\pi$
$409$	9.07180	0.448571	0.224286	+	0.974523i	$0.427995\pi$
$410$	0	0
$411$	14.3923i	0.709920i
$412$	0	0
$413$	− 33.9411i	− 1.67013i
$414$	0	0
$415$	−21.1117	−1.03633
$416$	0	0
$417$	6.92820	0.339276
$418$	0	0
$419$	− 0.392305i	− 0.0191653i	−0.999954	−	0.00958267i	$-0.996950\pi$
$419$	− 0.392305i	− 0.0191653i	0.999954	−	0.00958267i	$-0.00305031\pi$
$420$	0	0
$421$	− 19.6975i	− 0.959995i	−0.877270	−	0.479998i	$-0.840638\pi$
$421$	− 19.6975i	− 0.959995i	0.877270	−	0.479998i	$-0.159362\pi$
$422$	0	0
$423$	2.82843	0.137523
$424$	0	0
$425$	34.3923	1.66827
$426$	0	0
$427$	1.60770i	0.0778018i
$428$	0	0
$429$	− 6.21166i	− 0.299902i
$430$	0	0
$431$	4.34418	0.209252	0.104626	−	0.994512i	$-0.466636\pi$
$431$	4.34418	0.209252	0.104626	+	0.994512i	$0.466636\pi$
$432$	0	0
$433$	−29.8564	−1.43481	−0.717404	−	0.696658i	$-0.754670\pi$
$433$	−29.8564	−1.43481	−0.717404	+	0.696658i	$0.754670\pi$
$434$	0	0
$435$	− 33.8564i	− 1.62329i
$436$	0	0
$437$	21.1117i	1.00991i
$438$	0	0
$439$	−5.83272	−0.278381	−0.139190	−	0.990266i	$-0.544450\pi$
$439$	−5.83272	−0.278381	−0.139190	+	0.990266i	$0.544450\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	− 35.3205i	− 1.67813i	−0.544033	−	0.839064i	$-0.683103\pi$
$443$	− 35.3205i	− 1.67813i	0.544033	−	0.839064i	$-0.316897\pi$
$444$	0	0
$445$	− 19.0411i	− 0.902635i
$446$	0	0
$447$	5.37945	0.254439
$448$	0	0
$449$	−6.39230	−0.301672	−0.150836	−	0.988559i	$-0.548196\pi$
$449$	−6.39230	−0.301672	−0.150836	+	0.988559i	$0.548196\pi$
$450$	0	0
$451$	6.64102i	0.312713i
$452$	0	0
$453$	6.59059i	0.309653i
$454$	0	0
$455$	−40.1528	−1.88239
$456$	0	0
$457$	20.7846	0.972263	0.486132	−	0.873886i	$-0.338408\pi$
$457$	20.7846	0.972263	0.486132	+	0.873886i	$0.338408\pi$
$458$	0	0
$459$	− 3.46410i	− 0.161690i
$460$	0	0
$461$	31.1870i	1.45252i	0.687418	+	0.726262i	$0.258744\pi$
$461$	31.1870i	1.45252i	−0.687418	+	0.726262i	$0.741256\pi$
$462$	0	0
$463$	−29.2180	−1.35788	−0.678938	−	0.734196i	$-0.737560\pi$
$463$	−29.2180	−1.35788	−0.678938	+	0.734196i	$0.737560\pi$
$464$	0	0
$465$	−28.3923	−1.31666
$466$	0	0
$467$	30.2487i	1.39974i	0.714269	+	0.699872i	$0.246760\pi$
$467$	30.2487i	1.39974i	−0.714269	+	0.699872i	$0.753240\pi$
$468$	0	0
$469$	36.5665i	1.68848i
$470$	0	0
$471$	2.17209	0.100085
$472$	0	0
$473$	−5.07180	−0.233201
$474$	0	0
$475$	74.1051i	3.40018i
$476$	0	0
$477$	− 4.62158i	− 0.211608i
$478$	0	0
$479$	−29.0421	−1.32697	−0.663485	−	0.748190i	$-0.730923\pi$
$479$	−29.0421	−1.32697	−0.663485	+	0.748190i	$0.730923\pi$
$480$	0	0
$481$	2.78461	0.126967
$482$	0	0
$483$	− 6.92820i	− 0.315244i
$484$	0	0
$485$	− 4.14110i	− 0.188038i
$486$	0	0
$487$	−11.4896	−0.520642	−0.260321	−	0.965522i	$-0.583828\pi$
$487$	−11.4896	−0.520642	−0.260321	+	0.965522i	$0.583828\pi$
$488$	0	0
$489$	−4.53590	−0.205120
$490$	0	0
$491$	5.85641i	0.264296i	0.991230	+	0.132148i	$0.0421874\pi$
$491$	5.85641i	0.264296i	−0.991230	+	0.132148i	$0.957813\pi$
$492$	0	0
$493$	− 30.3548i	− 1.36711i
$494$	0	0
$495$	5.65685	0.254257
$496$	0	0
$497$	15.7128	0.704816
$498$	0	0
$499$	− 4.00000i	− 0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$	− 4.00000i	− 0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	0	0
$501$	− 7.72741i	− 0.345235i
$502$	0	0
$503$	−23.3853	−1.04270	−0.521349	−	0.853343i	$-0.674571\pi$
$503$	−23.3853	−1.04270	−0.521349	+	0.853343i	$0.674571\pi$
$504$	0	0
$505$	55.7128	2.47919
$506$	0	0
$507$	5.00000i	0.222058i
$508$	0	0
$509$	− 7.24693i	− 0.321215i	−0.987018	−	0.160607i	$-0.948655\pi$
$509$	− 7.24693i	− 0.321215i	0.987018	−	0.160607i	$-0.0513453\pi$
$510$	0	0
$511$	9.79796	0.433436
$512$	0	0
$513$	7.46410	0.329548
$514$	0	0
$515$	− 12.3923i	− 0.546070i
$516$	0	0
$517$	4.14110i	0.182126i
$518$	0	0
$519$	−17.2480	−0.757102
$520$	0	0
$521$	21.3205	0.934068	0.467034	−	0.884239i	$-0.345322\pi$
$521$	21.3205	0.934068	0.467034	+	0.884239i	$0.345322\pi$
$522$	0	0
$523$	31.4641i	1.37583i	0.725792	+	0.687915i	$0.241474\pi$
$523$	31.4641i	1.37583i	−0.725792	+	0.687915i	$0.758526\pi$
$524$	0	0
$525$	− 24.3190i	− 1.06137i
$526$	0	0
$527$	−25.4558	−1.10887
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	− 13.8564i	− 0.601317i
$532$	0	0
$533$	− 19.2442i	− 0.833558i
$534$	0	0
$535$	−15.4548	−0.668170
$536$	0	0
$537$	18.9282	0.816812
$538$	0	0
$539$	1.46410i	0.0630633i
$540$	0	0
$541$	− 28.1827i	− 1.21167i	−0.795590	−	0.605835i	$-0.792839\pi$
$541$	− 28.1827i	− 1.21167i	0.795590	−	0.605835i	$-0.207161\pi$
$542$	0	0
$543$	−7.07107	−0.303449
$544$	0	0
$545$	5.46410	0.234056
$546$	0	0
$547$	− 23.1769i	− 0.990973i	−0.868616	−	0.495487i	$-0.834990\pi$
$547$	− 23.1769i	− 0.990973i	0.868616	−	0.495487i	$-0.165010\pi$
$548$	0	0
$549$	0.656339i	0.0280119i
$550$	0	0
$551$	65.4056	2.78637
$552$	0	0
$553$	6.00000	0.255146
$554$	0	0
$555$	2.53590i	0.107643i
$556$	0	0
$557$	− 28.0069i	− 1.18669i	−0.804949	−	0.593345i	$-0.797807\pi$
$557$	− 28.0069i	− 1.18669i	0.804949	−	0.593345i	$-0.202193\pi$
$558$	0	0
$559$	14.6969	0.621614
$560$	0	0
$561$	5.07180	0.214131
$562$	0	0
$563$	− 1.46410i	− 0.0617045i	−0.999524	−	0.0308523i	$-0.990178\pi$
$563$	− 1.46410i	− 0.0617045i	0.999524	−	0.0308523i	$-0.00982214\pi$
$564$	0	0
$565$	− 69.5467i	− 2.92585i
$566$	0	0
$567$	−2.44949	−0.102869
$568$	0	0
$569$	3.46410	0.145223	0.0726113	−	0.997360i	$-0.476867\pi$
$569$	3.46410	0.145223	0.0726113	+	0.997360i	$0.476867\pi$
$570$	0	0
$571$	− 31.7128i	− 1.32714i	−0.748114	−	0.663570i	$-0.769041\pi$
$571$	− 31.7128i	− 1.32714i	0.748114	−	0.663570i	$-0.230959\pi$
$572$	0	0
$573$	7.72741i	0.322817i
$574$	0	0
$575$	28.0812	1.17107
$576$	0	0
$577$	−21.8564	−0.909894	−0.454947	−	0.890518i	$-0.650342\pi$
$577$	−21.8564	−0.909894	−0.454947	+	0.890518i	$0.650342\pi$
$578$	0	0
$579$	− 6.00000i	− 0.249351i
$580$	0	0
$581$	13.3843i	0.555273i
$582$	0	0
$583$	6.76646	0.280238
$584$	0	0
$585$	−16.3923	−0.677738
$586$	0	0
$587$	− 42.9282i	− 1.77184i	−0.463841	−	0.885918i	$-0.653529\pi$
$587$	− 42.9282i	− 1.77184i	0.463841	−	0.885918i	$-0.346471\pi$
$588$	0	0
$589$	− 54.8497i	− 2.26004i
$590$	0	0
$591$	8.76268	0.360449
$592$	0	0
$593$	37.7128	1.54868	0.774340	−	0.632770i	$-0.218082\pi$
$593$	37.7128	1.54868	0.774340	+	0.632770i	$0.218082\pi$
$594$	0	0
$595$	− 32.7846i	− 1.34404i
$596$	0	0
$597$	18.6622i	0.763792i
$598$	0	0
$599$	−6.96953	−0.284767	−0.142384	−	0.989812i	$-0.545477\pi$
$599$	−6.96953	−0.284767	−0.142384	+	0.989812i	$0.545477\pi$
$600$	0	0
$601$	−19.7128	−0.804102	−0.402051	−	0.915617i	$-0.631703\pi$
$601$	−19.7128	−0.804102	−0.402051	+	0.915617i	$0.631703\pi$
$602$	0	0
$603$	14.9282i	0.607923i
$604$	0	0
$605$	− 34.2185i	− 1.39118i
$606$	0	0
$607$	−31.8434	−1.29248	−0.646241	−	0.763133i	$-0.723660\pi$
$607$	−31.8434	−1.29248	−0.646241	+	0.763133i	$0.723660\pi$
$608$	0	0
$609$	−21.4641	−0.869769
$610$	0	0
$611$	− 12.0000i	− 0.485468i
$612$	0	0
$613$	6.11012i	0.246785i	0.992358	+	0.123393i	$0.0393775\pi$
$613$	6.11012i	0.246785i	−0.992358	+	0.123393i	$0.960623\pi$
$614$	0	0
$615$	17.5254	0.706691
$616$	0	0
$617$	39.8564	1.60456	0.802279	−	0.596949i	$-0.203621\pi$
$617$	39.8564	1.60456	0.802279	+	0.596949i	$0.203621\pi$
$618$	0	0
$619$	− 12.0000i	− 0.482321i	−0.970485	−	0.241160i	$-0.922472\pi$
$619$	− 12.0000i	− 0.482321i	0.970485	−	0.241160i	$-0.0775280\pi$
$620$	0	0
$621$	− 2.82843i	− 0.113501i
$622$	0	0
$623$	−12.0716	−0.483638
$624$	0	0
$625$	23.9282	0.957128
$626$	0	0
$627$	10.9282i	0.436430i
$628$	0	0
$629$	2.27362i	0.0906553i
$630$	0	0
$631$	23.5612	0.937955	0.468977	−	0.883210i	$-0.344623\pi$
$631$	23.5612	0.937955	0.468977	+	0.883210i	$0.344623\pi$
$632$	0	0
$633$	−6.92820	−0.275371
$634$	0	0
$635$	− 56.1051i	− 2.22646i
$636$	0	0
$637$	− 4.24264i	− 0.168100i
$638$	0	0
$639$	6.41473	0.253763
$640$	0	0
$641$	−25.6077	−1.01144	−0.505722	−	0.862697i	$-0.668774\pi$
$641$	−25.6077	−1.01144	−0.505722	+	0.862697i	$0.668774\pi$
$642$	0	0
$643$	28.5359i	1.12535i	0.826680	+	0.562673i	$0.190227\pi$
$643$	28.5359i	1.12535i	−0.826680	+	0.562673i	$0.809773\pi$
$644$	0	0
$645$	13.3843i	0.527005i
$646$	0	0
$647$	−40.9107	−1.60836	−0.804182	−	0.594383i	$-0.797396\pi$
$647$	−40.9107	−1.60836	−0.804182	+	0.594383i	$0.797396\pi$
$648$	0	0
$649$	20.2872	0.796342
$650$	0	0
$651$	18.0000i	0.705476i
$652$	0	0
$653$	− 24.9754i	− 0.977362i	−0.872463	−	0.488681i	$-0.837478\pi$
$653$	− 24.9754i	− 0.977362i	0.872463	−	0.488681i	$-0.162522\pi$
$654$	0	0
$655$	57.6781	2.25367
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	24.0000i	0.934907i	0.884018	+	0.467454i	$0.154829\pi$
$659$	24.0000i	0.934907i	−0.884018	+	0.467454i	$0.845171\pi$
$660$	0	0
$661$	− 6.51626i	− 0.253453i	−0.991938	−	0.126727i	$-0.959553\pi$
$661$	− 6.51626i	− 0.253453i	0.991938	−	0.126727i	$-0.0404471\pi$
$662$	0	0
$663$	−14.6969	−0.570782
$664$	0	0
$665$	70.6410	2.73934
$666$	0	0
$667$	− 24.7846i	− 0.959664i
$668$	0	0
$669$	14.5211i	0.561417i
$670$	0	0
$671$	−0.960947	−0.0370969
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	− 9.92820i	− 0.382137i
$676$	0	0
$677$	− 3.30890i	− 0.127171i	−0.997976	−	0.0635857i	$-0.979746\pi$
$677$	− 3.30890i	− 0.127171i	0.997976	−	0.0635857i	$-0.0202536\pi$
$678$	0	0
$679$	−2.62536	−0.100752
$680$	0	0
$681$	−11.3205	−0.433803
$682$	0	0
$683$	25.1769i	0.963368i	0.876345	+	0.481684i	$0.159975\pi$
$683$	25.1769i	0.963368i	−0.876345	+	0.481684i	$0.840025\pi$
$684$	0	0
$685$	55.6076i	2.12466i
$686$	0	0
$687$	16.8690	0.643594
$688$	0	0
$689$	−19.6077	−0.746994
$690$	0	0
$691$	− 26.3923i	− 1.00401i	−0.864865	−	0.502005i	$-0.832596\pi$
$691$	− 26.3923i	− 1.00401i	0.864865	−	0.502005i	$-0.167404\pi$
$692$	0	0
$693$	− 3.58630i	− 0.136232i
$694$	0	0
$695$	26.7685	1.01539
$696$	0	0
$697$	15.7128	0.595165
$698$	0	0
$699$	12.9282i	0.488990i
$700$	0	0
$701$	− 14.9743i	− 0.565573i	−0.959183	−	0.282787i	$-0.908741\pi$
$701$	− 14.9743i	− 0.565573i	0.959183	−	0.282787i	$-0.0912588\pi$
$702$	0	0
$703$	−4.89898	−0.184769
$704$	0	0
$705$	10.9282	0.411580
$706$	0	0
$707$	− 35.3205i	− 1.32836i
$708$	0	0
$709$	49.0913i	1.84366i	0.387590	+	0.921832i	$0.373308\pi$
$709$	49.0913i	1.84366i	−0.387590	+	0.921832i	$0.626692\pi$
$710$	0	0
$711$	2.44949	0.0918630
$712$	0	0
$713$	−20.7846	−0.778390
$714$	0	0
$715$	− 24.0000i	− 0.897549i
$716$	0	0
$717$	− 7.72741i	− 0.288585i
$718$	0	0
$719$	−2.27362	−0.0847919	−0.0423959	−	0.999101i	$-0.513499\pi$
$719$	−2.27362	−0.0847919	−0.0423959	+	0.999101i	$0.513499\pi$
$720$	0	0
$721$	−7.85641	−0.292588
$722$	0	0
$723$	26.7846i	0.996130i
$724$	0	0
$725$	− 86.9977i	− 3.23101i
$726$	0	0
$727$	8.10634	0.300648	0.150324	−	0.988637i	$-0.451968\pi$
$727$	8.10634	0.300648	0.150324	+	0.988637i	$0.451968\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	12.0000i	0.443836i
$732$	0	0
$733$	7.07107i	0.261176i	0.991437	+	0.130588i	$0.0416865\pi$
$733$	7.07107i	0.261176i	−0.991437	+	0.130588i	$0.958314\pi$
$734$	0	0
$735$	3.86370	0.142515
$736$	0	0
$737$	−21.8564	−0.805091
$738$	0	0
$739$	− 36.0000i	− 1.32428i	−0.749380	−	0.662141i	$-0.769648\pi$
$739$	− 36.0000i	− 1.32428i	0.749380	−	0.662141i	$-0.230352\pi$
$740$	0	0
$741$	− 31.6675i	− 1.16333i
$742$	0	0
$743$	−27.3233	−1.00240	−0.501198	−	0.865333i	$-0.667107\pi$
$743$	−27.3233	−1.00240	−0.501198	+	0.865333i	$0.667107\pi$
$744$	0	0
$745$	20.7846	0.761489
$746$	0	0
$747$	5.46410i	0.199921i
$748$	0	0
$749$	9.79796i	0.358010i
$750$	0	0
$751$	46.1886	1.68545	0.842723	−	0.538348i	$-0.180951\pi$
$751$	46.1886	1.68545	0.842723	+	0.538348i	$0.180951\pi$
$752$	0	0
$753$	−6.53590	−0.238181
$754$	0	0
$755$	25.4641i	0.926734i
$756$	0	0
$757$	33.6365i	1.22254i	0.791422	+	0.611270i	$0.209341\pi$
$757$	33.6365i	1.22254i	−0.791422	+	0.611270i	$0.790659\pi$
$758$	0	0
$759$	4.14110	0.150313
$760$	0	0
$761$	52.2487	1.89401	0.947007	−	0.321212i	$-0.104090\pi$
$761$	52.2487	1.89401	0.947007	+	0.321212i	$0.104090\pi$
$762$	0	0
$763$	− 3.46410i	− 0.125409i
$764$	0	0
$765$	− 13.3843i	− 0.483909i
$766$	0	0
$767$	−58.7878	−2.12270
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	22.7846i	0.820568i
$772$	0	0
$773$	− 48.9155i	− 1.75937i	−0.475560	−	0.879684i	$-0.657754\pi$
$773$	− 48.9155i	− 1.75937i	0.475560	−	0.879684i	$-0.342246\pi$
$774$	0	0
$775$	−72.9571	−2.62070
$776$	0	0
$777$	1.60770	0.0576757
$778$	0	0
$779$	33.8564i	1.21303i
$780$	0	0
$781$	9.39182i	0.336066i
$782$	0	0
$783$	−8.76268	−0.313153
$784$	0	0
$785$	8.39230	0.299534
$786$	0	0
$787$	23.4641i	0.836405i	0.908354	+	0.418202i	$0.137340\pi$
$787$	23.4641i	0.836405i	−0.908354	+	0.418202i	$0.862660\pi$
$788$	0	0
$789$	22.6274i	0.805557i
$790$	0	0
$791$	−44.0908	−1.56769
$792$	0	0
$793$	2.78461	0.0988844
$794$	0	0
$795$	− 17.8564i	− 0.633301i
$796$	0	0
$797$	21.9439i	0.777292i	0.921387	+	0.388646i	$0.127057\pi$
$797$	21.9439i	0.777292i	−0.921387	+	0.388646i	$0.872943\pi$
$798$	0	0
$799$	9.79796	0.346627
$800$	0	0
$801$	−4.92820	−0.174129
$802$	0	0
$803$	5.85641i	0.206668i
$804$	0	0
$805$	− 26.7685i	− 0.943466i
$806$	0	0
$807$	10.2784	0.361818
$808$	0	0
$809$	6.67949	0.234838	0.117419	−	0.993082i	$-0.462538\pi$
$809$	6.67949	0.234838	0.117419	+	0.993082i	$0.462538\pi$
$810$	0	0
$811$	54.3923i	1.90997i	0.296651	+	0.954986i	$0.404130\pi$
$811$	54.3923i	1.90997i	−0.296651	+	0.954986i	$0.595870\pi$
$812$	0	0
$813$	3.20736i	0.112487i
$814$	0	0
$815$	−17.5254	−0.613887
$816$	0	0
$817$	−25.8564	−0.904601
$818$	0	0
$819$	10.3923i	0.363137i
$820$	0	0
$821$	25.1784i	0.878734i	0.898308	+	0.439367i	$0.144797\pi$
$821$	25.1784i	0.878734i	−0.898308	+	0.439367i	$0.855203\pi$
$822$	0	0
$823$	−9.97382	−0.347666	−0.173833	−	0.984775i	$-0.555615\pi$
$823$	−9.97382	−0.347666	−0.173833	+	0.984775i	$0.555615\pi$
$824$	0	0
$825$	14.5359	0.506075
$826$	0	0
$827$	44.7846i	1.55731i	0.627450	+	0.778657i	$0.284099\pi$
$827$	44.7846i	1.55731i	−0.627450	+	0.778657i	$0.715901\pi$
$828$	0	0
$829$	− 1.61729i	− 0.0561706i	−0.999606	−	0.0280853i	$-0.991059\pi$
$829$	− 1.61729i	− 0.0561706i	0.999606	−	0.0280853i	$-0.00894101\pi$
$830$	0	0
$831$	−12.5249	−0.434482
$832$	0	0
$833$	3.46410	0.120024
$834$	0	0
$835$	− 29.8564i	− 1.03322i
$836$	0	0
$837$	7.34847i	0.254000i
$838$	0	0
$839$	26.0106	0.897987	0.448994	−	0.893535i	$-0.351783\pi$
$839$	26.0106	0.897987	0.448994	+	0.893535i	$0.351783\pi$
$840$	0	0
$841$	−47.7846	−1.64775
$842$	0	0
$843$	0.928203i	0.0319690i
$844$	0	0
$845$	19.3185i	0.664577i
$846$	0	0
$847$	−21.6937	−0.745404
$848$	0	0
$849$	9.85641	0.338271
$850$	0	0
$851$	1.85641i	0.0636368i
$852$	0	0
$853$	− 47.0208i	− 1.60996i	−0.593301	−	0.804980i	$-0.702176\pi$
$853$	− 47.0208i	− 1.60996i	0.593301	−	0.804980i	$-0.297824\pi$
$854$	0	0
$855$	28.8391	0.986276
$856$	0	0
$857$	−43.1769	−1.47490	−0.737448	−	0.675404i	$-0.763969\pi$
$857$	−43.1769	−1.47490	−0.737448	+	0.675404i	$0.763969\pi$
$858$	0	0
$859$	− 32.2487i	− 1.10031i	−0.835062	−	0.550156i	$-0.814568\pi$
$859$	− 32.2487i	− 1.10031i	0.835062	−	0.550156i	$-0.185432\pi$
$860$	0	0
$861$	− 11.1106i	− 0.378649i
$862$	0	0
$863$	−37.1213	−1.26362	−0.631812	−	0.775122i	$-0.717688\pi$
$863$	−37.1213	−1.26362	−0.631812	+	0.775122i	$0.717688\pi$
$864$	0	0
$865$	−66.6410	−2.26586
$866$	0	0
$867$	5.00000i	0.169809i
$868$	0	0
$869$	3.58630i	0.121657i
$870$	0	0
$871$	63.3350	2.14602
$872$	0	0
$873$	−1.07180	−0.0362748
$874$	0	0
$875$	− 46.6410i	− 1.57675i
$876$	0	0
$877$	− 35.7071i	− 1.20574i	−0.797839	−	0.602871i	$-0.794023\pi$
$877$	− 35.7071i	− 1.20574i	0.797839	−	0.602871i	$-0.205977\pi$
$878$	0	0
$879$	−10.8332	−0.365396
$880$	0	0
$881$	−36.6410	−1.23447	−0.617234	−	0.786780i	$-0.711747\pi$
$881$	−36.6410	−1.23447	−0.617234	+	0.786780i	$0.711747\pi$
$882$	0	0
$883$	5.60770i	0.188714i	0.995538	+	0.0943570i	$0.0300795\pi$
$883$	5.60770i	0.188714i	−0.995538	+	0.0943570i	$0.969920\pi$
$884$	0	0
$885$	− 53.5370i	− 1.79963i
$886$	0	0
$887$	36.5665	1.22778	0.613891	−	0.789391i	$-0.289603\pi$
$887$	36.5665	1.22778	0.613891	+	0.789391i	$0.289603\pi$
$888$	0	0
$889$	−35.5692	−1.19295
$890$	0	0
$891$	− 1.46410i	− 0.0490492i
$892$	0	0
$893$	21.1117i	0.706475i
$894$	0	0
$895$	73.1330	2.44457
$896$	0	0
$897$	−12.0000	−0.400668
$898$	0	0
$899$	64.3923i	2.14760i
$900$	0	0
$901$	− 16.0096i	− 0.533358i
$902$	0	0
$903$	8.48528	0.282372
$904$	0	0
$905$	−27.3205	−0.908164
$906$	0	0
$907$	− 42.3923i	− 1.40761i	−0.710392	−	0.703807i	$-0.751482\pi$
$907$	− 42.3923i	− 1.40761i	0.710392	−	0.703807i	$-0.248518\pi$
$908$	0	0
$909$	− 14.4195i	− 0.478266i
$910$	0	0
$911$	22.0726	0.731298	0.365649	−	0.930753i	$-0.380847\pi$
$911$	22.0726	0.731298	0.365649	+	0.930753i	$0.380847\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	2.53590i	0.0838342i
$916$	0	0
$917$	− 36.5665i	− 1.20753i
$918$	0	0
$919$	−0.582009	−0.0191987	−0.00959936	−	0.999954i	$-0.503056\pi$
$919$	−0.582009	−0.0191987	−0.00959936	+	0.999954i	$0.503056\pi$
$920$	0	0
$921$	−22.9282	−0.755510
$922$	0	0
$923$	− 27.2154i	− 0.895805i
$924$	0	0
$925$	6.51626i	0.214253i
$926$	0	0
$927$	−3.20736	−0.105344
$928$	0	0
$929$	−45.3205	−1.48692	−0.743459	−	0.668782i	$-0.766816\pi$
$929$	−45.3205	−1.48692	−0.743459	+	0.668782i	$0.766816\pi$
$930$	0	0
$931$	7.46410i	0.244626i
$932$	0	0
$933$	32.4254i	1.06156i
$934$	0	0
$935$	19.5959	0.640855
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	3.85641i	0.125849i
$940$	0	0
$941$	− 36.2891i	− 1.18299i	−0.806309	−	0.591495i	$-0.798538\pi$
$941$	− 36.2891i	− 1.18299i	0.806309	−	0.591495i	$-0.201462\pi$
$942$	0	0
$943$	12.8295	0.417785
$944$	0	0
$945$	−9.46410	−0.307867
$946$	0	0
$947$	− 29.0718i	− 0.944706i	−0.881409	−	0.472353i	$-0.843405\pi$
$947$	− 29.0718i	− 0.944706i	0.881409	−	0.472353i	$-0.156595\pi$
$948$	0	0
$949$	− 16.9706i	− 0.550888i
$950$	0	0
$951$	6.69213	0.217007
$952$	0	0
$953$	−46.3923	−1.50279	−0.751397	−	0.659850i	$-0.770620\pi$
$953$	−46.3923	−1.50279	−0.751397	+	0.659850i	$0.770620\pi$
$954$	0	0
$955$	29.8564i	0.966131i
$956$	0	0
$957$	− 12.8295i	− 0.414717i
$958$	0	0
$959$	35.2538	1.13840
$960$	0	0
$961$	23.0000	0.741935
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	− 23.1822i	− 0.746262i
$966$	0	0
$967$	10.3800	0.333797	0.166899	−	0.985974i	$-0.446625\pi$
$967$	10.3800	0.333797	0.166899	+	0.985974i	$0.446625\pi$
$968$	0	0
$969$	25.8564	0.830627
$970$	0	0
$971$	22.2487i	0.713995i	0.934105	+	0.356998i	$0.116200\pi$
$971$	22.2487i	0.713995i	−0.934105	+	0.356998i	$0.883800\pi$
$972$	0	0
$973$	− 16.9706i	− 0.544051i
$974$	0	0
$975$	−42.1218	−1.34898
$976$	0	0
$977$	−35.1769	−1.12541	−0.562705	−	0.826658i	$-0.690239\pi$
$977$	−35.1769	−1.12541	−0.562705	+	0.826658i	$0.690239\pi$
$978$	0	0
$979$	− 7.21539i	− 0.230605i
$980$	0	0
$981$	− 1.41421i	− 0.0451524i
$982$	0	0
$983$	23.7370	0.757093	0.378547	−	0.925582i	$-0.376424\pi$
$983$	23.7370	0.757093	0.378547	+	0.925582i	$0.376424\pi$
$984$	0	0
$985$	33.8564	1.07875
$986$	0	0
$987$	− 6.92820i	− 0.220527i
$988$	0	0
$989$	9.79796i	0.311557i
$990$	0	0
$991$	12.2474	0.389053	0.194527	−	0.980897i	$-0.437683\pi$
$991$	12.2474	0.389053	0.194527	+	0.980897i	$0.437683\pi$
$992$	0	0
$993$	1.07180	0.0340124
$994$	0	0
$995$	72.1051i	2.28589i
$996$	0	0
$997$	− 17.6269i	− 0.558250i	−0.960255	−	0.279125i	$-0.909956\pi$
$997$	− 17.6269i	− 0.558250i	0.960255	−	0.279125i	$-0.0900443\pi$
$998$	0	0
$999$	0.656339	0.0207656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.d.h.1537.1		8
4.3	odd	2	inner	3072.2.d.h.1537.5		8
8.3	odd	2	inner	3072.2.d.h.1537.4		8
8.5	even	2	inner	3072.2.d.h.1537.8		8
16.3	odd	4		3072.2.a.l.1.1		4
16.5	even	4		3072.2.a.l.1.4		4
16.11	odd	4		3072.2.a.r.1.4		4
16.13	even	4		3072.2.a.r.1.1		4
32.3	odd	8		768.2.j.f.193.2	yes	8
32.5	even	8		768.2.j.f.577.4	yes	8
32.11	odd	8		768.2.j.e.577.3	yes	8
32.13	even	8		768.2.j.e.193.1	✓	8
32.19	odd	8		768.2.j.e.193.3	yes	8
32.21	even	8		768.2.j.e.577.1	yes	8
32.27	odd	8		768.2.j.f.577.2	yes	8
32.29	even	8		768.2.j.f.193.4	yes	8
48.5	odd	4		9216.2.a.bi.1.1		4
48.11	even	4		9216.2.a.bc.1.1		4
48.29	odd	4		9216.2.a.bc.1.4		4
48.35	even	4		9216.2.a.bi.1.4		4
96.5	odd	8		2304.2.k.e.577.1		8
96.11	even	8		2304.2.k.l.577.4		8
96.29	odd	8		2304.2.k.e.1729.2		8
96.35	even	8		2304.2.k.e.1729.1		8
96.53	odd	8		2304.2.k.l.577.3		8
96.59	even	8		2304.2.k.e.577.2		8
96.77	odd	8		2304.2.k.l.1729.4		8
96.83	even	8		2304.2.k.l.1729.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.2.j.e.193.1	✓	8	32.13	even	8
768.2.j.e.193.3	yes	8	32.19	odd	8
768.2.j.e.577.1	yes	8	32.21	even	8
768.2.j.e.577.3	yes	8	32.11	odd	8
768.2.j.f.193.2	yes	8	32.3	odd	8
768.2.j.f.193.4	yes	8	32.29	even	8
768.2.j.f.577.2	yes	8	32.27	odd	8
768.2.j.f.577.4	yes	8	32.5	even	8
2304.2.k.e.577.1		8	96.5	odd	8
2304.2.k.e.577.2		8	96.59	even	8
2304.2.k.e.1729.1		8	96.35	even	8
2304.2.k.e.1729.2		8	96.29	odd	8
2304.2.k.l.577.3		8	96.53	odd	8
2304.2.k.l.577.4		8	96.11	even	8
2304.2.k.l.1729.3		8	96.83	even	8
2304.2.k.l.1729.4		8	96.77	odd	8
3072.2.a.l.1.1		4	16.3	odd	4
3072.2.a.l.1.4		4	16.5	even	4
3072.2.a.r.1.1		4	16.13	even	4
3072.2.a.r.1.4		4	16.11	odd	4
3072.2.d.h.1537.1		8	1.1	even	1	trivial
3072.2.d.h.1537.4		8	8.3	odd	2	inner
3072.2.d.h.1537.5		8	4.3	odd	2	inner
3072.2.d.h.1537.8		8	8.5	even	2	inner
9216.2.a.bc.1.1		4	48.11	even	4
9216.2.a.bc.1.4		4	48.29	odd	4
9216.2.a.bi.1.1		4	48.5	odd	4
9216.2.a.bi.1.4		4	48.35	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -3.86370i q^{5} -2.44949 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -3.86370i q^{5} -2.44949 q^{7} -1.00000 q^{9} -1.46410i q^{11} +4.24264i q^{13} -3.86370 q^{15} -3.46410 q^{17} -7.46410i q^{19} +2.44949i q^{21} -2.82843 q^{23} -9.92820 q^{25} +1.00000i q^{27} +8.76268i q^{29} +7.34847 q^{31} -1.46410 q^{33} +9.46410i q^{35} -0.656339i q^{37} +4.24264 q^{39} -4.53590 q^{41} -3.46410i q^{43} +3.86370i q^{45} -2.82843 q^{47} -1.00000 q^{49} +3.46410i q^{51} +4.62158i q^{53} -5.65685 q^{55} -7.46410 q^{57} +13.8564i q^{59} -0.656339i q^{61} +2.44949 q^{63} +16.3923 q^{65} -14.9282i q^{67} +2.82843i q^{69} -6.41473 q^{71} -4.00000 q^{73} +9.92820i q^{75} +3.58630i q^{77} -2.44949 q^{79} +1.00000 q^{81} -5.46410i q^{83} +13.3843i q^{85} +8.76268 q^{87} +4.92820 q^{89} -10.3923i q^{91} -7.34847i q^{93} -28.8391 q^{95} +1.07180 q^{97} +1.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 24 q^{25} + 16 q^{33} - 64 q^{41} - 8 q^{49} - 32 q^{57} + 48 q^{65} - 32 q^{73} + 8 q^{81} - 16 q^{89} + 64 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 24 * q^25 + 16 * q^33 - 64 * q^41 - 8 * q^49 - 32 * q^57 + 48 * q^65 - 32 * q^73 + 8 * q^81 - 16 * q^89 + 64 * q^97

Introduction

L-functions

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