LMFDB - Embedded newform 3072.2.d.e.1537.5

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			1537.5
Root			$-0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.e.1537.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.49661i q^{5} -0.917608 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.49661i q^{5} -0.917608 q^{7} -1.00000 q^{9} -3.69552i q^{11} +5.81204i q^{13} +2.49661 q^{15} -0.867091 q^{17} +6.52395i q^{19} -0.917608i q^{21} -4.00000 q^{23} -1.23304 q^{25} -1.00000i q^{27} -7.72286i q^{29} -2.14386 q^{31} +3.69552 q^{33} +2.29090i q^{35} +2.47568i q^{37} -5.81204 q^{39} +9.58541 q^{41} +9.58541i q^{43} +2.49661i q^{45} -1.65685 q^{47} -6.15800 q^{49} -0.867091i q^{51} -3.39329i q^{53} -9.22625 q^{55} -6.52395 q^{57} +12.7183i q^{59} +0.0231773i q^{61} +0.917608 q^{63} +14.5104 q^{65} +5.32729i q^{67} -4.00000i q^{69} -11.8216 q^{71} +15.2809 q^{73} -1.23304i q^{75} +3.39104i q^{77} +8.40968 q^{79} +1.00000 q^{81} +1.96134i q^{83} +2.16478i q^{85} +7.72286 q^{87} -2.79565 q^{89} -5.33317i q^{91} -2.14386i q^{93} +16.2877 q^{95} -2.26582 q^{97} +3.69552i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 2.49661i	− 1.11652i	−0.829667	−	0.558258i	$-0.811470\pi$
$5$	− 2.49661i	− 1.11652i	0.829667	−	0.558258i	$-0.188530\pi$
$6$	0	0
$7$	−0.917608	−0.346823	−0.173412	−	0.984849i	$-0.555479\pi$
$7$	−0.917608	−0.346823	−0.173412	+	0.984849i	$0.555479\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 3.69552i	− 1.11424i	−0.830432	−	0.557120i	$-0.811906\pi$
$11$	− 3.69552i	− 1.11424i	0.830432	−	0.557120i	$-0.188094\pi$
$12$	0	0
$13$	5.81204i	1.61197i	0.591936	+	0.805985i	$0.298364\pi$
$13$	5.81204i	1.61197i	−0.591936	+	0.805985i	$0.701636\pi$
$14$	0	0
$15$	2.49661	0.644621
$16$	0	0
$17$	−0.867091	−0.210300	−0.105150	−	0.994456i	$-0.533532\pi$
$17$	−0.867091	−0.210300	−0.105150	+	0.994456i	$0.533532\pi$
$18$	0	0
$19$	6.52395i	1.49670i	0.663306	+	0.748348i	$0.269153\pi$
$19$	6.52395i	1.49670i	−0.663306	+	0.748348i	$0.730847\pi$
$20$	0	0
$21$	− 0.917608i	− 0.200238i
$22$	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
$25$	−1.23304	−0.246608
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 7.72286i	− 1.43410i	−0.697022	−	0.717049i	$-0.745492\pi$
$29$	− 7.72286i	− 1.43410i	0.697022	−	0.717049i	$-0.254508\pi$
$30$	0	0
$31$	−2.14386	−0.385049	−0.192524	−	0.981292i	$-0.561667\pi$
$31$	−2.14386	−0.385049	−0.192524	+	0.981292i	$0.561667\pi$
$32$	0	0
$33$	3.69552	0.643307
$34$	0	0
$35$	2.29090i	0.387234i
$36$	0	0
$37$	2.47568i	0.406999i	0.979075	+	0.203500i	$0.0652316\pi$
$37$	2.47568i	0.406999i	−0.979075	+	0.203500i	$0.934768\pi$
$38$	0	0
$39$	−5.81204	−0.930671
$40$	0	0
$41$	9.58541	1.49699	0.748495	−	0.663140i	$-0.230777\pi$
$41$	9.58541	1.49699	0.748495	+	0.663140i	$0.230777\pi$
$42$	0	0
$43$	9.58541i	1.46176i	0.682505	+	0.730881i	$0.260891\pi$
$43$	9.58541i	1.46176i	−0.682505	+	0.730881i	$0.739109\pi$
$44$	0	0
$45$	2.49661i	0.372172i
$46$	0	0
$47$	−1.65685	−0.241677	−0.120839	−	0.992672i	$-0.538558\pi$
$47$	−1.65685	−0.241677	−0.120839	+	0.992672i	$0.538558\pi$
$48$	0	0
$49$	−6.15800	−0.879714
$50$	0	0
$51$	− 0.867091i	− 0.121417i
$52$	0	0
$53$	− 3.39329i	− 0.466104i	−0.972464	−	0.233052i	$-0.925129\pi$
$53$	− 3.39329i	− 0.466104i	0.972464	−	0.233052i	$-0.0748712\pi$
$54$	0	0
$55$	−9.22625	−1.24407
$56$	0	0
$57$	−6.52395	−0.864118
$58$	0	0
$59$	12.7183i	1.65578i	0.560887	+	0.827892i	$0.310460\pi$
$59$	12.7183i	1.65578i	−0.560887	+	0.827892i	$0.689540\pi$
$60$	0	0
$61$	0.0231773i	0.00296755i	0.999999	+	0.00148377i	$0.000472300\pi$
$61$	0.0231773i	0.00296755i	−0.999999	+	0.00148377i	$0.999528\pi$
$62$	0	0
$63$	0.917608	0.115608
$64$	0	0
$65$	14.5104	1.79979
$66$	0	0
$67$	5.32729i	0.650832i	0.945571	+	0.325416i	$0.105504\pi$
$67$	5.32729i	0.650832i	−0.945571	+	0.325416i	$0.894496\pi$
$68$	0	0
$69$	− 4.00000i	− 0.481543i
$70$	0	0
$71$	−11.8216	−1.40297	−0.701485	−	0.712684i	$-0.747479\pi$
$71$	−11.8216	−1.40297	−0.701485	+	0.712684i	$0.747479\pi$
$72$	0	0
$73$	15.2809	1.78850	0.894249	−	0.447570i	$-0.147711\pi$
$73$	15.2809	1.78850	0.894249	+	0.447570i	$0.147711\pi$
$74$	0	0
$75$	− 1.23304i	− 0.142379i
$76$	0	0
$77$	3.39104i	0.386444i
$78$	0	0
$79$	8.40968	0.946163	0.473081	−	0.881019i	$-0.343142\pi$
$79$	8.40968	0.946163	0.473081	+	0.881019i	$0.343142\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.96134i	0.215285i	0.994190	+	0.107642i	$0.0343301\pi$
$83$	1.96134i	0.215285i	−0.994190	+	0.107642i	$0.965670\pi$
$84$	0	0
$85$	2.16478i	0.234804i
$86$	0	0
$87$	7.72286	0.827977
$88$	0	0
$89$	−2.79565	−0.296338	−0.148169	−	0.988962i	$-0.547338\pi$
$89$	−2.79565	−0.296338	−0.148169	+	0.988962i	$0.547338\pi$
$90$	0	0
$91$	− 5.33317i	− 0.559068i
$92$	0	0
$93$	− 2.14386i	− 0.222308i
$94$	0	0
$95$	16.2877	1.67108
$96$	0	0
$97$	−2.26582	−0.230059	−0.115029	−	0.993362i	$-0.536696\pi$
$97$	−2.26582	−0.230059	−0.115029	+	0.993362i	$0.536696\pi$
$98$	0	0
$99$	3.69552i	0.371414i
$100$	0	0
$101$	9.98868i	0.993910i	0.867776	+	0.496955i	$0.165549\pi$
$101$	9.98868i	0.993910i	−0.867776	+	0.496955i	$0.834451\pi$
$102$	0	0
$103$	16.4180	1.61771	0.808857	−	0.588006i	$-0.200087\pi$
$103$	16.4180	1.61771	0.808857	+	0.588006i	$0.200087\pi$
$104$	0	0
$105$	−2.29090	−0.223569
$106$	0	0
$107$	12.2522i	1.18447i	0.805766	+	0.592234i	$0.201754\pi$
$107$	12.2522i	1.18447i	−0.805766	+	0.592234i	$0.798246\pi$
$108$	0	0
$109$	18.2973i	1.75257i	0.481797	+	0.876283i	$0.339984\pi$
$109$	18.2973i	1.75257i	−0.481797	+	0.876283i	$0.660016\pi$
$110$	0	0
$111$	−2.47568	−0.234981
$112$	0	0
$113$	3.65685	0.344008	0.172004	−	0.985096i	$-0.444976\pi$
$113$	3.65685	0.344008	0.172004	+	0.985096i	$0.444976\pi$
$114$	0	0
$115$	9.98642i	0.931239i
$116$	0	0
$117$	− 5.81204i	− 0.537323i
$118$	0	0
$119$	0.795649	0.0729371
$120$	0	0
$121$	−2.65685	−0.241532
$122$	0	0
$123$	9.58541i	0.864288i
$124$	0	0
$125$	− 9.40461i	− 0.841174i
$126$	0	0
$127$	20.6382	1.83135	0.915673	−	0.401925i	$-0.131659\pi$
$127$	20.6382	1.83135	0.915673	+	0.401925i	$0.131659\pi$
$128$	0	0
$129$	−9.58541	−0.843949
$130$	0	0
$131$	17.5140i	1.53020i	0.643910	+	0.765101i	$0.277311\pi$
$131$	17.5140i	1.53020i	−0.643910	+	0.765101i	$0.722689\pi$
$132$	0	0
$133$	− 5.98642i	− 0.519089i
$134$	0	0
$135$	−2.49661	−0.214874
$136$	0	0
$137$	−8.25813	−0.705539	−0.352770	−	0.935710i	$-0.614760\pi$
$137$	−8.25813	−0.705539	−0.352770	+	0.935710i	$0.614760\pi$
$138$	0	0
$139$	14.7047i	1.24724i	0.781728	+	0.623620i	$0.214339\pi$
$139$	14.7047i	1.24724i	−0.781728	+	0.623620i	$0.785661\pi$
$140$	0	0
$141$	− 1.65685i	− 0.139532i
$142$	0	0
$143$	21.4785	1.79612
$144$	0	0
$145$	−19.2809	−1.60119
$146$	0	0
$147$	− 6.15800i	− 0.507903i
$148$	0	0
$149$	16.1535i	1.32334i	0.749794	+	0.661672i	$0.230153\pi$
$149$	16.1535i	1.32334i	−0.749794	+	0.661672i	$0.769847\pi$
$150$	0	0
$151$	−19.2691	−1.56810	−0.784048	−	0.620701i	$-0.786848\pi$
$151$	−19.2691	−1.56810	−0.784048	+	0.620701i	$0.786848\pi$
$152$	0	0
$153$	0.867091	0.0701002
$154$	0	0
$155$	5.35237i	0.429913i
$156$	0	0
$157$	0.818827i	0.0653495i	0.999466	+	0.0326747i	$0.0104025\pi$
$157$	0.818827i	0.0653495i	−0.999466	+	0.0326747i	$0.989597\pi$
$158$	0	0
$159$	3.39329	0.269105
$160$	0	0
$161$	3.67043	0.289271
$162$	0	0
$163$	− 23.6492i	− 1.85235i	−0.377100	−	0.926173i	$-0.623079\pi$
$163$	− 23.6492i	− 1.85235i	0.377100	−	0.926173i	$-0.376921\pi$
$164$	0	0
$165$	− 9.22625i	− 0.718263i
$166$	0	0
$167$	−9.30358	−0.719933	−0.359966	−	0.932965i	$-0.617212\pi$
$167$	−9.30358	−0.719933	−0.359966	+	0.932965i	$0.617212\pi$
$168$	0	0
$169$	−20.7798	−1.59845
$170$	0	0
$171$	− 6.52395i	− 0.498899i
$172$	0	0
$173$	− 7.10557i	− 0.540226i	−0.962829	−	0.270113i	$-0.912939\pi$
$173$	− 7.10557i	− 0.540226i	0.962829	−	0.270113i	$-0.0870611\pi$
$174$	0	0
$175$	1.13145	0.0855294
$176$	0	0
$177$	−12.7183	−0.955968
$178$	0	0
$179$	− 2.67271i	− 0.199768i	−0.994999	−	0.0998840i	$-0.968153\pi$
$179$	− 2.67271i	− 0.199768i	0.994999	−	0.0998840i	$-0.0318472\pi$
$180$	0	0
$181$	0.640465i	0.0476054i	0.999717	+	0.0238027i	$0.00757735\pi$
$181$	0.640465i	0.0476054i	−0.999717	+	0.0238027i	$0.992423\pi$
$182$	0	0
$183$	−0.0231773	−0.00171332
$184$	0	0
$185$	6.18080	0.454421
$186$	0	0
$187$	3.20435i	0.234325i
$188$	0	0
$189$	0.917608i	0.0667461i
$190$	0	0
$191$	−14.6037	−1.05669	−0.528344	−	0.849031i	$-0.677187\pi$
$191$	−14.6037	−1.05669	−0.528344	+	0.849031i	$0.677187\pi$
$192$	0	0
$193$	−8.81485	−0.634507	−0.317253	−	0.948341i	$-0.602761\pi$
$193$	−8.81485	−0.634507	−0.317253	+	0.948341i	$0.602761\pi$
$194$	0	0
$195$	14.5104i	1.03911i
$196$	0	0
$197$	3.74551i	0.266856i	0.991058	+	0.133428i	$0.0425985\pi$
$197$	3.74551i	0.266856i	−0.991058	+	0.133428i	$0.957401\pi$
$198$	0	0
$199$	−14.8267	−1.05104	−0.525519	−	0.850782i	$-0.676129\pi$
$199$	−14.8267	−1.05104	−0.525519	+	0.850782i	$0.676129\pi$
$200$	0	0
$201$	−5.32729	−0.375758
$202$	0	0
$203$	7.08655i	0.497379i
$204$	0	0
$205$	− 23.9310i	− 1.67141i
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	24.1094	1.66768
$210$	0	0
$211$	11.3910i	0.784191i	0.919924	+	0.392096i	$0.128250\pi$
$211$	11.3910i	0.784191i	−0.919924	+	0.392096i	$0.871750\pi$
$212$	0	0
$213$	− 11.8216i	− 0.810005i
$214$	0	0
$215$	23.9310	1.63208
$216$	0	0
$217$	1.96722	0.133544
$218$	0	0
$219$	15.2809i	1.03259i
$220$	0	0
$221$	− 5.03957i	− 0.338998i
$222$	0	0
$223$	4.78110	0.320166	0.160083	−	0.987104i	$-0.448824\pi$
$223$	4.78110	0.320166	0.160083	+	0.987104i	$0.448824\pi$
$224$	0	0
$225$	1.23304	0.0822027
$226$	0	0
$227$	3.83840i	0.254764i	0.991854	+	0.127382i	$0.0406574\pi$
$227$	3.83840i	0.254764i	−0.991854	+	0.127382i	$0.959343\pi$
$228$	0	0
$229$	1.34596i	0.0889434i	0.999011	+	0.0444717i	$0.0141604\pi$
$229$	1.34596i	0.0889434i	−0.999011	+	0.0444717i	$0.985840\pi$
$230$	0	0
$231$	−3.39104	−0.223114
$232$	0	0
$233$	17.6433	1.15585	0.577925	−	0.816090i	$-0.303863\pi$
$233$	17.6433	1.15585	0.577925	+	0.816090i	$0.303863\pi$
$234$	0	0
$235$	4.13651i	0.269836i
$236$	0	0
$237$	8.40968i	0.546267i
$238$	0	0
$239$	−9.61500	−0.621943	−0.310971	−	0.950419i	$-0.600654\pi$
$239$	−9.61500	−0.621943	−0.310971	+	0.950419i	$0.600654\pi$
$240$	0	0
$241$	21.3583	1.37581	0.687903	−	0.725802i	$-0.258531\pi$
$241$	21.3583	1.37581	0.687903	+	0.725802i	$0.258531\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	15.3741i	0.982214i
$246$	0	0
$247$	−37.9174	−2.41263
$248$	0	0
$249$	−1.96134	−0.124295
$250$	0	0
$251$	− 4.30448i	− 0.271696i	−0.990730	−	0.135848i	$-0.956624\pi$
$251$	− 4.30448i	− 0.271696i	0.990730	−	0.135848i	$-0.0433760\pi$
$252$	0	0
$253$	14.7821i	0.929341i
$254$	0	0
$255$	−2.16478	−0.135564
$256$	0	0
$257$	−4.67271	−0.291476	−0.145738	−	0.989323i	$-0.546556\pi$
$257$	−4.67271	−0.291476	−0.145738	+	0.989323i	$0.546556\pi$
$258$	0	0
$259$	− 2.27170i	− 0.141157i
$260$	0	0
$261$	7.72286i	0.478033i
$262$	0	0
$263$	−15.4502	−0.952701	−0.476351	−	0.879255i	$-0.658041\pi$
$263$	−15.4502	−0.952701	−0.476351	+	0.879255i	$0.658041\pi$
$264$	0	0
$265$	−8.47170	−0.520413
$266$	0	0
$267$	− 2.79565i	− 0.171091i
$268$	0	0
$269$	− 31.2459i	− 1.90510i	−0.304388	−	0.952548i	$-0.598452\pi$
$269$	− 31.2459i	− 1.90510i	0.304388	−	0.952548i	$-0.401548\pi$
$270$	0	0
$271$	9.88817	0.600664	0.300332	−	0.953835i	$-0.402903\pi$
$271$	9.88817	0.600664	0.300332	+	0.953835i	$0.402903\pi$
$272$	0	0
$273$	5.33317	0.322778
$274$	0	0
$275$	4.55672i	0.274781i
$276$	0	0
$277$	− 22.9565i	− 1.37932i	−0.724133	−	0.689660i	$-0.757760\pi$
$277$	− 22.9565i	− 1.37932i	0.724133	−	0.689660i	$-0.242240\pi$
$278$	0	0
$279$	2.14386	0.128350
$280$	0	0
$281$	14.5754	0.869498	0.434749	−	0.900552i	$-0.356837\pi$
$281$	14.5754	0.869498	0.434749	+	0.900552i	$0.356837\pi$
$282$	0	0
$283$	− 4.71832i	− 0.280475i	−0.990118	−	0.140238i	$-0.955213\pi$
$283$	− 4.71832i	− 0.280475i	0.990118	−	0.140238i	$-0.0447866\pi$
$284$	0	0
$285$	16.2877i	0.964801i
$286$	0	0
$287$	−8.79565	−0.519191
$288$	0	0
$289$	−16.2482	−0.955774
$290$	0	0
$291$	− 2.26582i	− 0.132825i
$292$	0	0
$293$	23.9204i	1.39745i	0.715392	+	0.698723i	$0.246248\pi$
$293$	23.9204i	1.39745i	−0.715392	+	0.698723i	$0.753752\pi$
$294$	0	0
$295$	31.7526	1.84871
$296$	0	0
$297$	−3.69552	−0.214436
$298$	0	0
$299$	− 23.2482i	− 1.34448i
$300$	0	0
$301$	− 8.79565i	− 0.506973i
$302$	0	0
$303$	−9.98868	−0.573834
$304$	0	0
$305$	0.0578646	0.00331332
$306$	0	0
$307$	20.7685i	1.18532i	0.805453	+	0.592660i	$0.201922\pi$
$307$	20.7685i	1.18532i	−0.805453	+	0.592660i	$0.798078\pi$
$308$	0	0
$309$	16.4180i	0.933988i
$310$	0	0
$311$	−11.7798	−0.667971	−0.333985	−	0.942578i	$-0.608394\pi$
$311$	−11.7798	−0.667971	−0.333985	+	0.942578i	$0.608394\pi$
$312$	0	0
$313$	−11.1580	−0.630687	−0.315344	−	0.948978i	$-0.602120\pi$
$313$	−11.1580	−0.630687	−0.315344	+	0.948978i	$0.602120\pi$
$314$	0	0
$315$	− 2.29090i	− 0.129078i
$316$	0	0
$317$	15.9140i	0.893822i	0.894578	+	0.446911i	$0.147476\pi$
$317$	15.9140i	0.893822i	−0.894578	+	0.446911i	$0.852524\pi$
$318$	0	0
$319$	−28.5400	−1.59793
$320$	0	0
$321$	−12.2522	−0.683853
$322$	0	0
$323$	− 5.65685i	− 0.314756i
$324$	0	0
$325$	− 7.16648i	− 0.397525i
$326$	0	0
$327$	−18.2973	−1.01184
$328$	0	0
$329$	1.52034	0.0838192
$330$	0	0
$331$	2.73190i	0.150159i	0.997178	+	0.0750794i	$0.0239210\pi$
$331$	2.73190i	0.150159i	−0.997178	+	0.0750794i	$0.976079\pi$
$332$	0	0
$333$	− 2.47568i	− 0.135666i
$334$	0	0
$335$	13.3001	0.726664
$336$	0	0
$337$	−1.49657	−0.0815236	−0.0407618	−	0.999169i	$-0.512978\pi$
$337$	−1.49657	−0.0815236	−0.0407618	+	0.999169i	$0.512978\pi$
$338$	0	0
$339$	3.65685i	0.198613i
$340$	0	0
$341$	7.92267i	0.429037i
$342$	0	0
$343$	12.0739	0.651928
$344$	0	0
$345$	−9.98642	−0.537651
$346$	0	0
$347$	− 3.22944i	− 0.173365i	−0.996236	−	0.0866826i	$-0.972373\pi$
$347$	− 3.22944i	− 0.173365i	0.996236	−	0.0866826i	$-0.0276266\pi$
$348$	0	0
$349$	7.51074i	0.402041i	0.979587	+	0.201020i	$0.0644258\pi$
$349$	7.51074i	0.402041i	−0.979587	+	0.201020i	$0.935574\pi$
$350$	0	0
$351$	5.81204	0.310224
$352$	0	0
$353$	2.17491	0.115759	0.0578795	−	0.998324i	$-0.481566\pi$
$353$	2.17491	0.115759	0.0578795	+	0.998324i	$0.481566\pi$
$354$	0	0
$355$	29.5140i	1.56644i
$356$	0	0
$357$	0.795649i	0.0421102i
$358$	0	0
$359$	12.6828	0.669375	0.334687	−	0.942329i	$-0.391369\pi$
$359$	12.6828	0.669375	0.334687	+	0.942329i	$0.391369\pi$
$360$	0	0
$361$	−23.5619	−1.24010
$362$	0	0
$363$	− 2.65685i	− 0.139449i
$364$	0	0
$365$	− 38.1505i	− 1.99689i
$366$	0	0
$367$	−21.4576	−1.12008	−0.560038	−	0.828467i	$-0.689213\pi$
$367$	−21.4576	−1.12008	−0.560038	+	0.828467i	$0.689213\pi$
$368$	0	0
$369$	−9.58541	−0.498997
$370$	0	0
$371$	3.11371i	0.161656i
$372$	0	0
$373$	0.508459i	0.0263270i	0.999913	+	0.0131635i	$0.00419019\pi$
$373$	0.508459i	0.0263270i	−0.999913	+	0.0131635i	$0.995810\pi$
$374$	0	0
$375$	9.40461	0.485652
$376$	0	0
$377$	44.8855	2.31172
$378$	0	0
$379$	− 19.9286i	− 1.02366i	−0.859086	−	0.511831i	$-0.828968\pi$
$379$	− 19.9286i	− 1.02366i	0.859086	−	0.511831i	$-0.171032\pi$
$380$	0	0
$381$	20.6382i	1.05733i
$382$	0	0
$383$	29.3858	1.50154	0.750772	−	0.660562i	$-0.229682\pi$
$383$	29.3858	1.50154	0.750772	+	0.660562i	$0.229682\pi$
$384$	0	0
$385$	8.46608	0.431471
$386$	0	0
$387$	− 9.58541i	− 0.487254i
$388$	0	0
$389$	− 1.00379i	− 0.0508941i	−0.999676	−	0.0254470i	$-0.991899\pi$
$389$	− 1.00379i	− 0.0508941i	0.999676	−	0.0254470i	$-0.00810092\pi$
$390$	0	0
$391$	3.46836	0.175403
$392$	0	0
$393$	−17.5140	−0.883463
$394$	0	0
$395$	− 20.9956i	− 1.05641i
$396$	0	0
$397$	7.02546i	0.352598i	0.984337	+	0.176299i	$0.0564125\pi$
$397$	7.02546i	0.352598i	−0.984337	+	0.176299i	$0.943587\pi$
$398$	0	0
$399$	5.98642	0.299696
$400$	0	0
$401$	4.18080	0.208779	0.104390	−	0.994536i	$-0.466711\pi$
$401$	4.18080	0.208779	0.104390	+	0.994536i	$0.466711\pi$
$402$	0	0
$403$	− 12.4602i	− 0.620686i
$404$	0	0
$405$	− 2.49661i	− 0.124057i
$406$	0	0
$407$	9.14892	0.453495
$408$	0	0
$409$	−16.5818	−0.819918	−0.409959	−	0.912104i	$-0.634457\pi$
$409$	−16.5818	−0.819918	−0.409959	+	0.912104i	$0.634457\pi$
$410$	0	0
$411$	− 8.25813i	− 0.407343i
$412$	0	0
$413$	− 11.6704i	− 0.574264i
$414$	0	0
$415$	4.89668	0.240369
$416$	0	0
$417$	−14.7047	−0.720094
$418$	0	0
$419$	22.2574i	1.08734i	0.839298	+	0.543672i	$0.182966\pi$
$419$	22.2574i	1.08734i	−0.839298	+	0.543672i	$0.817034\pi$
$420$	0	0
$421$	− 28.4203i	− 1.38512i	−0.721361	−	0.692559i	$-0.756483\pi$
$421$	− 28.4203i	− 1.38512i	0.721361	−	0.692559i	$-0.243517\pi$
$422$	0	0
$423$	1.65685	0.0805590
$424$	0	0
$425$	1.06916	0.0518618
$426$	0	0
$427$	− 0.0212677i	− 0.00102921i
$428$	0	0
$429$	21.4785i	1.03699i
$430$	0	0
$431$	−36.6138	−1.76363	−0.881813	−	0.471599i	$-0.843677\pi$
$431$	−36.6138	−1.76363	−0.881813	+	0.471599i	$0.843677\pi$
$432$	0	0
$433$	27.7526	1.33371	0.666853	−	0.745189i	$-0.267641\pi$
$433$	27.7526	1.33371	0.666853	+	0.745189i	$0.267641\pi$
$434$	0	0
$435$	− 19.2809i	− 0.924450i
$436$	0	0
$437$	− 26.0958i	− 1.24833i
$438$	0	0
$439$	21.1461	1.00925	0.504625	−	0.863339i	$-0.331631\pi$
$439$	21.1461	1.00925	0.504625	+	0.863339i	$0.331631\pi$
$440$	0	0
$441$	6.15800	0.293238
$442$	0	0
$443$	7.30677i	0.347155i	0.984820	+	0.173577i	$0.0555327\pi$
$443$	7.30677i	0.347155i	−0.984820	+	0.173577i	$0.944467\pi$
$444$	0	0
$445$	6.97963i	0.330866i
$446$	0	0
$447$	−16.1535	−0.764032
$448$	0	0
$449$	−17.6356	−0.832275	−0.416137	−	0.909302i	$-0.636616\pi$
$449$	−17.6356	−0.832275	−0.416137	+	0.909302i	$0.636616\pi$
$450$	0	0
$451$	− 35.4231i	− 1.66801i
$452$	0	0
$453$	− 19.2691i	− 0.905340i
$454$	0	0
$455$	−13.3148	−0.624209
$456$	0	0
$457$	−17.7070	−0.828300	−0.414150	−	0.910209i	$-0.635921\pi$
$457$	−17.7070	−0.828300	−0.414150	+	0.910209i	$0.635921\pi$
$458$	0	0
$459$	0.867091i	0.0404723i
$460$	0	0
$461$	36.9811i	1.72238i	0.508280	+	0.861192i	$0.330281\pi$
$461$	36.9811i	1.72238i	−0.508280	+	0.861192i	$0.669719\pi$
$462$	0	0
$463$	10.9632	0.509504	0.254752	−	0.967006i	$-0.418006\pi$
$463$	10.9632	0.509504	0.254752	+	0.967006i	$0.418006\pi$
$464$	0	0
$465$	−5.35237	−0.248210
$466$	0	0
$467$	− 12.6661i	− 0.586116i	−0.956095	−	0.293058i	$-0.905327\pi$
$467$	− 12.6661i	− 0.586116i	0.956095	−	0.293058i	$-0.0946730\pi$
$468$	0	0
$469$	− 4.88836i	− 0.225723i
$470$	0	0
$471$	−0.818827	−0.0377295
$472$	0	0
$473$	35.4231	1.62875
$474$	0	0
$475$	− 8.04429i	− 0.369097i
$476$	0	0
$477$	3.39329i	0.155368i
$478$	0	0
$479$	−13.3036	−0.607856	−0.303928	−	0.952695i	$-0.598298\pi$
$479$	−13.3036	−0.607856	−0.303928	+	0.952695i	$0.598298\pi$
$480$	0	0
$481$	−14.3888	−0.656071
$482$	0	0
$483$	3.67043i	0.167010i
$484$	0	0
$485$	5.65685i	0.256865i
$486$	0	0
$487$	5.44924	0.246929	0.123464	−	0.992349i	$-0.460600\pi$
$487$	5.44924	0.246929	0.123464	+	0.992349i	$0.460600\pi$
$488$	0	0
$489$	23.6492	1.06945
$490$	0	0
$491$	21.7844i	0.983114i	0.870845	+	0.491557i	$0.163572\pi$
$491$	21.7844i	0.983114i	−0.870845	+	0.491557i	$0.836428\pi$
$492$	0	0
$493$	6.69642i	0.301592i
$494$	0	0
$495$	9.22625	0.414689
$496$	0	0
$497$	10.8476	0.486583
$498$	0	0
$499$	− 20.1595i	− 0.902465i	−0.892407	−	0.451232i	$-0.850985\pi$
$499$	− 20.1595i	− 0.902465i	0.892407	−	0.451232i	$-0.149015\pi$
$500$	0	0
$501$	− 9.30358i	− 0.415653i
$502$	0	0
$503$	−33.5879	−1.49761	−0.748804	−	0.662791i	$-0.769372\pi$
$503$	−33.5879	−1.49761	−0.748804	+	0.662791i	$0.769372\pi$
$504$	0	0
$505$	24.9378	1.10972
$506$	0	0
$507$	− 20.7798i	− 0.922863i
$508$	0	0
$509$	− 11.1957i	− 0.496242i	−0.968729	−	0.248121i	$-0.920187\pi$
$509$	− 11.1957i	− 0.496242i	0.968729	−	0.248121i	$-0.0798130\pi$
$510$	0	0
$511$	−14.0219	−0.620292
$512$	0	0
$513$	6.52395	0.288039
$514$	0	0
$515$	− 40.9893i	− 1.80620i
$516$	0	0
$517$	6.12293i	0.269286i
$518$	0	0
$519$	7.10557	0.311900
$520$	0	0
$521$	35.2151	1.54280	0.771401	−	0.636349i	$-0.219556\pi$
$521$	35.2151	1.54280	0.771401	+	0.636349i	$0.219556\pi$
$522$	0	0
$523$	12.9444i	0.566020i	0.959117	+	0.283010i	$0.0913329\pi$
$523$	12.9444i	0.566020i	−0.959117	+	0.283010i	$0.908667\pi$
$524$	0	0
$525$	1.13145i	0.0493804i
$526$	0	0
$527$	1.85892	0.0809759
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 12.7183i	− 0.551928i
$532$	0	0
$533$	55.7108i	2.41310i
$534$	0	0
$535$	30.5890	1.32248
$536$	0	0
$537$	2.67271	0.115336
$538$	0	0
$539$	22.7570i	0.980213i
$540$	0	0
$541$	17.8584i	0.767792i	0.923376	+	0.383896i	$0.125418\pi$
$541$	17.8584i	0.767792i	−0.923376	+	0.383896i	$0.874582\pi$
$542$	0	0
$543$	−0.640465	−0.0274850
$544$	0	0
$545$	45.6812	1.95677
$546$	0	0
$547$	− 14.9191i	− 0.637893i	−0.947773	−	0.318947i	$-0.896671\pi$
$547$	− 14.9191i	− 0.637893i	0.947773	−	0.318947i	$-0.103329\pi$
$548$	0	0
$549$	− 0.0231773i	0 0.000989183i
$550$	0	0
$551$	50.3835	2.14641
$552$	0	0
$553$	−7.71679	−0.328151
$554$	0	0
$555$	6.18080i	0.262360i
$556$	0	0
$557$	− 24.4649i	− 1.03661i	−0.855196	−	0.518305i	$-0.826563\pi$
$557$	− 24.4649i	− 1.03661i	0.855196	−	0.518305i	$-0.173437\pi$
$558$	0	0
$559$	−55.7108	−2.35632
$560$	0	0
$561$	−3.20435	−0.135288
$562$	0	0
$563$	− 40.4459i	− 1.70459i	−0.523061	−	0.852295i	$-0.675210\pi$
$563$	− 40.4459i	− 1.70459i	0.523061	−	0.852295i	$-0.324790\pi$
$564$	0	0
$565$	− 9.12972i	− 0.384090i
$566$	0	0
$567$	−0.917608	−0.0385359
$568$	0	0
$569$	−42.1017	−1.76499	−0.882497	−	0.470318i	$-0.844139\pi$
$569$	−42.1017	−1.76499	−0.882497	+	0.470318i	$0.844139\pi$
$570$	0	0
$571$	− 1.24996i	− 0.0523091i	−0.999658	−	0.0261546i	$-0.991674\pi$
$571$	− 1.24996i	− 0.0523091i	0.999658	−	0.0261546i	$-0.00832620\pi$
$572$	0	0
$573$	− 14.6037i	− 0.610079i
$574$	0	0
$575$	4.93216	0.205685
$576$	0	0
$577$	29.3767	1.22297	0.611484	−	0.791257i	$-0.290573\pi$
$577$	29.3767	1.22297	0.611484	+	0.791257i	$0.290573\pi$
$578$	0	0
$579$	− 8.81485i	− 0.366333i
$580$	0	0
$581$	− 1.79974i	− 0.0746657i
$582$	0	0
$583$	−12.5400	−0.519352
$584$	0	0
$585$	−14.5104	−0.599930
$586$	0	0
$587$	− 17.7070i	− 0.730847i	−0.930841	−	0.365424i	$-0.880924\pi$
$587$	− 17.7070i	− 0.730847i	0.930841	−	0.365424i	$-0.119076\pi$
$588$	0	0
$589$	− 13.9864i	− 0.576301i
$590$	0	0
$591$	−3.74551	−0.154070
$592$	0	0
$593$	17.1207	0.703061	0.351530	−	0.936176i	$-0.385661\pi$
$593$	17.1207	0.703061	0.351530	+	0.936176i	$0.385661\pi$
$594$	0	0
$595$	− 1.98642i	− 0.0814354i
$596$	0	0
$597$	− 14.8267i	− 0.606817i
$598$	0	0
$599$	25.1908	1.02927	0.514634	−	0.857410i	$-0.327928\pi$
$599$	25.1908	1.02927	0.514634	+	0.857410i	$0.327928\pi$
$600$	0	0
$601$	−22.5946	−0.921655	−0.460827	−	0.887490i	$-0.652447\pi$
$601$	−22.5946	−0.921655	−0.460827	+	0.887490i	$0.652447\pi$
$602$	0	0
$603$	− 5.32729i	− 0.216944i
$604$	0	0
$605$	6.63312i	0.269675i
$606$	0	0
$607$	8.88408	0.360594	0.180297	−	0.983612i	$-0.442294\pi$
$607$	8.88408	0.360594	0.180297	+	0.983612i	$0.442294\pi$
$608$	0	0
$609$	−7.08655	−0.287162
$610$	0	0
$611$	− 9.62970i	− 0.389576i
$612$	0	0
$613$	− 15.9960i	− 0.646073i	−0.946386	−	0.323037i	$-0.895296\pi$
$613$	− 15.9960i	− 0.646073i	0.946386	−	0.323037i	$-0.104704\pi$
$614$	0	0
$615$	23.9310	0.964991
$616$	0	0
$617$	2.99772	0.120684	0.0603418	−	0.998178i	$-0.480781\pi$
$617$	2.99772	0.120684	0.0603418	+	0.998178i	$0.480781\pi$
$618$	0	0
$619$	− 15.0279i	− 0.604024i	−0.953304	−	0.302012i	$-0.902342\pi$
$619$	− 15.0279i	− 0.604024i	0.953304	−	0.302012i	$-0.0976583\pi$
$620$	0	0
$621$	4.00000i	0.160514i
$622$	0	0
$623$	2.56531	0.102777
$624$	0	0
$625$	−29.6448	−1.18579
$626$	0	0
$627$	24.1094i	0.962835i
$628$	0	0
$629$	− 2.14664i	− 0.0855922i
$630$	0	0
$631$	19.3701	0.771112	0.385556	−	0.922684i	$-0.374010\pi$
$631$	19.3701	0.771112	0.385556	+	0.922684i	$0.374010\pi$
$632$	0	0
$633$	−11.3910	−0.452753
$634$	0	0
$635$	− 51.5255i	− 2.04473i
$636$	0	0
$637$	− 35.7905i	− 1.41807i
$638$	0	0
$639$	11.8216	0.467657
$640$	0	0
$641$	13.4489	0.531200	0.265600	−	0.964083i	$-0.414430\pi$
$641$	13.4489	0.531200	0.265600	+	0.964083i	$0.414430\pi$
$642$	0	0
$643$	− 2.44662i	− 0.0964852i	−0.998836	−	0.0482426i	$-0.984638\pi$
$643$	− 2.44662i	− 0.0964852i	0.998836	−	0.0482426i	$-0.0153621\pi$
$644$	0	0
$645$	23.9310i	0.942282i
$646$	0	0
$647$	16.1275	0.634038	0.317019	−	0.948419i	$-0.397318\pi$
$647$	16.1275	0.634038	0.317019	+	0.948419i	$0.397318\pi$
$648$	0	0
$649$	47.0008	1.84494
$650$	0	0
$651$	1.96722i	0.0771015i
$652$	0	0
$653$	28.6244i	1.12016i	0.828439	+	0.560080i	$0.189230\pi$
$653$	28.6244i	1.12016i	−0.828439	+	0.560080i	$0.810770\pi$
$654$	0	0
$655$	43.7255	1.70850
$656$	0	0
$657$	−15.2809	−0.596166
$658$	0	0
$659$	− 4.75185i	− 0.185106i	−0.995708	−	0.0925528i	$-0.970497\pi$
$659$	− 4.75185i	− 0.185106i	0.995708	−	0.0925528i	$-0.0295027\pi$
$660$	0	0
$661$	− 21.8086i	− 0.848256i	−0.905602	−	0.424128i	$-0.860581\pi$
$661$	− 21.8086i	− 0.848256i	0.905602	−	0.424128i	$-0.139419\pi$
$662$	0	0
$663$	5.03957	0.195721
$664$	0	0
$665$	−14.9457	−0.579571
$666$	0	0
$667$	30.8914i	1.19612i
$668$	0	0
$669$	4.78110i	0.184848i
$670$	0	0
$671$	0.0856521	0.00330656
$672$	0	0
$673$	6.34315	0.244510	0.122255	−	0.992499i	$-0.460987\pi$
$673$	6.34315	0.244510	0.122255	+	0.992499i	$0.460987\pi$
$674$	0	0
$675$	1.23304i	0.0474597i
$676$	0	0
$677$	45.6339i	1.75385i	0.480624	+	0.876927i	$0.340410\pi$
$677$	45.6339i	1.75385i	−0.480624	+	0.876927i	$0.659590\pi$
$678$	0	0
$679$	2.07913	0.0797898
$680$	0	0
$681$	−3.83840	−0.147088
$682$	0	0
$683$	− 37.9342i	− 1.45151i	−0.687953	−	0.725756i	$-0.741490\pi$
$683$	− 37.9342i	− 1.45151i	0.687953	−	0.725756i	$-0.258510\pi$
$684$	0	0
$685$	20.6173i	0.787746i
$686$	0	0
$687$	−1.34596	−0.0513515
$688$	0	0
$689$	19.7219	0.751345
$690$	0	0
$691$	− 33.3716i	− 1.26951i	−0.772712	−	0.634757i	$-0.781100\pi$
$691$	− 33.3716i	− 1.26951i	0.772712	−	0.634757i	$-0.218900\pi$
$692$	0	0
$693$	− 3.39104i	− 0.128815i
$694$	0	0
$695$	36.7120	1.39256
$696$	0	0
$697$	−8.31143	−0.314818
$698$	0	0
$699$	17.6433i	0.667330i
$700$	0	0
$701$	− 23.4480i	− 0.885618i	−0.896616	−	0.442809i	$-0.853982\pi$
$701$	− 23.4480i	− 0.885618i	0.896616	−	0.442809i	$-0.146018\pi$
$702$	0	0
$703$	−16.1512	−0.609154
$704$	0	0
$705$	−4.13651	−0.155790
$706$	0	0
$707$	− 9.16569i	− 0.344711i
$708$	0	0
$709$	12.1562i	0.456537i	0.973598	+	0.228269i	$0.0733064\pi$
$709$	12.1562i	0.456537i	−0.973598	+	0.228269i	$0.926694\pi$
$710$	0	0
$711$	−8.40968	−0.315388
$712$	0	0
$713$	8.57544	0.321153
$714$	0	0
$715$	− 53.6233i	− 2.00540i
$716$	0	0
$717$	− 9.61500i	− 0.359079i
$718$	0	0
$719$	−4.06900	−0.151748	−0.0758741	−	0.997117i	$-0.524175\pi$
$719$	−4.06900	−0.151748	−0.0758741	+	0.997117i	$0.524175\pi$
$720$	0	0
$721$	−15.0653	−0.561061
$722$	0	0
$723$	21.3583i	0.794322i
$724$	0	0
$725$	9.52259i	0.353660i
$726$	0	0
$727$	17.3211	0.642402	0.321201	−	0.947011i	$-0.395913\pi$
$727$	17.3211	0.642402	0.321201	+	0.947011i	$0.395913\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 8.31143i	− 0.307409i
$732$	0	0
$733$	− 21.4361i	− 0.791761i	−0.918302	−	0.395880i	$-0.870439\pi$
$733$	− 21.4361i	− 0.791761i	0.918302	−	0.395880i	$-0.129561\pi$
$734$	0	0
$735$	−15.3741	−0.567082
$736$	0	0
$737$	19.6871	0.725183
$738$	0	0
$739$	31.2545i	1.14972i	0.818253	+	0.574858i	$0.194943\pi$
$739$	31.2545i	1.14972i	−0.818253	+	0.574858i	$0.805057\pi$
$740$	0	0
$741$	− 37.9174i	− 1.39293i
$742$	0	0
$743$	17.7244	0.650244	0.325122	−	0.945672i	$-0.394595\pi$
$743$	17.7244	0.650244	0.325122	+	0.945672i	$0.394595\pi$
$744$	0	0
$745$	40.3288	1.47753
$746$	0	0
$747$	− 1.96134i	− 0.0717615i
$748$	0	0
$749$	− 11.2428i	− 0.410801i
$750$	0	0
$751$	7.38826	0.269601	0.134801	−	0.990873i	$-0.456961\pi$
$751$	7.38826	0.269601	0.134801	+	0.990873i	$0.456961\pi$
$752$	0	0
$753$	4.30448	0.156864
$754$	0	0
$755$	48.1073i	1.75080i
$756$	0	0
$757$	− 13.2804i	− 0.482684i	−0.970440	−	0.241342i	$-0.922412\pi$
$757$	− 13.2804i	− 0.482684i	0.970440	−	0.241342i	$-0.0775876\pi$
$758$	0	0
$759$	−14.7821	−0.536555
$760$	0	0
$761$	−28.1473	−1.02034	−0.510169	−	0.860074i	$-0.670417\pi$
$761$	−28.1473	−1.02034	−0.510169	+	0.860074i	$0.670417\pi$
$762$	0	0
$763$	− 16.7898i	− 0.607830i
$764$	0	0
$765$	− 2.16478i	− 0.0782679i
$766$	0	0
$767$	−73.9194	−2.66907
$768$	0	0
$769$	−11.9355	−0.430405	−0.215203	−	0.976569i	$-0.569041\pi$
$769$	−11.9355	−0.430405	−0.215203	+	0.976569i	$0.569041\pi$
$770$	0	0
$771$	− 4.67271i	− 0.168284i
$772$	0	0
$773$	− 14.5868i	− 0.524649i	−0.964980	−	0.262325i	$-0.915511\pi$
$773$	− 14.5868i	− 0.524649i	0.964980	−	0.262325i	$-0.0844891\pi$
$774$	0	0
$775$	2.64347	0.0949561
$776$	0	0
$777$	2.27170	0.0814969
$778$	0	0
$779$	62.5347i	2.24054i
$780$	0	0
$781$	43.6871i	1.56325i
$782$	0	0
$783$	−7.72286	−0.275992
$784$	0	0
$785$	2.04429	0.0729638
$786$	0	0
$787$	− 38.6835i	− 1.37892i	−0.724325	−	0.689459i	$-0.757849\pi$
$787$	− 38.6835i	− 1.37892i	0.724325	−	0.689459i	$-0.242151\pi$
$788$	0	0
$789$	− 15.4502i	− 0.550042i
$790$	0	0
$791$	−3.35556	−0.119310
$792$	0	0
$793$	−0.134707	−0.00478360
$794$	0	0
$795$	− 8.47170i	− 0.300460i
$796$	0	0
$797$	− 29.4107i	− 1.04178i	−0.853624	−	0.520890i	$-0.825600\pi$
$797$	− 29.4107i	− 1.04178i	0.853624	−	0.520890i	$-0.174400\pi$
$798$	0	0
$799$	1.43664	0.0508248
$800$	0	0
$801$	2.79565	0.0987794
$802$	0	0
$803$	− 56.4710i	− 1.99282i
$804$	0	0
$805$	− 9.16362i	− 0.322975i
$806$	0	0
$807$	31.2459	1.09991
$808$	0	0
$809$	−2.80334	−0.0985602	−0.0492801	−	0.998785i	$-0.515693\pi$
$809$	−2.80334	−0.0985602	−0.0492801	+	0.998785i	$0.515693\pi$
$810$	0	0
$811$	− 51.0722i	− 1.79339i	−0.442650	−	0.896694i	$-0.645962\pi$
$811$	− 51.0722i	− 1.79339i	0.442650	−	0.896694i	$-0.354038\pi$
$812$	0	0
$813$	9.88817i	0.346793i
$814$	0	0
$815$	−59.0426	−2.06817
$816$	0	0
$817$	−62.5347	−2.18781
$818$	0	0
$819$	5.33317i	0.186356i
$820$	0	0
$821$	− 17.4389i	− 0.608622i	−0.952573	−	0.304311i	$-0.901574\pi$
$821$	− 17.4389i	− 0.608622i	0.952573	−	0.304311i	$-0.0984261\pi$
$822$	0	0
$823$	−22.0665	−0.769191	−0.384595	−	0.923085i	$-0.625659\pi$
$823$	−22.0665	−0.769191	−0.384595	+	0.923085i	$0.625659\pi$
$824$	0	0
$825$	−4.55672	−0.158645
$826$	0	0
$827$	− 1.76621i	− 0.0614172i	−0.999528	−	0.0307086i	$-0.990224\pi$
$827$	− 1.76621i	− 0.0614172i	0.999528	−	0.0307086i	$-0.00977639\pi$
$828$	0	0
$829$	22.0514i	0.765879i	0.923774	+	0.382939i	$0.125088\pi$
$829$	22.0514i	0.765879i	−0.923774	+	0.382939i	$0.874912\pi$
$830$	0	0
$831$	22.9565	0.796351
$832$	0	0
$833$	5.33954	0.185004
$834$	0	0
$835$	23.2274i	0.803816i
$836$	0	0
$837$	2.14386i	0.0741026i
$838$	0	0
$839$	−22.1904	−0.766099	−0.383050	−	0.923728i	$-0.625126\pi$
$839$	−22.1904	−0.766099	−0.383050	+	0.923728i	$0.625126\pi$
$840$	0	0
$841$	−30.6425	−1.05664
$842$	0	0
$843$	14.5754i	0.502005i
$844$	0	0
$845$	51.8789i	1.78469i
$846$	0	0
$847$	2.43795	0.0837690
$848$	0	0
$849$	4.71832	0.161932
$850$	0	0
$851$	− 9.90272i	− 0.339461i
$852$	0	0
$853$	37.4972i	1.28388i	0.766756	+	0.641939i	$0.221870\pi$
$853$	37.4972i	1.28388i	−0.766756	+	0.641939i	$0.778130\pi$
$854$	0	0
$855$	−16.2877	−0.557028
$856$	0	0
$857$	17.4761	0.596971	0.298485	−	0.954414i	$-0.403519\pi$
$857$	17.4761	0.596971	0.298485	+	0.954414i	$0.403519\pi$
$858$	0	0
$859$	− 10.7442i	− 0.366586i	−0.983058	−	0.183293i	$-0.941324\pi$
$859$	− 10.7442i	− 0.366586i	0.983058	−	0.183293i	$-0.0586757\pi$
$860$	0	0
$861$	− 8.79565i	− 0.299755i
$862$	0	0
$863$	−23.1791	−0.789027	−0.394514	−	0.918890i	$-0.629087\pi$
$863$	−23.1791	−0.789027	−0.394514	+	0.918890i	$0.629087\pi$
$864$	0	0
$865$	−17.7398	−0.603171
$866$	0	0
$867$	− 16.2482i	− 0.551816i
$868$	0	0
$869$	− 31.0781i	− 1.05425i
$870$	0	0
$871$	−30.9624	−1.04912
$872$	0	0
$873$	2.26582	0.0766863
$874$	0	0
$875$	8.62975i	0.291739i
$876$	0	0
$877$	28.3694i	0.957966i	0.877824	+	0.478983i	$0.158995\pi$
$877$	28.3694i	0.957966i	−0.877824	+	0.478983i	$0.841005\pi$
$878$	0	0
$879$	−23.9204	−0.806816
$880$	0	0
$881$	−58.0822	−1.95684	−0.978420	−	0.206628i	$-0.933751\pi$
$881$	−58.0822	−1.95684	−0.978420	+	0.206628i	$0.933751\pi$
$882$	0	0
$883$	− 5.55826i	− 0.187050i	−0.995617	−	0.0935252i	$-0.970186\pi$
$883$	− 5.55826i	− 0.187050i	0.995617	−	0.0935252i	$-0.0298136\pi$
$884$	0	0
$885$	31.7526i	1.06735i
$886$	0	0
$887$	−9.24359	−0.310369	−0.155185	−	0.987885i	$-0.549597\pi$
$887$	−9.24359	−0.310369	−0.155185	+	0.987885i	$0.549597\pi$
$888$	0	0
$889$	−18.9378	−0.635153
$890$	0	0
$891$	− 3.69552i	− 0.123805i
$892$	0	0
$893$	− 10.8092i	− 0.361717i
$894$	0	0
$895$	−6.67271	−0.223044
$896$	0	0
$897$	23.2482	0.776233
$898$	0	0
$899$	16.5567i	0.552198i
$900$	0	0
$901$	2.94229i	0.0980219i
$902$	0	0
$903$	8.79565	0.292701
$904$	0	0
$905$	1.59899	0.0531522
$906$	0	0
$907$	− 15.4945i	− 0.514487i	−0.966347	−	0.257243i	$-0.917186\pi$
$907$	− 15.4945i	− 0.514487i	0.966347	−	0.257243i	$-0.0828142\pi$
$908$	0	0
$909$	− 9.98868i	− 0.331303i
$910$	0	0
$911$	43.7108	1.44820	0.724101	−	0.689693i	$-0.242255\pi$
$911$	43.7108	1.44820	0.724101	+	0.689693i	$0.242255\pi$
$912$	0	0
$913$	7.24815	0.239879
$914$	0	0
$915$	0.0578646i	0.00191294i
$916$	0	0
$917$	− 16.0710i	− 0.530710i
$918$	0	0
$919$	41.7955	1.37871	0.689353	−	0.724426i	$-0.257895\pi$
$919$	41.7955	1.37871	0.689353	+	0.724426i	$0.257895\pi$
$920$	0	0
$921$	−20.7685	−0.684345
$922$	0	0
$923$	− 68.7078i	− 2.26155i
$924$	0	0
$925$	− 3.05261i	− 0.100369i
$926$	0	0
$927$	−16.4180	−0.539238
$928$	0	0
$929$	24.1989	0.793942	0.396971	−	0.917831i	$-0.370061\pi$
$929$	24.1989	0.793942	0.396971	+	0.917831i	$0.370061\pi$
$930$	0	0
$931$	− 40.1744i	− 1.31666i
$932$	0	0
$933$	− 11.7798i	− 0.385653i
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	−15.3183	−0.500426	−0.250213	−	0.968191i	$-0.580501\pi$
$937$	−15.3183	−0.500426	−0.250213	+	0.968191i	$0.580501\pi$
$938$	0	0
$939$	− 11.1580i	− 0.364127i
$940$	0	0
$941$	− 35.3016i	− 1.15080i	−0.817872	−	0.575400i	$-0.804846\pi$
$941$	− 35.3016i	− 1.15080i	0.817872	−	0.575400i	$-0.195154\pi$
$942$	0	0
$943$	−38.3417	−1.24858
$944$	0	0
$945$	2.29090	0.0745231
$946$	0	0
$947$	14.2051i	0.461605i	0.973001	+	0.230802i	$0.0741351\pi$
$947$	14.2051i	0.461605i	−0.973001	+	0.230802i	$0.925865\pi$
$948$	0	0
$949$	88.8134i	2.88300i
$950$	0	0
$951$	−15.9140	−0.516048
$952$	0	0
$953$	19.6428	0.636292	0.318146	−	0.948042i	$-0.396940\pi$
$953$	19.6428	0.636292	0.318146	+	0.948042i	$0.396940\pi$
$954$	0	0
$955$	36.4597i	1.17981i
$956$	0	0
$957$	− 28.5400i	− 0.922566i
$958$	0	0
$959$	7.57772	0.244697
$960$	0	0
$961$	−26.4039	−0.851738
$962$	0	0
$963$	− 12.2522i	− 0.394823i
$964$	0	0
$965$	22.0072i	0.708437i
$966$	0	0
$967$	−44.5337	−1.43211	−0.716054	−	0.698045i	$-0.754054\pi$
$967$	−44.5337	−1.43211	−0.716054	+	0.698045i	$0.754054\pi$
$968$	0	0
$969$	5.65685	0.181724
$970$	0	0
$971$	1.40255i	0.0450099i	0.999747	+	0.0225049i	$0.00716415\pi$
$971$	1.40255i	0.0450099i	−0.999747	+	0.0225049i	$0.992836\pi$
$972$	0	0
$973$	− 13.4932i	− 0.432572i
$974$	0	0
$975$	7.16648	0.229511
$976$	0	0
$977$	−37.0794	−1.18628	−0.593138	−	0.805101i	$-0.702111\pi$
$977$	−37.0794	−1.18628	−0.593138	+	0.805101i	$0.702111\pi$
$978$	0	0
$979$	10.3314i	0.330192i
$980$	0	0
$981$	− 18.2973i	− 0.584188i
$982$	0	0
$983$	−46.6894	−1.48916	−0.744580	−	0.667534i	$-0.767350\pi$
$983$	−46.6894	−1.48916	−0.744580	+	0.667534i	$0.767350\pi$
$984$	0	0
$985$	9.35105	0.297949
$986$	0	0
$987$	1.52034i	0.0483930i
$988$	0	0
$989$	− 38.3417i	− 1.21919i
$990$	0	0
$991$	−11.8780	−0.377318	−0.188659	−	0.982043i	$-0.560414\pi$
$991$	−11.8780	−0.377318	−0.188659	+	0.982043i	$0.560414\pi$
$992$	0	0
$993$	−2.73190	−0.0866942
$994$	0	0
$995$	37.0164i	1.17350i
$996$	0	0
$997$	51.1515i	1.61998i	0.586441	+	0.809992i	$0.300528\pi$
$997$	51.1515i	1.61998i	−0.586441	+	0.809992i	$0.699472\pi$
$998$	0	0
$999$	2.47568	0.0783271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.d.e.1537.5		8
4.3	odd	2		3072.2.d.j.1537.1		8
8.3	odd	2		3072.2.d.j.1537.8		8
8.5	even	2	inner	3072.2.d.e.1537.4		8
16.3	odd	4		3072.2.a.p.1.1		4
16.5	even	4		3072.2.a.s.1.4		4
16.11	odd	4		3072.2.a.j.1.4		4
16.13	even	4		3072.2.a.m.1.1		4
32.3	odd	8		1536.2.j.i.385.4	yes	8
32.5	even	8		1536.2.j.j.1153.2	yes	8
32.11	odd	8		1536.2.j.f.1153.1	yes	8
32.13	even	8		1536.2.j.e.385.3	✓	8
32.19	odd	8		1536.2.j.f.385.1	yes	8
32.21	even	8		1536.2.j.e.1153.3	yes	8
32.27	odd	8		1536.2.j.i.1153.4	yes	8
32.29	even	8		1536.2.j.j.385.2	yes	8
48.5	odd	4		9216.2.a.bm.1.1		4
48.11	even	4		9216.2.a.ba.1.1		4
48.29	odd	4		9216.2.a.bl.1.4		4
48.35	even	4		9216.2.a.z.1.4		4
96.5	odd	8		4608.2.k.be.1153.2		8
96.11	even	8		4608.2.k.bj.1153.3		8
96.29	odd	8		4608.2.k.be.3457.2		8
96.35	even	8		4608.2.k.bc.3457.2		8
96.53	odd	8		4608.2.k.bh.1153.3		8
96.59	even	8		4608.2.k.bc.1153.2		8
96.77	odd	8		4608.2.k.bh.3457.3		8
96.83	even	8		4608.2.k.bj.3457.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.3	✓	8	32.13	even	8
1536.2.j.e.1153.3	yes	8	32.21	even	8
1536.2.j.f.385.1	yes	8	32.19	odd	8
1536.2.j.f.1153.1	yes	8	32.11	odd	8
1536.2.j.i.385.4	yes	8	32.3	odd	8
1536.2.j.i.1153.4	yes	8	32.27	odd	8
1536.2.j.j.385.2	yes	8	32.29	even	8
1536.2.j.j.1153.2	yes	8	32.5	even	8
3072.2.a.j.1.4		4	16.11	odd	4
3072.2.a.m.1.1		4	16.13	even	4
3072.2.a.p.1.1		4	16.3	odd	4
3072.2.a.s.1.4		4	16.5	even	4
3072.2.d.e.1537.4		8	8.5	even	2	inner
3072.2.d.e.1537.5		8	1.1	even	1	trivial
3072.2.d.j.1537.1		8	4.3	odd	2
3072.2.d.j.1537.8		8	8.3	odd	2
4608.2.k.bc.1153.2		8	96.59	even	8
4608.2.k.bc.3457.2		8	96.35	even	8
4608.2.k.be.1153.2		8	96.5	odd	8
4608.2.k.be.3457.2		8	96.29	odd	8
4608.2.k.bh.1153.3		8	96.53	odd	8
4608.2.k.bh.3457.3		8	96.77	odd	8
4608.2.k.bj.1153.3		8	96.11	even	8
4608.2.k.bj.3457.3		8	96.83	even	8
9216.2.a.z.1.4		4	48.35	even	4
9216.2.a.ba.1.1		4	48.11	even	4
9216.2.a.bl.1.4		4	48.29	odd	4
9216.2.a.bm.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.49661i q^{5} -0.917608 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.49661i q^{5} -0.917608 q^{7} -1.00000 q^{9} -3.69552i q^{11} +5.81204i q^{13} +2.49661 q^{15} -0.867091 q^{17} +6.52395i q^{19} -0.917608i q^{21} -4.00000 q^{23} -1.23304 q^{25} -1.00000i q^{27} -7.72286i q^{29} -2.14386 q^{31} +3.69552 q^{33} +2.29090i q^{35} +2.47568i q^{37} -5.81204 q^{39} +9.58541 q^{41} +9.58541i q^{43} +2.49661i q^{45} -1.65685 q^{47} -6.15800 q^{49} -0.867091i q^{51} -3.39329i q^{53} -9.22625 q^{55} -6.52395 q^{57} +12.7183i q^{59} +0.0231773i q^{61} +0.917608 q^{63} +14.5104 q^{65} +5.32729i q^{67} -4.00000i q^{69} -11.8216 q^{71} +15.2809 q^{73} -1.23304i q^{75} +3.39104i q^{77} +8.40968 q^{79} +1.00000 q^{81} +1.96134i q^{83} +2.16478i q^{85} +7.72286 q^{87} -2.79565 q^{89} -5.33317i q^{91} -2.14386i q^{93} +16.2877 q^{95} -2.26582 q^{97} +3.69552i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{7} - 8 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more