LMFDB - Embedded newform 3072.2.d.f.1537.4

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:	8.0.18939904.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 $ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.4
Root			$0.500000 + 0.691860i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.f.1537.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.79793i q^{5} -2.15894 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.79793i q^{5} -2.15894 q^{7} -1.00000 q^{9} +2.54266i q^{11} +1.95687i q^{13} +3.79793 q^{15} +0.224777 q^{17} +0.224777i q^{19} +2.15894i q^{21} +2.82843 q^{23} -9.42429 q^{25} +1.00000i q^{27} +2.62636i q^{29} +1.84106 q^{31} +2.54266 q^{33} -8.19951i q^{35} +5.18944i q^{37} +1.95687 q^{39} -5.88163 q^{41} -10.9670i q^{43} -3.79793i q^{45} +2.82843 q^{47} -2.33897 q^{49} -0.224777i q^{51} +10.6264i q^{53} -9.65685 q^{55} +0.224777 q^{57} -5.65685i q^{59} -8.46742i q^{61} +2.15894 q^{63} -7.43208 q^{65} +14.7422i q^{67} -2.82843i q^{69} +4.31788 q^{71} -5.97474 q^{73} +9.42429i q^{75} -5.48946i q^{77} -15.0075 q^{79} +1.00000 q^{81} -14.3059i q^{83} +0.853690i q^{85} +2.62636 q^{87} +1.42847 q^{89} -4.22478i q^{91} -1.84106i q^{93} -0.853690 q^{95} -16.3990 q^{97} -2.54266i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^7 - 8 * q^9	$ 8 q - 8 q^{7} - 8 q^{9} + 8 q^{15} - 8 q^{25} + 24 q^{31} - 16 q^{39} + 8 q^{49} - 32 q^{55} + 8 q^{63} - 16 q^{65} + 16 q^{71} + 16 q^{73} + 24 q^{79} + 8 q^{81} - 24 q^{87} + 16 q^{89} - 48 q^{95}+O(q^{100}) $ 8 * q - 8 * q^7 - 8 * q^9 + 8 * q^15 - 8 * q^25 + 24 * q^31 - 16 * q^39 + 8 * q^49 - 32 * q^55 + 8 * q^63 - 16 * q^65 + 16 * q^71 + 16 * q^73 + 24 * q^79 + 8 * q^81 - 24 * q^87 + 16 * q^89 - 48 * q^95

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
$5$	3.79793i	1.69849i	0.528001	+	0.849244i	$0.322942\pi$
$5$	3.79793i	1.69849i	−0.528001	+	0.849244i	$0.677058\pi$
$6$	0	0
$7$	−2.15894	−0.816003	−0.408002	−	0.912981i	$-0.633774\pi$
$7$	−2.15894	−0.816003	−0.408002	+	0.912981i	$0.633774\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.54266i	0.766641i	0.923615	+	0.383321i	$0.125220\pi$
$11$	2.54266i	0.766641i	−0.923615	+	0.383321i	$0.874780\pi$
$12$	0	0
$13$	1.95687i	0.542739i	0.962475	+	0.271370i	$0.0874766\pi$
$13$	1.95687i	0.542739i	−0.962475	+	0.271370i	$0.912523\pi$
$14$	0	0
$15$	3.79793	0.980622
$16$	0	0
$17$	0.224777	0.0545165	0.0272583	−	0.999628i	$-0.491322\pi$
$17$	0.224777	0.0545165	0.0272583	+	0.999628i	$0.491322\pi$
$18$	0	0
$19$	0.224777i	0.0515675i	0.999668	+	0.0257837i	$0.00820813\pi$
$19$	0.224777i	0.0515675i	−0.999668	+	0.0257837i	$0.991792\pi$
$20$	0	0
$21$	2.15894i	0.471120i
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	−9.42429	−1.88486
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	2.62636i	0.487703i	0.969813	+	0.243851i	$0.0784109\pi$
$29$	2.62636i	0.487703i	−0.969813	+	0.243851i	$0.921589\pi$
$30$	0	0
$31$	1.84106	0.330664	0.165332	−	0.986238i	$-0.447130\pi$
$31$	1.84106	0.330664	0.165332	+	0.986238i	$0.447130\pi$
$32$	0	0
$33$	2.54266	0.442620
$34$	0	0
$35$	− 8.19951i	− 1.38597i
$36$	0	0
$37$	5.18944i	0.853138i	0.904455	+	0.426569i	$0.140278\pi$
$37$	5.18944i	0.853138i	−0.904455	+	0.426569i	$0.859722\pi$
$38$	0	0
$39$	1.95687	0.313351
$40$	0	0
$41$	−5.88163	−0.918557	−0.459278	−	0.888292i	$-0.651892\pi$
$41$	−5.88163	−0.918557	−0.459278	+	0.888292i	$0.651892\pi$
$42$	0	0
$43$	− 10.9670i	− 1.67244i	−0.548391	−	0.836222i	$-0.684759\pi$
$43$	− 10.9670i	− 1.67244i	0.548391	−	0.836222i	$-0.315241\pi$
$44$	0	0
$45$	− 3.79793i	− 0.566162i
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	−2.33897	−0.334139
$50$	0	0
$51$	− 0.224777i	− 0.0314751i
$52$	0	0
$53$	10.6264i	1.45964i	0.683638	+	0.729821i	$0.260397\pi$
$53$	10.6264i	1.45964i	−0.683638	+	0.729821i	$0.739603\pi$
$54$	0	0
$55$	−9.65685	−1.30213
$56$	0	0
$57$	0.224777	0.0297725
$58$	0	0
$59$	− 5.65685i	− 0.736460i	−0.929735	−	0.368230i	$-0.879964\pi$
$59$	− 5.65685i	− 0.736460i	0.929735	−	0.368230i	$-0.120036\pi$
$60$	0	0
$61$	− 8.46742i	− 1.08414i	−0.840333	−	0.542071i	$-0.817640\pi$
$61$	− 8.46742i	− 1.08414i	0.840333	−	0.542071i	$-0.182360\pi$
$62$	0	0
$63$	2.15894	0.272001
$64$	0	0
$65$	−7.43208	−0.921836
$66$	0	0
$67$	14.7422i	1.80104i	0.434811	+	0.900522i	$0.356815\pi$
$67$	14.7422i	1.80104i	−0.434811	+	0.900522i	$0.643185\pi$
$68$	0	0
$69$	− 2.82843i	− 0.340503i
$70$	0	0
$71$	4.31788	0.512438	0.256219	−	0.966619i	$-0.417523\pi$
$71$	4.31788	0.512438	0.256219	+	0.966619i	$0.417523\pi$
$72$	0	0
$73$	−5.97474	−0.699290	−0.349645	−	0.936882i	$-0.613698\pi$
$73$	−5.97474	−0.699290	−0.349645	+	0.936882i	$0.613698\pi$
$74$	0	0
$75$	9.42429i	1.08822i
$76$	0	0
$77$	− 5.48946i	− 0.625582i
$78$	0	0
$79$	−15.0075	−1.68848	−0.844239	−	0.535966i	$-0.819947\pi$
$79$	−15.0075	−1.68848	−0.844239	+	0.535966i	$0.819947\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 14.3059i	− 1.57028i	−0.619319	−	0.785140i	$-0.712591\pi$
$83$	− 14.3059i	− 1.57028i	0.619319	−	0.785140i	$-0.287409\pi$
$84$	0	0
$85$	0.853690i	0.0925956i
$86$	0	0
$87$	2.62636	0.281575
$88$	0	0
$89$	1.42847	0.151417	0.0757086	−	0.997130i	$-0.475878\pi$
$89$	1.42847	0.151417	0.0757086	+	0.997130i	$0.475878\pi$
$90$	0	0
$91$	− 4.22478i	− 0.442877i
$92$	0	0
$93$	− 1.84106i	− 0.190909i
$94$	0	0
$95$	−0.853690	−0.0875867
$96$	0	0
$97$	−16.3990	−1.66507	−0.832535	−	0.553973i	$-0.813111\pi$
$97$	−16.3990	−1.66507	−0.832535	+	0.553973i	$0.813111\pi$
$98$	0	0
$99$	− 2.54266i	− 0.255547i
$100$	0	0
$101$	− 0.115816i	− 0.0115241i	−0.999983	−	0.00576206i	$-0.998166\pi$
$101$	− 0.115816i	− 0.0115241i	0.999983	−	0.00576206i	$-0.00183413\pi$
$102$	0	0
$103$	−13.3507	−1.31548	−0.657740	−	0.753245i	$-0.728488\pi$
$103$	−13.3507	−1.31548	−0.657740	+	0.753245i	$0.728488\pi$
$104$	0	0
$105$	−8.19951	−0.800191
$106$	0	0
$107$	− 10.2926i	− 0.995025i	−0.867457	−	0.497513i	$-0.834247\pi$
$107$	− 10.2926i	− 0.995025i	0.867457	−	0.497513i	$-0.165753\pi$
$108$	0	0
$109$	− 9.95687i	− 0.953696i	−0.878986	−	0.476848i	$-0.841779\pi$
$109$	− 9.95687i	− 0.953696i	0.878986	−	0.476848i	$-0.158221\pi$
$110$	0	0
$111$	5.18944	0.492559
$112$	0	0
$113$	−18.8486	−1.77313	−0.886563	−	0.462608i	$-0.846914\pi$
$113$	−18.8486	−1.77313	−0.886563	+	0.462608i	$0.846914\pi$
$114$	0	0
$115$	10.7422i	1.00171i
$116$	0	0
$117$	− 1.95687i	− 0.180913i
$118$	0	0
$119$	−0.485281	−0.0444857
$120$	0	0
$121$	4.53488	0.412261
$122$	0	0
$123$	5.88163i	0.530329i
$124$	0	0
$125$	− 16.8032i	− 1.50292i
$126$	0	0
$127$	−3.81580	−0.338597	−0.169299	−	0.985565i	$-0.554150\pi$
$127$	−3.81580	−0.338597	−0.169299	+	0.985565i	$0.554150\pi$
$128$	0	0
$129$	−10.9670	−0.965586
$130$	0	0
$131$	1.08532i	0.0948250i	0.998875	+	0.0474125i	$0.0150975\pi$
$131$	1.08532i	0.0948250i	−0.998875	+	0.0474125i	$0.984902\pi$
$132$	0	0
$133$	− 0.485281i	− 0.0420792i
$134$	0	0
$135$	−3.79793	−0.326874
$136$	0	0
$137$	5.31010	0.453672	0.226836	−	0.973933i	$-0.427162\pi$
$137$	5.31010	0.453672	0.226836	+	0.973933i	$0.427162\pi$
$138$	0	0
$139$	12.3990i	1.05167i	0.850586	+	0.525836i	$0.176247\pi$
$139$	12.3990i	1.05167i	−0.850586	+	0.525836i	$0.823753\pi$
$140$	0	0
$141$	− 2.82843i	− 0.238197i
$142$	0	0
$143$	−4.97567	−0.416086
$144$	0	0
$145$	−9.97474	−0.828357
$146$	0	0
$147$	2.33897i	0.192915i
$148$	0	0
$149$	− 1.45479i	− 0.119181i	−0.998223	−	0.0595904i	$-0.981021\pi$
$149$	− 1.45479i	− 0.119181i	0.998223	−	0.0595904i	$-0.0189795\pi$
$150$	0	0
$151$	−2.03696	−0.165766	−0.0828829	−	0.996559i	$-0.526413\pi$
$151$	−2.03696	−0.165766	−0.0828829	+	0.996559i	$0.526413\pi$
$152$	0	0
$153$	−0.224777	−0.0181722
$154$	0	0
$155$	6.99222i	0.561628i
$156$	0	0
$157$	− 8.61790i	− 0.687784i	−0.939009	−	0.343892i	$-0.888255\pi$
$157$	− 8.61790i	− 0.687784i	0.939009	−	0.343892i	$-0.111745\pi$
$158$	0	0
$159$	10.6264	0.842725
$160$	0	0
$161$	−6.10641	−0.481252
$162$	0	0
$163$	− 4.86054i	− 0.380707i	−0.981716	−	0.190354i	$-0.939037\pi$
$163$	− 4.86054i	− 0.380707i	0.981716	−	0.190354i	$-0.0609634\pi$
$164$	0	0
$165$	9.65685i	0.751785i
$166$	0	0
$167$	21.7023	1.67937	0.839686	−	0.543072i	$-0.182739\pi$
$167$	21.7023	1.67937	0.839686	+	0.543072i	$0.182739\pi$
$168$	0	0
$169$	9.17064	0.705434
$170$	0	0
$171$	− 0.224777i	− 0.0171892i
$172$	0	0
$173$	− 12.3695i	− 0.940433i	−0.882551	−	0.470217i	$-0.844176\pi$
$173$	− 12.3695i	− 0.940433i	0.882551	−	0.470217i	$-0.155824\pi$
$174$	0	0
$175$	20.3465	1.53805
$176$	0	0
$177$	−5.65685	−0.425195
$178$	0	0
$179$	11.6413i	0.870111i	0.900404	+	0.435055i	$0.143271\pi$
$179$	11.6413i	0.870111i	−0.900404	+	0.435055i	$0.856729\pi$
$180$	0	0
$181$	9.50732i	0.706673i	0.935496	+	0.353337i	$0.114953\pi$
$181$	9.50732i	0.706673i	−0.935496	+	0.353337i	$0.885047\pi$
$182$	0	0
$183$	−8.46742	−0.625930
$184$	0	0
$185$	−19.7091	−1.44904
$186$	0	0
$187$	0.571533i	0.0417946i
$188$	0	0
$189$	− 2.15894i	− 0.157040i
$190$	0	0
$191$	−20.8032	−1.50526	−0.752632	−	0.658441i	$-0.771216\pi$
$191$	−20.8032	−1.50526	−0.752632	+	0.658441i	$0.771216\pi$
$192$	0	0
$193$	14.1454	1.01821	0.509103	−	0.860705i	$-0.329977\pi$
$193$	14.1454	1.01821	0.509103	+	0.860705i	$0.329977\pi$
$194$	0	0
$195$	7.43208i	0.532222i
$196$	0	0
$197$	3.43463i	0.244707i	0.992487	+	0.122354i	$0.0390442\pi$
$197$	3.43463i	0.244707i	−0.992487	+	0.122354i	$0.960956\pi$
$198$	0	0
$199$	−0.306182	−0.0217047	−0.0108523	−	0.999941i	$-0.503454\pi$
$199$	−0.306182	−0.0217047	−0.0108523	+	0.999941i	$0.503454\pi$
$200$	0	0
$201$	14.7422	1.03983
$202$	0	0
$203$	− 5.67016i	− 0.397967i
$204$	0	0
$205$	− 22.3380i	− 1.56016i
$206$	0	0
$207$	−2.82843	−0.196589
$208$	0	0
$209$	−0.571533	−0.0395337
$210$	0	0
$211$	10.2284i	0.704151i	0.935972	+	0.352076i	$0.114524\pi$
$211$	10.2284i	0.704151i	−0.935972	+	0.352076i	$0.885476\pi$
$212$	0	0
$213$	− 4.31788i	− 0.295856i
$214$	0	0
$215$	41.6517	2.84063
$216$	0	0
$217$	−3.97474	−0.269823
$218$	0	0
$219$	5.97474i	0.403735i
$220$	0	0
$221$	0.439861i	0.0295883i
$222$	0	0
$223$	−1.71908	−0.115118	−0.0575591	−	0.998342i	$-0.518332\pi$
$223$	−1.71908	−0.115118	−0.0575591	+	0.998342i	$0.518332\pi$
$224$	0	0
$225$	9.42429	0.628286
$226$	0	0
$227$	− 14.3059i	− 0.949518i	−0.880116	−	0.474759i	$-0.842535\pi$
$227$	− 14.3059i	− 0.949518i	0.880116	−	0.474759i	$-0.157465\pi$
$228$	0	0
$229$	16.9981i	1.12327i	0.827386	+	0.561634i	$0.189827\pi$
$229$	16.9981i	1.12327i	−0.827386	+	0.561634i	$0.810173\pi$
$230$	0	0
$231$	−5.48946	−0.361180
$232$	0	0
$233$	−13.3779	−0.876418	−0.438209	−	0.898873i	$-0.644387\pi$
$233$	−13.3779	−0.876418	−0.438209	+	0.898873i	$0.644387\pi$
$234$	0	0
$235$	10.7422i	0.700742i
$236$	0	0
$237$	15.0075i	0.974844i
$238$	0	0
$239$	−13.3675	−0.864670	−0.432335	−	0.901713i	$-0.642310\pi$
$239$	−13.3675	−0.864670	−0.432335	+	0.901713i	$0.642310\pi$
$240$	0	0
$241$	−0.211474	−0.0136222	−0.00681112	−	0.999977i	$-0.502168\pi$
$241$	−0.211474	−0.0136222	−0.00681112	+	0.999977i	$0.502168\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 8.88325i	− 0.567530i
$246$	0	0
$247$	−0.439861	−0.0279877
$248$	0	0
$249$	−14.3059	−0.906601
$250$	0	0
$251$	− 14.7555i	− 0.931358i	−0.884954	−	0.465679i	$-0.845810\pi$
$251$	− 14.7555i	− 0.931358i	0.884954	−	0.465679i	$-0.154190\pi$
$252$	0	0
$253$	7.19173i	0.452140i
$254$	0	0
$255$	0.853690	0.0534601
$256$	0	0
$257$	−0.742176	−0.0462957	−0.0231478	−	0.999732i	$-0.507369\pi$
$257$	−0.742176	−0.0462957	−0.0231478	+	0.999732i	$0.507369\pi$
$258$	0	0
$259$	− 11.2037i	− 0.696163i
$260$	0	0
$261$	− 2.62636i	− 0.162568i
$262$	0	0
$263$	−5.48435	−0.338180	−0.169090	−	0.985601i	$-0.554083\pi$
$263$	−5.48435	−0.338180	−0.169090	+	0.985601i	$0.554083\pi$
$264$	0	0
$265$	−40.3582	−2.47918
$266$	0	0
$267$	− 1.42847i	− 0.0874208i
$268$	0	0
$269$	20.4694i	1.24804i	0.781407	+	0.624021i	$0.214502\pi$
$269$	20.4694i	1.24804i	−0.781407	+	0.624021i	$0.785498\pi$
$270$	0	0
$271$	14.0370	0.852685	0.426342	−	0.904562i	$-0.359802\pi$
$271$	14.0370	0.852685	0.426342	+	0.904562i	$0.359802\pi$
$272$	0	0
$273$	−4.22478	−0.255695
$274$	0	0
$275$	− 23.9628i	− 1.44501i
$276$	0	0
$277$	13.4211i	0.806394i	0.915113	+	0.403197i	$0.132101\pi$
$277$	13.4211i	0.806394i	−0.915113	+	0.403197i	$0.867899\pi$
$278$	0	0
$279$	−1.84106	−0.110221
$280$	0	0
$281$	−3.89359	−0.232272	−0.116136	−	0.993233i	$-0.537051\pi$
$281$	−3.89359	−0.232272	−0.116136	+	0.993233i	$0.537051\pi$
$282$	0	0
$283$	17.6569i	1.04959i	0.851228	+	0.524796i	$0.175858\pi$
$283$	17.6569i	1.04959i	−0.851228	+	0.524796i	$0.824142\pi$
$284$	0	0
$285$	0.853690i	0.0505682i
$286$	0	0
$287$	12.6981	0.749545
$288$	0	0
$289$	−16.9495	−0.997028
$290$	0	0
$291$	16.3990i	0.961328i
$292$	0	0
$293$	15.7759i	0.921639i	0.887494	+	0.460819i	$0.152444\pi$
$293$	15.7759i	0.921639i	−0.887494	+	0.460819i	$0.847556\pi$
$294$	0	0
$295$	21.4844	1.25087
$296$	0	0
$297$	−2.54266	−0.147540
$298$	0	0
$299$	5.53488i	0.320090i
$300$	0	0
$301$	23.6770i	1.36472i
$302$	0	0
$303$	−0.115816	−0.00665345
$304$	0	0
$305$	32.1587	1.84140
$306$	0	0
$307$	− 7.64129i	− 0.436111i	−0.975936	−	0.218056i	$-0.930029\pi$
$307$	− 7.64129i	− 0.436111i	0.975936	−	0.218056i	$-0.0699714\pi$
$308$	0	0
$309$	13.3507i	0.759493i
$310$	0	0
$311$	−24.1623	−1.37012	−0.685059	−	0.728488i	$-0.740224\pi$
$311$	−24.1623	−1.37012	−0.685059	+	0.728488i	$0.740224\pi$
$312$	0	0
$313$	16.6105	0.938881	0.469441	−	0.882964i	$-0.344456\pi$
$313$	16.6105	0.938881	0.469441	+	0.882964i	$0.344456\pi$
$314$	0	0
$315$	8.19951i	0.461990i
$316$	0	0
$317$	− 2.56213i	− 0.143903i	−0.997408	−	0.0719517i	$-0.977077\pi$
$317$	− 2.56213i	− 0.143903i	0.997408	−	0.0719517i	$-0.0229227\pi$
$318$	0	0
$319$	−6.67794	−0.373893
$320$	0	0
$321$	−10.2926	−0.574478
$322$	0	0
$323$	0.0505249i	0.00281128i
$324$	0	0
$325$	− 18.4422i	− 1.02299i
$326$	0	0
$327$	−9.95687	−0.550616
$328$	0	0
$329$	−6.10641	−0.336657
$330$	0	0
$331$	19.1275i	1.05134i	0.850688	+	0.525671i	$0.176186\pi$
$331$	19.1275i	1.05134i	−0.850688	+	0.525671i	$0.823814\pi$
$332$	0	0
$333$	− 5.18944i	− 0.284379i
$334$	0	0
$335$	−55.9898	−3.05905
$336$	0	0
$337$	1.12615	0.0613454	0.0306727	−	0.999529i	$-0.490235\pi$
$337$	1.12615	0.0613454	0.0306727	+	0.999529i	$0.490235\pi$
$338$	0	0
$339$	18.8486i	1.02371i
$340$	0	0
$341$	4.68119i	0.253500i
$342$	0	0
$343$	20.1623	1.08866
$344$	0	0
$345$	10.7422	0.578339
$346$	0	0
$347$	− 29.4068i	− 1.57864i	−0.613982	−	0.789320i	$-0.710433\pi$
$347$	− 29.4068i	− 1.57864i	0.613982	−	0.789320i	$-0.289567\pi$
$348$	0	0
$349$	27.2738i	1.45993i	0.683482	+	0.729967i	$0.260465\pi$
$349$	27.2738i	1.45993i	−0.683482	+	0.729967i	$0.739535\pi$
$350$	0	0
$351$	−1.95687	−0.104450
$352$	0	0
$353$	25.5908	1.36206	0.681029	−	0.732256i	$-0.261533\pi$
$353$	25.5908	1.36206	0.681029	+	0.732256i	$0.261533\pi$
$354$	0	0
$355$	16.3990i	0.870370i
$356$	0	0
$357$	0.485281i	0.0256838i
$358$	0	0
$359$	−3.77296	−0.199129	−0.0995645	−	0.995031i	$-0.531745\pi$
$359$	−3.77296	−0.199129	−0.0995645	+	0.995031i	$0.531745\pi$
$360$	0	0
$361$	18.9495	0.997341
$362$	0	0
$363$	− 4.53488i	− 0.238019i
$364$	0	0
$365$	− 22.6917i	− 1.18774i
$366$	0	0
$367$	27.4474	1.43274	0.716371	−	0.697720i	$-0.245802\pi$
$367$	27.4474	1.43274	0.716371	+	0.697720i	$0.245802\pi$
$368$	0	0
$369$	5.88163	0.306186
$370$	0	0
$371$	− 22.9417i	− 1.19107i
$372$	0	0
$373$	17.8518i	0.924332i	0.886794	+	0.462166i	$0.152928\pi$
$373$	17.8518i	0.924332i	−0.886794	+	0.462166i	$0.847072\pi$
$374$	0	0
$375$	−16.8032	−0.867712
$376$	0	0
$377$	−5.13946	−0.264695
$378$	0	0
$379$	16.5018i	0.847642i	0.905746	+	0.423821i	$0.139311\pi$
$379$	16.5018i	0.847642i	−0.905746	+	0.423821i	$0.860689\pi$
$380$	0	0
$381$	3.81580i	0.195489i
$382$	0	0
$383$	−17.1885	−0.878291	−0.439145	−	0.898416i	$-0.644719\pi$
$383$	−17.1885	−0.878291	−0.439145	+	0.898416i	$0.644719\pi$
$384$	0	0
$385$	20.8486	1.06254
$386$	0	0
$387$	10.9670i	0.557482i
$388$	0	0
$389$	2.66209i	0.134973i	0.997720	+	0.0674866i	$0.0214980\pi$
$389$	2.66209i	0.134973i	−0.997720	+	0.0674866i	$0.978502\pi$
$390$	0	0
$391$	0.635767	0.0321521
$392$	0	0
$393$	1.08532	0.0547472
$394$	0	0
$395$	− 56.9976i	− 2.86786i
$396$	0	0
$397$	11.8959i	0.597037i	0.954404	+	0.298519i	$0.0964925\pi$
$397$	11.8959i	0.597037i	−0.954404	+	0.298519i	$0.903507\pi$
$398$	0	0
$399$	−0.485281	−0.0242945
$400$	0	0
$401$	−1.12389	−0.0561242	−0.0280621	−	0.999606i	$-0.508934\pi$
$401$	−1.12389	−0.0561242	−0.0280621	+	0.999606i	$0.508934\pi$
$402$	0	0
$403$	3.60272i	0.179464i
$404$	0	0
$405$	3.79793i	0.188721i
$406$	0	0
$407$	−13.1950	−0.654051
$408$	0	0
$409$	−13.7211	−0.678464	−0.339232	−	0.940703i	$-0.610167\pi$
$409$	−13.7211	−0.678464	−0.339232	+	0.940703i	$0.610167\pi$
$410$	0	0
$411$	− 5.31010i	− 0.261928i
$412$	0	0
$413$	12.2128i	0.600953i
$414$	0	0
$415$	54.3329	2.66710
$416$	0	0
$417$	12.3990	0.607183
$418$	0	0
$419$	13.1629i	0.643048i	0.946901	+	0.321524i	$0.104195\pi$
$419$	13.1629i	0.643048i	−0.946901	+	0.321524i	$0.895805\pi$
$420$	0	0
$421$	− 11.9413i	− 0.581984i	−0.956726	−	0.290992i	$-0.906015\pi$
$421$	− 11.9413i	− 0.581984i	0.956726	−	0.290992i	$-0.0939852\pi$
$422$	0	0
$423$	−2.82843	−0.137523
$424$	0	0
$425$	−2.11837	−0.102756
$426$	0	0
$427$	18.2807i	0.884663i
$428$	0	0
$429$	4.97567i	0.240227i
$430$	0	0
$431$	30.6054	1.47421	0.737105	−	0.675778i	$-0.236192\pi$
$431$	30.6054	1.47421	0.737105	+	0.675778i	$0.236192\pi$
$432$	0	0
$433$	15.3137	0.735930	0.367965	−	0.929840i	$-0.380055\pi$
$433$	15.3137	0.735930	0.367965	+	0.929840i	$0.380055\pi$
$434$	0	0
$435$	9.97474i	0.478252i
$436$	0	0
$437$	0.635767i	0.0304128i
$438$	0	0
$439$	−33.3676	−1.59255	−0.796274	−	0.604936i	$-0.793199\pi$
$439$	−33.3676	−1.59255	−0.796274	+	0.604936i	$0.793199\pi$
$440$	0	0
$441$	2.33897	0.111380
$442$	0	0
$443$	− 3.23617i	− 0.153755i	−0.997041	−	0.0768776i	$-0.975505\pi$
$443$	− 3.23617i	− 0.153755i	0.997041	−	0.0768776i	$-0.0244951\pi$
$444$	0	0
$445$	5.42522i	0.257180i
$446$	0	0
$447$	−1.45479	−0.0688091
$448$	0	0
$449$	−27.4165	−1.29387	−0.646933	−	0.762547i	$-0.723948\pi$
$449$	−27.4165	−1.29387	−0.646933	+	0.762547i	$0.723948\pi$
$450$	0	0
$451$	− 14.9550i	− 0.704203i
$452$	0	0
$453$	2.03696i	0.0957049i
$454$	0	0
$455$	16.0454	0.752221
$456$	0	0
$457$	10.9147	0.510567	0.255284	−	0.966866i	$-0.417831\pi$
$457$	10.9147	0.510567	0.255284	+	0.966866i	$0.417831\pi$
$458$	0	0
$459$	0.224777i	0.0104917i
$460$	0	0
$461$	25.2181i	1.17452i	0.809398	+	0.587261i	$0.199794\pi$
$461$	25.2181i	1.17452i	−0.809398	+	0.587261i	$0.800206\pi$
$462$	0	0
$463$	−22.4937	−1.04537	−0.522686	−	0.852525i	$-0.675070\pi$
$463$	−22.4937	−1.04537	−0.522686	+	0.852525i	$0.675070\pi$
$464$	0	0
$465$	6.99222	0.324256
$466$	0	0
$467$	34.2482i	1.58482i	0.609991	+	0.792408i	$0.291173\pi$
$467$	34.2482i	1.58482i	−0.609991	+	0.792408i	$0.708827\pi$
$468$	0	0
$469$	− 31.8275i	− 1.46966i
$470$	0	0
$471$	−8.61790	−0.397092
$472$	0	0
$473$	27.8852	1.28216
$474$	0	0
$475$	− 2.11837i	− 0.0971974i
$476$	0	0
$477$	− 10.6264i	− 0.486548i
$478$	0	0
$479$	−36.2362	−1.65568	−0.827838	−	0.560968i	$-0.810429\pi$
$479$	−36.2362	−1.65568	−0.827838	+	0.560968i	$0.810429\pi$
$480$	0	0
$481$	−10.1551	−0.463032
$482$	0	0
$483$	6.10641i	0.277851i
$484$	0	0
$485$	− 62.2824i	− 2.82810i
$486$	0	0
$487$	−16.8200	−0.762186	−0.381093	−	0.924537i	$-0.624452\pi$
$487$	−16.8200	−0.762186	−0.381093	+	0.924537i	$0.624452\pi$
$488$	0	0
$489$	−4.86054	−0.219801
$490$	0	0
$491$	8.63577i	0.389727i	0.980830	+	0.194863i	$0.0624263\pi$
$491$	8.63577i	0.389727i	−0.980830	+	0.194863i	$0.937574\pi$
$492$	0	0
$493$	0.590346i	0.0265879i
$494$	0	0
$495$	9.65685	0.434043
$496$	0	0
$497$	−9.32206	−0.418151
$498$	0	0
$499$	− 27.8275i	− 1.24573i	−0.782329	−	0.622865i	$-0.785969\pi$
$499$	− 27.8275i	− 1.24573i	0.782329	−	0.622865i	$-0.214031\pi$
$500$	0	0
$501$	− 21.7023i	− 0.969586i
$502$	0	0
$503$	25.7308	1.14728	0.573639	−	0.819108i	$-0.305531\pi$
$503$	25.7308	1.14728	0.573639	+	0.819108i	$0.305531\pi$
$504$	0	0
$505$	0.439861	0.0195736
$506$	0	0
$507$	− 9.17064i	− 0.407283i
$508$	0	0
$509$	− 2.45386i	− 0.108765i	−0.998520	−	0.0543826i	$-0.982681\pi$
$509$	− 2.45386i	− 0.108765i	0.998520	−	0.0543826i	$-0.0173191\pi$
$510$	0	0
$511$	12.8991	0.570623
$512$	0	0
$513$	−0.224777	−0.00992417
$514$	0	0
$515$	− 50.7050i	− 2.23433i
$516$	0	0
$517$	7.19173i	0.316292i
$518$	0	0
$519$	−12.3695	−0.542959
$520$	0	0
$521$	33.5944	1.47180	0.735898	−	0.677092i	$-0.236760\pi$
$521$	33.5944	1.47180	0.735898	+	0.677092i	$0.236760\pi$
$522$	0	0
$523$	− 30.8522i	− 1.34907i	−0.738242	−	0.674536i	$-0.764344\pi$
$523$	− 30.8522i	− 1.34907i	0.738242	−	0.674536i	$-0.235656\pi$
$524$	0	0
$525$	− 20.3465i	− 0.887994i
$526$	0	0
$527$	0.413828	0.0180266
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	5.65685i	0.245487i
$532$	0	0
$533$	− 11.5096i	− 0.498537i
$534$	0	0
$535$	39.0907	1.69004
$536$	0	0
$537$	11.6413	0.502359
$538$	0	0
$539$	− 5.94721i	− 0.256164i
$540$	0	0
$541$	− 38.4825i	− 1.65449i	−0.561841	−	0.827245i	$-0.689907\pi$
$541$	− 38.4825i	− 1.65449i	0.561841	−	0.827245i	$-0.310093\pi$
$542$	0	0
$543$	9.50732	0.407998
$544$	0	0
$545$	37.8155	1.61984
$546$	0	0
$547$	9.61829i	0.411248i	0.978631	+	0.205624i	$0.0659224\pi$
$547$	9.61829i	0.411248i	−0.978631	+	0.205624i	$0.934078\pi$
$548$	0	0
$549$	8.46742i	0.361381i
$550$	0	0
$551$	−0.590346	−0.0251496
$552$	0	0
$553$	32.4004	1.37780
$554$	0	0
$555$	19.7091i	0.836606i
$556$	0	0
$557$	− 6.07174i	− 0.257268i	−0.991692	−	0.128634i	$-0.958941\pi$
$557$	− 6.07174i	− 0.257268i	0.991692	−	0.128634i	$-0.0410592\pi$
$558$	0	0
$559$	21.4609	0.907701
$560$	0	0
$561$	0.571533	0.0241301
$562$	0	0
$563$	− 14.2554i	− 0.600793i	−0.953814	−	0.300397i	$-0.902881\pi$
$563$	− 14.2554i	− 0.600793i	0.953814	−	0.300397i	$-0.0971191\pi$
$564$	0	0
$565$	− 71.5857i	− 3.01163i
$566$	0	0
$567$	−2.15894	−0.0906670
$568$	0	0
$569$	32.5018	1.36255	0.681274	−	0.732029i	$-0.261426\pi$
$569$	32.5018	1.36255	0.681274	+	0.732029i	$0.261426\pi$
$570$	0	0
$571$	12.9706i	0.542801i	0.962466	+	0.271401i	$0.0874868\pi$
$571$	12.9706i	0.542801i	−0.962466	+	0.271401i	$0.912513\pi$
$572$	0	0
$573$	20.8032i	0.869065i
$574$	0	0
$575$	−26.6559	−1.11163
$576$	0	0
$577$	11.7536	0.489308	0.244654	−	0.969611i	$-0.421326\pi$
$577$	11.7536	0.489308	0.244654	+	0.969611i	$0.421326\pi$
$578$	0	0
$579$	− 14.1454i	− 0.587862i
$580$	0	0
$581$	30.8857i	1.28135i
$582$	0	0
$583$	−27.0192	−1.11902
$584$	0	0
$585$	7.43208	0.307279
$586$	0	0
$587$	9.13585i	0.377077i	0.982066	+	0.188538i	$0.0603750\pi$
$587$	9.13585i	0.377077i	−0.982066	+	0.188538i	$0.939625\pi$
$588$	0	0
$589$	0.413828i	0.0170515i
$590$	0	0
$591$	3.43463	0.141282
$592$	0	0
$593$	−5.49270	−0.225558	−0.112779	−	0.993620i	$-0.535975\pi$
$593$	−5.49270	−0.225558	−0.112779	+	0.993620i	$0.535975\pi$
$594$	0	0
$595$	− 1.84307i	− 0.0755583i
$596$	0	0
$597$	0.306182i	0.0125312i
$598$	0	0
$599$	36.4348	1.48868	0.744342	−	0.667799i	$-0.232763\pi$
$599$	36.4348	1.48868	0.744342	+	0.667799i	$0.232763\pi$
$600$	0	0
$601$	9.97474	0.406878	0.203439	−	0.979088i	$-0.434788\pi$
$601$	9.97474	0.406878	0.203439	+	0.979088i	$0.434788\pi$
$602$	0	0
$603$	− 14.7422i	− 0.600348i
$604$	0	0
$605$	17.2232i	0.700221i
$606$	0	0
$607$	4.51900	0.183421	0.0917103	−	0.995786i	$-0.470767\pi$
$607$	4.51900	0.183421	0.0917103	+	0.995786i	$0.470767\pi$
$608$	0	0
$609$	−5.67016	−0.229766
$610$	0	0
$611$	5.53488i	0.223917i
$612$	0	0
$613$	11.9316i	0.481913i	0.970536	+	0.240957i	$0.0774612\pi$
$613$	11.9316i	0.481913i	−0.970536	+	0.240957i	$0.922539\pi$
$614$	0	0
$615$	−22.3380	−0.900757
$616$	0	0
$617$	32.1201	1.29311	0.646554	−	0.762869i	$-0.276210\pi$
$617$	32.1201	1.29311	0.646554	+	0.762869i	$0.276210\pi$
$618$	0	0
$619$	21.2715i	0.854975i	0.904021	+	0.427488i	$0.140601\pi$
$619$	21.2715i	0.854975i	−0.904021	+	0.427488i	$0.859399\pi$
$620$	0	0
$621$	2.82843i	0.113501i
$622$	0	0
$623$	−3.08398	−0.123557
$624$	0	0
$625$	16.6958	0.667833
$626$	0	0
$627$	0.571533i	0.0228248i
$628$	0	0
$629$	1.16647i	0.0465101i
$630$	0	0
$631$	−36.4685	−1.45179	−0.725894	−	0.687807i	$-0.758574\pi$
$631$	−36.4685	−1.45179	−0.725894	+	0.687807i	$0.758574\pi$
$632$	0	0
$633$	10.2284	0.406542
$634$	0	0
$635$	− 14.4921i	− 0.575103i
$636$	0	0
$637$	− 4.57707i	− 0.181350i
$638$	0	0
$639$	−4.31788	−0.170813
$640$	0	0
$641$	14.0036	0.553109	0.276555	−	0.960998i	$-0.410807\pi$
$641$	14.0036	0.553109	0.276555	+	0.960998i	$0.410807\pi$
$642$	0	0
$643$	− 23.4807i	− 0.925990i	−0.886361	−	0.462995i	$-0.846775\pi$
$643$	− 23.4807i	− 0.925990i	0.886361	−	0.462995i	$-0.153225\pi$
$644$	0	0
$645$	− 41.6517i	− 1.64004i
$646$	0	0
$647$	12.1908	0.479270	0.239635	−	0.970863i	$-0.422972\pi$
$647$	12.1908	0.479270	0.239635	+	0.970863i	$0.422972\pi$
$648$	0	0
$649$	14.3835	0.564600
$650$	0	0
$651$	3.97474i	0.155782i
$652$	0	0
$653$	1.39055i	0.0544165i	0.999630	+	0.0272083i	$0.00866173\pi$
$653$	1.39055i	0.0544165i	−0.999630	+	0.0272083i	$0.991338\pi$
$654$	0	0
$655$	−4.12198	−0.161059
$656$	0	0
$657$	5.97474	0.233097
$658$	0	0
$659$	− 25.5349i	− 0.994698i	−0.867551	−	0.497349i	$-0.834307\pi$
$659$	− 25.5349i	− 0.994698i	0.867551	−	0.497349i	$-0.165693\pi$
$660$	0	0
$661$	− 6.44726i	− 0.250769i	−0.992108	−	0.125385i	$-0.959983\pi$
$661$	− 6.44726i	− 0.250769i	0.992108	−	0.125385i	$-0.0400165\pi$
$662$	0	0
$663$	0.439861	0.0170828
$664$	0	0
$665$	1.84307	0.0714710
$666$	0	0
$667$	7.42847i	0.287631i
$668$	0	0
$669$	1.71908i	0.0664635i
$670$	0	0
$671$	21.5298	0.831148
$672$	0	0
$673$	−10.8569	−0.418504	−0.209252	−	0.977862i	$-0.567103\pi$
$673$	−10.8569	−0.418504	−0.209252	+	0.977862i	$0.567103\pi$
$674$	0	0
$675$	− 9.42429i	− 0.362741i
$676$	0	0
$677$	− 33.5262i	− 1.28852i	−0.764807	−	0.644259i	$-0.777166\pi$
$677$	− 33.5262i	− 1.28852i	0.764807	−	0.644259i	$-0.222834\pi$
$678$	0	0
$679$	35.4045	1.35870
$680$	0	0
$681$	−14.3059	−0.548204
$682$	0	0
$683$	− 25.2206i	− 0.965040i	−0.875885	−	0.482520i	$-0.839722\pi$
$683$	− 25.2206i	− 0.965040i	0.875885	−	0.482520i	$-0.160278\pi$
$684$	0	0
$685$	20.1674i	0.770557i
$686$	0	0
$687$	16.9981	0.648519
$688$	0	0
$689$	−20.7945	−0.792205
$690$	0	0
$691$	15.3523i	0.584028i	0.956414	+	0.292014i	$0.0943254\pi$
$691$	15.3523i	0.584028i	−0.956414	+	0.292014i	$0.905675\pi$
$692$	0	0
$693$	5.48946i	0.208527i
$694$	0	0
$695$	−47.0907	−1.78625
$696$	0	0
$697$	−1.32206	−0.0500765
$698$	0	0
$699$	13.3779i	0.506000i
$700$	0	0
$701$	8.61079i	0.325225i	0.986690	+	0.162613i	$0.0519921\pi$
$701$	8.61079i	0.325225i	−0.986690	+	0.162613i	$0.948008\pi$
$702$	0	0
$703$	−1.16647	−0.0439942
$704$	0	0
$705$	10.7422	0.404574
$706$	0	0
$707$	0.250040i	0.00940372i
$708$	0	0
$709$	− 32.3624i	− 1.21539i	−0.794169	−	0.607697i	$-0.792094\pi$
$709$	− 32.3624i	− 1.21539i	0.794169	−	0.607697i	$-0.207906\pi$
$710$	0	0
$711$	15.0075	0.562826
$712$	0	0
$713$	5.20730	0.195015
$714$	0	0
$715$	− 18.8972i	− 0.706717i
$716$	0	0
$717$	13.3675i	0.499218i
$718$	0	0
$719$	1.46744	0.0547262	0.0273631	−	0.999626i	$-0.491289\pi$
$719$	1.46744	0.0547262	0.0273631	+	0.999626i	$0.491289\pi$
$720$	0	0
$721$	28.8233	1.07344
$722$	0	0
$723$	0.211474i	0.00786481i
$724$	0	0
$725$	− 24.7516i	− 0.919251i
$726$	0	0
$727$	15.3928	0.570889	0.285445	−	0.958395i	$-0.407859\pi$
$727$	15.3928	0.570889	0.285445	+	0.958395i	$0.407859\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 2.46512i	− 0.0911759i
$732$	0	0
$733$	17.5624i	0.648683i	0.945940	+	0.324342i	$0.105143\pi$
$733$	17.5624i	0.648683i	−0.945940	+	0.324342i	$0.894857\pi$
$734$	0	0
$735$	−8.88325	−0.327664
$736$	0	0
$737$	−37.4844	−1.38075
$738$	0	0
$739$	20.7266i	0.762441i	0.924484	+	0.381220i	$0.124496\pi$
$739$	20.7266i	0.762441i	−0.924484	+	0.381220i	$0.875504\pi$
$740$	0	0
$741$	0.439861i	0.0161587i
$742$	0	0
$743$	31.7821	1.16597	0.582986	−	0.812482i	$-0.301884\pi$
$743$	31.7821	1.16597	0.582986	+	0.812482i	$0.301884\pi$
$744$	0	0
$745$	5.52518	0.202427
$746$	0	0
$747$	14.3059i	0.523426i
$748$	0	0
$749$	22.2212i	0.811944i
$750$	0	0
$751$	−29.7594	−1.08594	−0.542968	−	0.839753i	$-0.682699\pi$
$751$	−29.7594	−1.08594	−0.542968	+	0.839753i	$0.682699\pi$
$752$	0	0
$753$	−14.7555	−0.537720
$754$	0	0
$755$	− 7.73625i	− 0.281551i
$756$	0	0
$757$	22.1119i	0.803672i	0.915712	+	0.401836i	$0.131628\pi$
$757$	22.1119i	0.803672i	−0.915712	+	0.401836i	$0.868372\pi$
$758$	0	0
$759$	7.19173	0.261043
$760$	0	0
$761$	4.55957	0.165284	0.0826422	−	0.996579i	$-0.473664\pi$
$761$	4.55957	0.165284	0.0826422	+	0.996579i	$0.473664\pi$
$762$	0	0
$763$	21.4963i	0.778219i
$764$	0	0
$765$	− 0.853690i	− 0.0308652i
$766$	0	0
$767$	11.0698	0.399706
$768$	0	0
$769$	36.5794	1.31909	0.659543	−	0.751667i	$-0.270750\pi$
$769$	36.5794	1.31909	0.659543	+	0.751667i	$0.270750\pi$
$770$	0	0
$771$	0.742176i	0.0267288i
$772$	0	0
$773$	26.4611i	0.951739i	0.879516	+	0.475869i	$0.157867\pi$
$773$	26.4611i	0.951739i	−0.879516	+	0.475869i	$0.842133\pi$
$774$	0	0
$775$	−17.3507	−0.623255
$776$	0	0
$777$	−11.2037	−0.401930
$778$	0	0
$779$	− 1.32206i	− 0.0473676i
$780$	0	0
$781$	10.9789i	0.392856i
$782$	0	0
$783$	−2.62636	−0.0938584
$784$	0	0
$785$	32.7302	1.16819
$786$	0	0
$787$	18.9164i	0.674298i	0.941451	+	0.337149i	$0.109463\pi$
$787$	18.9164i	0.674298i	−0.941451	+	0.337149i	$0.890537\pi$
$788$	0	0
$789$	5.48435i	0.195248i
$790$	0	0
$791$	40.6930	1.44688
$792$	0	0
$793$	16.5697	0.588406
$794$	0	0
$795$	40.3582i	1.43136i
$796$	0	0
$797$	47.8065i	1.69339i	0.532075	+	0.846697i	$0.321412\pi$
$797$	47.8065i	1.69339i	−0.532075	+	0.846697i	$0.678588\pi$
$798$	0	0
$799$	0.635767	0.0224918
$800$	0	0
$801$	−1.42847	−0.0504724
$802$	0	0
$803$	− 15.1917i	− 0.536105i
$804$	0	0
$805$	− 23.1917i	− 0.817401i
$806$	0	0
$807$	20.4694	0.720558
$808$	0	0
$809$	−29.9862	−1.05426	−0.527129	−	0.849785i	$-0.676732\pi$
$809$	−29.9862	−1.05426	−0.527129	+	0.849785i	$0.676732\pi$
$810$	0	0
$811$	− 11.3899i	− 0.399954i	−0.979801	−	0.199977i	$-0.935913\pi$
$811$	− 11.3899i	− 0.399954i	0.979801	−	0.199977i	$-0.0640867\pi$
$812$	0	0
$813$	− 14.0370i	− 0.492298i
$814$	0	0
$815$	18.4600	0.646626
$816$	0	0
$817$	2.46512	0.0862438
$818$	0	0
$819$	4.22478i	0.147626i
$820$	0	0
$821$	19.6929i	0.687286i	0.939100	+	0.343643i	$0.111661\pi$
$821$	19.6929i	0.687286i	−0.939100	+	0.343643i	$0.888339\pi$
$822$	0	0
$823$	−22.4666	−0.783137	−0.391568	−	0.920149i	$-0.628067\pi$
$823$	−22.4666	−0.783137	−0.391568	+	0.920149i	$0.628067\pi$
$824$	0	0
$825$	−23.9628	−0.834277
$826$	0	0
$827$	− 10.7927i	− 0.375299i	−0.982236	−	0.187649i	$-0.939913\pi$
$827$	− 10.7927i	− 0.375299i	0.982236	−	0.187649i	$-0.0600869\pi$
$828$	0	0
$829$	45.9421i	1.59563i	0.602900	+	0.797817i	$0.294012\pi$
$829$	45.9421i	1.59563i	−0.602900	+	0.797817i	$0.705988\pi$
$830$	0	0
$831$	13.4211	0.465572
$832$	0	0
$833$	−0.525748	−0.0182161
$834$	0	0
$835$	82.4238i	2.85239i
$836$	0	0
$837$	1.84106i	0.0636363i
$838$	0	0
$839$	−22.9142	−0.791085	−0.395542	−	0.918448i	$-0.629443\pi$
$839$	−22.9142	−0.791085	−0.395542	+	0.918448i	$0.629443\pi$
$840$	0	0
$841$	22.1022	0.762146
$842$	0	0
$843$	3.89359i	0.134102i
$844$	0	0
$845$	34.8295i	1.19817i
$846$	0	0
$847$	−9.79053	−0.336407
$848$	0	0
$849$	17.6569	0.605982
$850$	0	0
$851$	14.6779i	0.503153i
$852$	0	0
$853$	55.0728i	1.88566i	0.333278	+	0.942829i	$0.391845\pi$
$853$	55.0728i	1.88566i	−0.333278	+	0.942829i	$0.608155\pi$
$854$	0	0
$855$	0.853690	0.0291956
$856$	0	0
$857$	1.79079	0.0611723	0.0305861	−	0.999532i	$-0.490263\pi$
$857$	1.79079	0.0611723	0.0305861	+	0.999532i	$0.490263\pi$
$858$	0	0
$859$	− 25.5468i	− 0.871647i	−0.900032	−	0.435823i	$-0.856457\pi$
$859$	− 25.5468i	− 0.871647i	0.900032	−	0.435823i	$-0.143543\pi$
$860$	0	0
$861$	− 12.6981i	− 0.432750i
$862$	0	0
$863$	42.1150	1.43361	0.716806	−	0.697273i	$-0.245603\pi$
$863$	42.1150	1.43361	0.716806	+	0.697273i	$0.245603\pi$
$864$	0	0
$865$	46.9784	1.59731
$866$	0	0
$867$	16.9495i	0.575634i
$868$	0	0
$869$	− 38.1590i	− 1.29446i
$870$	0	0
$871$	−28.8486	−0.977497
$872$	0	0
$873$	16.3990	0.555023
$874$	0	0
$875$	36.2771i	1.22639i
$876$	0	0
$877$	1.01044i	0.0341203i	0.999854	+	0.0170601i	$0.00543067\pi$
$877$	1.01044i	0.0341203i	−0.999854	+	0.0170601i	$0.994569\pi$
$878$	0	0
$879$	15.7759	0.532108
$880$	0	0
$881$	44.3972	1.49578	0.747889	−	0.663823i	$-0.231067\pi$
$881$	44.3972	1.49578	0.747889	+	0.663823i	$0.231067\pi$
$882$	0	0
$883$	1.82389i	0.0613787i	0.999529	+	0.0306894i	$0.00977026\pi$
$883$	1.82389i	0.0613787i	−0.999529	+	0.0306894i	$0.990230\pi$
$884$	0	0
$885$	− 21.4844i	− 0.722189i
$886$	0	0
$887$	−4.38532	−0.147245	−0.0736223	−	0.997286i	$-0.523456\pi$
$887$	−4.38532	−0.147245	−0.0736223	+	0.997286i	$0.523456\pi$
$888$	0	0
$889$	8.23808	0.276296
$890$	0	0
$891$	2.54266i	0.0851823i
$892$	0	0
$893$	0.635767i	0.0212751i
$894$	0	0
$895$	−44.2128	−1.47787
$896$	0	0
$897$	5.53488	0.184804
$898$	0	0
$899$	4.83528i	0.161266i
$900$	0	0
$901$	2.38857i	0.0795747i
$902$	0	0
$903$	23.6770	0.787922
$904$	0	0
$905$	−36.1082	−1.20028
$906$	0	0
$907$	− 4.06248i	− 0.134893i	−0.997723	−	0.0674463i	$-0.978515\pi$
$907$	− 4.06248i	− 0.134893i	0.997723	−	0.0674463i	$-0.0214851\pi$
$908$	0	0
$909$	0.115816i	0.00384137i
$910$	0	0
$911$	−42.3784	−1.40406	−0.702029	−	0.712149i	$-0.747722\pi$
$911$	−42.3784	−1.40406	−0.702029	+	0.712149i	$0.747722\pi$
$912$	0	0
$913$	36.3751	1.20384
$914$	0	0
$915$	− 32.1587i	− 1.06313i
$916$	0	0
$917$	− 2.34315i	− 0.0773775i
$918$	0	0
$919$	−44.8603	−1.47980	−0.739902	−	0.672715i	$-0.765128\pi$
$919$	−44.8603	−1.47980	−0.739902	+	0.672715i	$0.765128\pi$
$920$	0	0
$921$	−7.64129	−0.251789
$922$	0	0
$923$	8.44955i	0.278120i
$924$	0	0
$925$	− 48.9068i	− 1.60804i
$926$	0	0
$927$	13.3507	0.438494
$928$	0	0
$929$	−2.96695	−0.0973426	−0.0486713	−	0.998815i	$-0.515499\pi$
$929$	−2.96695	−0.0973426	−0.0486713	+	0.998815i	$0.515499\pi$
$930$	0	0
$931$	− 0.525748i	− 0.0172307i
$932$	0	0
$933$	24.1623i	0.791038i
$934$	0	0
$935$	−2.17064	−0.0709876
$936$	0	0
$937$	−54.7669	−1.78916	−0.894579	−	0.446910i	$-0.852524\pi$
$937$	−54.7669	−1.78916	−0.894579	+	0.446910i	$0.852524\pi$
$938$	0	0
$939$	− 16.6105i	− 0.542063i
$940$	0	0
$941$	− 8.89558i	− 0.289988i	−0.989433	−	0.144994i	$-0.953684\pi$
$941$	− 8.89558i	− 0.289988i	0.989433	−	0.144994i	$-0.0463162\pi$
$942$	0	0
$943$	−16.6358	−0.541735
$944$	0	0
$945$	8.19951	0.266730
$946$	0	0
$947$	− 15.8541i	− 0.515189i	−0.966253	−	0.257595i	$-0.917070\pi$
$947$	− 15.8541i	− 0.515189i	0.966253	−	0.257595i	$-0.0829299\pi$
$948$	0	0
$949$	− 11.6918i	− 0.379532i
$950$	0	0
$951$	−2.56213	−0.0830826
$952$	0	0
$953$	30.2807	0.980887	0.490443	−	0.871473i	$-0.336835\pi$
$953$	30.2807	0.980887	0.490443	+	0.871473i	$0.336835\pi$
$954$	0	0
$955$	− 79.0090i	− 2.55667i
$956$	0	0
$957$	6.67794i	0.215867i
$958$	0	0
$959$	−11.4642	−0.370198
$960$	0	0
$961$	−27.6105	−0.890661
$962$	0	0
$963$	10.2926i	0.331675i
$964$	0	0
$965$	53.7232i	1.72941i
$966$	0	0
$967$	10.5273	0.338537	0.169268	−	0.985570i	$-0.445860\pi$
$967$	10.5273	0.338537	0.169268	+	0.985570i	$0.445860\pi$
$968$	0	0
$969$	0.0505249	0.00162309
$970$	0	0
$971$	38.7050i	1.24210i	0.783771	+	0.621051i	$0.213294\pi$
$971$	38.7050i	1.24210i	−0.783771	+	0.621051i	$0.786706\pi$
$972$	0	0
$973$	− 26.7688i	− 0.858168i
$974$	0	0
$975$	−18.4422	−0.590622
$976$	0	0
$977$	16.9009	0.540706	0.270353	−	0.962761i	$-0.412860\pi$
$977$	16.9009	0.540706	0.270353	+	0.962761i	$0.412860\pi$
$978$	0	0
$979$	3.63211i	0.116083i
$980$	0	0
$981$	9.95687i	0.317899i
$982$	0	0
$983$	10.0798	0.321496	0.160748	−	0.986995i	$-0.448609\pi$
$983$	10.0798	0.321496	0.160748	+	0.986995i	$0.448609\pi$
$984$	0	0
$985$	−13.0445	−0.415632
$986$	0	0
$987$	6.10641i	0.194369i
$988$	0	0
$989$	− 31.0192i	− 0.986354i
$990$	0	0
$991$	−42.3446	−1.34512	−0.672561	−	0.740042i	$-0.734806\pi$
$991$	−42.3446	−1.34512	−0.672561	+	0.740042i	$0.734806\pi$
$992$	0	0
$993$	19.1275	0.606993
$994$	0	0
$995$	− 1.16286i	− 0.0368651i
$996$	0	0
$997$	47.6132i	1.50793i	0.656917	+	0.753963i	$0.271860\pi$
$997$	47.6132i	1.50793i	−0.656917	+	0.753963i	$0.728140\pi$
$998$	0	0
$999$	−5.18944	−0.164186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.d.f.1537.4		8
4.3	odd	2		3072.2.d.i.1537.8		8
8.3	odd	2		3072.2.d.i.1537.1		8
8.5	even	2	inner	3072.2.d.f.1537.5		8
16.3	odd	4		3072.2.a.n.1.4		4
16.5	even	4		3072.2.a.i.1.1		4
16.11	odd	4		3072.2.a.o.1.1		4
16.13	even	4		3072.2.a.t.1.4		4
32.3	odd	8		192.2.j.a.49.1		8
32.5	even	8		48.2.j.a.13.3	✓	8
32.11	odd	8		384.2.j.a.289.4		8
32.13	even	8		384.2.j.b.97.2		8
32.19	odd	8		384.2.j.a.97.4		8
32.21	even	8		384.2.j.b.289.2		8
32.27	odd	8		192.2.j.a.145.1		8
32.29	even	8		48.2.j.a.37.3	yes	8
48.5	odd	4		9216.2.a.bo.1.4		4
48.11	even	4		9216.2.a.bn.1.4		4
48.29	odd	4		9216.2.a.y.1.1		4
48.35	even	4		9216.2.a.x.1.1		4
96.5	odd	8		144.2.k.b.109.2		8
96.11	even	8		1152.2.k.f.289.1		8
96.29	odd	8		144.2.k.b.37.2		8
96.35	even	8		576.2.k.b.433.4		8
96.53	odd	8		1152.2.k.c.289.1		8
96.59	even	8		576.2.k.b.145.4		8
96.77	odd	8		1152.2.k.c.865.1		8
96.83	even	8		1152.2.k.f.865.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.3	✓	8	32.5	even	8
48.2.j.a.37.3	yes	8	32.29	even	8
144.2.k.b.37.2		8	96.29	odd	8
144.2.k.b.109.2		8	96.5	odd	8
192.2.j.a.49.1		8	32.3	odd	8
192.2.j.a.145.1		8	32.27	odd	8
384.2.j.a.97.4		8	32.19	odd	8
384.2.j.a.289.4		8	32.11	odd	8
384.2.j.b.97.2		8	32.13	even	8
384.2.j.b.289.2		8	32.21	even	8
576.2.k.b.145.4		8	96.59	even	8
576.2.k.b.433.4		8	96.35	even	8
1152.2.k.c.289.1		8	96.53	odd	8
1152.2.k.c.865.1		8	96.77	odd	8
1152.2.k.f.289.1		8	96.11	even	8
1152.2.k.f.865.1		8	96.83	even	8
3072.2.a.i.1.1		4	16.5	even	4
3072.2.a.n.1.4		4	16.3	odd	4
3072.2.a.o.1.1		4	16.11	odd	4
3072.2.a.t.1.4		4	16.13	even	4
3072.2.d.f.1537.4		8	1.1	even	1	trivial
3072.2.d.f.1537.5		8	8.5	even	2	inner
3072.2.d.i.1537.1		8	8.3	odd	2
3072.2.d.i.1537.8		8	4.3	odd	2
9216.2.a.x.1.1		4	48.35	even	4
9216.2.a.y.1.1		4	48.29	odd	4
9216.2.a.bn.1.4		4	48.11	even	4
9216.2.a.bo.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.79793i q^{5} -2.15894 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.79793i q^{5} -2.15894 q^{7} -1.00000 q^{9} +2.54266i q^{11} +1.95687i q^{13} +3.79793 q^{15} +0.224777 q^{17} +0.224777i q^{19} +2.15894i q^{21} +2.82843 q^{23} -9.42429 q^{25} +1.00000i q^{27} +2.62636i q^{29} +1.84106 q^{31} +2.54266 q^{33} -8.19951i q^{35} +5.18944i q^{37} +1.95687 q^{39} -5.88163 q^{41} -10.9670i q^{43} -3.79793i q^{45} +2.82843 q^{47} -2.33897 q^{49} -0.224777i q^{51} +10.6264i q^{53} -9.65685 q^{55} +0.224777 q^{57} -5.65685i q^{59} -8.46742i q^{61} +2.15894 q^{63} -7.43208 q^{65} +14.7422i q^{67} -2.82843i q^{69} +4.31788 q^{71} -5.97474 q^{73} +9.42429i q^{75} -5.48946i q^{77} -15.0075 q^{79} +1.00000 q^{81} -14.3059i q^{83} +0.853690i q^{85} +2.62636 q^{87} +1.42847 q^{89} -4.22478i q^{91} -1.84106i q^{93} -0.853690 q^{95} -16.3990 q^{97} -2.54266i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^7 - 8 * q^9	\( 8 q - 8 q^{7} - 8 q^{9} + 8 q^{15} - 8 q^{25} + 24 q^{31} - 16 q^{39} + 8 q^{49} - 32 q^{55} + 8 q^{63} - 16 q^{65} + 16 q^{71} + 16 q^{73} + 24 q^{79} + 8 q^{81} - 24 q^{87} + 16 q^{89} - 48 q^{95}+O(q^{100}) \) 8 * q - 8 * q^7 - 8 * q^9 + 8 * q^15 - 8 * q^25 + 24 * q^31 - 16 * q^39 + 8 * q^49 - 32 * q^55 + 8 * q^63 - 16 * q^65 + 16 * q^71 + 16 * q^73 + 24 * q^79 + 8 * q^81 - 24 * q^87 + 16 * q^89 - 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more