LMFDB - Embedded newform 3072.2.d.c.1537.3

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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.3
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.c.1537.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.41421i q^{5} +2.82843 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -1.41421i q^{5} +2.82843 q^{7} -1.00000 q^{9} -4.24264i q^{13} +1.41421 q^{15} +4.00000 q^{17} +8.00000i q^{19} +2.82843i q^{21} -5.65685 q^{23} +3.00000 q^{25} -1.00000i q^{27} -1.41421i q^{29} +2.82843 q^{31} -4.00000i q^{35} +4.24264i q^{37} +4.24264 q^{39} +4.00000 q^{41} +1.41421i q^{45} +11.3137 q^{47} +1.00000 q^{49} +4.00000i q^{51} -7.07107i q^{53} -8.00000 q^{57} -12.0000i q^{59} -1.41421i q^{61} -2.82843 q^{63} -6.00000 q^{65} -4.00000i q^{67} -5.65685i q^{69} +11.3137 q^{71} -14.0000 q^{73} +3.00000i q^{75} +8.48528 q^{79} +1.00000 q^{81} +16.0000i q^{83} -5.65685i q^{85} +1.41421 q^{87} -6.00000 q^{89} -12.0000i q^{91} +2.82843i q^{93} +11.3137 q^{95} +16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 16 q^{17} + 12 q^{25} + 16 q^{41} + 4 q^{49} - 32 q^{57} - 24 q^{65} - 56 q^{73} + 4 q^{81} - 24 q^{89} + 64 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 + 16 * q^17 + 12 * q^25 + 16 * q^41 + 4 * q^49 - 32 * q^57 - 24 * q^65 - 56 * q^73 + 4 * q^81 - 24 * 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
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 4.24264i	− 1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$	− 4.24264i	− 1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	8.00000i	1.83533i	0.397360	+	0.917663i	$0.369927\pi$
$19$	8.00000i	1.83533i	−0.397360	+	0.917663i	$0.630073\pi$
$20$	0	0
$21$	2.82843i	0.617213i
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 1.41421i	− 0.262613i	−0.991342	−	0.131306i	$-0.958083\pi$
$29$	− 1.41421i	− 0.262613i	0.991342	−	0.131306i	$-0.0419172\pi$
$30$	0	0
$31$	2.82843	0.508001	0.254000	−	0.967204i	$-0.418254\pi$
$31$	2.82843	0.508001	0.254000	+	0.967204i	$0.418254\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 4.00000i	− 0.676123i
$36$	0	0
$37$	4.24264i	0.697486i	0.937218	+	0.348743i	$0.113391\pi$
$37$	4.24264i	0.697486i	−0.937218	+	0.348743i	$0.886609\pi$
$38$	0	0
$39$	4.24264	0.679366
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	1.41421i	0.210819i
$46$	0	0
$47$	11.3137	1.65027	0.825137	−	0.564933i	$-0.191098\pi$
$47$	11.3137	1.65027	0.825137	+	0.564933i	$0.191098\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.00000i	0.560112i
$52$	0	0
$53$	− 7.07107i	− 0.971286i	−0.874157	−	0.485643i	$-0.838586\pi$
$53$	− 7.07107i	− 0.971286i	0.874157	−	0.485643i	$-0.161414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	0	0
$61$	− 1.41421i	− 0.181071i	−0.995893	−	0.0905357i	$-0.971142\pi$
$61$	− 1.41421i	− 0.181071i	0.995893	−	0.0905357i	$-0.0288579\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	− 5.65685i	− 0.681005i
$70$	0	0
$71$	11.3137	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	3.00000i	0.346410i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	0	0
$85$	− 5.65685i	− 0.613572i
$86$	0	0
$87$	1.41421	0.151620
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	− 12.0000i	− 1.25794i
$92$	0	0
$93$	2.82843i	0.293294i
$94$	0	0
$95$	11.3137	1.16076
$96$	0	0
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 15.5563i	− 1.54791i	−0.633238	−	0.773957i	$-0.718274\pi$
$101$	− 15.5563i	− 1.54791i	0.633238	−	0.773957i	$-0.281726\pi$
$102$	0	0
$103$	19.7990	1.95085	0.975426	−	0.220326i	$-0.0707122\pi$
$103$	19.7990	1.95085	0.975426	+	0.220326i	$0.0707122\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	4.24264i	0.406371i	0.979140	+	0.203186i	$0.0651295\pi$
$109$	4.24264i	0.406371i	−0.979140	+	0.203186i	$0.934871\pi$
$110$	0	0
$111$	−4.24264	−0.402694
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	8.00000i	0.746004i
$116$	0	0
$117$	4.24264i	0.392232i
$118$	0	0
$119$	11.3137	1.03713
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	4.00000i	0.360668i
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	−8.48528	−0.752947	−0.376473	−	0.926427i	$-0.622863\pi$
$127$	−8.48528	−0.752947	−0.376473	+	0.926427i	$0.622863\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.00000i	0.349482i	0.984614	+	0.174741i	$0.0559088\pi$
$131$	4.00000i	0.349482i	−0.984614	+	0.174741i	$0.944091\pi$
$132$	0	0
$133$	22.6274i	1.96205i
$134$	0	0
$135$	−1.41421	−0.121716
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	− 4.00000i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	− 4.00000i	− 0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	0	0
$141$	11.3137i	0.952786i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	12.7279i	1.04271i	0.853339	+	0.521356i	$0.174574\pi$
$149$	12.7279i	1.04271i	−0.853339	+	0.521356i	$0.825426\pi$
$150$	0	0
$151$	−2.82843	−0.230174	−0.115087	−	0.993355i	$-0.536715\pi$
$151$	−2.82843	−0.230174	−0.115087	+	0.993355i	$0.536715\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	− 4.00000i	− 0.321288i
$156$	0	0
$157$	21.2132i	1.69300i	0.532390	+	0.846499i	$0.321294\pi$
$157$	21.2132i	1.69300i	−0.532390	+	0.846499i	$0.678706\pi$
$158$	0	0
$159$	7.07107	0.560772
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	− 8.00000i	− 0.611775i
$172$	0	0
$173$	− 7.07107i	− 0.537603i	−0.963196	−	0.268802i	$-0.913372\pi$
$173$	− 7.07107i	− 0.537603i	0.963196	−	0.268802i	$-0.0866276\pi$
$174$	0	0
$175$	8.48528	0.641427
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	− 4.00000i	− 0.298974i	−0.988764	−	0.149487i	$-0.952238\pi$
$179$	− 4.00000i	− 0.298974i	0.988764	−	0.149487i	$-0.0477622\pi$
$180$	0	0
$181$	− 12.7279i	− 0.946059i	−0.881047	−	0.473029i	$-0.843160\pi$
$181$	− 12.7279i	− 0.946059i	0.881047	−	0.473029i	$-0.156840\pi$
$182$	0	0
$183$	1.41421	0.104542
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 2.82843i	− 0.205738i
$190$	0	0
$191$	−5.65685	−0.409316	−0.204658	−	0.978834i	$-0.565608\pi$
$191$	−5.65685	−0.409316	−0.204658	+	0.978834i	$0.565608\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	− 6.00000i	− 0.429669i
$196$	0	0
$197$	− 7.07107i	− 0.503793i	−0.967754	−	0.251896i	$-0.918946\pi$
$197$	− 7.07107i	− 0.503793i	0.967754	−	0.251896i	$-0.0810542\pi$
$198$	0	0
$199$	−8.48528	−0.601506	−0.300753	−	0.953702i	$-0.597238\pi$
$199$	−8.48528	−0.601506	−0.300753	+	0.953702i	$0.597238\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	− 4.00000i	− 0.280745i
$204$	0	0
$205$	− 5.65685i	− 0.395092i
$206$	0	0
$207$	5.65685	0.393179
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 12.0000i	− 0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	− 12.0000i	− 0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	0	0
$213$	11.3137i	0.775203i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	− 14.0000i	− 0.946032i
$220$	0	0
$221$	− 16.9706i	− 1.14156i
$222$	0	0
$223$	25.4558	1.70465	0.852325	−	0.523013i	$-0.175192\pi$
$223$	25.4558	1.70465	0.852325	+	0.523013i	$0.175192\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	8.00000i	0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$	8.00000i	0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$	0	0
$229$	24.0416i	1.58872i	0.607450	+	0.794358i	$0.292192\pi$
$229$	24.0416i	1.58872i	−0.607450	+	0.794358i	$0.707808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	− 16.0000i	− 1.04372i
$236$	0	0
$237$	8.48528i	0.551178i
$238$	0	0
$239$	−5.65685	−0.365911	−0.182956	−	0.983121i	$-0.558567\pi$
$239$	−5.65685	−0.365911	−0.182956	+	0.983121i	$0.558567\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 1.41421i	− 0.0903508i
$246$	0	0
$247$	33.9411	2.15962
$248$	0	0
$249$	−16.0000	−1.01396
$250$	0	0
$251$	8.00000i	0.504956i	0.967603	+	0.252478i	$0.0812455\pi$
$251$	8.00000i	0.504956i	−0.967603	+	0.252478i	$0.918755\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	5.65685	0.354246
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	12.0000i	0.745644i
$260$	0	0
$261$	1.41421i	0.0875376i
$262$	0	0
$263$	−28.2843	−1.74408	−0.872041	−	0.489432i	$-0.837204\pi$
$263$	−28.2843	−1.74408	−0.872041	+	0.489432i	$0.837204\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	0	0
$269$	− 21.2132i	− 1.29339i	−0.762748	−	0.646696i	$-0.776150\pi$
$269$	− 21.2132i	− 1.29339i	0.762748	−	0.646696i	$-0.223850\pi$
$270$	0	0
$271$	−8.48528	−0.515444	−0.257722	−	0.966219i	$-0.582972\pi$
$271$	−8.48528	−0.515444	−0.257722	+	0.966219i	$0.582972\pi$
$272$	0	0
$273$	12.0000	0.726273
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.41421i	0.0849719i	0.999097	+	0.0424859i	$0.0135278\pi$
$277$	1.41421i	0.0849719i	−0.999097	+	0.0424859i	$0.986472\pi$
$278$	0	0
$279$	−2.82843	−0.169334
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	11.3137i	0.670166i
$286$	0	0
$287$	11.3137	0.667827
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	16.0000i	0.937937i
$292$	0	0
$293$	− 18.3848i	− 1.07405i	−0.843566	−	0.537025i	$-0.819548\pi$
$293$	− 18.3848i	− 1.07405i	0.843566	−	0.537025i	$-0.180452\pi$
$294$	0	0
$295$	−16.9706	−0.988064
$296$	0	0
$297$	0	0
$298$	0	0
$299$	24.0000i	1.38796i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	15.5563	0.893689
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	19.7990i	1.12633i
$310$	0	0
$311$	28.2843	1.60385	0.801927	−	0.597422i	$-0.203808\pi$
$311$	28.2843	1.60385	0.801927	+	0.597422i	$0.203808\pi$
$312$	0	0
$313$	16.0000	0.904373	0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000	0.904373	0.452187	+	0.891923i	$0.350644\pi$
$314$	0	0
$315$	4.00000i	0.225374i
$316$	0	0
$317$	21.2132i	1.19145i	0.803188	+	0.595726i	$0.203136\pi$
$317$	21.2132i	1.19145i	−0.803188	+	0.595726i	$0.796864\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	32.0000i	1.78053i
$324$	0	0
$325$	− 12.7279i	− 0.706018i
$326$	0	0
$327$	−4.24264	−0.234619
$328$	0	0
$329$	32.0000	1.76422
$330$	0	0
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	0	0
$333$	− 4.24264i	− 0.232495i
$334$	0	0
$335$	−5.65685	−0.309067
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	− 18.0000i	− 0.977626i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	32.0000i	1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$	32.0000i	1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$	0	0
$349$	12.7279i	0.681310i	0.940188	+	0.340655i	$0.110649\pi$
$349$	12.7279i	0.681310i	−0.940188	+	0.340655i	$0.889351\pi$
$350$	0	0
$351$	−4.24264	−0.226455
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	− 16.0000i	− 0.849192i
$356$	0	0
$357$	11.3137i	0.598785i
$358$	0	0
$359$	11.3137	0.597115	0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137	0.597115	0.298557	+	0.954392i	$0.403495\pi$
$360$	0	0
$361$	−45.0000	−2.36842
$362$	0	0
$363$	11.0000i	0.577350i
$364$	0	0
$365$	19.7990i	1.03633i
$366$	0	0
$367$	2.82843	0.147643	0.0738213	−	0.997271i	$-0.476481\pi$
$367$	2.82843	0.147643	0.0738213	+	0.997271i	$0.476481\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	− 20.0000i	− 1.03835i
$372$	0	0
$373$	− 4.24264i	− 0.219676i	−0.993950	−	0.109838i	$-0.964967\pi$
$373$	− 4.24264i	− 0.219676i	0.993950	−	0.109838i	$-0.0350331\pi$
$374$	0	0
$375$	11.3137	0.584237
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	16.0000i	0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$	16.0000i	0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$	0	0
$381$	− 8.48528i	− 0.434714i
$382$	0	0
$383$	−28.2843	−1.44526	−0.722629	−	0.691236i	$-0.757067\pi$
$383$	−28.2843	−1.44526	−0.722629	+	0.691236i	$0.757067\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9.89949i	− 0.501924i	−0.967997	−	0.250962i	$-0.919253\pi$
$389$	− 9.89949i	− 0.501924i	0.967997	−	0.250962i	$-0.0807470\pi$
$390$	0	0
$391$	−22.6274	−1.14432
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	− 12.0000i	− 0.603786i
$396$	0	0
$397$	21.2132i	1.06466i	0.846537	+	0.532330i	$0.178683\pi$
$397$	21.2132i	1.06466i	−0.846537	+	0.532330i	$0.821317\pi$
$398$	0	0
$399$	−22.6274	−1.13279
$400$	0	0
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	0	0
$403$	− 12.0000i	− 0.597763i
$404$	0	0
$405$	− 1.41421i	− 0.0702728i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	12.0000i	0.591916i
$412$	0	0
$413$	− 33.9411i	− 1.67013i
$414$	0	0
$415$	22.6274	1.11074
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	8.00000i	0.390826i	0.980721	+	0.195413i	$0.0626047\pi$
$419$	8.00000i	0.390826i	−0.980721	+	0.195413i	$0.937395\pi$
$420$	0	0
$421$	12.7279i	0.620321i	0.950684	+	0.310160i	$0.100383\pi$
$421$	12.7279i	0.620321i	−0.950684	+	0.310160i	$0.899617\pi$
$422$	0	0
$423$	−11.3137	−0.550091
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.3137	−0.544962	−0.272481	−	0.962161i	$-0.587844\pi$
$431$	−11.3137	−0.544962	−0.272481	+	0.962161i	$0.587844\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	− 2.00000i	− 0.0958927i
$436$	0	0
$437$	− 45.2548i	− 2.16483i
$438$	0	0
$439$	2.82843	0.134993	0.0674967	−	0.997719i	$-0.478499\pi$
$439$	2.82843	0.134993	0.0674967	+	0.997719i	$0.478499\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	8.48528i	0.402241i
$446$	0	0
$447$	−12.7279	−0.602010
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 2.82843i	− 0.132891i
$454$	0	0
$455$	−16.9706	−0.795592
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	− 4.00000i	− 0.186704i
$460$	0	0
$461$	− 15.5563i	− 0.724531i	−0.932075	−	0.362266i	$-0.882003\pi$
$461$	− 15.5563i	− 0.724531i	0.932075	−	0.362266i	$-0.117997\pi$
$462$	0	0
$463$	−31.1127	−1.44593	−0.722965	−	0.690885i	$-0.757221\pi$
$463$	−31.1127	−1.44593	−0.722965	+	0.690885i	$0.757221\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	0	0
$469$	− 11.3137i	− 0.522419i
$470$	0	0
$471$	−21.2132	−0.977453
$472$	0	0
$473$	0	0
$474$	0	0
$475$	24.0000i	1.10120i
$476$	0	0
$477$	7.07107i	0.323762i
$478$	0	0
$479$	28.2843	1.29234	0.646171	−	0.763193i	$-0.276369\pi$
$479$	28.2843	1.29234	0.646171	+	0.763193i	$0.276369\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	0	0
$483$	− 16.0000i	− 0.728025i
$484$	0	0
$485$	− 22.6274i	− 1.02746i
$486$	0	0
$487$	−2.82843	−0.128168	−0.0640841	−	0.997944i	$-0.520413\pi$
$487$	−2.82843	−0.128168	−0.0640841	+	0.997944i	$0.520413\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	− 12.0000i	− 0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$	− 12.0000i	− 0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$	0	0
$493$	− 5.65685i	− 0.254772i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	32.0000	1.43540
$498$	0	0
$499$	− 28.0000i	− 1.25345i	−0.779240	−	0.626726i	$-0.784395\pi$
$499$	− 28.0000i	− 1.25345i	0.779240	−	0.626726i	$-0.215605\pi$
$500$	0	0
$501$	11.3137i	0.505459i
$502$	0	0
$503$	−11.3137	−0.504453	−0.252227	−	0.967668i	$-0.581163\pi$
$503$	−11.3137	−0.504453	−0.252227	+	0.967668i	$0.581163\pi$
$504$	0	0
$505$	−22.0000	−0.978987
$506$	0	0
$507$	− 5.00000i	− 0.222058i
$508$	0	0
$509$	− 32.5269i	− 1.44173i	−0.693075	−	0.720865i	$-0.743745\pi$
$509$	− 32.5269i	− 1.44173i	0.693075	−	0.720865i	$-0.256255\pi$
$510$	0	0
$511$	−39.5980	−1.75171
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	− 28.0000i	− 1.23383i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	7.07107	0.310385
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	8.00000i	0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	8.00000i	0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$	0	0
$525$	8.48528i	0.370328i
$526$	0	0
$527$	11.3137	0.492833
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	12.0000i	0.520756i
$532$	0	0
$533$	− 16.9706i	− 0.735077i
$534$	0	0
$535$	−16.9706	−0.733701
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	0	0
$540$	0	0
$541$	4.24264i	0.182405i	0.995832	+	0.0912027i	$0.0290711\pi$
$541$	4.24264i	0.182405i	−0.995832	+	0.0912027i	$0.970929\pi$
$542$	0	0
$543$	12.7279	0.546207
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	1.41421i	0.0603572i
$550$	0	0
$551$	11.3137	0.481980
$552$	0	0
$553$	24.0000	1.02058
$554$	0	0
$555$	6.00000i	0.254686i
$556$	0	0
$557$	− 29.6985i	− 1.25837i	−0.777258	−	0.629183i	$-0.783390\pi$
$557$	− 29.6985i	− 1.25837i	0.777258	−	0.629183i	$-0.216610\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.00000i	0.337160i	0.985688	+	0.168580i	$0.0539181\pi$
$563$	8.00000i	0.337160i	−0.985688	+	0.168580i	$0.946082\pi$
$564$	0	0
$565$	25.4558i	1.07094i
$566$	0	0
$567$	2.82843	0.118783
$568$	0	0
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	− 20.0000i	− 0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$	− 20.0000i	− 0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$	0	0
$573$	− 5.65685i	− 0.236318i
$574$	0	0
$575$	−16.9706	−0.707721
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	45.2548i	1.87749i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	4.00000i	0.165098i	0.996587	+	0.0825488i	$0.0263060\pi$
$587$	4.00000i	0.165098i	−0.996587	+	0.0825488i	$0.973694\pi$
$588$	0	0
$589$	22.6274i	0.932346i
$590$	0	0
$591$	7.07107	0.290865
$592$	0	0
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	− 16.0000i	− 0.655936i
$596$	0	0
$597$	− 8.48528i	− 0.347279i
$598$	0	0
$599$	−39.5980	−1.61793	−0.808965	−	0.587857i	$-0.799972\pi$
$599$	−39.5980	−1.61793	−0.808965	+	0.587857i	$0.799972\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	− 15.5563i	− 0.632456i
$606$	0	0
$607$	−42.4264	−1.72203	−0.861017	−	0.508576i	$-0.830172\pi$
$607$	−42.4264	−1.72203	−0.861017	+	0.508576i	$0.830172\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	− 48.0000i	− 1.94187i
$612$	0	0
$613$	− 15.5563i	− 0.628315i	−0.949371	−	0.314158i	$-0.898278\pi$
$613$	− 15.5563i	− 0.628315i	0.949371	−	0.314158i	$-0.101722\pi$
$614$	0	0
$615$	5.65685	0.228106
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	44.0000i	1.76851i	0.467005	+	0.884255i	$0.345333\pi$
$619$	44.0000i	1.76851i	−0.467005	+	0.884255i	$0.654667\pi$
$620$	0	0
$621$	5.65685i	0.227002i
$622$	0	0
$623$	−16.9706	−0.679911
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.9706i	0.676661i
$630$	0	0
$631$	8.48528	0.337794	0.168897	−	0.985634i	$-0.445980\pi$
$631$	8.48528	0.337794	0.168897	+	0.985634i	$0.445980\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	12.0000i	0.476205i
$636$	0	0
$637$	− 4.24264i	− 0.168100i
$638$	0	0
$639$	−11.3137	−0.447563
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.5980	−1.55676	−0.778379	−	0.627795i	$-0.783958\pi$
$647$	−39.5980	−1.55676	−0.778379	+	0.627795i	$0.783958\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	8.00000i	0.313545i
$652$	0	0
$653$	− 4.24264i	− 0.166027i	−0.996548	−	0.0830137i	$-0.973545\pi$
$653$	− 4.24264i	− 0.166027i	0.996548	−	0.0830137i	$-0.0264545\pi$
$654$	0	0
$655$	5.65685	0.221032
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	− 20.0000i	− 0.779089i	−0.921008	−	0.389545i	$-0.872632\pi$
$659$	− 20.0000i	− 0.779089i	0.921008	−	0.389545i	$-0.127368\pi$
$660$	0	0
$661$	26.8701i	1.04512i	0.852601	+	0.522562i	$0.175024\pi$
$661$	26.8701i	1.04512i	−0.852601	+	0.522562i	$0.824976\pi$
$662$	0	0
$663$	16.9706	0.659082
$664$	0	0
$665$	32.0000	1.24091
$666$	0	0
$667$	8.00000i	0.309761i
$668$	0	0
$669$	25.4558i	0.984180i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	0	0
$675$	− 3.00000i	− 0.115470i
$676$	0	0
$677$	− 32.5269i	− 1.25011i	−0.780580	−	0.625055i	$-0.785076\pi$
$677$	− 32.5269i	− 1.25011i	0.780580	−	0.625055i	$-0.214924\pi$
$678$	0	0
$679$	45.2548	1.73672
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	32.0000i	1.22445i	0.790685	+	0.612223i	$0.209725\pi$
$683$	32.0000i	1.22445i	−0.790685	+	0.612223i	$0.790275\pi$
$684$	0	0
$685$	− 16.9706i	− 0.648412i
$686$	0	0
$687$	−24.0416	−0.917245
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	40.0000i	1.52167i	0.648944	+	0.760836i	$0.275211\pi$
$691$	40.0000i	1.52167i	−0.648944	+	0.760836i	$0.724789\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.65685	−0.214577
$696$	0	0
$697$	16.0000	0.606043
$698$	0	0
$699$	10.0000i	0.378235i
$700$	0	0
$701$	46.6690i	1.76267i	0.472496	+	0.881333i	$0.343353\pi$
$701$	46.6690i	1.76267i	−0.472496	+	0.881333i	$0.656647\pi$
$702$	0	0
$703$	−33.9411	−1.28011
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	− 44.0000i	− 1.65479i
$708$	0	0
$709$	− 35.3553i	− 1.32780i	−0.747822	−	0.663899i	$-0.768901\pi$
$709$	− 35.3553i	− 1.32780i	0.747822	−	0.663899i	$-0.231099\pi$
$710$	0	0
$711$	−8.48528	−0.318223
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 5.65685i	− 0.211259i
$718$	0	0
$719$	−16.9706	−0.632895	−0.316448	−	0.948610i	$-0.602490\pi$
$719$	−16.9706	−0.632895	−0.316448	+	0.948610i	$0.602490\pi$
$720$	0	0
$721$	56.0000	2.08555
$722$	0	0
$723$	8.00000i	0.297523i
$724$	0	0
$725$	− 4.24264i	− 0.157568i
$726$	0	0
$727$	−42.4264	−1.57351	−0.786754	−	0.617266i	$-0.788240\pi$
$727$	−42.4264	−1.57351	−0.786754	+	0.617266i	$0.788240\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 4.24264i	− 0.156706i	−0.996926	−	0.0783528i	$-0.975034\pi$
$733$	− 4.24264i	− 0.156706i	0.996926	−	0.0783528i	$-0.0249660\pi$
$734$	0	0
$735$	1.41421	0.0521641
$736$	0	0
$737$	0	0
$738$	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
$741$	33.9411i	1.24686i
$742$	0	0
$743$	−11.3137	−0.415060	−0.207530	−	0.978229i	$-0.566542\pi$
$743$	−11.3137	−0.415060	−0.207530	+	0.978229i	$0.566542\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	− 16.0000i	− 0.585409i
$748$	0	0
$749$	− 33.9411i	− 1.24018i
$750$	0	0
$751$	−8.48528	−0.309632	−0.154816	−	0.987943i	$-0.549479\pi$
$751$	−8.48528	−0.309632	−0.154816	+	0.987943i	$0.549479\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	4.00000i	0.145575i
$756$	0	0
$757$	21.2132i	0.771007i	0.922706	+	0.385503i	$0.125972\pi$
$757$	21.2132i	0.771007i	−0.922706	+	0.385503i	$0.874028\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	0	0
$763$	12.0000i	0.434429i
$764$	0	0
$765$	5.65685i	0.204524i
$766$	0	0
$767$	−50.9117	−1.83831
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	2.00000i	0.0720282i
$772$	0	0
$773$	18.3848i	0.661254i	0.943761	+	0.330627i	$0.107260\pi$
$773$	18.3848i	0.661254i	−0.943761	+	0.330627i	$0.892740\pi$
$774$	0	0
$775$	8.48528	0.304800
$776$	0	0
$777$	−12.0000	−0.430498
$778$	0	0
$779$	32.0000i	1.14652i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1.41421	−0.0505399
$784$	0	0
$785$	30.0000	1.07075
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	0	0
$789$	− 28.2843i	− 1.00695i
$790$	0	0
$791$	−50.9117	−1.81021
$792$	0	0
$793$	−6.00000	−0.213066
$794$	0	0
$795$	− 10.0000i	− 0.354663i
$796$	0	0
$797$	− 49.4975i	− 1.75329i	−0.481137	−	0.876645i	$-0.659776\pi$
$797$	− 49.4975i	− 1.75329i	0.481137	−	0.876645i	$-0.340224\pi$
$798$	0	0
$799$	45.2548	1.60100
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	22.6274i	0.797512i
$806$	0	0
$807$	21.2132	0.746740
$808$	0	0
$809$	−4.00000	−0.140633	−0.0703163	−	0.997525i	$-0.522401\pi$
$809$	−4.00000	−0.140633	−0.0703163	+	0.997525i	$0.522401\pi$
$810$	0	0
$811$	40.0000i	1.40459i	0.711886	+	0.702295i	$0.247841\pi$
$811$	40.0000i	1.40459i	−0.711886	+	0.702295i	$0.752159\pi$
$812$	0	0
$813$	− 8.48528i	− 0.297592i
$814$	0	0
$815$	22.6274	0.792604
$816$	0	0
$817$	0	0
$818$	0	0
$819$	12.0000i	0.419314i
$820$	0	0
$821$	− 7.07107i	− 0.246782i	−0.992358	−	0.123391i	$-0.960623\pi$
$821$	− 7.07107i	− 0.246782i	0.992358	−	0.123391i	$-0.0393769\pi$
$822$	0	0
$823$	−25.4558	−0.887335	−0.443667	−	0.896191i	$-0.646323\pi$
$823$	−25.4558	−0.887335	−0.443667	+	0.896191i	$0.646323\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	0	0
$829$	− 52.3259i	− 1.81735i	−0.417500	−	0.908677i	$-0.637094\pi$
$829$	− 52.3259i	− 1.81735i	0.417500	−	0.908677i	$-0.362906\pi$
$830$	0	0
$831$	−1.41421	−0.0490585
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	− 16.0000i	− 0.553703i
$836$	0	0
$837$	− 2.82843i	− 0.0977647i
$838$	0	0
$839$	45.2548	1.56237	0.781185	−	0.624299i	$-0.214615\pi$
$839$	45.2548	1.56237	0.781185	+	0.624299i	$0.214615\pi$
$840$	0	0
$841$	27.0000	0.931034
$842$	0	0
$843$	22.0000i	0.757720i
$844$	0	0
$845$	7.07107i	0.243252i
$846$	0	0
$847$	31.1127	1.06904
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	− 24.0000i	− 0.822709i
$852$	0	0
$853$	− 15.5563i	− 0.532639i	−0.963885	−	0.266320i	$-0.914192\pi$
$853$	− 15.5563i	− 0.532639i	0.963885	−	0.266320i	$-0.0858077\pi$
$854$	0	0
$855$	−11.3137	−0.386921
$856$	0	0
$857$	20.0000	0.683187	0.341593	−	0.939848i	$-0.389033\pi$
$857$	20.0000	0.683187	0.341593	+	0.939848i	$0.389033\pi$
$858$	0	0
$859$	− 8.00000i	− 0.272956i	−0.990643	−	0.136478i	$-0.956422\pi$
$859$	− 8.00000i	− 0.272956i	0.990643	−	0.136478i	$-0.0435784\pi$
$860$	0	0
$861$	11.3137i	0.385570i
$862$	0	0
$863$	5.65685	0.192562	0.0962808	−	0.995354i	$-0.469305\pi$
$863$	5.65685	0.192562	0.0962808	+	0.995354i	$0.469305\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	− 1.00000i	− 0.0339618i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−16.9706	−0.575026
$872$	0	0
$873$	−16.0000	−0.541518
$874$	0	0
$875$	− 32.0000i	− 1.08180i
$876$	0	0
$877$	32.5269i	1.09836i	0.835705	+	0.549178i	$0.185059\pi$
$877$	32.5269i	1.09836i	−0.835705	+	0.549178i	$0.814941\pi$
$878$	0	0
$879$	18.3848	0.620103
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	− 24.0000i	− 0.807664i	−0.914833	−	0.403832i	$-0.867678\pi$
$883$	− 24.0000i	− 0.807664i	0.914833	−	0.403832i	$-0.132322\pi$
$884$	0	0
$885$	− 16.9706i	− 0.570459i
$886$	0	0
$887$	−16.9706	−0.569816	−0.284908	−	0.958555i	$-0.591963\pi$
$887$	−16.9706	−0.569816	−0.284908	+	0.958555i	$0.591963\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	0	0
$892$	0	0
$893$	90.5097i	3.02879i
$894$	0	0
$895$	−5.65685	−0.189088
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	− 4.00000i	− 0.133407i
$900$	0	0
$901$	− 28.2843i	− 0.942286i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.0000	−0.598340
$906$	0	0
$907$	− 32.0000i	− 1.06254i	−0.847202	−	0.531271i	$-0.821714\pi$
$907$	− 32.0000i	− 1.06254i	0.847202	−	0.531271i	$-0.178286\pi$
$908$	0	0
$909$	15.5563i	0.515972i
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 2.00000i	− 0.0661180i
$916$	0	0
$917$	11.3137i	0.373612i
$918$	0	0
$919$	−2.82843	−0.0933012	−0.0466506	−	0.998911i	$-0.514855\pi$
$919$	−2.82843	−0.0933012	−0.0466506	+	0.998911i	$0.514855\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	− 48.0000i	− 1.57994i
$924$	0	0
$925$	12.7279i	0.418491i
$926$	0	0
$927$	−19.7990	−0.650284
$928$	0	0
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	0	0
$931$	8.00000i	0.262189i
$932$	0	0
$933$	28.2843i	0.925985i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	16.0000i	0.522140i
$940$	0	0
$941$	52.3259i	1.70578i	0.522094	+	0.852888i	$0.325151\pi$
$941$	52.3259i	1.70578i	−0.522094	+	0.852888i	$0.674849\pi$
$942$	0	0
$943$	−22.6274	−0.736850
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	0	0
$949$	59.3970i	1.92811i
$950$	0	0
$951$	−21.2132	−0.687885
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	8.00000i	0.258874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	33.9411	1.09602
$960$	0	0
$961$	−23.0000	−0.741935
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−42.4264	−1.36434	−0.682171	−	0.731193i	$-0.738964\pi$
$967$	−42.4264	−1.36434	−0.682171	+	0.731193i	$0.738964\pi$
$968$	0	0
$969$	−32.0000	−1.02799
$970$	0	0
$971$	24.0000i	0.770197i	0.922876	+	0.385098i	$0.125832\pi$
$971$	24.0000i	0.770197i	−0.922876	+	0.385098i	$0.874168\pi$
$972$	0	0
$973$	− 11.3137i	− 0.362701i
$974$	0	0
$975$	12.7279	0.407620
$976$	0	0
$977$	−36.0000	−1.15174	−0.575871	−	0.817541i	$-0.695337\pi$
$977$	−36.0000	−1.15174	−0.575871	+	0.817541i	$0.695337\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 4.24264i	− 0.135457i
$982$	0	0
$983$	−16.9706	−0.541277	−0.270638	−	0.962681i	$-0.587235\pi$
$983$	−16.9706	−0.541277	−0.270638	+	0.962681i	$0.587235\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	32.0000i	1.01857i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	25.4558	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$991$	25.4558	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	12.0000i	0.380426i
$996$	0	0
$997$	− 7.07107i	− 0.223943i	−0.993711	−	0.111971i	$-0.964283\pi$
$997$	− 7.07107i	− 0.223943i	0.993711	−	0.111971i	$-0.0357165\pi$
$998$	0	0
$999$	4.24264	0.134231

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.d.c.1537.3		4
4.3	odd	2	inner	3072.2.d.c.1537.1		4
8.3	odd	2	inner	3072.2.d.c.1537.4		4
8.5	even	2	inner	3072.2.d.c.1537.2		4
16.3	odd	4		3072.2.a.h.1.1		2
16.5	even	4		3072.2.a.h.1.2		2
16.11	odd	4		3072.2.a.b.1.2		2
16.13	even	4		3072.2.a.b.1.1		2
32.3	odd	8		1536.2.j.c.385.2	yes	4
32.5	even	8		1536.2.j.c.1153.1	yes	4
32.11	odd	8		1536.2.j.b.1153.1	yes	4
32.13	even	8		1536.2.j.b.385.2	yes	4
32.19	odd	8		1536.2.j.b.385.1	✓	4
32.21	even	8		1536.2.j.b.1153.2	yes	4
32.27	odd	8		1536.2.j.c.1153.2	yes	4
32.29	even	8		1536.2.j.c.385.1	yes	4
48.5	odd	4		9216.2.a.i.1.1		2
48.11	even	4		9216.2.a.h.1.1		2
48.29	odd	4		9216.2.a.h.1.2		2
48.35	even	4		9216.2.a.i.1.2		2
96.5	odd	8		4608.2.k.y.1153.2		4
96.11	even	8		4608.2.k.bb.1153.1		4
96.29	odd	8		4608.2.k.y.3457.1		4
96.35	even	8		4608.2.k.y.3457.2		4
96.53	odd	8		4608.2.k.bb.1153.2		4
96.59	even	8		4608.2.k.y.1153.1		4
96.77	odd	8		4608.2.k.bb.3457.1		4
96.83	even	8		4608.2.k.bb.3457.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.b.385.1	✓	4	32.19	odd	8
1536.2.j.b.385.2	yes	4	32.13	even	8
1536.2.j.b.1153.1	yes	4	32.11	odd	8
1536.2.j.b.1153.2	yes	4	32.21	even	8
1536.2.j.c.385.1	yes	4	32.29	even	8
1536.2.j.c.385.2	yes	4	32.3	odd	8
1536.2.j.c.1153.1	yes	4	32.5	even	8
1536.2.j.c.1153.2	yes	4	32.27	odd	8
3072.2.a.b.1.1		2	16.13	even	4
3072.2.a.b.1.2		2	16.11	odd	4
3072.2.a.h.1.1		2	16.3	odd	4
3072.2.a.h.1.2		2	16.5	even	4
3072.2.d.c.1537.1		4	4.3	odd	2	inner
3072.2.d.c.1537.2		4	8.5	even	2	inner
3072.2.d.c.1537.3		4	1.1	even	1	trivial
3072.2.d.c.1537.4		4	8.3	odd	2	inner
4608.2.k.y.1153.1		4	96.59	even	8
4608.2.k.y.1153.2		4	96.5	odd	8
4608.2.k.y.3457.1		4	96.29	odd	8
4608.2.k.y.3457.2		4	96.35	even	8
4608.2.k.bb.1153.1		4	96.11	even	8
4608.2.k.bb.1153.2		4	96.53	odd	8
4608.2.k.bb.3457.1		4	96.77	odd	8
4608.2.k.bb.3457.2		4	96.83	even	8
9216.2.a.h.1.1		2	48.11	even	4
9216.2.a.h.1.2		2	48.29	odd	4
9216.2.a.i.1.1		2	48.5	odd	4
9216.2.a.i.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.41421i q^{5} +2.82843 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -1.41421i q^{5} +2.82843 q^{7} -1.00000 q^{9} -4.24264i q^{13} +1.41421 q^{15} +4.00000 q^{17} +8.00000i q^{19} +2.82843i q^{21} -5.65685 q^{23} +3.00000 q^{25} -1.00000i q^{27} -1.41421i q^{29} +2.82843 q^{31} -4.00000i q^{35} +4.24264i q^{37} +4.24264 q^{39} +4.00000 q^{41} +1.41421i q^{45} +11.3137 q^{47} +1.00000 q^{49} +4.00000i q^{51} -7.07107i q^{53} -8.00000 q^{57} -12.0000i q^{59} -1.41421i q^{61} -2.82843 q^{63} -6.00000 q^{65} -4.00000i q^{67} -5.65685i q^{69} +11.3137 q^{71} -14.0000 q^{73} +3.00000i q^{75} +8.48528 q^{79} +1.00000 q^{81} +16.0000i q^{83} -5.65685i q^{85} +1.41421 q^{87} -6.00000 q^{89} -12.0000i q^{91} +2.82843i q^{93} +11.3137 q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 16 q^{17} + 12 q^{25} + 16 q^{41} + 4 q^{49} - 32 q^{57} - 24 q^{65} - 56 q^{73} + 4 q^{81} - 24 q^{89} + 64 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 + 16 * q^17 + 12 * q^25 + 16 * q^41 + 4 * q^49 - 32 * q^57 - 24 * q^65 - 56 * q^73 + 4 * q^81 - 24 * q^89 + 64 * q^97

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