LMFDB - Embedded newform 3024.2.h.f.2591.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3024,2,Mod(2591,3024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3024.2591"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,12,0,4,0,0,0,0,0,0,0,0,0,36,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.195105024.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 $ x^8 - 4x^7 + 5x^6 + 4x^5 - 20x^4 + 12x^3 + 45x^2 - 108*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.3
Root			$0.560908 - 1.63871i$ of defining polynomial
Character	$\chi$	$=$	3024.2591
Dual form			3024.2.h.f.2591.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27743i q^{5} -1.00000i q^{7} +4.39924 q^{11} +1.78901 q^{13} +2.33438i q^{17} +0.155611i q^{19} +1.72257 q^{23} +3.36818 q^{25} -0.698256i q^{29} +2.67667i q^{31} -1.27743 q^{35} -2.05696 q^{37} -6.61181i q^{41} +11.6213i q^{43} +6.09865 q^{47} -1.00000 q^{49} -2.31797i q^{53} -5.61971i q^{55} +7.48842 q^{59} -12.9731 q^{61} -2.28533i q^{65} -0.480518i q^{67} +6.14771 q^{71} +5.88924 q^{73} -4.39924i q^{77} -1.79059i q^{79} +5.65771 q^{83} +2.98201 q^{85} +4.27743i q^{89} -1.78901i q^{91} +0.198782 q^{95} -9.08381 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{11} + 4 q^{13} + 36 q^{23} - 12 q^{25} + 12 q^{35} - 20 q^{37} + 24 q^{47} - 8 q^{49} + 48 q^{59} + 36 q^{61} + 36 q^{71} - 16 q^{73} + 72 q^{83} + 28 q^{85} + 84 q^{95} + 4 q^{97}+O(q^{100}) $ 8 * q + 12 * q^11 + 4 * q^13 + 36 * q^23 - 12 * q^25 + 12 * q^35 - 20 * q^37 + 24 * q^47 - 8 * q^49 + 48 * q^59 + 36 * q^61 + 36 * q^71 - 16 * q^73 + 72 * q^83 + 28 * q^85 + 84 * q^95 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 1.27743i	− 0.571283i	−0.958337	−	0.285641i	$-0.907793\pi$
$5$	− 1.27743i	− 0.571283i	0.958337	−	0.285641i	$-0.0922066\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.39924	1.32642	0.663211	−	0.748433i	$-0.269193\pi$
$11$	4.39924	1.32642	0.663211	+	0.748433i	$0.269193\pi$
$12$	0	0
$13$	1.78901	0.496182	0.248091	−	0.968737i	$-0.420197\pi$
$13$	1.78901	0.496182	0.248091	+	0.968737i	$0.420197\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.33438i	0.566171i	0.959095	+	0.283086i	$0.0913581\pi$
$17$	2.33438i	0.566171i	−0.959095	+	0.283086i	$0.908642\pi$
$18$	0	0
$19$	0.155611i	0.0356996i	0.999841	+	0.0178498i	$0.00568208\pi$
$19$	0.155611i	0.0356996i	−0.999841	+	0.0178498i	$0.994318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.72257	0.359181	0.179591	−	0.983741i	$-0.442523\pi$
$23$	1.72257	0.359181	0.179591	+	0.983741i	$0.442523\pi$
$24$	0	0
$25$	3.36818	0.673636
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.698256i	− 0.129663i	−0.997896	−	0.0648314i	$-0.979349\pi$
$29$	− 0.698256i	− 0.129663i	0.997896	−	0.0648314i	$-0.0206510\pi$
$30$	0	0
$31$	2.67667i	0.480744i	0.970681	+	0.240372i	$0.0772695\pi$
$31$	2.67667i	0.480744i	−0.970681	+	0.240372i	$0.922731\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.27743	−0.215925
$36$	0	0
$37$	−2.05696	−0.338162	−0.169081	−	0.985602i	$-0.554080\pi$
$37$	−2.05696	−0.338162	−0.169081	+	0.985602i	$0.554080\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.61181i	− 1.03259i	−0.856410	−	0.516296i	$-0.827311\pi$
$41$	− 6.61181i	− 1.03259i	0.856410	−	0.516296i	$-0.172689\pi$
$42$	0	0
$43$	11.6213i	1.77223i	0.463465	+	0.886115i	$0.346606\pi$
$43$	11.6213i	1.77223i	−0.463465	+	0.886115i	$0.653394\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.09865	0.889580	0.444790	−	0.895635i	$-0.353278\pi$
$47$	6.09865	0.889580	0.444790	+	0.895635i	$0.353278\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.31797i	− 0.318397i	−0.987247	−	0.159199i	$-0.949109\pi$
$53$	− 2.31797i	− 0.318397i	0.987247	−	0.159199i	$-0.0508911\pi$
$54$	0	0
$55$	− 5.61971i	− 0.757762i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.48842	0.974909	0.487455	−	0.873148i	$-0.337925\pi$
$59$	7.48842	0.974909	0.487455	+	0.873148i	$0.337925\pi$
$60$	0	0
$61$	−12.9731	−1.66103	−0.830515	−	0.556997i	$-0.811954\pi$
$61$	−12.9731	−1.66103	−0.830515	+	0.556997i	$0.811954\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.28533i	− 0.283460i
$66$	0	0
$67$	− 0.480518i	− 0.0587046i	−0.999569	−	0.0293523i	$-0.990656\pi$
$67$	− 0.480518i	− 0.0587046i	0.999569	−	0.0293523i	$-0.00934447\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.14771	0.729599	0.364799	−	0.931086i	$-0.381138\pi$
$71$	6.14771	0.729599	0.364799	+	0.931086i	$0.381138\pi$
$72$	0	0
$73$	5.88924	0.689283	0.344642	−	0.938734i	$-0.388000\pi$
$73$	5.88924	0.689283	0.344642	+	0.938734i	$0.388000\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.39924i	− 0.501340i
$78$	0	0
$79$	− 1.79059i	− 0.201457i	−0.994914	−	0.100728i	$-0.967883\pi$
$79$	− 1.79059i	− 0.201457i	0.994914	−	0.100728i	$-0.0321173\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.65771	0.621015	0.310507	−	0.950571i	$-0.399501\pi$
$83$	5.65771	0.621015	0.310507	+	0.950571i	$0.399501\pi$
$84$	0	0
$85$	2.98201	0.323444
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.27743i	0.453406i	0.973964	+	0.226703i	$0.0727947\pi$
$89$	4.27743i	0.453406i	−0.973964	+	0.226703i	$0.927205\pi$
$90$	0	0
$91$	− 1.78901i	− 0.187539i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.198782	0.0203946
$96$	0	0
$97$	−9.08381	−0.922322	−0.461161	−	0.887317i	$-0.652567\pi$
$97$	−9.08381	−0.922322	−0.461161	+	0.887317i	$0.652567\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 15.9050i	− 1.58261i	−0.611421	−	0.791305i	$-0.709402\pi$
$101$	− 15.9050i	− 1.58261i	0.611421	−	0.791305i	$-0.290598\pi$
$102$	0	0
$103$	− 9.87125i	− 0.972643i	−0.873780	−	0.486321i	$-0.838338\pi$
$103$	− 9.87125i	− 0.972643i	0.873780	−	0.486321i	$-0.161662\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.6846	1.22626	0.613132	−	0.789981i	$-0.289909\pi$
$107$	12.6846	1.22626	0.613132	+	0.789981i	$0.289909\pi$
$108$	0	0
$109$	0.985161	0.0943613	0.0471806	−	0.998886i	$-0.484976\pi$
$109$	0.985161	0.0943613	0.0471806	+	0.998886i	$0.484976\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12.9541i	− 1.21862i	−0.792932	−	0.609309i	$-0.791447\pi$
$113$	− 12.9541i	− 1.21862i	0.792932	−	0.609309i	$-0.208553\pi$
$114$	0	0
$115$	− 2.20046i	− 0.205194i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.33438	0.213993
$120$	0	0
$121$	8.35334	0.759395
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 10.6897i	− 0.956119i
$126$	0	0
$127$	5.36545i	0.476107i	0.971252	+	0.238053i	$0.0765093\pi$
$127$	5.36545i	0.476107i	−0.971252	+	0.238053i	$0.923491\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.96778	0.608778	0.304389	−	0.952548i	$-0.401548\pi$
$131$	6.96778	0.608778	0.304389	+	0.952548i	$0.401548\pi$
$132$	0	0
$133$	0.155611	0.0134932
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.2853i	1.22048i	0.792217	+	0.610239i	$0.208927\pi$
$137$	14.2853i	1.22048i	−0.792217	+	0.610239i	$0.791073\pi$
$138$	0	0
$139$	− 7.73363i	− 0.655958i	−0.944685	−	0.327979i	$-0.893633\pi$
$139$	− 7.73363i	− 0.655958i	0.944685	−	0.327979i	$-0.106367\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7.87028	0.658146
$144$	0	0
$145$	−0.891971	−0.0740741
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.08127i	0.170505i	0.996359	+	0.0852523i	$0.0271696\pi$
$149$	2.08127i	0.170505i	−0.996359	+	0.0852523i	$0.972830\pi$
$150$	0	0
$151$	1.90135i	0.154729i	0.997003	+	0.0773647i	$0.0246506\pi$
$151$	1.90135i	0.154729i	−0.997003	+	0.0773647i	$0.975349\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.41925	0.274641
$156$	0	0
$157$	12.9145	1.03069	0.515345	−	0.856983i	$-0.327664\pi$
$157$	12.9145	1.03069	0.515345	+	0.856983i	$0.327664\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.72257i	− 0.135758i
$162$	0	0
$163$	− 15.9294i	− 1.24768i	−0.781551	−	0.623842i	$-0.785571\pi$
$163$	− 15.9294i	− 1.24768i	0.781551	−	0.623842i	$-0.214429\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.50484	0.348595	0.174297	−	0.984693i	$-0.444235\pi$
$167$	4.50484	0.348595	0.174297	+	0.984693i	$0.444235\pi$
$168$	0	0
$169$	−9.79945	−0.753804
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 7.85641i	− 0.597312i	−0.954361	−	0.298656i	$-0.903462\pi$
$173$	− 7.85641i	− 0.597312i	0.954361	−	0.298656i	$-0.0965382\pi$
$174$	0	0
$175$	− 3.36818i	− 0.254610i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.9499	1.64061	0.820306	−	0.571925i	$-0.193803\pi$
$179$	21.9499	1.64061	0.820306	+	0.571925i	$0.193803\pi$
$180$	0	0
$181$	−0.0585348	−0.00435086	−0.00217543	−	0.999998i	$-0.500692\pi$
$181$	−0.0585348	−0.00435086	−0.00217543	+	0.999998i	$0.500692\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.62761i	0.193186i
$186$	0	0
$187$	10.2695i	0.750982i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.42514	0.320192	0.160096	−	0.987101i	$-0.448820\pi$
$191$	4.42514	0.320192	0.160096	+	0.987101i	$0.448820\pi$
$192$	0	0
$193$	19.1676	1.37972	0.689858	−	0.723945i	$-0.257673\pi$
$193$	19.1676	1.37972	0.689858	+	0.723945i	$0.257673\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.2843i	− 1.51644i	−0.651998	−	0.758221i	$-0.726069\pi$
$197$	− 21.2843i	− 1.51644i	0.651998	−	0.758221i	$-0.273931\pi$
$198$	0	0
$199$	5.25426i	0.372465i	0.982506	+	0.186232i	$0.0596278\pi$
$199$	5.25426i	0.372465i	−0.982506	+	0.186232i	$0.940372\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.698256	−0.0490079
$204$	0	0
$205$	−8.44611	−0.589902
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.684571i	0.0473528i
$210$	0	0
$211$	− 2.67236i	− 0.183973i	−0.995760	−	0.0919865i	$-0.970678\pi$
$211$	− 2.67236i	− 0.183973i	0.995760	−	0.0919865i	$-0.0293217\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14.8454	1.01244
$216$	0	0
$217$	2.67667	0.181704
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.17623i	0.280924i
$222$	0	0
$223$	18.3561i	1.22921i	0.788834	+	0.614607i	$0.210685\pi$
$223$	18.3561i	1.22921i	−0.788834	+	0.614607i	$0.789315\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.5605	−1.43102	−0.715512	−	0.698601i	$-0.753806\pi$
$227$	−21.5605	−1.43102	−0.715512	+	0.698601i	$0.753806\pi$
$228$	0	0
$229$	−8.62244	−0.569787	−0.284894	−	0.958559i	$-0.591958\pi$
$229$	−8.62244	−0.569787	−0.284894	+	0.958559i	$0.591958\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.8402i	0.710164i	0.934835	+	0.355082i	$0.115547\pi$
$233$	10.8402i	0.710164i	−0.934835	+	0.355082i	$0.884453\pi$
$234$	0	0
$235$	− 7.79059i	− 0.508202i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.53211	−0.422527	−0.211264	−	0.977429i	$-0.567758\pi$
$239$	−6.53211	−0.422527	−0.211264	+	0.977429i	$0.567758\pi$
$240$	0	0
$241$	−25.5900	−1.64840	−0.824200	−	0.566300i	$-0.808375\pi$
$241$	−25.5900	−1.64840	−0.824200	+	0.566300i	$0.808375\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.27743i	0.0816118i
$246$	0	0
$247$	0.278390i	0.0177135i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.8771	−1.00216	−0.501078	−	0.865402i	$-0.667063\pi$
$251$	−15.8771	−1.00216	−0.501078	+	0.865402i	$0.667063\pi$
$252$	0	0
$253$	7.57802	0.476426
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.2236i	1.19914i	0.800324	+	0.599568i	$0.204661\pi$
$257$	19.2236i	1.19914i	−0.800324	+	0.599568i	$0.795339\pi$
$258$	0	0
$259$	2.05696i	0.127813i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	25.7535	1.58803	0.794016	−	0.607897i	$-0.207987\pi$
$263$	25.7535	1.58803	0.794016	+	0.607897i	$0.207987\pi$
$264$	0	0
$265$	−2.96104	−0.181895
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 19.1180i	− 1.16565i	−0.812599	−	0.582823i	$-0.801948\pi$
$269$	− 19.1180i	− 1.16565i	0.812599	−	0.582823i	$-0.198052\pi$
$270$	0	0
$271$	23.5358i	1.42970i	0.699279	+	0.714849i	$0.253505\pi$
$271$	23.5358i	1.42970i	−0.699279	+	0.714849i	$0.746495\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	14.8174	0.893525
$276$	0	0
$277$	−28.8416	−1.73292	−0.866461	−	0.499245i	$-0.833611\pi$
$277$	−28.8416	−1.73292	−0.866461	+	0.499245i	$0.833611\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.9693i	1.37023i	0.728434	+	0.685116i	$0.240248\pi$
$281$	22.9693i	1.37023i	−0.728434	+	0.685116i	$0.759752\pi$
$282$	0	0
$283$	− 6.69151i	− 0.397769i	−0.980023	−	0.198884i	$-0.936268\pi$
$283$	− 6.69151i	− 0.397769i	0.980023	−	0.198884i	$-0.0637319\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.61181	−0.390283
$288$	0	0
$289$	11.5506	0.679450
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 20.6276i	− 1.20508i	−0.798089	−	0.602539i	$-0.794156\pi$
$293$	− 20.6276i	− 1.20508i	0.798089	−	0.602539i	$-0.205844\pi$
$294$	0	0
$295$	− 9.56591i	− 0.556949i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.08170	0.178219
$300$	0	0
$301$	11.6213	0.669840
$302$	0	0
$303$	0	0
$304$	0	0
$305$	16.5721i	0.948917i
$306$	0	0
$307$	6.28154i	0.358507i	0.983803	+	0.179253i	$0.0573682\pi$
$307$	6.28154i	0.358507i	−0.983803	+	0.179253i	$0.942632\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.745313	−0.0422628	−0.0211314	−	0.999777i	$-0.506727\pi$
$311$	−0.745313	−0.0422628	−0.0211314	+	0.999777i	$0.506727\pi$
$312$	0	0
$313$	3.91892	0.221510	0.110755	−	0.993848i	$-0.464673\pi$
$313$	3.91892	0.221510	0.110755	+	0.993848i	$0.464673\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 25.7488i	− 1.44620i	−0.690745	−	0.723098i	$-0.742717\pi$
$317$	− 25.7488i	− 1.44620i	0.690745	−	0.723098i	$-0.257283\pi$
$318$	0	0
$319$	− 3.07180i	− 0.171988i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.363256	−0.0202121
$324$	0	0
$325$	6.02570	0.334246
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 6.09865i	− 0.336230i
$330$	0	0
$331$	24.4684i	1.34491i	0.740140	+	0.672453i	$0.234759\pi$
$331$	24.4684i	1.34491i	−0.740140	+	0.672453i	$0.765241\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.613827	−0.0335369
$336$	0	0
$337$	−4.26910	−0.232553	−0.116276	−	0.993217i	$-0.537096\pi$
$337$	−4.26910	−0.232553	−0.116276	+	0.993217i	$0.537096\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11.7753i	0.637670i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.54792	0.136779	0.0683896	−	0.997659i	$-0.478214\pi$
$347$	2.54792	0.136779	0.0683896	+	0.997659i	$0.478214\pi$
$348$	0	0
$349$	−2.02570	−0.108433	−0.0542167	−	0.998529i	$-0.517266\pi$
$349$	−2.02570	−0.108433	−0.0542167	+	0.998529i	$0.517266\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 29.1329i	− 1.55059i	−0.631602	−	0.775293i	$-0.717602\pi$
$353$	− 29.1329i	− 1.55059i	0.631602	−	0.775293i	$-0.282398\pi$
$354$	0	0
$355$	− 7.85325i	− 0.416807i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.0517	−1.53329	−0.766645	−	0.642071i	$-0.778075\pi$
$359$	−29.0517	−1.53329	−0.766645	+	0.642071i	$0.778075\pi$
$360$	0	0
$361$	18.9758	0.998726
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 7.52307i	− 0.393776i
$366$	0	0
$367$	3.79332i	0.198010i	0.995087	+	0.0990048i	$0.0315659\pi$
$367$	3.79332i	0.198010i	−0.995087	+	0.0990048i	$0.968434\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.31797	−0.120343
$372$	0	0
$373$	27.8026	1.43956	0.719782	−	0.694200i	$-0.244242\pi$
$373$	27.8026	1.43956	0.719782	+	0.694200i	$0.244242\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.24919i	− 0.0643363i
$378$	0	0
$379$	− 15.7874i	− 0.810946i	−0.914107	−	0.405473i	$-0.867107\pi$
$379$	− 15.7874i	− 0.810946i	0.914107	−	0.405473i	$-0.132893\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.69247	0.393067	0.196533	−	0.980497i	$-0.437032\pi$
$383$	7.69247	0.393067	0.196533	+	0.980497i	$0.437032\pi$
$384$	0	0
$385$	−5.61971	−0.286407
$386$	0	0
$387$	0	0
$388$	0	0
$389$	38.8433i	1.96944i	0.174158	+	0.984718i	$0.444280\pi$
$389$	38.8433i	1.96944i	−0.174158	+	0.984718i	$0.555720\pi$
$390$	0	0
$391$	4.02115i	0.203358i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.28734	−0.115089
$396$	0	0
$397$	16.7364	0.839974	0.419987	−	0.907530i	$-0.362035\pi$
$397$	16.7364	0.839974	0.419987	+	0.907530i	$0.362035\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.0769i	1.15240i	0.817307	+	0.576202i	$0.195466\pi$
$401$	23.0769i	1.15240i	−0.817307	+	0.576202i	$0.804534\pi$
$402$	0	0
$403$	4.78859i	0.238536i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.04906	−0.448545
$408$	0	0
$409$	10.1950	0.504110	0.252055	−	0.967713i	$-0.418894\pi$
$409$	10.1950	0.504110	0.252055	+	0.967713i	$0.418894\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 7.48842i	− 0.368481i
$414$	0	0
$415$	− 7.22732i	− 0.354775i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.77522	−0.184432	−0.0922158	−	0.995739i	$-0.529395\pi$
$419$	−3.77522	−0.184432	−0.0922158	+	0.995739i	$0.529395\pi$
$420$	0	0
$421$	−16.7035	−0.814080	−0.407040	−	0.913410i	$-0.633439\pi$
$421$	−16.7035	−0.814080	−0.407040	+	0.913410i	$0.633439\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.86263i	0.381393i
$426$	0	0
$427$	12.9731i	0.627810i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.58453	−0.220829	−0.110415	−	0.993886i	$-0.535218\pi$
$431$	−4.58453	−0.220829	−0.110415	+	0.993886i	$0.535218\pi$
$432$	0	0
$433$	7.92547	0.380874	0.190437	−	0.981699i	$-0.439010\pi$
$433$	7.92547	0.380874	0.190437	+	0.981699i	$0.439010\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.268051i	0.0128226i
$438$	0	0
$439$	12.6021i	0.601467i	0.953708	+	0.300734i	$0.0972315\pi$
$439$	12.6021i	0.601467i	−0.953708	+	0.300734i	$0.902768\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.08066	0.0513437	0.0256719	−	0.999670i	$-0.491827\pi$
$443$	1.08066	0.0513437	0.0256719	+	0.999670i	$0.491827\pi$
$444$	0	0
$445$	5.46410	0.259023
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 4.15561i	− 0.196115i	−0.995181	−	0.0980577i	$-0.968737\pi$
$449$	− 4.15561i	− 0.196115i	0.995181	−	0.0980577i	$-0.0312630\pi$
$450$	0	0
$451$	− 29.0870i	− 1.36965i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.28533	−0.107138
$456$	0	0
$457$	−17.2246	−0.805732	−0.402866	−	0.915259i	$-0.631986\pi$
$457$	−17.2246	−0.805732	−0.402866	+	0.915259i	$0.631986\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 33.6266i	− 1.56615i	−0.621930	−	0.783073i	$-0.713651\pi$
$461$	− 33.6266i	− 1.56615i	0.621930	−	0.783073i	$-0.286349\pi$
$462$	0	0
$463$	− 36.4941i	− 1.69603i	−0.529976	−	0.848013i	$-0.677799\pi$
$463$	− 36.4941i	− 1.69603i	0.529976	−	0.848013i	$-0.322201\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−37.5380	−1.73705	−0.868526	−	0.495644i	$-0.834932\pi$
$467$	−37.5380	−1.73705	−0.868526	+	0.495644i	$0.834932\pi$
$468$	0	0
$469$	−0.480518	−0.0221883
$470$	0	0
$471$	0	0
$472$	0	0
$473$	51.1249i	2.35072i
$474$	0	0
$475$	0.524126i	0.0240486i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−19.3223	−0.882857	−0.441429	−	0.897296i	$-0.645528\pi$
$479$	−19.3223	−0.882857	−0.441429	+	0.897296i	$0.645528\pi$
$480$	0	0
$481$	−3.67991	−0.167790
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.6039i	0.526906i
$486$	0	0
$487$	− 11.0093i	− 0.498878i	−0.968390	−	0.249439i	$-0.919754\pi$
$487$	− 11.0093i	− 0.498878i	0.968390	−	0.249439i	$-0.0802463\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15.7147	−0.709193	−0.354597	−	0.935019i	$-0.615382\pi$
$491$	−15.7147	−0.709193	−0.354597	+	0.935019i	$0.615382\pi$
$492$	0	0
$493$	1.63000	0.0734114
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.14771i	− 0.275762i
$498$	0	0
$499$	18.7156i	0.837827i	0.908026	+	0.418913i	$0.137589\pi$
$499$	18.7156i	0.837827i	−0.908026	+	0.418913i	$0.862411\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−27.9614	−1.24674	−0.623368	−	0.781929i	$-0.714236\pi$
$503$	−27.9614	−1.24674	−0.623368	+	0.781929i	$0.714236\pi$
$504$	0	0
$505$	−20.3175	−0.904118
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3.00727i	− 0.133295i	−0.997777	−	0.0666475i	$-0.978770\pi$
$509$	− 3.00727i	− 0.133295i	0.997777	−	0.0666475i	$-0.0212303\pi$
$510$	0	0
$511$	− 5.88924i	− 0.260525i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.6098	−0.555654
$516$	0	0
$517$	26.8295	1.17996
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.26872i	0.0993944i	0.998764	+	0.0496972i	$0.0158256\pi$
$521$	2.26872i	0.0993944i	−0.998764	+	0.0496972i	$0.984174\pi$
$522$	0	0
$523$	3.38957i	0.148216i	0.997250	+	0.0741078i	$0.0236109\pi$
$523$	3.38957i	0.148216i	−0.997250	+	0.0741078i	$0.976389\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.24838	−0.272184
$528$	0	0
$529$	−20.0327	−0.870989
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 11.8286i	− 0.512353i
$534$	0	0
$535$	− 16.2036i	− 0.700543i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.39924	−0.189489
$540$	0	0
$541$	−9.32922	−0.401094	−0.200547	−	0.979684i	$-0.564272\pi$
$541$	−9.32922	−0.401094	−0.200547	+	0.979684i	$0.564272\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1.25847i	− 0.0539070i
$546$	0	0
$547$	16.7453i	0.715978i	0.933726	+	0.357989i	$0.116537\pi$
$547$	16.7453i	0.715978i	−0.933726	+	0.357989i	$0.883463\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.108656	0.00462892
$552$	0	0
$553$	−1.79059	−0.0761434
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.2369i	1.02695i	0.858105	+	0.513475i	$0.171642\pi$
$557$	24.2369i	1.02695i	−0.858105	+	0.513475i	$0.828358\pi$
$558$	0	0
$559$	20.7906i	0.879348i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.0405	−1.13962	−0.569810	−	0.821777i	$-0.692983\pi$
$563$	−27.0405	−1.13962	−0.569810	+	0.821777i	$0.692983\pi$
$564$	0	0
$565$	−16.5479	−0.696176
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.9406i	1.59055i	0.606246	+	0.795277i	$0.292675\pi$
$569$	37.9406i	1.59055i	−0.606246	+	0.795277i	$0.707325\pi$
$570$	0	0
$571$	12.3194i	0.515549i	0.966205	+	0.257774i	$0.0829891\pi$
$571$	12.3194i	0.515549i	−0.966205	+	0.257774i	$0.917011\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.80194	0.241957
$576$	0	0
$577$	−19.7785	−0.823389	−0.411694	−	0.911322i	$-0.635063\pi$
$577$	−19.7785	−0.823389	−0.411694	+	0.911322i	$0.635063\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 5.65771i	− 0.234722i
$582$	0	0
$583$	− 10.1973i	− 0.422329i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.3148	−0.425738	−0.212869	−	0.977081i	$-0.568281\pi$
$587$	−10.3148	−0.425738	−0.212869	+	0.977081i	$0.568281\pi$
$588$	0	0
$589$	−0.416520	−0.0171624
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25.7373i	1.05691i	0.848963	+	0.528453i	$0.177228\pi$
$593$	25.7373i	1.05691i	−0.848963	+	0.528453i	$0.822772\pi$
$594$	0	0
$595$	− 2.98201i	− 0.122250i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	28.5899	1.16815	0.584077	−	0.811698i	$-0.301457\pi$
$599$	28.5899	1.16815	0.584077	+	0.811698i	$0.301457\pi$
$600$	0	0
$601$	−21.4730	−0.875901	−0.437950	−	0.898999i	$-0.644295\pi$
$601$	−21.4730	−0.875901	−0.437950	+	0.898999i	$0.644295\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 10.6708i	− 0.433829i
$606$	0	0
$607$	− 40.7312i	− 1.65323i	−0.562768	−	0.826615i	$-0.690264\pi$
$607$	− 40.7312i	− 1.65323i	0.562768	−	0.826615i	$-0.309736\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.9105	0.441393
$612$	0	0
$613$	−25.3260	−1.02291	−0.511453	−	0.859311i	$-0.670893\pi$
$613$	−25.3260	−1.02291	−0.511453	+	0.859311i	$0.670893\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.1580i	0.529719i	0.964287	+	0.264860i	$0.0853257\pi$
$617$	13.1580i	0.529719i	−0.964287	+	0.264860i	$0.914674\pi$
$618$	0	0
$619$	20.5839i	0.827337i	0.910428	+	0.413668i	$0.135753\pi$
$619$	20.5839i	0.827337i	−0.910428	+	0.413668i	$0.864247\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.27743	0.171372
$624$	0	0
$625$	3.18553	0.127421
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 4.80173i	− 0.191458i
$630$	0	0
$631$	12.4878i	0.497132i	0.968615	+	0.248566i	$0.0799592\pi$
$631$	12.4878i	0.497132i	−0.968615	+	0.248566i	$0.920041\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.85397	0.271991
$636$	0	0
$637$	−1.78901	−0.0708831
$638$	0	0
$639$	0	0
$640$	0	0
$641$	32.2853i	1.27519i	0.770370	+	0.637597i	$0.220071\pi$
$641$	32.2853i	1.27519i	−0.770370	+	0.637597i	$0.779929\pi$
$642$	0	0
$643$	− 17.3565i	− 0.684473i	−0.939614	−	0.342237i	$-0.888816\pi$
$643$	− 17.3565i	− 0.684473i	0.939614	−	0.342237i	$-0.111184\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.5970	−0.691808	−0.345904	−	0.938270i	$-0.612428\pi$
$647$	−17.5970	−0.691808	−0.345904	+	0.938270i	$0.612428\pi$
$648$	0	0
$649$	32.9434	1.29314
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 5.04424i	− 0.197396i	−0.995117	−	0.0986981i	$-0.968532\pi$
$653$	− 5.04424i	− 0.197396i	0.995117	−	0.0986981i	$-0.0314678\pi$
$654$	0	0
$655$	− 8.90083i	− 0.347784i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−33.7322	−1.31402	−0.657009	−	0.753883i	$-0.728179\pi$
$659$	−33.7322	−1.31402	−0.657009	+	0.753883i	$0.728179\pi$
$660$	0	0
$661$	−9.08697	−0.353442	−0.176721	−	0.984261i	$-0.556549\pi$
$661$	−9.08697	−0.353442	−0.176721	+	0.984261i	$0.556549\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 0.198782i	− 0.00770843i
$666$	0	0
$667$	− 1.20280i	− 0.0465725i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−57.0716	−2.20323
$672$	0	0
$673$	−2.21299	−0.0853045	−0.0426523	−	0.999090i	$-0.513581\pi$
$673$	−2.21299	−0.0853045	−0.0426523	+	0.999090i	$0.513581\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.9735i	1.45944i	0.683746	+	0.729720i	$0.260349\pi$
$677$	37.9735i	1.45944i	−0.683746	+	0.729720i	$0.739651\pi$
$678$	0	0
$679$	9.08381i	0.348605i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.0625	0.576352	0.288176	−	0.957577i	$-0.406951\pi$
$683$	15.0625	0.576352	0.288176	+	0.957577i	$0.406951\pi$
$684$	0	0
$685$	18.2485	0.697238
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 4.14687i	− 0.157983i
$690$	0	0
$691$	35.5627i	1.35287i	0.736503	+	0.676434i	$0.236476\pi$
$691$	35.5627i	1.35287i	−0.736503	+	0.676434i	$0.763524\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.87915	−0.374737
$696$	0	0
$697$	15.4345	0.584624
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 23.8391i	− 0.900392i	−0.892930	−	0.450196i	$-0.851354\pi$
$701$	− 23.8391i	− 0.900392i	0.892930	−	0.450196i	$-0.148646\pi$
$702$	0	0
$703$	− 0.320085i	− 0.0120723i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15.9050	−0.598171
$708$	0	0
$709$	−19.6053	−0.736292	−0.368146	−	0.929768i	$-0.620007\pi$
$709$	−19.6053	−0.736292	−0.368146	+	0.929768i	$0.620007\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.61076i	0.172674i
$714$	0	0
$715$	− 10.0537i	− 0.375988i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.4126	0.910438	0.455219	−	0.890379i	$-0.349561\pi$
$719$	24.4126	0.910438	0.455219	+	0.890379i	$0.349561\pi$
$720$	0	0
$721$	−9.87125	−0.367624
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.35185i	− 0.0873455i
$726$	0	0
$727$	− 4.67036i	− 0.173214i	−0.996243	−	0.0866071i	$-0.972398\pi$
$727$	− 4.67036i	− 0.173214i	0.996243	−	0.0866071i	$-0.0276025\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−27.1286	−1.00339
$732$	0	0
$733$	12.3938	0.457775	0.228888	−	0.973453i	$-0.426491\pi$
$733$	12.3938	0.457775	0.228888	+	0.973453i	$0.426491\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.11392i	− 0.0778671i
$738$	0	0
$739$	− 34.3618i	− 1.26402i	−0.774961	−	0.632009i	$-0.782230\pi$
$739$	− 34.3618i	− 1.26402i	0.774961	−	0.632009i	$-0.217770\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.2263	−1.21895	−0.609477	−	0.792804i	$-0.708620\pi$
$743$	−33.2263	−1.21895	−0.609477	+	0.792804i	$0.708620\pi$
$744$	0	0
$745$	2.65868	0.0974064
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 12.6846i	− 0.463484i
$750$	0	0
$751$	5.32366i	0.194263i	0.995272	+	0.0971316i	$0.0309668\pi$
$751$	5.32366i	0.194263i	−0.995272	+	0.0971316i	$0.969033\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.42883	0.0883942
$756$	0	0
$757$	47.7784	1.73654	0.868268	−	0.496096i	$-0.165234\pi$
$757$	47.7784	1.73654	0.868268	+	0.496096i	$0.165234\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.9333i	1.23008i	0.788496	+	0.615040i	$0.210860\pi$
$761$	33.9333i	1.23008i	−0.788496	+	0.615040i	$0.789140\pi$
$762$	0	0
$763$	− 0.985161i	− 0.0356652i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.3968	0.483732
$768$	0	0
$769$	4.54136	0.163766	0.0818828	−	0.996642i	$-0.473907\pi$
$769$	4.54136	0.163766	0.0818828	+	0.996642i	$0.473907\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 19.6720i	− 0.707554i	−0.935330	−	0.353777i	$-0.884897\pi$
$773$	− 19.6720i	− 0.707554i	0.935330	−	0.353777i	$-0.115103\pi$
$774$	0	0
$775$	9.01551i	0.323847i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.02887	0.0368631
$780$	0	0
$781$	27.0453	0.967756
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 16.4974i	− 0.588816i
$786$	0	0
$787$	42.0240i	1.49800i	0.662573	+	0.748998i	$0.269465\pi$
$787$	42.0240i	1.49800i	−0.662573	+	0.748998i	$0.730535\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.9541	−0.460595
$792$	0	0
$793$	−23.2089	−0.824172
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 17.5073i	− 0.620139i	−0.950714	−	0.310070i	$-0.899648\pi$
$797$	− 17.5073i	− 0.620139i	0.950714	−	0.310070i	$-0.100352\pi$
$798$	0	0
$799$	14.2366i	0.503655i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	25.9082	0.914280
$804$	0	0
$805$	−2.20046	−0.0775561
$806$	0	0
$807$	0	0
$808$	0	0
$809$	48.7499i	1.71395i	0.515355	+	0.856977i	$0.327660\pi$
$809$	48.7499i	1.71395i	−0.515355	+	0.856977i	$0.672340\pi$
$810$	0	0
$811$	− 46.3624i	− 1.62800i	−0.580862	−	0.814002i	$-0.697285\pi$
$811$	− 46.3624i	− 1.62800i	0.580862	−	0.814002i	$-0.302715\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−20.3486	−0.712780
$816$	0	0
$817$	−1.80840	−0.0632680
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.2030i	1.08899i	0.838763	+	0.544496i	$0.183279\pi$
$821$	31.2030i	1.08899i	−0.838763	+	0.544496i	$0.816721\pi$
$822$	0	0
$823$	54.0205i	1.88304i	0.336960	+	0.941519i	$0.390601\pi$
$823$	54.0205i	1.88304i	−0.336960	+	0.941519i	$0.609399\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.89197	0.170110	0.0850552	−	0.996376i	$-0.472893\pi$
$827$	4.89197	0.170110	0.0850552	+	0.996376i	$0.472893\pi$
$828$	0	0
$829$	−1.73363	−0.0602114	−0.0301057	−	0.999547i	$-0.509584\pi$
$829$	−1.73363	−0.0602114	−0.0301057	+	0.999547i	$0.509584\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.33438i	− 0.0808816i
$834$	0	0
$835$	− 5.75460i	− 0.199146i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−40.1014	−1.38445	−0.692227	−	0.721679i	$-0.743371\pi$
$839$	−40.1014	−1.38445	−0.692227	+	0.721679i	$0.743371\pi$
$840$	0	0
$841$	28.5124	0.983188
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.5181i	0.430635i
$846$	0	0
$847$	− 8.35334i	− 0.287024i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.54326	−0.121461
$852$	0	0
$853$	7.20826	0.246806	0.123403	−	0.992357i	$-0.460619\pi$
$853$	7.20826	0.246806	0.123403	+	0.992357i	$0.460619\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.2866i	1.13705i	0.822666	+	0.568525i	$0.192486\pi$
$857$	33.2866i	1.13705i	−0.822666	+	0.568525i	$0.807514\pi$
$858$	0	0
$859$	− 56.9226i	− 1.94217i	−0.238729	−	0.971086i	$-0.576731\pi$
$859$	− 56.9226i	− 1.94217i	0.238729	−	0.971086i	$-0.423269\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.00286	−0.170299	−0.0851497	−	0.996368i	$-0.527137\pi$
$863$	−5.00286	−0.170299	−0.0851497	+	0.996368i	$0.527137\pi$
$864$	0	0
$865$	−10.0360	−0.341234
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 7.87722i	− 0.267216i
$870$	0	0
$871$	− 0.859651i	− 0.0291282i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10.6897	−0.361379
$876$	0	0
$877$	−13.4640	−0.454647	−0.227324	−	0.973819i	$-0.572998\pi$
$877$	−13.4640	−0.454647	−0.227324	+	0.973819i	$0.572998\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 14.8565i	− 0.500528i	−0.968178	−	0.250264i	$-0.919483\pi$
$881$	− 14.8565i	− 0.500528i	0.968178	−	0.250264i	$-0.0805174\pi$
$882$	0	0
$883$	33.0315i	1.11160i	0.831317	+	0.555799i	$0.187588\pi$
$883$	33.0315i	1.11160i	−0.831317	+	0.555799i	$0.812412\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.27364	0.177072	0.0885358	−	0.996073i	$-0.471781\pi$
$887$	5.27364	0.177072	0.0885358	+	0.996073i	$0.471781\pi$
$888$	0	0
$889$	5.36545	0.179951
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.949018i	0.0317577i
$894$	0	0
$895$	− 28.0394i	− 0.937253i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.86900	0.0623347
$900$	0	0
$901$	5.41103	0.180268
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.0747739i	0.00248557i
$906$	0	0
$907$	− 5.87167i	− 0.194966i	−0.995237	−	0.0974828i	$-0.968921\pi$
$907$	− 5.87167i	− 0.194966i	0.995237	−	0.0974828i	$-0.0310791\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−55.8316	−1.84978	−0.924891	−	0.380232i	$-0.875844\pi$
$911$	−55.8316	−1.84978	−0.924891	+	0.380232i	$0.875844\pi$
$912$	0	0
$913$	24.8897	0.823728
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.96778i	− 0.230096i
$918$	0	0
$919$	0.100963i	0.00333045i	0.999999	+	0.00166522i	$0.000530058\pi$
$919$	0.100963i	0.00333045i	−0.999999	+	0.00166522i	$0.999470\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10.9983	0.362014
$924$	0	0
$925$	−6.92820	−0.227798
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.0245i	0.328893i	0.986386	+	0.164447i	$0.0525839\pi$
$929$	10.0245i	0.328893i	−0.986386	+	0.164447i	$0.947416\pi$
$930$	0	0
$931$	− 0.155611i	− 0.00509995i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	13.1186	0.429023
$936$	0	0
$937$	−5.58688	−0.182515	−0.0912577	−	0.995827i	$-0.529089\pi$
$937$	−5.58688	−0.182515	−0.0912577	+	0.995827i	$0.529089\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 52.3440i	− 1.70636i	−0.521613	−	0.853182i	$-0.674669\pi$
$941$	− 52.3440i	− 1.70636i	0.521613	−	0.853182i	$-0.325331\pi$
$942$	0	0
$943$	− 11.3893i	− 0.370888i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.0156	0.747906	0.373953	−	0.927448i	$-0.378002\pi$
$947$	23.0156	0.747906	0.373953	+	0.927448i	$0.378002\pi$
$948$	0	0
$949$	10.5359	0.342010
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 26.0063i	− 0.842427i	−0.906962	−	0.421213i	$-0.861604\pi$
$953$	− 26.0063i	− 0.842427i	0.906962	−	0.421213i	$-0.138396\pi$
$954$	0	0
$955$	− 5.65279i	− 0.182920i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.2853	0.461297
$960$	0	0
$961$	23.8354	0.768885
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 24.4852i	− 0.788208i
$966$	0	0
$967$	6.45251i	0.207499i	0.994603	+	0.103749i	$0.0330840\pi$
$967$	6.45251i	0.207499i	−0.994603	+	0.103749i	$0.966916\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−46.4608	−1.49100	−0.745500	−	0.666506i	$-0.767789\pi$
$971$	−46.4608	−1.49100	−0.745500	+	0.666506i	$0.767789\pi$
$972$	0	0
$973$	−7.73363	−0.247929
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.2796i	1.51261i	0.654219	+	0.756305i	$0.272997\pi$
$977$	47.2796i	1.51261i	−0.654219	+	0.756305i	$0.727003\pi$
$978$	0	0
$979$	18.8174i	0.601408i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	53.4899	1.70606	0.853032	−	0.521859i	$-0.174761\pi$
$983$	53.4899	1.70606	0.853032	+	0.521859i	$0.174761\pi$
$984$	0	0
$985$	−27.1891	−0.866317
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20.0185i	0.636552i
$990$	0	0
$991$	41.3741i	1.31429i	0.753764	+	0.657145i	$0.228236\pi$
$991$	41.3741i	1.31429i	−0.753764	+	0.657145i	$0.771764\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.71194	0.212783
$996$	0	0
$997$	−12.2196	−0.386999	−0.193499	−	0.981100i	$-0.561984\pi$
$997$	−12.2196	−0.386999	−0.193499	+	0.981100i	$0.561984\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.h.f.2591.3	yes	8
3.2	odd	2		3024.2.h.a.2591.6	yes	8
4.3	odd	2		3024.2.h.a.2591.3	✓	8
12.11	even	2	inner	3024.2.h.f.2591.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3024.2.h.a.2591.3	✓	8	4.3	odd	2
3024.2.h.a.2591.6	yes	8	3.2	odd	2
3024.2.h.f.2591.3	yes	8	1.1	even	1	trivial
3024.2.h.f.2591.6	yes	8	12.11	even	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.27743i q^{5} -1.00000i q^{7} +4.39924 q^{11} +1.78901 q^{13} +2.33438i q^{17} +0.155611i q^{19} +1.72257 q^{23} +3.36818 q^{25} -0.698256i q^{29} +2.67667i q^{31} -1.27743 q^{35} -2.05696 q^{37} -6.61181i q^{41} +11.6213i q^{43} +6.09865 q^{47} -1.00000 q^{49} -2.31797i q^{53} -5.61971i q^{55} +7.48842 q^{59} -12.9731 q^{61} -2.28533i q^{65} -0.480518i q^{67} +6.14771 q^{71} +5.88924 q^{73} -4.39924i q^{77} -1.79059i q^{79} +5.65771 q^{83} +2.98201 q^{85} +4.27743i q^{89} -1.78901i q^{91} +0.198782 q^{95} -9.08381 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{11} + 4 q^{13} + 36 q^{23} - 12 q^{25} + 12 q^{35} - 20 q^{37} + 24 q^{47} - 8 q^{49} + 48 q^{59} + 36 q^{61} + 36 q^{71} - 16 q^{73} + 72 q^{83} + 28 q^{85} + 84 q^{95} + 4 q^{97}+O(q^{100}) \) 8 * q + 12 * q^11 + 4 * q^13 + 36 * q^23 - 12 * q^25 + 12 * q^35 - 20 * q^37 + 24 * q^47 - 8 * q^49 + 48 * q^59 + 36 * q^61 + 36 * q^71 - 16 * q^73 + 72 * q^83 + 28 * q^85 + 84 * q^95 + 4 * q^97

Introduction

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