LMFDB - Embedded newform 6048.2.h.f.2591.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(2591,6048)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("6048.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.h (of order $2$, degree $1$, 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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.f.2591.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.34607i q^{5} -1.00000i q^{7} +O(q^{10})$	$q-3.34607i q^{5} -1.00000i q^{7} -0.896575 q^{11} +4.00000 q^{13} +6.31319i q^{17} +1.00000i q^{19} +7.20977 q^{23} -6.19615 q^{25} +7.72741i q^{29} +3.73205i q^{31} -3.34607 q^{35} -11.7321 q^{37} +7.86611i q^{41} +7.46410i q^{43} +8.76268 q^{47} -1.00000 q^{49} +2.82843i q^{53} +3.00000i q^{55} -3.86370 q^{59} -12.3923 q^{61} -13.3843i q^{65} +10.9282i q^{67} +5.79555 q^{71} -14.3923 q^{73} +0.896575i q^{77} -7.46410i q^{79} -6.96953 q^{83} +21.1244 q^{85} +13.2456i q^{89} -4.00000i q^{91} +3.34607 q^{95} +2.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 32 q^{13} - 8 q^{25} - 80 q^{37} - 8 q^{49} - 16 q^{61} - 32 q^{73} + 72 q^{85} - 32 q^{97}+O(q^{100}) $ 8 * q + 32 * q^13 - 8 * q^25 - 80 * q^37 - 8 * q^49 - 16 * q^61 - 32 * q^73 + 72 * q^85 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$5$	− 3.34607i	− 1.49641i	−0.663470	−	0.748203i	$-0.730917\pi$
$5$	− 3.34607i	− 1.49641i	0.663470	−	0.748203i	$-0.269083\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.896575	−0.270328	−0.135164	−	0.990823i	$-0.543156\pi$
$11$	−0.896575	−0.270328	−0.135164	+	0.990823i	$0.543156\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.31319i	1.53117i	0.643332	+	0.765587i	$0.277551\pi$
$17$	6.31319i	1.53117i	−0.643332	+	0.765587i	$0.722449\pi$
$18$	0	0
$19$	1.00000i	0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$	1.00000i	0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.20977	1.50334	0.751670	−	0.659539i	$-0.229248\pi$
$23$	7.20977	1.50334	0.751670	+	0.659539i	$0.229248\pi$
$24$	0	0
$25$	−6.19615	−1.23923
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.72741i	1.43494i	0.696588	+	0.717472i	$0.254701\pi$
$29$	7.72741i	1.43494i	−0.696588	+	0.717472i	$0.745299\pi$
$30$	0	0
$31$	3.73205i	0.670296i	0.942165	+	0.335148i	$0.108786\pi$
$31$	3.73205i	0.670296i	−0.942165	+	0.335148i	$0.891214\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.34607	−0.565588
$36$	0	0
$37$	−11.7321	−1.92874	−0.964369	−	0.264562i	$-0.914773\pi$
$37$	−11.7321	−1.92874	−0.964369	+	0.264562i	$0.914773\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.86611i	1.22848i	0.789119	+	0.614240i	$0.210537\pi$
$41$	7.86611i	1.22848i	−0.789119	+	0.614240i	$0.789463\pi$
$42$	0	0
$43$	7.46410i	1.13826i	0.822246	+	0.569132i	$0.192721\pi$
$43$	7.46410i	1.13826i	−0.822246	+	0.569132i	$0.807279\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.76268	1.27817	0.639084	−	0.769137i	$-0.279313\pi$
$47$	8.76268	1.27817	0.639084	+	0.769137i	$0.279313\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.82843i	0.388514i	0.980951	+	0.194257i	$0.0622296\pi$
$53$	2.82843i	0.388514i	−0.980951	+	0.194257i	$0.937770\pi$
$54$	0	0
$55$	3.00000i	0.404520i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.86370	−0.503011	−0.251506	−	0.967856i	$-0.580926\pi$
$59$	−3.86370	−0.503011	−0.251506	+	0.967856i	$0.580926\pi$
$60$	0	0
$61$	−12.3923	−1.58667	−0.793336	−	0.608784i	$-0.791658\pi$
$61$	−12.3923	−1.58667	−0.793336	+	0.608784i	$0.791658\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 13.3843i	− 1.66011i
$66$	0	0
$67$	10.9282i	1.33509i	0.744568	+	0.667546i	$0.232655\pi$
$67$	10.9282i	1.33509i	−0.744568	+	0.667546i	$0.767345\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.79555	0.687806	0.343903	−	0.939005i	$-0.388251\pi$
$71$	5.79555	0.687806	0.343903	+	0.939005i	$0.388251\pi$
$72$	0	0
$73$	−14.3923	−1.68449	−0.842246	−	0.539093i	$-0.818767\pi$
$73$	−14.3923	−1.68449	−0.842246	+	0.539093i	$0.818767\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.896575i	0.102174i
$78$	0	0
$79$	− 7.46410i	− 0.839777i	−0.907576	−	0.419889i	$-0.862069\pi$
$79$	− 7.46410i	− 0.839777i	0.907576	−	0.419889i	$-0.137931\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.96953	−0.765006	−0.382503	−	0.923954i	$-0.624938\pi$
$83$	−6.96953	−0.765006	−0.382503	+	0.923954i	$0.624938\pi$
$84$	0	0
$85$	21.1244	2.29126
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.2456i	1.40403i	0.712164	+	0.702013i	$0.247715\pi$
$89$	13.2456i	1.40403i	−0.712164	+	0.702013i	$0.752285\pi$
$90$	0	0
$91$	− 4.00000i	− 0.419314i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.34607	0.343299
$96$	0	0
$97$	2.92820	0.297314	0.148657	−	0.988889i	$-0.452505\pi$
$97$	2.92820	0.297314	0.148657	+	0.988889i	$0.452505\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.656339i	0.0653082i	0.999467	+	0.0326541i	$0.0103960\pi$
$101$	0.656339i	0.0653082i	−0.999467	+	0.0326541i	$0.989604\pi$
$102$	0	0
$103$	18.1244i	1.78585i	0.450210	+	0.892923i	$0.351349\pi$
$103$	18.1244i	1.78585i	−0.450210	+	0.892923i	$0.648651\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.52004	0.436969	0.218484	−	0.975840i	$-0.429889\pi$
$107$	4.52004	0.436969	0.218484	+	0.975840i	$0.429889\pi$
$108$	0	0
$109$	−3.53590	−0.338678	−0.169339	−	0.985558i	$-0.554163\pi$
$109$	−3.53590	−0.338678	−0.169339	+	0.985558i	$0.554163\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 9.52056i	− 0.895619i	−0.894129	−	0.447809i	$-0.852204\pi$
$113$	− 9.52056i	− 0.895619i	0.894129	−	0.447809i	$-0.147796\pi$
$114$	0	0
$115$	− 24.1244i	− 2.24961i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.31319	0.578729
$120$	0	0
$121$	−10.1962	−0.926923
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.00240i	0.357986i
$126$	0	0
$127$	16.3923i	1.45458i	0.686329	+	0.727291i	$0.259221\pi$
$127$	16.3923i	1.45458i	−0.686329	+	0.727291i	$0.740779\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.37945	−0.470005	−0.235002	−	0.971995i	$-0.575510\pi$
$131$	−5.37945	−0.470005	−0.235002	+	0.971995i	$0.575510\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.31268i	0.112150i	0.998427	+	0.0560748i	$0.0178585\pi$
$137$	1.31268i	0.112150i	−0.998427	+	0.0560748i	$0.982141\pi$
$138$	0	0
$139$	0.535898i	0.0454543i	0.999742	+	0.0227272i	$0.00723490\pi$
$139$	0.535898i	0.0454543i	−0.999742	+	0.0227272i	$0.992765\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.58630	−0.299902
$144$	0	0
$145$	25.8564	2.14726
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.69213i	0.548241i	0.961695	+	0.274120i	$0.0883867\pi$
$149$	6.69213i	0.548241i	−0.961695	+	0.274120i	$0.911613\pi$
$150$	0	0
$151$	17.3205i	1.40952i	0.709444	+	0.704761i	$0.248946\pi$
$151$	17.3205i	1.40952i	−0.709444	+	0.704761i	$0.751054\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.4877	1.00304
$156$	0	0
$157$	−1.46410	−0.116848	−0.0584240	−	0.998292i	$-0.518608\pi$
$157$	−1.46410	−0.116848	−0.0584240	+	0.998292i	$0.518608\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 7.20977i	− 0.568209i
$162$	0	0
$163$	− 19.3205i	− 1.51330i	−0.653821	−	0.756649i	$-0.726835\pi$
$163$	− 19.3205i	− 1.51330i	0.653821	−	0.756649i	$-0.273165\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.8332	0.838301	0.419150	−	0.907917i	$-0.362328\pi$
$167$	10.8332	0.838301	0.419150	+	0.907917i	$0.362328\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.17398i	0.0892558i	0.999004	+	0.0446279i	$0.0142102\pi$
$173$	1.17398i	0.0892558i	−0.999004	+	0.0446279i	$0.985790\pi$
$174$	0	0
$175$	6.19615i	0.468385i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.10634	0.605897	0.302948	−	0.953007i	$-0.402029\pi$
$179$	8.10634	0.605897	0.302948	+	0.953007i	$0.402029\pi$
$180$	0	0
$181$	3.46410	0.257485	0.128742	−	0.991678i	$-0.458906\pi$
$181$	3.46410	0.257485	0.128742	+	0.991678i	$0.458906\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	39.2562i	2.88617i
$186$	0	0
$187$	− 5.66025i	− 0.413919i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	25.7704	1.86468	0.932341	−	0.361581i	$-0.117763\pi$
$191$	25.7704	1.86468	0.932341	+	0.361581i	$0.117763\pi$
$192$	0	0
$193$	14.9282	1.07456	0.537278	−	0.843405i	$-0.319453\pi$
$193$	14.9282	1.07456	0.537278	+	0.843405i	$0.319453\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.76268i	0.624315i	0.950030	+	0.312158i	$0.101052\pi$
$197$	8.76268i	0.624315i	−0.950030	+	0.312158i	$0.898948\pi$
$198$	0	0
$199$	− 26.4641i	− 1.87599i	−0.346648	−	0.937995i	$-0.612680\pi$
$199$	− 26.4641i	− 1.87599i	0.346648	−	0.937995i	$-0.387320\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.72741	0.542358
$204$	0	0
$205$	26.3205	1.83830
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 0.896575i	− 0.0620174i
$210$	0	0
$211$	− 13.3205i	− 0.917022i	−0.888689	−	0.458511i	$-0.848383\pi$
$211$	− 13.3205i	− 0.917022i	0.888689	−	0.458511i	$-0.151617\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	24.9754	1.70331
$216$	0	0
$217$	3.73205	0.253348
$218$	0	0
$219$	0	0
$220$	0	0
$221$	25.2528i	1.69869i
$222$	0	0
$223$	− 7.53590i	− 0.504641i	−0.967644	−	0.252321i	$-0.918806\pi$
$223$	− 7.53590i	− 0.504641i	0.967644	−	0.252321i	$-0.0811937\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.0764	1.33252	0.666258	−	0.745721i	$-0.267895\pi$
$227$	20.0764	1.33252	0.666258	+	0.745721i	$0.267895\pi$
$228$	0	0
$229$	−11.3205	−0.748080	−0.374040	−	0.927413i	$-0.622028\pi$
$229$	−11.3205	−0.748080	−0.374040	+	0.927413i	$0.622028\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.52056i	0.623712i	0.950129	+	0.311856i	$0.100951\pi$
$233$	9.52056i	0.623712i	−0.950129	+	0.311856i	$0.899049\pi$
$234$	0	0
$235$	− 29.3205i	− 1.91266i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.2485	−1.43913	−0.719567	−	0.694423i	$-0.755660\pi$
$239$	−22.2485	−1.43913	−0.719567	+	0.694423i	$0.755660\pi$
$240$	0	0
$241$	−12.9282	−0.832779	−0.416389	−	0.909186i	$-0.636705\pi$
$241$	−12.9282	−0.832779	−0.416389	+	0.909186i	$0.636705\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.34607i	0.213772i
$246$	0	0
$247$	4.00000i	0.254514i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.04008	0.570605	0.285303	−	0.958438i	$-0.407906\pi$
$251$	9.04008	0.570605	0.285303	+	0.958438i	$0.407906\pi$
$252$	0	0
$253$	−6.46410	−0.406395
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 12.8666i	− 0.802598i	−0.915947	−	0.401299i	$-0.868559\pi$
$257$	− 12.8666i	− 0.802598i	0.915947	−	0.401299i	$-0.131441\pi$
$258$	0	0
$259$	11.7321i	0.728994i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.7351	−1.52523	−0.762617	−	0.646850i	$-0.776086\pi$
$263$	−24.7351	−1.52523	−0.762617	+	0.646850i	$0.776086\pi$
$264$	0	0
$265$	9.46410	0.581375
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 27.3877i	− 1.66986i	−0.550358	−	0.834929i	$-0.685509\pi$
$269$	− 27.3877i	− 1.66986i	0.550358	−	0.834929i	$-0.314491\pi$
$270$	0	0
$271$	− 7.46410i	− 0.453412i	−0.973963	−	0.226706i	$-0.927204\pi$
$271$	− 7.46410i	− 0.453412i	0.973963	−	0.226706i	$-0.0727956\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.55532	0.334998
$276$	0	0
$277$	−0.803848	−0.0482985	−0.0241493	−	0.999708i	$-0.507688\pi$
$277$	−0.803848	−0.0482985	−0.0241493	+	0.999708i	$0.507688\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.17638i	0.308797i	0.988009	+	0.154398i	$0.0493440\pi$
$281$	5.17638i	0.308797i	−0.988009	+	0.154398i	$0.950656\pi$
$282$	0	0
$283$	− 17.8564i	− 1.06145i	−0.847543	−	0.530727i	$-0.821919\pi$
$283$	− 17.8564i	− 1.06145i	0.847543	−	0.530727i	$-0.178081\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.86611	0.464322
$288$	0	0
$289$	−22.8564	−1.34449
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.17209i	0.126895i	0.997985	+	0.0634474i	$0.0202095\pi$
$293$	2.17209i	0.126895i	−0.997985	+	0.0634474i	$0.979790\pi$
$294$	0	0
$295$	12.9282i	0.752709i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	28.8391	1.66781
$300$	0	0
$301$	7.46410	0.430224
$302$	0	0
$303$	0	0
$304$	0	0
$305$	41.4655i	2.37431i
$306$	0	0
$307$	5.00000i	0.285365i	0.989769	+	0.142683i	$0.0455728\pi$
$307$	5.00000i	0.285365i	−0.989769	+	0.142683i	$0.954427\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.4548	0.876362	0.438181	−	0.898887i	$-0.355623\pi$
$311$	15.4548	0.876362	0.438181	+	0.898887i	$0.355623\pi$
$312$	0	0
$313$	12.3923	0.700454	0.350227	−	0.936665i	$-0.386104\pi$
$313$	12.3923	0.700454	0.350227	+	0.936665i	$0.386104\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.4596i	1.31762i	0.752308	+	0.658812i	$0.228941\pi$
$317$	23.4596i	1.31762i	−0.752308	+	0.658812i	$0.771059\pi$
$318$	0	0
$319$	− 6.92820i	− 0.387905i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.31319	−0.351275
$324$	0	0
$325$	−24.7846	−1.37480
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 8.76268i	− 0.483102i
$330$	0	0
$331$	− 22.2487i	− 1.22290i	−0.791283	−	0.611450i	$-0.790587\pi$
$331$	− 22.2487i	− 1.22290i	0.791283	−	0.611450i	$-0.209413\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	36.5665	1.99784
$336$	0	0
$337$	22.7128	1.23725	0.618623	−	0.785688i	$-0.287691\pi$
$337$	22.7128	1.23725	0.618623	+	0.785688i	$0.287691\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3.34607i	− 0.181200i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.1755	−0.868348	−0.434174	−	0.900829i	$-0.642960\pi$
$347$	−16.1755	−0.868348	−0.434174	+	0.900829i	$0.642960\pi$
$348$	0	0
$349$	13.3205	0.713030	0.356515	−	0.934290i	$-0.383965\pi$
$349$	13.3205	0.713030	0.356515	+	0.934290i	$0.383965\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 11.1750i	− 0.594786i	−0.954755	−	0.297393i	$-0.903883\pi$
$353$	− 11.1750i	− 0.594786i	0.954755	−	0.297393i	$-0.0961171\pi$
$354$	0	0
$355$	− 19.3923i	− 1.02924i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.6622	−0.984952	−0.492476	−	0.870326i	$-0.663908\pi$
$359$	−18.6622	−0.984952	−0.492476	+	0.870326i	$0.663908\pi$
$360$	0	0
$361$	18.0000	0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	48.1576i	2.52068i
$366$	0	0
$367$	18.4641i	0.963818i	0.876221	+	0.481909i	$0.160056\pi$
$367$	18.4641i	0.963818i	−0.876221	+	0.481909i	$0.839944\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.82843	0.146845
$372$	0	0
$373$	15.3923	0.796983	0.398492	−	0.917172i	$-0.369534\pi$
$373$	15.3923	0.796983	0.398492	+	0.917172i	$0.369534\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	30.9096i	1.59193i
$378$	0	0
$379$	17.3205i	0.889695i	0.895606	+	0.444847i	$0.146742\pi$
$379$	17.3205i	0.889695i	−0.895606	+	0.444847i	$0.853258\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.1774	−0.775530	−0.387765	−	0.921758i	$-0.626753\pi$
$383$	−15.1774	−0.775530	−0.387765	+	0.921758i	$0.626753\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 35.2538i	− 1.78744i	−0.448626	−	0.893719i	$-0.648087\pi$
$389$	− 35.2538i	− 1.78744i	0.448626	−	0.893719i	$-0.351913\pi$
$390$	0	0
$391$	45.5167i	2.30188i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−24.9754	−1.25665
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.27362i	0.113539i	0.998387	+	0.0567697i	$0.0180801\pi$
$401$	2.27362i	0.113539i	−0.998387	+	0.0567697i	$0.981920\pi$
$402$	0	0
$403$	14.9282i	0.743627i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.5187	0.521391
$408$	0	0
$409$	−26.3923	−1.30502	−0.652508	−	0.757782i	$-0.726283\pi$
$409$	−26.3923	−1.30502	−0.652508	+	0.757782i	$0.726283\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.86370i	0.190120i
$414$	0	0
$415$	23.3205i	1.14476i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.45001	−0.363957	−0.181978	−	0.983303i	$-0.558250\pi$
$419$	−7.45001	−0.363957	−0.181978	+	0.983303i	$0.558250\pi$
$420$	0	0
$421$	−17.5885	−0.857209	−0.428604	−	0.903492i	$-0.640995\pi$
$421$	−17.5885	−0.857209	−0.428604	+	0.903492i	$0.640995\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 39.1175i	− 1.89748i
$426$	0	0
$427$	12.3923i	0.599706i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.5293	1.75956	0.879778	−	0.475385i	$-0.157691\pi$
$431$	36.5293	1.75956	0.879778	+	0.475385i	$0.157691\pi$
$432$	0	0
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.20977i	0.344890i
$438$	0	0
$439$	− 26.9282i	− 1.28521i	−0.766196	−	0.642607i	$-0.777853\pi$
$439$	− 26.9282i	− 1.28521i	0.766196	−	0.642607i	$-0.222147\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.76028	0.226168	0.113084	−	0.993585i	$-0.463927\pi$
$443$	4.76028	0.226168	0.113084	+	0.993585i	$0.463927\pi$
$444$	0	0
$445$	44.3205	2.10099
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.93426i	0.280055i	0.990148	+	0.140027i	$0.0447191\pi$
$449$	5.93426i	0.280055i	−0.990148	+	0.140027i	$0.955281\pi$
$450$	0	0
$451$	− 7.05256i	− 0.332092i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−13.3843	−0.627464
$456$	0	0
$457$	−3.78461	−0.177037	−0.0885183	−	0.996075i	$-0.528213\pi$
$457$	−3.78461	−0.177037	−0.0885183	+	0.996075i	$0.528213\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 37.7405i	− 1.75775i	−0.477054	−	0.878874i	$-0.658295\pi$
$461$	− 37.7405i	− 1.75775i	0.477054	−	0.878874i	$-0.341705\pi$
$462$	0	0
$463$	− 12.1436i	− 0.564361i	−0.959361	−	0.282180i	$-0.908942\pi$
$463$	− 12.1436i	− 0.564361i	0.959361	−	0.282180i	$-0.0910576\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.55103	−0.118047	−0.0590237	−	0.998257i	$-0.518799\pi$
$467$	−2.55103	−0.118047	−0.0590237	+	0.998257i	$0.518799\pi$
$468$	0	0
$469$	10.9282	0.504618
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 6.69213i	− 0.307704i
$474$	0	0
$475$	− 6.19615i	− 0.284299i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.62536	−0.119956	−0.0599778	−	0.998200i	$-0.519103\pi$
$479$	−2.62536	−0.119956	−0.0599778	+	0.998200i	$0.519103\pi$
$480$	0	0
$481$	−46.9282	−2.13974
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 9.79796i	− 0.444902i
$486$	0	0
$487$	29.7128i	1.34642i	0.739453	+	0.673208i	$0.235084\pi$
$487$	29.7128i	1.34642i	−0.739453	+	0.673208i	$0.764916\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.52245	−0.384613	−0.192306	−	0.981335i	$-0.561597\pi$
$491$	−8.52245	−0.384613	−0.192306	+	0.981335i	$0.561597\pi$
$492$	0	0
$493$	−48.7846	−2.19715
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.79555i	− 0.259966i
$498$	0	0
$499$	− 18.0000i	− 0.805791i	−0.915246	−	0.402895i	$-0.868004\pi$
$499$	− 18.0000i	− 0.805791i	0.915246	−	0.402895i	$-0.131996\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−38.8401	−1.73179	−0.865897	−	0.500222i	$-0.833252\pi$
$503$	−38.8401	−1.73179	−0.865897	+	0.500222i	$0.833252\pi$
$504$	0	0
$505$	2.19615	0.0977275
$506$	0	0
$507$	0	0
$508$	0	0
$509$	41.0122i	1.81783i	0.416978	+	0.908917i	$0.363089\pi$
$509$	41.0122i	1.81783i	−0.416978	+	0.908917i	$0.636911\pi$
$510$	0	0
$511$	14.3923i	0.636678i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	60.6453	2.67235
$516$	0	0
$517$	−7.85641	−0.345524
$518$	0	0
$519$	0	0
$520$	0	0
$521$	25.7704i	1.12902i	0.825425	+	0.564511i	$0.190935\pi$
$521$	25.7704i	1.12902i	−0.825425	+	0.564511i	$0.809065\pi$
$522$	0	0
$523$	− 28.6603i	− 1.25323i	−0.779331	−	0.626613i	$-0.784441\pi$
$523$	− 28.6603i	− 1.25323i	0.779331	−	0.626613i	$-0.215559\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23.5612	−1.02634
$528$	0	0
$529$	28.9808	1.26003
$530$	0	0
$531$	0	0
$532$	0	0
$533$	31.4644i	1.36288i
$534$	0	0
$535$	− 15.1244i	− 0.653883i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.896575	0.0386182
$540$	0	0
$541$	−17.3397	−0.745494	−0.372747	−	0.927933i	$-0.621584\pi$
$541$	−17.3397	−0.745494	−0.372747	+	0.927933i	$0.621584\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11.8313i	0.506799i
$546$	0	0
$547$	− 29.8564i	− 1.27657i	−0.769801	−	0.638284i	$-0.779645\pi$
$547$	− 29.8564i	− 1.27657i	0.769801	−	0.638284i	$-0.220355\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.72741	−0.329199
$552$	0	0
$553$	−7.46410	−0.317406
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.7942i	0.499736i	0.968280	+	0.249868i	$0.0803872\pi$
$557$	11.7942i	0.499736i	−0.968280	+	0.249868i	$0.919613\pi$
$558$	0	0
$559$	29.8564i	1.26279i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	27.5264	1.16010	0.580050	−	0.814581i	$-0.303033\pi$
$563$	27.5264	1.16010	0.580050	+	0.814581i	$0.303033\pi$
$564$	0	0
$565$	−31.8564	−1.34021
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 15.9353i	− 0.668042i	−0.942566	−	0.334021i	$-0.891594\pi$
$569$	− 15.9353i	− 0.668042i	0.942566	−	0.334021i	$-0.108406\pi$
$570$	0	0
$571$	5.60770i	0.234675i	0.993092	+	0.117337i	$0.0374359\pi$
$571$	5.60770i	0.234675i	−0.993092	+	0.117337i	$0.962564\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−44.6728	−1.86299
$576$	0	0
$577$	37.3205	1.55367	0.776837	−	0.629702i	$-0.216823\pi$
$577$	37.3205	1.55367	0.776837	+	0.629702i	$0.216823\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.96953i	0.289145i
$582$	0	0
$583$	− 2.53590i	− 0.105026i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.5312	1.46653	0.733265	−	0.679943i	$-0.237996\pi$
$587$	35.5312	1.46653	0.733265	+	0.679943i	$0.237996\pi$
$588$	0	0
$589$	−3.73205	−0.153776
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 10.3156i	− 0.423611i	−0.977312	−	0.211805i	$-0.932066\pi$
$593$	− 10.3156i	− 0.423611i	0.977312	−	0.211805i	$-0.0679343\pi$
$594$	0	0
$595$	− 21.1244i	− 0.866014i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7.69024	−0.314215	−0.157107	−	0.987582i	$-0.550217\pi$
$599$	−7.69024	−0.314215	−0.157107	+	0.987582i	$0.550217\pi$
$600$	0	0
$601$	−16.7846	−0.684659	−0.342329	−	0.939580i	$-0.611216\pi$
$601$	−16.7846	−0.684659	−0.342329	+	0.939580i	$0.611216\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	34.1170i	1.38705i
$606$	0	0
$607$	20.7846i	0.843621i	0.906684	+	0.421811i	$0.138605\pi$
$607$	20.7846i	0.843621i	−0.906684	+	0.421811i	$0.861395\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35.0507	1.41800
$612$	0	0
$613$	25.5359	1.03139	0.515693	−	0.856774i	$-0.327535\pi$
$613$	25.5359	1.03139	0.515693	+	0.856774i	$0.327535\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.0449i	0.686202i	0.939299	+	0.343101i	$0.111477\pi$
$617$	17.0449i	0.686202i	−0.939299	+	0.343101i	$0.888523\pi$
$618$	0	0
$619$	12.1244i	0.487319i	0.969861	+	0.243659i	$0.0783479\pi$
$619$	12.1244i	0.487319i	−0.969861	+	0.243659i	$0.921652\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.2456	0.530672
$624$	0	0
$625$	−17.5885	−0.703538
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 74.0667i	− 2.95323i
$630$	0	0
$631$	38.7846i	1.54399i	0.635628	+	0.771995i	$0.280741\pi$
$631$	38.7846i	1.54399i	−0.635628	+	0.771995i	$0.719259\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	54.8497	2.17664
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 10.2784i	− 0.405974i	−0.979181	−	0.202987i	$-0.934935\pi$
$641$	− 10.2784i	− 0.405974i	0.979181	−	0.202987i	$-0.0650649\pi$
$642$	0	0
$643$	− 23.9282i	− 0.943636i	−0.881696	−	0.471818i	$-0.843598\pi$
$643$	− 23.9282i	− 0.943636i	0.881696	−	0.471818i	$-0.156402\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.6932	−0.656276	−0.328138	−	0.944630i	$-0.606421\pi$
$647$	−16.6932	−0.656276	−0.328138	+	0.944630i	$0.606421\pi$
$648$	0	0
$649$	3.46410	0.135978
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.55103i	− 0.0998294i	−0.998753	−	0.0499147i	$-0.984105\pi$
$653$	− 2.55103i	− 0.0998294i	0.998753	−	0.0499147i	$-0.0158949\pi$
$654$	0	0
$655$	18.0000i	0.703318i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−30.5679	−1.19075	−0.595377	−	0.803446i	$-0.702997\pi$
$659$	−30.5679	−1.19075	−0.595377	+	0.803446i	$0.702997\pi$
$660$	0	0
$661$	−16.7846	−0.652846	−0.326423	−	0.945224i	$-0.605843\pi$
$661$	−16.7846	−0.652846	−0.326423	+	0.945224i	$0.605843\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.34607i	− 0.129755i
$666$	0	0
$667$	55.7128i	2.15721i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.1106	0.428921
$672$	0	0
$673$	20.0000	0.770943	0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000	0.770943	0.385472	+	0.922720i	$0.374039\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.6950i	− 0.603210i	−0.953433	−	0.301605i	$-0.902478\pi$
$677$	− 15.6950i	− 0.603210i	0.953433	−	0.301605i	$-0.0975223\pi$
$678$	0	0
$679$	− 2.92820i	− 0.112374i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.62398	−0.329988	−0.164994	−	0.986295i	$-0.552760\pi$
$683$	−8.62398	−0.329988	−0.164994	+	0.986295i	$0.552760\pi$
$684$	0	0
$685$	4.39230	0.167821
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.3137i	0.431018i
$690$	0	0
$691$	24.5359i	0.933390i	0.884418	+	0.466695i	$0.154555\pi$
$691$	24.5359i	0.933390i	−0.884418	+	0.466695i	$0.845445\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.79315	0.0680181
$696$	0	0
$697$	−49.6603	−1.88102
$698$	0	0
$699$	0	0
$700$	0	0
$701$	44.0908i	1.66529i	0.553809	+	0.832644i	$0.313174\pi$
$701$	44.0908i	1.66529i	−0.553809	+	0.832644i	$0.686826\pi$
$702$	0	0
$703$	− 11.7321i	− 0.442483i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.656339	0.0246842
$708$	0	0
$709$	0.607695	0.0228225	0.0114112	−	0.999935i	$-0.496368\pi$
$709$	0.607695	0.0228225	0.0114112	+	0.999935i	$0.496368\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	26.9072i	1.00768i
$714$	0	0
$715$	12.0000i	0.448775i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3.30890	−0.123401	−0.0617006	−	0.998095i	$-0.519652\pi$
$719$	−3.30890	−0.123401	−0.0617006	+	0.998095i	$0.519652\pi$
$720$	0	0
$721$	18.1244	0.674986
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 47.8802i	− 1.77823i
$726$	0	0
$727$	− 14.6795i	− 0.544432i	−0.962236	−	0.272216i	$-0.912243\pi$
$727$	− 14.6795i	− 0.544432i	0.962236	−	0.272216i	$-0.0877566\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−47.1223	−1.74288
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 9.79796i	− 0.360912i
$738$	0	0
$739$	35.8564i	1.31900i	0.751705	+	0.659500i	$0.229232\pi$
$739$	35.8564i	1.31900i	−0.751705	+	0.659500i	$0.770768\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.0778	−0.516463	−0.258232	−	0.966083i	$-0.583140\pi$
$743$	−14.0778	−0.516463	−0.258232	+	0.966083i	$0.583140\pi$
$744$	0	0
$745$	22.3923	0.820391
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 4.52004i	− 0.165159i
$750$	0	0
$751$	− 23.0718i	− 0.841902i	−0.907083	−	0.420951i	$-0.861696\pi$
$751$	− 23.0718i	− 0.841902i	0.907083	−	0.420951i	$-0.138304\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	57.9555	2.10922
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.92996i	0.106211i	0.998589	+	0.0531055i	$0.0169120\pi$
$761$	2.92996i	0.106211i	−0.998589	+	0.0531055i	$0.983088\pi$
$762$	0	0
$763$	3.53590i	0.128008i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15.4548	−0.558041
$768$	0	0
$769$	−38.2487	−1.37928	−0.689642	−	0.724151i	$-0.742232\pi$
$769$	−38.2487	−1.37928	−0.689642	+	0.724151i	$0.742232\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.3872i	0.805211i	0.915374	+	0.402605i	$0.131895\pi$
$773$	22.3872i	0.805211i	−0.915374	+	0.402605i	$0.868105\pi$
$774$	0	0
$775$	− 23.1244i	− 0.830651i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.86611	−0.281833
$780$	0	0
$781$	−5.19615	−0.185933
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.89898i	0.174852i
$786$	0	0
$787$	5.60770i	0.199893i	0.994993	+	0.0999464i	$0.0318671\pi$
$787$	5.60770i	0.199893i	−0.994993	+	0.0999464i	$0.968133\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.52056	−0.338512
$792$	0	0
$793$	−49.5692	−1.76025
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 47.1595i	− 1.67047i	−0.549890	−	0.835237i	$-0.685330\pi$
$797$	− 47.1595i	− 1.67047i	0.549890	−	0.835237i	$-0.314670\pi$
$798$	0	0
$799$	55.3205i	1.95710i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.9038	0.455365
$804$	0	0
$805$	−24.1244	−0.850272
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.03150i	0.106582i	0.998579	+	0.0532909i	$0.0169711\pi$
$809$	3.03150i	0.106582i	−0.998579	+	0.0532909i	$0.983029\pi$
$810$	0	0
$811$	1.48334i	0.0520871i	0.999661	+	0.0260435i	$0.00829086\pi$
$811$	1.48334i	0.0520871i	−0.999661	+	0.0260435i	$0.991709\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−64.6477	−2.26451
$816$	0	0
$817$	−7.46410	−0.261136
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 36.8439i	− 1.28586i	−0.765925	−	0.642930i	$-0.777719\pi$
$821$	− 36.8439i	− 1.28586i	0.765925	−	0.642930i	$-0.222281\pi$
$822$	0	0
$823$	37.7128i	1.31459i	0.753635	+	0.657293i	$0.228299\pi$
$823$	37.7128i	1.31459i	−0.753635	+	0.657293i	$0.771701\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39.4321	−1.37119	−0.685594	−	0.727984i	$-0.740457\pi$
$827$	−39.4321	−1.37119	−0.685594	+	0.727984i	$0.740457\pi$
$828$	0	0
$829$	47.1769	1.63852	0.819261	−	0.573421i	$-0.194384\pi$
$829$	47.1769	1.63852	0.819261	+	0.573421i	$0.194384\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 6.31319i	− 0.218739i
$834$	0	0
$835$	− 36.2487i	− 1.25444i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	45.8096	1.58152	0.790762	−	0.612124i	$-0.209684\pi$
$839$	45.8096	1.58152	0.790762	+	0.612124i	$0.209684\pi$
$840$	0	0
$841$	−30.7128	−1.05906
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 10.0382i	− 0.345324i
$846$	0	0
$847$	10.1962i	0.350344i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−84.5854	−2.89955
$852$	0	0
$853$	9.85641	0.337477	0.168738	−	0.985661i	$-0.446031\pi$
$853$	9.85641	0.337477	0.168738	+	0.985661i	$0.446031\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 47.7415i	− 1.63082i	−0.578885	−	0.815409i	$-0.696512\pi$
$857$	− 47.7415i	− 1.63082i	0.578885	−	0.815409i	$-0.303488\pi$
$858$	0	0
$859$	− 42.7128i	− 1.45734i	−0.684864	−	0.728671i	$-0.740138\pi$
$859$	− 42.7128i	− 1.45734i	0.684864	−	0.728671i	$-0.259862\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.89469	0.0644959	0.0322479	−	0.999480i	$-0.489733\pi$
$863$	1.89469	0.0644959	0.0322479	+	0.999480i	$0.489733\pi$
$864$	0	0
$865$	3.92820	0.133563
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.69213i	0.227015i
$870$	0	0
$871$	43.7128i	1.48115i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.00240	0.135306
$876$	0	0
$877$	50.7846	1.71487	0.857437	−	0.514589i	$-0.172055\pi$
$877$	50.7846	1.71487	0.857437	+	0.514589i	$0.172055\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 20.0120i	− 0.674222i	−0.941465	−	0.337111i	$-0.890550\pi$
$881$	− 20.0120i	− 0.674222i	0.941465	−	0.337111i	$-0.109450\pi$
$882$	0	0
$883$	0.679492i	0.0228667i	0.999935	+	0.0114334i	$0.00363943\pi$
$883$	0.679492i	0.0228667i	−0.999935	+	0.0114334i	$0.996361\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.9812	1.44317	0.721584	−	0.692327i	$-0.243414\pi$
$887$	42.9812	1.44317	0.721584	+	0.692327i	$0.243414\pi$
$888$	0	0
$889$	16.3923	0.549780
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.76268i	0.293232i
$894$	0	0
$895$	− 27.1244i	− 0.906667i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−28.8391	−0.961837
$900$	0	0
$901$	−17.8564	−0.594883
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 11.5911i	− 0.385302i
$906$	0	0
$907$	− 38.7846i	− 1.28782i	−0.765100	−	0.643911i	$-0.777311\pi$
$907$	− 38.7846i	− 1.28782i	0.765100	−	0.643911i	$-0.222689\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45.6338	1.51191	0.755957	−	0.654621i	$-0.227172\pi$
$911$	45.6338	1.51191	0.755957	+	0.654621i	$0.227172\pi$
$912$	0	0
$913$	6.24871	0.206802
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.37945i	0.177645i
$918$	0	0
$919$	50.9282i	1.67997i	0.542612	+	0.839983i	$0.317435\pi$
$919$	50.9282i	1.67997i	−0.542612	+	0.839983i	$0.682565\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	23.1822	0.763052
$924$	0	0
$925$	72.6936	2.39015
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 25.9091i	− 0.850050i	−0.905182	−	0.425025i	$-0.860265\pi$
$929$	− 25.9091i	− 0.850050i	0.905182	−	0.425025i	$-0.139735\pi$
$930$	0	0
$931$	− 1.00000i	− 0.0327737i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18.9396	−0.619390
$936$	0	0
$937$	−23.3205	−0.761848	−0.380924	−	0.924606i	$-0.624394\pi$
$937$	−23.3205	−0.761848	−0.380924	+	0.924606i	$0.624394\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13.0697i	0.426060i	0.977046	+	0.213030i	$0.0683332\pi$
$941$	13.0697i	0.426060i	−0.977046	+	0.213030i	$0.931667\pi$
$942$	0	0
$943$	56.7128i	1.84682i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.34984	−0.0438640	−0.0219320	−	0.999759i	$-0.506982\pi$
$947$	−1.34984	−0.0438640	−0.0219320	+	0.999759i	$0.506982\pi$
$948$	0	0
$949$	−57.5692	−1.86878
$950$	0	0
$951$	0	0
$952$	0	0
$953$	35.2538i	1.14198i	0.820956	+	0.570991i	$0.193441\pi$
$953$	35.2538i	1.14198i	−0.820956	+	0.570991i	$0.806559\pi$
$954$	0	0
$955$	− 86.2295i	− 2.79032i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.31268	0.0423886
$960$	0	0
$961$	17.0718	0.550703
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 49.9507i	− 1.60797i
$966$	0	0
$967$	− 12.5359i	− 0.403127i	−0.979475	−	0.201564i	$-0.935398\pi$
$967$	− 12.5359i	− 0.403127i	0.979475	−	0.201564i	$-0.0646023\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−33.7381	−1.08271	−0.541353	−	0.840796i	$-0.682088\pi$
$971$	−33.7381	−1.08271	−0.541353	+	0.840796i	$0.682088\pi$
$972$	0	0
$973$	0.535898	0.0171801
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34.4959i	1.10362i	0.833969	+	0.551811i	$0.186063\pi$
$977$	34.4959i	1.10362i	−0.833969	+	0.551811i	$0.813937\pi$
$978$	0	0
$979$	− 11.8756i	− 0.379547i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	54.5723	1.74059	0.870293	−	0.492534i	$-0.163929\pi$
$983$	54.5723	1.74059	0.870293	+	0.492534i	$0.163929\pi$
$984$	0	0
$985$	29.3205	0.934229
$986$	0	0
$987$	0	0
$988$	0	0
$989$	53.8144i	1.71120i
$990$	0	0
$991$	− 24.3923i	− 0.774847i	−0.921902	−	0.387424i	$-0.873365\pi$
$991$	− 24.3923i	− 0.774847i	0.921902	−	0.387424i	$-0.126635\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−88.5506	−2.80724
$996$	0	0
$997$	−36.2487	−1.14801	−0.574004	−	0.818852i	$-0.694611\pi$
$997$	−36.2487	−1.14801	−0.574004	+	0.818852i	$0.694611\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.f.2591.1	✓	8
3.2	odd	2	inner	6048.2.h.f.2591.7	yes	8
4.3	odd	2	inner	6048.2.h.f.2591.2	yes	8
12.11	even	2	inner	6048.2.h.f.2591.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.f.2591.1	✓	8	1.1	even	1	trivial
6048.2.h.f.2591.2	yes	8	4.3	odd	2	inner
6048.2.h.f.2591.7	yes	8	3.2	odd	2	inner
6048.2.h.f.2591.8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.34607i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-3.34607i q^{5} -1.00000i q^{7} -0.896575 q^{11} +4.00000 q^{13} +6.31319i q^{17} +1.00000i q^{19} +7.20977 q^{23} -6.19615 q^{25} +7.72741i q^{29} +3.73205i q^{31} -3.34607 q^{35} -11.7321 q^{37} +7.86611i q^{41} +7.46410i q^{43} +8.76268 q^{47} -1.00000 q^{49} +2.82843i q^{53} +3.00000i q^{55} -3.86370 q^{59} -12.3923 q^{61} -13.3843i q^{65} +10.9282i q^{67} +5.79555 q^{71} -14.3923 q^{73} +0.896575i q^{77} -7.46410i q^{79} -6.96953 q^{83} +21.1244 q^{85} +13.2456i q^{89} -4.00000i q^{91} +3.34607 q^{95} +2.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 32 q^{13} - 8 q^{25} - 80 q^{37} - 8 q^{49} - 16 q^{61} - 32 q^{73} + 72 q^{85} - 32 q^{97}+O(q^{100}) \) 8 * q + 32 * q^13 - 8 * q^25 - 80 * q^37 - 8 * q^49 - 16 * q^61 - 32 * q^73 + 72 * q^85 - 32 * q^97

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