LMFDB - Embedded newform 6048.2.c.d.3025.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(3025,6048)]

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([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6048.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.c (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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.8
Root			$0.987688 + 0.156434i$ of defining polynomial
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.d.3025.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0549306i q^{5} +1.00000 q^{7} +O(q^{10})$	$q-0.0549306i q^{5} +1.00000 q^{7} -2.63506i q^{11} +3.67853i q^{13} +3.16228 q^{17} +3.07768i q^{19} -2.86834 q^{23} +4.99698 q^{25} -10.1898i q^{29} -9.32437 q^{31} -0.0549306i q^{35} -0.774554i q^{37} +6.36183 q^{41} -9.98927i q^{43} +12.3126 q^{47} +1.00000 q^{49} +3.39291i q^{53} -0.144746 q^{55} +6.93263i q^{59} +8.35114i q^{61} +0.202064 q^{65} -8.93584i q^{67} -4.28255 q^{71} +8.38081 q^{73} -2.63506i q^{77} +3.03186 q^{79} +10.3255i q^{83} -0.173706i q^{85} +12.8155 q^{89} +3.67853i q^{91} +0.169059 q^{95} -10.1803 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{7}+O(q^{10}) $ 16 * q + 16 * q^7	$ 16 q + 16 q^{7} - 32 q^{25} - 40 q^{31} + 16 q^{49} + 72 q^{55} + 24 q^{73} - 24 q^{79} + 16 q^{97}+O(q^{100}) $ 16 * q + 16 * q^7 - 32 * q^25 - 40 * q^31 + 16 * q^49 + 72 * q^55 + 24 * q^73 - 24 * q^79 + 16 * 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
$5$	− 0.0549306i	− 0.0245657i	−0.999925	−	0.0122828i	$-0.996090\pi$
$5$	− 0.0549306i	− 0.0245657i	0.999925	−	0.0122828i	$-0.00390985\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.63506i	− 0.794502i	−0.917710	−	0.397251i	$-0.869964\pi$
$11$	− 2.63506i	− 0.794502i	0.917710	−	0.397251i	$-0.130036\pi$
$12$	0	0
$13$	3.67853i	1.02024i	0.860103	+	0.510121i	$0.170399\pi$
$13$	3.67853i	1.02024i	−0.860103	+	0.510121i	$0.829601\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.16228	0.766965	0.383482	−	0.923548i	$-0.374725\pi$
$17$	3.16228	0.766965	0.383482	+	0.923548i	$0.374725\pi$
$18$	0	0
$19$	3.07768i	0.706069i	0.935610	+	0.353035i	$0.114850\pi$
$19$	3.07768i	0.706069i	−0.935610	+	0.353035i	$0.885150\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.86834	−0.598090	−0.299045	−	0.954239i	$-0.596668\pi$
$23$	−2.86834	−0.598090	−0.299045	+	0.954239i	$0.596668\pi$
$24$	0	0
$25$	4.99698	0.999397
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 10.1898i	− 1.89219i	−0.323890	−	0.946095i	$-0.604991\pi$
$29$	− 10.1898i	− 1.89219i	0.323890	−	0.946095i	$-0.395009\pi$
$30$	0	0
$31$	−9.32437	−1.67471	−0.837353	−	0.546662i	$-0.815898\pi$
$31$	−9.32437	−1.67471	−0.837353	+	0.546662i	$0.815898\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.0549306i	− 0.00928496i
$36$	0	0
$37$	− 0.774554i	− 0.127336i	−0.997971	−	0.0636679i	$-0.979720\pi$
$37$	− 0.774554i	− 0.127336i	0.997971	−	0.0636679i	$-0.0202798\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.36183	0.993551	0.496775	−	0.867879i	$-0.334517\pi$
$41$	6.36183	0.993551	0.496775	+	0.867879i	$0.334517\pi$
$42$	0	0
$43$	− 9.98927i	− 1.52335i	−0.647960	−	0.761674i	$-0.724378\pi$
$43$	− 9.98927i	− 1.52335i	0.647960	−	0.761674i	$-0.275622\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.3126	1.79598	0.897990	−	0.440015i	$-0.145027\pi$
$47$	12.3126	1.79598	0.897990	+	0.440015i	$0.145027\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.39291i	0.466052i	0.972470	+	0.233026i	$0.0748628\pi$
$53$	3.39291i	0.466052i	−0.972470	+	0.233026i	$0.925137\pi$
$54$	0	0
$55$	−0.144746	−0.0195175
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.93263i	0.902552i	0.892384	+	0.451276i	$0.149031\pi$
$59$	6.93263i	0.902552i	−0.892384	+	0.451276i	$0.850969\pi$
$60$	0	0
$61$	8.35114i	1.06925i	0.845088	+	0.534627i	$0.179548\pi$
$61$	8.35114i	1.06925i	−0.845088	+	0.534627i	$0.820452\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.202064	0.0250629
$66$	0	0
$67$	− 8.93584i	− 1.09169i	−0.837887	−	0.545843i	$-0.816209\pi$
$67$	− 8.93584i	− 1.09169i	0.837887	−	0.545843i	$-0.183791\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.28255	−0.508245	−0.254123	−	0.967172i	$-0.581787\pi$
$71$	−4.28255	−0.508245	−0.254123	+	0.967172i	$0.581787\pi$
$72$	0	0
$73$	8.38081	0.980900	0.490450	−	0.871469i	$-0.336832\pi$
$73$	8.38081	0.980900	0.490450	+	0.871469i	$0.336832\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.63506i	− 0.300293i
$78$	0	0
$79$	3.03186	0.341111	0.170556	−	0.985348i	$-0.445444\pi$
$79$	3.03186	0.341111	0.170556	+	0.985348i	$0.445444\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.3255i	1.13338i	0.823932	+	0.566688i	$0.191776\pi$
$83$	10.3255i	1.13338i	−0.823932	+	0.566688i	$0.808224\pi$
$84$	0	0
$85$	− 0.173706i	− 0.0188410i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.8155	1.35844	0.679222	−	0.733933i	$-0.262317\pi$
$89$	12.8155	1.35844	0.679222	+	0.733933i	$0.262317\pi$
$90$	0	0
$91$	3.67853i	0.385615i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.169059	0.0173451
$96$	0	0
$97$	−10.1803	−1.03366	−0.516828	−	0.856089i	$-0.672887\pi$
$97$	−10.1803	−1.03366	−0.516828	+	0.856089i	$0.672887\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.02749i	− 0.102239i	−0.998693	−	0.0511194i	$-0.983721\pi$
$101$	− 1.02749i	− 0.102239i	0.998693	−	0.0511194i	$-0.0162789\pi$
$102$	0	0
$103$	−1.72905	−0.170368	−0.0851842	−	0.996365i	$-0.527148\pi$
$103$	−1.72905	−0.170368	−0.0851842	+	0.996365i	$0.527148\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.11985i	0.591629i	0.955245	+	0.295814i	$0.0955910\pi$
$107$	6.11985i	0.591629i	−0.955245	+	0.295814i	$0.904409\pi$
$108$	0	0
$109$	8.85597i	0.848248i	0.905604	+	0.424124i	$0.139418\pi$
$109$	8.85597i	0.848248i	−0.905604	+	0.424124i	$0.860582\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.02383	0.848891	0.424445	−	0.905454i	$-0.360469\pi$
$113$	9.02383	0.848891	0.424445	+	0.905454i	$0.360469\pi$
$114$	0	0
$115$	0.157559i	0.0146925i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.16228	0.289886
$120$	0	0
$121$	4.05644	0.368767
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 0.549140i	− 0.0491166i
$126$	0	0
$127$	−18.2656	−1.62081	−0.810406	−	0.585868i	$-0.800754\pi$
$127$	−18.2656	−1.62081	−0.810406	+	0.585868i	$0.800754\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 17.4470i	− 1.52435i	−0.647373	−	0.762174i	$-0.724132\pi$
$131$	− 17.4470i	− 1.52435i	0.647373	−	0.762174i	$-0.275868\pi$
$132$	0	0
$133$	3.07768i	0.266869i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.73613	0.660942	0.330471	−	0.943816i	$-0.392792\pi$
$137$	7.73613	0.660942	0.330471	+	0.943816i	$0.392792\pi$
$138$	0	0
$139$	0.802034i	0.0680276i	0.999421	+	0.0340138i	$0.0108290\pi$
$139$	0.802034i	0.0680276i	−0.999421	+	0.0340138i	$0.989171\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.69316	0.810583
$144$	0	0
$145$	−0.559729	−0.0464829
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 15.7165i	− 1.28755i	−0.765215	−	0.643775i	$-0.777367\pi$
$149$	− 15.7165i	− 1.28755i	0.765215	−	0.643775i	$-0.222633\pi$
$150$	0	0
$151$	−2.58129	−0.210062	−0.105031	−	0.994469i	$-0.533494\pi$
$151$	−2.58129	−0.210062	−0.105031	+	0.994469i	$0.533494\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.512193i	0.0411403i
$156$	0	0
$157$	0.680724i	0.0543277i	0.999631	+	0.0271638i	$0.00864758\pi$
$157$	0.680724i	0.0543277i	−0.999631	+	0.0271638i	$0.991352\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.86834	−0.226057
$162$	0	0
$163$	− 21.8339i	− 1.71016i	−0.518494	−	0.855081i	$-0.673507\pi$
$163$	− 21.8339i	− 1.71016i	0.518494	−	0.855081i	$-0.326493\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.2203	1.33255	0.666274	−	0.745707i	$-0.267888\pi$
$167$	17.2203	1.33255	0.666274	+	0.745707i	$0.267888\pi$
$168$	0	0
$169$	−0.531594	−0.0408918
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.57218i	− 0.195559i	−0.995208	−	0.0977797i	$-0.968826\pi$
$173$	− 2.57218i	− 0.195559i	0.995208	−	0.0977797i	$-0.0311741\pi$
$174$	0	0
$175$	4.99698	0.377736
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 5.26703i	− 0.393676i	−0.980436	−	0.196838i	$-0.936933\pi$
$179$	− 5.26703i	− 0.393676i	0.980436	−	0.196838i	$-0.0630673\pi$
$180$	0	0
$181$	− 16.0189i	− 1.19068i	−0.803475	−	0.595339i	$-0.797018\pi$
$181$	− 16.0189i	− 1.19068i	0.803475	−	0.595339i	$-0.202982\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.0425467	−0.00312809
$186$	0	0
$187$	− 8.33280i	− 0.609355i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.83730	0.132943	0.0664713	−	0.997788i	$-0.478826\pi$
$191$	1.83730	0.132943	0.0664713	+	0.997788i	$0.478826\pi$
$192$	0	0
$193$	15.3429	1.10441	0.552204	−	0.833709i	$-0.313787\pi$
$193$	15.3429	1.10441	0.552204	+	0.833709i	$0.313787\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.02447i	0.357979i	0.983851	+	0.178989i	$0.0572828\pi$
$197$	5.02447i	0.357979i	−0.983851	+	0.178989i	$0.942717\pi$
$198$	0	0
$199$	22.0333	1.56190	0.780949	−	0.624595i	$-0.214736\pi$
$199$	22.0333	1.56190	0.780949	+	0.624595i	$0.214736\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 10.1898i	− 0.715180i
$204$	0	0
$205$	− 0.349459i	− 0.0244073i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.10989	0.560973
$210$	0	0
$211$	8.50651i	0.585612i	0.956172	+	0.292806i	$0.0945890\pi$
$211$	8.50651i	0.585612i	−0.956172	+	0.292806i	$0.905411\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.548716	−0.0374221
$216$	0	0
$217$	−9.32437	−0.632980
$218$	0	0
$219$	0	0
$220$	0	0
$221$	11.6325i	0.782489i
$222$	0	0
$223$	13.5574	0.907872	0.453936	−	0.891034i	$-0.350019\pi$
$223$	13.5574	0.907872	0.453936	+	0.891034i	$0.350019\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.0721i	− 1.06674i	−0.845882	−	0.533370i	$-0.820925\pi$
$227$	− 16.0721i	− 1.06674i	0.845882	−	0.533370i	$-0.179075\pi$
$228$	0	0
$229$	5.50870i	0.364025i	0.983296	+	0.182013i	$0.0582612\pi$
$229$	5.50870i	0.364025i	−0.983296	+	0.182013i	$0.941739\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.6599	1.87757	0.938787	−	0.344498i	$-0.111951\pi$
$233$	28.6599	1.87757	0.938787	+	0.344498i	$0.111951\pi$
$234$	0	0
$235$	− 0.676339i	− 0.0441195i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.82998	−0.247741	−0.123870	−	0.992298i	$-0.539531\pi$
$239$	−3.82998	−0.247741	−0.123870	+	0.992298i	$0.539531\pi$
$240$	0	0
$241$	−1.72834	−0.111332	−0.0556660	−	0.998449i	$-0.517728\pi$
$241$	−1.72834	−0.111332	−0.0556660	+	0.998449i	$0.517728\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 0.0549306i	− 0.00350938i
$246$	0	0
$247$	−11.3214	−0.720361
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.245657i	0.0155057i	0.999970	+	0.00775286i	$0.00246784\pi$
$251$	0.245657i	0.0155057i	−0.999970	+	0.00775286i	$0.997532\pi$
$252$	0	0
$253$	7.55825i	0.475183i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.2120	−1.01128	−0.505638	−	0.862746i	$-0.668743\pi$
$257$	−16.2120	−1.01128	−0.505638	+	0.862746i	$0.668743\pi$
$258$	0	0
$259$	− 0.774554i	− 0.0481284i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.3265	1.43837	0.719187	−	0.694817i	$-0.244515\pi$
$263$	23.3265	1.43837	0.719187	+	0.694817i	$0.244515\pi$
$264$	0	0
$265$	0.186375	0.0114489
$266$	0	0
$267$	0	0
$268$	0	0
$269$	23.6947i	1.44469i	0.691534	+	0.722344i	$0.256935\pi$
$269$	23.6947i	1.44469i	−0.691534	+	0.722344i	$0.743065\pi$
$270$	0	0
$271$	−3.52786	−0.214302	−0.107151	−	0.994243i	$-0.534173\pi$
$271$	−3.52786	−0.214302	−0.107151	+	0.994243i	$0.534173\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 13.1674i	− 0.794022i
$276$	0	0
$277$	28.6014i	1.71849i	0.511561	+	0.859247i	$0.329068\pi$
$277$	28.6014i	1.71849i	−0.511561	+	0.859247i	$0.670932\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−25.6546	−1.53043	−0.765214	−	0.643776i	$-0.777367\pi$
$281$	−25.6546	−1.53043	−0.765214	+	0.643776i	$0.777367\pi$
$282$	0	0
$283$	14.8539i	0.882974i	0.897268	+	0.441487i	$0.145549\pi$
$283$	14.8539i	0.882974i	−0.897268	+	0.441487i	$0.854451\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.36183	0.375527
$288$	0	0
$289$	−7.00000	−0.411765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 21.6946i	− 1.26741i	−0.773574	−	0.633706i	$-0.781533\pi$
$293$	− 21.6946i	− 1.26741i	0.773574	−	0.633706i	$-0.218467\pi$
$294$	0	0
$295$	0.380813	0.0221718
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 10.5513i	− 0.610195i
$300$	0	0
$301$	− 9.98927i	− 0.575772i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.458733	0.0262670
$306$	0	0
$307$	− 0.422903i	− 0.0241363i	−0.999927	−	0.0120682i	$-0.996158\pi$
$307$	− 0.422903i	− 0.0241363i	0.999927	−	0.0120682i	$-0.00384151\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.6732	0.775338	0.387669	−	0.921799i	$-0.373280\pi$
$311$	13.6732	0.775338	0.387669	+	0.921799i	$0.373280\pi$
$312$	0	0
$313$	−7.68665	−0.434475	−0.217237	−	0.976119i	$-0.569705\pi$
$313$	−7.68665	−0.434475	−0.217237	+	0.976119i	$0.569705\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 24.2068i	− 1.35959i	−0.733401	−	0.679796i	$-0.762068\pi$
$317$	− 24.2068i	− 1.35959i	0.733401	−	0.679796i	$-0.237932\pi$
$318$	0	0
$319$	−26.8506	−1.50335
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.73249i	0.541530i
$324$	0	0
$325$	18.3816i	1.01963i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.3126	0.678817
$330$	0	0
$331$	9.15756i	0.503345i	0.967812	+	0.251672i	$0.0809806\pi$
$331$	9.15756i	0.503345i	−0.967812	+	0.251672i	$0.919019\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.490851	−0.0268180
$336$	0	0
$337$	33.2441	1.81092	0.905460	−	0.424432i	$-0.139526\pi$
$337$	33.2441	1.81092	0.905460	+	0.424432i	$0.139526\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24.5703i	1.33056i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.03418i	− 0.431297i	−0.976471	−	0.215649i	$-0.930813\pi$
$347$	− 8.03418i	− 0.431297i	0.976471	−	0.215649i	$-0.0691866\pi$
$348$	0	0
$349$	− 5.46493i	− 0.292531i	−0.989245	−	0.146265i	$-0.953275\pi$
$349$	− 5.46493i	− 0.292531i	0.989245	−	0.146265i	$-0.0467254\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.85159	−0.417898	−0.208949	−	0.977927i	$-0.567004\pi$
$353$	−7.85159	−0.417898	−0.208949	+	0.977927i	$0.567004\pi$
$354$	0	0
$355$	0.235243i	0.0124854i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.4616	1.60770	0.803851	−	0.594831i	$-0.202781\pi$
$359$	30.4616	1.60770	0.803851	+	0.594831i	$0.202781\pi$
$360$	0	0
$361$	9.52786	0.501467
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 0.460363i	− 0.0240965i
$366$	0	0
$367$	−8.97241	−0.468356	−0.234178	−	0.972194i	$-0.575240\pi$
$367$	−8.97241	−0.468356	−0.234178	+	0.972194i	$0.575240\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.39291i	0.176151i
$372$	0	0
$373$	− 19.6614i	− 1.01803i	−0.860759	−	0.509013i	$-0.830010\pi$
$373$	− 19.6614i	− 1.01803i	0.860759	−	0.509013i	$-0.169990\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	37.4833	1.93049
$378$	0	0
$379$	24.9871i	1.28350i	0.766914	+	0.641750i	$0.221791\pi$
$379$	24.9871i	1.28350i	−0.766914	+	0.641750i	$0.778209\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.3323	1.03893	0.519465	−	0.854492i	$-0.326131\pi$
$383$	20.3323	1.03893	0.519465	+	0.854492i	$0.326131\pi$
$384$	0	0
$385$	−0.144746	−0.00737691
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 2.92753i	− 0.148432i	−0.997242	−	0.0742159i	$-0.976355\pi$
$389$	− 2.92753i	− 0.148432i	0.997242	−	0.0742159i	$-0.0236454\pi$
$390$	0	0
$391$	−9.07048	−0.458714
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 0.166542i	− 0.00837964i
$396$	0	0
$397$	− 8.20287i	− 0.411690i	−0.978585	−	0.205845i	$-0.934006\pi$
$397$	− 8.20287i	− 0.411690i	0.978585	−	0.205845i	$-0.0659943\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.5590	1.82567	0.912834	−	0.408332i	$-0.133889\pi$
$401$	36.5590	1.82567	0.912834	+	0.408332i	$0.133889\pi$
$402$	0	0
$403$	− 34.3000i	− 1.70860i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.04100	−0.101169
$408$	0	0
$409$	−6.01783	−0.297562	−0.148781	−	0.988870i	$-0.547535\pi$
$409$	−6.01783	−0.297562	−0.148781	+	0.988870i	$0.547535\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.93263i	0.341133i
$414$	0	0
$415$	0.567188	0.0278422
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.25247i	0.110040i	0.998485	+	0.0550201i	$0.0175223\pi$
$419$	2.25247i	0.110040i	−0.998485	+	0.0550201i	$0.982478\pi$
$420$	0	0
$421$	2.87431i	0.140085i	0.997544	+	0.0700425i	$0.0223135\pi$
$421$	2.87431i	0.140085i	−0.997544	+	0.0700425i	$0.977686\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.8018	0.766502
$426$	0	0
$427$	8.35114i	0.404140i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.7211	0.757261	0.378630	−	0.925548i	$-0.376395\pi$
$431$	15.7211	0.757261	0.378630	+	0.925548i	$0.376395\pi$
$432$	0	0
$433$	−3.44027	−0.165329	−0.0826644	−	0.996577i	$-0.526343\pi$
$433$	−3.44027	−0.165329	−0.0826644	+	0.996577i	$0.526343\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.82783i	− 0.422292i
$438$	0	0
$439$	12.2955	0.586833	0.293417	−	0.955985i	$-0.405208\pi$
$439$	12.2955	0.586833	0.293417	+	0.955985i	$0.405208\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.8932i	0.755107i	0.925988	+	0.377553i	$0.123235\pi$
$443$	15.8932i	0.755107i	−0.925988	+	0.377553i	$0.876765\pi$
$444$	0	0
$445$	− 0.703964i	− 0.0333711i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.97450	0.423533	0.211766	−	0.977320i	$-0.432078\pi$
$449$	8.97450	0.423533	0.211766	+	0.977320i	$0.432078\pi$
$450$	0	0
$451$	− 16.7638i	− 0.789377i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.202064	0.00947290
$456$	0	0
$457$	−15.0037	−0.701845	−0.350922	−	0.936405i	$-0.614132\pi$
$457$	−15.0037	−0.701845	−0.350922	+	0.936405i	$0.614132\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.69884i	0.405145i	0.979267	+	0.202573i	$0.0649303\pi$
$461$	8.69884i	0.405145i	−0.979267	+	0.202573i	$0.935070\pi$
$462$	0	0
$463$	9.34522	0.434309	0.217155	−	0.976137i	$-0.430322\pi$
$463$	9.34522	0.434309	0.217155	+	0.976137i	$0.430322\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.23326i	0.427265i	0.976914	+	0.213632i	$0.0685294\pi$
$467$	9.23326i	0.427265i	−0.976914	+	0.213632i	$0.931471\pi$
$468$	0	0
$469$	− 8.93584i	− 0.412619i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−26.3223	−1.21030
$474$	0	0
$475$	15.3791i	0.705643i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.66050	−0.395708	−0.197854	−	0.980231i	$-0.563397\pi$
$479$	−8.66050	−0.395708	−0.197854	+	0.980231i	$0.563397\pi$
$480$	0	0
$481$	2.84922	0.129913
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.559212i	0.0253925i
$486$	0	0
$487$	−11.1424	−0.504912	−0.252456	−	0.967608i	$-0.581238\pi$
$487$	−11.1424	−0.504912	−0.252456	+	0.967608i	$0.581238\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 25.4479i	− 1.14845i	−0.818698	−	0.574224i	$-0.805304\pi$
$491$	− 25.4479i	− 1.14845i	0.818698	−	0.574224i	$-0.194696\pi$
$492$	0	0
$493$	− 32.2228i	− 1.45124i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.28255	−0.192099
$498$	0	0
$499$	− 4.17422i	− 0.186864i	−0.995626	−	0.0934318i	$-0.970216\pi$
$499$	− 4.17422i	− 0.186864i	0.995626	−	0.0934318i	$-0.0297837\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10.6106	0.473105	0.236552	−	0.971619i	$-0.423983\pi$
$503$	10.6106	0.473105	0.236552	+	0.971619i	$0.423983\pi$
$504$	0	0
$505$	−0.0564404	−0.00251156
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 24.1111i	− 1.06871i	−0.845262	−	0.534353i	$-0.820555\pi$
$509$	− 24.1111i	− 1.06871i	0.845262	−	0.534353i	$-0.179445\pi$
$510$	0	0
$511$	8.38081	0.370745
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.0949777i	0.00418522i
$516$	0	0
$517$	− 32.4445i	− 1.42691i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−32.5642	−1.42666	−0.713331	−	0.700827i	$-0.752815\pi$
$521$	−32.5642	−1.42666	−0.713331	+	0.700827i	$0.752815\pi$
$522$	0	0
$523$	− 18.6166i	− 0.814046i	−0.913418	−	0.407023i	$-0.866567\pi$
$523$	− 18.6166i	− 0.814046i	0.913418	−	0.407023i	$-0.133433\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.4863	−1.28444
$528$	0	0
$529$	−14.7726	−0.642289
$530$	0	0
$531$	0	0
$532$	0	0
$533$	23.4022i	1.01366i
$534$	0	0
$535$	0.336167	0.0145338
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.63506i	− 0.113500i
$540$	0	0
$541$	41.3556i	1.77802i	0.457891	+	0.889008i	$0.348605\pi$
$541$	41.3556i	1.77802i	−0.457891	+	0.889008i	$0.651395\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.486463	0.0208378
$546$	0	0
$547$	− 18.8135i	− 0.804408i	−0.915550	−	0.402204i	$-0.868244\pi$
$547$	− 18.8135i	− 0.804408i	0.915550	−	0.402204i	$-0.131756\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	31.3608	1.33602
$552$	0	0
$553$	3.03186	0.128928
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.05808i	0.171946i	0.996297	+	0.0859731i	$0.0273999\pi$
$557$	4.05808i	0.171946i	−0.996297	+	0.0859731i	$0.972600\pi$
$558$	0	0
$559$	36.7458	1.55418
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 45.2454i	− 1.90687i	−0.301605	−	0.953433i	$-0.597522\pi$
$563$	− 45.2454i	− 1.90687i	0.301605	−	0.953433i	$-0.402478\pi$
$564$	0	0
$565$	− 0.495684i	− 0.0208536i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.6255	0.571212	0.285606	−	0.958347i	$-0.407805\pi$
$569$	13.6255	0.571212	0.285606	+	0.958347i	$0.407805\pi$
$570$	0	0
$571$	30.9574i	1.29553i	0.761842	+	0.647763i	$0.224295\pi$
$571$	30.9574i	1.29553i	−0.761842	+	0.647763i	$0.775705\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−14.3330	−0.597729
$576$	0	0
$577$	−38.4675	−1.60142	−0.800712	−	0.599049i	$-0.795545\pi$
$577$	−38.4675	−1.60142	−0.800712	+	0.599049i	$0.795545\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.3255i	0.428376i
$582$	0	0
$583$	8.94054	0.370279
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 28.6131i	− 1.18099i	−0.807041	−	0.590495i	$-0.798932\pi$
$587$	− 28.6131i	− 1.18099i	0.807041	−	0.590495i	$-0.201068\pi$
$588$	0	0
$589$	− 28.6975i	− 1.18246i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.5554	0.926239	0.463119	−	0.886296i	$-0.346730\pi$
$593$	22.5554	0.926239	0.463119	+	0.886296i	$0.346730\pi$
$594$	0	0
$595$	− 0.173706i	− 0.00712124i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−42.1743	−1.72319	−0.861597	−	0.507593i	$-0.830535\pi$
$599$	−42.1743	−1.72319	−0.861597	+	0.507593i	$0.830535\pi$
$600$	0	0
$601$	−14.0155	−0.571705	−0.285853	−	0.958274i	$-0.592277\pi$
$601$	−14.0155	−0.571705	−0.285853	+	0.958274i	$0.592277\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 0.222823i	− 0.00905902i
$606$	0	0
$607$	21.5893	0.876282	0.438141	−	0.898906i	$-0.355637\pi$
$607$	21.5893	0.876282	0.438141	+	0.898906i	$0.355637\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	45.2924i	1.83233i
$612$	0	0
$613$	0.788508i	0.0318475i	0.999873	+	0.0159238i	$0.00506891\pi$
$613$	0.788508i	0.0318475i	−0.999873	+	0.0159238i	$0.994931\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−34.4936	−1.38866	−0.694331	−	0.719656i	$-0.744299\pi$
$617$	−34.4936	−1.38866	−0.694331	+	0.719656i	$0.744299\pi$
$618$	0	0
$619$	47.8005i	1.92127i	0.277822	+	0.960633i	$0.410388\pi$
$619$	47.8005i	1.92127i	−0.277822	+	0.960633i	$0.589612\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.8155	0.513443
$624$	0	0
$625$	24.9547	0.998190
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 2.44935i	− 0.0976621i
$630$	0	0
$631$	21.7200	0.864659	0.432330	−	0.901716i	$-0.357692\pi$
$631$	21.7200	0.864659	0.432330	+	0.901716i	$0.357692\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.00334i	0.0398164i
$636$	0	0
$637$	3.67853i	0.145749i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17.0166	0.672117	0.336058	−	0.941841i	$-0.390906\pi$
$641$	17.0166	0.672117	0.336058	+	0.941841i	$0.390906\pi$
$642$	0	0
$643$	38.4170i	1.51502i	0.652824	+	0.757509i	$0.273584\pi$
$643$	38.4170i	1.51502i	−0.652824	+	0.757509i	$0.726416\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.7497	−0.894382	−0.447191	−	0.894439i	$-0.647575\pi$
$647$	−22.7497	−0.894382	−0.447191	+	0.894439i	$0.647575\pi$
$648$	0	0
$649$	18.2679	0.717079
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 12.9198i	− 0.505591i	−0.967520	−	0.252795i	$-0.918650\pi$
$653$	− 12.9198i	− 0.505591i	0.967520	−	0.252795i	$-0.0813499\pi$
$654$	0	0
$655$	−0.958371	−0.0374466
$656$	0	0
$657$	0	0
$658$	0	0
$659$	47.1905i	1.83828i	0.393932	+	0.919140i	$0.371115\pi$
$659$	47.1905i	1.83828i	−0.393932	+	0.919140i	$0.628885\pi$
$660$	0	0
$661$	− 31.5648i	− 1.22773i	−0.789412	−	0.613864i	$-0.789614\pi$
$661$	− 31.5648i	− 1.22773i	0.789412	−	0.613864i	$-0.210386\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.169059	0.00655582
$666$	0	0
$667$	29.2276i	1.13170i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	22.0058	0.849524
$672$	0	0
$673$	−37.1186	−1.43082	−0.715408	−	0.698707i	$-0.753759\pi$
$673$	−37.1186	−1.43082	−0.715408	+	0.698707i	$0.753759\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.8912i	− 0.610750i	−0.952232	−	0.305375i	$-0.901218\pi$
$677$	− 15.8912i	− 0.610750i	0.952232	−	0.305375i	$-0.0987818\pi$
$678$	0	0
$679$	−10.1803	−0.390686
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.4319i	1.04965i	0.851210	+	0.524826i	$0.175869\pi$
$683$	27.4319i	1.04965i	−0.851210	+	0.524826i	$0.824131\pi$
$684$	0	0
$685$	− 0.424950i	− 0.0162365i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.4809	−0.475486
$690$	0	0
$691$	6.57770i	0.250227i	0.992142	+	0.125114i	$0.0399296\pi$
$691$	6.57770i	0.250227i	−0.992142	+	0.125114i	$0.960070\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.0440562	0.00167115
$696$	0	0
$697$	20.1179	0.762018
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.07153i	0.342627i	0.985217	+	0.171314i	$0.0548011\pi$
$701$	9.07153i	0.342627i	−0.985217	+	0.171314i	$0.945199\pi$
$702$	0	0
$703$	2.38383	0.0899079
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.02749i	− 0.0386426i
$708$	0	0
$709$	− 6.16889i	− 0.231678i	−0.993268	−	0.115839i	$-0.963044\pi$
$709$	− 6.16889i	− 0.231678i	0.993268	−	0.115839i	$-0.0369556\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	26.7454	1.00162
$714$	0	0
$715$	− 0.532451i	− 0.0199125i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30.5440	1.13910	0.569550	−	0.821956i	$-0.307117\pi$
$719$	30.5440	1.13910	0.569550	+	0.821956i	$0.307117\pi$
$720$	0	0
$721$	−1.72905	−0.0643932
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 50.9180i	− 1.89105i
$726$	0	0
$727$	21.3429	0.791565	0.395782	−	0.918344i	$-0.370473\pi$
$727$	21.3429	0.791565	0.395782	+	0.918344i	$0.370473\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 31.5888i	− 1.16836i
$732$	0	0
$733$	15.1235i	0.558599i	0.960204	+	0.279300i	$0.0901023\pi$
$733$	15.1235i	0.558599i	−0.960204	+	0.279300i	$0.909898\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−23.5465	−0.867347
$738$	0	0
$739$	− 38.7212i	− 1.42438i	−0.701985	−	0.712192i	$-0.747703\pi$
$739$	− 38.7212i	− 1.42438i	0.701985	−	0.712192i	$-0.252297\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−39.7258	−1.45740	−0.728698	−	0.684835i	$-0.759874\pi$
$743$	−39.7258	−1.45740	−0.728698	+	0.684835i	$0.759874\pi$
$744$	0	0
$745$	−0.863319	−0.0316296
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.11985i	0.223615i
$750$	0	0
$751$	2.96814	0.108309	0.0541544	−	0.998533i	$-0.482754\pi$
$751$	2.96814	0.108309	0.0541544	+	0.998533i	$0.482754\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.141792i	0.00516032i
$756$	0	0
$757$	− 47.1588i	− 1.71401i	−0.515305	−	0.857007i	$-0.672321\pi$
$757$	− 47.1588i	− 1.71401i	0.515305	−	0.857007i	$-0.327679\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.8000	1.47900	0.739499	−	0.673158i	$-0.235063\pi$
$761$	40.8000	1.47900	0.739499	+	0.673158i	$0.235063\pi$
$762$	0	0
$763$	8.85597i	0.320608i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25.5019	−0.920821
$768$	0	0
$769$	−52.9121	−1.90806	−0.954029	−	0.299714i	$-0.903109\pi$
$769$	−52.9121	−1.90806	−0.954029	+	0.299714i	$0.903109\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	39.0168i	1.40334i	0.712503	+	0.701669i	$0.247561\pi$
$773$	39.0168i	1.40334i	−0.712503	+	0.701669i	$0.752439\pi$
$774$	0	0
$775$	−46.5937	−1.67370
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.5797i	0.701515i
$780$	0	0
$781$	11.2848i	0.403802i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.0373925	0.00133460
$786$	0	0
$787$	1.01675i	0.0362431i	0.999836	+	0.0181215i	$0.00576858\pi$
$787$	1.01675i	0.0362431i	−0.999836	+	0.0181215i	$0.994231\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.02383	0.320851
$792$	0	0
$793$	−30.7199	−1.09090
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 50.9667i	− 1.80533i	−0.430339	−	0.902667i	$-0.641606\pi$
$797$	− 50.9667i	− 1.80533i	0.430339	−	0.902667i	$-0.358394\pi$
$798$	0	0
$799$	38.9359	1.37745
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 22.0840i	− 0.779327i
$804$	0	0
$805$	0.157559i	0.00555324i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−20.3800	−0.716522	−0.358261	−	0.933622i	$-0.616630\pi$
$809$	−20.3800	−0.716522	−0.358261	+	0.933622i	$0.616630\pi$
$810$	0	0
$811$	40.7638i	1.43141i	0.698402	+	0.715706i	$0.253895\pi$
$811$	40.7638i	1.43141i	−0.698402	+	0.715706i	$0.746105\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.19935	−0.0420113
$816$	0	0
$817$	30.7438	1.07559
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 35.8394i	− 1.25080i	−0.780303	−	0.625402i	$-0.784935\pi$
$821$	− 35.8394i	− 1.25080i	0.780303	−	0.625402i	$-0.215065\pi$
$822$	0	0
$823$	−34.8745	−1.21565	−0.607824	−	0.794071i	$-0.707958\pi$
$823$	−34.8745	−1.21565	−0.607824	+	0.794071i	$0.707958\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42.4727i	1.47692i	0.674296	+	0.738461i	$0.264447\pi$
$827$	42.4727i	1.47692i	−0.674296	+	0.738461i	$0.735553\pi$
$828$	0	0
$829$	− 33.1210i	− 1.15034i	−0.818034	−	0.575169i	$-0.804936\pi$
$829$	− 33.1210i	− 1.15034i	0.818034	−	0.575169i	$-0.195064\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.16228	0.109566
$834$	0	0
$835$	− 0.945922i	− 0.0327350i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−43.5817	−1.50461	−0.752304	−	0.658817i	$-0.771057\pi$
$839$	−43.5817	−1.50461	−0.752304	+	0.658817i	$0.771057\pi$
$840$	0	0
$841$	−74.8311	−2.58038
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.0292007i	0.00100454i
$846$	0	0
$847$	4.05644	0.139381
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.22168i	0.0761582i
$852$	0	0
$853$	36.8928i	1.26319i	0.775300	+	0.631593i	$0.217599\pi$
$853$	36.8928i	1.26319i	−0.775300	+	0.631593i	$0.782401\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19.2607	−0.657933	−0.328967	−	0.944342i	$-0.606700\pi$
$857$	−19.2607	−0.657933	−0.328967	+	0.944342i	$0.606700\pi$
$858$	0	0
$859$	− 19.4667i	− 0.664195i	−0.943245	−	0.332098i	$-0.892244\pi$
$859$	− 19.4667i	− 0.664195i	0.943245	−	0.332098i	$-0.107756\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21.4715	−0.730898	−0.365449	−	0.930831i	$-0.619085\pi$
$863$	−21.4715	−0.730898	−0.365449	+	0.930831i	$0.619085\pi$
$864$	0	0
$865$	−0.141291	−0.00480405
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 7.98916i	− 0.271014i
$870$	0	0
$871$	32.8708	1.11378
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 0.549140i	− 0.0185643i
$876$	0	0
$877$	23.8421i	0.805091i	0.915400	+	0.402545i	$0.131874\pi$
$877$	23.8421i	0.805091i	−0.915400	+	0.402545i	$0.868126\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−8.81134	−0.296862	−0.148431	−	0.988923i	$-0.547422\pi$
$881$	−8.81134	−0.296862	−0.148431	+	0.988923i	$0.547422\pi$
$882$	0	0
$883$	30.0935i	1.01273i	0.862320	+	0.506363i	$0.169011\pi$
$883$	30.0935i	1.01273i	−0.862320	+	0.506363i	$0.830989\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.0549	−0.438340	−0.219170	−	0.975687i	$-0.570335\pi$
$887$	−13.0549	−0.438340	−0.219170	+	0.975687i	$0.570335\pi$
$888$	0	0
$889$	−18.2656	−0.612609
$890$	0	0
$891$	0	0
$892$	0	0
$893$	37.8944i	1.26809i
$894$	0	0
$895$	−0.289321	−0.00967092
$896$	0	0
$897$	0	0
$898$	0	0
$899$	95.0131i	3.16886i
$900$	0	0
$901$	10.7293i	0.357446i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.879929	−0.0292498
$906$	0	0
$907$	− 35.8987i	− 1.19200i	−0.802985	−	0.595999i	$-0.796756\pi$
$907$	− 35.8987i	− 1.19200i	0.802985	−	0.595999i	$-0.203244\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−46.3926	−1.53705	−0.768527	−	0.639817i	$-0.779010\pi$
$911$	−46.3926	−1.53705	−0.768527	+	0.639817i	$0.779010\pi$
$912$	0	0
$913$	27.2085	0.900469
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 17.4470i	− 0.576149i
$918$	0	0
$919$	9.87878	0.325871	0.162935	−	0.986637i	$-0.447904\pi$
$919$	9.87878	0.325871	0.162935	+	0.986637i	$0.447904\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 15.7535i	− 0.518533i
$924$	0	0
$925$	− 3.87043i	− 0.127259i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.21688	−0.105543	−0.0527713	−	0.998607i	$-0.516805\pi$
$929$	−3.21688	−0.105543	−0.0527713	+	0.998607i	$0.516805\pi$
$930$	0	0
$931$	3.07768i	0.100867i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.457725	−0.0149692
$936$	0	0
$937$	54.7874	1.78983	0.894913	−	0.446241i	$-0.147237\pi$
$937$	54.7874	1.78983	0.894913	+	0.446241i	$0.147237\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 8.31449i	− 0.271045i	−0.990774	−	0.135522i	$-0.956729\pi$
$941$	− 8.31449i	− 0.271045i	0.990774	−	0.135522i	$-0.0432713\pi$
$942$	0	0
$943$	−18.2479	−0.594232
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 12.2157i	− 0.396958i	−0.980105	−	0.198479i	$-0.936400\pi$
$947$	− 12.2157i	− 0.396958i	0.980105	−	0.198479i	$-0.0636001\pi$
$948$	0	0
$949$	30.8291i	1.00075i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−50.2955	−1.62923	−0.814615	−	0.580002i	$-0.803052\pi$
$953$	−50.2955	−1.62923	−0.814615	+	0.580002i	$0.803052\pi$
$954$	0	0
$955$	− 0.100924i	− 0.00326583i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.73613	0.249813
$960$	0	0
$961$	55.9439	1.80464
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 0.842795i	− 0.0271305i
$966$	0	0
$967$	−6.17661	−0.198626	−0.0993132	−	0.995056i	$-0.531665\pi$
$967$	−6.17661	−0.198626	−0.0993132	+	0.995056i	$0.531665\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.74342i	0.120132i	0.998194	+	0.0600660i	$0.0191311\pi$
$971$	3.74342i	0.120132i	−0.998194	+	0.0600660i	$0.980869\pi$
$972$	0	0
$973$	0.802034i	0.0257120i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.1012	−1.37893	−0.689464	−	0.724320i	$-0.742154\pi$
$977$	−43.1012	−1.37893	−0.689464	+	0.724320i	$0.742154\pi$
$978$	0	0
$979$	− 33.7697i	− 1.07929i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−46.8079	−1.49294	−0.746470	−	0.665419i	$-0.768253\pi$
$983$	−46.8079	−1.49294	−0.746470	+	0.665419i	$0.768253\pi$
$984$	0	0
$985$	0.275997	0.00879399
$986$	0	0
$987$	0	0
$988$	0	0
$989$	28.6526i	0.911099i
$990$	0	0
$991$	44.6923	1.41970	0.709850	−	0.704353i	$-0.248763\pi$
$991$	44.6923	1.41970	0.709850	+	0.704353i	$0.248763\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.21030i	− 0.0383691i
$996$	0	0
$997$	− 41.1361i	− 1.30279i	−0.758737	−	0.651397i	$-0.774183\pi$
$997$	− 41.1361i	− 1.30279i	0.758737	−	0.651397i	$-0.225817\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.d.3025.8		16
3.2	odd	2	inner	6048.2.c.d.3025.10		16
4.3	odd	2		1512.2.c.d.757.10	yes	16
8.3	odd	2		1512.2.c.d.757.11	yes	16
8.5	even	2	inner	6048.2.c.d.3025.9		16
12.11	even	2		1512.2.c.d.757.7	yes	16
24.5	odd	2	inner	6048.2.c.d.3025.7		16
24.11	even	2		1512.2.c.d.757.6	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.d.757.6	✓	16	24.11	even	2
1512.2.c.d.757.7	yes	16	12.11	even	2
1512.2.c.d.757.10	yes	16	4.3	odd	2
1512.2.c.d.757.11	yes	16	8.3	odd	2
6048.2.c.d.3025.7		16	24.5	odd	2	inner
6048.2.c.d.3025.8		16	1.1	even	1	trivial
6048.2.c.d.3025.9		16	8.5	even	2	inner
6048.2.c.d.3025.10		16	3.2	odd	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.0549306i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-0.0549306i q^{5} +1.00000 q^{7} -2.63506i q^{11} +3.67853i q^{13} +3.16228 q^{17} +3.07768i q^{19} -2.86834 q^{23} +4.99698 q^{25} -10.1898i q^{29} -9.32437 q^{31} -0.0549306i q^{35} -0.774554i q^{37} +6.36183 q^{41} -9.98927i q^{43} +12.3126 q^{47} +1.00000 q^{49} +3.39291i q^{53} -0.144746 q^{55} +6.93263i q^{59} +8.35114i q^{61} +0.202064 q^{65} -8.93584i q^{67} -4.28255 q^{71} +8.38081 q^{73} -2.63506i q^{77} +3.03186 q^{79} +10.3255i q^{83} -0.173706i q^{85} +12.8155 q^{89} +3.67853i q^{91} +0.169059 q^{95} -10.1803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7}+O(q^{10}) \) 16 * q + 16 * q^7	\( 16 q + 16 q^{7} - 32 q^{25} - 40 q^{31} + 16 q^{49} + 72 q^{55} + 24 q^{73} - 24 q^{79} + 16 q^{97}+O(q^{100}) \) 16 * q + 16 * q^7 - 32 * q^25 - 40 * q^31 + 16 * q^49 + 72 * q^55 + 24 * q^73 - 24 * q^79 + 16 * q^97

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