LMFDB - Embedded newform 6048.2.h.d.2591.8

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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.14626i q^{5} +1.00000i q^{7} +O(q^{10})$	$q+3.14626i q^{5} +1.00000i q^{7} +2.04989 q^{11} +2.44949 q^{13} +3.14626i q^{17} +4.44949i q^{19} +2.82843 q^{23} -4.89898 q^{25} +0.778539i q^{29} +0.449490i q^{31} -3.14626 q^{35} -7.00000 q^{37} +2.51059i q^{41} +4.34847i q^{43} +7.38891 q^{47} -1.00000 q^{49} +6.29253i q^{53} +6.44949i q^{55} +5.83183 q^{59} -5.34847 q^{61} +7.70674i q^{65} +8.00000i q^{67} +12.8990 q^{73} +2.04989i q^{77} -13.4495i q^{79} +6.75323 q^{83} -9.89898 q^{85} -14.1421i q^{89} +2.44949i q^{91} -13.9993 q^{95} -12.2474 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 56 q^{37} - 8 q^{49} + 16 q^{61} + 64 q^{73} - 40 q^{85}+O(q^{100}) $ 8 * q - 56 * q^37 - 8 * q^49 + 16 * q^61 + 64 * q^73 - 40 * q^85

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
$5$	3.14626i	1.40705i	0.710669	+	0.703526i	$0.248392\pi$
$5$	3.14626i	1.40705i	−0.710669	+	0.703526i	$0.751608\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.04989	0.618065	0.309032	−	0.951052i	$-0.399995\pi$
$11$	2.04989	0.618065	0.309032	+	0.951052i	$0.399995\pi$
$12$	0	0
$13$	2.44949	0.679366	0.339683	−	0.940540i	$-0.389680\pi$
$13$	2.44949	0.679366	0.339683	+	0.940540i	$0.389680\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.14626i	0.763081i	0.924352	+	0.381541i	$0.124606\pi$
$17$	3.14626i	0.763081i	−0.924352	+	0.381541i	$0.875394\pi$
$18$	0	0
$19$	4.44949i	1.02078i	0.859942	+	0.510391i	$0.170499\pi$
$19$	4.44949i	1.02078i	−0.859942	+	0.510391i	$0.829501\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	−4.89898	−0.979796
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.778539i	0.144571i	0.997384	+	0.0722855i	$0.0230293\pi$
$29$	0.778539i	0.144571i	−0.997384	+	0.0722855i	$0.976971\pi$
$30$	0	0
$31$	0.449490i	0.0807307i	0.999185	+	0.0403654i	$0.0128522\pi$
$31$	0.449490i	0.0807307i	−0.999185	+	0.0403654i	$0.987148\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.14626	−0.531816
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.51059i	0.392088i	0.980595	+	0.196044i	$0.0628096\pi$
$41$	2.51059i	0.392088i	−0.980595	+	0.196044i	$0.937190\pi$
$42$	0	0
$43$	4.34847i	0.663135i	0.943431	+	0.331568i	$0.107578\pi$
$43$	4.34847i	0.663135i	−0.943431	+	0.331568i	$0.892422\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.38891	1.07778	0.538891	−	0.842375i	$-0.318843\pi$
$47$	7.38891	1.07778	0.538891	+	0.842375i	$0.318843\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.29253i	0.864345i	0.901791	+	0.432173i	$0.142253\pi$
$53$	6.29253i	0.864345i	−0.901791	+	0.432173i	$0.857747\pi$
$54$	0	0
$55$	6.44949i	0.869649i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.83183	0.759239	0.379620	−	0.925143i	$-0.376055\pi$
$59$	5.83183	0.759239	0.379620	+	0.925143i	$0.376055\pi$
$60$	0	0
$61$	−5.34847	−0.684801	−0.342401	−	0.939554i	$-0.611240\pi$
$61$	−5.34847	−0.684801	−0.342401	+	0.939554i	$0.611240\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.70674i	0.955904i
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	12.8990	1.50971	0.754856	−	0.655891i	$-0.227707\pi$
$73$	12.8990	1.50971	0.754856	+	0.655891i	$0.227707\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.04989i	0.233606i
$78$	0	0
$79$	− 13.4495i	− 1.51319i	−0.653886	−	0.756593i	$-0.726863\pi$
$79$	− 13.4495i	− 1.51319i	0.653886	−	0.756593i	$-0.273137\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.75323	0.741263	0.370632	−	0.928780i	$-0.379141\pi$
$83$	6.75323	0.741263	0.370632	+	0.928780i	$0.379141\pi$
$84$	0	0
$85$	−9.89898	−1.07370
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 14.1421i	− 1.49906i	−0.661968	−	0.749532i	$-0.730279\pi$
$89$	− 14.1421i	− 1.49906i	0.661968	−	0.749532i	$-0.269721\pi$
$90$	0	0
$91$	2.44949i	0.256776i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−13.9993	−1.43629
$96$	0	0
$97$	−12.2474	−1.24354	−0.621770	−	0.783200i	$-0.713586\pi$
$97$	−12.2474	−1.24354	−0.621770	+	0.783200i	$0.713586\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 18.2419i	− 1.81514i	−0.419903	−	0.907569i	$-0.637936\pi$
$101$	− 18.2419i	− 1.81514i	0.419903	−	0.907569i	$-0.362064\pi$
$102$	0	0
$103$	− 12.8990i	− 1.27097i	−0.772111	−	0.635487i	$-0.780799\pi$
$103$	− 12.8990i	− 1.27097i	0.772111	−	0.635487i	$-0.219201\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1421	1.36717	0.683586	−	0.729870i	$-0.260419\pi$
$107$	14.1421	1.36717	0.683586	+	0.729870i	$0.260419\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.1708i	1.05086i	0.850835	+	0.525432i	$0.176096\pi$
$113$	11.1708i	1.05086i	−0.850835	+	0.525432i	$0.823904\pi$
$114$	0	0
$115$	8.89898i	0.829834i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.14626	−0.288418
$120$	0	0
$121$	−6.79796	−0.617996
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0.317837i	0.0284282i
$126$	0	0
$127$	9.24745i	0.820578i	0.911955	+	0.410289i	$0.134572\pi$
$127$	9.24745i	0.820578i	−0.911955	+	0.410289i	$0.865428\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.73545	0.413738	0.206869	−	0.978369i	$-0.433673\pi$
$131$	4.73545	0.413738	0.206869	+	0.978369i	$0.433673\pi$
$132$	0	0
$133$	−4.44949	−0.385820
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.8065i	1.00870i	0.863500	+	0.504349i	$0.168268\pi$
$137$	11.8065i	1.00870i	−0.863500	+	0.504349i	$0.831732\pi$
$138$	0	0
$139$	12.2474i	1.03882i	0.854527	+	0.519408i	$0.173847\pi$
$139$	12.2474i	1.03882i	−0.854527	+	0.519408i	$0.826153\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.02118	0.419892
$144$	0	0
$145$	−2.44949	−0.203419
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.97129i	0.243418i	0.992566	+	0.121709i	$0.0388374\pi$
$149$	2.97129i	0.243418i	−0.992566	+	0.121709i	$0.961163\pi$
$150$	0	0
$151$	7.24745i	0.589789i	0.955530	+	0.294895i	$0.0952845\pi$
$151$	7.24745i	0.589789i	−0.955530	+	0.294895i	$0.904715\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.41421	−0.113592
$156$	0	0
$157$	0.651531	0.0519978	0.0259989	−	0.999662i	$-0.491723\pi$
$157$	0.651531	0.0519978	0.0259989	+	0.999662i	$0.491723\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.82843i	0.222911i
$162$	0	0
$163$	13.2474i	1.03762i	0.854889	+	0.518810i	$0.173625\pi$
$163$	13.2474i	1.03762i	−0.854889	+	0.518810i	$0.826375\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.2453	−1.64401	−0.822006	−	0.569479i	$-0.807145\pi$
$167$	−21.2453	−1.64401	−0.822006	+	0.569479i	$0.807145\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.19275i	− 0.166712i	−0.996520	−	0.0833559i	$-0.973436\pi$
$173$	− 2.19275i	− 0.166712i	0.996520	−	0.0833559i	$-0.0265638\pi$
$174$	0	0
$175$	− 4.89898i	− 0.370328i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.04989	0.153216	0.0766079	−	0.997061i	$-0.475591\pi$
$179$	2.04989	0.153216	0.0766079	+	0.997061i	$0.475591\pi$
$180$	0	0
$181$	−4.69694	−0.349121	−0.174560	−	0.984646i	$-0.555850\pi$
$181$	−4.69694	−0.349121	−0.174560	+	0.984646i	$0.555850\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 22.0239i	− 1.61923i
$186$	0	0
$187$	6.44949i	0.471633i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.73545	−0.342645	−0.171323	−	0.985215i	$-0.554804\pi$
$191$	−4.73545	−0.342645	−0.171323	+	0.985215i	$0.554804\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.70674i	− 0.549083i	−0.961575	−	0.274541i	$-0.911474\pi$
$197$	− 7.70674i	− 0.549083i	0.961575	−	0.274541i	$-0.0885260\pi$
$198$	0	0
$199$	10.0000i	0.708881i	0.935079	+	0.354441i	$0.115329\pi$
$199$	10.0000i	0.708881i	−0.935079	+	0.354441i	$0.884671\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.778539	−0.0546427
$204$	0	0
$205$	−7.89898	−0.551689
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.12096i	0.630910i
$210$	0	0
$211$	6.00000i	0.413057i	0.978441	+	0.206529i	$0.0662166\pi$
$211$	6.00000i	0.413057i	−0.978441	+	0.206529i	$0.933783\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−13.6814	−0.933066
$216$	0	0
$217$	−0.449490	−0.0305134
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.70674i	0.518412i
$222$	0	0
$223$	− 6.24745i	− 0.418360i	−0.977877	−	0.209180i	$-0.932921\pi$
$223$	− 6.24745i	− 0.418360i	0.977877	−	0.209180i	$-0.0670795\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.75663	0.647570	0.323785	−	0.946131i	$-0.395045\pi$
$227$	9.75663	0.647570	0.323785	+	0.946131i	$0.395045\pi$
$228$	0	0
$229$	−16.6969	−1.10336	−0.551682	−	0.834054i	$-0.686014\pi$
$229$	−16.6969	−1.10336	−0.551682	+	0.834054i	$0.686014\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 15.9063i	− 1.04206i	−0.853540	−	0.521028i	$-0.825549\pi$
$233$	− 15.9063i	− 1.04206i	0.853540	−	0.521028i	$-0.174451\pi$
$234$	0	0
$235$	23.2474i	1.51650i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−17.4634	−1.12961	−0.564806	−	0.825224i	$-0.691049\pi$
$239$	−17.4634	−1.12961	−0.564806	+	0.825224i	$0.691049\pi$
$240$	0	0
$241$	−28.2474	−1.81958	−0.909789	−	0.415071i	$-0.863757\pi$
$241$	−28.2474	−1.81958	−0.909789	+	0.415071i	$0.863757\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3.14626i	− 0.201007i
$246$	0	0
$247$	10.8990i	0.693485i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.7600	0.805406	0.402703	−	0.915331i	$-0.368071\pi$
$251$	12.7600	0.805406	0.402703	+	0.915331i	$0.368071\pi$
$252$	0	0
$253$	5.79796	0.364515
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 16.6848i	− 1.04077i	−0.853931	−	0.520386i	$-0.825788\pi$
$257$	− 16.6848i	− 1.04077i	0.853931	−	0.520386i	$-0.174212\pi$
$258$	0	0
$259$	− 7.00000i	− 0.434959i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.07107	−0.436021	−0.218010	−	0.975946i	$-0.569957\pi$
$263$	−7.07107	−0.436021	−0.218010	+	0.975946i	$0.569957\pi$
$264$	0	0
$265$	−19.7980	−1.21618
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 9.72452i	− 0.592915i	−0.955046	−	0.296457i	$-0.904195\pi$
$269$	− 9.72452i	− 0.592915i	0.955046	−	0.296457i	$-0.0958052\pi$
$270$	0	0
$271$	12.6969i	0.771284i	0.922648	+	0.385642i	$0.126020\pi$
$271$	12.6969i	0.771284i	−0.922648	+	0.385642i	$0.873980\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.0424	−0.605577
$276$	0	0
$277$	−10.7980	−0.648786	−0.324393	−	0.945922i	$-0.605160\pi$
$277$	−10.7980	−0.648786	−0.324393	+	0.945922i	$0.605160\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.48528i	0.506189i	0.967442	+	0.253095i	$0.0814484\pi$
$281$	8.48528i	0.506189i	−0.967442	+	0.253095i	$0.918552\pi$
$282$	0	0
$283$	− 7.75255i	− 0.460841i	−0.973091	−	0.230421i	$-0.925990\pi$
$283$	− 7.75255i	− 0.460841i	0.973091	−	0.230421i	$-0.0740102\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.51059	−0.148195
$288$	0	0
$289$	7.10102	0.417707
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.3602i	0.605249i	0.953110	+	0.302625i	$0.0978628\pi$
$293$	10.3602i	0.605249i	−0.953110	+	0.302625i	$0.902137\pi$
$294$	0	0
$295$	18.3485i	1.06829i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.92820	0.400668
$300$	0	0
$301$	−4.34847	−0.250642
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 16.8277i	− 0.963551i
$306$	0	0
$307$	− 4.89898i	− 0.279600i	−0.990180	−	0.139800i	$-0.955354\pi$
$307$	− 4.89898i	− 0.279600i	0.990180	−	0.139800i	$-0.0446459\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.0458	−0.739757	−0.369879	−	0.929080i	$-0.620601\pi$
$311$	−13.0458	−0.739757	−0.369879	+	0.929080i	$0.620601\pi$
$312$	0	0
$313$	−2.89898	−0.163860	−0.0819300	−	0.996638i	$-0.526108\pi$
$313$	−2.89898	−0.163860	−0.0819300	+	0.996638i	$0.526108\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 27.9985i	− 1.57255i	−0.617874	−	0.786277i	$-0.712006\pi$
$317$	− 27.9985i	− 1.57255i	0.617874	−	0.786277i	$-0.287994\pi$
$318$	0	0
$319$	1.59592i	0.0893543i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−13.9993	−0.778940
$324$	0	0
$325$	−12.0000	−0.665640
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.38891i	0.407364i
$330$	0	0
$331$	− 17.2474i	− 0.948006i	−0.880523	−	0.474003i	$-0.842809\pi$
$331$	− 17.2474i	− 0.948006i	0.880523	−	0.474003i	$-0.157191\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−25.1701	−1.37519
$336$	0	0
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.921404i	0.0498968i
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.4128	−1.57896	−0.789480	−	0.613777i	$-0.789650\pi$
$347$	−29.4128	−1.57896	−0.789480	+	0.613777i	$0.789650\pi$
$348$	0	0
$349$	33.7980	1.80916	0.904582	−	0.426300i	$-0.140183\pi$
$349$	33.7980	1.80916	0.904582	+	0.426300i	$0.140183\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 20.7525i	− 1.10454i	−0.833664	−	0.552272i	$-0.813761\pi$
$353$	− 20.7525i	− 1.10454i	0.833664	−	0.552272i	$-0.186239\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.0915	1.37706	0.688529	−	0.725209i	$-0.258257\pi$
$359$	26.0915	1.37706	0.688529	+	0.725209i	$0.258257\pi$
$360$	0	0
$361$	−0.797959	−0.0419978
$362$	0	0
$363$	0	0
$364$	0	0
$365$	40.5836i	2.12424i
$366$	0	0
$367$	16.6969i	0.871573i	0.900050	+	0.435787i	$0.143530\pi$
$367$	16.6969i	0.871573i	−0.900050	+	0.435787i	$0.856470\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.29253	−0.326692
$372$	0	0
$373$	−35.2929	−1.82739	−0.913697	−	0.406395i	$-0.866786\pi$
$373$	−35.2929	−1.82739	−0.913697	+	0.406395i	$0.866786\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.90702i	0.0982167i
$378$	0	0
$379$	11.0454i	0.567364i	0.958918	+	0.283682i	$0.0915561\pi$
$379$	11.0454i	0.567364i	−0.958918	+	0.283682i	$0.908444\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.4887	−0.587044	−0.293522	−	0.955952i	$-0.594827\pi$
$383$	−11.4887	−0.587044	−0.293522	+	0.955952i	$0.594827\pi$
$384$	0	0
$385$	−6.44949	−0.328696
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 7.07107i	− 0.358517i	−0.983802	−	0.179259i	$-0.942630\pi$
$389$	− 7.07107i	− 0.358517i	0.983802	−	0.179259i	$-0.0573699\pi$
$390$	0	0
$391$	8.89898i	0.450041i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	42.3157	2.12913
$396$	0	0
$397$	−24.8990	−1.24964	−0.624822	−	0.780767i	$-0.714828\pi$
$397$	−24.8990	−1.24964	−0.624822	+	0.780767i	$0.714828\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.9774i	1.14743i	0.819053	+	0.573717i	$0.194499\pi$
$401$	22.9774i	1.14743i	−0.819053	+	0.573717i	$0.805501\pi$
$402$	0	0
$403$	1.10102i	0.0548457i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−14.3492	−0.711264
$408$	0	0
$409$	13.7980	0.682265	0.341133	−	0.940015i	$-0.389189\pi$
$409$	13.7980	0.682265	0.341133	+	0.940015i	$0.389189\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.83183i	0.286965i
$414$	0	0
$415$	21.2474i	1.04300i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.63907	0.177780	0.0888902	−	0.996041i	$-0.471668\pi$
$419$	3.63907	0.177780	0.0888902	+	0.996041i	$0.471668\pi$
$420$	0	0
$421$	16.8990	0.823606	0.411803	−	0.911273i	$-0.364899\pi$
$421$	16.8990	0.823606	0.411803	+	0.911273i	$0.364899\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 15.4135i	− 0.747664i
$426$	0	0
$427$	− 5.34847i	− 0.258831i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−40.2979	−1.94108	−0.970540	−	0.240940i	$-0.922544\pi$
$431$	−40.2979	−1.94108	−0.970540	+	0.240940i	$0.922544\pi$
$432$	0	0
$433$	−37.8434	−1.81864	−0.909318	−	0.416102i	$-0.863396\pi$
$433$	−37.8434	−1.81864	−0.909318	+	0.416102i	$0.863396\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.5851i	0.602025i
$438$	0	0
$439$	2.00000i	0.0954548i	0.998860	+	0.0477274i	$0.0151979\pi$
$439$	2.00000i	0.0954548i	−0.998860	+	0.0477274i	$0.984802\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.6841	1.45785	0.728923	−	0.684596i	$-0.240021\pi$
$443$	30.6841	1.45785	0.728923	+	0.684596i	$0.240021\pi$
$444$	0	0
$445$	44.4949	2.10926
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 10.8209i	− 0.510670i	−0.966853	−	0.255335i	$-0.917814\pi$
$449$	− 10.8209i	− 0.510670i	0.966853	−	0.255335i	$-0.0821857\pi$
$450$	0	0
$451$	5.14643i	0.242336i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.70674	−0.361298
$456$	0	0
$457$	37.3939	1.74921	0.874606	−	0.484835i	$-0.161120\pi$
$457$	37.3939	1.74921	0.874606	+	0.484835i	$0.161120\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 0.953512i	− 0.0444095i	−0.999753	−	0.0222047i	$-0.992931\pi$
$461$	− 0.953512i	− 0.0444095i	0.999753	−	0.0222047i	$-0.00706857\pi$
$462$	0	0
$463$	34.8434i	1.61931i	0.586907	+	0.809654i	$0.300345\pi$
$463$	34.8434i	1.61931i	−0.586907	+	0.809654i	$0.699655\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.7272	1.23679	0.618393	−	0.785869i	$-0.287784\pi$
$467$	26.7272	1.23679	0.618393	+	0.785869i	$0.287784\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.91388i	0.409860i
$474$	0	0
$475$	− 21.7980i	− 1.00016i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.8024	1.04187	0.520934	−	0.853597i	$-0.325584\pi$
$479$	22.8024	1.04187	0.520934	+	0.853597i	$0.325584\pi$
$480$	0	0
$481$	−17.1464	−0.781810
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 38.5337i	− 1.74973i
$486$	0	0
$487$	− 25.7980i	− 1.16902i	−0.811388	−	0.584509i	$-0.801287\pi$
$487$	− 25.7980i	− 1.16902i	0.811388	−	0.584509i	$-0.198713\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.8851	0.491238	0.245619	−	0.969366i	$-0.421009\pi$
$491$	10.8851	0.491238	0.245619	+	0.969366i	$0.421009\pi$
$492$	0	0
$493$	−2.44949	−0.110319
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	16.5505i	0.740903i	0.928852	+	0.370451i	$0.120797\pi$
$499$	16.5505i	0.740903i	−0.928852	+	0.370451i	$0.879203\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.7956	0.748878	0.374439	−	0.927251i	$-0.377835\pi$
$503$	16.7956	0.748878	0.374439	+	0.927251i	$0.377835\pi$
$504$	0	0
$505$	57.3939	2.55399
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.9732i	1.50584i	0.658114	+	0.752919i	$0.271355\pi$
$509$	33.9732i	1.50584i	−0.658114	+	0.752919i	$0.728645\pi$
$510$	0	0
$511$	12.8990i	0.570617i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	40.5836	1.78833
$516$	0	0
$517$	15.1464	0.666139
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 18.9097i	− 0.828449i	−0.910175	−	0.414224i	$-0.864053\pi$
$521$	− 18.9097i	− 0.828449i	0.910175	−	0.414224i	$-0.135947\pi$
$522$	0	0
$523$	− 40.7423i	− 1.78154i	−0.454456	−	0.890769i	$-0.650166\pi$
$523$	− 40.7423i	− 1.78154i	0.454456	−	0.890769i	$-0.349834\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.41421	−0.0616041
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.14966i	0.266372i
$534$	0	0
$535$	44.4949i	1.92368i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.04989	−0.0882949
$540$	0	0
$541$	−36.5959	−1.57338	−0.786691	−	0.617347i	$-0.788207\pi$
$541$	−36.5959	−1.57338	−0.786691	+	0.617347i	$0.788207\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.0239i	0.943398i
$546$	0	0
$547$	3.04541i	0.130212i	0.997878	+	0.0651061i	$0.0207386\pi$
$547$	3.04541i	0.130212i	−0.997878	+	0.0651061i	$0.979261\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.46410	−0.147576
$552$	0	0
$553$	13.4495	0.571930
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.3991i	0.694852i	0.937707	+	0.347426i	$0.112944\pi$
$557$	16.3991i	0.694852i	−0.937707	+	0.347426i	$0.887056\pi$
$558$	0	0
$559$	10.6515i	0.450512i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.4838	1.53761	0.768805	−	0.639483i	$-0.220852\pi$
$563$	36.4838	1.53761	0.768805	+	0.639483i	$0.220852\pi$
$564$	0	0
$565$	−35.1464	−1.47862
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.5707i	0.568912i	0.958689	+	0.284456i	$0.0918130\pi$
$569$	13.5707i	0.568912i	−0.958689	+	0.284456i	$0.908187\pi$
$570$	0	0
$571$	− 7.44949i	− 0.311751i	−0.987777	−	0.155876i	$-0.950180\pi$
$571$	− 7.44949i	− 0.311751i	0.987777	−	0.155876i	$-0.0498199\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13.8564	−0.577852
$576$	0	0
$577$	−2.89898	−0.120686	−0.0603430	−	0.998178i	$-0.519219\pi$
$577$	−2.89898	−0.120686	−0.0603430	+	0.998178i	$0.519219\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.75323i	0.280171i
$582$	0	0
$583$	12.8990i	0.534221i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.48528	−0.350225	−0.175113	−	0.984548i	$-0.556029\pi$
$587$	−8.48528	−0.350225	−0.175113	+	0.984548i	$0.556029\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	35.8803i	1.47343i	0.676206	+	0.736713i	$0.263623\pi$
$593$	35.8803i	1.47343i	−0.676206	+	0.736713i	$0.736377\pi$
$594$	0	0
$595$	− 9.89898i	− 0.405819i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.3636	0.546022	0.273011	−	0.962011i	$-0.411980\pi$
$599$	13.3636	0.546022	0.273011	+	0.962011i	$0.411980\pi$
$600$	0	0
$601$	−36.2474	−1.47856	−0.739282	−	0.673396i	$-0.764835\pi$
$601$	−36.2474	−1.47856	−0.739282	+	0.673396i	$0.764835\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 21.3882i	− 0.869553i
$606$	0	0
$607$	− 23.8434i	− 0.967772i	−0.875131	−	0.483886i	$-0.839225\pi$
$607$	− 23.8434i	− 0.967772i	0.875131	−	0.483886i	$-0.160775\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18.0990	0.732209
$612$	0	0
$613$	−20.0000	−0.807792	−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000	−0.807792	−0.403896	+	0.914805i	$0.632344\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.5994i	0.466976i	0.972360	+	0.233488i	$0.0750139\pi$
$617$	11.5994i	0.466976i	−0.972360	+	0.233488i	$0.924986\pi$
$618$	0	0
$619$	31.1464i	1.25188i	0.779871	+	0.625940i	$0.215285\pi$
$619$	31.1464i	1.25188i	−0.779871	+	0.625940i	$0.784715\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.1421	0.566593
$624$	0	0
$625$	−25.4949	−1.01980
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 22.0239i	− 0.878148i
$630$	0	0
$631$	− 5.44949i	− 0.216941i	−0.994100	−	0.108470i	$-0.965405\pi$
$631$	− 5.44949i	− 0.216941i	0.994100	−	0.108470i	$-0.0345953\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−29.0949	−1.15460
$636$	0	0
$637$	−2.44949	−0.0970523
$638$	0	0
$639$	0	0
$640$	0	0
$641$	29.4128i	1.16173i	0.813998	+	0.580867i	$0.197286\pi$
$641$	29.4128i	1.16173i	−0.813998	+	0.580867i	$0.802714\pi$
$642$	0	0
$643$	− 15.3031i	− 0.603494i	−0.953388	−	0.301747i	$-0.902430\pi$
$643$	− 15.3031i	− 0.603494i	0.953388	−	0.301747i	$-0.0975698\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.0409	1.49554	0.747771	−	0.663957i	$-0.231124\pi$
$647$	38.0409	1.49554	0.747771	+	0.663957i	$0.231124\pi$
$648$	0	0
$649$	11.9546	0.469259
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 8.34242i	− 0.326464i	−0.986588	−	0.163232i	$-0.947808\pi$
$653$	− 8.34242i	− 0.326464i	0.986588	−	0.163232i	$-0.0521919\pi$
$654$	0	0
$655$	14.8990i	0.582151i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.90702	0.0742871	0.0371435	−	0.999310i	$-0.488174\pi$
$659$	1.90702	0.0742871	0.0371435	+	0.999310i	$0.488174\pi$
$660$	0	0
$661$	26.7423	1.04016	0.520078	−	0.854119i	$-0.325903\pi$
$661$	26.7423	1.04016	0.520078	+	0.854119i	$0.325903\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 13.9993i	− 0.542868i
$666$	0	0
$667$	2.20204i	0.0852634i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.9638	−0.423251
$672$	0	0
$673$	46.2929	1.78446	0.892229	−	0.451583i	$-0.149140\pi$
$673$	46.2929	1.78446	0.892229	+	0.451583i	$0.149140\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5.94258i	− 0.228392i	−0.993458	−	0.114196i	$-0.963571\pi$
$677$	− 5.94258i	− 0.228392i	0.993458	−	0.114196i	$-0.0364292\pi$
$678$	0	0
$679$	− 12.2474i	− 0.470014i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.2986	1.00629	0.503144	−	0.864203i	$-0.332177\pi$
$683$	26.2986	1.00629	0.503144	+	0.864203i	$0.332177\pi$
$684$	0	0
$685$	−37.1464	−1.41929
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15.4135i	0.587207i
$690$	0	0
$691$	28.2474i	1.07458i	0.843396	+	0.537292i	$0.180553\pi$
$691$	28.2474i	1.07458i	−0.843396	+	0.537292i	$0.819447\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−38.5337	−1.46167
$696$	0	0
$697$	−7.89898	−0.299195
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15.5563i	0.587555i	0.955874	+	0.293778i	$0.0949125\pi$
$701$	15.5563i	0.587555i	−0.955874	+	0.293778i	$0.905087\pi$
$702$	0	0
$703$	− 31.1464i	− 1.17471i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.2419	0.686058
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.27135i	0.0476124i
$714$	0	0
$715$	15.7980i	0.590810i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.23171	0.344285	0.172142	−	0.985072i	$-0.444931\pi$
$719$	9.23171	0.344285	0.172142	+	0.985072i	$0.444931\pi$
$720$	0	0
$721$	12.8990	0.480383
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.81405i	− 0.141650i
$726$	0	0
$727$	− 42.2474i	− 1.56687i	−0.621473	−	0.783436i	$-0.713465\pi$
$727$	− 42.2474i	− 1.56687i	0.621473	−	0.783436i	$-0.286535\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.6814	−0.506026
$732$	0	0
$733$	42.8990	1.58451	0.792255	−	0.610190i	$-0.208907\pi$
$733$	42.8990	1.58451	0.792255	+	0.610190i	$0.208907\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.3991i	0.604069i
$738$	0	0
$739$	− 39.5959i	− 1.45656i	−0.685280	−	0.728280i	$-0.740320\pi$
$739$	− 39.5959i	− 1.45656i	0.685280	−	0.728280i	$-0.259680\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−31.2556	−1.14666	−0.573328	−	0.819326i	$-0.694348\pi$
$743$	−31.2556	−1.14666	−0.573328	+	0.819326i	$0.694348\pi$
$744$	0	0
$745$	−9.34847	−0.342501
$746$	0	0
$747$	0	0
$748$	0	0
$749$	14.1421i	0.516742i
$750$	0	0
$751$	− 33.3939i	− 1.21856i	−0.792955	−	0.609280i	$-0.791459\pi$
$751$	− 33.3939i	− 1.21856i	0.792955	−	0.609280i	$-0.208541\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−22.8024	−0.829864
$756$	0	0
$757$	−24.3031	−0.883310	−0.441655	−	0.897185i	$-0.645608\pi$
$757$	−24.3031	−0.883310	−0.441655	+	0.897185i	$0.645608\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.79632i	0.101366i	0.998715	+	0.0506832i	$0.0161399\pi$
$761$	2.79632i	0.101366i	−0.998715	+	0.0506832i	$0.983860\pi$
$762$	0	0
$763$	7.00000i	0.253417i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.2850	0.515801
$768$	0	0
$769$	−25.1010	−0.905166	−0.452583	−	0.891722i	$-0.649497\pi$
$769$	−25.1010	−0.905166	−0.452583	+	0.891722i	$0.649497\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.0662i	1.15334i	0.816977	+	0.576671i	$0.195648\pi$
$773$	32.0662i	1.15334i	−0.816977	+	0.576671i	$0.804352\pi$
$774$	0	0
$775$	− 2.20204i	− 0.0790996i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.1708	−0.400237
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.04989i	0.0731636i
$786$	0	0
$787$	− 4.49490i	− 0.160226i	−0.996786	−	0.0801129i	$-0.974472\pi$
$787$	− 4.49490i	− 0.160226i	0.996786	−	0.0801129i	$-0.0255281\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11.1708	−0.397189
$792$	0	0
$793$	−13.1010	−0.465231
$794$	0	0
$795$	0	0
$796$	0	0
$797$	36.1981i	1.28220i	0.767456	+	0.641101i	$0.221522\pi$
$797$	36.1981i	1.28220i	−0.767456	+	0.641101i	$0.778478\pi$
$798$	0	0
$799$	23.2474i	0.822436i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	26.4415	0.933099
$804$	0	0
$805$	−8.89898	−0.313648
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 21.4203i	− 0.753097i	−0.926397	−	0.376549i	$-0.877111\pi$
$809$	− 21.4203i	− 0.753097i	0.926397	−	0.376549i	$-0.122889\pi$
$810$	0	0
$811$	− 36.7423i	− 1.29020i	−0.764099	−	0.645099i	$-0.776816\pi$
$811$	− 36.7423i	− 1.29020i	0.764099	−	0.645099i	$-0.223184\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−41.6800	−1.45999
$816$	0	0
$817$	−19.3485	−0.676917
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 22.0560i	− 0.769758i	−0.922967	−	0.384879i	$-0.874243\pi$
$821$	− 22.0560i	− 0.769758i	0.922967	−	0.384879i	$-0.125757\pi$
$822$	0	0
$823$	24.3485i	0.848734i	0.905490	+	0.424367i	$0.139503\pi$
$823$	24.3485i	0.848734i	−0.905490	+	0.424367i	$0.860497\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.9766	1.28580	0.642902	−	0.765949i	$-0.277730\pi$
$827$	36.9766	1.28580	0.642902	+	0.765949i	$0.277730\pi$
$828$	0	0
$829$	38.0454	1.32137	0.660686	−	0.750663i	$-0.270266\pi$
$829$	38.0454	1.32137	0.660686	+	0.750663i	$0.270266\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.14626i	− 0.109012i
$834$	0	0
$835$	− 66.8434i	− 2.31321i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	23.7238	0.819036	0.409518	−	0.912302i	$-0.365697\pi$
$839$	23.7238	0.819036	0.409518	+	0.912302i	$0.365697\pi$
$840$	0	0
$841$	28.3939	0.979099
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 22.0239i	− 0.757643i
$846$	0	0
$847$	− 6.79796i	− 0.233581i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−19.7990	−0.678701
$852$	0	0
$853$	37.1010	1.27031	0.635157	−	0.772383i	$-0.280935\pi$
$853$	37.1010	1.27031	0.635157	+	0.772383i	$0.280935\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 42.8085i	− 1.46231i	−0.682212	−	0.731155i	$-0.738982\pi$
$857$	− 42.8085i	− 1.46231i	0.682212	−	0.731155i	$-0.261018\pi$
$858$	0	0
$859$	− 22.4949i	− 0.767516i	−0.923434	−	0.383758i	$-0.874630\pi$
$859$	− 22.4949i	− 0.767516i	0.923434	−	0.383758i	$-0.125370\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.0915	−0.888166	−0.444083	−	0.895986i	$-0.646470\pi$
$863$	−26.0915	−0.888166	−0.444083	+	0.895986i	$0.646470\pi$
$864$	0	0
$865$	6.89898	0.234572
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 27.5699i	− 0.935246i
$870$	0	0
$871$	19.5959i	0.663982i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.317837	−0.0107449
$876$	0	0
$877$	−19.2020	−0.648407	−0.324203	−	0.945987i	$-0.605096\pi$
$877$	−19.2020	−0.648407	−0.324203	+	0.945987i	$0.605096\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 7.49966i	− 0.252670i	−0.991988	−	0.126335i	$-0.959679\pi$
$881$	− 7.49966i	− 0.252670i	0.991988	−	0.126335i	$-0.0403214\pi$
$882$	0	0
$883$	37.0454i	1.24668i	0.781952	+	0.623339i	$0.214224\pi$
$883$	37.0454i	1.24668i	−0.781952	+	0.623339i	$0.785776\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.0458	−0.438034	−0.219017	−	0.975721i	$-0.570285\pi$
$887$	−13.0458	−0.438034	−0.219017	+	0.975721i	$0.570285\pi$
$888$	0	0
$889$	−9.24745	−0.310149
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.8769i	1.10018i
$894$	0	0
$895$	6.44949i	0.215583i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.349945	−0.0116713
$900$	0	0
$901$	−19.7980	−0.659566
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 14.7778i	− 0.491231i
$906$	0	0
$907$	55.9444i	1.85760i	0.370578	+	0.928801i	$0.379160\pi$
$907$	55.9444i	1.85760i	−0.370578	+	0.928801i	$0.620840\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	7.70674	0.255336	0.127668	−	0.991817i	$-0.459251\pi$
$911$	7.70674	0.255336	0.127668	+	0.991817i	$0.459251\pi$
$912$	0	0
$913$	13.8434	0.458149
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.73545i	0.156378i
$918$	0	0
$919$	− 37.2474i	− 1.22868i	−0.789041	−	0.614340i	$-0.789422\pi$
$919$	− 37.2474i	− 1.22868i	0.789041	−	0.614340i	$-0.210578\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	34.2929	1.12754
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 13.8885i	− 0.455667i	−0.973700	−	0.227834i	$-0.926836\pi$
$929$	− 13.8885i	− 0.455667i	0.973700	−	0.227834i	$-0.0731643\pi$
$930$	0	0
$931$	− 4.44949i	− 0.145826i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−20.2918	−0.663613
$936$	0	0
$937$	0.651531	0.0212846	0.0106423	−	0.999943i	$-0.496612\pi$
$937$	0.651531	0.0212846	0.0106423	+	0.999943i	$0.496612\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	22.3096i	0.727272i	0.931541	+	0.363636i	$0.118465\pi$
$941$	22.3096i	0.727272i	−0.931541	+	0.363636i	$0.881535\pi$
$942$	0	0
$943$	7.10102i	0.231241i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.9842	0.941859	0.470929	−	0.882171i	$-0.343919\pi$
$947$	28.9842	0.941859	0.470929	+	0.882171i	$0.343919\pi$
$948$	0	0
$949$	31.5959	1.02565
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.25697i	0.0731104i	0.999332	+	0.0365552i	$0.0116385\pi$
$953$	2.25697i	0.0731104i	−0.999332	+	0.0365552i	$0.988362\pi$
$954$	0	0
$955$	− 14.8990i	− 0.482120i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11.8065	−0.381252
$960$	0	0
$961$	30.7980	0.993483
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 34.6089i	− 1.11410i
$966$	0	0
$967$	− 4.89898i	− 0.157541i	−0.996893	−	0.0787703i	$-0.974901\pi$
$967$	− 4.89898i	− 0.157541i	0.996893	−	0.0787703i	$-0.0250994\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	55.8221	1.79142	0.895708	−	0.444642i	$-0.146669\pi$
$971$	55.8221	1.79142	0.895708	+	0.444642i	$0.146669\pi$
$972$	0	0
$973$	−12.2474	−0.392635
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.7620i	1.40007i	0.714109	+	0.700035i	$0.246832\pi$
$977$	43.7620i	1.40007i	−0.714109	+	0.700035i	$0.753168\pi$
$978$	0	0
$979$	− 28.9898i	− 0.926518i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.1735	−0.898596	−0.449298	−	0.893382i	$-0.648326\pi$
$983$	−28.1735	−0.898596	−0.449298	+	0.893382i	$0.648326\pi$
$984$	0	0
$985$	24.2474	0.772588
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.2993i	0.391096i
$990$	0	0
$991$	14.5505i	0.462212i	0.972929	+	0.231106i	$0.0742344\pi$
$991$	14.5505i	0.462212i	−0.972929	+	0.231106i	$0.925766\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−31.4626	−0.997433
$996$	0	0
$997$	32.2474	1.02129	0.510643	−	0.859793i	$-0.329407\pi$
$997$	32.2474	1.02129	0.510643	+	0.859793i	$0.329407\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q+3.14626i q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+3.14626i q^{5} +1.00000i q^{7} +2.04989 q^{11} +2.44949 q^{13} +3.14626i q^{17} +4.44949i q^{19} +2.82843 q^{23} -4.89898 q^{25} +0.778539i q^{29} +0.449490i q^{31} -3.14626 q^{35} -7.00000 q^{37} +2.51059i q^{41} +4.34847i q^{43} +7.38891 q^{47} -1.00000 q^{49} +6.29253i q^{53} +6.44949i q^{55} +5.83183 q^{59} -5.34847 q^{61} +7.70674i q^{65} +8.00000i q^{67} +12.8990 q^{73} +2.04989i q^{77} -13.4495i q^{79} +6.75323 q^{83} -9.89898 q^{85} -14.1421i q^{89} +2.44949i q^{91} -13.9993 q^{95} -12.2474 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 56 q^{37} - 8 q^{49} + 16 q^{61} + 64 q^{73} - 40 q^{85}+O(q^{100}) \) 8 * q - 56 * q^37 - 8 * q^49 + 16 * q^61 + 64 * q^73 - 40 * q^85

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