LMFDB - Embedded newform 4200.2.t.q.1849.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4200,2,Mod(1849,4200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4200.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4200.t (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:	$33.5371688489$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4200.1849
Dual form			4200.2.t.q.1849.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} -2.00000i q^{13} +6.00000i q^{17} -4.00000 q^{19} -1.00000 q^{21} +8.00000i q^{23} +1.00000i q^{27} +2.00000 q^{29} -4.00000i q^{33} +2.00000i q^{37} -2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} -1.00000 q^{49} +6.00000 q^{51} +14.0000i q^{53} +4.00000i q^{57} -12.0000 q^{59} -2.00000 q^{61} +1.00000i q^{63} +4.00000i q^{67} +8.00000 q^{69} +2.00000i q^{73} -4.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{83} -2.00000i q^{87} +6.00000 q^{89} -2.00000 q^{91} +6.00000i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 8 q^{11} - 8 q^{19} - 2 q^{21} + 4 q^{29} - 4 q^{39} + 20 q^{41} - 2 q^{49} + 12 q^{51} - 24 q^{59} - 4 q^{61} + 16 q^{69} + 16 q^{79} + 2 q^{81} + 12 q^{89} - 4 q^{91} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 8 * q^11 - 8 * q^19 - 2 * q^21 + 4 * q^29 - 4 * q^39 + 20 * q^41 - 2 * q^49 + 12 * q^51 - 24 * q^59 - 4 * q^61 + 16 * q^69 + 16 * q^79 + 2 * q^81 + 12 * q^89 - 4 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4200\mathbb{Z}\right)^\times$.

$n$	$1177$	$2101$	$2801$	$3151$	$3601$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	− 4.00000i	− 0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	14.0000i	1.92305i	0.274721	+	0.961524i	$0.411414\pi$
$53$	14.0000i	1.92305i	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	1.00000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.00000i	− 0.214423i
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 20.0000i	− 1.93347i	−0.255774	−	0.966736i	$-0.582330\pi$
$107$	− 20.0000i	− 1.93347i	0.255774	−	0.966736i	$-0.417670\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	− 10.0000i	− 0.901670i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.0000i	1.87959i	0.341743	+	0.939793i	$0.388983\pi$
$137$	22.0000i	1.87959i	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	0	0
$159$	14.0000	1.11027
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	2.00000i	0.147844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24.0000i	1.75505i
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8.00000i	− 0.556038i
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 28.0000i	− 1.85843i	−0.369546	−	0.929213i	$-0.620487\pi$
$227$	− 28.0000i	− 1.85843i	0.369546	−	0.929213i	$-0.379513\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.00000i	0.509028i
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	32.0000i	2.01182i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−32.0000	−1.94386	−0.971931	−	0.235267i	$-0.924404\pi$
$271$	−32.0000	−1.94386	−0.971931	+	0.235267i	$0.924404\pi$
$272$	0	0
$273$	2.00000i	0.121046i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 10.0000i	− 0.590281i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.00000i	0.232104i
$298$	0	0
$299$	16.0000	0.925304
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	− 6.00000i	− 0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	2.00000i	0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$	2.00000i	0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.0000i	0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$	10.0000i	0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	−20.0000	−1.11629
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.0000i	0.774202i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 34.0000i	− 1.85210i	−0.377403	−	0.926049i	$-0.623183\pi$
$337$	− 34.0000i	− 1.85210i	0.377403	−	0.926049i	$-0.376817\pi$
$338$	0	0
$339$	−14.0000	−0.760376
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6.00000i	− 0.317554i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	14.0000	0.726844
$372$	0	0
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.00000i	− 0.206010i
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	0	0
$393$	− 4.00000i	− 0.201773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	34.0000i	1.70641i	0.521575	+	0.853206i	$0.325345\pi$
$397$	34.0000i	1.70641i	−0.521575	+	0.853206i	$0.674655\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.00000i	0.396545i
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	22.0000	1.08518
$412$	0	0
$413$	12.0000i	0.590481i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.0000i	− 0.979404i
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.00000i	0.0967868i
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	26.0000i	1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$	26.0000i	1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 32.0000i	− 1.53077i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	20.0000i	0.950229i	0.879924	+	0.475114i	$0.157593\pi$
$443$	20.0000i	0.950229i	−0.879924	+	0.475114i	$0.842407\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.00000i	0.283790i
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	0	0
$453$	− 16.0000i	− 0.751746i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.00000i	0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$	6.00000i	0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	0	0
$459$	−6.00000	−0.280056
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.00000i	0.185098i	0.995708	+	0.0925490i	$0.0295015\pi$
$467$	4.00000i	0.185098i	−0.995708	+	0.0925490i	$0.970499\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 14.0000i	− 0.641016i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	− 8.00000i	− 0.364013i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	24.0000i	1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$	24.0000i	1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	8.00000i	0.356702i	0.983967	+	0.178351i	$0.0570763\pi$
$503$	8.00000i	0.356702i	−0.983967	+	0.178351i	$0.942924\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	0	0
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.0000i	0.517838i
$538$	0	0
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	10.0000i	0.429141i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.0000i	0.423714i	0.977301	+	0.211857i	$0.0679510\pi$
$557$	10.0000i	0.423714i	−0.977301	+	0.211857i	$0.932049\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	− 8.00000i	− 0.334205i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.00000i	0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$	6.00000i	0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$	0	0
$579$	18.0000	0.748054
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	56.0000i	2.31928i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.0000i	1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$	38.0000i	1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$	0	0
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	0	0
$623$	− 6.00000i	− 0.240385i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.0000i	0.638978i
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	4.00000i	0.158986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	0	0
$663$	− 12.0000i	− 0.466041i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	−28.0000	−1.07296
$682$	0	0
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 26.0000i	− 0.991962i
$688$	0	0
$689$	28.0000	1.06672
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	4.00000i	0.151947i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	60.0000i	2.27266i
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	− 8.00000i	− 0.301726i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 24.0000i	− 0.896296i
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	− 2.00000i	− 0.0743808i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.0000i	0.589368i
$738$	0	0
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.00000i	0.146352i
$748$	0	0
$749$	−20.0000	−0.730784
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	− 12.0000i	− 0.437304i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 46.0000i	− 1.67190i	−0.548807	−	0.835949i	$-0.684918\pi$
$757$	− 46.0000i	− 1.67190i	0.548807	−	0.835949i	$-0.315082\pi$
$758$	0	0
$759$	32.0000	1.16153
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	14.0000i	0.506834i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.0000i	0.866590i
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	− 26.0000i	− 0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$	− 26.0000i	− 0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.00000i	− 0.0717496i
$778$	0	0
$779$	−40.0000	−1.43315
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.00000i	0.0714742i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	0	0
$789$	−8.00000	−0.284808
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	4.00000i	0.142044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 14.0000i	− 0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$	− 14.0000i	− 0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	8.00000i	0.282314i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 18.0000i	− 0.633630i
$808$	0	0
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	32.0000i	1.12229i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	56.0000i	1.95204i	0.217687	+	0.976019i	$0.430149\pi$
$823$	56.0000i	1.95204i	−0.217687	+	0.976019i	$0.569851\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 52.0000i	− 1.80822i	−0.427303	−	0.904109i	$-0.640536\pi$
$827$	− 52.0000i	− 1.80822i	0.427303	−	0.904109i	$-0.359464\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	− 26.0000i	− 0.895488i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.00000i	− 0.171802i
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	−16.0000	−0.548473
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.0000i	1.57133i	0.618652	+	0.785665i	$0.287679\pi$
$857$	46.0000i	1.57133i	−0.618652	+	0.785665i	$0.712321\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	−10.0000	−0.340799
$862$	0	0
$863$	16.0000i	0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$	16.0000i	0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	− 6.00000i	− 0.203069i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	0	0
$879$	−26.0000	−0.876958
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 16.0000i	− 0.534224i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−84.0000	−2.79845
$902$	0	0
$903$	− 4.00000i	− 0.133112i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−56.0000	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$912$	0	0
$913$	− 16.0000i	− 0.529523i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.00000i	− 0.132092i
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000i	0.262754i
$928$	0	0
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	0	0
$933$	8.00000i	0.261908i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.0000i	0.457360i	0.973502	+	0.228680i	$0.0734410\pi$
$937$	14.0000i	0.457360i	−0.973502	+	0.228680i	$0.926559\pi$
$938$	0	0
$939$	2.00000	0.0652675
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	80.0000i	2.60516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	10.0000	0.324272
$952$	0	0
$953$	42.0000i	1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$	42.0000i	1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8.00000i	− 0.258603i
$958$	0	0
$959$	22.0000	0.710417
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	20.0000i	0.644491i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.0000i	1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$	40.0000i	1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	40.0000i	1.27580i	0.770118	+	0.637901i	$0.220197\pi$
$983$	40.0000i	1.27580i	−0.770118	+	0.637901i	$0.779803\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	− 20.0000i	− 0.634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38.0000i	− 1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$	− 38.0000i	− 1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4200.2.t.q.1849.1		2
5.2	odd	4		840.2.a.f.1.1	✓	1
5.3	odd	4		4200.2.a.z.1.1		1
5.4	even	2	inner	4200.2.t.q.1849.2		2
15.2	even	4		2520.2.a.g.1.1		1
20.3	even	4		8400.2.a.q.1.1		1
20.7	even	4		1680.2.a.p.1.1		1
35.27	even	4		5880.2.a.z.1.1		1
40.27	even	4		6720.2.a.h.1.1		1
40.37	odd	4		6720.2.a.bn.1.1		1
60.47	odd	4		5040.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.a.f.1.1	✓	1	5.2	odd	4
1680.2.a.p.1.1		1	20.7	even	4
2520.2.a.g.1.1		1	15.2	even	4
4200.2.a.z.1.1		1	5.3	odd	4
4200.2.t.q.1849.1		2	1.1	even	1	trivial
4200.2.t.q.1849.2		2	5.4	even	2	inner
5040.2.a.j.1.1		1	60.47	odd	4
5880.2.a.z.1.1		1	35.27	even	4
6720.2.a.h.1.1		1	40.27	even	4
6720.2.a.bn.1.1		1	40.37	odd	4
8400.2.a.q.1.1		1	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4200.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} -2.00000i q^{13} +6.00000i q^{17} -4.00000 q^{19} -1.00000 q^{21} +8.00000i q^{23} +1.00000i q^{27} +2.00000 q^{29} -4.00000i q^{33} +2.00000i q^{37} -2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} -1.00000 q^{49} +6.00000 q^{51} +14.0000i q^{53} +4.00000i q^{57} -12.0000 q^{59} -2.00000 q^{61} +1.00000i q^{63} +4.00000i q^{67} +8.00000 q^{69} +2.00000i q^{73} -4.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{83} -2.00000i q^{87} +6.00000 q^{89} -2.00000 q^{91} +6.00000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 8 q^{11} - 8 q^{19} - 2 q^{21} + 4 q^{29} - 4 q^{39} + 20 q^{41} - 2 q^{49} + 12 q^{51} - 24 q^{59} - 4 q^{61} + 16 q^{69} + 16 q^{79} + 2 q^{81} + 12 q^{89} - 4 q^{91} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 8 * q^11 - 8 * q^19 - 2 * q^21 + 4 * q^29 - 4 * q^39 + 20 * q^41 - 2 * q^49 + 12 * q^51 - 24 * q^59 - 4 * q^61 + 16 * q^69 + 16 * q^79 + 2 * q^81 + 12 * q^89 - 4 * q^91 - 8 * q^99

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