LMFDB - Embedded newform 3200.2.d.l.1601.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(1601,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3200.1601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.d (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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1601.3
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	3200.1601
Dual form			3200.2.d.l.1601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949i q^{3} -2.44949 q^{7} -3.00000 q^{9} +O(q^{10})$	$q+2.44949i q^{3} -2.44949 q^{7} -3.00000 q^{9} +4.89898i q^{11} +4.00000 q^{17} +4.89898i q^{19} -6.00000i q^{21} -2.44949 q^{23} +8.00000i q^{29} -9.79796 q^{31} -12.0000 q^{33} -4.00000i q^{37} +8.00000 q^{41} -7.34847i q^{43} +12.2474 q^{47} -1.00000 q^{49} +9.79796i q^{51} -8.00000i q^{53} -12.0000 q^{57} +4.89898i q^{59} +6.00000i q^{61} +7.34847 q^{63} +2.44949i q^{67} -6.00000i q^{69} -9.79796 q^{71} -4.00000 q^{73} -12.0000i q^{77} -9.79796 q^{79} -9.00000 q^{81} -2.44949i q^{83} -19.5959 q^{87} -2.00000 q^{89} -24.0000i q^{93} -4.00000 q^{97} -14.6969i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 16 q^{17} - 48 q^{33} + 32 q^{41} - 4 q^{49} - 48 q^{57} - 16 q^{73} - 36 q^{81} - 8 q^{89} - 16 q^{97}+O(q^{100}) $ 4 * q - 12 * q^9 + 16 * q^17 - 48 * q^33 + 32 * q^41 - 4 * q^49 - 48 * q^57 - 16 * q^73 - 36 * q^81 - 8 * q^89 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.44949i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$3$	2.44949i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	4.89898i	1.47710i	0.674200	+	0.738549i	$0.264489\pi$
$11$	4.89898i	1.47710i	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	4.89898i	1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$	4.89898i	1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$	0	0
$21$	− 6.00000i	− 1.30931i
$22$	0	0
$23$	−2.44949	−0.510754	−0.255377	−	0.966842i	$-0.582200\pi$
$23$	−2.44949	−0.510754	−0.255377	+	0.966842i	$0.582200\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.00000i	1.48556i	0.669534	+	0.742781i	$0.266494\pi$
$29$	8.00000i	1.48556i	−0.669534	+	0.742781i	$0.733506\pi$
$30$	0	0
$31$	−9.79796	−1.75977	−0.879883	−	0.475191i	$-0.842379\pi$
$31$	−9.79796	−1.75977	−0.879883	+	0.475191i	$0.842379\pi$
$32$	0	0
$33$	−12.0000	−2.08893
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	− 7.34847i	− 1.12063i	−0.828279	−	0.560316i	$-0.810680\pi$
$43$	− 7.34847i	− 1.12063i	0.828279	−	0.560316i	$-0.189320\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.2474	1.78647	0.893237	−	0.449586i	$-0.148429\pi$
$47$	12.2474	1.78647	0.893237	+	0.449586i	$0.148429\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	9.79796i	1.37199i
$52$	0	0
$53$	− 8.00000i	− 1.09888i	−0.835532	−	0.549442i	$-0.814840\pi$
$53$	− 8.00000i	− 1.09888i	0.835532	−	0.549442i	$-0.185160\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−12.0000	−1.58944
$58$	0	0
$59$	4.89898i	0.637793i	0.947790	+	0.318896i	$0.103312\pi$
$59$	4.89898i	0.637793i	−0.947790	+	0.318896i	$0.896688\pi$
$60$	0	0
$61$	6.00000i	0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$	6.00000i	0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$	0	0
$63$	7.34847	0.925820
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.44949i	0.299253i	0.988743	+	0.149626i	$0.0478071\pi$
$67$	2.44949i	0.299253i	−0.988743	+	0.149626i	$0.952193\pi$
$68$	0	0
$69$	− 6.00000i	− 0.722315i
$70$	0	0
$71$	−9.79796	−1.16280	−0.581402	−	0.813617i	$-0.697496\pi$
$71$	−9.79796	−1.16280	−0.581402	+	0.813617i	$0.697496\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 12.0000i	− 1.36753i
$78$	0	0
$79$	−9.79796	−1.10236	−0.551178	−	0.834388i	$-0.685822\pi$
$79$	−9.79796	−1.10236	−0.551178	+	0.834388i	$0.685822\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	− 2.44949i	− 0.268866i	−0.990923	−	0.134433i	$-0.957079\pi$
$83$	− 2.44949i	− 0.268866i	0.990923	−	0.134433i	$-0.0429214\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−19.5959	−2.10090
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 24.0000i	− 2.48868i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	− 14.6969i	− 1.47710i
$100$	0	0
$101$	− 8.00000i	− 0.796030i	−0.917379	−	0.398015i	$-0.869699\pi$
$101$	− 8.00000i	− 0.796030i	0.917379	−	0.398015i	$-0.130301\pi$
$102$	0	0
$103$	−2.44949	−0.241355	−0.120678	−	0.992692i	$-0.538507\pi$
$103$	−2.44949	−0.241355	−0.120678	+	0.992692i	$0.538507\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.34847i	− 0.710403i	−0.934790	−	0.355202i	$-0.884412\pi$
$107$	− 7.34847i	− 0.710403i	0.934790	−	0.355202i	$-0.115588\pi$
$108$	0	0
$109$	− 10.0000i	− 0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$	− 10.0000i	− 0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$	0	0
$111$	9.79796	0.929981
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.79796	−0.898177
$120$	0	0
$121$	−13.0000	−1.18182
$122$	0	0
$123$	19.5959i	1.76690i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.34847	0.652071	0.326036	−	0.945357i	$-0.394287\pi$
$127$	7.34847	0.652071	0.326036	+	0.945357i	$0.394287\pi$
$128$	0	0
$129$	18.0000	1.58481
$130$	0	0
$131$	− 4.89898i	− 0.428026i	−0.976831	−	0.214013i	$-0.931347\pi$
$131$	− 4.89898i	− 0.428026i	0.976831	−	0.214013i	$-0.0686535\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	− 4.89898i	− 0.415526i	−0.978179	−	0.207763i	$-0.933382\pi$
$139$	− 4.89898i	− 0.415526i	0.978179	−	0.207763i	$-0.0666183\pi$
$140$	0	0
$141$	30.0000i	2.52646i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.44949i	− 0.202031i
$148$	0	0
$149$	− 2.00000i	− 0.163846i	−0.996639	−	0.0819232i	$-0.973894\pi$
$149$	− 2.00000i	− 0.163846i	0.996639	−	0.0819232i	$-0.0261062\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−12.0000	−0.970143
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.0000i	1.59617i	0.602542	+	0.798087i	$0.294154\pi$
$157$	20.0000i	1.59617i	−0.602542	+	0.798087i	$0.705846\pi$
$158$	0	0
$159$	19.5959	1.55406
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	2.44949i	0.191859i	0.995388	+	0.0959294i	$0.0305823\pi$
$163$	2.44949i	0.191859i	−0.995388	+	0.0959294i	$0.969418\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.44949	0.189547	0.0947736	−	0.995499i	$-0.469787\pi$
$167$	2.44949	0.189547	0.0947736	+	0.995499i	$0.469787\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	− 14.6969i	− 1.12390i
$172$	0	0
$173$	4.00000i	0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	4.00000i	0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	14.6969i	1.09850i	0.835658	+	0.549250i	$0.185087\pi$
$179$	14.6969i	1.09850i	−0.835658	+	0.549250i	$0.814913\pi$
$180$	0	0
$181$	− 8.00000i	− 0.594635i	−0.954779	−	0.297318i	$-0.903908\pi$
$181$	− 8.00000i	− 0.594635i	0.954779	−	0.297318i	$-0.0960920\pi$
$182$	0	0
$183$	−14.6969	−1.08643
$184$	0	0
$185$	0	0
$186$	0	0
$187$	19.5959i	1.43300i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.5959	1.41791	0.708955	−	0.705253i	$-0.249167\pi$
$191$	19.5959	1.41791	0.708955	+	0.705253i	$0.249167\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 8.00000i	− 0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$	− 8.00000i	− 0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$	0	0
$199$	9.79796	0.694559	0.347279	−	0.937762i	$-0.387106\pi$
$199$	9.79796	0.694559	0.347279	+	0.937762i	$0.387106\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	0	0
$203$	− 19.5959i	− 1.37536i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.34847	0.510754
$208$	0	0
$209$	−24.0000	−1.66011
$210$	0	0
$211$	− 24.4949i	− 1.68630i	−0.537680	−	0.843149i	$-0.680699\pi$
$211$	− 24.4949i	− 1.68630i	0.537680	−	0.843149i	$-0.319301\pi$
$212$	0	0
$213$	− 24.0000i	− 1.64445i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	24.0000	1.62923
$218$	0	0
$219$	− 9.79796i	− 0.662085i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−7.34847	−0.492090	−0.246045	−	0.969258i	$-0.579131\pi$
$223$	−7.34847	−0.492090	−0.246045	+	0.969258i	$0.579131\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.44949i	− 0.162578i	−0.996691	−	0.0812892i	$-0.974096\pi$
$227$	− 2.44949i	− 0.162578i	0.996691	−	0.0812892i	$-0.0259037\pi$
$228$	0	0
$229$	− 8.00000i	− 0.528655i	−0.964433	−	0.264327i	$-0.914850\pi$
$229$	− 8.00000i	− 0.528655i	0.964433	−	0.264327i	$-0.0851500\pi$
$230$	0	0
$231$	29.3939	1.93398
$232$	0	0
$233$	−20.0000	−1.31024	−0.655122	−	0.755523i	$-0.727383\pi$
$233$	−20.0000	−1.31024	−0.655122	+	0.755523i	$0.727383\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 24.0000i	− 1.55897i
$238$	0	0
$239$	9.79796	0.633777	0.316889	−	0.948463i	$-0.397362\pi$
$239$	9.79796	0.633777	0.316889	+	0.948463i	$0.397362\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	− 22.0454i	− 1.41421i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	4.89898i	0.309221i	0.987976	+	0.154610i	$0.0494122\pi$
$251$	4.89898i	0.309221i	−0.987976	+	0.154610i	$0.950588\pi$
$252$	0	0
$253$	− 12.0000i	− 0.754434i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	9.79796i	0.608816i
$260$	0	0
$261$	− 24.0000i	− 1.48556i
$262$	0	0
$263$	22.0454	1.35938	0.679689	−	0.733500i	$-0.262115\pi$
$263$	22.0454	1.35938	0.679689	+	0.733500i	$0.262115\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.89898i	− 0.299813i
$268$	0	0
$269$	26.0000i	1.58525i	0.609711	+	0.792624i	$0.291286\pi$
$269$	26.0000i	1.58525i	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	−9.79796	−0.595184	−0.297592	−	0.954693i	$-0.596183\pi$
$271$	−9.79796	−0.595184	−0.297592	+	0.954693i	$0.596183\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.0000i	0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$	12.0000i	0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$	0	0
$279$	29.3939	1.75977
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	26.9444i	1.60168i	0.598880	+	0.800839i	$0.295613\pi$
$283$	26.9444i	1.60168i	−0.598880	+	0.800839i	$0.704387\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−19.5959	−1.15671
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	− 9.79796i	− 0.574367i
$292$	0	0
$293$	20.0000i	1.16841i	0.811605	+	0.584206i	$0.198594\pi$
$293$	20.0000i	1.16841i	−0.811605	+	0.584206i	$0.801406\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	18.0000i	1.03750i
$302$	0	0
$303$	19.5959	1.12576
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 17.1464i	− 0.978598i	−0.872116	−	0.489299i	$-0.837253\pi$
$307$	− 17.1464i	− 0.978598i	0.872116	−	0.489299i	$-0.162747\pi$
$308$	0	0
$309$	− 6.00000i	− 0.341328i
$310$	0	0
$311$	19.5959	1.11118	0.555591	−	0.831456i	$-0.312492\pi$
$311$	19.5959	1.11118	0.555591	+	0.831456i	$0.312492\pi$
$312$	0	0
$313$	−24.0000	−1.35656	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	−24.0000	−1.35656	−0.678280	+	0.734803i	$0.737274\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.00000i	− 0.449325i	−0.974437	−	0.224662i	$-0.927872\pi$
$317$	− 8.00000i	− 0.449325i	0.974437	−	0.224662i	$-0.0721279\pi$
$318$	0	0
$319$	−39.1918	−2.19432
$320$	0	0
$321$	18.0000	1.00466
$322$	0	0
$323$	19.5959i	1.09035i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	24.4949	1.35457
$328$	0	0
$329$	−30.0000	−1.65395
$330$	0	0
$331$	− 4.89898i	− 0.269272i	−0.990895	−	0.134636i	$-0.957013\pi$
$331$	− 4.89898i	− 0.269272i	0.990895	−	0.134636i	$-0.0429866\pi$
$332$	0	0
$333$	12.0000i	0.657596i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	0	0
$339$	− 39.1918i	− 2.12861i
$340$	0	0
$341$	− 48.0000i	− 2.59935i
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.2474i	− 0.657477i	−0.944421	−	0.328739i	$-0.893376\pi$
$347$	− 12.2474i	− 0.657477i	0.944421	−	0.328739i	$-0.106624\pi$
$348$	0	0
$349$	24.0000i	1.28469i	0.766415	+	0.642345i	$0.222038\pi$
$349$	24.0000i	1.28469i	−0.766415	+	0.642345i	$0.777962\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.00000	−0.425797	−0.212899	−	0.977074i	$-0.568290\pi$
$353$	−8.00000	−0.425797	−0.212899	+	0.977074i	$0.568290\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 24.0000i	− 1.27021i
$358$	0	0
$359$	−19.5959	−1.03423	−0.517116	−	0.855915i	$-0.672995\pi$
$359$	−19.5959	−1.03423	−0.517116	+	0.855915i	$0.672995\pi$
$360$	0	0
$361$	−5.00000	−0.263158
$362$	0	0
$363$	− 31.8434i	− 1.67134i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−31.8434	−1.66221	−0.831105	−	0.556115i	$-0.812291\pi$
$367$	−31.8434	−1.66221	−0.831105	+	0.556115i	$0.812291\pi$
$368$	0	0
$369$	−24.0000	−1.24939
$370$	0	0
$371$	19.5959i	1.01737i
$372$	0	0
$373$	20.0000i	1.03556i	0.855514	+	0.517780i	$0.173242\pi$
$373$	20.0000i	1.03556i	−0.855514	+	0.517780i	$0.826758\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	14.6969i	0.754931i	0.926024	+	0.377466i	$0.123204\pi$
$379$	14.6969i	0.754931i	−0.926024	+	0.377466i	$0.876796\pi$
$380$	0	0
$381$	18.0000i	0.922168i
$382$	0	0
$383$	−26.9444	−1.37679	−0.688397	−	0.725334i	$-0.741685\pi$
$383$	−26.9444	−1.37679	−0.688397	+	0.725334i	$0.741685\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	22.0454i	1.12063i
$388$	0	0
$389$	− 2.00000i	− 0.101404i	−0.998714	−	0.0507020i	$-0.983854\pi$
$389$	− 2.00000i	− 0.101404i	0.998714	−	0.0507020i	$-0.0161459\pi$
$390$	0	0
$391$	−9.79796	−0.495504
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	0	0
$399$	29.3939	1.47153
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19.5959	0.971334
$408$	0	0
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	0	0
$411$	− 19.5959i	− 0.966595i
$412$	0	0
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	14.6969i	0.717992i	0.933339	+	0.358996i	$0.116881\pi$
$419$	14.6969i	0.717992i	−0.933339	+	0.358996i	$0.883119\pi$
$420$	0	0
$421$	2.00000i	0.0974740i	0.998812	+	0.0487370i	$0.0155196\pi$
$421$	2.00000i	0.0974740i	−0.998812	+	0.0487370i	$0.984480\pi$
$422$	0	0
$423$	−36.7423	−1.78647
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 14.6969i	− 0.711235i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−29.3939	−1.41585	−0.707927	−	0.706286i	$-0.750369\pi$
$431$	−29.3939	−1.41585	−0.707927	+	0.706286i	$0.750369\pi$
$432$	0	0
$433$	36.0000	1.73005	0.865025	−	0.501729i	$-0.167303\pi$
$433$	36.0000	1.73005	0.865025	+	0.501729i	$0.167303\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 12.0000i	− 0.574038i
$438$	0	0
$439$	−19.5959	−0.935262	−0.467631	−	0.883924i	$-0.654892\pi$
$439$	−19.5959	−0.935262	−0.467631	+	0.883924i	$0.654892\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	26.9444i	1.28017i	0.768306	+	0.640083i	$0.221100\pi$
$443$	26.9444i	1.28017i	−0.768306	+	0.640083i	$0.778900\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.89898	0.231714
$448$	0	0
$449$	32.0000	1.51017	0.755087	−	0.655625i	$-0.227595\pi$
$449$	32.0000	1.51017	0.755087	+	0.655625i	$0.227595\pi$
$450$	0	0
$451$	39.1918i	1.84547i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.00000i	0.372597i	0.982493	+	0.186299i	$0.0596492\pi$
$461$	8.00000i	0.372597i	−0.982493	+	0.186299i	$0.940351\pi$
$462$	0	0
$463$	−31.8434	−1.47989	−0.739943	−	0.672669i	$-0.765148\pi$
$463$	−31.8434	−1.47989	−0.739943	+	0.672669i	$0.765148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.7423i	1.70023i	0.526595	+	0.850117i	$0.323469\pi$
$467$	36.7423i	1.70023i	−0.526595	+	0.850117i	$0.676531\pi$
$468$	0	0
$469$	− 6.00000i	− 0.277054i
$470$	0	0
$471$	−48.9898	−2.25733
$472$	0	0
$473$	36.0000	1.65528
$474$	0	0
$475$	0	0
$476$	0	0
$477$	24.0000i	1.09888i
$478$	0	0
$479$	−19.5959	−0.895360	−0.447680	−	0.894194i	$-0.647750\pi$
$479$	−19.5959	−0.895360	−0.447680	+	0.894194i	$0.647750\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	14.6969i	0.668734i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	17.1464	0.776979	0.388489	−	0.921453i	$-0.372997\pi$
$487$	17.1464	0.776979	0.388489	+	0.921453i	$0.372997\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	14.6969i	0.663264i	0.943409	+	0.331632i	$0.107599\pi$
$491$	14.6969i	0.663264i	−0.943409	+	0.331632i	$0.892401\pi$
$492$	0	0
$493$	32.0000i	1.44121i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	− 4.89898i	− 0.219308i	−0.993970	−	0.109654i	$-0.965026\pi$
$499$	− 4.89898i	− 0.219308i	0.993970	−	0.109654i	$-0.0349744\pi$
$500$	0	0
$501$	6.00000i	0.268060i
$502$	0	0
$503$	17.1464	0.764521	0.382261	−	0.924055i	$-0.375146\pi$
$503$	17.1464	0.764521	0.382261	+	0.924055i	$0.375146\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	31.8434i	1.41421i
$508$	0	0
$509$	8.00000i	0.354594i	0.984157	+	0.177297i	$0.0567353\pi$
$509$	8.00000i	0.354594i	−0.984157	+	0.177297i	$0.943265\pi$
$510$	0	0
$511$	9.79796	0.433436
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	60.0000i	2.63880i
$518$	0	0
$519$	−9.79796	−0.430083
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	− 31.8434i	− 1.39241i	−0.717841	−	0.696207i	$-0.754870\pi$
$523$	− 31.8434i	− 1.39241i	0.717841	−	0.696207i	$-0.245130\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−39.1918	−1.70722
$528$	0	0
$529$	−17.0000	−0.739130
$530$	0	0
$531$	− 14.6969i	− 0.637793i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−36.0000	−1.55351
$538$	0	0
$539$	− 4.89898i	− 0.211014i
$540$	0	0
$541$	8.00000i	0.343947i	0.985102	+	0.171973i	$0.0550143\pi$
$541$	8.00000i	0.343947i	−0.985102	+	0.171973i	$0.944986\pi$
$542$	0	0
$543$	19.5959	0.840941
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 22.0454i	− 0.942594i	−0.881975	−	0.471297i	$-0.843786\pi$
$547$	− 22.0454i	− 0.942594i	0.881975	−	0.471297i	$-0.156214\pi$
$548$	0	0
$549$	− 18.0000i	− 0.768221i
$550$	0	0
$551$	−39.1918	−1.66963
$552$	0	0
$553$	24.0000	1.02058
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.00000i	0.169485i	0.996403	+	0.0847427i	$0.0270068\pi$
$557$	4.00000i	0.169485i	−0.996403	+	0.0847427i	$0.972993\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−48.0000	−2.02656
$562$	0	0
$563$	2.44949i	0.103234i	0.998667	+	0.0516168i	$0.0164375\pi$
$563$	2.44949i	0.103234i	−0.998667	+	0.0516168i	$0.983563\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	22.0454	0.925820
$568$	0	0
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	0	0
$571$	44.0908i	1.84514i	0.385826	+	0.922572i	$0.373917\pi$
$571$	44.0908i	1.84514i	−0.385826	+	0.922572i	$0.626083\pi$
$572$	0	0
$573$	48.0000i	2.00523i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	24.0000	0.999133	0.499567	−	0.866276i	$-0.333493\pi$
$577$	24.0000	0.999133	0.499567	+	0.866276i	$0.333493\pi$
$578$	0	0
$579$	29.3939i	1.22157i
$580$	0	0
$581$	6.00000i	0.248922i
$582$	0	0
$583$	39.1918	1.62316
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.2474i	− 0.505506i	−0.967531	−	0.252753i	$-0.918664\pi$
$587$	− 12.2474i	− 0.505506i	0.967531	−	0.252753i	$-0.0813361\pi$
$588$	0	0
$589$	− 48.0000i	− 1.97781i
$590$	0	0
$591$	19.5959	0.806068
$592$	0	0
$593$	−8.00000	−0.328521	−0.164260	−	0.986417i	$-0.552524\pi$
$593$	−8.00000	−0.328521	−0.164260	+	0.986417i	$0.552524\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	24.0000i	0.982255i
$598$	0	0
$599$	−9.79796	−0.400334	−0.200167	−	0.979762i	$-0.564148\pi$
$599$	−9.79796	−0.400334	−0.200167	+	0.979762i	$0.564148\pi$
$600$	0	0
$601$	16.0000	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$602$	0	0
$603$	− 7.34847i	− 0.299253i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.34847	−0.298265	−0.149133	−	0.988817i	$-0.547648\pi$
$607$	−7.34847	−0.298265	−0.149133	+	0.988817i	$0.547648\pi$
$608$	0	0
$609$	48.0000	1.94506
$610$	0	0
$611$	0	0
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28.0000	−1.12724	−0.563619	−	0.826035i	$-0.690591\pi$
$617$	−28.0000	−1.12724	−0.563619	+	0.826035i	$0.690591\pi$
$618$	0	0
$619$	24.4949i	0.984533i	0.870445	+	0.492267i	$0.163831\pi$
$619$	24.4949i	0.984533i	−0.870445	+	0.492267i	$0.836169\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.89898	0.196273
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 58.7878i	− 2.34776i
$628$	0	0
$629$	− 16.0000i	− 0.637962i
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	60.0000	2.38479
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	29.3939	1.16280
$640$	0	0
$641$	−16.0000	−0.631962	−0.315981	−	0.948766i	$-0.602334\pi$
$641$	−16.0000	−0.631962	−0.315981	+	0.948766i	$0.602334\pi$
$642$	0	0
$643$	− 2.44949i	− 0.0965984i	−0.998833	−	0.0482992i	$-0.984620\pi$
$643$	− 2.44949i	− 0.0965984i	0.998833	−	0.0482992i	$-0.0153801\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.0454	−0.866694	−0.433347	−	0.901227i	$-0.642668\pi$
$647$	−22.0454	−0.866694	−0.433347	+	0.901227i	$0.642668\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	58.7878i	2.30407i
$652$	0	0
$653$	− 16.0000i	− 0.626128i	−0.949732	−	0.313064i	$-0.898644\pi$
$653$	− 16.0000i	− 0.626128i	0.949732	−	0.313064i	$-0.101356\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.0000	0.468165
$658$	0	0
$659$	14.6969i	0.572511i	0.958153	+	0.286256i	$0.0924107\pi$
$659$	14.6969i	0.572511i	−0.958153	+	0.286256i	$0.907589\pi$
$660$	0	0
$661$	30.0000i	1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$	30.0000i	1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 19.5959i	− 0.758757i
$668$	0	0
$669$	− 18.0000i	− 0.695920i
$670$	0	0
$671$	−29.3939	−1.13474
$672$	0	0
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 32.0000i	− 1.22986i	−0.788582	−	0.614930i	$-0.789184\pi$
$677$	− 32.0000i	− 1.22986i	0.788582	−	0.614930i	$-0.210816\pi$
$678$	0	0
$679$	9.79796	0.376011
$680$	0	0
$681$	6.00000	0.229920
$682$	0	0
$683$	− 7.34847i	− 0.281181i	−0.990068	−	0.140591i	$-0.955100\pi$
$683$	− 7.34847i	− 0.281181i	0.990068	−	0.140591i	$-0.0449002\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	19.5959	0.747631
$688$	0	0
$689$	0	0
$690$	0	0
$691$	14.6969i	0.559098i	0.960131	+	0.279549i	$0.0901849\pi$
$691$	14.6969i	0.559098i	−0.960131	+	0.279549i	$0.909815\pi$
$692$	0	0
$693$	36.0000i	1.36753i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	32.0000	1.21209
$698$	0	0
$699$	− 48.9898i	− 1.85296i
$700$	0	0
$701$	26.0000i	0.982006i	0.871158	+	0.491003i	$0.163370\pi$
$701$	26.0000i	0.982006i	−0.871158	+	0.491003i	$0.836630\pi$
$702$	0	0
$703$	19.5959	0.739074
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.5959i	0.736980i
$708$	0	0
$709$	40.0000i	1.50223i	0.660171	+	0.751116i	$0.270484\pi$
$709$	40.0000i	1.50223i	−0.660171	+	0.751116i	$0.729516\pi$
$710$	0	0
$711$	29.3939	1.10236
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.0000i	0.896296i
$718$	0	0
$719$	−48.9898	−1.82701	−0.913506	−	0.406826i	$-0.866635\pi$
$719$	−48.9898	−1.82701	−0.913506	+	0.406826i	$0.866635\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	36.7423	1.36270	0.681349	−	0.731959i	$-0.261394\pi$
$727$	36.7423	1.36270	0.681349	+	0.731959i	$0.261394\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	− 29.3939i	− 1.08717i
$732$	0	0
$733$	12.0000i	0.443230i	0.975134	+	0.221615i	$0.0711328\pi$
$733$	12.0000i	0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	14.6969i	0.540636i	0.962771	+	0.270318i	$0.0871288\pi$
$739$	14.6969i	0.540636i	−0.962771	+	0.270318i	$0.912871\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.1464	−0.629041	−0.314521	−	0.949251i	$-0.601844\pi$
$743$	−17.1464	−0.629041	−0.314521	+	0.949251i	$0.601844\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.34847i	0.268866i
$748$	0	0
$749$	18.0000i	0.657706i
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 12.0000i	− 0.436147i	−0.975932	−	0.218074i	$-0.930023\pi$
$757$	− 12.0000i	− 0.436147i	0.975932	−	0.218074i	$-0.0699773\pi$
$758$	0	0
$759$	29.3939	1.06693
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	24.4949i	0.886775i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	− 19.5959i	− 0.705730i
$772$	0	0
$773$	− 8.00000i	− 0.287740i	−0.989597	−	0.143870i	$-0.954045\pi$
$773$	− 8.00000i	− 0.287740i	0.989597	−	0.143870i	$-0.0459547\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−24.0000	−0.860995
$778$	0	0
$779$	39.1918i	1.40419i
$780$	0	0
$781$	− 48.0000i	− 1.71758i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	41.6413i	1.48435i	0.670205	+	0.742176i	$0.266206\pi$
$787$	41.6413i	1.48435i	−0.670205	+	0.742176i	$0.733794\pi$
$788$	0	0
$789$	54.0000i	1.92245i
$790$	0	0
$791$	39.1918	1.39350
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 8.00000i	− 0.283375i	−0.989911	−	0.141687i	$-0.954747\pi$
$797$	− 8.00000i	− 0.283375i	0.989911	−	0.141687i	$-0.0452527\pi$
$798$	0	0
$799$	48.9898	1.73313
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	− 19.5959i	− 0.691525i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−63.6867	−2.24188
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	− 53.8888i	− 1.89229i	−0.323742	−	0.946145i	$-0.604941\pi$
$811$	− 53.8888i	− 1.89229i	0.323742	−	0.946145i	$-0.395059\pi$
$812$	0	0
$813$	− 24.0000i	− 0.841717i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	36.0000	1.25948
$818$	0	0
$819$	0	0
$820$	0	0
$821$	14.0000i	0.488603i	0.969699	+	0.244302i	$0.0785587\pi$
$821$	14.0000i	0.488603i	−0.969699	+	0.244302i	$0.921441\pi$
$822$	0	0
$823$	−22.0454	−0.768455	−0.384227	−	0.923239i	$-0.625532\pi$
$823$	−22.0454	−0.768455	−0.384227	+	0.923239i	$0.625532\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	46.5403i	1.61836i	0.587557	+	0.809182i	$0.300090\pi$
$827$	46.5403i	1.61836i	−0.587557	+	0.809182i	$0.699910\pi$
$828$	0	0
$829$	− 22.0000i	− 0.764092i	−0.924143	−	0.382046i	$-0.875220\pi$
$829$	− 22.0000i	− 0.764092i	0.924143	−	0.382046i	$-0.124780\pi$
$830$	0	0
$831$	−29.3939	−1.01966
$832$	0	0
$833$	−4.00000	−0.138592
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	39.1918	1.35305	0.676526	−	0.736419i	$-0.263485\pi$
$839$	39.1918	1.35305	0.676526	+	0.736419i	$0.263485\pi$
$840$	0	0
$841$	−35.0000	−1.20690
$842$	0	0
$843$	39.1918i	1.34984i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	31.8434	1.09415
$848$	0	0
$849$	−66.0000	−2.26511
$850$	0	0
$851$	9.79796i	0.335870i
$852$	0	0
$853$	32.0000i	1.09566i	0.836590	+	0.547830i	$0.184546\pi$
$853$	32.0000i	1.09566i	−0.836590	+	0.547830i	$0.815454\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−32.0000	−1.09310	−0.546550	−	0.837427i	$-0.684059\pi$
$857$	−32.0000	−1.09310	−0.546550	+	0.837427i	$0.684059\pi$
$858$	0	0
$859$	24.4949i	0.835755i	0.908503	+	0.417878i	$0.137226\pi$
$859$	24.4949i	0.835755i	−0.908503	+	0.417878i	$0.862774\pi$
$860$	0	0
$861$	− 48.0000i	− 1.63584i
$862$	0	0
$863$	−51.4393	−1.75101	−0.875507	−	0.483206i	$-0.839472\pi$
$863$	−51.4393	−1.75101	−0.875507	+	0.483206i	$0.839472\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.44949i	− 0.0831890i
$868$	0	0
$869$	− 48.0000i	− 1.62829i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28.0000i	0.945493i	0.881199	+	0.472746i	$0.156737\pi$
$877$	28.0000i	0.945493i	−0.881199	+	0.472746i	$0.843263\pi$
$878$	0	0
$879$	−48.9898	−1.65238
$880$	0	0
$881$	−8.00000	−0.269527	−0.134763	−	0.990878i	$-0.543027\pi$
$881$	−8.00000	−0.269527	−0.134763	+	0.990878i	$0.543027\pi$
$882$	0	0
$883$	− 36.7423i	− 1.23648i	−0.785990	−	0.618239i	$-0.787846\pi$
$883$	− 36.7423i	− 1.23648i	0.785990	−	0.618239i	$-0.212154\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.44949	−0.0822458	−0.0411229	−	0.999154i	$-0.513094\pi$
$887$	−2.44949	−0.0822458	−0.0411229	+	0.999154i	$0.513094\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	− 44.0908i	− 1.47710i
$892$	0	0
$893$	60.0000i	2.00782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 78.3837i	− 2.61424i
$900$	0	0
$901$	− 32.0000i	− 1.06607i
$902$	0	0
$903$	−44.0908	−1.46725
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12.2474i	0.406670i	0.979109	+	0.203335i	$0.0651780\pi$
$907$	12.2474i	0.406670i	−0.979109	+	0.203335i	$0.934822\pi$
$908$	0	0
$909$	24.0000i	0.796030i
$910$	0	0
$911$	29.3939	0.973863	0.486931	−	0.873440i	$-0.338116\pi$
$911$	29.3939	0.973863	0.486931	+	0.873440i	$0.338116\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.0000i	0.396275i
$918$	0	0
$919$	−48.9898	−1.61602	−0.808012	−	0.589166i	$-0.799456\pi$
$919$	−48.9898	−1.61602	−0.808012	+	0.589166i	$0.799456\pi$
$920$	0	0
$921$	42.0000	1.38395
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.34847	0.241355
$928$	0	0
$929$	16.0000	0.524943	0.262471	−	0.964940i	$-0.415462\pi$
$929$	16.0000	0.524943	0.262471	+	0.964940i	$0.415462\pi$
$930$	0	0
$931$	− 4.89898i	− 0.160558i
$932$	0	0
$933$	48.0000i	1.57145i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	60.0000	1.96011	0.980057	−	0.198715i	$-0.0636769\pi$
$937$	60.0000	1.96011	0.980057	+	0.198715i	$0.0636769\pi$
$938$	0	0
$939$	− 58.7878i	− 1.91847i
$940$	0	0
$941$	8.00000i	0.260793i	0.991462	+	0.130396i	$0.0416250\pi$
$941$	8.00000i	0.260793i	−0.991462	+	0.130396i	$0.958375\pi$
$942$	0	0
$943$	−19.5959	−0.638131
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 17.1464i	− 0.557184i	−0.960410	−	0.278592i	$-0.910132\pi$
$947$	− 17.1464i	− 0.557184i	0.960410	−	0.278592i	$-0.0898677\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	19.5959	0.635441
$952$	0	0
$953$	−32.0000	−1.03658	−0.518291	−	0.855204i	$-0.673432\pi$
$953$	−32.0000	−1.03658	−0.518291	+	0.855204i	$0.673432\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 96.0000i	− 3.10324i
$958$	0	0
$959$	19.5959	0.632785
$960$	0	0
$961$	65.0000	2.09677
$962$	0	0
$963$	22.0454i	0.710403i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17.1464	−0.551392	−0.275696	−	0.961245i	$-0.588908\pi$
$967$	−17.1464	−0.551392	−0.275696	+	0.961245i	$0.588908\pi$
$968$	0	0
$969$	−48.0000	−1.54198
$970$	0	0
$971$	− 24.4949i	− 0.786079i	−0.919522	−	0.393039i	$-0.871424\pi$
$971$	− 24.4949i	− 0.786079i	0.919522	−	0.393039i	$-0.128576\pi$
$972$	0	0
$973$	12.0000i	0.384702i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.00000	−0.127971	−0.0639857	−	0.997951i	$-0.520381\pi$
$977$	−4.00000	−0.127971	−0.0639857	+	0.997951i	$0.520381\pi$
$978$	0	0
$979$	− 9.79796i	− 0.313144i
$980$	0	0
$981$	30.0000i	0.957826i
$982$	0	0
$983$	22.0454	0.703139	0.351570	−	0.936162i	$-0.385648\pi$
$983$	22.0454	0.703139	0.351570	+	0.936162i	$0.385648\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 73.4847i	− 2.33904i
$988$	0	0
$989$	18.0000i	0.572367i
$990$	0	0
$991$	39.1918	1.24497	0.622485	−	0.782632i	$-0.286123\pi$
$991$	39.1918	1.24497	0.622485	+	0.782632i	$0.286123\pi$
$992$	0	0
$993$	12.0000	0.380808
$994$	0	0
$995$	0	0
$996$	0	0
$997$	56.0000i	1.77354i	0.462213	+	0.886769i	$0.347056\pi$
$997$	56.0000i	1.77354i	−0.462213	+	0.886769i	$0.652944\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.d.l.1601.3		4
4.3	odd	2	inner	3200.2.d.l.1601.2		4
5.2	odd	4		640.2.f.g.449.3	yes	4
5.3	odd	4		640.2.f.c.449.1	✓	4
5.4	even	2		3200.2.d.k.1601.2		4
8.3	odd	2	inner	3200.2.d.l.1601.4		4
8.5	even	2	inner	3200.2.d.l.1601.1		4
16.3	odd	4		6400.2.a.bw.1.2		2
16.5	even	4		6400.2.a.bx.1.2		2
16.11	odd	4		6400.2.a.bx.1.1		2
16.13	even	4		6400.2.a.bw.1.1		2
20.3	even	4		640.2.f.c.449.3	yes	4
20.7	even	4		640.2.f.g.449.1	yes	4
20.19	odd	2		3200.2.d.k.1601.3		4
40.3	even	4		640.2.f.g.449.2	yes	4
40.13	odd	4		640.2.f.g.449.4	yes	4
40.19	odd	2		3200.2.d.k.1601.1		4
40.27	even	4		640.2.f.c.449.4	yes	4
40.29	even	2		3200.2.d.k.1601.4		4
40.37	odd	4		640.2.f.c.449.2	yes	4
80.3	even	4		1280.2.c.e.769.4		4
80.13	odd	4		1280.2.c.e.769.2		4
80.19	odd	4		6400.2.a.bu.1.1		2
80.27	even	4		1280.2.c.m.769.4		4
80.29	even	4		6400.2.a.bu.1.2		2
80.37	odd	4		1280.2.c.m.769.2		4
80.43	even	4		1280.2.c.m.769.1		4
80.53	odd	4		1280.2.c.m.769.3		4
80.59	odd	4		6400.2.a.bv.1.2		2
80.67	even	4		1280.2.c.e.769.1		4
80.69	even	4		6400.2.a.bv.1.1		2
80.77	odd	4		1280.2.c.e.769.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.f.c.449.1	✓	4	5.3	odd	4
640.2.f.c.449.2	yes	4	40.37	odd	4
640.2.f.c.449.3	yes	4	20.3	even	4
640.2.f.c.449.4	yes	4	40.27	even	4
640.2.f.g.449.1	yes	4	20.7	even	4
640.2.f.g.449.2	yes	4	40.3	even	4
640.2.f.g.449.3	yes	4	5.2	odd	4
640.2.f.g.449.4	yes	4	40.13	odd	4
1280.2.c.e.769.1		4	80.67	even	4
1280.2.c.e.769.2		4	80.13	odd	4
1280.2.c.e.769.3		4	80.77	odd	4
1280.2.c.e.769.4		4	80.3	even	4
1280.2.c.m.769.1		4	80.43	even	4
1280.2.c.m.769.2		4	80.37	odd	4
1280.2.c.m.769.3		4	80.53	odd	4
1280.2.c.m.769.4		4	80.27	even	4
3200.2.d.k.1601.1		4	40.19	odd	2
3200.2.d.k.1601.2		4	5.4	even	2
3200.2.d.k.1601.3		4	20.19	odd	2
3200.2.d.k.1601.4		4	40.29	even	2
3200.2.d.l.1601.1		4	8.5	even	2	inner
3200.2.d.l.1601.2		4	4.3	odd	2	inner
3200.2.d.l.1601.3		4	1.1	even	1	trivial
3200.2.d.l.1601.4		4	8.3	odd	2	inner
6400.2.a.bu.1.1		2	80.19	odd	4
6400.2.a.bu.1.2		2	80.29	even	4
6400.2.a.bv.1.1		2	80.69	even	4
6400.2.a.bv.1.2		2	80.59	odd	4
6400.2.a.bw.1.1		2	16.13	even	4
6400.2.a.bw.1.2		2	16.3	odd	4
6400.2.a.bx.1.1		2	16.11	odd	4
6400.2.a.bx.1.2		2	16.5	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:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.44949i q^{3} -2.44949 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+2.44949i q^{3} -2.44949 q^{7} -3.00000 q^{9} +4.89898i q^{11} +4.00000 q^{17} +4.89898i q^{19} -6.00000i q^{21} -2.44949 q^{23} +8.00000i q^{29} -9.79796 q^{31} -12.0000 q^{33} -4.00000i q^{37} +8.00000 q^{41} -7.34847i q^{43} +12.2474 q^{47} -1.00000 q^{49} +9.79796i q^{51} -8.00000i q^{53} -12.0000 q^{57} +4.89898i q^{59} +6.00000i q^{61} +7.34847 q^{63} +2.44949i q^{67} -6.00000i q^{69} -9.79796 q^{71} -4.00000 q^{73} -12.0000i q^{77} -9.79796 q^{79} -9.00000 q^{81} -2.44949i q^{83} -19.5959 q^{87} -2.00000 q^{89} -24.0000i q^{93} -4.00000 q^{97} -14.6969i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 16 q^{17} - 48 q^{33} + 32 q^{41} - 4 q^{49} - 48 q^{57} - 16 q^{73} - 36 q^{81} - 8 q^{89} - 16 q^{97}+O(q^{100}) \) 4 * q - 12 * q^9 + 16 * q^17 - 48 * q^33 + 32 * q^41 - 4 * q^49 - 48 * q^57 - 16 * q^73 - 36 * q^81 - 8 * q^89 - 16 * q^97

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