LMFDB - Embedded newform 1600.2.c.j.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(449,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("1600.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.7760643234$
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 800)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1600.449
Dual form			1600.2.c.j.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +5.00000 q^{11} -5.00000i q^{17} -5.00000 q^{19} +2.00000 q^{21} -6.00000i q^{23} +5.00000i q^{27} +4.00000 q^{29} -10.0000 q^{31} +5.00000i q^{33} -10.0000i q^{37} +5.00000 q^{41} +4.00000i q^{43} -8.00000i q^{47} +3.00000 q^{49} +5.00000 q^{51} +10.0000i q^{53} -5.00000i q^{57} +10.0000 q^{61} -4.00000i q^{63} -3.00000i q^{67} +6.00000 q^{69} -5.00000i q^{73} -10.0000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -1.00000i q^{83} +4.00000i q^{87} +9.00000 q^{89} -10.0000i q^{93} -10.0000i q^{97} +10.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^9	$ 2 q + 4 q^{9} + 10 q^{11} - 10 q^{19} + 4 q^{21} + 8 q^{29} - 20 q^{31} + 10 q^{41} + 6 q^{49} + 10 q^{51} + 20 q^{61} + 12 q^{69} + 20 q^{79} + 2 q^{81} + 18 q^{89} + 20 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 + 10 * q^11 - 10 * q^19 + 4 * q^21 + 8 * q^29 - 20 * q^31 + 10 * q^41 + 6 * q^49 + 10 * q^51 + 20 * q^61 + 12 * q^69 + 20 * q^79 + 2 * q^81 + 18 * q^89 + 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.00000i	− 1.21268i	−0.795206	−	0.606339i	$-0.792637\pi$
$17$	− 5.00000i	− 1.21268i	0.795206	−	0.606339i	$-0.207363\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	5.00000i	0.870388i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	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
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	5.00000	0.700140
$52$	0	0
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 5.00000i	− 0.662266i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	− 4.00000i	− 0.503953i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 5.00000i	− 0.585206i	−0.956234	−	0.292603i	$-0.905479\pi$
$73$	− 5.00000i	− 0.585206i	0.956234	−	0.292603i	$-0.0945214\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.0000i	− 1.13961i
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 1.00000i	− 0.109764i	−0.998493	−	0.0548821i	$-0.982522\pi$
$83$	− 1.00000i	− 0.109764i	0.998493	−	0.0548821i	$-0.0174783\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.00000i	0.428845i
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 10.0000i	− 1.03695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	10.0000	1.00504
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	15.0000i	1.41108i	0.708669	+	0.705541i	$0.249296\pi$
$113$	15.0000i	1.41108i	−0.708669	+	0.705541i	$0.750704\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−10.0000	−0.916698
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	5.00000i	0.450835i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.0000i	1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$	18.0000i	1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	10.0000i	0.867110i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.00000i	− 0.427179i	−0.976924	−	0.213589i	$-0.931485\pi$
$137$	− 5.00000i	− 0.427179i	0.976924	−	0.213589i	$-0.0685155\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.00000i	0.247436i
$148$	0	0
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	− 10.0000i	− 0.808452i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	− 21.0000i	− 1.64485i	−0.568876	−	0.822423i	$-0.692621\pi$
$163$	− 21.0000i	− 1.64485i	0.568876	−	0.822423i	$-0.307379\pi$
$164$	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
$169$	13.0000	1.00000
$170$	0	0
$171$	−10.0000	−0.764719
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	10.0000i	0.739221i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 25.0000i	− 1.82818i
$188$	0	0
$189$	10.0000	0.727393
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	5.00000i	0.359908i	0.983675	+	0.179954i	$0.0575949\pi$
$193$	5.00000i	0.359908i	−0.983675	+	0.179954i	$0.942405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 10.0000i	− 0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$	− 10.0000i	− 0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	0	0
$203$	− 8.00000i	− 0.561490i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.0000i	− 0.834058i
$208$	0	0
$209$	−25.0000	−1.72929
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	20.0000i	1.35769i
$218$	0	0
$219$	5.00000	0.337869
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	10.0000	0.657952
$232$	0	0
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.0000i	0.649570i
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	−15.0000	−0.966235	−0.483117	−	0.875556i	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.00000	0.0633724
$250$	0	0
$251$	−5.00000	−0.315597	−0.157799	−	0.987471i	$-0.550440\pi$
$251$	−5.00000	−0.315597	−0.157799	+	0.987471i	$0.550440\pi$
$252$	0	0
$253$	− 30.0000i	− 1.88608i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.0000i	0.623783i	0.950118	+	0.311891i	$0.100963\pi$
$257$	10.0000i	0.623783i	−0.950118	+	0.311891i	$0.899037\pi$
$258$	0	0
$259$	−20.0000	−1.24274
$260$	0	0
$261$	8.00000	0.495188
$262$	0	0
$263$	− 14.0000i	− 0.863277i	−0.902047	−	0.431638i	$-0.857936\pi$
$263$	− 14.0000i	− 0.863277i	0.902047	−	0.431638i	$-0.142064\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.00000i	0.550791i
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.0000i	1.20168i	0.799368	+	0.600842i	$0.205168\pi$
$277$	20.0000i	1.20168i	−0.799368	+	0.600842i	$0.794832\pi$
$278$	0	0
$279$	−20.0000	−1.19737
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	− 29.0000i	− 1.72387i	−0.507018	−	0.861936i	$-0.669252\pi$
$283$	− 29.0000i	− 1.72387i	0.507018	−	0.861936i	$-0.330748\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 10.0000i	− 0.590281i
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	10.0000	0.586210
$292$	0	0
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	25.0000i	1.45065i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	− 2.00000i	− 0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	7.00000i	0.399511i	0.979846	+	0.199756i	$0.0640148\pi$
$307$	7.00000i	0.399511i	−0.979846	+	0.199756i	$0.935985\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	0	0
$313$	− 30.0000i	− 1.69570i	−0.530236	−	0.847850i	$-0.677897\pi$
$313$	− 30.0000i	− 1.69570i	0.530236	−	0.847850i	$-0.322103\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.0000i	1.12331i	0.827371	+	0.561656i	$0.189836\pi$
$317$	20.0000i	1.12331i	−0.827371	+	0.561656i	$0.810164\pi$
$318$	0	0
$319$	20.0000	1.11979
$320$	0	0
$321$	−3.00000	−0.167444
$322$	0	0
$323$	25.0000i	1.39104i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.0000i	0.553001i
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	25.0000	1.37412	0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000	1.37412	0.687062	+	0.726599i	$0.258900\pi$
$332$	0	0
$333$	− 20.0000i	− 1.09599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.00000i	0.272367i	0.990684	+	0.136184i	$0.0434837\pi$
$337$	5.00000i	0.272367i	−0.990684	+	0.136184i	$0.956516\pi$
$338$	0	0
$339$	−15.0000	−0.814688
$340$	0	0
$341$	−50.0000	−2.70765
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.0000i	1.44944i	0.689046	+	0.724718i	$0.258030\pi$
$347$	27.0000i	1.44944i	−0.689046	+	0.724718i	$0.741970\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 10.0000i	− 0.529256i
$358$	0	0
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	14.0000i	0.734809i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.0000i	0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$	12.0000i	0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	20.0000	1.03835
$372$	0	0
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−15.0000	−0.770498	−0.385249	−	0.922813i	$-0.625884\pi$
$379$	−15.0000	−0.770498	−0.385249	+	0.922813i	$0.625884\pi$
$380$	0	0
$381$	−18.0000	−0.922168
$382$	0	0
$383$	14.0000i	0.715367i	0.933843	+	0.357683i	$0.116433\pi$
$383$	14.0000i	0.715367i	−0.933843	+	0.357683i	$0.883567\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.00000i	0.406663i
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−30.0000	−1.51717
$392$	0	0
$393$	− 20.0000i	− 1.00887i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.0000i	1.00377i	0.864934	+	0.501886i	$0.167360\pi$
$397$	20.0000i	1.00377i	−0.864934	+	0.501886i	$0.832640\pi$
$398$	0	0
$399$	−10.0000	−0.500626
$400$	0	0
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 50.0000i	− 2.47841i
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	5.00000	0.246632
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 5.00000i	− 0.244851i
$418$	0	0
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	0	0
$423$	− 16.0000i	− 0.777947i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	0	0
$433$	5.00000i	0.240285i	0.992757	+	0.120142i	$0.0383351\pi$
$433$	5.00000i	0.240285i	−0.992757	+	0.120142i	$0.961665\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	30.0000i	1.43509i
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	39.0000i	1.85295i	0.376361	+	0.926473i	$0.377175\pi$
$443$	39.0000i	1.85295i	−0.376361	+	0.926473i	$0.622825\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 20.0000i	− 0.945968i
$448$	0	0
$449$	−25.0000	−1.17982	−0.589911	−	0.807468i	$-0.700837\pi$
$449$	−25.0000	−1.17982	−0.589911	+	0.807468i	$0.700837\pi$
$450$	0	0
$451$	25.0000	1.17720
$452$	0	0
$453$	10.0000i	0.469841i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.0000i	0.701670i	0.936437	+	0.350835i	$0.114102\pi$
$457$	15.0000i	0.701670i	−0.936437	+	0.350835i	$0.885898\pi$
$458$	0	0
$459$	25.0000	1.16690
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	−6.00000	−0.277054
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	20.0000i	0.919601i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	20.0000i	0.915737i
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 12.0000i	− 0.546019i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	0	0
$489$	21.0000	0.949653
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	− 20.0000i	− 0.900755i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	− 4.00000i	− 0.178351i	−0.996016	−	0.0891756i	$-0.971577\pi$
$503$	− 4.00000i	− 0.178351i	0.996016	−	0.0891756i	$-0.0284232\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13.0000i	0.577350i
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	− 25.0000i	− 1.10378i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 40.0000i	− 1.75920i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	1.00000i	0.0437269i	0.999761	+	0.0218635i	$0.00695991\pi$
$523$	1.00000i	0.0437269i	−0.999761	+	0.0218635i	$0.993040\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	50.0000i	2.17803i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.0000i	0.647298i
$538$	0	0
$539$	15.0000	0.646096
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	0	0
$543$	− 18.0000i	− 0.772454i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	23.0000i	0.983409i	0.870762	+	0.491704i	$0.163626\pi$
$547$	23.0000i	0.983409i	−0.870762	+	0.491704i	$0.836374\pi$
$548$	0	0
$549$	20.0000	0.853579
$550$	0	0
$551$	−20.0000	−0.852029
$552$	0	0
$553$	− 20.0000i	− 0.850487i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 40.0000i	− 1.69485i	−0.530912	−	0.847427i	$-0.678150\pi$
$557$	− 40.0000i	− 1.69485i	0.530912	−	0.847427i	$-0.321850\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	25.0000	1.05550
$562$	0	0
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	10.0000i	0.417756i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.00000i	0.208153i	0.994569	+	0.104076i	$0.0331886\pi$
$577$	5.00000i	0.208153i	−0.994569	+	0.104076i	$0.966811\pi$
$578$	0	0
$579$	−5.00000	−0.207793
$580$	0	0
$581$	−2.00000	−0.0829740
$582$	0	0
$583$	50.0000i	2.07079i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7.00000i	− 0.288921i	−0.989511	−	0.144460i	$-0.953855\pi$
$587$	− 7.00000i	− 0.288921i	0.989511	−	0.144460i	$-0.0461446\pi$
$588$	0	0
$589$	50.0000	2.06021
$590$	0	0
$591$	10.0000	0.411345
$592$	0	0
$593$	15.0000i	0.615976i	0.951390	+	0.307988i	$0.0996557\pi$
$593$	15.0000i	0.615976i	−0.951390	+	0.307988i	$0.900344\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.0000i	0.818546i
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	0	0
$603$	− 6.00000i	− 0.244339i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	0	0
$612$	0	0
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	30.0000	1.20386
$622$	0	0
$623$	− 18.0000i	− 0.721155i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 25.0000i	− 0.998404i
$628$	0	0
$629$	−50.0000	−1.99363
$630$	0	0
$631$	30.0000	1.19428	0.597141	−	0.802137i	$-0.296303\pi$
$631$	30.0000	1.19428	0.597141	+	0.802137i	$0.296303\pi$
$632$	0	0
$633$	− 5.00000i	− 0.198732i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	36.0000i	1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$	36.0000i	1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−20.0000	−0.783862
$652$	0	0
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	25.0000	0.973862	0.486931	−	0.873441i	$-0.338116\pi$
$659$	25.0000	0.973862	0.486931	+	0.873441i	$0.338116\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 24.0000i	− 0.929284i
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	50.0000	1.93023
$672$	0	0
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 40.0000i	− 1.53732i	−0.639655	−	0.768662i	$-0.720923\pi$
$677$	− 40.0000i	− 1.53732i	0.639655	−	0.768662i	$-0.279077\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	− 31.0000i	− 1.18618i	−0.805135	−	0.593091i	$-0.797907\pi$
$683$	− 31.0000i	− 1.18618i	0.805135	−	0.593091i	$-0.202093\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 4.00000i	− 0.152610i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−15.0000	−0.570627	−0.285313	−	0.958434i	$-0.592098\pi$
$691$	−15.0000	−0.570627	−0.285313	+	0.958434i	$0.592098\pi$
$692$	0	0
$693$	− 20.0000i	− 0.759737i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 25.0000i	− 0.946943i
$698$	0	0
$699$	10.0000	0.378235
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	50.0000i	1.88579i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.00000i	0.150435i
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	60.0000i	2.24702i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 20.0000i	− 0.746914i
$718$	0	0
$719$	−10.0000	−0.372937	−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000	−0.372937	−0.186469	+	0.982461i	$0.559704\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	0	0
$723$	− 15.0000i	− 0.557856i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.0000i	1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$	28.0000i	1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	20.0000	0.739727
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 15.0000i	− 0.552532i
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 6.00000i	− 0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$	− 6.00000i	− 0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2.00000i	− 0.0731762i
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	− 5.00000i	− 0.182210i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	40.0000i	1.45382i	0.686730	+	0.726912i	$0.259045\pi$
$757$	40.0000i	1.45382i	−0.686730	+	0.726912i	$0.740955\pi$
$758$	0	0
$759$	30.0000	1.08893
$760$	0	0
$761$	37.0000	1.34125	0.670624	−	0.741797i	$-0.266026\pi$
$761$	37.0000	1.34125	0.670624	+	0.741797i	$0.266026\pi$
$762$	0	0
$763$	− 20.0000i	− 0.724049i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	51.0000	1.83911	0.919554	−	0.392965i	$-0.128551\pi$
$769$	51.0000	1.83911	0.919554	+	0.392965i	$0.128551\pi$
$770$	0	0
$771$	−10.0000	−0.360141
$772$	0	0
$773$	20.0000i	0.719350i	0.933078	+	0.359675i	$0.117112\pi$
$773$	20.0000i	0.719350i	−0.933078	+	0.359675i	$0.882888\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 20.0000i	− 0.717496i
$778$	0	0
$779$	−25.0000	−0.895718
$780$	0	0
$781$	0	0
$782$	0	0
$783$	20.0000i	0.714742i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	0	0
$789$	14.0000	0.498413
$790$	0	0
$791$	30.0000	1.06668
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	− 25.0000i	− 0.882231i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	− 10.0000i	− 0.350715i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 20.0000i	− 0.699711i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	− 36.0000i	− 1.25488i	−0.778664	−	0.627441i	$-0.784103\pi$
$823$	− 36.0000i	− 1.25488i	0.778664	−	0.627441i	$-0.215897\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23.0000i	− 0.799788i	−0.916561	−	0.399894i	$-0.869047\pi$
$827$	− 23.0000i	− 0.799788i	0.916561	−	0.399894i	$-0.130953\pi$
$828$	0	0
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	0	0
$831$	−20.0000	−0.693792
$832$	0	0
$833$	− 15.0000i	− 0.519719i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 50.0000i	− 1.72825i
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	− 10.0000i	− 0.344418i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 28.0000i	− 0.962091i
$848$	0	0
$849$	29.0000	0.995277
$850$	0	0
$851$	−60.0000	−2.05677
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 45.0000i	− 1.53717i	−0.639747	−	0.768585i	$-0.720961\pi$
$857$	− 45.0000i	− 1.53717i	0.639747	−	0.768585i	$-0.279039\pi$
$858$	0	0
$859$	−5.00000	−0.170598	−0.0852989	−	0.996355i	$-0.527185\pi$
$859$	−5.00000	−0.170598	−0.0852989	+	0.996355i	$0.527185\pi$
$860$	0	0
$861$	10.0000	0.340799
$862$	0	0
$863$	4.00000i	0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$	4.00000i	0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8.00000i	− 0.271694i
$868$	0	0
$869$	50.0000	1.69613
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 20.0000i	− 0.676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 20.0000i	− 0.675352i	−0.941262	−	0.337676i	$-0.890359\pi$
$877$	− 20.0000i	− 0.675352i	0.941262	−	0.337676i	$-0.109641\pi$
$878$	0	0
$879$	−30.0000	−1.01187
$880$	0	0
$881$	−10.0000	−0.336909	−0.168454	−	0.985709i	$-0.553878\pi$
$881$	−10.0000	−0.336909	−0.168454	+	0.985709i	$0.553878\pi$
$882$	0	0
$883$	1.00000i	0.0336527i	0.999858	+	0.0168263i	$0.00535624\pi$
$883$	1.00000i	0.0336527i	−0.999858	+	0.0168263i	$0.994644\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.0000i	0.940148i	0.882627	+	0.470074i	$0.155773\pi$
$887$	28.0000i	0.940148i	−0.882627	+	0.470074i	$0.844227\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	40.0000i	1.33855i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−40.0000	−1.33407
$900$	0	0
$901$	50.0000	1.66574
$902$	0	0
$903$	8.00000i	0.266223i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	52.0000i	1.72663i	0.504664	+	0.863316i	$0.331616\pi$
$907$	52.0000i	1.72663i	−0.504664	+	0.863316i	$0.668384\pi$
$908$	0	0
$909$	−4.00000	−0.132672
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	− 5.00000i	− 0.165476i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40.0000i	1.32092i
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	−7.00000	−0.230658
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	32.0000i	1.05102i
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	−15.0000	−0.491605
$932$	0	0
$933$	− 10.0000i	− 0.327385i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.00000i	0.163343i	0.996659	+	0.0816714i	$0.0260258\pi$
$937$	5.00000i	0.163343i	−0.996659	+	0.0816714i	$0.973974\pi$
$938$	0	0
$939$	30.0000	0.979013
$940$	0	0
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	0	0
$943$	− 30.0000i	− 0.976934i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.0000i	1.03986i	0.854209	+	0.519930i	$0.174042\pi$
$947$	32.0000i	1.03986i	−0.854209	+	0.519930i	$0.825958\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−20.0000	−0.648544
$952$	0	0
$953$	25.0000i	0.809829i	0.914354	+	0.404915i	$0.132699\pi$
$953$	25.0000i	0.809829i	−0.914354	+	0.404915i	$0.867301\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	20.0000i	0.646508i
$958$	0	0
$959$	−10.0000	−0.322917
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	6.00000i	0.193347i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	0	0
$969$	−25.0000	−0.803116
$970$	0	0
$971$	45.0000	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$972$	0	0
$973$	10.0000i	0.320585i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.0000i	0.479893i	0.970786	+	0.239946i	$0.0771298\pi$
$977$	15.0000i	0.479893i	−0.970786	+	0.239946i	$0.922870\pi$
$978$	0	0
$979$	45.0000	1.43821
$980$	0	0
$981$	20.0000	0.638551
$982$	0	0
$983$	46.0000i	1.46717i	0.679597	+	0.733586i	$0.262155\pi$
$983$	46.0000i	1.46717i	−0.679597	+	0.733586i	$0.737845\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 16.0000i	− 0.509286i
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	0	0
$993$	25.0000i	0.793351i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 60.0000i	− 1.90022i	−0.311916	−	0.950110i	$-0.600971\pi$
$997$	− 60.0000i	− 1.90022i	0.311916	−	0.950110i	$-0.399029\pi$
$998$	0	0
$999$	50.0000	1.58193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.c.j.449.2		2
4.3	odd	2		1600.2.c.g.449.1		2
5.2	odd	4		1600.2.a.s.1.1		1
5.3	odd	4		1600.2.a.h.1.1		1
5.4	even	2	inner	1600.2.c.j.449.1		2
8.3	odd	2		800.2.c.d.449.2		2
8.5	even	2		800.2.c.c.449.1		2
20.3	even	4		1600.2.a.r.1.1		1
20.7	even	4		1600.2.a.g.1.1		1
20.19	odd	2		1600.2.c.g.449.2		2
24.5	odd	2		7200.2.f.bc.6049.1		2
24.11	even	2		7200.2.f.a.6049.2		2
40.3	even	4		800.2.a.c.1.1	yes	1
40.13	odd	4		800.2.a.g.1.1	yes	1
40.19	odd	2		800.2.c.d.449.1		2
40.27	even	4		800.2.a.h.1.1	yes	1
40.29	even	2		800.2.c.c.449.2		2
40.37	odd	4		800.2.a.b.1.1	✓	1
120.29	odd	2		7200.2.f.bc.6049.2		2
120.53	even	4		7200.2.a.o.1.1		1
120.59	even	2		7200.2.f.a.6049.1		2
120.77	even	4		7200.2.a.bq.1.1		1
120.83	odd	4		7200.2.a.bm.1.1		1
120.107	odd	4		7200.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.2.a.b.1.1	✓	1	40.37	odd	4
800.2.a.c.1.1	yes	1	40.3	even	4
800.2.a.g.1.1	yes	1	40.13	odd	4
800.2.a.h.1.1	yes	1	40.27	even	4
800.2.c.c.449.1		2	8.5	even	2
800.2.c.c.449.2		2	40.29	even	2
800.2.c.d.449.1		2	40.19	odd	2
800.2.c.d.449.2		2	8.3	odd	2
1600.2.a.g.1.1		1	20.7	even	4
1600.2.a.h.1.1		1	5.3	odd	4
1600.2.a.r.1.1		1	20.3	even	4
1600.2.a.s.1.1		1	5.2	odd	4
1600.2.c.g.449.1		2	4.3	odd	2
1600.2.c.g.449.2		2	20.19	odd	2
1600.2.c.j.449.1		2	5.4	even	2	inner
1600.2.c.j.449.2		2	1.1	even	1	trivial
7200.2.a.k.1.1		1	120.107	odd	4
7200.2.a.o.1.1		1	120.53	even	4
7200.2.a.bm.1.1		1	120.83	odd	4
7200.2.a.bq.1.1		1	120.77	even	4
7200.2.f.a.6049.1		2	120.59	even	2
7200.2.f.a.6049.2		2	24.11	even	2
7200.2.f.bc.6049.1		2	24.5	odd	2
7200.2.f.bc.6049.2		2	120.29	odd	2

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +5.00000 q^{11} -5.00000i q^{17} -5.00000 q^{19} +2.00000 q^{21} -6.00000i q^{23} +5.00000i q^{27} +4.00000 q^{29} -10.0000 q^{31} +5.00000i q^{33} -10.0000i q^{37} +5.00000 q^{41} +4.00000i q^{43} -8.00000i q^{47} +3.00000 q^{49} +5.00000 q^{51} +10.0000i q^{53} -5.00000i q^{57} +10.0000 q^{61} -4.00000i q^{63} -3.00000i q^{67} +6.00000 q^{69} -5.00000i q^{73} -10.0000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -1.00000i q^{83} +4.00000i q^{87} +9.00000 q^{89} -10.0000i q^{93} -10.0000i q^{97} +10.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^9	\( 2 q + 4 q^{9} + 10 q^{11} - 10 q^{19} + 4 q^{21} + 8 q^{29} - 20 q^{31} + 10 q^{41} + 6 q^{49} + 10 q^{51} + 20 q^{61} + 12 q^{69} + 20 q^{79} + 2 q^{81} + 18 q^{89} + 20 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 + 10 * q^11 - 10 * q^19 + 4 * q^21 + 8 * q^29 - 20 * q^31 + 10 * q^41 + 6 * q^49 + 10 * q^51 + 20 * q^61 + 12 * q^69 + 20 * q^79 + 2 * q^81 + 18 * q^89 + 20 * q^99

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