LMFDB - Embedded newform 800.2.c.g.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, 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("800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.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:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.2.c.g.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{3} -4.47214i q^{7} -2.00000 q^{9} +O(q^{10})$	$q+2.23607i q^{3} -4.47214i q^{7} -2.00000 q^{9} -2.23607 q^{11} -4.00000i q^{13} -7.00000i q^{17} -6.70820 q^{19} +10.0000 q^{21} -4.47214i q^{23} +2.23607i q^{27} -4.47214 q^{31} -5.00000i q^{33} +2.00000i q^{37} +8.94427 q^{39} +5.00000 q^{41} +8.94427i q^{47} -13.0000 q^{49} +15.6525 q^{51} -6.00000i q^{53} -15.0000i q^{57} +8.94427 q^{59} +10.0000 q^{61} +8.94427i q^{63} +2.23607i q^{67} +10.0000 q^{69} +8.94427 q^{71} +9.00000i q^{73} +10.0000i q^{77} -4.47214 q^{79} -11.0000 q^{81} -11.1803i q^{83} +5.00000 q^{89} -17.8885 q^{91} -10.0000i q^{93} +2.00000i q^{97} +4.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{9}+O(q^{10}) $ 4 * q - 8 * q^9	$ 4 q - 8 q^{9} + 40 q^{21} + 20 q^{41} - 52 q^{49} + 40 q^{61} + 40 q^{69} - 44 q^{81} + 20 q^{89}+O(q^{100}) $ 4 * q - 8 * q^9 + 40 * q^21 + 20 * q^41 - 52 * q^49 + 40 * q^61 + 40 * q^69 - 44 * q^81 + 20 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.23607i	1.29099i	0.763763	+	0.645497i	$0.223350\pi$
$3$	2.23607i	1.29099i	−0.763763	+	0.645497i	$0.776650\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.47214i	− 1.69031i	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	− 4.47214i	− 1.69031i	0.534522	−	0.845154i	$-0.320491\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.00000i	− 1.69775i	−0.528594	−	0.848875i	$-0.677281\pi$
$17$	− 7.00000i	− 1.69775i	0.528594	−	0.848875i	$-0.322719\pi$
$18$	0	0
$19$	−6.70820	−1.53897	−0.769484	−	0.638666i	$-0.779486\pi$
$19$	−6.70820	−1.53897	−0.769484	+	0.638666i	$0.779486\pi$
$20$	0	0
$21$	10.0000	2.18218
$22$	0	0
$23$	− 4.47214i	− 0.932505i	−0.884652	−	0.466252i	$-0.845604\pi$
$23$	− 4.47214i	− 0.932505i	0.884652	−	0.466252i	$-0.154396\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.23607i	0.430331i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.47214	−0.803219	−0.401610	−	0.915811i	$-0.631549\pi$
$31$	−4.47214	−0.803219	−0.401610	+	0.915811i	$0.631549\pi$
$32$	0	0
$33$	− 5.00000i	− 0.870388i
$34$	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
$39$	8.94427	1.43223
$40$	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
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.94427i	1.30466i	0.757937	+	0.652328i	$0.226208\pi$
$47$	8.94427i	1.30466i	−0.757937	+	0.652328i	$0.773792\pi$
$48$	0	0
$49$	−13.0000	−1.85714
$50$	0	0
$51$	15.6525	2.19179
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 15.0000i	− 1.98680i
$58$	0	0
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	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
$63$	8.94427i	1.12687i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.23607i	0.273179i	0.990628	+	0.136590i	$0.0436142\pi$
$67$	2.23607i	0.273179i	−0.990628	+	0.136590i	$0.956386\pi$
$68$	0	0
$69$	10.0000	1.20386
$70$	0	0
$71$	8.94427	1.06149	0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427	1.06149	0.530745	+	0.847532i	$0.321912\pi$
$72$	0	0
$73$	9.00000i	1.05337i	0.850060	+	0.526685i	$0.176565\pi$
$73$	9.00000i	1.05337i	−0.850060	+	0.526685i	$0.823435\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.0000i	1.13961i
$78$	0	0
$79$	−4.47214	−0.503155	−0.251577	−	0.967837i	$-0.580949\pi$
$79$	−4.47214	−0.503155	−0.251577	+	0.967837i	$0.580949\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 11.1803i	− 1.22720i	−0.789616	−	0.613601i	$-0.789720\pi$
$83$	− 11.1803i	− 1.22720i	0.789616	−	0.613601i	$-0.210280\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.00000	0.529999	0.264999	−	0.964249i	$-0.414628\pi$
$89$	5.00000	0.529999	0.264999	+	0.964249i	$0.414628\pi$
$90$	0	0
$91$	−17.8885	−1.87523
$92$	0	0
$93$	− 10.0000i	− 1.03695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	4.47214	0.449467
$100$	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
$103$	− 8.94427i	− 0.881305i	−0.897678	−	0.440653i	$-0.854747\pi$
$103$	− 8.94427i	− 0.881305i	0.897678	−	0.440653i	$-0.145253\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.23607i	− 0.216169i	−0.994142	−	0.108084i	$-0.965528\pi$
$107$	− 2.23607i	− 0.216169i	0.994142	−	0.108084i	$-0.0344717\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−4.47214	−0.424476
$112$	0	0
$113$	1.00000i	0.0940721i	0.998893	+	0.0470360i	$0.0149776\pi$
$113$	1.00000i	0.0940721i	−0.998893	+	0.0470360i	$0.985022\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.00000i	0.739600i
$118$	0	0
$119$	−31.3050	−2.86972
$120$	0	0
$121$	−6.00000	−0.545455
$122$	0	0
$123$	11.1803i	1.00810i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.47214i	− 0.396838i	−0.980117	−	0.198419i	$-0.936419\pi$
$127$	− 4.47214i	− 0.396838i	0.980117	−	0.198419i	$-0.0635807\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.8885	−1.56293	−0.781465	−	0.623949i	$-0.785527\pi$
$131$	−17.8885	−1.56293	−0.781465	+	0.623949i	$0.785527\pi$
$132$	0	0
$133$	30.0000i	2.60133i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	2.23607	0.189661	0.0948304	−	0.995493i	$-0.469769\pi$
$139$	2.23607	0.189661	0.0948304	+	0.995493i	$0.469769\pi$
$140$	0	0
$141$	−20.0000	−1.68430
$142$	0	0
$143$	8.94427i	0.747958i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 29.0689i	− 2.39756i
$148$	0	0
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	0	0
$151$	13.4164	1.09181	0.545906	−	0.837846i	$-0.316186\pi$
$151$	13.4164	1.09181	0.545906	+	0.837846i	$0.316186\pi$
$152$	0	0
$153$	14.0000i	1.13183i
$154$	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
$159$	13.4164	1.06399
$160$	0	0
$161$	−20.0000	−1.57622
$162$	0	0
$163$	− 2.23607i	− 0.175142i	−0.996158	−	0.0875712i	$-0.972089\pi$
$163$	− 2.23607i	− 0.175142i	0.996158	−	0.0875712i	$-0.0279105\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.94427i	− 0.692129i	−0.938211	−	0.346064i	$-0.887518\pi$
$167$	− 8.94427i	− 0.692129i	0.938211	−	0.346064i	$-0.112482\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	13.4164	1.02598
$172$	0	0
$173$	− 16.0000i	− 1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$	− 16.0000i	− 1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.0000i	1.50329i
$178$	0	0
$179$	2.23607	0.167132	0.0835658	−	0.996502i	$-0.473369\pi$
$179$	2.23607	0.167132	0.0835658	+	0.996502i	$0.473369\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	22.3607i	1.65295i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.6525i	1.14462i
$188$	0	0
$189$	10.0000	0.727393
$190$	0	0
$191$	−13.4164	−0.970777	−0.485389	−	0.874299i	$-0.661322\pi$
$191$	−13.4164	−0.970777	−0.485389	+	0.874299i	$0.661322\pi$
$192$	0	0
$193$	− 9.00000i	− 0.647834i	−0.946085	−	0.323917i	$-0.895000\pi$
$193$	− 9.00000i	− 0.647834i	0.946085	−	0.323917i	$-0.105000\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	8.94427	0.634043	0.317021	−	0.948418i	$-0.397317\pi$
$199$	8.94427	0.634043	0.317021	+	0.948418i	$0.397317\pi$
$200$	0	0
$201$	−5.00000	−0.352673
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.94427i	0.621670i
$208$	0	0
$209$	15.0000	1.03757
$210$	0	0
$211$	20.1246	1.38544	0.692718	−	0.721209i	$-0.256413\pi$
$211$	20.1246	1.38544	0.692718	+	0.721209i	$0.256413\pi$
$212$	0	0
$213$	20.0000i	1.37038i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	20.0000i	1.35769i
$218$	0	0
$219$	−20.1246	−1.35990
$220$	0	0
$221$	−28.0000	−1.88348
$222$	0	0
$223$	− 26.8328i	− 1.79686i	−0.439119	−	0.898429i	$-0.644709\pi$
$223$	− 26.8328i	− 1.79686i	0.439119	−	0.898429i	$-0.355291\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.8885i	1.18730i	0.804722	+	0.593652i	$0.202314\pi$
$227$	17.8885i	1.18730i	−0.804722	+	0.593652i	$0.797686\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	−22.3607	−1.47122
$232$	0	0
$233$	26.0000i	1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$	26.0000i	1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 10.0000i	− 0.649570i
$238$	0	0
$239$	17.8885	1.15711	0.578557	−	0.815642i	$-0.303616\pi$
$239$	17.8885	1.15711	0.578557	+	0.815642i	$0.303616\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	0	0
$243$	− 17.8885i	− 1.14755i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.8328i	1.70733i
$248$	0	0
$249$	25.0000	1.58431
$250$	0	0
$251$	11.1803	0.705697	0.352848	−	0.935681i	$-0.385213\pi$
$251$	11.1803	0.705697	0.352848	+	0.935681i	$0.385213\pi$
$252$	0	0
$253$	10.0000i	0.628695i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	0	0
$259$	8.94427	0.555770
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 22.3607i	− 1.37882i	−0.724372	−	0.689409i	$-0.757870\pi$
$263$	− 22.3607i	− 1.37882i	0.724372	−	0.689409i	$-0.242130\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.1803i	0.684226i
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	13.4164	0.814989	0.407494	−	0.913208i	$-0.366403\pi$
$271$	13.4164	0.814989	0.407494	+	0.913208i	$0.366403\pi$
$272$	0	0
$273$	− 40.0000i	− 2.42091i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	0	0
$279$	8.94427	0.535480
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	6.70820i	0.398761i	0.979922	+	0.199381i	$0.0638930\pi$
$283$	6.70820i	0.398761i	−0.979922	+	0.199381i	$0.936107\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 22.3607i	− 1.31991i
$288$	0	0
$289$	−32.0000	−1.88235
$290$	0	0
$291$	−4.47214	−0.262161
$292$	0	0
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.00000i	− 0.290129i
$298$	0	0
$299$	−17.8885	−1.03452
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 4.47214i	− 0.256917i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	15.6525i	0.893334i	0.894700	+	0.446667i	$0.147389\pi$
$307$	15.6525i	0.893334i	−0.894700	+	0.446667i	$0.852611\pi$
$308$	0	0
$309$	20.0000	1.13776
$310$	0	0
$311$	22.3607	1.26796	0.633979	−	0.773350i	$-0.281421\pi$
$311$	22.3607	1.26796	0.633979	+	0.773350i	$0.281421\pi$
$312$	0	0
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	5.00000	0.279073
$322$	0	0
$323$	46.9574i	2.61278i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 13.4164i	− 0.741929i
$328$	0	0
$329$	40.0000	2.20527
$330$	0	0
$331$	−11.1803	−0.614527	−0.307264	−	0.951624i	$-0.599413\pi$
$331$	−11.1803	−0.614527	−0.307264	+	0.951624i	$0.599413\pi$
$332$	0	0
$333$	− 4.00000i	− 0.219199i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	27.0000i	1.47078i	0.677642	+	0.735392i	$0.263002\pi$
$337$	27.0000i	1.47078i	−0.677642	+	0.735392i	$0.736998\pi$
$338$	0	0
$339$	−2.23607	−0.121447
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	26.8328i	1.44884i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.1246i	− 1.08035i	−0.841554	−	0.540173i	$-0.818359\pi$
$347$	− 20.1246i	− 1.08035i	0.841554	−	0.540173i	$-0.181641\pi$
$348$	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
$351$	8.94427	0.477410
$352$	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
$357$	− 70.0000i	− 3.70479i
$358$	0	0
$359$	31.3050	1.65221	0.826106	−	0.563515i	$-0.190551\pi$
$359$	31.3050	1.65221	0.826106	+	0.563515i	$0.190551\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0	0
$363$	− 13.4164i	− 0.704179i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 26.8328i	− 1.40066i	−0.713818	−	0.700331i	$-0.753036\pi$
$367$	− 26.8328i	− 1.40066i	0.713818	−	0.700331i	$-0.246964\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−26.8328	−1.39309
$372$	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
$377$	0	0
$378$	0	0
$379$	15.6525	0.804014	0.402007	−	0.915637i	$-0.368313\pi$
$379$	15.6525	0.804014	0.402007	+	0.915637i	$0.368313\pi$
$380$	0	0
$381$	10.0000	0.512316
$382$	0	0
$383$	− 4.47214i	− 0.228515i	−0.993451	−	0.114258i	$-0.963551\pi$
$383$	− 4.47214i	− 0.228515i	0.993451	−	0.114258i	$-0.0364490\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−31.3050	−1.58316
$392$	0	0
$393$	− 40.0000i	− 2.01773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 28.0000i	− 1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$	− 28.0000i	− 1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$	0	0
$399$	−67.0820	−3.35830
$400$	0	0
$401$	17.0000	0.848939	0.424470	−	0.905442i	$-0.360461\pi$
$401$	17.0000	0.848939	0.424470	+	0.905442i	$0.360461\pi$
$402$	0	0
$403$	17.8885i	0.891092i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4.47214i	− 0.221676i
$408$	0	0
$409$	−29.0000	−1.43396	−0.716979	−	0.697095i	$-0.754476\pi$
$409$	−29.0000	−1.43396	−0.716979	+	0.697095i	$0.754476\pi$
$410$	0	0
$411$	6.70820	0.330891
$412$	0	0
$413$	− 40.0000i	− 1.96827i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.00000i	0.244851i
$418$	0	0
$419$	−15.6525	−0.764673	−0.382337	−	0.924023i	$-0.624881\pi$
$419$	−15.6525	−0.764673	−0.382337	+	0.924023i	$0.624881\pi$
$420$	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
$423$	− 17.8885i	− 0.869771i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 44.7214i	− 2.16422i
$428$	0	0
$429$	−20.0000	−0.965609
$430$	0	0
$431$	4.47214	0.215415	0.107708	−	0.994183i	$-0.465649\pi$
$431$	4.47214	0.215415	0.107708	+	0.994183i	$0.465649\pi$
$432$	0	0
$433$	− 1.00000i	− 0.0480569i	−0.999711	−	0.0240285i	$-0.992351\pi$
$433$	− 1.00000i	− 0.0480569i	0.999711	−	0.0240285i	$-0.00764923\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	30.0000i	1.43509i
$438$	0	0
$439$	−35.7771	−1.70755	−0.853774	−	0.520644i	$-0.825692\pi$
$439$	−35.7771	−1.70755	−0.853774	+	0.520644i	$0.825692\pi$
$440$	0	0
$441$	26.0000	1.23810
$442$	0	0
$443$	− 38.0132i	− 1.80606i	−0.429578	−	0.903030i	$-0.641338\pi$
$443$	− 38.0132i	− 1.80606i	0.429578	−	0.903030i	$-0.358662\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	35.7771i	1.69220i
$448$	0	0
$449$	19.0000	0.896665	0.448333	−	0.893867i	$-0.352018\pi$
$449$	19.0000	0.896665	0.448333	+	0.893867i	$0.352018\pi$
$450$	0	0
$451$	−11.1803	−0.526462
$452$	0	0
$453$	30.0000i	1.40952i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7.00000i	− 0.327446i	−0.986506	−	0.163723i	$-0.947650\pi$
$457$	− 7.00000i	− 0.327446i	0.986506	−	0.163723i	$-0.0523504\pi$
$458$	0	0
$459$	15.6525	0.730595
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	26.8328i	1.24703i	0.781813	+	0.623513i	$0.214295\pi$
$463$	26.8328i	1.24703i	−0.781813	+	0.623513i	$0.785705\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.8885i	0.827783i	0.910326	+	0.413892i	$0.135831\pi$
$467$	17.8885i	0.827783i	−0.910326	+	0.413892i	$0.864169\pi$
$468$	0	0
$469$	10.0000	0.461757
$470$	0	0
$471$	−40.2492	−1.85459
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.0000i	0.549442i
$478$	0	0
$479$	13.4164	0.613011	0.306506	−	0.951869i	$-0.400840\pi$
$479$	13.4164	0.613011	0.306506	+	0.951869i	$0.400840\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	− 44.7214i	− 2.03489i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.47214i	− 0.202652i	−0.994853	−	0.101326i	$-0.967692\pi$
$487$	− 4.47214i	− 0.202652i	0.994853	−	0.101326i	$-0.0323085\pi$
$488$	0	0
$489$	5.00000	0.226108
$490$	0	0
$491$	35.7771	1.61460	0.807299	−	0.590143i	$-0.200929\pi$
$491$	35.7771	1.61460	0.807299	+	0.590143i	$0.200929\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 40.0000i	− 1.79425i
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	20.0000	0.893534
$502$	0	0
$503$	26.8328i	1.19642i	0.801341	+	0.598208i	$0.204120\pi$
$503$	26.8328i	1.19642i	−0.801341	+	0.598208i	$0.795880\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6.70820i	− 0.297922i
$508$	0	0
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	40.2492	1.78052
$512$	0	0
$513$	− 15.0000i	− 0.662266i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 20.0000i	− 0.879599i
$518$	0	0
$519$	35.7771	1.57044
$520$	0	0
$521$	13.0000	0.569540	0.284770	−	0.958596i	$-0.408083\pi$
$521$	13.0000	0.569540	0.284770	+	0.958596i	$0.408083\pi$
$522$	0	0
$523$	− 6.70820i	− 0.293329i	−0.989186	−	0.146665i	$-0.953146\pi$
$523$	− 6.70820i	− 0.293329i	0.989186	−	0.146665i	$-0.0468538\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31.3050i	1.36367i
$528$	0	0
$529$	3.00000	0.130435
$530$	0	0
$531$	−17.8885	−0.776297
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.00000i	0.215766i
$538$	0	0
$539$	29.0689	1.25209
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	4.47214i	0.191918i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 20.1246i	− 0.860466i	−0.902718	−	0.430233i	$-0.858431\pi$
$547$	− 20.1246i	− 0.860466i	0.902718	−	0.430233i	$-0.141569\pi$
$548$	0	0
$549$	−20.0000	−0.853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	20.0000i	0.850487i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 28.0000i	− 1.18640i	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	− 28.0000i	− 1.18640i	0.805056	−	0.593199i	$-0.202135\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−35.0000	−1.47770
$562$	0	0
$563$	44.7214i	1.88478i	0.334515	+	0.942390i	$0.391427\pi$
$563$	44.7214i	1.88478i	−0.334515	+	0.942390i	$0.608573\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	49.1935i	2.06593i
$568$	0	0
$569$	−1.00000	−0.0419222	−0.0209611	−	0.999780i	$-0.506673\pi$
$569$	−1.00000	−0.0419222	−0.0209611	+	0.999780i	$0.506673\pi$
$570$	0	0
$571$	17.8885	0.748612	0.374306	−	0.927305i	$-0.377881\pi$
$571$	17.8885	0.748612	0.374306	+	0.927305i	$0.377881\pi$
$572$	0	0
$573$	− 30.0000i	− 1.25327i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	43.0000i	1.79011i	0.445952	+	0.895057i	$0.352865\pi$
$577$	43.0000i	1.79011i	−0.445952	+	0.895057i	$0.647135\pi$
$578$	0	0
$579$	20.1246	0.836350
$580$	0	0
$581$	−50.0000	−2.07435
$582$	0	0
$583$	13.4164i	0.555651i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.1246i	0.830632i	0.909677	+	0.415316i	$0.136329\pi$
$587$	20.1246i	0.830632i	−0.909677	+	0.415316i	$0.863671\pi$
$588$	0	0
$589$	30.0000	1.23613
$590$	0	0
$591$	40.2492	1.65563
$592$	0	0
$593$	9.00000i	0.369586i	0.982777	+	0.184793i	$0.0591614\pi$
$593$	9.00000i	0.369586i	−0.982777	+	0.184793i	$0.940839\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.0000i	0.818546i
$598$	0	0
$599$	−4.47214	−0.182727	−0.0913633	−	0.995818i	$-0.529122\pi$
$599$	−4.47214	−0.182727	−0.0913633	+	0.995818i	$0.529122\pi$
$600$	0	0
$601$	−25.0000	−1.01977	−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000	−1.01977	−0.509886	+	0.860242i	$0.670312\pi$
$602$	0	0
$603$	− 4.47214i	− 0.182119i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 17.8885i	− 0.726074i	−0.931775	−	0.363037i	$-0.881740\pi$
$607$	− 17.8885i	− 0.726074i	0.931775	−	0.363037i	$-0.118260\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35.7771	1.44739
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.00000i	− 0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$	− 2.00000i	− 0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$	0	0
$619$	17.8885	0.719001	0.359501	−	0.933145i	$-0.382947\pi$
$619$	17.8885	0.719001	0.359501	+	0.933145i	$0.382947\pi$
$620$	0	0
$621$	10.0000	0.401286
$622$	0	0
$623$	− 22.3607i	− 0.895862i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	33.5410i	1.33950i
$628$	0	0
$629$	14.0000	0.558217
$630$	0	0
$631$	13.4164	0.534099	0.267049	−	0.963683i	$-0.413951\pi$
$631$	13.4164	0.534099	0.267049	+	0.963683i	$0.413951\pi$
$632$	0	0
$633$	45.0000i	1.78859i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	52.0000i	2.06032i
$638$	0	0
$639$	−17.8885	−0.707660
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	− 17.8885i	− 0.705455i	−0.935726	−	0.352728i	$-0.885254\pi$
$643$	− 17.8885i	− 0.705455i	0.935726	−	0.352728i	$-0.114746\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.94427i	− 0.351636i	−0.984423	−	0.175818i	$-0.943743\pi$
$647$	− 8.94427i	− 0.351636i	0.984423	−	0.175818i	$-0.0562570\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	−44.7214	−1.75277
$652$	0	0
$653$	34.0000i	1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$	34.0000i	1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 18.0000i	− 0.702247i
$658$	0	0
$659$	6.70820	0.261315	0.130657	−	0.991428i	$-0.458291\pi$
$659$	6.70820	0.261315	0.130657	+	0.991428i	$0.458291\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	0	0
$663$	− 62.6099i	− 2.43157i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	60.0000	2.31973
$670$	0	0
$671$	−22.3607	−0.863224
$672$	0	0
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.0000i	1.22986i	0.788582	+	0.614930i	$0.210816\pi$
$677$	32.0000i	1.22986i	−0.788582	+	0.614930i	$0.789184\pi$
$678$	0	0
$679$	8.94427	0.343250
$680$	0	0
$681$	−40.0000	−1.53280
$682$	0	0
$683$	11.1803i	0.427804i	0.976855	+	0.213902i	$0.0686173\pi$
$683$	11.1803i	0.427804i	−0.976855	+	0.213902i	$0.931383\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 44.7214i	− 1.70623i
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−38.0132	−1.44609	−0.723044	−	0.690802i	$-0.757258\pi$
$691$	−38.0132	−1.44609	−0.723044	+	0.690802i	$0.757258\pi$
$692$	0	0
$693$	− 20.0000i	− 0.759737i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 35.0000i	− 1.32572i
$698$	0	0
$699$	−58.1378	−2.19897
$700$	0	0
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	0	0
$703$	− 13.4164i	− 0.506009i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.94427i	0.336384i
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	8.94427	0.335436
$712$	0	0
$713$	20.0000i	0.749006i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	40.0000i	1.49383i
$718$	0	0
$719$	4.47214	0.166783	0.0833913	−	0.996517i	$-0.473425\pi$
$719$	4.47214	0.166783	0.0833913	+	0.996517i	$0.473425\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	0	0
$723$	11.1803i	0.415801i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	7.00000	0.259259
$730$	0	0
$731$	0	0
$732$	0	0
$733$	44.0000i	1.62518i	0.582838	+	0.812589i	$0.301942\pi$
$733$	44.0000i	1.62518i	−0.582838	+	0.812589i	$0.698058\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.00000i	− 0.184177i
$738$	0	0
$739$	−35.7771	−1.31608	−0.658041	−	0.752982i	$-0.728615\pi$
$739$	−35.7771	−1.31608	−0.658041	+	0.752982i	$0.728615\pi$
$740$	0	0
$741$	−60.0000	−2.20416
$742$	0	0
$743$	31.3050i	1.14847i	0.818691	+	0.574234i	$0.194700\pi$
$743$	31.3050i	1.14847i	−0.818691	+	0.574234i	$0.805300\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	22.3607i	0.818134i
$748$	0	0
$749$	−10.0000	−0.365392
$750$	0	0
$751$	−53.6656	−1.95829	−0.979143	−	0.203171i	$-0.934875\pi$
$751$	−53.6656	−1.95829	−0.979143	+	0.203171i	$0.934875\pi$
$752$	0	0
$753$	25.0000i	0.911051i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.00000i	0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$	8.00000i	0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$	0	0
$759$	−22.3607	−0.811641
$760$	0	0
$761$	−43.0000	−1.55875	−0.779374	−	0.626559i	$-0.784463\pi$
$761$	−43.0000	−1.55875	−0.779374	+	0.626559i	$0.784463\pi$
$762$	0	0
$763$	26.8328i	0.971413i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 35.7771i	− 1.29184i
$768$	0	0
$769$	15.0000	0.540914	0.270457	−	0.962732i	$-0.412825\pi$
$769$	15.0000	0.540914	0.270457	+	0.962732i	$0.412825\pi$
$770$	0	0
$771$	4.47214	0.161060
$772$	0	0
$773$	− 24.0000i	− 0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$	− 24.0000i	− 0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.0000i	0.717496i
$778$	0	0
$779$	−33.5410	−1.20173
$780$	0	0
$781$	−20.0000	−0.715656
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	50.0000	1.78005
$790$	0	0
$791$	4.47214	0.159011
$792$	0	0
$793$	− 40.0000i	− 1.42044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	62.6099	2.21498
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	− 20.1246i	− 0.710182i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 53.6656i	− 1.88912i
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	17.8885	0.628152	0.314076	−	0.949398i	$-0.398305\pi$
$811$	17.8885	0.628152	0.314076	+	0.949398i	$0.398305\pi$
$812$	0	0
$813$	30.0000i	1.05215i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	35.7771	1.25015
$820$	0	0
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	0	0
$823$	26.8328i	0.935333i	0.883905	+	0.467667i	$0.154905\pi$
$823$	26.8328i	0.935333i	−0.883905	+	0.467667i	$0.845095\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 51.4296i	− 1.78838i	−0.447687	−	0.894191i	$-0.647752\pi$
$827$	− 51.4296i	− 1.78838i	0.447687	−	0.894191i	$-0.352248\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	−17.8885	−0.620547
$832$	0	0
$833$	91.0000i	3.15296i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 10.0000i	− 0.345651i
$838$	0	0
$839$	−17.8885	−0.617581	−0.308791	−	0.951130i	$-0.599924\pi$
$839$	−17.8885	−0.617581	−0.308791	+	0.951130i	$0.599924\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	67.0820i	2.31043i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.8328i	0.921986i
$848$	0	0
$849$	−15.0000	−0.514799
$850$	0	0
$851$	8.94427	0.306606
$852$	0	0
$853$	− 34.0000i	− 1.16414i	−0.813139	−	0.582069i	$-0.802243\pi$
$853$	− 34.0000i	− 1.16414i	0.813139	−	0.582069i	$-0.197757\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.00000i	− 0.102478i	−0.998686	−	0.0512390i	$-0.983683\pi$
$857$	− 3.00000i	− 0.102478i	0.998686	−	0.0512390i	$-0.0163170\pi$
$858$	0	0
$859$	−33.5410	−1.14440	−0.572202	−	0.820112i	$-0.693911\pi$
$859$	−33.5410	−1.14440	−0.572202	+	0.820112i	$0.693911\pi$
$860$	0	0
$861$	50.0000	1.70400
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 71.5542i	− 2.43011i
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	8.94427	0.303065
$872$	0	0
$873$	− 4.00000i	− 0.135379i
$874$	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
$879$	31.3050	1.05589
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	2.23607i	0.0752497i	0.999292	+	0.0376248i	$0.0119792\pi$
$883$	2.23607i	0.0752497i	−0.999292	+	0.0376248i	$0.988021\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.94427i	0.300319i	0.988662	+	0.150160i	$0.0479788\pi$
$887$	8.94427i	0.300319i	−0.988662	+	0.150160i	$0.952021\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	24.5967	0.824022
$892$	0	0
$893$	− 60.0000i	− 2.00782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 40.0000i	− 1.33556i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−42.0000	−1.39922
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	53.6656i	1.78194i	0.454064	+	0.890969i	$0.349974\pi$
$907$	53.6656i	1.78194i	−0.454064	+	0.890969i	$0.650026\pi$
$908$	0	0
$909$	4.00000	0.132672
$910$	0	0
$911$	−44.7214	−1.48168	−0.740842	−	0.671679i	$-0.765573\pi$
$911$	−44.7214	−1.48168	−0.740842	+	0.671679i	$0.765573\pi$
$912$	0	0
$913$	25.0000i	0.827379i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	80.0000i	2.64183i
$918$	0	0
$919$	13.4164	0.442566	0.221283	−	0.975210i	$-0.428975\pi$
$919$	13.4164	0.442566	0.221283	+	0.975210i	$0.428975\pi$
$920$	0	0
$921$	−35.0000	−1.15329
$922$	0	0
$923$	− 35.7771i	− 1.17762i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	17.8885i	0.587537i
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	87.2067	2.85808
$932$	0	0
$933$	50.0000i	1.63693i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 17.0000i	− 0.555366i	−0.960673	−	0.277683i	$-0.910434\pi$
$937$	− 17.0000i	− 0.555366i	0.960673	−	0.277683i	$-0.0895665\pi$
$938$	0	0
$939$	−13.4164	−0.437828
$940$	0	0
$941$	−28.0000	−0.912774	−0.456387	−	0.889781i	$-0.650857\pi$
$941$	−28.0000	−0.912774	−0.456387	+	0.889781i	$0.650857\pi$
$942$	0	0
$943$	− 22.3607i	− 0.728164i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 26.8328i	− 0.871949i	−0.899959	−	0.435975i	$-0.856404\pi$
$947$	− 26.8328i	− 0.871949i	0.899959	−	0.435975i	$-0.143596\pi$
$948$	0	0
$949$	36.0000	1.16861
$950$	0	0
$951$	26.8328	0.870114
$952$	0	0
$953$	− 9.00000i	− 0.291539i	−0.989319	−	0.145769i	$-0.953434\pi$
$953$	− 9.00000i	− 0.291539i	0.989319	−	0.145769i	$-0.0465657\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.4164	−0.433238
$960$	0	0
$961$	−11.0000	−0.354839
$962$	0	0
$963$	4.47214i	0.144113i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	53.6656i	1.72577i	0.505400	+	0.862885i	$0.331345\pi$
$967$	53.6656i	1.72577i	−0.505400	+	0.862885i	$0.668655\pi$
$968$	0	0
$969$	−105.000	−3.37309
$970$	0	0
$971$	−55.9017	−1.79397	−0.896985	−	0.442060i	$-0.854248\pi$
$971$	−55.9017	−1.79397	−0.896985	+	0.442060i	$0.854248\pi$
$972$	0	0
$973$	− 10.0000i	− 0.320585i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 3.00000i	− 0.0959785i	−0.998848	−	0.0479893i	$-0.984719\pi$
$977$	− 3.00000i	− 0.0959785i	0.998848	−	0.0479893i	$-0.0152813\pi$
$978$	0	0
$979$	−11.1803	−0.357325
$980$	0	0
$981$	12.0000	0.383131
$982$	0	0
$983$	− 4.47214i	− 0.142639i	−0.997454	−	0.0713195i	$-0.977279\pi$
$983$	− 4.47214i	− 0.142639i	0.997454	−	0.0713195i	$-0.0227210\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	89.4427i	2.84699i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−13.4164	−0.426186	−0.213093	−	0.977032i	$-0.568354\pi$
$991$	−13.4164	−0.426186	−0.213093	+	0.977032i	$0.568354\pi$
$992$	0	0
$993$	− 25.0000i	− 0.793351i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 52.0000i	− 1.64686i	−0.567420	−	0.823428i	$-0.692059\pi$
$997$	− 52.0000i	− 1.64686i	0.567420	−	0.823428i	$-0.307941\pi$
$998$	0	0
$999$	−4.47214	−0.141492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.c.g.449.3		4
3.2	odd	2		7200.2.f.bg.6049.2		4
4.3	odd	2	inner	800.2.c.g.449.2		4
5.2	odd	4		800.2.a.k.1.2	yes	2
5.3	odd	4		800.2.a.l.1.1	yes	2
5.4	even	2	inner	800.2.c.g.449.1		4
8.3	odd	2		1600.2.c.o.449.3		4
8.5	even	2		1600.2.c.o.449.2		4
12.11	even	2		7200.2.f.bg.6049.3		4
15.2	even	4		7200.2.a.cf.1.2		2
15.8	even	4		7200.2.a.cn.1.1		2
15.14	odd	2		7200.2.f.bg.6049.4		4
20.3	even	4		800.2.a.l.1.2	yes	2
20.7	even	4		800.2.a.k.1.1	✓	2
20.19	odd	2	inner	800.2.c.g.449.4		4
40.3	even	4		1600.2.a.ba.1.1		2
40.13	odd	4		1600.2.a.ba.1.2		2
40.19	odd	2		1600.2.c.o.449.1		4
40.27	even	4		1600.2.a.bb.1.2		2
40.29	even	2		1600.2.c.o.449.4		4
40.37	odd	4		1600.2.a.bb.1.1		2
60.23	odd	4		7200.2.a.cn.1.2		2
60.47	odd	4		7200.2.a.cf.1.1		2
60.59	even	2		7200.2.f.bg.6049.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.2.a.k.1.1	✓	2	20.7	even	4
800.2.a.k.1.2	yes	2	5.2	odd	4
800.2.a.l.1.1	yes	2	5.3	odd	4
800.2.a.l.1.2	yes	2	20.3	even	4
800.2.c.g.449.1		4	5.4	even	2	inner
800.2.c.g.449.2		4	4.3	odd	2	inner
800.2.c.g.449.3		4	1.1	even	1	trivial
800.2.c.g.449.4		4	20.19	odd	2	inner
1600.2.a.ba.1.1		2	40.3	even	4
1600.2.a.ba.1.2		2	40.13	odd	4
1600.2.a.bb.1.1		2	40.37	odd	4
1600.2.a.bb.1.2		2	40.27	even	4
1600.2.c.o.449.1		4	40.19	odd	2
1600.2.c.o.449.2		4	8.5	even	2
1600.2.c.o.449.3		4	8.3	odd	2
1600.2.c.o.449.4		4	40.29	even	2
7200.2.a.cf.1.1		2	60.47	odd	4
7200.2.a.cf.1.2		2	15.2	even	4
7200.2.a.cn.1.1		2	15.8	even	4
7200.2.a.cn.1.2		2	60.23	odd	4
7200.2.f.bg.6049.1		4	60.59	even	2
7200.2.f.bg.6049.2		4	3.2	odd	2
7200.2.f.bg.6049.3		4	12.11	even	2
7200.2.f.bg.6049.4		4	15.14	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{3} -4.47214i q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+2.23607i q^{3} -4.47214i q^{7} -2.00000 q^{9} -2.23607 q^{11} -4.00000i q^{13} -7.00000i q^{17} -6.70820 q^{19} +10.0000 q^{21} -4.47214i q^{23} +2.23607i q^{27} -4.47214 q^{31} -5.00000i q^{33} +2.00000i q^{37} +8.94427 q^{39} +5.00000 q^{41} +8.94427i q^{47} -13.0000 q^{49} +15.6525 q^{51} -6.00000i q^{53} -15.0000i q^{57} +8.94427 q^{59} +10.0000 q^{61} +8.94427i q^{63} +2.23607i q^{67} +10.0000 q^{69} +8.94427 q^{71} +9.00000i q^{73} +10.0000i q^{77} -4.47214 q^{79} -11.0000 q^{81} -11.1803i q^{83} +5.00000 q^{89} -17.8885 q^{91} -10.0000i q^{93} +2.00000i q^{97} +4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{9}+O(q^{10}) \) 4 * q - 8 * q^9	\( 4 q - 8 q^{9} + 40 q^{21} + 20 q^{41} - 52 q^{49} + 40 q^{61} + 40 q^{69} - 44 q^{81} + 20 q^{89}+O(q^{100}) \) 4 * q - 8 * q^9 + 40 * q^21 + 20 * q^41 - 52 * q^49 + 40 * q^61 + 40 * q^69 - 44 * q^81 + 20 * q^89

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