LMFDB - Embedded newform 3800.2.d.m.3649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.4227136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.2
Root			$2.19869i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.m.3649.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83424i q^{3} -1.83424i q^{7} -0.364448 q^{9} +O(q^{10})$	$q-1.83424i q^{3} -1.83424i q^{7} -0.364448 q^{9} +0.834243 q^{11} +2.19869i q^{13} +2.56314i q^{17} +1.00000 q^{19} -3.36445 q^{21} -0.635552i q^{23} -4.83424i q^{27} +9.62901 q^{29} -6.59607 q^{31} -1.53020i q^{33} -5.23163i q^{37} +4.03293 q^{39} +4.43032 q^{41} -7.06587i q^{43} -9.86718i q^{47} +3.63555 q^{49} +4.70142 q^{51} +0.668486i q^{53} -1.83424i q^{57} -0.397382 q^{59} +2.26456 q^{61} +0.668486i q^{63} -2.43686i q^{67} -1.16576 q^{69} +4.12628 q^{71} -9.49073i q^{73} -1.53020i q^{77} -2.62901 q^{79} -9.96052 q^{81} +10.8277i q^{83} -17.6619i q^{87} +5.97252 q^{89} +4.03293 q^{91} +12.0988i q^{93} +10.5961i q^{97} -0.304038 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{9}+O(q^{10}) $ 6 * q - 2 * q^9	$ 6 q - 2 q^{9} - 6 q^{11} + 6 q^{19} - 20 q^{21} + 2 q^{29} - 6 q^{31} + 2 q^{39} - 18 q^{41} + 22 q^{49} - 16 q^{51} + 20 q^{59} - 42 q^{61} - 18 q^{69} + 2 q^{71} + 40 q^{79} - 26 q^{81} - 8 q^{89} + 2 q^{91} + 20 q^{99}+O(q^{100}) $ 6 * q - 2 * q^9 - 6 * q^11 + 6 * q^19 - 20 * q^21 + 2 * q^29 - 6 * q^31 + 2 * q^39 - 18 * q^41 + 22 * q^49 - 16 * q^51 + 20 * q^59 - 42 * q^61 - 18 * q^69 + 2 * q^71 + 40 * q^79 - 26 * q^81 - 8 * q^89 + 2 * q^91 + 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.83424i	− 1.05900i	−0.848310	−	0.529500i	$-0.822379\pi$
$3$	− 1.83424i	− 1.05900i	0.848310	−	0.529500i	$-0.177621\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.83424i	− 0.693279i	−0.937998	−	0.346639i	$-0.887323\pi$
$7$	− 1.83424i	− 0.693279i	0.937998	−	0.346639i	$-0.112677\pi$
$8$	0	0
$9$	−0.364448	−0.121483
$10$	0	0
$11$	0.834243	0.251534	0.125767	−	0.992060i	$-0.459861\pi$
$11$	0.834243	0.251534	0.125767	+	0.992060i	$0.459861\pi$
$12$	0	0
$13$	2.19869i	0.609807i	0.952383	+	0.304904i	$0.0986243\pi$
$13$	2.19869i	0.609807i	−0.952383	+	0.304904i	$0.901376\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.56314i	0.621653i	0.950467	+	0.310826i	$0.100606\pi$
$17$	2.56314i	0.621653i	−0.950467	+	0.310826i	$0.899394\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.36445	−0.734183
$22$	0	0
$23$	− 0.635552i	− 0.132522i	−0.997802	−	0.0662609i	$-0.978893\pi$
$23$	− 0.635552i	− 0.132522i	0.997802	−	0.0662609i	$-0.0211070\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.83424i	− 0.930351i
$28$	0	0
$29$	9.62901	1.78806	0.894031	−	0.448005i	$-0.147865\pi$
$29$	9.62901	1.78806	0.894031	+	0.448005i	$0.147865\pi$
$30$	0	0
$31$	−6.59607	−1.18469	−0.592345	−	0.805684i	$-0.701798\pi$
$31$	−6.59607	−1.18469	−0.592345	+	0.805684i	$0.701798\pi$
$32$	0	0
$33$	− 1.53020i	− 0.266374i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.23163i	− 0.860074i	−0.902811	−	0.430037i	$-0.858501\pi$
$37$	− 5.23163i	− 0.860074i	0.902811	−	0.430037i	$-0.141499\pi$
$38$	0	0
$39$	4.03293	0.645786
$40$	0	0
$41$	4.43032	0.691899	0.345950	−	0.938253i	$-0.387557\pi$
$41$	4.43032	0.691899	0.345950	+	0.938253i	$0.387557\pi$
$42$	0	0
$43$	− 7.06587i	− 1.07753i	−0.842455	−	0.538767i	$-0.818890\pi$
$43$	− 7.06587i	− 1.07753i	0.842455	−	0.538767i	$-0.181110\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.86718i	− 1.43928i	−0.694350	−	0.719638i	$-0.744308\pi$
$47$	− 9.86718i	− 1.43928i	0.694350	−	0.719638i	$-0.255692\pi$
$48$	0	0
$49$	3.63555	0.519365
$50$	0	0
$51$	4.70142	0.658331
$52$	0	0
$53$	0.668486i	0.0918237i	0.998945	+	0.0459118i	$0.0146193\pi$
$53$	0.668486i	0.0918237i	−0.998945	+	0.0459118i	$0.985381\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.83424i	− 0.242951i
$58$	0	0
$59$	−0.397382	−0.0517348	−0.0258674	−	0.999665i	$-0.508235\pi$
$59$	−0.397382	−0.0517348	−0.0258674	+	0.999665i	$0.508235\pi$
$60$	0	0
$61$	2.26456	0.289947	0.144974	−	0.989436i	$-0.453690\pi$
$61$	2.26456	0.289947	0.144974	+	0.989436i	$0.453690\pi$
$62$	0	0
$63$	0.668486i	0.0842214i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.43686i	− 0.297710i	−0.988859	−	0.148855i	$-0.952441\pi$
$67$	− 2.43686i	− 0.297710i	0.988859	−	0.148855i	$-0.0475588\pi$
$68$	0	0
$69$	−1.16576	−0.140341
$70$	0	0
$71$	4.12628	0.489699	0.244850	−	0.969561i	$-0.421261\pi$
$71$	4.12628	0.489699	0.244850	+	0.969561i	$0.421261\pi$
$72$	0	0
$73$	− 9.49073i	− 1.11081i	−0.831581	−	0.555403i	$-0.812564\pi$
$73$	− 9.49073i	− 1.11081i	0.831581	−	0.555403i	$-0.187436\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.53020i	− 0.174383i
$78$	0	0
$79$	−2.62901	−0.295787	−0.147893	−	0.989003i	$-0.547249\pi$
$79$	−2.62901	−0.295787	−0.147893	+	0.989003i	$0.547249\pi$
$80$	0	0
$81$	−9.96052	−1.10672
$82$	0	0
$83$	10.8277i	1.18849i	0.804282	+	0.594247i	$0.202550\pi$
$83$	10.8277i	1.18849i	−0.804282	+	0.594247i	$0.797450\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 17.6619i	− 1.89356i
$88$	0	0
$89$	5.97252	0.633086	0.316543	−	0.948578i	$-0.397478\pi$
$89$	5.97252	0.633086	0.316543	+	0.948578i	$0.397478\pi$
$90$	0	0
$91$	4.03293	0.422766
$92$	0	0
$93$	12.0988i	1.25459i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.5961i	1.07587i	0.842987	+	0.537934i	$0.180795\pi$
$97$	10.5961i	1.07587i	−0.842987	+	0.537934i	$0.819205\pi$
$98$	0	0
$99$	−0.304038	−0.0305570
$100$	0	0
$101$	−14.6290	−1.45564	−0.727820	−	0.685768i	$-0.759467\pi$
$101$	−14.6290	−1.45564	−0.727820	+	0.685768i	$0.759467\pi$
$102$	0	0
$103$	− 7.50273i	− 0.739266i	−0.929178	−	0.369633i	$-0.879483\pi$
$103$	− 7.50273i	− 0.739266i	0.929178	−	0.369633i	$-0.120517\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.7014i	− 1.42124i	−0.703576	−	0.710620i	$-0.748415\pi$
$107$	− 14.7014i	− 1.42124i	0.703576	−	0.710620i	$-0.251585\pi$
$108$	0	0
$109$	1.30404	0.124904	0.0624521	−	0.998048i	$-0.480108\pi$
$109$	1.30404	0.124904	0.0624521	+	0.998048i	$0.480108\pi$
$110$	0	0
$111$	−9.59607	−0.910819
$112$	0	0
$113$	− 1.13282i	− 0.106567i	−0.998579	−	0.0532835i	$-0.983031\pi$
$113$	− 1.13282i	− 0.106567i	0.998579	−	0.0532835i	$-0.0169687\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.801309i	− 0.0740810i
$118$	0	0
$119$	4.70142	0.430979
$120$	0	0
$121$	−10.3040	−0.936731
$122$	0	0
$123$	− 8.12628i	− 0.732722i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.88811i	0.433750i	0.976199	+	0.216875i	$0.0695863\pi$
$127$	4.88811i	0.433750i	−0.976199	+	0.216875i	$0.930414\pi$
$128$	0	0
$129$	−12.9605	−1.14111
$130$	0	0
$131$	−1.96707	−0.171863	−0.0859317	−	0.996301i	$-0.527387\pi$
$131$	−1.96707	−0.171863	−0.0859317	+	0.996301i	$0.527387\pi$
$132$	0	0
$133$	− 1.83424i	− 0.159049i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.49073i	− 0.298233i	−0.988820	−	0.149116i	$-0.952357\pi$
$137$	− 3.49073i	− 0.298233i	0.988820	−	0.149116i	$-0.0476429\pi$
$138$	0	0
$139$	15.2526	1.29371	0.646853	−	0.762615i	$-0.276085\pi$
$139$	15.2526	1.29371	0.646853	+	0.762615i	$0.276085\pi$
$140$	0	0
$141$	−18.0988	−1.52419
$142$	0	0
$143$	1.83424i	0.153387i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 6.66849i	− 0.550007i
$148$	0	0
$149$	−1.70142	−0.139386	−0.0696929	−	0.997568i	$-0.522202\pi$
$149$	−1.70142	−0.139386	−0.0696929	+	0.997568i	$0.522202\pi$
$150$	0	0
$151$	−4.80131	−0.390725	−0.195362	−	0.980731i	$-0.562588\pi$
$151$	−4.80131	−0.390725	−0.195362	+	0.980731i	$0.562588\pi$
$152$	0	0
$153$	− 0.934131i	− 0.0755200i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1.55114i	− 0.123794i	−0.998083	−	0.0618971i	$-0.980285\pi$
$157$	− 1.55114i	− 0.123794i	0.998083	−	0.0618971i	$-0.0197151\pi$
$158$	0	0
$159$	1.22617	0.0972413
$160$	0	0
$161$	−1.16576	−0.0918745
$162$	0	0
$163$	5.25910i	0.411925i	0.978560	+	0.205962i	$0.0660324\pi$
$163$	5.25910i	0.411925i	−0.978560	+	0.205962i	$0.933968\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.56968i	− 0.276230i	−0.990416	−	0.138115i	$-0.955896\pi$
$167$	− 3.56968i	− 0.276230i	0.990416	−	0.138115i	$-0.0441044\pi$
$168$	0	0
$169$	8.16576	0.628135
$170$	0	0
$171$	−0.364448	−0.0278700
$172$	0	0
$173$	− 19.9660i	− 1.51799i	−0.651099	−	0.758993i	$-0.725692\pi$
$173$	− 19.9660i	− 1.51799i	0.651099	−	0.758993i	$-0.274308\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.728896i	0.0547872i
$178$	0	0
$179$	−16.1921	−1.21026	−0.605129	−	0.796127i	$-0.706878\pi$
$179$	−16.1921	−1.21026	−0.605129	+	0.796127i	$0.706878\pi$
$180$	0	0
$181$	−13.2107	−0.981943	−0.490972	−	0.871176i	$-0.663358\pi$
$181$	−13.2107	−0.981943	−0.490972	+	0.871176i	$0.663358\pi$
$182$	0	0
$183$	− 4.15375i	− 0.307054i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.13828i	0.156367i
$188$	0	0
$189$	−8.86718	−0.644992
$190$	0	0
$191$	−1.37645	−0.0995965	−0.0497982	−	0.998759i	$-0.515858\pi$
$191$	−1.37645	−0.0995965	−0.0497982	+	0.998759i	$0.515858\pi$
$192$	0	0
$193$	2.21962i	0.159772i	0.996804	+	0.0798860i	$0.0254556\pi$
$193$	2.21962i	0.159772i	−0.996804	+	0.0798860i	$0.974544\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.0264i	1.14183i	0.821008	+	0.570917i	$0.193412\pi$
$197$	16.0264i	1.14183i	−0.821008	+	0.570917i	$0.806588\pi$
$198$	0	0
$199$	7.56314	0.536137	0.268068	−	0.963400i	$-0.413615\pi$
$199$	7.56314	0.536137	0.268068	+	0.963400i	$0.413615\pi$
$200$	0	0
$201$	−4.46980	−0.315275
$202$	0	0
$203$	− 17.6619i	− 1.23963i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.231626i	0.0160991i
$208$	0	0
$209$	0.834243	0.0577058
$210$	0	0
$211$	−7.18669	−0.494752	−0.247376	−	0.968920i	$-0.579568\pi$
$211$	−7.18669	−0.494752	−0.247376	+	0.968920i	$0.579568\pi$
$212$	0	0
$213$	− 7.56860i	− 0.518592i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0988i	0.821320i
$218$	0	0
$219$	−17.4083	−1.17634
$220$	0	0
$221$	−5.63555	−0.379088
$222$	0	0
$223$	− 17.6739i	− 1.18353i	−0.806109	−	0.591767i	$-0.798430\pi$
$223$	− 17.6739i	− 1.18353i	0.806109	−	0.591767i	$-0.201570\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.7618i	− 0.714288i	−0.934049	−	0.357144i	$-0.883751\pi$
$227$	− 10.7618i	− 0.714288i	0.934049	−	0.357144i	$-0.116249\pi$
$228$	0	0
$229$	7.66194	0.506315	0.253158	−	0.967425i	$-0.418531\pi$
$229$	7.66194	0.506315	0.253158	+	0.967425i	$0.418531\pi$
$230$	0	0
$231$	−2.80677	−0.184672
$232$	0	0
$233$	− 8.03948i	− 0.526684i	−0.964703	−	0.263342i	$-0.915175\pi$
$233$	− 8.03948i	− 0.526684i	0.964703	−	0.263342i	$-0.0848247\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.82224i	0.313238i
$238$	0	0
$239$	24.3304	1.57380	0.786902	−	0.617078i	$-0.211684\pi$
$239$	24.3304	1.57380	0.786902	+	0.617078i	$0.211684\pi$
$240$	0	0
$241$	5.15921	0.332334	0.166167	−	0.986098i	$-0.446861\pi$
$241$	5.15921	0.332334	0.166167	+	0.986098i	$0.446861\pi$
$242$	0	0
$243$	3.76729i	0.241672i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.19869i	0.139899i
$248$	0	0
$249$	19.8606	1.25862
$250$	0	0
$251$	7.95159	0.501900	0.250950	−	0.968000i	$-0.419257\pi$
$251$	7.95159	0.501900	0.250950	+	0.968000i	$0.419257\pi$
$252$	0	0
$253$	− 0.530205i	− 0.0333337i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 7.02639i	− 0.438294i	−0.975692	−	0.219147i	$-0.929673\pi$
$257$	− 7.02639i	− 0.438294i	0.975692	−	0.219147i	$-0.0703275\pi$
$258$	0	0
$259$	−9.59607	−0.596271
$260$	0	0
$261$	−3.50927	−0.217219
$262$	0	0
$263$	− 1.20415i	− 0.0742511i	−0.999311	−	0.0371255i	$-0.988180\pi$
$263$	− 1.20415i	− 0.0742511i	0.999311	−	0.0371255i	$-0.0118201\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 10.9551i	− 0.670439i
$268$	0	0
$269$	−9.74090	−0.593913	−0.296957	−	0.954891i	$-0.595972\pi$
$269$	−9.74090	−0.593913	−0.296957	+	0.954891i	$0.595972\pi$
$270$	0	0
$271$	−16.2371	−0.986333	−0.493166	−	0.869935i	$-0.664161\pi$
$271$	−16.2371	−0.986333	−0.493166	+	0.869935i	$0.664161\pi$
$272$	0	0
$273$	− 7.39738i	− 0.447710i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	19.6739i	1.18209i	0.806638	+	0.591046i	$0.201285\pi$
$277$	19.6739i	1.18209i	−0.806638	+	0.591046i	$0.798715\pi$
$278$	0	0
$279$	2.40393	0.143919
$280$	0	0
$281$	−12.5841	−0.750703	−0.375351	−	0.926883i	$-0.622478\pi$
$281$	−12.5841	−0.750703	−0.375351	+	0.926883i	$0.622478\pi$
$282$	0	0
$283$	14.0329i	0.834171i	0.908867	+	0.417086i	$0.136949\pi$
$283$	14.0329i	0.834171i	−0.908867	+	0.417086i	$0.863051\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 8.12628i	− 0.479679i
$288$	0	0
$289$	10.4303	0.613548
$290$	0	0
$291$	19.4358	1.13935
$292$	0	0
$293$	− 17.3095i	− 1.01123i	−0.862759	−	0.505616i	$-0.831265\pi$
$293$	− 17.3095i	− 1.01123i	0.862759	−	0.505616i	$-0.168735\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.03293i	− 0.234015i
$298$	0	0
$299$	1.39738	0.0808127
$300$	0	0
$301$	−12.9605	−0.747032
$302$	0	0
$303$	26.8332i	1.54152i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.0384i	0.915359i	0.889117	+	0.457680i	$0.151319\pi$
$307$	16.0384i	0.915359i	−0.889117	+	0.457680i	$0.848681\pi$
$308$	0	0
$309$	−13.7618	−0.782883
$310$	0	0
$311$	−20.5411	−1.16478	−0.582390	−	0.812909i	$-0.697882\pi$
$311$	−20.5411	−1.16478	−0.582390	+	0.812909i	$0.697882\pi$
$312$	0	0
$313$	− 9.44232i	− 0.533711i	−0.963736	−	0.266856i	$-0.914015\pi$
$313$	− 9.44232i	− 0.533711i	0.963736	−	0.266856i	$-0.0859847\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 0.0329344i	− 0.00184978i	−1.00000		0.000924891i	$-0.999706\pi$
$317$	− 0.0329344i	− 0.00184978i	1.00000		0.000924891i	$-0.000294402\pi$
$318$	0	0
$319$	8.03293	0.449758
$320$	0	0
$321$	−26.9660	−1.50509
$322$	0	0
$323$	2.56314i	0.142617i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.39192i	− 0.132274i
$328$	0	0
$329$	−18.0988	−0.997819
$330$	0	0
$331$	32.5226	1.78760	0.893801	−	0.448463i	$-0.148029\pi$
$331$	32.5226	1.78760	0.893801	+	0.448463i	$0.148029\pi$
$332$	0	0
$333$	1.90666i	0.104484i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	29.1252i	1.58655i	0.608863	+	0.793275i	$0.291626\pi$
$337$	29.1252i	1.58655i	−0.608863	+	0.793275i	$0.708374\pi$
$338$	0	0
$339$	−2.07787	−0.112855
$340$	0	0
$341$	−5.50273	−0.297990
$342$	0	0
$343$	− 19.5082i	− 1.05334i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.8266i	0.849617i	0.905283	+	0.424809i	$0.139659\pi$
$347$	15.8266i	0.849617i	−0.905283	+	0.424809i	$0.860341\pi$
$348$	0	0
$349$	−34.1911	−1.83021	−0.915103	−	0.403221i	$-0.867891\pi$
$349$	−34.1911	−1.83021	−0.915103	+	0.403221i	$0.867891\pi$
$350$	0	0
$351$	10.6290	0.567334
$352$	0	0
$353$	− 26.9330i	− 1.43350i	−0.697329	−	0.716751i	$-0.745629\pi$
$353$	− 26.9330i	− 1.43350i	0.697329	−	0.716751i	$-0.254371\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 8.62355i	− 0.456407i
$358$	0	0
$359$	8.22270	0.433977	0.216989	−	0.976174i	$-0.430377\pi$
$359$	8.22270	0.433977	0.216989	+	0.976174i	$0.430377\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	18.9001i	0.991999i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 30.3514i	− 1.58433i	−0.610308	−	0.792164i	$-0.708954\pi$
$367$	− 30.3514i	− 1.58433i	0.610308	−	0.792164i	$-0.291046\pi$
$368$	0	0
$369$	−1.61462	−0.0840538
$370$	0	0
$371$	1.22617	0.0636594
$372$	0	0
$373$	29.4018i	1.52237i	0.648538	+	0.761183i	$0.275381\pi$
$373$	29.4018i	1.52237i	−0.648538	+	0.761183i	$0.724619\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	21.1712i	1.09037i
$378$	0	0
$379$	13.8013	0.708926	0.354463	−	0.935070i	$-0.384664\pi$
$379$	13.8013	0.708926	0.354463	+	0.935070i	$0.384664\pi$
$380$	0	0
$381$	8.96598	0.459341
$382$	0	0
$383$	− 10.5961i	− 0.541434i	−0.962659	−	0.270717i	$-0.912739\pi$
$383$	− 10.5961i	− 0.541434i	0.962659	−	0.270717i	$-0.0872608\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.57514i	0.130902i
$388$	0	0
$389$	−1.84972	−0.0937843	−0.0468922	−	0.998900i	$-0.514932\pi$
$389$	−1.84972	−0.0937843	−0.0468922	+	0.998900i	$0.514932\pi$
$390$	0	0
$391$	1.62901	0.0823825
$392$	0	0
$393$	3.60808i	0.182003i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 0.998915i	− 0.0501341i	−0.999686	−	0.0250671i	$-0.992020\pi$
$397$	− 0.998915i	− 0.0501341i	0.999686	−	0.0250671i	$-0.00797993\pi$
$398$	0	0
$399$	−3.36445	−0.168433
$400$	0	0
$401$	17.2371	0.860779	0.430389	−	0.902643i	$-0.358376\pi$
$401$	17.2371	0.860779	0.430389	+	0.902643i	$0.358376\pi$
$402$	0	0
$403$	− 14.5027i	− 0.722432i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4.36445i	− 0.216338i
$408$	0	0
$409$	−8.43578	−0.417122	−0.208561	−	0.978009i	$-0.566878\pi$
$409$	−8.43578	−0.417122	−0.208561	+	0.978009i	$0.566878\pi$
$410$	0	0
$411$	−6.40284	−0.315829
$412$	0	0
$413$	0.728896i	0.0358666i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 27.9769i	− 1.37003i
$418$	0	0
$419$	−7.38538	−0.360799	−0.180400	−	0.983593i	$-0.557739\pi$
$419$	−7.38538	−0.360799	−0.180400	+	0.983593i	$0.557739\pi$
$420$	0	0
$421$	−32.7278	−1.59506	−0.797528	−	0.603282i	$-0.793859\pi$
$421$	−32.7278	−1.59506	−0.797528	+	0.603282i	$0.793859\pi$
$422$	0	0
$423$	3.59607i	0.174847i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.15375i	− 0.201014i
$428$	0	0
$429$	3.36445	0.162437
$430$	0	0
$431$	−18.2789	−0.880466	−0.440233	−	0.897884i	$-0.645104\pi$
$431$	−18.2789	−0.880466	−0.440233	+	0.897884i	$0.645104\pi$
$432$	0	0
$433$	− 19.4172i	− 0.933132i	−0.884486	−	0.466566i	$-0.845491\pi$
$433$	− 19.4172i	− 0.933132i	0.884486	−	0.466566i	$-0.154509\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 0.635552i	− 0.0304026i
$438$	0	0
$439$	21.4423	1.02339	0.511693	−	0.859168i	$-0.329019\pi$
$439$	21.4423	1.02339	0.511693	+	0.859168i	$0.329019\pi$
$440$	0	0
$441$	−1.32497	−0.0630938
$442$	0	0
$443$	− 11.4118i	− 0.542190i	−0.962553	−	0.271095i	$-0.912614\pi$
$443$	− 11.4118i	− 0.542190i	0.962553	−	0.271095i	$-0.0873857\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.12082i	0.147610i
$448$	0	0
$449$	−31.8122	−1.50131	−0.750656	−	0.660693i	$-0.770262\pi$
$449$	−31.8122	−1.50131	−0.750656	+	0.660693i	$0.770262\pi$
$450$	0	0
$451$	3.69596	0.174036
$452$	0	0
$453$	8.80677i	0.413778i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.5357i	0.913840i	0.889508	+	0.456920i	$0.151047\pi$
$457$	19.5357i	0.913840i	−0.889508	+	0.456920i	$0.848953\pi$
$458$	0	0
$459$	12.3908	0.578355
$460$	0	0
$461$	36.7333	1.71084	0.855419	−	0.517936i	$-0.173299\pi$
$461$	36.7333	1.71084	0.855419	+	0.517936i	$0.173299\pi$
$462$	0	0
$463$	6.00654i	0.279148i	0.990212	+	0.139574i	$0.0445733\pi$
$463$	6.00654i	0.279148i	−0.990212	+	0.139574i	$0.955427\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.4776i	1.54916i	0.632476	+	0.774580i	$0.282039\pi$
$467$	33.4776i	1.54916i	−0.632476	+	0.774580i	$0.717961\pi$
$468$	0	0
$469$	−4.46980	−0.206396
$470$	0	0
$471$	−2.84516	−0.131098
$472$	0	0
$473$	− 5.89465i	− 0.271036i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.243629i	− 0.0111550i
$478$	0	0
$479$	30.6609	1.40093	0.700465	−	0.713687i	$-0.252976\pi$
$479$	30.6609	1.40093	0.700465	+	0.713687i	$0.252976\pi$
$480$	0	0
$481$	11.5027	0.524479
$482$	0	0
$483$	2.13828i	0.0972952i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 9.11735i	− 0.413147i	−0.978431	−	0.206573i	$-0.933769\pi$
$487$	− 9.11735i	− 0.413147i	0.978431	−	0.206573i	$-0.0662312\pi$
$488$	0	0
$489$	9.64647	0.436228
$490$	0	0
$491$	7.36991	0.332599	0.166300	−	0.986075i	$-0.446818\pi$
$491$	7.36991	0.332599	0.166300	+	0.986075i	$0.446818\pi$
$492$	0	0
$493$	24.6805i	1.11155i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.56860i	− 0.339498i
$498$	0	0
$499$	13.1712	0.589625	0.294812	−	0.955555i	$-0.404743\pi$
$499$	13.1712	0.589625	0.294812	+	0.955555i	$0.404743\pi$
$500$	0	0
$501$	−6.54767	−0.292528
$502$	0	0
$503$	17.0024i	0.758099i	0.925376	+	0.379049i	$0.123749\pi$
$503$	17.0024i	0.758099i	−0.925376	+	0.379049i	$0.876251\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 14.9780i	− 0.665196i
$508$	0	0
$509$	−21.4489	−0.950704	−0.475352	−	0.879796i	$-0.657679\pi$
$509$	−21.4489	−0.950704	−0.475352	+	0.879796i	$0.657679\pi$
$510$	0	0
$511$	−17.4083	−0.770098
$512$	0	0
$513$	− 4.83424i	− 0.213437i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8.23163i	− 0.362026i
$518$	0	0
$519$	−36.6225	−1.60755
$520$	0	0
$521$	12.5182	0.548432	0.274216	−	0.961668i	$-0.411582\pi$
$521$	12.5182	0.548432	0.274216	+	0.961668i	$0.411582\pi$
$522$	0	0
$523$	31.5136i	1.37800i	0.724763	+	0.688998i	$0.241949\pi$
$523$	31.5136i	1.37800i	−0.724763	+	0.688998i	$0.758051\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 16.9067i	− 0.736465i
$528$	0	0
$529$	22.5961	0.982438
$530$	0	0
$531$	0.144825	0.00628488
$532$	0	0
$533$	9.74090i	0.421925i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	29.7003i	1.28166i
$538$	0	0
$539$	3.03293	0.130638
$540$	0	0
$541$	13.7189	0.589821	0.294910	−	0.955525i	$-0.404710\pi$
$541$	13.7189	0.589821	0.294910	+	0.955525i	$0.404710\pi$
$542$	0	0
$543$	24.2316i	1.03988i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4.12519i	− 0.176381i	−0.996104	−	0.0881903i	$-0.971892\pi$
$547$	− 4.12519i	− 0.176381i	0.996104	−	0.0881903i	$-0.0281084\pi$
$548$	0	0
$549$	−0.825315	−0.0352236
$550$	0	0
$551$	9.62901	0.410210
$552$	0	0
$553$	4.82224i	0.205063i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.10535i	− 0.301063i	−0.988605	−	0.150532i	$-0.951901\pi$
$557$	− 7.10535i	− 0.301063i	0.988605	−	0.150532i	$-0.0480985\pi$
$558$	0	0
$559$	15.5357	0.657089
$560$	0	0
$561$	3.92213	0.165592
$562$	0	0
$563$	37.7058i	1.58911i	0.607192	+	0.794555i	$0.292296\pi$
$563$	37.7058i	1.58911i	−0.607192	+	0.794555i	$0.707704\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	18.2700i	0.767269i
$568$	0	0
$569$	7.72344	0.323783	0.161892	−	0.986809i	$-0.448240\pi$
$569$	7.72344	0.323783	0.161892	+	0.986809i	$0.448240\pi$
$570$	0	0
$571$	−14.5621	−0.609403	−0.304702	−	0.952448i	$-0.598557\pi$
$571$	−14.5621	−0.609403	−0.304702	+	0.952448i	$0.598557\pi$
$572$	0	0
$573$	2.52475i	0.105473i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 18.0318i	− 0.750676i	−0.926888	−	0.375338i	$-0.877527\pi$
$577$	− 18.0318i	− 0.750676i	0.926888	−	0.375338i	$-0.122473\pi$
$578$	0	0
$579$	4.07133	0.169199
$580$	0	0
$581$	19.8606	0.823958
$582$	0	0
$583$	0.557680i	0.0230968i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 16.0923i	− 0.664199i	−0.943244	−	0.332099i	$-0.892243\pi$
$587$	− 16.0923i	− 0.664199i	0.943244	−	0.332099i	$-0.107757\pi$
$588$	0	0
$589$	−6.59607	−0.271786
$590$	0	0
$591$	29.3963	1.20920
$592$	0	0
$593$	− 31.0833i	− 1.27644i	−0.769854	−	0.638220i	$-0.779671\pi$
$593$	− 31.0833i	− 1.27644i	0.769854	−	0.638220i	$-0.220329\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 13.8726i	− 0.567769i
$598$	0	0
$599$	−28.5686	−1.16728	−0.583641	−	0.812012i	$-0.698372\pi$
$599$	−28.5686	−1.16728	−0.583641	+	0.812012i	$0.698372\pi$
$600$	0	0
$601$	32.8277	1.33907	0.669535	−	0.742781i	$-0.266493\pi$
$601$	32.8277	1.33907	0.669535	+	0.742781i	$0.266493\pi$
$602$	0	0
$603$	0.888109i	0.0361666i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.18322i	− 0.0480254i	−0.999712	−	0.0240127i	$-0.992356\pi$
$607$	− 1.18322i	− 0.0480254i	0.999712	−	0.0240127i	$-0.00764421\pi$
$608$	0	0
$609$	−32.3963	−1.31276
$610$	0	0
$611$	21.6949	0.877681
$612$	0	0
$613$	− 11.9385i	− 0.482192i	−0.970501	−	0.241096i	$-0.922493\pi$
$613$	− 11.9385i	− 0.482192i	0.970501	−	0.241096i	$-0.0775068\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.8552i	0.477271i	0.971109	+	0.238636i	$0.0767002\pi$
$617$	11.8552i	0.477271i	−0.971109	+	0.238636i	$0.923300\pi$
$618$	0	0
$619$	5.38299	0.216361	0.108180	−	0.994131i	$-0.465498\pi$
$619$	5.38299	0.216361	0.108180	+	0.994131i	$0.465498\pi$
$620$	0	0
$621$	−3.07241	−0.123292
$622$	0	0
$623$	− 10.9551i	− 0.438905i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.53020i	− 0.0611105i
$628$	0	0
$629$	13.4094	0.534667
$630$	0	0
$631$	25.4598	1.01354	0.506769	−	0.862082i	$-0.330840\pi$
$631$	25.4598	1.01354	0.506769	+	0.862082i	$0.330840\pi$
$632$	0	0
$633$	13.1821i	0.523943i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	7.99346i	0.316712i
$638$	0	0
$639$	−1.50381	−0.0594900
$640$	0	0
$641$	29.0899	1.14898	0.574490	−	0.818511i	$-0.305200\pi$
$641$	29.0899	1.14898	0.574490	+	0.818511i	$0.305200\pi$
$642$	0	0
$643$	30.7948i	1.21443i	0.794539	+	0.607213i	$0.207713\pi$
$643$	30.7948i	1.21443i	−0.794539	+	0.607213i	$0.792287\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 7.77275i	− 0.305578i	−0.988259	−	0.152789i	$-0.951174\pi$
$647$	− 7.77275i	− 0.305578i	0.988259	−	0.152789i	$-0.0488255\pi$
$648$	0	0
$649$	−0.331514	−0.0130130
$650$	0	0
$651$	22.1921	0.869779
$652$	0	0
$653$	29.1647i	1.14130i	0.821193	+	0.570651i	$0.193309\pi$
$653$	29.1647i	1.14130i	−0.821193	+	0.570651i	$0.806691\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.45888i	0.134944i
$658$	0	0
$659$	46.2569	1.80191	0.900957	−	0.433908i	$-0.142866\pi$
$659$	46.2569	1.80191	0.900957	+	0.433908i	$0.142866\pi$
$660$	0	0
$661$	−14.1143	−0.548982	−0.274491	−	0.961590i	$-0.588509\pi$
$661$	−14.1143	−0.548982	−0.274491	+	0.961590i	$0.588509\pi$
$662$	0	0
$663$	10.3370i	0.401455i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6.11973i	− 0.236957i
$668$	0	0
$669$	−32.4183	−1.25336
$670$	0	0
$671$	1.88919	0.0729315
$672$	0	0
$673$	− 10.6596i	− 0.410896i	−0.978668	−	0.205448i	$-0.934135\pi$
$673$	− 10.6596i	− 0.410896i	0.978668	−	0.205448i	$-0.0658651\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 28.5686i	− 1.09798i	−0.835829	−	0.548990i	$-0.815012\pi$
$677$	− 28.5686i	− 1.09798i	0.835829	−	0.548990i	$-0.184988\pi$
$678$	0	0
$679$	19.4358	0.745877
$680$	0	0
$681$	−19.7398	−0.756431
$682$	0	0
$683$	41.4478i	1.58596i	0.609251	+	0.792978i	$0.291470\pi$
$683$	41.4478i	1.58596i	−0.609251	+	0.792978i	$0.708530\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 14.0539i	− 0.536188i
$688$	0	0
$689$	−1.46980	−0.0559947
$690$	0	0
$691$	−2.79476	−0.106318	−0.0531589	−	0.998586i	$-0.516929\pi$
$691$	−2.79476	−0.106318	−0.0531589	+	0.998586i	$0.516929\pi$
$692$	0	0
$693$	0.557680i	0.0211845i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.3555i	0.430121i
$698$	0	0
$699$	−14.7464	−0.557758
$700$	0	0
$701$	50.3468	1.90157	0.950786	−	0.309847i	$-0.100278\pi$
$701$	50.3468	1.90157	0.950786	+	0.309847i	$0.100278\pi$
$702$	0	0
$703$	− 5.23163i	− 0.197314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	26.8332i	1.00916i
$708$	0	0
$709$	2.54112	0.0954339	0.0477169	−	0.998861i	$-0.484805\pi$
$709$	2.54112	0.0954339	0.0477169	+	0.998861i	$0.484805\pi$
$710$	0	0
$711$	0.958137	0.0359329
$712$	0	0
$713$	4.19215i	0.156997i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 44.6279i	− 1.66666i
$718$	0	0
$719$	−1.13828	−0.0424507	−0.0212254	−	0.999775i	$-0.506757\pi$
$719$	−1.13828	−0.0424507	−0.0212254	+	0.999775i	$0.506757\pi$
$720$	0	0
$721$	−13.7618	−0.512517
$722$	0	0
$723$	− 9.46325i	− 0.351942i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 19.1581i	− 0.710536i	−0.934765	−	0.355268i	$-0.884390\pi$
$727$	− 19.1581i	− 0.710536i	0.934765	−	0.355268i	$-0.115610\pi$
$728$	0	0
$729$	−22.9714	−0.850794
$730$	0	0
$731$	18.1108	0.669852
$732$	0	0
$733$	25.3974i	0.938074i	0.883179	+	0.469037i	$0.155399\pi$
$733$	25.3974i	0.938074i	−0.883179	+	0.469037i	$0.844601\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.03293i	− 0.0748841i
$738$	0	0
$739$	42.8122	1.57487	0.787437	−	0.616396i	$-0.211408\pi$
$739$	42.8122	1.57487	0.787437	+	0.616396i	$0.211408\pi$
$740$	0	0
$741$	4.03293	0.148154
$742$	0	0
$743$	− 46.9518i	− 1.72249i	−0.508186	−	0.861247i	$-0.669684\pi$
$743$	− 46.9518i	− 1.72249i	0.508186	−	0.861247i	$-0.330316\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.94613i	− 0.144381i
$748$	0	0
$749$	−26.9660	−0.985315
$750$	0	0
$751$	−10.6356	−0.388097	−0.194048	−	0.980992i	$-0.562162\pi$
$751$	−10.6356	−0.388097	−0.194048	+	0.980992i	$0.562162\pi$
$752$	0	0
$753$	− 14.5852i	− 0.531513i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35.0329i	1.27329i	0.771156	+	0.636647i	$0.219679\pi$
$757$	35.0329i	1.27329i	−0.771156	+	0.636647i	$0.780321\pi$
$758$	0	0
$759$	−0.972525	−0.0353004
$760$	0	0
$761$	23.8462	0.864426	0.432213	−	0.901772i	$-0.357733\pi$
$761$	23.8462	0.864426	0.432213	+	0.901772i	$0.357733\pi$
$762$	0	0
$763$	− 2.39192i	− 0.0865934i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 0.873721i	− 0.0315483i
$768$	0	0
$769$	38.1976	1.37744	0.688720	−	0.725027i	$-0.258173\pi$
$769$	38.1976	1.37744	0.688720	+	0.725027i	$0.258173\pi$
$770$	0	0
$771$	−12.8881	−0.464154
$772$	0	0
$773$	0.835328i	0.0300447i	0.999887	+	0.0150223i	$0.00478194\pi$
$773$	0.835328i	0.0300447i	−0.999887	+	0.0150223i	$0.995218\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	17.6015i	0.631451i
$778$	0	0
$779$	4.43032	0.158733
$780$	0	0
$781$	3.44232	0.123176
$782$	0	0
$783$	− 46.5490i	− 1.66352i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	50.7806i	1.81013i	0.425270	+	0.905066i	$0.360179\pi$
$787$	50.7806i	1.81013i	−0.425270	+	0.905066i	$0.639821\pi$
$788$	0	0
$789$	−2.20870	−0.0786320
$790$	0	0
$791$	−2.07787	−0.0738806
$792$	0	0
$793$	4.97907i	0.176812i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.2011i	0.963512i	0.876306	+	0.481756i	$0.160001\pi$
$797$	27.2011i	0.963512i	−0.876306	+	0.481756i	$0.839999\pi$
$798$	0	0
$799$	25.2910	0.894730
$800$	0	0
$801$	−2.17668	−0.0769090
$802$	0	0
$803$	− 7.91757i	− 0.279405i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	17.8672i	0.628954i
$808$	0	0
$809$	36.3304	1.27731	0.638655	−	0.769493i	$-0.279491\pi$
$809$	36.3304	1.27731	0.638655	+	0.769493i	$0.279491\pi$
$810$	0	0
$811$	54.3778	1.90946	0.954731	−	0.297472i	$-0.0961435\pi$
$811$	54.3778	1.90946	0.954731	+	0.297472i	$0.0961435\pi$
$812$	0	0
$813$	29.7828i	1.04453i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7.06587i	− 0.247203i
$818$	0	0
$819$	−1.46980	−0.0513588
$820$	0	0
$821$	−13.6225	−0.475427	−0.237714	−	0.971335i	$-0.576398\pi$
$821$	−13.6225	−0.475427	−0.237714	+	0.971335i	$0.576398\pi$
$822$	0	0
$823$	35.1725i	1.22604i	0.790069	+	0.613018i	$0.210045\pi$
$823$	35.1725i	1.22604i	−0.790069	+	0.613018i	$0.789955\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 9.86172i	− 0.342926i	−0.985191	−	0.171463i	$-0.945151\pi$
$827$	− 9.86172i	− 0.342926i	0.985191	−	0.171463i	$-0.0548493\pi$
$828$	0	0
$829$	−38.4873	−1.33672	−0.668359	−	0.743839i	$-0.733003\pi$
$829$	−38.4873	−1.33672	−0.668359	+	0.743839i	$0.733003\pi$
$830$	0	0
$831$	36.0868	1.25184
$832$	0	0
$833$	9.31843i	0.322864i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	31.8870i	1.10218i
$838$	0	0
$839$	−27.3908	−0.945637	−0.472818	−	0.881160i	$-0.656763\pi$
$839$	−27.3908	−0.945637	−0.472818	+	0.881160i	$0.656763\pi$
$840$	0	0
$841$	63.7178	2.19717
$842$	0	0
$843$	23.0822i	0.794995i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	18.9001i	0.649416i
$848$	0	0
$849$	25.7398	0.883388
$850$	0	0
$851$	−3.32497	−0.113978
$852$	0	0
$853$	47.7738i	1.63574i	0.575400	+	0.817872i	$0.304847\pi$
$853$	47.7738i	1.63574i	−0.575400	+	0.817872i	$0.695153\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.9571i	0.715879i	0.933745	+	0.357940i	$0.116521\pi$
$857$	20.9571i	0.715879i	−0.933745	+	0.357940i	$0.883479\pi$
$858$	0	0
$859$	−2.59300	−0.0884720	−0.0442360	−	0.999021i	$-0.514085\pi$
$859$	−2.59300	−0.0884720	−0.0442360	+	0.999021i	$0.514085\pi$
$860$	0	0
$861$	−14.9056	−0.507981
$862$	0	0
$863$	9.87372i	0.336105i	0.985778	+	0.168053i	$0.0537479\pi$
$863$	9.87372i	0.336105i	−0.985778	+	0.168053i	$0.946252\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 19.1317i	− 0.649748i
$868$	0	0
$869$	−2.19323	−0.0744003
$870$	0	0
$871$	5.35790	0.181546
$872$	0	0
$873$	− 3.86172i	− 0.130699i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.8421i	0.670020i	0.942215	+	0.335010i	$0.108740\pi$
$877$	19.8421i	0.670020i	−0.942215	+	0.335010i	$0.891260\pi$
$878$	0	0
$879$	−31.7498	−1.07090
$880$	0	0
$881$	13.9485	0.469938	0.234969	−	0.972003i	$-0.424501\pi$
$881$	13.9485	0.469938	0.234969	+	0.972003i	$0.424501\pi$
$882$	0	0
$883$	− 41.4766i	− 1.39580i	−0.716197	−	0.697899i	$-0.754119\pi$
$883$	− 41.4766i	− 1.39580i	0.716197	−	0.697899i	$-0.245881\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 13.1581i	− 0.441807i	−0.975296	−	0.220903i	$-0.929099\pi$
$887$	− 13.1581i	− 0.441807i	0.975296	−	0.220903i	$-0.0709005\pi$
$888$	0	0
$889$	8.96598	0.300709
$890$	0	0
$891$	−8.30950	−0.278379
$892$	0	0
$893$	− 9.86718i	− 0.330193i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.56314i	− 0.0855807i
$898$	0	0
$899$	−63.5136	−2.11830
$900$	0	0
$901$	−1.71342	−0.0570824
$902$	0	0
$903$	23.7727i	0.791108i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.23271i	0.173749i	0.996219	+	0.0868746i	$0.0276880\pi$
$907$	5.23271i	0.173749i	−0.996219	+	0.0868746i	$0.972312\pi$
$908$	0	0
$909$	5.33151	0.176835
$910$	0	0
$911$	25.6434	0.849604	0.424802	−	0.905286i	$-0.360344\pi$
$911$	25.6434	0.849604	0.424802	+	0.905286i	$0.360344\pi$
$912$	0	0
$913$	9.03293i	0.298946i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.60808i	0.119149i
$918$	0	0
$919$	19.2406	0.634687	0.317344	−	0.948311i	$-0.397209\pi$
$919$	19.2406	0.634687	0.317344	+	0.948311i	$0.397209\pi$
$920$	0	0
$921$	29.4183	0.969366
$922$	0	0
$923$	9.07241i	0.298622i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.73436i	0.0898080i
$928$	0	0
$929$	−29.1570	−0.956612	−0.478306	−	0.878193i	$-0.658749\pi$
$929$	−29.1570	−0.956612	−0.478306	+	0.878193i	$0.658749\pi$
$930$	0	0
$931$	3.63555	0.119150
$932$	0	0
$933$	37.6774i	1.23350i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.23163i	− 0.0729040i	−0.999335	−	0.0364520i	$-0.988394\pi$
$937$	− 2.23163i	− 0.0729040i	0.999335	−	0.0364520i	$-0.0116056\pi$
$938$	0	0
$939$	−17.3195	−0.565201
$940$	0	0
$941$	−49.8859	−1.62624	−0.813118	−	0.582099i	$-0.802231\pi$
$941$	−49.8859	−1.62624	−0.813118	+	0.582099i	$0.802231\pi$
$942$	0	0
$943$	− 2.81570i	− 0.0916917i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.8648i	0.385554i	0.981243	+	0.192777i	$0.0617494\pi$
$947$	11.8648i	0.385554i	−0.981243	+	0.192777i	$0.938251\pi$
$948$	0	0
$949$	20.8672	0.677377
$950$	0	0
$951$	−0.0604097	−0.00195892
$952$	0	0
$953$	− 46.4041i	− 1.50318i	−0.659632	−	0.751589i	$-0.729288\pi$
$953$	− 46.4041i	− 1.50318i	0.659632	−	0.751589i	$-0.270712\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 14.7344i	− 0.476294i
$958$	0	0
$959$	−6.40284	−0.206759
$960$	0	0
$961$	12.5082	0.403490
$962$	0	0
$963$	5.35790i	0.172656i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	36.2591i	1.16601i	0.812467	+	0.583007i	$0.198124\pi$
$967$	36.2591i	1.16601i	−0.812467	+	0.583007i	$0.801876\pi$
$968$	0	0
$969$	4.70142	0.151031
$970$	0	0
$971$	6.18669	0.198540	0.0992701	−	0.995061i	$-0.468349\pi$
$971$	6.18669	0.198540	0.0992701	+	0.995061i	$0.468349\pi$
$972$	0	0
$973$	− 27.9769i	− 0.896898i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 54.0277i	− 1.72850i	−0.503063	−	0.864249i	$-0.667794\pi$
$977$	− 54.0277i	− 1.72850i	0.503063	−	0.864249i	$-0.332206\pi$
$978$	0	0
$979$	4.98254	0.159243
$980$	0	0
$981$	−0.475254	−0.0151737
$982$	0	0
$983$	0.765300i	0.0244093i	0.999926	+	0.0122046i	$0.00388495\pi$
$983$	0.765300i	0.0244093i	−0.999926	+	0.0122046i	$0.996115\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	33.1976i	1.05669i
$988$	0	0
$989$	−4.49073	−0.142797
$990$	0	0
$991$	17.1173	0.543751	0.271875	−	0.962332i	$-0.412356\pi$
$991$	17.1173	0.543751	0.271875	+	0.962332i	$0.412356\pi$
$992$	0	0
$993$	− 59.6543i	− 1.89307i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 10.7673i	− 0.341003i	−0.985357	−	0.170502i	$-0.945461\pi$
$997$	− 10.7673i	− 0.341003i	0.985357	−	0.170502i	$-0.0545388\pi$
$998$	0	0
$999$	−25.2910	−0.800170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.m.3649.2		6
5.2	odd	4		3800.2.a.v.1.1	yes	3
5.3	odd	4		3800.2.a.u.1.3	✓	3
5.4	even	2	inner	3800.2.d.m.3649.5		6
20.3	even	4		7600.2.a.bt.1.1		3
20.7	even	4		7600.2.a.bu.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.u.1.3	✓	3	5.3	odd	4
3800.2.a.v.1.1	yes	3	5.2	odd	4
3800.2.d.m.3649.2		6	1.1	even	1	trivial
3800.2.d.m.3649.5		6	5.4	even	2	inner
7600.2.a.bt.1.1		3	20.3	even	4
7600.2.a.bu.1.3		3	20.7	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.83424i q^{3} -1.83424i q^{7} -0.364448 q^{9} +O(q^{10})\)	\(q-1.83424i q^{3} -1.83424i q^{7} -0.364448 q^{9} +0.834243 q^{11} +2.19869i q^{13} +2.56314i q^{17} +1.00000 q^{19} -3.36445 q^{21} -0.635552i q^{23} -4.83424i q^{27} +9.62901 q^{29} -6.59607 q^{31} -1.53020i q^{33} -5.23163i q^{37} +4.03293 q^{39} +4.43032 q^{41} -7.06587i q^{43} -9.86718i q^{47} +3.63555 q^{49} +4.70142 q^{51} +0.668486i q^{53} -1.83424i q^{57} -0.397382 q^{59} +2.26456 q^{61} +0.668486i q^{63} -2.43686i q^{67} -1.16576 q^{69} +4.12628 q^{71} -9.49073i q^{73} -1.53020i q^{77} -2.62901 q^{79} -9.96052 q^{81} +10.8277i q^{83} -17.6619i q^{87} +5.97252 q^{89} +4.03293 q^{91} +12.0988i q^{93} +10.5961i q^{97} -0.304038 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9}+O(q^{10}) \) 6 * q - 2 * q^9	\( 6 q - 2 q^{9} - 6 q^{11} + 6 q^{19} - 20 q^{21} + 2 q^{29} - 6 q^{31} + 2 q^{39} - 18 q^{41} + 22 q^{49} - 16 q^{51} + 20 q^{59} - 42 q^{61} - 18 q^{69} + 2 q^{71} + 40 q^{79} - 26 q^{81} - 8 q^{89} + 2 q^{91} + 20 q^{99}+O(q^{100}) \) 6 * q - 2 * q^9 - 6 * q^11 + 6 * q^19 - 20 * q^21 + 2 * q^29 - 6 * q^31 + 2 * q^39 - 18 * q^41 + 22 * q^49 - 16 * q^51 + 20 * q^59 - 42 * q^61 - 18 * q^69 + 2 * q^71 + 40 * q^79 - 26 * q^81 - 8 * q^89 + 2 * q^91 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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