LMFDB - Embedded newform 3800.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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, 0, 0]))

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

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

chi := DirichletCharacter("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.a (trivial)

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:	yes
Analytic conductor:	$30.3431527681$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.19869$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83424 q^{3} +1.83424 q^{7} +0.364448 q^{9} +O(q^{10})$	$q-1.83424 q^{3} +1.83424 q^{7} +0.364448 q^{9} +0.834243 q^{11} +2.19869 q^{13} -2.56314 q^{17} -1.00000 q^{19} -3.36445 q^{21} -0.635552 q^{23} +4.83424 q^{27} -9.62901 q^{29} -6.59607 q^{31} -1.53020 q^{33} +5.23163 q^{37} -4.03293 q^{39} +4.43032 q^{41} -7.06587 q^{43} +9.86718 q^{47} -3.63555 q^{49} +4.70142 q^{51} +0.668486 q^{53} +1.83424 q^{57} +0.397382 q^{59} +2.26456 q^{61} +0.668486 q^{63} +2.43686 q^{67} +1.16576 q^{69} +4.12628 q^{71} -9.49073 q^{73} +1.53020 q^{77} +2.62901 q^{79} -9.96052 q^{81} +10.8277 q^{83} +17.6619 q^{87} -5.97252 q^{89} +4.03293 q^{91} +12.0988 q^{93} -10.5961 q^{97} +0.304038 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{9}+O(q^{10}) $ 3 * q + q^9	$ 3 q + q^{9} - 3 q^{11} + q^{13} - 2 q^{17} - 3 q^{19} - 10 q^{21} - 2 q^{23} + 9 q^{27} - q^{29} - 3 q^{31} - 10 q^{33} - q^{37} - q^{39} - 9 q^{41} + q^{43} + 13 q^{47} - 11 q^{49} - 8 q^{51} - 9 q^{53} - 10 q^{59} - 21 q^{61} - 9 q^{63} + 13 q^{67} + 9 q^{69} + q^{71} - 17 q^{73} + 10 q^{77} - 20 q^{79} - 13 q^{81} - q^{83} + 14 q^{87} + 4 q^{89} + q^{91} + 3 q^{93} - 15 q^{97} - 10 q^{99}+O(q^{100}) $ 3 * q + q^9 - 3 * q^11 + q^13 - 2 * q^17 - 3 * q^19 - 10 * q^21 - 2 * q^23 + 9 * q^27 - q^29 - 3 * q^31 - 10 * q^33 - q^37 - q^39 - 9 * q^41 + q^43 + 13 * q^47 - 11 * q^49 - 8 * q^51 - 9 * q^53 - 10 * q^59 - 21 * q^61 - 9 * q^63 + 13 * q^67 + 9 * q^69 + q^71 - 17 * q^73 + 10 * q^77 - 20 * q^79 - 13 * q^81 - q^83 + 14 * q^87 + 4 * q^89 + q^91 + 3 * q^93 - 15 * q^97 - 10 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.83424	−1.05900	−0.529500	−	0.848310i	$-0.677621\pi$
$3$	−1.83424	−1.05900	−0.529500	+	0.848310i	$0.677621\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.83424	0.693279	0.346639	−	0.937998i	$-0.387323\pi$
$7$	1.83424	0.693279	0.346639	+	0.937998i	$0.387323\pi$
$8$	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
$13$	2.19869	0.609807	0.304904	−	0.952383i	$-0.401376\pi$
$13$	2.19869	0.609807	0.304904	+	0.952383i	$0.401376\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.56314	−0.621653	−0.310826	−	0.950467i	$-0.600606\pi$
$17$	−2.56314	−0.621653	−0.310826	+	0.950467i	$0.600606\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.36445	−0.734183
$22$	0	0
$23$	−0.635552	−0.132522	−0.0662609	−	0.997802i	$-0.521107\pi$
$23$	−0.635552	−0.132522	−0.0662609	+	0.997802i	$0.521107\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.83424	0.930351
$28$	0	0
$29$	−9.62901	−1.78806	−0.894031	−	0.448005i	$-0.852135\pi$
$29$	−9.62901	−1.78806	−0.894031	+	0.448005i	$0.852135\pi$
$30$	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
$33$	−1.53020	−0.266374
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.23163	0.860074	0.430037	−	0.902811i	$-0.358501\pi$
$37$	5.23163	0.860074	0.430037	+	0.902811i	$0.358501\pi$
$38$	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
$43$	−7.06587	−1.07753	−0.538767	−	0.842455i	$-0.681110\pi$
$43$	−7.06587	−1.07753	−0.538767	+	0.842455i	$0.681110\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.86718	1.43928	0.719638	−	0.694350i	$-0.244308\pi$
$47$	9.86718	1.43928	0.719638	+	0.694350i	$0.244308\pi$
$48$	0	0
$49$	−3.63555	−0.519365
$50$	0	0
$51$	4.70142	0.658331
$52$	0	0
$53$	0.668486	0.0918237	0.0459118	−	0.998945i	$-0.485381\pi$
$53$	0.668486	0.0918237	0.0459118	+	0.998945i	$0.485381\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.83424	0.242951
$58$	0	0
$59$	0.397382	0.0517348	0.0258674	−	0.999665i	$-0.491765\pi$
$59$	0.397382	0.0517348	0.0258674	+	0.999665i	$0.491765\pi$
$60$	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
$63$	0.668486	0.0842214
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.43686	0.297710	0.148855	−	0.988859i	$-0.452441\pi$
$67$	2.43686	0.297710	0.148855	+	0.988859i	$0.452441\pi$
$68$	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
$73$	−9.49073	−1.11081	−0.555403	−	0.831581i	$-0.687436\pi$
$73$	−9.49073	−1.11081	−0.555403	+	0.831581i	$0.687436\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.53020	0.174383
$78$	0	0
$79$	2.62901	0.295787	0.147893	−	0.989003i	$-0.452751\pi$
$79$	2.62901	0.295787	0.147893	+	0.989003i	$0.452751\pi$
$80$	0	0
$81$	−9.96052	−1.10672
$82$	0	0
$83$	10.8277	1.18849	0.594247	−	0.804282i	$-0.297450\pi$
$83$	10.8277	1.18849	0.594247	+	0.804282i	$0.297450\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	17.6619	1.89356
$88$	0	0
$89$	−5.97252	−0.633086	−0.316543	−	0.948578i	$-0.602522\pi$
$89$	−5.97252	−0.633086	−0.316543	+	0.948578i	$0.602522\pi$
$90$	0	0
$91$	4.03293	0.422766
$92$	0	0
$93$	12.0988	1.25459
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.5961	−1.07587	−0.537934	−	0.842987i	$-0.680795\pi$
$97$	−10.5961	−1.07587	−0.537934	+	0.842987i	$0.680795\pi$
$98$	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
$103$	−7.50273	−0.739266	−0.369633	−	0.929178i	$-0.620517\pi$
$103$	−7.50273	−0.739266	−0.369633	+	0.929178i	$0.620517\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.7014	1.42124	0.710620	−	0.703576i	$-0.248415\pi$
$107$	14.7014	1.42124	0.710620	+	0.703576i	$0.248415\pi$
$108$	0	0
$109$	−1.30404	−0.124904	−0.0624521	−	0.998048i	$-0.519892\pi$
$109$	−1.30404	−0.124904	−0.0624521	+	0.998048i	$0.519892\pi$
$110$	0	0
$111$	−9.59607	−0.910819
$112$	0	0
$113$	−1.13282	−0.106567	−0.0532835	−	0.998579i	$-0.516969\pi$
$113$	−1.13282	−0.106567	−0.0532835	+	0.998579i	$0.516969\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.801309	0.0740810
$118$	0	0
$119$	−4.70142	−0.430979
$120$	0	0
$121$	−10.3040	−0.936731
$122$	0	0
$123$	−8.12628	−0.732722
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.88811	−0.433750	−0.216875	−	0.976199i	$-0.569586\pi$
$127$	−4.88811	−0.433750	−0.216875	+	0.976199i	$0.569586\pi$
$128$	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
$133$	−1.83424	−0.159049
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.49073	0.298233	0.149116	−	0.988820i	$-0.452357\pi$
$137$	3.49073	0.298233	0.149116	+	0.988820i	$0.452357\pi$
$138$	0	0
$139$	−15.2526	−1.29371	−0.646853	−	0.762615i	$-0.723915\pi$
$139$	−15.2526	−1.29371	−0.646853	+	0.762615i	$0.723915\pi$
$140$	0	0
$141$	−18.0988	−1.52419
$142$	0	0
$143$	1.83424	0.153387
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.66849	0.550007
$148$	0	0
$149$	1.70142	0.139386	0.0696929	−	0.997568i	$-0.477798\pi$
$149$	1.70142	0.139386	0.0696929	+	0.997568i	$0.477798\pi$
$150$	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
$153$	−0.934131	−0.0755200
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.55114	0.123794	0.0618971	−	0.998083i	$-0.480285\pi$
$157$	1.55114	0.123794	0.0618971	+	0.998083i	$0.480285\pi$
$158$	0	0
$159$	−1.22617	−0.0972413
$160$	0	0
$161$	−1.16576	−0.0918745
$162$	0	0
$163$	5.25910	0.411925	0.205962	−	0.978560i	$-0.433968\pi$
$163$	5.25910	0.411925	0.205962	+	0.978560i	$0.433968\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.56968	0.276230	0.138115	−	0.990416i	$-0.455896\pi$
$167$	3.56968	0.276230	0.138115	+	0.990416i	$0.455896\pi$
$168$	0	0
$169$	−8.16576	−0.628135
$170$	0	0
$171$	−0.364448	−0.0278700
$172$	0	0
$173$	−19.9660	−1.51799	−0.758993	−	0.651099i	$-0.774308\pi$
$173$	−19.9660	−1.51799	−0.758993	+	0.651099i	$0.774308\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.728896	−0.0547872
$178$	0	0
$179$	16.1921	1.21026	0.605129	−	0.796127i	$-0.293122\pi$
$179$	16.1921	1.21026	0.605129	+	0.796127i	$0.293122\pi$
$180$	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
$183$	−4.15375	−0.307054
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.13828	−0.156367
$188$	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
$193$	2.21962	0.159772	0.0798860	−	0.996804i	$-0.474544\pi$
$193$	2.21962	0.159772	0.0798860	+	0.996804i	$0.474544\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.0264	−1.14183	−0.570917	−	0.821008i	$-0.693412\pi$
$197$	−16.0264	−1.14183	−0.570917	+	0.821008i	$0.693412\pi$
$198$	0	0
$199$	−7.56314	−0.536137	−0.268068	−	0.963400i	$-0.586385\pi$
$199$	−7.56314	−0.536137	−0.268068	+	0.963400i	$0.586385\pi$
$200$	0	0
$201$	−4.46980	−0.315275
$202$	0	0
$203$	−17.6619	−1.23963
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.231626	−0.0160991
$208$	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
$213$	−7.56860	−0.518592
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−12.0988	−0.821320
$218$	0	0
$219$	17.4083	1.17634
$220$	0	0
$221$	−5.63555	−0.379088
$222$	0	0
$223$	−17.6739	−1.18353	−0.591767	−	0.806109i	$-0.701570\pi$
$223$	−17.6739	−1.18353	−0.591767	+	0.806109i	$0.701570\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.7618	0.714288	0.357144	−	0.934049i	$-0.383751\pi$
$227$	10.7618	0.714288	0.357144	+	0.934049i	$0.383751\pi$
$228$	0	0
$229$	−7.66194	−0.506315	−0.253158	−	0.967425i	$-0.581469\pi$
$229$	−7.66194	−0.506315	−0.253158	+	0.967425i	$0.581469\pi$
$230$	0	0
$231$	−2.80677	−0.184672
$232$	0	0
$233$	−8.03948	−0.526684	−0.263342	−	0.964703i	$-0.584825\pi$
$233$	−8.03948	−0.526684	−0.263342	+	0.964703i	$0.584825\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.82224	−0.313238
$238$	0	0
$239$	−24.3304	−1.57380	−0.786902	−	0.617078i	$-0.788316\pi$
$239$	−24.3304	−1.57380	−0.786902	+	0.617078i	$0.788316\pi$
$240$	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
$243$	3.76729	0.241672
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.19869	−0.139899
$248$	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
$253$	−0.530205	−0.0333337
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.02639	0.438294	0.219147	−	0.975692i	$-0.429673\pi$
$257$	7.02639	0.438294	0.219147	+	0.975692i	$0.429673\pi$
$258$	0	0
$259$	9.59607	0.596271
$260$	0	0
$261$	−3.50927	−0.217219
$262$	0	0
$263$	−1.20415	−0.0742511	−0.0371255	−	0.999311i	$-0.511820\pi$
$263$	−1.20415	−0.0742511	−0.0371255	+	0.999311i	$0.511820\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.9551	0.670439
$268$	0	0
$269$	9.74090	0.593913	0.296957	−	0.954891i	$-0.404028\pi$
$269$	9.74090	0.593913	0.296957	+	0.954891i	$0.404028\pi$
$270$	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
$273$	−7.39738	−0.447710
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.6739	−1.18209	−0.591046	−	0.806638i	$-0.701285\pi$
$277$	−19.6739	−1.18209	−0.591046	+	0.806638i	$0.701285\pi$
$278$	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
$283$	14.0329	0.834171	0.417086	−	0.908867i	$-0.363051\pi$
$283$	14.0329	0.834171	0.417086	+	0.908867i	$0.363051\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.12628	0.479679
$288$	0	0
$289$	−10.4303	−0.613548
$290$	0	0
$291$	19.4358	1.13935
$292$	0	0
$293$	−17.3095	−1.01123	−0.505616	−	0.862759i	$-0.668735\pi$
$293$	−17.3095	−1.01123	−0.505616	+	0.862759i	$0.668735\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.03293	0.234015
$298$	0	0
$299$	−1.39738	−0.0808127
$300$	0	0
$301$	−12.9605	−0.747032
$302$	0	0
$303$	26.8332	1.54152
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−16.0384	−0.915359	−0.457680	−	0.889117i	$-0.651319\pi$
$307$	−16.0384	−0.915359	−0.457680	+	0.889117i	$0.651319\pi$
$308$	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
$313$	−9.44232	−0.533711	−0.266856	−	0.963736i	$-0.585985\pi$
$313$	−9.44232	−0.533711	−0.266856	+	0.963736i	$0.585985\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.0329344	0.00184978	0.000924891	−	1.00000i	$-0.499706\pi$
$317$	0.0329344	0.00184978	0.000924891		1.00000i	$0.499706\pi$
$318$	0	0
$319$	−8.03293	−0.449758
$320$	0	0
$321$	−26.9660	−1.50509
$322$	0	0
$323$	2.56314	0.142617
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.39192	0.132274
$328$	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
$333$	1.90666	0.104484
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.1252	−1.58655	−0.793275	−	0.608863i	$-0.791626\pi$
$337$	−29.1252	−1.58655	−0.793275	+	0.608863i	$0.791626\pi$
$338$	0	0
$339$	2.07787	0.112855
$340$	0	0
$341$	−5.50273	−0.297990
$342$	0	0
$343$	−19.5082	−1.05334
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.8266	−0.849617	−0.424809	−	0.905283i	$-0.639659\pi$
$347$	−15.8266	−0.849617	−0.424809	+	0.905283i	$0.639659\pi$
$348$	0	0
$349$	34.1911	1.83021	0.915103	−	0.403221i	$-0.132109\pi$
$349$	34.1911	1.83021	0.915103	+	0.403221i	$0.132109\pi$
$350$	0	0
$351$	10.6290	0.567334
$352$	0	0
$353$	−26.9330	−1.43350	−0.716751	−	0.697329i	$-0.754371\pi$
$353$	−26.9330	−1.43350	−0.716751	+	0.697329i	$0.754371\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.62355	0.456407
$358$	0	0
$359$	−8.22270	−0.433977	−0.216989	−	0.976174i	$-0.569623\pi$
$359$	−8.22270	−0.433977	−0.216989	+	0.976174i	$0.569623\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	18.9001	0.991999
$364$	0	0
$365$	0	0
$366$	0	0
$367$	30.3514	1.58433	0.792164	−	0.610308i	$-0.208954\pi$
$367$	30.3514	1.58433	0.792164	+	0.610308i	$0.208954\pi$
$368$	0	0
$369$	1.61462	0.0840538
$370$	0	0
$371$	1.22617	0.0636594
$372$	0	0
$373$	29.4018	1.52237	0.761183	−	0.648538i	$-0.224619\pi$
$373$	29.4018	1.52237	0.761183	+	0.648538i	$0.224619\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−21.1712	−1.09037
$378$	0	0
$379$	−13.8013	−0.708926	−0.354463	−	0.935070i	$-0.615336\pi$
$379$	−13.8013	−0.708926	−0.354463	+	0.935070i	$0.615336\pi$
$380$	0	0
$381$	8.96598	0.459341
$382$	0	0
$383$	−10.5961	−0.541434	−0.270717	−	0.962659i	$-0.587261\pi$
$383$	−10.5961	−0.541434	−0.270717	+	0.962659i	$0.587261\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.57514	−0.130902
$388$	0	0
$389$	1.84972	0.0937843	0.0468922	−	0.998900i	$-0.485068\pi$
$389$	1.84972	0.0937843	0.0468922	+	0.998900i	$0.485068\pi$
$390$	0	0
$391$	1.62901	0.0823825
$392$	0	0
$393$	3.60808	0.182003
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.998915	0.0501341	0.0250671	−	0.999686i	$-0.492020\pi$
$397$	0.998915	0.0501341	0.0250671	+	0.999686i	$0.492020\pi$
$398$	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
$403$	−14.5027	−0.722432
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.36445	0.216338
$408$	0	0
$409$	8.43578	0.417122	0.208561	−	0.978009i	$-0.433122\pi$
$409$	8.43578	0.417122	0.208561	+	0.978009i	$0.433122\pi$
$410$	0	0
$411$	−6.40284	−0.315829
$412$	0	0
$413$	0.728896	0.0358666
$414$	0	0
$415$	0	0
$416$	0	0
$417$	27.9769	1.37003
$418$	0	0
$419$	7.38538	0.360799	0.180400	−	0.983593i	$-0.442261\pi$
$419$	7.38538	0.360799	0.180400	+	0.983593i	$0.442261\pi$
$420$	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
$423$	3.59607	0.174847
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.15375	0.201014
$428$	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
$433$	−19.4172	−0.933132	−0.466566	−	0.884486i	$-0.654509\pi$
$433$	−19.4172	−0.933132	−0.466566	+	0.884486i	$0.654509\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.635552	0.0304026
$438$	0	0
$439$	−21.4423	−1.02339	−0.511693	−	0.859168i	$-0.670981\pi$
$439$	−21.4423	−1.02339	−0.511693	+	0.859168i	$0.670981\pi$
$440$	0	0
$441$	−1.32497	−0.0630938
$442$	0	0
$443$	−11.4118	−0.542190	−0.271095	−	0.962553i	$-0.587386\pi$
$443$	−11.4118	−0.542190	−0.271095	+	0.962553i	$0.587386\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.12082	−0.147610
$448$	0	0
$449$	31.8122	1.50131	0.750656	−	0.660693i	$-0.229738\pi$
$449$	31.8122	1.50131	0.750656	+	0.660693i	$0.229738\pi$
$450$	0	0
$451$	3.69596	0.174036
$452$	0	0
$453$	8.80677	0.413778
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.5357	−0.913840	−0.456920	−	0.889508i	$-0.651047\pi$
$457$	−19.5357	−0.913840	−0.456920	+	0.889508i	$0.651047\pi$
$458$	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
$463$	6.00654	0.279148	0.139574	−	0.990212i	$-0.455427\pi$
$463$	6.00654	0.279148	0.139574	+	0.990212i	$0.455427\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−33.4776	−1.54916	−0.774580	−	0.632476i	$-0.782039\pi$
$467$	−33.4776	−1.54916	−0.774580	+	0.632476i	$0.782039\pi$
$468$	0	0
$469$	4.46980	0.206396
$470$	0	0
$471$	−2.84516	−0.131098
$472$	0	0
$473$	−5.89465	−0.271036
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.243629	0.0111550
$478$	0	0
$479$	−30.6609	−1.40093	−0.700465	−	0.713687i	$-0.747024\pi$
$479$	−30.6609	−1.40093	−0.700465	+	0.713687i	$0.747024\pi$
$480$	0	0
$481$	11.5027	0.524479
$482$	0	0
$483$	2.13828	0.0972952
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9.11735	0.413147	0.206573	−	0.978431i	$-0.433769\pi$
$487$	9.11735	0.413147	0.206573	+	0.978431i	$0.433769\pi$
$488$	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
$493$	24.6805	1.11155
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.56860	0.339498
$498$	0	0
$499$	−13.1712	−0.589625	−0.294812	−	0.955555i	$-0.595257\pi$
$499$	−13.1712	−0.589625	−0.294812	+	0.955555i	$0.595257\pi$
$500$	0	0
$501$	−6.54767	−0.292528
$502$	0	0
$503$	17.0024	0.758099	0.379049	−	0.925376i	$-0.376251\pi$
$503$	17.0024	0.758099	0.379049	+	0.925376i	$0.376251\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.9780	0.665196
$508$	0	0
$509$	21.4489	0.950704	0.475352	−	0.879796i	$-0.342321\pi$
$509$	21.4489	0.950704	0.475352	+	0.879796i	$0.342321\pi$
$510$	0	0
$511$	−17.4083	−0.770098
$512$	0	0
$513$	−4.83424	−0.213437
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.23163	0.362026
$518$	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
$523$	31.5136	1.37800	0.688998	−	0.724763i	$-0.258051\pi$
$523$	31.5136	1.37800	0.688998	+	0.724763i	$0.258051\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.9067	0.736465
$528$	0	0
$529$	−22.5961	−0.982438
$530$	0	0
$531$	0.144825	0.00628488
$532$	0	0
$533$	9.74090	0.421925
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−29.7003	−1.28166
$538$	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
$543$	24.2316	1.03988
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.12519	0.176381	0.0881903	−	0.996104i	$-0.471892\pi$
$547$	4.12519	0.176381	0.0881903	+	0.996104i	$0.471892\pi$
$548$	0	0
$549$	0.825315	0.0352236
$550$	0	0
$551$	9.62901	0.410210
$552$	0	0
$553$	4.82224	0.205063
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.10535	0.301063	0.150532	−	0.988605i	$-0.451901\pi$
$557$	7.10535	0.301063	0.150532	+	0.988605i	$0.451901\pi$
$558$	0	0
$559$	−15.5357	−0.657089
$560$	0	0
$561$	3.92213	0.165592
$562$	0	0
$563$	37.7058	1.58911	0.794555	−	0.607192i	$-0.207704\pi$
$563$	37.7058	1.58911	0.794555	+	0.607192i	$0.207704\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−18.2700	−0.767269
$568$	0	0
$569$	−7.72344	−0.323783	−0.161892	−	0.986809i	$-0.551760\pi$
$569$	−7.72344	−0.323783	−0.161892	+	0.986809i	$0.551760\pi$
$570$	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
$573$	2.52475	0.105473
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.0318	0.750676	0.375338	−	0.926888i	$-0.377527\pi$
$577$	18.0318	0.750676	0.375338	+	0.926888i	$0.377527\pi$
$578$	0	0
$579$	−4.07133	−0.169199
$580$	0	0
$581$	19.8606	0.823958
$582$	0	0
$583$	0.557680	0.0230968
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.0923	0.664199	0.332099	−	0.943244i	$-0.392243\pi$
$587$	16.0923	0.664199	0.332099	+	0.943244i	$0.392243\pi$
$588$	0	0
$589$	6.59607	0.271786
$590$	0	0
$591$	29.3963	1.20920
$592$	0	0
$593$	−31.0833	−1.27644	−0.638220	−	0.769854i	$-0.720329\pi$
$593$	−31.0833	−1.27644	−0.638220	+	0.769854i	$0.720329\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13.8726	0.567769
$598$	0	0
$599$	28.5686	1.16728	0.583641	−	0.812012i	$-0.301628\pi$
$599$	28.5686	1.16728	0.583641	+	0.812012i	$0.301628\pi$
$600$	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
$603$	0.888109	0.0361666
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.18322	0.0480254	0.0240127	−	0.999712i	$-0.492356\pi$
$607$	1.18322	0.0480254	0.0240127	+	0.999712i	$0.492356\pi$
$608$	0	0
$609$	32.3963	1.31276
$610$	0	0
$611$	21.6949	0.877681
$612$	0	0
$613$	−11.9385	−0.482192	−0.241096	−	0.970501i	$-0.577507\pi$
$613$	−11.9385	−0.482192	−0.241096	+	0.970501i	$0.577507\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.8552	−0.477271	−0.238636	−	0.971109i	$-0.576700\pi$
$617$	−11.8552	−0.477271	−0.238636	+	0.971109i	$0.576700\pi$
$618$	0	0
$619$	−5.38299	−0.216361	−0.108180	−	0.994131i	$-0.534502\pi$
$619$	−5.38299	−0.216361	−0.108180	+	0.994131i	$0.534502\pi$
$620$	0	0
$621$	−3.07241	−0.123292
$622$	0	0
$623$	−10.9551	−0.438905
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.53020	0.0611105
$628$	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
$633$	13.1821	0.523943
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.99346	−0.316712
$638$	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
$643$	30.7948	1.21443	0.607213	−	0.794539i	$-0.292287\pi$
$643$	30.7948	1.21443	0.607213	+	0.794539i	$0.292287\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.77275	0.305578	0.152789	−	0.988259i	$-0.451174\pi$
$647$	7.77275	0.305578	0.152789	+	0.988259i	$0.451174\pi$
$648$	0	0
$649$	0.331514	0.0130130
$650$	0	0
$651$	22.1921	0.869779
$652$	0	0
$653$	29.1647	1.14130	0.570651	−	0.821193i	$-0.306691\pi$
$653$	29.1647	1.14130	0.570651	+	0.821193i	$0.306691\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.45888	−0.134944
$658$	0	0
$659$	−46.2569	−1.80191	−0.900957	−	0.433908i	$-0.857134\pi$
$659$	−46.2569	−1.80191	−0.900957	+	0.433908i	$0.857134\pi$
$660$	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
$663$	10.3370	0.401455
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.11973	0.236957
$668$	0	0
$669$	32.4183	1.25336
$670$	0	0
$671$	1.88919	0.0729315
$672$	0	0
$673$	−10.6596	−0.410896	−0.205448	−	0.978668i	$-0.565865\pi$
$673$	−10.6596	−0.410896	−0.205448	+	0.978668i	$0.565865\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.5686	1.09798	0.548990	−	0.835829i	$-0.315012\pi$
$677$	28.5686	1.09798	0.548990	+	0.835829i	$0.315012\pi$
$678$	0	0
$679$	−19.4358	−0.745877
$680$	0	0
$681$	−19.7398	−0.756431
$682$	0	0
$683$	41.4478	1.58596	0.792978	−	0.609251i	$-0.208530\pi$
$683$	41.4478	1.58596	0.792978	+	0.609251i	$0.208530\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14.0539	0.536188
$688$	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
$693$	0.557680	0.0211845
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−11.3555	−0.430121
$698$	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
$703$	−5.23163	−0.197314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−26.8332	−1.00916
$708$	0	0
$709$	−2.54112	−0.0954339	−0.0477169	−	0.998861i	$-0.515195\pi$
$709$	−2.54112	−0.0954339	−0.0477169	+	0.998861i	$0.515195\pi$
$710$	0	0
$711$	0.958137	0.0359329
$712$	0	0
$713$	4.19215	0.156997
$714$	0	0
$715$	0	0
$716$	0	0
$717$	44.6279	1.66666
$718$	0	0
$719$	1.13828	0.0424507	0.0212254	−	0.999775i	$-0.493243\pi$
$719$	1.13828	0.0424507	0.0212254	+	0.999775i	$0.493243\pi$
$720$	0	0
$721$	−13.7618	−0.512517
$722$	0	0
$723$	−9.46325	−0.351942
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19.1581	0.710536	0.355268	−	0.934765i	$-0.384390\pi$
$727$	19.1581	0.710536	0.355268	+	0.934765i	$0.384390\pi$
$728$	0	0
$729$	22.9714	0.850794
$730$	0	0
$731$	18.1108	0.669852
$732$	0	0
$733$	25.3974	0.938074	0.469037	−	0.883179i	$-0.344601\pi$
$733$	25.3974	0.938074	0.469037	+	0.883179i	$0.344601\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.03293	0.0748841
$738$	0	0
$739$	−42.8122	−1.57487	−0.787437	−	0.616396i	$-0.788592\pi$
$739$	−42.8122	−1.57487	−0.787437	+	0.616396i	$0.788592\pi$
$740$	0	0
$741$	4.03293	0.148154
$742$	0	0
$743$	−46.9518	−1.72249	−0.861247	−	0.508186i	$-0.830316\pi$
$743$	−46.9518	−1.72249	−0.861247	+	0.508186i	$0.830316\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.94613	0.144381
$748$	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
$753$	−14.5852	−0.531513
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−35.0329	−1.27329	−0.636647	−	0.771156i	$-0.719679\pi$
$757$	−35.0329	−1.27329	−0.636647	+	0.771156i	$0.719679\pi$
$758$	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
$763$	−2.39192	−0.0865934
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.873721	0.0315483
$768$	0	0
$769$	−38.1976	−1.37744	−0.688720	−	0.725027i	$-0.741827\pi$
$769$	−38.1976	−1.37744	−0.688720	+	0.725027i	$0.741827\pi$
$770$	0	0
$771$	−12.8881	−0.464154
$772$	0	0
$773$	0.835328	0.0300447	0.0150223	−	0.999887i	$-0.495218\pi$
$773$	0.835328	0.0300447	0.0150223	+	0.999887i	$0.495218\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−17.6015	−0.631451
$778$	0	0
$779$	−4.43032	−0.158733
$780$	0	0
$781$	3.44232	0.123176
$782$	0	0
$783$	−46.5490	−1.66352
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−50.7806	−1.81013	−0.905066	−	0.425270i	$-0.860179\pi$
$787$	−50.7806	−1.81013	−0.905066	+	0.425270i	$0.860179\pi$
$788$	0	0
$789$	2.20870	0.0786320
$790$	0	0
$791$	−2.07787	−0.0738806
$792$	0	0
$793$	4.97907	0.176812
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.2011	−0.963512	−0.481756	−	0.876306i	$-0.660001\pi$
$797$	−27.2011	−0.963512	−0.481756	+	0.876306i	$0.660001\pi$
$798$	0	0
$799$	−25.2910	−0.894730
$800$	0	0
$801$	−2.17668	−0.0769090
$802$	0	0
$803$	−7.91757	−0.279405
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−17.8672	−0.628954
$808$	0	0
$809$	−36.3304	−1.27731	−0.638655	−	0.769493i	$-0.720509\pi$
$809$	−36.3304	−1.27731	−0.638655	+	0.769493i	$0.720509\pi$
$810$	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
$813$	29.7828	1.04453
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.06587	0.247203
$818$	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
$823$	35.1725	1.22604	0.613018	−	0.790069i	$-0.289955\pi$
$823$	35.1725	1.22604	0.613018	+	0.790069i	$0.289955\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.86172	0.342926	0.171463	−	0.985191i	$-0.445151\pi$
$827$	9.86172	0.342926	0.171463	+	0.985191i	$0.445151\pi$
$828$	0	0
$829$	38.4873	1.33672	0.668359	−	0.743839i	$-0.266997\pi$
$829$	38.4873	1.33672	0.668359	+	0.743839i	$0.266997\pi$
$830$	0	0
$831$	36.0868	1.25184
$832$	0	0
$833$	9.31843	0.322864
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−31.8870	−1.10218
$838$	0	0
$839$	27.3908	0.945637	0.472818	−	0.881160i	$-0.343237\pi$
$839$	27.3908	0.945637	0.472818	+	0.881160i	$0.343237\pi$
$840$	0	0
$841$	63.7178	2.19717
$842$	0	0
$843$	23.0822	0.794995
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−18.9001	−0.649416
$848$	0	0
$849$	−25.7398	−0.883388
$850$	0	0
$851$	−3.32497	−0.113978
$852$	0	0
$853$	47.7738	1.63574	0.817872	−	0.575400i	$-0.195153\pi$
$853$	47.7738	1.63574	0.817872	+	0.575400i	$0.195153\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.9571	−0.715879	−0.357940	−	0.933745i	$-0.616521\pi$
$857$	−20.9571	−0.715879	−0.357940	+	0.933745i	$0.616521\pi$
$858$	0	0
$859$	2.59300	0.0884720	0.0442360	−	0.999021i	$-0.485915\pi$
$859$	2.59300	0.0884720	0.0442360	+	0.999021i	$0.485915\pi$
$860$	0	0
$861$	−14.9056	−0.507981
$862$	0	0
$863$	9.87372	0.336105	0.168053	−	0.985778i	$-0.446252\pi$
$863$	9.87372	0.336105	0.168053	+	0.985778i	$0.446252\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.1317	0.649748
$868$	0	0
$869$	2.19323	0.0744003
$870$	0	0
$871$	5.35790	0.181546
$872$	0	0
$873$	−3.86172	−0.130699
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19.8421	−0.670020	−0.335010	−	0.942215i	$-0.608740\pi$
$877$	−19.8421	−0.670020	−0.335010	+	0.942215i	$0.608740\pi$
$878$	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
$883$	−41.4766	−1.39580	−0.697899	−	0.716197i	$-0.745881\pi$
$883$	−41.4766	−1.39580	−0.697899	+	0.716197i	$0.745881\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.1581	0.441807	0.220903	−	0.975296i	$-0.429099\pi$
$887$	13.1581	0.441807	0.220903	+	0.975296i	$0.429099\pi$
$888$	0	0
$889$	−8.96598	−0.300709
$890$	0	0
$891$	−8.30950	−0.278379
$892$	0	0
$893$	−9.86718	−0.330193
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.56314	0.0855807
$898$	0	0
$899$	63.5136	2.11830
$900$	0	0
$901$	−1.71342	−0.0570824
$902$	0	0
$903$	23.7727	0.791108
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−5.23271	−0.173749	−0.0868746	−	0.996219i	$-0.527688\pi$
$907$	−5.23271	−0.173749	−0.0868746	+	0.996219i	$0.527688\pi$
$908$	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
$913$	9.03293	0.298946
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.60808	−0.119149
$918$	0	0
$919$	−19.2406	−0.634687	−0.317344	−	0.948311i	$-0.602791\pi$
$919$	−19.2406	−0.634687	−0.317344	+	0.948311i	$0.602791\pi$
$920$	0	0
$921$	29.4183	0.969366
$922$	0	0
$923$	9.07241	0.298622
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.73436	−0.0898080
$928$	0	0
$929$	29.1570	0.956612	0.478306	−	0.878193i	$-0.341251\pi$
$929$	29.1570	0.956612	0.478306	+	0.878193i	$0.341251\pi$
$930$	0	0
$931$	3.63555	0.119150
$932$	0	0
$933$	37.6774	1.23350
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.23163	0.0729040	0.0364520	−	0.999335i	$-0.488394\pi$
$937$	2.23163	0.0729040	0.0364520	+	0.999335i	$0.488394\pi$
$938$	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
$943$	−2.81570	−0.0916917
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.8648	−0.385554	−0.192777	−	0.981243i	$-0.561749\pi$
$947$	−11.8648	−0.385554	−0.192777	+	0.981243i	$0.561749\pi$
$948$	0	0
$949$	−20.8672	−0.677377
$950$	0	0
$951$	−0.0604097	−0.00195892
$952$	0	0
$953$	−46.4041	−1.50318	−0.751589	−	0.659632i	$-0.770712\pi$
$953$	−46.4041	−1.50318	−0.751589	+	0.659632i	$0.770712\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	14.7344	0.476294
$958$	0	0
$959$	6.40284	0.206759
$960$	0	0
$961$	12.5082	0.403490
$962$	0	0
$963$	5.35790	0.172656
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−36.2591	−1.16601	−0.583007	−	0.812467i	$-0.698124\pi$
$967$	−36.2591	−1.16601	−0.583007	+	0.812467i	$0.698124\pi$
$968$	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
$973$	−27.9769	−0.896898
$974$	0	0
$975$	0	0
$976$	0	0
$977$	54.0277	1.72850	0.864249	−	0.503063i	$-0.167794\pi$
$977$	54.0277	1.72850	0.864249	+	0.503063i	$0.167794\pi$
$978$	0	0
$979$	−4.98254	−0.159243
$980$	0	0
$981$	−0.475254	−0.0151737
$982$	0	0
$983$	0.765300	0.0244093	0.0122046	−	0.999926i	$-0.496115\pi$
$983$	0.765300	0.0244093	0.0122046	+	0.999926i	$0.496115\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−33.1976	−1.05669
$988$	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
$993$	−59.6543	−1.89307
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10.7673	0.341003	0.170502	−	0.985357i	$-0.445461\pi$
$997$	10.7673	0.341003	0.170502	+	0.985357i	$0.445461\pi$
$998$	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.a.v.1.1	yes	3
4.3	odd	2		7600.2.a.bu.1.3		3
5.2	odd	4		3800.2.d.m.3649.5		6
5.3	odd	4		3800.2.d.m.3649.2		6
5.4	even	2		3800.2.a.u.1.3	✓	3
20.19	odd	2		7600.2.a.bt.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.u.1.3	✓	3	5.4	even	2
3800.2.a.v.1.1	yes	3	1.1	even	1	trivial
3800.2.d.m.3649.2		6	5.3	odd	4
3800.2.d.m.3649.5		6	5.2	odd	4
7600.2.a.bt.1.1		3	20.19	odd	2
7600.2.a.bu.1.3		3	4.3	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.83424 q^{3} +1.83424 q^{7} +0.364448 q^{9} +O(q^{10})\)	\(q-1.83424 q^{3} +1.83424 q^{7} +0.364448 q^{9} +0.834243 q^{11} +2.19869 q^{13} -2.56314 q^{17} -1.00000 q^{19} -3.36445 q^{21} -0.635552 q^{23} +4.83424 q^{27} -9.62901 q^{29} -6.59607 q^{31} -1.53020 q^{33} +5.23163 q^{37} -4.03293 q^{39} +4.43032 q^{41} -7.06587 q^{43} +9.86718 q^{47} -3.63555 q^{49} +4.70142 q^{51} +0.668486 q^{53} +1.83424 q^{57} +0.397382 q^{59} +2.26456 q^{61} +0.668486 q^{63} +2.43686 q^{67} +1.16576 q^{69} +4.12628 q^{71} -9.49073 q^{73} +1.53020 q^{77} +2.62901 q^{79} -9.96052 q^{81} +10.8277 q^{83} +17.6619 q^{87} -5.97252 q^{89} +4.03293 q^{91} +12.0988 q^{93} -10.5961 q^{97} +0.304038 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{9}+O(q^{10}) \) 3 * q + q^9	\( 3 q + q^{9} - 3 q^{11} + q^{13} - 2 q^{17} - 3 q^{19} - 10 q^{21} - 2 q^{23} + 9 q^{27} - q^{29} - 3 q^{31} - 10 q^{33} - q^{37} - q^{39} - 9 q^{41} + q^{43} + 13 q^{47} - 11 q^{49} - 8 q^{51} - 9 q^{53} - 10 q^{59} - 21 q^{61} - 9 q^{63} + 13 q^{67} + 9 q^{69} + q^{71} - 17 q^{73} + 10 q^{77} - 20 q^{79} - 13 q^{81} - q^{83} + 14 q^{87} + 4 q^{89} + q^{91} + 3 q^{93} - 15 q^{97} - 10 q^{99}+O(q^{100}) \) 3 * q + q^9 - 3 * q^11 + q^13 - 2 * q^17 - 3 * q^19 - 10 * q^21 - 2 * q^23 + 9 * q^27 - q^29 - 3 * q^31 - 10 * q^33 - q^37 - q^39 - 9 * q^41 + q^43 + 13 * q^47 - 11 * q^49 - 8 * q^51 - 9 * q^53 - 10 * q^59 - 21 * q^61 - 9 * q^63 + 13 * q^67 + 9 * q^69 + q^71 - 17 * q^73 + 10 * q^77 - 20 * q^79 - 13 * q^81 - q^83 + 14 * q^87 + 4 * q^89 + q^91 + 3 * q^93 - 15 * q^97 - 10 * q^99

Introduction

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