LMFDB - Embedded newform 7600.2.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
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:	no (minimal twist has level 3800)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.19869$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-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	$-0.322379\pi$
$3$	1.83424	1.05900	0.529500	+	0.848310i	$0.322379\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.83424	−0.693279	−0.346639	−	0.937998i	$-0.612677\pi$
$7$	−1.83424	−0.693279	−0.346639	+	0.937998i	$0.612677\pi$
$8$	0	0
$9$	0.364448	0.121483
$10$	0	0
$11$	−0.834243	−0.251534	−0.125767	−	0.992060i	$-0.540139\pi$
$11$	−0.834243	−0.251534	−0.125767	+	0.992060i	$0.540139\pi$
$12$	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	$-0.478893\pi$
$23$	0.635552	0.132522	0.0662609	+	0.997802i	$0.478893\pi$
$24$	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	$-0.298202\pi$
$31$	6.59607	1.18469	0.592345	+	0.805684i	$0.298202\pi$
$32$	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	$-0.318890\pi$
$43$	7.06587	1.07753	0.538767	+	0.842455i	$0.318890\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.86718	−1.43928	−0.719638	−	0.694350i	$-0.755692\pi$
$47$	−9.86718	−1.43928	−0.719638	+	0.694350i	$0.755692\pi$
$48$	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	$-0.508235\pi$
$59$	−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
$63$	−0.668486	−0.0842214
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.43686	−0.297710	−0.148855	−	0.988859i	$-0.547559\pi$
$67$	−2.43686	−0.297710	−0.148855	+	0.988859i	$0.547559\pi$
$68$	0	0
$69$	1.16576	0.140341
$70$	0	0
$71$	−4.12628	−0.489699	−0.244850	−	0.969561i	$-0.578739\pi$
$71$	−4.12628	−0.489699	−0.244850	+	0.969561i	$0.578739\pi$
$72$	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	$-0.547249\pi$
$79$	−2.62901	−0.295787	−0.147893	+	0.989003i	$0.547249\pi$
$80$	0	0
$81$	−9.96052	−1.10672
$82$	0	0
$83$	−10.8277	−1.18849	−0.594247	−	0.804282i	$-0.702550\pi$
$83$	−10.8277	−1.18849	−0.594247	+	0.804282i	$0.702550\pi$
$84$	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	$-0.379483\pi$
$103$	7.50273	0.739266	0.369633	+	0.929178i	$0.379483\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.7014	−1.42124	−0.710620	−	0.703576i	$-0.751585\pi$
$107$	−14.7014	−1.42124	−0.710620	+	0.703576i	$0.751585\pi$
$108$	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	$-0.430414\pi$
$127$	4.88811	0.433750	0.216875	+	0.976199i	$0.430414\pi$
$128$	0	0
$129$	12.9605	1.14111
$130$	0	0
$131$	1.96707	0.171863	0.0859317	−	0.996301i	$-0.472613\pi$
$131$	1.96707	0.171863	0.0859317	+	0.996301i	$0.472613\pi$
$132$	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	$-0.276085\pi$
$139$	15.2526	1.29371	0.646853	+	0.762615i	$0.276085\pi$
$140$	0	0
$141$	−18.0988	−1.52419
$142$	0	0
$143$	−1.83424	−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	$-0.437412\pi$
$151$	4.80131	0.390725	0.195362	+	0.980731i	$0.437412\pi$
$152$	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	$-0.566032\pi$
$163$	−5.25910	−0.411925	−0.205962	+	0.978560i	$0.566032\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.56968	−0.276230	−0.138115	−	0.990416i	$-0.544104\pi$
$167$	−3.56968	−0.276230	−0.138115	+	0.990416i	$0.544104\pi$
$168$	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	$-0.706878\pi$
$179$	−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
$183$	4.15375	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	$-0.484142\pi$
$191$	1.37645	0.0995965	0.0497982	+	0.998759i	$0.484142\pi$
$192$	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	$-0.413615\pi$
$199$	7.56314	0.536137	0.268068	+	0.963400i	$0.413615\pi$
$200$	0	0
$201$	−4.46980	−0.315275
$202$	0	0
$203$	17.6619	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	$-0.420432\pi$
$211$	7.18669	0.494752	0.247376	+	0.968920i	$0.420432\pi$
$212$	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	$-0.298430\pi$
$223$	17.6739	1.18353	0.591767	+	0.806109i	$0.298430\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.7618	−0.714288	−0.357144	−	0.934049i	$-0.616249\pi$
$227$	−10.7618	−0.714288	−0.357144	+	0.934049i	$0.616249\pi$
$228$	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	$-0.211684\pi$
$239$	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
$243$	−3.76729	−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	$-0.580743\pi$
$251$	−7.95159	−0.501900	−0.250950	+	0.968000i	$0.580743\pi$
$252$	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	$-0.488180\pi$
$263$	1.20415	0.0742511	0.0371255	+	0.999311i	$0.488180\pi$
$264$	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	$-0.335839\pi$
$271$	16.2371	0.986333	0.493166	+	0.869935i	$0.335839\pi$
$272$	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	$-0.636949\pi$
$283$	−14.0329	−0.834171	−0.417086	+	0.908867i	$0.636949\pi$
$284$	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	$-0.348681\pi$
$307$	16.0384	0.915359	0.457680	+	0.889117i	$0.348681\pi$
$308$	0	0
$309$	13.7618	0.782883
$310$	0	0
$311$	20.5411	1.16478	0.582390	−	0.812909i	$-0.302118\pi$
$311$	20.5411	1.16478	0.582390	+	0.812909i	$0.302118\pi$
$312$	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	$-0.851971\pi$
$331$	−32.5226	−1.78760	−0.893801	+	0.448463i	$0.851971\pi$
$332$	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	$-0.360341\pi$
$347$	15.8266	0.849617	0.424809	+	0.905283i	$0.360341\pi$
$348$	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	$-0.430377\pi$
$359$	8.22270	0.433977	0.216989	+	0.976174i	$0.430377\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−18.9001	−0.991999
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−30.3514	−1.58433	−0.792164	−	0.610308i	$-0.791046\pi$
$367$	−30.3514	−1.58433	−0.792164	+	0.610308i	$0.791046\pi$
$368$	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	$-0.384664\pi$
$379$	13.8013	0.708926	0.354463	+	0.935070i	$0.384664\pi$
$380$	0	0
$381$	8.96598	0.459341
$382$	0	0
$383$	10.5961	0.541434	0.270717	−	0.962659i	$-0.412739\pi$
$383$	10.5961	0.541434	0.270717	+	0.962659i	$0.412739\pi$
$384$	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	$-0.557739\pi$
$419$	−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
$423$	−3.59607	−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	$-0.354896\pi$
$431$	18.2789	0.880466	0.440233	+	0.897884i	$0.354896\pi$
$432$	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	$-0.329019\pi$
$439$	21.4423	1.02339	0.511693	+	0.859168i	$0.329019\pi$
$440$	0	0
$441$	−1.32497	−0.0630938
$442$	0	0
$443$	11.4118	0.542190	0.271095	−	0.962553i	$-0.412614\pi$
$443$	11.4118	0.542190	0.271095	+	0.962553i	$0.412614\pi$
$444$	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	$-0.544573\pi$
$463$	−6.00654	−0.279148	−0.139574	+	0.990212i	$0.544573\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.4776	1.54916	0.774580	−	0.632476i	$-0.217961\pi$
$467$	33.4776	1.54916	0.774580	+	0.632476i	$0.217961\pi$
$468$	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	$-0.252976\pi$
$479$	30.6609	1.40093	0.700465	+	0.713687i	$0.252976\pi$
$480$	0	0
$481$	11.5027	0.524479
$482$	0	0
$483$	−2.13828	−0.0972952
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9.11735	−0.413147	−0.206573	−	0.978431i	$-0.566231\pi$
$487$	−9.11735	−0.413147	−0.206573	+	0.978431i	$0.566231\pi$
$488$	0	0
$489$	−9.64647	−0.436228
$490$	0	0
$491$	−7.36991	−0.332599	−0.166300	−	0.986075i	$-0.553182\pi$
$491$	−7.36991	−0.332599	−0.166300	+	0.986075i	$0.553182\pi$
$492$	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	$-0.404743\pi$
$499$	13.1712	0.589625	0.294812	+	0.955555i	$0.404743\pi$
$500$	0	0
$501$	−6.54767	−0.292528
$502$	0	0
$503$	−17.0024	−0.758099	−0.379049	−	0.925376i	$-0.623749\pi$
$503$	−17.0024	−0.758099	−0.379049	+	0.925376i	$0.623749\pi$
$504$	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	$-0.741949\pi$
$523$	−31.5136	−1.37800	−0.688998	+	0.724763i	$0.741949\pi$
$524$	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	$-0.528108\pi$
$547$	−4.12519	−0.176381	−0.0881903	+	0.996104i	$0.528108\pi$
$548$	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	$-0.792296\pi$
$563$	−37.7058	−1.58911	−0.794555	+	0.607192i	$0.792296\pi$
$564$	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	$-0.401443\pi$
$571$	14.5621	0.609403	0.304702	+	0.952448i	$0.401443\pi$
$572$	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	$-0.607757\pi$
$587$	−16.0923	−0.664199	−0.332099	+	0.943244i	$0.607757\pi$
$588$	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	$-0.698372\pi$
$599$	−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
$603$	−0.888109	−0.0361666
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.18322	−0.0480254	−0.0240127	−	0.999712i	$-0.507644\pi$
$607$	−1.18322	−0.0480254	−0.0240127	+	0.999712i	$0.507644\pi$
$608$	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	$-0.465498\pi$
$619$	5.38299	0.216361	0.108180	+	0.994131i	$0.465498\pi$
$620$	0	0
$621$	−3.07241	−0.123292
$622$	0	0
$623$	10.9551	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	$-0.669160\pi$
$631$	−25.4598	−1.01354	−0.506769	+	0.862082i	$0.669160\pi$
$632$	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	$-0.707713\pi$
$643$	−30.7948	−1.21443	−0.607213	+	0.794539i	$0.707713\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.77275	−0.305578	−0.152789	−	0.988259i	$-0.548826\pi$
$647$	−7.77275	−0.305578	−0.152789	+	0.988259i	$0.548826\pi$
$648$	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	$-0.142866\pi$
$659$	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
$663$	−10.3370	−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	$-0.791470\pi$
$683$	−41.4478	−1.58596	−0.792978	+	0.609251i	$0.791470\pi$
$684$	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	$-0.483071\pi$
$691$	2.79476	0.106318	0.0531589	+	0.998586i	$0.483071\pi$
$692$	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	$-0.506757\pi$
$719$	−1.13828	−0.0424507	−0.0212254	+	0.999775i	$0.506757\pi$
$720$	0	0
$721$	−13.7618	−0.512517
$722$	0	0
$723$	9.46325	0.351942
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−19.1581	−0.710536	−0.355268	−	0.934765i	$-0.615610\pi$
$727$	−19.1581	−0.710536	−0.355268	+	0.934765i	$0.615610\pi$
$728$	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	$-0.211408\pi$
$739$	42.8122	1.57487	0.787437	+	0.616396i	$0.211408\pi$
$740$	0	0
$741$	4.03293	0.148154
$742$	0	0
$743$	46.9518	1.72249	0.861247	−	0.508186i	$-0.169684\pi$
$743$	46.9518	1.72249	0.861247	+	0.508186i	$0.169684\pi$
$744$	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	$-0.437838\pi$
$751$	10.6356	0.388097	0.194048	+	0.980992i	$0.437838\pi$
$752$	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	$-0.139821\pi$
$787$	50.7806	1.81013	0.905066	+	0.425270i	$0.139821\pi$
$788$	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	$-0.903857\pi$
$811$	−54.3778	−1.90946	−0.954731	+	0.297472i	$0.903857\pi$
$812$	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	$-0.710045\pi$
$823$	−35.1725	−1.22604	−0.613018	+	0.790069i	$0.710045\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.86172	−0.342926	−0.171463	−	0.985191i	$-0.554849\pi$
$827$	−9.86172	−0.342926	−0.171463	+	0.985191i	$0.554849\pi$
$828$	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	$-0.656763\pi$
$839$	−27.3908	−0.945637	−0.472818	+	0.881160i	$0.656763\pi$
$840$	0	0
$841$	63.7178	2.19717
$842$	0	0
$843$	−23.0822	−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	$-0.514085\pi$
$859$	−2.59300	−0.0884720	−0.0442360	+	0.999021i	$0.514085\pi$
$860$	0	0
$861$	−14.9056	−0.507981
$862$	0	0
$863$	−9.87372	−0.336105	−0.168053	−	0.985778i	$-0.553748\pi$
$863$	−9.87372	−0.336105	−0.168053	+	0.985778i	$0.553748\pi$
$864$	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	$-0.254119\pi$
$883$	41.4766	1.39580	0.697899	+	0.716197i	$0.254119\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.1581	−0.441807	−0.220903	−	0.975296i	$-0.570901\pi$
$887$	−13.1581	−0.441807	−0.220903	+	0.975296i	$0.570901\pi$
$888$	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	$-0.472312\pi$
$907$	5.23271	0.173749	0.0868746	+	0.996219i	$0.472312\pi$
$908$	0	0
$909$	−5.33151	−0.176835
$910$	0	0
$911$	−25.6434	−0.849604	−0.424802	−	0.905286i	$-0.639656\pi$
$911$	−25.6434	−0.849604	−0.424802	+	0.905286i	$0.639656\pi$
$912$	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	$-0.397209\pi$
$919$	19.2406	0.634687	0.317344	+	0.948311i	$0.397209\pi$
$920$	0	0
$921$	29.4183	0.969366
$922$	0	0
$923$	−9.07241	−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	$-0.438251\pi$
$947$	11.8648	0.385554	0.192777	+	0.981243i	$0.438251\pi$
$948$	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	$-0.301876\pi$
$967$	36.2591	1.16601	0.583007	+	0.812467i	$0.301876\pi$
$968$	0	0
$969$	−4.70142	−0.151031
$970$	0	0
$971$	−6.18669	−0.198540	−0.0992701	−	0.995061i	$-0.531651\pi$
$971$	−6.18669	−0.198540	−0.0992701	+	0.995061i	$0.531651\pi$
$972$	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	$-0.503885\pi$
$983$	−0.765300	−0.0244093	−0.0122046	+	0.999926i	$0.503885\pi$
$984$	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	$-0.587644\pi$
$991$	−17.1173	−0.543751	−0.271875	+	0.962332i	$0.587644\pi$
$992$	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	7600.2.a.bu.1.3		3
4.3	odd	2		3800.2.a.v.1.1	yes	3
5.4	even	2		7600.2.a.bt.1.1		3
20.3	even	4		3800.2.d.m.3649.2		6
20.7	even	4		3800.2.d.m.3649.5		6
20.19	odd	2		3800.2.a.u.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.u.1.3	✓	3	20.19	odd	2
3800.2.a.v.1.1	yes	3	4.3	odd	2
3800.2.d.m.3649.2		6	20.3	even	4
3800.2.d.m.3649.5		6	20.7	even	4
7600.2.a.bt.1.1		3	5.4	even	2
7600.2.a.bu.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.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

L-functions

Modular forms

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Database

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