LMFDB - Embedded newform 7600.2.a.cb.1.1

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:	$0$
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 950)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-3.03293 q^{3} +2.46980 q^{7} +6.19869 q^{9} +O(q^{10})$	$q-3.03293 q^{3} +2.46980 q^{7} +6.19869 q^{9} -0.728896 q^{11} +6.23163 q^{13} -0.563139 q^{17} +1.00000 q^{19} -7.49073 q^{21} +4.63555 q^{23} -9.70142 q^{27} +10.2316 q^{29} -6.06587 q^{31} +2.21069 q^{33} +5.72890 q^{37} -18.9001 q^{39} +4.79476 q^{41} +8.06587 q^{43} +8.12628 q^{47} -0.900112 q^{49} +1.70796 q^{51} +1.53020 q^{53} -3.03293 q^{57} +5.76183 q^{59} +10.9396 q^{61} +15.3095 q^{63} +12.9330 q^{67} -14.0593 q^{69} +4.39738 q^{71} +4.09334 q^{73} -1.80022 q^{77} -15.3370 q^{79} +10.8277 q^{81} +7.85517 q^{83} -31.0318 q^{87} -10.0000 q^{89} +15.3908 q^{91} +18.3974 q^{93} -11.0055 q^{97} -4.51820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 2 * q^7 + 13 * q^9	$ 3 q + 2 q^{3} + 2 q^{7} + 13 q^{9} - 2 q^{11} + 2 q^{13} + 4 q^{17} + 3 q^{19} - 11 q^{21} + 14 q^{23} - 7 q^{27} + 14 q^{29} + 4 q^{31} - 4 q^{33} + 17 q^{37} - 29 q^{39} - 8 q^{41} + 2 q^{43} + 13 q^{47} + 25 q^{49} + 11 q^{51} + 10 q^{53} + 2 q^{57} + 6 q^{59} + 22 q^{61} + 2 q^{63} + 8 q^{69} + 2 q^{71} + 12 q^{73} + 50 q^{77} - 24 q^{79} - q^{81} + 12 q^{83} - 21 q^{87} - 30 q^{89} + 7 q^{91} + 44 q^{93} - 24 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^7 + 13 * q^9 - 2 * q^11 + 2 * q^13 + 4 * q^17 + 3 * q^19 - 11 * q^21 + 14 * q^23 - 7 * q^27 + 14 * q^29 + 4 * q^31 - 4 * q^33 + 17 * q^37 - 29 * q^39 - 8 * q^41 + 2 * q^43 + 13 * q^47 + 25 * q^49 + 11 * q^51 + 10 * q^53 + 2 * q^57 + 6 * q^59 + 22 * q^61 + 2 * q^63 + 8 * q^69 + 2 * q^71 + 12 * q^73 + 50 * q^77 - 24 * q^79 - q^81 + 12 * q^83 - 21 * q^87 - 30 * q^89 + 7 * q^91 + 44 * q^93 - 24 * 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
$3$	−3.03293	−1.75107	−0.875533	−	0.483159i	$-0.839489\pi$
$3$	−3.03293	−1.75107	−0.875533	+	0.483159i	$0.839489\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.46980	0.933495	0.466747	−	0.884391i	$-0.345426\pi$
$7$	2.46980	0.933495	0.466747	+	0.884391i	$0.345426\pi$
$8$	0	0
$9$	6.19869	2.06623
$10$	0	0
$11$	−0.728896	−0.219770	−0.109885	−	0.993944i	$-0.535048\pi$
$11$	−0.728896	−0.219770	−0.109885	+	0.993944i	$0.535048\pi$
$12$	0	0
$13$	6.23163	1.72834	0.864171	−	0.503198i	$-0.167843\pi$
$13$	6.23163	1.72834	0.864171	+	0.503198i	$0.167843\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.563139	−0.136581	−0.0682907	−	0.997665i	$-0.521755\pi$
$17$	−0.563139	−0.136581	−0.0682907	+	0.997665i	$0.521755\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−7.49073	−1.63461
$22$	0	0
$23$	4.63555	0.966579	0.483290	−	0.875460i	$-0.339442\pi$
$23$	4.63555	0.966579	0.483290	+	0.875460i	$0.339442\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−9.70142	−1.86704
$28$	0	0
$29$	10.2316	1.89997	0.949983	−	0.312303i	$-0.101100\pi$
$29$	10.2316	1.89997	0.949983	+	0.312303i	$0.101100\pi$
$30$	0	0
$31$	−6.06587	−1.08946	−0.544731	−	0.838611i	$-0.683368\pi$
$31$	−6.06587	−1.08946	−0.544731	+	0.838611i	$0.683368\pi$
$32$	0	0
$33$	2.21069	0.384832
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.72890	0.941825	0.470912	−	0.882180i	$-0.343925\pi$
$37$	5.72890	0.941825	0.470912	+	0.882180i	$0.343925\pi$
$38$	0	0
$39$	−18.9001	−3.02644
$40$	0	0
$41$	4.79476	0.748816	0.374408	−	0.927264i	$-0.377846\pi$
$41$	4.79476	0.748816	0.374408	+	0.927264i	$0.377846\pi$
$42$	0	0
$43$	8.06587	1.23003	0.615017	−	0.788514i	$-0.289149\pi$
$43$	8.06587	1.23003	0.615017	+	0.788514i	$0.289149\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.12628	1.18534	0.592670	−	0.805446i	$-0.298074\pi$
$47$	8.12628	1.18534	0.592670	+	0.805446i	$0.298074\pi$
$48$	0	0
$49$	−0.900112	−0.128587
$50$	0	0
$51$	1.70796	0.239163
$52$	0	0
$53$	1.53020	0.210190	0.105095	−	0.994462i	$-0.466485\pi$
$53$	1.53020	0.210190	0.105095	+	0.994462i	$0.466485\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.03293	−0.401722
$58$	0	0
$59$	5.76183	0.750126	0.375063	−	0.926999i	$-0.377621\pi$
$59$	5.76183	0.750126	0.375063	+	0.926999i	$0.377621\pi$
$60$	0	0
$61$	10.9396	1.40067	0.700336	−	0.713814i	$-0.253034\pi$
$61$	10.9396	1.40067	0.700336	+	0.713814i	$0.253034\pi$
$62$	0	0
$63$	15.3095	1.92882
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.9330	1.58002	0.790012	−	0.613092i	$-0.210074\pi$
$67$	12.9330	1.58002	0.790012	+	0.613092i	$0.210074\pi$
$68$	0	0
$69$	−14.0593	−1.69254
$70$	0	0
$71$	4.39738	0.521873	0.260937	−	0.965356i	$-0.415969\pi$
$71$	4.39738	0.521873	0.260937	+	0.965356i	$0.415969\pi$
$72$	0	0
$73$	4.09334	0.479090	0.239545	−	0.970885i	$-0.423002\pi$
$73$	4.09334	0.479090	0.239545	+	0.970885i	$0.423002\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.80022	−0.205155
$78$	0	0
$79$	−15.3370	−1.72554	−0.862772	−	0.505593i	$-0.831274\pi$
$79$	−15.3370	−1.72554	−0.862772	+	0.505593i	$0.831274\pi$
$80$	0	0
$81$	10.8277	1.20308
$82$	0	0
$83$	7.85517	0.862217	0.431109	−	0.902300i	$-0.358123\pi$
$83$	7.85517	0.862217	0.431109	+	0.902300i	$0.358123\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−31.0318	−3.32696
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	15.3908	1.61340
$92$	0	0
$93$	18.3974	1.90772
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.0055	−1.11744	−0.558718	−	0.829358i	$-0.688706\pi$
$97$	−11.0055	−1.11744	−0.558718	+	0.829358i	$0.688706\pi$
$98$	0	0
$99$	−4.51820	−0.454096
$100$	0	0
$101$	−9.73436	−0.968605	−0.484302	−	0.874901i	$-0.660926\pi$
$101$	−9.73436	−0.968605	−0.484302	+	0.874901i	$0.660926\pi$
$102$	0	0
$103$	−8.33151	−0.820928	−0.410464	−	0.911877i	$-0.634633\pi$
$103$	−8.33151	−0.820928	−0.410464	+	0.911877i	$0.634633\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.0264	−1.45266	−0.726328	−	0.687348i	$-0.758775\pi$
$107$	−15.0264	−1.45266	−0.726328	+	0.687348i	$0.758775\pi$
$108$	0	0
$109$	−13.6619	−1.30858	−0.654288	−	0.756245i	$-0.727032\pi$
$109$	−13.6619	−1.30858	−0.654288	+	0.756245i	$0.727032\pi$
$110$	0	0
$111$	−17.3754	−1.64920
$112$	0	0
$113$	−1.20524	−0.113379	−0.0566895	−	0.998392i	$-0.518055\pi$
$113$	−1.20524	−0.113379	−0.0566895	+	0.998392i	$0.518055\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	38.6279	3.57115
$118$	0	0
$119$	−1.39084	−0.127498
$120$	0	0
$121$	−10.4687	−0.951701
$122$	0	0
$123$	−14.5422	−1.31123
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.52366	0.845088	0.422544	−	0.906342i	$-0.361137\pi$
$127$	9.52366	0.845088	0.422544	+	0.906342i	$0.361137\pi$
$128$	0	0
$129$	−24.4633	−2.15387
$130$	0	0
$131$	17.4028	1.52049	0.760247	−	0.649635i	$-0.225078\pi$
$131$	17.4028	1.52049	0.760247	+	0.649635i	$0.225078\pi$
$132$	0	0
$133$	2.46980	0.214158
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.25910	−0.193008	−0.0965040	−	0.995333i	$-0.530766\pi$
$137$	−2.25910	−0.193008	−0.0965040	+	0.995333i	$0.530766\pi$
$138$	0	0
$139$	−16.8606	−1.43010	−0.715050	−	0.699073i	$-0.753596\pi$
$139$	−16.8606	−1.43010	−0.715050	+	0.699073i	$0.753596\pi$
$140$	0	0
$141$	−24.6465	−2.07561
$142$	0	0
$143$	−4.54221	−0.379838
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.72998	0.225165
$148$	0	0
$149$	−13.0055	−1.06545	−0.532724	−	0.846289i	$-0.678832\pi$
$149$	−13.0055	−1.06545	−0.532724	+	0.846289i	$0.678832\pi$
$150$	0	0
$151$	−17.5895	−1.43142	−0.715708	−	0.698400i	$-0.753896\pi$
$151$	−17.5895	−1.43142	−0.715708	+	0.698400i	$0.753896\pi$
$152$	0	0
$153$	−3.49073	−0.282209
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−9.52366	−0.760071	−0.380035	−	0.924972i	$-0.624088\pi$
$157$	−9.52366	−0.760071	−0.380035	+	0.924972i	$0.624088\pi$
$158$	0	0
$159$	−4.64101	−0.368056
$160$	0	0
$161$	11.4489	0.902297
$162$	0	0
$163$	13.1921	1.03329	0.516644	−	0.856200i	$-0.327181\pi$
$163$	13.1921	1.03329	0.516644	+	0.856200i	$0.327181\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.81331	0.140318	0.0701591	−	0.997536i	$-0.477649\pi$
$167$	1.81331	0.140318	0.0701591	+	0.997536i	$0.477649\pi$
$168$	0	0
$169$	25.8332	1.98717
$170$	0	0
$171$	6.19869	0.474026
$172$	0	0
$173$	12.5237	0.952156	0.476078	−	0.879403i	$-0.342058\pi$
$173$	12.5237	0.952156	0.476078	+	0.879403i	$0.342058\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−17.4753	−1.31352
$178$	0	0
$179$	−1.39738	−0.104445	−0.0522226	−	0.998635i	$-0.516631\pi$
$179$	−1.39738	−0.104445	−0.0522226	+	0.998635i	$0.516631\pi$
$180$	0	0
$181$	5.72890	0.425825	0.212913	−	0.977071i	$-0.431705\pi$
$181$	5.72890	0.425825	0.212913	+	0.977071i	$0.431705\pi$
$182$	0	0
$183$	−33.1791	−2.45267
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.410470	0.0300165
$188$	0	0
$189$	−23.9605	−1.74287
$190$	0	0
$191$	27.0198	1.95509	0.977544	−	0.210733i	$-0.0675849\pi$
$191$	27.0198	1.95509	0.977544	+	0.210733i	$0.0675849\pi$
$192$	0	0
$193$	23.0713	1.66071	0.830355	−	0.557234i	$-0.188138\pi$
$193$	23.0713	1.66071	0.830355	+	0.557234i	$0.188138\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.794765	−0.0566247	−0.0283123	−	0.999599i	$-0.509013\pi$
$197$	−0.794765	−0.0566247	−0.0283123	+	0.999599i	$0.509013\pi$
$198$	0	0
$199$	−8.07241	−0.572238	−0.286119	−	0.958194i	$-0.592365\pi$
$199$	−8.07241	−0.572238	−0.286119	+	0.958194i	$0.592365\pi$
$200$	0	0
$201$	−39.2251	−2.76672
$202$	0	0
$203$	25.2700	1.77361
$204$	0	0
$205$	0	0
$206$	0	0
$207$	28.7344	1.99718
$208$	0	0
$209$	−0.728896	−0.0504188
$210$	0	0
$211$	11.6410	0.801400	0.400700	−	0.916209i	$-0.368767\pi$
$211$	11.6410	0.801400	0.400700	+	0.916209i	$0.368767\pi$
$212$	0	0
$213$	−13.3370	−0.913834
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−14.9815	−1.01701
$218$	0	0
$219$	−12.4148	−0.838917
$220$	0	0
$221$	−3.50927	−0.236059
$222$	0	0
$223$	−15.6554	−1.04836	−0.524182	−	0.851607i	$-0.675629\pi$
$223$	−15.6554	−1.04836	−0.524182	+	0.851607i	$0.675629\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.80131	0.318674	0.159337	−	0.987224i	$-0.449064\pi$
$227$	4.80131	0.318674	0.159337	+	0.987224i	$0.449064\pi$
$228$	0	0
$229$	2.79476	0.184683	0.0923416	−	0.995727i	$-0.470565\pi$
$229$	2.79476	0.184683	0.0923416	+	0.995727i	$0.470565\pi$
$230$	0	0
$231$	5.45996	0.359239
$232$	0	0
$233$	−11.5422	−0.756155	−0.378078	−	0.925774i	$-0.623415\pi$
$233$	−11.5422	−0.756155	−0.378078	+	0.925774i	$0.623415\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	46.5160	3.02154
$238$	0	0
$239$	26.2251	1.69636	0.848180	−	0.529708i	$-0.177699\pi$
$239$	26.2251	1.69636	0.848180	+	0.529708i	$0.177699\pi$
$240$	0	0
$241$	−12.0659	−0.777231	−0.388615	−	0.921400i	$-0.627047\pi$
$241$	−12.0659	−0.777231	−0.388615	+	0.921400i	$0.627047\pi$
$242$	0	0
$243$	−3.73544	−0.239629
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.23163	0.396509
$248$	0	0
$249$	−23.8242	−1.50980
$250$	0	0
$251$	−13.5237	−0.853606	−0.426803	−	0.904345i	$-0.640360\pi$
$251$	−13.5237	−0.853606	−0.426803	+	0.904345i	$0.640360\pi$
$252$	0	0
$253$	−3.37884	−0.212426
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.7398	−1.41847	−0.709235	−	0.704972i	$-0.750960\pi$
$257$	−22.7398	−1.41847	−0.709235	+	0.704972i	$0.750960\pi$
$258$	0	0
$259$	14.1492	0.879188
$260$	0	0
$261$	63.4227	3.92577
$262$	0	0
$263$	−6.67395	−0.411533	−0.205767	−	0.978601i	$-0.565969\pi$
$263$	−6.67395	−0.411533	−0.205767	+	0.978601i	$0.565969\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	30.3293	1.85613
$268$	0	0
$269$	−29.7398	−1.81327	−0.906634	−	0.421917i	$-0.861357\pi$
$269$	−29.7398	−1.81327	−0.906634	+	0.421917i	$0.861357\pi$
$270$	0	0
$271$	−1.11189	−0.0675426	−0.0337713	−	0.999430i	$-0.510752\pi$
$271$	−1.11189	−0.0675426	−0.0337713	+	0.999430i	$0.510752\pi$
$272$	0	0
$273$	−46.6794	−2.82517
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.1263	−0.788682	−0.394341	−	0.918964i	$-0.629027\pi$
$277$	−13.1263	−0.788682	−0.394341	+	0.918964i	$0.629027\pi$
$278$	0	0
$279$	−37.6004	−2.25108
$280$	0	0
$281$	22.9265	1.36768	0.683840	−	0.729632i	$-0.260309\pi$
$281$	22.9265	1.36768	0.683840	+	0.729632i	$0.260309\pi$
$282$	0	0
$283$	0.860634	0.0511594	0.0255797	−	0.999673i	$-0.491857\pi$
$283$	0.860634	0.0511594	0.0255797	+	0.999673i	$0.491857\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.8421	0.699016
$288$	0	0
$289$	−16.6829	−0.981346
$290$	0	0
$291$	33.3788	1.95670
$292$	0	0
$293$	−19.8212	−1.15796	−0.578982	−	0.815340i	$-0.696550\pi$
$293$	−19.8212	−1.15796	−0.578982	+	0.815340i	$0.696550\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.07133	0.410320
$298$	0	0
$299$	28.8870	1.67058
$300$	0	0
$301$	19.9210	1.14823
$302$	0	0
$303$	29.5237	1.69609
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−16.8661	−0.962599	−0.481299	−	0.876556i	$-0.659835\pi$
$307$	−16.8661	−0.962599	−0.481299	+	0.876556i	$0.659835\pi$
$308$	0	0
$309$	25.2689	1.43750
$310$	0	0
$311$	−10.3250	−0.585475	−0.292738	−	0.956193i	$-0.594566\pi$
$311$	−10.3250	−0.585475	−0.292738	+	0.956193i	$0.594566\pi$
$312$	0	0
$313$	15.7684	0.891281	0.445641	−	0.895212i	$-0.352976\pi$
$313$	15.7684	0.891281	0.445641	+	0.895212i	$0.352976\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.17230	0.122009	0.0610043	−	0.998138i	$-0.480570\pi$
$317$	2.17230	0.122009	0.0610043	+	0.998138i	$0.480570\pi$
$318$	0	0
$319$	−7.45779	−0.417556
$320$	0	0
$321$	45.5741	2.54370
$322$	0	0
$323$	−0.563139	−0.0313339
$324$	0	0
$325$	0	0
$326$	0	0
$327$	41.4358	2.29140
$328$	0	0
$329$	20.0702	1.10651
$330$	0	0
$331$	−11.9791	−0.658429	−0.329215	−	0.944255i	$-0.606784\pi$
$331$	−11.9791	−0.658429	−0.329215	+	0.944255i	$0.606784\pi$
$332$	0	0
$333$	35.5117	1.94603
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.1921	0.609675	0.304838	−	0.952404i	$-0.401398\pi$
$337$	11.1921	0.609675	0.304838	+	0.952404i	$0.401398\pi$
$338$	0	0
$339$	3.65540	0.198534
$340$	0	0
$341$	4.42139	0.239432
$342$	0	0
$343$	−19.5117	−1.05353
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.3843	−1.09429	−0.547143	−	0.837039i	$-0.684285\pi$
$347$	−20.3843	−1.09429	−0.547143	+	0.837039i	$0.684285\pi$
$348$	0	0
$349$	−0.252557	−0.0135191	−0.00675954	−	0.999977i	$-0.502152\pi$
$349$	−0.252557	−0.0135191	−0.00675954	+	0.999977i	$0.502152\pi$
$350$	0	0
$351$	−60.4556	−3.22688
$352$	0	0
$353$	28.6434	1.52453	0.762267	−	0.647263i	$-0.224086\pi$
$353$	28.6434	1.52453	0.762267	+	0.647263i	$0.224086\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.21832	0.223257
$358$	0	0
$359$	18.5741	0.980301	0.490151	−	0.871638i	$-0.336942\pi$
$359$	18.5741	0.980301	0.490151	+	0.871638i	$0.336942\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	31.7509	1.66649
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.79476	−0.250285	−0.125142	−	0.992139i	$-0.539939\pi$
$367$	−4.79476	−0.250285	−0.125142	+	0.992139i	$0.539939\pi$
$368$	0	0
$369$	29.7213	1.54723
$370$	0	0
$371$	3.77929	0.196211
$372$	0	0
$373$	−15.9485	−0.825783	−0.412892	−	0.910780i	$-0.635481\pi$
$373$	−15.9485	−0.825783	−0.412892	+	0.910780i	$0.635481\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	63.7597	3.28379
$378$	0	0
$379$	16.8013	0.863025	0.431513	−	0.902107i	$-0.357980\pi$
$379$	16.8013	0.863025	0.431513	+	0.902107i	$0.357980\pi$
$380$	0	0
$381$	−28.8846	−1.47980
$382$	0	0
$383$	26.7948	1.36915	0.684574	−	0.728943i	$-0.259988\pi$
$383$	26.7948	1.36915	0.684574	+	0.728943i	$0.259988\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	49.9978	2.54153
$388$	0	0
$389$	18.3424	0.929998	0.464999	−	0.885311i	$-0.346055\pi$
$389$	18.3424	0.929998	0.464999	+	0.885311i	$0.346055\pi$
$390$	0	0
$391$	−2.61046	−0.132017
$392$	0	0
$393$	−52.7817	−2.66248
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−21.2162	−1.06481	−0.532404	−	0.846490i	$-0.678711\pi$
$397$	−21.2162	−1.06481	−0.532404	+	0.846490i	$0.678711\pi$
$398$	0	0
$399$	−7.49073	−0.375005
$400$	0	0
$401$	−27.8661	−1.39157	−0.695783	−	0.718252i	$-0.744943\pi$
$401$	−27.8661	−1.39157	−0.695783	+	0.718252i	$0.744943\pi$
$402$	0	0
$403$	−37.8002	−1.88296
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.17577	−0.206985
$408$	0	0
$409$	−18.0899	−0.894487	−0.447243	−	0.894412i	$-0.647594\pi$
$409$	−18.0899	−0.894487	−0.447243	+	0.894412i	$0.647594\pi$
$410$	0	0
$411$	6.85171	0.337970
$412$	0	0
$413$	14.2305	0.700239
$414$	0	0
$415$	0	0
$416$	0	0
$417$	51.1372	2.50420
$418$	0	0
$419$	15.3370	0.749260	0.374630	−	0.927174i	$-0.377770\pi$
$419$	15.3370	0.749260	0.374630	+	0.927174i	$0.377770\pi$
$420$	0	0
$421$	−8.42486	−0.410602	−0.205301	−	0.978699i	$-0.565817\pi$
$421$	−8.42486	−0.410602	−0.205301	+	0.978699i	$0.565817\pi$
$422$	0	0
$423$	50.3723	2.44918
$424$	0	0
$425$	0	0
$426$	0	0
$427$	27.0185	1.30752
$428$	0	0
$429$	13.7762	0.665122
$430$	0	0
$431$	−16.2766	−0.784014	−0.392007	−	0.919962i	$-0.628219\pi$
$431$	−16.2766	−0.784014	−0.392007	+	0.919962i	$0.628219\pi$
$432$	0	0
$433$	18.1976	0.874521	0.437261	−	0.899335i	$-0.355949\pi$
$433$	18.1976	0.874521	0.437261	+	0.899335i	$0.355949\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.63555	0.221749
$438$	0	0
$439$	−20.4214	−0.974660	−0.487330	−	0.873218i	$-0.662029\pi$
$439$	−20.4214	−0.974660	−0.487330	+	0.873218i	$0.662029\pi$
$440$	0	0
$441$	−5.57952	−0.265691
$442$	0	0
$443$	21.6685	1.02950	0.514750	−	0.857340i	$-0.327885\pi$
$443$	21.6685	1.02950	0.514750	+	0.857340i	$0.327885\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	39.4447	1.86567
$448$	0	0
$449$	23.8133	1.12382	0.561910	−	0.827198i	$-0.310067\pi$
$449$	23.8133	1.12382	0.561910	+	0.827198i	$0.310067\pi$
$450$	0	0
$451$	−3.49489	−0.164568
$452$	0	0
$453$	53.3479	2.50650
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9.15813	−0.428399	−0.214200	−	0.976790i	$-0.568714\pi$
$457$	−9.15813	−0.428399	−0.214200	+	0.976790i	$0.568714\pi$
$458$	0	0
$459$	5.46325	0.255003
$460$	0	0
$461$	−7.78931	−0.362784	−0.181392	−	0.983411i	$-0.558060\pi$
$461$	−7.78931	−0.362784	−0.181392	+	0.983411i	$0.558060\pi$
$462$	0	0
$463$	0.405011	0.0188225	0.00941123	−	0.999956i	$-0.497004\pi$
$463$	0.405011	0.0188225	0.00941123	+	0.999956i	$0.497004\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.95814	−0.0906118	−0.0453059	−	0.998973i	$-0.514426\pi$
$467$	−1.95814	−0.0906118	−0.0453059	+	0.998973i	$0.514426\pi$
$468$	0	0
$469$	31.9420	1.47494
$470$	0	0
$471$	28.8846	1.33093
$472$	0	0
$473$	−5.87918	−0.270325
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.48527	0.434301
$478$	0	0
$479$	22.9210	1.04729	0.523645	−	0.851937i	$-0.324572\pi$
$479$	22.9210	1.04729	0.523645	+	0.851937i	$0.324572\pi$
$480$	0	0
$481$	35.7003	1.62780
$482$	0	0
$483$	−34.7237	−1.57998
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−23.9450	−1.08505	−0.542527	−	0.840038i	$-0.682532\pi$
$487$	−23.9450	−1.08505	−0.542527	+	0.840038i	$0.682532\pi$
$488$	0	0
$489$	−40.0109	−1.80936
$490$	0	0
$491$	−3.86826	−0.174572	−0.0872861	−	0.996183i	$-0.527819\pi$
$491$	−3.86826	−0.174572	−0.0872861	+	0.996183i	$0.527819\pi$
$492$	0	0
$493$	−5.76183	−0.259500
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.8606	0.487166
$498$	0	0
$499$	7.92104	0.354595	0.177297	−	0.984157i	$-0.443265\pi$
$499$	7.92104	0.354595	0.177297	+	0.984157i	$0.443265\pi$
$500$	0	0
$501$	−5.49966	−0.245706
$502$	0	0
$503$	−16.6949	−0.744388	−0.372194	−	0.928155i	$-0.621394\pi$
$503$	−16.6949	−0.744388	−0.372194	+	0.928155i	$0.621394\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−78.3503	−3.47966
$508$	0	0
$509$	24.4028	1.08164	0.540818	−	0.841139i	$-0.318115\pi$
$509$	24.4028	1.08164	0.540818	+	0.841139i	$0.318115\pi$
$510$	0	0
$511$	10.1097	0.447228
$512$	0	0
$513$	−9.70142	−0.428328
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.92321	−0.260503
$518$	0	0
$519$	−37.9834	−1.66729
$520$	0	0
$521$	26.0659	1.14197	0.570983	−	0.820962i	$-0.306562\pi$
$521$	26.0659	1.14197	0.570983	+	0.820962i	$0.306562\pi$
$522$	0	0
$523$	17.8726	0.781516	0.390758	−	0.920493i	$-0.372213\pi$
$523$	17.8726	0.781516	0.390758	+	0.920493i	$0.372213\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.41593	0.148800
$528$	0	0
$529$	−1.51166	−0.0657243
$530$	0	0
$531$	35.7158	1.54993
$532$	0	0
$533$	29.8792	1.29421
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.23817	0.182891
$538$	0	0
$539$	0.656088	0.0282597
$540$	0	0
$541$	23.0604	0.991444	0.495722	−	0.868481i	$-0.334903\pi$
$541$	23.0604	0.991444	0.495722	+	0.868481i	$0.334903\pi$
$542$	0	0
$543$	−17.3754	−0.745648
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−27.7584	−1.18686	−0.593431	−	0.804885i	$-0.702227\pi$
$547$	−27.7584	−1.18686	−0.593431	+	0.804885i	$0.702227\pi$
$548$	0	0
$549$	67.8111	2.89411
$550$	0	0
$551$	10.2316	0.435882
$552$	0	0
$553$	−37.8792	−1.61079
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.60808	0.364736	0.182368	−	0.983230i	$-0.441624\pi$
$557$	8.60808	0.364736	0.182368	+	0.983230i	$0.441624\pi$
$558$	0	0
$559$	50.2635	2.12592
$560$	0	0
$561$	−1.24493	−0.0525609
$562$	0	0
$563$	7.93959	0.334614	0.167307	−	0.985905i	$-0.446493\pi$
$563$	7.93959	0.334614	0.167307	+	0.985905i	$0.446493\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	26.7422	1.12307
$568$	0	0
$569$	−1.65757	−0.0694889	−0.0347444	−	0.999396i	$-0.511062\pi$
$569$	−1.65757	−0.0694889	−0.0347444	+	0.999396i	$0.511062\pi$
$570$	0	0
$571$	14.0528	0.588091	0.294045	−	0.955791i	$-0.404998\pi$
$571$	14.0528	0.588091	0.294045	+	0.955791i	$0.404998\pi$
$572$	0	0
$573$	−81.9494	−3.42349
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.36445	0.0568027	0.0284014	−	0.999597i	$-0.490958\pi$
$577$	1.36445	0.0568027	0.0284014	+	0.999597i	$0.490958\pi$
$578$	0	0
$579$	−69.9738	−2.90801
$580$	0	0
$581$	19.4007	0.804876
$582$	0	0
$583$	−1.11536	−0.0461935
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.6609	−1.01786	−0.508931	−	0.860807i	$-0.669959\pi$
$587$	−24.6609	−1.01786	−0.508931	+	0.860807i	$0.669959\pi$
$588$	0	0
$589$	−6.06587	−0.249940
$590$	0	0
$591$	2.41047	0.0991535
$592$	0	0
$593$	0.747443	0.0306938	0.0153469	−	0.999882i	$-0.495115\pi$
$593$	0.747443	0.0306938	0.0153469	+	0.999882i	$0.495115\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	24.4831	1.00203
$598$	0	0
$599$	−1.40501	−0.0574072	−0.0287036	−	0.999588i	$-0.509138\pi$
$599$	−1.40501	−0.0574072	−0.0287036	+	0.999588i	$0.509138\pi$
$600$	0	0
$601$	29.7453	1.21334	0.606668	−	0.794956i	$-0.292506\pi$
$601$	29.7453	1.21334	0.606668	+	0.794956i	$0.292506\pi$
$602$	0	0
$603$	80.1680	3.26469
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.66849	0.230077	0.115038	−	0.993361i	$-0.463301\pi$
$607$	5.66849	0.230077	0.115038	+	0.993361i	$0.463301\pi$
$608$	0	0
$609$	−76.6423	−3.10570
$610$	0	0
$611$	50.6399	2.04867
$612$	0	0
$613$	2.99454	0.120948	0.0604742	−	0.998170i	$-0.480739\pi$
$613$	2.99454	0.120948	0.0604742	+	0.998170i	$0.480739\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.3370	1.30184	0.650919	−	0.759147i	$-0.274384\pi$
$617$	32.3370	1.30184	0.650919	+	0.759147i	$0.274384\pi$
$618$	0	0
$619$	20.9265	0.841107	0.420554	−	0.907268i	$-0.361836\pi$
$619$	20.9265	0.841107	0.420554	+	0.907268i	$0.361836\pi$
$620$	0	0
$621$	−44.9714	−1.80464
$622$	0	0
$623$	−24.6980	−0.989503
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.21069	0.0882866
$628$	0	0
$629$	−3.22617	−0.128636
$630$	0	0
$631$	−22.8002	−0.907663	−0.453831	−	0.891088i	$-0.649943\pi$
$631$	−22.8002	−0.907663	−0.453831	+	0.891088i	$0.649943\pi$
$632$	0	0
$633$	−35.3064	−1.40330
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−5.60916	−0.222243
$638$	0	0
$639$	27.2580	1.07831
$640$	0	0
$641$	36.3184	1.43449	0.717246	−	0.696820i	$-0.245402\pi$
$641$	36.3184	1.43449	0.717246	+	0.696820i	$0.245402\pi$
$642$	0	0
$643$	13.6135	0.536865	0.268433	−	0.963298i	$-0.413494\pi$
$643$	13.6135	0.536865	0.268433	+	0.963298i	$0.413494\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.0724126	−0.00284683	−0.00142342	−	0.999999i	$-0.500453\pi$
$647$	−0.0724126	−0.00284683	−0.00142342	+	0.999999i	$0.500453\pi$
$648$	0	0
$649$	−4.19978	−0.164856
$650$	0	0
$651$	45.4378	1.78085
$652$	0	0
$653$	43.5346	1.70364	0.851820	−	0.523835i	$-0.175499\pi$
$653$	43.5346	1.70364	0.851820	+	0.523835i	$0.175499\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	25.3734	0.989910
$658$	0	0
$659$	33.4512	1.30308	0.651538	−	0.758616i	$-0.274124\pi$
$659$	33.4512	1.30308	0.651538	+	0.758616i	$0.274124\pi$
$660$	0	0
$661$	−2.89465	−0.112589	−0.0562945	−	0.998414i	$-0.517929\pi$
$661$	−2.89465	−0.112589	−0.0562945	+	0.998414i	$0.517929\pi$
$662$	0	0
$663$	10.6434	0.413355
$664$	0	0
$665$	0	0
$666$	0	0
$667$	47.4292	1.83647
$668$	0	0
$669$	47.4818	1.83575
$670$	0	0
$671$	−7.97382	−0.307826
$672$	0	0
$673$	−42.8475	−1.65165	−0.825826	−	0.563925i	$-0.809291\pi$
$673$	−42.8475	−1.65165	−0.825826	+	0.563925i	$0.809291\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.4567	−1.32428	−0.662139	−	0.749381i	$-0.730351\pi$
$677$	−34.4567	−1.32428	−0.662139	+	0.749381i	$0.730351\pi$
$678$	0	0
$679$	−27.1812	−1.04312
$680$	0	0
$681$	−14.5621	−0.558019
$682$	0	0
$683$	−2.73436	−0.104627	−0.0523136	−	0.998631i	$-0.516660\pi$
$683$	−2.73436	−0.104627	−0.0523136	+	0.998631i	$0.516660\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.47634	−0.323393
$688$	0	0
$689$	9.53566	0.363280
$690$	0	0
$691$	4.74198	0.180394	0.0901968	−	0.995924i	$-0.471250\pi$
$691$	4.74198	0.180394	0.0901968	+	0.995924i	$0.471250\pi$
$692$	0	0
$693$	−11.1590	−0.423897
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2.70012	−0.102274
$698$	0	0
$699$	35.0068	1.32408
$700$	0	0
$701$	33.1372	1.25157	0.625787	−	0.779994i	$-0.284778\pi$
$701$	33.1372	1.25157	0.625787	+	0.779994i	$0.284778\pi$
$702$	0	0
$703$	5.72890	0.216069
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−24.0419	−0.904187
$708$	0	0
$709$	5.37884	0.202006	0.101003	−	0.994886i	$-0.467795\pi$
$709$	5.37884	0.202006	0.101003	+	0.994886i	$0.467795\pi$
$710$	0	0
$711$	−95.0692	−3.56537
$712$	0	0
$713$	−28.1187	−1.05305
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−79.5390	−2.97044
$718$	0	0
$719$	−33.0857	−1.23389	−0.616944	−	0.787007i	$-0.711630\pi$
$719$	−33.0857	−1.23389	−0.616944	+	0.787007i	$0.711630\pi$
$720$	0	0
$721$	−20.5771	−0.766332
$722$	0	0
$723$	36.5950	1.36098
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.62463	0.0973423	0.0486711	−	0.998815i	$-0.484501\pi$
$727$	2.62463	0.0973423	0.0486711	+	0.998815i	$0.484501\pi$
$728$	0	0
$729$	−21.1538	−0.783472
$730$	0	0
$731$	−4.54221	−0.168000
$732$	0	0
$733$	−0.608077	−0.0224598	−0.0112299	−	0.999937i	$-0.503575\pi$
$733$	−0.608077	−0.0224598	−0.0112299	+	0.999937i	$0.503575\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.42685	−0.347242
$738$	0	0
$739$	−36.3974	−1.33890	−0.669450	−	0.742857i	$-0.733470\pi$
$739$	−36.3974	−1.33890	−0.669450	+	0.742857i	$0.733470\pi$
$740$	0	0
$741$	−18.9001	−0.694313
$742$	0	0
$743$	−23.0713	−0.846405	−0.423202	−	0.906035i	$-0.639094\pi$
$743$	−23.0713	−0.846405	−0.423202	+	0.906035i	$0.639094\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	48.6918	1.78154
$748$	0	0
$749$	−37.1121	−1.35605
$750$	0	0
$751$	−4.75290	−0.173436	−0.0867179	−	0.996233i	$-0.527638\pi$
$751$	−4.75290	−0.173436	−0.0867179	+	0.996233i	$0.527638\pi$
$752$	0	0
$753$	41.0164	1.49472
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−26.8057	−0.974269	−0.487135	−	0.873327i	$-0.661958\pi$
$757$	−26.8057	−0.974269	−0.487135	+	0.873327i	$0.661958\pi$
$758$	0	0
$759$	10.2478	0.371971
$760$	0	0
$761$	25.4478	0.922481	0.461241	−	0.887275i	$-0.347404\pi$
$761$	25.4478	0.922481	0.461241	+	0.887275i	$0.347404\pi$
$762$	0	0
$763$	−33.7422	−1.22155
$764$	0	0
$765$	0	0
$766$	0	0
$767$	35.9056	1.29648
$768$	0	0
$769$	−14.6421	−0.528007	−0.264004	−	0.964522i	$-0.585043\pi$
$769$	−14.6421	−0.528007	−0.264004	+	0.964522i	$0.585043\pi$
$770$	0	0
$771$	68.9684	2.48384
$772$	0	0
$773$	16.8462	0.605917	0.302959	−	0.953004i	$-0.402026\pi$
$773$	16.8462	0.605917	0.302959	+	0.953004i	$0.402026\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−42.9136	−1.53952
$778$	0	0
$779$	4.79476	0.171790
$780$	0	0
$781$	−3.20524	−0.114692
$782$	0	0
$783$	−99.2613	−3.54731
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.7367	0.667893	0.333946	−	0.942592i	$-0.391620\pi$
$787$	18.7367	0.667893	0.333946	+	0.942592i	$0.391620\pi$
$788$	0	0
$789$	20.2416	0.720621
$790$	0	0
$791$	−2.97668	−0.105839
$792$	0	0
$793$	68.1714	2.42084
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.9900	1.34567	0.672837	−	0.739791i	$-0.265075\pi$
$797$	37.9900	1.34567	0.672837	+	0.739791i	$0.265075\pi$
$798$	0	0
$799$	−4.57623	−0.161895
$800$	0	0
$801$	−61.9869	−2.19020
$802$	0	0
$803$	−2.98362	−0.105290
$804$	0	0
$805$	0	0
$806$	0	0
$807$	90.1989	3.17515
$808$	0	0
$809$	21.8857	0.769461	0.384731	−	0.923029i	$-0.374294\pi$
$809$	21.8857	0.769461	0.384731	+	0.923029i	$0.374294\pi$
$810$	0	0
$811$	−37.8595	−1.32943	−0.664714	−	0.747098i	$-0.731447\pi$
$811$	−37.8595	−1.32943	−0.664714	+	0.747098i	$0.731447\pi$
$812$	0	0
$813$	3.37229	0.118271
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.06587	0.282189
$818$	0	0
$819$	95.4031	3.33365
$820$	0	0
$821$	−20.1976	−0.704901	−0.352451	−	0.935830i	$-0.614652\pi$
$821$	−20.1976	−0.704901	−0.352451	+	0.935830i	$0.614652\pi$
$822$	0	0
$823$	−4.34590	−0.151489	−0.0757443	−	0.997127i	$-0.524133\pi$
$823$	−4.34590	−0.151489	−0.0757443	+	0.997127i	$0.524133\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−56.8375	−1.97643	−0.988217	−	0.153057i	$-0.951088\pi$
$827$	−56.8375	−1.97643	−0.988217	+	0.153057i	$0.951088\pi$
$828$	0	0
$829$	−17.4543	−0.606214	−0.303107	−	0.952957i	$-0.598024\pi$
$829$	−17.4543	−0.606214	−0.303107	+	0.952957i	$0.598024\pi$
$830$	0	0
$831$	39.8111	1.38103
$832$	0	0
$833$	0.506888	0.0175626
$834$	0	0
$835$	0	0
$836$	0	0
$837$	58.8475	2.03407
$838$	0	0
$839$	−1.77622	−0.0613219	−0.0306609	−	0.999530i	$-0.509761\pi$
$839$	−1.77622	−0.0613219	−0.0306609	+	0.999530i	$0.509761\pi$
$840$	0	0
$841$	75.6862	2.60987
$842$	0	0
$843$	−69.5346	−2.39490
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−25.8556	−0.888408
$848$	0	0
$849$	−2.61025	−0.0895834
$850$	0	0
$851$	26.5566	0.910348
$852$	0	0
$853$	16.1187	0.551892	0.275946	−	0.961173i	$-0.411009\pi$
$853$	16.1187	0.551892	0.275946	+	0.961173i	$0.411009\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	47.2031	1.61243	0.806213	−	0.591625i	$-0.201514\pi$
$857$	47.2031	1.61243	0.806213	+	0.591625i	$0.201514\pi$
$858$	0	0
$859$	−37.3239	−1.27347	−0.636737	−	0.771081i	$-0.719716\pi$
$859$	−37.3239	−1.27347	−0.636737	+	0.771081i	$0.719716\pi$
$860$	0	0
$861$	−35.9163	−1.22402
$862$	0	0
$863$	−1.33697	−0.0455111	−0.0227555	−	0.999741i	$-0.507244\pi$
$863$	−1.33697	−0.0455111	−0.0227555	+	0.999741i	$0.507244\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	50.5981	1.71840
$868$	0	0
$869$	11.1791	0.379224
$870$	0	0
$871$	80.5939	2.73082
$872$	0	0
$873$	−68.2194	−2.30888
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.1647	−0.444539	−0.222270	−	0.974985i	$-0.571347\pi$
$877$	−13.1647	−0.444539	−0.222270	+	0.974985i	$0.571347\pi$
$878$	0	0
$879$	60.1163	2.02767
$880$	0	0
$881$	−10.2052	−0.343823	−0.171912	−	0.985112i	$-0.554994\pi$
$881$	−10.2052	−0.343823	−0.171912	+	0.985112i	$0.554994\pi$
$882$	0	0
$883$	1.49966	0.0504674	0.0252337	−	0.999682i	$-0.491967\pi$
$883$	1.49966	0.0504674	0.0252337	+	0.999682i	$0.491967\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−46.9505	−1.57644	−0.788222	−	0.615391i	$-0.788998\pi$
$887$	−46.9505	−1.57644	−0.788222	+	0.615391i	$0.788998\pi$
$888$	0	0
$889$	23.5215	0.788886
$890$	0	0
$891$	−7.89227	−0.264401
$892$	0	0
$893$	8.12628	0.271936
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−87.6125	−2.92529
$898$	0	0
$899$	−62.0637	−2.06994
$900$	0	0
$901$	−0.861719	−0.0287080
$902$	0	0
$903$	−60.4192	−2.01063
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.3668	0.543452	0.271726	−	0.962375i	$-0.412406\pi$
$907$	16.3668	0.543452	0.271726	+	0.962375i	$0.412406\pi$
$908$	0	0
$909$	−60.3403	−2.00136
$910$	0	0
$911$	−32.3293	−1.07112	−0.535559	−	0.844498i	$-0.679899\pi$
$911$	−32.3293	−1.07112	−0.535559	+	0.844498i	$0.679899\pi$
$912$	0	0
$913$	−5.72561	−0.189490
$914$	0	0
$915$	0	0
$916$	0	0
$917$	42.9815	1.41937
$918$	0	0
$919$	22.8157	0.752620	0.376310	−	0.926494i	$-0.377193\pi$
$919$	22.8157	0.752620	0.376310	+	0.926494i	$0.377193\pi$
$920$	0	0
$921$	51.1538	1.68557
$922$	0	0
$923$	27.4028	0.901976
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−51.6445	−1.69623
$928$	0	0
$929$	−27.2436	−0.893834	−0.446917	−	0.894575i	$-0.647478\pi$
$929$	−27.2436	−0.893834	−0.446917	+	0.894575i	$0.647478\pi$
$930$	0	0
$931$	−0.900112	−0.0295000
$932$	0	0
$933$	31.3150	1.02521
$934$	0	0
$935$	0	0
$936$	0	0
$937$	51.5006	1.68245	0.841225	−	0.540685i	$-0.181835\pi$
$937$	51.5006	1.68245	0.841225	+	0.540685i	$0.181835\pi$
$938$	0	0
$939$	−47.8244	−1.56069
$940$	0	0
$941$	−40.8541	−1.33181	−0.665903	−	0.746039i	$-0.731953\pi$
$941$	−40.8541	−1.33181	−0.665903	+	0.746039i	$0.731953\pi$
$942$	0	0
$943$	22.2264	0.723791
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.2526	1.24304	0.621521	−	0.783398i	$-0.286515\pi$
$947$	38.2526	1.24304	0.621521	+	0.783398i	$0.286515\pi$
$948$	0	0
$949$	25.5082	0.828031
$950$	0	0
$951$	−6.58845	−0.213645
$952$	0	0
$953$	−54.1187	−1.75308	−0.876538	−	0.481334i	$-0.840153\pi$
$953$	−54.1187	−1.75308	−0.876538	+	0.481334i	$0.840153\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	22.6190	0.731168
$958$	0	0
$959$	−5.57952	−0.180172
$960$	0	0
$961$	5.79476	0.186928
$962$	0	0
$963$	−93.1440	−3.00152
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28.4214	0.913970	0.456985	−	0.889474i	$-0.348929\pi$
$967$	28.4214	0.913970	0.456985	+	0.889474i	$0.348929\pi$
$968$	0	0
$969$	1.70796	0.0548677
$970$	0	0
$971$	−30.8057	−0.988601	−0.494301	−	0.869291i	$-0.664576\pi$
$971$	−30.8057	−0.988601	−0.494301	+	0.869291i	$0.664576\pi$
$972$	0	0
$973$	−41.6423	−1.33499
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37.6554	−1.20470	−0.602351	−	0.798231i	$-0.705769\pi$
$977$	−37.6554	−1.20470	−0.602351	+	0.798231i	$0.705769\pi$
$978$	0	0
$979$	7.28896	0.232956
$980$	0	0
$981$	−84.6862	−2.70382
$982$	0	0
$983$	−38.7948	−1.23736	−0.618680	−	0.785643i	$-0.712332\pi$
$983$	−38.7948	−1.23736	−0.618680	+	0.785643i	$0.712332\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−60.8717	−1.93757
$988$	0	0
$989$	37.3898	1.18893
$990$	0	0
$991$	15.0295	0.477427	0.238713	−	0.971090i	$-0.423274\pi$
$991$	15.0295	0.477427	0.238713	+	0.971090i	$0.423274\pi$
$992$	0	0
$993$	36.3317	1.15295
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.18123	−0.0374099	−0.0187050	−	0.999825i	$-0.505954\pi$
$997$	−1.18123	−0.0374099	−0.0187050	+	0.999825i	$0.505954\pi$
$998$	0	0
$999$	−55.5784	−1.75842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cb.1.1		3
4.3	odd	2		950.2.a.k.1.3	✓	3
5.4	even	2		7600.2.a.bm.1.3		3
12.11	even	2		8550.2.a.co.1.2		3
20.3	even	4		950.2.b.g.799.6		6
20.7	even	4		950.2.b.g.799.1		6
20.19	odd	2		950.2.a.m.1.1	yes	3
60.59	even	2		8550.2.a.cj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.k.1.3	✓	3	4.3	odd	2
950.2.a.m.1.1	yes	3	20.19	odd	2
950.2.b.g.799.1		6	20.7	even	4
950.2.b.g.799.6		6	20.3	even	4
7600.2.a.bm.1.3		3	5.4	even	2
7600.2.a.cb.1.1		3	1.1	even	1	trivial
8550.2.a.cj.1.2		3	60.59	even	2
8550.2.a.co.1.2		3	12.11	even	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-3.03293 q^{3} +2.46980 q^{7} +6.19869 q^{9} +O(q^{10})\)	\(q-3.03293 q^{3} +2.46980 q^{7} +6.19869 q^{9} -0.728896 q^{11} +6.23163 q^{13} -0.563139 q^{17} +1.00000 q^{19} -7.49073 q^{21} +4.63555 q^{23} -9.70142 q^{27} +10.2316 q^{29} -6.06587 q^{31} +2.21069 q^{33} +5.72890 q^{37} -18.9001 q^{39} +4.79476 q^{41} +8.06587 q^{43} +8.12628 q^{47} -0.900112 q^{49} +1.70796 q^{51} +1.53020 q^{53} -3.03293 q^{57} +5.76183 q^{59} +10.9396 q^{61} +15.3095 q^{63} +12.9330 q^{67} -14.0593 q^{69} +4.39738 q^{71} +4.09334 q^{73} -1.80022 q^{77} -15.3370 q^{79} +10.8277 q^{81} +7.85517 q^{83} -31.0318 q^{87} -10.0000 q^{89} +15.3908 q^{91} +18.3974 q^{93} -11.0055 q^{97} -4.51820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 2 * q^7 + 13 * q^9	\( 3 q + 2 q^{3} + 2 q^{7} + 13 q^{9} - 2 q^{11} + 2 q^{13} + 4 q^{17} + 3 q^{19} - 11 q^{21} + 14 q^{23} - 7 q^{27} + 14 q^{29} + 4 q^{31} - 4 q^{33} + 17 q^{37} - 29 q^{39} - 8 q^{41} + 2 q^{43} + 13 q^{47} + 25 q^{49} + 11 q^{51} + 10 q^{53} + 2 q^{57} + 6 q^{59} + 22 q^{61} + 2 q^{63} + 8 q^{69} + 2 q^{71} + 12 q^{73} + 50 q^{77} - 24 q^{79} - q^{81} + 12 q^{83} - 21 q^{87} - 30 q^{89} + 7 q^{91} + 44 q^{93} - 24 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^7 + 13 * q^9 - 2 * q^11 + 2 * q^13 + 4 * q^17 + 3 * q^19 - 11 * q^21 + 14 * q^23 - 7 * q^27 + 14 * q^29 + 4 * q^31 - 4 * q^33 + 17 * q^37 - 29 * q^39 - 8 * q^41 + 2 * q^43 + 13 * q^47 + 25 * q^49 + 11 * q^51 + 10 * q^53 + 2 * q^57 + 6 * q^59 + 22 * q^61 + 2 * q^63 + 8 * q^69 + 2 * q^71 + 12 * q^73 + 50 * q^77 - 24 * q^79 - q^81 + 12 * q^83 - 21 * q^87 - 30 * q^89 + 7 * q^91 + 44 * q^93 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more