LMFDB - Embedded newform 8550.2.a.cj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
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:	$\Z[a_1, \ldots, a_{11}]$
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			1.2
Root			$2.19869$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.46980 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +2.46980 q^{7} -1.00000 q^{8} -0.728896 q^{11} -6.23163 q^{13} -2.46980 q^{14} +1.00000 q^{16} -0.563139 q^{17} -1.00000 q^{19} +0.728896 q^{22} -4.63555 q^{23} +6.23163 q^{26} +2.46980 q^{28} -10.2316 q^{29} +6.06587 q^{31} -1.00000 q^{32} +0.563139 q^{34} -5.72890 q^{37} +1.00000 q^{38} -4.79476 q^{41} +8.06587 q^{43} -0.728896 q^{44} +4.63555 q^{46} -8.12628 q^{47} -0.900112 q^{49} -6.23163 q^{52} +1.53020 q^{53} -2.46980 q^{56} +10.2316 q^{58} +5.76183 q^{59} +10.9396 q^{61} -6.06587 q^{62} +1.00000 q^{64} +12.9330 q^{67} -0.563139 q^{68} +4.39738 q^{71} -4.09334 q^{73} +5.72890 q^{74} -1.00000 q^{76} -1.80022 q^{77} +15.3370 q^{79} +4.79476 q^{82} -7.85517 q^{83} -8.06587 q^{86} +0.728896 q^{88} +10.0000 q^{89} -15.3908 q^{91} -4.63555 q^{92} +8.12628 q^{94} +11.0055 q^{97} +0.900112 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 2 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 2 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 2 q^{7} - 3 q^{8} - 2 q^{11} - 2 q^{13} - 2 q^{14} + 3 q^{16} + 4 q^{17} - 3 q^{19} + 2 q^{22} - 14 q^{23} + 2 q^{26} + 2 q^{28} - 14 q^{29} - 4 q^{31} - 3 q^{32} - 4 q^{34} - 17 q^{37} + 3 q^{38} + 8 q^{41} + 2 q^{43} - 2 q^{44} + 14 q^{46} - 13 q^{47} + 25 q^{49} - 2 q^{52} + 10 q^{53} - 2 q^{56} + 14 q^{58} + 6 q^{59} + 22 q^{61} + 4 q^{62} + 3 q^{64} + 4 q^{68} + 2 q^{71} - 12 q^{73} + 17 q^{74} - 3 q^{76} + 50 q^{77} + 24 q^{79} - 8 q^{82} - 12 q^{83} - 2 q^{86} + 2 q^{88} + 30 q^{89} - 7 q^{91} - 14 q^{92} + 13 q^{94} - 25 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 2 * q^7 - 3 * q^8 - 2 * q^11 - 2 * q^13 - 2 * q^14 + 3 * q^16 + 4 * q^17 - 3 * q^19 + 2 * q^22 - 14 * q^23 + 2 * q^26 + 2 * q^28 - 14 * q^29 - 4 * q^31 - 3 * q^32 - 4 * q^34 - 17 * q^37 + 3 * q^38 + 8 * q^41 + 2 * q^43 - 2 * q^44 + 14 * q^46 - 13 * q^47 + 25 * q^49 - 2 * q^52 + 10 * q^53 - 2 * q^56 + 14 * q^58 + 6 * q^59 + 22 * q^61 + 4 * q^62 + 3 * q^64 + 4 * q^68 + 2 * q^71 - 12 * q^73 + 17 * q^74 - 3 * q^76 + 50 * q^77 + 24 * q^79 - 8 * q^82 - 12 * q^83 - 2 * q^86 + 2 * q^88 + 30 * q^89 - 7 * q^91 - 14 * q^92 + 13 * q^94 - 25 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	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	$0.345426\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	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	$-0.832157\pi$
$13$	−6.23163	−1.72834	−0.864171	+	0.503198i	$0.832157\pi$
$14$	−2.46980	−0.660081
$15$	0	0
$16$	1.00000	0.250000
$17$	−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
$21$	0	0
$22$	0.728896	0.155401
$23$	−4.63555	−0.966579	−0.483290	−	0.875460i	$-0.660558\pi$
$23$	−4.63555	−0.966579	−0.483290	+	0.875460i	$0.660558\pi$
$24$	0	0
$25$	0	0
$26$	6.23163	1.22212
$27$	0	0
$28$	2.46980	0.466747
$29$	−10.2316	−1.89997	−0.949983	−	0.312303i	$-0.898900\pi$
$29$	−10.2316	−1.89997	−0.949983	+	0.312303i	$0.898900\pi$
$30$	0	0
$31$	6.06587	1.08946	0.544731	−	0.838611i	$-0.316632\pi$
$31$	6.06587	1.08946	0.544731	+	0.838611i	$0.316632\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.563139	0.0965776
$35$	0	0
$36$	0	0
$37$	−5.72890	−0.941825	−0.470912	−	0.882180i	$-0.656075\pi$
$37$	−5.72890	−0.941825	−0.470912	+	0.882180i	$0.656075\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	−4.79476	−0.748816	−0.374408	−	0.927264i	$-0.622154\pi$
$41$	−4.79476	−0.748816	−0.374408	+	0.927264i	$0.622154\pi$
$42$	0	0
$43$	8.06587	1.23003	0.615017	−	0.788514i	$-0.289149\pi$
$43$	8.06587	1.23003	0.615017	+	0.788514i	$0.289149\pi$
$44$	−0.728896	−0.109885
$45$	0	0
$46$	4.63555	0.683475
$47$	−8.12628	−1.18534	−0.592670	−	0.805446i	$-0.701926\pi$
$47$	−8.12628	−1.18534	−0.592670	+	0.805446i	$0.701926\pi$
$48$	0	0
$49$	−0.900112	−0.128587
$50$	0	0
$51$	0	0
$52$	−6.23163	−0.864171
$53$	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
$56$	−2.46980	−0.330040
$57$	0	0
$58$	10.2316	1.34348
$59$	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	$0.253034\pi$
$62$	−6.06587	−0.770366
$63$	0	0
$64$	1.00000	0.125000
$65$	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	$0.210074\pi$
$68$	−0.563139	−0.0682907
$69$	0	0
$70$	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	$-0.576998\pi$
$73$	−4.09334	−0.479090	−0.239545	+	0.970885i	$0.576998\pi$
$74$	5.72890	0.665971
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−1.80022	−0.205155
$78$	0	0
$79$	15.3370	1.72554	0.862772	−	0.505593i	$-0.168726\pi$
$79$	15.3370	1.72554	0.862772	+	0.505593i	$0.168726\pi$
$80$	0	0
$81$	0	0
$82$	4.79476	0.529493
$83$	−7.85517	−0.862217	−0.431109	−	0.902300i	$-0.641877\pi$
$83$	−7.85517	−0.862217	−0.431109	+	0.902300i	$0.641877\pi$
$84$	0	0
$85$	0	0
$86$	−8.06587	−0.869765
$87$	0	0
$88$	0.728896	0.0777006
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−15.3908	−1.61340
$92$	−4.63555	−0.483290
$93$	0	0
$94$	8.12628	0.838162
$95$	0	0
$96$	0	0
$97$	11.0055	1.11744	0.558718	−	0.829358i	$-0.311294\pi$
$97$	11.0055	1.11744	0.558718	+	0.829358i	$0.311294\pi$
$98$	0.900112	0.0909250
$99$	0	0
$100$	0	0
$101$	9.73436	0.968605	0.484302	−	0.874901i	$-0.339074\pi$
$101$	9.73436	0.968605	0.484302	+	0.874901i	$0.339074\pi$
$102$	0	0
$103$	−8.33151	−0.820928	−0.410464	−	0.911877i	$-0.634633\pi$
$103$	−8.33151	−0.820928	−0.410464	+	0.911877i	$0.634633\pi$
$104$	6.23163	0.611061
$105$	0	0
$106$	−1.53020	−0.148627
$107$	15.0264	1.45266	0.726328	−	0.687348i	$-0.241225\pi$
$107$	15.0264	1.45266	0.726328	+	0.687348i	$0.241225\pi$
$108$	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
$111$	0	0
$112$	2.46980	0.233374
$113$	−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
$116$	−10.2316	−0.949983
$117$	0	0
$118$	−5.76183	−0.530419
$119$	−1.39084	−0.127498
$120$	0	0
$121$	−10.4687	−0.951701
$122$	−10.9396	−0.990424
$123$	0	0
$124$	6.06587	0.544731
$125$	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	$0.361137\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	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
$134$	−12.9330	−1.11725
$135$	0	0
$136$	0.563139	0.0482888
$137$	−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	$-0.246404\pi$
$139$	16.8606	1.43010	0.715050	+	0.699073i	$0.246404\pi$
$140$	0	0
$141$	0	0
$142$	−4.39738	−0.369020
$143$	4.54221	0.379838
$144$	0	0
$145$	0	0
$146$	4.09334	0.338768
$147$	0	0
$148$	−5.72890	−0.470912
$149$	13.0055	1.06545	0.532724	−	0.846289i	$-0.321168\pi$
$149$	13.0055	1.06545	0.532724	+	0.846289i	$0.321168\pi$
$150$	0	0
$151$	17.5895	1.43142	0.715708	−	0.698400i	$-0.246104\pi$
$151$	17.5895	1.43142	0.715708	+	0.698400i	$0.246104\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	1.80022	0.145066
$155$	0	0
$156$	0	0
$157$	9.52366	0.760071	0.380035	−	0.924972i	$-0.375912\pi$
$157$	9.52366	0.760071	0.380035	+	0.924972i	$0.375912\pi$
$158$	−15.3370	−1.22014
$159$	0	0
$160$	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	$0.327181\pi$
$164$	−4.79476	−0.374408
$165$	0	0
$166$	7.85517	0.609680
$167$	−1.81331	−0.140318	−0.0701591	−	0.997536i	$-0.522351\pi$
$167$	−1.81331	−0.140318	−0.0701591	+	0.997536i	$0.522351\pi$
$168$	0	0
$169$	25.8332	1.98717
$170$	0	0
$171$	0	0
$172$	8.06587	0.615017
$173$	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
$176$	−0.728896	−0.0549426
$177$	0	0
$178$	−10.0000	−0.749532
$179$	−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	$0.431705\pi$
$182$	15.3908	1.14084
$183$	0	0
$184$	4.63555	0.341737
$185$	0	0
$186$	0	0
$187$	0.410470	0.0300165
$188$	−8.12628	−0.592670
$189$	0	0
$190$	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	$-0.811862\pi$
$193$	−23.0713	−1.66071	−0.830355	+	0.557234i	$0.811862\pi$
$194$	−11.0055	−0.790146
$195$	0	0
$196$	−0.900112	−0.0642937
$197$	−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	$-0.407635\pi$
$199$	8.07241	0.572238	0.286119	+	0.958194i	$0.407635\pi$
$200$	0	0
$201$	0	0
$202$	−9.73436	−0.684907
$203$	−25.2700	−1.77361
$204$	0	0
$205$	0	0
$206$	8.33151	0.580484
$207$	0	0
$208$	−6.23163	−0.432085
$209$	0.728896	0.0504188
$210$	0	0
$211$	−11.6410	−0.801400	−0.400700	−	0.916209i	$-0.631233\pi$
$211$	−11.6410	−0.801400	−0.400700	+	0.916209i	$0.631233\pi$
$212$	1.53020	0.105095
$213$	0	0
$214$	−15.0264	−1.02718
$215$	0	0
$216$	0	0
$217$	14.9815	1.01701
$218$	13.6619	0.925304
$219$	0	0
$220$	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	$0.675629\pi$
$224$	−2.46980	−0.165020
$225$	0	0
$226$	1.20524	0.0801710
$227$	−4.80131	−0.318674	−0.159337	−	0.987224i	$-0.550936\pi$
$227$	−4.80131	−0.318674	−0.159337	+	0.987224i	$0.550936\pi$
$228$	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
$231$	0	0
$232$	10.2316	0.671739
$233$	−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
$236$	5.76183	0.375063
$237$	0	0
$238$	1.39084	0.0901547
$239$	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	$0.627047\pi$
$242$	10.4687	0.672954
$243$	0	0
$244$	10.9396	0.700336
$245$	0	0
$246$	0	0
$247$	6.23163	0.396509
$248$	−6.06587	−0.385183
$249$	0	0
$250$	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
$254$	−9.52366	−0.597568
$255$	0	0
$256$	1.00000	0.0625000
$257$	−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
$261$	0	0
$262$	−17.4028	−1.07515
$263$	6.67395	0.411533	0.205767	−	0.978601i	$-0.434031\pi$
$263$	6.67395	0.411533	0.205767	+	0.978601i	$0.434031\pi$
$264$	0	0
$265$	0	0
$266$	2.46980	0.151433
$267$	0	0
$268$	12.9330	0.790012
$269$	29.7398	1.81327	0.906634	−	0.421917i	$-0.138643\pi$
$269$	29.7398	1.81327	0.906634	+	0.421917i	$0.138643\pi$
$270$	0	0
$271$	1.11189	0.0675426	0.0337713	−	0.999430i	$-0.489248\pi$
$271$	1.11189	0.0675426	0.0337713	+	0.999430i	$0.489248\pi$
$272$	−0.563139	−0.0341453
$273$	0	0
$274$	2.25910	0.136477
$275$	0	0
$276$	0	0
$277$	13.1263	0.788682	0.394341	−	0.918964i	$-0.370973\pi$
$277$	13.1263	0.788682	0.394341	+	0.918964i	$0.370973\pi$
$278$	−16.8606	−1.01123
$279$	0	0
$280$	0	0
$281$	−22.9265	−1.36768	−0.683840	−	0.729632i	$-0.739691\pi$
$281$	−22.9265	−1.36768	−0.683840	+	0.729632i	$0.739691\pi$
$282$	0	0
$283$	0.860634	0.0511594	0.0255797	−	0.999673i	$-0.491857\pi$
$283$	0.860634	0.0511594	0.0255797	+	0.999673i	$0.491857\pi$
$284$	4.39738	0.260937
$285$	0	0
$286$	−4.54221	−0.268586
$287$	−11.8421	−0.699016
$288$	0	0
$289$	−16.6829	−0.981346
$290$	0	0
$291$	0	0
$292$	−4.09334	−0.239545
$293$	−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
$296$	5.72890	0.332985
$297$	0	0
$298$	−13.0055	−0.753386
$299$	28.8870	1.67058
$300$	0	0
$301$	19.9210	1.14823
$302$	−17.5895	−1.01216
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	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	$0.659835\pi$
$308$	−1.80022	−0.102577
$309$	0	0
$310$	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	$-0.647024\pi$
$313$	−15.7684	−0.891281	−0.445641	+	0.895212i	$0.647024\pi$
$314$	−9.52366	−0.537451
$315$	0	0
$316$	15.3370	0.862772
$317$	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
$321$	0	0
$322$	11.4489	0.638020
$323$	0.563139	0.0313339
$324$	0	0
$325$	0	0
$326$	−13.1921	−0.730645
$327$	0	0
$328$	4.79476	0.264747
$329$	−20.0702	−1.10651
$330$	0	0
$331$	11.9791	0.658429	0.329215	−	0.944255i	$-0.393216\pi$
$331$	11.9791	0.658429	0.329215	+	0.944255i	$0.393216\pi$
$332$	−7.85517	−0.431109
$333$	0	0
$334$	1.81331	0.0992200
$335$	0	0
$336$	0	0
$337$	−11.1921	−0.609675	−0.304838	−	0.952404i	$-0.598602\pi$
$337$	−11.1921	−0.609675	−0.304838	+	0.952404i	$0.598602\pi$
$338$	−25.8332	−1.40514
$339$	0	0
$340$	0	0
$341$	−4.42139	−0.239432
$342$	0	0
$343$	−19.5117	−1.05353
$344$	−8.06587	−0.434883
$345$	0	0
$346$	−12.5237	−0.673276
$347$	20.3843	1.09429	0.547143	−	0.837039i	$-0.315715\pi$
$347$	20.3843	1.09429	0.547143	+	0.837039i	$0.315715\pi$
$348$	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
$351$	0	0
$352$	0.728896	0.0388503
$353$	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
$356$	10.0000	0.529999
$357$	0	0
$358$	1.39738	0.0738540
$359$	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
$362$	−5.72890	−0.301104
$363$	0	0
$364$	−15.3908	−0.806699
$365$	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	$0.539939\pi$
$368$	−4.63555	−0.241645
$369$	0	0
$370$	0	0
$371$	3.77929	0.196211
$372$	0	0
$373$	15.9485	0.825783	0.412892	−	0.910780i	$-0.364519\pi$
$373$	15.9485	0.825783	0.412892	+	0.910780i	$0.364519\pi$
$374$	−0.410470	−0.0212249
$375$	0	0
$376$	8.12628	0.419081
$377$	63.7597	3.28379
$378$	0	0
$379$	−16.8013	−0.863025	−0.431513	−	0.902107i	$-0.642020\pi$
$379$	−16.8013	−0.863025	−0.431513	+	0.902107i	$0.642020\pi$
$380$	0	0
$381$	0	0
$382$	−27.0198	−1.38246
$383$	−26.7948	−1.36915	−0.684574	−	0.728943i	$-0.740012\pi$
$383$	−26.7948	−1.36915	−0.684574	+	0.728943i	$0.740012\pi$
$384$	0	0
$385$	0	0
$386$	23.0713	1.17430
$387$	0	0
$388$	11.0055	0.558718
$389$	−18.3424	−0.929998	−0.464999	−	0.885311i	$-0.653945\pi$
$389$	−18.3424	−0.929998	−0.464999	+	0.885311i	$0.653945\pi$
$390$	0	0
$391$	2.61046	0.132017
$392$	0.900112	0.0454625
$393$	0	0
$394$	0.794765	0.0400397
$395$	0	0
$396$	0	0
$397$	21.2162	1.06481	0.532404	−	0.846490i	$-0.321289\pi$
$397$	21.2162	1.06481	0.532404	+	0.846490i	$0.321289\pi$
$398$	−8.07241	−0.404633
$399$	0	0
$400$	0	0
$401$	27.8661	1.39157	0.695783	−	0.718252i	$-0.255057\pi$
$401$	27.8661	1.39157	0.695783	+	0.718252i	$0.255057\pi$
$402$	0	0
$403$	−37.8002	−1.88296
$404$	9.73436	0.484302
$405$	0	0
$406$	25.2700	1.25413
$407$	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
$411$	0	0
$412$	−8.33151	−0.410464
$413$	14.2305	0.700239
$414$	0	0
$415$	0	0
$416$	6.23163	0.305531
$417$	0	0
$418$	−0.728896	−0.0356515
$419$	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	$0.565817\pi$
$422$	11.6410	0.566676
$423$	0	0
$424$	−1.53020	−0.0743133
$425$	0	0
$426$	0	0
$427$	27.0185	1.30752
$428$	15.0264	0.726328
$429$	0	0
$430$	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	$-0.644051\pi$
$433$	−18.1976	−0.874521	−0.437261	+	0.899335i	$0.644051\pi$
$434$	−14.9815	−0.719133
$435$	0	0
$436$	−13.6619	−0.654288
$437$	4.63555	0.221749
$438$	0	0
$439$	20.4214	0.974660	0.487330	−	0.873218i	$-0.337971\pi$
$439$	20.4214	0.974660	0.487330	+	0.873218i	$0.337971\pi$
$440$	0	0
$441$	0	0
$442$	−3.50927	−0.166919
$443$	−21.6685	−1.02950	−0.514750	−	0.857340i	$-0.672115\pi$
$443$	−21.6685	−1.02950	−0.514750	+	0.857340i	$0.672115\pi$
$444$	0	0
$445$	0	0
$446$	15.6554	0.741305
$447$	0	0
$448$	2.46980	0.116687
$449$	−23.8133	−1.12382	−0.561910	−	0.827198i	$-0.689933\pi$
$449$	−23.8133	−1.12382	−0.561910	+	0.827198i	$0.689933\pi$
$450$	0	0
$451$	3.49489	0.164568
$452$	−1.20524	−0.0566895
$453$	0	0
$454$	4.80131	0.225337
$455$	0	0
$456$	0	0
$457$	9.15813	0.428399	0.214200	−	0.976790i	$-0.431286\pi$
$457$	9.15813	0.428399	0.214200	+	0.976790i	$0.431286\pi$
$458$	−2.79476	−0.130591
$459$	0	0
$460$	0	0
$461$	7.78931	0.362784	0.181392	−	0.983411i	$-0.441940\pi$
$461$	7.78931	0.362784	0.181392	+	0.983411i	$0.441940\pi$
$462$	0	0
$463$	0.405011	0.0188225	0.00941123	−	0.999956i	$-0.497004\pi$
$463$	0.405011	0.0188225	0.00941123	+	0.999956i	$0.497004\pi$
$464$	−10.2316	−0.474991
$465$	0	0
$466$	11.5422	0.534682
$467$	1.95814	0.0906118	0.0453059	−	0.998973i	$-0.485574\pi$
$467$	1.95814	0.0906118	0.0453059	+	0.998973i	$0.485574\pi$
$468$	0	0
$469$	31.9420	1.47494
$470$	0	0
$471$	0	0
$472$	−5.76183	−0.265210
$473$	−5.87918	−0.270325
$474$	0	0
$475$	0	0
$476$	−1.39084	−0.0637490
$477$	0	0
$478$	−26.2251	−1.19951
$479$	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
$482$	12.0659	0.549585
$483$	0	0
$484$	−10.4687	−0.475850
$485$	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	$0.682532\pi$
$488$	−10.9396	−0.495212
$489$	0	0
$490$	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
$494$	−6.23163	−0.280374
$495$	0	0
$496$	6.06587	0.272366
$497$	10.8606	0.487166
$498$	0	0
$499$	−7.92104	−0.354595	−0.177297	−	0.984157i	$-0.556735\pi$
$499$	−7.92104	−0.354595	−0.177297	+	0.984157i	$0.556735\pi$
$500$	0	0
$501$	0	0
$502$	13.5237	0.603591
$503$	16.6949	0.744388	0.372194	−	0.928155i	$-0.378606\pi$
$503$	16.6949	0.744388	0.372194	+	0.928155i	$0.378606\pi$
$504$	0	0
$505$	0	0
$506$	−3.37884	−0.150208
$507$	0	0
$508$	9.52366	0.422544
$509$	−24.4028	−1.08164	−0.540818	−	0.841139i	$-0.681885\pi$
$509$	−24.4028	−1.08164	−0.540818	+	0.841139i	$0.681885\pi$
$510$	0	0
$511$	−10.1097	−0.447228
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	22.7398	1.00301
$515$	0	0
$516$	0	0
$517$	5.92321	0.260503
$518$	14.1492	0.621680
$519$	0	0
$520$	0	0
$521$	−26.0659	−1.14197	−0.570983	−	0.820962i	$-0.693438\pi$
$521$	−26.0659	−1.14197	−0.570983	+	0.820962i	$0.693438\pi$
$522$	0	0
$523$	17.8726	0.781516	0.390758	−	0.920493i	$-0.372213\pi$
$523$	17.8726	0.781516	0.390758	+	0.920493i	$0.372213\pi$
$524$	17.4028	0.760247
$525$	0	0
$526$	−6.67395	−0.290998
$527$	−3.41593	−0.148800
$528$	0	0
$529$	−1.51166	−0.0657243
$530$	0	0
$531$	0	0
$532$	−2.46980	−0.107079
$533$	29.8792	1.29421
$534$	0	0
$535$	0	0
$536$	−12.9330	−0.558623
$537$	0	0
$538$	−29.7398	−1.28217
$539$	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	$0.334903\pi$
$542$	−1.11189	−0.0477598
$543$	0	0
$544$	0.563139	0.0241444
$545$	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	$0.702227\pi$
$548$	−2.25910	−0.0965040
$549$	0	0
$550$	0	0
$551$	10.2316	0.435882
$552$	0	0
$553$	37.8792	1.61079
$554$	−13.1263	−0.557682
$555$	0	0
$556$	16.8606	0.715050
$557$	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
$561$	0	0
$562$	22.9265	0.967096
$563$	−7.93959	−0.334614	−0.167307	−	0.985905i	$-0.553507\pi$
$563$	−7.93959	−0.334614	−0.167307	+	0.985905i	$0.553507\pi$
$564$	0	0
$565$	0	0
$566$	−0.860634	−0.0361751
$567$	0	0
$568$	−4.39738	−0.184510
$569$	1.65757	0.0694889	0.0347444	−	0.999396i	$-0.488938\pi$
$569$	1.65757	0.0694889	0.0347444	+	0.999396i	$0.488938\pi$
$570$	0	0
$571$	−14.0528	−0.588091	−0.294045	−	0.955791i	$-0.595002\pi$
$571$	−14.0528	−0.588091	−0.294045	+	0.955791i	$0.595002\pi$
$572$	4.54221	0.189919
$573$	0	0
$574$	11.8421	0.494279
$575$	0	0
$576$	0	0
$577$	−1.36445	−0.0568027	−0.0284014	−	0.999597i	$-0.509042\pi$
$577$	−1.36445	−0.0568027	−0.0284014	+	0.999597i	$0.509042\pi$
$578$	16.6829	0.693916
$579$	0	0
$580$	0	0
$581$	−19.4007	−0.804876
$582$	0	0
$583$	−1.11536	−0.0461935
$584$	4.09334	0.169384
$585$	0	0
$586$	19.8212	0.818804
$587$	24.6609	1.01786	0.508931	−	0.860807i	$-0.330041\pi$
$587$	24.6609	1.01786	0.508931	+	0.860807i	$0.330041\pi$
$588$	0	0
$589$	−6.06587	−0.249940
$590$	0	0
$591$	0	0
$592$	−5.72890	−0.235456
$593$	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
$596$	13.0055	0.532724
$597$	0	0
$598$	−28.8870	−1.18128
$599$	−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	$0.292506\pi$
$602$	−19.9210	−0.811921
$603$	0	0
$604$	17.5895	0.715708
$605$	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	$0.463301\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	50.6399	2.04867
$612$	0	0
$613$	−2.99454	−0.120948	−0.0604742	−	0.998170i	$-0.519261\pi$
$613$	−2.99454	−0.120948	−0.0604742	+	0.998170i	$0.519261\pi$
$614$	16.8661	0.680660
$615$	0	0
$616$	1.80022	0.0725331
$617$	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	$-0.638164\pi$
$619$	−20.9265	−0.841107	−0.420554	+	0.907268i	$0.638164\pi$
$620$	0	0
$621$	0	0
$622$	10.3250	0.413994
$623$	24.6980	0.989503
$624$	0	0
$625$	0	0
$626$	15.7684	0.630231
$627$	0	0
$628$	9.52366	0.380035
$629$	3.22617	0.128636
$630$	0	0
$631$	22.8002	0.907663	0.453831	−	0.891088i	$-0.350057\pi$
$631$	22.8002	0.907663	0.453831	+	0.891088i	$0.350057\pi$
$632$	−15.3370	−0.610072
$633$	0	0
$634$	−2.17230	−0.0862731
$635$	0	0
$636$	0	0
$637$	5.60916	0.222243
$638$	−7.45779	−0.295257
$639$	0	0
$640$	0	0
$641$	−36.3184	−1.43449	−0.717246	−	0.696820i	$-0.754598\pi$
$641$	−36.3184	−1.43449	−0.717246	+	0.696820i	$0.754598\pi$
$642$	0	0
$643$	13.6135	0.536865	0.268433	−	0.963298i	$-0.413494\pi$
$643$	13.6135	0.536865	0.268433	+	0.963298i	$0.413494\pi$
$644$	−11.4489	−0.451148
$645$	0	0
$646$	−0.563139	−0.0221564
$647$	0.0724126	0.00284683	0.00142342	−	0.999999i	$-0.499547\pi$
$647$	0.0724126	0.00284683	0.00142342	+	0.999999i	$0.499547\pi$
$648$	0	0
$649$	−4.19978	−0.164856
$650$	0	0
$651$	0	0
$652$	13.1921	0.516644
$653$	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
$656$	−4.79476	−0.187204
$657$	0	0
$658$	20.0702	0.782420
$659$	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	$0.517929\pi$
$662$	−11.9791	−0.465580
$663$	0	0
$664$	7.85517	0.304840
$665$	0	0
$666$	0	0
$667$	47.4292	1.83647
$668$	−1.81331	−0.0701591
$669$	0	0
$670$	0	0
$671$	−7.97382	−0.307826
$672$	0	0
$673$	42.8475	1.65165	0.825826	−	0.563925i	$-0.190709\pi$
$673$	42.8475	1.65165	0.825826	+	0.563925i	$0.190709\pi$
$674$	11.1921	0.431105
$675$	0	0
$676$	25.8332	0.993583
$677$	−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
$681$	0	0
$682$	4.42139	0.169304
$683$	2.73436	0.104627	0.0523136	−	0.998631i	$-0.483340\pi$
$683$	2.73436	0.104627	0.0523136	+	0.998631i	$0.483340\pi$
$684$	0	0
$685$	0	0
$686$	19.5117	0.744959
$687$	0	0
$688$	8.06587	0.307508
$689$	−9.53566	−0.363280
$690$	0	0
$691$	−4.74198	−0.180394	−0.0901968	−	0.995924i	$-0.528750\pi$
$691$	−4.74198	−0.180394	−0.0901968	+	0.995924i	$0.528750\pi$
$692$	12.5237	0.476078
$693$	0	0
$694$	−20.3843	−0.773777
$695$	0	0
$696$	0	0
$697$	2.70012	0.102274
$698$	0.252557	0.00955943
$699$	0	0
$700$	0	0
$701$	−33.1372	−1.25157	−0.625787	−	0.779994i	$-0.715222\pi$
$701$	−33.1372	−1.25157	−0.625787	+	0.779994i	$0.715222\pi$
$702$	0	0
$703$	5.72890	0.216069
$704$	−0.728896	−0.0274713
$705$	0	0
$706$	−28.6434	−1.07801
$707$	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
$711$	0	0
$712$	−10.0000	−0.374766
$713$	−28.1187	−1.05305
$714$	0	0
$715$	0	0
$716$	−1.39738	−0.0522226
$717$	0	0
$718$	−18.5741	−0.693178
$719$	−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
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	5.72890	0.212913
$725$	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	$0.484501\pi$
$728$	15.3908	0.570422
$729$	0	0
$730$	0	0
$731$	−4.54221	−0.168000
$732$	0	0
$733$	0.608077	0.0224598	0.0112299	−	0.999937i	$-0.496425\pi$
$733$	0.608077	0.0224598	0.0112299	+	0.999937i	$0.496425\pi$
$734$	4.79476	0.176978
$735$	0	0
$736$	4.63555	0.170869
$737$	−9.42685	−0.347242
$738$	0	0
$739$	36.3974	1.33890	0.669450	−	0.742857i	$-0.266530\pi$
$739$	36.3974	1.33890	0.669450	+	0.742857i	$0.266530\pi$
$740$	0	0
$741$	0	0
$742$	−3.77929	−0.138742
$743$	23.0713	0.846405	0.423202	−	0.906035i	$-0.360906\pi$
$743$	23.0713	0.846405	0.423202	+	0.906035i	$0.360906\pi$
$744$	0	0
$745$	0	0
$746$	−15.9485	−0.583917
$747$	0	0
$748$	0.410470	0.0150083
$749$	37.1121	1.35605
$750$	0	0
$751$	4.75290	0.173436	0.0867179	−	0.996233i	$-0.472362\pi$
$751$	4.75290	0.173436	0.0867179	+	0.996233i	$0.472362\pi$
$752$	−8.12628	−0.296335
$753$	0	0
$754$	−63.7597	−2.32199
$755$	0	0
$756$	0	0
$757$	26.8057	0.974269	0.487135	−	0.873327i	$-0.338042\pi$
$757$	26.8057	0.974269	0.487135	+	0.873327i	$0.338042\pi$
$758$	16.8013	0.610251
$759$	0	0
$760$	0	0
$761$	−25.4478	−0.922481	−0.461241	−	0.887275i	$-0.652596\pi$
$761$	−25.4478	−0.922481	−0.461241	+	0.887275i	$0.652596\pi$
$762$	0	0
$763$	−33.7422	−1.22155
$764$	27.0198	0.977544
$765$	0	0
$766$	26.7948	0.968134
$767$	−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
$771$	0	0
$772$	−23.0713	−0.830355
$773$	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
$776$	−11.0055	−0.395073
$777$	0	0
$778$	18.3424	0.657608
$779$	4.79476	0.171790
$780$	0	0
$781$	−3.20524	−0.114692
$782$	−2.61046	−0.0933499
$783$	0	0
$784$	−0.900112	−0.0321469
$785$	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	$0.391620\pi$
$788$	−0.794765	−0.0283123
$789$	0	0
$790$	0	0
$791$	−2.97668	−0.105839
$792$	0	0
$793$	−68.1714	−2.42084
$794$	−21.2162	−0.752933
$795$	0	0
$796$	8.07241	0.286119
$797$	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
$801$	0	0
$802$	−27.8661	−0.983986
$803$	2.98362	0.105290
$804$	0	0
$805$	0	0
$806$	37.8002	1.33146
$807$	0	0
$808$	−9.73436	−0.342453
$809$	−21.8857	−0.769461	−0.384731	−	0.923029i	$-0.625706\pi$
$809$	−21.8857	−0.769461	−0.384731	+	0.923029i	$0.625706\pi$
$810$	0	0
$811$	37.8595	1.32943	0.664714	−	0.747098i	$-0.268553\pi$
$811$	37.8595	1.32943	0.664714	+	0.747098i	$0.268553\pi$
$812$	−25.2700	−0.886804
$813$	0	0
$814$	−4.17577	−0.146361
$815$	0	0
$816$	0	0
$817$	−8.06587	−0.282189
$818$	18.0899	0.632498
$819$	0	0
$820$	0	0
$821$	20.1976	0.704901	0.352451	−	0.935830i	$-0.385348\pi$
$821$	20.1976	0.704901	0.352451	+	0.935830i	$0.385348\pi$
$822$	0	0
$823$	−4.34590	−0.151489	−0.0757443	−	0.997127i	$-0.524133\pi$
$823$	−4.34590	−0.151489	−0.0757443	+	0.997127i	$0.524133\pi$
$824$	8.33151	0.290242
$825$	0	0
$826$	−14.2305	−0.495144
$827$	56.8375	1.97643	0.988217	−	0.153057i	$-0.0489119\pi$
$827$	56.8375	1.97643	0.988217	+	0.153057i	$0.0489119\pi$
$828$	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
$831$	0	0
$832$	−6.23163	−0.216043
$833$	0.506888	0.0175626
$834$	0	0
$835$	0	0
$836$	0.728896	0.0252094
$837$	0	0
$838$	−15.3370	−0.529807
$839$	−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
$842$	8.42486	0.290340
$843$	0	0
$844$	−11.6410	−0.400700
$845$	0	0
$846$	0	0
$847$	−25.8556	−0.888408
$848$	1.53020	0.0525475
$849$	0	0
$850$	0	0
$851$	26.5566	0.910348
$852$	0	0
$853$	−16.1187	−0.551892	−0.275946	−	0.961173i	$-0.588991\pi$
$853$	−16.1187	−0.551892	−0.275946	+	0.961173i	$0.588991\pi$
$854$	−27.0185	−0.924556
$855$	0	0
$856$	−15.0264	−0.513591
$857$	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	$-0.280284\pi$
$859$	37.3239	1.27347	0.636737	+	0.771081i	$0.280284\pi$
$860$	0	0
$861$	0	0
$862$	16.2766	0.554382
$863$	1.33697	0.0455111	0.0227555	−	0.999741i	$-0.492756\pi$
$863$	1.33697	0.0455111	0.0227555	+	0.999741i	$0.492756\pi$
$864$	0	0
$865$	0	0
$866$	18.1976	0.618380
$867$	0	0
$868$	14.9815	0.508504
$869$	−11.1791	−0.379224
$870$	0	0
$871$	−80.5939	−2.73082
$872$	13.6619	0.462652
$873$	0	0
$874$	−4.63555	−0.156800
$875$	0	0
$876$	0	0
$877$	13.1647	0.444539	0.222270	−	0.974985i	$-0.428653\pi$
$877$	13.1647	0.444539	0.222270	+	0.974985i	$0.428653\pi$
$878$	−20.4214	−0.689188
$879$	0	0
$880$	0	0
$881$	10.2052	0.343823	0.171912	−	0.985112i	$-0.445006\pi$
$881$	10.2052	0.343823	0.171912	+	0.985112i	$0.445006\pi$
$882$	0	0
$883$	1.49966	0.0504674	0.0252337	−	0.999682i	$-0.491967\pi$
$883$	1.49966	0.0504674	0.0252337	+	0.999682i	$0.491967\pi$
$884$	3.50927	0.118030
$885$	0	0
$886$	21.6685	0.727967
$887$	46.9505	1.57644	0.788222	−	0.615391i	$-0.211002\pi$
$887$	46.9505	1.57644	0.788222	+	0.615391i	$0.211002\pi$
$888$	0	0
$889$	23.5215	0.788886
$890$	0	0
$891$	0	0
$892$	−15.6554	−0.524182
$893$	8.12628	0.271936
$894$	0	0
$895$	0	0
$896$	−2.46980	−0.0825101
$897$	0	0
$898$	23.8133	0.794661
$899$	−62.0637	−2.06994
$900$	0	0
$901$	−0.861719	−0.0287080
$902$	−3.49489	−0.116367
$903$	0	0
$904$	1.20524	0.0400855
$905$	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	$0.412406\pi$
$908$	−4.80131	−0.159337
$909$	0	0
$910$	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
$914$	−9.15813	−0.302924
$915$	0	0
$916$	2.79476	0.0923416
$917$	42.9815	1.41937
$918$	0	0
$919$	−22.8157	−0.752620	−0.376310	−	0.926494i	$-0.622807\pi$
$919$	−22.8157	−0.752620	−0.376310	+	0.926494i	$0.622807\pi$
$920$	0	0
$921$	0	0
$922$	−7.78931	−0.256527
$923$	−27.4028	−0.901976
$924$	0	0
$925$	0	0
$926$	−0.405011	−0.0133095
$927$	0	0
$928$	10.2316	0.335870
$929$	27.2436	0.893834	0.446917	−	0.894575i	$-0.352522\pi$
$929$	27.2436	0.893834	0.446917	+	0.894575i	$0.352522\pi$
$930$	0	0
$931$	0.900112	0.0295000
$932$	−11.5422	−0.378078
$933$	0	0
$934$	−1.95814	−0.0640722
$935$	0	0
$936$	0	0
$937$	−51.5006	−1.68245	−0.841225	−	0.540685i	$-0.818165\pi$
$937$	−51.5006	−1.68245	−0.841225	+	0.540685i	$0.818165\pi$
$938$	−31.9420	−1.04294
$939$	0	0
$940$	0	0
$941$	40.8541	1.33181	0.665903	−	0.746039i	$-0.268047\pi$
$941$	40.8541	1.33181	0.665903	+	0.746039i	$0.268047\pi$
$942$	0	0
$943$	22.2264	0.723791
$944$	5.76183	0.187532
$945$	0	0
$946$	5.87918	0.191149
$947$	−38.2526	−1.24304	−0.621521	−	0.783398i	$-0.713485\pi$
$947$	−38.2526	−1.24304	−0.621521	+	0.783398i	$0.713485\pi$
$948$	0	0
$949$	25.5082	0.828031
$950$	0	0
$951$	0	0
$952$	1.39084	0.0450773
$953$	−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
$956$	26.2251	0.848180
$957$	0	0
$958$	−22.9210	−0.740545
$959$	−5.57952	−0.180172
$960$	0	0
$961$	5.79476	0.186928
$962$	−35.7003	−1.15103
$963$	0	0
$964$	−12.0659	−0.388615
$965$	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	$0.348929\pi$
$968$	10.4687	0.336477
$969$	0	0
$970$	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
$974$	23.9450	0.767249
$975$	0	0
$976$	10.9396	0.350168
$977$	−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
$981$	0	0
$982$	3.86826	0.123441
$983$	38.7948	1.23736	0.618680	−	0.785643i	$-0.287668\pi$
$983$	38.7948	1.23736	0.618680	+	0.785643i	$0.287668\pi$
$984$	0	0
$985$	0	0
$986$	−5.76183	−0.183494
$987$	0	0
$988$	6.23163	0.198254
$989$	−37.3898	−1.18893
$990$	0	0
$991$	−15.0295	−0.477427	−0.238713	−	0.971090i	$-0.576726\pi$
$991$	−15.0295	−0.477427	−0.238713	+	0.971090i	$0.576726\pi$
$992$	−6.06587	−0.192592
$993$	0	0
$994$	−10.8606	−0.344478
$995$	0	0
$996$	0	0
$997$	1.18123	0.0374099	0.0187050	−	0.999825i	$-0.494046\pi$
$997$	1.18123	0.0374099	0.0187050	+	0.999825i	$0.494046\pi$
$998$	7.92104	0.250736
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.k.1.3	✓	3	15.14	odd	2
950.2.a.m.1.1	yes	3	3.2	odd	2
950.2.b.g.799.1		6	15.8	even	4
950.2.b.g.799.6		6	15.2	even	4
7600.2.a.bm.1.3		3	12.11	even	2
7600.2.a.cb.1.1		3	60.59	even	2
8550.2.a.cj.1.2		3	1.1	even	1	trivial
8550.2.a.co.1.2		3	5.4	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.46980 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.46980 q^{7} -1.00000 q^{8} -0.728896 q^{11} -6.23163 q^{13} -2.46980 q^{14} +1.00000 q^{16} -0.563139 q^{17} -1.00000 q^{19} +0.728896 q^{22} -4.63555 q^{23} +6.23163 q^{26} +2.46980 q^{28} -10.2316 q^{29} +6.06587 q^{31} -1.00000 q^{32} +0.563139 q^{34} -5.72890 q^{37} +1.00000 q^{38} -4.79476 q^{41} +8.06587 q^{43} -0.728896 q^{44} +4.63555 q^{46} -8.12628 q^{47} -0.900112 q^{49} -6.23163 q^{52} +1.53020 q^{53} -2.46980 q^{56} +10.2316 q^{58} +5.76183 q^{59} +10.9396 q^{61} -6.06587 q^{62} +1.00000 q^{64} +12.9330 q^{67} -0.563139 q^{68} +4.39738 q^{71} -4.09334 q^{73} +5.72890 q^{74} -1.00000 q^{76} -1.80022 q^{77} +15.3370 q^{79} +4.79476 q^{82} -7.85517 q^{83} -8.06587 q^{86} +0.728896 q^{88} +10.0000 q^{89} -15.3908 q^{91} -4.63555 q^{92} +8.12628 q^{94} +11.0055 q^{97} +0.900112 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 2 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 2 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 2 q^{7} - 3 q^{8} - 2 q^{11} - 2 q^{13} - 2 q^{14} + 3 q^{16} + 4 q^{17} - 3 q^{19} + 2 q^{22} - 14 q^{23} + 2 q^{26} + 2 q^{28} - 14 q^{29} - 4 q^{31} - 3 q^{32} - 4 q^{34} - 17 q^{37} + 3 q^{38} + 8 q^{41} + 2 q^{43} - 2 q^{44} + 14 q^{46} - 13 q^{47} + 25 q^{49} - 2 q^{52} + 10 q^{53} - 2 q^{56} + 14 q^{58} + 6 q^{59} + 22 q^{61} + 4 q^{62} + 3 q^{64} + 4 q^{68} + 2 q^{71} - 12 q^{73} + 17 q^{74} - 3 q^{76} + 50 q^{77} + 24 q^{79} - 8 q^{82} - 12 q^{83} - 2 q^{86} + 2 q^{88} + 30 q^{89} - 7 q^{91} - 14 q^{92} + 13 q^{94} - 25 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 2 * q^7 - 3 * q^8 - 2 * q^11 - 2 * q^13 - 2 * q^14 + 3 * q^16 + 4 * q^17 - 3 * q^19 + 2 * q^22 - 14 * q^23 + 2 * q^26 + 2 * q^28 - 14 * q^29 - 4 * q^31 - 3 * q^32 - 4 * q^34 - 17 * q^37 + 3 * q^38 + 8 * q^41 + 2 * q^43 - 2 * q^44 + 14 * q^46 - 13 * q^47 + 25 * q^49 - 2 * q^52 + 10 * q^53 - 2 * q^56 + 14 * q^58 + 6 * q^59 + 22 * q^61 + 4 * q^62 + 3 * q^64 + 4 * q^68 + 2 * q^71 - 12 * q^73 + 17 * q^74 - 3 * q^76 + 50 * q^77 + 24 * q^79 - 8 * q^82 - 12 * q^83 - 2 * q^86 + 2 * q^88 + 30 * q^89 - 7 * q^91 - 14 * q^92 + 13 * q^94 - 25 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more