LMFDB - Embedded newform 1288.2.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1288,2,Mod(1,1288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1288, 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("1288.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1288 = 2^{3} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1288.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:	$10.2847317803$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	1288.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21432 q^{3} +3.59210 q^{5} -1.00000 q^{7} +1.90321 q^{9} +O(q^{10})$	$q+2.21432 q^{3} +3.59210 q^{5} -1.00000 q^{7} +1.90321 q^{9} -3.52543 q^{11} -0.622216 q^{13} +7.95407 q^{15} +6.83654 q^{17} -2.21432 q^{21} +1.00000 q^{23} +7.90321 q^{25} -2.42864 q^{27} +1.09679 q^{29} +5.45875 q^{31} -7.80642 q^{33} -3.59210 q^{35} +7.37778 q^{37} -1.37778 q^{39} +9.61285 q^{41} +6.23506 q^{43} +6.83654 q^{45} +2.83654 q^{47} +1.00000 q^{49} +15.1383 q^{51} -13.6128 q^{53} -12.6637 q^{55} -13.2652 q^{59} -14.2558 q^{61} -1.90321 q^{63} -2.23506 q^{65} -4.76986 q^{67} +2.21432 q^{69} -8.85728 q^{71} -11.2859 q^{73} +17.5002 q^{75} +3.52543 q^{77} +6.28100 q^{79} -11.0874 q^{81} -2.62222 q^{83} +24.5575 q^{85} +2.42864 q^{87} -5.07160 q^{89} +0.622216 q^{91} +12.0874 q^{93} -17.3067 q^{97} -6.70964 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{5} - 3 q^{7} - q^{9}+O(q^{10}) $ 3 * q + 4 * q^5 - 3 * q^7 - q^9	$ 3 q + 4 q^{5} - 3 q^{7} - q^{9} - 4 q^{11} - 2 q^{13} + 4 q^{15} + 14 q^{17} + 3 q^{23} + 17 q^{25} + 6 q^{27} + 10 q^{29} + 10 q^{31} - 10 q^{33} - 4 q^{35} + 22 q^{37} - 4 q^{39} + 2 q^{41} - 8 q^{43} + 14 q^{45} + 2 q^{47} + 3 q^{49} + 12 q^{51} - 14 q^{53} + 2 q^{55} - 20 q^{59} + 4 q^{61} + q^{63} + 20 q^{65} - 8 q^{67} + 6 q^{73} + 6 q^{75} + 4 q^{77} + 12 q^{79} - 13 q^{81} - 8 q^{83} - 6 q^{87} + 18 q^{89} + 2 q^{91} + 16 q^{93} + 8 q^{97}+O(q^{100}) $ 3 * q + 4 * q^5 - 3 * q^7 - q^9 - 4 * q^11 - 2 * q^13 + 4 * q^15 + 14 * q^17 + 3 * q^23 + 17 * q^25 + 6 * q^27 + 10 * q^29 + 10 * q^31 - 10 * q^33 - 4 * q^35 + 22 * q^37 - 4 * q^39 + 2 * q^41 - 8 * q^43 + 14 * q^45 + 2 * q^47 + 3 * q^49 + 12 * q^51 - 14 * q^53 + 2 * q^55 - 20 * q^59 + 4 * q^61 + q^63 + 20 * q^65 - 8 * q^67 + 6 * q^73 + 6 * q^75 + 4 * q^77 + 12 * q^79 - 13 * q^81 - 8 * q^83 - 6 * q^87 + 18 * q^89 + 2 * q^91 + 16 * q^93 + 8 * q^97

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
$3$	2.21432	1.27844	0.639219	−	0.769025i	$-0.279258\pi$
$3$	2.21432	1.27844	0.639219	+	0.769025i	$0.279258\pi$
$4$	0	0
$5$	3.59210	1.60644	0.803219	−	0.595684i	$-0.203119\pi$
$5$	3.59210	1.60644	0.803219	+	0.595684i	$0.203119\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.90321	0.634404
$10$	0	0
$11$	−3.52543	−1.06296	−0.531478	−	0.847072i	$-0.678363\pi$
$11$	−3.52543	−1.06296	−0.531478	+	0.847072i	$0.678363\pi$
$12$	0	0
$13$	−0.622216	−0.172572	−0.0862858	−	0.996270i	$-0.527500\pi$
$13$	−0.622216	−0.172572	−0.0862858	+	0.996270i	$0.527500\pi$
$14$	0	0
$15$	7.95407	2.05373
$16$	0	0
$17$	6.83654	1.65810	0.829052	−	0.559172i	$-0.188881\pi$
$17$	6.83654	1.65810	0.829052	+	0.559172i	$0.188881\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.21432	−0.483204
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	7.90321	1.58064
$26$	0	0
$27$	−2.42864	−0.467392
$28$	0	0
$29$	1.09679	0.203668	0.101834	−	0.994801i	$-0.467529\pi$
$29$	1.09679	0.203668	0.101834	+	0.994801i	$0.467529\pi$
$30$	0	0
$31$	5.45875	0.980421	0.490210	−	0.871604i	$-0.336920\pi$
$31$	5.45875	0.980421	0.490210	+	0.871604i	$0.336920\pi$
$32$	0	0
$33$	−7.80642	−1.35892
$34$	0	0
$35$	−3.59210	−0.607176
$36$	0	0
$37$	7.37778	1.21290	0.606450	−	0.795122i	$-0.292593\pi$
$37$	7.37778	1.21290	0.606450	+	0.795122i	$0.292593\pi$
$38$	0	0
$39$	−1.37778	−0.220622
$40$	0	0
$41$	9.61285	1.50127	0.750637	−	0.660715i	$-0.229747\pi$
$41$	9.61285	1.50127	0.750637	+	0.660715i	$0.229747\pi$
$42$	0	0
$43$	6.23506	0.950838	0.475419	−	0.879759i	$-0.342296\pi$
$43$	6.23506	0.950838	0.475419	+	0.879759i	$0.342296\pi$
$44$	0	0
$45$	6.83654	1.01913
$46$	0	0
$47$	2.83654	0.413751	0.206876	−	0.978367i	$-0.433670\pi$
$47$	2.83654	0.413751	0.206876	+	0.978367i	$0.433670\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	15.1383	2.11978
$52$	0	0
$53$	−13.6128	−1.86987	−0.934934	−	0.354821i	$-0.884542\pi$
$53$	−13.6128	−1.86987	−0.934934	+	0.354821i	$0.884542\pi$
$54$	0	0
$55$	−12.6637	−1.70757
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.2652	−1.72698	−0.863489	−	0.504367i	$-0.831726\pi$
$59$	−13.2652	−1.72698	−0.863489	+	0.504367i	$0.831726\pi$
$60$	0	0
$61$	−14.2558	−1.82527	−0.912635	−	0.408777i	$-0.865956\pi$
$61$	−14.2558	−1.82527	−0.912635	+	0.408777i	$0.865956\pi$
$62$	0	0
$63$	−1.90321	−0.239782
$64$	0	0
$65$	−2.23506	−0.277225
$66$	0	0
$67$	−4.76986	−0.582731	−0.291366	−	0.956612i	$-0.594110\pi$
$67$	−4.76986	−0.582731	−0.291366	+	0.956612i	$0.594110\pi$
$68$	0	0
$69$	2.21432	0.266573
$70$	0	0
$71$	−8.85728	−1.05117	−0.525583	−	0.850742i	$-0.676153\pi$
$71$	−8.85728	−1.05117	−0.525583	+	0.850742i	$0.676153\pi$
$72$	0	0
$73$	−11.2859	−1.32092	−0.660458	−	0.750863i	$-0.729638\pi$
$73$	−11.2859	−1.32092	−0.660458	+	0.750863i	$0.729638\pi$
$74$	0	0
$75$	17.5002	2.02075
$76$	0	0
$77$	3.52543	0.401760
$78$	0	0
$79$	6.28100	0.706667	0.353334	−	0.935497i	$-0.385048\pi$
$79$	6.28100	0.706667	0.353334	+	0.935497i	$0.385048\pi$
$80$	0	0
$81$	−11.0874	−1.23194
$82$	0	0
$83$	−2.62222	−0.287826	−0.143913	−	0.989590i	$-0.545968\pi$
$83$	−2.62222	−0.287826	−0.143913	+	0.989590i	$0.545968\pi$
$84$	0	0
$85$	24.5575	2.66364
$86$	0	0
$87$	2.42864	0.260377
$88$	0	0
$89$	−5.07160	−0.537588	−0.268794	−	0.963198i	$-0.586625\pi$
$89$	−5.07160	−0.537588	−0.268794	+	0.963198i	$0.586625\pi$
$90$	0	0
$91$	0.622216	0.0652259
$92$	0	0
$93$	12.0874	1.25341
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−17.3067	−1.75723	−0.878613	−	0.477535i	$-0.841530\pi$
$97$	−17.3067	−1.75723	−0.878613	+	0.477535i	$0.841530\pi$
$98$	0	0
$99$	−6.70964	−0.674344
$100$	0	0
$101$	−1.18421	−0.117833	−0.0589166	−	0.998263i	$-0.518765\pi$
$101$	−1.18421	−0.117833	−0.0589166	+	0.998263i	$0.518765\pi$
$102$	0	0
$103$	−1.24443	−0.122617	−0.0613087	−	0.998119i	$-0.519527\pi$
$103$	−1.24443	−0.122617	−0.0613087	+	0.998119i	$0.519527\pi$
$104$	0	0
$105$	−7.95407	−0.776237
$106$	0	0
$107$	−15.0923	−1.45903	−0.729516	−	0.683964i	$-0.760255\pi$
$107$	−15.0923	−1.45903	−0.729516	+	0.683964i	$0.760255\pi$
$108$	0	0
$109$	12.5303	1.20019	0.600095	−	0.799929i	$-0.295129\pi$
$109$	12.5303	1.20019	0.600095	+	0.799929i	$0.295129\pi$
$110$	0	0
$111$	16.3368	1.55062
$112$	0	0
$113$	6.85728	0.645079	0.322539	−	0.946556i	$-0.395464\pi$
$113$	6.85728	0.645079	0.322539	+	0.946556i	$0.395464\pi$
$114$	0	0
$115$	3.59210	0.334965
$116$	0	0
$117$	−1.18421	−0.109480
$118$	0	0
$119$	−6.83654	−0.626704
$120$	0	0
$121$	1.42864	0.129876
$122$	0	0
$123$	21.2859	1.91929
$124$	0	0
$125$	10.4286	0.932766
$126$	0	0
$127$	−5.37778	−0.477201	−0.238601	−	0.971118i	$-0.576689\pi$
$127$	−5.37778	−0.477201	−0.238601	+	0.971118i	$0.576689\pi$
$128$	0	0
$129$	13.8064	1.21559
$130$	0	0
$131$	−3.82717	−0.334381	−0.167191	−	0.985925i	$-0.553470\pi$
$131$	−3.82717	−0.334381	−0.167191	+	0.985925i	$0.553470\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−8.72393	−0.750836
$136$	0	0
$137$	−2.13335	−0.182265	−0.0911323	−	0.995839i	$-0.529049\pi$
$137$	−2.13335	−0.182265	−0.0911323	+	0.995839i	$0.529049\pi$
$138$	0	0
$139$	−5.26517	−0.446586	−0.223293	−	0.974751i	$-0.571681\pi$
$139$	−5.26517	−0.446586	−0.223293	+	0.974751i	$0.571681\pi$
$140$	0	0
$141$	6.28100	0.528955
$142$	0	0
$143$	2.19358	0.183436
$144$	0	0
$145$	3.93978	0.327181
$146$	0	0
$147$	2.21432	0.182634
$148$	0	0
$149$	22.4701	1.84082	0.920412	−	0.390949i	$-0.127853\pi$
$149$	22.4701	1.84082	0.920412	+	0.390949i	$0.127853\pi$
$150$	0	0
$151$	7.74620	0.630377	0.315188	−	0.949029i	$-0.397932\pi$
$151$	7.74620	0.630377	0.315188	+	0.949029i	$0.397932\pi$
$152$	0	0
$153$	13.0114	1.05191
$154$	0	0
$155$	19.6084	1.57498
$156$	0	0
$157$	0.214320	0.0171046	0.00855229	−	0.999963i	$-0.497278\pi$
$157$	0.214320	0.0171046	0.00855229	+	0.999963i	$0.497278\pi$
$158$	0	0
$159$	−30.1432	−2.39051
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−5.53972	−0.433904	−0.216952	−	0.976182i	$-0.569612\pi$
$163$	−5.53972	−0.433904	−0.216952	+	0.976182i	$0.569612\pi$
$164$	0	0
$165$	−28.0415	−2.18303
$166$	0	0
$167$	−21.1131	−1.63378	−0.816890	−	0.576794i	$-0.804304\pi$
$167$	−21.1131	−1.63378	−0.816890	+	0.576794i	$0.804304\pi$
$168$	0	0
$169$	−12.6128	−0.970219
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.2444	1.15901	0.579506	−	0.814968i	$-0.303245\pi$
$173$	15.2444	1.15901	0.579506	+	0.814968i	$0.303245\pi$
$174$	0	0
$175$	−7.90321	−0.597427
$176$	0	0
$177$	−29.3733	−2.20784
$178$	0	0
$179$	7.47949	0.559044	0.279522	−	0.960139i	$-0.409824\pi$
$179$	7.47949	0.559044	0.279522	+	0.960139i	$0.409824\pi$
$180$	0	0
$181$	14.0622	1.04524	0.522619	−	0.852567i	$-0.324955\pi$
$181$	14.0622	1.04524	0.522619	+	0.852567i	$0.324955\pi$
$182$	0	0
$183$	−31.5669	−2.33349
$184$	0	0
$185$	26.5018	1.94845
$186$	0	0
$187$	−24.1017	−1.76249
$188$	0	0
$189$	2.42864	0.176658
$190$	0	0
$191$	−4.13335	−0.299079	−0.149539	−	0.988756i	$-0.547779\pi$
$191$	−4.13335	−0.299079	−0.149539	+	0.988756i	$0.547779\pi$
$192$	0	0
$193$	11.6731	0.840246	0.420123	−	0.907467i	$-0.361987\pi$
$193$	11.6731	0.840246	0.420123	+	0.907467i	$0.361987\pi$
$194$	0	0
$195$	−4.94914	−0.354416
$196$	0	0
$197$	4.10171	0.292235	0.146117	−	0.989267i	$-0.453322\pi$
$197$	4.10171	0.292235	0.146117	+	0.989267i	$0.453322\pi$
$198$	0	0
$199$	5.24443	0.371768	0.185884	−	0.982572i	$-0.440485\pi$
$199$	5.24443	0.371768	0.185884	+	0.982572i	$0.440485\pi$
$200$	0	0
$201$	−10.5620	−0.744986
$202$	0	0
$203$	−1.09679	−0.0769794
$204$	0	0
$205$	34.5303	2.41170
$206$	0	0
$207$	1.90321	0.132282
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.1334	−1.11067	−0.555333	−	0.831628i	$-0.687409\pi$
$211$	−16.1334	−1.11067	−0.555333	+	0.831628i	$0.687409\pi$
$212$	0	0
$213$	−19.6128	−1.34385
$214$	0	0
$215$	22.3970	1.52746
$216$	0	0
$217$	−5.45875	−0.370564
$218$	0	0
$219$	−24.9906	−1.68871
$220$	0	0
$221$	−4.25380	−0.286141
$222$	0	0
$223$	13.2652	0.888302	0.444151	−	0.895952i	$-0.353505\pi$
$223$	13.2652	0.888302	0.444151	+	0.895952i	$0.353505\pi$
$224$	0	0
$225$	15.0415	1.00277
$226$	0	0
$227$	15.0509	0.998960	0.499480	−	0.866325i	$-0.333524\pi$
$227$	15.0509	0.998960	0.499480	+	0.866325i	$0.333524\pi$
$228$	0	0
$229$	−13.3383	−0.881420	−0.440710	−	0.897650i	$-0.645273\pi$
$229$	−13.3383	−0.881420	−0.440710	+	0.897650i	$0.645273\pi$
$230$	0	0
$231$	7.80642	0.513625
$232$	0	0
$233$	−1.65878	−0.108670	−0.0543352	−	0.998523i	$-0.517304\pi$
$233$	−1.65878	−0.108670	−0.0543352	+	0.998523i	$0.517304\pi$
$234$	0	0
$235$	10.1891	0.664666
$236$	0	0
$237$	13.9081	0.903430
$238$	0	0
$239$	4.13335	0.267364	0.133682	−	0.991024i	$-0.457320\pi$
$239$	4.13335	0.267364	0.133682	+	0.991024i	$0.457320\pi$
$240$	0	0
$241$	7.69381	0.495602	0.247801	−	0.968811i	$-0.420292\pi$
$241$	7.69381	0.495602	0.247801	+	0.968811i	$0.420292\pi$
$242$	0	0
$243$	−17.2652	−1.10756
$244$	0	0
$245$	3.59210	0.229491
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−5.80642	−0.367967
$250$	0	0
$251$	−21.9813	−1.38745	−0.693723	−	0.720242i	$-0.744031\pi$
$251$	−21.9813	−1.38745	−0.693723	+	0.720242i	$0.744031\pi$
$252$	0	0
$253$	−3.52543	−0.221642
$254$	0	0
$255$	54.3783	3.40530
$256$	0	0
$257$	−12.1017	−0.754884	−0.377442	−	0.926033i	$-0.623196\pi$
$257$	−12.1017	−0.754884	−0.377442	+	0.926033i	$0.623196\pi$
$258$	0	0
$259$	−7.37778	−0.458433
$260$	0	0
$261$	2.08742	0.129208
$262$	0	0
$263$	8.20787	0.506119	0.253059	−	0.967451i	$-0.418563\pi$
$263$	8.20787	0.506119	0.253059	+	0.967451i	$0.418563\pi$
$264$	0	0
$265$	−48.8988	−3.00383
$266$	0	0
$267$	−11.2301	−0.687273
$268$	0	0
$269$	9.61285	0.586106	0.293053	−	0.956096i	$-0.405329\pi$
$269$	9.61285	0.586106	0.293053	+	0.956096i	$0.405329\pi$
$270$	0	0
$271$	27.6751	1.68114	0.840571	−	0.541702i	$-0.182220\pi$
$271$	27.6751	1.68114	0.840571	+	0.541702i	$0.182220\pi$
$272$	0	0
$273$	1.37778	0.0833873
$274$	0	0
$275$	−27.8622	−1.68015
$276$	0	0
$277$	−24.0558	−1.44537	−0.722686	−	0.691177i	$-0.757093\pi$
$277$	−24.0558	−1.44537	−0.722686	+	0.691177i	$0.757093\pi$
$278$	0	0
$279$	10.3892	0.621983
$280$	0	0
$281$	24.1432	1.44026	0.720131	−	0.693838i	$-0.244082\pi$
$281$	24.1432	1.44026	0.720131	+	0.693838i	$0.244082\pi$
$282$	0	0
$283$	5.15257	0.306288	0.153144	−	0.988204i	$-0.451060\pi$
$283$	5.15257	0.306288	0.153144	+	0.988204i	$0.451060\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.61285	−0.567428
$288$	0	0
$289$	29.7382	1.74931
$290$	0	0
$291$	−38.3225	−2.24650
$292$	0	0
$293$	−22.6844	−1.32524	−0.662620	−	0.748956i	$-0.730555\pi$
$293$	−22.6844	−1.32524	−0.662620	+	0.748956i	$0.730555\pi$
$294$	0	0
$295$	−47.6499	−2.77428
$296$	0	0
$297$	8.56199	0.496817
$298$	0	0
$299$	−0.622216	−0.0359837
$300$	0	0
$301$	−6.23506	−0.359383
$302$	0	0
$303$	−2.62222	−0.150642
$304$	0	0
$305$	−51.2083	−2.93218
$306$	0	0
$307$	−30.4494	−1.73784	−0.868919	−	0.494954i	$-0.835185\pi$
$307$	−30.4494	−1.73784	−0.868919	+	0.494954i	$0.835185\pi$
$308$	0	0
$309$	−2.75557	−0.156759
$310$	0	0
$311$	−1.10324	−0.0625591	−0.0312795	−	0.999511i	$-0.509958\pi$
$311$	−1.10324	−0.0625591	−0.0312795	+	0.999511i	$0.509958\pi$
$312$	0	0
$313$	25.3798	1.43455	0.717275	−	0.696790i	$-0.245389\pi$
$313$	25.3798	1.43455	0.717275	+	0.696790i	$0.245389\pi$
$314$	0	0
$315$	−6.83654	−0.385195
$316$	0	0
$317$	22.9862	1.29103	0.645516	−	0.763746i	$-0.276642\pi$
$317$	22.9862	1.29103	0.645516	+	0.763746i	$0.276642\pi$
$318$	0	0
$319$	−3.86665	−0.216491
$320$	0	0
$321$	−33.4193	−1.86528
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.91750	−0.272774
$326$	0	0
$327$	27.7462	1.53437
$328$	0	0
$329$	−2.83654	−0.156383
$330$	0	0
$331$	−28.0415	−1.54130	−0.770650	−	0.637259i	$-0.780068\pi$
$331$	−28.0415	−1.54130	−0.770650	+	0.637259i	$0.780068\pi$
$332$	0	0
$333$	14.0415	0.769469
$334$	0	0
$335$	−17.1338	−0.936121
$336$	0	0
$337$	5.95851	0.324581	0.162290	−	0.986743i	$-0.448112\pi$
$337$	5.95851	0.324581	0.162290	+	0.986743i	$0.448112\pi$
$338$	0	0
$339$	15.1842	0.824693
$340$	0	0
$341$	−19.2444	−1.04214
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.95407	0.428233
$346$	0	0
$347$	35.3590	1.89817	0.949087	−	0.315015i	$-0.102010\pi$
$347$	35.3590	1.89817	0.949087	+	0.315015i	$0.102010\pi$
$348$	0	0
$349$	−4.46028	−0.238753	−0.119377	−	0.992849i	$-0.538090\pi$
$349$	−4.46028	−0.238753	−0.119377	+	0.992849i	$0.538090\pi$
$350$	0	0
$351$	1.51114	0.0806586
$352$	0	0
$353$	−2.33677	−0.124374	−0.0621870	−	0.998065i	$-0.519808\pi$
$353$	−2.33677	−0.124374	−0.0621870	+	0.998065i	$0.519808\pi$
$354$	0	0
$355$	−31.8163	−1.68863
$356$	0	0
$357$	−15.1383	−0.801202
$358$	0	0
$359$	−7.52543	−0.397177	−0.198588	−	0.980083i	$-0.563636\pi$
$359$	−7.52543	−0.397177	−0.198588	+	0.980083i	$0.563636\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	3.16346	0.166039
$364$	0	0
$365$	−40.5402	−2.12197
$366$	0	0
$367$	26.6222	1.38967	0.694834	−	0.719170i	$-0.255478\pi$
$367$	26.6222	1.38967	0.694834	+	0.719170i	$0.255478\pi$
$368$	0	0
$369$	18.2953	0.952415
$370$	0	0
$371$	13.6128	0.706744
$372$	0	0
$373$	9.73329	0.503971	0.251985	−	0.967731i	$-0.418916\pi$
$373$	9.73329	0.503971	0.251985	+	0.967731i	$0.418916\pi$
$374$	0	0
$375$	23.0923	1.19248
$376$	0	0
$377$	−0.682439	−0.0351474
$378$	0	0
$379$	9.92242	0.509681	0.254840	−	0.966983i	$-0.417977\pi$
$379$	9.92242	0.509681	0.254840	+	0.966983i	$0.417977\pi$
$380$	0	0
$381$	−11.9081	−0.610072
$382$	0	0
$383$	34.4385	1.75972	0.879862	−	0.475229i	$-0.157635\pi$
$383$	34.4385	1.75972	0.879862	+	0.475229i	$0.157635\pi$
$384$	0	0
$385$	12.6637	0.645402
$386$	0	0
$387$	11.8666	0.603216
$388$	0	0
$389$	34.8988	1.76944	0.884719	−	0.466125i	$-0.154350\pi$
$389$	34.8988	1.76944	0.884719	+	0.466125i	$0.154350\pi$
$390$	0	0
$391$	6.83654	0.345738
$392$	0	0
$393$	−8.47457	−0.427486
$394$	0	0
$395$	22.5620	1.13522
$396$	0	0
$397$	−11.5812	−0.581244	−0.290622	−	0.956838i	$-0.593862\pi$
$397$	−11.5812	−0.581244	−0.290622	+	0.956838i	$0.593862\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.79706	−0.439304	−0.219652	−	0.975578i	$-0.570492\pi$
$401$	−8.79706	−0.439304	−0.219652	+	0.975578i	$0.570492\pi$
$402$	0	0
$403$	−3.39652	−0.169193
$404$	0	0
$405$	−39.8272	−1.97903
$406$	0	0
$407$	−26.0098	−1.28926
$408$	0	0
$409$	4.23506	0.209410	0.104705	−	0.994503i	$-0.466610\pi$
$409$	4.23506	0.209410	0.104705	+	0.994503i	$0.466610\pi$
$410$	0	0
$411$	−4.72393	−0.233014
$412$	0	0
$413$	13.2652	0.652737
$414$	0	0
$415$	−9.41927	−0.462374
$416$	0	0
$417$	−11.6588	−0.570933
$418$	0	0
$419$	19.4924	0.952266	0.476133	−	0.879373i	$-0.342038\pi$
$419$	19.4924	0.952266	0.476133	+	0.879373i	$0.342038\pi$
$420$	0	0
$421$	−21.8796	−1.06634	−0.533172	−	0.846007i	$-0.679000\pi$
$421$	−21.8796	−1.06634	−0.533172	+	0.846007i	$0.679000\pi$
$422$	0	0
$423$	5.39853	0.262485
$424$	0	0
$425$	54.0306	2.62087
$426$	0	0
$427$	14.2558	0.689887
$428$	0	0
$429$	4.85728	0.234512
$430$	0	0
$431$	−11.3461	−0.546524	−0.273262	−	0.961940i	$-0.588103\pi$
$431$	−11.3461	−0.546524	−0.273262	+	0.961940i	$0.588103\pi$
$432$	0	0
$433$	−15.2968	−0.735118	−0.367559	−	0.930000i	$-0.619806\pi$
$433$	−15.2968	−0.735118	−0.367559	+	0.930000i	$0.619806\pi$
$434$	0	0
$435$	8.72393	0.418280
$436$	0	0
$437$	0	0
$438$	0	0
$439$	36.1956	1.72752	0.863760	−	0.503903i	$-0.168103\pi$
$439$	36.1956	1.72752	0.863760	+	0.503903i	$0.168103\pi$
$440$	0	0
$441$	1.90321	0.0906291
$442$	0	0
$443$	0.682439	0.0324236	0.0162118	−	0.999869i	$-0.494839\pi$
$443$	0.682439	0.0324236	0.0162118	+	0.999869i	$0.494839\pi$
$444$	0	0
$445$	−18.2177	−0.863602
$446$	0	0
$447$	49.7560	2.35338
$448$	0	0
$449$	−4.23951	−0.200075	−0.100037	−	0.994984i	$-0.531896\pi$
$449$	−4.23951	−0.200075	−0.100037	+	0.994984i	$0.531896\pi$
$450$	0	0
$451$	−33.8894	−1.59579
$452$	0	0
$453$	17.1526	0.805898
$454$	0	0
$455$	2.23506	0.104781
$456$	0	0
$457$	−19.0321	−0.890285	−0.445142	−	0.895460i	$-0.646847\pi$
$457$	−19.0321	−0.890285	−0.445142	+	0.895460i	$0.646847\pi$
$458$	0	0
$459$	−16.6035	−0.774984
$460$	0	0
$461$	5.40943	0.251942	0.125971	−	0.992034i	$-0.459795\pi$
$461$	5.40943	0.251942	0.125971	+	0.992034i	$0.459795\pi$
$462$	0	0
$463$	−14.0602	−0.653434	−0.326717	−	0.945122i	$-0.605942\pi$
$463$	−14.0602	−0.653434	−0.326717	+	0.945122i	$0.605942\pi$
$464$	0	0
$465$	43.4193	2.01352
$466$	0	0
$467$	11.3461	0.525037	0.262518	−	0.964927i	$-0.415447\pi$
$467$	11.3461	0.525037	0.262518	+	0.964927i	$0.415447\pi$
$468$	0	0
$469$	4.76986	0.220252
$470$	0	0
$471$	0.474572	0.0218671
$472$	0	0
$473$	−21.9813	−1.01070
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−25.9081	−1.18625
$478$	0	0
$479$	34.5433	1.57832	0.789161	−	0.614187i	$-0.210516\pi$
$479$	34.5433	1.57832	0.789161	+	0.614187i	$0.210516\pi$
$480$	0	0
$481$	−4.59057	−0.209312
$482$	0	0
$483$	−2.21432	−0.100755
$484$	0	0
$485$	−62.1673	−2.82287
$486$	0	0
$487$	−12.0830	−0.547532	−0.273766	−	0.961796i	$-0.588269\pi$
$487$	−12.0830	−0.547532	−0.273766	+	0.961796i	$0.588269\pi$
$488$	0	0
$489$	−12.2667	−0.554720
$490$	0	0
$491$	9.12399	0.411760	0.205880	−	0.978577i	$-0.433994\pi$
$491$	9.12399	0.411760	0.205880	+	0.978577i	$0.433994\pi$
$492$	0	0
$493$	7.49823	0.337703
$494$	0	0
$495$	−24.1017	−1.08329
$496$	0	0
$497$	8.85728	0.397303
$498$	0	0
$499$	−19.6543	−0.879849	−0.439924	−	0.898035i	$-0.644995\pi$
$499$	−19.6543	−0.879849	−0.439924	+	0.898035i	$0.644995\pi$
$500$	0	0
$501$	−46.7511	−2.08869
$502$	0	0
$503$	31.9081	1.42271	0.711357	−	0.702831i	$-0.248081\pi$
$503$	31.9081	1.42271	0.711357	+	0.702831i	$0.248081\pi$
$504$	0	0
$505$	−4.25380	−0.189292
$506$	0	0
$507$	−27.9289	−1.24037
$508$	0	0
$509$	−15.8064	−0.700607	−0.350304	−	0.936636i	$-0.613922\pi$
$509$	−15.8064	−0.700607	−0.350304	+	0.936636i	$0.613922\pi$
$510$	0	0
$511$	11.2859	0.499260
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.47013	−0.196977
$516$	0	0
$517$	−10.0000	−0.439799
$518$	0	0
$519$	33.7560	1.48173
$520$	0	0
$521$	11.1635	0.489080	0.244540	−	0.969639i	$-0.421363\pi$
$521$	11.1635	0.489080	0.244540	+	0.969639i	$0.421363\pi$
$522$	0	0
$523$	10.0098	0.437700	0.218850	−	0.975759i	$-0.429769\pi$
$523$	10.0098	0.437700	0.218850	+	0.975759i	$0.429769\pi$
$524$	0	0
$525$	−17.5002	−0.763773
$526$	0	0
$527$	37.3189	1.62564
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−25.2464	−1.09560
$532$	0	0
$533$	−5.98126	−0.259077
$534$	0	0
$535$	−54.2133	−2.34384
$536$	0	0
$537$	16.5620	0.714703
$538$	0	0
$539$	−3.52543	−0.151851
$540$	0	0
$541$	−4.44293	−0.191016	−0.0955082	−	0.995429i	$-0.530448\pi$
$541$	−4.44293	−0.191016	−0.0955082	+	0.995429i	$0.530448\pi$
$542$	0	0
$543$	31.1383	1.33627
$544$	0	0
$545$	45.0103	1.92803
$546$	0	0
$547$	−17.3778	−0.743020	−0.371510	−	0.928429i	$-0.621160\pi$
$547$	−17.3778	−0.743020	−0.371510	+	0.928429i	$0.621160\pi$
$548$	0	0
$549$	−27.1318	−1.15796
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−6.28100	−0.267095
$554$	0	0
$555$	58.6834	2.49097
$556$	0	0
$557$	−7.98126	−0.338177	−0.169089	−	0.985601i	$-0.554082\pi$
$557$	−7.98126	−0.338177	−0.169089	+	0.985601i	$0.554082\pi$
$558$	0	0
$559$	−3.87955	−0.164088
$560$	0	0
$561$	−53.3689	−2.25324
$562$	0	0
$563$	38.6637	1.62948	0.814740	−	0.579826i	$-0.196879\pi$
$563$	38.6637	1.62948	0.814740	+	0.579826i	$0.196879\pi$
$564$	0	0
$565$	24.6321	1.03628
$566$	0	0
$567$	11.0874	0.465628
$568$	0	0
$569$	2.25380	0.0944842	0.0472421	−	0.998883i	$-0.484957\pi$
$569$	2.25380	0.0944842	0.0472421	+	0.998883i	$0.484957\pi$
$570$	0	0
$571$	13.1941	0.552154	0.276077	−	0.961136i	$-0.410966\pi$
$571$	13.1941	0.552154	0.276077	+	0.961136i	$0.410966\pi$
$572$	0	0
$573$	−9.15257	−0.382354
$574$	0	0
$575$	7.90321	0.329587
$576$	0	0
$577$	−41.8292	−1.74137	−0.870686	−	0.491840i	$-0.836325\pi$
$577$	−41.8292	−1.74137	−0.870686	+	0.491840i	$0.836325\pi$
$578$	0	0
$579$	25.8479	1.07420
$580$	0	0
$581$	2.62222	0.108788
$582$	0	0
$583$	47.9911	1.98759
$584$	0	0
$585$	−4.25380	−0.175873
$586$	0	0
$587$	27.5921	1.13885	0.569424	−	0.822044i	$-0.307166\pi$
$587$	27.5921	1.13885	0.569424	+	0.822044i	$0.307166\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	9.08250	0.373604
$592$	0	0
$593$	−16.6637	−0.684296	−0.342148	−	0.939646i	$-0.611154\pi$
$593$	−16.6637	−0.684296	−0.342148	+	0.939646i	$0.611154\pi$
$594$	0	0
$595$	−24.5575	−1.00676
$596$	0	0
$597$	11.6128	0.475282
$598$	0	0
$599$	−4.60348	−0.188093	−0.0940465	−	0.995568i	$-0.529980\pi$
$599$	−4.60348	−0.188093	−0.0940465	+	0.995568i	$0.529980\pi$
$600$	0	0
$601$	−13.0094	−0.530663	−0.265332	−	0.964157i	$-0.585481\pi$
$601$	−13.0094	−0.530663	−0.265332	+	0.964157i	$0.585481\pi$
$602$	0	0
$603$	−9.07805	−0.369687
$604$	0	0
$605$	5.13182	0.208638
$606$	0	0
$607$	5.32540	0.216151	0.108076	−	0.994143i	$-0.465531\pi$
$607$	5.32540	0.216151	0.108076	+	0.994143i	$0.465531\pi$
$608$	0	0
$609$	−2.42864	−0.0984134
$610$	0	0
$611$	−1.76494	−0.0714017
$612$	0	0
$613$	24.2636	0.979999	0.490000	−	0.871723i	$-0.336997\pi$
$613$	24.2636	0.979999	0.490000	+	0.871723i	$0.336997\pi$
$614$	0	0
$615$	76.4612	3.08321
$616$	0	0
$617$	−10.7239	−0.431729	−0.215864	−	0.976423i	$-0.569257\pi$
$617$	−10.7239	−0.431729	−0.215864	+	0.976423i	$0.569257\pi$
$618$	0	0
$619$	−16.6035	−0.667350	−0.333675	−	0.942688i	$-0.608289\pi$
$619$	−16.6035	−0.667350	−0.333675	+	0.942688i	$0.608289\pi$
$620$	0	0
$621$	−2.42864	−0.0974579
$622$	0	0
$623$	5.07160	0.203189
$624$	0	0
$625$	−2.05530	−0.0822120
$626$	0	0
$627$	0	0
$628$	0	0
$629$	50.4385	2.01111
$630$	0	0
$631$	−19.9037	−0.792353	−0.396177	−	0.918174i	$-0.629663\pi$
$631$	−19.9037	−0.792353	−0.396177	+	0.918174i	$0.629663\pi$
$632$	0	0
$633$	−35.7244	−1.41992
$634$	0	0
$635$	−19.3176	−0.766594
$636$	0	0
$637$	−0.622216	−0.0246531
$638$	0	0
$639$	−16.8573	−0.666864
$640$	0	0
$641$	47.3975	1.87209	0.936044	−	0.351883i	$-0.114459\pi$
$641$	47.3975	1.87209	0.936044	+	0.351883i	$0.114459\pi$
$642$	0	0
$643$	−37.2355	−1.46843	−0.734213	−	0.678919i	$-0.762449\pi$
$643$	−37.2355	−1.46843	−0.734213	+	0.678919i	$0.762449\pi$
$644$	0	0
$645$	49.5941	1.95277
$646$	0	0
$647$	34.4079	1.35271	0.676357	−	0.736574i	$-0.263558\pi$
$647$	34.4079	1.35271	0.676357	+	0.736574i	$0.263558\pi$
$648$	0	0
$649$	46.7654	1.83570
$650$	0	0
$651$	−12.0874	−0.473743
$652$	0	0
$653$	13.4924	0.527998	0.263999	−	0.964523i	$-0.414958\pi$
$653$	13.4924	0.527998	0.263999	+	0.964523i	$0.414958\pi$
$654$	0	0
$655$	−13.7476	−0.537163
$656$	0	0
$657$	−21.4795	−0.837995
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−6.73483	−0.261954	−0.130977	−	0.991385i	$-0.541811\pi$
$661$	−6.73483	−0.261954	−0.130977	+	0.991385i	$0.541811\pi$
$662$	0	0
$663$	−9.41927	−0.365814
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.09679	0.0424678
$668$	0	0
$669$	29.3733	1.13564
$670$	0	0
$671$	50.2578	1.94018
$672$	0	0
$673$	−29.7418	−1.14646	−0.573230	−	0.819394i	$-0.694310\pi$
$673$	−29.7418	−1.14646	−0.573230	+	0.819394i	$0.694310\pi$
$674$	0	0
$675$	−19.1941	−0.738779
$676$	0	0
$677$	15.0716	0.579249	0.289624	−	0.957140i	$-0.406470\pi$
$677$	15.0716	0.579249	0.289624	+	0.957140i	$0.406470\pi$
$678$	0	0
$679$	17.3067	0.664169
$680$	0	0
$681$	33.3274	1.27711
$682$	0	0
$683$	3.64143	0.139335	0.0696677	−	0.997570i	$-0.477806\pi$
$683$	3.64143	0.139335	0.0696677	+	0.997570i	$0.477806\pi$
$684$	0	0
$685$	−7.66323	−0.292797
$686$	0	0
$687$	−29.5353	−1.12684
$688$	0	0
$689$	8.47013	0.322686
$690$	0	0
$691$	14.8780	0.565987	0.282993	−	0.959122i	$-0.408673\pi$
$691$	14.8780	0.565987	0.282993	+	0.959122i	$0.408673\pi$
$692$	0	0
$693$	6.70964	0.254878
$694$	0	0
$695$	−18.9131	−0.717413
$696$	0	0
$697$	65.7186	2.48927
$698$	0	0
$699$	−3.67307	−0.138928
$700$	0	0
$701$	37.6829	1.42326	0.711632	−	0.702552i	$-0.247956\pi$
$701$	37.6829	1.42326	0.711632	+	0.702552i	$0.247956\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	22.5620	0.849734
$706$	0	0
$707$	1.18421	0.0445367
$708$	0	0
$709$	21.5714	0.810129	0.405065	−	0.914288i	$-0.367249\pi$
$709$	21.5714	0.810129	0.405065	+	0.914288i	$0.367249\pi$
$710$	0	0
$711$	11.9541	0.448313
$712$	0	0
$713$	5.45875	0.204432
$714$	0	0
$715$	7.87955	0.294679
$716$	0	0
$717$	9.15257	0.341809
$718$	0	0
$719$	−40.1639	−1.49786	−0.748931	−	0.662648i	$-0.769433\pi$
$719$	−40.1639	−1.49786	−0.748931	+	0.662648i	$0.769433\pi$
$720$	0	0
$721$	1.24443	0.0463450
$722$	0	0
$723$	17.0366	0.633597
$724$	0	0
$725$	8.66815	0.321927
$726$	0	0
$727$	−15.3461	−0.569157	−0.284578	−	0.958653i	$-0.591854\pi$
$727$	−15.3461	−0.569157	−0.284578	+	0.958653i	$0.591854\pi$
$728$	0	0
$729$	−4.96836	−0.184013
$730$	0	0
$731$	42.6262	1.57659
$732$	0	0
$733$	−21.8557	−0.807260	−0.403630	−	0.914922i	$-0.632252\pi$
$733$	−21.8557	−0.807260	−0.403630	+	0.914922i	$0.632252\pi$
$734$	0	0
$735$	7.95407	0.293390
$736$	0	0
$737$	16.8158	0.619418
$738$	0	0
$739$	−31.1052	−1.14423	−0.572113	−	0.820175i	$-0.693876\pi$
$739$	−31.1052	−1.14423	−0.572113	+	0.820175i	$0.693876\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.0178	0.844442	0.422221	−	0.906493i	$-0.361251\pi$
$743$	23.0178	0.844442	0.422221	+	0.906493i	$0.361251\pi$
$744$	0	0
$745$	80.7150	2.95717
$746$	0	0
$747$	−4.99063	−0.182598
$748$	0	0
$749$	15.0923	0.551462
$750$	0	0
$751$	−24.1160	−0.880005	−0.440003	−	0.897996i	$-0.645023\pi$
$751$	−24.1160	−0.880005	−0.440003	+	0.897996i	$0.645023\pi$
$752$	0	0
$753$	−48.6735	−1.77376
$754$	0	0
$755$	27.8252	1.01266
$756$	0	0
$757$	−23.6642	−0.860089	−0.430045	−	0.902808i	$-0.641502\pi$
$757$	−23.6642	−0.860089	−0.430045	+	0.902808i	$0.641502\pi$
$758$	0	0
$759$	−7.80642	−0.283355
$760$	0	0
$761$	54.1245	1.96201	0.981005	−	0.193982i	$-0.0621403\pi$
$761$	54.1245	1.96201	0.981005	+	0.193982i	$0.0621403\pi$
$762$	0	0
$763$	−12.5303	−0.453629
$764$	0	0
$765$	46.7382	1.68982
$766$	0	0
$767$	8.25380	0.298027
$768$	0	0
$769$	55.2148	1.99110	0.995548	−	0.0942534i	$-0.0300464\pi$
$769$	55.2148	1.99110	0.995548	+	0.0942534i	$0.0300464\pi$
$770$	0	0
$771$	−26.7971	−0.965072
$772$	0	0
$773$	−20.2272	−0.727523	−0.363761	−	0.931492i	$-0.618508\pi$
$773$	−20.2272	−0.727523	−0.363761	+	0.931492i	$0.618508\pi$
$774$	0	0
$775$	43.1417	1.54969
$776$	0	0
$777$	−16.3368	−0.586078
$778$	0	0
$779$	0	0
$780$	0	0
$781$	31.2257	1.11734
$782$	0	0
$783$	−2.66370	−0.0951930
$784$	0	0
$785$	0.769859	0.0274774
$786$	0	0
$787$	38.3466	1.36691	0.683455	−	0.729993i	$-0.260477\pi$
$787$	38.3466	1.36691	0.683455	+	0.729993i	$0.260477\pi$
$788$	0	0
$789$	18.1748	0.647041
$790$	0	0
$791$	−6.85728	−0.243817
$792$	0	0
$793$	8.87019	0.314990
$794$	0	0
$795$	−108.278	−3.84021
$796$	0	0
$797$	−27.2019	−0.963540	−0.481770	−	0.876298i	$-0.660006\pi$
$797$	−27.2019	−0.963540	−0.481770	+	0.876298i	$0.660006\pi$
$798$	0	0
$799$	19.3921	0.686042
$800$	0	0
$801$	−9.65233	−0.341048
$802$	0	0
$803$	39.7877	1.40408
$804$	0	0
$805$	−3.59210	−0.126605
$806$	0	0
$807$	21.2859	0.749300
$808$	0	0
$809$	−14.8158	−0.520895	−0.260448	−	0.965488i	$-0.583870\pi$
$809$	−14.8158	−0.520895	−0.260448	+	0.965488i	$0.583870\pi$
$810$	0	0
$811$	−20.6015	−0.723416	−0.361708	−	0.932292i	$-0.617806\pi$
$811$	−20.6015	−0.723416	−0.361708	+	0.932292i	$0.617806\pi$
$812$	0	0
$813$	61.2815	2.14924
$814$	0	0
$815$	−19.8992	−0.697040
$816$	0	0
$817$	0	0
$818$	0	0
$819$	1.18421	0.0413796
$820$	0	0
$821$	−19.8809	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$821$	−19.8809	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$822$	0	0
$823$	−10.3398	−0.360424	−0.180212	−	0.983628i	$-0.557678\pi$
$823$	−10.3398	−0.360424	−0.180212	+	0.983628i	$0.557678\pi$
$824$	0	0
$825$	−61.6958	−2.14797
$826$	0	0
$827$	−32.7940	−1.14036	−0.570179	−	0.821520i	$-0.693126\pi$
$827$	−32.7940	−1.14036	−0.570179	+	0.821520i	$0.693126\pi$
$828$	0	0
$829$	27.4193	0.952311	0.476155	−	0.879361i	$-0.342030\pi$
$829$	27.4193	0.952311	0.476155	+	0.879361i	$0.342030\pi$
$830$	0	0
$831$	−53.2672	−1.84782
$832$	0	0
$833$	6.83654	0.236872
$834$	0	0
$835$	−75.8404	−2.62457
$836$	0	0
$837$	−13.2573	−0.458241
$838$	0	0
$839$	−32.6035	−1.12560	−0.562799	−	0.826594i	$-0.690275\pi$
$839$	−32.6035	−1.12560	−0.562799	+	0.826594i	$0.690275\pi$
$840$	0	0
$841$	−27.7971	−0.958519
$842$	0	0
$843$	53.4608	1.84129
$844$	0	0
$845$	−45.3067	−1.55860
$846$	0	0
$847$	−1.42864	−0.0490886
$848$	0	0
$849$	11.4094	0.391571
$850$	0	0
$851$	7.37778	0.252907
$852$	0	0
$853$	−13.2761	−0.454564	−0.227282	−	0.973829i	$-0.572984\pi$
$853$	−13.2761	−0.454564	−0.227282	+	0.973829i	$0.572984\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.6351	−0.568245	−0.284122	−	0.958788i	$-0.591702\pi$
$857$	−16.6351	−0.568245	−0.284122	+	0.958788i	$0.591702\pi$
$858$	0	0
$859$	47.2335	1.61159	0.805794	−	0.592196i	$-0.201739\pi$
$859$	47.2335	1.61159	0.805794	+	0.592196i	$0.201739\pi$
$860$	0	0
$861$	−21.2859	−0.725422
$862$	0	0
$863$	44.7565	1.52353	0.761765	−	0.647854i	$-0.224333\pi$
$863$	44.7565	1.52353	0.761765	+	0.647854i	$0.224333\pi$
$864$	0	0
$865$	54.7596	1.86188
$866$	0	0
$867$	65.8499	2.23638
$868$	0	0
$869$	−22.1432	−0.751157
$870$	0	0
$871$	2.96788	0.100563
$872$	0	0
$873$	−32.9382	−1.11479
$874$	0	0
$875$	−10.4286	−0.352552
$876$	0	0
$877$	21.8336	0.737269	0.368634	−	0.929574i	$-0.379825\pi$
$877$	21.8336	0.737269	0.368634	+	0.929574i	$0.379825\pi$
$878$	0	0
$879$	−50.2306	−1.69424
$880$	0	0
$881$	−1.68397	−0.0567344	−0.0283672	−	0.999598i	$-0.509031\pi$
$881$	−1.68397	−0.0567344	−0.0283672	+	0.999598i	$0.509031\pi$
$882$	0	0
$883$	3.47949	0.117094	0.0585472	−	0.998285i	$-0.481353\pi$
$883$	3.47949	0.117094	0.0585472	+	0.998285i	$0.481353\pi$
$884$	0	0
$885$	−105.512	−3.54675
$886$	0	0
$887$	−3.29682	−0.110696	−0.0553482	−	0.998467i	$-0.517627\pi$
$887$	−3.29682	−0.110696	−0.0553482	+	0.998467i	$0.517627\pi$
$888$	0	0
$889$	5.37778	0.180365
$890$	0	0
$891$	39.0879	1.30949
$892$	0	0
$893$	0	0
$894$	0	0
$895$	26.8671	0.898069
$896$	0	0
$897$	−1.37778	−0.0460029
$898$	0	0
$899$	5.98709	0.199681
$900$	0	0
$901$	−93.0647	−3.10044
$902$	0	0
$903$	−13.8064	−0.459449
$904$	0	0
$905$	50.5130	1.67911
$906$	0	0
$907$	47.6128	1.58096	0.790479	−	0.612489i	$-0.209832\pi$
$907$	47.6128	1.58096	0.790479	+	0.612489i	$0.209832\pi$
$908$	0	0
$909$	−2.25380	−0.0747538
$910$	0	0
$911$	−56.4371	−1.86984	−0.934922	−	0.354853i	$-0.884531\pi$
$911$	−56.4371	−1.86984	−0.934922	+	0.354853i	$0.884531\pi$
$912$	0	0
$913$	9.24443	0.305946
$914$	0	0
$915$	−113.392	−3.74861
$916$	0	0
$917$	3.82717	0.126384
$918$	0	0
$919$	47.5165	1.56743	0.783713	−	0.621123i	$-0.213323\pi$
$919$	47.5165	1.56743	0.783713	+	0.621123i	$0.213323\pi$
$920$	0	0
$921$	−67.4247	−2.22172
$922$	0	0
$923$	5.51114	0.181401
$924$	0	0
$925$	58.3082	1.91716
$926$	0	0
$927$	−2.36842	−0.0777890
$928$	0	0
$929$	−15.5683	−0.510779	−0.255390	−	0.966838i	$-0.582204\pi$
$929$	−15.5683	−0.510779	−0.255390	+	0.966838i	$0.582204\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2.44293	−0.0799779
$934$	0	0
$935$	−86.5759	−2.83133
$936$	0	0
$937$	−48.1639	−1.57345	−0.786724	−	0.617305i	$-0.788224\pi$
$937$	−48.1639	−1.57345	−0.786724	+	0.617305i	$0.788224\pi$
$938$	0	0
$939$	56.1990	1.83398
$940$	0	0
$941$	−42.9324	−1.39956	−0.699778	−	0.714360i	$-0.746718\pi$
$941$	−42.9324	−1.39956	−0.699778	+	0.714360i	$0.746718\pi$
$942$	0	0
$943$	9.61285	0.313037
$944$	0	0
$945$	8.72393	0.283789
$946$	0	0
$947$	−10.2636	−0.333524	−0.166762	−	0.985997i	$-0.553331\pi$
$947$	−10.2636	−0.333524	−0.166762	+	0.985997i	$0.553331\pi$
$948$	0	0
$949$	7.02227	0.227953
$950$	0	0
$951$	50.8988	1.65051
$952$	0	0
$953$	−19.8064	−0.641593	−0.320797	−	0.947148i	$-0.603951\pi$
$953$	−19.8064	−0.641593	−0.320797	+	0.947148i	$0.603951\pi$
$954$	0	0
$955$	−14.8474	−0.480452
$956$	0	0
$957$	−8.56199	−0.276770
$958$	0	0
$959$	2.13335	0.0688896
$960$	0	0
$961$	−1.20204	−0.0387754
$962$	0	0
$963$	−28.7239	−0.925616
$964$	0	0
$965$	41.9309	1.34980
$966$	0	0
$967$	−25.6860	−0.826005	−0.413003	−	0.910730i	$-0.635520\pi$
$967$	−25.6860	−0.826005	−0.413003	+	0.910730i	$0.635520\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−34.7971	−1.11669	−0.558345	−	0.829609i	$-0.688564\pi$
$971$	−34.7971	−1.11669	−0.558345	+	0.829609i	$0.688564\pi$
$972$	0	0
$973$	5.26517	0.168794
$974$	0	0
$975$	−10.8889	−0.348725
$976$	0	0
$977$	52.1432	1.66821	0.834104	−	0.551607i	$-0.185985\pi$
$977$	52.1432	1.66821	0.834104	+	0.551607i	$0.185985\pi$
$978$	0	0
$979$	17.8796	0.571433
$980$	0	0
$981$	23.8479	0.761405
$982$	0	0
$983$	−58.4385	−1.86390	−0.931949	−	0.362589i	$-0.881893\pi$
$983$	−58.4385	−1.86390	−0.931949	+	0.362589i	$0.881893\pi$
$984$	0	0
$985$	14.7338	0.469457
$986$	0	0
$987$	−6.28100	−0.199926
$988$	0	0
$989$	6.23506	0.198263
$990$	0	0
$991$	43.0420	1.36727	0.683637	−	0.729823i	$-0.260397\pi$
$991$	43.0420	1.36727	0.683637	+	0.729823i	$0.260397\pi$
$992$	0	0
$993$	−62.0928	−1.97046
$994$	0	0
$995$	18.8385	0.597222
$996$	0	0
$997$	27.7146	0.877729	0.438864	−	0.898553i	$-0.355381\pi$
$997$	27.7146	0.877729	0.438864	+	0.898553i	$0.355381\pi$
$998$	0	0
$999$	−17.9180	−0.566900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1288.2.a.l.1.3	✓	3
4.3	odd	2		2576.2.a.y.1.1		3
7.6	odd	2		9016.2.a.z.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.a.l.1.3	✓	3	1.1	even	1	trivial
2576.2.a.y.1.1		3	4.3	odd	2
9016.2.a.z.1.1		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1288.a (trivial)

\(f(q)\)	\(=\)	\(q+2.21432 q^{3} +3.59210 q^{5} -1.00000 q^{7} +1.90321 q^{9} +O(q^{10})\)	\(q+2.21432 q^{3} +3.59210 q^{5} -1.00000 q^{7} +1.90321 q^{9} -3.52543 q^{11} -0.622216 q^{13} +7.95407 q^{15} +6.83654 q^{17} -2.21432 q^{21} +1.00000 q^{23} +7.90321 q^{25} -2.42864 q^{27} +1.09679 q^{29} +5.45875 q^{31} -7.80642 q^{33} -3.59210 q^{35} +7.37778 q^{37} -1.37778 q^{39} +9.61285 q^{41} +6.23506 q^{43} +6.83654 q^{45} +2.83654 q^{47} +1.00000 q^{49} +15.1383 q^{51} -13.6128 q^{53} -12.6637 q^{55} -13.2652 q^{59} -14.2558 q^{61} -1.90321 q^{63} -2.23506 q^{65} -4.76986 q^{67} +2.21432 q^{69} -8.85728 q^{71} -11.2859 q^{73} +17.5002 q^{75} +3.52543 q^{77} +6.28100 q^{79} -11.0874 q^{81} -2.62222 q^{83} +24.5575 q^{85} +2.42864 q^{87} -5.07160 q^{89} +0.622216 q^{91} +12.0874 q^{93} -17.3067 q^{97} -6.70964 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{5} - 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q + 4 * q^5 - 3 * q^7 - q^9	\( 3 q + 4 q^{5} - 3 q^{7} - q^{9} - 4 q^{11} - 2 q^{13} + 4 q^{15} + 14 q^{17} + 3 q^{23} + 17 q^{25} + 6 q^{27} + 10 q^{29} + 10 q^{31} - 10 q^{33} - 4 q^{35} + 22 q^{37} - 4 q^{39} + 2 q^{41} - 8 q^{43} + 14 q^{45} + 2 q^{47} + 3 q^{49} + 12 q^{51} - 14 q^{53} + 2 q^{55} - 20 q^{59} + 4 q^{61} + q^{63} + 20 q^{65} - 8 q^{67} + 6 q^{73} + 6 q^{75} + 4 q^{77} + 12 q^{79} - 13 q^{81} - 8 q^{83} - 6 q^{87} + 18 q^{89} + 2 q^{91} + 16 q^{93} + 8 q^{97}+O(q^{100}) \) 3 * q + 4 * q^5 - 3 * q^7 - q^9 - 4 * q^11 - 2 * q^13 + 4 * q^15 + 14 * q^17 + 3 * q^23 + 17 * q^25 + 6 * q^27 + 10 * q^29 + 10 * q^31 - 10 * q^33 - 4 * q^35 + 22 * q^37 - 4 * q^39 + 2 * q^41 - 8 * q^43 + 14 * q^45 + 2 * q^47 + 3 * q^49 + 12 * q^51 - 14 * q^53 + 2 * q^55 - 20 * q^59 + 4 * q^61 + q^63 + 20 * q^65 - 8 * q^67 + 6 * q^73 + 6 * q^75 + 4 * q^77 + 12 * q^79 - 13 * q^81 - 8 * q^83 - 6 * q^87 + 18 * q^89 + 2 * q^91 + 16 * q^93 + 8 * q^97

Introduction

L-functions

Modular forms

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Database

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