LMFDB - Embedded newform 1300.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,4,Mod(1,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1300.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1300.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:	$76.7024830075$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2785296.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 30x^{2} + 2x + 65 $ x^4 - 2x^3 - 30x^2 + 2*x + 65
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.48645$ of defining polynomial
Character	$\chi$	$=$	1300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.84775 q^{3} -1.86425 q^{7} +69.9781 q^{9} +O(q^{10})$	$q-9.84775 q^{3} -1.86425 q^{7} +69.9781 q^{9} -17.5494 q^{11} -13.0000 q^{13} -126.966 q^{17} +92.7773 q^{19} +18.3587 q^{21} +18.5879 q^{23} -423.237 q^{27} -122.291 q^{29} +106.822 q^{31} +172.822 q^{33} +268.422 q^{37} +128.021 q^{39} -100.687 q^{41} +220.593 q^{43} +546.232 q^{47} -339.525 q^{49} +1250.33 q^{51} -18.4284 q^{53} -913.647 q^{57} +129.501 q^{59} +792.160 q^{61} -130.457 q^{63} -383.263 q^{67} -183.049 q^{69} -312.554 q^{71} +933.120 q^{73} +32.7166 q^{77} +828.441 q^{79} +2278.52 q^{81} +147.683 q^{83} +1204.29 q^{87} -1604.84 q^{89} +24.2353 q^{91} -1051.96 q^{93} -654.169 q^{97} -1228.08 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 8 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 8 * q^7 + 36 * q^9	$ 4 q - 4 q^{3} + 8 q^{7} + 36 q^{9} + 28 q^{11} - 52 q^{13} - 200 q^{17} + 132 q^{19} + 160 q^{21} - 244 q^{23} - 496 q^{27} - 8 q^{29} + 292 q^{31} - 128 q^{33} - 168 q^{37} + 52 q^{39} + 280 q^{41} - 452 q^{43} + 280 q^{47} + 340 q^{49} + 792 q^{51} - 584 q^{53} - 1200 q^{57} - 708 q^{59} + 1128 q^{61} - 232 q^{63} - 176 q^{67} + 160 q^{69} - 1028 q^{71} - 664 q^{73} + 864 q^{77} - 728 q^{79} + 2244 q^{81} - 552 q^{83} + 2064 q^{87} - 2824 q^{89} - 104 q^{91} - 1392 q^{93} - 1160 q^{97} - 4220 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 8 * q^7 + 36 * q^9 + 28 * q^11 - 52 * q^13 - 200 * q^17 + 132 * q^19 + 160 * q^21 - 244 * q^23 - 496 * q^27 - 8 * q^29 + 292 * q^31 - 128 * q^33 - 168 * q^37 + 52 * q^39 + 280 * q^41 - 452 * q^43 + 280 * q^47 + 340 * q^49 + 792 * q^51 - 584 * q^53 - 1200 * q^57 - 708 * q^59 + 1128 * q^61 - 232 * q^63 - 176 * q^67 + 160 * q^69 - 1028 * q^71 - 664 * q^73 + 864 * q^77 - 728 * q^79 + 2244 * q^81 - 552 * q^83 + 2064 * q^87 - 2824 * q^89 - 104 * q^91 - 1392 * q^93 - 1160 * q^97 - 4220 * 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
$3$	−9.84775	−1.89520	−0.947600	−	0.319460i	$-0.896498\pi$
$3$	−9.84775	−1.89520	−0.947600	+	0.319460i	$0.896498\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.86425	−0.100660	−0.0503301	−	0.998733i	$-0.516027\pi$
$7$	−1.86425	−0.100660	−0.0503301	+	0.998733i	$0.516027\pi$
$8$	0	0
$9$	69.9781	2.59178
$10$	0	0
$11$	−17.5494	−0.481032	−0.240516	−	0.970645i	$-0.577317\pi$
$11$	−17.5494	−0.481032	−0.240516	+	0.970645i	$0.577317\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−126.966	−1.81140	−0.905701	−	0.423917i	$-0.860655\pi$
$17$	−126.966	−1.81140	−0.905701	+	0.423917i	$0.860655\pi$
$18$	0	0
$19$	92.7773	1.12024	0.560120	−	0.828411i	$-0.310755\pi$
$19$	92.7773	1.12024	0.560120	+	0.828411i	$0.310755\pi$
$20$	0	0
$21$	18.3587	0.190771
$22$	0	0
$23$	18.5879	0.168515	0.0842575	−	0.996444i	$-0.473148\pi$
$23$	18.5879	0.168515	0.0842575	+	0.996444i	$0.473148\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−423.237	−3.01674
$28$	0	0
$29$	−122.291	−0.783062	−0.391531	−	0.920165i	$-0.628054\pi$
$29$	−122.291	−0.783062	−0.391531	+	0.920165i	$0.628054\pi$
$30$	0	0
$31$	106.822	0.618898	0.309449	−	0.950916i	$-0.399855\pi$
$31$	106.822	0.618898	0.309449	+	0.950916i	$0.399855\pi$
$32$	0	0
$33$	172.822	0.911652
$34$	0	0
$35$	0	0
$36$	0	0
$37$	268.422	1.19266	0.596329	−	0.802740i	$-0.296625\pi$
$37$	268.422	1.19266	0.596329	+	0.802740i	$0.296625\pi$
$38$	0	0
$39$	128.021	0.525634
$40$	0	0
$41$	−100.687	−0.383529	−0.191764	−	0.981441i	$-0.561421\pi$
$41$	−100.687	−0.383529	−0.191764	+	0.981441i	$0.561421\pi$
$42$	0	0
$43$	220.593	0.782327	0.391164	−	0.920321i	$-0.372073\pi$
$43$	220.593	0.782327	0.391164	+	0.920321i	$0.372073\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	546.232	1.69524	0.847618	−	0.530606i	$-0.178036\pi$
$47$	546.232	1.69524	0.847618	+	0.530606i	$0.178036\pi$
$48$	0	0
$49$	−339.525	−0.989868
$50$	0	0
$51$	1250.33	3.43297
$52$	0	0
$53$	−18.4284	−0.0477609	−0.0238805	−	0.999715i	$-0.507602\pi$
$53$	−18.4284	−0.0477609	−0.0238805	+	0.999715i	$0.507602\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−913.647	−2.12308
$58$	0	0
$59$	129.501	0.285756	0.142878	−	0.989740i	$-0.454364\pi$
$59$	129.501	0.285756	0.142878	+	0.989740i	$0.454364\pi$
$60$	0	0
$61$	792.160	1.66272	0.831358	−	0.555737i	$-0.187564\pi$
$61$	792.160	1.66272	0.831358	+	0.555737i	$0.187564\pi$
$62$	0	0
$63$	−130.457	−0.260889
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−383.263	−0.698851	−0.349426	−	0.936964i	$-0.613623\pi$
$67$	−383.263	−0.698851	−0.349426	+	0.936964i	$0.613623\pi$
$68$	0	0
$69$	−183.049	−0.319370
$70$	0	0
$71$	−312.554	−0.522442	−0.261221	−	0.965279i	$-0.584125\pi$
$71$	−312.554	−0.522442	−0.261221	+	0.965279i	$0.584125\pi$
$72$	0	0
$73$	933.120	1.49607	0.748037	−	0.663657i	$-0.230997\pi$
$73$	933.120	1.49607	0.748037	+	0.663657i	$0.230997\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	32.7166	0.0484208
$78$	0	0
$79$	828.441	1.17983	0.589917	−	0.807464i	$-0.299160\pi$
$79$	828.441	1.17983	0.589917	+	0.807464i	$0.299160\pi$
$80$	0	0
$81$	2278.52	3.12555
$82$	0	0
$83$	147.683	0.195305	0.0976526	−	0.995221i	$-0.468867\pi$
$83$	147.683	0.195305	0.0976526	+	0.995221i	$0.468867\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1204.29	1.48406
$88$	0	0
$89$	−1604.84	−1.91138	−0.955691	−	0.294371i	$-0.904890\pi$
$89$	−1604.84	−1.91138	−0.955691	+	0.294371i	$0.904890\pi$
$90$	0	0
$91$	24.2353	0.0279181
$92$	0	0
$93$	−1051.96	−1.17293
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−654.169	−0.684750	−0.342375	−	0.939563i	$-0.611231\pi$
$97$	−654.169	−0.684750	−0.342375	+	0.939563i	$0.611231\pi$
$98$	0	0
$99$	−1228.08	−1.24673
$100$	0	0
$101$	329.496	0.324615	0.162308	−	0.986740i	$-0.448106\pi$
$101$	329.496	0.324615	0.162308	+	0.986740i	$0.448106\pi$
$102$	0	0
$103$	2036.17	1.94786	0.973929	−	0.226851i	$-0.0728430\pi$
$103$	2036.17	1.94786	0.973929	+	0.226851i	$0.0728430\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	522.563	0.472132	0.236066	−	0.971737i	$-0.424142\pi$
$107$	522.563	0.472132	0.236066	+	0.971737i	$0.424142\pi$
$108$	0	0
$109$	87.3349	0.0767447	0.0383723	−	0.999264i	$-0.487783\pi$
$109$	87.3349	0.0767447	0.0383723	+	0.999264i	$0.487783\pi$
$110$	0	0
$111$	−2643.35	−2.26033
$112$	0	0
$113$	−929.276	−0.773619	−0.386809	−	0.922160i	$-0.626423\pi$
$113$	−929.276	−0.773619	−0.386809	+	0.922160i	$0.626423\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−909.715	−0.718831
$118$	0	0
$119$	236.697	0.182336
$120$	0	0
$121$	−1023.02	−0.768608
$122$	0	0
$123$	991.540	0.726863
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−571.023	−0.398977	−0.199489	−	0.979900i	$-0.563928\pi$
$127$	−571.023	−0.398977	−0.199489	+	0.979900i	$0.563928\pi$
$128$	0	0
$129$	−2172.34	−1.48267
$130$	0	0
$131$	−1490.51	−0.994095	−0.497048	−	0.867723i	$-0.665583\pi$
$131$	−1490.51	−0.994095	−0.497048	+	0.867723i	$0.665583\pi$
$132$	0	0
$133$	−172.960	−0.112764
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2795.12	−1.74309	−0.871544	−	0.490317i	$-0.836881\pi$
$137$	−2795.12	−1.74309	−0.871544	+	0.490317i	$0.836881\pi$
$138$	0	0
$139$	−1792.92	−1.09405	−0.547026	−	0.837116i	$-0.684240\pi$
$139$	−1792.92	−1.09405	−0.547026	+	0.837116i	$0.684240\pi$
$140$	0	0
$141$	−5379.15	−3.21281
$142$	0	0
$143$	228.143	0.133414
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3343.55	1.87600
$148$	0	0
$149$	−2580.28	−1.41869	−0.709345	−	0.704862i	$-0.751009\pi$
$149$	−2580.28	−1.41869	−0.709345	+	0.704862i	$0.751009\pi$
$150$	0	0
$151$	1406.04	0.757764	0.378882	−	0.925445i	$-0.376309\pi$
$151$	1406.04	0.757764	0.378882	+	0.925445i	$0.376309\pi$
$152$	0	0
$153$	−8884.86	−4.69476
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−609.313	−0.309736	−0.154868	−	0.987935i	$-0.549495\pi$
$157$	−609.313	−0.309736	−0.154868	+	0.987935i	$0.549495\pi$
$158$	0	0
$159$	181.478	0.0905165
$160$	0	0
$161$	−34.6526	−0.0169628
$162$	0	0
$163$	−900.102	−0.432524	−0.216262	−	0.976335i	$-0.569387\pi$
$163$	−900.102	−0.432524	−0.216262	+	0.976335i	$0.569387\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2719.37	1.26007	0.630034	−	0.776568i	$-0.283041\pi$
$167$	2719.37	1.26007	0.630034	+	0.776568i	$0.283041\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	6492.38	2.90342
$172$	0	0
$173$	615.799	0.270626	0.135313	−	0.990803i	$-0.456796\pi$
$173$	615.799	0.270626	0.135313	+	0.990803i	$0.456796\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1275.29	−0.541565
$178$	0	0
$179$	3079.89	1.28604	0.643022	−	0.765848i	$-0.277680\pi$
$179$	3079.89	1.28604	0.643022	+	0.765848i	$0.277680\pi$
$180$	0	0
$181$	2789.10	1.14537	0.572685	−	0.819776i	$-0.305902\pi$
$181$	2789.10	1.14537	0.572685	+	0.819776i	$0.305902\pi$
$182$	0	0
$183$	−7800.99	−3.15118
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2228.19	0.871342
$188$	0	0
$189$	789.022	0.303666
$190$	0	0
$191$	−3155.07	−1.19525	−0.597625	−	0.801776i	$-0.703889\pi$
$191$	−3155.07	−1.19525	−0.597625	+	0.801776i	$0.703889\pi$
$192$	0	0
$193$	−4911.88	−1.83194	−0.915972	−	0.401242i	$-0.868579\pi$
$193$	−4911.88	−1.83194	−0.915972	+	0.401242i	$0.868579\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3038.52	1.09891	0.549456	−	0.835523i	$-0.314835\pi$
$197$	3038.52	1.09891	0.549456	+	0.835523i	$0.314835\pi$
$198$	0	0
$199$	−3226.63	−1.14939	−0.574697	−	0.818366i	$-0.694880\pi$
$199$	−3226.63	−1.14939	−0.574697	+	0.818366i	$0.694880\pi$
$200$	0	0
$201$	3774.28	1.32446
$202$	0	0
$203$	227.981	0.0788232
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1300.75	0.436754
$208$	0	0
$209$	−1628.19	−0.538872
$210$	0	0
$211$	−2238.69	−0.730415	−0.365208	−	0.930926i	$-0.619002\pi$
$211$	−2238.69	−0.730415	−0.365208	+	0.930926i	$0.619002\pi$
$212$	0	0
$213$	3077.96	0.990132
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−199.144	−0.0622984
$218$	0	0
$219$	−9189.12	−2.83536
$220$	0	0
$221$	1650.56	0.502393
$222$	0	0
$223$	−5433.01	−1.63148	−0.815742	−	0.578415i	$-0.803671\pi$
$223$	−5433.01	−1.63148	−0.815742	+	0.578415i	$0.803671\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1237.71	−0.361893	−0.180946	−	0.983493i	$-0.557916\pi$
$227$	−1237.71	−0.361893	−0.180946	+	0.983493i	$0.557916\pi$
$228$	0	0
$229$	−1286.37	−0.371204	−0.185602	−	0.982625i	$-0.559424\pi$
$229$	−1286.37	−0.371204	−0.185602	+	0.982625i	$0.559424\pi$
$230$	0	0
$231$	−322.185	−0.0917671
$232$	0	0
$233$	5507.41	1.54851	0.774253	−	0.632876i	$-0.218126\pi$
$233$	5507.41	1.54851	0.774253	+	0.632876i	$0.218126\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8158.28	−2.23602
$238$	0	0
$239$	−3378.14	−0.914282	−0.457141	−	0.889394i	$-0.651127\pi$
$239$	−3378.14	−0.914282	−0.457141	+	0.889394i	$0.651127\pi$
$240$	0	0
$241$	2968.24	0.793367	0.396683	−	0.917955i	$-0.370161\pi$
$241$	2968.24	0.793367	0.396683	+	0.917955i	$0.370161\pi$
$242$	0	0
$243$	−11010.9	−2.90680
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1206.10	−0.310699
$248$	0	0
$249$	−1454.35	−0.370142
$250$	0	0
$251$	2254.45	0.566931	0.283465	−	0.958983i	$-0.408516\pi$
$251$	2254.45	0.566931	0.283465	+	0.958983i	$0.408516\pi$
$252$	0	0
$253$	−326.207	−0.0810611
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6499.06	−1.57743	−0.788716	−	0.614757i	$-0.789254\pi$
$257$	−6499.06	−1.57743	−0.788716	+	0.614757i	$0.789254\pi$
$258$	0	0
$259$	−500.407	−0.120053
$260$	0	0
$261$	−8557.67	−2.02953
$262$	0	0
$263$	−2844.78	−0.666984	−0.333492	−	0.942753i	$-0.608227\pi$
$263$	−2844.78	−0.666984	−0.333492	+	0.942753i	$0.608227\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15804.1	3.62245
$268$	0	0
$269$	4660.99	1.05645	0.528225	−	0.849104i	$-0.322858\pi$
$269$	4660.99	1.05645	0.528225	+	0.849104i	$0.322858\pi$
$270$	0	0
$271$	−4633.26	−1.03856	−0.519281	−	0.854604i	$-0.673800\pi$
$271$	−4633.26	−1.03856	−0.519281	+	0.854604i	$0.673800\pi$
$272$	0	0
$273$	−238.663	−0.0529104
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1910.67	−0.414445	−0.207223	−	0.978294i	$-0.566442\pi$
$277$	−1910.67	−0.414445	−0.207223	+	0.978294i	$0.566442\pi$
$278$	0	0
$279$	7475.21	1.60405
$280$	0	0
$281$	−4810.85	−1.02132	−0.510660	−	0.859783i	$-0.670599\pi$
$281$	−4810.85	−1.02132	−0.510660	+	0.859783i	$0.670599\pi$
$282$	0	0
$283$	3954.01	0.830534	0.415267	−	0.909699i	$-0.363688\pi$
$283$	3954.01	0.830534	0.415267	+	0.909699i	$0.363688\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	187.706	0.0386061
$288$	0	0
$289$	11207.4	2.28118
$290$	0	0
$291$	6442.09	1.29774
$292$	0	0
$293$	6442.81	1.28462	0.642309	−	0.766446i	$-0.277977\pi$
$293$	6442.81	1.28462	0.642309	+	0.766446i	$0.277977\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7427.57	1.45115
$298$	0	0
$299$	−241.643	−0.0467377
$300$	0	0
$301$	−411.241	−0.0787493
$302$	0	0
$303$	−3244.80	−0.615210
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−10332.7	−1.92091	−0.960457	−	0.278427i	$-0.910187\pi$
$307$	−10332.7	−1.92091	−0.960457	+	0.278427i	$0.910187\pi$
$308$	0	0
$309$	−20051.7	−3.69158
$310$	0	0
$311$	2287.65	0.417109	0.208554	−	0.978011i	$-0.433124\pi$
$311$	2287.65	0.417109	0.208554	+	0.978011i	$0.433124\pi$
$312$	0	0
$313$	−3725.99	−0.672861	−0.336431	−	0.941708i	$-0.609220\pi$
$313$	−3725.99	−0.672861	−0.336431	+	0.941708i	$0.609220\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8533.31	−1.51192	−0.755960	−	0.654618i	$-0.772829\pi$
$317$	−8533.31	−1.51192	−0.755960	+	0.654618i	$0.772829\pi$
$318$	0	0
$319$	2146.13	0.376678
$320$	0	0
$321$	−5146.07	−0.894784
$322$	0	0
$323$	−11779.6	−2.02921
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−860.052	−0.145446
$328$	0	0
$329$	−1018.32	−0.170643
$330$	0	0
$331$	10582.4	1.75729	0.878645	−	0.477476i	$-0.158448\pi$
$331$	10582.4	1.75729	0.878645	+	0.477476i	$0.158448\pi$
$332$	0	0
$333$	18783.7	3.09111
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7118.09	−1.15058	−0.575292	−	0.817948i	$-0.695112\pi$
$337$	−7118.09	−1.15058	−0.575292	+	0.817948i	$0.695112\pi$
$338$	0	0
$339$	9151.27	1.46616
$340$	0	0
$341$	−1874.67	−0.297710
$342$	0	0
$343$	1272.40	0.200301
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4798.89	−0.742415	−0.371208	−	0.928550i	$-0.621056\pi$
$347$	−4798.89	−0.742415	−0.371208	+	0.928550i	$0.621056\pi$
$348$	0	0
$349$	−827.529	−0.126924	−0.0634622	−	0.997984i	$-0.520214\pi$
$349$	−827.529	−0.126924	−0.0634622	+	0.997984i	$0.520214\pi$
$350$	0	0
$351$	5502.09	0.836694
$352$	0	0
$353$	5325.99	0.803041	0.401521	−	0.915850i	$-0.368482\pi$
$353$	5325.99	0.803041	0.401521	+	0.915850i	$0.368482\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2330.94	−0.345564
$358$	0	0
$359$	637.847	0.0937724	0.0468862	−	0.998900i	$-0.485070\pi$
$359$	637.847	0.0937724	0.0468862	+	0.998900i	$0.485070\pi$
$360$	0	0
$361$	1748.63	0.254939
$362$	0	0
$363$	10074.4	1.45667
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2496.04	0.355020	0.177510	−	0.984119i	$-0.443196\pi$
$367$	2496.04	0.355020	0.177510	+	0.984119i	$0.443196\pi$
$368$	0	0
$369$	−7045.89	−0.994022
$370$	0	0
$371$	34.3551	0.00480763
$372$	0	0
$373$	−6650.68	−0.923215	−0.461608	−	0.887084i	$-0.652727\pi$
$373$	−6650.68	−0.923215	−0.461608	+	0.887084i	$0.652727\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1589.78	0.217182
$378$	0	0
$379$	−8689.59	−1.17772	−0.588858	−	0.808237i	$-0.700422\pi$
$379$	−8689.59	−1.17772	−0.588858	+	0.808237i	$0.700422\pi$
$380$	0	0
$381$	5623.29	0.756141
$382$	0	0
$383$	12703.7	1.69485	0.847427	−	0.530912i	$-0.178151\pi$
$383$	12703.7	1.69485	0.847427	+	0.530912i	$0.178151\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15436.7	2.02762
$388$	0	0
$389$	−2784.35	−0.362910	−0.181455	−	0.983399i	$-0.558081\pi$
$389$	−2784.35	−0.362910	−0.181455	+	0.983399i	$0.558081\pi$
$390$	0	0
$391$	−2360.04	−0.305249
$392$	0	0
$393$	14678.2	1.88401
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2596.32	−0.328226	−0.164113	−	0.986442i	$-0.552476\pi$
$397$	−2596.32	−0.328226	−0.164113	+	0.986442i	$0.552476\pi$
$398$	0	0
$399$	1703.27	0.213710
$400$	0	0
$401$	−2228.04	−0.277464	−0.138732	−	0.990330i	$-0.544303\pi$
$401$	−2228.04	−0.277464	−0.138732	+	0.990330i	$0.544303\pi$
$402$	0	0
$403$	−1388.69	−0.171651
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4710.66	−0.573707
$408$	0	0
$409$	1170.36	0.141493	0.0707464	−	0.997494i	$-0.477462\pi$
$409$	1170.36	0.141493	0.0707464	+	0.997494i	$0.477462\pi$
$410$	0	0
$411$	27525.6	3.30350
$412$	0	0
$413$	−241.423	−0.0287643
$414$	0	0
$415$	0	0
$416$	0	0
$417$	17656.2	2.07345
$418$	0	0
$419$	−11072.2	−1.29096	−0.645481	−	0.763777i	$-0.723343\pi$
$419$	−11072.2	−1.29096	−0.645481	+	0.763777i	$0.723343\pi$
$420$	0	0
$421$	7898.67	0.914389	0.457194	−	0.889367i	$-0.348854\pi$
$421$	7898.67	0.914389	0.457194	+	0.889367i	$0.348854\pi$
$422$	0	0
$423$	38224.3	4.39368
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1476.79	−0.167370
$428$	0	0
$429$	−2246.69	−0.252847
$430$	0	0
$431$	−11683.1	−1.30570	−0.652850	−	0.757487i	$-0.726427\pi$
$431$	−11683.1	−1.30570	−0.652850	+	0.757487i	$0.726427\pi$
$432$	0	0
$433$	−4501.36	−0.499588	−0.249794	−	0.968299i	$-0.580363\pi$
$433$	−4501.36	−0.499588	−0.249794	+	0.968299i	$0.580363\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1724.54	0.188777
$438$	0	0
$439$	9026.03	0.981296	0.490648	−	0.871358i	$-0.336760\pi$
$439$	9026.03	0.981296	0.490648	+	0.871358i	$0.336760\pi$
$440$	0	0
$441$	−23759.3	−2.56552
$442$	0	0
$443$	11118.7	1.19248	0.596239	−	0.802807i	$-0.296661\pi$
$443$	11118.7	1.19248	0.596239	+	0.802807i	$0.296661\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	25409.9	2.68870
$448$	0	0
$449$	3974.53	0.417749	0.208875	−	0.977942i	$-0.433020\pi$
$449$	3974.53	0.417749	0.208875	+	0.977942i	$0.433020\pi$
$450$	0	0
$451$	1767.00	0.184489
$452$	0	0
$453$	−13846.4	−1.43611
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6738.52	−0.689748	−0.344874	−	0.938649i	$-0.612078\pi$
$457$	−6738.52	−0.689748	−0.344874	+	0.938649i	$0.612078\pi$
$458$	0	0
$459$	53736.9	5.46454
$460$	0	0
$461$	18299.9	1.84883	0.924416	−	0.381385i	$-0.124553\pi$
$461$	18299.9	1.84883	0.924416	+	0.381385i	$0.124553\pi$
$462$	0	0
$463$	−9486.86	−0.952250	−0.476125	−	0.879378i	$-0.657959\pi$
$463$	−9486.86	−0.952250	−0.476125	+	0.879378i	$0.657959\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3220.12	0.319078	0.159539	−	0.987192i	$-0.448999\pi$
$467$	3220.12	0.319078	0.159539	+	0.987192i	$0.448999\pi$
$468$	0	0
$469$	714.500	0.0703465
$470$	0	0
$471$	6000.36	0.587011
$472$	0	0
$473$	−3871.28	−0.376324
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1289.58	−0.123786
$478$	0	0
$479$	−19935.4	−1.90161	−0.950803	−	0.309796i	$-0.899739\pi$
$479$	−19935.4	−1.90161	−0.950803	+	0.309796i	$0.899739\pi$
$480$	0	0
$481$	−3489.49	−0.330784
$482$	0	0
$483$	341.250	0.0321478
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−15349.4	−1.42823	−0.714116	−	0.700028i	$-0.753171\pi$
$487$	−15349.4	−1.42823	−0.714116	+	0.700028i	$0.753171\pi$
$488$	0	0
$489$	8863.98	0.819720
$490$	0	0
$491$	5310.90	0.488141	0.244071	−	0.969757i	$-0.421517\pi$
$491$	5310.90	0.488141	0.244071	+	0.969757i	$0.421517\pi$
$492$	0	0
$493$	15526.8	1.41844
$494$	0	0
$495$	0	0
$496$	0	0
$497$	582.681	0.0525892
$498$	0	0
$499$	−4524.09	−0.405864	−0.202932	−	0.979193i	$-0.565047\pi$
$499$	−4524.09	−0.405864	−0.202932	+	0.979193i	$0.565047\pi$
$500$	0	0
$501$	−26779.7	−2.38808
$502$	0	0
$503$	2911.77	0.258110	0.129055	−	0.991637i	$-0.458806\pi$
$503$	2911.77	0.258110	0.129055	+	0.991637i	$0.458806\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1664.27	−0.145785
$508$	0	0
$509$	5549.24	0.483234	0.241617	−	0.970372i	$-0.422322\pi$
$509$	5549.24	0.483234	0.241617	+	0.970372i	$0.422322\pi$
$510$	0	0
$511$	−1739.57	−0.150595
$512$	0	0
$513$	−39266.8	−3.37948
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9586.06	−0.815463
$518$	0	0
$519$	−6064.23	−0.512891
$520$	0	0
$521$	−6686.60	−0.562275	−0.281137	−	0.959668i	$-0.590712\pi$
$521$	−6686.60	−0.562275	−0.281137	+	0.959668i	$0.590712\pi$
$522$	0	0
$523$	19427.3	1.62428	0.812139	−	0.583463i	$-0.198303\pi$
$523$	19427.3	1.62428	0.812139	+	0.583463i	$0.198303\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13562.8	−1.12107
$528$	0	0
$529$	−11821.5	−0.971603
$530$	0	0
$531$	9062.24	0.740617
$532$	0	0
$533$	1308.93	0.106372
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−30330.0	−2.43731
$538$	0	0
$539$	5958.46	0.476158
$540$	0	0
$541$	17105.2	1.35935	0.679676	−	0.733512i	$-0.262120\pi$
$541$	17105.2	1.35935	0.679676	+	0.733512i	$0.262120\pi$
$542$	0	0
$543$	−27466.3	−2.17070
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13748.0	1.07463	0.537313	−	0.843383i	$-0.319439\pi$
$547$	13748.0	1.07463	0.537313	+	0.843383i	$0.319439\pi$
$548$	0	0
$549$	55433.9	4.30940
$550$	0	0
$551$	−11345.8	−0.877218
$552$	0	0
$553$	−1544.43	−0.118762
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15034.1	−1.14365	−0.571826	−	0.820375i	$-0.693765\pi$
$557$	−15034.1	−1.14365	−0.571826	+	0.820375i	$0.693765\pi$
$558$	0	0
$559$	−2867.70	−0.216979
$560$	0	0
$561$	−21942.6	−1.65137
$562$	0	0
$563$	−15079.8	−1.12884	−0.564421	−	0.825487i	$-0.690901\pi$
$563$	−15079.8	−1.12884	−0.564421	+	0.825487i	$0.690901\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4247.75	−0.314619
$568$	0	0
$569$	−14932.3	−1.10016	−0.550082	−	0.835111i	$-0.685403\pi$
$569$	−14932.3	−1.10016	−0.550082	+	0.835111i	$0.685403\pi$
$570$	0	0
$571$	−12254.6	−0.898143	−0.449071	−	0.893496i	$-0.648245\pi$
$571$	−12254.6	−0.898143	−0.449071	+	0.893496i	$0.648245\pi$
$572$	0	0
$573$	31070.3	2.26524
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−11175.4	−0.806306	−0.403153	−	0.915133i	$-0.632086\pi$
$577$	−11175.4	−0.806306	−0.403153	+	0.915133i	$0.632086\pi$
$578$	0	0
$579$	48371.0	3.47190
$580$	0	0
$581$	−275.319	−0.0196595
$582$	0	0
$583$	323.407	0.0229745
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1361.14	−0.0957077	−0.0478538	−	0.998854i	$-0.515238\pi$
$587$	−1361.14	−0.0957077	−0.0478538	+	0.998854i	$0.515238\pi$
$588$	0	0
$589$	9910.67	0.693314
$590$	0	0
$591$	−29922.6	−2.08266
$592$	0	0
$593$	−13577.2	−0.940220	−0.470110	−	0.882608i	$-0.655786\pi$
$593$	−13577.2	−0.940220	−0.470110	+	0.882608i	$0.655786\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	31775.0	2.17833
$598$	0	0
$599$	11875.3	0.810035	0.405017	−	0.914309i	$-0.367265\pi$
$599$	11875.3	0.810035	0.405017	+	0.914309i	$0.367265\pi$
$600$	0	0
$601$	−12056.6	−0.818300	−0.409150	−	0.912467i	$-0.634175\pi$
$601$	−12056.6	−0.818300	−0.409150	+	0.912467i	$0.634175\pi$
$602$	0	0
$603$	−26820.0	−1.81127
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4168.43	−0.278733	−0.139367	−	0.990241i	$-0.544507\pi$
$607$	−4168.43	−0.278733	−0.139367	+	0.990241i	$0.544507\pi$
$608$	0	0
$609$	−2245.10	−0.149386
$610$	0	0
$611$	−7101.02	−0.470174
$612$	0	0
$613$	−12221.3	−0.805243	−0.402622	−	0.915367i	$-0.631901\pi$
$613$	−12221.3	−0.805243	−0.402622	+	0.915367i	$0.631901\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7280.34	0.475033	0.237517	−	0.971383i	$-0.423667\pi$
$617$	7280.34	0.475033	0.237517	+	0.971383i	$0.423667\pi$
$618$	0	0
$619$	−17633.6	−1.14500	−0.572499	−	0.819905i	$-0.694026\pi$
$619$	−17633.6	−1.14500	−0.572499	+	0.819905i	$0.694026\pi$
$620$	0	0
$621$	−7867.09	−0.508367
$622$	0	0
$623$	2991.84	0.192400
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16034.0	1.02127
$628$	0	0
$629$	−34080.6	−2.16038
$630$	0	0
$631$	−12673.9	−0.799590	−0.399795	−	0.916605i	$-0.630919\pi$
$631$	−12673.9	−0.799590	−0.399795	+	0.916605i	$0.630919\pi$
$632$	0	0
$633$	22046.0	1.38428
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4413.82	0.274540
$638$	0	0
$639$	−21872.0	−1.35406
$640$	0	0
$641$	−23823.5	−1.46797	−0.733986	−	0.679164i	$-0.762342\pi$
$641$	−23823.5	−1.46797	−0.733986	+	0.679164i	$0.762342\pi$
$642$	0	0
$643$	15892.3	0.974700	0.487350	−	0.873207i	$-0.337964\pi$
$643$	15892.3	0.974700	0.487350	+	0.873207i	$0.337964\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2995.26	−0.182003	−0.0910014	−	0.995851i	$-0.529007\pi$
$647$	−2995.26	−0.182003	−0.0910014	+	0.995851i	$0.529007\pi$
$648$	0	0
$649$	−2272.67	−0.137458
$650$	0	0
$651$	1961.12	0.118068
$652$	0	0
$653$	−5064.90	−0.303529	−0.151765	−	0.988417i	$-0.548496\pi$
$653$	−5064.90	−0.303529	−0.151765	+	0.988417i	$0.548496\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	65297.9	3.87750
$658$	0	0
$659$	−3335.11	−0.197143	−0.0985716	−	0.995130i	$-0.531427\pi$
$659$	−3335.11	−0.197143	−0.0985716	+	0.995130i	$0.531427\pi$
$660$	0	0
$661$	16969.4	0.998538	0.499269	−	0.866447i	$-0.333602\pi$
$661$	16969.4	0.998538	0.499269	+	0.866447i	$0.333602\pi$
$662$	0	0
$663$	−16254.3	−0.952134
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2273.13	−0.131958
$668$	0	0
$669$	53502.9	3.09199
$670$	0	0
$671$	−13902.0	−0.799820
$672$	0	0
$673$	8512.32	0.487557	0.243779	−	0.969831i	$-0.421613\pi$
$673$	8512.32	0.487557	0.243779	+	0.969831i	$0.421613\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26671.7	−1.51415	−0.757073	−	0.653331i	$-0.773371\pi$
$677$	−26671.7	−1.51415	−0.757073	+	0.653331i	$0.773371\pi$
$678$	0	0
$679$	1219.54	0.0689271
$680$	0	0
$681$	12188.7	0.685859
$682$	0	0
$683$	20985.5	1.17568	0.587839	−	0.808978i	$-0.299979\pi$
$683$	20985.5	1.17568	0.587839	+	0.808978i	$0.299979\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	12667.8	0.703506
$688$	0	0
$689$	239.569	0.0132465
$690$	0	0
$691$	16350.7	0.900161	0.450080	−	0.892988i	$-0.351395\pi$
$691$	16350.7	0.900161	0.450080	+	0.892988i	$0.351395\pi$
$692$	0	0
$693$	2289.44	0.125496
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12783.9	0.694724
$698$	0	0
$699$	−54235.5	−2.93473
$700$	0	0
$701$	−13364.6	−0.720075	−0.360038	−	0.932938i	$-0.617236\pi$
$701$	−13364.6	−0.720075	−0.360038	+	0.932938i	$0.617236\pi$
$702$	0	0
$703$	24903.5	1.33606
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−614.265	−0.0326758
$708$	0	0
$709$	11560.7	0.612372	0.306186	−	0.951972i	$-0.400947\pi$
$709$	11560.7	0.612372	0.306186	+	0.951972i	$0.400947\pi$
$710$	0	0
$711$	57972.7	3.05787
$712$	0	0
$713$	1985.60	0.104294
$714$	0	0
$715$	0	0
$716$	0	0
$717$	33267.0	1.73275
$718$	0	0
$719$	20938.7	1.08607	0.543034	−	0.839711i	$-0.317276\pi$
$719$	20938.7	1.08607	0.543034	+	0.839711i	$0.317276\pi$
$720$	0	0
$721$	−3795.93	−0.196072
$722$	0	0
$723$	−29230.5	−1.50359
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14403.3	0.734783	0.367392	−	0.930066i	$-0.380251\pi$
$727$	14403.3	0.734783	0.367392	+	0.930066i	$0.380251\pi$
$728$	0	0
$729$	46912.6	2.38341
$730$	0	0
$731$	−28007.8	−1.41711
$732$	0	0
$733$	15053.8	0.758562	0.379281	−	0.925282i	$-0.376171\pi$
$733$	15053.8	0.758562	0.379281	+	0.925282i	$0.376171\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6726.05	0.336170
$738$	0	0
$739$	−11681.2	−0.581462	−0.290731	−	0.956805i	$-0.593898\pi$
$739$	−11681.2	−0.581462	−0.290731	+	0.956805i	$0.593898\pi$
$740$	0	0
$741$	11877.4	0.588836
$742$	0	0
$743$	−1608.30	−0.0794114	−0.0397057	−	0.999211i	$-0.512642\pi$
$743$	−1608.30	−0.0794114	−0.0397057	+	0.999211i	$0.512642\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10334.6	0.506188
$748$	0	0
$749$	−974.190	−0.0475249
$750$	0	0
$751$	20293.7	0.986053	0.493027	−	0.870014i	$-0.335891\pi$
$751$	20293.7	0.986053	0.493027	+	0.870014i	$0.335891\pi$
$752$	0	0
$753$	−22201.2	−1.07445
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−32422.7	−1.55670	−0.778352	−	0.627828i	$-0.783944\pi$
$757$	−32422.7	−1.55670	−0.778352	+	0.627828i	$0.783944\pi$
$758$	0	0
$759$	3212.40	0.153627
$760$	0	0
$761$	−30123.0	−1.43490	−0.717450	−	0.696610i	$-0.754691\pi$
$761$	−30123.0	−1.43490	−0.717450	+	0.696610i	$0.754691\pi$
$762$	0	0
$763$	−162.814	−0.00772514
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1683.51	−0.0792545
$768$	0	0
$769$	−22100.1	−1.03635	−0.518173	−	0.855276i	$-0.673388\pi$
$769$	−22100.1	−1.03635	−0.518173	+	0.855276i	$0.673388\pi$
$770$	0	0
$771$	64001.1	2.98955
$772$	0	0
$773$	−36290.0	−1.68856	−0.844281	−	0.535900i	$-0.819972\pi$
$773$	−36290.0	−1.68856	−0.844281	+	0.535900i	$0.819972\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4927.88	0.227525
$778$	0	0
$779$	−9341.47	−0.429644
$780$	0	0
$781$	5485.15	0.251311
$782$	0	0
$783$	51758.0	2.36230
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2614.32	0.118412	0.0592061	−	0.998246i	$-0.481143\pi$
$787$	2614.32	0.118412	0.0592061	+	0.998246i	$0.481143\pi$
$788$	0	0
$789$	28014.7	1.26407
$790$	0	0
$791$	1732.41	0.0778727
$792$	0	0
$793$	−10298.1	−0.461155
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18176.4	0.807833	0.403916	−	0.914796i	$-0.367649\pi$
$797$	18176.4	0.807833	0.403916	+	0.914796i	$0.367649\pi$
$798$	0	0
$799$	−69353.0	−3.07076
$800$	0	0
$801$	−112304.	−4.95388
$802$	0	0
$803$	−16375.7	−0.719659
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−45900.2	−2.00219
$808$	0	0
$809$	37765.8	1.64126	0.820628	−	0.571463i	$-0.193624\pi$
$809$	37765.8	1.64126	0.820628	+	0.571463i	$0.193624\pi$
$810$	0	0
$811$	−26863.8	−1.16315	−0.581575	−	0.813493i	$-0.697563\pi$
$811$	−26863.8	−1.16315	−0.581575	+	0.813493i	$0.697563\pi$
$812$	0	0
$813$	45627.1	1.96828
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20466.0	0.876395
$818$	0	0
$819$	1695.94	0.0723577
$820$	0	0
$821$	33769.1	1.43551	0.717754	−	0.696297i	$-0.245170\pi$
$821$	33769.1	1.43551	0.717754	+	0.696297i	$0.245170\pi$
$822$	0	0
$823$	43876.8	1.85838	0.929192	−	0.369597i	$-0.120504\pi$
$823$	43876.8	1.85838	0.929192	+	0.369597i	$0.120504\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31471.1	−1.32329	−0.661643	−	0.749819i	$-0.730140\pi$
$827$	−31471.1	−1.32329	−0.661643	+	0.749819i	$0.730140\pi$
$828$	0	0
$829$	14834.2	0.621489	0.310744	−	0.950494i	$-0.399422\pi$
$829$	14834.2	0.621489	0.310744	+	0.950494i	$0.399422\pi$
$830$	0	0
$831$	18815.8	0.785456
$832$	0	0
$833$	43108.2	1.79305
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−45211.1	−1.86706
$838$	0	0
$839$	14251.7	0.586438	0.293219	−	0.956045i	$-0.405273\pi$
$839$	14251.7	0.586438	0.293219	+	0.956045i	$0.405273\pi$
$840$	0	0
$841$	−9434.00	−0.386814
$842$	0	0
$843$	47376.0	1.93561
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1907.16	0.0773683
$848$	0	0
$849$	−38938.0	−1.57403
$850$	0	0
$851$	4989.41	0.200981
$852$	0	0
$853$	−8653.36	−0.347345	−0.173673	−	0.984803i	$-0.555563\pi$
$853$	−8653.36	−0.347345	−0.173673	+	0.984803i	$0.555563\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−33038.7	−1.31690	−0.658448	−	0.752626i	$-0.728787\pi$
$857$	−33038.7	−1.31690	−0.658448	+	0.752626i	$0.728787\pi$
$858$	0	0
$859$	38242.0	1.51898	0.759489	−	0.650520i	$-0.225449\pi$
$859$	38242.0	1.51898	0.759489	+	0.650520i	$0.225449\pi$
$860$	0	0
$861$	−1848.48	−0.0731662
$862$	0	0
$863$	−18929.8	−0.746673	−0.373337	−	0.927696i	$-0.621786\pi$
$863$	−18929.8	−0.746673	−0.373337	+	0.927696i	$0.621786\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−110368.	−4.32329
$868$	0	0
$869$	−14538.7	−0.567538
$870$	0	0
$871$	4982.42	0.193826
$872$	0	0
$873$	−45777.5	−1.77472
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16092.1	0.619601	0.309800	−	0.950802i	$-0.399738\pi$
$877$	16092.1	0.619601	0.309800	+	0.950802i	$0.399738\pi$
$878$	0	0
$879$	−63447.1	−2.43461
$880$	0	0
$881$	20937.9	0.800698	0.400349	−	0.916363i	$-0.368889\pi$
$881$	20937.9	0.800698	0.400349	+	0.916363i	$0.368889\pi$
$882$	0	0
$883$	−27975.8	−1.06621	−0.533104	−	0.846050i	$-0.678975\pi$
$883$	−27975.8	−1.06621	−0.533104	+	0.846050i	$0.678975\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14555.1	0.550974	0.275487	−	0.961305i	$-0.411161\pi$
$887$	14555.1	0.550974	0.275487	+	0.961305i	$0.411161\pi$
$888$	0	0
$889$	1064.53	0.0401611
$890$	0	0
$891$	−39986.8	−1.50349
$892$	0	0
$893$	50677.9	1.89907
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2379.64	0.0885772
$898$	0	0
$899$	−13063.4	−0.484635
$900$	0	0
$901$	2339.78	0.0865143
$902$	0	0
$903$	4049.79	0.149246
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1291.60	0.0472842	0.0236421	−	0.999720i	$-0.492474\pi$
$907$	1291.60	0.0472842	0.0236421	+	0.999720i	$0.492474\pi$
$908$	0	0
$909$	23057.5	0.841331
$910$	0	0
$911$	16998.6	0.618209	0.309105	−	0.951028i	$-0.399971\pi$
$911$	16998.6	0.618209	0.309105	+	0.951028i	$0.399971\pi$
$912$	0	0
$913$	−2591.75	−0.0939480
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2778.69	0.100066
$918$	0	0
$919$	−54171.0	−1.94443	−0.972217	−	0.234081i	$-0.924792\pi$
$919$	−54171.0	−1.94443	−0.972217	+	0.234081i	$0.924792\pi$
$920$	0	0
$921$	101754.	3.64052
$922$	0	0
$923$	4063.21	0.144899
$924$	0	0
$925$	0	0
$926$	0	0
$927$	142487.	5.04842
$928$	0	0
$929$	38630.5	1.36429	0.682145	−	0.731217i	$-0.261047\pi$
$929$	38630.5	1.36429	0.682145	+	0.731217i	$0.261047\pi$
$930$	0	0
$931$	−31500.2	−1.10889
$932$	0	0
$933$	−22528.2	−0.790505
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17689.1	0.616732	0.308366	−	0.951268i	$-0.400218\pi$
$937$	17689.1	0.616732	0.308366	+	0.951268i	$0.400218\pi$
$938$	0	0
$939$	36692.6	1.27521
$940$	0	0
$941$	−17709.0	−0.613492	−0.306746	−	0.951791i	$-0.599240\pi$
$941$	−17709.0	−0.613492	−0.306746	+	0.951791i	$0.599240\pi$
$942$	0	0
$943$	−1871.56	−0.0646303
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−49128.9	−1.68582	−0.842911	−	0.538053i	$-0.819160\pi$
$947$	−49128.9	−1.68582	−0.842911	+	0.538053i	$0.819160\pi$
$948$	0	0
$949$	−12130.6	−0.414936
$950$	0	0
$951$	84033.9	2.86539
$952$	0	0
$953$	31819.0	1.08155	0.540776	−	0.841167i	$-0.318131\pi$
$953$	31819.0	1.08155	0.540776	+	0.841167i	$0.318131\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−21134.6	−0.713880
$958$	0	0
$959$	5210.81	0.175460
$960$	0	0
$961$	−18380.0	−0.616966
$962$	0	0
$963$	36568.0	1.22366
$964$	0	0
$965$	0	0
$966$	0	0
$967$	45878.5	1.52570	0.762850	−	0.646575i	$-0.223799\pi$
$967$	45878.5	1.52570	0.762850	+	0.646575i	$0.223799\pi$
$968$	0	0
$969$	116002.	3.84575
$970$	0	0
$971$	−16620.5	−0.549309	−0.274654	−	0.961543i	$-0.588563\pi$
$971$	−16620.5	−0.549309	−0.274654	+	0.961543i	$0.588563\pi$
$972$	0	0
$973$	3342.45	0.110128
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30392.3	−0.995225	−0.497612	−	0.867400i	$-0.665790\pi$
$977$	−30392.3	−0.995225	−0.497612	+	0.867400i	$0.665790\pi$
$978$	0	0
$979$	28164.1	0.919436
$980$	0	0
$981$	6111.53	0.198905
$982$	0	0
$983$	−6108.94	−0.198215	−0.0991073	−	0.995077i	$-0.531599\pi$
$983$	−6108.94	−0.198215	−0.0991073	+	0.995077i	$0.531599\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	10028.1	0.323402
$988$	0	0
$989$	4100.36	0.131834
$990$	0	0
$991$	20654.6	0.662072	0.331036	−	0.943618i	$-0.392602\pi$
$991$	20654.6	0.662072	0.331036	+	0.943618i	$0.392602\pi$
$992$	0	0
$993$	−104213.	−3.33042
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22620.3	−0.718547	−0.359274	−	0.933232i	$-0.616975\pi$
$997$	−22620.3	−0.718547	−0.359274	+	0.933232i	$0.616975\pi$
$998$	0	0
$999$	−113606.	−3.59794

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.4.a.k.1.1		4
5.2	odd	4		1300.4.c.g.1249.8		8
5.3	odd	4		1300.4.c.g.1249.1		8
5.4	even	2		260.4.a.c.1.4	✓	4
15.14	odd	2		2340.4.a.m.1.3		4
20.19	odd	2		1040.4.a.u.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.4.a.c.1.4	✓	4	5.4	even	2
1040.4.a.u.1.1		4	20.19	odd	2
1300.4.a.k.1.1		4	1.1	even	1	trivial
1300.4.c.g.1249.1		8	5.3	odd	4
1300.4.c.g.1249.8		8	5.2	odd	4
2340.4.a.m.1.3		4	15.14	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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1300.a (trivial)

\(f(q)\)	\(=\)	\(q-9.84775 q^{3} -1.86425 q^{7} +69.9781 q^{9} +O(q^{10})\)	\(q-9.84775 q^{3} -1.86425 q^{7} +69.9781 q^{9} -17.5494 q^{11} -13.0000 q^{13} -126.966 q^{17} +92.7773 q^{19} +18.3587 q^{21} +18.5879 q^{23} -423.237 q^{27} -122.291 q^{29} +106.822 q^{31} +172.822 q^{33} +268.422 q^{37} +128.021 q^{39} -100.687 q^{41} +220.593 q^{43} +546.232 q^{47} -339.525 q^{49} +1250.33 q^{51} -18.4284 q^{53} -913.647 q^{57} +129.501 q^{59} +792.160 q^{61} -130.457 q^{63} -383.263 q^{67} -183.049 q^{69} -312.554 q^{71} +933.120 q^{73} +32.7166 q^{77} +828.441 q^{79} +2278.52 q^{81} +147.683 q^{83} +1204.29 q^{87} -1604.84 q^{89} +24.2353 q^{91} -1051.96 q^{93} -654.169 q^{97} -1228.08 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 8 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 8 * q^7 + 36 * q^9	\( 4 q - 4 q^{3} + 8 q^{7} + 36 q^{9} + 28 q^{11} - 52 q^{13} - 200 q^{17} + 132 q^{19} + 160 q^{21} - 244 q^{23} - 496 q^{27} - 8 q^{29} + 292 q^{31} - 128 q^{33} - 168 q^{37} + 52 q^{39} + 280 q^{41} - 452 q^{43} + 280 q^{47} + 340 q^{49} + 792 q^{51} - 584 q^{53} - 1200 q^{57} - 708 q^{59} + 1128 q^{61} - 232 q^{63} - 176 q^{67} + 160 q^{69} - 1028 q^{71} - 664 q^{73} + 864 q^{77} - 728 q^{79} + 2244 q^{81} - 552 q^{83} + 2064 q^{87} - 2824 q^{89} - 104 q^{91} - 1392 q^{93} - 1160 q^{97} - 4220 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 8 * q^7 + 36 * q^9 + 28 * q^11 - 52 * q^13 - 200 * q^17 + 132 * q^19 + 160 * q^21 - 244 * q^23 - 496 * q^27 - 8 * q^29 + 292 * q^31 - 128 * q^33 - 168 * q^37 + 52 * q^39 + 280 * q^41 - 452 * q^43 + 280 * q^47 + 340 * q^49 + 792 * q^51 - 584 * q^53 - 1200 * q^57 - 708 * q^59 + 1128 * q^61 - 232 * q^63 - 176 * q^67 + 160 * q^69 - 1028 * q^71 - 664 * q^73 + 864 * q^77 - 728 * q^79 + 2244 * q^81 - 552 * q^83 + 2064 * q^87 - 2824 * q^89 - 104 * q^91 - 1392 * q^93 - 1160 * q^97 - 4220 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more