LMFDB - Embedded newform 1568.4.a.t.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1568,4,Mod(1,1568)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1568.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1568.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,0,0,-40,0,0,0,-120,0,0,0,50,0,0,0,0,0,0,0,-248] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$92.5149948890$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	1568.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{3} +1.00000 q^{5} -20.0000 q^{9} -2.64575 q^{11} -60.0000 q^{13} +2.64575 q^{15} +25.0000 q^{17} +97.8928 q^{19} +156.099 q^{23} -124.000 q^{25} -124.350 q^{27} +52.0000 q^{29} -224.889 q^{31} -7.00000 q^{33} +95.0000 q^{37} -158.745 q^{39} +64.0000 q^{41} -201.077 q^{43} -20.0000 q^{45} +436.549 q^{47} +66.1438 q^{51} +203.000 q^{53} -2.64575 q^{55} +259.000 q^{57} +150.808 q^{59} +151.000 q^{61} -60.0000 q^{65} +891.618 q^{67} +413.000 q^{69} +888.972 q^{71} +335.000 q^{73} -328.073 q^{75} +7.93725 q^{79} +211.000 q^{81} -95.2470 q^{83} +25.0000 q^{85} +137.579 q^{87} +1263.00 q^{89} -595.000 q^{93} +97.8928 q^{95} -64.0000 q^{97} +52.9150 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 40 q^{9} - 120 q^{13} + 50 q^{17} - 248 q^{25} + 104 q^{29} - 14 q^{33} + 190 q^{37} + 128 q^{41} - 40 q^{45} + 406 q^{53} + 518 q^{57} + 302 q^{61} - 120 q^{65} + 826 q^{69} + 670 q^{73}+ \cdots - 128 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 40 * q^9 - 120 * q^13 + 50 * q^17 - 248 * q^25 + 104 * q^29 - 14 * q^33 + 190 * q^37 + 128 * q^41 - 40 * q^45 + 406 * q^53 + 518 * q^57 + 302 * q^61 - 120 * q^65 + 826 * q^69 + 670 * q^73 + 422 * q^81 + 50 * q^85 + 2526 * q^89 - 1190 * q^93 - 128 * q^97

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
$3$	2.64575	0.509175	0.254588	−	0.967050i	$-0.418060\pi$
$3$	2.64575	0.509175	0.254588	+	0.967050i	$0.418060\pi$
$4$	0	0
$5$	1.00000	0.0894427	0.0447214	−	0.998999i	$-0.485760\pi$
$5$	1.00000	0.0894427	0.0447214	+	0.998999i	$0.485760\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−20.0000	−0.740741
$10$	0	0
$11$	−2.64575	−0.0725204	−0.0362602	−	0.999342i	$-0.511545\pi$
$11$	−2.64575	−0.0725204	−0.0362602	+	0.999342i	$0.511545\pi$
$12$	0	0
$13$	−60.0000	−1.28008	−0.640039	−	0.768343i	$-0.721082\pi$
$13$	−60.0000	−1.28008	−0.640039	+	0.768343i	$0.721082\pi$
$14$	0	0
$15$	2.64575	0.0455420
$16$	0	0
$17$	25.0000	0.356670	0.178335	−	0.983970i	$-0.442929\pi$
$17$	25.0000	0.356670	0.178335	+	0.983970i	$0.442929\pi$
$18$	0	0
$19$	97.8928	1.18201	0.591004	−	0.806669i	$-0.298732\pi$
$19$	97.8928	1.18201	0.591004	+	0.806669i	$0.298732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	156.099	1.41517	0.707586	−	0.706627i	$-0.249784\pi$
$23$	156.099	1.41517	0.707586	+	0.706627i	$0.249784\pi$
$24$	0	0
$25$	−124.000	−0.992000
$26$	0	0
$27$	−124.350	−0.886342
$28$	0	0
$29$	52.0000	0.332971	0.166485	−	0.986044i	$-0.446758\pi$
$29$	52.0000	0.332971	0.166485	+	0.986044i	$0.446758\pi$
$30$	0	0
$31$	−224.889	−1.30294	−0.651471	−	0.758673i	$-0.725848\pi$
$31$	−224.889	−1.30294	−0.651471	+	0.758673i	$0.725848\pi$
$32$	0	0
$33$	−7.00000	−0.0369256
$34$	0	0
$35$	0	0
$36$	0	0
$37$	95.0000	0.422106	0.211053	−	0.977475i	$-0.432311\pi$
$37$	95.0000	0.422106	0.211053	+	0.977475i	$0.432311\pi$
$38$	0	0
$39$	−158.745	−0.651783
$40$	0	0
$41$	64.0000	0.243783	0.121892	−	0.992543i	$-0.461104\pi$
$41$	64.0000	0.243783	0.121892	+	0.992543i	$0.461104\pi$
$42$	0	0
$43$	−201.077	−0.713116	−0.356558	−	0.934273i	$-0.616050\pi$
$43$	−201.077	−0.713116	−0.356558	+	0.934273i	$0.616050\pi$
$44$	0	0
$45$	−20.0000	−0.0662539
$46$	0	0
$47$	436.549	1.35483	0.677417	−	0.735599i	$-0.263099\pi$
$47$	436.549	1.35483	0.677417	+	0.735599i	$0.263099\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	66.1438	0.181607
$52$	0	0
$53$	203.000	0.526117	0.263058	−	0.964780i	$-0.415269\pi$
$53$	203.000	0.526117	0.263058	+	0.964780i	$0.415269\pi$
$54$	0	0
$55$	−2.64575	−0.00648642
$56$	0	0
$57$	259.000	0.601849
$58$	0	0
$59$	150.808	0.332771	0.166386	−	0.986061i	$-0.446790\pi$
$59$	150.808	0.332771	0.166386	+	0.986061i	$0.446790\pi$
$60$	0	0
$61$	151.000	0.316944	0.158472	−	0.987363i	$-0.449343\pi$
$61$	151.000	0.316944	0.158472	+	0.987363i	$0.449343\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−60.0000	−0.114494
$66$	0	0
$67$	891.618	1.62580	0.812899	−	0.582404i	$-0.197888\pi$
$67$	891.618	1.62580	0.812899	+	0.582404i	$0.197888\pi$
$68$	0	0
$69$	413.000	0.720570
$70$	0	0
$71$	888.972	1.48594	0.742969	−	0.669326i	$-0.233417\pi$
$71$	888.972	1.48594	0.742969	+	0.669326i	$0.233417\pi$
$72$	0	0
$73$	335.000	0.537107	0.268553	−	0.963265i	$-0.413454\pi$
$73$	335.000	0.537107	0.268553	+	0.963265i	$0.413454\pi$
$74$	0	0
$75$	−328.073	−0.505102
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.93725	0.0113039	0.00565197	−	0.999984i	$-0.498201\pi$
$79$	7.93725	0.0113039	0.00565197	+	0.999984i	$0.498201\pi$
$80$	0	0
$81$	211.000	0.289438
$82$	0	0
$83$	−95.2470	−0.125961	−0.0629803	−	0.998015i	$-0.520061\pi$
$83$	−95.2470	−0.125961	−0.0629803	+	0.998015i	$0.520061\pi$
$84$	0	0
$85$	25.0000	0.0319015
$86$	0	0
$87$	137.579	0.169541
$88$	0	0
$89$	1263.00	1.50424	0.752122	−	0.659024i	$-0.229030\pi$
$89$	1263.00	1.50424	0.752122	+	0.659024i	$0.229030\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−595.000	−0.663426
$94$	0	0
$95$	97.8928	0.105722
$96$	0	0
$97$	−64.0000	−0.0669919	−0.0334960	−	0.999439i	$-0.510664\pi$
$97$	−64.0000	−0.0669919	−0.0334960	+	0.999439i	$0.510664\pi$
$98$	0	0
$99$	52.9150	0.0537188
$100$	0	0
$101$	1109.00	1.09257	0.546285	−	0.837599i	$-0.316041\pi$
$101$	1109.00	1.09257	0.546285	+	0.837599i	$0.316041\pi$
$102$	0	0
$103$	−1029.20	−0.984561	−0.492281	−	0.870437i	$-0.663837\pi$
$103$	−1029.20	−0.984561	−0.492281	+	0.870437i	$0.663837\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1595.39	−1.44142	−0.720710	−	0.693236i	$-0.756184\pi$
$107$	−1595.39	−1.44142	−0.720710	+	0.693236i	$0.756184\pi$
$108$	0	0
$109$	−1907.00	−1.67576	−0.837878	−	0.545857i	$-0.816204\pi$
$109$	−1907.00	−1.67576	−0.837878	+	0.545857i	$0.816204\pi$
$110$	0	0
$111$	251.346	0.214926
$112$	0	0
$113$	1288.00	1.07226	0.536128	−	0.844137i	$-0.319887\pi$
$113$	1288.00	1.07226	0.536128	+	0.844137i	$0.319887\pi$
$114$	0	0
$115$	156.099	0.126577
$116$	0	0
$117$	1200.00	0.948205
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1324.00	−0.994741
$122$	0	0
$123$	169.328	0.124128
$124$	0	0
$125$	−249.000	−0.178170
$126$	0	0
$127$	−105.830	−0.0739441	−0.0369720	−	0.999316i	$-0.511771\pi$
$127$	−105.830	−0.0739441	−0.0369720	+	0.999316i	$0.511771\pi$
$128$	0	0
$129$	−532.000	−0.363101
$130$	0	0
$131$	939.242	0.626427	0.313213	−	0.949683i	$-0.398594\pi$
$131$	939.242	0.626427	0.313213	+	0.949683i	$0.398594\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−124.350	−0.0792768
$136$	0	0
$137$	−255.000	−0.159023	−0.0795114	−	0.996834i	$-0.525336\pi$
$137$	−255.000	−0.159023	−0.0795114	+	0.996834i	$0.525336\pi$
$138$	0	0
$139$	497.401	0.303518	0.151759	−	0.988418i	$-0.451506\pi$
$139$	497.401	0.303518	0.151759	+	0.988418i	$0.451506\pi$
$140$	0	0
$141$	1155.00	0.689848
$142$	0	0
$143$	158.745	0.0928317
$144$	0	0
$145$	52.0000	0.0297818
$146$	0	0
$147$	0	0
$148$	0	0
$149$	777.000	0.427210	0.213605	−	0.976920i	$-0.431479\pi$
$149$	777.000	0.427210	0.213605	+	0.976920i	$0.431479\pi$
$150$	0	0
$151$	1997.54	1.07654	0.538270	−	0.842772i	$-0.319078\pi$
$151$	1997.54	1.07654	0.538270	+	0.842772i	$0.319078\pi$
$152$	0	0
$153$	−500.000	−0.264200
$154$	0	0
$155$	−224.889	−0.116539
$156$	0	0
$157$	2771.00	1.40860	0.704299	−	0.709903i	$-0.251261\pi$
$157$	2771.00	1.40860	0.704299	+	0.709903i	$0.251261\pi$
$158$	0	0
$159$	537.088	0.267886
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1891.71	0.909020	0.454510	−	0.890742i	$-0.349814\pi$
$163$	1891.71	0.909020	0.454510	+	0.890742i	$0.349814\pi$
$164$	0	0
$165$	−7.00000	−0.00330272
$166$	0	0
$167$	−841.349	−0.389853	−0.194927	−	0.980818i	$-0.562447\pi$
$167$	−841.349	−0.389853	−0.194927	+	0.980818i	$0.562447\pi$
$168$	0	0
$169$	1403.00	0.638598
$170$	0	0
$171$	−1957.86	−0.875561
$172$	0	0
$173$	3135.00	1.37774	0.688872	−	0.724883i	$-0.258106\pi$
$173$	3135.00	1.37774	0.688872	+	0.724883i	$0.258106\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	399.000	0.169439
$178$	0	0
$179$	−4616.84	−1.92781	−0.963907	−	0.266241i	$-0.914218\pi$
$179$	−4616.84	−1.92781	−0.963907	+	0.266241i	$0.914218\pi$
$180$	0	0
$181$	3074.00	1.26237	0.631184	−	0.775633i	$-0.282569\pi$
$181$	3074.00	1.26237	0.631184	+	0.775633i	$0.282569\pi$
$182$	0	0
$183$	399.508	0.161380
$184$	0	0
$185$	95.0000	0.0377543
$186$	0	0
$187$	−66.1438	−0.0258658
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3039.97	1.15165	0.575823	−	0.817574i	$-0.304682\pi$
$191$	3039.97	1.15165	0.575823	+	0.817574i	$0.304682\pi$
$192$	0	0
$193$	−1967.00	−0.733615	−0.366808	−	0.930297i	$-0.619549\pi$
$193$	−1967.00	−0.733615	−0.366808	+	0.930297i	$0.619549\pi$
$194$	0	0
$195$	−158.745	−0.0582973
$196$	0	0
$197$	3876.00	1.40179	0.700897	−	0.713262i	$-0.252783\pi$
$197$	3876.00	1.40179	0.700897	+	0.713262i	$0.252783\pi$
$198$	0	0
$199$	3796.65	1.35245	0.676225	−	0.736695i	$-0.263615\pi$
$199$	3796.65	1.35245	0.676225	+	0.736695i	$0.263615\pi$
$200$	0	0
$201$	2359.00	0.827816
$202$	0	0
$203$	0	0
$204$	0	0
$205$	64.0000	0.0218047
$206$	0	0
$207$	−3121.99	−1.04828
$208$	0	0
$209$	−259.000	−0.0857196
$210$	0	0
$211$	−3058.49	−0.997891	−0.498946	−	0.866633i	$-0.666279\pi$
$211$	−3058.49	−0.997891	−0.498946	+	0.866633i	$0.666279\pi$
$212$	0	0
$213$	2352.00	0.756603
$214$	0	0
$215$	−201.077	−0.0637830
$216$	0	0
$217$	0	0
$218$	0	0
$219$	886.327	0.273481
$220$	0	0
$221$	−1500.00	−0.456565
$222$	0	0
$223$	−3259.57	−0.978819	−0.489410	−	0.872054i	$-0.662788\pi$
$223$	−3259.57	−0.978819	−0.489410	+	0.872054i	$0.662788\pi$
$224$	0	0
$225$	2480.00	0.734815
$226$	0	0
$227$	−5780.97	−1.69029	−0.845146	−	0.534536i	$-0.820486\pi$
$227$	−5780.97	−1.69029	−0.845146	+	0.534536i	$0.820486\pi$
$228$	0	0
$229$	5865.00	1.69245	0.846223	−	0.532829i	$-0.178871\pi$
$229$	5865.00	1.69245	0.846223	+	0.532829i	$0.178871\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6695.00	−1.88242	−0.941210	−	0.337821i	$-0.890310\pi$
$233$	−6695.00	−1.88242	−0.941210	+	0.337821i	$0.890310\pi$
$234$	0	0
$235$	436.549	0.121180
$236$	0	0
$237$	21.0000	0.00575568
$238$	0	0
$239$	−3413.02	−0.923723	−0.461862	−	0.886952i	$-0.652818\pi$
$239$	−3413.02	−0.923723	−0.461862	+	0.886952i	$0.652818\pi$
$240$	0	0
$241$	1935.00	0.517196	0.258598	−	0.965985i	$-0.416739\pi$
$241$	1935.00	0.517196	0.258598	+	0.965985i	$0.416739\pi$
$242$	0	0
$243$	3915.71	1.03372
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5873.57	−1.51306
$248$	0	0
$249$	−252.000	−0.0641359
$250$	0	0
$251$	5529.62	1.39054	0.695272	−	0.718747i	$-0.255284\pi$
$251$	5529.62	1.39054	0.695272	+	0.718747i	$0.255284\pi$
$252$	0	0
$253$	−413.000	−0.102629
$254$	0	0
$255$	66.1438	0.0162435
$256$	0	0
$257$	4185.00	1.01577	0.507885	−	0.861425i	$-0.330427\pi$
$257$	4185.00	1.01577	0.507885	+	0.861425i	$0.330427\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1040.00	−0.246645
$262$	0	0
$263$	2277.99	0.534095	0.267048	−	0.963683i	$-0.413952\pi$
$263$	2277.99	0.534095	0.267048	+	0.963683i	$0.413952\pi$
$264$	0	0
$265$	203.000	0.0470573
$266$	0	0
$267$	3341.58	0.765924
$268$	0	0
$269$	5947.00	1.34794	0.673968	−	0.738760i	$-0.264588\pi$
$269$	5947.00	1.34794	0.673968	+	0.738760i	$0.264588\pi$
$270$	0	0
$271$	−3251.63	−0.728865	−0.364432	−	0.931230i	$-0.618737\pi$
$271$	−3251.63	−0.728865	−0.364432	+	0.931230i	$0.618737\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	328.073	0.0719402
$276$	0	0
$277$	−3285.00	−0.712551	−0.356275	−	0.934381i	$-0.615953\pi$
$277$	−3285.00	−0.712551	−0.356275	+	0.934381i	$0.615953\pi$
$278$	0	0
$279$	4497.78	0.965143
$280$	0	0
$281$	5520.00	1.17187	0.585935	−	0.810358i	$-0.300727\pi$
$281$	5520.00	1.17187	0.585935	+	0.810358i	$0.300727\pi$
$282$	0	0
$283$	8611.92	1.80892	0.904462	−	0.426554i	$-0.140273\pi$
$283$	8611.92	1.80892	0.904462	+	0.426554i	$0.140273\pi$
$284$	0	0
$285$	259.000	0.0538310
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4288.00	−0.872786
$290$	0	0
$291$	−169.328	−0.0341106
$292$	0	0
$293$	3630.00	0.723778	0.361889	−	0.932221i	$-0.382132\pi$
$293$	3630.00	0.723778	0.361889	+	0.932221i	$0.382132\pi$
$294$	0	0
$295$	150.808	0.0297640
$296$	0	0
$297$	329.000	0.0642778
$298$	0	0
$299$	−9365.96	−1.81153
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2934.14	0.556310
$304$	0	0
$305$	151.000	0.0283483
$306$	0	0
$307$	−4942.26	−0.918794	−0.459397	−	0.888231i	$-0.651935\pi$
$307$	−4942.26	−0.918794	−0.459397	+	0.888231i	$0.651935\pi$
$308$	0	0
$309$	−2723.00	−0.501314
$310$	0	0
$311$	−1828.21	−0.333339	−0.166670	−	0.986013i	$-0.553301\pi$
$311$	−1828.21	−0.333339	−0.166670	+	0.986013i	$0.553301\pi$
$312$	0	0
$313$	5335.00	0.963425	0.481713	−	0.876329i	$-0.340015\pi$
$313$	5335.00	0.963425	0.481713	+	0.876329i	$0.340015\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7135.00	1.26417	0.632084	−	0.774900i	$-0.282200\pi$
$317$	7135.00	1.26417	0.632084	+	0.774900i	$0.282200\pi$
$318$	0	0
$319$	−137.579	−0.0241472
$320$	0	0
$321$	−4221.00	−0.733935
$322$	0	0
$323$	2447.32	0.421587
$324$	0	0
$325$	7440.00	1.26984
$326$	0	0
$327$	−5045.45	−0.853254
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−10172.9	−1.68929	−0.844643	−	0.535329i	$-0.820187\pi$
$331$	−10172.9	−1.68929	−0.844643	+	0.535329i	$0.820187\pi$
$332$	0	0
$333$	−1900.00	−0.312671
$334$	0	0
$335$	891.618	0.145416
$336$	0	0
$337$	10384.0	1.67849	0.839247	−	0.543750i	$-0.182996\pi$
$337$	10384.0	1.67849	0.839247	+	0.543750i	$0.182996\pi$
$338$	0	0
$339$	3407.73	0.545966
$340$	0	0
$341$	595.000	0.0944899
$342$	0	0
$343$	0	0
$344$	0	0
$345$	413.000	0.0644498
$346$	0	0
$347$	1949.92	0.301663	0.150832	−	0.988559i	$-0.451805\pi$
$347$	1949.92	0.301663	0.150832	+	0.988559i	$0.451805\pi$
$348$	0	0
$349$	6348.00	0.973641	0.486820	−	0.873502i	$-0.338157\pi$
$349$	6348.00	0.973641	0.486820	+	0.873502i	$0.338157\pi$
$350$	0	0
$351$	7461.02	1.13459
$352$	0	0
$353$	−10527.0	−1.58724	−0.793620	−	0.608414i	$-0.791806\pi$
$353$	−10527.0	−1.58724	−0.793620	+	0.608414i	$0.791806\pi$
$354$	0	0
$355$	888.972	0.132906
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−833.412	−0.122523	−0.0612615	−	0.998122i	$-0.519512\pi$
$359$	−833.412	−0.122523	−0.0612615	+	0.998122i	$0.519512\pi$
$360$	0	0
$361$	2724.00	0.397142
$362$	0	0
$363$	−3502.97	−0.506497
$364$	0	0
$365$	335.000	0.0480403
$366$	0	0
$367$	−2611.36	−0.371422	−0.185711	−	0.982604i	$-0.559459\pi$
$367$	−2611.36	−0.371422	−0.185711	+	0.982604i	$0.559459\pi$
$368$	0	0
$369$	−1280.00	−0.180580
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4655.00	0.646184	0.323092	−	0.946368i	$-0.395278\pi$
$373$	4655.00	0.646184	0.323092	+	0.946368i	$0.395278\pi$
$374$	0	0
$375$	−658.792	−0.0907197
$376$	0	0
$377$	−3120.00	−0.426229
$378$	0	0
$379$	−7307.57	−0.990407	−0.495204	−	0.868777i	$-0.664907\pi$
$379$	−7307.57	−0.990407	−0.495204	+	0.868777i	$0.664907\pi$
$380$	0	0
$381$	−280.000	−0.0376505
$382$	0	0
$383$	12353.0	1.64807	0.824033	−	0.566541i	$-0.191719\pi$
$383$	12353.0	1.64807	0.824033	+	0.566541i	$0.191719\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4021.54	0.528234
$388$	0	0
$389$	−6955.00	−0.906510	−0.453255	−	0.891381i	$-0.649737\pi$
$389$	−6955.00	−0.906510	−0.453255	+	0.891381i	$0.649737\pi$
$390$	0	0
$391$	3902.48	0.504750
$392$	0	0
$393$	2485.00	0.318961
$394$	0	0
$395$	7.93725	0.00101105
$396$	0	0
$397$	241.000	0.0304671	0.0152336	−	0.999884i	$-0.495151\pi$
$397$	241.000	0.0304671	0.0152336	+	0.999884i	$0.495151\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8159.00	−1.01606	−0.508031	−	0.861339i	$-0.669627\pi$
$401$	−8159.00	−1.01606	−0.508031	+	0.861339i	$0.669627\pi$
$402$	0	0
$403$	13493.3	1.66787
$404$	0	0
$405$	211.000	0.0258881
$406$	0	0
$407$	−251.346	−0.0306112
$408$	0	0
$409$	−4327.00	−0.523121	−0.261560	−	0.965187i	$-0.584237\pi$
$409$	−4327.00	−0.523121	−0.261560	+	0.965187i	$0.584237\pi$
$410$	0	0
$411$	−674.667	−0.0809704
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−95.2470	−0.0112662
$416$	0	0
$417$	1316.00	0.154544
$418$	0	0
$419$	2381.18	0.277633	0.138816	−	0.990318i	$-0.455670\pi$
$419$	2381.18	0.277633	0.138816	+	0.990318i	$0.455670\pi$
$420$	0	0
$421$	−964.000	−0.111597	−0.0557987	−	0.998442i	$-0.517770\pi$
$421$	−964.000	−0.111597	−0.0557987	+	0.998442i	$0.517770\pi$
$422$	0	0
$423$	−8730.98	−1.00358
$424$	0	0
$425$	−3100.00	−0.353817
$426$	0	0
$427$	0	0
$428$	0	0
$429$	420.000	0.0472676
$430$	0	0
$431$	−9707.26	−1.08488	−0.542439	−	0.840095i	$-0.682499\pi$
$431$	−9707.26	−1.08488	−0.542439	+	0.840095i	$0.682499\pi$
$432$	0	0
$433$	−10392.0	−1.15337	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−10392.0	−1.15337	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	137.579	0.0151642
$436$	0	0
$437$	15281.0	1.67274
$438$	0	0
$439$	7632.99	0.829847	0.414924	−	0.909856i	$-0.363808\pi$
$439$	7632.99	0.829847	0.414924	+	0.909856i	$0.363808\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8543.13	0.916245	0.458122	−	0.888889i	$-0.348522\pi$
$443$	8543.13	0.916245	0.458122	+	0.888889i	$0.348522\pi$
$444$	0	0
$445$	1263.00	0.134544
$446$	0	0
$447$	2055.75	0.217525
$448$	0	0
$449$	−13608.0	−1.43029	−0.715146	−	0.698975i	$-0.753640\pi$
$449$	−13608.0	−1.43029	−0.715146	+	0.698975i	$0.753640\pi$
$450$	0	0
$451$	−169.328	−0.0176793
$452$	0	0
$453$	5285.00	0.548148
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2625.00	−0.268692	−0.134346	−	0.990934i	$-0.542893\pi$
$457$	−2625.00	−0.268692	−0.134346	+	0.990934i	$0.542893\pi$
$458$	0	0
$459$	−3108.76	−0.316132
$460$	0	0
$461$	5860.00	0.592033	0.296017	−	0.955183i	$-0.404342\pi$
$461$	5860.00	0.592033	0.296017	+	0.955183i	$0.404342\pi$
$462$	0	0
$463$	−11260.3	−1.13026	−0.565131	−	0.825001i	$-0.691174\pi$
$463$	−11260.3	−1.13026	−0.565131	+	0.825001i	$0.691174\pi$
$464$	0	0
$465$	−595.000	−0.0593386
$466$	0	0
$467$	8643.67	0.856491	0.428246	−	0.903662i	$-0.359132\pi$
$467$	8643.67	0.856491	0.428246	+	0.903662i	$0.359132\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	7331.38	0.717223
$472$	0	0
$473$	532.000	0.0517154
$474$	0	0
$475$	−12138.7	−1.17255
$476$	0	0
$477$	−4060.00	−0.389716
$478$	0	0
$479$	−5595.76	−0.533772	−0.266886	−	0.963728i	$-0.585995\pi$
$479$	−5595.76	−0.533772	−0.266886	+	0.963728i	$0.585995\pi$
$480$	0	0
$481$	−5700.00	−0.540328
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−64.0000	−0.00599194
$486$	0	0
$487$	542.379	0.0504672	0.0252336	−	0.999682i	$-0.491967\pi$
$487$	542.379	0.0504672	0.0252336	+	0.999682i	$0.491967\pi$
$488$	0	0
$489$	5005.00	0.462851
$490$	0	0
$491$	15001.4	1.37883	0.689414	−	0.724368i	$-0.257868\pi$
$491$	15001.4	1.37883	0.689414	+	0.724368i	$0.257868\pi$
$492$	0	0
$493$	1300.00	0.118761
$494$	0	0
$495$	52.9150	0.00480475
$496$	0	0
$497$	0	0
$498$	0	0
$499$	11077.8	0.993805	0.496902	−	0.867806i	$-0.334471\pi$
$499$	11077.8	0.993805	0.496902	+	0.867806i	$0.334471\pi$
$500$	0	0
$501$	−2226.00	−0.198504
$502$	0	0
$503$	−18477.9	−1.63795	−0.818976	−	0.573827i	$-0.805458\pi$
$503$	−18477.9	−1.63795	−0.818976	+	0.573827i	$0.805458\pi$
$504$	0	0
$505$	1109.00	0.0977225
$506$	0	0
$507$	3711.99	0.325158
$508$	0	0
$509$	1593.00	0.138720	0.0693600	−	0.997592i	$-0.477904\pi$
$509$	1593.00	0.138720	0.0693600	+	0.997592i	$0.477904\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−12173.0	−1.04766
$514$	0	0
$515$	−1029.20	−0.0880618
$516$	0	0
$517$	−1155.00	−0.0982531
$518$	0	0
$519$	8294.43	0.701513
$520$	0	0
$521$	−19585.0	−1.64690	−0.823450	−	0.567389i	$-0.807953\pi$
$521$	−19585.0	−1.64690	−0.823450	+	0.567389i	$0.807953\pi$
$522$	0	0
$523$	3082.30	0.257705	0.128852	−	0.991664i	$-0.458871\pi$
$523$	3082.30	0.257705	0.128852	+	0.991664i	$0.458871\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5622.22	−0.464721
$528$	0	0
$529$	12200.0	1.00271
$530$	0	0
$531$	−3016.16	−0.246497
$532$	0	0
$533$	−3840.00	−0.312062
$534$	0	0
$535$	−1595.39	−0.128925
$536$	0	0
$537$	−12215.0	−0.981594
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18529.0	1.47250	0.736251	−	0.676708i	$-0.236594\pi$
$541$	18529.0	1.47250	0.736251	+	0.676708i	$0.236594\pi$
$542$	0	0
$543$	8133.04	0.642766
$544$	0	0
$545$	−1907.00	−0.149884
$546$	0	0
$547$	7201.74	0.562932	0.281466	−	0.959571i	$-0.409179\pi$
$547$	7201.74	0.562932	0.281466	+	0.959571i	$0.409179\pi$
$548$	0	0
$549$	−3020.00	−0.234773
$550$	0	0
$551$	5090.43	0.393574
$552$	0	0
$553$	0	0
$554$	0	0
$555$	251.346	0.0192235
$556$	0	0
$557$	13221.0	1.00573	0.502865	−	0.864365i	$-0.332279\pi$
$557$	13221.0	1.00573	0.502865	+	0.864365i	$0.332279\pi$
$558$	0	0
$559$	12064.6	0.912843
$560$	0	0
$561$	−175.000	−0.0131702
$562$	0	0
$563$	−1775.30	−0.132895	−0.0664475	−	0.997790i	$-0.521167\pi$
$563$	−1775.30	−0.132895	−0.0664475	+	0.997790i	$0.521167\pi$
$564$	0	0
$565$	1288.00	0.0959054
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5415.00	0.398961	0.199480	−	0.979902i	$-0.436075\pi$
$569$	5415.00	0.398961	0.199480	+	0.979902i	$0.436075\pi$
$570$	0	0
$571$	−7410.75	−0.543135	−0.271568	−	0.962419i	$-0.587542\pi$
$571$	−7410.75	−0.543135	−0.271568	+	0.962419i	$0.587542\pi$
$572$	0	0
$573$	8043.00	0.586389
$574$	0	0
$575$	−19356.3	−1.40385
$576$	0	0
$577$	−16201.0	−1.16890	−0.584451	−	0.811429i	$-0.698690\pi$
$577$	−16201.0	−1.16890	−0.584451	+	0.811429i	$0.698690\pi$
$578$	0	0
$579$	−5204.19	−0.373539
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−537.088	−0.0381542
$584$	0	0
$585$	1200.00	0.0848101
$586$	0	0
$587$	24288.0	1.70779	0.853895	−	0.520445i	$-0.174234\pi$
$587$	24288.0	1.70779	0.853895	+	0.520445i	$0.174234\pi$
$588$	0	0
$589$	−22015.0	−1.54009
$590$	0	0
$591$	10254.9	0.713759
$592$	0	0
$593$	−24567.0	−1.70126	−0.850629	−	0.525767i	$-0.823778\pi$
$593$	−24567.0	−1.70126	−0.850629	+	0.525767i	$0.823778\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10045.0	0.688634
$598$	0	0
$599$	15358.6	1.04764	0.523819	−	0.851830i	$-0.324507\pi$
$599$	15358.6	1.04764	0.523819	+	0.851830i	$0.324507\pi$
$600$	0	0
$601$	−9680.00	−0.656997	−0.328499	−	0.944504i	$-0.606543\pi$
$601$	−9680.00	−0.656997	−0.328499	+	0.944504i	$0.606543\pi$
$602$	0	0
$603$	−17832.4	−1.20430
$604$	0	0
$605$	−1324.00	−0.0889723
$606$	0	0
$607$	−20179.1	−1.34933	−0.674667	−	0.738122i	$-0.735713\pi$
$607$	−20179.1	−1.34933	−0.674667	+	0.738122i	$0.735713\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−26192.9	−1.73429
$612$	0	0
$613$	605.000	0.0398625	0.0199313	−	0.999801i	$-0.493655\pi$
$613$	605.000	0.0398625	0.0199313	+	0.999801i	$0.493655\pi$
$614$	0	0
$615$	169.328	0.0111024
$616$	0	0
$617$	5400.00	0.352343	0.176172	−	0.984359i	$-0.443629\pi$
$617$	5400.00	0.352343	0.176172	+	0.984359i	$0.443629\pi$
$618$	0	0
$619$	−17586.3	−1.14193	−0.570964	−	0.820975i	$-0.693430\pi$
$619$	−17586.3	−1.14193	−0.570964	+	0.820975i	$0.693430\pi$
$620$	0	0
$621$	−19411.0	−1.25433
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15251.0	0.976064
$626$	0	0
$627$	−685.250	−0.0436463
$628$	0	0
$629$	2375.00	0.150552
$630$	0	0
$631$	11175.7	0.705065	0.352532	−	0.935800i	$-0.385321\pi$
$631$	11175.7	0.705065	0.352532	+	0.935800i	$0.385321\pi$
$632$	0	0
$633$	−8092.00	−0.508101
$634$	0	0
$635$	−105.830	−0.00661376
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−17779.4	−1.10070
$640$	0	0
$641$	10559.0	0.650632	0.325316	−	0.945605i	$-0.394529\pi$
$641$	10559.0	0.650632	0.325316	+	0.945605i	$0.394529\pi$
$642$	0	0
$643$	3756.97	0.230420	0.115210	−	0.993341i	$-0.463246\pi$
$643$	3756.97	0.230420	0.115210	+	0.993341i	$0.463246\pi$
$644$	0	0
$645$	−532.000	−0.0324767
$646$	0	0
$647$	18554.7	1.12745	0.563724	−	0.825963i	$-0.309368\pi$
$647$	18554.7	1.12745	0.563724	+	0.825963i	$0.309368\pi$
$648$	0	0
$649$	−399.000	−0.0241327
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4175.00	0.250200	0.125100	−	0.992144i	$-0.460075\pi$
$653$	4175.00	0.250200	0.125100	+	0.992144i	$0.460075\pi$
$654$	0	0
$655$	939.242	0.0560293
$656$	0	0
$657$	−6700.00	−0.397857
$658$	0	0
$659$	14530.5	0.858917	0.429459	−	0.903086i	$-0.358704\pi$
$659$	14530.5	0.858917	0.429459	+	0.903086i	$0.358704\pi$
$660$	0	0
$661$	12531.0	0.737367	0.368683	−	0.929555i	$-0.379809\pi$
$661$	12531.0	0.737367	0.368683	+	0.929555i	$0.379809\pi$
$662$	0	0
$663$	−3968.63	−0.232472
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8117.17	0.471211
$668$	0	0
$669$	−8624.00	−0.498390
$670$	0	0
$671$	−399.508	−0.0229849
$672$	0	0
$673$	−20800.0	−1.19135	−0.595677	−	0.803224i	$-0.703116\pi$
$673$	−20800.0	−1.19135	−0.595677	+	0.803224i	$0.703116\pi$
$674$	0	0
$675$	15419.4	0.879251
$676$	0	0
$677$	−24975.0	−1.41782	−0.708912	−	0.705297i	$-0.750814\pi$
$677$	−24975.0	−1.41782	−0.708912	+	0.705297i	$0.750814\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−15295.0	−0.860654
$682$	0	0
$683$	−12173.1	−0.681978	−0.340989	−	0.940067i	$-0.610762\pi$
$683$	−12173.1	−0.681978	−0.340989	+	0.940067i	$0.610762\pi$
$684$	0	0
$685$	−255.000	−0.0142234
$686$	0	0
$687$	15517.3	0.861751
$688$	0	0
$689$	−12180.0	−0.673470
$690$	0	0
$691$	−17200.0	−0.946917	−0.473459	−	0.880816i	$-0.656995\pi$
$691$	−17200.0	−0.946917	−0.473459	+	0.880816i	$0.656995\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	497.401	0.0271475
$696$	0	0
$697$	1600.00	0.0869502
$698$	0	0
$699$	−17713.3	−0.958482
$700$	0	0
$701$	−25386.0	−1.36778	−0.683892	−	0.729584i	$-0.739714\pi$
$701$	−25386.0	−1.36778	−0.683892	+	0.729584i	$0.739714\pi$
$702$	0	0
$703$	9299.82	0.498932
$704$	0	0
$705$	1155.00	0.0617019
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8543.00	0.452523	0.226262	−	0.974067i	$-0.427350\pi$
$709$	8543.00	0.452523	0.226262	+	0.974067i	$0.427350\pi$
$710$	0	0
$711$	−158.745	−0.00837328
$712$	0	0
$713$	−35105.0	−1.84389
$714$	0	0
$715$	158.745	0.00830312
$716$	0	0
$717$	−9030.00	−0.470337
$718$	0	0
$719$	−23528.7	−1.22040	−0.610202	−	0.792245i	$-0.708912\pi$
$719$	−23528.7	−1.22040	−0.610202	+	0.792245i	$0.708912\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	5119.53	0.263343
$724$	0	0
$725$	−6448.00	−0.330307
$726$	0	0
$727$	5418.50	0.276425	0.138213	−	0.990403i	$-0.455864\pi$
$727$	5418.50	0.276425	0.138213	+	0.990403i	$0.455864\pi$
$728$	0	0
$729$	4663.00	0.236905
$730$	0	0
$731$	−5026.93	−0.254347
$732$	0	0
$733$	−30953.0	−1.55972	−0.779860	−	0.625954i	$-0.784710\pi$
$733$	−30953.0	−1.55972	−0.779860	+	0.625954i	$0.784710\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2359.00	−0.117904
$738$	0	0
$739$	−4638.00	−0.230868	−0.115434	−	0.993315i	$-0.536826\pi$
$739$	−4638.00	−0.230868	−0.115434	+	0.993315i	$0.536826\pi$
$740$	0	0
$741$	−15540.0	−0.770413
$742$	0	0
$743$	−36045.7	−1.77980	−0.889898	−	0.456159i	$-0.849225\pi$
$743$	−36045.7	−1.77980	−0.889898	+	0.456159i	$0.849225\pi$
$744$	0	0
$745$	777.000	0.0382108
$746$	0	0
$747$	1904.94	0.0933041
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−3584.99	−0.174192	−0.0870960	−	0.996200i	$-0.527759\pi$
$751$	−3584.99	−0.174192	−0.0870960	+	0.996200i	$0.527759\pi$
$752$	0	0
$753$	14630.0	0.708030
$754$	0	0
$755$	1997.54	0.0962887
$756$	0	0
$757$	−8900.00	−0.427313	−0.213657	−	0.976909i	$-0.568537\pi$
$757$	−8900.00	−0.427313	−0.213657	+	0.976909i	$0.568537\pi$
$758$	0	0
$759$	−1092.70	−0.0522560
$760$	0	0
$761$	1593.00	0.0758820	0.0379410	−	0.999280i	$-0.487920\pi$
$761$	1593.00	0.0758820	0.0379410	+	0.999280i	$0.487920\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−500.000	−0.0236308
$766$	0	0
$767$	−9048.47	−0.425973
$768$	0	0
$769$	26240.0	1.23048	0.615240	−	0.788340i	$-0.289059\pi$
$769$	26240.0	1.23048	0.615240	+	0.788340i	$0.289059\pi$
$770$	0	0
$771$	11072.5	0.517205
$772$	0	0
$773$	−31515.0	−1.46639	−0.733193	−	0.680021i	$-0.761971\pi$
$773$	−31515.0	−1.46639	−0.733193	+	0.680021i	$0.761971\pi$
$774$	0	0
$775$	27886.2	1.29252
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6265.14	0.288154
$780$	0	0
$781$	−2352.00	−0.107761
$782$	0	0
$783$	−6466.22	−0.295126
$784$	0	0
$785$	2771.00	0.125989
$786$	0	0
$787$	−13353.1	−0.604812	−0.302406	−	0.953179i	$-0.597790\pi$
$787$	−13353.1	−0.604812	−0.302406	+	0.953179i	$0.597790\pi$
$788$	0	0
$789$	6027.00	0.271948
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9060.00	−0.405713
$794$	0	0
$795$	537.088	0.0239604
$796$	0	0
$797$	−2460.00	−0.109332	−0.0546660	−	0.998505i	$-0.517409\pi$
$797$	−2460.00	−0.109332	−0.0546660	+	0.998505i	$0.517409\pi$
$798$	0	0
$799$	10913.7	0.483229
$800$	0	0
$801$	−25260.0	−1.11425
$802$	0	0
$803$	−886.327	−0.0389512
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15734.3	0.686336
$808$	0	0
$809$	5143.00	0.223508	0.111754	−	0.993736i	$-0.464353\pi$
$809$	5143.00	0.223508	0.111754	+	0.993736i	$0.464353\pi$
$810$	0	0
$811$	40691.7	1.76187	0.880935	−	0.473236i	$-0.156914\pi$
$811$	40691.7	1.76187	0.880935	+	0.473236i	$0.156914\pi$
$812$	0	0
$813$	−8603.00	−0.371120
$814$	0	0
$815$	1891.71	0.0813053
$816$	0	0
$817$	−19684.0	−0.842908
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11345.0	0.482269	0.241135	−	0.970492i	$-0.422480\pi$
$821$	11345.0	0.482269	0.241135	+	0.970492i	$0.422480\pi$
$822$	0	0
$823$	−29132.4	−1.23389	−0.616944	−	0.787007i	$-0.711630\pi$
$823$	−29132.4	−1.23389	−0.616944	+	0.787007i	$0.711630\pi$
$824$	0	0
$825$	868.000	0.0366302
$826$	0	0
$827$	−37326.3	−1.56948	−0.784741	−	0.619824i	$-0.787204\pi$
$827$	−37326.3	−1.56948	−0.784741	+	0.619824i	$0.787204\pi$
$828$	0	0
$829$	17965.0	0.752654	0.376327	−	0.926487i	$-0.377187\pi$
$829$	17965.0	0.752654	0.376327	+	0.926487i	$0.377187\pi$
$830$	0	0
$831$	−8691.29	−0.362813
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−841.349	−0.0348696
$836$	0	0
$837$	27965.0	1.15485
$838$	0	0
$839$	−11662.5	−0.479897	−0.239948	−	0.970786i	$-0.577130\pi$
$839$	−11662.5	−0.479897	−0.239948	+	0.970786i	$0.577130\pi$
$840$	0	0
$841$	−21685.0	−0.889130
$842$	0	0
$843$	14604.5	0.596687
$844$	0	0
$845$	1403.00	0.0571179
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22785.0	0.921059
$850$	0	0
$851$	14829.4	0.597352
$852$	0	0
$853$	37092.0	1.48887	0.744435	−	0.667695i	$-0.232719\pi$
$853$	37092.0	1.48887	0.744435	+	0.667695i	$0.232719\pi$
$854$	0	0
$855$	−1957.86	−0.0783126
$856$	0	0
$857$	15831.0	0.631011	0.315506	−	0.948924i	$-0.397826\pi$
$857$	15831.0	0.631011	0.315506	+	0.948924i	$0.397826\pi$
$858$	0	0
$859$	42980.2	1.70718	0.853589	−	0.520946i	$-0.174421\pi$
$859$	42980.2	1.70718	0.853589	+	0.520946i	$0.174421\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12622.9	0.497900	0.248950	−	0.968516i	$-0.419914\pi$
$863$	12622.9	0.497900	0.248950	+	0.968516i	$0.419914\pi$
$864$	0	0
$865$	3135.00	0.123229
$866$	0	0
$867$	−11345.0	−0.444401
$868$	0	0
$869$	−21.0000	−0.000819765 0
$870$	0	0
$871$	−53497.1	−2.08115
$872$	0	0
$873$	1280.00	0.0496236
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6689.00	0.257550	0.128775	−	0.991674i	$-0.458895\pi$
$877$	6689.00	0.257550	0.128775	+	0.991674i	$0.458895\pi$
$878$	0	0
$879$	9604.08	0.368530
$880$	0	0
$881$	−1074.00	−0.0410715	−0.0205357	−	0.999789i	$-0.506537\pi$
$881$	−1074.00	−0.0410715	−0.0205357	+	0.999789i	$0.506537\pi$
$882$	0	0
$883$	−36246.8	−1.38143	−0.690714	−	0.723128i	$-0.742704\pi$
$883$	−36246.8	−1.38143	−0.690714	+	0.723128i	$0.742704\pi$
$884$	0	0
$885$	399.000	0.0151551
$886$	0	0
$887$	19253.1	0.728813	0.364406	−	0.931240i	$-0.381272\pi$
$887$	19253.1	0.728813	0.364406	+	0.931240i	$0.381272\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−558.254	−0.0209901
$892$	0	0
$893$	42735.0	1.60142
$894$	0	0
$895$	−4616.84	−0.172429
$896$	0	0
$897$	−24780.0	−0.922386
$898$	0	0
$899$	−11694.2	−0.433842
$900$	0	0
$901$	5075.00	0.187650
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3074.00	0.112910
$906$	0	0
$907$	−39281.5	−1.43806	−0.719030	−	0.694979i	$-0.755413\pi$
$907$	−39281.5	−1.43806	−0.719030	+	0.694979i	$0.755413\pi$
$908$	0	0
$909$	−22180.0	−0.809312
$910$	0	0
$911$	22922.8	0.833662	0.416831	−	0.908984i	$-0.363141\pi$
$911$	22922.8	0.833662	0.416831	+	0.908984i	$0.363141\pi$
$912$	0	0
$913$	252.000	0.00913470
$914$	0	0
$915$	399.508	0.0144343
$916$	0	0
$917$	0	0
$918$	0	0
$919$	20205.6	0.725268	0.362634	−	0.931932i	$-0.381878\pi$
$919$	20205.6	0.725268	0.362634	+	0.931932i	$0.381878\pi$
$920$	0	0
$921$	−13076.0	−0.467827
$922$	0	0
$923$	−53338.3	−1.90212
$924$	0	0
$925$	−11780.0	−0.418729
$926$	0	0
$927$	20583.9	0.729305
$928$	0	0
$929$	−45497.0	−1.60679	−0.803395	−	0.595446i	$-0.796975\pi$
$929$	−45497.0	−1.60679	−0.803395	+	0.595446i	$0.796975\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−4837.00	−0.169728
$934$	0	0
$935$	−66.1438	−0.00231351
$936$	0	0
$937$	34810.0	1.21365	0.606827	−	0.794834i	$-0.292442\pi$
$937$	34810.0	1.21365	0.606827	+	0.794834i	$0.292442\pi$
$938$	0	0
$939$	14115.1	0.490552
$940$	0	0
$941$	−43845.0	−1.51892	−0.759461	−	0.650552i	$-0.774537\pi$
$941$	−43845.0	−1.51892	−0.759461	+	0.650552i	$0.774537\pi$
$942$	0	0
$943$	9990.36	0.344996
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21941.2	−0.752897	−0.376449	−	0.926437i	$-0.622855\pi$
$947$	−21941.2	−0.752897	−0.376449	+	0.926437i	$0.622855\pi$
$948$	0	0
$949$	−20100.0	−0.687538
$950$	0	0
$951$	18877.4	0.643683
$952$	0	0
$953$	24120.0	0.819857	0.409928	−	0.912118i	$-0.365554\pi$
$953$	24120.0	0.819857	0.409928	+	0.912118i	$0.365554\pi$
$954$	0	0
$955$	3039.97	0.103006
$956$	0	0
$957$	−364.000	−0.0122951
$958$	0	0
$959$	0	0
$960$	0	0
$961$	20784.0	0.697660
$962$	0	0
$963$	31907.8	1.06772
$964$	0	0
$965$	−1967.00	−0.0656165
$966$	0	0
$967$	35643.6	1.18534	0.592668	−	0.805447i	$-0.298075\pi$
$967$	35643.6	1.18534	0.592668	+	0.805447i	$0.298075\pi$
$968$	0	0
$969$	6475.00	0.214661
$970$	0	0
$971$	−45731.8	−1.51144	−0.755718	−	0.654898i	$-0.772712\pi$
$971$	−45731.8	−1.51144	−0.755718	+	0.654898i	$0.772712\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	19684.4	0.646569
$976$	0	0
$977$	−8719.00	−0.285512	−0.142756	−	0.989758i	$-0.545596\pi$
$977$	−8719.00	−0.285512	−0.142756	+	0.989758i	$0.545596\pi$
$978$	0	0
$979$	−3341.58	−0.109088
$980$	0	0
$981$	38140.0	1.24130
$982$	0	0
$983$	12644.0	0.410257	0.205128	−	0.978735i	$-0.434239\pi$
$983$	12644.0	0.410257	0.205128	+	0.978735i	$0.434239\pi$
$984$	0	0
$985$	3876.00	0.125380
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−31388.0	−1.00918
$990$	0	0
$991$	−33958.2	−1.08851	−0.544257	−	0.838918i	$-0.683189\pi$
$991$	−33958.2	−1.08851	−0.544257	+	0.838918i	$0.683189\pi$
$992$	0	0
$993$	−26915.0	−0.860143
$994$	0	0
$995$	3796.65	0.120967
$996$	0	0
$997$	13819.0	0.438969	0.219485	−	0.975616i	$-0.429562\pi$
$997$	13819.0	0.438969	0.219485	+	0.975616i	$0.429562\pi$
$998$	0	0
$999$	−11813.3	−0.374130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.4.a.t.1.2		2
4.3	odd	2	inner	1568.4.a.t.1.1		2
7.2	even	3		224.4.i.a.193.1	yes	4
7.4	even	3		224.4.i.a.65.1	✓	4
7.6	odd	2		1568.4.a.q.1.1		2
28.11	odd	6		224.4.i.a.65.2	yes	4
28.23	odd	6		224.4.i.a.193.2	yes	4
28.27	even	2		1568.4.a.q.1.2		2
56.11	odd	6		448.4.i.i.65.1		4
56.37	even	6		448.4.i.i.193.2		4
56.51	odd	6		448.4.i.i.193.1		4
56.53	even	6		448.4.i.i.65.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.4.i.a.65.1	✓	4	7.4	even	3
224.4.i.a.65.2	yes	4	28.11	odd	6
224.4.i.a.193.1	yes	4	7.2	even	3
224.4.i.a.193.2	yes	4	28.23	odd	6
448.4.i.i.65.1		4	56.11	odd	6
448.4.i.i.65.2		4	56.53	even	6
448.4.i.i.193.1		4	56.51	odd	6
448.4.i.i.193.2		4	56.37	even	6
1568.4.a.q.1.1		2	7.6	odd	2
1568.4.a.q.1.2		2	28.27	even	2
1568.4.a.t.1.1		2	4.3	odd	2	inner
1568.4.a.t.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{3} +1.00000 q^{5} -20.0000 q^{9} -2.64575 q^{11} -60.0000 q^{13} +2.64575 q^{15} +25.0000 q^{17} +97.8928 q^{19} +156.099 q^{23} -124.000 q^{25} -124.350 q^{27} +52.0000 q^{29} -224.889 q^{31} -7.00000 q^{33} +95.0000 q^{37} -158.745 q^{39} +64.0000 q^{41} -201.077 q^{43} -20.0000 q^{45} +436.549 q^{47} +66.1438 q^{51} +203.000 q^{53} -2.64575 q^{55} +259.000 q^{57} +150.808 q^{59} +151.000 q^{61} -60.0000 q^{65} +891.618 q^{67} +413.000 q^{69} +888.972 q^{71} +335.000 q^{73} -328.073 q^{75} +7.93725 q^{79} +211.000 q^{81} -95.2470 q^{83} +25.0000 q^{85} +137.579 q^{87} +1263.00 q^{89} -595.000 q^{93} +97.8928 q^{95} -64.0000 q^{97} +52.9150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 40 q^{9} - 120 q^{13} + 50 q^{17} - 248 q^{25} + 104 q^{29} - 14 q^{33} + 190 q^{37} + 128 q^{41} - 40 q^{45} + 406 q^{53} + 518 q^{57} + 302 q^{61} - 120 q^{65} + 826 q^{69} + 670 q^{73}+ \cdots - 128 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 40 * q^9 - 120 * q^13 + 50 * q^17 - 248 * q^25 + 104 * q^29 - 14 * q^33 + 190 * q^37 + 128 * q^41 - 40 * q^45 + 406 * q^53 + 518 * q^57 + 302 * q^61 - 120 * q^65 + 826 * q^69 + 670 * q^73 + 422 * q^81 + 50 * q^85 + 2526 * q^89 - 1190 * q^93 - 128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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