LMFDB - Embedded newform 26.6.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [26,6,Mod(1,26)] 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("26.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(26, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 26 = 2 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	26.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.16997931514$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	26.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-4.00000 q^{2} +16.0000 q^{4} -14.0000 q^{5} -170.000 q^{7} -64.0000 q^{8} -243.000 q^{9} +56.0000 q^{10} -250.000 q^{11} -169.000 q^{13} +680.000 q^{14} +256.000 q^{16} +1062.00 q^{17} +972.000 q^{18} -78.0000 q^{19} -224.000 q^{20} +1000.00 q^{22} +1576.00 q^{23} -2929.00 q^{25} +676.000 q^{26} -2720.00 q^{28} +2578.00 q^{29} -8654.00 q^{31} -1024.00 q^{32} -4248.00 q^{34} +2380.00 q^{35} -3888.00 q^{36} +10986.0 q^{37} +312.000 q^{38} +896.000 q^{40} +1050.00 q^{41} -5900.00 q^{43} -4000.00 q^{44} +3402.00 q^{45} -6304.00 q^{46} -5962.00 q^{47} +12093.0 q^{49} +11716.0 q^{50} -2704.00 q^{52} +29046.0 q^{53} +3500.00 q^{55} +10880.0 q^{56} -10312.0 q^{58} -13922.0 q^{59} -32882.0 q^{61} +34616.0 q^{62} +41310.0 q^{63} +4096.00 q^{64} +2366.00 q^{65} -69566.0 q^{67} +16992.0 q^{68} -9520.00 q^{70} -50542.0 q^{71} +15552.0 q^{72} -46750.0 q^{73} -43944.0 q^{74} -1248.00 q^{76} +42500.0 q^{77} -19348.0 q^{79} -3584.00 q^{80} +59049.0 q^{81} -4200.00 q^{82} -87438.0 q^{83} -14868.0 q^{85} +23600.0 q^{86} +16000.0 q^{88} +94170.0 q^{89} -13608.0 q^{90} +28730.0 q^{91} +25216.0 q^{92} +23848.0 q^{94} +1092.00 q^{95} +182786. q^{97} -48372.0 q^{98} +60750.0 q^{99} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	16.0000	0.500000
$5$	−14.0000	−0.250440	−0.125220	−	0.992129i	$-0.539964\pi$
$5$	−14.0000	−0.250440	−0.125220	+	0.992129i	$0.539964\pi$
$6$	0	0
$7$	−170.000	−1.31131	−0.655653	−	0.755063i	$-0.727606\pi$
$7$	−170.000	−1.31131	−0.655653	+	0.755063i	$0.727606\pi$
$8$	−64.0000	−0.353553
$9$	−243.000	−1.00000
$10$	56.0000	0.177088
$11$	−250.000	−0.622957	−0.311479	−	0.950253i	$-0.600824\pi$
$11$	−250.000	−0.622957	−0.311479	+	0.950253i	$0.600824\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	680.000	0.927233
$15$	0	0
$16$	256.000	0.250000
$17$	1062.00	0.891255	0.445628	−	0.895218i	$-0.352981\pi$
$17$	1062.00	0.891255	0.445628	+	0.895218i	$0.352981\pi$
$18$	972.000	0.707107
$19$	−78.0000	−0.0495691	−0.0247845	−	0.999693i	$-0.507890\pi$
$19$	−78.0000	−0.0495691	−0.0247845	+	0.999693i	$0.507890\pi$
$20$	−224.000	−0.125220
$21$	0	0
$22$	1000.00	0.440497
$23$	1576.00	0.621207	0.310604	−	0.950539i	$-0.399469\pi$
$23$	1576.00	0.621207	0.310604	+	0.950539i	$0.399469\pi$
$24$	0	0
$25$	−2929.00	−0.937280
$26$	676.000	0.196116
$27$	0	0
$28$	−2720.00	−0.655653
$29$	2578.00	0.569230	0.284615	−	0.958642i	$-0.408134\pi$
$29$	2578.00	0.569230	0.284615	+	0.958642i	$0.408134\pi$
$30$	0	0
$31$	−8654.00	−1.61738	−0.808691	−	0.588234i	$-0.799824\pi$
$31$	−8654.00	−1.61738	−0.808691	+	0.588234i	$0.799824\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−4248.00	−0.630213
$35$	2380.00	0.328403
$36$	−3888.00	−0.500000
$37$	10986.0	1.31927	0.659637	−	0.751584i	$-0.270710\pi$
$37$	10986.0	1.31927	0.659637	+	0.751584i	$0.270710\pi$
$38$	312.000	0.0350506
$39$	0	0
$40$	896.000	0.0885438
$41$	1050.00	0.0975505	0.0487753	−	0.998810i	$-0.484468\pi$
$41$	1050.00	0.0975505	0.0487753	+	0.998810i	$0.484468\pi$
$42$	0	0
$43$	−5900.00	−0.486610	−0.243305	−	0.969950i	$-0.578232\pi$
$43$	−5900.00	−0.486610	−0.243305	+	0.969950i	$0.578232\pi$
$44$	−4000.00	−0.311479
$45$	3402.00	0.250440
$46$	−6304.00	−0.439260
$47$	−5962.00	−0.393684	−0.196842	−	0.980435i	$-0.563069\pi$
$47$	−5962.00	−0.393684	−0.196842	+	0.980435i	$0.563069\pi$
$48$	0	0
$49$	12093.0	0.719522
$50$	11716.0	0.662757
$51$	0	0
$52$	−2704.00	−0.138675
$53$	29046.0	1.42035	0.710177	−	0.704023i	$-0.248615\pi$
$53$	29046.0	1.42035	0.710177	+	0.704023i	$0.248615\pi$
$54$	0	0
$55$	3500.00	0.156013
$56$	10880.0	0.463616
$57$	0	0
$58$	−10312.0	−0.402507
$59$	−13922.0	−0.520681	−0.260340	−	0.965517i	$-0.583835\pi$
$59$	−13922.0	−0.520681	−0.260340	+	0.965517i	$0.583835\pi$
$60$	0	0
$61$	−32882.0	−1.13145	−0.565723	−	0.824596i	$-0.691403\pi$
$61$	−32882.0	−1.13145	−0.565723	+	0.824596i	$0.691403\pi$
$62$	34616.0	1.14366
$63$	41310.0	1.31131
$64$	4096.00	0.125000
$65$	2366.00	0.0694595
$66$	0	0
$67$	−69566.0	−1.89326	−0.946629	−	0.322324i	$-0.895536\pi$
$67$	−69566.0	−1.89326	−0.946629	+	0.322324i	$0.895536\pi$
$68$	16992.0	0.445628
$69$	0	0
$70$	−9520.00	−0.232216
$71$	−50542.0	−1.18989	−0.594945	−	0.803767i	$-0.702826\pi$
$71$	−50542.0	−1.18989	−0.594945	+	0.803767i	$0.702826\pi$
$72$	15552.0	0.353553
$73$	−46750.0	−1.02677	−0.513387	−	0.858157i	$-0.671609\pi$
$73$	−46750.0	−1.02677	−0.513387	+	0.858157i	$0.671609\pi$
$74$	−43944.0	−0.932868
$75$	0	0
$76$	−1248.00	−0.0247845
$77$	42500.0	0.816887
$78$	0	0
$79$	−19348.0	−0.348793	−0.174397	−	0.984675i	$-0.555798\pi$
$79$	−19348.0	−0.348793	−0.174397	+	0.984675i	$0.555798\pi$
$80$	−3584.00	−0.0626099
$81$	59049.0	1.00000
$82$	−4200.00	−0.0689786
$83$	−87438.0	−1.39317	−0.696586	−	0.717473i	$-0.745299\pi$
$83$	−87438.0	−1.39317	−0.696586	+	0.717473i	$0.745299\pi$
$84$	0	0
$85$	−14868.0	−0.223206
$86$	23600.0	0.344085
$87$	0	0
$88$	16000.0	0.220249
$89$	94170.0	1.26019	0.630097	−	0.776516i	$-0.283015\pi$
$89$	94170.0	1.26019	0.630097	+	0.776516i	$0.283015\pi$
$90$	−13608.0	−0.177088
$91$	28730.0	0.363691
$92$	25216.0	0.310604
$93$	0	0
$94$	23848.0	0.278376
$95$	1092.00	0.0124141
$96$	0	0
$97$	182786.	1.97248	0.986242	−	0.165307i	$-0.0528613\pi$
$97$	182786.	1.97248	0.986242	+	0.165307i	$0.0528613\pi$
$98$	−48372.0	−0.508779
$99$	60750.0	0.622957
$100$	−46864.0	−0.468640
$101$	−18514.0	−0.180591	−0.0902957	−	0.995915i	$-0.528781\pi$
$101$	−18514.0	−0.180591	−0.0902957	+	0.995915i	$0.528781\pi$
$102$	0	0
$103$	116056.	1.07789	0.538945	−	0.842341i	$-0.318823\pi$
$103$	116056.	1.07789	0.538945	+	0.842341i	$0.318823\pi$
$104$	10816.0	0.0980581
$105$	0	0
$106$	−116184.	−1.00434
$107$	153520.	1.29630	0.648150	−	0.761513i	$-0.275543\pi$
$107$	153520.	1.29630	0.648150	+	0.761513i	$0.275543\pi$
$108$	0	0
$109$	−178622.	−1.44002	−0.720010	−	0.693963i	$-0.755863\pi$
$109$	−178622.	−1.44002	−0.720010	+	0.693963i	$0.755863\pi$
$110$	−14000.0	−0.110318
$111$	0	0
$112$	−43520.0	−0.327826
$113$	−244754.	−1.80316	−0.901579	−	0.432615i	$-0.857591\pi$
$113$	−244754.	−1.80316	−0.901579	+	0.432615i	$0.857591\pi$
$114$	0	0
$115$	−22064.0	−0.155575
$116$	41248.0	0.284615
$117$	41067.0	0.277350
$118$	55688.0	0.368177
$119$	−180540.	−1.16871
$120$	0	0
$121$	−98551.0	−0.611924
$122$	131528.	0.800053
$123$	0	0
$124$	−138464.	−0.808691
$125$	84756.0	0.485172
$126$	−165240.	−0.927233
$127$	256600.	1.41172	0.705858	−	0.708353i	$-0.250562\pi$
$127$	256600.	1.41172	0.705858	+	0.708353i	$0.250562\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−9464.00	−0.0491152
$131$	−262736.	−1.33765	−0.668823	−	0.743421i	$-0.733202\pi$
$131$	−262736.	−1.33765	−0.668823	+	0.743421i	$0.733202\pi$
$132$	0	0
$133$	13260.0	0.0650002
$134$	278264.	1.33874
$135$	0	0
$136$	−67968.0	−0.315106
$137$	−38286.0	−0.174276	−0.0871382	−	0.996196i	$-0.527772\pi$
$137$	−38286.0	−0.174276	−0.0871382	+	0.996196i	$0.527772\pi$
$138$	0	0
$139$	−57776.0	−0.253636	−0.126818	−	0.991926i	$-0.540476\pi$
$139$	−57776.0	−0.253636	−0.126818	+	0.991926i	$0.540476\pi$
$140$	38080.0	0.164201
$141$	0	0
$142$	202168.	0.841379
$143$	42250.0	0.172777
$144$	−62208.0	−0.250000
$145$	−36092.0	−0.142558
$146$	187000.	0.726038
$147$	0	0
$148$	175776.	0.659637
$149$	28866.0	0.106517	0.0532587	−	0.998581i	$-0.483039\pi$
$149$	28866.0	0.106517	0.0532587	+	0.998581i	$0.483039\pi$
$150$	0	0
$151$	39870.0	0.142300	0.0711498	−	0.997466i	$-0.477333\pi$
$151$	39870.0	0.142300	0.0711498	+	0.997466i	$0.477333\pi$
$152$	4992.00	0.0175253
$153$	−258066.	−0.891255
$154$	−170000.	−0.577627
$155$	121156.	0.405057
$156$	0	0
$157$	161042.	0.521423	0.260711	−	0.965417i	$-0.416043\pi$
$157$	161042.	0.521423	0.260711	+	0.965417i	$0.416043\pi$
$158$	77392.0	0.246634
$159$	0	0
$160$	14336.0	0.0442719
$161$	−267920.	−0.814593
$162$	−236196.	−0.707107
$163$	312830.	0.922230	0.461115	−	0.887340i	$-0.347450\pi$
$163$	312830.	0.922230	0.461115	+	0.887340i	$0.347450\pi$
$164$	16800.0	0.0487753
$165$	0	0
$166$	349752.	0.985122
$167$	532926.	1.47869	0.739343	−	0.673329i	$-0.235136\pi$
$167$	532926.	1.47869	0.739343	+	0.673329i	$0.235136\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	59472.0	0.157830
$171$	18954.0	0.0495691
$172$	−94400.0	−0.243305
$173$	−630458.	−1.60155	−0.800776	−	0.598964i	$-0.795579\pi$
$173$	−630458.	−1.60155	−0.800776	+	0.598964i	$0.795579\pi$
$174$	0	0
$175$	497930.	1.22906
$176$	−64000.0	−0.155739
$177$	0	0
$178$	−376680.	−0.891092
$179$	−674916.	−1.57441	−0.787204	−	0.616693i	$-0.788472\pi$
$179$	−674916.	−1.57441	−0.787204	+	0.616693i	$0.788472\pi$
$180$	54432.0	0.125220
$181$	186282.	0.422644	0.211322	−	0.977417i	$-0.432223\pi$
$181$	186282.	0.422644	0.211322	+	0.977417i	$0.432223\pi$
$182$	−114920.	−0.257168
$183$	0	0
$184$	−100864.	−0.219630
$185$	−153804.	−0.330399
$186$	0	0
$187$	−265500.	−0.555214
$188$	−95392.0	−0.196842
$189$	0	0
$190$	−4368.00	−0.00877806
$191$	812180.	1.61090	0.805451	−	0.592663i	$-0.201923\pi$
$191$	812180.	1.61090	0.805451	+	0.592663i	$0.201923\pi$
$192$	0	0
$193$	−150142.	−0.290141	−0.145070	−	0.989421i	$-0.546341\pi$
$193$	−150142.	−0.290141	−0.145070	+	0.989421i	$0.546341\pi$
$194$	−731144.	−1.39476
$195$	0	0
$196$	193488.	0.359761
$197$	236394.	0.433981	0.216991	−	0.976174i	$-0.430376\pi$
$197$	236394.	0.433981	0.216991	+	0.976174i	$0.430376\pi$
$198$	−243000.	−0.440497
$199$	−39376.0	−0.0704854	−0.0352427	−	0.999379i	$-0.511220\pi$
$199$	−39376.0	−0.0704854	−0.0352427	+	0.999379i	$0.511220\pi$
$200$	187456.	0.331379
$201$	0	0
$202$	74056.0	0.127697
$203$	−438260.	−0.746435
$204$	0	0
$205$	−14700.0	−0.0244305
$206$	−464224.	−0.762183
$207$	−382968.	−0.621207
$208$	−43264.0	−0.0693375
$209$	19500.0	0.0308794
$210$	0	0
$211$	−410776.	−0.635183	−0.317592	−	0.948228i	$-0.602874\pi$
$211$	−410776.	−0.635183	−0.317592	+	0.948228i	$0.602874\pi$
$212$	464736.	0.710177
$213$	0	0
$214$	−614080.	−0.916623
$215$	82600.0	0.121866
$216$	0	0
$217$	1.47118e6	2.12088
$218$	714488.	1.01825
$219$	0	0
$220$	56000.0	0.0780066
$221$	−179478.	−0.247190
$222$	0	0
$223$	1.08688e6	1.46359	0.731796	−	0.681523i	$-0.238682\pi$
$223$	1.08688e6	1.46359	0.731796	+	0.681523i	$0.238682\pi$
$224$	174080.	0.231808
$225$	711747.	0.937280
$226$	979016.	1.27502
$227$	−256470.	−0.330348	−0.165174	−	0.986264i	$-0.552819\pi$
$227$	−256470.	−0.330348	−0.165174	+	0.986264i	$0.552819\pi$
$228$	0	0
$229$	−298110.	−0.375654	−0.187827	−	0.982202i	$-0.560144\pi$
$229$	−298110.	−0.375654	−0.187827	+	0.982202i	$0.560144\pi$
$230$	88256.0	0.110008
$231$	0	0
$232$	−164992.	−0.201253
$233$	−611926.	−0.738430	−0.369215	−	0.929344i	$-0.620373\pi$
$233$	−611926.	−0.738430	−0.369215	+	0.929344i	$0.620373\pi$
$234$	−164268.	−0.196116
$235$	83468.0	0.0985940
$236$	−222752.	−0.260340
$237$	0	0
$238$	722160.	0.826401
$239$	36570.0	0.0414124	0.0207062	−	0.999786i	$-0.493409\pi$
$239$	36570.0	0.0414124	0.0207062	+	0.999786i	$0.493409\pi$
$240$	0	0
$241$	380922.	0.422468	0.211234	−	0.977436i	$-0.432252\pi$
$241$	380922.	0.422468	0.211234	+	0.977436i	$0.432252\pi$
$242$	394204.	0.432696
$243$	0	0
$244$	−526112.	−0.565723
$245$	−169302.	−0.180197
$246$	0	0
$247$	13182.0	0.0137480
$248$	553856.	0.571831
$249$	0	0
$250$	−339024.	−0.343068
$251$	−1.22807e6	−1.23038	−0.615188	−	0.788380i	$-0.710920\pi$
$251$	−1.22807e6	−1.23038	−0.615188	+	0.788380i	$0.710920\pi$
$252$	660960.	0.655653
$253$	−394000.	−0.386986
$254$	−1.02640e6	−0.998234
$255$	0	0
$256$	65536.0	0.0625000
$257$	−439278.	−0.414865	−0.207432	−	0.978249i	$-0.566511\pi$
$257$	−439278.	−0.414865	−0.207432	+	0.978249i	$0.566511\pi$
$258$	0	0
$259$	−1.86762e6	−1.72997
$260$	37856.0	0.0347297
$261$	−626454.	−0.569230
$262$	1.05094e6	0.945859
$263$	−1.67987e6	−1.49757	−0.748783	−	0.662816i	$-0.769361\pi$
$263$	−1.67987e6	−1.49757	−0.748783	+	0.662816i	$0.769361\pi$
$264$	0	0
$265$	−406644.	−0.355713
$266$	−53040.0	−0.0459621
$267$	0	0
$268$	−1.11306e6	−0.946629
$269$	1.93840e6	1.63329	0.816645	−	0.577141i	$-0.195832\pi$
$269$	1.93840e6	1.63329	0.816645	+	0.577141i	$0.195832\pi$
$270$	0	0
$271$	−695498.	−0.575271	−0.287636	−	0.957740i	$-0.592869\pi$
$271$	−695498.	−0.575271	−0.287636	+	0.957740i	$0.592869\pi$
$272$	271872.	0.222814
$273$	0	0
$274$	153144.	0.123232
$275$	732250.	0.583885
$276$	0	0
$277$	−1.13138e6	−0.885948	−0.442974	−	0.896534i	$-0.646077\pi$
$277$	−1.13138e6	−0.885948	−0.442974	+	0.896534i	$0.646077\pi$
$278$	231104.	0.179348
$279$	2.10292e6	1.61738
$280$	−152320.	−0.116108
$281$	1.73122e6	1.30793	0.653967	−	0.756523i	$-0.273103\pi$
$281$	1.73122e6	1.30793	0.653967	+	0.756523i	$0.273103\pi$
$282$	0	0
$283$	−1.47124e6	−1.09199	−0.545995	−	0.837788i	$-0.683848\pi$
$283$	−1.47124e6	−1.09199	−0.545995	+	0.837788i	$0.683848\pi$
$284$	−808672.	−0.594945
$285$	0	0
$286$	−169000.	−0.122172
$287$	−178500.	−0.127919
$288$	248832.	0.176777
$289$	−292013.	−0.205664
$290$	144368.	0.100804
$291$	0	0
$292$	−748000.	−0.513387
$293$	2.88855e6	1.96567	0.982834	−	0.184491i	$-0.0590637\pi$
$293$	2.88855e6	1.96567	0.982834	+	0.184491i	$0.0590637\pi$
$294$	0	0
$295$	194908.	0.130399
$296$	−703104.	−0.466434
$297$	0	0
$298$	−115464.	−0.0753192
$299$	−266344.	−0.172292
$300$	0	0
$301$	1.00300e6	0.638094
$302$	−159480.	−0.100621
$303$	0	0
$304$	−19968.0	−0.0123923
$305$	460348.	0.283359
$306$	1.03226e6	0.630213
$307$	874118.	0.529327	0.264664	−	0.964341i	$-0.414739\pi$
$307$	874118.	0.529327	0.264664	+	0.964341i	$0.414739\pi$
$308$	680000.	0.408444
$309$	0	0
$310$	−484624.	−0.286418
$311$	2.68224e6	1.57252	0.786261	−	0.617895i	$-0.212014\pi$
$311$	2.68224e6	1.57252	0.786261	+	0.617895i	$0.212014\pi$
$312$	0	0
$313$	−1.34459e6	−0.775761	−0.387880	−	0.921710i	$-0.626793\pi$
$313$	−1.34459e6	−0.775761	−0.387880	+	0.921710i	$0.626793\pi$
$314$	−644168.	−0.368702
$315$	−578340.	−0.328403
$316$	−309568.	−0.174397
$317$	1.32074e6	0.738191	0.369095	−	0.929392i	$-0.379668\pi$
$317$	1.32074e6	0.738191	0.369095	+	0.929392i	$0.379668\pi$
$318$	0	0
$319$	−644500.	−0.354606
$320$	−57344.0	−0.0313050
$321$	0	0
$322$	1.07168e6	0.576004
$323$	−82836.0	−0.0441787
$324$	944784.	0.500000
$325$	495001.	0.259955
$326$	−1.25132e6	−0.652115
$327$	0	0
$328$	−67200.0	−0.0344893
$329$	1.01354e6	0.516239
$330$	0	0
$331$	−2.05728e6	−1.03210	−0.516051	−	0.856558i	$-0.672599\pi$
$331$	−2.05728e6	−1.03210	−0.516051	+	0.856558i	$0.672599\pi$
$332$	−1.39901e6	−0.696586
$333$	−2.66960e6	−1.31927
$334$	−2.13170e6	−1.04559
$335$	973924.	0.474147
$336$	0	0
$337$	453398.	0.217473	0.108736	−	0.994071i	$-0.465320\pi$
$337$	453398.	0.217473	0.108736	+	0.994071i	$0.465320\pi$
$338$	−114244.	−0.0543928
$339$	0	0
$340$	−237888.	−0.111603
$341$	2.16350e6	1.00756
$342$	−75816.0	−0.0350506
$343$	801380.	0.367793
$344$	377600.	0.172043
$345$	0	0
$346$	2.52183e6	1.13247
$347$	−1.23065e6	−0.548669	−0.274334	−	0.961634i	$-0.588457\pi$
$347$	−1.23065e6	−0.548669	−0.274334	+	0.961634i	$0.588457\pi$
$348$	0	0
$349$	−2.43825e6	−1.07155	−0.535777	−	0.844360i	$-0.679981\pi$
$349$	−2.43825e6	−1.07155	−0.535777	+	0.844360i	$0.679981\pi$
$350$	−1.99172e6	−0.869077
$351$	0	0
$352$	256000.	0.110124
$353$	−2.68315e6	−1.14606	−0.573031	−	0.819534i	$-0.694233\pi$
$353$	−2.68315e6	−1.14606	−0.573031	+	0.819534i	$0.694233\pi$
$354$	0	0
$355$	707588.	0.297995
$356$	1.50672e6	0.630097
$357$	0	0
$358$	2.69966e6	1.11327
$359$	1.58693e6	0.649864	0.324932	−	0.945737i	$-0.394659\pi$
$359$	1.58693e6	0.649864	0.324932	+	0.945737i	$0.394659\pi$
$360$	−217728.	−0.0885438
$361$	−2.47002e6	−0.997543
$362$	−745128.	−0.298854
$363$	0	0
$364$	459680.	0.181845
$365$	654500.	0.257145
$366$	0	0
$367$	−60052.0	−0.0232735	−0.0116368	−	0.999932i	$-0.503704\pi$
$367$	−60052.0	−0.0232735	−0.0116368	+	0.999932i	$0.503704\pi$
$368$	403456.	0.155302
$369$	−255150.	−0.0975505
$370$	615216.	0.233627
$371$	−4.93782e6	−1.86252
$372$	0	0
$373$	−4.01853e6	−1.49553	−0.747766	−	0.663963i	$-0.768873\pi$
$373$	−4.01853e6	−1.49553	−0.747766	+	0.663963i	$0.768873\pi$
$374$	1.06200e6	0.392596
$375$	0	0
$376$	381568.	0.139188
$377$	−435682.	−0.157876
$378$	0	0
$379$	1.67581e6	0.599276	0.299638	−	0.954053i	$-0.403134\pi$
$379$	1.67581e6	0.599276	0.299638	+	0.954053i	$0.403134\pi$
$380$	17472.0	0.00620703
$381$	0	0
$382$	−3.24872e6	−1.13908
$383$	687258.	0.239399	0.119700	−	0.992810i	$-0.461807\pi$
$383$	687258.	0.239399	0.119700	+	0.992810i	$0.461807\pi$
$384$	0	0
$385$	−595000.	−0.204581
$386$	600568.	0.205161
$387$	1.43370e6	0.486610
$388$	2.92458e6	0.986242
$389$	1.37611e6	0.461082	0.230541	−	0.973063i	$-0.425950\pi$
$389$	1.37611e6	0.461082	0.230541	+	0.973063i	$0.425950\pi$
$390$	0	0
$391$	1.67371e6	0.553655
$392$	−773952.	−0.254389
$393$	0	0
$394$	−945576.	−0.306871
$395$	270872.	0.0873517
$396$	972000.	0.311479
$397$	−721198.	−0.229656	−0.114828	−	0.993385i	$-0.536632\pi$
$397$	−721198.	−0.229656	−0.114828	+	0.993385i	$0.536632\pi$
$398$	157504.	0.0498407
$399$	0	0
$400$	−749824.	−0.234320
$401$	2.22681e6	0.691548	0.345774	−	0.938318i	$-0.387616\pi$
$401$	2.22681e6	0.691548	0.345774	+	0.938318i	$0.387616\pi$
$402$	0	0
$403$	1.46253e6	0.448581
$404$	−296224.	−0.0902957
$405$	−826686.	−0.250440
$406$	1.75304e6	0.527809
$407$	−2.74650e6	−0.821852
$408$	0	0
$409$	2.00783e6	0.593496	0.296748	−	0.954956i	$-0.404098\pi$
$409$	2.00783e6	0.593496	0.296748	+	0.954956i	$0.404098\pi$
$410$	58800.0	0.0172750
$411$	0	0
$412$	1.85690e6	0.538945
$413$	2.36674e6	0.682772
$414$	1.53187e6	0.439260
$415$	1.22413e6	0.348906
$416$	173056.	0.0490290
$417$	0	0
$418$	−78000.0	−0.0218350
$419$	5.99378e6	1.66788	0.833942	−	0.551852i	$-0.186079\pi$
$419$	5.99378e6	1.66788	0.833942	+	0.551852i	$0.186079\pi$
$420$	0	0
$421$	−5.32737e6	−1.46490	−0.732449	−	0.680822i	$-0.761623\pi$
$421$	−5.32737e6	−1.46490	−0.732449	+	0.680822i	$0.761623\pi$
$422$	1.64310e6	0.449142
$423$	1.44877e6	0.393684
$424$	−1.85894e6	−0.502171
$425$	−3.11060e6	−0.835356
$426$	0	0
$427$	5.58994e6	1.48367
$428$	2.45632e6	0.648150
$429$	0	0
$430$	−330400.	−0.0861725
$431$	−5.42972e6	−1.40794	−0.703970	−	0.710230i	$-0.748591\pi$
$431$	−5.42972e6	−1.40794	−0.703970	+	0.710230i	$0.748591\pi$
$432$	0	0
$433$	7.43979e6	1.90696	0.953479	−	0.301459i	$-0.0974737\pi$
$433$	7.43979e6	1.90696	0.953479	+	0.301459i	$0.0974737\pi$
$434$	−5.88472e6	−1.49969
$435$	0	0
$436$	−2.85795e6	−0.720010
$437$	−122928.	−0.0307927
$438$	0	0
$439$	6.86418e6	1.69991	0.849957	−	0.526852i	$-0.176628\pi$
$439$	6.86418e6	1.69991	0.849957	+	0.526852i	$0.176628\pi$
$440$	−224000.	−0.0551590
$441$	−2.93860e6	−0.719522
$442$	717912.	0.174790
$443$	−3.46630e6	−0.839182	−0.419591	−	0.907713i	$-0.637827\pi$
$443$	−3.46630e6	−0.839182	−0.419591	+	0.907713i	$0.637827\pi$
$444$	0	0
$445$	−1.31838e6	−0.315603
$446$	−4.34753e6	−1.03492
$447$	0	0
$448$	−696320.	−0.163913
$449$	−1.40426e6	−0.328725	−0.164362	−	0.986400i	$-0.552557\pi$
$449$	−1.40426e6	−0.328725	−0.164362	+	0.986400i	$0.552557\pi$
$450$	−2.84699e6	−0.662757
$451$	−262500.	−0.0607698
$452$	−3.91606e6	−0.901579
$453$	0	0
$454$	1.02588e6	0.233591
$455$	−402220.	−0.0910825
$456$	0	0
$457$	−5.95072e6	−1.33284	−0.666421	−	0.745575i	$-0.732175\pi$
$457$	−5.95072e6	−1.33284	−0.666421	+	0.745575i	$0.732175\pi$
$458$	1.19244e6	0.265627
$459$	0	0
$460$	−353024.	−0.0777875
$461$	−6.25465e6	−1.37073	−0.685363	−	0.728202i	$-0.740356\pi$
$461$	−6.25465e6	−1.37073	−0.685363	+	0.728202i	$0.740356\pi$
$462$	0	0
$463$	−1.55055e6	−0.336149	−0.168075	−	0.985774i	$-0.553755\pi$
$463$	−1.55055e6	−0.336149	−0.168075	+	0.985774i	$0.553755\pi$
$464$	659968.	0.142308
$465$	0	0
$466$	2.44770e6	0.522149
$467$	−1.80480e6	−0.382945	−0.191472	−	0.981498i	$-0.561326\pi$
$467$	−1.80480e6	−0.382945	−0.191472	+	0.981498i	$0.561326\pi$
$468$	657072.	0.138675
$469$	1.18262e7	2.48264
$470$	−333872.	−0.0697165
$471$	0	0
$472$	891008.	0.184088
$473$	1.47500e6	0.303137
$474$	0	0
$475$	228462.	0.0464601
$476$	−2.88864e6	−0.584354
$477$	−7.05818e6	−1.42035
$478$	−146280.	−0.0292830
$479$	2.21809e6	0.441712	0.220856	−	0.975306i	$-0.429115\pi$
$479$	2.21809e6	0.441712	0.220856	+	0.975306i	$0.429115\pi$
$480$	0	0
$481$	−1.85663e6	−0.365901
$482$	−1.52369e6	−0.298730
$483$	0	0
$484$	−1.57682e6	−0.305962
$485$	−2.55900e6	−0.493988
$486$	0	0
$487$	−6.14268e6	−1.17364	−0.586821	−	0.809717i	$-0.699621\pi$
$487$	−6.14268e6	−1.17364	−0.586821	+	0.809717i	$0.699621\pi$
$488$	2.10445e6	0.400026
$489$	0	0
$490$	677208.	0.127418
$491$	6.44486e6	1.20645	0.603226	−	0.797571i	$-0.293882\pi$
$491$	6.44486e6	1.20645	0.603226	+	0.797571i	$0.293882\pi$
$492$	0	0
$493$	2.73784e6	0.507330
$494$	−52728.0	−0.00972129
$495$	−850500.	−0.156013
$496$	−2.21542e6	−0.404346
$497$	8.59214e6	1.56031
$498$	0	0
$499$	4.25838e6	0.765584	0.382792	−	0.923835i	$-0.374963\pi$
$499$	4.25838e6	0.765584	0.382792	+	0.923835i	$0.374963\pi$
$500$	1.35610e6	0.242586
$501$	0	0
$502$	4.91227e6	0.870008
$503$	−3.56242e6	−0.627806	−0.313903	−	0.949455i	$-0.601637\pi$
$503$	−3.56242e6	−0.627806	−0.313903	+	0.949455i	$0.601637\pi$
$504$	−2.64384e6	−0.463616
$505$	259196.	0.0452272
$506$	1.57600e6	0.273640
$507$	0	0
$508$	4.10560e6	0.705858
$509$	4.23936e6	0.725281	0.362640	−	0.931929i	$-0.381875\pi$
$509$	4.23936e6	0.725281	0.362640	+	0.931929i	$0.381875\pi$
$510$	0	0
$511$	7.94750e6	1.34641
$512$	−262144.	−0.0441942
$513$	0	0
$514$	1.75711e6	0.293354
$515$	−1.62478e6	−0.269946
$516$	0	0
$517$	1.49050e6	0.245248
$518$	7.47048e6	1.22328
$519$	0	0
$520$	−151424.	−0.0245576
$521$	2.38657e6	0.385194	0.192597	−	0.981278i	$-0.438309\pi$
$521$	2.38657e6	0.385194	0.192597	+	0.981278i	$0.438309\pi$
$522$	2.50582e6	0.402507
$523$	−8.84129e6	−1.41339	−0.706694	−	0.707519i	$-0.749814\pi$
$523$	−8.84129e6	−1.41339	−0.706694	+	0.707519i	$0.749814\pi$
$524$	−4.20378e6	−0.668823
$525$	0	0
$526$	6.71947e6	1.05894
$527$	−9.19055e6	−1.44150
$528$	0	0
$529$	−3.95257e6	−0.614101
$530$	1.62658e6	0.251527
$531$	3.38305e6	0.520681
$532$	212160.	0.0325001
$533$	−177450.	−0.0270557
$534$	0	0
$535$	−2.14928e6	−0.324645
$536$	4.45222e6	0.669368
$537$	0	0
$538$	−7.75361e6	−1.15491
$539$	−3.02325e6	−0.448231
$540$	0	0
$541$	70058.0	0.0102912	0.00514558	−	0.999987i	$-0.498362\pi$
$541$	70058.0	0.0102912	0.00514558	+	0.999987i	$0.498362\pi$
$542$	2.78199e6	0.406778
$543$	0	0
$544$	−1.08749e6	−0.157553
$545$	2.50071e6	0.360638
$546$	0	0
$547$	−6.60752e6	−0.944213	−0.472107	−	0.881541i	$-0.656506\pi$
$547$	−6.60752e6	−0.944213	−0.472107	+	0.881541i	$0.656506\pi$
$548$	−612576.	−0.0871382
$549$	7.99033e6	1.13145
$550$	−2.92900e6	−0.412869
$551$	−201084.	−0.0282162
$552$	0	0
$553$	3.28916e6	0.457375
$554$	4.52551e6	0.626460
$555$	0	0
$556$	−924416.	−0.126818
$557$	−1.10726e7	−1.51221	−0.756107	−	0.654448i	$-0.772901\pi$
$557$	−1.10726e7	−1.51221	−0.756107	+	0.654448i	$0.772901\pi$
$558$	−8.41169e6	−1.14366
$559$	997100.	0.134961
$560$	609280.	0.0821007
$561$	0	0
$562$	−6.92487e6	−0.924849
$563$	−1.43532e6	−0.190843	−0.0954216	−	0.995437i	$-0.530420\pi$
$563$	−1.43532e6	−0.190843	−0.0954216	+	0.995437i	$0.530420\pi$
$564$	0	0
$565$	3.42656e6	0.451582
$566$	5.88498e6	0.772153
$567$	−1.00383e7	−1.31131
$568$	3.23469e6	0.420689
$569$	−1.17051e7	−1.51564	−0.757818	−	0.652466i	$-0.773734\pi$
$569$	−1.17051e7	−1.51564	−0.757818	+	0.652466i	$0.773734\pi$
$570$	0	0
$571$	4.81885e6	0.618519	0.309260	−	0.950978i	$-0.399919\pi$
$571$	4.81885e6	0.618519	0.309260	+	0.950978i	$0.399919\pi$
$572$	676000.	0.0863886
$573$	0	0
$574$	714000.	0.0904521
$575$	−4.61610e6	−0.582245
$576$	−995328.	−0.125000
$577$	−1.35572e6	−0.169523	−0.0847617	−	0.996401i	$-0.527013\pi$
$577$	−1.35572e6	−0.169523	−0.0847617	+	0.996401i	$0.527013\pi$
$578$	1.16805e6	0.145426
$579$	0	0
$580$	−577472.	−0.0712789
$581$	1.48645e7	1.82687
$582$	0	0
$583$	−7.26150e6	−0.884820
$584$	2.99200e6	0.363019
$585$	−574938.	−0.0694595
$586$	−1.15542e7	−1.38994
$587$	5.03941e6	0.603649	0.301824	−	0.953364i	$-0.402404\pi$
$587$	5.03941e6	0.603649	0.301824	+	0.953364i	$0.402404\pi$
$588$	0	0
$589$	675012.	0.0801721
$590$	−779632.	−0.0922061
$591$	0	0
$592$	2.81242e6	0.329819
$593$	9.16124e6	1.06984	0.534919	−	0.844904i	$-0.320342\pi$
$593$	9.16124e6	1.06984	0.534919	+	0.844904i	$0.320342\pi$
$594$	0	0
$595$	2.52756e6	0.292691
$596$	461856.	0.0532587
$597$	0	0
$598$	1.06538e6	0.121829
$599$	−6.46635e6	−0.736363	−0.368182	−	0.929754i	$-0.620020\pi$
$599$	−6.46635e6	−0.736363	−0.368182	+	0.929754i	$0.620020\pi$
$600$	0	0
$601$	−1.18021e7	−1.33282	−0.666411	−	0.745585i	$-0.732170\pi$
$601$	−1.18021e7	−1.33282	−0.666411	+	0.745585i	$0.732170\pi$
$602$	−4.01200e6	−0.451201
$603$	1.69045e7	1.89326
$604$	637920.	0.0711498
$605$	1.37971e6	0.153250
$606$	0	0
$607$	2.25748e6	0.248686	0.124343	−	0.992239i	$-0.460318\pi$
$607$	2.25748e6	0.248686	0.124343	+	0.992239i	$0.460318\pi$
$608$	79872.0	0.00876265
$609$	0	0
$610$	−1.84139e6	−0.200365
$611$	1.00758e6	0.109188
$612$	−4.12906e6	−0.445628
$613$	−2.75378e6	−0.295991	−0.147995	−	0.988988i	$-0.547282\pi$
$613$	−2.75378e6	−0.295991	−0.147995	+	0.988988i	$0.547282\pi$
$614$	−3.49647e6	−0.374291
$615$	0	0
$616$	−2.72000e6	−0.288813
$617$	3.41607e6	0.361255	0.180627	−	0.983552i	$-0.442187\pi$
$617$	3.41607e6	0.361255	0.180627	+	0.983552i	$0.442187\pi$
$618$	0	0
$619$	9.43169e6	0.989379	0.494690	−	0.869070i	$-0.335282\pi$
$619$	9.43169e6	0.989379	0.494690	+	0.869070i	$0.335282\pi$
$620$	1.93850e6	0.202528
$621$	0	0
$622$	−1.07290e7	−1.11194
$623$	−1.60089e7	−1.65250
$624$	0	0
$625$	7.96654e6	0.815774
$626$	5.37834e6	0.548546
$627$	0	0
$628$	2.57667e6	0.260711
$629$	1.16671e7	1.17581
$630$	2.31336e6	0.232216
$631$	−4.87474e6	−0.487391	−0.243696	−	0.969852i	$-0.578360\pi$
$631$	−4.87474e6	−0.487391	−0.243696	+	0.969852i	$0.578360\pi$
$632$	1.23827e6	0.123317
$633$	0	0
$634$	−5.28295e6	−0.521980
$635$	−3.59240e6	−0.353550
$636$	0	0
$637$	−2.04372e6	−0.199559
$638$	2.57800e6	0.250744
$639$	1.22817e7	1.18989
$640$	229376.	0.0221359
$641$	9.74279e6	0.936566	0.468283	−	0.883579i	$-0.344873\pi$
$641$	9.74279e6	0.936566	0.468283	+	0.883579i	$0.344873\pi$
$642$	0	0
$643$	1.63894e6	0.156327	0.0781637	−	0.996941i	$-0.475094\pi$
$643$	1.63894e6	0.156327	0.0781637	+	0.996941i	$0.475094\pi$
$644$	−4.28672e6	−0.407296
$645$	0	0
$646$	331344.	0.0312390
$647$	−1.59069e6	−0.149391	−0.0746955	−	0.997206i	$-0.523798\pi$
$647$	−1.59069e6	−0.149391	−0.0746955	+	0.997206i	$0.523798\pi$
$648$	−3.77914e6	−0.353553
$649$	3.48050e6	0.324362
$650$	−1.98000e6	−0.183816
$651$	0	0
$652$	5.00528e6	0.461115
$653$	1.59778e7	1.46634	0.733170	−	0.680045i	$-0.238040\pi$
$653$	1.59778e7	1.46634	0.733170	+	0.680045i	$0.238040\pi$
$654$	0	0
$655$	3.67830e6	0.335000
$656$	268800.	0.0243876
$657$	1.13602e7	1.02677
$658$	−4.05416e6	−0.365036
$659$	−6.02458e6	−0.540397	−0.270199	−	0.962805i	$-0.587089\pi$
$659$	−6.02458e6	−0.540397	−0.270199	+	0.962805i	$0.587089\pi$
$660$	0	0
$661$	−2.00705e7	−1.78671	−0.893355	−	0.449352i	$-0.851655\pi$
$661$	−2.00705e7	−1.78671	−0.893355	+	0.449352i	$0.851655\pi$
$662$	8.22911e6	0.729807
$663$	0	0
$664$	5.59603e6	0.492561
$665$	−185640.	−0.0162786
$666$	1.06784e7	0.932868
$667$	4.06293e6	0.353610
$668$	8.52682e6	0.739343
$669$	0	0
$670$	−3.89570e6	−0.335273
$671$	8.22050e6	0.704842
$672$	0	0
$673$	−5.48575e6	−0.466873	−0.233436	−	0.972372i	$-0.574997\pi$
$673$	−5.48575e6	−0.466873	−0.233436	+	0.972372i	$0.574997\pi$
$674$	−1.81359e6	−0.153776
$675$	0	0
$676$	456976.	0.0384615
$677$	−4.74926e6	−0.398248	−0.199124	−	0.979974i	$-0.563810\pi$
$677$	−4.74926e6	−0.398248	−0.199124	+	0.979974i	$0.563810\pi$
$678$	0	0
$679$	−3.10736e7	−2.58653
$680$	951552.	0.0789151
$681$	0	0
$682$	−8.65400e6	−0.712453
$683$	−6.13964e6	−0.503606	−0.251803	−	0.967778i	$-0.581024\pi$
$683$	−6.13964e6	−0.503606	−0.251803	+	0.967778i	$0.581024\pi$
$684$	303264.	0.0247845
$685$	536004.	0.0436457
$686$	−3.20552e6	−0.260069
$687$	0	0
$688$	−1.51040e6	−0.121652
$689$	−4.90877e6	−0.393935
$690$	0	0
$691$	1.57617e7	1.25577	0.627883	−	0.778308i	$-0.283922\pi$
$691$	1.57617e7	1.25577	0.627883	+	0.778308i	$0.283922\pi$
$692$	−1.00873e7	−0.800776
$693$	−1.03275e7	−0.816887
$694$	4.92259e6	0.387967
$695$	808864.	0.0635204
$696$	0	0
$697$	1.11510e6	0.0869424
$698$	9.75298e6	0.757703
$699$	0	0
$700$	7.96688e6	0.614530
$701$	−1.42036e7	−1.09170	−0.545851	−	0.837882i	$-0.683793\pi$
$701$	−1.42036e7	−1.09170	−0.545851	+	0.837882i	$0.683793\pi$
$702$	0	0
$703$	−856908.	−0.0653952
$704$	−1.02400e6	−0.0778697
$705$	0	0
$706$	1.07326e7	0.810388
$707$	3.14738e6	0.236810
$708$	0	0
$709$	1.60718e7	1.20074	0.600369	−	0.799723i	$-0.295020\pi$
$709$	1.60718e7	1.20074	0.600369	+	0.799723i	$0.295020\pi$
$710$	−2.83035e6	−0.210715
$711$	4.70156e6	0.348793
$712$	−6.02688e6	−0.445546
$713$	−1.36387e7	−1.00473
$714$	0	0
$715$	−591500.	−0.0432703
$716$	−1.07987e7	−0.787204
$717$	0	0
$718$	−6.34774e6	−0.459524
$719$	−2.07078e7	−1.49387	−0.746933	−	0.664900i	$-0.768474\pi$
$719$	−2.07078e7	−1.49387	−0.746933	+	0.664900i	$0.768474\pi$
$720$	870912.	0.0626099
$721$	−1.97295e7	−1.41344
$722$	9.88006e6	0.705369
$723$	0	0
$724$	2.98051e6	0.211322
$725$	−7.55096e6	−0.533528
$726$	0	0
$727$	5.04803e6	0.354231	0.177115	−	0.984190i	$-0.443323\pi$
$727$	5.04803e6	0.354231	0.177115	+	0.984190i	$0.443323\pi$
$728$	−1.83872e6	−0.128584
$729$	−1.43489e7	−1.00000
$730$	−2.61800e6	−0.181829
$731$	−6.26580e6	−0.433694
$732$	0	0
$733$	−2.10377e7	−1.44623	−0.723115	−	0.690728i	$-0.757290\pi$
$733$	−2.10377e7	−1.44623	−0.723115	+	0.690728i	$0.757290\pi$
$734$	240208.	0.0164569
$735$	0	0
$736$	−1.61382e6	−0.109815
$737$	1.73915e7	1.17942
$738$	1.02060e6	0.0689786
$739$	1.38992e7	0.936218	0.468109	−	0.883671i	$-0.344935\pi$
$739$	1.38992e7	0.936218	0.468109	+	0.883671i	$0.344935\pi$
$740$	−2.46086e6	−0.165199
$741$	0	0
$742$	1.97513e7	1.31700
$743$	1.23267e6	0.0819169	0.0409584	−	0.999161i	$-0.486959\pi$
$743$	1.23267e6	0.0819169	0.0409584	+	0.999161i	$0.486959\pi$
$744$	0	0
$745$	−404124.	−0.0266762
$746$	1.60741e7	1.05750
$747$	2.12474e7	1.39317
$748$	−4.24800e6	−0.277607
$749$	−2.60984e7	−1.69985
$750$	0	0
$751$	−1.62624e6	−0.105217	−0.0526084	−	0.998615i	$-0.516753\pi$
$751$	−1.62624e6	−0.105217	−0.0526084	+	0.998615i	$0.516753\pi$
$752$	−1.52627e6	−0.0984209
$753$	0	0
$754$	1.74273e6	0.111635
$755$	−558180.	−0.0356375
$756$	0	0
$757$	3.49882e6	0.221913	0.110956	−	0.993825i	$-0.464609\pi$
$757$	3.49882e6	0.221913	0.110956	+	0.993825i	$0.464609\pi$
$758$	−6.70324e6	−0.423752
$759$	0	0
$760$	−69888.0	−0.00438903
$761$	2.21713e7	1.38781	0.693905	−	0.720067i	$-0.255889\pi$
$761$	2.21713e7	1.38781	0.693905	+	0.720067i	$0.255889\pi$
$762$	0	0
$763$	3.03657e7	1.88831
$764$	1.29949e7	0.805451
$765$	3.61292e6	0.223206
$766$	−2.74903e6	−0.169281
$767$	2.35282e6	0.144411
$768$	0	0
$769$	1.08955e6	0.0664400	0.0332200	−	0.999448i	$-0.489424\pi$
$769$	1.08955e6	0.0664400	0.0332200	+	0.999448i	$0.489424\pi$
$770$	2.38000e6	0.144661
$771$	0	0
$772$	−2.40227e6	−0.145070
$773$	1.95219e6	0.117510	0.0587549	−	0.998272i	$-0.481287\pi$
$773$	1.95219e6	0.117510	0.0587549	+	0.998272i	$0.481287\pi$
$774$	−5.73480e6	−0.344085
$775$	2.53476e7	1.51594
$776$	−1.16983e7	−0.697379
$777$	0	0
$778$	−5.50442e6	−0.326034
$779$	−81900.0	−0.00483549
$780$	0	0
$781$	1.26355e7	0.741250
$782$	−6.69485e6	−0.391493
$783$	0	0
$784$	3.09581e6	0.179880
$785$	−2.25459e6	−0.130585
$786$	0	0
$787$	−1.44531e7	−0.831809	−0.415904	−	0.909408i	$-0.636535\pi$
$787$	−1.44531e7	−0.831809	−0.415904	+	0.909408i	$0.636535\pi$
$788$	3.78230e6	0.216991
$789$	0	0
$790$	−1.08349e6	−0.0617670
$791$	4.16082e7	2.36449
$792$	−3.88800e6	−0.220249
$793$	5.55706e6	0.313807
$794$	2.88479e6	0.162391
$795$	0	0
$796$	−630016.	−0.0352427
$797$	1.23500e7	0.688685	0.344343	−	0.938844i	$-0.388102\pi$
$797$	1.23500e7	0.688685	0.344343	+	0.938844i	$0.388102\pi$
$798$	0	0
$799$	−6.33164e6	−0.350873
$800$	2.99930e6	0.165689
$801$	−2.28833e7	−1.26019
$802$	−8.90724e6	−0.488998
$803$	1.16875e7	0.639636
$804$	0	0
$805$	3.75088e6	0.204006
$806$	−5.85010e6	−0.317195
$807$	0	0
$808$	1.18490e6	0.0638487
$809$	1.15968e7	0.622970	0.311485	−	0.950251i	$-0.399174\pi$
$809$	1.15968e7	0.622970	0.311485	+	0.950251i	$0.399174\pi$
$810$	3.30674e6	0.177088
$811$	−2.47534e7	−1.32155	−0.660774	−	0.750585i	$-0.729772\pi$
$811$	−2.47534e7	−1.32155	−0.660774	+	0.750585i	$0.729772\pi$
$812$	−7.01216e6	−0.373217
$813$	0	0
$814$	1.09860e7	0.581137
$815$	−4.37962e6	−0.230963
$816$	0	0
$817$	460200.	0.0241208
$818$	−8.03130e6	−0.419665
$819$	−6.98139e6	−0.363691
$820$	−235200.	−0.0122153
$821$	−2.47470e6	−0.128134	−0.0640671	−	0.997946i	$-0.520407\pi$
$821$	−2.47470e6	−0.128134	−0.0640671	+	0.997946i	$0.520407\pi$
$822$	0	0
$823$	−7.84754e6	−0.403863	−0.201932	−	0.979400i	$-0.564722\pi$
$823$	−7.84754e6	−0.403863	−0.201932	+	0.979400i	$0.564722\pi$
$824$	−7.42758e6	−0.381092
$825$	0	0
$826$	−9.46696e6	−0.482792
$827$	−2.26192e7	−1.15004	−0.575020	−	0.818140i	$-0.695006\pi$
$827$	−2.26192e7	−1.15004	−0.575020	+	0.818140i	$0.695006\pi$
$828$	−6.12749e6	−0.310604
$829$	−1.73912e7	−0.878907	−0.439454	−	0.898265i	$-0.644828\pi$
$829$	−1.73912e7	−0.878907	−0.439454	+	0.898265i	$0.644828\pi$
$830$	−4.89653e6	−0.246714
$831$	0	0
$832$	−692224.	−0.0346688
$833$	1.28428e7	0.641278
$834$	0	0
$835$	−7.46096e6	−0.370321
$836$	312000.	0.0154397
$837$	0	0
$838$	−2.39751e7	−1.17937
$839$	−3.43825e7	−1.68629	−0.843147	−	0.537684i	$-0.819299\pi$
$839$	−3.43825e7	−1.68629	−0.843147	+	0.537684i	$0.819299\pi$
$840$	0	0
$841$	−1.38651e7	−0.675977
$842$	2.13095e7	1.03584
$843$	0	0
$844$	−6.57242e6	−0.317592
$845$	−399854.	−0.0192646
$846$	−5.79506e6	−0.278376
$847$	1.67537e7	0.802419
$848$	7.43578e6	0.355089
$849$	0	0
$850$	1.24424e7	0.590686
$851$	1.73139e7	0.819543
$852$	0	0
$853$	2.31007e7	1.08706	0.543528	−	0.839391i	$-0.317088\pi$
$853$	2.31007e7	1.08706	0.543528	+	0.839391i	$0.317088\pi$
$854$	−2.23598e7	−1.04911
$855$	−265356.	−0.0124141
$856$	−9.82528e6	−0.458311
$857$	−7.02305e6	−0.326643	−0.163322	−	0.986573i	$-0.552221\pi$
$857$	−7.02305e6	−0.326643	−0.163322	+	0.986573i	$0.552221\pi$
$858$	0	0
$859$	8.82135e6	0.407899	0.203949	−	0.978981i	$-0.434622\pi$
$859$	8.82135e6	0.407899	0.203949	+	0.978981i	$0.434622\pi$
$860$	1.32160e6	0.0609332
$861$	0	0
$862$	2.17189e7	0.995564
$863$	2.39560e7	1.09493	0.547466	−	0.836828i	$-0.315592\pi$
$863$	2.39560e7	1.09493	0.547466	+	0.836828i	$0.315592\pi$
$864$	0	0
$865$	8.82641e6	0.401092
$866$	−2.97592e7	−1.34842
$867$	0	0
$868$	2.35389e7	1.06044
$869$	4.83700e6	0.217283
$870$	0	0
$871$	1.17567e7	0.525096
$872$	1.14318e7	0.509124
$873$	−4.44170e7	−1.97248
$874$	491712.	0.0217737
$875$	−1.44085e7	−0.636208
$876$	0	0
$877$	−5.79805e6	−0.254556	−0.127278	−	0.991867i	$-0.540624\pi$
$877$	−5.79805e6	−0.254556	−0.127278	+	0.991867i	$0.540624\pi$
$878$	−2.74567e7	−1.20202
$879$	0	0
$880$	896000.	0.0390033
$881$	−1.30527e7	−0.566580	−0.283290	−	0.959034i	$-0.591426\pi$
$881$	−1.30527e7	−0.566580	−0.283290	+	0.959034i	$0.591426\pi$
$882$	1.17544e7	0.508779
$883$	4.73009e6	0.204159	0.102079	−	0.994776i	$-0.467450\pi$
$883$	4.73009e6	0.204159	0.102079	+	0.994776i	$0.467450\pi$
$884$	−2.87165e6	−0.123595
$885$	0	0
$886$	1.38652e7	0.593392
$887$	2.80737e7	1.19809	0.599046	−	0.800714i	$-0.295547\pi$
$887$	2.80737e7	1.19809	0.599046	+	0.800714i	$0.295547\pi$
$888$	0	0
$889$	−4.36220e7	−1.85119
$890$	5.27352e6	0.223165
$891$	−1.47622e7	−0.622957
$892$	1.73901e7	0.731796
$893$	465036.	0.0195145
$894$	0	0
$895$	9.44882e6	0.394294
$896$	2.78528e6	0.115904
$897$	0	0
$898$	5.61705e6	0.232443
$899$	−2.23100e7	−0.920663
$900$	1.13880e7	0.468640
$901$	3.08469e7	1.26590
$902$	1.05000e6	0.0429708
$903$	0	0
$904$	1.56643e7	0.637512
$905$	−2.60795e6	−0.105847
$906$	0	0
$907$	2.28552e7	0.922500	0.461250	−	0.887270i	$-0.347401\pi$
$907$	2.28552e7	0.922500	0.461250	+	0.887270i	$0.347401\pi$
$908$	−4.10352e6	−0.165174
$909$	4.49890e6	0.180591
$910$	1.60888e6	0.0644051
$911$	−3.27335e7	−1.30676	−0.653381	−	0.757029i	$-0.726650\pi$
$911$	−3.27335e7	−1.30676	−0.653381	+	0.757029i	$0.726650\pi$
$912$	0	0
$913$	2.18595e7	0.867887
$914$	2.38029e7	0.942462
$915$	0	0
$916$	−4.76976e6	−0.187827
$917$	4.46651e7	1.75406
$918$	0	0
$919$	−1.27717e7	−0.498839	−0.249419	−	0.968396i	$-0.580240\pi$
$919$	−1.27717e7	−0.498839	−0.249419	+	0.968396i	$0.580240\pi$
$920$	1.41210e6	0.0550040
$921$	0	0
$922$	2.50186e7	0.969249
$923$	8.54160e6	0.330016
$924$	0	0
$925$	−3.21780e7	−1.23653
$926$	6.20218e6	0.237693
$927$	−2.82016e7	−1.07789
$928$	−2.63987e6	−0.100627
$929$	3.48297e7	1.32407	0.662034	−	0.749473i	$-0.269693\pi$
$929$	3.48297e7	1.32407	0.662034	+	0.749473i	$0.269693\pi$
$930$	0	0
$931$	−943254.	−0.0356660
$932$	−9.79082e6	−0.369215
$933$	0	0
$934$	7.21918e6	0.270783
$935$	3.71700e6	0.139048
$936$	−2.62829e6	−0.0980581
$937$	3.00172e7	1.11692	0.558459	−	0.829532i	$-0.311393\pi$
$937$	3.00172e7	1.11692	0.558459	+	0.829532i	$0.311393\pi$
$938$	−4.73049e7	−1.75549
$939$	0	0
$940$	1.33549e6	0.0492970
$941$	−4.50649e7	−1.65907	−0.829534	−	0.558457i	$-0.811394\pi$
$941$	−4.50649e7	−1.65907	−0.829534	+	0.558457i	$0.811394\pi$
$942$	0	0
$943$	1.65480e6	0.0605991
$944$	−3.56403e6	−0.130170
$945$	0	0
$946$	−5.90000e6	−0.214350
$947$	2.99276e7	1.08442	0.542210	−	0.840243i	$-0.317588\pi$
$947$	2.99276e7	1.08442	0.542210	+	0.840243i	$0.317588\pi$
$948$	0	0
$949$	7.90075e6	0.284776
$950$	−913848.	−0.0328522
$951$	0	0
$952$	1.15546e7	0.413201
$953$	−4.25147e7	−1.51638	−0.758188	−	0.652036i	$-0.773915\pi$
$953$	−4.25147e7	−1.51638	−0.758188	+	0.652036i	$0.773915\pi$
$954$	2.82327e7	1.00434
$955$	−1.13705e7	−0.403433
$956$	585120.	0.0207062
$957$	0	0
$958$	−8.87234e6	−0.312338
$959$	6.50862e6	0.228530
$960$	0	0
$961$	4.62626e7	1.61593
$962$	7.42654e6	0.258731
$963$	−3.73054e7	−1.29630
$964$	6.09475e6	0.211234
$965$	2.10199e6	0.0726628
$966$	0	0
$967$	−3.00251e7	−1.03257	−0.516284	−	0.856417i	$-0.672685\pi$
$967$	−3.00251e7	−1.03257	−0.516284	+	0.856417i	$0.672685\pi$
$968$	6.30726e6	0.216348
$969$	0	0
$970$	1.02360e7	0.349302
$971$	−4.00864e7	−1.36442	−0.682211	−	0.731155i	$-0.738982\pi$
$971$	−4.00864e7	−1.36442	−0.682211	+	0.731155i	$0.738982\pi$
$972$	0	0
$973$	9.82192e6	0.332594
$974$	2.45707e7	0.829890
$975$	0	0
$976$	−8.41779e6	−0.282861
$977$	5.12151e7	1.71657	0.858284	−	0.513174i	$-0.171531\pi$
$977$	5.12151e7	1.71657	0.858284	+	0.513174i	$0.171531\pi$
$978$	0	0
$979$	−2.35425e7	−0.785047
$980$	−2.70883e6	−0.0900984
$981$	4.34051e7	1.44002
$982$	−2.57794e7	−0.853090
$983$	1.82382e7	0.602004	0.301002	−	0.953624i	$-0.402679\pi$
$983$	1.82382e7	0.602004	0.301002	+	0.953624i	$0.402679\pi$
$984$	0	0
$985$	−3.30952e6	−0.108686
$986$	−1.09513e7	−0.358736
$987$	0	0
$988$	210912.	0.00687399
$989$	−9.29840e6	−0.302286
$990$	3.40200e6	0.110318
$991$	−3.24103e7	−1.04833	−0.524166	−	0.851616i	$-0.675623\pi$
$991$	−3.24103e7	−1.04833	−0.524166	+	0.851616i	$0.675623\pi$
$992$	8.86170e6	0.285915
$993$	0	0
$994$	−3.43686e7	−1.10330
$995$	551264.	0.0176523
$996$	0	0
$997$	−2.07867e7	−0.662289	−0.331145	−	0.943580i	$-0.607435\pi$
$997$	−2.07867e7	−0.662289	−0.331145	+	0.943580i	$0.607435\pi$
$998$	−1.70335e7	−0.541350
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	26.6.a.a.1.1	✓	1
3.2	odd	2		234.6.a.g.1.1		1
4.3	odd	2		208.6.a.b.1.1		1
5.2	odd	4		650.6.b.a.599.1		2
5.3	odd	4		650.6.b.a.599.2		2
5.4	even	2		650.6.a.a.1.1		1
8.3	odd	2		832.6.a.e.1.1		1
8.5	even	2		832.6.a.d.1.1		1
13.5	odd	4		338.6.b.a.337.2		2
13.8	odd	4		338.6.b.a.337.1		2
13.12	even	2		338.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.6.a.a.1.1	✓	1	1.1	even	1	trivial
208.6.a.b.1.1		1	4.3	odd	2
234.6.a.g.1.1		1	3.2	odd	2
338.6.a.d.1.1		1	13.12	even	2
338.6.b.a.337.1		2	13.8	odd	4
338.6.b.a.337.2		2	13.5	odd	4
650.6.a.a.1.1		1	5.4	even	2
650.6.b.a.599.1		2	5.2	odd	4
650.6.b.a.599.2		2	5.3	odd	4
832.6.a.d.1.1		1	8.5	even	2
832.6.a.e.1.1		1	8.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 26 = 2 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	26.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more