LMFDB - Embedded newform 1028.6.a.b.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1028,6,Mod(1,1028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1028, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1028.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1028 = 2^{2} \cdot 257 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1028.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:	$164.874566768$
Analytic rank:	$0$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	1028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-24.6173 q^{3} -7.87155 q^{5} +165.885 q^{7} +363.009 q^{9} +O(q^{10})$	$q-24.6173 q^{3} -7.87155 q^{5} +165.885 q^{7} +363.009 q^{9} -343.265 q^{11} -958.848 q^{13} +193.776 q^{15} -689.443 q^{17} +2895.85 q^{19} -4083.64 q^{21} +1503.79 q^{23} -3063.04 q^{25} -2954.29 q^{27} -1557.57 q^{29} +5227.12 q^{31} +8450.24 q^{33} -1305.77 q^{35} +13515.8 q^{37} +23604.2 q^{39} -10792.5 q^{41} -1788.04 q^{43} -2857.44 q^{45} -14407.4 q^{47} +10710.9 q^{49} +16972.2 q^{51} -15730.4 q^{53} +2702.03 q^{55} -71287.9 q^{57} -17082.7 q^{59} +26895.4 q^{61} +60217.8 q^{63} +7547.62 q^{65} +5959.33 q^{67} -37019.2 q^{69} -50516.4 q^{71} +23101.6 q^{73} +75403.6 q^{75} -56942.5 q^{77} +25526.2 q^{79} -15484.6 q^{81} +48933.7 q^{83} +5426.98 q^{85} +38343.1 q^{87} -123160. q^{89} -159059. q^{91} -128677. q^{93} -22794.8 q^{95} -125478. q^{97} -124608. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * 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
$3$	−24.6173	−1.57920	−0.789599	−	0.613624i	$-0.789711\pi$
$3$	−24.6173	−1.57920	−0.789599	+	0.613624i	$0.789711\pi$
$4$	0	0
$5$	−7.87155	−0.140811	−0.0704053	−	0.997518i	$-0.522429\pi$
$5$	−7.87155	−0.140811	−0.0704053	+	0.997518i	$0.522429\pi$
$6$	0	0
$7$	165.885	1.27957	0.639783	−	0.768556i	$-0.279024\pi$
$7$	165.885	1.27957	0.639783	+	0.768556i	$0.279024\pi$
$8$	0	0
$9$	363.009	1.49386
$10$	0	0
$11$	−343.265	−0.855358	−0.427679	−	0.903931i	$-0.640669\pi$
$11$	−343.265	−0.855358	−0.427679	+	0.903931i	$0.640669\pi$
$12$	0	0
$13$	−958.848	−1.57359	−0.786795	−	0.617215i	$-0.788261\pi$
$13$	−958.848	−1.57359	−0.786795	+	0.617215i	$0.788261\pi$
$14$	0	0
$15$	193.776	0.222368
$16$	0	0
$17$	−689.443	−0.578597	−0.289298	−	0.957239i	$-0.593422\pi$
$17$	−689.443	−0.578597	−0.289298	+	0.957239i	$0.593422\pi$
$18$	0	0
$19$	2895.85	1.84032	0.920158	−	0.391548i	$-0.128060\pi$
$19$	2895.85	1.84032	0.920158	+	0.391548i	$0.128060\pi$
$20$	0	0
$21$	−4083.64	−2.02069
$22$	0	0
$23$	1503.79	0.592744	0.296372	−	0.955073i	$-0.404223\pi$
$23$	1503.79	0.592744	0.296372	+	0.955073i	$0.404223\pi$
$24$	0	0
$25$	−3063.04	−0.980172
$26$	0	0
$27$	−2954.29	−0.779910
$28$	0	0
$29$	−1557.57	−0.343916	−0.171958	−	0.985104i	$-0.555009\pi$
$29$	−1557.57	−0.343916	−0.171958	+	0.985104i	$0.555009\pi$
$30$	0	0
$31$	5227.12	0.976918	0.488459	−	0.872587i	$-0.337559\pi$
$31$	5227.12	0.976918	0.488459	+	0.872587i	$0.337559\pi$
$32$	0	0
$33$	8450.24	1.35078
$34$	0	0
$35$	−1305.77	−0.180176
$36$	0	0
$37$	13515.8	1.62307	0.811535	−	0.584304i	$-0.198632\pi$
$37$	13515.8	1.62307	0.811535	+	0.584304i	$0.198632\pi$
$38$	0	0
$39$	23604.2	2.48501
$40$	0	0
$41$	−10792.5	−1.00268	−0.501338	−	0.865252i	$-0.667159\pi$
$41$	−10792.5	−1.00268	−0.501338	+	0.865252i	$0.667159\pi$
$42$	0	0
$43$	−1788.04	−0.147471	−0.0737353	−	0.997278i	$-0.523492\pi$
$43$	−1788.04	−0.147471	−0.0737353	+	0.997278i	$0.523492\pi$
$44$	0	0
$45$	−2857.44	−0.210352
$46$	0	0
$47$	−14407.4	−0.951353	−0.475676	−	0.879620i	$-0.657797\pi$
$47$	−14407.4	−0.951353	−0.475676	+	0.879620i	$0.657797\pi$
$48$	0	0
$49$	10710.9	0.637287
$50$	0	0
$51$	16972.2	0.913719
$52$	0	0
$53$	−15730.4	−0.769218	−0.384609	−	0.923080i	$-0.625664\pi$
$53$	−15730.4	−0.769218	−0.384609	+	0.923080i	$0.625664\pi$
$54$	0	0
$55$	2702.03	0.120443
$56$	0	0
$57$	−71287.9	−2.90622
$58$	0	0
$59$	−17082.7	−0.638890	−0.319445	−	0.947605i	$-0.603496\pi$
$59$	−17082.7	−0.638890	−0.319445	+	0.947605i	$0.603496\pi$
$60$	0	0
$61$	26895.4	0.925452	0.462726	−	0.886501i	$-0.346871\pi$
$61$	26895.4	0.925452	0.462726	+	0.886501i	$0.346871\pi$
$62$	0	0
$63$	60217.8	1.91150
$64$	0	0
$65$	7547.62	0.221578
$66$	0	0
$67$	5959.33	0.162185	0.0810924	−	0.996707i	$-0.474159\pi$
$67$	5959.33	0.162185	0.0810924	+	0.996707i	$0.474159\pi$
$68$	0	0
$69$	−37019.2	−0.936060
$70$	0	0
$71$	−50516.4	−1.18929	−0.594643	−	0.803990i	$-0.702707\pi$
$71$	−50516.4	−1.18929	−0.594643	+	0.803990i	$0.702707\pi$
$72$	0	0
$73$	23101.6	0.507381	0.253691	−	0.967285i	$-0.418356\pi$
$73$	23101.6	0.507381	0.253691	+	0.967285i	$0.418356\pi$
$74$	0	0
$75$	75403.6	1.54789
$76$	0	0
$77$	−56942.5	−1.09449
$78$	0	0
$79$	25526.2	0.460171	0.230085	−	0.973170i	$-0.426099\pi$
$79$	25526.2	0.460171	0.230085	+	0.973170i	$0.426099\pi$
$80$	0	0
$81$	−15484.6	−0.262233
$82$	0	0
$83$	48933.7	0.779673	0.389837	−	0.920884i	$-0.372531\pi$
$83$	48933.7	0.779673	0.389837	+	0.920884i	$0.372531\pi$
$84$	0	0
$85$	5426.98	0.0814725
$86$	0	0
$87$	38343.1	0.543112
$88$	0	0
$89$	−123160.	−1.64814	−0.824070	−	0.566488i	$-0.808302\pi$
$89$	−123160.	−1.64814	−0.824070	+	0.566488i	$0.808302\pi$
$90$	0	0
$91$	−159059.	−2.01351
$92$	0	0
$93$	−128677.	−1.54275
$94$	0	0
$95$	−22794.8	−0.259136
$96$	0	0
$97$	−125478.	−1.35406	−0.677032	−	0.735953i	$-0.736734\pi$
$97$	−125478.	−1.35406	−0.677032	+	0.735953i	$0.736734\pi$
$98$	0	0
$99$	−124608.	−1.27779
$100$	0	0
$101$	127975.	1.24831	0.624153	−	0.781302i	$-0.285444\pi$
$101$	127975.	1.24831	0.624153	+	0.781302i	$0.285444\pi$
$102$	0	0
$103$	−37940.5	−0.352379	−0.176190	−	0.984356i	$-0.556377\pi$
$103$	−37940.5	−0.352379	−0.176190	+	0.984356i	$0.556377\pi$
$104$	0	0
$105$	32144.5	0.284534
$106$	0	0
$107$	−52530.0	−0.443555	−0.221778	−	0.975097i	$-0.571186\pi$
$107$	−52530.0	−0.443555	−0.221778	+	0.975097i	$0.571186\pi$
$108$	0	0
$109$	52387.3	0.422338	0.211169	−	0.977450i	$-0.432273\pi$
$109$	52387.3	0.422338	0.211169	+	0.977450i	$0.432273\pi$
$110$	0	0
$111$	−332722.	−2.56315
$112$	0	0
$113$	−252932.	−1.86340	−0.931702	−	0.363224i	$-0.881676\pi$
$113$	−252932.	−1.86340	−0.931702	+	0.363224i	$0.881676\pi$
$114$	0	0
$115$	−11837.2	−0.0834647
$116$	0	0
$117$	−348071.	−2.35073
$118$	0	0
$119$	−114368.	−0.740352
$120$	0	0
$121$	−43220.2	−0.268364
$122$	0	0
$123$	265681.	1.58342
$124$	0	0
$125$	48709.4	0.278829
$126$	0	0
$127$	−99716.1	−0.548600	−0.274300	−	0.961644i	$-0.588446\pi$
$127$	−99716.1	−0.548600	−0.274300	+	0.961644i	$0.588446\pi$
$128$	0	0
$129$	44016.6	0.232885
$130$	0	0
$131$	−154662.	−0.787417	−0.393708	−	0.919235i	$-0.628808\pi$
$131$	−154662.	−0.787417	−0.393708	+	0.919235i	$0.628808\pi$
$132$	0	0
$133$	480379.	2.35480
$134$	0	0
$135$	23254.9	0.109819
$136$	0	0
$137$	234398.	1.06697	0.533484	−	0.845810i	$-0.320882\pi$
$137$	234398.	1.06697	0.533484	+	0.845810i	$0.320882\pi$
$138$	0	0
$139$	−214210.	−0.940378	−0.470189	−	0.882566i	$-0.655814\pi$
$139$	−214210.	−0.940378	−0.470189	+	0.882566i	$0.655814\pi$
$140$	0	0
$141$	354671.	1.50237
$142$	0	0
$143$	329139.	1.34598
$144$	0	0
$145$	12260.5	0.0484270
$146$	0	0
$147$	−263672.	−1.00640
$148$	0	0
$149$	269891.	0.995916	0.497958	−	0.867201i	$-0.334083\pi$
$149$	269891.	0.995916	0.497958	+	0.867201i	$0.334083\pi$
$150$	0	0
$151$	−127516.	−0.455118	−0.227559	−	0.973764i	$-0.573074\pi$
$151$	−127516.	−0.455118	−0.227559	+	0.973764i	$0.573074\pi$
$152$	0	0
$153$	−250274.	−0.864345
$154$	0	0
$155$	−41145.5	−0.137560
$156$	0	0
$157$	214195.	0.693523	0.346761	−	0.937953i	$-0.387281\pi$
$157$	214195.	0.693523	0.346761	+	0.937953i	$0.387281\pi$
$158$	0	0
$159$	387239.	1.21475
$160$	0	0
$161$	249456.	0.758455
$162$	0	0
$163$	−54488.5	−0.160633	−0.0803167	−	0.996769i	$-0.525593\pi$
$163$	−54488.5	−0.160633	−0.0803167	+	0.996769i	$0.525593\pi$
$164$	0	0
$165$	−66516.5	−0.190204
$166$	0	0
$167$	39141.5	0.108604	0.0543021	−	0.998525i	$-0.482707\pi$
$167$	39141.5	0.108604	0.0543021	+	0.998525i	$0.482707\pi$
$168$	0	0
$169$	548097.	1.47618
$170$	0	0
$171$	1.05122e6	2.74918
$172$	0	0
$173$	700367.	1.77914	0.889571	−	0.456797i	$-0.151003\pi$
$173$	700367.	1.77914	0.889571	+	0.456797i	$0.151003\pi$
$174$	0	0
$175$	−508113.	−1.25419
$176$	0	0
$177$	420529.	1.00893
$178$	0	0
$179$	−17131.8	−0.0399640	−0.0199820	−	0.999800i	$-0.506361\pi$
$179$	−17131.8	−0.0399640	−0.0199820	+	0.999800i	$0.506361\pi$
$180$	0	0
$181$	−167794.	−0.380699	−0.190349	−	0.981716i	$-0.560962\pi$
$181$	−167794.	−0.380699	−0.190349	+	0.981716i	$0.560962\pi$
$182$	0	0
$183$	−662091.	−1.46147
$184$	0	0
$185$	−106390.	−0.228545
$186$	0	0
$187$	236662.	0.494907
$188$	0	0
$189$	−490073.	−0.997945
$190$	0	0
$191$	−411741.	−0.816659	−0.408330	−	0.912835i	$-0.633889\pi$
$191$	−411741.	−0.816659	−0.408330	+	0.912835i	$0.633889\pi$
$192$	0	0
$193$	871963.	1.68502	0.842509	−	0.538682i	$-0.181078\pi$
$193$	871963.	1.68502	0.842509	+	0.538682i	$0.181078\pi$
$194$	0	0
$195$	−185802.	−0.349915
$196$	0	0
$197$	492620.	0.904370	0.452185	−	0.891924i	$-0.350645\pi$
$197$	492620.	0.904370	0.452185	+	0.891924i	$0.350645\pi$
$198$	0	0
$199$	232721.	0.416585	0.208293	−	0.978067i	$-0.433209\pi$
$199$	232721.	0.416585	0.208293	+	0.978067i	$0.433209\pi$
$200$	0	0
$201$	−146702.	−0.256122
$202$	0	0
$203$	−258378.	−0.440063
$204$	0	0
$205$	84953.3	0.141187
$206$	0	0
$207$	545889.	0.885480
$208$	0	0
$209$	−994044.	−1.57413
$210$	0	0
$211$	793596.	1.22714	0.613569	−	0.789641i	$-0.289733\pi$
$211$	793596.	1.22714	0.613569	+	0.789641i	$0.289733\pi$
$212$	0	0
$213$	1.24357e6	1.87812
$214$	0	0
$215$	14074.6	0.0207654
$216$	0	0
$217$	867101.	1.25003
$218$	0	0
$219$	−568697.	−0.801255
$220$	0	0
$221$	661071.	0.910474
$222$	0	0
$223$	1.21642e6	1.63803	0.819013	−	0.573774i	$-0.194521\pi$
$223$	1.21642e6	1.63803	0.819013	+	0.573774i	$0.194521\pi$
$224$	0	0
$225$	−1.11191e6	−1.46424
$226$	0	0
$227$	−93360.5	−0.120254	−0.0601269	−	0.998191i	$-0.519151\pi$
$227$	−93360.5	−0.120254	−0.0601269	+	0.998191i	$0.519151\pi$
$228$	0	0
$229$	198232.	0.249795	0.124898	−	0.992170i	$-0.460140\pi$
$229$	198232.	0.249795	0.124898	+	0.992170i	$0.460140\pi$
$230$	0	0
$231$	1.40177e6	1.72841
$232$	0	0
$233$	630020.	0.760264	0.380132	−	0.924932i	$-0.375879\pi$
$233$	630020.	0.760264	0.380132	+	0.924932i	$0.375879\pi$
$234$	0	0
$235$	113409.	0.133960
$236$	0	0
$237$	−628386.	−0.726700
$238$	0	0
$239$	−1.24954e6	−1.41499	−0.707497	−	0.706717i	$-0.750176\pi$
$239$	−1.24954e6	−1.41499	−0.707497	+	0.706717i	$0.750176\pi$
$240$	0	0
$241$	−1.08046e6	−1.19830	−0.599152	−	0.800635i	$-0.704495\pi$
$241$	−1.08046e6	−1.19830	−0.599152	+	0.800635i	$0.704495\pi$
$242$	0	0
$243$	1.09908e6	1.19403
$244$	0	0
$245$	−84311.2	−0.0897367
$246$	0	0
$247$	−2.77668e6	−2.89590
$248$	0	0
$249$	−1.20461e6	−1.23126
$250$	0	0
$251$	980231.	0.982074	0.491037	−	0.871139i	$-0.336618\pi$
$251$	980231.	0.982074	0.491037	+	0.871139i	$0.336618\pi$
$252$	0	0
$253$	−516198.	−0.507008
$254$	0	0
$255$	−133597.	−0.128661
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	2.24207e6	2.07682
$260$	0	0
$261$	−565412.	−0.513764
$262$	0	0
$263$	516915.	0.460818	0.230409	−	0.973094i	$-0.425994\pi$
$263$	516915.	0.460818	0.230409	+	0.973094i	$0.425994\pi$
$264$	0	0
$265$	123822.	0.108314
$266$	0	0
$267$	3.03186e6	2.60274
$268$	0	0
$269$	−1.21176e6	−1.02102	−0.510512	−	0.859871i	$-0.670544\pi$
$269$	−1.21176e6	−1.02102	−0.510512	+	0.859871i	$0.670544\pi$
$270$	0	0
$271$	−289170.	−0.239183	−0.119591	−	0.992823i	$-0.538158\pi$
$271$	−289170.	−0.239183	−0.119591	+	0.992823i	$0.538158\pi$
$272$	0	0
$273$	3.91559e6	3.17973
$274$	0	0
$275$	1.05143e6	0.838398
$276$	0	0
$277$	1.49724e6	1.17244	0.586222	−	0.810150i	$-0.300615\pi$
$277$	1.49724e6	1.17244	0.586222	+	0.810150i	$0.300615\pi$
$278$	0	0
$279$	1.89749e6	1.45938
$280$	0	0
$281$	2.37949e6	1.79770	0.898852	−	0.438252i	$-0.144402\pi$
$281$	2.37949e6	1.79770	0.898852	+	0.438252i	$0.144402\pi$
$282$	0	0
$283$	−1.43924e6	−1.06824	−0.534118	−	0.845410i	$-0.679356\pi$
$283$	−1.43924e6	−1.06824	−0.534118	+	0.845410i	$0.679356\pi$
$284$	0	0
$285$	561146.	0.409227
$286$	0	0
$287$	−1.79031e6	−1.28299
$288$	0	0
$289$	−944525.	−0.665226
$290$	0	0
$291$	3.08893e6	2.13834
$292$	0	0
$293$	592084.	0.402916	0.201458	−	0.979497i	$-0.435432\pi$
$293$	592084.	0.402916	0.201458	+	0.979497i	$0.435432\pi$
$294$	0	0
$295$	134467.	0.0899624
$296$	0	0
$297$	1.01411e6	0.667102
$298$	0	0
$299$	−1.44191e6	−0.932736
$300$	0	0
$301$	−296609.	−0.188698
$302$	0	0
$303$	−3.15039e6	−1.97132
$304$	0	0
$305$	−211709.	−0.130313
$306$	0	0
$307$	2.24785e6	1.36120	0.680598	−	0.732657i	$-0.261720\pi$
$307$	2.24785e6	1.36120	0.680598	+	0.732657i	$0.261720\pi$
$308$	0	0
$309$	933992.	0.556477
$310$	0	0
$311$	2.69417e6	1.57951	0.789757	−	0.613419i	$-0.210206\pi$
$311$	2.69417e6	1.57951	0.789757	+	0.613419i	$0.210206\pi$
$312$	0	0
$313$	426247.	0.245924	0.122962	−	0.992411i	$-0.460761\pi$
$313$	426247.	0.245924	0.122962	+	0.992411i	$0.460761\pi$
$314$	0	0
$315$	−474007.	−0.269159
$316$	0	0
$317$	−2.02844e6	−1.13374	−0.566870	−	0.823807i	$-0.691846\pi$
$317$	−2.02844e6	−1.13374	−0.566870	+	0.823807i	$0.691846\pi$
$318$	0	0
$319$	534659.	0.294171
$320$	0	0
$321$	1.29314e6	0.700462
$322$	0	0
$323$	−1.99652e6	−1.06480
$324$	0	0
$325$	2.93699e6	1.54239
$326$	0	0
$327$	−1.28963e6	−0.666955
$328$	0	0
$329$	−2.38998e6	−1.21732
$330$	0	0
$331$	335165.	0.168147	0.0840734	−	0.996460i	$-0.473207\pi$
$331$	335165.	0.168147	0.0840734	+	0.996460i	$0.473207\pi$
$332$	0	0
$333$	4.90635e6	2.42465
$334$	0	0
$335$	−46909.1	−0.0228373
$336$	0	0
$337$	−2.10238e6	−1.00841	−0.504203	−	0.863585i	$-0.668214\pi$
$337$	−2.10238e6	−1.00841	−0.504203	+	0.863585i	$0.668214\pi$
$338$	0	0
$339$	6.22648e6	2.94268
$340$	0	0
$341$	−1.79429e6	−0.835614
$342$	0	0
$343$	−1.01126e6	−0.464115
$344$	0	0
$345$	291398.	0.131807
$346$	0	0
$347$	−2.30875e6	−1.02932	−0.514662	−	0.857393i	$-0.672083\pi$
$347$	−2.30875e6	−1.02932	−0.514662	+	0.857393i	$0.672083\pi$
$348$	0	0
$349$	−3.04965e6	−1.34025	−0.670126	−	0.742248i	$-0.733760\pi$
$349$	−3.04965e6	−1.34025	−0.670126	+	0.742248i	$0.733760\pi$
$350$	0	0
$351$	2.83272e6	1.22726
$352$	0	0
$353$	33779.0	0.0144281	0.00721406	−	0.999974i	$-0.497704\pi$
$353$	33779.0	0.0144281	0.00721406	+	0.999974i	$0.497704\pi$
$354$	0	0
$355$	397642.	0.167464
$356$	0	0
$357$	2.81543e6	1.16916
$358$	0	0
$359$	2.58288e6	1.05771	0.528857	−	0.848711i	$-0.322621\pi$
$359$	2.58288e6	1.05771	0.528857	+	0.848711i	$0.322621\pi$
$360$	0	0
$361$	5.90986e6	2.38676
$362$	0	0
$363$	1.06396e6	0.423799
$364$	0	0
$365$	−181845.	−0.0714446
$366$	0	0
$367$	3.91525e6	1.51738	0.758690	−	0.651452i	$-0.225840\pi$
$367$	3.91525e6	1.51738	0.758690	+	0.651452i	$0.225840\pi$
$368$	0	0
$369$	−3.91776e6	−1.49786
$370$	0	0
$371$	−2.60944e6	−0.984265
$372$	0	0
$373$	−254089.	−0.0945615	−0.0472807	−	0.998882i	$-0.515056\pi$
$373$	−254089.	−0.0945615	−0.0472807	+	0.998882i	$0.515056\pi$
$374$	0	0
$375$	−1.19909e6	−0.440326
$376$	0	0
$377$	1.49347e6	0.541183
$378$	0	0
$379$	−232637.	−0.0831919	−0.0415960	−	0.999135i	$-0.513244\pi$
$379$	−232637.	−0.0831919	−0.0415960	+	0.999135i	$0.513244\pi$
$380$	0	0
$381$	2.45474e6	0.866348
$382$	0	0
$383$	−2.14043e6	−0.745598	−0.372799	−	0.927912i	$-0.621602\pi$
$383$	−2.14043e6	−0.745598	−0.372799	+	0.927912i	$0.621602\pi$
$384$	0	0
$385$	448226.	0.154115
$386$	0	0
$387$	−649074.	−0.220301
$388$	0	0
$389$	1.19760e6	0.401272	0.200636	−	0.979666i	$-0.435699\pi$
$389$	1.19760e6	0.401272	0.200636	+	0.979666i	$0.435699\pi$
$390$	0	0
$391$	−1.03678e6	−0.342960
$392$	0	0
$393$	3.80735e6	1.24349
$394$	0	0
$395$	−200931.	−0.0647969
$396$	0	0
$397$	−262873.	−0.0837085	−0.0418542	−	0.999124i	$-0.513327\pi$
$397$	−262873.	−0.0837085	−0.0418542	+	0.999124i	$0.513327\pi$
$398$	0	0
$399$	−1.18256e7	−3.71870
$400$	0	0
$401$	4.93436e6	1.53239	0.766196	−	0.642607i	$-0.222147\pi$
$401$	4.93436e6	1.53239	0.766196	+	0.642607i	$0.222147\pi$
$402$	0	0
$403$	−5.01201e6	−1.53727
$404$	0	0
$405$	121888.	0.0369252
$406$	0	0
$407$	−4.63950e6	−1.38830
$408$	0	0
$409$	−2.32565e6	−0.687443	−0.343721	−	0.939072i	$-0.611688\pi$
$409$	−2.32565e6	−0.687443	−0.343721	+	0.939072i	$0.611688\pi$
$410$	0	0
$411$	−5.77022e6	−1.68495
$412$	0	0
$413$	−2.83376e6	−0.817501
$414$	0	0
$415$	−385184.	−0.109786
$416$	0	0
$417$	5.27326e6	1.48504
$418$	0	0
$419$	1.23189e6	0.342798	0.171399	−	0.985202i	$-0.445171\pi$
$419$	1.23189e6	0.342798	0.171399	+	0.985202i	$0.445171\pi$
$420$	0	0
$421$	5.17989e6	1.42435	0.712173	−	0.702004i	$-0.247711\pi$
$421$	5.17989e6	1.42435	0.712173	+	0.702004i	$0.247711\pi$
$422$	0	0
$423$	−5.23002e6	−1.42119
$424$	0	0
$425$	2.11179e6	0.567125
$426$	0	0
$427$	4.46155e6	1.18418
$428$	0	0
$429$	−8.10250e6	−2.12557
$430$	0	0
$431$	−563995.	−0.146245	−0.0731227	−	0.997323i	$-0.523296\pi$
$431$	−563995.	−0.146245	−0.0731227	+	0.997323i	$0.523296\pi$
$432$	0	0
$433$	−2.91530e6	−0.747246	−0.373623	−	0.927581i	$-0.621885\pi$
$433$	−2.91530e6	−0.747246	−0.373623	+	0.927581i	$0.621885\pi$
$434$	0	0
$435$	−301820.	−0.0764759
$436$	0	0
$437$	4.35475e6	1.09084
$438$	0	0
$439$	3.22518e6	0.798717	0.399359	−	0.916795i	$-0.369233\pi$
$439$	3.22518e6	0.798717	0.399359	+	0.916795i	$0.369233\pi$
$440$	0	0
$441$	3.88815e6	0.952020
$442$	0	0
$443$	7.28947e6	1.76476	0.882382	−	0.470534i	$-0.155939\pi$
$443$	7.28947e6	1.76476	0.882382	+	0.470534i	$0.155939\pi$
$444$	0	0
$445$	969459.	0.232075
$446$	0	0
$447$	−6.64397e6	−1.57275
$448$	0	0
$449$	2.18664e6	0.511871	0.255936	−	0.966694i	$-0.417616\pi$
$449$	2.18664e6	0.511871	0.255936	+	0.966694i	$0.417616\pi$
$450$	0	0
$451$	3.70467e6	0.857646
$452$	0	0
$453$	3.13910e6	0.718721
$454$	0	0
$455$	1.25204e6	0.283523
$456$	0	0
$457$	1.27935e6	0.286549	0.143275	−	0.989683i	$-0.454237\pi$
$457$	1.27935e6	0.286549	0.143275	+	0.989683i	$0.454237\pi$
$458$	0	0
$459$	2.03682e6	0.451253
$460$	0	0
$461$	3.27113e6	0.716878	0.358439	−	0.933553i	$-0.383309\pi$
$461$	3.27113e6	0.716878	0.358439	+	0.933553i	$0.383309\pi$
$462$	0	0
$463$	2.53630e6	0.549856	0.274928	−	0.961465i	$-0.411346\pi$
$463$	2.53630e6	0.549856	0.274928	+	0.961465i	$0.411346\pi$
$464$	0	0
$465$	1.01289e6	0.217235
$466$	0	0
$467$	−4.27287e6	−0.906624	−0.453312	−	0.891352i	$-0.649758\pi$
$467$	−4.27287e6	−0.906624	−0.453312	+	0.891352i	$0.649758\pi$
$468$	0	0
$469$	988564.	0.207526
$470$	0	0
$471$	−5.27290e6	−1.09521
$472$	0	0
$473$	613771.	0.126140
$474$	0	0
$475$	−8.87011e6	−1.80383
$476$	0	0
$477$	−5.71027e6	−1.14911
$478$	0	0
$479$	−4.39482e6	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$479$	−4.39482e6	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$480$	0	0
$481$	−1.29596e7	−2.55405
$482$	0	0
$483$	−6.14093e6	−1.19775
$484$	0	0
$485$	987709.	0.190667
$486$	0	0
$487$	3.23331e6	0.617767	0.308883	−	0.951100i	$-0.400045\pi$
$487$	3.23331e6	0.617767	0.308883	+	0.951100i	$0.400045\pi$
$488$	0	0
$489$	1.34136e6	0.253672
$490$	0	0
$491$	1.80282e6	0.337481	0.168740	−	0.985661i	$-0.446030\pi$
$491$	1.80282e6	0.337481	0.168740	+	0.985661i	$0.446030\pi$
$492$	0	0
$493$	1.07386e6	0.198989
$494$	0	0
$495$	980860.	0.179926
$496$	0	0
$497$	−8.37991e6	−1.52177
$498$	0	0
$499$	7.96162e6	1.43136	0.715682	−	0.698426i	$-0.246116\pi$
$499$	7.96162e6	1.43136	0.715682	+	0.698426i	$0.246116\pi$
$500$	0	0
$501$	−963556.	−0.171507
$502$	0	0
$503$	9.35462e6	1.64857	0.824283	−	0.566178i	$-0.191579\pi$
$503$	9.35462e6	1.64857	0.824283	+	0.566178i	$0.191579\pi$
$504$	0	0
$505$	−1.00736e6	−0.175775
$506$	0	0
$507$	−1.34926e7	−2.33119
$508$	0	0
$509$	2.63441e6	0.450702	0.225351	−	0.974278i	$-0.427647\pi$
$509$	2.63441e6	0.450702	0.225351	+	0.974278i	$0.427647\pi$
$510$	0	0
$511$	3.83221e6	0.649227
$512$	0	0
$513$	−8.55520e6	−1.43528
$514$	0	0
$515$	298651.	0.0496187
$516$	0	0
$517$	4.94556e6	0.813747
$518$	0	0
$519$	−1.72411e7	−2.80962
$520$	0	0
$521$	1.21118e6	0.195485	0.0977427	−	0.995212i	$-0.468838\pi$
$521$	1.21118e6	0.195485	0.0977427	+	0.995212i	$0.468838\pi$
$522$	0	0
$523$	−1.05040e6	−0.167919	−0.0839594	−	0.996469i	$-0.526757\pi$
$523$	−1.05040e6	−0.167919	−0.0839594	+	0.996469i	$0.526757\pi$
$524$	0	0
$525$	1.25083e7	1.98062
$526$	0	0
$527$	−3.60380e6	−0.565242
$528$	0	0
$529$	−4.17496e6	−0.648654
$530$	0	0
$531$	−6.20117e6	−0.954415
$532$	0	0
$533$	1.03483e7	1.57780
$534$	0	0
$535$	413492.	0.0624573
$536$	0	0
$537$	421737.	0.0631111
$538$	0	0
$539$	−3.67667e6	−0.545108
$540$	0	0
$541$	6.14777e6	0.903076	0.451538	−	0.892252i	$-0.350876\pi$
$541$	6.14777e6	0.903076	0.451538	+	0.892252i	$0.350876\pi$
$542$	0	0
$543$	4.13064e6	0.601198
$544$	0	0
$545$	−412369.	−0.0594696
$546$	0	0
$547$	6.25765e6	0.894217	0.447109	−	0.894480i	$-0.352454\pi$
$547$	6.25765e6	0.894217	0.447109	+	0.894480i	$0.352454\pi$
$548$	0	0
$549$	9.76328e6	1.38250
$550$	0	0
$551$	−4.51049e6	−0.632915
$552$	0	0
$553$	4.23442e6	0.588818
$554$	0	0
$555$	2.61903e6	0.360918
$556$	0	0
$557$	−1.46464e6	−0.200029	−0.100014	−	0.994986i	$-0.531889\pi$
$557$	−1.46464e6	−0.200029	−0.100014	+	0.994986i	$0.531889\pi$
$558$	0	0
$559$	1.71446e6	0.232058
$560$	0	0
$561$	−5.82596e6	−0.781556
$562$	0	0
$563$	5.96043e6	0.792513	0.396257	−	0.918140i	$-0.370309\pi$
$563$	5.96043e6	0.792513	0.396257	+	0.918140i	$0.370309\pi$
$564$	0	0
$565$	1.99096e6	0.262387
$566$	0	0
$567$	−2.56867e6	−0.335545
$568$	0	0
$569$	−2.07350e6	−0.268487	−0.134243	−	0.990948i	$-0.542860\pi$
$569$	−2.07350e6	−0.268487	−0.134243	+	0.990948i	$0.542860\pi$
$570$	0	0
$571$	6.35393e6	0.815553	0.407776	−	0.913082i	$-0.366304\pi$
$571$	6.35393e6	0.815553	0.407776	+	0.913082i	$0.366304\pi$
$572$	0	0
$573$	1.01359e7	1.28967
$574$	0	0
$575$	−4.60617e6	−0.580992
$576$	0	0
$577$	−7.81886e6	−0.977696	−0.488848	−	0.872369i	$-0.662583\pi$
$577$	−7.81886e6	−0.977696	−0.488848	+	0.872369i	$0.662583\pi$
$578$	0	0
$579$	−2.14653e7	−2.66098
$580$	0	0
$581$	8.11737e6	0.997643
$582$	0	0
$583$	5.39969e6	0.657957
$584$	0	0
$585$	2.73985e6	0.331008
$586$	0	0
$587$	−5.30694e6	−0.635696	−0.317848	−	0.948142i	$-0.602960\pi$
$587$	−5.30694e6	−0.635696	−0.317848	+	0.948142i	$0.602960\pi$
$588$	0	0
$589$	1.51370e7	1.79784
$590$	0	0
$591$	−1.21269e7	−1.42818
$592$	0	0
$593$	−1.36104e7	−1.58941	−0.794704	−	0.606998i	$-0.792374\pi$
$593$	−1.36104e7	−1.58941	−0.794704	+	0.606998i	$0.792374\pi$
$594$	0	0
$595$	900256.	0.104249
$596$	0	0
$597$	−5.72896e6	−0.657870
$598$	0	0
$599$	−1.22058e7	−1.38995	−0.694977	−	0.719032i	$-0.744585\pi$
$599$	−1.22058e7	−1.38995	−0.694977	+	0.719032i	$0.744585\pi$
$600$	0	0
$601$	−4.51185e6	−0.509529	−0.254764	−	0.967003i	$-0.581998\pi$
$601$	−4.51185e6	−0.509529	−0.254764	+	0.967003i	$0.581998\pi$
$602$	0	0
$603$	2.16329e6	0.242282
$604$	0	0
$605$	340210.	0.0377884
$606$	0	0
$607$	1.39206e7	1.53351	0.766755	−	0.641939i	$-0.221870\pi$
$607$	1.39206e7	1.53351	0.766755	+	0.641939i	$0.221870\pi$
$608$	0	0
$609$	6.36055e6	0.694947
$610$	0	0
$611$	1.38145e7	1.49704
$612$	0	0
$613$	1.16901e7	1.25652	0.628258	−	0.778005i	$-0.283768\pi$
$613$	1.16901e7	1.25652	0.628258	+	0.778005i	$0.283768\pi$
$614$	0	0
$615$	−2.09132e6	−0.222963
$616$	0	0
$617$	1.35932e7	1.43751	0.718754	−	0.695265i	$-0.244713\pi$
$617$	1.35932e7	1.43751	0.718754	+	0.695265i	$0.244713\pi$
$618$	0	0
$619$	3.00847e6	0.315587	0.157793	−	0.987472i	$-0.449562\pi$
$619$	3.00847e6	0.315587	0.157793	+	0.987472i	$0.449562\pi$
$620$	0	0
$621$	−4.44264e6	−0.462287
$622$	0	0
$623$	−2.04304e7	−2.10890
$624$	0	0
$625$	9.18858e6	0.940910
$626$	0	0
$627$	2.44706e7	2.48586
$628$	0	0
$629$	−9.31837e6	−0.939103
$630$	0	0
$631$	−6.91884e6	−0.691767	−0.345884	−	0.938277i	$-0.612421\pi$
$631$	−6.91884e6	−0.691767	−0.345884	+	0.938277i	$0.612421\pi$
$632$	0	0
$633$	−1.95362e7	−1.93789
$634$	0	0
$635$	784920.	0.0772487
$636$	0	0
$637$	−1.02701e7	−1.00283
$638$	0	0
$639$	−1.83379e7	−1.77663
$640$	0	0
$641$	1.39099e7	1.33715	0.668575	−	0.743645i	$-0.266904\pi$
$641$	1.39099e7	1.33715	0.668575	+	0.743645i	$0.266904\pi$
$642$	0	0
$643$	−2.24545e6	−0.214178	−0.107089	−	0.994249i	$-0.534153\pi$
$643$	−2.24545e6	−0.214178	−0.107089	+	0.994249i	$0.534153\pi$
$644$	0	0
$645$	−346479.	−0.0327927
$646$	0	0
$647$	3.88795e6	0.365140	0.182570	−	0.983193i	$-0.441558\pi$
$647$	3.88795e6	0.365140	0.182570	+	0.983193i	$0.441558\pi$
$648$	0	0
$649$	5.86388e6	0.546479
$650$	0	0
$651$	−2.13457e7	−1.97404
$652$	0	0
$653$	1.89700e7	1.74094	0.870472	−	0.492218i	$-0.163814\pi$
$653$	1.89700e7	1.74094	0.870472	+	0.492218i	$0.163814\pi$
$654$	0	0
$655$	1.21743e6	0.110877
$656$	0	0
$657$	8.38608e6	0.757959
$658$	0	0
$659$	−7.47462e6	−0.670464	−0.335232	−	0.942136i	$-0.608815\pi$
$659$	−7.47462e6	−0.670464	−0.335232	+	0.942136i	$0.608815\pi$
$660$	0	0
$661$	1.45909e7	1.29890	0.649452	−	0.760402i	$-0.274998\pi$
$661$	1.45909e7	1.29890	0.649452	+	0.760402i	$0.274998\pi$
$662$	0	0
$663$	−1.62738e7	−1.43782
$664$	0	0
$665$	−3.78133e6	−0.331581
$666$	0	0
$667$	−2.34226e6	−0.203854
$668$	0	0
$669$	−2.99449e7	−2.58677
$670$	0	0
$671$	−9.23225e6	−0.791592
$672$	0	0
$673$	−1.17057e7	−0.996229	−0.498115	−	0.867111i	$-0.665974\pi$
$673$	−1.17057e7	−0.996229	−0.498115	+	0.867111i	$0.665974\pi$
$674$	0	0
$675$	9.04912e6	0.764446
$676$	0	0
$677$	7.24512e6	0.607538	0.303769	−	0.952746i	$-0.401755\pi$
$677$	7.24512e6	0.607538	0.303769	+	0.952746i	$0.401755\pi$
$678$	0	0
$679$	−2.08150e7	−1.73261
$680$	0	0
$681$	2.29828e6	0.189904
$682$	0	0
$683$	−1.37761e7	−1.12999	−0.564995	−	0.825095i	$-0.691122\pi$
$683$	−1.37761e7	−1.12999	−0.564995	+	0.825095i	$0.691122\pi$
$684$	0	0
$685$	−1.84507e6	−0.150240
$686$	0	0
$687$	−4.87992e6	−0.394476
$688$	0	0
$689$	1.50830e7	1.21043
$690$	0	0
$691$	1.26667e7	1.00918	0.504592	−	0.863358i	$-0.331643\pi$
$691$	1.26667e7	1.00918	0.504592	+	0.863358i	$0.331643\pi$
$692$	0	0
$693$	−2.06707e7	−1.63501
$694$	0	0
$695$	1.68616e6	0.132415
$696$	0	0
$697$	7.44078e6	0.580145
$698$	0	0
$699$	−1.55094e7	−1.20061
$700$	0	0
$701$	−6.23754e6	−0.479423	−0.239711	−	0.970844i	$-0.577053\pi$
$701$	−6.23754e6	−0.479423	−0.239711	+	0.970844i	$0.577053\pi$
$702$	0	0
$703$	3.91397e7	2.98696
$704$	0	0
$705$	−2.79181e6	−0.211550
$706$	0	0
$707$	2.12291e7	1.59729
$708$	0	0
$709$	−1.37403e7	−1.02655	−0.513275	−	0.858224i	$-0.671568\pi$
$709$	−1.37403e7	−1.02655	−0.513275	+	0.858224i	$0.671568\pi$
$710$	0	0
$711$	9.26625e6	0.687433
$712$	0	0
$713$	7.86049e6	0.579063
$714$	0	0
$715$	−2.59083e6	−0.189528
$716$	0	0
$717$	3.07602e7	2.23455
$718$	0	0
$719$	3.99820e6	0.288431	0.144216	−	0.989546i	$-0.453934\pi$
$719$	3.99820e6	0.288431	0.144216	+	0.989546i	$0.453934\pi$
$720$	0	0
$721$	−6.29377e6	−0.450892
$722$	0	0
$723$	2.65980e7	1.89236
$724$	0	0
$725$	4.77090e6	0.337097
$726$	0	0
$727$	1.28171e7	0.899404	0.449702	−	0.893179i	$-0.351530\pi$
$727$	1.28171e7	0.899404	0.449702	+	0.893179i	$0.351530\pi$
$728$	0	0
$729$	−2.32936e7	−1.62337
$730$	0	0
$731$	1.23275e6	0.0853261
$732$	0	0
$733$	7.69133e6	0.528739	0.264370	−	0.964421i	$-0.414836\pi$
$733$	7.69133e6	0.528739	0.264370	+	0.964421i	$0.414836\pi$
$734$	0	0
$735$	2.07551e6	0.141712
$736$	0	0
$737$	−2.04563e6	−0.138726
$738$	0	0
$739$	−2.38607e7	−1.60721	−0.803604	−	0.595165i	$-0.797087\pi$
$739$	−2.38607e7	−1.60721	−0.803604	+	0.595165i	$0.797087\pi$
$740$	0	0
$741$	6.83543e7	4.57320
$742$	0	0
$743$	1.02071e7	0.678310	0.339155	−	0.940730i	$-0.389859\pi$
$743$	1.02071e7	0.678310	0.339155	+	0.940730i	$0.389859\pi$
$744$	0	0
$745$	−2.12446e6	−0.140235
$746$	0	0
$747$	1.77634e7	1.16473
$748$	0	0
$749$	−8.71395e6	−0.567558
$750$	0	0
$751$	1.38152e7	0.893837	0.446919	−	0.894575i	$-0.352521\pi$
$751$	1.38152e7	0.893837	0.446919	+	0.894575i	$0.352521\pi$
$752$	0	0
$753$	−2.41306e7	−1.55089
$754$	0	0
$755$	1.00375e6	0.0640854
$756$	0	0
$757$	3.81747e6	0.242123	0.121062	−	0.992645i	$-0.461370\pi$
$757$	3.81747e6	0.242123	0.121062	+	0.992645i	$0.461370\pi$
$758$	0	0
$759$	1.27074e7	0.800666
$760$	0	0
$761$	−2.07219e7	−1.29708	−0.648542	−	0.761179i	$-0.724621\pi$
$761$	−2.07219e7	−1.29708	−0.648542	+	0.761179i	$0.724621\pi$
$762$	0	0
$763$	8.69028e6	0.540409
$764$	0	0
$765$	1.97004e6	0.121709
$766$	0	0
$767$	1.63797e7	1.00535
$768$	0	0
$769$	3.60940e6	0.220099	0.110050	−	0.993926i	$-0.464899\pi$
$769$	3.60940e6	0.220099	0.110050	+	0.993926i	$0.464899\pi$
$770$	0	0
$771$	1.62594e6	0.0985076
$772$	0	0
$773$	9.50053e6	0.571872	0.285936	−	0.958249i	$-0.407695\pi$
$773$	9.50053e6	0.571872	0.285936	+	0.958249i	$0.407695\pi$
$774$	0	0
$775$	−1.60109e7	−0.957548
$776$	0	0
$777$	−5.51936e7	−3.27971
$778$	0	0
$779$	−3.12533e7	−1.84524
$780$	0	0
$781$	1.73405e7	1.01726
$782$	0	0
$783$	4.60152e6	0.268224
$784$	0	0
$785$	−1.68605e6	−0.0976553
$786$	0	0
$787$	1.63265e7	0.939630	0.469815	−	0.882765i	$-0.344321\pi$
$787$	1.63265e7	0.939630	0.469815	+	0.882765i	$0.344321\pi$
$788$	0	0
$789$	−1.27250e7	−0.727723
$790$	0	0
$791$	−4.19576e7	−2.38435
$792$	0	0
$793$	−2.57886e7	−1.45628
$794$	0	0
$795$	−3.04817e6	−0.171049
$796$	0	0
$797$	2.02759e7	1.13067	0.565333	−	0.824863i	$-0.308748\pi$
$797$	2.02759e7	1.13067	0.565333	+	0.824863i	$0.308748\pi$
$798$	0	0
$799$	9.93309e6	0.550450
$800$	0	0
$801$	−4.47081e7	−2.46210
$802$	0	0
$803$	−7.92996e6	−0.433992
$804$	0	0
$805$	−1.96361e6	−0.106798
$806$	0	0
$807$	2.98302e7	1.61240
$808$	0	0
$809$	9.78137e6	0.525446	0.262723	−	0.964871i	$-0.415379\pi$
$809$	9.78137e6	0.525446	0.262723	+	0.964871i	$0.415379\pi$
$810$	0	0
$811$	3.23972e7	1.72964	0.864821	−	0.502081i	$-0.167432\pi$
$811$	3.23972e7	1.72964	0.864821	+	0.502081i	$0.167432\pi$
$812$	0	0
$813$	7.11856e6	0.377716
$814$	0	0
$815$	428909.	0.0226189
$816$	0	0
$817$	−5.17789e6	−0.271393
$818$	0	0
$819$	−5.77397e7	−3.00791
$820$	0	0
$821$	1.01128e7	0.523619	0.261809	−	0.965120i	$-0.415681\pi$
$821$	1.01128e7	0.523619	0.261809	+	0.965120i	$0.415681\pi$
$822$	0	0
$823$	2.07361e7	1.06716	0.533578	−	0.845751i	$-0.320847\pi$
$823$	2.07361e7	1.06716	0.533578	+	0.845751i	$0.320847\pi$
$824$	0	0
$825$	−2.58834e7	−1.32400
$826$	0	0
$827$	−8.63947e6	−0.439262	−0.219631	−	0.975583i	$-0.570485\pi$
$827$	−8.63947e6	−0.439262	−0.219631	+	0.975583i	$0.570485\pi$
$828$	0	0
$829$	8.08885e6	0.408790	0.204395	−	0.978888i	$-0.434477\pi$
$829$	8.08885e6	0.408790	0.204395	+	0.978888i	$0.434477\pi$
$830$	0	0
$831$	−3.68580e7	−1.85152
$832$	0	0
$833$	−7.38454e6	−0.368732
$834$	0	0
$835$	−308104.	−0.0152926
$836$	0	0
$837$	−1.54424e7	−0.761908
$838$	0	0
$839$	1.91942e7	0.941380	0.470690	−	0.882299i	$-0.344005\pi$
$839$	1.91942e7	0.941380	0.470690	+	0.882299i	$0.344005\pi$
$840$	0	0
$841$	−1.80851e7	−0.881722
$842$	0	0
$843$	−5.85765e7	−2.83893
$844$	0	0
$845$	−4.31437e6	−0.207862
$846$	0	0
$847$	−7.16959e6	−0.343389
$848$	0	0
$849$	3.54302e7	1.68696
$850$	0	0
$851$	2.03249e7	0.962065
$852$	0	0
$853$	−3.63036e7	−1.70835	−0.854175	−	0.519986i	$-0.825937\pi$
$853$	−3.63036e7	−1.70835	−0.854175	+	0.519986i	$0.825937\pi$
$854$	0	0
$855$	−8.27473e6	−0.387114
$856$	0	0
$857$	2.21156e7	1.02860	0.514300	−	0.857611i	$-0.328052\pi$
$857$	2.21156e7	1.02860	0.514300	+	0.857611i	$0.328052\pi$
$858$	0	0
$859$	−3.79857e7	−1.75646	−0.878229	−	0.478240i	$-0.841275\pi$
$859$	−3.79857e7	−1.75646	−0.878229	+	0.478240i	$0.841275\pi$
$860$	0	0
$861$	4.40724e7	2.02609
$862$	0	0
$863$	−3.77635e7	−1.72602	−0.863010	−	0.505187i	$-0.831424\pi$
$863$	−3.77635e7	−1.72602	−0.863010	+	0.505187i	$0.831424\pi$
$864$	0	0
$865$	−5.51297e6	−0.250522
$866$	0	0
$867$	2.32516e7	1.05052
$868$	0	0
$869$	−8.76226e6	−0.393610
$870$	0	0
$871$	−5.71409e6	−0.255212
$872$	0	0
$873$	−4.55498e7	−2.02279
$874$	0	0
$875$	8.08017e6	0.356780
$876$	0	0
$877$	−2.93230e7	−1.28739	−0.643694	−	0.765283i	$-0.722599\pi$
$877$	−2.93230e7	−1.28739	−0.643694	+	0.765283i	$0.722599\pi$
$878$	0	0
$879$	−1.45755e7	−0.636284
$880$	0	0
$881$	2.82660e7	1.22694	0.613472	−	0.789717i	$-0.289772\pi$
$881$	2.82660e7	1.22694	0.613472	+	0.789717i	$0.289772\pi$
$882$	0	0
$883$	9.38773e6	0.405190	0.202595	−	0.979263i	$-0.435062\pi$
$883$	9.38773e6	0.405190	0.202595	+	0.979263i	$0.435062\pi$
$884$	0	0
$885$	−3.31021e6	−0.142068
$886$	0	0
$887$	−1.65227e7	−0.705133	−0.352566	−	0.935787i	$-0.614691\pi$
$887$	−1.65227e7	−0.705133	−0.352566	+	0.935787i	$0.614691\pi$
$888$	0	0
$889$	−1.65414e7	−0.701970
$890$	0	0
$891$	5.31532e6	0.224303
$892$	0	0
$893$	−4.17218e7	−1.75079
$894$	0	0
$895$	134853.	0.00562736
$896$	0	0
$897$	3.54958e7	1.47297
$898$	0	0
$899$	−8.14161e6	−0.335978
$900$	0	0
$901$	1.08452e7	0.445067
$902$	0	0
$903$	7.30170e6	0.297992
$904$	0	0
$905$	1.32080e6	0.0536064
$906$	0	0
$907$	−2.17396e7	−0.877474	−0.438737	−	0.898615i	$-0.644574\pi$
$907$	−2.17396e7	−0.877474	−0.438737	+	0.898615i	$0.644574\pi$
$908$	0	0
$909$	4.64560e7	1.86480
$910$	0	0
$911$	4.03503e7	1.61083	0.805416	−	0.592710i	$-0.201942\pi$
$911$	4.03503e7	1.61083	0.805416	+	0.592710i	$0.201942\pi$
$912$	0	0
$913$	−1.67972e7	−0.666899
$914$	0	0
$915$	5.21168e6	0.205791
$916$	0	0
$917$	−2.56561e7	−1.00755
$918$	0	0
$919$	−2.03716e7	−0.795675	−0.397837	−	0.917456i	$-0.630239\pi$
$919$	−2.03716e7	−0.795675	−0.397837	+	0.917456i	$0.630239\pi$
$920$	0	0
$921$	−5.53358e7	−2.14960
$922$	0	0
$923$	4.84375e7	1.87145
$924$	0	0
$925$	−4.13994e7	−1.59089
$926$	0	0
$927$	−1.37728e7	−0.526407
$928$	0	0
$929$	−9.43818e6	−0.358797	−0.179399	−	0.983776i	$-0.557415\pi$
$929$	−9.43818e6	−0.358797	−0.179399	+	0.983776i	$0.557415\pi$
$930$	0	0
$931$	3.10171e7	1.17281
$932$	0	0
$933$	−6.63230e7	−2.49437
$934$	0	0
$935$	−1.86289e6	−0.0696881
$936$	0	0
$937$	1.97034e6	0.0733149	0.0366574	−	0.999328i	$-0.488329\pi$
$937$	1.97034e6	0.0733149	0.0366574	+	0.999328i	$0.488329\pi$
$938$	0	0
$939$	−1.04930e7	−0.388362
$940$	0	0
$941$	1.59544e6	0.0587365	0.0293682	−	0.999569i	$-0.490650\pi$
$941$	1.59544e6	0.0587365	0.0293682	+	0.999569i	$0.490650\pi$
$942$	0	0
$943$	−1.62296e7	−0.594330
$944$	0	0
$945$	3.85764e6	0.140521
$946$	0	0
$947$	−2.26194e7	−0.819608	−0.409804	−	0.912174i	$-0.634403\pi$
$947$	−2.26194e7	−0.819608	−0.409804	+	0.912174i	$0.634403\pi$
$948$	0	0
$949$	−2.21509e7	−0.798409
$950$	0	0
$951$	4.99345e7	1.79040
$952$	0	0
$953$	1.13871e7	0.406145	0.203073	−	0.979164i	$-0.434907\pi$
$953$	1.13871e7	0.406145	0.203073	+	0.979164i	$0.434907\pi$
$954$	0	0
$955$	3.24104e6	0.114994
$956$	0	0
$957$	−1.31618e7	−0.464555
$958$	0	0
$959$	3.88831e7	1.36526
$960$	0	0
$961$	−1.30638e6	−0.0456311
$962$	0	0
$963$	−1.90689e7	−0.662612
$964$	0	0
$965$	−6.86370e6	−0.237268
$966$	0	0
$967$	2.12164e7	0.729636	0.364818	−	0.931079i	$-0.381131\pi$
$967$	2.12164e7	0.729636	0.364818	+	0.931079i	$0.381131\pi$
$968$	0	0
$969$	4.91489e7	1.68153
$970$	0	0
$971$	−3.22909e7	−1.09909	−0.549543	−	0.835466i	$-0.685198\pi$
$971$	−3.22909e7	−1.09909	−0.549543	+	0.835466i	$0.685198\pi$
$972$	0	0
$973$	−3.55342e7	−1.20327
$974$	0	0
$975$	−7.23006e7	−2.43574
$976$	0	0
$977$	1.68190e7	0.563719	0.281859	−	0.959456i	$-0.409049\pi$
$977$	1.68190e7	0.563719	0.281859	+	0.959456i	$0.409049\pi$
$978$	0	0
$979$	4.22765e7	1.40975
$980$	0	0
$981$	1.90171e7	0.630916
$982$	0	0
$983$	4.77706e6	0.157680	0.0788400	−	0.996887i	$-0.474878\pi$
$983$	4.77706e6	0.157680	0.0788400	+	0.996887i	$0.474878\pi$
$984$	0	0
$985$	−3.87768e6	−0.127345
$986$	0	0
$987$	5.88347e7	1.92239
$988$	0	0
$989$	−2.68883e6	−0.0874124
$990$	0	0
$991$	−3.70448e7	−1.19824	−0.599119	−	0.800660i	$-0.704483\pi$
$991$	−3.70448e7	−1.19824	−0.599119	+	0.800660i	$0.704483\pi$
$992$	0	0
$993$	−8.25084e6	−0.265537
$994$	0	0
$995$	−1.83188e6	−0.0586596
$996$	0	0
$997$	−5.25510e7	−1.67434	−0.837169	−	0.546944i	$-0.815791\pi$
$997$	−5.25510e7	−1.67434	−0.837169	+	0.546944i	$0.815791\pi$
$998$	0	0
$999$	−3.99296e7	−1.26585

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.6	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.6	✓	57	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1028 = 2^{2} \cdot 257 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1028.a (trivial)

\(f(q)\)	\(=\)	\(q-24.6173 q^{3} -7.87155 q^{5} +165.885 q^{7} +363.009 q^{9} +O(q^{10})\)	\(q-24.6173 q^{3} -7.87155 q^{5} +165.885 q^{7} +363.009 q^{9} -343.265 q^{11} -958.848 q^{13} +193.776 q^{15} -689.443 q^{17} +2895.85 q^{19} -4083.64 q^{21} +1503.79 q^{23} -3063.04 q^{25} -2954.29 q^{27} -1557.57 q^{29} +5227.12 q^{31} +8450.24 q^{33} -1305.77 q^{35} +13515.8 q^{37} +23604.2 q^{39} -10792.5 q^{41} -1788.04 q^{43} -2857.44 q^{45} -14407.4 q^{47} +10710.9 q^{49} +16972.2 q^{51} -15730.4 q^{53} +2702.03 q^{55} -71287.9 q^{57} -17082.7 q^{59} +26895.4 q^{61} +60217.8 q^{63} +7547.62 q^{65} +5959.33 q^{67} -37019.2 q^{69} -50516.4 q^{71} +23101.6 q^{73} +75403.6 q^{75} -56942.5 q^{77} +25526.2 q^{79} -15484.6 q^{81} +48933.7 q^{83} +5426.98 q^{85} +38343.1 q^{87} -123160. q^{89} -159059. q^{91} -128677. q^{93} -22794.8 q^{95} -125478. q^{97} -124608. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Introduction

L-functions

Modular forms

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Database

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