LMFDB - Embedded newform 1028.6.a.a.1.14

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:	$1$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q-13.6293 q^{3} -91.2960 q^{5} -63.7695 q^{7} -57.2414 q^{9} +O(q^{10})$	$q-13.6293 q^{3} -91.2960 q^{5} -63.7695 q^{7} -57.2414 q^{9} -686.386 q^{11} -962.972 q^{13} +1244.30 q^{15} +1477.55 q^{17} -1326.61 q^{19} +869.136 q^{21} +2401.71 q^{23} +5209.95 q^{25} +4092.09 q^{27} -1737.95 q^{29} +4838.06 q^{31} +9354.97 q^{33} +5821.90 q^{35} +9146.16 q^{37} +13124.7 q^{39} -6861.74 q^{41} -6691.62 q^{43} +5225.91 q^{45} -5970.71 q^{47} -12740.4 q^{49} -20138.1 q^{51} +6464.50 q^{53} +62664.2 q^{55} +18080.7 q^{57} -12245.1 q^{59} -5532.01 q^{61} +3650.26 q^{63} +87915.5 q^{65} +27252.4 q^{67} -32733.7 q^{69} -18469.6 q^{71} -13979.0 q^{73} -71008.1 q^{75} +43770.5 q^{77} -69526.0 q^{79} -41862.8 q^{81} +95473.6 q^{83} -134895. q^{85} +23687.1 q^{87} +44472.6 q^{89} +61408.3 q^{91} -65939.5 q^{93} +121114. q^{95} +67595.7 q^{97} +39289.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * 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
$3$	−13.6293	−0.874322	−0.437161	−	0.899383i	$-0.644016\pi$
$3$	−13.6293	−0.874322	−0.437161	+	0.899383i	$0.644016\pi$
$4$	0	0
$5$	−91.2960	−1.63315	−0.816576	−	0.577238i	$-0.804130\pi$
$5$	−91.2960	−1.63315	−0.816576	+	0.577238i	$0.804130\pi$
$6$	0	0
$7$	−63.7695	−0.491890	−0.245945	−	0.969284i	$-0.579098\pi$
$7$	−63.7695	−0.491890	−0.245945	+	0.969284i	$0.579098\pi$
$8$	0	0
$9$	−57.2414	−0.235561
$10$	0	0
$11$	−686.386	−1.71036	−0.855178	−	0.518335i	$-0.826552\pi$
$11$	−686.386	−1.71036	−0.855178	+	0.518335i	$0.826552\pi$
$12$	0	0
$13$	−962.972	−1.58036	−0.790179	−	0.612877i	$-0.790012\pi$
$13$	−962.972	−1.58036	−0.790179	+	0.612877i	$0.790012\pi$
$14$	0	0
$15$	1244.30	1.42790
$16$	0	0
$17$	1477.55	1.24000	0.619999	−	0.784603i	$-0.287133\pi$
$17$	1477.55	1.24000	0.619999	+	0.784603i	$0.287133\pi$
$18$	0	0
$19$	−1326.61	−0.843059	−0.421529	−	0.906815i	$-0.638507\pi$
$19$	−1326.61	−0.843059	−0.421529	+	0.906815i	$0.638507\pi$
$20$	0	0
$21$	869.136	0.430070
$22$	0	0
$23$	2401.71	0.946674	0.473337	−	0.880881i	$-0.343049\pi$
$23$	2401.71	0.946674	0.473337	+	0.880881i	$0.343049\pi$
$24$	0	0
$25$	5209.95	1.66718
$26$	0	0
$27$	4092.09	1.08028
$28$	0	0
$29$	−1737.95	−0.383744	−0.191872	−	0.981420i	$-0.561456\pi$
$29$	−1737.95	−0.383744	−0.191872	+	0.981420i	$0.561456\pi$
$30$	0	0
$31$	4838.06	0.904205	0.452103	−	0.891966i	$-0.350674\pi$
$31$	4838.06	0.904205	0.452103	+	0.891966i	$0.350674\pi$
$32$	0	0
$33$	9354.97	1.49540
$34$	0	0
$35$	5821.90	0.803331
$36$	0	0
$37$	9146.16	1.09833	0.549167	−	0.835713i	$-0.314945\pi$
$37$	9146.16	1.09833	0.549167	+	0.835713i	$0.314945\pi$
$38$	0	0
$39$	13124.7	1.38174
$40$	0	0
$41$	−6861.74	−0.637492	−0.318746	−	0.947840i	$-0.603262\pi$
$41$	−6861.74	−0.637492	−0.318746	+	0.947840i	$0.603262\pi$
$42$	0	0
$43$	−6691.62	−0.551899	−0.275950	−	0.961172i	$-0.588992\pi$
$43$	−6691.62	−0.551899	−0.275950	+	0.961172i	$0.588992\pi$
$44$	0	0
$45$	5225.91	0.384708
$46$	0	0
$47$	−5970.71	−0.394259	−0.197129	−	0.980377i	$-0.563162\pi$
$47$	−5970.71	−0.394259	−0.197129	+	0.980377i	$0.563162\pi$
$48$	0	0
$49$	−12740.4	−0.758044
$50$	0	0
$51$	−20138.1	−1.08416
$52$	0	0
$53$	6464.50	0.316115	0.158058	−	0.987430i	$-0.449477\pi$
$53$	6464.50	0.316115	0.158058	+	0.987430i	$0.449477\pi$
$54$	0	0
$55$	62664.2	2.79327
$56$	0	0
$57$	18080.7	0.737105
$58$	0	0
$59$	−12245.1	−0.457964	−0.228982	−	0.973431i	$-0.573540\pi$
$59$	−12245.1	−0.457964	−0.228982	+	0.973431i	$0.573540\pi$
$60$	0	0
$61$	−5532.01	−0.190352	−0.0951762	−	0.995460i	$-0.530341\pi$
$61$	−5532.01	−0.190352	−0.0951762	+	0.995460i	$0.530341\pi$
$62$	0	0
$63$	3650.26	0.115870
$64$	0	0
$65$	87915.5	2.58096
$66$	0	0
$67$	27252.4	0.741683	0.370842	−	0.928696i	$-0.379069\pi$
$67$	27252.4	0.741683	0.370842	+	0.928696i	$0.379069\pi$
$68$	0	0
$69$	−32733.7	−0.827698
$70$	0	0
$71$	−18469.6	−0.434823	−0.217411	−	0.976080i	$-0.569761\pi$
$71$	−18469.6	−0.434823	−0.217411	+	0.976080i	$0.569761\pi$
$72$	0	0
$73$	−13979.0	−0.307022	−0.153511	−	0.988147i	$-0.549058\pi$
$73$	−13979.0	−0.307022	−0.153511	+	0.988147i	$0.549058\pi$
$74$	0	0
$75$	−71008.1	−1.45766
$76$	0	0
$77$	43770.5	0.841307
$78$	0	0
$79$	−69526.0	−1.25337	−0.626685	−	0.779272i	$-0.715589\pi$
$79$	−69526.0	−1.25337	−0.626685	+	0.779272i	$0.715589\pi$
$80$	0	0
$81$	−41862.8	−0.708949
$82$	0	0
$83$	95473.6	1.52121	0.760603	−	0.649218i	$-0.224904\pi$
$83$	95473.6	1.52121	0.760603	+	0.649218i	$0.224904\pi$
$84$	0	0
$85$	−134895.	−2.02511
$86$	0	0
$87$	23687.1	0.335516
$88$	0	0
$89$	44472.6	0.595138	0.297569	−	0.954700i	$-0.403824\pi$
$89$	44472.6	0.595138	0.297569	+	0.954700i	$0.403824\pi$
$90$	0	0
$91$	61408.3	0.777362
$92$	0	0
$93$	−65939.5	−0.790567
$94$	0	0
$95$	121114.	1.37684
$96$	0	0
$97$	67595.7	0.729441	0.364720	−	0.931117i	$-0.381165\pi$
$97$	67595.7	0.729441	0.364720	+	0.931117i	$0.381165\pi$
$98$	0	0
$99$	39289.7	0.402894
$100$	0	0
$101$	−2993.99	−0.0292043	−0.0146022	−	0.999893i	$-0.504648\pi$
$101$	−2993.99	−0.0292043	−0.0146022	+	0.999893i	$0.504648\pi$
$102$	0	0
$103$	153057.	1.42154	0.710771	−	0.703423i	$-0.248346\pi$
$103$	153057.	1.42154	0.710771	+	0.703423i	$0.248346\pi$
$104$	0	0
$105$	−79348.6	−0.702370
$106$	0	0
$107$	15068.7	0.127238	0.0636188	−	0.997974i	$-0.479736\pi$
$107$	15068.7	0.127238	0.0636188	+	0.997974i	$0.479736\pi$
$108$	0	0
$109$	147349.	1.18790	0.593952	−	0.804501i	$-0.297567\pi$
$109$	147349.	1.18790	0.593952	+	0.804501i	$0.297567\pi$
$110$	0	0
$111$	−124656.	−0.960297
$112$	0	0
$113$	153461.	1.13058	0.565291	−	0.824891i	$-0.308764\pi$
$113$	153461.	1.13058	0.565291	+	0.824891i	$0.308764\pi$
$114$	0	0
$115$	−219266.	−1.54606
$116$	0	0
$117$	55121.9	0.372271
$118$	0	0
$119$	−94222.9	−0.609943
$120$	0	0
$121$	310074.	1.92532
$122$	0	0
$123$	93520.9	0.557373
$124$	0	0
$125$	−190348.	−1.08961
$126$	0	0
$127$	14019.1	0.0771276	0.0385638	−	0.999256i	$-0.487722\pi$
$127$	14019.1	0.0771276	0.0385638	+	0.999256i	$0.487722\pi$
$128$	0	0
$129$	91202.2	0.482538
$130$	0	0
$131$	−353676.	−1.80064	−0.900320	−	0.435228i	$-0.856668\pi$
$131$	−353676.	−1.80064	−0.900320	+	0.435228i	$0.856668\pi$
$132$	0	0
$133$	84597.0	0.414692
$134$	0	0
$135$	−373591.	−1.76426
$136$	0	0
$137$	−119989.	−0.546184	−0.273092	−	0.961988i	$-0.588046\pi$
$137$	−119989.	−0.546184	−0.273092	+	0.961988i	$0.588046\pi$
$138$	0	0
$139$	12051.6	0.0529063	0.0264532	−	0.999650i	$-0.491579\pi$
$139$	12051.6	0.0529063	0.0264532	+	0.999650i	$0.491579\pi$
$140$	0	0
$141$	81376.8	0.344709
$142$	0	0
$143$	660970.	2.70297
$144$	0	0
$145$	158668.	0.626712
$146$	0	0
$147$	173644.	0.662774
$148$	0	0
$149$	−425024.	−1.56837	−0.784183	−	0.620529i	$-0.786918\pi$
$149$	−425024.	−1.56837	−0.784183	+	0.620529i	$0.786918\pi$
$150$	0	0
$151$	−183370.	−0.654466	−0.327233	−	0.944944i	$-0.606116\pi$
$151$	−183370.	−0.654466	−0.327233	+	0.944944i	$0.606116\pi$
$152$	0	0
$153$	−84577.3	−0.292096
$154$	0	0
$155$	−441695.	−1.47670
$156$	0	0
$157$	−139872.	−0.452879	−0.226439	−	0.974025i	$-0.572709\pi$
$157$	−139872.	−0.452879	−0.226439	+	0.974025i	$0.572709\pi$
$158$	0	0
$159$	−88106.8	−0.276386
$160$	0	0
$161$	−153156.	−0.465660
$162$	0	0
$163$	307010.	0.905073	0.452537	−	0.891746i	$-0.350519\pi$
$163$	307010.	0.905073	0.452537	+	0.891746i	$0.350519\pi$
$164$	0	0
$165$	−854071.	−2.44222
$166$	0	0
$167$	229343.	0.636349	0.318174	−	0.948032i	$-0.396930\pi$
$167$	229343.	0.636349	0.318174	+	0.948032i	$0.396930\pi$
$168$	0	0
$169$	556022.	1.49753
$170$	0	0
$171$	75936.8	0.198592
$172$	0	0
$173$	−621752.	−1.57944	−0.789718	−	0.613470i	$-0.789773\pi$
$173$	−621752.	−1.57944	−0.789718	+	0.613470i	$0.789773\pi$
$174$	0	0
$175$	−332236.	−0.820072
$176$	0	0
$177$	166892.	0.400408
$178$	0	0
$179$	−175206.	−0.408711	−0.204356	−	0.978897i	$-0.565510\pi$
$179$	−175206.	−0.408711	−0.204356	+	0.978897i	$0.565510\pi$
$180$	0	0
$181$	433174.	0.982803	0.491401	−	0.870933i	$-0.336485\pi$
$181$	433174.	0.982803	0.491401	+	0.870933i	$0.336485\pi$
$182$	0	0
$183$	75397.6	0.166429
$184$	0	0
$185$	−835007.	−1.79375
$186$	0	0
$187$	−1.01417e6	−2.12084
$188$	0	0
$189$	−260951.	−0.531378
$190$	0	0
$191$	−422260.	−0.837523	−0.418762	−	0.908096i	$-0.637536\pi$
$191$	−422260.	−0.837523	−0.418762	+	0.908096i	$0.637536\pi$
$192$	0	0
$193$	−611805.	−1.18228	−0.591140	−	0.806569i	$-0.701322\pi$
$193$	−611805.	−1.18228	−0.591140	+	0.806569i	$0.701322\pi$
$194$	0	0
$195$	−1.19823e6	−2.25659
$196$	0	0
$197$	191739.	0.352001	0.176001	−	0.984390i	$-0.443684\pi$
$197$	191739.	0.352001	0.176001	+	0.984390i	$0.443684\pi$
$198$	0	0
$199$	−348796.	−0.624366	−0.312183	−	0.950022i	$-0.601060\pi$
$199$	−348796.	−0.624366	−0.312183	+	0.950022i	$0.601060\pi$
$200$	0	0
$201$	−371433.	−0.648470
$202$	0	0
$203$	110828.	0.188760
$204$	0	0
$205$	626449.	1.04112
$206$	0	0
$207$	−137477.	−0.223000
$208$	0	0
$209$	910563.	1.44193
$210$	0	0
$211$	−512225.	−0.792054	−0.396027	−	0.918239i	$-0.629611\pi$
$211$	−512225.	−0.792054	−0.396027	+	0.918239i	$0.629611\pi$
$212$	0	0
$213$	251729.	0.380175
$214$	0	0
$215$	610917.	0.901335
$216$	0	0
$217$	−308521.	−0.444770
$218$	0	0
$219$	190525.	0.268436
$220$	0	0
$221$	−1.42284e6	−1.95964
$222$	0	0
$223$	258803.	0.348503	0.174252	−	0.984701i	$-0.444249\pi$
$223$	258803.	0.348503	0.174252	+	0.984701i	$0.444249\pi$
$224$	0	0
$225$	−298225.	−0.392724
$226$	0	0
$227$	337633.	0.434891	0.217446	−	0.976072i	$-0.430228\pi$
$227$	337633.	0.434891	0.217446	+	0.976072i	$0.430228\pi$
$228$	0	0
$229$	−55771.6	−0.0702789	−0.0351394	−	0.999382i	$-0.511188\pi$
$229$	−55771.6	−0.0702789	−0.0351394	+	0.999382i	$0.511188\pi$
$230$	0	0
$231$	−596562.	−0.735573
$232$	0	0
$233$	860759.	1.03870	0.519352	−	0.854561i	$-0.326173\pi$
$233$	860759.	1.03870	0.519352	+	0.854561i	$0.326173\pi$
$234$	0	0
$235$	545102.	0.643885
$236$	0	0
$237$	947593.	1.09585
$238$	0	0
$239$	−312130.	−0.353460	−0.176730	−	0.984259i	$-0.556552\pi$
$239$	−312130.	−0.353460	−0.176730	+	0.984259i	$0.556552\pi$
$240$	0	0
$241$	1.40832e6	1.56191	0.780957	−	0.624584i	$-0.214732\pi$
$241$	1.40832e6	1.56191	0.780957	+	0.624584i	$0.214732\pi$
$242$	0	0
$243$	−423816.	−0.460428
$244$	0	0
$245$	1.16315e6	1.23800
$246$	0	0
$247$	1.27748e6	1.33233
$248$	0	0
$249$	−1.30124e6	−1.33002
$250$	0	0
$251$	1.30415e6	1.30660	0.653298	−	0.757101i	$-0.273385\pi$
$251$	1.30415e6	1.30660	0.653298	+	0.757101i	$0.273385\pi$
$252$	0	0
$253$	−1.64850e6	−1.61915
$254$	0	0
$255$	1.83852e6	1.77059
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	−583246.	−0.540260
$260$	0	0
$261$	99482.6	0.0903953
$262$	0	0
$263$	2.02443e6	1.80473	0.902366	−	0.430971i	$-0.141829\pi$
$263$	2.02443e6	1.80473	0.902366	+	0.430971i	$0.141829\pi$
$264$	0	0
$265$	−590183.	−0.516264
$266$	0	0
$267$	−606132.	−0.520342
$268$	0	0
$269$	−964991.	−0.813097	−0.406549	−	0.913629i	$-0.633268\pi$
$269$	−964991.	−0.813097	−0.406549	+	0.913629i	$0.633268\pi$
$270$	0	0
$271$	−169128.	−0.139892	−0.0699460	−	0.997551i	$-0.522283\pi$
$271$	−169128.	−0.139892	−0.0699460	+	0.997551i	$0.522283\pi$
$272$	0	0
$273$	−836954.	−0.679665
$274$	0	0
$275$	−3.57604e6	−2.85148
$276$	0	0
$277$	−324658.	−0.254230	−0.127115	−	0.991888i	$-0.540572\pi$
$277$	−324658.	−0.254230	−0.127115	+	0.991888i	$0.540572\pi$
$278$	0	0
$279$	−276938.	−0.212996
$280$	0	0
$281$	1.13540e6	0.857796	0.428898	−	0.903353i	$-0.358902\pi$
$281$	1.13540e6	0.857796	0.428898	+	0.903353i	$0.358902\pi$
$282$	0	0
$283$	173988.	0.129138	0.0645691	−	0.997913i	$-0.479433\pi$
$283$	173988.	0.129138	0.0645691	+	0.997913i	$0.479433\pi$
$284$	0	0
$285$	−1.65070e6	−1.20380
$286$	0	0
$287$	437570.	0.313576
$288$	0	0
$289$	763309.	0.537595
$290$	0	0
$291$	−921284.	−0.637766
$292$	0	0
$293$	475682.	0.323704	0.161852	−	0.986815i	$-0.448253\pi$
$293$	475682.	0.323704	0.161852	+	0.986815i	$0.448253\pi$
$294$	0	0
$295$	1.11793e6	0.747925
$296$	0	0
$297$	−2.80875e6	−1.84766
$298$	0	0
$299$	−2.31278e6	−1.49608
$300$	0	0
$301$	426721.	0.271474
$302$	0	0
$303$	40806.1	0.0255340
$304$	0	0
$305$	505050.	0.310874
$306$	0	0
$307$	−1.89614e6	−1.14822	−0.574109	−	0.818779i	$-0.694652\pi$
$307$	−1.89614e6	−1.14822	−0.574109	+	0.818779i	$0.694652\pi$
$308$	0	0
$309$	−2.08606e6	−1.24289
$310$	0	0
$311$	−1.23876e6	−0.726249	−0.363125	−	0.931741i	$-0.618290\pi$
$311$	−1.23876e6	−0.726249	−0.363125	+	0.931741i	$0.618290\pi$
$312$	0	0
$313$	−513114.	−0.296042	−0.148021	−	0.988984i	$-0.547290\pi$
$313$	−513114.	−0.296042	−0.148021	+	0.988984i	$0.547290\pi$
$314$	0	0
$315$	−333254.	−0.189234
$316$	0	0
$317$	1.55650e6	0.869963	0.434982	−	0.900439i	$-0.356755\pi$
$317$	1.55650e6	0.869963	0.434982	+	0.900439i	$0.356755\pi$
$318$	0	0
$319$	1.19290e6	0.656339
$320$	0	0
$321$	−205376.	−0.111247
$322$	0	0
$323$	−1.96013e6	−1.04539
$324$	0	0
$325$	−5.01704e6	−2.63475
$326$	0	0
$327$	−2.00827e6	−1.03861
$328$	0	0
$329$	380750.	0.193932
$330$	0	0
$331$	−2.47093e6	−1.23963	−0.619813	−	0.784749i	$-0.712792\pi$
$331$	−2.47093e6	−1.23963	−0.619813	+	0.784749i	$0.712792\pi$
$332$	0	0
$333$	−523539.	−0.258725
$334$	0	0
$335$	−2.48804e6	−1.21128
$336$	0	0
$337$	51336.4	0.0246236	0.0123118	−	0.999924i	$-0.496081\pi$
$337$	51336.4	0.0246236	0.0123118	+	0.999924i	$0.496081\pi$
$338$	0	0
$339$	−2.09157e6	−0.988493
$340$	0	0
$341$	−3.32078e6	−1.54651
$342$	0	0
$343$	1.88423e6	0.864765
$344$	0	0
$345$	2.98845e6	1.35176
$346$	0	0
$347$	1.79922e6	0.802158	0.401079	−	0.916043i	$-0.368635\pi$
$347$	1.79922e6	0.802158	0.401079	+	0.916043i	$0.368635\pi$
$348$	0	0
$349$	4.03210e6	1.77202	0.886009	−	0.463668i	$-0.153467\pi$
$349$	4.03210e6	1.77202	0.886009	+	0.463668i	$0.153467\pi$
$350$	0	0
$351$	−3.94057e6	−1.70723
$352$	0	0
$353$	2.77628e6	1.18584	0.592921	−	0.805261i	$-0.297975\pi$
$353$	2.77628e6	1.18584	0.592921	+	0.805261i	$0.297975\pi$
$354$	0	0
$355$	1.68620e6	0.710132
$356$	0	0
$357$	1.28420e6	0.533286
$358$	0	0
$359$	3.87813e6	1.58813	0.794066	−	0.607831i	$-0.207960\pi$
$359$	3.87813e6	1.58813	0.794066	+	0.607831i	$0.207960\pi$
$360$	0	0
$361$	−716216.	−0.289252
$362$	0	0
$363$	−4.22610e6	−1.68335
$364$	0	0
$365$	1.27623e6	0.501413
$366$	0	0
$367$	−3.27280e6	−1.26839	−0.634197	−	0.773171i	$-0.718669\pi$
$367$	−3.27280e6	−1.26839	−0.634197	+	0.773171i	$0.718669\pi$
$368$	0	0
$369$	392776.	0.150169
$370$	0	0
$371$	−412238.	−0.155494
$372$	0	0
$373$	−4.57763e6	−1.70360	−0.851801	−	0.523865i	$-0.824490\pi$
$373$	−4.57763e6	−1.70360	−0.851801	+	0.523865i	$0.824490\pi$
$374$	0	0
$375$	2.59431e6	0.952673
$376$	0	0
$377$	1.67359e6	0.606453
$378$	0	0
$379$	1.18593e6	0.424095	0.212047	−	0.977259i	$-0.431987\pi$
$379$	1.18593e6	0.424095	0.212047	+	0.977259i	$0.431987\pi$
$380$	0	0
$381$	−191070.	−0.0674343
$382$	0	0
$383$	5.37561e6	1.87254	0.936269	−	0.351284i	$-0.114255\pi$
$383$	5.37561e6	1.87254	0.936269	+	0.351284i	$0.114255\pi$
$384$	0	0
$385$	−3.99607e6	−1.37398
$386$	0	0
$387$	383038.	0.130006
$388$	0	0
$389$	−2.66177e6	−0.891861	−0.445931	−	0.895068i	$-0.647127\pi$
$389$	−2.66177e6	−0.891861	−0.445931	+	0.895068i	$0.647127\pi$
$390$	0	0
$391$	3.54865e6	1.17387
$392$	0	0
$393$	4.82036e6	1.57434
$394$	0	0
$395$	6.34744e6	2.04694
$396$	0	0
$397$	−4.09168e6	−1.30294	−0.651471	−	0.758673i	$-0.725848\pi$
$397$	−4.09168e6	−1.30294	−0.651471	+	0.758673i	$0.725848\pi$
$398$	0	0
$399$	−1.15300e6	−0.362575
$400$	0	0
$401$	2.94310e6	0.913995	0.456998	−	0.889468i	$-0.348925\pi$
$401$	2.94310e6	0.913995	0.456998	+	0.889468i	$0.348925\pi$
$402$	0	0
$403$	−4.65892e6	−1.42897
$404$	0	0
$405$	3.82190e6	1.15782
$406$	0	0
$407$	−6.27779e6	−1.87854
$408$	0	0
$409$	117484.	0.0347273	0.0173636	−	0.999849i	$-0.494473\pi$
$409$	117484.	0.0347273	0.0173636	+	0.999849i	$0.494473\pi$
$410$	0	0
$411$	1.63536e6	0.477540
$412$	0	0
$413$	780863.	0.225268
$414$	0	0
$415$	−8.71635e6	−2.48436
$416$	0	0
$417$	−164255.	−0.0462572
$418$	0	0
$419$	−6.20269e6	−1.72602	−0.863008	−	0.505190i	$-0.831422\pi$
$419$	−6.20269e6	−1.72602	−0.863008	+	0.505190i	$0.831422\pi$
$420$	0	0
$421$	−5.84119e6	−1.60619	−0.803094	−	0.595853i	$-0.796814\pi$
$421$	−5.84119e6	−1.60619	−0.803094	+	0.595853i	$0.796814\pi$
$422$	0	0
$423$	341772.	0.0928722
$424$	0	0
$425$	7.69798e6	2.06731
$426$	0	0
$427$	352774.	0.0936325
$428$	0	0
$429$	−9.00858e6	−2.36327
$430$	0	0
$431$	2.18760e6	0.567251	0.283626	−	0.958935i	$-0.408463\pi$
$431$	2.18760e6	0.567251	0.283626	+	0.958935i	$0.408463\pi$
$432$	0	0
$433$	1.61963e6	0.415142	0.207571	−	0.978220i	$-0.433444\pi$
$433$	1.61963e6	0.415142	0.207571	+	0.978220i	$0.433444\pi$
$434$	0	0
$435$	−2.16253e6	−0.547948
$436$	0	0
$437$	−3.18612e6	−0.798102
$438$	0	0
$439$	−2.29703e6	−0.568861	−0.284430	−	0.958697i	$-0.591804\pi$
$439$	−2.29703e6	−0.568861	−0.284430	+	0.958697i	$0.591804\pi$
$440$	0	0
$441$	729281.	0.178566
$442$	0	0
$443$	3.91953e6	0.948910	0.474455	−	0.880280i	$-0.342645\pi$
$443$	3.91953e6	0.948910	0.474455	+	0.880280i	$0.342645\pi$
$444$	0	0
$445$	−4.06017e6	−0.971951
$446$	0	0
$447$	5.79279e6	1.37126
$448$	0	0
$449$	597659.	0.139906	0.0699532	−	0.997550i	$-0.477715\pi$
$449$	597659.	0.139906	0.0699532	+	0.997550i	$0.477715\pi$
$450$	0	0
$451$	4.70980e6	1.09034
$452$	0	0
$453$	2.49922e6	0.572214
$454$	0	0
$455$	−5.60633e6	−1.26955
$456$	0	0
$457$	−5.13093e6	−1.14923	−0.574613	−	0.818425i	$-0.694848\pi$
$457$	−5.13093e6	−1.14923	−0.574613	+	0.818425i	$0.694848\pi$
$458$	0	0
$459$	6.04628e6	1.33954
$460$	0	0
$461$	3.54688e6	0.777309	0.388655	−	0.921384i	$-0.372940\pi$
$461$	3.54688e6	0.777309	0.388655	+	0.921384i	$0.372940\pi$
$462$	0	0
$463$	6.45897e6	1.40027	0.700133	−	0.714012i	$-0.253124\pi$
$463$	6.45897e6	1.40027	0.700133	+	0.714012i	$0.253124\pi$
$464$	0	0
$465$	6.02001e6	1.29112
$466$	0	0
$467$	1.75147e6	0.371630	0.185815	−	0.982585i	$-0.440507\pi$
$467$	1.75147e6	0.371630	0.185815	+	0.982585i	$0.440507\pi$
$468$	0	0
$469$	−1.73788e6	−0.364827
$470$	0	0
$471$	1.90636e6	0.395962
$472$	0	0
$473$	4.59303e6	0.943944
$474$	0	0
$475$	−6.91155e6	−1.40553
$476$	0	0
$477$	−370037.	−0.0744646
$478$	0	0
$479$	−4.98673e6	−0.993063	−0.496532	−	0.868019i	$-0.665393\pi$
$479$	−4.98673e6	−0.993063	−0.496532	+	0.868019i	$0.665393\pi$
$480$	0	0
$481$	−8.80750e6	−1.73576
$482$	0	0
$483$	2.08741e6	0.407136
$484$	0	0
$485$	−6.17122e6	−1.19129
$486$	0	0
$487$	−626627.	−0.119726	−0.0598628	−	0.998207i	$-0.519066\pi$
$487$	−626627.	−0.119726	−0.0598628	+	0.998207i	$0.519066\pi$
$488$	0	0
$489$	−4.18434e6	−0.791325
$490$	0	0
$491$	1.31853e6	0.246824	0.123412	−	0.992356i	$-0.460616\pi$
$491$	1.31853e6	0.246824	0.123412	+	0.992356i	$0.460616\pi$
$492$	0	0
$493$	−2.56791e6	−0.475842
$494$	0	0
$495$	−3.58699e6	−0.657987
$496$	0	0
$497$	1.17780e6	0.213885
$498$	0	0
$499$	2.82596e6	0.508060	0.254030	−	0.967196i	$-0.418244\pi$
$499$	2.82596e6	0.508060	0.254030	+	0.967196i	$0.418244\pi$
$500$	0	0
$501$	−3.12580e6	−0.556374
$502$	0	0
$503$	1.01863e7	1.79513	0.897564	−	0.440885i	$-0.145335\pi$
$503$	1.01863e7	1.79513	0.897564	+	0.440885i	$0.145335\pi$
$504$	0	0
$505$	273339.	0.0476951
$506$	0	0
$507$	−7.57821e6	−1.30932
$508$	0	0
$509$	4.06518e6	0.695481	0.347741	−	0.937591i	$-0.386949\pi$
$509$	4.06518e6	0.695481	0.347741	+	0.937591i	$0.386949\pi$
$510$	0	0
$511$	891435.	0.151021
$512$	0	0
$513$	−5.42859e6	−0.910738
$514$	0	0
$515$	−1.39735e7	−2.32160
$516$	0	0
$517$	4.09821e6	0.674323
$518$	0	0
$519$	8.47406e6	1.38093
$520$	0	0
$521$	−7.68689e6	−1.24067	−0.620335	−	0.784337i	$-0.713004\pi$
$521$	−7.68689e6	−1.24067	−0.620335	+	0.784337i	$0.713004\pi$
$522$	0	0
$523$	9.37280e6	1.49836	0.749179	−	0.662368i	$-0.230448\pi$
$523$	9.37280e6	1.49836	0.749179	+	0.662368i	$0.230448\pi$
$524$	0	0
$525$	4.52816e6	0.717007
$526$	0	0
$527$	7.14850e6	1.12121
$528$	0	0
$529$	−668146.	−0.103808
$530$	0	0
$531$	700926.	0.107879
$532$	0	0
$533$	6.60766e6	1.00746
$534$	0	0
$535$	−1.37571e6	−0.207798
$536$	0	0
$537$	2.38794e6	0.357345
$538$	0	0
$539$	8.74486e6	1.29652
$540$	0	0
$541$	−9.50736e6	−1.39658	−0.698291	−	0.715814i	$-0.746056\pi$
$541$	−9.50736e6	−1.39658	−0.698291	+	0.715814i	$0.746056\pi$
$542$	0	0
$543$	−5.90387e6	−0.859286
$544$	0	0
$545$	−1.34524e7	−1.94003
$546$	0	0
$547$	1.23027e7	1.75805	0.879025	−	0.476776i	$-0.158195\pi$
$547$	1.23027e7	1.75805	0.879025	+	0.476776i	$0.158195\pi$
$548$	0	0
$549$	316660.	0.0448397
$550$	0	0
$551$	2.30557e6	0.323519
$552$	0	0
$553$	4.43364e6	0.616521
$554$	0	0
$555$	1.13806e7	1.56831
$556$	0	0
$557$	9.99762e6	1.36540	0.682698	−	0.730700i	$-0.260806\pi$
$557$	9.99762e6	1.36540	0.682698	+	0.730700i	$0.260806\pi$
$558$	0	0
$559$	6.44384e6	0.872198
$560$	0	0
$561$	1.38225e7	1.85429
$562$	0	0
$563$	9.90117e6	1.31648	0.658242	−	0.752807i	$-0.271300\pi$
$563$	9.90117e6	1.31648	0.658242	+	0.752807i	$0.271300\pi$
$564$	0	0
$565$	−1.40104e7	−1.84641
$566$	0	0
$567$	2.66957e6	0.348725
$568$	0	0
$569$	−5.67378e6	−0.734670	−0.367335	−	0.930089i	$-0.619730\pi$
$569$	−5.67378e6	−0.734670	−0.367335	+	0.930089i	$0.619730\pi$
$570$	0	0
$571$	−7.40380e6	−0.950308	−0.475154	−	0.879903i	$-0.657608\pi$
$571$	−7.40380e6	−0.950308	−0.475154	+	0.879903i	$0.657608\pi$
$572$	0	0
$573$	5.75512e6	0.732265
$574$	0	0
$575$	1.25128e7	1.57828
$576$	0	0
$577$	1.04084e6	0.130150	0.0650752	−	0.997880i	$-0.479271\pi$
$577$	1.04084e6	0.130150	0.0650752	+	0.997880i	$0.479271\pi$
$578$	0	0
$579$	8.33850e6	1.03369
$580$	0	0
$581$	−6.08830e6	−0.748266
$582$	0	0
$583$	−4.43714e6	−0.540669
$584$	0	0
$585$	−5.03241e6	−0.607975
$586$	0	0
$587$	2.66921e6	0.319734	0.159867	−	0.987139i	$-0.448894\pi$
$587$	2.66921e6	0.319734	0.159867	+	0.987139i	$0.448894\pi$
$588$	0	0
$589$	−6.41820e6	−0.762298
$590$	0	0
$591$	−2.61327e6	−0.307762
$592$	0	0
$593$	−9.09294e6	−1.06186	−0.530930	−	0.847415i	$-0.678157\pi$
$593$	−9.09294e6	−1.06186	−0.530930	+	0.847415i	$0.678157\pi$
$594$	0	0
$595$	8.60217e6	0.996129
$596$	0	0
$597$	4.75386e6	0.545897
$598$	0	0
$599$	7.06927e6	0.805022	0.402511	−	0.915415i	$-0.368138\pi$
$599$	7.06927e6	0.805022	0.402511	+	0.915415i	$0.368138\pi$
$600$	0	0
$601$	1.69462e6	0.191375	0.0956875	−	0.995411i	$-0.469495\pi$
$601$	1.69462e6	0.191375	0.0956875	+	0.995411i	$0.469495\pi$
$602$	0	0
$603$	−1.55997e6	−0.174712
$604$	0	0
$605$	−2.83085e7	−3.14433
$606$	0	0
$607$	1.94460e6	0.214219	0.107110	−	0.994247i	$-0.465840\pi$
$607$	1.94460e6	0.214219	0.107110	+	0.994247i	$0.465840\pi$
$608$	0	0
$609$	−1.51051e6	−0.165037
$610$	0	0
$611$	5.74963e6	0.623070
$612$	0	0
$613$	4.35920e6	0.468550	0.234275	−	0.972170i	$-0.424728\pi$
$613$	4.35920e6	0.468550	0.234275	+	0.972170i	$0.424728\pi$
$614$	0	0
$615$	−8.53808e6	−0.910275
$616$	0	0
$617$	−1.76284e7	−1.86423	−0.932114	−	0.362164i	$-0.882038\pi$
$617$	−1.76284e7	−1.86423	−0.932114	+	0.362164i	$0.882038\pi$
$618$	0	0
$619$	8.47540e6	0.889065	0.444532	−	0.895763i	$-0.353370\pi$
$619$	8.47540e6	0.889065	0.444532	+	0.895763i	$0.353370\pi$
$620$	0	0
$621$	9.82800e6	1.02267
$622$	0	0
$623$	−2.83600e6	−0.292743
$624$	0	0
$625$	1.09687e6	0.112319
$626$	0	0
$627$	−1.24104e7	−1.26071
$628$	0	0
$629$	1.35139e7	1.36193
$630$	0	0
$631$	1.39296e7	1.39272	0.696362	−	0.717690i	$-0.254801\pi$
$631$	1.39296e7	1.39272	0.696362	+	0.717690i	$0.254801\pi$
$632$	0	0
$633$	6.98128e6	0.692510
$634$	0	0
$635$	−1.27988e6	−0.125961
$636$	0	0
$637$	1.22687e7	1.19798
$638$	0	0
$639$	1.05723e6	0.102428
$640$	0	0
$641$	−8.16516e6	−0.784909	−0.392455	−	0.919771i	$-0.628374\pi$
$641$	−8.16516e6	−0.784909	−0.392455	+	0.919771i	$0.628374\pi$
$642$	0	0
$643$	−8.88134e6	−0.847132	−0.423566	−	0.905865i	$-0.639222\pi$
$643$	−8.88134e6	−0.847132	−0.423566	+	0.905865i	$0.639222\pi$
$644$	0	0
$645$	−8.32639e6	−0.788057
$646$	0	0
$647$	−4.33925e6	−0.407525	−0.203763	−	0.979020i	$-0.565317\pi$
$647$	−4.33925e6	−0.407525	−0.203763	+	0.979020i	$0.565317\pi$
$648$	0	0
$649$	8.40485e6	0.783282
$650$	0	0
$651$	4.20493e6	0.388872
$652$	0	0
$653$	9.08383e6	0.833655	0.416827	−	0.908986i	$-0.363142\pi$
$653$	9.08383e6	0.833655	0.416827	+	0.908986i	$0.363142\pi$
$654$	0	0
$655$	3.22892e7	2.94072
$656$	0	0
$657$	800179.	0.0723225
$658$	0	0
$659$	1.92066e7	1.72281	0.861405	−	0.507919i	$-0.169585\pi$
$659$	1.92066e7	1.72281	0.861405	+	0.507919i	$0.169585\pi$
$660$	0	0
$661$	−1.61844e7	−1.44076	−0.720381	−	0.693579i	$-0.756033\pi$
$661$	−1.61844e7	−1.44076	−0.720381	+	0.693579i	$0.756033\pi$
$662$	0	0
$663$	1.93924e7	1.71336
$664$	0	0
$665$	−7.72337e6	−0.677256
$666$	0	0
$667$	−4.17404e6	−0.363280
$668$	0	0
$669$	−3.52731e6	−0.304704
$670$	0	0
$671$	3.79709e6	0.325570
$672$	0	0
$673$	−1.94551e7	−1.65575	−0.827876	−	0.560911i	$-0.810451\pi$
$673$	−1.94551e7	−1.65575	−0.827876	+	0.560911i	$0.810451\pi$
$674$	0	0
$675$	2.13196e7	1.80102
$676$	0	0
$677$	−1.98691e6	−0.166612	−0.0833061	−	0.996524i	$-0.526548\pi$
$677$	−1.98691e6	−0.166612	−0.0833061	+	0.996524i	$0.526548\pi$
$678$	0	0
$679$	−4.31055e6	−0.358805
$680$	0	0
$681$	−4.60171e6	−0.380235
$682$	0	0
$683$	−1.93942e7	−1.59082	−0.795408	−	0.606074i	$-0.792744\pi$
$683$	−1.93942e7	−1.59082	−0.795408	+	0.606074i	$0.792744\pi$
$684$	0	0
$685$	1.09545e7	0.892001
$686$	0	0
$687$	760130.	0.0614463
$688$	0	0
$689$	−6.22513e6	−0.499575
$690$	0	0
$691$	3.79763e6	0.302564	0.151282	−	0.988491i	$-0.451660\pi$
$691$	3.79763e6	0.302564	0.151282	+	0.988491i	$0.451660\pi$
$692$	0	0
$693$	−2.50549e6	−0.198180
$694$	0	0
$695$	−1.10026e6	−0.0864041
$696$	0	0
$697$	−1.01386e7	−0.790489
$698$	0	0
$699$	−1.17316e7	−0.908161
$700$	0	0
$701$	8.94712e6	0.687683	0.343842	−	0.939028i	$-0.388272\pi$
$701$	8.94712e6	0.687683	0.343842	+	0.939028i	$0.388272\pi$
$702$	0	0
$703$	−1.21333e7	−0.925960
$704$	0	0
$705$	−7.42937e6	−0.562962
$706$	0	0
$707$	190925.	0.0143653
$708$	0	0
$709$	1.08335e7	0.809378	0.404689	−	0.914454i	$-0.367380\pi$
$709$	1.08335e7	0.809378	0.404689	+	0.914454i	$0.367380\pi$
$710$	0	0
$711$	3.97977e6	0.295246
$712$	0	0
$713$	1.16196e7	0.855988
$714$	0	0
$715$	−6.03439e7	−4.41436
$716$	0	0
$717$	4.25412e6	0.309038
$718$	0	0
$719$	1.97006e7	1.42121	0.710603	−	0.703593i	$-0.248422\pi$
$719$	1.97006e7	1.42121	0.710603	+	0.703593i	$0.248422\pi$
$720$	0	0
$721$	−9.76037e6	−0.699243
$722$	0	0
$723$	−1.91944e7	−1.36562
$724$	0	0
$725$	−9.05462e6	−0.639772
$726$	0	0
$727$	−1.46825e7	−1.03030	−0.515149	−	0.857101i	$-0.672263\pi$
$727$	−1.46825e7	−1.03030	−0.515149	+	0.857101i	$0.672263\pi$
$728$	0	0
$729$	1.59490e7	1.11151
$730$	0	0
$731$	−9.88722e6	−0.684354
$732$	0	0
$733$	1.67177e7	1.14925	0.574627	−	0.818415i	$-0.305147\pi$
$733$	1.67177e7	1.14925	0.574627	+	0.818415i	$0.305147\pi$
$734$	0	0
$735$	−1.58530e7	−1.08241
$736$	0	0
$737$	−1.87057e7	−1.26854
$738$	0	0
$739$	1.89691e7	1.27772	0.638860	−	0.769323i	$-0.279406\pi$
$739$	1.89691e7	1.27772	0.638860	+	0.769323i	$0.279406\pi$
$740$	0	0
$741$	−1.74113e7	−1.16489
$742$	0	0
$743$	2.26839e7	1.50746	0.753730	−	0.657184i	$-0.228252\pi$
$743$	2.26839e7	1.50746	0.753730	+	0.657184i	$0.228252\pi$
$744$	0	0
$745$	3.88030e7	2.56138
$746$	0	0
$747$	−5.46504e6	−0.358337
$748$	0	0
$749$	−960922.	−0.0625869
$750$	0	0
$751$	−812953.	−0.0525976	−0.0262988	−	0.999654i	$-0.508372\pi$
$751$	−812953.	−0.0525976	−0.0262988	+	0.999654i	$0.508372\pi$
$752$	0	0
$753$	−1.77746e7	−1.14239
$754$	0	0
$755$	1.67410e7	1.06884
$756$	0	0
$757$	−1.41529e7	−0.897646	−0.448823	−	0.893621i	$-0.648157\pi$
$757$	−1.41529e7	−0.897646	−0.448823	+	0.893621i	$0.648157\pi$
$758$	0	0
$759$	2.24679e7	1.41566
$760$	0	0
$761$	−1.08258e7	−0.677642	−0.338821	−	0.940851i	$-0.610028\pi$
$761$	−1.08258e7	−0.677642	−0.338821	+	0.940851i	$0.610028\pi$
$762$	0	0
$763$	−9.39638e6	−0.584318
$764$	0	0
$765$	7.72157e6	0.477037
$766$	0	0
$767$	1.17917e7	0.723748
$768$	0	0
$769$	1.89064e7	1.15290	0.576451	−	0.817132i	$-0.304437\pi$
$769$	1.89064e7	1.15290	0.576451	+	0.817132i	$0.304437\pi$
$770$	0	0
$771$	−900203.	−0.0545387
$772$	0	0
$773$	1.86479e7	1.12248	0.561242	−	0.827652i	$-0.310324\pi$
$773$	1.86479e7	1.12248	0.561242	+	0.827652i	$0.310324\pi$
$774$	0	0
$775$	2.52061e7	1.50748
$776$	0	0
$777$	7.94926e6	0.472361
$778$	0	0
$779$	9.10282e6	0.537443
$780$	0	0
$781$	1.26773e7	0.743702
$782$	0	0
$783$	−7.11183e6	−0.414550
$784$	0	0
$785$	1.27698e7	0.739620
$786$	0	0
$787$	−2.27126e6	−0.130716	−0.0653581	−	0.997862i	$-0.520819\pi$
$787$	−2.27126e6	−0.130716	−0.0653581	+	0.997862i	$0.520819\pi$
$788$	0	0
$789$	−2.75916e7	−1.57792
$790$	0	0
$791$	−9.78615e6	−0.556123
$792$	0	0
$793$	5.32717e6	0.300825
$794$	0	0
$795$	8.04380e6	0.451381
$796$	0	0
$797$	3.05098e7	1.70135	0.850674	−	0.525694i	$-0.176194\pi$
$797$	3.05098e7	1.70135	0.850674	+	0.525694i	$0.176194\pi$
$798$	0	0
$799$	−8.82205e6	−0.488880
$800$	0	0
$801$	−2.54568e6	−0.140192
$802$	0	0
$803$	9.59499e6	0.525117
$804$	0	0
$805$	1.39825e7	0.760493
$806$	0	0
$807$	1.31522e7	0.710908
$808$	0	0
$809$	−2.89873e7	−1.55717	−0.778587	−	0.627537i	$-0.784063\pi$
$809$	−2.89873e7	−1.55717	−0.778587	+	0.627537i	$0.784063\pi$
$810$	0	0
$811$	−1.48893e7	−0.794920	−0.397460	−	0.917619i	$-0.630108\pi$
$811$	−1.48893e7	−0.794920	−0.397460	+	0.917619i	$0.630108\pi$
$812$	0	0
$813$	2.30510e6	0.122311
$814$	0	0
$815$	−2.80288e7	−1.47812
$816$	0	0
$817$	8.87714e6	0.465284
$818$	0	0
$819$	−3.51510e6	−0.183117
$820$	0	0
$821$	−2.95982e7	−1.53253	−0.766263	−	0.642527i	$-0.777886\pi$
$821$	−2.95982e7	−1.53253	−0.766263	+	0.642527i	$0.777886\pi$
$822$	0	0
$823$	3.56220e6	0.183324	0.0916619	−	0.995790i	$-0.470782\pi$
$823$	3.56220e6	0.183324	0.0916619	+	0.995790i	$0.470782\pi$
$824$	0	0
$825$	4.87390e7	2.49311
$826$	0	0
$827$	−1.59875e7	−0.812861	−0.406430	−	0.913682i	$-0.633227\pi$
$827$	−1.59875e7	−0.812861	−0.406430	+	0.913682i	$0.633227\pi$
$828$	0	0
$829$	2.72235e7	1.37581	0.687903	−	0.725803i	$-0.258532\pi$
$829$	2.72235e7	1.37581	0.687903	+	0.725803i	$0.258532\pi$
$830$	0	0
$831$	4.42487e6	0.222279
$832$	0	0
$833$	−1.88247e7	−0.939973
$834$	0	0
$835$	−2.09381e7	−1.03925
$836$	0	0
$837$	1.97978e7	0.976794
$838$	0	0
$839$	−2.28204e6	−0.111923	−0.0559613	−	0.998433i	$-0.517822\pi$
$839$	−2.28204e6	−0.111923	−0.0559613	+	0.998433i	$0.517822\pi$
$840$	0	0
$841$	−1.74907e7	−0.852741
$842$	0	0
$843$	−1.54748e7	−0.749990
$844$	0	0
$845$	−5.07626e7	−2.44569
$846$	0	0
$847$	−1.97733e7	−0.947044
$848$	0	0
$849$	−2.37135e6	−0.112908
$850$	0	0
$851$	2.19664e7	1.03976
$852$	0	0
$853$	−2.41164e7	−1.13486	−0.567428	−	0.823423i	$-0.692061\pi$
$853$	−2.41164e7	−1.13486	−0.567428	+	0.823423i	$0.692061\pi$
$854$	0	0
$855$	−6.93272e6	−0.324331
$856$	0	0
$857$	1.40932e7	0.655477	0.327739	−	0.944768i	$-0.393713\pi$
$857$	1.40932e7	0.655477	0.327739	+	0.944768i	$0.393713\pi$
$858$	0	0
$859$	6.33600e6	0.292976	0.146488	−	0.989212i	$-0.453203\pi$
$859$	6.33600e6	0.292976	0.146488	+	0.989212i	$0.453203\pi$
$860$	0	0
$861$	−5.96379e6	−0.274166
$862$	0	0
$863$	3.68125e7	1.68255	0.841276	−	0.540606i	$-0.181805\pi$
$863$	3.68125e7	1.68255	0.841276	+	0.540606i	$0.181805\pi$
$864$	0	0
$865$	5.67634e7	2.57946
$866$	0	0
$867$	−1.04034e7	−0.470031
$868$	0	0
$869$	4.77216e7	2.14371
$870$	0	0
$871$	−2.62433e7	−1.17212
$872$	0	0
$873$	−3.86928e6	−0.171828
$874$	0	0
$875$	1.21384e7	0.535970
$876$	0	0
$877$	−3.82827e7	−1.68075	−0.840375	−	0.542005i	$-0.817666\pi$
$877$	−3.82827e7	−1.68075	−0.840375	+	0.542005i	$0.817666\pi$
$878$	0	0
$879$	−6.48323e6	−0.283021
$880$	0	0
$881$	−1.73986e7	−0.755222	−0.377611	−	0.925964i	$-0.623254\pi$
$881$	−1.73986e7	−0.755222	−0.377611	+	0.925964i	$0.623254\pi$
$882$	0	0
$883$	−5.56643e6	−0.240256	−0.120128	−	0.992758i	$-0.538331\pi$
$883$	−5.56643e6	−0.240256	−0.120128	+	0.992758i	$0.538331\pi$
$884$	0	0
$885$	−1.52366e7	−0.653928
$886$	0	0
$887$	2.71872e7	1.16026	0.580130	−	0.814524i	$-0.303002\pi$
$887$	2.71872e7	1.16026	0.580130	+	0.814524i	$0.303002\pi$
$888$	0	0
$889$	−893989.	−0.0379383
$890$	0	0
$891$	2.87340e7	1.21256
$892$	0	0
$893$	7.92078e6	0.332383
$894$	0	0
$895$	1.59956e7	0.667488
$896$	0	0
$897$	3.15216e7	1.30806
$898$	0	0
$899$	−8.40830e6	−0.346983
$900$	0	0
$901$	9.55165e6	0.391982
$902$	0	0
$903$	−5.81592e6	−0.237356
$904$	0	0
$905$	−3.95471e7	−1.60507
$906$	0	0
$907$	4.31540e7	1.74182	0.870908	−	0.491446i	$-0.163531\pi$
$907$	4.31540e7	1.74182	0.870908	+	0.491446i	$0.163531\pi$
$908$	0	0
$909$	171380.	0.00687941
$910$	0	0
$911$	1.26943e7	0.506771	0.253385	−	0.967365i	$-0.418456\pi$
$911$	1.26943e7	0.506771	0.253385	+	0.967365i	$0.418456\pi$
$912$	0	0
$913$	−6.55317e7	−2.60180
$914$	0	0
$915$	−6.88350e6	−0.271804
$916$	0	0
$917$	2.25537e7	0.885718
$918$	0	0
$919$	−4.05763e7	−1.58483	−0.792417	−	0.609979i	$-0.791178\pi$
$919$	−4.05763e7	−1.58483	−0.792417	+	0.609979i	$0.791178\pi$
$920$	0	0
$921$	2.58431e7	1.00391
$922$	0	0
$923$	1.77857e7	0.687176
$924$	0	0
$925$	4.76510e7	1.83113
$926$	0	0
$927$	−8.76120e6	−0.334861
$928$	0	0
$929$	3.94997e7	1.50160	0.750800	−	0.660530i	$-0.229668\pi$
$929$	3.94997e7	1.50160	0.750800	+	0.660530i	$0.229668\pi$
$930$	0	0
$931$	1.69015e7	0.639076
$932$	0	0
$933$	1.68834e7	0.634976
$934$	0	0
$935$	9.25898e7	3.46365
$936$	0	0
$937$	1.37657e7	0.512210	0.256105	−	0.966649i	$-0.417561\pi$
$937$	1.37657e7	0.512210	0.256105	+	0.966649i	$0.417561\pi$
$938$	0	0
$939$	6.99340e6	0.258836
$940$	0	0
$941$	9.99327e6	0.367903	0.183952	−	0.982935i	$-0.441111\pi$
$941$	9.99327e6	0.367903	0.183952	+	0.982935i	$0.441111\pi$
$942$	0	0
$943$	−1.64799e7	−0.603497
$944$	0	0
$945$	2.38237e7	0.867821
$946$	0	0
$947$	−2.29340e7	−0.831008	−0.415504	−	0.909591i	$-0.636395\pi$
$947$	−2.29340e7	−0.831008	−0.415504	+	0.909591i	$0.636395\pi$
$948$	0	0
$949$	1.34614e7	0.485204
$950$	0	0
$951$	−2.12140e7	−0.760628
$952$	0	0
$953$	−4.98754e7	−1.77891	−0.889455	−	0.457023i	$-0.848916\pi$
$953$	−4.98754e7	−1.77891	−0.889455	+	0.457023i	$0.848916\pi$
$954$	0	0
$955$	3.85507e7	1.36780
$956$	0	0
$957$	−1.62585e7	−0.573851
$958$	0	0
$959$	7.65162e6	0.268662
$960$	0	0
$961$	−5.22231e6	−0.182412
$962$	0	0
$963$	−862552.	−0.0299723
$964$	0	0
$965$	5.58554e7	1.93084
$966$	0	0
$967$	−2.19781e7	−0.755828	−0.377914	−	0.925841i	$-0.623359\pi$
$967$	−2.19781e7	−0.755828	−0.377914	+	0.925841i	$0.623359\pi$
$968$	0	0
$969$	2.67153e7	0.914008
$970$	0	0
$971$	5.05636e6	0.172104	0.0860518	−	0.996291i	$-0.472575\pi$
$971$	5.05636e6	0.172104	0.0860518	+	0.996291i	$0.472575\pi$
$972$	0	0
$973$	−768525.	−0.0260241
$974$	0	0
$975$	6.83788e7	2.30362
$976$	0	0
$977$	−1.37217e7	−0.459910	−0.229955	−	0.973201i	$-0.573858\pi$
$977$	−1.37217e7	−0.459910	−0.229955	+	0.973201i	$0.573858\pi$
$978$	0	0
$979$	−3.05254e7	−1.01790
$980$	0	0
$981$	−8.43447e6	−0.279824
$982$	0	0
$983$	−2.61409e7	−0.862852	−0.431426	−	0.902148i	$-0.641989\pi$
$983$	−2.61409e7	−0.862852	−0.431426	+	0.902148i	$0.641989\pi$
$984$	0	0
$985$	−1.75050e7	−0.574871
$986$	0	0
$987$	−5.18936e6	−0.169559
$988$	0	0
$989$	−1.60713e7	−0.522469
$990$	0	0
$991$	2.01361e7	0.651316	0.325658	−	0.945488i	$-0.394414\pi$
$991$	2.01361e7	0.651316	0.325658	+	0.945488i	$0.394414\pi$
$992$	0	0
$993$	3.36772e7	1.08383
$994$	0	0
$995$	3.18437e7	1.01968
$996$	0	0
$997$	3.15395e6	0.100489	0.0502443	−	0.998737i	$-0.484000\pi$
$997$	3.15395e6	0.100489	0.0502443	+	0.998737i	$0.484000\pi$
$998$	0	0
$999$	3.74269e7	1.18651

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.14	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.14	✓	49	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)

\(f(q)\)	\(=\)	\(q-13.6293 q^{3} -91.2960 q^{5} -63.7695 q^{7} -57.2414 q^{9} +O(q^{10})\)	\(q-13.6293 q^{3} -91.2960 q^{5} -63.7695 q^{7} -57.2414 q^{9} -686.386 q^{11} -962.972 q^{13} +1244.30 q^{15} +1477.55 q^{17} -1326.61 q^{19} +869.136 q^{21} +2401.71 q^{23} +5209.95 q^{25} +4092.09 q^{27} -1737.95 q^{29} +4838.06 q^{31} +9354.97 q^{33} +5821.90 q^{35} +9146.16 q^{37} +13124.7 q^{39} -6861.74 q^{41} -6691.62 q^{43} +5225.91 q^{45} -5970.71 q^{47} -12740.4 q^{49} -20138.1 q^{51} +6464.50 q^{53} +62664.2 q^{55} +18080.7 q^{57} -12245.1 q^{59} -5532.01 q^{61} +3650.26 q^{63} +87915.5 q^{65} +27252.4 q^{67} -32733.7 q^{69} -18469.6 q^{71} -13979.0 q^{73} -71008.1 q^{75} +43770.5 q^{77} -69526.0 q^{79} -41862.8 q^{81} +95473.6 q^{83} -134895. q^{85} +23687.1 q^{87} +44472.6 q^{89} +61408.3 q^{91} -65939.5 q^{93} +121114. q^{95} +67595.7 q^{97} +39289.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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