LMFDB - Embedded newform 1028.6.a.a.1.8

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			1.8
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)

$f(q)$	$=$	$q-20.5636 q^{3} +62.7910 q^{5} +15.3467 q^{7} +179.863 q^{9} +O(q^{10})$	$q-20.5636 q^{3} +62.7910 q^{5} +15.3467 q^{7} +179.863 q^{9} -96.4721 q^{11} -449.076 q^{13} -1291.21 q^{15} +1128.33 q^{17} +779.713 q^{19} -315.583 q^{21} +2344.45 q^{23} +817.707 q^{25} +1298.33 q^{27} -3396.57 q^{29} -8428.03 q^{31} +1983.82 q^{33} +963.632 q^{35} -4252.53 q^{37} +9234.64 q^{39} -551.918 q^{41} +4216.87 q^{43} +11293.8 q^{45} -232.672 q^{47} -16571.5 q^{49} -23202.6 q^{51} +8140.83 q^{53} -6057.58 q^{55} -16033.7 q^{57} -11282.6 q^{59} -14261.3 q^{61} +2760.29 q^{63} -28197.9 q^{65} +54973.0 q^{67} -48210.4 q^{69} -58983.8 q^{71} +32753.2 q^{73} -16815.0 q^{75} -1480.52 q^{77} +62716.9 q^{79} -70405.0 q^{81} +48533.8 q^{83} +70848.9 q^{85} +69845.9 q^{87} +13964.0 q^{89} -6891.82 q^{91} +173311. q^{93} +48958.9 q^{95} +182495. q^{97} -17351.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−20.5636	−1.31916	−0.659579	−	0.751636i	$-0.729265\pi$
$3$	−20.5636	−1.31916	−0.659579	+	0.751636i	$0.729265\pi$
$4$	0	0
$5$	62.7910	1.12324	0.561620	−	0.827396i	$-0.310178\pi$
$5$	62.7910	1.12324	0.561620	+	0.827396i	$0.310178\pi$
$6$	0	0
$7$	15.3467	0.118377	0.0591887	−	0.998247i	$-0.481149\pi$
$7$	15.3467	0.118377	0.0591887	+	0.998247i	$0.481149\pi$
$8$	0	0
$9$	179.863	0.740176
$10$	0	0
$11$	−96.4721	−0.240392	−0.120196	−	0.992750i	$-0.538352\pi$
$11$	−96.4721	−0.240392	−0.120196	+	0.992750i	$0.538352\pi$
$12$	0	0
$13$	−449.076	−0.736990	−0.368495	−	0.929630i	$-0.620127\pi$
$13$	−449.076	−0.736990	−0.368495	+	0.929630i	$0.620127\pi$
$14$	0	0
$15$	−1291.21	−1.48173
$16$	0	0
$17$	1128.33	0.946921	0.473461	−	0.880815i	$-0.343005\pi$
$17$	1128.33	0.946921	0.473461	+	0.880815i	$0.343005\pi$
$18$	0	0
$19$	779.713	0.495508	0.247754	−	0.968823i	$-0.420308\pi$
$19$	779.713	0.495508	0.247754	+	0.968823i	$0.420308\pi$
$20$	0	0
$21$	−315.583	−0.156158
$22$	0	0
$23$	2344.45	0.924106	0.462053	−	0.886852i	$-0.347113\pi$
$23$	2344.45	0.924106	0.462053	+	0.886852i	$0.347113\pi$
$24$	0	0
$25$	817.707	0.261666
$26$	0	0
$27$	1298.33	0.342749
$28$	0	0
$29$	−3396.57	−0.749974	−0.374987	−	0.927030i	$-0.622353\pi$
$29$	−3396.57	−0.749974	−0.374987	+	0.927030i	$0.622353\pi$
$30$	0	0
$31$	−8428.03	−1.57515	−0.787575	−	0.616219i	$-0.788664\pi$
$31$	−8428.03	−1.57515	−0.787575	+	0.616219i	$0.788664\pi$
$32$	0	0
$33$	1983.82	0.317115
$34$	0	0
$35$	963.632	0.132966
$36$	0	0
$37$	−4252.53	−0.510673	−0.255337	−	0.966852i	$-0.582186\pi$
$37$	−4252.53	−0.510673	−0.255337	+	0.966852i	$0.582186\pi$
$38$	0	0
$39$	9234.64	0.972206
$40$	0	0
$41$	−551.918	−0.0512761	−0.0256381	−	0.999671i	$-0.508162\pi$
$41$	−551.918	−0.0512761	−0.0256381	+	0.999671i	$0.508162\pi$
$42$	0	0
$43$	4216.87	0.347792	0.173896	−	0.984764i	$-0.444364\pi$
$43$	4216.87	0.347792	0.173896	+	0.984764i	$0.444364\pi$
$44$	0	0
$45$	11293.8	0.831394
$46$	0	0
$47$	−232.672	−0.0153639	−0.00768193	−	0.999970i	$-0.502445\pi$
$47$	−232.672	−0.0153639	−0.00768193	+	0.999970i	$0.502445\pi$
$48$	0	0
$49$	−16571.5	−0.985987
$50$	0	0
$51$	−23202.6	−1.24914
$52$	0	0
$53$	8140.83	0.398088	0.199044	−	0.979991i	$-0.436216\pi$
$53$	8140.83	0.398088	0.199044	+	0.979991i	$0.436216\pi$
$54$	0	0
$55$	−6057.58	−0.270018
$56$	0	0
$57$	−16033.7	−0.653653
$58$	0	0
$59$	−11282.6	−0.421968	−0.210984	−	0.977489i	$-0.567667\pi$
$59$	−11282.6	−0.421968	−0.210984	+	0.977489i	$0.567667\pi$
$60$	0	0
$61$	−14261.3	−0.490720	−0.245360	−	0.969432i	$-0.578906\pi$
$61$	−14261.3	−0.490720	−0.245360	+	0.969432i	$0.578906\pi$
$62$	0	0
$63$	2760.29	0.0876201
$64$	0	0
$65$	−28197.9	−0.827816
$66$	0	0
$67$	54973.0	1.49611	0.748053	−	0.663639i	$-0.230989\pi$
$67$	54973.0	1.49611	0.748053	+	0.663639i	$0.230989\pi$
$68$	0	0
$69$	−48210.4	−1.21904
$70$	0	0
$71$	−58983.8	−1.38863	−0.694316	−	0.719670i	$-0.744293\pi$
$71$	−58983.8	−1.38863	−0.694316	+	0.719670i	$0.744293\pi$
$72$	0	0
$73$	32753.2	0.719361	0.359680	−	0.933076i	$-0.382886\pi$
$73$	32753.2	0.719361	0.359680	+	0.933076i	$0.382886\pi$
$74$	0	0
$75$	−16815.0	−0.345179
$76$	0	0
$77$	−1480.52	−0.0284570
$78$	0	0
$79$	62716.9	1.13062	0.565310	−	0.824879i	$-0.308757\pi$
$79$	62716.9	1.13062	0.565310	+	0.824879i	$0.308757\pi$
$80$	0	0
$81$	−70405.0	−1.19232
$82$	0	0
$83$	48533.8	0.773302	0.386651	−	0.922226i	$-0.373632\pi$
$83$	48533.8	0.773302	0.386651	+	0.922226i	$0.373632\pi$
$84$	0	0
$85$	70848.9	1.06362
$86$	0	0
$87$	69845.9	0.989333
$88$	0	0
$89$	13964.0	0.186868	0.0934338	−	0.995625i	$-0.470216\pi$
$89$	13964.0	0.186868	0.0934338	+	0.995625i	$0.470216\pi$
$90$	0	0
$91$	−6891.82	−0.0872430
$92$	0	0
$93$	173311.	2.07787
$94$	0	0
$95$	48958.9	0.556574
$96$	0	0
$97$	182495.	1.96935	0.984673	−	0.174411i	$-0.0558021\pi$
$97$	182495.	1.96935	0.984673	+	0.174411i	$0.0558021\pi$
$98$	0	0
$99$	−17351.7	−0.177932
$100$	0	0
$101$	−113571.	−1.10781	−0.553903	−	0.832581i	$-0.686862\pi$
$101$	−113571.	−1.10781	−0.553903	+	0.832581i	$0.686862\pi$
$102$	0	0
$103$	−20659.4	−0.191878	−0.0959391	−	0.995387i	$-0.530585\pi$
$103$	−20659.4	−0.191878	−0.0959391	+	0.995387i	$0.530585\pi$
$104$	0	0
$105$	−19815.8	−0.175403
$106$	0	0
$107$	226442.	1.91204	0.956020	−	0.293302i	$-0.0947541\pi$
$107$	226442.	1.91204	0.956020	+	0.293302i	$0.0947541\pi$
$108$	0	0
$109$	−13442.1	−0.108368	−0.0541839	−	0.998531i	$-0.517256\pi$
$109$	−13442.1	−0.108368	−0.0541839	+	0.998531i	$0.517256\pi$
$110$	0	0
$111$	87447.4	0.673658
$112$	0	0
$113$	−90221.1	−0.664679	−0.332339	−	0.943160i	$-0.607838\pi$
$113$	−90221.1	−0.664679	−0.332339	+	0.943160i	$0.607838\pi$
$114$	0	0
$115$	147210.	1.03799
$116$	0	0
$117$	−80772.1	−0.545502
$118$	0	0
$119$	17316.1	0.112094
$120$	0	0
$121$	−151744.	−0.942212
$122$	0	0
$123$	11349.4	0.0676412
$124$	0	0
$125$	−144877.	−0.829325
$126$	0	0
$127$	19067.3	0.104901	0.0524504	−	0.998624i	$-0.483297\pi$
$127$	19067.3	0.104901	0.0524504	+	0.998624i	$0.483297\pi$
$128$	0	0
$129$	−86714.2	−0.458792
$130$	0	0
$131$	−238858.	−1.21608	−0.608039	−	0.793907i	$-0.708044\pi$
$131$	−238858.	−1.21608	−0.608039	+	0.793907i	$0.708044\pi$
$132$	0	0
$133$	11966.0	0.0586570
$134$	0	0
$135$	81523.5	0.384989
$136$	0	0
$137$	202134.	0.920105	0.460053	−	0.887892i	$-0.347831\pi$
$137$	202134.	0.920105	0.460053	+	0.887892i	$0.347831\pi$
$138$	0	0
$139$	247861.	1.08811	0.544053	−	0.839051i	$-0.316889\pi$
$139$	247861.	1.08811	0.544053	+	0.839051i	$0.316889\pi$
$140$	0	0
$141$	4784.59	0.0202673
$142$	0	0
$143$	43323.3	0.177167
$144$	0	0
$145$	−213274.	−0.842400
$146$	0	0
$147$	340770.	1.30067
$148$	0	0
$149$	122420.	0.451738	0.225869	−	0.974158i	$-0.427478\pi$
$149$	122420.	0.451738	0.225869	+	0.974158i	$0.427478\pi$
$150$	0	0
$151$	142222.	0.507603	0.253802	−	0.967256i	$-0.418319\pi$
$151$	142222.	0.507603	0.253802	+	0.967256i	$0.418319\pi$
$152$	0	0
$153$	202944.	0.700888
$154$	0	0
$155$	−529204.	−1.76927
$156$	0	0
$157$	−367480.	−1.18983	−0.594914	−	0.803789i	$-0.702814\pi$
$157$	−367480.	−1.18983	−0.594914	+	0.803789i	$0.702814\pi$
$158$	0	0
$159$	−167405.	−0.525141
$160$	0	0
$161$	35979.5	0.109393
$162$	0	0
$163$	−600793.	−1.77115	−0.885576	−	0.464494i	$-0.846236\pi$
$163$	−600793.	−1.77115	−0.885576	+	0.464494i	$0.846236\pi$
$164$	0	0
$165$	124566.	0.356196
$166$	0	0
$167$	603337.	1.67405	0.837026	−	0.547163i	$-0.184292\pi$
$167$	603337.	1.67405	0.837026	+	0.547163i	$0.184292\pi$
$168$	0	0
$169$	−169624.	−0.456846
$170$	0	0
$171$	140241.	0.366763
$172$	0	0
$173$	20389.5	0.0517955	0.0258977	−	0.999665i	$-0.491756\pi$
$173$	20389.5	0.0517955	0.0258977	+	0.999665i	$0.491756\pi$
$174$	0	0
$175$	12549.1	0.0309754
$176$	0	0
$177$	232012.	0.556643
$178$	0	0
$179$	−748611.	−1.74632	−0.873159	−	0.487435i	$-0.837933\pi$
$179$	−748611.	−1.74632	−0.873159	+	0.487435i	$0.837933\pi$
$180$	0	0
$181$	493467.	1.11960	0.559798	−	0.828629i	$-0.310879\pi$
$181$	493467.	1.11960	0.559798	+	0.828629i	$0.310879\pi$
$182$	0	0
$183$	293264.	0.647337
$184$	0	0
$185$	−267020.	−0.573608
$186$	0	0
$187$	−108852.	−0.227632
$188$	0	0
$189$	19925.1	0.0405737
$190$	0	0
$191$	73106.3	0.145001	0.0725006	−	0.997368i	$-0.476902\pi$
$191$	73106.3	0.145001	0.0725006	+	0.997368i	$0.476902\pi$
$192$	0	0
$193$	−946549.	−1.82915	−0.914577	−	0.404413i	$-0.867476\pi$
$193$	−946549.	−1.82915	−0.914577	+	0.404413i	$0.867476\pi$
$194$	0	0
$195$	579852.	1.09202
$196$	0	0
$197$	−725295.	−1.33152	−0.665762	−	0.746164i	$-0.731893\pi$
$197$	−725295.	−1.33152	−0.665762	+	0.746164i	$0.731893\pi$
$198$	0	0
$199$	115067.	0.205977	0.102989	−	0.994683i	$-0.467159\pi$
$199$	115067.	0.205977	0.102989	+	0.994683i	$0.467159\pi$
$200$	0	0
$201$	−1.13044e6	−1.97360
$202$	0	0
$203$	−52126.1	−0.0887799
$204$	0	0
$205$	−34655.5	−0.0575953
$206$	0	0
$207$	421679.	0.684001
$208$	0	0
$209$	−75220.5	−0.119116
$210$	0	0
$211$	521196.	0.805926	0.402963	−	0.915216i	$-0.367980\pi$
$211$	521196.	0.805926	0.402963	+	0.915216i	$0.367980\pi$
$212$	0	0
$213$	1.21292e6	1.83182
$214$	0	0
$215$	264782.	0.390653
$216$	0	0
$217$	−129342.	−0.186462
$218$	0	0
$219$	−673525.	−0.948950
$220$	0	0
$221$	−506706.	−0.697872
$222$	0	0
$223$	−1.32283e6	−1.78131	−0.890657	−	0.454676i	$-0.849755\pi$
$223$	−1.32283e6	−1.78131	−0.890657	+	0.454676i	$0.849755\pi$
$224$	0	0
$225$	147075.	0.193679
$226$	0	0
$227$	−974160.	−1.25477	−0.627387	−	0.778708i	$-0.715876\pi$
$227$	−974160.	−1.25477	−0.627387	+	0.778708i	$0.715876\pi$
$228$	0	0
$229$	530946.	0.669055	0.334528	−	0.942386i	$-0.391423\pi$
$229$	530946.	0.669055	0.334528	+	0.942386i	$0.391423\pi$
$230$	0	0
$231$	30445.0	0.0375392
$232$	0	0
$233$	−348511.	−0.420558	−0.210279	−	0.977641i	$-0.567437\pi$
$233$	−348511.	−0.420558	−0.210279	+	0.977641i	$0.567437\pi$
$234$	0	0
$235$	−14609.7	−0.0172573
$236$	0	0
$237$	−1.28969e6	−1.49147
$238$	0	0
$239$	1.65554e6	1.87476	0.937381	−	0.348307i	$-0.113243\pi$
$239$	1.65554e6	1.87476	0.937381	+	0.348307i	$0.113243\pi$
$240$	0	0
$241$	−840345.	−0.931998	−0.465999	−	0.884785i	$-0.654305\pi$
$241$	−840345.	−0.931998	−0.465999	+	0.884785i	$0.654305\pi$
$242$	0	0
$243$	1.13229e6	1.23010
$244$	0	0
$245$	−1.04054e6	−1.10750
$246$	0	0
$247$	−350151.	−0.365185
$248$	0	0
$249$	−998031.	−1.02011
$250$	0	0
$251$	−1.47958e6	−1.48237	−0.741183	−	0.671303i	$-0.765735\pi$
$251$	−1.47958e6	−1.48237	−0.741183	+	0.671303i	$0.765735\pi$
$252$	0	0
$253$	−226174.	−0.222148
$254$	0	0
$255$	−1.45691e6	−1.40308
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	−65262.1	−0.0604521
$260$	0	0
$261$	−610917.	−0.555112
$262$	0	0
$263$	−1.64880e6	−1.46987	−0.734936	−	0.678136i	$-0.762788\pi$
$263$	−1.64880e6	−1.46987	−0.734936	+	0.678136i	$0.762788\pi$
$264$	0	0
$265$	511171.	0.447148
$266$	0	0
$267$	−287150.	−0.246508
$268$	0	0
$269$	−1.12450e6	−0.947496	−0.473748	−	0.880660i	$-0.657099\pi$
$269$	−1.12450e6	−0.947496	−0.473748	+	0.880660i	$0.657099\pi$
$270$	0	0
$271$	2.02862e6	1.67794	0.838971	−	0.544176i	$-0.183158\pi$
$271$	2.02862e6	1.67794	0.838971	+	0.544176i	$0.183158\pi$
$272$	0	0
$273$	141721.	0.115087
$274$	0	0
$275$	−78885.9	−0.0629025
$276$	0	0
$277$	213392.	0.167101	0.0835504	−	0.996504i	$-0.473374\pi$
$277$	213392.	0.167101	0.0835504	+	0.996504i	$0.473374\pi$
$278$	0	0
$279$	−1.51589e6	−1.16589
$280$	0	0
$281$	−1.35087e6	−1.02058	−0.510290	−	0.860002i	$-0.670462\pi$
$281$	−1.35087e6	−1.02058	−0.510290	+	0.860002i	$0.670462\pi$
$282$	0	0
$283$	−2.42006e6	−1.79622	−0.898111	−	0.439769i	$-0.855060\pi$
$283$	−2.42006e6	−1.79622	−0.898111	+	0.439769i	$0.855060\pi$
$284$	0	0
$285$	−1.00677e6	−0.734209
$286$	0	0
$287$	−8470.10	−0.00606993
$288$	0	0
$289$	−146728.	−0.103340
$290$	0	0
$291$	−3.75276e6	−2.59788
$292$	0	0
$293$	−1.38360e6	−0.941543	−0.470771	−	0.882255i	$-0.656024\pi$
$293$	−1.38360e6	−0.941543	−0.470771	+	0.882255i	$0.656024\pi$
$294$	0	0
$295$	−708447.	−0.473971
$296$	0	0
$297$	−125253.	−0.0823942
$298$	0	0
$299$	−1.05284e6	−0.681057
$300$	0	0
$301$	64714.9	0.0411707
$302$	0	0
$303$	2.33543e6	1.46137
$304$	0	0
$305$	−895480.	−0.551196
$306$	0	0
$307$	312198.	0.189053	0.0945265	−	0.995522i	$-0.469866\pi$
$307$	312198.	0.189053	0.0945265	+	0.995522i	$0.469866\pi$
$308$	0	0
$309$	424833.	0.253118
$310$	0	0
$311$	1.81234e6	1.06252	0.531262	−	0.847207i	$-0.321718\pi$
$311$	1.81234e6	1.06252	0.531262	+	0.847207i	$0.321718\pi$
$312$	0	0
$313$	−2.96922e6	−1.71309	−0.856546	−	0.516070i	$-0.827394\pi$
$313$	−2.96922e6	−1.71309	−0.856546	+	0.516070i	$0.827394\pi$
$314$	0	0
$315$	173321.	0.0984183
$316$	0	0
$317$	−2.33674e6	−1.30606	−0.653028	−	0.757334i	$-0.726502\pi$
$317$	−2.33674e6	−1.30606	−0.653028	+	0.757334i	$0.726502\pi$
$318$	0	0
$319$	327675.	0.180288
$320$	0	0
$321$	−4.65646e6	−2.52228
$322$	0	0
$323$	879774.	0.469207
$324$	0	0
$325$	−367213.	−0.192845
$326$	0	0
$327$	276418.	0.142954
$328$	0	0
$329$	−3570.75	−0.00181873
$330$	0	0
$331$	810683.	0.406706	0.203353	−	0.979105i	$-0.434816\pi$
$331$	810683.	0.406706	0.203353	+	0.979105i	$0.434816\pi$
$332$	0	0
$333$	−764871.	−0.377988
$334$	0	0
$335$	3.45181e6	1.68048
$336$	0	0
$337$	541967.	0.259955	0.129977	−	0.991517i	$-0.458510\pi$
$337$	541967.	0.259955	0.129977	+	0.991517i	$0.458510\pi$
$338$	0	0
$339$	1.85527e6	0.876816
$340$	0	0
$341$	813070.	0.378653
$342$	0	0
$343$	−512248.	−0.235096
$344$	0	0
$345$	−3.02718e6	−1.36927
$346$	0	0
$347$	2.30413e6	1.02727	0.513633	−	0.858010i	$-0.328299\pi$
$347$	2.30413e6	1.02727	0.513633	+	0.858010i	$0.328299\pi$
$348$	0	0
$349$	2.86365e6	1.25851	0.629255	−	0.777199i	$-0.283360\pi$
$349$	2.86365e6	1.25851	0.629255	+	0.777199i	$0.283360\pi$
$350$	0	0
$351$	−583050.	−0.252603
$352$	0	0
$353$	1.31733e6	0.562676	0.281338	−	0.959609i	$-0.409222\pi$
$353$	1.31733e6	0.562676	0.281338	+	0.959609i	$0.409222\pi$
$354$	0	0
$355$	−3.70365e6	−1.55977
$356$	0	0
$357$	−356082.	−0.147870
$358$	0	0
$359$	−2.50546e6	−1.02601	−0.513005	−	0.858386i	$-0.671468\pi$
$359$	−2.50546e6	−1.02601	−0.513005	+	0.858386i	$0.671468\pi$
$360$	0	0
$361$	−1.86815e6	−0.754472
$362$	0	0
$363$	3.12041e6	1.24293
$364$	0	0
$365$	2.05661e6	0.808014
$366$	0	0
$367$	209665.	0.0812571	0.0406286	−	0.999174i	$-0.487064\pi$
$367$	209665.	0.0812571	0.0406286	+	0.999174i	$0.487064\pi$
$368$	0	0
$369$	−99269.5	−0.0379533
$370$	0	0
$371$	124935.	0.0471246
$372$	0	0
$373$	−333617.	−0.124158	−0.0620792	−	0.998071i	$-0.519773\pi$
$373$	−333617.	−0.124158	−0.0620792	+	0.998071i	$0.519773\pi$
$374$	0	0
$375$	2.97920e6	1.09401
$376$	0	0
$377$	1.52532e6	0.552723
$378$	0	0
$379$	2.85263e6	1.02011	0.510055	−	0.860142i	$-0.329625\pi$
$379$	2.85263e6	1.02011	0.510055	+	0.860142i	$0.329625\pi$
$380$	0	0
$381$	−392092.	−0.138381
$382$	0	0
$383$	−3.23018e6	−1.12520	−0.562600	−	0.826729i	$-0.690199\pi$
$383$	−3.23018e6	−1.12520	−0.562600	+	0.826729i	$0.690199\pi$
$384$	0	0
$385$	−92963.6	−0.0319640
$386$	0	0
$387$	758458.	0.257427
$388$	0	0
$389$	−1.07582e6	−0.360466	−0.180233	−	0.983624i	$-0.557685\pi$
$389$	−1.07582e6	−0.360466	−0.180233	+	0.983624i	$0.557685\pi$
$390$	0	0
$391$	2.64532e6	0.875055
$392$	0	0
$393$	4.91179e6	1.60420
$394$	0	0
$395$	3.93805e6	1.26996
$396$	0	0
$397$	−4.60960e6	−1.46787	−0.733934	−	0.679221i	$-0.762318\pi$
$397$	−4.60960e6	−1.46787	−0.733934	+	0.679221i	$0.762318\pi$
$398$	0	0
$399$	−246064.	−0.0773777
$400$	0	0
$401$	−3.10953e6	−0.965682	−0.482841	−	0.875708i	$-0.660395\pi$
$401$	−3.10953e6	−0.965682	−0.482841	+	0.875708i	$0.660395\pi$
$402$	0	0
$403$	3.78483e6	1.16087
$404$	0	0
$405$	−4.42080e6	−1.33926
$406$	0	0
$407$	410250.	0.122762
$408$	0	0
$409$	−1.62746e6	−0.481063	−0.240531	−	0.970641i	$-0.577322\pi$
$409$	−1.62746e6	−0.481063	−0.240531	+	0.970641i	$0.577322\pi$
$410$	0	0
$411$	−4.15660e6	−1.21376
$412$	0	0
$413$	−173150.	−0.0499515
$414$	0	0
$415$	3.04749e6	0.868603
$416$	0	0
$417$	−5.09693e6	−1.43538
$418$	0	0
$419$	−6.14657e6	−1.71040	−0.855200	−	0.518298i	$-0.826566\pi$
$419$	−6.14657e6	−1.71040	−0.855200	+	0.518298i	$0.826566\pi$
$420$	0	0
$421$	166951.	0.0459075	0.0229537	−	0.999737i	$-0.492693\pi$
$421$	166951.	0.0459075	0.0229537	+	0.999737i	$0.492693\pi$
$422$	0	0
$423$	−41849.1	−0.0113720
$424$	0	0
$425$	922643.	0.247777
$426$	0	0
$427$	−218863.	−0.0580902
$428$	0	0
$429$	−890885.	−0.233711
$430$	0	0
$431$	1.03155e6	0.267484	0.133742	−	0.991016i	$-0.457301\pi$
$431$	1.03155e6	0.267484	0.133742	+	0.991016i	$0.457301\pi$
$432$	0	0
$433$	1.50406e6	0.385518	0.192759	−	0.981246i	$-0.438256\pi$
$433$	1.50406e6	0.385518	0.192759	+	0.981246i	$0.438256\pi$
$434$	0	0
$435$	4.38569e6	1.11126
$436$	0	0
$437$	1.82800e6	0.457902
$438$	0	0
$439$	4.99894e6	1.23799	0.618995	−	0.785395i	$-0.287540\pi$
$439$	4.99894e6	1.23799	0.618995	+	0.785395i	$0.287540\pi$
$440$	0	0
$441$	−2.98059e6	−0.729803
$442$	0	0
$443$	6.63687e6	1.60677	0.803386	−	0.595459i	$-0.203030\pi$
$443$	6.63687e6	1.60677	0.803386	+	0.595459i	$0.203030\pi$
$444$	0	0
$445$	876811.	0.209897
$446$	0	0
$447$	−2.51740e6	−0.595913
$448$	0	0
$449$	−1.11355e6	−0.260671	−0.130335	−	0.991470i	$-0.541605\pi$
$449$	−1.11355e6	−0.260671	−0.130335	+	0.991470i	$0.541605\pi$
$450$	0	0
$451$	53244.7	0.0123264
$452$	0	0
$453$	−2.92460e6	−0.669609
$454$	0	0
$455$	−432744.	−0.0979947
$456$	0	0
$457$	5.27821e6	1.18221	0.591107	−	0.806593i	$-0.298691\pi$
$457$	5.27821e6	1.18221	0.591107	+	0.806593i	$0.298691\pi$
$458$	0	0
$459$	1.46495e6	0.324556
$460$	0	0
$461$	1.53319e6	0.336004	0.168002	−	0.985787i	$-0.446268\pi$
$461$	1.53319e6	0.336004	0.168002	+	0.985787i	$0.446268\pi$
$462$	0	0
$463$	−3.98186e6	−0.863243	−0.431621	−	0.902055i	$-0.642058\pi$
$463$	−3.98186e6	−0.863243	−0.431621	+	0.902055i	$0.642058\pi$
$464$	0	0
$465$	1.08824e7	2.33395
$466$	0	0
$467$	−5.05760e6	−1.07313	−0.536565	−	0.843859i	$-0.680278\pi$
$467$	−5.05760e6	−1.07313	−0.536565	+	0.843859i	$0.680278\pi$
$468$	0	0
$469$	843652.	0.177105
$470$	0	0
$471$	7.55672e6	1.56957
$472$	0	0
$473$	−406811.	−0.0836064
$474$	0	0
$475$	637577.	0.129658
$476$	0	0
$477$	1.46423e6	0.294655
$478$	0	0
$479$	−8.00610e6	−1.59434	−0.797172	−	0.603752i	$-0.793672\pi$
$479$	−8.00610e6	−1.59434	−0.797172	+	0.603752i	$0.793672\pi$
$480$	0	0
$481$	1.90971e6	0.376361
$482$	0	0
$483$	−739869.	−0.144307
$484$	0	0
$485$	1.14590e7	2.21205
$486$	0	0
$487$	161223.	0.0308039	0.0154019	−	0.999881i	$-0.495097\pi$
$487$	161223.	0.0308039	0.0154019	+	0.999881i	$0.495097\pi$
$488$	0	0
$489$	1.23545e7	2.33643
$490$	0	0
$491$	−1.37675e6	−0.257722	−0.128861	−	0.991663i	$-0.541132\pi$
$491$	−1.37675e6	−0.257722	−0.128861	+	0.991663i	$0.541132\pi$
$492$	0	0
$493$	−3.83246e6	−0.710166
$494$	0	0
$495$	−1.08953e6	−0.199861
$496$	0	0
$497$	−905205.	−0.164383
$498$	0	0
$499$	−2.79866e6	−0.503151	−0.251576	−	0.967838i	$-0.580949\pi$
$499$	−2.79866e6	−0.503151	−0.251576	+	0.967838i	$0.580949\pi$
$500$	0	0
$501$	−1.24068e7	−2.20834
$502$	0	0
$503$	107668.	0.0189744	0.00948718	−	0.999955i	$-0.496980\pi$
$503$	107668.	0.0189744	0.00948718	+	0.999955i	$0.496980\pi$
$504$	0	0
$505$	−7.13123e6	−1.24433
$506$	0	0
$507$	3.48808e6	0.602651
$508$	0	0
$509$	5.39790e6	0.923487	0.461743	−	0.887014i	$-0.347224\pi$
$509$	5.39790e6	0.923487	0.461743	+	0.887014i	$0.347224\pi$
$510$	0	0
$511$	502652.	0.0851560
$512$	0	0
$513$	1.01233e6	0.169835
$514$	0	0
$515$	−1.29723e6	−0.215525
$516$	0	0
$517$	22446.4	0.00369335
$518$	0	0
$519$	−419283.	−0.0683264
$520$	0	0
$521$	−7.37650e6	−1.19057	−0.595286	−	0.803514i	$-0.702961\pi$
$521$	−7.37650e6	−1.19057	−0.595286	+	0.803514i	$0.702961\pi$
$522$	0	0
$523$	4.57543e6	0.731438	0.365719	−	0.930725i	$-0.380823\pi$
$523$	4.57543e6	0.731438	0.365719	+	0.930725i	$0.380823\pi$
$524$	0	0
$525$	−258054.	−0.0408614
$526$	0	0
$527$	−9.50960e6	−1.49154
$528$	0	0
$529$	−939890.	−0.146029
$530$	0	0
$531$	−2.02932e6	−0.312331
$532$	0	0
$533$	247853.	0.0377900
$534$	0	0
$535$	1.42185e7	2.14768
$536$	0	0
$537$	1.53941e7	2.30367
$538$	0	0
$539$	1.59869e6	0.237023
$540$	0	0
$541$	−2.71195e6	−0.398372	−0.199186	−	0.979962i	$-0.563830\pi$
$541$	−2.71195e6	−0.398372	−0.199186	+	0.979962i	$0.563830\pi$
$542$	0	0
$543$	−1.01475e7	−1.47692
$544$	0	0
$545$	−844042.	−0.121723
$546$	0	0
$547$	−4.09826e6	−0.585640	−0.292820	−	0.956168i	$-0.594594\pi$
$547$	−4.09826e6	−0.585640	−0.292820	+	0.956168i	$0.594594\pi$
$548$	0	0
$549$	−2.56507e6	−0.363219
$550$	0	0
$551$	−2.64835e6	−0.371618
$552$	0	0
$553$	962495.	0.133840
$554$	0	0
$555$	5.49091e6	0.756679
$556$	0	0
$557$	−5.14211e6	−0.702269	−0.351134	−	0.936325i	$-0.614204\pi$
$557$	−5.14211e6	−0.702269	−0.351134	+	0.936325i	$0.614204\pi$
$558$	0	0
$559$	−1.89370e6	−0.256319
$560$	0	0
$561$	2.23840e6	0.300283
$562$	0	0
$563$	−4.68820e6	−0.623355	−0.311677	−	0.950188i	$-0.600891\pi$
$563$	−4.68820e6	−0.623355	−0.311677	+	0.950188i	$0.600891\pi$
$564$	0	0
$565$	−5.66507e6	−0.746593
$566$	0	0
$567$	−1.08048e6	−0.141143
$568$	0	0
$569$	1.31689e6	0.170517	0.0852586	−	0.996359i	$-0.472828\pi$
$569$	1.31689e6	0.170517	0.0852586	+	0.996359i	$0.472828\pi$
$570$	0	0
$571$	−3.29013e6	−0.422302	−0.211151	−	0.977453i	$-0.567721\pi$
$571$	−3.29013e6	−0.422302	−0.211151	+	0.977453i	$0.567721\pi$
$572$	0	0
$573$	−1.50333e6	−0.191279
$574$	0	0
$575$	1.91707e6	0.241807
$576$	0	0
$577$	−1.45150e7	−1.81501	−0.907503	−	0.420045i	$-0.862014\pi$
$577$	−1.45150e7	−1.81501	−0.907503	+	0.420045i	$0.862014\pi$
$578$	0	0
$579$	1.94645e7	2.41294
$580$	0	0
$581$	744832.	0.0915415
$582$	0	0
$583$	−785363.	−0.0956972
$584$	0	0
$585$	−5.07176e6	−0.612729
$586$	0	0
$587$	−1.11386e7	−1.33424	−0.667122	−	0.744948i	$-0.732474\pi$
$587$	−1.11386e7	−1.33424	−0.667122	+	0.744948i	$0.732474\pi$
$588$	0	0
$589$	−6.57144e6	−0.780499
$590$	0	0
$591$	1.49147e7	1.75649
$592$	0	0
$593$	−8.18992e6	−0.956407	−0.478204	−	0.878249i	$-0.658712\pi$
$593$	−8.18992e6	−0.956407	−0.478204	+	0.878249i	$0.658712\pi$
$594$	0	0
$595$	1.08729e6	0.125908
$596$	0	0
$597$	−2.36620e6	−0.271716
$598$	0	0
$599$	2.04734e6	0.233143	0.116572	−	0.993182i	$-0.462810\pi$
$599$	2.04734e6	0.233143	0.116572	+	0.993182i	$0.462810\pi$
$600$	0	0
$601$	816100.	0.0921631	0.0460816	−	0.998938i	$-0.485327\pi$
$601$	816100.	0.0921631	0.0460816	+	0.998938i	$0.485327\pi$
$602$	0	0
$603$	9.88759e6	1.10738
$604$	0	0
$605$	−9.52816e6	−1.05833
$606$	0	0
$607$	−1.68911e6	−0.186074	−0.0930370	−	0.995663i	$-0.529657\pi$
$607$	−1.68911e6	−0.186074	−0.0930370	+	0.995663i	$0.529657\pi$
$608$	0	0
$609$	1.07190e6	0.117115
$610$	0	0
$611$	104488.	0.0113230
$612$	0	0
$613$	3.87757e6	0.416781	0.208391	−	0.978046i	$-0.433177\pi$
$613$	3.87757e6	0.416781	0.208391	+	0.978046i	$0.433177\pi$
$614$	0	0
$615$	712642.	0.0759773
$616$	0	0
$617$	−1.05007e7	−1.11047	−0.555233	−	0.831695i	$-0.687371\pi$
$617$	−1.05007e7	−1.11047	−0.555233	+	0.831695i	$0.687371\pi$
$618$	0	0
$619$	−3.79402e6	−0.397990	−0.198995	−	0.980000i	$-0.563768\pi$
$619$	−3.79402e6	−0.397990	−0.198995	+	0.980000i	$0.563768\pi$
$620$	0	0
$621$	3.04388e6	0.316736
$622$	0	0
$623$	214300.	0.0221209
$624$	0	0
$625$	−1.16523e7	−1.19320
$626$	0	0
$627$	1.54681e6	0.157133
$628$	0	0
$629$	−4.79826e6	−0.483567
$630$	0	0
$631$	−1.04039e6	−0.104022	−0.0520109	−	0.998647i	$-0.516563\pi$
$631$	−1.04039e6	−0.104022	−0.0520109	+	0.998647i	$0.516563\pi$
$632$	0	0
$633$	−1.07177e7	−1.06314
$634$	0	0
$635$	1.19725e6	0.117829
$636$	0	0
$637$	7.44186e6	0.726663
$638$	0	0
$639$	−1.06090e7	−1.02783
$640$	0	0
$641$	−2.60432e6	−0.250351	−0.125175	−	0.992135i	$-0.539949\pi$
$641$	−2.60432e6	−0.250351	−0.125175	+	0.992135i	$0.539949\pi$
$642$	0	0
$643$	−5.68851e6	−0.542589	−0.271294	−	0.962496i	$-0.587452\pi$
$643$	−5.68851e6	−0.542589	−0.271294	+	0.962496i	$0.587452\pi$
$644$	0	0
$645$	−5.44487e6	−0.515333
$646$	0	0
$647$	1.95565e7	1.83667	0.918333	−	0.395809i	$-0.129536\pi$
$647$	1.95565e7	1.83667	0.918333	+	0.395809i	$0.129536\pi$
$648$	0	0
$649$	1.08846e6	0.101438
$650$	0	0
$651$	2.65974e6	0.245973
$652$	0	0
$653$	1.55955e7	1.43126	0.715628	−	0.698482i	$-0.246141\pi$
$653$	1.55955e7	1.43126	0.715628	+	0.698482i	$0.246141\pi$
$654$	0	0
$655$	−1.49981e7	−1.36595
$656$	0	0
$657$	5.89108e6	0.532453
$658$	0	0
$659$	−1.16032e7	−1.04080	−0.520398	−	0.853924i	$-0.674216\pi$
$659$	−1.16032e7	−1.04080	−0.520398	+	0.853924i	$0.674216\pi$
$660$	0	0
$661$	−2.72067e6	−0.242199	−0.121100	−	0.992640i	$-0.538642\pi$
$661$	−2.72067e6	−0.242199	−0.121100	+	0.992640i	$0.538642\pi$
$662$	0	0
$663$	1.04197e7	0.920602
$664$	0	0
$665$	751356.	0.0658858
$666$	0	0
$667$	−7.96310e6	−0.693055
$668$	0	0
$669$	2.72021e7	2.34983
$670$	0	0
$671$	1.37582e6	0.117965
$672$	0	0
$673$	1.82789e7	1.55566	0.777828	−	0.628478i	$-0.216322\pi$
$673$	1.82789e7	1.55566	0.777828	+	0.628478i	$0.216322\pi$
$674$	0	0
$675$	1.06166e6	0.0896859
$676$	0	0
$677$	−1.03326e7	−0.866441	−0.433221	−	0.901288i	$-0.642623\pi$
$677$	−1.03326e7	−0.866441	−0.433221	+	0.901288i	$0.642623\pi$
$678$	0	0
$679$	2.80069e6	0.233126
$680$	0	0
$681$	2.00323e7	1.65524
$682$	0	0
$683$	1.87505e7	1.53802	0.769009	−	0.639238i	$-0.220750\pi$
$683$	1.87505e7	1.53802	0.769009	+	0.639238i	$0.220750\pi$
$684$	0	0
$685$	1.26922e7	1.03350
$686$	0	0
$687$	−1.09182e7	−0.882589
$688$	0	0
$689$	−3.65585e6	−0.293387
$690$	0	0
$691$	−8.87506e6	−0.707092	−0.353546	−	0.935417i	$-0.615024\pi$
$691$	−8.87506e6	−0.707092	−0.353546	+	0.935417i	$0.615024\pi$
$692$	0	0
$693$	−266291.	−0.0210632
$694$	0	0
$695$	1.55634e7	1.22220
$696$	0	0
$697$	−622746.	−0.0485544
$698$	0	0
$699$	7.16664e6	0.554783
$700$	0	0
$701$	−1.82567e7	−1.40323	−0.701613	−	0.712558i	$-0.747536\pi$
$701$	−1.82567e7	−1.40323	−0.701613	+	0.712558i	$0.747536\pi$
$702$	0	0
$703$	−3.31575e6	−0.253043
$704$	0	0
$705$	300429.	0.0227651
$706$	0	0
$707$	−1.74293e6	−0.131139
$708$	0	0
$709$	1.30961e7	0.978426	0.489213	−	0.872164i	$-0.337284\pi$
$709$	1.30961e7	0.978426	0.489213	+	0.872164i	$0.337284\pi$
$710$	0	0
$711$	1.12804e7	0.836858
$712$	0	0
$713$	−1.97591e7	−1.45560
$714$	0	0
$715$	2.72031e6	0.199000
$716$	0	0
$717$	−3.40440e7	−2.47310
$718$	0	0
$719$	2.18572e6	0.157678	0.0788390	−	0.996887i	$-0.474879\pi$
$719$	2.18572e6	0.157678	0.0788390	+	0.996887i	$0.474879\pi$
$720$	0	0
$721$	−317054.	−0.0227140
$722$	0	0
$723$	1.72805e7	1.22945
$724$	0	0
$725$	−2.77740e6	−0.196243
$726$	0	0
$727$	1.67686e6	0.117669	0.0588345	−	0.998268i	$-0.481262\pi$
$727$	1.67686e6	0.117669	0.0588345	+	0.998268i	$0.481262\pi$
$728$	0	0
$729$	−6.17553e6	−0.430383
$730$	0	0
$731$	4.75802e6	0.329331
$732$	0	0
$733$	−4.78711e6	−0.329089	−0.164545	−	0.986370i	$-0.552615\pi$
$733$	−4.78711e6	−0.329089	−0.164545	+	0.986370i	$0.552615\pi$
$734$	0	0
$735$	2.13973e7	1.46097
$736$	0	0
$737$	−5.30336e6	−0.359652
$738$	0	0
$739$	−9.51856e6	−0.641151	−0.320575	−	0.947223i	$-0.603876\pi$
$739$	−9.51856e6	−0.641151	−0.320575	+	0.947223i	$0.603876\pi$
$740$	0	0
$741$	7.20036e6	0.481736
$742$	0	0
$743$	−2.49482e6	−0.165793	−0.0828967	−	0.996558i	$-0.526417\pi$
$743$	−2.49482e6	−0.165793	−0.0828967	+	0.996558i	$0.526417\pi$
$744$	0	0
$745$	7.68686e6	0.507409
$746$	0	0
$747$	8.72942e6	0.572380
$748$	0	0
$749$	3.47512e6	0.226342
$750$	0	0
$751$	2.10703e7	1.36324	0.681619	−	0.731707i	$-0.261276\pi$
$751$	2.10703e7	1.36324	0.681619	+	0.731707i	$0.261276\pi$
$752$	0	0
$753$	3.04256e7	1.95547
$754$	0	0
$755$	8.93026e6	0.570160
$756$	0	0
$757$	4.89163e6	0.310251	0.155126	−	0.987895i	$-0.450422\pi$
$757$	4.89163e6	0.310251	0.155126	+	0.987895i	$0.450422\pi$
$758$	0	0
$759$	4.65096e6	0.293048
$760$	0	0
$761$	2.63547e6	0.164967	0.0824833	−	0.996592i	$-0.473715\pi$
$761$	2.63547e6	0.164967	0.0824833	+	0.996592i	$0.473715\pi$
$762$	0	0
$763$	−206291.	−0.0128283
$764$	0	0
$765$	1.27431e7	0.787265
$766$	0	0
$767$	5.06676e6	0.310986
$768$	0	0
$769$	1.90152e7	1.15954	0.579769	−	0.814781i	$-0.303143\pi$
$769$	1.90152e7	1.15954	0.579769	+	0.814781i	$0.303143\pi$
$770$	0	0
$771$	−1.35821e6	−0.0822868
$772$	0	0
$773$	−3.18822e7	−1.91911	−0.959555	−	0.281522i	$-0.909161\pi$
$773$	−3.18822e7	−1.91911	−0.959555	+	0.281522i	$0.909161\pi$
$774$	0	0
$775$	−6.89166e6	−0.412164
$776$	0	0
$777$	1.34203e6	0.0797459
$778$	0	0
$779$	−430338.	−0.0254077
$780$	0	0
$781$	5.69030e6	0.333816
$782$	0	0
$783$	−4.40988e6	−0.257053
$784$	0	0
$785$	−2.30744e7	−1.33646
$786$	0	0
$787$	−3.30632e7	−1.90287	−0.951433	−	0.307857i	$-0.900388\pi$
$787$	−3.30632e7	−1.90287	−0.951433	+	0.307857i	$0.900388\pi$
$788$	0	0
$789$	3.39054e7	1.93899
$790$	0	0
$791$	−1.38459e6	−0.0786830
$792$	0	0
$793$	6.40441e6	0.361656
$794$	0	0
$795$	−1.05115e7	−0.589859
$796$	0	0
$797$	2.75564e7	1.53666	0.768328	−	0.640057i	$-0.221089\pi$
$797$	2.75564e7	1.53666	0.768328	+	0.640057i	$0.221089\pi$
$798$	0	0
$799$	−262531.	−0.0145484
$800$	0	0
$801$	2.51160e6	0.138315
$802$	0	0
$803$	−3.15977e6	−0.172929
$804$	0	0
$805$	2.25919e6	0.122875
$806$	0	0
$807$	2.31237e7	1.24990
$808$	0	0
$809$	1.25823e7	0.675912	0.337956	−	0.941162i	$-0.390264\pi$
$809$	1.25823e7	0.675912	0.337956	+	0.941162i	$0.390264\pi$
$810$	0	0
$811$	2.40136e7	1.28205	0.641025	−	0.767520i	$-0.278510\pi$
$811$	2.40136e7	1.28205	0.641025	+	0.767520i	$0.278510\pi$
$812$	0	0
$813$	−4.17157e7	−2.21347
$814$	0	0
$815$	−3.77244e7	−1.98943
$816$	0	0
$817$	3.28795e6	0.172334
$818$	0	0
$819$	−1.23958e6	−0.0645751
$820$	0	0
$821$	−2.92834e7	−1.51623	−0.758113	−	0.652124i	$-0.773878\pi$
$821$	−2.92834e7	−1.51623	−0.758113	+	0.652124i	$0.773878\pi$
$822$	0	0
$823$	−2.03535e7	−1.04747	−0.523733	−	0.851883i	$-0.675461\pi$
$823$	−2.03535e7	−1.04747	−0.523733	+	0.851883i	$0.675461\pi$
$824$	0	0
$825$	1.62218e6	0.0829783
$826$	0	0
$827$	2.74889e7	1.39763	0.698816	−	0.715301i	$-0.253711\pi$
$827$	2.74889e7	1.39763	0.698816	+	0.715301i	$0.253711\pi$
$828$	0	0
$829$	1.64618e7	0.831939	0.415970	−	0.909379i	$-0.363442\pi$
$829$	1.64618e7	0.831939	0.415970	+	0.909379i	$0.363442\pi$
$830$	0	0
$831$	−4.38811e6	−0.220432
$832$	0	0
$833$	−1.86981e7	−0.933652
$834$	0	0
$835$	3.78841e7	1.88036
$836$	0	0
$837$	−1.09424e7	−0.539881
$838$	0	0
$839$	−1.75961e7	−0.863001	−0.431501	−	0.902113i	$-0.642016\pi$
$839$	−1.75961e7	−0.863001	−0.431501	+	0.902113i	$0.642016\pi$
$840$	0	0
$841$	−8.97443e6	−0.437539
$842$	0	0
$843$	2.77788e7	1.34631
$844$	0	0
$845$	−1.06508e7	−0.513147
$846$	0	0
$847$	−2.32877e6	−0.111537
$848$	0	0
$849$	4.97652e7	2.36950
$850$	0	0
$851$	−9.96985e6	−0.471916
$852$	0	0
$853$	2.85920e6	0.134546	0.0672732	−	0.997735i	$-0.478570\pi$
$853$	2.85920e6	0.134546	0.0672732	+	0.997735i	$0.478570\pi$
$854$	0	0
$855$	8.80589e6	0.411963
$856$	0	0
$857$	−1.10233e7	−0.512698	−0.256349	−	0.966584i	$-0.582520\pi$
$857$	−1.10233e7	−0.512698	−0.256349	+	0.966584i	$0.582520\pi$
$858$	0	0
$859$	−1.46205e7	−0.676053	−0.338026	−	0.941137i	$-0.609759\pi$
$859$	−1.46205e7	−0.676053	−0.338026	+	0.941137i	$0.609759\pi$
$860$	0	0
$861$	174176.	0.00800719
$862$	0	0
$863$	−2.51312e7	−1.14865	−0.574324	−	0.818628i	$-0.694735\pi$
$863$	−2.51312e7	−1.14865	−0.574324	+	0.818628i	$0.694735\pi$
$864$	0	0
$865$	1.28028e6	0.0581787
$866$	0	0
$867$	3.01727e6	0.136322
$868$	0	0
$869$	−6.05043e6	−0.271792
$870$	0	0
$871$	−2.46871e7	−1.10262
$872$	0	0
$873$	3.28241e7	1.45766
$874$	0	0
$875$	−2.22338e6	−0.0981734
$876$	0	0
$877$	3.35725e7	1.47395	0.736977	−	0.675918i	$-0.236252\pi$
$877$	3.35725e7	1.47395	0.736977	+	0.675918i	$0.236252\pi$
$878$	0	0
$879$	2.84517e7	1.24204
$880$	0	0
$881$	2.24452e7	0.974278	0.487139	−	0.873324i	$-0.338040\pi$
$881$	2.24452e7	0.974278	0.487139	+	0.873324i	$0.338040\pi$
$882$	0	0
$883$	1.80688e7	0.779881	0.389940	−	0.920840i	$-0.372496\pi$
$883$	1.80688e7	0.779881	0.389940	+	0.920840i	$0.372496\pi$
$884$	0	0
$885$	1.45682e7	0.625243
$886$	0	0
$887$	2.17307e7	0.927393	0.463697	−	0.885994i	$-0.346523\pi$
$887$	2.17307e7	0.927393	0.463697	+	0.885994i	$0.346523\pi$
$888$	0	0
$889$	292619.	0.0124179
$890$	0	0
$891$	6.79212e6	0.286623
$892$	0	0
$893$	−181418.	−0.00761292
$894$	0	0
$895$	−4.70060e7	−1.96153
$896$	0	0
$897$	2.16502e7	0.898421
$898$	0	0
$899$	2.86264e7	1.18132
$900$	0	0
$901$	9.18555e6	0.376958
$902$	0	0
$903$	−1.33077e6	−0.0543106
$904$	0	0
$905$	3.09853e7	1.25757
$906$	0	0
$907$	−1.84646e7	−0.745282	−0.372641	−	0.927976i	$-0.621548\pi$
$907$	−1.84646e7	−0.745282	−0.372641	+	0.927976i	$0.621548\pi$
$908$	0	0
$909$	−2.04272e7	−0.819971
$910$	0	0
$911$	−1.85756e7	−0.741560	−0.370780	−	0.928721i	$-0.620910\pi$
$911$	−1.85756e7	−0.741560	−0.370780	+	0.928721i	$0.620910\pi$
$912$	0	0
$913$	−4.68216e6	−0.185896
$914$	0	0
$915$	1.84143e7	0.727115
$916$	0	0
$917$	−3.66567e6	−0.143956
$918$	0	0
$919$	−1.27567e7	−0.498251	−0.249125	−	0.968471i	$-0.580143\pi$
$919$	−1.27567e7	−0.498251	−0.249125	+	0.968471i	$0.580143\pi$
$920$	0	0
$921$	−6.41992e6	−0.249391
$922$	0	0
$923$	2.64882e7	1.02341
$924$	0	0
$925$	−3.47732e6	−0.133626
$926$	0	0
$927$	−3.71586e6	−0.142024
$928$	0	0
$929$	4.42755e7	1.68315	0.841577	−	0.540137i	$-0.181628\pi$
$929$	4.42755e7	1.68315	0.841577	+	0.540137i	$0.181628\pi$
$930$	0	0
$931$	−1.29210e7	−0.488564
$932$	0	0
$933$	−3.72683e7	−1.40164
$934$	0	0
$935$	−6.83495e6	−0.255686
$936$	0	0
$937$	2.85578e7	1.06262	0.531308	−	0.847179i	$-0.321701\pi$
$937$	2.85578e7	1.06262	0.531308	+	0.847179i	$0.321701\pi$
$938$	0	0
$939$	6.10578e7	2.25984
$940$	0	0
$941$	2.47508e7	0.911204	0.455602	−	0.890184i	$-0.349424\pi$
$941$	2.47508e7	0.911204	0.455602	+	0.890184i	$0.349424\pi$
$942$	0	0
$943$	−1.29395e6	−0.0473845
$944$	0	0
$945$	1.25111e6	0.0455740
$946$	0	0
$947$	−3.18615e7	−1.15449	−0.577247	−	0.816570i	$-0.695873\pi$
$947$	−3.18615e7	−1.15449	−0.577247	+	0.816570i	$0.695873\pi$
$948$	0	0
$949$	−1.47087e7	−0.530162
$950$	0	0
$951$	4.80518e7	1.72289
$952$	0	0
$953$	2.89583e7	1.03286	0.516428	−	0.856330i	$-0.327261\pi$
$953$	2.89583e7	1.03286	0.516428	+	0.856330i	$0.327261\pi$
$954$	0	0
$955$	4.59042e6	0.162871
$956$	0	0
$957$	−6.73818e6	−0.237828
$958$	0	0
$959$	3.10208e6	0.108920
$960$	0	0
$961$	4.24025e7	1.48110
$962$	0	0
$963$	4.07284e7	1.41525
$964$	0	0
$965$	−5.94348e7	−2.05458
$966$	0	0
$967$	2.80731e7	0.965437	0.482719	−	0.875775i	$-0.339649\pi$
$967$	2.80731e7	0.965437	0.482719	+	0.875775i	$0.339649\pi$
$968$	0	0
$969$	−1.80913e7	−0.618958
$970$	0	0
$971$	3.72812e7	1.26894	0.634472	−	0.772946i	$-0.281218\pi$
$971$	3.72812e7	1.26894	0.634472	+	0.772946i	$0.281218\pi$
$972$	0	0
$973$	3.80384e6	0.128807
$974$	0	0
$975$	7.55123e6	0.254393
$976$	0	0
$977$	1.48630e7	0.498161	0.249080	−	0.968483i	$-0.419872\pi$
$977$	1.48630e7	0.498161	0.249080	+	0.968483i	$0.419872\pi$
$978$	0	0
$979$	−1.34713e6	−0.0449215
$980$	0	0
$981$	−2.41773e6	−0.0802113
$982$	0	0
$983$	330625.	0.0109132	0.00545660	−	0.999985i	$-0.498263\pi$
$983$	330625.	0.0109132	0.00545660	+	0.999985i	$0.498263\pi$
$984$	0	0
$985$	−4.55420e7	−1.49562
$986$	0	0
$987$	73427.5	0.00239920
$988$	0	0
$989$	9.88626e6	0.321396
$990$	0	0
$991$	−5.07599e7	−1.64186	−0.820932	−	0.571027i	$-0.806545\pi$
$991$	−5.07599e7	−1.64186	−0.820932	+	0.571027i	$0.806545\pi$
$992$	0	0
$993$	−1.66706e7	−0.536509
$994$	0	0
$995$	7.22519e6	0.231362
$996$	0	0
$997$	−1.67020e7	−0.532146	−0.266073	−	0.963953i	$-0.585726\pi$
$997$	−1.67020e7	−0.532146	−0.266073	+	0.963953i	$0.585726\pi$
$998$	0	0
$999$	−5.52119e6	−0.175033

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.8	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-20.5636 q^{3} +62.7910 q^{5} +15.3467 q^{7} +179.863 q^{9} +O(q^{10})\)	\(q-20.5636 q^{3} +62.7910 q^{5} +15.3467 q^{7} +179.863 q^{9} -96.4721 q^{11} -449.076 q^{13} -1291.21 q^{15} +1128.33 q^{17} +779.713 q^{19} -315.583 q^{21} +2344.45 q^{23} +817.707 q^{25} +1298.33 q^{27} -3396.57 q^{29} -8428.03 q^{31} +1983.82 q^{33} +963.632 q^{35} -4252.53 q^{37} +9234.64 q^{39} -551.918 q^{41} +4216.87 q^{43} +11293.8 q^{45} -232.672 q^{47} -16571.5 q^{49} -23202.6 q^{51} +8140.83 q^{53} -6057.58 q^{55} -16033.7 q^{57} -11282.6 q^{59} -14261.3 q^{61} +2760.29 q^{63} -28197.9 q^{65} +54973.0 q^{67} -48210.4 q^{69} -58983.8 q^{71} +32753.2 q^{73} -16815.0 q^{75} -1480.52 q^{77} +62716.9 q^{79} -70405.0 q^{81} +48533.8 q^{83} +70848.9 q^{85} +69845.9 q^{87} +13964.0 q^{89} -6891.82 q^{91} +173311. q^{93} +48958.9 q^{95} +182495. q^{97} -17351.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 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

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