LMFDB - Embedded newform 1028.6.a.a.1.9

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			1.9
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)

$f(q)$	$=$	$q-19.7457 q^{3} -10.8196 q^{5} -70.7802 q^{7} +146.891 q^{9} +O(q^{10})$	$q-19.7457 q^{3} -10.8196 q^{5} -70.7802 q^{7} +146.891 q^{9} -539.605 q^{11} +766.155 q^{13} +213.640 q^{15} -703.216 q^{17} -276.074 q^{19} +1397.60 q^{21} +853.774 q^{23} -3007.94 q^{25} +1897.73 q^{27} -580.037 q^{29} -2735.26 q^{31} +10654.9 q^{33} +765.812 q^{35} +11448.6 q^{37} -15128.2 q^{39} -16864.0 q^{41} +17528.2 q^{43} -1589.30 q^{45} +23144.4 q^{47} -11797.2 q^{49} +13885.5 q^{51} -1206.94 q^{53} +5838.30 q^{55} +5451.25 q^{57} +33144.1 q^{59} -11745.9 q^{61} -10397.0 q^{63} -8289.48 q^{65} +1553.20 q^{67} -16858.3 q^{69} +38797.1 q^{71} +44398.6 q^{73} +59393.7 q^{75} +38193.4 q^{77} -18958.2 q^{79} -73166.5 q^{81} -21986.5 q^{83} +7608.50 q^{85} +11453.2 q^{87} +27961.7 q^{89} -54228.7 q^{91} +54009.5 q^{93} +2987.00 q^{95} -56138.5 q^{97} -79263.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−19.7457	−1.26668	−0.633342	−	0.773872i	$-0.718318\pi$
$3$	−19.7457	−1.26668	−0.633342	+	0.773872i	$0.718318\pi$
$4$	0	0
$5$	−10.8196	−0.193547	−0.0967733	−	0.995306i	$-0.530852\pi$
$5$	−10.8196	−0.193547	−0.0967733	+	0.995306i	$0.530852\pi$
$6$	0	0
$7$	−70.7802	−0.545968	−0.272984	−	0.962019i	$-0.588011\pi$
$7$	−70.7802	−0.545968	−0.272984	+	0.962019i	$0.588011\pi$
$8$	0	0
$9$	146.891	0.604491
$10$	0	0
$11$	−539.605	−1.34460	−0.672302	−	0.740277i	$-0.734694\pi$
$11$	−539.605	−1.34460	−0.672302	+	0.740277i	$0.734694\pi$
$12$	0	0
$13$	766.155	1.25736	0.628678	−	0.777665i	$-0.283596\pi$
$13$	766.155	1.25736	0.628678	+	0.777665i	$0.283596\pi$
$14$	0	0
$15$	213.640	0.245163
$16$	0	0
$17$	−703.216	−0.590155	−0.295078	−	0.955473i	$-0.595346\pi$
$17$	−703.216	−0.590155	−0.295078	+	0.955473i	$0.595346\pi$
$18$	0	0
$19$	−276.074	−0.175445	−0.0877224	−	0.996145i	$-0.527959\pi$
$19$	−276.074	−0.175445	−0.0877224	+	0.996145i	$0.527959\pi$
$20$	0	0
$21$	1397.60	0.691569
$22$	0	0
$23$	853.774	0.336530	0.168265	−	0.985742i	$-0.446184\pi$
$23$	853.774	0.336530	0.168265	+	0.985742i	$0.446184\pi$
$24$	0	0
$25$	−3007.94	−0.962540
$26$	0	0
$27$	1897.73	0.500986
$28$	0	0
$29$	−580.037	−0.128074	−0.0640369	−	0.997948i	$-0.520398\pi$
$29$	−580.037	−0.128074	−0.0640369	+	0.997948i	$0.520398\pi$
$30$	0	0
$31$	−2735.26	−0.511204	−0.255602	−	0.966782i	$-0.582274\pi$
$31$	−2735.26	−0.511204	−0.255602	+	0.966782i	$0.582274\pi$
$32$	0	0
$33$	10654.9	1.70319
$34$	0	0
$35$	765.812	0.105670
$36$	0	0
$37$	11448.6	1.37483	0.687413	−	0.726267i	$-0.258746\pi$
$37$	11448.6	1.37483	0.687413	+	0.726267i	$0.258746\pi$
$38$	0	0
$39$	−15128.2	−1.59267
$40$	0	0
$41$	−16864.0	−1.56676	−0.783379	−	0.621545i	$-0.786505\pi$
$41$	−16864.0	−1.56676	−0.783379	+	0.621545i	$0.786505\pi$
$42$	0	0
$43$	17528.2	1.44566	0.722829	−	0.691027i	$-0.242842\pi$
$43$	17528.2	1.44566	0.722829	+	0.691027i	$0.242842\pi$
$44$	0	0
$45$	−1589.30	−0.116997
$46$	0	0
$47$	23144.4	1.52827	0.764137	−	0.645054i	$-0.223165\pi$
$47$	23144.4	1.52827	0.764137	+	0.645054i	$0.223165\pi$
$48$	0	0
$49$	−11797.2	−0.701919
$50$	0	0
$51$	13885.5	0.747541
$52$	0	0
$53$	−1206.94	−0.0590196	−0.0295098	−	0.999564i	$-0.509395\pi$
$53$	−1206.94	−0.0590196	−0.0295098	+	0.999564i	$0.509395\pi$
$54$	0	0
$55$	5838.30	0.260243
$56$	0	0
$57$	5451.25	0.222233
$58$	0	0
$59$	33144.1	1.23959	0.619793	−	0.784766i	$-0.287217\pi$
$59$	33144.1	1.23959	0.619793	+	0.784766i	$0.287217\pi$
$60$	0	0
$61$	−11745.9	−0.404167	−0.202084	−	0.979368i	$-0.564771\pi$
$61$	−11745.9	−0.404167	−0.202084	+	0.979368i	$0.564771\pi$
$62$	0	0
$63$	−10397.0	−0.330032
$64$	0	0
$65$	−8289.48	−0.243357
$66$	0	0
$67$	1553.20	0.0422707	0.0211354	−	0.999777i	$-0.493272\pi$
$67$	1553.20	0.0422707	0.0211354	+	0.999777i	$0.493272\pi$
$68$	0	0
$69$	−16858.3	−0.426277
$70$	0	0
$71$	38797.1	0.913384	0.456692	−	0.889625i	$-0.349034\pi$
$71$	38797.1	0.913384	0.456692	+	0.889625i	$0.349034\pi$
$72$	0	0
$73$	44398.6	0.975129	0.487564	−	0.873087i	$-0.337885\pi$
$73$	44398.6	0.975129	0.487564	+	0.873087i	$0.337885\pi$
$74$	0	0
$75$	59393.7	1.21923
$76$	0	0
$77$	38193.4	0.734110
$78$	0	0
$79$	−18958.2	−0.341766	−0.170883	−	0.985291i	$-0.554662\pi$
$79$	−18958.2	−0.341766	−0.170883	+	0.985291i	$0.554662\pi$
$80$	0	0
$81$	−73166.5	−1.23908
$82$	0	0
$83$	−21986.5	−0.350317	−0.175159	−	0.984540i	$-0.556044\pi$
$83$	−21986.5	−0.350317	−0.175159	+	0.984540i	$0.556044\pi$
$84$	0	0
$85$	7608.50	0.114223
$86$	0	0
$87$	11453.2	0.162229
$88$	0	0
$89$	27961.7	0.374187	0.187093	−	0.982342i	$-0.440093\pi$
$89$	27961.7	0.374187	0.187093	+	0.982342i	$0.440093\pi$
$90$	0	0
$91$	−54228.7	−0.686476
$92$	0	0
$93$	54009.5	0.647534
$94$	0	0
$95$	2987.00	0.0339568
$96$	0	0
$97$	−56138.5	−0.605804	−0.302902	−	0.953022i	$-0.597955\pi$
$97$	−56138.5	−0.605804	−0.302902	+	0.953022i	$0.597955\pi$
$98$	0	0
$99$	−79263.2	−0.812800
$100$	0	0
$101$	82935.0	0.808974	0.404487	−	0.914544i	$-0.367450\pi$
$101$	82935.0	0.808974	0.404487	+	0.914544i	$0.367450\pi$
$102$	0	0
$103$	158210.	1.46940	0.734699	−	0.678393i	$-0.237323\pi$
$103$	158210.	1.46940	0.734699	+	0.678393i	$0.237323\pi$
$104$	0	0
$105$	−15121.5	−0.133851
$106$	0	0
$107$	166570.	1.40649	0.703246	−	0.710946i	$-0.251733\pi$
$107$	166570.	1.40649	0.703246	+	0.710946i	$0.251733\pi$
$108$	0	0
$109$	−133489.	−1.07616	−0.538082	−	0.842892i	$-0.680851\pi$
$109$	−133489.	−1.07616	−0.538082	+	0.842892i	$0.680851\pi$
$110$	0	0
$111$	−226060.	−1.74147
$112$	0	0
$113$	−37221.6	−0.274220	−0.137110	−	0.990556i	$-0.543781\pi$
$113$	−37221.6	−0.274220	−0.137110	+	0.990556i	$0.543781\pi$
$114$	0	0
$115$	−9237.48	−0.0651342
$116$	0	0
$117$	112541.	0.760060
$118$	0	0
$119$	49773.8	0.322206
$120$	0	0
$121$	130123.	0.807959
$122$	0	0
$123$	332991.	1.98459
$124$	0	0
$125$	66355.8	0.379843
$126$	0	0
$127$	225509.	1.24067	0.620334	−	0.784338i	$-0.286997\pi$
$127$	225509.	1.24067	0.620334	+	0.784338i	$0.286997\pi$
$128$	0	0
$129$	−346105.	−1.83119
$130$	0	0
$131$	253902.	1.29267	0.646335	−	0.763054i	$-0.276301\pi$
$131$	253902.	1.29267	0.646335	+	0.763054i	$0.276301\pi$
$132$	0	0
$133$	19540.5	0.0957872
$134$	0	0
$135$	−20532.7	−0.0969641
$136$	0	0
$137$	−265616.	−1.20907	−0.604537	−	0.796577i	$-0.706642\pi$
$137$	−265616.	−1.20907	−0.604537	+	0.796577i	$0.706642\pi$
$138$	0	0
$139$	−122167.	−0.536309	−0.268155	−	0.963376i	$-0.586414\pi$
$139$	−122167.	−0.536309	−0.268155	+	0.963376i	$0.586414\pi$
$140$	0	0
$141$	−457002.	−1.93584
$142$	0	0
$143$	−413421.	−1.69065
$144$	0	0
$145$	6275.75	0.0247882
$146$	0	0
$147$	232943.	0.889111
$148$	0	0
$149$	213602.	0.788206	0.394103	−	0.919066i	$-0.371055\pi$
$149$	213602.	0.788206	0.394103	+	0.919066i	$0.371055\pi$
$150$	0	0
$151$	214794.	0.766618	0.383309	−	0.923620i	$-0.374784\pi$
$151$	214794.	0.766618	0.383309	+	0.923620i	$0.374784\pi$
$152$	0	0
$153$	−103296.	−0.356743
$154$	0	0
$155$	29594.3	0.0989417
$156$	0	0
$157$	−359703.	−1.16465	−0.582325	−	0.812956i	$-0.697857\pi$
$157$	−359703.	−1.16465	−0.582325	+	0.812956i	$0.697857\pi$
$158$	0	0
$159$	23831.8	0.0747592
$160$	0	0
$161$	−60430.3	−0.183734
$162$	0	0
$163$	168711.	0.497364	0.248682	−	0.968585i	$-0.420003\pi$
$163$	168711.	0.497364	0.248682	+	0.968585i	$0.420003\pi$
$164$	0	0
$165$	−115281.	−0.329646
$166$	0	0
$167$	343781.	0.953873	0.476937	−	0.878938i	$-0.341747\pi$
$167$	343781.	0.953873	0.476937	+	0.878938i	$0.341747\pi$
$168$	0	0
$169$	215701.	0.580946
$170$	0	0
$171$	−40552.8	−0.106055
$172$	0	0
$173$	680803.	1.72944	0.864722	−	0.502251i	$-0.167495\pi$
$173$	680803.	1.72944	0.864722	+	0.502251i	$0.167495\pi$
$174$	0	0
$175$	212902.	0.525515
$176$	0	0
$177$	−654452.	−1.57016
$178$	0	0
$179$	−100857.	−0.235274	−0.117637	−	0.993057i	$-0.537532\pi$
$179$	−100857.	−0.235274	−0.117637	+	0.993057i	$0.537532\pi$
$180$	0	0
$181$	−827126.	−1.87661	−0.938307	−	0.345802i	$-0.887607\pi$
$181$	−827126.	−1.87661	−0.938307	+	0.345802i	$0.887607\pi$
$182$	0	0
$183$	231930.	0.511953
$184$	0	0
$185$	−123869.	−0.266093
$186$	0	0
$187$	379459.	0.793525
$188$	0	0
$189$	−134322.	−0.273522
$190$	0	0
$191$	−969792.	−1.92351	−0.961757	−	0.273906i	$-0.911684\pi$
$191$	−969792.	−1.92351	−0.961757	+	0.273906i	$0.911684\pi$
$192$	0	0
$193$	−587346.	−1.13501	−0.567506	−	0.823369i	$-0.692092\pi$
$193$	−587346.	−1.13501	−0.567506	+	0.823369i	$0.692092\pi$
$194$	0	0
$195$	163681.	0.308257
$196$	0	0
$197$	543194.	0.997217	0.498608	−	0.866827i	$-0.333845\pi$
$197$	543194.	0.997217	0.498608	+	0.866827i	$0.333845\pi$
$198$	0	0
$199$	−395481.	−0.707935	−0.353967	−	0.935258i	$-0.615168\pi$
$199$	−395481.	−0.707935	−0.353967	+	0.935258i	$0.615168\pi$
$200$	0	0
$201$	−30668.9	−0.0535437
$202$	0	0
$203$	41055.1	0.0699241
$204$	0	0
$205$	182462.	0.303240
$206$	0	0
$207$	125412.	0.203429
$208$	0	0
$209$	148971.	0.235904
$210$	0	0
$211$	−216576.	−0.334892	−0.167446	−	0.985881i	$-0.553552\pi$
$211$	−216576.	−0.334892	−0.167446	+	0.985881i	$0.553552\pi$
$212$	0	0
$213$	−766074.	−1.15697
$214$	0	0
$215$	−189647.	−0.279802
$216$	0	0
$217$	193602.	0.279101
$218$	0	0
$219$	−876680.	−1.23518
$220$	0	0
$221$	−538773.	−0.742036
$222$	0	0
$223$	−1.42492e6	−1.91880	−0.959398	−	0.282054i	$-0.908984\pi$
$223$	−1.42492e6	−1.91880	−0.959398	+	0.282054i	$0.908984\pi$
$224$	0	0
$225$	−441839.	−0.581846
$226$	0	0
$227$	−366177.	−0.471657	−0.235829	−	0.971795i	$-0.575780\pi$
$227$	−366177.	−0.471657	−0.235829	+	0.971795i	$0.575780\pi$
$228$	0	0
$229$	−1.07193e6	−1.35076	−0.675380	−	0.737470i	$-0.736020\pi$
$229$	−1.07193e6	−1.35076	−0.675380	+	0.737470i	$0.736020\pi$
$230$	0	0
$231$	−754153.	−0.929886
$232$	0	0
$233$	662541.	0.799508	0.399754	−	0.916623i	$-0.369096\pi$
$233$	662541.	0.799508	0.399754	+	0.916623i	$0.369096\pi$
$234$	0	0
$235$	−250413.	−0.295792
$236$	0	0
$237$	374342.	0.432910
$238$	0	0
$239$	−1.61634e6	−1.83037	−0.915183	−	0.403038i	$-0.867954\pi$
$239$	−1.61634e6	−1.83037	−0.915183	+	0.403038i	$0.867954\pi$
$240$	0	0
$241$	918061.	1.01819	0.509095	−	0.860710i	$-0.329980\pi$
$241$	918061.	1.01819	0.509095	+	0.860710i	$0.329980\pi$
$242$	0	0
$243$	983573.	1.06854
$244$	0	0
$245$	127640.	0.135854
$246$	0	0
$247$	−211515.	−0.220597
$248$	0	0
$249$	434139.	0.443742
$250$	0	0
$251$	−381379.	−0.382096	−0.191048	−	0.981581i	$-0.561189\pi$
$251$	−381379.	−0.382096	−0.191048	+	0.981581i	$0.561189\pi$
$252$	0	0
$253$	−460701.	−0.452499
$254$	0	0
$255$	−150235.	−0.144684
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	−810333.	−0.750610
$260$	0	0
$261$	−85202.3	−0.0774194
$262$	0	0
$263$	936197.	0.834599	0.417300	−	0.908769i	$-0.362977\pi$
$263$	936197.	0.834599	0.417300	+	0.908769i	$0.362977\pi$
$264$	0	0
$265$	13058.6	0.0114230
$266$	0	0
$267$	−552122.	−0.473977
$268$	0	0
$269$	−1.39746e6	−1.17750	−0.588748	−	0.808317i	$-0.700379\pi$
$269$	−1.39746e6	−1.17750	−0.588748	+	0.808317i	$0.700379\pi$
$270$	0	0
$271$	93411.5	0.0772640	0.0386320	−	0.999254i	$-0.487700\pi$
$271$	93411.5	0.0772640	0.0386320	+	0.999254i	$0.487700\pi$
$272$	0	0
$273$	1.07078e6	0.869549
$274$	0	0
$275$	1.62310e6	1.29423
$276$	0	0
$277$	−96426.2	−0.0755084	−0.0377542	−	0.999287i	$-0.512020\pi$
$277$	−96426.2	−0.0755084	−0.0377542	+	0.999287i	$0.512020\pi$
$278$	0	0
$279$	−401785.	−0.309018
$280$	0	0
$281$	1.81958e6	1.37469	0.687346	−	0.726331i	$-0.258776\pi$
$281$	1.81958e6	1.37469	0.687346	+	0.726331i	$0.258776\pi$
$282$	0	0
$283$	−2.07582e6	−1.54072	−0.770360	−	0.637609i	$-0.779924\pi$
$283$	−2.07582e6	−1.54072	−0.770360	+	0.637609i	$0.779924\pi$
$284$	0	0
$285$	−58980.3	−0.0430125
$286$	0	0
$287$	1.19364e6	0.855399
$288$	0	0
$289$	−925345.	−0.651717
$290$	0	0
$291$	1.10849e6	0.767362
$292$	0	0
$293$	−171742.	−0.116871	−0.0584357	−	0.998291i	$-0.518611\pi$
$293$	−171742.	−0.116871	−0.0584357	+	0.998291i	$0.518611\pi$
$294$	0	0
$295$	−358605.	−0.239917
$296$	0	0
$297$	−1.02403e6	−0.673627
$298$	0	0
$299$	654124.	0.423138
$300$	0	0
$301$	−1.24065e6	−0.789282
$302$	0	0
$303$	−1.63761e6	−1.02471
$304$	0	0
$305$	127086.	0.0782252
$306$	0	0
$307$	−1.63456e6	−0.989816	−0.494908	−	0.868945i	$-0.664798\pi$
$307$	−1.63456e6	−0.989816	−0.494908	+	0.868945i	$0.664798\pi$
$308$	0	0
$309$	−3.12395e6	−1.86126
$310$	0	0
$311$	−2.96942e6	−1.74089	−0.870443	−	0.492270i	$-0.836167\pi$
$311$	−2.96942e6	−1.74089	−0.870443	+	0.492270i	$0.836167\pi$
$312$	0	0
$313$	−1.82433e6	−1.05255	−0.526275	−	0.850314i	$-0.676412\pi$
$313$	−1.82433e6	−1.05255	−0.526275	+	0.850314i	$0.676412\pi$
$314$	0	0
$315$	112491.	0.0638766
$316$	0	0
$317$	−1.11069e6	−0.620792	−0.310396	−	0.950607i	$-0.600462\pi$
$317$	−1.11069e6	−0.620792	−0.310396	+	0.950607i	$0.600462\pi$
$318$	0	0
$319$	312991.	0.172209
$320$	0	0
$321$	−3.28903e6	−1.78158
$322$	0	0
$323$	194139.	0.103540
$324$	0	0
$325$	−2.30455e6	−1.21026
$326$	0	0
$327$	2.63582e6	1.36316
$328$	0	0
$329$	−1.63817e6	−0.834388
$330$	0	0
$331$	−3.11516e6	−1.56283	−0.781413	−	0.624015i	$-0.785500\pi$
$331$	−3.11516e6	−1.56283	−0.781413	+	0.624015i	$0.785500\pi$
$332$	0	0
$333$	1.68170e6	0.831069
$334$	0	0
$335$	−16805.0	−0.00818136
$336$	0	0
$337$	926933.	0.444604	0.222302	−	0.974978i	$-0.428643\pi$
$337$	926933.	0.444604	0.222302	+	0.974978i	$0.428643\pi$
$338$	0	0
$339$	734966.	0.347350
$340$	0	0
$341$	1.47596e6	0.687366
$342$	0	0
$343$	2.02461e6	0.929193
$344$	0	0
$345$	182400.	0.0825045
$346$	0	0
$347$	2.03678e6	0.908072	0.454036	−	0.890983i	$-0.349984\pi$
$347$	2.03678e6	0.908072	0.454036	+	0.890983i	$0.349984\pi$
$348$	0	0
$349$	−3.48633e6	−1.53216	−0.766082	−	0.642742i	$-0.777797\pi$
$349$	−3.48633e6	−1.53216	−0.766082	+	0.642742i	$0.777797\pi$
$350$	0	0
$351$	1.45396e6	0.629918
$352$	0	0
$353$	2.38081e6	1.01692	0.508462	−	0.861084i	$-0.330214\pi$
$353$	2.38081e6	1.01692	0.508462	+	0.861084i	$0.330214\pi$
$354$	0	0
$355$	−419768.	−0.176782
$356$	0	0
$357$	−982816.	−0.408133
$358$	0	0
$359$	−2.39130e6	−0.979261	−0.489631	−	0.871930i	$-0.662868\pi$
$359$	−2.39130e6	−0.979261	−0.489631	+	0.871930i	$0.662868\pi$
$360$	0	0
$361$	−2.39988e6	−0.969219
$362$	0	0
$363$	−2.56936e6	−1.02343
$364$	0	0
$365$	−480374.	−0.188733
$366$	0	0
$367$	2.72904e6	1.05766	0.528829	−	0.848729i	$-0.322631\pi$
$367$	2.72904e6	1.05766	0.528829	+	0.848729i	$0.322631\pi$
$368$	0	0
$369$	−2.47718e6	−0.947090
$370$	0	0
$371$	85427.5	0.0322228
$372$	0	0
$373$	−1.13720e6	−0.423218	−0.211609	−	0.977354i	$-0.567870\pi$
$373$	−1.13720e6	−0.423218	−0.211609	+	0.977354i	$0.567870\pi$
$374$	0	0
$375$	−1.31024e6	−0.481141
$376$	0	0
$377$	−444398.	−0.161034
$378$	0	0
$379$	2.10962e6	0.754409	0.377205	−	0.926130i	$-0.376885\pi$
$379$	2.10962e6	0.754409	0.377205	+	0.926130i	$0.376885\pi$
$380$	0	0
$381$	−4.45283e6	−1.57153
$382$	0	0
$383$	−4.16532e6	−1.45095	−0.725473	−	0.688251i	$-0.758379\pi$
$383$	−4.16532e6	−1.45095	−0.725473	+	0.688251i	$0.758379\pi$
$384$	0	0
$385$	−413236.	−0.142084
$386$	0	0
$387$	2.57473e6	0.873886
$388$	0	0
$389$	2.01686e6	0.675773	0.337887	−	0.941187i	$-0.390288\pi$
$389$	2.01686e6	0.675773	0.337887	+	0.941187i	$0.390288\pi$
$390$	0	0
$391$	−600388.	−0.198605
$392$	0	0
$393$	−5.01346e6	−1.63741
$394$	0	0
$395$	205120.	0.0661476
$396$	0	0
$397$	3.22674e6	1.02751	0.513757	−	0.857936i	$-0.328253\pi$
$397$	3.22674e6	1.02751	0.513757	+	0.857936i	$0.328253\pi$
$398$	0	0
$399$	−385841.	−0.121332
$400$	0	0
$401$	2.36047e6	0.733056	0.366528	−	0.930407i	$-0.380546\pi$
$401$	2.36047e6	0.733056	0.366528	+	0.930407i	$0.380546\pi$
$402$	0	0
$403$	−2.09563e6	−0.642765
$404$	0	0
$405$	791631.	0.239820
$406$	0	0
$407$	−6.17771e6	−1.84859
$408$	0	0
$409$	6.58421e6	1.94624	0.973118	−	0.230309i	$-0.0739737\pi$
$409$	6.58421e6	1.94624	0.973118	+	0.230309i	$0.0739737\pi$
$410$	0	0
$411$	5.24477e6	1.53152
$412$	0	0
$413$	−2.34595e6	−0.676773
$414$	0	0
$415$	237885.	0.0678027
$416$	0	0
$417$	2.41226e6	0.679335
$418$	0	0
$419$	288809.	0.0803667	0.0401834	−	0.999192i	$-0.487206\pi$
$419$	288809.	0.0803667	0.0401834	+	0.999192i	$0.487206\pi$
$420$	0	0
$421$	−947171.	−0.260449	−0.130225	−	0.991485i	$-0.541570\pi$
$421$	−947171.	−0.260449	−0.130225	+	0.991485i	$0.541570\pi$
$422$	0	0
$423$	3.39971e6	0.923828
$424$	0	0
$425$	2.11523e6	0.568048
$426$	0	0
$427$	831377.	0.220662
$428$	0	0
$429$	8.16328e6	2.14152
$430$	0	0
$431$	3.69053e6	0.956965	0.478483	−	0.878097i	$-0.341187\pi$
$431$	3.69053e6	0.956965	0.478483	+	0.878097i	$0.341187\pi$
$432$	0	0
$433$	3.11844e6	0.799315	0.399657	−	0.916665i	$-0.369129\pi$
$433$	3.11844e6	0.799315	0.399657	+	0.916665i	$0.369129\pi$
$434$	0	0
$435$	−123919.	−0.0313989
$436$	0	0
$437$	−235705.	−0.0590424
$438$	0	0
$439$	5.65229e6	1.39979	0.699896	−	0.714245i	$-0.253230\pi$
$439$	5.65229e6	1.39979	0.699896	+	0.714245i	$0.253230\pi$
$440$	0	0
$441$	−1.73290e6	−0.424304
$442$	0	0
$443$	−5.64247e6	−1.36603	−0.683014	−	0.730405i	$-0.739331\pi$
$443$	−5.64247e6	−1.36603	−0.683014	+	0.730405i	$0.739331\pi$
$444$	0	0
$445$	−302534.	−0.0724226
$446$	0	0
$447$	−4.21771e6	−0.998408
$448$	0	0
$449$	3.41036e6	0.798333	0.399166	−	0.916879i	$-0.369300\pi$
$449$	3.41036e6	0.798333	0.399166	+	0.916879i	$0.369300\pi$
$450$	0	0
$451$	9.09991e6	2.10667
$452$	0	0
$453$	−4.24124e6	−0.971064
$454$	0	0
$455$	586731.	0.132865
$456$	0	0
$457$	6.20942e6	1.39079	0.695393	−	0.718630i	$-0.255230\pi$
$457$	6.20942e6	1.39079	0.695393	+	0.718630i	$0.255230\pi$
$458$	0	0
$459$	−1.33451e6	−0.295659
$460$	0	0
$461$	−3.87995e6	−0.850302	−0.425151	−	0.905122i	$-0.639779\pi$
$461$	−3.87995e6	−0.850302	−0.425151	+	0.905122i	$0.639779\pi$
$462$	0	0
$463$	557296.	0.120819	0.0604093	−	0.998174i	$-0.480759\pi$
$463$	557296.	0.120819	0.0604093	+	0.998174i	$0.480759\pi$
$464$	0	0
$465$	−584360.	−0.125328
$466$	0	0
$467$	5.10781e6	1.08378	0.541891	−	0.840448i	$-0.317708\pi$
$467$	5.10781e6	1.08378	0.541891	+	0.840448i	$0.317708\pi$
$468$	0	0
$469$	−109936.	−0.0230784
$470$	0	0
$471$	7.10258e6	1.47524
$472$	0	0
$473$	−9.45828e6	−1.94384
$474$	0	0
$475$	830412.	0.168873
$476$	0	0
$477$	−177289.	−0.0356768
$478$	0	0
$479$	3.99686e6	0.795940	0.397970	−	0.917398i	$-0.369715\pi$
$479$	3.99686e6	0.795940	0.397970	+	0.917398i	$0.369715\pi$
$480$	0	0
$481$	8.77140e6	1.72865
$482$	0	0
$483$	1.19324e6	0.232734
$484$	0	0
$485$	607396.	0.117251
$486$	0	0
$487$	110937.	0.0211961	0.0105980	−	0.999944i	$-0.496626\pi$
$487$	110937.	0.0211961	0.0105980	+	0.999944i	$0.496626\pi$
$488$	0	0
$489$	−3.33131e6	−0.630004
$490$	0	0
$491$	2.54491e6	0.476397	0.238198	−	0.971217i	$-0.423443\pi$
$491$	2.54491e6	0.476397	0.238198	+	0.971217i	$0.423443\pi$
$492$	0	0
$493$	407891.	0.0755834
$494$	0	0
$495$	857595.	0.157315
$496$	0	0
$497$	−2.74607e6	−0.498678
$498$	0	0
$499$	5.94853e6	1.06944	0.534722	−	0.845028i	$-0.320416\pi$
$499$	5.94853e6	1.06944	0.534722	+	0.845028i	$0.320416\pi$
$500$	0	0
$501$	−6.78818e6	−1.20826
$502$	0	0
$503$	−1.96882e6	−0.346966	−0.173483	−	0.984837i	$-0.555502\pi$
$503$	−1.96882e6	−0.346966	−0.173483	+	0.984837i	$0.555502\pi$
$504$	0	0
$505$	−897322.	−0.156574
$506$	0	0
$507$	−4.25916e6	−0.735875
$508$	0	0
$509$	−3.28043e6	−0.561224	−0.280612	−	0.959821i	$-0.590537\pi$
$509$	−3.28043e6	−0.561224	−0.280612	+	0.959821i	$0.590537\pi$
$510$	0	0
$511$	−3.14254e6	−0.532389
$512$	0	0
$513$	−523914.	−0.0878954
$514$	0	0
$515$	−1.71176e6	−0.284397
$516$	0	0
$517$	−1.24888e7	−2.05492
$518$	0	0
$519$	−1.34429e7	−2.19066
$520$	0	0
$521$	−2.03028e6	−0.327689	−0.163844	−	0.986486i	$-0.552389\pi$
$521$	−2.03028e6	−0.327689	−0.163844	+	0.986486i	$0.552389\pi$
$522$	0	0
$523$	−3466.36	−0.000554139 0	−0.000277070	−	1.00000i	$-0.500088\pi$
$523$	−3466.36	−0.000554139 0	−0.000277070		1.00000i	$0.500088\pi$
$524$	0	0
$525$	−4.20390e6	−0.665662
$526$	0	0
$527$	1.92348e6	0.301689
$528$	0	0
$529$	−5.70741e6	−0.886748
$530$	0	0
$531$	4.86858e6	0.749317
$532$	0	0
$533$	−1.29205e7	−1.96997
$534$	0	0
$535$	−1.80222e6	−0.272222
$536$	0	0
$537$	1.99149e6	0.298018
$538$	0	0
$539$	6.36581e6	0.943803
$540$	0	0
$541$	7.15184e6	1.05057	0.525285	−	0.850927i	$-0.323959\pi$
$541$	7.15184e6	1.05057	0.525285	+	0.850927i	$0.323959\pi$
$542$	0	0
$543$	1.63321e7	2.37708
$544$	0	0
$545$	1.44429e6	0.208288
$546$	0	0
$547$	−2.77555e6	−0.396625	−0.198313	−	0.980139i	$-0.563546\pi$
$547$	−2.77555e6	−0.396625	−0.198313	+	0.980139i	$0.563546\pi$
$548$	0	0
$549$	−1.72537e6	−0.244315
$550$	0	0
$551$	160133.	0.0224699
$552$	0	0
$553$	1.34186e6	0.186593
$554$	0	0
$555$	2.44587e6	0.337056
$556$	0	0
$557$	−2.42083e6	−0.330618	−0.165309	−	0.986242i	$-0.552862\pi$
$557$	−2.42083e6	−0.330618	−0.165309	+	0.986242i	$0.552862\pi$
$558$	0	0
$559$	1.34293e7	1.81771
$560$	0	0
$561$	−7.49266e6	−1.00515
$562$	0	0
$563$	−9.18065e6	−1.22068	−0.610341	−	0.792139i	$-0.708968\pi$
$563$	−9.18065e6	−1.22068	−0.610341	+	0.792139i	$0.708968\pi$
$564$	0	0
$565$	402723.	0.0530744
$566$	0	0
$567$	5.17874e6	0.676498
$568$	0	0
$569$	−1.10924e7	−1.43629	−0.718147	−	0.695891i	$-0.755010\pi$
$569$	−1.10924e7	−1.43629	−0.718147	+	0.695891i	$0.755010\pi$
$570$	0	0
$571$	−4.75867e6	−0.610795	−0.305398	−	0.952225i	$-0.598789\pi$
$571$	−4.75867e6	−0.610795	−0.305398	+	0.952225i	$0.598789\pi$
$572$	0	0
$573$	1.91492e7	2.43649
$574$	0	0
$575$	−2.56810e6	−0.323923
$576$	0	0
$577$	−8.53778e6	−1.06759	−0.533796	−	0.845613i	$-0.679235\pi$
$577$	−8.53778e6	−1.06759	−0.533796	+	0.845613i	$0.679235\pi$
$578$	0	0
$579$	1.15975e7	1.43770
$580$	0	0
$581$	1.55621e6	0.191262
$582$	0	0
$583$	651271.	0.0793580
$584$	0	0
$585$	−1.21765e6	−0.147107
$586$	0	0
$587$	−7.21385e6	−0.864116	−0.432058	−	0.901846i	$-0.642212\pi$
$587$	−7.21385e6	−0.864116	−0.432058	+	0.901846i	$0.642212\pi$
$588$	0	0
$589$	755132.	0.0896881
$590$	0	0
$591$	−1.07257e7	−1.26316
$592$	0	0
$593$	−1.40561e7	−1.64145	−0.820727	−	0.571321i	$-0.806431\pi$
$593$	−1.40561e7	−1.64145	−0.820727	+	0.571321i	$0.806431\pi$
$594$	0	0
$595$	−538531.	−0.0623618
$596$	0	0
$597$	7.80904e6	0.896730
$598$	0	0
$599$	8.87231e6	1.01035	0.505173	−	0.863018i	$-0.331429\pi$
$599$	8.87231e6	1.01035	0.505173	+	0.863018i	$0.331429\pi$
$600$	0	0
$601$	−4.64997e6	−0.525127	−0.262564	−	0.964915i	$-0.584568\pi$
$601$	−4.64997e6	−0.525127	−0.262564	+	0.964915i	$0.584568\pi$
$602$	0	0
$603$	228151.	0.0255523
$604$	0	0
$605$	−1.40787e6	−0.156378
$606$	0	0
$607$	1.07004e7	1.17877	0.589385	−	0.807852i	$-0.299370\pi$
$607$	1.07004e7	1.17877	0.589385	+	0.807852i	$0.299370\pi$
$608$	0	0
$609$	−810660.	−0.0885719
$610$	0	0
$611$	1.77322e7	1.92159
$612$	0	0
$613$	−1.04785e7	−1.12628	−0.563141	−	0.826361i	$-0.690407\pi$
$613$	−1.04785e7	−1.12628	−0.563141	+	0.826361i	$0.690407\pi$
$614$	0	0
$615$	−3.60283e6	−0.384110
$616$	0	0
$617$	4.20652e6	0.444846	0.222423	−	0.974950i	$-0.428603\pi$
$617$	4.20652e6	0.444846	0.222423	+	0.974950i	$0.428603\pi$
$618$	0	0
$619$	1.28913e7	1.35229	0.676146	−	0.736767i	$-0.263649\pi$
$619$	1.28913e7	1.35229	0.676146	+	0.736767i	$0.263649\pi$
$620$	0	0
$621$	1.62024e6	0.168597
$622$	0	0
$623$	−1.97914e6	−0.204294
$624$	0	0
$625$	8.68186e6	0.889022
$626$	0	0
$627$	−2.94152e6	−0.298816
$628$	0	0
$629$	−8.05082e6	−0.811360
$630$	0	0
$631$	−1.67725e7	−1.67697	−0.838483	−	0.544927i	$-0.816557\pi$
$631$	−1.67725e7	−1.67697	−0.838483	+	0.544927i	$0.816557\pi$
$632$	0	0
$633$	4.27644e6	0.424202
$634$	0	0
$635$	−2.43992e6	−0.240127
$636$	0	0
$637$	−9.03846e6	−0.882563
$638$	0	0
$639$	5.69895e6	0.552132
$640$	0	0
$641$	1.33885e7	1.28703	0.643515	−	0.765434i	$-0.277476\pi$
$641$	1.33885e7	1.28703	0.643515	+	0.765434i	$0.277476\pi$
$642$	0	0
$643$	−8.29807e6	−0.791497	−0.395749	−	0.918359i	$-0.629515\pi$
$643$	−8.29807e6	−0.791497	−0.395749	+	0.918359i	$0.629515\pi$
$644$	0	0
$645$	3.74471e6	0.354421
$646$	0	0
$647$	8.49248e6	0.797579	0.398790	−	0.917042i	$-0.369430\pi$
$647$	8.49248e6	0.797579	0.398790	+	0.917042i	$0.369430\pi$
$648$	0	0
$649$	−1.78847e7	−1.66675
$650$	0	0
$651$	−3.82280e6	−0.353532
$652$	0	0
$653$	−3.09241e6	−0.283801	−0.141900	−	0.989881i	$-0.545321\pi$
$653$	−3.09241e6	−0.283801	−0.141900	+	0.989881i	$0.545321\pi$
$654$	0	0
$655$	−2.74711e6	−0.250192
$656$	0	0
$657$	6.52176e6	0.589456
$658$	0	0
$659$	−7.20637e6	−0.646402	−0.323201	−	0.946330i	$-0.604759\pi$
$659$	−7.20637e6	−0.646402	−0.323201	+	0.946330i	$0.604759\pi$
$660$	0	0
$661$	−1.25775e7	−1.11968	−0.559838	−	0.828602i	$-0.689137\pi$
$661$	−1.25775e7	−1.11968	−0.559838	+	0.828602i	$0.689137\pi$
$662$	0	0
$663$	1.06384e7	0.939925
$664$	0	0
$665$	−211421.	−0.0185393
$666$	0	0
$667$	−495220.	−0.0431007
$668$	0	0
$669$	2.81360e7	2.43051
$670$	0	0
$671$	6.33814e6	0.543445
$672$	0	0
$673$	1.03974e7	0.884888	0.442444	−	0.896796i	$-0.354111\pi$
$673$	1.03974e7	0.884888	0.442444	+	0.896796i	$0.354111\pi$
$674$	0	0
$675$	−5.70826e6	−0.482219
$676$	0	0
$677$	491250.	0.0411937	0.0205968	−	0.999788i	$-0.493443\pi$
$677$	491250.	0.0411937	0.0205968	+	0.999788i	$0.493443\pi$
$678$	0	0
$679$	3.97350e6	0.330749
$680$	0	0
$681$	7.23041e6	0.597441
$682$	0	0
$683$	−3.02325e6	−0.247983	−0.123991	−	0.992283i	$-0.539570\pi$
$683$	−3.02325e6	−0.247983	−0.123991	+	0.992283i	$0.539570\pi$
$684$	0	0
$685$	2.87386e6	0.234012
$686$	0	0
$687$	2.11660e7	1.71099
$688$	0	0
$689$	−924704.	−0.0742087
$690$	0	0
$691$	−1.20245e7	−0.958012	−0.479006	−	0.877812i	$-0.659003\pi$
$691$	−1.20245e7	−0.958012	−0.479006	+	0.877812i	$0.659003\pi$
$692$	0	0
$693$	5.61027e6	0.443762
$694$	0	0
$695$	1.32179e6	0.103801
$696$	0	0
$697$	1.18590e7	0.924630
$698$	0	0
$699$	−1.30823e7	−1.01272
$700$	0	0
$701$	2.02748e7	1.55834	0.779168	−	0.626815i	$-0.215642\pi$
$701$	2.02748e7	1.55834	0.779168	+	0.626815i	$0.215642\pi$
$702$	0	0
$703$	−3.16065e6	−0.241206
$704$	0	0
$705$	4.94457e6	0.374676
$706$	0	0
$707$	−5.87016e6	−0.441673
$708$	0	0
$709$	−6.00732e6	−0.448813	−0.224407	−	0.974496i	$-0.572044\pi$
$709$	−6.00732e6	−0.448813	−0.224407	+	0.974496i	$0.572044\pi$
$710$	0	0
$711$	−2.78479e6	−0.206594
$712$	0	0
$713$	−2.33529e6	−0.172035
$714$	0	0
$715$	4.47305e6	0.327219
$716$	0	0
$717$	3.19157e7	2.31850
$718$	0	0
$719$	1.12861e7	0.814182	0.407091	−	0.913388i	$-0.366543\pi$
$719$	1.12861e7	0.814182	0.407091	+	0.913388i	$0.366543\pi$
$720$	0	0
$721$	−1.11981e7	−0.802244
$722$	0	0
$723$	−1.81277e7	−1.28973
$724$	0	0
$725$	1.74471e6	0.123276
$726$	0	0
$727$	7.33010e6	0.514368	0.257184	−	0.966362i	$-0.417205\pi$
$727$	7.33010e6	0.514368	0.257184	+	0.966362i	$0.417205\pi$
$728$	0	0
$729$	−1.64183e6	−0.114422
$730$	0	0
$731$	−1.23261e7	−0.853162
$732$	0	0
$733$	2.19949e7	1.51204	0.756018	−	0.654551i	$-0.227142\pi$
$733$	2.19949e7	1.51204	0.756018	+	0.654551i	$0.227142\pi$
$734$	0	0
$735$	−2.52034e6	−0.172084
$736$	0	0
$737$	−838113.	−0.0568374
$738$	0	0
$739$	9.37986e6	0.631808	0.315904	−	0.948791i	$-0.397692\pi$
$739$	9.37986e6	0.631808	0.315904	+	0.948791i	$0.397692\pi$
$740$	0	0
$741$	4.17651e6	0.279427
$742$	0	0
$743$	−6.43778e6	−0.427823	−0.213911	−	0.976853i	$-0.568620\pi$
$743$	−6.43778e6	−0.427823	−0.213911	+	0.976853i	$0.568620\pi$
$744$	0	0
$745$	−2.31108e6	−0.152554
$746$	0	0
$747$	−3.22963e6	−0.211763
$748$	0	0
$749$	−1.17899e7	−0.767899
$750$	0	0
$751$	−4.90139e6	−0.317117	−0.158559	−	0.987350i	$-0.550685\pi$
$751$	−4.90139e6	−0.317117	−0.158559	+	0.987350i	$0.550685\pi$
$752$	0	0
$753$	7.53057e6	0.483995
$754$	0	0
$755$	−2.32398e6	−0.148376
$756$	0	0
$757$	−1.36991e7	−0.868867	−0.434433	−	0.900704i	$-0.643051\pi$
$757$	−1.36991e7	−0.868867	−0.434433	+	0.900704i	$0.643051\pi$
$758$	0	0
$759$	9.09685e6	0.573174
$760$	0	0
$761$	1.04112e7	0.651685	0.325843	−	0.945424i	$-0.394352\pi$
$761$	1.04112e7	0.651685	0.325843	+	0.945424i	$0.394352\pi$
$762$	0	0
$763$	9.44837e6	0.587551
$764$	0	0
$765$	1.11762e6	0.0690464
$766$	0	0
$767$	2.53935e7	1.55860
$768$	0	0
$769$	−2.99862e7	−1.82855	−0.914274	−	0.405097i	$-0.867238\pi$
$769$	−2.99862e7	−1.82855	−0.914274	+	0.405097i	$0.867238\pi$
$770$	0	0
$771$	−1.30418e6	−0.0790136
$772$	0	0
$773$	−1.00067e7	−0.602340	−0.301170	−	0.953570i	$-0.597377\pi$
$773$	−1.00067e7	−0.602340	−0.301170	+	0.953570i	$0.597377\pi$
$774$	0	0
$775$	8.22748e6	0.492054
$776$	0	0
$777$	1.60006e7	0.950786
$778$	0	0
$779$	4.65571e6	0.274880
$780$	0	0
$781$	−2.09351e7	−1.22814
$782$	0	0
$783$	−1.10075e6	−0.0641632
$784$	0	0
$785$	3.89184e6	0.225414
$786$	0	0
$787$	1.46008e7	0.840309	0.420155	−	0.907453i	$-0.361976\pi$
$787$	1.46008e7	0.840309	0.420155	+	0.907453i	$0.361976\pi$
$788$	0	0
$789$	−1.84858e7	−1.05717
$790$	0	0
$791$	2.63456e6	0.149715
$792$	0	0
$793$	−8.99918e6	−0.508183
$794$	0	0
$795$	−257851.	−0.0144694
$796$	0	0
$797$	−2.82663e7	−1.57624	−0.788122	−	0.615519i	$-0.788946\pi$
$797$	−2.82663e7	−1.57624	−0.788122	+	0.615519i	$0.788946\pi$
$798$	0	0
$799$	−1.62755e7	−0.901919
$800$	0	0
$801$	4.10733e6	0.226192
$802$	0	0
$803$	−2.39577e7	−1.31116
$804$	0	0
$805$	653831.	0.0355612
$806$	0	0
$807$	2.75938e7	1.49152
$808$	0	0
$809$	2.82039e7	1.51509	0.757543	−	0.652785i	$-0.226399\pi$
$809$	2.82039e7	1.51509	0.757543	+	0.652785i	$0.226399\pi$
$810$	0	0
$811$	−9.59042e6	−0.512018	−0.256009	−	0.966674i	$-0.582408\pi$
$811$	−9.59042e6	−0.512018	−0.256009	+	0.966674i	$0.582408\pi$
$812$	0	0
$813$	−1.84447e6	−0.0978691
$814$	0	0
$815$	−1.82538e6	−0.0962632
$816$	0	0
$817$	−4.83906e6	−0.253633
$818$	0	0
$819$	−7.96571e6	−0.414968
$820$	0	0
$821$	−2.94194e7	−1.52327	−0.761635	−	0.648007i	$-0.775603\pi$
$821$	−2.94194e7	−1.52327	−0.761635	+	0.648007i	$0.775603\pi$
$822$	0	0
$823$	1.73844e7	0.894666	0.447333	−	0.894367i	$-0.352374\pi$
$823$	1.73844e7	0.894666	0.447333	+	0.894367i	$0.352374\pi$
$824$	0	0
$825$	−3.20491e7	−1.63939
$826$	0	0
$827$	6.67799e6	0.339533	0.169766	−	0.985484i	$-0.445699\pi$
$827$	6.67799e6	0.339533	0.169766	+	0.985484i	$0.445699\pi$
$828$	0	0
$829$	2.45049e7	1.23842	0.619209	−	0.785226i	$-0.287453\pi$
$829$	2.45049e7	1.23842	0.619209	+	0.785226i	$0.287453\pi$
$830$	0	0
$831$	1.90400e6	0.0956454
$832$	0	0
$833$	8.29595e6	0.414241
$834$	0	0
$835$	−3.71957e6	−0.184619
$836$	0	0
$837$	−5.19078e6	−0.256106
$838$	0	0
$839$	2.52546e7	1.23861	0.619305	−	0.785150i	$-0.287414\pi$
$839$	2.52546e7	1.23861	0.619305	+	0.785150i	$0.287414\pi$
$840$	0	0
$841$	−2.01747e7	−0.983597
$842$	0	0
$843$	−3.59288e7	−1.74130
$844$	0	0
$845$	−2.33380e6	−0.112440
$846$	0	0
$847$	−9.21010e6	−0.441119
$848$	0	0
$849$	4.09885e7	1.95161
$850$	0	0
$851$	9.77451e6	0.462670
$852$	0	0
$853$	3.99114e7	1.87813	0.939063	−	0.343744i	$-0.111695\pi$
$853$	3.99114e7	1.87813	0.939063	+	0.343744i	$0.111695\pi$
$854$	0	0
$855$	438764.	0.0205265
$856$	0	0
$857$	3.36076e7	1.56309	0.781547	−	0.623846i	$-0.214431\pi$
$857$	3.36076e7	1.56309	0.781547	+	0.623846i	$0.214431\pi$
$858$	0	0
$859$	−9.08252e6	−0.419975	−0.209987	−	0.977704i	$-0.567342\pi$
$859$	−9.08252e6	−0.419975	−0.209987	+	0.977704i	$0.567342\pi$
$860$	0	0
$861$	−2.35692e7	−1.08352
$862$	0	0
$863$	−1.17795e7	−0.538393	−0.269196	−	0.963085i	$-0.586758\pi$
$863$	−1.17795e7	−0.538393	−0.269196	+	0.963085i	$0.586758\pi$
$864$	0	0
$865$	−7.36601e6	−0.334728
$866$	0	0
$867$	1.82715e7	0.825520
$868$	0	0
$869$	1.02299e7	0.459540
$870$	0	0
$871$	1.18999e6	0.0531494
$872$	0	0
$873$	−8.24626e6	−0.366203
$874$	0	0
$875$	−4.69668e6	−0.207382
$876$	0	0
$877$	4.92701e6	0.216314	0.108157	−	0.994134i	$-0.465505\pi$
$877$	4.92701e6	0.216314	0.108157	+	0.994134i	$0.465505\pi$
$878$	0	0
$879$	3.39117e6	0.148039
$880$	0	0
$881$	9.28193e6	0.402901	0.201451	−	0.979499i	$-0.435434\pi$
$881$	9.28193e6	0.402901	0.201451	+	0.979499i	$0.435434\pi$
$882$	0	0
$883$	3.44745e7	1.48798	0.743988	−	0.668193i	$-0.232932\pi$
$883$	3.44745e7	1.48798	0.743988	+	0.668193i	$0.232932\pi$
$884$	0	0
$885$	7.08090e6	0.303900
$886$	0	0
$887$	−3.57428e7	−1.52538	−0.762692	−	0.646762i	$-0.776123\pi$
$887$	−3.57428e7	−1.52538	−0.762692	+	0.646762i	$0.776123\pi$
$888$	0	0
$889$	−1.59616e7	−0.677364
$890$	0	0
$891$	3.94810e7	1.66607
$892$	0	0
$893$	−6.38956e6	−0.268128
$894$	0	0
$895$	1.09123e6	0.0455365
$896$	0	0
$897$	−1.29161e7	−0.535983
$898$	0	0
$899$	1.58655e6	0.0654718
$900$	0	0
$901$	848740.	0.0348307
$902$	0	0
$903$	2.44974e7	0.999771
$904$	0	0
$905$	8.94916e6	0.363212
$906$	0	0
$907$	1.73486e7	0.700238	0.350119	−	0.936705i	$-0.386141\pi$
$907$	1.73486e7	0.700238	0.350119	+	0.936705i	$0.386141\pi$
$908$	0	0
$909$	1.21824e7	0.489017
$910$	0	0
$911$	483305.	0.0192941	0.00964707	−	0.999953i	$-0.496929\pi$
$911$	483305.	0.0192941	0.00964707	+	0.999953i	$0.496929\pi$
$912$	0	0
$913$	1.18640e7	0.471038
$914$	0	0
$915$	−2.50939e6	−0.0990867
$916$	0	0
$917$	−1.79712e7	−0.705756
$918$	0	0
$919$	−2.08632e7	−0.814877	−0.407439	−	0.913233i	$-0.633578\pi$
$919$	−2.08632e7	−0.814877	−0.407439	+	0.913233i	$0.633578\pi$
$920$	0	0
$921$	3.22755e7	1.25379
$922$	0	0
$923$	2.97246e7	1.14845
$924$	0	0
$925$	−3.44366e7	−1.32332
$926$	0	0
$927$	2.32396e7	0.888237
$928$	0	0
$929$	−3.11918e6	−0.118577	−0.0592885	−	0.998241i	$-0.518883\pi$
$929$	−3.11918e6	−0.118577	−0.0592885	+	0.998241i	$0.518883\pi$
$930$	0	0
$931$	3.25688e6	0.123148
$932$	0	0
$933$	5.86331e7	2.20515
$934$	0	0
$935$	−4.10558e6	−0.153584
$936$	0	0
$937$	−1.19572e7	−0.444920	−0.222460	−	0.974942i	$-0.571409\pi$
$937$	−1.19572e7	−0.444920	−0.222460	+	0.974942i	$0.571409\pi$
$938$	0	0
$939$	3.60226e7	1.33325
$940$	0	0
$941$	−1.54713e7	−0.569576	−0.284788	−	0.958590i	$-0.591923\pi$
$941$	−1.54713e7	−0.569576	−0.284788	+	0.958590i	$0.591923\pi$
$942$	0	0
$943$	−1.43981e7	−0.527260
$944$	0	0
$945$	1.45331e6	0.0529393
$946$	0	0
$947$	4.80778e7	1.74208	0.871042	−	0.491208i	$-0.163444\pi$
$947$	4.80778e7	1.74208	0.871042	+	0.491208i	$0.163444\pi$
$948$	0	0
$949$	3.40162e7	1.22608
$950$	0	0
$951$	2.19314e7	0.786348
$952$	0	0
$953$	−2.46687e7	−0.879862	−0.439931	−	0.898032i	$-0.644997\pi$
$953$	−2.46687e7	−0.879862	−0.439931	+	0.898032i	$0.644997\pi$
$954$	0	0
$955$	1.04927e7	0.372289
$956$	0	0
$957$	−6.18021e6	−0.218134
$958$	0	0
$959$	1.88004e7	0.660116
$960$	0	0
$961$	−2.11475e7	−0.738671
$962$	0	0
$963$	2.44677e7	0.850211
$964$	0	0
$965$	6.35484e6	0.219678
$966$	0	0
$967$	−3.56696e6	−0.122668	−0.0613341	−	0.998117i	$-0.519536\pi$
$967$	−3.56696e6	−0.122668	−0.0613341	+	0.998117i	$0.519536\pi$
$968$	0	0
$969$	−3.83341e6	−0.131152
$970$	0	0
$971$	−1.51510e7	−0.515694	−0.257847	−	0.966186i	$-0.583013\pi$
$971$	−1.51510e7	−0.515694	−0.257847	+	0.966186i	$0.583013\pi$
$972$	0	0
$973$	8.64698e6	0.292808
$974$	0	0
$975$	4.55048e7	1.53301
$976$	0	0
$977$	2.45157e7	0.821689	0.410844	−	0.911705i	$-0.365234\pi$
$977$	2.45157e7	0.821689	0.410844	+	0.911705i	$0.365234\pi$
$978$	0	0
$979$	−1.50883e7	−0.503133
$980$	0	0
$981$	−1.96083e7	−0.650531
$982$	0	0
$983$	4.13681e7	1.36547	0.682734	−	0.730667i	$-0.260791\pi$
$983$	4.13681e7	1.36547	0.682734	+	0.730667i	$0.260791\pi$
$984$	0	0
$985$	−5.87714e6	−0.193008
$986$	0	0
$987$	3.23467e7	1.05691
$988$	0	0
$989$	1.49651e7	0.486507
$990$	0	0
$991$	−491825.	−0.0159084	−0.00795420	−	0.999968i	$-0.502532\pi$
$991$	−491825.	−0.0159084	−0.00795420	+	0.999968i	$0.502532\pi$
$992$	0	0
$993$	6.15109e7	1.97961
$994$	0	0
$995$	4.27894e6	0.137018
$996$	0	0
$997$	5.22616e7	1.66512	0.832559	−	0.553936i	$-0.186875\pi$
$997$	5.22616e7	1.66512	0.832559	+	0.553936i	$0.186875\pi$
$998$	0	0
$999$	2.17263e7	0.688768

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.9	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.9	✓	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)

\(f(q)\)	\(=\)	\(q-19.7457 q^{3} -10.8196 q^{5} -70.7802 q^{7} +146.891 q^{9} +O(q^{10})\)	\(q-19.7457 q^{3} -10.8196 q^{5} -70.7802 q^{7} +146.891 q^{9} -539.605 q^{11} +766.155 q^{13} +213.640 q^{15} -703.216 q^{17} -276.074 q^{19} +1397.60 q^{21} +853.774 q^{23} -3007.94 q^{25} +1897.73 q^{27} -580.037 q^{29} -2735.26 q^{31} +10654.9 q^{33} +765.812 q^{35} +11448.6 q^{37} -15128.2 q^{39} -16864.0 q^{41} +17528.2 q^{43} -1589.30 q^{45} +23144.4 q^{47} -11797.2 q^{49} +13885.5 q^{51} -1206.94 q^{53} +5838.30 q^{55} +5451.25 q^{57} +33144.1 q^{59} -11745.9 q^{61} -10397.0 q^{63} -8289.48 q^{65} +1553.20 q^{67} -16858.3 q^{69} +38797.1 q^{71} +44398.6 q^{73} +59393.7 q^{75} +38193.4 q^{77} -18958.2 q^{79} -73166.5 q^{81} -21986.5 q^{83} +7608.50 q^{85} +11453.2 q^{87} +27961.7 q^{89} -54228.7 q^{91} +54009.5 q^{93} +2987.00 q^{95} -56138.5 q^{97} -79263.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

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