LMFDB - Embedded newform 1028.6.a.b.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:	$0$
Dimension:	$57$
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)

$f(q)$	$=$	$q-22.4232 q^{3} +91.9325 q^{5} -225.282 q^{7} +259.802 q^{9} +O(q^{10})$	$q-22.4232 q^{3} +91.9325 q^{5} -225.282 q^{7} +259.802 q^{9} +403.222 q^{11} +1063.54 q^{13} -2061.43 q^{15} +1900.63 q^{17} +1593.11 q^{19} +5051.54 q^{21} -1696.84 q^{23} +5326.59 q^{25} -376.758 q^{27} +5890.02 q^{29} +1123.93 q^{31} -9041.55 q^{33} -20710.7 q^{35} -8764.03 q^{37} -23847.9 q^{39} -11618.0 q^{41} +19810.5 q^{43} +23884.3 q^{45} +7086.39 q^{47} +33944.8 q^{49} -42618.2 q^{51} +28599.4 q^{53} +37069.2 q^{55} -35722.6 q^{57} -19203.5 q^{59} +35197.4 q^{61} -58528.6 q^{63} +97773.6 q^{65} +18463.6 q^{67} +38048.8 q^{69} -35697.0 q^{71} -28924.5 q^{73} -119439. q^{75} -90838.6 q^{77} +59452.3 q^{79} -54683.8 q^{81} -95417.7 q^{83} +174729. q^{85} -132073. q^{87} -120695. q^{89} -239595. q^{91} -25202.1 q^{93} +146458. q^{95} +29116.4 q^{97} +104758. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−22.4232	−1.43845	−0.719226	−	0.694776i	$-0.755503\pi$
$3$	−22.4232	−1.43845	−0.719226	+	0.694776i	$0.755503\pi$
$4$	0	0
$5$	91.9325	1.64454	0.822269	−	0.569099i	$-0.192708\pi$
$5$	91.9325	1.64454	0.822269	+	0.569099i	$0.192708\pi$
$6$	0	0
$7$	−225.282	−1.73772	−0.868862	−	0.495055i	$-0.835148\pi$
$7$	−225.282	−1.73772	−0.868862	+	0.495055i	$0.835148\pi$
$8$	0	0
$9$	259.802	1.06914
$10$	0	0
$11$	403.222	1.00476	0.502381	−	0.864647i	$-0.332458\pi$
$11$	403.222	1.00476	0.502381	+	0.864647i	$0.332458\pi$
$12$	0	0
$13$	1063.54	1.74540	0.872698	−	0.488260i	$-0.162368\pi$
$13$	1063.54	1.74540	0.872698	+	0.488260i	$0.162368\pi$
$14$	0	0
$15$	−2061.43	−2.36559
$16$	0	0
$17$	1900.63	1.59505	0.797526	−	0.603285i	$-0.206142\pi$
$17$	1900.63	1.59505	0.797526	+	0.603285i	$0.206142\pi$
$18$	0	0
$19$	1593.11	1.01242	0.506210	−	0.862410i	$-0.331046\pi$
$19$	1593.11	1.01242	0.506210	+	0.862410i	$0.331046\pi$
$20$	0	0
$21$	5051.54	2.49963
$22$	0	0
$23$	−1696.84	−0.668840	−0.334420	−	0.942424i	$-0.608540\pi$
$23$	−1696.84	−0.668840	−0.334420	+	0.942424i	$0.608540\pi$
$24$	0	0
$25$	5326.59	1.70451
$26$	0	0
$27$	−376.758	−0.0994610
$28$	0	0
$29$	5890.02	1.30053	0.650267	−	0.759706i	$-0.274657\pi$
$29$	5890.02	1.30053	0.650267	+	0.759706i	$0.274657\pi$
$30$	0	0
$31$	1123.93	0.210055	0.105028	−	0.994469i	$-0.466507\pi$
$31$	1123.93	0.210055	0.105028	+	0.994469i	$0.466507\pi$
$32$	0	0
$33$	−9041.55	−1.44530
$34$	0	0
$35$	−20710.7	−2.85775
$36$	0	0
$37$	−8764.03	−1.05244	−0.526222	−	0.850347i	$-0.676392\pi$
$37$	−8764.03	−1.05244	−0.526222	+	0.850347i	$0.676392\pi$
$38$	0	0
$39$	−23847.9	−2.51067
$40$	0	0
$41$	−11618.0	−1.07938	−0.539688	−	0.841865i	$-0.681458\pi$
$41$	−11618.0	−1.07938	−0.539688	+	0.841865i	$0.681458\pi$
$42$	0	0
$43$	19810.5	1.63389	0.816947	−	0.576713i	$-0.195665\pi$
$43$	19810.5	1.63389	0.816947	+	0.576713i	$0.195665\pi$
$44$	0	0
$45$	23884.3	1.75825
$46$	0	0
$47$	7086.39	0.467930	0.233965	−	0.972245i	$-0.424830\pi$
$47$	7086.39	0.467930	0.233965	+	0.972245i	$0.424830\pi$
$48$	0	0
$49$	33944.8	2.01968
$50$	0	0
$51$	−42618.2	−2.29440
$52$	0	0
$53$	28599.4	1.39852	0.699258	−	0.714870i	$-0.253514\pi$
$53$	28599.4	1.39852	0.699258	+	0.714870i	$0.253514\pi$
$54$	0	0
$55$	37069.2	1.65237
$56$	0	0
$57$	−35722.6	−1.45632
$58$	0	0
$59$	−19203.5	−0.718208	−0.359104	−	0.933298i	$-0.616918\pi$
$59$	−19203.5	−0.718208	−0.359104	+	0.933298i	$0.616918\pi$
$60$	0	0
$61$	35197.4	1.21112	0.605558	−	0.795801i	$-0.292950\pi$
$61$	35197.4	1.21112	0.605558	+	0.795801i	$0.292950\pi$
$62$	0	0
$63$	−58528.6	−1.85788
$64$	0	0
$65$	97773.6	2.87037
$66$	0	0
$67$	18463.6	0.502493	0.251247	−	0.967923i	$-0.419159\pi$
$67$	18463.6	0.502493	0.251247	+	0.967923i	$0.419159\pi$
$68$	0	0
$69$	38048.8	0.962095
$70$	0	0
$71$	−35697.0	−0.840399	−0.420199	−	0.907432i	$-0.638040\pi$
$71$	−35697.0	−0.840399	−0.420199	+	0.907432i	$0.638040\pi$
$72$	0	0
$73$	−28924.5	−0.635271	−0.317635	−	0.948213i	$-0.602889\pi$
$73$	−28924.5	−0.635271	−0.317635	+	0.948213i	$0.602889\pi$
$74$	0	0
$75$	−119439.	−2.45185
$76$	0	0
$77$	−90838.6	−1.74600
$78$	0	0
$79$	59452.3	1.07177	0.535884	−	0.844291i	$-0.319978\pi$
$79$	59452.3	1.07177	0.535884	+	0.844291i	$0.319978\pi$
$80$	0	0
$81$	−54683.8	−0.926075
$82$	0	0
$83$	−95417.7	−1.52032	−0.760158	−	0.649739i	$-0.774878\pi$
$83$	−95417.7	−1.52032	−0.760158	+	0.649739i	$0.774878\pi$
$84$	0	0
$85$	174729.	2.62312
$86$	0	0
$87$	−132073.	−1.87076
$88$	0	0
$89$	−120695.	−1.61515	−0.807577	−	0.589762i	$-0.799221\pi$
$89$	−120695.	−1.61515	−0.807577	+	0.589762i	$0.799221\pi$
$90$	0	0
$91$	−239595.	−3.03302
$92$	0	0
$93$	−25202.1	−0.302155
$94$	0	0
$95$	146458.	1.66497
$96$	0	0
$97$	29116.4	0.314202	0.157101	−	0.987583i	$-0.449785\pi$
$97$	29116.4	0.314202	0.157101	+	0.987583i	$0.449785\pi$
$98$	0	0
$99$	104758.	1.07423
$100$	0	0
$101$	153221.	1.49457	0.747283	−	0.664506i	$-0.231358\pi$
$101$	153221.	1.49457	0.747283	+	0.664506i	$0.231358\pi$
$102$	0	0
$103$	38466.2	0.357261	0.178631	−	0.983916i	$-0.442833\pi$
$103$	38466.2	0.357261	0.178631	+	0.983916i	$0.442833\pi$
$104$	0	0
$105$	464401.	4.11074
$106$	0	0
$107$	−109431.	−0.924022	−0.462011	−	0.886874i	$-0.652872\pi$
$107$	−109431.	−0.924022	−0.462011	+	0.886874i	$0.652872\pi$
$108$	0	0
$109$	−208659.	−1.68217	−0.841086	−	0.540902i	$-0.818083\pi$
$109$	−208659.	−1.68217	−0.841086	+	0.540902i	$0.818083\pi$
$110$	0	0
$111$	196518.	1.51389
$112$	0	0
$113$	39483.5	0.290884	0.145442	−	0.989367i	$-0.453540\pi$
$113$	39483.5	0.290884	0.145442	+	0.989367i	$0.453540\pi$
$114$	0	0
$115$	−155995.	−1.09993
$116$	0	0
$117$	276309.	1.86608
$118$	0	0
$119$	−428176.	−2.77176
$120$	0	0
$121$	1537.25	0.00954510
$122$	0	0
$123$	260514.	1.55263
$124$	0	0
$125$	202397.	1.15859
$126$	0	0
$127$	−40667.3	−0.223736	−0.111868	−	0.993723i	$-0.535683\pi$
$127$	−40667.3	−0.223736	−0.111868	+	0.993723i	$0.535683\pi$
$128$	0	0
$129$	−444215.	−2.35028
$130$	0	0
$131$	−26456.1	−0.134694	−0.0673468	−	0.997730i	$-0.521453\pi$
$131$	−26456.1	−0.134694	−0.0673468	+	0.997730i	$0.521453\pi$
$132$	0	0
$133$	−358898.	−1.75931
$134$	0	0
$135$	−34636.3	−0.163567
$136$	0	0
$137$	−113932.	−0.518615	−0.259308	−	0.965795i	$-0.583494\pi$
$137$	−113932.	−0.518615	−0.259308	+	0.965795i	$0.583494\pi$
$138$	0	0
$139$	160473.	0.704474	0.352237	−	0.935911i	$-0.385421\pi$
$139$	160473.	0.704474	0.352237	+	0.935911i	$0.385421\pi$
$140$	0	0
$141$	−158900.	−0.673095
$142$	0	0
$143$	428842.	1.75371
$144$	0	0
$145$	541484.	2.13878
$146$	0	0
$147$	−761152.	−2.90522
$148$	0	0
$149$	−294376.	−1.08627	−0.543134	−	0.839646i	$-0.682762\pi$
$149$	−294376.	−1.08627	−0.543134	+	0.839646i	$0.682762\pi$
$150$	0	0
$151$	104480.	0.372900	0.186450	−	0.982464i	$-0.440302\pi$
$151$	104480.	0.372900	0.186450	+	0.982464i	$0.440302\pi$
$152$	0	0
$153$	493787.	1.70534
$154$	0	0
$155$	103325.	0.345444
$156$	0	0
$157$	−26487.3	−0.0857608	−0.0428804	−	0.999080i	$-0.513653\pi$
$157$	−26487.3	−0.0857608	−0.0428804	+	0.999080i	$0.513653\pi$
$158$	0	0
$159$	−641292.	−2.01170
$160$	0	0
$161$	382268.	1.16226
$162$	0	0
$163$	520198.	1.53356	0.766778	−	0.641913i	$-0.221859\pi$
$163$	520198.	1.53356	0.766778	+	0.641913i	$0.221859\pi$
$164$	0	0
$165$	−831213.	−2.37685
$166$	0	0
$167$	532208.	1.47669	0.738347	−	0.674421i	$-0.235607\pi$
$167$	532208.	1.47669	0.738347	+	0.674421i	$0.235607\pi$
$168$	0	0
$169$	759817.	2.04641
$170$	0	0
$171$	413893.	1.08242
$172$	0	0
$173$	405928.	1.03118	0.515589	−	0.856836i	$-0.327573\pi$
$173$	405928.	1.03118	0.515589	+	0.856836i	$0.327573\pi$
$174$	0	0
$175$	−1.19998e6	−2.96196
$176$	0	0
$177$	430605.	1.03311
$178$	0	0
$179$	−554701.	−1.29398	−0.646988	−	0.762500i	$-0.723972\pi$
$179$	−554701.	−1.29398	−0.646988	+	0.762500i	$0.723972\pi$
$180$	0	0
$181$	−710745.	−1.61257	−0.806283	−	0.591530i	$-0.798524\pi$
$181$	−710745.	−1.61257	−0.806283	+	0.591530i	$0.798524\pi$
$182$	0	0
$183$	−789240.	−1.74213
$184$	0	0
$185$	−805699.	−1.73079
$186$	0	0
$187$	766375.	1.60265
$188$	0	0
$189$	84876.6	0.172836
$190$	0	0
$191$	87005.9	0.172570	0.0862850	−	0.996270i	$-0.472500\pi$
$191$	87005.9	0.172570	0.0862850	+	0.996270i	$0.472500\pi$
$192$	0	0
$193$	−363116.	−0.701701	−0.350850	−	0.936432i	$-0.614107\pi$
$193$	−363116.	−0.701701	−0.350850	+	0.936432i	$0.614107\pi$
$194$	0	0
$195$	−2.19240e6	−4.12889
$196$	0	0
$197$	792836.	1.45552	0.727759	−	0.685833i	$-0.240562\pi$
$197$	792836.	1.45552	0.727759	+	0.685833i	$0.240562\pi$
$198$	0	0
$199$	593455.	1.06232	0.531160	−	0.847272i	$-0.321756\pi$
$199$	593455.	1.06232	0.531160	+	0.847272i	$0.321756\pi$
$200$	0	0
$201$	−414015.	−0.722813
$202$	0	0
$203$	−1.32691e6	−2.25997
$204$	0	0
$205$	−1.06808e6	−1.77508
$206$	0	0
$207$	−440844.	−0.715087
$208$	0	0
$209$	642377.	1.01724
$210$	0	0
$211$	−685820.	−1.06048	−0.530242	−	0.847846i	$-0.677899\pi$
$211$	−685820.	−1.06048	−0.530242	+	0.847846i	$0.677899\pi$
$212$	0	0
$213$	800442.	1.20887
$214$	0	0
$215$	1.82123e6	2.68700
$216$	0	0
$217$	−253200.	−0.365018
$218$	0	0
$219$	648581.	0.913807
$220$	0	0
$221$	2.02139e6	2.78400
$222$	0	0
$223$	113989.	0.153497	0.0767487	−	0.997050i	$-0.475546\pi$
$223$	113989.	0.153497	0.0767487	+	0.997050i	$0.475546\pi$
$224$	0	0
$225$	1.38386e6	1.82236
$226$	0	0
$227$	−1.11698e6	−1.43874	−0.719369	−	0.694629i	$-0.755569\pi$
$227$	−1.11698e6	−1.43874	−0.719369	+	0.694629i	$0.755569\pi$
$228$	0	0
$229$	202647.	0.255359	0.127679	−	0.991815i	$-0.459247\pi$
$229$	202647.	0.255359	0.127679	+	0.991815i	$0.459247\pi$
$230$	0	0
$231$	2.03690e6	2.51153
$232$	0	0
$233$	660894.	0.797521	0.398760	−	0.917055i	$-0.369441\pi$
$233$	660894.	0.797521	0.398760	+	0.917055i	$0.369441\pi$
$234$	0	0
$235$	651470.	0.769529
$236$	0	0
$237$	−1.33311e6	−1.54169
$238$	0	0
$239$	1.11221e6	1.25949	0.629743	−	0.776803i	$-0.283160\pi$
$239$	1.11221e6	1.25949	0.629743	+	0.776803i	$0.283160\pi$
$240$	0	0
$241$	−323590.	−0.358883	−0.179442	−	0.983769i	$-0.557429\pi$
$241$	−323590.	−0.358883	−0.179442	+	0.983769i	$0.557429\pi$
$242$	0	0
$243$	1.31774e6	1.43157
$244$	0	0
$245$	3.12063e6	3.32144
$246$	0	0
$247$	1.69433e6	1.76708
$248$	0	0
$249$	2.13957e6	2.18690
$250$	0	0
$251$	−879571.	−0.881225	−0.440612	−	0.897697i	$-0.645239\pi$
$251$	−879571.	−0.881225	−0.440612	+	0.897697i	$0.645239\pi$
$252$	0	0
$253$	−684206.	−0.672025
$254$	0	0
$255$	−3.91800e6	−3.77324
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	1.97437e6	1.82886
$260$	0	0
$261$	1.53024e6	1.39046
$262$	0	0
$263$	−607216.	−0.541320	−0.270660	−	0.962675i	$-0.587242\pi$
$263$	−607216.	−0.541320	−0.270660	+	0.962675i	$0.587242\pi$
$264$	0	0
$265$	2.62921e6	2.29991
$266$	0	0
$267$	2.70637e6	2.32332
$268$	0	0
$269$	2.06053e6	1.73619	0.868097	−	0.496395i	$-0.165343\pi$
$269$	2.06053e6	1.73619	0.868097	+	0.496395i	$0.165343\pi$
$270$	0	0
$271$	−1.69399e6	−1.40116	−0.700578	−	0.713576i	$-0.747074\pi$
$271$	−1.69399e6	−1.40116	−0.700578	+	0.713576i	$0.747074\pi$
$272$	0	0
$273$	5.37250e6	4.36285
$274$	0	0
$275$	2.14780e6	1.71262
$276$	0	0
$277$	1.20045e6	0.940036	0.470018	−	0.882657i	$-0.344247\pi$
$277$	1.20045e6	0.940036	0.470018	+	0.882657i	$0.344247\pi$
$278$	0	0
$279$	291999.	0.224580
$280$	0	0
$281$	226529.	0.171142	0.0855712	−	0.996332i	$-0.472728\pi$
$281$	226529.	0.171142	0.0855712	+	0.996332i	$0.472728\pi$
$282$	0	0
$283$	1.22404e6	0.908510	0.454255	−	0.890872i	$-0.349906\pi$
$283$	1.22404e6	0.908510	0.454255	+	0.890872i	$0.349906\pi$
$284$	0	0
$285$	−3.28407e6	−2.39497
$286$	0	0
$287$	2.61733e6	1.87566
$288$	0	0
$289$	2.19253e6	1.54419
$290$	0	0
$291$	−652884.	−0.451964
$292$	0	0
$293$	−667734.	−0.454396	−0.227198	−	0.973849i	$-0.572956\pi$
$293$	−667734.	−0.454396	−0.227198	+	0.973849i	$0.572956\pi$
$294$	0	0
$295$	−1.76543e6	−1.18112
$296$	0	0
$297$	−151917.	−0.0999346
$298$	0	0
$299$	−1.80466e6	−1.16739
$300$	0	0
$301$	−4.46294e6	−2.83926
$302$	0	0
$303$	−3.43571e6	−2.14986
$304$	0	0
$305$	3.23578e6	1.99173
$306$	0	0
$307$	830041.	0.502636	0.251318	−	0.967905i	$-0.419136\pi$
$307$	830041.	0.502636	0.251318	+	0.967905i	$0.419136\pi$
$308$	0	0
$309$	−862537.	−0.513903
$310$	0	0
$311$	1.18854e6	0.696809	0.348404	−	0.937344i	$-0.386724\pi$
$311$	1.18854e6	0.696809	0.348404	+	0.937344i	$0.386724\pi$
$312$	0	0
$313$	2.43040e6	1.40222	0.701110	−	0.713053i	$-0.252688\pi$
$313$	2.43040e6	1.40222	0.701110	+	0.713053i	$0.252688\pi$
$314$	0	0
$315$	−5.38068e6	−3.05535
$316$	0	0
$317$	−426999.	−0.238659	−0.119330	−	0.992855i	$-0.538075\pi$
$317$	−426999.	−0.238659	−0.119330	+	0.992855i	$0.538075\pi$
$318$	0	0
$319$	2.37499e6	1.30673
$320$	0	0
$321$	2.45381e6	1.32916
$322$	0	0
$323$	3.02790e6	1.61486
$324$	0	0
$325$	5.66502e6	2.97504
$326$	0	0
$327$	4.67881e6	2.41972
$328$	0	0
$329$	−1.59643e6	−0.813132
$330$	0	0
$331$	1.99753e6	1.00213	0.501063	−	0.865411i	$-0.332942\pi$
$331$	1.99753e6	1.00213	0.501063	+	0.865411i	$0.332942\pi$
$332$	0	0
$333$	−2.27691e6	−1.12522
$334$	0	0
$335$	1.69741e6	0.826370
$336$	0	0
$337$	−4.13443e6	−1.98309	−0.991543	−	0.129782i	$-0.958572\pi$
$337$	−4.13443e6	−1.98309	−0.991543	+	0.129782i	$0.958572\pi$
$338$	0	0
$339$	−885349.	−0.418423
$340$	0	0
$341$	453193.	0.211056
$342$	0	0
$343$	−3.86083e6	−1.77192
$344$	0	0
$345$	3.49792e6	1.58220
$346$	0	0
$347$	−2.84654e6	−1.26909	−0.634546	−	0.772885i	$-0.718813\pi$
$347$	−2.84654e6	−1.26909	−0.634546	+	0.772885i	$0.718813\pi$
$348$	0	0
$349$	−1.54342e6	−0.678299	−0.339150	−	0.940732i	$-0.610139\pi$
$349$	−1.54342e6	−0.678299	−0.339150	+	0.940732i	$0.610139\pi$
$350$	0	0
$351$	−400696.	−0.173599
$352$	0	0
$353$	2.45757e6	1.04971	0.524854	−	0.851192i	$-0.324120\pi$
$353$	2.45757e6	1.04971	0.524854	+	0.851192i	$0.324120\pi$
$354$	0	0
$355$	−3.28171e6	−1.38207
$356$	0	0
$357$	9.60110e6	3.98704
$358$	0	0
$359$	−1.41174e6	−0.578122	−0.289061	−	0.957311i	$-0.593343\pi$
$359$	−1.41174e6	−0.578122	−0.289061	+	0.957311i	$0.593343\pi$
$360$	0	0
$361$	61892.4	0.0249959
$362$	0	0
$363$	−34470.1	−0.0137302
$364$	0	0
$365$	−2.65910e6	−1.04473
$366$	0	0
$367$	2.63325e6	1.02053	0.510267	−	0.860016i	$-0.329547\pi$
$367$	2.63325e6	1.02053	0.510267	+	0.860016i	$0.329547\pi$
$368$	0	0
$369$	−3.01839e6	−1.15401
$370$	0	0
$371$	−6.44292e6	−2.43023
$372$	0	0
$373$	−4.40590e6	−1.63969	−0.819847	−	0.572583i	$-0.805942\pi$
$373$	−4.40590e6	−1.63969	−0.819847	+	0.572583i	$0.805942\pi$
$374$	0	0
$375$	−4.53840e6	−1.66658
$376$	0	0
$377$	6.26425e6	2.26995
$378$	0	0
$379$	−2.04091e6	−0.729838	−0.364919	−	0.931039i	$-0.618903\pi$
$379$	−2.04091e6	−0.729838	−0.364919	+	0.931039i	$0.618903\pi$
$380$	0	0
$381$	911894.	0.321834
$382$	0	0
$383$	729199.	0.254009	0.127004	−	0.991902i	$-0.459464\pi$
$383$	729199.	0.254009	0.127004	+	0.991902i	$0.459464\pi$
$384$	0	0
$385$	−8.35102e6	−2.87136
$386$	0	0
$387$	5.14681e6	1.74687
$388$	0	0
$389$	−5.30164e6	−1.77638	−0.888191	−	0.459475i	$-0.848038\pi$
$389$	−5.30164e6	−1.77638	−0.888191	+	0.459475i	$0.848038\pi$
$390$	0	0
$391$	−3.22507e6	−1.06683
$392$	0	0
$393$	593231.	0.193750
$394$	0	0
$395$	5.46560e6	1.76257
$396$	0	0
$397$	5.03936e6	1.60472	0.802359	−	0.596841i	$-0.203578\pi$
$397$	5.03936e6	1.60472	0.802359	+	0.596841i	$0.203578\pi$
$398$	0	0
$399$	8.04765e6	2.53068
$400$	0	0
$401$	4.89152e6	1.51909	0.759544	−	0.650456i	$-0.225422\pi$
$401$	4.89152e6	1.51909	0.759544	+	0.650456i	$0.225422\pi$
$402$	0	0
$403$	1.19534e6	0.366630
$404$	0	0
$405$	−5.02722e6	−1.52297
$406$	0	0
$407$	−3.53385e6	−1.05746
$408$	0	0
$409$	2.33051e6	0.688879	0.344439	−	0.938809i	$-0.388069\pi$
$409$	2.33051e6	0.688879	0.344439	+	0.938809i	$0.388069\pi$
$410$	0	0
$411$	2.55473e6	0.746003
$412$	0	0
$413$	4.32619e6	1.24805
$414$	0	0
$415$	−8.77199e6	−2.50022
$416$	0	0
$417$	−3.59833e6	−1.01335
$418$	0	0
$419$	1.51548e6	0.421712	0.210856	−	0.977517i	$-0.432375\pi$
$419$	1.51548e6	0.421712	0.210856	+	0.977517i	$0.432375\pi$
$420$	0	0
$421$	−5.39256e6	−1.48283	−0.741413	−	0.671049i	$-0.765844\pi$
$421$	−5.39256e6	−1.48283	−0.741413	+	0.671049i	$0.765844\pi$
$422$	0	0
$423$	1.84106e6	0.500284
$424$	0	0
$425$	1.01239e7	2.71878
$426$	0	0
$427$	−7.92932e6	−2.10458
$428$	0	0
$429$	−9.61602e6	−2.52262
$430$	0	0
$431$	3.29707e6	0.854938	0.427469	−	0.904030i	$-0.359405\pi$
$431$	3.29707e6	0.854938	0.427469	+	0.904030i	$0.359405\pi$
$432$	0	0
$433$	1.27953e6	0.327967	0.163983	−	0.986463i	$-0.447566\pi$
$433$	1.27953e6	0.327967	0.163983	+	0.986463i	$0.447566\pi$
$434$	0	0
$435$	−1.21418e7	−3.07653
$436$	0	0
$437$	−2.70326e6	−0.677148
$438$	0	0
$439$	1.04128e6	0.257873	0.128937	−	0.991653i	$-0.458844\pi$
$439$	1.04128e6	0.257873	0.128937	+	0.991653i	$0.458844\pi$
$440$	0	0
$441$	8.81893e6	2.15933
$442$	0	0
$443$	5.84943e6	1.41613	0.708067	−	0.706146i	$-0.249568\pi$
$443$	5.84943e6	1.41613	0.708067	+	0.706146i	$0.249568\pi$
$444$	0	0
$445$	−1.10958e7	−2.65618
$446$	0	0
$447$	6.60086e6	1.56254
$448$	0	0
$449$	−1.74751e6	−0.409075	−0.204538	−	0.978859i	$-0.565569\pi$
$449$	−1.74751e6	−0.409075	−0.204538	+	0.978859i	$0.565569\pi$
$450$	0	0
$451$	−4.68465e6	−1.08452
$452$	0	0
$453$	−2.34279e6	−0.536399
$454$	0	0
$455$	−2.20266e7	−4.98791
$456$	0	0
$457$	−2.24907e6	−0.503747	−0.251874	−	0.967760i	$-0.581047\pi$
$457$	−2.24907e6	−0.503747	−0.251874	+	0.967760i	$0.581047\pi$
$458$	0	0
$459$	−716076.	−0.158645
$460$	0	0
$461$	−2.14494e6	−0.470071	−0.235036	−	0.971987i	$-0.575521\pi$
$461$	−2.14494e6	−0.470071	−0.235036	+	0.971987i	$0.575521\pi$
$462$	0	0
$463$	−935329.	−0.202774	−0.101387	−	0.994847i	$-0.532328\pi$
$463$	−935329.	−0.202774	−0.101387	+	0.994847i	$0.532328\pi$
$464$	0	0
$465$	−2.31689e6	−0.496905
$466$	0	0
$467$	−798657.	−0.169460	−0.0847302	−	0.996404i	$-0.527003\pi$
$467$	−798657.	−0.169460	−0.0847302	+	0.996404i	$0.527003\pi$
$468$	0	0
$469$	−4.15952e6	−0.873194
$470$	0	0
$471$	593932.	0.123363
$472$	0	0
$473$	7.98803e6	1.64167
$474$	0	0
$475$	8.48582e6	1.72568
$476$	0	0
$477$	7.43018e6	1.49522
$478$	0	0
$479$	594775.	0.118444	0.0592221	−	0.998245i	$-0.481138\pi$
$479$	594775.	0.118444	0.0592221	+	0.998245i	$0.481138\pi$
$480$	0	0
$481$	−9.32086e6	−1.83693
$482$	0	0
$483$	−8.57169e6	−1.67185
$484$	0	0
$485$	2.67674e6	0.516717
$486$	0	0
$487$	−5.68750e6	−1.08667	−0.543337	−	0.839515i	$-0.682839\pi$
$487$	−5.68750e6	−1.08667	−0.543337	+	0.839515i	$0.682839\pi$
$488$	0	0
$489$	−1.16645e7	−2.20595
$490$	0	0
$491$	7.48957e6	1.40202	0.701008	−	0.713153i	$-0.252733\pi$
$491$	7.48957e6	1.40202	0.701008	+	0.713153i	$0.252733\pi$
$492$	0	0
$493$	1.11947e7	2.07442
$494$	0	0
$495$	9.63067e6	1.76662
$496$	0	0
$497$	8.04187e6	1.46038
$498$	0	0
$499$	6.09197e6	1.09523	0.547616	−	0.836730i	$-0.315535\pi$
$499$	6.09197e6	1.09523	0.547616	+	0.836730i	$0.315535\pi$
$500$	0	0
$501$	−1.19338e7	−2.12415
$502$	0	0
$503$	3.43717e6	0.605732	0.302866	−	0.953033i	$-0.402056\pi$
$503$	3.43717e6	0.605732	0.302866	+	0.953033i	$0.402056\pi$
$504$	0	0
$505$	1.40860e7	2.45787
$506$	0	0
$507$	−1.70376e7	−2.94366
$508$	0	0
$509$	−3.11711e6	−0.533282	−0.266641	−	0.963796i	$-0.585914\pi$
$509$	−3.11711e6	−0.533282	−0.266641	+	0.963796i	$0.585914\pi$
$510$	0	0
$511$	6.51616e6	1.10392
$512$	0	0
$513$	−600216.	−0.100696
$514$	0	0
$515$	3.53629e6	0.587530
$516$	0	0
$517$	2.85739e6	0.470158
$518$	0	0
$519$	−9.10222e6	−1.48330
$520$	0	0
$521$	−1.05848e7	−1.70840	−0.854199	−	0.519947i	$-0.825952\pi$
$521$	−1.05848e7	−1.70840	−0.854199	+	0.519947i	$0.825952\pi$
$522$	0	0
$523$	2.64695e6	0.423148	0.211574	−	0.977362i	$-0.432141\pi$
$523$	2.64695e6	0.423148	0.211574	+	0.977362i	$0.432141\pi$
$524$	0	0
$525$	2.69075e7	4.26064
$526$	0	0
$527$	2.13617e6	0.335049
$528$	0	0
$529$	−3.55706e6	−0.552652
$530$	0	0
$531$	−4.98911e6	−0.767868
$532$	0	0
$533$	−1.23562e7	−1.88394
$534$	0	0
$535$	−1.00603e7	−1.51959
$536$	0	0
$537$	1.24382e7	1.86132
$538$	0	0
$539$	1.36873e7	2.02930
$540$	0	0
$541$	−1.10109e7	−1.61744	−0.808720	−	0.588194i	$-0.799839\pi$
$541$	−1.10109e7	−1.61744	−0.808720	+	0.588194i	$0.799839\pi$
$542$	0	0
$543$	1.59372e7	2.31960
$544$	0	0
$545$	−1.91825e7	−2.76640
$546$	0	0
$547$	−152019.	−0.0217235	−0.0108618	−	0.999941i	$-0.503457\pi$
$547$	−152019.	−0.0217235	−0.0108618	+	0.999941i	$0.503457\pi$
$548$	0	0
$549$	9.14436e6	1.29486
$550$	0	0
$551$	9.38343e6	1.31669
$552$	0	0
$553$	−1.33935e7	−1.86244
$554$	0	0
$555$	1.80664e7	2.48965
$556$	0	0
$557$	375267.	0.0512510	0.0256255	−	0.999672i	$-0.491842\pi$
$557$	375267.	0.0512510	0.0256255	+	0.999672i	$0.491842\pi$
$558$	0	0
$559$	2.10692e7	2.85179
$560$	0	0
$561$	−1.71846e7	−2.30533
$562$	0	0
$563$	7.06327e6	0.939150	0.469575	−	0.882893i	$-0.344407\pi$
$563$	7.06327e6	0.939150	0.469575	+	0.882893i	$0.344407\pi$
$564$	0	0
$565$	3.62982e6	0.478370
$566$	0	0
$567$	1.23192e7	1.60926
$568$	0	0
$569$	−3.95594e6	−0.512235	−0.256117	−	0.966646i	$-0.582443\pi$
$569$	−3.95594e6	−0.512235	−0.256117	+	0.966646i	$0.582443\pi$
$570$	0	0
$571$	−4.95256e6	−0.635681	−0.317841	−	0.948144i	$-0.602958\pi$
$571$	−4.95256e6	−0.635681	−0.317841	+	0.948144i	$0.602958\pi$
$572$	0	0
$573$	−1.95096e6	−0.248234
$574$	0	0
$575$	−9.03839e6	−1.14004
$576$	0	0
$577$	4.24488e6	0.530794	0.265397	−	0.964139i	$-0.414497\pi$
$577$	4.24488e6	0.530794	0.265397	+	0.964139i	$0.414497\pi$
$578$	0	0
$579$	8.14224e6	1.00936
$580$	0	0
$581$	2.14958e7	2.64189
$582$	0	0
$583$	1.15319e7	1.40517
$584$	0	0
$585$	2.54018e7	3.06884
$586$	0	0
$587$	53074.1	0.00635751	0.00317876	−	0.999995i	$-0.498988\pi$
$587$	53074.1	0.00635751	0.00317876	+	0.999995i	$0.498988\pi$
$588$	0	0
$589$	1.79054e6	0.212665
$590$	0	0
$591$	−1.77780e7	−2.09369
$592$	0	0
$593$	1.28355e6	0.149891	0.0749453	−	0.997188i	$-0.476122\pi$
$593$	1.28355e6	0.149891	0.0749453	+	0.997188i	$0.476122\pi$
$594$	0	0
$595$	−3.93633e7	−4.55826
$596$	0	0
$597$	−1.33072e7	−1.52810
$598$	0	0
$599$	−3.99259e6	−0.454661	−0.227331	−	0.973818i	$-0.573000\pi$
$599$	−3.99259e6	−0.454661	−0.227331	+	0.973818i	$0.573000\pi$
$600$	0	0
$601$	1.62839e7	1.83896	0.919478	−	0.393140i	$-0.128611\pi$
$601$	1.62839e7	1.83896	0.919478	+	0.393140i	$0.128611\pi$
$602$	0	0
$603$	4.79689e6	0.537238
$604$	0	0
$605$	141323.	0.0156973
$606$	0	0
$607$	−4.24034e6	−0.467120	−0.233560	−	0.972342i	$-0.575038\pi$
$607$	−4.24034e6	−0.467120	−0.233560	+	0.972342i	$0.575038\pi$
$608$	0	0
$609$	2.97537e7	3.25085
$610$	0	0
$611$	7.53664e6	0.816723
$612$	0	0
$613$	1.91449e6	0.205780	0.102890	−	0.994693i	$-0.467191\pi$
$613$	1.91449e6	0.205780	0.102890	+	0.994693i	$0.467191\pi$
$614$	0	0
$615$	2.39497e7	2.55336
$616$	0	0
$617$	2.45233e6	0.259338	0.129669	−	0.991557i	$-0.458609\pi$
$617$	2.45233e6	0.259338	0.129669	+	0.991557i	$0.458609\pi$
$618$	0	0
$619$	−2.91102e6	−0.305364	−0.152682	−	0.988275i	$-0.548791\pi$
$619$	−2.91102e6	−0.305364	−0.152682	+	0.988275i	$0.548791\pi$
$620$	0	0
$621$	639299.	0.0665235
$622$	0	0
$623$	2.71903e7	2.80669
$624$	0	0
$625$	1.96131e6	0.200838
$626$	0	0
$627$	−1.44042e7	−1.46325
$628$	0	0
$629$	−1.66571e7	−1.67870
$630$	0	0
$631$	3.52255e6	0.352195	0.176098	−	0.984373i	$-0.443653\pi$
$631$	3.52255e6	0.352195	0.176098	+	0.984373i	$0.443653\pi$
$632$	0	0
$633$	1.53783e7	1.52546
$634$	0	0
$635$	−3.73865e6	−0.367943
$636$	0	0
$637$	3.61015e7	3.52515
$638$	0	0
$639$	−9.27415e6	−0.898508
$640$	0	0
$641$	1.07953e7	1.03775	0.518873	−	0.854852i	$-0.326352\pi$
$641$	1.07953e7	1.03775	0.518873	+	0.854852i	$0.326352\pi$
$642$	0	0
$643$	−8.86469e6	−0.845544	−0.422772	−	0.906236i	$-0.638943\pi$
$643$	−8.86469e6	−0.845544	−0.422772	+	0.906236i	$0.638943\pi$
$644$	0	0
$645$	−4.08378e7	−3.86512
$646$	0	0
$647$	−9.49189e6	−0.891440	−0.445720	−	0.895173i	$-0.647052\pi$
$647$	−9.49189e6	−0.891440	−0.445720	+	0.895173i	$0.647052\pi$
$648$	0	0
$649$	−7.74328e6	−0.721628
$650$	0	0
$651$	5.67757e6	0.525061
$652$	0	0
$653$	−1.37596e7	−1.26277	−0.631385	−	0.775470i	$-0.717513\pi$
$653$	−1.37596e7	−1.26277	−0.631385	+	0.775470i	$0.717513\pi$
$654$	0	0
$655$	−2.43217e6	−0.221509
$656$	0	0
$657$	−7.51465e6	−0.679196
$658$	0	0
$659$	1.13388e7	1.01707	0.508537	−	0.861040i	$-0.330186\pi$
$659$	1.13388e7	1.01707	0.508537	+	0.861040i	$0.330186\pi$
$660$	0	0
$661$	−1.27710e6	−0.113690	−0.0568450	−	0.998383i	$-0.518104\pi$
$661$	−1.27710e6	−0.113690	−0.0568450	+	0.998383i	$0.518104\pi$
$662$	0	0
$663$	−4.53261e7	−4.00465
$664$	0	0
$665$	−3.29944e7	−2.89325
$666$	0	0
$667$	−9.99444e6	−0.869849
$668$	0	0
$669$	−2.55601e6	−0.220799
$670$	0	0
$671$	1.41924e7	1.21688
$672$	0	0
$673$	−1.15531e7	−0.983243	−0.491621	−	0.870809i	$-0.663596\pi$
$673$	−1.15531e7	−0.983243	−0.491621	+	0.870809i	$0.663596\pi$
$674$	0	0
$675$	−2.00683e6	−0.169532
$676$	0	0
$677$	1.18245e7	0.991545	0.495773	−	0.868452i	$-0.334885\pi$
$677$	1.18245e7	0.991545	0.495773	+	0.868452i	$0.334885\pi$
$678$	0	0
$679$	−6.55939e6	−0.545995
$680$	0	0
$681$	2.50464e7	2.06955
$682$	0	0
$683$	−2.26869e6	−0.186090	−0.0930451	−	0.995662i	$-0.529660\pi$
$683$	−2.26869e6	−0.186090	−0.0930451	+	0.995662i	$0.529660\pi$
$684$	0	0
$685$	−1.04741e7	−0.852883
$686$	0	0
$687$	−4.54400e6	−0.367321
$688$	0	0
$689$	3.04165e7	2.44096
$690$	0	0
$691$	−1.33136e7	−1.06072	−0.530360	−	0.847772i	$-0.677943\pi$
$691$	−1.33136e7	−1.06072	−0.530360	+	0.847772i	$0.677943\pi$
$692$	0	0
$693$	−2.36000e7	−1.86672
$694$	0	0
$695$	1.47527e7	1.15853
$696$	0	0
$697$	−2.20816e7	−1.72166
$698$	0	0
$699$	−1.48194e7	−1.14720
$700$	0	0
$701$	−1.05227e7	−0.808785	−0.404392	−	0.914586i	$-0.632517\pi$
$701$	−1.05227e7	−0.808785	−0.404392	+	0.914586i	$0.632517\pi$
$702$	0	0
$703$	−1.39620e7	−1.06552
$704$	0	0
$705$	−1.46081e7	−1.10693
$706$	0	0
$707$	−3.45179e7	−2.59714
$708$	0	0
$709$	−1.04790e7	−0.782895	−0.391447	−	0.920200i	$-0.628026\pi$
$709$	−1.04790e7	−0.782895	−0.391447	+	0.920200i	$0.628026\pi$
$710$	0	0
$711$	1.54458e7	1.14588
$712$	0	0
$713$	−1.90713e6	−0.140494
$714$	0	0
$715$	3.94245e7	2.88404
$716$	0	0
$717$	−2.49394e7	−1.81171
$718$	0	0
$719$	1.63513e7	1.17959	0.589793	−	0.807554i	$-0.299209\pi$
$719$	1.63513e7	1.17959	0.589793	+	0.807554i	$0.299209\pi$
$720$	0	0
$721$	−8.66572e6	−0.620821
$722$	0	0
$723$	7.25595e6	0.516236
$724$	0	0
$725$	3.13737e7	2.21677
$726$	0	0
$727$	−1.84483e7	−1.29455	−0.647277	−	0.762255i	$-0.724092\pi$
$727$	−1.84483e7	−1.29455	−0.647277	+	0.762255i	$0.724092\pi$
$728$	0	0
$729$	−1.62599e7	−1.13318
$730$	0	0
$731$	3.76523e7	2.60614
$732$	0	0
$733$	2.61074e7	1.79475	0.897376	−	0.441267i	$-0.145471\pi$
$733$	2.61074e7	1.79475	0.897376	+	0.441267i	$0.145471\pi$
$734$	0	0
$735$	−6.99747e7	−4.77774
$736$	0	0
$737$	7.44495e6	0.504886
$738$	0	0
$739$	5.25997e6	0.354301	0.177150	−	0.984184i	$-0.443312\pi$
$739$	5.25997e6	0.354301	0.177150	+	0.984184i	$0.443312\pi$
$740$	0	0
$741$	−3.79923e7	−2.54185
$742$	0	0
$743$	−2.50500e7	−1.66470	−0.832350	−	0.554251i	$-0.813005\pi$
$743$	−2.50500e7	−1.66470	−0.832350	+	0.554251i	$0.813005\pi$
$744$	0	0
$745$	−2.70627e7	−1.78641
$746$	0	0
$747$	−2.47897e7	−1.62544
$748$	0	0
$749$	2.46529e7	1.60569
$750$	0	0
$751$	−1.87315e7	−1.21191	−0.605957	−	0.795497i	$-0.707210\pi$
$751$	−1.87315e7	−1.21191	−0.605957	+	0.795497i	$0.707210\pi$
$752$	0	0
$753$	1.97228e7	1.26760
$754$	0	0
$755$	9.60515e6	0.613249
$756$	0	0
$757$	3.22568e6	0.204589	0.102294	−	0.994754i	$-0.467382\pi$
$757$	3.22568e6	0.204589	0.102294	+	0.994754i	$0.467382\pi$
$758$	0	0
$759$	1.53421e7	0.966676
$760$	0	0
$761$	−1.20303e7	−0.753037	−0.376519	−	0.926409i	$-0.622879\pi$
$761$	−1.20303e7	−0.753037	−0.376519	+	0.926409i	$0.622879\pi$
$762$	0	0
$763$	4.70070e7	2.92315
$764$	0	0
$765$	4.53951e7	2.80450
$766$	0	0
$767$	−2.04236e7	−1.25356
$768$	0	0
$769$	7.59867e6	0.463364	0.231682	−	0.972792i	$-0.425577\pi$
$769$	7.59867e6	0.463364	0.231682	+	0.972792i	$0.425577\pi$
$770$	0	0
$771$	1.48103e6	0.0897282
$772$	0	0
$773$	4.83770e6	0.291199	0.145600	−	0.989344i	$-0.453489\pi$
$773$	4.83770e6	0.291199	0.145600	+	0.989344i	$0.453489\pi$
$774$	0	0
$775$	5.98670e6	0.358041
$776$	0	0
$777$	−4.42719e7	−2.63072
$778$	0	0
$779$	−1.85088e7	−1.09278
$780$	0	0
$781$	−1.43938e7	−0.844400
$782$	0	0
$783$	−2.21911e6	−0.129352
$784$	0	0
$785$	−2.43505e6	−0.141037
$786$	0	0
$787$	−1.10817e7	−0.637777	−0.318888	−	0.947792i	$-0.603310\pi$
$787$	−1.10817e7	−0.637777	−0.318888	+	0.947792i	$0.603310\pi$
$788$	0	0
$789$	1.36158e7	0.778663
$790$	0	0
$791$	−8.89491e6	−0.505476
$792$	0	0
$793$	3.74337e7	2.11388
$794$	0	0
$795$	−5.89555e7	−3.30831
$796$	0	0
$797$	2.56159e7	1.42845	0.714223	−	0.699918i	$-0.246780\pi$
$797$	2.56159e7	1.42845	0.714223	+	0.699918i	$0.246780\pi$
$798$	0	0
$799$	1.34686e7	0.746372
$800$	0	0
$801$	−3.13568e7	−1.72683
$802$	0	0
$803$	−1.16630e7	−0.638295
$804$	0	0
$805$	3.51428e7	1.91138
$806$	0	0
$807$	−4.62038e7	−2.49743
$808$	0	0
$809$	−3.58812e7	−1.92751	−0.963753	−	0.266798i	$-0.914034\pi$
$809$	−3.58812e7	−1.92751	−0.963753	+	0.266798i	$0.914034\pi$
$810$	0	0
$811$	−1.24113e7	−0.662618	−0.331309	−	0.943522i	$-0.607490\pi$
$811$	−1.24113e7	−0.662618	−0.331309	+	0.943522i	$0.607490\pi$
$812$	0	0
$813$	3.79847e7	2.01550
$814$	0	0
$815$	4.78231e7	2.52199
$816$	0	0
$817$	3.15602e7	1.65419
$818$	0	0
$819$	−6.22473e7	−3.24273
$820$	0	0
$821$	−1.34925e7	−0.698610	−0.349305	−	0.937009i	$-0.613582\pi$
$821$	−1.34925e7	−0.698610	−0.349305	+	0.937009i	$0.613582\pi$
$822$	0	0
$823$	2.89415e7	1.48943	0.744716	−	0.667381i	$-0.232585\pi$
$823$	2.89415e7	1.48943	0.744716	+	0.667381i	$0.232585\pi$
$824$	0	0
$825$	−4.81606e7	−2.46353
$826$	0	0
$827$	4.03120e6	0.204961	0.102480	−	0.994735i	$-0.467322\pi$
$827$	4.03120e6	0.204961	0.102480	+	0.994735i	$0.467322\pi$
$828$	0	0
$829$	3.46297e7	1.75010	0.875049	−	0.484034i	$-0.160829\pi$
$829$	3.46297e7	1.75010	0.875049	+	0.484034i	$0.160829\pi$
$830$	0	0
$831$	−2.69180e7	−1.35220
$832$	0	0
$833$	6.45164e7	3.22150
$834$	0	0
$835$	4.89273e7	2.42848
$836$	0	0
$837$	−423448.	−0.0208923
$838$	0	0
$839$	−2.78925e7	−1.36799	−0.683994	−	0.729488i	$-0.739758\pi$
$839$	−2.78925e7	−1.36799	−0.683994	+	0.729488i	$0.739758\pi$
$840$	0	0
$841$	1.41812e7	0.691387
$842$	0	0
$843$	−5.07951e6	−0.246180
$844$	0	0
$845$	6.98519e7	3.36540
$846$	0	0
$847$	−346314.	−0.0165867
$848$	0	0
$849$	−2.74470e7	−1.30685
$850$	0	0
$851$	1.48712e7	0.703918
$852$	0	0
$853$	−8.49969e6	−0.399973	−0.199986	−	0.979799i	$-0.564090\pi$
$853$	−8.49969e6	−0.399973	−0.199986	+	0.979799i	$0.564090\pi$
$854$	0	0
$855$	3.80502e7	1.78009
$856$	0	0
$857$	−3.17568e7	−1.47702	−0.738508	−	0.674245i	$-0.764469\pi$
$857$	−3.17568e7	−1.47702	−0.738508	+	0.674245i	$0.764469\pi$
$858$	0	0
$859$	1.25468e7	0.580164	0.290082	−	0.957002i	$-0.406317\pi$
$859$	1.25468e7	0.580164	0.290082	+	0.957002i	$0.406317\pi$
$860$	0	0
$861$	−5.86890e7	−2.69804
$862$	0	0
$863$	−3.34685e7	−1.52971	−0.764855	−	0.644202i	$-0.777190\pi$
$863$	−3.34685e7	−1.52971	−0.764855	+	0.644202i	$0.777190\pi$
$864$	0	0
$865$	3.73179e7	1.69581
$866$	0	0
$867$	−4.91636e7	−2.22124
$868$	0	0
$869$	2.39725e7	1.07687
$870$	0	0
$871$	1.96368e7	0.877050
$872$	0	0
$873$	7.56450e6	0.335927
$874$	0	0
$875$	−4.55964e7	−2.01331
$876$	0	0
$877$	−2.15080e7	−0.944282	−0.472141	−	0.881523i	$-0.656519\pi$
$877$	−2.15080e7	−0.944282	−0.472141	+	0.881523i	$0.656519\pi$
$878$	0	0
$879$	1.49728e7	0.653627
$880$	0	0
$881$	3.37210e7	1.46373	0.731864	−	0.681451i	$-0.238651\pi$
$881$	3.37210e7	1.46373	0.731864	+	0.681451i	$0.238651\pi$
$882$	0	0
$883$	−3.17115e7	−1.36872	−0.684361	−	0.729144i	$-0.739919\pi$
$883$	−3.17115e7	−1.36872	−0.684361	+	0.729144i	$0.739919\pi$
$884$	0	0
$885$	3.95866e7	1.69899
$886$	0	0
$887$	−2.56146e7	−1.09315	−0.546574	−	0.837411i	$-0.684068\pi$
$887$	−2.56146e7	−1.09315	−0.546574	+	0.837411i	$0.684068\pi$
$888$	0	0
$889$	9.16160e6	0.388792
$890$	0	0
$891$	−2.20497e7	−0.930484
$892$	0	0
$893$	1.12894e7	0.473742
$894$	0	0
$895$	−5.09950e7	−2.12799
$896$	0	0
$897$	4.04663e7	1.67924
$898$	0	0
$899$	6.61995e6	0.273184
$900$	0	0
$901$	5.43568e7	2.23070
$902$	0	0
$903$	1.00074e8	4.08413
$904$	0	0
$905$	−6.53406e7	−2.65193
$906$	0	0
$907$	1.21854e7	0.491838	0.245919	−	0.969290i	$-0.420910\pi$
$907$	1.21854e7	0.491838	0.245919	+	0.969290i	$0.420910\pi$
$908$	0	0
$909$	3.98071e7	1.59791
$910$	0	0
$911$	3.58147e7	1.42977	0.714883	−	0.699244i	$-0.246480\pi$
$911$	3.58147e7	1.42977	0.714883	+	0.699244i	$0.246480\pi$
$912$	0	0
$913$	−3.84745e7	−1.52755
$914$	0	0
$915$	−7.25568e7	−2.86500
$916$	0	0
$917$	5.96006e6	0.234060
$918$	0	0
$919$	−2.05842e7	−0.803981	−0.401991	−	0.915644i	$-0.631682\pi$
$919$	−2.05842e7	−0.803981	−0.401991	+	0.915644i	$0.631682\pi$
$920$	0	0
$921$	−1.86122e7	−0.723018
$922$	0	0
$923$	−3.79650e7	−1.46683
$924$	0	0
$925$	−4.66823e7	−1.79390
$926$	0	0
$927$	9.99359e6	0.381964
$928$	0	0
$929$	−9.89069e6	−0.375999	−0.188000	−	0.982169i	$-0.560200\pi$
$929$	−9.89069e6	−0.375999	−0.188000	+	0.982169i	$0.560200\pi$
$930$	0	0
$931$	5.40777e7	2.04477
$932$	0	0
$933$	−2.66510e7	−1.00233
$934$	0	0
$935$	7.04548e7	2.63561
$936$	0	0
$937$	−1.05116e7	−0.391130	−0.195565	−	0.980691i	$-0.562654\pi$
$937$	−1.05116e7	−0.391130	−0.195565	+	0.980691i	$0.562654\pi$
$938$	0	0
$939$	−5.44974e7	−2.01703
$940$	0	0
$941$	−3.82076e7	−1.40662	−0.703308	−	0.710885i	$-0.748295\pi$
$941$	−3.82076e7	−1.40662	−0.703308	+	0.710885i	$0.748295\pi$
$942$	0	0
$943$	1.97140e7	0.721931
$944$	0	0
$945$	7.80292e6	0.284235
$946$	0	0
$947$	−5.13525e6	−0.186074	−0.0930371	−	0.995663i	$-0.529658\pi$
$947$	−5.13525e6	−0.186074	−0.0930371	+	0.995663i	$0.529658\pi$
$948$	0	0
$949$	−3.07623e7	−1.10880
$950$	0	0
$951$	9.57470e6	0.343300
$952$	0	0
$953$	7.13509e6	0.254488	0.127244	−	0.991871i	$-0.459387\pi$
$953$	7.13509e6	0.254488	0.127244	+	0.991871i	$0.459387\pi$
$954$	0	0
$955$	7.99867e6	0.283798
$956$	0	0
$957$	−5.32549e7	−1.87966
$958$	0	0
$959$	2.56668e7	0.901210
$960$	0	0
$961$	−2.73659e7	−0.955877
$962$	0	0
$963$	−2.84305e7	−0.987913
$964$	0	0
$965$	−3.33821e7	−1.15397
$966$	0	0
$967$	4.66720e7	1.60506	0.802529	−	0.596613i	$-0.203487\pi$
$967$	4.66720e7	1.60506	0.802529	+	0.596613i	$0.203487\pi$
$968$	0	0
$969$	−6.78954e7	−2.32290
$970$	0	0
$971$	5.09118e7	1.73289	0.866443	−	0.499276i	$-0.166401\pi$
$971$	5.09118e7	1.73289	0.866443	+	0.499276i	$0.166401\pi$
$972$	0	0
$973$	−3.61516e7	−1.22418
$974$	0	0
$975$	−1.27028e8	−4.27945
$976$	0	0
$977$	1.29920e7	0.435452	0.217726	−	0.976010i	$-0.430136\pi$
$977$	1.29920e7	0.435452	0.217726	+	0.976010i	$0.430136\pi$
$978$	0	0
$979$	−4.86669e7	−1.62284
$980$	0	0
$981$	−5.42100e7	−1.79848
$982$	0	0
$983$	3.49749e7	1.15444	0.577222	−	0.816587i	$-0.304137\pi$
$983$	3.49749e7	1.15444	0.577222	+	0.816587i	$0.304137\pi$
$984$	0	0
$985$	7.28874e7	2.39366
$986$	0	0
$987$	3.57972e7	1.16965
$988$	0	0
$989$	−3.36153e7	−1.09281
$990$	0	0
$991$	4.84810e7	1.56815	0.784075	−	0.620666i	$-0.213138\pi$
$991$	4.84810e7	1.56815	0.784075	+	0.620666i	$0.213138\pi$
$992$	0	0
$993$	−4.47910e7	−1.44151
$994$	0	0
$995$	5.45578e7	1.74703
$996$	0	0
$997$	−2.57268e7	−0.819687	−0.409844	−	0.912156i	$-0.634417\pi$
$997$	−2.57268e7	−0.819687	−0.409844	+	0.912156i	$0.634417\pi$
$998$	0	0
$999$	3.30191e6	0.104677

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.8	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.8	✓	57	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-22.4232 q^{3} +91.9325 q^{5} -225.282 q^{7} +259.802 q^{9} +O(q^{10})\)	\(q-22.4232 q^{3} +91.9325 q^{5} -225.282 q^{7} +259.802 q^{9} +403.222 q^{11} +1063.54 q^{13} -2061.43 q^{15} +1900.63 q^{17} +1593.11 q^{19} +5051.54 q^{21} -1696.84 q^{23} +5326.59 q^{25} -376.758 q^{27} +5890.02 q^{29} +1123.93 q^{31} -9041.55 q^{33} -20710.7 q^{35} -8764.03 q^{37} -23847.9 q^{39} -11618.0 q^{41} +19810.5 q^{43} +23884.3 q^{45} +7086.39 q^{47} +33944.8 q^{49} -42618.2 q^{51} +28599.4 q^{53} +37069.2 q^{55} -35722.6 q^{57} -19203.5 q^{59} +35197.4 q^{61} -58528.6 q^{63} +97773.6 q^{65} +18463.6 q^{67} +38048.8 q^{69} -35697.0 q^{71} -28924.5 q^{73} -119439. q^{75} -90838.6 q^{77} +59452.3 q^{79} -54683.8 q^{81} -95417.7 q^{83} +174729. q^{85} -132073. q^{87} -120695. q^{89} -239595. q^{91} -25202.1 q^{93} +146458. q^{95} +29116.4 q^{97} +104758. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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