LMFDB - Embedded newform 1028.6.a.b.1.1

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			1.1
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)

$f(q)$	$=$	$q-29.5134 q^{3} -107.184 q^{5} -142.282 q^{7} +628.041 q^{9} +O(q^{10})$	$q-29.5134 q^{3} -107.184 q^{5} -142.282 q^{7} +628.041 q^{9} +553.782 q^{11} +856.139 q^{13} +3163.37 q^{15} -974.712 q^{17} -726.644 q^{19} +4199.21 q^{21} +3140.12 q^{23} +8363.41 q^{25} -11363.9 q^{27} +1058.37 q^{29} +9544.48 q^{31} -16344.0 q^{33} +15250.3 q^{35} +13503.7 q^{37} -25267.6 q^{39} -13895.8 q^{41} -4947.80 q^{43} -67316.0 q^{45} +20907.5 q^{47} +3437.03 q^{49} +28767.1 q^{51} +13615.7 q^{53} -59356.6 q^{55} +21445.7 q^{57} -34281.1 q^{59} +35954.9 q^{61} -89358.7 q^{63} -91764.4 q^{65} -771.481 q^{67} -92675.5 q^{69} -58564.5 q^{71} +23550.6 q^{73} -246833. q^{75} -78792.9 q^{77} +8662.76 q^{79} +182773. q^{81} +78910.3 q^{83} +104474. q^{85} -31236.0 q^{87} -13853.8 q^{89} -121813. q^{91} -281690. q^{93} +77884.6 q^{95} +588.420 q^{97} +347798. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−29.5134	−1.89329	−0.946643	−	0.322284i	$-0.895549\pi$
$3$	−29.5134	−1.89329	−0.946643	+	0.322284i	$0.895549\pi$
$4$	0	0
$5$	−107.184	−1.91737	−0.958683	−	0.284477i	$-0.908180\pi$
$5$	−107.184	−1.91737	−0.958683	+	0.284477i	$0.908180\pi$
$6$	0	0
$7$	−142.282	−1.09750	−0.548749	−	0.835987i	$-0.684896\pi$
$7$	−142.282	−1.09750	−0.548749	+	0.835987i	$0.684896\pi$
$8$	0	0
$9$	628.041	2.58453
$10$	0	0
$11$	553.782	1.37993	0.689965	−	0.723843i	$-0.257626\pi$
$11$	553.782	1.37993	0.689965	+	0.723843i	$0.257626\pi$
$12$	0	0
$13$	856.139	1.40503	0.702515	−	0.711669i	$-0.252060\pi$
$13$	856.139	1.40503	0.702515	+	0.711669i	$0.252060\pi$
$14$	0	0
$15$	3163.37	3.63012
$16$	0	0
$17$	−974.712	−0.818001	−0.409001	−	0.912534i	$-0.634123\pi$
$17$	−974.712	−0.818001	−0.409001	+	0.912534i	$0.634123\pi$
$18$	0	0
$19$	−726.644	−0.461783	−0.230891	−	0.972980i	$-0.574164\pi$
$19$	−726.644	−0.461783	−0.230891	+	0.972980i	$0.574164\pi$
$20$	0	0
$21$	4199.21	2.07788
$22$	0	0
$23$	3140.12	1.23773	0.618865	−	0.785497i	$-0.287593\pi$
$23$	3140.12	1.23773	0.618865	+	0.785497i	$0.287593\pi$
$24$	0	0
$25$	8363.41	2.67629
$26$	0	0
$27$	−11363.9	−2.99997
$28$	0	0
$29$	1058.37	0.233690	0.116845	−	0.993150i	$-0.462722\pi$
$29$	1058.37	0.233690	0.116845	+	0.993150i	$0.462722\pi$
$30$	0	0
$31$	9544.48	1.78381	0.891904	−	0.452225i	$-0.149369\pi$
$31$	9544.48	1.78381	0.891904	+	0.452225i	$0.149369\pi$
$32$	0	0
$33$	−16344.0	−2.61260
$34$	0	0
$35$	15250.3	2.10430
$36$	0	0
$37$	13503.7	1.62162	0.810808	−	0.585313i	$-0.199028\pi$
$37$	13503.7	1.62162	0.810808	+	0.585313i	$0.199028\pi$
$38$	0	0
$39$	−25267.6	−2.66012
$40$	0	0
$41$	−13895.8	−1.29099	−0.645496	−	0.763764i	$-0.723349\pi$
$41$	−13895.8	−1.29099	−0.645496	+	0.763764i	$0.723349\pi$
$42$	0	0
$43$	−4947.80	−0.408076	−0.204038	−	0.978963i	$-0.565407\pi$
$43$	−4947.80	−0.408076	−0.204038	+	0.978963i	$0.565407\pi$
$44$	0	0
$45$	−67316.0	−4.95549
$46$	0	0
$47$	20907.5	1.38057	0.690283	−	0.723540i	$-0.257486\pi$
$47$	20907.5	1.38057	0.690283	+	0.723540i	$0.257486\pi$
$48$	0	0
$49$	3437.03	0.204500
$50$	0	0
$51$	28767.1	1.54871
$52$	0	0
$53$	13615.7	0.665810	0.332905	−	0.942960i	$-0.391971\pi$
$53$	13615.7	0.665810	0.332905	+	0.942960i	$0.391971\pi$
$54$	0	0
$55$	−59356.6	−2.64583
$56$	0	0
$57$	21445.7	0.874287
$58$	0	0
$59$	−34281.1	−1.28211	−0.641055	−	0.767495i	$-0.721503\pi$
$59$	−34281.1	−1.28211	−0.641055	+	0.767495i	$0.721503\pi$
$60$	0	0
$61$	35954.9	1.23718	0.618591	−	0.785713i	$-0.287704\pi$
$61$	35954.9	1.23718	0.618591	+	0.785713i	$0.287704\pi$
$62$	0	0
$63$	−89358.7	−2.83652
$64$	0	0
$65$	−91764.4	−2.69396
$66$	0	0
$67$	−771.481	−0.0209961	−0.0104980	−	0.999945i	$-0.503342\pi$
$67$	−771.481	−0.0209961	−0.0104980	+	0.999945i	$0.503342\pi$
$68$	0	0
$69$	−92675.5	−2.34338
$70$	0	0
$71$	−58564.5	−1.37876	−0.689380	−	0.724400i	$-0.742117\pi$
$71$	−58564.5	−1.37876	−0.689380	+	0.724400i	$0.742117\pi$
$72$	0	0
$73$	23550.6	0.517243	0.258621	−	0.965979i	$-0.416732\pi$
$73$	23550.6	0.517243	0.258621	+	0.965979i	$0.416732\pi$
$74$	0	0
$75$	−246833.	−5.06699
$76$	0	0
$77$	−78792.9	−1.51447
$78$	0	0
$79$	8662.76	0.156167	0.0780834	−	0.996947i	$-0.475120\pi$
$79$	8662.76	0.156167	0.0780834	+	0.996947i	$0.475120\pi$
$80$	0	0
$81$	182773.	3.09527
$82$	0	0
$83$	78910.3	1.25730	0.628650	−	0.777689i	$-0.283608\pi$
$83$	78910.3	1.25730	0.628650	+	0.777689i	$0.283608\pi$
$84$	0	0
$85$	104474.	1.56841
$86$	0	0
$87$	−31236.0	−0.442442
$88$	0	0
$89$	−13853.8	−0.185394	−0.0926968	−	0.995694i	$-0.529549\pi$
$89$	−13853.8	−0.185394	−0.0926968	+	0.995694i	$0.529549\pi$
$90$	0	0
$91$	−121813.	−1.54202
$92$	0	0
$93$	−281690.	−3.37726
$94$	0	0
$95$	77884.6	0.885406
$96$	0	0
$97$	588.420	0.00634977	0.00317489	−	0.999995i	$-0.498989\pi$
$97$	588.420	0.00634977	0.00317489	+	0.999995i	$0.498989\pi$
$98$	0	0
$99$	347798.	3.56647
$100$	0	0
$101$	89325.3	0.871307	0.435653	−	0.900115i	$-0.356517\pi$
$101$	89325.3	0.871307	0.435653	+	0.900115i	$0.356517\pi$
$102$	0	0
$103$	−29028.5	−0.269607	−0.134804	−	0.990872i	$-0.543040\pi$
$103$	−29028.5	−0.269607	−0.134804	+	0.990872i	$0.543040\pi$
$104$	0	0
$105$	−450089.	−3.98405
$106$	0	0
$107$	170458.	1.43932	0.719659	−	0.694327i	$-0.244298\pi$
$107$	170458.	1.43932	0.719659	+	0.694327i	$0.244298\pi$
$108$	0	0
$109$	2590.87	0.0208871	0.0104436	−	0.999945i	$-0.496676\pi$
$109$	2590.87	0.0208871	0.0104436	+	0.999945i	$0.496676\pi$
$110$	0	0
$111$	−398540.	−3.07018
$112$	0	0
$113$	10727.7	0.0790332	0.0395166	−	0.999219i	$-0.487418\pi$
$113$	10727.7	0.0790332	0.0395166	+	0.999219i	$0.487418\pi$
$114$	0	0
$115$	−336570.	−2.37318
$116$	0	0
$117$	537690.	3.63135
$118$	0	0
$119$	138684.	0.897754
$120$	0	0
$121$	145623.	0.904207
$122$	0	0
$123$	410112.	2.44422
$124$	0	0
$125$	−561474.	−3.21407
$126$	0	0
$127$	201357.	1.10779	0.553896	−	0.832586i	$-0.313140\pi$
$127$	201357.	1.10779	0.553896	+	0.832586i	$0.313140\pi$
$128$	0	0
$129$	146027.	0.772605
$130$	0	0
$131$	−313287.	−1.59501	−0.797507	−	0.603309i	$-0.793848\pi$
$131$	−313287.	−1.59501	−0.797507	+	0.603309i	$0.793848\pi$
$132$	0	0
$133$	103388.	0.506805
$134$	0	0
$135$	1.21803e6	5.75204
$136$	0	0
$137$	246534.	1.12221	0.561106	−	0.827744i	$-0.310376\pi$
$137$	246534.	1.12221	0.561106	+	0.827744i	$0.310376\pi$
$138$	0	0
$139$	198548.	0.871620	0.435810	−	0.900039i	$-0.356462\pi$
$139$	198548.	0.871620	0.435810	+	0.900039i	$0.356462\pi$
$140$	0	0
$141$	−617051.	−2.61380
$142$	0	0
$143$	474114.	1.93884
$144$	0	0
$145$	−113440.	−0.448070
$146$	0	0
$147$	−101439.	−0.387177
$148$	0	0
$149$	417313.	1.53991	0.769956	−	0.638096i	$-0.220278\pi$
$149$	417313.	1.53991	0.769956	+	0.638096i	$0.220278\pi$
$150$	0	0
$151$	254873.	0.909664	0.454832	−	0.890577i	$-0.349699\pi$
$151$	254873.	0.909664	0.454832	+	0.890577i	$0.349699\pi$
$152$	0	0
$153$	−612159.	−2.11415
$154$	0	0
$155$	−1.02302e6	−3.42021
$156$	0	0
$157$	391615.	1.26797	0.633986	−	0.773344i	$-0.281417\pi$
$157$	391615.	1.26797	0.633986	+	0.773344i	$0.281417\pi$
$158$	0	0
$159$	−401846.	−1.26057
$160$	0	0
$161$	−446781.	−1.35841
$162$	0	0
$163$	−336469.	−0.991918	−0.495959	−	0.868346i	$-0.665183\pi$
$163$	−336469.	−0.991918	−0.495959	+	0.868346i	$0.665183\pi$
$164$	0	0
$165$	1.75181e6	5.00932
$166$	0	0
$167$	−161105.	−0.447010	−0.223505	−	0.974703i	$-0.571750\pi$
$167$	−161105.	−0.447010	−0.223505	+	0.974703i	$0.571750\pi$
$168$	0	0
$169$	361680.	0.974111
$170$	0	0
$171$	−456362.	−1.19349
$172$	0	0
$173$	−305983.	−0.777289	−0.388645	−	0.921388i	$-0.627057\pi$
$173$	−305983.	−0.777289	−0.388645	+	0.921388i	$0.627057\pi$
$174$	0	0
$175$	−1.18996e6	−2.93722
$176$	0	0
$177$	1.01175e6	2.42740
$178$	0	0
$179$	526962.	1.22927	0.614634	−	0.788812i	$-0.289304\pi$
$179$	526962.	1.22927	0.614634	+	0.788812i	$0.289304\pi$
$180$	0	0
$181$	416802.	0.945657	0.472829	−	0.881154i	$-0.343233\pi$
$181$	416802.	0.945657	0.472829	+	0.881154i	$0.343233\pi$
$182$	0	0
$183$	−1.06115e6	−2.34234
$184$	0	0
$185$	−1.44738e6	−3.10923
$186$	0	0
$187$	−539778.	−1.12878
$188$	0	0
$189$	1.61687e6	3.29246
$190$	0	0
$191$	−279538.	−0.554443	−0.277221	−	0.960806i	$-0.589414\pi$
$191$	−279538.	−0.554443	−0.277221	+	0.960806i	$0.589414\pi$
$192$	0	0
$193$	−939.007	−0.00181458	−0.000907289	−	1.00000i	$-0.500289\pi$
$193$	−939.007	−0.00181458	−0.000907289		1.00000i	$0.500289\pi$
$194$	0	0
$195$	2.70828e6	5.10043
$196$	0	0
$197$	−352278.	−0.646725	−0.323363	−	0.946275i	$-0.604813\pi$
$197$	−352278.	−0.646725	−0.323363	+	0.946275i	$0.604813\pi$
$198$	0	0
$199$	−367785.	−0.658357	−0.329179	−	0.944268i	$-0.606772\pi$
$199$	−367785.	−0.658357	−0.329179	+	0.944268i	$0.606772\pi$
$200$	0	0
$201$	22769.0	0.0397516
$202$	0	0
$203$	−150586.	−0.256474
$204$	0	0
$205$	1.48941e6	2.47530
$206$	0	0
$207$	1.97212e6	3.19895
$208$	0	0
$209$	−402402.	−0.637228
$210$	0	0
$211$	−681323.	−1.05353	−0.526765	−	0.850011i	$-0.676595\pi$
$211$	−681323.	−1.05353	−0.526765	+	0.850011i	$0.676595\pi$
$212$	0	0
$213$	1.72844e6	2.61039
$214$	0	0
$215$	530325.	0.782431
$216$	0	0
$217$	−1.35800e6	−1.95772
$218$	0	0
$219$	−695058.	−0.979289
$220$	0	0
$221$	−834489.	−1.14932
$222$	0	0
$223$	−73302.7	−0.0987093	−0.0493546	−	0.998781i	$-0.515716\pi$
$223$	−73302.7	−0.0987093	−0.0493546	+	0.998781i	$0.515716\pi$
$224$	0	0
$225$	5.25257e6	6.91696
$226$	0	0
$227$	304080.	0.391673	0.195836	−	0.980637i	$-0.437258\pi$
$227$	304080.	0.391673	0.195836	+	0.980637i	$0.437258\pi$
$228$	0	0
$229$	−742901.	−0.936142	−0.468071	−	0.883691i	$-0.655051\pi$
$229$	−742901.	−0.936142	−0.468071	+	0.883691i	$0.655051\pi$
$230$	0	0
$231$	2.32545e6	2.86732
$232$	0	0
$233$	1.09629e6	1.32293	0.661466	−	0.749975i	$-0.269935\pi$
$233$	1.09629e6	1.32293	0.661466	+	0.749975i	$0.269935\pi$
$234$	0	0
$235$	−2.24095e6	−2.64705
$236$	0	0
$237$	−255668.	−0.295668
$238$	0	0
$239$	−220395.	−0.249578	−0.124789	−	0.992183i	$-0.539825\pi$
$239$	−220395.	−0.249578	−0.124789	+	0.992183i	$0.539825\pi$
$240$	0	0
$241$	−1.34886e6	−1.49597	−0.747985	−	0.663716i	$-0.768978\pi$
$241$	−1.34886e6	−1.49597	−0.747985	+	0.663716i	$0.768978\pi$
$242$	0	0
$243$	−2.63283e6	−2.86027
$244$	0	0
$245$	−368395.	−0.392102
$246$	0	0
$247$	−622108.	−0.648819
$248$	0	0
$249$	−2.32891e6	−2.38043
$250$	0	0
$251$	327888.	0.328504	0.164252	−	0.986418i	$-0.447479\pi$
$251$	327888.	0.328504	0.164252	+	0.986418i	$0.447479\pi$
$252$	0	0
$253$	1.73894e6	1.70798
$254$	0	0
$255$	−3.08337e6	−2.96944
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	−1.92132e6	−1.77972
$260$	0	0
$261$	664697.	0.603980
$262$	0	0
$263$	−1.87750e6	−1.67375	−0.836876	−	0.547392i	$-0.815621\pi$
$263$	−1.87750e6	−1.67375	−0.836876	+	0.547392i	$0.815621\pi$
$264$	0	0
$265$	−1.45939e6	−1.27660
$266$	0	0
$267$	408873.	0.351003
$268$	0	0
$269$	1.73782e6	1.46428	0.732140	−	0.681154i	$-0.238522\pi$
$269$	1.73782e6	1.46428	0.732140	+	0.681154i	$0.238522\pi$
$270$	0	0
$271$	−370918.	−0.306800	−0.153400	−	0.988164i	$-0.549022\pi$
$271$	−370918.	−0.306800	−0.153400	+	0.988164i	$0.549022\pi$
$272$	0	0
$273$	3.59511e6	2.91948
$274$	0	0
$275$	4.63151e6	3.69310
$276$	0	0
$277$	915636.	0.717007	0.358504	−	0.933528i	$-0.383287\pi$
$277$	915636.	0.717007	0.358504	+	0.933528i	$0.383287\pi$
$278$	0	0
$279$	5.99433e6	4.61031
$280$	0	0
$281$	−687953.	−0.519748	−0.259874	−	0.965642i	$-0.583681\pi$
$281$	−687953.	−0.519748	−0.259874	+	0.965642i	$0.583681\pi$
$282$	0	0
$283$	−720057.	−0.534442	−0.267221	−	0.963635i	$-0.586105\pi$
$283$	−720057.	−0.534442	−0.267221	+	0.963635i	$0.586105\pi$
$284$	0	0
$285$	−2.29864e6	−1.67633
$286$	0	0
$287$	1.97711e6	1.41686
$288$	0	0
$289$	−469794.	−0.330874
$290$	0	0
$291$	−17366.3	−0.0120219
$292$	0	0
$293$	2.50105e6	1.70198	0.850988	−	0.525185i	$-0.176004\pi$
$293$	2.50105e6	1.70198	0.850988	+	0.525185i	$0.176004\pi$
$294$	0	0
$295$	3.67439e6	2.45827
$296$	0	0
$297$	−6.29311e6	−4.13975
$298$	0	0
$299$	2.68838e6	1.73905
$300$	0	0
$301$	703981.	0.447863
$302$	0	0
$303$	−2.63629e6	−1.64963
$304$	0	0
$305$	−3.85379e6	−2.37213
$306$	0	0
$307$	890959.	0.539526	0.269763	−	0.962927i	$-0.413055\pi$
$307$	890959.	0.539526	0.269763	+	0.962927i	$0.413055\pi$
$308$	0	0
$309$	856731.	0.510444
$310$	0	0
$311$	158483.	0.0929142	0.0464571	−	0.998920i	$-0.485207\pi$
$311$	158483.	0.0929142	0.0464571	+	0.998920i	$0.485207\pi$
$312$	0	0
$313$	45190.0	0.0260724	0.0130362	−	0.999915i	$-0.495850\pi$
$313$	45190.0	0.0260724	0.0130362	+	0.999915i	$0.495850\pi$
$314$	0	0
$315$	9.57782e6	5.43864
$316$	0	0
$317$	−517968.	−0.289504	−0.144752	−	0.989468i	$-0.546238\pi$
$317$	−517968.	−0.289504	−0.144752	+	0.989468i	$0.546238\pi$
$318$	0	0
$319$	586104.	0.322476
$320$	0	0
$321$	−5.03078e6	−2.72504
$322$	0	0
$323$	708269.	0.377739
$324$	0	0
$325$	7.16024e6	3.76027
$326$	0	0
$327$	−76465.3	−0.0395453
$328$	0	0
$329$	−2.97475e6	−1.51517
$330$	0	0
$331$	1.35614e6	0.680355	0.340178	−	0.940361i	$-0.389513\pi$
$331$	1.35614e6	0.680355	0.340178	+	0.940361i	$0.389513\pi$
$332$	0	0
$333$	8.48087e6	4.19112
$334$	0	0
$335$	82690.4	0.0402572
$336$	0	0
$337$	2.24146e6	1.07512	0.537559	−	0.843226i	$-0.319346\pi$
$337$	2.24146e6	1.07512	0.537559	+	0.843226i	$0.319346\pi$
$338$	0	0
$339$	−316610.	−0.149632
$340$	0	0
$341$	5.28556e6	2.46153
$342$	0	0
$343$	1.90230e6	0.873059
$344$	0	0
$345$	9.93334e6	4.49311
$346$	0	0
$347$	2.22588e6	0.992379	0.496189	−	0.868214i	$-0.334732\pi$
$347$	2.22588e6	0.992379	0.496189	+	0.868214i	$0.334732\pi$
$348$	0	0
$349$	−1.43930e6	−0.632538	−0.316269	−	0.948670i	$-0.602430\pi$
$349$	−1.43930e6	−0.632538	−0.316269	+	0.948670i	$0.602430\pi$
$350$	0	0
$351$	−9.72906e6	−4.21505
$352$	0	0
$353$	440453.	0.188132	0.0940660	−	0.995566i	$-0.470014\pi$
$353$	440453.	0.188132	0.0940660	+	0.995566i	$0.470014\pi$
$354$	0	0
$355$	6.27718e6	2.64359
$356$	0	0
$357$	−4.09302e6	−1.69971
$358$	0	0
$359$	−1.58024e6	−0.647121	−0.323561	−	0.946207i	$-0.604880\pi$
$359$	−1.58024e6	−0.647121	−0.323561	+	0.946207i	$0.604880\pi$
$360$	0	0
$361$	−1.94809e6	−0.786757
$362$	0	0
$363$	−4.29784e6	−1.71192
$364$	0	0
$365$	−2.52425e6	−0.991744
$366$	0	0
$367$	209027.	0.0810098	0.0405049	−	0.999179i	$-0.487103\pi$
$367$	209027.	0.0810098	0.0405049	+	0.999179i	$0.487103\pi$
$368$	0	0
$369$	−8.72713e6	−3.33661
$370$	0	0
$371$	−1.93726e6	−0.730725
$372$	0	0
$373$	−3.19924e6	−1.19062	−0.595312	−	0.803495i	$-0.702972\pi$
$373$	−3.19924e6	−1.19062	−0.595312	+	0.803495i	$0.702972\pi$
$374$	0	0
$375$	1.65710e7	6.08515
$376$	0	0
$377$	906107.	0.328342
$378$	0	0
$379$	−857016.	−0.306472	−0.153236	−	0.988190i	$-0.548969\pi$
$379$	−857016.	−0.306472	−0.153236	+	0.988190i	$0.548969\pi$
$380$	0	0
$381$	−5.94274e6	−2.09737
$382$	0	0
$383$	−4.23723e6	−1.47600	−0.737998	−	0.674803i	$-0.764229\pi$
$383$	−4.23723e6	−1.47600	−0.737998	+	0.674803i	$0.764229\pi$
$384$	0	0
$385$	8.44534e6	2.90379
$386$	0	0
$387$	−3.10742e6	−1.05469
$388$	0	0
$389$	−767746.	−0.257243	−0.128622	−	0.991694i	$-0.541055\pi$
$389$	−767746.	−0.257243	−0.128622	+	0.991694i	$0.541055\pi$
$390$	0	0
$391$	−3.06071e6	−1.01247
$392$	0	0
$393$	9.24618e6	3.01982
$394$	0	0
$395$	−928509.	−0.299429
$396$	0	0
$397$	2.85407e6	0.908841	0.454420	−	0.890787i	$-0.349846\pi$
$397$	2.85407e6	0.908841	0.454420	+	0.890787i	$0.349846\pi$
$398$	0	0
$399$	−3.05133e6	−0.959527
$400$	0	0
$401$	−3.96413e6	−1.23108	−0.615541	−	0.788105i	$-0.711063\pi$
$401$	−3.96413e6	−1.23108	−0.615541	+	0.788105i	$0.711063\pi$
$402$	0	0
$403$	8.17140e6	2.50631
$404$	0	0
$405$	−1.95903e7	−5.93477
$406$	0	0
$407$	7.47809e6	2.23772
$408$	0	0
$409$	2.27925e6	0.673728	0.336864	−	0.941553i	$-0.390634\pi$
$409$	2.27925e6	0.673728	0.336864	+	0.941553i	$0.390634\pi$
$410$	0	0
$411$	−7.27605e6	−2.12467
$412$	0	0
$413$	4.87757e6	1.40711
$414$	0	0
$415$	−8.45793e6	−2.41070
$416$	0	0
$417$	−5.85981e6	−1.65023
$418$	0	0
$419$	276232.	0.0768669	0.0384334	−	0.999261i	$-0.487763\pi$
$419$	276232.	0.0768669	0.0384334	+	0.999261i	$0.487763\pi$
$420$	0	0
$421$	−5.67886e6	−1.56155	−0.780776	−	0.624812i	$-0.785176\pi$
$421$	−5.67886e6	−1.56155	−0.780776	+	0.624812i	$0.785176\pi$
$422$	0	0
$423$	1.31308e7	3.56811
$424$	0	0
$425$	−8.15192e6	−2.18921
$426$	0	0
$427$	−5.11572e6	−1.35780
$428$	0	0
$429$	−1.39927e7	−3.67079
$430$	0	0
$431$	−3.72400e6	−0.965644	−0.482822	−	0.875719i	$-0.660388\pi$
$431$	−3.72400e6	−0.965644	−0.482822	+	0.875719i	$0.660388\pi$
$432$	0	0
$433$	1.04626e6	0.268176	0.134088	−	0.990969i	$-0.457189\pi$
$433$	1.04626e6	0.268176	0.134088	+	0.990969i	$0.457189\pi$
$434$	0	0
$435$	3.34800e6	0.848324
$436$	0	0
$437$	−2.28175e6	−0.571563
$438$	0	0
$439$	−5.02599e6	−1.24469	−0.622344	−	0.782744i	$-0.713820\pi$
$439$	−5.02599e6	−1.24469	−0.622344	+	0.782744i	$0.713820\pi$
$440$	0	0
$441$	2.15860e6	0.528537
$442$	0	0
$443$	−2.55911e6	−0.619555	−0.309777	−	0.950809i	$-0.600255\pi$
$443$	−2.55911e6	−0.619555	−0.309777	+	0.950809i	$0.600255\pi$
$444$	0	0
$445$	1.48491e6	0.355467
$446$	0	0
$447$	−1.23163e7	−2.91550
$448$	0	0
$449$	8.40957e6	1.96860	0.984301	−	0.176497i	$-0.0564765\pi$
$449$	8.40957e6	1.96860	0.984301	+	0.176497i	$0.0564765\pi$
$450$	0	0
$451$	−7.69524e6	−1.78148
$452$	0	0
$453$	−7.52216e6	−1.72225
$454$	0	0
$455$	1.30564e7	2.95661
$456$	0	0
$457$	6.35369e6	1.42310	0.711550	−	0.702636i	$-0.247994\pi$
$457$	6.35369e6	1.42310	0.711550	+	0.702636i	$0.247994\pi$
$458$	0	0
$459$	1.10765e7	2.45398
$460$	0	0
$461$	4.62544e6	1.01368	0.506840	−	0.862040i	$-0.330814\pi$
$461$	4.62544e6	1.01368	0.506840	+	0.862040i	$0.330814\pi$
$462$	0	0
$463$	6.61940e6	1.43505	0.717523	−	0.696534i	$-0.245276\pi$
$463$	6.61940e6	1.43505	0.717523	+	0.696534i	$0.245276\pi$
$464$	0	0
$465$	3.01927e7	6.47544
$466$	0	0
$467$	−1.47846e6	−0.313701	−0.156851	−	0.987622i	$-0.550134\pi$
$467$	−1.47846e6	−0.313701	−0.156851	+	0.987622i	$0.550134\pi$
$468$	0	0
$469$	109767.	0.0230431
$470$	0	0
$471$	−1.15579e7	−2.40064
$472$	0	0
$473$	−2.74000e6	−0.563117
$474$	0	0
$475$	−6.07722e6	−1.23587
$476$	0	0
$477$	8.55122e6	1.72081
$478$	0	0
$479$	−3.14959e6	−0.627213	−0.313607	−	0.949553i	$-0.601537\pi$
$479$	−3.14959e6	−0.627213	−0.313607	+	0.949553i	$0.601537\pi$
$480$	0	0
$481$	1.15610e7	2.27842
$482$	0	0
$483$	1.31860e7	2.57185
$484$	0	0
$485$	−63069.2	−0.0121748
$486$	0	0
$487$	−2.20529e6	−0.421350	−0.210675	−	0.977556i	$-0.567566\pi$
$487$	−2.20529e6	−0.421350	−0.210675	+	0.977556i	$0.567566\pi$
$488$	0	0
$489$	9.93034e6	1.87798
$490$	0	0
$491$	−8.43782e6	−1.57952	−0.789762	−	0.613413i	$-0.789796\pi$
$491$	−8.43782e6	−1.57952	−0.789762	+	0.613413i	$0.789796\pi$
$492$	0	0
$493$	−1.03160e6	−0.191159
$494$	0	0
$495$	−3.72784e7	−6.83824
$496$	0	0
$497$	8.33265e6	1.51319
$498$	0	0
$499$	−5.39653e6	−0.970204	−0.485102	−	0.874458i	$-0.661217\pi$
$499$	−5.39653e6	−0.970204	−0.485102	+	0.874458i	$0.661217\pi$
$500$	0	0
$501$	4.75475e6	0.846318
$502$	0	0
$503$	756947.	0.133397	0.0666984	−	0.997773i	$-0.478753\pi$
$503$	756947.	0.133397	0.0666984	+	0.997773i	$0.478753\pi$
$504$	0	0
$505$	−9.57424e6	−1.67061
$506$	0	0
$507$	−1.06744e7	−1.84427
$508$	0	0
$509$	1.44937e6	0.247962	0.123981	−	0.992285i	$-0.460434\pi$
$509$	1.44937e6	0.247962	0.123981	+	0.992285i	$0.460434\pi$
$510$	0	0
$511$	−3.35081e6	−0.567673
$512$	0	0
$513$	8.25750e6	1.38534
$514$	0	0
$515$	3.11139e6	0.516936
$516$	0	0
$517$	1.15782e7	1.90508
$518$	0	0
$519$	9.03061e6	1.47163
$520$	0	0
$521$	−8.18199e6	−1.32058	−0.660290	−	0.751011i	$-0.729566\pi$
$521$	−8.18199e6	−1.32058	−0.660290	+	0.751011i	$0.729566\pi$
$522$	0	0
$523$	567859.	0.0907793	0.0453896	−	0.998969i	$-0.485547\pi$
$523$	567859.	0.0907793	0.0453896	+	0.998969i	$0.485547\pi$
$524$	0	0
$525$	3.51198e7	5.56100
$526$	0	0
$527$	−9.30312e6	−1.45916
$528$	0	0
$529$	3.42399e6	0.531977
$530$	0	0
$531$	−2.15300e7	−3.31365
$532$	0	0
$533$	−1.18967e7	−1.81388
$534$	0	0
$535$	−1.82703e7	−2.75970
$536$	0	0
$537$	−1.55524e7	−2.32736
$538$	0	0
$539$	1.90337e6	0.282196
$540$	0	0
$541$	5.48471e6	0.805677	0.402838	−	0.915271i	$-0.368024\pi$
$541$	5.48471e6	0.805677	0.402838	+	0.915271i	$0.368024\pi$
$542$	0	0
$543$	−1.23013e7	−1.79040
$544$	0	0
$545$	−277699.	−0.0400483
$546$	0	0
$547$	−1.36329e6	−0.194814	−0.0974069	−	0.995245i	$-0.531055\pi$
$547$	−1.36329e6	−0.194814	−0.0974069	+	0.995245i	$0.531055\pi$
$548$	0	0
$549$	2.25812e7	3.19754
$550$	0	0
$551$	−769055.	−0.107914
$552$	0	0
$553$	−1.23255e6	−0.171393
$554$	0	0
$555$	4.27171e7	5.88666
$556$	0	0
$557$	−1.17090e7	−1.59912	−0.799561	−	0.600585i	$-0.794934\pi$
$557$	−1.17090e7	−1.59912	−0.799561	+	0.600585i	$0.794934\pi$
$558$	0	0
$559$	−4.23601e6	−0.573360
$560$	0	0
$561$	1.59307e7	2.13711
$562$	0	0
$563$	−1.27530e7	−1.69566	−0.847832	−	0.530265i	$-0.822093\pi$
$563$	−1.27530e7	−1.69566	−0.847832	+	0.530265i	$0.822093\pi$
$564$	0	0
$565$	−1.14984e6	−0.151536
$566$	0	0
$567$	−2.60052e7	−3.39705
$568$	0	0
$569$	1.39156e7	1.80187	0.900933	−	0.433958i	$-0.142883\pi$
$569$	1.39156e7	1.80187	0.900933	+	0.433958i	$0.142883\pi$
$570$	0	0
$571$	1.13086e7	1.45150	0.725750	−	0.687959i	$-0.241493\pi$
$571$	1.13086e7	1.45150	0.725750	+	0.687959i	$0.241493\pi$
$572$	0	0
$573$	8.25011e6	1.04972
$574$	0	0
$575$	2.62621e7	3.31253
$576$	0	0
$577$	−3.05689e6	−0.382244	−0.191122	−	0.981566i	$-0.561213\pi$
$577$	−3.05689e6	−0.382244	−0.191122	+	0.981566i	$0.561213\pi$
$578$	0	0
$579$	27713.3	0.00343551
$580$	0	0
$581$	−1.12275e7	−1.37988
$582$	0	0
$583$	7.54013e6	0.918771
$584$	0	0
$585$	−5.76318e7	−6.96262
$586$	0	0
$587$	1.11239e7	1.33249	0.666243	−	0.745735i	$-0.267902\pi$
$587$	1.11239e7	1.33249	0.666243	+	0.745735i	$0.267902\pi$
$588$	0	0
$589$	−6.93544e6	−0.823732
$590$	0	0
$591$	1.03969e7	1.22444
$592$	0	0
$593$	−1.00089e6	−0.116882	−0.0584412	−	0.998291i	$-0.518613\pi$
$593$	−1.00089e6	−0.116882	−0.0584412	+	0.998291i	$0.518613\pi$
$594$	0	0
$595$	−1.48647e7	−1.72132
$596$	0	0
$597$	1.08546e7	1.24646
$598$	0	0
$599$	−1.32654e7	−1.51061	−0.755305	−	0.655373i	$-0.772511\pi$
$599$	−1.32654e7	−1.51061	−0.755305	+	0.655373i	$0.772511\pi$
$600$	0	0
$601$	8.21318e6	0.927524	0.463762	−	0.885960i	$-0.346499\pi$
$601$	8.21318e6	0.927524	0.463762	+	0.885960i	$0.346499\pi$
$602$	0	0
$603$	−484522.	−0.0542650
$604$	0	0
$605$	−1.56085e7	−1.73370
$606$	0	0
$607$	−1.46241e7	−1.61101	−0.805503	−	0.592591i	$-0.798105\pi$
$607$	−1.46241e7	−1.61101	−0.805503	+	0.592591i	$0.798105\pi$
$608$	0	0
$609$	4.44430e6	0.485579
$610$	0	0
$611$	1.78997e7	1.93974
$612$	0	0
$613$	1.38549e7	1.48919	0.744596	−	0.667515i	$-0.232642\pi$
$613$	1.38549e7	1.48919	0.744596	+	0.667515i	$0.232642\pi$
$614$	0	0
$615$	−4.39575e7	−4.68646
$616$	0	0
$617$	−2.44331e6	−0.258384	−0.129192	−	0.991620i	$-0.541238\pi$
$617$	−2.44331e6	−0.258384	−0.129192	+	0.991620i	$0.541238\pi$
$618$	0	0
$619$	8.39103e6	0.880214	0.440107	−	0.897945i	$-0.354940\pi$
$619$	8.39103e6	0.880214	0.440107	+	0.897945i	$0.354940\pi$
$620$	0	0
$621$	−3.56839e7	−3.71316
$622$	0	0
$623$	1.97114e6	0.203469
$624$	0	0
$625$	3.40454e7	3.48625
$626$	0	0
$627$	1.18763e7	1.20645
$628$	0	0
$629$	−1.31622e7	−1.32648
$630$	0	0
$631$	−4.54075e6	−0.453999	−0.226999	−	0.973895i	$-0.572892\pi$
$631$	−4.54075e6	−0.453999	−0.226999	+	0.973895i	$0.572892\pi$
$632$	0	0
$633$	2.01082e7	1.99463
$634$	0	0
$635$	−2.15823e7	−2.12404
$636$	0	0
$637$	2.94258e6	0.287329
$638$	0	0
$639$	−3.67809e7	−3.56345
$640$	0	0
$641$	2.41022e6	0.231692	0.115846	−	0.993267i	$-0.463042\pi$
$641$	2.41022e6	0.231692	0.115846	+	0.993267i	$0.463042\pi$
$642$	0	0
$643$	1.68567e7	1.60785	0.803925	−	0.594731i	$-0.202742\pi$
$643$	1.68567e7	1.60785	0.803925	+	0.594731i	$0.202742\pi$
$644$	0	0
$645$	−1.56517e7	−1.48137
$646$	0	0
$647$	1.00907e7	0.947681	0.473841	−	0.880611i	$-0.342867\pi$
$647$	1.00907e7	0.947681	0.473841	+	0.880611i	$0.342867\pi$
$648$	0	0
$649$	−1.89843e7	−1.76922
$650$	0	0
$651$	4.00793e7	3.70653
$652$	0	0
$653$	−1.14156e6	−0.104765	−0.0523825	−	0.998627i	$-0.516681\pi$
$653$	−1.14156e6	−0.104765	−0.0523825	+	0.998627i	$0.516681\pi$
$654$	0	0
$655$	3.35794e7	3.05823
$656$	0	0
$657$	1.47907e7	1.33683
$658$	0	0
$659$	1.10448e7	0.990701	0.495350	−	0.868693i	$-0.335040\pi$
$659$	1.10448e7	0.990701	0.495350	+	0.868693i	$0.335040\pi$
$660$	0	0
$661$	2.03746e7	1.81379	0.906893	−	0.421360i	$-0.138447\pi$
$661$	2.03746e7	1.81379	0.906893	+	0.421360i	$0.138447\pi$
$662$	0	0
$663$	2.46286e7	2.17599
$664$	0	0
$665$	−1.10815e7	−0.971731
$666$	0	0
$667$	3.32339e6	0.289246
$668$	0	0
$669$	2.16341e6	0.186885
$670$	0	0
$671$	1.99112e7	1.70722
$672$	0	0
$673$	−4.48766e6	−0.381929	−0.190964	−	0.981597i	$-0.561161\pi$
$673$	−4.48766e6	−0.381929	−0.190964	+	0.981597i	$0.561161\pi$
$674$	0	0
$675$	−9.50408e7	−8.02880
$676$	0	0
$677$	−941194.	−0.0789237	−0.0394618	−	0.999221i	$-0.512564\pi$
$677$	−941194.	−0.0789237	−0.0394618	+	0.999221i	$0.512564\pi$
$678$	0	0
$679$	−83721.3	−0.00696886
$680$	0	0
$681$	−8.97444e6	−0.741548
$682$	0	0
$683$	−1.87838e7	−1.54075	−0.770376	−	0.637590i	$-0.779931\pi$
$683$	−1.87838e7	−1.54075	−0.770376	+	0.637590i	$0.779931\pi$
$684$	0	0
$685$	−2.64245e7	−2.15169
$686$	0	0
$687$	2.19255e7	1.77239
$688$	0	0
$689$	1.16569e7	0.935483
$690$	0	0
$691$	1.80279e7	1.43632	0.718158	−	0.695880i	$-0.244985\pi$
$691$	1.80279e7	1.43632	0.718158	+	0.695880i	$0.244985\pi$
$692$	0	0
$693$	−4.94852e7	−3.91419
$694$	0	0
$695$	−2.12811e7	−1.67122
$696$	0	0
$697$	1.35444e7	1.05603
$698$	0	0
$699$	−3.23554e7	−2.50469
$700$	0	0
$701$	1.50389e7	1.15590	0.577950	−	0.816072i	$-0.303853\pi$
$701$	1.50389e7	1.15590	0.577950	+	0.816072i	$0.303853\pi$
$702$	0	0
$703$	−9.81237e6	−0.748834
$704$	0	0
$705$	6.61380e7	5.01162
$706$	0	0
$707$	−1.27093e7	−0.956257
$708$	0	0
$709$	1.38935e7	1.03800	0.518999	−	0.854775i	$-0.326305\pi$
$709$	1.38935e7	1.03800	0.518999	+	0.854775i	$0.326305\pi$
$710$	0	0
$711$	5.44057e6	0.403618
$712$	0	0
$713$	2.99708e7	2.20787
$714$	0	0
$715$	−5.08175e7	−3.71747
$716$	0	0
$717$	6.50460e6	0.472523
$718$	0	0
$719$	−2.13131e7	−1.53753	−0.768765	−	0.639531i	$-0.779129\pi$
$719$	−2.13131e7	−1.53753	−0.768765	+	0.639531i	$0.779129\pi$
$720$	0	0
$721$	4.13022e6	0.295893
$722$	0	0
$723$	3.98093e7	2.83230
$724$	0	0
$725$	8.85155e6	0.625423
$726$	0	0
$727$	1.66321e6	0.116711	0.0583555	−	0.998296i	$-0.481414\pi$
$727$	1.66321e6	0.116711	0.0583555	+	0.998296i	$0.481414\pi$
$728$	0	0
$729$	3.32899e7	2.32003
$730$	0	0
$731$	4.82268e6	0.333807
$732$	0	0
$733$	7.92214e6	0.544606	0.272303	−	0.962212i	$-0.412215\pi$
$733$	7.92214e6	0.544606	0.272303	+	0.962212i	$0.412215\pi$
$734$	0	0
$735$	1.08726e7	0.742360
$736$	0	0
$737$	−427232.	−0.0289731
$738$	0	0
$739$	9.31692e6	0.627569	0.313784	−	0.949494i	$-0.398403\pi$
$739$	9.31692e6	0.627569	0.313784	+	0.949494i	$0.398403\pi$
$740$	0	0
$741$	1.83605e7	1.22840
$742$	0	0
$743$	−2.46653e7	−1.63913	−0.819566	−	0.572984i	$-0.805786\pi$
$743$	−2.46653e7	−1.63913	−0.819566	+	0.572984i	$0.805786\pi$
$744$	0	0
$745$	−4.47293e7	−2.95258
$746$	0	0
$747$	4.95589e7	3.24953
$748$	0	0
$749$	−2.42530e7	−1.57965
$750$	0	0
$751$	−2.40806e6	−0.155800	−0.0778999	−	0.996961i	$-0.524821\pi$
$751$	−2.40806e6	−0.155800	−0.0778999	+	0.996961i	$0.524821\pi$
$752$	0	0
$753$	−9.67708e6	−0.621952
$754$	0	0
$755$	−2.73183e7	−1.74416
$756$	0	0
$757$	2.96733e7	1.88203	0.941013	−	0.338370i	$-0.109876\pi$
$757$	2.96733e7	1.88203	0.941013	+	0.338370i	$0.109876\pi$
$758$	0	0
$759$	−5.13220e7	−3.23370
$760$	0	0
$761$	1.95285e6	0.122238	0.0611191	−	0.998130i	$-0.480533\pi$
$761$	1.95285e6	0.122238	0.0611191	+	0.998130i	$0.480533\pi$
$762$	0	0
$763$	−368632.	−0.0229236
$764$	0	0
$765$	6.56137e7	4.05360
$766$	0	0
$767$	−2.93494e7	−1.80140
$768$	0	0
$769$	5.40601e6	0.329656	0.164828	−	0.986322i	$-0.447293\pi$
$769$	5.40601e6	0.329656	0.164828	+	0.986322i	$0.447293\pi$
$770$	0	0
$771$	1.94933e6	0.118100
$772$	0	0
$773$	−1.35947e7	−0.818313	−0.409156	−	0.912464i	$-0.634177\pi$
$773$	−1.35947e7	−0.818313	−0.409156	+	0.912464i	$0.634177\pi$
$774$	0	0
$775$	7.98245e7	4.77399
$776$	0	0
$777$	5.67048e7	3.36952
$778$	0	0
$779$	1.00973e7	0.596158
$780$	0	0
$781$	−3.24320e7	−1.90259
$782$	0	0
$783$	−1.20271e7	−0.701064
$784$	0	0
$785$	−4.19748e7	−2.43117
$786$	0	0
$787$	−1.38334e7	−0.796146	−0.398073	−	0.917354i	$-0.630321\pi$
$787$	−1.38334e7	−0.796146	−0.398073	+	0.917354i	$0.630321\pi$
$788$	0	0
$789$	5.54115e7	3.16889
$790$	0	0
$791$	−1.52635e6	−0.0867387
$792$	0	0
$793$	3.07824e7	1.73828
$794$	0	0
$795$	4.30714e7	2.41697
$796$	0	0
$797$	2.14967e7	1.19874	0.599371	−	0.800471i	$-0.295418\pi$
$797$	2.14967e7	1.19874	0.599371	+	0.800471i	$0.295418\pi$
$798$	0	0
$799$	−2.03788e7	−1.12930
$800$	0	0
$801$	−8.70077e6	−0.479156
$802$	0	0
$803$	1.30419e7	0.713759
$804$	0	0
$805$	4.78877e7	2.60456
$806$	0	0
$807$	−5.12890e7	−2.77230
$808$	0	0
$809$	1.51738e7	0.815124	0.407562	−	0.913178i	$-0.366379\pi$
$809$	1.51738e7	0.815124	0.407562	+	0.913178i	$0.366379\pi$
$810$	0	0
$811$	−2.40572e7	−1.28438	−0.642188	−	0.766547i	$-0.721973\pi$
$811$	−2.40572e7	−1.28438	−0.642188	+	0.766547i	$0.721973\pi$
$812$	0	0
$813$	1.09471e7	0.580860
$814$	0	0
$815$	3.60641e7	1.90187
$816$	0	0
$817$	3.59529e6	0.188443
$818$	0	0
$819$	−7.65034e7	−3.98539
$820$	0	0
$821$	−1.59254e7	−0.824582	−0.412291	−	0.911052i	$-0.635271\pi$
$821$	−1.59254e7	−0.824582	−0.412291	+	0.911052i	$0.635271\pi$
$822$	0	0
$823$	−1.79433e7	−0.923427	−0.461714	−	0.887029i	$-0.652765\pi$
$823$	−1.79433e7	−0.923427	−0.461714	+	0.887029i	$0.652765\pi$
$824$	0	0
$825$	−1.36692e8	−6.99209
$826$	0	0
$827$	−1.88066e7	−0.956197	−0.478098	−	0.878306i	$-0.658674\pi$
$827$	−1.88066e7	−0.956197	−0.478098	+	0.878306i	$0.658674\pi$
$828$	0	0
$829$	3.34097e6	0.168844	0.0844221	−	0.996430i	$-0.473096\pi$
$829$	3.34097e6	0.168844	0.0844221	+	0.996430i	$0.473096\pi$
$830$	0	0
$831$	−2.70235e7	−1.35750
$832$	0	0
$833$	−3.35012e6	−0.167281
$834$	0	0
$835$	1.72679e7	0.857082
$836$	0	0
$837$	−1.08462e8	−5.35138
$838$	0	0
$839$	−3.60763e7	−1.76936	−0.884681	−	0.466197i	$-0.845624\pi$
$839$	−3.60763e7	−1.76936	−0.884681	+	0.466197i	$0.845624\pi$
$840$	0	0
$841$	−1.93910e7	−0.945389
$842$	0	0
$843$	2.03039e7	0.984033
$844$	0	0
$845$	−3.87664e7	−1.86773
$846$	0	0
$847$	−2.07195e7	−0.992365
$848$	0	0
$849$	2.12513e7	1.01185
$850$	0	0
$851$	4.24031e7	2.00712
$852$	0	0
$853$	−1.10135e7	−0.518268	−0.259134	−	0.965841i	$-0.583437\pi$
$853$	−1.10135e7	−0.518268	−0.259134	+	0.965841i	$0.583437\pi$
$854$	0	0
$855$	4.89148e7	2.28836
$856$	0	0
$857$	3.51754e6	0.163601	0.0818007	−	0.996649i	$-0.473933\pi$
$857$	3.51754e6	0.163601	0.0818007	+	0.996649i	$0.473933\pi$
$858$	0	0
$859$	2.45856e7	1.13684	0.568419	−	0.822740i	$-0.307555\pi$
$859$	2.45856e7	1.13684	0.568419	+	0.822740i	$0.307555\pi$
$860$	0	0
$861$	−5.83514e7	−2.68252
$862$	0	0
$863$	3.64259e7	1.66488	0.832441	−	0.554114i	$-0.186943\pi$
$863$	3.64259e7	1.66488	0.832441	+	0.554114i	$0.186943\pi$
$864$	0	0
$865$	3.27965e7	1.49035
$866$	0	0
$867$	1.38652e7	0.626439
$868$	0	0
$869$	4.79728e6	0.215499
$870$	0	0
$871$	−660495.	−0.0295001
$872$	0	0
$873$	369552.	0.0164112
$874$	0	0
$875$	7.98874e7	3.52743
$876$	0	0
$877$	1.67331e7	0.734644	0.367322	−	0.930094i	$-0.380275\pi$
$877$	1.67331e7	0.734644	0.367322	+	0.930094i	$0.380275\pi$
$878$	0	0
$879$	−7.38145e7	−3.22233
$880$	0	0
$881$	3.78395e7	1.64250	0.821250	−	0.570568i	$-0.193277\pi$
$881$	3.78395e7	1.64250	0.821250	+	0.570568i	$0.193277\pi$
$882$	0	0
$883$	1.39235e7	0.600963	0.300481	−	0.953788i	$-0.402853\pi$
$883$	1.39235e7	0.600963	0.300481	+	0.953788i	$0.402853\pi$
$884$	0	0
$885$	−1.08444e8	−4.65421
$886$	0	0
$887$	2.81019e7	1.19930	0.599648	−	0.800264i	$-0.295307\pi$
$887$	2.81019e7	1.19930	0.599648	+	0.800264i	$0.295307\pi$
$888$	0	0
$889$	−2.86494e7	−1.21580
$890$	0	0
$891$	1.01216e8	4.27126
$892$	0	0
$893$	−1.51923e7	−0.637521
$894$	0	0
$895$	−5.64819e7	−2.35696
$896$	0	0
$897$	−7.93431e7	−3.29252
$898$	0	0
$899$	1.01015e7	0.416859
$900$	0	0
$901$	−1.32714e7	−0.544633
$902$	0	0
$903$	−2.07769e7	−0.847932
$904$	0	0
$905$	−4.46746e7	−1.81317
$906$	0	0
$907$	9.09367e6	0.367046	0.183523	−	0.983015i	$-0.441250\pi$
$907$	9.09367e6	0.367046	0.183523	+	0.983015i	$0.441250\pi$
$908$	0	0
$909$	5.61000e7	2.25192
$910$	0	0
$911$	2.89135e7	1.15426	0.577131	−	0.816651i	$-0.304172\pi$
$911$	2.89135e7	1.15426	0.577131	+	0.816651i	$0.304172\pi$
$912$	0	0
$913$	4.36991e7	1.73498
$914$	0	0
$915$	1.13738e8	4.49112
$916$	0	0
$917$	4.45750e7	1.75052
$918$	0	0
$919$	2.08942e7	0.816086	0.408043	−	0.912963i	$-0.366211\pi$
$919$	2.08942e7	0.816086	0.408043	+	0.912963i	$0.366211\pi$
$920$	0	0
$921$	−2.62953e7	−1.02148
$922$	0	0
$923$	−5.01394e7	−1.93720
$924$	0	0
$925$	1.12937e8	4.33992
$926$	0	0
$927$	−1.82311e7	−0.696809
$928$	0	0
$929$	−3.32245e7	−1.26304	−0.631522	−	0.775358i	$-0.717569\pi$
$929$	−3.32245e7	−1.26304	−0.631522	+	0.775358i	$0.717569\pi$
$930$	0	0
$931$	−2.49750e6	−0.0944346
$932$	0	0
$933$	−4.67738e6	−0.175913
$934$	0	0
$935$	5.78556e7	2.16429
$936$	0	0
$937$	−1.63490e7	−0.608334	−0.304167	−	0.952619i	$-0.598378\pi$
$937$	−1.63490e7	−0.608334	−0.304167	+	0.952619i	$0.598378\pi$
$938$	0	0
$939$	−1.33371e6	−0.0493625
$940$	0	0
$941$	−3.42837e7	−1.26216	−0.631078	−	0.775719i	$-0.717387\pi$
$941$	−3.42837e7	−1.26216	−0.631078	+	0.775719i	$0.717387\pi$
$942$	0	0
$943$	−4.36344e7	−1.59790
$944$	0	0
$945$	−1.73303e8	−6.31285
$946$	0	0
$947$	1.10424e7	0.400118	0.200059	−	0.979784i	$-0.435887\pi$
$947$	1.10424e7	0.400118	0.200059	+	0.979784i	$0.435887\pi$
$948$	0	0
$949$	2.01626e7	0.726742
$950$	0	0
$951$	1.52870e7	0.548114
$952$	0	0
$953$	4.68483e7	1.67094	0.835471	−	0.549535i	$-0.185195\pi$
$953$	4.68483e7	1.67094	0.835471	+	0.549535i	$0.185195\pi$
$954$	0	0
$955$	2.99620e7	1.06307
$956$	0	0
$957$	−1.72979e7	−0.610540
$958$	0	0
$959$	−3.50772e7	−1.23162
$960$	0	0
$961$	6.24680e7	2.18197
$962$	0	0
$963$	1.07054e8	3.71996
$964$	0	0
$965$	100647.	0.00347921
$966$	0	0
$967$	−2.31587e7	−0.796430	−0.398215	−	0.917292i	$-0.630370\pi$
$967$	−2.31587e7	−0.796430	−0.398215	+	0.917292i	$0.630370\pi$
$968$	0	0
$969$	−2.09034e7	−0.715168
$970$	0	0
$971$	−3.68934e7	−1.25574	−0.627872	−	0.778317i	$-0.716074\pi$
$971$	−3.68934e7	−1.25574	−0.627872	+	0.778317i	$0.716074\pi$
$972$	0	0
$973$	−2.82496e7	−0.956601
$974$	0	0
$975$	−2.11323e8	−7.11927
$976$	0	0
$977$	−3.13268e7	−1.04998	−0.524989	−	0.851109i	$-0.675930\pi$
$977$	−3.13268e7	−1.04998	−0.524989	+	0.851109i	$0.675930\pi$
$978$	0	0
$979$	−7.67200e6	−0.255830
$980$	0	0
$981$	1.62717e6	0.0539834
$982$	0	0
$983$	−5.75211e7	−1.89864	−0.949322	−	0.314304i	$-0.898229\pi$
$983$	−5.75211e7	−1.89864	−0.949322	+	0.314304i	$0.898229\pi$
$984$	0	0
$985$	3.77586e7	1.24001
$986$	0	0
$987$	8.77949e7	2.86864
$988$	0	0
$989$	−1.55367e7	−0.505088
$990$	0	0
$991$	−1.60827e7	−0.520205	−0.260102	−	0.965581i	$-0.583756\pi$
$991$	−1.60827e7	−0.520205	−0.260102	+	0.965581i	$0.583756\pi$
$992$	0	0
$993$	−4.00244e7	−1.28811
$994$	0	0
$995$	3.94207e7	1.26231
$996$	0	0
$997$	2.79106e7	0.889264	0.444632	−	0.895713i	$-0.353334\pi$
$997$	2.79106e7	0.889264	0.444632	+	0.895713i	$0.353334\pi$
$998$	0	0
$999$	−1.53454e8	−4.86480

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-29.5134 q^{3} -107.184 q^{5} -142.282 q^{7} +628.041 q^{9} +O(q^{10})\)	\(q-29.5134 q^{3} -107.184 q^{5} -142.282 q^{7} +628.041 q^{9} +553.782 q^{11} +856.139 q^{13} +3163.37 q^{15} -974.712 q^{17} -726.644 q^{19} +4199.21 q^{21} +3140.12 q^{23} +8363.41 q^{25} -11363.9 q^{27} +1058.37 q^{29} +9544.48 q^{31} -16344.0 q^{33} +15250.3 q^{35} +13503.7 q^{37} -25267.6 q^{39} -13895.8 q^{41} -4947.80 q^{43} -67316.0 q^{45} +20907.5 q^{47} +3437.03 q^{49} +28767.1 q^{51} +13615.7 q^{53} -59356.6 q^{55} +21445.7 q^{57} -34281.1 q^{59} +35954.9 q^{61} -89358.7 q^{63} -91764.4 q^{65} -771.481 q^{67} -92675.5 q^{69} -58564.5 q^{71} +23550.6 q^{73} -246833. q^{75} -78792.9 q^{77} +8662.76 q^{79} +182773. q^{81} +78910.3 q^{83} +104474. q^{85} -31236.0 q^{87} -13853.8 q^{89} -121813. q^{91} -281690. q^{93} +77884.6 q^{95} +588.420 q^{97} +347798. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Introduction

L-functions

Modular forms

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