LMFDB - Embedded newform 1028.6.a.a.1.17

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			1.17
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)

$f(q)$	$=$	$q-9.78147 q^{3} -19.4476 q^{5} +160.887 q^{7} -147.323 q^{9} +O(q^{10})$	$q-9.78147 q^{3} -19.4476 q^{5} +160.887 q^{7} -147.323 q^{9} -771.004 q^{11} -30.0673 q^{13} +190.226 q^{15} +2118.85 q^{17} +2005.06 q^{19} -1573.71 q^{21} -4093.11 q^{23} -2746.79 q^{25} +3817.93 q^{27} +1493.03 q^{29} +7854.37 q^{31} +7541.55 q^{33} -3128.87 q^{35} -13114.2 q^{37} +294.103 q^{39} +7753.02 q^{41} -4732.69 q^{43} +2865.08 q^{45} +13678.9 q^{47} +9077.54 q^{49} -20725.5 q^{51} +23235.2 q^{53} +14994.2 q^{55} -19612.5 q^{57} -19673.3 q^{59} +6868.75 q^{61} -23702.3 q^{63} +584.738 q^{65} -7627.47 q^{67} +40036.7 q^{69} +10446.0 q^{71} -28970.5 q^{73} +26867.6 q^{75} -124044. q^{77} +48737.6 q^{79} -1545.55 q^{81} -88863.0 q^{83} -41206.6 q^{85} -14604.0 q^{87} +105897. q^{89} -4837.43 q^{91} -76827.3 q^{93} -38993.8 q^{95} -85340.5 q^{97} +113586. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−9.78147	−0.627482	−0.313741	−	0.949509i	$-0.601582\pi$
$3$	−9.78147	−0.627482	−0.313741	+	0.949509i	$0.601582\pi$
$4$	0	0
$5$	−19.4476	−0.347890	−0.173945	−	0.984755i	$-0.555651\pi$
$5$	−19.4476	−0.347890	−0.173945	+	0.984755i	$0.555651\pi$
$6$	0	0
$7$	160.887	1.24101	0.620505	−	0.784203i	$-0.286928\pi$
$7$	160.887	1.24101	0.620505	+	0.784203i	$0.286928\pi$
$8$	0	0
$9$	−147.323	−0.606267
$10$	0	0
$11$	−771.004	−1.92121	−0.960605	−	0.277917i	$-0.910356\pi$
$11$	−771.004	−1.92121	−0.960605	+	0.277917i	$0.910356\pi$
$12$	0	0
$13$	−30.0673	−0.0493442	−0.0246721	−	0.999696i	$-0.507854\pi$
$13$	−30.0673	−0.0493442	−0.0246721	+	0.999696i	$0.507854\pi$
$14$	0	0
$15$	190.226	0.218295
$16$	0	0
$17$	2118.85	1.77819	0.889094	−	0.457726i	$-0.151336\pi$
$17$	2118.85	1.77819	0.889094	+	0.457726i	$0.151336\pi$
$18$	0	0
$19$	2005.06	1.27422	0.637110	−	0.770773i	$-0.280130\pi$
$19$	2005.06	1.27422	0.637110	+	0.770773i	$0.280130\pi$
$20$	0	0
$21$	−1573.71	−0.778711
$22$	0	0
$23$	−4093.11	−1.61337	−0.806686	−	0.590981i	$-0.798741\pi$
$23$	−4093.11	−1.61337	−0.806686	+	0.590981i	$0.798741\pi$
$24$	0	0
$25$	−2746.79	−0.878973
$26$	0	0
$27$	3817.93	1.00790
$28$	0	0
$29$	1493.03	0.329665	0.164833	−	0.986322i	$-0.447292\pi$
$29$	1493.03	0.329665	0.164833	+	0.986322i	$0.447292\pi$
$30$	0	0
$31$	7854.37	1.46794	0.733968	−	0.679184i	$-0.237666\pi$
$31$	7854.37	1.46794	0.733968	+	0.679184i	$0.237666\pi$
$32$	0	0
$33$	7541.55	1.20552
$34$	0	0
$35$	−3128.87	−0.431735
$36$	0	0
$37$	−13114.2	−1.57485	−0.787423	−	0.616413i	$-0.788585\pi$
$37$	−13114.2	−1.57485	−0.787423	+	0.616413i	$0.788585\pi$
$38$	0	0
$39$	294.103	0.0309626
$40$	0	0
$41$	7753.02	0.720296	0.360148	−	0.932895i	$-0.382726\pi$
$41$	7753.02	0.720296	0.360148	+	0.932895i	$0.382726\pi$
$42$	0	0
$43$	−4732.69	−0.390335	−0.195167	−	0.980770i	$-0.562525\pi$
$43$	−4732.69	−0.390335	−0.195167	+	0.980770i	$0.562525\pi$
$44$	0	0
$45$	2865.08	0.210914
$46$	0	0
$47$	13678.9	0.903250	0.451625	−	0.892208i	$-0.350844\pi$
$47$	13678.9	0.903250	0.451625	+	0.892208i	$0.350844\pi$
$48$	0	0
$49$	9077.54	0.540105
$50$	0	0
$51$	−20725.5	−1.11578
$52$	0	0
$53$	23235.2	1.13620	0.568102	−	0.822958i	$-0.307678\pi$
$53$	23235.2	1.13620	0.568102	+	0.822958i	$0.307678\pi$
$54$	0	0
$55$	14994.2	0.668370
$56$	0	0
$57$	−19612.5	−0.799550
$58$	0	0
$59$	−19673.3	−0.735779	−0.367890	−	0.929869i	$-0.619920\pi$
$59$	−19673.3	−0.735779	−0.367890	+	0.929869i	$0.619920\pi$
$60$	0	0
$61$	6868.75	0.236348	0.118174	−	0.992993i	$-0.462296\pi$
$61$	6868.75	0.236348	0.118174	+	0.992993i	$0.462296\pi$
$62$	0	0
$63$	−23702.3	−0.752383
$64$	0	0
$65$	584.738	0.0171664
$66$	0	0
$67$	−7627.47	−0.207584	−0.103792	−	0.994599i	$-0.533098\pi$
$67$	−7627.47	−0.207584	−0.103792	+	0.994599i	$0.533098\pi$
$68$	0	0
$69$	40036.7	1.01236
$70$	0	0
$71$	10446.0	0.245925	0.122962	−	0.992411i	$-0.460761\pi$
$71$	10446.0	0.245925	0.122962	+	0.992411i	$0.460761\pi$
$72$	0	0
$73$	−28970.5	−0.636282	−0.318141	−	0.948043i	$-0.603058\pi$
$73$	−28970.5	−0.636282	−0.318141	+	0.948043i	$0.603058\pi$
$74$	0	0
$75$	26867.6	0.551539
$76$	0	0
$77$	−124044.	−2.38424
$78$	0	0
$79$	48737.6	0.878610	0.439305	−	0.898338i	$-0.355225\pi$
$79$	48737.6	0.878610	0.439305	+	0.898338i	$0.355225\pi$
$80$	0	0
$81$	−1545.55	−0.0261741
$82$	0	0
$83$	−88863.0	−1.41588	−0.707939	−	0.706274i	$-0.750375\pi$
$83$	−88863.0	−1.41588	−0.707939	+	0.706274i	$0.750375\pi$
$84$	0	0
$85$	−41206.6	−0.618613
$86$	0	0
$87$	−14604.0	−0.206859
$88$	0	0
$89$	105897.	1.41712	0.708561	−	0.705650i	$-0.249345\pi$
$89$	105897.	1.41712	0.708561	+	0.705650i	$0.249345\pi$
$90$	0	0
$91$	−4837.43	−0.0612367
$92$	0	0
$93$	−76827.3	−0.921103
$94$	0	0
$95$	−38993.8	−0.443288
$96$	0	0
$97$	−85340.5	−0.920928	−0.460464	−	0.887678i	$-0.652317\pi$
$97$	−85340.5	−0.920928	−0.460464	+	0.887678i	$0.652317\pi$
$98$	0	0
$99$	113586.	1.16477
$100$	0	0
$101$	−1334.15	−0.0130137	−0.00650683	−	0.999979i	$-0.502071\pi$
$101$	−1334.15	−0.0130137	−0.00650683	+	0.999979i	$0.502071\pi$
$102$	0	0
$103$	−79117.6	−0.734819	−0.367409	−	0.930059i	$-0.619755\pi$
$103$	−79117.6	−0.734819	−0.367409	+	0.930059i	$0.619755\pi$
$104$	0	0
$105$	30604.9	0.270906
$106$	0	0
$107$	211446.	1.78542	0.892708	−	0.450635i	$-0.148803\pi$
$107$	211446.	1.78542	0.892708	+	0.450635i	$0.148803\pi$
$108$	0	0
$109$	−30333.9	−0.244547	−0.122274	−	0.992496i	$-0.539019\pi$
$109$	−30333.9	−0.244547	−0.122274	+	0.992496i	$0.539019\pi$
$110$	0	0
$111$	128276.	0.988187
$112$	0	0
$113$	7824.87	0.0576476	0.0288238	−	0.999585i	$-0.490824\pi$
$113$	7824.87	0.0576476	0.0288238	+	0.999585i	$0.490824\pi$
$114$	0	0
$115$	79601.4	0.561276
$116$	0	0
$117$	4429.60	0.0299158
$118$	0	0
$119$	340895.	2.20675
$120$	0	0
$121$	433396.	2.69105
$122$	0	0
$123$	−75836.0	−0.451973
$124$	0	0
$125$	114192.	0.653676
$126$	0	0
$127$	104830.	0.576735	0.288368	−	0.957520i	$-0.406887\pi$
$127$	104830.	0.576735	0.288368	+	0.957520i	$0.406887\pi$
$128$	0	0
$129$	46292.7	0.244928
$130$	0	0
$131$	122910.	0.625763	0.312882	−	0.949792i	$-0.398706\pi$
$131$	122910.	0.625763	0.312882	+	0.949792i	$0.398706\pi$
$132$	0	0
$133$	322588.	1.58132
$134$	0	0
$135$	−74249.7	−0.350639
$136$	0	0
$137$	203406.	0.925897	0.462948	−	0.886385i	$-0.346792\pi$
$137$	203406.	0.925897	0.462948	+	0.886385i	$0.346792\pi$
$138$	0	0
$139$	−139851.	−0.613943	−0.306972	−	0.951719i	$-0.599316\pi$
$139$	−139851.	−0.613943	−0.306972	+	0.951719i	$0.599316\pi$
$140$	0	0
$141$	−133800.	−0.566773
$142$	0	0
$143$	23182.0	0.0948006
$144$	0	0
$145$	−29035.9	−0.114687
$146$	0	0
$147$	−88791.7	−0.338906
$148$	0	0
$149$	−167452.	−0.617910	−0.308955	−	0.951077i	$-0.599979\pi$
$149$	−167452.	−0.617910	−0.308955	+	0.951077i	$0.599979\pi$
$150$	0	0
$151$	−245495.	−0.876193	−0.438096	−	0.898928i	$-0.644347\pi$
$151$	−245495.	−0.876193	−0.438096	+	0.898928i	$0.644347\pi$
$152$	0	0
$153$	−312155.	−1.07806
$154$	0	0
$155$	−152749.	−0.510680
$156$	0	0
$157$	−387550.	−1.25481	−0.627406	−	0.778693i	$-0.715883\pi$
$157$	−387550.	−1.25481	−0.627406	+	0.778693i	$0.715883\pi$
$158$	0	0
$159$	−227274.	−0.712948
$160$	0	0
$161$	−658528.	−2.00221
$162$	0	0
$163$	−468479.	−1.38109	−0.690544	−	0.723291i	$-0.742629\pi$
$163$	−468479.	−1.38109	−0.690544	+	0.723291i	$0.742629\pi$
$164$	0	0
$165$	−146665.	−0.419390
$166$	0	0
$167$	55866.0	0.155009	0.0775044	−	0.996992i	$-0.475305\pi$
$167$	55866.0	0.155009	0.0775044	+	0.996992i	$0.475305\pi$
$168$	0	0
$169$	−370389.	−0.997565
$170$	0	0
$171$	−295392.	−0.772517
$172$	0	0
$173$	571385.	1.45149	0.725744	−	0.687965i	$-0.241496\pi$
$173$	571385.	1.45149	0.725744	+	0.687965i	$0.241496\pi$
$174$	0	0
$175$	−441922.	−1.09081
$176$	0	0
$177$	192434.	0.461688
$178$	0	0
$179$	−257516.	−0.600719	−0.300359	−	0.953826i	$-0.597107\pi$
$179$	−257516.	−0.600719	−0.300359	+	0.953826i	$0.597107\pi$
$180$	0	0
$181$	−415994.	−0.943823	−0.471912	−	0.881646i	$-0.656436\pi$
$181$	−415994.	−0.943823	−0.471912	+	0.881646i	$0.656436\pi$
$182$	0	0
$183$	−67186.4	−0.148304
$184$	0	0
$185$	255041.	0.547873
$186$	0	0
$187$	−1.63364e6	−3.41627
$188$	0	0
$189$	614255.	1.25082
$190$	0	0
$191$	230385.	0.456952	0.228476	−	0.973550i	$-0.426626\pi$
$191$	230385.	0.456952	0.228476	+	0.973550i	$0.426626\pi$
$192$	0	0
$193$	−171991.	−0.332362	−0.166181	−	0.986095i	$-0.553144\pi$
$193$	−171991.	−0.332362	−0.166181	+	0.986095i	$0.553144\pi$
$194$	0	0
$195$	−5719.60	−0.0107716
$196$	0	0
$197$	382677.	0.702533	0.351266	−	0.936276i	$-0.385751\pi$
$197$	382677.	0.702533	0.351266	+	0.936276i	$0.385751\pi$
$198$	0	0
$199$	−166130.	−0.297383	−0.148691	−	0.988884i	$-0.547506\pi$
$199$	−166130.	−0.297383	−0.148691	+	0.988884i	$0.547506\pi$
$200$	0	0
$201$	74607.9	0.130255
$202$	0	0
$203$	240209.	0.409118
$204$	0	0
$205$	−150778.	−0.250584
$206$	0	0
$207$	603009.	0.978133
$208$	0	0
$209$	−1.54591e6	−2.44804
$210$	0	0
$211$	−620961.	−0.960193	−0.480096	−	0.877216i	$-0.659398\pi$
$211$	−620961.	−0.960193	−0.480096	+	0.877216i	$0.659398\pi$
$212$	0	0
$213$	−102177.	−0.154313
$214$	0	0
$215$	92039.6	0.135793
$216$	0	0
$217$	1.26366e6	1.82172
$218$	0	0
$219$	283374.	0.399255
$220$	0	0
$221$	−63708.1	−0.0877433
$222$	0	0
$223$	−150457.	−0.202604	−0.101302	−	0.994856i	$-0.532301\pi$
$223$	−150457.	−0.202604	−0.101302	+	0.994856i	$0.532301\pi$
$224$	0	0
$225$	404665.	0.532892
$226$	0	0
$227$	323222.	0.416328	0.208164	−	0.978094i	$-0.433251\pi$
$227$	323222.	0.416328	0.208164	+	0.978094i	$0.433251\pi$
$228$	0	0
$229$	−60964.4	−0.0768223	−0.0384112	−	0.999262i	$-0.512230\pi$
$229$	−60964.4	−0.0768223	−0.0384112	+	0.999262i	$0.512230\pi$
$230$	0	0
$231$	1.21334e6	1.49607
$232$	0	0
$233$	−1.54948e6	−1.86980	−0.934900	−	0.354912i	$-0.884511\pi$
$233$	−1.54948e6	−1.86980	−0.934900	+	0.354912i	$0.884511\pi$
$234$	0	0
$235$	−266023.	−0.314232
$236$	0	0
$237$	−476725.	−0.551312
$238$	0	0
$239$	−846305.	−0.958367	−0.479184	−	0.877715i	$-0.659067\pi$
$239$	−846305.	−0.958367	−0.479184	+	0.877715i	$0.659067\pi$
$240$	0	0
$241$	−1.23857e6	−1.37366	−0.686829	−	0.726819i	$-0.740998\pi$
$241$	−1.23857e6	−1.37366	−0.686829	+	0.726819i	$0.740998\pi$
$242$	0	0
$243$	−912640.	−0.991479
$244$	0	0
$245$	−176537.	−0.187897
$246$	0	0
$247$	−60286.9	−0.0628754
$248$	0	0
$249$	869211.	0.888438
$250$	0	0
$251$	−1.31880e6	−1.32128	−0.660642	−	0.750701i	$-0.729716\pi$
$251$	−1.31880e6	−1.32128	−0.660642	+	0.750701i	$0.729716\pi$
$252$	0	0
$253$	3.15581e6	3.09963
$254$	0	0
$255$	403061.	0.388169
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	−2.10990e6	−1.95440
$260$	0	0
$261$	−219957.	−0.199865
$262$	0	0
$263$	−1.56202e6	−1.39250	−0.696251	−	0.717798i	$-0.745150\pi$
$263$	−1.56202e6	−1.39250	−0.696251	+	0.717798i	$0.745150\pi$
$264$	0	0
$265$	−451869.	−0.395274
$266$	0	0
$267$	−1.03582e6	−0.889218
$268$	0	0
$269$	−468613.	−0.394852	−0.197426	−	0.980318i	$-0.563258\pi$
$269$	−468613.	−0.394852	−0.197426	+	0.980318i	$0.563258\pi$
$270$	0	0
$271$	−1.21354e6	−1.00377	−0.501883	−	0.864935i	$-0.667359\pi$
$271$	−1.21354e6	−1.00377	−0.501883	+	0.864935i	$0.667359\pi$
$272$	0	0
$273$	47317.2	0.0384249
$274$	0	0
$275$	2.11779e6	1.68869
$276$	0	0
$277$	−1.96400e6	−1.53795	−0.768977	−	0.639277i	$-0.779234\pi$
$277$	−1.96400e6	−1.53795	−0.768977	+	0.639277i	$0.779234\pi$
$278$	0	0
$279$	−1.15713e6	−0.889960
$280$	0	0
$281$	57866.4	0.0437181	0.0218590	−	0.999761i	$-0.493042\pi$
$281$	57866.4	0.0437181	0.0218590	+	0.999761i	$0.493042\pi$
$282$	0	0
$283$	1.27177e6	0.943938	0.471969	−	0.881615i	$-0.343543\pi$
$283$	1.27177e6	0.943938	0.471969	+	0.881615i	$0.343543\pi$
$284$	0	0
$285$	381416.	0.278155
$286$	0	0
$287$	1.24736e6	0.893895
$288$	0	0
$289$	3.06966e6	2.16195
$290$	0	0
$291$	834756.	0.577866
$292$	0	0
$293$	1.98359e6	1.34984	0.674920	−	0.737891i	$-0.264178\pi$
$293$	1.98359e6	1.34984	0.674920	+	0.737891i	$0.264178\pi$
$294$	0	0
$295$	382599.	0.255970
$296$	0	0
$297$	−2.94364e6	−1.93639
$298$	0	0
$299$	123069.	0.0796105
$300$	0	0
$301$	−761427.	−0.484409
$302$	0	0
$303$	13049.9	0.00816584
$304$	0	0
$305$	−133581.	−0.0822232
$306$	0	0
$307$	−1.76692e6	−1.06997	−0.534984	−	0.844862i	$-0.679682\pi$
$307$	−1.76692e6	−1.06997	−0.534984	+	0.844862i	$0.679682\pi$
$308$	0	0
$309$	773887.	0.461085
$310$	0	0
$311$	1.13089e6	0.663008	0.331504	−	0.943454i	$-0.392444\pi$
$311$	1.13089e6	0.663008	0.331504	+	0.943454i	$0.392444\pi$
$312$	0	0
$313$	−2.06986e6	−1.19421	−0.597105	−	0.802163i	$-0.703683\pi$
$313$	−2.06986e6	−1.19421	−0.597105	+	0.802163i	$0.703683\pi$
$314$	0	0
$315$	460953.	0.261746
$316$	0	0
$317$	−3.57123e6	−1.99604	−0.998021	−	0.0628772i	$-0.979972\pi$
$317$	−3.57123e6	−1.99604	−0.998021	+	0.0628772i	$0.979972\pi$
$318$	0	0
$319$	−1.15113e6	−0.633357
$320$	0	0
$321$	−2.06825e6	−1.12032
$322$	0	0
$323$	4.24843e6	2.26580
$324$	0	0
$325$	82588.6	0.0433722
$326$	0	0
$327$	296711.	0.153449
$328$	0	0
$329$	2.20076e6	1.12094
$330$	0	0
$331$	−1.79790e6	−0.901976	−0.450988	−	0.892530i	$-0.648928\pi$
$331$	−1.79790e6	−0.901976	−0.450988	+	0.892530i	$0.648928\pi$
$332$	0	0
$333$	1.93202e6	0.954777
$334$	0	0
$335$	148336.	0.0722163
$336$	0	0
$337$	−1.67813e6	−0.804916	−0.402458	−	0.915438i	$-0.631844\pi$
$337$	−1.67813e6	−0.804916	−0.402458	+	0.915438i	$0.631844\pi$
$338$	0	0
$339$	−76538.7	−0.0361728
$340$	0	0
$341$	−6.05575e6	−2.82021
$342$	0	0
$343$	−1.24357e6	−0.570734
$344$	0	0
$345$	−778619.	−0.352190
$346$	0	0
$347$	−1.22751e6	−0.547271	−0.273636	−	0.961833i	$-0.588226\pi$
$347$	−1.22751e6	−0.547271	−0.273636	+	0.961833i	$0.588226\pi$
$348$	0	0
$349$	1.42936e6	0.628171	0.314085	−	0.949395i	$-0.398302\pi$
$349$	1.42936e6	0.628171	0.314085	+	0.949395i	$0.398302\pi$
$350$	0	0
$351$	−114795.	−0.0497342
$352$	0	0
$353$	3.20020e6	1.36691	0.683457	−	0.729991i	$-0.260476\pi$
$353$	3.20020e6	1.36691	0.683457	+	0.729991i	$0.260476\pi$
$354$	0	0
$355$	−203149.	−0.0855548
$356$	0	0
$357$	−3.33445e6	−1.38469
$358$	0	0
$359$	1.84200e6	0.754318	0.377159	−	0.926149i	$-0.376901\pi$
$359$	1.84200e6	0.754318	0.377159	+	0.926149i	$0.376901\pi$
$360$	0	0
$361$	1.54418e6	0.623636
$362$	0	0
$363$	−4.23925e6	−1.68858
$364$	0	0
$365$	563408.	0.221356
$366$	0	0
$367$	414075.	0.160477	0.0802387	−	0.996776i	$-0.474432\pi$
$367$	414075.	0.160477	0.0802387	+	0.996776i	$0.474432\pi$
$368$	0	0
$369$	−1.14220e6	−0.436692
$370$	0	0
$371$	3.73823e6	1.41004
$372$	0	0
$373$	−4.82622e6	−1.79612	−0.898060	−	0.439873i	$-0.855023\pi$
$373$	−4.82622e6	−1.79612	−0.898060	+	0.439873i	$0.855023\pi$
$374$	0	0
$375$	−1.11697e6	−0.410169
$376$	0	0
$377$	−44891.4	−0.0162671
$378$	0	0
$379$	3.37952e6	1.20853	0.604264	−	0.796784i	$-0.293467\pi$
$379$	3.37952e6	1.20853	0.604264	+	0.796784i	$0.293467\pi$
$380$	0	0
$381$	−1.02539e6	−0.361891
$382$	0	0
$383$	−3.51138e6	−1.22315	−0.611576	−	0.791185i	$-0.709464\pi$
$383$	−3.51138e6	−1.22315	−0.611576	+	0.791185i	$0.709464\pi$
$384$	0	0
$385$	2.41237e6	0.829453
$386$	0	0
$387$	697233.	0.236647
$388$	0	0
$389$	1.42309e6	0.476824	0.238412	−	0.971164i	$-0.423373\pi$
$389$	1.42309e6	0.476824	0.238412	+	0.971164i	$0.423373\pi$
$390$	0	0
$391$	−8.67269e6	−2.86888
$392$	0	0
$393$	−1.20224e6	−0.392655
$394$	0	0
$395$	−947831.	−0.305660
$396$	0	0
$397$	−3.77939e6	−1.20350	−0.601750	−	0.798685i	$-0.705529\pi$
$397$	−3.77939e6	−1.20350	−0.601750	+	0.798685i	$0.705529\pi$
$398$	0	0
$399$	−3.15539e6	−0.992249
$400$	0	0
$401$	−1.90507e6	−0.591629	−0.295814	−	0.955245i	$-0.595591\pi$
$401$	−1.90507e6	−0.591629	−0.295814	+	0.955245i	$0.595591\pi$
$402$	0	0
$403$	−236160.	−0.0724341
$404$	0	0
$405$	30057.4	0.00910570
$406$	0	0
$407$	1.01111e7	3.02561
$408$	0	0
$409$	−1.28998e6	−0.381307	−0.190654	−	0.981657i	$-0.561061\pi$
$409$	−1.28998e6	−0.381307	−0.190654	+	0.981657i	$0.561061\pi$
$410$	0	0
$411$	−1.98961e6	−0.580983
$412$	0	0
$413$	−3.16518e6	−0.913109
$414$	0	0
$415$	1.72818e6	0.492570
$416$	0	0
$417$	1.36795e6	0.385238
$418$	0	0
$419$	901428.	0.250840	0.125420	−	0.992104i	$-0.459972\pi$
$419$	901428.	0.250840	0.125420	+	0.992104i	$0.459972\pi$
$420$	0	0
$421$	−975599.	−0.268266	−0.134133	−	0.990963i	$-0.542825\pi$
$421$	−975599.	−0.268266	−0.134133	+	0.990963i	$0.542825\pi$
$422$	0	0
$423$	−2.01522e6	−0.547610
$424$	0	0
$425$	−5.82003e6	−1.56298
$426$	0	0
$427$	1.10509e6	0.293311
$428$	0	0
$429$	−226754.	−0.0594857
$430$	0	0
$431$	1.80814e6	0.468855	0.234428	−	0.972134i	$-0.424678\pi$
$431$	1.80814e6	0.468855	0.234428	+	0.972134i	$0.424678\pi$
$432$	0	0
$433$	1.16484e6	0.298570	0.149285	−	0.988794i	$-0.452303\pi$
$433$	1.16484e6	0.298570	0.149285	+	0.988794i	$0.452303\pi$
$434$	0	0
$435$	284014.	0.0719642
$436$	0	0
$437$	−8.20696e6	−2.05579
$438$	0	0
$439$	−1.46720e6	−0.363352	−0.181676	−	0.983358i	$-0.558152\pi$
$439$	−1.46720e6	−0.363352	−0.181676	+	0.983358i	$0.558152\pi$
$440$	0	0
$441$	−1.33733e6	−0.327448
$442$	0	0
$443$	4.11807e6	0.996976	0.498488	−	0.866897i	$-0.333889\pi$
$443$	4.11807e6	0.996976	0.498488	+	0.866897i	$0.333889\pi$
$444$	0	0
$445$	−2.05944e6	−0.493002
$446$	0	0
$447$	1.63793e6	0.387727
$448$	0	0
$449$	7.46174e6	1.74673	0.873363	−	0.487071i	$-0.161935\pi$
$449$	7.46174e6	1.74673	0.873363	+	0.487071i	$0.161935\pi$
$450$	0	0
$451$	−5.97761e6	−1.38384
$452$	0	0
$453$	2.40130e6	0.549795
$454$	0	0
$455$	94076.6	0.0213036
$456$	0	0
$457$	−3.62448e6	−0.811811	−0.405906	−	0.913915i	$-0.633044\pi$
$457$	−3.62448e6	−0.811811	−0.405906	+	0.913915i	$0.633044\pi$
$458$	0	0
$459$	8.08962e6	1.79224
$460$	0	0
$461$	−6.60509e6	−1.44753	−0.723763	−	0.690048i	$-0.757589\pi$
$461$	−6.60509e6	−1.44753	−0.723763	+	0.690048i	$0.757589\pi$
$462$	0	0
$463$	3.50836e6	0.760591	0.380296	−	0.924865i	$-0.375822\pi$
$463$	3.50836e6	0.760591	0.380296	+	0.924865i	$0.375822\pi$
$464$	0	0
$465$	1.49411e6	0.320442
$466$	0	0
$467$	3.62213e6	0.768549	0.384275	−	0.923219i	$-0.374452\pi$
$467$	3.62213e6	0.768549	0.384275	+	0.923219i	$0.374452\pi$
$468$	0	0
$469$	−1.22716e6	−0.257613
$470$	0	0
$471$	3.79081e6	0.787371
$472$	0	0
$473$	3.64892e6	0.749915
$474$	0	0
$475$	−5.50749e6	−1.12000
$476$	0	0
$477$	−3.42307e6	−0.688843
$478$	0	0
$479$	7.00794e6	1.39557	0.697785	−	0.716307i	$-0.254169\pi$
$479$	7.00794e6	1.39557	0.697785	+	0.716307i	$0.254169\pi$
$480$	0	0
$481$	394309.	0.0777096
$482$	0	0
$483$	6.44137e6	1.25635
$484$	0	0
$485$	1.65967e6	0.320382
$486$	0	0
$487$	5.93072e6	1.13314	0.566571	−	0.824013i	$-0.308270\pi$
$487$	5.93072e6	1.13314	0.566571	+	0.824013i	$0.308270\pi$
$488$	0	0
$489$	4.58241e6	0.866607
$490$	0	0
$491$	−2.91320e6	−0.545339	−0.272670	−	0.962108i	$-0.587907\pi$
$491$	−2.91320e6	−0.545339	−0.272670	+	0.962108i	$0.587907\pi$
$492$	0	0
$493$	3.16350e6	0.586207
$494$	0	0
$495$	−2.20899e6	−0.405210
$496$	0	0
$497$	1.68062e6	0.305195
$498$	0	0
$499$	−363003.	−0.0652617	−0.0326309	−	0.999467i	$-0.510389\pi$
$499$	−363003.	−0.0652617	−0.0326309	+	0.999467i	$0.510389\pi$
$500$	0	0
$501$	−546452.	−0.0972652
$502$	0	0
$503$	2.00628e6	0.353567	0.176783	−	0.984250i	$-0.443431\pi$
$503$	2.00628e6	0.353567	0.176783	+	0.984250i	$0.443431\pi$
$504$	0	0
$505$	25946.0	0.00452732
$506$	0	0
$507$	3.62295e6	0.625954
$508$	0	0
$509$	−9.45761e6	−1.61803	−0.809016	−	0.587787i	$-0.799999\pi$
$509$	−9.45761e6	−1.61803	−0.809016	+	0.587787i	$0.799999\pi$
$510$	0	0
$511$	−4.66097e6	−0.789631
$512$	0	0
$513$	7.65520e6	1.28429
$514$	0	0
$515$	1.53865e6	0.255636
$516$	0	0
$517$	−1.05465e7	−1.73533
$518$	0	0
$519$	−5.58899e6	−0.910782
$520$	0	0
$521$	−9.47555e6	−1.52936	−0.764680	−	0.644410i	$-0.777103\pi$
$521$	−9.47555e6	−1.52936	−0.764680	+	0.644410i	$0.777103\pi$
$522$	0	0
$523$	7.38844e6	1.18113	0.590566	−	0.806989i	$-0.298905\pi$
$523$	7.38844e6	1.18113	0.590566	+	0.806989i	$0.298905\pi$
$524$	0	0
$525$	4.32265e6	0.684466
$526$	0	0
$527$	1.66422e7	2.61026
$528$	0	0
$529$	1.03172e7	1.60297
$530$	0	0
$531$	2.89833e6	0.446078
$532$	0	0
$533$	−233113.	−0.0355425
$534$	0	0
$535$	−4.11212e6	−0.621128
$536$	0	0
$537$	2.51888e6	0.376940
$538$	0	0
$539$	−6.99882e6	−1.03766
$540$	0	0
$541$	−1.00389e7	−1.47466	−0.737328	−	0.675534i	$-0.763913\pi$
$541$	−1.00389e7	−1.47466	−0.737328	+	0.675534i	$0.763913\pi$
$542$	0	0
$543$	4.06904e6	0.592232
$544$	0	0
$545$	589923.	0.0850755
$546$	0	0
$547$	−1.04725e7	−1.49652	−0.748262	−	0.663403i	$-0.769111\pi$
$547$	−1.04725e7	−1.49652	−0.748262	+	0.663403i	$0.769111\pi$
$548$	0	0
$549$	−1.01192e6	−0.143290
$550$	0	0
$551$	2.99362e6	0.420066
$552$	0	0
$553$	7.84123e6	1.09036
$554$	0	0
$555$	−2.49467e6	−0.343780
$556$	0	0
$557$	−1.08554e6	−0.148254	−0.0741270	−	0.997249i	$-0.523617\pi$
$557$	−1.08554e6	−0.148254	−0.0741270	+	0.997249i	$0.523617\pi$
$558$	0	0
$559$	142299.	0.0192608
$560$	0	0
$561$	1.59794e7	2.14365
$562$	0	0
$563$	−353706.	−0.0470297	−0.0235148	−	0.999723i	$-0.507486\pi$
$563$	−353706.	−0.0470297	−0.0235148	+	0.999723i	$0.507486\pi$
$564$	0	0
$565$	−152175.	−0.0200550
$566$	0	0
$567$	−248659.	−0.0324823
$568$	0	0
$569$	−1.21426e7	−1.57228	−0.786140	−	0.618048i	$-0.787924\pi$
$569$	−1.21426e7	−1.57228	−0.786140	+	0.618048i	$0.787924\pi$
$570$	0	0
$571$	−8.98618e6	−1.15341	−0.576707	−	0.816951i	$-0.695662\pi$
$571$	−8.98618e6	−1.15341	−0.576707	+	0.816951i	$0.695662\pi$
$572$	0	0
$573$	−2.25350e6	−0.286729
$574$	0	0
$575$	1.12429e7	1.41811
$576$	0	0
$577$	−1.44577e6	−0.180784	−0.0903919	−	0.995906i	$-0.528812\pi$
$577$	−1.44577e6	−0.180784	−0.0903919	+	0.995906i	$0.528812\pi$
$578$	0	0
$579$	1.68232e6	0.208551
$580$	0	0
$581$	−1.42969e7	−1.75712
$582$	0	0
$583$	−1.79144e7	−2.18289
$584$	0	0
$585$	−86145.3	−0.0104074
$586$	0	0
$587$	2.07186e6	0.248179	0.124090	−	0.992271i	$-0.460399\pi$
$587$	2.07186e6	0.248179	0.124090	+	0.992271i	$0.460399\pi$
$588$	0	0
$589$	1.57485e7	1.87047
$590$	0	0
$591$	−3.74314e6	−0.440826
$592$	0	0
$593$	9.23826e6	1.07883	0.539416	−	0.842040i	$-0.318645\pi$
$593$	9.23826e6	1.07883	0.539416	+	0.842040i	$0.318645\pi$
$594$	0	0
$595$	−6.62959e6	−0.767705
$596$	0	0
$597$	1.62500e6	0.186602
$598$	0	0
$599$	1.25302e7	1.42689	0.713443	−	0.700713i	$-0.247135\pi$
$599$	1.25302e7	1.42689	0.713443	+	0.700713i	$0.247135\pi$
$600$	0	0
$601$	−6.29935e6	−0.711393	−0.355696	−	0.934602i	$-0.615756\pi$
$601$	−6.29935e6	−0.711393	−0.355696	+	0.934602i	$0.615756\pi$
$602$	0	0
$603$	1.12370e6	0.125851
$604$	0	0
$605$	−8.42853e6	−0.936189
$606$	0	0
$607$	1.36106e7	1.49935	0.749677	−	0.661804i	$-0.230209\pi$
$607$	1.36106e7	1.49935	0.749677	+	0.661804i	$0.230209\pi$
$608$	0	0
$609$	−2.34959e6	−0.256714
$610$	0	0
$611$	−411289.	−0.0445702
$612$	0	0
$613$	3.53237e6	0.379678	0.189839	−	0.981815i	$-0.439203\pi$
$613$	3.53237e6	0.379678	0.189839	+	0.981815i	$0.439203\pi$
$614$	0	0
$615$	1.47483e6	0.157237
$616$	0	0
$617$	3.13544e6	0.331578	0.165789	−	0.986161i	$-0.446983\pi$
$617$	3.13544e6	0.331578	0.165789	+	0.986161i	$0.446983\pi$
$618$	0	0
$619$	1.04099e7	1.09199	0.545995	−	0.837788i	$-0.316152\pi$
$619$	1.04099e7	1.09199	0.545995	+	0.837788i	$0.316152\pi$
$620$	0	0
$621$	−1.56272e7	−1.62612
$622$	0	0
$623$	1.70374e7	1.75866
$624$	0	0
$625$	6.36294e6	0.651566
$626$	0	0
$627$	1.51213e7	1.53610
$628$	0	0
$629$	−2.77870e7	−2.80037
$630$	0	0
$631$	5.24304e6	0.524215	0.262108	−	0.965039i	$-0.415582\pi$
$631$	5.24304e6	0.524215	0.262108	+	0.965039i	$0.415582\pi$
$632$	0	0
$633$	6.07392e6	0.602503
$634$	0	0
$635$	−2.03870e6	−0.200640
$636$	0	0
$637$	−272937.	−0.0266511
$638$	0	0
$639$	−1.53893e6	−0.149096
$640$	0	0
$641$	1.80441e7	1.73457	0.867283	−	0.497816i	$-0.165864\pi$
$641$	1.80441e7	1.73457	0.867283	+	0.497816i	$0.165864\pi$
$642$	0	0
$643$	1.95091e7	1.86084	0.930420	−	0.366496i	$-0.119442\pi$
$643$	1.95091e7	1.86084	0.930420	+	0.366496i	$0.119442\pi$
$644$	0	0
$645$	−900283.	−0.0852079
$646$	0	0
$647$	−1.34629e7	−1.26438	−0.632190	−	0.774814i	$-0.717844\pi$
$647$	−1.34629e7	−1.26438	−0.632190	+	0.774814i	$0.717844\pi$
$648$	0	0
$649$	1.51682e7	1.41359
$650$	0	0
$651$	−1.23605e7	−1.14310
$652$	0	0
$653$	−5.33048e6	−0.489197	−0.244598	−	0.969625i	$-0.578656\pi$
$653$	−5.33048e6	−0.489197	−0.244598	+	0.969625i	$0.578656\pi$
$654$	0	0
$655$	−2.39032e6	−0.217697
$656$	0	0
$657$	4.26802e6	0.385756
$658$	0	0
$659$	−6.91716e6	−0.620460	−0.310230	−	0.950661i	$-0.600406\pi$
$659$	−6.91716e6	−0.620460	−0.310230	+	0.950661i	$0.600406\pi$
$660$	0	0
$661$	3.28589e6	0.292516	0.146258	−	0.989246i	$-0.453277\pi$
$661$	3.28589e6	0.292516	0.146258	+	0.989246i	$0.453277\pi$
$662$	0	0
$663$	623159.	0.0550573
$664$	0	0
$665$	−6.27358e6	−0.550125
$666$	0	0
$667$	−6.11114e6	−0.531873
$668$	0	0
$669$	1.47169e6	0.127131
$670$	0	0
$671$	−5.29583e6	−0.454075
$672$	0	0
$673$	1.61815e7	1.37715	0.688574	−	0.725166i	$-0.258237\pi$
$673$	1.61815e7	1.37715	0.688574	+	0.725166i	$0.258237\pi$
$674$	0	0
$675$	−1.04871e7	−0.885919
$676$	0	0
$677$	−4.01620e6	−0.336778	−0.168389	−	0.985721i	$-0.553856\pi$
$677$	−4.01620e6	−0.336778	−0.168389	+	0.985721i	$0.553856\pi$
$678$	0	0
$679$	−1.37302e7	−1.14288
$680$	0	0
$681$	−3.16159e6	−0.261238
$682$	0	0
$683$	6.59184e6	0.540699	0.270349	−	0.962762i	$-0.412861\pi$
$683$	6.59184e6	0.540699	0.270349	+	0.962762i	$0.412861\pi$
$684$	0	0
$685$	−3.95577e6	−0.322110
$686$	0	0
$687$	596321.	0.0482046
$688$	0	0
$689$	−698620.	−0.0560651
$690$	0	0
$691$	1.13900e7	0.907460	0.453730	−	0.891139i	$-0.350093\pi$
$691$	1.13900e7	0.907460	0.453730	+	0.891139i	$0.350093\pi$
$692$	0	0
$693$	1.82746e7	1.44549
$694$	0	0
$695$	2.71977e6	0.213585
$696$	0	0
$697$	1.64275e7	1.28082
$698$	0	0
$699$	1.51562e7	1.17327
$700$	0	0
$701$	−9.67704e6	−0.743785	−0.371893	−	0.928276i	$-0.621291\pi$
$701$	−9.67704e6	−0.743785	−0.371893	+	0.928276i	$0.621291\pi$
$702$	0	0
$703$	−2.62949e7	−2.00670
$704$	0	0
$705$	2.60210e6	0.197175
$706$	0	0
$707$	−214646.	−0.0161501
$708$	0	0
$709$	−1.49465e7	−1.11667	−0.558334	−	0.829616i	$-0.688559\pi$
$709$	−1.49465e7	−1.11667	−0.558334	+	0.829616i	$0.688559\pi$
$710$	0	0
$711$	−7.18016e6	−0.532672
$712$	0	0
$713$	−3.21488e7	−2.36832
$714$	0	0
$715$	−450835.	−0.0329802
$716$	0	0
$717$	8.27811e6	0.601358
$718$	0	0
$719$	−1.15806e7	−0.835425	−0.417713	−	0.908579i	$-0.637168\pi$
$719$	−1.15806e7	−0.835425	−0.417713	+	0.908579i	$0.637168\pi$
$720$	0	0
$721$	−1.27290e7	−0.911917
$722$	0	0
$723$	1.21151e7	0.861945
$724$	0	0
$725$	−4.10104e6	−0.289767
$726$	0	0
$727$	1.75378e7	1.23066	0.615330	−	0.788270i	$-0.289023\pi$
$727$	1.75378e7	1.23066	0.615330	+	0.788270i	$0.289023\pi$
$728$	0	0
$729$	9.30253e6	0.648309
$730$	0	0
$731$	−1.00279e7	−0.694088
$732$	0	0
$733$	4.84518e6	0.333081	0.166541	−	0.986035i	$-0.446740\pi$
$733$	4.84518e6	0.333081	0.166541	+	0.986035i	$0.446740\pi$
$734$	0	0
$735$	1.72679e6	0.117902
$736$	0	0
$737$	5.88081e6	0.398812
$738$	0	0
$739$	9.15612e6	0.616737	0.308369	−	0.951267i	$-0.400217\pi$
$739$	9.15612e6	0.616737	0.308369	+	0.951267i	$0.400217\pi$
$740$	0	0
$741$	589695.	0.0394531
$742$	0	0
$743$	−3.43487e6	−0.228264	−0.114132	−	0.993466i	$-0.536409\pi$
$743$	−3.43487e6	−0.228264	−0.114132	+	0.993466i	$0.536409\pi$
$744$	0	0
$745$	3.25655e6	0.214965
$746$	0	0
$747$	1.30915e7	0.858400
$748$	0	0
$749$	3.40188e7	2.21572
$750$	0	0
$751$	−2.24740e7	−1.45405	−0.727025	−	0.686611i	$-0.759098\pi$
$751$	−2.24740e7	−1.45405	−0.727025	+	0.686611i	$0.759098\pi$
$752$	0	0
$753$	1.28998e7	0.829081
$754$	0	0
$755$	4.77429e6	0.304819
$756$	0	0
$757$	1.29889e7	0.823822	0.411911	−	0.911224i	$-0.364861\pi$
$757$	1.29889e7	0.823822	0.411911	+	0.911224i	$0.364861\pi$
$758$	0	0
$759$	−3.08684e7	−1.94496
$760$	0	0
$761$	−1.95550e7	−1.22404	−0.612021	−	0.790841i	$-0.709643\pi$
$761$	−1.95550e7	−1.22404	−0.612021	+	0.790841i	$0.709643\pi$
$762$	0	0
$763$	−4.88033e6	−0.303485
$764$	0	0
$765$	6.07067e6	0.375045
$766$	0	0
$767$	591524.	0.0363065
$768$	0	0
$769$	6.13777e6	0.374278	0.187139	−	0.982333i	$-0.440078\pi$
$769$	6.13777e6	0.374278	0.187139	+	0.982333i	$0.440078\pi$
$770$	0	0
$771$	−646056.	−0.0391412
$772$	0	0
$773$	1.25417e7	0.754930	0.377465	−	0.926024i	$-0.376796\pi$
$773$	1.25417e7	0.754930	0.377465	+	0.926024i	$0.376796\pi$
$774$	0	0
$775$	−2.15743e7	−1.29028
$776$	0	0
$777$	2.06380e7	1.22635
$778$	0	0
$779$	1.55453e7	0.917816
$780$	0	0
$781$	−8.05388e6	−0.472474
$782$	0	0
$783$	5.70028e6	0.332271
$784$	0	0
$785$	7.53693e6	0.436536
$786$	0	0
$787$	−140792.	−0.00810293	−0.00405146	−	0.999992i	$-0.501290\pi$
$787$	−140792.	−0.00810293	−0.00405146	+	0.999992i	$0.501290\pi$
$788$	0	0
$789$	1.52788e7	0.873770
$790$	0	0
$791$	1.25892e6	0.0715412
$792$	0	0
$793$	−206525.	−0.0116624
$794$	0	0
$795$	4.41995e6	0.248027
$796$	0	0
$797$	−2.87450e7	−1.60294	−0.801468	−	0.598037i	$-0.795947\pi$
$797$	−2.87450e7	−1.60294	−0.801468	+	0.598037i	$0.795947\pi$
$798$	0	0
$799$	2.89836e7	1.60615
$800$	0	0
$801$	−1.56010e7	−0.859153
$802$	0	0
$803$	2.23364e7	1.22243
$804$	0	0
$805$	1.28068e7	0.696548
$806$	0	0
$807$	4.58373e6	0.247762
$808$	0	0
$809$	−2.13421e7	−1.14648	−0.573240	−	0.819387i	$-0.694314\pi$
$809$	−2.13421e7	−1.14648	−0.573240	+	0.819387i	$0.694314\pi$
$810$	0	0
$811$	1.60757e6	0.0858257	0.0429129	−	0.999079i	$-0.486336\pi$
$811$	1.60757e6	0.0858257	0.0429129	+	0.999079i	$0.486336\pi$
$812$	0	0
$813$	1.18703e7	0.629845
$814$	0	0
$815$	9.11081e6	0.480466
$816$	0	0
$817$	−9.48935e6	−0.497372
$818$	0	0
$819$	712664.	0.0371257
$820$	0	0
$821$	2.42696e7	1.25662	0.628311	−	0.777962i	$-0.283746\pi$
$821$	2.42696e7	1.25662	0.628311	+	0.777962i	$0.283746\pi$
$822$	0	0
$823$	−2.82158e7	−1.45209	−0.726045	−	0.687647i	$-0.758644\pi$
$823$	−2.82158e7	−1.45209	−0.726045	+	0.687647i	$0.758644\pi$
$824$	0	0
$825$	−2.07151e7	−1.05962
$826$	0	0
$827$	−3.57673e7	−1.81854	−0.909268	−	0.416210i	$-0.863358\pi$
$827$	−3.57673e7	−1.81854	−0.909268	+	0.416210i	$0.863358\pi$
$828$	0	0
$829$	−3.75276e7	−1.89655	−0.948276	−	0.317447i	$-0.897174\pi$
$829$	−3.75276e7	−1.89655	−0.948276	+	0.317447i	$0.897174\pi$
$830$	0	0
$831$	1.92109e7	0.965038
$832$	0	0
$833$	1.92339e7	0.960408
$834$	0	0
$835$	−1.08646e6	−0.0539260
$836$	0	0
$837$	2.99874e7	1.47954
$838$	0	0
$839$	−1.43453e7	−0.703565	−0.351783	−	0.936082i	$-0.614424\pi$
$839$	−1.43453e7	−0.703565	−0.351783	+	0.936082i	$0.614424\pi$
$840$	0	0
$841$	−1.82820e7	−0.891321
$842$	0	0
$843$	−566019.	−0.0274323
$844$	0	0
$845$	7.20319e6	0.347043
$846$	0	0
$847$	6.97277e7	3.33962
$848$	0	0
$849$	−1.24398e7	−0.592304
$850$	0	0
$851$	5.36780e7	2.54081
$852$	0	0
$853$	−2.65788e7	−1.25073	−0.625365	−	0.780333i	$-0.715050\pi$
$853$	−2.65788e7	−1.25073	−0.625365	+	0.780333i	$0.715050\pi$
$854$	0	0
$855$	5.74467e6	0.268751
$856$	0	0
$857$	2.58892e7	1.20411	0.602057	−	0.798453i	$-0.294348\pi$
$857$	2.58892e7	1.20411	0.602057	+	0.798453i	$0.294348\pi$
$858$	0	0
$859$	2.23511e7	1.03351	0.516756	−	0.856132i	$-0.327139\pi$
$859$	2.23511e7	1.03351	0.516756	+	0.856132i	$0.327139\pi$
$860$	0	0
$861$	−1.22010e7	−0.560903
$862$	0	0
$863$	−1.59962e7	−0.731124	−0.365562	−	0.930787i	$-0.619123\pi$
$863$	−1.59962e7	−0.731124	−0.365562	+	0.930787i	$0.619123\pi$
$864$	0	0
$865$	−1.11121e7	−0.504958
$866$	0	0
$867$	−3.00258e7	−1.35658
$868$	0	0
$869$	−3.75769e7	−1.68799
$870$	0	0
$871$	229338.	0.0102431
$872$	0	0
$873$	1.25726e7	0.558328
$874$	0	0
$875$	1.83720e7	0.811218
$876$	0	0
$877$	−2.93238e7	−1.28742	−0.643712	−	0.765268i	$-0.722607\pi$
$877$	−2.93238e7	−1.28742	−0.643712	+	0.765268i	$0.722607\pi$
$878$	0	0
$879$	−1.94024e7	−0.847000
$880$	0	0
$881$	−3.44801e6	−0.149668	−0.0748340	−	0.997196i	$-0.523843\pi$
$881$	−3.44801e6	−0.149668	−0.0748340	+	0.997196i	$0.523843\pi$
$882$	0	0
$883$	1.84343e7	0.795657	0.397828	−	0.917460i	$-0.369764\pi$
$883$	1.84343e7	0.795657	0.397828	+	0.917460i	$0.369764\pi$
$884$	0	0
$885$	−3.74239e6	−0.160617
$886$	0	0
$887$	−2.93173e7	−1.25117	−0.625583	−	0.780157i	$-0.715139\pi$
$887$	−2.93173e7	−1.25117	−0.625583	+	0.780157i	$0.715139\pi$
$888$	0	0
$889$	1.68658e7	0.715734
$890$	0	0
$891$	1.19163e6	0.0502859
$892$	0	0
$893$	2.74272e7	1.15094
$894$	0	0
$895$	5.00807e6	0.208984
$896$	0	0
$897$	−1.20380e6	−0.0499542
$898$	0	0
$899$	1.17268e7	0.483928
$900$	0	0
$901$	4.92318e7	2.02038
$902$	0	0
$903$	7.44788e6	0.303958
$904$	0	0
$905$	8.09010e6	0.328347
$906$	0	0
$907$	4.27304e7	1.72472	0.862361	−	0.506294i	$-0.168985\pi$
$907$	4.27304e7	1.72472	0.862361	+	0.506294i	$0.168985\pi$
$908$	0	0
$909$	196550.	0.00788975
$910$	0	0
$911$	−2.13118e7	−0.850795	−0.425398	−	0.905006i	$-0.639866\pi$
$911$	−2.13118e7	−0.850795	−0.425398	+	0.905006i	$0.639866\pi$
$912$	0	0
$913$	6.85137e7	2.72020
$914$	0	0
$915$	1.30662e6	0.0515936
$916$	0	0
$917$	1.97746e7	0.776578
$918$	0	0
$919$	2.92575e7	1.14274	0.571371	−	0.820692i	$-0.306412\pi$
$919$	2.92575e7	1.14274	0.571371	+	0.820692i	$0.306412\pi$
$920$	0	0
$921$	1.72831e7	0.671385
$922$	0	0
$923$	−314082.	−0.0121350
$924$	0	0
$925$	3.60220e7	1.38425
$926$	0	0
$927$	1.16558e7	0.445496
$928$	0	0
$929$	3.47509e7	1.32107	0.660537	−	0.750793i	$-0.270329\pi$
$929$	3.47509e7	1.32107	0.660537	+	0.750793i	$0.270329\pi$
$930$	0	0
$931$	1.82011e7	0.688212
$932$	0	0
$933$	−1.10618e7	−0.416025
$934$	0	0
$935$	3.17704e7	1.18849
$936$	0	0
$937$	−1.96117e6	−0.0729736	−0.0364868	−	0.999334i	$-0.511617\pi$
$937$	−1.96117e6	−0.0729736	−0.0364868	+	0.999334i	$0.511617\pi$
$938$	0	0
$939$	2.02463e7	0.749345
$940$	0	0
$941$	−1.87510e7	−0.690319	−0.345160	−	0.938544i	$-0.612175\pi$
$941$	−1.87510e7	−0.690319	−0.345160	+	0.938544i	$0.612175\pi$
$942$	0	0
$943$	−3.17340e7	−1.16211
$944$	0	0
$945$	−1.19458e7	−0.435147
$946$	0	0
$947$	2.96074e7	1.07282	0.536408	−	0.843959i	$-0.319781\pi$
$947$	2.96074e7	1.07282	0.536408	+	0.843959i	$0.319781\pi$
$948$	0	0
$949$	871066.	0.0313968
$950$	0	0
$951$	3.49319e7	1.25248
$952$	0	0
$953$	−3.76053e6	−0.134127	−0.0670636	−	0.997749i	$-0.521363\pi$
$953$	−3.76053e6	−0.134127	−0.0670636	+	0.997749i	$0.521363\pi$
$954$	0	0
$955$	−4.48044e6	−0.158969
$956$	0	0
$957$	1.12598e7	0.397420
$958$	0	0
$959$	3.27253e7	1.14905
$960$	0	0
$961$	3.30619e7	1.15483
$962$	0	0
$963$	−3.11508e7	−1.08244
$964$	0	0
$965$	3.34481e6	0.115625
$966$	0	0
$967$	9.12302e6	0.313742	0.156871	−	0.987619i	$-0.449859\pi$
$967$	9.12302e6	0.313742	0.156871	+	0.987619i	$0.449859\pi$
$968$	0	0
$969$	−4.15559e7	−1.42175
$970$	0	0
$971$	5.24177e6	0.178414	0.0892072	−	0.996013i	$-0.471567\pi$
$971$	5.24177e6	0.178414	0.0892072	+	0.996013i	$0.471567\pi$
$972$	0	0
$973$	−2.25001e7	−0.761909
$974$	0	0
$975$	−807838.	−0.0272153
$976$	0	0
$977$	−4.36506e7	−1.46303	−0.731516	−	0.681825i	$-0.761187\pi$
$977$	−4.36506e7	−1.46303	−0.731516	+	0.681825i	$0.761187\pi$
$978$	0	0
$979$	−8.16467e7	−2.72259
$980$	0	0
$981$	4.46888e6	0.148261
$982$	0	0
$983$	−2.96433e7	−0.978459	−0.489229	−	0.872155i	$-0.662722\pi$
$983$	−2.96433e7	−0.978459	−0.489229	+	0.872155i	$0.662722\pi$
$984$	0	0
$985$	−7.44216e6	−0.244404
$986$	0	0
$987$	−2.15267e7	−0.703371
$988$	0	0
$989$	1.93714e7	0.629755
$990$	0	0
$991$	3.14157e7	1.01616	0.508081	−	0.861309i	$-0.330355\pi$
$991$	3.14157e7	1.01616	0.508081	+	0.861309i	$0.330355\pi$
$992$	0	0
$993$	1.75861e7	0.565973
$994$	0	0
$995$	3.23084e6	0.103456
$996$	0	0
$997$	−4.72565e7	−1.50565	−0.752824	−	0.658222i	$-0.771309\pi$
$997$	−4.72565e7	−1.50565	−0.752824	+	0.658222i	$0.771309\pi$
$998$	0	0
$999$	−5.00692e7	−1.58729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.17	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.17	✓	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)

\(f(q)\)	\(=\)	\(q-9.78147 q^{3} -19.4476 q^{5} +160.887 q^{7} -147.323 q^{9} +O(q^{10})\)	\(q-9.78147 q^{3} -19.4476 q^{5} +160.887 q^{7} -147.323 q^{9} -771.004 q^{11} -30.0673 q^{13} +190.226 q^{15} +2118.85 q^{17} +2005.06 q^{19} -1573.71 q^{21} -4093.11 q^{23} -2746.79 q^{25} +3817.93 q^{27} +1493.03 q^{29} +7854.37 q^{31} +7541.55 q^{33} -3128.87 q^{35} -13114.2 q^{37} +294.103 q^{39} +7753.02 q^{41} -4732.69 q^{43} +2865.08 q^{45} +13678.9 q^{47} +9077.54 q^{49} -20725.5 q^{51} +23235.2 q^{53} +14994.2 q^{55} -19612.5 q^{57} -19673.3 q^{59} +6868.75 q^{61} -23702.3 q^{63} +584.738 q^{65} -7627.47 q^{67} +40036.7 q^{69} +10446.0 q^{71} -28970.5 q^{73} +26867.6 q^{75} -124044. q^{77} +48737.6 q^{79} -1545.55 q^{81} -88863.0 q^{83} -41206.6 q^{85} -14604.0 q^{87} +105897. q^{89} -4837.43 q^{91} -76827.3 q^{93} -38993.8 q^{95} -85340.5 q^{97} +113586. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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