LMFDB - Embedded newform 1028.6.a.b.1.11

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			1.11
Character	$\chi$	$=$	1028.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.8847 q^{3} +110.347 q^{5} +192.813 q^{7} +193.172 q^{9} +O(q^{10})$	$q-20.8847 q^{3} +110.347 q^{5} +192.813 q^{7} +193.172 q^{9} +794.045 q^{11} -404.777 q^{13} -2304.57 q^{15} +426.850 q^{17} -1457.62 q^{19} -4026.85 q^{21} +3779.96 q^{23} +9051.47 q^{25} +1040.64 q^{27} -1888.95 q^{29} +5239.34 q^{31} -16583.4 q^{33} +21276.4 q^{35} +10898.0 q^{37} +8453.67 q^{39} +4855.04 q^{41} +16020.3 q^{43} +21316.0 q^{45} -12201.2 q^{47} +20369.9 q^{49} -8914.64 q^{51} -6173.22 q^{53} +87620.5 q^{55} +30442.0 q^{57} -37464.9 q^{59} +21659.6 q^{61} +37246.2 q^{63} -44666.0 q^{65} +12301.2 q^{67} -78943.6 q^{69} -15148.1 q^{71} -13003.2 q^{73} -189038. q^{75} +153102. q^{77} -38739.7 q^{79} -68674.3 q^{81} -98224.4 q^{83} +47101.6 q^{85} +39450.3 q^{87} +82375.6 q^{89} -78046.4 q^{91} -109422. q^{93} -160844. q^{95} -159472. q^{97} +153387. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−20.8847	−1.33976	−0.669878	−	0.742471i	$-0.733654\pi$
$3$	−20.8847	−1.33976	−0.669878	+	0.742471i	$0.733654\pi$
$4$	0	0
$5$	110.347	1.97395	0.986974	−	0.160881i	$-0.0514334\pi$
$5$	110.347	1.97395	0.986974	+	0.160881i	$0.0514334\pi$
$6$	0	0
$7$	192.813	1.48728	0.743638	−	0.668583i	$-0.233099\pi$
$7$	192.813	1.48728	0.743638	+	0.668583i	$0.233099\pi$
$8$	0	0
$9$	193.172	0.794948
$10$	0	0
$11$	794.045	1.97862	0.989312	−	0.145815i	$-0.0465803\pi$
$11$	794.045	1.97862	0.989312	+	0.145815i	$0.0465803\pi$
$12$	0	0
$13$	−404.777	−0.664290	−0.332145	−	0.943228i	$-0.607772\pi$
$13$	−404.777	−0.664290	−0.332145	+	0.943228i	$0.607772\pi$
$14$	0	0
$15$	−2304.57	−2.64461
$16$	0	0
$17$	426.850	0.358222	0.179111	−	0.983829i	$-0.442678\pi$
$17$	426.850	0.358222	0.179111	+	0.983829i	$0.442678\pi$
$18$	0	0
$19$	−1457.62	−0.926317	−0.463158	−	0.886276i	$-0.653284\pi$
$19$	−1457.62	−0.926317	−0.463158	+	0.886276i	$0.653284\pi$
$20$	0	0
$21$	−4026.85	−1.99259
$22$	0	0
$23$	3779.96	1.48994	0.744969	−	0.667099i	$-0.232464\pi$
$23$	3779.96	1.48994	0.744969	+	0.667099i	$0.232464\pi$
$24$	0	0
$25$	9051.47	2.89647
$26$	0	0
$27$	1040.64	0.274720
$28$	0	0
$29$	−1888.95	−0.417086	−0.208543	−	0.978013i	$-0.566872\pi$
$29$	−1888.95	−0.417086	−0.208543	+	0.978013i	$0.566872\pi$
$30$	0	0
$31$	5239.34	0.979202	0.489601	−	0.871947i	$-0.337143\pi$
$31$	5239.34	0.979202	0.489601	+	0.871947i	$0.337143\pi$
$32$	0	0
$33$	−16583.4	−2.65087
$34$	0	0
$35$	21276.4	2.93580
$36$	0	0
$37$	10898.0	1.30870	0.654351	−	0.756191i	$-0.272942\pi$
$37$	10898.0	1.30870	0.654351	+	0.756191i	$0.272942\pi$
$38$	0	0
$39$	8453.67	0.889987
$40$	0	0
$41$	4855.04	0.451059	0.225529	−	0.974236i	$-0.427589\pi$
$41$	4855.04	0.451059	0.225529	+	0.974236i	$0.427589\pi$
$42$	0	0
$43$	16020.3	1.32129	0.660647	−	0.750697i	$-0.270282\pi$
$43$	16020.3	1.32129	0.660647	+	0.750697i	$0.270282\pi$
$44$	0	0
$45$	21316.0	1.56919
$46$	0	0
$47$	−12201.2	−0.805673	−0.402837	−	0.915272i	$-0.631976\pi$
$47$	−12201.2	−0.805673	−0.402837	+	0.915272i	$0.631976\pi$
$48$	0	0
$49$	20369.9	1.21199
$50$	0	0
$51$	−8914.64	−0.479931
$52$	0	0
$53$	−6173.22	−0.301871	−0.150936	−	0.988544i	$-0.548229\pi$
$53$	−6173.22	−0.301871	−0.150936	+	0.988544i	$0.548229\pi$
$54$	0	0
$55$	87620.5	3.90570
$56$	0	0
$57$	30442.0	1.24104
$58$	0	0
$59$	−37464.9	−1.40118	−0.700591	−	0.713563i	$-0.747080\pi$
$59$	−37464.9	−1.40118	−0.700591	+	0.713563i	$0.747080\pi$
$60$	0	0
$61$	21659.6	0.745290	0.372645	−	0.927974i	$-0.378451\pi$
$61$	21659.6	0.745290	0.372645	+	0.927974i	$0.378451\pi$
$62$	0	0
$63$	37246.2	1.18231
$64$	0	0
$65$	−44666.0	−1.31127
$66$	0	0
$67$	12301.2	0.334780	0.167390	−	0.985891i	$-0.446466\pi$
$67$	12301.2	0.334780	0.167390	+	0.985891i	$0.446466\pi$
$68$	0	0
$69$	−78943.6	−1.99615
$70$	0	0
$71$	−15148.1	−0.356626	−0.178313	−	0.983974i	$-0.557064\pi$
$71$	−15148.1	−0.356626	−0.178313	+	0.983974i	$0.557064\pi$
$72$	0	0
$73$	−13003.2	−0.285590	−0.142795	−	0.989752i	$-0.545609\pi$
$73$	−13003.2	−0.285590	−0.142795	+	0.989752i	$0.545609\pi$
$74$	0	0
$75$	−189038.	−3.88056
$76$	0	0
$77$	153102.	2.94276
$78$	0	0
$79$	−38739.7	−0.698375	−0.349187	−	0.937053i	$-0.613542\pi$
$79$	−38739.7	−0.698375	−0.349187	+	0.937053i	$0.613542\pi$
$80$	0	0
$81$	−68674.3	−1.16301
$82$	0	0
$83$	−98224.4	−1.56503	−0.782517	−	0.622629i	$-0.786065\pi$
$83$	−98224.4	−1.56503	−0.782517	+	0.622629i	$0.786065\pi$
$84$	0	0
$85$	47101.6	0.707112
$86$	0	0
$87$	39450.3	0.558794
$88$	0	0
$89$	82375.6	1.10236	0.551180	−	0.834386i	$-0.314178\pi$
$89$	82375.6	1.10236	0.551180	+	0.834386i	$0.314178\pi$
$90$	0	0
$91$	−78046.4	−0.987983
$92$	0	0
$93$	−109422.	−1.31189
$94$	0	0
$95$	−160844.	−1.82850
$96$	0	0
$97$	−159472.	−1.72090	−0.860450	−	0.509535i	$-0.829817\pi$
$97$	−159472.	−1.72090	−0.860450	+	0.509535i	$0.829817\pi$
$98$	0	0
$99$	153387.	1.57290
$100$	0	0
$101$	115137.	1.12309	0.561543	−	0.827448i	$-0.310208\pi$
$101$	115137.	1.12309	0.561543	+	0.827448i	$0.310208\pi$
$102$	0	0
$103$	−134219.	−1.24658	−0.623289	−	0.781992i	$-0.714204\pi$
$103$	−134219.	−1.24658	−0.623289	+	0.781992i	$0.714204\pi$
$104$	0	0
$105$	−444351.	−3.93326
$106$	0	0
$107$	156602.	1.32233	0.661163	−	0.750243i	$-0.270063\pi$
$107$	156602.	1.32233	0.661163	+	0.750243i	$0.270063\pi$
$108$	0	0
$109$	−27368.8	−0.220642	−0.110321	−	0.993896i	$-0.535188\pi$
$109$	−27368.8	−0.220642	−0.110321	+	0.993896i	$0.535188\pi$
$110$	0	0
$111$	−227601.	−1.75334
$112$	0	0
$113$	120043.	0.884384	0.442192	−	0.896920i	$-0.354201\pi$
$113$	120043.	0.884384	0.442192	+	0.896920i	$0.354201\pi$
$114$	0	0
$115$	417108.	2.94106
$116$	0	0
$117$	−78191.8	−0.528076
$118$	0	0
$119$	82302.2	0.532775
$120$	0	0
$121$	469456.	2.91495
$122$	0	0
$123$	−101396.	−0.604309
$124$	0	0
$125$	653968.	3.74353
$126$	0	0
$127$	−294183.	−1.61849	−0.809243	−	0.587475i	$-0.800122\pi$
$127$	−294183.	−1.61849	−0.809243	+	0.587475i	$0.800122\pi$
$128$	0	0
$129$	−334580.	−1.77021
$130$	0	0
$131$	184270.	0.938160	0.469080	−	0.883156i	$-0.344586\pi$
$131$	184270.	0.938160	0.469080	+	0.883156i	$0.344586\pi$
$132$	0	0
$133$	−281048.	−1.37769
$134$	0	0
$135$	114831.	0.542283
$136$	0	0
$137$	−133573.	−0.608021	−0.304011	−	0.952669i	$-0.598326\pi$
$137$	−133573.	−0.608021	−0.304011	+	0.952669i	$0.598326\pi$
$138$	0	0
$139$	−137075.	−0.601757	−0.300879	−	0.953662i	$-0.597280\pi$
$139$	−137075.	−0.601757	−0.300879	+	0.953662i	$0.597280\pi$
$140$	0	0
$141$	254819.	1.07941
$142$	0	0
$143$	−321411.	−1.31438
$144$	0	0
$145$	−208440.	−0.823306
$146$	0	0
$147$	−425420.	−1.62377
$148$	0	0
$149$	−12315.5	−0.0454450	−0.0227225	−	0.999742i	$-0.507233\pi$
$149$	−12315.5	−0.0454450	−0.0227225	+	0.999742i	$0.507233\pi$
$150$	0	0
$151$	−278499.	−0.993987	−0.496993	−	0.867754i	$-0.665563\pi$
$151$	−278499.	−0.993987	−0.496993	+	0.867754i	$0.665563\pi$
$152$	0	0
$153$	82455.5	0.284768
$154$	0	0
$155$	578145.	1.93289
$156$	0	0
$157$	−373812.	−1.21033	−0.605165	−	0.796100i	$-0.706893\pi$
$157$	−373812.	−1.21033	−0.605165	+	0.796100i	$0.706893\pi$
$158$	0	0
$159$	128926.	0.404434
$160$	0	0
$161$	728827.	2.21595
$162$	0	0
$163$	−283288.	−0.835138	−0.417569	−	0.908645i	$-0.637118\pi$
$163$	−283288.	−0.835138	−0.417569	+	0.908645i	$0.637118\pi$
$164$	0	0
$165$	−1.82993e6	−5.23269
$166$	0	0
$167$	−285390.	−0.791857	−0.395929	−	0.918281i	$-0.629577\pi$
$167$	−285390.	−0.791857	−0.395929	+	0.918281i	$0.629577\pi$
$168$	0	0
$169$	−207448.	−0.558718
$170$	0	0
$171$	−281571.	−0.736373
$172$	0	0
$173$	−667502.	−1.69565	−0.847827	−	0.530272i	$-0.822090\pi$
$173$	−667502.	−1.69565	−0.847827	+	0.530272i	$0.822090\pi$
$174$	0	0
$175$	1.74524e6	4.30785
$176$	0	0
$177$	782445.	1.87724
$178$	0	0
$179$	496895.	1.15913	0.579564	−	0.814926i	$-0.303223\pi$
$179$	496895.	1.15913	0.579564	+	0.814926i	$0.303223\pi$
$180$	0	0
$181$	−472766.	−1.07263	−0.536315	−	0.844018i	$-0.680184\pi$
$181$	−472766.	−1.07263	−0.536315	+	0.844018i	$0.680184\pi$
$182$	0	0
$183$	−452354.	−0.998507
$184$	0	0
$185$	1.20256e6	2.58331
$186$	0	0
$187$	338938.	0.708787
$188$	0	0
$189$	200649.	0.408585
$190$	0	0
$191$	334336.	0.663131	0.331565	−	0.943432i	$-0.392423\pi$
$191$	334336.	0.663131	0.331565	+	0.943432i	$0.392423\pi$
$192$	0	0
$193$	396036.	0.765318	0.382659	−	0.923890i	$-0.375008\pi$
$193$	396036.	0.765318	0.382659	+	0.923890i	$0.375008\pi$
$194$	0	0
$195$	932838.	1.75679
$196$	0	0
$197$	−726938.	−1.33454	−0.667270	−	0.744816i	$-0.732537\pi$
$197$	−726938.	−1.33454	−0.667270	+	0.744816i	$0.732537\pi$
$198$	0	0
$199$	−504159.	−0.902474	−0.451237	−	0.892404i	$-0.649017\pi$
$199$	−504159.	−0.902474	−0.451237	+	0.892404i	$0.649017\pi$
$200$	0	0
$201$	−256906.	−0.448523
$202$	0	0
$203$	−364215.	−0.620322
$204$	0	0
$205$	535739.	0.890367
$206$	0	0
$207$	730184.	1.18442
$208$	0	0
$209$	−1.15741e6	−1.83283
$210$	0	0
$211$	−271940.	−0.420502	−0.210251	−	0.977647i	$-0.567428\pi$
$211$	−271940.	−0.420502	−0.210251	+	0.977647i	$0.567428\pi$
$212$	0	0
$213$	316364.	0.477791
$214$	0	0
$215$	1.76779e6	2.60817
$216$	0	0
$217$	1.01021e6	1.45634
$218$	0	0
$219$	271568.	0.382620
$220$	0	0
$221$	−172779.	−0.237964
$222$	0	0
$223$	−1.08792e6	−1.46499	−0.732496	−	0.680771i	$-0.761645\pi$
$223$	−1.08792e6	−1.46499	−0.732496	+	0.680771i	$0.761645\pi$
$224$	0	0
$225$	1.74849e6	2.30254
$226$	0	0
$227$	−100002.	−0.128809	−0.0644043	−	0.997924i	$-0.520515\pi$
$227$	−100002.	−0.128809	−0.0644043	+	0.997924i	$0.520515\pi$
$228$	0	0
$229$	1.11277e6	1.40223	0.701113	−	0.713050i	$-0.252687\pi$
$229$	1.11277e6	1.40223	0.701113	+	0.713050i	$0.252687\pi$
$230$	0	0
$231$	−3.19750e6	−3.94258
$232$	0	0
$233$	1.14661e6	1.38365	0.691827	−	0.722064i	$-0.256806\pi$
$233$	1.14661e6	1.38365	0.691827	+	0.722064i	$0.256806\pi$
$234$	0	0
$235$	−1.34637e6	−1.59036
$236$	0	0
$237$	809069.	0.935652
$238$	0	0
$239$	−1.41917e6	−1.60709	−0.803545	−	0.595244i	$-0.797055\pi$
$239$	−1.41917e6	−1.60709	−0.803545	+	0.595244i	$0.797055\pi$
$240$	0	0
$241$	259371.	0.287659	0.143830	−	0.989602i	$-0.454058\pi$
$241$	259371.	0.287659	0.143830	+	0.989602i	$0.454058\pi$
$242$	0	0
$243$	1.18137e6	1.28342
$244$	0	0
$245$	2.24776e6	2.39240
$246$	0	0
$247$	590011.	0.615343
$248$	0	0
$249$	2.05139e6	2.09677
$250$	0	0
$251$	−1.22005e6	−1.22234	−0.611170	−	0.791499i	$-0.709301\pi$
$251$	−1.22005e6	−1.22234	−0.611170	+	0.791499i	$0.709301\pi$
$252$	0	0
$253$	3.00146e6	2.94803
$254$	0	0
$255$	−983704.	−0.947358
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	2.10127e6	1.94640
$260$	0	0
$261$	−364893.	−0.331562
$262$	0	0
$263$	744297.	0.663525	0.331762	−	0.943363i	$-0.392357\pi$
$263$	744297.	0.663525	0.331762	+	0.943363i	$0.392357\pi$
$264$	0	0
$265$	−681197.	−0.595879
$266$	0	0
$267$	−1.72039e6	−1.47689
$268$	0	0
$269$	−1.09166e6	−0.919825	−0.459913	−	0.887964i	$-0.652119\pi$
$269$	−1.09166e6	−0.919825	−0.459913	+	0.887964i	$0.652119\pi$
$270$	0	0
$271$	−276500.	−0.228703	−0.114351	−	0.993440i	$-0.536479\pi$
$271$	−276500.	−0.228703	−0.114351	+	0.993440i	$0.536479\pi$
$272$	0	0
$273$	1.62998e6	1.32366
$274$	0	0
$275$	7.18727e6	5.73102
$276$	0	0
$277$	292738.	0.229234	0.114617	−	0.993410i	$-0.463436\pi$
$277$	292738.	0.229234	0.114617	+	0.993410i	$0.463436\pi$
$278$	0	0
$279$	1.01209e6	0.778414
$280$	0	0
$281$	2.25276e6	1.70196	0.850980	−	0.525199i	$-0.176009\pi$
$281$	2.25276e6	1.70196	0.850980	+	0.525199i	$0.176009\pi$
$282$	0	0
$283$	−472045.	−0.350362	−0.175181	−	0.984536i	$-0.556051\pi$
$283$	−472045.	−0.350362	−0.175181	+	0.984536i	$0.556051\pi$
$284$	0	0
$285$	3.35918e6	2.44975
$286$	0	0
$287$	936116.	0.670849
$288$	0	0
$289$	−1.23766e6	−0.871677
$290$	0	0
$291$	3.33054e6	2.30559
$292$	0	0
$293$	273462.	0.186092	0.0930460	−	0.995662i	$-0.470340\pi$
$293$	273462.	0.186092	0.0930460	+	0.995662i	$0.470340\pi$
$294$	0	0
$295$	−4.13414e6	−2.76586
$296$	0	0
$297$	826313.	0.543568
$298$	0	0
$299$	−1.53004e6	−0.989751
$300$	0	0
$301$	3.08892e6	1.96513
$302$	0	0
$303$	−2.40461e6	−1.50466
$304$	0	0
$305$	2.39007e6	1.47116
$306$	0	0
$307$	526185.	0.318634	0.159317	−	0.987227i	$-0.449071\pi$
$307$	526185.	0.318634	0.159317	+	0.987227i	$0.449071\pi$
$308$	0	0
$309$	2.80312e6	1.67011
$310$	0	0
$311$	1.18866e6	0.696877	0.348438	−	0.937332i	$-0.386712\pi$
$311$	1.18866e6	0.696877	0.348438	+	0.937332i	$0.386712\pi$
$312$	0	0
$313$	−1.12023e6	−0.646315	−0.323158	−	0.946345i	$-0.604744\pi$
$313$	−1.12023e6	−0.646315	−0.323158	+	0.946345i	$0.604744\pi$
$314$	0	0
$315$	4.11000e6	2.33381
$316$	0	0
$317$	446448.	0.249530	0.124765	−	0.992186i	$-0.460182\pi$
$317$	446448.	0.249530	0.124765	+	0.992186i	$0.460182\pi$
$318$	0	0
$319$	−1.49991e6	−0.825257
$320$	0	0
$321$	−3.27060e6	−1.77159
$322$	0	0
$323$	−622183.	−0.331827
$324$	0	0
$325$	−3.66383e6	−1.92410
$326$	0	0
$327$	571589.	0.295607
$328$	0	0
$329$	−2.35256e6	−1.19826
$330$	0	0
$331$	148961.	0.0747314	0.0373657	−	0.999302i	$-0.488103\pi$
$331$	148961.	0.0747314	0.0373657	+	0.999302i	$0.488103\pi$
$332$	0	0
$333$	2.10518e6	1.04035
$334$	0	0
$335$	1.35740e6	0.660837
$336$	0	0
$337$	1.86279e6	0.893489	0.446744	−	0.894662i	$-0.352583\pi$
$337$	1.86279e6	0.893489	0.446744	+	0.894662i	$0.352583\pi$
$338$	0	0
$339$	−2.50707e6	−1.18486
$340$	0	0
$341$	4.16027e6	1.93747
$342$	0	0
$343$	686976.	0.315287
$344$	0	0
$345$	−8.71119e6	−3.94030
$346$	0	0
$347$	−1.36567e6	−0.608866	−0.304433	−	0.952534i	$-0.598467\pi$
$347$	−1.36567e6	−0.608866	−0.304433	+	0.952534i	$0.598467\pi$
$348$	0	0
$349$	−1.35367e6	−0.594906	−0.297453	−	0.954737i	$-0.596137\pi$
$349$	−1.35367e6	−0.594906	−0.297453	+	0.954737i	$0.596137\pi$
$350$	0	0
$351$	−421227.	−0.182494
$352$	0	0
$353$	3.73757e6	1.59644	0.798219	−	0.602367i	$-0.205776\pi$
$353$	3.73757e6	1.59644	0.798219	+	0.602367i	$0.205776\pi$
$354$	0	0
$355$	−1.67155e6	−0.703960
$356$	0	0
$357$	−1.71886e6	−0.713789
$358$	0	0
$359$	−308057.	−0.126152	−0.0630760	−	0.998009i	$-0.520091\pi$
$359$	−308057.	−0.126152	−0.0630760	+	0.998009i	$0.520091\pi$
$360$	0	0
$361$	−351452.	−0.141938
$362$	0	0
$363$	−9.80447e6	−3.90533
$364$	0	0
$365$	−1.43486e6	−0.563739
$366$	0	0
$367$	−4.15689e6	−1.61103	−0.805515	−	0.592575i	$-0.798111\pi$
$367$	−4.15689e6	−1.61103	−0.805515	+	0.592575i	$0.798111\pi$
$368$	0	0
$369$	937859.	0.358568
$370$	0	0
$371$	−1.19028e6	−0.448966
$372$	0	0
$373$	164576.	0.0612484	0.0306242	−	0.999531i	$-0.490250\pi$
$373$	164576.	0.0612484	0.0306242	+	0.999531i	$0.490250\pi$
$374$	0	0
$375$	−1.36580e7	−5.01542
$376$	0	0
$377$	764605.	0.277066
$378$	0	0
$379$	1.03093e6	0.368663	0.184332	−	0.982864i	$-0.440988\pi$
$379$	1.03093e6	0.368663	0.184332	+	0.982864i	$0.440988\pi$
$380$	0	0
$381$	6.14394e6	2.16838
$382$	0	0
$383$	3.69550e6	1.28729	0.643645	−	0.765324i	$-0.277421\pi$
$383$	3.69550e6	1.28729	0.643645	+	0.765324i	$0.277421\pi$
$384$	0	0
$385$	1.68944e7	5.80885
$386$	0	0
$387$	3.09468e6	1.05036
$388$	0	0
$389$	−3.08466e6	−1.03356	−0.516778	−	0.856120i	$-0.672869\pi$
$389$	−3.08466e6	−1.03356	−0.516778	+	0.856120i	$0.672869\pi$
$390$	0	0
$391$	1.61348e6	0.533729
$392$	0	0
$393$	−3.84844e6	−1.25691
$394$	0	0
$395$	−4.27481e6	−1.37856
$396$	0	0
$397$	5.00594e6	1.59408	0.797039	−	0.603928i	$-0.206399\pi$
$397$	5.00594e6	1.59408	0.797039	+	0.603928i	$0.206399\pi$
$398$	0	0
$399$	5.86961e6	1.84577
$400$	0	0
$401$	1.38736e6	0.430853	0.215426	−	0.976520i	$-0.430886\pi$
$401$	1.38736e6	0.430853	0.215426	+	0.976520i	$0.430886\pi$
$402$	0	0
$403$	−2.12077e6	−0.650474
$404$	0	0
$405$	−7.57801e6	−2.29571
$406$	0	0
$407$	8.65347e6	2.58943
$408$	0	0
$409$	5.77382e6	1.70669	0.853346	−	0.521346i	$-0.174570\pi$
$409$	5.77382e6	1.70669	0.853346	+	0.521346i	$0.174570\pi$
$410$	0	0
$411$	2.78965e6	0.814600
$412$	0	0
$413$	−7.22373e6	−2.08395
$414$	0	0
$415$	−1.08388e7	−3.08930
$416$	0	0
$417$	2.86278e6	0.806209
$418$	0	0
$419$	3.75522e6	1.04496	0.522480	−	0.852651i	$-0.325007\pi$
$419$	3.75522e6	1.04496	0.522480	+	0.852651i	$0.325007\pi$
$420$	0	0
$421$	−1.91071e6	−0.525398	−0.262699	−	0.964878i	$-0.584613\pi$
$421$	−1.91071e6	−0.525398	−0.262699	+	0.964878i	$0.584613\pi$
$422$	0	0
$423$	−2.35694e6	−0.640468
$424$	0	0
$425$	3.86362e6	1.03758
$426$	0	0
$427$	4.17625e6	1.10845
$428$	0	0
$429$	6.71259e6	1.76095
$430$	0	0
$431$	−1.96783e6	−0.510263	−0.255131	−	0.966906i	$-0.582119\pi$
$431$	−1.96783e6	−0.510263	−0.255131	+	0.966906i	$0.582119\pi$
$432$	0	0
$433$	1.76747e6	0.453034	0.226517	−	0.974007i	$-0.427266\pi$
$433$	1.76747e6	0.453034	0.226517	+	0.974007i	$0.427266\pi$
$434$	0	0
$435$	4.35322e6	1.10303
$436$	0	0
$437$	−5.50974e6	−1.38015
$438$	0	0
$439$	−7.43488e6	−1.84125	−0.920625	−	0.390449i	$-0.872320\pi$
$439$	−7.43488e6	−1.84125	−0.920625	+	0.390449i	$0.872320\pi$
$440$	0	0
$441$	3.93490e6	0.963468
$442$	0	0
$443$	5.32537e6	1.28926	0.644630	−	0.764495i	$-0.277011\pi$
$443$	5.32537e6	1.28926	0.644630	+	0.764495i	$0.277011\pi$
$444$	0	0
$445$	9.08990e6	2.17600
$446$	0	0
$447$	257206.	0.0608852
$448$	0	0
$449$	−1.98961e6	−0.465749	−0.232875	−	0.972507i	$-0.574813\pi$
$449$	−1.98961e6	−0.465749	−0.232875	+	0.972507i	$0.574813\pi$
$450$	0	0
$451$	3.85512e6	0.892476
$452$	0	0
$453$	5.81637e6	1.33170
$454$	0	0
$455$	−8.61219e6	−1.95023
$456$	0	0
$457$	90415.6	0.0202513	0.0101257	−	0.999949i	$-0.496777\pi$
$457$	90415.6	0.0202513	0.0101257	+	0.999949i	$0.496777\pi$
$458$	0	0
$459$	444196.	0.0984109
$460$	0	0
$461$	−6.87067e6	−1.50573	−0.752865	−	0.658175i	$-0.771329\pi$
$461$	−6.87067e6	−1.50573	−0.752865	+	0.658175i	$0.771329\pi$
$462$	0	0
$463$	−808881.	−0.175361	−0.0876804	−	0.996149i	$-0.527945\pi$
$463$	−808881.	−0.175361	−0.0876804	+	0.996149i	$0.527945\pi$
$464$	0	0
$465$	−1.20744e7	−2.58961
$466$	0	0
$467$	−5.92999e6	−1.25824	−0.629118	−	0.777310i	$-0.716584\pi$
$467$	−5.92999e6	−1.25824	−0.629118	+	0.777310i	$0.716584\pi$
$468$	0	0
$469$	2.37182e6	0.497910
$470$	0	0
$471$	7.80696e6	1.62155
$472$	0	0
$473$	1.27208e7	2.61434
$474$	0	0
$475$	−1.31936e7	−2.68305
$476$	0	0
$477$	−1.19250e6	−0.239972
$478$	0	0
$479$	−6.96847e6	−1.38771	−0.693855	−	0.720115i	$-0.744089\pi$
$479$	−6.96847e6	−1.38771	−0.693855	+	0.720115i	$0.744089\pi$
$480$	0	0
$481$	−4.41125e6	−0.869358
$482$	0	0
$483$	−1.52214e7	−2.96883
$484$	0	0
$485$	−1.75973e7	−3.39697
$486$	0	0
$487$	3.42851e6	0.655062	0.327531	−	0.944840i	$-0.393783\pi$
$487$	3.42851e6	0.655062	0.327531	+	0.944840i	$0.393783\pi$
$488$	0	0
$489$	5.91639e6	1.11888
$490$	0	0
$491$	5.09346e6	0.953474	0.476737	−	0.879046i	$-0.341819\pi$
$491$	5.09346e6	0.953474	0.476737	+	0.879046i	$0.341819\pi$
$492$	0	0
$493$	−806298.	−0.149410
$494$	0	0
$495$	1.69258e7	3.10483
$496$	0	0
$497$	−2.92075e6	−0.530401
$498$	0	0
$499$	−2.61877e6	−0.470811	−0.235405	−	0.971897i	$-0.575642\pi$
$499$	−2.61877e6	−0.470811	−0.235405	+	0.971897i	$0.575642\pi$
$500$	0	0
$501$	5.96029e6	1.06090
$502$	0	0
$503$	−7.94806e6	−1.40069	−0.700344	−	0.713806i	$-0.746970\pi$
$503$	−7.94806e6	−1.40069	−0.700344	+	0.713806i	$0.746970\pi$
$504$	0	0
$505$	1.27051e7	2.21691
$506$	0	0
$507$	4.33250e6	0.748547
$508$	0	0
$509$	6.20695e6	1.06190	0.530950	−	0.847403i	$-0.321835\pi$
$509$	6.20695e6	1.06190	0.530950	+	0.847403i	$0.321835\pi$
$510$	0	0
$511$	−2.50718e6	−0.424750
$512$	0	0
$513$	−1.51685e6	−0.254478
$514$	0	0
$515$	−1.48106e7	−2.46068
$516$	0	0
$517$	−9.68832e6	−1.59412
$518$	0	0
$519$	1.39406e7	2.27176
$520$	0	0
$521$	−9.62731e6	−1.55385	−0.776927	−	0.629590i	$-0.783223\pi$
$521$	−9.62731e6	−1.55385	−0.776927	+	0.629590i	$0.783223\pi$
$522$	0	0
$523$	5.34167e6	0.853932	0.426966	−	0.904268i	$-0.359582\pi$
$523$	5.34167e6	0.853932	0.426966	+	0.904268i	$0.359582\pi$
$524$	0	0
$525$	−3.64489e7	−5.77147
$526$	0	0
$527$	2.23641e6	0.350772
$528$	0	0
$529$	7.85179e6	1.21991
$530$	0	0
$531$	−7.23719e6	−1.11387
$532$	0	0
$533$	−1.96521e6	−0.299634
$534$	0	0
$535$	1.72806e7	2.61020
$536$	0	0
$537$	−1.03775e7	−1.55295
$538$	0	0
$539$	1.61746e7	2.39807
$540$	0	0
$541$	72038.5	0.0105821	0.00529105	−	0.999986i	$-0.498316\pi$
$541$	72038.5	0.0105821	0.00529105	+	0.999986i	$0.498316\pi$
$542$	0	0
$543$	9.87359e6	1.43706
$544$	0	0
$545$	−3.02006e6	−0.435536
$546$	0	0
$547$	−2.82127e6	−0.403159	−0.201579	−	0.979472i	$-0.564607\pi$
$547$	−2.82127e6	−0.403159	−0.201579	+	0.979472i	$0.564607\pi$
$548$	0	0
$549$	4.18403e6	0.592467
$550$	0	0
$551$	2.75337e6	0.386354
$552$	0	0
$553$	−7.46952e6	−1.03868
$554$	0	0
$555$	−2.51151e7	−3.46101
$556$	0	0
$557$	6.16618e6	0.842128	0.421064	−	0.907031i	$-0.361657\pi$
$557$	6.16618e6	0.842128	0.421064	+	0.907031i	$0.361657\pi$
$558$	0	0
$559$	−6.48466e6	−0.877723
$560$	0	0
$561$	−7.07863e6	−0.949602
$562$	0	0
$563$	−6.58342e6	−0.875348	−0.437674	−	0.899134i	$-0.644198\pi$
$563$	−6.58342e6	−0.875348	−0.437674	+	0.899134i	$0.644198\pi$
$564$	0	0
$565$	1.32464e7	1.74573
$566$	0	0
$567$	−1.32413e7	−1.72971
$568$	0	0
$569$	1.35465e7	1.75407	0.877033	−	0.480430i	$-0.159519\pi$
$569$	1.35465e7	1.75407	0.877033	+	0.480430i	$0.159519\pi$
$570$	0	0
$571$	7.02418e6	0.901583	0.450791	−	0.892629i	$-0.351142\pi$
$571$	7.02418e6	0.901583	0.450791	+	0.892629i	$0.351142\pi$
$572$	0	0
$573$	−6.98251e6	−0.888434
$574$	0	0
$575$	3.42142e7	4.31556
$576$	0	0
$577$	−313473.	−0.0391977	−0.0195988	−	0.999808i	$-0.506239\pi$
$577$	−313473.	−0.0391977	−0.0195988	+	0.999808i	$0.506239\pi$
$578$	0	0
$579$	−8.27112e6	−1.02534
$580$	0	0
$581$	−1.89389e7	−2.32764
$582$	0	0
$583$	−4.90181e6	−0.597290
$584$	0	0
$585$	−8.62823e6	−1.04239
$586$	0	0
$587$	−2.05217e6	−0.245820	−0.122910	−	0.992418i	$-0.539223\pi$
$587$	−2.05217e6	−0.245820	−0.122910	+	0.992418i	$0.539223\pi$
$588$	0	0
$589$	−7.63695e6	−0.907051
$590$	0	0
$591$	1.51819e7	1.78796
$592$	0	0
$593$	2.52449e6	0.294806	0.147403	−	0.989076i	$-0.452909\pi$
$593$	2.52449e6	0.294806	0.147403	+	0.989076i	$0.452909\pi$
$594$	0	0
$595$	9.08181e6	1.05167
$596$	0	0
$597$	1.05292e7	1.20910
$598$	0	0
$599$	1.50781e7	1.71704	0.858520	−	0.512780i	$-0.171384\pi$
$599$	1.50781e7	1.71704	0.858520	+	0.512780i	$0.171384\pi$
$600$	0	0
$601$	−7.61765e6	−0.860271	−0.430135	−	0.902764i	$-0.641534\pi$
$601$	−7.61765e6	−0.860271	−0.430135	+	0.902764i	$0.641534\pi$
$602$	0	0
$603$	2.37624e6	0.266132
$604$	0	0
$605$	5.18031e7	5.75396
$606$	0	0
$607$	1.39216e7	1.53362	0.766811	−	0.641873i	$-0.221842\pi$
$607$	1.39216e7	1.53362	0.766811	+	0.641873i	$0.221842\pi$
$608$	0	0
$609$	7.60653e6	0.831081
$610$	0	0
$611$	4.93878e6	0.535201
$612$	0	0
$613$	−1.18835e7	−1.27730	−0.638651	−	0.769496i	$-0.720507\pi$
$613$	−1.18835e7	−1.27730	−0.638651	+	0.769496i	$0.720507\pi$
$614$	0	0
$615$	−1.11888e7	−1.19287
$616$	0	0
$617$	7.25035e6	0.766737	0.383368	−	0.923596i	$-0.374764\pi$
$617$	7.25035e6	0.766737	0.383368	+	0.923596i	$0.374764\pi$
$618$	0	0
$619$	3.28254e6	0.344337	0.172169	−	0.985067i	$-0.444923\pi$
$619$	3.28254e6	0.344337	0.172169	+	0.985067i	$0.444923\pi$
$620$	0	0
$621$	3.93358e6	0.409316
$622$	0	0
$623$	1.58831e7	1.63951
$624$	0	0
$625$	4.38776e7	4.49307
$626$	0	0
$627$	2.41723e7	2.45555
$628$	0	0
$629$	4.65179e6	0.468806
$630$	0	0
$631$	−2.91300e6	−0.291251	−0.145626	−	0.989340i	$-0.546519\pi$
$631$	−2.91300e6	−0.291251	−0.145626	+	0.989340i	$0.546519\pi$
$632$	0	0
$633$	5.67940e6	0.563370
$634$	0	0
$635$	−3.24623e7	−3.19481
$636$	0	0
$637$	−8.24528e6	−0.805113
$638$	0	0
$639$	−2.92619e6	−0.283499
$640$	0	0
$641$	−1.40125e7	−1.34701	−0.673503	−	0.739184i	$-0.735211\pi$
$641$	−1.40125e7	−1.34701	−0.673503	+	0.739184i	$0.735211\pi$
$642$	0	0
$643$	−1.03234e7	−0.984677	−0.492339	−	0.870404i	$-0.663858\pi$
$643$	−1.03234e7	−0.984677	−0.492339	+	0.870404i	$0.663858\pi$
$644$	0	0
$645$	−3.69199e7	−3.49431
$646$	0	0
$647$	1.54764e7	1.45348	0.726739	−	0.686913i	$-0.241035\pi$
$647$	1.54764e7	1.45348	0.726739	+	0.686913i	$0.241035\pi$
$648$	0	0
$649$	−2.97488e7	−2.77241
$650$	0	0
$651$	−2.10980e7	−1.95115
$652$	0	0
$653$	1.12287e6	0.103049	0.0515246	−	0.998672i	$-0.483592\pi$
$653$	1.12287e6	0.103049	0.0515246	+	0.998672i	$0.483592\pi$
$654$	0	0
$655$	2.03337e7	1.85188
$656$	0	0
$657$	−2.51185e6	−0.227029
$658$	0	0
$659$	−1.29384e7	−1.16056	−0.580281	−	0.814416i	$-0.697057\pi$
$659$	−1.29384e7	−1.16056	−0.580281	+	0.814416i	$0.697057\pi$
$660$	0	0
$661$	4.99794e6	0.444926	0.222463	−	0.974941i	$-0.428590\pi$
$661$	4.99794e6	0.444926	0.222463	+	0.974941i	$0.428590\pi$
$662$	0	0
$663$	3.60845e6	0.318813
$664$	0	0
$665$	−3.10128e7	−2.71948
$666$	0	0
$667$	−7.14017e6	−0.621432
$668$	0	0
$669$	2.27210e7	1.96273
$670$	0	0
$671$	1.71987e7	1.47465
$672$	0	0
$673$	8.08858e6	0.688391	0.344195	−	0.938898i	$-0.388152\pi$
$673$	8.08858e6	0.688391	0.344195	+	0.938898i	$0.388152\pi$
$674$	0	0
$675$	9.41931e6	0.795718
$676$	0	0
$677$	−4.39938e6	−0.368909	−0.184455	−	0.982841i	$-0.559052\pi$
$677$	−4.39938e6	−0.368909	−0.184455	+	0.982841i	$0.559052\pi$
$678$	0	0
$679$	−3.07483e7	−2.55945
$680$	0	0
$681$	2.08852e6	0.172572
$682$	0	0
$683$	1.49006e7	1.22223	0.611115	−	0.791542i	$-0.290721\pi$
$683$	1.49006e7	1.22223	0.611115	+	0.791542i	$0.290721\pi$
$684$	0	0
$685$	−1.47394e7	−1.20020
$686$	0	0
$687$	−2.32400e7	−1.87864
$688$	0	0
$689$	2.49878e6	0.200530
$690$	0	0
$691$	2.55047e6	0.203200	0.101600	−	0.994825i	$-0.467604\pi$
$691$	2.55047e6	0.203200	0.101600	+	0.994825i	$0.467604\pi$
$692$	0	0
$693$	2.95751e7	2.33934
$694$	0	0
$695$	−1.51258e7	−1.18784
$696$	0	0
$697$	2.07237e6	0.161579
$698$	0	0
$699$	−2.39467e7	−1.85376
$700$	0	0
$701$	1.23849e7	0.951910	0.475955	−	0.879470i	$-0.342102\pi$
$701$	1.23849e7	0.951910	0.475955	+	0.879470i	$0.342102\pi$
$702$	0	0
$703$	−1.58851e7	−1.21227
$704$	0	0
$705$	2.81186e7	2.13069
$706$	0	0
$707$	2.22000e7	1.67034
$708$	0	0
$709$	−5.83655e6	−0.436055	−0.218027	−	0.975943i	$-0.569962\pi$
$709$	−5.83655e6	−0.436055	−0.218027	+	0.975943i	$0.569962\pi$
$710$	0	0
$711$	−7.48344e6	−0.555172
$712$	0	0
$713$	1.98045e7	1.45895
$714$	0	0
$715$	−3.54668e7	−2.59452
$716$	0	0
$717$	2.96390e7	2.15311
$718$	0	0
$719$	−1.07585e7	−0.776125	−0.388062	−	0.921633i	$-0.626855\pi$
$719$	−1.07585e7	−0.776125	−0.388062	+	0.921633i	$0.626855\pi$
$720$	0	0
$721$	−2.58791e7	−1.85400
$722$	0	0
$723$	−5.41689e6	−0.385393
$724$	0	0
$725$	−1.70978e7	−1.20808
$726$	0	0
$727$	−4.80788e6	−0.337378	−0.168689	−	0.985669i	$-0.553953\pi$
$727$	−4.80788e6	−0.337378	−0.168689	+	0.985669i	$0.553953\pi$
$728$	0	0
$729$	−7.98475e6	−0.556471
$730$	0	0
$731$	6.83826e6	0.473317
$732$	0	0
$733$	1.64981e7	1.13416	0.567078	−	0.823664i	$-0.308074\pi$
$733$	1.64981e7	1.13416	0.567078	+	0.823664i	$0.308074\pi$
$734$	0	0
$735$	−4.69439e7	−3.20524
$736$	0	0
$737$	9.76767e6	0.662403
$738$	0	0
$739$	2.77693e7	1.87048	0.935242	−	0.354008i	$-0.115182\pi$
$739$	2.77693e7	1.87048	0.935242	+	0.354008i	$0.115182\pi$
$740$	0	0
$741$	−1.23222e7	−0.824410
$742$	0	0
$743$	197982.	0.0131569	0.00657844	−	0.999978i	$-0.497906\pi$
$743$	197982.	0.0131569	0.00657844	+	0.999978i	$0.497906\pi$
$744$	0	0
$745$	−1.35898e6	−0.0897060
$746$	0	0
$747$	−1.89742e7	−1.24412
$748$	0	0
$749$	3.01950e7	1.96666
$750$	0	0
$751$	1.83046e7	1.18430	0.592148	−	0.805829i	$-0.298280\pi$
$751$	1.83046e7	1.18430	0.592148	+	0.805829i	$0.298280\pi$
$752$	0	0
$753$	2.54804e7	1.63764
$754$	0	0
$755$	−3.07315e7	−1.96208
$756$	0	0
$757$	1.65525e7	1.04984	0.524921	−	0.851151i	$-0.324095\pi$
$757$	1.65525e7	1.04984	0.524921	+	0.851151i	$0.324095\pi$
$758$	0	0
$759$	−6.26847e7	−3.94964
$760$	0	0
$761$	2.36177e7	1.47835	0.739173	−	0.673516i	$-0.235217\pi$
$761$	2.36177e7	1.47835	0.739173	+	0.673516i	$0.235217\pi$
$762$	0	0
$763$	−5.27706e6	−0.328156
$764$	0	0
$765$	9.09872e6	0.562117
$766$	0	0
$767$	1.51650e7	0.930792
$768$	0	0
$769$	−1.31858e7	−0.804061	−0.402031	−	0.915626i	$-0.631696\pi$
$769$	−1.31858e7	−0.804061	−0.402031	+	0.915626i	$0.631696\pi$
$770$	0	0
$771$	1.37942e6	0.0835717
$772$	0	0
$773$	1.33093e7	0.801133	0.400567	−	0.916268i	$-0.368813\pi$
$773$	1.33093e7	0.801133	0.400567	+	0.916268i	$0.368813\pi$
$774$	0	0
$775$	4.74237e7	2.83623
$776$	0	0
$777$	−4.38845e7	−2.60770
$778$	0	0
$779$	−7.07679e6	−0.417823
$780$	0	0
$781$	−1.20283e7	−0.705628
$782$	0	0
$783$	−1.96572e6	−0.114582
$784$	0	0
$785$	−4.12490e7	−2.38913
$786$	0	0
$787$	2.07453e7	1.19394	0.596970	−	0.802264i	$-0.296371\pi$
$787$	2.07453e7	1.19394	0.596970	+	0.802264i	$0.296371\pi$
$788$	0	0
$789$	−1.55445e7	−0.888961
$790$	0	0
$791$	2.31459e7	1.31532
$792$	0	0
$793$	−8.76731e6	−0.495089
$794$	0	0
$795$	1.42266e7	0.798332
$796$	0	0
$797$	−2.62040e7	−1.46124	−0.730621	−	0.682783i	$-0.760769\pi$
$797$	−2.62040e7	−1.46124	−0.730621	+	0.682783i	$0.760769\pi$
$798$	0	0
$799$	−5.20809e6	−0.288610
$800$	0	0
$801$	1.59127e7	0.876319
$802$	0	0
$803$	−1.03251e7	−0.565074
$804$	0	0
$805$	8.04239e7	4.37417
$806$	0	0
$807$	2.27990e7	1.23234
$808$	0	0
$809$	2.23194e7	1.19898	0.599488	−	0.800383i	$-0.295371\pi$
$809$	2.23194e7	1.19898	0.599488	+	0.800383i	$0.295371\pi$
$810$	0	0
$811$	−2.22624e7	−1.18856	−0.594279	−	0.804259i	$-0.702562\pi$
$811$	−2.22624e7	−1.18856	−0.594279	+	0.804259i	$0.702562\pi$
$812$	0	0
$813$	5.77463e6	0.306406
$814$	0	0
$815$	−3.12599e7	−1.64852
$816$	0	0
$817$	−2.33515e7	−1.22394
$818$	0	0
$819$	−1.50764e7	−0.785395
$820$	0	0
$821$	−1.17866e7	−0.610280	−0.305140	−	0.952307i	$-0.598703\pi$
$821$	−1.17866e7	−0.610280	−0.305140	+	0.952307i	$0.598703\pi$
$822$	0	0
$823$	−1.67547e7	−0.862257	−0.431129	−	0.902291i	$-0.641884\pi$
$823$	−1.67547e7	−0.862257	−0.431129	+	0.902291i	$0.641884\pi$
$824$	0	0
$825$	−1.50104e8	−7.67818
$826$	0	0
$827$	3.24872e6	0.165177	0.0825883	−	0.996584i	$-0.473681\pi$
$827$	3.24872e6	0.165177	0.0825883	+	0.996584i	$0.473681\pi$
$828$	0	0
$829$	−3.08883e7	−1.56102	−0.780509	−	0.625144i	$-0.785040\pi$
$829$	−3.08883e7	−1.56102	−0.780509	+	0.625144i	$0.785040\pi$
$830$	0	0
$831$	−6.11376e6	−0.307118
$832$	0	0
$833$	8.69489e6	0.434162
$834$	0	0
$835$	−3.14919e7	−1.56308
$836$	0	0
$837$	5.45226e6	0.269006
$838$	0	0
$839$	−2.09150e7	−1.02578	−0.512888	−	0.858455i	$-0.671425\pi$
$839$	−2.09150e7	−1.02578	−0.512888	+	0.858455i	$0.671425\pi$
$840$	0	0
$841$	−1.69430e7	−0.826039
$842$	0	0
$843$	−4.70483e7	−2.28021
$844$	0	0
$845$	−2.28913e7	−1.10288
$846$	0	0
$847$	9.05173e7	4.33534
$848$	0	0
$849$	9.85854e6	0.469400
$850$	0	0
$851$	4.11939e7	1.94988
$852$	0	0
$853$	1.67213e7	0.786860	0.393430	−	0.919355i	$-0.371288\pi$
$853$	1.67213e7	0.786860	0.393430	+	0.919355i	$0.371288\pi$
$854$	0	0
$855$	−3.10706e7	−1.45356
$856$	0	0
$857$	2.68169e7	1.24726	0.623629	−	0.781721i	$-0.285658\pi$
$857$	2.68169e7	1.24726	0.623629	+	0.781721i	$0.285658\pi$
$858$	0	0
$859$	−3.05573e7	−1.41297	−0.706483	−	0.707730i	$-0.749719\pi$
$859$	−3.05573e7	−1.41297	−0.706483	+	0.707730i	$0.749719\pi$
$860$	0	0
$861$	−1.95505e7	−0.898774
$862$	0	0
$863$	−2.26266e7	−1.03417	−0.517085	−	0.855934i	$-0.672983\pi$
$863$	−2.26266e7	−1.03417	−0.517085	+	0.855934i	$0.672983\pi$
$864$	0	0
$865$	−7.36569e7	−3.34713
$866$	0	0
$867$	2.58481e7	1.16783
$868$	0	0
$869$	−3.07611e7	−1.38182
$870$	0	0
$871$	−4.97923e6	−0.222391
$872$	0	0
$873$	−3.08056e7	−1.36803
$874$	0	0
$875$	1.26094e8	5.56767
$876$	0	0
$877$	−4.12251e6	−0.180993	−0.0904967	−	0.995897i	$-0.528845\pi$
$877$	−4.12251e6	−0.180993	−0.0904967	+	0.995897i	$0.528845\pi$
$878$	0	0
$879$	−5.71118e6	−0.249318
$880$	0	0
$881$	−2.64021e7	−1.14604	−0.573018	−	0.819543i	$-0.694228\pi$
$881$	−2.64021e7	−1.14604	−0.573018	+	0.819543i	$0.694228\pi$
$882$	0	0
$883$	1.31956e7	0.569544	0.284772	−	0.958595i	$-0.408082\pi$
$883$	1.31956e7	0.569544	0.284772	+	0.958595i	$0.408082\pi$
$884$	0	0
$885$	8.63405e7	3.70558
$886$	0	0
$887$	1.11848e7	0.477331	0.238666	−	0.971102i	$-0.423290\pi$
$887$	1.11848e7	0.477331	0.238666	+	0.971102i	$0.423290\pi$
$888$	0	0
$889$	−5.67224e7	−2.40713
$890$	0	0
$891$	−5.45305e7	−2.30115
$892$	0	0
$893$	1.77847e7	0.746308
$894$	0	0
$895$	5.48308e7	2.28806
$896$	0	0
$897$	3.19546e7	1.32603
$898$	0	0
$899$	−9.89685e6	−0.408411
$900$	0	0
$901$	−2.63504e6	−0.108137
$902$	0	0
$903$	−6.45114e7	−2.63279
$904$	0	0
$905$	−5.21683e7	−2.11731
$906$	0	0
$907$	2.48247e7	1.00200	0.500998	−	0.865449i	$-0.332967\pi$
$907$	2.48247e7	1.00200	0.500998	+	0.865449i	$0.332967\pi$
$908$	0	0
$909$	2.22413e7	0.892794
$910$	0	0
$911$	−1.93536e7	−0.772619	−0.386310	−	0.922369i	$-0.626250\pi$
$911$	−1.93536e7	−0.772619	−0.386310	+	0.922369i	$0.626250\pi$
$912$	0	0
$913$	−7.79945e7	−3.09662
$914$	0	0
$915$	−4.99160e7	−1.97100
$916$	0	0
$917$	3.55297e7	1.39530
$918$	0	0
$919$	−2.99383e7	−1.16933	−0.584666	−	0.811274i	$-0.698774\pi$
$919$	−2.99383e7	−1.16933	−0.584666	+	0.811274i	$0.698774\pi$
$920$	0	0
$921$	−1.09892e7	−0.426892
$922$	0	0
$923$	6.13161e6	0.236903
$924$	0	0
$925$	9.86425e7	3.79062
$926$	0	0
$927$	−2.59273e7	−0.990964
$928$	0	0
$929$	2.51196e7	0.954935	0.477467	−	0.878649i	$-0.341555\pi$
$929$	2.51196e7	0.954935	0.477467	+	0.878649i	$0.341555\pi$
$930$	0	0
$931$	−2.96915e7	−1.12269
$932$	0	0
$933$	−2.48248e7	−0.933645
$934$	0	0
$935$	3.74008e7	1.39911
$936$	0	0
$937$	3.83706e7	1.42774	0.713872	−	0.700277i	$-0.246940\pi$
$937$	3.83706e7	1.42774	0.713872	+	0.700277i	$0.246940\pi$
$938$	0	0
$939$	2.33956e7	0.865905
$940$	0	0
$941$	−4.09088e7	−1.50606	−0.753031	−	0.657985i	$-0.771409\pi$
$941$	−4.09088e7	−1.50606	−0.753031	+	0.657985i	$0.771409\pi$
$942$	0	0
$943$	1.83519e7	0.672050
$944$	0	0
$945$	2.21410e7	0.806525
$946$	0	0
$947$	−1.78552e7	−0.646979	−0.323489	−	0.946232i	$-0.604856\pi$
$947$	−1.78552e7	−0.646979	−0.323489	+	0.946232i	$0.604856\pi$
$948$	0	0
$949$	5.26339e6	0.189714
$950$	0	0
$951$	−9.32394e6	−0.334309
$952$	0	0
$953$	−3.42787e7	−1.22262	−0.611311	−	0.791391i	$-0.709357\pi$
$953$	−3.42787e7	−1.22262	−0.611311	+	0.791391i	$0.709357\pi$
$954$	0	0
$955$	3.68929e7	1.30899
$956$	0	0
$957$	3.13253e7	1.10564
$958$	0	0
$959$	−2.57547e7	−0.904295
$960$	0	0
$961$	−1.17850e6	−0.0411643
$962$	0	0
$963$	3.02512e7	1.05118
$964$	0	0
$965$	4.37014e7	1.51070
$966$	0	0
$967$	−9.90057e6	−0.340482	−0.170241	−	0.985402i	$-0.554455\pi$
$967$	−9.90057e6	−0.340482	−0.170241	+	0.985402i	$0.554455\pi$
$968$	0	0
$969$	1.29941e7	0.444568
$970$	0	0
$971$	5.37946e7	1.83101	0.915504	−	0.402309i	$-0.131792\pi$
$971$	5.37946e7	1.83101	0.915504	+	0.402309i	$0.131792\pi$
$972$	0	0
$973$	−2.64299e7	−0.894979
$974$	0	0
$975$	7.65181e7	2.57782
$976$	0	0
$977$	−2.49602e7	−0.836589	−0.418295	−	0.908311i	$-0.637372\pi$
$977$	−2.49602e7	−0.836589	−0.418295	+	0.908311i	$0.637372\pi$
$978$	0	0
$979$	6.54099e7	2.18116
$980$	0	0
$981$	−5.28689e6	−0.175399
$982$	0	0
$983$	−5.73829e6	−0.189408	−0.0947041	−	0.995505i	$-0.530190\pi$
$983$	−5.73829e6	−0.189408	−0.0947041	+	0.995505i	$0.530190\pi$
$984$	0	0
$985$	−8.02154e7	−2.63431
$986$	0	0
$987$	4.91325e7	1.60537
$988$	0	0
$989$	6.05562e7	1.96865
$990$	0	0
$991$	4.27533e6	0.138288	0.0691442	−	0.997607i	$-0.477973\pi$
$991$	4.27533e6	0.138288	0.0691442	+	0.997607i	$0.477973\pi$
$992$	0	0
$993$	−3.11101e6	−0.100122
$994$	0	0
$995$	−5.56324e7	−1.78144
$996$	0	0
$997$	−1.38984e7	−0.442818	−0.221409	−	0.975181i	$-0.571066\pi$
$997$	−1.38984e7	−0.442818	−0.221409	+	0.975181i	$0.571066\pi$
$998$	0	0
$999$	1.13408e7	0.359527

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.11	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.11	✓	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)

\(f(q)\)	\(=\)	\(q-20.8847 q^{3} +110.347 q^{5} +192.813 q^{7} +193.172 q^{9} +O(q^{10})\)	\(q-20.8847 q^{3} +110.347 q^{5} +192.813 q^{7} +193.172 q^{9} +794.045 q^{11} -404.777 q^{13} -2304.57 q^{15} +426.850 q^{17} -1457.62 q^{19} -4026.85 q^{21} +3779.96 q^{23} +9051.47 q^{25} +1040.64 q^{27} -1888.95 q^{29} +5239.34 q^{31} -16583.4 q^{33} +21276.4 q^{35} +10898.0 q^{37} +8453.67 q^{39} +4855.04 q^{41} +16020.3 q^{43} +21316.0 q^{45} -12201.2 q^{47} +20369.9 q^{49} -8914.64 q^{51} -6173.22 q^{53} +87620.5 q^{55} +30442.0 q^{57} -37464.9 q^{59} +21659.6 q^{61} +37246.2 q^{63} -44666.0 q^{65} +12301.2 q^{67} -78943.6 q^{69} -15148.1 q^{71} -13003.2 q^{73} -189038. q^{75} +153102. q^{77} -38739.7 q^{79} -68674.3 q^{81} -98224.4 q^{83} +47101.6 q^{85} +39450.3 q^{87} +82375.6 q^{89} -78046.4 q^{91} -109422. q^{93} -160844. q^{95} -159472. q^{97} +153387. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 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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