LMFDB - Embedded newform 256.6.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,6,Mod(1,256)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, 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("256.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	256.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:	$41.0582578721$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{29})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 17x^{2} + 18x + 23 $ x^4 - 2x^3 - 17x^2 + 18*x + 23
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 128)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.60680$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.2315 q^{3} -43.0813 q^{5} +22.6274 q^{7} -11.0000 q^{9} +O(q^{10})$	$q+15.2315 q^{3} -43.0813 q^{5} +22.6274 q^{7} -11.0000 q^{9} -258.936 q^{11} +990.870 q^{13} -656.195 q^{15} -218.000 q^{17} -1995.33 q^{19} +344.651 q^{21} +3416.74 q^{23} -1269.00 q^{25} -3868.81 q^{27} -4868.19 q^{29} -8688.93 q^{31} -3944.00 q^{33} -974.819 q^{35} +3231.10 q^{37} +15092.5 q^{39} -8410.00 q^{41} -685.420 q^{43} +473.895 q^{45} -18147.2 q^{47} -16295.0 q^{49} -3320.48 q^{51} -26581.2 q^{53} +11155.3 q^{55} -30392.0 q^{57} +33798.8 q^{59} +27356.6 q^{61} -248.902 q^{63} -42688.0 q^{65} +66729.4 q^{67} +52042.2 q^{69} -21609.2 q^{71} -43006.0 q^{73} -19328.8 q^{75} -5859.06 q^{77} +64759.7 q^{79} -56255.0 q^{81} -70659.1 q^{83} +9391.73 q^{85} -74150.0 q^{87} -93870.0 q^{89} +22420.8 q^{91} -132346. q^{93} +85961.6 q^{95} -81706.0 q^{97} +2848.30 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 44 q^{9}+O(q^{10}) $ 4 * q - 44 * q^9	$ 4 q - 44 q^{9} - 872 q^{17} - 5076 q^{25} - 15776 q^{33} - 33640 q^{41} - 65180 q^{49} - 121568 q^{57} - 170752 q^{65} - 172024 q^{73} - 225020 q^{81} - 375480 q^{89} - 326824 q^{97}+O(q^{100}) $ 4 * q - 44 * q^9 - 872 * q^17 - 5076 * q^25 - 15776 * q^33 - 33640 * q^41 - 65180 * q^49 - 121568 * q^57 - 170752 * q^65 - 172024 * q^73 - 225020 * q^81 - 375480 * q^89 - 326824 * q^97

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
$3$	15.2315	0.977104	0.488552	−	0.872535i	$-0.337525\pi$
$3$	15.2315	0.977104	0.488552	+	0.872535i	$0.337525\pi$
$4$	0	0
$5$	−43.0813	−0.770662	−0.385331	−	0.922778i	$-0.625913\pi$
$5$	−43.0813	−0.770662	−0.385331	+	0.922778i	$0.625913\pi$
$6$	0	0
$7$	22.6274	0.174538	0.0872690	−	0.996185i	$-0.472186\pi$
$7$	22.6274	0.174538	0.0872690	+	0.996185i	$0.472186\pi$
$8$	0	0
$9$	−11.0000	−0.0452675
$10$	0	0
$11$	−258.936	−0.645225	−0.322613	−	0.946531i	$-0.604561\pi$
$11$	−258.936	−0.645225	−0.322613	+	0.946531i	$0.604561\pi$
$12$	0	0
$13$	990.870	1.62614	0.813071	−	0.582165i	$-0.197794\pi$
$13$	990.870	1.62614	0.813071	+	0.582165i	$0.197794\pi$
$14$	0	0
$15$	−656.195	−0.753017
$16$	0	0
$17$	−218.000	−0.182951	−0.0914754	−	0.995807i	$-0.529158\pi$
$17$	−218.000	−0.182951	−0.0914754	+	0.995807i	$0.529158\pi$
$18$	0	0
$19$	−1995.33	−1.26804	−0.634018	−	0.773319i	$-0.718595\pi$
$19$	−1995.33	−1.26804	−0.634018	+	0.773319i	$0.718595\pi$
$20$	0	0
$21$	344.651	0.170542
$22$	0	0
$23$	3416.74	1.34677	0.673383	−	0.739294i	$-0.264840\pi$
$23$	3416.74	1.34677	0.673383	+	0.739294i	$0.264840\pi$
$24$	0	0
$25$	−1269.00	−0.406080
$26$	0	0
$27$	−3868.81	−1.02134
$28$	0	0
$29$	−4868.19	−1.07491	−0.537455	−	0.843292i	$-0.680614\pi$
$29$	−4868.19	−1.07491	−0.537455	+	0.843292i	$0.680614\pi$
$30$	0	0
$31$	−8688.93	−1.62391	−0.811955	−	0.583720i	$-0.801597\pi$
$31$	−8688.93	−1.62391	−0.811955	+	0.583720i	$0.801597\pi$
$32$	0	0
$33$	−3944.00	−0.630452
$34$	0	0
$35$	−974.819	−0.134510
$36$	0	0
$37$	3231.10	0.388013	0.194006	−	0.981000i	$-0.437852\pi$
$37$	3231.10	0.388013	0.194006	+	0.981000i	$0.437852\pi$
$38$	0	0
$39$	15092.5	1.58891
$40$	0	0
$41$	−8410.00	−0.781333	−0.390667	−	0.920532i	$-0.627755\pi$
$41$	−8410.00	−0.781333	−0.390667	+	0.920532i	$0.627755\pi$
$42$	0	0
$43$	−685.420	−0.0565308	−0.0282654	−	0.999600i	$-0.508998\pi$
$43$	−685.420	−0.0565308	−0.0282654	+	0.999600i	$0.508998\pi$
$44$	0	0
$45$	473.895	0.0348859
$46$	0	0
$47$	−18147.2	−1.19830	−0.599149	−	0.800638i	$-0.704494\pi$
$47$	−18147.2	−1.19830	−0.599149	+	0.800638i	$0.704494\pi$
$48$	0	0
$49$	−16295.0	−0.969537
$50$	0	0
$51$	−3320.48	−0.178762
$52$	0	0
$53$	−26581.2	−1.29982	−0.649912	−	0.760010i	$-0.725194\pi$
$53$	−26581.2	−1.29982	−0.649912	+	0.760010i	$0.725194\pi$
$54$	0	0
$55$	11155.3	0.497250
$56$	0	0
$57$	−30392.0	−1.23900
$58$	0	0
$59$	33798.8	1.26407	0.632035	−	0.774940i	$-0.282220\pi$
$59$	33798.8	1.26407	0.632035	+	0.774940i	$0.282220\pi$
$60$	0	0
$61$	27356.6	0.941322	0.470661	−	0.882314i	$-0.344016\pi$
$61$	27356.6	0.941322	0.470661	+	0.882314i	$0.344016\pi$
$62$	0	0
$63$	−248.902	−0.00790090
$64$	0	0
$65$	−42688.0	−1.25321
$66$	0	0
$67$	66729.4	1.81606	0.908030	−	0.418905i	$-0.137586\pi$
$67$	66729.4	1.81606	0.908030	+	0.418905i	$0.137586\pi$
$68$	0	0
$69$	52042.2	1.31593
$70$	0	0
$71$	−21609.2	−0.508736	−0.254368	−	0.967108i	$-0.581867\pi$
$71$	−21609.2	−0.508736	−0.254368	+	0.967108i	$0.581867\pi$
$72$	0	0
$73$	−43006.0	−0.944543	−0.472272	−	0.881453i	$-0.656566\pi$
$73$	−43006.0	−0.944543	−0.472272	+	0.881453i	$0.656566\pi$
$74$	0	0
$75$	−19328.8	−0.396782
$76$	0	0
$77$	−5859.06	−0.112616
$78$	0	0
$79$	64759.7	1.16745	0.583723	−	0.811953i	$-0.301595\pi$
$79$	64759.7	1.16745	0.583723	+	0.811953i	$0.301595\pi$
$80$	0	0
$81$	−56255.0	−0.952683
$82$	0	0
$83$	−70659.1	−1.12583	−0.562915	−	0.826515i	$-0.690320\pi$
$83$	−70659.1	−1.12583	−0.562915	+	0.826515i	$0.690320\pi$
$84$	0	0
$85$	9391.73	0.140993
$86$	0	0
$87$	−74150.0	−1.05030
$88$	0	0
$89$	−93870.0	−1.25618	−0.628090	−	0.778141i	$-0.716163\pi$
$89$	−93870.0	−1.25618	−0.628090	+	0.778141i	$0.716163\pi$
$90$	0	0
$91$	22420.8	0.283823
$92$	0	0
$93$	−132346.	−1.58673
$94$	0	0
$95$	85961.6	0.977227
$96$	0	0
$97$	−81706.0	−0.881708	−0.440854	−	0.897579i	$-0.645324\pi$
$97$	−81706.0	−0.881708	−0.440854	+	0.897579i	$0.645324\pi$
$98$	0	0
$99$	2848.30	0.0292077
$100$	0	0
$101$	−87412.0	−0.852644	−0.426322	−	0.904572i	$-0.640191\pi$
$101$	−87412.0	−0.852644	−0.426322	+	0.904572i	$0.640191\pi$
$102$	0	0
$103$	−132665.	−1.23214	−0.616072	−	0.787689i	$-0.711277\pi$
$103$	−132665.	−1.23214	−0.616072	+	0.787689i	$0.711277\pi$
$104$	0	0
$105$	−14848.0	−0.131430
$106$	0	0
$107$	−87870.8	−0.741968	−0.370984	−	0.928639i	$-0.620980\pi$
$107$	−87870.8	−0.741968	−0.370984	+	0.928639i	$0.620980\pi$
$108$	0	0
$109$	121791.	0.981858	0.490929	−	0.871200i	$-0.336657\pi$
$109$	121791.	0.981858	0.490929	+	0.871200i	$0.336657\pi$
$110$	0	0
$111$	49214.6	0.379129
$112$	0	0
$113$	−29614.0	−0.218173	−0.109086	−	0.994032i	$-0.534793\pi$
$113$	−29614.0	−0.218173	−0.109086	+	0.994032i	$0.534793\pi$
$114$	0	0
$115$	−147198.	−1.03790
$116$	0	0
$117$	−10899.6	−0.0736114
$118$	0	0
$119$	−4932.78	−0.0319318
$120$	0	0
$121$	−94003.0	−0.583685
$122$	0	0
$123$	−128097.	−0.763444
$124$	0	0
$125$	189299.	1.08361
$126$	0	0
$127$	60551.0	0.333128	0.166564	−	0.986031i	$-0.446733\pi$
$127$	60551.0	0.333128	0.166564	+	0.986031i	$0.446733\pi$
$128$	0	0
$129$	−10440.0	−0.0552365
$130$	0	0
$131$	186754.	0.950805	0.475403	−	0.879768i	$-0.342302\pi$
$131$	186754.	0.950805	0.475403	+	0.879768i	$0.342302\pi$
$132$	0	0
$133$	−45149.2	−0.221320
$134$	0	0
$135$	166674.	0.787104
$136$	0	0
$137$	182054.	0.828703	0.414351	−	0.910117i	$-0.364008\pi$
$137$	182054.	0.828703	0.414351	+	0.910117i	$0.364008\pi$
$138$	0	0
$139$	45131.1	0.198125	0.0990624	−	0.995081i	$-0.468416\pi$
$139$	45131.1	0.198125	0.0990624	+	0.995081i	$0.468416\pi$
$140$	0	0
$141$	−276410.	−1.17086
$142$	0	0
$143$	−256572.	−1.04923
$144$	0	0
$145$	209728.	0.828393
$146$	0	0
$147$	−248198.	−0.947338
$148$	0	0
$149$	262408.	0.968304	0.484152	−	0.874984i	$-0.339128\pi$
$149$	262408.	0.968304	0.484152	+	0.874984i	$0.339128\pi$
$150$	0	0
$151$	113839.	0.406300	0.203150	−	0.979148i	$-0.434882\pi$
$151$	113839.	0.406300	0.203150	+	0.979148i	$0.434882\pi$
$152$	0	0
$153$	2398.00	0.00828172
$154$	0	0
$155$	374330.	1.25149
$156$	0	0
$157$	114898.	0.372017	0.186009	−	0.982548i	$-0.440445\pi$
$157$	114898.	0.372017	0.186009	+	0.982548i	$0.440445\pi$
$158$	0	0
$159$	−404872.	−1.27006
$160$	0	0
$161$	77312.0	0.235062
$162$	0	0
$163$	284784.	0.839551	0.419775	−	0.907628i	$-0.362109\pi$
$163$	284784.	0.839551	0.419775	+	0.907628i	$0.362109\pi$
$164$	0	0
$165$	169913.	0.485865
$166$	0	0
$167$	425871.	1.18164	0.590822	−	0.806802i	$-0.298804\pi$
$167$	425871.	1.18164	0.590822	+	0.806802i	$0.298804\pi$
$168$	0	0
$169$	610531.	1.64434
$170$	0	0
$171$	21948.7	0.0574008
$172$	0	0
$173$	−165131.	−0.419481	−0.209741	−	0.977757i	$-0.567262\pi$
$173$	−165131.	−0.419481	−0.209741	+	0.977757i	$0.567262\pi$
$174$	0	0
$175$	−28714.2	−0.0708764
$176$	0	0
$177$	514808.	1.23513
$178$	0	0
$179$	724671.	1.69047	0.845237	−	0.534392i	$-0.179459\pi$
$179$	724671.	1.69047	0.845237	+	0.534392i	$0.179459\pi$
$180$	0	0
$181$	−45709.3	−0.103707	−0.0518535	−	0.998655i	$-0.516513\pi$
$181$	−45709.3	−0.103707	−0.0518535	+	0.998655i	$0.516513\pi$
$182$	0	0
$183$	416684.	0.919769
$184$	0	0
$185$	−139200.	−0.299027
$186$	0	0
$187$	56448.1	0.118044
$188$	0	0
$189$	−87541.2	−0.178262
$190$	0	0
$191$	−434537.	−0.861873	−0.430936	−	0.902382i	$-0.641817\pi$
$191$	−434537.	−0.861873	−0.430936	+	0.902382i	$0.641817\pi$
$192$	0	0
$193$	277878.	0.536983	0.268492	−	0.963282i	$-0.413475\pi$
$193$	277878.	0.536983	0.268492	+	0.963282i	$0.413475\pi$
$194$	0	0
$195$	−650204.	−1.22451
$196$	0	0
$197$	−1.00539e6	−1.84573	−0.922866	−	0.385122i	$-0.874159\pi$
$197$	−1.00539e6	−1.84573	−0.922866	+	0.385122i	$0.874159\pi$
$198$	0	0
$199$	−477506.	−0.854765	−0.427382	−	0.904071i	$-0.640564\pi$
$199$	−477506.	−0.854765	−0.427382	+	0.904071i	$0.640564\pi$
$200$	0	0
$201$	1.01639e6	1.77448
$202$	0	0
$203$	−110155.	−0.187613
$204$	0	0
$205$	362314.	0.602144
$206$	0	0
$207$	−37584.1	−0.0609647
$208$	0	0
$209$	516664.	0.818168
$210$	0	0
$211$	−18871.9	−0.0291816	−0.0145908	−	0.999894i	$-0.504645\pi$
$211$	−18871.9	−0.0291816	−0.0145908	+	0.999894i	$0.504645\pi$
$212$	0	0
$213$	−329141.	−0.497088
$214$	0	0
$215$	29528.8	0.0435662
$216$	0	0
$217$	−196608.	−0.283434
$218$	0	0
$219$	−655048.	−0.922917
$220$	0	0
$221$	−216010.	−0.297504
$222$	0	0
$223$	1.23636e6	1.66488	0.832441	−	0.554114i	$-0.186943\pi$
$223$	1.23636e6	1.66488	0.832441	+	0.554114i	$0.186943\pi$
$224$	0	0
$225$	13959.0	0.0183822
$226$	0	0
$227$	−802261.	−1.03336	−0.516679	−	0.856179i	$-0.672832\pi$
$227$	−802261.	−1.03336	−0.516679	+	0.856179i	$0.672832\pi$
$228$	0	0
$229$	1.39579e6	1.75886	0.879431	−	0.476026i	$-0.157923\pi$
$229$	1.39579e6	1.75886	0.879431	+	0.476026i	$0.157923\pi$
$230$	0	0
$231$	−89242.5	−0.110038
$232$	0	0
$233$	812898.	0.980949	0.490474	−	0.871456i	$-0.336824\pi$
$233$	812898.	0.980949	0.490474	+	0.871456i	$0.336824\pi$
$234$	0	0
$235$	781805.	0.923483
$236$	0	0
$237$	986390.	1.14072
$238$	0	0
$239$	403537.	0.456971	0.228486	−	0.973547i	$-0.426623\pi$
$239$	403537.	0.456971	0.228486	+	0.973547i	$0.426623\pi$
$240$	0	0
$241$	560262.	0.621368	0.310684	−	0.950513i	$-0.399442\pi$
$241$	560262.	0.621368	0.310684	+	0.950513i	$0.399442\pi$
$242$	0	0
$243$	83270.9	0.0904643
$244$	0	0
$245$	702010.	0.747185
$246$	0	0
$247$	−1.97712e6	−2.06201
$248$	0	0
$249$	−1.07625e6	−1.10005
$250$	0	0
$251$	−901784.	−0.903479	−0.451740	−	0.892150i	$-0.649196\pi$
$251$	−901784.	−0.903479	−0.451740	+	0.892150i	$0.649196\pi$
$252$	0	0
$253$	−884718.	−0.868968
$254$	0	0
$255$	143051.	0.137765
$256$	0	0
$257$	−870558.	−0.822176	−0.411088	−	0.911596i	$-0.634851\pi$
$257$	−870558.	−0.822176	−0.411088	+	0.911596i	$0.634851\pi$
$258$	0	0
$259$	73111.4	0.0677229
$260$	0	0
$261$	53550.1	0.0486585
$262$	0	0
$263$	−231954.	−0.206782	−0.103391	−	0.994641i	$-0.532969\pi$
$263$	−231954.	−0.206782	−0.103391	+	0.994641i	$0.532969\pi$
$264$	0	0
$265$	1.14515e6	1.00172
$266$	0	0
$267$	−1.42979e6	−1.22742
$268$	0	0
$269$	−1.43396e6	−1.20825	−0.604125	−	0.796890i	$-0.706477\pi$
$269$	−1.43396e6	−1.20825	−0.604125	+	0.796890i	$0.706477\pi$
$270$	0	0
$271$	1.59627e6	1.32034	0.660168	−	0.751118i	$-0.270485\pi$
$271$	1.59627e6	1.32034	0.660168	+	0.751118i	$0.270485\pi$
$272$	0	0
$273$	341504.	0.277325
$274$	0	0
$275$	328590.	0.262013
$276$	0	0
$277$	−1.38278e6	−1.08281	−0.541407	−	0.840761i	$-0.682108\pi$
$277$	−1.38278e6	−1.08281	−0.541407	+	0.840761i	$0.682108\pi$
$278$	0	0
$279$	95578.2	0.0735103
$280$	0	0
$281$	−2.39301e6	−1.80791	−0.903957	−	0.427622i	$-0.859351\pi$
$281$	−2.39301e6	−1.80791	−0.903957	+	0.427622i	$0.859351\pi$
$282$	0	0
$283$	597671.	0.443604	0.221802	−	0.975092i	$-0.428806\pi$
$283$	597671.	0.443604	0.221802	+	0.975092i	$0.428806\pi$
$284$	0	0
$285$	1.30933e6	0.954852
$286$	0	0
$287$	−190297.	−0.136372
$288$	0	0
$289$	−1.37233e6	−0.966529
$290$	0	0
$291$	−1.24451e6	−0.861520
$292$	0	0
$293$	874680.	0.595224	0.297612	−	0.954687i	$-0.403810\pi$
$293$	874680.	0.595224	0.297612	+	0.954687i	$0.403810\pi$
$294$	0	0
$295$	−1.45610e6	−0.974171
$296$	0	0
$297$	1.00178e6	0.658991
$298$	0	0
$299$	3.38555e6	2.19003
$300$	0	0
$301$	−15509.3	−0.00986678
$302$	0	0
$303$	−1.33142e6	−0.833122
$304$	0	0
$305$	−1.17856e6	−0.725441
$306$	0	0
$307$	2.70313e6	1.63689	0.818447	−	0.574582i	$-0.194835\pi$
$307$	2.70313e6	1.63689	0.818447	+	0.574582i	$0.194835\pi$
$308$	0	0
$309$	−2.02069e6	−1.20393
$310$	0	0
$311$	2.49642e6	1.46358	0.731789	−	0.681531i	$-0.238686\pi$
$311$	2.49642e6	1.46358	0.731789	+	0.681531i	$0.238686\pi$
$312$	0	0
$313$	−2.92331e6	−1.68661	−0.843303	−	0.537439i	$-0.819392\pi$
$313$	−2.92331e6	−1.68661	−0.843303	+	0.537439i	$0.819392\pi$
$314$	0	0
$315$	10723.0	0.00608892
$316$	0	0
$317$	1.61671e6	0.903618	0.451809	−	0.892115i	$-0.350779\pi$
$317$	1.61671e6	0.903618	0.451809	+	0.892115i	$0.350779\pi$
$318$	0	0
$319$	1.26055e6	0.693559
$320$	0	0
$321$	−1.33841e6	−0.724980
$322$	0	0
$323$	434982.	0.231988
$324$	0	0
$325$	−1.25741e6	−0.660344
$326$	0	0
$327$	1.85506e6	0.959377
$328$	0	0
$329$	−410624.	−0.209148
$330$	0	0
$331$	−1.14774e6	−0.575804	−0.287902	−	0.957660i	$-0.592958\pi$
$331$	−1.14774e6	−0.575804	−0.287902	+	0.957660i	$0.592958\pi$
$332$	0	0
$333$	−35542.1	−0.0175644
$334$	0	0
$335$	−2.87479e6	−1.39957
$336$	0	0
$337$	371378.	0.178132	0.0890659	−	0.996026i	$-0.471612\pi$
$337$	371378.	0.178132	0.0890659	+	0.996026i	$0.471612\pi$
$338$	0	0
$339$	−451067.	−0.213178
$340$	0	0
$341$	2.24988e6	1.04779
$342$	0	0
$343$	−749013.	−0.343759
$344$	0	0
$345$	−2.24205e6	−1.01414
$346$	0	0
$347$	1.13912e6	0.507863	0.253931	−	0.967222i	$-0.418276\pi$
$347$	1.13912e6	0.507863	0.253931	+	0.967222i	$0.418276\pi$
$348$	0	0
$349$	−4.08105e6	−1.79353	−0.896764	−	0.442508i	$-0.854089\pi$
$349$	−4.08105e6	−1.79353	−0.896764	+	0.442508i	$0.854089\pi$
$350$	0	0
$351$	−3.83349e6	−1.66084
$352$	0	0
$353$	−3.90953e6	−1.66989	−0.834946	−	0.550332i	$-0.814501\pi$
$353$	−3.90953e6	−1.66989	−0.834946	+	0.550332i	$0.814501\pi$
$354$	0	0
$355$	930952.	0.392064
$356$	0	0
$357$	−75133.8	−0.0312007
$358$	0	0
$359$	2.62028e6	1.07303	0.536514	−	0.843891i	$-0.319741\pi$
$359$	2.62028e6	1.07303	0.536514	+	0.843891i	$0.319741\pi$
$360$	0	0
$361$	1.50525e6	0.607913
$362$	0	0
$363$	−1.43181e6	−0.570321
$364$	0	0
$365$	1.85276e6	0.727924
$366$	0	0
$367$	1.65266e6	0.640499	0.320250	−	0.947333i	$-0.396233\pi$
$367$	1.65266e6	0.640499	0.320250	+	0.947333i	$0.396233\pi$
$368$	0	0
$369$	92510.0	0.0353690
$370$	0	0
$371$	−601463.	−0.226869
$372$	0	0
$373$	3.09630e6	1.15231	0.576157	−	0.817339i	$-0.304552\pi$
$373$	3.09630e6	1.15231	0.576157	+	0.817339i	$0.304552\pi$
$374$	0	0
$375$	2.88332e6	1.05880
$376$	0	0
$377$	−4.82374e6	−1.74796
$378$	0	0
$379$	−647478.	−0.231540	−0.115770	−	0.993276i	$-0.536934\pi$
$379$	−647478.	−0.231540	−0.115770	+	0.993276i	$0.536934\pi$
$380$	0	0
$381$	922285.	0.325501
$382$	0	0
$383$	−2.42168e6	−0.843566	−0.421783	−	0.906697i	$-0.638596\pi$
$383$	−2.42168e6	−0.843566	−0.421783	+	0.906697i	$0.638596\pi$
$384$	0	0
$385$	252416.	0.0867891
$386$	0	0
$387$	7539.62	0.00255901
$388$	0	0
$389$	1.06441e6	0.356644	0.178322	−	0.983972i	$-0.442933\pi$
$389$	1.06441e6	0.356644	0.178322	+	0.983972i	$0.442933\pi$
$390$	0	0
$391$	−744849.	−0.246392
$392$	0	0
$393$	2.84455e6	0.929036
$394$	0	0
$395$	−2.78993e6	−0.899706
$396$	0	0
$397$	734235.	0.233808	0.116904	−	0.993143i	$-0.462703\pi$
$397$	734235.	0.233808	0.116904	+	0.993143i	$0.462703\pi$
$398$	0	0
$399$	−687692.	−0.216253
$400$	0	0
$401$	−448218.	−0.139197	−0.0695983	−	0.997575i	$-0.522172\pi$
$401$	−448218.	−0.139197	−0.0695983	+	0.997575i	$0.522172\pi$
$402$	0	0
$403$	−8.60960e6	−2.64071
$404$	0	0
$405$	2.42354e6	0.734197
$406$	0	0
$407$	−836649.	−0.250356
$408$	0	0
$409$	84854.0	0.0250821	0.0125411	−	0.999921i	$-0.496008\pi$
$409$	84854.0	0.0250821	0.0125411	+	0.999921i	$0.496008\pi$
$410$	0	0
$411$	2.77296e6	0.809729
$412$	0	0
$413$	764780.	0.220628
$414$	0	0
$415$	3.04409e6	0.867635
$416$	0	0
$417$	687416.	0.193588
$418$	0	0
$419$	2.75650e6	0.767048	0.383524	−	0.923531i	$-0.374710\pi$
$419$	2.75650e6	0.767048	0.383524	+	0.923531i	$0.374710\pi$
$420$	0	0
$421$	−6.94647e6	−1.91011	−0.955057	−	0.296423i	$-0.904206\pi$
$421$	−6.94647e6	−1.91011	−0.955057	+	0.296423i	$0.904206\pi$
$422$	0	0
$423$	199619.	0.0542439
$424$	0	0
$425$	276642.	0.0742926
$426$	0	0
$427$	619010.	0.164296
$428$	0	0
$429$	−3.90799e6	−1.02520
$430$	0	0
$431$	−1.08227e6	−0.280635	−0.140318	−	0.990107i	$-0.544812\pi$
$431$	−1.08227e6	−0.280635	−0.140318	+	0.990107i	$0.544812\pi$
$432$	0	0
$433$	2.26093e6	0.579520	0.289760	−	0.957099i	$-0.406425\pi$
$433$	2.26093e6	0.579520	0.289760	+	0.957099i	$0.406425\pi$
$434$	0	0
$435$	3.19448e6	0.809426
$436$	0	0
$437$	−6.81753e6	−1.70775
$438$	0	0
$439$	−6.16645e6	−1.52712	−0.763561	−	0.645736i	$-0.776551\pi$
$439$	−6.16645e6	−1.52712	−0.763561	+	0.645736i	$0.776551\pi$
$440$	0	0
$441$	179245.	0.0438885
$442$	0	0
$443$	−6.61587e6	−1.60169	−0.800843	−	0.598874i	$-0.795615\pi$
$443$	−6.61587e6	−1.60169	−0.800843	+	0.598874i	$0.795615\pi$
$444$	0	0
$445$	4.04404e6	0.968090
$446$	0	0
$447$	3.99688e6	0.946134
$448$	0	0
$449$	−807882.	−0.189118	−0.0945588	−	0.995519i	$-0.530144\pi$
$449$	−807882.	−0.189118	−0.0945588	+	0.995519i	$0.530144\pi$
$450$	0	0
$451$	2.17765e6	0.504136
$452$	0	0
$453$	1.73394e6	0.396998
$454$	0	0
$455$	−965919.	−0.218732
$456$	0	0
$457$	6.20353e6	1.38947	0.694733	−	0.719267i	$-0.255522\pi$
$457$	6.20353e6	1.38947	0.694733	+	0.719267i	$0.255522\pi$
$458$	0	0
$459$	843401.	0.186854
$460$	0	0
$461$	−1.93853e6	−0.424835	−0.212417	−	0.977179i	$-0.568134\pi$
$461$	−1.93853e6	−0.424835	−0.212417	+	0.977179i	$0.568134\pi$
$462$	0	0
$463$	6.96413e6	1.50978	0.754891	−	0.655850i	$-0.227690\pi$
$463$	6.96413e6	1.50978	0.754891	+	0.655850i	$0.227690\pi$
$464$	0	0
$465$	5.70163e6	1.22283
$466$	0	0
$467$	−4.65298e6	−0.987277	−0.493638	−	0.869667i	$-0.664333\pi$
$467$	−4.65298e6	−0.987277	−0.493638	+	0.869667i	$0.664333\pi$
$468$	0	0
$469$	1.50991e6	0.316971
$470$	0	0
$471$	1.75007e6	0.363499
$472$	0	0
$473$	177480.	0.0364751
$474$	0	0
$475$	2.53208e6	0.514924
$476$	0	0
$477$	292393.	0.0588398
$478$	0	0
$479$	2.10743e6	0.419676	0.209838	−	0.977736i	$-0.432706\pi$
$479$	2.10743e6	0.419676	0.209838	+	0.977736i	$0.432706\pi$
$480$	0	0
$481$	3.20160e6	0.630964
$482$	0	0
$483$	1.17758e6	0.229680
$484$	0	0
$485$	3.52000e6	0.679499
$486$	0	0
$487$	−9.02334e6	−1.72403	−0.862015	−	0.506883i	$-0.830798\pi$
$487$	−9.02334e6	−1.72403	−0.862015	+	0.506883i	$0.830798\pi$
$488$	0	0
$489$	4.33770e6	0.820328
$490$	0	0
$491$	2.69262e6	0.504047	0.252023	−	0.967721i	$-0.418904\pi$
$491$	2.69262e6	0.504047	0.252023	+	0.967721i	$0.418904\pi$
$492$	0	0
$493$	1.06127e6	0.196656
$494$	0	0
$495$	−122708.	−0.0225093
$496$	0	0
$497$	−488960.	−0.0887937
$498$	0	0
$499$	4.32456e6	0.777482	0.388741	−	0.921347i	$-0.372910\pi$
$499$	4.32456e6	0.777482	0.388741	+	0.921347i	$0.372910\pi$
$500$	0	0
$501$	6.48667e6	1.15459
$502$	0	0
$503$	−2.58525e6	−0.455599	−0.227799	−	0.973708i	$-0.573153\pi$
$503$	−2.58525e6	−0.455599	−0.227799	+	0.973708i	$0.573153\pi$
$504$	0	0
$505$	3.76582e6	0.657100
$506$	0	0
$507$	9.29933e6	1.60669
$508$	0	0
$509$	−4.02746e6	−0.689027	−0.344514	−	0.938781i	$-0.611956\pi$
$509$	−4.02746e6	−0.689027	−0.344514	+	0.938781i	$0.611956\pi$
$510$	0	0
$511$	−973115.	−0.164859
$512$	0	0
$513$	7.71957e6	1.29509
$514$	0	0
$515$	5.71536e6	0.949567
$516$	0	0
$517$	4.69897e6	0.773172
$518$	0	0
$519$	−2.51520e6	−0.409877
$520$	0	0
$521$	−2.11324e6	−0.341079	−0.170539	−	0.985351i	$-0.554551\pi$
$521$	−2.11324e6	−0.341079	−0.170539	+	0.985351i	$0.554551\pi$
$522$	0	0
$523$	−2.72110e6	−0.435001	−0.217501	−	0.976060i	$-0.569790\pi$
$523$	−2.72110e6	−0.435001	−0.217501	+	0.976060i	$0.569790\pi$
$524$	0	0
$525$	−437362.	−0.0692536
$526$	0	0
$527$	1.89419e6	0.297096
$528$	0	0
$529$	5.23777e6	0.813780
$530$	0	0
$531$	−371787.	−0.0572213
$532$	0	0
$533$	−8.33322e6	−1.27056
$534$	0	0
$535$	3.78559e6	0.571806
$536$	0	0
$537$	1.10379e7	1.65177
$538$	0	0
$539$	4.21937e6	0.625569
$540$	0	0
$541$	1.05683e6	0.155243	0.0776213	−	0.996983i	$-0.475267\pi$
$541$	1.05683e6	0.155243	0.0776213	+	0.996983i	$0.475267\pi$
$542$	0	0
$543$	−696223.	−0.101333
$544$	0	0
$545$	−5.24691e6	−0.756680
$546$	0	0
$547$	−3.14920e6	−0.450020	−0.225010	−	0.974356i	$-0.572241\pi$
$547$	−3.14920e6	−0.450020	−0.225010	+	0.974356i	$0.572241\pi$
$548$	0	0
$549$	−300923.	−0.0426113
$550$	0	0
$551$	9.71366e6	1.36302
$552$	0	0
$553$	1.46534e6	0.203764
$554$	0	0
$555$	−2.12023e6	−0.292180
$556$	0	0
$557$	3.91932e6	0.535270	0.267635	−	0.963520i	$-0.413758\pi$
$557$	3.91932e6	0.535270	0.267635	+	0.963520i	$0.413758\pi$
$558$	0	0
$559$	−679162.	−0.0919272
$560$	0	0
$561$	859792.	0.115342
$562$	0	0
$563$	−2.62499e6	−0.349025	−0.174513	−	0.984655i	$-0.555835\pi$
$563$	−2.62499e6	−0.349025	−0.174513	+	0.984655i	$0.555835\pi$
$564$	0	0
$565$	1.27581e6	0.168138
$566$	0	0
$567$	−1.27291e6	−0.166279
$568$	0	0
$569$	−6.56020e6	−0.849448	−0.424724	−	0.905323i	$-0.639629\pi$
$569$	−6.56020e6	−0.849448	−0.424724	+	0.905323i	$0.639629\pi$
$570$	0	0
$571$	−1.14800e7	−1.47350	−0.736751	−	0.676165i	$-0.763641\pi$
$571$	−1.14800e7	−1.47350	−0.736751	+	0.676165i	$0.763641\pi$
$572$	0	0
$573$	−6.61867e6	−0.842140
$574$	0	0
$575$	−4.33584e6	−0.546895
$576$	0	0
$577$	1.04493e7	1.30662	0.653308	−	0.757092i	$-0.273381\pi$
$577$	1.04493e7	1.30662	0.653308	+	0.757092i	$0.273381\pi$
$578$	0	0
$579$	4.23251e6	0.524689
$580$	0	0
$581$	−1.59883e6	−0.196500
$582$	0	0
$583$	6.88283e6	0.838679
$584$	0	0
$585$	469568.	0.0567295
$586$	0	0
$587$	−6.24714e6	−0.748318	−0.374159	−	0.927365i	$-0.622069\pi$
$587$	−6.24714e6	−0.748318	−0.374159	+	0.927365i	$0.622069\pi$
$588$	0	0
$589$	1.73373e7	2.05917
$590$	0	0
$591$	−1.53136e7	−1.80347
$592$	0	0
$593$	−6.31288e6	−0.737209	−0.368605	−	0.929586i	$-0.620164\pi$
$593$	−6.31288e6	−0.737209	−0.368605	+	0.929586i	$0.620164\pi$
$594$	0	0
$595$	212511.	0.0246087
$596$	0	0
$597$	−7.27316e6	−0.835194
$598$	0	0
$599$	1.21797e6	0.138697	0.0693487	−	0.997592i	$-0.477908\pi$
$599$	1.21797e6	0.138697	0.0693487	+	0.997592i	$0.477908\pi$
$600$	0	0
$601$	1.05226e7	1.18833	0.594167	−	0.804342i	$-0.297482\pi$
$601$	1.05226e7	1.18833	0.594167	+	0.804342i	$0.297482\pi$
$602$	0	0
$603$	−734023.	−0.0822085
$604$	0	0
$605$	4.04977e6	0.449824
$606$	0	0
$607$	−2.18789e6	−0.241020	−0.120510	−	0.992712i	$-0.538453\pi$
$607$	−2.18789e6	−0.241020	−0.120510	+	0.992712i	$0.538453\pi$
$608$	0	0
$609$	−1.67782e6	−0.183317
$610$	0	0
$611$	−1.79815e7	−1.94860
$612$	0	0
$613$	−3.98593e6	−0.428428	−0.214214	−	0.976787i	$-0.568719\pi$
$613$	−3.98593e6	−0.428428	−0.214214	+	0.976787i	$0.568719\pi$
$614$	0	0
$615$	5.51860e6	0.588357
$616$	0	0
$617$	−1.42965e7	−1.51188	−0.755938	−	0.654643i	$-0.772819\pi$
$617$	−1.42965e7	−1.51188	−0.755938	+	0.654643i	$0.772819\pi$
$618$	0	0
$619$	1.32078e7	1.38549	0.692746	−	0.721182i	$-0.256401\pi$
$619$	1.32078e7	1.38549	0.692746	+	0.721182i	$0.256401\pi$
$620$	0	0
$621$	−1.32187e7	−1.37550
$622$	0	0
$623$	−2.12404e6	−0.219251
$624$	0	0
$625$	−4.18964e6	−0.429019
$626$	0	0
$627$	7.86959e6	0.799435
$628$	0	0
$629$	−704380.	−0.0709872
$630$	0	0
$631$	1.51047e7	1.51021	0.755107	−	0.655602i	$-0.227585\pi$
$631$	1.51047e7	1.51021	0.755107	+	0.655602i	$0.227585\pi$
$632$	0	0
$633$	−287448.	−0.0285135
$634$	0	0
$635$	−2.60862e6	−0.256729
$636$	0	0
$637$	−1.61462e7	−1.57660
$638$	0	0
$639$	237701.	0.0230292
$640$	0	0
$641$	−6.88071e6	−0.661437	−0.330718	−	0.943729i	$-0.607291\pi$
$641$	−6.88071e6	−0.661437	−0.330718	+	0.943729i	$0.607291\pi$
$642$	0	0
$643$	−1.44561e7	−1.37887	−0.689437	−	0.724346i	$-0.742142\pi$
$643$	−1.44561e7	−1.37887	−0.689437	+	0.724346i	$0.742142\pi$
$644$	0	0
$645$	449769.	0.0425687
$646$	0	0
$647$	−4.83310e6	−0.453905	−0.226953	−	0.973906i	$-0.572876\pi$
$647$	−4.83310e6	−0.453905	−0.226953	+	0.973906i	$0.572876\pi$
$648$	0	0
$649$	−8.75174e6	−0.815610
$650$	0	0
$651$	−2.99464e6	−0.276944
$652$	0	0
$653$	−8.22289e6	−0.754643	−0.377321	−	0.926082i	$-0.623155\pi$
$653$	−8.22289e6	−0.754643	−0.377321	+	0.926082i	$0.623155\pi$
$654$	0	0
$655$	−8.04561e6	−0.732750
$656$	0	0
$657$	473066.	0.0427571
$658$	0	0
$659$	5.93447e6	0.532315	0.266157	−	0.963930i	$-0.414246\pi$
$659$	5.93447e6	0.532315	0.266157	+	0.963930i	$0.414246\pi$
$660$	0	0
$661$	6.07425e6	0.540741	0.270370	−	0.962756i	$-0.412854\pi$
$661$	6.07425e6	0.540741	0.270370	+	0.962756i	$0.412854\pi$
$662$	0	0
$663$	−3.29016e6	−0.290692
$664$	0	0
$665$	1.94509e6	0.170563
$666$	0	0
$667$	−1.66333e7	−1.44765
$668$	0	0
$669$	1.88317e7	1.62676
$670$	0	0
$671$	−7.08363e6	−0.607364
$672$	0	0
$673$	−4.01847e6	−0.341998	−0.170999	−	0.985271i	$-0.554699\pi$
$673$	−4.01847e6	−0.341998	−0.170999	+	0.985271i	$0.554699\pi$
$674$	0	0
$675$	4.90952e6	0.414744
$676$	0	0
$677$	1.84827e7	1.54986	0.774932	−	0.632044i	$-0.217784\pi$
$677$	1.84827e7	1.54986	0.774932	+	0.632044i	$0.217784\pi$
$678$	0	0
$679$	−1.84880e6	−0.153891
$680$	0	0
$681$	−1.22197e7	−1.00970
$682$	0	0
$683$	1.40610e6	0.115336	0.0576679	−	0.998336i	$-0.481634\pi$
$683$	1.40610e6	0.115336	0.0576679	+	0.998336i	$0.481634\pi$
$684$	0	0
$685$	−7.84313e6	−0.638650
$686$	0	0
$687$	2.12601e7	1.71859
$688$	0	0
$689$	−2.63385e7	−2.11370
$690$	0	0
$691$	7.01640e6	0.559009	0.279505	−	0.960144i	$-0.409830\pi$
$691$	7.01640e6	0.559009	0.279505	+	0.960144i	$0.409830\pi$
$692$	0	0
$693$	64449.7	0.00509786
$694$	0	0
$695$	−1.94431e6	−0.152687
$696$	0	0
$697$	1.83338e6	0.142946
$698$	0	0
$699$	1.23817e7	0.958489
$700$	0	0
$701$	1.28008e7	0.983879	0.491940	−	0.870629i	$-0.336288\pi$
$701$	1.28008e7	0.983879	0.491940	+	0.870629i	$0.336288\pi$
$702$	0	0
$703$	−6.44712e6	−0.492014
$704$	0	0
$705$	1.19081e7	0.902339
$706$	0	0
$707$	−1.97791e6	−0.148819
$708$	0	0
$709$	3.05442e6	0.228199	0.114099	−	0.993469i	$-0.463602\pi$
$709$	3.05442e6	0.228199	0.114099	+	0.993469i	$0.463602\pi$
$710$	0	0
$711$	−712356.	−0.0528474
$712$	0	0
$713$	−2.96878e7	−2.18703
$714$	0	0
$715$	1.10535e7	0.808600
$716$	0	0
$717$	6.14650e6	0.446509
$718$	0	0
$719$	1.99890e7	1.44201	0.721006	−	0.692928i	$-0.243680\pi$
$719$	1.99890e7	1.44201	0.721006	+	0.692928i	$0.243680\pi$
$720$	0	0
$721$	−3.00186e6	−0.215056
$722$	0	0
$723$	8.53366e6	0.607141
$724$	0	0
$725$	6.17773e6	0.436500
$726$	0	0
$727$	−1.34254e7	−0.942088	−0.471044	−	0.882110i	$-0.656123\pi$
$727$	−1.34254e7	−0.942088	−0.471044	+	0.882110i	$0.656123\pi$
$728$	0	0
$729$	1.49383e7	1.04108
$730$	0	0
$731$	149421.	0.0103424
$732$	0	0
$733$	1.89485e6	0.130261	0.0651304	−	0.997877i	$-0.479254\pi$
$733$	1.89485e6	0.130261	0.0651304	+	0.997877i	$0.479254\pi$
$734$	0	0
$735$	1.06927e7	0.730078
$736$	0	0
$737$	−1.72787e7	−1.17177
$738$	0	0
$739$	1.52333e7	1.02608	0.513041	−	0.858364i	$-0.328519\pi$
$739$	1.52333e7	1.02608	0.513041	+	0.858364i	$0.328519\pi$
$740$	0	0
$741$	−3.01145e7	−2.01479
$742$	0	0
$743$	−1.79078e7	−1.19006	−0.595031	−	0.803703i	$-0.702860\pi$
$743$	−1.79078e7	−1.19006	−0.595031	+	0.803703i	$0.702860\pi$
$744$	0	0
$745$	−1.13049e7	−0.746235
$746$	0	0
$747$	777251.	0.0509635
$748$	0	0
$749$	−1.98829e6	−0.129502
$750$	0	0
$751$	−2.08382e7	−1.34822	−0.674109	−	0.738632i	$-0.735472\pi$
$751$	−2.08382e7	−1.34822	−0.674109	+	0.738632i	$0.735472\pi$
$752$	0	0
$753$	−1.37356e7	−0.882793
$754$	0	0
$755$	−4.90431e6	−0.313120
$756$	0	0
$757$	3.00548e6	0.190623	0.0953113	−	0.995448i	$-0.469615\pi$
$757$	3.00548e6	0.190623	0.0953113	+	0.995448i	$0.469615\pi$
$758$	0	0
$759$	−1.34756e7	−0.849072
$760$	0	0
$761$	−1.28926e7	−0.807013	−0.403507	−	0.914977i	$-0.632209\pi$
$761$	−1.28926e7	−0.807013	−0.403507	+	0.914977i	$0.632209\pi$
$762$	0	0
$763$	2.75581e6	0.171371
$764$	0	0
$765$	−103309.	−0.00638241
$766$	0	0
$767$	3.34902e7	2.05556
$768$	0	0
$769$	1.07309e7	0.654364	0.327182	−	0.944961i	$-0.393901\pi$
$769$	1.07309e7	0.654364	0.327182	+	0.944961i	$0.393901\pi$
$770$	0	0
$771$	−1.32599e7	−0.803352
$772$	0	0
$773$	−1.83952e7	−1.10728	−0.553639	−	0.832757i	$-0.686761\pi$
$773$	−1.83952e7	−1.10728	−0.553639	+	0.832757i	$0.686761\pi$
$774$	0	0
$775$	1.10262e7	0.659437
$776$	0	0
$777$	1.11360e6	0.0661724
$778$	0	0
$779$	1.67807e7	0.990758
$780$	0	0
$781$	5.59540e6	0.328249
$782$	0	0
$783$	1.88341e7	1.09784
$784$	0	0
$785$	−4.94995e6	−0.286699
$786$	0	0
$787$	4.26448e6	0.245431	0.122716	−	0.992442i	$-0.460840\pi$
$787$	4.26448e6	0.245431	0.122716	+	0.992442i	$0.460840\pi$
$788$	0	0
$789$	−3.53301e6	−0.202047
$790$	0	0
$791$	−670088.	−0.0380795
$792$	0	0
$793$	2.71069e7	1.53072
$794$	0	0
$795$	1.74424e7	0.978790
$796$	0	0
$797$	−2.96002e7	−1.65063	−0.825313	−	0.564676i	$-0.809001\pi$
$797$	−2.96002e7	−1.65063	−0.825313	+	0.564676i	$0.809001\pi$
$798$	0	0
$799$	3.95609e6	0.219229
$800$	0	0
$801$	1.03257e6	0.0568641
$802$	0	0
$803$	1.11358e7	0.609443
$804$	0	0
$805$	−3.33070e6	−0.181153
$806$	0	0
$807$	−2.18415e7	−1.18059
$808$	0	0
$809$	−3.37728e7	−1.81424	−0.907121	−	0.420870i	$-0.861725\pi$
$809$	−3.37728e7	−1.81424	−0.907121	+	0.420870i	$0.861725\pi$
$810$	0	0
$811$	−2.05658e7	−1.09798	−0.548988	−	0.835830i	$-0.684987\pi$
$811$	−2.05658e7	−1.09798	−0.548988	+	0.835830i	$0.684987\pi$
$812$	0	0
$813$	2.43137e7	1.29010
$814$	0	0
$815$	−1.22689e7	−0.647010
$816$	0	0
$817$	1.36764e6	0.0716831
$818$	0	0
$819$	−246629.	−0.0128480
$820$	0	0
$821$	1.78535e7	0.924414	0.462207	−	0.886772i	$-0.347058\pi$
$821$	1.78535e7	0.924414	0.462207	+	0.886772i	$0.347058\pi$
$822$	0	0
$823$	8.13336e6	0.418572	0.209286	−	0.977854i	$-0.432886\pi$
$823$	8.13336e6	0.418572	0.209286	+	0.977854i	$0.432886\pi$
$824$	0	0
$825$	5.00494e6	0.256014
$826$	0	0
$827$	−2.12039e7	−1.07808	−0.539042	−	0.842279i	$-0.681213\pi$
$827$	−2.12039e7	−1.07808	−0.539042	+	0.842279i	$0.681213\pi$
$828$	0	0
$829$	−3.73037e6	−0.188523	−0.0942617	−	0.995547i	$-0.530049\pi$
$829$	−3.73037e6	−0.188523	−0.0942617	+	0.995547i	$0.530049\pi$
$830$	0	0
$831$	−2.10619e7	−1.05802
$832$	0	0
$833$	3.55231e6	0.177377
$834$	0	0
$835$	−1.83471e7	−0.910648
$836$	0	0
$837$	3.36158e7	1.65856
$838$	0	0
$839$	−1.64106e7	−0.804861	−0.402430	−	0.915451i	$-0.631834\pi$
$839$	−1.64106e7	−0.804861	−0.402430	+	0.915451i	$0.631834\pi$
$840$	0	0
$841$	3.18812e6	0.155433
$842$	0	0
$843$	−3.64492e7	−1.76652
$844$	0	0
$845$	−2.63025e7	−1.26723
$846$	0	0
$847$	−2.12705e6	−0.101875
$848$	0	0
$849$	9.10345e6	0.433448
$850$	0	0
$851$	1.10398e7	0.522563
$852$	0	0
$853$	8.74236e6	0.411392	0.205696	−	0.978616i	$-0.434054\pi$
$853$	8.74236e6	0.411392	0.205696	+	0.978616i	$0.434054\pi$
$854$	0	0
$855$	−945577.	−0.0442366
$856$	0	0
$857$	3.31459e7	1.54162	0.770810	−	0.637065i	$-0.219852\pi$
$857$	3.31459e7	1.54162	0.770810	+	0.637065i	$0.219852\pi$
$858$	0	0
$859$	−898006.	−0.0415237	−0.0207619	−	0.999784i	$-0.506609\pi$
$859$	−898006.	−0.0415237	−0.0207619	+	0.999784i	$0.506609\pi$
$860$	0	0
$861$	−2.89851e6	−0.133250
$862$	0	0
$863$	1.63551e6	0.0747526	0.0373763	−	0.999301i	$-0.488100\pi$
$863$	1.63551e6	0.0747526	0.0373763	+	0.999301i	$0.488100\pi$
$864$	0	0
$865$	7.11405e6	0.323278
$866$	0	0
$867$	−2.09028e7	−0.944400
$868$	0	0
$869$	−1.67686e7	−0.753266
$870$	0	0
$871$	6.61202e7	2.95317
$872$	0	0
$873$	898766.	0.0399127
$874$	0	0
$875$	4.28335e6	0.189132
$876$	0	0
$877$	−1.42532e7	−0.625770	−0.312885	−	0.949791i	$-0.601295\pi$
$877$	−1.42532e7	−0.625770	−0.312885	+	0.949791i	$0.601295\pi$
$878$	0	0
$879$	1.33227e7	0.581595
$880$	0	0
$881$	−1.24428e7	−0.540105	−0.270053	−	0.962846i	$-0.587041\pi$
$881$	−1.24428e7	−0.540105	−0.270053	+	0.962846i	$0.587041\pi$
$882$	0	0
$883$	−4.34630e7	−1.87593	−0.937967	−	0.346725i	$-0.887294\pi$
$883$	−4.34630e7	−1.87593	−0.937967	+	0.346725i	$0.887294\pi$
$884$	0	0
$885$	−2.21786e7	−0.951867
$886$	0	0
$887$	−4.26973e6	−0.182218	−0.0911089	−	0.995841i	$-0.529041\pi$
$887$	−4.26973e6	−0.182218	−0.0911089	+	0.995841i	$0.529041\pi$
$888$	0	0
$889$	1.37011e6	0.0581436
$890$	0	0
$891$	1.45665e7	0.614695
$892$	0	0
$893$	3.62097e7	1.51948
$894$	0	0
$895$	−3.12198e7	−1.30278
$896$	0	0
$897$	5.15671e7	2.13989
$898$	0	0
$899$	4.22993e7	1.74556
$900$	0	0
$901$	5.79470e6	0.237804
$902$	0	0
$903$	−236230.	−0.00964087
$904$	0	0
$905$	1.96922e6	0.0799230
$906$	0	0
$907$	3.88728e7	1.56902	0.784508	−	0.620119i	$-0.212916\pi$
$907$	3.88728e7	1.56902	0.784508	+	0.620119i	$0.212916\pi$
$908$	0	0
$909$	961532.	0.0385970
$910$	0	0
$911$	−2.73840e7	−1.09320	−0.546602	−	0.837393i	$-0.684079\pi$
$911$	−2.73840e7	−1.09320	−0.546602	+	0.837393i	$0.684079\pi$
$912$	0	0
$913$	1.82962e7	0.726414
$914$	0	0
$915$	−1.79513e7	−0.708831
$916$	0	0
$917$	4.22576e6	0.165952
$918$	0	0
$919$	3.14731e7	1.22928	0.614640	−	0.788808i	$-0.289301\pi$
$919$	3.14731e7	1.22928	0.614640	+	0.788808i	$0.289301\pi$
$920$	0	0
$921$	4.11728e7	1.59942
$922$	0	0
$923$	−2.14119e7	−0.827277
$924$	0	0
$925$	−4.10026e6	−0.157564
$926$	0	0
$927$	1.45931e6	0.0557761
$928$	0	0
$929$	−2.58150e7	−0.981371	−0.490685	−	0.871337i	$-0.663254\pi$
$929$	−2.58150e7	−0.981371	−0.490685	+	0.871337i	$0.663254\pi$
$930$	0	0
$931$	3.25139e7	1.22941
$932$	0	0
$933$	3.80243e7	1.43007
$934$	0	0
$935$	−2.43186e6	−0.0909723
$936$	0	0
$937$	1.77379e7	0.660014	0.330007	−	0.943979i	$-0.392949\pi$
$937$	1.77379e7	0.660014	0.330007	+	0.943979i	$0.392949\pi$
$938$	0	0
$939$	−4.45265e7	−1.64799
$940$	0	0
$941$	3.65458e7	1.34544	0.672719	−	0.739898i	$-0.265126\pi$
$941$	3.65458e7	1.34544	0.672719	+	0.739898i	$0.265126\pi$
$942$	0	0
$943$	−2.87348e7	−1.05227
$944$	0	0
$945$	3.77139e6	0.137380
$946$	0	0
$947$	−4.00234e7	−1.45024	−0.725119	−	0.688623i	$-0.758215\pi$
$947$	−4.00234e7	−1.45024	−0.725119	+	0.688623i	$0.758215\pi$
$948$	0	0
$949$	−4.26134e7	−1.53596
$950$	0	0
$951$	2.46250e7	0.882928
$952$	0	0
$953$	−1.61525e7	−0.576114	−0.288057	−	0.957613i	$-0.593009\pi$
$953$	−1.61525e7	−0.576114	−0.288057	+	0.957613i	$0.593009\pi$
$954$	0	0
$955$	1.87204e7	0.664213
$956$	0	0
$957$	1.92001e7	0.677680
$958$	0	0
$959$	4.11941e6	0.144640
$960$	0	0
$961$	4.68683e7	1.63708
$962$	0	0
$963$	966579.	0.0335870
$964$	0	0
$965$	−1.19714e7	−0.413833
$966$	0	0
$967$	−1.46946e7	−0.505349	−0.252674	−	0.967551i	$-0.581310\pi$
$967$	−1.46946e7	−0.505349	−0.252674	+	0.967551i	$0.581310\pi$
$968$	0	0
$969$	6.62546e6	0.226676
$970$	0	0
$971$	2.33873e7	0.796034	0.398017	−	0.917378i	$-0.369698\pi$
$971$	2.33873e7	0.796034	0.398017	+	0.917378i	$0.369698\pi$
$972$	0	0
$973$	1.02120e6	0.0345803
$974$	0	0
$975$	−1.91524e7	−0.645225
$976$	0	0
$977$	3.68638e7	1.23556	0.617779	−	0.786351i	$-0.288032\pi$
$977$	3.68638e7	1.23556	0.617779	+	0.786351i	$0.288032\pi$
$978$	0	0
$979$	2.43063e7	0.810519
$980$	0	0
$981$	−1.33970e6	−0.0444462
$982$	0	0
$983$	1.35629e7	0.447682	0.223841	−	0.974626i	$-0.428140\pi$
$983$	1.35629e7	0.447682	0.223841	+	0.974626i	$0.428140\pi$
$984$	0	0
$985$	4.33135e7	1.42243
$986$	0	0
$987$	−6.25444e6	−0.204360
$988$	0	0
$989$	−2.34190e6	−0.0761338
$990$	0	0
$991$	−3.33321e7	−1.07815	−0.539074	−	0.842258i	$-0.681226\pi$
$991$	−3.33321e7	−1.07815	−0.539074	+	0.842258i	$0.681226\pi$
$992$	0	0
$993$	−1.74819e7	−0.562620
$994$	0	0
$995$	2.05716e7	0.658735
$996$	0	0
$997$	−3.52048e6	−0.112167	−0.0560833	−	0.998426i	$-0.517861\pi$
$997$	−3.52048e6	−0.112167	−0.0560833	+	0.998426i	$0.517861\pi$
$998$	0	0
$999$	−1.25005e7	−0.396291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.6.a.m.1.3		4
4.3	odd	2	inner	256.6.a.m.1.1		4
8.3	odd	2	inner	256.6.a.m.1.4		4
8.5	even	2	inner	256.6.a.m.1.2		4
16.3	odd	4		128.6.b.d.65.2	yes	4
16.5	even	4		128.6.b.d.65.1	✓	4
16.11	odd	4		128.6.b.d.65.3	yes	4
16.13	even	4		128.6.b.d.65.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.6.b.d.65.1	✓	4	16.5	even	4
128.6.b.d.65.2	yes	4	16.3	odd	4
128.6.b.d.65.3	yes	4	16.11	odd	4
128.6.b.d.65.4	yes	4	16.13	even	4
256.6.a.m.1.1		4	4.3	odd	2	inner
256.6.a.m.1.2		4	8.5	even	2	inner
256.6.a.m.1.3		4	1.1	even	1	trivial
256.6.a.m.1.4		4	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

\(f(q)\)	\(=\)	\(q+15.2315 q^{3} -43.0813 q^{5} +22.6274 q^{7} -11.0000 q^{9} +O(q^{10})\)	\(q+15.2315 q^{3} -43.0813 q^{5} +22.6274 q^{7} -11.0000 q^{9} -258.936 q^{11} +990.870 q^{13} -656.195 q^{15} -218.000 q^{17} -1995.33 q^{19} +344.651 q^{21} +3416.74 q^{23} -1269.00 q^{25} -3868.81 q^{27} -4868.19 q^{29} -8688.93 q^{31} -3944.00 q^{33} -974.819 q^{35} +3231.10 q^{37} +15092.5 q^{39} -8410.00 q^{41} -685.420 q^{43} +473.895 q^{45} -18147.2 q^{47} -16295.0 q^{49} -3320.48 q^{51} -26581.2 q^{53} +11155.3 q^{55} -30392.0 q^{57} +33798.8 q^{59} +27356.6 q^{61} -248.902 q^{63} -42688.0 q^{65} +66729.4 q^{67} +52042.2 q^{69} -21609.2 q^{71} -43006.0 q^{73} -19328.8 q^{75} -5859.06 q^{77} +64759.7 q^{79} -56255.0 q^{81} -70659.1 q^{83} +9391.73 q^{85} -74150.0 q^{87} -93870.0 q^{89} +22420.8 q^{91} -132346. q^{93} +85961.6 q^{95} -81706.0 q^{97} +2848.30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 44 q^{9}+O(q^{10}) \) 4 * q - 44 * q^9	\( 4 q - 44 q^{9} - 872 q^{17} - 5076 q^{25} - 15776 q^{33} - 33640 q^{41} - 65180 q^{49} - 121568 q^{57} - 170752 q^{65} - 172024 q^{73} - 225020 q^{81} - 375480 q^{89} - 326824 q^{97}+O(q^{100}) \) 4 * q - 44 * q^9 - 872 * q^17 - 5076 * q^25 - 15776 * q^33 - 33640 * q^41 - 65180 * q^49 - 121568 * q^57 - 170752 * q^65 - 172024 * q^73 - 225020 * q^81 - 375480 * q^89 - 326824 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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