LMFDB - Embedded newform 456.6.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [456,6,Mod(1,456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(456, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("456.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 456 = 2^{3} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	456.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:	$73.1350218347$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 4725x^{4} + 92430x^{3} + 1610577x^{2} - 16081740x - 24661341 $ x^6 - 2x^5 - 4725x^4 + 92430x^3 + 1610577x^2 - 16081740*x - 24661341
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$49.1493$ of defining polynomial
Character	$\chi$	$=$	456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +28.9245 q^{5} +210.915 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +28.9245 q^{5} +210.915 q^{7} +81.0000 q^{9} +169.342 q^{11} -1019.58 q^{13} -260.320 q^{15} -1848.05 q^{17} +361.000 q^{19} -1898.23 q^{21} -1415.83 q^{23} -2288.37 q^{25} -729.000 q^{27} +2799.23 q^{29} +61.3714 q^{31} -1524.08 q^{33} +6100.61 q^{35} +2247.44 q^{37} +9176.19 q^{39} +997.106 q^{41} -3731.90 q^{43} +2342.88 q^{45} -16075.6 q^{47} +27678.1 q^{49} +16632.5 q^{51} +39562.9 q^{53} +4898.14 q^{55} -3249.00 q^{57} -35454.3 q^{59} -56219.8 q^{61} +17084.1 q^{63} -29490.7 q^{65} -6653.47 q^{67} +12742.5 q^{69} -8888.19 q^{71} -12509.4 q^{73} +20595.4 q^{75} +35716.8 q^{77} -63220.1 q^{79} +6561.00 q^{81} -37207.9 q^{83} -53454.0 q^{85} -25193.1 q^{87} +46963.0 q^{89} -215044. q^{91} -552.343 q^{93} +10441.7 q^{95} +46146.9 q^{97} +13716.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9}+O(q^{10}) $ 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9	$ 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9} - 203 q^{11} - 298 q^{13} + 585 q^{15} + 1319 q^{17} + 2166 q^{19} + 1341 q^{21} + 1234 q^{23} + 4589 q^{25} - 4374 q^{27} + 7356 q^{29} + 1632 q^{31} + 1827 q^{33} + 4383 q^{35} + 14204 q^{37} + 2682 q^{39} + 14734 q^{41} - 4693 q^{43} - 5265 q^{45} - 10955 q^{47} + 38561 q^{49} - 11871 q^{51} + 47500 q^{53} + 769 q^{55} - 19494 q^{57} - 61744 q^{59} - 81581 q^{61} - 12069 q^{63} - 59686 q^{65} - 45756 q^{67} - 11106 q^{69} - 10416 q^{71} - 54615 q^{73} - 41301 q^{75} - 29515 q^{77} - 145594 q^{79} + 39366 q^{81} - 160548 q^{83} - 53947 q^{85} - 66204 q^{87} - 97728 q^{89} - 418294 q^{91} - 14688 q^{93} - 23465 q^{95} - 760 q^{97} - 16443 q^{99}+O(q^{100}) $ 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9 - 203 * q^11 - 298 * q^13 + 585 * q^15 + 1319 * q^17 + 2166 * q^19 + 1341 * q^21 + 1234 * q^23 + 4589 * q^25 - 4374 * q^27 + 7356 * q^29 + 1632 * q^31 + 1827 * q^33 + 4383 * q^35 + 14204 * q^37 + 2682 * q^39 + 14734 * q^41 - 4693 * q^43 - 5265 * q^45 - 10955 * q^47 + 38561 * q^49 - 11871 * q^51 + 47500 * q^53 + 769 * q^55 - 19494 * q^57 - 61744 * q^59 - 81581 * q^61 - 12069 * q^63 - 59686 * q^65 - 45756 * q^67 - 11106 * q^69 - 10416 * q^71 - 54615 * q^73 - 41301 * q^75 - 29515 * q^77 - 145594 * q^79 + 39366 * q^81 - 160548 * q^83 - 53947 * q^85 - 66204 * q^87 - 97728 * q^89 - 418294 * q^91 - 14688 * q^93 - 23465 * q^95 - 760 * q^97 - 16443 * 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	28.9245	0.517417	0.258708	−	0.965955i	$-0.416703\pi$
$5$	28.9245	0.517417	0.258708	+	0.965955i	$0.416703\pi$
$6$	0	0
$7$	210.915	1.62691	0.813453	−	0.581631i	$-0.197585\pi$
$7$	210.915	1.62691	0.813453	+	0.581631i	$0.197585\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	169.342	0.421972	0.210986	−	0.977489i	$-0.432333\pi$
$11$	169.342	0.421972	0.210986	+	0.977489i	$0.432333\pi$
$12$	0	0
$13$	−1019.58	−1.67325	−0.836626	−	0.547774i	$-0.815475\pi$
$13$	−1019.58	−1.67325	−0.836626	+	0.547774i	$0.815475\pi$
$14$	0	0
$15$	−260.320	−0.298731
$16$	0	0
$17$	−1848.05	−1.55093	−0.775465	−	0.631390i	$-0.782485\pi$
$17$	−1848.05	−1.55093	−0.775465	+	0.631390i	$0.782485\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	−1898.23	−0.939294
$22$	0	0
$23$	−1415.83	−0.558074	−0.279037	−	0.960280i	$-0.590015\pi$
$23$	−1415.83	−0.558074	−0.279037	+	0.960280i	$0.590015\pi$
$24$	0	0
$25$	−2288.37	−0.732280
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	2799.23	0.618079	0.309039	−	0.951049i	$-0.399993\pi$
$29$	2799.23	0.618079	0.309039	+	0.951049i	$0.399993\pi$
$30$	0	0
$31$	61.3714	0.0114700	0.00573498	−	0.999984i	$-0.498174\pi$
$31$	61.3714	0.0114700	0.00573498	+	0.999984i	$0.498174\pi$
$32$	0	0
$33$	−1524.08	−0.243626
$34$	0	0
$35$	6100.61	0.841788
$36$	0	0
$37$	2247.44	0.269888	0.134944	−	0.990853i	$-0.456914\pi$
$37$	2247.44	0.269888	0.134944	+	0.990853i	$0.456914\pi$
$38$	0	0
$39$	9176.19	0.966053
$40$	0	0
$41$	997.106	0.0926364	0.0463182	−	0.998927i	$-0.485251\pi$
$41$	997.106	0.0926364	0.0463182	+	0.998927i	$0.485251\pi$
$42$	0	0
$43$	−3731.90	−0.307793	−0.153897	−	0.988087i	$-0.549182\pi$
$43$	−3731.90	−0.307793	−0.153897	+	0.988087i	$0.549182\pi$
$44$	0	0
$45$	2342.88	0.172472
$46$	0	0
$47$	−16075.6	−1.06151	−0.530754	−	0.847526i	$-0.678091\pi$
$47$	−16075.6	−1.06151	−0.530754	+	0.847526i	$0.678091\pi$
$48$	0	0
$49$	27678.1	1.64682
$50$	0	0
$51$	16632.5	0.895430
$52$	0	0
$53$	39562.9	1.93463	0.967317	−	0.253570i	$-0.0816047\pi$
$53$	39562.9	1.93463	0.967317	+	0.253570i	$0.0816047\pi$
$54$	0	0
$55$	4898.14	0.218335
$56$	0	0
$57$	−3249.00	−0.132453
$58$	0	0
$59$	−35454.3	−1.32599	−0.662993	−	0.748626i	$-0.730714\pi$
$59$	−35454.3	−1.32599	−0.662993	+	0.748626i	$0.730714\pi$
$60$	0	0
$61$	−56219.8	−1.93448	−0.967241	−	0.253859i	$-0.918300\pi$
$61$	−56219.8	−1.93448	−0.967241	+	0.253859i	$0.918300\pi$
$62$	0	0
$63$	17084.1	0.542302
$64$	0	0
$65$	−29490.7	−0.865769
$66$	0	0
$67$	−6653.47	−0.181076	−0.0905381	−	0.995893i	$-0.528859\pi$
$67$	−6653.47	−0.181076	−0.0905381	+	0.995893i	$0.528859\pi$
$68$	0	0
$69$	12742.5	0.322204
$70$	0	0
$71$	−8888.19	−0.209251	−0.104626	−	0.994512i	$-0.533364\pi$
$71$	−8888.19	−0.209251	−0.104626	+	0.994512i	$0.533364\pi$
$72$	0	0
$73$	−12509.4	−0.274746	−0.137373	−	0.990519i	$-0.543866\pi$
$73$	−12509.4	−0.274746	−0.137373	+	0.990519i	$0.543866\pi$
$74$	0	0
$75$	20595.4	0.422782
$76$	0	0
$77$	35716.8	0.686508
$78$	0	0
$79$	−63220.1	−1.13969	−0.569846	−	0.821752i	$-0.692997\pi$
$79$	−63220.1	−1.13969	−0.569846	+	0.821752i	$0.692997\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−37207.9	−0.592843	−0.296422	−	0.955057i	$-0.595793\pi$
$83$	−37207.9	−0.592843	−0.296422	+	0.955057i	$0.595793\pi$
$84$	0	0
$85$	−53454.0	−0.802478
$86$	0	0
$87$	−25193.1	−0.356848
$88$	0	0
$89$	46963.0	0.628464	0.314232	−	0.949346i	$-0.398253\pi$
$89$	46963.0	0.628464	0.314232	+	0.949346i	$0.398253\pi$
$90$	0	0
$91$	−215044.	−2.72222
$92$	0	0
$93$	−552.343	−0.00662219
$94$	0	0
$95$	10441.7	0.118704
$96$	0	0
$97$	46146.9	0.497981	0.248991	−	0.968506i	$-0.419901\pi$
$97$	46146.9	0.497981	0.248991	+	0.968506i	$0.419901\pi$
$98$	0	0
$99$	13716.7	0.140657
$100$	0	0
$101$	2530.94	0.0246875	0.0123438	−	0.999924i	$-0.496071\pi$
$101$	2530.94	0.0246875	0.0123438	+	0.999924i	$0.496071\pi$
$102$	0	0
$103$	93712.2	0.870369	0.435184	−	0.900341i	$-0.356683\pi$
$103$	93712.2	0.870369	0.435184	+	0.900341i	$0.356683\pi$
$104$	0	0
$105$	−54905.5	−0.486007
$106$	0	0
$107$	−153860.	−1.29917	−0.649585	−	0.760289i	$-0.725057\pi$
$107$	−153860.	−1.29917	−0.649585	+	0.760289i	$0.725057\pi$
$108$	0	0
$109$	−93199.1	−0.751355	−0.375678	−	0.926750i	$-0.622590\pi$
$109$	−93199.1	−0.751355	−0.375678	+	0.926750i	$0.622590\pi$
$110$	0	0
$111$	−20227.0	−0.155820
$112$	0	0
$113$	88014.1	0.648419	0.324210	−	0.945985i	$-0.394902\pi$
$113$	88014.1	0.648419	0.324210	+	0.945985i	$0.394902\pi$
$114$	0	0
$115$	−40952.2	−0.288757
$116$	0	0
$117$	−82585.7	−0.557751
$118$	0	0
$119$	−389782.	−2.52322
$120$	0	0
$121$	−132374.	−0.821940
$122$	0	0
$123$	−8973.95	−0.0534836
$124$	0	0
$125$	−156579.	−0.896311
$126$	0	0
$127$	137253.	0.755114	0.377557	−	0.925986i	$-0.376764\pi$
$127$	137253.	0.755114	0.377557	+	0.925986i	$0.376764\pi$
$128$	0	0
$129$	33587.1	0.177705
$130$	0	0
$131$	−97239.0	−0.495065	−0.247532	−	0.968880i	$-0.579620\pi$
$131$	−97239.0	−0.495065	−0.247532	+	0.968880i	$0.579620\pi$
$132$	0	0
$133$	76140.3	0.373238
$134$	0	0
$135$	−21086.0	−0.0995769
$136$	0	0
$137$	244529.	1.11309	0.556543	−	0.830819i	$-0.312128\pi$
$137$	244529.	1.11309	0.556543	+	0.830819i	$0.312128\pi$
$138$	0	0
$139$	−4472.71	−0.0196351	−0.00981756	−	0.999952i	$-0.503125\pi$
$139$	−4472.71	−0.0196351	−0.00981756	+	0.999952i	$0.503125\pi$
$140$	0	0
$141$	144681.	0.612862
$142$	0	0
$143$	−172657.	−0.706066
$144$	0	0
$145$	80966.3	0.319804
$146$	0	0
$147$	−249103.	−0.950792
$148$	0	0
$149$	−261625.	−0.965415	−0.482707	−	0.875782i	$-0.660347\pi$
$149$	−261625.	−0.965415	−0.482707	+	0.875782i	$0.660347\pi$
$150$	0	0
$151$	245463.	0.876080	0.438040	−	0.898956i	$-0.355673\pi$
$151$	245463.	0.876080	0.438040	+	0.898956i	$0.355673\pi$
$152$	0	0
$153$	−149692.	−0.516977
$154$	0	0
$155$	1775.14	0.00593475
$156$	0	0
$157$	−448479.	−1.45209	−0.726045	−	0.687647i	$-0.758644\pi$
$157$	−448479.	−1.45209	−0.726045	+	0.687647i	$0.758644\pi$
$158$	0	0
$159$	−356066.	−1.11696
$160$	0	0
$161$	−298620.	−0.907933
$162$	0	0
$163$	439435.	1.29547	0.647733	−	0.761867i	$-0.275717\pi$
$163$	439435.	1.29547	0.647733	+	0.761867i	$0.275717\pi$
$164$	0	0
$165$	−44083.2	−0.126056
$166$	0	0
$167$	−171577.	−0.476067	−0.238033	−	0.971257i	$-0.576503\pi$
$167$	−171577.	−0.476067	−0.238033	+	0.971257i	$0.576503\pi$
$168$	0	0
$169$	668244.	1.79977
$170$	0	0
$171$	29241.0	0.0764719
$172$	0	0
$173$	20210.0	0.0513395	0.0256698	−	0.999670i	$-0.491828\pi$
$173$	20210.0	0.0513395	0.0256698	+	0.999670i	$0.491828\pi$
$174$	0	0
$175$	−482652.	−1.19135
$176$	0	0
$177$	319089.	0.765558
$178$	0	0
$179$	−207989.	−0.485185	−0.242593	−	0.970128i	$-0.577998\pi$
$179$	−207989.	−0.485185	−0.242593	+	0.970128i	$0.577998\pi$
$180$	0	0
$181$	−799903.	−1.81485	−0.907425	−	0.420214i	$-0.861955\pi$
$181$	−799903.	−1.81485	−0.907425	+	0.420214i	$0.861955\pi$
$182$	0	0
$183$	505978.	1.11687
$184$	0	0
$185$	65006.1	0.139645
$186$	0	0
$187$	−312954.	−0.654449
$188$	0	0
$189$	−153757.	−0.313098
$190$	0	0
$191$	580340.	1.15106	0.575531	−	0.817780i	$-0.304795\pi$
$191$	580340.	1.15106	0.575531	+	0.817780i	$0.304795\pi$
$192$	0	0
$193$	−99920.5	−0.193091	−0.0965453	−	0.995329i	$-0.530779\pi$
$193$	−99920.5	−0.193091	−0.0965453	+	0.995329i	$0.530779\pi$
$194$	0	0
$195$	265417.	0.499852
$196$	0	0
$197$	−333306.	−0.611897	−0.305948	−	0.952048i	$-0.598973\pi$
$197$	−333306.	−0.611897	−0.305948	+	0.952048i	$0.598973\pi$
$198$	0	0
$199$	−840198.	−1.50400	−0.752002	−	0.659161i	$-0.770912\pi$
$199$	−840198.	−1.50400	−0.752002	+	0.659161i	$0.770912\pi$
$200$	0	0
$201$	59881.2	0.104544
$202$	0	0
$203$	590399.	1.00556
$204$	0	0
$205$	28840.8	0.0479316
$206$	0	0
$207$	−114682.	−0.186025
$208$	0	0
$209$	61132.5	0.0968070
$210$	0	0
$211$	−819075.	−1.26654	−0.633268	−	0.773933i	$-0.718287\pi$
$211$	−819075.	−1.26654	−0.633268	+	0.773933i	$0.718287\pi$
$212$	0	0
$213$	79993.8	0.120811
$214$	0	0
$215$	−107943.	−0.159257
$216$	0	0
$217$	12944.1	0.0186605
$218$	0	0
$219$	112585.	0.158624
$220$	0	0
$221$	1.88423e6	2.59510
$222$	0	0
$223$	−471476.	−0.634888	−0.317444	−	0.948277i	$-0.602825\pi$
$223$	−471476.	−0.634888	−0.317444	+	0.948277i	$0.602825\pi$
$224$	0	0
$225$	−185358.	−0.244093
$226$	0	0
$227$	669991.	0.862987	0.431494	−	0.902116i	$-0.357987\pi$
$227$	669991.	0.862987	0.431494	+	0.902116i	$0.357987\pi$
$228$	0	0
$229$	−1.04824e6	−1.32091	−0.660453	−	0.750867i	$-0.729636\pi$
$229$	−1.04824e6	−1.32091	−0.660453	+	0.750867i	$0.729636\pi$
$230$	0	0
$231$	−321451.	−0.396356
$232$	0	0
$233$	−456245.	−0.550564	−0.275282	−	0.961363i	$-0.588771\pi$
$233$	−456245.	−0.550564	−0.275282	+	0.961363i	$0.588771\pi$
$234$	0	0
$235$	−464979.	−0.549242
$236$	0	0
$237$	568981.	0.658001
$238$	0	0
$239$	−87877.9	−0.0995142	−0.0497571	−	0.998761i	$-0.515845\pi$
$239$	−87877.9	−0.0995142	−0.0497571	+	0.998761i	$0.515845\pi$
$240$	0	0
$241$	996858.	1.10558	0.552791	−	0.833320i	$-0.313563\pi$
$241$	996858.	1.10558	0.552791	+	0.833320i	$0.313563\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	800575.	0.852093
$246$	0	0
$247$	−368067.	−0.383870
$248$	0	0
$249$	334871.	0.342278
$250$	0	0
$251$	651472.	0.652696	0.326348	−	0.945250i	$-0.394182\pi$
$251$	651472.	0.652696	0.326348	+	0.945250i	$0.394182\pi$
$252$	0	0
$253$	−239760.	−0.235491
$254$	0	0
$255$	481086.	0.463311
$256$	0	0
$257$	1.96985e6	1.86037	0.930185	−	0.367090i	$-0.119646\pi$
$257$	1.96985e6	1.86037	0.930185	+	0.367090i	$0.119646\pi$
$258$	0	0
$259$	474019.	0.439083
$260$	0	0
$261$	226738.	0.206026
$262$	0	0
$263$	−1.13782e6	−1.01434	−0.507171	−	0.861846i	$-0.669309\pi$
$263$	−1.13782e6	−1.01434	−0.507171	+	0.861846i	$0.669309\pi$
$264$	0	0
$265$	1.14434e6	1.00101
$266$	0	0
$267$	−422667.	−0.362844
$268$	0	0
$269$	−641291.	−0.540349	−0.270174	−	0.962811i	$-0.587081\pi$
$269$	−641291.	−0.540349	−0.270174	+	0.962811i	$0.587081\pi$
$270$	0	0
$271$	−938680.	−0.776416	−0.388208	−	0.921572i	$-0.626906\pi$
$271$	−938680.	−0.776416	−0.388208	+	0.921572i	$0.626906\pi$
$272$	0	0
$273$	1.93540e6	1.57168
$274$	0	0
$275$	−387518.	−0.309001
$276$	0	0
$277$	−1.02519e6	−0.802795	−0.401398	−	0.915904i	$-0.631475\pi$
$277$	−1.02519e6	−0.802795	−0.401398	+	0.915904i	$0.631475\pi$
$278$	0	0
$279$	4971.09	0.00382332
$280$	0	0
$281$	−841492.	−0.635747	−0.317873	−	0.948133i	$-0.602969\pi$
$281$	−841492.	−0.635747	−0.317873	+	0.948133i	$0.602969\pi$
$282$	0	0
$283$	−1.57993e6	−1.17266	−0.586330	−	0.810073i	$-0.699428\pi$
$283$	−1.57993e6	−1.17266	−0.586330	+	0.810073i	$0.699428\pi$
$284$	0	0
$285$	−93975.7	−0.0685335
$286$	0	0
$287$	210305.	0.150711
$288$	0	0
$289$	1.99545e6	1.40539
$290$	0	0
$291$	−415322.	−0.287510
$292$	0	0
$293$	2.28841e6	1.55727	0.778635	−	0.627477i	$-0.215912\pi$
$293$	2.28841e6	1.55727	0.778635	+	0.627477i	$0.215912\pi$
$294$	0	0
$295$	−1.02550e6	−0.686088
$296$	0	0
$297$	−123450.	−0.0812085
$298$	0	0
$299$	1.44355e6	0.933799
$300$	0	0
$301$	−787114.	−0.500751
$302$	0	0
$303$	−22778.4	−0.0142534
$304$	0	0
$305$	−1.62613e6	−1.00093
$306$	0	0
$307$	−1.93279e6	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$307$	−1.93279e6	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$308$	0	0
$309$	−843410.	−0.502507
$310$	0	0
$311$	−2.50607e6	−1.46924	−0.734619	−	0.678479i	$-0.762639\pi$
$311$	−2.50607e6	−1.46924	−0.734619	+	0.678479i	$0.762639\pi$
$312$	0	0
$313$	−744655.	−0.429629	−0.214815	−	0.976655i	$-0.568915\pi$
$313$	−744655.	−0.429629	−0.214815	+	0.976655i	$0.568915\pi$
$314$	0	0
$315$	494149.	0.280596
$316$	0	0
$317$	1.95216e6	1.09111	0.545554	−	0.838076i	$-0.316319\pi$
$317$	1.95216e6	1.09111	0.545554	+	0.838076i	$0.316319\pi$
$318$	0	0
$319$	474028.	0.260812
$320$	0	0
$321$	1.38474e6	0.750076
$322$	0	0
$323$	−667147.	−0.355808
$324$	0	0
$325$	2.33317e6	1.22529
$326$	0	0
$327$	838792.	0.433795
$328$	0	0
$329$	−3.39059e6	−1.72697
$330$	0	0
$331$	2.09353e6	1.05029	0.525145	−	0.851013i	$-0.324011\pi$
$331$	2.09353e6	1.05029	0.525145	+	0.851013i	$0.324011\pi$
$332$	0	0
$333$	182043.	0.0899628
$334$	0	0
$335$	−192448.	−0.0936919
$336$	0	0
$337$	−939240.	−0.450507	−0.225254	−	0.974300i	$-0.572321\pi$
$337$	−939240.	−0.450507	−0.225254	+	0.974300i	$0.572321\pi$
$338$	0	0
$339$	−792127.	−0.374365
$340$	0	0
$341$	10392.8	0.00484000
$342$	0	0
$343$	2.29288e6	1.05231
$344$	0	0
$345$	368570.	0.166714
$346$	0	0
$347$	1.82627e6	0.814219	0.407109	−	0.913379i	$-0.366537\pi$
$347$	1.82627e6	0.814219	0.407109	+	0.913379i	$0.366537\pi$
$348$	0	0
$349$	−2.03096e6	−0.892561	−0.446280	−	0.894893i	$-0.647252\pi$
$349$	−2.03096e6	−0.892561	−0.446280	+	0.894893i	$0.647252\pi$
$350$	0	0
$351$	743271.	0.322018
$352$	0	0
$353$	3.63876e6	1.55424	0.777118	−	0.629355i	$-0.216681\pi$
$353$	3.63876e6	1.55424	0.777118	+	0.629355i	$0.216681\pi$
$354$	0	0
$355$	−257086.	−0.108270
$356$	0	0
$357$	3.50804e6	1.45678
$358$	0	0
$359$	737834.	0.302150	0.151075	−	0.988522i	$-0.451726\pi$
$359$	737834.	0.302150	0.151075	+	0.988522i	$0.451726\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	1.19137e6	0.474547
$364$	0	0
$365$	−361829.	−0.142158
$366$	0	0
$367$	834883.	0.323564	0.161782	−	0.986826i	$-0.448276\pi$
$367$	834883.	0.323564	0.161782	+	0.986826i	$0.448276\pi$
$368$	0	0
$369$	80765.6	0.0308788
$370$	0	0
$371$	8.34441e6	3.14747
$372$	0	0
$373$	4.71333e6	1.75410	0.877052	−	0.480395i	$-0.159507\pi$
$373$	4.71333e6	1.75410	0.877052	+	0.480395i	$0.159507\pi$
$374$	0	0
$375$	1.40921e6	0.517485
$376$	0	0
$377$	−2.85403e6	−1.03420
$378$	0	0
$379$	−5.30886e6	−1.89847	−0.949234	−	0.314572i	$-0.898139\pi$
$379$	−5.30886e6	−1.89847	−0.949234	+	0.314572i	$0.898139\pi$
$380$	0	0
$381$	−1.23528e6	−0.435965
$382$	0	0
$383$	5.06430e6	1.76410	0.882048	−	0.471159i	$-0.156164\pi$
$383$	5.06430e6	1.76410	0.882048	+	0.471159i	$0.156164\pi$
$384$	0	0
$385$	1.03309e6	0.355211
$386$	0	0
$387$	−302284.	−0.102598
$388$	0	0
$389$	1.65007e6	0.552877	0.276438	−	0.961032i	$-0.410846\pi$
$389$	1.65007e6	0.552877	0.276438	+	0.961032i	$0.410846\pi$
$390$	0	0
$391$	2.61653e6	0.865534
$392$	0	0
$393$	875151.	0.285826
$394$	0	0
$395$	−1.82861e6	−0.589696
$396$	0	0
$397$	3.64454e6	1.16056	0.580279	−	0.814418i	$-0.302944\pi$
$397$	3.64454e6	1.16056	0.580279	+	0.814418i	$0.302944\pi$
$398$	0	0
$399$	−685263.	−0.215489
$400$	0	0
$401$	3.54899e6	1.10216	0.551079	−	0.834453i	$-0.314216\pi$
$401$	3.54899e6	1.10216	0.551079	+	0.834453i	$0.314216\pi$
$402$	0	0
$403$	−62572.9	−0.0191921
$404$	0	0
$405$	189774.	0.0574908
$406$	0	0
$407$	380587.	0.113885
$408$	0	0
$409$	−4.24114e6	−1.25364	−0.626822	−	0.779163i	$-0.715645\pi$
$409$	−4.24114e6	−1.25364	−0.626822	+	0.779163i	$0.715645\pi$
$410$	0	0
$411$	−2.20076e6	−0.642640
$412$	0	0
$413$	−7.47784e6	−2.15725
$414$	0	0
$415$	−1.07622e6	−0.306747
$416$	0	0
$417$	40254.4	0.0113363
$418$	0	0
$419$	513024.	0.142759	0.0713793	−	0.997449i	$-0.477260\pi$
$419$	513024.	0.142759	0.0713793	+	0.997449i	$0.477260\pi$
$420$	0	0
$421$	1.30360e6	0.358460	0.179230	−	0.983807i	$-0.442639\pi$
$421$	1.30360e6	0.358460	0.179230	+	0.983807i	$0.442639\pi$
$422$	0	0
$423$	−1.30213e6	−0.353836
$424$	0	0
$425$	4.22904e6	1.13571
$426$	0	0
$427$	−1.18576e7	−3.14722
$428$	0	0
$429$	1.55392e6	0.407647
$430$	0	0
$431$	6.04300e6	1.56697	0.783483	−	0.621413i	$-0.213441\pi$
$431$	6.04300e6	1.56697	0.783483	+	0.621413i	$0.213441\pi$
$432$	0	0
$433$	4.01535e6	1.02921	0.514605	−	0.857427i	$-0.327939\pi$
$433$	4.01535e6	1.02921	0.514605	+	0.857427i	$0.327939\pi$
$434$	0	0
$435$	−728697.	−0.184639
$436$	0	0
$437$	−511115.	−0.128031
$438$	0	0
$439$	−5.54934e6	−1.37429	−0.687147	−	0.726518i	$-0.741137\pi$
$439$	−5.54934e6	−1.37429	−0.687147	+	0.726518i	$0.741137\pi$
$440$	0	0
$441$	2.24193e6	0.548940
$442$	0	0
$443$	−6.39785e6	−1.54890	−0.774452	−	0.632632i	$-0.781974\pi$
$443$	−6.39785e6	−1.54890	−0.774452	+	0.632632i	$0.781974\pi$
$444$	0	0
$445$	1.35838e6	0.325178
$446$	0	0
$447$	2.35463e6	0.557383
$448$	0	0
$449$	−273835.	−0.0641023	−0.0320511	−	0.999486i	$-0.510204\pi$
$449$	−273835.	−0.0641023	−0.0320511	+	0.999486i	$0.510204\pi$
$450$	0	0
$451$	168852.	0.0390900
$452$	0	0
$453$	−2.20917e6	−0.505805
$454$	0	0
$455$	−6.22004e6	−1.40852
$456$	0	0
$457$	606862.	0.135925	0.0679626	−	0.997688i	$-0.478350\pi$
$457$	606862.	0.135925	0.0679626	+	0.997688i	$0.478350\pi$
$458$	0	0
$459$	1.34723e6	0.298477
$460$	0	0
$461$	6.65261e6	1.45794	0.728970	−	0.684545i	$-0.239999\pi$
$461$	6.65261e6	1.45794	0.728970	+	0.684545i	$0.239999\pi$
$462$	0	0
$463$	6.17901e6	1.33957	0.669786	−	0.742554i	$-0.266386\pi$
$463$	6.17901e6	1.33957	0.669786	+	0.742554i	$0.266386\pi$
$464$	0	0
$465$	−15976.2	−0.00342643
$466$	0	0
$467$	−4.11852e6	−0.873874	−0.436937	−	0.899492i	$-0.643937\pi$
$467$	−4.11852e6	−0.873874	−0.436937	+	0.899492i	$0.643937\pi$
$468$	0	0
$469$	−1.40332e6	−0.294594
$470$	0	0
$471$	4.03632e6	0.838364
$472$	0	0
$473$	−631969.	−0.129880
$474$	0	0
$475$	−826103.	−0.167996
$476$	0	0
$477$	3.20460e6	0.644878
$478$	0	0
$479$	2.35240e6	0.468459	0.234230	−	0.972181i	$-0.424743\pi$
$479$	2.35240e6	0.468459	0.234230	+	0.972181i	$0.424743\pi$
$480$	0	0
$481$	−2.29144e6	−0.451591
$482$	0	0
$483$	2.68758e6	0.524195
$484$	0	0
$485$	1.33477e6	0.257664
$486$	0	0
$487$	−2.77278e6	−0.529776	−0.264888	−	0.964279i	$-0.585335\pi$
$487$	−2.77278e6	−0.529776	−0.264888	+	0.964279i	$0.585335\pi$
$488$	0	0
$489$	−3.95492e6	−0.747938
$490$	0	0
$491$	7.43178e6	1.39120	0.695600	−	0.718430i	$-0.255139\pi$
$491$	7.43178e6	1.39120	0.695600	+	0.718430i	$0.255139\pi$
$492$	0	0
$493$	−5.17313e6	−0.958597
$494$	0	0
$495$	396749.	0.0727785
$496$	0	0
$497$	−1.87465e6	−0.340432
$498$	0	0
$499$	−1.43910e6	−0.258726	−0.129363	−	0.991597i	$-0.541293\pi$
$499$	−1.43910e6	−0.258726	−0.129363	+	0.991597i	$0.541293\pi$
$500$	0	0
$501$	1.54419e6	0.274857
$502$	0	0
$503$	−7.91258e6	−1.39444	−0.697218	−	0.716860i	$-0.745579\pi$
$503$	−7.91258e6	−1.39444	−0.697218	+	0.716860i	$0.745579\pi$
$504$	0	0
$505$	73206.0	0.0127737
$506$	0	0
$507$	−6.01419e6	−1.03910
$508$	0	0
$509$	4.37164e6	0.747910	0.373955	−	0.927447i	$-0.378001\pi$
$509$	4.37164e6	0.747910	0.373955	+	0.927447i	$0.378001\pi$
$510$	0	0
$511$	−2.63843e6	−0.446985
$512$	0	0
$513$	−263169.	−0.0441511
$514$	0	0
$515$	2.71058e6	0.450343
$516$	0	0
$517$	−2.72228e6	−0.447927
$518$	0	0
$519$	−181890.	−0.0296409
$520$	0	0
$521$	−4.31485e6	−0.696420	−0.348210	−	0.937416i	$-0.613210\pi$
$521$	−4.31485e6	−0.696420	−0.348210	+	0.937416i	$0.613210\pi$
$522$	0	0
$523$	−5.04725e6	−0.806864	−0.403432	−	0.915010i	$-0.632183\pi$
$523$	−5.04725e6	−0.806864	−0.403432	+	0.915010i	$0.632183\pi$
$524$	0	0
$525$	4.34387e6	0.687826
$526$	0	0
$527$	−113418.	−0.0177891
$528$	0	0
$529$	−4.43177e6	−0.688554
$530$	0	0
$531$	−2.87180e6	−0.441995
$532$	0	0
$533$	−1.01663e6	−0.155004
$534$	0	0
$535$	−4.45032e6	−0.672213
$536$	0	0
$537$	1.87190e6	0.280122
$538$	0	0
$539$	4.68707e6	0.694912
$540$	0	0
$541$	−9.77646e6	−1.43611	−0.718056	−	0.695985i	$-0.754968\pi$
$541$	−9.77646e6	−1.43611	−0.718056	+	0.695985i	$0.754968\pi$
$542$	0	0
$543$	7.19913e6	1.04780
$544$	0	0
$545$	−2.69573e6	−0.388764
$546$	0	0
$547$	−7.32800e6	−1.04717	−0.523585	−	0.851974i	$-0.675406\pi$
$547$	−7.32800e6	−1.04717	−0.523585	+	0.851974i	$0.675406\pi$
$548$	0	0
$549$	−4.55381e6	−0.644828
$550$	0	0
$551$	1.01052e6	0.141797
$552$	0	0
$553$	−1.33341e7	−1.85417
$554$	0	0
$555$	−585055.	−0.0806240
$556$	0	0
$557$	−7.91105e6	−1.08043	−0.540214	−	0.841527i	$-0.681657\pi$
$557$	−7.91105e6	−1.08043	−0.540214	+	0.841527i	$0.681657\pi$
$558$	0	0
$559$	3.80496e6	0.515016
$560$	0	0
$561$	2.81658e6	0.377846
$562$	0	0
$563$	−207075.	−0.0275333	−0.0137666	−	0.999905i	$-0.504382\pi$
$563$	−207075.	−0.0275333	−0.0137666	+	0.999905i	$0.504382\pi$
$564$	0	0
$565$	2.54576e6	0.335503
$566$	0	0
$567$	1.38381e6	0.180767
$568$	0	0
$569$	1.15559e7	1.49631	0.748156	−	0.663523i	$-0.230939\pi$
$569$	1.15559e7	1.49631	0.748156	+	0.663523i	$0.230939\pi$
$570$	0	0
$571$	1.42478e7	1.82876	0.914379	−	0.404858i	$-0.132679\pi$
$571$	1.42478e7	1.82876	0.914379	+	0.404858i	$0.132679\pi$
$572$	0	0
$573$	−5.22306e6	−0.664566
$574$	0	0
$575$	3.23995e6	0.408666
$576$	0	0
$577$	−3.50790e6	−0.438639	−0.219320	−	0.975653i	$-0.570384\pi$
$577$	−3.50790e6	−0.438639	−0.219320	+	0.975653i	$0.570384\pi$
$578$	0	0
$579$	899285.	0.111481
$580$	0	0
$581$	−7.84770e6	−0.964499
$582$	0	0
$583$	6.69968e6	0.816361
$584$	0	0
$585$	−2.38875e6	−0.288590
$586$	0	0
$587$	−1.36708e6	−0.163756	−0.0818780	−	0.996642i	$-0.526092\pi$
$587$	−1.36708e6	−0.163756	−0.0818780	+	0.996642i	$0.526092\pi$
$588$	0	0
$589$	22155.1	0.00263139
$590$	0	0
$591$	2.99976e6	0.353279
$592$	0	0
$593$	9.26761e6	1.08226	0.541129	−	0.840939i	$-0.317997\pi$
$593$	9.26761e6	1.08226	0.541129	+	0.840939i	$0.317997\pi$
$594$	0	0
$595$	−1.12742e7	−1.30555
$596$	0	0
$597$	7.56178e6	0.868337
$598$	0	0
$599$	−1.19662e6	−0.136267	−0.0681333	−	0.997676i	$-0.521704\pi$
$599$	−1.19662e6	−0.136267	−0.0681333	+	0.997676i	$0.521704\pi$
$600$	0	0
$601$	−5.19257e6	−0.586403	−0.293201	−	0.956051i	$-0.594721\pi$
$601$	−5.19257e6	−0.586403	−0.293201	+	0.956051i	$0.594721\pi$
$602$	0	0
$603$	−538931.	−0.0603587
$604$	0	0
$605$	−3.82886e6	−0.425286
$606$	0	0
$607$	−6.94465e6	−0.765031	−0.382515	−	0.923949i	$-0.624942\pi$
$607$	−6.94465e6	−0.765031	−0.382515	+	0.923949i	$0.624942\pi$
$608$	0	0
$609$	−5.31360e6	−0.580558
$610$	0	0
$611$	1.63903e7	1.77617
$612$	0	0
$613$	−1.78506e7	−1.91868	−0.959339	−	0.282257i	$-0.908917\pi$
$613$	−1.78506e7	−1.91868	−0.959339	+	0.282257i	$0.908917\pi$
$614$	0	0
$615$	−259567.	−0.0276733
$616$	0	0
$617$	−1.92760e6	−0.203847	−0.101923	−	0.994792i	$-0.532500\pi$
$617$	−1.92760e6	−0.203847	−0.101923	+	0.994792i	$0.532500\pi$
$618$	0	0
$619$	−8.17886e6	−0.857958	−0.428979	−	0.903315i	$-0.641127\pi$
$619$	−8.17886e6	−0.857958	−0.428979	+	0.903315i	$0.641127\pi$
$620$	0	0
$621$	1.03214e6	0.107401
$622$	0	0
$623$	9.90519e6	1.02245
$624$	0	0
$625$	2.62220e6	0.268513
$626$	0	0
$627$	−550193.	−0.0558915
$628$	0	0
$629$	−4.15339e6	−0.418578
$630$	0	0
$631$	3.39940e6	0.339883	0.169941	−	0.985454i	$-0.445642\pi$
$631$	3.39940e6	0.339883	0.169941	+	0.985454i	$0.445642\pi$
$632$	0	0
$633$	7.37167e6	0.731234
$634$	0	0
$635$	3.96997e6	0.390709
$636$	0	0
$637$	−2.82200e7	−2.75555
$638$	0	0
$639$	−719944.	−0.0697504
$640$	0	0
$641$	−1.56117e6	−0.150074	−0.0750371	−	0.997181i	$-0.523908\pi$
$641$	−1.56117e6	−0.150074	−0.0750371	+	0.997181i	$0.523908\pi$
$642$	0	0
$643$	1.07265e7	1.02313	0.511567	−	0.859243i	$-0.329065\pi$
$643$	1.07265e7	1.02313	0.511567	+	0.859243i	$0.329065\pi$
$644$	0	0
$645$	971490.	0.0919474
$646$	0	0
$647$	−4.28656e6	−0.402576	−0.201288	−	0.979532i	$-0.564513\pi$
$647$	−4.28656e6	−0.402576	−0.201288	+	0.979532i	$0.564513\pi$
$648$	0	0
$649$	−6.00391e6	−0.559529
$650$	0	0
$651$	−116497.	−0.0107737
$652$	0	0
$653$	−2.00405e7	−1.83918	−0.919592	−	0.392875i	$-0.871480\pi$
$653$	−2.00405e7	−1.83918	−0.919592	+	0.392875i	$0.871480\pi$
$654$	0	0
$655$	−2.81259e6	−0.256155
$656$	0	0
$657$	−1.01327e6	−0.0915819
$658$	0	0
$659$	4.00472e6	0.359218	0.179609	−	0.983738i	$-0.442517\pi$
$659$	4.00472e6	0.359218	0.179609	+	0.983738i	$0.442517\pi$
$660$	0	0
$661$	−7.40685e6	−0.659371	−0.329685	−	0.944091i	$-0.606943\pi$
$661$	−7.40685e6	−0.659371	−0.329685	+	0.944091i	$0.606943\pi$
$662$	0	0
$663$	−1.69581e7	−1.49828
$664$	0	0
$665$	2.20232e6	0.193119
$666$	0	0
$667$	−3.96324e6	−0.344933
$668$	0	0
$669$	4.24328e6	0.366553
$670$	0	0
$671$	−9.52039e6	−0.816297
$672$	0	0
$673$	1.33052e7	1.13236	0.566180	−	0.824282i	$-0.308421\pi$
$673$	1.33052e7	1.13236	0.566180	+	0.824282i	$0.308421\pi$
$674$	0	0
$675$	1.66822e6	0.140927
$676$	0	0
$677$	−1.07656e7	−0.902751	−0.451375	−	0.892334i	$-0.649066\pi$
$677$	−1.07656e7	−0.902751	−0.451375	+	0.892334i	$0.649066\pi$
$678$	0	0
$679$	9.73306e6	0.810168
$680$	0	0
$681$	−6.02992e6	−0.498246
$682$	0	0
$683$	−1.55246e7	−1.27341	−0.636706	−	0.771107i	$-0.719703\pi$
$683$	−1.55246e7	−1.27341	−0.636706	+	0.771107i	$0.719703\pi$
$684$	0	0
$685$	7.07287e6	0.575929
$686$	0	0
$687$	9.43416e6	0.762626
$688$	0	0
$689$	−4.03375e7	−3.23713
$690$	0	0
$691$	−5.48083e6	−0.436668	−0.218334	−	0.975874i	$-0.570062\pi$
$691$	−5.48083e6	−0.436668	−0.218334	+	0.975874i	$0.570062\pi$
$692$	0	0
$693$	2.89306e6	0.228836
$694$	0	0
$695$	−129371.	−0.0101595
$696$	0	0
$697$	−1.84271e6	−0.143673
$698$	0	0
$699$	4.10620e6	0.317869
$700$	0	0
$701$	−1.35701e7	−1.04301	−0.521504	−	0.853249i	$-0.674629\pi$
$701$	−1.35701e7	−1.04301	−0.521504	+	0.853249i	$0.674629\pi$
$702$	0	0
$703$	811327.	0.0619166
$704$	0	0
$705$	4.18481e6	0.317105
$706$	0	0
$707$	533812.	0.0401643
$708$	0	0
$709$	1.08328e7	0.809332	0.404666	−	0.914465i	$-0.367388\pi$
$709$	1.08328e7	0.809332	0.404666	+	0.914465i	$0.367388\pi$
$710$	0	0
$711$	−5.12083e6	−0.379897
$712$	0	0
$713$	−86891.5	−0.00640109
$714$	0	0
$715$	−4.99403e6	−0.365330
$716$	0	0
$717$	790901.	0.0574546
$718$	0	0
$719$	1.84578e7	1.33155	0.665774	−	0.746154i	$-0.268102\pi$
$719$	1.84578e7	1.33155	0.665774	+	0.746154i	$0.268102\pi$
$720$	0	0
$721$	1.97653e7	1.41601
$722$	0	0
$723$	−8.97172e6	−0.638308
$724$	0	0
$725$	−6.40569e6	−0.452606
$726$	0	0
$727$	2.36232e7	1.65769	0.828843	−	0.559482i	$-0.189000\pi$
$727$	2.36232e7	1.65769	0.828843	+	0.559482i	$0.189000\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	6.89676e6	0.477366
$732$	0	0
$733$	2.32984e7	1.60165	0.800823	−	0.598901i	$-0.204396\pi$
$733$	2.32984e7	1.60165	0.800823	+	0.598901i	$0.204396\pi$
$734$	0	0
$735$	−7.20517e6	−0.491956
$736$	0	0
$737$	−1.12671e6	−0.0764091
$738$	0	0
$739$	2.76787e7	1.86438	0.932190	−	0.361969i	$-0.117895\pi$
$739$	2.76787e7	1.86438	0.932190	+	0.361969i	$0.117895\pi$
$740$	0	0
$741$	3.31260e6	0.221628
$742$	0	0
$743$	2.15463e7	1.43186	0.715930	−	0.698172i	$-0.246003\pi$
$743$	2.15463e7	1.43186	0.715930	+	0.698172i	$0.246003\pi$
$744$	0	0
$745$	−7.56738e6	−0.499522
$746$	0	0
$747$	−3.01384e6	−0.197614
$748$	0	0
$749$	−3.24513e7	−2.11363
$750$	0	0
$751$	1.62799e6	0.105330	0.0526649	−	0.998612i	$-0.483228\pi$
$751$	1.62799e6	0.105330	0.0526649	+	0.998612i	$0.483228\pi$
$752$	0	0
$753$	−5.86324e6	−0.376834
$754$	0	0
$755$	7.09989e6	0.453298
$756$	0	0
$757$	3.12417e7	1.98151	0.990753	−	0.135678i	$-0.0433214\pi$
$757$	3.12417e7	1.98151	0.990753	+	0.135678i	$0.0433214\pi$
$758$	0	0
$759$	2.15784e6	0.135961
$760$	0	0
$761$	2.26372e7	1.41697	0.708485	−	0.705726i	$-0.249379\pi$
$761$	2.26372e7	1.41697	0.708485	+	0.705726i	$0.249379\pi$
$762$	0	0
$763$	−1.96571e7	−1.22238
$764$	0	0
$765$	−4.32977e6	−0.267493
$766$	0	0
$767$	3.61484e7	2.21871
$768$	0	0
$769$	1.46708e6	0.0894618	0.0447309	−	0.998999i	$-0.485757\pi$
$769$	1.46708e6	0.0894618	0.0447309	+	0.998999i	$0.485757\pi$
$770$	0	0
$771$	−1.77286e7	−1.07409
$772$	0	0
$773$	−3.85297e6	−0.231925	−0.115962	−	0.993254i	$-0.536995\pi$
$773$	−3.85297e6	−0.231925	−0.115962	+	0.993254i	$0.536995\pi$
$774$	0	0
$775$	−140441.	−0.00839922
$776$	0	0
$777$	−4.26617e6	−0.253505
$778$	0	0
$779$	359955.	0.0212522
$780$	0	0
$781$	−1.50515e6	−0.0882981
$782$	0	0
$783$	−2.04064e6	−0.118949
$784$	0	0
$785$	−1.29720e7	−0.751336
$786$	0	0
$787$	−2.50359e7	−1.44088	−0.720439	−	0.693519i	$-0.756059\pi$
$787$	−2.50359e7	−1.44088	−0.720439	+	0.693519i	$0.756059\pi$
$788$	0	0
$789$	1.02404e7	0.585630
$790$	0	0
$791$	1.85635e7	1.05492
$792$	0	0
$793$	5.73204e7	3.23688
$794$	0	0
$795$	−1.02990e7	−0.577935
$796$	0	0
$797$	3.36793e7	1.87809	0.939047	−	0.343788i	$-0.111710\pi$
$797$	3.36793e7	1.87809	0.939047	+	0.343788i	$0.111710\pi$
$798$	0	0
$799$	2.97086e7	1.64633
$800$	0	0
$801$	3.80400e6	0.209488
$802$	0	0
$803$	−2.11838e6	−0.115935
$804$	0	0
$805$	−8.63742e6	−0.469780
$806$	0	0
$807$	5.77162e6	0.311971
$808$	0	0
$809$	−1.99641e7	−1.07245	−0.536227	−	0.844074i	$-0.680151\pi$
$809$	−1.99641e7	−1.07245	−0.536227	+	0.844074i	$0.680151\pi$
$810$	0	0
$811$	−8.00386e6	−0.427314	−0.213657	−	0.976909i	$-0.568538\pi$
$811$	−8.00386e6	−0.427314	−0.213657	+	0.976909i	$0.568538\pi$
$812$	0	0
$813$	8.44812e6	0.448264
$814$	0	0
$815$	1.27104e7	0.670296
$816$	0	0
$817$	−1.34722e6	−0.0706126
$818$	0	0
$819$	−1.74186e7	−0.907408
$820$	0	0
$821$	1.19433e7	0.618397	0.309199	−	0.950998i	$-0.399939\pi$
$821$	1.19433e7	0.618397	0.309199	+	0.950998i	$0.399939\pi$
$822$	0	0
$823$	7.36634e6	0.379098	0.189549	−	0.981871i	$-0.439297\pi$
$823$	7.36634e6	0.379098	0.189549	+	0.981871i	$0.439297\pi$
$824$	0	0
$825$	3.48767e6	0.178402
$826$	0	0
$827$	−8.37344e6	−0.425736	−0.212868	−	0.977081i	$-0.568280\pi$
$827$	−8.37344e6	−0.425736	−0.212868	+	0.977081i	$0.568280\pi$
$828$	0	0
$829$	3.71149e7	1.87569	0.937846	−	0.347052i	$-0.112817\pi$
$829$	3.71149e7	1.87569	0.937846	+	0.347052i	$0.112817\pi$
$830$	0	0
$831$	9.22670e6	0.463494
$832$	0	0
$833$	−5.11506e7	−2.55410
$834$	0	0
$835$	−4.96278e6	−0.246325
$836$	0	0
$837$	−44739.8	−0.00220740
$838$	0	0
$839$	−1.03736e7	−0.508774	−0.254387	−	0.967102i	$-0.581874\pi$
$839$	−1.03736e7	−0.508774	−0.254387	+	0.967102i	$0.581874\pi$
$840$	0	0
$841$	−1.26755e7	−0.617979
$842$	0	0
$843$	7.57343e6	0.367049
$844$	0	0
$845$	1.93286e7	0.931234
$846$	0	0
$847$	−2.79197e7	−1.33722
$848$	0	0
$849$	1.42194e7	0.677035
$850$	0	0
$851$	−3.18200e6	−0.150618
$852$	0	0
$853$	2.88586e7	1.35801	0.679004	−	0.734135i	$-0.262412\pi$
$853$	2.88586e7	1.35801	0.679004	+	0.734135i	$0.262412\pi$
$854$	0	0
$855$	845781.	0.0395679
$856$	0	0
$857$	2.87680e7	1.33800	0.669002	−	0.743261i	$-0.266722\pi$
$857$	2.87680e7	1.33800	0.669002	+	0.743261i	$0.266722\pi$
$858$	0	0
$859$	2.81738e6	0.130275	0.0651376	−	0.997876i	$-0.479251\pi$
$859$	2.81738e6	0.130275	0.0651376	+	0.997876i	$0.479251\pi$
$860$	0	0
$861$	−1.89274e6	−0.0870128
$862$	0	0
$863$	2.34321e7	1.07099	0.535493	−	0.844539i	$-0.320126\pi$
$863$	2.34321e7	1.07099	0.535493	+	0.844539i	$0.320126\pi$
$864$	0	0
$865$	584565.	0.0265639
$866$	0	0
$867$	−1.79590e7	−0.811399
$868$	0	0
$869$	−1.07058e7	−0.480918
$870$	0	0
$871$	6.78372e6	0.302986
$872$	0	0
$873$	3.73790e6	0.165994
$874$	0	0
$875$	−3.30249e7	−1.45821
$876$	0	0
$877$	−4.12225e7	−1.80982	−0.904909	−	0.425604i	$-0.860062\pi$
$877$	−4.12225e7	−1.80982	−0.904909	+	0.425604i	$0.860062\pi$
$878$	0	0
$879$	−2.05957e7	−0.899091
$880$	0	0
$881$	1.30218e7	0.565238	0.282619	−	0.959232i	$-0.408797\pi$
$881$	1.30218e7	0.565238	0.282619	+	0.959232i	$0.408797\pi$
$882$	0	0
$883$	−406852.	−0.0175604	−0.00878021	−	0.999961i	$-0.502795\pi$
$883$	−406852.	−0.0175604	−0.00878021	+	0.999961i	$0.502795\pi$
$884$	0	0
$885$	9.22948e6	0.396113
$886$	0	0
$887$	−2.48049e7	−1.05859	−0.529296	−	0.848437i	$-0.677544\pi$
$887$	−2.48049e7	−1.05859	−0.529296	+	0.848437i	$0.677544\pi$
$888$	0	0
$889$	2.89487e7	1.22850
$890$	0	0
$891$	1.11105e6	0.0468858
$892$	0	0
$893$	−5.80330e6	−0.243527
$894$	0	0
$895$	−6.01597e6	−0.251043
$896$	0	0
$897$	−1.29919e7	−0.539129
$898$	0	0
$899$	171793.	0.00708934
$900$	0	0
$901$	−7.31144e7	−3.00048
$902$	0	0
$903$	7.08403e6	0.289108
$904$	0	0
$905$	−2.31368e7	−0.939034
$906$	0	0
$907$	2.35678e6	0.0951265	0.0475633	−	0.998868i	$-0.484854\pi$
$907$	2.35678e6	0.0951265	0.0475633	+	0.998868i	$0.484854\pi$
$908$	0	0
$909$	205006.	0.00822918
$910$	0	0
$911$	3.11062e7	1.24180	0.620898	−	0.783891i	$-0.286768\pi$
$911$	3.11062e7	1.24180	0.620898	+	0.783891i	$0.286768\pi$
$912$	0	0
$913$	−6.30087e6	−0.250163
$914$	0	0
$915$	1.46352e7	0.577890
$916$	0	0
$917$	−2.05092e7	−0.805424
$918$	0	0
$919$	1.17745e6	0.0459890	0.0229945	−	0.999736i	$-0.492680\pi$
$919$	1.17745e6	0.0459890	0.0229945	+	0.999736i	$0.492680\pi$
$920$	0	0
$921$	1.73951e7	0.675737
$922$	0	0
$923$	9.06220e6	0.350130
$924$	0	0
$925$	−5.14299e6	−0.197634
$926$	0	0
$927$	7.59069e6	0.290123
$928$	0	0
$929$	−3.01092e7	−1.14462	−0.572308	−	0.820039i	$-0.693952\pi$
$929$	−3.01092e7	−1.14462	−0.572308	+	0.820039i	$0.693952\pi$
$930$	0	0
$931$	9.99180e6	0.377806
$932$	0	0
$933$	2.25546e7	0.848265
$934$	0	0
$935$	−9.05202e6	−0.338623
$936$	0	0
$937$	1.22792e7	0.456902	0.228451	−	0.973555i	$-0.426634\pi$
$937$	1.22792e7	0.456902	0.228451	+	0.973555i	$0.426634\pi$
$938$	0	0
$939$	6.70189e6	0.248047
$940$	0	0
$941$	4.46062e7	1.64218	0.821090	−	0.570799i	$-0.193366\pi$
$941$	4.46062e7	1.64218	0.821090	+	0.570799i	$0.193366\pi$
$942$	0	0
$943$	−1.41173e6	−0.0516979
$944$	0	0
$945$	−4.44734e6	−0.162002
$946$	0	0
$947$	5.16996e7	1.87332	0.936661	−	0.350238i	$-0.113899\pi$
$947$	5.16996e7	1.87332	0.936661	+	0.350238i	$0.113899\pi$
$948$	0	0
$949$	1.27543e7	0.459719
$950$	0	0
$951$	−1.75694e7	−0.629951
$952$	0	0
$953$	−3.83133e7	−1.36652	−0.683262	−	0.730173i	$-0.739439\pi$
$953$	−3.83133e7	−1.36652	−0.683262	+	0.730173i	$0.739439\pi$
$954$	0	0
$955$	1.67860e7	0.595579
$956$	0	0
$957$	−4.26625e6	−0.150580
$958$	0	0
$959$	5.15748e7	1.81088
$960$	0	0
$961$	−2.86254e7	−0.999868
$962$	0	0
$963$	−1.24627e7	−0.433057
$964$	0	0
$965$	−2.89015e6	−0.0999084
$966$	0	0
$967$	−1.46233e7	−0.502896	−0.251448	−	0.967871i	$-0.580907\pi$
$967$	−1.46233e7	−0.502896	−0.251448	+	0.967871i	$0.580907\pi$
$968$	0	0
$969$	6.00433e6	0.205426
$970$	0	0
$971$	1.06002e7	0.360800	0.180400	−	0.983593i	$-0.442261\pi$
$971$	1.06002e7	0.360800	0.180400	+	0.983593i	$0.442261\pi$
$972$	0	0
$973$	−943361.	−0.0319445
$974$	0	0
$975$	−2.09986e7	−0.707421
$976$	0	0
$977$	1.93574e7	0.648799	0.324400	−	0.945920i	$-0.394838\pi$
$977$	1.93574e7	0.648799	0.324400	+	0.945920i	$0.394838\pi$
$978$	0	0
$979$	7.95281e6	0.265194
$980$	0	0
$981$	−7.54912e6	−0.250452
$982$	0	0
$983$	−3.37569e7	−1.11424	−0.557120	−	0.830432i	$-0.688094\pi$
$983$	−3.37569e7	−1.11424	−0.557120	+	0.830432i	$0.688094\pi$
$984$	0	0
$985$	−9.64072e6	−0.316606
$986$	0	0
$987$	3.05153e7	0.997069
$988$	0	0
$989$	5.28374e6	0.171771
$990$	0	0
$991$	4.24463e7	1.37295	0.686477	−	0.727152i	$-0.259156\pi$
$991$	4.24463e7	1.37295	0.686477	+	0.727152i	$0.259156\pi$
$992$	0	0
$993$	−1.88418e7	−0.606385
$994$	0	0
$995$	−2.43023e7	−0.778197
$996$	0	0
$997$	5.54861e7	1.76785	0.883927	−	0.467624i	$-0.154890\pi$
$997$	5.54861e7	1.76785	0.883927	+	0.467624i	$0.154890\pi$
$998$	0	0
$999$	−1.63839e6	−0.0519400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.6.a.f.1.4	✓	6
4.3	odd	2		912.6.a.y.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.6.a.f.1.4	✓	6	1.1	even	1	trivial
912.6.a.y.1.4		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	456.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +28.9245 q^{5} +210.915 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +28.9245 q^{5} +210.915 q^{7} +81.0000 q^{9} +169.342 q^{11} -1019.58 q^{13} -260.320 q^{15} -1848.05 q^{17} +361.000 q^{19} -1898.23 q^{21} -1415.83 q^{23} -2288.37 q^{25} -729.000 q^{27} +2799.23 q^{29} +61.3714 q^{31} -1524.08 q^{33} +6100.61 q^{35} +2247.44 q^{37} +9176.19 q^{39} +997.106 q^{41} -3731.90 q^{43} +2342.88 q^{45} -16075.6 q^{47} +27678.1 q^{49} +16632.5 q^{51} +39562.9 q^{53} +4898.14 q^{55} -3249.00 q^{57} -35454.3 q^{59} -56219.8 q^{61} +17084.1 q^{63} -29490.7 q^{65} -6653.47 q^{67} +12742.5 q^{69} -8888.19 q^{71} -12509.4 q^{73} +20595.4 q^{75} +35716.8 q^{77} -63220.1 q^{79} +6561.00 q^{81} -37207.9 q^{83} -53454.0 q^{85} -25193.1 q^{87} +46963.0 q^{89} -215044. q^{91} -552.343 q^{93} +10441.7 q^{95} +46146.9 q^{97} +13716.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9}+O(q^{10}) \) 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9	\( 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9} - 203 q^{11} - 298 q^{13} + 585 q^{15} + 1319 q^{17} + 2166 q^{19} + 1341 q^{21} + 1234 q^{23} + 4589 q^{25} - 4374 q^{27} + 7356 q^{29} + 1632 q^{31} + 1827 q^{33} + 4383 q^{35} + 14204 q^{37} + 2682 q^{39} + 14734 q^{41} - 4693 q^{43} - 5265 q^{45} - 10955 q^{47} + 38561 q^{49} - 11871 q^{51} + 47500 q^{53} + 769 q^{55} - 19494 q^{57} - 61744 q^{59} - 81581 q^{61} - 12069 q^{63} - 59686 q^{65} - 45756 q^{67} - 11106 q^{69} - 10416 q^{71} - 54615 q^{73} - 41301 q^{75} - 29515 q^{77} - 145594 q^{79} + 39366 q^{81} - 160548 q^{83} - 53947 q^{85} - 66204 q^{87} - 97728 q^{89} - 418294 q^{91} - 14688 q^{93} - 23465 q^{95} - 760 q^{97} - 16443 q^{99}+O(q^{100}) \) 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9 - 203 * q^11 - 298 * q^13 + 585 * q^15 + 1319 * q^17 + 2166 * q^19 + 1341 * q^21 + 1234 * q^23 + 4589 * q^25 - 4374 * q^27 + 7356 * q^29 + 1632 * q^31 + 1827 * q^33 + 4383 * q^35 + 14204 * q^37 + 2682 * q^39 + 14734 * q^41 - 4693 * q^43 - 5265 * q^45 - 10955 * q^47 + 38561 * q^49 - 11871 * q^51 + 47500 * q^53 + 769 * q^55 - 19494 * q^57 - 61744 * q^59 - 81581 * q^61 - 12069 * q^63 - 59686 * q^65 - 45756 * q^67 - 11106 * q^69 - 10416 * q^71 - 54615 * q^73 - 41301 * q^75 - 29515 * q^77 - 145594 * q^79 + 39366 * q^81 - 160548 * q^83 - 53947 * q^85 - 66204 * q^87 - 97728 * q^89 - 418294 * q^91 - 14688 * q^93 - 23465 * q^95 - 760 * q^97 - 16443 * q^99

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