LMFDB - Embedded newform 27.5.b.b.26.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,5,Mod(26,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("27.26");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	27.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$2.79098900326$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 6 $ x^2 + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.2
Root			$2.44949i$ of defining polynomial
Character	$\chi$	$=$	27.26
Dual form			27.5.b.b.26.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.34847i q^{2} -38.0000 q^{4} +14.6969i q^{5} +17.0000 q^{7} -161.666i q^{8} +O(q^{10})$	\(q+7.34847i q^{2} -38.0000 q^{4} +14.6969i q^{5} +17.0000 q^{7} -161.666i q^{8} -108.000 q^{10} +161.666i q^{11} +95.0000 q^{13} +124.924i q^{14} +580.000 q^{16} +308.636i q^{17} +209.000 q^{19} -558.484i q^{20} -1188.00 q^{22} -867.119i q^{23} +409.000 q^{25} +698.105i q^{26} -646.000 q^{28} -323.333i q^{29} +950.000 q^{31} +1675.45i q^{32} -2268.00 q^{34} +249.848i q^{35} -1177.00 q^{37} +1535.83i q^{38} +2376.00 q^{40} -2145.75i q^{41} +1430.00 q^{43} -6143.32i q^{44} +6372.00 q^{46} +1572.57i q^{47} -2112.00 q^{49} +3005.52i q^{50} -3610.00 q^{52} +2909.99i q^{53} -2376.00 q^{55} -2748.33i q^{56} +2376.00 q^{58} +2131.06i q^{59} -1441.00 q^{61} +6981.05i q^{62} -3032.00 q^{64} +1396.21i q^{65} +3497.00 q^{67} -11728.2i q^{68} -1836.00 q^{70} -1940.00i q^{71} -9025.00 q^{73} -8649.15i q^{74} -7942.00 q^{76} +2748.33i q^{77} +5273.00 q^{79} +8524.22i q^{80} +15768.0 q^{82} -6143.32i q^{83} -4536.00 q^{85} +10508.3i q^{86} +26136.0 q^{88} -11155.0i q^{89} +1615.00 q^{91} +32950.5i q^{92} -11556.0 q^{94} +3071.66i q^{95} -2809.00 q^{97} -15520.0i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 76 q^{4} + 34 q^{7}+O(q^{10}) $ 2 * q - 76 * q^4 + 34 * q^7	$ 2 q - 76 q^{4} + 34 q^{7} - 216 q^{10} + 190 q^{13} + 1160 q^{16} + 418 q^{19} - 2376 q^{22} + 818 q^{25} - 1292 q^{28} + 1900 q^{31} - 4536 q^{34} - 2354 q^{37} + 4752 q^{40} + 2860 q^{43} + 12744 q^{46} - 4224 q^{49} - 7220 q^{52} - 4752 q^{55} + 4752 q^{58} - 2882 q^{61} - 6064 q^{64} + 6994 q^{67} - 3672 q^{70} - 18050 q^{73} - 15884 q^{76} + 10546 q^{79} + 31536 q^{82} - 9072 q^{85} + 52272 q^{88} + 3230 q^{91} - 23112 q^{94} - 5618 q^{97}+O(q^{100}) $ 2 * q - 76 * q^4 + 34 * q^7 - 216 * q^10 + 190 * q^13 + 1160 * q^16 + 418 * q^19 - 2376 * q^22 + 818 * q^25 - 1292 * q^28 + 1900 * q^31 - 4536 * q^34 - 2354 * q^37 + 4752 * q^40 + 2860 * q^43 + 12744 * q^46 - 4224 * q^49 - 7220 * q^52 - 4752 * q^55 + 4752 * q^58 - 2882 * q^61 - 6064 * q^64 + 6994 * q^67 - 3672 * q^70 - 18050 * q^73 - 15884 * q^76 + 10546 * q^79 + 31536 * q^82 - 9072 * q^85 + 52272 * q^88 + 3230 * q^91 - 23112 * q^94 - 5618 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	7.34847i	1.83712i	0.395285	+	0.918559i	$0.370646\pi$
$2$	7.34847i	1.83712i	−0.395285	+	0.918559i	$0.629354\pi$
$3$	0	0
$3$	0	0
$4$	−38.0000	−2.37500
$5$	14.6969i	0.587878i	0.955824	+	0.293939i	$0.0949662\pi$
$5$	14.6969i	0.587878i	−0.955824	+	0.293939i	$0.905034\pi$
$6$	0	0
$7$	17.0000	0.346939	0.173469	−	0.984839i	$-0.444502\pi$
$7$	17.0000	0.346939	0.173469	+	0.984839i	$0.444502\pi$
$8$	− 161.666i	− 2.52604i
$9$	0	0
$10$	−108.000	−1.08000
$11$	161.666i	1.33609i	0.744123	+	0.668043i	$0.232868\pi$
$11$	161.666i	1.33609i	−0.744123	+	0.668043i	$0.767132\pi$
$12$	0	0
$13$	95.0000	0.562130	0.281065	−	0.959689i	$-0.409312\pi$
$13$	95.0000	0.562130	0.281065	+	0.959689i	$0.409312\pi$
$14$	124.924i	0.637367i
$15$	0	0
$16$	580.000	2.26562
$17$	308.636i	1.06794i	0.845502	+	0.533972i	$0.179301\pi$
$17$	308.636i	1.06794i	−0.845502	+	0.533972i	$0.820699\pi$
$18$	0	0
$19$	209.000	0.578947	0.289474	−	0.957186i	$-0.406520\pi$
$19$	209.000	0.578947	0.289474	+	0.957186i	$0.406520\pi$
$20$	− 558.484i	− 1.39621i
$21$	0	0
$22$	−1188.00	−2.45455
$23$	− 867.119i	− 1.63917i	−0.572960	−	0.819584i	$-0.694205\pi$
$23$	− 867.119i	− 1.63917i	0.572960	−	0.819584i	$-0.305795\pi$
$24$	0	0
$25$	409.000	0.654400
$26$	698.105i	1.03270i
$27$	0	0
$28$	−646.000	−0.823980
$29$	− 323.333i	− 0.384462i	−0.981350	−	0.192231i	$-0.938428\pi$
$29$	− 323.333i	− 0.384462i	0.981350	−	0.192231i	$-0.0615723\pi$
$30$	0	0
$31$	950.000	0.988554	0.494277	−	0.869305i	$-0.335433\pi$
$31$	950.000	0.988554	0.494277	+	0.869305i	$0.335433\pi$
$32$	1675.45i	1.63618i
$33$	0	0
$34$	−2268.00	−1.96194
$35$	249.848i	0.203958i
$36$	0	0
$37$	−1177.00	−0.859752	−0.429876	−	0.902888i	$-0.641443\pi$
$37$	−1177.00	−0.859752	−0.429876	+	0.902888i	$0.641443\pi$
$38$	1535.83i	1.06359i
$39$	0	0
$40$	2376.00	1.48500
$41$	− 2145.75i	− 1.27647i	−0.769840	−	0.638237i	$-0.779664\pi$
$41$	− 2145.75i	− 1.27647i	0.769840	−	0.638237i	$-0.220336\pi$
$42$	0	0
$43$	1430.00	0.773391	0.386696	−	0.922207i	$-0.373616\pi$
$43$	1430.00	0.773391	0.386696	+	0.922207i	$0.373616\pi$
$44$	− 6143.32i	− 3.17320i
$45$	0	0
$46$	6372.00	3.01134
$47$	1572.57i	0.711893i	0.934506	+	0.355947i	$0.115842\pi$
$47$	1572.57i	0.711893i	−0.934506	+	0.355947i	$0.884158\pi$
$48$	0	0
$49$	−2112.00	−0.879633
$50$	3005.52i	1.20221i
$51$	0	0
$52$	−3610.00	−1.33506
$53$	2909.99i	1.03595i	0.855395	+	0.517977i	$0.173315\pi$
$53$	2909.99i	1.03595i	−0.855395	+	0.517977i	$0.826685\pi$
$54$	0	0
$55$	−2376.00	−0.785455
$56$	− 2748.33i	− 0.876380i
$57$	0	0
$58$	2376.00	0.706302
$59$	2131.06i	0.612197i	0.952000	+	0.306098i	$0.0990236\pi$
$59$	2131.06i	0.612197i	−0.952000	+	0.306098i	$0.900976\pi$
$60$	0	0
$61$	−1441.00	−0.387261	−0.193631	−	0.981074i	$-0.562026\pi$
$61$	−1441.00	−0.387261	−0.193631	+	0.981074i	$0.562026\pi$
$62$	6981.05i	1.81609i
$63$	0	0
$64$	−3032.00	−0.740234
$65$	1396.21i	0.330464i
$66$	0	0
$67$	3497.00	0.779015	0.389508	−	0.921023i	$-0.372645\pi$
$67$	3497.00	0.779015	0.389508	+	0.921023i	$0.372645\pi$
$68$	− 11728.2i	− 2.53637i
$69$	0	0
$70$	−1836.00	−0.374694
$71$	− 1940.00i	− 0.384843i	−0.981312	−	0.192422i	$-0.938366\pi$
$71$	− 1940.00i	− 0.384843i	0.981312	−	0.192422i	$-0.0616342\pi$
$72$	0	0
$73$	−9025.00	−1.69356	−0.846782	−	0.531940i	$-0.821463\pi$
$73$	−9025.00	−1.69356	−0.846782	+	0.531940i	$0.821463\pi$
$74$	− 8649.15i	− 1.57946i
$75$	0	0
$76$	−7942.00	−1.37500
$77$	2748.33i	0.463540i
$78$	0	0
$79$	5273.00	0.844897	0.422448	−	0.906387i	$-0.361171\pi$
$79$	5273.00	0.844897	0.422448	+	0.906387i	$0.361171\pi$
$80$	8524.22i	1.33191i
$81$	0	0
$82$	15768.0	2.34503
$83$	− 6143.32i	− 0.891758i	−0.895093	−	0.445879i	$-0.852891\pi$
$83$	− 6143.32i	− 0.891758i	0.895093	−	0.445879i	$-0.147109\pi$
$84$	0	0
$85$	−4536.00	−0.627820
$86$	10508.3i	1.42081i
$87$	0	0
$88$	26136.0	3.37500
$89$	− 11155.0i	− 1.40828i	−0.710062	−	0.704139i	$-0.751333\pi$
$89$	− 11155.0i	− 1.40828i	0.710062	−	0.704139i	$-0.248667\pi$
$90$	0	0
$91$	1615.00	0.195025
$92$	32950.5i	3.89302i
$93$	0	0
$94$	−11556.0	−1.30783
$95$	3071.66i	0.340350i
$96$	0	0
$97$	−2809.00	−0.298544	−0.149272	−	0.988796i	$-0.547693\pi$
$97$	−2809.00	−0.298544	−0.149272	+	0.988796i	$0.547693\pi$
$98$	− 15520.0i	− 1.61599i
$99$	0	0
$100$	−15542.0	−1.55420
$101$	− 6231.50i	− 0.610872i	−0.952213	−	0.305436i	$-0.901198\pi$
$101$	− 6231.50i	− 0.610872i	0.952213	−	0.305436i	$-0.0988021\pi$
$102$	0	0
$103$	16385.0	1.54444	0.772222	−	0.635353i	$-0.219145\pi$
$103$	16385.0	1.54444	0.772222	+	0.635353i	$0.219145\pi$
$104$	− 15358.3i	− 1.41996i
$105$	0	0
$106$	−21384.0	−1.90317
$107$	− 7274.98i	− 0.635425i	−0.948187	−	0.317713i	$-0.897085\pi$
$107$	− 7274.98i	− 0.635425i	0.948187	−	0.317713i	$-0.102915\pi$
$108$	0	0
$109$	1946.00	0.163791	0.0818955	−	0.996641i	$-0.473903\pi$
$109$	1946.00	0.163791	0.0818955	+	0.996641i	$0.473903\pi$
$110$	− 17460.0i	− 1.44297i
$111$	0	0
$112$	9860.00	0.786033
$113$	15358.3i	1.20278i	0.798956	+	0.601390i	$0.205386\pi$
$113$	15358.3i	1.20278i	−0.798956	+	0.601390i	$0.794614\pi$
$114$	0	0
$115$	12744.0	0.963629
$116$	12286.6i	0.913098i
$117$	0	0
$118$	−15660.0	−1.12468
$119$	5246.81i	0.370511i
$120$	0	0
$121$	−11495.0	−0.785124
$122$	− 10589.1i	− 0.711445i
$123$	0	0
$124$	−36100.0	−2.34781
$125$	15196.6i	0.972585i
$126$	0	0
$127$	6878.00	0.426437	0.213218	−	0.977005i	$-0.431605\pi$
$127$	6878.00	0.426437	0.213218	+	0.977005i	$0.431605\pi$
$128$	4526.66i	0.276285i
$129$	0	0
$130$	−10260.0	−0.607101
$131$	6143.32i	0.357981i	0.983851	+	0.178991i	$0.0572832\pi$
$131$	6143.32i	0.357981i	−0.983851	+	0.178991i	$0.942717\pi$
$132$	0	0
$133$	3553.00	0.200859
$134$	25697.6i	1.43114i
$135$	0	0
$136$	49896.0	2.69766
$137$	1925.30i	0.102579i	0.998684	+	0.0512893i	$0.0163331\pi$
$137$	1925.30i	0.102579i	−0.998684	+	0.0512893i	$0.983667\pi$
$138$	0	0
$139$	10049.0	0.520108	0.260054	−	0.965594i	$-0.416260\pi$
$139$	10049.0	0.520108	0.260054	+	0.965594i	$0.416260\pi$
$140$	− 9494.22i	− 0.484399i
$141$	0	0
$142$	14256.0	0.707003
$143$	15358.3i	0.751054i
$144$	0	0
$145$	4752.00	0.226017
$146$	− 66319.9i	− 3.11127i
$147$	0	0
$148$	44726.0	2.04191
$149$	− 25690.2i	− 1.15717i	−0.815623	−	0.578583i	$-0.803606\pi$
$149$	− 25690.2i	− 1.15717i	0.815623	−	0.578583i	$-0.196394\pi$
$150$	0	0
$151$	−13015.0	−0.570808	−0.285404	−	0.958407i	$-0.592128\pi$
$151$	−13015.0	−0.570808	−0.285404	+	0.958407i	$0.592128\pi$
$152$	− 33788.3i	− 1.46244i
$153$	0	0
$154$	−20196.0	−0.851577
$155$	13962.1i	0.581148i
$156$	0	0
$157$	−18742.0	−0.760355	−0.380178	−	0.924913i	$-0.624137\pi$
$157$	−18742.0	−0.760355	−0.380178	+	0.924913i	$0.624137\pi$
$158$	38748.5i	1.55217i
$159$	0	0
$160$	−24624.0	−0.961875
$161$	− 14741.0i	− 0.568691i
$162$	0	0
$163$	−13471.0	−0.507019	−0.253510	−	0.967333i	$-0.581585\pi$
$163$	−13471.0	−0.507019	−0.253510	+	0.967333i	$0.581585\pi$
$164$	81538.6i	3.03163i
$165$	0	0
$166$	45144.0	1.63826
$167$	− 42841.6i	− 1.53615i	−0.640362	−	0.768073i	$-0.721216\pi$
$167$	− 42841.6i	− 1.53615i	0.640362	−	0.768073i	$-0.278784\pi$
$168$	0	0
$169$	−19536.0	−0.684010
$170$	− 33332.7i	− 1.15338i
$171$	0	0
$172$	−54340.0	−1.83680
$173$	13256.6i	0.442936i	0.975168	+	0.221468i	$0.0710849\pi$
$173$	13256.6i	0.442936i	−0.975168	+	0.221468i	$0.928915\pi$
$174$	0	0
$175$	6953.00	0.227037
$176$	93766.5i	3.02707i
$177$	0	0
$178$	81972.0	2.58717
$179$	21340.0i	0.666020i	0.942923	+	0.333010i	$0.108064\pi$
$179$	21340.0i	0.666020i	−0.942923	+	0.333010i	$0.891936\pi$
$180$	0	0
$181$	57071.0	1.74204	0.871020	−	0.491247i	$-0.163459\pi$
$181$	57071.0	1.74204	0.871020	+	0.491247i	$0.163459\pi$
$182$	11867.8i	0.358283i
$183$	0	0
$184$	−140184.	−4.14060
$185$	− 17298.3i	− 0.505429i
$186$	0	0
$187$	−49896.0	−1.42686
$188$	− 59757.8i	− 1.69075i
$189$	0	0
$190$	−22572.0	−0.625263
$191$	− 41195.5i	− 1.12923i	−0.825354	−	0.564616i	$-0.809024\pi$
$191$	− 41195.5i	− 1.12923i	0.825354	−	0.564616i	$-0.190976\pi$
$192$	0	0
$193$	−59449.0	−1.59599	−0.797995	−	0.602665i	$-0.794106\pi$
$193$	−59449.0	−1.59599	−0.797995	+	0.602665i	$0.794106\pi$
$194$	− 20641.9i	− 0.548460i
$195$	0	0
$196$	80256.0	2.08913
$197$	− 19620.4i	− 0.505563i	−0.967523	−	0.252782i	$-0.918655\pi$
$197$	− 19620.4i	− 0.505563i	0.967523	−	0.252782i	$-0.0813455\pi$
$198$	0	0
$199$	−17215.0	−0.434711	−0.217356	−	0.976092i	$-0.569743\pi$
$199$	−17215.0	−0.434711	−0.217356	+	0.976092i	$0.569743\pi$
$200$	− 66121.5i	− 1.65304i
$201$	0	0
$202$	45792.0	1.12224
$203$	− 5496.65i	− 0.133385i
$204$	0	0
$205$	31536.0	0.750410
$206$	120405.i	2.83732i
$207$	0	0
$208$	55100.0	1.27358
$209$	33788.3i	0.773523i
$210$	0	0
$211$	20273.0	0.455358	0.227679	−	0.973736i	$-0.426886\pi$
$211$	20273.0	0.455358	0.227679	+	0.973736i	$0.426886\pi$
$212$	− 110580.i	− 2.46039i
$213$	0	0
$214$	53460.0	1.16735
$215$	21016.6i	0.454659i
$216$	0	0
$217$	16150.0	0.342968
$218$	14300.1i	0.300903i
$219$	0	0
$220$	90288.0	1.86545
$221$	29320.4i	0.600323i
$222$	0	0
$223$	−47962.0	−0.964467	−0.482234	−	0.876043i	$-0.660174\pi$
$223$	−47962.0	−0.964467	−0.482234	+	0.876043i	$0.660174\pi$
$224$	28482.7i	0.567655i
$225$	0	0
$226$	−112860.	−2.20965
$227$	− 1293.33i	− 0.0250991i	−0.999921	−	0.0125495i	$-0.996005\pi$
$227$	− 1293.33i	− 0.0250991i	0.999921	−	0.0125495i	$-0.00399475\pi$
$228$	0	0
$229$	24530.0	0.467764	0.233882	−	0.972265i	$-0.424857\pi$
$229$	24530.0	0.467764	0.233882	+	0.972265i	$0.424857\pi$
$230$	93648.9i	1.77030i
$231$	0	0
$232$	−52272.0	−0.971165
$233$	102820.i	1.89393i	0.321331	+	0.946967i	$0.395870\pi$
$233$	102820.i	1.89393i	−0.321331	+	0.946967i	$0.604130\pi$
$234$	0	0
$235$	−23112.0	−0.418506
$236$	− 80980.1i	− 1.45397i
$237$	0	0
$238$	−38556.0	−0.680672
$239$	73073.2i	1.27927i	0.768679	+	0.639635i	$0.220914\pi$
$239$	73073.2i	1.27927i	−0.768679	+	0.639635i	$0.779086\pi$
$240$	0	0
$241$	38807.0	0.668153	0.334077	−	0.942546i	$-0.391576\pi$
$241$	38807.0	0.668153	0.334077	+	0.942546i	$0.391576\pi$
$242$	− 84470.7i	− 1.44236i
$243$	0	0
$244$	54758.0	0.919746
$245$	− 31039.9i	− 0.517117i
$246$	0	0
$247$	19855.0	0.325444
$248$	− 153583.i	− 2.49712i
$249$	0	0
$250$	−111672.	−1.78675
$251$	− 106788.i	− 1.69502i	−0.530779	−	0.847510i	$-0.678101\pi$
$251$	− 106788.i	− 1.69502i	0.530779	−	0.847510i	$-0.321899\pi$
$252$	0	0
$253$	140184.	2.19007
$254$	50542.8i	0.783415i
$255$	0	0
$256$	−81776.0	−1.24780
$257$	− 17489.4i	− 0.264794i	−0.991197	−	0.132397i	$-0.957733\pi$
$257$	− 17489.4i	− 0.264794i	0.991197	−	0.132397i	$-0.0422673\pi$
$258$	0	0
$259$	−20009.0	−0.298281
$260$	− 53055.9i	− 0.784851i
$261$	0	0
$262$	−45144.0	−0.657654
$263$	90591.9i	1.30972i	0.755751	+	0.654859i	$0.227272\pi$
$263$	90591.9i	1.30972i	−0.755751	+	0.654859i	$0.772728\pi$
$264$	0	0
$265$	−42768.0	−0.609014
$266$	26109.1i	0.369002i
$267$	0	0
$268$	−132886.	−1.85016
$269$	− 121558.i	− 1.67989i	−0.542673	−	0.839944i	$-0.682588\pi$
$269$	− 121558.i	− 1.67989i	0.542673	−	0.839944i	$-0.317412\pi$
$270$	0	0
$271$	−73519.0	−1.00106	−0.500531	−	0.865719i	$-0.666862\pi$
$271$	−73519.0	−1.00106	−0.500531	+	0.865719i	$0.666862\pi$
$272$	179009.i	2.41956i
$273$	0	0
$274$	−14148.0	−0.188449
$275$	66121.5i	0.874334i
$276$	0	0
$277$	139850.	1.82265	0.911324	−	0.411689i	$-0.135061\pi$
$277$	139850.	1.82265	0.911324	+	0.411689i	$0.135061\pi$
$278$	73844.8i	0.955499i
$279$	0	0
$280$	40392.0	0.515204
$281$	− 120603.i	− 1.52738i	−0.645586	−	0.763688i	$-0.723387\pi$
$281$	− 120603.i	− 1.52738i	0.645586	−	0.763688i	$-0.276613\pi$
$282$	0	0
$283$	52934.0	0.660940	0.330470	−	0.943817i	$-0.392793\pi$
$283$	52934.0	0.660940	0.330470	+	0.943817i	$0.392793\pi$
$284$	73719.8i	0.914003i
$285$	0	0
$286$	−112860.	−1.37977
$287$	− 36477.8i	− 0.442858i
$288$	0	0
$289$	−11735.0	−0.140504
$290$	34919.9i	0.415219i
$291$	0	0
$292$	342950.	4.02221
$293$	89078.1i	1.03761i	0.854891	+	0.518807i	$0.173624\pi$
$293$	89078.1i	1.03761i	−0.854891	+	0.518807i	$0.826376\pi$
$294$	0	0
$295$	−31320.0	−0.359897
$296$	190281.i	2.17176i
$297$	0	0
$298$	188784.	2.12585
$299$	− 82376.3i	− 0.921425i
$300$	0	0
$301$	24310.0	0.268319
$302$	− 95640.3i	− 1.04864i
$303$	0	0
$304$	121220.	1.31168
$305$	− 21178.3i	− 0.227662i
$306$	0	0
$307$	−135178.	−1.43426	−0.717132	−	0.696937i	$-0.754546\pi$
$307$	−135178.	−1.43426	−0.717132	+	0.696937i	$0.754546\pi$
$308$	− 104436.i	− 1.10091i
$309$	0	0
$310$	−102600.	−1.06764
$311$	− 160211.i	− 1.65643i	−0.560412	−	0.828214i	$-0.689357\pi$
$311$	− 160211.i	− 1.65643i	0.560412	−	0.828214i	$-0.310643\pi$
$312$	0	0
$313$	−62857.0	−0.641601	−0.320800	−	0.947147i	$-0.603952\pi$
$313$	−62857.0	−0.641601	−0.320800	+	0.947147i	$0.603952\pi$
$314$	− 137725.i	− 1.39686i
$315$	0	0
$316$	−200374.	−2.00663
$317$	49058.4i	0.488197i	0.969751	+	0.244098i	$0.0784919\pi$
$317$	49058.4i	0.488197i	−0.969751	+	0.244098i	$0.921508\pi$
$318$	0	0
$319$	52272.0	0.513674
$320$	− 44561.1i	− 0.435167i
$321$	0	0
$322$	108324.	1.04475
$323$	64504.9i	0.618283i
$324$	0	0
$325$	38855.0	0.367858
$326$	− 98991.2i	− 0.931454i
$327$	0	0
$328$	−346896.	−3.22442
$329$	26733.7i	0.246983i
$330$	0	0
$331$	81665.0	0.745384	0.372692	−	0.927955i	$-0.378435\pi$
$331$	81665.0	0.745384	0.372692	+	0.927955i	$0.378435\pi$
$332$	233446.i	2.11793i
$333$	0	0
$334$	314820.	2.82208
$335$	51395.2i	0.457966i
$336$	0	0
$337$	−145705.	−1.28296	−0.641482	−	0.767138i	$-0.721680\pi$
$337$	−145705.	−1.28296	−0.641482	+	0.767138i	$0.721680\pi$
$338$	− 143560.i	− 1.25661i
$339$	0	0
$340$	172368.	1.49107
$341$	153583.i	1.32079i
$342$	0	0
$343$	−76721.0	−0.652118
$344$	− 231183.i	− 1.95361i
$345$	0	0
$346$	−97416.0	−0.813726
$347$	105877.i	0.879309i	0.898167	+	0.439655i	$0.144899\pi$
$347$	105877.i	0.879309i	−0.898167	+	0.439655i	$0.855101\pi$
$348$	0	0
$349$	40535.0	0.332797	0.166398	−	0.986059i	$-0.446786\pi$
$349$	40535.0	0.332797	0.166398	+	0.986059i	$0.446786\pi$
$350$	51093.9i	0.417093i
$351$	0	0
$352$	−270864.	−2.18608
$353$	8406.65i	0.0674642i	0.999431	+	0.0337321i	$0.0107393\pi$
$353$	8406.65i	0.0674642i	−0.999431	+	0.0337321i	$0.989261\pi$
$354$	0	0
$355$	28512.0	0.226241
$356$	423889.i	3.34466i
$357$	0	0
$358$	−156816.	−1.22356
$359$	19267.7i	0.149500i	0.997202	+	0.0747499i	$0.0238158\pi$
$359$	19267.7i	0.149500i	−0.997202	+	0.0747499i	$0.976184\pi$
$360$	0	0
$361$	−86640.0	−0.664820
$362$	419384.i	3.20033i
$363$	0	0
$364$	−61370.0	−0.463184
$365$	− 132640.i	− 0.995608i
$366$	0	0
$367$	−119719.	−0.888855	−0.444428	−	0.895815i	$-0.646593\pi$
$367$	−119719.	−0.888855	−0.444428	+	0.895815i	$0.646593\pi$
$368$	− 502929.i	− 3.71374i
$369$	0	0
$370$	127116.	0.928532
$371$	49469.9i	0.359412i
$372$	0	0
$373$	1319.00	0.00948041	0.00474021	−	0.999989i	$-0.498491\pi$
$373$	1319.00	0.00948041	0.00474021	+	0.999989i	$0.498491\pi$
$374$	− 366659.i	− 2.62132i
$375$	0	0
$376$	254232.	1.79827
$377$	− 30716.6i	− 0.216118i
$378$	0	0
$379$	−164887.	−1.14791	−0.573955	−	0.818887i	$-0.694592\pi$
$379$	−164887.	−1.14791	−0.573955	+	0.818887i	$0.694592\pi$
$380$	− 116723.i	− 0.808332i
$381$	0	0
$382$	302724.	2.07453
$383$	29511.5i	0.201184i	0.994928	+	0.100592i	$0.0320736\pi$
$383$	29511.5i	0.201184i	−0.994928	+	0.100592i	$0.967926\pi$
$384$	0	0
$385$	−40392.0	−0.272505
$386$	− 436859.i	− 2.93202i
$387$	0	0
$388$	106742.	0.709042
$389$	168941.i	1.11644i	0.829692	+	0.558222i	$0.188516\pi$
$389$	168941.i	1.11644i	−0.829692	+	0.558222i	$0.811484\pi$
$390$	0	0
$391$	267624.	1.75054
$392$	341439.i	2.22199i
$393$	0	0
$394$	144180.	0.928779
$395$	77497.0i	0.496696i
$396$	0	0
$397$	21242.0	0.134777	0.0673883	−	0.997727i	$-0.478533\pi$
$397$	21242.0	0.134777	0.0673883	+	0.997727i	$0.478533\pi$
$398$	− 126504.i	− 0.798616i
$399$	0	0
$400$	237220.	1.48263
$401$	− 109463.i	− 0.680735i	−0.940293	−	0.340367i	$-0.889449\pi$
$401$	− 109463.i	− 0.680735i	0.940293	−	0.340367i	$-0.110551\pi$
$402$	0	0
$403$	90250.0	0.555696
$404$	236797.i	1.45082i
$405$	0	0
$406$	40392.0	0.245044
$407$	− 190281.i	− 1.14870i
$408$	0	0
$409$	−302665.	−1.80932	−0.904660	−	0.426133i	$-0.859875\pi$
$409$	−302665.	−1.80932	−0.904660	+	0.426133i	$0.859875\pi$
$410$	231741.i	1.37859i
$411$	0	0
$412$	−622630.	−3.66805
$413$	36228.0i	0.212395i
$414$	0	0
$415$	90288.0	0.524244
$416$	159168.i	0.919748i
$417$	0	0
$418$	−248292.	−1.42105
$419$	− 299626.i	− 1.70668i	−0.521355	−	0.853340i	$-0.674573\pi$
$419$	− 299626.i	− 1.70668i	0.521355	−	0.853340i	$-0.325427\pi$
$420$	0	0
$421$	195479.	1.10290	0.551450	−	0.834208i	$-0.314075\pi$
$421$	195479.	1.10290	0.551450	+	0.834208i	$0.314075\pi$
$422$	148976.i	0.836546i
$423$	0	0
$424$	470448.	2.61686
$425$	126232.i	0.698862i
$426$	0	0
$427$	−24497.0	−0.134356
$428$	276449.i	1.50914i
$429$	0	0
$430$	−154440.	−0.835262
$431$	36374.9i	0.195816i	0.995195	+	0.0979079i	$0.0312150\pi$
$431$	36374.9i	0.195816i	−0.995195	+	0.0979079i	$0.968785\pi$
$432$	0	0
$433$	−173014.	−0.922795	−0.461398	−	0.887193i	$-0.652652\pi$
$433$	−173014.	−0.922795	−0.461398	+	0.887193i	$0.652652\pi$
$434$	118678.i	0.630072i
$435$	0	0
$436$	−73948.0	−0.389003
$437$	− 181228.i	− 0.948991i
$438$	0	0
$439$	115334.	0.598451	0.299225	−	0.954182i	$-0.403272\pi$
$439$	115334.	0.598451	0.299225	+	0.954182i	$0.403272\pi$
$440$	384119.i	1.98409i
$441$	0	0
$442$	−215460.	−1.10286
$443$	− 9494.22i	− 0.0483784i	−0.999707	−	0.0241892i	$-0.992300\pi$
$443$	− 9494.22i	− 0.0483784i	0.999707	−	0.0241892i	$-0.00770042\pi$
$444$	0	0
$445$	163944.	0.827895
$446$	− 352447.i	− 1.77184i
$447$	0	0
$448$	−51544.0	−0.256816
$449$	− 165296.i	− 0.819919i	−0.912104	−	0.409959i	$-0.865543\pi$
$449$	− 165296.i	− 0.819919i	0.912104	−	0.409959i	$-0.134457\pi$
$450$	0	0
$451$	346896.	1.70548
$452$	− 583615.i	− 2.85660i
$453$	0	0
$454$	9504.00	0.0461100
$455$	23735.6i	0.114651i
$456$	0	0
$457$	−255574.	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$457$	−255574.	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$458$	180258.i	0.859337i
$459$	0	0
$460$	−484272.	−2.28862
$461$	− 234578.i	− 1.10379i	−0.833915	−	0.551893i	$-0.813906\pi$
$461$	− 234578.i	− 1.10379i	0.833915	−	0.551893i	$-0.186094\pi$
$462$	0	0
$463$	155849.	0.727013	0.363506	−	0.931592i	$-0.381579\pi$
$463$	155849.	0.727013	0.363506	+	0.931592i	$0.381579\pi$
$464$	− 187533.i	− 0.871047i
$465$	0	0
$466$	−755568.	−3.47938
$467$	52864.9i	0.242401i	0.992628	+	0.121200i	$0.0386743\pi$
$467$	52864.9i	0.242401i	−0.992628	+	0.121200i	$0.961326\pi$
$468$	0	0
$469$	59449.0	0.270271
$470$	− 169838.i	− 0.768845i
$471$	0	0
$472$	344520.	1.54643
$473$	231183.i	1.03332i
$474$	0	0
$475$	85481.0	0.378863
$476$	− 199379.i	− 0.879964i
$477$	0	0
$478$	−536976.	−2.35017
$479$	87152.8i	0.379849i	0.981799	+	0.189924i	$0.0608243\pi$
$479$	87152.8i	0.379849i	−0.981799	+	0.189924i	$0.939176\pi$
$480$	0	0
$481$	−111815.	−0.483292
$482$	285172.i	1.22748i
$483$	0	0
$484$	436810.	1.86467
$485$	− 41283.7i	− 0.175507i
$486$	0	0
$487$	109001.	0.459592	0.229796	−	0.973239i	$-0.426194\pi$
$487$	109001.	0.459592	0.229796	+	0.973239i	$0.426194\pi$
$488$	232961.i	0.978237i
$489$	0	0
$490$	228096.	0.950004
$491$	191281.i	0.793429i	0.917942	+	0.396714i	$0.129850\pi$
$491$	191281.i	0.793429i	−0.917942	+	0.396714i	$0.870150\pi$
$492$	0	0
$493$	99792.0	0.410584
$494$	145904.i	0.597878i
$495$	0	0
$496$	551000.	2.23969
$497$	− 32979.9i	− 0.133517i
$498$	0	0
$499$	43214.0	0.173550	0.0867748	−	0.996228i	$-0.472344\pi$
$499$	43214.0	0.173550	0.0867748	+	0.996228i	$0.472344\pi$
$500$	− 577472.i	− 2.30989i
$501$	0	0
$502$	784728.	3.11395
$503$	80465.7i	0.318035i	0.987276	+	0.159018i	$0.0508326\pi$
$503$	80465.7i	0.318035i	−0.987276	+	0.159018i	$0.949167\pi$
$504$	0	0
$505$	91584.0	0.359118
$506$	1.03014e6i	4.02341i
$507$	0	0
$508$	−261364.	−1.01279
$509$	493567.i	1.90507i	0.304430	+	0.952535i	$0.401534\pi$
$509$	493567.i	1.90507i	−0.304430	+	0.952535i	$0.598466\pi$
$510$	0	0
$511$	−153425.	−0.587563
$512$	− 528502.i	− 2.01607i
$513$	0	0
$514$	128520.	0.486457
$515$	240809.i	0.907944i
$516$	0	0
$517$	−254232.	−0.951150
$518$	− 147036.i	− 0.547978i
$519$	0	0
$520$	225720.	0.834763
$521$	− 101365.i	− 0.373432i	−0.982414	−	0.186716i	$-0.940216\pi$
$521$	− 101365.i	− 0.373432i	0.982414	−	0.186716i	$-0.0597844\pi$
$522$	0	0
$523$	38945.0	0.142380	0.0711899	−	0.997463i	$-0.477320\pi$
$523$	38945.0	0.142380	0.0711899	+	0.997463i	$0.477320\pi$
$524$	− 233446.i	− 0.850206i
$525$	0	0
$526$	−665712.	−2.40611
$527$	293204.i	1.05572i
$528$	0	0
$529$	−472055.	−1.68687
$530$	− 314279.i	− 1.11883i
$531$	0	0
$532$	−135014.	−0.477041
$533$	− 203847.i	− 0.717545i
$534$	0	0
$535$	106920.	0.373552
$536$	− 565347.i	− 1.96782i
$537$	0	0
$538$	893268.	3.08615
$539$	− 341439.i	− 1.17527i
$540$	0	0
$541$	55271.0	0.188844	0.0944219	−	0.995532i	$-0.469900\pi$
$541$	55271.0	0.188844	0.0944219	+	0.995532i	$0.469900\pi$
$542$	− 540252.i	− 1.83907i
$543$	0	0
$544$	−517104.	−1.74735
$545$	28600.2i	0.0962890i
$546$	0	0
$547$	297449.	0.994118	0.497059	−	0.867717i	$-0.334413\pi$
$547$	297449.	0.994118	0.497059	+	0.867717i	$0.334413\pi$
$548$	− 73161.4i	− 0.243624i
$549$	0	0
$550$	−485892.	−1.60625
$551$	− 67576.5i	− 0.222583i
$552$	0	0
$553$	89641.0	0.293127
$554$	1.02768e6i	3.34842i
$555$	0	0
$556$	−381862.	−1.23526
$557$	175437.i	0.565473i	0.959198	+	0.282736i	$0.0912422\pi$
$557$	175437.i	0.565473i	−0.959198	+	0.282736i	$0.908758\pi$
$558$	0	0
$559$	135850.	0.434746
$560$	144912.i	0.462091i
$561$	0	0
$562$	886248.	2.80597
$563$	− 324538.i	− 1.02388i	−0.859022	−	0.511939i	$-0.828927\pi$
$563$	− 324538.i	− 1.02388i	0.859022	−	0.511939i	$-0.171073\pi$
$564$	0	0
$565$	−225720.	−0.707087
$566$	388984.i	1.21422i
$567$	0	0
$568$	−313632.	−0.972129
$569$	137901.i	0.425936i	0.977059	+	0.212968i	$0.0683130\pi$
$569$	137901.i	0.425936i	−0.977059	+	0.212968i	$0.931687\pi$
$570$	0	0
$571$	−412015.	−1.26369	−0.631845	−	0.775094i	$-0.717702\pi$
$571$	−412015.	−1.26369	−0.631845	+	0.775094i	$0.717702\pi$
$572$	− 583615.i	− 1.78375i
$573$	0	0
$574$	268056.	0.813583
$575$	− 354652.i	− 1.07267i
$576$	0	0
$577$	−108097.	−0.324685	−0.162342	−	0.986734i	$-0.551905\pi$
$577$	−108097.	−0.324685	−0.162342	+	0.986734i	$0.551905\pi$
$578$	− 86234.3i	− 0.258122i
$579$	0	0
$580$	−180576.	−0.536790
$581$	− 104436.i	− 0.309385i
$582$	0	0
$583$	−470448.	−1.38412
$584$	1.45904e6i	4.27800i
$585$	0	0
$586$	−654588.	−1.90622
$587$	− 380430.i	− 1.10408i	−0.833819	−	0.552038i	$-0.813850\pi$
$587$	− 380430.i	− 1.10408i	0.833819	−	0.552038i	$-0.186150\pi$
$588$	0	0
$589$	198550.	0.572320
$590$	− 230154.i	− 0.661172i
$591$	0	0
$592$	−682660.	−1.94787
$593$	185975.i	0.528866i	0.964404	+	0.264433i	$0.0851847\pi$
$593$	185975.i	0.528866i	−0.964404	+	0.264433i	$0.914815\pi$
$594$	0	0
$595$	−77112.0	−0.217815
$596$	976229.i	2.74827i
$597$	0	0
$598$	605340.	1.69277
$599$	237179.i	0.661033i	0.943800	+	0.330516i	$0.107223\pi$
$599$	237179.i	0.661033i	−0.943800	+	0.330516i	$0.892777\pi$
$600$	0	0
$601$	198170.	0.548642	0.274321	−	0.961638i	$-0.411547\pi$
$601$	198170.	0.548642	0.274321	+	0.961638i	$0.411547\pi$
$602$	178641.i	0.492934i
$603$	0	0
$604$	494570.	1.35567
$605$	− 168941.i	− 0.461557i
$606$	0	0
$607$	−432175.	−1.17296	−0.586479	−	0.809965i	$-0.699486\pi$
$607$	−432175.	−1.17296	−0.586479	+	0.809965i	$0.699486\pi$
$608$	350169.i	0.947264i
$609$	0	0
$610$	155628.	0.418242
$611$	149394.i	0.400177i
$612$	0	0
$613$	107831.	0.286961	0.143480	−	0.989653i	$-0.454171\pi$
$613$	107831.	0.286961	0.143480	+	0.989653i	$0.454171\pi$
$614$	− 993351.i	− 2.63491i
$615$	0	0
$616$	444312.	1.17092
$617$	− 329961.i	− 0.866747i	−0.901214	−	0.433373i	$-0.857323\pi$
$617$	− 329961.i	− 0.866747i	0.901214	−	0.433373i	$-0.142677\pi$
$618$	0	0
$619$	63593.0	0.165969	0.0829847	−	0.996551i	$-0.473555\pi$
$619$	63593.0	0.165969	0.0829847	+	0.996551i	$0.473555\pi$
$620$	− 530559.i	− 1.38023i
$621$	0	0
$622$	1.17731e6	3.04305
$623$	− 189635.i	− 0.488587i
$624$	0	0
$625$	32281.0	0.0826394
$626$	− 461903.i	− 1.17870i
$627$	0	0
$628$	712196.	1.80584
$629$	− 363264.i	− 0.918166i
$630$	0	0
$631$	530777.	1.33307	0.666536	−	0.745473i	$-0.267776\pi$
$631$	530777.	1.33307	0.666536	+	0.745473i	$0.267776\pi$
$632$	− 852467.i	− 2.13424i
$633$	0	0
$634$	−360504.	−0.896874
$635$	101086.i	0.250693i
$636$	0	0
$637$	−200640.	−0.494469
$638$	384119.i	0.943680i
$639$	0	0
$640$	−66528.0	−0.162422
$641$	419686.i	1.02143i	0.859750	+	0.510715i	$0.170619\pi$
$641$	419686.i	1.02143i	−0.859750	+	0.510715i	$0.829381\pi$
$642$	0	0
$643$	−494266.	−1.19547	−0.597735	−	0.801694i	$-0.703933\pi$
$643$	−494266.	−1.19547	−0.597735	+	0.801694i	$0.703933\pi$
$644$	560159.i	1.35064i
$645$	0	0
$646$	−474012.	−1.13586
$647$	91268.0i	0.218027i	0.994040	+	0.109013i	$0.0347691\pi$
$647$	91268.0i	0.218027i	−0.994040	+	0.109013i	$0.965231\pi$
$648$	0	0
$649$	−344520.	−0.817947
$650$	285525.i	0.675798i
$651$	0	0
$652$	511898.	1.20417
$653$	− 595608.i	− 1.39680i	−0.715708	−	0.698400i	$-0.753896\pi$
$653$	− 595608.i	− 1.39680i	0.715708	−	0.698400i	$-0.246104\pi$
$654$	0	0
$655$	−90288.0	−0.210449
$656$	− 1.24454e6i	− 2.89201i
$657$	0	0
$658$	−196452.	−0.453737
$659$	452930.i	1.04294i	0.853269	+	0.521471i	$0.174617\pi$
$659$	452930.i	1.04294i	−0.853269	+	0.521471i	$0.825383\pi$
$660$	0	0
$661$	804695.	1.84174	0.920870	−	0.389869i	$-0.127480\pi$
$661$	804695.	1.84174	0.920870	+	0.389869i	$0.127480\pi$
$662$	600113.i	1.36936i
$663$	0	0
$664$	−993168.	−2.25261
$665$	52218.2i	0.118081i
$666$	0	0
$667$	−280368.	−0.630198
$668$	1.62798e6i	3.64835i
$669$	0	0
$670$	−377676.	−0.841337
$671$	− 232961.i	− 0.517414i
$672$	0	0
$673$	308807.	0.681800	0.340900	−	0.940100i	$-0.389268\pi$
$673$	308807.	0.681800	0.340900	+	0.940100i	$0.389268\pi$
$674$	− 1.07071e6i	− 2.35696i
$675$	0	0
$676$	742368.	1.62452
$677$	510381.i	1.11357i	0.830657	+	0.556784i	$0.187965\pi$
$677$	510381.i	1.11357i	−0.830657	+	0.556784i	$0.812035\pi$
$678$	0	0
$679$	−47753.0	−0.103576
$680$	733318.i	1.58590i
$681$	0	0
$682$	−1.12860e6	−2.42645
$683$	796633.i	1.70772i	0.520502	+	0.853860i	$0.325745\pi$
$683$	796633.i	1.70772i	−0.520502	+	0.853860i	$0.674255\pi$
$684$	0	0
$685$	−28296.0	−0.0603037
$686$	− 563782.i	− 1.19802i
$687$	0	0
$688$	829400.	1.75221
$689$	276449.i	0.582341i
$690$	0	0
$691$	759590.	1.59083	0.795414	−	0.606067i	$-0.207254\pi$
$691$	759590.	1.59083	0.795414	+	0.606067i	$0.207254\pi$
$692$	− 503752.i	− 1.05197i
$693$	0	0
$694$	−778032.	−1.61539
$695$	147690.i	0.305760i
$696$	0	0
$697$	662256.	1.36320
$698$	297870.i	0.611387i
$699$	0	0
$700$	−264214.	−0.539212
$701$	650913.i	1.32461i	0.749236	+	0.662303i	$0.230421\pi$
$701$	650913.i	1.32461i	−0.749236	+	0.662303i	$0.769579\pi$
$702$	0	0
$703$	−245993.	−0.497751
$704$	− 490172.i	− 0.989016i
$705$	0	0
$706$	−61776.0	−0.123940
$707$	− 105936.i	− 0.211935i
$708$	0	0
$709$	183455.	0.364953	0.182477	−	0.983210i	$-0.441589\pi$
$709$	183455.	0.364953	0.182477	+	0.983210i	$0.441589\pi$
$710$	209520.i	0.415631i
$711$	0	0
$712$	−1.80338e6	−3.55736
$713$	− 823763.i	− 1.62040i
$714$	0	0
$715$	−225720.	−0.441528
$716$	− 810918.i	− 1.58180i
$717$	0	0
$718$	−141588.	−0.274649
$719$	− 412249.i	− 0.797447i	−0.917071	−	0.398724i	$-0.869453\pi$
$719$	− 412249.i	− 0.797447i	0.917071	−	0.398724i	$-0.130547\pi$
$720$	0	0
$721$	278545.	0.535827
$722$	− 636671.i	− 1.22135i
$723$	0	0
$724$	−2.16870e6	−4.13735
$725$	− 132243.i	− 0.251592i
$726$	0	0
$727$	−446842.	−0.845445	−0.422722	−	0.906259i	$-0.638925\pi$
$727$	−446842.	−0.845445	−0.422722	+	0.906259i	$0.638925\pi$
$728$	− 261091.i	− 0.492640i
$729$	0	0
$730$	974700.	1.82905
$731$	441349.i	0.825938i
$732$	0	0
$733$	204314.	0.380268	0.190134	−	0.981758i	$-0.439108\pi$
$733$	204314.	0.380268	0.190134	+	0.981758i	$0.439108\pi$
$734$	− 879751.i	− 1.63293i
$735$	0	0
$736$	1.45282e6	2.68198
$737$	565347.i	1.04083i
$738$	0	0
$739$	140870.	0.257946	0.128973	−	0.991648i	$-0.458832\pi$
$739$	140870.	0.257946	0.128973	+	0.991648i	$0.458832\pi$
$740$	657335.i	1.20039i
$741$	0	0
$742$	−363528.	−0.660283
$743$	− 284547.i	− 0.515439i	−0.966220	−	0.257719i	$-0.917029\pi$
$743$	− 284547.i	− 0.515439i	0.966220	−	0.257719i	$-0.0829710\pi$
$744$	0	0
$745$	377568.	0.680272
$746$	9692.63i	0.0174166i
$747$	0	0
$748$	1.89605e6	3.38880
$749$	− 123675.i	− 0.220454i
$750$	0	0
$751$	−315823.	−0.559969	−0.279984	−	0.960005i	$-0.590329\pi$
$751$	−315823.	−0.559969	−0.279984	+	0.960005i	$0.590329\pi$
$752$	912092.i	1.61288i
$753$	0	0
$754$	225720.	0.397034
$755$	− 191281.i	− 0.335565i
$756$	0	0
$757$	−436561.	−0.761821	−0.380911	−	0.924612i	$-0.624389\pi$
$757$	−436561.	−0.761821	−0.380911	+	0.924612i	$0.624389\pi$
$758$	− 1.21167e6i	− 2.10885i
$759$	0	0
$760$	496584.	0.859737
$761$	120118.i	0.207414i	0.994608	+	0.103707i	$0.0330705\pi$
$761$	120118.i	0.207414i	−0.994608	+	0.103707i	$0.966929\pi$
$762$	0	0
$763$	33082.0	0.0568254
$764$	1.56543e6i	2.68193i
$765$	0	0
$766$	−216864.	−0.369598
$767$	202450.i	0.344134i
$768$	0	0
$769$	697775.	1.17995	0.589974	−	0.807422i	$-0.299138\pi$
$769$	697775.	1.17995	0.589974	+	0.807422i	$0.299138\pi$
$770$	− 296819.i	− 0.500623i
$771$	0	0
$772$	2.25906e6	3.79047
$773$	− 679969.i	− 1.13797i	−0.822349	−	0.568984i	$-0.807337\pi$
$773$	− 679969.i	− 1.13797i	0.822349	−	0.568984i	$-0.192663\pi$
$774$	0	0
$775$	388550.	0.646909
$776$	454121.i	0.754133i
$777$	0	0
$778$	−1.24146e6	−2.05104
$779$	− 448462.i	− 0.739011i
$780$	0	0
$781$	313632.	0.514184
$782$	1.96663e6i	3.21594i
$783$	0	0
$784$	−1.22496e6	−1.99292
$785$	− 275450.i	− 0.446996i
$786$	0	0
$787$	−765823.	−1.23646	−0.618228	−	0.785999i	$-0.712149\pi$
$787$	−765823.	−1.23646	−0.618228	+	0.785999i	$0.712149\pi$
$788$	745576.i	1.20071i
$789$	0	0
$790$	−569484.	−0.912488
$791$	261091.i	0.417291i
$792$	0	0
$793$	−136895.	−0.217691
$794$	156096.i	0.247600i
$795$	0	0
$796$	654170.	1.03244
$797$	621916.i	0.979072i	0.871983	+	0.489536i	$0.162834\pi$
$797$	621916.i	0.979072i	−0.871983	+	0.489536i	$0.837166\pi$
$798$	0	0
$799$	−485352.	−0.760262
$800$	685259.i	1.07072i
$801$	0	0
$802$	804384.	1.25059
$803$	− 1.45904e6i	− 2.26275i
$804$	0	0
$805$	216648.	0.334320
$806$	663199.i	1.02088i
$807$	0	0
$808$	−1.00742e6	−1.54308
$809$	− 611099.i	− 0.933715i	−0.884333	−	0.466857i	$-0.845386\pi$
$809$	− 611099.i	− 0.933715i	0.884333	−	0.466857i	$-0.154614\pi$
$810$	0	0
$811$	−1.19631e6	−1.81888	−0.909439	−	0.415838i	$-0.863488\pi$
$811$	−1.19631e6	−1.81888	−0.909439	+	0.415838i	$0.863488\pi$
$812$	208873.i	0.316789i
$813$	0	0
$814$	1.39828e6	2.11030
$815$	− 197982.i	− 0.298065i
$816$	0	0
$817$	298870.	0.447753
$818$	− 2.22412e6i	− 3.32393i
$819$	0	0
$820$	−1.19837e6	−1.78222
$821$	635863.i	0.943360i	0.881770	+	0.471680i	$0.156352\pi$
$821$	635863.i	0.943360i	−0.881770	+	0.471680i	$0.843648\pi$
$822$	0	0
$823$	573089.	0.846101	0.423051	−	0.906106i	$-0.360959\pi$
$823$	573089.	0.846101	0.423051	+	0.906106i	$0.360959\pi$
$824$	− 2.64890e6i	− 3.90132i
$825$	0	0
$826$	−266220.	−0.390194
$827$	− 702852.i	− 1.02767i	−0.857890	−	0.513834i	$-0.828225\pi$
$827$	− 702852.i	− 1.02767i	0.857890	−	0.513834i	$-0.171775\pi$
$828$	0	0
$829$	−590281.	−0.858914	−0.429457	−	0.903087i	$-0.641295\pi$
$829$	−590281.	−0.858914	−0.429457	+	0.903087i	$0.641295\pi$
$830$	663479.i	0.963099i
$831$	0	0
$832$	−288040.	−0.416108
$833$	− 651839.i	− 0.939399i
$834$	0	0
$835$	629640.	0.903066
$836$	− 1.28395e6i	− 1.83712i
$837$	0	0
$838$	2.20180e6	3.13537
$839$	617124.i	0.876696i	0.898805	+	0.438348i	$0.144436\pi$
$839$	617124.i	0.876696i	−0.898805	+	0.438348i	$0.855564\pi$
$840$	0	0
$841$	602737.	0.852189
$842$	1.43647e6i	2.02616i
$843$	0	0
$844$	−770374.	−1.08148
$845$	− 287119.i	− 0.402114i
$846$	0	0
$847$	−195415.	−0.272390
$848$	1.68780e6i	2.34708i
$849$	0	0
$850$	−927612.	−1.28389
$851$	1.02060e6i	1.40928i
$852$	0	0
$853$	−1.07179e6	−1.47303	−0.736517	−	0.676419i	$-0.763531\pi$
$853$	−1.07179e6	−1.47303	−0.736517	+	0.676419i	$0.763531\pi$
$854$	− 180015.i	− 0.246828i
$855$	0	0
$856$	−1.17612e6	−1.60511
$857$	− 78187.7i	− 0.106458i	−0.998582	−	0.0532288i	$-0.983049\pi$
$857$	− 78187.7i	− 0.106458i	0.998582	−	0.0532288i	$-0.0169513\pi$
$858$	0	0
$859$	−1.19862e6	−1.62440	−0.812201	−	0.583378i	$-0.801731\pi$
$859$	−1.19862e6	−1.62440	−0.812201	+	0.583378i	$0.801731\pi$
$860$	− 798632.i	− 1.07982i
$861$	0	0
$862$	−267300.	−0.359736
$863$	705673.i	0.947507i	0.880658	+	0.473753i	$0.157101\pi$
$863$	705673.i	0.947507i	−0.880658	+	0.473753i	$0.842899\pi$
$864$	0	0
$865$	−194832.	−0.260392
$866$	− 1.27139e6i	− 1.69528i
$867$	0	0
$868$	−613700.	−0.814548
$869$	852467.i	1.12885i
$870$	0	0
$871$	332215.	0.437908
$872$	− 314603.i	− 0.413742i
$873$	0	0
$874$	1.33175e6	1.74341
$875$	258343.i	0.337427i
$876$	0	0
$877$	−639265.	−0.831154	−0.415577	−	0.909558i	$-0.636420\pi$
$877$	−639265.	−0.831154	−0.415577	+	0.909558i	$0.636420\pi$
$878$	847528.i	1.09942i
$879$	0	0
$880$	−1.37808e6	−1.77955
$881$	− 230110.i	− 0.296472i	−0.988952	−	0.148236i	$-0.952640\pi$
$881$	− 230110.i	− 0.296472i	0.988952	−	0.148236i	$-0.0473595\pi$
$882$	0	0
$883$	711929.	0.913094	0.456547	−	0.889699i	$-0.349086\pi$
$883$	711929.	0.913094	0.456547	+	0.889699i	$0.349086\pi$
$884$	− 1.11417e6i	− 1.42577i
$885$	0	0
$886$	69768.0	0.0888769
$887$	1.11353e6i	1.41532i	0.706554	+	0.707659i	$0.250249\pi$
$887$	1.11353e6i	1.41532i	−0.706554	+	0.707659i	$0.749751\pi$
$888$	0	0
$889$	116926.	0.147947
$890$	1.20474e6i	1.52094i
$891$	0	0
$892$	1.82256e6	2.29061
$893$	328668.i	0.412149i
$894$	0	0
$895$	−313632.	−0.391538
$896$	76953.2i	0.0958541i
$897$	0	0
$898$	1.21468e6	1.50629
$899$	− 307166.i	− 0.380061i
$900$	0	0
$901$	−898128.	−1.10634
$902$	2.54915e6i	3.13316i
$903$	0	0
$904$	2.48292e6	3.03827
$905$	838769.i	1.02411i
$906$	0	0
$907$	762641.	0.927055	0.463528	−	0.886082i	$-0.346583\pi$
$907$	762641.	0.927055	0.463528	+	0.886082i	$0.346583\pi$
$908$	49146.6i	0.0596103i
$909$	0	0
$910$	−174420.	−0.210627
$911$	− 927641.i	− 1.11775i	−0.829253	−	0.558873i	$-0.811234\pi$
$911$	− 927641.i	− 1.11775i	0.829253	−	0.558873i	$-0.188766\pi$
$912$	0	0
$913$	993168.	1.19146
$914$	− 1.87808e6i	− 2.24813i
$915$	0	0
$916$	−932140.	−1.11094
$917$	104436.i	0.124198i
$918$	0	0
$919$	615062.	0.728262	0.364131	−	0.931348i	$-0.381366\pi$
$919$	615062.	0.728262	0.364131	+	0.931348i	$0.381366\pi$
$920$	− 2.06028e6i	− 2.43416i
$921$	0	0
$922$	1.72379e6	2.02779
$923$	− 184300.i	− 0.216332i
$924$	0	0
$925$	−481393.	−0.562621
$926$	1.14525e6i	1.33561i
$927$	0	0
$928$	541728.	0.629050
$929$	324626.i	0.376142i	0.982155	+	0.188071i	$0.0602235\pi$
$929$	324626.i	0.376142i	−0.982155	+	0.188071i	$0.939776\pi$
$930$	0	0
$931$	−441408.	−0.509261
$932$	− 3.90715e6i	− 4.49809i
$933$	0	0
$934$	−388476.	−0.445318
$935$	− 733318.i	− 0.838821i
$936$	0	0
$937$	−176425.	−0.200947	−0.100473	−	0.994940i	$-0.532036\pi$
$937$	−176425.	−0.200947	−0.100473	+	0.994940i	$0.532036\pi$
$938$	436859.i	0.496519i
$939$	0	0
$940$	878256.	0.993952
$941$	10508.3i	0.0118673i	0.999982	+	0.00593367i	$0.00188876\pi$
$941$	10508.3i	0.0118673i	−0.999982	+	0.00593367i	$0.998111\pi$
$942$	0	0
$943$	−1.86062e6	−2.09235
$944$	1.23601e6i	1.38701i
$945$	0	0
$946$	−1.69884e6	−1.89832
$947$	123351.i	0.137545i	0.997632	+	0.0687724i	$0.0219082\pi$
$947$	123351.i	0.137545i	−0.997632	+	0.0687724i	$0.978092\pi$
$948$	0	0
$949$	−857375.	−0.952003
$950$	628154.i	0.696016i
$951$	0	0
$952$	848232.	0.935924
$953$	1.01158e6i	1.11381i	0.830575	+	0.556907i	$0.188012\pi$
$953$	1.01158e6i	1.11381i	−0.830575	+	0.556907i	$0.811988\pi$
$954$	0	0
$955$	605448.	0.663850
$956$	− 2.77678e6i	− 3.03827i
$957$	0	0
$958$	−640440.	−0.697826
$959$	32730.1i	0.0355885i
$960$	0	0
$961$	−21021.0	−0.0227618
$962$	− 821669.i	− 0.887865i
$963$	0	0
$964$	−1.47467e6	−1.58686
$965$	− 873718.i	− 0.938246i
$966$	0	0
$967$	1.59902e6	1.71002	0.855008	−	0.518615i	$-0.173552\pi$
$967$	1.59902e6	1.71002	0.855008	+	0.518615i	$0.173552\pi$
$968$	1.85835e6i	1.98325i
$969$	0	0
$970$	303372.	0.322427
$971$	1.21148e6i	1.28493i	0.766316	+	0.642464i	$0.222088\pi$
$971$	1.21148e6i	1.28493i	−0.766316	+	0.642464i	$0.777912\pi$
$972$	0	0
$973$	170833.	0.180446
$974$	800990.i	0.844325i
$975$	0	0
$976$	−835780.	−0.877389
$977$	− 1.10353e6i	− 1.15610i	−0.816000	−	0.578052i	$-0.803813\pi$
$977$	− 1.10353e6i	− 1.15610i	0.816000	−	0.578052i	$-0.196187\pi$
$978$	0	0
$979$	1.80338e6	1.88158
$980$	1.17952e6i	1.22815i
$981$	0	0
$982$	−1.40562e6	−1.45762
$983$	− 881566.i	− 0.912322i	−0.889897	−	0.456161i	$-0.849224\pi$
$983$	− 881566.i	− 0.912322i	0.889897	−	0.456161i	$-0.150776\pi$
$984$	0	0
$985$	288360.	0.297209
$986$	733318.i	0.754291i
$987$	0	0
$988$	−754490.	−0.772929
$989$	− 1.23998e6i	− 1.26772i
$990$	0	0
$991$	750377.	0.764068	0.382034	−	0.924148i	$-0.375224\pi$
$991$	750377.	0.764068	0.382034	+	0.924148i	$0.375224\pi$
$992$	1.59168e6i	1.61745i
$993$	0	0
$994$	242352.	0.245287
$995$	− 253008.i	− 0.255557i
$996$	0	0
$997$	−208918.	−0.210177	−0.105089	−	0.994463i	$-0.533513\pi$
$997$	−208918.	−0.210177	−0.105089	+	0.994463i	$0.533513\pi$
$998$	317557.i	0.318831i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.5.b.b.26.2	yes	2
3.2	odd	2	inner	27.5.b.b.26.1	✓	2
4.3	odd	2		432.5.e.c.161.2		2
5.2	odd	4		675.5.d.g.674.2		4
5.3	odd	4		675.5.d.g.674.3		4
5.4	even	2		675.5.c.d.26.1		2
9.2	odd	6		81.5.d.d.53.1		4
9.4	even	3		81.5.d.d.26.1		4
9.5	odd	6		81.5.d.d.26.2		4
9.7	even	3		81.5.d.d.53.2		4
12.11	even	2		432.5.e.c.161.1		2
15.2	even	4		675.5.d.g.674.4		4
15.8	even	4		675.5.d.g.674.1		4
15.14	odd	2		675.5.c.d.26.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.b.b.26.1	✓	2	3.2	odd	2	inner
27.5.b.b.26.2	yes	2	1.1	even	1	trivial
81.5.d.d.26.1		4	9.4	even	3
81.5.d.d.26.2		4	9.5	odd	6
81.5.d.d.53.1		4	9.2	odd	6
81.5.d.d.53.2		4	9.7	even	3
432.5.e.c.161.1		2	12.11	even	2
432.5.e.c.161.2		2	4.3	odd	2
675.5.c.d.26.1		2	5.4	even	2
675.5.c.d.26.2		2	15.14	odd	2
675.5.d.g.674.1		4	15.8	even	4
675.5.d.g.674.2		4	5.2	odd	4
675.5.d.g.674.3		4	5.3	odd	4
675.5.d.g.674.4		4	15.2	even	4

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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	27.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.34847i q^{2} -38.0000 q^{4} +14.6969i q^{5} +17.0000 q^{7} -161.666i q^{8} +O(q^{10})\)	\(q+7.34847i q^{2} -38.0000 q^{4} +14.6969i q^{5} +17.0000 q^{7} -161.666i q^{8} -108.000 q^{10} +161.666i q^{11} +95.0000 q^{13} +124.924i q^{14} +580.000 q^{16} +308.636i q^{17} +209.000 q^{19} -558.484i q^{20} -1188.00 q^{22} -867.119i q^{23} +409.000 q^{25} +698.105i q^{26} -646.000 q^{28} -323.333i q^{29} +950.000 q^{31} +1675.45i q^{32} -2268.00 q^{34} +249.848i q^{35} -1177.00 q^{37} +1535.83i q^{38} +2376.00 q^{40} -2145.75i q^{41} +1430.00 q^{43} -6143.32i q^{44} +6372.00 q^{46} +1572.57i q^{47} -2112.00 q^{49} +3005.52i q^{50} -3610.00 q^{52} +2909.99i q^{53} -2376.00 q^{55} -2748.33i q^{56} +2376.00 q^{58} +2131.06i q^{59} -1441.00 q^{61} +6981.05i q^{62} -3032.00 q^{64} +1396.21i q^{65} +3497.00 q^{67} -11728.2i q^{68} -1836.00 q^{70} -1940.00i q^{71} -9025.00 q^{73} -8649.15i q^{74} -7942.00 q^{76} +2748.33i q^{77} +5273.00 q^{79} +8524.22i q^{80} +15768.0 q^{82} -6143.32i q^{83} -4536.00 q^{85} +10508.3i q^{86} +26136.0 q^{88} -11155.0i q^{89} +1615.00 q^{91} +32950.5i q^{92} -11556.0 q^{94} +3071.66i q^{95} -2809.00 q^{97} -15520.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 76 q^{4} + 34 q^{7}+O(q^{10}) \) 2 * q - 76 * q^4 + 34 * q^7	\( 2 q - 76 q^{4} + 34 q^{7} - 216 q^{10} + 190 q^{13} + 1160 q^{16} + 418 q^{19} - 2376 q^{22} + 818 q^{25} - 1292 q^{28} + 1900 q^{31} - 4536 q^{34} - 2354 q^{37} + 4752 q^{40} + 2860 q^{43} + 12744 q^{46} - 4224 q^{49} - 7220 q^{52} - 4752 q^{55} + 4752 q^{58} - 2882 q^{61} - 6064 q^{64} + 6994 q^{67} - 3672 q^{70} - 18050 q^{73} - 15884 q^{76} + 10546 q^{79} + 31536 q^{82} - 9072 q^{85} + 52272 q^{88} + 3230 q^{91} - 23112 q^{94} - 5618 q^{97}+O(q^{100}) \) 2 * q - 76 * q^4 + 34 * q^7 - 216 * q^10 + 190 * q^13 + 1160 * q^16 + 418 * q^19 - 2376 * q^22 + 818 * q^25 - 1292 * q^28 + 1900 * q^31 - 4536 * q^34 - 2354 * q^37 + 4752 * q^40 + 2860 * q^43 + 12744 * q^46 - 4224 * q^49 - 7220 * q^52 - 4752 * q^55 + 4752 * q^58 - 2882 * q^61 - 6064 * q^64 + 6994 * q^67 - 3672 * q^70 - 18050 * q^73 - 15884 * q^76 + 10546 * q^79 + 31536 * q^82 - 9072 * q^85 + 52272 * q^88 + 3230 * q^91 - 23112 * q^94 - 5618 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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