LMFDB - Embedded newform 8379.2.a.cs.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8379,2,Mod(1,8379)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8379, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8379.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.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:	$66.9066518536$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 57x^{6} - 98x^{5} - 93x^{4} + 152x^{3} + 39x^{2} - 58x - 7 $ x^10 - 2x^9 - 13x^8 + 24x^7 + 57x^6 - 98x^5 - 93x^4 + 152x^3 + 39x^2 - 58*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 931)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.20114$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.20114 q^{2} +2.84501 q^{4} -4.23731 q^{5} +1.86000 q^{8} +O(q^{10})$	$q+2.20114 q^{2} +2.84501 q^{4} -4.23731 q^{5} +1.86000 q^{8} -9.32690 q^{10} +4.54922 q^{11} +0.667049 q^{13} -1.59592 q^{16} +1.86633 q^{17} +1.00000 q^{19} -12.0552 q^{20} +10.0135 q^{22} -9.29601 q^{23} +12.9548 q^{25} +1.46827 q^{26} +1.31070 q^{29} -3.90360 q^{31} -7.23283 q^{32} +4.10806 q^{34} +3.00492 q^{37} +2.20114 q^{38} -7.88137 q^{40} -5.48606 q^{41} +4.44567 q^{43} +12.9426 q^{44} -20.4618 q^{46} -5.81459 q^{47} +28.5152 q^{50} +1.89777 q^{52} +9.95130 q^{53} -19.2764 q^{55} +2.88503 q^{58} +2.07940 q^{59} -10.1390 q^{61} -8.59238 q^{62} -12.7286 q^{64} -2.82649 q^{65} -0.523489 q^{67} +5.30974 q^{68} +4.03304 q^{71} -5.38666 q^{73} +6.61425 q^{74} +2.84501 q^{76} -2.64530 q^{79} +6.76240 q^{80} -12.0756 q^{82} +2.83415 q^{83} -7.90822 q^{85} +9.78554 q^{86} +8.46152 q^{88} -14.0986 q^{89} -26.4473 q^{92} -12.7987 q^{94} -4.23731 q^{95} +3.71220 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 10 q^{4} - 16 q^{5} + 6 q^{8}+O(q^{10}) $ 10 * q + 2 * q^2 + 10 * q^4 - 16 * q^5 + 6 * q^8	$ 10 q + 2 q^{2} + 10 q^{4} - 16 q^{5} + 6 q^{8} - 12 q^{10} + 12 q^{13} + 2 q^{16} - 16 q^{17} + 10 q^{19} - 32 q^{20} + 4 q^{22} - 12 q^{23} + 14 q^{25} - 24 q^{26} + 12 q^{29} + 8 q^{31} + 34 q^{32} + 16 q^{34} + 4 q^{37} + 2 q^{38} - 20 q^{40} - 40 q^{41} + 4 q^{43} + 20 q^{44} - 32 q^{46} - 16 q^{47} + 34 q^{50} - 40 q^{52} + 16 q^{55} - 8 q^{58} - 36 q^{59} + 16 q^{61} + 16 q^{62} + 18 q^{64} - 8 q^{65} - 28 q^{67} - 40 q^{68} + 12 q^{71} - 24 q^{74} + 10 q^{76} - 8 q^{79} - 8 q^{80} - 8 q^{82} + 40 q^{85} + 52 q^{86} - 4 q^{88} - 48 q^{89} - 28 q^{92} - 36 q^{94} - 16 q^{95} - 16 q^{97}+O(q^{100}) $ 10 * q + 2 * q^2 + 10 * q^4 - 16 * q^5 + 6 * q^8 - 12 * q^10 + 12 * q^13 + 2 * q^16 - 16 * q^17 + 10 * q^19 - 32 * q^20 + 4 * q^22 - 12 * q^23 + 14 * q^25 - 24 * q^26 + 12 * q^29 + 8 * q^31 + 34 * q^32 + 16 * q^34 + 4 * q^37 + 2 * q^38 - 20 * q^40 - 40 * q^41 + 4 * q^43 + 20 * q^44 - 32 * q^46 - 16 * q^47 + 34 * q^50 - 40 * q^52 + 16 * q^55 - 8 * q^58 - 36 * q^59 + 16 * q^61 + 16 * q^62 + 18 * q^64 - 8 * q^65 - 28 * q^67 - 40 * q^68 + 12 * q^71 - 24 * q^74 + 10 * q^76 - 8 * q^79 - 8 * q^80 - 8 * q^82 + 40 * q^85 + 52 * q^86 - 4 * q^88 - 48 * q^89 - 28 * q^92 - 36 * q^94 - 16 * q^95 - 16 * 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 $

$2$	2.20114	1.55644	0.778220	−	0.627991i	$-0.216123\pi$
$2$	2.20114	1.55644	0.778220	+	0.627991i	$0.216123\pi$
$3$	0	0
$3$	0	0
$4$	2.84501	1.42251
$5$	−4.23731	−1.89498	−0.947490	−	0.319784i	$-0.896390\pi$
$5$	−4.23731	−1.89498	−0.947490	+	0.319784i	$0.896390\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.86000	0.657608
$9$	0	0
$10$	−9.32690	−2.94942
$11$	4.54922	1.37164	0.685821	−	0.727771i	$-0.259444\pi$
$11$	4.54922	1.37164	0.685821	+	0.727771i	$0.259444\pi$
$12$	0	0
$13$	0.667049	0.185006	0.0925031	−	0.995712i	$-0.470513\pi$
$13$	0.667049	0.185006	0.0925031	+	0.995712i	$0.470513\pi$
$14$	0	0
$15$	0	0
$16$	−1.59592	−0.398980
$17$	1.86633	0.452652	0.226326	−	0.974052i	$-0.427329\pi$
$17$	1.86633	0.452652	0.226326	+	0.974052i	$0.427329\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−12.0552	−2.69562
$21$	0	0
$22$	10.0135	2.13488
$23$	−9.29601	−1.93835	−0.969176	−	0.246369i	$-0.920763\pi$
$23$	−9.29601	−1.93835	−0.969176	+	0.246369i	$0.920763\pi$
$24$	0	0
$25$	12.9548	2.59095
$26$	1.46827	0.287951
$27$	0	0
$28$	0	0
$29$	1.31070	0.243390	0.121695	−	0.992568i	$-0.461167\pi$
$29$	1.31070	0.243390	0.121695	+	0.992568i	$0.461167\pi$
$30$	0	0
$31$	−3.90360	−0.701108	−0.350554	−	0.936542i	$-0.614007\pi$
$31$	−3.90360	−0.701108	−0.350554	+	0.936542i	$0.614007\pi$
$32$	−7.23283	−1.27860
$33$	0	0
$34$	4.10806	0.704526
$35$	0	0
$36$	0	0
$37$	3.00492	0.494006	0.247003	−	0.969015i	$-0.420554\pi$
$37$	3.00492	0.494006	0.247003	+	0.969015i	$0.420554\pi$
$38$	2.20114	0.357072
$39$	0	0
$40$	−7.88137	−1.24615
$41$	−5.48606	−0.856779	−0.428390	−	0.903594i	$-0.640919\pi$
$41$	−5.48606	−0.856779	−0.428390	+	0.903594i	$0.640919\pi$
$42$	0	0
$43$	4.44567	0.677958	0.338979	−	0.940794i	$-0.389918\pi$
$43$	4.44567	0.677958	0.338979	+	0.940794i	$0.389918\pi$
$44$	12.9426	1.95117
$45$	0	0
$46$	−20.4618	−3.01693
$47$	−5.81459	−0.848146	−0.424073	−	0.905628i	$-0.639400\pi$
$47$	−5.81459	−0.848146	−0.424073	+	0.905628i	$0.639400\pi$
$48$	0	0
$49$	0	0
$50$	28.5152	4.03266
$51$	0	0
$52$	1.89777	0.263173
$53$	9.95130	1.36692	0.683458	−	0.729990i	$-0.260475\pi$
$53$	9.95130	1.36692	0.683458	+	0.729990i	$0.260475\pi$
$54$	0	0
$55$	−19.2764	−2.59923
$56$	0	0
$57$	0	0
$58$	2.88503	0.378823
$59$	2.07940	0.270714	0.135357	−	0.990797i	$-0.456782\pi$
$59$	2.07940	0.270714	0.135357	+	0.990797i	$0.456782\pi$
$60$	0	0
$61$	−10.1390	−1.29816	−0.649080	−	0.760720i	$-0.724846\pi$
$61$	−10.1390	−1.29816	−0.649080	+	0.760720i	$0.724846\pi$
$62$	−8.59238	−1.09123
$63$	0	0
$64$	−12.7286	−1.59108
$65$	−2.82649	−0.350583
$66$	0	0
$67$	−0.523489	−0.0639544	−0.0319772	−	0.999489i	$-0.510180\pi$
$67$	−0.523489	−0.0639544	−0.0319772	+	0.999489i	$0.510180\pi$
$68$	5.30974	0.643901
$69$	0	0
$70$	0	0
$71$	4.03304	0.478634	0.239317	−	0.970942i	$-0.423077\pi$
$71$	4.03304	0.478634	0.239317	+	0.970942i	$0.423077\pi$
$72$	0	0
$73$	−5.38666	−0.630461	−0.315230	−	0.949015i	$-0.602082\pi$
$73$	−5.38666	−0.630461	−0.315230	+	0.949015i	$0.602082\pi$
$74$	6.61425	0.768891
$75$	0	0
$76$	2.84501	0.326346
$77$	0	0
$78$	0	0
$79$	−2.64530	−0.297619	−0.148810	−	0.988866i	$-0.547544\pi$
$79$	−2.64530	−0.297619	−0.148810	+	0.988866i	$0.547544\pi$
$80$	6.76240	0.756060
$81$	0	0
$82$	−12.0756	−1.33353
$83$	2.83415	0.311088	0.155544	−	0.987829i	$-0.450287\pi$
$83$	2.83415	0.311088	0.155544	+	0.987829i	$0.450287\pi$
$84$	0	0
$85$	−7.90822	−0.857767
$86$	9.78554	1.05520
$87$	0	0
$88$	8.46152	0.902002
$89$	−14.0986	−1.49445	−0.747225	−	0.664572i	$-0.768614\pi$
$89$	−14.0986	−1.49445	−0.747225	+	0.664572i	$0.768614\pi$
$90$	0	0
$91$	0	0
$92$	−26.4473	−2.75732
$93$	0	0
$94$	−12.7987	−1.32009
$95$	−4.23731	−0.434738
$96$	0	0
$97$	3.71220	0.376917	0.188459	−	0.982081i	$-0.439651\pi$
$97$	3.71220	0.376917	0.188459	+	0.982081i	$0.439651\pi$
$98$	0	0
$99$	0	0
$100$	36.8565	3.68565
$101$	−3.14167	−0.312608	−0.156304	−	0.987709i	$-0.549958\pi$
$101$	−3.14167	−0.312608	−0.156304	+	0.987709i	$0.549958\pi$
$102$	0	0
$103$	−11.4811	−1.13127	−0.565633	−	0.824657i	$-0.691368\pi$
$103$	−11.4811	−1.13127	−0.565633	+	0.824657i	$0.691368\pi$
$104$	1.24071	0.121662
$105$	0	0
$106$	21.9042	2.12752
$107$	−2.13866	−0.206752	−0.103376	−	0.994642i	$-0.532965\pi$
$107$	−2.13866	−0.206752	−0.103376	+	0.994642i	$0.532965\pi$
$108$	0	0
$109$	−7.02420	−0.672796	−0.336398	−	0.941720i	$-0.609209\pi$
$109$	−7.02420	−0.672796	−0.336398	+	0.941720i	$0.609209\pi$
$110$	−42.4301	−4.04555
$111$	0	0
$112$	0	0
$113$	−6.54272	−0.615487	−0.307744	−	0.951469i	$-0.599574\pi$
$113$	−6.54272	−0.615487	−0.307744	+	0.951469i	$0.599574\pi$
$114$	0	0
$115$	39.3900	3.67314
$116$	3.72895	0.346225
$117$	0	0
$118$	4.57704	0.421351
$119$	0	0
$120$	0	0
$121$	9.69539	0.881399
$122$	−22.3173	−2.02051
$123$	0	0
$124$	−11.1058	−0.997332
$125$	−33.7067	−3.01482
$126$	0	0
$127$	−19.8356	−1.76012	−0.880061	−	0.474860i	$-0.842499\pi$
$127$	−19.8356	−1.76012	−0.880061	+	0.474860i	$0.842499\pi$
$128$	−13.5518	−1.19782
$129$	0	0
$130$	−6.22150	−0.545662
$131$	10.7068	0.935456	0.467728	−	0.883872i	$-0.345073\pi$
$131$	10.7068	0.935456	0.467728	+	0.883872i	$0.345073\pi$
$132$	0	0
$133$	0	0
$134$	−1.15227	−0.0995412
$135$	0	0
$136$	3.47137	0.297667
$137$	−18.2869	−1.56235	−0.781177	−	0.624310i	$-0.785380\pi$
$137$	−18.2869	−1.56235	−0.781177	+	0.624310i	$0.785380\pi$
$138$	0	0
$139$	−6.71739	−0.569762	−0.284881	−	0.958563i	$-0.591954\pi$
$139$	−6.71739	−0.569762	−0.284881	+	0.958563i	$0.591954\pi$
$140$	0	0
$141$	0	0
$142$	8.87728	0.744965
$143$	3.03455	0.253762
$144$	0	0
$145$	−5.55383	−0.461220
$146$	−11.8568	−0.981275
$147$	0	0
$148$	8.54905	0.702727
$149$	−15.9572	−1.30727	−0.653633	−	0.756812i	$-0.726756\pi$
$149$	−15.9572	−1.30727	−0.653633	+	0.756812i	$0.726756\pi$
$150$	0	0
$151$	−19.1129	−1.55539	−0.777694	−	0.628643i	$-0.783611\pi$
$151$	−19.1129	−1.55539	−0.777694	+	0.628643i	$0.783611\pi$
$152$	1.86000	0.150866
$153$	0	0
$154$	0	0
$155$	16.5408	1.32859
$156$	0	0
$157$	3.10148	0.247525	0.123763	−	0.992312i	$-0.460504\pi$
$157$	3.10148	0.247525	0.123763	+	0.992312i	$0.460504\pi$
$158$	−5.82267	−0.463226
$159$	0	0
$160$	30.6477	2.42292
$161$	0	0
$162$	0	0
$163$	−4.45275	−0.348766	−0.174383	−	0.984678i	$-0.555793\pi$
$163$	−4.45275	−0.348766	−0.174383	+	0.984678i	$0.555793\pi$
$164$	−15.6079	−1.21877
$165$	0	0
$166$	6.23835	0.484190
$167$	18.9814	1.46883	0.734413	−	0.678703i	$-0.237458\pi$
$167$	18.9814	1.46883	0.734413	+	0.678703i	$0.237458\pi$
$168$	0	0
$169$	−12.5550	−0.965773
$170$	−17.4071	−1.33506
$171$	0	0
$172$	12.6480	0.964400
$173$	4.37027	0.332266	0.166133	−	0.986103i	$-0.446872\pi$
$173$	4.37027	0.332266	0.166133	+	0.986103i	$0.446872\pi$
$174$	0	0
$175$	0	0
$176$	−7.26019	−0.547258
$177$	0	0
$178$	−31.0330	−2.32602
$179$	5.77456	0.431611	0.215806	−	0.976436i	$-0.430762\pi$
$179$	5.77456	0.431611	0.215806	+	0.976436i	$0.430762\pi$
$180$	0	0
$181$	8.85069	0.657867	0.328933	−	0.944353i	$-0.393311\pi$
$181$	8.85069	0.657867	0.328933	+	0.944353i	$0.393311\pi$
$182$	0	0
$183$	0	0
$184$	−17.2905	−1.27468
$185$	−12.7328	−0.936132
$186$	0	0
$187$	8.49035	0.620876
$188$	−16.5426	−1.20649
$189$	0	0
$190$	−9.32690	−0.676644
$191$	−5.50278	−0.398167	−0.199084	−	0.979983i	$-0.563797\pi$
$191$	−5.50278	−0.398167	−0.199084	+	0.979983i	$0.563797\pi$
$192$	0	0
$193$	−18.5258	−1.33352	−0.666758	−	0.745274i	$-0.732319\pi$
$193$	−18.5258	−1.33352	−0.666758	+	0.745274i	$0.732319\pi$
$194$	8.17108	0.586649
$195$	0	0
$196$	0	0
$197$	−7.47529	−0.532593	−0.266296	−	0.963891i	$-0.585800\pi$
$197$	−7.47529	−0.532593	−0.266296	+	0.963891i	$0.585800\pi$
$198$	0	0
$199$	−22.1007	−1.56668	−0.783338	−	0.621596i	$-0.786485\pi$
$199$	−22.1007	−1.56668	−0.783338	+	0.621596i	$0.786485\pi$
$200$	24.0958	1.70383
$201$	0	0
$202$	−6.91525	−0.486555
$203$	0	0
$204$	0	0
$205$	23.2461	1.62358
$206$	−25.2715	−1.76075
$207$	0	0
$208$	−1.06456	−0.0738138
$209$	4.54922	0.314676
$210$	0	0
$211$	10.3344	0.711449	0.355724	−	0.934591i	$-0.384234\pi$
$211$	10.3344	0.711449	0.355724	+	0.934591i	$0.384234\pi$
$212$	28.3116	1.94445
$213$	0	0
$214$	−4.70749	−0.321797
$215$	−18.8377	−1.28472
$216$	0	0
$217$	0	0
$218$	−15.4612	−1.04717
$219$	0	0
$220$	−54.8417	−3.69743
$221$	1.24494	0.0837434
$222$	0	0
$223$	−17.9428	−1.20154	−0.600768	−	0.799423i	$-0.705138\pi$
$223$	−17.9428	−1.20154	−0.600768	+	0.799423i	$0.705138\pi$
$224$	0	0
$225$	0	0
$226$	−14.4014	−0.957970
$227$	14.5072	0.962875	0.481437	−	0.876481i	$-0.340115\pi$
$227$	14.5072	0.962875	0.481437	+	0.876481i	$0.340115\pi$
$228$	0	0
$229$	4.81034	0.317876	0.158938	−	0.987289i	$-0.449193\pi$
$229$	4.81034	0.317876	0.158938	+	0.987289i	$0.449193\pi$
$230$	86.7030	5.71702
$231$	0	0
$232$	2.43789	0.160055
$233$	19.2611	1.26184	0.630920	−	0.775848i	$-0.282678\pi$
$233$	19.2611	1.26184	0.630920	+	0.775848i	$0.282678\pi$
$234$	0	0
$235$	24.6382	1.60722
$236$	5.91592	0.385093
$237$	0	0
$238$	0	0
$239$	−6.84407	−0.442706	−0.221353	−	0.975194i	$-0.571047\pi$
$239$	−6.84407	−0.442706	−0.221353	+	0.975194i	$0.571047\pi$
$240$	0	0
$241$	−0.589968	−0.0380032	−0.0190016	−	0.999819i	$-0.506049\pi$
$241$	−0.589968	−0.0380032	−0.0190016	+	0.999819i	$0.506049\pi$
$242$	21.3409	1.37185
$243$	0	0
$244$	−28.8455	−1.84664
$245$	0	0
$246$	0	0
$247$	0.667049	0.0424433
$248$	−7.26069	−0.461054
$249$	0	0
$250$	−74.1932	−4.69239
$251$	−7.77752	−0.490913	−0.245456	−	0.969408i	$-0.578938\pi$
$251$	−7.77752	−0.490913	−0.245456	+	0.969408i	$0.578938\pi$
$252$	0	0
$253$	−42.2896	−2.65872
$254$	−43.6609	−2.73953
$255$	0	0
$256$	−4.37220	−0.273263
$257$	5.16184	0.321987	0.160993	−	0.986955i	$-0.448530\pi$
$257$	5.16184	0.321987	0.160993	+	0.986955i	$0.448530\pi$
$258$	0	0
$259$	0	0
$260$	−8.04141	−0.498707
$261$	0	0
$262$	23.5671	1.45598
$263$	−21.2984	−1.31331	−0.656657	−	0.754190i	$-0.728030\pi$
$263$	−21.2984	−1.31331	−0.656657	+	0.754190i	$0.728030\pi$
$264$	0	0
$265$	−42.1667	−2.59028
$266$	0	0
$267$	0	0
$268$	−1.48933	−0.0909756
$269$	−10.2840	−0.627025	−0.313512	−	0.949584i	$-0.601506\pi$
$269$	−10.2840	−0.627025	−0.313512	+	0.949584i	$0.601506\pi$
$270$	0	0
$271$	25.2222	1.53214	0.766069	−	0.642758i	$-0.222210\pi$
$271$	25.2222	1.53214	0.766069	+	0.642758i	$0.222210\pi$
$272$	−2.97852	−0.180599
$273$	0	0
$274$	−40.2520	−2.43171
$275$	58.9340	3.55386
$276$	0	0
$277$	11.2653	0.676864	0.338432	−	0.940991i	$-0.390104\pi$
$277$	11.2653	0.676864	0.338432	+	0.940991i	$0.390104\pi$
$278$	−14.7859	−0.886800
$279$	0	0
$280$	0	0
$281$	7.07464	0.422038	0.211019	−	0.977482i	$-0.432322\pi$
$281$	7.07464	0.422038	0.211019	+	0.977482i	$0.432322\pi$
$282$	0	0
$283$	24.3703	1.44866	0.724331	−	0.689453i	$-0.242149\pi$
$283$	24.3703	1.44866	0.724331	+	0.689453i	$0.242149\pi$
$284$	11.4741	0.680860
$285$	0	0
$286$	6.67948	0.394966
$287$	0	0
$288$	0	0
$289$	−13.5168	−0.795106
$290$	−12.2247	−0.717862
$291$	0	0
$292$	−15.3251	−0.896835
$293$	−16.0761	−0.939177	−0.469588	−	0.882886i	$-0.655598\pi$
$293$	−16.0761	−0.939177	−0.469588	+	0.882886i	$0.655598\pi$
$294$	0	0
$295$	−8.81104	−0.512999
$296$	5.58914	0.324862
$297$	0	0
$298$	−35.1241	−2.03468
$299$	−6.20090	−0.358607
$300$	0	0
$301$	0	0
$302$	−42.0702	−2.42087
$303$	0	0
$304$	−1.59592	−0.0915323
$305$	42.9619	2.45999
$306$	0	0
$307$	−13.1085	−0.748142	−0.374071	−	0.927400i	$-0.622038\pi$
$307$	−13.1085	−0.748142	−0.374071	+	0.927400i	$0.622038\pi$
$308$	0	0
$309$	0	0
$310$	36.4085	2.06787
$311$	28.0359	1.58977	0.794885	−	0.606760i	$-0.207531\pi$
$311$	28.0359	1.58977	0.794885	+	0.606760i	$0.207531\pi$
$312$	0	0
$313$	9.53490	0.538945	0.269472	−	0.963008i	$-0.413151\pi$
$313$	9.53490	0.538945	0.269472	+	0.963008i	$0.413151\pi$
$314$	6.82680	0.385258
$315$	0	0
$316$	−7.52591	−0.423365
$317$	11.1347	0.625389	0.312694	−	0.949854i	$-0.398768\pi$
$317$	11.1347	0.625389	0.312694	+	0.949854i	$0.398768\pi$
$318$	0	0
$319$	5.96265	0.333844
$320$	53.9351	3.01506
$321$	0	0
$322$	0	0
$323$	1.86633	0.103845
$324$	0	0
$325$	8.64146	0.479342
$326$	−9.80113	−0.542834
$327$	0	0
$328$	−10.2040	−0.563424
$329$	0	0
$330$	0	0
$331$	−0.894117	−0.0491451	−0.0245726	−	0.999698i	$-0.507822\pi$
$331$	−0.894117	−0.0491451	−0.0245726	+	0.999698i	$0.507822\pi$
$332$	8.06319	0.442525
$333$	0	0
$334$	41.7807	2.28614
$335$	2.21818	0.121192
$336$	0	0
$337$	−10.4880	−0.571315	−0.285658	−	0.958332i	$-0.592212\pi$
$337$	−10.4880	−0.571315	−0.285658	+	0.958332i	$0.592212\pi$
$338$	−27.6354	−1.50317
$339$	0	0
$340$	−22.4990	−1.22018
$341$	−17.7584	−0.961669
$342$	0	0
$343$	0	0
$344$	8.26892	0.445830
$345$	0	0
$346$	9.61958	0.517152
$347$	−9.44400	−0.506981	−0.253490	−	0.967338i	$-0.581579\pi$
$347$	−9.44400	−0.506981	−0.253490	+	0.967338i	$0.581579\pi$
$348$	0	0
$349$	21.5964	1.15603	0.578015	−	0.816026i	$-0.303827\pi$
$349$	21.5964	1.15603	0.578015	+	0.816026i	$0.303827\pi$
$350$	0	0
$351$	0	0
$352$	−32.9037	−1.75378
$353$	0.973443	0.0518112	0.0259056	−	0.999664i	$-0.491753\pi$
$353$	0.973443	0.0518112	0.0259056	+	0.999664i	$0.491753\pi$
$354$	0	0
$355$	−17.0892	−0.907002
$356$	−40.1107	−2.12586
$357$	0	0
$358$	12.7106	0.671777
$359$	4.13501	0.218238	0.109119	−	0.994029i	$-0.465197\pi$
$359$	4.13501	0.218238	0.109119	+	0.994029i	$0.465197\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	19.4816	1.02393
$363$	0	0
$364$	0	0
$365$	22.8249	1.19471
$366$	0	0
$367$	31.0923	1.62301	0.811503	−	0.584348i	$-0.198650\pi$
$367$	31.0923	1.62301	0.811503	+	0.584348i	$0.198650\pi$
$368$	14.8357	0.773364
$369$	0	0
$370$	−28.0266	−1.45703
$371$	0	0
$372$	0	0
$373$	−16.6976	−0.864569	−0.432285	−	0.901737i	$-0.642292\pi$
$373$	−16.6976	−0.864569	−0.432285	+	0.901737i	$0.642292\pi$
$374$	18.6884	0.966356
$375$	0	0
$376$	−10.8151	−0.557747
$377$	0.874300	0.0450287
$378$	0	0
$379$	−6.93613	−0.356285	−0.178142	−	0.984005i	$-0.557009\pi$
$379$	−6.93613	−0.356285	−0.178142	+	0.984005i	$0.557009\pi$
$380$	−12.0552	−0.618419
$381$	0	0
$382$	−12.1124	−0.619723
$383$	−26.4779	−1.35296	−0.676478	−	0.736463i	$-0.736495\pi$
$383$	−26.4779	−1.35296	−0.676478	+	0.736463i	$0.736495\pi$
$384$	0	0
$385$	0	0
$386$	−40.7779	−2.07554
$387$	0	0
$388$	10.5613	0.536168
$389$	−19.4236	−0.984813	−0.492407	−	0.870365i	$-0.663883\pi$
$389$	−19.4236	−0.984813	−0.492407	+	0.870365i	$0.663883\pi$
$390$	0	0
$391$	−17.3494	−0.877399
$392$	0	0
$393$	0	0
$394$	−16.4542	−0.828949
$395$	11.2089	0.563982
$396$	0	0
$397$	6.44353	0.323392	0.161696	−	0.986841i	$-0.448304\pi$
$397$	6.44353	0.323392	0.161696	+	0.986841i	$0.448304\pi$
$398$	−48.6467	−2.43844
$399$	0	0
$400$	−20.6748	−1.03374
$401$	19.1953	0.958570	0.479285	−	0.877659i	$-0.340896\pi$
$401$	19.1953	0.958570	0.479285	+	0.877659i	$0.340896\pi$
$402$	0	0
$403$	−2.60390	−0.129709
$404$	−8.93810	−0.444687
$405$	0	0
$406$	0	0
$407$	13.6700	0.677599
$408$	0	0
$409$	21.7829	1.07709	0.538547	−	0.842595i	$-0.318973\pi$
$409$	21.7829	1.07709	0.538547	+	0.842595i	$0.318973\pi$
$410$	51.1680	2.52701
$411$	0	0
$412$	−32.6639	−1.60924
$413$	0	0
$414$	0	0
$415$	−12.0091	−0.589506
$416$	−4.82466	−0.236548
$417$	0	0
$418$	10.0135	0.489775
$419$	12.7698	0.623844	0.311922	−	0.950108i	$-0.399027\pi$
$419$	12.7698	0.623844	0.311922	+	0.950108i	$0.399027\pi$
$420$	0	0
$421$	30.1544	1.46963	0.734817	−	0.678266i	$-0.237268\pi$
$421$	30.1544	1.46963	0.734817	+	0.678266i	$0.237268\pi$
$422$	22.7474	1.10733
$423$	0	0
$424$	18.5094	0.898895
$425$	24.1779	1.17280
$426$	0	0
$427$	0	0
$428$	−6.08452	−0.294106
$429$	0	0
$430$	−41.4643	−1.99959
$431$	−2.95699	−0.142433	−0.0712165	−	0.997461i	$-0.522688\pi$
$431$	−2.95699	−0.142433	−0.0712165	+	0.997461i	$0.522688\pi$
$432$	0	0
$433$	−1.44760	−0.0695674	−0.0347837	−	0.999395i	$-0.511074\pi$
$433$	−1.44760	−0.0695674	−0.0347837	+	0.999395i	$0.511074\pi$
$434$	0	0
$435$	0	0
$436$	−19.9839	−0.957057
$437$	−9.29601	−0.444689
$438$	0	0
$439$	2.08826	0.0996671	0.0498336	−	0.998758i	$-0.484131\pi$
$439$	2.08826	0.0996671	0.0498336	+	0.998758i	$0.484131\pi$
$440$	−35.8541	−1.70928
$441$	0	0
$442$	2.74028	0.130342
$443$	27.9091	1.32600	0.663001	−	0.748618i	$-0.269282\pi$
$443$	27.9091	1.32600	0.663001	+	0.748618i	$0.269282\pi$
$444$	0	0
$445$	59.7401	2.83195
$446$	−39.4945	−1.87012
$447$	0	0
$448$	0	0
$449$	−34.9394	−1.64889	−0.824446	−	0.565941i	$-0.808513\pi$
$449$	−34.9394	−1.64889	−0.824446	+	0.565941i	$0.808513\pi$
$450$	0	0
$451$	−24.9573	−1.17519
$452$	−18.6141	−0.875535
$453$	0	0
$454$	31.9323	1.49866
$455$	0	0
$456$	0	0
$457$	−19.1491	−0.895759	−0.447879	−	0.894094i	$-0.647821\pi$
$457$	−19.1491	−0.895759	−0.447879	+	0.894094i	$0.647821\pi$
$458$	10.5882	0.494755
$459$	0	0
$460$	112.065	5.22507
$461$	−30.7840	−1.43375	−0.716877	−	0.697200i	$-0.754429\pi$
$461$	−30.7840	−1.43375	−0.716877	+	0.697200i	$0.754429\pi$
$462$	0	0
$463$	−9.88109	−0.459213	−0.229607	−	0.973284i	$-0.573744\pi$
$463$	−9.88109	−0.459213	−0.229607	+	0.973284i	$0.573744\pi$
$464$	−2.09177	−0.0971079
$465$	0	0
$466$	42.3965	1.96398
$467$	10.5101	0.486349	0.243174	−	0.969983i	$-0.421811\pi$
$467$	10.5101	0.486349	0.243174	+	0.969983i	$0.421811\pi$
$468$	0	0
$469$	0	0
$470$	54.2321	2.50154
$471$	0	0
$472$	3.86767	0.178024
$473$	20.2243	0.929915
$474$	0	0
$475$	12.9548	0.594405
$476$	0	0
$477$	0	0
$478$	−15.0648	−0.689046
$479$	21.1885	0.968127	0.484063	−	0.875033i	$-0.339160\pi$
$479$	21.1885	0.968127	0.484063	+	0.875033i	$0.339160\pi$
$480$	0	0
$481$	2.00443	0.0913942
$482$	−1.29860	−0.0591497
$483$	0	0
$484$	27.5835	1.25380
$485$	−15.7297	−0.714251
$486$	0	0
$487$	−2.06595	−0.0936171	−0.0468085	−	0.998904i	$-0.514905\pi$
$487$	−2.06595	−0.0936171	−0.0468085	+	0.998904i	$0.514905\pi$
$488$	−18.8584	−0.853680
$489$	0	0
$490$	0	0
$491$	−17.6735	−0.797594	−0.398797	−	0.917039i	$-0.630572\pi$
$491$	−17.6735	−0.797594	−0.398797	+	0.917039i	$0.630572\pi$
$492$	0	0
$493$	2.44620	0.110171
$494$	1.46827	0.0660605
$495$	0	0
$496$	6.22984	0.279728
$497$	0	0
$498$	0	0
$499$	8.94463	0.400417	0.200208	−	0.979753i	$-0.435838\pi$
$499$	8.94463	0.400417	0.200208	+	0.979753i	$0.435838\pi$
$500$	−95.8962	−4.28861
$501$	0	0
$502$	−17.1194	−0.764077
$503$	−18.8431	−0.840171	−0.420085	−	0.907485i	$-0.638000\pi$
$503$	−18.8431	−0.840171	−0.420085	+	0.907485i	$0.638000\pi$
$504$	0	0
$505$	13.3122	0.592386
$506$	−93.0853	−4.13815
$507$	0	0
$508$	−56.4325	−2.50379
$509$	10.7558	0.476743	0.238371	−	0.971174i	$-0.423386\pi$
$509$	10.7558	0.476743	0.238371	+	0.971174i	$0.423386\pi$
$510$	0	0
$511$	0	0
$512$	17.4798	0.772507
$513$	0	0
$514$	11.3619	0.501153
$515$	48.6489	2.14373
$516$	0	0
$517$	−26.4519	−1.16335
$518$	0	0
$519$	0	0
$520$	−5.25726	−0.230546
$521$	−3.90860	−0.171239	−0.0856194	−	0.996328i	$-0.527287\pi$
$521$	−3.90860	−0.171239	−0.0856194	+	0.996328i	$0.527287\pi$
$522$	0	0
$523$	−34.5912	−1.51257	−0.756284	−	0.654243i	$-0.772987\pi$
$523$	−34.5912	−1.51257	−0.756284	+	0.654243i	$0.772987\pi$
$524$	30.4610	1.33069
$525$	0	0
$526$	−46.8807	−2.04409
$527$	−7.28542	−0.317358
$528$	0	0
$529$	63.4158	2.75721
$530$	−92.8148	−4.03162
$531$	0	0
$532$	0	0
$533$	−3.65947	−0.158509
$534$	0	0
$535$	9.06216	0.391791
$536$	−0.973687	−0.0420569
$537$	0	0
$538$	−22.6365	−0.975927
$539$	0	0
$540$	0	0
$541$	0.256694	0.0110361	0.00551807	−	0.999985i	$-0.498244\pi$
$541$	0.256694	0.0110361	0.00551807	+	0.999985i	$0.498244\pi$
$542$	55.5175	2.38468
$543$	0	0
$544$	−13.4989	−0.578759
$545$	29.7637	1.27494
$546$	0	0
$547$	−8.24778	−0.352650	−0.176325	−	0.984332i	$-0.556421\pi$
$547$	−8.24778	−0.352650	−0.176325	+	0.984332i	$0.556421\pi$
$548$	−52.0265	−2.22246
$549$	0	0
$550$	129.722	5.53137
$551$	1.31070	0.0558376
$552$	0	0
$553$	0	0
$554$	24.7964	1.05350
$555$	0	0
$556$	−19.1111	−0.810490
$557$	3.18844	0.135098	0.0675492	−	0.997716i	$-0.478482\pi$
$557$	3.18844	0.135098	0.0675492	+	0.997716i	$0.478482\pi$
$558$	0	0
$559$	2.96548	0.125426
$560$	0	0
$561$	0	0
$562$	15.5723	0.656876
$563$	−32.5758	−1.37291	−0.686453	−	0.727175i	$-0.740833\pi$
$563$	−32.5758	−1.37291	−0.686453	+	0.727175i	$0.740833\pi$
$564$	0	0
$565$	27.7235	1.16634
$566$	53.6423	2.25476
$567$	0	0
$568$	7.50143	0.314753
$569$	39.5612	1.65849	0.829246	−	0.558884i	$-0.188770\pi$
$569$	39.5612	1.65849	0.829246	+	0.558884i	$0.188770\pi$
$570$	0	0
$571$	−6.55004	−0.274111	−0.137055	−	0.990563i	$-0.543764\pi$
$571$	−6.55004	−0.274111	−0.137055	+	0.990563i	$0.543764\pi$
$572$	8.63335	0.360978
$573$	0	0
$574$	0	0
$575$	−120.428	−5.02218
$576$	0	0
$577$	−17.5603	−0.731045	−0.365522	−	0.930803i	$-0.619110\pi$
$577$	−17.5603	−0.731045	−0.365522	+	0.930803i	$0.619110\pi$
$578$	−29.7524	−1.23754
$579$	0	0
$580$	−15.8007	−0.656089
$581$	0	0
$582$	0	0
$583$	45.2706	1.87492
$584$	−10.0192	−0.414596
$585$	0	0
$586$	−35.3858	−1.46177
$587$	30.2766	1.24965	0.624825	−	0.780765i	$-0.285170\pi$
$587$	30.2766	1.24965	0.624825	+	0.780765i	$0.285170\pi$
$588$	0	0
$589$	−3.90360	−0.160845
$590$	−19.3943	−0.798452
$591$	0	0
$592$	−4.79562	−0.197099
$593$	−24.8433	−1.02019	−0.510097	−	0.860117i	$-0.670390\pi$
$593$	−24.8433	−1.02019	−0.510097	+	0.860117i	$0.670390\pi$
$594$	0	0
$595$	0	0
$596$	−45.3985	−1.85960
$597$	0	0
$598$	−13.6490	−0.558151
$599$	29.8211	1.21846	0.609228	−	0.792995i	$-0.291479\pi$
$599$	29.8211	1.21846	0.609228	+	0.792995i	$0.291479\pi$
$600$	0	0
$601$	43.8767	1.78977	0.894884	−	0.446298i	$-0.147258\pi$
$601$	43.8767	1.78977	0.894884	+	0.446298i	$0.147258\pi$
$602$	0	0
$603$	0	0
$604$	−54.3765	−2.21255
$605$	−41.0823	−1.67023
$606$	0	0
$607$	6.47053	0.262631	0.131315	−	0.991341i	$-0.458080\pi$
$607$	6.47053	0.262631	0.131315	+	0.991341i	$0.458080\pi$
$608$	−7.23283	−0.293330
$609$	0	0
$610$	94.5651	3.82883
$611$	−3.87862	−0.156912
$612$	0	0
$613$	−40.0247	−1.61658	−0.808291	−	0.588783i	$-0.799607\pi$
$613$	−40.0247	−1.61658	−0.808291	+	0.588783i	$0.799607\pi$
$614$	−28.8536	−1.16444
$615$	0	0
$616$	0	0
$617$	17.7730	0.715513	0.357757	−	0.933815i	$-0.383542\pi$
$617$	17.7730	0.715513	0.357757	+	0.933815i	$0.383542\pi$
$618$	0	0
$619$	20.5165	0.824628	0.412314	−	0.911042i	$-0.364721\pi$
$619$	20.5165	0.824628	0.412314	+	0.911042i	$0.364721\pi$
$620$	47.0587	1.88992
$621$	0	0
$622$	61.7109	2.47438
$623$	0	0
$624$	0	0
$625$	78.0520	3.12208
$626$	20.9877	0.838835
$627$	0	0
$628$	8.82376	0.352106
$629$	5.60818	0.223613
$630$	0	0
$631$	−2.78970	−0.111056	−0.0555281	−	0.998457i	$-0.517684\pi$
$631$	−2.78970	−0.111056	−0.0555281	+	0.998457i	$0.517684\pi$
$632$	−4.92024	−0.195717
$633$	0	0
$634$	24.5091	0.973380
$635$	84.0494	3.33540
$636$	0	0
$637$	0	0
$638$	13.1246	0.519609
$639$	0	0
$640$	57.4233	2.26985
$641$	34.5824	1.36592	0.682961	−	0.730455i	$-0.260692\pi$
$641$	34.5824	1.36592	0.682961	+	0.730455i	$0.260692\pi$
$642$	0	0
$643$	−20.5081	−0.808761	−0.404381	−	0.914591i	$-0.632513\pi$
$643$	−20.5081	−0.808761	−0.404381	+	0.914591i	$0.632513\pi$
$644$	0	0
$645$	0	0
$646$	4.10806	0.161629
$647$	18.5977	0.731153	0.365576	−	0.930781i	$-0.380872\pi$
$647$	18.5977	0.731153	0.365576	+	0.930781i	$0.380872\pi$
$648$	0	0
$649$	9.45963	0.371323
$650$	19.0211	0.746068
$651$	0	0
$652$	−12.6681	−0.496123
$653$	42.3665	1.65793	0.828964	−	0.559302i	$-0.188931\pi$
$653$	42.3665	1.65793	0.828964	+	0.559302i	$0.188931\pi$
$654$	0	0
$655$	−45.3679	−1.77267
$656$	8.75532	0.341838
$657$	0	0
$658$	0	0
$659$	−37.3476	−1.45486	−0.727429	−	0.686183i	$-0.759285\pi$
$659$	−37.3476	−1.45486	−0.727429	+	0.686183i	$0.759285\pi$
$660$	0	0
$661$	−41.0123	−1.59519	−0.797596	−	0.603192i	$-0.793895\pi$
$661$	−41.0123	−1.59519	−0.797596	+	0.603192i	$0.793895\pi$
$662$	−1.96808	−0.0764915
$663$	0	0
$664$	5.27150	0.204574
$665$	0	0
$666$	0	0
$667$	−12.1843	−0.471776
$668$	54.0024	2.08942
$669$	0	0
$670$	4.88253	0.188629
$671$	−46.1243	−1.78061
$672$	0	0
$673$	42.3029	1.63066	0.815329	−	0.578998i	$-0.196556\pi$
$673$	42.3029	1.63066	0.815329	+	0.578998i	$0.196556\pi$
$674$	−23.0855	−0.889219
$675$	0	0
$676$	−35.7193	−1.37382
$677$	36.5041	1.40297	0.701483	−	0.712686i	$-0.252522\pi$
$677$	36.5041	1.40297	0.701483	+	0.712686i	$0.252522\pi$
$678$	0	0
$679$	0	0
$680$	−14.7092	−0.564074
$681$	0	0
$682$	−39.0886	−1.49678
$683$	−27.0271	−1.03416	−0.517082	−	0.855936i	$-0.672982\pi$
$683$	−27.0271	−1.03416	−0.517082	+	0.855936i	$0.672982\pi$
$684$	0	0
$685$	77.4871	2.96063
$686$	0	0
$687$	0	0
$688$	−7.09493	−0.270492
$689$	6.63801	0.252888
$690$	0	0
$691$	−5.49793	−0.209151	−0.104576	−	0.994517i	$-0.533348\pi$
$691$	−5.49793	−0.209151	−0.104576	+	0.994517i	$0.533348\pi$
$692$	12.4335	0.472650
$693$	0	0
$694$	−20.7876	−0.789085
$695$	28.4636	1.07969
$696$	0	0
$697$	−10.2388	−0.387823
$698$	47.5368	1.79929
$699$	0	0
$700$	0	0
$701$	32.8214	1.23965	0.619824	−	0.784741i	$-0.287204\pi$
$701$	32.8214	1.23965	0.619824	+	0.784741i	$0.287204\pi$
$702$	0	0
$703$	3.00492	0.113333
$704$	−57.9053	−2.18239
$705$	0	0
$706$	2.14268	0.0806410
$707$	0	0
$708$	0	0
$709$	4.93049	0.185168	0.0925842	−	0.995705i	$-0.470487\pi$
$709$	4.93049	0.185168	0.0925842	+	0.995705i	$0.470487\pi$
$710$	−37.6158	−1.41169
$711$	0	0
$712$	−26.2233	−0.982761
$713$	36.2880	1.35899
$714$	0	0
$715$	−12.8583	−0.480874
$716$	16.4287	0.613970
$717$	0	0
$718$	9.10174	0.339674
$719$	−15.6062	−0.582014	−0.291007	−	0.956721i	$-0.593990\pi$
$719$	−15.6062	−0.582014	−0.291007	+	0.956721i	$0.593990\pi$
$720$	0	0
$721$	0	0
$722$	2.20114	0.0819179
$723$	0	0
$724$	25.1803	0.935820
$725$	16.9798	0.630613
$726$	0	0
$727$	26.5642	0.985213	0.492606	−	0.870252i	$-0.336044\pi$
$727$	26.5642	0.985213	0.492606	+	0.870252i	$0.336044\pi$
$728$	0	0
$729$	0	0
$730$	50.2408	1.85950
$731$	8.29709	0.306879
$732$	0	0
$733$	37.8191	1.39688	0.698440	−	0.715668i	$-0.253878\pi$
$733$	37.8191	1.39688	0.698440	+	0.715668i	$0.253878\pi$
$734$	68.4385	2.52611
$735$	0	0
$736$	67.2365	2.47837
$737$	−2.38147	−0.0877224
$738$	0	0
$739$	42.8341	1.57568	0.787839	−	0.615881i	$-0.211200\pi$
$739$	42.8341	1.57568	0.787839	+	0.615881i	$0.211200\pi$
$740$	−36.2249	−1.33165
$741$	0	0
$742$	0	0
$743$	35.0205	1.28478	0.642389	−	0.766379i	$-0.277943\pi$
$743$	35.0205	1.28478	0.642389	+	0.766379i	$0.277943\pi$
$744$	0	0
$745$	67.6156	2.47724
$746$	−36.7538	−1.34565
$747$	0	0
$748$	24.1552	0.883201
$749$	0	0
$750$	0	0
$751$	−11.6655	−0.425679	−0.212839	−	0.977087i	$-0.568271\pi$
$751$	−11.6655	−0.425679	−0.212839	+	0.977087i	$0.568271\pi$
$752$	9.27963	0.338393
$753$	0	0
$754$	1.92446	0.0700846
$755$	80.9873	2.94743
$756$	0	0
$757$	−13.5376	−0.492031	−0.246016	−	0.969266i	$-0.579121\pi$
$757$	−13.5376	−0.492031	−0.246016	+	0.969266i	$0.579121\pi$
$758$	−15.2674	−0.554536
$759$	0	0
$760$	−7.88137	−0.285887
$761$	−46.4300	−1.68309	−0.841543	−	0.540190i	$-0.818352\pi$
$761$	−46.4300	−1.68309	−0.841543	+	0.540190i	$0.818352\pi$
$762$	0	0
$763$	0	0
$764$	−15.6555	−0.566396
$765$	0	0
$766$	−58.2815	−2.10580
$767$	1.38706	0.0500839
$768$	0	0
$769$	−12.0896	−0.435963	−0.217981	−	0.975953i	$-0.569947\pi$
$769$	−12.0896	−0.435963	−0.217981	+	0.975953i	$0.569947\pi$
$770$	0	0
$771$	0	0
$772$	−52.7062	−1.89694
$773$	−41.3224	−1.48627	−0.743133	−	0.669144i	$-0.766661\pi$
$773$	−41.3224	−1.48627	−0.743133	+	0.669144i	$0.766661\pi$
$774$	0	0
$775$	−50.5703	−1.81654
$776$	6.90468	0.247864
$777$	0	0
$778$	−42.7540	−1.53280
$779$	−5.48606	−0.196559
$780$	0	0
$781$	18.3472	0.656514
$782$	−38.1885	−1.36562
$783$	0	0
$784$	0	0
$785$	−13.1419	−0.469056
$786$	0	0
$787$	−12.6708	−0.451664	−0.225832	−	0.974166i	$-0.572510\pi$
$787$	−12.6708	−0.451664	−0.225832	+	0.974166i	$0.572510\pi$
$788$	−21.2673	−0.757617
$789$	0	0
$790$	24.6724	0.877805
$791$	0	0
$792$	0	0
$793$	−6.76319	−0.240168
$794$	14.1831	0.503340
$795$	0	0
$796$	−62.8768	−2.22861
$797$	−8.94088	−0.316702	−0.158351	−	0.987383i	$-0.550618\pi$
$797$	−8.94088	−0.316702	−0.158351	+	0.987383i	$0.550618\pi$
$798$	0	0
$799$	−10.8520	−0.383915
$800$	−93.6996	−3.31278
$801$	0	0
$802$	42.2516	1.49196
$803$	−24.5051	−0.864766
$804$	0	0
$805$	0	0
$806$	−5.73154	−0.201885
$807$	0	0
$808$	−5.84349	−0.205573
$809$	−19.5233	−0.686404	−0.343202	−	0.939262i	$-0.611512\pi$
$809$	−19.5233	−0.686404	−0.343202	+	0.939262i	$0.611512\pi$
$810$	0	0
$811$	22.4029	0.786671	0.393336	−	0.919395i	$-0.371321\pi$
$811$	22.4029	0.786671	0.393336	+	0.919395i	$0.371321\pi$
$812$	0	0
$813$	0	0
$814$	30.0897	1.05464
$815$	18.8677	0.660906
$816$	0	0
$817$	4.44567	0.155534
$818$	47.9472	1.67643
$819$	0	0
$820$	66.1356	2.30955
$821$	−52.6094	−1.83608	−0.918040	−	0.396487i	$-0.870229\pi$
$821$	−52.6094	−1.83608	−0.918040	+	0.396487i	$0.870229\pi$
$822$	0	0
$823$	9.72070	0.338842	0.169421	−	0.985544i	$-0.445810\pi$
$823$	9.72070	0.338842	0.169421	+	0.985544i	$0.445810\pi$
$824$	−21.3548	−0.743930
$825$	0	0
$826$	0	0
$827$	−52.0766	−1.81088	−0.905440	−	0.424473i	$-0.860459\pi$
$827$	−52.0766	−1.81088	−0.905440	+	0.424473i	$0.860459\pi$
$828$	0	0
$829$	6.02902	0.209396	0.104698	−	0.994504i	$-0.466612\pi$
$829$	6.02902	0.209396	0.104698	+	0.994504i	$0.466612\pi$
$830$	−26.4338	−0.917531
$831$	0	0
$832$	−8.49063	−0.294360
$833$	0	0
$834$	0	0
$835$	−80.4300	−2.78340
$836$	12.9426	0.447629
$837$	0	0
$838$	28.1080	0.970976
$839$	−52.1776	−1.80137	−0.900685	−	0.434473i	$-0.856935\pi$
$839$	−52.1776	−1.80137	−0.900685	+	0.434473i	$0.856935\pi$
$840$	0	0
$841$	−27.2821	−0.940761
$842$	66.3740	2.28740
$843$	0	0
$844$	29.4015	1.01204
$845$	53.1996	1.83012
$846$	0	0
$847$	0	0
$848$	−15.8815	−0.545372
$849$	0	0
$850$	53.2189	1.82539
$851$	−27.9338	−0.957558
$852$	0	0
$853$	32.7787	1.12232	0.561161	−	0.827707i	$-0.310355\pi$
$853$	32.7787	1.12232	0.561161	+	0.827707i	$0.310355\pi$
$854$	0	0
$855$	0	0
$856$	−3.97790	−0.135962
$857$	52.7222	1.80095	0.900477	−	0.434903i	$-0.143217\pi$
$857$	52.7222	1.80095	0.900477	+	0.434903i	$0.143217\pi$
$858$	0	0
$859$	−38.9499	−1.32895	−0.664476	−	0.747309i	$-0.731345\pi$
$859$	−38.9499	−1.32895	−0.664476	+	0.747309i	$0.731345\pi$
$860$	−53.5934	−1.82752
$861$	0	0
$862$	−6.50874	−0.221689
$863$	42.4625	1.44544	0.722721	−	0.691140i	$-0.242891\pi$
$863$	42.4625	1.44544	0.722721	+	0.691140i	$0.242891\pi$
$864$	0	0
$865$	−18.5182	−0.629637
$866$	−3.18638	−0.108278
$867$	0	0
$868$	0	0
$869$	−12.0340	−0.408227
$870$	0	0
$871$	−0.349193	−0.0118320
$872$	−13.0650	−0.442436
$873$	0	0
$874$	−20.4618	−0.692131
$875$	0	0
$876$	0	0
$877$	34.0068	1.14833	0.574163	−	0.818741i	$-0.305327\pi$
$877$	34.0068	1.14833	0.574163	+	0.818741i	$0.305327\pi$
$878$	4.59655	0.155126
$879$	0	0
$880$	30.7637	1.03704
$881$	19.0580	0.642079	0.321040	−	0.947066i	$-0.395968\pi$
$881$	19.0580	0.642079	0.321040	+	0.947066i	$0.395968\pi$
$882$	0	0
$883$	−40.7499	−1.37134	−0.685671	−	0.727911i	$-0.740491\pi$
$883$	−40.7499	−1.37134	−0.685671	+	0.727911i	$0.740491\pi$
$884$	3.54186	0.119126
$885$	0	0
$886$	61.4319	2.06384
$887$	−6.28833	−0.211141	−0.105571	−	0.994412i	$-0.533667\pi$
$887$	−6.28833	−0.211141	−0.105571	+	0.994412i	$0.533667\pi$
$888$	0	0
$889$	0	0
$890$	131.496	4.40777
$891$	0	0
$892$	−51.0474	−1.70919
$893$	−5.81459	−0.194578
$894$	0	0
$895$	−24.4686	−0.817895
$896$	0	0
$897$	0	0
$898$	−76.9064	−2.56640
$899$	−5.11644	−0.170643
$900$	0	0
$901$	18.5724	0.618737
$902$	−54.9345	−1.82912
$903$	0	0
$904$	−12.1694	−0.404749
$905$	−37.5031	−1.24664
$906$	0	0
$907$	−25.7603	−0.855355	−0.427678	−	0.903931i	$-0.640668\pi$
$907$	−25.7603	−0.855355	−0.427678	+	0.903931i	$0.640668\pi$
$908$	41.2731	1.36970
$909$	0	0
$910$	0	0
$911$	30.3836	1.00665	0.503327	−	0.864096i	$-0.332109\pi$
$911$	30.3836	1.00665	0.503327	+	0.864096i	$0.332109\pi$
$912$	0	0
$913$	12.8932	0.426701
$914$	−42.1499	−1.39420
$915$	0	0
$916$	13.6855	0.452181
$917$	0	0
$918$	0	0
$919$	16.3148	0.538174	0.269087	−	0.963116i	$-0.413278\pi$
$919$	16.3148	0.538174	0.269087	+	0.963116i	$0.413278\pi$
$920$	73.2653	2.41548
$921$	0	0
$922$	−67.7599	−2.23155
$923$	2.69024	0.0885502
$924$	0	0
$925$	38.9280	1.27995
$926$	−21.7497	−0.714738
$927$	0	0
$928$	−9.48006	−0.311198
$929$	−35.3308	−1.15917	−0.579583	−	0.814913i	$-0.696784\pi$
$929$	−35.3308	−1.15917	−0.579583	+	0.814913i	$0.696784\pi$
$930$	0	0
$931$	0	0
$932$	54.7982	1.79498
$933$	0	0
$934$	23.1342	0.756973
$935$	−35.9762	−1.17655
$936$	0	0
$937$	28.5431	0.932461	0.466231	−	0.884663i	$-0.345612\pi$
$937$	28.5431	0.932461	0.466231	+	0.884663i	$0.345612\pi$
$938$	0	0
$939$	0	0
$940$	70.0961	2.28628
$941$	−51.9012	−1.69193	−0.845966	−	0.533237i	$-0.820975\pi$
$941$	−51.9012	−1.69193	−0.845966	+	0.533237i	$0.820975\pi$
$942$	0	0
$943$	50.9985	1.66074
$944$	−3.31855	−0.108010
$945$	0	0
$946$	44.5165	1.44736
$947$	−29.9686	−0.973848	−0.486924	−	0.873444i	$-0.661881\pi$
$947$	−29.9686	−0.973848	−0.486924	+	0.873444i	$0.661881\pi$
$948$	0	0
$949$	−3.59317	−0.116639
$950$	28.5152	0.925156
$951$	0	0
$952$	0	0
$953$	−10.6441	−0.344796	−0.172398	−	0.985027i	$-0.555152\pi$
$953$	−10.6441	−0.344796	−0.172398	+	0.985027i	$0.555152\pi$
$954$	0	0
$955$	23.3170	0.754519
$956$	−19.4715	−0.629753
$957$	0	0
$958$	46.6388	1.50683
$959$	0	0
$960$	0	0
$961$	−15.7619	−0.508447
$962$	4.41203	0.142250
$963$	0	0
$964$	−1.67847	−0.0540598
$965$	78.4995	2.52699
$966$	0	0
$967$	45.3775	1.45924	0.729620	−	0.683852i	$-0.239697\pi$
$967$	45.3775	1.45924	0.729620	+	0.683852i	$0.239697\pi$
$968$	18.0334	0.579615
$969$	0	0
$970$	−34.6234	−1.11169
$971$	10.6259	0.341000	0.170500	−	0.985358i	$-0.445462\pi$
$971$	10.6259	0.341000	0.170500	+	0.985358i	$0.445462\pi$
$972$	0	0
$973$	0	0
$974$	−4.54744	−0.145709
$975$	0	0
$976$	16.1810	0.517940
$977$	−34.7509	−1.11178	−0.555890	−	0.831256i	$-0.687622\pi$
$977$	−34.7509	−1.11178	−0.555890	+	0.831256i	$0.687622\pi$
$978$	0	0
$979$	−64.1376	−2.04985
$980$	0	0
$981$	0	0
$982$	−38.9019	−1.24141
$983$	31.8461	1.01573	0.507867	−	0.861436i	$-0.330434\pi$
$983$	31.8461	1.01573	0.507867	+	0.861436i	$0.330434\pi$
$984$	0	0
$985$	31.6751	1.00925
$986$	5.38442	0.171475
$987$	0	0
$988$	1.89777	0.0603760
$989$	−41.3270	−1.31412
$990$	0	0
$991$	−20.2572	−0.643490	−0.321745	−	0.946826i	$-0.604269\pi$
$991$	−20.2572	−0.643490	−0.321745	+	0.946826i	$0.604269\pi$
$992$	28.2341	0.896434
$993$	0	0
$994$	0	0
$995$	93.6473	2.96882
$996$	0	0
$997$	−28.8629	−0.914096	−0.457048	−	0.889442i	$-0.651093\pi$
$997$	−28.8629	−0.914096	−0.457048	+	0.889442i	$0.651093\pi$
$998$	19.6884	0.623225
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.cs.1.9		10
3.2	odd	2		931.2.a.q.1.2	yes	10
7.6	odd	2		8379.2.a.ct.1.9		10
21.2	odd	6		931.2.f.q.704.9		20
21.5	even	6		931.2.f.r.704.9		20
21.11	odd	6		931.2.f.q.324.9		20
21.17	even	6		931.2.f.r.324.9		20
21.20	even	2		931.2.a.p.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
931.2.a.p.1.2	✓	10	21.20	even	2
931.2.a.q.1.2	yes	10	3.2	odd	2
931.2.f.q.324.9		20	21.11	odd	6
931.2.f.q.704.9		20	21.2	odd	6
931.2.f.r.324.9		20	21.17	even	6
931.2.f.r.704.9		20	21.5	even	6
8379.2.a.cs.1.9		10	1.1	even	1	trivial
8379.2.a.ct.1.9		10	7.6	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:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q+2.20114 q^{2} +2.84501 q^{4} -4.23731 q^{5} +1.86000 q^{8} +O(q^{10})\)	\(q+2.20114 q^{2} +2.84501 q^{4} -4.23731 q^{5} +1.86000 q^{8} -9.32690 q^{10} +4.54922 q^{11} +0.667049 q^{13} -1.59592 q^{16} +1.86633 q^{17} +1.00000 q^{19} -12.0552 q^{20} +10.0135 q^{22} -9.29601 q^{23} +12.9548 q^{25} +1.46827 q^{26} +1.31070 q^{29} -3.90360 q^{31} -7.23283 q^{32} +4.10806 q^{34} +3.00492 q^{37} +2.20114 q^{38} -7.88137 q^{40} -5.48606 q^{41} +4.44567 q^{43} +12.9426 q^{44} -20.4618 q^{46} -5.81459 q^{47} +28.5152 q^{50} +1.89777 q^{52} +9.95130 q^{53} -19.2764 q^{55} +2.88503 q^{58} +2.07940 q^{59} -10.1390 q^{61} -8.59238 q^{62} -12.7286 q^{64} -2.82649 q^{65} -0.523489 q^{67} +5.30974 q^{68} +4.03304 q^{71} -5.38666 q^{73} +6.61425 q^{74} +2.84501 q^{76} -2.64530 q^{79} +6.76240 q^{80} -12.0756 q^{82} +2.83415 q^{83} -7.90822 q^{85} +9.78554 q^{86} +8.46152 q^{88} -14.0986 q^{89} -26.4473 q^{92} -12.7987 q^{94} -4.23731 q^{95} +3.71220 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 10 q^{4} - 16 q^{5} + 6 q^{8}+O(q^{10}) \) 10 * q + 2 * q^2 + 10 * q^4 - 16 * q^5 + 6 * q^8	\( 10 q + 2 q^{2} + 10 q^{4} - 16 q^{5} + 6 q^{8} - 12 q^{10} + 12 q^{13} + 2 q^{16} - 16 q^{17} + 10 q^{19} - 32 q^{20} + 4 q^{22} - 12 q^{23} + 14 q^{25} - 24 q^{26} + 12 q^{29} + 8 q^{31} + 34 q^{32} + 16 q^{34} + 4 q^{37} + 2 q^{38} - 20 q^{40} - 40 q^{41} + 4 q^{43} + 20 q^{44} - 32 q^{46} - 16 q^{47} + 34 q^{50} - 40 q^{52} + 16 q^{55} - 8 q^{58} - 36 q^{59} + 16 q^{61} + 16 q^{62} + 18 q^{64} - 8 q^{65} - 28 q^{67} - 40 q^{68} + 12 q^{71} - 24 q^{74} + 10 q^{76} - 8 q^{79} - 8 q^{80} - 8 q^{82} + 40 q^{85} + 52 q^{86} - 4 q^{88} - 48 q^{89} - 28 q^{92} - 36 q^{94} - 16 q^{95} - 16 q^{97}+O(q^{100}) \) 10 * q + 2 * q^2 + 10 * q^4 - 16 * q^5 + 6 * q^8 - 12 * q^10 + 12 * q^13 + 2 * q^16 - 16 * q^17 + 10 * q^19 - 32 * q^20 + 4 * q^22 - 12 * q^23 + 14 * q^25 - 24 * q^26 + 12 * q^29 + 8 * q^31 + 34 * q^32 + 16 * q^34 + 4 * q^37 + 2 * q^38 - 20 * q^40 - 40 * q^41 + 4 * q^43 + 20 * q^44 - 32 * q^46 - 16 * q^47 + 34 * q^50 - 40 * q^52 + 16 * q^55 - 8 * q^58 - 36 * q^59 + 16 * q^61 + 16 * q^62 + 18 * q^64 - 8 * q^65 - 28 * q^67 - 40 * q^68 + 12 * q^71 - 24 * q^74 + 10 * q^76 - 8 * q^79 - 8 * q^80 - 8 * q^82 + 40 * q^85 + 52 * q^86 - 4 * q^88 - 48 * q^89 - 28 * q^92 - 36 * q^94 - 16 * q^95 - 16 * q^97

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