LMFDB - Embedded newform 279.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [279,2,Mod(1,279)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(279, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("279.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 279 = 3^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	279.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:	$2.22782621639$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 93)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	279.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.86081 q^{2} +1.46260 q^{4} +3.32340 q^{5} -1.32340 q^{7} +1.00000 q^{8} -6.18421 q^{10} -0.925197 q^{11} +2.92520 q^{13} +2.46260 q^{14} -4.78600 q^{16} -4.64681 q^{17} +6.11982 q^{19} +4.86081 q^{20} +1.72161 q^{22} +5.72161 q^{23} +6.04502 q^{25} -5.44322 q^{26} -1.93561 q^{28} +9.57201 q^{29} -1.00000 q^{31} +6.90582 q^{32} +8.64681 q^{34} -4.39821 q^{35} +3.72161 q^{37} -11.3878 q^{38} +3.32340 q^{40} +8.11982 q^{41} -4.36842 q^{43} -1.35319 q^{44} -10.6468 q^{46} -11.4432 q^{47} -5.24860 q^{49} -11.2486 q^{50} +4.27839 q^{52} +1.20359 q^{53} -3.07480 q^{55} -1.32340 q^{56} -17.8116 q^{58} -7.60179 q^{59} +12.0900 q^{61} +1.86081 q^{62} -3.27839 q^{64} +9.72161 q^{65} +4.00000 q^{67} -6.79641 q^{68} +8.18421 q^{70} -10.4882 q^{71} -15.1648 q^{73} -6.92520 q^{74} +8.95084 q^{76} +1.22441 q^{77} -2.64681 q^{79} -15.9058 q^{80} -15.1094 q^{82} -1.35319 q^{83} -15.4432 q^{85} +8.12878 q^{86} -0.925197 q^{88} +6.00000 q^{89} -3.87122 q^{91} +8.36842 q^{92} +21.2936 q^{94} +20.3386 q^{95} -5.04502 q^{97} +9.76663 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{4} + 2 q^{5} + 4 q^{7} + 3 q^{8} - 5 q^{10} + 2 q^{11} + 4 q^{13} + 5 q^{14} - 4 q^{16} + 2 q^{17} + 4 q^{19} + 9 q^{20} - 6 q^{22} + 6 q^{23} - q^{25} + 6 q^{26} - 5 q^{28} + 8 q^{29} - 3 q^{31}+ \cdots - q^{98}+O(q^{100}) $ 3 * q + 2 * q^4 + 2 * q^5 + 4 * q^7 + 3 * q^8 - 5 * q^10 + 2 * q^11 + 4 * q^13 + 5 * q^14 - 4 * q^16 + 2 * q^17 + 4 * q^19 + 9 * q^20 - 6 * q^22 + 6 * q^23 - q^25 + 6 * q^26 - 5 * q^28 + 8 * q^29 - 3 * q^31 - 4 * q^32 + 10 * q^34 - 10 * q^35 - 27 * q^38 + 2 * q^40 + 10 * q^41 + 14 * q^43 - 20 * q^44 - 16 * q^46 - 12 * q^47 - 3 * q^49 - 21 * q^50 + 24 * q^52 + 10 * q^53 - 14 * q^55 + 4 * q^56 - 4 * q^58 - 26 * q^59 - 2 * q^61 - 21 * q^64 + 18 * q^65 + 12 * q^67 - 14 * q^68 + 11 * q^70 + 10 * q^71 - 12 * q^73 - 16 * q^74 - 17 * q^76 + 18 * q^77 + 8 * q^79 - 23 * q^80 - 27 * q^82 - 20 * q^83 - 24 * q^85 + 26 * q^86 + 2 * q^88 + 18 * q^89 - 10 * q^91 - 2 * q^92 + 32 * q^94 + 10 * q^95 + 4 * q^97 - q^98

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 $

$2$	−1.86081	−1.31579	−0.657894	−	0.753110i	$-0.728553\pi$
$2$	−1.86081	−1.31579	−0.657894	+	0.753110i	$0.728553\pi$
$3$	0	0
$3$	0	0
$4$	1.46260	0.731299
$5$	3.32340	1.48627	0.743136	−	0.669141i	$-0.233338\pi$
$5$	3.32340	1.48627	0.743136	+	0.669141i	$0.233338\pi$
$6$	0	0
$7$	−1.32340	−0.500200	−0.250100	−	0.968220i	$-0.580463\pi$
$7$	−1.32340	−0.500200	−0.250100	+	0.968220i	$0.580463\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−6.18421	−1.95562
$11$	−0.925197	−0.278957	−0.139479	−	0.990225i	$-0.544543\pi$
$11$	−0.925197	−0.278957	−0.139479	+	0.990225i	$0.544543\pi$
$12$	0	0
$13$	2.92520	0.811304	0.405652	−	0.914028i	$-0.367045\pi$
$13$	2.92520	0.811304	0.405652	+	0.914028i	$0.367045\pi$
$14$	2.46260	0.658157
$15$	0	0
$16$	−4.78600	−1.19650
$17$	−4.64681	−1.12702	−0.563508	−	0.826110i	$-0.690549\pi$
$17$	−4.64681	−1.12702	−0.563508	+	0.826110i	$0.690549\pi$
$18$	0	0
$19$	6.11982	1.40398	0.701991	−	0.712185i	$-0.252294\pi$
$19$	6.11982	1.40398	0.701991	+	0.712185i	$0.252294\pi$
$20$	4.86081	1.08691
$21$	0	0
$22$	1.72161	0.367049
$23$	5.72161	1.19304	0.596519	−	0.802599i	$-0.296550\pi$
$23$	5.72161	1.19304	0.596519	+	0.802599i	$0.296550\pi$
$24$	0	0
$25$	6.04502	1.20900
$26$	−5.44322	−1.06750
$27$	0	0
$28$	−1.93561	−0.365796
$29$	9.57201	1.77748	0.888738	−	0.458415i	$-0.151583\pi$
$29$	9.57201	1.77748	0.888738	+	0.458415i	$0.151583\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	6.90582	1.22079
$33$	0	0
$34$	8.64681	1.48292
$35$	−4.39821	−0.743433
$36$	0	0
$37$	3.72161	0.611829	0.305915	−	0.952059i	$-0.401038\pi$
$37$	3.72161	0.611829	0.305915	+	0.952059i	$0.401038\pi$
$38$	−11.3878	−1.84734
$39$	0	0
$40$	3.32340	0.525476
$41$	8.11982	1.26810	0.634051	−	0.773291i	$-0.281391\pi$
$41$	8.11982	1.26810	0.634051	+	0.773291i	$0.281391\pi$
$42$	0	0
$43$	−4.36842	−0.666178	−0.333089	−	0.942895i	$-0.608091\pi$
$43$	−4.36842	−0.666178	−0.333089	+	0.942895i	$0.608091\pi$
$44$	−1.35319	−0.204001
$45$	0	0
$46$	−10.6468	−1.56979
$47$	−11.4432	−1.66917	−0.834583	−	0.550882i	$-0.814291\pi$
$47$	−11.4432	−1.66917	−0.834583	+	0.550882i	$0.814291\pi$
$48$	0	0
$49$	−5.24860	−0.749800
$50$	−11.2486	−1.59079
$51$	0	0
$52$	4.27839	0.593306
$53$	1.20359	0.165325	0.0826626	−	0.996578i	$-0.473658\pi$
$53$	1.20359	0.165325	0.0826626	+	0.996578i	$0.473658\pi$
$54$	0	0
$55$	−3.07480	−0.414606
$56$	−1.32340	−0.176847
$57$	0	0
$58$	−17.8116	−2.33878
$59$	−7.60179	−0.989669	−0.494835	−	0.868987i	$-0.664771\pi$
$59$	−7.60179	−0.989669	−0.494835	+	0.868987i	$0.664771\pi$
$60$	0	0
$61$	12.0900	1.54797	0.773985	−	0.633204i	$-0.218261\pi$
$61$	12.0900	1.54797	0.773985	+	0.633204i	$0.218261\pi$
$62$	1.86081	0.236323
$63$	0	0
$64$	−3.27839	−0.409799
$65$	9.72161	1.20582
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−6.79641	−0.824186
$69$	0	0
$70$	8.18421	0.978200
$71$	−10.4882	−1.24473	−0.622363	−	0.782729i	$-0.713827\pi$
$71$	−10.4882	−1.24473	−0.622363	+	0.782729i	$0.713827\pi$
$72$	0	0
$73$	−15.1648	−1.77491	−0.887455	−	0.460895i	$-0.847529\pi$
$73$	−15.1648	−1.77491	−0.887455	+	0.460895i	$0.847529\pi$
$74$	−6.92520	−0.805038
$75$	0	0
$76$	8.95084	1.02673
$77$	1.22441	0.139534
$78$	0	0
$79$	−2.64681	−0.297789	−0.148895	−	0.988853i	$-0.547572\pi$
$79$	−2.64681	−0.297789	−0.148895	+	0.988853i	$0.547572\pi$
$80$	−15.9058	−1.77832
$81$	0	0
$82$	−15.1094	−1.66855
$83$	−1.35319	−0.148532	−0.0742660	−	0.997238i	$-0.523661\pi$
$83$	−1.35319	−0.148532	−0.0742660	+	0.997238i	$0.523661\pi$
$84$	0	0
$85$	−15.4432	−1.67505
$86$	8.12878	0.876549
$87$	0	0
$88$	−0.925197	−0.0986263
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−3.87122	−0.405814
$92$	8.36842	0.872468
$93$	0	0
$94$	21.2936	2.19627
$95$	20.3386	2.08670
$96$	0	0
$97$	−5.04502	−0.512244	−0.256122	−	0.966644i	$-0.582445\pi$
$97$	−5.04502	−0.512244	−0.256122	+	0.966644i	$0.582445\pi$
$98$	9.76663	0.986578
$99$	0	0
$100$	8.84143	0.884143
$101$	13.1738	1.31084	0.655421	−	0.755264i	$-0.272491\pi$
$101$	13.1738	1.31084	0.655421	+	0.755264i	$0.272491\pi$
$102$	0	0
$103$	−10.1198	−0.997135	−0.498568	−	0.866851i	$-0.666140\pi$
$103$	−10.1198	−0.997135	−0.498568	+	0.866851i	$0.666140\pi$
$104$	2.92520	0.286839
$105$	0	0
$106$	−2.23964	−0.217533
$107$	−10.2486	−0.990770	−0.495385	−	0.868674i	$-0.664973\pi$
$107$	−10.2486	−0.990770	−0.495385	+	0.868674i	$0.664973\pi$
$108$	0	0
$109$	3.45219	0.330659	0.165330	−	0.986238i	$-0.447131\pi$
$109$	3.45219	0.330659	0.165330	+	0.986238i	$0.447131\pi$
$110$	5.72161	0.545534
$111$	0	0
$112$	6.33382	0.598489
$113$	−9.41344	−0.885542	−0.442771	−	0.896635i	$-0.646004\pi$
$113$	−9.41344	−0.885542	−0.442771	+	0.896635i	$0.646004\pi$
$114$	0	0
$115$	19.0152	1.77318
$116$	14.0000	1.29987
$117$	0	0
$118$	14.1455	1.30220
$119$	6.14961	0.563733
$120$	0	0
$121$	−10.1440	−0.922183
$122$	−22.4972	−2.03680
$123$	0	0
$124$	−1.46260	−0.131345
$125$	3.47301	0.310636
$126$	0	0
$127$	6.51803	0.578381	0.289191	−	0.957272i	$-0.406614\pi$
$127$	6.51803	0.578381	0.289191	+	0.957272i	$0.406614\pi$
$128$	−7.71120	−0.681580
$129$	0	0
$130$	−18.0900	−1.58660
$131$	1.29362	0.113024	0.0565119	−	0.998402i	$-0.482002\pi$
$131$	1.29362	0.113024	0.0565119	+	0.998402i	$0.482002\pi$
$132$	0	0
$133$	−8.09899	−0.702272
$134$	−7.44322	−0.642997
$135$	0	0
$136$	−4.64681	−0.398461
$137$	−6.66763	−0.569654	−0.284827	−	0.958579i	$-0.591936\pi$
$137$	−6.66763	−0.569654	−0.284827	+	0.958579i	$0.591936\pi$
$138$	0	0
$139$	13.9612	1.18418	0.592089	−	0.805873i	$-0.298303\pi$
$139$	13.9612	1.18418	0.592089	+	0.805873i	$0.298303\pi$
$140$	−6.43281	−0.543672
$141$	0	0
$142$	19.5166	1.63779
$143$	−2.70638	−0.226319
$144$	0	0
$145$	31.8116	2.64181
$146$	28.2188	2.33541
$147$	0	0
$148$	5.44322	0.447430
$149$	0.149606	0.0122562	0.00612811	−	0.999981i	$-0.498049\pi$
$149$	0.149606	0.0122562	0.00612811	+	0.999981i	$0.498049\pi$
$150$	0	0
$151$	−4.66763	−0.379847	−0.189923	−	0.981799i	$-0.560824\pi$
$151$	−4.66763	−0.379847	−0.189923	+	0.981799i	$0.560824\pi$
$152$	6.11982	0.496383
$153$	0	0
$154$	−2.27839	−0.183598
$155$	−3.32340	−0.266942
$156$	0	0
$157$	5.04502	0.402636	0.201318	−	0.979526i	$-0.435478\pi$
$157$	5.04502	0.402636	0.201318	+	0.979526i	$0.435478\pi$
$158$	4.92520	0.391828
$159$	0	0
$160$	22.9508	1.81442
$161$	−7.57201	−0.596758
$162$	0	0
$163$	4.52699	0.354581	0.177291	−	0.984159i	$-0.443267\pi$
$163$	4.52699	0.354581	0.177291	+	0.984159i	$0.443267\pi$
$164$	11.8760	0.927363
$165$	0	0
$166$	2.51803	0.195437
$167$	−5.72161	−0.442752	−0.221376	−	0.975189i	$-0.571055\pi$
$167$	−5.72161	−0.442752	−0.221376	+	0.975189i	$0.571055\pi$
$168$	0	0
$169$	−4.44322	−0.341786
$170$	28.7368	2.20402
$171$	0	0
$172$	−6.38924	−0.487175
$173$	−19.2936	−1.46687	−0.733433	−	0.679761i	$-0.762083\pi$
$173$	−19.2936	−1.46687	−0.733433	+	0.679761i	$0.762083\pi$
$174$	0	0
$175$	−8.00000	−0.604743
$176$	4.42799	0.333773
$177$	0	0
$178$	−11.1648	−0.836840
$179$	7.44322	0.556333	0.278166	−	0.960533i	$-0.410273\pi$
$179$	7.44322	0.556333	0.278166	+	0.960533i	$0.410273\pi$
$180$	0	0
$181$	−19.5333	−1.45190	−0.725948	−	0.687750i	$-0.758599\pi$
$181$	−19.5333	−1.45190	−0.725948	+	0.687750i	$0.758599\pi$
$182$	7.20359	0.533965
$183$	0	0
$184$	5.72161	0.421803
$185$	12.3684	0.909344
$186$	0	0
$187$	4.29921	0.314390
$188$	−16.7368	−1.22066
$189$	0	0
$190$	−37.8462	−2.74566
$191$	−8.09899	−0.586023	−0.293011	−	0.956109i	$-0.594657\pi$
$191$	−8.09899	−0.586023	−0.293011	+	0.956109i	$0.594657\pi$
$192$	0	0
$193$	0.248601	0.0178947	0.00894735	−	0.999960i	$-0.497152\pi$
$193$	0.248601	0.0178947	0.00894735	+	0.999960i	$0.497152\pi$
$194$	9.38780	0.674004
$195$	0	0
$196$	−7.67660	−0.548328
$197$	10.3684	0.738719	0.369360	−	0.929287i	$-0.379577\pi$
$197$	10.3684	0.738719	0.369360	+	0.929287i	$0.379577\pi$
$198$	0	0
$199$	−18.2188	−1.29150	−0.645749	−	0.763550i	$-0.723455\pi$
$199$	−18.2188	−1.29150	−0.645749	+	0.763550i	$0.723455\pi$
$200$	6.04502	0.427447
$201$	0	0
$202$	−24.5139	−1.72479
$203$	−12.6676	−0.889094
$204$	0	0
$205$	26.9854	1.88474
$206$	18.8310	1.31202
$207$	0	0
$208$	−14.0000	−0.970725
$209$	−5.66204	−0.391651
$210$	0	0
$211$	−25.5630	−1.75983	−0.879916	−	0.475129i	$-0.842401\pi$
$211$	−25.5630	−1.75983	−0.879916	+	0.475129i	$0.842401\pi$
$212$	1.76036	0.120902
$213$	0	0
$214$	19.0707	1.30364
$215$	−14.5180	−0.990121
$216$	0	0
$217$	1.32340	0.0898385
$218$	−6.42385	−0.435078
$219$	0	0
$220$	−4.49720	−0.303201
$221$	−13.5928	−0.914353
$222$	0	0
$223$	−2.27839	−0.152572	−0.0762861	−	0.997086i	$-0.524306\pi$
$223$	−2.27839	−0.152572	−0.0762861	+	0.997086i	$0.524306\pi$
$224$	−9.13919	−0.610638
$225$	0	0
$226$	17.5166	1.16519
$227$	−21.5928	−1.43317	−0.716583	−	0.697502i	$-0.754295\pi$
$227$	−21.5928	−1.43317	−0.716583	+	0.697502i	$0.754295\pi$
$228$	0	0
$229$	18.8656	1.24668	0.623338	−	0.781953i	$-0.285776\pi$
$229$	18.8656	1.24668	0.623338	+	0.781953i	$0.285776\pi$
$230$	−35.3836	−2.33313
$231$	0	0
$232$	9.57201	0.628433
$233$	4.61702	0.302471	0.151236	−	0.988498i	$-0.451675\pi$
$233$	4.61702	0.302471	0.151236	+	0.988498i	$0.451675\pi$
$234$	0	0
$235$	−38.0305	−2.48083
$236$	−11.1184	−0.723744
$237$	0	0
$238$	−11.4432	−0.741754
$239$	10.3892	0.672024	0.336012	−	0.941858i	$-0.390922\pi$
$239$	10.3892	0.672024	0.336012	+	0.941858i	$0.390922\pi$
$240$	0	0
$241$	−12.4585	−0.802519	−0.401260	−	0.915964i	$-0.631428\pi$
$241$	−12.4585	−0.802519	−0.401260	+	0.915964i	$0.631428\pi$
$242$	18.8760	1.21340
$243$	0	0
$244$	17.6829	1.13203
$245$	−17.4432	−1.11441
$246$	0	0
$247$	17.9017	1.13906
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	−6.46260	−0.408731
$251$	10.5180	0.663892	0.331946	−	0.943298i	$-0.392295\pi$
$251$	10.5180	0.663892	0.331946	+	0.943298i	$0.392295\pi$
$252$	0	0
$253$	−5.29362	−0.332807
$254$	−12.1288	−0.761027
$255$	0	0
$256$	20.9058	1.30661
$257$	2.76663	0.172577	0.0862887	−	0.996270i	$-0.472499\pi$
$257$	2.76663	0.172577	0.0862887	+	0.996270i	$0.472499\pi$
$258$	0	0
$259$	−4.92520	−0.306037
$260$	14.2188	0.881813
$261$	0	0
$262$	−2.40717	−0.148715
$263$	−3.44322	−0.212318	−0.106159	−	0.994349i	$-0.533855\pi$
$263$	−3.44322	−0.212318	−0.106159	+	0.994349i	$0.533855\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	15.0707	0.924041
$267$	0	0
$268$	5.85039	0.357370
$269$	2.36842	0.144405	0.0722026	−	0.997390i	$-0.476997\pi$
$269$	2.36842	0.144405	0.0722026	+	0.997390i	$0.476997\pi$
$270$	0	0
$271$	25.5333	1.55103	0.775517	−	0.631326i	$-0.217489\pi$
$271$	25.5333	1.55103	0.775517	+	0.631326i	$0.217489\pi$
$272$	22.2396	1.34848
$273$	0	0
$274$	12.4072	0.749545
$275$	−5.59283	−0.337260
$276$	0	0
$277$	−19.8504	−1.19269	−0.596347	−	0.802727i	$-0.703382\pi$
$277$	−19.8504	−1.19269	−0.596347	+	0.802727i	$0.703382\pi$
$278$	−25.9792	−1.55813
$279$	0	0
$280$	−4.39821	−0.262843
$281$	0.916234	0.0546579	0.0273290	−	0.999626i	$-0.491300\pi$
$281$	0.916234	0.0546579	0.0273290	+	0.999626i	$0.491300\pi$
$282$	0	0
$283$	20.9944	1.24799	0.623994	−	0.781429i	$-0.285509\pi$
$283$	20.9944	1.24799	0.623994	+	0.781429i	$0.285509\pi$
$284$	−15.3401	−0.910266
$285$	0	0
$286$	5.03605	0.297788
$287$	−10.7458	−0.634305
$288$	0	0
$289$	4.59283	0.270166
$290$	−59.1953	−3.47607
$291$	0	0
$292$	−22.1801	−1.29799
$293$	−5.26316	−0.307477	−0.153739	−	0.988112i	$-0.549131\pi$
$293$	−5.26316	−0.307477	−0.153739	+	0.988112i	$0.549131\pi$
$294$	0	0
$295$	−25.2638	−1.47092
$296$	3.72161	0.216314
$297$	0	0
$298$	−0.278388	−0.0161266
$299$	16.7368	0.967916
$300$	0	0
$301$	5.78119	0.333222
$302$	8.68556	0.499798
$303$	0	0
$304$	−29.2895	−1.67987
$305$	40.1801	2.30070
$306$	0	0
$307$	31.1142	1.77578	0.887891	−	0.460053i	$-0.152170\pi$
$307$	31.1142	1.77578	0.887891	+	0.460053i	$0.152170\pi$
$308$	1.79082	0.102041
$309$	0	0
$310$	6.18421	0.351240
$311$	18.5478	1.05175	0.525875	−	0.850562i	$-0.323738\pi$
$311$	18.5478	1.05175	0.525875	+	0.850562i	$0.323738\pi$
$312$	0	0
$313$	23.2340	1.31327	0.656633	−	0.754210i	$-0.271980\pi$
$313$	23.2340	1.31327	0.656633	+	0.754210i	$0.271980\pi$
$314$	−9.38780	−0.529784
$315$	0	0
$316$	−3.87122	−0.217773
$317$	6.76663	0.380052	0.190026	−	0.981779i	$-0.439143\pi$
$317$	6.76663	0.380052	0.190026	+	0.981779i	$0.439143\pi$
$318$	0	0
$319$	−8.85599	−0.495840
$320$	−10.8954	−0.609072
$321$	0	0
$322$	14.0900	0.785207
$323$	−28.4376	−1.58231
$324$	0	0
$325$	17.6829	0.980869
$326$	−8.42385	−0.466554
$327$	0	0
$328$	8.11982	0.448342
$329$	15.1440	0.834916
$330$	0	0
$331$	6.38924	0.351185	0.175592	−	0.984463i	$-0.443816\pi$
$331$	6.38924	0.351185	0.175592	+	0.984463i	$0.443816\pi$
$332$	−1.97918	−0.108621
$333$	0	0
$334$	10.6468	0.582567
$335$	13.2936	0.726308
$336$	0	0
$337$	−19.2936	−1.05099	−0.525495	−	0.850797i	$-0.676120\pi$
$337$	−19.2936	−1.05099	−0.525495	+	0.850797i	$0.676120\pi$
$338$	8.26798	0.449719
$339$	0	0
$340$	−22.5872	−1.22496
$341$	0.925197	0.0501022
$342$	0	0
$343$	16.2099	0.875250
$344$	−4.36842	−0.235529
$345$	0	0
$346$	35.9017	1.93009
$347$	16.8656	0.905394	0.452697	−	0.891664i	$-0.350462\pi$
$347$	16.8656	0.905394	0.452697	+	0.891664i	$0.350462\pi$
$348$	0	0
$349$	−26.9944	−1.44498	−0.722489	−	0.691383i	$-0.757002\pi$
$349$	−26.9944	−1.44498	−0.722489	+	0.691383i	$0.757002\pi$
$350$	14.8864	0.795714
$351$	0	0
$352$	−6.38924	−0.340548
$353$	−3.96125	−0.210836	−0.105418	−	0.994428i	$-0.533618\pi$
$353$	−3.96125	−0.210836	−0.105418	+	0.994428i	$0.533618\pi$
$354$	0	0
$355$	−34.8567	−1.85000
$356$	8.77559	0.465105
$357$	0	0
$358$	−13.8504	−0.732016
$359$	32.0990	1.69412	0.847060	−	0.531497i	$-0.178370\pi$
$359$	32.0990	1.69412	0.847060	+	0.531497i	$0.178370\pi$
$360$	0	0
$361$	18.4522	0.971168
$362$	36.3476	1.91039
$363$	0	0
$364$	−5.66204	−0.296771
$365$	−50.3989	−2.63800
$366$	0	0
$367$	−10.3892	−0.542314	−0.271157	−	0.962535i	$-0.587406\pi$
$367$	−10.3892	−0.542314	−0.271157	+	0.962535i	$0.587406\pi$
$368$	−27.3836	−1.42747
$369$	0	0
$370$	−23.0152	−1.19650
$371$	−1.59283	−0.0826956
$372$	0	0
$373$	24.4882	1.26795	0.633976	−	0.773352i	$-0.281422\pi$
$373$	24.4882	1.26795	0.633976	+	0.773352i	$0.281422\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	−11.4432	−0.590139
$377$	28.0000	1.44207
$378$	0	0
$379$	11.1440	0.572429	0.286215	−	0.958166i	$-0.407603\pi$
$379$	11.1440	0.572429	0.286215	+	0.958166i	$0.407603\pi$
$380$	29.7473	1.52600
$381$	0	0
$382$	15.0707	0.771082
$383$	14.0900	0.719967	0.359984	−	0.932959i	$-0.382782\pi$
$383$	14.0900	0.719967	0.359984	+	0.932959i	$0.382782\pi$
$384$	0	0
$385$	4.06921	0.207386
$386$	−0.462598	−0.0235456
$387$	0	0
$388$	−7.37883	−0.374603
$389$	−19.9196	−1.00996	−0.504982	−	0.863130i	$-0.668501\pi$
$389$	−19.9196	−1.00996	−0.504982	+	0.863130i	$0.668501\pi$
$390$	0	0
$391$	−26.5872	−1.34457
$392$	−5.24860	−0.265094
$393$	0	0
$394$	−19.2936	−0.971998
$395$	−8.79641	−0.442596
$396$	0	0
$397$	16.8054	0.843438	0.421719	−	0.906727i	$-0.361427\pi$
$397$	16.8054	0.843438	0.421719	+	0.906727i	$0.361427\pi$
$398$	33.9017	1.69934
$399$	0	0
$400$	−28.9315	−1.44657
$401$	−21.7521	−1.08625	−0.543123	−	0.839653i	$-0.682758\pi$
$401$	−21.7521	−1.08625	−0.543123	+	0.839653i	$0.682758\pi$
$402$	0	0
$403$	−2.92520	−0.145714
$404$	19.2680	0.958618
$405$	0	0
$406$	23.5720	1.16986
$407$	−3.44322	−0.170674
$408$	0	0
$409$	14.9252	0.738003	0.369002	−	0.929429i	$-0.379700\pi$
$409$	14.9252	0.738003	0.369002	+	0.929429i	$0.379700\pi$
$410$	−50.2147	−2.47993
$411$	0	0
$412$	−14.8012	−0.729204
$413$	10.0602	0.495032
$414$	0	0
$415$	−4.49720	−0.220759
$416$	20.2009	0.990430
$417$	0	0
$418$	10.5360	0.515330
$419$	24.8775	1.21534	0.607672	−	0.794188i	$-0.292103\pi$
$419$	24.8775	1.21534	0.607672	+	0.794188i	$0.292103\pi$
$420$	0	0
$421$	−3.49383	−0.170279	−0.0851395	−	0.996369i	$-0.527134\pi$
$421$	−3.49383	−0.170279	−0.0851395	+	0.996369i	$0.527134\pi$
$422$	47.5679	2.31557
$423$	0	0
$424$	1.20359	0.0584513
$425$	−28.0900	−1.36257
$426$	0	0
$427$	−16.0000	−0.774294
$428$	−14.9896	−0.724549
$429$	0	0
$430$	27.0152	1.30279
$431$	−29.0361	−1.39862	−0.699309	−	0.714820i	$-0.746509\pi$
$431$	−29.0361	−1.39862	−0.699309	+	0.714820i	$0.746509\pi$
$432$	0	0
$433$	−39.7312	−1.90936	−0.954681	−	0.297631i	$-0.903803\pi$
$433$	−39.7312	−1.90936	−0.954681	+	0.297631i	$0.903803\pi$
$434$	−2.46260	−0.118209
$435$	0	0
$436$	5.04916	0.241811
$437$	35.0152	1.67501
$438$	0	0
$439$	−13.8206	−0.659622	−0.329811	−	0.944047i	$-0.606985\pi$
$439$	−13.8206	−0.659622	−0.329811	+	0.944047i	$0.606985\pi$
$440$	−3.07480	−0.146585
$441$	0	0
$442$	25.2936	1.20309
$443$	0.457782	0.0217499	0.0108749	−	0.999941i	$-0.496538\pi$
$443$	0.457782	0.0217499	0.0108749	+	0.999941i	$0.496538\pi$
$444$	0	0
$445$	19.9404	0.945267
$446$	4.23964	0.200753
$447$	0	0
$448$	4.33863	0.204981
$449$	19.9792	0.942876	0.471438	−	0.881899i	$-0.343735\pi$
$449$	19.9792	0.942876	0.471438	+	0.881899i	$0.343735\pi$
$450$	0	0
$451$	−7.51243	−0.353747
$452$	−13.7681	−0.647596
$453$	0	0
$454$	40.1801	1.88574
$455$	−12.8656	−0.603150
$456$	0	0
$457$	16.5180	0.772681	0.386340	−	0.922356i	$-0.373739\pi$
$457$	16.5180	0.772681	0.386340	+	0.922356i	$0.373739\pi$
$458$	−35.1053	−1.64036
$459$	0	0
$460$	27.8116	1.29672
$461$	−19.2936	−0.898593	−0.449297	−	0.893383i	$-0.648325\pi$
$461$	−19.2936	−0.898593	−0.449297	+	0.893383i	$0.648325\pi$
$462$	0	0
$463$	−17.7908	−0.826809	−0.413405	−	0.910547i	$-0.635661\pi$
$463$	−17.7908	−0.826809	−0.413405	+	0.910547i	$0.635661\pi$
$464$	−45.8116	−2.12675
$465$	0	0
$466$	−8.59138	−0.397988
$467$	−14.0090	−0.648257	−0.324129	−	0.946013i	$-0.605071\pi$
$467$	−14.0090	−0.648257	−0.324129	+	0.946013i	$0.605071\pi$
$468$	0	0
$469$	−5.29362	−0.244437
$470$	70.7673	3.26425
$471$	0	0
$472$	−7.60179	−0.349901
$473$	4.04165	0.185835
$474$	0	0
$475$	36.9944	1.69742
$476$	8.99440	0.412258
$477$	0	0
$478$	−19.3324	−0.884242
$479$	−28.8775	−1.31945	−0.659723	−	0.751509i	$-0.729326\pi$
$479$	−28.8775	−1.31945	−0.659723	+	0.751509i	$0.729326\pi$
$480$	0	0
$481$	10.8864	0.496379
$482$	23.1828	1.05595
$483$	0	0
$484$	−14.8366	−0.674392
$485$	−16.7666	−0.761333
$486$	0	0
$487$	−4.86562	−0.220482	−0.110241	−	0.993905i	$-0.535162\pi$
$487$	−4.86562	−0.220482	−0.110241	+	0.993905i	$0.535162\pi$
$488$	12.0900	0.547290
$489$	0	0
$490$	32.4585	1.46632
$491$	−0.608059	−0.0274413	−0.0137206	−	0.999906i	$-0.504368\pi$
$491$	−0.608059	−0.0274413	−0.0137206	+	0.999906i	$0.504368\pi$
$492$	0	0
$493$	−44.4793	−2.00325
$494$	−33.3115	−1.49876
$495$	0	0
$496$	4.78600	0.214898
$497$	13.8802	0.622611
$498$	0	0
$499$	−36.6773	−1.64190	−0.820950	−	0.571000i	$-0.806556\pi$
$499$	−36.6773	−1.64190	−0.820950	+	0.571000i	$0.806556\pi$
$500$	5.07962	0.227168
$501$	0	0
$502$	−19.5720	−0.873541
$503$	11.5422	0.514642	0.257321	−	0.966326i	$-0.417160\pi$
$503$	11.5422	0.514642	0.257321	+	0.966326i	$0.417160\pi$
$504$	0	0
$505$	43.7819	1.94827
$506$	9.85039	0.437903
$507$	0	0
$508$	9.53326	0.422970
$509$	14.8656	0.658907	0.329454	−	0.944172i	$-0.393135\pi$
$509$	14.8656	0.658907	0.329454	+	0.944172i	$0.393135\pi$
$510$	0	0
$511$	20.0692	0.887809
$512$	−23.4793	−1.03765
$513$	0	0
$514$	−5.14816	−0.227075
$515$	−33.6323	−1.48201
$516$	0	0
$517$	10.5872	0.465626
$518$	9.16484	0.402680
$519$	0	0
$520$	9.72161	0.426321
$521$	5.14401	0.225363	0.112682	−	0.993631i	$-0.464056\pi$
$521$	5.14401	0.225363	0.112682	+	0.993631i	$0.464056\pi$
$522$	0	0
$523$	8.98477	0.392877	0.196438	−	0.980516i	$-0.437062\pi$
$523$	8.98477	0.392877	0.196438	+	0.980516i	$0.437062\pi$
$524$	1.89204	0.0826543
$525$	0	0
$526$	6.40717	0.279366
$527$	4.64681	0.202418
$528$	0	0
$529$	9.73684	0.423341
$530$	−7.44322	−0.323313
$531$	0	0
$532$	−11.8456	−0.513571
$533$	23.7521	1.02882
$534$	0	0
$535$	−34.0602	−1.47255
$536$	4.00000	0.172774
$537$	0	0
$538$	−4.40717	−0.190007
$539$	4.85599	0.209162
$540$	0	0
$541$	14.1586	0.608724	0.304362	−	0.952556i	$-0.401557\pi$
$541$	14.1586	0.608724	0.304362	+	0.952556i	$0.401557\pi$
$542$	−47.5124	−2.04083
$543$	0	0
$544$	−32.0900	−1.37585
$545$	11.4730	0.491450
$546$	0	0
$547$	6.97581	0.298264	0.149132	−	0.988817i	$-0.452352\pi$
$547$	6.97581	0.298264	0.149132	+	0.988817i	$0.452352\pi$
$548$	−9.75207	−0.416588
$549$	0	0
$550$	10.4072	0.443763
$551$	58.5789	2.49555
$552$	0	0
$553$	3.50280	0.148954
$554$	36.9377	1.56933
$555$	0	0
$556$	20.4197	0.865988
$557$	1.63158	0.0691323	0.0345661	−	0.999402i	$-0.488995\pi$
$557$	1.63158	0.0691323	0.0345661	+	0.999402i	$0.488995\pi$
$558$	0	0
$559$	−12.7785	−0.540473
$560$	21.0498	0.889518
$561$	0	0
$562$	−1.70493	−0.0719183
$563$	−19.3026	−0.813507	−0.406753	−	0.913538i	$-0.633339\pi$
$563$	−19.3026	−0.813507	−0.406753	+	0.913538i	$0.633339\pi$
$564$	0	0
$565$	−31.2847	−1.31616
$566$	−39.0665	−1.64209
$567$	0	0
$568$	−10.4882	−0.440077
$569$	34.4972	1.44620	0.723099	−	0.690744i	$-0.242717\pi$
$569$	34.4972	1.44620	0.723099	+	0.690744i	$0.242717\pi$
$570$	0	0
$571$	1.46405	0.0612685	0.0306342	−	0.999531i	$-0.490247\pi$
$571$	1.46405	0.0612685	0.0306342	+	0.999531i	$0.490247\pi$
$572$	−3.95835	−0.165507
$573$	0	0
$574$	19.9959	0.834611
$575$	34.5872	1.44239
$576$	0	0
$577$	−2.55678	−0.106440	−0.0532200	−	0.998583i	$-0.516948\pi$
$577$	−2.55678	−0.106440	−0.0532200	+	0.998583i	$0.516948\pi$
$578$	−8.54636	−0.355482
$579$	0	0
$580$	46.5277	1.93196
$581$	1.79082	0.0742957
$582$	0	0
$583$	−1.11355	−0.0461187
$584$	−15.1648	−0.627525
$585$	0	0
$586$	9.79372	0.404575
$587$	1.03605	0.0427625	0.0213812	−	0.999771i	$-0.493194\pi$
$587$	1.03605	0.0427625	0.0213812	+	0.999771i	$0.493194\pi$
$588$	0	0
$589$	−6.11982	−0.252163
$590$	47.0111	1.93542
$591$	0	0
$592$	−17.8116	−0.732054
$593$	5.41344	0.222303	0.111152	−	0.993803i	$-0.464546\pi$
$593$	5.41344	0.222303	0.111152	+	0.993803i	$0.464546\pi$
$594$	0	0
$595$	20.4376	0.837861
$596$	0.218814	0.00896297
$597$	0	0
$598$	−31.1440	−1.27357
$599$	−32.0990	−1.31153	−0.655765	−	0.754965i	$-0.727654\pi$
$599$	−32.0990	−1.31153	−0.655765	+	0.754965i	$0.727654\pi$
$600$	0	0
$601$	−16.7577	−0.683559	−0.341780	−	0.939780i	$-0.611030\pi$
$601$	−16.7577	−0.683559	−0.341780	+	0.939780i	$0.611030\pi$
$602$	−10.7577	−0.438450
$603$	0	0
$604$	−6.82687	−0.277782
$605$	−33.7126	−1.37061
$606$	0	0
$607$	15.8809	0.644584	0.322292	−	0.946640i	$-0.395547\pi$
$607$	15.8809	0.644584	0.322292	+	0.946640i	$0.395547\pi$
$608$	42.2624	1.71397
$609$	0	0
$610$	−74.7673	−3.02724
$611$	−33.4737	−1.35420
$612$	0	0
$613$	−5.95835	−0.240656	−0.120328	−	0.992734i	$-0.538395\pi$
$613$	−5.95835	−0.240656	−0.120328	+	0.992734i	$0.538395\pi$
$614$	−57.8975	−2.33655
$615$	0	0
$616$	1.22441	0.0493329
$617$	−0.149606	−0.00602292	−0.00301146	−	0.999995i	$-0.500959\pi$
$617$	−0.149606	−0.00602292	−0.00301146	+	0.999995i	$0.500959\pi$
$618$	0	0
$619$	33.4916	1.34614	0.673071	−	0.739578i	$-0.264975\pi$
$619$	33.4916	1.34614	0.673071	+	0.739578i	$0.264975\pi$
$620$	−4.86081	−0.195215
$621$	0	0
$622$	−34.5139	−1.38388
$623$	−7.94043	−0.318126
$624$	0	0
$625$	−18.6829	−0.747314
$626$	−43.2340	−1.72798
$627$	0	0
$628$	7.37883	0.294447
$629$	−17.2936	−0.689542
$630$	0	0
$631$	32.7964	1.30560	0.652802	−	0.757528i	$-0.273593\pi$
$631$	32.7964	1.30560	0.652802	+	0.757528i	$0.273593\pi$
$632$	−2.64681	−0.105284
$633$	0	0
$634$	−12.5914	−0.500068
$635$	21.6620	0.859632
$636$	0	0
$637$	−15.3532	−0.608316
$638$	16.4793	0.652421
$639$	0	0
$640$	−25.6274	−1.01301
$641$	37.4253	1.47821	0.739105	−	0.673590i	$-0.235249\pi$
$641$	37.4253	1.47821	0.739105	+	0.673590i	$0.235249\pi$
$642$	0	0
$643$	22.6468	0.893103	0.446551	−	0.894758i	$-0.352652\pi$
$643$	22.6468	0.893103	0.446551	+	0.894758i	$0.352652\pi$
$644$	−11.0748	−0.436408
$645$	0	0
$646$	52.9169	2.08199
$647$	−15.5720	−0.612199	−0.306099	−	0.952000i	$-0.599024\pi$
$647$	−15.5720	−0.612199	−0.306099	+	0.952000i	$0.599024\pi$
$648$	0	0
$649$	7.03315	0.276075
$650$	−32.9044	−1.29062
$651$	0	0
$652$	6.62117	0.259305
$653$	−42.1801	−1.65063	−0.825317	−	0.564670i	$-0.809003\pi$
$653$	−42.1801	−1.65063	−0.825317	+	0.564670i	$0.809003\pi$
$654$	0	0
$655$	4.29921	0.167984
$656$	−38.8615	−1.51729
$657$	0	0
$658$	−28.1801	−1.09857
$659$	12.3566	0.481343	0.240672	−	0.970607i	$-0.422632\pi$
$659$	12.3566	0.481343	0.240672	+	0.970607i	$0.422632\pi$
$660$	0	0
$661$	−18.0990	−0.703969	−0.351985	−	0.936006i	$-0.614493\pi$
$661$	−18.0990	−0.703969	−0.351985	+	0.936006i	$0.614493\pi$
$662$	−11.8891	−0.462085
$663$	0	0
$664$	−1.35319	−0.0525140
$665$	−26.9162	−1.04377
$666$	0	0
$667$	54.7673	2.12060
$668$	−8.36842	−0.323784
$669$	0	0
$670$	−24.7368	−0.955668
$671$	−11.1857	−0.431818
$672$	0	0
$673$	23.9792	0.924329	0.462165	−	0.886794i	$-0.347073\pi$
$673$	23.9792	0.924329	0.462165	+	0.886794i	$0.347073\pi$
$674$	35.9017	1.38288
$675$	0	0
$676$	−6.49865	−0.249948
$677$	−37.1953	−1.42953	−0.714766	−	0.699364i	$-0.753467\pi$
$677$	−37.1953	−1.42953	−0.714766	+	0.699364i	$0.753467\pi$
$678$	0	0
$679$	6.67660	0.256224
$680$	−15.4432	−0.592221
$681$	0	0
$682$	−1.72161	−0.0659239
$683$	24.5962	0.941147	0.470574	−	0.882361i	$-0.344047\pi$
$683$	24.5962	0.941147	0.470574	+	0.882361i	$0.344047\pi$
$684$	0	0
$685$	−22.1592	−0.846661
$686$	−30.1634	−1.15164
$687$	0	0
$688$	20.9073	0.797082
$689$	3.52072	0.134129
$690$	0	0
$691$	−37.5214	−1.42738	−0.713691	−	0.700461i	$-0.752978\pi$
$691$	−37.5214	−1.42738	−0.713691	+	0.700461i	$0.752978\pi$
$692$	−28.2188	−1.07272
$693$	0	0
$694$	−31.3836	−1.19131
$695$	46.3989	1.76001
$696$	0	0
$697$	−37.7312	−1.42917
$698$	50.2313	1.90128
$699$	0	0
$700$	−11.7008	−0.442248
$701$	−35.5035	−1.34095	−0.670474	−	0.741933i	$-0.733909\pi$
$701$	−35.5035	−1.34095	−0.670474	+	0.741933i	$0.733909\pi$
$702$	0	0
$703$	22.7756	0.858998
$704$	3.03315	0.114316
$705$	0	0
$706$	7.37112	0.277416
$707$	−17.4343	−0.655683
$708$	0	0
$709$	−34.7964	−1.30681	−0.653403	−	0.757010i	$-0.726659\pi$
$709$	−34.7964	−1.30681	−0.653403	+	0.757010i	$0.726659\pi$
$710$	64.8615	2.43421
$711$	0	0
$712$	6.00000	0.224860
$713$	−5.72161	−0.214276
$714$	0	0
$715$	−8.99440	−0.336372
$716$	10.8864	0.406846
$717$	0	0
$718$	−59.7300	−2.22910
$719$	−25.5333	−0.952230	−0.476115	−	0.879383i	$-0.657955\pi$
$719$	−25.5333	−0.952230	−0.476115	+	0.879383i	$0.657955\pi$
$720$	0	0
$721$	13.3926	0.498767
$722$	−34.3359	−1.27785
$723$	0	0
$724$	−28.5693	−1.06177
$725$	57.8629	2.14898
$726$	0	0
$727$	0.586564	0.0217544	0.0108772	−	0.999941i	$-0.496538\pi$
$727$	0.586564	0.0217544	0.0108772	+	0.999941i	$0.496538\pi$
$728$	−3.87122	−0.143477
$729$	0	0
$730$	93.7825	3.47105
$731$	20.2992	0.750793
$732$	0	0
$733$	28.2486	1.04339	0.521693	−	0.853133i	$-0.325301\pi$
$733$	28.2486	1.04339	0.521693	+	0.853133i	$0.325301\pi$
$734$	19.3324	0.713571
$735$	0	0
$736$	39.5124	1.45645
$737$	−3.70079	−0.136320
$738$	0	0
$739$	13.7037	0.504098	0.252049	−	0.967714i	$-0.418896\pi$
$739$	13.7037	0.504098	0.252049	+	0.967714i	$0.418896\pi$
$740$	18.0900	0.665003
$741$	0	0
$742$	2.96395	0.108810
$743$	41.1053	1.50801	0.754003	−	0.656871i	$-0.228120\pi$
$743$	41.1053	1.50801	0.754003	+	0.656871i	$0.228120\pi$
$744$	0	0
$745$	0.497202	0.0182161
$746$	−45.5679	−1.66836
$747$	0	0
$748$	6.28802	0.229913
$749$	13.5630	0.495583
$750$	0	0
$751$	−24.2694	−0.885604	−0.442802	−	0.896619i	$-0.646016\pi$
$751$	−24.2694	−0.885604	−0.442802	+	0.896619i	$0.646016\pi$
$752$	54.7673	1.99716
$753$	0	0
$754$	−52.1026	−1.89746
$755$	−15.5124	−0.564555
$756$	0	0
$757$	−11.5415	−0.419485	−0.209742	−	0.977757i	$-0.567263\pi$
$757$	−11.5415	−0.419485	−0.209742	+	0.977757i	$0.567263\pi$
$758$	−20.7368	−0.753196
$759$	0	0
$760$	20.3386	0.737760
$761$	−29.0152	−1.05180	−0.525901	−	0.850546i	$-0.676272\pi$
$761$	−29.0152	−1.05180	−0.525901	+	0.850546i	$0.676272\pi$
$762$	0	0
$763$	−4.56864	−0.165396
$764$	−11.8456	−0.428558
$765$	0	0
$766$	−26.2188	−0.947324
$767$	−22.2367	−0.802922
$768$	0	0
$769$	2.69742	0.0972715	0.0486357	−	0.998817i	$-0.484513\pi$
$769$	2.69742	0.0972715	0.0486357	+	0.998817i	$0.484513\pi$
$770$	−7.57201	−0.272876
$771$	0	0
$772$	0.363604	0.0130864
$773$	0.835165	0.0300388	0.0150194	−	0.999887i	$-0.495219\pi$
$773$	0.835165	0.0300388	0.0150194	+	0.999887i	$0.495219\pi$
$774$	0	0
$775$	−6.04502	−0.217143
$776$	−5.04502	−0.181106
$777$	0	0
$778$	37.0665	1.32890
$779$	49.6918	1.78039
$780$	0	0
$781$	9.70369	0.347225
$782$	49.4737	1.76918
$783$	0	0
$784$	25.1198	0.897136
$785$	16.7666	0.598427
$786$	0	0
$787$	−31.3836	−1.11871	−0.559353	−	0.828929i	$-0.688950\pi$
$787$	−31.3836	−1.11871	−0.559353	+	0.828929i	$0.688950\pi$
$788$	15.1648	0.540225
$789$	0	0
$790$	16.3684	0.582362
$791$	12.4578	0.442948
$792$	0	0
$793$	35.3657	1.25587
$794$	−31.2715	−1.10979
$795$	0	0
$796$	−26.6468	−0.944471
$797$	26.6773	0.944957	0.472479	−	0.881342i	$-0.343359\pi$
$797$	26.6773	0.944957	0.472479	+	0.881342i	$0.343359\pi$
$798$	0	0
$799$	53.1745	1.88118
$800$	41.7458	1.47594
$801$	0	0
$802$	40.4764	1.42927
$803$	14.0305	0.495124
$804$	0	0
$805$	−25.1648	−0.886944
$806$	5.44322	0.191729
$807$	0	0
$808$	13.1738	0.463453
$809$	−22.5485	−0.792762	−0.396381	−	0.918086i	$-0.629734\pi$
$809$	−22.5485	−0.792762	−0.396381	+	0.918086i	$0.629734\pi$
$810$	0	0
$811$	17.2936	0.607261	0.303631	−	0.952790i	$-0.401801\pi$
$811$	17.2936	0.607261	0.303631	+	0.952790i	$0.401801\pi$
$812$	−18.5277	−0.650193
$813$	0	0
$814$	6.40717	0.224571
$815$	15.0450	0.527004
$816$	0	0
$817$	−26.7339	−0.935302
$818$	−27.7729	−0.971056
$819$	0	0
$820$	39.4689	1.37831
$821$	17.7424	0.619215	0.309608	−	0.950864i	$-0.399802\pi$
$821$	17.7424	0.619215	0.309608	+	0.950864i	$0.399802\pi$
$822$	0	0
$823$	50.4681	1.75921	0.879603	−	0.475708i	$-0.157808\pi$
$823$	50.4681	1.75921	0.879603	+	0.475708i	$0.157808\pi$
$824$	−10.1198	−0.352541
$825$	0	0
$826$	−18.7202	−0.651358
$827$	20.0513	0.697251	0.348626	−	0.937262i	$-0.386648\pi$
$827$	20.0513	0.697251	0.348626	+	0.937262i	$0.386648\pi$
$828$	0	0
$829$	−24.1496	−0.838750	−0.419375	−	0.907813i	$-0.637751\pi$
$829$	−24.1496	−0.838750	−0.419375	+	0.907813i	$0.637751\pi$
$830$	8.36842	0.290472
$831$	0	0
$832$	−9.58993	−0.332471
$833$	24.3892	0.845037
$834$	0	0
$835$	−19.0152	−0.658049
$836$	−8.28129	−0.286414
$837$	0	0
$838$	−46.2922	−1.59914
$839$	−11.1857	−0.386172	−0.193086	−	0.981182i	$-0.561850\pi$
$839$	−11.1857	−0.386172	−0.193086	+	0.981182i	$0.561850\pi$
$840$	0	0
$841$	62.6233	2.15942
$842$	6.50135	0.224051
$843$	0	0
$844$	−37.3885	−1.28696
$845$	−14.7666	−0.507987
$846$	0	0
$847$	13.4246	0.461276
$848$	−5.76036	−0.197812
$849$	0	0
$850$	52.2701	1.79285
$851$	21.2936	0.729936
$852$	0	0
$853$	−21.9584	−0.751840	−0.375920	−	0.926652i	$-0.622673\pi$
$853$	−21.9584	−0.751840	−0.375920	+	0.926652i	$0.622673\pi$
$854$	29.7729	1.01881
$855$	0	0
$856$	−10.2486	−0.350290
$857$	24.3297	0.831086	0.415543	−	0.909574i	$-0.363592\pi$
$857$	24.3297	0.831086	0.415543	+	0.909574i	$0.363592\pi$
$858$	0	0
$859$	11.3144	0.386044	0.193022	−	0.981194i	$-0.438171\pi$
$859$	11.3144	0.386044	0.193022	+	0.981194i	$0.438171\pi$
$860$	−21.2340	−0.724075
$861$	0	0
$862$	54.0305	1.84028
$863$	41.5333	1.41381	0.706904	−	0.707309i	$-0.250091\pi$
$863$	41.5333	1.41381	0.706904	+	0.707309i	$0.250091\pi$
$864$	0	0
$865$	−64.1205	−2.18016
$866$	73.9321	2.51232
$867$	0	0
$868$	1.93561	0.0656989
$869$	2.44882	0.0830705
$870$	0	0
$871$	11.7008	0.396466
$872$	3.45219	0.116906
$873$	0	0
$874$	−65.1565	−2.20395
$875$	−4.59620	−0.155380
$876$	0	0
$877$	−0.0506115	−0.00170903	−0.000854514	−	1.00000i	$-0.500272\pi$
$877$	−0.0506115	−0.00170903	−0.000854514		1.00000i	$0.500272\pi$
$878$	25.7175	0.867922
$879$	0	0
$880$	14.7160	0.496077
$881$	−1.46115	−0.0492274	−0.0246137	−	0.999697i	$-0.507836\pi$
$881$	−1.46115	−0.0492274	−0.0246137	+	0.999697i	$0.507836\pi$
$882$	0	0
$883$	7.70079	0.259152	0.129576	−	0.991569i	$-0.458638\pi$
$883$	7.70079	0.259152	0.129576	+	0.991569i	$0.458638\pi$
$884$	−19.8809	−0.668665
$885$	0	0
$886$	−0.851843	−0.0286182
$887$	−52.5366	−1.76401	−0.882004	−	0.471243i	$-0.843806\pi$
$887$	−52.5366	−1.76401	−0.882004	+	0.471243i	$0.843806\pi$
$888$	0	0
$889$	−8.62598	−0.289306
$890$	−37.1053	−1.24377
$891$	0	0
$892$	−3.33237	−0.111576
$893$	−70.0305	−2.34348
$894$	0	0
$895$	24.7368	0.826861
$896$	10.2050	0.340926
$897$	0	0
$898$	−37.1774	−1.24062
$899$	−9.57201	−0.319244
$900$	0	0
$901$	−5.59283	−0.186324
$902$	13.9792	0.465456
$903$	0	0
$904$	−9.41344	−0.313086
$905$	−64.9169	−2.15791
$906$	0	0
$907$	−11.4134	−0.378977	−0.189488	−	0.981883i	$-0.560683\pi$
$907$	−11.4134	−0.378977	−0.189488	+	0.981883i	$0.560683\pi$
$908$	−31.5816	−1.04807
$909$	0	0
$910$	23.9404	0.793617
$911$	10.2188	0.338564	0.169282	−	0.985568i	$-0.445855\pi$
$911$	10.2188	0.338564	0.169282	+	0.985568i	$0.445855\pi$
$912$	0	0
$913$	1.25197	0.0414341
$914$	−30.7368	−1.01668
$915$	0	0
$916$	27.5928	0.911693
$917$	−1.71198	−0.0565345
$918$	0	0
$919$	−21.7729	−0.718221	−0.359111	−	0.933295i	$-0.616920\pi$
$919$	−21.7729	−0.718221	−0.359111	+	0.933295i	$0.616920\pi$
$920$	19.0152	0.626914
$921$	0	0
$922$	35.9017	1.18236
$923$	−30.6802	−1.00985
$924$	0	0
$925$	22.4972	0.739703
$926$	33.1053	1.08791
$927$	0	0
$928$	66.1026	2.16992
$929$	41.8116	1.37180	0.685898	−	0.727698i	$-0.259410\pi$
$929$	41.8116	1.37180	0.685898	+	0.727698i	$0.259410\pi$
$930$	0	0
$931$	−32.1205	−1.05271
$932$	6.75285	0.221197
$933$	0	0
$934$	26.0680	0.852970
$935$	14.2880	0.467268
$936$	0	0
$937$	−30.4793	−0.995715	−0.497857	−	0.867259i	$-0.665880\pi$
$937$	−30.4793	−0.995715	−0.497857	+	0.867259i	$0.665880\pi$
$938$	9.85039	0.321627
$939$	0	0
$940$	−55.6233	−1.81423
$941$	14.1801	0.462257	0.231128	−	0.972923i	$-0.425758\pi$
$941$	14.1801	0.462257	0.231128	+	0.972923i	$0.425758\pi$
$942$	0	0
$943$	46.4585	1.51290
$944$	36.3822	1.18414
$945$	0	0
$946$	−7.52072	−0.244520
$947$	42.1413	1.36941	0.684704	−	0.728821i	$-0.259931\pi$
$947$	42.1413	1.36941	0.684704	+	0.728821i	$0.259931\pi$
$948$	0	0
$949$	−44.3601	−1.43999
$950$	−68.8394	−2.23345
$951$	0	0
$952$	6.14961	0.199310
$953$	35.9709	1.16521	0.582606	−	0.812755i	$-0.302033\pi$
$953$	35.9709	1.16521	0.582606	+	0.812755i	$0.302033\pi$
$954$	0	0
$955$	−26.9162	−0.870989
$956$	15.1953	0.491451
$957$	0	0
$958$	53.7354	1.73611
$959$	8.82397	0.284941
$960$	0	0
$961$	1.00000	0.0322581
$962$	−20.2576	−0.653130
$963$	0	0
$964$	−18.2217	−0.586882
$965$	0.826202	0.0265964
$966$	0	0
$967$	−17.1648	−0.551984	−0.275992	−	0.961160i	$-0.589006\pi$
$967$	−17.1648	−0.551984	−0.275992	+	0.961160i	$0.589006\pi$
$968$	−10.1440	−0.326041
$969$	0	0
$970$	31.1994	1.00175
$971$	−17.2936	−0.554979	−0.277489	−	0.960729i	$-0.589502\pi$
$971$	−17.2936	−0.554979	−0.277489	+	0.960729i	$0.589502\pi$
$972$	0	0
$973$	−18.4764	−0.592325
$974$	9.05398	0.290108
$975$	0	0
$976$	−57.8629	−1.85215
$977$	11.4439	0.366123	0.183061	−	0.983102i	$-0.441399\pi$
$977$	11.4439	0.366123	0.183061	+	0.983102i	$0.441399\pi$
$978$	0	0
$979$	−5.55118	−0.177417
$980$	−25.5124	−0.814965
$981$	0	0
$982$	1.13148	0.0361069
$983$	15.8891	0.506785	0.253392	−	0.967364i	$-0.418454\pi$
$983$	15.8891	0.506785	0.253392	+	0.967364i	$0.418454\pi$
$984$	0	0
$985$	34.4585	1.09794
$986$	82.7673	2.63585
$987$	0	0
$988$	26.1830	0.832991
$989$	−24.9944	−0.794776
$990$	0	0
$991$	5.29362	0.168157	0.0840786	−	0.996459i	$-0.473205\pi$
$991$	5.29362	0.168157	0.0840786	+	0.996459i	$0.473205\pi$
$992$	−6.90582	−0.219260
$993$	0	0
$994$	−25.8283	−0.819225
$995$	−60.5485	−1.91952
$996$	0	0
$997$	36.6683	1.16130	0.580648	−	0.814154i	$-0.302799\pi$
$997$	36.6683	1.16130	0.580648	+	0.814154i	$0.302799\pi$
$998$	68.2493	2.16039
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	279.2.a.c.1.1		3
3.2	odd	2		93.2.a.b.1.3	✓	3
4.3	odd	2		4464.2.a.bq.1.3		3
5.4	even	2		6975.2.a.bb.1.3		3
12.11	even	2		1488.2.a.t.1.1		3
15.2	even	4		2325.2.c.n.1024.5		6
15.8	even	4		2325.2.c.n.1024.2		6
15.14	odd	2		2325.2.a.s.1.1		3
21.20	even	2		4557.2.a.v.1.3		3
24.5	odd	2		5952.2.a.bz.1.3		3
24.11	even	2		5952.2.a.cf.1.3		3
31.30	odd	2		8649.2.a.p.1.1		3
93.92	even	2		2883.2.a.f.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.a.b.1.3	✓	3	3.2	odd	2
279.2.a.c.1.1		3	1.1	even	1	trivial
1488.2.a.t.1.1		3	12.11	even	2
2325.2.a.s.1.1		3	15.14	odd	2
2325.2.c.n.1024.2		6	15.8	even	4
2325.2.c.n.1024.5		6	15.2	even	4
2883.2.a.f.1.3		3	93.92	even	2
4464.2.a.bq.1.3		3	4.3	odd	2
4557.2.a.v.1.3		3	21.20	even	2
5952.2.a.bz.1.3		3	24.5	odd	2
5952.2.a.cf.1.3		3	24.11	even	2
6975.2.a.bb.1.3		3	5.4	even	2
8649.2.a.p.1.1		3	31.30	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:	\( N \)	\(=\)	\( 279 = 3^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	279.a (trivial)

\(f(q)\)	\(=\)	\(q-1.86081 q^{2} +1.46260 q^{4} +3.32340 q^{5} -1.32340 q^{7} +1.00000 q^{8} -6.18421 q^{10} -0.925197 q^{11} +2.92520 q^{13} +2.46260 q^{14} -4.78600 q^{16} -4.64681 q^{17} +6.11982 q^{19} +4.86081 q^{20} +1.72161 q^{22} +5.72161 q^{23} +6.04502 q^{25} -5.44322 q^{26} -1.93561 q^{28} +9.57201 q^{29} -1.00000 q^{31} +6.90582 q^{32} +8.64681 q^{34} -4.39821 q^{35} +3.72161 q^{37} -11.3878 q^{38} +3.32340 q^{40} +8.11982 q^{41} -4.36842 q^{43} -1.35319 q^{44} -10.6468 q^{46} -11.4432 q^{47} -5.24860 q^{49} -11.2486 q^{50} +4.27839 q^{52} +1.20359 q^{53} -3.07480 q^{55} -1.32340 q^{56} -17.8116 q^{58} -7.60179 q^{59} +12.0900 q^{61} +1.86081 q^{62} -3.27839 q^{64} +9.72161 q^{65} +4.00000 q^{67} -6.79641 q^{68} +8.18421 q^{70} -10.4882 q^{71} -15.1648 q^{73} -6.92520 q^{74} +8.95084 q^{76} +1.22441 q^{77} -2.64681 q^{79} -15.9058 q^{80} -15.1094 q^{82} -1.35319 q^{83} -15.4432 q^{85} +8.12878 q^{86} -0.925197 q^{88} +6.00000 q^{89} -3.87122 q^{91} +8.36842 q^{92} +21.2936 q^{94} +20.3386 q^{95} -5.04502 q^{97} +9.76663 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{4} + 2 q^{5} + 4 q^{7} + 3 q^{8} - 5 q^{10} + 2 q^{11} + 4 q^{13} + 5 q^{14} - 4 q^{16} + 2 q^{17} + 4 q^{19} + 9 q^{20} - 6 q^{22} + 6 q^{23} - q^{25} + 6 q^{26} - 5 q^{28} + 8 q^{29} - 3 q^{31}+ \cdots - q^{98}+O(q^{100}) \) 3 * q + 2 * q^4 + 2 * q^5 + 4 * q^7 + 3 * q^8 - 5 * q^10 + 2 * q^11 + 4 * q^13 + 5 * q^14 - 4 * q^16 + 2 * q^17 + 4 * q^19 + 9 * q^20 - 6 * q^22 + 6 * q^23 - q^25 + 6 * q^26 - 5 * q^28 + 8 * q^29 - 3 * q^31 - 4 * q^32 + 10 * q^34 - 10 * q^35 - 27 * q^38 + 2 * q^40 + 10 * q^41 + 14 * q^43 - 20 * q^44 - 16 * q^46 - 12 * q^47 - 3 * q^49 - 21 * q^50 + 24 * q^52 + 10 * q^53 - 14 * q^55 + 4 * q^56 - 4 * q^58 - 26 * q^59 - 2 * q^61 - 21 * q^64 + 18 * q^65 + 12 * q^67 - 14 * q^68 + 11 * q^70 + 10 * q^71 - 12 * q^73 - 16 * q^74 - 17 * q^76 + 18 * q^77 + 8 * q^79 - 23 * q^80 - 27 * q^82 - 20 * q^83 - 24 * q^85 + 26 * q^86 + 2 * q^88 + 18 * q^89 - 10 * q^91 - 2 * q^92 + 32 * q^94 + 10 * q^95 + 4 * q^97 - q^98

Introduction

L-functions

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