LMFDB - Embedded newform 855.2.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [855,2,Mod(1,855)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("855.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,10,-2,0,-2,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.64575 q^{2} +5.00000 q^{4} -1.00000 q^{5} +1.64575 q^{7} +7.93725 q^{8} -2.64575 q^{10} -0.354249 q^{11} -0.354249 q^{13} +4.35425 q^{14} +11.0000 q^{16} +4.00000 q^{17} -1.00000 q^{19} -5.00000 q^{20} -0.937254 q^{22} -9.29150 q^{23} +1.00000 q^{25} -0.937254 q^{26} +8.22876 q^{28} -8.93725 q^{29} +6.00000 q^{31} +13.2288 q^{32} +10.5830 q^{34} -1.64575 q^{35} +3.64575 q^{37} -2.64575 q^{38} -7.93725 q^{40} +9.64575 q^{41} +5.64575 q^{43} -1.77124 q^{44} -24.5830 q^{46} +1.29150 q^{47} -4.29150 q^{49} +2.64575 q^{50} -1.77124 q^{52} -11.2915 q^{53} +0.354249 q^{55} +13.0627 q^{56} -23.6458 q^{58} -11.2915 q^{59} -11.2915 q^{61} +15.8745 q^{62} +13.0000 q^{64} +0.354249 q^{65} -6.58301 q^{67} +20.0000 q^{68} -4.35425 q^{70} -7.29150 q^{71} +10.0000 q^{73} +9.64575 q^{74} -5.00000 q^{76} -0.583005 q^{77} +6.58301 q^{79} -11.0000 q^{80} +25.5203 q^{82} -6.00000 q^{83} -4.00000 q^{85} +14.9373 q^{86} -2.81176 q^{88} +16.9373 q^{89} -0.583005 q^{91} -46.4575 q^{92} +3.41699 q^{94} +1.00000 q^{95} -2.93725 q^{97} -11.3542 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{11} - 6 q^{13} + 14 q^{14} + 22 q^{16} + 8 q^{17} - 2 q^{19} - 10 q^{20} + 14 q^{22} - 8 q^{23} + 2 q^{25} + 14 q^{26} - 10 q^{28} - 2 q^{29} + 12 q^{31} + 2 q^{35}+ \cdots - 28 q^{98}+O(q^{100}) $ 2 * q + 10 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^11 - 6 * q^13 + 14 * q^14 + 22 * q^16 + 8 * q^17 - 2 * q^19 - 10 * q^20 + 14 * q^22 - 8 * q^23 + 2 * q^25 + 14 * q^26 - 10 * q^28 - 2 * q^29 + 12 * q^31 + 2 * q^35 + 2 * q^37 + 14 * q^41 + 6 * q^43 - 30 * q^44 - 28 * q^46 - 8 * q^47 + 2 * q^49 - 30 * q^52 - 12 * q^53 + 6 * q^55 + 42 * q^56 - 42 * q^58 - 12 * q^59 - 12 * q^61 + 26 * q^64 + 6 * q^65 + 8 * q^67 + 40 * q^68 - 14 * q^70 - 4 * q^71 + 20 * q^73 + 14 * q^74 - 10 * q^76 + 20 * q^77 - 8 * q^79 - 22 * q^80 + 14 * q^82 - 12 * q^83 - 8 * q^85 + 14 * q^86 + 42 * q^88 + 18 * q^89 + 20 * q^91 - 40 * q^92 + 28 * q^94 + 2 * q^95 + 10 * q^97 - 28 * 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 $

$2$	2.64575	1.87083	0.935414	−	0.353553i	$-0.115027\pi$
$2$	2.64575	1.87083	0.935414	+	0.353553i	$0.115027\pi$
$3$	0	0
$3$	0	0
$4$	5.00000	2.50000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.64575	0.622036	0.311018	−	0.950404i	$-0.399330\pi$
$7$	1.64575	0.622036	0.311018	+	0.950404i	$0.399330\pi$
$8$	7.93725	2.80624
$9$	0	0
$10$	−2.64575	−0.836660
$11$	−0.354249	−0.106810	−0.0534050	−	0.998573i	$-0.517007\pi$
$11$	−0.354249	−0.106810	−0.0534050	+	0.998573i	$0.517007\pi$
$12$	0	0
$13$	−0.354249	−0.0982509	−0.0491255	−	0.998793i	$-0.515643\pi$
$13$	−0.354249	−0.0982509	−0.0491255	+	0.998793i	$0.515643\pi$
$14$	4.35425	1.16372
$15$	0	0
$16$	11.0000	2.75000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−5.00000	−1.11803
$21$	0	0
$22$	−0.937254	−0.199823
$23$	−9.29150	−1.93741	−0.968706	−	0.248211i	$-0.920158\pi$
$23$	−9.29150	−1.93741	−0.968706	+	0.248211i	$0.920158\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.937254	−0.183811
$27$	0	0
$28$	8.22876	1.55509
$29$	−8.93725	−1.65961	−0.829803	−	0.558056i	$-0.811547\pi$
$29$	−8.93725	−1.65961	−0.829803	+	0.558056i	$0.811547\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	13.2288	2.33854
$33$	0	0
$34$	10.5830	1.81497
$35$	−1.64575	−0.278183
$36$	0	0
$37$	3.64575	0.599358	0.299679	−	0.954040i	$-0.403120\pi$
$37$	3.64575	0.599358	0.299679	+	0.954040i	$0.403120\pi$
$38$	−2.64575	−0.429198
$39$	0	0
$40$	−7.93725	−1.25499
$41$	9.64575	1.50641	0.753207	−	0.657784i	$-0.228506\pi$
$41$	9.64575	1.50641	0.753207	+	0.657784i	$0.228506\pi$
$42$	0	0
$43$	5.64575	0.860969	0.430485	−	0.902598i	$-0.358343\pi$
$43$	5.64575	0.860969	0.430485	+	0.902598i	$0.358343\pi$
$44$	−1.77124	−0.267025
$45$	0	0
$46$	−24.5830	−3.62457
$47$	1.29150	0.188385	0.0941925	−	0.995554i	$-0.469973\pi$
$47$	1.29150	0.188385	0.0941925	+	0.995554i	$0.469973\pi$
$48$	0	0
$49$	−4.29150	−0.613072
$50$	2.64575	0.374166
$51$	0	0
$52$	−1.77124	−0.245627
$53$	−11.2915	−1.55101	−0.775504	−	0.631343i	$-0.782504\pi$
$53$	−11.2915	−1.55101	−0.775504	+	0.631343i	$0.782504\pi$
$54$	0	0
$55$	0.354249	0.0477669
$56$	13.0627	1.74558
$57$	0	0
$58$	−23.6458	−3.10484
$59$	−11.2915	−1.47003	−0.735014	−	0.678052i	$-0.762825\pi$
$59$	−11.2915	−1.47003	−0.735014	+	0.678052i	$0.762825\pi$
$60$	0	0
$61$	−11.2915	−1.44573	−0.722864	−	0.690990i	$-0.757175\pi$
$61$	−11.2915	−1.44573	−0.722864	+	0.690990i	$0.757175\pi$
$62$	15.8745	2.01606
$63$	0	0
$64$	13.0000	1.62500
$65$	0.354249	0.0439391
$66$	0	0
$67$	−6.58301	−0.804242	−0.402121	−	0.915587i	$-0.631727\pi$
$67$	−6.58301	−0.804242	−0.402121	+	0.915587i	$0.631727\pi$
$68$	20.0000	2.42536
$69$	0	0
$70$	−4.35425	−0.520432
$71$	−7.29150	−0.865342	−0.432671	−	0.901552i	$-0.642429\pi$
$71$	−7.29150	−0.865342	−0.432671	+	0.901552i	$0.642429\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	9.64575	1.12130
$75$	0	0
$76$	−5.00000	−0.573539
$77$	−0.583005	−0.0664396
$78$	0	0
$79$	6.58301	0.740646	0.370323	−	0.928903i	$-0.379247\pi$
$79$	6.58301	0.740646	0.370323	+	0.928903i	$0.379247\pi$
$80$	−11.0000	−1.22984
$81$	0	0
$82$	25.5203	2.81824
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	14.9373	1.61073
$87$	0	0
$88$	−2.81176	−0.299735
$89$	16.9373	1.79535	0.897673	−	0.440663i	$-0.145257\pi$
$89$	16.9373	1.79535	0.897673	+	0.440663i	$0.145257\pi$
$90$	0	0
$91$	−0.583005	−0.0611156
$92$	−46.4575	−4.84353
$93$	0	0
$94$	3.41699	0.352436
$95$	1.00000	0.102598
$96$	0	0
$97$	−2.93725	−0.298233	−0.149116	−	0.988820i	$-0.547643\pi$
$97$	−2.93725	−0.298233	−0.149116	+	0.988820i	$0.547643\pi$
$98$	−11.3542	−1.14695
$99$	0	0
$100$	5.00000	0.500000
$101$	−9.29150	−0.924539	−0.462270	−	0.886739i	$-0.652965\pi$
$101$	−9.29150	−0.924539	−0.462270	+	0.886739i	$0.652965\pi$
$102$	0	0
$103$	−10.5830	−1.04277	−0.521387	−	0.853320i	$-0.674585\pi$
$103$	−10.5830	−1.04277	−0.521387	+	0.853320i	$0.674585\pi$
$104$	−2.81176	−0.275716
$105$	0	0
$106$	−29.8745	−2.90167
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	5.29150	0.506834	0.253417	−	0.967357i	$-0.418446\pi$
$109$	5.29150	0.506834	0.253417	+	0.967357i	$0.418446\pi$
$110$	0.937254	0.0893637
$111$	0	0
$112$	18.1033	1.71060
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	9.29150	0.866437
$116$	−44.6863	−4.14902
$117$	0	0
$118$	−29.8745	−2.75017
$119$	6.58301	0.603463
$120$	0	0
$121$	−10.8745	−0.988592
$122$	−29.8745	−2.70471
$123$	0	0
$124$	30.0000	2.69408
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	7.93725	0.701561
$129$	0	0
$130$	0.937254	0.0822026
$131$	14.2288	1.24317	0.621586	−	0.783346i	$-0.286489\pi$
$131$	14.2288	1.24317	0.621586	+	0.783346i	$0.286489\pi$
$132$	0	0
$133$	−1.64575	−0.142705
$134$	−17.4170	−1.50460
$135$	0	0
$136$	31.7490	2.72246
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	17.8745	1.51610	0.758048	−	0.652199i	$-0.226153\pi$
$139$	17.8745	1.51610	0.758048	+	0.652199i	$0.226153\pi$
$140$	−8.22876	−0.695457
$141$	0	0
$142$	−19.2915	−1.61891
$143$	0.125492	0.0104942
$144$	0	0
$145$	8.93725	0.742199
$146$	26.4575	2.18964
$147$	0	0
$148$	18.2288	1.49839
$149$	20.5830	1.68623	0.843113	−	0.537737i	$-0.180721\pi$
$149$	20.5830	1.68623	0.843113	+	0.537737i	$0.180721\pi$
$150$	0	0
$151$	−8.58301	−0.698475	−0.349238	−	0.937034i	$-0.613559\pi$
$151$	−8.58301	−0.698475	−0.349238	+	0.937034i	$0.613559\pi$
$152$	−7.93725	−0.643796
$153$	0	0
$154$	−1.54249	−0.124297
$155$	−6.00000	−0.481932
$156$	0	0
$157$	−13.2915	−1.06078	−0.530389	−	0.847755i	$-0.677954\pi$
$157$	−13.2915	−1.06078	−0.530389	+	0.847755i	$0.677954\pi$
$158$	17.4170	1.38562
$159$	0	0
$160$	−13.2288	−1.04583
$161$	−15.2915	−1.20514
$162$	0	0
$163$	−2.35425	−0.184399	−0.0921995	−	0.995741i	$-0.529390\pi$
$163$	−2.35425	−0.184399	−0.0921995	+	0.995741i	$0.529390\pi$
$164$	48.2288	3.76603
$165$	0	0
$166$	−15.8745	−1.23210
$167$	21.2915	1.64759	0.823793	−	0.566891i	$-0.191854\pi$
$167$	21.2915	1.64759	0.823793	+	0.566891i	$0.191854\pi$
$168$	0	0
$169$	−12.8745	−0.990347
$170$	−10.5830	−0.811679
$171$	0	0
$172$	28.2288	2.15242
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	1.64575	0.124407
$176$	−3.89674	−0.293727
$177$	0	0
$178$	44.8118	3.35878
$179$	7.29150	0.544992	0.272496	−	0.962157i	$-0.412151\pi$
$179$	7.29150	0.544992	0.272496	+	0.962157i	$0.412151\pi$
$180$	0	0
$181$	17.2915	1.28527	0.642634	−	0.766174i	$-0.277842\pi$
$181$	17.2915	1.28527	0.642634	+	0.766174i	$0.277842\pi$
$182$	−1.54249	−0.114337
$183$	0	0
$184$	−73.7490	−5.43685
$185$	−3.64575	−0.268041
$186$	0	0
$187$	−1.41699	−0.103621
$188$	6.45751	0.470963
$189$	0	0
$190$	2.64575	0.191943
$191$	−6.22876	−0.450697	−0.225349	−	0.974278i	$-0.572352\pi$
$191$	−6.22876	−0.450697	−0.225349	+	0.974278i	$0.572352\pi$
$192$	0	0
$193$	−11.6458	−0.838280	−0.419140	−	0.907922i	$-0.637668\pi$
$193$	−11.6458	−0.838280	−0.419140	+	0.907922i	$0.637668\pi$
$194$	−7.77124	−0.557943
$195$	0	0
$196$	−21.4575	−1.53268
$197$	25.1660	1.79300	0.896502	−	0.443040i	$-0.146100\pi$
$197$	25.1660	1.79300	0.896502	+	0.443040i	$0.146100\pi$
$198$	0	0
$199$	7.29150	0.516881	0.258440	−	0.966027i	$-0.416791\pi$
$199$	7.29150	0.516881	0.258440	+	0.966027i	$0.416791\pi$
$200$	7.93725	0.561249
$201$	0	0
$202$	−24.5830	−1.72965
$203$	−14.7085	−1.03233
$204$	0	0
$205$	−9.64575	−0.673688
$206$	−28.0000	−1.95085
$207$	0	0
$208$	−3.89674	−0.270190
$209$	0.354249	0.0245039
$210$	0	0
$211$	2.58301	0.177821	0.0889107	−	0.996040i	$-0.471661\pi$
$211$	2.58301	0.177821	0.0889107	+	0.996040i	$0.471661\pi$
$212$	−56.4575	−3.87752
$213$	0	0
$214$	0	0
$215$	−5.64575	−0.385037
$216$	0	0
$217$	9.87451	0.670325
$218$	14.0000	0.948200
$219$	0	0
$220$	1.77124	0.119417
$221$	−1.41699	−0.0953174
$222$	0	0
$223$	2.58301	0.172971	0.0864854	−	0.996253i	$-0.472436\pi$
$223$	2.58301	0.172971	0.0864854	+	0.996253i	$0.472436\pi$
$224$	21.7712	1.45465
$225$	0	0
$226$	10.5830	0.703971
$227$	10.7085	0.710748	0.355374	−	0.934724i	$-0.384354\pi$
$227$	10.7085	0.710748	0.355374	+	0.934724i	$0.384354\pi$
$228$	0	0
$229$	8.70850	0.575474	0.287737	−	0.957710i	$-0.407097\pi$
$229$	8.70850	0.575474	0.287737	+	0.957710i	$0.407097\pi$
$230$	24.5830	1.62096
$231$	0	0
$232$	−70.9373	−4.65726
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	−1.29150	−0.0842483
$236$	−56.4575	−3.67507
$237$	0	0
$238$	17.4170	1.12898
$239$	−15.6458	−1.01204	−0.506020	−	0.862522i	$-0.668884\pi$
$239$	−15.6458	−1.01204	−0.506020	+	0.862522i	$0.668884\pi$
$240$	0	0
$241$	16.5830	1.06821	0.534103	−	0.845420i	$-0.320650\pi$
$241$	16.5830	1.06821	0.534103	+	0.845420i	$0.320650\pi$
$242$	−28.7712	−1.84949
$243$	0	0
$244$	−56.4575	−3.61432
$245$	4.29150	0.274174
$246$	0	0
$247$	0.354249	0.0225403
$248$	47.6235	3.02410
$249$	0	0
$250$	−2.64575	−0.167332
$251$	16.3542	1.03227	0.516136	−	0.856507i	$-0.327370\pi$
$251$	16.3542	1.03227	0.516136	+	0.856507i	$0.327370\pi$
$252$	0	0
$253$	3.29150	0.206935
$254$	10.5830	0.664037
$255$	0	0
$256$	−5.00000	−0.312500
$257$	5.41699	0.337903	0.168951	−	0.985624i	$-0.445962\pi$
$257$	5.41699	0.337903	0.168951	+	0.985624i	$0.445962\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	1.77124	0.109848
$261$	0	0
$262$	37.6458	2.32576
$263$	−4.58301	−0.282600	−0.141300	−	0.989967i	$-0.545128\pi$
$263$	−4.58301	−0.282600	−0.141300	+	0.989967i	$0.545128\pi$
$264$	0	0
$265$	11.2915	0.693631
$266$	−4.35425	−0.266976
$267$	0	0
$268$	−32.9150	−2.01061
$269$	−4.22876	−0.257832	−0.128916	−	0.991656i	$-0.541150\pi$
$269$	−4.22876	−0.257832	−0.128916	+	0.991656i	$0.541150\pi$
$270$	0	0
$271$	−4.70850	−0.286021	−0.143010	−	0.989721i	$-0.545678\pi$
$271$	−4.70850	−0.286021	−0.143010	+	0.989721i	$0.545678\pi$
$272$	44.0000	2.66789
$273$	0	0
$274$	−15.8745	−0.959014
$275$	−0.354249	−0.0213620
$276$	0	0
$277$	0.583005	0.0350294	0.0175147	−	0.999847i	$-0.494425\pi$
$277$	0.583005	0.0350294	0.0175147	+	0.999847i	$0.494425\pi$
$278$	47.2915	2.83636
$279$	0	0
$280$	−13.0627	−0.780648
$281$	32.2288	1.92261	0.961303	−	0.275493i	$-0.0888409\pi$
$281$	32.2288	1.92261	0.961303	+	0.275493i	$0.0888409\pi$
$282$	0	0
$283$	−11.5203	−0.684808	−0.342404	−	0.939553i	$-0.611241\pi$
$283$	−11.5203	−0.684808	−0.342404	+	0.939553i	$0.611241\pi$
$284$	−36.4575	−2.16336
$285$	0	0
$286$	0.332021	0.0196328
$287$	15.8745	0.937043
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	23.6458	1.38853
$291$	0	0
$292$	50.0000	2.92603
$293$	−5.41699	−0.316464	−0.158232	−	0.987402i	$-0.550579\pi$
$293$	−5.41699	−0.316464	−0.158232	+	0.987402i	$0.550579\pi$
$294$	0	0
$295$	11.2915	0.657417
$296$	28.9373	1.68194
$297$	0	0
$298$	54.4575	3.15464
$299$	3.29150	0.190353
$300$	0	0
$301$	9.29150	0.535553
$302$	−22.7085	−1.30673
$303$	0	0
$304$	−11.0000	−0.630893
$305$	11.2915	0.646550
$306$	0	0
$307$	−24.4575	−1.39586	−0.697932	−	0.716164i	$-0.745896\pi$
$307$	−24.4575	−1.39586	−0.697932	+	0.716164i	$0.745896\pi$
$308$	−2.91503	−0.166099
$309$	0	0
$310$	−15.8745	−0.901611
$311$	−22.9373	−1.30065	−0.650326	−	0.759655i	$-0.725368\pi$
$311$	−22.9373	−1.30065	−0.650326	+	0.759655i	$0.725368\pi$
$312$	0	0
$313$	17.2915	0.977374	0.488687	−	0.872459i	$-0.337476\pi$
$313$	17.2915	0.977374	0.488687	+	0.872459i	$0.337476\pi$
$314$	−35.1660	−1.98453
$315$	0	0
$316$	32.9150	1.85161
$317$	−20.4575	−1.14901	−0.574504	−	0.818502i	$-0.694805\pi$
$317$	−20.4575	−1.14901	−0.574504	+	0.818502i	$0.694805\pi$
$318$	0	0
$319$	3.16601	0.177263
$320$	−13.0000	−0.726722
$321$	0	0
$322$	−40.4575	−2.25461
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−0.354249	−0.0196502
$326$	−6.22876	−0.344979
$327$	0	0
$328$	76.5608	4.22736
$329$	2.12549	0.117182
$330$	0	0
$331$	11.4170	0.627535	0.313767	−	0.949500i	$-0.398409\pi$
$331$	11.4170	0.627535	0.313767	+	0.949500i	$0.398409\pi$
$332$	−30.0000	−1.64646
$333$	0	0
$334$	56.3320	3.08235
$335$	6.58301	0.359668
$336$	0	0
$337$	14.9373	0.813684	0.406842	−	0.913499i	$-0.366630\pi$
$337$	14.9373	0.813684	0.406842	+	0.913499i	$0.366630\pi$
$338$	−34.0627	−1.85277
$339$	0	0
$340$	−20.0000	−1.08465
$341$	−2.12549	−0.115102
$342$	0	0
$343$	−18.5830	−1.00339
$344$	44.8118	2.41609
$345$	0	0
$346$	0	0
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	0	0
$349$	23.1660	1.24005	0.620024	−	0.784583i	$-0.287123\pi$
$349$	23.1660	1.24005	0.620024	+	0.784583i	$0.287123\pi$
$350$	4.35425	0.232744
$351$	0	0
$352$	−4.68627	−0.249779
$353$	−0.583005	−0.0310302	−0.0155151	−	0.999880i	$-0.504939\pi$
$353$	−0.583005	−0.0310302	−0.0155151	+	0.999880i	$0.504939\pi$
$354$	0	0
$355$	7.29150	0.386993
$356$	84.6863	4.48836
$357$	0	0
$358$	19.2915	1.01959
$359$	36.1033	1.90546	0.952729	−	0.303822i	$-0.0982629\pi$
$359$	36.1033	1.90546	0.952729	+	0.303822i	$0.0982629\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	45.7490	2.40451
$363$	0	0
$364$	−2.91503	−0.152789
$365$	−10.0000	−0.523424
$366$	0	0
$367$	1.64575	0.0859075	0.0429538	−	0.999077i	$-0.486323\pi$
$367$	1.64575	0.0859075	0.0429538	+	0.999077i	$0.486323\pi$
$368$	−102.207	−5.32788
$369$	0	0
$370$	−9.64575	−0.501459
$371$	−18.5830	−0.964782
$372$	0	0
$373$	5.06275	0.262139	0.131070	−	0.991373i	$-0.458159\pi$
$373$	5.06275	0.262139	0.131070	+	0.991373i	$0.458159\pi$
$374$	−3.74902	−0.193857
$375$	0	0
$376$	10.2510	0.528654
$377$	3.16601	0.163058
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	5.00000	0.256495
$381$	0	0
$382$	−16.4797	−0.843177
$383$	18.5830	0.949547	0.474774	−	0.880108i	$-0.342530\pi$
$383$	18.5830	0.949547	0.474774	+	0.880108i	$0.342530\pi$
$384$	0	0
$385$	0.583005	0.0297127
$386$	−30.8118	−1.56828
$387$	0	0
$388$	−14.6863	−0.745582
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−37.1660	−1.87957
$392$	−34.0627	−1.72043
$393$	0	0
$394$	66.5830	3.35440
$395$	−6.58301	−0.331227
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	19.2915	0.966996
$399$	0	0
$400$	11.0000	0.550000
$401$	4.93725	0.246555	0.123277	−	0.992372i	$-0.460660\pi$
$401$	4.93725	0.246555	0.123277	+	0.992372i	$0.460660\pi$
$402$	0	0
$403$	−2.12549	−0.105878
$404$	−46.4575	−2.31135
$405$	0	0
$406$	−38.9150	−1.93132
$407$	−1.29150	−0.0640174
$408$	0	0
$409$	−6.70850	−0.331714	−0.165857	−	0.986150i	$-0.553039\pi$
$409$	−6.70850	−0.331714	−0.165857	+	0.986150i	$0.553039\pi$
$410$	−25.5203	−1.26036
$411$	0	0
$412$	−52.9150	−2.60694
$413$	−18.5830	−0.914410
$414$	0	0
$415$	6.00000	0.294528
$416$	−4.68627	−0.229763
$417$	0	0
$418$	0.937254	0.0458426
$419$	−38.9373	−1.90221	−0.951105	−	0.308869i	$-0.900050\pi$
$419$	−38.9373	−1.90221	−0.951105	+	0.308869i	$0.900050\pi$
$420$	0	0
$421$	28.5830	1.39305	0.696525	−	0.717532i	$-0.254728\pi$
$421$	28.5830	1.39305	0.696525	+	0.717532i	$0.254728\pi$
$422$	6.83399	0.332673
$423$	0	0
$424$	−89.6235	−4.35250
$425$	4.00000	0.194029
$426$	0	0
$427$	−18.5830	−0.899295
$428$	0	0
$429$	0	0
$430$	−14.9373	−0.720338
$431$	−3.29150	−0.158546	−0.0792731	−	0.996853i	$-0.525260\pi$
$431$	−3.29150	−0.158546	−0.0792731	+	0.996853i	$0.525260\pi$
$432$	0	0
$433$	−27.6458	−1.32857	−0.664285	−	0.747479i	$-0.731264\pi$
$433$	−27.6458	−1.32857	−0.664285	+	0.747479i	$0.731264\pi$
$434$	26.1255	1.25406
$435$	0	0
$436$	26.4575	1.26709
$437$	9.29150	0.444473
$438$	0	0
$439$	−1.41699	−0.0676295	−0.0338147	−	0.999428i	$-0.510766\pi$
$439$	−1.41699	−0.0676295	−0.0338147	+	0.999428i	$0.510766\pi$
$440$	2.81176	0.134045
$441$	0	0
$442$	−3.74902	−0.178322
$443$	−26.7085	−1.26896	−0.634480	−	0.772940i	$-0.718786\pi$
$443$	−26.7085	−1.26896	−0.634480	+	0.772940i	$0.718786\pi$
$444$	0	0
$445$	−16.9373	−0.802903
$446$	6.83399	0.323599
$447$	0	0
$448$	21.3948	1.01081
$449$	36.2288	1.70974	0.854870	−	0.518842i	$-0.173637\pi$
$449$	36.2288	1.70974	0.854870	+	0.518842i	$0.173637\pi$
$450$	0	0
$451$	−3.41699	−0.160900
$452$	20.0000	0.940721
$453$	0	0
$454$	28.3320	1.32969
$455$	0.583005	0.0273317
$456$	0	0
$457$	0.125492	0.00587027	0.00293514	−	0.999996i	$-0.499066\pi$
$457$	0.125492	0.00587027	0.00293514	+	0.999996i	$0.499066\pi$
$458$	23.0405	1.07661
$459$	0	0
$460$	46.4575	2.16609
$461$	37.7490	1.75815	0.879073	−	0.476686i	$-0.158162\pi$
$461$	37.7490	1.75815	0.879073	+	0.476686i	$0.158162\pi$
$462$	0	0
$463$	−35.5203	−1.65077	−0.825383	−	0.564573i	$-0.809041\pi$
$463$	−35.5203	−1.65077	−0.825383	+	0.564573i	$0.809041\pi$
$464$	−98.3098	−4.56392
$465$	0	0
$466$	47.6235	2.20612
$467$	−19.8745	−0.919683	−0.459841	−	0.888001i	$-0.652094\pi$
$467$	−19.8745	−0.919683	−0.459841	+	0.888001i	$0.652094\pi$
$468$	0	0
$469$	−10.8340	−0.500267
$470$	−3.41699	−0.157614
$471$	0	0
$472$	−89.6235	−4.12526
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	32.9150	1.50866
$477$	0	0
$478$	−41.3948	−1.89335
$479$	−4.35425	−0.198951	−0.0994754	−	0.995040i	$-0.531716\pi$
$479$	−4.35425	−0.198951	−0.0994754	+	0.995040i	$0.531716\pi$
$480$	0	0
$481$	−1.29150	−0.0588875
$482$	43.8745	1.99843
$483$	0	0
$484$	−54.3725	−2.47148
$485$	2.93725	0.133374
$486$	0	0
$487$	17.8745	0.809971	0.404986	−	0.914323i	$-0.367277\pi$
$487$	17.8745	0.809971	0.404986	+	0.914323i	$0.367277\pi$
$488$	−89.6235	−4.05707
$489$	0	0
$490$	11.3542	0.512933
$491$	5.77124	0.260453	0.130226	−	0.991484i	$-0.458430\pi$
$491$	5.77124	0.260453	0.130226	+	0.991484i	$0.458430\pi$
$492$	0	0
$493$	−35.7490	−1.61005
$494$	0.937254	0.0421690
$495$	0	0
$496$	66.0000	2.96349
$497$	−12.0000	−0.538274
$498$	0	0
$499$	−31.0405	−1.38956	−0.694782	−	0.719220i	$-0.744499\pi$
$499$	−31.0405	−1.38956	−0.694782	+	0.719220i	$0.744499\pi$
$500$	−5.00000	−0.223607
$501$	0	0
$502$	43.2693	1.93120
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	9.29150	0.413466
$506$	8.70850	0.387140
$507$	0	0
$508$	20.0000	0.887357
$509$	34.1033	1.51160	0.755800	−	0.654802i	$-0.227248\pi$
$509$	34.1033	1.51160	0.755800	+	0.654802i	$0.227248\pi$
$510$	0	0
$511$	16.4575	0.728038
$512$	−29.1033	−1.28619
$513$	0	0
$514$	14.3320	0.632158
$515$	10.5830	0.466343
$516$	0	0
$517$	−0.457513	−0.0201214
$518$	15.8745	0.697486
$519$	0	0
$520$	2.81176	0.123304
$521$	−22.1033	−0.968362	−0.484181	−	0.874968i	$-0.660882\pi$
$521$	−22.1033	−0.968362	−0.484181	+	0.874968i	$0.660882\pi$
$522$	0	0
$523$	−23.2915	−1.01847	−0.509233	−	0.860629i	$-0.670071\pi$
$523$	−23.2915	−1.01847	−0.509233	+	0.860629i	$0.670071\pi$
$524$	71.1438	3.10793
$525$	0	0
$526$	−12.1255	−0.528697
$527$	24.0000	1.04546
$528$	0	0
$529$	63.3320	2.75357
$530$	29.8745	1.29767
$531$	0	0
$532$	−8.22876	−0.356762
$533$	−3.41699	−0.148006
$534$	0	0
$535$	0	0
$536$	−52.2510	−2.25690
$537$	0	0
$538$	−11.1882	−0.482359
$539$	1.52026	0.0654822
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−12.4575	−0.535096
$543$	0	0
$544$	52.9150	2.26871
$545$	−5.29150	−0.226663
$546$	0	0
$547$	1.87451	0.0801482	0.0400741	−	0.999197i	$-0.487241\pi$
$547$	1.87451	0.0801482	0.0400741	+	0.999197i	$0.487241\pi$
$548$	−30.0000	−1.28154
$549$	0	0
$550$	−0.937254	−0.0399646
$551$	8.93725	0.380740
$552$	0	0
$553$	10.8340	0.460708
$554$	1.54249	0.0655340
$555$	0	0
$556$	89.3725	3.79024
$557$	11.4170	0.483754	0.241877	−	0.970307i	$-0.422237\pi$
$557$	11.4170	0.483754	0.241877	+	0.970307i	$0.422237\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	−18.1033	−0.765003
$561$	0	0
$562$	85.2693	3.59687
$563$	−8.12549	−0.342449	−0.171224	−	0.985232i	$-0.554772\pi$
$563$	−8.12549	−0.342449	−0.171224	+	0.985232i	$0.554772\pi$
$564$	0	0
$565$	−4.00000	−0.168281
$566$	−30.4797	−1.28116
$567$	0	0
$568$	−57.8745	−2.42836
$569$	7.06275	0.296086	0.148043	−	0.988981i	$-0.452703\pi$
$569$	7.06275	0.296086	0.148043	+	0.988981i	$0.452703\pi$
$570$	0	0
$571$	16.7085	0.699229	0.349614	−	0.936894i	$-0.386313\pi$
$571$	16.7085	0.699229	0.349614	+	0.936894i	$0.386313\pi$
$572$	0.627461	0.0262354
$573$	0	0
$574$	42.0000	1.75305
$575$	−9.29150	−0.387482
$576$	0	0
$577$	−13.2915	−0.553332	−0.276666	−	0.960966i	$-0.589230\pi$
$577$	−13.2915	−0.553332	−0.276666	+	0.960966i	$0.589230\pi$
$578$	−2.64575	−0.110049
$579$	0	0
$580$	44.6863	1.85550
$581$	−9.87451	−0.409664
$582$	0	0
$583$	4.00000	0.165663
$584$	79.3725	3.28446
$585$	0	0
$586$	−14.3320	−0.592050
$587$	33.2915	1.37409	0.687044	−	0.726616i	$-0.258908\pi$
$587$	33.2915	1.37409	0.687044	+	0.726616i	$0.258908\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	29.8745	1.22991
$591$	0	0
$592$	40.1033	1.64823
$593$	−3.41699	−0.140319	−0.0701596	−	0.997536i	$-0.522351\pi$
$593$	−3.41699	−0.140319	−0.0701596	+	0.997536i	$0.522351\pi$
$594$	0	0
$595$	−6.58301	−0.269877
$596$	102.915	4.21556
$597$	0	0
$598$	8.70850	0.356117
$599$	−9.41699	−0.384768	−0.192384	−	0.981320i	$-0.561622\pi$
$599$	−9.41699	−0.384768	−0.192384	+	0.981320i	$0.561622\pi$
$600$	0	0
$601$	22.7085	0.926299	0.463149	−	0.886280i	$-0.346719\pi$
$601$	22.7085	0.926299	0.463149	+	0.886280i	$0.346719\pi$
$602$	24.5830	1.00193
$603$	0	0
$604$	−42.9150	−1.74619
$605$	10.8745	0.442112
$606$	0	0
$607$	−43.0405	−1.74696	−0.873480	−	0.486859i	$-0.838142\pi$
$607$	−43.0405	−1.74696	−0.873480	+	0.486859i	$0.838142\pi$
$608$	−13.2288	−0.536497
$609$	0	0
$610$	29.8745	1.20958
$611$	−0.457513	−0.0185090
$612$	0	0
$613$	−30.4575	−1.23017	−0.615084	−	0.788462i	$-0.710878\pi$
$613$	−30.4575	−1.23017	−0.615084	+	0.788462i	$0.710878\pi$
$614$	−64.7085	−2.61142
$615$	0	0
$616$	−4.62746	−0.186446
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	23.2915	0.936165	0.468082	−	0.883685i	$-0.344945\pi$
$619$	23.2915	0.936165	0.468082	+	0.883685i	$0.344945\pi$
$620$	−30.0000	−1.20483
$621$	0	0
$622$	−60.6863	−2.43330
$623$	27.8745	1.11677
$624$	0	0
$625$	1.00000	0.0400000
$626$	45.7490	1.82850
$627$	0	0
$628$	−66.4575	−2.65194
$629$	14.5830	0.581462
$630$	0	0
$631$	−14.5830	−0.580540	−0.290270	−	0.956945i	$-0.593745\pi$
$631$	−14.5830	−0.580540	−0.290270	+	0.956945i	$0.593745\pi$
$632$	52.2510	2.07843
$633$	0	0
$634$	−54.1255	−2.14960
$635$	−4.00000	−0.158735
$636$	0	0
$637$	1.52026	0.0602349
$638$	8.37648	0.331628
$639$	0	0
$640$	−7.93725	−0.313748
$641$	16.9373	0.668981	0.334491	−	0.942399i	$-0.391436\pi$
$641$	16.9373	0.668981	0.334491	+	0.942399i	$0.391436\pi$
$642$	0	0
$643$	−13.6458	−0.538136	−0.269068	−	0.963121i	$-0.586716\pi$
$643$	−13.6458	−0.538136	−0.269068	+	0.963121i	$0.586716\pi$
$644$	−76.4575	−3.01285
$645$	0	0
$646$	−10.5830	−0.416383
$647$	3.41699	0.134336	0.0671680	−	0.997742i	$-0.478604\pi$
$647$	3.41699	0.134336	0.0671680	+	0.997742i	$0.478604\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	−0.937254	−0.0367621
$651$	0	0
$652$	−11.7712	−0.460997
$653$	−9.41699	−0.368515	−0.184258	−	0.982878i	$-0.558988\pi$
$653$	−9.41699	−0.368515	−0.184258	+	0.982878i	$0.558988\pi$
$654$	0	0
$655$	−14.2288	−0.555964
$656$	106.103	4.14264
$657$	0	0
$658$	5.62352	0.219228
$659$	−25.4170	−0.990106	−0.495053	−	0.868863i	$-0.664851\pi$
$659$	−25.4170	−0.990106	−0.495053	+	0.868863i	$0.664851\pi$
$660$	0	0
$661$	1.29150	0.0502336	0.0251168	−	0.999685i	$-0.492004\pi$
$661$	1.29150	0.0502336	0.0251168	+	0.999685i	$0.492004\pi$
$662$	30.2065	1.17401
$663$	0	0
$664$	−47.6235	−1.84815
$665$	1.64575	0.0638195
$666$	0	0
$667$	83.0405	3.21534
$668$	106.458	4.11896
$669$	0	0
$670$	17.4170	0.672877
$671$	4.00000	0.154418
$672$	0	0
$673$	−13.0627	−0.503532	−0.251766	−	0.967788i	$-0.581011\pi$
$673$	−13.0627	−0.503532	−0.251766	+	0.967788i	$0.581011\pi$
$674$	39.5203	1.52226
$675$	0	0
$676$	−64.3725	−2.47587
$677$	−40.4575	−1.55491	−0.777454	−	0.628939i	$-0.783489\pi$
$677$	−40.4575	−1.55491	−0.777454	+	0.628939i	$0.783489\pi$
$678$	0	0
$679$	−4.83399	−0.185511
$680$	−31.7490	−1.21752
$681$	0	0
$682$	−5.62352	−0.215336
$683$	−19.7490	−0.755675	−0.377838	−	0.925872i	$-0.623332\pi$
$683$	−19.7490	−0.755675	−0.377838	+	0.925872i	$0.623332\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	−49.1660	−1.87717
$687$	0	0
$688$	62.1033	2.36766
$689$	4.00000	0.152388
$690$	0	0
$691$	−43.0405	−1.63734	−0.818669	−	0.574265i	$-0.805288\pi$
$691$	−43.0405	−1.63734	−0.818669	+	0.574265i	$0.805288\pi$
$692$	0	0
$693$	0	0
$694$	−68.7895	−2.61122
$695$	−17.8745	−0.678019
$696$	0	0
$697$	38.5830	1.46144
$698$	61.2915	2.31992
$699$	0	0
$700$	8.22876	0.311018
$701$	−16.5830	−0.626331	−0.313166	−	0.949698i	$-0.601390\pi$
$701$	−16.5830	−0.626331	−0.313166	+	0.949698i	$0.601390\pi$
$702$	0	0
$703$	−3.64575	−0.137502
$704$	−4.60523	−0.173566
$705$	0	0
$706$	−1.54249	−0.0580523
$707$	−15.2915	−0.575096
$708$	0	0
$709$	−4.70850	−0.176831	−0.0884157	−	0.996084i	$-0.528180\pi$
$709$	−4.70850	−0.176831	−0.0884157	+	0.996084i	$0.528180\pi$
$710$	19.2915	0.723997
$711$	0	0
$712$	134.435	5.03818
$713$	−55.7490	−2.08782
$714$	0	0
$715$	−0.125492	−0.00469314
$716$	36.4575	1.36248
$717$	0	0
$718$	95.5203	3.56478
$719$	−32.8118	−1.22367	−0.611836	−	0.790985i	$-0.709569\pi$
$719$	−32.8118	−1.22367	−0.611836	+	0.790985i	$0.709569\pi$
$720$	0	0
$721$	−17.4170	−0.648643
$722$	2.64575	0.0984647
$723$	0	0
$724$	86.4575	3.21317
$725$	−8.93725	−0.331921
$726$	0	0
$727$	14.1033	0.523061	0.261531	−	0.965195i	$-0.415773\pi$
$727$	14.1033	0.523061	0.261531	+	0.965195i	$0.415773\pi$
$728$	−4.62746	−0.171505
$729$	0	0
$730$	−26.4575	−0.979236
$731$	22.5830	0.835263
$732$	0	0
$733$	7.41699	0.273953	0.136976	−	0.990574i	$-0.456262\pi$
$733$	7.41699	0.273953	0.136976	+	0.990574i	$0.456262\pi$
$734$	4.35425	0.160718
$735$	0	0
$736$	−122.915	−4.53071
$737$	2.33202	0.0859011
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−18.2288	−0.670102
$741$	0	0
$742$	−49.1660	−1.80494
$743$	10.5830	0.388253	0.194126	−	0.980977i	$-0.437813\pi$
$743$	10.5830	0.388253	0.194126	+	0.980977i	$0.437813\pi$
$744$	0	0
$745$	−20.5830	−0.754103
$746$	13.3948	0.490417
$747$	0	0
$748$	−7.08497	−0.259052
$749$	0	0
$750$	0	0
$751$	−44.5830	−1.62686	−0.813428	−	0.581665i	$-0.802401\pi$
$751$	−44.5830	−1.62686	−0.813428	+	0.581665i	$0.802401\pi$
$752$	14.2065	0.518059
$753$	0	0
$754$	8.37648	0.305053
$755$	8.58301	0.312368
$756$	0	0
$757$	−23.8745	−0.867734	−0.433867	−	0.900977i	$-0.642851\pi$
$757$	−23.8745	−0.867734	−0.433867	+	0.900977i	$0.642851\pi$
$758$	26.4575	0.960980
$759$	0	0
$760$	7.93725	0.287914
$761$	−25.7490	−0.933401	−0.466701	−	0.884415i	$-0.654557\pi$
$761$	−25.7490	−0.933401	−0.466701	+	0.884415i	$0.654557\pi$
$762$	0	0
$763$	8.70850	0.315269
$764$	−31.1438	−1.12674
$765$	0	0
$766$	49.1660	1.77644
$767$	4.00000	0.144432
$768$	0	0
$769$	−17.7490	−0.640046	−0.320023	−	0.947410i	$-0.603691\pi$
$769$	−17.7490	−0.640046	−0.320023	+	0.947410i	$0.603691\pi$
$770$	1.54249	0.0555874
$771$	0	0
$772$	−58.2288	−2.09570
$773$	8.70850	0.313223	0.156611	−	0.987660i	$-0.449943\pi$
$773$	8.70850	0.313223	0.156611	+	0.987660i	$0.449943\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	−23.3137	−0.836914
$777$	0	0
$778$	15.8745	0.569129
$779$	−9.64575	−0.345595
$780$	0	0
$781$	2.58301	0.0924272
$782$	−98.3320	−3.51635
$783$	0	0
$784$	−47.2065	−1.68595
$785$	13.2915	0.474394
$786$	0	0
$787$	37.8745	1.35008	0.675040	−	0.737781i	$-0.264126\pi$
$787$	37.8745	1.35008	0.675040	+	0.737781i	$0.264126\pi$
$788$	125.830	4.48251
$789$	0	0
$790$	−17.4170	−0.619669
$791$	6.58301	0.234065
$792$	0	0
$793$	4.00000	0.142044
$794$	−5.29150	−0.187788
$795$	0	0
$796$	36.4575	1.29220
$797$	−40.0000	−1.41687	−0.708436	−	0.705775i	$-0.750599\pi$
$797$	−40.0000	−1.41687	−0.708436	+	0.705775i	$0.750599\pi$
$798$	0	0
$799$	5.16601	0.182760
$800$	13.2288	0.467707
$801$	0	0
$802$	13.0627	0.461262
$803$	−3.54249	−0.125012
$804$	0	0
$805$	15.2915	0.538955
$806$	−5.62352	−0.198080
$807$	0	0
$808$	−73.7490	−2.59448
$809$	19.4170	0.682665	0.341333	−	0.939943i	$-0.389122\pi$
$809$	19.4170	0.682665	0.341333	+	0.939943i	$0.389122\pi$
$810$	0	0
$811$	38.3320	1.34602	0.673010	−	0.739634i	$-0.265001\pi$
$811$	38.3320	1.34602	0.673010	+	0.739634i	$0.265001\pi$
$812$	−73.5425	−2.58084
$813$	0	0
$814$	−3.41699	−0.119766
$815$	2.35425	0.0824657
$816$	0	0
$817$	−5.64575	−0.197520
$818$	−17.7490	−0.620580
$819$	0	0
$820$	−48.2288	−1.68422
$821$	−47.6235	−1.66207	−0.831036	−	0.556218i	$-0.812252\pi$
$821$	−47.6235	−1.66207	−0.831036	+	0.556218i	$0.812252\pi$
$822$	0	0
$823$	4.22876	0.147405	0.0737026	−	0.997280i	$-0.476518\pi$
$823$	4.22876	0.147405	0.0737026	+	0.997280i	$0.476518\pi$
$824$	−84.0000	−2.92628
$825$	0	0
$826$	−49.1660	−1.71070
$827$	−30.4575	−1.05911	−0.529556	−	0.848275i	$-0.677641\pi$
$827$	−30.4575	−1.05911	−0.529556	+	0.848275i	$0.677641\pi$
$828$	0	0
$829$	2.70850	0.0940700	0.0470350	−	0.998893i	$-0.485023\pi$
$829$	2.70850	0.0940700	0.0470350	+	0.998893i	$0.485023\pi$
$830$	15.8745	0.551012
$831$	0	0
$832$	−4.60523	−0.159658
$833$	−17.1660	−0.594767
$834$	0	0
$835$	−21.2915	−0.736823
$836$	1.77124	0.0612597
$837$	0	0
$838$	−103.018	−3.55871
$839$	12.4575	0.430081	0.215041	−	0.976605i	$-0.431012\pi$
$839$	12.4575	0.430081	0.215041	+	0.976605i	$0.431012\pi$
$840$	0	0
$841$	50.8745	1.75429
$842$	75.6235	2.60616
$843$	0	0
$844$	12.9150	0.444554
$845$	12.8745	0.442897
$846$	0	0
$847$	−17.8967	−0.614939
$848$	−124.207	−4.26527
$849$	0	0
$850$	10.5830	0.362994
$851$	−33.8745	−1.16120
$852$	0	0
$853$	14.7085	0.503609	0.251805	−	0.967778i	$-0.418976\pi$
$853$	14.7085	0.503609	0.251805	+	0.967778i	$0.418976\pi$
$854$	−49.1660	−1.68243
$855$	0	0
$856$	0	0
$857$	−31.0405	−1.06032	−0.530162	−	0.847896i	$-0.677869\pi$
$857$	−31.0405	−1.06032	−0.530162	+	0.847896i	$0.677869\pi$
$858$	0	0
$859$	−34.3320	−1.17139	−0.585697	−	0.810530i	$-0.699179\pi$
$859$	−34.3320	−1.17139	−0.585697	+	0.810530i	$0.699179\pi$
$860$	−28.2288	−0.962593
$861$	0	0
$862$	−8.70850	−0.296613
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	0	0
$865$	0	0
$866$	−73.1438	−2.48553
$867$	0	0
$868$	49.3725	1.67581
$869$	−2.33202	−0.0791084
$870$	0	0
$871$	2.33202	0.0790175
$872$	42.0000	1.42230
$873$	0	0
$874$	24.5830	0.831533
$875$	−1.64575	−0.0556365
$876$	0	0
$877$	−25.0627	−0.846309	−0.423154	−	0.906058i	$-0.639077\pi$
$877$	−25.0627	−0.846309	−0.423154	+	0.906058i	$0.639077\pi$
$878$	−3.74902	−0.126523
$879$	0	0
$880$	3.89674	0.131359
$881$	0.125492	0.00422794	0.00211397	−	0.999998i	$-0.499327\pi$
$881$	0.125492	0.00422794	0.00211397	+	0.999998i	$0.499327\pi$
$882$	0	0
$883$	10.8118	0.363845	0.181922	−	0.983313i	$-0.441768\pi$
$883$	10.8118	0.363845	0.181922	+	0.983313i	$0.441768\pi$
$884$	−7.08497	−0.238293
$885$	0	0
$886$	−70.6640	−2.37400
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	6.58301	0.220787
$890$	−44.8118	−1.50209
$891$	0	0
$892$	12.9150	0.432427
$893$	−1.29150	−0.0432185
$894$	0	0
$895$	−7.29150	−0.243728
$896$	13.0627	0.436396
$897$	0	0
$898$	95.8523	3.19863
$899$	−53.6235	−1.78844
$900$	0	0
$901$	−45.1660	−1.50470
$902$	−9.04052	−0.301016
$903$	0	0
$904$	31.7490	1.05596
$905$	−17.2915	−0.574789
$906$	0	0
$907$	−47.0405	−1.56195	−0.780977	−	0.624559i	$-0.785279\pi$
$907$	−47.0405	−1.56195	−0.780977	+	0.624559i	$0.785279\pi$
$908$	53.5425	1.77687
$909$	0	0
$910$	1.54249	0.0511329
$911$	−26.5830	−0.880734	−0.440367	−	0.897818i	$-0.645152\pi$
$911$	−26.5830	−0.880734	−0.440367	+	0.897818i	$0.645152\pi$
$912$	0	0
$913$	2.12549	0.0703435
$914$	0.332021	0.0109823
$915$	0	0
$916$	43.5425	1.43868
$917$	23.4170	0.773297
$918$	0	0
$919$	14.8340	0.489328	0.244664	−	0.969608i	$-0.421322\pi$
$919$	14.8340	0.489328	0.244664	+	0.969608i	$0.421322\pi$
$920$	73.7490	2.43143
$921$	0	0
$922$	99.8745	3.28919
$923$	2.58301	0.0850207
$924$	0	0
$925$	3.64575	0.119872
$926$	−93.9778	−3.08830
$927$	0	0
$928$	−118.229	−3.88105
$929$	11.8745	0.389590	0.194795	−	0.980844i	$-0.437596\pi$
$929$	11.8745	0.389590	0.194795	+	0.980844i	$0.437596\pi$
$930$	0	0
$931$	4.29150	0.140648
$932$	90.0000	2.94805
$933$	0	0
$934$	−52.5830	−1.72057
$935$	1.41699	0.0463407
$936$	0	0
$937$	20.1255	0.657471	0.328736	−	0.944422i	$-0.393378\pi$
$937$	20.1255	0.657471	0.328736	+	0.944422i	$0.393378\pi$
$938$	−28.6640	−0.935914
$939$	0	0
$940$	−6.45751	−0.210621
$941$	11.7712	0.383732	0.191866	−	0.981421i	$-0.438546\pi$
$941$	11.7712	0.383732	0.191866	+	0.981421i	$0.438546\pi$
$942$	0	0
$943$	−89.6235	−2.91854
$944$	−124.207	−4.04258
$945$	0	0
$946$	−5.29150	−0.172042
$947$	−11.4170	−0.371002	−0.185501	−	0.982644i	$-0.559391\pi$
$947$	−11.4170	−0.371002	−0.185501	+	0.982644i	$0.559391\pi$
$948$	0	0
$949$	−3.54249	−0.114994
$950$	−2.64575	−0.0858395
$951$	0	0
$952$	52.2510	1.69346
$953$	10.5830	0.342817	0.171409	−	0.985200i	$-0.445168\pi$
$953$	10.5830	0.342817	0.171409	+	0.985200i	$0.445168\pi$
$954$	0	0
$955$	6.22876	0.201558
$956$	−78.2288	−2.53010
$957$	0	0
$958$	−11.5203	−0.372203
$959$	−9.87451	−0.318864
$960$	0	0
$961$	5.00000	0.161290
$962$	−3.41699	−0.110168
$963$	0	0
$964$	82.9150	2.67051
$965$	11.6458	0.374890
$966$	0	0
$967$	−41.6458	−1.33924	−0.669619	−	0.742705i	$-0.733542\pi$
$967$	−41.6458	−1.33924	−0.669619	+	0.742705i	$0.733542\pi$
$968$	−86.3137	−2.77423
$969$	0	0
$970$	7.77124	0.249520
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	29.4170	0.943066
$974$	47.2915	1.51532
$975$	0	0
$976$	−124.207	−3.97575
$977$	51.0405	1.63293	0.816465	−	0.577394i	$-0.195930\pi$
$977$	51.0405	1.63293	0.816465	+	0.577394i	$0.195930\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	21.4575	0.685435
$981$	0	0
$982$	15.2693	0.487262
$983$	−34.4575	−1.09902	−0.549512	−	0.835486i	$-0.685186\pi$
$983$	−34.4575	−1.09902	−0.549512	+	0.835486i	$0.685186\pi$
$984$	0	0
$985$	−25.1660	−0.801856
$986$	−94.5830	−3.01214
$987$	0	0
$988$	1.77124	0.0563508
$989$	−52.4575	−1.66805
$990$	0	0
$991$	−35.7490	−1.13560	−0.567802	−	0.823165i	$-0.692206\pi$
$991$	−35.7490	−1.13560	−0.567802	+	0.823165i	$0.692206\pi$
$992$	79.3725	2.52008
$993$	0	0
$994$	−31.7490	−1.00702
$995$	−7.29150	−0.231156
$996$	0	0
$997$	26.7085	0.845867	0.422933	−	0.906161i	$-0.361000\pi$
$997$	26.7085	0.845867	0.422933	+	0.906161i	$0.361000\pi$
$998$	−82.1255	−2.59964
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.g.1.2		2
3.2	odd	2		285.2.a.d.1.1	✓	2
5.4	even	2		4275.2.a.u.1.1		2
12.11	even	2		4560.2.a.bo.1.1		2
15.2	even	4		1425.2.c.i.799.2		4
15.8	even	4		1425.2.c.i.799.3		4
15.14	odd	2		1425.2.a.p.1.2		2
57.56	even	2		5415.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.d.1.1	✓	2	3.2	odd	2
855.2.a.g.1.2		2	1.1	even	1	trivial
1425.2.a.p.1.2		2	15.14	odd	2
1425.2.c.i.799.2		4	15.2	even	4
1425.2.c.i.799.3		4	15.8	even	4
4275.2.a.u.1.1		2	5.4	even	2
4560.2.a.bo.1.1		2	12.11	even	2
5415.2.a.s.1.2		2	57.56	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{2} +5.00000 q^{4} -1.00000 q^{5} +1.64575 q^{7} +7.93725 q^{8} -2.64575 q^{10} -0.354249 q^{11} -0.354249 q^{13} +4.35425 q^{14} +11.0000 q^{16} +4.00000 q^{17} -1.00000 q^{19} -5.00000 q^{20} -0.937254 q^{22} -9.29150 q^{23} +1.00000 q^{25} -0.937254 q^{26} +8.22876 q^{28} -8.93725 q^{29} +6.00000 q^{31} +13.2288 q^{32} +10.5830 q^{34} -1.64575 q^{35} +3.64575 q^{37} -2.64575 q^{38} -7.93725 q^{40} +9.64575 q^{41} +5.64575 q^{43} -1.77124 q^{44} -24.5830 q^{46} +1.29150 q^{47} -4.29150 q^{49} +2.64575 q^{50} -1.77124 q^{52} -11.2915 q^{53} +0.354249 q^{55} +13.0627 q^{56} -23.6458 q^{58} -11.2915 q^{59} -11.2915 q^{61} +15.8745 q^{62} +13.0000 q^{64} +0.354249 q^{65} -6.58301 q^{67} +20.0000 q^{68} -4.35425 q^{70} -7.29150 q^{71} +10.0000 q^{73} +9.64575 q^{74} -5.00000 q^{76} -0.583005 q^{77} +6.58301 q^{79} -11.0000 q^{80} +25.5203 q^{82} -6.00000 q^{83} -4.00000 q^{85} +14.9373 q^{86} -2.81176 q^{88} +16.9373 q^{89} -0.583005 q^{91} -46.4575 q^{92} +3.41699 q^{94} +1.00000 q^{95} -2.93725 q^{97} -11.3542 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{11} - 6 q^{13} + 14 q^{14} + 22 q^{16} + 8 q^{17} - 2 q^{19} - 10 q^{20} + 14 q^{22} - 8 q^{23} + 2 q^{25} + 14 q^{26} - 10 q^{28} - 2 q^{29} + 12 q^{31} + 2 q^{35}+ \cdots - 28 q^{98}+O(q^{100}) \) 2 * q + 10 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^11 - 6 * q^13 + 14 * q^14 + 22 * q^16 + 8 * q^17 - 2 * q^19 - 10 * q^20 + 14 * q^22 - 8 * q^23 + 2 * q^25 + 14 * q^26 - 10 * q^28 - 2 * q^29 + 12 * q^31 + 2 * q^35 + 2 * q^37 + 14 * q^41 + 6 * q^43 - 30 * q^44 - 28 * q^46 - 8 * q^47 + 2 * q^49 - 30 * q^52 - 12 * q^53 + 6 * q^55 + 42 * q^56 - 42 * q^58 - 12 * q^59 - 12 * q^61 + 26 * q^64 + 6 * q^65 + 8 * q^67 + 40 * q^68 - 14 * q^70 - 4 * q^71 + 20 * q^73 + 14 * q^74 - 10 * q^76 + 20 * q^77 - 8 * q^79 - 22 * q^80 + 14 * q^82 - 12 * q^83 - 8 * q^85 + 14 * q^86 + 42 * q^88 + 18 * q^89 + 20 * q^91 - 40 * q^92 + 28 * q^94 + 2 * q^95 + 10 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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