LMFDB - Embedded newform 855.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.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:	$6.82720937282$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ 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)

$f(q)$	$=$	$q-2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} -0.539189 q^{7} -1.53919 q^{8} +O(q^{10})$	\(q-2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} -0.539189 q^{7} -1.53919 q^{8} +2.17009 q^{10} -3.17009 q^{11} +4.87936 q^{13} +1.17009 q^{14} -2.07838 q^{16} -1.36910 q^{17} -1.00000 q^{19} -2.70928 q^{20} +6.87936 q^{22} +2.78765 q^{23} +1.00000 q^{25} -10.5886 q^{26} -1.46081 q^{28} -3.90829 q^{29} -2.44748 q^{31} +7.58864 q^{32} +2.97107 q^{34} +0.539189 q^{35} -4.14116 q^{37} +2.17009 q^{38} +1.53919 q^{40} -3.01333 q^{41} +5.95774 q^{43} -8.58864 q^{44} -6.04945 q^{46} +4.04945 q^{47} -6.70928 q^{49} -2.17009 q^{50} +13.2195 q^{52} -6.63090 q^{53} +3.17009 q^{55} +0.829914 q^{56} +8.48133 q^{58} +12.4391 q^{59} -9.31124 q^{61} +5.31124 q^{62} -12.3112 q^{64} -4.87936 q^{65} -7.75872 q^{67} -3.70928 q^{68} -1.17009 q^{70} -2.18342 q^{71} -7.60197 q^{73} +8.98667 q^{74} -2.70928 q^{76} +1.70928 q^{77} -15.0205 q^{79} +2.07838 q^{80} +6.53919 q^{82} -2.78765 q^{83} +1.36910 q^{85} -12.9288 q^{86} +4.87936 q^{88} -0.829914 q^{89} -2.63090 q^{91} +7.55252 q^{92} -8.78765 q^{94} +1.00000 q^{95} -5.37629 q^{97} +14.5597 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8	$ 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8} + q^{10} - 4 q^{11} + 2 q^{13} - 2 q^{14} - 3 q^{16} - 8 q^{17} - 3 q^{19} - q^{20} + 8 q^{22} - 2 q^{23} + 3 q^{25} - 12 q^{26} - 6 q^{28} - 14 q^{29} - 8 q^{31} + 3 q^{32} - 6 q^{34} + 8 q^{37} + q^{38} + 3 q^{40} - 10 q^{41} + 2 q^{43} - 6 q^{44} - 6 q^{47} - 13 q^{49} - q^{50} + 16 q^{52} - 16 q^{53} + 4 q^{55} + 8 q^{56} - 6 q^{58} - 10 q^{59} - 2 q^{61} - 10 q^{62} - 11 q^{64} - 2 q^{65} + 2 q^{67} - 4 q^{68} + 2 q^{70} - 2 q^{71} - 4 q^{73} + 26 q^{74} - q^{76} - 2 q^{77} - 12 q^{79} + 3 q^{80} + 18 q^{82} + 2 q^{83} + 8 q^{85} - 8 q^{86} + 2 q^{88} - 8 q^{89} - 4 q^{91} + 22 q^{92} - 16 q^{94} + 3 q^{95} + 14 q^{97} + 9 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8 + q^10 - 4 * q^11 + 2 * q^13 - 2 * q^14 - 3 * q^16 - 8 * q^17 - 3 * q^19 - q^20 + 8 * q^22 - 2 * q^23 + 3 * q^25 - 12 * q^26 - 6 * q^28 - 14 * q^29 - 8 * q^31 + 3 * q^32 - 6 * q^34 + 8 * q^37 + q^38 + 3 * q^40 - 10 * q^41 + 2 * q^43 - 6 * q^44 - 6 * q^47 - 13 * q^49 - q^50 + 16 * q^52 - 16 * q^53 + 4 * q^55 + 8 * q^56 - 6 * q^58 - 10 * q^59 - 2 * q^61 - 10 * q^62 - 11 * q^64 - 2 * q^65 + 2 * q^67 - 4 * q^68 + 2 * q^70 - 2 * q^71 - 4 * q^73 + 26 * q^74 - q^76 - 2 * q^77 - 12 * q^79 + 3 * q^80 + 18 * q^82 + 2 * q^83 + 8 * q^85 - 8 * q^86 + 2 * q^88 - 8 * q^89 - 4 * q^91 + 22 * q^92 - 16 * q^94 + 3 * q^95 + 14 * q^97 + 9 * q^98

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 $

$2$	−2.17009	−1.53448	−0.767241	−	0.641358i	$-0.778371\pi$
$2$	−2.17009	−1.53448	−0.767241	+	0.641358i	$0.778371\pi$
$3$	0	0
$3$	0	0
$4$	2.70928	1.35464
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.539189	−0.203794	−0.101897	−	0.994795i	$-0.532491\pi$
$7$	−0.539189	−0.203794	−0.101897	+	0.994795i	$0.532491\pi$
$8$	−1.53919	−0.544185
$9$	0	0
$10$	2.17009	0.686242
$11$	−3.17009	−0.955817	−0.477909	−	0.878410i	$-0.658605\pi$
$11$	−3.17009	−0.955817	−0.477909	+	0.878410i	$0.658605\pi$
$12$	0	0
$13$	4.87936	1.35329	0.676646	−	0.736309i	$-0.263433\pi$
$13$	4.87936	1.35329	0.676646	+	0.736309i	$0.263433\pi$
$14$	1.17009	0.312719
$15$	0	0
$16$	−2.07838	−0.519594
$17$	−1.36910	−0.332056	−0.166028	−	0.986121i	$-0.553094\pi$
$17$	−1.36910	−0.332056	−0.166028	+	0.986121i	$0.553094\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.70928	−0.605812
$21$	0	0
$22$	6.87936	1.46668
$23$	2.78765	0.581266	0.290633	−	0.956835i	$-0.406134\pi$
$23$	2.78765	0.581266	0.290633	+	0.956835i	$0.406134\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−10.5886	−2.07660
$27$	0	0
$28$	−1.46081	−0.276067
$29$	−3.90829	−0.725751	−0.362876	−	0.931838i	$-0.618205\pi$
$29$	−3.90829	−0.725751	−0.362876	+	0.931838i	$0.618205\pi$
$30$	0	0
$31$	−2.44748	−0.439580	−0.219790	−	0.975547i	$-0.570537\pi$
$31$	−2.44748	−0.439580	−0.219790	+	0.975547i	$0.570537\pi$
$32$	7.58864	1.34149
$33$	0	0
$34$	2.97107	0.509534
$35$	0.539189	0.0911396
$36$	0	0
$37$	−4.14116	−0.680802	−0.340401	−	0.940280i	$-0.610563\pi$
$37$	−4.14116	−0.680802	−0.340401	+	0.940280i	$0.610563\pi$
$38$	2.17009	0.352035
$39$	0	0
$40$	1.53919	0.243367
$41$	−3.01333	−0.470603	−0.235302	−	0.971922i	$-0.575608\pi$
$41$	−3.01333	−0.470603	−0.235302	+	0.971922i	$0.575608\pi$
$42$	0	0
$43$	5.95774	0.908547	0.454273	−	0.890862i	$-0.349899\pi$
$43$	5.95774	0.908547	0.454273	+	0.890862i	$0.349899\pi$
$44$	−8.58864	−1.29479
$45$	0	0
$46$	−6.04945	−0.891942
$47$	4.04945	0.590673	0.295336	−	0.955393i	$-0.404568\pi$
$47$	4.04945	0.590673	0.295336	+	0.955393i	$0.404568\pi$
$48$	0	0
$49$	−6.70928	−0.958468
$50$	−2.17009	−0.306897
$51$	0	0
$52$	13.2195	1.83322
$53$	−6.63090	−0.910824	−0.455412	−	0.890281i	$-0.650508\pi$
$53$	−6.63090	−0.910824	−0.455412	+	0.890281i	$0.650508\pi$
$54$	0	0
$55$	3.17009	0.427454
$56$	0.829914	0.110902
$57$	0	0
$58$	8.48133	1.11365
$59$	12.4391	1.61943	0.809714	−	0.586824i	$-0.199622\pi$
$59$	12.4391	1.61943	0.809714	+	0.586824i	$0.199622\pi$
$60$	0	0
$61$	−9.31124	−1.19218	−0.596091	−	0.802917i	$-0.703280\pi$
$61$	−9.31124	−1.19218	−0.596091	+	0.802917i	$0.703280\pi$
$62$	5.31124	0.674529
$63$	0	0
$64$	−12.3112	−1.53891
$65$	−4.87936	−0.605210
$66$	0	0
$67$	−7.75872	−0.947879	−0.473939	−	0.880557i	$-0.657168\pi$
$67$	−7.75872	−0.947879	−0.473939	+	0.880557i	$0.657168\pi$
$68$	−3.70928	−0.449816
$69$	0	0
$70$	−1.17009	−0.139852
$71$	−2.18342	−0.259124	−0.129562	−	0.991571i	$-0.541357\pi$
$71$	−2.18342	−0.259124	−0.129562	+	0.991571i	$0.541357\pi$
$72$	0	0
$73$	−7.60197	−0.889743	−0.444872	−	0.895594i	$-0.646751\pi$
$73$	−7.60197	−0.889743	−0.444872	+	0.895594i	$0.646751\pi$
$74$	8.98667	1.04468
$75$	0	0
$76$	−2.70928	−0.310775
$77$	1.70928	0.194790
$78$	0	0
$79$	−15.0205	−1.68994	−0.844970	−	0.534813i	$-0.820382\pi$
$79$	−15.0205	−1.68994	−0.844970	+	0.534813i	$0.820382\pi$
$80$	2.07838	0.232370
$81$	0	0
$82$	6.53919	0.722133
$83$	−2.78765	−0.305985	−0.152992	−	0.988227i	$-0.548891\pi$
$83$	−2.78765	−0.305985	−0.152992	+	0.988227i	$0.548891\pi$
$84$	0	0
$85$	1.36910	0.148500
$86$	−12.9288	−1.39415
$87$	0	0
$88$	4.87936	0.520142
$89$	−0.829914	−0.0879707	−0.0439853	−	0.999032i	$-0.514005\pi$
$89$	−0.829914	−0.0879707	−0.0439853	+	0.999032i	$0.514005\pi$
$90$	0	0
$91$	−2.63090	−0.275793
$92$	7.55252	0.787405
$93$	0	0
$94$	−8.78765	−0.906377
$95$	1.00000	0.102598
$96$	0	0
$97$	−5.37629	−0.545880	−0.272940	−	0.962031i	$-0.587996\pi$
$97$	−5.37629	−0.545880	−0.272940	+	0.962031i	$0.587996\pi$
$98$	14.5597	1.47075
$99$	0	0
$100$	2.70928	0.270928
$101$	−17.3340	−1.72480	−0.862400	−	0.506227i	$-0.831040\pi$
$101$	−17.3340	−1.72480	−0.862400	+	0.506227i	$0.831040\pi$
$102$	0	0
$103$	−2.52359	−0.248657	−0.124328	−	0.992241i	$-0.539678\pi$
$103$	−2.52359	−0.248657	−0.124328	+	0.992241i	$0.539678\pi$
$104$	−7.51026	−0.736442
$105$	0	0
$106$	14.3896	1.39764
$107$	−10.6537	−1.02993	−0.514965	−	0.857211i	$-0.672195\pi$
$107$	−10.6537	−1.02993	−0.514965	+	0.857211i	$0.672195\pi$
$108$	0	0
$109$	−5.81658	−0.557128	−0.278564	−	0.960418i	$-0.589858\pi$
$109$	−5.81658	−0.557128	−0.278564	+	0.960418i	$0.589858\pi$
$110$	−6.87936	−0.655921
$111$	0	0
$112$	1.12064	0.105890
$113$	−19.6248	−1.84614	−0.923071	−	0.384630i	$-0.874329\pi$
$113$	−19.6248	−1.84614	−0.923071	+	0.384630i	$0.874329\pi$
$114$	0	0
$115$	−2.78765	−0.259950
$116$	−10.5886	−0.983130
$117$	0	0
$118$	−26.9939	−2.48499
$119$	0.738205	0.0676711
$120$	0	0
$121$	−0.950552	−0.0864138
$122$	20.2062	1.82938
$123$	0	0
$124$	−6.63090	−0.595472
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.340173	−0.0301855	−0.0150927	−	0.999886i	$-0.504804\pi$
$127$	−0.340173	−0.0301855	−0.0150927	+	0.999886i	$0.504804\pi$
$128$	11.5392	1.01993
$129$	0	0
$130$	10.5886	0.928685
$131$	−12.0072	−1.04907	−0.524536	−	0.851388i	$-0.675761\pi$
$131$	−12.0072	−1.04907	−0.524536	+	0.851388i	$0.675761\pi$
$132$	0	0
$133$	0.539189	0.0467536
$134$	16.8371	1.45450
$135$	0	0
$136$	2.10731	0.180700
$137$	19.1773	1.63843	0.819213	−	0.573490i	$-0.194411\pi$
$137$	19.1773	1.63843	0.819213	+	0.573490i	$0.194411\pi$
$138$	0	0
$139$	15.3340	1.30062	0.650308	−	0.759671i	$-0.274640\pi$
$139$	15.3340	1.30062	0.650308	+	0.759671i	$0.274640\pi$
$140$	1.46081	0.123461
$141$	0	0
$142$	4.73820	0.397621
$143$	−15.4680	−1.29350
$144$	0	0
$145$	3.90829	0.324566
$146$	16.4969	1.36530
$147$	0	0
$148$	−11.2195	−0.922240
$149$	10.5236	0.862126	0.431063	−	0.902322i	$-0.358139\pi$
$149$	10.5236	0.862126	0.431063	+	0.902322i	$0.358139\pi$
$150$	0	0
$151$	−4.47414	−0.364101	−0.182050	−	0.983289i	$-0.558273\pi$
$151$	−4.47414	−0.364101	−0.182050	+	0.983289i	$0.558273\pi$
$152$	1.53919	0.124845
$153$	0	0
$154$	−3.70928	−0.298902
$155$	2.44748	0.196586
$156$	0	0
$157$	19.2039	1.53264	0.766320	−	0.642458i	$-0.222085\pi$
$157$	19.2039	1.53264	0.766320	+	0.642458i	$0.222085\pi$
$158$	32.5958	2.59318
$159$	0	0
$160$	−7.58864	−0.599934
$161$	−1.50307	−0.118459
$162$	0	0
$163$	−4.53919	−0.355537	−0.177768	−	0.984072i	$-0.556888\pi$
$163$	−4.53919	−0.355537	−0.177768	+	0.984072i	$0.556888\pi$
$164$	−8.16394	−0.637497
$165$	0	0
$166$	6.04945	0.469528
$167$	5.12783	0.396803	0.198402	−	0.980121i	$-0.436425\pi$
$167$	5.12783	0.396803	0.198402	+	0.980121i	$0.436425\pi$
$168$	0	0
$169$	10.8082	0.831398
$170$	−2.97107	−0.227871
$171$	0	0
$172$	16.1412	1.23075
$173$	−4.47414	−0.340163	−0.170081	−	0.985430i	$-0.554403\pi$
$173$	−4.47414	−0.340163	−0.170081	+	0.985430i	$0.554403\pi$
$174$	0	0
$175$	−0.539189	−0.0407588
$176$	6.58864	0.496637
$177$	0	0
$178$	1.80098	0.134989
$179$	8.34017	0.623374	0.311687	−	0.950185i	$-0.399106\pi$
$179$	8.34017	0.623374	0.311687	+	0.950185i	$0.399106\pi$
$180$	0	0
$181$	−9.60197	−0.713709	−0.356854	−	0.934160i	$-0.616151\pi$
$181$	−9.60197	−0.713709	−0.356854	+	0.934160i	$0.616151\pi$
$182$	5.70928	0.423200
$183$	0	0
$184$	−4.29072	−0.316316
$185$	4.14116	0.304464
$186$	0	0
$187$	4.34017	0.317385
$188$	10.9711	0.800148
$189$	0	0
$190$	−2.17009	−0.157435
$191$	6.48974	0.469581	0.234791	−	0.972046i	$-0.424560\pi$
$191$	6.48974	0.469581	0.234791	+	0.972046i	$0.424560\pi$
$192$	0	0
$193$	−9.19287	−0.661717	−0.330859	−	0.943680i	$-0.607338\pi$
$193$	−9.19287	−0.661717	−0.330859	+	0.943680i	$0.607338\pi$
$194$	11.6670	0.837643
$195$	0	0
$196$	−18.1773	−1.29838
$197$	3.52586	0.251207	0.125603	−	0.992081i	$-0.459913\pi$
$197$	3.52586	0.251207	0.125603	+	0.992081i	$0.459913\pi$
$198$	0	0
$199$	−21.1773	−1.50122	−0.750609	−	0.660747i	$-0.770240\pi$
$199$	−21.1773	−1.50122	−0.750609	+	0.660747i	$0.770240\pi$
$200$	−1.53919	−0.108837
$201$	0	0
$202$	37.6163	2.64668
$203$	2.10731	0.147904
$204$	0	0
$205$	3.01333	0.210460
$206$	5.47641	0.381560
$207$	0	0
$208$	−10.1412	−0.703163
$209$	3.17009	0.219279
$210$	0	0
$211$	11.0205	0.758684	0.379342	−	0.925257i	$-0.376150\pi$
$211$	11.0205	0.758684	0.379342	+	0.925257i	$0.376150\pi$
$212$	−17.9649	−1.23384
$213$	0	0
$214$	23.1194	1.58041
$215$	−5.95774	−0.406314
$216$	0	0
$217$	1.31965	0.0895840
$218$	12.6225	0.854903
$219$	0	0
$220$	8.58864	0.579046
$221$	−6.68035	−0.449369
$222$	0	0
$223$	11.2039	0.750271	0.375136	−	0.926970i	$-0.377596\pi$
$223$	11.2039	0.750271	0.375136	+	0.926970i	$0.377596\pi$
$224$	−4.09171	−0.273389
$225$	0	0
$226$	42.5874	2.83287
$227$	−0.546377	−0.0362643	−0.0181322	−	0.999836i	$-0.505772\pi$
$227$	−0.546377	−0.0362643	−0.0181322	+	0.999836i	$0.505772\pi$
$228$	0	0
$229$	22.4885	1.48608	0.743042	−	0.669245i	$-0.233382\pi$
$229$	22.4885	1.48608	0.743042	+	0.669245i	$0.233382\pi$
$230$	6.04945	0.398889
$231$	0	0
$232$	6.01560	0.394943
$233$	10.7792	0.706172	0.353086	−	0.935591i	$-0.385132\pi$
$233$	10.7792	0.706172	0.353086	+	0.935591i	$0.385132\pi$
$234$	0	0
$235$	−4.04945	−0.264157
$236$	33.7009	2.19374
$237$	0	0
$238$	−1.60197	−0.103840
$239$	20.6875	1.33817	0.669083	−	0.743188i	$-0.266687\pi$
$239$	20.6875	1.33817	0.669083	+	0.743188i	$0.266687\pi$
$240$	0	0
$241$	17.5174	1.12840	0.564199	−	0.825639i	$-0.309185\pi$
$241$	17.5174	1.12840	0.564199	+	0.825639i	$0.309185\pi$
$242$	2.06278	0.132600
$243$	0	0
$244$	−25.2267	−1.61498
$245$	6.70928	0.428640
$246$	0	0
$247$	−4.87936	−0.310466
$248$	3.76713	0.239213
$249$	0	0
$250$	2.17009	0.137248
$251$	14.7454	0.930721	0.465360	−	0.885121i	$-0.345925\pi$
$251$	14.7454	0.930721	0.465360	+	0.885121i	$0.345925\pi$
$252$	0	0
$253$	−8.83710	−0.555584
$254$	0.738205	0.0463191
$255$	0	0
$256$	−0.418551	−0.0261594
$257$	−25.8660	−1.61348	−0.806739	−	0.590907i	$-0.798770\pi$
$257$	−25.8660	−1.61348	−0.806739	+	0.590907i	$0.798770\pi$
$258$	0	0
$259$	2.23287	0.138744
$260$	−13.2195	−0.819841
$261$	0	0
$262$	26.0566	1.60978
$263$	6.19183	0.381804	0.190902	−	0.981609i	$-0.438859\pi$
$263$	6.19183	0.381804	0.190902	+	0.981609i	$0.438859\pi$
$264$	0	0
$265$	6.63090	0.407333
$266$	−1.17009	−0.0717426
$267$	0	0
$268$	−21.0205	−1.28403
$269$	−19.6670	−1.19912	−0.599560	−	0.800330i	$-0.704658\pi$
$269$	−19.6670	−1.19912	−0.599560	+	0.800330i	$0.704658\pi$
$270$	0	0
$271$	10.1256	0.615084	0.307542	−	0.951535i	$-0.400494\pi$
$271$	10.1256	0.615084	0.307542	+	0.951535i	$0.400494\pi$
$272$	2.84551	0.172535
$273$	0	0
$274$	−41.6163	−2.51414
$275$	−3.17009	−0.191163
$276$	0	0
$277$	4.73820	0.284691	0.142346	−	0.989817i	$-0.454536\pi$
$277$	4.73820	0.284691	0.142346	+	0.989817i	$0.454536\pi$
$278$	−33.2762	−1.99577
$279$	0	0
$280$	−0.829914	−0.0495968
$281$	−13.9083	−0.829699	−0.414850	−	0.909890i	$-0.636166\pi$
$281$	−13.9083	−0.829699	−0.414850	+	0.909890i	$0.636166\pi$
$282$	0	0
$283$	23.9265	1.42229	0.711143	−	0.703048i	$-0.248178\pi$
$283$	23.9265	1.42229	0.711143	+	0.703048i	$0.248178\pi$
$284$	−5.91548	−0.351019
$285$	0	0
$286$	33.5669	1.98485
$287$	1.62475	0.0959062
$288$	0	0
$289$	−15.1256	−0.889739
$290$	−8.48133	−0.498041
$291$	0	0
$292$	−20.5958	−1.20528
$293$	−13.6514	−0.797524	−0.398762	−	0.917054i	$-0.630560\pi$
$293$	−13.6514	−0.797524	−0.398762	+	0.917054i	$0.630560\pi$
$294$	0	0
$295$	−12.4391	−0.724231
$296$	6.37402	0.370483
$297$	0	0
$298$	−22.8371	−1.32292
$299$	13.6020	0.786622
$300$	0	0
$301$	−3.21235	−0.185157
$302$	9.70928	0.558706
$303$	0	0
$304$	2.07838	0.119203
$305$	9.31124	0.533160
$306$	0	0
$307$	22.9672	1.31081	0.655404	−	0.755279i	$-0.272499\pi$
$307$	22.9672	1.31081	0.655404	+	0.755279i	$0.272499\pi$
$308$	4.63090	0.263870
$309$	0	0
$310$	−5.31124	−0.301658
$311$	−19.0589	−1.08073	−0.540366	−	0.841430i	$-0.681714\pi$
$311$	−19.0589	−1.08073	−0.540366	+	0.841430i	$0.681714\pi$
$312$	0	0
$313$	−3.10504	−0.175507	−0.0877536	−	0.996142i	$-0.527969\pi$
$313$	−3.10504	−0.175507	−0.0877536	+	0.996142i	$0.527969\pi$
$314$	−41.6742	−2.35181
$315$	0	0
$316$	−40.6947	−2.28926
$317$	−9.96493	−0.559686	−0.279843	−	0.960046i	$-0.590282\pi$
$317$	−9.96493	−0.559686	−0.279843	+	0.960046i	$0.590282\pi$
$318$	0	0
$319$	12.3896	0.693686
$320$	12.3112	0.688219
$321$	0	0
$322$	3.26180	0.181773
$323$	1.36910	0.0761789
$324$	0	0
$325$	4.87936	0.270658
$326$	9.85043	0.545565
$327$	0	0
$328$	4.63809	0.256095
$329$	−2.18342	−0.120376
$330$	0	0
$331$	−24.6719	−1.35609	−0.678046	−	0.735019i	$-0.737173\pi$
$331$	−24.6719	−1.35609	−0.678046	+	0.735019i	$0.737173\pi$
$332$	−7.55252	−0.414498
$333$	0	0
$334$	−11.1278	−0.608888
$335$	7.75872	0.423904
$336$	0	0
$337$	35.9721	1.95953	0.979763	−	0.200161i	$-0.0641466\pi$
$337$	35.9721	1.95953	0.979763	+	0.200161i	$0.0641466\pi$
$338$	−23.4547	−1.27577
$339$	0	0
$340$	3.70928	0.201164
$341$	7.75872	0.420158
$342$	0	0
$343$	7.39189	0.399124
$344$	−9.17009	−0.494418
$345$	0	0
$346$	9.70928	0.521974
$347$	−33.4101	−1.79355	−0.896775	−	0.442487i	$-0.854096\pi$
$347$	−33.4101	−1.79355	−0.896775	+	0.442487i	$0.854096\pi$
$348$	0	0
$349$	19.2762	1.03183	0.515915	−	0.856640i	$-0.327452\pi$
$349$	19.2762	1.03183	0.515915	+	0.856640i	$0.327452\pi$
$350$	1.17009	0.0625438
$351$	0	0
$352$	−24.0566	−1.28222
$353$	−1.68649	−0.0897628	−0.0448814	−	0.998992i	$-0.514291\pi$
$353$	−1.68649	−0.0897628	−0.0448814	+	0.998992i	$0.514291\pi$
$354$	0	0
$355$	2.18342	0.115884
$356$	−2.24846	−0.119168
$357$	0	0
$358$	−18.0989	−0.956556
$359$	−17.0277	−0.898688	−0.449344	−	0.893359i	$-0.648342\pi$
$359$	−17.0277	−0.898688	−0.449344	+	0.893359i	$0.648342\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	20.8371	1.09517
$363$	0	0
$364$	−7.12783	−0.373600
$365$	7.60197	0.397905
$366$	0	0
$367$	−32.0300	−1.67195	−0.835976	−	0.548767i	$-0.815098\pi$
$367$	−32.0300	−1.67195	−0.835976	+	0.548767i	$0.815098\pi$
$368$	−5.79380	−0.302023
$369$	0	0
$370$	−8.98667	−0.467195
$371$	3.57531	0.185621
$372$	0	0
$373$	19.3041	0.999526	0.499763	−	0.866162i	$-0.333420\pi$
$373$	19.3041	0.999526	0.499763	+	0.866162i	$0.333420\pi$
$374$	−9.41855	−0.487022
$375$	0	0
$376$	−6.23287	−0.321436
$377$	−19.0700	−0.982153
$378$	0	0
$379$	−31.1422	−1.59967	−0.799834	−	0.600222i	$-0.795079\pi$
$379$	−31.1422	−1.59967	−0.799834	+	0.600222i	$0.795079\pi$
$380$	2.70928	0.138983
$381$	0	0
$382$	−14.0833	−0.720564
$383$	−10.7382	−0.548697	−0.274348	−	0.961630i	$-0.588462\pi$
$383$	−10.7382	−0.548697	−0.274348	+	0.961630i	$0.588462\pi$
$384$	0	0
$385$	−1.70928	−0.0871127
$386$	19.9493	1.01539
$387$	0	0
$388$	−14.5659	−0.739469
$389$	−23.0472	−1.16854	−0.584269	−	0.811560i	$-0.698619\pi$
$389$	−23.0472	−1.16854	−0.584269	+	0.811560i	$0.698619\pi$
$390$	0	0
$391$	−3.81658	−0.193013
$392$	10.3268	0.521584
$393$	0	0
$394$	−7.65142	−0.385473
$395$	15.0205	0.755764
$396$	0	0
$397$	−5.05172	−0.253538	−0.126769	−	0.991932i	$-0.540461\pi$
$397$	−5.05172	−0.253538	−0.126769	+	0.991932i	$0.540461\pi$
$398$	45.9565	2.30359
$399$	0	0
$400$	−2.07838	−0.103919
$401$	−17.2546	−0.861654	−0.430827	−	0.902435i	$-0.641778\pi$
$401$	−17.2546	−0.861654	−0.430827	+	0.902435i	$0.641778\pi$
$402$	0	0
$403$	−11.9421	−0.594880
$404$	−46.9627	−2.33648
$405$	0	0
$406$	−4.57304	−0.226956
$407$	13.1278	0.650722
$408$	0	0
$409$	−19.0472	−0.941822	−0.470911	−	0.882181i	$-0.656075\pi$
$409$	−19.0472	−0.941822	−0.470911	+	0.882181i	$0.656075\pi$
$410$	−6.53919	−0.322948
$411$	0	0
$412$	−6.83710	−0.336840
$413$	−6.70701	−0.330030
$414$	0	0
$415$	2.78765	0.136841
$416$	37.0277	1.81543
$417$	0	0
$418$	−6.87936	−0.336481
$419$	11.1389	0.544170	0.272085	−	0.962273i	$-0.412287\pi$
$419$	11.1389	0.544170	0.272085	+	0.962273i	$0.412287\pi$
$420$	0	0
$421$	−29.3340	−1.42965	−0.714827	−	0.699302i	$-0.753494\pi$
$421$	−29.3340	−1.42965	−0.714827	+	0.699302i	$0.753494\pi$
$422$	−23.9155	−1.16419
$423$	0	0
$424$	10.2062	0.495657
$425$	−1.36910	−0.0664112
$426$	0	0
$427$	5.02052	0.242960
$428$	−28.8638	−1.39518
$429$	0	0
$430$	12.9288	0.623483
$431$	24.2823	1.16964	0.584819	−	0.811164i	$-0.301166\pi$
$431$	24.2823	1.16964	0.584819	+	0.811164i	$0.301166\pi$
$432$	0	0
$433$	29.2606	1.40617	0.703087	−	0.711104i	$-0.251805\pi$
$433$	29.2606	1.40617	0.703087	+	0.711104i	$0.251805\pi$
$434$	−2.86376	−0.137465
$435$	0	0
$436$	−15.7587	−0.754706
$437$	−2.78765	−0.133352
$438$	0	0
$439$	21.3074	1.01695	0.508473	−	0.861078i	$-0.330210\pi$
$439$	21.3074	1.01695	0.508473	+	0.861078i	$0.330210\pi$
$440$	−4.87936	−0.232614
$441$	0	0
$442$	14.4969	0.689549
$443$	13.1812	0.626255	0.313128	−	0.949711i	$-0.398623\pi$
$443$	13.1812	0.626255	0.313128	+	0.949711i	$0.398623\pi$
$444$	0	0
$445$	0.829914	0.0393417
$446$	−24.3135	−1.15128
$447$	0	0
$448$	6.63809	0.313620
$449$	32.0216	1.51119	0.755596	−	0.655038i	$-0.227348\pi$
$449$	32.0216	1.51119	0.755596	+	0.655038i	$0.227348\pi$
$450$	0	0
$451$	9.55252	0.449811
$452$	−53.1689	−2.50085
$453$	0	0
$454$	1.18568	0.0556469
$455$	2.63090	0.123338
$456$	0	0
$457$	−37.4063	−1.74979	−0.874896	−	0.484312i	$-0.839070\pi$
$457$	−37.4063	−1.74979	−0.874896	+	0.484312i	$0.839070\pi$
$458$	−48.8020	−2.28037
$459$	0	0
$460$	−7.55252	−0.352138
$461$	31.0349	1.44544	0.722720	−	0.691141i	$-0.242892\pi$
$461$	31.0349	1.44544	0.722720	+	0.691141i	$0.242892\pi$
$462$	0	0
$463$	17.8588	0.829971	0.414985	−	0.909828i	$-0.363787\pi$
$463$	17.8588	0.829971	0.414985	+	0.909828i	$0.363787\pi$
$464$	8.12291	0.377096
$465$	0	0
$466$	−23.3919	−1.08361
$467$	1.34244	0.0621207	0.0310604	−	0.999518i	$-0.490112\pi$
$467$	1.34244	0.0621207	0.0310604	+	0.999518i	$0.490112\pi$
$468$	0	0
$469$	4.18342	0.193172
$470$	8.78765	0.405344
$471$	0	0
$472$	−19.1461	−0.881270
$473$	−18.8865	−0.868404
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	2.00000	0.0916698
$477$	0	0
$478$	−44.8937	−2.05339
$479$	13.9350	0.636704	0.318352	−	0.947973i	$-0.396871\pi$
$479$	13.9350	0.636704	0.318352	+	0.947973i	$0.396871\pi$
$480$	0	0
$481$	−20.2062	−0.921324
$482$	−38.0144	−1.73151
$483$	0	0
$484$	−2.57531	−0.117059
$485$	5.37629	0.244125
$486$	0	0
$487$	39.8720	1.80677	0.903386	−	0.428828i	$-0.141073\pi$
$487$	39.8720	1.80677	0.903386	+	0.428828i	$0.141073\pi$
$488$	14.3318	0.648768
$489$	0	0
$490$	−14.5597	−0.657741
$491$	21.3268	0.962467	0.481233	−	0.876592i	$-0.340189\pi$
$491$	21.3268	0.962467	0.481233	+	0.876592i	$0.340189\pi$
$492$	0	0
$493$	5.35085	0.240990
$494$	10.5886	0.476405
$495$	0	0
$496$	5.08679	0.228404
$497$	1.17727	0.0528080
$498$	0	0
$499$	12.8638	0.575861	0.287931	−	0.957651i	$-0.407033\pi$
$499$	12.8638	0.575861	0.287931	+	0.957651i	$0.407033\pi$
$500$	−2.70928	−0.121162
$501$	0	0
$502$	−31.9988	−1.42818
$503$	19.3835	0.864267	0.432133	−	0.901810i	$-0.357761\pi$
$503$	19.3835	0.864267	0.432133	+	0.901810i	$0.357761\pi$
$504$	0	0
$505$	17.3340	0.771354
$506$	19.1773	0.852534
$507$	0	0
$508$	−0.921622	−0.0408904
$509$	7.21112	0.319627	0.159814	−	0.987147i	$-0.448911\pi$
$509$	7.21112	0.319627	0.159814	+	0.987147i	$0.448911\pi$
$510$	0	0
$511$	4.09890	0.181325
$512$	−22.1701	−0.979789
$513$	0	0
$514$	56.1315	2.47586
$515$	2.52359	0.111203
$516$	0	0
$517$	−12.8371	−0.564575
$518$	−4.84551	−0.212900
$519$	0	0
$520$	7.51026	0.329347
$521$	−29.1578	−1.27743	−0.638713	−	0.769445i	$-0.720533\pi$
$521$	−29.1578	−1.27743	−0.638713	+	0.769445i	$0.720533\pi$
$522$	0	0
$523$	20.1568	0.881393	0.440697	−	0.897656i	$-0.354731\pi$
$523$	20.1568	0.881393	0.440697	+	0.897656i	$0.354731\pi$
$524$	−32.5308	−1.42111
$525$	0	0
$526$	−13.4368	−0.585872
$527$	3.35085	0.145965
$528$	0	0
$529$	−15.2290	−0.662130
$530$	−14.3896	−0.625045
$531$	0	0
$532$	1.46081	0.0633342
$533$	−14.7031	−0.636863
$534$	0	0
$535$	10.6537	0.460599
$536$	11.9421	0.515822
$537$	0	0
$538$	42.6791	1.84003
$539$	21.2690	0.916120
$540$	0	0
$541$	−15.3607	−0.660408	−0.330204	−	0.943910i	$-0.607117\pi$
$541$	−15.3607	−0.660408	−0.330204	+	0.943910i	$0.607117\pi$
$542$	−21.9733	−0.943836
$543$	0	0
$544$	−10.3896	−0.445451
$545$	5.81658	0.249155
$546$	0	0
$547$	−2.92162	−0.124920	−0.0624598	−	0.998047i	$-0.519895\pi$
$547$	−2.92162	−0.124920	−0.0624598	+	0.998047i	$0.519895\pi$
$548$	51.9565	2.21947
$549$	0	0
$550$	6.87936	0.293337
$551$	3.90829	0.166499
$552$	0	0
$553$	8.09890	0.344400
$554$	−10.2823	−0.436854
$555$	0	0
$556$	41.5441	1.76186
$557$	11.0784	0.469406	0.234703	−	0.972067i	$-0.424588\pi$
$557$	11.0784	0.469406	0.234703	+	0.972067i	$0.424588\pi$
$558$	0	0
$559$	29.0700	1.22953
$560$	−1.12064	−0.0473556
$561$	0	0
$562$	30.1822	1.27316
$563$	−25.5936	−1.07864	−0.539320	−	0.842101i	$-0.681319\pi$
$563$	−25.5936	−1.07864	−0.539320	+	0.842101i	$0.681319\pi$
$564$	0	0
$565$	19.6248	0.825620
$566$	−51.9227	−2.18247
$567$	0	0
$568$	3.36069	0.141011
$569$	−0.0605119	−0.00253679	−0.00126840	−	0.999999i	$-0.500404\pi$
$569$	−0.0605119	−0.00253679	−0.00126840	+	0.999999i	$0.500404\pi$
$570$	0	0
$571$	−14.7382	−0.616775	−0.308387	−	0.951261i	$-0.599789\pi$
$571$	−14.7382	−0.616775	−0.308387	+	0.951261i	$0.599789\pi$
$572$	−41.9071	−1.75222
$573$	0	0
$574$	−3.52586	−0.147166
$575$	2.78765	0.116253
$576$	0	0
$577$	−22.0410	−0.917580	−0.458790	−	0.888545i	$-0.651717\pi$
$577$	−22.0410	−0.917580	−0.458790	+	0.888545i	$0.651717\pi$
$578$	32.8238	1.36529
$579$	0	0
$580$	10.5886	0.439669
$581$	1.50307	0.0623579
$582$	0	0
$583$	21.0205	0.870581
$584$	11.7009	0.484185
$585$	0	0
$586$	29.6248	1.22379
$587$	24.2329	1.00020	0.500099	−	0.865968i	$-0.333297\pi$
$587$	24.2329	1.00020	0.500099	+	0.865968i	$0.333297\pi$
$588$	0	0
$589$	2.44748	0.100847
$590$	26.9939	1.11132
$591$	0	0
$592$	8.60689	0.353741
$593$	−38.1133	−1.56512	−0.782562	−	0.622572i	$-0.786088\pi$
$593$	−38.1133	−1.56512	−0.782562	+	0.622572i	$0.786088\pi$
$594$	0	0
$595$	−0.738205	−0.0302634
$596$	28.5113	1.16787
$597$	0	0
$598$	−29.5174	−1.20706
$599$	48.5523	1.98379	0.991897	−	0.127041i	$-0.0405480\pi$
$599$	48.5523	1.98379	0.991897	+	0.127041i	$0.0405480\pi$
$600$	0	0
$601$	34.7936	1.41926	0.709631	−	0.704574i	$-0.248862\pi$
$601$	34.7936	1.41926	0.709631	+	0.704574i	$0.248862\pi$
$602$	6.97107	0.284120
$603$	0	0
$604$	−12.1217	−0.493224
$605$	0.950552	0.0386454
$606$	0	0
$607$	−45.2039	−1.83477	−0.917386	−	0.398000i	$-0.869705\pi$
$607$	−45.2039	−1.83477	−0.917386	+	0.398000i	$0.869705\pi$
$608$	−7.58864	−0.307760
$609$	0	0
$610$	−20.2062	−0.818125
$611$	19.7587	0.799352
$612$	0	0
$613$	38.8203	1.56794	0.783968	−	0.620801i	$-0.213193\pi$
$613$	38.8203	1.56794	0.783968	+	0.620801i	$0.213193\pi$
$614$	−49.8408	−2.01141
$615$	0	0
$616$	−2.63090	−0.106002
$617$	−31.5259	−1.26918	−0.634592	−	0.772848i	$-0.718832\pi$
$617$	−31.5259	−1.26918	−0.634592	+	0.772848i	$0.718832\pi$
$618$	0	0
$619$	−10.9360	−0.439555	−0.219777	−	0.975550i	$-0.570533\pi$
$619$	−10.9360	−0.439555	−0.219777	+	0.975550i	$0.570533\pi$
$620$	6.63090	0.266303
$621$	0	0
$622$	41.3595	1.65836
$623$	0.447480	0.0179279
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.73820	0.269313
$627$	0	0
$628$	52.0288	2.07617
$629$	5.66967	0.226064
$630$	0	0
$631$	−19.5318	−0.777550	−0.388775	−	0.921333i	$-0.627102\pi$
$631$	−19.5318	−0.777550	−0.388775	+	0.921333i	$0.627102\pi$
$632$	23.1194	0.919641
$633$	0	0
$634$	21.6248	0.858829
$635$	0.340173	0.0134994
$636$	0	0
$637$	−32.7370	−1.29709
$638$	−26.8865	−1.06445
$639$	0	0
$640$	−11.5392	−0.456126
$641$	−8.87549	−0.350561	−0.175280	−	0.984519i	$-0.556083\pi$
$641$	−8.87549	−0.350561	−0.175280	+	0.984519i	$0.556083\pi$
$642$	0	0
$643$	28.6647	1.13043	0.565214	−	0.824945i	$-0.308794\pi$
$643$	28.6647	1.13043	0.565214	+	0.824945i	$0.308794\pi$
$644$	−4.07223	−0.160469
$645$	0	0
$646$	−2.97107	−0.116895
$647$	−43.6802	−1.71724	−0.858622	−	0.512609i	$-0.828679\pi$
$647$	−43.6802	−1.71724	−0.858622	+	0.512609i	$0.828679\pi$
$648$	0	0
$649$	−39.4329	−1.54788
$650$	−10.5886	−0.415321
$651$	0	0
$652$	−12.2979	−0.481623
$653$	−12.0905	−0.473137	−0.236569	−	0.971615i	$-0.576023\pi$
$653$	−12.0905	−0.473137	−0.236569	+	0.971615i	$0.576023\pi$
$654$	0	0
$655$	12.0072	0.469160
$656$	6.26284	0.244523
$657$	0	0
$658$	4.73820	0.184714
$659$	−49.2327	−1.91783	−0.958917	−	0.283688i	$-0.908442\pi$
$659$	−49.2327	−1.91783	−0.958917	+	0.283688i	$0.908442\pi$
$660$	0	0
$661$	−1.90110	−0.0739443	−0.0369722	−	0.999316i	$-0.511771\pi$
$661$	−1.90110	−0.0739443	−0.0369722	+	0.999316i	$0.511771\pi$
$662$	53.5402	2.08090
$663$	0	0
$664$	4.29072	0.166512
$665$	−0.539189	−0.0209088
$666$	0	0
$667$	−10.8950	−0.421855
$668$	13.8927	0.537524
$669$	0	0
$670$	−16.8371	−0.650474
$671$	29.5174	1.13951
$672$	0	0
$673$	−19.3763	−0.746901	−0.373451	−	0.927650i	$-0.621825\pi$
$673$	−19.3763	−0.746901	−0.373451	+	0.927650i	$0.621825\pi$
$674$	−78.0626	−3.00686
$675$	0	0
$676$	29.2823	1.12624
$677$	0.237401	0.00912405	0.00456202	−	0.999990i	$-0.498548\pi$
$677$	0.237401	0.00912405	0.00456202	+	0.999990i	$0.498548\pi$
$678$	0	0
$679$	2.89884	0.111247
$680$	−2.10731	−0.0808115
$681$	0	0
$682$	−16.8371	−0.644726
$683$	48.0144	1.83722	0.918610	−	0.395166i	$-0.129313\pi$
$683$	48.0144	1.83722	0.918610	+	0.395166i	$0.129313\pi$
$684$	0	0
$685$	−19.1773	−0.732726
$686$	−16.0410	−0.612450
$687$	0	0
$688$	−12.3824	−0.472076
$689$	−32.3545	−1.23261
$690$	0	0
$691$	−29.5753	−1.12510	−0.562549	−	0.826764i	$-0.690179\pi$
$691$	−29.5753	−1.12510	−0.562549	+	0.826764i	$0.690179\pi$
$692$	−12.1217	−0.460797
$693$	0	0
$694$	72.5029	2.75217
$695$	−15.3340	−0.581653
$696$	0	0
$697$	4.12556	0.156267
$698$	−41.8310	−1.58333
$699$	0	0
$700$	−1.46081	−0.0552135
$701$	41.4596	1.56591	0.782954	−	0.622080i	$-0.213712\pi$
$701$	41.4596	1.56591	0.782954	+	0.622080i	$0.213712\pi$
$702$	0	0
$703$	4.14116	0.156187
$704$	39.0277	1.47091
$705$	0	0
$706$	3.65983	0.137739
$707$	9.34632	0.351504
$708$	0	0
$709$	−2.52747	−0.0949210	−0.0474605	−	0.998873i	$-0.515113\pi$
$709$	−2.52747	−0.0949210	−0.0474605	+	0.998873i	$0.515113\pi$
$710$	−4.73820	−0.177822
$711$	0	0
$712$	1.27739	0.0478724
$713$	−6.82273	−0.255513
$714$	0	0
$715$	15.4680	0.578470
$716$	22.5958	0.844446
$717$	0	0
$718$	36.9516	1.37902
$719$	49.8504	1.85911	0.929554	−	0.368687i	$-0.120192\pi$
$719$	49.8504	1.85911	0.929554	+	0.368687i	$0.120192\pi$
$720$	0	0
$721$	1.36069	0.0506748
$722$	−2.17009	−0.0807623
$723$	0	0
$724$	−26.0144	−0.966817
$725$	−3.90829	−0.145150
$726$	0	0
$727$	−20.4547	−0.758622	−0.379311	−	0.925269i	$-0.623839\pi$
$727$	−20.4547	−0.758622	−0.379311	+	0.925269i	$0.623839\pi$
$728$	4.04945	0.150083
$729$	0	0
$730$	−16.4969	−0.610579
$731$	−8.15676	−0.301688
$732$	0	0
$733$	−30.6369	−1.13160	−0.565799	−	0.824543i	$-0.691432\pi$
$733$	−30.6369	−1.13160	−0.565799	+	0.824543i	$0.691432\pi$
$734$	69.5078	2.56558
$735$	0	0
$736$	21.1545	0.779765
$737$	24.5958	0.905999
$738$	0	0
$739$	29.4908	1.08484	0.542418	−	0.840109i	$-0.317509\pi$
$739$	29.4908	1.08484	0.542418	+	0.840109i	$0.317509\pi$
$740$	11.2195	0.412438
$741$	0	0
$742$	−7.75872	−0.284832
$743$	−41.5129	−1.52296	−0.761481	−	0.648188i	$-0.775527\pi$
$743$	−41.5129	−1.52296	−0.761481	+	0.648188i	$0.775527\pi$
$744$	0	0
$745$	−10.5236	−0.385555
$746$	−41.8915	−1.53376
$747$	0	0
$748$	11.7587	0.429942
$749$	5.74435	0.209894
$750$	0	0
$751$	−19.5792	−0.714454	−0.357227	−	0.934018i	$-0.616278\pi$
$751$	−19.5792	−0.714454	−0.357227	+	0.934018i	$0.616278\pi$
$752$	−8.41628	−0.306910
$753$	0	0
$754$	41.3835	1.50710
$755$	4.47414	0.162831
$756$	0	0
$757$	51.7875	1.88225	0.941124	−	0.338062i	$-0.109771\pi$
$757$	51.7875	1.88225	0.941124	+	0.338062i	$0.109771\pi$
$758$	67.5813	2.45466
$759$	0	0
$760$	−1.53919	−0.0558322
$761$	−3.62863	−0.131538	−0.0657689	−	0.997835i	$-0.520950\pi$
$761$	−3.62863	−0.131538	−0.0657689	+	0.997835i	$0.520950\pi$
$762$	0	0
$763$	3.13624	0.113539
$764$	17.5825	0.636112
$765$	0	0
$766$	23.3028	0.841966
$767$	60.6947	2.19156
$768$	0	0
$769$	23.7275	0.855637	0.427818	−	0.903865i	$-0.359282\pi$
$769$	23.7275	0.855637	0.427818	+	0.903865i	$0.359282\pi$
$770$	3.70928	0.133673
$771$	0	0
$772$	−24.9060	−0.896387
$773$	40.3584	1.45159	0.725796	−	0.687910i	$-0.241472\pi$
$773$	40.3584	1.45159	0.725796	+	0.687910i	$0.241472\pi$
$774$	0	0
$775$	−2.44748	−0.0879161
$776$	8.27513	0.297060
$777$	0	0
$778$	50.0144	1.79310
$779$	3.01333	0.107964
$780$	0	0
$781$	6.92162	0.247675
$782$	8.28231	0.296175
$783$	0	0
$784$	13.9444	0.498015
$785$	−19.2039	−0.685418
$786$	0	0
$787$	−32.5236	−1.15934	−0.579670	−	0.814851i	$-0.696819\pi$
$787$	−32.5236	−1.15934	−0.579670	+	0.814851i	$0.696819\pi$
$788$	9.55252	0.340294
$789$	0	0
$790$	−32.5958	−1.15971
$791$	10.5814	0.376233
$792$	0	0
$793$	−45.4329	−1.61337
$794$	10.9627	0.389050
$795$	0	0
$796$	−57.3751	−2.03361
$797$	32.4885	1.15080	0.575401	−	0.817871i	$-0.304846\pi$
$797$	32.4885	1.15080	0.575401	+	0.817871i	$0.304846\pi$
$798$	0	0
$799$	−5.54411	−0.196136
$800$	7.58864	0.268299
$801$	0	0
$802$	37.4440	1.32219
$803$	24.0989	0.850432
$804$	0	0
$805$	1.50307	0.0529763
$806$	25.9155	0.912834
$807$	0	0
$808$	26.6803	0.938611
$809$	1.16290	0.0408853	0.0204427	−	0.999791i	$-0.493492\pi$
$809$	1.16290	0.0408853	0.0204427	+	0.999791i	$0.493492\pi$
$810$	0	0
$811$	37.7275	1.32479	0.662396	−	0.749154i	$-0.269540\pi$
$811$	37.7275	1.32479	0.662396	+	0.749154i	$0.269540\pi$
$812$	5.70928	0.200356
$813$	0	0
$814$	−28.4885	−0.998522
$815$	4.53919	0.159001
$816$	0	0
$817$	−5.95774	−0.208435
$818$	41.3340	1.44521
$819$	0	0
$820$	8.16394	0.285097
$821$	−24.0554	−0.839540	−0.419770	−	0.907631i	$-0.637889\pi$
$821$	−24.0554	−0.839540	−0.419770	+	0.907631i	$0.637889\pi$
$822$	0	0
$823$	19.6032	0.683324	0.341662	−	0.939823i	$-0.389010\pi$
$823$	19.6032	0.683324	0.341662	+	0.939823i	$0.389010\pi$
$824$	3.88428	0.135315
$825$	0	0
$826$	14.5548	0.506426
$827$	4.54638	0.158093	0.0790465	−	0.996871i	$-0.474812\pi$
$827$	4.54638	0.158093	0.0790465	+	0.996871i	$0.474812\pi$
$828$	0	0
$829$	−49.4473	−1.71738	−0.858688	−	0.512499i	$-0.828720\pi$
$829$	−49.4473	−1.71738	−0.858688	+	0.512499i	$0.828720\pi$
$830$	−6.04945	−0.209979
$831$	0	0
$832$	−60.0710	−2.08259
$833$	9.18568	0.318265
$834$	0	0
$835$	−5.12783	−0.177456
$836$	8.58864	0.297044
$837$	0	0
$838$	−24.1724	−0.835020
$839$	−25.3874	−0.876469	−0.438234	−	0.898861i	$-0.644396\pi$
$839$	−25.3874	−0.876469	−0.438234	+	0.898861i	$0.644396\pi$
$840$	0	0
$841$	−13.7253	−0.473285
$842$	63.6574	2.19378
$843$	0	0
$844$	29.8576	1.02774
$845$	−10.8082	−0.371812
$846$	0	0
$847$	0.512527	0.0176106
$848$	13.7815	0.473259
$849$	0	0
$850$	2.97107	0.101907
$851$	−11.5441	−0.395727
$852$	0	0
$853$	1.71769	0.0588124	0.0294062	−	0.999568i	$-0.490638\pi$
$853$	1.71769	0.0588124	0.0294062	+	0.999568i	$0.490638\pi$
$854$	−10.8950	−0.372818
$855$	0	0
$856$	16.3980	0.560473
$857$	−45.8804	−1.56724	−0.783622	−	0.621237i	$-0.786630\pi$
$857$	−45.8804	−1.56724	−0.783622	+	0.621237i	$0.786630\pi$
$858$	0	0
$859$	−4.96266	−0.169324	−0.0846619	−	0.996410i	$-0.526981\pi$
$859$	−4.96266	−0.169324	−0.0846619	+	0.996410i	$0.526981\pi$
$860$	−16.1412	−0.550409
$861$	0	0
$862$	−52.6947	−1.79479
$863$	33.5441	1.14185	0.570927	−	0.821001i	$-0.306584\pi$
$863$	33.5441	1.14185	0.570927	+	0.821001i	$0.306584\pi$
$864$	0	0
$865$	4.47414	0.152125
$866$	−63.4980	−2.15775
$867$	0	0
$868$	3.57531	0.121354
$869$	47.6163	1.61527
$870$	0	0
$871$	−37.8576	−1.28276
$872$	8.95282	0.303181
$873$	0	0
$874$	6.04945	0.204626
$875$	0.539189	0.0182279
$876$	0	0
$877$	40.5991	1.37094	0.685468	−	0.728103i	$-0.259598\pi$
$877$	40.5991	1.37094	0.685468	+	0.728103i	$0.259598\pi$
$878$	−46.2388	−1.56048
$879$	0	0
$880$	−6.58864	−0.222103
$881$	−5.13170	−0.172891	−0.0864457	−	0.996257i	$-0.527551\pi$
$881$	−5.13170	−0.172891	−0.0864457	+	0.996257i	$0.527551\pi$
$882$	0	0
$883$	−47.4752	−1.59767	−0.798834	−	0.601552i	$-0.794549\pi$
$883$	−47.4752	−1.59767	−0.798834	+	0.601552i	$0.794549\pi$
$884$	−18.0989	−0.608732
$885$	0	0
$886$	−28.6042	−0.960978
$887$	39.4017	1.32298	0.661490	−	0.749954i	$-0.269924\pi$
$887$	39.4017	1.32298	0.661490	+	0.749954i	$0.269924\pi$
$888$	0	0
$889$	0.183417	0.00615162
$890$	−1.80098	−0.0603691
$891$	0	0
$892$	30.3545	1.01635
$893$	−4.04945	−0.135510
$894$	0	0
$895$	−8.34017	−0.278781
$896$	−6.22180	−0.207856
$897$	0	0
$898$	−69.4896	−2.31890
$899$	9.56547	0.319026
$900$	0	0
$901$	9.07838	0.302445
$902$	−20.7298	−0.690227
$903$	0	0
$904$	30.2062	1.00464
$905$	9.60197	0.319180
$906$	0	0
$907$	−10.0533	−0.333815	−0.166908	−	0.985973i	$-0.553378\pi$
$907$	−10.0533	−0.333815	−0.166908	+	0.985973i	$0.553378\pi$
$908$	−1.48029	−0.0491250
$909$	0	0
$910$	−5.70928	−0.189261
$911$	5.16290	0.171054	0.0855272	−	0.996336i	$-0.472743\pi$
$911$	5.16290	0.171054	0.0855272	+	0.996336i	$0.472743\pi$
$912$	0	0
$913$	8.83710	0.292465
$914$	81.1748	2.68502
$915$	0	0
$916$	60.9276	2.01310
$917$	6.47414	0.213795
$918$	0	0
$919$	36.2290	1.19508	0.597542	−	0.801838i	$-0.296144\pi$
$919$	36.2290	1.19508	0.597542	+	0.801838i	$0.296144\pi$
$920$	4.29072	0.141461
$921$	0	0
$922$	−67.3484	−2.21800
$923$	−10.6537	−0.350670
$924$	0	0
$925$	−4.14116	−0.136160
$926$	−38.7552	−1.27358
$927$	0	0
$928$	−29.6586	−0.973591
$929$	−18.0845	−0.593334	−0.296667	−	0.954981i	$-0.595875\pi$
$929$	−18.0845	−0.593334	−0.296667	+	0.954981i	$0.595875\pi$
$930$	0	0
$931$	6.70928	0.219888
$932$	29.2039	0.956607
$933$	0	0
$934$	−2.91321	−0.0953232
$935$	−4.34017	−0.141939
$936$	0	0
$937$	−27.1773	−0.887843	−0.443921	−	0.896066i	$-0.646413\pi$
$937$	−27.1773	−0.887843	−0.443921	+	0.896066i	$0.646413\pi$
$938$	−9.07838	−0.296420
$939$	0	0
$940$	−10.9711	−0.357837
$941$	10.8488	0.353662	0.176831	−	0.984241i	$-0.443415\pi$
$941$	10.8488	0.353662	0.176831	+	0.984241i	$0.443415\pi$
$942$	0	0
$943$	−8.40012	−0.273546
$944$	−25.8531	−0.841446
$945$	0	0
$946$	40.9854	1.33255
$947$	12.6842	0.412182	0.206091	−	0.978533i	$-0.433926\pi$
$947$	12.6842	0.412182	0.206091	+	0.978533i	$0.433926\pi$
$948$	0	0
$949$	−37.0928	−1.20408
$950$	2.17009	0.0704069
$951$	0	0
$952$	−1.13624	−0.0368256
$953$	0.573039	0.0185625	0.00928127	−	0.999957i	$-0.497046\pi$
$953$	0.573039	0.0185625	0.00928127	+	0.999957i	$0.497046\pi$
$954$	0	0
$955$	−6.48974	−0.210003
$956$	56.0482	1.81273
$957$	0	0
$958$	−30.2401	−0.977012
$959$	−10.3402	−0.333902
$960$	0	0
$961$	−25.0098	−0.806769
$962$	43.8492	1.41376
$963$	0	0
$964$	47.4596	1.52857
$965$	9.19287	0.295929
$966$	0	0
$967$	−30.5659	−0.982931	−0.491466	−	0.870897i	$-0.663539\pi$
$967$	−30.5659	−0.982931	−0.491466	+	0.870897i	$0.663539\pi$
$968$	1.46308	0.0470251
$969$	0	0
$970$	−11.6670	−0.374605
$971$	−61.8310	−1.98425	−0.992125	−	0.125252i	$-0.960026\pi$
$971$	−61.8310	−1.98425	−0.992125	+	0.125252i	$0.960026\pi$
$972$	0	0
$973$	−8.26794	−0.265058
$974$	−86.5257	−2.77246
$975$	0	0
$976$	19.3523	0.619451
$977$	−10.5464	−0.337408	−0.168704	−	0.985667i	$-0.553958\pi$
$977$	−10.5464	−0.337408	−0.168704	+	0.985667i	$0.553958\pi$
$978$	0	0
$979$	2.63090	0.0840839
$980$	18.1773	0.580652
$981$	0	0
$982$	−46.2811	−1.47689
$983$	−4.95878	−0.158161	−0.0790803	−	0.996868i	$-0.525198\pi$
$983$	−4.95878	−0.158161	−0.0790803	+	0.996868i	$0.525198\pi$
$984$	0	0
$985$	−3.52586	−0.112343
$986$	−11.6118	−0.369795
$987$	0	0
$988$	−13.2195	−0.420569
$989$	16.6081	0.528107
$990$	0	0
$991$	−5.78992	−0.183923	−0.0919614	−	0.995763i	$-0.529314\pi$
$991$	−5.78992	−0.183923	−0.0919614	+	0.995763i	$0.529314\pi$
$992$	−18.5730	−0.589695
$993$	0	0
$994$	−2.55479	−0.0810329
$995$	21.1773	0.671365
$996$	0	0
$997$	23.7321	0.751602	0.375801	−	0.926700i	$-0.377368\pi$
$997$	23.7321	0.751602	0.375801	+	0.926700i	$0.377368\pi$
$998$	−27.9155	−0.883649
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.h.1.1	✓	3
3.2	odd	2		855.2.a.l.1.3	yes	3
5.4	even	2		4275.2.a.bj.1.3		3
15.14	odd	2		4275.2.a.bb.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.2.a.h.1.1	✓	3	1.1	even	1	trivial
855.2.a.l.1.3	yes	3	3.2	odd	2
4275.2.a.bb.1.1		3	15.14	odd	2
4275.2.a.bj.1.3		3	5.4	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} -0.539189 q^{7} -1.53919 q^{8} +O(q^{10})\)	\(q-2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} -0.539189 q^{7} -1.53919 q^{8} +2.17009 q^{10} -3.17009 q^{11} +4.87936 q^{13} +1.17009 q^{14} -2.07838 q^{16} -1.36910 q^{17} -1.00000 q^{19} -2.70928 q^{20} +6.87936 q^{22} +2.78765 q^{23} +1.00000 q^{25} -10.5886 q^{26} -1.46081 q^{28} -3.90829 q^{29} -2.44748 q^{31} +7.58864 q^{32} +2.97107 q^{34} +0.539189 q^{35} -4.14116 q^{37} +2.17009 q^{38} +1.53919 q^{40} -3.01333 q^{41} +5.95774 q^{43} -8.58864 q^{44} -6.04945 q^{46} +4.04945 q^{47} -6.70928 q^{49} -2.17009 q^{50} +13.2195 q^{52} -6.63090 q^{53} +3.17009 q^{55} +0.829914 q^{56} +8.48133 q^{58} +12.4391 q^{59} -9.31124 q^{61} +5.31124 q^{62} -12.3112 q^{64} -4.87936 q^{65} -7.75872 q^{67} -3.70928 q^{68} -1.17009 q^{70} -2.18342 q^{71} -7.60197 q^{73} +8.98667 q^{74} -2.70928 q^{76} +1.70928 q^{77} -15.0205 q^{79} +2.07838 q^{80} +6.53919 q^{82} -2.78765 q^{83} +1.36910 q^{85} -12.9288 q^{86} +4.87936 q^{88} -0.829914 q^{89} -2.63090 q^{91} +7.55252 q^{92} -8.78765 q^{94} +1.00000 q^{95} -5.37629 q^{97} +14.5597 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8	\( 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8} + q^{10} - 4 q^{11} + 2 q^{13} - 2 q^{14} - 3 q^{16} - 8 q^{17} - 3 q^{19} - q^{20} + 8 q^{22} - 2 q^{23} + 3 q^{25} - 12 q^{26} - 6 q^{28} - 14 q^{29} - 8 q^{31} + 3 q^{32} - 6 q^{34} + 8 q^{37} + q^{38} + 3 q^{40} - 10 q^{41} + 2 q^{43} - 6 q^{44} - 6 q^{47} - 13 q^{49} - q^{50} + 16 q^{52} - 16 q^{53} + 4 q^{55} + 8 q^{56} - 6 q^{58} - 10 q^{59} - 2 q^{61} - 10 q^{62} - 11 q^{64} - 2 q^{65} + 2 q^{67} - 4 q^{68} + 2 q^{70} - 2 q^{71} - 4 q^{73} + 26 q^{74} - q^{76} - 2 q^{77} - 12 q^{79} + 3 q^{80} + 18 q^{82} + 2 q^{83} + 8 q^{85} - 8 q^{86} + 2 q^{88} - 8 q^{89} - 4 q^{91} + 22 q^{92} - 16 q^{94} + 3 q^{95} + 14 q^{97} + 9 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8 + q^10 - 4 * q^11 + 2 * q^13 - 2 * q^14 - 3 * q^16 - 8 * q^17 - 3 * q^19 - q^20 + 8 * q^22 - 2 * q^23 + 3 * q^25 - 12 * q^26 - 6 * q^28 - 14 * q^29 - 8 * q^31 + 3 * q^32 - 6 * q^34 + 8 * q^37 + q^38 + 3 * q^40 - 10 * q^41 + 2 * q^43 - 6 * q^44 - 6 * q^47 - 13 * q^49 - q^50 + 16 * q^52 - 16 * q^53 + 4 * q^55 + 8 * q^56 - 6 * q^58 - 10 * q^59 - 2 * q^61 - 10 * q^62 - 11 * q^64 - 2 * q^65 + 2 * q^67 - 4 * q^68 + 2 * q^70 - 2 * q^71 - 4 * q^73 + 26 * q^74 - q^76 - 2 * q^77 - 12 * q^79 + 3 * q^80 + 18 * q^82 + 2 * q^83 + 8 * q^85 - 8 * q^86 + 2 * q^88 - 8 * q^89 - 4 * q^91 + 22 * q^92 - 16 * q^94 + 3 * q^95 + 14 * q^97 + 9 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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