LMFDB - Embedded newform 855.2.a.m.1.2

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.51658$ 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)

$f(q)$	$=$	$q-0.816594 q^{2} -1.33317 q^{4} +1.00000 q^{5} +5.03316 q^{7} +2.72185 q^{8} +O(q^{10})$	\(q-0.816594 q^{2} -1.33317 q^{4} +1.00000 q^{5} +5.03316 q^{7} +2.72185 q^{8} -0.816594 q^{10} +3.03316 q^{11} -4.57160 q^{13} -4.11005 q^{14} +0.443701 q^{16} +1.07689 q^{17} +1.00000 q^{19} -1.33317 q^{20} -2.47686 q^{22} -4.11005 q^{23} +1.00000 q^{25} +3.73315 q^{26} -6.71008 q^{28} +1.07689 q^{29} +5.58946 q^{31} -5.80602 q^{32} -0.879381 q^{34} +5.03316 q^{35} +0.0947438 q^{37} -0.816594 q^{38} +2.72185 q^{40} -10.6663 q^{41} +5.03316 q^{43} -4.04373 q^{44} +3.35624 q^{46} +12.2995 q^{47} +18.3327 q^{49} -0.816594 q^{50} +6.09474 q^{52} +4.09474 q^{53} +3.03316 q^{55} +13.6995 q^{56} -0.879381 q^{58} +1.39997 q^{59} -5.69951 q^{61} -4.56432 q^{62} +3.85376 q^{64} -4.57160 q^{65} +5.28168 q^{67} -1.43568 q^{68} -4.11005 q^{70} +5.67692 q^{71} +9.07689 q^{73} -0.0773672 q^{74} -1.33317 q^{76} +15.2664 q^{77} -5.39997 q^{79} +0.443701 q^{80} +8.71008 q^{82} -1.95627 q^{83} +1.07689 q^{85} -4.11005 q^{86} +8.25581 q^{88} +2.18949 q^{89} -23.0096 q^{91} +5.47941 q^{92} -10.0437 q^{94} +1.00000 q^{95} -2.16106 q^{97} -14.9704 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8	$ 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{13} + 8 q^{14} + 4 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 4 q^{26} - 8 q^{28} - 4 q^{29} + 4 q^{31} + 6 q^{32} - 4 q^{34} + 4 q^{35} - 6 q^{37} + 2 q^{38} + 12 q^{40} - 16 q^{41} + 4 q^{43} - 24 q^{44} + 12 q^{47} + 20 q^{49} + 2 q^{50} + 18 q^{52} + 10 q^{53} - 4 q^{55} + 12 q^{56} - 4 q^{58} + 20 q^{61} - 20 q^{62} - 4 q^{64} + 2 q^{65} - 18 q^{67} - 4 q^{68} + 8 q^{70} + 20 q^{71} + 28 q^{73} - 32 q^{74} + 8 q^{76} + 40 q^{77} - 16 q^{79} + 4 q^{80} + 16 q^{82} - 4 q^{85} + 8 q^{86} - 12 q^{88} - 4 q^{89} - 36 q^{91} + 28 q^{92} - 48 q^{94} + 4 q^{95} + 30 q^{97} - 38 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^13 + 8 * q^14 + 4 * q^16 - 4 * q^17 + 4 * q^19 + 8 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 - 4 * q^26 - 8 * q^28 - 4 * q^29 + 4 * q^31 + 6 * q^32 - 4 * q^34 + 4 * q^35 - 6 * q^37 + 2 * q^38 + 12 * q^40 - 16 * q^41 + 4 * q^43 - 24 * q^44 + 12 * q^47 + 20 * q^49 + 2 * q^50 + 18 * q^52 + 10 * q^53 - 4 * q^55 + 12 * q^56 - 4 * q^58 + 20 * q^61 - 20 * q^62 - 4 * q^64 + 2 * q^65 - 18 * q^67 - 4 * q^68 + 8 * q^70 + 20 * q^71 + 28 * q^73 - 32 * q^74 + 8 * q^76 + 40 * q^77 - 16 * q^79 + 4 * q^80 + 16 * q^82 - 4 * q^85 + 8 * q^86 - 12 * q^88 - 4 * q^89 - 36 * q^91 + 28 * q^92 - 48 * q^94 + 4 * q^95 + 30 * q^97 - 38 * 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 $

$2$	−0.816594	−0.577419	−0.288710	−	0.957417i	$-0.593226\pi$
$2$	−0.816594	−0.577419	−0.288710	+	0.957417i	$0.593226\pi$
$3$	0	0
$3$	0	0
$4$	−1.33317	−0.666587
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	5.03316	1.90236	0.951178	−	0.308644i	$-0.0998750\pi$
$7$	5.03316	1.90236	0.951178	+	0.308644i	$0.0998750\pi$
$8$	2.72185	0.962319
$9$	0	0
$10$	−0.816594	−0.258230
$11$	3.03316	0.914532	0.457266	−	0.889330i	$-0.348829\pi$
$11$	3.03316	0.914532	0.457266	+	0.889330i	$0.348829\pi$
$12$	0	0
$13$	−4.57160	−1.26793	−0.633967	−	0.773360i	$-0.718575\pi$
$13$	−4.57160	−1.26793	−0.633967	+	0.773360i	$0.718575\pi$
$14$	−4.11005	−1.09846
$15$	0	0
$16$	0.443701	0.110925
$17$	1.07689	0.261184	0.130592	−	0.991436i	$-0.458312\pi$
$17$	1.07689	0.261184	0.130592	+	0.991436i	$0.458312\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.33317	−0.298107
$21$	0	0
$22$	−2.47686	−0.528068
$23$	−4.11005	−0.857004	−0.428502	−	0.903541i	$-0.640959\pi$
$23$	−4.11005	−0.857004	−0.428502	+	0.903541i	$0.640959\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.73315	0.732130
$27$	0	0
$28$	−6.71008	−1.26809
$29$	1.07689	0.199973	0.0999866	−	0.994989i	$-0.468120\pi$
$29$	1.07689	0.199973	0.0999866	+	0.994989i	$0.468120\pi$
$30$	0	0
$31$	5.58946	1.00390	0.501948	−	0.864898i	$-0.332617\pi$
$31$	5.58946	1.00390	0.501948	+	0.864898i	$0.332617\pi$
$32$	−5.80602	−1.02637
$33$	0	0
$34$	−0.879381	−0.150813
$35$	5.03316	0.850759
$36$	0	0
$37$	0.0947438	0.0155758	0.00778789	−	0.999970i	$-0.497521\pi$
$37$	0.0947438	0.0155758	0.00778789	+	0.999970i	$0.497521\pi$
$38$	−0.816594	−0.132469
$39$	0	0
$40$	2.72185	0.430362
$41$	−10.6663	−1.66580	−0.832902	−	0.553421i	$-0.813322\pi$
$41$	−10.6663	−1.66580	−0.832902	+	0.553421i	$0.813322\pi$
$42$	0	0
$43$	5.03316	0.767550	0.383775	−	0.923427i	$-0.374624\pi$
$43$	5.03316	0.767550	0.383775	+	0.923427i	$0.374624\pi$
$44$	−4.04373	−0.609615
$45$	0	0
$46$	3.35624	0.494851
$47$	12.2995	1.79407	0.897036	−	0.441958i	$-0.145716\pi$
$47$	12.2995	1.79407	0.897036	+	0.441958i	$0.145716\pi$
$48$	0	0
$49$	18.3327	2.61896
$50$	−0.816594	−0.115484
$51$	0	0
$52$	6.09474	0.845189
$53$	4.09474	0.562456	0.281228	−	0.959641i	$-0.409258\pi$
$53$	4.09474	0.562456	0.281228	+	0.959641i	$0.409258\pi$
$54$	0	0
$55$	3.03316	0.408991
$56$	13.6995	1.83067
$57$	0	0
$58$	−0.879381	−0.115468
$59$	1.39997	0.182261	0.0911304	−	0.995839i	$-0.470952\pi$
$59$	1.39997	0.182261	0.0911304	+	0.995839i	$0.470952\pi$
$60$	0	0
$61$	−5.69951	−0.729747	−0.364874	−	0.931057i	$-0.618888\pi$
$61$	−5.69951	−0.729747	−0.364874	+	0.931057i	$0.618888\pi$
$62$	−4.56432	−0.579669
$63$	0	0
$64$	3.85376	0.481720
$65$	−4.57160	−0.567038
$66$	0	0
$67$	5.28168	0.645260	0.322630	−	0.946525i	$-0.395433\pi$
$67$	5.28168	0.645260	0.322630	+	0.946525i	$0.395433\pi$
$68$	−1.43568	−0.174102
$69$	0	0
$70$	−4.11005	−0.491245
$71$	5.67692	0.673726	0.336863	−	0.941554i	$-0.390634\pi$
$71$	5.67692	0.673726	0.336863	+	0.941554i	$0.390634\pi$
$72$	0	0
$73$	9.07689	1.06237	0.531185	−	0.847256i	$-0.321747\pi$
$73$	9.07689	1.06237	0.531185	+	0.847256i	$0.321747\pi$
$74$	−0.0773672	−0.00899375
$75$	0	0
$76$	−1.33317	−0.152926
$77$	15.2664	1.73977
$78$	0	0
$79$	−5.39997	−0.607544	−0.303772	−	0.952745i	$-0.598246\pi$
$79$	−5.39997	−0.607544	−0.303772	+	0.952745i	$0.598246\pi$
$80$	0.443701	0.0496073
$81$	0	0
$82$	8.71008	0.961867
$83$	−1.95627	−0.214729	−0.107364	−	0.994220i	$-0.534241\pi$
$83$	−1.95627	−0.214729	−0.107364	+	0.994220i	$0.534241\pi$
$84$	0	0
$85$	1.07689	0.116805
$86$	−4.11005	−0.443198
$87$	0	0
$88$	8.25581	0.880072
$89$	2.18949	0.232085	0.116043	−	0.993244i	$-0.462979\pi$
$89$	2.18949	0.232085	0.116043	+	0.993244i	$0.462979\pi$
$90$	0	0
$91$	−23.0096	−2.41206
$92$	5.47941	0.571268
$93$	0	0
$94$	−10.0437	−1.03593
$95$	1.00000	0.102598
$96$	0	0
$97$	−2.16106	−0.219423	−0.109711	−	0.993963i	$-0.534993\pi$
$97$	−2.16106	−0.219423	−0.109711	+	0.993963i	$0.534993\pi$
$98$	−14.9704	−1.51224
$99$	0	0
$100$	−1.33317	−0.133317
$101$	−12.5869	−1.25244	−0.626222	−	0.779645i	$-0.715400\pi$
$101$	−12.5869	−1.25244	−0.626222	+	0.779645i	$0.715400\pi$
$102$	0	0
$103$	−6.20479	−0.611376	−0.305688	−	0.952132i	$-0.598887\pi$
$103$	−6.20479	−0.611376	−0.305688	+	0.952132i	$0.598887\pi$
$104$	−12.4432	−1.22016
$105$	0	0
$106$	−3.34374	−0.324773
$107$	12.5481	1.21307	0.606533	−	0.795058i	$-0.292560\pi$
$107$	12.5481	1.21307	0.606533	+	0.795058i	$0.292560\pi$
$108$	0	0
$109$	−15.8096	−1.51428	−0.757140	−	0.653252i	$-0.773404\pi$
$109$	−15.8096	−1.51428	−0.757140	+	0.653252i	$0.773404\pi$
$110$	−2.47686	−0.236159
$111$	0	0
$112$	2.23322	0.211019
$113$	5.49472	0.516899	0.258450	−	0.966025i	$-0.416788\pi$
$113$	5.49472	0.516899	0.258450	+	0.966025i	$0.416788\pi$
$114$	0	0
$115$	−4.11005	−0.383264
$116$	−1.43568	−0.133300
$117$	0	0
$118$	−1.14321	−0.105241
$119$	5.42015	0.496865
$120$	0	0
$121$	−1.79994	−0.163631
$122$	4.65418	0.421370
$123$	0	0
$124$	−7.45172	−0.669184
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.61533	−0.764487	−0.382244	−	0.924062i	$-0.624848\pi$
$127$	−8.61533	−0.764487	−0.382244	+	0.924062i	$0.624848\pi$
$128$	8.46509	0.748215
$129$	0	0
$130$	3.73315	0.327419
$131$	2.15378	0.188176	0.0940882	−	0.995564i	$-0.470006\pi$
$131$	2.15378	0.188176	0.0940882	+	0.995564i	$0.470006\pi$
$132$	0	0
$133$	5.03316	0.436430
$134$	−4.31299	−0.372586
$135$	0	0
$136$	2.93113	0.251342
$137$	−2.18949	−0.187061	−0.0935303	−	0.995616i	$-0.529815\pi$
$137$	−2.18949	−0.187061	−0.0935303	+	0.995616i	$0.529815\pi$
$138$	0	0
$139$	22.5196	1.91009	0.955045	−	0.296460i	$-0.0958062\pi$
$139$	22.5196	1.91009	0.955045	+	0.296460i	$0.0958062\pi$
$140$	−6.71008	−0.567105
$141$	0	0
$142$	−4.63574	−0.389022
$143$	−13.8664	−1.15957
$144$	0	0
$145$	1.07689	0.0894308
$146$	−7.41213	−0.613433
$147$	0	0
$148$	−0.126310	−0.0103826
$149$	−9.78697	−0.801780	−0.400890	−	0.916126i	$-0.631299\pi$
$149$	−9.78697	−0.801780	−0.400890	+	0.916126i	$0.631299\pi$
$150$	0	0
$151$	5.87683	0.478250	0.239125	−	0.970989i	$-0.423139\pi$
$151$	5.87683	0.478250	0.239125	+	0.970989i	$0.423139\pi$
$152$	2.72185	0.220771
$153$	0	0
$154$	−12.4664	−1.00457
$155$	5.58946	0.448956
$156$	0	0
$157$	10.1895	0.813210	0.406605	−	0.913604i	$-0.366713\pi$
$157$	10.1895	0.813210	0.406605	+	0.913604i	$0.366713\pi$
$158$	4.40959	0.350808
$159$	0	0
$160$	−5.80602	−0.459007
$161$	−20.6865	−1.63033
$162$	0	0
$163$	−10.3659	−0.811916	−0.405958	−	0.913892i	$-0.633062\pi$
$163$	−10.3659	−0.811916	−0.405958	+	0.913892i	$0.633062\pi$
$164$	14.2201	1.11040
$165$	0	0
$166$	1.59748	0.123988
$167$	15.6048	1.20753	0.603766	−	0.797161i	$-0.293666\pi$
$167$	15.6048	1.20753	0.603766	+	0.797161i	$0.293666\pi$
$168$	0	0
$169$	7.89956	0.607659
$170$	−0.879381	−0.0674455
$171$	0	0
$172$	−6.71008	−0.511639
$173$	−0.571604	−0.0434583	−0.0217291	−	0.999764i	$-0.506917\pi$
$173$	−0.571604	−0.0434583	−0.0217291	+	0.999764i	$0.506917\pi$
$174$	0	0
$175$	5.03316	0.380471
$176$	1.34582	0.101445
$177$	0	0
$178$	−1.78792	−0.134010
$179$	−24.8864	−1.86010	−0.930050	−	0.367433i	$-0.880237\pi$
$179$	−24.8864	−1.86010	−0.930050	+	0.367433i	$0.880237\pi$
$180$	0	0
$181$	6.95372	0.516866	0.258433	−	0.966029i	$-0.416794\pi$
$181$	6.95372	0.516866	0.258433	+	0.966029i	$0.416794\pi$
$182$	18.7895	1.39277
$183$	0	0
$184$	−11.1869	−0.824712
$185$	0.0947438	0.00696570
$186$	0	0
$187$	3.26638	0.238861
$188$	−16.3974	−1.19590
$189$	0	0
$190$	−0.816594	−0.0592420
$191$	3.91254	0.283102	0.141551	−	0.989931i	$-0.454791\pi$
$191$	3.91254	0.283102	0.141551	+	0.989931i	$0.454791\pi$
$192$	0	0
$193$	−9.20734	−0.662759	−0.331379	−	0.943498i	$-0.607514\pi$
$193$	−9.20734	−0.662759	−0.331379	+	0.943498i	$0.607514\pi$
$194$	1.76471	0.126699
$195$	0	0
$196$	−24.4407	−1.74576
$197$	3.84622	0.274032	0.137016	−	0.990569i	$-0.456249\pi$
$197$	3.84622	0.274032	0.137016	+	0.990569i	$0.456249\pi$
$198$	0	0
$199$	−18.2864	−1.29629	−0.648145	−	0.761517i	$-0.724455\pi$
$199$	−18.2864	−1.29629	−0.648145	+	0.761517i	$0.724455\pi$
$200$	2.72185	0.192464
$201$	0	0
$202$	10.2784	0.723185
$203$	5.42015	0.380420
$204$	0	0
$205$	−10.6663	−0.744970
$206$	5.06680	0.353020
$207$	0	0
$208$	−2.02842	−0.140646
$209$	3.03316	0.209808
$210$	0	0
$211$	−19.2970	−1.32846	−0.664230	−	0.747529i	$-0.731240\pi$
$211$	−19.2970	−1.32846	−0.664230	+	0.747529i	$0.731240\pi$
$212$	−5.45901	−0.374926
$213$	0	0
$214$	−10.2467	−0.700448
$215$	5.03316	0.343259
$216$	0	0
$217$	28.1326	1.90977
$218$	12.9100	0.874375
$219$	0	0
$220$	−4.04373	−0.272628
$221$	−4.92311	−0.331164
$222$	0	0
$223$	0.615334	0.0412058	0.0206029	−	0.999788i	$-0.493441\pi$
$223$	0.615334	0.0412058	0.0206029	+	0.999788i	$0.493441\pi$
$224$	−29.2226	−1.95252
$225$	0	0
$226$	−4.48695	−0.298468
$227$	−17.4712	−1.15960	−0.579801	−	0.814758i	$-0.696870\pi$
$227$	−17.4712	−1.15960	−0.579801	+	0.814758i	$0.696870\pi$
$228$	0	0
$229$	9.69951	0.640962	0.320481	−	0.947255i	$-0.396156\pi$
$229$	9.69951	0.640962	0.320481	+	0.947255i	$0.396156\pi$
$230$	3.35624	0.221304
$231$	0	0
$232$	2.93113	0.192438
$233$	8.15378	0.534172	0.267086	−	0.963673i	$-0.413939\pi$
$233$	8.15378	0.534172	0.267086	+	0.963673i	$0.413939\pi$
$234$	0	0
$235$	12.2995	0.802333
$236$	−1.86641	−0.121493
$237$	0	0
$238$	−4.42607	−0.286899
$239$	−24.5991	−1.59118	−0.795591	−	0.605834i	$-0.792839\pi$
$239$	−24.5991	−1.59118	−0.795591	+	0.605834i	$0.792839\pi$
$240$	0	0
$241$	25.7738	1.66024	0.830120	−	0.557585i	$-0.188272\pi$
$241$	25.7738	1.66024	0.830120	+	0.557585i	$0.188272\pi$
$242$	1.46982	0.0944838
$243$	0	0
$244$	7.59844	0.486440
$245$	18.3327	1.17123
$246$	0	0
$247$	−4.57160	−0.290884
$248$	15.2137	0.966069
$249$	0	0
$250$	−0.816594	−0.0516459
$251$	−12.5991	−0.795246	−0.397623	−	0.917549i	$-0.630165\pi$
$251$	−12.5991	−0.795246	−0.397623	+	0.917549i	$0.630165\pi$
$252$	0	0
$253$	−12.4664	−0.783758
$254$	7.03523	0.441430
$255$	0	0
$256$	−14.6201	−0.913754
$257$	−0.182203	−0.0113655	−0.00568275	−	0.999984i	$-0.501809\pi$
$257$	−0.182203	−0.0113655	−0.00568275	+	0.999984i	$0.501809\pi$
$258$	0	0
$259$	0.476860	0.0296307
$260$	6.09474	0.377980
$261$	0	0
$262$	−1.75876	−0.108657
$263$	−7.37643	−0.454850	−0.227425	−	0.973796i	$-0.573031\pi$
$263$	−7.37643	−0.454850	−0.227425	+	0.973796i	$0.573031\pi$
$264$	0	0
$265$	4.09474	0.251538
$266$	−4.11005	−0.252003
$267$	0	0
$268$	−7.04140	−0.430122
$269$	−4.70206	−0.286689	−0.143345	−	0.989673i	$-0.545786\pi$
$269$	−4.70206	−0.286689	−0.143345	+	0.989673i	$0.545786\pi$
$270$	0	0
$271$	9.92056	0.602631	0.301316	−	0.953524i	$-0.402574\pi$
$271$	9.92056	0.602631	0.301316	+	0.953524i	$0.402574\pi$
$272$	0.477817	0.0289719
$273$	0	0
$274$	1.78792	0.108012
$275$	3.03316	0.182906
$276$	0	0
$277$	−22.6297	−1.35969	−0.679843	−	0.733358i	$-0.737952\pi$
$277$	−22.6297	−1.35969	−0.679843	+	0.733358i	$0.737952\pi$
$278$	−18.3894	−1.10292
$279$	0	0
$280$	13.6995	0.818702
$281$	3.95386	0.235868	0.117934	−	0.993021i	$-0.462373\pi$
$281$	3.95386	0.235868	0.117934	+	0.993021i	$0.462373\pi$
$282$	0	0
$283$	−13.0638	−0.776561	−0.388280	−	0.921541i	$-0.626931\pi$
$283$	−13.0638	−0.776561	−0.388280	+	0.921541i	$0.626931\pi$
$284$	−7.56832	−0.449097
$285$	0	0
$286$	11.3232	0.669556
$287$	−53.6854	−3.16895
$288$	0	0
$289$	−15.8403	−0.931783
$290$	−0.879381	−0.0516391
$291$	0	0
$292$	−12.1011	−0.708162
$293$	9.69463	0.566366	0.283183	−	0.959066i	$-0.408610\pi$
$293$	9.69463	0.566366	0.283183	+	0.959066i	$0.408610\pi$
$294$	0	0
$295$	1.39997	0.0815095
$296$	0.257878	0.0149889
$297$	0	0
$298$	7.99198	0.462963
$299$	18.7895	1.08663
$300$	0	0
$301$	25.3327	1.46015
$302$	−4.79899	−0.276151
$303$	0	0
$304$	0.443701	0.0254480
$305$	−5.69951	−0.326353
$306$	0	0
$307$	−28.5481	−1.62932	−0.814662	−	0.579936i	$-0.803077\pi$
$307$	−28.5481	−1.62932	−0.814662	+	0.579936i	$0.803077\pi$
$308$	−20.3527	−1.15970
$309$	0	0
$310$	−4.56432	−0.259236
$311$	−25.4070	−1.44070	−0.720350	−	0.693610i	$-0.756019\pi$
$311$	−25.4070	−1.44070	−0.720350	+	0.693610i	$0.756019\pi$
$312$	0	0
$313$	16.7999	0.949589	0.474794	−	0.880097i	$-0.342522\pi$
$313$	16.7999	0.949589	0.474794	+	0.880097i	$0.342522\pi$
$314$	−8.32068	−0.469563
$315$	0	0
$316$	7.19910	0.404981
$317$	31.7505	1.78329	0.891643	−	0.452738i	$-0.149553\pi$
$317$	31.7505	1.78329	0.891643	+	0.452738i	$0.149553\pi$
$318$	0	0
$319$	3.26638	0.182882
$320$	3.85376	0.215432
$321$	0	0
$322$	16.8925	0.941382
$323$	1.07689	0.0599197
$324$	0	0
$325$	−4.57160	−0.253587
$326$	8.46470	0.468816
$327$	0	0
$328$	−29.0322	−1.60304
$329$	61.9055	3.41296
$330$	0	0
$331$	−9.96429	−0.547687	−0.273843	−	0.961774i	$-0.588295\pi$
$331$	−9.96429	−0.547687	−0.273843	+	0.961774i	$0.588295\pi$
$332$	2.60805	0.143135
$333$	0	0
$334$	−12.7428	−0.697253
$335$	5.28168	0.288569
$336$	0	0
$337$	23.1918	1.26334	0.631669	−	0.775238i	$-0.282370\pi$
$337$	23.1918	1.26334	0.631669	+	0.775238i	$0.282370\pi$
$338$	−6.45074	−0.350874
$339$	0	0
$340$	−1.43568	−0.0778607
$341$	16.9537	0.918095
$342$	0	0
$343$	57.0393	3.07983
$344$	13.6995	0.738628
$345$	0	0
$346$	0.466769	0.0250936
$347$	−21.1352	−1.13460	−0.567298	−	0.823512i	$-0.692011\pi$
$347$	−21.1352	−1.13460	−0.567298	+	0.823512i	$0.692011\pi$
$348$	0	0
$349$	−5.04628	−0.270121	−0.135061	−	0.990837i	$-0.543123\pi$
$349$	−5.04628	−0.270121	−0.135061	+	0.990837i	$0.543123\pi$
$350$	−4.11005	−0.219691
$351$	0	0
$352$	−17.6106	−0.938648
$353$	−12.5634	−0.668680	−0.334340	−	0.942452i	$-0.608513\pi$
$353$	−12.5634	−0.668680	−0.334340	+	0.942452i	$0.608513\pi$
$354$	0	0
$355$	5.67692	0.301300
$356$	−2.91897	−0.154705
$357$	0	0
$358$	20.3221	1.07406
$359$	17.6534	0.931709	0.465855	−	0.884861i	$-0.345747\pi$
$359$	17.6534	0.931709	0.465855	+	0.884861i	$0.345747\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.67837	−0.298448
$363$	0	0
$364$	30.6758	1.60785
$365$	9.07689	0.475106
$366$	0	0
$367$	4.43409	0.231457	0.115729	−	0.993281i	$-0.463080\pi$
$367$	4.43409	0.231457	0.115729	+	0.993281i	$0.463080\pi$
$368$	−1.82363	−0.0950634
$369$	0	0
$370$	−0.0773672	−0.00402213
$371$	20.6095	1.06999
$372$	0	0
$373$	−20.8735	−1.08079	−0.540396	−	0.841411i	$-0.681725\pi$
$373$	−20.8735	−1.08079	−0.540396	+	0.841411i	$0.681725\pi$
$374$	−2.66730	−0.137923
$375$	0	0
$376$	33.4775	1.72647
$377$	−4.92311	−0.253553
$378$	0	0
$379$	−6.79994	−0.349290	−0.174645	−	0.984632i	$-0.555878\pi$
$379$	−6.79994	−0.349290	−0.174645	+	0.984632i	$0.555878\pi$
$380$	−1.33317	−0.0683904
$381$	0	0
$382$	−3.19496	−0.163468
$383$	10.9078	0.557363	0.278681	−	0.960384i	$-0.410103\pi$
$383$	10.9078	0.557363	0.278681	+	0.960384i	$0.410103\pi$
$384$	0	0
$385$	15.2664	0.778047
$386$	7.51866	0.382690
$387$	0	0
$388$	2.88107	0.146264
$389$	−20.9326	−1.06132	−0.530662	−	0.847584i	$-0.678057\pi$
$389$	−20.9326	−1.06132	−0.530662	+	0.847584i	$0.678057\pi$
$390$	0	0
$391$	−4.42607	−0.223836
$392$	49.8989	2.52027
$393$	0	0
$394$	−3.14080	−0.158231
$395$	−5.39997	−0.271702
$396$	0	0
$397$	13.1221	0.658578	0.329289	−	0.944229i	$-0.393191\pi$
$397$	13.1221	0.658578	0.329289	+	0.944229i	$0.393191\pi$
$398$	14.9326	0.748503
$399$	0	0
$400$	0.443701	0.0221850
$401$	11.5528	0.576919	0.288459	−	0.957492i	$-0.406857\pi$
$401$	11.5528	0.576919	0.288459	+	0.957492i	$0.406857\pi$
$402$	0	0
$403$	−25.5528	−1.27288
$404$	16.7805	0.834863
$405$	0	0
$406$	−4.42607	−0.219662
$407$	0.287373	0.0142445
$408$	0	0
$409$	−18.9433	−0.936686	−0.468343	−	0.883547i	$-0.655149\pi$
$409$	−18.9433	−0.936686	−0.468343	+	0.883547i	$0.655149\pi$
$410$	8.71008	0.430160
$411$	0	0
$412$	8.27207	0.407536
$413$	7.04628	0.346725
$414$	0	0
$415$	−1.95627	−0.0960295
$416$	26.5428	1.30137
$417$	0	0
$418$	−2.47686	−0.121147
$419$	−27.2664	−1.33205	−0.666025	−	0.745930i	$-0.732006\pi$
$419$	−27.2664	−1.33205	−0.666025	+	0.745930i	$0.732006\pi$
$420$	0	0
$421$	6.66635	0.324898	0.162449	−	0.986717i	$-0.448061\pi$
$421$	6.66635	0.324898	0.162449	+	0.986717i	$0.448061\pi$
$422$	15.7578	0.767078
$423$	0	0
$424$	11.1453	0.541263
$425$	1.07689	0.0522368
$426$	0	0
$427$	−28.6865	−1.38824
$428$	−16.7287	−0.808614
$429$	0	0
$430$	−4.11005	−0.198204
$431$	32.4548	1.56329	0.781646	−	0.623723i	$-0.214381\pi$
$431$	32.4548	1.56329	0.781646	+	0.623723i	$0.214381\pi$
$432$	0	0
$433$	33.8168	1.62513	0.812567	−	0.582868i	$-0.198070\pi$
$433$	33.8168	1.62513	0.812567	+	0.582868i	$0.198070\pi$
$434$	−22.9729	−1.10274
$435$	0	0
$436$	21.0769	1.00940
$437$	−4.11005	−0.196610
$438$	0	0
$439$	−27.8453	−1.32898	−0.664491	−	0.747296i	$-0.731352\pi$
$439$	−27.8453	−1.32898	−0.664491	+	0.747296i	$0.731352\pi$
$440$	8.25581	0.393580
$441$	0	0
$442$	4.02018	0.191221
$443$	−22.7754	−1.08209	−0.541047	−	0.840992i	$-0.681972\pi$
$443$	−22.7754	−1.08209	−0.541047	+	0.840992i	$0.681972\pi$
$444$	0	0
$445$	2.18949	0.103792
$446$	−0.502478	−0.0237930
$447$	0	0
$448$	19.3966	0.916404
$449$	−24.8507	−1.17278	−0.586389	−	0.810029i	$-0.699451\pi$
$449$	−24.8507	−1.17278	−0.586389	+	0.810029i	$0.699451\pi$
$450$	0	0
$451$	−32.3527	−1.52343
$452$	−7.32541	−0.344558
$453$	0	0
$454$	14.2669	0.669577
$455$	−23.0096	−1.07871
$456$	0	0
$457$	−7.33270	−0.343009	−0.171505	−	0.985183i	$-0.554863\pi$
$457$	−7.33270	−0.343009	−0.171505	+	0.985183i	$0.554863\pi$
$458$	−7.92056	−0.370104
$459$	0	0
$460$	5.47941	0.255479
$461$	−10.3076	−0.480071	−0.240035	−	0.970764i	$-0.577159\pi$
$461$	−10.3076	−0.480071	−0.240035	+	0.970764i	$0.577159\pi$
$462$	0	0
$463$	7.80249	0.362613	0.181306	−	0.983427i	$-0.441967\pi$
$463$	7.80249	0.362613	0.181306	+	0.983427i	$0.441967\pi$
$464$	0.477817	0.0221821
$465$	0	0
$466$	−6.65833	−0.308441
$467$	7.37643	0.341340	0.170670	−	0.985328i	$-0.445407\pi$
$467$	7.37643	0.341340	0.170670	+	0.985328i	$0.445407\pi$
$468$	0	0
$469$	26.5835	1.22751
$470$	−10.0437	−0.463283
$471$	0	0
$472$	3.81051	0.175393
$473$	15.2664	0.701949
$474$	0	0
$475$	1.00000	0.0458831
$476$	−7.22601	−0.331204
$477$	0	0
$478$	20.0875	0.918779
$479$	16.1458	0.737719	0.368859	−	0.929485i	$-0.379748\pi$
$479$	16.1458	0.737719	0.368859	+	0.929485i	$0.379748\pi$
$480$	0	0
$481$	−0.433131	−0.0197491
$482$	−21.0468	−0.958654
$483$	0	0
$484$	2.39964	0.109074
$485$	−2.16106	−0.0981288
$486$	0	0
$487$	−8.40566	−0.380897	−0.190448	−	0.981697i	$-0.560994\pi$
$487$	−8.40566	−0.380897	−0.190448	+	0.981697i	$0.560994\pi$
$488$	−15.5132	−0.702250
$489$	0	0
$490$	−14.9704	−0.676292
$491$	36.0855	1.62852	0.814259	−	0.580502i	$-0.197144\pi$
$491$	36.0855	1.62852	0.814259	+	0.580502i	$0.197144\pi$
$492$	0	0
$493$	1.15969	0.0522298
$494$	3.73315	0.167962
$495$	0	0
$496$	2.48005	0.111357
$497$	28.5728	1.28167
$498$	0	0
$499$	−3.30035	−0.147744	−0.0738720	−	0.997268i	$-0.523536\pi$
$499$	−3.30035	−0.147744	−0.0738720	+	0.997268i	$0.523536\pi$
$500$	−1.33317	−0.0596214
$501$	0	0
$502$	10.2883	0.459191
$503$	−3.53530	−0.157631	−0.0788157	−	0.996889i	$-0.525114\pi$
$503$	−3.53530	−0.157631	−0.0788157	+	0.996889i	$0.525114\pi$
$504$	0	0
$505$	−12.5869	−0.560110
$506$	10.1800	0.452557
$507$	0	0
$508$	11.4857	0.509597
$509$	−22.4096	−0.993287	−0.496644	−	0.867955i	$-0.665434\pi$
$509$	−22.4096	−0.993287	−0.496644	+	0.867955i	$0.665434\pi$
$510$	0	0
$511$	45.6854	2.02100
$512$	−4.99151	−0.220596
$513$	0	0
$514$	0.148786	0.00656266
$515$	−6.20479	−0.273416
$516$	0	0
$517$	37.3065	1.64074
$518$	−0.389401	−0.0171093
$519$	0	0
$520$	−12.4432	−0.545671
$521$	−34.4402	−1.50885	−0.754426	−	0.656385i	$-0.772085\pi$
$521$	−34.4402	−1.50885	−0.754426	+	0.656385i	$0.772085\pi$
$522$	0	0
$523$	−30.4451	−1.33127	−0.665635	−	0.746277i	$-0.731839\pi$
$523$	−30.4451	−1.33127	−0.665635	+	0.746277i	$0.731839\pi$
$524$	−2.87136	−0.125436
$525$	0	0
$526$	6.02355	0.262639
$527$	6.01923	0.262202
$528$	0	0
$529$	−6.10750	−0.265543
$530$	−3.34374	−0.145243
$531$	0	0
$532$	−6.71008	−0.290919
$533$	48.7623	2.11213
$534$	0	0
$535$	12.5481	0.542500
$536$	14.3759	0.620946
$537$	0	0
$538$	3.83967	0.165540
$539$	55.6060	2.39512
$540$	0	0
$541$	−12.2794	−0.527931	−0.263965	−	0.964532i	$-0.585030\pi$
$541$	−12.2794	−0.527931	−0.263965	+	0.964532i	$0.585030\pi$
$542$	−8.10107	−0.347971
$543$	0	0
$544$	−6.25244	−0.268071
$545$	−15.8096	−0.677207
$546$	0	0
$547$	0.250928	0.0107289	0.00536446	−	0.999986i	$-0.498292\pi$
$547$	0.250928	0.0107289	0.00536446	+	0.999986i	$0.498292\pi$
$548$	2.91897	0.124692
$549$	0	0
$550$	−2.47686	−0.105614
$551$	1.07689	0.0458770
$552$	0	0
$553$	−27.1789	−1.15577
$554$	18.4793	0.785109
$555$	0	0
$556$	−30.0226	−1.27324
$557$	−1.67596	−0.0710128	−0.0355064	−	0.999369i	$-0.511304\pi$
$557$	−1.67596	−0.0710128	−0.0355064	+	0.999369i	$0.511304\pi$
$558$	0	0
$559$	−23.0096	−0.973203
$560$	2.23322	0.0943706
$561$	0	0
$562$	−3.22870	−0.136195
$563$	−20.6605	−0.870737	−0.435368	−	0.900252i	$-0.643382\pi$
$563$	−20.6605	−0.870737	−0.435368	+	0.900252i	$0.643382\pi$
$564$	0	0
$565$	5.49472	0.231164
$566$	10.6678	0.448401
$567$	0	0
$568$	15.4517	0.648340
$569$	−40.9673	−1.71744	−0.858720	−	0.512445i	$-0.828740\pi$
$569$	−40.9673	−1.71744	−0.858720	+	0.512445i	$0.828740\pi$
$570$	0	0
$571$	24.2070	1.01303	0.506515	−	0.862231i	$-0.330933\pi$
$571$	24.2070	1.01303	0.506515	+	0.862231i	$0.330933\pi$
$572$	18.4863	0.772952
$573$	0	0
$574$	43.8392	1.82981
$575$	−4.11005	−0.171401
$576$	0	0
$577$	40.2864	1.67715	0.838573	−	0.544790i	$-0.183391\pi$
$577$	40.2864	1.67715	0.838573	+	0.544790i	$0.183391\pi$
$578$	12.9351	0.538029
$579$	0	0
$580$	−1.43568	−0.0596134
$581$	−9.84622	−0.408490
$582$	0	0
$583$	12.4200	0.514384
$584$	24.7059	1.02234
$585$	0	0
$586$	−7.91658	−0.327031
$587$	34.7754	1.43534	0.717668	−	0.696385i	$-0.245210\pi$
$587$	34.7754	1.43534	0.717668	+	0.696385i	$0.245210\pi$
$588$	0	0
$589$	5.58946	0.230310
$590$	−1.14321	−0.0470651
$591$	0	0
$592$	0.0420379	0.00172775
$593$	35.0864	1.44082	0.720412	−	0.693546i	$-0.243953\pi$
$593$	35.0864	1.44082	0.720412	+	0.693546i	$0.243953\pi$
$594$	0	0
$595$	5.42015	0.222205
$596$	13.0477	0.534456
$597$	0	0
$598$	−15.3434	−0.627439
$599$	−30.8201	−1.25928	−0.629638	−	0.776889i	$-0.716797\pi$
$599$	−30.8201	−1.25928	−0.629638	+	0.776889i	$0.716797\pi$
$600$	0	0
$601$	−7.10654	−0.289882	−0.144941	−	0.989440i	$-0.546299\pi$
$601$	−7.10654	−0.289882	−0.144941	+	0.989440i	$0.546299\pi$
$602$	−20.6865	−0.843120
$603$	0	0
$604$	−7.83484	−0.318795
$605$	−1.79994	−0.0731781
$606$	0	0
$607$	2.01531	0.0817987	0.0408994	−	0.999163i	$-0.486978\pi$
$607$	2.01531	0.0817987	0.0408994	+	0.999163i	$0.486978\pi$
$608$	−5.80602	−0.235465
$609$	0	0
$610$	4.65418	0.188442
$611$	−56.2286	−2.27477
$612$	0	0
$613$	−11.6096	−0.468909	−0.234455	−	0.972127i	$-0.575330\pi$
$613$	−11.6096	−0.468909	−0.234455	+	0.972127i	$0.575330\pi$
$614$	23.3122	0.940803
$615$	0	0
$616$	41.5528	1.67421
$617$	−12.4307	−0.500442	−0.250221	−	0.968189i	$-0.580503\pi$
$617$	−12.4307	−0.500442	−0.250221	+	0.968189i	$0.580503\pi$
$618$	0	0
$619$	3.85424	0.154915	0.0774575	−	0.996996i	$-0.475320\pi$
$619$	3.85424	0.154915	0.0774575	+	0.996996i	$0.475320\pi$
$620$	−7.45172	−0.299268
$621$	0	0
$622$	20.7472	0.831888
$623$	11.0200	0.441509
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.7187	−0.548311
$627$	0	0
$628$	−13.5844	−0.542075
$629$	0.102029	0.00406814
$630$	0	0
$631$	−16.8794	−0.671958	−0.335979	−	0.941870i	$-0.609067\pi$
$631$	−16.8794	−0.671958	−0.335979	+	0.941870i	$0.609067\pi$
$632$	−14.6979	−0.584652
$633$	0	0
$634$	−25.9273	−1.02970
$635$	−8.61533	−0.341889
$636$	0	0
$637$	−83.8098	−3.32067
$638$	−2.66730	−0.105600
$639$	0	0
$640$	8.46509	0.334612
$641$	−43.4855	−1.71757	−0.858787	−	0.512332i	$-0.828782\pi$
$641$	−43.4855	−1.71757	−0.858787	+	0.512332i	$0.828782\pi$
$642$	0	0
$643$	11.0689	0.436514	0.218257	−	0.975891i	$-0.429963\pi$
$643$	11.0689	0.436514	0.218257	+	0.975891i	$0.429963\pi$
$644$	27.5787	1.08675
$645$	0	0
$646$	−0.879381	−0.0345988
$647$	−5.61766	−0.220853	−0.110427	−	0.993884i	$-0.535222\pi$
$647$	−5.61766	−0.220853	−0.110427	+	0.993884i	$0.535222\pi$
$648$	0	0
$649$	4.24634	0.166683
$650$	3.73315	0.146426
$651$	0	0
$652$	13.8195	0.541213
$653$	−44.5211	−1.74224	−0.871122	−	0.491066i	$-0.836607\pi$
$653$	−44.5211	−1.74224	−0.871122	+	0.491066i	$0.836607\pi$
$654$	0	0
$655$	2.15378	0.0841551
$656$	−4.73267	−0.184780
$657$	0	0
$658$	−50.5517	−1.97071
$659$	6.89154	0.268456	0.134228	−	0.990950i	$-0.457144\pi$
$659$	6.89154	0.268456	0.134228	+	0.990950i	$0.457144\pi$
$660$	0	0
$661$	−44.3989	−1.72692	−0.863458	−	0.504421i	$-0.831706\pi$
$661$	−44.3989	−1.72692	−0.863458	+	0.504421i	$0.831706\pi$
$662$	8.13678	0.316245
$663$	0	0
$664$	−5.32468	−0.206637
$665$	5.03316	0.195178
$666$	0	0
$667$	−4.42607	−0.171378
$668$	−20.8039	−0.804926
$669$	0	0
$670$	−4.31299	−0.166625
$671$	−17.2875	−0.667377
$672$	0	0
$673$	38.6214	1.48875	0.744374	−	0.667763i	$-0.232748\pi$
$673$	38.6214	1.48875	0.744374	+	0.667763i	$0.232748\pi$
$674$	−18.9383	−0.729476
$675$	0	0
$676$	−10.5315	−0.405057
$677$	−43.3917	−1.66768	−0.833840	−	0.552006i	$-0.813862\pi$
$677$	−43.3917	−1.66768	−0.833840	+	0.552006i	$0.813862\pi$
$678$	0	0
$679$	−10.8770	−0.417420
$680$	2.93113	0.112404
$681$	0	0
$682$	−13.8443	−0.530126
$683$	34.7123	1.32823	0.664114	−	0.747631i	$-0.268809\pi$
$683$	34.7123	1.32823	0.664114	+	0.747631i	$0.268809\pi$
$684$	0	0
$685$	−2.18949	−0.0836560
$686$	−46.5779	−1.77835
$687$	0	0
$688$	2.23322	0.0851406
$689$	−18.7195	−0.713158
$690$	0	0
$691$	2.94570	0.112060	0.0560299	−	0.998429i	$-0.482156\pi$
$691$	2.94570	0.112060	0.0560299	+	0.998429i	$0.482156\pi$
$692$	0.762048	0.0289687
$693$	0	0
$694$	17.2589	0.655138
$695$	22.5196	0.854218
$696$	0	0
$697$	−11.4865	−0.435081
$698$	4.12076	0.155973
$699$	0	0
$700$	−6.71008	−0.253617
$701$	−18.5920	−0.702210	−0.351105	−	0.936336i	$-0.614194\pi$
$701$	−18.5920	−0.702210	−0.351105	+	0.936336i	$0.614194\pi$
$702$	0	0
$703$	0.0947438	0.00357333
$704$	11.6891	0.440549
$705$	0	0
$706$	10.2592	0.386109
$707$	−63.3519	−2.38259
$708$	0	0
$709$	39.8443	1.49638	0.748192	−	0.663482i	$-0.230922\pi$
$709$	39.8443	1.49638	0.748192	+	0.663482i	$0.230922\pi$
$710$	−4.63574	−0.173976
$711$	0	0
$712$	5.95946	0.223340
$713$	−22.9729	−0.860344
$714$	0	0
$715$	−13.8664	−0.518574
$716$	33.1780	1.23992
$717$	0	0
$718$	−14.4156	−0.537987
$719$	−21.2321	−0.791824	−0.395912	−	0.918288i	$-0.629572\pi$
$719$	−21.2321	−0.791824	−0.395912	+	0.918288i	$0.629572\pi$
$720$	0	0
$721$	−31.2297	−1.16306
$722$	−0.816594	−0.0303905
$723$	0	0
$724$	−9.27052	−0.344536
$725$	1.07689	0.0399947
$726$	0	0
$727$	−49.1658	−1.82346	−0.911729	−	0.410792i	$-0.865252\pi$
$727$	−49.1658	−1.82346	−0.911729	+	0.410792i	$0.865252\pi$
$728$	−62.6287	−2.32118
$729$	0	0
$730$	−7.41213	−0.274335
$731$	5.42015	0.200472
$732$	0	0
$733$	1.60498	0.0592815	0.0296407	−	0.999561i	$-0.490564\pi$
$733$	1.60498	0.0592815	0.0296407	+	0.999561i	$0.490564\pi$
$734$	−3.62085	−0.133648
$735$	0	0
$736$	23.8630	0.879603
$737$	16.0202	0.590111
$738$	0	0
$739$	8.13264	0.299164	0.149582	−	0.988749i	$-0.452207\pi$
$739$	8.13264	0.299164	0.149582	+	0.988749i	$0.452207\pi$
$740$	−0.126310	−0.00464324
$741$	0	0
$742$	−16.8296	−0.617834
$743$	42.9261	1.57481	0.787403	−	0.616439i	$-0.211425\pi$
$743$	42.9261	1.57481	0.787403	+	0.616439i	$0.211425\pi$
$744$	0	0
$745$	−9.78697	−0.358567
$746$	17.0452	0.624070
$747$	0	0
$748$	−4.35465	−0.159222
$749$	63.1564	2.30768
$750$	0	0
$751$	4.05685	0.148037	0.0740183	−	0.997257i	$-0.476418\pi$
$751$	4.05685	0.148037	0.0740183	+	0.997257i	$0.476418\pi$
$752$	5.45731	0.199008
$753$	0	0
$754$	4.02018	0.146406
$755$	5.87683	0.213880
$756$	0	0
$757$	22.6151	0.821960	0.410980	−	0.911644i	$-0.365187\pi$
$757$	22.6151	0.821960	0.410980	+	0.911644i	$0.365187\pi$
$758$	5.55279	0.201687
$759$	0	0
$760$	2.72185	0.0987319
$761$	41.3609	1.49933	0.749666	−	0.661817i	$-0.230214\pi$
$761$	41.3609	1.49933	0.749666	+	0.661817i	$0.230214\pi$
$762$	0	0
$763$	−79.5720	−2.88070
$764$	−5.21610	−0.188712
$765$	0	0
$766$	−8.90725	−0.321832
$767$	−6.40011	−0.231095
$768$	0	0
$769$	17.9196	0.646197	0.323099	−	0.946365i	$-0.395275\pi$
$769$	17.9196	0.646197	0.323099	+	0.946365i	$0.395275\pi$
$770$	−12.4664	−0.449259
$771$	0	0
$772$	12.2750	0.441787
$773$	21.9043	0.787843	0.393921	−	0.919144i	$-0.371118\pi$
$773$	21.9043	0.787843	0.393921	+	0.919144i	$0.371118\pi$
$774$	0	0
$775$	5.58946	0.200779
$776$	−5.88209	−0.211155
$777$	0	0
$778$	17.0934	0.612829
$779$	−10.6663	−0.382162
$780$	0	0
$781$	17.2190	0.616144
$782$	3.61430	0.129247
$783$	0	0
$784$	8.13423	0.290508
$785$	10.1895	0.363678
$786$	0	0
$787$	−31.6354	−1.12768	−0.563840	−	0.825884i	$-0.690676\pi$
$787$	−31.6354	−1.12768	−0.563840	+	0.825884i	$0.690676\pi$
$788$	−5.12768	−0.182666
$789$	0	0
$790$	4.40959	0.156886
$791$	27.6558	0.983326
$792$	0	0
$793$	26.0559	0.925272
$794$	−10.7154	−0.380275
$795$	0	0
$796$	24.3790	0.864090
$797$	15.4486	0.547217	0.273608	−	0.961841i	$-0.411783\pi$
$797$	15.4486	0.547217	0.273608	+	0.961841i	$0.411783\pi$
$798$	0	0
$799$	13.2452	0.468583
$800$	−5.80602	−0.205274
$801$	0	0
$802$	−9.43394	−0.333124
$803$	27.5317	0.971571
$804$	0	0
$805$	−20.6865	−0.729104
$806$	20.8663	0.734983
$807$	0	0
$808$	−34.2597	−1.20525
$809$	32.9326	1.15785	0.578924	−	0.815382i	$-0.303473\pi$
$809$	32.9326	1.15785	0.578924	+	0.815382i	$0.303473\pi$
$810$	0	0
$811$	−25.5895	−0.898567	−0.449284	−	0.893389i	$-0.648321\pi$
$811$	−25.5895	−0.898567	−0.449284	+	0.893389i	$0.648321\pi$
$812$	−7.22601	−0.253583
$813$	0	0
$814$	−0.234667	−0.00822508
$815$	−10.3659	−0.363100
$816$	0	0
$817$	5.03316	0.176088
$818$	15.4690	0.540860
$819$	0	0
$820$	14.2201	0.496587
$821$	−17.8674	−0.623575	−0.311788	−	0.950152i	$-0.600928\pi$
$821$	−17.8674	−0.623575	−0.311788	+	0.950152i	$0.600928\pi$
$822$	0	0
$823$	−18.4533	−0.643242	−0.321621	−	0.946868i	$-0.604228\pi$
$823$	−18.4533	−0.643242	−0.321621	+	0.946868i	$0.604228\pi$
$824$	−16.8885	−0.588339
$825$	0	0
$826$	−5.75395	−0.200206
$827$	−47.5460	−1.65334	−0.826668	−	0.562690i	$-0.809767\pi$
$827$	−47.5460	−1.65334	−0.826668	+	0.562690i	$0.809767\pi$
$828$	0	0
$829$	19.0308	0.660965	0.330483	−	0.943812i	$-0.392788\pi$
$829$	19.0308	0.660965	0.330483	+	0.943812i	$0.392788\pi$
$830$	1.59748	0.0554493
$831$	0	0
$832$	−17.6179	−0.610790
$833$	19.7423	0.684029
$834$	0	0
$835$	15.6048	0.540025
$836$	−4.04373	−0.139855
$837$	0	0
$838$	22.2656	0.769151
$839$	−3.04022	−0.104960	−0.0524801	−	0.998622i	$-0.516713\pi$
$839$	−3.04022	−0.104960	−0.0524801	+	0.998622i	$0.516713\pi$
$840$	0	0
$841$	−27.8403	−0.960011
$842$	−5.44370	−0.187602
$843$	0	0
$844$	25.7262	0.885534
$845$	7.89956	0.271753
$846$	0	0
$847$	−9.05940	−0.311285
$848$	1.81684	0.0623906
$849$	0	0
$850$	−0.879381	−0.0301625
$851$	−0.389401	−0.0133485
$852$	0	0
$853$	−18.2201	−0.623844	−0.311922	−	0.950108i	$-0.600973\pi$
$853$	−18.2201	−0.623844	−0.311922	+	0.950108i	$0.600973\pi$
$854$	23.4253	0.801596
$855$	0	0
$856$	34.1539	1.16736
$857$	−19.3611	−0.661363	−0.330682	−	0.943742i	$-0.607279\pi$
$857$	−19.3611	−0.661363	−0.330682	+	0.943742i	$0.607279\pi$
$858$	0	0
$859$	3.44129	0.117415	0.0587077	−	0.998275i	$-0.481302\pi$
$859$	3.44129	0.117415	0.0587077	+	0.998275i	$0.481302\pi$
$860$	−6.71008	−0.228812
$861$	0	0
$862$	−26.5024	−0.902674
$863$	26.5471	0.903674	0.451837	−	0.892101i	$-0.350769\pi$
$863$	26.5471	0.903674	0.451837	+	0.892101i	$0.350769\pi$
$864$	0	0
$865$	−0.571604	−0.0194351
$866$	−27.6146	−0.938383
$867$	0	0
$868$	−37.5057	−1.27303
$869$	−16.3790	−0.555619
$870$	0	0
$871$	−24.1458	−0.818148
$872$	−43.0312	−1.45722
$873$	0	0
$874$	3.35624	0.113527
$875$	5.03316	0.170152
$876$	0	0
$877$	20.5495	0.693908	0.346954	−	0.937882i	$-0.387216\pi$
$877$	20.5495	0.693908	0.346954	+	0.937882i	$0.387216\pi$
$878$	22.7383	0.767380
$879$	0	0
$880$	1.34582	0.0453674
$881$	31.1911	1.05085	0.525427	−	0.850839i	$-0.323906\pi$
$881$	31.1911	1.05085	0.525427	+	0.850839i	$0.323906\pi$
$882$	0	0
$883$	14.1861	0.477401	0.238701	−	0.971093i	$-0.423279\pi$
$883$	14.1861	0.477401	0.238701	+	0.971093i	$0.423279\pi$
$884$	6.56336	0.220750
$885$	0	0
$886$	18.5983	0.624822
$887$	−46.6046	−1.56483	−0.782415	−	0.622757i	$-0.786012\pi$
$887$	−46.6046	−1.56483	−0.782415	+	0.622757i	$0.786012\pi$
$888$	0	0
$889$	−43.3623	−1.45433
$890$	−1.78792	−0.0599313
$891$	0	0
$892$	−0.820347	−0.0274672
$893$	12.2995	0.411588
$894$	0	0
$895$	−24.8864	−0.831862
$896$	42.6061	1.42337
$897$	0	0
$898$	20.2930	0.677185
$899$	6.01923	0.200752
$900$	0	0
$901$	4.40959	0.146905
$902$	26.4191	0.879658
$903$	0	0
$904$	14.9558	0.497422
$905$	6.95372	0.231150
$906$	0	0
$907$	−0.754028	−0.0250371	−0.0125185	−	0.999922i	$-0.503985\pi$
$907$	−0.754028	−0.0250371	−0.0125185	+	0.999922i	$0.503985\pi$
$908$	23.2921	0.772976
$909$	0	0
$910$	18.7895	0.622866
$911$	53.1413	1.76065	0.880325	−	0.474371i	$-0.157325\pi$
$911$	53.1413	1.76065	0.880325	+	0.474371i	$0.157325\pi$
$912$	0	0
$913$	−5.93368	−0.196376
$914$	5.98784	0.198060
$915$	0	0
$916$	−12.9311	−0.427257
$917$	10.8403	0.357979
$918$	0	0
$919$	37.4202	1.23438	0.617189	−	0.786815i	$-0.288272\pi$
$919$	37.4202	1.23438	0.617189	+	0.786815i	$0.288272\pi$
$920$	−11.1869	−0.368822
$921$	0	0
$922$	8.41709	0.277202
$923$	−25.9526	−0.854241
$924$	0	0
$925$	0.0947438	0.00311516
$926$	−6.37147	−0.209379
$927$	0	0
$928$	−6.25244	−0.205247
$929$	19.1981	0.629871	0.314935	−	0.949113i	$-0.398017\pi$
$929$	19.1981	0.629871	0.314935	+	0.949113i	$0.398017\pi$
$930$	0	0
$931$	18.3327	0.600830
$932$	−10.8704	−0.356072
$933$	0	0
$934$	−6.02355	−0.197096
$935$	3.26638	0.106822
$936$	0	0
$937$	−20.0095	−0.653681	−0.326840	−	0.945080i	$-0.605984\pi$
$937$	−20.0095	−0.653681	−0.326840	+	0.945080i	$0.605984\pi$
$938$	−21.7080	−0.708790
$939$	0	0
$940$	−16.3974	−0.534825
$941$	−15.3327	−0.499832	−0.249916	−	0.968268i	$-0.580403\pi$
$941$	−15.3327	−0.499832	−0.249916	+	0.968268i	$0.580403\pi$
$942$	0	0
$943$	43.8392	1.42760
$944$	0.621168	0.0202173
$945$	0	0
$946$	−12.4664	−0.405319
$947$	29.8217	0.969076	0.484538	−	0.874770i	$-0.338988\pi$
$947$	29.8217	0.969076	0.484538	+	0.874770i	$0.338988\pi$
$948$	0	0
$949$	−41.4959	−1.34702
$950$	−0.816594	−0.0264938
$951$	0	0
$952$	14.7529	0.478143
$953$	−31.4938	−1.02018	−0.510091	−	0.860120i	$-0.670388\pi$
$953$	−31.4938	−1.02018	−0.510091	+	0.860120i	$0.670388\pi$
$954$	0	0
$955$	3.91254	0.126607
$956$	32.7948	1.06066
$957$	0	0
$958$	−13.1845	−0.425973
$959$	−11.0200	−0.355856
$960$	0	0
$961$	0.242050	0.00780806
$962$	0.353692	0.0114035
$963$	0	0
$964$	−34.3610	−1.10669
$965$	−9.20734	−0.296395
$966$	0	0
$967$	1.16143	0.0373490	0.0186745	−	0.999826i	$-0.494055\pi$
$967$	1.16143	0.0373490	0.0186745	+	0.999826i	$0.494055\pi$
$968$	−4.89917	−0.157465
$969$	0	0
$970$	1.76471	0.0566615
$971$	−18.4557	−0.592272	−0.296136	−	0.955146i	$-0.595698\pi$
$971$	−18.4557	−0.592272	−0.296136	+	0.955146i	$0.595698\pi$
$972$	0	0
$973$	113.345	3.63367
$974$	6.86401	0.219937
$975$	0	0
$976$	−2.52888	−0.0809474
$977$	−5.62735	−0.180035	−0.0900175	−	0.995940i	$-0.528692\pi$
$977$	−5.62735	−0.180035	−0.0900175	+	0.995940i	$0.528692\pi$
$978$	0	0
$979$	6.64107	0.212249
$980$	−24.4407	−0.780729
$981$	0	0
$982$	−29.4672	−0.940338
$983$	25.1228	0.801293	0.400647	−	0.916233i	$-0.368786\pi$
$983$	25.1228	0.801293	0.400647	+	0.916233i	$0.368786\pi$
$984$	0	0
$985$	3.84622	0.122551
$986$	−0.946996	−0.0301585
$987$	0	0
$988$	6.09474	0.193900
$989$	−20.6865	−0.657793
$990$	0	0
$991$	−41.8749	−1.33020	−0.665100	−	0.746754i	$-0.731611\pi$
$991$	−41.8749	−1.33020	−0.665100	+	0.746754i	$0.731611\pi$
$992$	−32.4525	−1.03037
$993$	0	0
$994$	−23.3324	−0.740059
$995$	−18.2864	−0.579718
$996$	0	0
$997$	−37.0346	−1.17290	−0.586449	−	0.809986i	$-0.699475\pi$
$997$	−37.0346	−1.17290	−0.586449	+	0.809986i	$0.699475\pi$
$998$	2.69505	0.0853102
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.m.1.2		4
3.2	odd	2		95.2.a.b.1.3	✓	4
5.4	even	2		4275.2.a.bo.1.3		4
12.11	even	2		1520.2.a.t.1.2		4
15.2	even	4		475.2.b.e.324.5		8
15.8	even	4		475.2.b.e.324.4		8
15.14	odd	2		475.2.a.i.1.2		4
21.20	even	2		4655.2.a.y.1.3		4
24.5	odd	2		6080.2.a.cc.1.2		4
24.11	even	2		6080.2.a.ch.1.3		4
57.56	even	2		1805.2.a.p.1.2		4
60.59	even	2		7600.2.a.cf.1.3		4
285.284	even	2		9025.2.a.bf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.3	✓	4	3.2	odd	2
475.2.a.i.1.2		4	15.14	odd	2
475.2.b.e.324.4		8	15.8	even	4
475.2.b.e.324.5		8	15.2	even	4
855.2.a.m.1.2		4	1.1	even	1	trivial
1520.2.a.t.1.2		4	12.11	even	2
1805.2.a.p.1.2		4	57.56	even	2
4275.2.a.bo.1.3		4	5.4	even	2
4655.2.a.y.1.3		4	21.20	even	2
6080.2.a.cc.1.2		4	24.5	odd	2
6080.2.a.ch.1.3		4	24.11	even	2
7600.2.a.cf.1.3		4	60.59	even	2
9025.2.a.bf.1.3		4	285.284	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)

\(f(q)\)	\(=\)	\(q-0.816594 q^{2} -1.33317 q^{4} +1.00000 q^{5} +5.03316 q^{7} +2.72185 q^{8} +O(q^{10})\)	\(q-0.816594 q^{2} -1.33317 q^{4} +1.00000 q^{5} +5.03316 q^{7} +2.72185 q^{8} -0.816594 q^{10} +3.03316 q^{11} -4.57160 q^{13} -4.11005 q^{14} +0.443701 q^{16} +1.07689 q^{17} +1.00000 q^{19} -1.33317 q^{20} -2.47686 q^{22} -4.11005 q^{23} +1.00000 q^{25} +3.73315 q^{26} -6.71008 q^{28} +1.07689 q^{29} +5.58946 q^{31} -5.80602 q^{32} -0.879381 q^{34} +5.03316 q^{35} +0.0947438 q^{37} -0.816594 q^{38} +2.72185 q^{40} -10.6663 q^{41} +5.03316 q^{43} -4.04373 q^{44} +3.35624 q^{46} +12.2995 q^{47} +18.3327 q^{49} -0.816594 q^{50} +6.09474 q^{52} +4.09474 q^{53} +3.03316 q^{55} +13.6995 q^{56} -0.879381 q^{58} +1.39997 q^{59} -5.69951 q^{61} -4.56432 q^{62} +3.85376 q^{64} -4.57160 q^{65} +5.28168 q^{67} -1.43568 q^{68} -4.11005 q^{70} +5.67692 q^{71} +9.07689 q^{73} -0.0773672 q^{74} -1.33317 q^{76} +15.2664 q^{77} -5.39997 q^{79} +0.443701 q^{80} +8.71008 q^{82} -1.95627 q^{83} +1.07689 q^{85} -4.11005 q^{86} +8.25581 q^{88} +2.18949 q^{89} -23.0096 q^{91} +5.47941 q^{92} -10.0437 q^{94} +1.00000 q^{95} -2.16106 q^{97} -14.9704 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8	\( 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{13} + 8 q^{14} + 4 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 4 q^{26} - 8 q^{28} - 4 q^{29} + 4 q^{31} + 6 q^{32} - 4 q^{34} + 4 q^{35} - 6 q^{37} + 2 q^{38} + 12 q^{40} - 16 q^{41} + 4 q^{43} - 24 q^{44} + 12 q^{47} + 20 q^{49} + 2 q^{50} + 18 q^{52} + 10 q^{53} - 4 q^{55} + 12 q^{56} - 4 q^{58} + 20 q^{61} - 20 q^{62} - 4 q^{64} + 2 q^{65} - 18 q^{67} - 4 q^{68} + 8 q^{70} + 20 q^{71} + 28 q^{73} - 32 q^{74} + 8 q^{76} + 40 q^{77} - 16 q^{79} + 4 q^{80} + 16 q^{82} - 4 q^{85} + 8 q^{86} - 12 q^{88} - 4 q^{89} - 36 q^{91} + 28 q^{92} - 48 q^{94} + 4 q^{95} + 30 q^{97} - 38 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^13 + 8 * q^14 + 4 * q^16 - 4 * q^17 + 4 * q^19 + 8 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 - 4 * q^26 - 8 * q^28 - 4 * q^29 + 4 * q^31 + 6 * q^32 - 4 * q^34 + 4 * q^35 - 6 * q^37 + 2 * q^38 + 12 * q^40 - 16 * q^41 + 4 * q^43 - 24 * q^44 + 12 * q^47 + 20 * q^49 + 2 * q^50 + 18 * q^52 + 10 * q^53 - 4 * q^55 + 12 * q^56 - 4 * q^58 + 20 * q^61 - 20 * q^62 - 4 * q^64 + 2 * q^65 - 18 * q^67 - 4 * q^68 + 8 * q^70 + 20 * q^71 + 28 * q^73 - 32 * q^74 + 8 * q^76 + 40 * q^77 - 16 * q^79 + 4 * q^80 + 16 * q^82 - 4 * q^85 + 8 * q^86 - 12 * q^88 - 4 * q^89 - 36 * q^91 + 28 * q^92 - 48 * q^94 + 4 * q^95 + 30 * q^97 - 38 * q^98

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