LMFDB - Embedded newform 2254.2.a.x.1.2

Error: no document with id 226792884 found in table mf_hecke_traces.

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2254,2,Mod(1,2254)] 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("2254.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2254, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2254.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4,-3,4,-7,-3,0,4,-1,-7,2] 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:	$17.9982806156$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1957.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} - x + 1 $ x^4 - 4*x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.693822$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.74747 q^{3} +1.00000 q^{4} -4.26608 q^{5} -1.74747 q^{6} +1.00000 q^{8} +0.0536460 q^{9} -4.26608 q^{10} +3.18876 q^{11} -1.74747 q^{12} +4.60948 q^{13} +7.45484 q^{15} +1.00000 q^{16} -3.15879 q^{17} +0.0536460 q^{18} +4.53631 q^{19} -4.26608 q^{20} +3.18876 q^{22} -1.00000 q^{23} -1.74747 q^{24} +13.1994 q^{25} +4.60948 q^{26} +5.14866 q^{27} -3.70737 q^{29} +7.45484 q^{30} -3.23611 q^{31} +1.00000 q^{32} -5.57226 q^{33} -3.15879 q^{34} +0.0536460 q^{36} -7.24655 q^{37} +4.53631 q^{38} -8.05492 q^{39} -4.26608 q^{40} -5.11454 q^{41} -2.29605 q^{43} +3.18876 q^{44} -0.228858 q^{45} -1.00000 q^{46} -5.69797 q^{47} -1.74747 q^{48} +13.1994 q^{50} +5.51988 q^{51} +4.60948 q^{52} -12.1994 q^{53} +5.14866 q^{54} -13.6035 q^{55} -7.92705 q^{57} -3.70737 q^{58} -0.918533 q^{59} +7.45484 q^{60} -13.4890 q^{61} -3.23611 q^{62} +1.00000 q^{64} -19.6644 q^{65} -5.57226 q^{66} +4.75576 q^{67} -3.15879 q^{68} +1.74747 q^{69} +3.61960 q^{71} +0.0536460 q^{72} +13.3705 q^{73} -7.24655 q^{74} -23.0656 q^{75} +4.53631 q^{76} -8.05492 q^{78} -7.42072 q^{79} -4.26608 q^{80} -9.15806 q^{81} -5.11454 q^{82} +9.36524 q^{83} +13.4756 q^{85} -2.29605 q^{86} +6.47851 q^{87} +3.18876 q^{88} +5.22310 q^{89} -0.228858 q^{90} -1.00000 q^{92} +5.65499 q^{93} -5.69797 q^{94} -19.3522 q^{95} -1.74747 q^{96} +5.74459 q^{97} +0.171064 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 3 q^{3} + 4 q^{4} - 7 q^{5} - 3 q^{6} + 4 q^{8} - q^{9} - 7 q^{10} + 2 q^{11} - 3 q^{12} - q^{13} + 9 q^{15} + 4 q^{16} - 5 q^{17} - q^{18} - 11 q^{19} - 7 q^{20} + 2 q^{22} - 4 q^{23}+ \cdots + 13 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 - 3 * q^3 + 4 * q^4 - 7 * q^5 - 3 * q^6 + 4 * q^8 - q^9 - 7 * q^10 + 2 * q^11 - 3 * q^12 - q^13 + 9 * q^15 + 4 * q^16 - 5 * q^17 - q^18 - 11 * q^19 - 7 * q^20 + 2 * q^22 - 4 * q^23 - 3 * q^24 + 3 * q^25 - q^26 - 3 * q^27 + 2 * q^29 + 9 * q^30 - 6 * q^31 + 4 * q^32 - 15 * q^33 - 5 * q^34 - q^36 - 8 * q^37 - 11 * q^38 + 3 * q^39 - 7 * q^40 - 9 * q^41 + 4 * q^43 + 2 * q^44 - 3 * q^45 - 4 * q^46 - 11 * q^47 - 3 * q^48 + 3 * q^50 - 18 * q^51 - q^52 + q^53 - 3 * q^54 - 10 * q^55 + 3 * q^57 + 2 * q^58 - 12 * q^59 + 9 * q^60 - 21 * q^61 - 6 * q^62 + 4 * q^64 - 24 * q^65 - 15 * q^66 - 3 * q^67 - 5 * q^68 + 3 * q^69 + 11 * q^71 - q^72 + 16 * q^73 - 8 * q^74 - 18 * q^75 - 11 * q^76 + 3 * q^78 - 21 * q^79 - 7 * q^80 - 8 * q^81 - 9 * q^82 - 4 * q^83 + 10 * q^85 + 4 * q^86 + 7 * q^87 + 2 * q^88 - 27 * q^89 - 3 * q^90 - 4 * q^92 - 27 * q^93 - 11 * q^94 + 5 * q^95 - 3 * q^96 - 6 * q^97 + 13 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.74747	−1.00890	−0.504451	−	0.863441i	$-0.668305\pi$
$3$	−1.74747	−1.00890	−0.504451	+	0.863441i	$0.668305\pi$
$4$	1.00000	0.500000
$5$	−4.26608	−1.90785	−0.953924	−	0.300048i	$-0.902997\pi$
$5$	−4.26608	−1.90785	−0.953924	+	0.300048i	$0.902997\pi$
$6$	−1.74747	−0.713401
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0.0536460	0.0178820
$10$	−4.26608	−1.34905
$11$	3.18876	0.961447	0.480724	−	0.876872i	$-0.340374\pi$
$11$	3.18876	0.961447	0.480724	+	0.876872i	$0.340374\pi$
$12$	−1.74747	−0.504451
$13$	4.60948	1.27844	0.639219	−	0.769024i	$-0.279258\pi$
$13$	4.60948	1.27844	0.639219	+	0.769024i	$0.279258\pi$
$14$	0	0
$15$	7.45484	1.92483
$16$	1.00000	0.250000
$17$	−3.15879	−0.766118	−0.383059	−	0.923724i	$-0.625129\pi$
$17$	−3.15879	−0.766118	−0.383059	+	0.923724i	$0.625129\pi$
$18$	0.0536460	0.0126445
$19$	4.53631	1.04070	0.520350	−	0.853953i	$-0.325802\pi$
$19$	4.53631	1.04070	0.520350	+	0.853953i	$0.325802\pi$
$20$	−4.26608	−0.953924
$21$	0	0
$22$	3.18876	0.679846
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.74747	−0.356701
$25$	13.1994	2.63989
$26$	4.60948	0.903993
$27$	5.14866	0.990860
$28$	0	0
$29$	−3.70737	−0.688441	−0.344221	−	0.938889i	$-0.611857\pi$
$29$	−3.70737	−0.688441	−0.344221	+	0.938889i	$0.611857\pi$
$30$	7.45484	1.36106
$31$	−3.23611	−0.581222	−0.290611	−	0.956841i	$-0.593859\pi$
$31$	−3.23611	−0.581222	−0.290611	+	0.956841i	$0.593859\pi$
$32$	1.00000	0.176777
$33$	−5.57226	−0.970005
$34$	−3.15879	−0.541727
$35$	0	0
$36$	0.0536460	0.00894100
$37$	−7.24655	−1.19133	−0.595663	−	0.803234i	$-0.703111\pi$
$37$	−7.24655	−1.19133	−0.595663	+	0.803234i	$0.703111\pi$
$38$	4.53631	0.735886
$39$	−8.05492	−1.28982
$40$	−4.26608	−0.674526
$41$	−5.11454	−0.798757	−0.399378	−	0.916786i	$-0.630774\pi$
$41$	−5.11454	−0.798757	−0.399378	+	0.916786i	$0.630774\pi$
$42$	0	0
$43$	−2.29605	−0.350145	−0.175072	−	0.984556i	$-0.556016\pi$
$43$	−2.29605	−0.350145	−0.175072	+	0.984556i	$0.556016\pi$
$44$	3.18876	0.480724
$45$	−0.228858	−0.0341161
$46$	−1.00000	−0.147442
$47$	−5.69797	−0.831134	−0.415567	−	0.909562i	$-0.636417\pi$
$47$	−5.69797	−0.831134	−0.415567	+	0.909562i	$0.636417\pi$
$48$	−1.74747	−0.252225
$49$	0	0
$50$	13.1994	1.86668
$51$	5.51988	0.772938
$52$	4.60948	0.639219
$53$	−12.1994	−1.67572	−0.837860	−	0.545885i	$-0.816194\pi$
$53$	−12.1994	−1.67572	−0.837860	+	0.545885i	$0.816194\pi$
$54$	5.14866	0.700644
$55$	−13.6035	−1.83430
$56$	0	0
$57$	−7.92705	−1.04996
$58$	−3.70737	−0.486801
$59$	−0.918533	−0.119583	−0.0597914	−	0.998211i	$-0.519044\pi$
$59$	−0.918533	−0.119583	−0.0597914	+	0.998211i	$0.519044\pi$
$60$	7.45484	0.962415
$61$	−13.4890	−1.72708	−0.863542	−	0.504277i	$-0.831759\pi$
$61$	−13.4890	−1.72708	−0.863542	+	0.504277i	$0.831759\pi$
$62$	−3.23611	−0.410986
$63$	0	0
$64$	1.00000	0.125000
$65$	−19.6644	−2.43907
$66$	−5.57226	−0.685897
$67$	4.75576	0.581009	0.290505	−	0.956874i	$-0.406177\pi$
$67$	4.75576	0.581009	0.290505	+	0.956874i	$0.406177\pi$
$68$	−3.15879	−0.383059
$69$	1.74747	0.210370
$70$	0	0
$71$	3.61960	0.429568	0.214784	−	0.976662i	$-0.431095\pi$
$71$	3.61960	0.429568	0.214784	+	0.976662i	$0.431095\pi$
$72$	0.0536460	0.00632224
$73$	13.3705	1.56490	0.782449	−	0.622715i	$-0.213970\pi$
$73$	13.3705	1.56490	0.782449	+	0.622715i	$0.213970\pi$
$74$	−7.24655	−0.842395
$75$	−23.0656	−2.66338
$76$	4.53631	0.520350
$77$	0	0
$78$	−8.05492	−0.912040
$79$	−7.42072	−0.834896	−0.417448	−	0.908701i	$-0.637075\pi$
$79$	−7.42072	−0.834896	−0.417448	+	0.908701i	$0.637075\pi$
$80$	−4.26608	−0.476962
$81$	−9.15806	−1.01756
$82$	−5.11454	−0.564806
$83$	9.36524	1.02797	0.513984	−	0.857800i	$-0.328169\pi$
$83$	9.36524	1.02797	0.513984	+	0.857800i	$0.328169\pi$
$84$	0	0
$85$	13.4756	1.46164
$86$	−2.29605	−0.247590
$87$	6.47851	0.694569
$88$	3.18876	0.339923
$89$	5.22310	0.553648	0.276824	−	0.960921i	$-0.410718\pi$
$89$	5.22310	0.553648	0.276824	+	0.960921i	$0.410718\pi$
$90$	−0.228858	−0.0241238
$91$	0	0
$92$	−1.00000	−0.104257
$93$	5.65499	0.586396
$94$	−5.69797	−0.587701
$95$	−19.3522	−1.98550
$96$	−1.74747	−0.178350
$97$	5.74459	0.583275	0.291637	−	0.956529i	$-0.405800\pi$
$97$	5.74459	0.583275	0.291637	+	0.956529i	$0.405800\pi$
$98$	0	0
$99$	0.171064	0.0171926
$100$	13.1994	1.31994
$101$	−12.9191	−1.28550	−0.642748	−	0.766078i	$-0.722206\pi$
$101$	−12.9191	−1.28550	−0.642748	+	0.766078i	$0.722206\pi$
$102$	5.51988	0.546550
$103$	−5.46082	−0.538070	−0.269035	−	0.963130i	$-0.586705\pi$
$103$	−5.46082	−0.538070	−0.269035	+	0.963130i	$0.586705\pi$
$104$	4.60948	0.451996
$105$	0	0
$106$	−12.1994	−1.18491
$107$	−10.2312	−0.989091	−0.494545	−	0.869152i	$-0.664665\pi$
$107$	−10.2312	−0.989091	−0.494545	+	0.869152i	$0.664665\pi$
$108$	5.14866	0.495430
$109$	8.83131	0.845886	0.422943	−	0.906156i	$-0.360997\pi$
$109$	8.83131	0.845886	0.422943	+	0.906156i	$0.360997\pi$
$110$	−13.6035	−1.29704
$111$	12.6631	1.20193
$112$	0	0
$113$	−18.6201	−1.75164	−0.875818	−	0.482641i	$-0.839677\pi$
$113$	−18.6201	−1.75164	−0.875818	+	0.482641i	$0.839677\pi$
$114$	−7.92705	−0.742436
$115$	4.26608	0.397814
$116$	−3.70737	−0.344221
$117$	0.247280	0.0228610
$118$	−0.918533	−0.0845578
$119$	0	0
$120$	7.45484	0.680531
$121$	−0.831814	−0.0756194
$122$	−13.4890	−1.22123
$123$	8.93750	0.805867
$124$	−3.23611	−0.290611
$125$	−34.9794	−3.12865
$126$	0	0
$127$	1.03070	0.0914598	0.0457299	−	0.998954i	$-0.485439\pi$
$127$	1.03070	0.0914598	0.0457299	+	0.998954i	$0.485439\pi$
$128$	1.00000	0.0883883
$129$	4.01228	0.353261
$130$	−19.6644	−1.72468
$131$	7.57878	0.662161	0.331080	−	0.943603i	$-0.392587\pi$
$131$	7.57878	0.662161	0.331080	+	0.943603i	$0.392587\pi$
$132$	−5.57226	−0.485003
$133$	0	0
$134$	4.75576	0.410835
$135$	−21.9646	−1.89041
$136$	−3.15879	−0.270864
$137$	12.4583	1.06438	0.532190	−	0.846625i	$-0.321369\pi$
$137$	12.4583	1.06438	0.532190	+	0.846625i	$0.321369\pi$
$138$	1.74747	0.148754
$139$	−16.3739	−1.38882	−0.694409	−	0.719581i	$-0.744334\pi$
$139$	−16.3739	−1.38882	−0.694409	+	0.719581i	$0.744334\pi$
$140$	0	0
$141$	9.95702	0.838533
$142$	3.61960	0.303750
$143$	14.6985	1.22915
$144$	0.0536460	0.00447050
$145$	15.8159	1.31344
$146$	13.3705	1.10655
$147$	0	0
$148$	−7.24655	−0.595663
$149$	−9.70522	−0.795082	−0.397541	−	0.917584i	$-0.630136\pi$
$149$	−9.70522	−0.795082	−0.397541	+	0.917584i	$0.630136\pi$
$150$	−23.0656	−1.88330
$151$	−12.0784	−0.982924	−0.491462	−	0.870899i	$-0.663537\pi$
$151$	−12.0784	−0.982924	−0.491462	+	0.870899i	$0.663537\pi$
$152$	4.53631	0.367943
$153$	−0.169456	−0.0136997
$154$	0	0
$155$	13.8055	1.10888
$156$	−8.05492	−0.644909
$157$	1.17809	0.0940218	0.0470109	−	0.998894i	$-0.485030\pi$
$157$	1.17809	0.0940218	0.0470109	+	0.998894i	$0.485030\pi$
$158$	−7.42072	−0.590361
$159$	21.3181	1.69064
$160$	−4.26608	−0.337263
$161$	0	0
$162$	−9.15806	−0.719525
$163$	14.4645	1.13294	0.566472	−	0.824081i	$-0.308308\pi$
$163$	14.4645	1.13294	0.566472	+	0.824081i	$0.308308\pi$
$164$	−5.11454	−0.399378
$165$	23.7717	1.85062
$166$	9.36524	0.726884
$167$	6.95702	0.538351	0.269175	−	0.963091i	$-0.413249\pi$
$167$	6.95702	0.538351	0.269175	+	0.963091i	$0.413249\pi$
$168$	0	0
$169$	8.24728	0.634406
$170$	13.4756	1.03353
$171$	0.243355	0.0186098
$172$	−2.29605	−0.175072
$173$	3.35672	0.255207	0.127603	−	0.991825i	$-0.459272\pi$
$173$	3.35672	0.255207	0.127603	+	0.991825i	$0.459272\pi$
$174$	6.47851	0.491135
$175$	0	0
$176$	3.18876	0.240362
$177$	1.60511	0.120647
$178$	5.22310	0.391488
$179$	−19.0417	−1.42324	−0.711621	−	0.702563i	$-0.752039\pi$
$179$	−19.0417	−1.42324	−0.711621	+	0.702563i	$0.752039\pi$
$180$	−0.228858	−0.0170581
$181$	−9.25380	−0.687830	−0.343915	−	0.939001i	$-0.611753\pi$
$181$	−9.25380	−0.687830	−0.343915	+	0.939001i	$0.611753\pi$
$182$	0	0
$183$	23.5715	1.74246
$184$	−1.00000	−0.0737210
$185$	30.9144	2.27287
$186$	5.65499	0.414644
$187$	−10.0726	−0.736582
$188$	−5.69797	−0.415567
$189$	0	0
$190$	−19.3522	−1.40396
$191$	15.3705	1.11217	0.556085	−	0.831125i	$-0.312303\pi$
$191$	15.3705	1.11217	0.556085	+	0.831125i	$0.312303\pi$
$192$	−1.74747	−0.126113
$193$	0.565957	0.0407385	0.0203693	−	0.999793i	$-0.493516\pi$
$193$	0.565957	0.0407385	0.0203693	+	0.999793i	$0.493516\pi$
$194$	5.74459	0.412438
$195$	34.3629	2.46078
$196$	0	0
$197$	−8.71335	−0.620800	−0.310400	−	0.950606i	$-0.600463\pi$
$197$	−8.71335	−0.620800	−0.310400	+	0.950606i	$0.600463\pi$
$198$	0.171064	0.0121570
$199$	−13.0995	−0.928602	−0.464301	−	0.885678i	$-0.653694\pi$
$199$	−13.0995	−0.928602	−0.464301	+	0.885678i	$0.653694\pi$
$200$	13.1994	0.933341
$201$	−8.31055	−0.586181
$202$	−12.9191	−0.908983
$203$	0	0
$204$	5.51988	0.386469
$205$	21.8190	1.52391
$206$	−5.46082	−0.380473
$207$	−0.0536460	−0.00372865
$208$	4.60948	0.319610
$209$	14.4652	1.00058
$210$	0	0
$211$	6.67125	0.459268	0.229634	−	0.973277i	$-0.426247\pi$
$211$	6.67125	0.459268	0.229634	+	0.973277i	$0.426247\pi$
$212$	−12.1994	−0.837860
$213$	−6.32514	−0.433392
$214$	−10.2312	−0.699393
$215$	9.79514	0.668023
$216$	5.14866	0.350322
$217$	0	0
$218$	8.83131	0.598132
$219$	−23.3645	−1.57883
$220$	−13.6035	−0.917148
$221$	−14.5604	−0.979436
$222$	12.6631	0.849893
$223$	−16.9305	−1.13375	−0.566874	−	0.823804i	$-0.691847\pi$
$223$	−16.9305	−1.13375	−0.566874	+	0.823804i	$0.691847\pi$
$224$	0	0
$225$	0.708097	0.0472064
$226$	−18.6201	−1.23859
$227$	−18.2753	−1.21298	−0.606488	−	0.795093i	$-0.707422\pi$
$227$	−18.2753	−1.21298	−0.606488	+	0.795093i	$0.707422\pi$
$228$	−7.92705	−0.524982
$229$	−3.19076	−0.210851	−0.105426	−	0.994427i	$-0.533620\pi$
$229$	−3.19076	−0.210851	−0.105426	+	0.994427i	$0.533620\pi$
$230$	4.26608	0.281297
$231$	0	0
$232$	−3.70737	−0.243401
$233$	3.99890	0.261976	0.130988	−	0.991384i	$-0.458185\pi$
$233$	3.99890	0.261976	0.130988	+	0.991384i	$0.458185\pi$
$234$	0.247280	0.0161652
$235$	24.3080	1.58568
$236$	−0.918533	−0.0597914
$237$	12.9675	0.842328
$238$	0	0
$239$	19.5912	1.26725	0.633625	−	0.773640i	$-0.281566\pi$
$239$	19.5912	1.26725	0.633625	+	0.773640i	$0.281566\pi$
$240$	7.45484	0.481208
$241$	−8.30435	−0.534930	−0.267465	−	0.963568i	$-0.586186\pi$
$241$	−8.30435	−0.534930	−0.267465	+	0.963568i	$0.586186\pi$
$242$	−0.831814	−0.0534710
$243$	0.557439	0.0357597
$244$	−13.4890	−0.863542
$245$	0	0
$246$	8.93750	0.569834
$247$	20.9100	1.33047
$248$	−3.23611	−0.205493
$249$	−16.3655	−1.03712
$250$	−34.9794	−2.21229
$251$	−22.1789	−1.39992	−0.699958	−	0.714184i	$-0.746798\pi$
$251$	−22.1789	−1.39992	−0.699958	+	0.714184i	$0.746798\pi$
$252$	0	0
$253$	−3.18876	−0.200476
$254$	1.03070	0.0646718
$255$	−23.5482	−1.47465
$256$	1.00000	0.0625000
$257$	8.97705	0.559973	0.279987	−	0.960004i	$-0.409670\pi$
$257$	8.97705	0.559973	0.279987	+	0.960004i	$0.409670\pi$
$258$	4.01228	0.249793
$259$	0	0
$260$	−19.6644	−1.21953
$261$	−0.198886	−0.0123107
$262$	7.57878	0.468218
$263$	−15.7829	−0.973213	−0.486606	−	0.873621i	$-0.661765\pi$
$263$	−15.7829	−0.973213	−0.486606	+	0.873621i	$0.661765\pi$
$264$	−5.57226	−0.342949
$265$	52.0437	3.19702
$266$	0	0
$267$	−9.12721	−0.558576
$268$	4.75576	0.290505
$269$	6.24440	0.380728	0.190364	−	0.981714i	$-0.439033\pi$
$269$	6.24440	0.380728	0.190364	+	0.981714i	$0.439033\pi$
$270$	−21.9646	−1.33672
$271$	28.6485	1.74027	0.870137	−	0.492811i	$-0.164030\pi$
$271$	28.6485	1.74027	0.870137	+	0.492811i	$0.164030\pi$
$272$	−3.15879	−0.191530
$273$	0	0
$274$	12.4583	0.752631
$275$	42.0898	2.53811
$276$	1.74747	0.105185
$277$	−14.8563	−0.892626	−0.446313	−	0.894877i	$-0.647263\pi$
$277$	−14.8563	−0.892626	−0.446313	+	0.894877i	$0.647263\pi$
$278$	−16.3739	−0.982042
$279$	−0.173604	−0.0103934
$280$	0	0
$281$	−1.93983	−0.115721	−0.0578604	−	0.998325i	$-0.518428\pi$
$281$	−1.93983	−0.115721	−0.0578604	+	0.998325i	$0.518428\pi$
$282$	9.95702	0.592932
$283$	−25.4194	−1.51103	−0.755514	−	0.655132i	$-0.772613\pi$
$283$	−25.4194	−1.51103	−0.755514	+	0.655132i	$0.772613\pi$
$284$	3.61960	0.214784
$285$	33.8174	2.00317
$286$	14.6985	0.869141
$287$	0	0
$288$	0.0536460	0.00316112
$289$	−7.02207	−0.413063
$290$	15.8159	0.928743
$291$	−10.0385	−0.588467
$292$	13.3705	0.782449
$293$	19.8987	1.16249	0.581246	−	0.813728i	$-0.302565\pi$
$293$	19.8987	1.16249	0.581246	+	0.813728i	$0.302565\pi$
$294$	0	0
$295$	3.91853	0.228146
$296$	−7.24655	−0.421197
$297$	16.4178	0.952660
$298$	−9.70522	−0.562208
$299$	−4.60948	−0.266573
$300$	−23.0656	−1.33169
$301$	0	0
$302$	−12.0784	−0.695032
$303$	22.5757	1.29694
$304$	4.53631	0.260175
$305$	57.5450	3.29501
$306$	−0.169456	−0.00968717
$307$	−7.30872	−0.417131	−0.208565	−	0.978008i	$-0.566879\pi$
$307$	−7.30872	−0.417131	−0.208565	+	0.978008i	$0.566879\pi$
$308$	0	0
$309$	9.54260	0.542860
$310$	13.8055	0.784099
$311$	−28.1727	−1.59752	−0.798762	−	0.601647i	$-0.794512\pi$
$311$	−28.1727	−1.59752	−0.798762	+	0.601647i	$0.794512\pi$
$312$	−8.05492	−0.456020
$313$	−10.3739	−0.586368	−0.293184	−	0.956056i	$-0.594715\pi$
$313$	−10.3739	−0.586368	−0.293184	+	0.956056i	$0.594715\pi$
$314$	1.17809	0.0664835
$315$	0	0
$316$	−7.42072	−0.417448
$317$	15.1465	0.850713	0.425356	−	0.905026i	$-0.360149\pi$
$317$	15.1465	0.850713	0.425356	+	0.905026i	$0.360149\pi$
$318$	21.3181	1.19546
$319$	−11.8219	−0.661900
$320$	−4.26608	−0.238481
$321$	17.8788	0.997895
$322$	0	0
$323$	−14.3292	−0.797299
$324$	−9.15806	−0.508781
$325$	60.8425	3.37493
$326$	14.4645	0.801112
$327$	−15.4324	−0.853416
$328$	−5.11454	−0.282403
$329$	0	0
$330$	23.7717	1.30859
$331$	−18.0343	−0.991257	−0.495629	−	0.868535i	$-0.665062\pi$
$331$	−18.0343	−0.991257	−0.495629	+	0.868535i	$0.665062\pi$
$332$	9.36524	0.513984
$333$	−0.388749	−0.0213033
$334$	6.95702	0.380671
$335$	−20.2885	−1.10848
$336$	0	0
$337$	24.1008	1.31285	0.656427	−	0.754389i	$-0.272067\pi$
$337$	24.1008	1.31285	0.656427	+	0.754389i	$0.272067\pi$
$338$	8.24728	0.448593
$339$	32.5381	1.76723
$340$	13.4756	0.730819
$341$	−10.3192	−0.558814
$342$	0.243355	0.0131591
$343$	0	0
$344$	−2.29605	−0.123795
$345$	−7.45484	−0.401355
$346$	3.35672	0.180459
$347$	−22.6975	−1.21846	−0.609232	−	0.792992i	$-0.708522\pi$
$347$	−22.6975	−1.21846	−0.609232	+	0.792992i	$0.708522\pi$
$348$	6.47851	0.347285
$349$	−22.9733	−1.22973	−0.614865	−	0.788632i	$-0.710790\pi$
$349$	−22.9733	−1.22973	−0.614865	+	0.788632i	$0.710790\pi$
$350$	0	0
$351$	23.7326	1.26675
$352$	3.18876	0.169961
$353$	−17.5852	−0.935968	−0.467984	−	0.883737i	$-0.655020\pi$
$353$	−17.5852	−0.935968	−0.467984	+	0.883737i	$0.655020\pi$
$354$	1.60511	0.0853105
$355$	−15.4415	−0.819550
$356$	5.22310	0.276824
$357$	0	0
$358$	−19.0417	−1.00638
$359$	−3.46474	−0.182862	−0.0914310	−	0.995811i	$-0.529144\pi$
$359$	−3.46474	−0.182862	−0.0914310	+	0.995811i	$0.529144\pi$
$360$	−0.228858	−0.0120619
$361$	1.57807	0.0830562
$362$	−9.25380	−0.486369
$363$	1.45357	0.0762925
$364$	0	0
$365$	−57.0396	−2.98559
$366$	23.5715	1.23210
$367$	−1.10077	−0.0574598	−0.0287299	−	0.999587i	$-0.509146\pi$
$367$	−1.10077	−0.0574598	−0.0287299	+	0.999587i	$0.509146\pi$
$368$	−1.00000	−0.0521286
$369$	−0.274375	−0.0142834
$370$	30.9144	1.60716
$371$	0	0
$372$	5.65499	0.293198
$373$	−23.7420	−1.22932	−0.614658	−	0.788794i	$-0.710706\pi$
$373$	−23.7420	−1.22932	−0.614658	+	0.788794i	$0.710706\pi$
$374$	−10.0726	−0.520842
$375$	61.1254	3.15650
$376$	−5.69797	−0.293850
$377$	−17.0890	−0.880130
$378$	0	0
$379$	−3.03959	−0.156133	−0.0780667	−	0.996948i	$-0.524875\pi$
$379$	−3.03959	−0.156133	−0.0780667	+	0.996948i	$0.524875\pi$
$380$	−19.3522	−0.992749
$381$	−1.80111	−0.0922739
$382$	15.3705	0.786423
$383$	14.5407	0.742994	0.371497	−	0.928434i	$-0.378845\pi$
$383$	14.5407	0.742994	0.371497	+	0.928434i	$0.378845\pi$
$384$	−1.74747	−0.0891751
$385$	0	0
$386$	0.565957	0.0288065
$387$	−0.123174	−0.00626128
$388$	5.74459	0.291637
$389$	31.3174	1.58785	0.793927	−	0.608013i	$-0.208033\pi$
$389$	31.3174	1.58785	0.793927	+	0.608013i	$0.208033\pi$
$390$	34.3629	1.74003
$391$	3.15879	0.159747
$392$	0	0
$393$	−13.2437	−0.668055
$394$	−8.71335	−0.438972
$395$	31.6574	1.59286
$396$	0.171064	0.00859630
$397$	17.6190	0.884274	0.442137	−	0.896947i	$-0.354220\pi$
$397$	17.6190	0.884274	0.442137	+	0.896947i	$0.354220\pi$
$398$	−13.0995	−0.656621
$399$	0	0
$400$	13.1994	0.659971
$401$	−0.966200	−0.0482497	−0.0241249	−	0.999709i	$-0.507680\pi$
$401$	−0.966200	−0.0482497	−0.0241249	+	0.999709i	$0.507680\pi$
$402$	−8.31055	−0.414492
$403$	−14.9168	−0.743057
$404$	−12.9191	−0.642748
$405$	39.0690	1.94135
$406$	0	0
$407$	−23.1075	−1.14540
$408$	5.51988	0.273275
$409$	33.0657	1.63499	0.817496	−	0.575934i	$-0.195361\pi$
$409$	33.0657	1.63499	0.817496	+	0.575934i	$0.195361\pi$
$410$	21.8190	1.07757
$411$	−21.7704	−1.07386
$412$	−5.46082	−0.269035
$413$	0	0
$414$	−0.0536460	−0.00263656
$415$	−39.9529	−1.96121
$416$	4.60948	0.225998
$417$	28.6129	1.40118
$418$	14.4652	0.707515
$419$	−30.6704	−1.49835	−0.749173	−	0.662375i	$-0.769549\pi$
$419$	−30.6704	−1.49835	−0.749173	+	0.662375i	$0.769549\pi$
$420$	0	0
$421$	3.14483	0.153270	0.0766349	−	0.997059i	$-0.475582\pi$
$421$	3.14483	0.153270	0.0766349	+	0.997059i	$0.475582\pi$
$422$	6.67125	0.324751
$423$	−0.305673	−0.0148623
$424$	−12.1994	−0.592457
$425$	−41.6942	−2.02246
$426$	−6.32514	−0.306454
$427$	0	0
$428$	−10.2312	−0.494545
$429$	−25.6852	−1.24009
$430$	9.79514	0.472363
$431$	−6.68265	−0.321892	−0.160946	−	0.986963i	$-0.551454\pi$
$431$	−6.68265	−0.321892	−0.160946	+	0.986963i	$0.551454\pi$
$432$	5.14866	0.247715
$433$	14.1421	0.679628	0.339814	−	0.940493i	$-0.389636\pi$
$433$	14.1421	0.679628	0.339814	+	0.940493i	$0.389636\pi$
$434$	0	0
$435$	−27.6378	−1.32513
$436$	8.83131	0.422943
$437$	−4.53631	−0.217001
$438$	−23.3645	−1.11640
$439$	1.48554	0.0709008	0.0354504	−	0.999371i	$-0.488713\pi$
$439$	1.48554	0.0709008	0.0354504	+	0.999371i	$0.488713\pi$
$440$	−13.6035	−0.648521
$441$	0	0
$442$	−14.5604	−0.692566
$443$	−13.0907	−0.621959	−0.310979	−	0.950417i	$-0.600657\pi$
$443$	−13.0907	−0.621959	−0.310979	+	0.950417i	$0.600657\pi$
$444$	12.6631	0.600965
$445$	−22.2822	−1.05628
$446$	−16.9305	−0.801681
$447$	16.9596	0.802160
$448$	0	0
$449$	−28.0004	−1.32142	−0.660710	−	0.750641i	$-0.729745\pi$
$449$	−28.0004	−1.32142	−0.660710	+	0.750641i	$0.729745\pi$
$450$	0.708097	0.0333800
$451$	−16.3090	−0.767963
$452$	−18.6201	−0.875818
$453$	21.1066	0.991673
$454$	−18.2753	−0.857703
$455$	0	0
$456$	−7.92705	−0.371218
$457$	−3.88479	−0.181723	−0.0908614	−	0.995864i	$-0.528962\pi$
$457$	−3.88479	−0.181723	−0.0908614	+	0.995864i	$0.528962\pi$
$458$	−3.19076	−0.149094
$459$	−16.2635	−0.759116
$460$	4.26608	0.198907
$461$	21.7186	1.01154	0.505768	−	0.862670i	$-0.331209\pi$
$461$	21.7186	1.01154	0.505768	+	0.862670i	$0.331209\pi$
$462$	0	0
$463$	4.97234	0.231084	0.115542	−	0.993303i	$-0.463139\pi$
$463$	4.97234	0.231084	0.115542	+	0.993303i	$0.463139\pi$
$464$	−3.70737	−0.172110
$465$	−24.1246	−1.11875
$466$	3.99890	0.185245
$467$	6.84929	0.316947	0.158474	−	0.987363i	$-0.449343\pi$
$467$	6.84929	0.316947	0.158474	+	0.987363i	$0.449343\pi$
$468$	0.247280	0.0114305
$469$	0	0
$470$	24.3080	1.12124
$471$	−2.05868	−0.0948587
$472$	−0.918533	−0.0422789
$473$	−7.32156	−0.336645
$474$	12.9675	0.595616
$475$	59.8766	2.74733
$476$	0	0
$477$	−0.654451	−0.0299652
$478$	19.5912	0.896082
$479$	−10.0047	−0.457126	−0.228563	−	0.973529i	$-0.573403\pi$
$479$	−10.0047	−0.457126	−0.228563	+	0.973529i	$0.573403\pi$
$480$	7.45484	0.340265
$481$	−33.4028	−1.52304
$482$	−8.30435	−0.378253
$483$	0	0
$484$	−0.831814	−0.0378097
$485$	−24.5069	−1.11280
$486$	0.557439	0.0252860
$487$	26.9160	1.21968	0.609840	−	0.792525i	$-0.291234\pi$
$487$	26.9160	1.21968	0.609840	+	0.792525i	$0.291234\pi$
$488$	−13.4890	−0.610616
$489$	−25.2762	−1.14303
$490$	0	0
$491$	−17.6787	−0.797827	−0.398914	−	0.916989i	$-0.630613\pi$
$491$	−17.6787	−0.797827	−0.398914	+	0.916989i	$0.630613\pi$
$492$	8.93750	0.402934
$493$	11.7108	0.527427
$494$	20.9100	0.940785
$495$	−0.729773	−0.0328009
$496$	−3.23611	−0.145305
$497$	0	0
$498$	−16.3655	−0.733354
$499$	21.5441	0.964446	0.482223	−	0.876048i	$-0.339829\pi$
$499$	21.5441	0.964446	0.482223	+	0.876048i	$0.339829\pi$
$500$	−34.9794	−1.56433
$501$	−12.1572	−0.543143
$502$	−22.1789	−0.989891
$503$	30.0934	1.34180	0.670900	−	0.741548i	$-0.265908\pi$
$503$	30.0934	1.34180	0.670900	+	0.741548i	$0.265908\pi$
$504$	0	0
$505$	55.1138	2.45253
$506$	−3.18876	−0.141758
$507$	−14.4119	−0.640053
$508$	1.03070	0.0457299
$509$	17.6680	0.783120	0.391560	−	0.920153i	$-0.371936\pi$
$509$	17.6680	0.783120	0.391560	+	0.920153i	$0.371936\pi$
$510$	−23.5482	−1.04273
$511$	0	0
$512$	1.00000	0.0441942
$513$	23.3559	1.03119
$514$	8.97705	0.395961
$515$	23.2963	1.02656
$516$	4.01228	0.176631
$517$	−18.1695	−0.799092
$518$	0	0
$519$	−5.86577	−0.257479
$520$	−19.6644	−0.862341
$521$	14.1499	0.619920	0.309960	−	0.950750i	$-0.399684\pi$
$521$	14.1499	0.619920	0.309960	+	0.950750i	$0.399684\pi$
$522$	−0.198886	−0.00870498
$523$	30.3429	1.32680	0.663401	−	0.748264i	$-0.269113\pi$
$523$	30.3429	1.32680	0.663401	+	0.748264i	$0.269113\pi$
$524$	7.57878	0.331080
$525$	0	0
$526$	−15.7829	−0.688165
$527$	10.2222	0.445285
$528$	−5.57226	−0.242501
$529$	1.00000	0.0434783
$530$	52.0437	2.26063
$531$	−0.0492756	−0.00213838
$532$	0	0
$533$	−23.5754	−1.02116
$534$	−9.12721	−0.394973
$535$	43.6472	1.88703
$536$	4.75576	0.205418
$537$	33.2748	1.43591
$538$	6.24440	0.269215
$539$	0	0
$540$	−21.9646	−0.945206
$541$	24.4127	1.04959	0.524793	−	0.851230i	$-0.324143\pi$
$541$	24.4127	1.04959	0.524793	+	0.851230i	$0.324143\pi$
$542$	28.6485	1.23056
$543$	16.1707	0.693952
$544$	−3.15879	−0.135432
$545$	−37.6751	−1.61382
$546$	0	0
$547$	19.8203	0.847455	0.423727	−	0.905790i	$-0.360721\pi$
$547$	19.8203	0.847455	0.423727	+	0.905790i	$0.360721\pi$
$548$	12.4583	0.532190
$549$	−0.723629	−0.0308837
$550$	42.0898	1.79472
$551$	−16.8178	−0.716461
$552$	1.74747	0.0743772
$553$	0	0
$554$	−14.8563	−0.631182
$555$	−54.0219	−2.29310
$556$	−16.3739	−0.694409
$557$	1.78050	0.0754423	0.0377212	−	0.999288i	$-0.487990\pi$
$557$	1.78050	0.0754423	0.0377212	+	0.999288i	$0.487990\pi$
$558$	−0.173604	−0.00734925
$559$	−10.5836	−0.447638
$560$	0	0
$561$	17.6016	0.743139
$562$	−1.93983	−0.0818269
$563$	−34.4227	−1.45074	−0.725372	−	0.688357i	$-0.758332\pi$
$563$	−34.4227	−1.45074	−0.725372	+	0.688357i	$0.758332\pi$
$564$	9.95702	0.419266
$565$	79.4350	3.34186
$566$	−25.4194	−1.06846
$567$	0	0
$568$	3.61960	0.151875
$569$	13.0547	0.547284	0.273642	−	0.961832i	$-0.411772\pi$
$569$	13.0547	0.547284	0.273642	+	0.961832i	$0.411772\pi$
$570$	33.8174	1.41646
$571$	−25.4010	−1.06300	−0.531500	−	0.847059i	$-0.678371\pi$
$571$	−25.4010	−1.06300	−0.531500	+	0.847059i	$0.678371\pi$
$572$	14.6985	0.614576
$573$	−26.8595	−1.12207
$574$	0	0
$575$	−13.1994	−0.550454
$576$	0.0536460	0.00223525
$577$	41.9277	1.74547	0.872736	−	0.488192i	$-0.162343\pi$
$577$	41.9277	1.74547	0.872736	+	0.488192i	$0.162343\pi$
$578$	−7.02207	−0.292079
$579$	−0.988993	−0.0411011
$580$	15.8159	0.656721
$581$	0	0
$582$	−10.0385	−0.416109
$583$	−38.9010	−1.61112
$584$	13.3705	0.553275
$585$	−1.05492	−0.0436154
$586$	19.8987	0.822006
$587$	−27.1223	−1.11946	−0.559729	−	0.828676i	$-0.689095\pi$
$587$	−27.1223	−1.11946	−0.559729	+	0.828676i	$0.689095\pi$
$588$	0	0
$589$	−14.6800	−0.604878
$590$	3.91853	0.161323
$591$	15.2263	0.626326
$592$	−7.24655	−0.297832
$593$	1.56811	0.0643945	0.0321972	−	0.999482i	$-0.489750\pi$
$593$	1.56811	0.0643945	0.0321972	+	0.999482i	$0.489750\pi$
$594$	16.4178	0.673632
$595$	0	0
$596$	−9.70522	−0.397541
$597$	22.8910	0.936868
$598$	−4.60948	−0.188496
$599$	42.2777	1.72742	0.863711	−	0.503988i	$-0.168134\pi$
$599$	42.2777	1.72742	0.863711	+	0.503988i	$0.168134\pi$
$600$	−23.0656	−0.941649
$601$	29.1651	1.18967	0.594834	−	0.803848i	$-0.297218\pi$
$601$	29.1651	1.18967	0.594834	+	0.803848i	$0.297218\pi$
$602$	0	0
$603$	0.255128	0.0103896
$604$	−12.0784	−0.491462
$605$	3.54858	0.144270
$606$	22.5757	0.917074
$607$	−10.5927	−0.429943	−0.214972	−	0.976620i	$-0.568966\pi$
$607$	−10.5927	−0.429943	−0.214972	+	0.976620i	$0.568966\pi$
$608$	4.53631	0.183971
$609$	0	0
$610$	57.5450	2.32993
$611$	−26.2647	−1.06255
$612$	−0.169456	−0.00684986
$613$	−10.8146	−0.436798	−0.218399	−	0.975860i	$-0.570083\pi$
$613$	−10.8146	−0.436798	−0.218399	+	0.975860i	$0.570083\pi$
$614$	−7.30872	−0.294956
$615$	−38.1281	−1.53747
$616$	0	0
$617$	−18.2246	−0.733693	−0.366846	−	0.930281i	$-0.619563\pi$
$617$	−18.2246	−0.733693	−0.366846	+	0.930281i	$0.619563\pi$
$618$	9.54260	0.383860
$619$	−32.6877	−1.31383	−0.656914	−	0.753966i	$-0.728139\pi$
$619$	−32.6877	−1.31383	−0.656914	+	0.753966i	$0.728139\pi$
$620$	13.8055	0.554442
$621$	−5.14866	−0.206609
$622$	−28.1727	−1.12962
$623$	0	0
$624$	−8.05492	−0.322455
$625$	83.2278	3.32911
$626$	−10.3739	−0.414625
$627$	−25.2775	−1.00948
$628$	1.17809	0.0470109
$629$	22.8903	0.912697
$630$	0	0
$631$	−16.7672	−0.667492	−0.333746	−	0.942663i	$-0.608313\pi$
$631$	−16.7672	−0.667492	−0.333746	+	0.942663i	$0.608313\pi$
$632$	−7.42072	−0.295180
$633$	−11.6578	−0.463356
$634$	15.1465	0.601545
$635$	−4.39704	−0.174491
$636$	21.3181	0.845318
$637$	0	0
$638$	−11.8219	−0.468034
$639$	0.194177	0.00768153
$640$	−4.26608	−0.168632
$641$	−32.5905	−1.28725	−0.643624	−	0.765342i	$-0.722570\pi$
$641$	−32.5905	−1.28725	−0.643624	+	0.765342i	$0.722570\pi$
$642$	17.8788	0.705618
$643$	6.41984	0.253174	0.126587	−	0.991956i	$-0.459598\pi$
$643$	6.41984	0.253174	0.126587	+	0.991956i	$0.459598\pi$
$644$	0	0
$645$	−17.1167	−0.673969
$646$	−14.3292	−0.563776
$647$	16.2710	0.639677	0.319839	−	0.947472i	$-0.396371\pi$
$647$	16.2710	0.639677	0.319839	+	0.947472i	$0.396371\pi$
$648$	−9.15806	−0.359763
$649$	−2.92898	−0.114973
$650$	60.8425	2.38644
$651$	0	0
$652$	14.4645	0.566472
$653$	23.7977	0.931277	0.465639	−	0.884975i	$-0.345825\pi$
$653$	23.7977	0.931277	0.465639	+	0.884975i	$0.345825\pi$
$654$	−15.4324	−0.603456
$655$	−32.3317	−1.26330
$656$	−5.11454	−0.199689
$657$	0.717273	0.0279835
$658$	0	0
$659$	42.1039	1.64013	0.820067	−	0.572268i	$-0.193936\pi$
$659$	42.1039	1.64013	0.820067	+	0.572268i	$0.193936\pi$
$660$	23.7717	0.925312
$661$	−50.8363	−1.97730	−0.988652	−	0.150225i	$-0.952000\pi$
$661$	−50.8363	−1.97730	−0.988652	+	0.150225i	$0.952000\pi$
$662$	−18.0343	−0.700925
$663$	25.4438	0.988154
$664$	9.36524	0.363442
$665$	0	0
$666$	−0.388749	−0.0150637
$667$	3.70737	0.143550
$668$	6.95702	0.269175
$669$	29.5855	1.14384
$670$	−20.2885	−0.783812
$671$	−43.0130	−1.66050
$672$	0	0
$673$	20.5556	0.792360	0.396180	−	0.918173i	$-0.370336\pi$
$673$	20.5556	0.792360	0.396180	+	0.918173i	$0.370336\pi$
$674$	24.1008	0.928328
$675$	67.9594	2.61576
$676$	8.24728	0.317203
$677$	38.1726	1.46709	0.733546	−	0.679640i	$-0.237864\pi$
$677$	38.1726	1.46709	0.733546	+	0.679640i	$0.237864\pi$
$678$	32.5381	1.24962
$679$	0	0
$680$	13.4756	0.516767
$681$	31.9356	1.22377
$682$	−10.3192	−0.395141
$683$	2.64737	0.101299	0.0506494	−	0.998716i	$-0.483871\pi$
$683$	2.64737	0.101299	0.0506494	+	0.998716i	$0.483871\pi$
$684$	0.243355	0.00930490
$685$	−53.1479	−2.03068
$686$	0	0
$687$	5.57575	0.212728
$688$	−2.29605	−0.0875361
$689$	−56.2330	−2.14231
$690$	−7.45484	−0.283801
$691$	−44.8548	−1.70636	−0.853178	−	0.521619i	$-0.825328\pi$
$691$	−44.8548	−1.70636	−0.853178	+	0.521619i	$0.825328\pi$
$692$	3.35672	0.127603
$693$	0	0
$694$	−22.6975	−0.861584
$695$	69.8524	2.64965
$696$	6.47851	0.245567
$697$	16.1557	0.611942
$698$	−22.9733	−0.869551
$699$	−6.98795	−0.264308
$700$	0	0
$701$	−3.67590	−0.138837	−0.0694185	−	0.997588i	$-0.522114\pi$
$701$	−3.67590	−0.138837	−0.0694185	+	0.997588i	$0.522114\pi$
$702$	23.7326	0.895731
$703$	−32.8726	−1.23981
$704$	3.18876	0.120181
$705$	−42.4774	−1.59979
$706$	−17.5852	−0.661829
$707$	0	0
$708$	1.60511	0.0603236
$709$	−7.08855	−0.266216	−0.133108	−	0.991102i	$-0.542496\pi$
$709$	−7.08855	−0.266216	−0.133108	+	0.991102i	$0.542496\pi$
$710$	−15.4415	−0.579510
$711$	−0.398092	−0.0149296
$712$	5.22310	0.195744
$713$	3.23611	0.121193
$714$	0	0
$715$	−62.7050	−2.34503
$716$	−19.0417	−0.711621
$717$	−34.2350	−1.27853
$718$	−3.46474	−0.129303
$719$	14.8732	0.554678	0.277339	−	0.960772i	$-0.410548\pi$
$719$	14.8732	0.554678	0.277339	+	0.960772i	$0.410548\pi$
$720$	−0.228858	−0.00852904
$721$	0	0
$722$	1.57807	0.0587296
$723$	14.5116	0.539692
$724$	−9.25380	−0.343915
$725$	−48.9352	−1.81741
$726$	1.45357	0.0539470
$727$	−14.9667	−0.555086	−0.277543	−	0.960713i	$-0.589520\pi$
$727$	−14.9667	−0.555086	−0.277543	+	0.960713i	$0.589520\pi$
$728$	0	0
$729$	26.5001	0.981484
$730$	−57.0396	−2.11113
$731$	7.25274	0.268252
$732$	23.5715	0.871229
$733$	−46.4686	−1.71636	−0.858178	−	0.513352i	$-0.828404\pi$
$733$	−46.4686	−1.71636	−0.858178	+	0.513352i	$0.828404\pi$
$734$	−1.10077	−0.0406302
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	15.1650	0.558610
$738$	−0.274375	−0.0100999
$739$	34.8786	1.28303	0.641516	−	0.767110i	$-0.278306\pi$
$739$	34.8786	1.28303	0.641516	+	0.767110i	$0.278306\pi$
$740$	30.9144	1.13643
$741$	−36.5396	−1.34231
$742$	0	0
$743$	5.81935	0.213491	0.106746	−	0.994286i	$-0.465957\pi$
$743$	5.81935	0.213491	0.106746	+	0.994286i	$0.465957\pi$
$744$	5.65499	0.207322
$745$	41.4032	1.51690
$746$	−23.7420	−0.869258
$747$	0.502408	0.0183821
$748$	−10.0726	−0.368291
$749$	0	0
$750$	61.1254	2.23198
$751$	−50.2390	−1.83325	−0.916623	−	0.399753i	$-0.869096\pi$
$751$	−50.2390	−1.83325	−0.916623	+	0.399753i	$0.869096\pi$
$752$	−5.69797	−0.207784
$753$	38.7569	1.41238
$754$	−17.0890	−0.622346
$755$	51.5273	1.87527
$756$	0	0
$757$	45.5514	1.65559	0.827797	−	0.561028i	$-0.189594\pi$
$757$	45.5514	1.65559	0.827797	+	0.561028i	$0.189594\pi$
$758$	−3.03959	−0.110403
$759$	5.57226	0.202260
$760$	−19.3522	−0.701979
$761$	19.2863	0.699128	0.349564	−	0.936913i	$-0.386330\pi$
$761$	19.2863	0.699128	0.349564	+	0.936913i	$0.386330\pi$
$762$	−1.80111	−0.0652475
$763$	0	0
$764$	15.3705	0.556085
$765$	0.722914	0.0261370
$766$	14.5407	0.525376
$767$	−4.23396	−0.152879
$768$	−1.74747	−0.0630563
$769$	37.7235	1.36034	0.680171	−	0.733053i	$-0.261905\pi$
$769$	37.7235	1.36034	0.680171	+	0.733053i	$0.261905\pi$
$770$	0	0
$771$	−15.6871	−0.564958
$772$	0.565957	0.0203693
$773$	17.4594	0.627970	0.313985	−	0.949428i	$-0.398336\pi$
$773$	17.4594	0.627970	0.313985	+	0.949428i	$0.398336\pi$
$774$	−0.123174	−0.00442740
$775$	−42.7148	−1.53436
$776$	5.74459	0.206219
$777$	0	0
$778$	31.3174	1.12278
$779$	−23.2011	−0.831266
$780$	34.3629	1.23039
$781$	11.5420	0.413007
$782$	3.15879	0.112958
$783$	−19.0880	−0.682149
$784$	0	0
$785$	−5.02582	−0.179379
$786$	−13.2437	−0.472386
$787$	13.4288	0.478686	0.239343	−	0.970935i	$-0.423068\pi$
$787$	13.4288	0.478686	0.239343	+	0.970935i	$0.423068\pi$
$788$	−8.71335	−0.310400
$789$	27.5800	0.981876
$790$	31.6574	1.12632
$791$	0	0
$792$	0.171064	0.00607850
$793$	−62.1770	−2.20797
$794$	17.6190	0.625276
$795$	−90.9448	−3.22548
$796$	−13.0995	−0.464301
$797$	−21.3322	−0.755624	−0.377812	−	0.925882i	$-0.623323\pi$
$797$	−21.3322	−0.755624	−0.377812	+	0.925882i	$0.623323\pi$
$798$	0	0
$799$	17.9987	0.636747
$800$	13.1994	0.466670
$801$	0.280199	0.00990033
$802$	−0.966200	−0.0341177
$803$	42.6353	1.50457
$804$	−8.31055	−0.293090
$805$	0	0
$806$	−14.9168	−0.525420
$807$	−10.9119	−0.384117
$808$	−12.9191	−0.454492
$809$	−42.2976	−1.48710	−0.743552	−	0.668678i	$-0.766860\pi$
$809$	−42.2976	−1.48710	−0.743552	+	0.668678i	$0.766860\pi$
$810$	39.0690	1.37274
$811$	23.5400	0.826603	0.413301	−	0.910594i	$-0.364376\pi$
$811$	23.5400	0.826603	0.413301	+	0.910594i	$0.364376\pi$
$812$	0	0
$813$	−50.0624	−1.75576
$814$	−23.1075	−0.809918
$815$	−61.7065	−2.16149
$816$	5.51988	0.193234
$817$	−10.4156	−0.364395
$818$	33.0657	1.15611
$819$	0	0
$820$	21.8190	0.761954
$821$	22.3094	0.778605	0.389302	−	0.921110i	$-0.372716\pi$
$821$	22.3094	0.778605	0.389302	+	0.921110i	$0.372716\pi$
$822$	−21.7704	−0.759330
$823$	13.6087	0.474371	0.237185	−	0.971464i	$-0.423775\pi$
$823$	13.6087	0.474371	0.237185	+	0.971464i	$0.423775\pi$
$824$	−5.46082	−0.190237
$825$	−73.5506	−2.56070
$826$	0	0
$827$	−15.3089	−0.532344	−0.266172	−	0.963926i	$-0.585759\pi$
$827$	−15.3089	−0.532344	−0.266172	+	0.963926i	$0.585759\pi$
$828$	−0.0536460	−0.00186433
$829$	−24.9010	−0.864847	−0.432424	−	0.901671i	$-0.642341\pi$
$829$	−24.9010	−0.864847	−0.432424	+	0.901671i	$0.642341\pi$
$830$	−39.9529	−1.38678
$831$	25.9608	0.900571
$832$	4.60948	0.159805
$833$	0	0
$834$	28.6129	0.990784
$835$	−29.6792	−1.02709
$836$	14.4652	0.500289
$837$	−16.6616	−0.575910
$838$	−30.6704	−1.05949
$839$	−45.3719	−1.56641	−0.783205	−	0.621763i	$-0.786417\pi$
$839$	−45.3719	−1.56641	−0.783205	+	0.621763i	$0.786417\pi$
$840$	0	0
$841$	−15.2554	−0.526049
$842$	3.14483	0.108378
$843$	3.38980	0.116751
$844$	6.67125	0.229634
$845$	−35.1835	−1.21035
$846$	−0.305673	−0.0105093
$847$	0	0
$848$	−12.1994	−0.418930
$849$	44.4197	1.52448
$850$	−41.6942	−1.43010
$851$	7.24655	0.248409
$852$	−6.32514	−0.216696
$853$	−1.29412	−0.0443099	−0.0221550	−	0.999755i	$-0.507053\pi$
$853$	−1.29412	−0.0443099	−0.0221550	+	0.999755i	$0.507053\pi$
$854$	0	0
$855$	−1.03817	−0.0355047
$856$	−10.2312	−0.349696
$857$	19.0333	0.650166	0.325083	−	0.945685i	$-0.394608\pi$
$857$	19.0333	0.650166	0.325083	+	0.945685i	$0.394608\pi$
$858$	−25.6852	−0.876878
$859$	29.3201	1.00039	0.500195	−	0.865913i	$-0.333262\pi$
$859$	29.3201	1.00039	0.500195	+	0.865913i	$0.333262\pi$
$860$	9.79514	0.334011
$861$	0	0
$862$	−6.68265	−0.227612
$863$	23.9405	0.814943	0.407472	−	0.913218i	$-0.366411\pi$
$863$	23.9405	0.814943	0.407472	+	0.913218i	$0.366411\pi$
$864$	5.14866	0.175161
$865$	−14.3200	−0.486896
$866$	14.1421	0.480569
$867$	12.2708	0.416740
$868$	0	0
$869$	−23.6629	−0.802708
$870$	−27.6378	−0.937011
$871$	21.9216	0.742785
$872$	8.83131	0.299066
$873$	0.308174	0.0104301
$874$	−4.53631	−0.153443
$875$	0	0
$876$	−23.3645	−0.789414
$877$	−42.0081	−1.41851	−0.709256	−	0.704951i	$-0.750969\pi$
$877$	−42.0081	−1.41851	−0.709256	+	0.704951i	$0.750969\pi$
$878$	1.48554	0.0501345
$879$	−34.7723	−1.17284
$880$	−13.6035	−0.458574
$881$	37.3864	1.25958	0.629790	−	0.776766i	$-0.283141\pi$
$881$	37.3864	1.25958	0.629790	+	0.776766i	$0.283141\pi$
$882$	0	0
$883$	−38.6925	−1.30211	−0.651053	−	0.759032i	$-0.725672\pi$
$883$	−38.6925	−1.30211	−0.651053	+	0.759032i	$0.725672\pi$
$884$	−14.5604	−0.489718
$885$	−6.84751	−0.230177
$886$	−13.0907	−0.439791
$887$	−23.7538	−0.797573	−0.398786	−	0.917044i	$-0.630569\pi$
$887$	−23.7538	−0.797573	−0.398786	+	0.917044i	$0.630569\pi$
$888$	12.6631	0.424947
$889$	0	0
$890$	−22.2822	−0.746900
$891$	−29.2028	−0.978332
$892$	−16.9305	−0.566874
$893$	−25.8477	−0.864961
$894$	16.9596	0.567213
$895$	81.2333	2.71533
$896$	0	0
$897$	8.05492	0.268946
$898$	−28.0004	−0.934385
$899$	11.9974	0.400137
$900$	0.708097	0.0236032
$901$	38.5354	1.28380
$902$	−16.3090	−0.543032
$903$	0	0
$904$	−18.6201	−0.619297
$905$	39.4774	1.31227
$906$	21.1066	0.701219
$907$	25.1859	0.836283	0.418141	−	0.908382i	$-0.362682\pi$
$907$	25.1859	0.836283	0.418141	+	0.908382i	$0.362682\pi$
$908$	−18.2753	−0.606488
$909$	−0.693057	−0.0229872
$910$	0	0
$911$	−33.9550	−1.12498	−0.562490	−	0.826804i	$-0.690156\pi$
$911$	−33.9550	−1.12498	−0.562490	+	0.826804i	$0.690156\pi$
$912$	−7.92705	−0.262491
$913$	29.8635	0.988338
$914$	−3.88479	−0.128497
$915$	−100.558	−3.32435
$916$	−3.19076	−0.105426
$917$	0	0
$918$	−16.2635	−0.536776
$919$	33.8739	1.11740	0.558699	−	0.829370i	$-0.311301\pi$
$919$	33.8739	1.11740	0.558699	+	0.829370i	$0.311301\pi$
$920$	4.26608	0.140648
$921$	12.7718	0.420844
$922$	21.7186	0.715263
$923$	16.6845	0.549176
$924$	0	0
$925$	−95.6504	−3.14496
$926$	4.97234	0.163401
$927$	−0.292951	−0.00962177
$928$	−3.70737	−0.121700
$929$	14.8890	0.488494	0.244247	−	0.969713i	$-0.421459\pi$
$929$	14.8890	0.488494	0.244247	+	0.969713i	$0.421459\pi$
$930$	−24.1246	−0.791078
$931$	0	0
$932$	3.99890	0.130988
$933$	49.2308	1.61174
$934$	6.84929	0.224115
$935$	42.9706	1.40529
$936$	0.247280	0.00808260
$937$	−10.1786	−0.332520	−0.166260	−	0.986082i	$-0.553169\pi$
$937$	−10.1786	−0.332520	−0.166260	+	0.986082i	$0.553169\pi$
$938$	0	0
$939$	18.1281	0.591588
$940$	24.3080	0.792839
$941$	−29.3398	−0.956449	−0.478224	−	0.878238i	$-0.658719\pi$
$941$	−29.3398	−0.956449	−0.478224	+	0.878238i	$0.658719\pi$
$942$	−2.05868	−0.0670753
$943$	5.11454	0.166552
$944$	−0.918533	−0.0298957
$945$	0	0
$946$	−7.32156	−0.238044
$947$	40.0527	1.30154	0.650770	−	0.759275i	$-0.274446\pi$
$947$	40.0527	1.30154	0.650770	+	0.759275i	$0.274446\pi$
$948$	12.9675	0.421164
$949$	61.6310	2.00063
$950$	59.8766	1.94265
$951$	−26.4680	−0.858285
$952$	0	0
$953$	53.5369	1.73423	0.867115	−	0.498107i	$-0.165971\pi$
$953$	53.5369	1.73423	0.867115	+	0.498107i	$0.165971\pi$
$954$	−0.654451	−0.0211886
$955$	−65.5717	−2.12185
$956$	19.5912	0.633625
$957$	20.6584	0.667792
$958$	−10.0047	−0.323237
$959$	0	0
$960$	7.45484	0.240604
$961$	−20.5276	−0.662181
$962$	−33.4028	−1.07695
$963$	−0.548865	−0.0176869
$964$	−8.30435	−0.267465
$965$	−2.41442	−0.0777229
$966$	0	0
$967$	−10.1189	−0.325400	−0.162700	−	0.986676i	$-0.552020\pi$
$967$	−10.1189	−0.325400	−0.162700	+	0.986676i	$0.552020\pi$
$968$	−0.831814	−0.0267355
$969$	25.0399	0.804396
$970$	−24.5069	−0.786868
$971$	−29.0462	−0.932138	−0.466069	−	0.884748i	$-0.654330\pi$
$971$	−29.0462	−0.932138	−0.466069	+	0.884748i	$0.654330\pi$
$972$	0.557439	0.0178799
$973$	0	0
$974$	26.9160	0.862443
$975$	−106.320	−3.40497
$976$	−13.4890	−0.431771
$977$	13.9144	0.445162	0.222581	−	0.974914i	$-0.428552\pi$
$977$	13.9144	0.445162	0.222581	+	0.974914i	$0.428552\pi$
$978$	−25.2762	−0.808243
$979$	16.6552	0.532303
$980$	0	0
$981$	0.473764	0.0151261
$982$	−17.6787	−0.564149
$983$	−29.9336	−0.954733	−0.477366	−	0.878704i	$-0.658409\pi$
$983$	−29.9336	−0.954733	−0.477366	+	0.878704i	$0.658409\pi$
$984$	8.93750	0.284917
$985$	37.1718	1.18439
$986$	11.7108	0.372948
$987$	0	0
$988$	20.9100	0.665236
$989$	2.29605	0.0730102
$990$	−0.729773	−0.0231937
$991$	−42.0609	−1.33611	−0.668055	−	0.744112i	$-0.732873\pi$
$991$	−42.0609	−1.33611	−0.668055	+	0.744112i	$0.732873\pi$
$992$	−3.23611	−0.102746
$993$	31.5144	1.00008
$994$	0	0
$995$	55.8837	1.77163
$996$	−16.3655	−0.518560
$997$	9.84084	0.311662	0.155831	−	0.987784i	$-0.450194\pi$
$997$	9.84084	0.311662	0.155831	+	0.987784i	$0.450194\pi$
$998$	21.5441	0.681966
$999$	−37.3100	−1.18044

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.2.a.x.1.2		4
7.3	odd	6		322.2.e.a.93.2	✓	8
7.5	odd	6		322.2.e.a.277.2	yes	8
7.6	odd	2		2254.2.a.z.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.e.a.93.2	✓	8	7.3	odd	6
322.2.e.a.277.2	yes	8	7.5	odd	6
2254.2.a.x.1.2		4	1.1	even	1	trivial
2254.2.a.z.1.3		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.74747 q^{3} +1.00000 q^{4} -4.26608 q^{5} -1.74747 q^{6} +1.00000 q^{8} +0.0536460 q^{9} -4.26608 q^{10} +3.18876 q^{11} -1.74747 q^{12} +4.60948 q^{13} +7.45484 q^{15} +1.00000 q^{16} -3.15879 q^{17} +0.0536460 q^{18} +4.53631 q^{19} -4.26608 q^{20} +3.18876 q^{22} -1.00000 q^{23} -1.74747 q^{24} +13.1994 q^{25} +4.60948 q^{26} +5.14866 q^{27} -3.70737 q^{29} +7.45484 q^{30} -3.23611 q^{31} +1.00000 q^{32} -5.57226 q^{33} -3.15879 q^{34} +0.0536460 q^{36} -7.24655 q^{37} +4.53631 q^{38} -8.05492 q^{39} -4.26608 q^{40} -5.11454 q^{41} -2.29605 q^{43} +3.18876 q^{44} -0.228858 q^{45} -1.00000 q^{46} -5.69797 q^{47} -1.74747 q^{48} +13.1994 q^{50} +5.51988 q^{51} +4.60948 q^{52} -12.1994 q^{53} +5.14866 q^{54} -13.6035 q^{55} -7.92705 q^{57} -3.70737 q^{58} -0.918533 q^{59} +7.45484 q^{60} -13.4890 q^{61} -3.23611 q^{62} +1.00000 q^{64} -19.6644 q^{65} -5.57226 q^{66} +4.75576 q^{67} -3.15879 q^{68} +1.74747 q^{69} +3.61960 q^{71} +0.0536460 q^{72} +13.3705 q^{73} -7.24655 q^{74} -23.0656 q^{75} +4.53631 q^{76} -8.05492 q^{78} -7.42072 q^{79} -4.26608 q^{80} -9.15806 q^{81} -5.11454 q^{82} +9.36524 q^{83} +13.4756 q^{85} -2.29605 q^{86} +6.47851 q^{87} +3.18876 q^{88} +5.22310 q^{89} -0.228858 q^{90} -1.00000 q^{92} +5.65499 q^{93} -5.69797 q^{94} -19.3522 q^{95} -1.74747 q^{96} +5.74459 q^{97} +0.171064 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 3 q^{3} + 4 q^{4} - 7 q^{5} - 3 q^{6} + 4 q^{8} - q^{9} - 7 q^{10} + 2 q^{11} - 3 q^{12} - q^{13} + 9 q^{15} + 4 q^{16} - 5 q^{17} - q^{18} - 11 q^{19} - 7 q^{20} + 2 q^{22} - 4 q^{23}+ \cdots + 13 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 3 * q^3 + 4 * q^4 - 7 * q^5 - 3 * q^6 + 4 * q^8 - q^9 - 7 * q^10 + 2 * q^11 - 3 * q^12 - q^13 + 9 * q^15 + 4 * q^16 - 5 * q^17 - q^18 - 11 * q^19 - 7 * q^20 + 2 * q^22 - 4 * q^23 - 3 * q^24 + 3 * q^25 - q^26 - 3 * q^27 + 2 * q^29 + 9 * q^30 - 6 * q^31 + 4 * q^32 - 15 * q^33 - 5 * q^34 - q^36 - 8 * q^37 - 11 * q^38 + 3 * q^39 - 7 * q^40 - 9 * q^41 + 4 * q^43 + 2 * q^44 - 3 * q^45 - 4 * q^46 - 11 * q^47 - 3 * q^48 + 3 * q^50 - 18 * q^51 - q^52 + q^53 - 3 * q^54 - 10 * q^55 + 3 * q^57 + 2 * q^58 - 12 * q^59 + 9 * q^60 - 21 * q^61 - 6 * q^62 + 4 * q^64 - 24 * q^65 - 15 * q^66 - 3 * q^67 - 5 * q^68 + 3 * q^69 + 11 * q^71 - q^72 + 16 * q^73 - 8 * q^74 - 18 * q^75 - 11 * q^76 + 3 * q^78 - 21 * q^79 - 7 * q^80 - 8 * q^81 - 9 * q^82 - 4 * q^83 + 10 * q^85 + 4 * q^86 + 7 * q^87 + 2 * q^88 - 27 * q^89 - 3 * q^90 - 4 * q^92 - 27 * q^93 - 11 * q^94 + 5 * q^95 - 3 * q^96 - 6 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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