Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1881,2,Mod(1,1881)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1881.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1881, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1881.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,1,0,15,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.0198606202\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.03821\) of defining polynomial
Character \(\chi\) \(=\) 1881.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.03821 q^{2} +2.15429 q^{4} +3.24760 q^{5} +1.92338 q^{7} -0.314472 q^{8} -6.61928 q^{10} +1.00000 q^{11} +2.85122 q^{13} -3.92024 q^{14} -3.66762 q^{16} +2.33033 q^{17} +1.00000 q^{19} +6.99626 q^{20} -2.03821 q^{22} +2.74653 q^{23} +5.54689 q^{25} -5.81138 q^{26} +4.14350 q^{28} +0.972965 q^{29} -0.00551178 q^{31} +8.10431 q^{32} -4.74970 q^{34} +6.24635 q^{35} +9.67124 q^{37} -2.03821 q^{38} -1.02128 q^{40} -6.65137 q^{41} +7.99413 q^{43} +2.15429 q^{44} -5.59800 q^{46} -3.46982 q^{47} -3.30063 q^{49} -11.3057 q^{50} +6.14236 q^{52} -10.5493 q^{53} +3.24760 q^{55} -0.604847 q^{56} -1.98311 q^{58} +13.7814 q^{59} +3.74608 q^{61} +0.0112342 q^{62} -9.18303 q^{64} +9.25963 q^{65} -3.97172 q^{67} +5.02021 q^{68} -12.7314 q^{70} -14.2688 q^{71} -13.2263 q^{73} -19.7120 q^{74} +2.15429 q^{76} +1.92338 q^{77} -1.87656 q^{79} -11.9109 q^{80} +13.5569 q^{82} +10.9619 q^{83} +7.56799 q^{85} -16.2937 q^{86} -0.314472 q^{88} -15.0195 q^{89} +5.48397 q^{91} +5.91682 q^{92} +7.07220 q^{94} +3.24760 q^{95} -7.57248 q^{97} +6.72736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 15 q^{4} - 2 q^{5} + 10 q^{7} + 9 q^{8} - 6 q^{10} + 7 q^{11} - 4 q^{13} - 6 q^{14} + 27 q^{16} - 2 q^{17} + 7 q^{19} + 4 q^{20} + q^{22} - 10 q^{23} + 9 q^{25} + 8 q^{26} + 26 q^{28} + 18 q^{29}+ \cdots - 19 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.03821 −1.44123 −0.720615 − 0.693335i \(-0.756140\pi\)
−0.720615 + 0.693335i \(0.756140\pi\)
\(3\) 0 0
\(4\) 2.15429 1.07714
\(5\) 3.24760 1.45237 0.726185 − 0.687499i \(-0.241292\pi\)
0.726185 + 0.687499i \(0.241292\pi\)
\(6\) 0 0
\(7\) 1.92338 0.726967 0.363484 − 0.931601i \(-0.381587\pi\)
0.363484 + 0.931601i \(0.381587\pi\)
\(8\) −0.314472 −0.111183
\(9\) 0 0
\(10\) −6.61928 −2.09320
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.85122 0.790787 0.395394 − 0.918512i \(-0.370608\pi\)
0.395394 + 0.918512i \(0.370608\pi\)
\(14\) −3.92024 −1.04773
\(15\) 0 0
\(16\) −3.66762 −0.916905
\(17\) 2.33033 0.565189 0.282594 − 0.959239i \(-0.408805\pi\)
0.282594 + 0.959239i \(0.408805\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 6.99626 1.56441
\(21\) 0 0
\(22\) −2.03821 −0.434547
\(23\) 2.74653 0.572691 0.286346 − 0.958126i \(-0.407559\pi\)
0.286346 + 0.958126i \(0.407559\pi\)
\(24\) 0 0
\(25\) 5.54689 1.10938
\(26\) −5.81138 −1.13971
\(27\) 0 0
\(28\) 4.14350 0.783049
\(29\) 0.972965 0.180675 0.0903376 − 0.995911i \(-0.471205\pi\)
0.0903376 + 0.995911i \(0.471205\pi\)
\(30\) 0 0
\(31\) −0.00551178 −0.000989945 0 −0.000494973 − 1.00000i \(-0.500158\pi\)
−0.000494973 1.00000i \(0.500158\pi\)
\(32\) 8.10431 1.43265
\(33\) 0 0
\(34\) −4.74970 −0.814567
\(35\) 6.24635 1.05583
\(36\) 0 0
\(37\) 9.67124 1.58994 0.794971 − 0.606647i \(-0.207486\pi\)
0.794971 + 0.606647i \(0.207486\pi\)
\(38\) −2.03821 −0.330641
\(39\) 0 0
\(40\) −1.02128 −0.161478
\(41\) −6.65137 −1.03877 −0.519385 − 0.854540i \(-0.673839\pi\)
−0.519385 + 0.854540i \(0.673839\pi\)
\(42\) 0 0
\(43\) 7.99413 1.21909 0.609547 − 0.792750i \(-0.291352\pi\)
0.609547 + 0.792750i \(0.291352\pi\)
\(44\) 2.15429 0.324771
\(45\) 0 0
\(46\) −5.59800 −0.825380
\(47\) −3.46982 −0.506125 −0.253062 − 0.967450i \(-0.581438\pi\)
−0.253062 + 0.967450i \(0.581438\pi\)
\(48\) 0 0
\(49\) −3.30063 −0.471518
\(50\) −11.3057 −1.59887
\(51\) 0 0
\(52\) 6.14236 0.851792
\(53\) −10.5493 −1.44905 −0.724526 − 0.689247i \(-0.757942\pi\)
−0.724526 + 0.689247i \(0.757942\pi\)
\(54\) 0 0
\(55\) 3.24760 0.437906
\(56\) −0.604847 −0.0808261
\(57\) 0 0
\(58\) −1.98311 −0.260394
\(59\) 13.7814 1.79419 0.897096 − 0.441836i \(-0.145673\pi\)
0.897096 + 0.441836i \(0.145673\pi\)
\(60\) 0 0
\(61\) 3.74608 0.479636 0.239818 − 0.970818i \(-0.422912\pi\)
0.239818 + 0.970818i \(0.422912\pi\)
\(62\) 0.0112342 0.00142674
\(63\) 0 0
\(64\) −9.18303 −1.14788
\(65\) 9.25963 1.14852
\(66\) 0 0
\(67\) −3.97172 −0.485223 −0.242612 − 0.970124i \(-0.578004\pi\)
−0.242612 + 0.970124i \(0.578004\pi\)
\(68\) 5.02021 0.608790
\(69\) 0 0
\(70\) −12.7314 −1.52169
\(71\) −14.2688 −1.69339 −0.846695 − 0.532078i \(-0.821411\pi\)
−0.846695 + 0.532078i \(0.821411\pi\)
\(72\) 0 0
\(73\) −13.2263 −1.54803 −0.774013 − 0.633170i \(-0.781753\pi\)
−0.774013 + 0.633170i \(0.781753\pi\)
\(74\) −19.7120 −2.29147
\(75\) 0 0
\(76\) 2.15429 0.247114
\(77\) 1.92338 0.219189
\(78\) 0 0
\(79\) −1.87656 −0.211130 −0.105565 − 0.994412i \(-0.533665\pi\)
−0.105565 + 0.994412i \(0.533665\pi\)
\(80\) −11.9109 −1.33168
\(81\) 0 0
\(82\) 13.5569 1.49711
\(83\) 10.9619 1.20322 0.601612 − 0.798789i \(-0.294526\pi\)
0.601612 + 0.798789i \(0.294526\pi\)
\(84\) 0 0
\(85\) 7.56799 0.820863
\(86\) −16.2937 −1.75699
\(87\) 0 0
\(88\) −0.314472 −0.0335228
\(89\) −15.0195 −1.59207 −0.796034 − 0.605253i \(-0.793072\pi\)
−0.796034 + 0.605253i \(0.793072\pi\)
\(90\) 0 0
\(91\) 5.48397 0.574876
\(92\) 5.91682 0.616871
\(93\) 0 0
\(94\) 7.07220 0.729442
\(95\) 3.24760 0.333196
\(96\) 0 0
\(97\) −7.57248 −0.768869 −0.384434 − 0.923152i \(-0.625604\pi\)
−0.384434 + 0.923152i \(0.625604\pi\)
\(98\) 6.72736 0.679566
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1881.2.a.p.1.2 7
3.2 odd 2 209.2.a.d.1.6 ✓ 7
12.11 even 2 3344.2.a.ba.1.2 7
15.14 odd 2 5225.2.a.n.1.2 7
33.32 even 2 2299.2.a.q.1.2 7
57.56 even 2 3971.2.a.i.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.6 ✓ 7 3.2 odd 2
1881.2.a.p.1.2 7 1.1 even 1 trivial
2299.2.a.q.1.2 7 33.32 even 2
3344.2.a.ba.1.2 7 12.11 even 2
3971.2.a.i.1.2 7 57.56 even 2
5225.2.a.n.1.2 7 15.14 odd 2