Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3971,2,Mod(1,3971)] 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("3971.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3971, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3971 = 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3971.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.7085946427\)
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: \(\Z[a_1, a_2, a_3]\)
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 \(\chi\) \(=\) 3971.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.03821 q^{2} -1.87275 q^{3} +2.15429 q^{4} -3.24760 q^{5} +3.81704 q^{6} +1.92338 q^{7} -0.314472 q^{8} +0.507178 q^{9} +6.61928 q^{10} -1.00000 q^{11} -4.03444 q^{12} -2.85122 q^{13} -3.92024 q^{14} +6.08193 q^{15} -3.66762 q^{16} -2.33033 q^{17} -1.03373 q^{18} -6.99626 q^{20} -3.60199 q^{21} +2.03821 q^{22} -2.74653 q^{23} +0.588926 q^{24} +5.54689 q^{25} +5.81138 q^{26} +4.66842 q^{27} +4.14350 q^{28} +0.972965 q^{29} -12.3962 q^{30} +0.00551178 q^{31} +8.10431 q^{32} +1.87275 q^{33} +4.74970 q^{34} -6.24635 q^{35} +1.09261 q^{36} -9.67124 q^{37} +5.33962 q^{39} +1.02128 q^{40} -6.65137 q^{41} +7.34161 q^{42} +7.99413 q^{43} -2.15429 q^{44} -1.64711 q^{45} +5.59800 q^{46} +3.46982 q^{47} +6.86852 q^{48} -3.30063 q^{49} -11.3057 q^{50} +4.36412 q^{51} -6.14236 q^{52} -10.5493 q^{53} -9.51521 q^{54} +3.24760 q^{55} -0.604847 q^{56} -1.98311 q^{58} +13.7814 q^{59} +13.1022 q^{60} +3.74608 q^{61} -0.0112342 q^{62} +0.975494 q^{63} -9.18303 q^{64} +9.25963 q^{65} -3.81704 q^{66} +3.97172 q^{67} -5.02021 q^{68} +5.14356 q^{69} +12.7314 q^{70} -14.2688 q^{71} -0.159493 q^{72} -13.2263 q^{73} +19.7120 q^{74} -10.3879 q^{75} -1.92338 q^{77} -10.8832 q^{78} +1.87656 q^{79} +11.9109 q^{80} -10.2643 q^{81} +13.5569 q^{82} -10.9619 q^{83} -7.75973 q^{84} +7.56799 q^{85} -16.2937 q^{86} -1.82212 q^{87} +0.314472 q^{88} -15.0195 q^{89} +3.35715 q^{90} -5.48397 q^{91} -5.91682 q^{92} -0.0103222 q^{93} -7.07220 q^{94} -15.1773 q^{96} +7.57248 q^{97} +6.72736 q^{98} -0.507178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} - 2 q^{3} + 15 q^{4} + 2 q^{5} - 2 q^{6} + 10 q^{7} + 9 q^{8} + 11 q^{9} + 6 q^{10} - 7 q^{11} + 16 q^{12} + 4 q^{13} - 6 q^{14} - 12 q^{15} + 27 q^{16} + 2 q^{17} - 9 q^{18} - 4 q^{20} + 14 q^{21}+ \cdots - 11 q^{99}+O(q^{100}) \) 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 \(0.756140\pi\)
\(3\) −1.87275 −1.08123 −0.540615 − 0.841270i \(-0.681809\pi\)
−0.540615 + 0.841270i \(0.681809\pi\)
\(4\) 2.15429 1.07714
\(5\) −3.24760 −1.45237 −0.726185 − 0.687499i \(-0.758708\pi\)
−0.726185 + 0.687499i \(0.758708\pi\)
\(6\) 3.81704 1.55830
\(7\) 1.92338 0.726967 0.363484 − 0.931601i \(-0.381587\pi\)
0.363484 + 0.931601i \(0.381587\pi\)
\(8\) −0.314472 −0.111183
\(9\) 0.507178 0.169059
\(10\) 6.61928 2.09320
\(11\) −1.00000 −0.301511
\(12\) −4.03444 −1.16464
\(13\) −2.85122 −0.790787 −0.395394 − 0.918512i \(-0.629392\pi\)
−0.395394 + 0.918512i \(0.629392\pi\)
\(14\) −3.92024 −1.04773
\(15\) 6.08193 1.57035
\(16\) −3.66762 −0.916905
\(17\) −2.33033 −0.565189 −0.282594 − 0.959239i \(-0.591195\pi\)
−0.282594 + 0.959239i \(0.591195\pi\)
\(18\) −1.03373 −0.243653
\(19\) 0 0
\(20\) −6.99626 −1.56441
\(21\) −3.60199 −0.786019
\(22\) 2.03821 0.434547
\(23\) −2.74653 −0.572691 −0.286346 − 0.958126i \(-0.592441\pi\)
−0.286346 + 0.958126i \(0.592441\pi\)
\(24\) 0.588926 0.120214
\(25\) 5.54689 1.10938
\(26\) 5.81138 1.13971
\(27\) 4.66842 0.898438
\(28\) 4.14350 0.783049
\(29\) 0.972965 0.180675 0.0903376 − 0.995911i \(-0.471205\pi\)
0.0903376 + 0.995911i \(0.471205\pi\)
\(30\) −12.3962 −2.26323
\(31\) 0.00551178 0.000989945 0 0.000494973 − 1.00000i \(-0.499842\pi\)
0.000494973 1.00000i \(0.499842\pi\)
\(32\) 8.10431 1.43265
\(33\) 1.87275 0.326003
\(34\) 4.74970 0.814567
\(35\) −6.24635 −1.05583
\(36\) 1.09261 0.182101
\(37\) −9.67124 −1.58994 −0.794971 − 0.606647i \(-0.792514\pi\)
−0.794971 + 0.606647i \(0.792514\pi\)
\(38\) 0 0
\(39\) 5.33962 0.855023
\(40\) 1.02128 0.161478
\(41\) −6.65137 −1.03877 −0.519385 − 0.854540i \(-0.673839\pi\)
−0.519385 + 0.854540i \(0.673839\pi\)
\(42\) 7.34161 1.13283
\(43\) 7.99413 1.21909 0.609547 − 0.792750i \(-0.291352\pi\)
0.609547 + 0.792750i \(0.291352\pi\)
\(44\) −2.15429 −0.324771
\(45\) −1.64711 −0.245537
\(46\) 5.59800 0.825380
\(47\) 3.46982 0.506125 0.253062 − 0.967450i \(-0.418562\pi\)
0.253062 + 0.967450i \(0.418562\pi\)
\(48\) 6.86852 0.991385
\(49\) −3.30063 −0.471518
\(50\) −11.3057 −1.59887
\(51\) 4.36412 0.611100
\(52\) −6.14236 −0.851792
\(53\) −10.5493 −1.44905 −0.724526 − 0.689247i \(-0.757942\pi\)
−0.724526 + 0.689247i \(0.757942\pi\)
\(54\) −9.51521 −1.29486
\(55\) 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 \(0.145673\pi\)
\(60\) 13.1022 1.69149
\(61\) 3.74608 0.479636 0.239818 − 0.970818i \(-0.422912\pi\)
0.239818 + 0.970818i \(0.422912\pi\)
\(62\) −0.0112342 −0.00142674
\(63\) 0.975494 0.122901
\(64\) −9.18303 −1.14788
\(65\) 9.25963 1.14852
\(66\) −3.81704 −0.469846
\(67\) 3.97172 0.485223 0.242612 − 0.970124i \(-0.421996\pi\)
0.242612 + 0.970124i \(0.421996\pi\)
\(68\) −5.02021 −0.608790
\(69\) 5.14356 0.619211
\(70\) 12.7314 1.52169
\(71\) −14.2688 −1.69339 −0.846695 − 0.532078i \(-0.821411\pi\)
−0.846695 + 0.532078i \(0.821411\pi\)
\(72\) −0.159493 −0.0187965
\(73\) −13.2263 −1.54803 −0.774013 − 0.633170i \(-0.781753\pi\)
−0.774013 + 0.633170i \(0.781753\pi\)
\(74\) 19.7120 2.29147
\(75\) −10.3879 −1.19949
\(76\) 0 0
\(77\) −1.92338 −0.219189
\(78\) −10.8832 −1.23229
\(79\) 1.87656 0.211130 0.105565 − 0.994412i \(-0.466335\pi\)
0.105565 + 0.994412i \(0.466335\pi\)
\(80\) 11.9109 1.33168
\(81\) −10.2643 −1.14048
\(82\) 13.5569 1.49711
\(83\) −10.9619 −1.20322 −0.601612 − 0.798789i \(-0.705474\pi\)
−0.601612 + 0.798789i \(0.705474\pi\)
\(84\) −7.75973 −0.846656
\(85\) 7.56799 0.820863
\(86\) −16.2937 −1.75699
\(87\) −1.82212 −0.195351
\(88\) 0.314472 0.0335228
\(89\) −15.0195 −1.59207 −0.796034 − 0.605253i \(-0.793072\pi\)
−0.796034 + 0.605253i \(0.793072\pi\)
\(90\) 3.35715 0.353875
\(91\) −5.48397 −0.574876
\(92\) −5.91682 −0.616871
\(93\) −0.0103222 −0.00107036
\(94\) −7.07220 −0.729442
\(95\) 0 0
\(96\) −15.1773 −1.54903
\(97\) 7.57248 0.768869 0.384434 − 0.923152i \(-0.374396\pi\)
0.384434 + 0.923152i \(0.374396\pi\)
\(98\) 6.72736 0.679566
\(99\) −0.507178 −0.0509733
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 3971.2.a.i.1.2 7
19.18 odd 2 209.2.a.d.1.6 ✓ 7
57.56 even 2 1881.2.a.p.1.2 7
76.75 even 2 3344.2.a.ba.1.2 7
95.94 odd 2 5225.2.a.n.1.2 7
209.208 even 2 2299.2.a.q.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.6 ✓ 7 19.18 odd 2
1881.2.a.p.1.2 7 57.56 even 2
2299.2.a.q.1.2 7 209.208 even 2
3344.2.a.ba.1.2 7 76.75 even 2
3971.2.a.i.1.2 7 1.1 even 1 trivial
5225.2.a.n.1.2 7 95.94 odd 2