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Properties

Label 841.2.a.k.1.11
Level $841$
Weight $2$
Character 841.1
Self dual yes
Analytic conductor $6.715$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [841,2,Mod(1,841)] 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("841.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(841, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 841 = 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 841.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.71541880999\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.12.32268092290502656.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 78x^{8} - 169x^{6} + 148x^{4} - 36x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-0.572821\) of defining polynomial
Character \(\chi\) \(=\) 841.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.26775 q^{2} +2.84057 q^{3} +3.14268 q^{4} +0.0578828 q^{5} +6.44169 q^{6} -1.56196 q^{7} +2.59131 q^{8} +5.06883 q^{9} +0.131264 q^{10} -3.97289 q^{11} +8.92699 q^{12} +0.404986 q^{13} -3.54213 q^{14} +0.164420 q^{15} -0.408929 q^{16} +5.16843 q^{17} +11.4948 q^{18} -3.45088 q^{19} +0.181907 q^{20} -4.43685 q^{21} -9.00950 q^{22} +0.230185 q^{23} +7.36078 q^{24} -4.99665 q^{25} +0.918405 q^{26} +5.87665 q^{27} -4.90874 q^{28} +0.372863 q^{30} +4.31832 q^{31} -6.10996 q^{32} -11.2853 q^{33} +11.7207 q^{34} -0.0904106 q^{35} +15.9297 q^{36} +5.37070 q^{37} -7.82573 q^{38} +1.15039 q^{39} +0.149992 q^{40} -1.46294 q^{41} -10.0617 q^{42} +6.48316 q^{43} -12.4855 q^{44} +0.293398 q^{45} +0.522001 q^{46} -9.57399 q^{47} -1.16159 q^{48} -4.56028 q^{49} -11.3311 q^{50} +14.6813 q^{51} +1.27274 q^{52} -1.78879 q^{53} +13.3268 q^{54} -0.229962 q^{55} -4.04752 q^{56} -9.80247 q^{57} -1.05216 q^{59} +0.516719 q^{60} -3.18937 q^{61} +9.79287 q^{62} -7.91731 q^{63} -13.0380 q^{64} +0.0234417 q^{65} -25.5921 q^{66} +10.8726 q^{67} +16.2427 q^{68} +0.653856 q^{69} -0.205028 q^{70} +13.4529 q^{71} +13.1349 q^{72} -7.01563 q^{73} +12.1794 q^{74} -14.1933 q^{75} -10.8450 q^{76} +6.20549 q^{77} +2.60879 q^{78} +4.78464 q^{79} -0.0236700 q^{80} +1.48654 q^{81} -3.31758 q^{82} +8.42345 q^{83} -13.9436 q^{84} +0.299163 q^{85} +14.7022 q^{86} -10.2950 q^{88} -1.13267 q^{89} +0.665353 q^{90} -0.632571 q^{91} +0.723397 q^{92} +12.2665 q^{93} -21.7114 q^{94} -0.199747 q^{95} -17.3558 q^{96} -17.2131 q^{97} -10.3416 q^{98} -20.1379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4} + 8 q^{5} + 24 q^{6} + 10 q^{7} + 10 q^{9} + 12 q^{13} + 16 q^{16} + 24 q^{20} - 38 q^{22} + 30 q^{23} + 10 q^{24} - 8 q^{25} - 12 q^{28} + 2 q^{30} - 4 q^{33} + 6 q^{34} + 44 q^{35} + 16 q^{36}+ \cdots - 58 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.26775 1.60354 0.801770 − 0.597633i \(-0.203892\pi\)
0.801770 + 0.597633i \(0.203892\pi\)
\(3\) 2.84057 1.64000 0.820001 − 0.572361i \(-0.193973\pi\)
0.820001 + 0.572361i \(0.193973\pi\)
\(4\) 3.14268 1.57134
\(5\) 0.0578828 0.0258860 0.0129430 − 0.999916i \(-0.495880\pi\)
0.0129430 + 0.999916i \(0.495880\pi\)
\(6\) 6.44169 2.62981
\(7\) −1.56196 −0.590365 −0.295183 − 0.955441i \(-0.595381\pi\)
−0.295183 + 0.955441i \(0.595381\pi\)
\(8\) 2.59131 0.916165
\(9\) 5.06883 1.68961
\(10\) 0.131264 0.0415092
\(11\) −3.97289 −1.19787 −0.598935 − 0.800798i \(-0.704409\pi\)
−0.598935 + 0.800798i \(0.704409\pi\)
\(12\) 8.92699 2.57700
\(13\) 0.404986 0.112323 0.0561614 − 0.998422i \(-0.482114\pi\)
0.0561614 + 0.998422i \(0.482114\pi\)
\(14\) −3.54213 −0.946674
\(15\) 0.164420 0.0424531
\(16\) −0.408929 −0.102232
\(17\) 5.16843 1.25353 0.626764 − 0.779209i \(-0.284379\pi\)
0.626764 + 0.779209i \(0.284379\pi\)
\(18\) 11.4948 2.70936
\(19\) −3.45088 −0.791687 −0.395844 − 0.918318i \(-0.629548\pi\)
−0.395844 + 0.918318i \(0.629548\pi\)
\(20\) 0.181907 0.0406757
\(21\) −4.43685 −0.968201
\(22\) −9.00950 −1.92083
\(23\) 0.230185 0.0479968 0.0239984 − 0.999712i \(-0.492360\pi\)
0.0239984 + 0.999712i \(0.492360\pi\)
\(24\) 7.36078 1.50251
\(25\) −4.99665 −0.999330
\(26\) 0.918405 0.180114
\(27\) 5.87665 1.13096
\(28\) −4.90874 −0.927664
\(29\) 0 0
\(30\) 0.372863 0.0680752
\(31\) 4.31832 0.775594 0.387797 − 0.921745i \(-0.373236\pi\)
0.387797 + 0.921745i \(0.373236\pi\)
\(32\) −6.10996 −1.08010
\(33\) −11.2853 −1.96451
\(34\) 11.7207 2.01008
\(35\) −0.0904106 −0.0152822
\(36\) 15.9297 2.65495
\(37\) 5.37070 0.882937 0.441469 − 0.897277i \(-0.354458\pi\)
0.441469 + 0.897277i \(0.354458\pi\)
\(38\) −7.82573 −1.26950
\(39\) 1.15039 0.184210
\(40\) 0.149992 0.0237158
\(41\) −1.46294 −0.228473 −0.114237 − 0.993454i \(-0.536442\pi\)
−0.114237 + 0.993454i \(0.536442\pi\)
\(42\) −10.0617 −1.55255
\(43\) 6.48316 0.988672 0.494336 − 0.869271i \(-0.335411\pi\)
0.494336 + 0.869271i \(0.335411\pi\)
\(44\) −12.4855 −1.88226
\(45\) 0.293398 0.0437372
\(46\) 0.522001 0.0769648
\(47\) −9.57399 −1.39651 −0.698255 − 0.715849i \(-0.746040\pi\)
−0.698255 + 0.715849i \(0.746040\pi\)
\(48\) −1.16159 −0.167661
\(49\) −4.56028 −0.651469
\(50\) −11.3311 −1.60247
\(51\) 14.6813 2.05579
\(52\) 1.27274 0.176497
\(53\) −1.78879 −0.245710 −0.122855 − 0.992425i \(-0.539205\pi\)
−0.122855 + 0.992425i \(0.539205\pi\)
\(54\) 13.3268 1.81354
\(55\) −0.229962 −0.0310080
\(56\) −4.04752 −0.540872
\(57\) −9.80247 −1.29837
\(58\) 0 0
\(59\) −1.05216 −0.136980 −0.0684900 − 0.997652i \(-0.521818\pi\)
−0.0684900 + 0.997652i \(0.521818\pi\)
\(60\) 0.516719 0.0667082
\(61\) −3.18937 −0.408357 −0.204178 − 0.978934i \(-0.565452\pi\)
−0.204178 + 0.978934i \(0.565452\pi\)
\(62\) 9.79287 1.24370
\(63\) −7.91731 −0.997487
\(64\) −13.0380 −1.62975
\(65\) 0.0234417 0.00290759
\(66\) −25.5921 −3.15017
\(67\) 10.8726 1.32830 0.664148 − 0.747601i \(-0.268795\pi\)
0.664148 + 0.747601i \(0.268795\pi\)
\(68\) 16.2427 1.96972
\(69\) 0.653856 0.0787150
\(70\) −0.205028 −0.0245056
\(71\) 13.4529 1.59656 0.798281 − 0.602285i \(-0.205743\pi\)
0.798281 + 0.602285i \(0.205743\pi\)
\(72\) 13.1349 1.54796
\(73\) −7.01563 −0.821118 −0.410559 − 0.911834i \(-0.634666\pi\)
−0.410559 + 0.911834i \(0.634666\pi\)
\(74\) 12.1794 1.41582
\(75\) −14.1933 −1.63890
\(76\) −10.8450 −1.24401
\(77\) 6.20549 0.707181
\(78\) 2.60879 0.295388
\(79\) 4.78464 0.538314 0.269157 − 0.963096i \(-0.413255\pi\)
0.269157 + 0.963096i \(0.413255\pi\)
\(80\) −0.0236700 −0.00264638
\(81\) 1.48654 0.165171
\(82\) −3.31758 −0.366366
\(83\) 8.42345 0.924594 0.462297 − 0.886725i \(-0.347025\pi\)
0.462297 + 0.886725i \(0.347025\pi\)
\(84\) −13.9436 −1.52137
\(85\) 0.299163 0.0324488
\(86\) 14.7022 1.58537
\(87\) 0 0
\(88\) −10.2950 −1.09745
\(89\) −1.13267 −0.120063 −0.0600314 − 0.998196i \(-0.519120\pi\)
−0.0600314 + 0.998196i \(0.519120\pi\)
\(90\) 0.665353 0.0701343
\(91\) −0.632571 −0.0663115
\(92\) 0.723397 0.0754193
\(93\) 12.2665 1.27198
\(94\) −21.7114 −2.23936
\(95\) −0.199747 −0.0204936
\(96\) −17.3558 −1.77136
\(97\) −17.2131 −1.74773 −0.873864 − 0.486170i \(-0.838393\pi\)
−0.873864 + 0.486170i \(0.838393\pi\)
\(98\) −10.3416 −1.04466
\(99\) −20.1379 −2.02393
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 841.2.a.k.1.11 12
3.2 odd 2 7569.2.a.bp.1.2 12
29.2 odd 28 841.2.e.i.236.2 12
29.3 odd 28 841.2.e.h.270.2 12
29.4 even 14 841.2.d.l.190.4 24
29.5 even 14 841.2.d.m.605.4 24
29.6 even 14 841.2.d.m.645.4 24
29.7 even 7 841.2.d.l.571.1 24
29.8 odd 28 841.2.e.e.267.2 12
29.9 even 14 841.2.d.k.574.1 24
29.10 odd 28 841.2.e.h.651.2 12
29.11 odd 28 841.2.e.e.63.2 12
29.12 odd 4 841.2.b.e.840.11 12
29.13 even 14 841.2.d.k.778.1 24
29.14 odd 28 29.2.e.a.22.1 yes 12
29.15 odd 28 841.2.e.i.196.2 12
29.16 even 7 841.2.d.k.778.4 24
29.17 odd 4 841.2.b.e.840.2 12
29.18 odd 28 841.2.e.f.63.1 12
29.19 odd 28 841.2.e.a.651.1 12
29.20 even 7 841.2.d.k.574.4 24
29.21 odd 28 841.2.e.f.267.1 12
29.22 even 14 841.2.d.l.571.4 24
29.23 even 7 841.2.d.m.645.1 24
29.24 even 7 841.2.d.m.605.1 24
29.25 even 7 841.2.d.l.190.1 24
29.26 odd 28 841.2.e.a.270.1 12
29.27 odd 28 29.2.e.a.4.1 ✓ 12
29.28 even 2 inner 841.2.a.k.1.2 12
87.14 even 28 261.2.o.a.109.2 12
87.56 even 28 261.2.o.a.91.2 12
87.86 odd 2 7569.2.a.bp.1.11 12
116.27 even 28 464.2.y.d.33.2 12
116.43 even 28 464.2.y.d.225.2 12
145.14 odd 28 725.2.q.a.51.2 12
145.27 even 28 725.2.p.a.149.4 24
145.43 even 28 725.2.p.a.399.4 24
145.72 even 28 725.2.p.a.399.1 24
145.114 odd 28 725.2.q.a.526.2 12
145.143 even 28 725.2.p.a.149.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.2.e.a.4.1 ✓ 12 29.27 odd 28
29.2.e.a.22.1 yes 12 29.14 odd 28
261.2.o.a.91.2 12 87.56 even 28
261.2.o.a.109.2 12 87.14 even 28
464.2.y.d.33.2 12 116.27 even 28
464.2.y.d.225.2 12 116.43 even 28
725.2.p.a.149.1 24 145.143 even 28
725.2.p.a.149.4 24 145.27 even 28
725.2.p.a.399.1 24 145.72 even 28
725.2.p.a.399.4 24 145.43 even 28
725.2.q.a.51.2 12 145.14 odd 28
725.2.q.a.526.2 12 145.114 odd 28
841.2.a.k.1.2 12 29.28 even 2 inner
841.2.a.k.1.11 12 1.1 even 1 trivial
841.2.b.e.840.2 12 29.17 odd 4
841.2.b.e.840.11 12 29.12 odd 4
841.2.d.k.574.1 24 29.9 even 14
841.2.d.k.574.4 24 29.20 even 7
841.2.d.k.778.1 24 29.13 even 14
841.2.d.k.778.4 24 29.16 even 7
841.2.d.l.190.1 24 29.25 even 7
841.2.d.l.190.4 24 29.4 even 14
841.2.d.l.571.1 24 29.7 even 7
841.2.d.l.571.4 24 29.22 even 14
841.2.d.m.605.1 24 29.24 even 7
841.2.d.m.605.4 24 29.5 even 14
841.2.d.m.645.1 24 29.23 even 7
841.2.d.m.645.4 24 29.6 even 14
841.2.e.a.270.1 12 29.26 odd 28
841.2.e.a.651.1 12 29.19 odd 28
841.2.e.e.63.2 12 29.11 odd 28
841.2.e.e.267.2 12 29.8 odd 28
841.2.e.f.63.1 12 29.18 odd 28
841.2.e.f.267.1 12 29.21 odd 28
841.2.e.h.270.2 12 29.3 odd 28
841.2.e.h.651.2 12 29.10 odd 28
841.2.e.i.196.2 12 29.15 odd 28
841.2.e.i.236.2 12 29.2 odd 28
7569.2.a.bp.1.2 12 3.2 odd 2
7569.2.a.bp.1.11 12 87.86 odd 2