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Properties

Label 28.8.a.b.1.2
Level $28$
Weight $8$
Character 28.1
Self dual yes
Analytic conductor $8.747$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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] = [28,8,Mod(1,28)] 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("28.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(28, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 28.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,14] 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: \(8.74678071356\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1009}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 252 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-15.3824\) of defining polynomial
Character \(\chi\) \(=\) 28.1

$q$-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)
 
\(f(q)\) \(=\) \(q+38.7648 q^{3} -496.412 q^{5} -343.000 q^{7} -684.293 q^{9} +4035.19 q^{11} -14469.7 q^{13} -19243.3 q^{15} -26750.1 q^{17} +36610.0 q^{19} -13296.3 q^{21} -34920.1 q^{23} +168300. q^{25} -111305. q^{27} +35874.3 q^{29} +198598. q^{31} +156423. q^{33} +170269. q^{35} -114413. q^{37} -560915. q^{39} +245990. q^{41} +30224.0 q^{43} +339692. q^{45} -305233. q^{47} +117649. q^{49} -1.03696e6 q^{51} -1.21053e6 q^{53} -2.00312e6 q^{55} +1.41918e6 q^{57} -1.53557e6 q^{59} -45086.4 q^{61} +234713. q^{63} +7.18295e6 q^{65} +3.08501e6 q^{67} -1.35367e6 q^{69} -1.69051e6 q^{71} -3.86932e6 q^{73} +6.52412e6 q^{75} -1.38407e6 q^{77} +2.00910e6 q^{79} -2.81816e6 q^{81} -8.28720e6 q^{83} +1.32791e7 q^{85} +1.39066e6 q^{87} -6.50624e6 q^{89} +4.96311e6 q^{91} +7.69862e6 q^{93} -1.81737e7 q^{95} -1.14882e7 q^{97} -2.76125e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{3} - 294 q^{5} - 686 q^{7} - 2258 q^{9} - 3492 q^{11} - 16170 q^{13} - 24256 q^{15} - 29232 q^{17} - 3206 q^{19} - 4802 q^{21} - 9360 q^{23} + 131146 q^{25} - 18172 q^{27} + 184704 q^{29} + 165060 q^{31}+ \cdots + 9084332 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 38.7648 0.828920 0.414460 − 0.910067i \(-0.363970\pi\)
0.414460 + 0.910067i \(0.363970\pi\)
\(4\) 0 0
\(5\) −496.412 −1.77602 −0.888009 − 0.459825i \(-0.847912\pi\)
−0.888009 + 0.459825i \(0.847912\pi\)
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 0 0
\(9\) −684.293 −0.312891
\(10\) 0 0
\(11\) 4035.19 0.914091 0.457045 − 0.889443i \(-0.348908\pi\)
0.457045 + 0.889443i \(0.348908\pi\)
\(12\) 0 0
\(13\) −14469.7 −1.82666 −0.913331 − 0.407217i \(-0.866499\pi\)
−0.913331 + 0.407217i \(0.866499\pi\)
\(14\) 0 0
\(15\) −19243.3 −1.47218
\(16\) 0 0
\(17\) −26750.1 −1.32055 −0.660275 − 0.751024i \(-0.729560\pi\)
−0.660275 + 0.751024i \(0.729560\pi\)
\(18\) 0 0
\(19\) 36610.0 1.22451 0.612255 − 0.790661i \(-0.290263\pi\)
0.612255 + 0.790661i \(0.290263\pi\)
\(20\) 0 0
\(21\) −13296.3 −0.313302
\(22\) 0 0
\(23\) −34920.1 −0.598449 −0.299225 − 0.954183i \(-0.596728\pi\)
−0.299225 + 0.954183i \(0.596728\pi\)
\(24\) 0 0
\(25\) 168300. 2.15424
\(26\) 0 0
\(27\) −111305. −1.08828
\(28\) 0 0
\(29\) 35874.3 0.273143 0.136571 − 0.990630i \(-0.456392\pi\)
0.136571 + 0.990630i \(0.456392\pi\)
\(30\) 0 0
\(31\) 198598. 1.19732 0.598660 − 0.801004i \(-0.295700\pi\)
0.598660 + 0.801004i \(0.295700\pi\)
\(32\) 0 0
\(33\) 156423. 0.757708
\(34\) 0 0
\(35\) 170269. 0.671272
\(36\) 0 0
\(37\) −114413. −0.371338 −0.185669 − 0.982612i \(-0.559445\pi\)
−0.185669 + 0.982612i \(0.559445\pi\)
\(38\) 0 0
\(39\) −560915. −1.51416
\(40\) 0 0
\(41\) 245990. 0.557409 0.278704 − 0.960377i \(-0.410095\pi\)
0.278704 + 0.960377i \(0.410095\pi\)
\(42\) 0 0
\(43\) 30224.0 0.0579711 0.0289856 − 0.999580i \(-0.490772\pi\)
0.0289856 + 0.999580i \(0.490772\pi\)
\(44\) 0 0
\(45\) 339692. 0.555701
\(46\) 0 0
\(47\) −305233. −0.428833 −0.214417 − 0.976742i \(-0.568785\pi\)
−0.214417 + 0.976742i \(0.568785\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) −1.03696e6 −1.09463
\(52\) 0 0
\(53\) −1.21053e6 −1.11689 −0.558444 − 0.829542i \(-0.688601\pi\)
−0.558444 + 0.829542i \(0.688601\pi\)
\(54\) 0 0
\(55\) −2.00312e6 −1.62344
\(56\) 0 0
\(57\) 1.41918e6 1.01502
\(58\) 0 0
\(59\) −1.53557e6 −0.973391 −0.486696 − 0.873572i \(-0.661798\pi\)
−0.486696 + 0.873572i \(0.661798\pi\)
\(60\) 0 0
\(61\) −45086.4 −0.0254326 −0.0127163 − 0.999919i \(-0.504048\pi\)
−0.0127163 + 0.999919i \(0.504048\pi\)
\(62\) 0 0
\(63\) 234713. 0.118262
\(64\) 0 0
\(65\) 7.18295e6 3.24419
\(66\) 0 0
\(67\) 3.08501e6 1.25312 0.626562 − 0.779371i \(-0.284461\pi\)
0.626562 + 0.779371i \(0.284461\pi\)
\(68\) 0 0
\(69\) −1.35367e6 −0.496067
\(70\) 0 0
\(71\) −1.69051e6 −0.560550 −0.280275 − 0.959920i \(-0.590426\pi\)
−0.280275 + 0.959920i \(0.590426\pi\)
\(72\) 0 0
\(73\) −3.86932e6 −1.16414 −0.582069 − 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(74\) 0 0
\(75\) 6.52412e6 1.78570
\(76\) 0 0
\(77\) −1.38407e6 −0.345494
\(78\) 0 0
\(79\) 2.00910e6 0.458464 0.229232 − 0.973372i \(-0.426378\pi\)
0.229232 + 0.973372i \(0.426378\pi\)
\(80\) 0 0
\(81\) −2.81816e6 −0.589208
\(82\) 0 0
\(83\) −8.28720e6 −1.59087 −0.795435 − 0.606039i \(-0.792758\pi\)
−0.795435 + 0.606039i \(0.792758\pi\)
\(84\) 0 0
\(85\) 1.32791e7 2.34532
\(86\) 0 0
\(87\) 1.39066e6 0.226414
\(88\) 0 0
\(89\) −6.50624e6 −0.978285 −0.489142 − 0.872204i \(-0.662690\pi\)
−0.489142 + 0.872204i \(0.662690\pi\)
\(90\) 0 0
\(91\) 4.96311e6 0.690414
\(92\) 0 0
\(93\) 7.69862e6 0.992482
\(94\) 0 0
\(95\) −1.81737e7 −2.17475
\(96\) 0 0
\(97\) −1.14882e7 −1.27806 −0.639028 − 0.769183i \(-0.720663\pi\)
−0.639028 + 0.769183i \(0.720663\pi\)
\(98\) 0 0
\(99\) −2.76125e6 −0.286011
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 28.8.a.b.1.2 ✓ 2
3.2 odd 2 252.8.a.f.1.2 2
4.3 odd 2 112.8.a.h.1.1 2
7.2 even 3 196.8.e.b.165.1 4
7.3 odd 6 196.8.e.c.177.2 4
7.4 even 3 196.8.e.b.177.1 4
7.5 odd 6 196.8.e.c.165.2 4
7.6 odd 2 196.8.a.a.1.1 2
8.3 odd 2 448.8.a.q.1.2 2
8.5 even 2 448.8.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.8.a.b.1.2 ✓ 2 1.1 even 1 trivial
112.8.a.h.1.1 2 4.3 odd 2
196.8.a.a.1.1 2 7.6 odd 2
196.8.e.b.165.1 4 7.2 even 3
196.8.e.b.177.1 4 7.4 even 3
196.8.e.c.165.2 4 7.5 odd 6
196.8.e.c.177.2 4 7.3 odd 6
252.8.a.f.1.2 2 3.2 odd 2
448.8.a.o.1.1 2 8.5 even 2
448.8.a.q.1.2 2 8.3 odd 2