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Properties

Label 471.2.q.a
Level $471$
Weight $2$
Character orbit 471.q
Analytic conductor $3.761$
Analytic rank $0$
Dimension $312$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [471,2,Mod(19,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([0, 62]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("471.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.q (of order \(39\), degree \(24\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.76095393520\)
Analytic rank: \(0\)
Dimension: \(312\)
Relative dimension: \(13\) over \(\Q(\zeta_{39})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{39}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 2 q^{2} - 13 q^{3} - 30 q^{4} + 4 q^{5} - q^{6} + 4 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 312 q - 2 q^{2} - 13 q^{3} - 30 q^{4} + 4 q^{5} - q^{6} + 4 q^{7} + 13 q^{9} - 3 q^{10} - 4 q^{11} + 154 q^{12} + 21 q^{13} + 14 q^{14} + 22 q^{15} - 38 q^{16} - 7 q^{17} + q^{18} + 3 q^{19} + 7 q^{20} + 2 q^{21} + 14 q^{22} + 15 q^{23} - 83 q^{25} - 128 q^{26} + 26 q^{27} - 40 q^{28} + 43 q^{29} + 3 q^{30} - 24 q^{31} - 32 q^{32} - 35 q^{33} - 43 q^{34} + 63 q^{35} + 15 q^{36} - 20 q^{37} - 97 q^{38} - 10 q^{39} + 49 q^{40} - 94 q^{41} + 7 q^{42} + 15 q^{43} + 12 q^{44} + 5 q^{45} + 88 q^{46} - 15 q^{47} - 19 q^{48} - 36 q^{49} - 78 q^{50} - 97 q^{51} - 17 q^{52} - 40 q^{53} + 2 q^{54} - 67 q^{55} + 82 q^{56} - 3 q^{57} - 250 q^{58} - 34 q^{59} - 7 q^{60} - 34 q^{61} - 74 q^{62} + 24 q^{63} + 36 q^{65} - 40 q^{66} + 36 q^{67} + 164 q^{68} + q^{69} + 25 q^{70} - 41 q^{71} - 33 q^{73} + 67 q^{74} + 42 q^{75} - 370 q^{76} - 26 q^{77} - 61 q^{78} + 33 q^{79} + 64 q^{80} + 13 q^{81} - 24 q^{82} - 172 q^{83} - 85 q^{84} + 54 q^{85} - 104 q^{86} - 11 q^{87} + 289 q^{88} + 74 q^{89} + 6 q^{90} + 98 q^{91} + 174 q^{92} + 30 q^{93} + 16 q^{94} + 54 q^{95} + 114 q^{96} - 92 q^{97} - 142 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.59192 − 0.638851i 0.845190 + 0.534466i 4.53901 + 2.38225i 1.57482 − 1.18340i −1.84922 − 1.92524i −3.23692 + 1.69887i −6.24655 − 5.53396i 0.428693 + 0.903450i −4.83781 + 2.06120i
19.2 −2.34426 − 0.577808i 0.845190 + 0.534466i 3.39078 + 1.77962i −0.779207 + 0.585536i −1.67253 − 1.74128i 2.58662 − 1.35756i −3.30615 − 2.92899i 0.428693 + 0.903450i 2.16499 − 0.922417i
19.3 −1.50072 − 0.369893i 0.845190 + 0.534466i 0.344413 + 0.180762i 3.18589 − 2.39404i −1.07069 − 1.11471i 3.69330 − 1.93839i 1.86384 + 1.65121i 0.428693 + 0.903450i −5.66666 + 2.41434i
19.4 −1.44152 − 0.355302i 0.845190 + 0.534466i 0.180823 + 0.0949032i 1.68491 − 1.26612i −1.02846 − 1.07074i −4.31474 + 2.26455i 1.99563 + 1.76797i 0.428693 + 0.903450i −2.87868 + 1.22649i
19.5 −1.11565 − 0.274982i 0.845190 + 0.534466i −0.601859 − 0.315880i −0.231662 + 0.174082i −0.795965 − 0.828688i −0.525028 + 0.275556i 2.30473 + 2.04181i 0.428693 + 0.903450i 0.306322 − 0.130512i
19.6 −0.438971 − 0.108197i 0.845190 + 0.534466i −1.58992 − 0.834456i −3.37704 + 2.53768i −0.313186 − 0.326062i 1.55737 − 0.817372i 1.28446 + 1.13793i 0.428693 + 0.903450i 1.75699 − 0.748584i
19.7 −0.0298113 − 0.00734782i 0.845190 + 0.534466i −1.77008 − 0.929008i −0.439412 + 0.330197i −0.0212690 − 0.0221434i −1.93630 + 1.01625i 0.0919058 + 0.0814214i 0.428693 + 0.903450i 0.0155257 − 0.00661487i
19.8 0.261260 + 0.0643949i 0.845190 + 0.534466i −1.70680 − 0.895799i 1.41999 − 1.06705i 0.186398 + 0.194061i 0.229128 − 0.120256i −0.791053 − 0.700811i 0.428693 + 0.903450i 0.439700 − 0.187339i
19.9 0.973981 + 0.240065i 0.845190 + 0.534466i −0.879904 − 0.461809i 1.13063 − 0.849613i 0.694893 + 0.723460i 4.16689 − 2.18695i −2.24785 − 1.99142i 0.428693 + 0.903450i 1.30517 − 0.556082i
19.10 1.07093 + 0.263961i 0.845190 + 0.534466i −0.693690 − 0.364077i −2.43903 + 1.83281i 0.764064 + 0.795475i −2.27220 + 1.19255i −2.29798 − 2.03584i 0.428693 + 0.903450i −3.09583 + 1.31901i
19.11 1.90593 + 0.469770i 0.845190 + 0.534466i 1.64099 + 0.861257i 2.52921 − 1.90058i 1.35980 + 1.41570i −1.65168 + 0.866869i −0.215594 − 0.191000i 0.428693 + 0.903450i 5.71335 − 2.43423i
19.12 1.90789 + 0.470253i 0.845190 + 0.534466i 1.64801 + 0.864940i −1.10668 + 0.831620i 1.36120 + 1.41716i 2.68594 − 1.40969i −0.204157 − 0.180867i 0.428693 + 0.903450i −2.50251 + 1.06622i
19.13 2.37190 + 0.584621i 0.845190 + 0.534466i 3.51321 + 1.84388i 0.460331 − 0.345916i 1.69225 + 1.76181i 0.0791391 − 0.0415354i 3.59796 + 3.18752i 0.428693 + 0.903450i 1.29409 − 0.551360i
37.1 −2.16827 + 1.13800i 0.996757 − 0.0804666i 2.27024 − 3.28901i 2.83776 − 2.95442i −2.06967 + 1.30878i 0.347375 + 0.503260i −0.589282 + 4.85318i 0.987050 − 0.160411i −2.79091 + 9.63534i
37.2 −2.02361 + 1.06207i 0.996757 − 0.0804666i 1.83088 − 2.65248i −0.358199 + 0.372924i −1.93159 + 1.22146i −1.05990 − 1.53553i −0.336906 + 2.77467i 0.987050 − 0.160411i 0.328782 − 1.13509i
37.3 −1.65962 + 0.871036i 0.996757 − 0.0804666i 0.859508 − 1.24521i 0.123581 − 0.128661i −1.58415 + 1.00176i 2.61423 + 3.78737i 0.110014 − 0.906050i 0.987050 − 0.160411i −0.0930289 + 0.321173i
37.4 −1.29545 + 0.679907i 0.996757 − 0.0804666i 0.0797997 − 0.115610i −0.481131 + 0.500910i −1.23654 + 0.781943i −0.737427 − 1.06835i 0.327926 − 2.70072i 0.987050 − 0.160411i 0.282711 − 0.976031i
37.5 −0.536361 + 0.281504i 0.996757 − 0.0804666i −0.927691 + 1.34399i 0.547692 − 0.570208i −0.511970 + 0.323751i 2.07705 + 3.00912i 0.265267 − 2.18467i 0.987050 − 0.160411i −0.133245 + 0.460015i
37.6 −0.477414 + 0.250566i 0.996757 − 0.0804666i −0.970989 + 1.40672i −2.17892 + 2.26849i −0.455704 + 0.288170i −1.99492 − 2.89014i 0.241067 − 1.98537i 0.987050 − 0.160411i 0.471838 − 1.62897i
37.7 −0.377261 + 0.198002i 0.996757 − 0.0804666i −1.03301 + 1.49657i 2.39889 − 2.49750i −0.360105 + 0.227717i −1.52702 − 2.21227i 0.196103 − 1.61505i 0.987050 − 0.160411i −0.410495 + 1.41719i
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
157.i even 39 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.q.a ✓ 312
157.i even 39 1 inner 471.2.q.a ✓ 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.q.a ✓ 312 1.a even 1 1 trivial
471.2.q.a ✓ 312 157.i even 39 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{312} + 2 T_{2}^{311} + 43 T_{2}^{310} + 86 T_{2}^{309} + 1035 T_{2}^{308} + 2078 T_{2}^{307} + \cdots + 69\!\cdots\!09 \) acting on \(S_{2}^{\mathrm{new}}(471, [\chi])\). Copy content Toggle raw display