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Properties

Label 51.7.b.a
Level $51$
Weight $7$
Character orbit 51.b
Analytic conductor $11.733$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [51,7,Mod(35,51)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(51, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("51.35");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 51.b (of order \(2\), degree \(1\), 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: \(11.7327582646\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{3} - 1024 q^{4} + 286 q^{6} + 568 q^{7} - 912 q^{9} - 744 q^{10} + 194 q^{12} - 2312 q^{13} - 6240 q^{15} + 13208 q^{16} + 2936 q^{18} + 7936 q^{19} - 21688 q^{21} + 13176 q^{22} + 18282 q^{24}+ \cdots + 1619864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 − 14.7209i −19.0657 + 19.1180i −152.704 46.7702i 281.434 + 280.664i −79.2824 1305.80i −1.99511 − 728.997i 688.499
35.2 − 14.2771i −8.35589 − 25.6745i −139.834 127.328i −366.556 + 119.298i 222.133 1082.69i −589.358 + 429.066i 1817.87
35.3 − 13.9722i 23.6370 + 13.0496i −131.222 76.5124i 182.331 − 330.261i −548.925 939.239i 388.417 + 616.906i 1069.05
35.4 − 13.8546i 20.4999 − 17.5713i −127.950 − 188.083i −243.444 − 284.019i 207.865 886.004i 111.495 − 720.423i −2605.82
35.5 − 12.5103i 10.7847 + 24.7526i −92.5084 − 102.267i 309.663 − 134.920i 385.286 356.650i −496.380 + 533.899i −1279.40
35.6 − 11.7246i −22.8808 − 14.3342i −73.4666 − 127.663i −168.063 + 268.269i −425.386 110.992i 318.063 + 655.955i −1496.80
35.7 − 9.79650i 26.5144 − 5.09762i −31.9714 157.131i −49.9388 − 259.748i 301.822 − 313.768i 677.029 − 270.321i 1539.33
35.8 − 8.98275i −26.9994 − 0.177085i −16.6898 127.230i −1.59071 + 242.529i 363.441 − 424.976i 728.937 + 9.56240i 1142.88
35.9 − 8.88211i −19.4565 + 18.7201i −14.8920 − 143.656i 166.274 + 172.815i 49.5075 − 436.183i 28.1129 − 728.458i −1275.97
35.10 − 7.50579i 8.40901 − 25.6571i 7.66314 24.7073i −192.577 − 63.1162i −318.597 − 537.888i −587.577 − 431.502i 185.448
35.11 − 6.35177i −6.87539 + 26.1099i 23.6551 214.273i 165.844 + 43.6709i −487.138 − 556.764i −634.458 − 359.032i 1361.01
35.12 − 6.15394i 8.83860 + 25.5123i 26.1290 10.7972i 157.001 − 54.3922i 137.101 − 554.649i −572.758 + 450.987i 66.4455
35.13 − 4.81455i 26.1801 + 6.60322i 40.8201 − 186.576i 31.7916 − 126.045i −455.638 − 504.662i 641.795 + 345.746i −898.281
35.14 − 4.48185i −14.6696 − 22.6672i 43.9130 − 175.097i −101.591 + 65.7468i 666.818 − 483.650i −298.607 + 665.037i −784.758
35.15 − 0.853586i −12.6020 − 23.8786i 63.2714 153.242i −20.3825 + 10.7569i −31.3758 − 108.637i −411.378 + 601.838i 130.806
35.16 − 0.461474i 22.0416 − 15.5939i 63.7870 − 70.0200i −7.19617 − 10.1716i 296.369 − 58.9705i 242.663 − 687.427i −32.3124
35.17 0.461474i 22.0416 + 15.5939i 63.7870 70.0200i −7.19617 + 10.1716i 296.369 58.9705i 242.663 + 687.427i −32.3124
35.18 0.853586i −12.6020 + 23.8786i 63.2714 − 153.242i −20.3825 − 10.7569i −31.3758 108.637i −411.378 − 601.838i 130.806
35.19 4.48185i −14.6696 + 22.6672i 43.9130 175.097i −101.591 − 65.7468i 666.818 483.650i −298.607 − 665.037i −784.758
35.20 4.81455i 26.1801 − 6.60322i 40.8201 186.576i 31.7916 + 126.045i −455.638 504.662i 641.795 − 345.746i −898.281
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.7.b.a ✓ 32
3.b odd 2 1 inner 51.7.b.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.7.b.a ✓ 32 1.a even 1 1 trivial
51.7.b.a ✓ 32 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(51, [\chi])\).