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Properties

Label 315.10.b.a
Level $315$
Weight $10$
Character orbit 315.b
Analytic conductor $162.236$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,10,Mod(251,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.251");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(162.236288392\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12288 q^{4} - 30000 q^{5} + 1824 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12288 q^{4} - 30000 q^{5} + 1824 q^{7} + 442908 q^{14} + 3258948 q^{16} + 7680000 q^{20} - 2860668 q^{22} + 18750000 q^{25} - 5432976 q^{26} - 3685092 q^{28} - 1140000 q^{35} + 7750344 q^{37} + 17423136 q^{38} - 70626600 q^{41} + 16073832 q^{43} - 15083820 q^{46} - 33200856 q^{47} - 39367500 q^{49} - 318486852 q^{56} + 293819940 q^{58} + 342990864 q^{59} + 896801400 q^{62} - 1639789824 q^{64} - 980229936 q^{67} - 438669576 q^{68} - 276817500 q^{70} - 900236484 q^{77} + 325977144 q^{79} - 2036842500 q^{80} + 1415370096 q^{83} + 3319557300 q^{88} - 1132967016 q^{89} - 1162515060 q^{91} - 2546372484 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 − 44.9460i 0 −1508.14 −625.000 0 429.891 + 6337.89i 44772.6i 0 28091.2i
251.2 − 43.4603i 0 −1376.80 −625.000 0 5681.54 − 2841.43i 37584.3i 0 27162.7i
251.3 − 41.7518i 0 −1231.22 −625.000 0 −6112.87 + 1728.13i 30028.6i 0 26094.9i
251.4 − 38.7119i 0 −986.613 −625.000 0 −2221.76 − 5951.25i 18373.2i 0 24195.0i
251.5 − 38.5224i 0 −971.978 −625.000 0 5534.46 − 3118.23i 17719.5i 0 24076.5i
251.6 − 38.0944i 0 −939.186 −625.000 0 −1841.41 + 6079.71i 16273.4i 0 23809.0i
251.7 − 33.3124i 0 −597.718 −625.000 0 −1777.21 − 6098.78i 2855.49i 0 20820.3i
251.8 − 33.1306i 0 −585.638 −625.000 0 1836.41 + 6081.22i 2439.66i 0 20706.6i
251.9 − 31.2804i 0 −466.463 −625.000 0 4497.64 + 4486.07i − 1424.41i 0 19550.2i
251.10 − 28.7659i 0 −315.478 −625.000 0 −6144.87 + 1610.66i − 5653.15i 0 17978.7i
251.11 − 27.0678i 0 −220.664 −625.000 0 −2939.90 − 5631.22i − 7885.83i 0 16917.3i
251.12 − 27.0056i 0 −217.303 −625.000 0 6343.29 + 341.079i − 7958.48i 0 16878.5i
251.13 − 25.1545i 0 −120.747 −625.000 0 −559.056 + 6327.80i − 9841.77i 0 15721.5i
251.14 − 23.5339i 0 −41.8461 −625.000 0 −3605.99 − 5229.77i − 11064.6i 0 14708.7i
251.15 − 22.4652i 0 7.31397 −625.000 0 6124.53 − 1686.34i − 11666.5i 0 14040.8i
251.16 − 20.3785i 0 96.7172 −625.000 0 −6209.18 + 1341.52i − 12404.7i 0 12736.6i
251.17 − 16.2824i 0 246.883 −625.000 0 −4032.57 + 4908.36i − 12356.4i 0 10176.5i
251.18 − 13.2885i 0 335.417 −625.000 0 4209.35 − 4757.62i − 11260.9i 0 8305.29i
251.19 − 10.2080i 0 407.798 −625.000 0 −6351.60 + 103.891i − 9389.26i 0 6379.98i
251.20 − 9.87231i 0 414.537 −625.000 0 −1874.58 + 6069.56i − 9147.07i 0 6170.19i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.10.b.a ✓ 48
3.b odd 2 1 315.10.b.b yes 48
7.b odd 2 1 315.10.b.b yes 48
21.c even 2 1 inner 315.10.b.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.10.b.a ✓ 48 1.a even 1 1 trivial
315.10.b.a ✓ 48 21.c even 2 1 inner
315.10.b.b yes 48 3.b odd 2 1
315.10.b.b yes 48 7.b odd 2 1