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Properties

Label 112.7.s.e
Level $112$
Weight $7$
Character orbit 112.s
Analytic conductor $25.766$
Analytic rank $0$
Dimension $24$
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] = [112,7,Mod(17,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.17");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 112.s (of order \(6\), degree \(2\), not 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: \(25.7660573654\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 564 q^{7} + 2300 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 564 q^{7} + 2300 q^{9} - 812 q^{11} + 11752 q^{15} - 840 q^{17} + 2268 q^{19} + 68 q^{21} - 1392 q^{23} + 28104 q^{25} + 51464 q^{29} - 139860 q^{31} - 124572 q^{33} + 87564 q^{35} - 22548 q^{37} - 45476 q^{39} + 221472 q^{43} + 15204 q^{45} + 124740 q^{47} - 83184 q^{49} - 224772 q^{51} + 143692 q^{53} + 227176 q^{57} + 378840 q^{59} + 345492 q^{61} + 166556 q^{63} + 351380 q^{65} + 744 q^{67} - 1821328 q^{71} - 918036 q^{73} + 1826496 q^{75} - 826704 q^{77} + 756960 q^{79} - 188640 q^{81} - 1589448 q^{85} - 1642452 q^{87} + 1809444 q^{89} - 3269160 q^{91} + 1843780 q^{93} + 1423224 q^{95} - 68552 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −41.2415 − 23.8108i 0 −37.5506 + 21.6798i 0 −342.808 − 11.4676i 0 769.408 + 1332.65i 0
17.2 0 −40.7707 − 23.5390i 0 −60.4339 + 34.8915i 0 225.250 − 258.673i 0 743.668 + 1288.07i 0
17.3 0 −22.1092 − 12.7648i 0 122.892 − 70.9516i 0 −36.1745 + 341.087i 0 −38.6222 − 66.8956i 0
17.4 0 −13.7930 − 7.96340i 0 −47.1456 + 27.2195i 0 316.915 + 131.201i 0 −237.669 − 411.654i 0
17.5 0 −9.53773 − 5.50661i 0 183.292 − 105.824i 0 −46.8195 − 339.790i 0 −303.854 − 526.291i 0
17.6 0 −7.70960 − 4.45114i 0 −213.648 + 123.350i 0 −325.918 + 106.894i 0 −324.875 − 562.700i 0
17.7 0 −1.35117 − 0.780100i 0 14.0845 − 8.13166i 0 123.633 − 319.944i 0 −363.283 − 629.224i 0
17.8 0 11.6484 + 6.72521i 0 −43.5745 + 25.1577i 0 −292.362 + 179.369i 0 −274.043 − 474.657i 0
17.9 0 26.2731 + 15.1688i 0 102.414 − 59.1289i 0 −322.679 − 116.307i 0 95.6838 + 165.729i 0
17.10 0 30.7831 + 17.7726i 0 −131.199 + 75.7478i 0 2.86620 − 342.988i 0 267.231 + 462.858i 0
17.11 0 30.8184 + 17.7930i 0 −54.4036 + 31.4099i 0 295.298 + 174.493i 0 268.681 + 465.369i 0
17.12 0 36.9900 + 21.3562i 0 165.273 − 95.4202i 0 120.800 + 321.024i 0 547.675 + 948.600i 0
33.1 0 −41.2415 + 23.8108i 0 −37.5506 − 21.6798i 0 −342.808 + 11.4676i 0 769.408 − 1332.65i 0
33.2 0 −40.7707 + 23.5390i 0 −60.4339 − 34.8915i 0 225.250 + 258.673i 0 743.668 − 1288.07i 0
33.3 0 −22.1092 + 12.7648i 0 122.892 + 70.9516i 0 −36.1745 − 341.087i 0 −38.6222 + 66.8956i 0
33.4 0 −13.7930 + 7.96340i 0 −47.1456 − 27.2195i 0 316.915 − 131.201i 0 −237.669 + 411.654i 0
33.5 0 −9.53773 + 5.50661i 0 183.292 + 105.824i 0 −46.8195 + 339.790i 0 −303.854 + 526.291i 0
33.6 0 −7.70960 + 4.45114i 0 −213.648 − 123.350i 0 −325.918 − 106.894i 0 −324.875 + 562.700i 0
33.7 0 −1.35117 + 0.780100i 0 14.0845 + 8.13166i 0 123.633 + 319.944i 0 −363.283 + 629.224i 0
33.8 0 11.6484 − 6.72521i 0 −43.5745 − 25.1577i 0 −292.362 − 179.369i 0 −274.043 + 474.657i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.7.s.e 24
4.b odd 2 1 56.7.o.a ✓ 24
7.d odd 6 1 inner 112.7.s.e 24
28.f even 6 1 56.7.o.a ✓ 24
28.f even 6 1 392.7.c.c 24
28.g odd 6 1 392.7.c.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.7.o.a ✓ 24 4.b odd 2 1
56.7.o.a ✓ 24 28.f even 6 1
112.7.s.e 24 1.a even 1 1 trivial
112.7.s.e 24 7.d odd 6 1 inner
392.7.c.c 24 28.f even 6 1
392.7.c.c 24 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 5524 T_{3}^{22} + 19750619 T_{3}^{20} - 71413104 T_{3}^{19} - 41664600932 T_{3}^{18} + \cdots + 94\!\cdots\!29 \) acting on \(S_{7}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display