Properties

Label 112.2.v.a
Level 112
Weight 2
Character orbit 112.v
Analytic conductor 0.894
Analytic rank 0
Dimension 56
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 112.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56q - 2q^{2} - 6q^{3} - 4q^{4} - 6q^{5} - 8q^{7} + 4q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q - 2q^{2} - 6q^{3} - 4q^{4} - 6q^{5} - 8q^{7} + 4q^{8} - 24q^{10} + 2q^{11} - 6q^{12} + 16q^{14} + 8q^{16} - 12q^{17} - 30q^{18} - 6q^{19} - 10q^{21} - 28q^{22} - 12q^{23} - 6q^{24} - 6q^{26} + 26q^{28} - 24q^{29} - 18q^{30} - 12q^{32} - 12q^{33} - 2q^{35} + 16q^{36} + 6q^{37} - 6q^{38} - 4q^{39} - 66q^{40} + 70q^{42} + 26q^{44} + 12q^{45} + 16q^{46} - 8q^{49} - 34q^{51} + 84q^{52} + 6q^{53} + 42q^{54} + 16q^{56} + 18q^{58} + 42q^{59} + 78q^{60} - 6q^{61} - 16q^{64} - 4q^{65} + 126q^{66} + 6q^{67} + 24q^{68} - 80q^{70} - 80q^{71} - 4q^{72} + 62q^{74} + 24q^{75} + 10q^{77} + 4q^{78} + 12q^{80} - 8q^{81} + 42q^{82} - 152q^{84} - 28q^{85} - 12q^{87} + 30q^{88} + 16q^{91} - 20q^{92} + 10q^{93} - 42q^{94} + 36q^{96} - 108q^{98} - 16q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.41419 0.00846089i 1.95777 0.524583i 1.99986 + 0.0239306i 0.666898 2.48890i −2.77309 + 0.725295i −2.38027 + 1.15512i −2.82797 0.0507629i 0.959602 0.554026i −0.964178 + 3.51413i
3.2 −1.25731 0.647437i −3.08091 + 0.825526i 1.16165 + 1.62806i 0.501060 1.86998i 4.40813 + 0.956753i 2.17953 + 1.49988i −0.406487 2.79907i 6.21241 3.58674i −1.84068 + 2.02674i
3.3 −1.25445 + 0.652965i −1.05433 + 0.282507i 1.14727 1.63822i −0.374919 + 1.39922i 1.13814 1.04283i −0.298614 + 2.62885i −0.369496 + 2.80419i −1.56627 + 0.904287i −0.443324 2.00005i
3.4 −0.946521 + 1.05076i 2.77508 0.743580i −0.208197 1.98913i −0.907491 + 3.38680i −1.84535 + 3.61976i 0.0791552 2.64457i 2.28717 + 1.66399i 4.55008 2.62699i −2.69976 4.15923i
3.5 −0.910025 1.08252i 1.67893 0.449868i −0.343708 + 1.97024i 0.195879 0.731029i −2.01486 1.40809i 2.52163 0.800849i 2.44562 1.42090i 0.0183525 0.0105958i −0.969610 + 0.453212i
3.6 −0.799555 1.16650i −0.763582 + 0.204601i −0.721425 + 1.86535i −1.02188 + 3.81370i 0.849192 + 0.727125i −2.64575 + 0.00379639i 2.75275 0.649913i −2.05688 + 1.18754i 5.26571 1.85724i
3.7 −0.252874 + 1.39142i −2.61463 + 0.700587i −1.87211 0.703710i 0.120751 0.450647i −0.313640 3.81521i −2.37326 1.16946i 1.45257 2.42694i 3.74737 2.16355i 0.596506 + 0.281972i
3.8 0.149759 1.40626i −1.44888 + 0.388227i −1.95514 0.421199i 0.755106 2.81809i 0.328966 + 2.09565i −1.47726 2.19493i −0.885116 + 2.68637i −0.649537 + 0.375010i −3.84990 1.48391i
3.9 0.187641 + 1.40171i 0.198087 0.0530773i −1.92958 + 0.526037i −0.487744 + 1.82029i 0.111568 + 0.267701i 1.84933 + 1.89208i −1.09942 2.60601i −2.56165 + 1.47897i −2.64303 0.342115i
3.10 0.438321 1.34457i 2.29647 0.615336i −1.61575 1.17871i −0.223959 + 0.835825i 0.179226 3.35748i −1.25761 + 2.32775i −2.29308 + 1.65584i 2.29704 1.32620i 1.02566 + 0.667489i
3.11 0.898888 + 1.09179i 1.30734 0.350301i −0.384000 + 1.96279i 0.294400 1.09872i 1.55761 + 1.11246i −1.56831 2.13082i −2.48812 + 1.34508i −1.01165 + 0.584076i 1.46420 0.666200i
3.12 1.06970 0.925063i −0.246909 + 0.0661591i 0.288515 1.97908i −0.133797 + 0.499339i −0.202917 + 0.299177i 2.45011 0.998472i −1.52215 2.38392i −2.54149 + 1.46733i 0.318797 + 0.657913i
3.13 1.31418 + 0.522432i −2.48480 + 0.665801i 1.45413 + 1.37314i −0.837076 + 3.12401i −3.61331 0.423158i 1.56215 2.13534i 1.19362 + 2.56423i 3.13287 1.80877i −2.73215 + 3.66819i
3.14 1.41041 + 0.103712i −0.885661 + 0.237312i 1.97849 + 0.292553i 0.818796 3.05579i −1.27375 + 0.242853i −0.640837 + 2.56697i 2.76013 + 0.617811i −1.87000 + 1.07964i 1.47176 4.22498i
19.1 −1.38359 + 0.292689i 0.615336 2.29647i 1.82867 0.809925i −0.835825 + 0.223959i −0.179226 + 3.35748i −1.25761 2.32775i −2.29308 + 1.65584i −2.29704 1.32620i 1.09089 0.554505i
19.2 −1.33598 0.463855i −0.0661591 + 0.246909i 1.56968 + 1.23940i −0.499339 + 0.133797i 0.202917 0.299177i 2.45011 + 0.998472i −1.52215 2.38392i 2.54149 + 1.46733i 0.729168 + 0.0528705i
19.3 −1.29274 + 0.573436i −0.388227 + 1.44888i 1.34234 1.48261i 2.81809 0.755106i −0.328966 2.09565i −1.47726 + 2.19493i −0.885116 + 2.68637i 0.649537 + 0.375010i −3.21005 + 2.59215i
19.4 −0.615385 1.27330i −0.237312 + 0.885661i −1.24260 + 1.56714i 3.05579 0.818796i 1.27375 0.242853i −0.640837 2.56697i 2.76013 + 0.617811i 1.87000 + 1.07964i −2.92306 3.38707i
19.5 −0.610438 + 1.27568i −0.204601 + 0.763582i −1.25473 1.55745i −3.81370 + 1.02188i −0.849192 0.727125i −2.64575 0.00379639i 2.75275 0.649913i 2.05688 + 1.18754i 1.02443 5.48886i
19.6 −0.482479 + 1.32937i 0.449868 1.67893i −1.53443 1.28278i 0.731029 0.195879i 2.01486 + 1.40809i 2.52163 + 0.800849i 2.44562 1.42090i −0.0183525 0.0105958i −0.0923119 + 1.06631i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
16.f odd 4 1 inner
112.v even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.2.v.a 56
4.b odd 2 1 448.2.z.a 56
7.b odd 2 1 784.2.w.f 56
7.c even 3 1 784.2.j.a 56
7.c even 3 1 784.2.w.f 56
7.d odd 6 1 inner 112.2.v.a 56
7.d odd 6 1 784.2.j.a 56
8.b even 2 1 896.2.z.b 56
8.d odd 2 1 896.2.z.a 56
16.e even 4 1 448.2.z.a 56
16.e even 4 1 896.2.z.a 56
16.f odd 4 1 inner 112.2.v.a 56
16.f odd 4 1 896.2.z.b 56
28.f even 6 1 448.2.z.a 56
56.j odd 6 1 896.2.z.b 56
56.m even 6 1 896.2.z.a 56
112.j even 4 1 784.2.w.f 56
112.u odd 12 1 784.2.j.a 56
112.u odd 12 1 784.2.w.f 56
112.v even 12 1 inner 112.2.v.a 56
112.v even 12 1 784.2.j.a 56
112.v even 12 1 896.2.z.b 56
112.x odd 12 1 448.2.z.a 56
112.x odd 12 1 896.2.z.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.v.a 56 1.a even 1 1 trivial
112.2.v.a 56 7.d odd 6 1 inner
112.2.v.a 56 16.f odd 4 1 inner
112.2.v.a 56 112.v even 12 1 inner
448.2.z.a 56 4.b odd 2 1
448.2.z.a 56 16.e even 4 1
448.2.z.a 56 28.f even 6 1
448.2.z.a 56 112.x odd 12 1
784.2.j.a 56 7.c even 3 1
784.2.j.a 56 7.d odd 6 1
784.2.j.a 56 112.u odd 12 1
784.2.j.a 56 112.v even 12 1
784.2.w.f 56 7.b odd 2 1
784.2.w.f 56 7.c even 3 1
784.2.w.f 56 112.j even 4 1
784.2.w.f 56 112.u odd 12 1
896.2.z.a 56 8.d odd 2 1
896.2.z.a 56 16.e even 4 1
896.2.z.a 56 56.m even 6 1
896.2.z.a 56 112.x odd 12 1
896.2.z.b 56 8.b even 2 1
896.2.z.b 56 16.f odd 4 1
896.2.z.b 56 56.j odd 6 1
896.2.z.b 56 112.v even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(112, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database