Properties

Label 231.2.e.a
Level 231
Weight 2
Character orbit 231.e
Analytic conductor 1.845
Analytic rank 0
Dimension 28
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 231.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(28\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28q - 32q^{4} - 8q^{7} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q - 32q^{4} - 8q^{7} - 8q^{9} - 20q^{15} + 40q^{16} - 12q^{18} - 10q^{21} + 36q^{25} + 12q^{28} - 4q^{30} + 24q^{36} - 24q^{37} + 16q^{39} - 40q^{43} - 16q^{46} + 4q^{49} - 8q^{51} - 4q^{57} - 44q^{58} + 52q^{60} + 6q^{63} - 68q^{64} + 40q^{67} + 20q^{70} + 24q^{72} - 28q^{78} + 56q^{79} + 32q^{81} + 100q^{84} - 8q^{85} + 12q^{88} + 8q^{91} - 36q^{93} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1 2.73839i −0.280799 1.70914i −5.49880 2.40492 −4.68029 + 0.768937i −1.28455 2.31299i 9.58108i −2.84230 + 0.959847i 6.58561i
188.2 2.73839i 0.280799 + 1.70914i −5.49880 −2.40492 4.68029 0.768937i −1.28455 + 2.31299i 9.58108i −2.84230 + 0.959847i 6.58561i
188.3 2.33290i −1.41254 + 1.00236i −3.44244 4.22171 2.33841 + 3.29532i 2.13159 + 1.56727i 3.36508i 0.990548 2.83175i 9.84884i
188.4 2.33290i 1.41254 1.00236i −3.44244 −4.22171 −2.33841 3.29532i 2.13159 1.56727i 3.36508i 0.990548 2.83175i 9.84884i
188.5 2.06094i −1.71028 0.273753i −2.24749 −1.49849 −0.564190 + 3.52479i −2.62644 + 0.319041i 0.510055i 2.85012 + 0.936390i 3.08830i
188.6 2.06094i 1.71028 + 0.273753i −2.24749 1.49849 0.564190 3.52479i −2.62644 0.319041i 0.510055i 2.85012 + 0.936390i 3.08830i
188.7 1.68989i −1.02930 + 1.39303i −0.855732 −1.45819 2.35408 + 1.73940i 0.366782 2.62020i 1.93369i −0.881093 2.86770i 2.46419i
188.8 1.68989i 1.02930 1.39303i −0.855732 1.45819 −2.35408 1.73940i 0.366782 + 2.62020i 1.93369i −0.881093 2.86770i 2.46419i
188.9 1.12088i −0.799601 1.53644i 0.743632 0.911250 −1.72216 + 0.896255i 2.18571 1.49086i 3.07528i −1.72128 + 2.45707i 1.02140i
188.10 1.12088i 0.799601 + 1.53644i 0.743632 −0.911250 1.72216 0.896255i 2.18571 + 1.49086i 3.07528i −1.72128 + 2.45707i 1.02140i
188.11 0.776672i −0.233960 1.71618i 1.39678 −3.58495 −1.33291 + 0.181710i −2.64177 + 0.145081i 2.63818i −2.89053 + 0.803035i 2.78433i
188.12 0.776672i 0.233960 + 1.71618i 1.39678 3.58495 1.33291 0.181710i −2.64177 0.145081i 2.63818i −2.89053 + 0.803035i 2.78433i
188.13 0.309769i −1.65749 0.502726i 1.90404 1.52955 −0.155729 + 0.513438i −0.131314 + 2.64249i 1.20935i 2.49453 + 1.66652i 0.473807i
188.14 0.309769i 1.65749 + 0.502726i 1.90404 −1.52955 0.155729 0.513438i −0.131314 2.64249i 1.20935i 2.49453 + 1.66652i 0.473807i
188.15 0.309769i −1.65749 + 0.502726i 1.90404 1.52955 −0.155729 0.513438i −0.131314 2.64249i 1.20935i 2.49453 1.66652i 0.473807i
188.16 0.309769i 1.65749 0.502726i 1.90404 −1.52955 0.155729 + 0.513438i −0.131314 + 2.64249i 1.20935i 2.49453 1.66652i 0.473807i
188.17 0.776672i −0.233960 + 1.71618i 1.39678 −3.58495 −1.33291 0.181710i −2.64177 0.145081i 2.63818i −2.89053 0.803035i 2.78433i
188.18 0.776672i 0.233960 1.71618i 1.39678 3.58495 1.33291 + 0.181710i −2.64177 + 0.145081i 2.63818i −2.89053 0.803035i 2.78433i
188.19 1.12088i −0.799601 + 1.53644i 0.743632 0.911250 −1.72216 0.896255i 2.18571 + 1.49086i 3.07528i −1.72128 2.45707i 1.02140i
188.20 1.12088i 0.799601 1.53644i 0.743632 −0.911250 1.72216 + 0.896255i 2.18571 1.49086i 3.07528i −1.72128 2.45707i 1.02140i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
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 231.2.e.a 28
3.b odd 2 1 inner 231.2.e.a 28
7.b odd 2 1 inner 231.2.e.a 28
21.c even 2 1 inner 231.2.e.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.e.a 28 1.a even 1 1 trivial
231.2.e.a 28 3.b odd 2 1 inner
231.2.e.a 28 7.b odd 2 1 inner
231.2.e.a 28 21.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(231, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database