Properties

Label 180.2.k.e
Level $180$
Weight $2$
Character orbit 180.k
Analytic conductor $1.437$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.426337261060096.1
Defining polynomial: \(x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -1 - \beta_{1} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{5} + ( \beta_{1} - \beta_{10} ) q^{7} + ( 1 + \beta_{4} + \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -1 - \beta_{1} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{5} + ( \beta_{1} - \beta_{10} ) q^{7} + ( 1 + \beta_{4} + \beta_{5} ) q^{8} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{10} + ( 1 - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{11} + ( -1 - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{13} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{14} + ( \beta_{2} - 2 \beta_{3} + \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{16} + ( 2 + \beta_{3} - 2 \beta_{4} - \beta_{8} + \beta_{9} ) q^{17} + ( -\beta_{1} - \beta_{3} - \beta_{5} + \beta_{10} ) q^{19} + ( -2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{20} + ( 3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{22} + ( -1 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{23} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{25} + ( -2 - \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{7} + \beta_{10} ) q^{26} + ( 1 + \beta_{2} - 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{11} ) q^{28} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} ) q^{29} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{11} ) q^{31} + ( -1 + 2 \beta_{1} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - \beta_{10} ) q^{32} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{10} ) q^{34} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{11} ) q^{35} + ( -\beta_{3} + \beta_{8} - \beta_{9} ) q^{37} + ( -2 - 2 \beta_{2} - 2 \beta_{9} + 2 \beta_{11} ) q^{38} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{40} + ( -1 + \beta_{2} - 2 \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{41} + ( -2 - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{11} ) q^{43} + ( 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + 4 \beta_{6} + \beta_{10} ) q^{44} + ( -5 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{46} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{47} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{49} + ( 2 - \beta_{1} - 2 \beta_{2} + 4 \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{10} ) q^{50} + ( -2 - 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{52} + ( -\beta_{3} - \beta_{8} - \beta_{9} ) q^{53} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{55} + ( 4 - \beta_{5} + 2 \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{56} + ( -1 + \beta_{2} - 2 \beta_{4} - \beta_{8} - \beta_{11} ) q^{58} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{59} + ( 1 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{6} - 3 \beta_{7} - 5 \beta_{8} - 3 \beta_{11} ) q^{61} + ( 6 - 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{8} - 2 \beta_{11} ) q^{62} + ( 1 + 2 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{64} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 4 \beta_{4} + 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{65} + ( -1 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{67} + ( 2 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} ) q^{68} + ( 4 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} - 2 \beta_{11} ) q^{70} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} + 3 \beta_{10} - \beta_{11} ) q^{71} + ( 3 + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{8} + 2 \beta_{9} ) q^{73} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{10} ) q^{74} + ( -2 - 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{76} + ( -3 + 2 \beta_{1} - \beta_{2} + 4 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} ) q^{77} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 5 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{79} + ( -1 - 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{10} ) q^{80} + ( 2 - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} ) q^{82} + ( -1 - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{83} + ( 3 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} - 3 \beta_{9} + \beta_{11} ) q^{85} + ( -9 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + 3 \beta_{7} + 5 \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{86} + ( 5 - \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{88} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{89} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - 3 \beta_{11} ) q^{91} + ( -4 - 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{8} + 2 \beta_{9} ) q^{92} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{94} + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{9} - \beta_{10} ) q^{95} + ( -3 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} ) q^{97} + ( 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{8} + O(q^{10}) \) \( 12q + 12q^{8} - 8q^{10} - 4q^{13} + 12q^{16} + 20q^{17} - 20q^{20} + 12q^{22} - 20q^{25} - 16q^{26} - 4q^{28} - 20q^{32} + 4q^{37} - 16q^{38} - 8q^{40} - 16q^{41} - 40q^{46} + 16q^{50} - 8q^{52} - 4q^{53} + 64q^{56} - 20q^{58} - 32q^{61} + 56q^{62} - 20q^{65} + 16q^{68} + 44q^{70} + 44q^{73} + 8q^{76} - 48q^{77} - 4q^{80} + 16q^{82} + 44q^{85} - 64q^{86} + 60q^{88} - 56q^{92} - 20q^{97} - 24q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{11} - 8 \nu^{9} - 4 \nu^{8} + 11 \nu^{7} + 12 \nu^{6} + 16 \nu^{5} - 48 \nu^{4} - 60 \nu^{3} + 16 \nu^{2} + 64 \nu + 192 \)\()/80\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{11} + \nu^{10} - 4 \nu^{8} - 5 \nu^{7} + \nu^{6} + 12 \nu^{5} + 8 \nu^{4} + 12 \nu^{3} - 36 \nu^{2} - 16 \nu - 64 \)\()/80\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{11} - \nu^{10} + 4 \nu^{8} + 5 \nu^{7} - \nu^{6} - 12 \nu^{5} - 8 \nu^{4} + 68 \nu^{3} + 36 \nu^{2} + 16 \nu - 16 \)\()/80\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} + 12 \nu^{10} + 4 \nu^{9} + 4 \nu^{8} - 13 \nu^{7} - 24 \nu^{6} - 4 \nu^{5} + 40 \nu^{4} + 44 \nu^{3} - 80 \nu^{2} - 144 \nu - 64 \)\()/160\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{11} - 4 \nu^{10} - 12 \nu^{9} - 20 \nu^{8} - 11 \nu^{7} + 64 \nu^{6} - 4 \nu^{5} - 24 \nu^{4} - 108 \nu^{3} - 112 \nu^{2} + 240 \nu + 64 \)\()/160\)
\(\beta_{8}\)\(=\)\((\)\( -3 \nu^{11} - 2 \nu^{10} - 6 \nu^{9} + 17 \nu^{7} + 2 \nu^{6} + 18 \nu^{5} - 12 \nu^{4} - 44 \nu^{3} + 24 \nu^{2} + 40 \nu + 192 \)\()/80\)
\(\beta_{9}\)\(=\)\((\)\( 3 \nu^{11} + \nu^{10} + 5 \nu^{9} - 4 \nu^{8} - 5 \nu^{7} + \nu^{6} - 3 \nu^{5} + 28 \nu^{4} + 12 \nu^{3} - 56 \nu^{2} - 36 \nu - 144 \)\()/40\)
\(\beta_{10}\)\(=\)\((\)\( -5 \nu^{11} - 7 \nu^{10} + 8 \nu^{9} + 12 \nu^{8} + 19 \nu^{7} - 39 \nu^{6} - 60 \nu^{5} - 8 \nu^{4} + 76 \nu^{3} + 156 \nu^{2} + 48 \nu - 144 \)\()/80\)
\(\beta_{11}\)\(=\)\((\)\( 3 \nu^{11} + 10 \nu^{10} + 14 \nu^{9} - 8 \nu^{8} - 33 \nu^{7} - 26 \nu^{6} + 22 \nu^{5} + 84 \nu^{4} + 60 \nu^{3} - 88 \nu^{2} - 232 \nu - 176 \)\()/80\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{3} + \beta_{2}\)
\(\nu^{5}\)\(=\)\(-\beta_{10} - 2 \beta_{7} - 2 \beta_{6} + \beta_{4} + 2 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-\beta_{11} - 2 \beta_{10} + 2 \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{2} + 1\)
\(\nu^{7}\)\(=\)\(-2 \beta_{11} + 4 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3\)
\(\nu^{8}\)\(=\)\(-6 \beta_{11} - 4 \beta_{10} - 4 \beta_{8} - 7 \beta_{7} + \beta_{6} - 4 \beta_{5} - 6 \beta_{3} + \beta_{2} + 4 \beta_{1} + 6\)
\(\nu^{9}\)\(=\)\(-3 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 10 \beta_{7} - 10 \beta_{6} - 13 \beta_{4} + 8 \beta_{3} + 10 \beta_{1} + 13\)
\(\nu^{10}\)\(=\)\(-3 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + 13 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} + 8 \beta_{1} + 3\)
\(\nu^{11}\)\(=\)\(-10 \beta_{11} + 4 \beta_{9} - 6 \beta_{8} - 6 \beta_{7} + 6 \beta_{6} - 9 \beta_{5} + 23 \beta_{4} - 2 \beta_{3} + 10 \beta_{2} + 33\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-\beta_{4}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−1.35818 0.394157i
−0.760198 + 1.19252i
−0.394157 1.35818i
−0.0912546 + 1.41127i
1.19252 0.760198i
1.41127 0.0912546i
−1.35818 + 0.394157i
−0.760198 1.19252i
−0.394157 + 1.35818i
−0.0912546 1.41127i
1.19252 + 0.760198i
1.41127 + 0.0912546i
−1.35818 0.394157i 0 1.68928 + 1.07067i 1.75233 + 1.38900i 0 −2.47817 + 2.47817i −1.87233 2.12000i 0 −1.83249 2.57720i
127.2 −0.760198 + 1.19252i 0 −0.844199 1.81310i −0.432320 2.19388i 0 0.611393 0.611393i 2.80391 + 0.371591i 0 2.94489 + 1.15223i
127.3 −0.394157 1.35818i 0 −1.68928 + 1.07067i 1.75233 + 1.38900i 0 2.47817 2.47817i 2.12000 + 1.87233i 0 1.19582 2.92746i
127.4 −0.0912546 + 1.41127i 0 −1.98335 0.257569i −1.32001 + 1.80487i 0 −1.86678 + 1.86678i 0.544488 2.77552i 0 −2.42670 2.02759i
127.5 1.19252 0.760198i 0 0.844199 1.81310i −0.432320 2.19388i 0 −0.611393 + 0.611393i −0.371591 2.80391i 0 −2.18333 2.28759i
127.6 1.41127 0.0912546i 0 1.98335 0.257569i −1.32001 + 1.80487i 0 1.86678 1.86678i 2.77552 0.544488i 0 −1.69819 + 2.66761i
163.1 −1.35818 + 0.394157i 0 1.68928 1.07067i 1.75233 1.38900i 0 −2.47817 2.47817i −1.87233 + 2.12000i 0 −1.83249 + 2.57720i
163.2 −0.760198 1.19252i 0 −0.844199 + 1.81310i −0.432320 + 2.19388i 0 0.611393 + 0.611393i 2.80391 0.371591i 0 2.94489 1.15223i
163.3 −0.394157 + 1.35818i 0 −1.68928 1.07067i 1.75233 1.38900i 0 2.47817 + 2.47817i 2.12000 1.87233i 0 1.19582 + 2.92746i
163.4 −0.0912546 1.41127i 0 −1.98335 + 0.257569i −1.32001 1.80487i 0 −1.86678 1.86678i 0.544488 + 2.77552i 0 −2.42670 + 2.02759i
163.5 1.19252 + 0.760198i 0 0.844199 + 1.81310i −0.432320 + 2.19388i 0 −0.611393 0.611393i −0.371591 + 2.80391i 0 −2.18333 + 2.28759i
163.6 1.41127 + 0.0912546i 0 1.98335 + 0.257569i −1.32001 1.80487i 0 1.86678 + 1.86678i 2.77552 + 0.544488i 0 −1.69819 2.66761i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.2.k.e 12
3.b odd 2 1 60.2.j.a 12
4.b odd 2 1 inner 180.2.k.e 12
5.b even 2 1 900.2.k.n 12
5.c odd 4 1 inner 180.2.k.e 12
5.c odd 4 1 900.2.k.n 12
12.b even 2 1 60.2.j.a 12
15.d odd 2 1 300.2.j.d 12
15.e even 4 1 60.2.j.a 12
15.e even 4 1 300.2.j.d 12
20.d odd 2 1 900.2.k.n 12
20.e even 4 1 inner 180.2.k.e 12
20.e even 4 1 900.2.k.n 12
24.f even 2 1 960.2.w.g 12
24.h odd 2 1 960.2.w.g 12
60.h even 2 1 300.2.j.d 12
60.l odd 4 1 60.2.j.a 12
60.l odd 4 1 300.2.j.d 12
120.q odd 4 1 960.2.w.g 12
120.w even 4 1 960.2.w.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.j.a 12 3.b odd 2 1
60.2.j.a 12 12.b even 2 1
60.2.j.a 12 15.e even 4 1
60.2.j.a 12 60.l odd 4 1
180.2.k.e 12 1.a even 1 1 trivial
180.2.k.e 12 4.b odd 2 1 inner
180.2.k.e 12 5.c odd 4 1 inner
180.2.k.e 12 20.e even 4 1 inner
300.2.j.d 12 15.d odd 2 1
300.2.j.d 12 15.e even 4 1
300.2.j.d 12 60.h even 2 1
300.2.j.d 12 60.l odd 4 1
900.2.k.n 12 5.b even 2 1
900.2.k.n 12 5.c odd 4 1
900.2.k.n 12 20.d odd 2 1
900.2.k.n 12 20.e even 4 1
960.2.w.g 12 24.f even 2 1
960.2.w.g 12 24.h odd 2 1
960.2.w.g 12 120.q odd 4 1
960.2.w.g 12 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(180, [\chi])\):

\( T_{7}^{12} + 200 T_{7}^{8} + 7440 T_{7}^{4} + 4096 \)
\( T_{13}^{6} + 2 T_{13}^{5} + 2 T_{13}^{4} - 32 T_{13}^{3} + 144 T_{13}^{2} - 96 T_{13} + 32 \)
\( T_{17}^{6} - 10 T_{17}^{5} + 50 T_{17}^{4} - 80 T_{17}^{3} + 16 T_{17}^{2} + 160 T_{17} + 800 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 64 - 32 T^{3} - 12 T^{4} + 8 T^{5} + 8 T^{6} + 4 T^{7} - 3 T^{8} - 4 T^{9} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( ( 125 + 25 T^{2} - 8 T^{3} + 5 T^{4} + T^{6} )^{2} \)
$7$ \( 4096 + 7440 T^{4} + 200 T^{8} + T^{12} \)
$11$ \( ( 128 + 260 T^{2} + 36 T^{4} + T^{6} )^{2} \)
$13$ \( ( 32 - 96 T + 144 T^{2} - 32 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$17$ \( ( 800 + 160 T + 16 T^{2} - 80 T^{3} + 50 T^{4} - 10 T^{5} + T^{6} )^{2} \)
$19$ \( ( -512 + 400 T^{2} - 40 T^{4} + T^{6} )^{2} \)
$23$ \( 65536 + 37120 T^{4} + 4640 T^{8} + T^{12} \)
$29$ \( ( 64 + 100 T^{2} + 20 T^{4} + T^{6} )^{2} \)
$31$ \( ( 32768 + 3648 T^{2} + 112 T^{4} + T^{6} )^{2} \)
$37$ \( ( 32 + 96 T + 144 T^{2} + 32 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$41$ \( ( -64 - 20 T + 4 T^{2} + T^{3} )^{4} \)
$43$ \( 15352201216 + 24350720 T^{4} + 9600 T^{8} + T^{12} \)
$47$ \( 40960000 + 901376 T^{4} + 4896 T^{8} + T^{12} \)
$53$ \( ( 128 + 256 T + 256 T^{2} - 16 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$59$ \( ( -512 + 1860 T^{2} - 100 T^{4} + T^{6} )^{2} \)
$61$ \( ( 176 - 100 T + 8 T^{2} + T^{3} )^{4} \)
$67$ \( 15352201216 + 24350720 T^{4} + 9600 T^{8} + T^{12} \)
$71$ \( ( 204800 + 15616 T^{2} + 256 T^{4} + T^{6} )^{2} \)
$73$ \( ( 55112 + 4648 T + 196 T^{2} - 640 T^{3} + 242 T^{4} - 22 T^{5} + T^{6} )^{2} \)
$79$ \( ( -2048 + 12864 T^{2} - 304 T^{4} + T^{6} )^{2} \)
$83$ \( 65536 + 10264832 T^{4} + 16672 T^{8} + T^{12} \)
$89$ \( ( 1024 + 1040 T^{2} + 72 T^{4} + T^{6} )^{2} \)
$97$ \( ( 35912 - 47704 T + 31684 T^{2} - 2048 T^{3} + 50 T^{4} + 10 T^{5} + T^{6} )^{2} \)
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