Properties

Label 735.2.q.f
Level 735
Weight 2
Character orbit 735.q
Analytic conductor 5.869
Analytic rank 0
Dimension 12
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 735.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.89539436150784.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{3} - \beta_{4} ) q^{2} + ( \beta_{5} + \beta_{7} ) q^{3} + ( 1 - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{4} + ( \beta_{3} + \beta_{4} - \beta_{9} - \beta_{10} ) q^{5} -\beta_{2} q^{6} + ( -\beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{10} ) q^{8} + \beta_{8} q^{9} +O(q^{10})\) \( q + ( -\beta_{3} - \beta_{4} ) q^{2} + ( \beta_{5} + \beta_{7} ) q^{3} + ( 1 - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{4} + ( \beta_{3} + \beta_{4} - \beta_{9} - \beta_{10} ) q^{5} -\beta_{2} q^{6} + ( -\beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{10} ) q^{8} + \beta_{8} q^{9} + ( -2 - \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{10} + ( -2 + 2 \beta_{8} ) q^{11} + ( \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{12} + ( 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{10} ) q^{13} + ( \beta_{2} - \beta_{6} ) q^{15} + ( -4 \beta_{1} + 4 \beta_{2} - 3 \beta_{8} ) q^{16} + ( 2 \beta_{3} - \beta_{6} - \beta_{9} + \beta_{11} ) q^{17} -\beta_{3} q^{18} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{19} + ( 4 - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{20} + 2 \beta_{4} q^{22} + ( -2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{23} + ( 2 - \beta_{1} - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{24} + ( 1 + 2 \beta_{1} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{25} + ( -4 \beta_{1} + 4 \beta_{2} + 4 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{26} + \beta_{5} q^{27} + ( -2 - 2 \beta_{6} - 2 \beta_{10} ) q^{29} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} - \beta_{11} ) q^{30} + ( 2 + 2 \beta_{1} + \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{31} + ( \beta_{3} - 8 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} + 2 \beta_{9} - 2 \beta_{11} ) q^{32} -2 \beta_{7} q^{33} + ( -4 - 2 \beta_{2} + \beta_{6} + \beta_{10} ) q^{34} + ( 1 - \beta_{6} - \beta_{10} ) q^{36} + ( 4 \beta_{3} + 4 \beta_{4} ) q^{37} + ( -4 \beta_{5} + \beta_{6} - 4 \beta_{7} + \beta_{9} - \beta_{11} ) q^{38} + ( 2 + 2 \beta_{1} - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{39} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 5 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{40} + ( 4 \beta_{2} - \beta_{6} - \beta_{10} ) q^{41} + ( 4 \beta_{4} + 4 \beta_{6} - 4 \beta_{10} ) q^{43} + ( 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{44} + ( \beta_{3} - \beta_{9} ) q^{45} + ( 2 - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{46} + ( -4 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{47} + ( -4 \beta_{4} - 3 \beta_{5} ) q^{48} + ( -2 - \beta_{4} + 6 \beta_{5} - \beta_{6} + 3 \beta_{10} ) q^{50} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{51} + ( -2 \beta_{3} - 6 \beta_{5} + \beta_{6} - 6 \beta_{7} + \beta_{9} - \beta_{11} ) q^{52} + ( 2 \beta_{3} + 6 \beta_{5} + \beta_{6} + 6 \beta_{7} + \beta_{9} - \beta_{11} ) q^{53} + ( \beta_{1} - \beta_{2} ) q^{54} + ( -2 \beta_{4} + 2 \beta_{10} ) q^{55} + ( -2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{10} ) q^{57} + ( -2 \beta_{3} - 2 \beta_{4} + 4 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{58} + ( -4 - 4 \beta_{1} + 4 \beta_{6} + 4 \beta_{8} - 4 \beta_{9} - 4 \beta_{11} ) q^{59} + ( 2 + \beta_{1} - 2 \beta_{3} + 4 \beta_{5} + 4 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{60} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{61} + ( 8 \beta_{5} - 3 \beta_{6} + 3 \beta_{10} ) q^{62} + ( -1 + 4 \beta_{2} + 3 \beta_{6} + 3 \beta_{10} ) q^{64} + ( 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{7} - 6 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{65} + 2 \beta_{1} q^{66} + ( -4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} ) q^{67} + ( 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{68} + ( -2 - \beta_{6} - \beta_{10} ) q^{69} + 2 q^{71} + ( -\beta_{3} - \beta_{4} + 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{72} + ( -2 \beta_{3} + 6 \beta_{5} + \beta_{6} + 6 \beta_{7} + \beta_{9} - \beta_{11} ) q^{73} + ( -12 + 4 \beta_{6} + 12 \beta_{8} - 4 \beta_{9} - 4 \beta_{11} ) q^{74} + ( 2 \beta_{3} + 2 \beta_{4} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{75} + ( 2 + 2 \beta_{2} + 3 \beta_{6} + 3 \beta_{10} ) q^{76} + ( -4 \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{10} ) q^{78} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{79} + ( 4 - 4 \beta_{1} + \beta_{3} + 8 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + 4 \beta_{11} ) q^{80} + ( -1 + \beta_{8} ) q^{81} + ( -2 \beta_{3} - 2 \beta_{4} - 10 \beta_{7} + 3 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{82} + ( -4 \beta_{4} + 4 \beta_{5} ) q^{83} + ( 6 + 2 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{10} ) q^{85} + ( -8 \beta_{1} + 8 \beta_{2} + 4 \beta_{8} ) q^{86} + ( -2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} ) q^{87} + ( -2 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} ) q^{88} + ( -4 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{89} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{10} ) q^{90} + ( -4 \beta_{4} - 6 \beta_{5} - \beta_{6} + \beta_{10} ) q^{92} + ( 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{93} + ( 4 - 2 \beta_{6} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{94} + ( -2 + 6 \beta_{5} - \beta_{6} + 6 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + \beta_{11} ) q^{95} + ( -\beta_{1} + \beta_{2} - 8 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{96} + ( -2 \beta_{4} + 10 \beta_{5} + 3 \beta_{6} - 3 \beta_{10} ) q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 10q^{4} + 2q^{5} + 4q^{6} + 6q^{9} + O(q^{10}) \) \( 12q + 10q^{4} + 2q^{5} + 4q^{6} + 6q^{9} - 12q^{10} - 12q^{11} - 26q^{16} - 12q^{19} + 60q^{20} + 18q^{24} + 2q^{25} + 20q^{26} - 8q^{29} + 10q^{30} + 4q^{31} - 48q^{34} + 20q^{36} + 12q^{39} + 4q^{40} - 8q^{41} + 20q^{44} - 2q^{45} + 16q^{46} - 32q^{50} + 2q^{54} - 8q^{55} - 32q^{59} + 8q^{60} - 12q^{61} - 52q^{64} - 32q^{65} - 4q^{66} - 16q^{69} + 24q^{71} - 88q^{74} + 8q^{75} - 8q^{76} + 24q^{79} + 46q^{80} - 6q^{81} + 64q^{85} + 8q^{86} - 28q^{89} - 24q^{90} + 32q^{94} - 4q^{95} - 58q^{96} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} + 2 x^{10} - 8 x^{9} + 4 x^{8} + 16 x^{7} - 8 x^{6} + 20 x^{5} + 20 x^{4} - 24 x^{3} + 8 x^{2} - 8 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} + 3 \nu^{8} - 62 \nu^{7} + 112 \nu^{6} - 276 \nu^{5} + 338 \nu^{4} + 482 \nu^{3} - 170 \nu^{2} + 164 \nu - 164 \)\()/460\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{11} + 2 \nu^{10} - 8 \nu^{9} + 33 \nu^{8} - 13 \nu^{7} + 21 \nu^{6} - 96 \nu^{5} - 154 \nu^{4} + 150 \nu^{3} - 50 \nu^{2} + 50 \nu - 68 \)\()/230\)
\(\beta_{3}\)\(=\)\((\)\( -8 \nu^{11} + 21 \nu^{10} - 21 \nu^{9} + 61 \nu^{8} - 80 \nu^{7} - 130 \nu^{6} + 230 \nu^{5} - 58 \nu^{4} + 18 \nu^{3} - 298 \nu^{2} - 8 \nu + 8 \)\()/460\)
\(\beta_{4}\)\(=\)\((\)\( 10 \nu^{11} - 10 \nu^{10} - 6 \nu^{9} - 50 \nu^{8} - 50 \nu^{7} + 263 \nu^{6} + 20 \nu^{5} + 80 \nu^{4} + 216 \nu^{3} + 20 \nu^{2} - 20 \nu - 28 \)\()/230\)
\(\beta_{5}\)\(=\)\((\)\( 31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + 524 \nu^{4} + 1148 \nu^{3} + 154 \nu^{2} - 154 \nu - 142 \)\()/230\)
\(\beta_{6}\)\(=\)\((\)\( 19 \nu^{11} - 19 \nu^{10} + 7 \nu^{9} - 118 \nu^{8} - 72 \nu^{7} + 318 \nu^{6} + 130 \nu^{5} + 290 \nu^{4} + 967 \nu^{3} + 84 \nu^{2} - 84 \nu + 48 \)\()/115\)
\(\beta_{7}\)\(=\)\((\)\( -74 \nu^{11} + 105 \nu^{10} - 61 \nu^{9} + 505 \nu^{8} + 52 \nu^{7} - 1384 \nu^{6} - 78 \nu^{5} - 1158 \nu^{4} - 2246 \nu^{3} + 1062 \nu^{2} + 296 \nu + 296 \)\()/460\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + 86 \nu^{3} - 38 \nu^{2} + 32 \nu - 12 \)\()/20\)
\(\beta_{9}\)\(=\)\((\)\( 85 \nu^{11} - 88 \nu^{10} + 88 \nu^{9} - 674 \nu^{8} - 162 \nu^{7} + 1042 \nu^{6} + 966 \nu^{5} + 2232 \nu^{4} + 2908 \nu^{3} + 688 \nu^{2} + 1028 \nu - 1028 \)\()/460\)
\(\beta_{10}\)\(=\)\((\)\( -43 \nu^{11} + 43 \nu^{10} - 34 \nu^{9} + 330 \nu^{8} + 100 \nu^{7} - 572 \nu^{6} - 546 \nu^{5} - 1034 \nu^{4} - 1490 \nu^{3} - 316 \nu^{2} + 316 \nu + 516 \)\()/230\)
\(\beta_{11}\)\(=\)\((\)\( 225 \nu^{11} - 378 \nu^{10} + 330 \nu^{9} - 1680 \nu^{8} + 374 \nu^{7} + 3630 \nu^{6} - 630 \nu^{5} + 3948 \nu^{4} + 6540 \nu^{3} - 2940 \nu^{2} + 756 \nu - 900 \)\()/460\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} - \beta_{9} + 4 \beta_{7} - \beta_{6} + 4 \beta_{5} + 2 \beta_{3}\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(-2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 7 \beta_{8} + 5 \beta_{2} - 5 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-5 \beta_{9} + 9 \beta_{8} + 9 \beta_{7} + 9 \beta_{5} + 8 \beta_{3} - 8 \beta_{1} - 9\)
\(\nu^{6}\)\(=\)\(8 \beta_{10} - 8 \beta_{6} + 28 \beta_{5} - 22 \beta_{4}\)
\(\nu^{7}\)\(=\)\(-22 \beta_{11} + 39 \beta_{8} - 39 \beta_{7} - 33 \beta_{4} - 33 \beta_{3} + 33 \beta_{2} - 33 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-33 \beta_{11} - 33 \beta_{9} + 116 \beta_{8} + 33 \beta_{6} - 94 \beta_{1} - 116\)
\(\nu^{9}\)\(=\)\(94 \beta_{10} + 166 \beta_{5} - 138 \beta_{4} - 138 \beta_{2} - 166\)
\(\nu^{10}\)\(=\)\(-138 \beta_{11} + 138 \beta_{10} + 138 \beta_{9} - 486 \beta_{7} - 398 \beta_{4} - 398 \beta_{3}\)
\(\nu^{11}\)\(=\)\(-398 \beta_{11} + 702 \beta_{8} - 702 \beta_{7} + 398 \beta_{6} - 702 \beta_{5} - 580 \beta_{3} - 580 \beta_{1} - 702\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-\beta_{8}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.312819 + 1.16746i
1.98293 0.531325i
−0.147520 0.550552i
0.550552 0.147520i
−0.531325 1.98293i
−1.16746 + 0.312819i
0.312819 1.16746i
1.98293 + 0.531325i
−0.147520 + 0.550552i
0.550552 + 0.147520i
−0.531325 + 1.98293i
−1.16746 0.312819i
−2.34630 1.35464i −0.866025 + 0.500000i 2.67009 + 4.62473i 1.55199 1.60976i 2.70928 0 9.04945i 0.500000 0.866025i −5.82208 + 1.67458i
79.2 −1.64823 0.951606i 0.866025 0.500000i 0.811108 + 1.40488i 2.07321 + 0.837733i −1.90321 0 0.719004i 0.500000 0.866025i −2.61994 3.35366i
79.3 −0.167954 0.0969683i −0.866025 + 0.500000i −0.981194 1.69948i 0.710109 + 2.12032i 0.193937 0 0.768452i 0.500000 0.866025i 0.0863379 0.424974i
79.4 0.167954 + 0.0969683i 0.866025 0.500000i −0.981194 1.69948i −2.19130 + 0.445186i 0.193937 0 0.768452i 0.500000 0.866025i −0.411207 0.137716i
79.5 1.64823 + 0.951606i −0.866025 + 0.500000i 0.811108 + 1.40488i −1.76210 1.37659i −1.90321 0 0.719004i 0.500000 0.866025i −1.59438 3.94576i
79.6 2.34630 + 1.35464i 0.866025 0.500000i 2.67009 + 4.62473i 0.618092 2.14894i 2.70928 0 9.04945i 0.500000 0.866025i 4.36127 4.20478i
214.1 −2.34630 + 1.35464i −0.866025 0.500000i 2.67009 4.62473i 1.55199 + 1.60976i 2.70928 0 9.04945i 0.500000 + 0.866025i −5.82208 1.67458i
214.2 −1.64823 + 0.951606i 0.866025 + 0.500000i 0.811108 1.40488i 2.07321 0.837733i −1.90321 0 0.719004i 0.500000 + 0.866025i −2.61994 + 3.35366i
214.3 −0.167954 + 0.0969683i −0.866025 0.500000i −0.981194 + 1.69948i 0.710109 2.12032i 0.193937 0 0.768452i 0.500000 + 0.866025i 0.0863379 + 0.424974i
214.4 0.167954 0.0969683i 0.866025 + 0.500000i −0.981194 + 1.69948i −2.19130 0.445186i 0.193937 0 0.768452i 0.500000 + 0.866025i −0.411207 + 0.137716i
214.5 1.64823 0.951606i −0.866025 0.500000i 0.811108 1.40488i −1.76210 + 1.37659i −1.90321 0 0.719004i 0.500000 + 0.866025i −1.59438 + 3.94576i
214.6 2.34630 1.35464i 0.866025 + 0.500000i 2.67009 4.62473i 0.618092 + 2.14894i 2.70928 0 9.04945i 0.500000 + 0.866025i 4.36127 + 4.20478i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 214.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.q.f 12
5.b even 2 1 inner 735.2.q.f 12
7.b odd 2 1 735.2.q.e 12
7.c even 3 1 735.2.d.b 6
7.c even 3 1 inner 735.2.q.f 12
7.d odd 6 1 105.2.d.b 6
7.d odd 6 1 735.2.q.e 12
21.g even 6 1 315.2.d.e 6
21.h odd 6 1 2205.2.d.l 6
28.f even 6 1 1680.2.t.k 6
35.c odd 2 1 735.2.q.e 12
35.i odd 6 1 105.2.d.b 6
35.i odd 6 1 735.2.q.e 12
35.j even 6 1 735.2.d.b 6
35.j even 6 1 inner 735.2.q.f 12
35.k even 12 1 525.2.a.j 3
35.k even 12 1 525.2.a.k 3
35.l odd 12 1 3675.2.a.bi 3
35.l odd 12 1 3675.2.a.bj 3
84.j odd 6 1 5040.2.t.v 6
105.o odd 6 1 2205.2.d.l 6
105.p even 6 1 315.2.d.e 6
105.w odd 12 1 1575.2.a.w 3
105.w odd 12 1 1575.2.a.x 3
140.s even 6 1 1680.2.t.k 6
140.x odd 12 1 8400.2.a.dg 3
140.x odd 12 1 8400.2.a.dj 3
420.be odd 6 1 5040.2.t.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 7.d odd 6 1
105.2.d.b 6 35.i odd 6 1
315.2.d.e 6 21.g even 6 1
315.2.d.e 6 105.p even 6 1
525.2.a.j 3 35.k even 12 1
525.2.a.k 3 35.k even 12 1
735.2.d.b 6 7.c even 3 1
735.2.d.b 6 35.j even 6 1
735.2.q.e 12 7.b odd 2 1
735.2.q.e 12 7.d odd 6 1
735.2.q.e 12 35.c odd 2 1
735.2.q.e 12 35.i odd 6 1
735.2.q.f 12 1.a even 1 1 trivial
735.2.q.f 12 5.b even 2 1 inner
735.2.q.f 12 7.c even 3 1 inner
735.2.q.f 12 35.j even 6 1 inner
1575.2.a.w 3 105.w odd 12 1
1575.2.a.x 3 105.w odd 12 1
1680.2.t.k 6 28.f even 6 1
1680.2.t.k 6 140.s even 6 1
2205.2.d.l 6 21.h odd 6 1
2205.2.d.l 6 105.o odd 6 1
3675.2.a.bi 3 35.l odd 12 1
3675.2.a.bj 3 35.l odd 12 1
5040.2.t.v 6 84.j odd 6 1
5040.2.t.v 6 420.be odd 6 1
8400.2.a.dg 3 140.x odd 12 1
8400.2.a.dj 3 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{12} - 11 T_{2}^{10} + 94 T_{2}^{8} - 295 T_{2}^{6} + 718 T_{2}^{4} - 27 T_{2}^{2} + 1 \)
\( T_{19}^{6} + 6 T_{19}^{5} + 40 T_{19}^{4} + 56 T_{19}^{3} + 256 T_{19}^{2} + 160 T_{19} + 1600 \)
\( T_{73}^{12} - 140 T_{73}^{10} + 14880 T_{73}^{8} - 639168 T_{73}^{6} + 20764160 T_{73}^{4} - 51051520 T_{73}^{2} + 116985856 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} + 2 T^{4} - 7 T^{6} + 2 T^{8} + 21 T^{10} + 117 T^{12} + 84 T^{14} + 32 T^{16} - 448 T^{18} + 512 T^{20} + 1024 T^{22} + 4096 T^{24} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{3} \)
$5$ \( 1 - 2 T + T^{2} + 18 T^{3} - 30 T^{4} - 26 T^{5} + 249 T^{6} - 130 T^{7} - 750 T^{8} + 2250 T^{9} + 625 T^{10} - 6250 T^{11} + 15625 T^{12} \)
$7$ 1
$11$ \( ( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{6} \)
$13$ \( ( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 60671 T^{8} - 971074 T^{10} + 4826809 T^{12} )^{2} \)
$17$ \( 1 + 70 T^{2} + 2485 T^{4} + 66610 T^{6} + 1548890 T^{8} + 31831870 T^{10} + 576997613 T^{12} + 9199410430 T^{14} + 129364841690 T^{16} + 1607803471090 T^{18} + 17334757240885 T^{20} + 141119573031430 T^{22} + 582622237229761 T^{24} \)
$19$ \( ( 1 + 6 T - 17 T^{2} - 58 T^{3} + 674 T^{4} + 46 T^{5} - 17305 T^{6} + 874 T^{7} + 243314 T^{8} - 397822 T^{9} - 2215457 T^{10} + 14856594 T^{11} + 47045881 T^{12} )^{2} \)
$23$ \( 1 + 106 T^{2} + 6053 T^{4} + 248702 T^{6} + 8184794 T^{8} + 227653746 T^{10} + 5545513725 T^{12} + 120428831634 T^{14} + 2290440937754 T^{16} + 36816821666078 T^{18} + 474016393905893 T^{20} + 4391210188646794 T^{22} + 21914624432020321 T^{24} \)
$29$ \( ( 1 + 2 T + 35 T^{2} + 76 T^{3} + 1015 T^{4} + 1682 T^{5} + 24389 T^{6} )^{4} \)
$31$ \( ( 1 - 2 T - 37 T^{2} - 202 T^{3} + 530 T^{4} + 5622 T^{5} + 7227 T^{6} + 174282 T^{7} + 509330 T^{8} - 6017782 T^{9} - 34170277 T^{10} - 57258302 T^{11} + 887503681 T^{12} )^{2} \)
$37$ \( 1 + 46 T^{2} + 717 T^{4} - 85222 T^{6} - 3398278 T^{8} + 14638566 T^{10} + 4079564917 T^{12} + 20040196854 T^{14} - 6368920094758 T^{16} - 218656336027798 T^{18} + 2518447768461357 T^{20} + 221194881131221054 T^{22} + 6582952005840035281 T^{24} \)
$41$ \( ( 1 + 2 T + 63 T^{2} - 36 T^{3} + 2583 T^{4} + 3362 T^{5} + 68921 T^{6} )^{4} \)
$43$ \( ( 1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 5249311 T^{8} + 157264846 T^{10} + 6321363049 T^{12} )^{2} \)
$47$ \( 1 + 154 T^{2} + 11573 T^{4} + 565358 T^{6} + 20169434 T^{8} + 410981154 T^{10} + 5646494445 T^{12} + 907857369186 T^{14} + 98420403870554 T^{16} + 6094115619972782 T^{18} + 275568020536560053 T^{20} + 8100266364317827546 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 146 T^{2} + 7213 T^{4} + 289382 T^{6} + 30094394 T^{8} + 2063064026 T^{10} + 100360862165 T^{12} + 5795146849034 T^{14} + 237459244063514 T^{16} + 6413967152232278 T^{18} + 449079146937146893 T^{20} + 25533570673364905154 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( ( 1 + 16 T + 143 T^{2} + 592 T^{3} - 3626 T^{4} - 88944 T^{5} - 864085 T^{6} - 5247696 T^{7} - 12622106 T^{8} + 121584368 T^{9} + 1732782623 T^{10} + 11438788784 T^{11} + 42180533641 T^{12} )^{2} \)
$61$ \( ( 1 + 6 T - 95 T^{2} - 182 T^{3} + 6266 T^{4} - 10162 T^{5} - 492559 T^{6} - 619882 T^{7} + 23315786 T^{8} - 41310542 T^{9} - 1315354895 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} )^{2} \)
$67$ \( 1 + 274 T^{2} + 38973 T^{4} + 3966278 T^{6} + 329505914 T^{8} + 23681216154 T^{10} + 1596172229125 T^{12} + 106304979315306 T^{14} + 6639913543229594 T^{16} + 358783091112496982 T^{18} + 15825675597414969693 T^{20} + \)\(49\!\cdots\!26\)\( T^{22} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( ( 1 - 2 T + 71 T^{2} )^{12} \)
$73$ \( 1 + 298 T^{2} + 45029 T^{4} + 5080382 T^{6} + 496253018 T^{8} + 43754080818 T^{10} + 3427677334845 T^{12} + 233165496679122 T^{14} + 14092712802141338 T^{16} + 768835679222562398 T^{18} + 36314091477898573349 T^{20} + \)\(12\!\cdots\!02\)\( T^{22} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( ( 1 - 12 T - 77 T^{2} + 500 T^{3} + 12470 T^{4} - 4172 T^{5} - 1287289 T^{6} - 329588 T^{7} + 77825270 T^{8} + 246519500 T^{9} - 2999156237 T^{10} - 36924676788 T^{11} + 243087455521 T^{12} )^{2} \)
$83$ \( ( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 329177087 T^{8} - 14522246226 T^{10} + 326940373369 T^{12} )^{2} \)
$89$ \( ( 1 + 14 T - 123 T^{2} - 598 T^{3} + 37922 T^{4} + 123654 T^{5} - 2788283 T^{6} + 11005206 T^{7} + 300380162 T^{8} - 421571462 T^{9} - 7717295643 T^{10} + 78176832286 T^{11} + 496981290961 T^{12} )^{2} \)
$97$ \( ( 1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 82037071 T^{8} - 2301761306 T^{10} + 832972004929 T^{12} )^{2} \)
show more
show less