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 $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(79,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.89539436150784.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3}) q^{2} + (\beta_{7} + \beta_{5}) q^{3} + (\beta_{11} + \beta_{9} - \beta_{8} - \beta_{6} + 1) q^{4} + ( - \beta_{10} - \beta_{9} + \beta_{4} + \beta_{3}) q^{5} - \beta_{2} q^{6} + ( - \beta_{10} + \beta_{6} - 2 \beta_{5} - \beta_{4}) q^{8} + \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{3}) q^{2} + (\beta_{7} + \beta_{5}) q^{3} + (\beta_{11} + \beta_{9} - \beta_{8} - \beta_{6} + 1) q^{4} + ( - \beta_{10} - \beta_{9} + \beta_{4} + \beta_{3}) q^{5} - \beta_{2} q^{6} + ( - \beta_{10} + \beta_{6} - 2 \beta_{5} - \beta_{4}) q^{8} + \beta_{8} q^{9} + ( - \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{5} - \beta_{3} - \beta_1 - 2) q^{10} + (2 \beta_{8} - 2) q^{11} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{7}) q^{12} + ( - \beta_{10} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4}) q^{13} + ( - \beta_{6} + \beta_{2}) q^{15} + ( - 3 \beta_{8} + 4 \beta_{2} - 4 \beta_1) q^{16} + (\beta_{11} - \beta_{9} - \beta_{6} + 2 \beta_{3}) q^{17} - \beta_{3} q^{18} + ( - \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} + 2 \beta_{2} - 2 \beta_1) q^{19} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{2} + 4) q^{20} + 2 \beta_{4} q^{22} + (\beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{7}) q^{23} + (\beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{6} - \beta_1 + 2) q^{24} + ( - \beta_{11} + \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_1 + 1) q^{25} + ( - \beta_{11} - \beta_{10} - \beta_{9} + 4 \beta_{8} + 4 \beta_{2} - 4 \beta_1) q^{26} + \beta_{5} q^{27} + ( - 2 \beta_{10} - 2 \beta_{6} - 2) q^{29} + ( - \beta_{11} + \beta_{8} - 2 \beta_{7} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{30} + ( - \beta_{11} - \beta_{9} - 2 \beta_{8} + \beta_{6} + 2 \beta_1 + 2) q^{31} + ( - 2 \beta_{11} + 2 \beta_{9} - 8 \beta_{7} + 2 \beta_{6} - 8 \beta_{5} + \beta_{3}) q^{32} - 2 \beta_{7} q^{33} + (\beta_{10} + \beta_{6} - 2 \beta_{2} - 4) q^{34} + ( - \beta_{10} - \beta_{6} + 1) q^{36} + (4 \beta_{4} + 4 \beta_{3}) q^{37} + ( - \beta_{11} + \beta_{9} - 4 \beta_{7} + \beta_{6} - 4 \beta_{5}) q^{38} + (\beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{6} + 2 \beta_1 + 2) q^{39} + (\beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 5 \beta_{7} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1) q^{40} + ( - \beta_{10} - \beta_{6} + 4 \beta_{2}) q^{41} + ( - 4 \beta_{10} + 4 \beta_{6} + 4 \beta_{4}) q^{43} + ( - 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8}) q^{44} + ( - \beta_{9} + \beta_{3}) q^{45} + (\beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{6} + 2) q^{46} + (2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 4 \beta_{7}) q^{47} + ( - 3 \beta_{5} - 4 \beta_{4}) q^{48} + (3 \beta_{10} - \beta_{6} + 6 \beta_{5} - \beta_{4} - 2) q^{50} + ( - \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{2} - 2 \beta_1) q^{51} + ( - \beta_{11} + \beta_{9} - 6 \beta_{7} + \beta_{6} - 6 \beta_{5} - 2 \beta_{3}) q^{52} + ( - \beta_{11} + \beta_{9} + 6 \beta_{7} + \beta_{6} + 6 \beta_{5} + 2 \beta_{3}) q^{53} + ( - \beta_{2} + \beta_1) q^{54} + (2 \beta_{10} - 2 \beta_{4}) q^{55} + (\beta_{10} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4}) q^{57} + (2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 4 \beta_{7} - 2 \beta_{4} - 2 \beta_{3}) q^{58} + ( - 4 \beta_{11} - 4 \beta_{9} + 4 \beta_{8} + 4 \beta_{6} - 4 \beta_1 - 4) q^{59} + ( - \beta_{9} - 2 \beta_{8} + 4 \beta_{7} + 4 \beta_{5} - 2 \beta_{3} + \beta_1 + 2) q^{60} + ( - 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + 4 \beta_{2} - 4 \beta_1) q^{61} + (3 \beta_{10} - 3 \beta_{6} + 8 \beta_{5}) q^{62} + (3 \beta_{10} + 3 \beta_{6} + 4 \beta_{2} - 1) q^{64} + (\beta_{11} + \beta_{10} + \beta_{9} - 6 \beta_{8} + 2 \beta_{7} - 4 \beta_{2} + 4 \beta_1) q^{65} + 2 \beta_1 q^{66} + (2 \beta_{11} - 2 \beta_{9} - 4 \beta_{7} - 2 \beta_{6} - 4 \beta_{5}) q^{67} + ( - \beta_{11} + \beta_{10} + \beta_{9} + 4 \beta_{7} + 2 \beta_{4} + 2 \beta_{3}) q^{68} + ( - \beta_{10} - \beta_{6} - 2) q^{69} + 2 q^{71} + (\beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{7} - \beta_{4} - \beta_{3}) q^{72} + ( - \beta_{11} + \beta_{9} + 6 \beta_{7} + \beta_{6} + 6 \beta_{5} - 2 \beta_{3}) q^{73} + ( - 4 \beta_{11} - 4 \beta_{9} + 12 \beta_{8} + 4 \beta_{6} - 12) q^{74} + (\beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{7} + 2 \beta_{4} + 2 \beta_{3}) q^{75} + (3 \beta_{10} + 3 \beta_{6} + 2 \beta_{2} + 2) q^{76} + (\beta_{10} - \beta_{6} + 4 \beta_{5} - 4 \beta_{4}) q^{78} + (2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 4 \beta_{2} + 4 \beta_1) q^{79} + (4 \beta_{11} + 3 \beta_{9} - 4 \beta_{8} + 8 \beta_{7} - 4 \beta_{6} + 8 \beta_{5} + \cdots + 4) q^{80}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 10 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{9} - 12 q^{10} - 12 q^{11} - 26 q^{16} - 12 q^{19} + 60 q^{20} + 18 q^{24} + 2 q^{25} + 20 q^{26} - 8 q^{29} + 10 q^{30} + 4 q^{31} - 48 q^{34} + 20 q^{36} + 12 q^{39} + 4 q^{40} - 8 q^{41} + 20 q^{44} - 2 q^{45} + 16 q^{46} - 32 q^{50} + 2 q^{54} - 8 q^{55} - 32 q^{59} + 8 q^{60} - 12 q^{61} - 52 q^{64} - 32 q^{65} - 4 q^{66} - 16 q^{69} + 24 q^{71} - 88 q^{74} + 8 q^{75} - 8 q^{76} + 24 q^{79} + 46 q^{80} - 6 q^{81} + 64 q^{85} + 8 q^{86} - 28 q^{89} - 24 q^{90} + 32 q^{94} - 4 q^{95} - 58 q^{96} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} + 3 \nu^{8} - 62 \nu^{7} + 112 \nu^{6} - 276 \nu^{5} + 338 \nu^{4} + 482 \nu^{3} - 170 \nu^{2} + 164 \nu - 164 ) / 460 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2 \nu^{11} + 2 \nu^{10} - 8 \nu^{9} + 33 \nu^{8} - 13 \nu^{7} + 21 \nu^{6} - 96 \nu^{5} - 154 \nu^{4} + 150 \nu^{3} - 50 \nu^{2} + 50 \nu - 68 ) / 230 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8 \nu^{11} + 21 \nu^{10} - 21 \nu^{9} + 61 \nu^{8} - 80 \nu^{7} - 130 \nu^{6} + 230 \nu^{5} - 58 \nu^{4} + 18 \nu^{3} - 298 \nu^{2} - 8 \nu + 8 ) / 460 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10 \nu^{11} - 10 \nu^{10} - 6 \nu^{9} - 50 \nu^{8} - 50 \nu^{7} + 263 \nu^{6} + 20 \nu^{5} + 80 \nu^{4} + 216 \nu^{3} + 20 \nu^{2} - 20 \nu - 28 ) / 230 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + 524 \nu^{4} + 1148 \nu^{3} + 154 \nu^{2} - 154 \nu - 142 ) / 230 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19 \nu^{11} - 19 \nu^{10} + 7 \nu^{9} - 118 \nu^{8} - 72 \nu^{7} + 318 \nu^{6} + 130 \nu^{5} + 290 \nu^{4} + 967 \nu^{3} + 84 \nu^{2} - 84 \nu + 48 ) / 115 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 74 \nu^{11} + 105 \nu^{10} - 61 \nu^{9} + 505 \nu^{8} + 52 \nu^{7} - 1384 \nu^{6} - 78 \nu^{5} - 1158 \nu^{4} - 2246 \nu^{3} + 1062 \nu^{2} + 296 \nu + 296 ) / 460 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + 86 \nu^{3} - 38 \nu^{2} + 32 \nu - 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 85 \nu^{11} - 88 \nu^{10} + 88 \nu^{9} - 674 \nu^{8} - 162 \nu^{7} + 1042 \nu^{6} + 966 \nu^{5} + 2232 \nu^{4} + 2908 \nu^{3} + 688 \nu^{2} + 1028 \nu - 1028 ) / 460 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 43 \nu^{11} + 43 \nu^{10} - 34 \nu^{9} + 330 \nu^{8} + 100 \nu^{7} - 572 \nu^{6} - 546 \nu^{5} - 1034 \nu^{4} - 1490 \nu^{3} - 316 \nu^{2} + 316 \nu + 516 ) / 230 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 225 \nu^{11} - 378 \nu^{10} + 330 \nu^{9} - 1680 \nu^{8} + 374 \nu^{7} + 3630 \nu^{6} - 630 \nu^{5} + 3948 \nu^{4} + 6540 \nu^{3} - 2940 \nu^{2} + 756 \nu - 900 ) / 460 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{9} + 4\beta_{7} - \beta_{6} + 4\beta_{5} + 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 2\beta_{5} - 2\beta_{4} + 2\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{11} - 2\beta_{10} - 2\beta_{9} + 7\beta_{8} + 5\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{9} + 9\beta_{8} + 9\beta_{7} + 9\beta_{5} + 8\beta_{3} - 8\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{10} - 8\beta_{6} + 28\beta_{5} - 22\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -22\beta_{11} + 39\beta_{8} - 39\beta_{7} - 33\beta_{4} - 33\beta_{3} + 33\beta_{2} - 33\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -33\beta_{11} - 33\beta_{9} + 116\beta_{8} + 33\beta_{6} - 94\beta _1 - 116 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 94\beta_{10} + 166\beta_{5} - 138\beta_{4} - 138\beta_{2} - 166 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -138\beta_{11} + 138\beta_{10} + 138\beta_{9} - 486\beta_{7} - 398\beta_{4} - 398\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -398\beta_{11} + 702\beta_{8} - 702\beta_{7} + 398\beta_{6} - 702\beta_{5} - 580\beta_{3} - 580\beta _1 - 702 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 79.6
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 \(S_{2}^{\mathrm{new}}(735, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 11 T^{10} + 94 T^{8} - 295 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} - 2 T^{11} + T^{10} + 18 T^{9} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{6} \) Copy content Toggle raw display
$13$ \( (T^{6} + 44 T^{4} + 112 T^{2} + 64)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 32 T^{10} + 768 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( (T^{6} + 6 T^{5} + 40 T^{4} + 56 T^{3} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 32 T^{10} + 832 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( (T^{3} + 2 T^{2} - 52 T - 40)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} - 2 T^{5} + 56 T^{4} - 264 T^{3} + \cdots + 33856)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 176 T^{10} + \cdots + 16777216 \) Copy content Toggle raw display
$41$ \( (T^{3} + 2 T^{2} - 60 T - 200)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 304 T^{4} + 27392 T^{2} + \cdots + 692224)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 128 T^{10} + \cdots + 268435456 \) Copy content Toggle raw display
$53$ \( T^{12} - 172 T^{10} + \cdots + 7676563456 \) Copy content Toggle raw display
$59$ \( (T^{6} + 16 T^{5} + 320 T^{4} + \cdots + 1638400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 6 T^{5} + 88 T^{4} + 184 T^{3} + \cdots + 61504)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 128 T^{10} + \cdots + 268435456 \) Copy content Toggle raw display
$71$ \( (T - 2)^{12} \) Copy content Toggle raw display
$73$ \( T^{12} - 140 T^{10} + \cdots + 116985856 \) Copy content Toggle raw display
$79$ \( (T^{6} - 12 T^{5} + 160 T^{4} + \cdots + 102400)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 192 T^{4} + 8192 T^{2} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 14 T^{5} + 144 T^{4} + 648 T^{3} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 556 T^{4} + 83312 T^{2} + \cdots + 3474496)^{2} \) Copy content Toggle raw display
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