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Properties

Label 420.2.bh.b
Level $420$
Weight $2$
Character orbit 420.bh
Analytic conductor $3.354$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.bh (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.29471584693248.1
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 13x^{6} - 36x^{5} + 39x^{4} - 36x^{3} + 54x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + (\beta_{7} + 1) q^{5} + (\beta_{9} - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{7} + (\beta_{9} - \beta_{7} + \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + (\beta_{7} + 1) q^{5} + (\beta_{9} - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{7} + (\beta_{9} - \beta_{7} + \beta_{3} - \beta_{2}) q^{9} + (\beta_{8} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_1 + 1) q^{11} + (\beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{2} + 1) q^{13} + \beta_1 q^{15} + ( - 2 \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{8} + \beta_{6}) q^{19} + (2 \beta_{8} + 2 \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + 3) q^{21} + (2 \beta_{9} + \beta_{7} - \beta_{4} + 2 \beta_{2} - 3) q^{23} + \beta_{7} q^{25} + ( - 2 \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} - 2 \beta_{2}) q^{27} + ( - 2 \beta_{9} - \beta_{8} + 4 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \cdots + 2) q^{29}+ \cdots + ( - \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{3} - 2 \beta_{2} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} + 5 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} + 5 q^{5} - 5 q^{7} + 3 q^{9} + 6 q^{11} + 2 q^{15} - 6 q^{17} + 3 q^{19} + 12 q^{21} - 24 q^{23} - 5 q^{25} - 8 q^{27} + 15 q^{31} - 4 q^{33} - q^{35} - q^{37} - 21 q^{39} + 8 q^{41} - 26 q^{43} + 3 q^{45} - 14 q^{47} - 13 q^{49} + 40 q^{51} + 24 q^{53} + 18 q^{57} + 42 q^{61} - 49 q^{63} - 9 q^{65} + 7 q^{67} + 14 q^{69} - 3 q^{73} + q^{75} + 26 q^{77} + q^{79} - 13 q^{81} + 8 q^{83} - 12 q^{85} + 8 q^{87} - 28 q^{89} - 11 q^{91} + 25 q^{93} + 3 q^{95} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 13x^{6} - 36x^{5} + 39x^{4} - 36x^{3} + 54x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 2\nu^{5} + 4\nu^{4} - 4\nu^{3} + 18\nu^{2} - 21\nu + 18 ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - \nu^{8} - 9\nu^{7} + 7\nu^{6} - 18\nu^{5} + 22\nu^{4} - 51\nu^{3} + 21\nu^{2} - 171\nu + 135 ) / 216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{9} - 4\nu^{8} + 7\nu^{7} + 28\nu^{6} + 32\nu^{5} - 30\nu^{4} + 123\nu^{3} - 216\nu^{2} - 243\nu - 486 ) / 648 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{9} + 2\nu^{8} - \nu^{7} + 4\nu^{6} + 16\nu^{5} - 58\nu^{4} + 9\nu^{3} + 6\nu^{2} + 9\nu - 486 ) / 216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{9} - \nu^{7} + 6\nu^{6} - 20\nu^{5} + 22\nu^{4} - 15\nu^{3} + 48\nu^{2} - 63\nu + 216 ) / 72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{9} - 7\nu^{8} + 16\nu^{7} - 23\nu^{6} + 50\nu^{5} - 108\nu^{4} + 114\nu^{3} - 153\nu^{2} - 729 ) / 648 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - 2\nu^{8} + 2\nu^{7} - 4\nu^{6} + 13\nu^{5} - 36\nu^{4} + 39\nu^{3} - 36\nu^{2} + 54\nu - 162 ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13 \nu^{9} + 5 \nu^{8} + 25 \nu^{7} - 35 \nu^{6} - 94 \nu^{5} + 186 \nu^{4} + 231 \nu^{3} - 297 \nu^{2} + 243 \nu + 1701 ) / 648 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 2\beta_{8} + 3\beta_{7} - 2\beta_{6} - 2\beta_{5} + \beta_{4} + 2\beta_{2} + 2\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{9} - \beta_{8} - 2\beta_{7} - 4\beta_{6} - 4\beta_{5} + 2\beta_{4} - 4\beta_{3} + 2\beta_{2} - 2\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{9} - 8\beta_{7} + 4\beta_{6} + 12\beta_{4} - 4\beta_{3} - 2\beta_{2} + 4\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -8\beta_{9} - 10\beta_{8} + 18\beta_{7} + 12\beta_{6} - 4\beta_{5} + 6\beta_{4} - 18\beta_{3} - 8\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{9} - 16 \beta_{8} - 7 \beta_{7} - 10 \beta_{6} + 38 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 3 \beta_{2} - 8 \beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16\beta_{9} - 29\beta_{8} + 88\beta_{7} - 9\beta_{6} + 25\beta_{5} + 26\beta_{3} + 18\beta_{2} + 46\beta _1 + 48 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(1\) \(1 + \beta_{7}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−1.31611 − 1.12599i
0.527154 − 1.64988i
−1.08831 + 1.34743i
1.72689 + 0.133595i
1.15038 + 1.29484i
−1.31611 + 1.12599i
0.527154 + 1.64988i
−1.08831 − 1.34743i
1.72689 − 0.133595i
1.15038 − 1.29484i
0 −1.63319 + 0.576792i 0 0.500000 + 0.866025i 0 −1.73439 − 1.99797i 0 2.33462 − 1.88402i 0
101.2 0 −1.16526 − 1.28147i 0 0.500000 + 0.866025i 0 −1.80025 + 1.93884i 0 −0.284326 + 2.98650i 0
101.3 0 0.622752 + 1.61622i 0 0.500000 + 0.866025i 0 2.57325 − 0.615143i 0 −2.22436 + 2.01301i 0
101.4 0 0.979142 − 1.42873i 0 0.500000 + 0.866025i 0 −0.456468 − 2.60608i 0 −1.08256 − 2.79787i 0
101.5 0 1.69656 − 0.348838i 0 0.500000 + 0.866025i 0 −1.08214 + 2.41433i 0 2.75662 − 1.18365i 0
341.1 0 −1.63319 − 0.576792i 0 0.500000 − 0.866025i 0 −1.73439 + 1.99797i 0 2.33462 + 1.88402i 0
341.2 0 −1.16526 + 1.28147i 0 0.500000 − 0.866025i 0 −1.80025 − 1.93884i 0 −0.284326 − 2.98650i 0
341.3 0 0.622752 − 1.61622i 0 0.500000 − 0.866025i 0 2.57325 + 0.615143i 0 −2.22436 − 2.01301i 0
341.4 0 0.979142 + 1.42873i 0 0.500000 − 0.866025i 0 −0.456468 + 2.60608i 0 −1.08256 + 2.79787i 0
341.5 0 1.69656 + 0.348838i 0 0.500000 − 0.866025i 0 −1.08214 − 2.41433i 0 2.75662 + 1.18365i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.bh.b yes 10
3.b odd 2 1 420.2.bh.a ✓ 10
5.b even 2 1 2100.2.bi.j 10
5.c odd 4 2 2100.2.bo.g 20
7.c even 3 1 2940.2.d.a 10
7.d odd 6 1 420.2.bh.a ✓ 10
7.d odd 6 1 2940.2.d.b 10
15.d odd 2 1 2100.2.bi.k 10
15.e even 4 2 2100.2.bo.h 20
21.g even 6 1 inner 420.2.bh.b yes 10
21.g even 6 1 2940.2.d.a 10
21.h odd 6 1 2940.2.d.b 10
35.i odd 6 1 2100.2.bi.k 10
35.k even 12 2 2100.2.bo.h 20
105.p even 6 1 2100.2.bi.j 10
105.w odd 12 2 2100.2.bo.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bh.a ✓ 10 3.b odd 2 1
420.2.bh.a ✓ 10 7.d odd 6 1
420.2.bh.b yes 10 1.a even 1 1 trivial
420.2.bh.b yes 10 21.g even 6 1 inner
2100.2.bi.j 10 5.b even 2 1
2100.2.bi.j 10 105.p even 6 1
2100.2.bi.k 10 15.d odd 2 1
2100.2.bi.k 10 35.i odd 6 1
2100.2.bo.g 20 5.c odd 4 2
2100.2.bo.g 20 105.w odd 12 2
2100.2.bo.h 20 15.e even 4 2
2100.2.bo.h 20 35.k even 12 2
2940.2.d.a 10 7.c even 3 1
2940.2.d.a 10 21.g even 6 1
2940.2.d.b 10 7.d odd 6 1
2940.2.d.b 10 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{10} - 6 T_{11}^{9} - 4 T_{11}^{8} + 96 T_{11}^{7} + 84 T_{11}^{6} - 2280 T_{11}^{5} + 8156 T_{11}^{4} - 14352 T_{11}^{3} + 14128 T_{11}^{2} - 7488 T_{11} + 1728 \) acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} - T^{8} + 4 T^{7} + T^{6} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + 5 T^{9} + 19 T^{8} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} - 6 T^{9} - 4 T^{8} + 96 T^{7} + \cdots + 1728 \) Copy content Toggle raw display
$13$ \( T^{10} + 81 T^{8} + 2183 T^{6} + \cdots + 3888 \) Copy content Toggle raw display
$17$ \( T^{10} + 6 T^{9} + 70 T^{8} + \cdots + 69696 \) Copy content Toggle raw display
$19$ \( T^{10} - 3 T^{9} - 12 T^{8} + 45 T^{7} + \cdots + 192 \) Copy content Toggle raw display
$23$ \( T^{10} + 24 T^{9} + 223 T^{8} + \cdots + 2462508 \) Copy content Toggle raw display
$29$ \( T^{10} + 142 T^{8} + 6537 T^{6} + \cdots + 8748 \) Copy content Toggle raw display
$31$ \( T^{10} - 15 T^{9} + 12 T^{8} + \cdots + 1978032 \) Copy content Toggle raw display
$37$ \( T^{10} + T^{9} + 68 T^{8} - 65 T^{7} + \cdots + 12544 \) Copy content Toggle raw display
$41$ \( (T^{5} - 4 T^{4} - 115 T^{3} + 64 T^{2} + \cdots - 1338)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 13 T^{4} - 42 T^{3} - 568 T^{2} + \cdots + 1559)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 14 T^{9} + \cdots + 295289856 \) Copy content Toggle raw display
$53$ \( T^{10} - 24 T^{9} + \cdots + 113246208 \) Copy content Toggle raw display
$59$ \( T^{10} + 160 T^{8} + \cdots + 125081856 \) Copy content Toggle raw display
$61$ \( T^{10} - 42 T^{9} + 807 T^{8} + \cdots + 1338672 \) Copy content Toggle raw display
$67$ \( T^{10} - 7 T^{9} + 241 T^{8} + \cdots + 45684081 \) Copy content Toggle raw display
$71$ \( T^{10} + 288 T^{8} + \cdots + 88259328 \) Copy content Toggle raw display
$73$ \( T^{10} + 3 T^{9} - 250 T^{8} + \cdots + 1572528 \) Copy content Toggle raw display
$79$ \( T^{10} - T^{9} + 194 T^{8} + \cdots + 138485824 \) Copy content Toggle raw display
$83$ \( (T^{5} - 4 T^{4} - 379 T^{3} + 1024 T^{2} + \cdots + 28794)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 28 T^{9} + 563 T^{8} + \cdots + 272484 \) Copy content Toggle raw display
$97$ \( T^{10} + 212 T^{8} + 11040 T^{6} + \cdots + 1051392 \) Copy content Toggle raw display
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