Properties

Label 363.2.f.g
Level $363$
Weight $2$
Character orbit 363.f
Analytic conductor $2.899$
Analytic rank $0$
Dimension $16$
CM discriminant -3
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.f (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.6879707136000000000000.7
Defining polynomial: \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 11^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{13} - \beta_{10} + \beta_{5} + \beta_1) q^{3} + 2 \beta_{4} q^{4} - \beta_{14} q^{7} + ( - 3 \beta_{12} + 3 \beta_{7} + 3 \beta_{4} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} - \beta_{10} + \beta_{5} + \beta_1) q^{3} + 2 \beta_{4} q^{4} - \beta_{14} q^{7} + ( - 3 \beta_{12} + 3 \beta_{7} + 3 \beta_{4} - 3) q^{9} + 2 \beta_1 q^{12} + \beta_{3} q^{13} - 4 \beta_{7} q^{16} + (\beta_{8} + \beta_{6}) q^{19} + (\beta_{11} + \beta_{9}) q^{21} + 5 \beta_{12} q^{25} + 3 \beta_{10} q^{27} + ( - 2 \beta_{14} + 2 \beta_{9} - 2 \beta_{6} + 2 \beta_{2}) q^{28} - \beta_{13} q^{31} - 6 \beta_{12} q^{36} + 3 \beta_{5} q^{37} + ( - \beta_{15} + 2 \beta_{14} - 2 \beta_{9} + 2 \beta_{6} - 2 \beta_{2}) q^{39} + ( - \beta_{11} - 2 \beta_{9}) q^{43} - 4 \beta_{5} q^{48} + ( - 5 \beta_{10} + 7 \beta_{7}) q^{49} - 2 \beta_{8} q^{52} + ( - 2 \beta_{3} + \beta_{2}) q^{57} + (2 \beta_{15} + \beta_{14} - \beta_{9} + \beta_{6} - \beta_{2}) q^{61} - 3 \beta_{6} q^{63} + (8 \beta_{12} - 8 \beta_{7} - 8 \beta_{4} + 8) q^{64} - 7 \beta_1 q^{67} + ( - \beta_{15} + 3 \beta_{14} - \beta_{11} - \beta_{8} - \beta_{3}) q^{73} + 5 \beta_{13} q^{75} + ( - 2 \beta_{11} + 2 \beta_{9}) q^{76} + ( - \beta_{3} - 3 \beta_{2}) q^{79} - 9 \beta_{4} q^{81} + (2 \beta_{15} + 2 \beta_{14} + 2 \beta_{11} + 2 \beta_{8} + 2 \beta_{3}) q^{84} + (9 \beta_{13} - \beta_{12} + 9 \beta_{10} - 9 \beta_{5} - 9 \beta_1) q^{91} + 3 \beta_{7} q^{93} + (5 \beta_{12} - 5 \beta_{7} - 5 \beta_{4} + 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 12 q^{9} - 16 q^{16} - 20 q^{25} + 24 q^{36} + 28 q^{49} + 32 q^{64} - 36 q^{81} + 4 q^{91} + 12 q^{93} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} + 362 ) / 209 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{11} - 933\nu ) / 209 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{11} + 2549\nu ) / 209 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} + 780\nu^{2} ) / 209 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{12} - 1351\nu^{2} ) / 209 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{13} - 3482\nu^{3} ) / 209 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{14} - 2911\nu^{4} ) / 209 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -13\nu^{13} - 9513\nu^{3} ) / 209 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -18\nu^{15} - 12995\nu^{5} ) / 209 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -7\nu^{14} - 5042\nu^{4} ) / 209 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49\nu^{15} + 35503\nu^{5} ) / 209 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -268\nu^{15} + 1005\nu^{13} - 3752\nu^{11} + 14003\nu^{9} - 52250\nu^{7} + 1005\nu^{5} - 268\nu^{3} + 67\nu ) / 209 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 683 \nu^{15} - 2732 \nu^{13} + 10245 \nu^{11} - 38247 \nu^{9} + 142747 \nu^{7} - 38248 \nu^{5} + 10245 \nu^{3} - 2732 \nu ) / 209 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{3} + 3\beta_{2} ) / 11 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{8} + 13\beta_{6} ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{10} - 7\beta_{7} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18\beta_{11} + 49\beta_{9} ) / 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{13} + 26\beta_{12} - 26\beta_{7} - 26\beta_{4} + 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 67\beta_{15} + 183\beta_{14} + 67\beta_{11} + 67\beta_{8} + 67\beta_{3} ) / 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -56\beta_{13} + 97\beta_{12} - 56\beta_{10} + 56\beta_{5} + 56\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 250\beta_{15} + 683\beta_{14} - 683\beta_{9} + 683\beta_{6} - 683\beta_{2} ) / 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 209\beta _1 - 362 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -933\beta_{3} - 2549\beta_{2} ) / 11 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -780\beta_{5} - 1351\beta_{4} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3482\beta_{8} - 9513\beta_{6} ) / 11 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -2911\beta_{10} + 5042\beta_{7} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -12995\beta_{11} - 35503\beta_{9} ) / 11 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.304260 + 0.418778i
0.304260 0.418778i
1.13551 1.56290i
−1.13551 + 1.56290i
−1.83730 + 0.596975i
1.83730 0.596975i
0.492303 0.159959i
−0.492303 + 0.159959i
−1.83730 0.596975i
1.83730 + 0.596975i
0.492303 + 0.159959i
−0.492303 0.159959i
−0.304260 0.418778i
0.304260 + 0.418778i
1.13551 + 1.56290i
−1.13551 1.56290i
0 −0.535233 + 1.64728i −0.618034 1.90211i 0 0 −2.19769 + 0.714073i 0 −2.42705 1.76336i 0
161.2 0 −0.535233 + 1.64728i −0.618034 1.90211i 0 0 2.19769 0.714073i 0 −2.42705 1.76336i 0
161.3 0 0.535233 1.64728i −0.618034 1.90211i 0 0 −4.52729 + 1.47101i 0 −2.42705 1.76336i 0
161.4 0 0.535233 1.64728i −0.618034 1.90211i 0 0 4.52729 1.47101i 0 −2.42705 1.76336i 0
215.1 0 −1.40126 1.01807i 1.61803 1.17557i 0 0 −2.79802 3.85115i 0 0.927051 + 2.85317i 0
215.2 0 −1.40126 1.01807i 1.61803 1.17557i 0 0 2.79802 + 3.85115i 0 0.927051 + 2.85317i 0
215.3 0 1.40126 + 1.01807i 1.61803 1.17557i 0 0 −1.35825 1.86947i 0 0.927051 + 2.85317i 0
215.4 0 1.40126 + 1.01807i 1.61803 1.17557i 0 0 1.35825 + 1.86947i 0 0.927051 + 2.85317i 0
233.1 0 −1.40126 + 1.01807i 1.61803 + 1.17557i 0 0 −2.79802 + 3.85115i 0 0.927051 2.85317i 0
233.2 0 −1.40126 + 1.01807i 1.61803 + 1.17557i 0 0 2.79802 3.85115i 0 0.927051 2.85317i 0
233.3 0 1.40126 1.01807i 1.61803 + 1.17557i 0 0 −1.35825 + 1.86947i 0 0.927051 2.85317i 0
233.4 0 1.40126 1.01807i 1.61803 + 1.17557i 0 0 1.35825 1.86947i 0 0.927051 2.85317i 0
239.1 0 −0.535233 1.64728i −0.618034 + 1.90211i 0 0 −2.19769 0.714073i 0 −2.42705 + 1.76336i 0
239.2 0 −0.535233 1.64728i −0.618034 + 1.90211i 0 0 2.19769 + 0.714073i 0 −2.42705 + 1.76336i 0
239.3 0 0.535233 + 1.64728i −0.618034 + 1.90211i 0 0 −4.52729 1.47101i 0 −2.42705 + 1.76336i 0
239.4 0 0.535233 + 1.64728i −0.618034 + 1.90211i 0 0 4.52729 + 1.47101i 0 −2.42705 + 1.76336i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.f.g 16
3.b odd 2 1 CM 363.2.f.g 16
11.b odd 2 1 inner 363.2.f.g 16
11.c even 5 1 363.2.d.d 4
11.c even 5 3 inner 363.2.f.g 16
11.d odd 10 1 363.2.d.d 4
11.d odd 10 3 inner 363.2.f.g 16
33.d even 2 1 inner 363.2.f.g 16
33.f even 10 1 363.2.d.d 4
33.f even 10 3 inner 363.2.f.g 16
33.h odd 10 1 363.2.d.d 4
33.h odd 10 3 inner 363.2.f.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.d.d 4 11.c even 5 1
363.2.d.d 4 11.d odd 10 1
363.2.d.d 4 33.f even 10 1
363.2.d.d 4 33.h odd 10 1
363.2.f.g 16 1.a even 1 1 trivial
363.2.f.g 16 3.b odd 2 1 CM
363.2.f.g 16 11.b odd 2 1 inner
363.2.f.g 16 11.c even 5 3 inner
363.2.f.g 16 11.d odd 10 3 inner
363.2.f.g 16 33.d even 2 1 inner
363.2.f.g 16 33.f even 10 3 inner
363.2.f.g 16 33.h odd 10 3 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{16} - 28 T_{7}^{14} + 663 T_{7}^{12} - 15176 T_{7}^{10} + 344705 T_{7}^{8} - 1836296 T_{7}^{6} + 9706983 T_{7}^{4} - 49603708 T_{7}^{2} + 214358881 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 28 T^{14} + \cdots + 214358881 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 52 T^{14} + \cdots + 54875873536 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} - 76 T^{14} + \cdots + 214358881 \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 27 T^{6} + 729 T^{4} + \cdots + 531441)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{4} + 172 T^{2} + 484)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} - 244 T^{14} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( (T^{2} - 147)^{8} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 292 T^{14} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{16} - 316 T^{14} + \cdots + 214358881 \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( (T^{4} - 5 T^{3} + 25 T^{2} - 125 T + 625)^{4} \) Copy content Toggle raw display
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