Properties

Label 363.2.f.h
Level $363$
Weight $2$
Character orbit 363.f
Analytic conductor $2.899$
Analytic rank $0$
Dimension $16$
CM no
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: 16.0.26873856000000000000.1
Defining polynomial: \( x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{13} + \beta_{12} - \beta_{8} - 2 \beta_{6} + \beta_{2}) q^{2} + ( - \beta_{15} + 2 \beta_{5}) q^{3} + (4 \beta_{15} + 4 \beta_{9} + 4 \beta_{7} - 4 \beta_{3} - 4 \beta_1 - 4) q^{4} + (\beta_{15} + \beta_{11} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{3} - \beta_1) q^{5} - 3 \beta_{14} q^{6} - \beta_{8} q^{7} + (2 \beta_{13} + 4 \beta_{10}) q^{8} + (3 \beta_{11} - 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{13} + \beta_{12} - \beta_{8} - 2 \beta_{6} + \beta_{2}) q^{2} + ( - \beta_{15} + 2 \beta_{5}) q^{3} + (4 \beta_{15} + 4 \beta_{9} + 4 \beta_{7} - 4 \beta_{3} - 4 \beta_1 - 4) q^{4} + (\beta_{15} + \beta_{11} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{3} - \beta_1) q^{5} - 3 \beta_{14} q^{6} - \beta_{8} q^{7} + (2 \beta_{13} + 4 \beta_{10}) q^{8} + (3 \beta_{11} - 3 \beta_1) q^{9} + 3 \beta_{12} q^{10} + ( - 4 \beta_{9} - 4) q^{12} + ( - 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{8} - 2 \beta_{2}) q^{13} + ( - 4 \beta_{15} + 2 \beta_{5}) q^{14} + ( - 3 \beta_{15} - 3 \beta_{9} - 3 \beta_{7} + 3 \beta_1 + 3) q^{15} + ( - 4 \beta_{15} - 4 \beta_{11} - 4 \beta_{9} - 4 \beta_{7} + 4 \beta_{5} + 4 \beta_{3} + \cdots + 4 \beta_1) q^{16}+ \cdots + (5 \beta_{12} - 10 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{3} - 16 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{3} - 16 q^{4} - 6 q^{9} - 96 q^{12} + 6 q^{15} - 16 q^{16} - 8 q^{25} - 12 q^{31} + 96 q^{34} - 24 q^{36} - 12 q^{37} + 12 q^{42} + 72 q^{45} + 24 q^{48} - 20 q^{49} + 48 q^{58} + 24 q^{60} + 32 q^{64} - 48 q^{67} - 30 q^{69} + 24 q^{70} + 12 q^{75} - 96 q^{78} + 18 q^{81} + 96 q^{82} + 16 q^{91} + 18 q^{93} - 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{10} ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{11} ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{12} ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -\nu^{11} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{13} - 32\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{14} - 4\nu^{10} - 8\nu^{8} + 64\nu^{2} + 128 ) / 128 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{13} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{5} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16\beta_{7} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16\beta_{8} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{9} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 32\beta_{10} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 64\beta_{11} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 64\beta_{14} + 64\beta_{2} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 128\beta_{15} + 128\beta_{9} + 128\beta_{7} - 128\beta _1 - 128 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 128\beta_{12} \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(1 + \beta_{1} + \beta_{3} - \beta_{7} - \beta_{9} - \beta_{15}\)

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
−0.294032 1.38331i
1.05097 0.946294i
0.294032 + 1.38331i
−1.05097 + 0.946294i
1.40647 + 0.147826i
0.575212 + 1.29195i
−1.40647 0.147826i
−0.575212 1.29195i
1.40647 0.147826i
0.575212 1.29195i
−1.40647 + 0.147826i
−0.575212 + 1.29195i
−0.294032 + 1.38331i
1.05097 + 0.946294i
0.294032 1.38331i
−1.05097 0.946294i
−1.98168 1.43977i −1.28716 + 1.15897i 1.23607 + 3.80423i 1.01807 + 1.40126i 4.21940 0.443477i 1.34500 0.437016i 1.51387 4.65921i 0.313585 2.98357i 4.24264i
161.2 −1.98168 1.43977i 0.360114 + 1.69420i 1.23607 + 3.80423i −1.01807 1.40126i 1.72564 3.87585i −1.34500 + 0.437016i 1.51387 4.65921i −2.74064 + 1.22021i 4.24264i
161.3 1.98168 + 1.43977i −1.28716 + 1.15897i 1.23607 + 3.80423i 1.01807 + 1.40126i −4.21940 + 0.443477i −1.34500 + 0.437016i −1.51387 + 4.65921i 0.313585 2.98357i 4.24264i
161.4 1.98168 + 1.43977i 0.360114 + 1.69420i 1.23607 + 3.80423i −1.01807 1.40126i −1.72564 + 3.87585i 1.34500 0.437016i −1.51387 + 4.65921i −2.74064 + 1.22021i 4.24264i
215.1 −0.756934 2.32960i 0.704489 + 1.58231i −3.23607 + 2.35114i −1.64728 0.535233i 3.15290 2.83888i −0.831254 1.14412i 3.96336 + 2.87955i −2.00739 + 2.22943i 4.24264i
215.2 −0.756934 2.32960i 1.72256 + 0.181049i −3.23607 + 2.35114i 1.64728 + 0.535233i −0.882095 4.14993i 0.831254 + 1.14412i 3.96336 + 2.87955i 2.93444 + 0.623735i 4.24264i
215.3 0.756934 + 2.32960i 0.704489 + 1.58231i −3.23607 + 2.35114i −1.64728 0.535233i −3.15290 + 2.83888i 0.831254 + 1.14412i −3.96336 2.87955i −2.00739 + 2.22943i 4.24264i
215.4 0.756934 + 2.32960i 1.72256 + 0.181049i −3.23607 + 2.35114i 1.64728 + 0.535233i 0.882095 + 4.14993i −0.831254 1.14412i −3.96336 2.87955i 2.93444 + 0.623735i 4.24264i
233.1 −0.756934 + 2.32960i 0.704489 1.58231i −3.23607 2.35114i −1.64728 + 0.535233i 3.15290 + 2.83888i −0.831254 + 1.14412i 3.96336 2.87955i −2.00739 2.22943i 4.24264i
233.2 −0.756934 + 2.32960i 1.72256 0.181049i −3.23607 2.35114i 1.64728 0.535233i −0.882095 + 4.14993i 0.831254 1.14412i 3.96336 2.87955i 2.93444 0.623735i 4.24264i
233.3 0.756934 2.32960i 0.704489 1.58231i −3.23607 2.35114i −1.64728 + 0.535233i −3.15290 2.83888i 0.831254 1.14412i −3.96336 + 2.87955i −2.00739 2.22943i 4.24264i
233.4 0.756934 2.32960i 1.72256 0.181049i −3.23607 2.35114i 1.64728 0.535233i 0.882095 4.14993i −0.831254 + 1.14412i −3.96336 + 2.87955i 2.93444 0.623735i 4.24264i
239.1 −1.98168 + 1.43977i −1.28716 1.15897i 1.23607 3.80423i 1.01807 1.40126i 4.21940 + 0.443477i 1.34500 + 0.437016i 1.51387 + 4.65921i 0.313585 + 2.98357i 4.24264i
239.2 −1.98168 + 1.43977i 0.360114 1.69420i 1.23607 3.80423i −1.01807 + 1.40126i 1.72564 + 3.87585i −1.34500 0.437016i 1.51387 + 4.65921i −2.74064 1.22021i 4.24264i
239.3 1.98168 1.43977i −1.28716 1.15897i 1.23607 3.80423i 1.01807 1.40126i −4.21940 0.443477i −1.34500 0.437016i −1.51387 4.65921i 0.313585 + 2.98357i 4.24264i
239.4 1.98168 1.43977i 0.360114 1.69420i 1.23607 3.80423i −1.01807 + 1.40126i −1.72564 3.87585i 1.34500 + 0.437016i −1.51387 4.65921i −2.74064 1.22021i 4.24264i
\(n\): 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 inner
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.h 16
3.b odd 2 1 inner 363.2.f.h 16
11.b odd 2 1 inner 363.2.f.h 16
11.c even 5 1 363.2.d.c 4
11.c even 5 3 inner 363.2.f.h 16
11.d odd 10 1 363.2.d.c 4
11.d odd 10 3 inner 363.2.f.h 16
33.d even 2 1 inner 363.2.f.h 16
33.f even 10 1 363.2.d.c 4
33.f even 10 3 inner 363.2.f.h 16
33.h odd 10 1 363.2.d.c 4
33.h odd 10 3 inner 363.2.f.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.d.c 4 11.c even 5 1
363.2.d.c 4 11.d odd 10 1
363.2.d.c 4 33.f even 10 1
363.2.d.c 4 33.h odd 10 1
363.2.f.h 16 1.a even 1 1 trivial
363.2.f.h 16 3.b odd 2 1 inner
363.2.f.h 16 11.b odd 2 1 inner
363.2.f.h 16 11.c even 5 3 inner
363.2.f.h 16 11.d odd 10 3 inner
363.2.f.h 16 33.d even 2 1 inner
363.2.f.h 16 33.f even 10 3 inner
363.2.f.h 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 \(S_{2}^{\mathrm{new}}(363, [\chi])\):

\( T_{2}^{8} + 6T_{2}^{6} + 36T_{2}^{4} + 216T_{2}^{2} + 1296 \) Copy content Toggle raw display
\( T_{5}^{8} - 3T_{5}^{6} + 9T_{5}^{4} - 27T_{5}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{8} - 2T_{7}^{6} + 4T_{7}^{4} - 8T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 6 T^{6} + 36 T^{4} + 216 T^{2} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - 3 T^{7} + 6 T^{6} - 9 T^{5} + 9 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 3 T^{6} + 9 T^{4} - 27 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} - 8 T^{6} + 64 T^{4} - 512 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 6 T^{6} + 36 T^{4} + 216 T^{2} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 8 T^{6} + 64 T^{4} - 512 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 75)^{8} \) Copy content Toggle raw display
$29$ \( (T^{8} + 24 T^{6} + 576 T^{4} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 96 T^{6} + 9216 T^{4} + \cdots + 84934656)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 98)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} - 48 T^{6} + 2304 T^{4} + \cdots + 5308416)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 12 T^{6} + 144 T^{4} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 3 T^{6} + 9 T^{4} - 27 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 8 T^{6} + 64 T^{4} - 512 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$67$ \( (T + 3)^{16} \) Copy content Toggle raw display
$71$ \( (T^{8} - 3 T^{6} + 9 T^{4} - 27 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 32 T^{6} + 1024 T^{4} + \cdots + 1048576)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 98 T^{6} + 9604 T^{4} + \cdots + 92236816)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 6 T^{6} + 36 T^{4} + 216 T^{2} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 75)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 13 T^{3} + 169 T^{2} + \cdots + 28561)^{4} \) Copy content Toggle raw display
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