Properties

Label 361.3.f.d
Level $361$
Weight $3$
Character orbit 361.f
Analytic conductor $9.837$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $12$

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Newspace parameters

Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 361.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.83653754341\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.7659539263855005696.1
Defining polynomial: \(x^{12} - 2197 x^{6} + 4826809\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{7} ) q^{3} + 9 \beta_{2} q^{4} + ( -4 \beta_{4} + 4 \beta_{10} ) q^{5} + ( 13 \beta_{2} - 13 \beta_{8} ) q^{6} + 5 \beta_{6} q^{7} + 5 \beta_{3} q^{8} -4 \beta_{8} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{7} ) q^{3} + 9 \beta_{2} q^{4} + ( -4 \beta_{4} + 4 \beta_{10} ) q^{5} + ( 13 \beta_{2} - 13 \beta_{8} ) q^{6} + 5 \beta_{6} q^{7} + 5 \beta_{3} q^{8} -4 \beta_{8} q^{9} + ( -4 \beta_{5} + 4 \beta_{11} ) q^{10} + ( 10 - 10 \beta_{6} ) q^{11} + ( 9 \beta_{3} - 9 \beta_{9} ) q^{12} + \beta_{5} q^{13} + 5 \beta_{7} q^{14} + 4 \beta_{11} q^{15} + 29 \beta_{4} q^{16} -15 \beta_{10} q^{17} -4 \beta_{9} q^{18} -36 q^{20} + 5 \beta_{1} q^{21} + ( 10 \beta_{1} - 10 \beta_{7} ) q^{22} -35 \beta_{2} q^{23} + ( 65 \beta_{4} - 65 \beta_{10} ) q^{24} + ( -9 \beta_{2} + 9 \beta_{8} ) q^{25} + 13 \beta_{6} q^{26} + 5 \beta_{3} q^{27} + 45 \beta_{8} q^{28} + ( -5 \beta_{5} + 5 \beta_{11} ) q^{29} + ( -52 + 52 \beta_{6} ) q^{30} + ( -10 \beta_{3} + 10 \beta_{9} ) q^{31} + 9 \beta_{5} q^{32} -10 \beta_{7} q^{33} -15 \beta_{11} q^{34} -20 \beta_{4} q^{35} -36 \beta_{10} q^{36} -6 \beta_{9} q^{37} + 13 q^{39} -20 \beta_{1} q^{40} + ( -10 \beta_{1} + 10 \beta_{7} ) q^{41} + 65 \beta_{2} q^{42} + ( 20 \beta_{4} - 20 \beta_{10} ) q^{43} + ( 90 \beta_{2} - 90 \beta_{8} ) q^{44} + 16 \beta_{6} q^{45} -35 \beta_{3} q^{46} + 10 \beta_{8} q^{47} + ( 29 \beta_{5} - 29 \beta_{11} ) q^{48} + ( 24 - 24 \beta_{6} ) q^{49} + ( -9 \beta_{3} + 9 \beta_{9} ) q^{50} -15 \beta_{5} q^{51} + 9 \beta_{7} q^{52} + 21 \beta_{11} q^{53} + 65 \beta_{4} q^{54} + 40 \beta_{10} q^{55} + 25 \beta_{9} q^{56} -65 q^{58} + 5 \beta_{1} q^{59} + ( -36 \beta_{1} + 36 \beta_{7} ) q^{60} + 40 \beta_{2} q^{61} + ( -130 \beta_{4} + 130 \beta_{10} ) q^{62} + ( 20 \beta_{2} - 20 \beta_{8} ) q^{63} + \beta_{6} q^{64} -4 \beta_{3} q^{65} -130 \beta_{8} q^{66} + ( -11 \beta_{5} + 11 \beta_{11} ) q^{67} + ( 135 - 135 \beta_{6} ) q^{68} + ( -35 \beta_{3} + 35 \beta_{9} ) q^{69} -20 \beta_{5} q^{70} -30 \beta_{7} q^{71} -20 \beta_{11} q^{72} + 105 \beta_{4} q^{73} -78 \beta_{10} q^{74} + 9 \beta_{9} q^{75} + 50 q^{77} + 13 \beta_{1} q^{78} + ( 10 \beta_{1} - 10 \beta_{7} ) q^{79} -116 \beta_{2} q^{80} + ( 101 \beta_{4} - 101 \beta_{10} ) q^{81} + ( -130 \beta_{2} + 130 \beta_{8} ) q^{82} + 40 \beta_{6} q^{83} + 45 \beta_{3} q^{84} + 60 \beta_{8} q^{85} + ( 20 \beta_{5} - 20 \beta_{11} ) q^{86} + ( -65 + 65 \beta_{6} ) q^{87} + ( 50 \beta_{3} - 50 \beta_{9} ) q^{88} + 16 \beta_{7} q^{90} + 5 \beta_{11} q^{91} -315 \beta_{4} q^{92} + 130 \beta_{10} q^{93} + 10 \beta_{9} q^{94} + 117 q^{96} + 34 \beta_{1} q^{97} + ( 24 \beta_{1} - 24 \beta_{7} ) q^{98} -40 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 30q^{7} + O(q^{10}) \) \( 12q + 30q^{7} + 60q^{11} - 432q^{20} + 78q^{26} - 312q^{30} + 156q^{39} + 96q^{45} + 144q^{49} - 780q^{58} + 6q^{64} + 810q^{68} + 600q^{77} + 240q^{83} - 390q^{87} + 1404q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2197 x^{6} + 4826809\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/13\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/13\)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/169\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/169\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/2197\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/2197\)
\(\beta_{8}\)\(=\)\( \nu^{8} \)\(/28561\)
\(\beta_{9}\)\(=\)\( \nu^{9} \)\(/28561\)
\(\beta_{10}\)\(=\)\( \nu^{10} \)\(/371293\)
\(\beta_{11}\)\(=\)\( \nu^{11} \)\(/371293\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(13 \beta_{2}\)
\(\nu^{3}\)\(=\)\(13 \beta_{3}\)
\(\nu^{4}\)\(=\)\(169 \beta_{4}\)
\(\nu^{5}\)\(=\)\(169 \beta_{5}\)
\(\nu^{6}\)\(=\)\(2197 \beta_{6}\)
\(\nu^{7}\)\(=\)\(2197 \beta_{7}\)
\(\nu^{8}\)\(=\)\(28561 \beta_{8}\)
\(\nu^{9}\)\(=\)\(28561 \beta_{9}\)
\(\nu^{10}\)\(=\)\(371293 \beta_{10}\)
\(\nu^{11}\)\(=\)\(371293 \beta_{11}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−3.55077 0.626097i
3.55077 + 0.626097i
−2.31760 2.76201i
2.31760 + 2.76201i
−1.23317 + 3.38811i
1.23317 3.38811i
−1.23317 3.38811i
1.23317 + 3.38811i
−2.31760 + 2.76201i
2.31760 2.76201i
−3.55077 + 0.626097i
3.55077 0.626097i
−3.55077 0.626097i −2.31760 + 2.76201i 8.45723 + 3.07818i −3.75877 + 1.36808i 9.95858 8.35624i 2.50000 + 4.33013i −15.6125 9.01388i −0.694593 3.93923i 14.2031 2.50439i
116.2 3.55077 + 0.626097i 2.31760 2.76201i 8.45723 + 3.07818i −3.75877 + 1.36808i 9.95858 8.35624i 2.50000 + 4.33013i 15.6125 + 9.01388i −0.694593 3.93923i −14.2031 + 2.50439i
127.1 −2.31760 2.76201i 1.23317 3.38811i −1.56283 + 8.86327i 0.694593 + 3.93923i −12.2160 + 4.44626i 2.50000 4.33013i 15.6125 9.01388i −3.06418 2.57115i 9.27041 11.0481i
127.2 2.31760 + 2.76201i −1.23317 + 3.38811i −1.56283 + 8.86327i 0.694593 + 3.93923i −12.2160 + 4.44626i 2.50000 4.33013i −15.6125 + 9.01388i −3.06418 2.57115i −9.27041 + 11.0481i
262.1 −1.23317 + 3.38811i −3.55077 + 0.626097i −6.89440 5.78509i 3.06418 2.57115i 2.25743 12.8025i 2.50000 4.33013i 15.6125 9.01388i 3.75877 1.36808i 4.93268 + 13.5524i
262.2 1.23317 3.38811i 3.55077 0.626097i −6.89440 5.78509i 3.06418 2.57115i 2.25743 12.8025i 2.50000 4.33013i −15.6125 + 9.01388i 3.75877 1.36808i −4.93268 13.5524i
299.1 −1.23317 3.38811i −3.55077 0.626097i −6.89440 + 5.78509i 3.06418 + 2.57115i 2.25743 + 12.8025i 2.50000 + 4.33013i 15.6125 + 9.01388i 3.75877 + 1.36808i 4.93268 13.5524i
299.2 1.23317 + 3.38811i 3.55077 + 0.626097i −6.89440 + 5.78509i 3.06418 + 2.57115i 2.25743 + 12.8025i 2.50000 + 4.33013i −15.6125 9.01388i 3.75877 + 1.36808i −4.93268 + 13.5524i
307.1 −2.31760 + 2.76201i 1.23317 + 3.38811i −1.56283 8.86327i 0.694593 3.93923i −12.2160 4.44626i 2.50000 + 4.33013i 15.6125 + 9.01388i −3.06418 + 2.57115i 9.27041 + 11.0481i
307.2 2.31760 2.76201i −1.23317 3.38811i −1.56283 8.86327i 0.694593 3.93923i −12.2160 4.44626i 2.50000 + 4.33013i −15.6125 9.01388i −3.06418 + 2.57115i −9.27041 11.0481i
333.1 −3.55077 + 0.626097i −2.31760 2.76201i 8.45723 3.07818i −3.75877 1.36808i 9.95858 + 8.35624i 2.50000 4.33013i −15.6125 + 9.01388i −0.694593 + 3.93923i 14.2031 + 2.50439i
333.2 3.55077 0.626097i 2.31760 + 2.76201i 8.45723 3.07818i −3.75877 1.36808i 9.95858 + 8.35624i 2.50000 4.33013i 15.6125 9.01388i −0.694593 + 3.93923i −14.2031 2.50439i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 333.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner
19.c even 3 2 inner
19.d odd 6 2 inner
19.e even 9 3 inner
19.f odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.3.f.d 12
19.b odd 2 1 inner 361.3.f.d 12
19.c even 3 2 inner 361.3.f.d 12
19.d odd 6 2 inner 361.3.f.d 12
19.e even 9 1 19.3.b.b 2
19.e even 9 2 361.3.d.b 4
19.e even 9 3 inner 361.3.f.d 12
19.f odd 18 1 19.3.b.b 2
19.f odd 18 2 361.3.d.b 4
19.f odd 18 3 inner 361.3.f.d 12
57.j even 18 1 171.3.c.b 2
57.l odd 18 1 171.3.c.b 2
76.k even 18 1 304.3.e.d 2
76.l odd 18 1 304.3.e.d 2
95.o odd 18 1 475.3.c.b 2
95.p even 18 1 475.3.c.b 2
95.q odd 36 2 475.3.d.b 4
95.r even 36 2 475.3.d.b 4
152.s odd 18 1 1216.3.e.g 2
152.t even 18 1 1216.3.e.g 2
152.u odd 18 1 1216.3.e.h 2
152.v even 18 1 1216.3.e.h 2
228.u odd 18 1 2736.3.o.d 2
228.v even 18 1 2736.3.o.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 19.e even 9 1
19.3.b.b 2 19.f odd 18 1
171.3.c.b 2 57.j even 18 1
171.3.c.b 2 57.l odd 18 1
304.3.e.d 2 76.k even 18 1
304.3.e.d 2 76.l odd 18 1
361.3.d.b 4 19.e even 9 2
361.3.d.b 4 19.f odd 18 2
361.3.f.d 12 1.a even 1 1 trivial
361.3.f.d 12 19.b odd 2 1 inner
361.3.f.d 12 19.c even 3 2 inner
361.3.f.d 12 19.d odd 6 2 inner
361.3.f.d 12 19.e even 9 3 inner
361.3.f.d 12 19.f odd 18 3 inner
475.3.c.b 2 95.o odd 18 1
475.3.c.b 2 95.p even 18 1
475.3.d.b 4 95.q odd 36 2
475.3.d.b 4 95.r even 36 2
1216.3.e.g 2 152.s odd 18 1
1216.3.e.g 2 152.t even 18 1
1216.3.e.h 2 152.u odd 18 1
1216.3.e.h 2 152.v even 18 1
2736.3.o.d 2 228.u odd 18 1
2736.3.o.d 2 228.v even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 2197 T_{2}^{6} + 4826809 \) acting on \(S_{3}^{\mathrm{new}}(361, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4826809 - 2197 T^{6} + T^{12} \)
$3$ \( 4826809 - 2197 T^{6} + T^{12} \)
$5$ \( ( 4096 + 64 T^{3} + T^{6} )^{2} \)
$7$ \( ( 25 - 5 T + T^{2} )^{6} \)
$11$ \( ( 100 - 10 T + T^{2} )^{6} \)
$13$ \( 4826809 - 2197 T^{6} + T^{12} \)
$17$ \( ( 11390625 + 3375 T^{3} + T^{6} )^{2} \)
$19$ \( T^{12} \)
$23$ \( ( 1838265625 + 42875 T^{3} + T^{6} )^{2} \)
$29$ \( 1178420166015625 - 34328125 T^{6} + T^{12} \)
$31$ \( ( 1690000 - 1300 T^{2} + T^{4} )^{3} \)
$37$ \( ( 468 + T^{2} )^{6} \)
$41$ \( 4826809000000000000 - 2197000000 T^{6} + T^{12} \)
$43$ \( ( 64000000 - 8000 T^{3} + T^{6} )^{2} \)
$47$ \( ( 1000000 + 1000 T^{3} + T^{6} )^{2} \)
$53$ \( \)\(35\!\cdots\!69\)\( - 188428167837 T^{6} + T^{12} \)
$59$ \( 1178420166015625 - 34328125 T^{6} + T^{12} \)
$61$ \( ( 4096000000 - 64000 T^{3} + T^{6} )^{2} \)
$67$ \( 15148594334612313289 - 3892119517 T^{6} + T^{12} \)
$71$ \( \)\(25\!\cdots\!00\)\( - 1601613000000 T^{6} + T^{12} \)
$73$ \( ( 1340095640625 + 1157625 T^{3} + T^{6} )^{2} \)
$79$ \( 4826809000000000000 - 2197000000 T^{6} + T^{12} \)
$83$ \( ( 1600 - 40 T + T^{2} )^{6} \)
$89$ \( T^{12} \)
$97$ \( \)\(11\!\cdots\!04\)\( - 3393935301952 T^{6} + T^{12} \)
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