Properties

Label 361.2.c.i
Level $361$
Weight $2$
Character orbit 361.c
Analytic conductor $2.883$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.88259951297\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{2} + ( - \beta_{2} + \beta_1) q^{3} + (2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + (\beta_{2} + \beta_1) q^{5} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{6} + ( - \beta_{4} - \beta_{3}) q^{7} + (2 \beta_{4} - \beta_{3} - 2) q^{8} + ( - \beta_{5} + 3 \beta_{3} - 3 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{2} + ( - \beta_{2} + \beta_1) q^{3} + (2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + (\beta_{2} + \beta_1) q^{5} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{6} + ( - \beta_{4} - \beta_{3}) q^{7} + (2 \beta_{4} - \beta_{3} - 2) q^{8} + ( - \beta_{5} + 3 \beta_{3} - 3 \beta_{2}) q^{9} + (2 \beta_{5} - \beta_{3} + \beta_{2}) q^{10} + (\beta_{4} + 2 \beta_{3}) q^{11} + (\beta_{4} - 1) q^{12} + (2 \beta_{5} + \beta_{3} - \beta_{2}) q^{13} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{14} + (\beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{15} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - \beta_1) q^{16} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{17} + ( - 4 \beta_{4} - 1) q^{18} + (3 \beta_{4} + 2 \beta_{3} - 1) q^{20} + \beta_1 q^{21} - 3 \beta_1 q^{22} + ( - 2 \beta_{5} - 2 \beta_1 + 2) q^{23} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{24} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{25} + (\beta_{4} - 5) q^{26} + ( - 3 \beta_{4} - 2) q^{27} + (2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{28} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 + 5) q^{29} + (\beta_{4} - \beta_{3}) q^{30} + (\beta_{4} + 3 \beta_{3} - 3) q^{31} - 3 \beta_{5} q^{32} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{33} + ( - 4 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{34} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_1) q^{35} + (5 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{36} + (2 \beta_{4} - \beta_{3}) q^{37} + ( - \beta_{4} + \beta_{3} - 4) q^{39} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{40} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{41} + (\beta_{5} + \beta_1 - 1) q^{42} + (5 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 3 \beta_{2}) q^{43} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{44} + (\beta_{4} + \beta_{3} + 5) q^{45} + ( - 4 \beta_{4} - 2 \beta_{3} + 6) q^{46} + (\beta_{5} - 3 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 2) q^{47} + ( - 3 \beta_{5} + 5 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 2) q^{48} + (\beta_{3} - 5) q^{49} + ( - 2 \beta_{4} - 2 \beta_{3} + 5) q^{50} + ( - \beta_{5} - \beta_1 + 1) q^{51} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} - 6 \beta_1) q^{52} + (3 \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{53} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 6 \beta_{2} + \beta_1) q^{54} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_1) q^{55} + (5 \beta_{4} + 3 \beta_{3} - 1) q^{56} + ( - 4 \beta_{4} - 5 \beta_{3} + 6) q^{58} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 7 \beta_1) q^{59} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{60} + ( - 4 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 3) q^{61} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 7 \beta_1) q^{62} + ( - 3 \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{63} + (3 \beta_{4} + 4) q^{64} + (5 \beta_{4} + 5 \beta_{3} + 4) q^{65} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{66} + ( - 6 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 6) q^{67} + ( - 7 \beta_{4} - 3 \beta_{3} + 5) q^{68} + ( - 2 \beta_{3} + 4) q^{69} + (\beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{70} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 10 \beta_1) q^{71} + (2 \beta_{5} + \beta_{3} - \beta_{2} + 13 \beta_1 - 13) q^{72} + ( - 4 \beta_{5} + 4 \beta_{4} + 4 \beta_{3}) q^{73} + ( - 5 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - \beta_1) q^{74} + (\beta_{4} - 3 \beta_{3} + 5) q^{75} + ( - \beta_{4} - 2 \beta_{3} - 3) q^{77} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{78} + ( - 7 \beta_{5} + 7 \beta_{4} + 7 \beta_{3} - 6 \beta_{2} + 3 \beta_1) q^{79} + ( - 3 \beta_{5} + \beta_{3} - \beta_{2} - 4 \beta_1 + 4) q^{80} + ( - \beta_{2} - 5 \beta_1) q^{81} + (11 \beta_{5} - 7 \beta_{3} + 7 \beta_{2} + 6 \beta_1 - 6) q^{82} + ( - 6 \beta_{4} + 3 \beta_{3}) q^{83} + (2 \beta_{4} + \beta_{3} - 1) q^{84} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{85} + (8 \beta_{5} - 8 \beta_{3} + 8 \beta_{2} + 7 \beta_1 - 7) q^{86} + (\beta_{4} - 6 \beta_{3} + 7) q^{87} + ( - 6 \beta_{4} - 3 \beta_{3} - 3) q^{88} + (\beta_{5} - 3 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 5) q^{89} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - \beta_{2} + 3 \beta_1) q^{90} + ( - \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 - 5) q^{91} + (8 \beta_{5} - 8 \beta_{4} - 8 \beta_{3} + 6 \beta_{2} + 8 \beta_1) q^{92} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 7 \beta_{2} - 8 \beta_1) q^{93} + (2 \beta_{4} - 2 \beta_{3} + 3) q^{94} + 3 q^{96} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1) q^{97} + ( - 4 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + \beta_{2} - 6 \beta_1) q^{98} + (5 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 3 q^{3} - 3 q^{4} + 3 q^{5} - 6 q^{6} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} + 3 q^{3} - 3 q^{4} + 3 q^{5} - 6 q^{6} - 12 q^{8} - 6 q^{12} + 6 q^{14} + 3 q^{15} - 3 q^{16} - 6 q^{17} - 6 q^{18} - 6 q^{20} + 3 q^{21} - 9 q^{22} + 6 q^{23} + 6 q^{24} + 6 q^{25} - 30 q^{26} - 12 q^{27} - 9 q^{28} + 15 q^{29} - 18 q^{31} - 9 q^{33} + 9 q^{34} - 3 q^{35} + 9 q^{36} - 24 q^{39} - 18 q^{40} + 12 q^{41} - 3 q^{42} + 9 q^{44} + 30 q^{45} + 36 q^{46} + 6 q^{47} - 6 q^{48} - 30 q^{49} + 30 q^{50} + 3 q^{51} - 18 q^{52} + 6 q^{53} + 3 q^{54} + 9 q^{55} - 6 q^{56} + 36 q^{58} + 21 q^{59} - 6 q^{60} - 9 q^{61} - 21 q^{62} - 3 q^{63} + 24 q^{64} + 24 q^{65} + 9 q^{66} - 18 q^{67} + 30 q^{68} + 24 q^{69} - 9 q^{70} + 30 q^{71} - 39 q^{72} - 3 q^{74} + 30 q^{75} - 18 q^{77} - 12 q^{78} + 9 q^{79} + 12 q^{80} - 15 q^{81} - 18 q^{82} - 6 q^{84} + 9 q^{85} - 21 q^{86} + 42 q^{87} - 18 q^{88} + 15 q^{89} + 9 q^{90} - 15 q^{91} + 24 q^{92} - 24 q^{93} + 18 q^{94} + 18 q^{96} - 15 q^{97} - 18 q^{98} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{18}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{18}^{5} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{18}^{5} + \zeta_{18}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \) Copy content Toggle raw display
\(\zeta_{18}\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{2}\)\(=\) \( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{18}^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{5}\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.939693 0.342020i
−0.173648 + 0.984808i
−0.766044 0.642788i
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 + 0.642788i
−0.439693 + 0.761570i −0.266044 + 0.460802i 0.613341 + 1.06234i 1.26604 2.19285i −0.233956 0.405223i −1.87939 −2.83750 1.35844 + 2.35289i 1.11334 + 1.92836i
68.2 0.673648 1.16679i 1.43969 2.49362i 0.0923963 + 0.160035i −0.439693 + 0.761570i −1.93969 3.35965i 0.347296 2.94356 −2.64543 4.58202i 0.592396 + 1.02606i
68.3 1.26604 2.19285i 0.326352 0.565258i −2.20574 3.82045i 0.673648 1.16679i −0.826352 1.43128i 1.53209 −6.10607 1.28699 + 2.22913i −1.70574 2.95442i
292.1 −0.439693 0.761570i −0.266044 0.460802i 0.613341 1.06234i 1.26604 + 2.19285i −0.233956 + 0.405223i −1.87939 −2.83750 1.35844 2.35289i 1.11334 1.92836i
292.2 0.673648 + 1.16679i 1.43969 + 2.49362i 0.0923963 0.160035i −0.439693 0.761570i −1.93969 + 3.35965i 0.347296 2.94356 −2.64543 + 4.58202i 0.592396 1.02606i
292.3 1.26604 + 2.19285i 0.326352 + 0.565258i −2.20574 + 3.82045i 0.673648 + 1.16679i −0.826352 + 1.43128i 1.53209 −6.10607 1.28699 2.22913i −1.70574 + 2.95442i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 292.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.2.c.i 6
19.b odd 2 1 361.2.c.h 6
19.c even 3 1 361.2.a.g 3
19.c even 3 1 inner 361.2.c.i 6
19.d odd 6 1 361.2.a.h 3
19.d odd 6 1 361.2.c.h 6
19.e even 9 2 19.2.e.a 6
19.e even 9 2 361.2.e.f 6
19.e even 9 2 361.2.e.g 6
19.f odd 18 2 361.2.e.a 6
19.f odd 18 2 361.2.e.b 6
19.f odd 18 2 361.2.e.h 6
57.f even 6 1 3249.2.a.s 3
57.h odd 6 1 3249.2.a.z 3
57.l odd 18 2 171.2.u.c 6
76.f even 6 1 5776.2.a.bi 3
76.g odd 6 1 5776.2.a.br 3
76.l odd 18 2 304.2.u.b 6
95.h odd 6 1 9025.2.a.x 3
95.i even 6 1 9025.2.a.bd 3
95.p even 18 2 475.2.l.a 6
95.q odd 36 4 475.2.u.a 12
133.u even 9 2 931.2.v.b 6
133.w even 9 2 931.2.x.a 6
133.x odd 18 2 931.2.v.a 6
133.y odd 18 2 931.2.w.a 6
133.z odd 18 2 931.2.x.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 19.e even 9 2
171.2.u.c 6 57.l odd 18 2
304.2.u.b 6 76.l odd 18 2
361.2.a.g 3 19.c even 3 1
361.2.a.h 3 19.d odd 6 1
361.2.c.h 6 19.b odd 2 1
361.2.c.h 6 19.d odd 6 1
361.2.c.i 6 1.a even 1 1 trivial
361.2.c.i 6 19.c even 3 1 inner
361.2.e.a 6 19.f odd 18 2
361.2.e.b 6 19.f odd 18 2
361.2.e.f 6 19.e even 9 2
361.2.e.g 6 19.e even 9 2
361.2.e.h 6 19.f odd 18 2
475.2.l.a 6 95.p even 18 2
475.2.u.a 12 95.q odd 36 4
931.2.v.a 6 133.x odd 18 2
931.2.v.b 6 133.u even 9 2
931.2.w.a 6 133.y odd 18 2
931.2.x.a 6 133.w even 9 2
931.2.x.b 6 133.z odd 18 2
3249.2.a.s 3 57.f even 6 1
3249.2.a.z 3 57.h odd 6 1
5776.2.a.bi 3 76.f even 6 1
5776.2.a.br 3 76.g odd 6 1
9025.2.a.x 3 95.h odd 6 1
9025.2.a.bd 3 95.i even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(361, [\chi])\):

\( T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 6T_{2}^{3} + 9T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{6} - 3T_{3}^{5} + 9T_{3}^{4} - 2T_{3}^{3} + 3T_{3}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} + 9 T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + 9 T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 9 T - 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 21 T^{4} + 74 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$17$ \( T^{6} + 6 T^{5} + 27 T^{4} + 48 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} + 36 T^{4} - 48 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$29$ \( T^{6} - 15 T^{5} + 153 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$31$ \( (T^{3} + 9 T^{2} + 6 T - 53)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 21 T - 17)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 12 T^{5} + 135 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$43$ \( T^{6} + 57 T^{4} + 326 T^{3} + \cdots + 26569 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + 45 T^{4} + 60 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( T^{6} - 6 T^{5} + 45 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$59$ \( T^{6} - 21 T^{5} + 306 T^{4} + \cdots + 71289 \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} + 102 T^{4} + \cdots + 32761 \) Copy content Toggle raw display
$67$ \( T^{6} + 18 T^{5} + 300 T^{4} + \cdots + 179776 \) Copy content Toggle raw display
$71$ \( T^{6} - 30 T^{5} + 612 T^{4} + \cdots + 788544 \) Copy content Toggle raw display
$73$ \( T^{6} + 48 T^{4} + 128 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( T^{6} - 9 T^{5} + 183 T^{4} + \cdots + 654481 \) Copy content Toggle raw display
$83$ \( (T^{3} - 189 T + 459)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 15 T^{5} + 171 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$97$ \( T^{6} + 15 T^{5} + 186 T^{4} + \cdots + 16129 \) Copy content Toggle raw display
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