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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [361,3,Mod(116,361)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(361, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("361.116");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 361.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2197x^{6} + 4826809 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{7} + \beta_1) q^{3} + 9 \beta_{2} q^{4} + (4 \beta_{10} - 4 \beta_{4}) q^{5} + ( - 13 \beta_{8} + 13 \beta_{2}) q^{6} + 5 \beta_{6} q^{7} + 5 \beta_{3} q^{8} - 4 \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{7} + \beta_1) q^{3} + 9 \beta_{2} q^{4} + (4 \beta_{10} - 4 \beta_{4}) q^{5} + ( - 13 \beta_{8} + 13 \beta_{2}) q^{6} + 5 \beta_{6} q^{7} + 5 \beta_{3} q^{8} - 4 \beta_{8} q^{9} + (4 \beta_{11} - 4 \beta_{5}) q^{10} + ( - 10 \beta_{6} + 10) q^{11} + ( - 9 \beta_{9} + 9 \beta_{3}) 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_{7} + 10 \beta_1) q^{22} - 35 \beta_{2} q^{23} + ( - 65 \beta_{10} + 65 \beta_{4}) q^{24} + (9 \beta_{8} - 9 \beta_{2}) q^{25} + 13 \beta_{6} q^{26} + 5 \beta_{3} q^{27} + 45 \beta_{8} q^{28} + (5 \beta_{11} - 5 \beta_{5}) q^{29} + (52 \beta_{6} - 52) q^{30} + (10 \beta_{9} - 10 \beta_{3}) 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_{7} - 10 \beta_1) q^{41} + 65 \beta_{2} q^{42} + ( - 20 \beta_{10} + 20 \beta_{4}) q^{43} + ( - 90 \beta_{8} + 90 \beta_{2}) q^{44} + 16 \beta_{6} q^{45} - 35 \beta_{3} q^{46} + 10 \beta_{8} q^{47} + ( - 29 \beta_{11} + 29 \beta_{5}) q^{48} + ( - 24 \beta_{6} + 24) q^{49} + (9 \beta_{9} - 9 \beta_{3}) 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_{7} - 36 \beta_1) q^{60} + 40 \beta_{2} q^{61} + (130 \beta_{10} - 130 \beta_{4}) q^{62} + ( - 20 \beta_{8} + 20 \beta_{2}) q^{63} + \beta_{6} q^{64} - 4 \beta_{3} q^{65} - 130 \beta_{8} q^{66} + (11 \beta_{11} - 11 \beta_{5}) q^{67} + ( - 135 \beta_{6} + 135) q^{68} + (35 \beta_{9} - 35 \beta_{3}) 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_{7} + 10 \beta_1) q^{79} - 116 \beta_{2} q^{80} + ( - 101 \beta_{10} + 101 \beta_{4}) q^{81} + (130 \beta_{8} - 130 \beta_{2}) q^{82} + 40 \beta_{6} q^{83} + 45 \beta_{3} q^{84} + 60 \beta_{8} q^{85} + ( - 20 \beta_{11} + 20 \beta_{5}) q^{86} + (65 \beta_{6} - 65) q^{87} + ( - 50 \beta_{9} + 50 \beta_{3}) 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_{7} + 24 \beta_1) q^{98} - 40 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 30 q^{7} + 60 q^{11} - 432 q^{20} + 78 q^{26} - 312 q^{30} + 156 q^{39} + 96 q^{45} + 144 q^{49} - 780 q^{58} + 6 q^{64} + 810 q^{68} + 600 q^{77} + 240 q^{83} - 390 q^{87} + 1404 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 169 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 169 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 2197 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 2197 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} ) / 28561 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} ) / 28561 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} ) / 371293 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} ) / 371293 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 13\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 13\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 169\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 169\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2197\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2197\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 28561\beta_{8} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 28561\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 371293\beta_{10} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 371293\beta_{11} \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 116.2
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 \( T_{2}^{12} - 2197T_{2}^{6} + 4826809 \) acting on \(S_{3}^{\mathrm{new}}(361, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2197 T^{6} + 4826809 \) Copy content Toggle raw display
$3$ \( T^{12} - 2197 T^{6} + 4826809 \) Copy content Toggle raw display
$5$ \( (T^{6} + 64 T^{3} + 4096)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T + 25)^{6} \) Copy content Toggle raw display
$11$ \( (T^{2} - 10 T + 100)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} - 2197 T^{6} + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} + 3375 T^{3} + 11390625)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} + 42875 T^{3} + 1838265625)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( (T^{4} - 1300 T^{2} + 1690000)^{3} \) Copy content Toggle raw display
$37$ \( (T^{2} + 468)^{6} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} - 8000 T^{3} + 64000000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 1000 T^{3} + 1000000)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 35\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{6} - 64000 T^{3} + 4096000000)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15\!\cdots\!89 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 1340095640625)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{2} - 40 T + 1600)^{6} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
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