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Properties

Label 845.2.l.c
Level $845$
Weight $2$
Character orbit 845.l
Analytic conductor $6.747$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{3}) q^{3} + (\beta_{6} + \beta_{4} + \beta_{2}) q^{4} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1) q^{5} + (\beta_{6} - \beta_{2} + 1) q^{6} + ( - 2 \beta_{5} - \beta_{3}) q^{7} + (\beta_{5} + 2 \beta_{3}) q^{8} + (2 \beta_{4} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{3}) q^{3} + (\beta_{6} + \beta_{4} + \beta_{2}) q^{4} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1) q^{5} + (\beta_{6} - \beta_{2} + 1) q^{6} + ( - 2 \beta_{5} - \beta_{3}) q^{7} + (\beta_{5} + 2 \beta_{3}) q^{8} + (2 \beta_{4} - 2) q^{9} + (2 \beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_1) q^{10} + (\beta_{4} + 2 \beta_{2}) q^{11} + (\beta_{7} + \beta_{5} + \beta_1) q^{12} + (\beta_{6} - \beta_{4} - 2 \beta_{2} + 2) q^{14} + (\beta_{5} + \beta_{2} + \beta_1 + 1) q^{15} + (2 \beta_{6} - \beta_{4} - \beta_{2} + 1) q^{16} + ( - 2 \beta_{7} + 2 \beta_{5} + 3 \beta_{3} + 4 \beta_1) q^{17} + (2 \beta_{7} - 2 \beta_1) q^{18} + (\beta_{4} + 1) q^{19} + (\beta_{7} - 2 \beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 3) q^{20} + ( - 2 \beta_{4} + 1) q^{21} + (\beta_{7} - \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{22} + (4 \beta_{7} - \beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{23} + (\beta_{4} - 2) q^{24} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 4 \beta_{2} - 1) q^{25} + 5 \beta_{5} q^{27} + (2 \beta_{5} + \beta_{3} + 3 \beta_1) q^{28} + ( - 4 \beta_{6} + \beta_{4} + 2 \beta_{2} - 2) q^{29} + (\beta_{6} + 3 \beta_{4} + \beta_{3} + \beta_{2} - 2) q^{30} + ( - 6 \beta_{6} - 2 \beta_{4} - 2) q^{31} + (\beta_{7} + 7 \beta_{5} + 3 \beta_{3}) q^{32} + (2 \beta_{7} - \beta_{3}) q^{33} + (3 \beta_{6} + 7 \beta_{4} - 2) q^{34} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{35} + ( - 4 \beta_{6} + 2 \beta_{2} - 2) q^{36} + (3 \beta_{5} + 3 \beta_{3} + 6 \beta_1) q^{37} + (\beta_{7} + 2 \beta_{5} + \beta_1) q^{38} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{2} - \beta_1 - 1) q^{40} + ( - \beta_{4} - 2 \beta_{2}) q^{41} + ( - 2 \beta_{7} - \beta_{5} + \beta_1) q^{42} + ( - 2 \beta_{7} + 2 \beta_{5} + 9 \beta_{3} + 4 \beta_1) q^{43} + ( - \beta_{6} - 7 \beta_{4} + 3) q^{44} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{45} + ( - 3 \beta_{6} - 2 \beta_{4} + 3 \beta_{2} - 5) q^{46} + ( - 4 \beta_{7} + 4 \beta_1) q^{47} + ( - \beta_{7} + \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{48} + 4 \beta_{4} q^{49} + ( - 5 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 2) q^{50} + (2 \beta_{6} - 2 \beta_{4} - 4 \beta_{2} + 1) q^{51} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{53} + (5 \beta_{4} + 5 \beta_{2} - 5) q^{54} + (2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{55} + (3 \beta_{4} - 3) q^{56} + ( - \beta_{5} - 2 \beta_{3}) q^{57} + ( - 3 \beta_{7} - 7 \beta_{5} - 2 \beta_{3}) q^{58} + ( - 4 \beta_{6} + 7 \beta_{4} + 4 \beta_{2} + 3) q^{59} + (2 \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 2) q^{60} + ( - 2 \beta_{6} - 7 \beta_{4} - 2 \beta_{2} + 5) q^{61} + ( - 8 \beta_{7} - 10 \beta_{5} - 6 \beta_{3} + 4 \beta_1) q^{62} + (2 \beta_{5} - 2 \beta_{3}) q^{63} + ( - 2 \beta_{6} + 2 \beta_{4} + 4 \beta_{2} - 9) q^{64} + ( - \beta_{6} + \beta_{4} + 2 \beta_{2} - 4) q^{66} + ( - 7 \beta_{5} + \beta_{3} - 6 \beta_1) q^{67} + (2 \beta_{7} + 10 \beta_{5} + 9 \beta_{3} - \beta_1) q^{68} + (2 \beta_{6} + \beta_{4} + 2 \beta_{2} + 1) q^{69} + (3 \beta_{6} + \beta_{5} + 5 \beta_{4} + 2 \beta_{3} - 1) q^{70} + (4 \beta_{6} - 3 \beta_{4} - 4 \beta_{2} + 1) q^{71} + ( - 4 \beta_{5} - 2 \beta_{3}) q^{72} + (3 \beta_{6} + 12 \beta_{4} + 3 \beta_{2} - 9) q^{74} + (4 \beta_{7} - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{75} + (2 \beta_{4} + 3 \beta_{2} - 1) q^{76} + (2 \beta_{7} + \beta_{5} + 2 \beta_1) q^{77} + 6 q^{79} + ( - \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{80} - \beta_{4} q^{81} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{82} + ( - 2 \beta_{7} - 4 \beta_{5} - 8 \beta_{3} + 2 \beta_1) q^{83} + (3 \beta_{6} - \beta_{4} - 3 \beta_{2} + 2) q^{84} + (3 \beta_{7} + 3 \beta_{6} + 7 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + \cdots - 2) q^{85}+ \cdots + ( - 4 \beta_{6} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{4} + 6 q^{6} - 8 q^{9} - 4 q^{10} + 12 q^{14} + 6 q^{15} - 2 q^{16} + 12 q^{19} - 24 q^{20} - 12 q^{24} - 10 q^{30} + 6 q^{35} - 4 q^{36} - 12 q^{40} - 12 q^{45} - 42 q^{46} + 16 q^{49} + 12 q^{50} - 30 q^{54} - 14 q^{55} - 12 q^{56} + 60 q^{59} + 24 q^{61} - 64 q^{64} - 28 q^{66} - 12 q^{71} - 42 q^{74} - 8 q^{75} - 6 q^{76} + 48 q^{79} - 18 q^{80} - 4 q^{81} + 6 q^{84} - 42 q^{85} - 48 q^{89} + 16 q^{90} + 40 q^{94} - 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 5\nu^{4} - 5\nu^{2} - 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 5\nu^{5} - 5\nu^{3} - 12\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 36 ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 7 ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 3\nu^{5} + 5\nu^{3} + 12\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{5} - \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{6} - 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10\beta_{5} - 10\beta_{3} - 7\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(1 - \beta_{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
654.1
−1.09445 + 0.895644i
−0.228425 + 1.39564i
0.228425 − 1.39564i
1.09445 − 0.895644i
−1.09445 − 0.895644i
−0.228425 − 1.39564i
0.228425 + 1.39564i
1.09445 + 0.895644i
−1.09445 + 1.89564i −0.866025 − 0.500000i −1.39564 − 2.41733i 0.456850 − 2.18890i 1.89564 − 1.09445i −0.866025 − 1.50000i 1.73205 −1.00000 − 1.73205i 3.64938 + 3.26167i
654.2 −0.228425 + 0.395644i 0.866025 + 0.500000i 0.895644 + 1.55130i 2.18890 + 0.456850i −0.395644 + 0.228425i 0.866025 + 1.50000i −1.73205 −1.00000 − 1.73205i −0.680750 + 0.761669i
654.3 0.228425 − 0.395644i −0.866025 − 0.500000i 0.895644 + 1.55130i −2.18890 + 0.456850i −0.395644 + 0.228425i −0.866025 − 1.50000i 1.73205 −1.00000 − 1.73205i −0.319250 + 0.970381i
654.4 1.09445 − 1.89564i 0.866025 + 0.500000i −1.39564 − 2.41733i −0.456850 − 2.18890i 1.89564 − 1.09445i 0.866025 + 1.50000i −1.73205 −1.00000 − 1.73205i −4.64938 − 1.52962i
699.1 −1.09445 − 1.89564i −0.866025 + 0.500000i −1.39564 + 2.41733i 0.456850 + 2.18890i 1.89564 + 1.09445i −0.866025 + 1.50000i 1.73205 −1.00000 + 1.73205i 3.64938 − 3.26167i
699.2 −0.228425 − 0.395644i 0.866025 − 0.500000i 0.895644 − 1.55130i 2.18890 − 0.456850i −0.395644 − 0.228425i 0.866025 − 1.50000i −1.73205 −1.00000 + 1.73205i −0.680750 − 0.761669i
699.3 0.228425 + 0.395644i −0.866025 + 0.500000i 0.895644 − 1.55130i −2.18890 − 0.456850i −0.395644 − 0.228425i −0.866025 + 1.50000i 1.73205 −1.00000 + 1.73205i −0.319250 − 0.970381i
699.4 1.09445 + 1.89564i 0.866025 − 0.500000i −1.39564 + 2.41733i −0.456850 + 2.18890i 1.89564 + 1.09445i 0.866025 − 1.50000i −1.73205 −1.00000 + 1.73205i −4.64938 + 1.52962i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 699.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.e even 6 1 inner
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.l.c 8
5.b even 2 1 inner 845.2.l.c 8
13.b even 2 1 65.2.l.a ✓ 8
13.c even 3 1 65.2.l.a ✓ 8
13.c even 3 1 845.2.d.c 8
13.d odd 4 1 845.2.n.c 8
13.d odd 4 1 845.2.n.d 8
13.e even 6 1 845.2.d.c 8
13.e even 6 1 inner 845.2.l.c 8
13.f odd 12 2 845.2.b.f 8
13.f odd 12 1 845.2.n.c 8
13.f odd 12 1 845.2.n.d 8
39.d odd 2 1 585.2.bf.a 8
39.i odd 6 1 585.2.bf.a 8
52.b odd 2 1 1040.2.df.b 8
52.j odd 6 1 1040.2.df.b 8
65.d even 2 1 65.2.l.a ✓ 8
65.g odd 4 1 845.2.n.c 8
65.g odd 4 1 845.2.n.d 8
65.h odd 4 1 325.2.n.b 4
65.h odd 4 1 325.2.n.c 4
65.l even 6 1 845.2.d.c 8
65.l even 6 1 inner 845.2.l.c 8
65.n even 6 1 65.2.l.a ✓ 8
65.n even 6 1 845.2.d.c 8
65.o even 12 1 4225.2.a.bj 4
65.o even 12 1 4225.2.a.bk 4
65.q odd 12 1 325.2.n.b 4
65.q odd 12 1 325.2.n.c 4
65.s odd 12 2 845.2.b.f 8
65.s odd 12 1 845.2.n.c 8
65.s odd 12 1 845.2.n.d 8
65.t even 12 1 4225.2.a.bj 4
65.t even 12 1 4225.2.a.bk 4
195.e odd 2 1 585.2.bf.a 8
195.x odd 6 1 585.2.bf.a 8
260.g odd 2 1 1040.2.df.b 8
260.v odd 6 1 1040.2.df.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.l.a ✓ 8 13.b even 2 1
65.2.l.a ✓ 8 13.c even 3 1
65.2.l.a ✓ 8 65.d even 2 1
65.2.l.a ✓ 8 65.n even 6 1
325.2.n.b 4 65.h odd 4 1
325.2.n.b 4 65.q odd 12 1
325.2.n.c 4 65.h odd 4 1
325.2.n.c 4 65.q odd 12 1
585.2.bf.a 8 39.d odd 2 1
585.2.bf.a 8 39.i odd 6 1
585.2.bf.a 8 195.e odd 2 1
585.2.bf.a 8 195.x odd 6 1
845.2.b.f 8 13.f odd 12 2
845.2.b.f 8 65.s odd 12 2
845.2.d.c 8 13.c even 3 1
845.2.d.c 8 13.e even 6 1
845.2.d.c 8 65.l even 6 1
845.2.d.c 8 65.n even 6 1
845.2.l.c 8 1.a even 1 1 trivial
845.2.l.c 8 5.b even 2 1 inner
845.2.l.c 8 13.e even 6 1 inner
845.2.l.c 8 65.l even 6 1 inner
845.2.n.c 8 13.d odd 4 1
845.2.n.c 8 13.f odd 12 1
845.2.n.c 8 65.g odd 4 1
845.2.n.c 8 65.s odd 12 1
845.2.n.d 8 13.d odd 4 1
845.2.n.d 8 13.f odd 12 1
845.2.n.d 8 65.g odd 4 1
845.2.n.d 8 65.s odd 12 1
1040.2.df.b 8 52.b odd 2 1
1040.2.df.b 8 52.j odd 6 1
1040.2.df.b 8 260.g odd 2 1
1040.2.df.b 8 260.v odd 6 1
4225.2.a.bj 4 65.o even 12 1
4225.2.a.bj 4 65.t even 12 1
4225.2.a.bk 4 65.o even 12 1
4225.2.a.bk 4 65.t even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 5T_{2}^{6} + 24T_{2}^{4} + 5T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{6} + 24 T^{4} + 5 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 34T^{4} + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 21 T^{2} + 441)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 21 T^{2} + 441)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 21 T^{2} + 441)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 132 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 63 T^{2} + 3969)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 114 T^{6} + 12771 T^{4} + \cdots + 50625 \) Copy content Toggle raw display
$47$ \( (T^{4} - 80 T^{2} + 256)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 60 T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 30 T^{3} + 347 T^{2} - 1410 T + 2209)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 12 T^{3} + 129 T^{2} - 180 T + 225)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 222 T^{6} + 49059 T^{4} + \cdots + 50625 \) Copy content Toggle raw display
$71$ \( (T^{4} + 6 T^{3} - 13 T^{2} - 150 T + 625)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T - 6)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 164 T^{2} + 4624)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 24 T^{3} + 233 T^{2} + 984 T + 1681)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 150 T^{6} + 19899 T^{4} + \cdots + 6765201 \) Copy content Toggle raw display
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