Properties

Label 845.2.m.f
Level $845$
Weight $2$
Character orbit 845.m
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.m (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: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3} - \beta_1) q^{2} - \beta_{4} q^{3} + ( - 2 \beta_{7} + 2 \beta_{4} - \beta_{2} + 1) q^{4} + \beta_{3} q^{5} + (\beta_{6} - 2 \beta_1) q^{6} + ( - 2 \beta_{6} + 2 \beta_1) q^{7} + ( - \beta_{6} + \beta_{5} + 3 \beta_{3}) q^{8} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3} - \beta_1) q^{2} - \beta_{4} q^{3} + ( - 2 \beta_{7} + 2 \beta_{4} - \beta_{2} + 1) q^{4} + \beta_{3} q^{5} + (\beta_{6} - 2 \beta_1) q^{6} + ( - 2 \beta_{6} + 2 \beta_1) q^{7} + ( - \beta_{6} + \beta_{5} + 3 \beta_{3}) q^{8} + ( - \beta_{2} + 1) q^{9} + (\beta_{4} - \beta_{2}) q^{10} + (\beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{11} + ( - \beta_{7} + 4) q^{12} + (4 \beta_{7} - 6) q^{14} + \beta_{5} q^{15} - 3 \beta_{2} q^{16} + ( - 2 \beta_{7} + 2 \beta_{4} + 2 \beta_{2} - 2) q^{17} + ( - \beta_{6} + \beta_{5} + \beta_{3}) q^{18} + ( - \beta_{6} - 2 \beta_1) q^{19} + ( - 2 \beta_{6} + \beta_1) q^{20} + ( - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{3}) q^{21} + ( - 3 \beta_{7} + 3 \beta_{4} - 4 \beta_{2} + 4) q^{22} - \beta_{4} q^{23} + (3 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{24} - q^{25} - 4 \beta_{7} q^{27} + ( - 6 \beta_{5} - 10 \beta_{3} + 10 \beta_1) q^{28} - 4 \beta_{4} q^{29} + ( - \beta_{7} + \beta_{4} - 2 \beta_{2} + 2) q^{30} + (3 \beta_{6} - 3 \beta_{5} + 6 \beta_{3}) q^{31} + ( - \beta_{6} - 3 \beta_1) q^{32} + (2 \beta_{6} - 2 \beta_1) q^{33} + 2 \beta_{3} q^{34} + (2 \beta_{7} - 2 \beta_{4} + 2 \beta_{2} - 2) q^{35} + (2 \beta_{4} - \beta_{2}) q^{36} - 6 \beta_{5} q^{37} - \beta_{7} q^{38} + (\beta_{7} - 3) q^{40} + ( - 2 \beta_{5} + 6 \beta_{3} - 6 \beta_1) q^{41} + (6 \beta_{4} - 8 \beta_{2}) q^{42} + (5 \beta_{7} - 5 \beta_{4} + 4 \beta_{2} - 4) q^{43} + ( - 5 \beta_{6} + 5 \beta_{5} + 6 \beta_{3}) q^{44} + \beta_1 q^{45} + (\beta_{6} - 2 \beta_1) q^{46} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{47} + ( - 3 \beta_{7} + 3 \beta_{4}) q^{48} + ( - 8 \beta_{4} + 5 \beta_{2}) q^{49} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{50} + (2 \beta_{7} + 4) q^{51} + ( - 6 \beta_{7} - 6) q^{53} + (4 \beta_{5} + 8 \beta_{3} - 8 \beta_1) q^{54} + (\beta_{4} - 2 \beta_{2}) q^{55} + (8 \beta_{7} - 8 \beta_{4} + 10 \beta_{2} - 10) q^{56} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3}) q^{57} + (4 \beta_{6} - 8 \beta_1) q^{58} + (3 \beta_{6} + 6 \beta_1) q^{59} + ( - \beta_{6} + \beta_{5} + 4 \beta_{3}) q^{60} + ( - 8 \beta_{2} + 8) q^{61} + 3 \beta_{4} q^{62} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{63} + ( - 2 \beta_{7} + 7) q^{64} + ( - 4 \beta_{7} + 6) q^{66} + (2 \beta_{3} - 2 \beta_1) q^{67} + ( - 2 \beta_{4} - 6 \beta_{2}) q^{68} + (2 \beta_{2} - 2) q^{69} + (4 \beta_{6} - 4 \beta_{5} - 6 \beta_{3}) q^{70} + (7 \beta_{6} - 2 \beta_1) q^{71} + ( - \beta_{6} + 3 \beta_1) q^{72} + (6 \beta_{6} - 6 \beta_{5}) q^{73} + (6 \beta_{7} - 6 \beta_{4} + 12 \beta_{2} - 12) q^{74} + \beta_{4} q^{75} + (3 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{76} + (6 \beta_{7} - 8) q^{77} + 6 \beta_{7} q^{79} + ( - 3 \beta_{3} + 3 \beta_1) q^{80} + 5 \beta_{2} q^{81} + ( - 4 \beta_{7} + 4 \beta_{4} - 2 \beta_{2} + 2) q^{82} + ( - 2 \beta_{6} + 2 \beta_{5} - 6 \beta_{3}) q^{83} + ( - 10 \beta_{6} + 12 \beta_1) q^{84} + ( - 2 \beta_{6} - 2 \beta_1) q^{85} + (9 \beta_{6} - 9 \beta_{5} - 14 \beta_{3}) q^{86} + (8 \beta_{2} - 8) q^{87} + (5 \beta_{4} - 8 \beta_{2}) q^{88} + (6 \beta_{3} - 6 \beta_1) q^{89} + (\beta_{7} - 1) q^{90} + ( - \beta_{7} + 4) q^{92} + (6 \beta_{5} - 6 \beta_{3} + 6 \beta_1) q^{93} + (4 \beta_{4} - 6 \beta_{2}) q^{94} + (\beta_{7} - \beta_{4} - 2 \beta_{2} + 2) q^{95} + (3 \beta_{6} - 3 \beta_{5} + 2 \beta_{3}) q^{96} + ( - 4 \beta_{6} + 2 \beta_1) q^{97} + (13 \beta_{6} - 21 \beta_1) q^{98} + ( - \beta_{6} + \beta_{5} + 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{9} - 4 q^{10} + 32 q^{12} - 48 q^{14} - 12 q^{16} - 8 q^{17} + 16 q^{22} - 8 q^{25} + 8 q^{30} - 8 q^{35} - 4 q^{36} - 24 q^{40} - 32 q^{42} - 16 q^{43} + 20 q^{49} + 32 q^{51} - 48 q^{53} - 8 q^{55} - 40 q^{56} + 32 q^{61} + 56 q^{64} + 48 q^{66} - 24 q^{68} - 8 q^{69} - 48 q^{74} - 64 q^{77} + 20 q^{81} + 8 q^{82} - 32 q^{87} - 32 q^{88} - 8 q^{90} + 32 q^{92} - 24 q^{94} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
−0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 0.965926i
−2.09077 1.20711i 0.707107 1.22474i 1.91421 + 3.31552i 1.00000i −2.95680 + 1.70711i 4.18154 2.41421i 4.41421i 0.500000 + 0.866025i −1.20711 + 2.09077i
316.2 −0.358719 0.207107i −0.707107 + 1.22474i −0.914214 1.58346i 1.00000i 0.507306 0.292893i 0.717439 0.414214i 1.58579i 0.500000 + 0.866025i 0.207107 0.358719i
316.3 0.358719 + 0.207107i −0.707107 + 1.22474i −0.914214 1.58346i 1.00000i −0.507306 + 0.292893i −0.717439 + 0.414214i 1.58579i 0.500000 + 0.866025i 0.207107 0.358719i
316.4 2.09077 + 1.20711i 0.707107 1.22474i 1.91421 + 3.31552i 1.00000i 2.95680 1.70711i −4.18154 + 2.41421i 4.41421i 0.500000 + 0.866025i −1.20711 + 2.09077i
361.1 −2.09077 + 1.20711i 0.707107 + 1.22474i 1.91421 3.31552i 1.00000i −2.95680 1.70711i 4.18154 + 2.41421i 4.41421i 0.500000 0.866025i −1.20711 2.09077i
361.2 −0.358719 + 0.207107i −0.707107 1.22474i −0.914214 + 1.58346i 1.00000i 0.507306 + 0.292893i 0.717439 + 0.414214i 1.58579i 0.500000 0.866025i 0.207107 + 0.358719i
361.3 0.358719 0.207107i −0.707107 1.22474i −0.914214 + 1.58346i 1.00000i −0.507306 0.292893i −0.717439 0.414214i 1.58579i 0.500000 0.866025i 0.207107 + 0.358719i
361.4 2.09077 1.20711i 0.707107 + 1.22474i 1.91421 3.31552i 1.00000i 2.95680 + 1.70711i −4.18154 2.41421i 4.41421i 0.500000 0.866025i −1.20711 2.09077i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e 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.m.f 8
13.b even 2 1 inner 845.2.m.f 8
13.c even 3 1 845.2.c.b 4
13.c even 3 1 inner 845.2.m.f 8
13.d odd 4 1 845.2.e.c 4
13.d odd 4 1 845.2.e.h 4
13.e even 6 1 845.2.c.b 4
13.e even 6 1 inner 845.2.m.f 8
13.f odd 12 1 65.2.a.b 2
13.f odd 12 1 845.2.a.g 2
13.f odd 12 1 845.2.e.c 4
13.f odd 12 1 845.2.e.h 4
39.k even 12 1 585.2.a.m 2
39.k even 12 1 7605.2.a.x 2
52.l even 12 1 1040.2.a.j 2
65.o even 12 1 325.2.b.f 4
65.s odd 12 1 325.2.a.i 2
65.s odd 12 1 4225.2.a.r 2
65.t even 12 1 325.2.b.f 4
91.bc even 12 1 3185.2.a.j 2
104.u even 12 1 4160.2.a.z 2
104.x odd 12 1 4160.2.a.bf 2
143.o even 12 1 7865.2.a.j 2
156.v odd 12 1 9360.2.a.cd 2
195.bc odd 12 1 2925.2.c.r 4
195.bh even 12 1 2925.2.a.u 2
195.bn odd 12 1 2925.2.c.r 4
260.bc even 12 1 5200.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 13.f odd 12 1
325.2.a.i 2 65.s odd 12 1
325.2.b.f 4 65.o even 12 1
325.2.b.f 4 65.t even 12 1
585.2.a.m 2 39.k even 12 1
845.2.a.g 2 13.f odd 12 1
845.2.c.b 4 13.c even 3 1
845.2.c.b 4 13.e even 6 1
845.2.e.c 4 13.d odd 4 1
845.2.e.c 4 13.f odd 12 1
845.2.e.h 4 13.d odd 4 1
845.2.e.h 4 13.f odd 12 1
845.2.m.f 8 1.a even 1 1 trivial
845.2.m.f 8 13.b even 2 1 inner
845.2.m.f 8 13.c even 3 1 inner
845.2.m.f 8 13.e even 6 1 inner
1040.2.a.j 2 52.l even 12 1
2925.2.a.u 2 195.bh even 12 1
2925.2.c.r 4 195.bc odd 12 1
2925.2.c.r 4 195.bn odd 12 1
3185.2.a.j 2 91.bc even 12 1
4160.2.a.z 2 104.u even 12 1
4160.2.a.bf 2 104.x odd 12 1
4225.2.a.r 2 65.s odd 12 1
5200.2.a.bu 2 260.bc even 12 1
7605.2.a.x 2 39.k even 12 1
7865.2.a.j 2 143.o even 12 1
9360.2.a.cd 2 156.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6T_{2}^{6} + 35T_{2}^{4} - 6T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{6} + 35 T^{4} - 6 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 24 T^{6} + 560 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{6} + 140 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 12 T^{6} + 140 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 108 T^{2} + 324)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 88 T^{6} + 6960 T^{4} + \cdots + 614656 \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} + 98 T^{2} - 272 T + 1156)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 24 T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T - 36)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} - 108 T^{6} + 11340 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8 T + 64)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 204 T^{6} + \cdots + 78074896 \) Copy content Toggle raw display
$73$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 72)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 88 T^{2} + 784)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 72 T^{6} + 4400 T^{4} + \cdots + 614656 \) Copy content Toggle raw display
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