Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−0.207107 0.358719i −0.707107 1.22474i 0.914214 1.58346i 1.00000 −0.292893 + 0.507306i 0.414214 0.717439i −1.58579 0.500000 0.866025i −0.207107 0.358719i
146.2 1.20711 + 2.09077i 0.707107 + 1.22474i −1.91421 + 3.31552i 1.00000 −1.70711 + 2.95680i −2.41421 + 4.18154i −4.41421 0.500000 0.866025i 1.20711 + 2.09077i
191.1 −0.207107 + 0.358719i −0.707107 + 1.22474i 0.914214 + 1.58346i 1.00000 −0.292893 0.507306i 0.414214 + 0.717439i −1.58579 0.500000 + 0.866025i −0.207107 + 0.358719i
191.2 1.20711 2.09077i 0.707107 1.22474i −1.91421 3.31552i 1.00000 −1.70711 2.95680i −2.41421 4.18154i −4.41421 0.500000 + 0.866025i 1.20711 2.09077i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 4T_{7}^{3} + 20T_{7}^{2} - 16T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + 14 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + 14 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$23$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$29$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$31$ \( (T^{2} - 12 T + 18)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 72T^{2} + 5184 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + 116 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + 98 T^{2} + \cdots + 1156 \) Copy content Toggle raw display
$47$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T - 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 4 T^{3} + 110 T^{2} + \cdots + 8836 \) Copy content Toggle raw display
$73$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784 \) Copy content Toggle raw display
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