Properties

Label 845.2.a.l
Level $845$
Weight $2$
Character orbit 845.a
Self dual yes
Analytic conductor $6.747$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.74735897080\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.49551
1.21969
−0.219687
−1.49551
−2.49551 −2.82684 4.22756 1.00000 7.05440 −1.90521 −5.55889 4.99102 −2.49551
1.2 −1.21969 2.33225 −0.512364 1.00000 −2.84461 −3.60020 3.06430 2.43937 −1.21969
1.3 0.219687 −1.60020 −1.95174 1.00000 −0.351542 0.332247 −0.868145 −0.439374 0.219687
1.4 1.49551 0.0947876 0.236543 1.00000 0.141756 −4.82684 −2.63726 −2.99102 1.49551
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.a.l 4
3.b odd 2 1 7605.2.a.cj 4
5.b even 2 1 4225.2.a.bl 4
13.b even 2 1 845.2.a.m 4
13.c even 3 2 845.2.e.n 8
13.d odd 4 2 845.2.c.g 8
13.e even 6 2 845.2.e.m 8
13.f odd 12 2 65.2.m.a 8
13.f odd 12 2 845.2.m.g 8
39.d odd 2 1 7605.2.a.cf 4
39.k even 12 2 585.2.bu.c 8
52.l even 12 2 1040.2.da.b 8
65.d even 2 1 4225.2.a.bi 4
65.o even 12 2 325.2.m.c 8
65.s odd 12 2 325.2.n.d 8
65.t even 12 2 325.2.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 13.f odd 12 2
325.2.m.b 8 65.t even 12 2
325.2.m.c 8 65.o even 12 2
325.2.n.d 8 65.s odd 12 2
585.2.bu.c 8 39.k even 12 2
845.2.a.l 4 1.a even 1 1 trivial
845.2.a.m 4 13.b even 2 1
845.2.c.g 8 13.d odd 4 2
845.2.e.m 8 13.e even 6 2
845.2.e.n 8 13.c even 3 2
845.2.m.g 8 13.f odd 12 2
1040.2.da.b 8 52.l even 12 2
4225.2.a.bi 4 65.d even 2 1
4225.2.a.bl 4 5.b even 2 1
7605.2.a.cf 4 39.d odd 2 1
7605.2.a.cj 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2 T_{2}^{3} - 3 T_{2}^{2} - 4 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(845))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T - 3 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( 1 - 10 T - 6 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( -11 + 22 T + 30 T^{2} + 10 T^{3} + T^{4} \)
$11$ \( 33 - 30 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 13 - 10 T - 18 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( ( 13 + 8 T + T^{2} )^{2} \)
$23$ \( -299 - 146 T + 6 T^{2} + 10 T^{3} + T^{4} \)
$29$ \( 1 + 40 T - 18 T^{2} - 8 T^{3} + T^{4} \)
$31$ \( ( -8 + 4 T + T^{2} )^{2} \)
$37$ \( 1 - 38 T - 54 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( ( 1 + 4 T + T^{2} )^{2} \)
$43$ \( 13 - 10 T - 18 T^{2} + 2 T^{3} + T^{4} \)
$47$ \( -1328 - 736 T - 72 T^{2} + 8 T^{3} + T^{4} \)
$53$ \( -48 + 36 T^{2} + 12 T^{3} + T^{4} \)
$59$ \( -3 + 12 T + 30 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( 1261 - 964 T + 258 T^{2} - 28 T^{3} + T^{4} \)
$67$ \( 2769 + 1578 T + 330 T^{2} + 30 T^{3} + T^{4} \)
$71$ \( 10477 - 428 T - 210 T^{2} + 4 T^{3} + T^{4} \)
$73$ \( -1712 + 832 T - 84 T^{2} - 8 T^{3} + T^{4} \)
$79$ \( 4432 - 640 T - 132 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( -192 + 288 T - 24 T^{2} - 12 T^{3} + T^{4} \)
$89$ \( 8853 + 2148 T - 234 T^{2} - 12 T^{3} + T^{4} \)
$97$ \( -443 + 374 T - 90 T^{2} + 2 T^{3} + T^{4} \)
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