Properties

Label 825.2.n.b
Level $825$
Weight $2$
Character orbit 825.n
Analytic conductor $6.588$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{2} -\zeta_{10}^{2} q^{3} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{4} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{6} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{7} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{2} -\zeta_{10}^{2} q^{3} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{4} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{6} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{7} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + ( 3 - 2 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{11} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{12} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{13} + ( -4 \zeta_{10} + 6 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{14} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{16} -2 \zeta_{10} q^{17} + ( 1 - \zeta_{10} ) q^{18} + 5 \zeta_{10}^{2} q^{19} + ( 2 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{21} + ( -4 + 3 \zeta_{10} - \zeta_{10}^{2} ) q^{22} + ( 3 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{23} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{24} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{26} + \zeta_{10} q^{27} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{28} + ( -3 + 3 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{29} + ( -7 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{31} + ( 5 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{32} + ( 2 - 4 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{33} + ( 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{34} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{36} + ( -4 + 4 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{37} + ( -5 + 5 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{38} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{39} + ( -4 \zeta_{10} + 9 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{41} + ( 2 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{42} + ( 1 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{43} + ( -3 + 3 \zeta_{10} - 2 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{44} + ( 1 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{46} + ( 5 \zeta_{10} - 3 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{47} + ( -3 + 3 \zeta_{10} ) q^{48} + ( 12 - 13 \zeta_{10} + 12 \zeta_{10}^{2} ) q^{49} + 2 \zeta_{10}^{3} q^{51} + 2 \zeta_{10}^{2} q^{52} + ( 7 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{53} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{54} + ( -2 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{56} + ( 5 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{57} + ( -\zeta_{10} - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{58} + ( -6 + 6 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{59} -7 \zeta_{10} q^{61} + ( 7 - 7 \zeta_{10} ) q^{62} + ( 2 \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{63} + ( -1 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( -1 + \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{66} + ( -10 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{67} + ( 2 - 2 \zeta_{10}^{3} ) q^{68} + ( \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{69} + 8 \zeta_{10} q^{71} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{72} + ( -4 + 4 \zeta_{10} + 11 \zeta_{10}^{3} ) q^{73} + ( -\zeta_{10} - 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{74} + ( -5 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{76} + ( -2 - 6 \zeta_{10} + 12 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{77} + ( 2 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{78} + ( 5 - 5 \zeta_{10}^{3} ) q^{79} -\zeta_{10}^{3} q^{81} + ( -9 + 13 \zeta_{10} - 9 \zeta_{10}^{2} ) q^{82} + ( 6 - 4 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{83} + ( -2 + 2 \zeta_{10} ) q^{84} + ( 5 - 11 \zeta_{10} + 11 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{86} + ( 4 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{87} + ( -2 + 2 \zeta_{10} - 5 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{88} + ( 9 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{89} + ( 8 \zeta_{10} - 12 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{91} + ( -3 + 3 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{92} + 7 \zeta_{10} q^{93} + ( 3 - 8 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{94} + ( -\zeta_{10} - 4 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{96} + ( -9 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{97} + ( -12 + 13 \zeta_{10}^{2} - 13 \zeta_{10}^{3} ) q^{98} + ( -1 - \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + q^{3} - 2 q^{4} + 2 q^{6} + 8 q^{7} - 5 q^{8} - q^{9} + O(q^{10}) \) \( 4 q - 2 q^{2} + q^{3} - 2 q^{4} + 2 q^{6} + 8 q^{7} - 5 q^{8} - q^{9} + 4 q^{11} + 2 q^{12} + 4 q^{13} - 14 q^{14} - 6 q^{16} - 2 q^{17} + 3 q^{18} - 5 q^{19} + 12 q^{21} - 12 q^{22} + 14 q^{23} - 5 q^{24} + 8 q^{26} + q^{27} - 4 q^{28} - 5 q^{29} - 7 q^{31} + 18 q^{32} + q^{33} - 4 q^{34} - 2 q^{36} - 7 q^{37} - 10 q^{38} - 4 q^{39} - 17 q^{41} - 6 q^{42} + 14 q^{43} - 2 q^{44} - 7 q^{46} + 13 q^{47} - 9 q^{48} + 23 q^{49} + 2 q^{51} - 2 q^{52} + 9 q^{53} + 2 q^{54} - 20 q^{56} + 5 q^{57} - 15 q^{59} - 7 q^{61} + 21 q^{62} + 8 q^{63} + 3 q^{64} - 8 q^{66} - 42 q^{67} + 6 q^{68} + 6 q^{69} + 8 q^{71} + 5 q^{72} - q^{73} + q^{74} - 10 q^{76} - 32 q^{77} + 12 q^{78} + 15 q^{79} - q^{81} - 14 q^{82} + 14 q^{83} - 6 q^{84} - 7 q^{86} + 10 q^{87} - 5 q^{88} + 30 q^{89} + 28 q^{91} - 7 q^{92} + 7 q^{93} + q^{94} + 2 q^{96} - 17 q^{97} - 74 q^{98} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.500000 0.363271i −0.309017 + 0.951057i −0.500000 1.53884i 0 0.500000 0.363271i −0.236068 0.726543i −0.690983 + 2.12663i −0.809017 0.587785i 0
526.1 −0.500000 + 1.53884i 0.809017 0.587785i −0.500000 0.363271i 0 0.500000 + 1.53884i 4.23607 + 3.07768i −1.80902 + 1.31433i 0.309017 0.951057i 0
676.1 −0.500000 1.53884i 0.809017 + 0.587785i −0.500000 + 0.363271i 0 0.500000 1.53884i 4.23607 3.07768i −1.80902 1.31433i 0.309017 + 0.951057i 0
751.1 −0.500000 + 0.363271i −0.309017 0.951057i −0.500000 + 1.53884i 0 0.500000 + 0.363271i −0.236068 + 0.726543i −0.690983 2.12663i −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.n.b 4
5.b even 2 1 825.2.n.d yes 4
5.c odd 4 2 825.2.bx.c 8
11.c even 5 1 inner 825.2.n.b 4
11.c even 5 1 9075.2.a.z 2
11.d odd 10 1 9075.2.a.bt 2
55.h odd 10 1 9075.2.a.bc 2
55.j even 10 1 825.2.n.d yes 4
55.j even 10 1 9075.2.a.by 2
55.k odd 20 2 825.2.bx.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.b 4 1.a even 1 1 trivial
825.2.n.b 4 11.c even 5 1 inner
825.2.n.d yes 4 5.b even 2 1
825.2.n.d yes 4 55.j even 10 1
825.2.bx.c 8 5.c odd 4 2
825.2.bx.c 8 55.k odd 20 2
9075.2.a.z 2 11.c even 5 1
9075.2.a.bc 2 55.h odd 10 1
9075.2.a.bt 2 11.d odd 10 1
9075.2.a.by 2 55.j even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{4} + 2 T_{2}^{3} + 4 T_{2}^{2} + 3 T_{2} + 1 \)
\( T_{13}^{4} - 4 T_{13}^{3} + 16 T_{13}^{2} - 24 T_{13} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 + 8 T + 24 T^{2} - 8 T^{3} + T^{4} \)
$11$ \( 121 - 44 T + 6 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( 16 - 24 T + 16 T^{2} - 4 T^{3} + T^{4} \)
$17$ \( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( 625 + 125 T + 25 T^{2} + 5 T^{3} + T^{4} \)
$23$ \( ( 11 - 7 T + T^{2} )^{2} \)
$29$ \( 25 + 50 T + 40 T^{2} + 5 T^{3} + T^{4} \)
$31$ \( 2401 + 343 T + 49 T^{2} + 7 T^{3} + T^{4} \)
$37$ \( 121 + 143 T + 69 T^{2} + 7 T^{3} + T^{4} \)
$41$ \( 841 - 87 T + 109 T^{2} + 17 T^{3} + T^{4} \)
$43$ \( ( -19 - 7 T + T^{2} )^{2} \)
$47$ \( 961 - 372 T + 94 T^{2} - 13 T^{3} + T^{4} \)
$53$ \( 1681 - 164 T + 46 T^{2} - 9 T^{3} + T^{4} \)
$59$ \( 2025 + 675 T + 135 T^{2} + 15 T^{3} + T^{4} \)
$61$ \( 2401 + 343 T + 49 T^{2} + 7 T^{3} + T^{4} \)
$67$ \( ( 109 + 21 T + T^{2} )^{2} \)
$71$ \( 4096 - 512 T + 64 T^{2} - 8 T^{3} + T^{4} \)
$73$ \( 3721 - 1159 T + 141 T^{2} + T^{3} + T^{4} \)
$79$ \( 625 - 250 T + 100 T^{2} - 15 T^{3} + T^{4} \)
$83$ \( 1936 - 704 T + 136 T^{2} - 14 T^{3} + T^{4} \)
$89$ \( ( 45 - 15 T + T^{2} )^{2} \)
$97$ \( 841 - 87 T + 109 T^{2} + 17 T^{3} + T^{4} \)
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