Properties

Label 825.2.n.d
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: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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.d yes 4
5.b even 2 1 825.2.n.b 4
5.c odd 4 2 825.2.bx.c 8
11.c even 5 1 inner 825.2.n.d yes 4
11.c even 5 1 9075.2.a.by 2
11.d odd 10 1 9075.2.a.bc 2
55.h odd 10 1 9075.2.a.bt 2
55.j even 10 1 825.2.n.b 4
55.j even 10 1 9075.2.a.z 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 5.b even 2 1
825.2.n.b 4 55.j even 10 1
825.2.n.d yes 4 1.a even 1 1 trivial
825.2.n.d yes 4 11.c even 5 1 inner
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 55.j even 10 1
9075.2.a.bc 2 11.d odd 10 1
9075.2.a.bt 2 55.h odd 10 1
9075.2.a.by 2 11.c even 5 1

Hecke kernels

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

\( T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} + 4T_{13}^{3} + 16T_{13}^{2} + 24T_{13} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

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