Properties

Label 825.2.n.e
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{2} q^{3} + ( 1 - \zeta_{10} ) q^{4} + ( -1 - \zeta_{10}^{2} ) q^{6} + 3 \zeta_{10}^{3} q^{7} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{2} q^{3} + ( 1 - \zeta_{10} ) q^{4} + ( -1 - \zeta_{10}^{2} ) q^{6} + 3 \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} + ( -2 + 4 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{11} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{12} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{13} + ( 3 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{14} + ( 3 + 3 \zeta_{10}^{2} ) q^{16} + ( 4 + \zeta_{10} + 4 \zeta_{10}^{2} ) q^{17} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{18} + ( 2 \zeta_{10} - 6 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{19} + 3 q^{21} + ( 1 + 2 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{22} + ( -7 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{23} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{24} + 3 \zeta_{10}^{3} q^{26} + \zeta_{10} q^{27} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{28} + ( 1 - \zeta_{10} - 3 \zeta_{10}^{3} ) q^{29} + ( 7 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{31} + ( 4 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{32} + ( 1 + \zeta_{10} + \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{33} + ( 9 + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{34} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{36} + ( -3 + 3 \zeta_{10} + 7 \zeta_{10}^{3} ) q^{37} + ( -4 + 2 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{38} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{39} + ( -2 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{41} + ( 3 - 3 \zeta_{10}^{3} ) q^{42} + ( -3 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( 4 \zeta_{10} - 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{44} + ( -8 + \zeta_{10} - \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{46} + ( \zeta_{10} - 11 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{47} + ( 3 - 3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{48} -2 \zeta_{10} q^{49} + ( 4 - 4 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{51} + ( -3 \zeta_{10} + 6 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{52} + ( 2 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{53} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{54} + ( -3 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{56} + ( -4 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{57} + ( -3 \zeta_{10} - \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{58} + ( 3 - 3 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{59} + ( -2 + 9 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{61} + ( 7 - 7 \zeta_{10} - 10 \zeta_{10}^{3} ) q^{62} -3 \zeta_{10}^{2} q^{63} + ( -3 + \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{64} + ( 3 - 3 \zeta_{10} + 2 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{66} + ( 5 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{67} + ( 4 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{68} + ( \zeta_{10} + 6 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{69} + ( -4 + 5 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{71} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{72} + ( 3 - 3 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{73} + ( 7 \zeta_{10} + 3 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{74} + ( 2 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{76} + ( -9 + 6 \zeta_{10} - 12 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{77} + 3 q^{78} + ( 2 + 7 \zeta_{10} - 7 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{79} -\zeta_{10}^{3} q^{81} + ( 1 - 2 \zeta_{10} + \zeta_{10}^{2} ) q^{82} + ( -1 + 12 \zeta_{10} - \zeta_{10}^{2} ) q^{83} + ( 3 - 3 \zeta_{10} ) q^{84} + ( -1 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{86} + ( -3 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{87} + ( -9 + 5 \zeta_{10} - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{88} + ( -3 + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{89} + ( -9 \zeta_{10} + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{91} + ( -6 + 6 \zeta_{10} + \zeta_{10}^{3} ) q^{92} + ( -3 - 4 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{93} + ( -10 + \zeta_{10} - 10 \zeta_{10}^{2} ) q^{94} + ( \zeta_{10} - 5 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{96} + ( -3 - 9 \zeta_{10} + 9 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{97} + ( -2 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{98} + ( -2 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{7} + 5 q^{8} - q^{9} + O(q^{10}) \) \( 4 q + 3 q^{2} + q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{7} + 5 q^{8} - q^{9} - q^{11} + 2 q^{12} - 6 q^{13} + 6 q^{14} + 9 q^{16} + 13 q^{17} - 2 q^{18} + 10 q^{19} + 12 q^{21} + 3 q^{22} - 26 q^{23} + 5 q^{24} + 3 q^{26} + q^{27} + 6 q^{28} + 13 q^{31} + 18 q^{32} + q^{33} + 26 q^{34} + 3 q^{36} - 2 q^{37} - 10 q^{38} + 6 q^{39} - 7 q^{41} + 9 q^{42} - 16 q^{43} + 8 q^{44} - 22 q^{46} + 13 q^{47} + 6 q^{48} - 2 q^{49} + 7 q^{51} - 12 q^{52} - 6 q^{53} + 2 q^{54} - 5 q^{58} + 15 q^{59} + 3 q^{61} + 11 q^{62} + 3 q^{63} - 7 q^{64} + 2 q^{66} + 8 q^{67} + 6 q^{68} - 4 q^{69} - 7 q^{71} - 5 q^{72} + 4 q^{73} + 11 q^{74} + 20 q^{76} - 12 q^{77} + 12 q^{78} + 20 q^{79} - q^{81} + q^{82} + 9 q^{83} + 9 q^{84} - 7 q^{86} - 10 q^{87} - 25 q^{88} - 30 q^{89} - 27 q^{91} - 17 q^{92} - 13 q^{93} - 29 q^{94} + 7 q^{96} - 27 q^{97} - 4 q^{98} - 11 q^{99} + O(q^{100}) \)

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
1.30902 + 0.951057i −0.309017 + 0.951057i 0.190983 + 0.587785i 0 −1.30902 + 0.951057i −0.927051 2.85317i 0.690983 2.12663i −0.809017 0.587785i 0
526.1 0.190983 0.587785i 0.809017 0.587785i 1.30902 + 0.951057i 0 −0.190983 0.587785i 2.42705 + 1.76336i 1.80902 1.31433i 0.309017 0.951057i 0
676.1 0.190983 + 0.587785i 0.809017 + 0.587785i 1.30902 0.951057i 0 −0.190983 + 0.587785i 2.42705 1.76336i 1.80902 + 1.31433i 0.309017 + 0.951057i 0
751.1 1.30902 0.951057i −0.309017 0.951057i 0.190983 0.587785i 0 −1.30902 0.951057i −0.927051 + 2.85317i 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.e yes 4
5.b even 2 1 825.2.n.a 4
5.c odd 4 2 825.2.bx.a 8
11.c even 5 1 inner 825.2.n.e yes 4
11.c even 5 1 9075.2.a.y 2
11.d odd 10 1 9075.2.a.bu 2
55.h odd 10 1 9075.2.a.bb 2
55.j even 10 1 825.2.n.a 4
55.j even 10 1 9075.2.a.bz 2
55.k odd 20 2 825.2.bx.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.a 4 5.b even 2 1
825.2.n.a 4 55.j even 10 1
825.2.n.e yes 4 1.a even 1 1 trivial
825.2.n.e yes 4 11.c even 5 1 inner
825.2.bx.a 8 5.c odd 4 2
825.2.bx.a 8 55.k odd 20 2
9075.2.a.y 2 11.c even 5 1
9075.2.a.bb 2 55.h odd 10 1
9075.2.a.bu 2 11.d odd 10 1
9075.2.a.bz 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 \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{4} - 3 T_{2}^{3} + 4 T_{2}^{2} - 2 T_{2} + 1 \)
\( T_{13}^{4} + 6 T_{13}^{3} + 36 T_{13}^{2} + 81 T_{13} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} \)
$11$ \( 121 + 11 T + 21 T^{2} + T^{3} + T^{4} \)
$13$ \( 81 + 81 T + 36 T^{2} + 6 T^{3} + T^{4} \)
$17$ \( 121 - 77 T + 69 T^{2} - 13 T^{3} + T^{4} \)
$19$ \( 400 + 40 T^{2} - 10 T^{3} + T^{4} \)
$23$ \( ( 41 + 13 T + T^{2} )^{2} \)
$29$ \( 25 + 25 T + 10 T^{2} + T^{4} \)
$31$ \( 361 + 38 T + 64 T^{2} - 13 T^{3} + T^{4} \)
$37$ \( 361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( 1 + 3 T + 19 T^{2} + 7 T^{3} + T^{4} \)
$43$ \( ( 11 + 8 T + T^{2} )^{2} \)
$47$ \( 11881 - 872 T + 114 T^{2} - 13 T^{3} + T^{4} \)
$53$ \( 16 + 56 T + 76 T^{2} + 6 T^{3} + T^{4} \)
$59$ \( 2025 + 90 T^{2} - 15 T^{3} + T^{4} \)
$61$ \( 3481 - 767 T + 79 T^{2} - 3 T^{3} + T^{4} \)
$67$ \( ( -41 - 4 T + T^{2} )^{2} \)
$71$ \( 121 + 143 T + 69 T^{2} + 7 T^{3} + T^{4} \)
$73$ \( 1 + 11 T + 46 T^{2} - 4 T^{3} + T^{4} \)
$79$ \( 9025 - 2375 T + 310 T^{2} - 20 T^{3} + T^{4} \)
$83$ \( 17161 - 1834 T + 136 T^{2} - 9 T^{3} + T^{4} \)
$89$ \( ( -45 + 15 T + T^{2} )^{2} \)
$97$ \( 29241 + 5643 T + 549 T^{2} + 27 T^{3} + T^{4} \)
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