Properties

Label 825.2.n.k
Level $825$
Weight $2$
Character orbit 825.n
Analytic conductor $6.588$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) 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 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{2} + (\beta_{7} + \beta_{4} - \beta_{3} + 1) q^{3} + (\beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{4} + (\beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{6} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{2} - 1) q^{7} + (3 \beta_{7} + \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{8} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{2} + (\beta_{7} + \beta_{4} - \beta_{3} + 1) q^{3} + (\beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{4} + (\beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{6} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{2} - 1) q^{7} + (3 \beta_{7} + \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{8} - \beta_{3} q^{9} + (\beta_{7} - 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{11} + (\beta_{5} - 2 \beta_{2} - 2 \beta_1 - 1) q^{12} + ( - \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 2) q^{13} + (2 \beta_{7} + 3 \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 1) q^{14} + (5 \beta_{6} - 5 \beta_{2} - 5 \beta_1) q^{16} - 5 \beta_{7} q^{17} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + 1) q^{18} + ( - \beta_{7} - 3 \beta_{6} - 5 \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 1) q^{19} + ( - \beta_{7} - 3 \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{21} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 1) q^{22} + ( - 2 \beta_{7} + 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{23} + ( - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_1 - 2) q^{24} + ( - \beta_{7} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 1) q^{26} - \beta_{7} q^{27} + (4 \beta_{4} - 4 \beta_{3} - 5 \beta_{2}) q^{28} + (\beta_{7} - \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 4 \beta_{2} + 1) q^{29} + ( - \beta_{4} + 3 \beta_1 - 1) q^{31} + (\beta_{7} - 4 \beta_{5} - \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 3) q^{32} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + 3 \beta_1 - 2) q^{33} + (5 \beta_{7} - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 + 5) q^{34} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{36} + (4 \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{2} + 4) q^{37} + ( - \beta_{7} + 3 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 7 \beta_{2} - 3 \beta_1) q^{38} + ( - \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{39} + ( - 6 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} - 6 \beta_{4} + 7 \beta_{3} - 4 \beta_{2} + \cdots - 7) q^{41}+ \cdots + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 2 q^{4} - q^{7} - 5 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 2 q^{4} - q^{7} - 5 q^{8} - 2 q^{9} - 3 q^{11} - 18 q^{12} - 6 q^{13} - 10 q^{14} - 20 q^{16} + 10 q^{17} + 5 q^{18} + 6 q^{19} - 4 q^{21} + 25 q^{22} + 10 q^{23} - 20 q^{24} - 8 q^{26} + 2 q^{27} - 31 q^{28} + 3 q^{31} - 60 q^{32} - 2 q^{33} + 50 q^{34} - 2 q^{36} + 19 q^{37} + 28 q^{38} + 6 q^{39} - 25 q^{41} - 15 q^{42} + 4 q^{43} + 7 q^{44} - 6 q^{46} - 15 q^{47} - 5 q^{48} + 21 q^{49} - 10 q^{51} - 6 q^{52} - 7 q^{53} + 10 q^{54} + 20 q^{56} + 9 q^{57} + 2 q^{58} + 35 q^{59} + 21 q^{61} + 19 q^{62} - q^{63} - 77 q^{64} + 25 q^{66} + 26 q^{67} + 35 q^{68} - 5 q^{69} + 25 q^{71} + 20 q^{72} - q^{73} - 29 q^{74} - 14 q^{76} + 61 q^{77} - 12 q^{78} + 30 q^{79} - 2 q^{81} - 57 q^{82} - 11 q^{83} - 34 q^{84} - 34 q^{86} - 10 q^{87} + 85 q^{88} + 32 q^{89} + 37 q^{91} + 10 q^{92} - 3 q^{93} - 39 q^{94} + 10 q^{96} - 5 q^{97} - 50 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 \) Copy content Toggle raw display

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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
−0.386111 0.280526i
1.69513 + 1.23158i
0.418926 1.28932i
−0.227943 + 0.701538i
0.418926 + 1.28932i
−0.227943 0.701538i
−0.386111 + 0.280526i
1.69513 1.23158i
−1.12474 0.817172i −0.309017 + 0.951057i −0.0207616 0.0638975i 0 1.12474 0.817172i −0.394797 1.21506i −0.888090 + 2.73326i −0.809017 0.587785i 0
301.2 2.24278 + 1.62947i −0.309017 + 0.951057i 1.75683 + 5.40697i 0 −2.24278 + 1.62947i 0.703814 + 2.16612i −3.15700 + 9.71623i −0.809017 0.587785i 0
526.1 −0.758911 + 2.33569i 0.809017 0.587785i −3.26145 2.36959i 0 0.758911 + 2.33569i 2.65911 + 1.93196i 4.03606 2.93237i 0.309017 0.951057i 0
526.2 −0.359123 + 1.10527i 0.809017 0.587785i 0.525387 + 0.381716i 0 0.359123 + 1.10527i −3.46813 2.51974i −2.49097 + 1.80980i 0.309017 0.951057i 0
676.1 −0.758911 2.33569i 0.809017 + 0.587785i −3.26145 + 2.36959i 0 0.758911 2.33569i 2.65911 1.93196i 4.03606 + 2.93237i 0.309017 + 0.951057i 0
676.2 −0.359123 1.10527i 0.809017 + 0.587785i 0.525387 0.381716i 0 0.359123 1.10527i −3.46813 + 2.51974i −2.49097 1.80980i 0.309017 + 0.951057i 0
751.1 −1.12474 + 0.817172i −0.309017 0.951057i −0.0207616 + 0.0638975i 0 1.12474 + 0.817172i −0.394797 + 1.21506i −0.888090 2.73326i −0.809017 + 0.587785i 0
751.2 2.24278 1.62947i −0.309017 0.951057i 1.75683 5.40697i 0 −2.24278 1.62947i 0.703814 2.16612i −3.15700 9.71623i −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.2
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.k 8
5.b even 2 1 165.2.m.a 8
5.c odd 4 2 825.2.bx.h 16
11.c even 5 1 inner 825.2.n.k 8
11.c even 5 1 9075.2.a.cl 4
11.d odd 10 1 9075.2.a.dj 4
15.d odd 2 1 495.2.n.d 8
55.h odd 10 1 1815.2.a.o 4
55.j even 10 1 165.2.m.a 8
55.j even 10 1 1815.2.a.x 4
55.k odd 20 2 825.2.bx.h 16
165.o odd 10 1 495.2.n.d 8
165.o odd 10 1 5445.2.a.be 4
165.r even 10 1 5445.2.a.bv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.a 8 5.b even 2 1
165.2.m.a 8 55.j even 10 1
495.2.n.d 8 15.d odd 2 1
495.2.n.d 8 165.o odd 10 1
825.2.n.k 8 1.a even 1 1 trivial
825.2.n.k 8 11.c even 5 1 inner
825.2.bx.h 16 5.c odd 4 2
825.2.bx.h 16 55.k odd 20 2
1815.2.a.o 4 55.h odd 10 1
1815.2.a.x 4 55.j even 10 1
5445.2.a.be 4 165.o odd 10 1
5445.2.a.bv 4 165.r even 10 1
9075.2.a.cl 4 11.c even 5 1
9075.2.a.dj 4 11.d odd 10 1

Hecke kernels

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

\( T_{2}^{8} + 3T_{2}^{6} - 5T_{2}^{5} + 24T_{2}^{4} + 85T_{2}^{3} + 177T_{2}^{2} + 165T_{2} + 121 \) Copy content Toggle raw display
\( T_{13}^{8} + 6T_{13}^{7} + 39T_{13}^{6} + 87T_{13}^{5} + 94T_{13}^{4} + 39T_{13}^{3} + 11T_{13}^{2} + 3T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{6} - 5 T^{5} + 24 T^{4} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} - 3 T^{6} - 7 T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$11$ \( T^{8} + 3 T^{7} + 8 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + 39 T^{6} + 87 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{4} - 5 T^{3} + 25 T^{2} - 125 T + 625)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + 9 T^{6} + 123 T^{5} + \cdots + 961 \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{3} - T^{2} + 5 T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 17 T^{6} + 95 T^{5} + \cdots + 290521 \) Copy content Toggle raw display
$31$ \( T^{8} - 3 T^{7} + 11 T^{6} + \cdots + 19321 \) Copy content Toggle raw display
$37$ \( T^{8} - 19 T^{7} + 255 T^{6} + \cdots + 185761 \) Copy content Toggle raw display
$41$ \( T^{8} + 25 T^{7} + 327 T^{6} + \cdots + 4289041 \) Copy content Toggle raw display
$43$ \( (T^{4} - 2 T^{3} - 92 T^{2} + 63 T + 1861)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 15 T^{7} + 123 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{8} + 7 T^{7} - 14 T^{6} + \cdots + 1615441 \) Copy content Toggle raw display
$59$ \( T^{8} - 35 T^{7} + 633 T^{6} + \cdots + 5285401 \) Copy content Toggle raw display
$61$ \( T^{8} - 21 T^{7} + 242 T^{6} + \cdots + 3575881 \) Copy content Toggle raw display
$67$ \( (T^{4} - 13 T^{3} - 136 T^{2} + 1768 T - 3379)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 25 T^{7} + 347 T^{6} + \cdots + 5527201 \) Copy content Toggle raw display
$73$ \( T^{8} + T^{7} + 30 T^{6} + \cdots + 84621601 \) Copy content Toggle raw display
$79$ \( T^{8} - 30 T^{7} + 487 T^{6} + \cdots + 1437601 \) Copy content Toggle raw display
$83$ \( T^{8} + 11 T^{7} + \cdots + 149352841 \) Copy content Toggle raw display
$89$ \( (T^{4} - 16 T^{3} + 45 T^{2} + 132 T - 271)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 5 T^{7} - 20 T^{6} - 225 T^{5} + \cdots + 625 \) Copy content Toggle raw display
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