Properties

Label 825.2.bx.h
Level $825$
Weight $2$
Character orbit 825.bx
Analytic conductor $6.588$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.bx (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - x^{14} + 15x^{12} - 59x^{10} + 104x^{8} - 59x^{6} + 15x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{15} + \beta_{13} - \beta_{8}) q^{2} - \beta_{14} q^{3} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + 4 \beta_{3} - 2 \beta_{2} - 1) q^{4} + ( - \beta_{10} - 2 \beta_{7} - \beta_{6} - \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{14} + 3 \beta_{13} + \beta_{4} + \beta_1) q^{7} + ( - 4 \beta_{14} + 3 \beta_{11} + 3 \beta_{9} - 3 \beta_{8} + \beta_{4} + 3 \beta_1) q^{8} - \beta_{12} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{15} + \beta_{13} - \beta_{8}) q^{2} - \beta_{14} q^{3} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + 4 \beta_{3} - 2 \beta_{2} - 1) q^{4} + ( - \beta_{10} - 2 \beta_{7} - \beta_{6} - \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{14} + 3 \beta_{13} + \beta_{4} + \beta_1) q^{7} + ( - 4 \beta_{14} + 3 \beta_{11} + 3 \beta_{9} - 3 \beta_{8} + \beta_{4} + 3 \beta_1) q^{8} - \beta_{12} q^{9} + (\beta_{12} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - 3 \beta_{2}) q^{11} + (4 \beta_{15} - 3 \beta_{14} - 2 \beta_{13} - 3 \beta_{11} - 4 \beta_{9} + \beta_{8} - 2 \beta_{4}) q^{12} + (\beta_{14} + \beta_{13} + \beta_{9} - \beta_{8} + 2 \beta_1) q^{13} + (\beta_{12} - 2 \beta_{10} + 3 \beta_{7} - 2 \beta_{5} + 3 \beta_{3} - 5 \beta_{2} + 1) q^{14} + (5 \beta_{12} - 5 \beta_{10} - 10 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} - 5 \beta_{3}) q^{16} + 5 \beta_{9} q^{17} + ( - 2 \beta_{11} - \beta_{4} - \beta_1) q^{18} + ( - \beta_{12} + 2 \beta_{10} - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} + 5 \beta_{2} + \cdots - 1) q^{19}+ \cdots + ( - \beta_{12} + 3 \beta_{10} + \beta_{7} + \beta_{5} + 2 \beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} + 4 q^{9} - 6 q^{11} + 20 q^{14} - 40 q^{16} - 12 q^{19} - 8 q^{21} + 40 q^{24} - 16 q^{26} + 6 q^{31} - 100 q^{34} - 4 q^{36} - 12 q^{39} - 50 q^{41} - 14 q^{44} - 12 q^{46} - 42 q^{49} - 20 q^{51} - 20 q^{54} + 40 q^{56} - 70 q^{59} + 42 q^{61} + 154 q^{64} + 50 q^{66} + 10 q^{69} + 50 q^{71} + 58 q^{74} - 28 q^{76} - 60 q^{79} - 4 q^{81} + 68 q^{84} - 68 q^{86} - 64 q^{89} + 74 q^{91} + 78 q^{94} + 20 q^{96} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{14} + 15x^{12} - 59x^{10} + 104x^{8} - 59x^{6} + 15x^{4} - x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 21\nu^{14} + 2\nu^{12} + 289\nu^{10} - 908\nu^{8} + 772\nu^{6} + 1045\nu^{4} - 622\nu^{2} - 63 ) / 384 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\nu^{14} - 22\nu^{12} + 329\nu^{10} - 1260\nu^{8} + 2436\nu^{6} - 1995\nu^{4} + 1498\nu^{2} - 119 ) / 384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -21\nu^{15} + 22\nu^{13} - 329\nu^{11} + 1260\nu^{9} - 2436\nu^{7} + 1995\nu^{5} - 1498\nu^{3} + 119\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{14} - 11\nu^{12} + 76\nu^{10} - 384\nu^{8} + 804\nu^{6} - 671\nu^{4} + 137\nu^{2} - 40 ) / 96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -21\nu^{14} + 10\nu^{12} - 309\nu^{10} + 1084\nu^{8} - 1604\nu^{6} + 475\nu^{4} - 246\nu^{2} + 91 ) / 192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{14} - 17\nu^{12} + 138\nu^{10} - 648\nu^{8} + 1332\nu^{6} - 1107\nu^{4} + 155\nu^{2} + 30 ) / 96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9\nu^{15} - 17\nu^{13} + 138\nu^{11} - 648\nu^{9} + 1332\nu^{7} - 1107\nu^{5} + 155\nu^{3} + 30\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -9\nu^{15} + 23\nu^{13} - 148\nu^{11} + 736\nu^{9} - 1748\nu^{7} + 1867\nu^{5} - 685\nu^{3} - 16\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 57\nu^{14} - 30\nu^{12} + 845\nu^{10} - 2972\nu^{8} + 4564\nu^{6} - 1511\nu^{4} + 466\nu^{2} + 77 ) / 384 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -63\nu^{15} + 42\nu^{13} - 947\nu^{11} + 3428\nu^{9} - 5644\nu^{7} + 2945\nu^{5} - 1990\nu^{3} + 685\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -17\nu^{14} + 5\nu^{12} - 246\nu^{10} + 824\nu^{8} - 1108\nu^{6} - 101\nu^{4} + 201\nu^{2} - 58 ) / 96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -17\nu^{15} + 5\nu^{13} - 246\nu^{11} + 824\nu^{9} - 1108\nu^{7} - 101\nu^{5} + 201\nu^{3} - 58\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 77\nu^{15} - 134\nu^{13} + 1185\nu^{11} - 5388\nu^{9} + 10980\nu^{7} - 9107\nu^{5} + 2666\nu^{3} - 543\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 40\nu^{15} - 35\nu^{13} + 589\nu^{11} - 2284\nu^{9} + 3776\nu^{7} - 1556\nu^{5} - 71\nu^{3} + 97\nu ) / 96 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{8} - 3\beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + 5\beta_{10} - 3\beta_{7} + 4\beta_{6} + 4\beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{15} - \beta_{14} + 6\beta_{13} - \beta_{11} - 5\beta_{9} - 12\beta_{8} + 6\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6\beta_{12} - 16\beta_{10} - 23\beta_{6} - 6\beta_{3} - 16\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -16\beta_{15} + 16\beta_{14} - 22\beta_{13} - 7\beta_{11} + 29\beta_{4} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -22\beta_{12} - 67\beta_{10} + 51\beta_{7} - 67\beta_{5} + 51\beta_{3} + 36\beta_{2} - 22 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -67\beta_{15} - 89\beta_{13} + 67\beta_{11} + 103\beta_{9} + 221\beta_{8} - 221\beta_{4} - 89\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 132\beta_{12} + 456\beta_{10} - 132\beta_{7} + 456\beta_{6} + 168\beta_{5} + 168\beta_{2} - 89 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 456\beta_{15} - 288\beta_{14} + 588\beta_{13} - 288\beta_{9} - 588\beta_{8} - 377\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -588\beta_{7} - 1253\beta_{6} + 756\beta_{5} - 965\beta_{3} - 1253\beta_{2} + 588 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 756\beta_{14} - 1253\beta_{11} - 1253\beta_{9} - 2597\beta_{8} + 4227\beta_{4} + 2597\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -2597\beta_{12} - 8833\beta_{10} + 4227\beta_{7} - 5480\beta_{6} - 5480\beta_{5} + 2597\beta_{3} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 8833 \beta_{15} + 3353 \beta_{14} - 11430 \beta_{13} + 3353 \beta_{11} + 8833 \beta_{9} + 18540 \beta_{8} - 11430 \beta_{4} \) Copy content Toggle raw display

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.701538 + 0.227943i
1.28932 0.418926i
−1.28932 + 0.418926i
0.701538 0.227943i
0.280526 + 0.386111i
1.23158 + 1.69513i
−1.23158 1.69513i
−0.280526 0.386111i
0.280526 0.386111i
1.23158 1.69513i
−1.23158 + 1.69513i
−0.280526 + 0.386111i
−0.701538 0.227943i
1.28932 + 0.418926i
−1.28932 0.418926i
0.701538 + 0.227943i
−2.33569 + 0.758911i −0.587785 + 0.809017i 3.26145 2.36959i 0 0.758911 2.33569i −1.93196 2.65911i −2.93237 + 4.03606i −0.309017 0.951057i 0
49.2 −1.10527 + 0.359123i −0.587785 + 0.809017i −0.525387 + 0.381716i 0 0.359123 1.10527i 2.51974 + 3.46813i 1.80980 2.49097i −0.309017 0.951057i 0
49.3 1.10527 0.359123i 0.587785 0.809017i −0.525387 + 0.381716i 0 0.359123 1.10527i −2.51974 3.46813i −1.80980 + 2.49097i −0.309017 0.951057i 0
49.4 2.33569 0.758911i 0.587785 0.809017i 3.26145 2.36959i 0 0.758911 2.33569i 1.93196 + 2.65911i 2.93237 4.03606i −0.309017 0.951057i 0
124.1 −1.62947 2.24278i 0.951057 0.309017i −1.75683 + 5.40697i 0 −2.24278 1.62947i −2.16612 0.703814i 9.71623 3.15700i 0.809017 0.587785i 0
124.2 −0.817172 1.12474i −0.951057 + 0.309017i 0.0207616 0.0638975i 0 1.12474 + 0.817172i −1.21506 0.394797i −2.73326 + 0.888090i 0.809017 0.587785i 0
124.3 0.817172 + 1.12474i 0.951057 0.309017i 0.0207616 0.0638975i 0 1.12474 + 0.817172i 1.21506 + 0.394797i 2.73326 0.888090i 0.809017 0.587785i 0
124.4 1.62947 + 2.24278i −0.951057 + 0.309017i −1.75683 + 5.40697i 0 −2.24278 1.62947i 2.16612 + 0.703814i −9.71623 + 3.15700i 0.809017 0.587785i 0
499.1 −1.62947 + 2.24278i 0.951057 + 0.309017i −1.75683 5.40697i 0 −2.24278 + 1.62947i −2.16612 + 0.703814i 9.71623 + 3.15700i 0.809017 + 0.587785i 0
499.2 −0.817172 + 1.12474i −0.951057 0.309017i 0.0207616 + 0.0638975i 0 1.12474 0.817172i −1.21506 + 0.394797i −2.73326 0.888090i 0.809017 + 0.587785i 0
499.3 0.817172 1.12474i 0.951057 + 0.309017i 0.0207616 + 0.0638975i 0 1.12474 0.817172i 1.21506 0.394797i 2.73326 + 0.888090i 0.809017 + 0.587785i 0
499.4 1.62947 2.24278i −0.951057 0.309017i −1.75683 5.40697i 0 −2.24278 + 1.62947i 2.16612 0.703814i −9.71623 3.15700i 0.809017 + 0.587785i 0
724.1 −2.33569 0.758911i −0.587785 0.809017i 3.26145 + 2.36959i 0 0.758911 + 2.33569i −1.93196 + 2.65911i −2.93237 4.03606i −0.309017 + 0.951057i 0
724.2 −1.10527 0.359123i −0.587785 0.809017i −0.525387 0.381716i 0 0.359123 + 1.10527i 2.51974 3.46813i 1.80980 + 2.49097i −0.309017 + 0.951057i 0
724.3 1.10527 + 0.359123i 0.587785 + 0.809017i −0.525387 0.381716i 0 0.359123 + 1.10527i −2.51974 + 3.46813i −1.80980 2.49097i −0.309017 + 0.951057i 0
724.4 2.33569 + 0.758911i 0.587785 + 0.809017i 3.26145 + 2.36959i 0 0.758911 + 2.33569i 1.93196 2.65911i 2.93237 + 4.03606i −0.309017 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 724.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.bx.h 16
5.b even 2 1 inner 825.2.bx.h 16
5.c odd 4 1 165.2.m.a 8
5.c odd 4 1 825.2.n.k 8
11.c even 5 1 inner 825.2.bx.h 16
15.e even 4 1 495.2.n.d 8
55.j even 10 1 inner 825.2.bx.h 16
55.k odd 20 1 165.2.m.a 8
55.k odd 20 1 825.2.n.k 8
55.k odd 20 1 1815.2.a.x 4
55.k odd 20 1 9075.2.a.cl 4
55.l even 20 1 1815.2.a.o 4
55.l even 20 1 9075.2.a.dj 4
165.u odd 20 1 5445.2.a.bv 4
165.v even 20 1 495.2.n.d 8
165.v even 20 1 5445.2.a.be 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.a 8 5.c odd 4 1
165.2.m.a 8 55.k odd 20 1
495.2.n.d 8 15.e even 4 1
495.2.n.d 8 165.v even 20 1
825.2.n.k 8 5.c odd 4 1
825.2.n.k 8 55.k odd 20 1
825.2.bx.h 16 1.a even 1 1 trivial
825.2.bx.h 16 5.b even 2 1 inner
825.2.bx.h 16 11.c even 5 1 inner
825.2.bx.h 16 55.j even 10 1 inner
1815.2.a.o 4 55.l even 20 1
1815.2.a.x 4 55.k odd 20 1
5445.2.a.be 4 165.v even 20 1
5445.2.a.bv 4 165.u odd 20 1
9075.2.a.cl 4 55.k odd 20 1
9075.2.a.dj 4 55.l even 20 1

Hecke kernels

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

\( T_{2}^{16} - 6T_{2}^{14} + 57T_{2}^{12} - 473T_{2}^{10} + 2730T_{2}^{8} - 3647T_{2}^{6} + 9087T_{2}^{4} - 15609T_{2}^{2} + 14641 \) Copy content Toggle raw display
\( T_{13}^{16} - 42T_{13}^{14} + 665T_{13}^{12} + 683T_{13}^{10} + 2874T_{13}^{8} - 103T_{13}^{6} + 75T_{13}^{4} - 13T_{13}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 6 T^{14} + 57 T^{12} + \cdots + 14641 \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 7 T^{14} + 383 T^{12} + \cdots + 2825761 \) Copy content Toggle raw display
$11$ \( (T^{8} + 3 T^{7} + 8 T^{6} + T^{5} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 42 T^{14} + 665 T^{12} + 683 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{8} - 25 T^{6} + 625 T^{4} + \cdots + 390625)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 6 T^{7} + 9 T^{6} - 123 T^{5} + \cdots + 961)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 27 T^{6} + 49 T^{4} + 23 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 17 T^{6} - 95 T^{5} + \cdots + 290521)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 3 T^{7} + 11 T^{6} + T^{5} + \cdots + 19321)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 149 T^{14} + \cdots + 34507149121 \) Copy content Toggle raw display
$41$ \( (T^{8} + 25 T^{7} + 327 T^{6} + \cdots + 4289041)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 188 T^{6} + 12438 T^{4} + \cdots + 3463321)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 21 T^{14} + 15567 T^{12} + \cdots + 14641 \) Copy content Toggle raw display
$53$ \( T^{16} + 77 T^{14} + \cdots + 2609649624481 \) Copy content Toggle raw display
$59$ \( (T^{8} + 35 T^{7} + 633 T^{6} + \cdots + 5285401)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 21 T^{7} + 242 T^{6} + \cdots + 3575881)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 441 T^{6} + 57706 T^{4} + \cdots + 11417641)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 25 T^{7} + 347 T^{6} + \cdots + 5527201)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} - 59 T^{14} + \cdots + 71\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( (T^{8} + 30 T^{7} + 487 T^{6} + \cdots + 1437601)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 597 T^{14} + \cdots + 22\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{4} + 16 T^{3} + 45 T^{2} - 132 T - 271)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + 65 T^{14} + 8000 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
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