Properties

Label 825.4.c.j
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -3 \beta_{2} q^{3} + ( 3 + \beta_{3} ) q^{4} + ( -3 + 3 \beta_{3} ) q^{6} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{7} + ( 11 \beta_{1} + 4 \beta_{2} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -3 \beta_{2} q^{3} + ( 3 + \beta_{3} ) q^{4} + ( -3 + 3 \beta_{3} ) q^{6} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{7} + ( 11 \beta_{1} + 4 \beta_{2} ) q^{8} -9 q^{9} -11 q^{11} + ( -3 \beta_{1} - 12 \beta_{2} ) q^{12} + ( -2 \beta_{1} + 44 \beta_{2} ) q^{13} + 16 q^{14} + ( -27 + 15 \beta_{3} ) q^{16} + ( -44 \beta_{1} - 30 \beta_{2} ) q^{17} -9 \beta_{1} q^{18} + ( 96 - 22 \beta_{3} ) q^{19} -12 \beta_{3} q^{21} -11 \beta_{1} q^{22} + ( -60 \beta_{1} + 32 \beta_{2} ) q^{23} + ( -21 + 33 \beta_{3} ) q^{24} + ( 54 - 46 \beta_{3} ) q^{26} + 27 \beta_{2} q^{27} + ( -16 \beta_{1} - 32 \beta_{2} ) q^{28} + ( 62 + 34 \beta_{3} ) q^{29} + ( 24 + 12 \beta_{3} ) q^{31} + ( 61 \beta_{1} + 92 \beta_{2} ) q^{32} + 33 \beta_{2} q^{33} + ( 190 - 14 \beta_{3} ) q^{34} + ( -27 - 9 \beta_{3} ) q^{36} + ( 112 \beta_{1} - 130 \beta_{2} ) q^{37} + ( 96 \beta_{1} - 88 \beta_{2} ) q^{38} + ( 138 - 6 \beta_{3} ) q^{39} + ( -58 + 154 \beta_{3} ) q^{41} -48 \beta_{2} q^{42} + ( -124 \beta_{1} + 196 \beta_{2} ) q^{43} + ( -33 - 11 \beta_{3} ) q^{44} + ( 332 - 92 \beta_{3} ) q^{46} + ( -216 \beta_{1} + 4 \beta_{2} ) q^{47} + ( -45 \beta_{1} + 36 \beta_{2} ) q^{48} + ( 279 - 16 \beta_{3} ) q^{49} + ( 42 - 132 \beta_{3} ) q^{51} + ( 38 \beta_{1} + 168 \beta_{2} ) q^{52} + ( -196 \beta_{1} - 334 \beta_{2} ) q^{53} + ( 27 - 27 \beta_{3} ) q^{54} + ( 176 + 16 \beta_{3} ) q^{56} + ( 66 \beta_{1} - 222 \beta_{2} ) q^{57} + ( 62 \beta_{1} + 136 \beta_{2} ) q^{58} + ( -244 + 240 \beta_{3} ) q^{59} + ( 218 - 364 \beta_{3} ) q^{61} + ( 24 \beta_{1} + 48 \beta_{2} ) q^{62} + ( 36 \beta_{1} + 36 \beta_{2} ) q^{63} + ( -429 + 89 \beta_{3} ) q^{64} + ( 33 - 33 \beta_{3} ) q^{66} + ( -16 \beta_{1} - 380 \beta_{2} ) q^{67} + ( -162 \beta_{1} - 296 \beta_{2} ) q^{68} + ( 276 - 180 \beta_{3} ) q^{69} + ( 1052 - 44 \beta_{3} ) q^{71} + ( -99 \beta_{1} - 36 \beta_{2} ) q^{72} + ( 58 \beta_{1} + 272 \beta_{2} ) q^{73} + ( -690 + 242 \beta_{3} ) q^{74} + ( 200 + 8 \beta_{3} ) q^{76} + ( 44 \beta_{1} + 44 \beta_{2} ) q^{77} + ( 138 \beta_{1} - 24 \beta_{2} ) q^{78} + ( -168 - 306 \beta_{3} ) q^{79} + 81 q^{81} + ( -58 \beta_{1} + 616 \beta_{2} ) q^{82} + ( -426 \beta_{1} - 70 \beta_{2} ) q^{83} + ( -48 - 48 \beta_{3} ) q^{84} + ( 816 - 320 \beta_{3} ) q^{86} + ( -102 \beta_{1} - 288 \beta_{2} ) q^{87} + ( -121 \beta_{1} - 44 \beta_{2} ) q^{88} + ( -58 - 128 \beta_{3} ) q^{89} + ( -32 + 176 \beta_{3} ) q^{91} + ( -148 \beta_{1} - 112 \beta_{2} ) q^{92} + ( -36 \beta_{1} - 108 \beta_{2} ) q^{93} + ( 1084 - 220 \beta_{3} ) q^{94} + ( 93 + 183 \beta_{3} ) q^{96} + ( -428 \beta_{1} - 298 \beta_{2} ) q^{97} + ( 279 \beta_{1} - 64 \beta_{2} ) q^{98} + 99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 14q^{4} - 6q^{6} - 36q^{9} + O(q^{10}) \) \( 4q + 14q^{4} - 6q^{6} - 36q^{9} - 44q^{11} + 64q^{14} - 78q^{16} + 340q^{19} - 24q^{21} - 18q^{24} + 124q^{26} + 316q^{29} + 120q^{31} + 732q^{34} - 126q^{36} + 540q^{39} + 76q^{41} - 154q^{44} + 1144q^{46} + 1084q^{49} - 96q^{51} + 54q^{54} + 736q^{56} - 496q^{59} + 144q^{61} - 1538q^{64} + 66q^{66} + 744q^{69} + 4120q^{71} - 2276q^{74} + 816q^{76} - 1284q^{79} + 324q^{81} - 288q^{84} + 2624q^{86} - 488q^{89} + 224q^{91} + 3896q^{94} + 738q^{96} + 396q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 5\)
\(\nu^{3}\)\(=\)\(4 \beta_{2} - 5 \beta_{1}\)

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
2.56155i
1.56155i
1.56155i
2.56155i
2.56155i 3.00000i 1.43845 0 −7.68466 6.24621i 24.1771i −9.00000 0
199.2 1.56155i 3.00000i 5.56155 0 4.68466 10.2462i 21.1771i −9.00000 0
199.3 1.56155i 3.00000i 5.56155 0 4.68466 10.2462i 21.1771i −9.00000 0
199.4 2.56155i 3.00000i 1.43845 0 −7.68466 6.24621i 24.1771i −9.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.4.c.j 4
5.b even 2 1 inner 825.4.c.j 4
5.c odd 4 1 165.4.a.c 2
5.c odd 4 1 825.4.a.m 2
15.e even 4 1 495.4.a.d 2
15.e even 4 1 2475.4.a.n 2
55.e even 4 1 1815.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.c 2 5.c odd 4 1
495.4.a.d 2 15.e even 4 1
825.4.a.m 2 5.c odd 4 1
825.4.c.j 4 1.a even 1 1 trivial
825.4.c.j 4 5.b even 2 1 inner
1815.4.a.n 2 55.e even 4 1
2475.4.a.n 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{4} + 9 T_{2}^{2} + 16 \)
\( T_{7}^{4} + 144 T_{7}^{2} + 4096 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 9 T^{2} + T^{4} \)
$3$ \( ( 9 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 4096 + 144 T^{2} + T^{4} \)
$11$ \( ( 11 + T )^{4} \)
$13$ \( 4032064 + 4084 T^{2} + T^{4} \)
$17$ \( 66650896 + 16584 T^{2} + T^{4} \)
$19$ \( ( 5168 - 170 T + T^{2} )^{2} \)
$23$ \( 131239936 + 38288 T^{2} + T^{4} \)
$29$ \( ( 1328 - 158 T + T^{2} )^{2} \)
$31$ \( ( 288 - 60 T + T^{2} )^{2} \)
$37$ \( 350288656 + 175816 T^{2} + T^{4} \)
$41$ \( ( -100432 - 38 T + T^{2} )^{2} \)
$43$ \( 1478656 + 263824 T^{2} + T^{4} \)
$47$ \( 34500833536 + 421664 T^{2} + T^{4} \)
$53$ \( 11571735184 + 437928 T^{2} + T^{4} \)
$59$ \( ( -229424 + 248 T + T^{2} )^{2} \)
$61$ \( ( -561812 - 72 T + T^{2} )^{2} \)
$67$ \( 18850191616 + 278944 T^{2} + T^{4} \)
$71$ \( ( 1052672 - 2060 T + T^{2} )^{2} \)
$73$ \( 2002741504 + 146692 T^{2} + T^{4} \)
$79$ \( ( -294912 + 642 T + T^{2} )^{2} \)
$83$ \( 563736678976 + 1583444 T^{2} + T^{4} \)
$89$ \( ( -54748 + 244 T + T^{2} )^{2} \)
$97$ \( 595175218576 + 1571176 T^{2} + T^{4} \)
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