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: \( x^{4} + 9x^{2} + 16 \) 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: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 \beta_{2} q^{3} + (\beta_{3} + 3) q^{4} + (3 \beta_{3} - 3) q^{6} + ( - 4 \beta_{2} - 4 \beta_1) q^{7} + (4 \beta_{2} + 11 \beta_1) q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 \beta_{2} q^{3} + (\beta_{3} + 3) q^{4} + (3 \beta_{3} - 3) q^{6} + ( - 4 \beta_{2} - 4 \beta_1) q^{7} + (4 \beta_{2} + 11 \beta_1) q^{8} - 9 q^{9} - 11 q^{11} + ( - 12 \beta_{2} - 3 \beta_1) q^{12} + (44 \beta_{2} - 2 \beta_1) q^{13} + 16 q^{14} + (15 \beta_{3} - 27) q^{16} + ( - 30 \beta_{2} - 44 \beta_1) q^{17} - 9 \beta_1 q^{18} + ( - 22 \beta_{3} + 96) q^{19} - 12 \beta_{3} q^{21} - 11 \beta_1 q^{22} + (32 \beta_{2} - 60 \beta_1) q^{23} + (33 \beta_{3} - 21) q^{24} + ( - 46 \beta_{3} + 54) q^{26} + 27 \beta_{2} q^{27} + ( - 32 \beta_{2} - 16 \beta_1) q^{28} + (34 \beta_{3} + 62) q^{29} + (12 \beta_{3} + 24) q^{31} + (92 \beta_{2} + 61 \beta_1) q^{32} + 33 \beta_{2} q^{33} + ( - 14 \beta_{3} + 190) q^{34} + ( - 9 \beta_{3} - 27) q^{36} + ( - 130 \beta_{2} + 112 \beta_1) q^{37} + ( - 88 \beta_{2} + 96 \beta_1) q^{38} + ( - 6 \beta_{3} + 138) q^{39} + (154 \beta_{3} - 58) q^{41} - 48 \beta_{2} q^{42} + (196 \beta_{2} - 124 \beta_1) q^{43} + ( - 11 \beta_{3} - 33) q^{44} + ( - 92 \beta_{3} + 332) q^{46} + (4 \beta_{2} - 216 \beta_1) q^{47} + (36 \beta_{2} - 45 \beta_1) q^{48} + ( - 16 \beta_{3} + 279) q^{49} + ( - 132 \beta_{3} + 42) q^{51} + (168 \beta_{2} + 38 \beta_1) q^{52} + ( - 334 \beta_{2} - 196 \beta_1) q^{53} + ( - 27 \beta_{3} + 27) q^{54} + (16 \beta_{3} + 176) q^{56} + ( - 222 \beta_{2} + 66 \beta_1) q^{57} + (136 \beta_{2} + 62 \beta_1) q^{58} + (240 \beta_{3} - 244) q^{59} + ( - 364 \beta_{3} + 218) q^{61} + (48 \beta_{2} + 24 \beta_1) q^{62} + (36 \beta_{2} + 36 \beta_1) q^{63} + (89 \beta_{3} - 429) q^{64} + ( - 33 \beta_{3} + 33) q^{66} + ( - 380 \beta_{2} - 16 \beta_1) q^{67} + ( - 296 \beta_{2} - 162 \beta_1) q^{68} + ( - 180 \beta_{3} + 276) q^{69} + ( - 44 \beta_{3} + 1052) q^{71} + ( - 36 \beta_{2} - 99 \beta_1) q^{72} + (272 \beta_{2} + 58 \beta_1) q^{73} + (242 \beta_{3} - 690) q^{74} + (8 \beta_{3} + 200) q^{76} + (44 \beta_{2} + 44 \beta_1) q^{77} + ( - 24 \beta_{2} + 138 \beta_1) q^{78} + ( - 306 \beta_{3} - 168) q^{79} + 81 q^{81} + (616 \beta_{2} - 58 \beta_1) q^{82} + ( - 70 \beta_{2} - 426 \beta_1) q^{83} + ( - 48 \beta_{3} - 48) q^{84} + ( - 320 \beta_{3} + 816) q^{86} + ( - 288 \beta_{2} - 102 \beta_1) q^{87} + ( - 44 \beta_{2} - 121 \beta_1) q^{88} + ( - 128 \beta_{3} - 58) q^{89} + (176 \beta_{3} - 32) q^{91} + ( - 112 \beta_{2} - 148 \beta_1) q^{92} + ( - 108 \beta_{2} - 36 \beta_1) q^{93} + ( - 220 \beta_{3} + 1084) q^{94} + (183 \beta_{3} + 93) q^{96} + ( - 298 \beta_{2} - 428 \beta_1) q^{97} + ( - 64 \beta_{2} + 279 \beta_1) q^{98} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{4} - 6 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 14 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 64 q^{14} - 78 q^{16} + 340 q^{19} - 24 q^{21} - 18 q^{24} + 124 q^{26} + 316 q^{29} + 120 q^{31} + 732 q^{34} - 126 q^{36} + 540 q^{39} + 76 q^{41} - 154 q^{44} + 1144 q^{46} + 1084 q^{49} - 96 q^{51} + 54 q^{54} + 736 q^{56} - 496 q^{59} + 144 q^{61} - 1538 q^{64} + 66 q^{66} + 744 q^{69} + 4120 q^{71} - 2276 q^{74} + 816 q^{76} - 1284 q^{79} + 324 q^{81} - 288 q^{84} + 2624 q^{86} - 488 q^{89} + 224 q^{91} + 3896 q^{94} + 738 q^{96} + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

Character values

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

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

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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{4} + 9T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{4} + 144T_{7}^{2} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

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