Properties

Label 825.4.c.i
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: \(\Q(i, \sqrt{33})\)
Defining polynomial: \(x^{4} + 17 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{3} + ( -1 + \beta_{3} ) q^{4} + ( 3 - 3 \beta_{3} ) q^{6} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 7 \beta_{1} + 8 \beta_{2} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{3} + ( -1 + \beta_{3} ) q^{4} + ( 3 - 3 \beta_{3} ) q^{6} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 7 \beta_{1} + 8 \beta_{2} ) q^{8} -9 q^{9} + 11 q^{11} + 3 \beta_{1} q^{12} + ( -8 \beta_{1} - 42 \beta_{2} ) q^{13} + 16 q^{14} + ( -63 + 7 \beta_{3} ) q^{16} + ( 30 \beta_{1} + 28 \beta_{2} ) q^{17} -9 \beta_{1} q^{18} + ( 36 - 18 \beta_{3} ) q^{19} + 6 \beta_{3} q^{21} + 11 \beta_{1} q^{22} + 112 \beta_{2} q^{23} + ( -3 - 21 \beta_{3} ) q^{24} + ( 30 + 34 \beta_{3} ) q^{26} -27 \beta_{2} q^{27} -16 \beta_{2} q^{28} + ( -134 + 46 \beta_{3} ) q^{29} + ( 32 - 104 \beta_{3} ) q^{31} + ( -7 \beta_{1} + 120 \beta_{2} ) q^{32} + 33 \beta_{2} q^{33} + ( -242 + 2 \beta_{3} ) q^{34} + ( 9 - 9 \beta_{3} ) q^{36} + ( 44 \beta_{1} + 46 \beta_{2} ) q^{37} + ( 36 \beta_{1} - 144 \beta_{2} ) q^{38} + ( 102 + 24 \beta_{3} ) q^{39} + ( -246 - 2 \beta_{3} ) q^{41} + 48 \beta_{2} q^{42} + ( 86 \beta_{1} + 10 \beta_{2} ) q^{43} + ( -11 + 11 \beta_{3} ) q^{44} + ( 112 - 112 \beta_{3} ) q^{46} + ( -48 \beta_{1} + 8 \beta_{2} ) q^{47} + ( 21 \beta_{1} - 168 \beta_{2} ) q^{48} + ( 311 - 4 \beta_{3} ) q^{49} + ( 6 - 90 \beta_{3} ) q^{51} + ( -34 \beta_{1} - 64 \beta_{2} ) q^{52} + ( 128 \beta_{1} + 22 \beta_{2} ) q^{53} + ( -27 + 27 \beta_{3} ) q^{54} + ( 112 + 16 \beta_{3} ) q^{56} + ( -54 \beta_{1} + 54 \beta_{2} ) q^{57} + ( -134 \beta_{1} + 368 \beta_{2} ) q^{58} -196 q^{59} + ( -578 + 52 \beta_{3} ) q^{61} + ( 32 \beta_{1} - 832 \beta_{2} ) q^{62} + ( 18 \beta_{1} + 18 \beta_{2} ) q^{63} + ( -321 - 71 \beta_{3} ) q^{64} + ( 33 - 33 \beta_{3} ) q^{66} + ( 152 \beta_{1} - 388 \beta_{2} ) q^{67} + ( -2 \beta_{1} + 240 \beta_{2} ) q^{68} -336 q^{69} + ( 320 - 184 \beta_{3} ) q^{71} + ( -63 \beta_{1} - 72 \beta_{2} ) q^{72} + ( 252 \beta_{1} - 170 \beta_{2} ) q^{73} + ( -350 - 2 \beta_{3} ) q^{74} + ( -180 + 36 \beta_{3} ) q^{76} + ( -22 \beta_{1} - 22 \beta_{2} ) q^{77} + ( 102 \beta_{1} + 192 \beta_{2} ) q^{78} + ( 152 - 74 \beta_{3} ) q^{79} + 81 q^{81} + ( -246 \beta_{1} - 16 \beta_{2} ) q^{82} + ( 324 \beta_{1} + 336 \beta_{2} ) q^{83} + 48 q^{84} + ( -764 + 76 \beta_{3} ) q^{86} + ( 138 \beta_{1} - 264 \beta_{2} ) q^{87} + ( 77 \beta_{1} + 88 \beta_{2} ) q^{88} + ( -490 + 8 \beta_{3} ) q^{89} + ( -128 - 84 \beta_{3} ) q^{91} + 112 \beta_{1} q^{92} + ( -312 \beta_{1} - 216 \beta_{2} ) q^{93} + ( 440 - 56 \beta_{3} ) q^{94} + ( -381 + 21 \beta_{3} ) q^{96} + ( 420 \beta_{1} + 802 \beta_{2} ) q^{97} + ( 311 \beta_{1} - 32 \beta_{2} ) q^{98} -99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{4} + 6q^{6} - 36q^{9} + O(q^{10}) \) \( 4q - 2q^{4} + 6q^{6} - 36q^{9} + 44q^{11} + 64q^{14} - 238q^{16} + 108q^{19} + 12q^{21} - 54q^{24} + 188q^{26} - 444q^{29} - 80q^{31} - 964q^{34} + 18q^{36} + 456q^{39} - 988q^{41} - 22q^{44} + 224q^{46} + 1236q^{49} - 156q^{51} - 54q^{54} + 480q^{56} - 784q^{59} - 2208q^{61} - 1426q^{64} + 66q^{66} - 1344q^{69} + 912q^{71} - 1404q^{74} - 648q^{76} + 460q^{79} + 324q^{81} + 192q^{84} - 2904q^{86} - 1944q^{89} - 680q^{91} + 1648q^{94} - 1482q^{96} - 396q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 17 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 9 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 9\)
\(\nu^{3}\)\(=\)\(8 \beta_{2} - 9 \beta_{1}\)

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
3.37228i
2.37228i
2.37228i
3.37228i
3.37228i 3.00000i −3.37228 0 10.1168 4.74456i 15.6060i −9.00000 0
199.2 2.37228i 3.00000i 2.37228 0 −7.11684 6.74456i 24.6060i −9.00000 0
199.3 2.37228i 3.00000i 2.37228 0 −7.11684 6.74456i 24.6060i −9.00000 0
199.4 3.37228i 3.00000i −3.37228 0 10.1168 4.74456i 15.6060i −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.i 4
5.b even 2 1 inner 825.4.c.i 4
5.c odd 4 1 33.4.a.d 2
5.c odd 4 1 825.4.a.k 2
15.e even 4 1 99.4.a.e 2
15.e even 4 1 2475.4.a.o 2
20.e even 4 1 528.4.a.o 2
35.f even 4 1 1617.4.a.j 2
40.i odd 4 1 2112.4.a.ba 2
40.k even 4 1 2112.4.a.bh 2
55.e even 4 1 363.4.a.j 2
60.l odd 4 1 1584.4.a.x 2
165.l odd 4 1 1089.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 5.c odd 4 1
99.4.a.e 2 15.e even 4 1
363.4.a.j 2 55.e even 4 1
528.4.a.o 2 20.e even 4 1
825.4.a.k 2 5.c odd 4 1
825.4.c.i 4 1.a even 1 1 trivial
825.4.c.i 4 5.b even 2 1 inner
1089.4.a.t 2 165.l odd 4 1
1584.4.a.x 2 60.l odd 4 1
1617.4.a.j 2 35.f even 4 1
2112.4.a.ba 2 40.i odd 4 1
2112.4.a.bh 2 40.k even 4 1
2475.4.a.o 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{4} + 17 T_{2}^{2} + 64 \)
\( T_{7}^{4} + 68 T_{7}^{2} + 1024 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 64 + 17 T^{2} + T^{4} \)
$3$ \( ( 9 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 1024 + 68 T^{2} + T^{4} \)
$11$ \( ( -11 + T )^{4} \)
$13$ \( 839056 + 3944 T^{2} + T^{4} \)
$17$ \( 52649536 + 15188 T^{2} + T^{4} \)
$19$ \( ( -1944 - 54 T + T^{2} )^{2} \)
$23$ \( ( 12544 + T^{2} )^{2} \)
$29$ \( ( -5136 + 222 T + T^{2} )^{2} \)
$31$ \( ( -88832 + 40 T + T^{2} )^{2} \)
$37$ \( 237036816 + 33096 T^{2} + T^{4} \)
$41$ \( ( 60976 + 494 T + T^{2} )^{2} \)
$43$ \( 3591365184 + 124212 T^{2} + T^{4} \)
$47$ \( 323424256 + 40064 T^{2} + T^{4} \)
$53$ \( 17796627216 + 273864 T^{2} + T^{4} \)
$59$ \( ( 196 + T )^{4} \)
$61$ \( ( 282396 + 1104 T + T^{2} )^{2} \)
$67$ \( 609497344 + 811808 T^{2} + T^{4} \)
$71$ \( ( -227328 - 456 T + T^{2} )^{2} \)
$73$ \( 190350709264 + 1223048 T^{2} + T^{4} \)
$79$ \( ( -31952 - 230 T + T^{2} )^{2} \)
$83$ \( 698521522176 + 1792656 T^{2} + T^{4} \)
$89$ \( ( 235668 + 972 T + T^{2} )^{2} \)
$97$ \( 1220662586896 + 3611528 T^{2} + T^{4} \)
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