Properties

Label 825.4.c.o
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.9935104.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 6 x^{3} + 16 x^{2} - 8 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + 3 \beta_{3} q^{3} + ( 2 - \beta_{4} ) q^{4} + 3 \beta_{1} q^{6} + ( -2 \beta_{2} - 3 \beta_{3} + 5 \beta_{5} ) q^{7} + ( 7 \beta_{2} - \beta_{3} + \beta_{5} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + 3 \beta_{3} q^{3} + ( 2 - \beta_{4} ) q^{4} + 3 \beta_{1} q^{6} + ( -2 \beta_{2} - 3 \beta_{3} + 5 \beta_{5} ) q^{7} + ( 7 \beta_{2} - \beta_{3} + \beta_{5} ) q^{8} -9 q^{9} -11 q^{11} + ( 6 \beta_{3} + 3 \beta_{5} ) q^{12} + ( 20 \beta_{2} + 11 \beta_{3} + 11 \beta_{5} ) q^{13} + ( 17 - 18 \beta_{1} + 7 \beta_{4} ) q^{14} + ( -25 - 4 \beta_{1} - 14 \beta_{4} ) q^{16} + ( 14 \beta_{2} - 19 \beta_{3} - 9 \beta_{5} ) q^{17} -9 \beta_{2} q^{18} + ( 88 - 26 \beta_{1} - 10 \beta_{4} ) q^{19} + ( 9 - 6 \beta_{1} + 15 \beta_{4} ) q^{21} -11 \beta_{2} q^{22} + ( -36 \beta_{2} + 54 \beta_{3} + 14 \beta_{5} ) q^{23} + ( 3 + 21 \beta_{1} + 3 \beta_{4} ) q^{24} + ( -109 - 22 \beta_{1} - 9 \beta_{4} ) q^{26} -27 \beta_{3} q^{27} + ( 22 \beta_{2} + 91 \beta_{3} + 15 \beta_{5} ) q^{28} + ( 88 + 18 \beta_{1} + 44 \beta_{4} ) q^{29} + ( -134 - 32 \beta_{1} - 22 \beta_{4} ) q^{31} + ( -11 \beta_{2} + 2 \beta_{3} + 18 \beta_{5} ) q^{32} -33 \beta_{3} q^{33} + ( -93 + 8 \beta_{1} - 23 \beta_{4} ) q^{34} + ( -18 + 9 \beta_{4} ) q^{36} + ( -56 \beta_{2} + 198 \beta_{3} - 32 \beta_{5} ) q^{37} + ( 58 \beta_{2} + 146 \beta_{3} - 16 \beta_{5} ) q^{38} + ( -33 + 60 \beta_{1} + 33 \beta_{4} ) q^{39} + ( -272 - 110 \beta_{1} + 44 \beta_{4} ) q^{41} + ( 54 \beta_{2} + 51 \beta_{3} - 21 \beta_{5} ) q^{42} + ( 22 \beta_{2} - 13 \beta_{3} - 85 \beta_{5} ) q^{43} + ( -22 + 11 \beta_{4} ) q^{44} + ( 230 + 12 \beta_{1} + 50 \beta_{4} ) q^{46} + ( 112 \beta_{2} + 38 \beta_{3} - 38 \beta_{5} ) q^{47} + ( 12 \beta_{2} - 75 \beta_{3} + 42 \beta_{5} ) q^{48} + ( -185 + 172 \beta_{1} - 4 \beta_{4} ) q^{49} + ( 57 + 42 \beta_{1} - 27 \beta_{4} ) q^{51} + ( 24 \beta_{2} + 211 \beta_{3} + 75 \beta_{5} ) q^{52} + ( 60 \beta_{2} + 24 \beta_{3} + 38 \beta_{5} ) q^{53} -27 \beta_{1} q^{54} + ( 19 - 98 \beta_{1} + 49 \beta_{4} ) q^{56} + ( 78 \beta_{2} + 264 \beta_{3} + 30 \beta_{5} ) q^{57} + ( 220 \beta_{2} - 64 \beta_{3} - 26 \beta_{5} ) q^{58} + ( 174 - 196 \beta_{1} - 106 \beta_{4} ) q^{59} + ( -168 + 100 \beta_{1} + 22 \beta_{4} ) q^{61} + ( -200 \beta_{2} + 170 \beta_{3} - 10 \beta_{5} ) q^{62} + ( 18 \beta_{2} + 27 \beta_{3} - 45 \beta_{5} ) q^{63} + ( -116 - 84 \beta_{1} - 83 \beta_{4} ) q^{64} -33 \beta_{1} q^{66} + ( 272 \beta_{2} + 242 \beta_{3} + 58 \beta_{5} ) q^{67} + ( -50 \beta_{2} - 223 \beta_{3} - 41 \beta_{5} ) q^{68} + ( -162 - 108 \beta_{1} + 42 \beta_{4} ) q^{69} + ( -546 - 96 \beta_{1} - 74 \beta_{4} ) q^{71} + ( -63 \beta_{2} + 9 \beta_{3} - 9 \beta_{5} ) q^{72} + ( -16 \beta_{2} - 235 \beta_{3} + 17 \beta_{5} ) q^{73} + ( 304 + 294 \beta_{1} + 24 \beta_{4} ) q^{74} + ( 340 - 14 \beta_{1} - 154 \beta_{4} ) q^{76} + ( 22 \beta_{2} + 33 \beta_{3} - 55 \beta_{5} ) q^{77} + ( 66 \beta_{2} - 327 \beta_{3} + 27 \beta_{5} ) q^{78} + ( -92 - 130 \beta_{1} + 22 \beta_{4} ) q^{79} + 81 q^{81} + ( -140 \beta_{2} + 704 \beta_{3} - 154 \beta_{5} ) q^{82} + ( 148 \beta_{2} + 359 \beta_{3} - 7 \beta_{5} ) q^{83} + ( -273 + 66 \beta_{1} + 45 \beta_{4} ) q^{84} + ( -217 + 242 \beta_{1} - 107 \beta_{4} ) q^{86} + ( -54 \beta_{2} + 264 \beta_{3} - 132 \beta_{5} ) q^{87} + ( -77 \beta_{2} + 11 \beta_{3} - 11 \beta_{5} ) q^{88} + ( -402 - 236 \beta_{1} + 132 \beta_{4} ) q^{89} + ( -694 - 96 \beta_{1} + 250 \beta_{4} ) q^{91} + ( 92 \beta_{2} + 410 \beta_{3} + 74 \beta_{5} ) q^{92} + ( 96 \beta_{2} - 402 \beta_{3} + 66 \beta_{5} ) q^{93} + ( -710 + 152 \beta_{1} - 150 \beta_{4} ) q^{94} + ( -6 - 33 \beta_{1} + 54 \beta_{4} ) q^{96} + ( -16 \beta_{2} + 68 \beta_{3} + 278 \beta_{5} ) q^{97} + ( -197 \beta_{2} - 1036 \beta_{3} + 176 \beta_{5} ) q^{98} + 99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 10q^{4} - 6q^{6} - 54q^{9} + O(q^{10}) \) \( 6q + 10q^{4} - 6q^{6} - 54q^{9} - 66q^{11} + 152q^{14} - 170q^{16} + 560q^{19} + 96q^{21} - 18q^{24} - 628q^{26} + 580q^{29} - 784q^{31} - 620q^{34} - 90q^{36} - 252q^{39} - 1324q^{41} - 110q^{44} + 1456q^{46} - 1462q^{49} + 204q^{51} + 54q^{54} + 408q^{56} + 1224q^{59} - 1164q^{61} - 694q^{64} + 66q^{66} - 672q^{69} - 3232q^{71} + 1284q^{74} + 1760q^{76} - 248q^{79} + 486q^{81} - 1680q^{84} - 2000q^{86} - 1676q^{89} - 3472q^{91} - 4864q^{94} + 138q^{96} + 594q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 6 x^{3} + 16 x^{2} - 8 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 12 \nu^{3} - 6 \nu^{2} + 4 \nu - 19 \)\()/33\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{5} + 7 \nu^{4} + 3 \nu^{3} - 48 \nu^{2} - 56 \nu + 13 \)\()/33\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{5} + 12 \nu^{4} - 9 \nu^{3} - 54 \nu^{2} - 118 \nu + 27 \)\()/33\)
\(\beta_{4}\)\(=\)\((\)\( 6 \nu^{5} - 26 \nu^{4} + 36 \nu^{3} + 18 \nu^{2} - 12 \nu - 185 \)\()/33\)
\(\beta_{5}\)\(=\)\((\)\( -39 \nu^{5} + 70 \nu^{4} - 69 \nu^{3} - 216 \nu^{2} - 714 \nu + 163 \)\()/33\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - 7 \beta_{3} + 2 \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} - \beta_{4} - 9 \beta_{3} + 6 \beta_{2} - 6 \beta_{1} - 9\)\()/2\)
\(\nu^{4}\)\(=\)\(-3 \beta_{4} - 9 \beta_{1} - 22\)
\(\nu^{5}\)\(=\)\((\)\(-9 \beta_{5} - 9 \beta_{4} + 73 \beta_{3} - 40 \beta_{2} - 40 \beta_{1} - 73\)\()/2\)

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
−1.13973 1.13973i
1.91004 1.91004i
0.229681 + 0.229681i
0.229681 0.229681i
1.91004 + 1.91004i
−1.13973 + 1.13973i
3.27945i 3.00000i −2.75481 0 −9.83836 33.3329i 17.2014i −9.00000 0
199.2 2.82009i 3.00000i 0.0470959 0 8.46027 7.12434i 22.6935i −9.00000 0
199.3 0.540637i 3.00000i 7.70771 0 −1.62191 24.4573i 8.49217i −9.00000 0
199.4 0.540637i 3.00000i 7.70771 0 −1.62191 24.4573i 8.49217i −9.00000 0
199.5 2.82009i 3.00000i 0.0470959 0 8.46027 7.12434i 22.6935i −9.00000 0
199.6 3.27945i 3.00000i −2.75481 0 −9.83836 33.3329i 17.2014i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
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.o 6
5.b even 2 1 inner 825.4.c.o 6
5.c odd 4 1 165.4.a.f 3
5.c odd 4 1 825.4.a.n 3
15.e even 4 1 495.4.a.g 3
15.e even 4 1 2475.4.a.w 3
55.e even 4 1 1815.4.a.p 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.f 3 5.c odd 4 1
495.4.a.g 3 15.e even 4 1
825.4.a.n 3 5.c odd 4 1
825.4.c.o 6 1.a even 1 1 trivial
825.4.c.o 6 5.b even 2 1 inner
1815.4.a.p 3 55.e even 4 1
2475.4.a.w 3 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{6} + 19 T_{2}^{4} + 91 T_{2}^{2} + 25 \)
\( T_{7}^{6} + 1760 T_{7}^{4} + 751360 T_{7}^{2} + 33732864 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 + 91 T^{2} + 19 T^{4} + T^{6} \)
$3$ \( ( 9 + T^{2} )^{3} \)
$5$ \( T^{6} \)
$7$ \( 33732864 + 751360 T^{2} + 1760 T^{4} + T^{6} \)
$11$ \( ( 11 + T )^{6} \)
$13$ \( 18883157056 + 38874928 T^{2} + 12220 T^{4} + T^{6} \)
$17$ \( 32063916096 + 33891952 T^{2} + 10476 T^{4} + T^{6} \)
$19$ \( ( 97056 + 18624 T - 280 T^{2} + T^{3} )^{2} \)
$23$ \( 3672159698944 + 719065088 T^{2} + 46592 T^{4} + T^{6} \)
$29$ \( ( 9251496 - 26500 T - 290 T^{2} + T^{3} )^{2} \)
$31$ \( ( -316800 + 32160 T + 392 T^{2} + T^{3} )^{2} \)
$37$ \( 1080818061376 + 4898854960 T^{2} + 203020 T^{4} + T^{6} \)
$41$ \( ( -68561784 - 55908 T + 662 T^{2} + T^{3} )^{2} \)
$43$ \( 419883375995136 + 54486628992 T^{2} + 459376 T^{4} + T^{6} \)
$47$ \( 199419586560000 + 19629054976 T^{2} + 398848 T^{4} + T^{6} \)
$53$ \( 5779177536064 + 3794177840 T^{2} + 124460 T^{4} + T^{6} \)
$59$ \( ( 162128320 - 424784 T - 612 T^{2} + T^{3} )^{2} \)
$61$ \( ( -21355000 + 20044 T + 582 T^{2} + T^{3} )^{2} \)
$67$ \( 155628861363097600 + 893357593344 T^{2} + 1662576 T^{4} + T^{6} \)
$71$ \( ( 40198784 + 671200 T + 1616 T^{2} + T^{3} )^{2} \)
$73$ \( 145112476357696 + 10736115760 T^{2} + 206908 T^{4} + T^{6} \)
$79$ \( ( -26871328 - 185872 T + 124 T^{2} + T^{3} )^{2} \)
$83$ \( 245122860960000 + 124520155200 T^{2} + 931504 T^{4} + T^{6} \)
$89$ \( ( -831946232 - 1004532 T + 838 T^{2} + T^{3} )^{2} \)
$97$ \( 854091475345124416 + 5439528120880 T^{2} + 4600780 T^{4} + T^{6} \)
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