Properties

Label 825.6.a.m
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} + \beta_1 + 13) q^{4} + ( - 9 \beta_1 - 9) q^{6} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + 4 \beta_1 - 11) q^{7} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 11 \beta_1 - 25) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} + \beta_1 + 13) q^{4} + ( - 9 \beta_1 - 9) q^{6} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + 4 \beta_1 - 11) q^{7} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 11 \beta_1 - 25) q^{8} + 81 q^{9} + 121 q^{11} + (9 \beta_{2} + 9 \beta_1 + 117) q^{12} + (5 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 42 \beta_1 + 4) q^{13} + ( - 2 \beta_{6} + 5 \beta_{5} - 13 \beta_{4} - 6 \beta_{3} - 19 \beta_{2} + \cdots - 173) q^{14}+ \cdots + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9} + 847 q^{11} + 855 q^{12} + 113 q^{13} - 1212 q^{14} + 499 q^{16} - 1030 q^{17} - 729 q^{18} - 3803 q^{19} - 585 q^{21} - 1089 q^{22} - 514 q^{23} - 1863 q^{24} - 12111 q^{26} + 5103 q^{27} + 342 q^{28} - 2698 q^{29} - 17233 q^{31} - 9943 q^{32} + 7623 q^{33} + 4090 q^{34} + 7695 q^{36} - 23182 q^{37} + 11943 q^{38} + 1017 q^{39} - 16158 q^{41} - 10908 q^{42} + 4249 q^{43} + 11495 q^{44} - 28769 q^{46} - 7580 q^{47} + 4491 q^{48} + 37140 q^{49} - 9270 q^{51} + 23887 q^{52} - 20574 q^{53} - 6561 q^{54} - 73276 q^{56} - 34227 q^{57} - 8733 q^{58} - 364 q^{59} - 28127 q^{61} + 71917 q^{62} - 5265 q^{63} + 43379 q^{64} - 9801 q^{66} + 21493 q^{67} - 160660 q^{68} - 4626 q^{69} - 177084 q^{71} - 16767 q^{72} - 78670 q^{73} + 196750 q^{74} - 32701 q^{76} - 7865 q^{77} - 108999 q^{78} - 187432 q^{79} + 45927 q^{81} - 179552 q^{82} + 44592 q^{83} + 3078 q^{84} - 110433 q^{86} - 24282 q^{87} - 25047 q^{88} - 151168 q^{89} - 230153 q^{91} + 44767 q^{92} - 155097 q^{93} + 54040 q^{94} - 89487 q^{96} + 55589 q^{97} - 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + \nu^{4} - 123\nu^{3} - 57\nu^{2} + 2642\nu + 464 ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 11\nu^{4} + 171\nu^{3} - 1275\nu^{2} - 5642\nu + 21136 ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 3\nu^{5} + 133\nu^{4} + 351\nu^{3} - 3716\nu^{2} - 9276\nu + 14048 ) / 192 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 148\nu^{4} + 66\nu^{3} + 5315\nu^{2} - 2754\nu - 37040 ) / 192 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + 75\beta _1 - 44 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{6} - 8\beta_{5} + 8\beta_{4} - 4\beta_{3} + 115\beta_{2} - 161\beta _1 + 3260 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 254\beta_{6} + 254\beta_{5} + 238\beta_{4} + 298\beta_{3} - 181\beta_{2} + 6687\beta _1 - 6628 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1124\beta_{6} - 1316\beta_{5} + 1052\beta_{4} - 724\beta_{3} + 11771\beta_{2} - 20709\beta _1 + 288564 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
9.48549
6.35371
4.42881
1.60892
−3.33276
−6.33680
−10.2074
−10.4855 9.00000 77.9456 0 −94.3694 152.972 −481.762 81.0000 0
1.2 −7.35371 9.00000 22.0770 0 −66.1834 −251.987 72.9708 81.0000 0
1.3 −5.42881 9.00000 −2.52798 0 −48.8593 29.0935 187.446 81.0000 0
1.4 −2.60892 9.00000 −25.1935 0 −23.4803 174.969 149.214 81.0000 0
1.5 2.33276 9.00000 −26.5582 0 20.9949 −146.260 −136.602 81.0000 0
1.6 5.33680 9.00000 −3.51852 0 48.0312 73.9998 −189.555 81.0000 0
1.7 9.20737 9.00000 52.7757 0 82.8663 −97.7871 191.289 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.6.a.m 7
5.b even 2 1 825.6.a.o yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.6.a.m 7 1.a even 1 1 trivial
825.6.a.o yes 7 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + 9T_{2}^{6} - 119T_{2}^{5} - 1053T_{2}^{4} + 2894T_{2}^{3} + 27140T_{2}^{2} - 9288T_{2} - 125184 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + 9 T^{6} - 119 T^{5} + \cdots - 125184 \) Copy content Toggle raw display
$3$ \( (T - 9)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots + 207675630158484 \) Copy content Toggle raw display
$11$ \( (T - 121)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} - 113 T^{6} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{7} + 1030 T^{6} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{7} + 3803 T^{6} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{7} + 514 T^{6} + \cdots + 56\!\cdots\!54 \) Copy content Toggle raw display
$29$ \( T^{7} + 2698 T^{6} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{7} + 17233 T^{6} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{7} + 23182 T^{6} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{7} + 16158 T^{6} + \cdots - 11\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{7} - 4249 T^{6} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{7} + 7580 T^{6} + \cdots - 20\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{7} + 20574 T^{6} + \cdots - 27\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{7} + 364 T^{6} + \cdots - 47\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{7} + 28127 T^{6} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{7} - 21493 T^{6} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{7} + 177084 T^{6} + \cdots - 27\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{7} + 78670 T^{6} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{7} + 187432 T^{6} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} - 44592 T^{6} + \cdots - 36\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{7} + 151168 T^{6} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} - 55589 T^{6} + \cdots - 15\!\cdots\!23 \) Copy content Toggle raw display
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