Properties

Label 400.10.a.ba
Level 400
Weight 10
Character orbit 400.a
Self dual yes
Analytic conductor 206.014
Analytic rank 1
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(206.014334466\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.49740556.1
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 49 \beta_{1} + 21 \beta_{2} ) q^{7} + ( -2907 + 9 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 49 \beta_{1} + 21 \beta_{2} ) q^{7} + ( -2907 + 9 \beta_{3} ) q^{9} + ( -27492 - 10 \beta_{3} ) q^{11} + ( 110 \beta_{1} - 507 \beta_{2} ) q^{13} + ( -1632 \beta_{1} + 2270 \beta_{2} ) q^{17} + ( -159220 - 238 \beta_{3} ) q^{19} + ( 880992 + 399 \beta_{3} ) q^{21} + ( 5199 \beta_{1} - 5551 \beta_{2} ) q^{23} + ( -7362 \beta_{1} - 6966 \beta_{2} ) q^{27} + ( 882930 - 988 \beta_{3} ) q^{29} + ( 2646928 - 380 \beta_{3} ) q^{31} + ( -44412 \beta_{1} + 7740 \beta_{2} ) q^{33} + ( -51378 \beta_{1} + 96687 \beta_{2} ) q^{37} + ( 421704 + 2004 \beta_{3} ) q^{39} + ( -4197138 - 12445 \beta_{3} ) q^{41} + ( -132643 \beta_{1} - 146622 \beta_{2} ) q^{43} + ( -214259 \beta_{1} - 199463 \beta_{2} ) q^{47} + ( 11730257 + 13965 \beta_{3} ) q^{49} + ( -21004272 - 19228 \beta_{3} ) q^{51} + ( 259794 \beta_{1} - 402365 \beta_{2} ) q^{53} + ( -561916 \beta_{1} + 184212 \beta_{2} ) q^{57} + ( -115207260 + 26486 \beta_{3} ) q^{59} + ( 90122642 + 21575 \beta_{3} ) q^{61} + ( 591633 \beta_{1} - 722169 \beta_{2} ) q^{63} + ( -1448953 \beta_{1} + 2029650 \beta_{2} ) q^{67} + ( 71631216 + 57893 \beta_{3} ) q^{69} + ( 11902968 - 45600 \beta_{3} ) q^{71} + ( -90564 \beta_{1} - 439344 \beta_{2} ) q^{73} + ( -2162748 \beta_{1} + 157248 \beta_{2} ) q^{77} + ( -182010880 + 133988 \beta_{3} ) q^{79} + ( -85846959 - 229473 \beta_{3} ) q^{81} + ( 1180353 \beta_{1} + 2889326 \beta_{2} ) q^{83} + ( -788766 \beta_{1} + 764712 \beta_{2} ) q^{87} + ( 395675190 + 92796 \beta_{3} ) q^{89} + ( -118330632 + 178752 \beta_{3} ) q^{91} + ( 2003968 \beta_{1} + 294120 \beta_{2} ) q^{93} + ( -1954216 \beta_{1} - 3208062 \beta_{2} ) q^{97} + ( -182196756 - 218358 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 11628q^{9} + O(q^{10}) \) \( 4q - 11628q^{9} - 109968q^{11} - 636880q^{19} + 3523968q^{21} + 3531720q^{29} + 10587712q^{31} + 1686816q^{39} - 16788552q^{41} + 46921028q^{49} - 84017088q^{51} - 460829040q^{59} + 360490568q^{61} + 286524864q^{69} + 47611872q^{71} - 728043520q^{79} - 343387836q^{81} + 1582700760q^{89} - 473322528q^{91} - 728787024q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 45 x^{2} + 304\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3 \nu^{3} - 51 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} - 74 \nu \)
\(\beta_{3}\)\(=\)\( 120 \nu^{2} - 2700 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{2} + 4 \beta_{1}\)\()/120\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 2700\)\()/120\)
\(\nu^{3}\)\(=\)\((\)\(-51 \beta_{2} + 148 \beta_{1}\)\()/120\)

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
−6.05982
2.87724
−2.87724
6.05982
0 −179.263 0 0 0 −8712.99 0 12452.2 0
1.2 0 −37.6407 0 0 0 −5315.22 0 −18266.2 0
1.3 0 37.6407 0 0 0 5315.22 0 −18266.2 0
1.4 0 179.263 0 0 0 8712.99 0 12452.2 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 400.10.a.ba 4
4.b odd 2 1 25.10.a.e 4
5.b even 2 1 inner 400.10.a.ba 4
5.c odd 4 2 80.10.c.c 4
12.b even 2 1 225.10.a.s 4
20.d odd 2 1 25.10.a.e 4
20.e even 4 2 5.10.b.a 4
60.h even 2 1 225.10.a.s 4
60.l odd 4 2 45.10.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.b.a 4 20.e even 4 2
25.10.a.e 4 4.b odd 2 1
25.10.a.e 4 20.d odd 2 1
45.10.b.b 4 60.l odd 4 2
80.10.c.c 4 5.c odd 4 2
225.10.a.s 4 12.b even 2 1
225.10.a.s 4 60.h even 2 1
400.10.a.ba 4 1.a even 1 1 trivial
400.10.a.ba 4 5.b even 2 1 inner

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 33552 T_{3}^{2} + 45529776 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(400))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 45180 T^{2} + 1049244678 T^{4} + 17503657693020 T^{6} + 150094635296999121 T^{8} \)
$5$ 1
$7$ \( 1 + 57246700 T^{2} + 3508143545353398 T^{4} + \)\(93\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 + 54984 T + 5180465446 T^{2} + 129649395841944 T^{3} + 5559917313492231481 T^{4} )^{2} \)
$13$ \( 1 + 35613791860 T^{2} + \)\(53\!\cdots\!58\)\( T^{4} + \)\(40\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \)
$17$ \( 1 + 285780369220 T^{2} + \)\(48\!\cdots\!18\)\( T^{4} + \)\(40\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \)
$19$ \( ( 1 + 318440 T + 505756418358 T^{2} + 102756670480744760 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \)
$23$ \( 1 + 5779790962540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} + \)\(18\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 - 1765860 T + 26950935551038 T^{2} - 25617588792948032340 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$31$ \( ( 1 - 5293856 T + 59464921598526 T^{2} - \)\(13\!\cdots\!76\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \)
$37$ \( 1 + 231603274936660 T^{2} + \)\(44\!\cdots\!58\)\( T^{4} + \)\(39\!\cdots\!40\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \)
$41$ \( ( 1 + 8394276 T + 221313076168966 T^{2} + \)\(27\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \)
$43$ \( 1 + 614109141147100 T^{2} + \)\(57\!\cdots\!98\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + 1368976020813580 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} + \)\(17\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 + 7684297973864980 T^{2} + \)\(36\!\cdots\!78\)\( T^{4} + \)\(83\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \)
$59$ \( ( 1 + 230414520 T + 28555631923987078 T^{2} + \)\(19\!\cdots\!80\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \)
$61$ \( ( 1 - 180245284 T + 30154717014478446 T^{2} - \)\(21\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \)
$67$ \( 1 - 41160407446058180 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} - \)\(30\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \)
$71$ \( ( 1 - 23805936 T + 85782754020107086 T^{2} - \)\(10\!\cdots\!16\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$73$ \( 1 + 229489314868712740 T^{2} + \)\(20\!\cdots\!38\)\( T^{4} + \)\(79\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \)
$79$ \( ( 1 + 364021760 T + 220545463862625438 T^{2} + \)\(43\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \)
$83$ \( 1 + 434569632367965820 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} + \)\(15\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \)
$89$ \( ( 1 - 791350380 T + 832192702699668118 T^{2} - \)\(27\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \)
$97$ \( 1 + 2561123777205326980 T^{2} + \)\(27\!\cdots\!78\)\( T^{4} + \)\(14\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \)
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