Properties

Label 400.6.c.l
Level 400
Weight 6
Character orbit 400.c
Analytic conductor 64.154
Analytic rank 0
Dimension 4
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{129})\)
Defining polynomial: \(x^{4} + 65 x^{2} + 1024\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 40)
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.

\(f(q)\) \(=\) \( q + ( -3 \beta_{1} + \beta_{2} ) q^{3} + ( -13 \beta_{1} - 3 \beta_{2} ) q^{7} + ( -309 + 6 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -3 \beta_{1} + \beta_{2} ) q^{3} + ( -13 \beta_{1} - 3 \beta_{2} ) q^{7} + ( -309 + 6 \beta_{3} ) q^{9} + ( -280 + 3 \beta_{3} ) q^{11} + ( -347 \beta_{1} + 12 \beta_{2} ) q^{13} + ( 37 \beta_{1} + 84 \beta_{2} ) q^{17} + ( -500 - 18 \beta_{3} ) q^{19} + ( 1392 + 4 \beta_{3} ) q^{21} + ( -613 \beta_{1} - 123 \beta_{2} ) q^{23} + ( 3294 \beta_{1} - 138 \beta_{2} ) q^{27} + ( -670 + 156 \beta_{3} ) q^{29} + ( 1124 + 27 \beta_{3} ) q^{31} + ( 2388 \beta_{1} - 316 \beta_{2} ) q^{33} + ( -1485 \beta_{1} - 216 \beta_{2} ) q^{37} + ( -10356 + 383 \beta_{3} ) q^{39} + ( 11538 + 78 \beta_{3} ) q^{41} + ( 4421 \beta_{1} - 339 \beta_{2} ) q^{43} + ( 727 \beta_{1} + 885 \beta_{2} ) q^{47} + ( 11487 - 78 \beta_{3} ) q^{49} + ( -42900 + 215 \beta_{3} ) q^{51} + ( 1353 \beta_{1} - 540 \beta_{2} ) q^{53} + ( -7788 \beta_{1} - 284 \beta_{2} ) q^{57} + ( 31292 + 252 \beta_{3} ) q^{59} + ( 7054 + 552 \beta_{3} ) q^{61} + ( -5271 \beta_{1} + 615 \beta_{2} ) q^{63} + ( 21353 \beta_{1} - 543 \beta_{2} ) q^{67} + ( 56112 + 244 \beta_{3} ) q^{69} + ( -23604 - 273 \beta_{3} ) q^{71} + ( 16863 \beta_{1} + 1308 \beta_{2} ) q^{73} + ( -1004 \beta_{1} + 684 \beta_{2} ) q^{77} + ( -32952 - 1254 \beta_{3} ) q^{79} + ( 35649 - 2250 \beta_{3} ) q^{81} + ( 27181 \beta_{1} - 711 \beta_{2} ) q^{83} + ( 82506 \beta_{1} - 2542 \beta_{2} ) q^{87} + ( 27510 - 732 \beta_{3} ) q^{89} + ( 532 - 885 \beta_{3} ) q^{91} + ( 10560 \beta_{1} + 800 \beta_{2} ) q^{93} + ( 36917 \beta_{1} + 4620 \beta_{2} ) q^{97} + ( 123672 - 2607 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 1236q^{9} + O(q^{10}) \) \( 4q - 1236q^{9} - 1120q^{11} - 2000q^{19} + 5568q^{21} - 2680q^{29} + 4496q^{31} - 41424q^{39} + 46152q^{41} + 45948q^{49} - 171600q^{51} + 125168q^{59} + 28216q^{61} + 224448q^{69} - 94416q^{71} - 131808q^{79} + 142596q^{81} + 110040q^{89} + 2128q^{91} + 494688q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 65 x^{2} + 1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 33 \nu \)\()/16\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 97 \nu \)\()/16\)
\(\beta_{3}\)\(=\)\( 8 \nu^{2} + 260 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 260\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-33 \beta_{2} + 97 \beta_{1}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
6.17891i
5.17891i
5.17891i
6.17891i
0 28.7156i 0 0 0 42.1469i 0 −581.588 0
49.2 0 16.7156i 0 0 0 94.1469i 0 −36.4124 0
49.3 0 16.7156i 0 0 0 94.1469i 0 −36.4124 0
49.4 0 28.7156i 0 0 0 42.1469i 0 −581.588 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 400.6.c.l 4
4.b odd 2 1 200.6.c.e 4
5.b even 2 1 inner 400.6.c.l 4
5.c odd 4 1 80.6.a.i 2
5.c odd 4 1 400.6.a.q 2
15.e even 4 1 720.6.a.z 2
20.d odd 2 1 200.6.c.e 4
20.e even 4 1 40.6.a.d 2
20.e even 4 1 200.6.a.g 2
40.i odd 4 1 320.6.a.q 2
40.k even 4 1 320.6.a.w 2
60.l odd 4 1 360.6.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.d 2 20.e even 4 1
80.6.a.i 2 5.c odd 4 1
200.6.a.g 2 20.e even 4 1
200.6.c.e 4 4.b odd 2 1
200.6.c.e 4 20.d odd 2 1
320.6.a.q 2 40.i odd 4 1
320.6.a.w 2 40.k even 4 1
360.6.a.l 2 60.l odd 4 1
400.6.a.q 2 5.c odd 4 1
400.6.c.l 4 1.a even 1 1 trivial
400.6.c.l 4 5.b even 2 1 inner
720.6.a.z 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 1104 T_{3}^{2} + 230400 \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 132 T^{2} + 48150 T^{4} + 7794468 T^{6} + 3486784401 T^{8} \)
$5$ 1
$7$ \( 1 - 56588 T^{2} + 1352943558 T^{4} - 15984709390412 T^{6} + 79792266297612001 T^{8} \)
$11$ \( ( 1 + 560 T + 381926 T^{2} + 90188560 T^{3} + 25937424601 T^{4} )^{2} \)
$13$ \( 1 - 373292 T^{2} + 167403787638 T^{4} - 51461472139296908 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \)
$17$ \( 1 + 1613316 T^{2} + 4602934743878 T^{4} + 3252435215496778884 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} \)
$19$ \( ( 1 + 1000 T + 4533462 T^{2} + 2476099000 T^{3} + 6131066257801 T^{4} )^{2} \)
$23$ \( 1 - 7126092 T^{2} + 48612883261958 T^{4} - \)\(29\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \)
$29$ \( ( 1 + 1340 T - 8758306 T^{2} + 27484939660 T^{3} + 420707233300201 T^{4} )^{2} \)
$31$ \( ( 1 - 2248 T + 57017022 T^{2} - 64358331448 T^{3} + 819628286980801 T^{4} )^{2} \)
$37$ \( 1 - 211585036 T^{2} + 19959790722550422 T^{4} - \)\(10\!\cdots\!64\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} \)
$41$ \( ( 1 - 23076 T + 352280470 T^{2} - 2673497694276 T^{3} + 13422659310152401 T^{4} )^{2} \)
$43$ \( 1 - 313073372 T^{2} + 49182412950657078 T^{4} - \)\(67\!\cdots\!28\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 104863596 T^{2} + 104529727880710502 T^{4} - \)\(55\!\cdots\!04\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} \)
$53$ \( 1 - 1357205900 T^{2} + 805869805574922198 T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \)
$59$ \( ( 1 - 62584 T + 2277965606 T^{2} - 44742822328616 T^{3} + 511116753300641401 T^{4} )^{2} \)
$61$ \( ( 1 - 14108 T + 1110042462 T^{2} - 11915564614508 T^{3} + 713342911662882601 T^{4} )^{2} \)
$67$ \( 1 - 1448611388 T^{2} + 3060385933795078038 T^{4} - \)\(26\!\cdots\!12\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \)
$71$ \( ( 1 + 47208 T + 4011779662 T^{2} + 85174059202008 T^{3} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( 1 - 4251788572 T^{2} + 9098112686809466598 T^{4} - \)\(18\!\cdots\!28\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} \)
$79$ \( ( 1 + 65904 T + 3994274078 T^{2} + 202790324919696 T^{3} + 9468276082626847201 T^{4} )^{2} \)
$83$ \( 1 - 9324010812 T^{2} + 49682906641812448598 T^{4} - \)\(14\!\cdots\!88\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \)
$89$ \( ( 1 - 55020 T + 10818978262 T^{2} - 307234950883980 T^{3} + 31181719929966183601 T^{4} )^{2} \)
$97$ \( 1 - 1419021116 T^{2} - 92174956617487206138 T^{4} - \)\(10\!\cdots\!84\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} \)
show more
show less