Properties

Label 400.6.a.t
Level 400
Weight 6
Character orbit 400.a
Self dual yes
Analytic conductor 64.154
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1535279252\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \(x^{2} - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta q^{3} -9 \beta q^{7} + 153 q^{9} +O(q^{10})\) \( q + 3 \beta q^{3} -9 \beta q^{7} + 153 q^{9} -252 q^{11} -18 \beta q^{13} + 104 \beta q^{17} -220 q^{19} -1188 q^{21} -367 \beta q^{23} -270 \beta q^{27} + 6930 q^{29} -6752 q^{31} -756 \beta q^{33} -2106 \beta q^{37} -2376 q^{39} -198 q^{41} + 63 \beta q^{43} -1589 \beta q^{47} -13243 q^{49} + 13728 q^{51} -878 \beta q^{53} -660 \beta q^{57} -24660 q^{59} -5698 q^{61} -1377 \beta q^{63} -6579 \beta q^{67} -48444 q^{69} -53352 q^{71} + 10692 \beta q^{73} + 2268 \beta q^{77} + 51920 q^{79} -72819 q^{81} + 9323 \beta q^{83} + 20790 \beta q^{87} + 9990 q^{89} + 7128 q^{91} -20256 \beta q^{93} + 15264 \beta q^{97} -38556 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 306q^{9} + O(q^{10}) \) \( 2q + 306q^{9} - 504q^{11} - 440q^{19} - 2376q^{21} + 13860q^{29} - 13504q^{31} - 4752q^{39} - 396q^{41} - 26486q^{49} + 27456q^{51} - 49320q^{59} - 11396q^{61} - 96888q^{69} - 106704q^{71} + 103840q^{79} - 145638q^{81} + 19980q^{89} + 14256q^{91} - 77112q^{99} + O(q^{100}) \)

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
−3.31662
3.31662
0 −19.8997 0 0 0 59.6992 0 153.000 0
1.2 0 19.8997 0 0 0 −59.6992 0 153.000 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.a.t 2
4.b odd 2 1 25.6.a.c 2
5.b even 2 1 inner 400.6.a.t 2
5.c odd 4 2 80.6.c.a 2
12.b even 2 1 225.6.a.n 2
15.e even 4 2 720.6.f.f 2
20.d odd 2 1 25.6.a.c 2
20.e even 4 2 5.6.b.a 2
40.i odd 4 2 320.6.c.g 2
40.k even 4 2 320.6.c.f 2
60.h even 2 1 225.6.a.n 2
60.l odd 4 2 45.6.b.b 2
140.j odd 4 2 245.6.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.b.a 2 20.e even 4 2
25.6.a.c 2 4.b odd 2 1
25.6.a.c 2 20.d odd 2 1
45.6.b.b 2 60.l odd 4 2
80.6.c.a 2 5.c odd 4 2
225.6.a.n 2 12.b even 2 1
225.6.a.n 2 60.h even 2 1
245.6.b.a 2 140.j odd 4 2
320.6.c.f 2 40.k even 4 2
320.6.c.g 2 40.i odd 4 2
400.6.a.t 2 1.a even 1 1 trivial
400.6.a.t 2 5.b even 2 1 inner
720.6.f.f 2 15.e even 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 396 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 90 T^{2} + 59049 T^{4} \)
$5$ 1
$7$ \( 1 + 30050 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 + 252 T + 161051 T^{2} )^{2} \)
$13$ \( 1 + 728330 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 + 2363810 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 + 220 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 + 6946370 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 - 6930 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 + 6752 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 56462470 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 + 198 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 + 293842250 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 + 347593490 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 + 802472090 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 + 24660 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 + 5698 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 + 795787610 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 + 53352 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 - 883886830 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 - 51920 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 + 4053674810 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 - 9990 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 + 6923133890 T^{2} + 73742412689492826049 T^{4} \)
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