Properties

Label 400.6.a.w
Level $400$
Weight $6$
Character orbit 400.a
Self dual yes
Analytic conductor $64.154$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
Defining polynomial: \(x^{2} - x - 60\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{241}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 10 - \beta ) q^{3} + ( 100 - 2 \beta ) q^{7} + ( 98 - 20 \beta ) q^{9} +O(q^{10})\) \( q + ( 10 - \beta ) q^{3} + ( 100 - 2 \beta ) q^{7} + ( 98 - 20 \beta ) q^{9} + ( 98 + 25 \beta ) q^{11} + ( 180 + 16 \beta ) q^{13} + ( -745 - 68 \beta ) q^{17} + ( 1590 - 35 \beta ) q^{19} + ( 1482 - 120 \beta ) q^{21} + ( 780 - 6 \beta ) q^{23} + ( 3370 - 55 \beta ) q^{27} + ( -1960 + 40 \beta ) q^{29} + ( 548 - 550 \beta ) q^{31} + ( -5045 + 152 \beta ) q^{33} + ( -1010 + 192 \beta ) q^{37} + ( -2056 - 20 \beta ) q^{39} + ( 13877 - 200 \beta ) q^{41} + ( -1500 + 1064 \beta ) q^{43} + ( 12880 - 772 \beta ) q^{47} + ( -5843 - 400 \beta ) q^{49} + ( 8938 + 65 \beta ) q^{51} + ( 13490 + 376 \beta ) q^{53} + ( 24335 - 1940 \beta ) q^{57} + ( -5980 - 980 \beta ) q^{59} + ( -12198 - 1000 \beta ) q^{61} + ( 19440 - 2196 \beta ) q^{63} + ( -20030 + 793 \beta ) q^{67} + ( 9246 - 840 \beta ) q^{69} + ( 43648 + 500 \beta ) q^{71} + ( 35145 + 556 \beta ) q^{73} + ( -2250 + 2304 \beta ) q^{77} + ( -32740 + 2510 \beta ) q^{79} + ( 23141 + 940 \beta ) q^{81} + ( 46290 + 429 \beta ) q^{83} + ( -29240 + 2360 \beta ) q^{87} + ( -36405 + 5220 \beta ) q^{89} + ( 10288 + 1240 \beta ) q^{91} + ( 138030 - 6048 \beta ) q^{93} + ( 63070 + 5472 \beta ) q^{97} + ( -110896 + 490 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 20q^{3} + 200q^{7} + 196q^{9} + O(q^{10}) \) \( 2q + 20q^{3} + 200q^{7} + 196q^{9} + 196q^{11} + 360q^{13} - 1490q^{17} + 3180q^{19} + 2964q^{21} + 1560q^{23} + 6740q^{27} - 3920q^{29} + 1096q^{31} - 10090q^{33} - 2020q^{37} - 4112q^{39} + 27754q^{41} - 3000q^{43} + 25760q^{47} - 11686q^{49} + 17876q^{51} + 26980q^{53} + 48670q^{57} - 11960q^{59} - 24396q^{61} + 38880q^{63} - 40060q^{67} + 18492q^{69} + 87296q^{71} + 70290q^{73} - 4500q^{77} - 65480q^{79} + 46282q^{81} + 92580q^{83} - 58480q^{87} - 72810q^{89} + 20576q^{91} + 276060q^{93} + 126140q^{97} - 221792q^{99} + O(q^{100}) \)

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
8.26209
−7.26209
0 −5.52417 0 0 0 68.9517 0 −212.483 0
1.2 0 25.5242 0 0 0 131.048 0 408.483 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 400.6.a.w 2
4.b odd 2 1 25.6.a.b 2
5.b even 2 1 400.6.a.o 2
5.c odd 4 2 400.6.c.n 4
12.b even 2 1 225.6.a.s 2
20.d odd 2 1 25.6.a.d yes 2
20.e even 4 2 25.6.b.b 4
60.h even 2 1 225.6.a.l 2
60.l odd 4 2 225.6.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.a.b 2 4.b odd 2 1
25.6.a.d yes 2 20.d odd 2 1
25.6.b.b 4 20.e even 4 2
225.6.a.l 2 60.h even 2 1
225.6.a.s 2 12.b even 2 1
225.6.b.i 4 60.l odd 4 2
400.6.a.o 2 5.b even 2 1
400.6.a.w 2 1.a even 1 1 trivial
400.6.c.n 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 20 T_{3} - 141 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -141 - 20 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 9036 - 200 T + T^{2} \)
$11$ \( -141021 - 196 T + T^{2} \)
$13$ \( -29296 - 360 T + T^{2} \)
$17$ \( -559359 + 1490 T + T^{2} \)
$19$ \( 2232875 - 3180 T + T^{2} \)
$23$ \( 599724 - 1560 T + T^{2} \)
$29$ \( 3456000 + 3920 T + T^{2} \)
$31$ \( -72602196 - 1096 T + T^{2} \)
$37$ \( -7864124 + 2020 T + T^{2} \)
$41$ \( 182931129 - 27754 T + T^{2} \)
$43$ \( -270585136 + 3000 T + T^{2} \)
$47$ \( 22262256 - 25760 T + T^{2} \)
$53$ \( 147908484 - 26980 T + T^{2} \)
$59$ \( -195696000 + 11960 T + T^{2} \)
$61$ \( -92208796 + 24396 T + T^{2} \)
$67$ \( 249648291 + 40060 T + T^{2} \)
$71$ \( 1844897904 - 87296 T + T^{2} \)
$73$ \( 1160669249 - 70290 T + T^{2} \)
$79$ \( -446416500 + 65480 T + T^{2} \)
$83$ \( 2098410219 - 92580 T + T^{2} \)
$89$ \( -5241540375 + 72810 T + T^{2} \)
$97$ \( -3238386044 - 126140 T + T^{2} \)
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