Properties

Label 400.2.u.f
Level $400$
Weight $2$
Character orbit 400.u
Analytic conductor $3.194$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 3 x^{11} + 13 x^{10} - 24 x^{9} + 93 x^{8} - 6 x^{7} + 342 x^{6} + 786 x^{5} + 1473 x^{4} + 84 x^{3} + 688 x^{2} + 168 x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{6} + \beta_{5} - \beta_1 + 1) q^{7} + (\beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{6} + \beta_{5} - \beta_1 + 1) q^{7} + (\beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{9} + ( - \beta_{11} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{11} + ( - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - \beta_{3} + 1) q^{13} + (\beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{6} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{15} + ( - 3 \beta_{8} + 3 \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + 2) q^{17} + ( - \beta_{11} + \beta_{9} + \beta_{5} + \beta_{4} + \beta_{2} + 2) q^{19} + (\beta_{10} - 2 \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 1) q^{21} + ( - \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 1) q^{23} + ( - \beta_{10} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_1 - 1) q^{25} + (\beta_{11} - 2 \beta_{8} + \beta_{7} - 3 \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1 + 2) q^{27} + (2 \beta_{11} + \beta_{10} + \beta_{9} - 3 \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \cdots - 2) q^{29}+ \cdots + ( - \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{7} - 7 \beta_{6} - 7 \beta_{5} - 6 \beta_{4} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 4 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} - 4 q^{5} + 2 q^{7} - 3 q^{9} + 5 q^{11} - 2 q^{13} - 18 q^{15} + q^{17} + 8 q^{19} + 2 q^{21} + 6 q^{23} - 26 q^{25} + 34 q^{27} - 18 q^{29} - 12 q^{31} - 35 q^{33} + 3 q^{35} + 13 q^{37} - 22 q^{39} - 23 q^{41} - 50 q^{43} + 71 q^{45} - q^{47} + 34 q^{49} - 14 q^{51} + 21 q^{53} - 5 q^{55} + 72 q^{57} - 9 q^{59} - 26 q^{61} + 32 q^{63} - 18 q^{65} + 37 q^{67} - 44 q^{69} - 21 q^{71} + 18 q^{73} + 73 q^{75} - 60 q^{77} + 24 q^{79} + 18 q^{81} + 46 q^{83} - 16 q^{85} - 57 q^{87} - 2 q^{89} + 32 q^{91} + 22 q^{93} - 6 q^{95} - 7 q^{97} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 13 x^{10} - 24 x^{9} + 93 x^{8} - 6 x^{7} + 342 x^{6} + 786 x^{5} + 1473 x^{4} + 84 x^{3} + 688 x^{2} + 168 x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 260325772 \nu^{11} - 576792708 \nu^{10} + 3020952123 \nu^{9} - 4900209927 \nu^{8} + 25045472937 \nu^{7} + 3519097812 \nu^{6} + \cdots + 112273499910 ) / 438461548785 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3457672033 \nu^{11} - 11783661403 \nu^{10} + 48283397793 \nu^{9} - 97783045044 \nu^{8} + 339987222837 \nu^{7} + \cdots + 178985069048 ) / 1753846195140 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5369632561 \nu^{11} - 15433734165 \nu^{10} + 67292419131 \nu^{9} - 119576878026 \nu^{8} + 481102476429 \nu^{7} + \cdots + 391154012184 ) / 1753846195140 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22373133631 \nu^{11} - 74034744959 \nu^{10} + 314418060009 \nu^{9} - 633522002730 \nu^{8} + 2276267517771 \nu^{7} + \cdots - 273874062464 ) / 3507692390280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29615632063 \nu^{11} + 87105581309 \nu^{10} - 382317412959 \nu^{9} + 696924743820 \nu^{8} - 2749867795311 \nu^{7} + \cdots - 5593310307856 ) / 3507692390280 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99321102659 \nu^{11} - 361617020679 \nu^{10} + 1484218111509 \nu^{9} - 3218484118998 \nu^{8} + 10803833121471 \nu^{7} + \cdots - 17878470653712 ) / 8769230975700 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 48894251523 \nu^{11} - 157422019691 \nu^{10} + 666492738129 \nu^{9} - 1308046874814 \nu^{8} + 4786319147691 \nu^{7} + \cdots + 2898411243400 ) / 3507692390280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 167923324379 \nu^{11} + 478101391109 \nu^{10} - 2090376253539 \nu^{9} + 3656650156068 \nu^{8} - 14810975764191 \nu^{7} + \cdots - 37003359759088 ) / 8769230975700 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 542891618887 \nu^{11} - 1717504490237 \nu^{10} + 7294073305527 \nu^{9} - 14065746308484 \nu^{8} + 52158480741663 \nu^{7} + \cdots + 77595685775224 ) / 17538461951400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 572643183207 \nu^{11} - 1777970044487 \nu^{10} + 7631251682697 \nu^{9} - 14590245523914 \nu^{8} + 54915725330643 \nu^{7} + \cdots + 26266096925344 ) / 17538461951400 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 4\beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 2\beta_{10} + \beta_{9} - 6\beta_{8} - \beta_{7} + 4\beta_{6} + 5\beta_{5} + 7\beta_{4} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{11} + 8 \beta_{10} + 8 \beta_{9} - 2 \beta_{8} - 16 \beta_{7} + 24 \beta_{6} + 10 \beta_{5} - 8 \beta_{4} + 4 \beta_{3} + 15 \beta_{2} - 4 \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{11} + 3 \beta_{10} - 12 \beta_{9} - 12 \beta_{7} + 29 \beta_{6} + 29 \beta_{5} - 69 \beta_{4} + 69 \beta_{3} + 78 \beta_{2} - 75 \beta _1 - 47 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 75 \beta_{11} - 66 \beta_{10} - 132 \beta_{9} + 54 \beta_{8} + 57 \beta_{7} + 66 \beta_{6} + 177 \beta_{5} - 71 \beta_{4} + 66 \beta_{3} + 137 \beta_{2} - 244 \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 256 \beta_{11} - 178 \beta_{10} - 128 \beta_{9} + 844 \beta_{8} + 78 \beta_{7} - 220 \beta_{6} - 256 \beta_{5} - 184 \beta_{4} - 491 \beta_{3} + 128 \beta_{2} - 312 \beta _1 - 220 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 753 \beta_{11} - 368 \beta_{10} + 385 \beta_{9} + 3112 \beta_{8} + 184 \beta_{7} - 2744 \beta_{6} - 2973 \beta_{5} - 1037 \beta_{4} - 1357 \beta_{3} - 788 \beta_{2} - 184 \beta _1 - 1852 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 652 \beta_{11} - 2009 \beta_{10} + 705 \beta_{9} + 4658 \beta_{8} + 1304 \beta_{7} - 7165 \beta_{6} - 5310 \beta_{5} + 652 \beta_{4} - 3405 \beta_{3} - 7800 \beta_{2} + 2048 \beta _1 + 2009 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5100 \beta_{11} - 5100 \beta_{10} + 2700 \beta_{9} + 2700 \beta_{7} - 7110 \beta_{6} - 7110 \beta_{5} + 12915 \beta_{4} - 12915 \beta_{3} - 27435 \beta_{2} + 22335 \beta _1 + 25394 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 22335 \beta_{11} + 7815 \beta_{10} + 15630 \beta_{9} - 53280 \beta_{8} + 6705 \beta_{7} - 7815 \beta_{6} - 23925 \beta_{5} + 41550 \beta_{4} - 7815 \beta_{3} - 49365 \beta_{2} + 90074 \beta _1 + 45465 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-\beta_{8}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.838695 + 2.58124i
0.213831 + 0.658105i
−0.861543 2.65156i
2.67982 1.94700i
−0.120753 + 0.0877319i
−1.25005 + 0.908212i
2.67982 + 1.94700i
−0.120753 0.0877319i
−1.25005 0.908212i
0.838695 2.58124i
0.213831 0.658105i
−0.861543 + 2.65156i
0 −2.19573 + 1.59529i 0 0.548020 + 2.16787i 0 4.40288 0 1.34923 4.15250i 0
81.2 0 −0.559818 + 0.406731i 0 −0.463031 2.18760i 0 −3.25686 0 −0.779086 + 2.39778i 0
81.3 0 2.25555 1.63875i 0 −2.20302 + 0.383000i 0 2.70809 0 1.47494 4.53940i 0
161.1 0 −1.02360 + 3.15031i 0 −1.34720 + 1.78467i 0 −1.69984 0 −6.44966 4.68596i 0
161.2 0 0.0461234 0.141953i 0 0.383646 2.20291i 0 −3.58696 0 2.40903 + 1.75026i 0
161.3 0 0.477475 1.46952i 0 1.08159 + 1.95708i 0 2.43270 0 0.495552 + 0.360039i 0
241.1 0 −1.02360 3.15031i 0 −1.34720 1.78467i 0 −1.69984 0 −6.44966 + 4.68596i 0
241.2 0 0.0461234 + 0.141953i 0 0.383646 + 2.20291i 0 −3.58696 0 2.40903 1.75026i 0
241.3 0 0.477475 + 1.46952i 0 1.08159 1.95708i 0 2.43270 0 0.495552 0.360039i 0
321.1 0 −2.19573 1.59529i 0 0.548020 2.16787i 0 4.40288 0 1.34923 + 4.15250i 0
321.2 0 −0.559818 0.406731i 0 −0.463031 + 2.18760i 0 −3.25686 0 −0.779086 2.39778i 0
321.3 0 2.25555 + 1.63875i 0 −2.20302 0.383000i 0 2.70809 0 1.47494 + 4.53940i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.u.f 12
4.b odd 2 1 100.2.g.a 12
12.b even 2 1 900.2.n.c 12
20.d odd 2 1 500.2.g.a 12
20.e even 4 2 500.2.i.b 24
25.d even 5 1 inner 400.2.u.f 12
25.d even 5 1 10000.2.a.bc 6
25.e even 10 1 10000.2.a.bd 6
100.h odd 10 1 500.2.g.a 12
100.h odd 10 1 2500.2.a.c 6
100.j odd 10 1 100.2.g.a 12
100.j odd 10 1 2500.2.a.d 6
100.l even 20 2 500.2.i.b 24
100.l even 20 2 2500.2.c.c 12
300.n even 10 1 900.2.n.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.g.a 12 4.b odd 2 1
100.2.g.a 12 100.j odd 10 1
400.2.u.f 12 1.a even 1 1 trivial
400.2.u.f 12 25.d even 5 1 inner
500.2.g.a 12 20.d odd 2 1
500.2.g.a 12 100.h odd 10 1
500.2.i.b 24 20.e even 4 2
500.2.i.b 24 100.l even 20 2
900.2.n.c 12 12.b even 2 1
900.2.n.c 12 300.n even 10 1
2500.2.a.c 6 100.h odd 10 1
2500.2.a.d 6 100.j odd 10 1
2500.2.c.c 12 100.l even 20 2
10000.2.a.bc 6 25.d even 5 1
10000.2.a.bd 6 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 2 T_{3}^{11} + 8 T_{3}^{10} - 4 T_{3}^{9} + 23 T_{3}^{8} + 124 T_{3}^{7} + 637 T_{3}^{6} + 281 T_{3}^{5} + 1403 T_{3}^{4} + 1414 T_{3}^{3} + 608 T_{3}^{2} - 32 T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + 8 T^{10} - 4 T^{9} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{11} + 21 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} - T^{5} - 29 T^{4} + 18 T^{3} + \cdots - 576)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 5 T^{11} + 40 T^{10} + \cdots + 160000 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + 23 T^{10} + \cdots + 32761 \) Copy content Toggle raw display
$17$ \( T^{12} - T^{11} + 42 T^{10} - 187 T^{9} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{11} + 51 T^{10} + \cdots + 10471696 \) Copy content Toggle raw display
$23$ \( T^{12} - 6 T^{11} + 82 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( T^{12} + 18 T^{11} + 241 T^{10} + \cdots + 1042441 \) Copy content Toggle raw display
$31$ \( T^{12} + 12 T^{11} + \cdots + 102495376 \) Copy content Toggle raw display
$37$ \( T^{12} - 13 T^{11} + 78 T^{10} + \cdots + 2128681 \) Copy content Toggle raw display
$41$ \( T^{12} + 23 T^{11} + 306 T^{10} + \cdots + 41422096 \) Copy content Toggle raw display
$43$ \( (T^{6} + 25 T^{5} + 185 T^{4} + 140 T^{3} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + T^{11} + 72 T^{10} - 43 T^{9} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{12} - 21 T^{11} + 272 T^{10} + \cdots + 1745041 \) Copy content Toggle raw display
$59$ \( T^{12} + 9 T^{11} + 184 T^{10} + \cdots + 130051216 \) Copy content Toggle raw display
$61$ \( T^{12} + 26 T^{11} + \cdots + 1661296081 \) Copy content Toggle raw display
$67$ \( T^{12} - 37 T^{11} + \cdots + 58285547776 \) Copy content Toggle raw display
$71$ \( T^{12} + 21 T^{11} + \cdots + 5547866256 \) Copy content Toggle raw display
$73$ \( T^{12} - 18 T^{11} + \cdots + 4336354201 \) Copy content Toggle raw display
$79$ \( T^{12} - 24 T^{11} + \cdots + 56261942416 \) Copy content Toggle raw display
$83$ \( T^{12} - 46 T^{11} + 1162 T^{10} + \cdots + 36048016 \) Copy content Toggle raw display
$89$ \( T^{12} + 2 T^{11} + 301 T^{10} + \cdots + 63744256 \) Copy content Toggle raw display
$97$ \( T^{12} + 7 T^{11} + \cdots + 121807282081 \) Copy content Toggle raw display
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