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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [700,5,Mod(549,700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("700.549"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 700.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(72.3589741587\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.32905425960566784.37
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 36x^{10} + 432x^{8} + 2040x^{6} + 3780x^{4} + 2592x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{6}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + (\beta_{8} + \beta_{2} + 2 \beta_1) q^{3} + (\beta_{11} - 3 \beta_{8} + \cdots - 9 \beta_1) q^{7} + (2 \beta_{10} - \beta_{9} + \cdots - 30 \beta_{3}) q^{9} + (\beta_{10} - 2 \beta_{9} - 4 \beta_{6} + \cdots + 45) q^{11}+ \cdots + ( - 9 \beta_{10} - 9 \beta_{9} + \cdots - 1350) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 180 q^{9} + 270 q^{11} + 1494 q^{19} + 4338 q^{21} + 1080 q^{29} - 10710 q^{31} - 13176 q^{39} - 16860 q^{49} - 1782 q^{51} + 5886 q^{59} + 8082 q^{61} + 4536 q^{71} + 15546 q^{79} - 14958 q^{81}+ \cdots - 16200 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 36x^{10} + 432x^{8} + 2040x^{6} + 3780x^{4} + 2592x^{2} + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{11} + 104\nu^{9} + 1176\nu^{7} + 5112\nu^{5} + 9564\nu^{3} + 9360\nu ) / 4032 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3 \nu^{11} - 69 \nu^{10} - 104 \nu^{9} - 2448 \nu^{8} - 1176 \nu^{7} - 28560 \nu^{6} + \cdots - 76176 ) / 8064 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{11} + 320\nu^{9} + 3744\nu^{7} + 16632\nu^{5} + 25956\nu^{3} + 9936\nu + 384 ) / 768 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{10} - 386\nu^{8} - 4410\nu^{6} - 18744\nu^{4} - 27816\nu^{2} - 11808 ) / 168 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2 \nu^{11} + 3 \nu^{10} + 71 \nu^{9} + 108 \nu^{8} + 828 \nu^{7} + 1290 \nu^{6} + 3654 \nu^{5} + \cdots + 4032 ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2 \nu^{11} + 3 \nu^{10} - 71 \nu^{9} + 108 \nu^{8} - 828 \nu^{7} + 1290 \nu^{6} - 3654 \nu^{5} + \cdots + 4032 ) / 48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -89\nu^{10} - 3132\nu^{8} - 35868\nu^{6} - 151656\nu^{4} - 211212\nu^{2} - 66672 ) / 336 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 34 \nu^{11} - 11 \nu^{10} + 1209 \nu^{9} - 386 \nu^{8} + 14154 \nu^{7} - 4410 \nu^{6} + 63102 \nu^{5} + \cdots - 11808 ) / 336 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29 \nu^{11} - 60 \nu^{10} - 1020 \nu^{9} - 2112 \nu^{8} - 11688 \nu^{7} - 24264 \nu^{6} + \cdots - 51840 ) / 192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29 \nu^{11} - 60 \nu^{10} + 1020 \nu^{9} - 2112 \nu^{8} + 11688 \nu^{7} - 24264 \nu^{6} + 49632 \nu^{5} + \cdots - 51840 ) / 192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 255 \nu^{11} - 89 \nu^{10} + 8994 \nu^{9} - 3132 \nu^{8} + 103656 \nu^{7} - 35868 \nu^{6} + \cdots - 66672 ) / 672 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{11} - 2\beta_{10} + 2\beta_{9} - 4\beta_{8} - \beta_{7} - 3\beta_{6} + 3\beta_{5} + 2\beta_{4} ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - \beta_{9} + 3\beta_{7} + 5\beta_{6} + 5\beta_{5} - 15\beta_{4} + 168\beta_{2} + 84\beta _1 - 252 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 10 \beta_{11} + 9 \beta_{10} - 9 \beta_{9} + 6 \beta_{8} + 5 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + \cdots - 28 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14 \beta_{10} + 14 \beta_{9} - 27 \beta_{7} - 42 \beta_{6} - 42 \beta_{5} + 93 \beta_{4} - 1512 \beta_{2} + \cdots + 1512 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 76 \beta_{11} - 67 \beta_{10} + 67 \beta_{9} + 2 \beta_{8} - 38 \beta_{7} - 20 \beta_{6} + 20 \beta_{5} + \cdots + 420 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 93 \beta_{10} - 93 \beta_{9} + 165 \beta_{7} + 213 \beta_{6} + 213 \beta_{5} - 405 \beta_{4} + \cdots - 7140 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1178 \beta_{11} + 1027 \beta_{10} - 1027 \beta_{9} - 430 \beta_{8} + 589 \beta_{7} + 459 \beta_{6} + \cdots - 8820 ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1694 \beta_{10} + 1694 \beta_{9} - 2952 \beta_{7} - 3178 \beta_{6} - 3178 \beta_{5} + 5688 \beta_{4} + \cdots + 107352 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 18540 \beta_{11} - 16092 \beta_{10} + 16092 \beta_{9} + 10800 \beta_{8} - 9270 \beta_{7} - 9774 \beta_{6} + \cdots + 164976 ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 29690 \beta_{10} - 29690 \beta_{9} + 51510 \beta_{7} + 47874 \beta_{6} + 47874 \beta_{5} + \cdots - 1664712 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 295548 \beta_{11} + 256134 \beta_{10} - 256134 \beta_{9} - 216732 \beta_{8} + 147774 \beta_{7} + \cdots - 2932776 ) / 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
549.1
0.689786i
− 1.42101i
− 0.802567i
− 4.06501i
3.37523i
2.22358i
− 0.689786i
1.42101i
0.802567i
4.06501i
− 3.37523i
− 2.22358i
0 −7.89268 + 13.6705i 0 0 0 −29.2993 − 39.2754i 0 −84.0888 − 145.646i 0
549.2 0 −4.50998 + 7.81152i 0 0 0 5.65972 − 48.6720i 0 −0.179888 − 0.311574i 0
549.3 0 −0.784623 + 1.35901i 0 0 0 −24.5667 + 42.3967i 0 39.2687 + 68.0154i 0
549.4 0 0.784623 − 1.35901i 0 0 0 24.5667 − 42.3967i 0 39.2687 + 68.0154i 0
549.5 0 4.50998 − 7.81152i 0 0 0 −5.65972 + 48.6720i 0 −0.179888 − 0.311574i 0
549.6 0 7.89268 − 13.6705i 0 0 0 29.2993 + 39.2754i 0 −84.0888 − 145.646i 0
649.1 0 −7.89268 − 13.6705i 0 0 0 −29.2993 + 39.2754i 0 −84.0888 + 145.646i 0
649.2 0 −4.50998 − 7.81152i 0 0 0 5.65972 + 48.6720i 0 −0.179888 + 0.311574i 0
649.3 0 −0.784623 − 1.35901i 0 0 0 −24.5667 − 42.3967i 0 39.2687 − 68.0154i 0
649.4 0 0.784623 + 1.35901i 0 0 0 24.5667 + 42.3967i 0 39.2687 − 68.0154i 0
649.5 0 4.50998 + 7.81152i 0 0 0 −5.65972 − 48.6720i 0 −0.179888 + 0.311574i 0
649.6 0 7.89268 + 13.6705i 0 0 0 29.2993 − 39.2754i 0 −84.0888 + 145.646i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 549.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.5.o.a 12
5.b even 2 1 inner 700.5.o.a 12
5.c odd 4 1 28.5.h.a ✓ 6
5.c odd 4 1 700.5.s.a 6
7.d odd 6 1 inner 700.5.o.a 12
15.e even 4 1 252.5.z.f 6
20.e even 4 1 112.5.s.c 6
35.f even 4 1 196.5.h.c 6
35.i odd 6 1 inner 700.5.o.a 12
35.k even 12 1 28.5.h.a ✓ 6
35.k even 12 1 196.5.b.a 6
35.k even 12 1 700.5.s.a 6
35.l odd 12 1 196.5.b.a 6
35.l odd 12 1 196.5.h.c 6
105.w odd 12 1 252.5.z.f 6
140.w even 12 1 784.5.c.e 6
140.x odd 12 1 112.5.s.c 6
140.x odd 12 1 784.5.c.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.5.h.a ✓ 6 5.c odd 4 1
28.5.h.a ✓ 6 35.k even 12 1
112.5.s.c 6 20.e even 4 1
112.5.s.c 6 140.x odd 12 1
196.5.b.a 6 35.k even 12 1
196.5.b.a 6 35.l odd 12 1
196.5.h.c 6 35.f even 4 1
196.5.h.c 6 35.l odd 12 1
252.5.z.f 6 15.e even 4 1
252.5.z.f 6 105.w odd 12 1
700.5.o.a 12 1.a even 1 1 trivial
700.5.o.a 12 5.b even 2 1 inner
700.5.o.a 12 7.d odd 6 1 inner
700.5.o.a 12 35.i odd 6 1 inner
700.5.s.a 6 5.c odd 4 1
700.5.s.a 6 35.k even 12 1
784.5.c.e 6 140.w even 12 1
784.5.c.e 6 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 333T_{3}^{10} + 89802T_{3}^{8} + 6922125T_{3}^{6} + 428037210T_{3}^{4} + 1052726301T_{3}^{2} + 2492305929 \) acting on \(S_{5}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 2492305929 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{6} - 135 T^{5} + \cdots + 1920280041)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 65933832204288)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 75\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 163848759776883)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 34\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{3} - 270 T^{2} + \cdots - 401089968)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 81\!\cdots\!63)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 38\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 30\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 85\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 14\!\cdots\!29 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 40\!\cdots\!87)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 15\!\cdots\!47)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 92\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{3} - 1134 T^{2} + \cdots + 66080643048)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 26\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 32\!\cdots\!81)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 21\!\cdots\!72)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 12\!\cdots\!07)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 22\!\cdots\!32)^{2} \) Copy content Toggle raw display
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