Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [700,2,Mod(157,700)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(700, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("700.157");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 700.bc (of order \(12\), degree \(4\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(5.58952814149\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 802 x^{12} - 2264 x^{11} + 5402 x^{10} - 10642 x^{9} + \cdots + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 140) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 802 x^{12} - 2264 x^{11} + 5402 x^{10} - 10642 x^{9} + \cdots + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu + 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 604 \nu^{15} + 29211 \nu^{14} - 215333 \nu^{13} + 1327712 \nu^{12} - 5547843 \nu^{11} + \cdots + 26970188 ) / 933548 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 51 \nu^{14} - 357 \nu^{13} + 2190 \nu^{12} - 8499 \nu^{11} + 27923 \nu^{10} - 70216 \nu^{9} + \cdots + 5992 ) / 196 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 921 \nu^{15} + 17949 \nu^{14} - 126600 \nu^{13} + 689883 \nu^{12} - 2686219 \nu^{11} + \cdots + 3222296 ) / 84868 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 24739 \nu^{15} + 163243 \nu^{14} - 976832 \nu^{13} + 3587415 \nu^{12} - 11211849 \nu^{11} + \cdots - 9046016 ) / 933548 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2249 \nu^{15} - 5826 \nu^{14} + 25703 \nu^{13} + 60501 \nu^{12} - 480220 \nu^{11} + 2553517 \nu^{10} + \cdots + 1543192 ) / 84868 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 24739 \nu^{15} + 207842 \nu^{14} - 1289025 \nu^{13} + 5512533 \nu^{12} - 18704048 \nu^{11} + \cdots + 1768592 ) / 933548 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 25343 \nu^{15} - 143092 \nu^{14} + 846631 \nu^{13} - 2675651 \nu^{12} + 7814778 \nu^{11} + \cdots + 8093932 ) / 933548 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 44374 \nu^{15} - 244473 \nu^{14} + 1553681 \nu^{13} - 5234976 \nu^{12} + 17420997 \nu^{11} + \cdots - 21177296 ) / 933548 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 4698 \nu^{15} + 45627 \nu^{14} - 297041 \nu^{13} + 1376018 \nu^{12} - 4971131 \nu^{11} + \cdots + 2750916 ) / 84868 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 59085 \nu^{15} - 321681 \nu^{14} + 2007692 \nu^{13} - 6639865 \nu^{12} + 21512783 \nu^{11} + \cdots - 5605768 ) / 933548 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 69113 \nu^{15} - 485223 \nu^{14} + 3073062 \nu^{13} - 12240493 \nu^{12} + 42088321 \nu^{11} + \cdots - 14774312 ) / 933548 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 852 \nu^{15} - 6390 \nu^{14} + 40676 \nu^{13} - 167479 \nu^{12} + 580296 \nu^{11} - 1562572 \nu^{10} + \cdots - 175506 ) / 9526 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 86329 \nu^{15} - 637292 \nu^{14} + 4122399 \nu^{13} - 16915591 \nu^{12} + 59542930 \nu^{11} + \cdots - 25063192 ) / 933548 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 86329 \nu^{15} + 657643 \nu^{14} - 4264856 \nu^{13} + 17961719 \nu^{12} - 63967757 \nu^{11} + \cdots + 39442256 ) / 933548 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{3} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{3} + \beta _1 - 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{15} - \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} + 2 \beta_{9} + \beta_{8} + \cdots - 9 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4 \beta_{15} - 4 \beta_{13} + 10 \beta_{12} + 8 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 9 \beta_{7} + \cdots + 17 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 10 \beta_{14} + 20 \beta_{13} - 2 \beta_{12} - 24 \beta_{11} + 10 \beta_{10} - 22 \beta_{9} - 10 \beta_{8} + \cdots + 71 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 29 \beta_{15} + 11 \beta_{14} + 70 \beta_{13} - 104 \beta_{12} - 92 \beta_{11} - 34 \beta_{10} + \cdots - 33 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 32 \beta_{15} - 66 \beta_{14} - 116 \beta_{13} - 114 \beta_{12} + 94 \beta_{11} - 154 \beta_{10} + \cdots - 519 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 168 \beta_{15} - 164 \beta_{14} - 800 \beta_{13} + 800 \beta_{12} + 824 \beta_{11} + 148 \beta_{10} + \cdots - 331 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 357 \beta_{15} + 291 \beta_{14} + 156 \beta_{13} + 1892 \beta_{12} + 76 \beta_{11} + 1632 \beta_{10} + \cdots + 3427 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 898 \beta_{15} + 1562 \beta_{14} + 7290 \beta_{13} - 4784 \beta_{12} - 6484 \beta_{11} + 520 \beta_{10} + \cdots + 6173 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 3180 \beta_{15} - 296 \beta_{14} + 6610 \beta_{13} - 21084 \beta_{12} - 7034 \beta_{11} + \cdots - 19277 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 4259 \beta_{15} - 11921 \beta_{14} - 55336 \beta_{13} + 17646 \beta_{12} + 44872 \beta_{11} + \cdots - 67705 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 26642 \beta_{15} - 11380 \beta_{14} - 115110 \beta_{13} + 193012 \beta_{12} + 100934 \beta_{11} + \cdots + 77217 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 12568 \beta_{15} + 76802 \beta_{14} + 341246 \beta_{13} + 55446 \beta_{12} - 259102 \beta_{11} + \cdots + 606325 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 214359 \beta_{15} + 170355 \beta_{14} + 1314998 \beta_{13} - 1522274 \beta_{12} - 1067652 \beta_{11} + \cdots + 26483 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(351\) | \(477\) |
| \(\chi(n)\) | \(1 + \beta_{13}\) | \(1\) | \(\beta_{5} + \beta_{7}\) |
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 157.1 |
|
0 | −2.42519 | + | 0.649827i | 0 | 0 | 0 | −2.59537 | + | 0.513853i | 0 | 2.86119 | − | 1.65191i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 157.2 | 0 | −1.52691 | + | 0.409133i | 0 | 0 | 0 | 2.59572 | + | 0.512081i | 0 | −0.434025 | + | 0.250584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 157.3 | 0 | 0.827625 | − | 0.221762i | 0 | 0 | 0 | −2.22829 | − | 1.42643i | 0 | −1.96229 | + | 1.13293i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 157.4 | 0 | 3.12447 | − | 0.837199i | 0 | 0 | 0 | −0.870132 | + | 2.49857i | 0 | 6.46333 | − | 3.73161i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.1 | 0 | −0.649827 | + | 2.42519i | 0 | 0 | 0 | 0.513853 | − | 2.59537i | 0 | −2.86119 | − | 1.65191i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.2 | 0 | −0.409133 | + | 1.52691i | 0 | 0 | 0 | 0.512081 | + | 2.59572i | 0 | 0.434025 | + | 0.250584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.3 | 0 | 0.221762 | − | 0.827625i | 0 | 0 | 0 | −1.42643 | − | 2.22829i | 0 | 1.96229 | + | 1.13293i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.4 | 0 | 0.837199 | − | 3.12447i | 0 | 0 | 0 | 2.49857 | − | 0.870132i | 0 | −6.46333 | − | 3.73161i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 493.1 | 0 | −0.649827 | − | 2.42519i | 0 | 0 | 0 | 0.513853 | + | 2.59537i | 0 | −2.86119 | + | 1.65191i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 493.2 | 0 | −0.409133 | − | 1.52691i | 0 | 0 | 0 | 0.512081 | − | 2.59572i | 0 | 0.434025 | − | 0.250584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 493.3 | 0 | 0.221762 | + | 0.827625i | 0 | 0 | 0 | −1.42643 | + | 2.22829i | 0 | 1.96229 | − | 1.13293i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 493.4 | 0 | 0.837199 | + | 3.12447i | 0 | 0 | 0 | 2.49857 | + | 0.870132i | 0 | −6.46333 | + | 3.73161i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 593.1 | 0 | −2.42519 | − | 0.649827i | 0 | 0 | 0 | −2.59537 | − | 0.513853i | 0 | 2.86119 | + | 1.65191i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 593.2 | 0 | −1.52691 | − | 0.409133i | 0 | 0 | 0 | 2.59572 | − | 0.512081i | 0 | −0.434025 | − | 0.250584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 593.3 | 0 | 0.827625 | + | 0.221762i | 0 | 0 | 0 | −2.22829 | + | 1.42643i | 0 | −1.96229 | − | 1.13293i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 593.4 | 0 | 3.12447 | + | 0.837199i | 0 | 0 | 0 | −0.870132 | − | 2.49857i | 0 | 6.46333 | + | 3.73161i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 35.k | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 700.2.bc.b | 16 | |
| 5.b | even | 2 | 1 | 140.2.u.a | ✓ | 16 | |
| 5.c | odd | 4 | 1 | 140.2.u.a | ✓ | 16 | |
| 5.c | odd | 4 | 1 | inner | 700.2.bc.b | 16 | |
| 7.d | odd | 6 | 1 | inner | 700.2.bc.b | 16 | |
| 15.d | odd | 2 | 1 | 1260.2.dq.a | 16 | ||
| 15.e | even | 4 | 1 | 1260.2.dq.a | 16 | ||
| 20.d | odd | 2 | 1 | 560.2.ci.d | 16 | ||
| 20.e | even | 4 | 1 | 560.2.ci.d | 16 | ||
| 35.c | odd | 2 | 1 | 980.2.v.a | 16 | ||
| 35.f | even | 4 | 1 | 980.2.v.a | 16 | ||
| 35.i | odd | 6 | 1 | 140.2.u.a | ✓ | 16 | |
| 35.i | odd | 6 | 1 | 980.2.m.a | 16 | ||
| 35.j | even | 6 | 1 | 980.2.m.a | 16 | ||
| 35.j | even | 6 | 1 | 980.2.v.a | 16 | ||
| 35.k | even | 12 | 1 | 140.2.u.a | ✓ | 16 | |
| 35.k | even | 12 | 1 | inner | 700.2.bc.b | 16 | |
| 35.k | even | 12 | 1 | 980.2.m.a | 16 | ||
| 35.l | odd | 12 | 1 | 980.2.m.a | 16 | ||
| 35.l | odd | 12 | 1 | 980.2.v.a | 16 | ||
| 105.p | even | 6 | 1 | 1260.2.dq.a | 16 | ||
| 105.w | odd | 12 | 1 | 1260.2.dq.a | 16 | ||
| 140.s | even | 6 | 1 | 560.2.ci.d | 16 | ||
| 140.x | odd | 12 | 1 | 560.2.ci.d | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.2.u.a | ✓ | 16 | 5.b | even | 2 | 1 | |
| 140.2.u.a | ✓ | 16 | 5.c | odd | 4 | 1 | |
| 140.2.u.a | ✓ | 16 | 35.i | odd | 6 | 1 | |
| 140.2.u.a | ✓ | 16 | 35.k | even | 12 | 1 | |
| 560.2.ci.d | 16 | 20.d | odd | 2 | 1 | ||
| 560.2.ci.d | 16 | 20.e | even | 4 | 1 | ||
| 560.2.ci.d | 16 | 140.s | even | 6 | 1 | ||
| 560.2.ci.d | 16 | 140.x | odd | 12 | 1 | ||
| 700.2.bc.b | 16 | 1.a | even | 1 | 1 | trivial | |
| 700.2.bc.b | 16 | 5.c | odd | 4 | 1 | inner | |
| 700.2.bc.b | 16 | 7.d | odd | 6 | 1 | inner | |
| 700.2.bc.b | 16 | 35.k | even | 12 | 1 | inner | |
| 980.2.m.a | 16 | 35.i | odd | 6 | 1 | ||
| 980.2.m.a | 16 | 35.j | even | 6 | 1 | ||
| 980.2.m.a | 16 | 35.k | even | 12 | 1 | ||
| 980.2.m.a | 16 | 35.l | odd | 12 | 1 | ||
| 980.2.v.a | 16 | 35.c | odd | 2 | 1 | ||
| 980.2.v.a | 16 | 35.f | even | 4 | 1 | ||
| 980.2.v.a | 16 | 35.j | even | 6 | 1 | ||
| 980.2.v.a | 16 | 35.l | odd | 12 | 1 | ||
| 1260.2.dq.a | 16 | 15.d | odd | 2 | 1 | ||
| 1260.2.dq.a | 16 | 15.e | even | 4 | 1 | ||
| 1260.2.dq.a | 16 | 105.p | even | 6 | 1 | ||
| 1260.2.dq.a | 16 | 105.w | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 78 T_{3}^{12} - 120 T_{3}^{11} + 1068 T_{3}^{9} + 6443 T_{3}^{8} + 9360 T_{3}^{7} + \cdots + 14641 \)
acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} - 78 T^{12} + \cdots + 14641 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + 2 T^{15} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{8} + 13 T^{6} + \cdots + 100)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 268435456 \)
|
| $17$ |
\( T^{16} + 18 T^{15} + \cdots + 256 \)
|
| $19$ |
\( T^{16} + \cdots + 4711998736 \)
|
| $23$ |
\( T^{16} - 16 T^{15} + \cdots + 7890481 \)
|
| $29$ |
\( (T^{8} + 162 T^{6} + \cdots + 16384)^{2} \)
|
| $31$ |
\( (T^{8} + 6 T^{7} + \cdots + 414736)^{2} \)
|
| $37$ |
\( T^{16} - 14 T^{15} + \cdots + 4096 \)
|
| $41$ |
\( (T^{8} + 170 T^{6} + \cdots + 12544)^{2} \)
|
| $43$ |
\( (T^{8} + 14 T^{7} + \cdots + 10432900)^{2} \)
|
| $47$ |
\( T^{16} - 6 T^{15} + \cdots + 10000 \)
|
| $53$ |
\( T^{16} + \cdots + 35477982736 \)
|
| $59$ |
\( T^{16} + \cdots + 6263627420176 \)
|
| $61$ |
\( (T^{8} - 30 T^{7} + \cdots + 4092529)^{2} \)
|
| $67$ |
\( T^{16} + 8 T^{15} + \cdots + 2401 \)
|
| $71$ |
\( (T^{4} + 2 T^{3} + \cdots + 448)^{4} \)
|
| $73$ |
\( T^{16} + \cdots + 3841600000000 \)
|
| $79$ |
\( T^{16} + \cdots + 3647809685776 \)
|
| $83$ |
\( T^{16} + \cdots + 5006411536 \)
|
| $89$ |
\( T^{16} + \cdots + 33243864241 \)
|
| $97$ |
\( T^{16} + \cdots + 47698139955456 \)
|
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