Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,3,Mod(148,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.148");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.g (of order \(4\), degree \(2\), 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: | \(6.67576647683\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4 x^{11} + 8 x^{10} + 8 x^{9} + 70 x^{8} - 248 x^{7} + 464 x^{6} + 432 x^{5} + 1129 x^{4} + \cdots + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4 x^{11} + 8 x^{10} + 8 x^{9} + 70 x^{8} - 248 x^{7} + 464 x^{6} + 432 x^{5} + 1129 x^{4} + \cdots + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 216885938 \nu^{11} + 1060104974 \nu^{10} - 3418831332 \nu^{9} + 4776047991 \nu^{8} + \cdots + 9734414919250 ) / 2042321889134 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 650167512 \nu^{11} - 2167388474 \nu^{10} + 4094056596 \nu^{9} + 5876565360 \nu^{8} + \cdots + 412177775220 ) / 2042321889134 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 20608888761 \nu^{11} + 85686392604 \nu^{10} - 175708052458 \nu^{9} + \cdots - 8569191768820 ) / 10211609445670 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 62789472393 \nu^{11} + 265477896952 \nu^{10} - 559679834849 \nu^{9} + \cdots + 4818810061550 ) / 20423218891340 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 137676730578 \nu^{11} + 596855018097 \nu^{10} - 1287825684554 \nu^{9} + \cdots + 10093227434800 ) / 40846437782680 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 150422124598 \nu^{11} - 616051382677 \nu^{10} + 1215756736104 \nu^{9} + \cdots + 126398845752920 ) / 40846437782680 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 86748937073 \nu^{11} + 338454191902 \nu^{10} - 673792264554 \nu^{9} + \cdots - 70443542020170 ) / 20423218891340 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 360571142916 \nu^{11} + 1533377436449 \nu^{10} - 3341673499113 \nu^{9} + \cdots + 64434080701620 ) / 40846437782680 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 19958721249 \nu^{11} - 83519004130 \nu^{10} + 171613995862 \nu^{9} + 138524261748 \nu^{8} + \cdots + 8157013993600 ) / 2042321889134 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 448601700821 \nu^{11} - 1754253899729 \nu^{10} + 3449654172798 \nu^{9} + \cdots + 409254215744460 ) / 40846437782680 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} + 5\beta_{4} + \beta_{3} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} + 2\beta_{8} + 2\beta_{4} + 9\beta_{3} + \beta_{2} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{8} - 2\beta_{7} - 2\beta_{6} - 2\beta_{5} + 13\beta_{3} + 11\beta_{2} - 13\beta _1 - 41 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 15 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 6 \beta_{6} - 28 \beta_{5} - 38 \beta_{4} + \cdots - 42 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 8 \beta_{11} - 115 \beta_{10} + 8 \beta_{9} - 44 \beta_{8} + 32 \beta_{7} - 32 \beta_{6} + \cdots - 159 \beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 80 \beta_{11} - 195 \beta_{10} - 342 \beta_{8} + 136 \beta_{7} + 40 \beta_{6} - 40 \beta_{5} + \cdots + 630 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 176 \beta_{11} - 176 \beta_{9} - 526 \beta_{8} + 598 \beta_{7} + 598 \beta_{6} + 526 \beta_{5} + \cdots + 4401 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2451 \beta_{10} - 1196 \beta_{9} + 598 \beta_{8} + 598 \beta_{7} + 2178 \beta_{6} + 4112 \beta_{5} + \cdots + 8450 \)
|
| \(\nu^{10}\) | \(=\) |
\( 2776 \beta_{11} + 14083 \beta_{10} - 2776 \beta_{9} + 9856 \beta_{8} - 5308 \beta_{7} + \cdots + 23323 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( 16168 \beta_{11} + 30403 \beta_{10} + 49642 \beta_{8} - 30572 \beta_{7} - 8084 \beta_{6} + 8084 \beta_{5} + \cdots - 108230 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(1\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 148.1 |
|
−2.47480 | + | 2.47480i | −2.98009 | − | 2.98009i | − | 8.24929i | 4.25027 | − | 2.63347i | 14.7503 | 0 | 10.5161 | + | 10.5161i | 8.76185i | −4.00127 | + | 17.0359i | |||||||||||||||||||||||||||||||||||||||||||
| 148.2 | −1.90845 | + | 1.90845i | 3.30412 | + | 3.30412i | − | 3.28438i | 3.50673 | + | 3.56410i | −12.6115 | 0 | −1.36573 | − | 1.36573i | 12.8345i | −13.4943 | − | 0.109487i | ||||||||||||||||||||||||||||||||||||||||||||
| 148.3 | −0.950261 | + | 0.950261i | −0.269488 | − | 0.269488i | 2.19401i | −4.00169 | − | 2.99774i | 0.512169 | 0 | −5.88593 | − | 5.88593i | − | 8.85475i | 6.65129 | − | 0.954016i | ||||||||||||||||||||||||||||||||||||||||||||
| 148.4 | 0.109772 | − | 0.109772i | −1.67281 | − | 1.67281i | 3.97590i | 0.861142 | + | 4.92529i | −0.367256 | 0 | 0.875529 | + | 0.875529i | − | 3.40339i | 0.635186 | + | 0.446128i | ||||||||||||||||||||||||||||||||||||||||||||
| 148.5 | 1.36503 | − | 1.36503i | 3.78207 | + | 3.78207i | 0.273387i | −4.98225 | − | 0.420986i | 10.3253 | 0 | 5.83330 | + | 5.83330i | 19.6082i | −7.37557 | + | 6.22626i | |||||||||||||||||||||||||||||||||||||||||||||
| 148.6 | 1.85871 | − | 1.85871i | −0.163806 | − | 0.163806i | − | 2.90963i | 4.36579 | − | 2.43719i | −0.608937 | 0 | 2.02669 | + | 2.02669i | − | 8.94634i | 3.58471 | − | 12.6448i | |||||||||||||||||||||||||||||||||||||||||||
| 197.1 | −2.47480 | − | 2.47480i | −2.98009 | + | 2.98009i | 8.24929i | 4.25027 | + | 2.63347i | 14.7503 | 0 | 10.5161 | − | 10.5161i | − | 8.76185i | −4.00127 | − | 17.0359i | ||||||||||||||||||||||||||||||||||||||||||||
| 197.2 | −1.90845 | − | 1.90845i | 3.30412 | − | 3.30412i | 3.28438i | 3.50673 | − | 3.56410i | −12.6115 | 0 | −1.36573 | + | 1.36573i | − | 12.8345i | −13.4943 | + | 0.109487i | ||||||||||||||||||||||||||||||||||||||||||||
| 197.3 | −0.950261 | − | 0.950261i | −0.269488 | + | 0.269488i | − | 2.19401i | −4.00169 | + | 2.99774i | 0.512169 | 0 | −5.88593 | + | 5.88593i | 8.85475i | 6.65129 | + | 0.954016i | ||||||||||||||||||||||||||||||||||||||||||||
| 197.4 | 0.109772 | + | 0.109772i | −1.67281 | + | 1.67281i | − | 3.97590i | 0.861142 | − | 4.92529i | −0.367256 | 0 | 0.875529 | − | 0.875529i | 3.40339i | 0.635186 | − | 0.446128i | ||||||||||||||||||||||||||||||||||||||||||||
| 197.5 | 1.36503 | + | 1.36503i | 3.78207 | − | 3.78207i | − | 0.273387i | −4.98225 | + | 0.420986i | 10.3253 | 0 | 5.83330 | − | 5.83330i | − | 19.6082i | −7.37557 | − | 6.22626i | |||||||||||||||||||||||||||||||||||||||||||
| 197.6 | 1.85871 | + | 1.85871i | −0.163806 | + | 0.163806i | 2.90963i | 4.36579 | + | 2.43719i | −0.608937 | 0 | 2.02669 | − | 2.02669i | 8.94634i | 3.58471 | + | 12.6448i | |||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 245.3.g.a | 12 | |
| 5.c | odd | 4 | 1 | inner | 245.3.g.a | 12 | |
| 7.b | odd | 2 | 1 | 35.3.g.a | ✓ | 12 | |
| 7.c | even | 3 | 2 | 245.3.m.c | 24 | ||
| 7.d | odd | 6 | 2 | 245.3.m.d | 24 | ||
| 21.c | even | 2 | 1 | 315.3.o.a | 12 | ||
| 28.d | even | 2 | 1 | 560.3.bh.e | 12 | ||
| 35.c | odd | 2 | 1 | 175.3.g.b | 12 | ||
| 35.f | even | 4 | 1 | 35.3.g.a | ✓ | 12 | |
| 35.f | even | 4 | 1 | 175.3.g.b | 12 | ||
| 35.k | even | 12 | 2 | 245.3.m.d | 24 | ||
| 35.l | odd | 12 | 2 | 245.3.m.c | 24 | ||
| 105.k | odd | 4 | 1 | 315.3.o.a | 12 | ||
| 140.j | odd | 4 | 1 | 560.3.bh.e | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.3.g.a | ✓ | 12 | 7.b | odd | 2 | 1 | |
| 35.3.g.a | ✓ | 12 | 35.f | even | 4 | 1 | |
| 175.3.g.b | 12 | 35.c | odd | 2 | 1 | ||
| 175.3.g.b | 12 | 35.f | even | 4 | 1 | ||
| 245.3.g.a | 12 | 1.a | even | 1 | 1 | trivial | |
| 245.3.g.a | 12 | 5.c | odd | 4 | 1 | inner | |
| 245.3.m.c | 24 | 7.c | even | 3 | 2 | ||
| 245.3.m.c | 24 | 35.l | odd | 12 | 2 | ||
| 245.3.m.d | 24 | 7.d | odd | 6 | 2 | ||
| 245.3.m.d | 24 | 35.k | even | 12 | 2 | ||
| 315.3.o.a | 12 | 21.c | even | 2 | 1 | ||
| 315.3.o.a | 12 | 105.k | odd | 4 | 1 | ||
| 560.3.bh.e | 12 | 28.d | even | 2 | 1 | ||
| 560.3.bh.e | 12 | 140.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(245, [\chi])\):
|
\( T_{2}^{12} + 4 T_{2}^{11} + 8 T_{2}^{10} - 8 T_{2}^{9} + 70 T_{2}^{8} + 248 T_{2}^{7} + 464 T_{2}^{6} + \cdots + 100 \)
|
|
\( T_{3}^{12} - 4 T_{3}^{11} + 8 T_{3}^{10} + 68 T_{3}^{9} + 517 T_{3}^{8} - 1192 T_{3}^{7} + 2944 T_{3}^{6} + \cdots + 484 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 4 T^{11} + \cdots + 100 \)
|
| $3$ |
\( T^{12} - 4 T^{11} + \cdots + 484 \)
|
| $5$ |
\( T^{12} + \cdots + 244140625 \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( (T^{6} + 6 T^{5} + \cdots + 255536)^{2} \)
|
| $13$ |
\( T^{12} + \cdots + 7014062500 \)
|
| $17$ |
\( T^{12} + \cdots + 70807081216 \)
|
| $19$ |
\( T^{12} + \cdots + 2061234490000 \)
|
| $23$ |
\( T^{12} + \cdots + 133532630919424 \)
|
| $29$ |
\( T^{12} + \cdots + 196000000 \)
|
| $31$ |
\( (T^{6} - 48 T^{5} + \cdots + 116022400)^{2} \)
|
| $37$ |
\( T^{12} + \cdots + 37\!\cdots\!64 \)
|
| $41$ |
\( (T^{6} - 104 T^{5} + \cdots + 38452576)^{2} \)
|
| $43$ |
\( T^{12} + \cdots + 159385575040000 \)
|
| $47$ |
\( T^{12} + \cdots + 11\!\cdots\!64 \)
|
| $53$ |
\( T^{12} + \cdots + 91\!\cdots\!76 \)
|
| $59$ |
\( T^{12} + \cdots + 69\!\cdots\!00 \)
|
| $61$ |
\( (T^{6} + 140 T^{5} + \cdots + 35862860384)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 72\!\cdots\!16 \)
|
| $71$ |
\( (T^{6} - 72 T^{5} + \cdots + 15911820320)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 14\!\cdots\!24 \)
|
| $79$ |
\( T^{12} + \cdots + 89\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots + 20\!\cdots\!96 \)
|
| $89$ |
\( T^{12} + \cdots + 87\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots + 36\!\cdots\!96 \)
|
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