Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [450,6,Mod(107,450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(450, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("450.107");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 450.f (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: | \(72.1727189158\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 252 x^{14} + 27174 x^{12} - 1635700 x^{10} + 60061815 x^{8} - 1376564028 x^{6} + \cdots + 498214340649 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{8}\cdot 5^{16} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 252 x^{14} + 27174 x^{12} - 1635700 x^{10} + 60061815 x^{8} - 1376564028 x^{6} + \cdots + 498214340649 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 1243 \nu^{14} - 299263 \nu^{12} + 29877040 \nu^{10} - 1593636821 \nu^{8} + \cdots - 29261576101626 ) / 112420373040 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 24977 \nu^{14} - 5856811 \nu^{12} + 573008898 \nu^{10} - 30267472987 \nu^{8} + \cdots - 629954627005836 ) / 19448681904 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 163510355 \nu^{14} - 40270160735 \nu^{12} + 4115640904256 \nu^{10} - 225374410231405 \nu^{8} + \cdots - 50\!\cdots\!54 ) / 51286174180848 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13005769 \nu^{15} - 2922257905 \nu^{13} + 274239495022 \nu^{11} - 13908239540510 \nu^{9} + \cdots - 25\!\cdots\!23 \nu ) / 34\!\cdots\!68 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4342753 \nu^{15} - 939402004 \nu^{13} + 83950770889 \nu^{11} - 4002387230393 \nu^{9} + \cdots - 51\!\cdots\!17 \nu ) / 10\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 35476130685 \nu^{15} - 763584094480 \nu^{14} + 8478443606160 \nu^{13} + \cdots + 11\!\cdots\!20 ) / 10\!\cdots\!24 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 35476130685 \nu^{15} - 763584094480 \nu^{14} - 8478443606160 \nu^{13} + \cdots + 11\!\cdots\!20 ) / 10\!\cdots\!24 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 2072048995 \nu^{15} + 19070701455 \nu^{14} - 491744708950 \nu^{13} - 4207239550965 \nu^{12} + \cdots - 29\!\cdots\!40 ) / 27\!\cdots\!44 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 2072048995 \nu^{15} + 19070701455 \nu^{14} + 491744708950 \nu^{13} + \cdots - 29\!\cdots\!40 ) / 27\!\cdots\!44 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 429018357807 \nu^{15} + 10599864475589 \nu^{14} - 93430716258888 \nu^{13} + \cdots - 19\!\cdots\!42 ) / 40\!\cdots\!96 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 429018357807 \nu^{15} - 10599864475589 \nu^{14} - 93430716258888 \nu^{13} + \cdots + 19\!\cdots\!42 ) / 40\!\cdots\!96 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 3041597975 \nu^{15} + 52730001315 \nu^{14} - 738765803630 \nu^{13} - 11534685140745 \nu^{12} + \cdots - 70\!\cdots\!00 ) / 27\!\cdots\!44 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 3041597975 \nu^{15} + 52730001315 \nu^{14} + 738765803630 \nu^{13} + \cdots - 70\!\cdots\!00 ) / 27\!\cdots\!44 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 11804228221 \nu^{15} + 2602717150930 \nu^{13} - 238555634297431 \nu^{11} + \cdots + 17\!\cdots\!71 \nu ) / 13\!\cdots\!72 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 710861580817 \nu^{15} + 156264690672860 \nu^{13} + \cdots + 10\!\cdots\!93 \nu ) / 44\!\cdots\!44 \)
|
| \(\nu\) | \(=\) |
\( ( -5\beta_{11} - 5\beta_{10} - 2\beta_{7} + 2\beta_{6} + 150\beta_{5} + 150\beta_{4} ) / 300 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 10 \beta_{13} + 10 \beta_{12} - 5 \beta_{11} + 5 \beta_{10} - 50 \beta_{9} - 50 \beta_{8} + \cdots + 9450 ) / 300 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 45 \beta_{15} - 15 \beta_{13} + 15 \beta_{12} - 160 \beta_{11} - 160 \beta_{10} + \cdots + 14250 \beta_{4} ) / 300 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 620 \beta_{13} + 620 \beta_{12} - 325 \beta_{11} + 325 \beta_{10} - 3136 \beta_{9} - 3136 \beta_{8} + \cdots + 343350 ) / 300 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2925 \beta_{15} + 450 \beta_{14} - 1575 \beta_{13} + 1575 \beta_{12} - 5870 \beta_{11} + \cdots + 882150 \beta_{4} ) / 300 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 6428 \beta_{13} + 6428 \beta_{12} - 3685 \beta_{11} + 3685 \beta_{10} - 31510 \beta_{9} + \cdots + 2749230 ) / 60 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 151380 \beta_{15} + 59850 \beta_{14} - 118020 \beta_{13} + 118020 \beta_{12} + \cdots + 51215700 \beta_{4} ) / 300 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 1593440 \beta_{13} + 1593440 \beta_{12} - 1030585 \beta_{11} + 1030585 \beta_{10} - 7401064 \beta_{9} + \cdots + 581929650 ) / 300 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 7255935 \beta_{15} + 5118930 \beta_{14} - 7885245 \beta_{13} + 7885245 \beta_{12} + \cdots + 2929711200 \beta_{4} ) / 300 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 76776310 \beta_{13} + 76776310 \beta_{12} - 57761270 \beta_{11} + 57761270 \beta_{10} + \cdots + 25247590200 ) / 300 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 333273195 \beta_{15} + 361958850 \beta_{14} - 495866745 \beta_{13} + 495866745 \beta_{12} + \cdots + 166290586950 \beta_{4} ) / 300 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 893550485 \beta_{13} + 893550485 \beta_{12} - 808648375 \beta_{11} + 808648375 \beta_{10} + \cdots + 273843706725 ) / 75 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 14730106860 \beta_{15} + 23086418940 \beta_{14} - 29871401820 \beta_{13} + \cdots + 9353669469450 \beta_{4} ) / 300 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 158932875970 \beta_{13} + 158932875970 \beta_{12} - 180043217735 \beta_{11} + 180043217735 \beta_{10} + \cdots + 46466673841350 ) / 300 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 620186330205 \beta_{15} + 1381665070800 \beta_{14} - 1738273372215 \beta_{13} + \cdots + 520115803250250 \beta_{4} ) / 300 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 107.1 |
|
−2.82843 | + | 2.82843i | 0 | − | 16.0000i | 0 | 0 | −117.170 | − | 117.170i | 45.2548 | + | 45.2548i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.2 | −2.82843 | + | 2.82843i | 0 | − | 16.0000i | 0 | 0 | −50.1584 | − | 50.1584i | 45.2548 | + | 45.2548i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.3 | −2.82843 | + | 2.82843i | 0 | − | 16.0000i | 0 | 0 | 140.653 | + | 140.653i | 45.2548 | + | 45.2548i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.4 | −2.82843 | + | 2.82843i | 0 | − | 16.0000i | 0 | 0 | 158.675 | + | 158.675i | 45.2548 | + | 45.2548i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.5 | 2.82843 | − | 2.82843i | 0 | − | 16.0000i | 0 | 0 | −117.170 | − | 117.170i | −45.2548 | − | 45.2548i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.6 | 2.82843 | − | 2.82843i | 0 | − | 16.0000i | 0 | 0 | −50.1584 | − | 50.1584i | −45.2548 | − | 45.2548i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.7 | 2.82843 | − | 2.82843i | 0 | − | 16.0000i | 0 | 0 | 140.653 | + | 140.653i | −45.2548 | − | 45.2548i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.8 | 2.82843 | − | 2.82843i | 0 | − | 16.0000i | 0 | 0 | 158.675 | + | 158.675i | −45.2548 | − | 45.2548i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.1 | −2.82843 | − | 2.82843i | 0 | 16.0000i | 0 | 0 | −117.170 | + | 117.170i | 45.2548 | − | 45.2548i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.2 | −2.82843 | − | 2.82843i | 0 | 16.0000i | 0 | 0 | −50.1584 | + | 50.1584i | 45.2548 | − | 45.2548i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.3 | −2.82843 | − | 2.82843i | 0 | 16.0000i | 0 | 0 | 140.653 | − | 140.653i | 45.2548 | − | 45.2548i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.4 | −2.82843 | − | 2.82843i | 0 | 16.0000i | 0 | 0 | 158.675 | − | 158.675i | 45.2548 | − | 45.2548i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.5 | 2.82843 | + | 2.82843i | 0 | 16.0000i | 0 | 0 | −117.170 | + | 117.170i | −45.2548 | + | 45.2548i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.6 | 2.82843 | + | 2.82843i | 0 | 16.0000i | 0 | 0 | −50.1584 | + | 50.1584i | −45.2548 | + | 45.2548i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.7 | 2.82843 | + | 2.82843i | 0 | 16.0000i | 0 | 0 | 140.653 | − | 140.653i | −45.2548 | + | 45.2548i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.8 | 2.82843 | + | 2.82843i | 0 | 16.0000i | 0 | 0 | 158.675 | − | 158.675i | −45.2548 | + | 45.2548i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 450.6.f.g | yes | 16 |
| 3.b | odd | 2 | 1 | inner | 450.6.f.g | yes | 16 |
| 5.b | even | 2 | 1 | 450.6.f.f | ✓ | 16 | |
| 5.c | odd | 4 | 1 | 450.6.f.f | ✓ | 16 | |
| 5.c | odd | 4 | 1 | inner | 450.6.f.g | yes | 16 |
| 15.d | odd | 2 | 1 | 450.6.f.f | ✓ | 16 | |
| 15.e | even | 4 | 1 | 450.6.f.f | ✓ | 16 | |
| 15.e | even | 4 | 1 | inner | 450.6.f.g | yes | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 450.6.f.f | ✓ | 16 | 5.b | even | 2 | 1 | |
| 450.6.f.f | ✓ | 16 | 5.c | odd | 4 | 1 | |
| 450.6.f.f | ✓ | 16 | 15.d | odd | 2 | 1 | |
| 450.6.f.f | ✓ | 16 | 15.e | even | 4 | 1 | |
| 450.6.f.g | yes | 16 | 1.a | even | 1 | 1 | trivial |
| 450.6.f.g | yes | 16 | 3.b | odd | 2 | 1 | inner |
| 450.6.f.g | yes | 16 | 5.c | odd | 4 | 1 | inner |
| 450.6.f.g | yes | 16 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 264 T_{7}^{7} + 34848 T_{7}^{6} + 3657240 T_{7}^{5} + 867535506 T_{7}^{4} + \cdots + 27\!\cdots\!81 \)
acting on \(S_{6}^{\mathrm{new}}(450, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 256)^{4} \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( (T^{8} + \cdots + 27\!\cdots\!81)^{2} \)
|
| $11$ |
\( (T^{8} + \cdots + 22\!\cdots\!56)^{2} \)
|
| $13$ |
\( (T^{8} + \cdots + 58\!\cdots\!61)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 26\!\cdots\!96 \)
|
| $19$ |
\( (T^{8} + \cdots + 50\!\cdots\!61)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 18\!\cdots\!96 \)
|
| $29$ |
\( (T^{8} + \cdots + 14\!\cdots\!56)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots - 32693228586959)^{4} \)
|
| $37$ |
\( (T^{8} + \cdots + 93\!\cdots\!56)^{2} \)
|
| $41$ |
\( (T^{8} + \cdots + 56\!\cdots\!76)^{2} \)
|
| $43$ |
\( (T^{8} + \cdots + 17\!\cdots\!25)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 26\!\cdots\!00 \)
|
| $53$ |
\( T^{16} + \cdots + 57\!\cdots\!76 \)
|
| $59$ |
\( (T^{8} + \cdots + 71\!\cdots\!76)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots - 16\!\cdots\!75)^{4} \)
|
| $67$ |
\( (T^{8} + \cdots + 54\!\cdots\!61)^{2} \)
|
| $71$ |
\( (T^{8} + \cdots + 97\!\cdots\!56)^{2} \)
|
| $73$ |
\( (T^{8} + \cdots + 11\!\cdots\!36)^{2} \)
|
| $79$ |
\( (T^{8} + \cdots + 34\!\cdots\!16)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 58\!\cdots\!36 \)
|
| $89$ |
\( (T^{8} + \cdots + 12\!\cdots\!96)^{2} \)
|
| $97$ |
\( (T^{8} + \cdots + 11\!\cdots\!01)^{2} \)
|
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