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Properties

Label 448.3.o.a
Level $448$
Weight $3$
Character orbit 448.o
Analytic conductor $12.207$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,3,Mod(95,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.95"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.2071158433\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 34 x^{18} + 755 x^{16} - 9698 x^{14} + 89921 x^{12} - 522048 x^{10} + 2189920 x^{8} + \cdots + 7311616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{11} q^{3} + (\beta_{6} - \beta_{5}) q^{5} + ( - \beta_{19} - \beta_{9} - \beta_{4}) q^{7} + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots - 2) q^{9} + ( - \beta_{19} - \beta_{17} + \cdots - 2 \beta_{12}) q^{11}+ \cdots + (5 \beta_{19} + 5 \beta_{18} + \cdots + 23 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 6 q^{5} - 28 q^{9} + 46 q^{17} + 114 q^{21} + 36 q^{25} + 94 q^{33} - 114 q^{37} - 160 q^{41} + 708 q^{45} - 92 q^{49} + 6 q^{53} + 308 q^{57} - 90 q^{61} + 212 q^{65} + 314 q^{73} + 198 q^{77} - 322 q^{81}+ \cdots - 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 34 x^{18} + 755 x^{16} - 9698 x^{14} + 89921 x^{12} - 522048 x^{10} + 2189920 x^{8} + \cdots + 7311616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6178891093061 \nu^{18} + 202194994815850 \nu^{16} + \cdots + 21\!\cdots\!24 ) / 29\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 74793058692353 \nu^{18} + \cdots - 63\!\cdots\!36 ) / 20\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7536626957017 \nu^{18} - 160634970287730 \nu^{16} + \cdots + 11\!\cdots\!00 ) / 96\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 131763721003 \nu^{19} + 3873553393682 \nu^{17} - 83302075295446 \nu^{15} + \cdots - 32\!\cdots\!16 \nu ) / 19\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 156849163360857 \nu^{18} + \cdots + 15\!\cdots\!04 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 530520981186385 \nu^{18} + \cdots + 13\!\cdots\!76 ) / 20\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22319680131651 \nu^{18} + 695686864779370 \nu^{16} + \cdots + 38\!\cdots\!68 ) / 78\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 109071631282621 \nu^{18} + \cdots + 42\!\cdots\!64 ) / 29\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 36381780868686 \nu^{19} + \cdots + 90\!\cdots\!52 \nu ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 990559116329657 \nu^{18} + \cdots - 85\!\cdots\!56 ) / 20\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 93940477944073 \nu^{19} + \cdots + 39\!\cdots\!52 \nu ) / 20\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2988257823 \nu^{19} + 96267694836 \nu^{17} - 2091233989993 \nu^{15} + \cdots + 98\!\cdots\!24 \nu ) / 62\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 912490276567931 \nu^{18} + \cdots - 64\!\cdots\!08 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!30 \nu^{19} + \cdots + 29\!\cdots\!08 \nu ) / 13\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 35\!\cdots\!11 \nu^{19} + \cdots - 14\!\cdots\!24 \nu ) / 13\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 18\!\cdots\!19 \nu^{19} + \cdots + 14\!\cdots\!72 \nu ) / 66\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 46\!\cdots\!47 \nu^{19} + \cdots + 13\!\cdots\!12 \nu ) / 13\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 68\!\cdots\!63 \nu^{19} + \cdots + 15\!\cdots\!40 \nu ) / 13\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 40\!\cdots\!50 \nu^{19} + \cdots - 36\!\cdots\!24 \nu ) / 66\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{19} - 4 \beta_{18} - 2 \beta_{17} + \beta_{16} + 2 \beta_{15} + 4 \beta_{14} + \cdots + 5 \beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{13} + \beta_{10} - 3\beta_{8} - 3\beta_{7} - 2\beta_{6} + 4\beta_{5} + 6\beta_{2} - 78\beta _1 + 3 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 18 \beta_{19} - 18 \beta_{18} - 48 \beta_{17} + 14 \beta_{16} + 7 \beta_{15} + 30 \beta_{14} + \cdots + 48 \beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 13 \beta_{13} - 13 \beta_{10} + 35 \beta_{7} + 34 \beta_{6} + 34 \beta_{5} - 63 \beta_{3} + \cdots - 806 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 624 \beta_{19} + 414 \beta_{18} - 210 \beta_{17} + 55 \beta_{16} - 55 \beta_{15} - 210 \beta_{14} + \cdots - 639 \beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 185 \beta_{13} - 370 \beta_{10} + 1059 \beta_{8} + 686 \beta_{7} + 916 \beta_{6} - 458 \beta_{5} + \cdots - 10229 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5654 \beta_{19} + 8320 \beta_{18} + 5654 \beta_{17} - 459 \beta_{16} - 918 \beta_{15} + \cdots - 18467 \beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5578 \beta_{13} - 2789 \beta_{10} + 16599 \beta_{8} + 3083 \beta_{7} + 5730 \beta_{6} - 11460 \beta_{5} + \cdots - 3083 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 35170 \beta_{19} + 35170 \beta_{18} + 112496 \beta_{17} - 7550 \beta_{16} - 3775 \beta_{15} + \cdots - 112496 \beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 42257 \beta_{13} + 42257 \beta_{10} - 24255 \beta_{7} - 70138 \beta_{6} - 70138 \beta_{5} + \cdots + 1728078 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1536928 \beta_{19} - 1062950 \beta_{18} + 473978 \beta_{17} - 27459 \beta_{16} + \cdots + 2422939 \beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 633549 \beta_{13} + 1267098 \beta_{10} - 3746799 \beta_{8} - 265286 \beta_{7} - 1717604 \beta_{6} + \cdots + 23659529 ) / 12 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 4899338 \beta_{19} - 7057392 \beta_{18} - 4899338 \beta_{17} + 41277 \beta_{16} + \cdots + 19187893 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 18770866 \beta_{13} + 9385433 \beta_{10} - 55059651 \beta_{8} + 415433 \beta_{7} - 10625162 \beta_{6} + \cdots - 415433 ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 89260042 \beta_{19} - 89260042 \beta_{18} - 293611520 \beta_{17} - 2088394 \beta_{16} + \cdots + 293611520 \beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 137646837 \beta_{13} - 137646837 \beta_{10} - 30886597 \beta_{7} + 133409282 \beta_{6} + \cdots - 4484747702 ) / 12 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 4093536560 \beta_{19} + 2854391086 \beta_{18} - 1239145474 \beta_{17} - 44857681 \beta_{16} + \cdots - 7898588231 \beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 2003123937 \beta_{13} - 4006247874 \beta_{10} + 11611199931 \beta_{8} - 1420843298 \beta_{7} + \cdots - 61792976989 ) / 12 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 40021349702 \beta_{19} + 57314524768 \beta_{18} + 40021349702 \beta_{17} + \cdots - 172178767979 \beta_{4} ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
1.31764 − 0.760743i
−2.85584 + 1.64882i
1.05174 − 0.607222i
3.26468 − 1.88486i
1.96054 − 1.13192i
−1.96054 + 1.13192i
−3.26468 + 1.88486i
−1.05174 + 0.607222i
2.85584 − 1.64882i
−1.31764 + 0.760743i
1.31764 + 0.760743i
−2.85584 − 1.64882i
1.05174 + 0.607222i
3.26468 + 1.88486i
1.96054 + 1.13192i
−1.96054 − 1.13192i
−3.26468 − 1.88486i
−1.05174 − 0.607222i
2.85584 + 1.64882i
−1.31764 − 0.760743i
0 −2.75361 + 4.76940i 0 −4.79340 + 2.76747i 0 4.79446 − 5.10031i 0 −10.6648 − 18.4719i 0
95.2 0 −2.20080 + 3.81190i 0 −3.13426 + 1.80957i 0 −4.05500 − 5.70587i 0 −5.18704 − 8.98421i 0
95.3 0 −1.33885 + 2.31895i 0 1.82811 − 1.05546i 0 −4.25503 + 5.55830i 0 0.914971 + 1.58478i 0
95.4 0 −0.565829 + 0.980045i 0 −3.20158 + 1.84843i 0 2.72101 + 6.44951i 0 3.85967 + 6.68515i 0
95.5 0 −0.459807 + 0.796409i 0 7.80113 − 4.50398i 0 −6.78682 + 1.71436i 0 4.07716 + 7.06184i 0
95.6 0 0.459807 − 0.796409i 0 7.80113 − 4.50398i 0 6.78682 − 1.71436i 0 4.07716 + 7.06184i 0
95.7 0 0.565829 − 0.980045i 0 −3.20158 + 1.84843i 0 −2.72101 − 6.44951i 0 3.85967 + 6.68515i 0
95.8 0 1.33885 − 2.31895i 0 1.82811 − 1.05546i 0 4.25503 − 5.55830i 0 0.914971 + 1.58478i 0
95.9 0 2.20080 − 3.81190i 0 −3.13426 + 1.80957i 0 4.05500 + 5.70587i 0 −5.18704 − 8.98421i 0
95.10 0 2.75361 − 4.76940i 0 −4.79340 + 2.76747i 0 −4.79446 + 5.10031i 0 −10.6648 − 18.4719i 0
415.1 0 −2.75361 − 4.76940i 0 −4.79340 − 2.76747i 0 4.79446 + 5.10031i 0 −10.6648 + 18.4719i 0
415.2 0 −2.20080 − 3.81190i 0 −3.13426 − 1.80957i 0 −4.05500 + 5.70587i 0 −5.18704 + 8.98421i 0
415.3 0 −1.33885 − 2.31895i 0 1.82811 + 1.05546i 0 −4.25503 − 5.55830i 0 0.914971 − 1.58478i 0
415.4 0 −0.565829 − 0.980045i 0 −3.20158 − 1.84843i 0 2.72101 − 6.44951i 0 3.85967 − 6.68515i 0
415.5 0 −0.459807 − 0.796409i 0 7.80113 + 4.50398i 0 −6.78682 − 1.71436i 0 4.07716 − 7.06184i 0
415.6 0 0.459807 + 0.796409i 0 7.80113 + 4.50398i 0 6.78682 + 1.71436i 0 4.07716 − 7.06184i 0
415.7 0 0.565829 + 0.980045i 0 −3.20158 − 1.84843i 0 −2.72101 + 6.44951i 0 3.85967 − 6.68515i 0
415.8 0 1.33885 + 2.31895i 0 1.82811 + 1.05546i 0 4.25503 + 5.55830i 0 0.914971 − 1.58478i 0
415.9 0 2.20080 + 3.81190i 0 −3.13426 − 1.80957i 0 4.05500 − 5.70587i 0 −5.18704 + 8.98421i 0
415.10 0 2.75361 + 4.76940i 0 −4.79340 − 2.76747i 0 −4.79446 − 5.10031i 0 −10.6648 + 18.4719i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.k odd 6 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.3.o.a ✓ 20
4.b odd 2 1 inner 448.3.o.a ✓ 20
7.c even 3 1 448.3.o.b yes 20
8.b even 2 1 448.3.o.b yes 20
8.d odd 2 1 448.3.o.b yes 20
28.g odd 6 1 448.3.o.b yes 20
56.k odd 6 1 inner 448.3.o.a ✓ 20
56.p even 6 1 inner 448.3.o.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.3.o.a ✓ 20 1.a even 1 1 trivial
448.3.o.a ✓ 20 4.b odd 2 1 inner
448.3.o.a ✓ 20 56.k odd 6 1 inner
448.3.o.a ✓ 20 56.p even 6 1 inner
448.3.o.b yes 20 7.c even 3 1
448.3.o.b yes 20 8.b even 2 1
448.3.o.b yes 20 8.d odd 2 1
448.3.o.b yes 20 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{20} + 59 T_{3}^{18} + 2415 T_{3}^{16} + 50330 T_{3}^{14} + 755737 T_{3}^{12} + 5523417 T_{3}^{10} + \cdots + 20820969 \) Copy content Toggle raw display
\( T_{5}^{10} + 3 T_{5}^{9} - 67 T_{5}^{8} - 210 T_{5}^{7} + 4213 T_{5}^{6} + 31581 T_{5}^{5} + \cdots + 1982907 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 59 T^{18} + \cdots + 20820969 \) Copy content Toggle raw display
$5$ \( (T^{10} + 3 T^{9} + \cdots + 1982907)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 54\!\cdots\!89 \) Copy content Toggle raw display
$13$ \( (T^{10} + 764 T^{8} + \cdots + 3713741568)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} - 23 T^{9} + \cdots + 3674889)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 20\!\cdots\!69 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 144130588939008)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 24\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 45849920147307)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + 40 T^{4} + \cdots + 1361664)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 15\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 13\!\cdots\!81 \) Copy content Toggle raw display
$53$ \( (T^{10} - 3 T^{9} + \cdots + 7271664867)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 30\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 74386589397363)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 39\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 64\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 71\!\cdots\!69)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 27\!\cdots\!88)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 13\!\cdots\!01)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + 56 T^{4} + \cdots - 259913408)^{4} \) Copy content Toggle raw display
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