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Properties

Label 4896.2.l.b
Level $4896$
Weight $2$
Character orbit 4896.l
Analytic conductor $39.095$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4896,2,Mod(3025,4896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4896, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4896.3025");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4896 = 2^{5} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4896.l (of order \(2\), degree \(1\), not 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: \(39.0947568296\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 8x^{14} + 20x^{12} + 36x^{10} + 240x^{8} - 156x^{6} + 268x^{4} + 136x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{5} + \beta_{10} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{5} + \beta_{10} q^{7} - \beta_{8} q^{11} + \beta_{5} q^{13} + ( - \beta_{13} + \beta_{2}) q^{17} + \beta_{9} q^{19} + ( - \beta_{13} - \beta_{12}) q^{23} + ( - \beta_{4} + \beta_{3}) q^{25} + (\beta_{14} - \beta_{11} - \beta_{8}) q^{29} - \beta_{15} q^{31} + ( - \beta_{6} - \beta_{5}) q^{35} + (\beta_{14} - \beta_{11} + \beta_{8}) q^{37} + (\beta_{15} - \beta_{13} - \beta_{12}) q^{41} + ( - \beta_{9} + \beta_{6} - \beta_{5}) q^{43} + (\beta_{4} + \beta_{2} - 2) q^{47} + (\beta_{3} - 1) q^{49} + (\beta_{9} - 2 \beta_{6} + \cdots + \beta_1) q^{53}+ \cdots + (\beta_{15} - \beta_{13} + \cdots - 2 \beta_{10}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{47} - 8 q^{49} + 8 q^{55} + 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 8x^{14} + 20x^{12} + 36x^{10} + 240x^{8} - 156x^{6} + 268x^{4} + 136x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 179923 \nu^{14} + 1562779 \nu^{12} + 4653533 \nu^{10} + 8736515 \nu^{8} + 45525263 \nu^{6} + \cdots + 21519666 ) / 48503473 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 331582 \nu^{14} + 2871705 \nu^{12} + 8109943 \nu^{10} + 14276644 \nu^{8} + 85224509 \nu^{6} + \cdots + 123528909 ) / 48503473 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1093891 \nu^{14} + 9853951 \nu^{12} + 30748020 \nu^{10} + 59667608 \nu^{8} + 285004618 \nu^{6} + \cdots + 93695856 ) / 97006946 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1397209 \nu^{14} + 12471803 \nu^{12} + 37660840 \nu^{10} + 70747866 \nu^{8} + 364403110 \nu^{6} + \cdots + 491728234 ) / 97006946 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2092075 \nu^{14} - 15798824 \nu^{12} - 34434792 \nu^{10} - 57835638 \nu^{8} - 476961490 \nu^{6} + \cdots - 105189352 ) / 97006946 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3320746 \nu^{14} + 25991529 \nu^{12} + 62949376 \nu^{10} + 117712646 \nu^{8} + 802852150 \nu^{6} + \cdots + 243445064 ) / 97006946 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3391627 \nu^{15} + 26976512 \nu^{13} + 68170444 \nu^{11} + 131353936 \nu^{9} + \cdots - 49963176 \nu ) / 388027784 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3710839 \nu^{15} - 33502210 \nu^{13} - 107383932 \nu^{11} - 227489312 \nu^{9} + \cdots - 14322168 \nu ) / 388027784 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4623809 \nu^{14} + 34724438 \nu^{12} + 72937230 \nu^{10} + 109588766 \nu^{8} + 996562316 \nu^{6} + \cdots + 184332700 ) / 97006946 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2689779 \nu^{15} - 22018279 \nu^{13} - 58720920 \nu^{11} - 117192984 \nu^{9} + \cdots - 870945096 \nu ) / 194013892 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7976257 \nu^{15} + 58299578 \nu^{13} + 116440672 \nu^{11} + 185835896 \nu^{9} + \cdots - 205818936 \nu ) / 388027784 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10100635 \nu^{15} - 82840258 \nu^{13} - 216828580 \nu^{11} - 388102144 \nu^{9} + \cdots - 2412849528 \nu ) / 388027784 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10838191 \nu^{15} + 87123872 \nu^{13} + 214521736 \nu^{11} + 348654544 \nu^{9} + \cdots + 1783490648 \nu ) / 388027784 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18532339 \nu^{15} - 141202776 \nu^{13} - 315675880 \nu^{11} - 536513664 \nu^{9} + \cdots + 373725384 \nu ) / 388027784 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10411324 \nu^{15} - 84304175 \nu^{13} - 213892650 \nu^{11} - 379357740 \nu^{9} + \cdots - 2339759696 \nu ) / 194013892 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 2\beta_{10} + \beta_{8} + \beta_{7} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} - 2\beta_{13} - \beta_{12} + \beta_{11} + 2\beta_{10} - \beta_{8} - 3\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{9} + 2\beta_{6} + 5\beta_{5} - 5\beta_{4} + 3\beta_{3} + 6\beta_{2} - 3\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{15} - 5\beta_{14} + 5\beta_{13} + \beta_{12} - 14\beta_{11} - 7\beta_{10} + 10\beta_{8} + 17\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -9\beta_{9} - 12\beta_{6} - 35\beta_{5} + 11\beta_{4} - 11\beta_{3} - 4\beta_{2} + 21\beta _1 - 35 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{15} + 19\beta_{14} - 5\beta_{13} + 7\beta_{12} + 69\beta_{11} + 4\beta_{10} - 61\beta_{8} - 125\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 18\beta_{9} + 38\beta_{6} + 86\beta_{5} + 9\beta_{4} - 3\beta_{3} - 14\beta_{2} - 82\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 65 \beta_{15} - 81 \beta_{14} - 79 \beta_{13} - 3 \beta_{12} - 339 \beta_{11} + 98 \beta_{10} + \cdots + 697 \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -93\beta_{9} - 183\beta_{6} - 430\beta_{5} - 241\beta_{4} + 216\beta_{3} + 153\beta_{2} + 381\beta _1 + 624 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 414 \beta_{15} + 225 \beta_{14} + 501 \beta_{13} + 7 \beta_{12} + 812 \beta_{11} - 609 \beta_{10} + \cdots - 1457 \beta_{7} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 408\beta_{9} + 688\beta_{6} + 1754\beta_{5} + 2266\beta_{4} - 1908\beta_{3} - 1652\beta_{2} - 1348\beta _1 - 5438 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3129 \beta_{15} - 763 \beta_{14} - 3915 \beta_{13} - 331 \beta_{12} - 2669 \beta_{11} + \cdots + 4629 \beta_{7} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 890 \beta_{9} - 1602 \beta_{6} - 3942 \beta_{5} - 15464 \beta_{4} + 12742 \beta_{3} + 11748 \beta_{2} + \cdots + 36164 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 19189 \beta_{15} + 369 \beta_{14} + 23969 \beta_{13} + 1945 \beta_{12} + 1693 \beta_{11} + \cdots - 3725 \beta_{7} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4896\mathbb{Z}\right)^\times\).

\(n\) \(613\) \(2143\) \(3809\) \(4321\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3025.1
0.476789 + 2.28924i
0.476789 − 2.28924i
1.32216 − 1.07919i
1.32216 + 1.07919i
−0.970055 + 0.510012i
−0.970055 − 0.510012i
−0.289081 + 0.578468i
−0.289081 − 0.578468i
0.289081 + 0.578468i
0.289081 − 0.578468i
0.970055 + 0.510012i
0.970055 − 0.510012i
−1.32216 − 1.07919i
−1.32216 + 1.07919i
−0.476789 + 2.28924i
−0.476789 − 2.28924i
0 0 0 −3.30694 0 − 1.93801i 0 0 0
3025.2 0 0 0 −3.30694 0 1.93801i 0 0 0
3025.3 0 0 0 −2.41361 0 − 2.57100i 0 0 0
3025.4 0 0 0 −2.41361 0 2.57100i 0 0 0
3025.5 0 0 0 −1.54992 0 − 3.57366i 0 0 0
3025.6 0 0 0 −1.54992 0 3.57366i 0 0 0
3025.7 0 0 0 −0.914541 0 − 2.61974i 0 0 0
3025.8 0 0 0 −0.914541 0 2.61974i 0 0 0
3025.9 0 0 0 0.914541 0 − 2.61974i 0 0 0
3025.10 0 0 0 0.914541 0 2.61974i 0 0 0
3025.11 0 0 0 1.54992 0 − 3.57366i 0 0 0
3025.12 0 0 0 1.54992 0 3.57366i 0 0 0
3025.13 0 0 0 2.41361 0 − 2.57100i 0 0 0
3025.14 0 0 0 2.41361 0 2.57100i 0 0 0
3025.15 0 0 0 3.30694 0 − 1.93801i 0 0 0
3025.16 0 0 0 3.30694 0 1.93801i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3025.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
17.b even 2 1 inner
136.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4896.2.l.b 16
3.b odd 2 1 544.2.h.a 16
4.b odd 2 1 1224.2.l.b 16
8.b even 2 1 inner 4896.2.l.b 16
8.d odd 2 1 1224.2.l.b 16
12.b even 2 1 136.2.h.a ✓ 16
17.b even 2 1 inner 4896.2.l.b 16
24.f even 2 1 136.2.h.a ✓ 16
24.h odd 2 1 544.2.h.a 16
51.c odd 2 1 544.2.h.a 16
68.d odd 2 1 1224.2.l.b 16
136.e odd 2 1 1224.2.l.b 16
136.h even 2 1 inner 4896.2.l.b 16
204.h even 2 1 136.2.h.a ✓ 16
408.b odd 2 1 544.2.h.a 16
408.h even 2 1 136.2.h.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.h.a ✓ 16 12.b even 2 1
136.2.h.a ✓ 16 24.f even 2 1
136.2.h.a ✓ 16 204.h even 2 1
136.2.h.a ✓ 16 408.h even 2 1
544.2.h.a 16 3.b odd 2 1
544.2.h.a 16 24.h odd 2 1
544.2.h.a 16 51.c odd 2 1
544.2.h.a 16 408.b odd 2 1
1224.2.l.b 16 4.b odd 2 1
1224.2.l.b 16 8.d odd 2 1
1224.2.l.b 16 68.d odd 2 1
1224.2.l.b 16 136.e odd 2 1
4896.2.l.b 16 1.a even 1 1 trivial
4896.2.l.b 16 8.b even 2 1 inner
4896.2.l.b 16 17.b even 2 1 inner
4896.2.l.b 16 136.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 20T_{5}^{6} + 120T_{5}^{4} - 240T_{5}^{2} + 128 \) acting on \(S_{2}^{\mathrm{new}}(4896, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 20 T^{6} + \cdots + 128)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 30 T^{6} + \cdots + 2176)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 42 T^{6} + \cdots + 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 56 T^{6} + \cdots + 4352)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 12 T^{6} + \cdots + 83521)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 80 T^{6} + \cdots + 4352)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 106 T^{6} + \cdots + 8704)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 132 T^{6} + \cdots + 445568)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 178 T^{6} + \cdots + 8704)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 164 T^{6} + \cdots + 682112)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 264 T^{6} + \cdots + 8912896)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 124 T^{6} + \cdots + 17408)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 6 T^{3} + \cdots - 256)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} + 208 T^{6} + \cdots + 69632)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 188 T^{6} + \cdots + 17408)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 164 T^{6} + \cdots + 15488)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 224 T^{6} + \cdots + 526592)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 182 T^{6} + \cdots + 34816)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 464 T^{6} + \cdots + 67403776)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 266 T^{6} + \cdots + 263296)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 268 T^{6} + \cdots + 5030912)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 6 T^{3} - 96 T^{2} + \cdots + 8)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 424 T^{6} + \cdots + 117121024)^{2} \) Copy content Toggle raw display
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