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Properties

Label 845.2.d.d
Level $845$
Weight $2$
Character orbit 845.d
Analytic conductor $6.747$
Analytic rank $0$
Dimension $12$
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] = [845,2,Mod(844,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.844");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.d (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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 22x^{10} + 147x^{8} + 390x^{6} + 413x^{4} + 128x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{11} - \beta_{3}) q^{3} + ( - \beta_{2} + 1) q^{4} - \beta_{9} q^{5} + ( - \beta_{10} - 2 \beta_{4}) q^{6} + ( - \beta_{6} - \beta_1) q^{7} + ( - \beta_{9} - \beta_{6} + \cdots + \beta_1) q^{8}+ \cdots + ( - \beta_{8} - \beta_{7} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{11} - \beta_{3}) q^{3} + ( - \beta_{2} + 1) q^{4} - \beta_{9} q^{5} + ( - \beta_{10} - 2 \beta_{4}) q^{6} + ( - \beta_{6} - \beta_1) q^{7} + ( - \beta_{9} - \beta_{6} + \cdots + \beta_1) q^{8}+ \cdots + ( - 2 \beta_{10} + 8 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{4} - 12 q^{9} + 14 q^{10} - 44 q^{14} + 32 q^{16} + 2 q^{25} - 36 q^{29} + 8 q^{30} - 20 q^{35} - 4 q^{36} + 70 q^{40} + 12 q^{49} - 24 q^{51} + 52 q^{55} - 32 q^{56} - 12 q^{61} + 12 q^{64} + 4 q^{66} - 48 q^{69} - 16 q^{74} + 4 q^{75} - 104 q^{79} - 28 q^{81} - 62 q^{90} - 112 q^{94} + 40 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 22x^{10} + 147x^{8} + 390x^{6} + 413x^{4} + 128x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{10} - 77\nu^{8} - 372\nu^{6} - 452\nu^{4} + 67\nu^{2} + 124 ) / 102 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{10} + 77\nu^{8} + 372\nu^{6} + 452\nu^{4} - 16\nu^{2} + 80 ) / 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{11} + 250\nu^{9} + 1771\nu^{7} + 5034\nu^{5} + 5447\nu^{3} + 1478\nu ) / 204 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{11} + 250\nu^{9} + 1771\nu^{7} + 5034\nu^{5} + 5447\nu^{3} + 1274\nu ) / 204 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 25 \nu^{11} + 30 \nu^{10} - 528 \nu^{9} + 620 \nu^{8} - 3243 \nu^{7} + 3606 \nu^{6} - 7568 \nu^{5} + \cdots + 872 ) / 408 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -12\nu^{10} - 248\nu^{8} - 1439\nu^{6} - 2869\nu^{4} - 1635\nu^{2} + 32 ) / 102 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33 \nu^{11} - 8 \nu^{10} + 716 \nu^{9} - 188 \nu^{8} + 4667 \nu^{7} - 1424 \nu^{6} + 12076 \nu^{5} + \cdots - 1180 ) / 408 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 33 \nu^{11} - 8 \nu^{10} - 716 \nu^{9} - 188 \nu^{8} - 4667 \nu^{7} - 1424 \nu^{6} - 12076 \nu^{5} + \cdots - 772 ) / 408 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25 \nu^{11} + 30 \nu^{10} + 528 \nu^{9} + 620 \nu^{8} + 3243 \nu^{7} + 3606 \nu^{6} + 7568 \nu^{5} + \cdots + 872 ) / 408 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -15\nu^{11} - 327\nu^{9} - 2143\nu^{7} - 5486\nu^{5} - 5380\nu^{3} - 1048\nu ) / 102 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -35\nu^{11} - 746\nu^{9} - 4649\nu^{7} - 10772\nu^{5} - 8717\nu^{3} - 1210\nu ) / 204 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 3\beta_{10} - \beta_{8} + \beta_{7} + 10\beta_{4} - 8\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{9} + \beta_{8} + \beta_{7} + 4\beta_{6} + 4\beta_{5} - 13\beta_{2} - 24\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17\beta_{11} - 45\beta_{10} + 5\beta_{9} + 18\beta_{8} - 18\beta_{7} - 5\beta_{5} - 111\beta_{4} + 85\beta_{3} - 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -68\beta_{9} - 23\beta_{8} - 23\beta_{7} - 62\beta_{6} - 68\beta_{5} + 166\beta_{2} + 286\beta _1 - 408 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 228 \beta_{11} + 588 \beta_{10} - 91 \beta_{9} - 257 \beta_{8} + 257 \beta_{7} + 91 \beta_{5} + \cdots + 257 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 936\beta_{9} + 348\beta_{8} + 348\beta_{7} + 816\beta_{6} + 936\beta_{5} - 2105\beta_{2} - 3496\beta _1 + 4960 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2921 \beta_{11} - 7473 \beta_{10} + 1284 \beta_{9} + 3389 \beta_{8} - 3389 \beta_{7} - 1284 \beta_{5} + \cdots - 3389 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 12146 \beta_{9} - 4673 \beta_{8} - 4673 \beta_{7} - 10394 \beta_{6} - 12146 \beta_{5} + 26569 \beta_{2} + \cdots - 61640 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 36963 \beta_{11} + 94281 \beta_{10} - 16819 \beta_{9} - 43388 \beta_{8} + 43388 \beta_{7} + \cdots + 43388 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
844.1
3.54574i
− 3.54574i
2.18733i
− 2.18733i
− 1.33084i
1.33084i
0.669163i
− 0.669163i
0.187333i
− 0.187333i
1.54574i
− 1.54574i
−2.54574 − 2.15293i 4.48079 −2.08125 − 0.817544i 5.48079i 2.93855 −6.31544 −1.63509 5.29833 + 2.08125i
844.2 −2.54574 2.15293i 4.48079 −2.08125 + 0.817544i − 5.48079i 2.93855 −6.31544 −1.63509 5.29833 − 2.08125i
844.3 −1.18733 − 0.345110i −0.590239 1.71029 + 1.44045i 0.409761i 2.02956 3.07548 2.88090 −2.03069 − 1.71029i
844.4 −1.18733 0.345110i −0.590239 1.71029 − 1.44045i − 0.409761i 2.02956 3.07548 2.88090 −2.03069 + 1.71029i
844.5 −0.330837 − 2.69180i −1.89055 −0.702335 + 2.12291i 0.890547i 3.35348 1.28714 −4.24581 0.232358 − 0.702335i
844.6 −0.330837 2.69180i −1.89055 −0.702335 − 2.12291i − 0.890547i 3.35348 1.28714 −4.24581 0.232358 + 0.702335i
844.7 0.330837 − 2.69180i −1.89055 0.702335 − 2.12291i − 0.890547i −3.35348 −1.28714 −4.24581 0.232358 − 0.702335i
844.8 0.330837 2.69180i −1.89055 0.702335 + 2.12291i 0.890547i −3.35348 −1.28714 −4.24581 0.232358 + 0.702335i
844.9 1.18733 − 0.345110i −0.590239 −1.71029 − 1.44045i − 0.409761i −2.02956 −3.07548 2.88090 −2.03069 − 1.71029i
844.10 1.18733 0.345110i −0.590239 −1.71029 + 1.44045i 0.409761i −2.02956 −3.07548 2.88090 −2.03069 + 1.71029i
844.11 2.54574 − 2.15293i 4.48079 2.08125 + 0.817544i − 5.48079i −2.93855 6.31544 −1.63509 5.29833 + 2.08125i
844.12 2.54574 2.15293i 4.48079 2.08125 − 0.817544i 5.48079i −2.93855 6.31544 −1.63509 5.29833 − 2.08125i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 844.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.b even 2 1 inner
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.d.d 12
5.b even 2 1 inner 845.2.d.d 12
13.b even 2 1 inner 845.2.d.d 12
13.c even 3 2 845.2.l.f 24
13.d odd 4 1 845.2.b.d 6
13.d odd 4 1 845.2.b.e 6
13.e even 6 2 845.2.l.f 24
13.f odd 12 2 65.2.n.a ✓ 12
13.f odd 12 2 845.2.n.e 12
39.k even 12 2 585.2.bs.a 12
52.l even 12 2 1040.2.dh.a 12
65.d even 2 1 inner 845.2.d.d 12
65.f even 4 1 4225.2.a.bq 6
65.f even 4 1 4225.2.a.br 6
65.g odd 4 1 845.2.b.d 6
65.g odd 4 1 845.2.b.e 6
65.k even 4 1 4225.2.a.bq 6
65.k even 4 1 4225.2.a.br 6
65.l even 6 2 845.2.l.f 24
65.n even 6 2 845.2.l.f 24
65.o even 12 2 325.2.e.e 12
65.s odd 12 2 65.2.n.a ✓ 12
65.s odd 12 2 845.2.n.e 12
65.t even 12 2 325.2.e.e 12
195.bh even 12 2 585.2.bs.a 12
260.bc even 12 2 1040.2.dh.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a ✓ 12 13.f odd 12 2
65.2.n.a ✓ 12 65.s odd 12 2
325.2.e.e 12 65.o even 12 2
325.2.e.e 12 65.t even 12 2
585.2.bs.a 12 39.k even 12 2
585.2.bs.a 12 195.bh even 12 2
845.2.b.d 6 13.d odd 4 1
845.2.b.d 6 65.g odd 4 1
845.2.b.e 6 13.d odd 4 1
845.2.b.e 6 65.g odd 4 1
845.2.d.d 12 1.a even 1 1 trivial
845.2.d.d 12 5.b even 2 1 inner
845.2.d.d 12 13.b even 2 1 inner
845.2.d.d 12 65.d even 2 1 inner
845.2.l.f 24 13.c even 3 2
845.2.l.f 24 13.e even 6 2
845.2.l.f 24 65.l even 6 2
845.2.l.f 24 65.n even 6 2
845.2.n.e 12 13.f odd 12 2
845.2.n.e 12 65.s odd 12 2
1040.2.dh.a 12 52.l even 12 2
1040.2.dh.a 12 260.bc even 12 2
4225.2.a.bq 6 65.f even 4 1
4225.2.a.bq 6 65.k even 4 1
4225.2.a.br 6 65.f even 4 1
4225.2.a.br 6 65.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 8T_{2}^{4} + 10T_{2}^{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 8 T^{4} + 10 T^{2} - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} + 12 T^{4} + 35 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} - 24 T^{4} + \cdots - 400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 26 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 35 T^{4} + \cdots + 169)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 38 T^{4} + \cdots + 100)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 12 T^{4} + 35 T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 3)^{12} \) Copy content Toggle raw display
$31$ \( (T^{6} + 96 T^{4} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 35 T^{4} + \cdots - 169)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 107 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 80 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 236 T^{4} + \cdots - 270400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 171 T^{4} + \cdots + 400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 114 T^{4} + \cdots + 18496)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 3 T^{2} + \cdots - 115)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 100 T^{4} + \cdots - 20164)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 38 T^{4} + \cdots + 676)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 215 T^{4} + \cdots - 250000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 26 T^{2} + \cdots + 160)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 276 T^{4} + \cdots - 640000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 414 T^{4} + \cdots + 2515396)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 280 T^{4} + \cdots - 204304)^{2} \) Copy content Toggle raw display
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