Properties

Label 845.2.e.n
Level $845$
Weight $2$
Character orbit 845.e
Analytic conductor $6.747$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [845,2,Mod(146,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.146");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.e (of order \(3\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_1 + 1) q^{2} + (\beta_{7} + \beta_{6} + \beta_{4}) q^{3} + (\beta_{7} - \beta_{5} - \beta_{2} - \beta_1 + 1) q^{4} + q^{5} + (2 \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{6} + (3 \beta_{5} + \beta_{4} - \beta_{3}) q^{7} + ( - 2 \beta_{6} - \beta_{3} - 1) q^{8} + ( - 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_1 + 1) q^{2} + (\beta_{7} + \beta_{6} + \beta_{4}) q^{3} + (\beta_{7} - \beta_{5} - \beta_{2} - \beta_1 + 1) q^{4} + q^{5} + (2 \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{6} + (3 \beta_{5} + \beta_{4} - \beta_{3}) q^{7} + ( - 2 \beta_{6} - \beta_{3} - 1) q^{8} + ( - 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 2) q^{9} + ( - \beta_{5} - \beta_1 + 1) q^{10} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_1 + 2) q^{11} + ( - 2 \beta_{6} - 3 \beta_{2} - 1) q^{12} + ( - \beta_{6} + 3 \beta_{2} - 1) q^{14} + (\beta_{7} + \beta_{6} + \beta_{4}) q^{15} + ( - \beta_{7} - \beta_{6} - 2 \beta_{4} + \beta_1) q^{16} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{17} + ( - 2 \beta_{6} - 2 \beta_{2} - 4) q^{18} + ( - \beta_{7} + 4 \beta_{5}) q^{19} + (\beta_{7} - \beta_{5} - \beta_{2} - \beta_1 + 1) q^{20} + (3 \beta_{6} + 2 \beta_{3} - 2) q^{21} + ( - \beta_{7} - 4 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{22} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_1 + 2) q^{23} + ( - \beta_{7} - \beta_{6} + 8 \beta_{5} + 4 \beta_1 - 8) q^{24} + q^{25} + ( - \beta_{6} + \beta_{3} - 2 \beta_{2}) q^{27} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_1 + 2) q^{28} + ( - 2 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + 2 \beta_1 - 3) q^{29} + (2 \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{30} + (2 \beta_{6} - 2) q^{31} + (2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{32} + (4 \beta_{7} + 5 \beta_{5} + \beta_{4} - \beta_{3}) q^{33} + ( - \beta_{3} - \beta_{2} + 2) q^{34} + (3 \beta_{5} + \beta_{4} - \beta_{3}) q^{35} + ( - 4 \beta_{7} - 4 \beta_{6} + 6 \beta_{5} - 2 \beta_{4} + 4 \beta_1 - 6) q^{36} + (2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 2 \beta_1 - 3) q^{37} + (\beta_{6} + \beta_{3} + 5 \beta_{2} - 1) q^{38} + ( - 2 \beta_{6} - \beta_{3} - 1) q^{40} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} + 2) q^{41} + (5 \beta_{7} + 5 \beta_{6} + 3 \beta_{4} - \beta_1) q^{42} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{43} + ( - \beta_{6} - 3 \beta_{3} + \beta_{2} - 2) q^{44} + ( - 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 2) q^{45} + ( - 4 \beta_{7} + 3 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{46} + (2 \beta_{6} + 4 \beta_{2} - 4) q^{47} + ( - \beta_{7} + 8 \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 3) q^{48} + (4 \beta_{5} + 4 \beta_{4} - 2 \beta_1 - 4) q^{49} + ( - \beta_{5} - \beta_1 + 1) q^{50} + (2 \beta_{3} - 2 \beta_{2} + 1) q^{51} + ( - 2 \beta_{2} - 2) q^{53} + ( - 2 \beta_{7} - 2 \beta_{6} + 5 \beta_{5} - \beta_{4} + 3 \beta_1 - 5) q^{54} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_1 + 2) q^{55} + (6 \beta_{7} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{56} + (4 \beta_{6} + 3 \beta_{3} + 2 \beta_{2} + 1) q^{57} + ( - 4 \beta_{7} + 7 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 5) q^{58} + ( - \beta_{7} + 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 2) q^{59} + ( - 2 \beta_{6} - 3 \beta_{2} - 1) q^{60} + ( - 5 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{61} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2) q^{62} + ( - 2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} - 6 \beta_1 + 4) q^{63} + ( - 2 \beta_{6} - 4 \beta_{3} + 3) q^{64} + ( - 5 \beta_{6} - 4 \beta_{3} + \beta_{2} + 3) q^{66} + ( - 7 \beta_{5} + \beta_{4} + 7) q^{67} + (2 \beta_{5} - 2 \beta_{4} - \beta_1 - 2) q^{68} + (7 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 4) q^{69} + ( - \beta_{6} + 3 \beta_{2} - 1) q^{70} + ( - 3 \beta_{7} - 2 \beta_{5} - 6 \beta_{4} + 6 \beta_{3}) q^{71} + ( - 6 \beta_{7} + 4 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 6) q^{72} + ( - 2 \beta_{6} + 2 \beta_{3} + 2 \beta_{2}) q^{73} + (3 \beta_{7} + 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{74} + (\beta_{7} + \beta_{6} + \beta_{4}) q^{75} + (5 \beta_{7} + 5 \beta_{6} - 7 \beta_{5} + \beta_{4} - 5 \beta_1 + 7) q^{76} + ( - 5 \beta_{6} - 3 \beta_{3} + 4 \beta_{2} + 4) q^{77} + (2 \beta_{6} + 2 \beta_{3} - 6 \beta_{2}) q^{79} + ( - \beta_{7} - \beta_{6} - 2 \beta_{4} + \beta_1) q^{80} + ( - 4 \beta_{7} - 4 \beta_{6} - 3 \beta_{5} - 2 \beta_1 + 3) q^{81} + ( - \beta_{7} - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{82} + ( - 4 \beta_{6} - 2 \beta_{3} + 4) q^{83} + (3 \beta_{7} - 9 \beta_{5} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 5) q^{84} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{85} + ( - \beta_{3} - \beta_{2} + 2) q^{86} + ( - 5 \beta_{7} + 8 \beta_{5} - \beta_{4} + \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 6) q^{87} + ( - 5 \beta_{7} - 5 \beta_{6} - 6 \beta_{5} - 3 \beta_{4} + 6) q^{88} + (3 \beta_{7} + 3 \beta_{6} + 8 \beta_{5} + 2 \beta_{4} + 8 \beta_1 - 8) q^{89} + ( - 2 \beta_{6} - 2 \beta_{2} - 4) q^{90} + (2 \beta_{6} + 2 \beta_{3} + 3 \beta_{2} + 3) q^{92} + ( - 2 \beta_{7} - 2 \beta_{6} - 6 \beta_{5} - 4 \beta_{4} - 4 \beta_1 + 6) q^{93} + (6 \beta_{7} + 6 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} - 2 \beta_1 + 8) q^{94} + ( - \beta_{7} + 4 \beta_{5}) q^{95} + (3 \beta_{6} + \beta_{3} + \beta_{2} - 2) q^{96} + (4 \beta_{7} + 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{97} + (6 \beta_{7} - 4 \beta_{5} + 4 \beta_{2} + 4 \beta_1 - 4) q^{98} + (2 \beta_{6} + 2 \beta_{3} - 2 \beta_{2} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 8 q^{5} - 4 q^{6} + 10 q^{7} - 12 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 8 q^{5} - 4 q^{6} + 10 q^{7} - 12 q^{8} - 4 q^{9} + 2 q^{10} - 20 q^{12} + 4 q^{14} + 2 q^{15} - 2 q^{16} + 2 q^{17} - 40 q^{18} + 16 q^{19} - 2 q^{20} - 8 q^{21} - 12 q^{22} + 10 q^{23} - 24 q^{24} + 8 q^{25} - 4 q^{27} + 8 q^{28} - 8 q^{29} - 4 q^{30} - 16 q^{31} + 4 q^{32} + 18 q^{33} + 8 q^{34} + 10 q^{35} - 20 q^{36} - 2 q^{37} + 16 q^{38} - 12 q^{40} + 8 q^{41} + 4 q^{42} + 2 q^{43} - 24 q^{44} - 4 q^{45} + 16 q^{46} - 16 q^{47} + 28 q^{48} - 12 q^{49} + 2 q^{50} + 8 q^{51} - 24 q^{53} - 16 q^{54} - 12 q^{56} + 28 q^{57} + 22 q^{58} + 12 q^{59} - 20 q^{60} - 28 q^{61} - 4 q^{62} + 4 q^{63} + 8 q^{64} + 12 q^{66} + 30 q^{67} - 14 q^{68} + 16 q^{69} + 4 q^{70} + 4 q^{71} + 12 q^{72} + 16 q^{73} + 10 q^{74} + 2 q^{75} + 20 q^{76} + 36 q^{77} - 16 q^{79} - 2 q^{80} + 8 q^{81} - 4 q^{82} + 24 q^{83} - 28 q^{84} + 2 q^{85} + 8 q^{86} + 22 q^{87} + 18 q^{88} - 12 q^{89} - 40 q^{90} + 44 q^{92} + 8 q^{93} + 32 q^{94} + 16 q^{95} - 8 q^{96} + 2 q^{97} - 24 q^{98} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 10\nu^{2} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - \nu^{5} + 4\nu^{4} - \nu^{3} - 6\nu^{2} + 10\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} - 8\nu^{6} + 3\nu^{5} + 10\nu^{4} - 13\nu^{3} - 8\nu^{2} + 32\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} - 14\nu^{6} + 9\nu^{5} + 16\nu^{4} - 27\nu^{3} - 14\nu^{2} + 76\nu - 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 7\nu^{6} + 3\nu^{5} + 11\nu^{4} - 15\nu^{3} - 11\nu^{2} + 40\nu - 28 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 9\nu^{6} + 5\nu^{5} + 13\nu^{4} - 21\nu^{3} - 13\nu^{2} + 54\nu - 40 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} - 24\nu^{6} + 13\nu^{5} + 38\nu^{4} - 51\nu^{3} - 32\nu^{2} + 132\nu - 104 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{3} + \beta_{2} - \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - 3\beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} - 3\beta_{5} + 3\beta_{4} - 3\beta_{3} + 3\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{7} - 2\beta_{6} - 3\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_{2} - \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{7} - 4\beta_{6} - 3\beta_{5} + 4\beta_{4} - 5\beta_{3} - 2\beta_{2} + 5\beta _1 + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{7} - 5\beta_{6} + 9\beta_{5} - 12\beta_{3} - 3\beta_{2} + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -4\beta_{7} - 14\beta_{6} + 24\beta_{5} + 5\beta_{4} - 4\beta_{3} - 7\beta_{2} + \beta _1 + 4 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
0.665665 1.24775i
−1.27597 + 0.609843i
1.40994 0.109843i
1.20036 + 0.747754i
0.665665 + 1.24775i
−1.27597 0.609843i
1.40994 + 0.109843i
1.20036 0.747754i
−0.747754 1.29515i −0.0473938 0.0820885i −0.118272 + 0.204852i 1.00000 −0.0708778 + 0.122764i 2.41342 4.18016i −2.63726 1.49551 2.59030i −0.747754 1.29515i
146.2 −0.109843 0.190254i 0.800098 + 1.38581i 0.975869 1.69025i 1.00000 0.175771 0.304444i −0.166123 + 0.287734i −0.868145 0.219687 0.380509i −0.109843 0.190254i
146.3 0.609843 + 1.05628i −1.16612 2.01978i 0.256182 0.443720i 1.00000 1.42231 2.46350i 1.80010 3.11786i 3.06430 −1.21969 + 2.11256i 0.609843 + 1.05628i
146.4 1.24775 + 2.16117i 1.41342 + 2.44811i −2.11378 + 3.66117i 1.00000 −3.52720 + 6.10929i 0.952606 1.64996i −5.55889 −2.49551 + 4.32235i 1.24775 + 2.16117i
191.1 −0.747754 + 1.29515i −0.0473938 + 0.0820885i −0.118272 0.204852i 1.00000 −0.0708778 0.122764i 2.41342 + 4.18016i −2.63726 1.49551 + 2.59030i −0.747754 + 1.29515i
191.2 −0.109843 + 0.190254i 0.800098 1.38581i 0.975869 + 1.69025i 1.00000 0.175771 + 0.304444i −0.166123 0.287734i −0.868145 0.219687 + 0.380509i −0.109843 + 0.190254i
191.3 0.609843 1.05628i −1.16612 + 2.01978i 0.256182 + 0.443720i 1.00000 1.42231 + 2.46350i 1.80010 + 3.11786i 3.06430 −1.21969 2.11256i 0.609843 1.05628i
191.4 1.24775 2.16117i 1.41342 2.44811i −2.11378 3.66117i 1.00000 −3.52720 6.10929i 0.952606 + 1.64996i −5.55889 −2.49551 4.32235i 1.24775 2.16117i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.e.n 8
13.b even 2 1 845.2.e.m 8
13.c even 3 1 845.2.a.l 4
13.c even 3 1 inner 845.2.e.n 8
13.d odd 4 1 65.2.m.a 8
13.d odd 4 1 845.2.m.g 8
13.e even 6 1 845.2.a.m 4
13.e even 6 1 845.2.e.m 8
13.f odd 12 1 65.2.m.a 8
13.f odd 12 2 845.2.c.g 8
13.f odd 12 1 845.2.m.g 8
39.f even 4 1 585.2.bu.c 8
39.h odd 6 1 7605.2.a.cf 4
39.i odd 6 1 7605.2.a.cj 4
39.k even 12 1 585.2.bu.c 8
52.f even 4 1 1040.2.da.b 8
52.l even 12 1 1040.2.da.b 8
65.f even 4 1 325.2.m.b 8
65.g odd 4 1 325.2.n.d 8
65.k even 4 1 325.2.m.c 8
65.l even 6 1 4225.2.a.bi 4
65.n even 6 1 4225.2.a.bl 4
65.o even 12 1 325.2.m.c 8
65.s odd 12 1 325.2.n.d 8
65.t even 12 1 325.2.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 13.d odd 4 1
65.2.m.a 8 13.f odd 12 1
325.2.m.b 8 65.f even 4 1
325.2.m.b 8 65.t even 12 1
325.2.m.c 8 65.k even 4 1
325.2.m.c 8 65.o even 12 1
325.2.n.d 8 65.g odd 4 1
325.2.n.d 8 65.s odd 12 1
585.2.bu.c 8 39.f even 4 1
585.2.bu.c 8 39.k even 12 1
845.2.a.l 4 13.c even 3 1
845.2.a.m 4 13.e even 6 1
845.2.c.g 8 13.f odd 12 2
845.2.e.m 8 13.b even 2 1
845.2.e.m 8 13.e even 6 1
845.2.e.n 8 1.a even 1 1 trivial
845.2.e.n 8 13.c even 3 1 inner
845.2.m.g 8 13.d odd 4 1
845.2.m.g 8 13.f odd 12 1
1040.2.da.b 8 52.f even 4 1
1040.2.da.b 8 52.l even 12 1
4225.2.a.bi 4 65.l even 6 1
4225.2.a.bl 4 65.n even 6 1
7605.2.a.cf 4 39.h odd 6 1
7605.2.a.cj 4 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{8} - 2T_{2}^{7} + 7T_{2}^{6} - 2T_{2}^{5} + 16T_{2}^{4} - 8T_{2}^{3} + 19T_{2}^{2} + 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} - 10T_{7}^{7} + 70T_{7}^{6} - 256T_{7}^{5} + 691T_{7}^{4} - 880T_{7}^{3} + 814T_{7}^{2} + 242T_{7} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + 7 T^{6} - 2 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + 10 T^{6} - 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 10 T^{7} + 70 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{8} + 30 T^{6} + 867 T^{4} + \cdots + 1089 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + 22 T^{6} + 16 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$19$ \( (T^{4} - 8 T^{3} + 51 T^{2} - 104 T + 169)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 10 T^{7} + 94 T^{6} + \cdots + 89401 \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + 82 T^{6} - 64 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 8)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 2 T^{7} + 58 T^{6} - 184 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 2 T^{7} + 22 T^{6} + 16 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} - 72 T^{2} - 736 T - 1328)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 12 T^{3} + 36 T^{2} - 48)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 12 T^{7} + 114 T^{6} - 336 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$61$ \( T^{8} + 28 T^{7} + 526 T^{6} + \cdots + 1590121 \) Copy content Toggle raw display
$67$ \( T^{8} - 30 T^{7} + 570 T^{6} + \cdots + 7667361 \) Copy content Toggle raw display
$71$ \( T^{8} - 4 T^{7} + 226 T^{6} + \cdots + 109767529 \) Copy content Toggle raw display
$73$ \( (T^{4} - 8 T^{3} - 84 T^{2} + 832 T - 1712)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} - 132 T^{2} - 640 T + 4432)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 12 T^{3} - 24 T^{2} + 288 T - 192)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 12 T^{7} + 378 T^{6} + \cdots + 78375609 \) Copy content Toggle raw display
$97$ \( T^{8} - 2 T^{7} + 94 T^{6} + \cdots + 196249 \) Copy content Toggle raw display
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