Properties

Label 325.2.m.b
Level $325$
Weight $2$
Character orbit 325.m
Analytic conductor $2.595$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
Defining polynomial: \(x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} - \beta_{3} - \beta_{6} ) q^{2} + ( 2 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{3} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{4} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{6} + ( 2 + \beta_{1} + \beta_{4} - 3 \beta_{6} ) q^{7} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{8} + ( 4 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} - \beta_{3} - \beta_{6} ) q^{2} + ( 2 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{3} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{4} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{6} + ( 2 + \beta_{1} + \beta_{4} - 3 \beta_{6} ) q^{7} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{8} + ( 4 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{9} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{6} ) q^{11} + ( -5 + 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 4 \beta_{6} + 4 \beta_{7} ) q^{12} + ( \beta_{1} - \beta_{3} - 3 \beta_{5} - 2 \beta_{7} ) q^{13} + ( -2 - \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{14} + ( 1 - \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{16} + ( 2 - \beta_{1} + \beta_{4} + \beta_{6} ) q^{17} + ( -6 + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{18} + ( -1 - \beta_{6} - 4 \beta_{7} ) q^{19} + ( 5 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} ) q^{21} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{22} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{23} + ( 6 - 4 \beta_{1} - 8 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{24} + ( -1 + \beta_{1} + 5 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 6 \beta_{7} ) q^{26} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{27} + ( -1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{28} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{6} + 4 \beta_{7} ) q^{29} + ( -2 - 2 \beta_{2} + 4 \beta_{6} - 2 \beta_{7} ) q^{31} + ( -2 + \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{32} + ( -4 - \beta_{1} - 8 \beta_{2} - \beta_{4} + 5 \beta_{6} - 4 \beta_{7} ) q^{33} + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{34} + ( -4 + 4 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 8 \beta_{7} ) q^{36} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{37} + ( -7 + 6 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} + 5 \beta_{4} - \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{38} + ( -2 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{39} + ( 2 + 2 \beta_{2} - \beta_{6} ) q^{41} + ( -11 + \beta_{1} + 3 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} ) q^{42} + ( -2 + \beta_{1} - \beta_{4} - \beta_{6} ) q^{43} + ( -3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{44} + ( -3 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{46} + ( -2 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{47} + ( -\beta_{1} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} - 8 \beta_{7} ) q^{48} + ( -2 + 2 \beta_{1} + 6 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} ) q^{49} + ( -1 + 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{51} + ( 8 - 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} ) q^{52} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{7} ) q^{53} + ( -7 + 3 \beta_{1} + 5 \beta_{2} + \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} ) q^{54} + ( 1 - 12 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - \beta_{6} - 6 \beta_{7} ) q^{56} + ( -6 + 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - 4 \beta_{7} ) q^{57} + ( 14 - 7 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} - 7 \beta_{4} + 5 \beta_{5} - 7 \beta_{6} - 4 \beta_{7} ) q^{58} + ( 1 - 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{59} + ( -5 - 2 \beta_{3} + 2 \beta_{5} + 5 \beta_{6} ) q^{61} + ( 4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{62} + ( 6 - 6 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 4 \beta_{6} + 4 \beta_{7} ) q^{63} + ( -1 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} - 2 \beta_{7} ) q^{64} + ( 4 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} - \beta_{4} - 3 \beta_{5} - 5 \beta_{7} ) q^{66} + ( -\beta_{3} - \beta_{4} - \beta_{5} - 7 \beta_{6} ) q^{67} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{68} + ( 9 - 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 7 \beta_{6} ) q^{69} + ( -3 + 6 \beta_{1} - 6 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} ) q^{71} + ( 14 - 10 \beta_{1} - 12 \beta_{2} - 6 \beta_{3} - 10 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} ) q^{72} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -3 - \beta_{1} - 6 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{74} + ( 15 - 5 \beta_{1} - 7 \beta_{2} - \beta_{3} - 6 \beta_{4} + 6 \beta_{5} - 5 \beta_{6} ) q^{76} + ( -4 - \beta_{1} - 8 \beta_{2} - \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + 10 \beta_{6} - 8 \beta_{7} ) q^{77} + ( -2 + 3 \beta_{1} + 6 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} + 3 \beta_{6} + 14 \beta_{7} ) q^{78} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} ) q^{79} + ( -2 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 3 \beta_{6} - 8 \beta_{7} ) q^{81} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{82} + ( 2 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{7} ) q^{83} + ( 2 + \beta_{1} + 5 \beta_{3} - \beta_{4} + 5 \beta_{5} + 3 \beta_{6} + 9 \beta_{7} ) q^{84} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{86} + ( 4 + \beta_{1} + 6 \beta_{3} - \beta_{4} + 6 \beta_{5} + 5 \beta_{6} + 8 \beta_{7} ) q^{87} + ( 10 + 6 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 5 \beta_{6} ) q^{88} + ( 2 - 8 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} ) q^{89} + ( -1 + 8 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{91} + ( -7 + 5 \beta_{1} + 6 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 6 \beta_{7} ) q^{92} + ( -4 + 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} ) q^{93} + ( 2 - 2 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} - 12 \beta_{7} ) q^{94} + ( -5 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 6 \beta_{6} - \beta_{7} ) q^{96} + ( -2 - \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{97} + ( 4 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} ) q^{98} + ( -2 - 8 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{2} + 6q^{3} - 2q^{4} - 18q^{6} + 10q^{7} + 12q^{8} + 4q^{9} + O(q^{10}) \) \( 8q - 2q^{2} + 6q^{3} - 2q^{4} - 18q^{6} + 10q^{7} + 12q^{8} + 4q^{9} + 8q^{13} - 4q^{14} - 2q^{16} + 18q^{17} - 40q^{18} - 12q^{19} + 6q^{22} + 6q^{23} + 12q^{24} + 10q^{26} + 8q^{28} + 8q^{29} - 4q^{32} - 18q^{33} + 20q^{36} - 2q^{37} + 12q^{41} - 42q^{42} - 18q^{43} - 42q^{46} - 16q^{47} - 6q^{48} - 12q^{49} - 8q^{51} + 16q^{52} - 18q^{54} + 12q^{56} - 28q^{57} + 22q^{58} + 12q^{59} - 28q^{61} + 12q^{62} - 4q^{63} + 8q^{64} + 12q^{66} - 30q^{67} + 12q^{68} + 16q^{69} + 12q^{72} - 16q^{73} - 10q^{74} + 54q^{76} + 18q^{78} + 16q^{79} + 8q^{81} - 6q^{82} + 24q^{83} + 30q^{84} + 54q^{87} + 42q^{88} - 24q^{89} + 28q^{91} + 8q^{93} - 32q^{94} - 2q^{97} + 24q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} + \nu^{5} + 4 \nu^{4} - 3 \nu^{3} - 2 \nu^{2} + 8 \nu - 8 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} - \nu^{5} - 4 \nu^{4} + 3 \nu^{3} + 10 \nu^{2} - 16 \nu + 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 7 \nu^{3} - 3 \nu^{2} + 18 \nu - 16 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{7} - 5 \nu^{6} + 2 \nu^{5} + 7 \nu^{4} - 8 \nu^{3} - 9 \nu^{2} + 28 \nu - 20 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} + 7 \nu^{6} - 3 \nu^{5} - 11 \nu^{4} + 15 \nu^{3} + 11 \nu^{2} - 40 \nu + 32 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 7 \nu^{7} - 20 \nu^{6} + 11 \nu^{5} + 30 \nu^{4} - 45 \nu^{3} - 28 \nu^{2} + 116 \nu - 88 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + 2 \beta_{2} + \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + \beta_{1} - 1\)
\(\nu^{5}\)\(=\)\(\beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 1\)
\(\nu^{6}\)\(=\)\(-\beta_{7} - 3 \beta_{6} - 5 \beta_{5} + \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 4 \beta_{1} - 1\)
\(\nu^{7}\)\(=\)\(-6 \beta_{7} - 8 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{2} + \beta_{1} + 6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1 - \beta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.20036 + 0.747754i
1.40994 0.109843i
−1.27597 + 0.609843i
0.665665 1.24775i
1.20036 0.747754i
1.40994 + 0.109843i
−1.27597 0.609843i
0.665665 + 1.24775i
−1.24775 2.16117i 2.44811 1.41342i −2.11378 + 3.66117i 0 −6.10929 3.52720i 0.952606 1.64996i 5.55889 2.49551 4.32235i 0
49.2 −0.609843 1.05628i 2.01978 1.16612i 0.256182 0.443720i 0 −2.46350 1.42231i 1.80010 3.11786i −3.06430 1.21969 2.11256i 0
49.3 0.109843 + 0.190254i −1.38581 + 0.800098i 0.975869 1.69025i 0 −0.304444 0.175771i −0.166123 + 0.287734i 0.868145 −0.219687 + 0.380509i 0
49.4 0.747754 + 1.29515i −0.0820885 + 0.0473938i −0.118272 + 0.204852i 0 −0.122764 0.0708778i 2.41342 4.18016i 2.63726 −1.49551 + 2.59030i 0
199.1 −1.24775 + 2.16117i 2.44811 + 1.41342i −2.11378 3.66117i 0 −6.10929 + 3.52720i 0.952606 + 1.64996i 5.55889 2.49551 + 4.32235i 0
199.2 −0.609843 + 1.05628i 2.01978 + 1.16612i 0.256182 + 0.443720i 0 −2.46350 + 1.42231i 1.80010 + 3.11786i −3.06430 1.21969 + 2.11256i 0
199.3 0.109843 0.190254i −1.38581 0.800098i 0.975869 + 1.69025i 0 −0.304444 + 0.175771i −0.166123 0.287734i 0.868145 −0.219687 0.380509i 0
199.4 0.747754 1.29515i −0.0820885 0.0473938i −0.118272 0.204852i 0 −0.122764 + 0.0708778i 2.41342 + 4.18016i 2.63726 −1.49551 2.59030i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.m.b 8
5.b even 2 1 325.2.m.c 8
5.c odd 4 1 65.2.m.a 8
5.c odd 4 1 325.2.n.d 8
13.e even 6 1 325.2.m.c 8
15.e even 4 1 585.2.bu.c 8
20.e even 4 1 1040.2.da.b 8
65.f even 4 1 845.2.e.n 8
65.h odd 4 1 845.2.m.g 8
65.k even 4 1 845.2.e.m 8
65.l even 6 1 inner 325.2.m.b 8
65.o even 12 1 845.2.a.m 4
65.o even 12 1 845.2.e.m 8
65.o even 12 1 4225.2.a.bl 4
65.q odd 12 1 845.2.c.g 8
65.q odd 12 1 845.2.m.g 8
65.r odd 12 1 65.2.m.a 8
65.r odd 12 1 325.2.n.d 8
65.r odd 12 1 845.2.c.g 8
65.t even 12 1 845.2.a.l 4
65.t even 12 1 845.2.e.n 8
65.t even 12 1 4225.2.a.bi 4
195.bc odd 12 1 7605.2.a.cj 4
195.bf even 12 1 585.2.bu.c 8
195.bn odd 12 1 7605.2.a.cf 4
260.bg even 12 1 1040.2.da.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 5.c odd 4 1
65.2.m.a 8 65.r odd 12 1
325.2.m.b 8 1.a even 1 1 trivial
325.2.m.b 8 65.l even 6 1 inner
325.2.m.c 8 5.b even 2 1
325.2.m.c 8 13.e even 6 1
325.2.n.d 8 5.c odd 4 1
325.2.n.d 8 65.r odd 12 1
585.2.bu.c 8 15.e even 4 1
585.2.bu.c 8 195.bf even 12 1
845.2.a.l 4 65.t even 12 1
845.2.a.m 4 65.o even 12 1
845.2.c.g 8 65.q odd 12 1
845.2.c.g 8 65.r odd 12 1
845.2.e.m 8 65.k even 4 1
845.2.e.m 8 65.o even 12 1
845.2.e.n 8 65.f even 4 1
845.2.e.n 8 65.t even 12 1
845.2.m.g 8 65.h odd 4 1
845.2.m.g 8 65.q odd 12 1
1040.2.da.b 8 20.e even 4 1
1040.2.da.b 8 260.bg even 12 1
4225.2.a.bi 4 65.t even 12 1
4225.2.a.bl 4 65.o even 12 1
7605.2.a.cf 4 195.bn odd 12 1
7605.2.a.cj 4 195.bc odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 19 T^{2} + 8 T^{3} + 16 T^{4} + 2 T^{5} + 7 T^{6} + 2 T^{7} + T^{8} \)
$3$ \( 1 + 18 T + 106 T^{2} - 36 T^{3} - 33 T^{4} + 12 T^{5} + 10 T^{6} - 6 T^{7} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( 121 + 242 T + 814 T^{2} - 880 T^{3} + 691 T^{4} - 256 T^{5} + 70 T^{6} - 10 T^{7} + T^{8} \)
$11$ \( 1089 - 990 T^{2} + 867 T^{4} - 30 T^{6} + T^{8} \)
$13$ \( 28561 - 17576 T + 2704 T^{2} + 520 T^{3} - 290 T^{4} + 40 T^{5} + 16 T^{6} - 8 T^{7} + T^{8} \)
$17$ \( 169 - 858 T + 1894 T^{2} - 2244 T^{3} + 1539 T^{4} - 612 T^{5} + 142 T^{6} - 18 T^{7} + T^{8} \)
$19$ \( ( 169 - 78 T - T^{2} + 6 T^{3} + T^{4} )^{2} \)
$23$ \( 89401 - 113022 T + 58990 T^{2} - 14364 T^{3} + 987 T^{4} + 228 T^{5} - 26 T^{6} - 6 T^{7} + T^{8} \)
$29$ \( 1 + 40 T + 1618 T^{2} - 704 T^{3} + 643 T^{4} + 64 T^{5} + 82 T^{6} - 8 T^{7} + T^{8} \)
$31$ \( ( 64 + 32 T^{2} + T^{4} )^{2} \)
$37$ \( 1 + 38 T + 1498 T^{2} - 2056 T^{3} + 2839 T^{4} - 184 T^{5} + 58 T^{6} + 2 T^{7} + T^{8} \)
$41$ \( ( 1 + 6 T + 11 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$43$ \( 169 + 858 T + 1894 T^{2} + 2244 T^{3} + 1539 T^{4} + 612 T^{5} + 142 T^{6} + 18 T^{7} + T^{8} \)
$47$ \( ( -1328 - 736 T - 72 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$53$ \( 2304 + 3456 T^{2} + 1200 T^{4} + 72 T^{6} + T^{8} \)
$59$ \( 9 - 108 T + 486 T^{2} - 648 T^{3} + 183 T^{4} + 216 T^{5} + 30 T^{6} - 12 T^{7} + T^{8} \)
$61$ \( 1590121 + 1215604 T + 603958 T^{2} + 178096 T^{3} + 38311 T^{4} + 5296 T^{5} + 526 T^{6} + 28 T^{7} + T^{8} \)
$67$ \( 7667361 + 4369482 T + 1576314 T^{2} + 354600 T^{3} + 58791 T^{4} + 6744 T^{5} + 570 T^{6} + 30 T^{7} + T^{8} \)
$71$ \( 109767529 + 2263032 T - 2268434 T^{2} - 47088 T^{3} + 37047 T^{4} - 218 T^{6} + T^{8} \)
$73$ \( ( -1712 - 832 T - 84 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$79$ \( ( 4432 + 640 T - 132 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$83$ \( ( -192 + 288 T - 24 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$89$ \( 78375609 - 34420464 T + 3179718 T^{2} + 816480 T^{3} + 4143 T^{4} - 5040 T^{5} - 18 T^{6} + 24 T^{7} + T^{8} \)
$97$ \( 196249 - 165682 T + 100006 T^{2} - 31888 T^{3} + 7795 T^{4} - 928 T^{5} + 94 T^{6} + 2 T^{7} + T^{8} \)
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