Properties

Label 2015.1.h.e
Level 2015
Weight 1
Character orbit 2015.h
Self dual Yes
Analytic conductor 1.006
Analytic rank 0
Dimension 6
Projective image \(D_{13}\)
CM disc. -2015
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 2015 = 5 \cdot 13 \cdot 31 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2015.h (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(1.00561600046\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{26})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{13}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{13} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + \beta_{3} q^{3} \) \( + ( 1 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + ( \beta_{2} + \beta_{4} ) q^{6} \) \( -\beta_{4} q^{7} \) \( + ( \beta_{1} + \beta_{3} ) q^{8} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + \beta_{3} q^{3} \) \( + ( 1 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( + ( \beta_{2} + \beta_{4} ) q^{6} \) \( -\beta_{4} q^{7} \) \( + ( \beta_{1} + \beta_{3} ) q^{8} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} \) \( -\beta_{1} q^{10} \) \( -\beta_{5} q^{11} \) \( + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{12} \) \(- q^{13}\) \( + ( -\beta_{3} - \beta_{5} ) q^{14} \) \( -\beta_{3} q^{15} \) \( + ( 1 + \beta_{2} + \beta_{4} ) q^{16} \) \( -\beta_{2} q^{17} \) \( + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} ) q^{18} \) \( + ( -1 - \beta_{2} ) q^{20} \) \( + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{21} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{22} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{23} \) \( + ( 1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{24} \) \(+ q^{25}\) \( -\beta_{1} q^{26} \) \( + ( \beta_{3} - \beta_{4} ) q^{27} \) \( + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} ) q^{28} \) \( + ( -\beta_{2} - \beta_{4} ) q^{30} \) \(+ q^{31}\) \( + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{32} \) \( + ( -\beta_{2} + \beta_{5} ) q^{33} \) \( + ( -\beta_{1} - \beta_{3} ) q^{34} \) \( + \beta_{4} q^{35} \) \( + ( \beta_{1} + \beta_{3} ) q^{36} \) \( -\beta_{3} q^{39} \) \( + ( -\beta_{1} - \beta_{3} ) q^{40} \) \( + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{42} \) \( + \beta_{1} q^{43} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{44} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{45} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{46} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{47} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} ) q^{48} \) \( + ( 1 - \beta_{5} ) q^{49} \) \( + \beta_{1} q^{50} \) \( + ( -\beta_{1} - \beta_{5} ) q^{51} \) \( + ( -1 - \beta_{2} ) q^{52} \) \( + \beta_{5} q^{53} \) \( + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{54} \) \( + \beta_{5} q^{55} \) \( + ( -1 - \beta_{2} - \beta_{4} ) q^{56} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{60} \) \( + \beta_{1} q^{62} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{4} ) q^{63} \) \( + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{64} \) \(+ q^{65}\) \( + ( -1 - \beta_{2} + \beta_{5} ) q^{66} \) \( + \beta_{5} q^{67} \) \( + ( -2 - \beta_{2} - \beta_{4} ) q^{68} \) \( + ( -\beta_{3} + \beta_{4} ) q^{69} \) \( + ( \beta_{3} + \beta_{5} ) q^{70} \) \( + ( 1 + \beta_{2} + \beta_{3} ) q^{72} \) \( + \beta_{3} q^{75} \) \( + ( \beta_{1} - \beta_{4} ) q^{77} \) \( + ( -\beta_{2} - \beta_{4} ) q^{78} \) \( + ( -1 - \beta_{2} - \beta_{4} ) q^{80} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{81} \) \( + ( -1 - \beta_{1} - \beta_{2} ) q^{84} \) \( + \beta_{2} q^{85} \) \( + ( 2 + \beta_{2} ) q^{86} \) \( + ( 1 - \beta_{1} - \beta_{3} ) q^{88} \) \( + \beta_{4} q^{89} \) \( + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{90} \) \( + \beta_{4} q^{91} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{92} \) \( + \beta_{3} q^{93} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{94} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{96} \) \( + \beta_{3} q^{97} \) \( + ( 1 + \beta_{2} - \beta_{3} - \beta_{5} ) q^{98} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 5q^{9} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut +\mathstrut 3q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 2q^{14} \) \(\mathstrut -\mathstrut q^{15} \) \(\mathstrut +\mathstrut 4q^{16} \) \(\mathstrut +\mathstrut q^{17} \) \(\mathstrut +\mathstrut 3q^{18} \) \(\mathstrut -\mathstrut 5q^{20} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut 2q^{22} \) \(\mathstrut +\mathstrut q^{23} \) \(\mathstrut +\mathstrut 9q^{24} \) \(\mathstrut +\mathstrut 6q^{25} \) \(\mathstrut -\mathstrut q^{26} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut +\mathstrut 3q^{28} \) \(\mathstrut +\mathstrut 2q^{30} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut +\mathstrut 3q^{32} \) \(\mathstrut +\mathstrut 2q^{33} \) \(\mathstrut -\mathstrut 2q^{34} \) \(\mathstrut -\mathstrut q^{35} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut q^{39} \) \(\mathstrut -\mathstrut 2q^{40} \) \(\mathstrut -\mathstrut 9q^{42} \) \(\mathstrut +\mathstrut q^{43} \) \(\mathstrut -\mathstrut 3q^{44} \) \(\mathstrut -\mathstrut 5q^{45} \) \(\mathstrut -\mathstrut 2q^{46} \) \(\mathstrut +\mathstrut q^{47} \) \(\mathstrut +\mathstrut 5q^{48} \) \(\mathstrut +\mathstrut 5q^{49} \) \(\mathstrut +\mathstrut q^{50} \) \(\mathstrut -\mathstrut 2q^{51} \) \(\mathstrut -\mathstrut 5q^{52} \) \(\mathstrut +\mathstrut q^{53} \) \(\mathstrut -\mathstrut 4q^{54} \) \(\mathstrut +\mathstrut q^{55} \) \(\mathstrut -\mathstrut 4q^{56} \) \(\mathstrut -\mathstrut 3q^{60} \) \(\mathstrut +\mathstrut q^{62} \) \(\mathstrut +\mathstrut 3q^{63} \) \(\mathstrut +\mathstrut 3q^{64} \) \(\mathstrut +\mathstrut 6q^{65} \) \(\mathstrut -\mathstrut 4q^{66} \) \(\mathstrut +\mathstrut q^{67} \) \(\mathstrut -\mathstrut 10q^{68} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut 2q^{70} \) \(\mathstrut +\mathstrut 6q^{72} \) \(\mathstrut +\mathstrut q^{75} \) \(\mathstrut +\mathstrut 2q^{77} \) \(\mathstrut +\mathstrut 2q^{78} \) \(\mathstrut -\mathstrut 4q^{80} \) \(\mathstrut +\mathstrut 4q^{81} \) \(\mathstrut -\mathstrut 6q^{84} \) \(\mathstrut -\mathstrut q^{85} \) \(\mathstrut +\mathstrut 11q^{86} \) \(\mathstrut +\mathstrut 4q^{88} \) \(\mathstrut -\mathstrut q^{89} \) \(\mathstrut -\mathstrut 3q^{90} \) \(\mathstrut -\mathstrut q^{91} \) \(\mathstrut +\mathstrut 3q^{92} \) \(\mathstrut +\mathstrut q^{93} \) \(\mathstrut -\mathstrut 2q^{94} \) \(\mathstrut +\mathstrut 7q^{96} \) \(\mathstrut +\mathstrut q^{97} \) \(\mathstrut +\mathstrut 3q^{98} \) \(\mathstrut -\mathstrut 3q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of \(\nu = \zeta_{26} + \zeta_{26}^{-1}\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 3 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 4 \nu^{2} + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 5 \nu^{3} + 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(6\)
\(\nu^{5}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2015\mathbb{Z}\right)^\times\).

\(n\) \(716\) \(807\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2014.1
−1.77091
−1.13613
−0.241073
0.709210
1.49702
1.94188
−1.77091 −0.241073 2.13613 −1.00000 0.426920 0.709210 −2.01199 −0.941884 1.77091
2014.2 −1.13613 1.94188 0.290790 −1.00000 −2.20623 1.49702 0.805754 2.77091 1.13613
2014.3 −0.241073 0.709210 −0.941884 −1.00000 −0.170972 −1.77091 0.468136 −0.497021 0.241073
2014.4 0.709210 −1.77091 −0.497021 −1.00000 −1.25595 −0.241073 −1.06170 2.13613 −0.709210
2014.5 1.49702 −1.13613 1.24107 −1.00000 −1.70081 1.94188 0.360892 0.290790 −1.49702
2014.6 1.94188 1.49702 2.77091 −1.00000 2.90704 −1.13613 3.43891 1.24107 −1.94188
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2014.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
2015.h Odd 1 CM by \(\Q(\sqrt{-2015}) \) yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2015, [\chi])\):

\(T_{2}^{6} \) \(\mathstrut -\mathstrut T_{2}^{5} \) \(\mathstrut -\mathstrut 5 T_{2}^{4} \) \(\mathstrut +\mathstrut 4 T_{2}^{3} \) \(\mathstrut +\mathstrut 6 T_{2}^{2} \) \(\mathstrut -\mathstrut 3 T_{2} \) \(\mathstrut -\mathstrut 1 \)
\(T_{3}^{6} \) \(\mathstrut -\mathstrut T_{3}^{5} \) \(\mathstrut -\mathstrut 5 T_{3}^{4} \) \(\mathstrut +\mathstrut 4 T_{3}^{3} \) \(\mathstrut +\mathstrut 6 T_{3}^{2} \) \(\mathstrut -\mathstrut 3 T_{3} \) \(\mathstrut -\mathstrut 1 \)