Properties

Label 300.3.c.c
Level $300$
Weight $3$
Character orbit 300.c
Analytic conductor $8.174$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + ( 4 - 2 \zeta_{6} ) q^{6} + ( 6 - 12 \zeta_{6} ) q^{7} -8 q^{8} -3 q^{9} +O(q^{10})\) \( q + 2 \zeta_{6} q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + ( 4 - 2 \zeta_{6} ) q^{6} + ( 6 - 12 \zeta_{6} ) q^{7} -8 q^{8} -3 q^{9} + ( 6 - 12 \zeta_{6} ) q^{11} + ( 4 + 4 \zeta_{6} ) q^{12} + 18 q^{13} + ( 24 - 12 \zeta_{6} ) q^{14} -16 \zeta_{6} q^{16} + 10 q^{17} -6 \zeta_{6} q^{18} + ( 8 - 16 \zeta_{6} ) q^{19} -18 q^{21} + ( 24 - 12 \zeta_{6} ) q^{22} + ( 4 - 8 \zeta_{6} ) q^{23} + ( -8 + 16 \zeta_{6} ) q^{24} + 36 \zeta_{6} q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 24 + 24 \zeta_{6} ) q^{28} -36 q^{29} + ( -4 + 8 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} -18 q^{33} + 20 \zeta_{6} q^{34} + ( 12 - 12 \zeta_{6} ) q^{36} + 54 q^{37} + ( 32 - 16 \zeta_{6} ) q^{38} + ( 18 - 36 \zeta_{6} ) q^{39} + 18 q^{41} -36 \zeta_{6} q^{42} + ( 12 - 24 \zeta_{6} ) q^{43} + ( 24 + 24 \zeta_{6} ) q^{44} + ( 16 - 8 \zeta_{6} ) q^{46} + ( -32 + 16 \zeta_{6} ) q^{48} -59 q^{49} + ( 10 - 20 \zeta_{6} ) q^{51} + ( -72 + 72 \zeta_{6} ) q^{52} -26 q^{53} + ( -12 + 6 \zeta_{6} ) q^{54} + ( -48 + 96 \zeta_{6} ) q^{56} -24 q^{57} -72 \zeta_{6} q^{58} + ( -18 + 36 \zeta_{6} ) q^{59} -74 q^{61} + ( -16 + 8 \zeta_{6} ) q^{62} + ( -18 + 36 \zeta_{6} ) q^{63} + 64 q^{64} -36 \zeta_{6} q^{66} + ( 24 - 48 \zeta_{6} ) q^{67} + ( -40 + 40 \zeta_{6} ) q^{68} -12 q^{69} + ( -60 + 120 \zeta_{6} ) q^{71} + 24 q^{72} + 36 q^{73} + 108 \zeta_{6} q^{74} + ( 32 + 32 \zeta_{6} ) q^{76} -108 q^{77} + ( 72 - 36 \zeta_{6} ) q^{78} + ( 52 - 104 \zeta_{6} ) q^{79} + 9 q^{81} + 36 \zeta_{6} q^{82} + ( -52 + 104 \zeta_{6} ) q^{83} + ( 72 - 72 \zeta_{6} ) q^{84} + ( 48 - 24 \zeta_{6} ) q^{86} + ( -36 + 72 \zeta_{6} ) q^{87} + ( -48 + 96 \zeta_{6} ) q^{88} -18 q^{89} + ( 108 - 216 \zeta_{6} ) q^{91} + ( 16 + 16 \zeta_{6} ) q^{92} + 12 q^{93} + ( -32 - 32 \zeta_{6} ) q^{96} -72 q^{97} -118 \zeta_{6} q^{98} + ( -18 + 36 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{4} + 6q^{6} - 16q^{8} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{4} + 6q^{6} - 16q^{8} - 6q^{9} + 12q^{12} + 36q^{13} + 36q^{14} - 16q^{16} + 20q^{17} - 6q^{18} - 36q^{21} + 36q^{22} + 36q^{26} + 72q^{28} - 72q^{29} + 32q^{32} - 36q^{33} + 20q^{34} + 12q^{36} + 108q^{37} + 48q^{38} + 36q^{41} - 36q^{42} + 72q^{44} + 24q^{46} - 48q^{48} - 118q^{49} - 72q^{52} - 52q^{53} - 18q^{54} - 48q^{57} - 72q^{58} - 148q^{61} - 24q^{62} + 128q^{64} - 36q^{66} - 40q^{68} - 24q^{69} + 48q^{72} + 72q^{73} + 108q^{74} + 96q^{76} - 216q^{77} + 108q^{78} + 18q^{81} + 36q^{82} + 72q^{84} + 72q^{86} - 36q^{89} + 48q^{92} + 24q^{93} - 96q^{96} - 144q^{97} - 118q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 1.73205i 1.73205i −2.00000 3.46410i 0 3.00000 + 1.73205i 10.3923i −8.00000 −3.00000 0
151.2 1.00000 + 1.73205i 1.73205i −2.00000 + 3.46410i 0 3.00000 1.73205i 10.3923i −8.00000 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.c.c 2
3.b odd 2 1 900.3.c.f 2
4.b odd 2 1 inner 300.3.c.c 2
5.b even 2 1 300.3.c.a 2
5.c odd 4 2 60.3.f.a 4
12.b even 2 1 900.3.c.f 2
15.d odd 2 1 900.3.c.j 2
15.e even 4 2 180.3.f.e 4
20.d odd 2 1 300.3.c.a 2
20.e even 4 2 60.3.f.a 4
40.i odd 4 2 960.3.j.b 4
40.k even 4 2 960.3.j.b 4
60.h even 2 1 900.3.c.j 2
60.l odd 4 2 180.3.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.a 4 5.c odd 4 2
60.3.f.a 4 20.e even 4 2
180.3.f.e 4 15.e even 4 2
180.3.f.e 4 60.l odd 4 2
300.3.c.a 2 5.b even 2 1
300.3.c.a 2 20.d odd 2 1
300.3.c.c 2 1.a even 1 1 trivial
300.3.c.c 2 4.b odd 2 1 inner
900.3.c.f 2 3.b odd 2 1
900.3.c.f 2 12.b even 2 1
900.3.c.j 2 15.d odd 2 1
900.3.c.j 2 60.h even 2 1
960.3.j.b 4 40.i odd 4 2
960.3.j.b 4 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{2} + 108 \)
\( T_{13} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T + T^{2} \)
$3$ \( 3 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 108 + T^{2} \)
$11$ \( 108 + T^{2} \)
$13$ \( ( -18 + T )^{2} \)
$17$ \( ( -10 + T )^{2} \)
$19$ \( 192 + T^{2} \)
$23$ \( 48 + T^{2} \)
$29$ \( ( 36 + T )^{2} \)
$31$ \( 48 + T^{2} \)
$37$ \( ( -54 + T )^{2} \)
$41$ \( ( -18 + T )^{2} \)
$43$ \( 432 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( 26 + T )^{2} \)
$59$ \( 972 + T^{2} \)
$61$ \( ( 74 + T )^{2} \)
$67$ \( 1728 + T^{2} \)
$71$ \( 10800 + T^{2} \)
$73$ \( ( -36 + T )^{2} \)
$79$ \( 8112 + T^{2} \)
$83$ \( 8112 + T^{2} \)
$89$ \( ( 18 + T )^{2} \)
$97$ \( ( 72 + T )^{2} \)
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