Properties

Label 3800.1.cq.b
Level $3800$
Weight $1$
Character orbit 3800.cq
Analytic conductor $1.896$
Analytic rank $0$
Dimension $12$
Projective image $D_{9}$
CM discriminant -8
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.cq (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Defining polynomial: \(x^{12} - x^{6} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.69564674215936.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{36} q^{2} + ( \zeta_{36}^{3} - \zeta_{36}^{5} ) q^{3} + \zeta_{36}^{2} q^{4} + ( \zeta_{36}^{4} - \zeta_{36}^{6} ) q^{6} + \zeta_{36}^{3} q^{8} + ( \zeta_{36}^{6} - \zeta_{36}^{8} + \zeta_{36}^{10} ) q^{9} +O(q^{10})\) \( q + \zeta_{36} q^{2} + ( \zeta_{36}^{3} - \zeta_{36}^{5} ) q^{3} + \zeta_{36}^{2} q^{4} + ( \zeta_{36}^{4} - \zeta_{36}^{6} ) q^{6} + \zeta_{36}^{3} q^{8} + ( \zeta_{36}^{6} - \zeta_{36}^{8} + \zeta_{36}^{10} ) q^{9} + ( \zeta_{36}^{8} + \zeta_{36}^{16} ) q^{11} + ( \zeta_{36}^{5} - \zeta_{36}^{7} ) q^{12} + \zeta_{36}^{4} q^{16} -\zeta_{36} q^{17} + ( \zeta_{36}^{7} - \zeta_{36}^{9} + \zeta_{36}^{11} ) q^{18} + \zeta_{36}^{2} q^{19} + ( \zeta_{36}^{9} + \zeta_{36}^{17} ) q^{22} + ( \zeta_{36}^{6} - \zeta_{36}^{8} ) q^{24} + ( \zeta_{36}^{9} - \zeta_{36}^{11} + \zeta_{36}^{13} - \zeta_{36}^{15} ) q^{27} + \zeta_{36}^{5} q^{32} + ( -\zeta_{36} + \zeta_{36}^{3} + \zeta_{36}^{11} - \zeta_{36}^{13} ) q^{33} -\zeta_{36}^{2} q^{34} + ( \zeta_{36}^{8} - \zeta_{36}^{10} + \zeta_{36}^{12} ) q^{36} + \zeta_{36}^{3} q^{38} + ( -\zeta_{36}^{2} - \zeta_{36}^{6} ) q^{41} -\zeta_{36}^{7} q^{43} + ( -1 + \zeta_{36}^{10} ) q^{44} + ( \zeta_{36}^{7} - \zeta_{36}^{9} ) q^{48} -\zeta_{36}^{12} q^{49} + ( -\zeta_{36}^{4} + \zeta_{36}^{6} ) q^{51} + ( \zeta_{36}^{10} - \zeta_{36}^{12} + \zeta_{36}^{14} - \zeta_{36}^{16} ) q^{54} + ( \zeta_{36}^{5} - \zeta_{36}^{7} ) q^{57} + ( \zeta_{36}^{6} + \zeta_{36}^{14} ) q^{59} + \zeta_{36}^{6} q^{64} + ( -\zeta_{36}^{2} + \zeta_{36}^{4} + \zeta_{36}^{12} - \zeta_{36}^{14} ) q^{66} + ( \zeta_{36} - \zeta_{36}^{15} ) q^{67} -\zeta_{36}^{3} q^{68} + ( \zeta_{36}^{9} - \zeta_{36}^{11} + \zeta_{36}^{13} ) q^{72} + ( -\zeta_{36}^{9} - \zeta_{36}^{17} ) q^{73} + \zeta_{36}^{4} q^{76} + ( 1 - \zeta_{36}^{2} + \zeta_{36}^{12} - \zeta_{36}^{14} + \zeta_{36}^{16} ) q^{81} + ( -\zeta_{36}^{3} - \zeta_{36}^{7} ) q^{82} + ( -\zeta_{36}^{13} - \zeta_{36}^{17} ) q^{83} -\zeta_{36}^{8} q^{86} + ( -\zeta_{36} + \zeta_{36}^{11} ) q^{88} -\zeta_{36}^{14} q^{89} + ( \zeta_{36}^{8} - \zeta_{36}^{10} ) q^{96} + ( -\zeta_{36}^{3} + \zeta_{36}^{17} ) q^{97} -\zeta_{36}^{13} q^{98} + ( -1 - \zeta_{36}^{4} + \zeta_{36}^{6} - \zeta_{36}^{8} + \zeta_{36}^{14} - \zeta_{36}^{16} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{6} + 6 q^{9} + O(q^{10}) \) \( 12 q - 6 q^{6} + 6 q^{9} + 6 q^{24} - 6 q^{36} - 6 q^{41} - 12 q^{44} + 6 q^{49} + 6 q^{51} + 6 q^{54} + 6 q^{59} + 6 q^{64} - 6 q^{66} + 6 q^{81} - 6 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(-\zeta_{36}^{2}\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−0.342020 + 0.939693i
0.342020 0.939693i
−0.342020 0.939693i
0.342020 + 0.939693i
−0.642788 0.766044i
0.642788 + 0.766044i
−0.642788 + 0.766044i
0.642788 0.766044i
−0.984808 0.173648i
0.984808 + 0.173648i
−0.984808 + 0.173648i
0.984808 0.173648i
−0.342020 + 0.939693i 1.85083 0.326352i −0.766044 0.642788i 0 −0.326352 + 1.85083i 0 0.866025 0.500000i 2.37939 0.866025i 0
99.2 0.342020 0.939693i −1.85083 + 0.326352i −0.766044 0.642788i 0 −0.326352 + 1.85083i 0 −0.866025 + 0.500000i 2.37939 0.866025i 0
499.1 −0.342020 0.939693i 1.85083 + 0.326352i −0.766044 + 0.642788i 0 −0.326352 1.85083i 0 0.866025 + 0.500000i 2.37939 + 0.866025i 0
499.2 0.342020 + 0.939693i −1.85083 0.326352i −0.766044 + 0.642788i 0 −0.326352 1.85083i 0 −0.866025 0.500000i 2.37939 + 0.866025i 0
899.1 −0.642788 0.766044i 0.524005 1.43969i −0.173648 + 0.984808i 0 −1.43969 + 0.524005i 0 0.866025 0.500000i −1.03209 0.866025i 0
899.2 0.642788 + 0.766044i −0.524005 + 1.43969i −0.173648 + 0.984808i 0 −1.43969 + 0.524005i 0 −0.866025 + 0.500000i −1.03209 0.866025i 0
1099.1 −0.642788 + 0.766044i 0.524005 + 1.43969i −0.173648 0.984808i 0 −1.43969 0.524005i 0 0.866025 + 0.500000i −1.03209 + 0.866025i 0
1099.2 0.642788 0.766044i −0.524005 1.43969i −0.173648 0.984808i 0 −1.43969 0.524005i 0 −0.866025 0.500000i −1.03209 + 0.866025i 0
1499.1 −0.984808 0.173648i −0.223238 + 0.266044i 0.939693 + 0.342020i 0 0.266044 0.223238i 0 −0.866025 0.500000i 0.152704 + 0.866025i 0
1499.2 0.984808 + 0.173648i 0.223238 0.266044i 0.939693 + 0.342020i 0 0.266044 0.223238i 0 0.866025 + 0.500000i 0.152704 + 0.866025i 0
2099.1 −0.984808 + 0.173648i −0.223238 0.266044i 0.939693 0.342020i 0 0.266044 + 0.223238i 0 −0.866025 + 0.500000i 0.152704 0.866025i 0
2099.2 0.984808 0.173648i 0.223238 + 0.266044i 0.939693 0.342020i 0 0.266044 + 0.223238i 0 0.866025 0.500000i 0.152704 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2099.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
5.b even 2 1 inner
19.e even 9 1 inner
40.e odd 2 1 inner
95.p even 18 1 inner
152.u odd 18 1 inner
760.bz odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.cq.b 12
5.b even 2 1 inner 3800.1.cq.b 12
5.c odd 4 1 152.1.u.a 6
5.c odd 4 1 3800.1.cv.c 6
8.d odd 2 1 CM 3800.1.cq.b 12
15.e even 4 1 1368.1.eh.a 6
19.e even 9 1 inner 3800.1.cq.b 12
20.e even 4 1 608.1.bg.a 6
40.e odd 2 1 inner 3800.1.cq.b 12
40.i odd 4 1 608.1.bg.a 6
40.k even 4 1 152.1.u.a 6
40.k even 4 1 3800.1.cv.c 6
95.g even 4 1 2888.1.u.e 6
95.l even 12 1 2888.1.u.a 6
95.l even 12 1 2888.1.u.f 6
95.m odd 12 1 2888.1.u.b 6
95.m odd 12 1 2888.1.u.g 6
95.p even 18 1 inner 3800.1.cq.b 12
95.q odd 36 1 152.1.u.a 6
95.q odd 36 1 2888.1.f.d 3
95.q odd 36 2 2888.1.k.b 6
95.q odd 36 1 2888.1.u.b 6
95.q odd 36 1 2888.1.u.g 6
95.q odd 36 1 3800.1.cv.c 6
95.r even 36 1 2888.1.f.c 3
95.r even 36 2 2888.1.k.c 6
95.r even 36 1 2888.1.u.a 6
95.r even 36 1 2888.1.u.e 6
95.r even 36 1 2888.1.u.f 6
120.q odd 4 1 1368.1.eh.a 6
152.u odd 18 1 inner 3800.1.cq.b 12
285.bi even 36 1 1368.1.eh.a 6
380.bj even 36 1 608.1.bg.a 6
760.y odd 4 1 2888.1.u.e 6
760.bu odd 12 1 2888.1.u.a 6
760.bu odd 12 1 2888.1.u.f 6
760.bw even 12 1 2888.1.u.b 6
760.bw even 12 1 2888.1.u.g 6
760.bz odd 18 1 inner 3800.1.cq.b 12
760.cn odd 36 1 2888.1.f.c 3
760.cn odd 36 2 2888.1.k.c 6
760.cn odd 36 1 2888.1.u.a 6
760.cn odd 36 1 2888.1.u.e 6
760.cn odd 36 1 2888.1.u.f 6
760.cp even 36 1 152.1.u.a 6
760.cp even 36 1 2888.1.f.d 3
760.cp even 36 2 2888.1.k.b 6
760.cp even 36 1 2888.1.u.b 6
760.cp even 36 1 2888.1.u.g 6
760.cp even 36 1 3800.1.cv.c 6
760.cq odd 36 1 608.1.bg.a 6
2280.fl odd 36 1 1368.1.eh.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a 6 5.c odd 4 1
152.1.u.a 6 40.k even 4 1
152.1.u.a 6 95.q odd 36 1
152.1.u.a 6 760.cp even 36 1
608.1.bg.a 6 20.e even 4 1
608.1.bg.a 6 40.i odd 4 1
608.1.bg.a 6 380.bj even 36 1
608.1.bg.a 6 760.cq odd 36 1
1368.1.eh.a 6 15.e even 4 1
1368.1.eh.a 6 120.q odd 4 1
1368.1.eh.a 6 285.bi even 36 1
1368.1.eh.a 6 2280.fl odd 36 1
2888.1.f.c 3 95.r even 36 1
2888.1.f.c 3 760.cn odd 36 1
2888.1.f.d 3 95.q odd 36 1
2888.1.f.d 3 760.cp even 36 1
2888.1.k.b 6 95.q odd 36 2
2888.1.k.b 6 760.cp even 36 2
2888.1.k.c 6 95.r even 36 2
2888.1.k.c 6 760.cn odd 36 2
2888.1.u.a 6 95.l even 12 1
2888.1.u.a 6 95.r even 36 1
2888.1.u.a 6 760.bu odd 12 1
2888.1.u.a 6 760.cn odd 36 1
2888.1.u.b 6 95.m odd 12 1
2888.1.u.b 6 95.q odd 36 1
2888.1.u.b 6 760.bw even 12 1
2888.1.u.b 6 760.cp even 36 1
2888.1.u.e 6 95.g even 4 1
2888.1.u.e 6 95.r even 36 1
2888.1.u.e 6 760.y odd 4 1
2888.1.u.e 6 760.cn odd 36 1
2888.1.u.f 6 95.l even 12 1
2888.1.u.f 6 95.r even 36 1
2888.1.u.f 6 760.bu odd 12 1
2888.1.u.f 6 760.cn odd 36 1
2888.1.u.g 6 95.m odd 12 1
2888.1.u.g 6 95.q odd 36 1
2888.1.u.g 6 760.bw even 12 1
2888.1.u.g 6 760.cp even 36 1
3800.1.cq.b 12 1.a even 1 1 trivial
3800.1.cq.b 12 5.b even 2 1 inner
3800.1.cq.b 12 8.d odd 2 1 CM
3800.1.cq.b 12 19.e even 9 1 inner
3800.1.cq.b 12 40.e odd 2 1 inner
3800.1.cq.b 12 95.p even 18 1 inner
3800.1.cq.b 12 152.u odd 18 1 inner
3800.1.cq.b 12 760.bz odd 18 1 inner
3800.1.cv.c 6 5.c odd 4 1
3800.1.cv.c 6 40.k even 4 1
3800.1.cv.c 6 95.q odd 36 1
3800.1.cv.c 6 760.cp even 36 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 3 T_{3}^{10} - 6 T_{3}^{8} + 8 T_{3}^{6} + 69 T_{3}^{4} + 3 T_{3}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3800, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{6} + T^{12} \)
$3$ \( 1 + 3 T^{2} + 69 T^{4} + 8 T^{6} - 6 T^{8} - 3 T^{10} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( T^{12} \)
$11$ \( ( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} )^{2} \)
$13$ \( T^{12} \)
$17$ \( 1 - T^{6} + T^{12} \)
$19$ \( ( 1 - T^{3} + T^{6} )^{2} \)
$23$ \( T^{12} \)
$29$ \( T^{12} \)
$31$ \( T^{12} \)
$37$ \( T^{12} \)
$41$ \( ( 1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$43$ \( 1 - T^{6} + T^{12} \)
$47$ \( T^{12} \)
$53$ \( T^{12} \)
$59$ \( ( 1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} )^{2} \)
$61$ \( T^{12} \)
$67$ \( 1 + 3 T^{2} + 69 T^{4} + 8 T^{6} - 6 T^{8} - 3 T^{10} + T^{12} \)
$71$ \( T^{12} \)
$73$ \( 1 - 15 T^{2} + 60 T^{4} + 35 T^{6} + 21 T^{8} + 6 T^{10} + T^{12} \)
$79$ \( T^{12} \)
$83$ \( 1 - 9 T^{2} + 75 T^{4} - 52 T^{6} + 27 T^{8} - 6 T^{10} + T^{12} \)
$89$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$97$ \( 1 + 3 T^{2} + 69 T^{4} + 8 T^{6} - 6 T^{8} - 3 T^{10} + T^{12} \)
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