Properties

Label 3800.1.cv.c
Level $3800$
Weight $1$
Character orbit 3800.cv
Analytic conductor $1.896$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.cv (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
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.

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

Character values

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

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(-\zeta_{18}\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−0.766044 0.642788i
−0.766044 + 0.642788i
−0.173648 + 0.984808i
−0.173648 0.984808i
0.939693 + 0.342020i
0.939693 0.342020i
0.939693 + 0.342020i 0.326352 + 1.85083i 0.766044 + 0.642788i 0 −0.326352 + 1.85083i 0 0.500000 + 0.866025i −2.37939 + 0.866025i 0
651.1 0.939693 0.342020i 0.326352 1.85083i 0.766044 0.642788i 0 −0.326352 1.85083i 0 0.500000 0.866025i −2.37939 0.866025i 0
1051.1 −0.766044 + 0.642788i 1.43969 + 0.524005i 0.173648 0.984808i 0 −1.43969 + 0.524005i 0 0.500000 + 0.866025i 1.03209 + 0.866025i 0
1251.1 −0.766044 0.642788i 1.43969 0.524005i 0.173648 + 0.984808i 0 −1.43969 0.524005i 0 0.500000 0.866025i 1.03209 0.866025i 0
1651.1 −0.173648 + 0.984808i −0.266044 0.223238i −0.939693 0.342020i 0 0.266044 0.223238i 0 0.500000 0.866025i −0.152704 0.866025i 0
2251.1 −0.173648 0.984808i −0.266044 + 0.223238i −0.939693 + 0.342020i 0 0.266044 + 0.223238i 0 0.500000 + 0.866025i −0.152704 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2251.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
19.e even 9 1 inner
152.u 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.cv.c 6
5.b even 2 1 152.1.u.a 6
5.c odd 4 2 3800.1.cq.b 12
8.d odd 2 1 CM 3800.1.cv.c 6
15.d odd 2 1 1368.1.eh.a 6
19.e even 9 1 inner 3800.1.cv.c 6
20.d odd 2 1 608.1.bg.a 6
40.e odd 2 1 152.1.u.a 6
40.f even 2 1 608.1.bg.a 6
40.k even 4 2 3800.1.cq.b 12
95.d odd 2 1 2888.1.u.e 6
95.h odd 6 1 2888.1.u.a 6
95.h odd 6 1 2888.1.u.f 6
95.i even 6 1 2888.1.u.b 6
95.i even 6 1 2888.1.u.g 6
95.o odd 18 1 2888.1.f.c 3
95.o odd 18 2 2888.1.k.c 6
95.o odd 18 1 2888.1.u.a 6
95.o odd 18 1 2888.1.u.e 6
95.o odd 18 1 2888.1.u.f 6
95.p even 18 1 152.1.u.a 6
95.p even 18 1 2888.1.f.d 3
95.p even 18 2 2888.1.k.b 6
95.p even 18 1 2888.1.u.b 6
95.p even 18 1 2888.1.u.g 6
95.q odd 36 2 3800.1.cq.b 12
120.m even 2 1 1368.1.eh.a 6
152.u odd 18 1 inner 3800.1.cv.c 6
285.bd odd 18 1 1368.1.eh.a 6
380.ba odd 18 1 608.1.bg.a 6
760.p even 2 1 2888.1.u.e 6
760.bf even 6 1 2888.1.u.a 6
760.bf even 6 1 2888.1.u.f 6
760.bm odd 6 1 2888.1.u.b 6
760.bm odd 6 1 2888.1.u.g 6
760.bx even 18 1 2888.1.f.c 3
760.bx even 18 2 2888.1.k.c 6
760.bx even 18 1 2888.1.u.a 6
760.bx even 18 1 2888.1.u.e 6
760.bx even 18 1 2888.1.u.f 6
760.bz odd 18 1 152.1.u.a 6
760.bz odd 18 1 2888.1.f.d 3
760.bz odd 18 2 2888.1.k.b 6
760.bz odd 18 1 2888.1.u.b 6
760.bz odd 18 1 2888.1.u.g 6
760.cj even 18 1 608.1.bg.a 6
760.cp even 36 2 3800.1.cq.b 12
2280.eg even 18 1 1368.1.eh.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a 6 5.b even 2 1
152.1.u.a 6 40.e odd 2 1
152.1.u.a 6 95.p even 18 1
152.1.u.a 6 760.bz odd 18 1
608.1.bg.a 6 20.d odd 2 1
608.1.bg.a 6 40.f even 2 1
608.1.bg.a 6 380.ba odd 18 1
608.1.bg.a 6 760.cj even 18 1
1368.1.eh.a 6 15.d odd 2 1
1368.1.eh.a 6 120.m even 2 1
1368.1.eh.a 6 285.bd odd 18 1
1368.1.eh.a 6 2280.eg even 18 1
2888.1.f.c 3 95.o odd 18 1
2888.1.f.c 3 760.bx even 18 1
2888.1.f.d 3 95.p even 18 1
2888.1.f.d 3 760.bz odd 18 1
2888.1.k.b 6 95.p even 18 2
2888.1.k.b 6 760.bz odd 18 2
2888.1.k.c 6 95.o odd 18 2
2888.1.k.c 6 760.bx even 18 2
2888.1.u.a 6 95.h odd 6 1
2888.1.u.a 6 95.o odd 18 1
2888.1.u.a 6 760.bf even 6 1
2888.1.u.a 6 760.bx even 18 1
2888.1.u.b 6 95.i even 6 1
2888.1.u.b 6 95.p even 18 1
2888.1.u.b 6 760.bm odd 6 1
2888.1.u.b 6 760.bz odd 18 1
2888.1.u.e 6 95.d odd 2 1
2888.1.u.e 6 95.o odd 18 1
2888.1.u.e 6 760.p even 2 1
2888.1.u.e 6 760.bx even 18 1
2888.1.u.f 6 95.h odd 6 1
2888.1.u.f 6 95.o odd 18 1
2888.1.u.f 6 760.bf even 6 1
2888.1.u.f 6 760.bx even 18 1
2888.1.u.g 6 95.i even 6 1
2888.1.u.g 6 95.p even 18 1
2888.1.u.g 6 760.bm odd 6 1
2888.1.u.g 6 760.bz odd 18 1
3800.1.cq.b 12 5.c odd 4 2
3800.1.cq.b 12 40.k even 4 2
3800.1.cq.b 12 95.q odd 36 2
3800.1.cq.b 12 760.cp even 36 2
3800.1.cv.c 6 1.a even 1 1 trivial
3800.1.cv.c 6 8.d odd 2 1 CM
3800.1.cv.c 6 19.e even 9 1 inner
3800.1.cv.c 6 152.u odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 3T_{3}^{5} + 6T_{3}^{4} - 8T_{3}^{3} + 3T_{3}^{2} + 3T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3800, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$19$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$97$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
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