Properties

Label 10.3.c.a
Level $10$
Weight $3$
Character orbit 10.c
Analytic conductor $0.272$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.272480264360\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - i - 1) q^{2} + (2 i - 2) q^{3} + 2 i q^{4} - 5 i q^{5} + 4 q^{6} + (2 i + 2) q^{7} + ( - 2 i + 2) q^{8} + i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i - 1) q^{2} + (2 i - 2) q^{3} + 2 i q^{4} - 5 i q^{5} + 4 q^{6} + (2 i + 2) q^{7} + ( - 2 i + 2) q^{8} + i q^{9} + (5 i - 5) q^{10} - 8 q^{11} + ( - 4 i - 4) q^{12} + ( - 3 i + 3) q^{13} - 4 i q^{14} + (10 i + 10) q^{15} - 4 q^{16} + (7 i + 7) q^{17} + ( - i + 1) q^{18} - 20 i q^{19} + 10 q^{20} - 8 q^{21} + (8 i + 8) q^{22} + (2 i - 2) q^{23} + 8 i q^{24} - 25 q^{25} - 6 q^{26} + ( - 20 i - 20) q^{27} + (4 i - 4) q^{28} + 40 i q^{29} - 20 i q^{30} + 52 q^{31} + (4 i + 4) q^{32} + ( - 16 i + 16) q^{33} - 14 i q^{34} + ( - 10 i + 10) q^{35} - 2 q^{36} + ( - 3 i - 3) q^{37} + (20 i - 20) q^{38} + 12 i q^{39} + ( - 10 i - 10) q^{40} - 8 q^{41} + (8 i + 8) q^{42} + (42 i - 42) q^{43} - 16 i q^{44} + 5 q^{45} + 4 q^{46} + ( - 18 i - 18) q^{47} + ( - 8 i + 8) q^{48} - 41 i q^{49} + (25 i + 25) q^{50} - 28 q^{51} + (6 i + 6) q^{52} + ( - 53 i + 53) q^{53} + 40 i q^{54} + 40 i q^{55} + 8 q^{56} + (40 i + 40) q^{57} + ( - 40 i + 40) q^{58} - 20 i q^{59} + (20 i - 20) q^{60} - 48 q^{61} + ( - 52 i - 52) q^{62} + (2 i - 2) q^{63} - 8 i q^{64} + ( - 15 i - 15) q^{65} - 32 q^{66} + (62 i + 62) q^{67} + (14 i - 14) q^{68} - 8 i q^{69} - 20 q^{70} - 28 q^{71} + (2 i + 2) q^{72} + (47 i - 47) q^{73} + 6 i q^{74} + ( - 50 i + 50) q^{75} + 40 q^{76} + ( - 16 i - 16) q^{77} + ( - 12 i + 12) q^{78} + 20 i q^{80} + 71 q^{81} + (8 i + 8) q^{82} + ( - 18 i + 18) q^{83} - 16 i q^{84} + ( - 35 i + 35) q^{85} + 84 q^{86} + ( - 80 i - 80) q^{87} + (16 i - 16) q^{88} + 80 i q^{89} + ( - 5 i - 5) q^{90} + 12 q^{91} + ( - 4 i - 4) q^{92} + (104 i - 104) q^{93} + 36 i q^{94} - 100 q^{95} - 16 q^{96} + ( - 63 i - 63) q^{97} + (41 i - 41) q^{98} - 8 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} + 8 q^{6} + 4 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{3} + 8 q^{6} + 4 q^{7} + 4 q^{8} - 10 q^{10} - 16 q^{11} - 8 q^{12} + 6 q^{13} + 20 q^{15} - 8 q^{16} + 14 q^{17} + 2 q^{18} + 20 q^{20} - 16 q^{21} + 16 q^{22} - 4 q^{23} - 50 q^{25} - 12 q^{26} - 40 q^{27} - 8 q^{28} + 104 q^{31} + 8 q^{32} + 32 q^{33} + 20 q^{35} - 4 q^{36} - 6 q^{37} - 40 q^{38} - 20 q^{40} - 16 q^{41} + 16 q^{42} - 84 q^{43} + 10 q^{45} + 8 q^{46} - 36 q^{47} + 16 q^{48} + 50 q^{50} - 56 q^{51} + 12 q^{52} + 106 q^{53} + 16 q^{56} + 80 q^{57} + 80 q^{58} - 40 q^{60} - 96 q^{61} - 104 q^{62} - 4 q^{63} - 30 q^{65} - 64 q^{66} + 124 q^{67} - 28 q^{68} - 40 q^{70} - 56 q^{71} + 4 q^{72} - 94 q^{73} + 100 q^{75} + 80 q^{76} - 32 q^{77} + 24 q^{78} + 142 q^{81} + 16 q^{82} + 36 q^{83} + 70 q^{85} + 168 q^{86} - 160 q^{87} - 32 q^{88} - 10 q^{90} + 24 q^{91} - 8 q^{92} - 208 q^{93} - 200 q^{95} - 32 q^{96} - 126 q^{97} - 82 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.00000i
1.00000i
−1.00000 + 1.00000i −2.00000 2.00000i 2.00000i 5.00000i 4.00000 2.00000 2.00000i 2.00000 + 2.00000i 1.00000i −5.00000 5.00000i
7.1 −1.00000 1.00000i −2.00000 + 2.00000i 2.00000i 5.00000i 4.00000 2.00000 + 2.00000i 2.00000 2.00000i 1.00000i −5.00000 + 5.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.3.c.a 2
3.b odd 2 1 90.3.g.b 2
4.b odd 2 1 80.3.p.c 2
5.b even 2 1 50.3.c.c 2
5.c odd 4 1 inner 10.3.c.a 2
5.c odd 4 1 50.3.c.c 2
7.b odd 2 1 490.3.f.b 2
8.b even 2 1 320.3.p.h 2
8.d odd 2 1 320.3.p.a 2
12.b even 2 1 720.3.bh.c 2
15.d odd 2 1 450.3.g.b 2
15.e even 4 1 90.3.g.b 2
15.e even 4 1 450.3.g.b 2
20.d odd 2 1 400.3.p.b 2
20.e even 4 1 80.3.p.c 2
20.e even 4 1 400.3.p.b 2
35.f even 4 1 490.3.f.b 2
40.i odd 4 1 320.3.p.h 2
40.k even 4 1 320.3.p.a 2
60.l odd 4 1 720.3.bh.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.3.c.a 2 1.a even 1 1 trivial
10.3.c.a 2 5.c odd 4 1 inner
50.3.c.c 2 5.b even 2 1
50.3.c.c 2 5.c odd 4 1
80.3.p.c 2 4.b odd 2 1
80.3.p.c 2 20.e even 4 1
90.3.g.b 2 3.b odd 2 1
90.3.g.b 2 15.e even 4 1
320.3.p.a 2 8.d odd 2 1
320.3.p.a 2 40.k even 4 1
320.3.p.h 2 8.b even 2 1
320.3.p.h 2 40.i odd 4 1
400.3.p.b 2 20.d odd 2 1
400.3.p.b 2 20.e even 4 1
450.3.g.b 2 15.d odd 2 1
450.3.g.b 2 15.e even 4 1
490.3.f.b 2 7.b odd 2 1
490.3.f.b 2 35.f even 4 1
720.3.bh.c 2 12.b even 2 1
720.3.bh.c 2 60.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(10, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$5$ \( T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$11$ \( (T + 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$17$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$19$ \( T^{2} + 400 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 1600 \) Copy content Toggle raw display
$31$ \( (T - 52)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$41$ \( (T + 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 84T + 3528 \) Copy content Toggle raw display
$47$ \( T^{2} + 36T + 648 \) Copy content Toggle raw display
$53$ \( T^{2} - 106T + 5618 \) Copy content Toggle raw display
$59$ \( T^{2} + 400 \) Copy content Toggle raw display
$61$ \( (T + 48)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 124T + 7688 \) Copy content Toggle raw display
$71$ \( (T + 28)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 94T + 4418 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 36T + 648 \) Copy content Toggle raw display
$89$ \( T^{2} + 6400 \) Copy content Toggle raw display
$97$ \( T^{2} + 126T + 7938 \) Copy content Toggle raw display
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