Properties

Label 35.3.d.a
Level $35$
Weight $3$
Character orbit 35.d
Analytic conductor $0.954$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.953680925261\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + 2 \beta q^{3} - 3 q^{4} + \beta q^{5} - 2 \beta q^{6} + 7 q^{7} + 7 q^{8} - 11 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + 2 \beta q^{3} - 3 q^{4} + \beta q^{5} - 2 \beta q^{6} + 7 q^{7} + 7 q^{8} - 11 q^{9} - \beta q^{10} + 2 q^{11} - 6 \beta q^{12} - 6 \beta q^{13} - 7 q^{14} - 10 q^{15} + 5 q^{16} + 12 \beta q^{17} + 11 q^{18} - 6 \beta q^{19} - 3 \beta q^{20} + 14 \beta q^{21} - 2 q^{22} + 26 q^{23} + 14 \beta q^{24} - 5 q^{25} + 6 \beta q^{26} - 4 \beta q^{27} - 21 q^{28} - 22 q^{29} + 10 q^{30} - 24 \beta q^{31} - 33 q^{32} + 4 \beta q^{33} - 12 \beta q^{34} + 7 \beta q^{35} + 33 q^{36} + 14 q^{37} + 6 \beta q^{38} + 60 q^{39} + 7 \beta q^{40} - 12 \beta q^{41} - 14 \beta q^{42} - 34 q^{43} - 6 q^{44} - 11 \beta q^{45} - 26 q^{46} - 12 \beta q^{47} + 10 \beta q^{48} + 49 q^{49} + 5 q^{50} - 120 q^{51} + 18 \beta q^{52} - 34 q^{53} + 4 \beta q^{54} + 2 \beta q^{55} + 49 q^{56} + 60 q^{57} + 22 q^{58} - 18 \beta q^{59} + 30 q^{60} + 42 \beta q^{61} + 24 \beta q^{62} - 77 q^{63} + 13 q^{64} + 30 q^{65} - 4 \beta q^{66} + 14 q^{67} - 36 \beta q^{68} + 52 \beta q^{69} - 7 \beta q^{70} + 62 q^{71} - 77 q^{72} - 24 \beta q^{73} - 14 q^{74} - 10 \beta q^{75} + 18 \beta q^{76} + 14 q^{77} - 60 q^{78} + 38 q^{79} + 5 \beta q^{80} - 59 q^{81} + 12 \beta q^{82} + 18 \beta q^{83} - 42 \beta q^{84} - 60 q^{85} + 34 q^{86} - 44 \beta q^{87} + 14 q^{88} - 12 \beta q^{89} + 11 \beta q^{90} - 42 \beta q^{91} - 78 q^{92} + 240 q^{93} + 12 \beta q^{94} + 30 q^{95} - 66 \beta q^{96} - 12 \beta q^{97} - 49 q^{98} - 22 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 6 q^{4} + 14 q^{7} + 14 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 6 q^{4} + 14 q^{7} + 14 q^{8} - 22 q^{9} + 4 q^{11} - 14 q^{14} - 20 q^{15} + 10 q^{16} + 22 q^{18} - 4 q^{22} + 52 q^{23} - 10 q^{25} - 42 q^{28} - 44 q^{29} + 20 q^{30} - 66 q^{32} + 66 q^{36} + 28 q^{37} + 120 q^{39} - 68 q^{43} - 12 q^{44} - 52 q^{46} + 98 q^{49} + 10 q^{50} - 240 q^{51} - 68 q^{53} + 98 q^{56} + 120 q^{57} + 44 q^{58} + 60 q^{60} - 154 q^{63} + 26 q^{64} + 60 q^{65} + 28 q^{67} + 124 q^{71} - 154 q^{72} - 28 q^{74} + 28 q^{77} - 120 q^{78} + 76 q^{79} - 118 q^{81} - 120 q^{85} + 68 q^{86} + 28 q^{88} - 156 q^{92} + 480 q^{93} + 60 q^{95} - 98 q^{98} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
2.23607i
2.23607i
−1.00000 4.47214i −3.00000 2.23607i 4.47214i 7.00000 7.00000 −11.0000 2.23607i
6.2 −1.00000 4.47214i −3.00000 2.23607i 4.47214i 7.00000 7.00000 −11.0000 2.23607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.3.d.a 2
3.b odd 2 1 315.3.h.b 2
4.b odd 2 1 560.3.f.a 2
5.b even 2 1 175.3.d.f 2
5.c odd 4 2 175.3.c.d 4
7.b odd 2 1 inner 35.3.d.a 2
7.c even 3 2 245.3.h.b 4
7.d odd 6 2 245.3.h.b 4
21.c even 2 1 315.3.h.b 2
28.d even 2 1 560.3.f.a 2
35.c odd 2 1 175.3.d.f 2
35.f even 4 2 175.3.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.a 2 1.a even 1 1 trivial
35.3.d.a 2 7.b odd 2 1 inner
175.3.c.d 4 5.c odd 4 2
175.3.c.d 4 35.f even 4 2
175.3.d.f 2 5.b even 2 1
175.3.d.f 2 35.c odd 2 1
245.3.h.b 4 7.c even 3 2
245.3.h.b 4 7.d odd 6 2
315.3.h.b 2 3.b odd 2 1
315.3.h.b 2 21.c even 2 1
560.3.f.a 2 4.b odd 2 1
560.3.f.a 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(35, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 20 \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 180 \) Copy content Toggle raw display
$17$ \( T^{2} + 720 \) Copy content Toggle raw display
$19$ \( T^{2} + 180 \) Copy content Toggle raw display
$23$ \( (T - 26)^{2} \) Copy content Toggle raw display
$29$ \( (T + 22)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2880 \) Copy content Toggle raw display
$37$ \( (T - 14)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 720 \) Copy content Toggle raw display
$43$ \( (T + 34)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 720 \) Copy content Toggle raw display
$53$ \( (T + 34)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 1620 \) Copy content Toggle raw display
$61$ \( T^{2} + 8820 \) Copy content Toggle raw display
$67$ \( (T - 14)^{2} \) Copy content Toggle raw display
$71$ \( (T - 62)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2880 \) Copy content Toggle raw display
$79$ \( (T - 38)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1620 \) Copy content Toggle raw display
$89$ \( T^{2} + 720 \) Copy content Toggle raw display
$97$ \( T^{2} + 720 \) Copy content Toggle raw display
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