Properties

Label 35.3.d.b
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + \beta q^{3} - \beta q^{5} + 2 \beta q^{6} + ( - 3 \beta - 2) q^{7} - 8 q^{8} + 4 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + \beta q^{3} - \beta q^{5} + 2 \beta q^{6} + ( - 3 \beta - 2) q^{7} - 8 q^{8} + 4 q^{9} - 2 \beta q^{10} - q^{11} + 9 \beta q^{13} + ( - 6 \beta - 4) q^{14} + 5 q^{15} - 16 q^{16} + 3 \beta q^{17} + 8 q^{18} - 6 \beta q^{19} + ( - 2 \beta + 15) q^{21} - 2 q^{22} + 8 q^{23} - 8 \beta q^{24} - 5 q^{25} + 18 \beta q^{26} + 13 \beta q^{27} + 41 q^{29} + 10 q^{30} - 18 \beta q^{31} - \beta q^{33} + 6 \beta q^{34} + (2 \beta - 15) q^{35} - 28 q^{37} - 12 \beta q^{38} - 45 q^{39} + 8 \beta q^{40} - 6 \beta q^{41} + ( - 4 \beta + 30) q^{42} - 82 q^{43} - 4 \beta q^{45} + 16 q^{46} - 9 \beta q^{47} - 16 \beta q^{48} + (12 \beta - 41) q^{49} - 10 q^{50} - 15 q^{51} + 74 q^{53} + 26 \beta q^{54} + \beta q^{55} + (24 \beta + 16) q^{56} + 30 q^{57} + 82 q^{58} + 42 \beta q^{59} - 36 \beta q^{61} - 36 \beta q^{62} + ( - 12 \beta - 8) q^{63} + 64 q^{64} + 45 q^{65} - 2 \beta q^{66} + 2 q^{67} + 8 \beta q^{69} + (4 \beta - 30) q^{70} + 14 q^{71} - 32 q^{72} + 30 \beta q^{73} - 56 q^{74} - 5 \beta q^{75} + (3 \beta + 2) q^{77} - 90 q^{78} - 19 q^{79} + 16 \beta q^{80} - 29 q^{81} - 12 \beta q^{82} - 42 \beta q^{83} + 15 q^{85} - 164 q^{86} + 41 \beta q^{87} + 8 q^{88} - 48 \beta q^{89} - 8 \beta q^{90} + ( - 18 \beta + 135) q^{91} + 90 q^{93} - 18 \beta q^{94} - 30 q^{95} + 27 \beta q^{97} + (24 \beta - 82) q^{98} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 4 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 4 q^{7} - 16 q^{8} + 8 q^{9} - 2 q^{11} - 8 q^{14} + 10 q^{15} - 32 q^{16} + 16 q^{18} + 30 q^{21} - 4 q^{22} + 16 q^{23} - 10 q^{25} + 82 q^{29} + 20 q^{30} - 30 q^{35} - 56 q^{37} - 90 q^{39} + 60 q^{42} - 164 q^{43} + 32 q^{46} - 82 q^{49} - 20 q^{50} - 30 q^{51} + 148 q^{53} + 32 q^{56} + 60 q^{57} + 164 q^{58} - 16 q^{63} + 128 q^{64} + 90 q^{65} + 4 q^{67} - 60 q^{70} + 28 q^{71} - 64 q^{72} - 112 q^{74} + 4 q^{77} - 180 q^{78} - 38 q^{79} - 58 q^{81} + 30 q^{85} - 328 q^{86} + 16 q^{88} + 270 q^{91} + 180 q^{93} - 60 q^{95} - 164 q^{98} - 8 q^{99}+O(q^{100}) \) 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
2.00000 2.23607i 0 2.23607i 4.47214i −2.00000 + 6.70820i −8.00000 4.00000 4.47214i
6.2 2.00000 2.23607i 0 2.23607i 4.47214i −2.00000 6.70820i −8.00000 4.00000 4.47214i
\(n\): 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.b 2
3.b odd 2 1 315.3.h.a 2
4.b odd 2 1 560.3.f.b 2
5.b even 2 1 175.3.d.c 2
5.c odd 4 2 175.3.c.c 4
7.b odd 2 1 inner 35.3.d.b 2
7.c even 3 2 245.3.h.a 4
7.d odd 6 2 245.3.h.a 4
21.c even 2 1 315.3.h.a 2
28.d even 2 1 560.3.f.b 2
35.c odd 2 1 175.3.d.c 2
35.f even 4 2 175.3.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.b 2 1.a even 1 1 trivial
35.3.d.b 2 7.b odd 2 1 inner
175.3.c.c 4 5.c odd 4 2
175.3.c.c 4 35.f even 4 2
175.3.d.c 2 5.b even 2 1
175.3.d.c 2 35.c odd 2 1
245.3.h.a 4 7.c even 3 2
245.3.h.a 4 7.d odd 6 2
315.3.h.a 2 3.b odd 2 1
315.3.h.a 2 21.c even 2 1
560.3.f.b 2 4.b odd 2 1
560.3.f.b 2 28.d even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5 \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 49 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 405 \) Copy content Toggle raw display
$17$ \( T^{2} + 45 \) Copy content Toggle raw display
$19$ \( T^{2} + 180 \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T - 41)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1620 \) Copy content Toggle raw display
$37$ \( (T + 28)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 180 \) Copy content Toggle raw display
$43$ \( (T + 82)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 405 \) Copy content Toggle raw display
$53$ \( (T - 74)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8820 \) Copy content Toggle raw display
$61$ \( T^{2} + 6480 \) Copy content Toggle raw display
$67$ \( (T - 2)^{2} \) Copy content Toggle raw display
$71$ \( (T - 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4500 \) Copy content Toggle raw display
$79$ \( (T + 19)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 8820 \) Copy content Toggle raw display
$89$ \( T^{2} + 11520 \) Copy content Toggle raw display
$97$ \( T^{2} + 3645 \) Copy content Toggle raw display
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