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: \(x^{2} + 5\)
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.

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

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 \(S_{3}^{\mathrm{new}}(35, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + 4 T^{2} )^{2} \)
$3$ \( ( 1 - 4 T + 9 T^{2} )( 1 + 4 T + 9 T^{2} ) \)
$5$ \( 1 + 5 T^{2} \)
$7$ \( ( 1 - 7 T )^{2} \)
$11$ \( ( 1 - 2 T + 121 T^{2} )^{2} \)
$13$ \( 1 - 158 T^{2} + 28561 T^{4} \)
$17$ \( 1 + 142 T^{2} + 83521 T^{4} \)
$19$ \( 1 - 542 T^{2} + 130321 T^{4} \)
$23$ \( ( 1 - 26 T + 529 T^{2} )^{2} \)
$29$ \( ( 1 + 22 T + 841 T^{2} )^{2} \)
$31$ \( 1 + 958 T^{2} + 923521 T^{4} \)
$37$ \( ( 1 - 14 T + 1369 T^{2} )^{2} \)
$41$ \( 1 - 2642 T^{2} + 2825761 T^{4} \)
$43$ \( ( 1 + 34 T + 1849 T^{2} )^{2} \)
$47$ \( 1 - 3698 T^{2} + 4879681 T^{4} \)
$53$ \( ( 1 + 34 T + 2809 T^{2} )^{2} \)
$59$ \( 1 - 5342 T^{2} + 12117361 T^{4} \)
$61$ \( 1 + 1378 T^{2} + 13845841 T^{4} \)
$67$ \( ( 1 - 14 T + 4489 T^{2} )^{2} \)
$71$ \( ( 1 - 62 T + 5041 T^{2} )^{2} \)
$73$ \( 1 - 7778 T^{2} + 28398241 T^{4} \)
$79$ \( ( 1 - 38 T + 6241 T^{2} )^{2} \)
$83$ \( 1 - 12158 T^{2} + 47458321 T^{4} \)
$89$ \( 1 - 15122 T^{2} + 62742241 T^{4} \)
$97$ \( 1 - 18098 T^{2} + 88529281 T^{4} \)
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