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

\(f(q)\) \(=\) \( q + 2 q^{2} + \beta q^{3} -\beta q^{5} + 2 \beta q^{6} + ( -2 - 3 \beta ) q^{7} -8 q^{8} + 4 q^{9} +O(q^{10})\) \( q + 2 q^{2} + \beta q^{3} -\beta q^{5} + 2 \beta q^{6} + ( -2 - 3 \beta ) q^{7} -8 q^{8} + 4 q^{9} -2 \beta q^{10} - q^{11} + 9 \beta q^{13} + ( -4 - 6 \beta ) q^{14} + 5 q^{15} -16 q^{16} + 3 \beta q^{17} + 8 q^{18} -6 \beta q^{19} + ( 15 - 2 \beta ) q^{21} -2 q^{22} + 8 q^{23} -8 \beta q^{24} -5 q^{25} + 18 \beta q^{26} + 13 \beta q^{27} + 41 q^{29} + 10 q^{30} -18 \beta q^{31} -\beta q^{33} + 6 \beta q^{34} + ( -15 + 2 \beta ) q^{35} -28 q^{37} -12 \beta q^{38} -45 q^{39} + 8 \beta q^{40} -6 \beta q^{41} + ( 30 - 4 \beta ) q^{42} -82 q^{43} -4 \beta q^{45} + 16 q^{46} -9 \beta q^{47} -16 \beta q^{48} + ( -41 + 12 \beta ) q^{49} -10 q^{50} -15 q^{51} + 74 q^{53} + 26 \beta q^{54} + \beta q^{55} + ( 16 + 24 \beta ) q^{56} + 30 q^{57} + 82 q^{58} + 42 \beta q^{59} -36 \beta q^{61} -36 \beta q^{62} + ( -8 - 12 \beta ) q^{63} + 64 q^{64} + 45 q^{65} -2 \beta q^{66} + 2 q^{67} + 8 \beta q^{69} + ( -30 + 4 \beta ) q^{70} + 14 q^{71} -32 q^{72} + 30 \beta q^{73} -56 q^{74} -5 \beta q^{75} + ( 2 + 3 \beta ) q^{77} -90 q^{78} -19 q^{79} + 16 \beta q^{80} -29 q^{81} -12 \beta q^{82} -42 \beta q^{83} + 15 q^{85} -164 q^{86} + 41 \beta q^{87} + 8 q^{88} -48 \beta q^{89} -8 \beta q^{90} + ( 135 - 18 \beta ) q^{91} + 90 q^{93} -18 \beta q^{94} -30 q^{95} + 27 \beta q^{97} + ( -82 + 24 \beta ) q^{98} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} - 4q^{7} - 16q^{8} + 8q^{9} + O(q^{10}) \) \( 2q + 4q^{2} - 4q^{7} - 16q^{8} + 8q^{9} - 2q^{11} - 8q^{14} + 10q^{15} - 32q^{16} + 16q^{18} + 30q^{21} - 4q^{22} + 16q^{23} - 10q^{25} + 82q^{29} + 20q^{30} - 30q^{35} - 56q^{37} - 90q^{39} + 60q^{42} - 164q^{43} + 32q^{46} - 82q^{49} - 20q^{50} - 30q^{51} + 148q^{53} + 32q^{56} + 60q^{57} + 164q^{58} - 16q^{63} + 128q^{64} + 90q^{65} + 4q^{67} - 60q^{70} + 28q^{71} - 64q^{72} - 112q^{74} + 4q^{77} - 180q^{78} - 38q^{79} - 58q^{81} + 30q^{85} - 328q^{86} + 16q^{88} + 270q^{91} + 180q^{93} - 60q^{95} - 164q^{98} - 8q^{99} + O(q^{100}) \)

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 4 T^{2} )^{2} \)
$3$ \( 1 - 13 T^{2} + 81 T^{4} \)
$5$ \( 1 + 5 T^{2} \)
$7$ \( 1 + 4 T + 49 T^{2} \)
$11$ \( ( 1 + T + 121 T^{2} )^{2} \)
$13$ \( 1 + 67 T^{2} + 28561 T^{4} \)
$17$ \( 1 - 533 T^{2} + 83521 T^{4} \)
$19$ \( 1 - 542 T^{2} + 130321 T^{4} \)
$23$ \( ( 1 - 8 T + 529 T^{2} )^{2} \)
$29$ \( ( 1 - 41 T + 841 T^{2} )^{2} \)
$31$ \( 1 - 302 T^{2} + 923521 T^{4} \)
$37$ \( ( 1 + 28 T + 1369 T^{2} )^{2} \)
$41$ \( 1 - 3182 T^{2} + 2825761 T^{4} \)
$43$ \( ( 1 + 82 T + 1849 T^{2} )^{2} \)
$47$ \( 1 - 4013 T^{2} + 4879681 T^{4} \)
$53$ \( ( 1 - 74 T + 2809 T^{2} )^{2} \)
$59$ \( 1 + 1858 T^{2} + 12117361 T^{4} \)
$61$ \( 1 - 962 T^{2} + 13845841 T^{4} \)
$67$ \( ( 1 - 2 T + 4489 T^{2} )^{2} \)
$71$ \( ( 1 - 14 T + 5041 T^{2} )^{2} \)
$73$ \( 1 - 6158 T^{2} + 28398241 T^{4} \)
$79$ \( ( 1 + 19 T + 6241 T^{2} )^{2} \)
$83$ \( 1 - 4958 T^{2} + 47458321 T^{4} \)
$89$ \( ( 1 - 142 T + 7921 T^{2} )( 1 + 142 T + 7921 T^{2} ) \)
$97$ \( 1 - 15173 T^{2} + 88529281 T^{4} \)
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