Properties

Label 35.2.b.a
Level 35
Weight 2
Character orbit 35.b
Analytic conductor 0.279
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.279476407074\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{2} -i q^{3} -2 q^{4} + ( -2 - i ) q^{5} + 2 q^{6} -i q^{7} + 2 q^{9} +O(q^{10})\) \( q + 2 i q^{2} -i q^{3} -2 q^{4} + ( -2 - i ) q^{5} + 2 q^{6} -i q^{7} + 2 q^{9} + ( 2 - 4 i ) q^{10} -3 q^{11} + 2 i q^{12} -i q^{13} + 2 q^{14} + ( -1 + 2 i ) q^{15} -4 q^{16} + 7 i q^{17} + 4 i q^{18} + ( 4 + 2 i ) q^{20} - q^{21} -6 i q^{22} -6 i q^{23} + ( 3 + 4 i ) q^{25} + 2 q^{26} -5 i q^{27} + 2 i q^{28} + 5 q^{29} + ( -4 - 2 i ) q^{30} + 2 q^{31} -8 i q^{32} + 3 i q^{33} -14 q^{34} + ( -1 + 2 i ) q^{35} -4 q^{36} + 2 i q^{37} - q^{39} + 2 q^{41} -2 i q^{42} + 4 i q^{43} + 6 q^{44} + ( -4 - 2 i ) q^{45} + 12 q^{46} -3 i q^{47} + 4 i q^{48} - q^{49} + ( -8 + 6 i ) q^{50} + 7 q^{51} + 2 i q^{52} -6 i q^{53} + 10 q^{54} + ( 6 + 3 i ) q^{55} + 10 i q^{58} -10 q^{59} + ( 2 - 4 i ) q^{60} -8 q^{61} + 4 i q^{62} -2 i q^{63} + 8 q^{64} + ( -1 + 2 i ) q^{65} -6 q^{66} + 2 i q^{67} -14 i q^{68} -6 q^{69} + ( -4 - 2 i ) q^{70} -8 q^{71} -6 i q^{73} -4 q^{74} + ( 4 - 3 i ) q^{75} + 3 i q^{77} -2 i q^{78} + 5 q^{79} + ( 8 + 4 i ) q^{80} + q^{81} + 4 i q^{82} + 4 i q^{83} + 2 q^{84} + ( 7 - 14 i ) q^{85} -8 q^{86} -5 i q^{87} + ( 4 - 8 i ) q^{90} - q^{91} + 12 i q^{92} -2 i q^{93} + 6 q^{94} -8 q^{96} + 7 i q^{97} -2 i q^{98} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} - 4q^{5} + 4q^{6} + 4q^{9} + O(q^{10}) \) \( 2q - 4q^{4} - 4q^{5} + 4q^{6} + 4q^{9} + 4q^{10} - 6q^{11} + 4q^{14} - 2q^{15} - 8q^{16} + 8q^{20} - 2q^{21} + 6q^{25} + 4q^{26} + 10q^{29} - 8q^{30} + 4q^{31} - 28q^{34} - 2q^{35} - 8q^{36} - 2q^{39} + 4q^{41} + 12q^{44} - 8q^{45} + 24q^{46} - 2q^{49} - 16q^{50} + 14q^{51} + 20q^{54} + 12q^{55} - 20q^{59} + 4q^{60} - 16q^{61} + 16q^{64} - 2q^{65} - 12q^{66} - 12q^{69} - 8q^{70} - 16q^{71} - 8q^{74} + 8q^{75} + 10q^{79} + 16q^{80} + 2q^{81} + 4q^{84} + 14q^{85} - 16q^{86} + 8q^{90} - 2q^{91} + 12q^{94} - 16q^{96} - 12q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
1.00000i
1.00000i
2.00000i 1.00000i −2.00000 −2.00000 + 1.00000i 2.00000 1.00000i 0 2.00000 2.00000 + 4.00000i
29.2 2.00000i 1.00000i −2.00000 −2.00000 1.00000i 2.00000 1.00000i 0 2.00000 2.00000 4.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.2.b.a 2
3.b odd 2 1 315.2.d.a 2
4.b odd 2 1 560.2.g.b 2
5.b even 2 1 inner 35.2.b.a 2
5.c odd 4 1 175.2.a.a 1
5.c odd 4 1 175.2.a.c 1
7.b odd 2 1 245.2.b.a 2
7.c even 3 2 245.2.j.e 4
7.d odd 6 2 245.2.j.d 4
8.b even 2 1 2240.2.g.h 2
8.d odd 2 1 2240.2.g.g 2
12.b even 2 1 5040.2.t.p 2
15.d odd 2 1 315.2.d.a 2
15.e even 4 1 1575.2.a.a 1
15.e even 4 1 1575.2.a.k 1
20.d odd 2 1 560.2.g.b 2
20.e even 4 1 2800.2.a.l 1
20.e even 4 1 2800.2.a.w 1
21.c even 2 1 2205.2.d.b 2
35.c odd 2 1 245.2.b.a 2
35.f even 4 1 1225.2.a.a 1
35.f even 4 1 1225.2.a.i 1
35.i odd 6 2 245.2.j.d 4
35.j even 6 2 245.2.j.e 4
40.e odd 2 1 2240.2.g.g 2
40.f even 2 1 2240.2.g.h 2
60.h even 2 1 5040.2.t.p 2
105.g even 2 1 2205.2.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 1.a even 1 1 trivial
35.2.b.a 2 5.b even 2 1 inner
175.2.a.a 1 5.c odd 4 1
175.2.a.c 1 5.c odd 4 1
245.2.b.a 2 7.b odd 2 1
245.2.b.a 2 35.c odd 2 1
245.2.j.d 4 7.d odd 6 2
245.2.j.d 4 35.i odd 6 2
245.2.j.e 4 7.c even 3 2
245.2.j.e 4 35.j even 6 2
315.2.d.a 2 3.b odd 2 1
315.2.d.a 2 15.d odd 2 1
560.2.g.b 2 4.b odd 2 1
560.2.g.b 2 20.d odd 2 1
1225.2.a.a 1 35.f even 4 1
1225.2.a.i 1 35.f even 4 1
1575.2.a.a 1 15.e even 4 1
1575.2.a.k 1 15.e even 4 1
2205.2.d.b 2 21.c even 2 1
2205.2.d.b 2 105.g even 2 1
2240.2.g.g 2 8.d odd 2 1
2240.2.g.g 2 40.e odd 2 1
2240.2.g.h 2 8.b even 2 1
2240.2.g.h 2 40.f even 2 1
2800.2.a.l 1 20.e even 4 1
2800.2.a.w 1 20.e even 4 1
5040.2.t.p 2 12.b even 2 1
5040.2.t.p 2 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(35, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} ) \)
$3$ \( 1 - 5 T^{2} + 9 T^{4} \)
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + 3 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 25 T^{2} + 169 T^{4} \)
$17$ \( 1 + 15 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 - 10 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 5 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 2 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 85 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 10 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 8 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 130 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 8 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \)
$79$ \( ( 1 - 5 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 150 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 89 T^{2} )^{2} \)
$97$ \( 1 - 145 T^{2} + 9409 T^{4} \)
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