Properties

Label 70.2.c.a
Level 70
Weight 2
Character orbit 70.c
Analytic conductor 0.559
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 70.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.558952814149\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} - q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{6} + \beta_{2} q^{7} -\beta_{2} q^{8} -3 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} - q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{6} + \beta_{2} q^{7} -\beta_{2} q^{8} -3 q^{9} + ( 1 + \beta_{2} - \beta_{3} ) q^{10} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{11} + ( -\beta_{1} - \beta_{3} ) q^{12} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{13} - q^{14} + ( 3 + 2 \beta_{1} - 3 \beta_{2} ) q^{15} + q^{16} -2 \beta_{2} q^{17} -3 \beta_{2} q^{18} + ( -4 - \beta_{1} + \beta_{3} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} ) q^{20} + ( -\beta_{1} + \beta_{3} ) q^{21} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{22} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{23} + ( \beta_{1} - \beta_{3} ) q^{24} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{25} + ( -2 + \beta_{1} - \beta_{3} ) q^{26} -\beta_{2} q^{28} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{29} + ( 3 + 3 \beta_{2} + 2 \beta_{3} ) q^{30} + ( 4 - 2 \beta_{1} + 2 \beta_{3} ) q^{31} + \beta_{2} q^{32} + 12 \beta_{2} q^{33} + 2 q^{34} + ( 1 + \beta_{2} - \beta_{3} ) q^{35} + 3 q^{36} -2 \beta_{2} q^{37} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{38} + ( 6 - 2 \beta_{1} + 2 \beta_{3} ) q^{39} + ( -1 - \beta_{2} + \beta_{3} ) q^{40} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{41} + ( -\beta_{1} - \beta_{3} ) q^{42} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{44} + ( -3 + 3 \beta_{1} + 3 \beta_{2} ) q^{45} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{46} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{47} + ( \beta_{1} + \beta_{3} ) q^{48} - q^{49} + ( -1 - 2 \beta_{1} - 2 \beta_{3} ) q^{50} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{51} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{52} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -6 - 6 \beta_{2} - 4 \beta_{3} ) q^{55} + q^{56} + ( -4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{57} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{58} + ( 4 + \beta_{1} - \beta_{3} ) q^{59} + ( -3 - 2 \beta_{1} + 3 \beta_{2} ) q^{60} + ( 6 + \beta_{1} - \beta_{3} ) q^{61} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{62} -3 \beta_{2} q^{63} - q^{64} + ( -1 - 2 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{65} -12 q^{66} + 8 \beta_{2} q^{67} + 2 \beta_{2} q^{68} + ( 12 + 2 \beta_{1} - 2 \beta_{3} ) q^{69} + ( -1 + \beta_{1} + \beta_{2} ) q^{70} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{71} + 3 \beta_{2} q^{72} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{73} + 2 q^{74} + ( -\beta_{1} - 12 \beta_{2} + \beta_{3} ) q^{75} + ( 4 + \beta_{1} - \beta_{3} ) q^{76} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{77} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{78} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{79} + ( 1 - \beta_{1} - \beta_{2} ) q^{80} -9 q^{81} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{82} + ( \beta_{1} + \beta_{3} ) q^{83} + ( \beta_{1} - \beta_{3} ) q^{84} + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{86} + ( -2 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{87} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{88} + 10 q^{89} + ( -3 - 3 \beta_{2} + 3 \beta_{3} ) q^{90} + ( -2 + \beta_{1} - \beta_{3} ) q^{91} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 4 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} ) q^{93} + ( 4 - 2 \beta_{1} + 2 \beta_{3} ) q^{94} + ( -1 + 4 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -\beta_{1} + \beta_{3} ) q^{96} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{97} -\beta_{2} q^{98} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{5} - 12q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{5} - 12q^{9} + 4q^{10} - 4q^{14} + 12q^{15} + 4q^{16} - 16q^{19} - 4q^{20} - 8q^{26} - 8q^{29} + 12q^{30} + 16q^{31} + 8q^{34} + 4q^{35} + 12q^{36} + 24q^{39} - 4q^{40} - 24q^{41} - 12q^{45} + 8q^{46} - 4q^{49} - 4q^{50} - 24q^{55} + 4q^{56} + 16q^{59} - 12q^{60} + 24q^{61} - 4q^{64} - 4q^{65} - 48q^{66} + 48q^{69} - 4q^{70} - 24q^{71} + 8q^{74} + 16q^{76} - 8q^{79} + 4q^{80} - 36q^{81} - 8q^{85} - 16q^{86} + 40q^{89} - 12q^{90} - 8q^{91} + 16q^{94} - 4q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(57\)
\(\chi(n)\) \(1\) \(-1\)

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
1.22474 1.22474i
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
1.00000i 2.44949i −1.00000 −0.224745 + 2.22474i −2.44949 1.00000i 1.00000i −3.00000 2.22474 + 0.224745i
29.2 1.00000i 2.44949i −1.00000 2.22474 0.224745i 2.44949 1.00000i 1.00000i −3.00000 −0.224745 2.22474i
29.3 1.00000i 2.44949i −1.00000 2.22474 + 0.224745i 2.44949 1.00000i 1.00000i −3.00000 −0.224745 + 2.22474i
29.4 1.00000i 2.44949i −1.00000 −0.224745 2.22474i −2.44949 1.00000i 1.00000i −3.00000 2.22474 0.224745i
\(n\): 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 70.2.c.a 4
3.b odd 2 1 630.2.g.g 4
4.b odd 2 1 560.2.g.e 4
5.b even 2 1 inner 70.2.c.a 4
5.c odd 4 1 350.2.a.g 2
5.c odd 4 1 350.2.a.h 2
7.b odd 2 1 490.2.c.e 4
7.c even 3 2 490.2.i.c 8
7.d odd 6 2 490.2.i.f 8
8.b even 2 1 2240.2.g.j 4
8.d odd 2 1 2240.2.g.i 4
12.b even 2 1 5040.2.t.t 4
15.d odd 2 1 630.2.g.g 4
15.e even 4 1 3150.2.a.bs 2
15.e even 4 1 3150.2.a.bt 2
20.d odd 2 1 560.2.g.e 4
20.e even 4 1 2800.2.a.bl 2
20.e even 4 1 2800.2.a.bm 2
35.c odd 2 1 490.2.c.e 4
35.f even 4 1 2450.2.a.bl 2
35.f even 4 1 2450.2.a.bq 2
35.i odd 6 2 490.2.i.f 8
35.j even 6 2 490.2.i.c 8
40.e odd 2 1 2240.2.g.i 4
40.f even 2 1 2240.2.g.j 4
60.h even 2 1 5040.2.t.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 1.a even 1 1 trivial
70.2.c.a 4 5.b even 2 1 inner
350.2.a.g 2 5.c odd 4 1
350.2.a.h 2 5.c odd 4 1
490.2.c.e 4 7.b odd 2 1
490.2.c.e 4 35.c odd 2 1
490.2.i.c 8 7.c even 3 2
490.2.i.c 8 35.j even 6 2
490.2.i.f 8 7.d odd 6 2
490.2.i.f 8 35.i odd 6 2
560.2.g.e 4 4.b odd 2 1
560.2.g.e 4 20.d odd 2 1
630.2.g.g 4 3.b odd 2 1
630.2.g.g 4 15.d odd 2 1
2240.2.g.i 4 8.d odd 2 1
2240.2.g.i 4 40.e odd 2 1
2240.2.g.j 4 8.b even 2 1
2240.2.g.j 4 40.f even 2 1
2450.2.a.bl 2 35.f even 4 1
2450.2.a.bq 2 35.f even 4 1
2800.2.a.bl 2 20.e even 4 1
2800.2.a.bm 2 20.e even 4 1
3150.2.a.bs 2 15.e even 4 1
3150.2.a.bt 2 15.e even 4 1
5040.2.t.t 4 12.b even 2 1
5040.2.t.t 4 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(70, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + 9 T^{4} )^{2} \)
$5$ \( 1 - 4 T + 8 T^{2} - 20 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \)
$13$ \( 1 - 32 T^{2} + 498 T^{4} - 5408 T^{6} + 28561 T^{8} \)
$17$ \( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 8 T + 48 T^{2} + 152 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 - 36 T^{2} + 998 T^{4} - 19044 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 4 T + 38 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 8 T + 54 T^{2} - 248 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 12 T + 94 T^{2} + 492 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 - 92 T^{2} + 4278 T^{4} - 170108 T^{6} + 3418801 T^{8} \)
$47$ \( 1 - 108 T^{2} + 5798 T^{4} - 238572 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 92 T^{2} + 4278 T^{4} - 258428 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 8 T + 128 T^{2} - 472 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 12 T + 152 T^{2} - 732 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 12 T + 154 T^{2} + 852 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 236 T^{2} + 24198 T^{4} - 1257644 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 + 4 T + 138 T^{2} + 316 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 10 T + 89 T^{2} )^{4} \)
$97$ \( 1 - 124 T^{2} + 8838 T^{4} - 1166716 T^{6} + 88529281 T^{8} \)
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