Properties

Label 70.4.c.a
Level $70$
Weight $4$
Character orbit 70.c
Analytic conductor $4.130$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 70.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.13013370040\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{2} + 7 i q^{3} - 4 q^{4} + (5 i + 10) q^{5} - 14 q^{6} + 7 i q^{7} - 8 i q^{8} - 22 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} + 7 i q^{3} - 4 q^{4} + (5 i + 10) q^{5} - 14 q^{6} + 7 i q^{7} - 8 i q^{8} - 22 q^{9} + (20 i - 10) q^{10} - 37 q^{11} - 28 i q^{12} - 51 i q^{13} - 14 q^{14} + (70 i - 35) q^{15} + 16 q^{16} + 41 i q^{17} - 44 i q^{18} + 108 q^{19} + ( - 20 i - 40) q^{20} - 49 q^{21} - 74 i q^{22} + 70 i q^{23} + 56 q^{24} + (100 i + 75) q^{25} + 102 q^{26} + 35 i q^{27} - 28 i q^{28} + 249 q^{29} + ( - 70 i - 140) q^{30} - 134 q^{31} + 32 i q^{32} - 259 i q^{33} - 82 q^{34} + (70 i - 35) q^{35} + 88 q^{36} - 334 i q^{37} + 216 i q^{38} + 357 q^{39} + ( - 80 i + 40) q^{40} + 206 q^{41} - 98 i q^{42} + 376 i q^{43} + 148 q^{44} + ( - 110 i - 220) q^{45} - 140 q^{46} - 287 i q^{47} + 112 i q^{48} - 49 q^{49} + (150 i - 200) q^{50} - 287 q^{51} + 204 i q^{52} + 6 i q^{53} - 70 q^{54} + ( - 185 i - 370) q^{55} + 56 q^{56} + 756 i q^{57} + 498 i q^{58} + 2 q^{59} + ( - 280 i + 140) q^{60} - 940 q^{61} - 268 i q^{62} - 154 i q^{63} - 64 q^{64} + ( - 510 i + 255) q^{65} + 518 q^{66} + 106 i q^{67} - 164 i q^{68} - 490 q^{69} + ( - 70 i - 140) q^{70} + 456 q^{71} + 176 i q^{72} - 650 i q^{73} + 668 q^{74} + (525 i - 700) q^{75} - 432 q^{76} - 259 i q^{77} + 714 i q^{78} + 1239 q^{79} + (80 i + 160) q^{80} - 839 q^{81} + 412 i q^{82} - 428 i q^{83} + 196 q^{84} + (410 i - 205) q^{85} - 752 q^{86} + 1743 i q^{87} + 296 i q^{88} + 220 q^{89} + ( - 440 i + 220) q^{90} + 357 q^{91} - 280 i q^{92} - 938 i q^{93} + 574 q^{94} + (540 i + 1080) q^{95} - 224 q^{96} - 1055 i q^{97} - 98 i q^{98} + 814 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 20 q^{5} - 28 q^{6} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 20 q^{5} - 28 q^{6} - 44 q^{9} - 20 q^{10} - 74 q^{11} - 28 q^{14} - 70 q^{15} + 32 q^{16} + 216 q^{19} - 80 q^{20} - 98 q^{21} + 112 q^{24} + 150 q^{25} + 204 q^{26} + 498 q^{29} - 280 q^{30} - 268 q^{31} - 164 q^{34} - 70 q^{35} + 176 q^{36} + 714 q^{39} + 80 q^{40} + 412 q^{41} + 296 q^{44} - 440 q^{45} - 280 q^{46} - 98 q^{49} - 400 q^{50} - 574 q^{51} - 140 q^{54} - 740 q^{55} + 112 q^{56} + 4 q^{59} + 280 q^{60} - 1880 q^{61} - 128 q^{64} + 510 q^{65} + 1036 q^{66} - 980 q^{69} - 280 q^{70} + 912 q^{71} + 1336 q^{74} - 1400 q^{75} - 864 q^{76} + 2478 q^{79} + 320 q^{80} - 1678 q^{81} + 392 q^{84} - 410 q^{85} - 1504 q^{86} + 440 q^{89} + 440 q^{90} + 714 q^{91} + 1148 q^{94} + 2160 q^{95} - 448 q^{96} + 1628 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
1.00000i
1.00000i
2.00000i 7.00000i −4.00000 10.0000 5.00000i −14.0000 7.00000i 8.00000i −22.0000 −10.0000 20.0000i
29.2 2.00000i 7.00000i −4.00000 10.0000 + 5.00000i −14.0000 7.00000i 8.00000i −22.0000 −10.0000 + 20.0000i
\(n\): 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 70.4.c.a 2
3.b odd 2 1 630.4.g.a 2
4.b odd 2 1 560.4.g.c 2
5.b even 2 1 inner 70.4.c.a 2
5.c odd 4 1 350.4.a.i 1
5.c odd 4 1 350.4.a.m 1
7.b odd 2 1 490.4.c.a 2
15.d odd 2 1 630.4.g.a 2
20.d odd 2 1 560.4.g.c 2
35.c odd 2 1 490.4.c.a 2
35.f even 4 1 2450.4.a.c 1
35.f even 4 1 2450.4.a.bn 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.c.a 2 1.a even 1 1 trivial
70.4.c.a 2 5.b even 2 1 inner
350.4.a.i 1 5.c odd 4 1
350.4.a.m 1 5.c odd 4 1
490.4.c.a 2 7.b odd 2 1
490.4.c.a 2 35.c odd 2 1
560.4.g.c 2 4.b odd 2 1
560.4.g.c 2 20.d odd 2 1
630.4.g.a 2 3.b odd 2 1
630.4.g.a 2 15.d odd 2 1
2450.4.a.c 1 35.f even 4 1
2450.4.a.bn 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 49 \) acting on \(S_{4}^{\mathrm{new}}(70, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 49 \) Copy content Toggle raw display
$5$ \( T^{2} - 20T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T + 37)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2601 \) Copy content Toggle raw display
$17$ \( T^{2} + 1681 \) Copy content Toggle raw display
$19$ \( (T - 108)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4900 \) Copy content Toggle raw display
$29$ \( (T - 249)^{2} \) Copy content Toggle raw display
$31$ \( (T + 134)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 111556 \) Copy content Toggle raw display
$41$ \( (T - 206)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 141376 \) Copy content Toggle raw display
$47$ \( T^{2} + 82369 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 2)^{2} \) Copy content Toggle raw display
$61$ \( (T + 940)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 11236 \) Copy content Toggle raw display
$71$ \( (T - 456)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 422500 \) Copy content Toggle raw display
$79$ \( (T - 1239)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 183184 \) Copy content Toggle raw display
$89$ \( (T - 220)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1113025 \) Copy content Toggle raw display
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