Properties

Label 630.4.g.a
Level $630$
Weight $4$
Character orbit 630.g
Analytic conductor $37.171$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 630.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.1712033036\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} - 4 q^{4} + (5 i - 10) q^{5} - 7 i q^{7} - 8 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - 4 q^{4} + (5 i - 10) q^{5} - 7 i q^{7} - 8 i q^{8} + ( - 20 i - 10) q^{10} + 37 q^{11} + 51 i q^{13} + 14 q^{14} + 16 q^{16} + 41 i q^{17} + 108 q^{19} + ( - 20 i + 40) q^{20} + 74 i q^{22} + 70 i q^{23} + ( - 100 i + 75) q^{25} - 102 q^{26} + 28 i q^{28} - 249 q^{29} - 134 q^{31} + 32 i q^{32} - 82 q^{34} + (70 i + 35) q^{35} + 334 i q^{37} + 216 i q^{38} + (80 i + 40) q^{40} - 206 q^{41} - 376 i q^{43} - 148 q^{44} - 140 q^{46} - 287 i q^{47} - 49 q^{49} + (150 i + 200) q^{50} - 204 i q^{52} + 6 i q^{53} + (185 i - 370) q^{55} - 56 q^{56} - 498 i q^{58} - 2 q^{59} - 940 q^{61} - 268 i q^{62} - 64 q^{64} + ( - 510 i - 255) q^{65} - 106 i q^{67} - 164 i q^{68} + (70 i - 140) q^{70} - 456 q^{71} + 650 i q^{73} - 668 q^{74} - 432 q^{76} - 259 i q^{77} + 1239 q^{79} + (80 i - 160) q^{80} - 412 i q^{82} - 428 i q^{83} + ( - 410 i - 205) q^{85} + 752 q^{86} - 296 i q^{88} - 220 q^{89} + 357 q^{91} - 280 i q^{92} + 574 q^{94} + (540 i - 1080) q^{95} + 1055 i q^{97} - 98 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 20 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 20 q^{5} - 20 q^{10} + 74 q^{11} + 28 q^{14} + 32 q^{16} + 216 q^{19} + 80 q^{20} + 150 q^{25} - 204 q^{26} - 498 q^{29} - 268 q^{31} - 164 q^{34} + 70 q^{35} + 80 q^{40} - 412 q^{41} - 296 q^{44} - 280 q^{46} - 98 q^{49} + 400 q^{50} - 740 q^{55} - 112 q^{56} - 4 q^{59} - 1880 q^{61} - 128 q^{64} - 510 q^{65} - 280 q^{70} - 912 q^{71} - 1336 q^{74} - 864 q^{76} + 2478 q^{79} - 320 q^{80} - 410 q^{85} + 1504 q^{86} - 440 q^{89} + 714 q^{91} + 1148 q^{94} - 2160 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
1.00000i
1.00000i
2.00000i 0 −4.00000 −10.0000 5.00000i 0 7.00000i 8.00000i 0 −10.0000 + 20.0000i
379.2 2.00000i 0 −4.00000 −10.0000 + 5.00000i 0 7.00000i 8.00000i 0 −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 630.4.g.a 2
3.b odd 2 1 70.4.c.a 2
5.b even 2 1 inner 630.4.g.a 2
12.b even 2 1 560.4.g.c 2
15.d odd 2 1 70.4.c.a 2
15.e even 4 1 350.4.a.i 1
15.e even 4 1 350.4.a.m 1
21.c even 2 1 490.4.c.a 2
60.h even 2 1 560.4.g.c 2
105.g even 2 1 490.4.c.a 2
105.k odd 4 1 2450.4.a.c 1
105.k odd 4 1 2450.4.a.bn 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.c.a 2 3.b odd 2 1
70.4.c.a 2 15.d odd 2 1
350.4.a.i 1 15.e even 4 1
350.4.a.m 1 15.e even 4 1
490.4.c.a 2 21.c even 2 1
490.4.c.a 2 105.g even 2 1
560.4.g.c 2 12.b even 2 1
560.4.g.c 2 60.h even 2 1
630.4.g.a 2 1.a even 1 1 trivial
630.4.g.a 2 5.b even 2 1 inner
2450.4.a.c 1 105.k odd 4 1
2450.4.a.bn 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 37 \) acting on \(S_{4}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) 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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