Properties

Label 630.2.m.a
Level 630
Weight 2
Character orbit 630.m
Analytic conductor 5.031
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( -2 - \zeta_{8}^{2} ) q^{5} -\zeta_{8}^{3} q^{7} + \zeta_{8}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( -2 - \zeta_{8}^{2} ) q^{5} -\zeta_{8}^{3} q^{7} + \zeta_{8}^{3} q^{8} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{10} + ( \zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{11} + ( 4 + 4 \zeta_{8}^{2} ) q^{13} + q^{14} - q^{16} + ( 2 + 2 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{17} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{19} + ( 1 - 2 \zeta_{8}^{2} ) q^{20} + ( -1 + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{22} + ( 2 - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{23} + ( 3 + 4 \zeta_{8}^{2} ) q^{25} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{26} + \zeta_{8} q^{28} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( 6 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{31} -\zeta_{8} q^{32} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{34} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{35} + ( 2 - 2 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{37} + ( -2 + 2 \zeta_{8}^{2} ) q^{38} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{40} + ( -2 \zeta_{8} - 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{41} + ( -1 - 10 \zeta_{8} - \zeta_{8}^{2} ) q^{43} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{44} + ( -4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{46} + ( 5 - 6 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{47} -\zeta_{8}^{2} q^{49} + ( 3 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{50} + ( -4 + 4 \zeta_{8}^{2} ) q^{52} + ( -4 + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{53} + ( 2 - \zeta_{8} - 4 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{55} + \zeta_{8}^{2} q^{56} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{58} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{59} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{61} + ( 3 + 6 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{62} -\zeta_{8}^{2} q^{64} + ( -4 - 12 \zeta_{8}^{2} ) q^{65} + ( 7 - 7 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{67} + ( -2 + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{68} + ( -2 - \zeta_{8}^{2} ) q^{70} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{71} -10 \zeta_{8} q^{73} + ( -6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{74} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{76} + ( 1 + 2 \zeta_{8} + \zeta_{8}^{2} ) q^{77} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{79} + ( 2 + \zeta_{8}^{2} ) q^{80} + ( 2 - 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{82} + ( 2 - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{83} + ( -2 - 4 \zeta_{8} - 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{85} + ( -\zeta_{8} - 10 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{86} + ( -1 - 2 \zeta_{8} - \zeta_{8}^{2} ) q^{88} + ( 6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{89} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{91} + ( 2 - 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{92} + ( 5 \zeta_{8} - 6 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{94} + ( -2 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{95} + ( -6 + 6 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{97} -\zeta_{8}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{5} + O(q^{10}) \) \( 4q - 8q^{5} + 16q^{13} + 4q^{14} - 4q^{16} + 8q^{17} + 4q^{20} - 4q^{22} + 8q^{23} + 12q^{25} - 8q^{29} + 24q^{31} + 8q^{37} - 8q^{38} - 4q^{43} - 8q^{44} - 16q^{46} + 20q^{47} - 16q^{52} - 16q^{53} + 8q^{55} - 8q^{58} + 12q^{62} - 16q^{65} + 28q^{67} - 8q^{68} - 8q^{70} - 24q^{74} + 4q^{77} + 8q^{80} + 8q^{82} + 8q^{83} - 8q^{85} - 4q^{88} + 24q^{89} + 8q^{92} - 24q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i 0 1.00000i −2.00000 + 1.00000i 0 −0.707107 0.707107i 0.707107 + 0.707107i 0 0.707107 2.12132i
197.2 0.707107 0.707107i 0 1.00000i −2.00000 + 1.00000i 0 0.707107 + 0.707107i −0.707107 0.707107i 0 −0.707107 + 2.12132i
323.1 −0.707107 0.707107i 0 1.00000i −2.00000 1.00000i 0 −0.707107 + 0.707107i 0.707107 0.707107i 0 0.707107 + 2.12132i
323.2 0.707107 + 0.707107i 0 1.00000i −2.00000 1.00000i 0 0.707107 0.707107i −0.707107 + 0.707107i 0 −0.707107 2.12132i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.m.a 4
3.b odd 2 1 630.2.m.b yes 4
5.b even 2 1 3150.2.m.b 4
5.c odd 4 1 630.2.m.b yes 4
5.c odd 4 1 3150.2.m.a 4
15.d odd 2 1 3150.2.m.a 4
15.e even 4 1 inner 630.2.m.a 4
15.e even 4 1 3150.2.m.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.m.a 4 1.a even 1 1 trivial
630.2.m.a 4 15.e even 4 1 inner
630.2.m.b yes 4 3.b odd 2 1
630.2.m.b yes 4 5.c odd 4 1
3150.2.m.a 4 5.c odd 4 1
3150.2.m.a 4 15.d odd 2 1
3150.2.m.b 4 5.b even 2 1
3150.2.m.b 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{4} + 12 T_{11}^{2} + 4 \)
\( T_{17}^{4} - 8 T_{17}^{3} + 32 T_{17}^{2} - 32 T_{17} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ 1
$5$ \( ( 1 + 4 T + 5 T^{2} )^{2} \)
$7$ \( 1 + T^{4} \)
$11$ \( 1 - 32 T^{2} + 466 T^{4} - 3872 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )^{2} \)
$17$ \( 1 - 8 T + 32 T^{2} - 168 T^{3} + 866 T^{4} - 2856 T^{5} + 9248 T^{6} - 39304 T^{7} + 83521 T^{8} \)
$19$ \( ( 1 - 30 T^{2} + 361 T^{4} )^{2} \)
$23$ \( 1 - 8 T + 32 T^{2} - 120 T^{3} + 386 T^{4} - 2760 T^{5} + 16928 T^{6} - 97336 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 + 4 T + 54 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 12 T + 80 T^{2} - 372 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 - 8 T + 32 T^{2} - 72 T^{3} - 622 T^{4} - 2664 T^{5} + 43808 T^{6} - 405224 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 76 T^{2} + 3654 T^{4} - 127756 T^{6} + 2825761 T^{8} \)
$43$ \( 1 + 4 T + 8 T^{2} - 220 T^{3} - 3554 T^{4} - 9460 T^{5} + 14792 T^{6} + 318028 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 20 T + 200 T^{2} - 1220 T^{3} + 7246 T^{4} - 57340 T^{5} + 441800 T^{6} - 2076460 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 16 T + 128 T^{2} + 1296 T^{3} + 12338 T^{4} + 68688 T^{5} + 359552 T^{6} + 2382032 T^{7} + 7890481 T^{8} \)
$59$ \( ( 1 + 110 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 24 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( 1 - 28 T + 392 T^{2} - 4508 T^{3} + 43006 T^{4} - 302036 T^{5} + 1759688 T^{6} - 8421364 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 236 T^{2} + 23494 T^{4} - 1189676 T^{6} + 25411681 T^{8} \)
$73$ \( 1 - 8542 T^{4} + 28398241 T^{8} \)
$79$ \( ( 1 - 126 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 8 T + 32 T^{2} - 600 T^{3} + 11186 T^{4} - 49800 T^{5} + 220448 T^{6} - 4574296 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 - 12 T + 206 T^{2} - 1068 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 + 24 T + 288 T^{2} + 3960 T^{3} + 49826 T^{4} + 384120 T^{5} + 2709792 T^{6} + 21904152 T^{7} + 88529281 T^{8} \)
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