Properties

Label 3150.2.m.c
Level 3150
Weight 2
Character orbit 3150.m
Analytic conductor 25.153
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1457.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0
1457.2 0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i −0.707107 0.707107i 0 0
2843.1 −0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i 0.707107 0.707107i 0 0
2843.2 0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0
\(n\): 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 3150.2.m.c 4
3.b odd 2 1 3150.2.m.d yes 4
5.b even 2 1 3150.2.m.e yes 4
5.c odd 4 1 3150.2.m.d yes 4
5.c odd 4 1 3150.2.m.f yes 4
15.d odd 2 1 3150.2.m.f yes 4
15.e even 4 1 inner 3150.2.m.c 4
15.e even 4 1 3150.2.m.e yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3150.2.m.c 4 1.a even 1 1 trivial
3150.2.m.c 4 15.e even 4 1 inner
3150.2.m.d yes 4 3.b odd 2 1
3150.2.m.d yes 4 5.c odd 4 1
3150.2.m.e yes 4 5.b even 2 1
3150.2.m.e yes 4 15.e even 4 1
3150.2.m.f yes 4 5.c odd 4 1
3150.2.m.f yes 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3150, [\chi])\):

\( T_{11}^{4} + 24 T_{11}^{2} + 16 \)
\( T_{13}^{4} + 8 T_{13}^{3} + 32 T_{13}^{2} + 32 T_{13} + 16 \)
\( T_{17}^{4} + 12 T_{17}^{3} + 72 T_{17}^{2} + 168 T_{17} + 196 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ 1
$5$ 1
$7$ \( 1 + T^{4} \)
$11$ \( 1 - 20 T^{2} + 214 T^{4} - 2420 T^{6} + 14641 T^{8} \)
$13$ \( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 1768 T^{5} + 5408 T^{6} + 17576 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 12 T + 72 T^{2} + 372 T^{3} + 1726 T^{4} + 6324 T^{5} + 20808 T^{6} + 58956 T^{7} + 83521 T^{8} \)
$19$ \( ( 1 + 12 T^{2} + 361 T^{4} )^{2} \)
$23$ \( 1 + 8 T + 32 T^{2} + 120 T^{3} + 386 T^{4} + 2760 T^{5} + 16928 T^{6} + 97336 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 - 12 T + 86 T^{2} - 348 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 12 T + 90 T^{2} + 372 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 - 4 T + 8 T^{2} + 244 T^{3} - 2162 T^{4} + 9028 T^{5} + 10952 T^{6} - 202612 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 140 T^{2} + 8134 T^{4} - 235340 T^{6} + 2825761 T^{8} \)
$43$ \( 1 - 20 T + 200 T^{2} - 1780 T^{3} + 13726 T^{4} - 76540 T^{5} + 369800 T^{6} - 1590140 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 20 T + 200 T^{2} + 1860 T^{3} + 15182 T^{4} + 87420 T^{5} + 441800 T^{6} + 2076460 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 8 T + 32 T^{2} - 360 T^{3} + 3986 T^{4} - 19080 T^{5} + 89888 T^{6} - 1191016 T^{7} + 7890481 T^{8} \)
$59$ \( ( 1 - 8 T + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 16 T + 184 T^{2} - 976 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 4 T + 8 T^{2} + 260 T^{3} + 8446 T^{4} + 17420 T^{5} + 35912 T^{6} + 1203052 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 120 T^{2} + 9074 T^{4} - 604920 T^{6} + 25411681 T^{8} \)
$73$ \( ( 1 + 12 T + 72 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 - 52 T^{2} + 11110 T^{4} - 324532 T^{6} + 38950081 T^{8} \)
$83$ \( 1 - 16 T + 128 T^{2} - 816 T^{3} + 4178 T^{4} - 67728 T^{5} + 881792 T^{6} - 9148592 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 - 4 T + 54 T^{2} - 356 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 + 16 T + 128 T^{2} + 1488 T^{3} + 17282 T^{4} + 144336 T^{5} + 1204352 T^{6} + 14602768 T^{7} + 88529281 T^{8} \)
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