Properties

Label 150.7.b.a
Level $150$
Weight $7$
Character orbit 150.b
Analytic conductor $34.508$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.5081125430\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{2} + ( -12 \zeta_{8} - 21 \zeta_{8}^{2} + 12 \zeta_{8}^{3} ) q^{3} + 32 q^{4} + ( -96 - 84 \zeta_{8} - 84 \zeta_{8}^{3} ) q^{6} + 2 \zeta_{8}^{2} q^{7} + ( 128 \zeta_{8} - 128 \zeta_{8}^{3} ) q^{8} + ( -153 + 504 \zeta_{8} + 504 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{2} + ( -12 \zeta_{8} - 21 \zeta_{8}^{2} + 12 \zeta_{8}^{3} ) q^{3} + 32 q^{4} + ( -96 - 84 \zeta_{8} - 84 \zeta_{8}^{3} ) q^{6} + 2 \zeta_{8}^{2} q^{7} + ( 128 \zeta_{8} - 128 \zeta_{8}^{3} ) q^{8} + ( -153 + 504 \zeta_{8} + 504 \zeta_{8}^{3} ) q^{9} + ( 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{11} + ( -384 \zeta_{8} - 672 \zeta_{8}^{2} + 384 \zeta_{8}^{3} ) q^{12} + 2950 \zeta_{8}^{2} q^{13} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{14} + 1024 q^{16} + ( 3168 \zeta_{8} - 3168 \zeta_{8}^{3} ) q^{17} + ( -612 \zeta_{8} + 4032 \zeta_{8}^{2} + 612 \zeta_{8}^{3} ) q^{18} -5258 q^{19} + ( 42 - 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{21} + 192 \zeta_{8}^{2} q^{22} + ( 7248 \zeta_{8} - 7248 \zeta_{8}^{3} ) q^{23} + ( -3072 - 2688 \zeta_{8} - 2688 \zeta_{8}^{3} ) q^{24} + ( 11800 \zeta_{8} + 11800 \zeta_{8}^{3} ) q^{26} + ( 12420 \zeta_{8} - 8883 \zeta_{8}^{2} - 12420 \zeta_{8}^{3} ) q^{27} + 64 \zeta_{8}^{2} q^{28} + ( 1560 \zeta_{8} + 1560 \zeta_{8}^{3} ) q^{29} + 22898 q^{31} + ( 4096 \zeta_{8} - 4096 \zeta_{8}^{3} ) q^{32} + ( 504 \zeta_{8} - 576 \zeta_{8}^{2} - 504 \zeta_{8}^{3} ) q^{33} + 25344 q^{34} + ( -4896 + 16128 \zeta_{8} + 16128 \zeta_{8}^{3} ) q^{36} + 34058 \zeta_{8}^{2} q^{37} + ( -21032 \zeta_{8} + 21032 \zeta_{8}^{3} ) q^{38} + ( 61950 - 35400 \zeta_{8} - 35400 \zeta_{8}^{3} ) q^{39} + ( 11856 \zeta_{8} + 11856 \zeta_{8}^{3} ) q^{41} + ( 168 \zeta_{8} - 192 \zeta_{8}^{2} - 168 \zeta_{8}^{3} ) q^{42} + 6406 \zeta_{8}^{2} q^{43} + ( 768 \zeta_{8} + 768 \zeta_{8}^{3} ) q^{44} + 57984 q^{46} + ( 127200 \zeta_{8} - 127200 \zeta_{8}^{3} ) q^{47} + ( -12288 \zeta_{8} - 21504 \zeta_{8}^{2} + 12288 \zeta_{8}^{3} ) q^{48} + 117645 q^{49} + ( -76032 - 66528 \zeta_{8} - 66528 \zeta_{8}^{3} ) q^{51} + 94400 \zeta_{8}^{2} q^{52} + ( 136152 \zeta_{8} - 136152 \zeta_{8}^{3} ) q^{53} + ( 99360 - 35532 \zeta_{8} - 35532 \zeta_{8}^{3} ) q^{54} + ( 256 \zeta_{8} + 256 \zeta_{8}^{3} ) q^{56} + ( 63096 \zeta_{8} + 110418 \zeta_{8}^{2} - 63096 \zeta_{8}^{3} ) q^{57} + 12480 \zeta_{8}^{2} q^{58} + ( -231096 \zeta_{8} - 231096 \zeta_{8}^{3} ) q^{59} -62566 q^{61} + ( 91592 \zeta_{8} - 91592 \zeta_{8}^{3} ) q^{62} + ( -1008 \zeta_{8} - 306 \zeta_{8}^{2} + 1008 \zeta_{8}^{3} ) q^{63} + 32768 q^{64} + ( 4032 - 2304 \zeta_{8} - 2304 \zeta_{8}^{3} ) q^{66} + 438698 \zeta_{8}^{2} q^{67} + ( 101376 \zeta_{8} - 101376 \zeta_{8}^{3} ) q^{68} + ( -173952 - 152208 \zeta_{8} - 152208 \zeta_{8}^{3} ) q^{69} + ( 48240 \zeta_{8} + 48240 \zeta_{8}^{3} ) q^{71} + ( -19584 \zeta_{8} + 129024 \zeta_{8}^{2} + 19584 \zeta_{8}^{3} ) q^{72} + 730510 \zeta_{8}^{2} q^{73} + ( 136232 \zeta_{8} + 136232 \zeta_{8}^{3} ) q^{74} -168256 q^{76} + ( -48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{77} + ( 247800 \zeta_{8} - 283200 \zeta_{8}^{2} - 247800 \zeta_{8}^{3} ) q^{78} -340562 q^{79} + ( -484623 - 154224 \zeta_{8} - 154224 \zeta_{8}^{3} ) q^{81} + 94848 \zeta_{8}^{2} q^{82} + ( -350904 \zeta_{8} + 350904 \zeta_{8}^{3} ) q^{83} + ( 1344 - 768 \zeta_{8} - 768 \zeta_{8}^{3} ) q^{84} + ( 25624 \zeta_{8} + 25624 \zeta_{8}^{3} ) q^{86} + ( 32760 \zeta_{8} - 37440 \zeta_{8}^{2} - 32760 \zeta_{8}^{3} ) q^{87} + 6144 \zeta_{8}^{2} q^{88} + ( -273456 \zeta_{8} - 273456 \zeta_{8}^{3} ) q^{89} -5900 q^{91} + ( 231936 \zeta_{8} - 231936 \zeta_{8}^{3} ) q^{92} + ( -274776 \zeta_{8} - 480858 \zeta_{8}^{2} + 274776 \zeta_{8}^{3} ) q^{93} + 1017600 q^{94} + ( -98304 - 86016 \zeta_{8} - 86016 \zeta_{8}^{3} ) q^{96} -281086 \zeta_{8}^{2} q^{97} + ( 470580 \zeta_{8} - 470580 \zeta_{8}^{3} ) q^{98} + ( -24192 - 3672 \zeta_{8} - 3672 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 128 q^{4} - 384 q^{6} - 612 q^{9} + O(q^{10}) \) \( 4 q + 128 q^{4} - 384 q^{6} - 612 q^{9} + 4096 q^{16} - 21032 q^{19} + 168 q^{21} - 12288 q^{24} + 91592 q^{31} + 101376 q^{34} - 19584 q^{36} + 247800 q^{39} + 231936 q^{46} + 470580 q^{49} - 304128 q^{51} + 397440 q^{54} - 250264 q^{61} + 131072 q^{64} + 16128 q^{66} - 695808 q^{69} - 673024 q^{76} - 1362248 q^{79} - 1938492 q^{81} + 5376 q^{84} - 23600 q^{91} + 4070400 q^{94} - 393216 q^{96} - 96768 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−5.65685 16.9706 21.0000i 32.0000 0 −96.0000 + 118.794i 2.00000i −181.019 −153.000 712.764i 0
149.2 −5.65685 16.9706 + 21.0000i 32.0000 0 −96.0000 118.794i 2.00000i −181.019 −153.000 + 712.764i 0
149.3 5.65685 −16.9706 21.0000i 32.0000 0 −96.0000 118.794i 2.00000i 181.019 −153.000 + 712.764i 0
149.4 5.65685 −16.9706 + 21.0000i 32.0000 0 −96.0000 + 118.794i 2.00000i 181.019 −153.000 712.764i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.7.b.a 4
3.b odd 2 1 inner 150.7.b.a 4
5.b even 2 1 inner 150.7.b.a 4
5.c odd 4 1 6.7.b.a 2
5.c odd 4 1 150.7.d.a 2
15.d odd 2 1 inner 150.7.b.a 4
15.e even 4 1 6.7.b.a 2
15.e even 4 1 150.7.d.a 2
20.e even 4 1 48.7.e.b 2
35.f even 4 1 294.7.b.a 2
40.i odd 4 1 192.7.e.c 2
40.k even 4 1 192.7.e.f 2
45.k odd 12 2 162.7.d.b 4
45.l even 12 2 162.7.d.b 4
60.l odd 4 1 48.7.e.b 2
105.k odd 4 1 294.7.b.a 2
120.q odd 4 1 192.7.e.f 2
120.w even 4 1 192.7.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a 2 5.c odd 4 1
6.7.b.a 2 15.e even 4 1
48.7.e.b 2 20.e even 4 1
48.7.e.b 2 60.l odd 4 1
150.7.b.a 4 1.a even 1 1 trivial
150.7.b.a 4 3.b odd 2 1 inner
150.7.b.a 4 5.b even 2 1 inner
150.7.b.a 4 15.d odd 2 1 inner
150.7.d.a 2 5.c odd 4 1
150.7.d.a 2 15.e even 4 1
162.7.d.b 4 45.k odd 12 2
162.7.d.b 4 45.l even 12 2
192.7.e.c 2 40.i odd 4 1
192.7.e.c 2 120.w even 4 1
192.7.e.f 2 40.k even 4 1
192.7.e.f 2 120.q odd 4 1
294.7.b.a 2 35.f even 4 1
294.7.b.a 2 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4 \) acting on \(S_{7}^{\mathrm{new}}(150, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -32 + T^{2} )^{2} \)
$3$ \( 531441 + 306 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 4 + T^{2} )^{2} \)
$11$ \( ( 1152 + T^{2} )^{2} \)
$13$ \( ( 8702500 + T^{2} )^{2} \)
$17$ \( ( -20072448 + T^{2} )^{2} \)
$19$ \( ( 5258 + T )^{4} \)
$23$ \( ( -105067008 + T^{2} )^{2} \)
$29$ \( ( 4867200 + T^{2} )^{2} \)
$31$ \( ( -22898 + T )^{4} \)
$37$ \( ( 1159947364 + T^{2} )^{2} \)
$41$ \( ( 281129472 + T^{2} )^{2} \)
$43$ \( ( 41036836 + T^{2} )^{2} \)
$47$ \( ( -32359680000 + T^{2} )^{2} \)
$53$ \( ( -37074734208 + T^{2} )^{2} \)
$59$ \( ( 106810722432 + T^{2} )^{2} \)
$61$ \( ( 62566 + T )^{4} \)
$67$ \( ( 192455935204 + T^{2} )^{2} \)
$71$ \( ( 4654195200 + T^{2} )^{2} \)
$73$ \( ( 533644860100 + T^{2} )^{2} \)
$79$ \( ( 340562 + T )^{4} \)
$83$ \( ( -246267234432 + T^{2} )^{2} \)
$89$ \( ( 149556367872 + T^{2} )^{2} \)
$97$ \( ( 79009339396 + T^{2} )^{2} \)
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