Properties

Label 350.3.k.a
Level $350$
Weight $3$
Character orbit 350.k
Analytic conductor $9.537$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{6} + ( -3 - 4 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{7} -2 \beta_{3} q^{8} + 6 \beta_{1} q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{6} + ( -3 - 4 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{7} -2 \beta_{3} q^{8} + 6 \beta_{1} q^{9} + ( -3 \beta_{1} - 9 \beta_{2} - 3 \beta_{3} ) q^{11} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{12} + ( -6 + 4 \beta_{1} - 12 \beta_{2} + 2 \beta_{3} ) q^{13} + ( -10 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{14} + ( -4 - 4 \beta_{2} ) q^{16} + ( 10 + 2 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{17} -12 \beta_{2} q^{18} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{19} + ( 8 - 8 \beta_{1} - 11 \beta_{2} - 10 \beta_{3} ) q^{21} + ( -6 + 9 \beta_{3} ) q^{22} + ( -15 + 9 \beta_{1} - 15 \beta_{2} ) q^{23} + ( 8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{24} + ( 4 + 6 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} ) q^{26} + ( 3 - 6 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{27} + ( 4 + 10 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{28} + ( 12 - 6 \beta_{3} ) q^{29} + ( -14 + 15 \beta_{1} - 7 \beta_{2} - 15 \beta_{3} ) q^{31} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( 15 - 12 \beta_{1} - 15 \beta_{2} - 24 \beta_{3} ) q^{33} + ( -4 - 10 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} ) q^{34} + 12 \beta_{3} q^{36} + ( 31 + 24 \beta_{1} + 31 \beta_{2} ) q^{37} + ( 4 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{38} + ( -12 \beta_{1} - 6 \beta_{2} - 12 \beta_{3} ) q^{39} + ( -2 + 20 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} ) q^{41} + ( -20 - 8 \beta_{1} - 4 \beta_{2} + 11 \beta_{3} ) q^{42} + ( 2 + 6 \beta_{3} ) q^{43} + ( 18 + 6 \beta_{1} + 18 \beta_{2} ) q^{44} + ( 15 \beta_{1} - 18 \beta_{2} + 15 \beta_{3} ) q^{46} + ( -29 - \beta_{1} + 29 \beta_{2} - 2 \beta_{3} ) q^{47} + ( -4 - 8 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{48} + ( -25 + 4 \beta_{1} - 40 \beta_{2} + 26 \beta_{3} ) q^{49} + ( 27 + 21 \beta_{1} + 27 \beta_{2} ) q^{51} + ( 24 - 4 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} ) q^{52} + ( 12 \beta_{1} - 39 \beta_{2} + 12 \beta_{3} ) q^{53} + ( -6 - 3 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{54} + ( 4 - 4 \beta_{1} - 16 \beta_{2} + 2 \beta_{3} ) q^{56} -3 q^{57} + ( -12 - 12 \beta_{1} - 12 \beta_{2} ) q^{58} + ( -26 - 25 \beta_{1} - 13 \beta_{2} + 25 \beta_{3} ) q^{59} + ( -7 - 32 \beta_{1} + 7 \beta_{2} - 64 \beta_{3} ) q^{61} + ( -30 + 14 \beta_{1} - 60 \beta_{2} + 7 \beta_{3} ) q^{62} + ( 60 - 18 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{63} + 8 q^{64} + ( -48 - 15 \beta_{1} - 24 \beta_{2} + 15 \beta_{3} ) q^{66} + ( -45 \beta_{1} - 29 \beta_{2} - 45 \beta_{3} ) q^{67} + ( -10 + 4 \beta_{1} + 10 \beta_{2} + 8 \beta_{3} ) q^{68} + ( 3 - 12 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{69} + ( -6 + 30 \beta_{3} ) q^{71} + ( 24 + 24 \beta_{2} ) q^{72} + ( -106 - 16 \beta_{1} - 53 \beta_{2} + 16 \beta_{3} ) q^{73} + ( -31 \beta_{1} - 48 \beta_{2} - 31 \beta_{3} ) q^{74} + ( 2 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{76} + ( -42 - 42 \beta_{1} - 21 \beta_{2} ) q^{77} + ( -24 + 6 \beta_{3} ) q^{78} + ( 55 + 15 \beta_{1} + 55 \beta_{2} ) q^{79} + ( -54 \beta_{1} - 9 \beta_{2} - 54 \beta_{3} ) q^{81} + ( 20 + 2 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} ) q^{82} + ( 68 - 8 \beta_{1} + 136 \beta_{2} - 4 \beta_{3} ) q^{83} + ( 22 + 20 \beta_{1} + 38 \beta_{2} + 4 \beta_{3} ) q^{84} + ( 12 - 2 \beta_{1} + 12 \beta_{2} ) q^{86} + ( 48 + 18 \beta_{1} + 24 \beta_{2} - 18 \beta_{3} ) q^{87} + ( -18 \beta_{1} - 12 \beta_{2} - 18 \beta_{3} ) q^{88} + ( -63 + 24 \beta_{1} + 63 \beta_{2} + 48 \beta_{3} ) q^{89} + ( 30 - 44 \beta_{1} + 48 \beta_{2} + 8 \beta_{3} ) q^{91} + ( 30 + 18 \beta_{3} ) q^{92} + ( 69 + 24 \beta_{1} + 69 \beta_{2} ) q^{93} + ( -4 + 29 \beta_{1} - 2 \beta_{2} - 29 \beta_{3} ) q^{94} + ( -8 + 4 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{96} + ( -22 + 52 \beta_{1} - 44 \beta_{2} + 26 \beta_{3} ) q^{97} + ( 52 + 25 \beta_{1} + 44 \beta_{2} + 40 \beta_{3} ) q^{98} + ( 36 - 54 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 4 q^{4} - 8 q^{7} + O(q^{10}) \) \( 4 q + 6 q^{3} - 4 q^{4} - 8 q^{7} + 18 q^{11} - 12 q^{12} - 36 q^{14} - 8 q^{16} + 30 q^{17} + 24 q^{18} + 6 q^{19} + 54 q^{21} - 24 q^{22} - 30 q^{23} + 24 q^{24} + 24 q^{26} + 20 q^{28} + 48 q^{29} - 42 q^{31} + 90 q^{33} + 62 q^{37} + 12 q^{38} + 12 q^{39} - 72 q^{42} + 8 q^{43} + 36 q^{44} + 36 q^{46} - 174 q^{47} - 20 q^{49} + 54 q^{51} + 72 q^{52} + 78 q^{53} - 36 q^{54} + 48 q^{56} - 12 q^{57} - 24 q^{58} - 78 q^{59} - 42 q^{61} + 216 q^{63} + 32 q^{64} - 144 q^{66} + 58 q^{67} - 60 q^{68} - 24 q^{71} + 48 q^{72} - 318 q^{73} + 96 q^{74} - 126 q^{77} - 96 q^{78} + 110 q^{79} + 18 q^{81} + 120 q^{82} + 12 q^{84} + 24 q^{86} + 144 q^{87} + 24 q^{88} - 378 q^{89} + 24 q^{91} + 120 q^{92} + 138 q^{93} - 12 q^{94} - 48 q^{96} + 120 q^{98} + 144 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1 + \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
−0.707107 1.22474i 3.62132 + 2.09077i −1.00000 + 1.73205i 0 5.91359i 2.24264 6.63103i 2.82843 4.24264 + 7.34847i 0
101.2 0.707107 + 1.22474i −0.621320 0.358719i −1.00000 + 1.73205i 0 1.01461i −6.24264 + 3.16693i −2.82843 −4.24264 7.34847i 0
201.1 −0.707107 + 1.22474i 3.62132 2.09077i −1.00000 1.73205i 0 5.91359i 2.24264 + 6.63103i 2.82843 4.24264 7.34847i 0
201.2 0.707107 1.22474i −0.621320 + 0.358719i −1.00000 1.73205i 0 1.01461i −6.24264 3.16693i −2.82843 −4.24264 + 7.34847i 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
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.k.a 4
5.b even 2 1 14.3.d.a 4
5.c odd 4 2 350.3.i.a 8
7.d odd 6 1 inner 350.3.k.a 4
15.d odd 2 1 126.3.n.c 4
20.d odd 2 1 112.3.s.b 4
35.c odd 2 1 98.3.d.a 4
35.i odd 6 1 14.3.d.a 4
35.i odd 6 1 98.3.b.b 4
35.j even 6 1 98.3.b.b 4
35.j even 6 1 98.3.d.a 4
35.k even 12 2 350.3.i.a 8
40.e odd 2 1 448.3.s.c 4
40.f even 2 1 448.3.s.d 4
60.h even 2 1 1008.3.cg.l 4
105.g even 2 1 882.3.n.b 4
105.o odd 6 1 882.3.c.f 4
105.o odd 6 1 882.3.n.b 4
105.p even 6 1 126.3.n.c 4
105.p even 6 1 882.3.c.f 4
140.c even 2 1 784.3.s.c 4
140.p odd 6 1 784.3.c.e 4
140.p odd 6 1 784.3.s.c 4
140.s even 6 1 112.3.s.b 4
140.s even 6 1 784.3.c.e 4
280.ba even 6 1 448.3.s.c 4
280.bk odd 6 1 448.3.s.d 4
420.be odd 6 1 1008.3.cg.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a 4 5.b even 2 1
14.3.d.a 4 35.i odd 6 1
98.3.b.b 4 35.i odd 6 1
98.3.b.b 4 35.j even 6 1
98.3.d.a 4 35.c odd 2 1
98.3.d.a 4 35.j even 6 1
112.3.s.b 4 20.d odd 2 1
112.3.s.b 4 140.s even 6 1
126.3.n.c 4 15.d odd 2 1
126.3.n.c 4 105.p even 6 1
350.3.i.a 8 5.c odd 4 2
350.3.i.a 8 35.k even 12 2
350.3.k.a 4 1.a even 1 1 trivial
350.3.k.a 4 7.d odd 6 1 inner
448.3.s.c 4 40.e odd 2 1
448.3.s.c 4 280.ba even 6 1
448.3.s.d 4 40.f even 2 1
448.3.s.d 4 280.bk odd 6 1
784.3.c.e 4 140.p odd 6 1
784.3.c.e 4 140.s even 6 1
784.3.s.c 4 140.c even 2 1
784.3.s.c 4 140.p odd 6 1
882.3.c.f 4 105.o odd 6 1
882.3.c.f 4 105.p even 6 1
882.3.n.b 4 105.g even 2 1
882.3.n.b 4 105.o odd 6 1
1008.3.cg.l 4 60.h even 2 1
1008.3.cg.l 4 420.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 6 T_{3}^{3} + 9 T_{3}^{2} + 18 T_{3} + 9 \) acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T^{2} + T^{4} \)
$3$ \( 9 + 18 T + 9 T^{2} - 6 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 2401 + 392 T + 42 T^{2} + 8 T^{3} + T^{4} \)
$11$ \( 3969 - 1134 T + 261 T^{2} - 18 T^{3} + T^{4} \)
$13$ \( 7056 + 264 T^{2} + T^{4} \)
$17$ \( 2601 - 1530 T + 351 T^{2} - 30 T^{3} + T^{4} \)
$19$ \( 9 + 18 T + 9 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( 3969 + 1890 T + 837 T^{2} + 30 T^{3} + T^{4} \)
$29$ \( ( 72 - 24 T + T^{2} )^{2} \)
$31$ \( 1447209 - 50526 T - 615 T^{2} + 42 T^{3} + T^{4} \)
$37$ \( 36481 + 11842 T + 4035 T^{2} - 62 T^{3} + T^{4} \)
$41$ \( 345744 + 1224 T^{2} + T^{4} \)
$43$ \( ( -68 - 4 T + T^{2} )^{2} \)
$47$ \( 6335289 + 437958 T + 12609 T^{2} + 174 T^{3} + T^{4} \)
$53$ \( 1520289 - 96174 T + 4851 T^{2} - 78 T^{3} + T^{4} \)
$59$ \( 10517049 - 252954 T - 1215 T^{2} + 78 T^{3} + T^{4} \)
$61$ \( 35964009 - 251874 T - 5409 T^{2} + 42 T^{3} + T^{4} \)
$67$ \( 10297681 + 186122 T + 6573 T^{2} - 58 T^{3} + T^{4} \)
$71$ \( ( -1764 + 12 T + T^{2} )^{2} \)
$73$ \( 47485881 + 2191338 T + 40599 T^{2} + 318 T^{3} + T^{4} \)
$79$ \( 6630625 - 283250 T + 9525 T^{2} - 110 T^{3} + T^{4} \)
$83$ \( 189778176 + 27936 T^{2} + T^{4} \)
$89$ \( 71419401 + 3194478 T + 56079 T^{2} + 378 T^{3} + T^{4} \)
$97$ \( 6780816 + 11016 T^{2} + T^{4} \)
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