Properties

Label 350.4.j.d
Level 350
Weight 4
Character orbit 350.j
Analytic conductor 20.651
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{2} + \zeta_{12} q^{3} + ( 4 - 4 \zeta_{12}^{2} ) q^{4} + 2 q^{6} + ( -18 \zeta_{12} + 19 \zeta_{12}^{3} ) q^{7} -8 \zeta_{12}^{3} q^{8} -26 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{2} + \zeta_{12} q^{3} + ( 4 - 4 \zeta_{12}^{2} ) q^{4} + 2 q^{6} + ( -18 \zeta_{12} + 19 \zeta_{12}^{3} ) q^{7} -8 \zeta_{12}^{3} q^{8} -26 \zeta_{12}^{2} q^{9} + ( -35 + 35 \zeta_{12}^{2} ) q^{11} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{12} + 66 \zeta_{12}^{3} q^{13} + ( -36 + 38 \zeta_{12}^{2} ) q^{14} -16 \zeta_{12}^{2} q^{16} + 59 \zeta_{12} q^{17} -52 \zeta_{12} q^{18} + 137 \zeta_{12}^{2} q^{19} + ( -19 + \zeta_{12}^{2} ) q^{21} + 70 \zeta_{12}^{3} q^{22} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{23} + ( 8 - 8 \zeta_{12}^{2} ) q^{24} + 132 \zeta_{12}^{2} q^{26} -53 \zeta_{12}^{3} q^{27} + ( 4 \zeta_{12} + 72 \zeta_{12}^{3} ) q^{28} -106 q^{29} + ( -75 + 75 \zeta_{12}^{2} ) q^{31} -32 \zeta_{12} q^{32} + ( -35 \zeta_{12} + 35 \zeta_{12}^{3} ) q^{33} + 118 q^{34} -104 q^{36} + ( -11 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{37} + 274 \zeta_{12} q^{38} + ( -66 + 66 \zeta_{12}^{2} ) q^{39} -498 q^{41} + ( -36 \zeta_{12} + 38 \zeta_{12}^{3} ) q^{42} + 260 \zeta_{12}^{3} q^{43} + 140 \zeta_{12}^{2} q^{44} + ( -14 + 14 \zeta_{12}^{2} ) q^{46} + ( 171 \zeta_{12} - 171 \zeta_{12}^{3} ) q^{47} -16 \zeta_{12}^{3} q^{48} + ( 323 - 360 \zeta_{12}^{2} ) q^{49} + 59 \zeta_{12}^{2} q^{51} + 264 \zeta_{12} q^{52} + 417 \zeta_{12} q^{53} -106 \zeta_{12}^{2} q^{54} + ( 8 + 144 \zeta_{12}^{2} ) q^{56} + 137 \zeta_{12}^{3} q^{57} + ( -212 \zeta_{12} + 212 \zeta_{12}^{3} ) q^{58} + ( -17 + 17 \zeta_{12}^{2} ) q^{59} -51 \zeta_{12}^{2} q^{61} + 150 \zeta_{12}^{3} q^{62} + ( 494 \zeta_{12} - 26 \zeta_{12}^{3} ) q^{63} -64 q^{64} + ( -70 + 70 \zeta_{12}^{2} ) q^{66} + 439 \zeta_{12} q^{67} + ( 236 \zeta_{12} - 236 \zeta_{12}^{3} ) q^{68} -7 q^{69} -784 q^{71} + ( -208 \zeta_{12} + 208 \zeta_{12}^{3} ) q^{72} -295 \zeta_{12} q^{73} + ( -22 + 22 \zeta_{12}^{2} ) q^{74} + 548 q^{76} + ( -35 \zeta_{12} - 630 \zeta_{12}^{3} ) q^{77} + 132 \zeta_{12}^{3} q^{78} -495 \zeta_{12}^{2} q^{79} + ( -649 + 649 \zeta_{12}^{2} ) q^{81} + ( -996 \zeta_{12} + 996 \zeta_{12}^{3} ) q^{82} + 932 \zeta_{12}^{3} q^{83} + ( -72 + 76 \zeta_{12}^{2} ) q^{84} + 520 \zeta_{12}^{2} q^{86} -106 \zeta_{12} q^{87} + 280 \zeta_{12} q^{88} -873 \zeta_{12}^{2} q^{89} + ( -66 - 1188 \zeta_{12}^{2} ) q^{91} + 28 \zeta_{12}^{3} q^{92} + ( -75 \zeta_{12} + 75 \zeta_{12}^{3} ) q^{93} + ( 342 - 342 \zeta_{12}^{2} ) q^{94} -32 \zeta_{12}^{2} q^{96} + 290 \zeta_{12}^{3} q^{97} + ( -74 \zeta_{12} - 646 \zeta_{12}^{3} ) q^{98} + 910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + 8q^{6} - 52q^{9} + O(q^{10}) \) \( 4q + 8q^{4} + 8q^{6} - 52q^{9} - 70q^{11} - 68q^{14} - 32q^{16} + 274q^{19} - 74q^{21} + 16q^{24} + 264q^{26} - 424q^{29} - 150q^{31} + 472q^{34} - 416q^{36} - 132q^{39} - 1992q^{41} + 280q^{44} - 28q^{46} + 572q^{49} + 118q^{51} - 212q^{54} + 320q^{56} - 34q^{59} - 102q^{61} - 256q^{64} - 140q^{66} - 28q^{69} - 3136q^{71} - 44q^{74} + 2192q^{76} - 990q^{79} - 1298q^{81} - 136q^{84} + 1040q^{86} - 1746q^{89} - 2640q^{91} + 684q^{94} - 64q^{96} + 3640q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−1.73205 1.00000i −0.866025 + 0.500000i 2.00000 + 3.46410i 0 2.00000 15.5885 + 10.0000i 8.00000i −13.0000 + 22.5167i 0
149.2 1.73205 + 1.00000i 0.866025 0.500000i 2.00000 + 3.46410i 0 2.00000 −15.5885 10.0000i 8.00000i −13.0000 + 22.5167i 0
249.1 −1.73205 + 1.00000i −0.866025 0.500000i 2.00000 3.46410i 0 2.00000 15.5885 10.0000i 8.00000i −13.0000 22.5167i 0
249.2 1.73205 1.00000i 0.866025 + 0.500000i 2.00000 3.46410i 0 2.00000 −15.5885 + 10.0000i 8.00000i −13.0000 22.5167i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.4.j.d 4
5.b even 2 1 inner 350.4.j.d 4
5.c odd 4 1 14.4.c.b 2
5.c odd 4 1 350.4.e.b 2
7.c even 3 1 inner 350.4.j.d 4
15.e even 4 1 126.4.g.c 2
20.e even 4 1 112.4.i.b 2
35.f even 4 1 98.4.c.e 2
35.j even 6 1 inner 350.4.j.d 4
35.k even 12 1 98.4.a.c 1
35.k even 12 1 98.4.c.e 2
35.k even 12 1 2450.4.a.bf 1
35.l odd 12 1 14.4.c.b 2
35.l odd 12 1 98.4.a.b 1
35.l odd 12 1 350.4.e.b 2
35.l odd 12 1 2450.4.a.bh 1
40.i odd 4 1 448.4.i.c 2
40.k even 4 1 448.4.i.d 2
105.k odd 4 1 882.4.g.d 2
105.w odd 12 1 882.4.a.p 1
105.w odd 12 1 882.4.g.d 2
105.x even 12 1 126.4.g.c 2
105.x even 12 1 882.4.a.k 1
140.w even 12 1 112.4.i.b 2
140.w even 12 1 784.4.a.l 1
140.x odd 12 1 784.4.a.j 1
280.br even 12 1 448.4.i.d 2
280.bt odd 12 1 448.4.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 5.c odd 4 1
14.4.c.b 2 35.l odd 12 1
98.4.a.b 1 35.l odd 12 1
98.4.a.c 1 35.k even 12 1
98.4.c.e 2 35.f even 4 1
98.4.c.e 2 35.k even 12 1
112.4.i.b 2 20.e even 4 1
112.4.i.b 2 140.w even 12 1
126.4.g.c 2 15.e even 4 1
126.4.g.c 2 105.x even 12 1
350.4.e.b 2 5.c odd 4 1
350.4.e.b 2 35.l odd 12 1
350.4.j.d 4 1.a even 1 1 trivial
350.4.j.d 4 5.b even 2 1 inner
350.4.j.d 4 7.c even 3 1 inner
350.4.j.d 4 35.j even 6 1 inner
448.4.i.c 2 40.i odd 4 1
448.4.i.c 2 280.bt odd 12 1
448.4.i.d 2 40.k even 4 1
448.4.i.d 2 280.br even 12 1
784.4.a.j 1 140.x odd 12 1
784.4.a.l 1 140.w even 12 1
882.4.a.k 1 105.x even 12 1
882.4.a.p 1 105.w odd 12 1
882.4.g.d 2 105.k odd 4 1
882.4.g.d 2 105.w odd 12 1
2450.4.a.bf 1 35.k even 12 1
2450.4.a.bh 1 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{4} - T_{3}^{2} + 1 \)
\( T_{11}^{2} + 35 T_{11} + 1225 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T^{2} + 16 T^{4} \)
$3$ \( 1 + 53 T^{2} + 2080 T^{4} + 38637 T^{6} + 531441 T^{8} \)
$5$ 1
$7$ \( 1 - 286 T^{2} + 117649 T^{4} \)
$11$ \( ( 1 + 35 T - 106 T^{2} + 46585 T^{3} + 1771561 T^{4} )^{2} \)
$13$ \( ( 1 - 38 T^{2} + 4826809 T^{4} )^{2} \)
$17$ \( 1 + 6345 T^{2} + 16121456 T^{4} + 153152875305 T^{6} + 582622237229761 T^{8} \)
$19$ \( ( 1 - 137 T + 11910 T^{2} - 939683 T^{3} + 47045881 T^{4} )^{2} \)
$23$ \( 1 + 24285 T^{2} + 441725336 T^{4} + 3595051564365 T^{6} + 21914624432020321 T^{8} \)
$29$ \( ( 1 + 106 T + 24389 T^{2} )^{4} \)
$31$ \( ( 1 + 75 T - 24166 T^{2} + 2234325 T^{3} + 887503681 T^{4} )^{2} \)
$37$ \( 1 + 101185 T^{2} + 7672677816 T^{4} + 259613026694665 T^{6} + 6582952005840035281 T^{8} \)
$41$ \( ( 1 + 498 T + 68921 T^{2} )^{4} \)
$43$ \( ( 1 - 91414 T^{2} + 6321363049 T^{4} )^{2} \)
$47$ \( 1 + 178405 T^{2} + 21049128696 T^{4} + 1923065910770245 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 + 123865 T^{2} - 6821822904 T^{4} + 2745388591243585 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \)
$59$ \( ( 1 + 17 T - 205090 T^{2} + 3491443 T^{3} + 42180533641 T^{4} )^{2} \)
$61$ \( ( 1 + 51 T - 224380 T^{2} + 11576031 T^{3} + 51520374361 T^{4} )^{2} \)
$67$ \( 1 + 408805 T^{2} + 76663145856 T^{4} + 36979838922598045 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \)
$71$ \( ( 1 + 784 T + 357911 T^{2} )^{4} \)
$73$ \( 1 + 691009 T^{2} + 326159211792 T^{4} + 104573312373735601 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \)
$79$ \( ( 1 + 495 T - 248014 T^{2} + 244054305 T^{3} + 243087455521 T^{4} )^{2} \)
$83$ \( ( 1 - 274950 T^{2} + 326940373369 T^{4} )^{2} \)
$89$ \( ( 1 + 873 T + 57160 T^{2} + 615437937 T^{3} + 496981290961 T^{4} )^{2} \)
$97$ \( ( 1 - 1741246 T^{2} + 832972004929 T^{4} )^{2} \)
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