Properties

Label 350.4.j.b
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: \(\Q(\zeta_{12})\)
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.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{2} + 5 \zeta_{12} q^{3} + ( 4 - 4 \zeta_{12}^{2} ) q^{4} -10 q^{6} + ( 14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{7} + 8 \zeta_{12}^{3} q^{8} -2 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{2} + 5 \zeta_{12} q^{3} + ( 4 - 4 \zeta_{12}^{2} ) q^{4} -10 q^{6} + ( 14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{7} + 8 \zeta_{12}^{3} q^{8} -2 \zeta_{12}^{2} q^{9} + ( 57 - 57 \zeta_{12}^{2} ) q^{11} + ( 20 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{12} -70 \zeta_{12}^{3} q^{13} + ( -28 - 14 \zeta_{12}^{2} ) q^{14} -16 \zeta_{12}^{2} q^{16} + 51 \zeta_{12} q^{17} + 4 \zeta_{12} q^{18} + 5 \zeta_{12}^{2} q^{19} + ( -35 + 105 \zeta_{12}^{2} ) q^{21} + 114 \zeta_{12}^{3} q^{22} + ( 69 \zeta_{12} - 69 \zeta_{12}^{3} ) q^{23} + ( -40 + 40 \zeta_{12}^{2} ) q^{24} + 140 \zeta_{12}^{2} q^{26} -145 \zeta_{12}^{3} q^{27} + ( 84 \zeta_{12} - 56 \zeta_{12}^{3} ) q^{28} -114 q^{29} + ( -23 + 23 \zeta_{12}^{2} ) q^{31} + 32 \zeta_{12} q^{32} + ( 285 \zeta_{12} - 285 \zeta_{12}^{3} ) q^{33} -102 q^{34} -8 q^{36} + ( 253 \zeta_{12} - 253 \zeta_{12}^{3} ) q^{37} -10 \zeta_{12} q^{38} + ( 350 - 350 \zeta_{12}^{2} ) q^{39} -42 q^{41} + ( -140 \zeta_{12} - 70 \zeta_{12}^{3} ) q^{42} -124 \zeta_{12}^{3} q^{43} -228 \zeta_{12}^{2} q^{44} + ( -138 + 138 \zeta_{12}^{2} ) q^{46} + ( -201 \zeta_{12} + 201 \zeta_{12}^{3} ) q^{47} -80 \zeta_{12}^{3} q^{48} + ( -245 + 392 \zeta_{12}^{2} ) q^{49} + 255 \zeta_{12}^{2} q^{51} -280 \zeta_{12} q^{52} + 393 \zeta_{12} q^{53} + 290 \zeta_{12}^{2} q^{54} + ( -168 + 112 \zeta_{12}^{2} ) q^{56} + 25 \zeta_{12}^{3} q^{57} + ( 228 \zeta_{12} - 228 \zeta_{12}^{3} ) q^{58} + ( 219 - 219 \zeta_{12}^{2} ) q^{59} + 709 \zeta_{12}^{2} q^{61} -46 \zeta_{12}^{3} q^{62} + ( 14 \zeta_{12} - 42 \zeta_{12}^{3} ) q^{63} -64 q^{64} + ( -570 + 570 \zeta_{12}^{2} ) q^{66} + 419 \zeta_{12} q^{67} + ( 204 \zeta_{12} - 204 \zeta_{12}^{3} ) q^{68} + 345 q^{69} -96 q^{71} + ( 16 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{72} + 313 \zeta_{12} q^{73} + ( -506 + 506 \zeta_{12}^{2} ) q^{74} + 20 q^{76} + ( 1197 \zeta_{12} - 798 \zeta_{12}^{3} ) q^{77} + 700 \zeta_{12}^{3} q^{78} + 461 \zeta_{12}^{2} q^{79} + ( 671 - 671 \zeta_{12}^{2} ) q^{81} + ( 84 \zeta_{12} - 84 \zeta_{12}^{3} ) q^{82} -588 \zeta_{12}^{3} q^{83} + ( 280 + 140 \zeta_{12}^{2} ) q^{84} + 248 \zeta_{12}^{2} q^{86} -570 \zeta_{12} q^{87} + 456 \zeta_{12} q^{88} -1017 \zeta_{12}^{2} q^{89} + ( 1470 - 980 \zeta_{12}^{2} ) q^{91} -276 \zeta_{12}^{3} q^{92} + ( -115 \zeta_{12} + 115 \zeta_{12}^{3} ) q^{93} + ( 402 - 402 \zeta_{12}^{2} ) q^{94} + 160 \zeta_{12}^{2} q^{96} + 1834 \zeta_{12}^{3} q^{97} + ( -294 \zeta_{12} - 490 \zeta_{12}^{3} ) q^{98} -114 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} - 40q^{6} - 4q^{9} + O(q^{10}) \) \( 4q + 8q^{4} - 40q^{6} - 4q^{9} + 114q^{11} - 140q^{14} - 32q^{16} + 10q^{19} + 70q^{21} - 80q^{24} + 280q^{26} - 456q^{29} - 46q^{31} - 408q^{34} - 32q^{36} + 700q^{39} - 168q^{41} - 456q^{44} - 276q^{46} - 196q^{49} + 510q^{51} + 580q^{54} - 448q^{56} + 438q^{59} + 1418q^{61} - 256q^{64} - 1140q^{66} + 1380q^{69} - 384q^{71} - 1012q^{74} + 80q^{76} + 922q^{79} + 1342q^{81} + 1400q^{84} + 496q^{86} - 2034q^{89} + 3920q^{91} + 804q^{94} + 320q^{96} - 456q^{99} + O(q^{100}) \)

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 4.33013 2.50000i 2.00000 + 3.46410i 0 −10.0000 12.1244 14.0000i 8.00000i −1.00000 + 1.73205i 0
149.2 1.73205 + 1.00000i −4.33013 + 2.50000i 2.00000 + 3.46410i 0 −10.0000 −12.1244 + 14.0000i 8.00000i −1.00000 + 1.73205i 0
249.1 −1.73205 + 1.00000i 4.33013 + 2.50000i 2.00000 3.46410i 0 −10.0000 12.1244 + 14.0000i 8.00000i −1.00000 1.73205i 0
249.2 1.73205 1.00000i −4.33013 2.50000i 2.00000 3.46410i 0 −10.0000 −12.1244 14.0000i 8.00000i −1.00000 1.73205i 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.b 4
5.b even 2 1 inner 350.4.j.b 4
5.c odd 4 1 14.4.c.a 2
5.c odd 4 1 350.4.e.e 2
7.c even 3 1 inner 350.4.j.b 4
15.e even 4 1 126.4.g.d 2
20.e even 4 1 112.4.i.a 2
35.f even 4 1 98.4.c.a 2
35.j even 6 1 inner 350.4.j.b 4
35.k even 12 1 98.4.a.f 1
35.k even 12 1 98.4.c.a 2
35.k even 12 1 2450.4.a.d 1
35.l odd 12 1 14.4.c.a 2
35.l odd 12 1 98.4.a.d 1
35.l odd 12 1 350.4.e.e 2
35.l odd 12 1 2450.4.a.q 1
40.i odd 4 1 448.4.i.b 2
40.k even 4 1 448.4.i.e 2
105.k odd 4 1 882.4.g.u 2
105.w odd 12 1 882.4.a.c 1
105.w odd 12 1 882.4.g.u 2
105.x even 12 1 126.4.g.d 2
105.x even 12 1 882.4.a.f 1
140.w even 12 1 112.4.i.a 2
140.w even 12 1 784.4.a.p 1
140.x odd 12 1 784.4.a.c 1
280.br even 12 1 448.4.i.e 2
280.bt odd 12 1 448.4.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 5.c odd 4 1
14.4.c.a 2 35.l odd 12 1
98.4.a.d 1 35.l odd 12 1
98.4.a.f 1 35.k even 12 1
98.4.c.a 2 35.f even 4 1
98.4.c.a 2 35.k even 12 1
112.4.i.a 2 20.e even 4 1
112.4.i.a 2 140.w even 12 1
126.4.g.d 2 15.e even 4 1
126.4.g.d 2 105.x even 12 1
350.4.e.e 2 5.c odd 4 1
350.4.e.e 2 35.l odd 12 1
350.4.j.b 4 1.a even 1 1 trivial
350.4.j.b 4 5.b even 2 1 inner
350.4.j.b 4 7.c even 3 1 inner
350.4.j.b 4 35.j even 6 1 inner
448.4.i.b 2 40.i odd 4 1
448.4.i.b 2 280.bt odd 12 1
448.4.i.e 2 40.k even 4 1
448.4.i.e 2 280.br even 12 1
784.4.a.c 1 140.x odd 12 1
784.4.a.p 1 140.w even 12 1
882.4.a.c 1 105.w odd 12 1
882.4.a.f 1 105.x even 12 1
882.4.g.u 2 105.k odd 4 1
882.4.g.u 2 105.w odd 12 1
2450.4.a.d 1 35.k even 12 1
2450.4.a.q 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 \(S_{4}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{4} - 25 T_{3}^{2} + 625 \)
\( T_{11}^{2} - 57 T_{11} + 3249 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T^{2} + 16 T^{4} \)
$3$ \( 1 + 29 T^{2} + 112 T^{4} + 21141 T^{6} + 531441 T^{8} \)
$5$ \( \)
$7$ \( 1 + 98 T^{2} + 117649 T^{4} \)
$11$ \( ( 1 - 57 T + 1918 T^{2} - 75867 T^{3} + 1771561 T^{4} )^{2} \)
$13$ \( ( 1 + 506 T^{2} + 4826809 T^{4} )^{2} \)
$17$ \( 1 + 7225 T^{2} + 28063056 T^{4} + 174393936025 T^{6} + 582622237229761 T^{8} \)
$19$ \( ( 1 - 5 T - 6834 T^{2} - 34295 T^{3} + 47045881 T^{4} )^{2} \)
$23$ \( 1 + 19573 T^{2} + 235066440 T^{4} + 2897506455397 T^{6} + 21914624432020321 T^{8} \)
$29$ \( ( 1 + 114 T + 24389 T^{2} )^{4} \)
$31$ \( ( 1 + 23 T - 29262 T^{2} + 685193 T^{3} + 887503681 T^{4} )^{2} \)
$37$ \( 1 + 37297 T^{2} - 1174660200 T^{4} + 95693897876473 T^{6} + 6582952005840035281 T^{8} \)
$41$ \( ( 1 + 42 T + 68921 T^{2} )^{4} \)
$43$ \( ( 1 - 143638 T^{2} + 6321363049 T^{4} )^{2} \)
$47$ \( 1 + 167245 T^{2} + 17191674696 T^{4} + 1802769867698605 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 + 143305 T^{2} - 1628038104 T^{4} + 3176263771591345 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \)
$59$ \( ( 1 - 219 T - 157418 T^{2} - 44978001 T^{3} + 42180533641 T^{4} )^{2} \)
$61$ \( ( 1 - 709 T + 275700 T^{2} - 160929529 T^{3} + 51520374361 T^{4} )^{2} \)
$67$ \( 1 + 425965 T^{2} + 90987799056 T^{4} + 38532104760618085 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \)
$71$ \( ( 1 + 96 T + 357911 T^{2} )^{4} \)
$73$ \( 1 + 680065 T^{2} + 311154177936 T^{4} + 102917110601228785 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \)
$79$ \( ( 1 - 461 T - 280518 T^{2} - 227290979 T^{3} + 243087455521 T^{4} )^{2} \)
$83$ \( ( 1 - 797830 T^{2} + 326940373369 T^{4} )^{2} \)
$89$ \( ( 1 + 1017 T + 329320 T^{2} + 716953473 T^{3} + 496981290961 T^{4} )^{2} \)
$97$ \( ( 1 + 1538210 T^{2} + 832972004929 T^{4} )^{2} \)
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