Properties

Label 350.4.e.e
Level 350
Weight 4
Character orbit 350.e
Analytic conductor 20.651
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 350.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + ( -5 + 5 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -10 q^{6} + ( 21 - 14 \zeta_{6} ) q^{7} -8 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + 2 \zeta_{6} q^{2} + ( -5 + 5 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -10 q^{6} + ( 21 - 14 \zeta_{6} ) q^{7} -8 q^{8} + 2 \zeta_{6} q^{9} + ( 57 - 57 \zeta_{6} ) q^{11} -20 \zeta_{6} q^{12} + 70 q^{13} + ( 28 + 14 \zeta_{6} ) q^{14} -16 \zeta_{6} q^{16} + ( 51 - 51 \zeta_{6} ) q^{17} + ( -4 + 4 \zeta_{6} ) q^{18} -5 \zeta_{6} q^{19} + ( -35 + 105 \zeta_{6} ) q^{21} + 114 q^{22} + 69 \zeta_{6} q^{23} + ( 40 - 40 \zeta_{6} ) q^{24} + 140 \zeta_{6} q^{26} -145 q^{27} + ( -28 + 84 \zeta_{6} ) q^{28} + 114 q^{29} + ( -23 + 23 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} + 285 \zeta_{6} q^{33} + 102 q^{34} -8 q^{36} -253 \zeta_{6} q^{37} + ( 10 - 10 \zeta_{6} ) q^{38} + ( -350 + 350 \zeta_{6} ) q^{39} -42 q^{41} + ( -210 + 140 \zeta_{6} ) q^{42} + 124 q^{43} + 228 \zeta_{6} q^{44} + ( -138 + 138 \zeta_{6} ) q^{46} + 201 \zeta_{6} q^{47} + 80 q^{48} + ( 245 - 392 \zeta_{6} ) q^{49} + 255 \zeta_{6} q^{51} + ( -280 + 280 \zeta_{6} ) q^{52} + ( -393 + 393 \zeta_{6} ) q^{53} -290 \zeta_{6} q^{54} + ( -168 + 112 \zeta_{6} ) q^{56} + 25 q^{57} + 228 \zeta_{6} q^{58} + ( -219 + 219 \zeta_{6} ) q^{59} + 709 \zeta_{6} q^{61} -46 q^{62} + ( 28 + 14 \zeta_{6} ) q^{63} + 64 q^{64} + ( -570 + 570 \zeta_{6} ) q^{66} + ( 419 - 419 \zeta_{6} ) q^{67} + 204 \zeta_{6} q^{68} -345 q^{69} -96 q^{71} -16 \zeta_{6} q^{72} + ( -313 + 313 \zeta_{6} ) q^{73} + ( 506 - 506 \zeta_{6} ) q^{74} + 20 q^{76} + ( 399 - 1197 \zeta_{6} ) q^{77} -700 q^{78} -461 \zeta_{6} q^{79} + ( 671 - 671 \zeta_{6} ) q^{81} -84 \zeta_{6} q^{82} + 588 q^{83} + ( -280 - 140 \zeta_{6} ) q^{84} + 248 \zeta_{6} q^{86} + ( -570 + 570 \zeta_{6} ) q^{87} + ( -456 + 456 \zeta_{6} ) q^{88} + 1017 \zeta_{6} q^{89} + ( 1470 - 980 \zeta_{6} ) q^{91} -276 q^{92} -115 \zeta_{6} q^{93} + ( -402 + 402 \zeta_{6} ) q^{94} + 160 \zeta_{6} q^{96} + 1834 q^{97} + ( 784 - 294 \zeta_{6} ) q^{98} + 114 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 5q^{3} - 4q^{4} - 20q^{6} + 28q^{7} - 16q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 5q^{3} - 4q^{4} - 20q^{6} + 28q^{7} - 16q^{8} + 2q^{9} + 57q^{11} - 20q^{12} + 140q^{13} + 70q^{14} - 16q^{16} + 51q^{17} - 4q^{18} - 5q^{19} + 35q^{21} + 228q^{22} + 69q^{23} + 40q^{24} + 140q^{26} - 290q^{27} + 28q^{28} + 228q^{29} - 23q^{31} + 32q^{32} + 285q^{33} + 204q^{34} - 16q^{36} - 253q^{37} + 10q^{38} - 350q^{39} - 84q^{41} - 280q^{42} + 248q^{43} + 228q^{44} - 138q^{46} + 201q^{47} + 160q^{48} + 98q^{49} + 255q^{51} - 280q^{52} - 393q^{53} - 290q^{54} - 224q^{56} + 50q^{57} + 228q^{58} - 219q^{59} + 709q^{61} - 92q^{62} + 70q^{63} + 128q^{64} - 570q^{66} + 419q^{67} + 204q^{68} - 690q^{69} - 192q^{71} - 16q^{72} - 313q^{73} + 506q^{74} + 40q^{76} - 399q^{77} - 1400q^{78} - 461q^{79} + 671q^{81} - 84q^{82} + 1176q^{83} - 700q^{84} + 248q^{86} - 570q^{87} - 456q^{88} + 1017q^{89} + 1960q^{91} - 552q^{92} - 115q^{93} - 402q^{94} + 160q^{96} + 3668q^{97} + 1274q^{98} + 228q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 1.73205i −2.50000 4.33013i −2.00000 3.46410i 0 −10.0000 14.0000 + 12.1244i −8.00000 1.00000 1.73205i 0
151.1 1.00000 + 1.73205i −2.50000 + 4.33013i −2.00000 + 3.46410i 0 −10.0000 14.0000 12.1244i −8.00000 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
7.c even 3 1 inner

Twists

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

Hecke kernels

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

\( T_{3}^{2} + 5 T_{3} + 25 \)
\( T_{11}^{2} - 57 T_{11} + 3249 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} \)
$3$ \( 1 + 5 T - 2 T^{2} + 135 T^{3} + 729 T^{4} \)
$5$ \( \)
$7$ \( 1 - 28 T + 343 T^{2} \)
$11$ \( 1 - 57 T + 1918 T^{2} - 75867 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 - 70 T + 2197 T^{2} )^{2} \)
$17$ \( 1 - 51 T - 2312 T^{2} - 250563 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 5 T - 6834 T^{2} + 34295 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 69 T - 7406 T^{2} - 839523 T^{3} + 148035889 T^{4} \)
$29$ \( ( 1 - 114 T + 24389 T^{2} )^{2} \)
$31$ \( 1 + 23 T - 29262 T^{2} + 685193 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 253 T + 13356 T^{2} + 12815209 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 + 42 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 - 124 T + 79507 T^{2} )^{2} \)
$47$ \( 1 - 201 T - 63422 T^{2} - 20868423 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 393 T + 5572 T^{2} + 58508661 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 219 T - 157418 T^{2} + 44978001 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 709 T + 275700 T^{2} - 160929529 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 419 T - 125202 T^{2} - 126019697 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 + 96 T + 357911 T^{2} )^{2} \)
$73$ \( 1 + 313 T - 291048 T^{2} + 121762321 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 461 T - 280518 T^{2} + 227290979 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 - 588 T + 571787 T^{2} )^{2} \)
$89$ \( 1 - 1017 T + 329320 T^{2} - 716953473 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 - 1834 T + 912673 T^{2} )^{2} \)
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