Properties

Label 350.4.e.b
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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + 2 q^{6} + ( 1 + 18 \zeta_{6} ) q^{7} + 8 q^{8} + 26 \zeta_{6} q^{9} +O(q^{10})\) \( q -2 \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + 2 q^{6} + ( 1 + 18 \zeta_{6} ) q^{7} + 8 q^{8} + 26 \zeta_{6} q^{9} + ( -35 + 35 \zeta_{6} ) q^{11} -4 \zeta_{6} q^{12} -66 q^{13} + ( 36 - 38 \zeta_{6} ) q^{14} -16 \zeta_{6} q^{16} + ( 59 - 59 \zeta_{6} ) q^{17} + ( 52 - 52 \zeta_{6} ) q^{18} -137 \zeta_{6} q^{19} + ( -19 + \zeta_{6} ) q^{21} + 70 q^{22} -7 \zeta_{6} q^{23} + ( -8 + 8 \zeta_{6} ) q^{24} + 132 \zeta_{6} q^{26} -53 q^{27} + ( -76 + 4 \zeta_{6} ) q^{28} + 106 q^{29} + ( -75 + 75 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} -35 \zeta_{6} q^{33} -118 q^{34} -104 q^{36} + 11 \zeta_{6} q^{37} + ( -274 + 274 \zeta_{6} ) q^{38} + ( 66 - 66 \zeta_{6} ) q^{39} -498 q^{41} + ( 2 + 36 \zeta_{6} ) q^{42} -260 q^{43} -140 \zeta_{6} q^{44} + ( -14 + 14 \zeta_{6} ) q^{46} -171 \zeta_{6} q^{47} + 16 q^{48} + ( -323 + 360 \zeta_{6} ) q^{49} + 59 \zeta_{6} q^{51} + ( 264 - 264 \zeta_{6} ) q^{52} + ( -417 + 417 \zeta_{6} ) q^{53} + 106 \zeta_{6} q^{54} + ( 8 + 144 \zeta_{6} ) q^{56} + 137 q^{57} -212 \zeta_{6} q^{58} + ( 17 - 17 \zeta_{6} ) q^{59} -51 \zeta_{6} q^{61} + 150 q^{62} + ( -468 + 494 \zeta_{6} ) q^{63} + 64 q^{64} + ( -70 + 70 \zeta_{6} ) q^{66} + ( 439 - 439 \zeta_{6} ) q^{67} + 236 \zeta_{6} q^{68} + 7 q^{69} -784 q^{71} + 208 \zeta_{6} q^{72} + ( 295 - 295 \zeta_{6} ) q^{73} + ( 22 - 22 \zeta_{6} ) q^{74} + 548 q^{76} + ( -665 + 35 \zeta_{6} ) q^{77} -132 q^{78} + 495 \zeta_{6} q^{79} + ( -649 + 649 \zeta_{6} ) q^{81} + 996 \zeta_{6} q^{82} -932 q^{83} + ( 72 - 76 \zeta_{6} ) q^{84} + 520 \zeta_{6} q^{86} + ( -106 + 106 \zeta_{6} ) q^{87} + ( -280 + 280 \zeta_{6} ) q^{88} + 873 \zeta_{6} q^{89} + ( -66 - 1188 \zeta_{6} ) q^{91} + 28 q^{92} -75 \zeta_{6} q^{93} + ( -342 + 342 \zeta_{6} ) q^{94} -32 \zeta_{6} q^{96} + 290 q^{97} + ( 720 - 74 \zeta_{6} ) q^{98} -910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - q^{3} - 4q^{4} + 4q^{6} + 20q^{7} + 16q^{8} + 26q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - q^{3} - 4q^{4} + 4q^{6} + 20q^{7} + 16q^{8} + 26q^{9} - 35q^{11} - 4q^{12} - 132q^{13} + 34q^{14} - 16q^{16} + 59q^{17} + 52q^{18} - 137q^{19} - 37q^{21} + 140q^{22} - 7q^{23} - 8q^{24} + 132q^{26} - 106q^{27} - 148q^{28} + 212q^{29} - 75q^{31} - 32q^{32} - 35q^{33} - 236q^{34} - 208q^{36} + 11q^{37} - 274q^{38} + 66q^{39} - 996q^{41} + 40q^{42} - 520q^{43} - 140q^{44} - 14q^{46} - 171q^{47} + 32q^{48} - 286q^{49} + 59q^{51} + 264q^{52} - 417q^{53} + 106q^{54} + 160q^{56} + 274q^{57} - 212q^{58} + 17q^{59} - 51q^{61} + 300q^{62} - 442q^{63} + 128q^{64} - 70q^{66} + 439q^{67} + 236q^{68} + 14q^{69} - 1568q^{71} + 208q^{72} + 295q^{73} + 22q^{74} + 1096q^{76} - 1295q^{77} - 264q^{78} + 495q^{79} - 649q^{81} + 996q^{82} - 1864q^{83} + 68q^{84} + 520q^{86} - 106q^{87} - 280q^{88} + 873q^{89} - 1320q^{91} + 56q^{92} - 75q^{93} - 342q^{94} - 32q^{96} + 580q^{97} + 1366q^{98} - 1820q^{99} + O(q^{100}) \)

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 −0.500000 0.866025i −2.00000 3.46410i 0 2.00000 10.0000 15.5885i 8.00000 13.0000 22.5167i 0
151.1 −1.00000 1.73205i −0.500000 + 0.866025i −2.00000 + 3.46410i 0 2.00000 10.0000 + 15.5885i 8.00000 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
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.b 2
5.b even 2 1 14.4.c.b 2
5.c odd 4 2 350.4.j.d 4
7.c even 3 1 inner 350.4.e.b 2
7.c even 3 1 2450.4.a.bh 1
7.d odd 6 1 2450.4.a.bf 1
15.d odd 2 1 126.4.g.c 2
20.d odd 2 1 112.4.i.b 2
35.c odd 2 1 98.4.c.e 2
35.i odd 6 1 98.4.a.c 1
35.i odd 6 1 98.4.c.e 2
35.j even 6 1 14.4.c.b 2
35.j even 6 1 98.4.a.b 1
35.l odd 12 2 350.4.j.d 4
40.e odd 2 1 448.4.i.d 2
40.f even 2 1 448.4.i.c 2
105.g even 2 1 882.4.g.d 2
105.o odd 6 1 126.4.g.c 2
105.o odd 6 1 882.4.a.k 1
105.p even 6 1 882.4.a.p 1
105.p even 6 1 882.4.g.d 2
140.p odd 6 1 112.4.i.b 2
140.p odd 6 1 784.4.a.l 1
140.s even 6 1 784.4.a.j 1
280.bf even 6 1 448.4.i.c 2
280.bi odd 6 1 448.4.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 5.b even 2 1
14.4.c.b 2 35.j even 6 1
98.4.a.b 1 35.j even 6 1
98.4.a.c 1 35.i odd 6 1
98.4.c.e 2 35.c odd 2 1
98.4.c.e 2 35.i odd 6 1
112.4.i.b 2 20.d odd 2 1
112.4.i.b 2 140.p odd 6 1
126.4.g.c 2 15.d odd 2 1
126.4.g.c 2 105.o odd 6 1
350.4.e.b 2 1.a even 1 1 trivial
350.4.e.b 2 7.c even 3 1 inner
350.4.j.d 4 5.c odd 4 2
350.4.j.d 4 35.l odd 12 2
448.4.i.c 2 40.f even 2 1
448.4.i.c 2 280.bf even 6 1
448.4.i.d 2 40.e odd 2 1
448.4.i.d 2 280.bi odd 6 1
784.4.a.j 1 140.s even 6 1
784.4.a.l 1 140.p odd 6 1
882.4.a.k 1 105.o odd 6 1
882.4.a.p 1 105.p even 6 1
882.4.g.d 2 105.g even 2 1
882.4.g.d 2 105.p even 6 1
2450.4.a.bf 1 7.d odd 6 1
2450.4.a.bh 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 \(S_{4}^{\mathrm{new}}(350, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 4 T^{2} \)
$3$ \( 1 + T - 26 T^{2} + 27 T^{3} + 729 T^{4} \)
$5$ 1
$7$ \( 1 - 20 T + 343 T^{2} \)
$11$ \( 1 + 35 T - 106 T^{2} + 46585 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 + 66 T + 2197 T^{2} )^{2} \)
$17$ \( 1 - 59 T - 1432 T^{2} - 289867 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 137 T + 11910 T^{2} + 939683 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 7 T - 12118 T^{2} + 85169 T^{3} + 148035889 T^{4} \)
$29$ \( ( 1 - 106 T + 24389 T^{2} )^{2} \)
$31$ \( 1 + 75 T - 24166 T^{2} + 2234325 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 11 T - 50532 T^{2} - 557183 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 + 498 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 + 260 T + 79507 T^{2} )^{2} \)
$47$ \( 1 + 171 T - 74582 T^{2} + 17753733 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 417 T + 25012 T^{2} + 62081709 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 17 T - 205090 T^{2} - 3491443 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 51 T - 224380 T^{2} + 11576031 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 439 T - 108042 T^{2} - 132034957 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 + 784 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 295 T - 301992 T^{2} - 114760015 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 495 T - 248014 T^{2} - 244054305 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 + 932 T + 571787 T^{2} )^{2} \)
$89$ \( 1 - 873 T + 57160 T^{2} - 615437937 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 - 290 T + 912673 T^{2} )^{2} \)
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