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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{4}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 35T_{11} + 1225 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 20T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 35T + 1225 \) Copy content Toggle raw display
$13$ \( (T + 66)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 59T + 3481 \) Copy content Toggle raw display
$19$ \( T^{2} + 137T + 18769 \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$29$ \( (T - 106)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 75T + 5625 \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$41$ \( (T + 498)^{2} \) Copy content Toggle raw display
$43$ \( (T + 260)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 171T + 29241 \) Copy content Toggle raw display
$53$ \( T^{2} + 417T + 173889 \) Copy content Toggle raw display
$59$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$61$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$67$ \( T^{2} - 439T + 192721 \) Copy content Toggle raw display
$71$ \( (T + 784)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 295T + 87025 \) Copy content Toggle raw display
$79$ \( T^{2} - 495T + 245025 \) Copy content Toggle raw display
$83$ \( (T + 932)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 873T + 762129 \) Copy content Toggle raw display
$97$ \( (T - 290)^{2} \) Copy content Toggle raw display
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