Properties

Label 350.4.c.g
Level $350$
Weight $4$
Character orbit 350.c
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{2} + 2 i q^{3} -4 q^{4} -4 q^{6} + 7 i q^{7} -8 i q^{8} + 23 q^{9} +O(q^{10})\) \( q + 2 i q^{2} + 2 i q^{3} -4 q^{4} -4 q^{6} + 7 i q^{7} -8 i q^{8} + 23 q^{9} + 48 q^{11} -8 i q^{12} -56 i q^{13} -14 q^{14} + 16 q^{16} -114 i q^{17} + 46 i q^{18} -2 q^{19} -14 q^{21} + 96 i q^{22} + 120 i q^{23} + 16 q^{24} + 112 q^{26} + 100 i q^{27} -28 i q^{28} + 54 q^{29} + 236 q^{31} + 32 i q^{32} + 96 i q^{33} + 228 q^{34} -92 q^{36} + 146 i q^{37} -4 i q^{38} + 112 q^{39} + 126 q^{41} -28 i q^{42} + 376 i q^{43} -192 q^{44} -240 q^{46} -12 i q^{47} + 32 i q^{48} -49 q^{49} + 228 q^{51} + 224 i q^{52} -174 i q^{53} -200 q^{54} + 56 q^{56} -4 i q^{57} + 108 i q^{58} -138 q^{59} + 380 q^{61} + 472 i q^{62} + 161 i q^{63} -64 q^{64} -192 q^{66} -484 i q^{67} + 456 i q^{68} -240 q^{69} + 576 q^{71} -184 i q^{72} + 1150 i q^{73} -292 q^{74} + 8 q^{76} + 336 i q^{77} + 224 i q^{78} -776 q^{79} + 421 q^{81} + 252 i q^{82} -378 i q^{83} + 56 q^{84} -752 q^{86} + 108 i q^{87} -384 i q^{88} + 390 q^{89} + 392 q^{91} -480 i q^{92} + 472 i q^{93} + 24 q^{94} -64 q^{96} -1330 i q^{97} -98 i q^{98} + 1104 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} - 8q^{6} + 46q^{9} + O(q^{10}) \) \( 2q - 8q^{4} - 8q^{6} + 46q^{9} + 96q^{11} - 28q^{14} + 32q^{16} - 4q^{19} - 28q^{21} + 32q^{24} + 224q^{26} + 108q^{29} + 472q^{31} + 456q^{34} - 184q^{36} + 224q^{39} + 252q^{41} - 384q^{44} - 480q^{46} - 98q^{49} + 456q^{51} - 400q^{54} + 112q^{56} - 276q^{59} + 760q^{61} - 128q^{64} - 384q^{66} - 480q^{69} + 1152q^{71} - 584q^{74} + 16q^{76} - 1552q^{79} + 842q^{81} + 112q^{84} - 1504q^{86} + 780q^{89} + 784q^{91} + 48q^{94} - 128q^{96} + 2208q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
2.00000i 2.00000i −4.00000 0 −4.00000 7.00000i 8.00000i 23.0000 0
99.2 2.00000i 2.00000i −4.00000 0 −4.00000 7.00000i 8.00000i 23.0000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.4.c.g 2
5.b even 2 1 inner 350.4.c.g 2
5.c odd 4 1 14.4.a.b 1
5.c odd 4 1 350.4.a.f 1
15.e even 4 1 126.4.a.d 1
20.e even 4 1 112.4.a.e 1
35.f even 4 1 98.4.a.e 1
35.f even 4 1 2450.4.a.i 1
35.k even 12 2 98.4.c.b 2
35.l odd 12 2 98.4.c.c 2
40.i odd 4 1 448.4.a.k 1
40.k even 4 1 448.4.a.g 1
55.e even 4 1 1694.4.a.b 1
60.l odd 4 1 1008.4.a.r 1
65.h odd 4 1 2366.4.a.c 1
105.k odd 4 1 882.4.a.b 1
105.w odd 12 2 882.4.g.v 2
105.x even 12 2 882.4.g.p 2
140.j odd 4 1 784.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 5.c odd 4 1
98.4.a.e 1 35.f even 4 1
98.4.c.b 2 35.k even 12 2
98.4.c.c 2 35.l odd 12 2
112.4.a.e 1 20.e even 4 1
126.4.a.d 1 15.e even 4 1
350.4.a.f 1 5.c odd 4 1
350.4.c.g 2 1.a even 1 1 trivial
350.4.c.g 2 5.b even 2 1 inner
448.4.a.g 1 40.k even 4 1
448.4.a.k 1 40.i odd 4 1
784.4.a.h 1 140.j odd 4 1
882.4.a.b 1 105.k odd 4 1
882.4.g.p 2 105.x even 12 2
882.4.g.v 2 105.w odd 12 2
1008.4.a.r 1 60.l odd 4 1
1694.4.a.b 1 55.e even 4 1
2366.4.a.c 1 65.h odd 4 1
2450.4.a.i 1 35.f even 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \)
\( T_{11} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 49 + T^{2} \)
$11$ \( ( -48 + T )^{2} \)
$13$ \( 3136 + T^{2} \)
$17$ \( 12996 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 14400 + T^{2} \)
$29$ \( ( -54 + T )^{2} \)
$31$ \( ( -236 + T )^{2} \)
$37$ \( 21316 + T^{2} \)
$41$ \( ( -126 + T )^{2} \)
$43$ \( 141376 + T^{2} \)
$47$ \( 144 + T^{2} \)
$53$ \( 30276 + T^{2} \)
$59$ \( ( 138 + T )^{2} \)
$61$ \( ( -380 + T )^{2} \)
$67$ \( 234256 + T^{2} \)
$71$ \( ( -576 + T )^{2} \)
$73$ \( 1322500 + T^{2} \)
$79$ \( ( 776 + T )^{2} \)
$83$ \( 142884 + T^{2} \)
$89$ \( ( -390 + T )^{2} \)
$97$ \( 1768900 + T^{2} \)
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