Properties

Label 700.4.e.a
Level $700$
Weight $4$
Character orbit 700.e
Analytic conductor $41.301$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 700.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(41.3013370040\)
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 28)
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.

\(f(q)\) \(=\) \( q + 10 i q^{3} -7 i q^{7} -73 q^{9} +O(q^{10})\) \( q + 10 i q^{3} -7 i q^{7} -73 q^{9} -40 q^{11} + 12 i q^{13} -58 i q^{17} -26 q^{19} + 70 q^{21} + 64 i q^{23} -460 i q^{27} + 62 q^{29} + 252 q^{31} -400 i q^{33} + 26 i q^{37} -120 q^{39} + 6 q^{41} -416 i q^{43} -396 i q^{47} -49 q^{49} + 580 q^{51} + 450 i q^{53} -260 i q^{57} -274 q^{59} -576 q^{61} + 511 i q^{63} -476 i q^{67} -640 q^{69} -448 q^{71} + 158 i q^{73} + 280 i q^{77} + 936 q^{79} + 2629 q^{81} -530 i q^{83} + 620 i q^{87} + 390 q^{89} + 84 q^{91} + 2520 i q^{93} + 214 i q^{97} + 2920 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 146q^{9} + O(q^{10}) \) \( 2q - 146q^{9} - 80q^{11} - 52q^{19} + 140q^{21} + 124q^{29} + 504q^{31} - 240q^{39} + 12q^{41} - 98q^{49} + 1160q^{51} - 548q^{59} - 1152q^{61} - 1280q^{69} - 896q^{71} + 1872q^{79} + 5258q^{81} + 780q^{89} + 168q^{91} + 5840q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 10.0000i 0 0 0 7.00000i 0 −73.0000 0
449.2 0 10.0000i 0 0 0 7.00000i 0 −73.0000 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 700.4.e.a 2
5.b even 2 1 inner 700.4.e.a 2
5.c odd 4 1 28.4.a.a 1
5.c odd 4 1 700.4.a.n 1
15.e even 4 1 252.4.a.d 1
20.e even 4 1 112.4.a.g 1
35.f even 4 1 196.4.a.d 1
35.k even 12 2 196.4.e.a 2
35.l odd 12 2 196.4.e.f 2
40.i odd 4 1 448.4.a.p 1
40.k even 4 1 448.4.a.a 1
60.l odd 4 1 1008.4.a.o 1
105.k odd 4 1 1764.4.a.c 1
105.w odd 12 2 1764.4.k.m 2
105.x even 12 2 1764.4.k.d 2
140.j odd 4 1 784.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 5.c odd 4 1
112.4.a.g 1 20.e even 4 1
196.4.a.d 1 35.f even 4 1
196.4.e.a 2 35.k even 12 2
196.4.e.f 2 35.l odd 12 2
252.4.a.d 1 15.e even 4 1
448.4.a.a 1 40.k even 4 1
448.4.a.p 1 40.i odd 4 1
700.4.a.n 1 5.c odd 4 1
700.4.e.a 2 1.a even 1 1 trivial
700.4.e.a 2 5.b even 2 1 inner
784.4.a.a 1 140.j odd 4 1
1008.4.a.o 1 60.l odd 4 1
1764.4.a.c 1 105.k odd 4 1
1764.4.k.d 2 105.x even 12 2
1764.4.k.m 2 105.w odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{2} + 100 \)
\( T_{11} + 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 100 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 49 + T^{2} \)
$11$ \( ( 40 + T )^{2} \)
$13$ \( 144 + T^{2} \)
$17$ \( 3364 + T^{2} \)
$19$ \( ( 26 + T )^{2} \)
$23$ \( 4096 + T^{2} \)
$29$ \( ( -62 + T )^{2} \)
$31$ \( ( -252 + T )^{2} \)
$37$ \( 676 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 173056 + T^{2} \)
$47$ \( 156816 + T^{2} \)
$53$ \( 202500 + T^{2} \)
$59$ \( ( 274 + T )^{2} \)
$61$ \( ( 576 + T )^{2} \)
$67$ \( 226576 + T^{2} \)
$71$ \( ( 448 + T )^{2} \)
$73$ \( 24964 + T^{2} \)
$79$ \( ( -936 + T )^{2} \)
$83$ \( 280900 + T^{2} \)
$89$ \( ( -390 + T )^{2} \)
$97$ \( 45796 + T^{2} \)
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