Properties

Label 700.6.e.g
Level $700$
Weight $6$
Character orbit 700.e
Analytic conductor $112.269$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 700.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(112.268673869\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Defining polynomial: \( x^{6} + 998x^{4} + 249001x^{2} + 44100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + \beta_1) q^{3} + 49 \beta_{2} q^{7} + (\beta_{4} - 5 \beta_{3} - 94) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} + \beta_1) q^{3} + 49 \beta_{2} q^{7} + (\beta_{4} - 5 \beta_{3} - 94) q^{9} + (\beta_{4} + 11 \beta_{3} - 5) q^{11} + ( - 2 \beta_{5} + 2 \beta_{2} + 13 \beta_1) q^{13} + (2 \beta_{5} + 14 \beta_{2} - 31 \beta_1) q^{17} + (9 \beta_{4} - 6 \beta_{3} - 779) q^{19} + (49 \beta_{3} + 98) q^{21} + ( - \beta_{5} - 1225 \beta_{2} - 70 \beta_1) q^{23} + ( - 6 \beta_{5} + 1244 \beta_{2} - 31 \beta_1) q^{27} + ( - 15 \beta_{4} + 99 \beta_{3} - 1359) q^{29} + (17 \beta_{4} - 248 \beta_{3} + 1957) q^{31} + (10 \beta_{5} - 3776 \beta_{2} - 137 \beta_1) q^{33} + ( - \beta_{5} + 3793 \beta_{2} - 64 \beta_1) q^{37} + (11 \beta_{4} + 293 \beta_{3} - 4571) q^{39} + ( - 31 \beta_{4} + 142 \beta_{3} + 3827) q^{41} + ( - 31 \beta_{5} - 6171 \beta_{2} - 322 \beta_1) q^{43} + (16 \beta_{5} + 7246 \beta_{2} - 113 \beta_1) q^{47} - 2401 q^{49} + ( - 29 \beta_{4} - 223 \beta_{3} + 10597) q^{51} + ( - 6 \beta_{5} - 2496 \beta_{2} + 594 \beta_1) q^{53} + ( - 15 \beta_{5} + 2449 \beta_{2} - 2282 \beta_1) q^{57} + ( - 64 \beta_{4} + 1840 \beta_{3} - 4108) q^{59} + (71 \beta_{4} - 110 \beta_{3} + 9037) q^{61} + (49 \beta_{5} - 4606 \beta_{2} + 245 \beta_1) q^{63} + ( - 8 \beta_{5} - 2556 \beta_{2} - 2228 \beta_1) q^{67} + ( - 71 \beta_{4} - 850 \beta_{3} + 20737) q^{69} + ( - 184 \beta_{4} + 136 \beta_{3} + 27392) q^{71} + (250 \beta_{5} - 160 \beta_{2} - 1772 \beta_1) q^{73} + (49 \beta_{5} - 245 \beta_{2} - 539 \beta_1) q^{77} + (95 \beta_{4} + 3919 \beta_{3} + 5277) q^{79} + (206 \beta_{4} + 1112 \beta_{3} - 10769) q^{81} + ( - 138 \beta_{5} + 28746 \beta_{2} - 960 \beta_1) q^{83} + (114 \beta_{5} - 28404 \beta_{2} + 1413 \beta_1) q^{87} + (127 \beta_{4} + 2870 \beta_{3} + 31861) q^{89} + (98 \beta_{4} + 637 \beta_{3} - 98) q^{91} + ( - 265 \beta_{5} + 76579 \beta_{2} - 1592 \beta_1) q^{93} + ( - 322 \beta_{5} - 58622 \beta_{2} - 931 \beta_1) q^{97} + (116 \beta_{4} - 2342 \beta_{3} + 38084) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 562 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 562 q^{9} - 28 q^{11} - 4656 q^{19} + 588 q^{21} - 8184 q^{29} + 11776 q^{31} - 27404 q^{39} + 22900 q^{41} - 14406 q^{49} + 63524 q^{51} - 24776 q^{59} + 54364 q^{61} + 124280 q^{69} + 163984 q^{71} + 31852 q^{79} - 64202 q^{81} + 191420 q^{89} - 392 q^{91} + 228736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 998x^{4} + 249001x^{2} + 44100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 499\nu ) / 210 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 499\nu^{2} ) / 210 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 709\nu^{2} + 69930 ) / 210 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 832\nu^{3} + 165957\nu ) / 210 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 333 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 210\beta_{2} - 499\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -499\beta_{4} + 709\beta_{3} + 166167 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 210\beta_{5} - 174720\beta_{2} + 249211\beta_1 \) Copy content Toggle raw display

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
22.5458i
22.1248i
0.420991i
0.420991i
22.1248i
22.5458i
0 24.5458i 0 0 0 49.0000i 0 −359.498 0
449.2 0 20.1248i 0 0 0 49.0000i 0 −162.009 0
449.3 0 1.57901i 0 0 0 49.0000i 0 240.507 0
449.4 0 1.57901i 0 0 0 49.0000i 0 240.507 0
449.5 0 20.1248i 0 0 0 49.0000i 0 −162.009 0
449.6 0 24.5458i 0 0 0 49.0000i 0 −359.498 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.6
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 700.6.e.g 6
5.b even 2 1 inner 700.6.e.g 6
5.c odd 4 1 140.6.a.d 3
5.c odd 4 1 700.6.a.i 3
20.e even 4 1 560.6.a.t 3
35.f even 4 1 980.6.a.h 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.a.d 3 5.c odd 4 1
560.6.a.t 3 20.e even 4 1
700.6.a.i 3 5.c odd 4 1
700.6.e.g 6 1.a even 1 1 trivial
700.6.e.g 6 5.b even 2 1 inner
980.6.a.h 3 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 1010T_{3}^{4} + 246529T_{3}^{2} + 608400 \) acting on \(S_{6}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 1010 T^{4} + 246529 T^{2} + \cdots + 608400 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2401)^{3} \) Copy content Toggle raw display
$11$ \( (T^{3} + 14 T^{2} - 147231 T + 12436740)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 850758 T^{4} + \cdots + 37708614058756 \) Copy content Toggle raw display
$17$ \( T^{6} + 1668310 T^{4} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{3} + 2328 T^{2} + \cdots - 11429521264)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 9508744 T^{4} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{3} + 4092 T^{2} + \cdots - 17580868722)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 5888 T^{2} + \cdots + 211802104832)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 47359404 T^{4} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{3} - 11450 T^{2} + \cdots + 478579953600)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 370031384 T^{4} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{6} + 214248754 T^{4} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{6} + 379440036 T^{4} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{3} + 12388 T^{2} + \cdots - 41176040028480)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 27182 T^{2} + \cdots - 169955356480)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 4971185168 T^{4} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{3} - 81992 T^{2} + \cdots + 113425504819200)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 13818436332 T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{3} - 15926 T^{2} + \cdots + 274092525845520)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 6449648688 T^{4} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{3} - 95710 T^{2} + \cdots + 305457269205600)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 28176239622 T^{4} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
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