Properties

Label 49.10.c.e
Level $49$
Weight $10$
Character orbit 49.c
Analytic conductor $25.237$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.2367559720\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - x^{5} + 427x^{4} - 3606x^{3} + 183492x^{2} - 858816x + 4064256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + 7 \beta_{3}) q^{2} + ( - \beta_{5} - \beta_{4} + 28 \beta_{3} - \beta_{2} - \beta_1 + 28) q^{3} + (8 \beta_{5} - 7 \beta_{4} - 519 \beta_{3} - 7 \beta_{2} + 8 \beta_1 - 519) q^{4} + ( - 43 \beta_{5} + 13 \beta_{4} - 518 \beta_{3}) q^{5} + (6 \beta_{2} - 36 \beta_1 - 1638) q^{6} + (147 \beta_{2} - 470 \beta_1 + 4685) q^{8} + (90 \beta_{5} - 126 \beta_{4} - 8667 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + 7 \beta_{3}) q^{2} + ( - \beta_{5} - \beta_{4} + 28 \beta_{3} - \beta_{2} - \beta_1 + 28) q^{3} + (8 \beta_{5} - 7 \beta_{4} - 519 \beta_{3} - 7 \beta_{2} + 8 \beta_1 - 519) q^{4} + ( - 43 \beta_{5} + 13 \beta_{4} - 518 \beta_{3}) q^{5} + (6 \beta_{2} - 36 \beta_1 - 1638) q^{6} + (147 \beta_{2} - 470 \beta_1 + 4685) q^{8} + (90 \beta_{5} - 126 \beta_{4} - 8667 \beta_{3}) q^{9} + (370 \beta_{5} - 470 \beta_{4} - 32620 \beta_{3} - 470 \beta_{2} + \cdots - 32620) q^{10}+ \cdots + (376362 \beta_{2} - 7689150 \beta_1 - 633659724) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 21 q^{2} + 84 q^{3} - 1557 q^{4} + 1554 q^{5} - 9828 q^{6} + 28110 q^{8} + 26001 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 21 q^{2} + 84 q^{3} - 1557 q^{4} + 1554 q^{5} - 9828 q^{6} + 28110 q^{8} + 26001 q^{9} - 97860 q^{10} + 3444 q^{11} - 106386 q^{12} + 39564 q^{13} + 400608 q^{15} - 482961 q^{16} + 1016694 q^{17} + 273267 q^{18} + 222852 q^{19} + 3844176 q^{20} - 5694096 q^{22} - 1885632 q^{23} - 1449630 q^{24} - 3073221 q^{25} - 8785056 q^{26} - 1103760 q^{27} + 8163636 q^{29} - 8053200 q^{30} + 2869440 q^{31} - 25221951 q^{32} - 20259792 q^{33} + 7963284 q^{34} - 70792758 q^{36} - 1395618 q^{37} + 43479870 q^{38} + 8990688 q^{39} - 82859280 q^{40} + 28841316 q^{41} - 123262344 q^{43} - 97011984 q^{44} + 29774682 q^{45} - 89747664 q^{46} - 10368960 q^{47} - 33597564 q^{48} + 146650110 q^{50} + 26146728 q^{51} - 80908044 q^{52} - 67502610 q^{53} - 117879300 q^{54} + 211646064 q^{55} + 16942224 q^{57} + 159163830 q^{58} - 42590100 q^{59} + 179551008 q^{60} + 191746842 q^{61} + 93966936 q^{62} + 15704322 q^{64} - 364283220 q^{65} - 8057952 q^{66} + 255175788 q^{67} + 743485806 q^{68} - 515807712 q^{69} + 593029008 q^{71} + 609314265 q^{72} + 344213310 q^{73} + 690696462 q^{74} + 279031116 q^{75} - 1457679972 q^{76} + 560265552 q^{78} + 960412656 q^{79} - 1333333344 q^{80} + 35827677 q^{81} + 562675302 q^{82} + 2201034360 q^{83} + 876358824 q^{85} + 880982256 q^{86} - 621821592 q^{87} - 1206124800 q^{88} + 506816478 q^{89} - 4606905240 q^{90} + 1382246976 q^{92} - 1693258512 q^{93} + 1388004828 q^{94} + 2203071072 q^{95} + 333385794 q^{96} + 1294996500 q^{97} - 3801958344 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 427x^{4} - 3606x^{3} + 183492x^{2} - 858816x + 4064256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 415\nu^{5} - 177205\nu^{4} - 1825733\nu^{3} - 76149180\nu^{2} + 356408640\nu - 18279850176 ) / 464953608 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -451\nu^{5} + 192577\nu^{4} - 4738111\nu^{3} + 82754892\nu^{2} - 387326016\nu + 30551662512 ) / 464953608 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4331\nu^{5} + 4283\nu^{4} - 1828841\nu^{3} + 6865794\nu^{2} - 785896236\nu - 41319936 ) / 3719628864 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -97909\nu^{5} - 629099\nu^{4} - 41343799\nu^{3} + 155211966\nu^{2} - 24361417764\nu - 934101504 ) / 1859814432 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 710\nu^{5} + 4339\nu^{4} + 299810\nu^{3} - 1125540\nu^{2} + 97141857\nu + 6773760 ) / 12915378 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} - \beta_{4} - 2\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -25\beta_{5} + 11\beta_{4} - 1706\beta_{3} + 11\beta_{2} - 25\beta _1 - 1706 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -415\beta_{2} - 451\beta _1 + 9538 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9085\beta_{5} - 6287\beta_{4} + 713186\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 150811\beta_{5} + 192679\beta_{4} - 6789298\beta_{3} + 192679\beta_{2} + 150811\beta _1 - 6789298 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
2.48064 + 4.29659i
−11.1179 19.2567i
9.13724 + 15.8262i
2.48064 4.29659i
−11.1179 + 19.2567i
9.13724 15.8262i
−20.9010 36.2015i 0.116170 0.201212i −617.701 + 1069.89i −895.944 1551.82i −9.71222 0 30239.6 9841.47 + 17045.9i −37452.2 + 64869.1i
18.2 −6.68036 11.5707i 81.7073 141.521i 166.746 288.812i 961.094 + 1664.66i −2183.34 0 −11296.4 −3510.66 6080.64i 12840.9 22241.1i
18.3 17.0813 + 29.5857i −39.8234 + 68.9762i −327.544 + 567.323i 711.850 + 1232.96i −2720.95 0 −4888.28 6669.69 + 11552.2i −24318.7 + 42121.2i
30.1 −20.9010 + 36.2015i 0.116170 + 0.201212i −617.701 1069.89i −895.944 + 1551.82i −9.71222 0 30239.6 9841.47 17045.9i −37452.2 64869.1i
30.2 −6.68036 + 11.5707i 81.7073 + 141.521i 166.746 + 288.812i 961.094 1664.66i −2183.34 0 −11296.4 −3510.66 + 6080.64i 12840.9 + 22241.1i
30.3 17.0813 29.5857i −39.8234 68.9762i −327.544 567.323i 711.850 1232.96i −2720.95 0 −4888.28 6669.69 11552.2i −24318.7 42121.2i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 30.3
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 49.10.c.e 6
7.b odd 2 1 49.10.c.d 6
7.c even 3 1 49.10.a.c 3
7.c even 3 1 inner 49.10.c.e 6
7.d odd 6 1 7.10.a.b 3
7.d odd 6 1 49.10.c.d 6
21.g even 6 1 63.10.a.e 3
28.f even 6 1 112.10.a.h 3
35.i odd 6 1 175.10.a.d 3
35.k even 12 2 175.10.b.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.b 3 7.d odd 6 1
49.10.a.c 3 7.c even 3 1
49.10.c.d 6 7.b odd 2 1
49.10.c.d 6 7.d odd 6 1
49.10.c.e 6 1.a even 1 1 trivial
49.10.c.e 6 7.c even 3 1 inner
63.10.a.e 3 21.g even 6 1
112.10.a.h 3 28.f even 6 1
175.10.a.d 3 35.i odd 6 1
175.10.b.d 6 35.k even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{6} + 21T_{2}^{5} + 1767T_{2}^{4} + 10314T_{2}^{3} + 2158956T_{2}^{2} + 25300080T_{2} + 364046400 \) Copy content Toggle raw display
\( T_{3}^{6} - 84T_{3}^{5} + 20052T_{3}^{4} + 1085616T_{3}^{3} + 169150032T_{3}^{2} - 39299904T_{3} + 9144576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 21 T^{5} + \cdots + 364046400 \) Copy content Toggle raw display
$3$ \( T^{6} - 84 T^{5} + 20052 T^{4} + \cdots + 9144576 \) Copy content Toggle raw display
$5$ \( T^{6} - 1554 T^{5} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 3444 T^{5} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T^{3} - 19782 T^{2} + \cdots + 41548412541440)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 1016694 T^{5} + \cdots + 48\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{6} - 222852 T^{5} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + 1885632 T^{5} + \cdots + 94\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{3} - 4081818 T^{2} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 2869440 T^{5} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{6} + 1395618 T^{5} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{3} - 14420658 T^{2} + \cdots + 19\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 61631172 T^{2} + \cdots + 68\!\cdots\!80)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 10368960 T^{5} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{6} + 67502610 T^{5} + \cdots + 57\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{6} + 42590100 T^{5} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} - 191746842 T^{5} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{6} - 255175788 T^{5} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{3} - 296514504 T^{2} + \cdots + 16\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 344213310 T^{5} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{6} - 960412656 T^{5} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{3} - 1100517180 T^{2} + \cdots - 18\!\cdots\!48)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 506816478 T^{5} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{3} - 647498250 T^{2} + \cdots + 49\!\cdots\!16)^{2} \) Copy content Toggle raw display
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