Properties

Label 49.6.c.f
Level $49$
Weight $6$
Character orbit 49.c
Analytic conductor $7.859$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{37})\)
Defining polynomial: \( x^{4} - x^{3} + 10x^{2} + 9x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{3} + (2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 6) q^{4} + ( - 10 \beta_{2} - 19 \beta_1) q^{5} + ( - 5 \beta_{3} + 41) q^{6} + (24 \beta_{3} + 48) q^{8} + ( - 8 \beta_{2} + 190 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{3} + (2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 6) q^{4} + ( - 10 \beta_{2} - 19 \beta_1) q^{5} + ( - 5 \beta_{3} + 41) q^{6} + (24 \beta_{3} + 48) q^{8} + ( - 8 \beta_{2} + 190 \beta_1) q^{9} + (29 \beta_{3} + 29 \beta_{2} + 389 \beta_1 - 389) q^{10} + (23 \beta_{3} + 23 \beta_{2} + 212 \beta_1 - 212) q^{11} + ( - 14 \beta_{2} - 98 \beta_1) q^{12} + (28 \beta_{3} + 462) q^{13} + ( - 59 \beta_{3} + 446) q^{15} + (40 \beta_{2} + 1032 \beta_1) q^{16} + ( - 132 \beta_{3} - 132 \beta_{2} + 1173 \beta_1 - 1173) q^{17} + ( - 182 \beta_{3} - 182 \beta_{2} + 106 \beta_1 - 106) q^{18} + (277 \beta_{2} - 180 \beta_1) q^{19} + ( - 98 \beta_{3} + 854) q^{20} + ( - 235 \beta_{3} + 1063) q^{22} + (69 \beta_{2} + 6 \beta_1) q^{23} + ( - 48 \beta_{3} - 48 \beta_{2} - 696 \beta_1 + 696) q^{24} + (380 \beta_{3} + 380 \beta_{2} + 936 \beta_1 - 936) q^{25} + ( - 434 \beta_{2} + 574 \beta_1) q^{26} + (401 \beta_{3} - 1436) q^{27} + (700 \beta_{3} - 3526) q^{29} + ( - 505 \beta_{2} - 2629 \beta_1) q^{30} + (715 \beta_{3} + 715 \beta_{2} - 1774 \beta_1 + 1774) q^{31} + ( - 304 \beta_{3} - 304 \beta_{2} - 4048 \beta_1 + 4048) q^{32} + ( - 304 \beta_{2} - 1699 \beta_1) q^{33} + ( - 1041 \beta_{3} - 3711) q^{34} + (332 \beta_{3} - 548) q^{36} + (790 \beta_{2} - 5545 \beta_1) q^{37} + ( - 97 \beta_{3} - 97 \beta_{2} - 10069 \beta_1 + 10069) q^{38} + (350 \beta_{3} + 350 \beta_{2} + 812 \beta_1 - 812) q^{39} + ( - 24 \beta_{2} + 7968 \beta_1) q^{40} + (868 \beta_{3} - 1750) q^{41} + ( - 1344 \beta_{3} - 6340) q^{43} + ( - 562 \beta_{2} - 2974 \beta_1) q^{44} + ( - 1748 \beta_{3} - 1748 \beta_{2} - 650 \beta_1 + 650) q^{45} + ( - 75 \beta_{3} - 75 \beta_{2} - 2559 \beta_1 + 2559) q^{46} + (1635 \beta_{2} - 11478 \beta_1) q^{47} + (1192 \beta_{3} - 5608) q^{48} + ( - 1316 \beta_{3} + 14996) q^{50} + ( - 645 \beta_{2} + 192 \beta_1) q^{51} + (756 \beta_{3} + 756 \beta_{2} + 700 \beta_1 - 700) q^{52} + ( - 1818 \beta_{3} - 1818 \beta_{2} + 1521 \beta_1 - 1521) q^{53} + (1837 \beta_{2} + 16273 \beta_1) q^{54} + ( - 2557 \beta_{3} + 12538) q^{55} + (928 \beta_{3} - 9529) q^{57} + (4226 \beta_{2} + 29426 \beta_1) q^{58} + ( - 531 \beta_{3} - 531 \beta_{2} + 32904 \beta_1 - 32904) q^{59} + (1246 \beta_{3} + 1246 \beta_{2} + 7042 \beta_1 - 7042) q^{60} + ( - 4154 \beta_{2} - 21243 \beta_1) q^{61} + (1059 \beta_{3} + 24681) q^{62} + (3072 \beta_{3} + 17728) q^{64} + ( - 4088 \beta_{2} + 1582 \beta_1) q^{65} + (2003 \beta_{3} + 2003 \beta_{2} + 12947 \beta_1 - 12947) q^{66} + (919 \beta_{3} + 919 \beta_{2} + 21156 \beta_1 - 21156) q^{67} + ( - 1554 \beta_{2} + 2730 \beta_1) q^{68} + (282 \beta_{3} - 2577) q^{69} + (2184 \beta_{3} - 1104) q^{71} + ( - 4944 \beta_{2} + 16224 \beta_1) q^{72} + (7372 \beta_{3} + 7372 \beta_{2} + 25253 \beta_1 - 25253) q^{73} + (4755 \beta_{3} + 4755 \beta_{2} - 23685 \beta_1 + 23685) q^{74} + ( - 2456 \beta_{2} - 17804 \beta_1) q^{75} + (1302 \beta_{3} - 19418) q^{76} + ( - 1162 \beta_{3} + 13762) q^{78} + (5193 \beta_{2} - 4502 \beta_1) q^{79} + ( - 11080 \beta_{3} - 11080 \beta_{2} - 34408 \beta_1 + 34408) q^{80} + ( - 4984 \beta_{3} - 4984 \beta_{2} + 25589 \beta_1 - 25589) q^{81} + (2618 \beta_{2} + 33866 \beta_1) q^{82} + ( - 4536 \beta_{3} + 52164) q^{83} + ( - 9222 \beta_{3} - 26553) q^{85} + (4996 \beta_{2} - 43388 \beta_1) q^{86} + ( - 6326 \beta_{3} - 6326 \beta_{2} - 40004 \beta_1 + 40004) q^{87} + ( - 3984 \beta_{3} - 3984 \beta_{2} - 10248 \beta_1 + 10248) q^{88} + (9356 \beta_{2} - 13333 \beta_1) q^{89} + (2398 \beta_{3} - 65326) q^{90} + (426 \beta_{3} - 5142) q^{92} + ( - 1086 \beta_{2} - 19359 \beta_1) q^{93} + (9843 \beta_{3} + 9843 \beta_{2} - 49017 \beta_1 + 49017) q^{94} + ( - 3463 \beta_{3} - 3463 \beta_{2} - 99070 \beta_1 + 99070) q^{95} + (5264 \beta_{2} + 27440 \beta_1) q^{96} + ( - 196 \beta_{3} - 104566) q^{97} + (2674 \beta_{3} - 33472) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 8 q^{3} - 12 q^{4} - 38 q^{5} + 164 q^{6} + 192 q^{8} + 380 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 8 q^{3} - 12 q^{4} - 38 q^{5} + 164 q^{6} + 192 q^{8} + 380 q^{9} - 778 q^{10} - 424 q^{11} - 196 q^{12} + 1848 q^{13} + 1784 q^{15} + 2064 q^{16} - 2346 q^{17} - 212 q^{18} - 360 q^{19} + 3416 q^{20} + 4252 q^{22} + 12 q^{23} + 1392 q^{24} - 1872 q^{25} + 1148 q^{26} - 5744 q^{27} - 14104 q^{29} - 5258 q^{30} + 3548 q^{31} + 8096 q^{32} - 3398 q^{33} - 14844 q^{34} - 2192 q^{36} - 11090 q^{37} + 20138 q^{38} - 1624 q^{39} + 15936 q^{40} - 7000 q^{41} - 25360 q^{43} - 5948 q^{44} + 1300 q^{45} + 5118 q^{46} - 22956 q^{47} - 22432 q^{48} + 59984 q^{50} + 384 q^{51} - 1400 q^{52} - 3042 q^{53} + 32546 q^{54} + 50152 q^{55} - 38116 q^{57} + 58852 q^{58} - 65808 q^{59} - 14084 q^{60} - 42486 q^{61} + 98724 q^{62} + 70912 q^{64} + 3164 q^{65} - 25894 q^{66} - 42312 q^{67} + 5460 q^{68} - 10308 q^{69} - 4416 q^{71} + 32448 q^{72} - 50506 q^{73} + 47370 q^{74} - 35608 q^{75} - 77672 q^{76} + 55048 q^{78} - 9004 q^{79} + 68816 q^{80} - 51178 q^{81} + 67732 q^{82} + 208656 q^{83} - 106212 q^{85} - 86776 q^{86} + 80008 q^{87} + 20496 q^{88} - 26666 q^{89} - 261304 q^{90} - 20568 q^{92} - 38718 q^{93} + 98034 q^{94} + 198140 q^{95} + 54880 q^{96} - 418264 q^{97} - 133888 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 10x^{2} + 9x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 10\nu^{2} - 10\nu + 81 ) / 90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 10\nu^{2} + 190\nu - 81 ) / 90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 14 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 19\beta _1 - 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 14 \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
1.77069 + 3.06693i
−1.27069 2.20090i
1.77069 3.06693i
−1.27069 + 2.20090i
−3.54138 6.13385i −5.04138 + 8.73193i −9.08276 + 15.7318i −39.9138 69.1328i 71.4138 0 −97.9863 70.6689 + 122.402i −282.700 + 489.651i
18.2 2.54138 + 4.40180i 1.04138 1.80373i 3.08276 5.33950i 20.9138 + 36.2238i 10.5862 0 193.986 119.331 + 206.687i −106.300 + 184.117i
30.1 −3.54138 + 6.13385i −5.04138 8.73193i −9.08276 15.7318i −39.9138 + 69.1328i 71.4138 0 −97.9863 70.6689 122.402i −282.700 489.651i
30.2 2.54138 4.40180i 1.04138 + 1.80373i 3.08276 + 5.33950i 20.9138 36.2238i 10.5862 0 193.986 119.331 206.687i −106.300 184.117i
\(n\): 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 49.6.c.f 4
7.b odd 2 1 7.6.c.a 4
7.c even 3 1 49.6.a.e 2
7.c even 3 1 inner 49.6.c.f 4
7.d odd 6 1 7.6.c.a 4
7.d odd 6 1 49.6.a.d 2
21.c even 2 1 63.6.e.d 4
21.g even 6 1 63.6.e.d 4
21.g even 6 1 441.6.a.n 2
21.h odd 6 1 441.6.a.m 2
28.d even 2 1 112.6.i.c 4
28.f even 6 1 112.6.i.c 4
28.f even 6 1 784.6.a.ba 2
28.g odd 6 1 784.6.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.c.a 4 7.b odd 2 1
7.6.c.a 4 7.d odd 6 1
49.6.a.d 2 7.d odd 6 1
49.6.a.e 2 7.c even 3 1
49.6.c.f 4 1.a even 1 1 trivial
49.6.c.f 4 7.c even 3 1 inner
63.6.e.d 4 21.c even 2 1
63.6.e.d 4 21.g even 6 1
112.6.i.c 4 28.d even 2 1
112.6.i.c 4 28.f even 6 1
441.6.a.m 2 21.h odd 6 1
441.6.a.n 2 21.g even 6 1
784.6.a.t 2 28.g odd 6 1
784.6.a.ba 2 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{4} + 2T_{2}^{3} + 40T_{2}^{2} - 72T_{2} + 1296 \) Copy content Toggle raw display
\( T_{3}^{4} + 8T_{3}^{3} + 85T_{3}^{2} - 168T_{3} + 441 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 40 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$3$ \( T^{4} + 8 T^{3} + 85 T^{2} - 168 T + 441 \) Copy content Toggle raw display
$5$ \( T^{4} + 38 T^{3} + 4783 T^{2} + \cdots + 11148921 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 424 T^{3} + \cdots + 643687641 \) Copy content Toggle raw display
$13$ \( (T^{2} - 924 T + 184436)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2346 T^{3} + \cdots + 534713400081 \) Copy content Toggle raw display
$19$ \( T^{4} + 360 T^{3} + \cdots + 7876852004329 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + \cdots + 31018606641 \) Copy content Toggle raw display
$29$ \( (T^{2} + 7052 T - 5697324)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 248637676526001 \) Copy content Toggle raw display
$37$ \( T^{4} + 11090 T^{3} + \cdots + 58604000855625 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3500 T - 24814188)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 12680 T - 26638832)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 22956 T^{3} + \cdots + 10\!\cdots\!81 \) Copy content Toggle raw display
$53$ \( T^{4} + 3042 T^{3} + \cdots + 14\!\cdots\!09 \) Copy content Toggle raw display
$59$ \( T^{4} + 65808 T^{3} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{4} + 42486 T^{3} + \cdots + 35\!\cdots\!49 \) Copy content Toggle raw display
$67$ \( T^{4} + 42312 T^{3} + \cdots + 17\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{2} + 2208 T - 175265856)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 50506 T^{3} + \cdots + 18\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{4} + 9004 T^{3} + \cdots + 95\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( (T^{2} - 104328 T + 1959796944)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 26666 T^{3} + \cdots + 93\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( (T^{2} + 209132 T + 10932626964)^{2} \) Copy content Toggle raw display
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