Properties

Label 49.12.c.b
Level $49$
Weight $12$
Character orbit 49.c
Analytic conductor $37.649$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.6488158474\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 24 \zeta_{6} q^{2} + (252 \zeta_{6} - 252) q^{3} + ( - 1472 \zeta_{6} + 1472) q^{4} - 4830 \zeta_{6} q^{5} - 6048 q^{6} + 84480 q^{8} + 113643 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 24 \zeta_{6} q^{2} + (252 \zeta_{6} - 252) q^{3} + ( - 1472 \zeta_{6} + 1472) q^{4} - 4830 \zeta_{6} q^{5} - 6048 q^{6} + 84480 q^{8} + 113643 \zeta_{6} q^{9} + ( - 115920 \zeta_{6} + 115920) q^{10} + (534612 \zeta_{6} - 534612) q^{11} + 370944 \zeta_{6} q^{12} - 577738 q^{13} + 1217160 q^{15} - 987136 \zeta_{6} q^{16} + ( - 6905934 \zeta_{6} + 6905934) q^{17} + (2727432 \zeta_{6} - 2727432) q^{18} - 10661420 \zeta_{6} q^{19} - 7109760 q^{20} - 12830688 q^{22} - 18643272 \zeta_{6} q^{23} + (21288960 \zeta_{6} - 21288960) q^{24} + ( - 25499225 \zeta_{6} + 25499225) q^{25} - 13865712 \zeta_{6} q^{26} - 73279080 q^{27} + 128406630 q^{29} + 29211840 \zeta_{6} q^{30} + ( - 52843168 \zeta_{6} + 52843168) q^{31} + ( - 196706304 \zeta_{6} + 196706304) q^{32} - 134722224 \zeta_{6} q^{33} + 165742416 q^{34} + 167282496 q^{36} + 182213314 \zeta_{6} q^{37} + ( - 255874080 \zeta_{6} + 255874080) q^{38} + ( - 145589976 \zeta_{6} + 145589976) q^{39} - 408038400 \zeta_{6} q^{40} + 308120442 q^{41} - 17125708 q^{43} + 786948864 \zeta_{6} q^{44} + ( - 548895690 \zeta_{6} + 548895690) q^{45} + ( - 447438528 \zeta_{6} + 447438528) q^{46} - 2687348496 \zeta_{6} q^{47} + 248758272 q^{48} + 611981400 q^{50} + 1740295368 \zeta_{6} q^{51} + (850430336 \zeta_{6} - 850430336) q^{52} + ( - 1596055698 \zeta_{6} + 1596055698) q^{53} - 1758697920 \zeta_{6} q^{54} + 2582175960 q^{55} + 2686677840 q^{57} + 3081759120 \zeta_{6} q^{58} + ( - 5189203740 \zeta_{6} + 5189203740) q^{59} + ( - 1791659520 \zeta_{6} + 1791659520) q^{60} - 6956478662 \zeta_{6} q^{61} + 1268236032 q^{62} + 2699296768 q^{64} + 2790474540 \zeta_{6} q^{65} + ( - 3233333376 \zeta_{6} + 3233333376) q^{66} + ( - 15481826884 \zeta_{6} + 15481826884) q^{67} - 10165534848 \zeta_{6} q^{68} + 4698104544 q^{69} + 9791485272 q^{71} + 9600560640 \zeta_{6} q^{72} + (1463791322 \zeta_{6} - 1463791322) q^{73} + (4373119536 \zeta_{6} - 4373119536) q^{74} + 6425804700 \zeta_{6} q^{75} - 15693610240 q^{76} + 3494159424 q^{78} - 38116845680 \zeta_{6} q^{79} + (4767866880 \zeta_{6} - 4767866880) q^{80} + (1665188361 \zeta_{6} - 1665188361) q^{81} + 7394890608 \zeta_{6} q^{82} - 29335099668 q^{83} - 33355661220 q^{85} - 411016992 \zeta_{6} q^{86} + (32358470760 \zeta_{6} - 32358470760) q^{87} + (45164021760 \zeta_{6} - 45164021760) q^{88} + 24992917110 \zeta_{6} q^{89} + 13173496560 q^{90} - 27442896384 q^{92} + 13316478336 \zeta_{6} q^{93} + ( - 64496363904 \zeta_{6} + 64496363904) q^{94} + (51494658600 \zeta_{6} - 51494658600) q^{95} + 49569988608 \zeta_{6} q^{96} + 75013568546 q^{97} - 60754911516 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{2} - 252 q^{3} + 1472 q^{4} - 4830 q^{5} - 12096 q^{6} + 168960 q^{8} + 113643 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 24 q^{2} - 252 q^{3} + 1472 q^{4} - 4830 q^{5} - 12096 q^{6} + 168960 q^{8} + 113643 q^{9} + 115920 q^{10} - 534612 q^{11} + 370944 q^{12} - 1155476 q^{13} + 2434320 q^{15} - 987136 q^{16} + 6905934 q^{17} - 2727432 q^{18} - 10661420 q^{19} - 14219520 q^{20} - 25661376 q^{22} - 18643272 q^{23} - 21288960 q^{24} + 25499225 q^{25} - 13865712 q^{26} - 146558160 q^{27} + 256813260 q^{29} + 29211840 q^{30} + 52843168 q^{31} + 196706304 q^{32} - 134722224 q^{33} + 331484832 q^{34} + 334564992 q^{36} + 182213314 q^{37} + 255874080 q^{38} + 145589976 q^{39} - 408038400 q^{40} + 616240884 q^{41} - 34251416 q^{43} + 786948864 q^{44} + 548895690 q^{45} + 447438528 q^{46} - 2687348496 q^{47} + 497516544 q^{48} + 1223962800 q^{50} + 1740295368 q^{51} - 850430336 q^{52} + 1596055698 q^{53} - 1758697920 q^{54} + 5164351920 q^{55} + 5373355680 q^{57} + 3081759120 q^{58} + 5189203740 q^{59} + 1791659520 q^{60} - 6956478662 q^{61} + 2536472064 q^{62} + 5398593536 q^{64} + 2790474540 q^{65} + 3233333376 q^{66} + 15481826884 q^{67} - 10165534848 q^{68} + 9396209088 q^{69} + 19582970544 q^{71} + 9600560640 q^{72} - 1463791322 q^{73} - 4373119536 q^{74} + 6425804700 q^{75} - 31387220480 q^{76} + 6988318848 q^{78} - 38116845680 q^{79} - 4767866880 q^{80} - 1665188361 q^{81} + 7394890608 q^{82} - 58670199336 q^{83} - 66711322440 q^{85} - 411016992 q^{86} - 32358470760 q^{87} - 45164021760 q^{88} + 24992917110 q^{89} + 26346993120 q^{90} - 54885792768 q^{92} + 13316478336 q^{93} + 64496363904 q^{94} - 51494658600 q^{95} + 49569988608 q^{96} + 150027137092 q^{97} - 121509823032 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
12.0000 + 20.7846i −126.000 + 218.238i 736.000 1274.79i −2415.00 4182.90i −6048.00 0 84480.0 56821.5 + 98417.7i 57960.0 100390.i
30.1 12.0000 20.7846i −126.000 218.238i 736.000 + 1274.79i −2415.00 + 4182.90i −6048.00 0 84480.0 56821.5 98417.7i 57960.0 + 100390.i
\(n\): 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 49.12.c.b 2
7.b odd 2 1 49.12.c.c 2
7.c even 3 1 1.12.a.a 1
7.c even 3 1 inner 49.12.c.b 2
7.d odd 6 1 49.12.a.a 1
7.d odd 6 1 49.12.c.c 2
21.h odd 6 1 9.12.a.b 1
28.g odd 6 1 16.12.a.a 1
35.j even 6 1 25.12.a.b 1
35.l odd 12 2 25.12.b.b 2
56.k odd 6 1 64.12.a.f 1
56.p even 6 1 64.12.a.b 1
63.g even 3 1 81.12.c.d 2
63.h even 3 1 81.12.c.d 2
63.j odd 6 1 81.12.c.b 2
63.n odd 6 1 81.12.c.b 2
77.h odd 6 1 121.12.a.b 1
84.n even 6 1 144.12.a.d 1
91.r even 6 1 169.12.a.a 1
105.o odd 6 1 225.12.a.b 1
105.x even 12 2 225.12.b.d 2
112.u odd 12 2 256.12.b.c 2
112.w even 12 2 256.12.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 7.c even 3 1
9.12.a.b 1 21.h odd 6 1
16.12.a.a 1 28.g odd 6 1
25.12.a.b 1 35.j even 6 1
25.12.b.b 2 35.l odd 12 2
49.12.a.a 1 7.d odd 6 1
49.12.c.b 2 1.a even 1 1 trivial
49.12.c.b 2 7.c even 3 1 inner
49.12.c.c 2 7.b odd 2 1
49.12.c.c 2 7.d odd 6 1
64.12.a.b 1 56.p even 6 1
64.12.a.f 1 56.k odd 6 1
81.12.c.b 2 63.j odd 6 1
81.12.c.b 2 63.n odd 6 1
81.12.c.d 2 63.g even 3 1
81.12.c.d 2 63.h even 3 1
121.12.a.b 1 77.h odd 6 1
144.12.a.d 1 84.n even 6 1
169.12.a.a 1 91.r even 6 1
225.12.a.b 1 105.o odd 6 1
225.12.b.d 2 105.x even 12 2
256.12.b.c 2 112.u odd 12 2
256.12.b.e 2 112.w even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{2} - 24T_{2} + 576 \) Copy content Toggle raw display
\( T_{3}^{2} + 252T_{3} + 63504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 24T + 576 \) Copy content Toggle raw display
$3$ \( T^{2} + 252T + 63504 \) Copy content Toggle raw display
$5$ \( T^{2} + 4830 T + 23328900 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 534612 T + 285809990544 \) Copy content Toggle raw display
$13$ \( (T + 577738)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6905934 T + 47691924412356 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 113665876416400 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 347571590865984 \) Copy content Toggle raw display
$29$ \( (T - 128406630)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 52843168 T + 27\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{2} - 182213314 T + 33\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T - 308120442)^{2} \) Copy content Toggle raw display
$43$ \( (T + 17125708)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2687348496 T + 72\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{2} - 1596055698 T + 25\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} - 5189203740 T + 26\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + 6956478662 T + 48\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{2} - 15481826884 T + 23\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T - 9791485272)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1463791322 T + 21\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{2} + 38116845680 T + 14\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T + 29335099668)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 24992917110 T + 62\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T - 75013568546)^{2} \) Copy content Toggle raw display
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