Properties

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

Related objects

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: $$x^{2} - x + 1$$ 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.

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

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
 0.5 + 0.866025i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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.c 2
7.b odd 2 1 49.12.c.b 2
7.c even 3 1 49.12.a.a 1
7.c even 3 1 inner 49.12.c.c 2
7.d odd 6 1 1.12.a.a 1
7.d odd 6 1 49.12.c.b 2
21.g even 6 1 9.12.a.b 1
28.f even 6 1 16.12.a.a 1
35.i odd 6 1 25.12.a.b 1
35.k even 12 2 25.12.b.b 2
56.j odd 6 1 64.12.a.b 1
56.m even 6 1 64.12.a.f 1
63.i even 6 1 81.12.c.b 2
63.k odd 6 1 81.12.c.d 2
63.s even 6 1 81.12.c.b 2
63.t odd 6 1 81.12.c.d 2
77.i even 6 1 121.12.a.b 1
84.j odd 6 1 144.12.a.d 1
91.s odd 6 1 169.12.a.a 1
105.p even 6 1 225.12.a.b 1
105.w odd 12 2 225.12.b.d 2
112.v even 12 2 256.12.b.c 2
112.x odd 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.d odd 6 1
9.12.a.b 1 21.g even 6 1
16.12.a.a 1 28.f even 6 1
25.12.a.b 1 35.i odd 6 1
25.12.b.b 2 35.k even 12 2
49.12.a.a 1 7.c even 3 1
49.12.c.b 2 7.b odd 2 1
49.12.c.b 2 7.d odd 6 1
49.12.c.c 2 1.a even 1 1 trivial
49.12.c.c 2 7.c even 3 1 inner
64.12.a.b 1 56.j odd 6 1
64.12.a.f 1 56.m even 6 1
81.12.c.b 2 63.i even 6 1
81.12.c.b 2 63.s even 6 1
81.12.c.d 2 63.k odd 6 1
81.12.c.d 2 63.t odd 6 1
121.12.a.b 1 77.i even 6 1
144.12.a.d 1 84.j odd 6 1
169.12.a.a 1 91.s odd 6 1
225.12.a.b 1 105.p even 6 1
225.12.b.d 2 105.w odd 12 2
256.12.b.c 2 112.v even 12 2
256.12.b.e 2 112.x odd 12 2

Hecke kernels

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

 $$T_{2}^{2} - 24 T_{2} + 576$$ $$T_{3}^{2} - 252 T_{3} + 63504$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$576 - 24 T + T^{2}$$
$3$ $$63504 - 252 T + T^{2}$$
$5$ $$23328900 - 4830 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$285809990544 + 534612 T + T^{2}$$
$13$ $$( -577738 + T )^{2}$$
$17$ $$47691924412356 + 6905934 T + T^{2}$$
$19$ $$113665876416400 - 10661420 T + T^{2}$$
$23$ $$347571590865984 + 18643272 T + T^{2}$$
$29$ $$( -128406630 + T )^{2}$$
$31$ $$2792400404276224 + 52843168 T + T^{2}$$
$37$ $$33201691798862596 - 182213314 T + T^{2}$$
$41$ $$( 308120442 + T )^{2}$$
$43$ $$( 17125708 + T )^{2}$$
$47$ $$7221841938953462016 - 2687348496 T + T^{2}$$
$53$ $$2547393791118267204 - 1596055698 T + T^{2}$$
$59$ $$26927835455229987600 + 5189203740 T + T^{2}$$
$61$ $$48392595374861310244 - 6956478662 T + T^{2}$$
$67$ $$23\!\cdots\!56$$$$- 15481826884 T + T^{2}$$
$71$ $$( -9791485272 + T )^{2}$$
$73$ $$2142685034362507684 - 1463791322 T + T^{2}$$
$79$ $$14\!\cdots\!00$$$$+ 38116845680 T + T^{2}$$
$83$ $$( -29335099668 + T )^{2}$$
$89$ $$62\!\cdots\!00$$$$+ 24992917110 T + T^{2}$$
$97$ $$( 75013568546 + T )^{2}$$