# 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 $2$

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

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

## 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.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 $$S_{12}^{\mathrm{new}}(49, [\chi])$$:

 $$T_{2}^{2} - 24T_{2} + 576$$ T2^2 - 24*T2 + 576 $$T_{3}^{2} + 252T_{3} + 63504$$ T3^2 + 252*T3 + 63504

## Hecke characteristic polynomials

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