# Properties

 Label 7.4.c.a Level $7$ Weight $4$ Character orbit 7.c Analytic conductor $0.413$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 7.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.413013370040$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} - 7 \zeta_{6} q^{3} + 4 \zeta_{6} q^{4} + (7 \zeta_{6} - 7) q^{5} + 14 q^{6} + ( - 14 \zeta_{6} + 21) q^{7} - 24 q^{8} + (22 \zeta_{6} - 22) q^{9} +O(q^{10})$$ q + (2*z - 2) * q^2 - 7*z * q^3 + 4*z * q^4 + (7*z - 7) * q^5 + 14 * q^6 + (-14*z + 21) * q^7 - 24 * q^8 + (22*z - 22) * q^9 $$q + (2 \zeta_{6} - 2) q^{2} - 7 \zeta_{6} q^{3} + 4 \zeta_{6} q^{4} + (7 \zeta_{6} - 7) q^{5} + 14 q^{6} + ( - 14 \zeta_{6} + 21) q^{7} - 24 q^{8} + (22 \zeta_{6} - 22) q^{9} - 14 \zeta_{6} q^{10} + 5 \zeta_{6} q^{11} + ( - 28 \zeta_{6} + 28) q^{12} - 14 q^{13} + (42 \zeta_{6} - 14) q^{14} + 49 q^{15} + ( - 16 \zeta_{6} + 16) q^{16} + 21 \zeta_{6} q^{17} - 44 \zeta_{6} q^{18} + (49 \zeta_{6} - 49) q^{19} - 28 q^{20} + ( - 49 \zeta_{6} - 98) q^{21} - 10 q^{22} + ( - 159 \zeta_{6} + 159) q^{23} + 168 \zeta_{6} q^{24} + 76 \zeta_{6} q^{25} + ( - 28 \zeta_{6} + 28) q^{26} - 35 q^{27} + (28 \zeta_{6} + 56) q^{28} + 58 q^{29} + (98 \zeta_{6} - 98) q^{30} - 147 \zeta_{6} q^{31} - 160 \zeta_{6} q^{32} + ( - 35 \zeta_{6} + 35) q^{33} - 42 q^{34} + (147 \zeta_{6} - 49) q^{35} - 88 q^{36} + (219 \zeta_{6} - 219) q^{37} - 98 \zeta_{6} q^{38} + 98 \zeta_{6} q^{39} + ( - 168 \zeta_{6} + 168) q^{40} + 350 q^{41} + ( - 196 \zeta_{6} + 294) q^{42} - 124 q^{43} + (20 \zeta_{6} - 20) q^{44} - 154 \zeta_{6} q^{45} + 318 \zeta_{6} q^{46} + (525 \zeta_{6} - 525) q^{47} - 112 q^{48} + ( - 392 \zeta_{6} + 245) q^{49} - 152 q^{50} + ( - 147 \zeta_{6} + 147) q^{51} - 56 \zeta_{6} q^{52} - 303 \zeta_{6} q^{53} + ( - 70 \zeta_{6} + 70) q^{54} - 35 q^{55} + (336 \zeta_{6} - 504) q^{56} + 343 q^{57} + (116 \zeta_{6} - 116) q^{58} + 105 \zeta_{6} q^{59} + 196 \zeta_{6} q^{60} + ( - 413 \zeta_{6} + 413) q^{61} + 294 q^{62} + (462 \zeta_{6} - 154) q^{63} + 448 q^{64} + ( - 98 \zeta_{6} + 98) q^{65} + 70 \zeta_{6} q^{66} - 415 \zeta_{6} q^{67} + (84 \zeta_{6} - 84) q^{68} - 1113 q^{69} + ( - 98 \zeta_{6} - 196) q^{70} - 432 q^{71} + ( - 528 \zeta_{6} + 528) q^{72} + 1113 \zeta_{6} q^{73} - 438 \zeta_{6} q^{74} + ( - 532 \zeta_{6} + 532) q^{75} - 196 q^{76} + (35 \zeta_{6} + 70) q^{77} - 196 q^{78} + ( - 103 \zeta_{6} + 103) q^{79} + 112 \zeta_{6} q^{80} + 839 \zeta_{6} q^{81} + (700 \zeta_{6} - 700) q^{82} + 1092 q^{83} + ( - 588 \zeta_{6} + 196) q^{84} - 147 q^{85} + ( - 248 \zeta_{6} + 248) q^{86} - 406 \zeta_{6} q^{87} - 120 \zeta_{6} q^{88} + ( - 329 \zeta_{6} + 329) q^{89} + 308 q^{90} + (196 \zeta_{6} - 294) q^{91} + 636 q^{92} + (1029 \zeta_{6} - 1029) q^{93} - 1050 \zeta_{6} q^{94} - 343 \zeta_{6} q^{95} + (1120 \zeta_{6} - 1120) q^{96} - 882 q^{97} + (490 \zeta_{6} + 294) q^{98} - 110 q^{99} +O(q^{100})$$ q + (2*z - 2) * q^2 - 7*z * q^3 + 4*z * q^4 + (7*z - 7) * q^5 + 14 * q^6 + (-14*z + 21) * q^7 - 24 * q^8 + (22*z - 22) * q^9 - 14*z * q^10 + 5*z * q^11 + (-28*z + 28) * q^12 - 14 * q^13 + (42*z - 14) * q^14 + 49 * q^15 + (-16*z + 16) * q^16 + 21*z * q^17 - 44*z * q^18 + (49*z - 49) * q^19 - 28 * q^20 + (-49*z - 98) * q^21 - 10 * q^22 + (-159*z + 159) * q^23 + 168*z * q^24 + 76*z * q^25 + (-28*z + 28) * q^26 - 35 * q^27 + (28*z + 56) * q^28 + 58 * q^29 + (98*z - 98) * q^30 - 147*z * q^31 - 160*z * q^32 + (-35*z + 35) * q^33 - 42 * q^34 + (147*z - 49) * q^35 - 88 * q^36 + (219*z - 219) * q^37 - 98*z * q^38 + 98*z * q^39 + (-168*z + 168) * q^40 + 350 * q^41 + (-196*z + 294) * q^42 - 124 * q^43 + (20*z - 20) * q^44 - 154*z * q^45 + 318*z * q^46 + (525*z - 525) * q^47 - 112 * q^48 + (-392*z + 245) * q^49 - 152 * q^50 + (-147*z + 147) * q^51 - 56*z * q^52 - 303*z * q^53 + (-70*z + 70) * q^54 - 35 * q^55 + (336*z - 504) * q^56 + 343 * q^57 + (116*z - 116) * q^58 + 105*z * q^59 + 196*z * q^60 + (-413*z + 413) * q^61 + 294 * q^62 + (462*z - 154) * q^63 + 448 * q^64 + (-98*z + 98) * q^65 + 70*z * q^66 - 415*z * q^67 + (84*z - 84) * q^68 - 1113 * q^69 + (-98*z - 196) * q^70 - 432 * q^71 + (-528*z + 528) * q^72 + 1113*z * q^73 - 438*z * q^74 + (-532*z + 532) * q^75 - 196 * q^76 + (35*z + 70) * q^77 - 196 * q^78 + (-103*z + 103) * q^79 + 112*z * q^80 + 839*z * q^81 + (700*z - 700) * q^82 + 1092 * q^83 + (-588*z + 196) * q^84 - 147 * q^85 + (-248*z + 248) * q^86 - 406*z * q^87 - 120*z * q^88 + (-329*z + 329) * q^89 + 308 * q^90 + (196*z - 294) * q^91 + 636 * q^92 + (1029*z - 1029) * q^93 - 1050*z * q^94 - 343*z * q^95 + (1120*z - 1120) * q^96 - 882 * q^97 + (490*z + 294) * q^98 - 110 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 7 q^{3} + 4 q^{4} - 7 q^{5} + 28 q^{6} + 28 q^{7} - 48 q^{8} - 22 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 7 * q^3 + 4 * q^4 - 7 * q^5 + 28 * q^6 + 28 * q^7 - 48 * q^8 - 22 * q^9 $$2 q - 2 q^{2} - 7 q^{3} + 4 q^{4} - 7 q^{5} + 28 q^{6} + 28 q^{7} - 48 q^{8} - 22 q^{9} - 14 q^{10} + 5 q^{11} + 28 q^{12} - 28 q^{13} + 14 q^{14} + 98 q^{15} + 16 q^{16} + 21 q^{17} - 44 q^{18} - 49 q^{19} - 56 q^{20} - 245 q^{21} - 20 q^{22} + 159 q^{23} + 168 q^{24} + 76 q^{25} + 28 q^{26} - 70 q^{27} + 140 q^{28} + 116 q^{29} - 98 q^{30} - 147 q^{31} - 160 q^{32} + 35 q^{33} - 84 q^{34} + 49 q^{35} - 176 q^{36} - 219 q^{37} - 98 q^{38} + 98 q^{39} + 168 q^{40} + 700 q^{41} + 392 q^{42} - 248 q^{43} - 20 q^{44} - 154 q^{45} + 318 q^{46} - 525 q^{47} - 224 q^{48} + 98 q^{49} - 304 q^{50} + 147 q^{51} - 56 q^{52} - 303 q^{53} + 70 q^{54} - 70 q^{55} - 672 q^{56} + 686 q^{57} - 116 q^{58} + 105 q^{59} + 196 q^{60} + 413 q^{61} + 588 q^{62} + 154 q^{63} + 896 q^{64} + 98 q^{65} + 70 q^{66} - 415 q^{67} - 84 q^{68} - 2226 q^{69} - 490 q^{70} - 864 q^{71} + 528 q^{72} + 1113 q^{73} - 438 q^{74} + 532 q^{75} - 392 q^{76} + 175 q^{77} - 392 q^{78} + 103 q^{79} + 112 q^{80} + 839 q^{81} - 700 q^{82} + 2184 q^{83} - 196 q^{84} - 294 q^{85} + 248 q^{86} - 406 q^{87} - 120 q^{88} + 329 q^{89} + 616 q^{90} - 392 q^{91} + 1272 q^{92} - 1029 q^{93} - 1050 q^{94} - 343 q^{95} - 1120 q^{96} - 1764 q^{97} + 1078 q^{98} - 220 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 7 * q^3 + 4 * q^4 - 7 * q^5 + 28 * q^6 + 28 * q^7 - 48 * q^8 - 22 * q^9 - 14 * q^10 + 5 * q^11 + 28 * q^12 - 28 * q^13 + 14 * q^14 + 98 * q^15 + 16 * q^16 + 21 * q^17 - 44 * q^18 - 49 * q^19 - 56 * q^20 - 245 * q^21 - 20 * q^22 + 159 * q^23 + 168 * q^24 + 76 * q^25 + 28 * q^26 - 70 * q^27 + 140 * q^28 + 116 * q^29 - 98 * q^30 - 147 * q^31 - 160 * q^32 + 35 * q^33 - 84 * q^34 + 49 * q^35 - 176 * q^36 - 219 * q^37 - 98 * q^38 + 98 * q^39 + 168 * q^40 + 700 * q^41 + 392 * q^42 - 248 * q^43 - 20 * q^44 - 154 * q^45 + 318 * q^46 - 525 * q^47 - 224 * q^48 + 98 * q^49 - 304 * q^50 + 147 * q^51 - 56 * q^52 - 303 * q^53 + 70 * q^54 - 70 * q^55 - 672 * q^56 + 686 * q^57 - 116 * q^58 + 105 * q^59 + 196 * q^60 + 413 * q^61 + 588 * q^62 + 154 * q^63 + 896 * q^64 + 98 * q^65 + 70 * q^66 - 415 * q^67 - 84 * q^68 - 2226 * q^69 - 490 * q^70 - 864 * q^71 + 528 * q^72 + 1113 * q^73 - 438 * q^74 + 532 * q^75 - 392 * q^76 + 175 * q^77 - 392 * q^78 + 103 * q^79 + 112 * q^80 + 839 * q^81 - 700 * q^82 + 2184 * q^83 - 196 * q^84 - 294 * q^85 + 248 * q^86 - 406 * q^87 - 120 * q^88 + 329 * q^89 + 616 * q^90 - 392 * q^91 + 1272 * q^92 - 1029 * q^93 - 1050 * q^94 - 343 * q^95 - 1120 * q^96 - 1764 * q^97 + 1078 * q^98 - 220 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 + 1.73205i −3.50000 6.06218i 2.00000 + 3.46410i −3.50000 + 6.06218i 14.0000 14.0000 12.1244i −24.0000 −11.0000 + 19.0526i −7.00000 12.1244i
4.1 −1.00000 1.73205i −3.50000 + 6.06218i 2.00000 3.46410i −3.50000 6.06218i 14.0000 14.0000 + 12.1244i −24.0000 −11.0000 19.0526i −7.00000 + 12.1244i
 $$n$$: 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 7.4.c.a 2
3.b odd 2 1 63.4.e.b 2
4.b odd 2 1 112.4.i.c 2
5.b even 2 1 175.4.e.a 2
5.c odd 4 2 175.4.k.a 4
7.b odd 2 1 49.4.c.a 2
7.c even 3 1 inner 7.4.c.a 2
7.c even 3 1 49.4.a.d 1
7.d odd 6 1 49.4.a.c 1
7.d odd 6 1 49.4.c.a 2
8.b even 2 1 448.4.i.f 2
8.d odd 2 1 448.4.i.a 2
21.c even 2 1 441.4.e.k 2
21.g even 6 1 441.4.a.e 1
21.g even 6 1 441.4.e.k 2
21.h odd 6 1 63.4.e.b 2
21.h odd 6 1 441.4.a.d 1
28.f even 6 1 784.4.a.r 1
28.g odd 6 1 112.4.i.c 2
28.g odd 6 1 784.4.a.b 1
35.i odd 6 1 1225.4.a.d 1
35.j even 6 1 175.4.e.a 2
35.j even 6 1 1225.4.a.c 1
35.l odd 12 2 175.4.k.a 4
56.k odd 6 1 448.4.i.a 2
56.p even 6 1 448.4.i.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 1.a even 1 1 trivial
7.4.c.a 2 7.c even 3 1 inner
49.4.a.c 1 7.d odd 6 1
49.4.a.d 1 7.c even 3 1
49.4.c.a 2 7.b odd 2 1
49.4.c.a 2 7.d odd 6 1
63.4.e.b 2 3.b odd 2 1
63.4.e.b 2 21.h odd 6 1
112.4.i.c 2 4.b odd 2 1
112.4.i.c 2 28.g odd 6 1
175.4.e.a 2 5.b even 2 1
175.4.e.a 2 35.j even 6 1
175.4.k.a 4 5.c odd 4 2
175.4.k.a 4 35.l odd 12 2
441.4.a.d 1 21.h odd 6 1
441.4.a.e 1 21.g even 6 1
441.4.e.k 2 21.c even 2 1
441.4.e.k 2 21.g even 6 1
448.4.i.a 2 8.d odd 2 1
448.4.i.a 2 56.k odd 6 1
448.4.i.f 2 8.b even 2 1
448.4.i.f 2 56.p even 6 1
784.4.a.b 1 28.g odd 6 1
784.4.a.r 1 28.f even 6 1
1225.4.a.c 1 35.j even 6 1
1225.4.a.d 1 35.i odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(7, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} + 7T + 49$$
$5$ $$T^{2} + 7T + 49$$
$7$ $$T^{2} - 28T + 343$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$(T + 14)^{2}$$
$17$ $$T^{2} - 21T + 441$$
$19$ $$T^{2} + 49T + 2401$$
$23$ $$T^{2} - 159T + 25281$$
$29$ $$(T - 58)^{2}$$
$31$ $$T^{2} + 147T + 21609$$
$37$ $$T^{2} + 219T + 47961$$
$41$ $$(T - 350)^{2}$$
$43$ $$(T + 124)^{2}$$
$47$ $$T^{2} + 525T + 275625$$
$53$ $$T^{2} + 303T + 91809$$
$59$ $$T^{2} - 105T + 11025$$
$61$ $$T^{2} - 413T + 170569$$
$67$ $$T^{2} + 415T + 172225$$
$71$ $$(T + 432)^{2}$$
$73$ $$T^{2} - 1113 T + 1238769$$
$79$ $$T^{2} - 103T + 10609$$
$83$ $$(T - 1092)^{2}$$
$89$ $$T^{2} - 329T + 108241$$
$97$ $$(T + 882)^{2}$$