# 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 $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.41301337004$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

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

## 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.c Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{4}^{\mathrm{new}}(7, [\chi])$$.