# Properties

 Label 7.6.c.a Level 7 Weight 6 Character orbit 7.c Analytic conductor 1.123 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$7$$ Weight: $$k$$ = $$6$$ Character orbit: $$[\chi]$$ = 7.c (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.12268673869$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{37})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$2^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 4 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( -6 + 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{4} + ( 19 \beta_{1} + 10 \beta_{2} ) q^{5} + ( -41 + 5 \beta_{3} ) q^{6} + ( -14 - 56 \beta_{1} - 14 \beta_{2} - 21 \beta_{3} ) q^{7} + ( 48 + 24 \beta_{3} ) q^{8} + ( 190 \beta_{1} - 8 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 4 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( -6 + 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{4} + ( 19 \beta_{1} + 10 \beta_{2} ) q^{5} + ( -41 + 5 \beta_{3} ) q^{6} + ( -14 - 56 \beta_{1} - 14 \beta_{2} - 21 \beta_{3} ) q^{7} + ( 48 + 24 \beta_{3} ) q^{8} + ( 190 \beta_{1} - 8 \beta_{2} ) q^{9} + ( 389 - 389 \beta_{1} - 29 \beta_{2} - 29 \beta_{3} ) q^{10} + ( -212 + 212 \beta_{1} + 23 \beta_{2} + 23 \beta_{3} ) q^{11} + ( 98 \beta_{1} + 14 \beta_{2} ) q^{12} + ( -462 - 28 \beta_{3} ) q^{13} + ( -574 - 189 \beta_{1} + 63 \beta_{2} + 70 \beta_{3} ) q^{14} + ( 446 - 59 \beta_{3} ) q^{15} + ( 1032 \beta_{1} + 40 \beta_{2} ) q^{16} + ( 1173 - 1173 \beta_{1} + 132 \beta_{2} + 132 \beta_{3} ) q^{17} + ( -106 + 106 \beta_{1} - 182 \beta_{2} - 182 \beta_{3} ) q^{18} + ( 180 \beta_{1} - 277 \beta_{2} ) q^{19} + ( -854 + 98 \beta_{3} ) q^{20} + ( -21 - 721 \beta_{1} - 70 \beta_{2} + 42 \beta_{3} ) q^{21} + ( 1063 - 235 \beta_{3} ) q^{22} + ( 6 \beta_{1} + 69 \beta_{2} ) q^{23} + ( -696 + 696 \beta_{1} + 48 \beta_{2} + 48 \beta_{3} ) q^{24} + ( -936 + 936 \beta_{1} + 380 \beta_{2} + 380 \beta_{3} ) q^{25} + ( -574 \beta_{1} + 434 \beta_{2} ) q^{26} + ( 1436 - 401 \beta_{3} ) q^{27} + ( -98 + 1470 \beta_{1} + 98 \beta_{2} - 98 \beta_{3} ) q^{28} + ( -3526 + 700 \beta_{3} ) q^{29} + ( -2629 \beta_{1} - 505 \beta_{2} ) q^{30} + ( -1774 + 1774 \beta_{1} - 715 \beta_{2} - 715 \beta_{3} ) q^{31} + ( 4048 - 4048 \beta_{1} - 304 \beta_{2} - 304 \beta_{3} ) q^{32} + ( 1699 \beta_{1} + 304 \beta_{2} ) q^{33} + ( 3711 + 1041 \beta_{3} ) q^{34} + ( 6244 + 1260 \beta_{1} - 567 \beta_{2} - 826 \beta_{3} ) q^{35} + ( -548 + 332 \beta_{3} ) q^{36} + ( -5545 \beta_{1} + 790 \beta_{2} ) q^{37} + ( -10069 + 10069 \beta_{1} + 97 \beta_{2} + 97 \beta_{3} ) q^{38} + ( -812 + 812 \beta_{1} + 350 \beta_{2} + 350 \beta_{3} ) q^{39} + ( -7968 \beta_{1} + 24 \beta_{2} ) q^{40} + ( 1750 - 868 \beta_{3} ) q^{41} + ( -3311 + 4886 \beta_{1} + 854 \beta_{2} + 791 \beta_{3} ) q^{42} + ( -6340 - 1344 \beta_{3} ) q^{43} + ( -2974 \beta_{1} - 562 \beta_{2} ) q^{44} + ( -650 + 650 \beta_{1} + 1748 \beta_{2} + 1748 \beta_{3} ) q^{45} + ( 2559 - 2559 \beta_{1} - 75 \beta_{2} - 75 \beta_{3} ) q^{46} + ( 11478 \beta_{1} - 1635 \beta_{2} ) q^{47} + ( 5608 - 1192 \beta_{3} ) q^{48} + ( 6125 - 9800 \beta_{1} - 392 \beta_{2} + 2156 \beta_{3} ) q^{49} + ( 14996 - 1316 \beta_{3} ) q^{50} + ( 192 \beta_{1} - 645 \beta_{2} ) q^{51} + ( 700 - 700 \beta_{1} - 756 \beta_{2} - 756 \beta_{3} ) q^{52} + ( -1521 + 1521 \beta_{1} - 1818 \beta_{2} - 1818 \beta_{3} ) q^{53} + ( -16273 \beta_{1} - 1837 \beta_{2} ) q^{54} + ( -12538 + 2557 \beta_{3} ) q^{55} + ( -19320 + 9744 \beta_{1} + 672 \beta_{2} - 1344 \beta_{3} ) q^{56} + ( -9529 + 928 \beta_{3} ) q^{57} + ( 29426 \beta_{1} + 4226 \beta_{2} ) q^{58} + ( 32904 - 32904 \beta_{1} + 531 \beta_{2} + 531 \beta_{3} ) q^{59} + ( -7042 + 7042 \beta_{1} + 1246 \beta_{2} + 1246 \beta_{3} ) q^{60} + ( 21243 \beta_{1} + 4154 \beta_{2} ) q^{61} + ( -24681 - 1059 \beta_{3} ) q^{62} + ( 6496 - 15372 \beta_{1} + 1890 \beta_{2} - 2212 \beta_{3} ) q^{63} + ( 17728 + 3072 \beta_{3} ) q^{64} + ( 1582 \beta_{1} - 4088 \beta_{2} ) q^{65} + ( 12947 - 12947 \beta_{1} - 2003 \beta_{2} - 2003 \beta_{3} ) q^{66} + ( -21156 + 21156 \beta_{1} + 919 \beta_{2} + 919 \beta_{3} ) q^{67} + ( -2730 \beta_{1} + 1554 \beta_{2} ) q^{68} + ( 2577 - 282 \beta_{3} ) q^{69} + ( -19719 - 17087 \beta_{1} - 7763 \beta_{2} - 693 \beta_{3} ) q^{70} + ( -1104 + 2184 \beta_{3} ) q^{71} + ( 16224 \beta_{1} - 4944 \beta_{2} ) q^{72} + ( 25253 - 25253 \beta_{1} - 7372 \beta_{2} - 7372 \beta_{3} ) q^{73} + ( 23685 - 23685 \beta_{1} + 4755 \beta_{2} + 4755 \beta_{3} ) q^{74} + ( 17804 \beta_{1} + 2456 \beta_{2} ) q^{75} + ( 19418 - 1302 \beta_{3} ) q^{76} + ( 8883 + 14903 \beta_{1} + 4130 \beta_{2} - 126 \beta_{3} ) q^{77} + ( 13762 - 1162 \beta_{3} ) q^{78} + ( -4502 \beta_{1} + 5193 \beta_{2} ) q^{79} + ( -34408 + 34408 \beta_{1} + 11080 \beta_{2} + 11080 \beta_{3} ) q^{80} + ( -25589 + 25589 \beta_{1} - 4984 \beta_{2} - 4984 \beta_{3} ) q^{81} + ( -33866 \beta_{1} - 2618 \beta_{2} ) q^{82} + ( -52164 + 4536 \beta_{3} ) q^{83} + ( 12740 - 3234 \beta_{1} - 294 \beta_{2} - 2156 \beta_{3} ) q^{84} + ( -26553 - 9222 \beta_{3} ) q^{85} + ( -43388 \beta_{1} + 4996 \beta_{2} ) q^{86} + ( -40004 + 40004 \beta_{1} + 6326 \beta_{2} + 6326 \beta_{3} ) q^{87} + ( 10248 - 10248 \beta_{1} - 3984 \beta_{2} - 3984 \beta_{3} ) q^{88} + ( 13333 \beta_{1} - 9356 \beta_{2} ) q^{89} + ( 65326 - 2398 \beta_{3} ) q^{90} + ( 28224 + 11368 \beta_{1} + 4900 \beta_{2} + 10094 \beta_{3} ) q^{91} + ( -5142 + 426 \beta_{3} ) q^{92} + ( -19359 \beta_{1} - 1086 \beta_{2} ) q^{93} + ( -49017 + 49017 \beta_{1} - 9843 \beta_{2} - 9843 \beta_{3} ) q^{94} + ( 99070 - 99070 \beta_{1} - 3463 \beta_{2} - 3463 \beta_{3} ) q^{95} + ( -27440 \beta_{1} - 5264 \beta_{2} ) q^{96} + ( 104566 + 196 \beta_{3} ) q^{97} + ( -24304 + 97951 \beta_{1} + 6223 \beta_{2} + 10192 \beta_{3} ) q^{98} + ( -33472 + 2674 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 8q^{3} - 12q^{4} + 38q^{5} - 164q^{6} - 168q^{7} + 192q^{8} + 380q^{9} + O(q^{10})$$ $$4q - 2q^{2} + 8q^{3} - 12q^{4} + 38q^{5} - 164q^{6} - 168q^{7} + 192q^{8} + 380q^{9} + 778q^{10} - 424q^{11} + 196q^{12} - 1848q^{13} - 2674q^{14} + 1784q^{15} + 2064q^{16} + 2346q^{17} - 212q^{18} + 360q^{19} - 3416q^{20} - 1526q^{21} + 4252q^{22} + 12q^{23} - 1392q^{24} - 1872q^{25} - 1148q^{26} + 5744q^{27} + 2548q^{28} - 14104q^{29} - 5258q^{30} - 3548q^{31} + 8096q^{32} + 3398q^{33} + 14844q^{34} + 27496q^{35} - 2192q^{36} - 11090q^{37} - 20138q^{38} - 1624q^{39} - 15936q^{40} + 7000q^{41} - 3472q^{42} - 25360q^{43} - 5948q^{44} - 1300q^{45} + 5118q^{46} + 22956q^{47} + 22432q^{48} + 4900q^{49} + 59984q^{50} + 384q^{51} + 1400q^{52} - 3042q^{53} - 32546q^{54} - 50152q^{55} - 57792q^{56} - 38116q^{57} + 58852q^{58} + 65808q^{59} - 14084q^{60} + 42486q^{61} - 98724q^{62} - 4760q^{63} + 70912q^{64} + 3164q^{65} + 25894q^{66} - 42312q^{67} - 5460q^{68} + 10308q^{69} - 113050q^{70} - 4416q^{71} + 32448q^{72} + 50506q^{73} + 47370q^{74} + 35608q^{75} + 77672q^{76} + 65338q^{77} + 55048q^{78} - 9004q^{79} - 68816q^{80} - 51178q^{81} - 67732q^{82} - 208656q^{83} + 44492q^{84} - 106212q^{85} - 86776q^{86} - 80008q^{87} + 20496q^{88} + 26666q^{89} + 261304q^{90} + 135632q^{91} - 20568q^{92} - 38718q^{93} - 98034q^{94} + 198140q^{95} - 54880q^{96} + 418264q^{97} + 98686q^{98} - 133888q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 10 x^{2} + 9 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 10 \nu^{2} - 10 \nu + 81$$$$)/90$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 10 \nu^{2} + 190 \nu - 81$$$$)/90$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 14$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 19 \beta_{1} - 19$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - 14$$

## Character Values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$-\beta_{1}$$

## 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
 1.77069 − 3.06693i −1.27069 + 2.20090i 1.77069 + 3.06693i −1.27069 − 2.20090i
−3.54138 + 6.13385i 5.04138 + 8.73193i −9.08276 15.7318i 39.9138 69.1328i −71.4138 43.1587 + 122.247i −97.9863 70.6689 122.402i 282.700 + 489.651i
2.2 2.54138 4.40180i −1.04138 1.80373i 3.08276 + 5.33950i −20.9138 + 36.2238i −10.5862 −127.159 25.2522i 193.986 119.331 206.687i 106.300 + 184.117i
4.1 −3.54138 6.13385i 5.04138 8.73193i −9.08276 + 15.7318i 39.9138 + 69.1328i −71.4138 43.1587 122.247i −97.9863 70.6689 + 122.402i 282.700 489.651i
4.2 2.54138 + 4.40180i −1.04138 + 1.80373i 3.08276 5.33950i −20.9138 36.2238i −10.5862 −127.159 + 25.2522i 193.986 119.331 + 206.687i 106.300 184.117i
 $$n$$: 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 $$S_{6}^{\mathrm{new}}(7, [\chi])$$.