# 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.

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

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

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

## 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 $$S_{6}^{\mathrm{new}}(7, [\chi])$$.