# Properties

 Label 7.5.d.a Level 7 Weight 5 Character orbit 7.d Analytic conductor 0.724 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$7$$ Weight: $$k$$ = $$5$$ Character orbit: $$[\chi]$$ = 7.d (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.723589741587$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{22})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -2 + \beta_{1} - 2 \beta_{2} ) q^{2}$$ $$+ ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{3}$$ $$+ ( -4 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} ) q^{4}$$ $$+ ( -5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{5}$$ $$+ ( -24 + 6 \beta_{1} - 48 \beta_{2} + 3 \beta_{3} ) q^{6}$$ $$+ ( 21 + 42 \beta_{2} - 7 \beta_{3} ) q^{7}$$ $$+ ( 76 + 2 \beta_{3} ) q^{8}$$ $$+ ( -12 - 6 \beta_{1} - 12 \beta_{2} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -2 + \beta_{1} - 2 \beta_{2} ) q^{2}$$ $$+ ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{3}$$ $$+ ( -4 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} ) q^{4}$$ $$+ ( -5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{5}$$ $$+ ( -24 + 6 \beta_{1} - 48 \beta_{2} + 3 \beta_{3} ) q^{6}$$ $$+ ( 21 + 42 \beta_{2} - 7 \beta_{3} ) q^{7}$$ $$+ ( 76 + 2 \beta_{3} ) q^{8}$$ $$+ ( -12 - 6 \beta_{1} - 12 \beta_{2} ) q^{9}$$ $$+ ( -68 - \beta_{1} - 34 \beta_{2} + \beta_{3} ) q^{10}$$ $$+ ( 17 \beta_{1} + 29 \beta_{2} + 17 \beta_{3} ) q^{11}$$ $$+ ( -98 - 14 \beta_{1} + 98 \beta_{2} - 28 \beta_{3} ) q^{12}$$ $$+ ( -70 - 28 \beta_{1} - 140 \beta_{2} - 14 \beta_{3} ) q^{13}$$ $$+ ( 196 + 7 \beta_{1} + 112 \beta_{2} + 42 \beta_{3} ) q^{14}$$ $$+ ( 117 - 9 \beta_{3} ) q^{15}$$ $$+ ( -36 + 16 \beta_{1} - 36 \beta_{2} ) q^{16}$$ $$+ ( -82 + 34 \beta_{1} - 41 \beta_{2} - 34 \beta_{3} ) q^{17}$$ $$-108 \beta_{2} q^{18}$$ $$+ ( 107 + 37 \beta_{1} - 107 \beta_{2} + 74 \beta_{3} ) q^{19}$$ $$+ ( 126 + 252 \beta_{2} ) q^{20}$$ $$+ ( -308 - 56 \beta_{1} - 91 \beta_{2} - 70 \beta_{3} ) q^{21}$$ $$+ ( -316 - 5 \beta_{3} ) q^{22}$$ $$+ ( 145 - 41 \beta_{1} + 145 \beta_{2} ) q^{23}$$ $$+ ( 240 - 78 \beta_{1} + 120 \beta_{2} + 78 \beta_{3} ) q^{24}$$ $$+ ( -60 \beta_{1} + 286 \beta_{2} - 60 \beta_{3} ) q^{25}$$ $$+ ( 168 - 42 \beta_{1} - 168 \beta_{2} - 84 \beta_{3} ) q^{26}$$ $$+ ( 39 + 174 \beta_{1} + 78 \beta_{2} + 87 \beta_{3} ) q^{27}$$ $$+ ( -420 + 154 \beta_{1} - 826 \beta_{2} - 14 \beta_{3} ) q^{28}$$ $$+ ( -544 + 70 \beta_{3} ) q^{29}$$ $$+ ( -36 + 99 \beta_{1} - 36 \beta_{2} ) q^{30}$$ $$+ ( 1206 - 29 \beta_{1} + 603 \beta_{2} + 29 \beta_{3} ) q^{31}$$ $$+ ( -36 \beta_{1} - 792 \beta_{2} - 36 \beta_{3} ) q^{32}$$ $$+ ( 345 - 12 \beta_{1} - 345 \beta_{2} - 24 \beta_{3} ) q^{33}$$ $$+ ( 830 - 218 \beta_{1} + 1660 \beta_{2} - 109 \beta_{3} ) q^{34}$$ $$+ ( -623 - 91 \beta_{1} - 7 \beta_{2} + 70 \beta_{3} ) q^{35}$$ $$+ ( -408 - 12 \beta_{3} ) q^{36}$$ $$+ ( -135 - 104 \beta_{1} - 135 \beta_{2} ) q^{37}$$ $$+ ( -2056 + 181 \beta_{1} - 1028 \beta_{2} - 181 \beta_{3} ) q^{38}$$ $$+ ( 168 \beta_{1} + 714 \beta_{2} + 168 \beta_{3} ) q^{39}$$ $$+ ( -292 + 142 \beta_{1} + 292 \beta_{2} + 284 \beta_{3} ) q^{40}$$ $$+ ( -798 - 84 \beta_{1} - 1596 \beta_{2} - 42 \beta_{3} ) q^{41}$$ $$+ ( 1974 - 336 \beta_{1} + 924 \beta_{2} + 21 \beta_{3} ) q^{42}$$ $$+ ( 618 - 350 \beta_{3} ) q^{43}$$ $$+ ( 1206 - 54 \beta_{1} + 1206 \beta_{2} ) q^{44}$$ $$+ ( 648 + 54 \beta_{1} + 324 \beta_{2} - 54 \beta_{3} ) q^{45}$$ $$+ ( 227 \beta_{1} - 1192 \beta_{2} + 227 \beta_{3} ) q^{46}$$ $$+ ( -257 - 187 \beta_{1} + 257 \beta_{2} - 374 \beta_{3} ) q^{47}$$ $$+ ( -388 + 104 \beta_{1} - 776 \beta_{2} + 52 \beta_{3} ) q^{48}$$ $$+ ( -245 + 588 \beta_{1} + 294 \beta_{3} ) q^{49}$$ $$+ ( 1892 + 406 \beta_{3} ) q^{50}$$ $$+ ( -2367 + 225 \beta_{1} - 2367 \beta_{2} ) q^{51}$$ $$+ ( -1064 - 140 \beta_{1} - 532 \beta_{2} + 140 \beta_{3} ) q^{52}$$ $$+ ( -340 \beta_{1} + 2255 \beta_{2} - 340 \beta_{3} ) q^{53}$$ $$+ ( -1836 - 135 \beta_{1} + 1836 \beta_{2} - 270 \beta_{3} ) q^{54}$$ $$+ ( -893 - 286 \beta_{1} - 1786 \beta_{2} - 143 \beta_{3} ) q^{55}$$ $$+ ( 1288 - 84 \beta_{1} + 3192 \beta_{2} - 574 \beta_{3} ) q^{56}$$ $$+ ( 2763 + 432 \beta_{3} ) q^{57}$$ $$+ ( -452 - 404 \beta_{1} - 452 \beta_{2} ) q^{58}$$ $$+ ( 842 - 449 \beta_{1} + 421 \beta_{2} + 449 \beta_{3} ) q^{59}$$ $$+ ( -378 \beta_{1} + 378 \beta_{2} - 378 \beta_{3} ) q^{60}$$ $$+ ( -47 + 240 \beta_{1} + 47 \beta_{2} + 480 \beta_{3} ) q^{61}$$ $$+ ( -1844 + 1322 \beta_{1} - 3688 \beta_{2} + 661 \beta_{3} ) q^{62}$$ $$+ ( -672 - 210 \beta_{1} - 1176 \beta_{2} - 252 \beta_{3} ) q^{63}$$ $$+ ( -1368 - 976 \beta_{3} ) q^{64}$$ $$+ ( 2898 + 630 \beta_{1} + 2898 \beta_{2} ) q^{65}$$ $$+ ( -852 + 321 \beta_{1} - 426 \beta_{2} - 321 \beta_{3} ) q^{66}$$ $$+ ( 45 \beta_{1} + 659 \beta_{2} + 45 \beta_{3} ) q^{67}$$ $$+ ( 3402 + 504 \beta_{1} - 3402 \beta_{2} + 1008 \beta_{3} ) q^{68}$$ $$+ ( 1047 - 372 \beta_{1} + 2094 \beta_{2} - 186 \beta_{3} ) q^{69}$$ $$+ ( -308 - 301 \beta_{1} - 2296 \beta_{2} + 175 \beta_{3} ) q^{70}$$ $$+ ( -2602 + 238 \beta_{3} ) q^{71}$$ $$+ ( -648 - 432 \beta_{1} - 648 \beta_{2} ) q^{72}$$ $$+ ( 1738 + 272 \beta_{1} + 869 \beta_{2} - 272 \beta_{3} ) q^{73}$$ $$+ ( 73 \beta_{1} - 2018 \beta_{2} + 73 \beta_{3} ) q^{74}$$ $$+ ( -1606 - 346 \beta_{1} + 1606 \beta_{2} - 692 \beta_{3} ) q^{75}$$ $$+ ( 4326 - 1596 \beta_{1} + 8652 \beta_{2} - 798 \beta_{3} ) q^{76}$$ $$+ ( -1218 - 154 \beta_{1} + 2009 \beta_{2} + 560 \beta_{3} ) q^{77}$$ $$+ ( -2268 + 378 \beta_{3} ) q^{78}$$ $$+ ( -4055 + 351 \beta_{1} - 4055 \beta_{2} ) q^{79}$$ $$+ ( -1048 - 8 \beta_{1} - 524 \beta_{2} + 8 \beta_{3} ) q^{80}$$ $$+ ( 630 \beta_{1} - 4653 \beta_{2} + 630 \beta_{3} ) q^{81}$$ $$+ ( -672 - 714 \beta_{1} + 672 \beta_{2} - 1428 \beta_{3} ) q^{82}$$ $$+ ( 1932 - 168 \beta_{1} + 3864 \beta_{2} - 84 \beta_{3} ) q^{83}$$ $$+ ( -4018 + 1568 \beta_{1} - 8330 \beta_{2} + 1372 \beta_{3} ) q^{84}$$ $$+ ( -3873 + 264 \beta_{3} ) q^{85}$$ $$+ ( 6464 - 82 \beta_{1} + 6464 \beta_{2} ) q^{86}$$ $$+ ( 1992 + 474 \beta_{1} + 996 \beta_{2} - 474 \beta_{3} ) q^{87}$$ $$+ ( 1234 \beta_{1} + 1456 \beta_{2} + 1234 \beta_{3} ) q^{88}$$ $$+ ( 5665 - 376 \beta_{1} - 5665 \beta_{2} - 752 \beta_{3} ) q^{89}$$ $$+ ( 540 + 432 \beta_{1} + 1080 \beta_{2} + 216 \beta_{3} ) q^{90}$$ $$+ ( 2254 - 980 \beta_{1} - 4312 \beta_{2} - 1372 \beta_{3} ) q^{91}$$ $$+ ( -5058 - 990 \beta_{3} ) q^{92}$$ $$+ ( 3723 - 1896 \beta_{1} + 3723 \beta_{2} ) q^{93}$$ $$+ ( 9256 - 631 \beta_{1} + 4628 \beta_{2} + 631 \beta_{3} ) q^{94}$$ $$+ ( 87 \beta_{1} - 3279 \beta_{2} + 87 \beta_{3} ) q^{95}$$ $$+ ( 756 \beta_{1} + 1512 \beta_{3} ) q^{96}$$ $$+ ( -686 + 2548 \beta_{1} - 1372 \beta_{2} + 1274 \beta_{3} ) q^{97}$$ $$+ ( -5978 - 833 \beta_{1} + 6958 \beta_{2} - 1176 \beta_{3} ) q^{98}$$ $$+ ( 2592 - 378 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 4q^{2}$$ $$\mathstrut +\mathstrut 6q^{3}$$ $$\mathstrut -\mathstrut 20q^{4}$$ $$\mathstrut -\mathstrut 30q^{5}$$ $$\mathstrut +\mathstrut 304q^{8}$$ $$\mathstrut -\mathstrut 24q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 4q^{2}$$ $$\mathstrut +\mathstrut 6q^{3}$$ $$\mathstrut -\mathstrut 20q^{4}$$ $$\mathstrut -\mathstrut 30q^{5}$$ $$\mathstrut +\mathstrut 304q^{8}$$ $$\mathstrut -\mathstrut 24q^{9}$$ $$\mathstrut -\mathstrut 204q^{10}$$ $$\mathstrut -\mathstrut 58q^{11}$$ $$\mathstrut -\mathstrut 588q^{12}$$ $$\mathstrut +\mathstrut 560q^{14}$$ $$\mathstrut +\mathstrut 468q^{15}$$ $$\mathstrut -\mathstrut 72q^{16}$$ $$\mathstrut -\mathstrut 246q^{17}$$ $$\mathstrut +\mathstrut 216q^{18}$$ $$\mathstrut +\mathstrut 642q^{19}$$ $$\mathstrut -\mathstrut 1050q^{21}$$ $$\mathstrut -\mathstrut 1264q^{22}$$ $$\mathstrut +\mathstrut 290q^{23}$$ $$\mathstrut +\mathstrut 720q^{24}$$ $$\mathstrut -\mathstrut 572q^{25}$$ $$\mathstrut +\mathstrut 1008q^{26}$$ $$\mathstrut -\mathstrut 28q^{28}$$ $$\mathstrut -\mathstrut 2176q^{29}$$ $$\mathstrut -\mathstrut 72q^{30}$$ $$\mathstrut +\mathstrut 3618q^{31}$$ $$\mathstrut +\mathstrut 1584q^{32}$$ $$\mathstrut +\mathstrut 2070q^{33}$$ $$\mathstrut -\mathstrut 2478q^{35}$$ $$\mathstrut -\mathstrut 1632q^{36}$$ $$\mathstrut -\mathstrut 270q^{37}$$ $$\mathstrut -\mathstrut 6168q^{38}$$ $$\mathstrut -\mathstrut 1428q^{39}$$ $$\mathstrut -\mathstrut 1752q^{40}$$ $$\mathstrut +\mathstrut 6048q^{42}$$ $$\mathstrut +\mathstrut 2472q^{43}$$ $$\mathstrut +\mathstrut 2412q^{44}$$ $$\mathstrut +\mathstrut 1944q^{45}$$ $$\mathstrut +\mathstrut 2384q^{46}$$ $$\mathstrut -\mathstrut 1542q^{47}$$ $$\mathstrut -\mathstrut 980q^{49}$$ $$\mathstrut +\mathstrut 7568q^{50}$$ $$\mathstrut -\mathstrut 4734q^{51}$$ $$\mathstrut -\mathstrut 3192q^{52}$$ $$\mathstrut -\mathstrut 4510q^{53}$$ $$\mathstrut -\mathstrut 11016q^{54}$$ $$\mathstrut -\mathstrut 1232q^{56}$$ $$\mathstrut +\mathstrut 11052q^{57}$$ $$\mathstrut -\mathstrut 904q^{58}$$ $$\mathstrut +\mathstrut 2526q^{59}$$ $$\mathstrut -\mathstrut 756q^{60}$$ $$\mathstrut -\mathstrut 282q^{61}$$ $$\mathstrut -\mathstrut 336q^{63}$$ $$\mathstrut -\mathstrut 5472q^{64}$$ $$\mathstrut +\mathstrut 5796q^{65}$$ $$\mathstrut -\mathstrut 2556q^{66}$$ $$\mathstrut -\mathstrut 1318q^{67}$$ $$\mathstrut +\mathstrut 20412q^{68}$$ $$\mathstrut +\mathstrut 3360q^{70}$$ $$\mathstrut -\mathstrut 10408q^{71}$$ $$\mathstrut -\mathstrut 1296q^{72}$$ $$\mathstrut +\mathstrut 5214q^{73}$$ $$\mathstrut +\mathstrut 4036q^{74}$$ $$\mathstrut -\mathstrut 9636q^{75}$$ $$\mathstrut -\mathstrut 8890q^{77}$$ $$\mathstrut -\mathstrut 9072q^{78}$$ $$\mathstrut -\mathstrut 8110q^{79}$$ $$\mathstrut -\mathstrut 3144q^{80}$$ $$\mathstrut +\mathstrut 9306q^{81}$$ $$\mathstrut -\mathstrut 4032q^{82}$$ $$\mathstrut +\mathstrut 588q^{84}$$ $$\mathstrut -\mathstrut 15492q^{85}$$ $$\mathstrut +\mathstrut 12928q^{86}$$ $$\mathstrut +\mathstrut 5976q^{87}$$ $$\mathstrut -\mathstrut 2912q^{88}$$ $$\mathstrut +\mathstrut 33990q^{89}$$ $$\mathstrut +\mathstrut 17640q^{91}$$ $$\mathstrut -\mathstrut 20232q^{92}$$ $$\mathstrut +\mathstrut 7446q^{93}$$ $$\mathstrut +\mathstrut 27768q^{94}$$ $$\mathstrut +\mathstrut 6558q^{95}$$ $$\mathstrut -\mathstrut 37828q^{98}$$ $$\mathstrut +\mathstrut 10368q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$22$$ $$x^{2}\mathstrut +\mathstrut$$ $$484$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/22$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/22$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$22$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$22$$ $$\beta_{3}$$

## Character Values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −2.34521 − 4.06202i 2.34521 + 4.06202i −2.34521 + 4.06202i 2.34521 − 4.06202i
−3.34521 5.79407i 8.53562 + 4.92804i −14.3808 + 24.9083i 6.57125 3.79391i 65.9413i −32.8329 + 36.3731i 85.3808 8.07125 + 13.9798i −43.9644 25.3828i
3.2 1.34521 + 2.32997i −5.53562 3.19599i 4.38083 7.58782i −21.5712 + 12.4542i 17.1971i 32.8329 + 36.3731i 66.6192 −20.0712 34.7644i −58.0356 33.5069i
5.1 −3.34521 + 5.79407i 8.53562 4.92804i −14.3808 24.9083i 6.57125 + 3.79391i 65.9413i −32.8329 36.3731i 85.3808 8.07125 13.9798i −43.9644 + 25.3828i
5.2 1.34521 2.32997i −5.53562 + 3.19599i 4.38083 + 7.58782i −21.5712 12.4542i 17.1971i 32.8329 36.3731i 66.6192 −20.0712 + 34.7644i −58.0356 + 33.5069i
 $$n$$: 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.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{5}^{\mathrm{new}}(7, [\chi])$$.