# Properties

 Label 7.9.b.b Level 7 Weight 9 Character orbit 7.b Analytic conductor 2.852 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$7$$ Weight: $$k$$ = $$9$$ Character orbit: $$[\chi]$$ = 7.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.85165027043$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}\cdot 3\cdot 7$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 8 + \beta_{3} ) q^{2}$$ $$-\beta_{1} q^{3}$$ $$+ ( -8 + 16 \beta_{3} ) q^{4}$$ $$+ ( 2 \beta_{1} + \beta_{2} ) q^{5}$$ $$+ ( -17 \beta_{1} - \beta_{2} ) q^{6}$$ $$+ ( 357 + 16 \beta_{1} - \beta_{2} + 119 \beta_{3} ) q^{7}$$ $$+ ( 832 - 136 \beta_{3} ) q^{8}$$ $$+ ( -2031 - 426 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 8 + \beta_{3} ) q^{2}$$ $$-\beta_{1} q^{3}$$ $$+ ( -8 + 16 \beta_{3} ) q^{4}$$ $$+ ( 2 \beta_{1} + \beta_{2} ) q^{5}$$ $$+ ( -17 \beta_{1} - \beta_{2} ) q^{6}$$ $$+ ( 357 + 16 \beta_{1} - \beta_{2} + 119 \beta_{3} ) q^{7}$$ $$+ ( 832 - 136 \beta_{3} ) q^{8}$$ $$+ ( -2031 - 426 \beta_{3} ) q^{9}$$ $$+ ( 137 \beta_{1} + \beta_{2} ) q^{10}$$ $$+ ( -5542 - 68 \beta_{3} ) q^{11}$$ $$+ ( -136 \beta_{1} - 16 \beta_{2} ) q^{12}$$ $$+ ( -272 \beta_{1} + 17 \beta_{2} ) q^{13}$$ $$+ ( 24752 + 169 \beta_{1} + 17 \beta_{2} + 1309 \beta_{3} ) q^{14}$$ $$+ ( 18240 + 5610 \beta_{3} ) q^{15}$$ $$+ ( -16320 - 4352 \beta_{3} ) q^{16}$$ $$+ ( -630 \beta_{1} - 18 \beta_{2} ) q^{17}$$ $$+ ( -94632 - 5439 \beta_{3} ) q^{18}$$ $$+ ( 969 \beta_{1} + 102 \beta_{2} ) q^{19}$$ $$+ ( 1920 \beta_{1} - 120 \beta_{2} ) q^{20}$$ $$+ ( 136416 - 1428 \beta_{1} - 119 \beta_{2} + 2058 \beta_{3} ) q^{21}$$ $$+ ( -56848 - 6086 \beta_{3} ) q^{22}$$ $$+ ( 227018 - 5984 \beta_{3} ) q^{23}$$ $$+ ( 392 \beta_{1} + 136 \beta_{2} ) q^{24}$$ $$+ ( -513935 + 21030 \beta_{3} ) q^{25}$$ $$+ ( -2873 \beta_{1} - 289 \beta_{2} ) q^{26}$$ $$+ ( -696 \beta_{1} + 426 \beta_{2} ) q^{27}$$ $$+ ( 347480 + 528 \beta_{1} + 408 \beta_{2} + 4760 \beta_{3} ) q^{28}$$ $$+ ( -368254 - 23324 \beta_{3} ) q^{29}$$ $$+ ( 1178160 + 63120 \beta_{3} ) q^{30}$$ $$+ ( -2582 \beta_{1} - 526 \beta_{2} ) q^{31}$$ $$+ ( -1144320 - 16320 \beta_{3} ) q^{32}$$ $$+ ( 6154 \beta_{1} + 68 \beta_{2} ) q^{33}$$ $$+ ( -12564 \beta_{1} - 612 \beta_{2} ) q^{34}$$ $$+ ( 576240 + 15113 \beta_{1} - 476 \beta_{2} - 122010 \beta_{3} ) q^{35}$$ $$+ ( -1237896 - 29088 \beta_{3} ) q^{36}$$ $$+ ( 1678818 + 29716 \beta_{3} ) q^{37}$$ $$+ ( 26979 \beta_{1} + 867 \beta_{2} ) q^{38}$$ $$+ ( -2319072 - 34986 \beta_{3} ) q^{39}$$ $$+ ( -14792 \beta_{1} + 1784 \beta_{2} ) q^{40}$$ $$+ ( 5456 \beta_{1} - 782 \beta_{2} ) q^{41}$$ $$+ ( 1470000 - 36533 \beta_{1} - 1309 \beta_{2} + 152880 \beta_{3} ) q^{42}$$ $$+ ( 1437018 + 173656 \beta_{3} ) q^{43}$$ $$+ ( -155856 - 88128 \beta_{3} ) q^{44}$$ $$+ ( -55608 \beta_{1} + 951 \beta_{2} ) q^{45}$$ $$+ ( 715088 + 179146 \beta_{3} ) q^{46}$$ $$+ ( 16966 \beta_{1} - 3910 \beta_{2} ) q^{47}$$ $$+ ( 55488 \beta_{1} + 4352 \beta_{2} ) q^{48}$$ $$+ ( -298655 + 21182 \beta_{1} + 5236 \beta_{2} + 169932 \beta_{3} ) q^{49}$$ $$+ ( -241960 - 345695 \beta_{3} ) q^{50}$$ $$+ ( -5431968 - 354024 \beta_{3} ) q^{51}$$ $$+ ( -8976 \beta_{1} - 6936 \beta_{2} ) q^{52}$$ $$+ ( 1687394 - 379832 \beta_{3} ) q^{53}$$ $$+ ( 32046 \beta_{1} - 1122 \beta_{2} ) q^{54}$$ $$+ ( -19312 \beta_{1} - 5066 \beta_{2} ) q^{55}$$ $$+ ( -2680832 + 7736 \beta_{1} - 4232 \beta_{2} + 50456 \beta_{3} ) q^{56}$$ $$+ ( 8433360 + 898110 \beta_{3} ) q^{57}$$ $$+ ( -7237648 - 554846 \beta_{3} ) q^{58}$$ $$+ ( 67847 \beta_{1} + 9826 \beta_{2} ) q^{59}$$ $$+ ( 16369920 + 246960 \beta_{3} ) q^{60}$$ $$+ ( -67474 \beta_{1} + 13243 \beta_{2} ) q^{61}$$ $$+ ( -98072 \beta_{1} - 2056 \beta_{2} ) q^{62}$$ $$+ ( -10052763 - 49962 \beta_{1} - 8619 \beta_{2} - 393771 \beta_{3} ) q^{63}$$ $$+ ( -7979520 - 160768 \beta_{3} ) q^{64}$$ $$+ ( -9796080 + 2074170 \beta_{3} ) q^{65}$$ $$+ ( 111622 \beta_{1} + 6086 \beta_{2} ) q^{66}$$ $$+ ( 17506778 - 580704 \beta_{3} ) q^{67}$$ $$+ ( -115344 \beta_{1} - 7344 \beta_{2} ) q^{68}$$ $$+ ( -173162 \beta_{1} + 5984 \beta_{2} ) q^{69}$$ $$+ ( -17839920 + 207893 \beta_{1} + 15589 \beta_{2} - 399840 \beta_{3} ) q^{70}$$ $$+ ( 12475178 - 1287818 \beta_{3} ) q^{71}$$ $$+ ( 8970432 - 78216 \beta_{3} ) q^{72}$$ $$+ ( -49794 \beta_{1} - 29784 \beta_{2} ) q^{73}$$ $$+ ( 18898288 + 1916546 \beta_{3} ) q^{74}$$ $$+ ( 324665 \beta_{1} - 21030 \beta_{2} ) q^{75}$$ $$+ 299880 \beta_{1} q^{76}$$ $$+ ( -3467422 - 91460 \beta_{1} + 3842 \beta_{2} - 683774 \beta_{3} ) q^{77}$$ $$+ ( -24990000 - 2598960 \beta_{3} ) q^{78}$$ $$+ ( -20541814 - 1054374 \beta_{3} ) q^{79}$$ $$+ ( -559232 \beta_{1} + 14144 \beta_{2} ) q^{80}$$ $$+ ( -18855567 - 1064574 \beta_{3} ) q^{81}$$ $$+ ( 12206 \beta_{1} + 6238 \beta_{2} ) q^{82}$$ $$+ ( 597295 \beta_{1} + 18428 \beta_{2} ) q^{83}$$ $$+ ( 4967424 - 390320 \beta_{1} - 4760 \beta_{2} + 2166192 \beta_{3} ) q^{84}$$ $$+ ( 27116640 + 2953800 \beta_{3} ) q^{85}$$ $$+ ( 43448848 + 2826266 \beta_{3} ) q^{86}$$ $$+ ( 578170 \beta_{1} + 23324 \beta_{2} ) q^{87}$$ $$+ ( -2909312 + 697136 \beta_{3} ) q^{88}$$ $$+ ( -801618 \beta_{1} + 46716 \beta_{2} ) q^{89}$$ $$+ ( -847383 \beta_{1} - 56559 \beta_{2} ) q^{90}$$ $$+ ( 51539376 - 180047 \beta_{1} - 44506 \beta_{2} - 1444422 \beta_{3} ) q^{91}$$ $$+ ( -19433040 + 3680160 \beta_{3} ) q^{92}$$ $$+ ( -22740000 - 3602640 \beta_{3} ) q^{93}$$ $$+ ( -114308 \beta_{1} + 20876 \beta_{2} ) q^{94}$$ $$+ ( -106218720 - 2146590 \beta_{3} ) q^{95}$$ $$+ ( 1291200 \beta_{1} + 16320 \beta_{2} ) q^{96}$$ $$+ ( 203798 \beta_{1} + 13006 \beta_{2} ) q^{97}$$ $$+ ( 28878248 + 899402 \beta_{1} + 15946 \beta_{2} + 1060801 \beta_{3} ) q^{98}$$ $$+ ( 16585914 + 2499000 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut 32q^{2}$$ $$\mathstrut -\mathstrut 32q^{4}$$ $$\mathstrut +\mathstrut 1428q^{7}$$ $$\mathstrut +\mathstrut 3328q^{8}$$ $$\mathstrut -\mathstrut 8124q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut 32q^{2}$$ $$\mathstrut -\mathstrut 32q^{4}$$ $$\mathstrut +\mathstrut 1428q^{7}$$ $$\mathstrut +\mathstrut 3328q^{8}$$ $$\mathstrut -\mathstrut 8124q^{9}$$ $$\mathstrut -\mathstrut 22168q^{11}$$ $$\mathstrut +\mathstrut 99008q^{14}$$ $$\mathstrut +\mathstrut 72960q^{15}$$ $$\mathstrut -\mathstrut 65280q^{16}$$ $$\mathstrut -\mathstrut 378528q^{18}$$ $$\mathstrut +\mathstrut 545664q^{21}$$ $$\mathstrut -\mathstrut 227392q^{22}$$ $$\mathstrut +\mathstrut 908072q^{23}$$ $$\mathstrut -\mathstrut 2055740q^{25}$$ $$\mathstrut +\mathstrut 1389920q^{28}$$ $$\mathstrut -\mathstrut 1473016q^{29}$$ $$\mathstrut +\mathstrut 4712640q^{30}$$ $$\mathstrut -\mathstrut 4577280q^{32}$$ $$\mathstrut +\mathstrut 2304960q^{35}$$ $$\mathstrut -\mathstrut 4951584q^{36}$$ $$\mathstrut +\mathstrut 6715272q^{37}$$ $$\mathstrut -\mathstrut 9276288q^{39}$$ $$\mathstrut +\mathstrut 5880000q^{42}$$ $$\mathstrut +\mathstrut 5748072q^{43}$$ $$\mathstrut -\mathstrut 623424q^{44}$$ $$\mathstrut +\mathstrut 2860352q^{46}$$ $$\mathstrut -\mathstrut 1194620q^{49}$$ $$\mathstrut -\mathstrut 967840q^{50}$$ $$\mathstrut -\mathstrut 21727872q^{51}$$ $$\mathstrut +\mathstrut 6749576q^{53}$$ $$\mathstrut -\mathstrut 10723328q^{56}$$ $$\mathstrut +\mathstrut 33733440q^{57}$$ $$\mathstrut -\mathstrut 28950592q^{58}$$ $$\mathstrut +\mathstrut 65479680q^{60}$$ $$\mathstrut -\mathstrut 40211052q^{63}$$ $$\mathstrut -\mathstrut 31918080q^{64}$$ $$\mathstrut -\mathstrut 39184320q^{65}$$ $$\mathstrut +\mathstrut 70027112q^{67}$$ $$\mathstrut -\mathstrut 71359680q^{70}$$ $$\mathstrut +\mathstrut 49900712q^{71}$$ $$\mathstrut +\mathstrut 35881728q^{72}$$ $$\mathstrut +\mathstrut 75593152q^{74}$$ $$\mathstrut -\mathstrut 13869688q^{77}$$ $$\mathstrut -\mathstrut 99960000q^{78}$$ $$\mathstrut -\mathstrut 82167256q^{79}$$ $$\mathstrut -\mathstrut 75422268q^{81}$$ $$\mathstrut +\mathstrut 19869696q^{84}$$ $$\mathstrut +\mathstrut 108466560q^{85}$$ $$\mathstrut +\mathstrut 173795392q^{86}$$ $$\mathstrut -\mathstrut 11637248q^{88}$$ $$\mathstrut +\mathstrut 206157504q^{91}$$ $$\mathstrut -\mathstrut 77732160q^{92}$$ $$\mathstrut -\mathstrut 90960000q^{93}$$ $$\mathstrut -\mathstrut 424874880q^{95}$$ $$\mathstrut +\mathstrut 115512992q^{98}$$ $$\mathstrut +\mathstrut 66343656q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$1016$$ $$x^{2}\mathstrut +\mathstrut$$ $$51570$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{3} + 3506 \nu$$$$)/201$$ $$\beta_{2}$$ $$=$$ $$($$$$8 \nu^{3} + 15454 \nu$$$$)/201$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{2} + 1016$$$$)/67$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{1}$$$$)/42$$ $$\nu^{2}$$ $$=$$ $$($$$$67$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1016$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$1753$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$7727$$ $$\beta_{1}$$$$)/84$$

## Character Values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 − 31.0228i 31.0228i 7.32010i − 7.32010i
−5.56466 53.0420i −225.035 1090.79i 295.161i −1257.19 + 2045.55i 2676.79 3747.55 6069.88i
6.2 −5.56466 53.0420i −225.035 1090.79i 295.161i −1257.19 2045.55i 2676.79 3747.55 6069.88i
6.3 21.5647 119.877i 209.035 786.953i 2585.11i 1971.19 + 1370.84i −1012.79 −7809.55 16970.4i
6.4 21.5647 119.877i 209.035 786.953i 2585.11i 1971.19 1370.84i −1012.79 −7809.55 16970.4i
 $$n$$: 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.b Odd 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{2}$$ $$\mathstrut -\mathstrut 16 T_{2}$$ $$\mathstrut -\mathstrut 120$$ acting on $$S_{9}^{\mathrm{new}}(7, [\chi])$$.