# Properties

 Label 128.4.e.a Level 128 Weight 4 Character orbit 128.e Analytic conductor 7.552 Analytic rank 0 Dimension 10 CM No Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$128 = 2^{7}$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 128.e (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$7.55224448073$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{20}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{5} q^{3}$$ $$-\beta_{3} q^{5}$$ $$+ ( 3 \beta_{1} - \beta_{4} ) q^{7}$$ $$+ ( 5 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{9} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{5} q^{3}$$ $$-\beta_{3} q^{5}$$ $$+ ( 3 \beta_{1} - \beta_{4} ) q^{7}$$ $$+ ( 5 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{9} ) q^{9}$$ $$+ ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{7} - \beta_{9} ) q^{11}$$ $$+ ( \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{13}$$ $$+ ( 13 + 2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{15}$$ $$+ ( -4 + 5 \beta_{2} - 2 \beta_{3} + 5 \beta_{5} + 2 \beta_{6} + \beta_{7} - 3 \beta_{8} ) q^{17}$$ $$+ ( -1 - \beta_{1} - 2 \beta_{4} - \beta_{5} - 6 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{19}$$ $$+ ( -8 + 8 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{21}$$ $$+ ( -23 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} - \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - 2 \beta_{9} ) q^{23}$$ $$+ ( -\beta_{1} + 10 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} ) q^{25}$$ $$+ ( 19 - 19 \beta_{1} + 6 \beta_{3} - 2 \beta_{4} + \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{27}$$ $$+ ( 16 + 16 \beta_{1} - 2 \beta_{4} + 20 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{29}$$ $$+ ( -38 - 4 \beta_{2} - 6 \beta_{3} - 4 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} ) q^{31}$$ $$+ ( 4 - 15 \beta_{2} - 4 \beta_{3} - 15 \beta_{5} + 4 \beta_{6} + \beta_{7} + \beta_{8} ) q^{33}$$ $$+ ( 46 + 46 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 8 \beta_{6} + 2 \beta_{8} ) q^{35}$$ $$+ ( 8 - 8 \beta_{1} - 30 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{37}$$ $$+ ( 71 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} ) q^{39}$$ $$+ ( -8 \beta_{1} - 24 \beta_{2} - 2 \beta_{3} + 24 \beta_{5} - 2 \beta_{6} - 4 \beta_{9} ) q^{41}$$ $$+ ( -84 + 84 \beta_{1} - 3 \beta_{2} - 8 \beta_{3} - 4 \beta_{7} + 4 \beta_{9} ) q^{43}$$ $$+ ( -8 - 8 \beta_{1} - \beta_{4} - 46 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + \beta_{8} - 5 \beta_{9} ) q^{45}$$ $$+ ( 98 + 6 \beta_{3} - 6 \beta_{6} - 10 \beta_{7} - 4 \beta_{8} ) q^{47}$$ $$+ ( 7 + 26 \beta_{2} + 12 \beta_{3} + 26 \beta_{5} - 12 \beta_{6} - 6 \beta_{7} + 10 \beta_{8} ) q^{49}$$ $$+ ( -153 - 153 \beta_{1} + 12 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} - 5 \beta_{7} - 12 \beta_{8} - 5 \beta_{9} ) q^{51}$$ $$+ ( 32 - 32 \beta_{1} + 42 \beta_{2} + 7 \beta_{3} + 13 \beta_{4} + 7 \beta_{7} + 13 \beta_{8} - 7 \beta_{9} ) q^{53}$$ $$+ ( -157 \beta_{1} - 28 \beta_{2} + 12 \beta_{3} + 3 \beta_{4} + 28 \beta_{5} + 12 \beta_{6} + 12 \beta_{9} ) q^{55}$$ $$+ ( -4 \beta_{1} + 25 \beta_{2} + 16 \beta_{3} - 7 \beta_{4} - 25 \beta_{5} + 16 \beta_{6} - 9 \beta_{9} ) q^{57}$$ $$+ ( 174 - 174 \beta_{1} + \beta_{2} + 14 \beta_{4} - 4 \beta_{7} + 14 \beta_{8} + 4 \beta_{9} ) q^{59}$$ $$+ ( -96 - 96 \beta_{1} + 9 \beta_{4} + 30 \beta_{5} - 3 \beta_{6} - 11 \beta_{7} - 9 \beta_{8} - 11 \beta_{9} ) q^{61}$$ $$+ ( -271 + 32 \beta_{2} + 6 \beta_{3} + 32 \beta_{5} - 6 \beta_{6} - 10 \beta_{7} + \beta_{8} ) q^{63}$$ $$+ ( -22 - 40 \beta_{2} + 26 \beta_{3} - 40 \beta_{5} - 26 \beta_{6} - 4 \beta_{7} ) q^{65}$$ $$+ ( 189 + 189 \beta_{1} + 12 \beta_{4} + 11 \beta_{5} + 2 \beta_{6} + \beta_{7} - 12 \beta_{8} + \beta_{9} ) q^{67}$$ $$+ ( -48 + 48 \beta_{1} - 50 \beta_{2} + 6 \beta_{3} - \beta_{4} + 5 \beta_{7} - \beta_{8} - 5 \beta_{9} ) q^{69}$$ $$+ ( 337 \beta_{1} - 28 \beta_{2} - 6 \beta_{3} + 7 \beta_{4} + 28 \beta_{5} - 6 \beta_{6} - 2 \beta_{9} ) q^{71}$$ $$+ ( -50 \beta_{1} - 31 \beta_{2} + 12 \beta_{3} + 17 \beta_{4} + 31 \beta_{5} + 12 \beta_{6} - \beta_{9} ) q^{73}$$ $$+ ( -288 + 288 \beta_{1} - 9 \beta_{2} + 28 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{75}$$ $$+ ( 32 + 32 \beta_{1} - 15 \beta_{4} - 18 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} + 15 \beta_{8} + 5 \beta_{9} ) q^{77}$$ $$+ ( 428 - 4 \beta_{2} - 32 \beta_{3} - 4 \beta_{5} + 32 \beta_{6} + 8 \beta_{7} - 4 \beta_{8} ) q^{79}$$ $$+ ( 35 + 9 \beta_{2} - 24 \beta_{3} + 9 \beta_{5} + 24 \beta_{6} + 9 \beta_{7} + 9 \beta_{8} ) q^{81}$$ $$+ ( -260 - 260 \beta_{1} - 22 \beta_{4} + 11 \beta_{5} + 28 \beta_{6} + 2 \beta_{7} + 22 \beta_{8} + 2 \beta_{9} ) q^{83}$$ $$+ ( 8 - 8 \beta_{1} - 14 \beta_{2} + 10 \beta_{3} - 7 \beta_{4} - 17 \beta_{7} - 7 \beta_{8} + 17 \beta_{9} ) q^{85}$$ $$+ ( -575 \beta_{1} + 76 \beta_{2} + 10 \beta_{3} + 7 \beta_{4} - 76 \beta_{5} + 10 \beta_{6} - 18 \beta_{9} ) q^{87}$$ $$+ ( 18 \beta_{1} + 15 \beta_{2} - 52 \beta_{3} - \beta_{4} - 15 \beta_{5} - 52 \beta_{6} + \beta_{9} ) q^{89}$$ $$+ ( 320 - 320 \beta_{1} - 2 \beta_{2} - 36 \beta_{3} - 34 \beta_{4} - 2 \beta_{7} - 34 \beta_{8} + 2 \beta_{9} ) q^{91}$$ $$+ ( 216 + 216 \beta_{1} - 10 \beta_{4} + 20 \beta_{5} - 8 \beta_{6} + 14 \beta_{7} + 10 \beta_{8} + 14 \beta_{9} ) q^{93}$$ $$+ ( -641 - 104 \beta_{2} + 24 \beta_{3} - 104 \beta_{5} - 24 \beta_{6} + 8 \beta_{7} - 3 \beta_{8} ) q^{95}$$ $$+ ( -36 + 25 \beta_{2} - 58 \beta_{3} + 25 \beta_{5} + 58 \beta_{6} - 3 \beta_{7} - 15 \beta_{8} ) q^{97}$$ $$+ ( 518 + 518 \beta_{1} - 22 \beta_{4} - 21 \beta_{5} - 64 \beta_{6} - 4 \beta_{7} + 22 \beta_{8} - 4 \beta_{9} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 2q^{5}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$10q$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 2q^{5}$$ $$\mathstrut +\mathstrut 18q^{11}$$ $$\mathstrut +\mathstrut 2q^{13}$$ $$\mathstrut +\mathstrut 124q^{15}$$ $$\mathstrut -\mathstrut 4q^{17}$$ $$\mathstrut -\mathstrut 26q^{19}$$ $$\mathstrut -\mathstrut 52q^{21}$$ $$\mathstrut +\mathstrut 184q^{27}$$ $$\mathstrut +\mathstrut 202q^{29}$$ $$\mathstrut -\mathstrut 368q^{31}$$ $$\mathstrut -\mathstrut 4q^{33}$$ $$\mathstrut +\mathstrut 476q^{35}$$ $$\mathstrut +\mathstrut 10q^{37}$$ $$\mathstrut -\mathstrut 838q^{43}$$ $$\mathstrut -\mathstrut 194q^{45}$$ $$\mathstrut +\mathstrut 944q^{47}$$ $$\mathstrut +\mathstrut 94q^{49}$$ $$\mathstrut -\mathstrut 1500q^{51}$$ $$\mathstrut +\mathstrut 378q^{53}$$ $$\mathstrut +\mathstrut 1706q^{59}$$ $$\mathstrut -\mathstrut 910q^{61}$$ $$\mathstrut -\mathstrut 2628q^{63}$$ $$\mathstrut -\mathstrut 492q^{65}$$ $$\mathstrut +\mathstrut 1942q^{67}$$ $$\mathstrut -\mathstrut 580q^{69}$$ $$\mathstrut -\mathstrut 2954q^{75}$$ $$\mathstrut +\mathstrut 268q^{77}$$ $$\mathstrut +\mathstrut 4416q^{79}$$ $$\mathstrut +\mathstrut 482q^{81}$$ $$\mathstrut -\mathstrut 2562q^{83}$$ $$\mathstrut +\mathstrut 12q^{85}$$ $$\mathstrut +\mathstrut 3332q^{91}$$ $$\mathstrut +\mathstrut 2192q^{93}$$ $$\mathstrut -\mathstrut 6900q^{95}$$ $$\mathstrut -\mathstrut 4q^{97}$$ $$\mathstrut +\mathstrut 4958q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10}\mathstrut -\mathstrut$$ $$2$$ $$x^{9}\mathstrut -\mathstrut$$ $$x^{8}\mathstrut +\mathstrut$$ $$6$$ $$x^{7}\mathstrut +\mathstrut$$ $$14$$ $$x^{6}\mathstrut -\mathstrut$$ $$80$$ $$x^{5}\mathstrut +\mathstrut$$ $$56$$ $$x^{4}\mathstrut +\mathstrut$$ $$96$$ $$x^{3}\mathstrut -\mathstrut$$ $$64$$ $$x^{2}\mathstrut -\mathstrut$$ $$512$$ $$x\mathstrut +\mathstrut$$ $$1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{9} + 14 \nu^{8} - 7 \nu^{7} - 82 \nu^{6} + 170 \nu^{5} + 120 \nu^{4} - 536 \nu^{3} - 384 \nu^{2} + 2752 \nu - 3072$$$$)/1280$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{9} + 2 \nu^{8} - 101 \nu^{7} + 114 \nu^{6} - 210 \nu^{5} + 120 \nu^{4} - 8 \nu^{3} + 3008 \nu^{2} - 3264 \nu + 7424$$$$)/1280$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{9} - 122 \nu^{8} + 341 \nu^{7} - 634 \nu^{6} + 130 \nu^{5} + 120 \nu^{4} + 3848 \nu^{3} - 9728 \nu^{2} + 16064 \nu - 2304$$$$)/1280$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{9} - 18 \nu^{8} + 49 \nu^{7} + 174 \nu^{6} - 870 \nu^{5} + 1240 \nu^{4} + 2152 \nu^{3} - 5632 \nu^{2} - 15424 \nu + 31744$$$$)/1280$$ $$\beta_{5}$$ $$=$$ $$($$$$-17 \nu^{9} + 38 \nu^{8} - 39 \nu^{7} + 86 \nu^{6} + 90 \nu^{5} + 520 \nu^{4} - 792 \nu^{3} + 2752 \nu^{2} - 1856 \nu - 4864$$$$)/1280$$ $$\beta_{6}$$ $$=$$ $$($$$$-37 \nu^{9} + 198 \nu^{8} - 419 \nu^{7} + 166 \nu^{6} + 50 \nu^{5} + 1560 \nu^{4} - 7992 \nu^{3} + 11392 \nu^{2} - 6976 \nu + 256$$$$)/1280$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{9} + 2 \nu^{8} + \nu^{7} - 6 \nu^{6} - 14 \nu^{5} + 80 \nu^{4} - 56 \nu^{3} - 96 \nu^{2} + 320 \nu + 416$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{9} + 10 \nu^{8} + 3 \nu^{7} - 6 \nu^{6} - 90 \nu^{5} + 184 \nu^{4} - 56 \nu^{3} - 128 \nu^{2} - 896 \nu + 1728$$$$)/64$$ $$\beta_{9}$$ $$=$$ $$($$$$39 \nu^{9} - 66 \nu^{8} - 47 \nu^{7} + 414 \nu^{6} - 294 \nu^{5} - 1288 \nu^{4} + 1704 \nu^{3} + 2944 \nu^{2} - 11072 \nu + 11264$$$$)/256$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$9$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$2$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$15$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$33$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$9$$$$)/16$$ $$\nu^{4}$$ $$=$$ $$($$$$3$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$23$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$43$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$117$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$6$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$43$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$13$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$67$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$301$$$$)/16$$ $$\nu^{6}$$ $$=$$ $$($$$$-$$$$3$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$28$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$27$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$17$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$49$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$43$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$99$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$117$$$$)/16$$ $$\nu^{7}$$ $$=$$ $$($$$$24$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$22$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$15$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$33$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$133$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$45$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$61$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$233$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$155$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$859$$$$)/16$$ $$\nu^{8}$$ $$=$$ $$($$$$11$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$119$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$197$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$17$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$122$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$191$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$91$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1043$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$987$$$$)/16$$ $$\nu^{9}$$ $$=$$ $$($$$$208$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$146$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$121$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$31$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$155$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$75$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$211$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$71$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$3019$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$4187$$$$)/16$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$127$$ $$\chi(n)$$ $$-\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 1.97476 + 0.316760i −1.56339 − 1.24732i −1.62580 + 1.16481i 0.932438 − 1.76934i 1.28199 + 1.53509i 1.97476 − 0.316760i −1.56339 + 1.24732i −1.62580 − 1.16481i 0.932438 + 1.76934i 1.28199 − 1.53509i
0 −5.96513 5.96513i 0 −8.67959 + 8.67959i 0 1.63924i 0 44.1656i 0
33.2 0 −3.27139 3.27139i 0 12.6449 12.6449i 0 13.8754i 0 5.59607i 0
33.3 0 0.756776 + 0.756776i 0 −8.22587 + 8.22587i 0 2.67171i 0 25.8546i 0
33.4 0 1.98356 + 1.98356i 0 0.596848 0.596848i 0 29.0828i 0 19.1310i 0
33.5 0 5.49618 + 5.49618i 0 4.66372 4.66372i 0 24.8965i 0 33.4160i 0
97.1 0 −5.96513 + 5.96513i 0 −8.67959 8.67959i 0 1.63924i 0 44.1656i 0
97.2 0 −3.27139 + 3.27139i 0 12.6449 + 12.6449i 0 13.8754i 0 5.59607i 0
97.3 0 0.756776 0.756776i 0 −8.22587 8.22587i 0 2.67171i 0 25.8546i 0
97.4 0 1.98356 1.98356i 0 0.596848 + 0.596848i 0 29.0828i 0 19.1310i 0
97.5 0 5.49618 5.49618i 0 4.66372 + 4.66372i 0 24.8965i 0 33.4160i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.5 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
16.e Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{10} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(128, [\chi])$$.