# Properties

 Label 354.6.a.c Level 354 Weight 6 Character orbit 354.a Self dual Yes Analytic conductor 56.776 Analytic rank 1 Dimension 5 CM No Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$6$$ Character orbit: $$[\chi]$$ = 354.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$56.7758722138$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 q^{2} + 9 q^{3} + 16 q^{4} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} -36 q^{6} + ( -31 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{7} -64 q^{8} + 81 q^{9} +O(q^{10})$$ $$q -4 q^{2} + 9 q^{3} + 16 q^{4} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} -36 q^{6} + ( -31 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{7} -64 q^{8} + 81 q^{9} + ( 8 - 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{10} + ( -44 + \beta_{1} - 3 \beta_{2} + 5 \beta_{3} ) q^{11} + 144 q^{12} + ( -84 - 4 \beta_{1} + 10 \beta_{2} - 9 \beta_{3} + 11 \beta_{4} ) q^{13} + ( 124 + 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} + 16 \beta_{4} ) q^{14} + ( -18 + 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{15} + 256 q^{16} + ( 255 + 7 \beta_{1} - 8 \beta_{2} + 14 \beta_{4} ) q^{17} -324 q^{18} + ( 53 - 5 \beta_{1} - 2 \beta_{2} + 12 \beta_{3} + 47 \beta_{4} ) q^{19} + ( -32 + 16 \beta_{1} + 16 \beta_{2} - 16 \beta_{3} ) q^{20} + ( -279 - 9 \beta_{1} - 18 \beta_{2} + 9 \beta_{3} - 36 \beta_{4} ) q^{21} + ( 176 - 4 \beta_{1} + 12 \beta_{2} - 20 \beta_{3} ) q^{22} + ( -21 - 143 \beta_{1} - 4 \beta_{2} + 62 \beta_{3} + 2 \beta_{4} ) q^{23} -576 q^{24} + ( -715 + 97 \beta_{1} - 23 \beta_{2} + 13 \beta_{3} + 45 \beta_{4} ) q^{25} + ( 336 + 16 \beta_{1} - 40 \beta_{2} + 36 \beta_{3} - 44 \beta_{4} ) q^{26} + 729 q^{27} + ( -496 - 16 \beta_{1} - 32 \beta_{2} + 16 \beta_{3} - 64 \beta_{4} ) q^{28} + ( -967 - 32 \beta_{1} - 109 \beta_{2} - 63 \beta_{3} - 4 \beta_{4} ) q^{29} + ( 72 - 36 \beta_{1} - 36 \beta_{2} + 36 \beta_{3} ) q^{30} + ( -2787 + 75 \beta_{1} + 102 \beta_{2} + 143 \beta_{3} - 29 \beta_{4} ) q^{31} -1024 q^{32} + ( -396 + 9 \beta_{1} - 27 \beta_{2} + 45 \beta_{3} ) q^{33} + ( -1020 - 28 \beta_{1} + 32 \beta_{2} - 56 \beta_{4} ) q^{34} + ( -1143 - 222 \beta_{1} - 103 \beta_{2} + 37 \beta_{3} - 79 \beta_{4} ) q^{35} + 1296 q^{36} + ( -4485 - 189 \beta_{1} - 46 \beta_{2} - 228 \beta_{3} + 25 \beta_{4} ) q^{37} + ( -212 + 20 \beta_{1} + 8 \beta_{2} - 48 \beta_{3} - 188 \beta_{4} ) q^{38} + ( -756 - 36 \beta_{1} + 90 \beta_{2} - 81 \beta_{3} + 99 \beta_{4} ) q^{39} + ( 128 - 64 \beta_{1} - 64 \beta_{2} + 64 \beta_{3} ) q^{40} + ( -2909 + 276 \beta_{1} + 89 \beta_{2} + 167 \beta_{3} - 120 \beta_{4} ) q^{41} + ( 1116 + 36 \beta_{1} + 72 \beta_{2} - 36 \beta_{3} + 144 \beta_{4} ) q^{42} + ( -4350 + 431 \beta_{1} + 17 \beta_{2} - \beta_{3} + 51 \beta_{4} ) q^{43} + ( -704 + 16 \beta_{1} - 48 \beta_{2} + 80 \beta_{3} ) q^{44} + ( -162 + 81 \beta_{1} + 81 \beta_{2} - 81 \beta_{3} ) q^{45} + ( 84 + 572 \beta_{1} + 16 \beta_{2} - 248 \beta_{3} - 8 \beta_{4} ) q^{46} + ( -1001 - 332 \beta_{1} - 93 \beta_{2} + 33 \beta_{3} - 122 \beta_{4} ) q^{47} + 2304 q^{48} + ( 562 - 94 \beta_{1} + 153 \beta_{2} - 406 \beta_{3} + 196 \beta_{4} ) q^{49} + ( 2860 - 388 \beta_{1} + 92 \beta_{2} - 52 \beta_{3} - 180 \beta_{4} ) q^{50} + ( 2295 + 63 \beta_{1} - 72 \beta_{2} + 126 \beta_{4} ) q^{51} + ( -1344 - 64 \beta_{1} + 160 \beta_{2} - 144 \beta_{3} + 176 \beta_{4} ) q^{52} + ( 2850 + 897 \beta_{1} + 161 \beta_{2} + 121 \beta_{3} - 132 \beta_{4} ) q^{53} -2916 q^{54} + ( -8110 - 297 \beta_{1} + 139 \beta_{2} - 67 \beta_{3} - 105 \beta_{4} ) q^{55} + ( 1984 + 64 \beta_{1} + 128 \beta_{2} - 64 \beta_{3} + 256 \beta_{4} ) q^{56} + ( 477 - 45 \beta_{1} - 18 \beta_{2} + 108 \beta_{3} + 423 \beta_{4} ) q^{57} + ( 3868 + 128 \beta_{1} + 436 \beta_{2} + 252 \beta_{3} + 16 \beta_{4} ) q^{58} + 3481 q^{59} + ( -288 + 144 \beta_{1} + 144 \beta_{2} - 144 \beta_{3} ) q^{60} + ( -3797 - 1643 \beta_{1} + 164 \beta_{2} - 129 \beta_{3} - 272 \beta_{4} ) q^{61} + ( 11148 - 300 \beta_{1} - 408 \beta_{2} - 572 \beta_{3} + 116 \beta_{4} ) q^{62} + ( -2511 - 81 \beta_{1} - 162 \beta_{2} + 81 \beta_{3} - 324 \beta_{4} ) q^{63} + 4096 q^{64} + ( 13624 + 670 \beta_{1} - 256 \beta_{2} + 317 \beta_{3} + 316 \beta_{4} ) q^{65} + ( 1584 - 36 \beta_{1} + 108 \beta_{2} - 180 \beta_{3} ) q^{66} + ( -6150 + 913 \beta_{1} + 227 \beta_{2} + 571 \beta_{3} - 5 \beta_{4} ) q^{67} + ( 4080 + 112 \beta_{1} - 128 \beta_{2} + 224 \beta_{4} ) q^{68} + ( -189 - 1287 \beta_{1} - 36 \beta_{2} + 558 \beta_{3} + 18 \beta_{4} ) q^{69} + ( 4572 + 888 \beta_{1} + 412 \beta_{2} - 148 \beta_{3} + 316 \beta_{4} ) q^{70} + ( 2936 - 666 \beta_{1} - 1088 \beta_{2} - 153 \beta_{3} - 284 \beta_{4} ) q^{71} -5184 q^{72} + ( -7693 + 1057 \beta_{1} + 16 \beta_{2} + 63 \beta_{3} - 214 \beta_{4} ) q^{73} + ( 17940 + 756 \beta_{1} + 184 \beta_{2} + 912 \beta_{3} - 100 \beta_{4} ) q^{74} + ( -6435 + 873 \beta_{1} - 207 \beta_{2} + 117 \beta_{3} + 405 \beta_{4} ) q^{75} + ( 848 - 80 \beta_{1} - 32 \beta_{2} + 192 \beta_{3} + 752 \beta_{4} ) q^{76} + ( 6481 + 100 \beta_{1} + 463 \beta_{2} - 99 \beta_{3} + 645 \beta_{4} ) q^{77} + ( 3024 + 144 \beta_{1} - 360 \beta_{2} + 324 \beta_{3} - 396 \beta_{4} ) q^{78} + ( -10496 - 28 \beta_{1} - 980 \beta_{2} - 923 \beta_{3} - 795 \beta_{4} ) q^{79} + ( -512 + 256 \beta_{1} + 256 \beta_{2} - 256 \beta_{3} ) q^{80} + 6561 q^{81} + ( 11636 - 1104 \beta_{1} - 356 \beta_{2} - 668 \beta_{3} + 480 \beta_{4} ) q^{82} + ( 2013 - 1250 \beta_{1} - 589 \beta_{2} + 1621 \beta_{3} - 568 \beta_{4} ) q^{83} + ( -4464 - 144 \beta_{1} - 288 \beta_{2} + 144 \beta_{3} - 576 \beta_{4} ) q^{84} + ( -18489 + 140 \beta_{1} + 1045 \beta_{2} - 698 \beta_{3} - 156 \beta_{4} ) q^{85} + ( 17400 - 1724 \beta_{1} - 68 \beta_{2} + 4 \beta_{3} - 204 \beta_{4} ) q^{86} + ( -8703 - 288 \beta_{1} - 981 \beta_{2} - 567 \beta_{3} - 36 \beta_{4} ) q^{87} + ( 2816 - 64 \beta_{1} + 192 \beta_{2} - 320 \beta_{3} ) q^{88} + ( 29 - 468 \beta_{1} + 599 \beta_{2} - 2137 \beta_{3} - 39 \beta_{4} ) q^{89} + ( 648 - 324 \beta_{1} - 324 \beta_{2} + 324 \beta_{3} ) q^{90} + ( -51913 + 1253 \beta_{1} - 456 \beta_{2} + 1115 \beta_{3} - 575 \beta_{4} ) q^{91} + ( -336 - 2288 \beta_{1} - 64 \beta_{2} + 992 \beta_{3} + 32 \beta_{4} ) q^{92} + ( -25083 + 675 \beta_{1} + 918 \beta_{2} + 1287 \beta_{3} - 261 \beta_{4} ) q^{93} + ( 4004 + 1328 \beta_{1} + 372 \beta_{2} - 132 \beta_{3} + 488 \beta_{4} ) q^{94} + ( -44005 + 16 \beta_{1} + 1599 \beta_{2} - 544 \beta_{3} - 123 \beta_{4} ) q^{95} -9216 q^{96} + ( -57046 - 430 \beta_{1} - 54 \beta_{2} - 1145 \beta_{3} + 1241 \beta_{4} ) q^{97} + ( -2248 + 376 \beta_{1} - 612 \beta_{2} + 1624 \beta_{3} - 784 \beta_{4} ) q^{98} + ( -3564 + 81 \beta_{1} - 243 \beta_{2} + 405 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 20q^{2} + 45q^{3} + 80q^{4} - 10q^{5} - 180q^{6} - 162q^{7} - 320q^{8} + 405q^{9} + O(q^{10})$$ $$5q - 20q^{2} + 45q^{3} + 80q^{4} - 10q^{5} - 180q^{6} - 162q^{7} - 320q^{8} + 405q^{9} + 40q^{10} - 228q^{11} + 720q^{12} - 386q^{13} + 648q^{14} - 90q^{15} + 1280q^{16} + 1304q^{17} - 1620q^{18} + 342q^{19} - 160q^{20} - 1458q^{21} + 912q^{22} - 78q^{23} - 2880q^{24} - 3585q^{25} + 1544q^{26} + 3645q^{27} - 2592q^{28} - 4576q^{29} + 360q^{30} - 14456q^{31} - 5120q^{32} - 2052q^{33} - 5216q^{34} - 5622q^{35} + 6480q^{36} - 21684q^{37} - 1368q^{38} - 3474q^{39} + 640q^{40} - 15484q^{41} + 5832q^{42} - 22094q^{43} - 3648q^{44} - 810q^{45} + 312q^{46} - 4890q^{47} + 11520q^{48} + 3955q^{49} + 14340q^{50} + 11736q^{51} - 6176q^{52} + 12686q^{53} - 14580q^{54} - 40468q^{55} + 10368q^{56} + 3078q^{57} + 18304q^{58} + 17405q^{59} - 1440q^{60} - 17792q^{61} + 57824q^{62} - 13122q^{63} + 20480q^{64} + 67704q^{65} + 8208q^{66} - 33042q^{67} + 20864q^{68} - 702q^{69} + 22488q^{70} + 16172q^{71} - 25920q^{72} - 40092q^{73} + 86736q^{74} - 32265q^{75} + 5472q^{76} + 33330q^{77} + 13896q^{78} - 51216q^{79} - 2560q^{80} + 32805q^{81} + 61936q^{82} + 7526q^{83} - 23328q^{84} - 92546q^{85} + 88376q^{86} - 41184q^{87} + 14592q^{88} + 4210q^{89} + 3240q^{90} - 263742q^{91} - 1248q^{92} - 130104q^{93} + 19560q^{94} - 220798q^{95} - 46080q^{96} - 279974q^{97} - 15820q^{98} - 18468q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 290 x^{3} - 616 x^{2} + 4720 x + 11900$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-29 \nu^{4} + 189 \nu^{3} + 7600 \nu^{2} - 16386 \nu - 96440$$$$)/4500$$ $$\beta_{2}$$ $$=$$ $$($$$$29 \nu^{4} - 189 \nu^{3} - 7600 \nu^{2} + 25386 \nu + 93440$$$$)/1500$$ $$\beta_{3}$$ $$=$$ $$($$$$-47 \nu^{4} + 177 \nu^{3} + 12925 \nu^{2} - 6798 \nu - 159545$$$$)/1125$$ $$\beta_{4}$$ $$=$$ $$($$$$-89 \nu^{4} + 399 \nu^{3} + 24850 \nu^{2} - 33426 \nu - 374540$$$$)/2250$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 3 \beta_{1} + 2$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$10 \beta_{4} - 7 \beta_{3} + 8 \beta_{2} + 8 \beta_{1} + 345$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$47 \beta_{4} - 59 \beta_{3} + 114 \beta_{2} + 436 \beta_{1} + 1699$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$2927 \beta_{4} - 2219 \beta_{3} + 2557 \beta_{2} + 3625 \beta_{1} + 90945$$$$)/3$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.80991 −14.6687 4.22515 −2.79263 18.0461
−4.00000 9.00000 16.0000 −83.8793 −36.0000 −13.6774 −64.0000 81.0000 335.517
1.2 −4.00000 9.00000 16.0000 −8.96408 −36.0000 −187.945 −64.0000 81.0000 35.8563
1.3 −4.00000 9.00000 16.0000 −3.84671 −36.0000 56.0281 −64.0000 81.0000 15.3869
1.4 −4.00000 9.00000 16.0000 19.3501 −36.0000 148.650 −64.0000 81.0000 −77.4004
1.5 −4.00000 9.00000 16.0000 67.3400 −36.0000 −165.056 −64.0000 81.0000 −269.360
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$59$$ $$-1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{5}^{5} + 10 T_{5}^{4} - 5970 T_{5}^{3} + 32740 T_{5}^{2} + 1194385 T_{5} + 3768834$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(354))$$.