# Properties

 Label 4008.2.a.j Level $4008$ Weight $2$ Character orbit 4008.a Self dual yes Analytic conductor $32.004$ Analytic rank $1$ Dimension $10$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4008 = 2^{3} \cdot 3 \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4008.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0040411301$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 26 x^{8} - 3 x^{7} + 220 x^{6} + 42 x^{5} - 675 x^{4} - 67 x^{3} + 628 x^{2} - 48 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{3} + ( -1 - \beta_{1} ) q^{5} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( -1 - \beta_{1} ) q^{5} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{7} + q^{9} + ( -\beta_{4} - \beta_{7} ) q^{11} + ( -1 + \beta_{1} + \beta_{8} ) q^{13} + ( 1 + \beta_{1} ) q^{15} + ( \beta_{3} - \beta_{8} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{19} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{21} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{23} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{25} - q^{27} + ( -1 - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{29} + ( 2 - \beta_{2} - \beta_{5} + \beta_{7} + \beta_{9} ) q^{31} + ( \beta_{4} + \beta_{7} ) q^{33} + ( -1 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{35} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{7} ) q^{37} + ( 1 - \beta_{1} - \beta_{8} ) q^{39} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{41} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - \beta_{9} ) q^{43} + ( -1 - \beta_{1} ) q^{45} + ( 1 - \beta_{2} - \beta_{4} ) q^{47} + ( 3 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} ) q^{49} + ( -\beta_{3} + \beta_{8} ) q^{51} + ( -2 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{53} + ( 3 + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{57} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{59} + ( 1 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{61} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{63} + ( -1 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{65} + ( -2 + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{67} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{69} + ( 3 + \beta_{1} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{71} + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{73} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{75} + ( -4 - \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} ) q^{77} + ( 3 - \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{79} + q^{81} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{83} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} + 2 \beta_{8} ) q^{85} + ( 1 + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{87} + ( -5 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{9} ) q^{89} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{91} + ( -2 + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{9} ) q^{93} + ( \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{95} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{97} + ( -\beta_{4} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 10q^{3} - 10q^{5} + q^{7} + 10q^{9} + O(q^{10})$$ $$10q - 10q^{3} - 10q^{5} + q^{7} + 10q^{9} - q^{11} - 6q^{13} + 10q^{15} - 9q^{17} + 2q^{19} - q^{21} + 7q^{23} + 12q^{25} - 10q^{27} - 13q^{29} + 23q^{31} + q^{33} + q^{35} - 6q^{37} + 6q^{39} - 12q^{41} - 10q^{45} + 10q^{47} + 7q^{49} + 9q^{51} - 26q^{53} + 11q^{55} - 2q^{57} - 10q^{59} - 10q^{61} + q^{63} - 22q^{65} - 5q^{67} - 7q^{69} + 25q^{71} - 8q^{73} - 12q^{75} - 46q^{77} + 26q^{79} + 10q^{81} - 14q^{83} + 9q^{85} + 13q^{87} - 31q^{89} - 3q^{91} - 23q^{93} - 5q^{95} - 32q^{97} - q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 26 x^{8} - 3 x^{7} + 220 x^{6} + 42 x^{5} - 675 x^{4} - 67 x^{3} + 628 x^{2} - 48 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$31274 \nu^{9} + 188403 \nu^{8} - 799085 \nu^{7} - 4445172 \nu^{6} + 5651462 \nu^{5} + 31352276 \nu^{4} - 4209233 \nu^{3} - 62769314 \nu^{2} - 26831938 \nu + 17959625$$$$)/6386707$$ $$\beta_{3}$$ $$=$$ $$($$$$45076 \nu^{9} - 26608 \nu^{8} - 1096194 \nu^{7} + 535650 \nu^{6} + 9006576 \nu^{5} - 3054363 \nu^{4} - 31921253 \nu^{3} + 4461694 \nu^{2} + 45128857 \nu + 3445799$$$$)/6386707$$ $$\beta_{4}$$ $$=$$ $$($$$$-140074 \nu^{9} - 197290 \nu^{8} + 4018521 \nu^{7} + 4959639 \nu^{6} - 37660725 \nu^{5} - 36535738 \nu^{4} + 125932814 \nu^{3} + 66704449 \nu^{2} - 128029606 \nu - 3074005$$$$)/6386707$$ $$\beta_{5}$$ $$=$$ $$($$$$-252127 \nu^{9} - 49534 \nu^{8} + 6474592 \nu^{7} + 2278935 \nu^{6} - 53041455 \nu^{5} - 25138542 \nu^{4} + 150476284 \nu^{3} + 61255546 \nu^{2} - 122806757 \nu - 20051371$$$$)/6386707$$ $$\beta_{6}$$ $$=$$ $$($$$$-400908 \nu^{9} + 156741 \nu^{8} + 10079448 \nu^{7} - 2045962 \nu^{6} - 81378952 \nu^{5} + 2450805 \nu^{4} + 233226318 \nu^{3} - 5489909 \nu^{2} - 195374082 \nu + 15210925$$$$)/6386707$$ $$\beta_{7}$$ $$=$$ $$($$$$-446372 \nu^{9} + 13553 \nu^{8} + 11479221 \nu^{7} + 1022842 \nu^{6} - 95483893 \nu^{5} - 17429855 \nu^{4} + 282874842 \nu^{3} + 36533942 \nu^{2} - 233347950 \nu + 3313559$$$$)/6386707$$ $$\beta_{8}$$ $$=$$ $$($$$$-581351 \nu^{9} + 4817 \nu^{8} + 14786968 \nu^{7} + 1678767 \nu^{6} - 120466276 \nu^{5} - 24247147 \nu^{4} + 344299563 \nu^{3} + 37019495 \nu^{2} - 291434843 \nu + 24062029$$$$)/6386707$$ $$\beta_{9}$$ $$=$$ $$($$$$676598 \nu^{9} + 130521 \nu^{8} - 17690130 \nu^{7} - 4911181 \nu^{6} + 151157770 \nu^{5} + 47856867 \nu^{4} - 472650162 \nu^{3} - 87928296 \nu^{2} + 458021977 \nu - 18894784$$$$)/6386707$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{7} + \beta_{4} - 2 \beta_{3} - \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 9 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$12 \beta_{9} + \beta_{8} + 9 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 14 \beta_{4} - 24 \beta_{3} - \beta_{2} - 6 \beta_{1} + 33$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{9} + 18 \beta_{8} - 11 \beta_{7} - 18 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} - 13 \beta_{3} + 17 \beta_{2} + 93 \beta_{1} + 3$$ $$\nu^{6}$$ $$=$$ $$132 \beta_{9} + 6 \beta_{8} + 81 \beta_{7} + 59 \beta_{6} - 40 \beta_{5} + 167 \beta_{4} - 268 \beta_{3} - 20 \beta_{2} - 40 \beta_{1} + 341$$ $$\nu^{7}$$ $$=$$ $$81 \beta_{9} + 246 \beta_{8} - 103 \beta_{7} - 253 \beta_{6} + 212 \beta_{5} + 51 \beta_{4} - 139 \beta_{3} + 245 \beta_{2} + 1024 \beta_{1} + 45$$ $$\nu^{8}$$ $$=$$ $$1477 \beta_{9} - 11 \beta_{8} + 784 \beta_{7} + 876 \beta_{6} - 447 \beta_{5} + 1957 \beta_{4} - 2992 \beta_{3} - 265 \beta_{2} - 319 \beta_{1} + 3816$$ $$\nu^{9}$$ $$=$$ $$1188 \beta_{9} + 3084 \beta_{8} - 1004 \beta_{7} - 3245 \beta_{6} + 2675 \beta_{5} + 861 \beta_{4} - 1359 \beta_{3} + 3283 \beta_{2} + 11672 \beta_{1} + 459$$

## 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
 3.44741 3.20371 1.76045 1.15141 0.108417 −0.0299246 −1.39234 −2.25873 −2.53098 −3.45943
0 −1.00000 0 −4.44741 0 2.42708 0 1.00000 0
1.2 0 −1.00000 0 −4.20371 0 −3.93441 0 1.00000 0
1.3 0 −1.00000 0 −2.76045 0 3.28542 0 1.00000 0
1.4 0 −1.00000 0 −2.15141 0 −2.22742 0 1.00000 0
1.5 0 −1.00000 0 −1.10842 0 −3.58606 0 1.00000 0
1.6 0 −1.00000 0 −0.970075 0 4.67245 0 1.00000 0
1.7 0 −1.00000 0 0.392343 0 0.0476950 0 1.00000 0
1.8 0 −1.00000 0 1.25873 0 −0.924269 0 1.00000 0
1.9 0 −1.00000 0 1.53098 0 1.95354 0 1.00000 0
1.10 0 −1.00000 0 2.45943 0 −0.714036 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$167$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4008.2.a.j 10
4.b odd 2 1 8016.2.a.bd 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.j 10 1.a even 1 1 trivial
8016.2.a.bd 10 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4008))$$:

 $$T_{5}^{10} + \cdots$$ $$T_{7}^{10} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$( 1 + T )^{10}$$
$5$ $$222 - 262 T - 961 T^{2} + 116 T^{3} + 910 T^{4} + 137 T^{5} - 277 T^{6} - 85 T^{7} + 19 T^{8} + 10 T^{9} + T^{10}$$
$7$ $$-72 + 1364 T + 3051 T^{2} + 134 T^{3} - 2125 T^{4} - 186 T^{5} + 467 T^{6} + 25 T^{7} - 38 T^{8} - T^{9} + T^{10}$$
$11$ $$4656 + 10432 T + 2548 T^{2} - 7060 T^{3} - 3290 T^{4} + 1395 T^{5} + 781 T^{6} - 76 T^{7} - 53 T^{8} + T^{9} + T^{10}$$
$13$ $$-96 - 2320 T + 20900 T^{2} - 5540 T^{3} - 8328 T^{4} + 2443 T^{5} + 920 T^{6} - 224 T^{7} - 47 T^{8} + 6 T^{9} + T^{10}$$
$17$ $$-3336 - 1280 T + 20696 T^{2} - 17866 T^{3} - 4186 T^{4} + 6077 T^{5} + 317 T^{6} - 540 T^{7} - 52 T^{8} + 9 T^{9} + T^{10}$$
$19$ $$184480 - 228432 T - 74316 T^{2} + 153696 T^{3} - 25656 T^{4} - 13305 T^{5} + 3188 T^{6} + 340 T^{7} - 105 T^{8} - 2 T^{9} + T^{10}$$
$23$ $$1580032 - 2652800 T + 589420 T^{2} + 596328 T^{3} - 118734 T^{4} - 37379 T^{5} + 6379 T^{6} + 880 T^{7} - 135 T^{8} - 7 T^{9} + T^{10}$$
$29$ $$-15617040 - 25967696 T - 12803820 T^{2} - 1466876 T^{3} + 506058 T^{4} + 129591 T^{5} - 871 T^{6} - 2454 T^{7} - 131 T^{8} + 13 T^{9} + T^{10}$$
$31$ $$1637000 + 998172 T - 997167 T^{2} - 366545 T^{3} + 208400 T^{4} + 16558 T^{5} - 16709 T^{6} + 1930 T^{7} + 71 T^{8} - 23 T^{9} + T^{10}$$
$37$ $$4012928 - 405168 T - 2325113 T^{2} - 463148 T^{3} + 251036 T^{4} + 96471 T^{5} + 5647 T^{6} - 1557 T^{7} - 177 T^{8} + 6 T^{9} + T^{10}$$
$41$ $$-317520 + 100856 T + 1655700 T^{2} - 612746 T^{3} - 146696 T^{4} + 48649 T^{5} + 5688 T^{6} - 1313 T^{7} - 116 T^{8} + 12 T^{9} + T^{10}$$
$43$ $$-7185208 + 5045504 T + 2266464 T^{2} - 710998 T^{3} - 251224 T^{4} + 28195 T^{5} + 10878 T^{6} - 279 T^{7} - 178 T^{8} + T^{10}$$
$47$ $$780 - 9008 T - 1913 T^{2} + 11951 T^{3} - 630 T^{4} - 3707 T^{5} + 351 T^{6} + 370 T^{7} - 29 T^{8} - 10 T^{9} + T^{10}$$
$53$ $$8344626 + 13865104 T - 11345473 T^{2} - 952517 T^{3} + 1043452 T^{4} + 107387 T^{5} - 24483 T^{6} - 3640 T^{7} + 47 T^{8} + 26 T^{9} + T^{10}$$
$59$ $$-13097044 + 878956 T + 4372051 T^{2} - 553469 T^{3} - 410590 T^{4} + 58491 T^{5} + 12527 T^{6} - 1414 T^{7} - 177 T^{8} + 10 T^{9} + T^{10}$$
$61$ $$-51200 + 199680 T - 82176 T^{2} - 173328 T^{3} - 6496 T^{4} + 21983 T^{5} + 1612 T^{6} - 874 T^{7} - 79 T^{8} + 10 T^{9} + T^{10}$$
$67$ $$-11149482 - 1512956 T + 12886459 T^{2} - 2451429 T^{3} - 1223423 T^{4} + 89264 T^{5} + 32123 T^{6} - 1122 T^{7} - 314 T^{8} + 5 T^{9} + T^{10}$$
$71$ $$424928 + 1802288 T - 10093732 T^{2} + 610684 T^{3} + 877462 T^{4} - 102779 T^{5} - 18409 T^{6} + 3049 T^{7} + 48 T^{8} - 25 T^{9} + T^{10}$$
$73$ $$-2097200 + 1305584 T + 762324 T^{2} - 350304 T^{3} - 103648 T^{4} + 31089 T^{5} + 6528 T^{6} - 1014 T^{7} - 173 T^{8} + 8 T^{9} + T^{10}$$
$79$ $$-7804352 - 7475104 T + 4072664 T^{2} + 4089678 T^{3} + 89434 T^{4} - 205611 T^{5} - 5726 T^{6} + 3775 T^{7} - 8 T^{8} - 26 T^{9} + T^{10}$$
$83$ $$-374292 - 30368128 T - 21033021 T^{2} - 2600273 T^{3} + 1034400 T^{4} + 253933 T^{5} - 3285 T^{6} - 4658 T^{7} - 253 T^{8} + 14 T^{9} + T^{10}$$
$89$ $$-195650252 - 111425896 T - 4947799 T^{2} + 8405415 T^{3} + 1706703 T^{4} - 47054 T^{5} - 42311 T^{6} - 3012 T^{7} + 194 T^{8} + 31 T^{9} + T^{10}$$
$97$ $$-476686580 + 797311508 T - 131253009 T^{2} - 63271276 T^{3} + 7885481 T^{4} + 1641029 T^{5} - 58731 T^{6} - 15141 T^{7} - 230 T^{8} + 32 T^{9} + T^{10}$$