# Properties

 Label 14.9.b.a Level $14$ Weight $9$ Character orbit 14.b Analytic conductor $5.703$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 14.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.70330054086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.3520512.3 Defining polynomial: $$x^{4} + 120 x^{2} + 3438$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{3} q^{2} + ( 2 \beta_{1} - \beta_{2} ) q^{3} + 128 q^{4} + ( 7 \beta_{1} + 2 \beta_{2} ) q^{5} + ( 6 \beta_{1} + 14 \beta_{2} ) q^{6} + ( -1519 + 49 \beta_{1} + 147 \beta_{3} ) q^{7} + 256 \beta_{3} q^{8} + ( -3423 + 1062 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + 2 \beta_{3} q^{2} + ( 2 \beta_{1} - \beta_{2} ) q^{3} + 128 q^{4} + ( 7 \beta_{1} + 2 \beta_{2} ) q^{5} + ( 6 \beta_{1} + 14 \beta_{2} ) q^{6} + ( -1519 + 49 \beta_{1} + 147 \beta_{3} ) q^{7} + 256 \beta_{3} q^{8} + ( -3423 + 1062 \beta_{3} ) q^{9} + ( 98 \beta_{1} - 6 \beta_{2} ) q^{10} + ( -3390 + 1308 \beta_{3} ) q^{11} + ( 256 \beta_{1} - 128 \beta_{2} ) q^{12} + ( -229 \beta_{1} + 360 \beta_{2} ) q^{13} + ( 9408 + 490 \beta_{1} + 98 \beta_{2} - 3038 \beta_{3} ) q^{14} + ( -3264 - 3774 \beta_{3} ) q^{15} + 16384 q^{16} + ( -1658 \beta_{1} - 622 \beta_{2} ) q^{17} + ( 67968 - 6846 \beta_{3} ) q^{18} + ( -2888 \beta_{1} - 2391 \beta_{2} ) q^{19} + ( 896 \beta_{1} + 256 \beta_{2} ) q^{20} + ( -103488 - 2597 \beta_{1} + 2548 \beta_{2} - 7350 \beta_{3} ) q^{21} + ( 83712 - 6780 \beta_{3} ) q^{22} + ( -223518 + 11952 \beta_{3} ) q^{23} + ( 768 \beta_{1} + 1792 \beta_{2} ) q^{24} + ( 304225 - 11658 \beta_{3} ) q^{25} + ( 2750 \beta_{1} - 4058 \beta_{2} ) q^{26} + ( 9462 \beta_{1} + 4296 \beta_{2} ) q^{27} + ( -194432 + 6272 \beta_{1} + 18816 \beta_{3} ) q^{28} + ( 79266 + 135396 \beta_{3} ) q^{29} + ( -241536 - 6528 \beta_{3} ) q^{30} + ( 24378 \beta_{1} - 8790 \beta_{2} ) q^{31} + 32768 \beta_{3} q^{32} + ( -2856 \beta_{1} + 12546 \beta_{2} ) q^{33} + ( -25288 \beta_{1} + 2904 \beta_{2} ) q^{34} + ( -413952 - 3430 \beta_{1} - 3479 \beta_{2} - 85554 \beta_{3} ) q^{35} + ( -438144 + 135936 \beta_{3} ) q^{36} + ( -623774 - 89676 \beta_{3} ) q^{37} + ( -62354 \beta_{1} + 18134 \beta_{2} ) q^{38} + ( 2557248 - 455970 \beta_{3} ) q^{39} + ( 12544 \beta_{1} - 768 \beta_{2} ) q^{40} + ( -12442 \beta_{1} - 39488 \beta_{2} ) q^{41} + ( -470400 + 9702 \beta_{1} - 30674 \beta_{2} - 206976 \beta_{3} ) q^{42} + ( 2296642 + 291720 \beta_{3} ) q^{43} + ( -433920 + 167424 \beta_{3} ) q^{44} + ( 28077 \beta_{1} - 10032 \beta_{2} ) q^{45} + ( 764928 - 447036 \beta_{3} ) q^{46} + ( -5030 \beta_{1} + 73286 \beta_{2} ) q^{47} + ( 32768 \beta_{1} - 16384 \beta_{2} ) q^{48} + ( 232897 - 76832 \beta_{1} + 14406 \beta_{2} - 893172 \beta_{3} ) q^{49} + ( -746112 + 608450 \beta_{3} ) q^{50} + ( -81024 + 1095864 \beta_{3} ) q^{51} + ( -29312 \beta_{1} + 46080 \beta_{2} ) q^{52} + ( -9681822 - 300696 \beta_{3} ) q^{53} + ( 154764 \beta_{1} - 24036 \beta_{2} ) q^{54} + ( 40362 \beta_{1} - 10704 \beta_{2} ) q^{55} + ( 1204224 + 62720 \beta_{1} + 12544 \beta_{2} - 388864 \beta_{3} ) q^{56} + ( -7672704 + 3689742 \beta_{3} ) q^{57} + ( 8665344 + 158532 \beta_{3} ) q^{58} + ( 219680 \beta_{1} - 32825 \beta_{2} ) q^{59} + ( -417792 - 483072 \beta_{3} ) q^{60} + ( -315443 \beta_{1} - 127506 \beta_{2} ) q^{61} + ( 120720 \beta_{1} + 136656 \beta_{2} ) q^{62} + ( 10195185 + 92463 \beta_{1} + 52038 \beta_{2} - 2116359 \beta_{3} ) q^{63} + 2097152 q^{64} + ( -2972928 + 501354 \beta_{3} ) q^{65} + ( 147084 \beta_{1} - 131172 \beta_{2} ) q^{66} + ( -3080062 - 6305424 \beta_{3} ) q^{67} + ( -212224 \beta_{1} - 79616 \beta_{2} ) q^{68} + ( -411180 \beta_{1} + 307182 \beta_{2} ) q^{69} + ( -5475456 - 83006 \beta_{1} + 27930 \beta_{2} - 827904 \beta_{3} ) q^{70} + ( 15542178 + 5125614 \beta_{3} ) q^{71} + ( 8699904 - 876288 \beta_{3} ) q^{72} + ( -959492 \beta_{1} - 58842 \beta_{2} ) q^{73} + ( -5739264 - 1247548 \beta_{3} ) q^{74} + ( 573476 \beta_{1} - 385831 \beta_{2} ) q^{75} + ( -369664 \beta_{1} - 306048 \beta_{2} ) q^{76} + ( 11302242 + 154350 \beta_{1} + 64092 \beta_{2} - 2485182 \beta_{3} ) q^{77} + ( -29182080 + 5114496 \beta_{3} ) q^{78} + ( 6222434 - 1672830 \beta_{3} ) q^{79} + ( 114688 \beta_{1} + 32768 \beta_{2} ) q^{80} + ( -17697087 - 302670 \beta_{3} ) q^{81} + ( -677252 \beta_{1} + 369996 \beta_{2} ) q^{82} + ( 1668614 \beta_{1} - 261617 \beta_{2} ) q^{83} + ( -13246464 - 332416 \beta_{1} + 326144 \beta_{2} - 940800 \beta_{3} ) q^{84} + ( 22485888 + 2719464 \beta_{3} ) q^{85} + ( 18670080 + 4593284 \beta_{3} ) q^{86} + ( 564720 \beta_{1} + 868506 \beta_{2} ) q^{87} + ( 10715136 - 867840 \beta_{3} ) q^{88} + ( 1588536 \beta_{1} + 474774 \beta_{2} ) q^{89} + ( 140322 \beta_{1} + 156474 \beta_{2} ) q^{90} + ( 9539712 + 549976 \beta_{1} - 845103 \beta_{2} - 1828974 \beta_{3} ) q^{91} + ( -28610304 + 1529856 \beta_{3} ) q^{92} + ( -102116736 + 8315280 \beta_{3} ) q^{93} + ( 975704 \beta_{1} - 742920 \beta_{2} ) q^{94} + ( 56991936 + 4368186 \beta_{3} ) q^{95} + ( 98304 \beta_{1} + 229376 \beta_{2} ) q^{96} + ( -2464114 \beta_{1} - 369330 \beta_{2} ) q^{97} + ( -57163008 - 566636 \beta_{1} - 297724 \beta_{2} + 465794 \beta_{3} ) q^{98} + ( 56055042 - 8077464 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 512q^{4} - 6076q^{7} - 13692q^{9} + O(q^{10})$$ $$4q + 512q^{4} - 6076q^{7} - 13692q^{9} - 13560q^{11} + 37632q^{14} - 13056q^{15} + 65536q^{16} + 271872q^{18} - 413952q^{21} + 334848q^{22} - 894072q^{23} + 1216900q^{25} - 777728q^{28} + 317064q^{29} - 966144q^{30} - 1655808q^{35} - 1752576q^{36} - 2495096q^{37} + 10228992q^{39} - 1881600q^{42} + 9186568q^{43} - 1735680q^{44} + 3059712q^{46} + 931588q^{49} - 2984448q^{50} - 324096q^{51} - 38727288q^{53} + 4816896q^{56} - 30690816q^{57} + 34661376q^{58} - 1671168q^{60} + 40780740q^{63} + 8388608q^{64} - 11891712q^{65} - 12320248q^{67} - 21901824q^{70} + 62168712q^{71} + 34799616q^{72} - 22957056q^{74} + 45208968q^{77} - 116728320q^{78} + 24889736q^{79} - 70788348q^{81} - 52985856q^{84} + 89943552q^{85} + 74680320q^{86} + 42860544q^{88} + 38158848q^{91} - 114441216q^{92} - 408466944q^{93} + 227967744q^{95} - 228652032q^{98} + 224220168q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 120 x^{2} + 3438$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{3} + 156 \nu$$$$)/9$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{3} + 108 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{2} + 240$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + 3 \beta_{1}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$9 \beta_{3} - 240$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$39 \beta_{2} - 81 \beta_{1}$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/14\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 − 8.52807i 8.52807i − 6.87547i 6.87547i
−11.3137 126.458i 128.000 143.012i 1430.71i −2350.56 489.571i −1448.15 −9430.58 1617.99i
13.2 −11.3137 126.458i 128.000 143.012i 1430.71i −2350.56 + 489.571i −1448.15 −9430.58 1617.99i
13.3 11.3137 63.0589i 128.000 390.317i 713.430i −687.442 2300.48i 1448.15 2584.58 4415.94i
13.4 11.3137 63.0589i 128.000 390.317i 713.430i −687.442 + 2300.48i 1448.15 2584.58 4415.94i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.9.b.a 4
3.b odd 2 1 126.9.c.a 4
4.b odd 2 1 112.9.c.c 4
5.b even 2 1 350.9.b.a 4
5.c odd 4 2 350.9.d.a 8
7.b odd 2 1 inner 14.9.b.a 4
7.c even 3 2 98.9.d.a 8
7.d odd 6 2 98.9.d.a 8
21.c even 2 1 126.9.c.a 4
28.d even 2 1 112.9.c.c 4
35.c odd 2 1 350.9.b.a 4
35.f even 4 2 350.9.d.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.9.b.a 4 1.a even 1 1 trivial
14.9.b.a 4 7.b odd 2 1 inner
98.9.d.a 8 7.c even 3 2
98.9.d.a 8 7.d odd 6 2
112.9.c.c 4 4.b odd 2 1
112.9.c.c 4 28.d even 2 1
126.9.c.a 4 3.b odd 2 1
126.9.c.a 4 21.c even 2 1
350.9.b.a 4 5.b even 2 1
350.9.b.a 4 35.c odd 2 1
350.9.d.a 8 5.c odd 4 2
350.9.d.a 8 35.f even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -128 + T^{2} )^{2}$$
$3$ $$63589248 + 19968 T^{2} + T^{4}$$
$5$ $$3115873152 + 172800 T^{2} + T^{4}$$
$7$ $$33232930569601 + 35026930876 T + 17993094 T^{2} + 6076 T^{3} + T^{4}$$
$11$ $$( -43255548 + 6780 T + T^{2} )^{2}$$
$13$ $$202796647224192 + 1650034944 T^{2} + T^{4}$$
$17$ $$23216380602145800192 + 11879663616 T^{2} + T^{4}$$
$19$ $$22\!\cdots\!88$$$$+ 94768846848 T^{2} + T^{4}$$
$23$ $$( 45389086596 + 447036 T + T^{2} )^{2}$$
$29$ $$( -580343359356 - 158532 T + T^{2} )^{2}$$
$31$ $$10\!\cdots\!92$$$$+ 2154089189376 T^{2} + T^{4}$$
$37$ $$( 131756883844 + 1247548 T + T^{2} )^{2}$$
$41$ $$54\!\cdots\!12$$$$+ 19894697542656 T^{2} + T^{4}$$
$43$ $$( 2551346607364 - 4593284 T + T^{2} )^{2}$$
$47$ $$25\!\cdots\!88$$$$+ 65772041361408 T^{2} + T^{4}$$
$53$ $$( 90844298538372 + 19363644 T + T^{2} )^{2}$$
$59$ $$28\!\cdots\!00$$$$+ 118891469721600 T^{2} + T^{4}$$
$61$ $$38\!\cdots\!48$$$$+ 459923108721408 T^{2} + T^{4}$$
$67$ $$( -1262781116308988 + 6160124 T + T^{2} )^{2}$$
$71$ $$( -599142107080188 - 31084356 T + T^{2} )^{2}$$
$73$ $$24\!\cdots\!32$$$$+ 2207025575331840 T^{2} + T^{4}$$
$79$ $$( -50828841800444 - 12444868 T + T^{2} )^{2}$$
$83$ $$97\!\cdots\!88$$$$+ 6920736406405632 T^{2} + T^{4}$$
$89$ $$95\!\cdots\!32$$$$+ 9163085627879424 T^{2} + T^{4}$$
$97$ $$34\!\cdots\!12$$$$+ 16364635408134144 T^{2} + T^{4}$$