# Properties

 Label 14.7.b.a Level $14$ Weight $7$ Character orbit 14.b Analytic conductor $3.221$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.22075717068$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.211968.1 Defining polynomial: $$x^{4} + 30 x^{2} + 207$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{1} q^{3} + 32 q^{4} -5 \beta_{2} q^{5} + ( -5 \beta_{1} - \beta_{2} ) q^{6} + ( 77 + 14 \beta_{1} - 7 \beta_{2} - 21 \beta_{3} ) q^{7} -32 \beta_{3} q^{8} + ( 273 + 78 \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{1} q^{3} + 32 q^{4} -5 \beta_{2} q^{5} + ( -5 \beta_{1} - \beta_{2} ) q^{6} + ( 77 + 14 \beta_{1} - 7 \beta_{2} - 21 \beta_{3} ) q^{7} -32 \beta_{3} q^{8} + ( 273 + 78 \beta_{3} ) q^{9} + ( -35 \beta_{1} + 25 \beta_{2} ) q^{10} + ( -1110 - 12 \beta_{3} ) q^{11} -32 \beta_{1} q^{12} + ( 94 \beta_{1} + 43 \beta_{2} ) q^{13} + ( 672 + 21 \beta_{1} + 49 \beta_{2} - 77 \beta_{3} ) q^{14} + ( -1080 + 330 \beta_{3} ) q^{15} + 1024 q^{16} + ( 150 \beta_{1} - 134 \beta_{2} ) q^{17} + ( -2496 - 273 \beta_{3} ) q^{18} + ( -283 \beta_{1} + 50 \beta_{2} ) q^{19} -160 \beta_{2} q^{20} + ( 4872 - 182 \beta_{1} - 21 \beta_{2} - 630 \beta_{3} ) q^{21} + ( 384 + 1110 \beta_{3} ) q^{22} + ( 10146 - 264 \beta_{3} ) q^{23} + ( -160 \beta_{1} - 32 \beta_{2} ) q^{24} + ( -10175 - 2850 \beta_{3} ) q^{25} + ( 771 \beta_{1} - 121 \beta_{2} ) q^{26} + ( -612 \beta_{1} + 78 \beta_{2} ) q^{27} + ( 2464 + 448 \beta_{1} - 224 \beta_{2} - 672 \beta_{3} ) q^{28} + ( -4566 + 6564 \beta_{3} ) q^{29} + ( -10560 + 1080 \beta_{3} ) q^{30} + ( 54 \beta_{1} + 1038 \beta_{2} ) q^{31} -1024 \beta_{3} q^{32} + ( 1050 \beta_{1} - 12 \beta_{2} ) q^{33} + ( -188 \beta_{1} + 820 \beta_{2} ) q^{34} + ( -21000 - 735 \beta_{1} + 140 \beta_{2} - 8610 \beta_{3} ) q^{35} + ( 8736 + 2496 \beta_{3} ) q^{36} + ( -5798 + 7668 \beta_{3} ) q^{37} + ( -1065 \beta_{1} - 533 \beta_{2} ) q^{38} + ( 52152 - 10170 \beta_{3} ) q^{39} + ( -1120 \beta_{1} + 800 \beta_{2} ) q^{40} + ( 1980 \beta_{1} - 2098 \beta_{2} ) q^{41} + ( 20160 - 1057 \beta_{1} - 77 \beta_{2} - 4872 \beta_{3} ) q^{42} + ( -11174 + 18240 \beta_{3} ) q^{43} + ( -35520 - 384 \beta_{3} ) q^{44} + ( 2730 \beta_{1} - 3315 \beta_{2} ) q^{45} + ( 8448 - 10146 \beta_{3} ) q^{46} + ( -4014 \beta_{1} - 986 \beta_{2} ) q^{47} -1024 \beta_{1} q^{48} + ( -77567 + 3038 \beta_{1} + 980 \beta_{2} - 6468 \beta_{3} ) q^{49} + ( 91200 + 10175 \beta_{3} ) q^{50} + ( 39456 - 2856 \beta_{3} ) q^{51} + ( 3008 \beta_{1} + 1376 \beta_{2} ) q^{52} + ( 62154 - 10968 \beta_{3} ) q^{53} + ( -2514 \beta_{1} - 1002 \beta_{2} ) q^{54} + ( -420 \beta_{1} + 5850 \beta_{2} ) q^{55} + ( 21504 + 672 \beta_{1} + 1568 \beta_{2} - 2464 \beta_{3} ) q^{56} + ( -118248 + 18774 \beta_{3} ) q^{57} + ( -210048 + 4566 \beta_{3} ) q^{58} + ( -13485 \beta_{1} + 6710 \beta_{2} ) q^{59} + ( -34560 + 10560 \beta_{3} ) q^{60} + ( 9368 \beta_{1} + 2945 \beta_{2} ) q^{61} + ( 7536 \beta_{1} - 5136 \beta_{2} ) q^{62} + ( -31395 + 2184 \beta_{1} - 5733 \beta_{2} + 273 \beta_{3} ) q^{63} + 32768 q^{64} + ( 323400 - 6510 \beta_{3} ) q^{65} + ( 5166 \beta_{1} + 1110 \beta_{2} ) q^{66} + ( -108694 - 408 \beta_{3} ) q^{67} + ( 4800 \beta_{1} - 4288 \beta_{2} ) q^{68} + ( -11466 \beta_{1} - 264 \beta_{2} ) q^{69} + ( 275520 - 2695 \beta_{1} - 1435 \beta_{2} + 21000 \beta_{3} ) q^{70} + ( -112902 - 58146 \beta_{3} ) q^{71} + ( -79872 - 8736 \beta_{3} ) q^{72} + ( 3494 \beta_{1} - 11008 \beta_{2} ) q^{73} + ( -245376 + 5798 \beta_{3} ) q^{74} + ( -4075 \beta_{1} - 2850 \beta_{2} ) q^{75} + ( -9056 \beta_{1} + 1600 \beta_{2} ) q^{76} + ( -77406 - 15288 \beta_{1} + 8358 \beta_{2} + 22386 \beta_{3} ) q^{77} + ( 325440 - 52152 \beta_{3} ) q^{78} + ( 523226 + 12690 \beta_{3} ) q^{79} -5120 \beta_{2} q^{80} + ( -63207 + 99450 \beta_{3} ) q^{81} + ( -4786 \beta_{1} + 12470 \beta_{2} ) q^{82} + ( 16983 \beta_{1} - 14380 \beta_{2} ) q^{83} + ( 155904 - 5824 \beta_{1} - 672 \beta_{2} - 20160 \beta_{3} ) q^{84} + ( -529440 - 125880 \beta_{3} ) q^{85} + ( -583680 + 11174 \beta_{3} ) q^{86} + ( 37386 \beta_{1} + 6564 \beta_{2} ) q^{87} + ( 12288 + 35520 \beta_{3} ) q^{88} + ( 8430 \beta_{1} + 8076 \beta_{2} ) q^{89} + ( -9555 \beta_{1} + 19305 \beta_{2} ) q^{90} + ( -277368 + 23429 \beta_{1} + 770 \beta_{2} + 133266 \beta_{3} ) q^{91} + ( 324672 - 8448 \beta_{3} ) q^{92} + ( 248832 - 72720 \beta_{3} ) q^{93} + ( -26972 \beta_{1} + 916 \beta_{2} ) q^{94} + ( -47640 + 121890 \beta_{3} ) q^{95} + ( -5120 \beta_{1} - 1024 \beta_{2} ) q^{96} + ( -6254 \beta_{1} + 3562 \beta_{2} ) q^{97} + ( 206976 + 22050 \beta_{1} - 1862 \beta_{2} + 77567 \beta_{3} ) q^{98} + ( -332982 - 89856 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 128q^{4} + 308q^{7} + 1092q^{9} + O(q^{10})$$ $$4q + 128q^{4} + 308q^{7} + 1092q^{9} - 4440q^{11} + 2688q^{14} - 4320q^{15} + 4096q^{16} - 9984q^{18} + 19488q^{21} + 1536q^{22} + 40584q^{23} - 40700q^{25} + 9856q^{28} - 18264q^{29} - 42240q^{30} - 84000q^{35} + 34944q^{36} - 23192q^{37} + 208608q^{39} + 80640q^{42} - 44696q^{43} - 142080q^{44} + 33792q^{46} - 310268q^{49} + 364800q^{50} + 157824q^{51} + 248616q^{53} + 86016q^{56} - 472992q^{57} - 840192q^{58} - 138240q^{60} - 125580q^{63} + 131072q^{64} + 1293600q^{65} - 434776q^{67} + 1102080q^{70} - 451608q^{71} - 319488q^{72} - 981504q^{74} - 309624q^{77} + 1301760q^{78} + 2092904q^{79} - 252828q^{81} + 623616q^{84} - 2117760q^{85} - 2334720q^{86} + 49152q^{88} - 1109472q^{91} + 1298688q^{92} + 995328q^{93} - 190560q^{95} + 827904q^{98} - 1331928q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 30 x^{2} + 207$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{3} + 18 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 34 \nu$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{2} + 60$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 3 \beta_{1}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{3} - 60$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{2} + 51 \beta_{1}$$$$)/16$$

## 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
 − 3.27984i 3.27984i − 4.38664i 4.38664i
−5.65685 3.84257i 32.0000 204.749i 21.7369i −41.7939 + 340.444i −181.019 714.235 1158.23i
13.2 −5.65685 3.84257i 32.0000 204.749i 21.7369i −41.7939 340.444i −181.019 714.235 1158.23i
13.3 5.65685 29.9539i 32.0000 98.3767i 169.445i 195.794 + 281.627i 181.019 −168.235 556.502i
13.4 5.65685 29.9539i 32.0000 98.3767i 169.445i 195.794 281.627i 181.019 −168.235 556.502i
 $$n$$: 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 14.7.b.a 4
3.b odd 2 1 126.7.c.a 4
4.b odd 2 1 112.7.c.c 4
5.b even 2 1 350.7.b.a 4
5.c odd 4 2 350.7.d.a 8
7.b odd 2 1 inner 14.7.b.a 4
7.c even 3 2 98.7.d.b 8
7.d odd 6 2 98.7.d.b 8
8.b even 2 1 448.7.c.h 4
8.d odd 2 1 448.7.c.e 4
21.c even 2 1 126.7.c.a 4
28.d even 2 1 112.7.c.c 4
35.c odd 2 1 350.7.b.a 4
35.f even 4 2 350.7.d.a 8
56.e even 2 1 448.7.c.e 4
56.h odd 2 1 448.7.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.7.b.a 4 1.a even 1 1 trivial
14.7.b.a 4 7.b odd 2 1 inner
98.7.d.b 8 7.c even 3 2
98.7.d.b 8 7.d odd 6 2
112.7.c.c 4 4.b odd 2 1
112.7.c.c 4 28.d even 2 1
126.7.c.a 4 3.b odd 2 1
126.7.c.a 4 21.c even 2 1
350.7.b.a 4 5.b even 2 1
350.7.b.a 4 35.c odd 2 1
350.7.d.a 8 5.c odd 4 2
350.7.d.a 8 35.f even 4 2
448.7.c.e 4 8.d odd 2 1
448.7.c.e 4 56.e even 2 1
448.7.c.h 4 8.b even 2 1
448.7.c.h 4 56.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -32 + T^{2} )^{2}$$
$3$ $$13248 + 912 T^{2} + T^{4}$$
$5$ $$405720000 + 51600 T^{2} + T^{4}$$
$7$ $$13841287201 - 36235892 T + 202566 T^{2} - 308 T^{3} + T^{4}$$
$11$ $$( 1227492 + 2220 T + T^{2} )^{2}$$
$13$ $$26266196601792 + 15367056 T^{2} + T^{4}$$
$17$ $$126735731638272 + 40214784 T^{2} + T^{4}$$
$19$ $$551809934313408 + 65975568 T^{2} + T^{4}$$
$23$ $$( 100711044 - 20292 T + T^{2} )^{2}$$
$29$ $$( -1357906716 + 9132 T + T^{2} )^{2}$$
$31$ $$869139471230042112 + 2274932736 T^{2} + T^{4}$$
$37$ $$( -1847926364 + 11596 T + T^{2} )^{2}$$
$41$ $$2843152747207621632 + 9071224896 T^{2} + T^{4}$$
$43$ $$( -10521464924 + 22348 T + T^{2} )^{2}$$
$47$ $$12139758672035954688 + 20120477952 T^{2} + T^{4}$$
$53$ $$( 13614948 - 124308 T + T^{2} )^{2}$$
$59$ $$78\!\cdots\!00$$$$+ 180594109200 T^{2} + T^{4}$$
$61$ $$82\!\cdots\!48$$$$+ 121774406928 T^{2} + T^{4}$$
$67$ $$( 11809058788 + 217388 T + T^{2} )^{2}$$
$71$ $$( -95443772508 + 225804 T + T^{2} )^{2}$$
$73$ $$26\!\cdots\!72$$$$+ 228009998400 T^{2} + T^{4}$$
$79$ $$( 268612291876 - 1046452 T + T^{2} )^{2}$$
$83$ $$21\!\cdots\!08$$$$+ 478841902608 T^{2} + T^{4}$$
$89$ $$15\!\cdots\!52$$$$+ 258250641984 T^{2} + T^{4}$$
$97$ $$39\!\cdots\!12$$$$+ 42611214336 T^{2} + T^{4}$$