# Properties

 Label 14.5.d.a Level 14 Weight 5 Character orbit 14.d Analytic conductor 1.447 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$14 = 2 \cdot 7$$ Weight: $$k$$ = $$5$$ Character orbit: $$[\chi]$$ = 14.d (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.44717948317$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -6 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{3}$$ $$+ 8 \beta_{2} q^{4}$$ $$+ ( 9 + 2 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{5}$$ $$+ ( 16 - 6 \beta_{1} + 32 \beta_{2} - 3 \beta_{3} ) q^{6}$$ $$+ ( -35 - 56 \beta_{2} ) q^{7}$$ $$+ 8 \beta_{3} q^{8}$$ $$+ ( 42 - 36 \beta_{1} + 42 \beta_{2} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -6 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{3}$$ $$+ 8 \beta_{2} q^{4}$$ $$+ ( 9 + 2 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{5}$$ $$+ ( 16 - 6 \beta_{1} + 32 \beta_{2} - 3 \beta_{3} ) q^{6}$$ $$+ ( -35 - 56 \beta_{2} ) q^{7}$$ $$+ 8 \beta_{3} q^{8}$$ $$+ ( 42 - 36 \beta_{1} + 42 \beta_{2} ) q^{9}$$ $$+ ( -32 + 9 \beta_{1} - 16 \beta_{2} - 9 \beta_{3} ) q^{10}$$ $$+ ( -6 \beta_{1} + 27 \beta_{2} - 6 \beta_{3} ) q^{11}$$ $$+ ( 24 + 16 \beta_{1} - 24 \beta_{2} + 32 \beta_{3} ) q^{12}$$ $$+ ( 8 + 72 \beta_{1} + 16 \beta_{2} + 36 \beta_{3} ) q^{13}$$ $$+ ( -35 \beta_{1} - 56 \beta_{3} ) q^{14}$$ $$+ ( -177 - 72 \beta_{3} ) q^{15}$$ $$+ ( -64 - 64 \beta_{2} ) q^{16}$$ $$+ ( 306 + 32 \beta_{1} + 153 \beta_{2} - 32 \beta_{3} ) q^{17}$$ $$+ ( 42 \beta_{1} - 288 \beta_{2} + 42 \beta_{3} ) q^{18}$$ $$+ ( 5 + 90 \beta_{1} - 5 \beta_{2} + 180 \beta_{3} ) q^{19}$$ $$+ ( 72 - 32 \beta_{1} + 144 \beta_{2} - 16 \beta_{3} ) q^{20}$$ $$+ ( 42 - 182 \beta_{1} + 273 \beta_{2} - 154 \beta_{3} ) q^{21}$$ $$+ ( 48 + 27 \beta_{3} ) q^{22}$$ $$+ ( -243 - 240 \beta_{1} - 243 \beta_{2} ) q^{23}$$ $$+ ( -256 + 24 \beta_{1} - 128 \beta_{2} - 24 \beta_{3} ) q^{24}$$ $$+ ( 108 \beta_{1} + 286 \beta_{2} + 108 \beta_{3} ) q^{25}$$ $$+ ( -288 + 8 \beta_{1} + 288 \beta_{2} + 16 \beta_{3} ) q^{26}$$ $$+ ( -459 + 60 \beta_{1} - 918 \beta_{2} + 30 \beta_{3} ) q^{27}$$ $$+ ( 448 + 168 \beta_{2} ) q^{28}$$ $$+ ( 810 - 24 \beta_{3} ) q^{29}$$ $$+ ( 576 - 177 \beta_{1} + 576 \beta_{2} ) q^{30}$$ $$+ ( -182 + 180 \beta_{1} - 91 \beta_{2} - 180 \beta_{3} ) q^{31}$$ $$+ ( -64 \beta_{1} - 64 \beta_{3} ) q^{32}$$ $$+ ( 177 + 72 \beta_{1} - 177 \beta_{2} + 144 \beta_{3} ) q^{33}$$ $$+ ( 256 + 306 \beta_{1} + 512 \beta_{2} + 153 \beta_{3} ) q^{34}$$ $$+ ( -819 + 154 \beta_{1} - 693 \beta_{2} - 28 \beta_{3} ) q^{35}$$ $$+ ( -336 - 288 \beta_{3} ) q^{36}$$ $$+ ( -223 + 270 \beta_{1} - 223 \beta_{2} ) q^{37}$$ $$+ ( -1440 + 5 \beta_{1} - 720 \beta_{2} - 5 \beta_{3} ) q^{38}$$ $$+ ( -276 \beta_{1} + 1656 \beta_{2} - 276 \beta_{3} ) q^{39}$$ $$+ ( 128 + 72 \beta_{1} - 128 \beta_{2} + 144 \beta_{3} ) q^{40}$$ $$+ ( 72 - 416 \beta_{1} + 144 \beta_{2} - 208 \beta_{3} ) q^{41}$$ $$+ ( 1232 + 42 \beta_{1} - 224 \beta_{2} + 273 \beta_{3} ) q^{42}$$ $$+ ( 586 + 648 \beta_{3} ) q^{43}$$ $$+ ( -216 + 48 \beta_{1} - 216 \beta_{2} ) q^{44}$$ $$+ ( 1908 - 408 \beta_{1} + 954 \beta_{2} + 408 \beta_{3} ) q^{45}$$ $$+ ( -243 \beta_{1} - 1920 \beta_{2} - 243 \beta_{3} ) q^{46}$$ $$+ ( 117 - 316 \beta_{1} - 117 \beta_{2} - 632 \beta_{3} ) q^{47}$$ $$+ ( 192 - 256 \beta_{1} + 384 \beta_{2} - 128 \beta_{3} ) q^{48}$$ $$+ ( -1911 + 784 \beta_{2} ) q^{49}$$ $$+ ( -864 + 286 \beta_{3} ) q^{50}$$ $$+ ( 159 + 630 \beta_{1} + 159 \beta_{2} ) q^{51}$$ $$+ ( -128 - 288 \beta_{1} - 64 \beta_{2} + 288 \beta_{3} ) q^{52}$$ $$+ ( 774 \beta_{1} - 1377 \beta_{2} + 774 \beta_{3} ) q^{53}$$ $$+ ( -240 - 459 \beta_{1} + 240 \beta_{2} - 918 \beta_{3} ) q^{54}$$ $$+ ( 339 - 216 \beta_{1} + 678 \beta_{2} - 108 \beta_{3} ) q^{55}$$ $$+ ( 448 \beta_{1} + 168 \beta_{3} ) q^{56}$$ $$+ ( -4365 - 840 \beta_{3} ) q^{57}$$ $$+ ( 192 + 810 \beta_{1} + 192 \beta_{2} ) q^{58}$$ $$+ ( 4122 - 146 \beta_{1} + 2061 \beta_{2} + 146 \beta_{3} ) q^{59}$$ $$+ ( 576 \beta_{1} - 1416 \beta_{2} + 576 \beta_{3} ) q^{60}$$ $$+ ( 1281 - 738 \beta_{1} - 1281 \beta_{2} - 1476 \beta_{3} ) q^{61}$$ $$+ ( 1440 - 182 \beta_{1} + 2880 \beta_{2} - 91 \beta_{3} ) q^{62}$$ $$+ ( 882 + 1260 \beta_{1} - 1470 \beta_{2} + 2016 \beta_{3} ) q^{63}$$ $$+ 512 q^{64}$$ $$+ ( -1512 + 924 \beta_{1} - 1512 \beta_{2} ) q^{65}$$ $$+ ( -1152 + 177 \beta_{1} - 576 \beta_{2} - 177 \beta_{3} ) q^{66}$$ $$+ ( -1674 \beta_{1} + 2531 \beta_{2} - 1674 \beta_{3} ) q^{67}$$ $$+ ( -1224 + 256 \beta_{1} + 1224 \beta_{2} + 512 \beta_{3} ) q^{68}$$ $$+ ( -3111 + 468 \beta_{1} - 6222 \beta_{2} + 234 \beta_{3} ) q^{69}$$ $$+ ( 224 - 819 \beta_{1} + 1456 \beta_{2} - 693 \beta_{3} ) q^{70}$$ $$+ ( 4698 + 456 \beta_{3} ) q^{71}$$ $$+ ( 2304 - 336 \beta_{1} + 2304 \beta_{2} ) q^{72}$$ $$+ ( -5758 - 900 \beta_{1} - 2879 \beta_{2} + 900 \beta_{3} ) q^{73}$$ $$+ ( -223 \beta_{1} + 2160 \beta_{2} - 223 \beta_{3} ) q^{74}$$ $$+ ( -870 + 248 \beta_{1} + 870 \beta_{2} + 496 \beta_{3} ) q^{75}$$ $$+ ( 40 - 1440 \beta_{1} + 80 \beta_{2} - 720 \beta_{3} ) q^{76}$$ $$+ ( 1512 - 126 \beta_{1} + 567 \beta_{2} + 210 \beta_{3} ) q^{77}$$ $$+ ( 2208 + 1656 \beta_{3} ) q^{78}$$ $$+ ( 397 - 972 \beta_{1} + 397 \beta_{2} ) q^{79}$$ $$+ ( -1152 + 128 \beta_{1} - 576 \beta_{2} - 128 \beta_{3} ) q^{80}$$ $$+ ( -108 \beta_{1} + 2169 \beta_{2} - 108 \beta_{3} ) q^{81}$$ $$+ ( 1664 + 72 \beta_{1} - 1664 \beta_{2} + 144 \beta_{3} ) q^{82}$$ $$+ ( 2448 - 200 \beta_{1} + 4896 \beta_{2} - 100 \beta_{3} ) q^{83}$$ $$+ ( -2184 + 1232 \beta_{1} - 1848 \beta_{2} - 224 \beta_{3} ) q^{84}$$ $$+ ( 2595 + 54 \beta_{3} ) q^{85}$$ $$+ ( -5184 + 586 \beta_{1} - 5184 \beta_{2} ) q^{86}$$ $$+ ( -4092 + 1548 \beta_{1} - 2046 \beta_{2} - 1548 \beta_{3} ) q^{87}$$ $$+ ( -216 \beta_{1} + 384 \beta_{2} - 216 \beta_{3} ) q^{88}$$ $$+ ( -2079 + 564 \beta_{1} + 2079 \beta_{2} + 1128 \beta_{3} ) q^{89}$$ $$+ ( -3264 + 1908 \beta_{1} - 6528 \beta_{2} + 954 \beta_{3} ) q^{90}$$ $$+ ( 616 - 504 \beta_{1} - 112 \beta_{2} - 3276 \beta_{3} ) q^{91}$$ $$+ ( 1944 - 1920 \beta_{3} ) q^{92}$$ $$+ ( 9459 - 2166 \beta_{1} + 9459 \beta_{2} ) q^{93}$$ $$+ ( 5056 + 117 \beta_{1} + 2528 \beta_{2} - 117 \beta_{3} ) q^{94}$$ $$+ ( 2460 \beta_{1} - 4455 \beta_{2} + 2460 \beta_{3} ) q^{95}$$ $$+ ( 1024 + 192 \beta_{1} - 1024 \beta_{2} + 384 \beta_{3} ) q^{96}$$ $$+ ( 216 + 2448 \beta_{1} + 432 \beta_{2} + 1224 \beta_{3} ) q^{97}$$ $$+ ( -1911 \beta_{1} + 784 \beta_{3} ) q^{98}$$ $$+ ( -2862 - 1224 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 18q^{3}$$ $$\mathstrut -\mathstrut 16q^{4}$$ $$\mathstrut +\mathstrut 54q^{5}$$ $$\mathstrut -\mathstrut 28q^{7}$$ $$\mathstrut +\mathstrut 84q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 18q^{3}$$ $$\mathstrut -\mathstrut 16q^{4}$$ $$\mathstrut +\mathstrut 54q^{5}$$ $$\mathstrut -\mathstrut 28q^{7}$$ $$\mathstrut +\mathstrut 84q^{9}$$ $$\mathstrut -\mathstrut 96q^{10}$$ $$\mathstrut -\mathstrut 54q^{11}$$ $$\mathstrut +\mathstrut 144q^{12}$$ $$\mathstrut -\mathstrut 708q^{15}$$ $$\mathstrut -\mathstrut 128q^{16}$$ $$\mathstrut +\mathstrut 918q^{17}$$ $$\mathstrut +\mathstrut 576q^{18}$$ $$\mathstrut +\mathstrut 30q^{19}$$ $$\mathstrut -\mathstrut 378q^{21}$$ $$\mathstrut +\mathstrut 192q^{22}$$ $$\mathstrut -\mathstrut 486q^{23}$$ $$\mathstrut -\mathstrut 768q^{24}$$ $$\mathstrut -\mathstrut 572q^{25}$$ $$\mathstrut -\mathstrut 1728q^{26}$$ $$\mathstrut +\mathstrut 1456q^{28}$$ $$\mathstrut +\mathstrut 3240q^{29}$$ $$\mathstrut +\mathstrut 1152q^{30}$$ $$\mathstrut -\mathstrut 546q^{31}$$ $$\mathstrut +\mathstrut 1062q^{33}$$ $$\mathstrut -\mathstrut 1890q^{35}$$ $$\mathstrut -\mathstrut 1344q^{36}$$ $$\mathstrut -\mathstrut 446q^{37}$$ $$\mathstrut -\mathstrut 4320q^{38}$$ $$\mathstrut -\mathstrut 3312q^{39}$$ $$\mathstrut +\mathstrut 768q^{40}$$ $$\mathstrut +\mathstrut 5376q^{42}$$ $$\mathstrut +\mathstrut 2344q^{43}$$ $$\mathstrut -\mathstrut 432q^{44}$$ $$\mathstrut +\mathstrut 5724q^{45}$$ $$\mathstrut +\mathstrut 3840q^{46}$$ $$\mathstrut +\mathstrut 702q^{47}$$ $$\mathstrut -\mathstrut 9212q^{49}$$ $$\mathstrut -\mathstrut 3456q^{50}$$ $$\mathstrut +\mathstrut 318q^{51}$$ $$\mathstrut -\mathstrut 384q^{52}$$ $$\mathstrut +\mathstrut 2754q^{53}$$ $$\mathstrut -\mathstrut 1440q^{54}$$ $$\mathstrut -\mathstrut 17460q^{57}$$ $$\mathstrut +\mathstrut 384q^{58}$$ $$\mathstrut +\mathstrut 12366q^{59}$$ $$\mathstrut +\mathstrut 2832q^{60}$$ $$\mathstrut +\mathstrut 7686q^{61}$$ $$\mathstrut +\mathstrut 6468q^{63}$$ $$\mathstrut +\mathstrut 2048q^{64}$$ $$\mathstrut -\mathstrut 3024q^{65}$$ $$\mathstrut -\mathstrut 3456q^{66}$$ $$\mathstrut -\mathstrut 5062q^{67}$$ $$\mathstrut -\mathstrut 7344q^{68}$$ $$\mathstrut -\mathstrut 2016q^{70}$$ $$\mathstrut +\mathstrut 18792q^{71}$$ $$\mathstrut +\mathstrut 4608q^{72}$$ $$\mathstrut -\mathstrut 17274q^{73}$$ $$\mathstrut -\mathstrut 4320q^{74}$$ $$\mathstrut -\mathstrut 5220q^{75}$$ $$\mathstrut +\mathstrut 4914q^{77}$$ $$\mathstrut +\mathstrut 8832q^{78}$$ $$\mathstrut +\mathstrut 794q^{79}$$ $$\mathstrut -\mathstrut 3456q^{80}$$ $$\mathstrut -\mathstrut 4338q^{81}$$ $$\mathstrut +\mathstrut 9984q^{82}$$ $$\mathstrut -\mathstrut 5040q^{84}$$ $$\mathstrut +\mathstrut 10380q^{85}$$ $$\mathstrut -\mathstrut 10368q^{86}$$ $$\mathstrut -\mathstrut 12276q^{87}$$ $$\mathstrut -\mathstrut 768q^{88}$$ $$\mathstrut -\mathstrut 12474q^{89}$$ $$\mathstrut +\mathstrut 2688q^{91}$$ $$\mathstrut +\mathstrut 7776q^{92}$$ $$\mathstrut +\mathstrut 18918q^{93}$$ $$\mathstrut +\mathstrut 15168q^{94}$$ $$\mathstrut +\mathstrut 8910q^{95}$$ $$\mathstrut +\mathstrut 6144q^{96}$$ $$\mathstrut -\mathstrut 11448q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$2$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}$$

## Character Values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−1.41421 2.44949i −12.9853 7.49706i −4.00000 + 6.92820i 21.9853 12.6932i 42.4098i −7.00000 48.4974i 22.6274 71.9117 + 124.555i −62.1838 35.9018i
3.2 1.41421 + 2.44949i 3.98528 + 2.30090i −4.00000 + 6.92820i 5.01472 2.89525i 13.0159i −7.00000 48.4974i −22.6274 −29.9117 51.8086i 14.1838 + 8.18900i
5.1 −1.41421 + 2.44949i −12.9853 + 7.49706i −4.00000 6.92820i 21.9853 + 12.6932i 42.4098i −7.00000 + 48.4974i 22.6274 71.9117 124.555i −62.1838 + 35.9018i
5.2 1.41421 2.44949i 3.98528 2.30090i −4.00000 6.92820i 5.01472 + 2.89525i 13.0159i −7.00000 + 48.4974i −22.6274 −29.9117 + 51.8086i 14.1838 8.18900i
 $$n$$: e.g. 2-40 or 990-1000 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
7.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{5}^{\mathrm{new}}(14, [\chi])$$.