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

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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:

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 \(S_{5}^{\mathrm{new}}(14, [\chi])\).