↑

Properties

Label 14.8.c.b.9.2
Level $14$
Weight $8$
Character 14.9
Analytic conductor $4.373$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [14,8,Mod(9,14)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("14.9"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(14, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.37339035678\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{949})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 238x^{2} + 237x + 56169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 9.2
Root \(7.95146 - 13.7723i\) of defining polynomial
Character \(\chi\) \(=\) 14.9
Dual form 14.8.c.b.11.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.00000 - 6.92820i) q^{2} +(1.40292 + 2.42993i) q^{3} +(-32.0000 - 55.4256i) q^{4} +(219.141 - 379.563i) q^{5} +22.4467 q^{6} +(-893.282 - 159.971i) q^{7} -512.000 q^{8} +(1089.56 - 1887.18i) q^{9} +(-1753.13 - 3036.51i) q^{10} +(2740.43 + 4746.57i) q^{11} +(89.7870 - 155.516i) q^{12} +4006.54 q^{13} +(-4681.44 + 5548.96i) q^{14} +1229.75 q^{15} +(-2048.00 + 3547.24i) q^{16} +(14011.5 + 24268.6i) q^{17} +(-8716.51 - 15097.4i) q^{18} +(-11920.6 + 20647.0i) q^{19} -28050.0 q^{20} +(-864.487 - 2395.04i) q^{21} +43846.9 q^{22} +(36877.3 - 63873.3i) q^{23} +(-718.296 - 1244.13i) q^{24} +(-56983.0 - 98697.4i) q^{25} +(16026.2 - 27758.1i) q^{26} +12250.7 q^{27} +(19718.5 + 54629.8i) q^{28} -98721.3 q^{29} +(4919.00 - 8519.96i) q^{30} +(23743.4 + 41124.8i) q^{31} +(16384.0 + 28377.9i) q^{32} +(-7689.23 + 13318.1i) q^{33} +224184. q^{34} +(-256474. + 304001. i) q^{35} -139464. q^{36} +(50031.2 - 86656.7i) q^{37} +(95364.6 + 165176. i) q^{38} +(5620.87 + 9735.63i) q^{39} +(-112200. + 194336. i) q^{40} +489123. q^{41} +(-20051.3 - 3590.82i) q^{42} +299600. q^{43} +(175388. - 303781. i) q^{44} +(-477536. - 827116. i) q^{45} +(-295018. - 510986. i) q^{46} +(-481369. + 833756. i) q^{47} -11492.7 q^{48} +(772362. + 285798. i) q^{49} -911728. q^{50} +(-39314.1 + 68093.9i) q^{51} +(-128209. - 222065. i) q^{52} +(-918933. - 1.59164e6i) q^{53} +(49002.7 - 84875.1i) q^{54} +2.40216e6 q^{55} +(457360. + 81905.0i) q^{56} -66894.5 q^{57} +(-394885. + 683961. i) q^{58} +(7255.29 + 12566.5i) q^{59} +(-39352.0 - 68159.7i) q^{60} +(-1.01469e6 + 1.75749e6i) q^{61} +379895. q^{62} +(-1.27518e6 + 1.51148e6i) q^{63} +262144. q^{64} +(877997. - 1.52074e6i) q^{65} +(61513.8 + 106545. i) q^{66} +(1.48449e6 + 2.57121e6i) q^{67} +(896736. - 1.55319e6i) q^{68} +206944. q^{69} +(1.08028e6 + 2.99290e6i) q^{70} -4.34296e6 q^{71} +(-557857. + 966236. i) q^{72} +(-750529. - 1.29995e6i) q^{73} +(-400250. - 693253. i) q^{74} +(159885. - 276929. i) q^{75} +1.52583e6 q^{76} +(-1.68867e6 - 4.67841e6i) q^{77} +89933.9 q^{78} +(886182. - 1.53491e6i) q^{79} +(897601. + 1.55469e6i) q^{80} +(-2.36569e6 - 4.09749e6i) q^{81} +(1.95649e6 - 3.38874e6i) q^{82} -1.57509e6 q^{83} +(-105083. + 124556. i) q^{84} +1.22820e7 q^{85} +(1.19840e6 - 2.07569e6i) q^{86} +(-138498. - 239886. i) q^{87} +(-1.40310e6 - 2.43024e6i) q^{88} +(-4.39727e6 + 7.61629e6i) q^{89} -7.64057e6 q^{90} +(-3.57897e6 - 640929. i) q^{91} -4.72029e6 q^{92} +(-66620.4 + 115390. i) q^{93} +(3.85096e6 + 6.67005e6i) q^{94} +(5.22457e6 + 9.04922e6i) q^{95} +(-45970.9 + 79624.0i) q^{96} -1.03493e7 q^{97} +(5.06951e6 - 4.20789e6i) q^{98} +1.19435e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{2} - 56 q^{3} - 128 q^{4} + 14 q^{5} - 896 q^{6} - 1848 q^{7} - 2048 q^{8} + 908 q^{9} - 112 q^{10} - 2408 q^{11} - 3584 q^{12} + 21448 q^{13} - 22176 q^{14} + 52360 q^{15} - 8192 q^{16} + 35098 q^{17}+ \cdots + 43942528 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 − 6.92820i 0.353553 − 0.612372i
\(3\) 1.40292 + 2.42993i 0.0299992 + 0.0519601i 0.880635 − 0.473795i \(-0.157116\pi\)
−0.850636 + 0.525755i \(0.823783\pi\)
\(4\) −32.0000 − 55.4256i −0.250000 − 0.433013i
\(5\) 219.141 − 379.563i 0.784022 − 1.35797i −0.145559 − 0.989350i \(-0.546498\pi\)
0.929581 − 0.368617i \(-0.120169\pi\)
\(6\) 22.4467 0.0424252
\(7\) −893.282 − 159.971i −0.984340 − 0.176278i
\(8\) −512.000 −0.353553
\(9\) 1089.56 − 1887.18i 0.498200 − 0.862908i
\(10\) −1753.13 − 3036.51i −0.554388 − 0.960227i
\(11\) 2740.43 + 4746.57i 0.620790 + 1.07524i 0.989339 + 0.145632i \(0.0465215\pi\)
−0.368549 + 0.929609i \(0.620145\pi\)
\(12\) 89.7870 − 155.516i 0.0149996 − 0.0259800i
\(13\) 4006.54 0.505787 0.252894 − 0.967494i \(-0.418618\pi\)
0.252894 + 0.967494i \(0.418618\pi\)
\(14\) −4681.44 + 5548.96i −0.455964 + 0.540459i
\(15\) 1229.75 0.0940800
\(16\) −2048.00 + 3547.24i −0.125000 + 0.216506i
\(17\) 14011.5 + 24268.6i 0.691693 + 1.19805i 0.971283 + 0.237927i \(0.0764681\pi\)
−0.279590 + 0.960119i \(0.590199\pi\)
\(18\) −8716.51 − 15097.4i −0.352281 − 0.610168i
\(19\) −11920.6 + 20647.0i −0.398712 + 0.690590i −0.993567 − 0.113243i \(-0.963876\pi\)
0.594855 + 0.803833i \(0.297209\pi\)
\(20\) −28050.0 −0.784022
\(21\) −864.487 − 2395.04i −0.0203700 − 0.0564346i
\(22\) 43846.9 0.877930
\(23\) 36877.3 − 63873.3i 0.631992 − 1.09464i −0.355152 − 0.934808i \(-0.615571\pi\)
0.987144 − 0.159833i \(-0.0510956\pi\)
\(24\) −718.296 − 1244.13i −0.0106063 − 0.0183707i
\(25\) −56983.0 − 98697.4i −0.729382 − 1.26333i
\(26\) 16026.2 − 27758.1i 0.178823 − 0.309730i
\(27\) 12250.7 0.119781
\(28\) 19718.5 + 54629.8i 0.169755 + 0.470301i
\(29\) −98721.3 −0.751653 −0.375827 − 0.926690i \(-0.622641\pi\)
−0.375827 + 0.926690i \(0.622641\pi\)
\(30\) 4919.00 − 8519.96i 0.0332623 − 0.0576120i
\(31\) 23743.4 + 41124.8i 0.143145 + 0.247935i 0.928680 − 0.370883i \(-0.120945\pi\)
−0.785534 + 0.618818i \(0.787612\pi\)
\(32\) 16384.0 + 28377.9i 0.0883883 + 0.153093i
\(33\) −7689.23 + 13318.1i −0.0372464 + 0.0645126i
\(34\) 224184. 0.978201
\(35\) −256474. + 304001.i −1.01112 + 1.19850i
\(36\) −139464. −0.498200
\(37\) 50031.2 − 86656.7i 0.162381 − 0.281252i −0.773341 − 0.633990i \(-0.781416\pi\)
0.935722 + 0.352738i \(0.114749\pi\)
\(38\) 95364.6 + 165176.i 0.281932 + 0.488321i
\(39\) 5620.87 + 9735.63i 0.0151732 + 0.0262808i
\(40\) −112200. + 194336.i −0.277194 + 0.480114i
\(41\) 489123. 1.10834 0.554172 − 0.832402i \(-0.313035\pi\)
0.554172 + 0.832402i \(0.313035\pi\)
\(42\) −20051.3 − 3590.82i −0.0417609 − 0.00747862i
\(43\) 299600. 0.574649 0.287324 − 0.957833i \(-0.407234\pi\)
0.287324 + 0.957833i \(0.407234\pi\)
\(44\) 175388. − 303781.i 0.310395 − 0.537620i
\(45\) −477536. − 827116.i −0.781200 − 1.35308i
\(46\) −295018. − 510986.i −0.446886 − 0.774029i
\(47\) −481369. + 833756.i −0.676295 + 1.17138i 0.299794 + 0.954004i \(0.403082\pi\)
−0.976089 + 0.217373i \(0.930251\pi\)
\(48\) −11492.7 −0.0149996
\(49\) 772362. + 285798.i 0.937852 + 0.347034i
\(50\) −911728. −1.03150
\(51\) −39314.1 + 68093.9i −0.0415004 + 0.0718808i
\(52\) −128209. − 222065.i −0.126447 − 0.219012i
\(53\) −918933. − 1.59164e6i −0.847849 − 1.46852i −0.883124 − 0.469139i \(-0.844564\pi\)
0.0352758 − 0.999378i \(-0.488769\pi\)
\(54\) 49002.7 − 84875.1i 0.0423489 − 0.0733504i
\(55\) 2.40216e6 1.94685
\(56\) 457360. + 81905.0i 0.348017 + 0.0623235i
\(57\) −66894.5 −0.0478441
\(58\) −394885. + 683961.i −0.265750 + 0.460292i
\(59\) 7255.29 + 12566.5i 0.00459910 + 0.00796587i 0.868316 − 0.496012i \(-0.165203\pi\)
−0.863717 + 0.503978i \(0.831869\pi\)
\(60\) −39352.0 − 68159.7i −0.0235200 − 0.0407379i
\(61\) −1.01469e6 + 1.75749e6i −0.572370 + 0.991374i 0.423952 + 0.905685i \(0.360643\pi\)
−0.996322 + 0.0856896i \(0.972691\pi\)
\(62\) 379895. 0.202438
\(63\) −1.27518e6 + 1.51148e6i −0.642510 + 0.761574i
\(64\) 262144. 0.125000
\(65\) 877997. − 1.52074e6i 0.396549 − 0.686842i
\(66\) 61513.8 + 106545.i 0.0263372 + 0.0456173i
\(67\) 1.48449e6 + 2.57121e6i 0.602997 + 1.04442i 0.992365 + 0.123339i \(0.0393602\pi\)
−0.389368 + 0.921082i \(0.627306\pi\)
\(68\) 896736. − 1.55319e6i 0.345846 − 0.599023i
\(69\) 206944. 0.0758369
\(70\) 1.08028e6 + 2.99290e6i 0.376439 + 1.04292i
\(71\) −4.34296e6 −1.44006 −0.720031 − 0.693942i \(-0.755872\pi\)
−0.720031 + 0.693942i \(0.755872\pi\)
\(72\) −557857. + 966236.i −0.176140 + 0.305084i
\(73\) −750529. − 1.29995e6i −0.225807 − 0.391109i 0.730754 − 0.682641i \(-0.239169\pi\)
−0.956561 + 0.291531i \(0.905835\pi\)
\(74\) −400250. − 693253.i −0.114821 − 0.198875i
\(75\) 159885. − 276929.i 0.0437617 − 0.0757975i
\(76\) 1.52583e6 0.398712
\(77\) −1.68867e6 − 4.67841e6i −0.421528 − 1.16783i
\(78\) 89933.9 0.0214581
\(79\) 886182. − 1.53491e6i 0.202222 − 0.350259i −0.747022 − 0.664799i \(-0.768517\pi\)
0.949244 + 0.314541i \(0.101850\pi\)
\(80\) 897601. + 1.55469e6i 0.196006 + 0.339492i
\(81\) −2.36569e6 − 4.09749e6i −0.494607 − 0.856684i
\(82\) 1.95649e6 − 3.38874e6i 0.391859 − 0.678720i
\(83\) −1.57509e6 −0.302366 −0.151183 − 0.988506i \(-0.548308\pi\)
−0.151183 + 0.988506i \(0.548308\pi\)
\(84\) −105083. + 124556.i −0.0193444 + 0.0229291i
\(85\) 1.22820e7 2.16921
\(86\) 1.19840e6 − 2.07569e6i 0.203169 − 0.351899i
\(87\) −138498. − 239886.i −0.0225490 − 0.0390560i
\(88\) −1.40310e6 − 2.43024e6i −0.219483 − 0.380155i
\(89\) −4.39727e6 + 7.61629e6i −0.661177 + 1.14519i 0.319130 + 0.947711i \(0.396609\pi\)
−0.980307 + 0.197481i \(0.936724\pi\)
\(90\) −7.64057e6 −1.10478
\(91\) −3.57897e6 − 640929.i −0.497867 − 0.0891590i
\(92\) −4.72029e6 −0.631992
\(93\) −66620.4 + 115390.i −0.00858849 + 0.0148757i
\(94\) 3.85096e6 + 6.67005e6i 0.478213 + 0.828288i
\(95\) 5.22457e6 + 9.04922e6i 0.625199 + 1.08288i
\(96\) −45970.9 + 79624.0i −0.00530315 + 0.00918533i
\(97\) −1.03493e7 −1.15135 −0.575676 − 0.817678i \(-0.695261\pi\)
−0.575676 + 0.817678i \(0.695261\pi\)
\(98\) 5.06951e6 − 4.20789e6i 0.544095 − 0.451620i
\(99\) 1.19435e7 1.23711
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 14.8.c.b.9.2 ✓ 4
3.2 odd 2 126.8.g.d.37.1 4
4.3 odd 2 112.8.i.b.65.1 4
7.2 even 3 98.8.a.f.1.1 2
7.3 odd 6 98.8.c.m.67.1 4
7.4 even 3 inner 14.8.c.b.11.2 yes 4
7.5 odd 6 98.8.a.d.1.2 2
7.6 odd 2 98.8.c.m.79.1 4
21.11 odd 6 126.8.g.d.109.1 4
28.11 odd 6 112.8.i.b.81.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.c.b.9.2 ✓ 4 1.1 even 1 trivial
14.8.c.b.11.2 yes 4 7.4 even 3 inner
98.8.a.d.1.2 2 7.5 odd 6
98.8.a.f.1.1 2 7.2 even 3
98.8.c.m.67.1 4 7.3 odd 6
98.8.c.m.79.1 4 7.6 odd 2
112.8.i.b.65.1 4 4.3 odd 2
112.8.i.b.81.1 4 28.11 odd 6
126.8.g.d.37.1 4 3.2 odd 2
126.8.g.d.109.1 4 21.11 odd 6