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Properties

Label 336.2.bq.b.109.24
Level $336$
Weight $2$
Character 336.109
Analytic conductor $2.683$
Analytic rank $0$
Dimension $120$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,2,Mod(37,336)] 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("336.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bq (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 109.24
Character \(\chi\) \(=\) 336.109
Dual form 336.2.bq.b.37.24

$q$-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+(1.07147 + 0.923008i) q^{2} +(0.965926 - 0.258819i) q^{3} +(0.296112 + 1.97796i) q^{4} +(-1.04287 - 0.279436i) q^{5} +(1.27386 + 0.614240i) q^{6} +(-0.0520571 + 2.64524i) q^{7} +(-1.50840 + 2.39264i) q^{8} +(0.866025 - 0.500000i) q^{9} +(-0.859485 - 1.26198i) q^{10} +(0.0302178 + 0.112774i) q^{11} +(0.797955 + 1.83392i) q^{12} +(3.16113 + 3.16113i) q^{13} +(-2.49735 + 2.78626i) q^{14} -1.07966 q^{15} +(-3.82464 + 1.17139i) q^{16} +(2.36989 - 4.10477i) q^{17} +(1.38943 + 0.263612i) q^{18} +(0.640273 - 2.38953i) q^{19} +(0.243906 - 2.14550i) q^{20} +(0.634355 + 2.56858i) q^{21} +(-0.0717140 + 0.148726i) q^{22} +(2.25577 - 1.30237i) q^{23} +(-0.837736 + 2.70152i) q^{24} +(-3.32064 - 1.91717i) q^{25} +(0.469319 + 6.30481i) q^{26} +(0.707107 - 0.707107i) q^{27} +(-5.24759 + 0.680320i) q^{28} +(-1.88062 - 1.88062i) q^{29} +(-1.15682 - 0.996533i) q^{30} +(2.82877 - 4.89958i) q^{31} +(-5.17920 - 2.27505i) q^{32} +(0.0583763 + 0.101111i) q^{33} +(6.32801 - 2.21072i) q^{34} +(0.793463 - 2.74409i) q^{35} +(1.24542 + 1.56491i) q^{36} +(-1.30502 - 0.349678i) q^{37} +(2.89159 - 1.96934i) q^{38} +(3.87157 + 2.23525i) q^{39} +(2.24165 - 2.07371i) q^{40} -12.7143i q^{41} +(-1.69112 + 3.33768i) q^{42} +(-8.99443 + 8.99443i) q^{43} +(-0.214115 + 0.0931633i) q^{44} +(-1.04287 + 0.279436i) q^{45} +(3.61910 + 0.686641i) q^{46} +(3.95118 + 6.84364i) q^{47} +(-3.39113 + 2.12137i) q^{48} +(-6.99458 - 0.275407i) q^{49} +(-1.78841 - 5.11917i) q^{50} +(1.22674 - 4.57827i) q^{51} +(-5.31653 + 7.18863i) q^{52} +(-1.52260 - 5.68244i) q^{53} +(1.41031 - 0.104981i) q^{54} -0.126053i q^{55} +(-6.25059 - 4.11462i) q^{56} -2.47382i q^{57} +(-0.279208 - 3.75087i) q^{58} +(0.863085 + 3.22108i) q^{59} +(-0.319700 - 2.13552i) q^{60} +(-0.633383 + 2.36382i) q^{61} +(7.55330 - 2.63879i) q^{62} +(1.27754 + 2.31687i) q^{63} +(-3.44949 - 7.21810i) q^{64} +(-2.41331 - 4.17997i) q^{65} +(-0.0307773 + 0.162219i) q^{66} +(7.79545 - 2.08878i) q^{67} +(8.82081 + 3.47207i) q^{68} +(1.84183 - 1.84183i) q^{69} +(3.38299 - 2.20785i) q^{70} +2.29386i q^{71} +(-0.109987 + 2.82629i) q^{72} +(1.73205 + 1.00000i) q^{73} +(-1.07553 - 1.57921i) q^{74} +(-3.70369 - 0.992400i) q^{75} +(4.91598 + 0.558864i) q^{76} +(-0.299888 + 0.0740626i) q^{77} +(2.08513 + 5.96851i) q^{78} +(5.83087 + 10.0994i) q^{79} +(4.31592 - 0.152870i) q^{80} +(0.500000 - 0.866025i) q^{81} +(11.7354 - 13.6231i) q^{82} +(-6.30368 - 6.30368i) q^{83} +(-4.89270 + 2.01531i) q^{84} +(-3.61850 + 3.61850i) q^{85} +(-17.9392 + 1.33536i) q^{86} +(-2.30328 - 1.32980i) q^{87} +(-0.315409 - 0.0978078i) q^{88} +(1.47496 - 0.851568i) q^{89} +(-1.37533 - 0.663168i) q^{90} +(-8.52650 + 8.19738i) q^{91} +(3.24400 + 4.07618i) q^{92} +(1.46428 - 5.46477i) q^{93} +(-2.08315 + 10.9798i) q^{94} +(-1.33544 + 2.31305i) q^{95} +(-5.59155 - 0.857055i) q^{96} +18.5880 q^{97} +(-7.24031 - 6.75115i) q^{98} +(0.0825565 + 0.0825565i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8} - 8 q^{10} + 4 q^{11} + 8 q^{13} + 32 q^{14} + 8 q^{15} + 12 q^{16} - 16 q^{17} - 4 q^{18} - 8 q^{19} - 64 q^{20} + 12 q^{21} + 24 q^{22} - 8 q^{24} - 8 q^{26}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.07147 + 0.923008i 0.757646 + 0.652665i
\(3\) 0.965926 − 0.258819i 0.557678 − 0.149429i
\(4\) 0.296112 + 1.97796i 0.148056 + 0.988979i
\(5\) −1.04287 − 0.279436i −0.466385 − 0.124968i 0.0179709 − 0.999839i \(-0.494279\pi\)
−0.484356 + 0.874871i \(0.660946\pi\)
\(6\) 1.27386 + 0.614240i 0.520050 + 0.250762i
\(7\) −0.0520571 + 2.64524i −0.0196757 + 0.999806i
\(8\) −1.50840 + 2.39264i −0.533298 + 0.845927i
\(9\) 0.866025 − 0.500000i 0.288675 − 0.166667i
\(10\) −0.859485 − 1.26198i −0.271793 − 0.399075i
\(11\) 0.0302178 + 0.112774i 0.00911101 + 0.0340027i 0.970332 − 0.241778i \(-0.0777304\pi\)
−0.961221 + 0.275780i \(0.911064\pi\)
\(12\) 0.797955 + 1.83392i 0.230350 + 0.529407i
\(13\) 3.16113 + 3.16113i 0.876739 + 0.876739i 0.993196 − 0.116457i \(-0.0371536\pi\)
−0.116457 + 0.993196i \(0.537154\pi\)
\(14\) −2.49735 + 2.78626i −0.667446 + 0.744658i
\(15\) −1.07966 −0.278766
\(16\) −3.82464 + 1.17139i −0.956159 + 0.292849i
\(17\) 2.36989 − 4.10477i 0.574782 − 0.995552i −0.421283 − 0.906929i \(-0.638420\pi\)
0.996065 − 0.0886231i \(-0.0282467\pi\)
\(18\) 1.38943 + 0.263612i 0.327491 + 0.0621339i
\(19\) 0.640273 − 2.38953i 0.146889 − 0.548196i −0.852775 − 0.522278i \(-0.825082\pi\)
0.999664 − 0.0259182i \(-0.00825094\pi\)
\(20\) 0.243906 − 2.14550i 0.0545391 − 0.479747i
\(21\) 0.634355 + 2.56858i 0.138428 + 0.560510i
\(22\) −0.0717140 + 0.148726i −0.0152895 + 0.0317085i
\(23\) 2.25577 − 1.30237i 0.470361 − 0.271563i −0.246030 − 0.969262i \(-0.579126\pi\)
0.716391 + 0.697699i \(0.245793\pi\)
\(24\) −0.837736 + 2.70152i −0.171002 + 0.551445i
\(25\) −3.32064 − 1.91717i −0.664127 − 0.383434i
\(26\) 0.469319 + 6.30481i 0.0920410 + 1.23648i
\(27\) 0.707107 − 0.707107i 0.136083 − 0.136083i
\(28\) −5.24759 + 0.680320i −0.991701 + 0.128568i
\(29\) −1.88062 − 1.88062i −0.349223 − 0.349223i 0.510597 − 0.859820i \(-0.329424\pi\)
−0.859820 + 0.510597i \(0.829424\pi\)
\(30\) −1.15682 − 0.996533i −0.211206 − 0.181941i
\(31\) 2.82877 − 4.89958i 0.508062 − 0.879990i −0.491894 − 0.870655i \(-0.663695\pi\)
0.999956 − 0.00933478i \(-0.00297140\pi\)
\(32\) −5.17920 − 2.27505i −0.915562 − 0.402176i
\(33\) 0.0583763 + 0.101111i 0.0101620 + 0.0176011i
\(34\) 6.32801 − 2.21072i 1.08524 − 0.379136i
\(35\) 0.793463 − 2.74409i 0.134120 − 0.463836i
\(36\) 1.24542 + 1.56491i 0.207570 + 0.260818i
\(37\) −1.30502 − 0.349678i −0.214543 − 0.0574867i 0.149946 − 0.988694i \(-0.452090\pi\)
−0.364490 + 0.931207i \(0.618757\pi\)
\(38\) 2.89159 − 1.96934i 0.469078 − 0.319470i
\(39\) 3.87157 + 2.23525i 0.619948 + 0.357927i
\(40\) 2.24165 − 2.07371i 0.354436 − 0.327883i
\(41\) − 12.7143i − 1.98564i −0.119608 − 0.992821i \(-0.538164\pi\)
0.119608 − 0.992821i \(-0.461836\pi\)
\(42\) −1.69112 + 3.33768i −0.260946 + 0.515015i
\(43\) −8.99443 + 8.99443i −1.37164 + 1.37164i −0.513619 + 0.858018i \(0.671696\pi\)
−0.858018 + 0.513619i \(0.828304\pi\)
\(44\) −0.214115 + 0.0931633i −0.0322791 + 0.0140449i
\(45\) −1.04287 + 0.279436i −0.155462 + 0.0416558i
\(46\) 3.61910 + 0.686641i 0.533608 + 0.101240i
\(47\) 3.95118 + 6.84364i 0.576339 + 0.998248i 0.995895 + 0.0905184i \(0.0288524\pi\)
−0.419556 + 0.907729i \(0.637814\pi\)
\(48\) −3.39113 + 2.12137i −0.489468 + 0.306193i
\(49\) −6.99458 − 0.275407i −0.999226 − 0.0393438i
\(50\) −1.78841 − 5.11917i −0.252919 − 0.723960i
\(51\) 1.22674 − 4.57827i 0.171779 − 0.641087i
\(52\) −5.31653 + 7.18863i −0.737270 + 0.996883i
\(53\) −1.52260 − 5.68244i −0.209146 − 0.780543i −0.988146 − 0.153519i \(-0.950939\pi\)
0.779000 − 0.627024i \(-0.215727\pi\)
\(54\) 1.41031 − 0.104981i 0.191919 − 0.0142861i
\(55\) − 0.126053i − 0.0169970i
\(56\) −6.25059 − 4.11462i −0.835271 − 0.549839i
\(57\) − 2.47382i − 0.327666i
\(58\) −0.279208 − 3.75087i −0.0366618 − 0.492513i
\(59\) 0.863085 + 3.22108i 0.112364 + 0.419348i 0.999076 − 0.0429735i \(-0.0136831\pi\)
−0.886712 + 0.462322i \(0.847016\pi\)
\(60\) −0.319700 − 2.13552i −0.0412730 − 0.275694i
\(61\) −0.633383 + 2.36382i −0.0810963 + 0.302656i −0.994546 − 0.104295i \(-0.966741\pi\)
0.913450 + 0.406951i \(0.133408\pi\)
\(62\) 7.55330 − 2.63879i 0.959270 − 0.335126i
\(63\) 1.27754 + 2.31687i 0.160955 + 0.291899i
\(64\) −3.44949 − 7.21810i −0.431186 − 0.902263i
\(65\) −2.41331 − 4.17997i −0.299334 − 0.518462i
\(66\) −0.0307773 + 0.162219i −0.00378843 + 0.0199678i
\(67\) 7.79545 − 2.08878i 0.952365 − 0.255186i 0.251000 − 0.967987i \(-0.419241\pi\)
0.701366 + 0.712802i \(0.252574\pi\)
\(68\) 8.82081 + 3.47207i 1.06968 + 0.421050i
\(69\) 1.84183 − 1.84183i 0.221731 − 0.221731i
\(70\) 3.38299 − 2.20785i 0.404345 − 0.263888i
\(71\) 2.29386i 0.272231i 0.990693 + 0.136115i \(0.0434618\pi\)
−0.990693 + 0.136115i \(0.956538\pi\)
\(72\) −0.109987 + 2.82629i −0.0129620 + 0.333081i
\(73\) 1.73205 + 1.00000i 0.202722 + 0.117041i 0.597924 − 0.801553i \(-0.295992\pi\)
−0.395203 + 0.918594i \(0.629326\pi\)
\(74\) −1.07553 − 1.57921i −0.125028 − 0.183579i
\(75\) −3.70369 − 0.992400i −0.427665 − 0.114593i
\(76\) 4.91598 + 0.558864i 0.563902 + 0.0641061i
\(77\) −0.299888 + 0.0740626i −0.0341754 + 0.00844021i
\(78\) 2.08513 + 5.96851i 0.236095 + 0.675801i
\(79\) 5.83087 + 10.0994i 0.656024 + 1.13627i 0.981636 + 0.190764i \(0.0610964\pi\)
−0.325612 + 0.945504i \(0.605570\pi\)
\(80\) 4.31592 − 0.152870i 0.482535 − 0.0170914i
\(81\) 0.500000 − 0.866025i 0.0555556 − 0.0962250i
\(82\) 11.7354 − 13.6231i 1.29596 − 1.50441i
\(83\) −6.30368 − 6.30368i −0.691919 − 0.691919i 0.270735 − 0.962654i \(-0.412733\pi\)
−0.962654 + 0.270735i \(0.912733\pi\)
\(84\) −4.89270 + 2.01531i −0.533837 + 0.219889i
\(85\) −3.61850 + 3.61850i −0.392482 + 0.392482i
\(86\) −17.9392 + 1.33536i −1.93444 + 0.143996i
\(87\) −2.30328 − 1.32980i −0.246938 − 0.142570i
\(88\) −0.315409 − 0.0978078i −0.0336227 − 0.0104263i
\(89\) 1.47496 − 0.851568i 0.156345 − 0.0902660i −0.419786 − 0.907623i \(-0.637895\pi\)
0.576132 + 0.817357i \(0.304562\pi\)
\(90\) −1.37533 − 0.663168i −0.144972 − 0.0699041i
\(91\) −8.52650 + 8.19738i −0.893820 + 0.859319i
\(92\) 3.24400 + 4.07618i 0.338210 + 0.424971i
\(93\) 1.46428 − 5.46477i 0.151839 − 0.566670i
\(94\) −2.08315 + 10.9798i −0.214861 + 1.13248i
\(95\) −1.33544 + 2.31305i −0.137013 + 0.237314i
\(96\) −5.59155 − 0.857055i −0.570685 − 0.0874728i
\(97\) 18.5880 1.88732 0.943661 − 0.330913i \(-0.107357\pi\)
0.943661 + 0.330913i \(0.107357\pi\)
\(98\) −7.24031 − 6.75115i −0.731381 − 0.681969i
\(99\) 0.0825565 + 0.0825565i 0.00829724 + 0.00829724i
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 336.2.bq.b.109.24 yes 120
7.2 even 3 inner 336.2.bq.b.205.18 yes 120
16.5 even 4 inner 336.2.bq.b.277.18 yes 120
112.37 even 12 inner 336.2.bq.b.37.24 ✓ 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bq.b.37.24 ✓ 120 112.37 even 12 inner
336.2.bq.b.109.24 yes 120 1.1 even 1 trivial
336.2.bq.b.205.18 yes 120 7.2 even 3 inner
336.2.bq.b.277.18 yes 120 16.5 even 4 inner