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Properties

Label 336.2.bq.b.109.25
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 109.25
Character \(\chi\) \(=\) 336.109
Dual form 336.2.bq.b.37.25

$q$-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+(1.18402 + 0.773373i) q^{2} +(-0.965926 + 0.258819i) q^{3} +(0.803789 + 1.83137i) q^{4} +(-4.06164 - 1.08831i) q^{5} +(-1.34384 - 0.440575i) q^{6} +(-2.62820 + 0.304251i) q^{7} +(-0.464633 + 2.79000i) q^{8} +(0.866025 - 0.500000i) q^{9} +(-3.96738 - 4.42974i) q^{10} +(-0.970020 - 3.62016i) q^{11} +(-1.25039 - 1.56093i) q^{12} +(0.0588814 + 0.0588814i) q^{13} +(-3.34713 - 1.67234i) q^{14} +4.20492 q^{15} +(-2.70785 + 2.94407i) q^{16} +(-2.36571 + 4.09753i) q^{17} +(1.41207 + 0.0777523i) q^{18} +(0.709823 - 2.64909i) q^{19} +(-1.27160 - 8.31315i) q^{20} +(2.45990 - 0.974112i) q^{21} +(1.65122 - 5.03652i) q^{22} +(-6.09208 + 3.51726i) q^{23} +(-0.273305 - 2.81519i) q^{24} +(10.9824 + 6.34068i) q^{25} +(0.0241793 + 0.115254i) q^{26} +(-0.707107 + 0.707107i) q^{27} +(-2.66971 - 4.56866i) q^{28} +(1.33075 + 1.33075i) q^{29} +(4.97869 + 3.25197i) q^{30} +(-4.34953 + 7.53361i) q^{31} +(-5.48300 + 1.39166i) q^{32} +(1.87394 + 3.24575i) q^{33} +(-5.96996 + 3.02197i) q^{34} +(11.0059 + 1.62455i) q^{35} +(1.61179 + 1.18412i) q^{36} +(3.23201 + 0.866013i) q^{37} +(2.88918 - 2.58761i) q^{38} +(-0.0721147 - 0.0416354i) q^{39} +(4.92357 - 10.8263i) q^{40} -6.05310i q^{41} +(3.66591 + 0.749055i) q^{42} +(-1.31081 + 1.31081i) q^{43} +(5.85018 - 4.68632i) q^{44} +(-4.06164 + 1.08831i) q^{45} +(-9.93328 - 0.546950i) q^{46} +(-1.56674 - 2.71367i) q^{47} +(1.85360 - 3.54460i) q^{48} +(6.81486 - 1.59926i) q^{49} +(8.09961 + 16.0009i) q^{50} +(1.22458 - 4.57020i) q^{51} +(-0.0605055 + 0.155162i) q^{52} +(-0.818053 - 3.05301i) q^{53} +(-1.38408 + 0.290369i) q^{54} +15.7595i q^{55} +(0.372288 - 7.47405i) q^{56} +2.74254i q^{57} +(0.546464 + 2.60479i) q^{58} +(-1.60481 - 5.98923i) q^{59} +(3.37987 + 7.70077i) q^{60} +(-0.0796808 + 0.297373i) q^{61} +(-10.9762 + 5.55611i) q^{62} +(-2.12396 + 1.57759i) q^{63} +(-7.56823 - 2.59266i) q^{64} +(-0.175074 - 0.303237i) q^{65} +(-0.291405 + 5.29227i) q^{66} +(-12.5143 + 3.35320i) q^{67} +(-9.40563 - 1.03894i) q^{68} +(4.97416 - 4.97416i) q^{69} +(11.7748 + 10.4352i) q^{70} -4.95669i q^{71} +(0.992617 + 2.64853i) q^{72} +(10.5223 + 6.07505i) q^{73} +(3.15700 + 3.52492i) q^{74} +(-12.2493 - 3.28218i) q^{75} +(5.42202 - 0.829364i) q^{76} +(3.65084 + 9.21938i) q^{77} +(-0.0531853 - 0.105069i) q^{78} +(-3.19808 - 5.53924i) q^{79} +(14.2024 - 9.01079i) q^{80} +(0.500000 - 0.866025i) q^{81} +(4.68130 - 7.16697i) q^{82} +(1.77457 + 1.77457i) q^{83} +(3.76120 + 3.72201i) q^{84} +(14.0681 - 14.0681i) q^{85} +(-2.56577 + 0.538277i) q^{86} +(-1.62983 - 0.940982i) q^{87} +(10.5510 - 1.02431i) q^{88} +(0.469248 - 0.270920i) q^{89} +(-5.65072 - 1.85258i) q^{90} +(-0.172667 - 0.136837i) q^{91} +(-11.3382 - 8.32972i) q^{92} +(2.25148 - 8.40265i) q^{93} +(0.243634 - 4.42469i) q^{94} +(-5.76609 + 9.98716i) q^{95} +(4.93598 - 2.76334i) q^{96} -6.14973 q^{97} +(9.30574 + 3.37687i) q^{98} +(-2.65014 - 2.65014i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.18402 + 0.773373i 0.837226 + 0.546857i
\(3\) −0.965926 + 0.258819i −0.557678 + 0.149429i
\(4\) 0.803789 + 1.83137i 0.401895 + 0.915686i
\(5\) −4.06164 − 1.08831i −1.81642 − 0.486709i −0.820086 − 0.572241i \(-0.806074\pi\)
−0.996335 + 0.0855320i \(0.972741\pi\)
\(6\) −1.34384 − 0.440575i −0.548619 − 0.179864i
\(7\) −2.62820 + 0.304251i −0.993366 + 0.114996i
\(8\) −0.464633 + 2.79000i −0.164273 + 0.986415i
\(9\) 0.866025 − 0.500000i 0.288675 − 0.166667i
\(10\) −3.96738 − 4.42974i −1.25459 − 1.40081i
\(11\) −0.970020 − 3.62016i −0.292472 − 1.09152i −0.943204 − 0.332213i \(-0.892205\pi\)
0.650732 − 0.759307i \(-0.274462\pi\)
\(12\) −1.25039 − 1.56093i −0.360958 − 0.450603i
\(13\) 0.0588814 + 0.0588814i 0.0163308 + 0.0163308i 0.715225 − 0.698894i \(-0.246324\pi\)
−0.698894 + 0.715225i \(0.746324\pi\)
\(14\) −3.34713 − 1.67234i −0.894558 − 0.446952i
\(15\) 4.20492 1.08571
\(16\) −2.70785 + 2.94407i −0.676961 + 0.736019i
\(17\) −2.36571 + 4.09753i −0.573769 + 0.993797i 0.422405 + 0.906407i \(0.361186\pi\)
−0.996174 + 0.0873900i \(0.972147\pi\)
\(18\) 1.41207 + 0.0777523i 0.332829 + 0.0183264i
\(19\) 0.709823 − 2.64909i 0.162844 − 0.607744i −0.835461 − 0.549550i \(-0.814799\pi\)
0.998305 − 0.0581939i \(-0.0185342\pi\)
\(20\) −1.27160 − 8.31315i −0.284338 − 1.85888i
\(21\) 2.45990 − 0.974112i 0.536794 − 0.212569i
\(22\) 1.65122 − 5.03652i 0.352041 − 1.07379i
\(23\) −6.09208 + 3.51726i −1.27029 + 0.733400i −0.975042 − 0.222021i \(-0.928735\pi\)
−0.295245 + 0.955422i \(0.595401\pi\)
\(24\) −0.273305 − 2.81519i −0.0557881 − 0.574649i
\(25\) 10.9824 + 6.34068i 2.19648 + 1.26814i
\(26\) 0.0241793 + 0.115254i 0.00474194 + 0.0226031i
\(27\) −0.707107 + 0.707107i −0.136083 + 0.136083i
\(28\) −2.66971 − 4.56866i −0.504529 − 0.863395i
\(29\) 1.33075 + 1.33075i 0.247114 + 0.247114i 0.819785 − 0.572671i \(-0.194093\pi\)
−0.572671 + 0.819785i \(0.694093\pi\)
\(30\) 4.97869 + 3.25197i 0.908981 + 0.593726i
\(31\) −4.34953 + 7.53361i −0.781199 + 1.35308i 0.150045 + 0.988679i \(0.452058\pi\)
−0.931244 + 0.364397i \(0.881275\pi\)
\(32\) −5.48300 + 1.39166i −0.969267 + 0.246013i
\(33\) 1.87394 + 3.24575i 0.326210 + 0.565013i
\(34\) −5.96996 + 3.02197i −1.02384 + 0.518263i
\(35\) 11.0059 + 1.62455i 1.86034 + 0.274599i
\(36\) 1.61179 + 1.18412i 0.268631 + 0.197353i
\(37\) 3.23201 + 0.866013i 0.531339 + 0.142372i 0.514506 − 0.857487i \(-0.327975\pi\)
0.0168322 + 0.999858i \(0.494642\pi\)
\(38\) 2.88918 − 2.58761i 0.468687 − 0.419766i
\(39\) −0.0721147 − 0.0416354i −0.0115476 − 0.00666700i
\(40\) 4.92357 − 10.8263i 0.778485 − 1.71179i
\(41\) − 6.05310i − 0.945336i −0.881241 − 0.472668i \(-0.843291\pi\)
0.881241 − 0.472668i \(-0.156709\pi\)
\(42\) 3.66591 + 0.749055i 0.565663 + 0.115582i
\(43\) −1.31081 + 1.31081i −0.199897 + 0.199897i −0.799956 − 0.600059i \(-0.795144\pi\)
0.600059 + 0.799956i \(0.295144\pi\)
\(44\) 5.85018 − 4.68632i 0.881947 − 0.706489i
\(45\) −4.06164 + 1.08831i −0.605474 + 0.162236i
\(46\) −9.93328 − 0.546950i −1.46458 − 0.0806434i
\(47\) −1.56674 − 2.71367i −0.228532 − 0.395829i 0.728841 − 0.684683i \(-0.240059\pi\)
−0.957373 + 0.288854i \(0.906726\pi\)
\(48\) 1.85360 − 3.54460i 0.267543 − 0.511619i
\(49\) 6.81486 − 1.59926i 0.973552 − 0.228466i
\(50\) 8.09961 + 16.0009i 1.14546 + 2.26287i
\(51\) 1.22458 − 4.57020i 0.171476 − 0.639956i
\(52\) −0.0605055 + 0.155162i −0.00839060 + 0.0215171i
\(53\) −0.818053 − 3.05301i −0.112368 − 0.419364i 0.886708 − 0.462329i \(-0.152986\pi\)
−0.999077 + 0.0429654i \(0.986319\pi\)
\(54\) −1.38408 + 0.290369i −0.188350 + 0.0395142i
\(55\) 15.7595i 2.12501i
\(56\) 0.372288 − 7.47405i 0.0497491 − 0.998762i
\(57\) 2.74254i 0.363259i
\(58\) 0.546464 + 2.60479i 0.0717542 + 0.342026i
\(59\) −1.60481 − 5.98923i −0.208929 − 0.779732i −0.988216 − 0.153065i \(-0.951086\pi\)
0.779288 − 0.626667i \(-0.215581\pi\)
\(60\) 3.37987 + 7.70077i 0.436339 + 0.994166i
\(61\) −0.0796808 + 0.297373i −0.0102021 + 0.0380747i −0.970839 − 0.239731i \(-0.922941\pi\)
0.960637 + 0.277806i \(0.0896073\pi\)
\(62\) −10.9762 + 5.55611i −1.39398 + 0.705627i
\(63\) −2.12396 + 1.57759i −0.267594 + 0.198757i
\(64\) −7.56823 − 2.59266i −0.946029 − 0.324082i
\(65\) −0.175074 − 0.303237i −0.0217152 − 0.0376119i
\(66\) −0.291405 + 5.29227i −0.0358695 + 0.651434i
\(67\) −12.5143 + 3.35320i −1.52887 + 0.409659i −0.922650 − 0.385639i \(-0.873981\pi\)
−0.606219 + 0.795298i \(0.707314\pi\)
\(68\) −9.40563 − 1.03894i −1.14060 − 0.125990i
\(69\) 4.97416 − 4.97416i 0.598819 − 0.598819i
\(70\) 11.7748 + 10.4352i 1.40736 + 1.24724i
\(71\) − 4.95669i − 0.588251i −0.955767 − 0.294126i \(-0.904972\pi\)
0.955767 − 0.294126i \(-0.0950284\pi\)
\(72\) 0.992617 + 2.64853i 0.116981 + 0.312132i
\(73\) 10.5223 + 6.07505i 1.23154 + 0.711031i 0.967351 − 0.253440i \(-0.0815619\pi\)
0.264190 + 0.964471i \(0.414895\pi\)
\(74\) 3.15700 + 3.52492i 0.366993 + 0.409764i
\(75\) −12.2493 − 3.28218i −1.41442 − 0.378993i
\(76\) 5.42202 − 0.829364i 0.621949 − 0.0951346i
\(77\) 3.65084 + 9.21938i 0.416052 + 1.05065i
\(78\) −0.0531853 − 0.105069i −0.00602204 − 0.0118967i
\(79\) −3.19808 − 5.53924i −0.359812 − 0.623213i 0.628117 − 0.778119i \(-0.283826\pi\)
−0.987929 + 0.154906i \(0.950493\pi\)
\(80\) 14.2024 − 9.01079i 1.58787 − 1.00744i
\(81\) 0.500000 − 0.866025i 0.0555556 − 0.0962250i
\(82\) 4.68130 − 7.16697i 0.516964 − 0.791460i
\(83\) 1.77457 + 1.77457i 0.194785 + 0.194785i 0.797760 − 0.602975i \(-0.206018\pi\)
−0.602975 + 0.797760i \(0.706018\pi\)
\(84\) 3.76120 + 3.72201i 0.410381 + 0.406105i
\(85\) 14.0681 − 14.0681i 1.52590 − 1.52590i
\(86\) −2.56577 + 0.538277i −0.276674 + 0.0580439i
\(87\) −1.62983 − 0.940982i −0.174736 − 0.100884i
\(88\) 10.5510 − 1.02431i 1.12474 − 0.109192i
\(89\) 0.469248 − 0.270920i 0.0497402 − 0.0287175i −0.474924 − 0.880027i \(-0.657524\pi\)
0.524664 + 0.851309i \(0.324191\pi\)
\(90\) −5.65072 − 1.85258i −0.595638 − 0.195279i
\(91\) −0.172667 − 0.136837i −0.0181004 − 0.0143444i
\(92\) −11.3382 − 8.32972i −1.18209 − 0.868434i
\(93\) 2.25148 − 8.40265i 0.233468 − 0.871314i
\(94\) 0.243634 − 4.42469i 0.0251290 − 0.456372i
\(95\) −5.76609 + 9.98716i −0.591588 + 1.02466i
\(96\) 4.93598 − 2.76334i 0.503777 − 0.282033i
\(97\) −6.14973 −0.624411 −0.312205 − 0.950015i \(-0.601068\pi\)
−0.312205 + 0.950015i \(0.601068\pi\)
\(98\) 9.30574 + 3.37687i 0.940021 + 0.341116i
\(99\) −2.65014 − 2.65014i −0.266350 − 0.266350i
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.109.25 yes 120
7.2 even 3 inner 336.2.bq.b.205.15 yes 120
16.5 even 4 inner 336.2.bq.b.277.15 yes 120
112.37 even 12 inner 336.2.bq.b.37.25 ✓ 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bq.b.37.25 ✓ 120 112.37 even 12 inner
336.2.bq.b.109.25 yes 120 1.1 even 1 trivial
336.2.bq.b.205.15 yes 120 7.2 even 3 inner
336.2.bq.b.277.15 yes 120 16.5 even 4 inner