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Properties

Label 384.2.v.a.13.1
Level $384$
Weight $2$
Character 384.13
Analytic conductor $3.066$
Analytic rank $0$
Dimension $512$
Inner twists $2$

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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] = [384,2,Mod(13,384)] 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("384.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(32)) chi = DirichletCharacter(H, H._module([0, 15, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.v (of order \(32\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(512\)
Relative dimension: \(32\) over \(\Q(\zeta_{32})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label 13.1
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.1

$q$-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+(-1.40051 - 0.196427i) q^{2} +(0.290285 + 0.956940i) q^{3} +(1.92283 + 0.550195i) q^{4} +(0.0502132 + 0.509823i) q^{5} +(-0.218576 - 1.39722i) q^{6} +(-1.66106 + 2.48595i) q^{7} +(-2.58487 - 1.14825i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(0.0298193 - 0.723874i) q^{10} +(-2.39179 + 1.27844i) q^{11} +(0.0316653 + 1.99975i) q^{12} +(-0.147254 - 0.0145033i) q^{13} +(2.81463 - 3.15532i) q^{14} +(-0.473294 + 0.196045i) q^{15} +(3.39457 + 2.11586i) q^{16} +(-5.02760 - 2.08250i) q^{17} +(1.27361 - 0.614756i) q^{18} +(-0.869829 + 0.713850i) q^{19} +(-0.183951 + 1.00793i) q^{20} +(-2.86109 - 0.867902i) q^{21} +(3.60084 - 1.32065i) q^{22} +(-0.0792843 + 0.398589i) q^{23} +(0.348458 - 2.80688i) q^{24} +(4.64653 - 0.924252i) q^{25} +(0.203382 + 0.0492367i) q^{26} +(-0.773010 - 0.634393i) q^{27} +(-4.56170 + 3.86617i) q^{28} +(1.28926 - 2.41204i) q^{29} +(0.701360 - 0.181594i) q^{30} +(-3.00006 + 3.00006i) q^{31} +(-4.33850 - 3.63007i) q^{32} +(-1.91769 - 1.91769i) q^{33} +(6.63212 + 3.90411i) q^{34} +(-1.35080 - 0.722020i) q^{35} +(-1.90445 + 0.610798i) q^{36} +(-5.49304 + 6.69329i) q^{37} +(1.35842 - 0.828893i) q^{38} +(-0.0288669 - 0.145124i) q^{39} +(0.455609 - 1.37548i) q^{40} +(-7.61357 - 1.51443i) q^{41} +(3.83649 + 1.77750i) q^{42} +(-1.01220 + 3.33679i) q^{43} +(-5.30241 + 1.14227i) q^{44} +(-0.324993 - 0.396006i) q^{45} +(0.189332 - 0.542652i) q^{46} +(-3.94790 + 9.53108i) q^{47} +(-1.03936 + 3.86261i) q^{48} +(-0.742058 - 1.79149i) q^{49} +(-6.68904 + 0.381716i) q^{50} +(0.533393 - 5.41563i) q^{51} +(-0.275166 - 0.108906i) q^{52} +(4.05063 + 7.57820i) q^{53} +(0.957994 + 1.04031i) q^{54} +(-0.771878 - 1.15520i) q^{55} +(7.14811 - 4.51855i) q^{56} +(-0.935610 - 0.625154i) q^{57} +(-2.27941 + 3.12483i) q^{58} +(6.40642 - 0.630977i) q^{59} +(-1.01793 + 0.116558i) q^{60} +(9.35612 - 2.83815i) q^{61} +(4.79090 - 3.61231i) q^{62} -2.98983i q^{63} +(5.36306 + 5.93613i) q^{64} -0.0758019i q^{65} +(2.30905 + 3.06243i) q^{66} +(-0.0861833 + 0.0261434i) q^{67} +(-8.52145 - 6.77046i) q^{68} +(-0.404441 + 0.0398339i) q^{69} +(1.74999 + 1.27653i) q^{70} +(8.16418 + 5.45513i) q^{71} +(2.78717 - 0.481341i) q^{72} +(-9.01176 - 13.4871i) q^{73} +(9.00778 - 8.29501i) q^{74} +(2.23327 + 4.17815i) q^{75} +(-2.06529 + 0.894039i) q^{76} +(0.794772 - 8.06946i) q^{77} +(0.0119220 + 0.208917i) q^{78} +(1.41245 + 3.40995i) q^{79} +(-0.908265 + 1.83688i) q^{80} +(0.382683 - 0.923880i) q^{81} +(10.3654 + 3.61648i) q^{82} +(2.88556 + 3.51607i) q^{83} +(-5.02388 - 3.24299i) q^{84} +(0.809255 - 2.66776i) q^{85} +(2.07303 - 4.47437i) q^{86} +(2.68243 + 0.533569i) q^{87} +(7.65043 - 0.558225i) q^{88} +(-0.951785 - 4.78495i) q^{89} +(0.377369 + 0.618446i) q^{90} +(0.280653 - 0.341977i) q^{91} +(-0.371752 + 0.722798i) q^{92} +(-3.74175 - 2.00001i) q^{93} +(7.40122 - 12.5729i) q^{94} +(-0.407614 - 0.407614i) q^{95} +(2.21436 - 5.20544i) q^{96} +(8.57980 - 8.57980i) q^{97} +(0.687360 + 2.65475i) q^{98} +(1.27844 - 2.39179i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 512 q - 96 q^{50} - 96 q^{52} - 32 q^{54} - 224 q^{56} - 192 q^{60} - 192 q^{62} - 192 q^{64} - 192 q^{66} - 192 q^{68} - 192 q^{70} - 224 q^{74} - 32 q^{76} - 96 q^{78} - 96 q^{80}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(e\left(\frac{15}{32}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40051 − 0.196427i −0.990307 − 0.138895i
\(3\) 0.290285 + 0.956940i 0.167596 + 0.552490i
\(4\) 1.92283 + 0.550195i 0.961416 + 0.275097i
\(5\) 0.0502132 + 0.509823i 0.0224560 + 0.228000i 0.999862 + 0.0166280i \(0.00529312\pi\)
−0.977406 + 0.211372i \(0.932207\pi\)
\(6\) −0.218576 − 1.39722i −0.0892334 − 0.570413i
\(7\) −1.66106 + 2.48595i −0.627822 + 0.939602i 0.372113 + 0.928188i \(0.378633\pi\)
−0.999935 + 0.0114147i \(0.996367\pi\)
\(8\) −2.58487 − 1.14825i −0.913888 − 0.405967i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) 0.0298193 − 0.723874i 0.00942968 − 0.228909i
\(11\) −2.39179 + 1.27844i −0.721153 + 0.385464i −0.790773 − 0.612109i \(-0.790321\pi\)
0.0696203 + 0.997574i \(0.477821\pi\)
\(12\) 0.0316653 + 1.99975i 0.00914100 + 0.577278i
\(13\) −0.147254 − 0.0145033i −0.0408410 − 0.00402249i 0.0775760 − 0.996986i \(-0.475282\pi\)
−0.118417 + 0.992964i \(0.537782\pi\)
\(14\) 2.81463 − 3.15532i 0.752243 − 0.843293i
\(15\) −0.473294 + 0.196045i −0.122204 + 0.0506186i
\(16\) 3.39457 + 2.11586i 0.848643 + 0.528966i
\(17\) −5.02760 − 2.08250i −1.21937 − 0.505080i −0.322162 − 0.946685i \(-0.604409\pi\)
−0.897210 + 0.441604i \(0.854409\pi\)
\(18\) 1.27361 − 0.614756i 0.300192 − 0.144899i
\(19\) −0.869829 + 0.713850i −0.199552 + 0.163768i −0.728863 − 0.684660i \(-0.759951\pi\)
0.529310 + 0.848428i \(0.322451\pi\)
\(20\) −0.183951 + 1.00793i −0.0411326 + 0.225380i
\(21\) −2.86109 − 0.867902i −0.624341 − 0.189392i
\(22\) 3.60084 − 1.32065i 0.767702 − 0.281563i
\(23\) −0.0792843 + 0.398589i −0.0165319 + 0.0831115i −0.988170 − 0.153359i \(-0.950991\pi\)
0.971639 + 0.236471i \(0.0759908\pi\)
\(24\) 0.348458 − 2.80688i 0.0711286 − 0.572952i
\(25\) 4.64653 − 0.924252i 0.929306 − 0.184850i
\(26\) 0.203382 + 0.0492367i 0.0398864 + 0.00965611i
\(27\) −0.773010 − 0.634393i −0.148766 − 0.122089i
\(28\) −4.56170 + 3.86617i −0.862081 + 0.730637i
\(29\) 1.28926 − 2.41204i 0.239410 − 0.447905i −0.733307 − 0.679897i \(-0.762024\pi\)
0.972718 + 0.231992i \(0.0745244\pi\)
\(30\) 0.701360 − 0.181594i 0.128050 − 0.0331544i
\(31\) −3.00006 + 3.00006i −0.538827 + 0.538827i −0.923184 − 0.384358i \(-0.874423\pi\)
0.384358 + 0.923184i \(0.374423\pi\)
\(32\) −4.33850 − 3.63007i −0.766946 − 0.641711i
\(33\) −1.91769 − 1.91769i −0.333827 − 0.333827i
\(34\) 6.63212 + 3.90411i 1.13740 + 0.669549i
\(35\) −1.35080 − 0.722020i −0.228328 − 0.122044i
\(36\) −1.90445 + 0.610798i −0.317408 + 0.101800i
\(37\) −5.49304 + 6.69329i −0.903051 + 1.10037i 0.0914675 + 0.995808i \(0.470844\pi\)
−0.994518 + 0.104563i \(0.966656\pi\)
\(38\) 1.35842 − 0.828893i 0.220365 − 0.134464i
\(39\) −0.0288669 − 0.145124i −0.00462240 − 0.0232384i
\(40\) 0.455609 − 1.37548i 0.0720381 − 0.217483i
\(41\) −7.61357 − 1.51443i −1.18904 − 0.236515i −0.439357 − 0.898312i \(-0.644794\pi\)
−0.749682 + 0.661798i \(0.769794\pi\)
\(42\) 3.83649 + 1.77750i 0.591984 + 0.274274i
\(43\) −1.01220 + 3.33679i −0.154360 + 0.508855i −0.999677 − 0.0254088i \(-0.991911\pi\)
0.845318 + 0.534264i \(0.179411\pi\)
\(44\) −5.30241 + 1.14227i −0.799368 + 0.172204i
\(45\) −0.324993 − 0.396006i −0.0484472 − 0.0590330i
\(46\) 0.189332 − 0.542652i 0.0279154 − 0.0800097i
\(47\) −3.94790 + 9.53108i −0.575861 + 1.39025i 0.320637 + 0.947202i \(0.396103\pi\)
−0.896498 + 0.443048i \(0.853897\pi\)
\(48\) −1.03936 + 3.86261i −0.150019 + 0.557519i
\(49\) −0.742058 − 1.79149i −0.106008 − 0.255927i
\(50\) −6.68904 + 0.381716i −0.945973 + 0.0539828i
\(51\) 0.533393 − 5.41563i 0.0746899 − 0.758340i
\(52\) −0.275166 − 0.108906i −0.0381586 − 0.0151025i
\(53\) 4.05063 + 7.57820i 0.556397 + 1.04095i 0.990163 + 0.139920i \(0.0446845\pi\)
−0.433766 + 0.901026i \(0.642815\pi\)
\(54\) 0.957994 + 1.04031i 0.130366 + 0.141568i
\(55\) −0.771878 − 1.15520i −0.104080 − 0.155767i
\(56\) 7.14811 − 4.51855i 0.955206 − 0.603816i
\(57\) −0.935610 − 0.625154i −0.123925 − 0.0828037i
\(58\) −2.27941 + 3.12483i −0.299301 + 0.410311i
\(59\) 6.40642 − 0.630977i 0.834045 − 0.0821463i 0.328026 − 0.944669i \(-0.393617\pi\)
0.506019 + 0.862522i \(0.331117\pi\)
\(60\) −1.01793 + 0.116558i −0.131414 + 0.0150475i
\(61\) 9.35612 − 2.83815i 1.19793 − 0.363388i 0.372574 − 0.928003i \(-0.378475\pi\)
0.825354 + 0.564615i \(0.190975\pi\)
\(62\) 4.79090 − 3.61231i 0.608444 − 0.458764i
\(63\) − 2.98983i − 0.376683i
\(64\) 5.36306 + 5.93613i 0.670382 + 0.742016i
\(65\) − 0.0758019i − 0.00940207i
\(66\) 2.30905 + 3.06243i 0.284225 + 0.376958i
\(67\) −0.0861833 + 0.0261434i −0.0105290 + 0.00319393i −0.295545 − 0.955329i \(-0.595501\pi\)
0.285016 + 0.958523i \(0.408001\pi\)
\(68\) −8.52145 − 6.77046i −1.03338 − 0.821038i
\(69\) −0.404441 + 0.0398339i −0.0486889 + 0.00479544i
\(70\) 1.74999 + 1.27653i 0.209163 + 0.152574i
\(71\) 8.16418 + 5.45513i 0.968910 + 0.647405i 0.935980 − 0.352052i \(-0.114516\pi\)
0.0329301 + 0.999458i \(0.489516\pi\)
\(72\) 2.78717 − 0.481341i 0.328471 − 0.0567266i
\(73\) −9.01176 − 13.4871i −1.05475 − 1.57854i −0.788896 − 0.614527i \(-0.789347\pi\)
−0.265851 − 0.964014i \(-0.585653\pi\)
\(74\) 9.00778 − 8.29501i 1.04713 − 0.964276i
\(75\) 2.23327 + 4.17815i 0.257876 + 0.482452i
\(76\) −2.06529 + 0.894039i −0.236905 + 0.102553i
\(77\) 0.794772 − 8.06946i 0.0905727 − 0.919600i
\(78\) 0.0119220 + 0.208917i 0.00134990 + 0.0236552i
\(79\) 1.41245 + 3.40995i 0.158913 + 0.383650i 0.983202 − 0.182519i \(-0.0584251\pi\)
−0.824289 + 0.566169i \(0.808425\pi\)
\(80\) −0.908265 + 1.83688i −0.101547 + 0.205369i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) 10.3654 + 3.61648i 1.14466 + 0.399374i
\(83\) 2.88556 + 3.51607i 0.316732 + 0.385939i 0.906903 − 0.421339i \(-0.138440\pi\)
−0.590171 + 0.807278i \(0.700940\pi\)
\(84\) −5.02388 − 3.24299i −0.548151 − 0.353839i
\(85\) 0.809255 − 2.66776i 0.0877760 − 0.289359i
\(86\) 2.07303 − 4.47437i 0.223541 − 0.482483i
\(87\) 2.68243 + 0.533569i 0.287587 + 0.0572046i
\(88\) 7.65043 − 0.558225i 0.815538 − 0.0595070i
\(89\) −0.951785 − 4.78495i −0.100889 − 0.507203i −0.997876 − 0.0651459i \(-0.979249\pi\)
0.896987 − 0.442058i \(-0.145751\pi\)
\(90\) 0.377369 + 0.618446i 0.0397782 + 0.0651899i
\(91\) 0.280653 − 0.341977i 0.0294204 − 0.0358489i
\(92\) −0.371752 + 0.722798i −0.0387578 + 0.0753569i
\(93\) −3.74175 − 2.00001i −0.388001 − 0.207391i
\(94\) 7.40122 − 12.5729i 0.763378 − 1.29679i
\(95\) −0.407614 − 0.407614i −0.0418203 − 0.0418203i
\(96\) 2.21436 − 5.20544i 0.226002 − 0.531278i
\(97\) 8.57980 − 8.57980i 0.871146 − 0.871146i −0.121451 − 0.992597i \(-0.538755\pi\)
0.992597 + 0.121451i \(0.0387547\pi\)
\(98\) 0.687360 + 2.65475i 0.0694338 + 0.268170i
\(99\) 1.27844 − 2.39179i 0.128488 − 0.240384i
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 384.2.v.a.13.1 ✓ 512
128.69 even 32 inner 384.2.v.a.325.1 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.1 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.1 yes 512 128.69 even 32 inner