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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(13,380)] 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("380.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([0, 27, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.bh (of order \(36\), degree \(12\), 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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 13.8
Character \(\chi\) \(=\) 380.13
Dual form 380.2.bh.a.117.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.80092 - 0.839782i) q^{3} +(-1.74340 - 1.40020i) q^{5} +(-0.755503 - 2.81958i) q^{7} +(0.609708 - 0.726622i) q^{9} +(1.17270 + 2.03117i) q^{11} +(2.20631 - 4.73144i) q^{13} +(-4.31558 - 1.05758i) q^{15} +(0.125014 - 1.42892i) q^{17} +(-3.05201 - 3.11211i) q^{19} +(-3.72843 - 4.44337i) q^{21} +(2.21427 - 1.55045i) q^{23} +(1.07886 + 4.88222i) q^{25} +(-1.05506 + 3.93755i) q^{27} +(5.56573 + 4.67020i) q^{29} +(2.34635 + 1.35467i) q^{31} +(3.81768 + 2.67317i) q^{33} +(-2.63084 + 5.97350i) q^{35} +(-3.32665 - 3.32665i) q^{37} -10.3738i q^{39} +(-1.41908 + 3.89890i) q^{41} +(-3.51529 + 5.02035i) q^{43} +(-2.08038 + 0.413075i) q^{45} +(8.34312 - 0.729929i) q^{47} +(-1.31705 + 0.760401i) q^{49} +(-0.974838 - 2.67835i) q^{51} +(2.51445 + 3.59100i) q^{53} +(0.799577 - 5.18316i) q^{55} +(-8.10991 - 3.04164i) q^{57} +(7.59535 - 6.37326i) q^{59} +(-1.68347 + 9.54743i) q^{61} +(-2.50940 - 1.17015i) q^{63} +(-10.4715 + 5.15950i) q^{65} +(-0.534287 - 6.10693i) q^{67} +(2.68568 - 4.65173i) q^{69} +(4.76199 - 0.839668i) q^{71} +(5.54088 + 11.8824i) q^{73} +(6.04294 + 7.88646i) q^{75} +(4.84108 - 4.84108i) q^{77} +(-16.5155 - 6.01117i) q^{79} +(1.90073 + 10.7796i) q^{81} +(13.1282 - 3.51769i) q^{83} +(-2.21872 + 2.31612i) q^{85} +(13.9454 + 3.73665i) q^{87} +(-4.65929 + 1.69584i) q^{89} +(-15.0075 - 2.64623i) q^{91} +(5.36321 + 0.469220i) q^{93} +(0.963270 + 9.69908i) q^{95} +(5.32688 + 0.466041i) q^{97} +(2.19090 + 0.386315i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.80092 − 0.839782i 1.03976 − 0.484848i 0.173706 − 0.984798i \(-0.444426\pi\)
0.866055 + 0.499949i \(0.166648\pi\)
\(4\) 0 0
\(5\) −1.74340 − 1.40020i −0.779671 − 0.626190i
\(6\) 0 0
\(7\) −0.755503 − 2.81958i −0.285553 − 1.06570i −0.948434 − 0.316975i \(-0.897333\pi\)
0.662880 − 0.748725i \(-0.269334\pi\)
\(8\) 0 0
\(9\) 0.609708 − 0.726622i 0.203236 − 0.242207i
\(10\) 0 0
\(11\) 1.17270 + 2.03117i 0.353582 + 0.612422i 0.986874 − 0.161491i \(-0.0516302\pi\)
−0.633292 + 0.773913i \(0.718297\pi\)
\(12\) 0 0
\(13\) 2.20631 − 4.73144i 0.611920 − 1.31227i −0.319247 − 0.947672i \(-0.603430\pi\)
0.931166 − 0.364594i \(-0.118792\pi\)
\(14\) 0 0
\(15\) −4.31558 − 1.05758i −1.11428 − 0.273065i
\(16\) 0 0
\(17\) 0.125014 − 1.42892i 0.0303204 − 0.346563i −0.965848 − 0.259109i \(-0.916571\pi\)
0.996169 − 0.0874545i \(-0.0278732\pi\)
\(18\) 0 0
\(19\) −3.05201 − 3.11211i −0.700179 − 0.713968i
\(20\) 0 0
\(21\) −3.72843 − 4.44337i −0.813610 − 0.969622i
\(22\) 0 0
\(23\) 2.21427 − 1.55045i 0.461707 − 0.323291i −0.319474 − 0.947595i \(-0.603506\pi\)
0.781181 + 0.624304i \(0.214617\pi\)
\(24\) 0 0
\(25\) 1.07886 + 4.88222i 0.215773 + 0.976444i
\(26\) 0 0
\(27\) −1.05506 + 3.93755i −0.203047 + 0.757781i
\(28\) 0 0
\(29\) 5.56573 + 4.67020i 1.03353 + 0.867235i 0.991267 − 0.131871i \(-0.0420986\pi\)
0.0422637 + 0.999106i \(0.486543\pi\)
\(30\) 0 0
\(31\) 2.34635 + 1.35467i 0.421418 + 0.243306i 0.695684 − 0.718348i \(-0.255102\pi\)
−0.274266 + 0.961654i \(0.588435\pi\)
\(32\) 0 0
\(33\) 3.81768 + 2.67317i 0.664572 + 0.465339i
\(34\) 0 0
\(35\) −2.63084 + 5.97350i −0.444693 + 1.00971i
\(36\) 0 0
\(37\) −3.32665 − 3.32665i −0.546898 − 0.546898i 0.378644 − 0.925542i \(-0.376390\pi\)
−0.925542 + 0.378644i \(0.876390\pi\)
\(38\) 0 0
\(39\) − 10.3738i − 1.66113i
\(40\) 0 0
\(41\) −1.41908 + 3.89890i −0.221623 + 0.608905i −0.999817 − 0.0191188i \(-0.993914\pi\)
0.778194 + 0.628024i \(0.216136\pi\)
\(42\) 0 0
\(43\) −3.51529 + 5.02035i −0.536076 + 0.765596i −0.992245 − 0.124297i \(-0.960332\pi\)
0.456169 + 0.889893i \(0.349221\pi\)
\(44\) 0 0
\(45\) −2.08038 + 0.413075i −0.310125 + 0.0615776i
\(46\) 0 0
\(47\) 8.34312 − 0.729929i 1.21697 − 0.106471i 0.539477 − 0.842000i \(-0.318622\pi\)
0.677493 + 0.735529i \(0.263066\pi\)
\(48\) 0 0
\(49\) −1.31705 + 0.760401i −0.188150 + 0.108629i
\(50\) 0 0
\(51\) −0.974838 − 2.67835i −0.136505 − 0.375043i
\(52\) 0 0
\(53\) 2.51445 + 3.59100i 0.345386 + 0.493262i 0.953835 − 0.300330i \(-0.0970970\pi\)
−0.608449 + 0.793593i \(0.708208\pi\)
\(54\) 0 0
\(55\) 0.799577 − 5.18316i 0.107815 − 0.698897i
\(56\) 0 0
\(57\) −8.10991 − 3.04164i −1.07418 − 0.402875i
\(58\) 0 0
\(59\) 7.59535 − 6.37326i 0.988830 − 0.829727i 0.00343235 − 0.999994i \(-0.498907\pi\)
0.985398 + 0.170267i \(0.0544630\pi\)
\(60\) 0 0
\(61\) −1.68347 + 9.54743i −0.215546 + 1.22242i 0.664410 + 0.747368i \(0.268683\pi\)
−0.879956 + 0.475055i \(0.842428\pi\)
\(62\) 0 0
\(63\) −2.50940 − 1.17015i −0.316155 − 0.147425i
\(64\) 0 0
\(65\) −10.4715 + 5.15950i −1.29882 + 0.639958i
\(66\) 0 0
\(67\) −0.534287 − 6.10693i −0.0652736 − 0.746081i −0.956349 − 0.292226i \(-0.905604\pi\)
0.891076 − 0.453855i \(-0.149951\pi\)
\(68\) 0 0
\(69\) 2.68568 − 4.65173i 0.323318 − 0.560002i
\(70\) 0 0
\(71\) 4.76199 − 0.839668i 0.565145 − 0.0996502i 0.116226 − 0.993223i \(-0.462920\pi\)
0.448919 + 0.893573i \(0.351809\pi\)
\(72\) 0 0
\(73\) 5.54088 + 11.8824i 0.648510 + 1.39074i 0.905511 + 0.424324i \(0.139488\pi\)
−0.257000 + 0.966411i \(0.582734\pi\)
\(74\) 0 0
\(75\) 6.04294 + 7.88646i 0.697779 + 0.910650i
\(76\) 0 0
\(77\) 4.84108 − 4.84108i 0.551692 − 0.551692i
\(78\) 0 0
\(79\) −16.5155 − 6.01117i −1.85814 − 0.676309i −0.980348 − 0.197274i \(-0.936791\pi\)
−0.877796 − 0.479035i \(-0.840987\pi\)
\(80\) 0 0
\(81\) 1.90073 + 10.7796i 0.211193 + 1.19773i
\(82\) 0 0
\(83\) 13.1282 − 3.51769i 1.44101 − 0.386117i 0.548121 − 0.836399i \(-0.315343\pi\)
0.892886 + 0.450282i \(0.148676\pi\)
\(84\) 0 0
\(85\) −2.21872 + 2.31612i −0.240654 + 0.251219i
\(86\) 0 0
\(87\) 13.9454 + 3.73665i 1.49510 + 0.400611i
\(88\) 0 0
\(89\) −4.65929 + 1.69584i −0.493884 + 0.179759i −0.576941 − 0.816786i \(-0.695754\pi\)
0.0830572 + 0.996545i \(0.473532\pi\)
\(90\) 0 0
\(91\) −15.0075 − 2.64623i −1.57322 − 0.277401i
\(92\) 0 0
\(93\) 5.36321 + 0.469220i 0.556139 + 0.0486559i
\(94\) 0 0
\(95\) 0.963270 + 9.69908i 0.0988294 + 0.995104i
\(96\) 0 0
\(97\) 5.32688 + 0.466041i 0.540862 + 0.0473193i 0.354314 − 0.935126i \(-0.384714\pi\)
0.186548 + 0.982446i \(0.440270\pi\)
\(98\) 0 0
\(99\) 2.19090 + 0.386315i 0.220194 + 0.0388261i
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 380.2.bh.a.13.8 ✓ 120
5.2 odd 4 inner 380.2.bh.a.317.3 yes 120
19.3 odd 18 inner 380.2.bh.a.193.3 yes 120
95.22 even 36 inner 380.2.bh.a.117.8 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.8 ✓ 120 1.1 even 1 trivial
380.2.bh.a.117.8 yes 120 95.22 even 36 inner
380.2.bh.a.193.3 yes 120 19.3 odd 18 inner
380.2.bh.a.317.3 yes 120 5.2 odd 4 inner