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Properties

Label 371.2.w.a.3.8
Level $371$
Weight $2$
Character 371.3
Analytic conductor $2.962$
Analytic rank $0$
Dimension $1632$
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] = [371,2,Mod(3,371)] 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("371.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(371, base_ring=CyclotomicField(156)) chi = DirichletCharacter(H, H._module([26, 51])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 371.w (of order \(156\), degree \(48\), 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: \(2.96244991499\)
Analytic rank: \(0\)
Dimension: \(1632\)
Relative dimension: \(34\) over \(\Q(\zeta_{156})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{156}]$

Embedding invariants

Embedding label 3.8
Character \(\chi\) \(=\) 371.3
Dual form 371.2.w.a.124.8

$q$-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+(-0.881669 + 1.60069i) q^{2} +(-0.111072 + 0.130580i) q^{3} +(-0.715952 - 1.13219i) q^{4} +(-1.75950 - 1.26756i) q^{5} +(-0.111090 - 0.292921i) q^{6} +(0.726100 - 2.54417i) q^{7} +(-1.20471 + 0.0728715i) q^{8} +(0.476520 + 2.93214i) q^{9} +(3.58027 - 1.69886i) q^{10} +(-0.552050 + 1.29571i) q^{11} +(0.227364 + 0.0322652i) q^{12} +(-3.27543 - 6.24081i) q^{13} +(3.43225 + 3.40538i) q^{14} +(0.360950 - 0.0889662i) q^{15} +(2.09403 - 4.41308i) q^{16} +(-1.03442 - 5.06694i) q^{17} +(-5.11360 - 1.82242i) q^{18} +(7.64942 - 1.72279i) q^{19} +(-0.175393 + 2.89960i) q^{20} +(0.251568 + 0.377400i) q^{21} +(-1.58731 - 2.02605i) q^{22} +(1.07317 + 0.287554i) q^{23} +(0.124294 - 0.165405i) q^{24} +(-0.0941902 - 0.282134i) q^{25} +(12.8775 + 0.259367i) q^{26} +(-0.875928 - 0.529517i) q^{27} +(-3.40032 + 0.999418i) q^{28} +(1.57198 + 0.190873i) q^{29} +(-0.175831 + 0.656210i) q^{30} +(-7.95309 + 1.12862i) q^{31} +(3.80682 + 5.28426i) q^{32} +(-0.107877 - 0.216004i) q^{33} +(9.02265 + 2.81157i) q^{34} +(-4.50246 + 3.55609i) q^{35} +(2.97857 - 2.63878i) q^{36} +(1.57648 + 0.748047i) q^{37} +(-3.98660 + 13.7633i) q^{38} +(1.17874 + 0.265473i) q^{39} +(2.21206 + 1.39882i) q^{40} +(3.05601 + 2.39423i) q^{41} +(-0.825903 + 0.0699420i) q^{42} +(-0.943630 - 0.651341i) q^{43} +(1.86223 - 0.302641i) q^{44} +(2.87822 - 5.76313i) q^{45} +(-1.40646 + 1.46428i) q^{46} +(-2.06900 - 1.68929i) q^{47} +(0.343673 + 0.763610i) q^{48} +(-5.94556 - 3.69464i) q^{49} +(0.534655 + 0.0979792i) q^{50} +(0.776538 + 0.427721i) q^{51} +(-4.72072 + 8.17652i) q^{52} +(0.456340 - 7.26579i) q^{53} +(1.61987 - 0.935235i) q^{54} +(2.61372 - 1.58005i) q^{55} +(-0.689343 + 3.11789i) q^{56} +(-0.624675 + 1.19022i) q^{57} +(-1.69149 + 2.34797i) q^{58} +(-10.5428 - 1.71338i) q^{59} +(-0.359149 - 0.344967i) q^{60} +(-3.01568 + 1.99312i) q^{61} +(5.20541 - 13.7255i) q^{62} +(7.80586 + 0.916686i) q^{63} +(-2.11669 + 0.257012i) q^{64} +(-2.14746 + 15.1325i) q^{65} +(0.440868 + 0.0177664i) q^{66} +(-0.0623736 + 0.276947i) q^{67} +(-4.99613 + 4.79885i) q^{68} +(-0.156748 + 0.108195i) q^{69} +(-1.72255 - 10.3424i) q^{70} +(-1.17538 - 6.41383i) q^{71} +(-0.787737 - 3.49765i) q^{72} +(9.29569 + 6.14371i) q^{73} +(-2.58733 + 1.86393i) q^{74} +(0.0473031 + 0.0190379i) q^{75} +(-7.42714 - 7.42714i) q^{76} +(2.89566 + 2.34532i) q^{77} +(-1.46420 + 1.65274i) q^{78} +(0.207308 - 10.2928i) q^{79} +(-9.27830 + 5.11052i) q^{80} +(-8.28676 + 2.76653i) q^{81} +(-6.52683 + 2.78082i) q^{82} +(4.51244 - 4.51244i) q^{83} +(0.247177 - 0.555023i) q^{84} +(-4.60257 + 10.2265i) q^{85} +(1.87457 - 0.936196i) q^{86} +(-0.199527 + 0.184069i) q^{87} +(0.570640 - 1.60118i) q^{88} +(-0.669848 + 2.00644i) q^{89} +(6.68738 + 9.68834i) q^{90} +(-18.2560 + 3.80178i) q^{91} +(-0.442770 - 1.42090i) q^{92} +(0.735990 - 1.16388i) q^{93} +(4.52821 - 1.82245i) q^{94} +(-15.6429 - 6.66483i) q^{95} +(-1.11285 - 0.0898387i) q^{96} +(-11.6191 - 4.40653i) q^{97} +(11.1560 - 6.25957i) q^{98} +(-4.06227 - 1.00126i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 1632 q - 26 q^{2} - 72 q^{3} - 26 q^{4} - 72 q^{5} - 52 q^{7} - 92 q^{8} - 26 q^{9} - 66 q^{10} - 26 q^{11} - 90 q^{12} - 62 q^{14} - 104 q^{15} - 98 q^{16} - 78 q^{17} - 22 q^{18} - 84 q^{19} - 56 q^{21}+ \cdots - 408 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/371\mathbb{Z}\right)^\times\).

\(n\) \(213\) \(267\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{52}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.881669 + 1.60069i −0.623434 + 1.13186i 0.356830 + 0.934169i \(0.383858\pi\)
−0.980264 + 0.197693i \(0.936655\pi\)
\(3\) −0.111072 + 0.130580i −0.0641275 + 0.0753906i −0.792843 − 0.609426i \(-0.791400\pi\)
0.728715 + 0.684817i \(0.240118\pi\)
\(4\) −0.715952 − 1.13219i −0.357976 − 0.566093i
\(5\) −1.75950 − 1.26756i −0.786874 − 0.566869i 0.117768 − 0.993041i \(-0.462426\pi\)
−0.904642 + 0.426172i \(0.859862\pi\)
\(6\) −0.111090 − 0.292921i −0.0453525 − 0.119585i
\(7\) 0.726100 − 2.54417i 0.274440 − 0.961604i
\(8\) −1.20471 + 0.0728715i −0.425929 + 0.0257640i
\(9\) 0.476520 + 2.93214i 0.158840 + 0.977381i
\(10\) 3.58027 − 1.69886i 1.13218 − 0.537227i
\(11\) −0.552050 + 1.29571i −0.166449 + 0.390671i −0.982228 − 0.187694i \(-0.939899\pi\)
0.815778 + 0.578365i \(0.196309\pi\)
\(12\) 0.227364 + 0.0322652i 0.0656342 + 0.00931416i
\(13\) −3.27543 − 6.24081i −0.908441 − 1.73089i −0.641404 − 0.767204i \(-0.721648\pi\)
−0.267038 − 0.963686i \(-0.586045\pi\)
\(14\) 3.43225 + 3.40538i 0.917308 + 0.910126i
\(15\) 0.360950 − 0.0889662i 0.0931969 − 0.0229710i
\(16\) 2.09403 − 4.41308i 0.523508 − 1.10327i
\(17\) −1.03442 − 5.06694i −0.250885 − 1.22891i −0.888615 − 0.458654i \(-0.848332\pi\)
0.637730 − 0.770260i \(-0.279873\pi\)
\(18\) −5.11360 − 1.82242i −1.20529 − 0.429548i
\(19\) 7.64942 − 1.72279i 1.75490 − 0.395236i 0.781418 − 0.624008i \(-0.214497\pi\)
0.973480 + 0.228772i \(0.0734711\pi\)
\(20\) −0.175393 + 2.89960i −0.0392191 + 0.648370i
\(21\) 0.251568 + 0.377400i 0.0548967 + 0.0823555i
\(22\) −1.58731 − 2.02605i −0.338416 − 0.431956i
\(23\) 1.07317 + 0.287554i 0.223770 + 0.0599591i 0.368962 − 0.929444i \(-0.379713\pi\)
−0.145192 + 0.989404i \(0.546380\pi\)
\(24\) 0.124294 − 0.165405i 0.0253714 − 0.0337632i
\(25\) −0.0941902 − 0.282134i −0.0188380 − 0.0564268i
\(26\) 12.8775 + 0.259367i 2.52548 + 0.0508661i
\(27\) −0.875928 − 0.529517i −0.168572 − 0.101906i
\(28\) −3.40032 + 0.999418i −0.642601 + 0.188872i
\(29\) 1.57198 + 0.190873i 0.291909 + 0.0354442i 0.265181 − 0.964199i \(-0.414568\pi\)
0.0267277 + 0.999643i \(0.491491\pi\)
\(30\) −0.175831 + 0.656210i −0.0321022 + 0.119807i
\(31\) −7.95309 + 1.12862i −1.42842 + 0.202707i −0.811514 − 0.584333i \(-0.801356\pi\)
−0.616902 + 0.787040i \(0.711613\pi\)
\(32\) 3.80682 + 5.28426i 0.672957 + 0.934135i
\(33\) −0.107877 − 0.216004i −0.0187789 − 0.0376015i
\(34\) 9.02265 + 2.81157i 1.54737 + 0.482180i
\(35\) −4.50246 + 3.55609i −0.761054 + 0.601090i
\(36\) 2.97857 − 2.63878i 0.496428 − 0.439797i
\(37\) 1.57648 + 0.748047i 0.259171 + 0.122978i 0.553845 − 0.832620i \(-0.313160\pi\)
−0.294674 + 0.955598i \(0.595211\pi\)
\(38\) −3.98660 + 13.7633i −0.646711 + 2.23271i
\(39\) 1.17874 + 0.265473i 0.188749 + 0.0425098i
\(40\) 2.21206 + 1.39882i 0.349757 + 0.221173i
\(41\) 3.05601 + 2.39423i 0.477269 + 0.373916i 0.826332 − 0.563183i \(-0.190423\pi\)
−0.349063 + 0.937099i \(0.613500\pi\)
\(42\) −0.825903 + 0.0699420i −0.127440 + 0.0107923i
\(43\) −0.943630 − 0.651341i −0.143902 − 0.0993285i 0.493938 − 0.869497i \(-0.335557\pi\)
−0.637841 + 0.770168i \(0.720172\pi\)
\(44\) 1.86223 − 0.302641i 0.280741 − 0.0456249i
\(45\) 2.87822 − 5.76313i 0.429060 − 0.859117i
\(46\) −1.40646 + 1.46428i −0.207372 + 0.215897i
\(47\) −2.06900 − 1.68929i −0.301795 − 0.246408i 0.469431 − 0.882969i \(-0.344459\pi\)
−0.771226 + 0.636561i \(0.780356\pi\)
\(48\) 0.343673 + 0.763610i 0.0496049 + 0.110218i
\(49\) −5.94556 − 3.69464i −0.849365 − 0.527806i
\(50\) 0.534655 + 0.0979792i 0.0756117 + 0.0138564i
\(51\) 0.776538 + 0.427721i 0.108737 + 0.0598929i
\(52\) −4.72072 + 8.17652i −0.654645 + 1.13388i
\(53\) 0.456340 − 7.26579i 0.0626831 − 0.998033i
\(54\) 1.61987 − 0.935235i 0.220437 − 0.127269i
\(55\) 2.61372 − 1.58005i 0.352434 − 0.213054i
\(56\) −0.689343 + 3.11789i −0.0921173 + 0.416646i
\(57\) −0.624675 + 1.19022i −0.0827402 + 0.157648i
\(58\) −1.69149 + 2.34797i −0.222104 + 0.308304i
\(59\) −10.5428 − 1.71338i −1.37256 − 0.223062i −0.570986 − 0.820960i \(-0.693439\pi\)
−0.801573 + 0.597897i \(0.796003\pi\)
\(60\) −0.359149 − 0.344967i −0.0463659 − 0.0445351i
\(61\) −3.01568 + 1.99312i −0.386118 + 0.255193i −0.729469 − 0.684014i \(-0.760233\pi\)
0.343351 + 0.939207i \(0.388438\pi\)
\(62\) 5.20541 − 13.7255i 0.661088 − 1.74314i
\(63\) 7.80586 + 0.916686i 0.983446 + 0.115492i
\(64\) −2.11669 + 0.257012i −0.264586 + 0.0321265i
\(65\) −2.14746 + 15.1325i −0.266360 + 1.87696i
\(66\) 0.440868 + 0.0177664i 0.0542671 + 0.00218689i
\(67\) −0.0623736 + 0.276947i −0.00762014 + 0.0338344i −0.979227 − 0.202765i \(-0.935007\pi\)
0.971607 + 0.236599i \(0.0760329\pi\)
\(68\) −4.99613 + 4.79885i −0.605869 + 0.581946i
\(69\) −0.156748 + 0.108195i −0.0188702 + 0.0130252i
\(70\) −1.72255 − 10.3424i −0.205884 − 1.23615i
\(71\) −1.17538 − 6.41383i −0.139492 − 0.761182i −0.976751 − 0.214376i \(-0.931228\pi\)
0.837259 − 0.546806i \(-0.184156\pi\)
\(72\) −0.787737 − 3.49765i −0.0928357 − 0.412202i
\(73\) 9.29569 + 6.14371i 1.08798 + 0.719067i 0.962023 − 0.272969i \(-0.0880056\pi\)
0.125955 + 0.992036i \(0.459801\pi\)
\(74\) −2.58733 + 1.86393i −0.300770 + 0.216677i
\(75\) 0.0473031 + 0.0190379i 0.00546209 + 0.00219830i
\(76\) −7.42714 − 7.42714i −0.851951 − 0.851951i
\(77\) 2.89566 + 2.34532i 0.329991 + 0.267274i
\(78\) −1.46420 + 1.65274i −0.165788 + 0.187136i
\(79\) 0.207308 − 10.2928i 0.0233240 − 1.15803i −0.803407 − 0.595430i \(-0.796982\pi\)
0.826731 − 0.562597i \(-0.190198\pi\)
\(80\) −9.27830 + 5.11052i −1.03735 + 0.571374i
\(81\) −8.28676 + 2.76653i −0.920752 + 0.307392i
\(82\) −6.52683 + 2.78082i −0.720768 + 0.307090i
\(83\) 4.51244 − 4.51244i 0.495304 − 0.495304i −0.414668 − 0.909973i \(-0.636102\pi\)
0.909973 + 0.414668i \(0.136102\pi\)
\(84\) 0.247177 − 0.555023i 0.0269692 − 0.0605580i
\(85\) −4.60257 + 10.2265i −0.499219 + 1.10922i
\(86\) 1.87457 − 0.936196i 0.202140 − 0.100953i
\(87\) −0.199527 + 0.184069i −0.0213916 + 0.0197342i
\(88\) 0.570640 − 1.60118i 0.0608304 − 0.170687i
\(89\) −0.669848 + 2.00644i −0.0710037 + 0.212682i −0.978058 − 0.208331i \(-0.933197\pi\)
0.907055 + 0.421013i \(0.138325\pi\)
\(90\) 6.68738 + 9.68834i 0.704911 + 1.02124i
\(91\) −18.2560 + 3.80178i −1.91374 + 0.398535i
\(92\) −0.442770 − 1.42090i −0.0461619 − 0.148139i
\(93\) 0.735990 − 1.16388i 0.0763186 − 0.120688i
\(94\) 4.52821 − 1.82245i 0.467049 − 0.187971i
\(95\) −15.6429 − 6.66483i −1.60493 − 0.683797i
\(96\) −1.11285 − 0.0898387i −0.113580 − 0.00916913i
\(97\) −11.6191 − 4.40653i −1.17974 − 0.447415i −0.314733 − 0.949180i \(-0.601915\pi\)
−0.865004 + 0.501765i \(0.832684\pi\)
\(98\) 11.1560 − 6.25957i 1.12693 − 0.632312i
\(99\) −4.06227 − 1.00126i −0.408273 − 0.100630i
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 371.2.w.a.3.8 ✓ 1632
7.5 odd 6 inner 371.2.w.a.215.27 yes 1632
53.18 odd 52 inner 371.2.w.a.283.27 yes 1632
371.124 even 156 inner 371.2.w.a.124.8 yes 1632
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
371.2.w.a.3.8 ✓ 1632 1.1 even 1 trivial
371.2.w.a.124.8 yes 1632 371.124 even 156 inner
371.2.w.a.215.27 yes 1632 7.5 odd 6 inner
371.2.w.a.283.27 yes 1632 53.18 odd 52 inner