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Properties

Label 371.2.w.a.215.27
Level $371$
Weight $2$
Character 371.215
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 215.27
Character \(\chi\) \(=\) 371.215
Dual form 371.2.w.a.283.27

$q$-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+(1.82708 - 0.0367994i) q^{2} +(0.0575498 + 0.161481i) q^{3} +(1.33848 - 0.0539388i) q^{4} +(0.217986 - 2.15755i) q^{5} +(0.111090 + 0.292921i) q^{6} +(0.539403 - 2.59018i) q^{7} +(-1.20471 + 0.0728715i) q^{8} +(2.30105 - 1.87875i) q^{9} +(0.318880 - 3.95004i) q^{10} +(-0.846092 - 1.12594i) q^{11} +(0.0857393 + 0.213035i) q^{12} +(3.27543 + 6.24081i) q^{13} +(0.890213 - 4.75231i) q^{14} +(0.360950 - 0.0889662i) q^{15} +(-4.86886 + 0.393055i) q^{16} +(-4.90531 - 1.63763i) q^{17} +(4.13506 - 3.51730i) q^{18} +(5.31669 + 5.76320i) q^{19} +(0.175393 - 2.89960i) q^{20} +(0.449309 - 0.0619610i) q^{21} +(-1.58731 - 2.02605i) q^{22} +(-0.287554 - 1.07317i) q^{23} +(-0.0810982 - 0.190344i) q^{24} +(0.291431 + 0.0594960i) q^{25} +(6.21412 + 11.2819i) q^{26} +(0.875928 + 0.529517i) q^{27} +(0.582267 - 3.49600i) q^{28} +(1.57198 + 0.190873i) q^{29} +(0.656210 - 0.175831i) q^{30} +(-2.99913 + 7.45189i) q^{31} +(-6.47972 + 0.654670i) q^{32} +(0.133127 - 0.201426i) q^{33} +(-9.02265 - 2.81157i) q^{34} +(-5.47088 - 1.72841i) q^{35} +(2.97857 - 2.63878i) q^{36} +(-0.140410 - 1.73929i) q^{37} +(9.92609 + 10.3342i) q^{38} +(-0.819275 + 0.888079i) q^{39} +(-0.105385 + 2.61511i) q^{40} +(-3.05601 - 2.39423i) q^{41} +(0.818642 - 0.129742i) q^{42} +(-0.943630 - 0.651341i) q^{43} +(-1.19321 - 1.46141i) q^{44} +(-3.55191 - 5.37418i) q^{45} +(-0.564875 - 1.95017i) q^{46} +(0.428465 - 2.63645i) q^{47} +(-0.343673 - 0.763610i) q^{48} +(-6.41809 - 2.79430i) q^{49} +(0.534655 + 0.0979792i) q^{50} +(-0.0178523 - 0.886362i) q^{51} +(4.72072 + 8.17652i) q^{52} +(6.06419 + 4.02810i) q^{53} +(1.61987 + 0.935235i) q^{54} +(-2.61372 + 1.58005i) q^{55} +(-0.461073 + 3.15972i) q^{56} +(-0.624675 + 1.19022i) q^{57} +(2.87915 + 0.290891i) q^{58} +(-6.75524 + 8.27366i) q^{59} +(0.478325 - 0.138549i) q^{60} +(-3.23393 - 1.61509i) q^{61} +(-5.20541 + 13.7255i) q^{62} +(-3.62511 - 6.97354i) q^{63} +(-2.11669 + 0.257012i) q^{64} +(14.1789 - 5.70651i) q^{65} +(0.235820 - 0.372920i) q^{66} +(-0.208656 - 0.192490i) q^{67} +(-6.65399 - 1.92735i) q^{68} +(0.156748 - 0.108195i) q^{69} +(-10.0593 - 2.95662i) q^{70} +(-1.17538 - 6.41383i) q^{71} +(-2.63519 + 2.43103i) q^{72} +(9.96845 - 4.97845i) q^{73} +(-0.320545 - 3.17265i) q^{74} +(0.00716428 + 0.0504846i) q^{75} +(7.42714 + 7.42714i) q^{76} +(-3.37279 + 1.58420i) q^{77} +(-1.46420 + 1.65274i) q^{78} +(-9.01746 + 4.96685i) q^{79} +(-0.213305 + 10.5905i) q^{80} +(1.74750 - 8.55981i) q^{81} +(-5.67168 - 4.26199i) q^{82} +(-4.51244 + 4.51244i) q^{83} +(0.598048 - 0.107169i) q^{84} +(-4.60257 + 10.2265i) q^{85} +(-1.74805 - 1.15532i) q^{86} +(0.0596446 + 0.264830i) q^{87} +(1.10134 + 1.29478i) q^{88} +(-2.07255 + 0.423114i) q^{89} +(-6.68738 - 9.68834i) q^{90} +(17.9316 - 5.11765i) q^{91} +(-0.442770 - 1.42090i) q^{92} +(-1.37594 - 0.0554485i) q^{93} +(0.685819 - 4.83277i) q^{94} +(13.5934 - 10.2148i) q^{95} +(-0.478624 - 1.00868i) q^{96} +(11.6191 + 4.40653i) q^{97} +(-11.8292 - 4.86922i) 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 \(\left(\mathbb{Z}/371\mathbb{Z}\right)^\times\).

\(n\) \(213\) \(267\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{17}{52}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82708 − 0.0367994i 1.29194 − 0.0260211i 0.630600 − 0.776108i \(-0.282809\pi\)
0.661339 + 0.750087i \(0.269989\pi\)
\(3\) 0.0575498 + 0.161481i 0.0332264 + 0.0932314i 0.957426 − 0.288678i \(-0.0932155\pi\)
−0.924200 + 0.381909i \(0.875267\pi\)
\(4\) 1.33848 − 0.0539388i 0.669239 − 0.0269694i
\(5\) 0.217986 − 2.15755i 0.0974862 − 0.964888i −0.821397 − 0.570357i \(-0.806805\pi\)
0.918883 − 0.394530i \(-0.129093\pi\)
\(6\) 0.111090 + 0.292921i 0.0453525 + 0.119585i
\(7\) 0.539403 − 2.59018i 0.203875 − 0.978997i
\(8\) −1.20471 + 0.0728715i −0.425929 + 0.0257640i
\(9\) 2.30105 − 1.87875i 0.767017 − 0.626250i
\(10\) 0.318880 − 3.95004i 0.100839 − 1.24911i
\(11\) −0.846092 − 1.12594i −0.255106 − 0.339485i 0.653661 − 0.756787i \(-0.273232\pi\)
−0.908768 + 0.417302i \(0.862976\pi\)
\(12\) 0.0857393 + 0.213035i 0.0247508 + 0.0614980i
\(13\) 3.27543 + 6.24081i 0.908441 + 1.73089i 0.641404 + 0.767204i \(0.278352\pi\)
0.267038 + 0.963686i \(0.413955\pi\)
\(14\) 0.890213 − 4.75231i 0.237919 − 1.27011i
\(15\) 0.360950 − 0.0889662i 0.0931969 − 0.0229710i
\(16\) −4.86886 + 0.393055i −1.21721 + 0.0982637i
\(17\) −4.90531 − 1.63763i −1.18971 − 0.397185i −0.348199 − 0.937421i \(-0.613207\pi\)
−0.841514 + 0.540236i \(0.818335\pi\)
\(18\) 4.13506 − 3.51730i 0.974643 − 0.829035i
\(19\) 5.31669 + 5.76320i 1.21973 + 1.32217i 0.931116 + 0.364724i \(0.118837\pi\)
0.288617 + 0.957445i \(0.406804\pi\)
\(20\) 0.175393 − 2.89960i 0.0392191 − 0.648370i
\(21\) 0.449309 − 0.0619610i 0.0980472 − 0.0135210i
\(22\) −1.58731 − 2.02605i −0.338416 − 0.431956i
\(23\) −0.287554 − 1.07317i −0.0599591 − 0.223770i 0.929444 − 0.368962i \(-0.120287\pi\)
−0.989404 + 0.145192i \(0.953620\pi\)
\(24\) −0.0810982 − 0.190344i −0.0165541 − 0.0388539i
\(25\) 0.291431 + 0.0594960i 0.0582861 + 0.0118992i
\(26\) 6.21412 + 11.2819i 1.21869 + 2.21256i
\(27\) 0.875928 + 0.529517i 0.168572 + 0.101906i
\(28\) 0.582267 − 3.49600i 0.110038 − 0.660681i
\(29\) 1.57198 + 0.190873i 0.291909 + 0.0354442i 0.265181 − 0.964199i \(-0.414568\pi\)
0.0267277 + 0.999643i \(0.491491\pi\)
\(30\) 0.656210 − 0.175831i 0.119807 − 0.0321022i
\(31\) −2.99913 + 7.45189i −0.538659 + 1.33840i 0.373145 + 0.927773i \(0.378279\pi\)
−0.911804 + 0.410625i \(0.865310\pi\)
\(32\) −6.47972 + 0.654670i −1.14546 + 0.115730i
\(33\) 0.133127 − 0.201426i 0.0231744 − 0.0350638i
\(34\) −9.02265 − 2.81157i −1.54737 − 0.482180i
\(35\) −5.47088 − 1.72841i −0.924747 − 0.292155i
\(36\) 2.97857 − 2.63878i 0.496428 − 0.439797i
\(37\) −0.140410 − 1.73929i −0.0230833 − 0.285938i −0.997992 − 0.0633344i \(-0.979827\pi\)
0.974909 − 0.222603i \(-0.0714555\pi\)
\(38\) 9.92609 + 10.3342i 1.61022 + 1.67642i
\(39\) −0.819275 + 0.888079i −0.131189 + 0.142206i
\(40\) −0.105385 + 2.61511i −0.0166629 + 0.413485i
\(41\) −3.05601 − 2.39423i −0.477269 − 0.373916i 0.349063 − 0.937099i \(-0.386500\pi\)
−0.826332 + 0.563183i \(0.809577\pi\)
\(42\) 0.818642 − 0.129742i 0.126319 − 0.0200196i
\(43\) −0.943630 − 0.651341i −0.143902 − 0.0993285i 0.493938 − 0.869497i \(-0.335557\pi\)
−0.637841 + 0.770168i \(0.720172\pi\)
\(44\) −1.19321 − 1.46141i −0.179883 − 0.220317i
\(45\) −3.55191 − 5.37418i −0.529487 − 0.801136i
\(46\) −0.564875 − 1.95017i −0.0832862 − 0.287537i
\(47\) 0.428465 − 2.63645i 0.0624981 − 0.384566i −0.936891 − 0.349621i \(-0.886310\pi\)
0.999389 − 0.0349450i \(-0.0111256\pi\)
\(48\) −0.343673 − 0.763610i −0.0496049 − 0.110218i
\(49\) −6.41809 − 2.79430i −0.916870 − 0.399186i
\(50\) 0.534655 + 0.0979792i 0.0756117 + 0.0138564i
\(51\) −0.0178523 − 0.886362i −0.00249983 − 0.124116i
\(52\) 4.72072 + 8.17652i 0.654645 + 1.13388i
\(53\) 6.06419 + 4.02810i 0.832981 + 0.553302i
\(54\) 1.61987 + 0.935235i 0.220437 + 0.127269i
\(55\) −2.61372 + 1.58005i −0.352434 + 0.213054i
\(56\) −0.461073 + 3.15972i −0.0616134 + 0.422236i
\(57\) −0.624675 + 1.19022i −0.0827402 + 0.157648i
\(58\) 2.87915 + 0.290891i 0.378051 + 0.0381959i
\(59\) −6.75524 + 8.27366i −0.879457 + 1.07714i 0.117008 + 0.993131i \(0.462670\pi\)
−0.996465 + 0.0840083i \(0.973228\pi\)
\(60\) 0.478325 − 0.138549i 0.0617515 − 0.0178865i
\(61\) −3.23393 − 1.61509i −0.414063 − 0.206791i 0.227639 − 0.973746i \(-0.426899\pi\)
−0.641702 + 0.766954i \(0.721771\pi\)
\(62\) −5.20541 + 13.7255i −0.661088 + 1.74314i
\(63\) −3.62511 − 6.97354i −0.456721 − 0.878584i
\(64\) −2.11669 + 0.257012i −0.264586 + 0.0321265i
\(65\) 14.1789 − 5.70651i 1.75867 − 0.707806i
\(66\) 0.235820 − 0.372920i 0.0290275 − 0.0459033i
\(67\) −0.208656 − 0.192490i −0.0254914 − 0.0235165i 0.665213 − 0.746653i \(-0.268341\pi\)
−0.690705 + 0.723137i \(0.742700\pi\)
\(68\) −6.65399 − 1.92735i −0.806914 − 0.233726i
\(69\) 0.156748 − 0.108195i 0.0188702 − 0.0130252i
\(70\) −10.0593 − 2.95662i −1.20232 − 0.353384i
\(71\) −1.17538 − 6.41383i −0.139492 − 0.761182i −0.976751 − 0.214376i \(-0.931228\pi\)
0.837259 − 0.546806i \(-0.184156\pi\)
\(72\) −2.63519 + 2.43103i −0.310560 + 0.286499i
\(73\) 9.96845 − 4.97845i 1.16672 − 0.582683i 0.244613 − 0.969621i \(-0.421339\pi\)
0.922106 + 0.386938i \(0.126467\pi\)
\(74\) −0.320545 − 3.17265i −0.0372626 − 0.368813i
\(75\) 0.00716428 + 0.0504846i 0.000827260 + 0.00582946i
\(76\) 7.42714 + 7.42714i 0.851951 + 0.851951i
\(77\) −3.37279 + 1.58420i −0.384365 + 0.180536i
\(78\) −1.46420 + 1.65274i −0.165788 + 0.187136i
\(79\) −9.01746 + 4.96685i −1.01454 + 0.558815i −0.900589 − 0.434671i \(-0.856865\pi\)
−0.113954 + 0.993486i \(0.536352\pi\)
\(80\) −0.213305 + 10.5905i −0.0238482 + 1.18405i
\(81\) 1.74750 − 8.55981i 0.194166 − 0.951090i
\(82\) −5.67168 − 4.26199i −0.626332 − 0.470658i
\(83\) −4.51244 + 4.51244i −0.495304 + 0.495304i −0.909973 − 0.414668i \(-0.863898\pi\)
0.414668 + 0.909973i \(0.363898\pi\)
\(84\) 0.598048 − 0.107169i 0.0652524 − 0.0116931i
\(85\) −4.60257 + 10.2265i −0.499219 + 1.10922i
\(86\) −1.74805 − 1.15532i −0.188497 − 0.124582i
\(87\) 0.0596446 + 0.264830i 0.00639458 + 0.0283927i
\(88\) 1.10134 + 1.29478i 0.117404 + 0.138024i
\(89\) −2.07255 + 0.423114i −0.219690 + 0.0448500i −0.308610 − 0.951189i \(-0.599864\pi\)
0.0889197 + 0.996039i \(0.471659\pi\)
\(90\) −6.68738 − 9.68834i −0.704911 − 1.02124i
\(91\) 17.9316 − 5.11765i 1.87974 − 0.536476i
\(92\) −0.442770 − 1.42090i −0.0461619 − 0.148139i
\(93\) −1.37594 − 0.0554485i −0.142678 − 0.00574974i
\(94\) 0.685819 − 4.83277i 0.0707369 − 0.498462i
\(95\) 13.5934 − 10.2148i 1.39465 − 1.04801i
\(96\) −0.478624 − 1.00868i −0.0488493 − 0.102948i
\(97\) 11.6191 + 4.40653i 1.17974 + 0.447415i 0.865004 − 0.501765i \(-0.167316\pi\)
0.314733 + 0.949180i \(0.398085\pi\)
\(98\) −11.8292 − 4.86922i −1.19493 − 0.491866i
\(99\) −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.215.27 yes 1632
7.3 odd 6 inner 371.2.w.a.3.8 ✓ 1632
53.18 odd 52 inner 371.2.w.a.124.8 yes 1632
371.283 even 156 inner 371.2.w.a.283.27 yes 1632
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
371.2.w.a.3.8 ✓ 1632 7.3 odd 6 inner
371.2.w.a.124.8 yes 1632 53.18 odd 52 inner
371.2.w.a.215.27 yes 1632 1.1 even 1 trivial
371.2.w.a.283.27 yes 1632 371.283 even 156 inner