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Properties

Label 371.2.t.a.305.7
Level $371$
Weight $2$
Character 371.305
Analytic conductor $2.962$
Analytic rank $0$
Dimension $816$
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(4,371)] 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("371.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(371, base_ring=CyclotomicField(78)) chi = DirichletCharacter(H, H._module([52, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 371.t (of order \(78\), degree \(24\), 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: \(816\)
Relative dimension: \(34\) over \(\Q(\zeta_{78})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label 305.7
Character \(\chi\) \(=\) 371.305
Dual form 371.2.t.a.163.7

$q$-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+(-1.24608 - 1.65823i) q^{2} +(-3.18013 - 0.128155i) q^{3} +(-0.640577 + 2.21153i) q^{4} +(-0.245392 + 3.03972i) q^{5} +(3.75018 + 5.43307i) q^{6} +(-1.36947 + 2.26375i) q^{7} +(0.586548 - 0.222448i) q^{8} +(7.10654 + 0.573699i) q^{9} +(5.34633 - 3.38081i) q^{10} +(-0.815612 - 3.99513i) q^{11} +(2.32054 - 6.95085i) q^{12} +(-4.89281 - 1.20597i) q^{13} +(5.46028 - 0.549906i) q^{14} +(1.16993 - 9.63526i) q^{15} +(2.79223 + 1.76570i) q^{16} +(2.97864 + 3.64817i) q^{17} +(-7.90397 - 12.4991i) q^{18} +(-3.84428 + 1.11351i) q^{19} +(-6.56523 - 2.48986i) q^{20} +(4.64521 - 7.02350i) q^{21} +(-5.60853 + 6.33072i) q^{22} +(2.04314 - 1.17961i) q^{23} +(-1.89381 + 0.632246i) q^{24} +(-4.24443 - 0.689787i) q^{25} +(4.09705 + 9.61614i) q^{26} +(-13.0477 - 1.58427i) q^{27} +(-4.12908 - 4.47873i) q^{28} +(2.21481 + 1.96215i) q^{29} +(-17.4353 + 10.0663i) q^{30} +(-1.48154 - 4.43775i) q^{31} +(-0.652359 - 8.08091i) q^{32} +(2.08176 + 12.8096i) q^{33} +(2.33789 - 9.48518i) q^{34} +(-6.54510 - 4.71832i) q^{35} +(-5.82103 + 15.3488i) q^{36} +(-0.945772 - 0.598070i) q^{37} +(6.63672 + 4.98717i) q^{38} +(15.4052 + 4.46218i) q^{39} +(0.532247 + 1.83753i) q^{40} +(-3.40458 - 3.84298i) q^{41} +(-17.4349 + 1.04901i) q^{42} +(1.22934 + 0.645207i) q^{43} +(9.35781 + 0.755440i) q^{44} +(-3.48777 + 21.4611i) q^{45} +(-4.50197 - 1.91811i) q^{46} +(5.20448 - 10.9682i) q^{47} +(-8.65337 - 5.97299i) q^{48} +(-3.24909 - 6.20027i) q^{49} +(4.14507 + 7.89777i) q^{50} +(-9.00494 - 11.9834i) q^{51} +(5.80126 - 10.0481i) q^{52} +(3.12594 - 6.57484i) q^{53} +(13.6313 + 23.6102i) q^{54} +(12.3442 - 1.49886i) q^{55} +(-0.299695 + 1.63243i) q^{56} +(12.3680 - 3.04844i) q^{57} +(0.493868 - 6.11765i) q^{58} +(6.83777 - 0.552001i) q^{59} +(20.5592 + 8.75946i) q^{60} +(0.164610 + 0.134400i) q^{61} +(-5.51269 + 7.98651i) q^{62} +(-11.0309 + 15.3017i) q^{63} +(-7.64144 + 6.76973i) q^{64} +(4.86647 - 14.5769i) q^{65} +(18.6472 - 19.4137i) q^{66} +(-10.1440 - 2.93825i) q^{67} +(-9.97609 + 4.25041i) q^{68} +(-6.64863 + 3.48947i) q^{69} +(0.331654 + 16.7327i) q^{70} +(0.171711 - 0.327168i) q^{71} +(4.29594 - 1.24434i) q^{72} +(-1.98376 + 1.61969i) q^{73} +(0.186769 + 2.31355i) q^{74} +(13.4094 + 2.73756i) q^{75} -9.21501i q^{76} +(10.1609 + 3.62488i) q^{77} +(-11.7968 - 31.1056i) q^{78} +(3.68486 + 8.64869i) q^{79} +(-6.05242 + 8.05431i) q^{80} +(20.1783 + 3.27929i) q^{81} +(-2.13016 + 10.4342i) q^{82} +10.1570i q^{83} +(12.5571 + 14.7721i) q^{84} +(-11.8204 + 8.15901i) q^{85} +(-0.461952 - 2.84251i) q^{86} +(-6.79193 - 6.52373i) q^{87} +(-1.36711 - 2.16191i) q^{88} +(6.61955 - 1.07578i) q^{89} +(39.9334 - 20.9587i) q^{90} +(9.43058 - 9.42454i) q^{91} +(1.29995 + 5.27409i) q^{92} +(4.14276 + 14.3025i) q^{93} +(-24.6730 + 5.03702i) q^{94} +(-2.44140 - 11.9588i) q^{95} +(1.03898 + 25.7820i) q^{96} +(1.82908 - 2.64988i) q^{97} +(-6.23285 + 13.1138i) q^{98} +(-3.50417 - 28.8595i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 816 q - 13 q^{2} - 13 q^{3} - 45 q^{4} - 13 q^{5} - 28 q^{6} - 26 q^{7} - 41 q^{9} - 3 q^{10} - 11 q^{11} - 13 q^{12} - 52 q^{13} - 26 q^{14} - 28 q^{15} + 7 q^{16} + 18 q^{17} - 13 q^{18} - 13 q^{19} - 52 q^{20}+ \cdots + 196 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(213\) \(267\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{25}{26}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.24608 − 1.65823i −0.881110 − 1.17254i −0.984056 − 0.177858i \(-0.943083\pi\)
0.102946 − 0.994687i \(-0.467173\pi\)
\(3\) −3.18013 − 0.128155i −1.83605 − 0.0739903i −0.902128 − 0.431469i \(-0.857996\pi\)
−0.933921 + 0.357478i \(0.883637\pi\)
\(4\) −0.640577 + 2.21153i −0.320288 + 1.10576i
\(5\) −0.245392 + 3.03972i −0.109742 + 1.35940i 0.674190 + 0.738558i \(0.264493\pi\)
−0.783932 + 0.620846i \(0.786789\pi\)
\(6\) 3.75018 + 5.43307i 1.53100 + 2.21804i
\(7\) −1.36947 + 2.26375i −0.517612 + 0.855616i
\(8\) 0.586548 − 0.222448i 0.207376 − 0.0786474i
\(9\) 7.10654 + 0.573699i 2.36885 + 0.191233i
\(10\) 5.34633 − 3.38081i 1.69066 − 1.06911i
\(11\) −0.815612 − 3.99513i −0.245916 − 1.20458i −0.895770 − 0.444518i \(-0.853375\pi\)
0.649854 − 0.760059i \(-0.274830\pi\)
\(12\) 2.32054 − 6.95085i 0.669881 − 2.00654i
\(13\) −4.89281 − 1.20597i −1.35702 − 0.334476i −0.507329 − 0.861753i \(-0.669367\pi\)
−0.849694 + 0.527277i \(0.823213\pi\)
\(14\) 5.46028 − 0.549906i 1.45932 − 0.146969i
\(15\) 1.16993 − 9.63526i 0.302075 − 2.48781i
\(16\) 2.79223 + 1.76570i 0.698058 + 0.441425i
\(17\) 2.97864 + 3.64817i 0.722427 + 0.884812i 0.997051 − 0.0767393i \(-0.0244509\pi\)
−0.274624 + 0.961552i \(0.588553\pi\)
\(18\) −7.90397 − 12.4991i −1.86298 − 2.94607i
\(19\) −3.84428 + 1.11351i −0.881938 + 0.255456i −0.688134 − 0.725584i \(-0.741570\pi\)
−0.193804 + 0.981040i \(0.562083\pi\)
\(20\) −6.56523 − 2.48986i −1.46803 − 0.556751i
\(21\) 4.64521 − 7.02350i 1.01367 − 1.53265i
\(22\) −5.60853 + 6.33072i −1.19574 + 1.34971i
\(23\) 2.04314 − 1.17961i 0.426025 − 0.245965i −0.271627 − 0.962403i \(-0.587562\pi\)
0.697652 + 0.716437i \(0.254228\pi\)
\(24\) −1.89381 + 0.632246i −0.386572 + 0.129057i
\(25\) −4.24443 − 0.689787i −0.848886 − 0.137957i
\(26\) 4.09705 + 9.61614i 0.803498 + 1.88588i
\(27\) −13.0477 − 1.58427i −2.51103 − 0.304894i
\(28\) −4.12908 − 4.47873i −0.780324 − 0.846400i
\(29\) 2.21481 + 1.96215i 0.411280 + 0.364362i 0.843246 − 0.537528i \(-0.180642\pi\)
−0.431966 + 0.901890i \(0.642180\pi\)
\(30\) −17.4353 + 10.0663i −3.18323 + 1.83784i
\(31\) −1.48154 − 4.43775i −0.266092 − 0.797043i −0.993803 − 0.111159i \(-0.964544\pi\)
0.727711 − 0.685884i \(-0.240584\pi\)
\(32\) −0.652359 − 8.08091i −0.115322 − 1.42852i
\(33\) 2.08176 + 12.8096i 0.362387 + 2.22986i
\(34\) 2.33789 − 9.48518i 0.400944 − 1.62669i
\(35\) −6.54510 − 4.71832i −1.10632 − 0.797541i
\(36\) −5.82103 + 15.3488i −0.970172 + 2.55813i
\(37\) −0.945772 − 0.598070i −0.155484 − 0.0983221i 0.454458 − 0.890768i \(-0.349833\pi\)
−0.609942 + 0.792446i \(0.708807\pi\)
\(38\) 6.63672 + 4.98717i 1.07662 + 0.809026i
\(39\) 15.4052 + 4.46218i 2.46681 + 0.714521i
\(40\) 0.532247 + 1.83753i 0.0841556 + 0.290539i
\(41\) −3.40458 − 3.84298i −0.531707 − 0.600173i 0.419870 − 0.907584i \(-0.362076\pi\)
−0.951577 + 0.307412i \(0.900537\pi\)
\(42\) −17.4349 + 1.04901i −2.69026 + 0.161866i
\(43\) 1.22934 + 0.645207i 0.187473 + 0.0983932i 0.555848 − 0.831284i \(-0.312394\pi\)
−0.368376 + 0.929677i \(0.620086\pi\)
\(44\) 9.35781 + 0.755440i 1.41074 + 0.113887i
\(45\) −3.48777 + 21.4611i −0.519926 + 3.19923i
\(46\) −4.50197 − 1.91811i −0.663780 − 0.282810i
\(47\) 5.20448 − 10.9682i 0.759151 − 1.59988i −0.0403309 − 0.999186i \(-0.512841\pi\)
0.799482 − 0.600690i \(-0.205108\pi\)
\(48\) −8.65337 − 5.97299i −1.24901 − 0.862127i
\(49\) −3.24909 − 6.20027i −0.464156 − 0.885753i
\(50\) 4.14507 + 7.89777i 0.586201 + 1.11691i
\(51\) −9.00494 − 11.9834i −1.26094 − 1.67801i
\(52\) 5.80126 − 10.0481i 0.804490 − 1.39342i
\(53\) 3.12594 − 6.57484i 0.429381 − 0.903123i
\(54\) 13.6313 + 23.6102i 1.85499 + 3.21293i
\(55\) 12.3442 − 1.49886i 1.66450 − 0.202106i
\(56\) −0.299695 + 1.63243i −0.0400484 + 0.218143i
\(57\) 12.3680 − 3.04844i 1.63818 − 0.403776i
\(58\) 0.493868 − 6.11765i 0.0648480 − 0.803287i
\(59\) 6.83777 − 0.552001i 0.890201 − 0.0718645i 0.373102 − 0.927790i \(-0.378294\pi\)
0.517099 + 0.855926i \(0.327012\pi\)
\(60\) 20.5592 + 8.75946i 2.65418 + 1.13084i
\(61\) 0.164610 + 0.134400i 0.0210761 + 0.0172081i 0.642925 − 0.765929i \(-0.277721\pi\)
−0.621849 + 0.783137i \(0.713618\pi\)
\(62\) −5.51269 + 7.98651i −0.700112 + 1.01429i
\(63\) −11.0309 + 15.3017i −1.38976 + 1.92784i
\(64\) −7.64144 + 6.76973i −0.955180 + 0.846216i
\(65\) 4.86647 − 14.5769i 0.603611 − 1.80804i
\(66\) 18.6472 − 19.4137i 2.29531 − 2.38967i
\(67\) −10.1440 − 2.93825i −1.23929 − 0.358965i −0.407063 − 0.913400i \(-0.633447\pi\)
−0.832227 + 0.554436i \(0.812934\pi\)
\(68\) −9.97609 + 4.25041i −1.20978 + 0.515438i
\(69\) −6.64863 + 3.48947i −0.800401 + 0.420083i
\(70\) 0.331654 + 16.7327i 0.0396403 + 1.99994i
\(71\) 0.171711 − 0.327168i 0.0203784 − 0.0388277i −0.875054 − 0.484025i \(-0.839174\pi\)
0.895432 + 0.445198i \(0.146867\pi\)
\(72\) 4.29594 − 1.24434i 0.506282 − 0.146646i
\(73\) −1.98376 + 1.61969i −0.232181 + 0.189570i −0.741393 − 0.671071i \(-0.765835\pi\)
0.509212 + 0.860641i \(0.329937\pi\)
\(74\) 0.186769 + 2.31355i 0.0217114 + 0.268945i
\(75\) 13.4094 + 2.73756i 1.54839 + 0.316106i
\(76\) − 9.21501i − 1.05703i
\(77\) 10.1609 + 3.62488i 1.15794 + 0.413094i
\(78\) −11.7968 − 31.1056i −1.33573 − 3.52202i
\(79\) 3.68486 + 8.64869i 0.414579 + 0.973053i 0.988093 + 0.153857i \(0.0491697\pi\)
−0.573514 + 0.819196i \(0.694420\pi\)
\(80\) −6.05242 + 8.05431i −0.676681 + 0.900499i
\(81\) 20.1783 + 3.27929i 2.24203 + 0.364366i
\(82\) −2.13016 + 10.4342i −0.235237 + 1.15227i
\(83\) 10.1570i 1.11488i 0.830217 + 0.557440i \(0.188216\pi\)
−0.830217 + 0.557440i \(0.811784\pi\)
\(84\) 12.5571 + 14.7721i 1.37009 + 1.61177i
\(85\) −11.8204 + 8.15901i −1.28210 + 0.884969i
\(86\) −0.461952 − 2.84251i −0.0498136 − 0.306515i
\(87\) −6.79193 − 6.52373i −0.728171 − 0.699418i
\(88\) −1.36711 − 2.16191i −0.145734 − 0.230460i
\(89\) 6.61955 − 1.07578i 0.701670 − 0.114033i 0.200899 − 0.979612i \(-0.435614\pi\)
0.500772 + 0.865579i \(0.333050\pi\)
\(90\) 39.9334 − 20.9587i 4.20935 − 2.20924i
\(91\) 9.43058 − 9.42454i 0.988594 − 0.987961i
\(92\) 1.29995 + 5.27409i 0.135529 + 0.549862i
\(93\) 4.14276 + 14.3025i 0.429585 + 1.48310i
\(94\) −24.6730 + 5.03702i −2.54482 + 0.519529i
\(95\) −2.44140 − 11.9588i −0.250483 − 1.22694i
\(96\) 1.03898 + 25.7820i 0.106040 + 2.63136i
\(97\) 1.82908 − 2.64988i 0.185715 − 0.269054i −0.719057 − 0.694951i \(-0.755426\pi\)
0.904772 + 0.425897i \(0.140041\pi\)
\(98\) −6.23285 + 13.1138i −0.629613 + 1.32469i
\(99\) −3.50417 − 28.8595i −0.352183 − 2.90049i
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.t.a.305.7 yes 816
7.2 even 3 inner 371.2.t.a.93.7 yes 816
53.4 even 26 inner 371.2.t.a.4.7 ✓ 816
371.163 even 78 inner 371.2.t.a.163.7 yes 816
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
371.2.t.a.4.7 ✓ 816 53.4 even 26 inner
371.2.t.a.93.7 yes 816 7.2 even 3 inner
371.2.t.a.163.7 yes 816 371.163 even 78 inner
371.2.t.a.305.7 yes 816 1.1 even 1 trivial