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Properties

Label 371.2.t.a.4.7
Level $371$
Weight $2$
Character 371.4
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 4.7
Character \(\chi\) \(=\) 371.4
Dual form 371.2.t.a.93.7

$q$-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+(-0.813029 - 1.90825i) q^{2} +(1.70105 + 2.69000i) q^{3} +(-1.59495 + 1.66052i) q^{4} +(2.75517 + 1.30734i) q^{5} +(3.75018 - 5.43307i) q^{6} +(1.87143 - 1.87023i) q^{7} +(0.586548 + 0.222448i) q^{8} +(-3.05643 + 6.44129i) q^{9} +(0.254706 - 6.32046i) q^{10} +(3.86769 - 1.29123i) q^{11} +(-7.17988 - 1.46578i) q^{12} +(-4.89281 + 1.20597i) q^{13} +(-5.09039 - 2.05060i) q^{14} +(1.16993 + 9.63526i) q^{15} +(0.133025 + 3.30099i) q^{16} +(-4.64873 - 0.755493i) q^{17} +(14.7766 + 0.595475i) q^{18} +(2.88647 - 2.77249i) q^{19} +(-6.56523 + 2.48986i) q^{20} +(8.21430 + 1.85278i) q^{21} +(-5.60853 - 6.33072i) q^{22} +(-2.04314 + 1.17961i) q^{23} +(0.399363 + 1.95621i) q^{24} +(2.71959 + 3.33089i) q^{25} +(6.27929 + 8.35622i) q^{26} +(-13.0477 + 1.58427i) q^{27} +(0.120718 + 6.09047i) q^{28} +(2.21481 - 1.96215i) q^{29} +(17.4353 - 10.0663i) q^{30} +(4.58397 - 0.935824i) q^{31} +(7.32446 - 3.47550i) q^{32} +(10.0525 + 8.20764i) q^{33} +(2.33789 + 9.48518i) q^{34} +(7.60114 - 2.70620i) q^{35} +(-5.82103 - 15.3488i) q^{36} +(-0.0450578 - 1.11810i) q^{37} +(-7.63738 - 3.25398i) q^{38} +(-11.5670 - 11.1102i) q^{39} +(1.32522 + 1.37970i) q^{40} +(-3.40458 + 3.84298i) q^{41} +(-3.14290 - 17.1813i) q^{42} +(1.22934 - 0.645207i) q^{43} +(-4.02467 + 8.48182i) q^{44} +(-16.8420 + 13.7510i) q^{45} +(3.91212 + 2.93977i) q^{46} +(-12.1010 - 0.976892i) q^{47} +(-8.65337 + 5.97299i) q^{48} +(0.00448333 - 7.00000i) q^{49} +(4.14507 - 7.89777i) q^{50} +(-5.87546 - 13.7902i) q^{51} +(5.80126 - 10.0481i) q^{52} +(4.13100 - 5.99456i) q^{53} +(13.6313 + 23.6102i) q^{54} +(12.3442 + 1.49886i) q^{55} +(1.51371 - 0.680684i) q^{56} +(12.3680 + 3.04844i) q^{57} +(-5.54498 - 2.63112i) q^{58} +(-2.94084 - 6.19768i) q^{59} +(-17.8655 - 13.4251i) q^{60} +(0.0340887 + 0.209756i) q^{61} +(-5.51269 - 7.98651i) q^{62} +(6.32680 + 17.7706i) q^{63} +(-7.64144 - 6.76973i) q^{64} +(-15.0572 - 3.07394i) q^{65} +(7.48922 - 25.8558i) q^{66} +(7.61661 + 7.31585i) q^{67} +(8.66901 - 6.51434i) q^{68} +(-6.64863 - 3.48947i) q^{69} +(-11.3441 - 12.3046i) q^{70} +(0.171711 + 0.327168i) q^{71} +(-3.22560 + 3.09823i) q^{72} +(-0.410812 + 2.52783i) q^{73} +(-2.09698 + 0.995027i) q^{74} +(-4.33393 + 12.9817i) q^{75} +9.21501i q^{76} +(4.82322 - 9.64991i) q^{77} +(-11.7968 + 31.1056i) q^{78} +(5.64755 + 7.51553i) q^{79} +(-3.94903 + 9.26871i) q^{80} +(-12.9291 - 15.8353i) q^{81} +(10.1014 + 3.37234i) q^{82} -10.1570i q^{83} +(-16.1780 + 10.6849i) q^{84} +(-11.8204 - 8.15901i) q^{85} +(-2.23071 - 1.82131i) q^{86} +(9.04568 + 2.62011i) q^{87} +(2.55582 + 0.102996i) q^{88} +(-4.24143 + 5.19480i) q^{89} +(39.9334 + 20.9587i) q^{90} +(-6.90111 + 11.4076i) q^{91} +(1.29995 - 5.27409i) q^{92} +(10.3149 + 10.7390i) q^{93} +(7.97429 + 23.8859i) q^{94} +(11.5773 - 3.86507i) q^{95} +(21.8083 + 13.7908i) q^{96} +(1.82908 + 2.64988i) q^{97} +(-13.3614 + 5.68265i) 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 \(\left(\mathbb{Z}/371\mathbb{Z}\right)^\times\).

\(n\) \(213\) \(267\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{26}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.813029 − 1.90825i −0.574898 − 1.34934i −0.912897 − 0.408190i \(-0.866160\pi\)
0.337999 − 0.941147i \(-0.390250\pi\)
\(3\) 1.70105 + 2.69000i 0.982102 + 1.55307i 0.824727 + 0.565531i \(0.191329\pi\)
0.157375 + 0.987539i \(0.449697\pi\)
\(4\) −1.59495 + 1.66052i −0.797475 + 0.830260i
\(5\) 2.75517 + 1.30734i 1.23215 + 0.584662i 0.929635 − 0.368482i \(-0.120122\pi\)
0.302515 + 0.953145i \(0.402174\pi\)
\(6\) 3.75018 − 5.43307i 1.53100 − 2.21804i
\(7\) 1.87143 − 1.87023i 0.707333 − 0.706880i
\(8\) 0.586548 + 0.222448i 0.207376 + 0.0786474i
\(9\) −3.05643 + 6.44129i −1.01881 + 2.14710i
\(10\) 0.254706 − 6.32046i 0.0805451 − 1.99871i
\(11\) 3.86769 − 1.29123i 1.16615 − 0.389319i 0.333304 − 0.942819i \(-0.391836\pi\)
0.832849 + 0.553500i \(0.186708\pi\)
\(12\) −7.17988 − 1.46578i −2.07265 − 0.423135i
\(13\) −4.89281 + 1.20597i −1.35702 + 0.334476i −0.849694 − 0.527277i \(-0.823213\pi\)
−0.507329 + 0.861753i \(0.669367\pi\)
\(14\) −5.09039 − 2.05060i −1.36046 − 0.548046i
\(15\) 1.16993 + 9.63526i 0.302075 + 2.48781i
\(16\) 0.133025 + 3.30099i 0.0332564 + 0.825248i
\(17\) −4.64873 − 0.755493i −1.12748 − 0.183234i −0.432067 − 0.901841i \(-0.642216\pi\)
−0.695416 + 0.718607i \(0.744780\pi\)
\(18\) 14.7766 + 0.595475i 3.48287 + 0.140355i
\(19\) 2.88647 − 2.77249i 0.662201 − 0.636052i −0.284307 − 0.958733i \(-0.591764\pi\)
0.946508 + 0.322681i \(0.104584\pi\)
\(20\) −6.56523 + 2.48986i −1.46803 + 0.556751i
\(21\) 8.21430 + 1.85278i 1.79251 + 0.404309i
\(22\) −5.60853 − 6.33072i −1.19574 − 1.34971i
\(23\) −2.04314 + 1.17961i −0.426025 + 0.245965i −0.697652 − 0.716437i \(-0.745772\pi\)
0.271627 + 0.962403i \(0.412438\pi\)
\(24\) 0.399363 + 1.95621i 0.0815196 + 0.399309i
\(25\) 2.71959 + 3.33089i 0.543918 + 0.666178i
\(26\) 6.27929 + 8.35622i 1.23147 + 1.63879i
\(27\) −13.0477 + 1.58427i −2.51103 + 0.304894i
\(28\) 0.120718 + 6.09047i 0.0228135 + 1.15099i
\(29\) 2.21481 − 1.96215i 0.411280 − 0.364362i −0.431966 − 0.901890i \(-0.642180\pi\)
0.843246 + 0.537528i \(0.180642\pi\)
\(30\) 17.4353 − 10.0663i 3.18323 − 1.83784i
\(31\) 4.58397 − 0.935824i 0.823305 − 0.168079i 0.230138 − 0.973158i \(-0.426082\pi\)
0.593168 + 0.805079i \(0.297877\pi\)
\(32\) 7.32446 − 3.47550i 1.29479 − 0.614387i
\(33\) 10.0525 + 8.20764i 1.74992 + 1.42877i
\(34\) 2.33789 + 9.48518i 0.400944 + 1.62669i
\(35\) 7.60114 − 2.70620i 1.28483 − 0.457431i
\(36\) −5.82103 − 15.3488i −0.970172 − 2.55813i
\(37\) −0.0450578 − 1.11810i −0.00740746 − 0.183814i −0.998657 − 0.0518118i \(-0.983500\pi\)
0.991249 − 0.132002i \(-0.0421406\pi\)
\(38\) −7.63738 − 3.25398i −1.23895 − 0.527866i
\(39\) −11.5670 − 11.1102i −1.85220 − 1.77906i
\(40\) 1.32522 + 1.37970i 0.209536 + 0.218150i
\(41\) −3.40458 + 3.84298i −0.531707 + 0.600173i −0.951577 − 0.307412i \(-0.900537\pi\)
0.419870 + 0.907584i \(0.362076\pi\)
\(42\) −3.14290 − 17.1813i −0.484961 − 2.65113i
\(43\) 1.22934 − 0.645207i 0.187473 − 0.0983932i −0.368376 − 0.929677i \(-0.620086\pi\)
0.555848 + 0.831284i \(0.312394\pi\)
\(44\) −4.02467 + 8.48182i −0.606742 + 1.27868i
\(45\) −16.8420 + 13.7510i −2.51065 + 2.04989i
\(46\) 3.91212 + 2.93977i 0.576811 + 0.433445i
\(47\) −12.1010 − 0.976892i −1.76511 − 0.142494i −0.845155 − 0.534521i \(-0.820492\pi\)
−0.919954 + 0.392026i \(0.871774\pi\)
\(48\) −8.65337 + 5.97299i −1.24901 + 0.862127i
\(49\) 0.00448333 − 7.00000i 0.000640476 − 1.00000i
\(50\) 4.14507 − 7.89777i 0.586201 − 1.11691i
\(51\) −5.87546 − 13.7902i −0.822729 − 1.93102i
\(52\) 5.80126 − 10.0481i 0.804490 − 1.39342i
\(53\) 4.13100 − 5.99456i 0.567437 − 0.823417i
\(54\) 13.6313 + 23.6102i 1.85499 + 3.21293i
\(55\) 12.3442 + 1.49886i 1.66450 + 0.202106i
\(56\) 1.51371 − 0.680684i 0.202278 − 0.0909602i
\(57\) 12.3680 + 3.04844i 1.63818 + 0.403776i
\(58\) −5.54498 − 2.63112i −0.728091 − 0.345484i
\(59\) −2.94084 − 6.19768i −0.382864 − 0.806869i −0.999803 − 0.0198579i \(-0.993679\pi\)
0.616939 − 0.787011i \(-0.288373\pi\)
\(60\) −17.8655 − 13.4251i −2.30643 − 1.73317i
\(61\) 0.0340887 + 0.209756i 0.00436461 + 0.0268565i 0.989141 − 0.146968i \(-0.0469515\pi\)
−0.984777 + 0.173825i \(0.944387\pi\)
\(62\) −5.51269 − 7.98651i −0.700112 − 1.01429i
\(63\) 6.32680 + 17.7706i 0.797102 + 2.23889i
\(64\) −7.64144 − 6.76973i −0.955180 − 0.846216i
\(65\) −15.0572 − 3.07394i −1.86761 − 0.381275i
\(66\) 7.48922 − 25.8558i 0.921859 − 3.18263i
\(67\) 7.61661 + 7.31585i 0.930517 + 0.893774i 0.994559 − 0.104173i \(-0.0332197\pi\)
−0.0640421 + 0.997947i \(0.520399\pi\)
\(68\) 8.66901 − 6.51434i 1.05127 − 0.789980i
\(69\) −6.64863 − 3.48947i −0.800401 − 0.420083i
\(70\) −11.3441 − 12.3046i −1.35587 − 1.47069i
\(71\) 0.171711 + 0.327168i 0.0203784 + 0.0388277i 0.895432 − 0.445198i \(-0.146867\pi\)
−0.875054 + 0.484025i \(0.839174\pi\)
\(72\) −3.22560 + 3.09823i −0.380140 + 0.365130i
\(73\) −0.410812 + 2.52783i −0.0480820 + 0.295860i −0.999943 − 0.0106694i \(-0.996604\pi\)
0.951861 + 0.306530i \(0.0991678\pi\)
\(74\) −2.09698 + 0.995027i −0.243769 + 0.115670i
\(75\) −4.33393 + 12.9817i −0.500439 + 1.49900i
\(76\) 9.21501i 1.05703i
\(77\) 4.82322 − 9.64991i 0.549657 − 1.09971i
\(78\) −11.7968 + 31.1056i −1.33573 + 3.52202i
\(79\) 5.64755 + 7.51553i 0.635399 + 0.845563i 0.996201 − 0.0870795i \(-0.0277534\pi\)
−0.360802 + 0.932642i \(0.617497\pi\)
\(80\) −3.94903 + 9.26871i −0.441515 + 1.03627i
\(81\) −12.9291 − 15.8353i −1.43657 − 1.75947i
\(82\) 10.1014 + 3.37234i 1.11551 + 0.372413i
\(83\) − 10.1570i − 1.11488i −0.830217 − 0.557440i \(-0.811784\pi\)
0.830217 − 0.557440i \(-0.188216\pi\)
\(84\) −16.1780 + 10.6849i −1.76516 + 1.16582i
\(85\) −11.8204 − 8.15901i −1.28210 − 0.884969i
\(86\) −2.23071 − 1.82131i −0.240543 − 0.196397i
\(87\) 9.04568 + 2.62011i 0.969799 + 0.280906i
\(88\) 2.55582 + 0.102996i 0.272451 + 0.0109794i
\(89\) −4.24143 + 5.19480i −0.449590 + 0.550648i −0.948818 − 0.315822i \(-0.897720\pi\)
0.499228 + 0.866471i \(0.333617\pi\)
\(90\) 39.9334 + 20.9587i 4.20935 + 2.20924i
\(91\) −6.90111 + 11.4076i −0.723432 + 1.19584i
\(92\) 1.29995 − 5.27409i 0.135529 − 0.549862i
\(93\) 10.3149 + 10.7390i 1.06961 + 1.11358i
\(94\) 7.97429 + 23.8859i 0.822486 + 2.46365i
\(95\) 11.5773 − 3.86507i 1.18781 − 0.396548i
\(96\) 21.8083 + 13.7908i 2.22581 + 1.40751i
\(97\) 1.82908 + 2.64988i 0.185715 + 0.269054i 0.904772 − 0.425897i \(-0.140041\pi\)
−0.719057 + 0.694951i \(0.755426\pi\)
\(98\) −13.3614 + 5.68265i −1.34970 + 0.574034i
\(99\) −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.4.7 ✓ 816
7.2 even 3 inner 371.2.t.a.163.7 yes 816
53.40 even 26 inner 371.2.t.a.305.7 yes 816
371.93 even 78 inner 371.2.t.a.93.7 yes 816
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
371.2.t.a.4.7 ✓ 816 1.1 even 1 trivial
371.2.t.a.93.7 yes 816 371.93 even 78 inner
371.2.t.a.163.7 yes 816 7.2 even 3 inner
371.2.t.a.305.7 yes 816 53.40 even 26 inner