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Properties

Label 371.2.g.e.76.4
Level $371$
Weight $2$
Character 371.76
Analytic conductor $2.962$
Analytic rank $0$
Dimension $52$
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(76,371)] 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("371.76"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(371, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 371.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [52,-8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.96244991499\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 76.4
Character \(\chi\) \(=\) 371.76
Dual form 371.2.g.e.83.4

$q$-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+(-1.64562 + 1.64562i) q^{2} +(2.31665 + 2.31665i) q^{3} -3.41613i q^{4} +(-0.901960 + 0.901960i) q^{5} -7.62466 q^{6} +(-1.63697 + 2.07854i) q^{7} +(2.33042 + 2.33042i) q^{8} +7.73375i q^{9} -2.96857i q^{10} -3.52181i q^{11} +(7.91399 - 7.91399i) q^{12} -1.06594i q^{13} +(-0.726646 - 6.11432i) q^{14} -4.17905 q^{15} -0.837700 q^{16} -5.58409 q^{17} +(-12.7268 - 12.7268i) q^{18} +(4.07045 + 4.07045i) q^{19} +(3.08122 + 3.08122i) q^{20} +(-8.60754 + 1.02295i) q^{21} +(5.79556 + 5.79556i) q^{22} +(0.625005 + 0.625005i) q^{23} +10.7975i q^{24} +3.37294i q^{25} +(1.75413 + 1.75413i) q^{26} +(-10.9664 + 10.9664i) q^{27} +(7.10056 + 5.59212i) q^{28} -10.0157i q^{29} +(6.87714 - 6.87714i) q^{30} +(3.17863 + 3.17863i) q^{31} +(-3.28230 + 3.28230i) q^{32} +(8.15880 - 8.15880i) q^{33} +(9.18929 - 9.18929i) q^{34} +(-0.398273 - 3.35124i) q^{35} +26.4195 q^{36} +0.262457i q^{37} -13.3968 q^{38} +(2.46940 - 2.46940i) q^{39} -4.20389 q^{40} +(5.14990 + 5.14990i) q^{41} +(12.4814 - 15.8481i) q^{42} -0.117505i q^{43} -12.0310 q^{44} +(-6.97553 - 6.97553i) q^{45} -2.05704 q^{46} +9.01257i q^{47} +(-1.94066 - 1.94066i) q^{48} +(-1.64063 - 6.80502i) q^{49} +(-5.55057 - 5.55057i) q^{50} +(-12.9364 - 12.9364i) q^{51} -3.64138 q^{52} +(2.73567 - 6.74656i) q^{53} -36.0932i q^{54} +(3.17653 + 3.17653i) q^{55} +(-8.65869 + 1.02903i) q^{56} +18.8596i q^{57} +(16.4821 + 16.4821i) q^{58} +2.99831 q^{59} +14.2762i q^{60} +(0.879606 - 0.879606i) q^{61} -10.4616 q^{62} +(-16.0749 - 12.6599i) q^{63} -12.4782i q^{64} +(0.961431 + 0.961431i) q^{65} +26.8526i q^{66} +(2.01509 + 2.01509i) q^{67} +19.0760i q^{68} +2.89584i q^{69} +(6.17028 + 4.85947i) q^{70} +(1.21184 + 1.21184i) q^{71} +(-18.0229 + 18.0229i) q^{72} +(1.28235 + 1.28235i) q^{73} +(-0.431905 - 0.431905i) q^{74} +(-7.81392 + 7.81392i) q^{75} +(13.9052 - 13.9052i) q^{76} +(7.32021 + 5.76511i) q^{77} +8.12739i q^{78} +(6.52163 - 6.52163i) q^{79} +(0.755572 - 0.755572i) q^{80} -27.6096 q^{81} -16.9496 q^{82} +(4.41752 + 4.41752i) q^{83} +(3.49453 + 29.4045i) q^{84} +(5.03662 - 5.03662i) q^{85} +(0.193369 + 0.193369i) q^{86} +(23.2029 - 23.2029i) q^{87} +(8.20729 - 8.20729i) q^{88} +12.8192i q^{89} +22.9582 q^{90} +(2.21559 + 1.74491i) q^{91} +(2.13510 - 2.13510i) q^{92} +14.7276i q^{93} +(-14.8313 - 14.8313i) q^{94} -7.34277 q^{95} -15.2079 q^{96} +3.75594i q^{97} +(13.8983 + 8.49862i) q^{98} +27.2368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 52 q - 8 q^{2} + 20 q^{8} - 50 q^{14} - 24 q^{15} - 64 q^{16} - 8 q^{18} - 30 q^{21} + 40 q^{22} - 20 q^{23} + 80 q^{30} + 12 q^{32} - 24 q^{35} - 16 q^{36} + 4 q^{39} - 80 q^{42} + 64 q^{44} + 88 q^{46}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(213\) \(267\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.64562 + 1.64562i −1.16363 + 1.16363i −0.179954 + 0.983675i \(0.557595\pi\)
−0.983675 + 0.179954i \(0.942405\pi\)
\(3\) 2.31665 + 2.31665i 1.33752 + 1.33752i 0.898456 + 0.439063i \(0.144690\pi\)
0.439063 + 0.898456i \(0.355310\pi\)
\(4\) − 3.41613i − 1.70807i
\(5\) −0.901960 + 0.901960i −0.403369 + 0.403369i −0.879418 − 0.476050i \(-0.842068\pi\)
0.476050 + 0.879418i \(0.342068\pi\)
\(6\) −7.62466 −3.11275
\(7\) −1.63697 + 2.07854i −0.618718 + 0.785613i
\(8\) 2.33042 + 2.33042i 0.823927 + 0.823927i
\(9\) 7.73375i 2.57792i
\(10\) − 2.96857i − 0.938744i
\(11\) − 3.52181i − 1.06187i −0.847414 − 0.530933i \(-0.821842\pi\)
0.847414 − 0.530933i \(-0.178158\pi\)
\(12\) 7.91399 − 7.91399i 2.28457 − 2.28457i
\(13\) − 1.06594i − 0.295637i −0.989014 − 0.147819i \(-0.952775\pi\)
0.989014 − 0.147819i \(-0.0472252\pi\)
\(14\) −0.726646 − 6.11432i −0.194204 − 1.63412i
\(15\) −4.17905 −1.07903
\(16\) −0.837700 −0.209425
\(17\) −5.58409 −1.35434 −0.677170 − 0.735827i \(-0.736794\pi\)
−0.677170 + 0.735827i \(0.736794\pi\)
\(18\) −12.7268 − 12.7268i −2.99974 − 2.99974i
\(19\) 4.07045 + 4.07045i 0.933825 + 0.933825i 0.997942 − 0.0641171i \(-0.0204231\pi\)
−0.0641171 + 0.997942i \(0.520423\pi\)
\(20\) 3.08122 + 3.08122i 0.688981 + 0.688981i
\(21\) −8.60754 + 1.02295i −1.87832 + 0.223226i
\(22\) 5.79556 + 5.79556i 1.23562 + 1.23562i
\(23\) 0.625005 + 0.625005i 0.130323 + 0.130323i 0.769259 − 0.638937i \(-0.220625\pi\)
−0.638937 + 0.769259i \(0.720625\pi\)
\(24\) 10.7975i 2.20404i
\(25\) 3.37294i 0.674587i
\(26\) 1.75413 + 1.75413i 0.344012 + 0.344012i
\(27\) −10.9664 + 10.9664i −2.11049 + 2.11049i
\(28\) 7.10056 + 5.59212i 1.34188 + 1.05681i
\(29\) − 10.0157i − 1.85987i −0.367722 − 0.929936i \(-0.619862\pi\)
0.367722 − 0.929936i \(-0.380138\pi\)
\(30\) 6.87714 − 6.87714i 1.25559 − 1.25559i
\(31\) 3.17863 + 3.17863i 0.570899 + 0.570899i 0.932380 − 0.361480i \(-0.117729\pi\)
−0.361480 + 0.932380i \(0.617729\pi\)
\(32\) −3.28230 + 3.28230i −0.580234 + 0.580234i
\(33\) 8.15880 − 8.15880i 1.42027 − 1.42027i
\(34\) 9.18929 − 9.18929i 1.57595 − 1.57595i
\(35\) −0.398273 − 3.35124i −0.0673204 − 0.566463i
\(36\) 26.4195 4.40325
\(37\) 0.262457i 0.0431477i 0.999767 + 0.0215738i \(0.00686770\pi\)
−0.999767 + 0.0215738i \(0.993132\pi\)
\(38\) −13.3968 −2.17325
\(39\) 2.46940 − 2.46940i 0.395421 − 0.395421i
\(40\) −4.20389 −0.664693
\(41\) 5.14990 + 5.14990i 0.804280 + 0.804280i 0.983761 − 0.179482i \(-0.0574420\pi\)
−0.179482 + 0.983761i \(0.557442\pi\)
\(42\) 12.4814 − 15.8481i 1.92592 − 2.44542i
\(43\) − 0.117505i − 0.0179193i −0.999960 − 0.00895967i \(-0.997148\pi\)
0.999960 − 0.00895967i \(-0.00285199\pi\)
\(44\) −12.0310 −1.81374
\(45\) −6.97553 − 6.97553i −1.03985 − 1.03985i
\(46\) −2.05704 −0.303294
\(47\) 9.01257i 1.31462i 0.753621 + 0.657310i \(0.228306\pi\)
−0.753621 + 0.657310i \(0.771694\pi\)
\(48\) −1.94066 − 1.94066i −0.280110 − 0.280110i
\(49\) −1.64063 − 6.80502i −0.234376 − 0.972146i
\(50\) −5.55057 − 5.55057i −0.784970 − 0.784970i
\(51\) −12.9364 − 12.9364i −1.81146 − 1.81146i
\(52\) −3.64138 −0.504968
\(53\) 2.73567 − 6.74656i 0.375774 − 0.926711i
\(54\) − 36.0932i − 4.91166i
\(55\) 3.17653 + 3.17653i 0.428323 + 0.428323i
\(56\) −8.65869 + 1.02903i −1.15707 + 0.137510i
\(57\) 18.8596i 2.49802i
\(58\) 16.4821 + 16.4821i 2.16420 + 2.16420i
\(59\) 2.99831 0.390347 0.195174 − 0.980769i \(-0.437473\pi\)
0.195174 + 0.980769i \(0.437473\pi\)
\(60\) 14.2762i 1.84305i
\(61\) 0.879606 − 0.879606i 0.112622 − 0.112622i −0.648550 − 0.761172i \(-0.724624\pi\)
0.761172 + 0.648550i \(0.224624\pi\)
\(62\) −10.4616 −1.32863
\(63\) −16.0749 − 12.6599i −2.02524 − 1.59500i
\(64\) − 12.4782i − 1.55978i
\(65\) 0.961431 + 0.961431i 0.119251 + 0.119251i
\(66\) 26.8526i 3.30532i
\(67\) 2.01509 + 2.01509i 0.246182 + 0.246182i 0.819402 − 0.573220i \(-0.194306\pi\)
−0.573220 + 0.819402i \(0.694306\pi\)
\(68\) 19.0760i 2.31330i
\(69\) 2.89584i 0.348618i
\(70\) 6.17028 + 4.85947i 0.737489 + 0.580817i
\(71\) 1.21184 + 1.21184i 0.143819 + 0.143819i 0.775351 − 0.631531i \(-0.217573\pi\)
−0.631531 + 0.775351i \(0.717573\pi\)
\(72\) −18.0229 + 18.0229i −2.12401 + 2.12401i
\(73\) 1.28235 + 1.28235i 0.150088 + 0.150088i 0.778157 − 0.628069i \(-0.216155\pi\)
−0.628069 + 0.778157i \(0.716155\pi\)
\(74\) −0.431905 − 0.431905i −0.0502079 − 0.0502079i
\(75\) −7.81392 + 7.81392i −0.902273 + 0.902273i
\(76\) 13.9052 − 13.9052i 1.59504 − 1.59504i
\(77\) 7.32021 + 5.76511i 0.834215 + 0.656995i
\(78\) 8.12739i 0.920246i
\(79\) 6.52163 − 6.52163i 0.733741 − 0.733741i −0.237618 − 0.971359i \(-0.576366\pi\)
0.971359 + 0.237618i \(0.0763665\pi\)
\(80\) 0.755572 − 0.755572i 0.0844756 − 0.0844756i
\(81\) −27.6096 −3.06773
\(82\) −16.9496 −1.87177
\(83\) 4.41752 + 4.41752i 0.484886 + 0.484886i 0.906688 − 0.421802i \(-0.138602\pi\)
−0.421802 + 0.906688i \(0.638602\pi\)
\(84\) 3.49453 + 29.4045i 0.381284 + 3.20830i
\(85\) 5.03662 − 5.03662i 0.546298 − 0.546298i
\(86\) 0.193369 + 0.193369i 0.0208515 + 0.0208515i
\(87\) 23.2029 − 23.2029i 2.48761 − 2.48761i
\(88\) 8.20729 − 8.20729i 0.874900 − 0.874900i
\(89\) 12.8192i 1.35883i 0.733753 + 0.679416i \(0.237767\pi\)
−0.733753 + 0.679416i \(0.762233\pi\)
\(90\) 22.9582 2.42000
\(91\) 2.21559 + 1.74491i 0.232257 + 0.182916i
\(92\) 2.13510 − 2.13510i 0.222600 − 0.222600i
\(93\) 14.7276i 1.52718i
\(94\) −14.8313 − 14.8313i −1.52973 − 1.52973i
\(95\) −7.34277 −0.753352
\(96\) −15.2079 −1.55215
\(97\) 3.75594i 0.381358i 0.981652 + 0.190679i \(0.0610689\pi\)
−0.981652 + 0.190679i \(0.938931\pi\)
\(98\) 13.8983 + 8.49862i 1.40394 + 0.858490i
\(99\) 27.2368 2.73740
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.g.e.76.4 yes 52
7.6 odd 2 inner 371.2.g.e.76.3 ✓ 52
53.30 odd 4 inner 371.2.g.e.83.3 yes 52
371.83 even 4 inner 371.2.g.e.83.4 yes 52
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
371.2.g.e.76.3 ✓ 52 7.6 odd 2 inner
371.2.g.e.76.4 yes 52 1.1 even 1 trivial
371.2.g.e.83.3 yes 52 53.30 odd 4 inner
371.2.g.e.83.4 yes 52 371.83 even 4 inner