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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(227,380)] 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("380.227"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.j (of order \(4\), degree \(2\), 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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(232\)
Relative dimension: \(116\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 227.4
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.4

$q$-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+(-1.99490 + 0.142778i) q^{2} +(1.34489 - 1.34489i) q^{3} +(3.95923 - 0.569656i) q^{4} +(-4.96914 - 0.554657i) q^{5} +(-2.49090 + 2.87494i) q^{6} +(4.10715 - 4.10715i) q^{7} +(-7.81692 + 1.70170i) q^{8} +5.38253i q^{9} +(9.99212 + 0.396998i) q^{10} +13.8814i q^{11} +(4.55861 - 6.09086i) q^{12} +(-6.27479 - 6.27479i) q^{13} +(-7.60693 + 8.77975i) q^{14} +(-7.42891 + 5.93700i) q^{15} +(15.3510 - 4.51080i) q^{16} +(15.0594 + 15.0594i) q^{17} +(-0.768508 - 10.7376i) q^{18} +(18.3224 + 5.02889i) q^{19} +(-19.9899 + 0.634686i) q^{20} -11.0473i q^{21} +(-1.98197 - 27.6920i) q^{22} +(-10.1791 - 10.1791i) q^{23} +(-8.22432 + 12.8015i) q^{24} +(24.3847 + 5.51234i) q^{25} +(13.4135 + 11.6217i) q^{26} +(19.3430 + 19.3430i) q^{27} +(13.9215 - 18.6008i) q^{28} +12.5488 q^{29} +(13.9722 - 12.9044i) q^{30} -19.7872 q^{31} +(-29.9796 + 11.1904i) q^{32} +(18.6690 + 18.6690i) q^{33} +(-32.1920 - 27.8917i) q^{34} +(-22.6871 + 18.1309i) q^{35} +(3.06619 + 21.3107i) q^{36} +(33.1652 - 33.1652i) q^{37} +(-37.2693 - 7.41608i) q^{38} -16.8778 q^{39} +(39.7872 - 4.12026i) q^{40} -55.2228i q^{41} +(1.57732 + 22.0383i) q^{42} +(27.5934 + 27.5934i) q^{43} +(7.90764 + 54.9598i) q^{44} +(2.98546 - 26.7465i) q^{45} +(21.7597 + 18.8530i) q^{46} +(21.6976 - 21.6976i) q^{47} +(14.5789 - 26.7120i) q^{48} +15.2627i q^{49} +(-49.4320 - 7.51494i) q^{50} +40.5064 q^{51} +(-28.4178 - 21.2689i) q^{52} +(72.2683 + 72.2683i) q^{53} +(-41.3490 - 35.8254i) q^{54} +(7.69944 - 68.9788i) q^{55} +(-25.1161 + 39.0944i) q^{56} +(31.4050 - 17.8783i) q^{57} +(-25.0336 + 1.79170i) q^{58} +111.357i q^{59} +(-26.0307 + 27.7379i) q^{60} -47.2031 q^{61} +(39.4735 - 2.82519i) q^{62} +(22.1068 + 22.1068i) q^{63} +(58.2085 - 26.6041i) q^{64} +(27.7000 + 34.6607i) q^{65} +(-39.9083 - 34.5773i) q^{66} +(-10.4691 - 10.4691i) q^{67} +(68.2021 + 51.0448i) q^{68} -27.3797 q^{69} +(42.6696 - 39.4086i) q^{70} +130.709 q^{71} +(-9.15943 - 42.0748i) q^{72} +(100.965 - 100.965i) q^{73} +(-61.4259 + 70.8964i) q^{74} +(40.2083 - 25.3813i) q^{75} +(75.4073 + 9.47306i) q^{76} +(57.0131 + 57.0131i) q^{77} +(33.6696 - 2.40979i) q^{78} +109.995i q^{79} +(-78.7831 + 13.9003i) q^{80} +3.58562 q^{81} +(7.88461 + 110.164i) q^{82} +(31.3130 + 31.3130i) q^{83} +(-6.29319 - 43.7390i) q^{84} +(-66.4793 - 83.1848i) q^{85} +(-58.9857 - 51.1063i) q^{86} +(16.8768 - 16.8768i) q^{87} +(-23.6220 - 108.510i) q^{88} -59.3913 q^{89} +(-2.13686 + 53.7829i) q^{90} -51.5430 q^{91} +(-46.1001 - 34.5029i) q^{92} +(-26.6117 + 26.6117i) q^{93} +(-40.1865 + 46.3824i) q^{94} +(-88.2573 - 35.1519i) q^{95} +(-25.2695 + 55.3692i) q^{96} +(-2.65389 + 2.65389i) q^{97} +(-2.17918 - 30.4475i) q^{98} -74.7172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 8 q^{5} - 32 q^{6} - 64 q^{16} - 8 q^{17} + 80 q^{20} - 8 q^{25} + 40 q^{26} + 40 q^{28} - 104 q^{30} + 104 q^{36} + 184 q^{38} + 192 q^{42} + 192 q^{45} + 280 q^{58} + 112 q^{61} - 280 q^{62} + 216 q^{66}+ \cdots + 504 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99490 + 0.142778i −0.997449 + 0.0713892i
\(3\) 1.34489 − 1.34489i 0.448297 − 0.448297i −0.446491 − 0.894788i \(-0.647326\pi\)
0.894788 + 0.446491i \(0.147326\pi\)
\(4\) 3.95923 − 0.569656i 0.989807 − 0.142414i
\(5\) −4.96914 − 0.554657i −0.993828 − 0.110931i
\(6\) −2.49090 + 2.87494i −0.415150 + 0.479157i
\(7\) 4.10715 − 4.10715i 0.586735 − 0.586735i −0.350010 − 0.936746i \(-0.613822\pi\)
0.936746 + 0.350010i \(0.113822\pi\)
\(8\) −7.81692 + 1.70170i −0.977115 + 0.212712i
\(9\) 5.38253i 0.598059i
\(10\) 9.99212 + 0.396998i 0.999212 + 0.0396998i
\(11\) 13.8814i 1.26195i 0.775804 + 0.630974i \(0.217345\pi\)
−0.775804 + 0.630974i \(0.782655\pi\)
\(12\) 4.55861 − 6.09086i 0.379884 − 0.507572i
\(13\) −6.27479 − 6.27479i −0.482676 − 0.482676i 0.423309 − 0.905985i \(-0.360868\pi\)
−0.905985 + 0.423309i \(0.860868\pi\)
\(14\) −7.60693 + 8.77975i −0.543352 + 0.627125i
\(15\) −7.42891 + 5.93700i −0.495261 + 0.395800i
\(16\) 15.3510 − 4.51080i 0.959436 − 0.281925i
\(17\) 15.0594 + 15.0594i 0.885844 + 0.885844i 0.994121 − 0.108276i \(-0.0345332\pi\)
−0.108276 + 0.994121i \(0.534533\pi\)
\(18\) −0.768508 − 10.7376i −0.0426949 − 0.596533i
\(19\) 18.3224 + 5.02889i 0.964337 + 0.264678i
\(20\) −19.9899 + 0.634686i −0.999496 + 0.0317343i
\(21\) − 11.0473i − 0.526064i
\(22\) −1.98197 − 27.6920i −0.0900895 − 1.25873i
\(23\) −10.1791 − 10.1791i −0.442571 − 0.442571i 0.450304 − 0.892875i \(-0.351316\pi\)
−0.892875 + 0.450304i \(0.851316\pi\)
\(24\) −8.22432 + 12.8015i −0.342680 + 0.533396i
\(25\) 24.3847 + 5.51234i 0.975388 + 0.220494i
\(26\) 13.4135 + 11.6217i 0.515903 + 0.446987i
\(27\) 19.3430 + 19.3430i 0.716406 + 0.716406i
\(28\) 13.9215 − 18.6008i 0.497196 − 0.664314i
\(29\) 12.5488 0.432718 0.216359 − 0.976314i \(-0.430582\pi\)
0.216359 + 0.976314i \(0.430582\pi\)
\(30\) 13.9722 − 12.9044i 0.465741 − 0.430147i
\(31\) −19.7872 −0.638298 −0.319149 − 0.947705i \(-0.603397\pi\)
−0.319149 + 0.947705i \(0.603397\pi\)
\(32\) −29.9796 + 11.1904i −0.936862 + 0.349699i
\(33\) 18.6690 + 18.6690i 0.565728 + 0.565728i
\(34\) −32.1920 − 27.8917i −0.946824 − 0.820344i
\(35\) −22.6871 + 18.1309i −0.648202 + 0.518027i
\(36\) 3.06619 + 21.3107i 0.0851720 + 0.591963i
\(37\) 33.1652 − 33.1652i 0.896357 − 0.896357i −0.0987552 − 0.995112i \(-0.531486\pi\)
0.995112 + 0.0987552i \(0.0314861\pi\)
\(38\) −37.2693 − 7.41608i −0.980771 − 0.195160i
\(39\) −16.8778 −0.432765
\(40\) 39.7872 − 4.12026i 0.994681 − 0.103007i
\(41\) − 55.2228i − 1.34690i −0.739234 − 0.673448i \(-0.764812\pi\)
0.739234 − 0.673448i \(-0.235188\pi\)
\(42\) 1.57732 + 22.0383i 0.0375553 + 0.524722i
\(43\) 27.5934 + 27.5934i 0.641707 + 0.641707i 0.950975 − 0.309268i \(-0.100084\pi\)
−0.309268 + 0.950975i \(0.600084\pi\)
\(44\) 7.90764 + 54.9598i 0.179719 + 1.24909i
\(45\) 2.98546 − 26.7465i 0.0663435 − 0.594368i
\(46\) 21.7597 + 18.8530i 0.473036 + 0.409847i
\(47\) 21.6976 − 21.6976i 0.461651 − 0.461651i −0.437545 − 0.899196i \(-0.644152\pi\)
0.899196 + 0.437545i \(0.144152\pi\)
\(48\) 14.5789 − 26.7120i 0.303727 − 0.556499i
\(49\) 15.2627i 0.311483i
\(50\) −49.4320 − 7.51494i −0.988641 − 0.150299i
\(51\) 40.5064 0.794244
\(52\) −28.4178 − 21.2689i −0.546496 − 0.409017i
\(53\) 72.2683 + 72.2683i 1.36355 + 1.36355i 0.869342 + 0.494212i \(0.164543\pi\)
0.494212 + 0.869342i \(0.335457\pi\)
\(54\) −41.3490 − 35.8254i −0.765721 − 0.663434i
\(55\) 7.69944 − 68.9788i 0.139990 − 1.25416i
\(56\) −25.1161 + 39.0944i −0.448502 + 0.698114i
\(57\) 31.4050 − 17.8783i 0.550964 − 0.313655i
\(58\) −25.0336 + 1.79170i −0.431614 + 0.0308914i
\(59\) 111.357i 1.88741i 0.330791 + 0.943704i \(0.392684\pi\)
−0.330791 + 0.943704i \(0.607316\pi\)
\(60\) −26.0307 + 27.7379i −0.433845 + 0.462298i
\(61\) −47.2031 −0.773821 −0.386911 − 0.922117i \(-0.626458\pi\)
−0.386911 + 0.922117i \(0.626458\pi\)
\(62\) 39.4735 − 2.82519i 0.636669 − 0.0455675i
\(63\) 22.1068 + 22.1068i 0.350902 + 0.350902i
\(64\) 58.2085 − 26.6041i 0.909507 − 0.415688i
\(65\) 27.7000 + 34.6607i 0.426153 + 0.533241i
\(66\) −39.9083 − 34.5773i −0.604672 − 0.523898i
\(67\) −10.4691 − 10.4691i −0.156255 − 0.156255i 0.624650 − 0.780905i \(-0.285242\pi\)
−0.780905 + 0.624650i \(0.785242\pi\)
\(68\) 68.2021 + 51.0448i 1.00297 + 0.750658i
\(69\) −27.3797 −0.396807
\(70\) 42.6696 − 39.4086i 0.609566 − 0.562980i
\(71\) 130.709 1.84098 0.920489 − 0.390769i \(-0.127791\pi\)
0.920489 + 0.390769i \(0.127791\pi\)
\(72\) −9.15943 − 42.0748i −0.127214 − 0.584372i
\(73\) 100.965 − 100.965i 1.38308 − 1.38308i 0.543988 − 0.839093i \(-0.316914\pi\)
0.839093 − 0.543988i \(-0.183086\pi\)
\(74\) −61.4259 + 70.8964i −0.830079 + 0.958060i
\(75\) 40.2083 − 25.3813i 0.536111 − 0.338417i
\(76\) 75.4073 + 9.47306i 0.992201 + 0.124646i
\(77\) 57.0131 + 57.0131i 0.740430 + 0.740430i
\(78\) 33.6696 − 2.40979i 0.431661 − 0.0308947i
\(79\) 109.995i 1.39235i 0.717874 + 0.696173i \(0.245115\pi\)
−0.717874 + 0.696173i \(0.754885\pi\)
\(80\) −78.7831 + 13.9003i −0.984789 + 0.173753i
\(81\) 3.58562 0.0442669
\(82\) 7.88461 + 110.164i 0.0961538 + 1.34346i
\(83\) 31.3130 + 31.3130i 0.377265 + 0.377265i 0.870114 − 0.492850i \(-0.164045\pi\)
−0.492850 + 0.870114i \(0.664045\pi\)
\(84\) −6.29319 − 43.7390i −0.0749189 − 0.520702i
\(85\) −66.4793 − 83.1848i −0.782109 − 0.978645i
\(86\) −58.9857 − 51.1063i −0.685881 − 0.594259i
\(87\) 16.8768 − 16.8768i 0.193986 − 0.193986i
\(88\) −23.6220 − 108.510i −0.268432 − 1.23307i
\(89\) −59.3913 −0.667318 −0.333659 − 0.942694i \(-0.608283\pi\)
−0.333659 + 0.942694i \(0.608283\pi\)
\(90\) −2.13686 + 53.7829i −0.0237428 + 0.597587i
\(91\) −51.5430 −0.566407
\(92\) −46.1001 − 34.5029i −0.501088 − 0.375032i
\(93\) −26.6117 + 26.6117i −0.286147 + 0.286147i
\(94\) −40.1865 + 46.3824i −0.427516 + 0.493430i
\(95\) −88.2573 − 35.1519i −0.929024 − 0.370020i
\(96\) −25.2695 + 55.3692i −0.263224 + 0.576762i
\(97\) −2.65389 + 2.65389i −0.0273597 + 0.0273597i −0.720654 − 0.693295i \(-0.756158\pi\)
0.693295 + 0.720654i \(0.256158\pi\)
\(98\) −2.17918 − 30.4475i −0.0222365 − 0.310688i
\(99\) −74.7172 −0.754719
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 380.3.j.a.227.4 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.55 yes 232
5.3 odd 4 inner 380.3.j.a.303.62 yes 232
19.18 odd 2 inner 380.3.j.a.227.113 yes 232
20.3 even 4 inner 380.3.j.a.303.113 yes 232
76.75 even 2 inner 380.3.j.a.227.62 yes 232
95.18 even 4 inner 380.3.j.a.303.55 yes 232
380.303 odd 4 inner 380.3.j.a.303.4 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.4 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.55 yes 232 4.3 odd 2 inner
380.3.j.a.227.62 yes 232 76.75 even 2 inner
380.3.j.a.227.113 yes 232 19.18 odd 2 inner
380.3.j.a.303.4 yes 232 380.303 odd 4 inner
380.3.j.a.303.55 yes 232 95.18 even 4 inner
380.3.j.a.303.62 yes 232 5.3 odd 4 inner
380.3.j.a.303.113 yes 232 20.3 even 4 inner