Show commands: Magma / Pari/GP / SageMath

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 227.6
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.6

$q$-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+(-1.98524 - 0.242507i) q^{2} +(0.169672 - 0.169672i) q^{3} +(3.88238 + 0.962870i) q^{4} +(0.857699 - 4.92589i) q^{5} +(-0.377988 + 0.295694i) q^{6} +(1.67200 - 1.67200i) q^{7} +(-7.47397 - 2.85303i) q^{8} +8.94242i q^{9} +(-2.89730 + 9.57108i) q^{10} -16.5628i q^{11} +(0.822105 - 0.495360i) q^{12} +(-9.85537 - 9.85537i) q^{13} +(-3.72479 + 2.91385i) q^{14} +(-0.690259 - 0.981315i) q^{15} +(14.1458 + 7.47645i) q^{16} +(-9.91816 - 9.91816i) q^{17} +(2.16860 - 17.7529i) q^{18} +(8.86008 + 16.8077i) q^{19} +(8.07290 - 18.2983i) q^{20} -0.567383i q^{21} +(-4.01659 + 32.8812i) q^{22} +(13.0113 + 13.0113i) q^{23} +(-1.75221 + 0.784045i) q^{24} +(-23.5287 - 8.44985i) q^{25} +(17.1753 + 21.9553i) q^{26} +(3.04433 + 3.04433i) q^{27} +(8.10124 - 4.88141i) q^{28} -44.6723 q^{29} +(1.13236 + 2.11554i) q^{30} +23.6260 q^{31} +(-26.2697 - 18.2730i) q^{32} +(-2.81025 - 2.81025i) q^{33} +(17.2847 + 22.0952i) q^{34} +(-6.80200 - 9.67014i) q^{35} +(-8.61039 + 34.7179i) q^{36} +(10.6340 - 10.6340i) q^{37} +(-13.5134 - 35.5160i) q^{38} -3.34437 q^{39} +(-20.4641 + 34.3689i) q^{40} -38.5464i q^{41} +(-0.137594 + 1.12639i) q^{42} +(-46.2625 - 46.2625i) q^{43} +(15.9478 - 64.3031i) q^{44} +(44.0494 + 7.66991i) q^{45} +(-22.6752 - 28.9858i) q^{46} +(-3.82359 + 3.82359i) q^{47} +(3.66869 - 1.13160i) q^{48} +43.4089i q^{49} +(44.6611 + 22.4809i) q^{50} -3.36568 q^{51} +(-28.7729 - 47.7517i) q^{52} +(-34.7444 - 34.7444i) q^{53} +(-5.30547 - 6.78202i) q^{54} +(-81.5865 - 14.2059i) q^{55} +(-17.2667 + 7.72619i) q^{56} +(4.35511 + 1.34849i) q^{57} +(88.6854 + 10.8333i) q^{58} -77.8483i q^{59} +(-1.73497 - 4.47447i) q^{60} -117.455 q^{61} +(-46.9033 - 5.72946i) q^{62} +(14.9517 + 14.9517i) q^{63} +(47.7204 + 42.6470i) q^{64} +(-56.9994 + 40.0935i) q^{65} +(4.89753 + 6.26054i) q^{66} +(-70.7739 - 70.7739i) q^{67} +(-28.9562 - 48.0560i) q^{68} +4.41530 q^{69} +(11.1585 + 20.8471i) q^{70} +53.9724 q^{71} +(25.5130 - 66.8354i) q^{72} +(15.1243 - 15.1243i) q^{73} +(-23.6898 + 18.5322i) q^{74} +(-5.42588 + 2.55847i) q^{75} +(18.2146 + 73.7850i) q^{76} +(-27.6930 - 27.6930i) q^{77} +(6.63938 + 0.811032i) q^{78} +130.726i q^{79} +(48.9610 - 63.2679i) q^{80} -79.4487 q^{81} +(-9.34776 + 76.5240i) q^{82} +(-49.7289 - 49.7289i) q^{83} +(0.546316 - 2.20280i) q^{84} +(-57.3625 + 40.3489i) q^{85} +(80.6234 + 103.061i) q^{86} +(-7.57966 + 7.57966i) q^{87} +(-47.2543 + 123.790i) q^{88} +144.601 q^{89} +(-85.5887 - 25.9089i) q^{90} -32.9563 q^{91} +(37.9865 + 63.0428i) q^{92} +(4.00868 - 4.00868i) q^{93} +(8.51801 - 6.66352i) q^{94} +(90.3921 - 29.2278i) q^{95} +(-7.55767 + 1.35681i) q^{96} +(0.560108 - 0.560108i) q^{97} +(10.5269 - 86.1771i) q^{98} +148.112 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.98524 − 0.242507i −0.992622 − 0.121253i
\(3\) 0.169672 − 0.169672i 0.0565575 − 0.0565575i −0.678262 − 0.734820i \(-0.737267\pi\)
0.734820 + 0.678262i \(0.237267\pi\)
\(4\) 3.88238 + 0.962870i 0.970595 + 0.240717i
\(5\) 0.857699 − 4.92589i 0.171540 − 0.985177i
\(6\) −0.377988 + 0.295694i −0.0629980 + 0.0492824i
\(7\) 1.67200 − 1.67200i 0.238857 − 0.238857i −0.577520 − 0.816377i \(-0.695979\pi\)
0.816377 + 0.577520i \(0.195979\pi\)
\(8\) −7.47397 − 2.85303i −0.934246 − 0.356629i
\(9\) 8.94242i 0.993603i
\(10\) −2.89730 + 9.57108i −0.289730 + 0.957108i
\(11\) − 16.5628i − 1.50571i −0.658186 − 0.752855i \(-0.728676\pi\)
0.658186 − 0.752855i \(-0.271324\pi\)
\(12\) 0.822105 − 0.495360i 0.0685088 − 0.0412800i
\(13\) −9.85537 − 9.85537i −0.758105 − 0.758105i 0.217872 − 0.975977i \(-0.430088\pi\)
−0.975977 + 0.217872i \(0.930088\pi\)
\(14\) −3.72479 + 2.91385i −0.266056 + 0.208132i
\(15\) −0.690259 − 0.981315i −0.0460173 − 0.0654210i
\(16\) 14.1458 + 7.47645i 0.884110 + 0.467278i
\(17\) −9.91816 − 9.91816i −0.583421 − 0.583421i 0.352420 − 0.935842i \(-0.385359\pi\)
−0.935842 + 0.352420i \(0.885359\pi\)
\(18\) 2.16860 − 17.7529i 0.120478 − 0.986271i
\(19\) 8.86008 + 16.8077i 0.466320 + 0.884616i
\(20\) 8.07290 − 18.2983i 0.403645 − 0.914916i
\(21\) − 0.567383i − 0.0270183i
\(22\) −4.01659 + 32.8812i −0.182572 + 1.49460i
\(23\) 13.0113 + 13.0113i 0.565707 + 0.565707i 0.930923 − 0.365216i \(-0.119005\pi\)
−0.365216 + 0.930923i \(0.619005\pi\)
\(24\) −1.75221 + 0.784045i −0.0730086 + 0.0326685i
\(25\) −23.5287 − 8.44985i −0.941148 − 0.337994i
\(26\) 17.1753 + 21.9553i 0.660589 + 0.844435i
\(27\) 3.04433 + 3.04433i 0.112753 + 0.112753i
\(28\) 8.10124 − 4.88141i 0.289330 − 0.174336i
\(29\) −44.6723 −1.54043 −0.770213 − 0.637787i \(-0.779850\pi\)
−0.770213 + 0.637787i \(0.779850\pi\)
\(30\) 1.13236 + 2.11554i 0.0377452 + 0.0705180i
\(31\) 23.6260 0.762128 0.381064 − 0.924549i \(-0.375558\pi\)
0.381064 + 0.924549i \(0.375558\pi\)
\(32\) −26.2697 − 18.2730i −0.820928 − 0.571032i
\(33\) −2.81025 − 2.81025i −0.0851592 − 0.0851592i
\(34\) 17.2847 + 22.0952i 0.508375 + 0.649858i
\(35\) −6.80200 − 9.67014i −0.194343 − 0.276290i
\(36\) −8.61039 + 34.7179i −0.239177 + 0.964386i
\(37\) 10.6340 − 10.6340i 0.287405 − 0.287405i −0.548648 − 0.836053i \(-0.684857\pi\)
0.836053 + 0.548648i \(0.184857\pi\)
\(38\) −13.5134 − 35.5160i −0.355616 − 0.934632i
\(39\) −3.34437 −0.0857530
\(40\) −20.4641 + 34.3689i −0.511603 + 0.859222i
\(41\) − 38.5464i − 0.940156i −0.882625 − 0.470078i \(-0.844226\pi\)
0.882625 − 0.470078i \(-0.155774\pi\)
\(42\) −0.137594 + 1.12639i −0.00327606 + 0.0268189i
\(43\) −46.2625 − 46.2625i −1.07587 − 1.07587i −0.996875 − 0.0789981i \(-0.974828\pi\)
−0.0789981 − 0.996875i \(-0.525172\pi\)
\(44\) 15.9478 − 64.3031i 0.362451 − 1.46144i
\(45\) 44.0494 + 7.66991i 0.978875 + 0.170442i
\(46\) −22.6752 − 28.9858i −0.492939 − 0.630127i
\(47\) −3.82359 + 3.82359i −0.0813531 + 0.0813531i −0.746612 − 0.665259i \(-0.768321\pi\)
0.665259 + 0.746612i \(0.268321\pi\)
\(48\) 3.66869 − 1.13160i 0.0764311 − 0.0235750i
\(49\) 43.4089i 0.885895i
\(50\) 44.6611 + 22.4809i 0.893221 + 0.449618i
\(51\) −3.36568 −0.0659937
\(52\) −28.7729 − 47.7517i −0.553324 − 0.918303i
\(53\) −34.7444 − 34.7444i −0.655555 − 0.655555i 0.298770 − 0.954325i \(-0.403424\pi\)
−0.954325 + 0.298770i \(0.903424\pi\)
\(54\) −5.30547 − 6.78202i −0.0982495 − 0.125593i
\(55\) −81.5865 − 14.2059i −1.48339 − 0.258289i
\(56\) −17.2667 + 7.72619i −0.308334 + 0.137968i
\(57\) 4.35511 + 1.34849i 0.0764055 + 0.0236578i
\(58\) 88.6854 + 10.8333i 1.52906 + 0.186782i
\(59\) − 77.8483i − 1.31946i −0.751501 − 0.659732i \(-0.770670\pi\)
0.751501 − 0.659732i \(-0.229330\pi\)
\(60\) −1.73497 − 4.47447i −0.0289162 − 0.0745745i
\(61\) −117.455 −1.92549 −0.962744 − 0.270413i \(-0.912840\pi\)
−0.962744 + 0.270413i \(0.912840\pi\)
\(62\) −46.9033 − 5.72946i −0.756505 − 0.0924106i
\(63\) 14.9517 + 14.9517i 0.237329 + 0.237329i
\(64\) 47.7204 + 42.6470i 0.745631 + 0.666359i
\(65\) −56.9994 + 40.0935i −0.876913 + 0.616823i
\(66\) 4.89753 + 6.26054i 0.0742050 + 0.0948567i
\(67\) −70.7739 − 70.7739i −1.05633 − 1.05633i −0.998316 − 0.0580113i \(-0.981524\pi\)
−0.0580113 − 0.998316i \(-0.518476\pi\)
\(68\) −28.9562 − 48.0560i −0.425826 − 0.706706i
\(69\) 4.41530 0.0639899
\(70\) 11.1585 + 20.8471i 0.159408 + 0.297816i
\(71\) 53.9724 0.760174 0.380087 − 0.924951i \(-0.375894\pi\)
0.380087 + 0.924951i \(0.375894\pi\)
\(72\) 25.5130 − 66.8354i 0.354348 − 0.928269i
\(73\) 15.1243 − 15.1243i 0.207182 − 0.207182i −0.595886 − 0.803069i \(-0.703199\pi\)
0.803069 + 0.595886i \(0.203199\pi\)
\(74\) −23.6898 + 18.5322i −0.320133 + 0.250435i
\(75\) −5.42588 + 2.55847i −0.0723451 + 0.0341129i
\(76\) 18.2146 + 73.7850i 0.239665 + 0.970856i
\(77\) −27.6930 − 27.6930i −0.359649 − 0.359649i
\(78\) 6.63938 + 0.811032i 0.0851203 + 0.0103978i
\(79\) 130.726i 1.65476i 0.561642 + 0.827380i \(0.310170\pi\)
−0.561642 + 0.827380i \(0.689830\pi\)
\(80\) 48.9610 − 63.2679i 0.612012 − 0.790848i
\(81\) −79.4487 −0.980848
\(82\) −9.34776 + 76.5240i −0.113997 + 0.933219i
\(83\) −49.7289 − 49.7289i −0.599144 − 0.599144i 0.340941 − 0.940085i \(-0.389254\pi\)
−0.940085 + 0.340941i \(0.889254\pi\)
\(84\) 0.546316 − 2.20280i 0.00650377 − 0.0262238i
\(85\) −57.3625 + 40.3489i −0.674853 + 0.474693i
\(86\) 80.6234 + 103.061i 0.937481 + 1.19839i
\(87\) −7.57966 + 7.57966i −0.0871226 + 0.0871226i
\(88\) −47.2543 + 123.790i −0.536980 + 1.40670i
\(89\) 144.601 1.62472 0.812362 − 0.583153i \(-0.198181\pi\)
0.812362 + 0.583153i \(0.198181\pi\)
\(90\) −85.5887 − 25.9089i −0.950985 − 0.287877i
\(91\) −32.9563 −0.362157
\(92\) 37.9865 + 63.0428i 0.412897 + 0.685248i
\(93\) 4.00868 − 4.00868i 0.0431041 − 0.0431041i
\(94\) 8.51801 − 6.66352i 0.0906172 − 0.0708885i
\(95\) 90.3921 − 29.2278i 0.951496 − 0.307661i
\(96\) −7.55767 + 1.35681i −0.0787257 + 0.0141335i
\(97\) 0.560108 − 0.560108i 0.00577431 − 0.00577431i −0.704214 − 0.709988i \(-0.748700\pi\)
0.709988 + 0.704214i \(0.248700\pi\)
\(98\) 10.5269 − 86.1771i 0.107418 − 0.879358i
\(99\) 148.112 1.49608
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.227.6 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.64 yes 232
5.3 odd 4 inner 380.3.j.a.303.53 yes 232
19.18 odd 2 inner 380.3.j.a.227.111 yes 232
20.3 even 4 inner 380.3.j.a.303.111 yes 232
76.75 even 2 inner 380.3.j.a.227.53 yes 232
95.18 even 4 inner 380.3.j.a.303.64 yes 232
380.303 odd 4 inner 380.3.j.a.303.6 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.6 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.53 yes 232 76.75 even 2 inner
380.3.j.a.227.64 yes 232 4.3 odd 2 inner
380.3.j.a.227.111 yes 232 19.18 odd 2 inner
380.3.j.a.303.6 yes 232 380.303 odd 4 inner
380.3.j.a.303.53 yes 232 5.3 odd 4 inner
380.3.j.a.303.64 yes 232 95.18 even 4 inner
380.3.j.a.303.111 yes 232 20.3 even 4 inner