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Properties

Label 420.2.bf.a.11.17
Level $420$
Weight $2$
Character 420.11
Analytic conductor $3.354$
Analytic rank $0$
Dimension $128$
Inner twists $8$

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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] = [420,2,Mod(11,420)] 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("420.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.bf (of order \(6\), 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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(64\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.17
Character \(\chi\) \(=\) 420.11
Dual form 420.2.bf.a.191.17

$q$-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+(-0.974322 - 1.02504i) q^{2} +(-1.46311 - 0.926981i) q^{3} +(-0.101394 + 1.99743i) q^{4} +(-0.866025 + 0.500000i) q^{5} +(0.475356 + 2.40292i) q^{6} +(1.76764 - 1.96862i) q^{7} +(2.14622 - 1.84220i) q^{8} +(1.28141 + 2.71256i) q^{9} +(1.35630 + 0.400546i) q^{10} +(0.295541 - 0.511892i) q^{11} +(1.99993 - 2.82848i) q^{12} +4.09332 q^{13} +(-3.74015 + 0.106176i) q^{14} +(1.73059 + 0.0712321i) q^{15} +(-3.97944 - 0.405056i) q^{16} +(-4.60893 - 2.66097i) q^{17} +(1.53196 - 3.95640i) q^{18} +(1.61063 - 0.929899i) q^{19} +(-0.910904 - 1.78052i) q^{20} +(-4.41113 + 1.24175i) q^{21} +(-0.812659 + 0.195808i) q^{22} +(-2.17076 - 3.75987i) q^{23} +(-4.84786 + 0.705847i) q^{24} +(0.500000 - 0.866025i) q^{25} +(-3.98821 - 4.19580i) q^{26} +(0.639643 - 5.15663i) q^{27} +(3.75294 + 3.73034i) q^{28} -1.84032i q^{29} +(-1.61313 - 1.84331i) q^{30} +(-6.10975 - 3.52747i) q^{31} +(3.46206 + 4.47372i) q^{32} +(-0.906925 + 0.474996i) q^{33} +(1.76300 + 7.31696i) q^{34} +(-0.546511 + 2.58869i) q^{35} +(-5.54807 + 2.28449i) q^{36} +(0.430200 + 0.745129i) q^{37} +(-2.52245 - 0.744934i) q^{38} +(-5.98900 - 3.79443i) q^{39} +(-0.937583 + 2.66851i) q^{40} -1.19595i q^{41} +(5.57069 + 3.31170i) q^{42} +0.837533i q^{43} +(0.992501 + 0.642225i) q^{44} +(-2.46601 - 1.70844i) q^{45} +(-1.73898 + 5.88843i) q^{46} +(-6.50419 - 11.2656i) q^{47} +(5.44690 + 4.28151i) q^{48} +(-0.750912 - 6.95961i) q^{49} +(-1.37487 + 0.331270i) q^{50} +(4.27673 + 8.16570i) q^{51} +(-0.415040 + 8.17612i) q^{52} +(11.5625 + 6.67564i) q^{53} +(-5.90895 + 4.36856i) q^{54} +0.591082i q^{55} +(0.167151 - 7.48145i) q^{56} +(-3.21854 - 0.132477i) q^{57} +(-1.88640 + 1.79307i) q^{58} +(-4.38882 + 7.60166i) q^{59} +(-0.317752 + 3.44950i) q^{60} +(-5.22617 - 9.05199i) q^{61} +(2.33709 + 9.69960i) q^{62} +(7.60507 + 2.27222i) q^{63} +(1.21256 - 7.90757i) q^{64} +(-3.54492 + 2.04666i) q^{65} +(1.37052 + 0.466831i) q^{66} +(-4.21436 - 2.43316i) q^{67} +(5.78241 - 8.93620i) q^{68} +(-0.309256 + 7.51337i) q^{69} +(3.18598 - 1.96203i) q^{70} +8.54722 q^{71} +(7.74729 + 3.46114i) q^{72} +(7.18369 - 12.4425i) q^{73} +(0.344630 - 1.16697i) q^{74} +(-1.53435 + 0.803604i) q^{75} +(1.69410 + 3.31141i) q^{76} +(-0.485310 - 1.48665i) q^{77} +(1.94579 + 9.83594i) q^{78} +(6.25053 - 3.60875i) q^{79} +(3.64882 - 1.63893i) q^{80} +(-5.71597 + 6.95181i) q^{81} +(-1.22589 + 1.16524i) q^{82} +4.77256 q^{83} +(-2.03304 - 8.93682i) q^{84} +5.32194 q^{85} +(0.858501 - 0.816027i) q^{86} +(-1.70595 + 2.69260i) q^{87} +(-0.308713 - 1.64308i) q^{88} +(12.2176 - 7.05385i) q^{89} +(0.651479 + 4.19232i) q^{90} +(7.23551 - 8.05819i) q^{91} +(7.73017 - 3.95471i) q^{92} +(5.66937 + 10.8247i) q^{93} +(-5.21045 + 17.6433i) q^{94} +(-0.929899 + 1.61063i) q^{95} +(-0.918333 - 9.75483i) q^{96} -9.39079 q^{97} +(-6.40221 + 7.55061i) q^{98} +(1.76725 + 0.145729i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 10 q^{12} + 16 q^{13} + 16 q^{16} + 10 q^{18} - 4 q^{21} - 56 q^{22} + 20 q^{24} + 64 q^{25} - 12 q^{28} - 48 q^{34} - 40 q^{36} + 8 q^{37} - 54 q^{42} - 4 q^{45} - 16 q^{46} - 36 q^{48} - 24 q^{49}+ \cdots - 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.974322 − 1.02504i −0.688950 − 0.724809i
\(3\) −1.46311 − 0.926981i −0.844730 − 0.535193i
\(4\) −0.101394 + 1.99743i −0.0506971 + 0.998714i
\(5\) −0.866025 + 0.500000i −0.387298 + 0.223607i
\(6\) 0.475356 + 2.40292i 0.194063 + 0.980989i
\(7\) 1.76764 − 1.96862i 0.668104 − 0.744068i
\(8\) 2.14622 − 1.84220i 0.758805 − 0.651318i
\(9\) 1.28141 + 2.71256i 0.427137 + 0.904187i
\(10\) 1.35630 + 0.400546i 0.428901 + 0.126664i
\(11\) 0.295541 − 0.511892i 0.0891089 − 0.154341i −0.818026 − 0.575182i \(-0.804931\pi\)
0.907135 + 0.420840i \(0.138265\pi\)
\(12\) 1.99993 − 2.82848i 0.577330 − 0.816511i
\(13\) 4.09332 1.13528 0.567642 − 0.823276i \(-0.307856\pi\)
0.567642 + 0.823276i \(0.307856\pi\)
\(14\) −3.74015 + 0.106176i −0.999597 + 0.0283767i
\(15\) 1.73059 + 0.0712321i 0.446835 + 0.0183920i
\(16\) −3.97944 − 0.405056i −0.994860 − 0.101264i
\(17\) −4.60893 − 2.66097i −1.11783 − 0.645380i −0.176984 − 0.984214i \(-0.556634\pi\)
−0.940846 + 0.338834i \(0.889967\pi\)
\(18\) 1.53196 − 3.95640i 0.361087 − 0.932532i
\(19\) 1.61063 − 0.929899i 0.369505 − 0.213334i −0.303737 − 0.952756i \(-0.598235\pi\)
0.673242 + 0.739422i \(0.264901\pi\)
\(20\) −0.910904 − 1.78052i −0.203684 − 0.398137i
\(21\) −4.41113 + 1.24175i −0.962587 + 0.270971i
\(22\) −0.812659 + 0.195808i −0.173260 + 0.0417463i
\(23\) −2.17076 − 3.75987i −0.452635 − 0.783987i 0.545914 − 0.837841i \(-0.316183\pi\)
−0.998549 + 0.0538546i \(0.982849\pi\)
\(24\) −4.84786 + 0.705847i −0.989566 + 0.144080i
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) −3.98821 − 4.19580i −0.782153 − 0.822864i
\(27\) 0.639643 − 5.15663i 0.123099 − 0.992394i
\(28\) 3.75294 + 3.73034i 0.709240 + 0.704967i
\(29\) − 1.84032i − 0.341739i −0.985294 − 0.170870i \(-0.945342\pi\)
0.985294 − 0.170870i \(-0.0546577\pi\)
\(30\) −1.61313 − 1.84331i −0.294516 − 0.336542i
\(31\) −6.10975 − 3.52747i −1.09734 − 0.633552i −0.161822 − 0.986820i \(-0.551737\pi\)
−0.935522 + 0.353268i \(0.885070\pi\)
\(32\) 3.46206 + 4.47372i 0.612011 + 0.790849i
\(33\) −0.906925 + 0.474996i −0.157875 + 0.0826861i
\(34\) 1.76300 + 7.31696i 0.302351 + 1.25485i
\(35\) −0.546511 + 2.58869i −0.0923771 + 0.437569i
\(36\) −5.54807 + 2.28449i −0.924679 + 0.380748i
\(37\) 0.430200 + 0.745129i 0.0707245 + 0.122498i 0.899219 − 0.437499i \(-0.144136\pi\)
−0.828495 + 0.559997i \(0.810802\pi\)
\(38\) −2.52245 − 0.744934i −0.409196 − 0.120844i
\(39\) −5.98900 − 3.79443i −0.959008 − 0.607596i
\(40\) −0.937583 + 2.66851i −0.148245 + 0.421928i
\(41\) − 1.19595i − 0.186777i −0.995630 − 0.0933883i \(-0.970230\pi\)
0.995630 − 0.0933883i \(-0.0297698\pi\)
\(42\) 5.57069 + 3.31170i 0.859577 + 0.511007i
\(43\) 0.837533i 0.127723i 0.997959 + 0.0638613i \(0.0203415\pi\)
−0.997959 + 0.0638613i \(0.979658\pi\)
\(44\) 0.992501 + 0.642225i 0.149625 + 0.0968190i
\(45\) −2.46601 − 1.70844i −0.367612 − 0.254679i
\(46\) −1.73898 + 5.88843i −0.256398 + 0.868201i
\(47\) −6.50419 − 11.2656i −0.948734 − 1.64326i −0.748096 − 0.663591i \(-0.769032\pi\)
−0.200638 − 0.979665i \(-0.564302\pi\)
\(48\) 5.44690 + 4.28151i 0.786192 + 0.617983i
\(49\) −0.750912 − 6.95961i −0.107273 − 0.994230i
\(50\) −1.37487 + 0.331270i −0.194436 + 0.0468486i
\(51\) 4.27673 + 8.16570i 0.598862 + 1.14343i
\(52\) −0.415040 + 8.17612i −0.0575556 + 1.13382i
\(53\) 11.5625 + 6.67564i 1.58824 + 0.916969i 0.993598 + 0.112976i \(0.0360385\pi\)
0.594639 + 0.803993i \(0.297295\pi\)
\(54\) −5.90895 + 4.36856i −0.804106 + 0.594486i
\(55\) 0.591082i 0.0797015i
\(56\) 0.167151 − 7.48145i 0.0223365 − 0.999751i
\(57\) −3.21854 − 0.132477i −0.426306 − 0.0175470i
\(58\) −1.88640 + 1.79307i −0.247696 + 0.235441i
\(59\) −4.38882 + 7.60166i −0.571376 + 0.989652i 0.425049 + 0.905170i \(0.360257\pi\)
−0.996425 + 0.0844820i \(0.973076\pi\)
\(60\) −0.317752 + 3.44950i −0.0410217 + 0.445328i
\(61\) −5.22617 − 9.05199i −0.669142 − 1.15899i −0.978144 − 0.207927i \(-0.933328\pi\)
0.309002 − 0.951061i \(-0.400005\pi\)
\(62\) 2.33709 + 9.69960i 0.296810 + 1.23185i
\(63\) 7.60507 + 2.27222i 0.958148 + 0.286273i
\(64\) 1.21256 − 7.90757i 0.151570 − 0.988446i
\(65\) −3.54492 + 2.04666i −0.439693 + 0.253857i
\(66\) 1.37052 + 0.466831i 0.168700 + 0.0574629i
\(67\) −4.21436 − 2.43316i −0.514866 − 0.297258i 0.219966 − 0.975508i \(-0.429405\pi\)
−0.734832 + 0.678250i \(0.762739\pi\)
\(68\) 5.78241 − 8.93620i 0.701220 − 1.08367i
\(69\) −0.309256 + 7.51337i −0.0372300 + 0.904504i
\(70\) 3.18598 − 1.96203i 0.380797 − 0.234507i
\(71\) 8.54722 1.01437 0.507184 − 0.861838i \(-0.330686\pi\)
0.507184 + 0.861838i \(0.330686\pi\)
\(72\) 7.74729 + 3.46114i 0.913027 + 0.407900i
\(73\) 7.18369 − 12.4425i 0.840787 − 1.45629i −0.0484428 − 0.998826i \(-0.515426\pi\)
0.889230 − 0.457460i \(-0.151241\pi\)
\(74\) 0.344630 − 1.16697i 0.0400624 − 0.135657i
\(75\) −1.53435 + 0.803604i −0.177171 + 0.0927922i
\(76\) 1.69410 + 3.31141i 0.194326 + 0.379845i
\(77\) −0.485310 − 1.48665i −0.0553062 − 0.169419i
\(78\) 1.94579 + 9.83594i 0.220317 + 1.11370i
\(79\) 6.25053 − 3.60875i 0.703240 − 0.406016i −0.105313 − 0.994439i \(-0.533584\pi\)
0.808553 + 0.588423i \(0.200251\pi\)
\(80\) 3.64882 − 1.63893i 0.407951 − 0.183238i
\(81\) −5.71597 + 6.95181i −0.635108 + 0.772423i
\(82\) −1.22589 + 1.16524i −0.135377 + 0.128680i
\(83\) 4.77256 0.523857 0.261928 − 0.965087i \(-0.415642\pi\)
0.261928 + 0.965087i \(0.415642\pi\)
\(84\) −2.03304 − 8.93682i −0.221822 − 0.975087i
\(85\) 5.32194 0.577245
\(86\) 0.858501 − 0.816027i 0.0925746 − 0.0879945i
\(87\) −1.70595 + 2.69260i −0.182897 + 0.288677i
\(88\) −0.308713 − 1.64308i −0.0329089 − 0.175153i
\(89\) 12.2176 − 7.05385i 1.29507 − 0.747707i 0.315518 − 0.948919i \(-0.397822\pi\)
0.979548 + 0.201213i \(0.0644882\pi\)
\(90\) 0.651479 + 4.19232i 0.0686719 + 0.441910i
\(91\) 7.23551 − 8.05819i 0.758488 − 0.844728i
\(92\) 7.73017 − 3.95471i 0.805926 − 0.412307i
\(93\) 5.66937 + 10.8247i 0.587887 + 1.12247i
\(94\) −5.21045 + 17.6433i −0.537417 + 1.81977i
\(95\) −0.929899 + 1.61063i −0.0954057 + 0.165247i
\(96\) −0.918333 − 9.75483i −0.0937270 − 0.995598i
\(97\) −9.39079 −0.953491 −0.476745 − 0.879041i \(-0.658184\pi\)
−0.476745 + 0.879041i \(0.658184\pi\)
\(98\) −6.40221 + 7.55061i −0.646721 + 0.762727i
\(99\) 1.76725 + 0.145729i 0.177615 + 0.0146463i
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 420.2.bf.a.11.17 ✓ 128
3.2 odd 2 inner 420.2.bf.a.11.48 yes 128
4.3 odd 2 inner 420.2.bf.a.11.38 yes 128
7.2 even 3 inner 420.2.bf.a.191.27 yes 128
12.11 even 2 inner 420.2.bf.a.11.27 yes 128
21.2 odd 6 inner 420.2.bf.a.191.38 yes 128
28.23 odd 6 inner 420.2.bf.a.191.48 yes 128
84.23 even 6 inner 420.2.bf.a.191.17 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bf.a.11.17 ✓ 128 1.1 even 1 trivial
420.2.bf.a.11.27 yes 128 12.11 even 2 inner
420.2.bf.a.11.38 yes 128 4.3 odd 2 inner
420.2.bf.a.11.48 yes 128 3.2 odd 2 inner
420.2.bf.a.191.17 yes 128 84.23 even 6 inner
420.2.bf.a.191.27 yes 128 7.2 even 3 inner
420.2.bf.a.191.38 yes 128 21.2 odd 6 inner
420.2.bf.a.191.48 yes 128 28.23 odd 6 inner