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Properties

Label 420.2.c.b.391.4
Level $420$
Weight $2$
Character 420.391
Analytic conductor $3.354$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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(391,420)] 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("420.391"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,2,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 3 x^{12} + 2 x^{11} - 7 x^{10} + 12 x^{9} - 28 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 391.4
Root \(-0.947441 + 1.04993i\) of defining polynomial
Character \(\chi\) \(=\) 420.391
Dual form 420.2.c.b.391.3

$q$-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+(-0.947441 + 1.04993i) q^{2} +1.00000 q^{3} +(-0.204711 - 1.98950i) q^{4} +1.00000i q^{5} +(-0.947441 + 1.04993i) q^{6} +(2.29670 - 1.31346i) q^{7} +(2.28279 + 1.67000i) q^{8} +1.00000 q^{9} +(-1.04993 - 0.947441i) q^{10} +0.477147i q^{11} +(-0.204711 - 1.98950i) q^{12} -2.96271i q^{13} +(-0.796939 + 3.65580i) q^{14} +1.00000i q^{15} +(-3.91619 + 0.814543i) q^{16} -3.83353i q^{17} +(-0.947441 + 1.04993i) q^{18} +5.31262 q^{19} +(1.98950 - 0.204711i) q^{20} +(2.29670 - 1.31346i) q^{21} +(-0.500972 - 0.452069i) q^{22} +7.60808i q^{23} +(2.28279 + 1.67000i) q^{24} -1.00000 q^{25} +(3.11064 + 2.80699i) q^{26} +1.00000 q^{27} +(-3.08329 - 4.30039i) q^{28} +6.17752 q^{29} +(-1.04993 - 0.947441i) q^{30} +3.38789 q^{31} +(2.85514 - 4.88346i) q^{32} +0.477147i q^{33} +(4.02494 + 3.63204i) q^{34} +(1.31346 + 2.29670i) q^{35} +(-0.204711 - 1.98950i) q^{36} -8.62867 q^{37} +(-5.03339 + 5.57788i) q^{38} -2.96271i q^{39} +(-1.67000 + 2.28279i) q^{40} +1.01125i q^{41} +(-0.796939 + 3.65580i) q^{42} +6.85412i q^{43} +(0.949282 - 0.0976772i) q^{44} +1.00000i q^{45} +(-7.98796 - 7.20820i) q^{46} +6.21838 q^{47} +(-3.91619 + 0.814543i) q^{48} +(3.54963 - 6.03325i) q^{49} +(0.947441 - 1.04993i) q^{50} -3.83353i q^{51} +(-5.89430 + 0.606499i) q^{52} -9.30380 q^{53} +(-0.947441 + 1.04993i) q^{54} -0.477147 q^{55} +(7.43634 + 0.837125i) q^{56} +5.31262 q^{57} +(-5.85284 + 6.48597i) q^{58} -4.88854 q^{59} +(1.98950 - 0.204711i) q^{60} +4.75818i q^{61} +(-3.20982 + 3.55705i) q^{62} +(2.29670 - 1.31346i) q^{63} +(2.42221 + 7.62449i) q^{64} +2.96271 q^{65} +(-0.500972 - 0.452069i) q^{66} -1.30610i q^{67} +(-7.62679 + 0.784764i) q^{68} +7.60808i q^{69} +(-3.65580 - 0.796939i) q^{70} +9.18700i q^{71} +(2.28279 + 1.67000i) q^{72} -4.49766i q^{73} +(8.17516 - 9.05951i) q^{74} -1.00000 q^{75} +(-1.08755 - 10.5694i) q^{76} +(0.626715 + 1.09586i) q^{77} +(3.11064 + 2.80699i) q^{78} -8.80833i q^{79} +(-0.814543 - 3.91619i) q^{80} +1.00000 q^{81} +(-1.06174 - 0.958097i) q^{82} -10.9520 q^{83} +(-3.08329 - 4.30039i) q^{84} +3.83353 q^{85} +(-7.19635 - 6.49388i) q^{86} +6.17752 q^{87} +(-0.796835 + 1.08922i) q^{88} -13.1208i q^{89} +(-1.04993 - 0.947441i) q^{90} +(-3.89141 - 6.80445i) q^{91} +(15.1362 - 1.55746i) q^{92} +3.38789 q^{93} +(-5.89154 + 6.52887i) q^{94} +5.31262i q^{95} +(2.85514 - 4.88346i) q^{96} -1.60612i q^{97} +(2.97143 + 9.44302i) q^{98} +0.477147i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 16 q^{3} - 2 q^{4} + 2 q^{6} + 4 q^{7} + 2 q^{8} + 16 q^{9} - 2 q^{12} + 10 q^{14} + 6 q^{16} + 2 q^{18} + 24 q^{19} + 4 q^{21} - 12 q^{22} + 2 q^{24} - 16 q^{25} + 12 q^{26} + 16 q^{27}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.947441 + 1.04993i −0.669942 + 0.742413i
\(3\) 1.00000 0.577350
\(4\) −0.204711 − 1.98950i −0.102355 − 0.994748i
\(5\) 1.00000i 0.447214i
\(6\) −0.947441 + 1.04993i −0.386791 + 0.428633i
\(7\) 2.29670 − 1.31346i 0.868070 − 0.496442i
\(8\) 2.28279 + 1.67000i 0.807086 + 0.590433i
\(9\) 1.00000 0.333333
\(10\) −1.04993 − 0.947441i −0.332017 − 0.299607i
\(11\) 0.477147i 0.143865i 0.997409 + 0.0719326i \(0.0229167\pi\)
−0.997409 + 0.0719326i \(0.977083\pi\)
\(12\) −0.204711 − 1.98950i −0.0590949 − 0.574318i
\(13\) − 2.96271i − 0.821708i −0.911701 − 0.410854i \(-0.865231\pi\)
0.911701 − 0.410854i \(-0.134769\pi\)
\(14\) −0.796939 + 3.65580i −0.212991 + 0.977054i
\(15\) 1.00000i 0.258199i
\(16\) −3.91619 + 0.814543i −0.979047 + 0.203636i
\(17\) − 3.83353i − 0.929767i −0.885372 − 0.464883i \(-0.846096\pi\)
0.885372 − 0.464883i \(-0.153904\pi\)
\(18\) −0.947441 + 1.04993i −0.223314 + 0.247471i
\(19\) 5.31262 1.21880 0.609399 − 0.792864i \(-0.291411\pi\)
0.609399 + 0.792864i \(0.291411\pi\)
\(20\) 1.98950 − 0.204711i 0.444865 − 0.0457747i
\(21\) 2.29670 − 1.31346i 0.501180 − 0.286621i
\(22\) −0.500972 − 0.452069i −0.106808 − 0.0963814i
\(23\) 7.60808i 1.58639i 0.608965 + 0.793197i \(0.291585\pi\)
−0.608965 + 0.793197i \(0.708415\pi\)
\(24\) 2.28279 + 1.67000i 0.465972 + 0.340887i
\(25\) −1.00000 −0.200000
\(26\) 3.11064 + 2.80699i 0.610047 + 0.550497i
\(27\) 1.00000 0.192450
\(28\) −3.08329 − 4.30039i −0.582687 − 0.812697i
\(29\) 6.17752 1.14714 0.573569 − 0.819158i \(-0.305559\pi\)
0.573569 + 0.819158i \(0.305559\pi\)
\(30\) −1.04993 − 0.947441i −0.191690 − 0.172978i
\(31\) 3.38789 0.608482 0.304241 − 0.952595i \(-0.401597\pi\)
0.304241 + 0.952595i \(0.401597\pi\)
\(32\) 2.85514 − 4.88346i 0.504723 − 0.863282i
\(33\) 0.477147i 0.0830607i
\(34\) 4.02494 + 3.63204i 0.690271 + 0.622890i
\(35\) 1.31346 + 2.29670i 0.222016 + 0.388213i
\(36\) −0.204711 − 1.98950i −0.0341185 − 0.331583i
\(37\) −8.62867 −1.41854 −0.709272 − 0.704935i \(-0.750976\pi\)
−0.709272 + 0.704935i \(0.750976\pi\)
\(38\) −5.03339 + 5.57788i −0.816524 + 0.904852i
\(39\) − 2.96271i − 0.474413i
\(40\) −1.67000 + 2.28279i −0.264050 + 0.360940i
\(41\) 1.01125i 0.157930i 0.996877 + 0.0789651i \(0.0251616\pi\)
−0.996877 + 0.0789651i \(0.974838\pi\)
\(42\) −0.796939 + 3.65580i −0.122970 + 0.564102i
\(43\) 6.85412i 1.04524i 0.852565 + 0.522622i \(0.175046\pi\)
−0.852565 + 0.522622i \(0.824954\pi\)
\(44\) 0.949282 − 0.0976772i 0.143110 − 0.0147254i
\(45\) 1.00000i 0.149071i
\(46\) −7.98796 − 7.20820i −1.17776 − 1.06279i
\(47\) 6.21838 0.907043 0.453522 − 0.891245i \(-0.350167\pi\)
0.453522 + 0.891245i \(0.350167\pi\)
\(48\) −3.91619 + 0.814543i −0.565253 + 0.117569i
\(49\) 3.54963 − 6.03325i 0.507090 − 0.861893i
\(50\) 0.947441 − 1.04993i 0.133988 − 0.148483i
\(51\) − 3.83353i − 0.536801i
\(52\) −5.89430 + 0.606499i −0.817392 + 0.0841063i
\(53\) −9.30380 −1.27797 −0.638987 − 0.769217i \(-0.720646\pi\)
−0.638987 + 0.769217i \(0.720646\pi\)
\(54\) −0.947441 + 1.04993i −0.128930 + 0.142878i
\(55\) −0.477147 −0.0643385
\(56\) 7.43634 + 0.837125i 0.993723 + 0.111866i
\(57\) 5.31262 0.703673
\(58\) −5.85284 + 6.48597i −0.768515 + 0.851650i
\(59\) −4.88854 −0.636433 −0.318217 − 0.948018i \(-0.603084\pi\)
−0.318217 + 0.948018i \(0.603084\pi\)
\(60\) 1.98950 − 0.204711i 0.256843 − 0.0264281i
\(61\) 4.75818i 0.609222i 0.952477 + 0.304611i \(0.0985264\pi\)
−0.952477 + 0.304611i \(0.901474\pi\)
\(62\) −3.20982 + 3.55705i −0.407648 + 0.451745i
\(63\) 2.29670 − 1.31346i 0.289357 − 0.165481i
\(64\) 2.42221 + 7.62449i 0.302777 + 0.953061i
\(65\) 2.96271 0.367479
\(66\) −0.500972 − 0.452069i −0.0616654 − 0.0556458i
\(67\) − 1.30610i − 0.159565i −0.996812 − 0.0797826i \(-0.974577\pi\)
0.996812 − 0.0797826i \(-0.0254226\pi\)
\(68\) −7.62679 + 0.784764i −0.924884 + 0.0951667i
\(69\) 7.60808i 0.915905i
\(70\) −3.65580 − 0.796939i −0.436952 − 0.0952525i
\(71\) 9.18700i 1.09030i 0.838340 + 0.545148i \(0.183527\pi\)
−0.838340 + 0.545148i \(0.816473\pi\)
\(72\) 2.28279 + 1.67000i 0.269029 + 0.196811i
\(73\) − 4.49766i − 0.526411i −0.964740 − 0.263206i \(-0.915220\pi\)
0.964740 − 0.263206i \(-0.0847797\pi\)
\(74\) 8.17516 − 9.05951i 0.950343 − 1.05315i
\(75\) −1.00000 −0.115470
\(76\) −1.08755 − 10.5694i −0.124751 − 1.21240i
\(77\) 0.626715 + 1.09586i 0.0714208 + 0.124885i
\(78\) 3.11064 + 2.80699i 0.352211 + 0.317829i
\(79\) − 8.80833i − 0.991015i −0.868604 − 0.495507i \(-0.834982\pi\)
0.868604 − 0.495507i \(-0.165018\pi\)
\(80\) −0.814543 − 3.91619i −0.0910686 − 0.437843i
\(81\) 1.00000 0.111111
\(82\) −1.06174 − 0.958097i −0.117249 − 0.105804i
\(83\) −10.9520 −1.20214 −0.601068 − 0.799198i \(-0.705258\pi\)
−0.601068 + 0.799198i \(0.705258\pi\)
\(84\) −3.08329 − 4.30039i −0.336414 − 0.469211i
\(85\) 3.83353 0.415804
\(86\) −7.19635 − 6.49388i −0.776003 − 0.700253i
\(87\) 6.17752 0.662300
\(88\) −0.796835 + 1.08922i −0.0849429 + 0.116112i
\(89\) − 13.1208i − 1.39080i −0.718621 − 0.695402i \(-0.755226\pi\)
0.718621 − 0.695402i \(-0.244774\pi\)
\(90\) −1.04993 − 0.947441i −0.110672 − 0.0998691i
\(91\) −3.89141 − 6.80445i −0.407931 − 0.713300i
\(92\) 15.1362 − 1.55746i 1.57806 − 0.162376i
\(93\) 3.38789 0.351307
\(94\) −5.89154 + 6.52887i −0.607666 + 0.673401i
\(95\) 5.31262i 0.545063i
\(96\) 2.85514 − 4.88346i 0.291402 − 0.498416i
\(97\) − 1.60612i − 0.163077i −0.996670 − 0.0815384i \(-0.974017\pi\)
0.996670 − 0.0815384i \(-0.0259833\pi\)
\(98\) 2.97143 + 9.44302i 0.300160 + 0.953889i
\(99\) 0.477147i 0.0479551i
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.c.b.391.4 yes 16
3.2 odd 2 1260.2.c.e.811.13 16
4.3 odd 2 420.2.c.a.391.3 ✓ 16
7.6 odd 2 420.2.c.a.391.4 yes 16
12.11 even 2 1260.2.c.d.811.14 16
21.20 even 2 1260.2.c.d.811.13 16
28.27 even 2 inner 420.2.c.b.391.3 yes 16
84.83 odd 2 1260.2.c.e.811.14 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.c.a.391.3 ✓ 16 4.3 odd 2
420.2.c.a.391.4 yes 16 7.6 odd 2
420.2.c.b.391.3 yes 16 28.27 even 2 inner
420.2.c.b.391.4 yes 16 1.1 even 1 trivial
1260.2.c.d.811.13 16 21.20 even 2
1260.2.c.d.811.14 16 12.11 even 2
1260.2.c.e.811.13 16 3.2 odd 2
1260.2.c.e.811.14 16 84.83 odd 2