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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [441,2,Mod(62,441)] 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("441.62"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([7, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.w (of order \(14\), degree \(6\), 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 251.9
Character \(\chi\) \(=\) 441.251
Dual form 441.2.w.a.188.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.203165 - 0.162019i) q^{2} +(-0.430016 - 1.88402i) q^{4} +(-2.46158 - 1.18543i) q^{5} +(-1.53249 + 2.15673i) q^{7} +(-0.443379 + 0.920685i) q^{8} +(0.308044 + 0.639659i) q^{10} +(-2.00878 - 1.60195i) q^{11} +(4.63632 + 3.69734i) q^{13} +(0.660778 - 0.189879i) q^{14} +(-3.24295 + 1.56172i) q^{16} +(-1.48891 + 6.52334i) q^{17} -0.418798i q^{19} +(-1.17487 + 5.14742i) q^{20} +(0.148568 + 0.650919i) q^{22} +(-5.32955 + 1.21643i) q^{23} +(1.53666 + 1.92691i) q^{25} +(-0.342899 - 1.50234i) q^{26} +(4.72232 + 1.95982i) q^{28} +(2.32890 + 0.531557i) q^{29} -5.40375i q^{31} +(2.90441 + 0.662912i) q^{32} +(1.35940 - 1.08408i) q^{34} +(6.32900 - 3.49228i) q^{35} +(-1.37636 + 6.03022i) q^{37} +(-0.0678530 + 0.0850850i) q^{38} +(2.18282 - 1.74074i) q^{40} +(-5.97322 - 2.87655i) q^{41} +(-7.06131 + 3.40055i) q^{43} +(-2.15430 + 4.47345i) q^{44} +(1.27986 + 0.616349i) q^{46} +(2.90399 - 3.64149i) q^{47} +(-2.30293 - 6.61033i) q^{49} -0.640448i q^{50} +(4.97219 - 10.3249i) q^{52} +(-13.5007 + 3.08145i) q^{53} +(3.04576 + 6.32459i) q^{55} +(-1.30619 - 2.36719i) q^{56} +(-0.387029 - 0.485319i) q^{58} +(5.52357 - 2.66001i) q^{59} +(-9.77982 - 2.23218i) q^{61} +(-0.875508 + 1.09785i) q^{62} +(4.00571 + 5.02301i) q^{64} +(-7.02971 - 14.5973i) q^{65} -8.37721 q^{67} +12.9304 q^{68} +(-1.85164 - 0.315907i) q^{70} +(6.34846 - 1.44900i) q^{71} +(-5.32281 + 4.24480i) q^{73} +(1.25664 - 1.00213i) q^{74} +(-0.789025 + 0.180090i) q^{76} +(6.53340 - 1.87741i) q^{77} +0.883489 q^{79} +9.83409 q^{80} +(0.747494 + 1.55219i) q^{82} +(6.46744 + 8.10992i) q^{83} +(11.3980 - 14.2927i) q^{85} +(1.98556 + 0.453192i) q^{86} +(2.36554 - 1.13918i) q^{88} +(-1.58193 - 1.98368i) q^{89} +(-15.0793 + 4.33313i) q^{91} +(4.58358 + 9.51791i) q^{92} +(-1.17998 + 0.269322i) q^{94} +(-0.496457 + 1.03090i) q^{95} -3.88426i q^{97} +(-0.603122 + 1.71611i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 24 q^{4} - 32 q^{16} - 44 q^{22} - 4 q^{25} - 56 q^{28} + 112 q^{34} - 76 q^{37} + 28 q^{40} + 8 q^{43} - 40 q^{46} - 84 q^{49} - 140 q^{52} + 12 q^{58} - 84 q^{61} + 24 q^{64} + 16 q^{67} + 112 q^{70}+ \cdots - 112 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{9}{14}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.203165 − 0.162019i −0.143659 − 0.114564i 0.549023 − 0.835807i \(-0.315000\pi\)
−0.692682 + 0.721243i \(0.743571\pi\)
\(3\) 0 0
\(4\) −0.430016 − 1.88402i −0.215008 − 0.942011i
\(5\) −2.46158 − 1.18543i −1.10085 − 0.530142i −0.206925 − 0.978357i \(-0.566346\pi\)
−0.893926 + 0.448215i \(0.852060\pi\)
\(6\) 0 0
\(7\) −1.53249 + 2.15673i −0.579228 + 0.815166i
\(8\) −0.443379 + 0.920685i −0.156758 + 0.325511i
\(9\) 0 0
\(10\) 0.308044 + 0.639659i 0.0974120 + 0.202278i
\(11\) −2.00878 − 1.60195i −0.605670 − 0.483005i 0.271983 − 0.962302i \(-0.412320\pi\)
−0.877653 + 0.479297i \(0.840892\pi\)
\(12\) 0 0
\(13\) 4.63632 + 3.69734i 1.28588 + 1.02546i 0.997693 + 0.0678861i \(0.0216255\pi\)
0.288192 + 0.957573i \(0.406946\pi\)
\(14\) 0.660778 − 0.189879i 0.176600 − 0.0507472i
\(15\) 0 0
\(16\) −3.24295 + 1.56172i −0.810738 + 0.390431i
\(17\) −1.48891 + 6.52334i −0.361114 + 1.58214i 0.389260 + 0.921128i \(0.372731\pi\)
−0.750373 + 0.661014i \(0.770126\pi\)
\(18\) 0 0
\(19\) − 0.418798i − 0.0960788i −0.998845 − 0.0480394i \(-0.984703\pi\)
0.998845 − 0.0480394i \(-0.0152973\pi\)
\(20\) −1.17487 + 5.14742i −0.262708 + 1.15100i
\(21\) 0 0
\(22\) 0.148568 + 0.650919i 0.0316748 + 0.138776i
\(23\) −5.32955 + 1.21643i −1.11129 + 0.253644i −0.738487 − 0.674267i \(-0.764460\pi\)
−0.372800 + 0.927912i \(0.621602\pi\)
\(24\) 0 0
\(25\) 1.53666 + 1.92691i 0.307332 + 0.385382i
\(26\) −0.342899 − 1.50234i −0.0672481 − 0.294633i
\(27\) 0 0
\(28\) 4.72232 + 1.95982i 0.892434 + 0.370372i
\(29\) 2.32890 + 0.531557i 0.432466 + 0.0987076i 0.433212 − 0.901292i \(-0.357380\pi\)
−0.000745584 1.00000i \(0.500237\pi\)
\(30\) 0 0
\(31\) − 5.40375i − 0.970542i −0.874364 − 0.485271i \(-0.838721\pi\)
0.874364 − 0.485271i \(-0.161279\pi\)
\(32\) 2.90441 + 0.662912i 0.513431 + 0.117187i
\(33\) 0 0
\(34\) 1.35940 − 1.08408i 0.233135 − 0.185919i
\(35\) 6.32900 − 3.49228i 1.06980 − 0.590303i
\(36\) 0 0
\(37\) −1.37636 + 6.03022i −0.226272 + 0.991362i 0.726379 + 0.687295i \(0.241202\pi\)
−0.952651 + 0.304067i \(0.901655\pi\)
\(38\) −0.0678530 + 0.0850850i −0.0110072 + 0.0138026i
\(39\) 0 0
\(40\) 2.18282 − 1.74074i 0.345134 − 0.275235i
\(41\) −5.97322 − 2.87655i −0.932861 − 0.449242i −0.0952155 − 0.995457i \(-0.530354\pi\)
−0.837645 + 0.546215i \(0.816068\pi\)
\(42\) 0 0
\(43\) −7.06131 + 3.40055i −1.07684 + 0.518579i −0.886305 − 0.463102i \(-0.846736\pi\)
−0.190535 + 0.981680i \(0.561022\pi\)
\(44\) −2.15430 + 4.47345i −0.324773 + 0.674398i
\(45\) 0 0
\(46\) 1.27986 + 0.616349i 0.188705 + 0.0908757i
\(47\) 2.90399 − 3.64149i 0.423590 − 0.531165i −0.523546 − 0.851997i \(-0.675391\pi\)
0.947136 + 0.320832i \(0.103963\pi\)
\(48\) 0 0
\(49\) −2.30293 − 6.61033i −0.328990 − 0.944333i
\(50\) − 0.640448i − 0.0905730i
\(51\) 0 0
\(52\) 4.97219 − 10.3249i 0.689519 − 1.43180i
\(53\) −13.5007 + 3.08145i −1.85447 + 0.423270i −0.995981 − 0.0895695i \(-0.971451\pi\)
−0.858485 + 0.512839i \(0.828594\pi\)
\(54\) 0 0
\(55\) 3.04576 + 6.32459i 0.410690 + 0.852807i
\(56\) −1.30619 − 2.36719i −0.174547 − 0.316329i
\(57\) 0 0
\(58\) −0.387029 − 0.485319i −0.0508194 − 0.0637255i
\(59\) 5.52357 − 2.66001i 0.719107 − 0.346304i −0.0382823 − 0.999267i \(-0.512189\pi\)
0.757390 + 0.652963i \(0.226474\pi\)
\(60\) 0 0
\(61\) −9.77982 − 2.23218i −1.25218 − 0.285801i −0.455517 − 0.890227i \(-0.650545\pi\)
−0.796661 + 0.604426i \(0.793403\pi\)
\(62\) −0.875508 + 1.09785i −0.111190 + 0.139427i
\(63\) 0 0
\(64\) 4.00571 + 5.02301i 0.500714 + 0.627876i
\(65\) −7.02971 − 14.5973i −0.871928 − 1.81058i
\(66\) 0 0
\(67\) −8.37721 −1.02344 −0.511719 − 0.859153i \(-0.670991\pi\)
−0.511719 + 0.859153i \(0.670991\pi\)
\(68\) 12.9304 1.56804
\(69\) 0 0
\(70\) −1.85164 − 0.315907i −0.221314 − 0.0377582i
\(71\) 6.34846 − 1.44900i 0.753424 − 0.171964i 0.171474 − 0.985189i \(-0.445147\pi\)
0.581950 + 0.813225i \(0.302290\pi\)
\(72\) 0 0
\(73\) −5.32281 + 4.24480i −0.622987 + 0.496816i −0.883362 − 0.468692i \(-0.844725\pi\)
0.260374 + 0.965508i \(0.416154\pi\)
\(74\) 1.25664 − 1.00213i 0.146081 − 0.116496i
\(75\) 0 0
\(76\) −0.789025 + 0.180090i −0.0905074 + 0.0206577i
\(77\) 6.53340 − 1.87741i 0.744550 − 0.213951i
\(78\) 0 0
\(79\) 0.883489 0.0994002 0.0497001 − 0.998764i \(-0.484173\pi\)
0.0497001 + 0.998764i \(0.484173\pi\)
\(80\) 9.83409 1.09948
\(81\) 0 0
\(82\) 0.747494 + 1.55219i 0.0825469 + 0.171410i
\(83\) 6.46744 + 8.10992i 0.709894 + 0.890179i 0.997719 − 0.0675041i \(-0.0215036\pi\)
−0.287825 + 0.957683i \(0.592932\pi\)
\(84\) 0 0
\(85\) 11.3980 − 14.2927i 1.23629 − 1.55026i
\(86\) 1.98556 + 0.453192i 0.214109 + 0.0488689i
\(87\) 0 0
\(88\) 2.36554 − 1.13918i 0.252167 − 0.121437i
\(89\) −1.58193 − 1.98368i −0.167685 − 0.210270i 0.690888 − 0.722962i \(-0.257220\pi\)
−0.858573 + 0.512692i \(0.828648\pi\)
\(90\) 0 0
\(91\) −15.0793 + 4.33313i −1.58074 + 0.454235i
\(92\) 4.58358 + 9.51791i 0.477871 + 0.992310i
\(93\) 0 0
\(94\) −1.17998 + 0.269322i −0.121705 + 0.0277784i
\(95\) −0.496457 + 1.03090i −0.0509354 + 0.105768i
\(96\) 0 0
\(97\) − 3.88426i − 0.394387i −0.980365 − 0.197193i \(-0.936817\pi\)
0.980365 − 0.197193i \(-0.0631827\pi\)
\(98\) −0.603122 + 1.71611i −0.0609245 + 0.173353i
\(99\) 0 0
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 441.2.w.a.251.9 yes 120
3.2 odd 2 inner 441.2.w.a.251.12 yes 120
49.41 odd 14 inner 441.2.w.a.188.12 yes 120
147.41 even 14 inner 441.2.w.a.188.9 ✓ 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.w.a.188.9 ✓ 120 147.41 even 14 inner
441.2.w.a.188.12 yes 120 49.41 odd 14 inner
441.2.w.a.251.9 yes 120 1.1 even 1 trivial
441.2.w.a.251.12 yes 120 3.2 odd 2 inner