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Properties

Label 1210.3.d.c.241.9
Level $1210$
Weight $3$
Character 1210.241
Analytic conductor $32.970$
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] = [1210,3,Mod(241,1210)] 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("1210.241"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1210, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1210.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,-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: \(32.9701119876\)
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: \( x^{16} - 8 x^{15} + 92 x^{14} - 504 x^{13} + 3078 x^{12} - 12280 x^{11} + 49836 x^{10} - 147672 x^{9} + \cdots + 339856 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 241.9
Root \(0.500000 - 0.322806i\) of defining polynomial
Character \(\chi\) \(=\) 1210.241
Dual form 1210.3.d.c.241.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -5.52516 q^{3} -2.00000 q^{4} -2.23607 q^{5} -7.81376i q^{6} -11.7293i q^{7} -2.82843i q^{8} +21.5274 q^{9} -3.16228i q^{10} +11.0503 q^{12} -12.1462i q^{13} +16.5877 q^{14} +12.3546 q^{15} +4.00000 q^{16} -10.4122i q^{17} +30.4444i q^{18} -5.07289i q^{19} +4.47214 q^{20} +64.8063i q^{21} +30.9230 q^{23} +15.6275i q^{24} +5.00000 q^{25} +17.1773 q^{26} -69.2160 q^{27} +23.4586i q^{28} -15.4160i q^{29} +17.4721i q^{30} +13.1631 q^{31} +5.65685i q^{32} +14.7251 q^{34} +26.2275i q^{35} -43.0548 q^{36} +6.49462 q^{37} +7.17414 q^{38} +67.1098i q^{39} +6.32456i q^{40} +48.6595i q^{41} -91.6499 q^{42} -35.6140i q^{43} -48.1368 q^{45} +43.7318i q^{46} +33.8830 q^{47} -22.1006 q^{48} -88.5766 q^{49} +7.07107i q^{50} +57.5292i q^{51} +24.2924i q^{52} -55.5414 q^{53} -97.8862i q^{54} -33.1755 q^{56} +28.0285i q^{57} +21.8015 q^{58} +53.8551 q^{59} -24.7093 q^{60} -43.7977i q^{61} +18.6154i q^{62} -252.502i q^{63} -8.00000 q^{64} +27.1597i q^{65} +117.778 q^{67} +20.8244i q^{68} -170.855 q^{69} -37.0913 q^{70} +92.8887 q^{71} -60.8887i q^{72} -1.27031i q^{73} +9.18477i q^{74} -27.6258 q^{75} +10.1458i q^{76} -94.9075 q^{78} -112.311i q^{79} -8.94427 q^{80} +188.683 q^{81} -68.8149 q^{82} -129.374i q^{83} -129.613i q^{84} +23.2824i q^{85} +50.3659 q^{86} +85.1759i q^{87} -4.92279 q^{89} -68.0757i q^{90} -142.467 q^{91} -61.8461 q^{92} -72.7280 q^{93} +47.9178i q^{94} +11.3433i q^{95} -31.2550i q^{96} -17.9600 q^{97} -125.266i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{3} - 32 q^{4} + 40 q^{9} + 32 q^{12} + 16 q^{14} + 40 q^{15} + 64 q^{16} + 108 q^{23} + 80 q^{25} - 292 q^{27} - 268 q^{31} - 16 q^{34} - 80 q^{36} + 44 q^{37} - 280 q^{38} - 16 q^{42} - 476 q^{47}+ \cdots + 296 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1210\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1091\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 0.707107i
\(3\) −5.52516 −1.84172 −0.920860 − 0.389893i \(-0.872512\pi\)
−0.920860 + 0.389893i \(0.872512\pi\)
\(4\) −2.00000 −0.500000
\(5\) −2.23607 −0.447214
\(6\) − 7.81376i − 1.30229i
\(7\) − 11.7293i − 1.67561i −0.545966 − 0.837807i \(-0.683837\pi\)
0.545966 − 0.837807i \(-0.316163\pi\)
\(8\) − 2.82843i − 0.353553i
\(9\) 21.5274 2.39193
\(10\) − 3.16228i − 0.316228i
\(11\) 0 0
\(12\) 11.0503 0.920860
\(13\) − 12.1462i − 0.934324i −0.884172 − 0.467162i \(-0.845277\pi\)
0.884172 − 0.467162i \(-0.154723\pi\)
\(14\) 16.5877 1.18484
\(15\) 12.3546 0.823642
\(16\) 4.00000 0.250000
\(17\) − 10.4122i − 0.612484i −0.951954 − 0.306242i \(-0.900928\pi\)
0.951954 − 0.306242i \(-0.0990716\pi\)
\(18\) 30.4444i 1.69135i
\(19\) − 5.07289i − 0.266994i −0.991049 − 0.133497i \(-0.957379\pi\)
0.991049 − 0.133497i \(-0.0426206\pi\)
\(20\) 4.47214 0.223607
\(21\) 64.8063i 3.08601i
\(22\) 0 0
\(23\) 30.9230 1.34448 0.672240 − 0.740333i \(-0.265332\pi\)
0.672240 + 0.740333i \(0.265332\pi\)
\(24\) 15.6275i 0.651147i
\(25\) 5.00000 0.200000
\(26\) 17.1773 0.660667
\(27\) −69.2160 −2.56355
\(28\) 23.4586i 0.837807i
\(29\) − 15.4160i − 0.531586i −0.964030 − 0.265793i \(-0.914366\pi\)
0.964030 − 0.265793i \(-0.0856338\pi\)
\(30\) 17.4721i 0.582403i
\(31\) 13.1631 0.424615 0.212307 − 0.977203i \(-0.431902\pi\)
0.212307 + 0.977203i \(0.431902\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 0 0
\(34\) 14.7251 0.433091
\(35\) 26.2275i 0.749358i
\(36\) −43.0548 −1.19597
\(37\) 6.49462 0.175530 0.0877651 − 0.996141i \(-0.472028\pi\)
0.0877651 + 0.996141i \(0.472028\pi\)
\(38\) 7.17414 0.188793
\(39\) 67.1098i 1.72076i
\(40\) 6.32456i 0.158114i
\(41\) 48.6595i 1.18682i 0.804902 + 0.593408i \(0.202218\pi\)
−0.804902 + 0.593408i \(0.797782\pi\)
\(42\) −91.6499 −2.18214
\(43\) − 35.6140i − 0.828234i −0.910224 − 0.414117i \(-0.864091\pi\)
0.910224 − 0.414117i \(-0.135909\pi\)
\(44\) 0 0
\(45\) −48.1368 −1.06971
\(46\) 43.7318i 0.950691i
\(47\) 33.8830 0.720915 0.360457 − 0.932776i \(-0.382621\pi\)
0.360457 + 0.932776i \(0.382621\pi\)
\(48\) −22.1006 −0.460430
\(49\) −88.5766 −1.80769
\(50\) 7.07107i 0.141421i
\(51\) 57.5292i 1.12802i
\(52\) 24.2924i 0.467162i
\(53\) −55.5414 −1.04795 −0.523975 − 0.851734i \(-0.675552\pi\)
−0.523975 + 0.851734i \(0.675552\pi\)
\(54\) − 97.8862i − 1.81271i
\(55\) 0 0
\(56\) −33.1755 −0.592419
\(57\) 28.0285i 0.491728i
\(58\) 21.8015 0.375888
\(59\) 53.8551 0.912798 0.456399 − 0.889775i \(-0.349139\pi\)
0.456399 + 0.889775i \(0.349139\pi\)
\(60\) −24.7093 −0.411821
\(61\) − 43.7977i − 0.717995i −0.933339 − 0.358997i \(-0.883119\pi\)
0.933339 − 0.358997i \(-0.116881\pi\)
\(62\) 18.6154i 0.300248i
\(63\) − 252.502i − 4.00796i
\(64\) −8.00000 −0.125000
\(65\) 27.1597i 0.417842i
\(66\) 0 0
\(67\) 117.778 1.75788 0.878938 − 0.476937i \(-0.158253\pi\)
0.878938 + 0.476937i \(0.158253\pi\)
\(68\) 20.8244i 0.306242i
\(69\) −170.855 −2.47616
\(70\) −37.0913 −0.529876
\(71\) 92.8887 1.30829 0.654146 − 0.756368i \(-0.273028\pi\)
0.654146 + 0.756368i \(0.273028\pi\)
\(72\) − 60.8887i − 0.845677i
\(73\) − 1.27031i − 0.0174015i −0.999962 − 0.00870076i \(-0.997230\pi\)
0.999962 − 0.00870076i \(-0.00276957\pi\)
\(74\) 9.18477i 0.124119i
\(75\) −27.6258 −0.368344
\(76\) 10.1458i 0.133497i
\(77\) 0 0
\(78\) −94.9075 −1.21676
\(79\) − 112.311i − 1.42166i −0.703363 − 0.710831i \(-0.748319\pi\)
0.703363 − 0.710831i \(-0.251681\pi\)
\(80\) −8.94427 −0.111803
\(81\) 188.683 2.32942
\(82\) −68.8149 −0.839206
\(83\) − 129.374i − 1.55873i −0.626573 − 0.779363i \(-0.715543\pi\)
0.626573 − 0.779363i \(-0.284457\pi\)
\(84\) − 129.613i − 1.54301i
\(85\) 23.2824i 0.273911i
\(86\) 50.3659 0.585650
\(87\) 85.1759i 0.979033i
\(88\) 0 0
\(89\) −4.92279 −0.0553122 −0.0276561 − 0.999617i \(-0.508804\pi\)
−0.0276561 + 0.999617i \(0.508804\pi\)
\(90\) − 68.0757i − 0.756396i
\(91\) −142.467 −1.56557
\(92\) −61.8461 −0.672240
\(93\) −72.7280 −0.782022
\(94\) 47.9178i 0.509764i
\(95\) 11.3433i 0.119403i
\(96\) − 31.2550i − 0.325573i
\(97\) −17.9600 −0.185154 −0.0925771 − 0.995706i \(-0.529510\pi\)
−0.0925771 + 0.995706i \(0.529510\pi\)
\(98\) − 125.266i − 1.27823i
\(99\) 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 1210.3.d.c.241.9 16
11.3 even 5 110.3.h.b.101.4 yes 16
11.7 odd 10 110.3.h.b.61.4 ✓ 16
11.10 odd 2 inner 1210.3.d.c.241.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.h.b.61.4 ✓ 16 11.7 odd 10
110.3.h.b.101.4 yes 16 11.3 even 5
1210.3.d.c.241.1 16 11.10 odd 2 inner
1210.3.d.c.241.9 16 1.1 even 1 trivial