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Properties

Label 3420.3.h.e.2089.7
Level $3420$
Weight $3$
Character 3420.2089
Analytic conductor $93.188$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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] = [3420,3,Mod(2089,3420)] 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("3420.2089"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3420.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(93.1882504112\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 44x^{8} - 270x^{6} + 36676x^{4} - 71664x^{2} + 687241 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{19}\cdot 3 \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2089.7
Root \(2.80718 - 1.46321i\) of defining polynomial
Character \(\chi\) \(=\) 3420.2089
Dual form 3420.3.h.e.2089.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.48938 + 4.77302i) q^{5} -7.03410i q^{7} +1.02125 q^{11} -11.3480 q^{13} -16.5801i q^{17} +(-9.54227 + 16.4300i) q^{19} +35.1705i q^{23} +(-20.5635 - 14.2177i) q^{25} -6.63193i q^{29} +25.3890i q^{31} +(33.5739 + 10.4764i) q^{35} +47.8401 q^{37} -12.1251i q^{41} +50.0927i q^{43} -78.3252i q^{47} -0.478526 q^{49} -45.0342 q^{53} +(-1.52102 + 4.87444i) q^{55} +9.44897i q^{59} -43.1487 q^{61} +(16.9015 - 54.1643i) q^{65} +112.523 q^{67} -40.3311i q^{71} -45.4211i q^{73} -7.18356i q^{77} -144.704i q^{79} +56.8706i q^{83} +(79.1374 + 24.6941i) q^{85} -81.8009i q^{89} +79.8230i q^{91} +(-64.2088 - 70.0159i) q^{95} -103.506 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{5} + 24 q^{11} + 68 q^{19} - 76 q^{25} + 32 q^{35} + 212 q^{49} + 176 q^{55} - 600 q^{61} + 408 q^{85} - 124 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.48938 + 4.77302i −0.297875 + 0.954605i
\(6\) 0 0
\(7\) − 7.03410i − 1.00487i −0.864615 − 0.502436i \(-0.832437\pi\)
0.864615 − 0.502436i \(-0.167563\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.02125 0.0928407 0.0464204 − 0.998922i \(-0.485219\pi\)
0.0464204 + 0.998922i \(0.485219\pi\)
\(12\) 0 0
\(13\) −11.3480 −0.872924 −0.436462 − 0.899723i \(-0.643769\pi\)
−0.436462 + 0.899723i \(0.643769\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 16.5801i − 0.975303i −0.873038 − 0.487651i \(-0.837854\pi\)
0.873038 − 0.487651i \(-0.162146\pi\)
\(18\) 0 0
\(19\) −9.54227 + 16.4300i −0.502225 + 0.864737i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 35.1705i 1.52915i 0.644534 + 0.764576i \(0.277051\pi\)
−0.644534 + 0.764576i \(0.722949\pi\)
\(24\) 0 0
\(25\) −20.5635 − 14.2177i −0.822541 − 0.568706i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.63193i − 0.228687i −0.993441 − 0.114344i \(-0.963523\pi\)
0.993441 − 0.114344i \(-0.0364765\pi\)
\(30\) 0 0
\(31\) 25.3890i 0.818998i 0.912310 + 0.409499i \(0.134297\pi\)
−0.912310 + 0.409499i \(0.865703\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 33.5739 + 10.4764i 0.959255 + 0.299326i
\(36\) 0 0
\(37\) 47.8401 1.29297 0.646487 − 0.762925i \(-0.276237\pi\)
0.646487 + 0.762925i \(0.276237\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 12.1251i − 0.295734i −0.989007 − 0.147867i \(-0.952759\pi\)
0.989007 − 0.147867i \(-0.0472407\pi\)
\(42\) 0 0
\(43\) 50.0927i 1.16495i 0.812850 + 0.582474i \(0.197915\pi\)
−0.812850 + 0.582474i \(0.802085\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 78.3252i − 1.66649i −0.552901 − 0.833247i \(-0.686479\pi\)
0.552901 − 0.833247i \(-0.313521\pi\)
\(48\) 0 0
\(49\) −0.478526 −0.00976584
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −45.0342 −0.849701 −0.424851 − 0.905263i \(-0.639673\pi\)
−0.424851 + 0.905263i \(0.639673\pi\)
\(54\) 0 0
\(55\) −1.52102 + 4.87444i −0.0276549 + 0.0886262i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.44897i 0.160152i 0.996789 + 0.0800760i \(0.0255163\pi\)
−0.996789 + 0.0800760i \(0.974484\pi\)
\(60\) 0 0
\(61\) −43.1487 −0.707356 −0.353678 − 0.935367i \(-0.615069\pi\)
−0.353678 + 0.935367i \(0.615069\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.9015 − 54.1643i 0.260022 − 0.833297i
\(66\) 0 0
\(67\) 112.523 1.67945 0.839726 − 0.543011i \(-0.182716\pi\)
0.839726 + 0.543011i \(0.182716\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 40.3311i − 0.568043i −0.958818 − 0.284022i \(-0.908331\pi\)
0.958818 − 0.284022i \(-0.0916687\pi\)
\(72\) 0 0
\(73\) − 45.4211i − 0.622207i −0.950376 − 0.311104i \(-0.899301\pi\)
0.950376 − 0.311104i \(-0.100699\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 7.18356i − 0.0932930i
\(78\) 0 0
\(79\) − 144.704i − 1.83170i −0.401526 − 0.915848i \(-0.631520\pi\)
0.401526 − 0.915848i \(-0.368480\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 56.8706i 0.685188i 0.939484 + 0.342594i \(0.111306\pi\)
−0.939484 + 0.342594i \(0.888694\pi\)
\(84\) 0 0
\(85\) 79.1374 + 24.6941i 0.931029 + 0.290518i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 81.8009i − 0.919111i −0.888149 − 0.459556i \(-0.848009\pi\)
0.888149 − 0.459556i \(-0.151991\pi\)
\(90\) 0 0
\(91\) 79.8230i 0.877176i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −64.2088 − 70.0159i −0.675882 − 0.737010i
\(96\) 0 0
\(97\) −103.506 −1.06708 −0.533539 − 0.845776i \(-0.679138\pi\)
−0.533539 + 0.845776i \(0.679138\pi\)
\(98\) 0 0
\(99\) 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 3420.3.h.e.2089.7 12
3.2 odd 2 380.3.g.c.189.11 yes 12
5.4 even 2 inner 3420.3.h.e.2089.6 12
15.2 even 4 1900.3.e.e.1101.2 12
15.8 even 4 1900.3.e.e.1101.11 12
15.14 odd 2 380.3.g.c.189.2 yes 12
19.18 odd 2 inner 3420.3.h.e.2089.8 12
57.56 even 2 380.3.g.c.189.1 ✓ 12
95.94 odd 2 inner 3420.3.h.e.2089.5 12
285.113 odd 4 1900.3.e.e.1101.1 12
285.227 odd 4 1900.3.e.e.1101.12 12
285.284 even 2 380.3.g.c.189.12 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.g.c.189.1 ✓ 12 57.56 even 2
380.3.g.c.189.2 yes 12 15.14 odd 2
380.3.g.c.189.11 yes 12 3.2 odd 2
380.3.g.c.189.12 yes 12 285.284 even 2
1900.3.e.e.1101.1 12 285.113 odd 4
1900.3.e.e.1101.2 12 15.2 even 4
1900.3.e.e.1101.11 12 15.8 even 4
1900.3.e.e.1101.12 12 285.227 odd 4
3420.3.h.e.2089.5 12 95.94 odd 2 inner
3420.3.h.e.2089.6 12 5.4 even 2 inner
3420.3.h.e.2089.7 12 1.1 even 1 trivial
3420.3.h.e.2089.8 12 19.18 odd 2 inner