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Properties

Label 380.3.g.c.189.5
Level $380$
Weight $3$
Character 380.189
Analytic conductor $10.354$
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] = [380,3,Mod(189,380)] 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("380.189"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.g (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: \(10.3542500457\)
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: \( 2^{19} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 189.5
Root \(0.975196 - 4.19028i\) of defining polynomial
Character \(\chi\) \(=\) 380.189
Dual form 380.3.g.c.189.6

$q$-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)
 
\(f(q)\) \(=\) \(q-1.95039 q^{3} +(-2.75473 - 4.17271i) q^{5} +2.18749i q^{7} -5.19597 q^{9} -9.50946 q^{11} +20.4976 q^{13} +(5.37280 + 8.13841i) q^{15} +6.15792i q^{17} +(9.68651 + 16.3454i) q^{19} -4.26647i q^{21} +10.9375i q^{23} +(-9.82294 + 22.9893i) q^{25} +27.6877 q^{27} -28.4243i q^{29} +24.8098i q^{31} +18.5472 q^{33} +(9.12776 - 6.02594i) q^{35} +37.4397 q^{37} -39.9783 q^{39} +81.6583i q^{41} +70.8743i q^{43} +(14.3135 + 21.6813i) q^{45} -46.1193i q^{47} +44.2149 q^{49} -12.0104i q^{51} -32.1999 q^{53} +(26.1960 + 39.6802i) q^{55} +(-18.8925 - 31.8799i) q^{57} +94.3206i q^{59} -102.566 q^{61} -11.3661i q^{63} +(-56.4652 - 85.5303i) q^{65} +69.0282 q^{67} -21.3323i q^{69} +40.5718i q^{71} -102.209i q^{73} +(19.1586 - 44.8382i) q^{75} -20.8018i q^{77} -73.9145i q^{79} -7.23816 q^{81} +91.9574i q^{83} +(25.6952 - 16.9634i) q^{85} +55.4385i q^{87} -57.3634i q^{89} +44.8382i q^{91} -48.3888i q^{93} +(41.5207 - 85.4461i) q^{95} -84.3229 q^{97} +49.4109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{5} + 76 q^{9} - 24 q^{11} + 68 q^{19} - 76 q^{25} - 32 q^{35} - 264 q^{39} + 220 q^{45} + 212 q^{49} + 176 q^{55} - 600 q^{61} + 492 q^{81} + 408 q^{85} + 124 q^{95} + 136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −1.95039 −0.650131 −0.325065 − 0.945692i \(-0.605386\pi\)
−0.325065 + 0.945692i \(0.605386\pi\)
\(4\) 0 0
\(5\) −2.75473 − 4.17271i −0.550946 − 0.834541i
\(6\) 0 0
\(7\) 2.18749i 0.312499i 0.987718 + 0.156249i \(0.0499403\pi\)
−0.987718 + 0.156249i \(0.950060\pi\)
\(8\) 0 0
\(9\) −5.19597 −0.577330
\(10\) 0 0
\(11\) −9.50946 −0.864496 −0.432248 − 0.901755i \(-0.642280\pi\)
−0.432248 + 0.901755i \(0.642280\pi\)
\(12\) 0 0
\(13\) 20.4976 1.57674 0.788368 − 0.615204i \(-0.210926\pi\)
0.788368 + 0.615204i \(0.210926\pi\)
\(14\) 0 0
\(15\) 5.37280 + 8.13841i 0.358187 + 0.542561i
\(16\) 0 0
\(17\) 6.15792i 0.362231i 0.983462 + 0.181115i \(0.0579707\pi\)
−0.983462 + 0.181115i \(0.942029\pi\)
\(18\) 0 0
\(19\) 9.68651 + 16.3454i 0.509816 + 0.860283i
\(20\) 0 0
\(21\) − 4.26647i − 0.203165i
\(22\) 0 0
\(23\) 10.9375i 0.475541i 0.971321 + 0.237771i \(0.0764167\pi\)
−0.971321 + 0.237771i \(0.923583\pi\)
\(24\) 0 0
\(25\) −9.82294 + 22.9893i −0.392918 + 0.919574i
\(26\) 0 0
\(27\) 27.6877 1.02547
\(28\) 0 0
\(29\) − 28.4243i − 0.980148i −0.871681 − 0.490074i \(-0.836970\pi\)
0.871681 − 0.490074i \(-0.163030\pi\)
\(30\) 0 0
\(31\) 24.8098i 0.800315i 0.916446 + 0.400157i \(0.131045\pi\)
−0.916446 + 0.400157i \(0.868955\pi\)
\(32\) 0 0
\(33\) 18.5472 0.562035
\(34\) 0 0
\(35\) 9.12776 − 6.02594i 0.260793 − 0.172170i
\(36\) 0 0
\(37\) 37.4397 1.01188 0.505941 − 0.862568i \(-0.331145\pi\)
0.505941 + 0.862568i \(0.331145\pi\)
\(38\) 0 0
\(39\) −39.9783 −1.02508
\(40\) 0 0
\(41\) 81.6583i 1.99167i 0.0911903 + 0.995833i \(0.470933\pi\)
−0.0911903 + 0.995833i \(0.529067\pi\)
\(42\) 0 0
\(43\) 70.8743i 1.64824i 0.566415 + 0.824120i \(0.308330\pi\)
−0.566415 + 0.824120i \(0.691670\pi\)
\(44\) 0 0
\(45\) 14.3135 + 21.6813i 0.318077 + 0.481806i
\(46\) 0 0
\(47\) − 46.1193i − 0.981261i −0.871368 − 0.490631i \(-0.836766\pi\)
0.871368 − 0.490631i \(-0.163234\pi\)
\(48\) 0 0
\(49\) 44.2149 0.902345
\(50\) 0 0
\(51\) − 12.0104i − 0.235497i
\(52\) 0 0
\(53\) −32.1999 −0.607546 −0.303773 − 0.952745i \(-0.598246\pi\)
−0.303773 + 0.952745i \(0.598246\pi\)
\(54\) 0 0
\(55\) 26.1960 + 39.6802i 0.476290 + 0.721458i
\(56\) 0 0
\(57\) −18.8925 − 31.8799i −0.331447 − 0.559297i
\(58\) 0 0
\(59\) 94.3206i 1.59865i 0.600896 + 0.799327i \(0.294811\pi\)
−0.600896 + 0.799327i \(0.705189\pi\)
\(60\) 0 0
\(61\) −102.566 −1.68141 −0.840707 − 0.541491i \(-0.817860\pi\)
−0.840707 + 0.541491i \(0.817860\pi\)
\(62\) 0 0
\(63\) − 11.3661i − 0.180415i
\(64\) 0 0
\(65\) −56.4652 − 85.5303i −0.868696 − 1.31585i
\(66\) 0 0
\(67\) 69.0282 1.03027 0.515136 − 0.857109i \(-0.327742\pi\)
0.515136 + 0.857109i \(0.327742\pi\)
\(68\) 0 0
\(69\) − 21.3323i − 0.309164i
\(70\) 0 0
\(71\) 40.5718i 0.571433i 0.958314 + 0.285717i \(0.0922316\pi\)
−0.958314 + 0.285717i \(0.907768\pi\)
\(72\) 0 0
\(73\) − 102.209i − 1.40012i −0.714082 − 0.700062i \(-0.753156\pi\)
0.714082 − 0.700062i \(-0.246844\pi\)
\(74\) 0 0
\(75\) 19.1586 − 44.8382i 0.255448 − 0.597843i
\(76\) 0 0
\(77\) − 20.8018i − 0.270154i
\(78\) 0 0
\(79\) − 73.9145i − 0.935626i −0.883827 − 0.467813i \(-0.845042\pi\)
0.883827 − 0.467813i \(-0.154958\pi\)
\(80\) 0 0
\(81\) −7.23816 −0.0893600
\(82\) 0 0
\(83\) 91.9574i 1.10792i 0.832543 + 0.553960i \(0.186884\pi\)
−0.832543 + 0.553960i \(0.813116\pi\)
\(84\) 0 0
\(85\) 25.6952 − 16.9634i 0.302296 − 0.199569i
\(86\) 0 0
\(87\) 55.4385i 0.637224i
\(88\) 0 0
\(89\) − 57.3634i − 0.644533i −0.946649 − 0.322266i \(-0.895555\pi\)
0.946649 − 0.322266i \(-0.104445\pi\)
\(90\) 0 0
\(91\) 44.8382i 0.492728i
\(92\) 0 0
\(93\) − 48.3888i − 0.520309i
\(94\) 0 0
\(95\) 41.5207 − 85.4461i 0.437060 − 0.899432i
\(96\) 0 0
\(97\) −84.3229 −0.869308 −0.434654 − 0.900598i \(-0.643129\pi\)
−0.434654 + 0.900598i \(0.643129\pi\)
\(98\) 0 0
\(99\) 49.4109 0.499100
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 380.3.g.c.189.5 ✓ 12
3.2 odd 2 3420.3.h.e.2089.12 12
5.2 odd 4 1900.3.e.e.1101.7 12
5.3 odd 4 1900.3.e.e.1101.6 12
5.4 even 2 inner 380.3.g.c.189.8 yes 12
15.14 odd 2 3420.3.h.e.2089.9 12
19.18 odd 2 inner 380.3.g.c.189.7 yes 12
57.56 even 2 3420.3.h.e.2089.11 12
95.18 even 4 1900.3.e.e.1101.8 12
95.37 even 4 1900.3.e.e.1101.5 12
95.94 odd 2 inner 380.3.g.c.189.6 yes 12
285.284 even 2 3420.3.h.e.2089.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.g.c.189.5 ✓ 12 1.1 even 1 trivial
380.3.g.c.189.6 yes 12 95.94 odd 2 inner
380.3.g.c.189.7 yes 12 19.18 odd 2 inner
380.3.g.c.189.8 yes 12 5.4 even 2 inner
1900.3.e.e.1101.5 12 95.37 even 4
1900.3.e.e.1101.6 12 5.3 odd 4
1900.3.e.e.1101.7 12 5.2 odd 4
1900.3.e.e.1101.8 12 95.18 even 4
3420.3.h.e.2089.9 12 15.14 odd 2
3420.3.h.e.2089.10 12 285.284 even 2
3420.3.h.e.2089.11 12 57.56 even 2
3420.3.h.e.2089.12 12 3.2 odd 2