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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [378,2,Mod(127,378)] 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("378.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(378, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,0,-1,2] 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: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 127.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 378.127
Dual form 378.2.f.b.253.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.00000 + 1.73205i) q^{5} +(0.500000 - 0.866025i) q^{7} +1.00000 q^{8} -2.00000 q^{10} +(0.500000 - 0.866025i) q^{11} +(3.00000 + 5.19615i) q^{13} +(0.500000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +5.00000 q^{17} -7.00000 q^{19} +(1.00000 - 1.73205i) q^{20} +(0.500000 + 0.866025i) q^{22} +(2.00000 + 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} -6.00000 q^{26} -1.00000 q^{28} +(-2.00000 + 3.46410i) q^{29} +(3.00000 + 5.19615i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(-2.50000 + 4.33013i) q^{34} +2.00000 q^{35} +2.00000 q^{37} +(3.50000 - 6.06218i) q^{38} +(1.00000 + 1.73205i) q^{40} +(1.50000 + 2.59808i) q^{41} +(0.500000 - 0.866025i) q^{43} -1.00000 q^{44} -4.00000 q^{46} +(-0.500000 - 0.866025i) q^{49} +(0.500000 + 0.866025i) q^{50} +(3.00000 - 5.19615i) q^{52} -12.0000 q^{53} +2.00000 q^{55} +(0.500000 - 0.866025i) q^{56} +(-2.00000 - 3.46410i) q^{58} +(-3.50000 - 6.06218i) q^{59} +(6.00000 - 10.3923i) q^{61} -6.00000 q^{62} +1.00000 q^{64} +(-6.00000 + 10.3923i) q^{65} +(-6.50000 - 11.2583i) q^{67} +(-2.50000 - 4.33013i) q^{68} +(-1.00000 + 1.73205i) q^{70} +8.00000 q^{71} +1.00000 q^{73} +(-1.00000 + 1.73205i) q^{74} +(3.50000 + 6.06218i) q^{76} +(-0.500000 - 0.866025i) q^{77} +(3.00000 - 5.19615i) q^{79} -2.00000 q^{80} -3.00000 q^{82} +(8.00000 - 13.8564i) q^{83} +(5.00000 + 8.66025i) q^{85} +(0.500000 + 0.866025i) q^{86} +(0.500000 - 0.866025i) q^{88} +6.00000 q^{89} +6.00000 q^{91} +(2.00000 - 3.46410i) q^{92} +(-7.00000 - 12.1244i) q^{95} +(2.50000 - 4.33013i) q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 2 q^{8} - 4 q^{10} + q^{11} + 6 q^{13} + q^{14} - q^{16} + 10 q^{17} - 14 q^{19} + 2 q^{20} + q^{22} + 4 q^{23} + q^{25} - 12 q^{26} - 2 q^{28} - 4 q^{29}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.353553 + 0.612372i
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 1.00000 + 1.73205i 0.447214 + 0.774597i 0.998203 − 0.0599153i \(-0.0190830\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 0 0
\(7\) 0.500000 − 0.866025i 0.188982 − 0.327327i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 0.500000 − 0.866025i 0.150756 − 0.261116i −0.780750 − 0.624844i \(-0.785163\pi\)
0.931505 + 0.363727i \(0.118496\pi\)
\(12\) 0 0
\(13\) 3.00000 + 5.19615i 0.832050 + 1.44115i 0.896410 + 0.443227i \(0.146166\pi\)
−0.0643593 + 0.997927i \(0.520500\pi\)
\(14\) 0.500000 + 0.866025i 0.133631 + 0.231455i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 1.00000 − 1.73205i 0.223607 − 0.387298i
\(21\) 0 0
\(22\) 0.500000 + 0.866025i 0.106600 + 0.184637i
\(23\) 2.00000 + 3.46410i 0.417029 + 0.722315i 0.995639 − 0.0932891i \(-0.0297381\pi\)
−0.578610 + 0.815604i \(0.696405\pi\)
\(24\) 0 0
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −2.00000 + 3.46410i −0.371391 + 0.643268i −0.989780 − 0.142605i \(-0.954452\pi\)
0.618389 + 0.785872i \(0.287786\pi\)
\(30\) 0 0
\(31\) 3.00000 + 5.19615i 0.538816 + 0.933257i 0.998968 + 0.0454165i \(0.0144615\pi\)
−0.460152 + 0.887840i \(0.652205\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −2.50000 + 4.33013i −0.428746 + 0.742611i
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 3.50000 − 6.06218i 0.567775 − 0.983415i
\(39\) 0 0
\(40\) 1.00000 + 1.73205i 0.158114 + 0.273861i
\(41\) 1.50000 + 2.59808i 0.234261 + 0.405751i 0.959058 − 0.283211i \(-0.0913998\pi\)
−0.724797 + 0.688963i \(0.758066\pi\)
\(42\) 0 0
\(43\) 0.500000 − 0.866025i 0.0762493 − 0.132068i −0.825380 − 0.564578i \(-0.809039\pi\)
0.901629 + 0.432511i \(0.142372\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) 0.500000 + 0.866025i 0.0707107 + 0.122474i
\(51\) 0 0
\(52\) 3.00000 − 5.19615i 0.416025 − 0.720577i
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0.500000 − 0.866025i 0.0668153 − 0.115728i
\(57\) 0 0
\(58\) −2.00000 − 3.46410i −0.262613 − 0.454859i
\(59\) −3.50000 − 6.06218i −0.455661 − 0.789228i 0.543065 − 0.839691i \(-0.317264\pi\)
−0.998726 + 0.0504625i \(0.983930\pi\)
\(60\) 0 0
\(61\) 6.00000 − 10.3923i 0.768221 − 1.33060i −0.170305 − 0.985391i \(-0.554475\pi\)
0.938527 − 0.345207i \(-0.112191\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 + 10.3923i −0.744208 + 1.28901i
\(66\) 0 0
\(67\) −6.50000 − 11.2583i −0.794101 − 1.37542i −0.923408 − 0.383819i \(-0.874609\pi\)
0.129307 − 0.991605i \(-0.458725\pi\)
\(68\) −2.50000 − 4.33013i −0.303170 − 0.525105i
\(69\) 0 0
\(70\) −1.00000 + 1.73205i −0.119523 + 0.207020i
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) −1.00000 + 1.73205i −0.116248 + 0.201347i
\(75\) 0 0
\(76\) 3.50000 + 6.06218i 0.401478 + 0.695379i
\(77\) −0.500000 − 0.866025i −0.0569803 − 0.0986928i
\(78\) 0 0
\(79\) 3.00000 − 5.19615i 0.337526 − 0.584613i −0.646440 − 0.762964i \(-0.723743\pi\)
0.983967 + 0.178352i \(0.0570765\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) 8.00000 − 13.8564i 0.878114 − 1.52094i 0.0247060 − 0.999695i \(-0.492135\pi\)
0.853408 − 0.521243i \(-0.174532\pi\)
\(84\) 0 0
\(85\) 5.00000 + 8.66025i 0.542326 + 0.939336i
\(86\) 0.500000 + 0.866025i 0.0539164 + 0.0933859i
\(87\) 0 0
\(88\) 0.500000 − 0.866025i 0.0533002 − 0.0923186i
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 2.00000 − 3.46410i 0.208514 − 0.361158i
\(93\) 0 0
\(94\) 0 0
\(95\) −7.00000 − 12.1244i −0.718185 − 1.24393i
\(96\) 0 0
\(97\) 2.50000 − 4.33013i 0.253837 − 0.439658i −0.710742 − 0.703452i \(-0.751641\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) 1.00000 0.101015
\(99\) 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 378.2.f.b.127.1 2
3.2 odd 2 126.2.f.b.43.1 ✓ 2
4.3 odd 2 3024.2.r.c.2017.1 2
7.2 even 3 2646.2.e.i.2125.1 2
7.3 odd 6 2646.2.h.c.667.1 2
7.4 even 3 2646.2.h.b.667.1 2
7.5 odd 6 2646.2.e.h.2125.1 2
7.6 odd 2 2646.2.f.b.883.1 2
9.2 odd 6 1134.2.a.c.1.1 1
9.4 even 3 inner 378.2.f.b.253.1 2
9.5 odd 6 126.2.f.b.85.1 yes 2
9.7 even 3 1134.2.a.f.1.1 1
12.11 even 2 1008.2.r.a.673.1 2
21.2 odd 6 882.2.e.a.655.1 2
21.5 even 6 882.2.e.e.655.1 2
21.11 odd 6 882.2.h.h.79.1 2
21.17 even 6 882.2.h.g.79.1 2
21.20 even 2 882.2.f.f.295.1 2
36.7 odd 6 9072.2.a.f.1.1 1
36.11 even 6 9072.2.a.t.1.1 1
36.23 even 6 1008.2.r.a.337.1 2
36.31 odd 6 3024.2.r.c.1009.1 2
63.4 even 3 2646.2.e.i.1549.1 2
63.5 even 6 882.2.h.g.67.1 2
63.13 odd 6 2646.2.f.b.1765.1 2
63.20 even 6 7938.2.a.e.1.1 1
63.23 odd 6 882.2.h.h.67.1 2
63.31 odd 6 2646.2.e.h.1549.1 2
63.32 odd 6 882.2.e.a.373.1 2
63.34 odd 6 7938.2.a.bb.1.1 1
63.40 odd 6 2646.2.h.c.361.1 2
63.41 even 6 882.2.f.f.589.1 2
63.58 even 3 2646.2.h.b.361.1 2
63.59 even 6 882.2.e.e.373.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.b.43.1 ✓ 2 3.2 odd 2
126.2.f.b.85.1 yes 2 9.5 odd 6
378.2.f.b.127.1 2 1.1 even 1 trivial
378.2.f.b.253.1 2 9.4 even 3 inner
882.2.e.a.373.1 2 63.32 odd 6
882.2.e.a.655.1 2 21.2 odd 6
882.2.e.e.373.1 2 63.59 even 6
882.2.e.e.655.1 2 21.5 even 6
882.2.f.f.295.1 2 21.20 even 2
882.2.f.f.589.1 2 63.41 even 6
882.2.h.g.67.1 2 63.5 even 6
882.2.h.g.79.1 2 21.17 even 6
882.2.h.h.67.1 2 63.23 odd 6
882.2.h.h.79.1 2 21.11 odd 6
1008.2.r.a.337.1 2 36.23 even 6
1008.2.r.a.673.1 2 12.11 even 2
1134.2.a.c.1.1 1 9.2 odd 6
1134.2.a.f.1.1 1 9.7 even 3
2646.2.e.h.1549.1 2 63.31 odd 6
2646.2.e.h.2125.1 2 7.5 odd 6
2646.2.e.i.1549.1 2 63.4 even 3
2646.2.e.i.2125.1 2 7.2 even 3
2646.2.f.b.883.1 2 7.6 odd 2
2646.2.f.b.1765.1 2 63.13 odd 6
2646.2.h.b.361.1 2 63.58 even 3
2646.2.h.b.667.1 2 7.4 even 3
2646.2.h.c.361.1 2 63.40 odd 6
2646.2.h.c.667.1 2 7.3 odd 6
3024.2.r.c.1009.1 2 36.31 odd 6
3024.2.r.c.2017.1 2 4.3 odd 2
7938.2.a.e.1.1 1 63.20 even 6
7938.2.a.bb.1.1 1 63.34 odd 6
9072.2.a.f.1.1 1 36.7 odd 6
9072.2.a.t.1.1 1 36.11 even 6