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Properties

Label 30.11.f.b.13.6
Level $30$
Weight $11$
Character 30.13
Analytic conductor $19.061$
Analytic rank $0$
Dimension $12$
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] = [30,11,Mod(7,30)] 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("30.7"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(30, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 30.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.0607175802\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
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: \( x^{12} - 4 x^{11} + 8 x^{10} - 43756 x^{9} + 15246847 x^{8} - 166544168 x^{7} + 1501495664 x^{6} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{24}\cdot 5^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.6
Root \(7.78443 - 7.78443i\) of defining polynomial
Character \(\chi\) \(=\) 30.13
Dual form 30.11.f.b.7.6

$q$-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+(16.0000 - 16.0000i) q^{2} +(99.2043 + 99.2043i) q^{3} -512.000i q^{4} +(2015.56 + 2388.12i) q^{5} +3174.54 q^{6} +(-17871.8 + 17871.8i) q^{7} +(-8192.00 - 8192.00i) q^{8} +19683.0i q^{9} +(70459.0 + 5960.93i) q^{10} -167706. q^{11} +(50792.6 - 50792.6i) q^{12} +(15865.7 + 15865.7i) q^{13} +571899. i q^{14} +(-36959.4 + 436865. i) q^{15} -262144. q^{16} +(-1.45213e6 + 1.45213e6i) q^{17} +(314928. + 314928. i) q^{18} -2.48441e6i q^{19} +(1.22272e6 - 1.03197e6i) q^{20} -3.54593e6 q^{21} +(-2.68329e6 + 2.68329e6i) q^{22} +(7.56139e6 + 7.56139e6i) q^{23} -1.62536e6i q^{24} +(-1.64063e6 + 9.62683e6i) q^{25} +507702. q^{26} +(-1.95264e6 + 1.95264e6i) q^{27} +(9.15038e6 + 9.15038e6i) q^{28} +1.19239e7i q^{29} +(6.39849e6 + 7.58119e6i) q^{30} +3.53790e7 q^{31} +(-4.19430e6 + 4.19430e6i) q^{32} +(-1.66371e7 - 1.66371e7i) q^{33} +4.64681e7i q^{34} +(-7.87020e7 - 6.65830e6i) q^{35} +1.00777e7 q^{36} +(-4.85402e7 + 4.85402e7i) q^{37} +(-3.97505e7 - 3.97505e7i) q^{38} +3.14789e6i q^{39} +(3.05200e6 - 3.60750e7i) q^{40} +3.76547e7 q^{41} +(-5.67348e7 + 5.67348e7i) q^{42} +(2.19359e7 + 2.19359e7i) q^{43} +8.58654e7i q^{44} +(-4.70054e7 + 3.96723e7i) q^{45} +2.41964e8 q^{46} +(2.71165e8 - 2.71165e8i) q^{47} +(-2.60058e7 - 2.60058e7i) q^{48} -3.56330e8i q^{49} +(1.27779e8 + 1.80279e8i) q^{50} -2.88115e8 q^{51} +(8.12323e6 - 8.12323e6i) q^{52} +(-3.39462e8 - 3.39462e8i) q^{53} +6.24844e7i q^{54} +(-3.38022e8 - 4.00502e8i) q^{55} +2.92812e8 q^{56} +(2.46464e8 - 2.46464e8i) q^{57} +(1.90782e8 + 1.90782e8i) q^{58} -9.69202e8i q^{59} +(2.23675e8 + 1.89232e7i) q^{60} +5.12127e8 q^{61} +(5.66065e8 - 5.66065e8i) q^{62} +(-3.51771e8 - 3.51771e8i) q^{63} +1.34218e8i q^{64} +(-5.91089e6 + 6.98675e7i) q^{65} -5.32389e8 q^{66} +(-1.09457e9 + 1.09457e9i) q^{67} +(7.43489e8 + 7.43489e8i) q^{68} +1.50024e9i q^{69} +(-1.36576e9 + 1.15270e9i) q^{70} +5.61233e8 q^{71} +(1.61243e8 - 1.61243e8i) q^{72} +(-1.08186e9 - 1.08186e9i) q^{73} +1.55329e9i q^{74} +(-1.11778e9 + 7.92265e8i) q^{75} -1.27202e9 q^{76} +(2.99721e9 - 2.99721e9i) q^{77} +(5.03662e7 + 5.03662e7i) q^{78} +5.51831e9i q^{79} +(-5.28368e8 - 6.26032e8i) q^{80} -3.87420e8 q^{81} +(6.02475e8 - 6.02475e8i) q^{82} +(3.59432e9 + 3.59432e9i) q^{83} +1.81551e9i q^{84} +(-6.39471e9 - 5.41002e8i) q^{85} +7.01949e8 q^{86} +(-1.18290e9 + 1.18290e9i) q^{87} +(1.37385e9 + 1.37385e9i) q^{88} -1.27453e9i q^{89} +(-1.17329e8 + 1.38684e9i) q^{90} -5.67098e8 q^{91} +(3.87143e9 - 3.87143e9i) q^{92} +(3.50975e9 + 3.50975e9i) q^{93} -8.67728e9i q^{94} +(5.93307e9 - 5.00749e9i) q^{95} -8.32186e8 q^{96} +(-3.20605e9 + 3.20605e9i) q^{97} +(-5.70128e9 - 5.70128e9i) q^{98} -3.30095e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 192 q^{2} - 5880 q^{5} + 11352 q^{7} - 98304 q^{8} - 94656 q^{10} + 406032 q^{11} - 112404 q^{13} - 21384 q^{15} - 3145728 q^{16} - 4207428 q^{17} + 3779136 q^{18} - 18432 q^{20} - 5917536 q^{21}+ \cdots - 30689842752 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 − 16.0000i 0.500000 − 0.500000i
\(3\) 99.2043 + 99.2043i 0.408248 + 0.408248i
\(4\) − 512.000i − 0.500000i
\(5\) 2015.56 + 2388.12i 0.644980 + 0.764199i
\(6\) 3174.54 0.408248
\(7\) −17871.8 + 17871.8i −1.06336 + 1.06336i −0.0655044 + 0.997852i \(0.520866\pi\)
−0.997852 + 0.0655044i \(0.979134\pi\)
\(8\) −8192.00 − 8192.00i −0.250000 − 0.250000i
\(9\) 19683.0i 0.333333i
\(10\) 70459.0 + 5960.93i 0.704590 + 0.0596093i
\(11\) −167706. −1.04132 −0.520661 − 0.853764i \(-0.674314\pi\)
−0.520661 + 0.853764i \(0.674314\pi\)
\(12\) 50792.6 − 50792.6i 0.204124 − 0.204124i
\(13\) 15865.7 + 15865.7i 0.0427309 + 0.0427309i 0.728149 − 0.685418i \(-0.240381\pi\)
−0.685418 + 0.728149i \(0.740381\pi\)
\(14\) 571899.i 1.06336i
\(15\) −36959.4 + 436865.i −0.0486708 + 0.575295i
\(16\) −262144. −0.250000
\(17\) −1.45213e6 + 1.45213e6i −1.02273 + 1.02273i −0.0229924 + 0.999736i \(0.507319\pi\)
−0.999736 + 0.0229924i \(0.992681\pi\)
\(18\) 314928. + 314928.i 0.166667 + 0.166667i
\(19\) − 2.48441e6i − 1.00336i −0.865054 − 0.501678i \(-0.832716\pi\)
0.865054 − 0.501678i \(-0.167284\pi\)
\(20\) 1.22272e6 − 1.03197e6i 0.382100 − 0.322490i
\(21\) −3.54593e6 −0.868227
\(22\) −2.68329e6 + 2.68329e6i −0.520661 + 0.520661i
\(23\) 7.56139e6 + 7.56139e6i 1.17480 + 1.17480i 0.981052 + 0.193743i \(0.0620629\pi\)
0.193743 + 0.981052i \(0.437937\pi\)
\(24\) − 1.62536e6i − 0.204124i
\(25\) −1.64063e6 + 9.62683e6i −0.168000 + 0.985787i
\(26\) 507702. 0.0427309
\(27\) −1.95264e6 + 1.95264e6i −0.136083 + 0.136083i
\(28\) 9.15038e6 + 9.15038e6i 0.531678 + 0.531678i
\(29\) 1.19239e7i 0.581335i 0.956824 + 0.290668i \(0.0938774\pi\)
−0.956824 + 0.290668i \(0.906123\pi\)
\(30\) 6.39849e6 + 7.58119e6i 0.263312 + 0.311983i
\(31\) 3.53790e7 1.23577 0.617885 − 0.786269i \(-0.287990\pi\)
0.617885 + 0.786269i \(0.287990\pi\)
\(32\) −4.19430e6 + 4.19430e6i −0.125000 + 0.125000i
\(33\) −1.66371e7 − 1.66371e7i −0.425118 − 0.425118i
\(34\) 4.64681e7i 1.02273i
\(35\) −7.87020e7 − 6.65830e6i −1.49846 − 0.126772i
\(36\) 1.00777e7 0.166667
\(37\) −4.85402e7 + 4.85402e7i −0.699992 + 0.699992i −0.964409 − 0.264417i \(-0.914821\pi\)
0.264417 + 0.964409i \(0.414821\pi\)
\(38\) −3.97505e7 − 3.97505e7i −0.501678 − 0.501678i
\(39\) 3.14789e6i 0.0348896i
\(40\) 3.05200e6 − 3.60750e7i 0.0298047 − 0.352295i
\(41\) 3.76547e7 0.325012 0.162506 − 0.986708i \(-0.448042\pi\)
0.162506 + 0.986708i \(0.448042\pi\)
\(42\) −5.67348e7 + 5.67348e7i −0.434114 + 0.434114i
\(43\) 2.19359e7 + 2.19359e7i 0.149215 + 0.149215i 0.777767 − 0.628552i \(-0.216352\pi\)
−0.628552 + 0.777767i \(0.716352\pi\)
\(44\) 8.58654e7i 0.520661i
\(45\) −4.70054e7 + 3.96723e7i −0.254733 + 0.214993i
\(46\) 2.41964e8 1.17480
\(47\) 2.71165e8 − 2.71165e8i 1.18235 − 1.18235i 0.203210 − 0.979135i \(-0.434862\pi\)
0.979135 − 0.203210i \(-0.0651376\pi\)
\(48\) −2.60058e7 − 2.60058e7i −0.102062 − 0.102062i
\(49\) − 3.56330e8i − 1.26146i
\(50\) 1.27779e8 + 1.80279e8i 0.408893 + 0.576894i
\(51\) −2.88115e8 −0.835054
\(52\) 8.12323e6 − 8.12323e6i 0.0213654 − 0.0213654i
\(53\) −3.39462e8 − 3.39462e8i −0.811730 − 0.811730i 0.173163 − 0.984893i \(-0.444601\pi\)
−0.984893 + 0.173163i \(0.944601\pi\)
\(54\) 6.24844e7i 0.136083i
\(55\) −3.38022e8 − 4.00502e8i −0.671632 − 0.795777i
\(56\) 2.92812e8 0.531678
\(57\) 2.46464e8 − 2.46464e8i 0.409618 − 0.409618i
\(58\) 1.90782e8 + 1.90782e8i 0.290668 + 0.290668i
\(59\) − 9.69202e8i − 1.35567i −0.735213 − 0.677836i \(-0.762918\pi\)
0.735213 − 0.677836i \(-0.237082\pi\)
\(60\) 2.23675e8 + 1.89232e7i 0.287648 + 0.0243354i
\(61\) 5.12127e8 0.606357 0.303178 − 0.952934i \(-0.401952\pi\)
0.303178 + 0.952934i \(0.401952\pi\)
\(62\) 5.66065e8 − 5.66065e8i 0.617885 − 0.617885i
\(63\) −3.51771e8 − 3.51771e8i −0.354452 − 0.354452i
\(64\) 1.34218e8i 0.125000i
\(65\) −5.91089e6 + 6.98675e7i −0.00509432 + 0.0602155i
\(66\) −5.32389e8 −0.425118
\(67\) −1.09457e9 + 1.09457e9i −0.810714 + 0.810714i −0.984741 − 0.174027i \(-0.944322\pi\)
0.174027 + 0.984741i \(0.444322\pi\)
\(68\) 7.43489e8 + 7.43489e8i 0.511364 + 0.511364i
\(69\) 1.50024e9i 0.959217i
\(70\) −1.36576e9 + 1.15270e9i −0.812616 + 0.685844i
\(71\) 5.61233e8 0.311065 0.155532 − 0.987831i \(-0.450291\pi\)
0.155532 + 0.987831i \(0.450291\pi\)
\(72\) 1.61243e8 − 1.61243e8i 0.0833333 − 0.0833333i
\(73\) −1.08186e9 − 1.08186e9i −0.521864 − 0.521864i 0.396270 − 0.918134i \(-0.370304\pi\)
−0.918134 + 0.396270i \(0.870304\pi\)
\(74\) 1.55329e9i 0.699992i
\(75\) −1.11778e9 + 7.92265e8i −0.471032 + 0.333860i
\(76\) −1.27202e9 −0.501678
\(77\) 2.99721e9 − 2.99721e9i 1.10730 − 1.10730i
\(78\) 5.03662e7 + 5.03662e7i 0.0174448 + 0.0174448i
\(79\) 5.51831e9i 1.79337i 0.442667 + 0.896686i \(0.354032\pi\)
−0.442667 + 0.896686i \(0.645968\pi\)
\(80\) −5.28368e8 − 6.26032e8i −0.161245 − 0.191050i
\(81\) −3.87420e8 −0.111111
\(82\) 6.02475e8 − 6.02475e8i 0.162506 − 0.162506i
\(83\) 3.59432e9 + 3.59432e9i 0.912487 + 0.912487i 0.996467 − 0.0839801i \(-0.0267632\pi\)
−0.0839801 + 0.996467i \(0.526763\pi\)
\(84\) 1.81551e9i 0.434114i
\(85\) −6.39471e9 − 5.41002e8i −1.44121 − 0.121928i
\(86\) 7.01949e8 0.149215
\(87\) −1.18290e9 + 1.18290e9i −0.237329 + 0.237329i
\(88\) 1.37385e9 + 1.37385e9i 0.260330 + 0.260330i
\(89\) − 1.27453e9i − 0.228245i −0.993467 − 0.114122i \(-0.963594\pi\)
0.993467 − 0.114122i \(-0.0364056\pi\)
\(90\) −1.17329e8 + 1.38684e9i −0.0198698 + 0.234863i
\(91\) −5.67098e8 −0.0908764
\(92\) 3.87143e9 − 3.87143e9i 0.587398 − 0.587398i
\(93\) 3.50975e9 + 3.50975e9i 0.504501 + 0.504501i
\(94\) − 8.67728e9i − 1.18235i
\(95\) 5.93307e9 − 5.00749e9i 0.766764 − 0.647145i
\(96\) −8.32186e8 −0.102062
\(97\) −3.20605e9 + 3.20605e9i −0.373346 + 0.373346i −0.868694 − 0.495349i \(-0.835040\pi\)
0.495349 + 0.868694i \(0.335040\pi\)
\(98\) −5.70128e9 − 5.70128e9i −0.630728 − 0.630728i
\(99\) − 3.30095e9i − 0.347107i
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 30.11.f.b.13.6 yes 12
3.2 odd 2 90.11.g.g.73.1 12
5.2 odd 4 inner 30.11.f.b.7.6 ✓ 12
5.3 odd 4 150.11.f.h.7.3 12
5.4 even 2 150.11.f.h.43.3 12
15.2 even 4 90.11.g.g.37.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.11.f.b.7.6 ✓ 12 5.2 odd 4 inner
30.11.f.b.13.6 yes 12 1.1 even 1 trivial
90.11.g.g.37.1 12 15.2 even 4
90.11.g.g.73.1 12 3.2 odd 2
150.11.f.h.7.3 12 5.3 odd 4
150.11.f.h.43.3 12 5.4 even 2