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Properties

Label 1950.2.bc.g.751.1
Level $1950$
Weight $2$
Character 1950.751
Analytic conductor $15.571$
Analytic rank $0$
Dimension $8$
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] = [1950,2,Mod(751,1950)] 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("1950.751"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1950, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.bc (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,4,4,0,0,0,0,-4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.17284886784.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 30x^{5} + 185x^{4} + 36x^{3} + 8x^{2} + 208x + 2704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 751.1
Root \(1.33404 - 1.33404i\) of defining polynomial
Character \(\chi\) \(=\) 1950.751
Dual form 1950.2.bc.g.901.1

$q$-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+(-0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.866025 - 0.500000i) q^{6} +(-4.02239 - 2.32233i) q^{7} +1.00000i q^{8} +(-0.500000 + 0.866025i) q^{9} +(3.81062 - 2.20006i) q^{11} +1.00000 q^{12} +(3.35432 - 1.32233i) q^{13} +4.64466 q^{14} +(-0.500000 - 0.866025i) q^{16} +(-2.00000 + 3.46410i) q^{17} -1.00000i q^{18} +(-6.96699 - 4.02239i) q^{19} -4.64466i q^{21} +(-2.20006 + 3.81062i) q^{22} +(0.488292 + 0.845746i) q^{23} +(-0.866025 + 0.500000i) q^{24} +(-2.24376 + 2.82233i) q^{26} -1.00000 q^{27} +(-4.02239 + 2.32233i) q^{28} +(2.15637 + 3.73494i) q^{29} +6.44069i q^{31} +(0.866025 + 0.500000i) q^{32} +(3.81062 + 2.20006i) q^{33} -4.00000i q^{34} +(0.500000 + 0.866025i) q^{36} +(-3.28745 + 1.89801i) q^{37} +8.04479 q^{38} +(2.82233 + 2.24376i) q^{39} +(-6.31274 + 3.64466i) q^{41} +(2.32233 + 4.02239i) q^{42} +(-0.358228 + 0.620469i) q^{43} -4.40013i q^{44} +(-0.845746 - 0.488292i) q^{46} -9.75342i q^{47} +(0.500000 - 0.866025i) q^{48} +(7.28643 + 12.6205i) q^{49} -4.00000 q^{51} +(0.531987 - 3.56609i) q^{52} -13.5089 q^{53} +(0.866025 - 0.500000i) q^{54} +(2.32233 - 4.02239i) q^{56} -8.04479i q^{57} +(-3.73494 - 2.15637i) q^{58} +(-1.88842 - 1.09028i) q^{59} +(-3.73205 + 6.46410i) q^{61} +(-3.22034 - 5.57780i) q^{62} +(4.02239 - 2.32233i) q^{63} -1.00000 q^{64} -4.40013 q^{66} +(1.58068 - 0.912609i) q^{67} +(2.00000 + 3.46410i) q^{68} +(-0.488292 + 0.845746i) q^{69} +(-6.88764 - 3.97658i) q^{71} +(-0.866025 - 0.500000i) q^{72} +4.36112i q^{73} +(1.89801 - 3.28745i) q^{74} +(-6.96699 + 4.02239i) q^{76} -20.4371 q^{77} +(-3.56609 - 0.531987i) q^{78} -14.9340 q^{79} +(-0.500000 - 0.866025i) q^{81} +(3.64466 - 6.31274i) q^{82} -3.51093i q^{83} +(-4.02239 - 2.32233i) q^{84} -0.716456i q^{86} +(-2.15637 + 3.73494i) q^{87} +(2.20006 + 3.81062i) q^{88} +(7.07780 - 4.08637i) q^{89} +(-16.5633 - 2.47090i) q^{91} +0.976584 q^{92} +(-5.57780 + 3.22034i) q^{93} +(4.87671 + 8.44671i) q^{94} +1.00000i q^{96} +(-11.9730 - 6.91261i) q^{97} +(-12.6205 - 7.28643i) q^{98} +4.40013i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{4} - 4 q^{9} + 6 q^{11} + 8 q^{12} + 12 q^{13} + 4 q^{14} - 4 q^{16} - 16 q^{17} - 6 q^{19} - 2 q^{22} - 4 q^{23} - 12 q^{26} - 8 q^{27} - 8 q^{29} + 6 q^{33} + 4 q^{36} - 30 q^{37}+ \cdots - 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 0 0
\(6\) −0.866025 − 0.500000i −0.353553 − 0.204124i
\(7\) −4.02239 − 2.32233i −1.52032 − 0.877758i −0.999713 − 0.0239629i \(-0.992372\pi\)
−0.520609 − 0.853795i \(-0.674295\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 3.81062 − 2.20006i 1.14895 − 0.663344i 0.200316 − 0.979731i \(-0.435803\pi\)
0.948630 + 0.316387i \(0.102470\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.35432 − 1.32233i 0.930320 − 0.366748i
\(14\) 4.64466 1.24134
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −2.00000 + 3.46410i −0.485071 + 0.840168i −0.999853 − 0.0171533i \(-0.994540\pi\)
0.514782 + 0.857321i \(0.327873\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −6.96699 − 4.02239i −1.59834 − 0.922800i −0.991808 − 0.127739i \(-0.959228\pi\)
−0.606529 − 0.795061i \(-0.707439\pi\)
\(20\) 0 0
\(21\) − 4.64466i − 1.01355i
\(22\) −2.20006 + 3.81062i −0.469055 + 0.812427i
\(23\) 0.488292 + 0.845746i 0.101816 + 0.176350i 0.912433 − 0.409226i \(-0.134201\pi\)
−0.810617 + 0.585577i \(0.800868\pi\)
\(24\) −0.866025 + 0.500000i −0.176777 + 0.102062i
\(25\) 0 0
\(26\) −2.24376 + 2.82233i −0.440037 + 0.553504i
\(27\) −1.00000 −0.192450
\(28\) −4.02239 + 2.32233i −0.760161 + 0.438879i
\(29\) 2.15637 + 3.73494i 0.400427 + 0.693561i 0.993777 − 0.111384i \(-0.0355283\pi\)
−0.593350 + 0.804945i \(0.702195\pi\)
\(30\) 0 0
\(31\) 6.44069i 1.15678i 0.815760 + 0.578391i \(0.196319\pi\)
−0.815760 + 0.578391i \(0.803681\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 3.81062 + 2.20006i 0.663344 + 0.382982i
\(34\) − 4.00000i − 0.685994i
\(35\) 0 0
\(36\) 0.500000 + 0.866025i 0.0833333 + 0.144338i
\(37\) −3.28745 + 1.89801i −0.540454 + 0.312031i −0.745263 − 0.666771i \(-0.767676\pi\)
0.204809 + 0.978802i \(0.434343\pi\)
\(38\) 8.04479 1.30504
\(39\) 2.82233 + 2.24376i 0.451934 + 0.359289i
\(40\) 0 0
\(41\) −6.31274 + 3.64466i −0.985884 + 0.569200i −0.904041 − 0.427445i \(-0.859414\pi\)
−0.0818424 + 0.996645i \(0.526080\pi\)
\(42\) 2.32233 + 4.02239i 0.358343 + 0.620669i
\(43\) −0.358228 + 0.620469i −0.0546293 + 0.0946207i −0.892047 − 0.451943i \(-0.850731\pi\)
0.837418 + 0.546564i \(0.184064\pi\)
\(44\) − 4.40013i − 0.663344i
\(45\) 0 0
\(46\) −0.845746 − 0.488292i −0.124698 − 0.0719947i
\(47\) − 9.75342i − 1.42268i −0.702847 − 0.711341i \(-0.748088\pi\)
0.702847 − 0.711341i \(-0.251912\pi\)
\(48\) 0.500000 − 0.866025i 0.0721688 − 0.125000i
\(49\) 7.28643 + 12.6205i 1.04092 + 1.80292i
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0.531987 − 3.56609i 0.0737734 − 0.494528i
\(53\) −13.5089 −1.85559 −0.927794 − 0.373092i \(-0.878297\pi\)
−0.927794 + 0.373092i \(0.878297\pi\)
\(54\) 0.866025 − 0.500000i 0.117851 − 0.0680414i
\(55\) 0 0
\(56\) 2.32233 − 4.02239i 0.310334 − 0.537515i
\(57\) − 8.04479i − 1.06556i
\(58\) −3.73494 − 2.15637i −0.490421 − 0.283145i
\(59\) −1.88842 − 1.09028i −0.245851 − 0.141942i 0.372012 − 0.928228i \(-0.378668\pi\)
−0.617863 + 0.786286i \(0.712001\pi\)
\(60\) 0 0
\(61\) −3.73205 + 6.46410i −0.477840 + 0.827643i −0.999677 − 0.0254017i \(-0.991914\pi\)
0.521837 + 0.853045i \(0.325247\pi\)
\(62\) −3.22034 − 5.57780i −0.408984 − 0.708381i
\(63\) 4.02239 − 2.32233i 0.506774 − 0.292586i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.40013 −0.541618
\(67\) 1.58068 − 0.912609i 0.193111 − 0.111493i −0.400327 − 0.916372i \(-0.631103\pi\)
0.593438 + 0.804879i \(0.297770\pi\)
\(68\) 2.00000 + 3.46410i 0.242536 + 0.420084i
\(69\) −0.488292 + 0.845746i −0.0587834 + 0.101816i
\(70\) 0 0
\(71\) −6.88764 − 3.97658i −0.817413 − 0.471934i 0.0321105 − 0.999484i \(-0.489777\pi\)
−0.849524 + 0.527551i \(0.823110\pi\)
\(72\) −0.866025 − 0.500000i −0.102062 − 0.0589256i
\(73\) 4.36112i 0.510430i 0.966884 + 0.255215i \(0.0821463\pi\)
−0.966884 + 0.255215i \(0.917854\pi\)
\(74\) 1.89801 − 3.28745i 0.220640 − 0.382159i
\(75\) 0 0
\(76\) −6.96699 + 4.02239i −0.799168 + 0.461400i
\(77\) −20.4371 −2.32902
\(78\) −3.56609 − 0.531987i −0.403780 − 0.0602357i
\(79\) −14.9340 −1.68020 −0.840102 − 0.542429i \(-0.817505\pi\)
−0.840102 + 0.542429i \(0.817505\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 3.64466 − 6.31274i 0.402485 − 0.697125i
\(83\) − 3.51093i − 0.385375i −0.981260 − 0.192688i \(-0.938280\pi\)
0.981260 − 0.192688i \(-0.0617204\pi\)
\(84\) −4.02239 − 2.32233i −0.438879 − 0.253387i
\(85\) 0 0
\(86\) − 0.716456i − 0.0772575i
\(87\) −2.15637 + 3.73494i −0.231187 + 0.400427i
\(88\) 2.20006 + 3.81062i 0.234528 + 0.406214i
\(89\) 7.07780 − 4.08637i 0.750245 − 0.433154i −0.0755374 − 0.997143i \(-0.524067\pi\)
0.825782 + 0.563989i \(0.190734\pi\)
\(90\) 0 0
\(91\) −16.5633 − 2.47090i −1.73630 − 0.259021i
\(92\) 0.976584 0.101816
\(93\) −5.57780 + 3.22034i −0.578391 + 0.333934i
\(94\) 4.87671 + 8.44671i 0.502994 + 0.871212i
\(95\) 0 0
\(96\) 1.00000i 0.102062i
\(97\) −11.9730 − 6.91261i −1.21567 − 0.701869i −0.251683 − 0.967810i \(-0.580984\pi\)
−0.963989 + 0.265941i \(0.914318\pi\)
\(98\) −12.6205 − 7.28643i −1.27486 − 0.736041i
\(99\) 4.40013i 0.442229i
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 1950.2.bc.g.751.1 8
5.2 odd 4 1950.2.y.j.49.4 8
5.3 odd 4 1950.2.y.k.49.1 8
5.4 even 2 390.2.bb.c.361.4 yes 8
13.4 even 6 inner 1950.2.bc.g.901.1 8
15.14 odd 2 1170.2.bs.f.361.2 8
65.4 even 6 390.2.bb.c.121.4 ✓ 8
65.17 odd 12 1950.2.y.k.199.1 8
65.24 odd 12 5070.2.a.bz.1.1 4
65.29 even 6 5070.2.b.ba.1351.5 8
65.43 odd 12 1950.2.y.j.199.4 8
65.49 even 6 5070.2.b.ba.1351.4 8
65.54 odd 12 5070.2.a.ca.1.4 4
195.134 odd 6 1170.2.bs.f.901.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.c.121.4 ✓ 8 65.4 even 6
390.2.bb.c.361.4 yes 8 5.4 even 2
1170.2.bs.f.361.2 8 15.14 odd 2
1170.2.bs.f.901.2 8 195.134 odd 6
1950.2.y.j.49.4 8 5.2 odd 4
1950.2.y.j.199.4 8 65.43 odd 12
1950.2.y.k.49.1 8 5.3 odd 4
1950.2.y.k.199.1 8 65.17 odd 12
1950.2.bc.g.751.1 8 1.1 even 1 trivial
1950.2.bc.g.901.1 8 13.4 even 6 inner
5070.2.a.bz.1.1 4 65.24 odd 12
5070.2.a.ca.1.4 4 65.54 odd 12
5070.2.b.ba.1351.4 8 65.49 even 6
5070.2.b.ba.1351.5 8 65.29 even 6