↑

Properties

Label 925.2.b.f.149.4
Level $925$
Weight $2$
Character 925.149
Analytic conductor $7.386$
Analytic rank $0$
Dimension $10$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(149,925)] 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("925.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-20,0,-12,0,0,-12,0,-10] 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: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.60703296077824.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 20x^{8} + 142x^{6} + 420x^{4} + 457x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.4
Root \(1.13359i\) of defining polynomial
Character \(\chi\) \(=\) 925.149
Dual form 925.2.b.f.149.7

$q$-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)
 
\(f(q)\) \(=\) \(q-1.13359i q^{2} +1.10563i q^{3} +0.714970 q^{4} +1.25333 q^{6} +2.46164i q^{7} -3.07767i q^{8} +1.77758 q^{9} +1.71497 q^{11} +0.790492i q^{12} -6.49255i q^{13} +2.79049 q^{14} -2.05888 q^{16} +3.32980i q^{17} -2.01505i q^{18} -0.734568 q^{19} -2.72166 q^{21} -1.94408i q^{22} +2.08603i q^{23} +3.40276 q^{24} -7.35990 q^{26} +5.28224i q^{27} +1.76000i q^{28} +4.21126 q^{29} +7.46459 q^{31} -3.82141i q^{32} +1.89612i q^{33} +3.77463 q^{34} +1.27092 q^{36} +1.00000i q^{37} +0.832700i q^{38} +7.17836 q^{39} +1.71497 q^{41} +3.08525i q^{42} -1.81885i q^{43} +1.22615 q^{44} +2.36471 q^{46} -0.882270i q^{47} -2.27636i q^{48} +0.940340 q^{49} -3.68152 q^{51} -4.64198i q^{52} +7.03066i q^{53} +5.98790 q^{54} +7.57610 q^{56} -0.812159i q^{57} -4.77385i q^{58} -0.387867 q^{59} -11.8224 q^{61} -8.46180i q^{62} +4.37577i q^{63} -8.44967 q^{64} +2.14943 q^{66} +12.1086i q^{67} +2.38070i q^{68} -2.30638 q^{69} +13.7486 q^{71} -5.47081i q^{72} -16.6719i q^{73} +1.13359 q^{74} -0.525194 q^{76} +4.22163i q^{77} -8.13733i q^{78} -8.23253 q^{79} -0.507447 q^{81} -1.94408i q^{82} +4.80275i q^{83} -1.94590 q^{84} -2.06183 q^{86} +4.65609i q^{87} -5.27811i q^{88} +1.52506 q^{89} +15.9823 q^{91} +1.49145i q^{92} +8.25307i q^{93} -1.00013 q^{94} +4.22506 q^{96} -18.1588i q^{97} -1.06596i q^{98} +3.04850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 20 q^{4} - 12 q^{6} - 12 q^{9} - 10 q^{11} + 16 q^{14} + 32 q^{16} + 8 q^{19} + 6 q^{21} + 84 q^{24} - 8 q^{26} + 8 q^{29} + 16 q^{31} + 64 q^{34} + 32 q^{36} - 4 q^{39} - 10 q^{41} + 92 q^{44} - 44 q^{49}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.13359i − 0.801570i −0.916172 − 0.400785i \(-0.868737\pi\)
0.916172 − 0.400785i \(-0.131263\pi\)
\(3\) 1.10563i 0.638335i 0.947698 + 0.319168i \(0.103403\pi\)
−0.947698 + 0.319168i \(0.896597\pi\)
\(4\) 0.714970 0.357485
\(5\) 0 0
\(6\) 1.25333 0.511671
\(7\) 2.46164i 0.930412i 0.885203 + 0.465206i \(0.154020\pi\)
−0.885203 + 0.465206i \(0.845980\pi\)
\(8\) − 3.07767i − 1.08812i
\(9\) 1.77758 0.592528
\(10\) 0 0
\(11\) 1.71497 0.517083 0.258541 − 0.966000i \(-0.416758\pi\)
0.258541 + 0.966000i \(0.416758\pi\)
\(12\) 0.790492i 0.228195i
\(13\) − 6.49255i − 1.80071i −0.435156 − 0.900355i \(-0.643307\pi\)
0.435156 − 0.900355i \(-0.356693\pi\)
\(14\) 2.79049 0.745790
\(15\) 0 0
\(16\) −2.05888 −0.514719
\(17\) 3.32980i 0.807594i 0.914849 + 0.403797i \(0.132310\pi\)
−0.914849 + 0.403797i \(0.867690\pi\)
\(18\) − 2.01505i − 0.474953i
\(19\) −0.734568 −0.168521 −0.0842607 − 0.996444i \(-0.526853\pi\)
−0.0842607 + 0.996444i \(0.526853\pi\)
\(20\) 0 0
\(21\) −2.72166 −0.593915
\(22\) − 1.94408i − 0.414478i
\(23\) 2.08603i 0.434968i 0.976064 + 0.217484i \(0.0697850\pi\)
−0.976064 + 0.217484i \(0.930215\pi\)
\(24\) 3.40276 0.694585
\(25\) 0 0
\(26\) −7.35990 −1.44340
\(27\) 5.28224i 1.01657i
\(28\) 1.76000i 0.332608i
\(29\) 4.21126 0.782011 0.391006 − 0.920388i \(-0.372127\pi\)
0.391006 + 0.920388i \(0.372127\pi\)
\(30\) 0 0
\(31\) 7.46459 1.34068 0.670340 − 0.742054i \(-0.266148\pi\)
0.670340 + 0.742054i \(0.266148\pi\)
\(32\) − 3.82141i − 0.675536i
\(33\) 1.89612i 0.330072i
\(34\) 3.77463 0.647344
\(35\) 0 0
\(36\) 1.27092 0.211820
\(37\) 1.00000i 0.164399i
\(38\) 0.832700i 0.135082i
\(39\) 7.17836 1.14946
\(40\) 0 0
\(41\) 1.71497 0.267833 0.133917 − 0.990993i \(-0.457245\pi\)
0.133917 + 0.990993i \(0.457245\pi\)
\(42\) 3.08525i 0.476064i
\(43\) − 1.81885i − 0.277372i −0.990336 − 0.138686i \(-0.955712\pi\)
0.990336 − 0.138686i \(-0.0442878\pi\)
\(44\) 1.22615 0.184849
\(45\) 0 0
\(46\) 2.36471 0.348657
\(47\) − 0.882270i − 0.128692i −0.997928 − 0.0643462i \(-0.979504\pi\)
0.997928 − 0.0643462i \(-0.0204962\pi\)
\(48\) − 2.27636i − 0.328564i
\(49\) 0.940340 0.134334
\(50\) 0 0
\(51\) −3.68152 −0.515516
\(52\) − 4.64198i − 0.643727i
\(53\) 7.03066i 0.965735i 0.875693 + 0.482867i \(0.160405\pi\)
−0.875693 + 0.482867i \(0.839595\pi\)
\(54\) 5.98790 0.814850
\(55\) 0 0
\(56\) 7.57610 1.01240
\(57\) − 0.812159i − 0.107573i
\(58\) − 4.77385i − 0.626837i
\(59\) −0.387867 −0.0504959 −0.0252480 − 0.999681i \(-0.508038\pi\)
−0.0252480 + 0.999681i \(0.508038\pi\)
\(60\) 0 0
\(61\) −11.8224 −1.51370 −0.756848 − 0.653590i \(-0.773262\pi\)
−0.756848 + 0.653590i \(0.773262\pi\)
\(62\) − 8.46180i − 1.07465i
\(63\) 4.37577i 0.551295i
\(64\) −8.44967 −1.05621
\(65\) 0 0
\(66\) 2.14943 0.264576
\(67\) 12.1086i 1.47930i 0.672992 + 0.739649i \(0.265009\pi\)
−0.672992 + 0.739649i \(0.734991\pi\)
\(68\) 2.38070i 0.288703i
\(69\) −2.30638 −0.277655
\(70\) 0 0
\(71\) 13.7486 1.63166 0.815828 − 0.578295i \(-0.196282\pi\)
0.815828 + 0.578295i \(0.196282\pi\)
\(72\) − 5.47081i − 0.644741i
\(73\) − 16.6719i − 1.95129i −0.219349 − 0.975646i \(-0.570393\pi\)
0.219349 − 0.975646i \(-0.429607\pi\)
\(74\) 1.13359 0.131777
\(75\) 0 0
\(76\) −0.525194 −0.0602439
\(77\) 4.22163i 0.481100i
\(78\) − 8.13733i − 0.921371i
\(79\) −8.23253 −0.926232 −0.463116 − 0.886298i \(-0.653269\pi\)
−0.463116 + 0.886298i \(0.653269\pi\)
\(80\) 0 0
\(81\) −0.507447 −0.0563829
\(82\) − 1.94408i − 0.214687i
\(83\) 4.80275i 0.527171i 0.964636 + 0.263585i \(0.0849050\pi\)
−0.964636 + 0.263585i \(0.915095\pi\)
\(84\) −1.94590 −0.212316
\(85\) 0 0
\(86\) −2.06183 −0.222333
\(87\) 4.65609i 0.499185i
\(88\) − 5.27811i − 0.562648i
\(89\) 1.52506 0.161656 0.0808280 − 0.996728i \(-0.474244\pi\)
0.0808280 + 0.996728i \(0.474244\pi\)
\(90\) 0 0
\(91\) 15.9823 1.67540
\(92\) 1.49145i 0.155494i
\(93\) 8.25307i 0.855804i
\(94\) −1.00013 −0.103156
\(95\) 0 0
\(96\) 4.22506 0.431218
\(97\) − 18.1588i − 1.84375i −0.387487 − 0.921875i \(-0.626657\pi\)
0.387487 − 0.921875i \(-0.373343\pi\)
\(98\) − 1.06596i − 0.107678i
\(99\) 3.04850 0.306386
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 925.2.b.f.149.4 10
5.2 odd 4 925.2.a.f.1.4 5
5.3 odd 4 185.2.a.e.1.2 ✓ 5
5.4 even 2 inner 925.2.b.f.149.7 10
15.2 even 4 8325.2.a.ch.1.2 5
15.8 even 4 1665.2.a.p.1.4 5
20.3 even 4 2960.2.a.w.1.4 5
35.13 even 4 9065.2.a.k.1.2 5
185.73 odd 4 6845.2.a.f.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.2 ✓ 5 5.3 odd 4
925.2.a.f.1.4 5 5.2 odd 4
925.2.b.f.149.4 10 1.1 even 1 trivial
925.2.b.f.149.7 10 5.4 even 2 inner
1665.2.a.p.1.4 5 15.8 even 4
2960.2.a.w.1.4 5 20.3 even 4
6845.2.a.f.1.4 5 185.73 odd 4
8325.2.a.ch.1.2 5 15.2 even 4
9065.2.a.k.1.2 5 35.13 even 4