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Properties

Label 925.2.b.f.149.7
Level $925$
Weight $2$
Character 925.149
Analytic conductor $7.386$
Analytic rank $0$
Dimension $10$
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] = [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 149.7
Root \(-1.13359i\) of defining polynomial
Character \(\chi\) \(=\) 925.149
Dual form 925.2.b.f.149.4

$q$-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 \(0.131263\pi\)
−0.916172 + 0.400785i \(0.868737\pi\)
\(3\) − 1.10563i − 0.638335i −0.947698 − 0.319168i \(-0.896597\pi\)
0.947698 − 0.319168i \(-0.103403\pi\)
\(4\) 0.714970 0.357485
\(5\) 0 0
\(6\) 1.25333 0.511671
\(7\) − 2.46164i − 0.930412i −0.885203 − 0.465206i \(-0.845980\pi\)
0.885203 − 0.465206i \(-0.154020\pi\)
\(8\) 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 \(0.356693\pi\)
−0.435156 + 0.900355i \(0.643307\pi\)
\(14\) 2.79049 0.745790
\(15\) 0 0
\(16\) −2.05888 −0.514719
\(17\) − 3.32980i − 0.807594i −0.914849 − 0.403797i \(-0.867690\pi\)
0.914849 − 0.403797i \(-0.132310\pi\)
\(18\) 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 \(-0.930215\pi\)
0.976064 − 0.217484i \(-0.0697850\pi\)
\(24\) 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 \(0.0442878\pi\)
−0.990336 + 0.138686i \(0.955712\pi\)
\(44\) 1.22615 0.184849
\(45\) 0 0
\(46\) 2.36471 0.348657
\(47\) 0.882270i 0.128692i 0.997928 + 0.0643462i \(0.0204962\pi\)
−0.997928 + 0.0643462i \(0.979504\pi\)
\(48\) 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 \(-0.839595\pi\)
0.875693 − 0.482867i \(-0.160405\pi\)
\(54\) 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 \(-0.734991\pi\)
0.672992 − 0.739649i \(-0.265009\pi\)
\(68\) − 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 \(0.429607\pi\)
−0.219349 + 0.975646i \(0.570393\pi\)
\(74\) 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 \(-0.915095\pi\)
0.964636 − 0.263585i \(-0.0849050\pi\)
\(84\) −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 \(0.373343\pi\)
−0.387487 + 0.921875i \(0.626657\pi\)
\(98\) 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.149.7 10
5.2 odd 4 185.2.a.e.1.2 ✓ 5
5.3 odd 4 925.2.a.f.1.4 5
5.4 even 2 inner 925.2.b.f.149.4 10
15.2 even 4 1665.2.a.p.1.4 5
15.8 even 4 8325.2.a.ch.1.2 5
20.7 even 4 2960.2.a.w.1.4 5
35.27 even 4 9065.2.a.k.1.2 5
185.147 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 5.2 odd 4
925.2.a.f.1.4 5 5.3 odd 4
925.2.b.f.149.4 10 5.4 even 2 inner
925.2.b.f.149.7 10 1.1 even 1 trivial
1665.2.a.p.1.4 5 15.2 even 4
2960.2.a.w.1.4 5 20.7 even 4
6845.2.a.f.1.4 5 185.147 odd 4
8325.2.a.ch.1.2 5 15.8 even 4
9065.2.a.k.1.2 5 35.27 even 4