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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7,7,Mod(6,7)] 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("7.6"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 7.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: \(1.61037858534\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-510}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 510 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 6.1
Root \(-22.5832i\) of defining polynomial
Character \(\chi\) \(=\) 7.6
Dual form 7.7.b.b.6.2

$q$-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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} -45.1664i q^{3} +45.1664i q^{5} +361.331i q^{6} +(133.000 - 316.165i) q^{7} +512.000 q^{8} -1311.00 q^{9} -361.331i q^{10} +874.000 q^{11} -2213.15i q^{13} +(-1064.00 + 2529.32i) q^{14} +2040.00 q^{15} -4096.00 q^{16} +5961.96i q^{17} +10488.0 q^{18} -3116.48i q^{19} +(-14280.0 - 6007.13i) q^{21} -6992.00 q^{22} +4738.00 q^{23} -23125.2i q^{24} +13585.0 q^{25} +17705.2i q^{26} +26286.8i q^{27} +11146.0 q^{29} -16320.0 q^{30} +27461.1i q^{31} -39475.4i q^{33} -47695.7i q^{34} +(14280.0 + 6007.13i) q^{35} +3002.00 q^{37} +24931.8i q^{38} -99960.0 q^{39} +23125.2i q^{40} -57541.9i q^{41} +(114240. + 48057.0i) q^{42} +31418.0 q^{43} -59213.1i q^{45} -37904.0 q^{46} +72446.8i q^{47} +185001. i q^{48} +(-82271.0 - 84099.8i) q^{49} -108680. q^{50} +269280. q^{51} -76406.0 q^{53} -210295. i q^{54} +39475.4i q^{55} +(68096.0 - 161876. i) q^{56} -140760. q^{57} -89168.0 q^{58} -113232. i q^{59} +275108. i q^{61} -219689. i q^{62} +(-174363. + 414492. i) q^{63} +262144. q^{64} +99960.0 q^{65} +315803. i q^{66} +495242. q^{67} -213998. i q^{69} +(-114240. - 48057.0i) q^{70} -184406. q^{71} -671232. q^{72} +60974.6i q^{73} -24016.0 q^{74} -613585. i q^{75} +(116242. - 276328. i) q^{77} +799680. q^{78} -534934. q^{79} -185001. i q^{80} +231561. q^{81} +460336. i q^{82} +714848. i q^{83} -269280. q^{85} -251344. q^{86} -503424. i q^{87} +447488. q^{88} +629529. i q^{89} +473705. i q^{90} +(-699720. - 294349. i) q^{91} +1.24032e6 q^{93} -579575. i q^{94} +140760. q^{95} -814440. i q^{97} +(658168. + 672798. i) q^{98} -1.14581e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} + 266 q^{7} + 1024 q^{8} - 2622 q^{9} + 1748 q^{11} - 2128 q^{14} + 4080 q^{15} - 8192 q^{16} + 20976 q^{18} - 28560 q^{21} - 13984 q^{22} + 9476 q^{23} + 27170 q^{25} + 22292 q^{29} - 32640 q^{30}+ \cdots - 2291628 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) − 45.1664i − 1.67283i −0.548098 − 0.836414i \(-0.684648\pi\)
0.548098 − 0.836414i \(-0.315352\pi\)
\(4\) 0 0
\(5\) 45.1664i 0.361331i 0.983545 + 0.180665i \(0.0578251\pi\)
−0.983545 + 0.180665i \(0.942175\pi\)
\(6\) 361.331i 1.67283i
\(7\) 133.000 − 316.165i 0.387755 − 0.921762i
\(8\) 512.000 1.00000
\(9\) −1311.00 −1.79835
\(10\) − 361.331i − 0.361331i
\(11\) 874.000 0.656649 0.328325 − 0.944565i \(-0.393516\pi\)
0.328325 + 0.944565i \(0.393516\pi\)
\(12\) 0 0
\(13\) − 2213.15i − 1.00735i −0.863893 − 0.503676i \(-0.831981\pi\)
0.863893 − 0.503676i \(-0.168019\pi\)
\(14\) −1064.00 + 2529.32i −0.387755 + 0.921762i
\(15\) 2040.00 0.604444
\(16\) −4096.00 −1.00000
\(17\) 5961.96i 1.21351i 0.794890 + 0.606753i \(0.207528\pi\)
−0.794890 + 0.606753i \(0.792472\pi\)
\(18\) 10488.0 1.79835
\(19\) − 3116.48i − 0.454363i −0.973852 − 0.227182i \(-0.927049\pi\)
0.973852 − 0.227182i \(-0.0729511\pi\)
\(20\) 0 0
\(21\) −14280.0 − 6007.13i −1.54195 − 0.648648i
\(22\) −6992.00 −0.656649
\(23\) 4738.00 0.389414 0.194707 − 0.980861i \(-0.437624\pi\)
0.194707 + 0.980861i \(0.437624\pi\)
\(24\) − 23125.2i − 1.67283i
\(25\) 13585.0 0.869440
\(26\) 17705.2i 1.00735i
\(27\) 26286.8i 1.33551i
\(28\) 0 0
\(29\) 11146.0 0.457009 0.228505 − 0.973543i \(-0.426616\pi\)
0.228505 + 0.973543i \(0.426616\pi\)
\(30\) −16320.0 −0.604444
\(31\) 27461.1i 0.921793i 0.887454 + 0.460897i \(0.152472\pi\)
−0.887454 + 0.460897i \(0.847528\pi\)
\(32\) 0 0
\(33\) − 39475.4i − 1.09846i
\(34\) − 47695.7i − 1.21351i
\(35\) 14280.0 + 6007.13i 0.333061 + 0.140108i
\(36\) 0 0
\(37\) 3002.00 0.0592660 0.0296330 − 0.999561i \(-0.490566\pi\)
0.0296330 + 0.999561i \(0.490566\pi\)
\(38\) 24931.8i 0.454363i
\(39\) −99960.0 −1.68513
\(40\) 23125.2i 0.361331i
\(41\) − 57541.9i − 0.834897i −0.908701 − 0.417449i \(-0.862924\pi\)
0.908701 − 0.417449i \(-0.137076\pi\)
\(42\) 114240. + 48057.0i 1.54195 + 0.648648i
\(43\) 31418.0 0.395160 0.197580 − 0.980287i \(-0.436692\pi\)
0.197580 + 0.980287i \(0.436692\pi\)
\(44\) 0 0
\(45\) − 59213.1i − 0.649801i
\(46\) −37904.0 −0.389414
\(47\) 72446.8i 0.697792i 0.937161 + 0.348896i \(0.113443\pi\)
−0.937161 + 0.348896i \(0.886557\pi\)
\(48\) 185001.i 1.67283i
\(49\) −82271.0 − 84099.8i −0.699292 − 0.714836i
\(50\) −108680. −0.869440
\(51\) 269280. 2.02999
\(52\) 0 0
\(53\) −76406.0 −0.513216 −0.256608 − 0.966516i \(-0.582605\pi\)
−0.256608 + 0.966516i \(0.582605\pi\)
\(54\) − 210295.i − 1.33551i
\(55\) 39475.4i 0.237268i
\(56\) 68096.0 − 161876.i 0.387755 − 0.921762i
\(57\) −140760. −0.760072
\(58\) −89168.0 −0.457009
\(59\) − 113232.i − 0.551332i −0.961253 − 0.275666i \(-0.911102\pi\)
0.961253 − 0.275666i \(-0.0888984\pi\)
\(60\) 0 0
\(61\) 275108.i 1.21203i 0.795452 + 0.606016i \(0.207233\pi\)
−0.795452 + 0.606016i \(0.792767\pi\)
\(62\) − 219689.i − 0.921793i
\(63\) −174363. + 414492.i −0.697321 + 1.65766i
\(64\) 262144. 1.00000
\(65\) 99960.0 0.363987
\(66\) 315803.i 1.09846i
\(67\) 495242. 1.64662 0.823309 − 0.567593i \(-0.192125\pi\)
0.823309 + 0.567593i \(0.192125\pi\)
\(68\) 0 0
\(69\) − 213998.i − 0.651423i
\(70\) −114240. − 48057.0i −0.333061 − 0.140108i
\(71\) −184406. −0.515229 −0.257614 − 0.966248i \(-0.582936\pi\)
−0.257614 + 0.966248i \(0.582936\pi\)
\(72\) −671232. −1.79835
\(73\) 60974.6i 0.156740i 0.996924 + 0.0783701i \(0.0249716\pi\)
−0.996924 + 0.0783701i \(0.975028\pi\)
\(74\) −24016.0 −0.0592660
\(75\) − 613585.i − 1.45442i
\(76\) 0 0
\(77\) 116242. − 276328.i 0.254619 − 0.605275i
\(78\) 799680. 1.68513
\(79\) −534934. −1.08497 −0.542486 − 0.840065i \(-0.682517\pi\)
−0.542486 + 0.840065i \(0.682517\pi\)
\(80\) − 185001.i − 0.361331i
\(81\) 231561. 0.435723
\(82\) 460336.i 0.834897i
\(83\) 714848.i 1.25020i 0.780545 + 0.625100i \(0.214942\pi\)
−0.780545 + 0.625100i \(0.785058\pi\)
\(84\) 0 0
\(85\) −269280. −0.438478
\(86\) −251344. −0.395160
\(87\) − 503424.i − 0.764498i
\(88\) 447488. 0.656649
\(89\) 629529.i 0.892988i 0.894787 + 0.446494i \(0.147328\pi\)
−0.894787 + 0.446494i \(0.852672\pi\)
\(90\) 473705.i 0.649801i
\(91\) −699720. − 294349.i −0.928539 − 0.390606i
\(92\) 0 0
\(93\) 1.24032e6 1.54200
\(94\) − 579575.i − 0.697792i
\(95\) 140760. 0.164176
\(96\) 0 0
\(97\) − 814440.i − 0.892368i −0.894941 − 0.446184i \(-0.852783\pi\)
0.894941 − 0.446184i \(-0.147217\pi\)
\(98\) 658168. + 672798.i 0.699292 + 0.714836i
\(99\) −1.14581e6 −1.18089
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 7.7.b.b.6.1 ✓ 2
3.2 odd 2 63.7.d.d.55.1 2
4.3 odd 2 112.7.c.b.97.2 2
5.2 odd 4 175.7.c.c.174.1 4
5.3 odd 4 175.7.c.c.174.4 4
5.4 even 2 175.7.d.e.76.2 2
7.2 even 3 49.7.d.d.31.2 4
7.3 odd 6 49.7.d.d.19.2 4
7.4 even 3 49.7.d.d.19.1 4
7.5 odd 6 49.7.d.d.31.1 4
7.6 odd 2 inner 7.7.b.b.6.2 yes 2
8.3 odd 2 448.7.c.c.321.1 2
8.5 even 2 448.7.c.d.321.2 2
21.20 even 2 63.7.d.d.55.2 2
28.27 even 2 112.7.c.b.97.1 2
35.13 even 4 175.7.c.c.174.3 4
35.27 even 4 175.7.c.c.174.2 4
35.34 odd 2 175.7.d.e.76.1 2
56.13 odd 2 448.7.c.d.321.1 2
56.27 even 2 448.7.c.c.321.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.b.b.6.1 ✓ 2 1.1 even 1 trivial
7.7.b.b.6.2 yes 2 7.6 odd 2 inner
49.7.d.d.19.1 4 7.4 even 3
49.7.d.d.19.2 4 7.3 odd 6
49.7.d.d.31.1 4 7.5 odd 6
49.7.d.d.31.2 4 7.2 even 3
63.7.d.d.55.1 2 3.2 odd 2
63.7.d.d.55.2 2 21.20 even 2
112.7.c.b.97.1 2 28.27 even 2
112.7.c.b.97.2 2 4.3 odd 2
175.7.c.c.174.1 4 5.2 odd 4
175.7.c.c.174.2 4 35.27 even 4
175.7.c.c.174.3 4 35.13 even 4
175.7.c.c.174.4 4 5.3 odd 4
175.7.d.e.76.1 2 35.34 odd 2
175.7.d.e.76.2 2 5.4 even 2
448.7.c.c.321.1 2 8.3 odd 2
448.7.c.c.321.2 2 56.27 even 2
448.7.c.d.321.1 2 56.13 odd 2
448.7.c.d.321.2 2 8.5 even 2