↑

Properties

Label 17.4.d.a.2.3
Level $17$
Weight $4$
Character 17.2
Analytic conductor $1.003$
Analytic rank $0$
Dimension $12$
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] = [17,4,Mod(2,17)] 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("17.2"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(17, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([7])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.00303247010\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{8})\)
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: \( x^{12} + 54x^{10} + 1085x^{8} + 9836x^{6} + 38276x^{4} + 49664x^{2} + 16384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 2.3
Root \(4.15292i\) of defining polynomial
Character \(\chi\) \(=\) 17.2
Dual form 17.4.d.a.9.3

$q$-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+(2.22945 + 2.22945i) q^{2} +(-1.83980 + 0.762069i) q^{3} +1.94089i q^{4} +(-1.91633 - 4.62643i) q^{5} +(-5.80073 - 2.40274i) q^{6} +(1.06584 - 2.57316i) q^{7} +(13.5085 - 13.5085i) q^{8} +(-16.2878 + 16.2878i) q^{9} +(6.04203 - 14.5867i) q^{10} +(-25.1714 - 10.4263i) q^{11} +(-1.47909 - 3.57084i) q^{12} +59.7352i q^{13} +(8.11295 - 3.36049i) q^{14} +(7.05131 + 7.05131i) q^{15} +75.7601 q^{16} +(70.0883 + 0.790881i) q^{17} -72.6256 q^{18} +(23.5187 + 23.5187i) q^{19} +(8.97939 - 3.71938i) q^{20} +5.54633i q^{21} +(-32.8734 - 79.3634i) q^{22} +(-194.831 - 80.7017i) q^{23} +(-14.5585 + 35.1473i) q^{24} +(70.6568 - 70.6568i) q^{25} +(-133.177 + 133.177i) q^{26} +(38.1297 - 92.0531i) q^{27} +(4.99421 + 2.06867i) q^{28} +(7.67362 + 18.5258i) q^{29} +31.4411i q^{30} +(123.485 - 51.1492i) q^{31} +(60.8354 + 60.8354i) q^{32} +54.2559 q^{33} +(154.495 + 158.022i) q^{34} -13.9470 q^{35} +(-31.6128 - 31.6128i) q^{36} +(141.143 - 58.4634i) q^{37} +104.868i q^{38} +(-45.5223 - 109.901i) q^{39} +(-88.3827 - 36.6093i) q^{40} +(-100.202 + 241.908i) q^{41} +(-12.3653 + 12.3653i) q^{42} +(-224.025 + 224.025i) q^{43} +(20.2364 - 48.8550i) q^{44} +(106.567 + 44.1415i) q^{45} +(-254.446 - 614.286i) q^{46} -329.443i q^{47} +(-139.383 + 57.7344i) q^{48} +(237.052 + 237.052i) q^{49} +315.052 q^{50} +(-129.551 + 51.9571i) q^{51} -115.939 q^{52} +(-219.585 - 219.585i) q^{53} +(290.236 - 120.220i) q^{54} +136.434i q^{55} +(-20.3616 - 49.1573i) q^{56} +(-61.1926 - 25.3468i) q^{57} +(-24.1943 + 58.4102i) q^{58} +(38.7062 - 38.7062i) q^{59} +(-13.6858 + 13.6858i) q^{60} +(-313.322 + 756.427i) q^{61} +(389.338 + 161.269i) q^{62} +(24.5509 + 59.2711i) q^{63} -334.822i q^{64} +(276.360 - 114.472i) q^{65} +(120.961 + 120.961i) q^{66} -731.181 q^{67} +(-1.53501 + 136.034i) q^{68} +419.950 q^{69} +(-31.0942 - 31.0942i) q^{70} +(581.286 - 240.777i) q^{71} +440.046i q^{72} +(-189.995 - 458.689i) q^{73} +(445.012 + 184.330i) q^{74} +(-76.1489 + 183.840i) q^{75} +(-45.6473 + 45.6473i) q^{76} +(-53.6572 + 53.6572i) q^{77} +(143.528 - 346.507i) q^{78} +(83.1733 + 34.4515i) q^{79} +(-145.181 - 350.498i) q^{80} -423.512i q^{81} +(-762.717 + 315.928i) q^{82} +(-257.404 - 257.404i) q^{83} -10.7648 q^{84} +(-130.653 - 325.774i) q^{85} -998.907 q^{86} +(-28.2358 - 28.2358i) q^{87} +(-480.872 + 199.184i) q^{88} -192.079i q^{89} +(139.174 + 335.997i) q^{90} +(153.708 + 63.6679i) q^{91} +(156.633 - 378.146i) q^{92} +(-188.208 + 188.208i) q^{93} +(734.476 - 734.476i) q^{94} +(63.7380 - 153.877i) q^{95} +(-158.286 - 65.5640i) q^{96} +(516.698 + 1247.42i) q^{97} +1056.99i q^{98} +(579.808 - 240.165i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} - 4 q^{3} - 20 q^{5} + 20 q^{6} - 4 q^{7} + 28 q^{8} - 64 q^{9} - 116 q^{10} + 40 q^{11} + 56 q^{12} - 132 q^{14} + 244 q^{15} + 184 q^{16} + 52 q^{17} - 12 q^{19} + 572 q^{20} - 620 q^{22}+ \cdots + 3280 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.22945 + 2.22945i 0.788229 + 0.788229i 0.981204 − 0.192974i \(-0.0618134\pi\)
−0.192974 + 0.981204i \(0.561813\pi\)
\(3\) −1.83980 + 0.762069i −0.354069 + 0.146660i −0.552626 − 0.833429i \(-0.686374\pi\)
0.198557 + 0.980089i \(0.436374\pi\)
\(4\) 1.94089i 0.242611i
\(5\) −1.91633 − 4.62643i −0.171402 − 0.413800i 0.814713 − 0.579864i \(-0.196894\pi\)
−0.986115 + 0.166064i \(0.946894\pi\)
\(6\) −5.80073 − 2.40274i −0.394690 − 0.163486i
\(7\) 1.06584 − 2.57316i 0.0575498 − 0.138937i −0.892489 − 0.451069i \(-0.851043\pi\)
0.950039 + 0.312131i \(0.101043\pi\)
\(8\) 13.5085 − 13.5085i 0.596996 − 0.596996i
\(9\) −16.2878 + 16.2878i −0.603251 + 0.603251i
\(10\) 6.04203 − 14.5867i 0.191066 − 0.461273i
\(11\) −25.1714 − 10.4263i −0.689952 − 0.285787i 0.0100284 − 0.999950i \(-0.496808\pi\)
−0.699980 + 0.714162i \(0.746808\pi\)
\(12\) −1.47909 − 3.57084i −0.0355814 − 0.0859011i
\(13\) 59.7352i 1.27443i 0.770687 + 0.637214i \(0.219913\pi\)
−0.770687 + 0.637214i \(0.780087\pi\)
\(14\) 8.11295 − 3.36049i 0.154877 − 0.0641521i
\(15\) 7.05131 + 7.05131i 0.121376 + 0.121376i
\(16\) 75.7601 1.18375
\(17\) 70.0883 + 0.790881i 0.999936 + 0.0112833i
\(18\) −72.6256 −0.951000
\(19\) 23.5187 + 23.5187i 0.283977 + 0.283977i 0.834693 − 0.550716i \(-0.185645\pi\)
−0.550716 + 0.834693i \(0.685645\pi\)
\(20\) 8.97939 − 3.71938i 0.100393 − 0.0415840i
\(21\) 5.54633i 0.0576337i
\(22\) −32.8734 − 79.3634i −0.318574 − 0.769106i
\(23\) −194.831 − 80.7017i −1.76631 − 0.731629i −0.995522 − 0.0945353i \(-0.969863\pi\)
−0.770787 − 0.637093i \(-0.780137\pi\)
\(24\) −14.5585 + 35.1473i −0.123822 + 0.298933i
\(25\) 70.6568 − 70.6568i 0.565255 − 0.565255i
\(26\) −133.177 + 133.177i −1.00454 + 1.00454i
\(27\) 38.1297 − 92.0531i 0.271780 − 0.656135i
\(28\) 4.99421 + 2.06867i 0.0337078 + 0.0139622i
\(29\) 7.67362 + 18.5258i 0.0491364 + 0.118626i 0.946542 − 0.322581i \(-0.104551\pi\)
−0.897405 + 0.441207i \(0.854551\pi\)
\(30\) 31.4411i 0.191344i
\(31\) 123.485 − 51.1492i 0.715438 − 0.296344i 0.00488535 − 0.999988i \(-0.498445\pi\)
0.710553 + 0.703644i \(0.248445\pi\)
\(32\) 60.8354 + 60.8354i 0.336071 + 0.336071i
\(33\) 54.2559 0.286204
\(34\) 154.495 + 158.022i 0.779285 + 0.797073i
\(35\) −13.9470 −0.0673565
\(36\) −31.6128 − 31.6128i −0.146355 − 0.146355i
\(37\) 141.143 − 58.4634i 0.627129 − 0.259765i −0.0464037 − 0.998923i \(-0.514776\pi\)
0.673533 + 0.739157i \(0.264776\pi\)
\(38\) 104.868i 0.447678i
\(39\) −45.5223 − 109.901i −0.186908 − 0.451235i
\(40\) −88.3827 − 36.6093i −0.349363 − 0.144711i
\(41\) −100.202 + 241.908i −0.381680 + 0.921456i 0.609962 + 0.792431i \(0.291185\pi\)
−0.991641 + 0.129025i \(0.958815\pi\)
\(42\) −12.3653 + 12.3653i −0.0454286 + 0.0454286i
\(43\) −224.025 + 224.025i −0.794501 + 0.794501i −0.982222 − 0.187721i \(-0.939890\pi\)
0.187721 + 0.982222i \(0.439890\pi\)
\(44\) 20.2364 − 48.8550i 0.0693352 − 0.167390i
\(45\) 106.567 + 44.1415i 0.353024 + 0.146227i
\(46\) −254.446 − 614.286i −0.815565 − 1.96895i
\(47\) − 329.443i − 1.02243i −0.859453 − 0.511215i \(-0.829196\pi\)
0.859453 − 0.511215i \(-0.170804\pi\)
\(48\) −139.383 + 57.7344i −0.419130 + 0.173609i
\(49\) 237.052 + 237.052i 0.691115 + 0.691115i
\(50\) 315.052 0.891101
\(51\) −129.551 + 51.9571i −0.355701 + 0.142656i
\(52\) −115.939 −0.309190
\(53\) −219.585 − 219.585i −0.569100 − 0.569100i 0.362776 − 0.931876i \(-0.381829\pi\)
−0.931876 + 0.362776i \(0.881829\pi\)
\(54\) 290.236 − 120.220i 0.731409 − 0.302960i
\(55\) 136.434i 0.334487i
\(56\) −20.3616 − 49.1573i −0.0485881 − 0.117302i
\(57\) −61.1926 − 25.3468i −0.142196 − 0.0588994i
\(58\) −24.1943 + 58.4102i −0.0547735 + 0.132235i
\(59\) 38.7062 − 38.7062i 0.0854087 − 0.0854087i −0.663112 − 0.748520i \(-0.730765\pi\)
0.748520 + 0.663112i \(0.230765\pi\)
\(60\) −13.6858 + 13.6858i −0.0294472 + 0.0294472i
\(61\) −313.322 + 756.427i −0.657653 + 1.58771i 0.143767 + 0.989612i \(0.454079\pi\)
−0.801419 + 0.598103i \(0.795921\pi\)
\(62\) 389.338 + 161.269i 0.797517 + 0.330342i
\(63\) 24.5509 + 59.2711i 0.0490972 + 0.118531i
\(64\) − 334.822i − 0.653948i
\(65\) 276.360 − 114.472i 0.527358 − 0.218439i
\(66\) 120.961 + 120.961i 0.225595 + 0.225595i
\(67\) −731.181 −1.33325 −0.666627 − 0.745392i \(-0.732262\pi\)
−0.666627 + 0.745392i \(0.732262\pi\)
\(68\) −1.53501 + 136.034i −0.00273747 + 0.242596i
\(69\) 419.950 0.732696
\(70\) −31.0942 − 31.0942i −0.0530923 − 0.0530923i
\(71\) 581.286 − 240.777i 0.971633 − 0.402464i 0.160313 − 0.987066i \(-0.448750\pi\)
0.811320 + 0.584603i \(0.198750\pi\)
\(72\) 440.046i 0.720277i
\(73\) −189.995 − 458.689i −0.304620 − 0.735417i −0.999862 − 0.0166414i \(-0.994703\pi\)
0.695242 − 0.718776i \(-0.255297\pi\)
\(74\) 445.012 + 184.330i 0.699076 + 0.289567i
\(75\) −76.1489 + 183.840i −0.117239 + 0.283040i
\(76\) −45.6473 + 45.6473i −0.0688960 + 0.0688960i
\(77\) −53.6572 + 53.6572i −0.0794131 + 0.0794131i
\(78\) 143.528 − 346.507i 0.208351 − 0.503003i
\(79\) 83.1733 + 34.4515i 0.118452 + 0.0490646i 0.441122 − 0.897447i \(-0.354580\pi\)
−0.322670 + 0.946511i \(0.604580\pi\)
\(80\) −145.181 − 350.498i −0.202897 − 0.489836i
\(81\) − 423.512i − 0.580950i
\(82\) −762.717 + 315.928i −1.02717 + 0.425468i
\(83\) −257.404 − 257.404i −0.340407 − 0.340407i 0.516113 − 0.856520i \(-0.327378\pi\)
−0.856520 + 0.516113i \(0.827378\pi\)
\(84\) −10.7648 −0.0139826
\(85\) −130.653 − 325.774i −0.166722 − 0.415708i
\(86\) −998.907 −1.25250
\(87\) −28.2358 − 28.2358i −0.0347954 − 0.0347954i
\(88\) −480.872 + 199.184i −0.582512 + 0.241285i
\(89\) − 192.079i − 0.228767i −0.993437 − 0.114384i \(-0.963511\pi\)
0.993437 − 0.114384i \(-0.0364893\pi\)
\(90\) 139.174 + 335.997i 0.163003 + 0.393524i
\(91\) 153.708 + 63.6679i 0.177066 + 0.0733430i
\(92\) 156.633 − 378.146i 0.177501 − 0.428526i
\(93\) −188.208 + 188.208i −0.209853 + 0.209853i
\(94\) 734.476 − 734.476i 0.805909 − 0.805909i
\(95\) 63.7380 − 153.877i 0.0688356 − 0.166184i
\(96\) −158.286 − 65.5640i −0.168281 − 0.0697042i
\(97\) 516.698 + 1247.42i 0.540853 + 1.30574i 0.924122 + 0.382098i \(0.124798\pi\)
−0.383268 + 0.923637i \(0.625202\pi\)
\(98\) 1056.99i 1.08951i
\(99\) 579.808 − 240.165i 0.588616 − 0.243813i
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 17.4.d.a.2.3 ✓ 12
3.2 odd 2 153.4.l.a.19.1 12
17.3 odd 16 289.4.a.g.1.11 12
17.5 odd 16 289.4.b.e.288.1 12
17.9 even 8 inner 17.4.d.a.9.3 yes 12
17.12 odd 16 289.4.b.e.288.2 12
17.14 odd 16 289.4.a.g.1.12 12
51.26 odd 8 153.4.l.a.145.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.d.a.2.3 ✓ 12 1.1 even 1 trivial
17.4.d.a.9.3 yes 12 17.9 even 8 inner
153.4.l.a.19.1 12 3.2 odd 2
153.4.l.a.145.1 12 51.26 odd 8
289.4.a.g.1.11 12 17.3 odd 16
289.4.a.g.1.12 12 17.14 odd 16
289.4.b.e.288.1 12 17.5 odd 16
289.4.b.e.288.2 12 17.12 odd 16