LMFDB - Embedded newform 475.2.b.d.324.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.2
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.d.324.5

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.48119i q^{2} -0.806063i q^{3} -0.193937 q^{4} -1.19394 q^{6} +3.35026i q^{7} -2.67513i q^{8} +2.35026 q^{9} +O(q^{10})$	\(q-1.48119i q^{2} -0.806063i q^{3} -0.193937 q^{4} -1.19394 q^{6} +3.35026i q^{7} -2.67513i q^{8} +2.35026 q^{9} +0.962389 q^{11} +0.156325i q^{12} -6.15633i q^{13} +4.96239 q^{14} -4.35026 q^{16} -6.31265i q^{17} -3.48119i q^{18} +1.00000 q^{19} +2.70052 q^{21} -1.42548i q^{22} +4.96239i q^{23} -2.15633 q^{24} -9.11871 q^{26} -4.31265i q^{27} -0.649738i q^{28} +3.61213 q^{29} -5.92478 q^{31} +1.09332i q^{32} -0.775746i q^{33} -9.35026 q^{34} -0.455802 q^{36} +10.1563i q^{37} -1.48119i q^{38} -4.96239 q^{39} +6.31265 q^{41} -4.00000i q^{42} +4.12601i q^{43} -0.186642 q^{44} +7.35026 q^{46} +3.35026i q^{47} +3.50659i q^{48} -4.22425 q^{49} -5.08840 q^{51} +1.19394i q^{52} -1.84367i q^{53} -6.38787 q^{54} +8.96239 q^{56} -0.806063i q^{57} -5.35026i q^{58} +6.38787 q^{59} -11.2750 q^{61} +8.77575i q^{62} +7.87399i q^{63} -7.08110 q^{64} -1.14903 q^{66} -6.73084i q^{67} +1.22425i q^{68} +4.00000 q^{69} -0.775746 q^{71} -6.28726i q^{72} -0.387873i q^{73} +15.0435 q^{74} -0.193937 q^{76} +3.22425i q^{77} +7.35026i q^{78} +0.836381 q^{79} +3.57452 q^{81} -9.35026i q^{82} +7.03761i q^{83} -0.523730 q^{84} +6.11142 q^{86} -2.91160i q^{87} -2.57452i q^{88} -7.08840 q^{89} +20.6253 q^{91} -0.962389i q^{92} +4.77575i q^{93} +4.96239 q^{94} +0.881286 q^{96} +10.9927i q^{97} +6.25694i q^{98} +2.26187 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9	$ 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} - 16 q^{11} + 8 q^{14} - 6 q^{16} + 6 q^{19} - 24 q^{21} + 8 q^{24} - 12 q^{26} + 20 q^{29} + 8 q^{31} - 36 q^{34} - 22 q^{36} - 8 q^{39} - 4 q^{41} + 24 q^{44} + 24 q^{46} - 22 q^{49} + 8 q^{51} - 40 q^{54} + 32 q^{56} + 40 q^{59} - 4 q^{61} + 22 q^{64} + 40 q^{66} + 24 q^{69} - 8 q^{71} + 4 q^{74} - 2 q^{76} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 4 q^{89} + 40 q^{91} + 8 q^{94} + 48 q^{96} + 32 q^{99}+O(q^{100}) $ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 - 16 * q^11 + 8 * q^14 - 6 * q^16 + 6 * q^19 - 24 * q^21 + 8 * q^24 - 12 * q^26 + 20 * q^29 + 8 * q^31 - 36 * q^34 - 22 * q^36 - 8 * q^39 - 4 * q^41 + 24 * q^44 + 24 * q^46 - 22 * q^49 + 8 * q^51 - 40 * q^54 + 32 * q^56 + 40 * q^59 - 4 * q^61 + 22 * q^64 + 40 * q^66 + 24 * q^69 - 8 * q^71 + 4 * q^74 - 2 * q^76 - 2 * q^81 - 40 * q^84 - 32 * q^86 - 4 * q^89 + 40 * q^91 + 8 * q^94 + 48 * q^96 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.48119i	− 1.04736i	−0.851914	−	0.523681i	$-0.824558\pi$
$2$	− 1.48119i	− 1.04736i	0.851914	−	0.523681i	$-0.175442\pi$
$3$	− 0.806063i	− 0.465381i	−0.972551	−	0.232690i	$-0.925247\pi$
$3$	− 0.806063i	− 0.465381i	0.972551	−	0.232690i	$-0.0747529\pi$
$4$	−0.193937	−0.0969683
$5$	0	0
$5$	0	0
$6$	−1.19394	−0.487423
$7$	3.35026i	1.26628i	0.774037	+	0.633140i	$0.218234\pi$
$7$	3.35026i	1.26628i	−0.774037	+	0.633140i	$0.781766\pi$
$8$	− 2.67513i	− 0.945802i
$9$	2.35026	0.783421
$10$	0	0
$11$	0.962389	0.290171	0.145086	−	0.989419i	$-0.453654\pi$
$11$	0.962389	0.290171	0.145086	+	0.989419i	$0.453654\pi$
$12$	0.156325i	0.0451272i
$13$	− 6.15633i	− 1.70746i	−0.520718	−	0.853729i	$-0.674336\pi$
$13$	− 6.15633i	− 1.70746i	0.520718	−	0.853729i	$-0.325664\pi$
$14$	4.96239	1.32625
$15$	0	0
$16$	−4.35026	−1.08757
$17$	− 6.31265i	− 1.53104i	−0.643411	−	0.765521i	$-0.722481\pi$
$17$	− 6.31265i	− 1.53104i	0.643411	−	0.765521i	$-0.277519\pi$
$18$	− 3.48119i	− 0.820525i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.70052	0.589303
$22$	− 1.42548i	− 0.303914i
$23$	4.96239i	1.03473i	0.855765	+	0.517365i	$0.173087\pi$
$23$	4.96239i	1.03473i	−0.855765	+	0.517365i	$0.826913\pi$
$24$	−2.15633	−0.440158
$25$	0	0
$26$	−9.11871	−1.78833
$27$	− 4.31265i	− 0.829970i
$28$	− 0.649738i	− 0.122789i
$29$	3.61213	0.670755	0.335378	−	0.942084i	$-0.391136\pi$
$29$	3.61213	0.670755	0.335378	+	0.942084i	$0.391136\pi$
$30$	0	0
$31$	−5.92478	−1.06412	−0.532061	−	0.846706i	$-0.678582\pi$
$31$	−5.92478	−1.06412	−0.532061	+	0.846706i	$0.678582\pi$
$32$	1.09332i	0.193274i
$33$	− 0.775746i	− 0.135040i
$34$	−9.35026	−1.60356
$35$	0	0
$36$	−0.455802	−0.0759669
$37$	10.1563i	1.66969i	0.550485	+	0.834845i	$0.314443\pi$
$37$	10.1563i	1.66969i	−0.550485	+	0.834845i	$0.685557\pi$
$38$	− 1.48119i	− 0.240281i
$39$	−4.96239	−0.794618
$40$	0	0
$41$	6.31265	0.985870	0.492935	−	0.870066i	$-0.335924\pi$
$41$	6.31265	0.985870	0.492935	+	0.870066i	$0.335924\pi$
$42$	− 4.00000i	− 0.617213i
$43$	4.12601i	0.629210i	0.949223	+	0.314605i	$0.101872\pi$
$43$	4.12601i	0.629210i	−0.949223	+	0.314605i	$0.898128\pi$
$44$	−0.186642	−0.0281374
$45$	0	0
$46$	7.35026	1.08374
$47$	3.35026i	0.488686i	0.969689	+	0.244343i	$0.0785723\pi$
$47$	3.35026i	0.488686i	−0.969689	+	0.244343i	$0.921428\pi$
$48$	3.50659i	0.506132i
$49$	−4.22425	−0.603465
$50$	0	0
$51$	−5.08840	−0.712518
$52$	1.19394i	0.165569i
$53$	− 1.84367i	− 0.253248i	−0.991951	−	0.126624i	$-0.959586\pi$
$53$	− 1.84367i	− 0.253248i	0.991951	−	0.126624i	$-0.0404142\pi$
$54$	−6.38787	−0.869279
$55$	0	0
$56$	8.96239	1.19765
$57$	− 0.806063i	− 0.106766i
$58$	− 5.35026i	− 0.702524i
$59$	6.38787	0.831630	0.415815	−	0.909449i	$-0.363496\pi$
$59$	6.38787	0.831630	0.415815	+	0.909449i	$0.363496\pi$
$60$	0	0
$61$	−11.2750	−1.44362	−0.721810	−	0.692091i	$-0.756690\pi$
$61$	−11.2750	−1.44362	−0.721810	+	0.692091i	$0.756690\pi$
$62$	8.77575i	1.11452i
$63$	7.87399i	0.992030i
$64$	−7.08110	−0.885138
$65$	0	0
$66$	−1.14903	−0.141436
$67$	− 6.73084i	− 0.822303i	−0.911567	−	0.411152i	$-0.865127\pi$
$67$	− 6.73084i	− 0.822303i	0.911567	−	0.411152i	$-0.134873\pi$
$68$	1.22425i	0.148463i
$69$	4.00000	0.481543
$70$	0	0
$71$	−0.775746	−0.0920641	−0.0460321	−	0.998940i	$-0.514658\pi$
$71$	−0.775746	−0.0920641	−0.0460321	+	0.998940i	$0.514658\pi$
$72$	− 6.28726i	− 0.740960i
$73$	− 0.387873i	− 0.0453971i	−0.999742	−	0.0226986i	$-0.992774\pi$
$73$	− 0.387873i	− 0.0453971i	0.999742	−	0.0226986i	$-0.00722580\pi$
$74$	15.0435	1.74877
$75$	0	0
$76$	−0.193937	−0.0222460
$77$	3.22425i	0.367438i
$78$	7.35026i	0.832253i
$79$	0.836381	0.0941002	0.0470501	−	0.998893i	$-0.485018\pi$
$79$	0.836381	0.0941002	0.0470501	+	0.998893i	$0.485018\pi$
$80$	0	0
$81$	3.57452	0.397168
$82$	− 9.35026i	− 1.03256i
$83$	7.03761i	0.772478i	0.922399	+	0.386239i	$0.126226\pi$
$83$	7.03761i	0.772478i	−0.922399	+	0.386239i	$0.873774\pi$
$84$	−0.523730	−0.0571437
$85$	0	0
$86$	6.11142	0.659011
$87$	− 2.91160i	− 0.312157i
$88$	− 2.57452i	− 0.274444i
$89$	−7.08840	−0.751369	−0.375684	−	0.926748i	$-0.622592\pi$
$89$	−7.08840	−0.751369	−0.375684	+	0.926748i	$0.622592\pi$
$90$	0	0
$91$	20.6253	2.16212
$92$	− 0.962389i	− 0.100336i
$93$	4.77575i	0.495222i
$94$	4.96239	0.511831
$95$	0	0
$96$	0.881286	0.0899459
$97$	10.9927i	1.11614i	0.829794	+	0.558070i	$0.188458\pi$
$97$	10.9927i	1.11614i	−0.829794	+	0.558070i	$0.811542\pi$
$98$	6.25694i	0.632046i
$99$	2.26187	0.227326
$100$	0	0
$101$	−2.64974	−0.263659	−0.131829	−	0.991272i	$-0.542085\pi$
$101$	−2.64974	−0.263659	−0.131829	+	0.991272i	$0.542085\pi$
$102$	7.53690i	0.746265i
$103$	10.7308i	1.05734i	0.848827	+	0.528671i	$0.177309\pi$
$103$	10.7308i	1.05734i	−0.848827	+	0.528671i	$0.822691\pi$
$104$	−16.4690	−1.61492
$105$	0	0
$106$	−2.73084	−0.265243
$107$	4.80606i	0.464620i	0.972642	+	0.232310i	$0.0746283\pi$
$107$	4.80606i	0.464620i	−0.972642	+	0.232310i	$0.925372\pi$
$108$	0.836381i	0.0804808i
$109$	2.77575	0.265868	0.132934	−	0.991125i	$-0.457560\pi$
$109$	2.77575	0.265868	0.132934	+	0.991125i	$0.457560\pi$
$110$	0	0
$111$	8.18664	0.777042
$112$	− 14.5745i	− 1.37716i
$113$	− 6.99271i	− 0.657818i	−0.944362	−	0.328909i	$-0.893319\pi$
$113$	− 6.99271i	− 0.657818i	0.944362	−	0.328909i	$-0.106681\pi$
$114$	−1.19394	−0.111822
$115$	0	0
$116$	−0.700523	−0.0650420
$117$	− 14.4690i	− 1.33766i
$118$	− 9.46168i	− 0.871018i
$119$	21.1490	1.93873
$120$	0	0
$121$	−10.0738	−0.915801
$122$	16.7005i	1.51199i
$123$	− 5.08840i	− 0.458805i
$124$	1.14903	0.103186
$125$	0	0
$126$	11.6629	1.03901
$127$	13.4314i	1.19184i	0.803043	+	0.595920i	$0.203213\pi$
$127$	13.4314i	1.19184i	−0.803043	+	0.595920i	$0.796787\pi$
$128$	12.6751i	1.12033i
$129$	3.32582	0.292822
$130$	0	0
$131$	20.6253	1.80204	0.901020	−	0.433777i	$-0.142819\pi$
$131$	20.6253	1.80204	0.901020	+	0.433777i	$0.142819\pi$
$132$	0.150446i	0.0130946i
$133$	3.35026i	0.290505i
$134$	−9.96968	−0.861249
$135$	0	0
$136$	−16.8872	−1.44806
$137$	20.2374i	1.72900i	0.502633	+	0.864500i	$0.332365\pi$
$137$	20.2374i	1.72900i	−0.502633	+	0.864500i	$0.667635\pi$
$138$	− 5.92478i	− 0.504351i
$139$	17.5877	1.49177	0.745884	−	0.666076i	$-0.232027\pi$
$139$	17.5877	1.49177	0.745884	+	0.666076i	$0.232027\pi$
$140$	0	0
$141$	2.70052	0.227425
$142$	1.14903i	0.0964245i
$143$	− 5.92478i	− 0.495455i
$144$	−10.2243	−0.852021
$145$	0	0
$146$	−0.574515	−0.0475472
$147$	3.40502i	0.280841i
$148$	− 1.96968i	− 0.161907i
$149$	7.42548	0.608319	0.304160	−	0.952621i	$-0.401624\pi$
$149$	7.42548	0.608319	0.304160	+	0.952621i	$0.401624\pi$
$150$	0	0
$151$	−1.61213	−0.131193	−0.0655965	−	0.997846i	$-0.520895\pi$
$151$	−1.61213	−0.131193	−0.0655965	+	0.997846i	$0.520895\pi$
$152$	− 2.67513i	− 0.216982i
$153$	− 14.8364i	− 1.19945i
$154$	4.77575	0.384841
$155$	0	0
$156$	0.962389	0.0770528
$157$	4.38787i	0.350190i	0.984552	+	0.175095i	$0.0560233\pi$
$157$	4.38787i	0.350190i	−0.984552	+	0.175095i	$0.943977\pi$
$158$	− 1.23884i	− 0.0985570i
$159$	−1.48612	−0.117857
$160$	0	0
$161$	−16.6253	−1.31026
$162$	− 5.29455i	− 0.415979i
$163$	0.649738i	0.0508914i	0.999676	+	0.0254457i	$0.00810050\pi$
$163$	0.649738i	0.0508914i	−0.999676	+	0.0254457i	$0.991900\pi$
$164$	−1.22425	−0.0955982
$165$	0	0
$166$	10.4241	0.809065
$167$	− 15.3561i	− 1.18829i	−0.804357	−	0.594147i	$-0.797490\pi$
$167$	− 15.3561i	− 1.18829i	0.804357	−	0.594147i	$-0.202510\pi$
$168$	− 7.22425i	− 0.557363i
$169$	−24.9003	−1.91541
$170$	0	0
$171$	2.35026	0.179729
$172$	− 0.800184i	− 0.0610134i
$173$	− 3.24472i	− 0.246692i	−0.992364	−	0.123346i	$-0.960638\pi$
$173$	− 3.24472i	− 0.246692i	0.992364	−	0.123346i	$-0.0393624\pi$
$174$	−4.31265	−0.326941
$175$	0	0
$176$	−4.18664	−0.315580
$177$	− 5.14903i	− 0.387025i
$178$	10.4993i	0.786955i
$179$	−15.0132	−1.12214	−0.561069	−	0.827769i	$-0.689610\pi$
$179$	−15.0132	−1.12214	−0.561069	+	0.827769i	$0.689610\pi$
$180$	0	0
$181$	9.22425	0.685633	0.342817	−	0.939402i	$-0.388619\pi$
$181$	9.22425	0.685633	0.342817	+	0.939402i	$0.388619\pi$
$182$	− 30.5501i	− 2.26452i
$183$	9.08840i	0.671834i
$184$	13.2750	0.978649
$185$	0	0
$186$	7.07381	0.518677
$187$	− 6.07522i	− 0.444264i
$188$	− 0.649738i	− 0.0473870i
$189$	14.4485	1.05097
$190$	0	0
$191$	−21.7743	−1.57554	−0.787768	−	0.615972i	$-0.788763\pi$
$191$	−21.7743	−1.57554	−0.787768	+	0.615972i	$0.788763\pi$
$192$	5.70782i	0.411926i
$193$	− 12.5442i	− 0.902951i	−0.892283	−	0.451476i	$-0.850898\pi$
$193$	− 12.5442i	− 0.902951i	0.892283	−	0.451476i	$-0.149102\pi$
$194$	16.2823	1.16900
$195$	0	0
$196$	0.819237	0.0585169
$197$	24.5501i	1.74912i	0.484917	+	0.874560i	$0.338850\pi$
$197$	24.5501i	1.74912i	−0.484917	+	0.874560i	$0.661150\pi$
$198$	− 3.35026i	− 0.238093i
$199$	−23.0738	−1.63566	−0.817829	−	0.575461i	$-0.804823\pi$
$199$	−23.0738	−1.63566	−0.817829	+	0.575461i	$0.804823\pi$
$200$	0	0
$201$	−5.42548	−0.382684
$202$	3.92478i	0.276146i
$203$	12.1016i	0.849364i
$204$	0.986826	0.0690917
$205$	0	0
$206$	15.8945	1.10742
$207$	11.6629i	0.810628i
$208$	26.7816i	1.85697i
$209$	0.962389	0.0665698
$210$	0	0
$211$	−20.9380	−1.44143	−0.720714	−	0.693233i	$-0.756186\pi$
$211$	−20.9380	−1.44143	−0.720714	+	0.693233i	$0.756186\pi$
$212$	0.357556i	0.0245570i
$213$	0.625301i	0.0428449i
$214$	7.11871	0.486625
$215$	0	0
$216$	−11.5369	−0.784987
$217$	− 19.8496i	− 1.34748i
$218$	− 4.11142i	− 0.278460i
$219$	−0.312650	−0.0211270
$220$	0	0
$221$	−38.8627	−2.61419
$222$	− 12.1260i	− 0.813844i
$223$	0.0303172i	0.00203019i	0.999999	+	0.00101509i	$0.000323114\pi$
$223$	0.0303172i	0.00203019i	−0.999999	+	0.00101509i	$0.999677\pi$
$224$	−3.66291	−0.244739
$225$	0	0
$226$	−10.3576	−0.688974
$227$	4.80606i	0.318990i	0.987199	+	0.159495i	$0.0509865\pi$
$227$	4.80606i	0.318990i	−0.987199	+	0.159495i	$0.949013\pi$
$228$	0.156325i	0.0103529i
$229$	−1.87399	−0.123837	−0.0619184	−	0.998081i	$-0.519722\pi$
$229$	−1.87399	−0.123837	−0.0619184	+	0.998081i	$0.519722\pi$
$230$	0	0
$231$	2.59895	0.170999
$232$	− 9.66291i	− 0.634401i
$233$	11.1490i	0.730397i	0.930930	+	0.365199i	$0.118999\pi$
$233$	11.1490i	0.730397i	−0.930930	+	0.365199i	$0.881001\pi$
$234$	−21.4314	−1.40101
$235$	0	0
$236$	−1.23884	−0.0806418
$237$	− 0.674176i	− 0.0437924i
$238$	− 31.3258i	− 2.03055i
$239$	−9.29948	−0.601533	−0.300767	−	0.953698i	$-0.597243\pi$
$239$	−9.29948	−0.601533	−0.300767	+	0.953698i	$0.597243\pi$
$240$	0	0
$241$	−2.31265	−0.148971	−0.0744855	−	0.997222i	$-0.523731\pi$
$241$	−2.31265	−0.148971	−0.0744855	+	0.997222i	$0.523731\pi$
$242$	14.9213i	0.959175i
$243$	− 15.8192i	− 1.01480i
$244$	2.18664	0.139985
$245$	0	0
$246$	−7.53690	−0.480535
$247$	− 6.15633i	− 0.391718i
$248$	15.8496i	1.00645i
$249$	5.67276	0.359497
$250$	0	0
$251$	24.1016	1.52128	0.760639	−	0.649175i	$-0.224886\pi$
$251$	24.1016	1.52128	0.760639	+	0.649175i	$0.224886\pi$
$252$	− 1.52705i	− 0.0961954i
$253$	4.77575i	0.300249i
$254$	19.8945	1.24829
$255$	0	0
$256$	4.61213	0.288258
$257$	− 13.3199i	− 0.830875i	−0.909622	−	0.415438i	$-0.863628\pi$
$257$	− 13.3199i	− 0.830875i	0.909622	−	0.415438i	$-0.136372\pi$
$258$	− 4.92619i	− 0.306691i
$259$	−34.0263	−2.11429
$260$	0	0
$261$	8.48944	0.525483
$262$	− 30.5501i	− 1.88739i
$263$	− 12.9624i	− 0.799295i	−0.916669	−	0.399648i	$-0.869133\pi$
$263$	− 12.9624i	− 0.799295i	0.916669	−	0.399648i	$-0.130867\pi$
$264$	−2.07522	−0.127721
$265$	0	0
$266$	4.96239	0.304264
$267$	5.71370i	0.349673i
$268$	1.30536i	0.0797373i
$269$	11.4010	0.695134	0.347567	−	0.937655i	$-0.387008\pi$
$269$	11.4010	0.695134	0.347567	+	0.937655i	$0.387008\pi$
$270$	0	0
$271$	16.8119	1.02125	0.510626	−	0.859803i	$-0.329414\pi$
$271$	16.8119	1.02125	0.510626	+	0.859803i	$0.329414\pi$
$272$	27.4617i	1.66511i
$273$	− 16.6253i	− 1.00621i
$274$	29.9756	1.81089
$275$	0	0
$276$	−0.775746	−0.0466944
$277$	− 29.7889i	− 1.78984i	−0.446224	−	0.894921i	$-0.647231\pi$
$277$	− 29.7889i	− 1.78984i	0.446224	−	0.894921i	$-0.352769\pi$
$278$	− 26.0508i	− 1.56242i
$279$	−13.9248	−0.833655
$280$	0	0
$281$	−11.6121	−0.692721	−0.346361	−	0.938101i	$-0.612583\pi$
$281$	−11.6121	−0.692721	−0.346361	+	0.938101i	$0.612583\pi$
$282$	− 4.00000i	− 0.238197i
$283$	− 2.26187i	− 0.134454i	−0.997738	−	0.0672270i	$-0.978585\pi$
$283$	− 2.26187i	− 0.134454i	0.997738	−	0.0672270i	$-0.0214152\pi$
$284$	0.150446	0.00892730
$285$	0	0
$286$	−8.77575	−0.518921
$287$	21.1490i	1.24839i
$288$	2.56959i	0.151415i
$289$	−22.8496	−1.34409
$290$	0	0
$291$	8.86082	0.519430
$292$	0.0752228i	0.00440208i
$293$	− 1.84367i	− 0.107709i	−0.998549	−	0.0538543i	$-0.982849\pi$
$293$	− 1.84367i	− 0.107709i	0.998549	−	0.0538543i	$-0.0171507\pi$
$294$	5.04349	0.294142
$295$	0	0
$296$	27.1695	1.57920
$297$	− 4.15045i	− 0.240833i
$298$	− 10.9986i	− 0.637131i
$299$	30.5501	1.76676
$300$	0	0
$301$	−13.8232	−0.796756
$302$	2.38787i	0.137407i
$303$	2.13586i	0.122702i
$304$	−4.35026	−0.249505
$305$	0	0
$306$	−21.9756	−1.25626
$307$	26.2071i	1.49572i	0.663857	+	0.747859i	$0.268918\pi$
$307$	26.2071i	1.49572i	−0.663857	+	0.747859i	$0.731082\pi$
$308$	− 0.625301i	− 0.0356298i
$309$	8.64974	0.492066
$310$	0	0
$311$	6.51388	0.369368	0.184684	−	0.982798i	$-0.440874\pi$
$311$	6.51388	0.369368	0.184684	+	0.982798i	$0.440874\pi$
$312$	13.2750i	0.751551i
$313$	− 16.0752i	− 0.908625i	−0.890842	−	0.454313i	$-0.849885\pi$
$313$	− 16.0752i	− 0.908625i	0.890842	−	0.454313i	$-0.150115\pi$
$314$	6.49929	0.366776
$315$	0	0
$316$	−0.162205	−0.00912473
$317$	− 5.69323i	− 0.319764i	−0.987136	−	0.159882i	$-0.948889\pi$
$317$	− 5.69323i	− 0.319764i	0.987136	−	0.159882i	$-0.0511113\pi$
$318$	2.20123i	0.123439i
$319$	3.47627	0.194634
$320$	0	0
$321$	3.87399	0.216225
$322$	24.6253i	1.37231i
$323$	− 6.31265i	− 0.351245i
$324$	−0.693229	−0.0385127
$325$	0	0
$326$	0.962389	0.0533018
$327$	− 2.23743i	− 0.123730i
$328$	− 16.8872i	− 0.932438i
$329$	−11.2243	−0.618813
$330$	0	0
$331$	−12.3127	−0.676764	−0.338382	−	0.941009i	$-0.609880\pi$
$331$	−12.3127	−0.676764	−0.338382	+	0.941009i	$0.609880\pi$
$332$	− 1.36485i	− 0.0749059i
$333$	23.8700i	1.30807i
$334$	−22.7454	−1.24457
$335$	0	0
$336$	−11.7480	−0.640905
$337$	3.76845i	0.205281i	0.994719	+	0.102640i	$0.0327291\pi$
$337$	3.76845i	0.205281i	−0.994719	+	0.102640i	$0.967271\pi$
$338$	36.8822i	2.00613i
$339$	−5.63656	−0.306136
$340$	0	0
$341$	−5.70194	−0.308777
$342$	− 3.48119i	− 0.188241i
$343$	9.29948i	0.502125i
$344$	11.0376	0.595108
$345$	0	0
$346$	−4.80606	−0.258376
$347$	− 22.3634i	− 1.20053i	−0.799800	−	0.600266i	$-0.795061\pi$
$347$	− 22.3634i	− 1.20053i	0.799800	−	0.600266i	$-0.204939\pi$
$348$	0.564666i	0.0302693i
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−26.5501	−1.41714
$352$	1.05220i	0.0560824i
$353$	− 5.53690i	− 0.294700i	−0.989084	−	0.147350i	$-0.952926\pi$
$353$	− 5.53690i	− 0.294700i	0.989084	−	0.147350i	$-0.0470743\pi$
$354$	−7.62672	−0.405355
$355$	0	0
$356$	1.37470	0.0728589
$357$	− 17.0475i	− 0.902247i
$358$	22.2374i	1.17528i
$359$	10.3634	0.546961	0.273481	−	0.961878i	$-0.411825\pi$
$359$	10.3634	0.546961	0.273481	+	0.961878i	$0.411825\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 13.6629i	− 0.718107i
$363$	8.12013i	0.426196i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	13.4617	0.703653
$367$	3.35026i	0.174882i	0.996170	+	0.0874411i	$0.0278690\pi$
$367$	3.35026i	0.174882i	−0.996170	+	0.0874411i	$0.972131\pi$
$368$	− 21.5877i	− 1.12534i
$369$	14.8364	0.772351
$370$	0	0
$371$	6.17679	0.320683
$372$	− 0.926192i	− 0.0480208i
$373$	12.6048i	0.652653i	0.945257	+	0.326327i	$0.105811\pi$
$373$	12.6048i	0.652653i	−0.945257	+	0.326327i	$0.894189\pi$
$374$	−8.99859	−0.465306
$375$	0	0
$376$	8.96239	0.462200
$377$	− 22.2374i	− 1.14529i
$378$	− 21.4010i	− 1.10075i
$379$	37.2506	1.91343	0.956717	−	0.291018i	$-0.0939941\pi$
$379$	37.2506	1.91343	0.956717	+	0.291018i	$0.0939941\pi$
$380$	0	0
$381$	10.8265	0.554660
$382$	32.2520i	1.65016i
$383$	− 30.8324i	− 1.57546i	−0.616019	−	0.787731i	$-0.711256\pi$
$383$	− 30.8324i	− 1.57546i	0.616019	−	0.787731i	$-0.288744\pi$
$384$	10.2170	0.521382
$385$	0	0
$386$	−18.5804	−0.945717
$387$	9.69720i	0.492936i
$388$	− 2.13189i	− 0.108230i
$389$	1.37470	0.0697000	0.0348500	−	0.999393i	$-0.488905\pi$
$389$	1.37470	0.0697000	0.0348500	+	0.999393i	$0.488905\pi$
$390$	0	0
$391$	31.3258	1.58422
$392$	11.3004i	0.570758i
$393$	− 16.6253i	− 0.838635i
$394$	36.3634	1.83196
$395$	0	0
$396$	−0.438658	−0.0220434
$397$	− 9.38646i	− 0.471093i	−0.971863	−	0.235546i	$-0.924312\pi$
$397$	− 9.38646i	− 0.471093i	0.971863	−	0.235546i	$-0.0756880\pi$
$398$	34.1768i	1.71313i
$399$	2.70052	0.135195
$400$	0	0
$401$	14.1016	0.704199	0.352099	−	0.935963i	$-0.385468\pi$
$401$	14.1016	0.704199	0.352099	+	0.935963i	$0.385468\pi$
$402$	8.03620i	0.400809i
$403$	36.4749i	1.81694i
$404$	0.513881	0.0255665
$405$	0	0
$406$	17.9248	0.889592
$407$	9.77433i	0.484496i
$408$	13.6121i	0.673901i
$409$	−35.1490	−1.73801	−0.869004	−	0.494805i	$-0.835239\pi$
$409$	−35.1490	−1.73801	−0.869004	+	0.494805i	$0.835239\pi$
$410$	0	0
$411$	16.3127	0.804644
$412$	− 2.08110i	− 0.102529i
$413$	21.4010i	1.05308i
$414$	17.2750	0.849022
$415$	0	0
$416$	6.73084	0.330007
$417$	− 14.1768i	− 0.694241i
$418$	− 1.42548i	− 0.0697227i
$419$	−9.02776	−0.441035	−0.220518	−	0.975383i	$-0.570775\pi$
$419$	−9.02776	−0.441035	−0.220518	+	0.975383i	$0.570775\pi$
$420$	0	0
$421$	15.2097	0.741274	0.370637	−	0.928778i	$-0.379139\pi$
$421$	15.2097	0.741274	0.370637	+	0.928778i	$0.379139\pi$
$422$	31.0132i	1.50970i
$423$	7.87399i	0.382847i
$424$	−4.93207	−0.239523
$425$	0	0
$426$	0.926192	0.0448741
$427$	− 37.7743i	− 1.82803i
$428$	− 0.932071i	− 0.0450534i
$429$	−4.77575	−0.230575
$430$	0	0
$431$	16.3127	0.785753	0.392876	−	0.919591i	$-0.371480\pi$
$431$	16.3127	0.785753	0.392876	+	0.919591i	$0.371480\pi$
$432$	18.7612i	0.902647i
$433$	− 11.1432i	− 0.535506i	−0.963488	−	0.267753i	$-0.913719\pi$
$433$	− 11.1432i	− 0.535506i	0.963488	−	0.267753i	$-0.0862811\pi$
$434$	−29.4010	−1.41130
$435$	0	0
$436$	−0.538319	−0.0257808
$437$	4.96239i	0.237383i
$438$	0.463096i	0.0221276i
$439$	−27.3865	−1.30708	−0.653542	−	0.756890i	$-0.726718\pi$
$439$	−27.3865	−1.30708	−0.653542	+	0.756890i	$0.726718\pi$
$440$	0	0
$441$	−9.92810	−0.472767
$442$	57.5633i	2.73800i
$443$	− 19.5125i	− 0.927065i	−0.886080	−	0.463533i	$-0.846582\pi$
$443$	− 19.5125i	− 0.927065i	0.886080	−	0.463533i	$-0.153418\pi$
$444$	−1.58769	−0.0753484
$445$	0	0
$446$	0.0449056	0.00212634
$447$	− 5.98541i	− 0.283100i
$448$	− 23.7235i	− 1.12083i
$449$	22.1016	1.04304	0.521519	−	0.853240i	$-0.325366\pi$
$449$	22.1016	1.04304	0.521519	+	0.853240i	$0.325366\pi$
$450$	0	0
$451$	6.07522	0.286071
$452$	1.35614i	0.0637875i
$453$	1.29948i	0.0610547i
$454$	7.11871	0.334098
$455$	0	0
$456$	−2.15633	−0.100979
$457$	− 17.8496i	− 0.834967i	−0.908685	−	0.417483i	$-0.862912\pi$
$457$	− 17.8496i	− 0.834967i	0.908685	−	0.417483i	$-0.137088\pi$
$458$	2.77575i	0.129702i
$459$	−27.2243	−1.27072
$460$	0	0
$461$	−35.2506	−1.64178	−0.820892	−	0.571083i	$-0.806523\pi$
$461$	−35.2506	−1.64178	−0.820892	+	0.571083i	$0.806523\pi$
$462$	− 3.84955i	− 0.179097i
$463$	26.3634i	1.22521i	0.790388	+	0.612606i	$0.209879\pi$
$463$	26.3634i	1.22521i	−0.790388	+	0.612606i	$0.790121\pi$
$464$	−15.7137	−0.729490
$465$	0	0
$466$	16.5139	0.764991
$467$	− 6.78560i	− 0.314000i	−0.987599	−	0.157000i	$-0.949818\pi$
$467$	− 6.78560i	− 0.314000i	0.987599	−	0.157000i	$-0.0501823\pi$
$468$	2.80606i	0.129710i
$469$	22.5501	1.04127
$470$	0	0
$471$	3.53690	0.162972
$472$	− 17.0884i	− 0.786557i
$473$	3.97082i	0.182579i
$474$	−0.998585	−0.0458665
$475$	0	0
$476$	−4.10157	−0.187995
$477$	− 4.33312i	− 0.198400i
$478$	13.7743i	0.630023i
$479$	−12.7104	−0.580752	−0.290376	−	0.956913i	$-0.593780\pi$
$479$	−12.7104	−0.580752	−0.290376	+	0.956913i	$0.593780\pi$
$480$	0	0
$481$	62.5256	2.85092
$482$	3.42548i	0.156027i
$483$	13.4010i	0.609769i
$484$	1.95368	0.0888036
$485$	0	0
$486$	−23.4314	−1.06287
$487$	− 15.7586i	− 0.714090i	−0.934087	−	0.357045i	$-0.883784\pi$
$487$	− 15.7586i	− 0.714090i	0.934087	−	0.357045i	$-0.116216\pi$
$488$	30.1622i	1.36538i
$489$	0.523730	0.0236839
$490$	0	0
$491$	−14.5501	−0.656636	−0.328318	−	0.944567i	$-0.606482\pi$
$491$	−14.5501	−0.656636	−0.328318	+	0.944567i	$0.606482\pi$
$492$	0.986826i	0.0444896i
$493$	− 22.8021i	− 1.02695i
$494$	−9.11871	−0.410270
$495$	0	0
$496$	25.7743	1.15730
$497$	− 2.59895i	− 0.116579i
$498$	− 8.40246i	− 0.376523i
$499$	5.48612	0.245592	0.122796	−	0.992432i	$-0.460814\pi$
$499$	5.48612	0.245592	0.122796	+	0.992432i	$0.460814\pi$
$500$	0	0
$501$	−12.3780	−0.553009
$502$	− 35.6991i	− 1.59333i
$503$	36.6615i	1.63466i	0.576173	+	0.817328i	$0.304545\pi$
$503$	36.6615i	1.63466i	−0.576173	+	0.817328i	$0.695455\pi$
$504$	21.0640	0.938263
$505$	0	0
$506$	7.07381	0.314469
$507$	20.0713i	0.891396i
$508$	− 2.60483i	− 0.115571i
$509$	−39.1900	−1.73706	−0.868532	−	0.495632i	$-0.834936\pi$
$509$	−39.1900	−1.73706	−0.868532	+	0.495632i	$0.834936\pi$
$510$	0	0
$511$	1.29948	0.0574855
$512$	18.5188i	0.818423i
$513$	− 4.31265i	− 0.190408i
$514$	−19.7294	−0.870228
$515$	0	0
$516$	−0.644999	−0.0283945
$517$	3.22425i	0.141803i
$518$	50.3996i	2.21443i
$519$	−2.61545	−0.114806
$520$	0	0
$521$	−17.7283	−0.776690	−0.388345	−	0.921514i	$-0.626953\pi$
$521$	−17.7283	−0.776690	−0.388345	+	0.921514i	$0.626953\pi$
$522$	− 12.5745i	− 0.550372i
$523$	40.7572i	1.78219i	0.453819	+	0.891094i	$0.350061\pi$
$523$	40.7572i	1.78219i	−0.453819	+	0.891094i	$0.649939\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−19.1998	−0.837152
$527$	37.4010i	1.62922i
$528$	3.37470i	0.146865i
$529$	−1.62530	−0.0706652
$530$	0	0
$531$	15.0132	0.651516
$532$	− 0.649738i	− 0.0281697i
$533$	− 38.8627i	− 1.68333i
$534$	8.46310	0.366234
$535$	0	0
$536$	−18.0059	−0.777736
$537$	12.1016i	0.522221i
$538$	− 16.8872i	− 0.728057i
$539$	−4.06537	−0.175108
$540$	0	0
$541$	23.9003	1.02756	0.513778	−	0.857923i	$-0.328246\pi$
$541$	23.9003	1.02756	0.513778	+	0.857923i	$0.328246\pi$
$542$	− 24.9018i	− 1.06962i
$543$	− 7.43533i	− 0.319081i
$544$	6.90175	0.295910
$545$	0	0
$546$	−24.6253	−1.05387
$547$	8.55405i	0.365745i	0.983137	+	0.182872i	$0.0585395\pi$
$547$	8.55405i	0.365745i	−0.983137	+	0.182872i	$0.941461\pi$
$548$	− 3.92478i	− 0.167658i
$549$	−26.4993	−1.13096
$550$	0	0
$551$	3.61213	0.153882
$552$	− 10.7005i	− 0.455445i
$553$	2.80209i	0.119157i
$554$	−44.1232	−1.87461
$555$	0	0
$556$	−3.41090	−0.144654
$557$	− 4.23743i	− 0.179546i	−0.995962	−	0.0897728i	$-0.971386\pi$
$557$	− 4.23743i	− 0.179546i	0.995962	−	0.0897728i	$-0.0286141\pi$
$558$	20.6253i	0.873139i
$559$	25.4010	1.07435
$560$	0	0
$561$	−4.89701	−0.206752
$562$	17.1998i	0.725530i
$563$	− 16.4934i	− 0.695114i	−0.937659	−	0.347557i	$-0.887011\pi$
$563$	− 16.4934i	− 0.695114i	0.937659	−	0.347557i	$-0.112989\pi$
$564$	−0.523730	−0.0220530
$565$	0	0
$566$	−3.35026	−0.140822
$567$	11.9756i	0.502926i
$568$	2.07522i	0.0870744i
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−26.2619	−1.09902	−0.549512	−	0.835486i	$-0.685186\pi$
$571$	−26.2619	−1.09902	−0.549512	+	0.835486i	$0.685186\pi$
$572$	1.14903i	0.0480434i
$573$	17.5515i	0.733224i
$574$	31.3258	1.30751
$575$	0	0
$576$	−16.6424	−0.693435
$577$	30.1016i	1.25314i	0.779363	+	0.626572i	$0.215543\pi$
$577$	30.1016i	1.25314i	−0.779363	+	0.626572i	$0.784457\pi$
$578$	33.8446i	1.40775i
$579$	−10.1114	−0.420216
$580$	0	0
$581$	−23.5778	−0.978174
$582$	− 13.1246i	− 0.544032i
$583$	− 1.77433i	− 0.0734853i
$584$	−1.03761	−0.0429367
$585$	0	0
$586$	−2.73084	−0.112810
$587$	− 35.1392i	− 1.45035i	−0.688565	−	0.725175i	$-0.741759\pi$
$587$	− 35.1392i	− 1.45035i	0.688565	−	0.725175i	$-0.258241\pi$
$588$	− 0.660357i	− 0.0272327i
$589$	−5.92478	−0.244126
$590$	0	0
$591$	19.7889	0.814007
$592$	− 44.1827i	− 1.81590i
$593$	− 34.3244i	− 1.40953i	−0.709439	−	0.704767i	$-0.751051\pi$
$593$	− 34.3244i	− 1.40953i	0.709439	−	0.704767i	$-0.248949\pi$
$594$	−6.14762	−0.252240
$595$	0	0
$596$	−1.44007	−0.0589877
$597$	18.5990i	0.761204i
$598$	− 45.2506i	− 1.85043i
$599$	−14.6107	−0.596978	−0.298489	−	0.954413i	$-0.596483\pi$
$599$	−14.6107	−0.596978	−0.298489	+	0.954413i	$0.596483\pi$
$600$	0	0
$601$	23.5633	0.961165	0.480583	−	0.876949i	$-0.340425\pi$
$601$	23.5633	0.961165	0.480583	+	0.876949i	$0.340425\pi$
$602$	20.4749i	0.834493i
$603$	− 15.8192i	− 0.644209i
$604$	0.312650	0.0127216
$605$	0	0
$606$	3.16362	0.128513
$607$	− 8.80606i	− 0.357427i	−0.983901	−	0.178714i	$-0.942806\pi$
$607$	− 8.80606i	− 0.357427i	0.983901	−	0.178714i	$-0.0571935\pi$
$608$	1.09332i	0.0443400i
$609$	9.75463	0.395278
$610$	0	0
$611$	20.6253	0.834410
$612$	2.87732i	0.116309i
$613$	− 10.4142i	− 0.420626i	−0.977634	−	0.210313i	$-0.932552\pi$
$613$	− 10.4142i	− 0.420626i	0.977634	−	0.210313i	$-0.0674484\pi$
$614$	38.8178	1.56656
$615$	0	0
$616$	8.62530	0.347523
$617$	− 17.2849i	− 0.695863i	−0.937520	−	0.347932i	$-0.886884\pi$
$617$	− 17.2849i	− 0.695863i	0.937520	−	0.347932i	$-0.113116\pi$
$618$	− 12.8119i	− 0.515372i
$619$	10.6351	0.427463	0.213731	−	0.976892i	$-0.431438\pi$
$619$	10.6351	0.427463	0.213731	+	0.976892i	$0.431438\pi$
$620$	0	0
$621$	21.4010	0.858794
$622$	− 9.64832i	− 0.386863i
$623$	− 23.7480i	− 0.951443i
$624$	21.5877	0.864199
$625$	0	0
$626$	−23.8105	−0.951660
$627$	− 0.775746i	− 0.0309803i
$628$	− 0.850969i	− 0.0339574i
$629$	64.1133	2.55637
$630$	0	0
$631$	−16.5599	−0.659240	−0.329620	−	0.944114i	$-0.606921\pi$
$631$	−16.5599	−0.659240	−0.329620	+	0.944114i	$0.606921\pi$
$632$	− 2.23743i	− 0.0890001i
$633$	16.8773i	0.670813i
$634$	−8.43278	−0.334908
$635$	0	0
$636$	0.288213	0.0114284
$637$	26.0059i	1.03039i
$638$	− 5.14903i	− 0.203852i
$639$	−1.82321	−0.0721249
$640$	0	0
$641$	−16.7612	−0.662026	−0.331013	−	0.943626i	$-0.607390\pi$
$641$	−16.7612	−0.662026	−0.331013	+	0.943626i	$0.607390\pi$
$642$	− 5.73813i	− 0.226466i
$643$	− 5.73813i	− 0.226290i	−0.993578	−	0.113145i	$-0.963908\pi$
$643$	− 5.73813i	− 0.226290i	0.993578	−	0.113145i	$-0.0360925\pi$
$644$	3.22425	0.127053
$645$	0	0
$646$	−9.35026	−0.367881
$647$	− 37.2144i	− 1.46305i	−0.681815	−	0.731525i	$-0.738809\pi$
$647$	− 37.2144i	− 1.46305i	0.681815	−	0.731525i	$-0.261191\pi$
$648$	− 9.56230i	− 0.375642i
$649$	6.14762	0.241315
$650$	0	0
$651$	−16.0000	−0.627089
$652$	− 0.126008i	− 0.00493485i
$653$	− 11.7626i	− 0.460305i	−0.973155	−	0.230153i	$-0.926077\pi$
$653$	− 11.7626i	− 0.460305i	0.973155	−	0.230153i	$-0.0739225\pi$
$654$	−3.31406	−0.129590
$655$	0	0
$656$	−27.4617	−1.07220
$657$	− 0.911603i	− 0.0355650i
$658$	16.6253i	0.648122i
$659$	−1.23884	−0.0482584	−0.0241292	−	0.999709i	$-0.507681\pi$
$659$	−1.23884	−0.0482584	−0.0241292	+	0.999709i	$0.507681\pi$
$660$	0	0
$661$	−9.53690	−0.370943	−0.185471	−	0.982650i	$-0.559381\pi$
$661$	−9.53690	−0.370943	−0.185471	+	0.982650i	$0.559381\pi$
$662$	18.2374i	0.708818i
$663$	31.3258i	1.21659i
$664$	18.8265	0.730611
$665$	0	0
$666$	35.3561	1.37002
$667$	17.9248i	0.694050i
$668$	2.97812i	0.115227i
$669$	0.0244376	0.000944811 0
$670$	0	0
$671$	−10.8510	−0.418897
$672$	2.95254i	0.113897i
$673$	− 39.9307i	− 1.53921i	−0.638518	−	0.769607i	$-0.720452\pi$
$673$	− 39.9307i	− 1.53921i	0.638518	−	0.769607i	$-0.279548\pi$
$674$	5.58181	0.215003
$675$	0	0
$676$	4.82909	0.185734
$677$	− 3.05334i	− 0.117349i	−0.998277	−	0.0586747i	$-0.981313\pi$
$677$	− 3.05334i	− 0.117349i	0.998277	−	0.0586747i	$-0.0186875\pi$
$678$	8.34885i	0.320636i
$679$	−36.8284	−1.41335
$680$	0	0
$681$	3.87399	0.148452
$682$	8.44568i	0.323402i
$683$	− 29.2692i	− 1.11995i	−0.828508	−	0.559977i	$-0.810810\pi$
$683$	− 29.2692i	− 1.11995i	0.828508	−	0.559977i	$-0.189190\pi$
$684$	−0.455802	−0.0174280
$685$	0	0
$686$	13.7743	0.525906
$687$	1.51056i	0.0576313i
$688$	− 17.9492i	− 0.684307i
$689$	−11.3503	−0.432411
$690$	0	0
$691$	2.63515	0.100246	0.0501229	−	0.998743i	$-0.484039\pi$
$691$	2.63515	0.100246	0.0501229	+	0.998743i	$0.484039\pi$
$692$	0.629270i	0.0239213i
$693$	7.57784i	0.287858i
$694$	−33.1246	−1.25739
$695$	0	0
$696$	−7.78892	−0.295238
$697$	− 39.8496i	− 1.50941i
$698$	− 14.8119i	− 0.560640i
$699$	8.98683	0.339913
$700$	0	0
$701$	−25.0494	−0.946102	−0.473051	−	0.881035i	$-0.656847\pi$
$701$	−25.0494	−0.946102	−0.473051	+	0.881035i	$0.656847\pi$
$702$	39.3258i	1.48426i
$703$	10.1563i	0.383053i
$704$	−6.81477	−0.256841
$705$	0	0
$706$	−8.20123	−0.308657
$707$	− 8.87732i	− 0.333866i
$708$	0.998585i	0.0375291i
$709$	41.6991	1.56604	0.783021	−	0.621995i	$-0.213677\pi$
$709$	41.6991	1.56604	0.783021	+	0.621995i	$0.213677\pi$
$710$	0	0
$711$	1.96571	0.0737200
$712$	18.9624i	0.710646i
$713$	− 29.4010i	− 1.10108i
$714$	−25.2506	−0.944980
$715$	0	0
$716$	2.91160	0.108812
$717$	7.49597i	0.279942i
$718$	− 15.3503i	− 0.572867i
$719$	−30.6351	−1.14250	−0.571249	−	0.820777i	$-0.693541\pi$
$719$	−30.6351	−1.14250	−0.571249	+	0.820777i	$0.693541\pi$
$720$	0	0
$721$	−35.9511	−1.33889
$722$	− 1.48119i	− 0.0551243i
$723$	1.86414i	0.0693282i
$724$	−1.78892	−0.0664847
$725$	0	0
$726$	12.0275	0.446382
$727$	7.50071i	0.278186i	0.990279	+	0.139093i	$0.0444187\pi$
$727$	7.50071i	0.278186i	−0.990279	+	0.139093i	$0.955581\pi$
$728$	− 55.1754i	− 2.04494i
$729$	−2.02776	−0.0751023
$730$	0	0
$731$	26.0460	0.963348
$732$	− 1.76257i	− 0.0651466i
$733$	− 9.84955i	− 0.363802i	−0.983317	−	0.181901i	$-0.941775\pi$
$733$	− 9.84955i	− 0.363802i	0.983317	−	0.181901i	$-0.0582250\pi$
$734$	4.96239	0.183165
$735$	0	0
$736$	−5.42548	−0.199986
$737$	− 6.47768i	− 0.238609i
$738$	− 21.9756i	− 0.808932i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−4.96239	−0.182298
$742$	− 9.14903i	− 0.335871i
$743$	15.7177i	0.576625i	0.957536	+	0.288313i	$0.0930942\pi$
$743$	15.7177i	0.576625i	−0.957536	+	0.288313i	$0.906906\pi$
$744$	12.7757	0.468382
$745$	0	0
$746$	18.6702	0.683565
$747$	16.5402i	0.605175i
$748$	1.17821i	0.0430795i
$749$	−16.1016	−0.588339
$750$	0	0
$751$	−43.7400	−1.59610	−0.798048	−	0.602593i	$-0.794134\pi$
$751$	−43.7400	−1.59610	−0.798048	+	0.602593i	$0.794134\pi$
$752$	− 14.5745i	− 0.531478i
$753$	− 19.4274i	− 0.707974i
$754$	−32.9380	−1.19953
$755$	0	0
$756$	−2.80209	−0.101911
$757$	− 15.7743i	− 0.573328i	−0.958031	−	0.286664i	$-0.907454\pi$
$757$	− 15.7743i	− 0.573328i	0.958031	−	0.286664i	$-0.0925463\pi$
$758$	− 55.1754i	− 2.00406i
$759$	3.84955	0.139730
$760$	0	0
$761$	43.2262	1.56695	0.783474	−	0.621425i	$-0.213446\pi$
$761$	43.2262	1.56695	0.783474	+	0.621425i	$0.213446\pi$
$762$	− 16.0362i	− 0.580930i
$763$	9.29948i	0.336664i
$764$	4.22284	0.152777
$765$	0	0
$766$	−45.6688	−1.65008
$767$	− 39.3258i	− 1.41997i
$768$	− 3.71767i	− 0.134150i
$769$	19.1246	0.689650	0.344825	−	0.938667i	$-0.387938\pi$
$769$	19.1246	0.689650	0.344825	+	0.938667i	$0.387938\pi$
$770$	0	0
$771$	−10.7367	−0.386674
$772$	2.43278i	0.0875576i
$773$	− 25.8846i	− 0.931005i	−0.885047	−	0.465502i	$-0.845874\pi$
$773$	− 25.8846i	− 0.931005i	0.885047	−	0.465502i	$-0.154126\pi$
$774$	14.3634	0.516283
$775$	0	0
$776$	29.4069	1.05565
$777$	27.4274i	0.983952i
$778$	− 2.03620i	− 0.0730012i
$779$	6.31265	0.226174
$780$	0	0
$781$	−0.746569	−0.0267144
$782$	− 46.3996i	− 1.65925i
$783$	− 15.5778i	− 0.556707i
$784$	18.3766	0.656307
$785$	0	0
$786$	−24.6253	−0.878355
$787$	19.9814i	0.712261i	0.934436	+	0.356131i	$0.115904\pi$
$787$	19.9814i	0.712261i	−0.934436	+	0.356131i	$0.884096\pi$
$788$	− 4.76116i	− 0.169609i
$789$	−10.4485	−0.371977
$790$	0	0
$791$	23.4274	0.832982
$792$	− 6.05079i	− 0.215005i
$793$	69.4128i	2.46492i
$794$	−13.9032	−0.493405
$795$	0	0
$796$	4.47486	0.158607
$797$	28.6458i	1.01469i	0.861744	+	0.507343i	$0.169372\pi$
$797$	28.6458i	1.01469i	−0.861744	+	0.507343i	$0.830628\pi$
$798$	− 4.00000i	− 0.141598i
$799$	21.1490	0.748199
$800$	0	0
$801$	−16.6596	−0.588638
$802$	− 20.8872i	− 0.737551i
$803$	− 0.373285i	− 0.0131729i
$804$	1.05220	0.0371082
$805$	0	0
$806$	54.0263	1.90300
$807$	− 9.18997i	− 0.323502i
$808$	7.08840i	0.249369i
$809$	17.2243	0.605573	0.302786	−	0.953058i	$-0.402083\pi$
$809$	17.2243	0.605573	0.302786	+	0.953058i	$0.402083\pi$
$810$	0	0
$811$	−15.6267	−0.548728	−0.274364	−	0.961626i	$-0.588467\pi$
$811$	−15.6267	−0.548728	−0.274364	+	0.961626i	$0.588467\pi$
$812$	− 2.34694i	− 0.0823613i
$813$	− 13.5515i	− 0.475272i
$814$	14.4777	0.507443
$815$	0	0
$816$	22.1359	0.774910
$817$	4.12601i	0.144351i
$818$	52.0625i	1.82032i
$819$	48.4749	1.69385
$820$	0	0
$821$	39.2506	1.36986	0.684928	−	0.728611i	$-0.259834\pi$
$821$	39.2506	1.36986	0.684928	+	0.728611i	$0.259834\pi$
$822$	− 24.1622i	− 0.842754i
$823$	45.5271i	1.58697i	0.608588	+	0.793487i	$0.291736\pi$
$823$	45.5271i	1.58697i	−0.608588	+	0.793487i	$0.708264\pi$
$824$	28.7064	1.00003
$825$	0	0
$826$	31.6991	1.10295
$827$	17.0698i	0.593576i	0.954943	+	0.296788i	$0.0959155\pi$
$827$	17.0698i	0.593576i	−0.954943	+	0.296788i	$0.904084\pi$
$828$	− 2.26187i	− 0.0786052i
$829$	−1.69911	−0.0590125	−0.0295062	−	0.999565i	$-0.509393\pi$
$829$	−1.69911	−0.0590125	−0.0295062	+	0.999565i	$0.509393\pi$
$830$	0	0
$831$	−24.0118	−0.832959
$832$	43.5936i	1.51134i
$833$	26.6662i	0.923930i
$834$	−20.9986	−0.727122
$835$	0	0
$836$	−0.186642	−0.00645516
$837$	25.5515i	0.883189i
$838$	13.3719i	0.461924i
$839$	50.5910	1.74660	0.873298	−	0.487187i	$-0.161977\pi$
$839$	50.5910	1.74660	0.873298	+	0.487187i	$0.161977\pi$
$840$	0	0
$841$	−15.9525	−0.550088
$842$	− 22.5285i	− 0.776382i
$843$	9.36011i	0.322379i
$844$	4.06063	0.139773
$845$	0	0
$846$	11.6629	0.400979
$847$	− 33.7499i	− 1.15966i
$848$	8.02047i	0.275424i
$849$	−1.82321	−0.0625723
$850$	0	0
$851$	−50.3996	−1.72768
$852$	− 0.121269i	− 0.00415460i
$853$	22.5237i	0.771198i	0.922667	+	0.385599i	$0.126005\pi$
$853$	22.5237i	0.771198i	−0.922667	+	0.385599i	$0.873995\pi$
$854$	−55.9511	−1.91461
$855$	0	0
$856$	12.8568	0.439438
$857$	− 23.6180i	− 0.806776i	−0.915029	−	0.403388i	$-0.867833\pi$
$857$	− 23.6180i	− 0.806776i	0.915029	−	0.403388i	$-0.132167\pi$
$858$	7.07381i	0.241496i
$859$	−15.1754	−0.517777	−0.258889	−	0.965907i	$-0.583356\pi$
$859$	−15.1754	−0.517777	−0.258889	+	0.965907i	$0.583356\pi$
$860$	0	0
$861$	17.0475	0.580976
$862$	− 24.1622i	− 0.822968i
$863$	− 30.1055i	− 1.02480i	−0.858746	−	0.512402i	$-0.828756\pi$
$863$	− 30.1055i	− 1.02480i	0.858746	−	0.512402i	$-0.171244\pi$
$864$	4.71511	0.160411
$865$	0	0
$866$	−16.5052	−0.560869
$867$	18.4182i	0.625515i
$868$	3.84955i	0.130662i
$869$	0.804923	0.0273051
$870$	0	0
$871$	−41.4372	−1.40405
$872$	− 7.42548i	− 0.251459i
$873$	25.8357i	0.874407i
$874$	7.35026	0.248626
$875$	0	0
$876$	0.0606343	0.00204864
$877$	− 5.53102i	− 0.186769i	−0.995630	−	0.0933847i	$-0.970231\pi$
$877$	− 5.53102i	− 0.186769i	0.995630	−	0.0933847i	$-0.0297687\pi$
$878$	40.5647i	1.36899i
$879$	−1.48612	−0.0501255
$880$	0	0
$881$	20.8265	0.701664	0.350832	−	0.936438i	$-0.385899\pi$
$881$	20.8265	0.701664	0.350832	+	0.936438i	$0.385899\pi$
$882$	14.7054i	0.495158i
$883$	− 43.1509i	− 1.45214i	−0.687618	−	0.726072i	$-0.741344\pi$
$883$	− 43.1509i	− 1.45214i	0.687618	−	0.726072i	$-0.258656\pi$
$884$	7.53690	0.253494
$885$	0	0
$886$	−28.9018	−0.970973
$887$	− 44.5461i	− 1.49571i	−0.663861	−	0.747856i	$-0.731083\pi$
$887$	− 44.5461i	− 1.49571i	0.663861	−	0.747856i	$-0.268917\pi$
$888$	− 21.9003i	− 0.734927i
$889$	−44.9986	−1.50920
$890$	0	0
$891$	3.44007	0.115247
$892$	− 0.00587961i	0 0.000196864i
$893$	3.35026i	0.112112i
$894$	−8.86556	−0.296509
$895$	0	0
$896$	−42.4650	−1.41866
$897$	− 24.6253i	− 0.822215i
$898$	− 32.7367i	− 1.09244i
$899$	−21.4010	−0.713765
$900$	0	0
$901$	−11.6385	−0.387734
$902$	− 8.99859i	− 0.299620i
$903$	11.1424i	0.370795i
$904$	−18.7064	−0.622166
$905$	0	0
$906$	1.92478	0.0639464
$907$	53.7558i	1.78493i	0.451115	+	0.892466i	$0.351026\pi$
$907$	53.7558i	1.78493i	−0.451115	+	0.892466i	$0.648974\pi$
$908$	− 0.932071i	− 0.0309319i
$909$	−6.22758	−0.206556
$910$	0	0
$911$	−2.28630	−0.0757486	−0.0378743	−	0.999283i	$-0.512059\pi$
$911$	−2.28630	−0.0757486	−0.0378743	+	0.999283i	$0.512059\pi$
$912$	3.50659i	0.116115i
$913$	6.77292i	0.224151i
$914$	−26.4387	−0.874513
$915$	0	0
$916$	0.363436	0.0120082
$917$	69.1002i	2.28189i
$918$	40.3244i	1.33090i
$919$	34.8510	1.14963	0.574814	−	0.818284i	$-0.305075\pi$
$919$	34.8510	1.14963	0.574814	+	0.818284i	$0.305075\pi$
$920$	0	0
$921$	21.1246	0.696079
$922$	52.2130i	1.71954i
$923$	4.77575i	0.157196i
$924$	−0.504032	−0.0165814
$925$	0	0
$926$	39.0494	1.28324
$927$	25.2203i	0.828343i
$928$	3.94921i	0.129639i
$929$	−41.6991	−1.36810	−0.684052	−	0.729434i	$-0.739784\pi$
$929$	−41.6991	−1.36810	−0.684052	+	0.729434i	$0.739784\pi$
$930$	0	0
$931$	−4.22425	−0.138444
$932$	− 2.16220i	− 0.0708254i
$933$	− 5.25060i	− 0.171897i
$934$	−10.0508	−0.328872
$935$	0	0
$936$	−38.7064	−1.26516
$937$	21.9102i	0.715775i	0.933765	+	0.357887i	$0.116503\pi$
$937$	21.9102i	0.715775i	−0.933765	+	0.357887i	$0.883497\pi$
$938$	− 33.4010i	− 1.09058i
$939$	−12.9576	−0.422857
$940$	0	0
$941$	−3.55149	−0.115775	−0.0578877	−	0.998323i	$-0.518437\pi$
$941$	−3.55149	−0.115775	−0.0578877	+	0.998323i	$0.518437\pi$
$942$	− 5.23884i	− 0.170691i
$943$	31.3258i	1.02011i
$944$	−27.7889	−0.904452
$945$	0	0
$946$	5.88156	0.191226
$947$	38.3634i	1.24664i	0.781965	+	0.623322i	$0.214217\pi$
$947$	38.3634i	1.24664i	−0.781965	+	0.623322i	$0.785783\pi$
$948$	0.130747i	0.00424648i
$949$	−2.38787	−0.0775136
$950$	0	0
$951$	−4.58910	−0.148812
$952$	− 56.5764i	− 1.83365i
$953$	22.8714i	0.740879i	0.928857	+	0.370439i	$0.120793\pi$
$953$	22.8714i	0.740879i	−0.928857	+	0.370439i	$0.879207\pi$
$954$	−6.41819	−0.207797
$955$	0	0
$956$	1.80351	0.0583296
$957$	− 2.80209i	− 0.0905788i
$958$	18.8265i	0.608258i
$959$	−67.8007	−2.18940
$960$	0	0
$961$	4.10299	0.132354
$962$	− 92.6126i	− 2.98595i
$963$	11.2955i	0.363993i
$964$	0.448507	0.0144455
$965$	0	0
$966$	19.8496	0.638649
$967$	27.6629i	0.889579i	0.895635	+	0.444790i	$0.146722\pi$
$967$	27.6629i	0.889579i	−0.895635	+	0.444790i	$0.853278\pi$
$968$	26.9488i	0.866166i
$969$	−5.08840	−0.163463
$970$	0	0
$971$	−42.1768	−1.35352	−0.676759	−	0.736205i	$-0.736616\pi$
$971$	−42.1768	−1.35352	−0.676759	+	0.736205i	$0.736616\pi$
$972$	3.06793i	0.0984039i
$973$	58.9234i	1.88900i
$974$	−23.3416	−0.747912
$975$	0	0
$976$	49.0494	1.57003
$977$	0.856849i	0.0274130i	0.999906	+	0.0137065i	$0.00436305\pi$
$977$	0.856849i	0.0274130i	−0.999906	+	0.0137065i	$0.995637\pi$
$978$	− 0.775746i	− 0.0248056i
$979$	−6.82179	−0.218025
$980$	0	0
$981$	6.52373	0.208287
$982$	21.5515i	0.687736i
$983$	− 45.2809i	− 1.44424i	−0.691769	−	0.722119i	$-0.743169\pi$
$983$	− 45.2809i	− 1.44424i	0.691769	−	0.722119i	$-0.256831\pi$
$984$	−13.6121	−0.433939
$985$	0	0
$986$	−33.7743	−1.07559
$987$	9.04746i	0.287984i
$988$	1.19394i	0.0379842i
$989$	−20.4749	−0.651063
$990$	0	0
$991$	47.0132	1.49342	0.746711	−	0.665148i	$-0.231632\pi$
$991$	47.0132	1.49342	0.746711	+	0.665148i	$0.231632\pi$
$992$	− 6.47768i	− 0.205667i
$993$	9.92478i	0.314953i
$994$	−3.84955	−0.122100
$995$	0	0
$996$	−1.10016	−0.0348598
$997$	13.6873i	0.433483i	0.976229	+	0.216741i	$0.0695428\pi$
$997$	13.6873i	0.433483i	−0.976229	+	0.216741i	$0.930457\pi$
$998$	− 8.12601i	− 0.257224i
$999$	43.8007	1.38579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.d.324.2		6
5.2	odd	4		475.2.a.f.1.3		3
5.3	odd	4		95.2.a.a.1.1	✓	3
5.4	even	2	inner	475.2.b.d.324.5		6
15.2	even	4		4275.2.a.bk.1.1		3
15.8	even	4		855.2.a.i.1.3		3
20.3	even	4		1520.2.a.p.1.2		3
20.7	even	4		7600.2.a.bx.1.2		3
35.13	even	4		4655.2.a.u.1.1		3
40.3	even	4		6080.2.a.by.1.2		3
40.13	odd	4		6080.2.a.bo.1.2		3
95.18	even	4		1805.2.a.f.1.3		3
95.37	even	4		9025.2.a.bb.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.a.1.1	✓	3	5.3	odd	4
475.2.a.f.1.3		3	5.2	odd	4
475.2.b.d.324.2		6	1.1	even	1	trivial
475.2.b.d.324.5		6	5.4	even	2	inner
855.2.a.i.1.3		3	15.8	even	4
1520.2.a.p.1.2		3	20.3	even	4
1805.2.a.f.1.3		3	95.18	even	4
4275.2.a.bk.1.1		3	15.2	even	4
4655.2.a.u.1.1		3	35.13	even	4
6080.2.a.bo.1.2		3	40.13	odd	4
6080.2.a.by.1.2		3	40.3	even	4
7600.2.a.bx.1.2		3	20.7	even	4
9025.2.a.bb.1.1		3	95.37	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.48119i q^{2} -0.806063i q^{3} -0.193937 q^{4} -1.19394 q^{6} +3.35026i q^{7} -2.67513i q^{8} +2.35026 q^{9} +O(q^{10})\)	\(q-1.48119i q^{2} -0.806063i q^{3} -0.193937 q^{4} -1.19394 q^{6} +3.35026i q^{7} -2.67513i q^{8} +2.35026 q^{9} +0.962389 q^{11} +0.156325i q^{12} -6.15633i q^{13} +4.96239 q^{14} -4.35026 q^{16} -6.31265i q^{17} -3.48119i q^{18} +1.00000 q^{19} +2.70052 q^{21} -1.42548i q^{22} +4.96239i q^{23} -2.15633 q^{24} -9.11871 q^{26} -4.31265i q^{27} -0.649738i q^{28} +3.61213 q^{29} -5.92478 q^{31} +1.09332i q^{32} -0.775746i q^{33} -9.35026 q^{34} -0.455802 q^{36} +10.1563i q^{37} -1.48119i q^{38} -4.96239 q^{39} +6.31265 q^{41} -4.00000i q^{42} +4.12601i q^{43} -0.186642 q^{44} +7.35026 q^{46} +3.35026i q^{47} +3.50659i q^{48} -4.22425 q^{49} -5.08840 q^{51} +1.19394i q^{52} -1.84367i q^{53} -6.38787 q^{54} +8.96239 q^{56} -0.806063i q^{57} -5.35026i q^{58} +6.38787 q^{59} -11.2750 q^{61} +8.77575i q^{62} +7.87399i q^{63} -7.08110 q^{64} -1.14903 q^{66} -6.73084i q^{67} +1.22425i q^{68} +4.00000 q^{69} -0.775746 q^{71} -6.28726i q^{72} -0.387873i q^{73} +15.0435 q^{74} -0.193937 q^{76} +3.22425i q^{77} +7.35026i q^{78} +0.836381 q^{79} +3.57452 q^{81} -9.35026i q^{82} +7.03761i q^{83} -0.523730 q^{84} +6.11142 q^{86} -2.91160i q^{87} -2.57452i q^{88} -7.08840 q^{89} +20.6253 q^{91} -0.962389i q^{92} +4.77575i q^{93} +4.96239 q^{94} +0.881286 q^{96} +10.9927i q^{97} +6.25694i q^{98} +2.26187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9	\( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} - 16 q^{11} + 8 q^{14} - 6 q^{16} + 6 q^{19} - 24 q^{21} + 8 q^{24} - 12 q^{26} + 20 q^{29} + 8 q^{31} - 36 q^{34} - 22 q^{36} - 8 q^{39} - 4 q^{41} + 24 q^{44} + 24 q^{46} - 22 q^{49} + 8 q^{51} - 40 q^{54} + 32 q^{56} + 40 q^{59} - 4 q^{61} + 22 q^{64} + 40 q^{66} + 24 q^{69} - 8 q^{71} + 4 q^{74} - 2 q^{76} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 4 q^{89} + 40 q^{91} + 8 q^{94} + 48 q^{96} + 32 q^{99}+O(q^{100}) \) 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 - 16 * q^11 + 8 * q^14 - 6 * q^16 + 6 * q^19 - 24 * q^21 + 8 * q^24 - 12 * q^26 + 20 * q^29 + 8 * q^31 - 36 * q^34 - 22 * q^36 - 8 * q^39 - 4 * q^41 + 24 * q^44 + 24 * q^46 - 22 * q^49 + 8 * q^51 - 40 * q^54 + 32 * q^56 + 40 * q^59 - 4 * q^61 + 22 * q^64 + 40 * q^66 + 24 * q^69 - 8 * q^71 + 4 * q^74 - 2 * q^76 - 2 * q^81 - 40 * q^84 - 32 * q^86 - 4 * q^89 + 40 * q^91 + 8 * q^94 + 48 * q^96 + 32 * q^99

Introduction

L-functions

Modular forms

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Database

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