LMFDB - Embedded newform 950.2.b.f.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,2,Mod(799,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, 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("950.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	950.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:	$7.58578819202$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	950.799
Dual form			950.2.b.f.799.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +2.56155i q^{3} -1.00000 q^{4} +2.56155 q^{6} -2.56155i q^{7} +1.00000i q^{8} -3.56155 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +2.56155i q^{3} -1.00000 q^{4} +2.56155 q^{6} -2.56155i q^{7} +1.00000i q^{8} -3.56155 q^{9} +4.00000 q^{11} -2.56155i q^{12} -5.68466i q^{13} -2.56155 q^{14} +1.00000 q^{16} +3.43845i q^{17} +3.56155i q^{18} +1.00000 q^{19} +6.56155 q^{21} -4.00000i q^{22} -7.68466i q^{23} -2.56155 q^{24} -5.68466 q^{26} -1.43845i q^{27} +2.56155i q^{28} +5.68466 q^{29} -5.12311 q^{31} -1.00000i q^{32} +10.2462i q^{33} +3.43845 q^{34} +3.56155 q^{36} -6.00000i q^{37} -1.00000i q^{38} +14.5616 q^{39} +12.2462 q^{41} -6.56155i q^{42} +2.87689i q^{43} -4.00000 q^{44} -7.68466 q^{46} +6.24621i q^{47} +2.56155i q^{48} +0.438447 q^{49} -8.80776 q^{51} +5.68466i q^{52} +4.56155i q^{53} -1.43845 q^{54} +2.56155 q^{56} +2.56155i q^{57} -5.68466i q^{58} -2.56155 q^{59} +11.1231 q^{61} +5.12311i q^{62} +9.12311i q^{63} -1.00000 q^{64} +10.2462 q^{66} -2.56155i q^{67} -3.43845i q^{68} +19.6847 q^{69} +10.2462 q^{71} -3.56155i q^{72} +1.68466i q^{73} -6.00000 q^{74} -1.00000 q^{76} -10.2462i q^{77} -14.5616i q^{78} +5.12311 q^{79} -7.00000 q^{81} -12.2462i q^{82} -2.87689i q^{83} -6.56155 q^{84} +2.87689 q^{86} +14.5616i q^{87} +4.00000i q^{88} -2.00000 q^{89} -14.5616 q^{91} +7.68466i q^{92} -13.1231i q^{93} +6.24621 q^{94} +2.56155 q^{96} +6.00000i q^{97} -0.438447i q^{98} -14.2462 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 2 q^{6} - 6 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 2 * q^6 - 6 * q^9	$ 4 q - 4 q^{4} + 2 q^{6} - 6 q^{9} + 16 q^{11} - 2 q^{14} + 4 q^{16} + 4 q^{19} + 18 q^{21} - 2 q^{24} + 2 q^{26} - 2 q^{29} - 4 q^{31} + 22 q^{34} + 6 q^{36} + 50 q^{39} + 16 q^{41} - 16 q^{44} - 6 q^{46} + 10 q^{49} + 6 q^{51} - 14 q^{54} + 2 q^{56} - 2 q^{59} + 28 q^{61} - 4 q^{64} + 8 q^{66} + 54 q^{69} + 8 q^{71} - 24 q^{74} - 4 q^{76} + 4 q^{79} - 28 q^{81} - 18 q^{84} + 28 q^{86} - 8 q^{89} - 50 q^{91} - 8 q^{94} + 2 q^{96} - 24 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 2 * q^6 - 6 * q^9 + 16 * q^11 - 2 * q^14 + 4 * q^16 + 4 * q^19 + 18 * q^21 - 2 * q^24 + 2 * q^26 - 2 * q^29 - 4 * q^31 + 22 * q^34 + 6 * q^36 + 50 * q^39 + 16 * q^41 - 16 * q^44 - 6 * q^46 + 10 * q^49 + 6 * q^51 - 14 * q^54 + 2 * q^56 - 2 * q^59 + 28 * q^61 - 4 * q^64 + 8 * q^66 + 54 * q^69 + 8 * q^71 - 24 * q^74 - 4 * q^76 + 4 * q^79 - 28 * q^81 - 18 * q^84 + 28 * q^86 - 8 * q^89 - 50 * q^91 - 8 * q^94 + 2 * q^96 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$.

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

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	2.56155i	1.47891i	0.673204	+	0.739457i	$0.264917\pi$
$3$	2.56155i	1.47891i	−0.673204	+	0.739457i	$0.735083\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.56155	1.04575
$7$	− 2.56155i	− 0.968176i	−0.875019	−	0.484088i	$-0.839151\pi$
$7$	− 2.56155i	− 0.968176i	0.875019	−	0.484088i	$-0.160849\pi$
$8$	1.00000i	0.353553i
$9$	−3.56155	−1.18718
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	− 2.56155i	− 0.739457i
$13$	− 5.68466i	− 1.57664i	−0.615265	−	0.788320i	$-0.710951\pi$
$13$	− 5.68466i	− 1.57664i	0.615265	−	0.788320i	$-0.289049\pi$
$14$	−2.56155	−0.684604
$15$	0	0
$16$	1.00000	0.250000
$17$	3.43845i	0.833946i	0.908919	+	0.416973i	$0.136909\pi$
$17$	3.43845i	0.833946i	−0.908919	+	0.416973i	$0.863091\pi$
$18$	3.56155i	0.839466i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	6.56155	1.43185
$22$	− 4.00000i	− 0.852803i
$23$	− 7.68466i	− 1.60236i	−0.598422	−	0.801181i	$-0.704205\pi$
$23$	− 7.68466i	− 1.60236i	0.598422	−	0.801181i	$-0.295795\pi$
$24$	−2.56155	−0.522875
$25$	0	0
$26$	−5.68466	−1.11485
$27$	− 1.43845i	− 0.276829i
$28$	2.56155i	0.484088i
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	−5.12311	−0.920137	−0.460068	−	0.887883i	$-0.652175\pi$
$31$	−5.12311	−0.920137	−0.460068	+	0.887883i	$0.652175\pi$
$32$	− 1.00000i	− 0.176777i
$33$	10.2462i	1.78364i
$34$	3.43845	0.589689
$35$	0	0
$36$	3.56155	0.593592
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	− 1.00000i	− 0.162221i
$39$	14.5616	2.33171
$40$	0	0
$41$	12.2462	1.91254	0.956268	−	0.292490i	$-0.0944840\pi$
$41$	12.2462	1.91254	0.956268	+	0.292490i	$0.0944840\pi$
$42$	− 6.56155i	− 1.01247i
$43$	2.87689i	0.438722i	0.975644	+	0.219361i	$0.0703973\pi$
$43$	2.87689i	0.438722i	−0.975644	+	0.219361i	$0.929603\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−7.68466	−1.13304
$47$	6.24621i	0.911104i	0.890209	+	0.455552i	$0.150558\pi$
$47$	6.24621i	0.911104i	−0.890209	+	0.455552i	$0.849442\pi$
$48$	2.56155i	0.369728i
$49$	0.438447	0.0626353
$50$	0	0
$51$	−8.80776	−1.23333
$52$	5.68466i	0.788320i
$53$	4.56155i	0.626577i	0.949658	+	0.313289i	$0.101431\pi$
$53$	4.56155i	0.626577i	−0.949658	+	0.313289i	$0.898569\pi$
$54$	−1.43845	−0.195748
$55$	0	0
$56$	2.56155	0.342302
$57$	2.56155i	0.339286i
$58$	− 5.68466i	− 0.746432i
$59$	−2.56155	−0.333486	−0.166743	−	0.986000i	$-0.553325\pi$
$59$	−2.56155	−0.333486	−0.166743	+	0.986000i	$0.553325\pi$
$60$	0	0
$61$	11.1231	1.42417	0.712084	−	0.702094i	$-0.247752\pi$
$61$	11.1231	1.42417	0.712084	+	0.702094i	$0.247752\pi$
$62$	5.12311i	0.650635i
$63$	9.12311i	1.14940i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	10.2462	1.26122
$67$	− 2.56155i	− 0.312943i	−0.987682	−	0.156472i	$-0.949988\pi$
$67$	− 2.56155i	− 0.312943i	0.987682	−	0.156472i	$-0.0500120\pi$
$68$	− 3.43845i	− 0.416973i
$69$	19.6847	2.36975
$70$	0	0
$71$	10.2462	1.21600	0.608001	−	0.793936i	$-0.291972\pi$
$71$	10.2462	1.21600	0.608001	+	0.793936i	$0.291972\pi$
$72$	− 3.56155i	− 0.419733i
$73$	1.68466i	0.197174i	0.995128	+	0.0985872i	$0.0314323\pi$
$73$	1.68466i	0.197174i	−0.995128	+	0.0985872i	$0.968568\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−1.00000	−0.114708
$77$	− 10.2462i	− 1.16766i
$78$	− 14.5616i	− 1.64877i
$79$	5.12311	0.576394	0.288197	−	0.957571i	$-0.406944\pi$
$79$	5.12311	0.576394	0.288197	+	0.957571i	$0.406944\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	− 12.2462i	− 1.35237i
$83$	− 2.87689i	− 0.315780i	−0.987457	−	0.157890i	$-0.949531\pi$
$83$	− 2.87689i	− 0.315780i	0.987457	−	0.157890i	$-0.0504692\pi$
$84$	−6.56155	−0.715924
$85$	0	0
$86$	2.87689	0.310223
$87$	14.5616i	1.56116i
$88$	4.00000i	0.426401i
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−14.5616	−1.52647
$92$	7.68466i	0.801181i
$93$	− 13.1231i	− 1.36080i
$94$	6.24621	0.644247
$95$	0	0
$96$	2.56155	0.261437
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	− 0.438447i	− 0.0442899i
$99$	−14.2462	−1.43180
$100$	0	0
$101$	−17.3693	−1.72831	−0.864156	−	0.503224i	$-0.832147\pi$
$101$	−17.3693	−1.72831	−0.864156	+	0.503224i	$0.832147\pi$
$102$	8.80776i	0.872099i
$103$	− 2.24621i	− 0.221326i	−0.993858	−	0.110663i	$-0.964703\pi$
$103$	− 2.24621i	− 0.221326i	0.993858	−	0.110663i	$-0.0352974\pi$
$104$	5.68466	0.557427
$105$	0	0
$106$	4.56155	0.443057
$107$	− 5.43845i	− 0.525755i	−0.964829	−	0.262877i	$-0.915329\pi$
$107$	− 5.43845i	− 0.525755i	0.964829	−	0.262877i	$-0.0846714\pi$
$108$	1.43845i	0.138415i
$109$	0.561553	0.0537870	0.0268935	−	0.999638i	$-0.491439\pi$
$109$	0.561553	0.0537870	0.0268935	+	0.999638i	$0.491439\pi$
$110$	0	0
$111$	15.3693	1.45879
$112$	− 2.56155i	− 0.242044i
$113$	− 8.87689i	− 0.835068i	−0.908661	−	0.417534i	$-0.862894\pi$
$113$	− 8.87689i	− 0.835068i	0.908661	−	0.417534i	$-0.137106\pi$
$114$	2.56155	0.239911
$115$	0	0
$116$	−5.68466	−0.527807
$117$	20.2462i	1.87176i
$118$	2.56155i	0.235810i
$119$	8.80776	0.807406
$120$	0	0
$121$	5.00000	0.454545
$122$	− 11.1231i	− 1.00704i
$123$	31.3693i	2.82848i
$124$	5.12311	0.460068
$125$	0	0
$126$	9.12311	0.812751
$127$	13.1231i	1.16449i	0.813014	+	0.582244i	$0.197825\pi$
$127$	13.1231i	1.16449i	−0.813014	+	0.582244i	$0.802175\pi$
$128$	1.00000i	0.0883883i
$129$	−7.36932	−0.648832
$130$	0	0
$131$	−16.4924	−1.44095	−0.720475	−	0.693481i	$-0.756076\pi$
$131$	−16.4924	−1.44095	−0.720475	+	0.693481i	$0.756076\pi$
$132$	− 10.2462i	− 0.891818i
$133$	− 2.56155i	− 0.222115i
$134$	−2.56155	−0.221284
$135$	0	0
$136$	−3.43845	−0.294844
$137$	− 14.8078i	− 1.26511i	−0.774514	−	0.632556i	$-0.782006\pi$
$137$	− 14.8078i	− 1.26511i	0.774514	−	0.632556i	$-0.217994\pi$
$138$	− 19.6847i	− 1.67567i
$139$	−16.4924	−1.39887	−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924	−1.39887	−0.699435	+	0.714697i	$0.746565\pi$
$140$	0	0
$141$	−16.0000	−1.34744
$142$	− 10.2462i	− 0.859843i
$143$	− 22.7386i	− 1.90150i
$144$	−3.56155	−0.296796
$145$	0	0
$146$	1.68466	0.139423
$147$	1.12311i	0.0926322i
$148$	6.00000i	0.493197i
$149$	−13.3693	−1.09526	−0.547629	−	0.836722i	$-0.684469\pi$
$149$	−13.3693	−1.09526	−0.547629	+	0.836722i	$0.684469\pi$
$150$	0	0
$151$	5.12311	0.416912	0.208456	−	0.978032i	$-0.433156\pi$
$151$	5.12311	0.416912	0.208456	+	0.978032i	$0.433156\pi$
$152$	1.00000i	0.0811107i
$153$	− 12.2462i	− 0.990048i
$154$	−10.2462	−0.825663
$155$	0	0
$156$	−14.5616	−1.16586
$157$	− 20.2462i	− 1.61582i	−0.589303	−	0.807912i	$-0.700598\pi$
$157$	− 20.2462i	− 1.61582i	0.589303	−	0.807912i	$-0.299402\pi$
$158$	− 5.12311i	− 0.407572i
$159$	−11.6847	−0.926654
$160$	0	0
$161$	−19.6847	−1.55137
$162$	7.00000i	0.549972i
$163$	15.3693i	1.20382i	0.798565	+	0.601909i	$0.205593\pi$
$163$	15.3693i	1.20382i	−0.798565	+	0.601909i	$0.794407\pi$
$164$	−12.2462	−0.956268
$165$	0	0
$166$	−2.87689	−0.223290
$167$	− 7.36932i	− 0.570255i	−0.958490	−	0.285127i	$-0.907964\pi$
$167$	− 7.36932i	− 0.570255i	0.958490	−	0.285127i	$-0.0920359\pi$
$168$	6.56155i	0.506235i
$169$	−19.3153	−1.48580
$170$	0	0
$171$	−3.56155	−0.272359
$172$	− 2.87689i	− 0.219361i
$173$	− 20.2462i	− 1.53929i	−0.638471	−	0.769645i	$-0.720433\pi$
$173$	− 20.2462i	− 1.53929i	0.638471	−	0.769645i	$-0.279567\pi$
$174$	14.5616	1.10391
$175$	0	0
$176$	4.00000	0.301511
$177$	− 6.56155i	− 0.493197i
$178$	2.00000i	0.149906i
$179$	22.2462	1.66276	0.831380	−	0.555704i	$-0.187551\pi$
$179$	22.2462	1.66276	0.831380	+	0.555704i	$0.187551\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	14.5616i	1.07937i
$183$	28.4924i	2.10622i
$184$	7.68466	0.566521
$185$	0	0
$186$	−13.1231	−0.962233
$187$	13.7538i	1.00578i
$188$	− 6.24621i	− 0.455552i
$189$	−3.68466	−0.268019
$190$	0	0
$191$	−3.68466	−0.266613	−0.133306	−	0.991075i	$-0.542559\pi$
$191$	−3.68466	−0.266613	−0.133306	+	0.991075i	$0.542559\pi$
$192$	− 2.56155i	− 0.184864i
$193$	14.4924i	1.04319i	0.853194	+	0.521594i	$0.174662\pi$
$193$	14.4924i	1.04319i	−0.853194	+	0.521594i	$0.825338\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−0.438447	−0.0313177
$197$	− 20.2462i	− 1.44248i	−0.692684	−	0.721241i	$-0.743572\pi$
$197$	− 20.2462i	− 1.44248i	0.692684	−	0.721241i	$-0.256428\pi$
$198$	14.2462i	1.01243i
$199$	−16.8078	−1.19147	−0.595735	−	0.803181i	$-0.703139\pi$
$199$	−16.8078	−1.19147	−0.595735	+	0.803181i	$0.703139\pi$
$200$	0	0
$201$	6.56155	0.462816
$202$	17.3693i	1.22210i
$203$	− 14.5616i	− 1.02202i
$204$	8.80776	0.616667
$205$	0	0
$206$	−2.24621	−0.156501
$207$	27.3693i	1.90230i
$208$	− 5.68466i	− 0.394160i
$209$	4.00000	0.276686
$210$	0	0
$211$	8.31534	0.572452	0.286226	−	0.958162i	$-0.407599\pi$
$211$	8.31534	0.572452	0.286226	+	0.958162i	$0.407599\pi$
$212$	− 4.56155i	− 0.313289i
$213$	26.2462i	1.79836i
$214$	−5.43845	−0.371765
$215$	0	0
$216$	1.43845	0.0978739
$217$	13.1231i	0.890854i
$218$	− 0.561553i	− 0.0380332i
$219$	−4.31534	−0.291604
$220$	0	0
$221$	19.5464	1.31483
$222$	− 15.3693i	− 1.03152i
$223$	23.3693i	1.56493i	0.622698	+	0.782463i	$0.286037\pi$
$223$	23.3693i	1.56493i	−0.622698	+	0.782463i	$0.713963\pi$
$224$	−2.56155	−0.171151
$225$	0	0
$226$	−8.87689	−0.590482
$227$	25.9309i	1.72109i	0.509373	+	0.860546i	$0.329878\pi$
$227$	25.9309i	1.72109i	−0.509373	+	0.860546i	$0.670122\pi$
$228$	− 2.56155i	− 0.169643i
$229$	14.4924	0.957686	0.478843	−	0.877900i	$-0.341056\pi$
$229$	14.4924	0.957686	0.478843	+	0.877900i	$0.341056\pi$
$230$	0	0
$231$	26.2462	1.72687
$232$	5.68466i	0.373216i
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	20.2462	1.32354
$235$	0	0
$236$	2.56155	0.166743
$237$	13.1231i	0.852437i
$238$	− 8.80776i	− 0.570923i
$239$	1.43845	0.0930454	0.0465227	−	0.998917i	$-0.485186\pi$
$239$	1.43845	0.0930454	0.0465227	+	0.998917i	$0.485186\pi$
$240$	0	0
$241$	23.1231	1.48949	0.744745	−	0.667349i	$-0.232571\pi$
$241$	23.1231	1.48949	0.744745	+	0.667349i	$0.232571\pi$
$242$	− 5.00000i	− 0.321412i
$243$	− 22.2462i	− 1.42710i
$244$	−11.1231	−0.712084
$245$	0	0
$246$	31.3693	2.00003
$247$	− 5.68466i	− 0.361706i
$248$	− 5.12311i	− 0.325318i
$249$	7.36932	0.467011
$250$	0	0
$251$	6.24621	0.394257	0.197129	−	0.980378i	$-0.436838\pi$
$251$	6.24621	0.394257	0.197129	+	0.980378i	$0.436838\pi$
$252$	− 9.12311i	− 0.574702i
$253$	− 30.7386i	− 1.93252i
$254$	13.1231	0.823417
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	7.36932i	0.458794i
$259$	−15.3693	−0.955003
$260$	0	0
$261$	−20.2462	−1.25321
$262$	16.4924i	1.01891i
$263$	22.2462i	1.37176i	0.727715	+	0.685880i	$0.240583\pi$
$263$	22.2462i	1.37176i	−0.727715	+	0.685880i	$0.759417\pi$
$264$	−10.2462	−0.630611
$265$	0	0
$266$	−2.56155	−0.157059
$267$	− 5.12311i	− 0.313529i
$268$	2.56155i	0.156472i
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	−21.9309	−1.33221	−0.666103	−	0.745860i	$-0.732039\pi$
$271$	−21.9309	−1.33221	−0.666103	+	0.745860i	$0.732039\pi$
$272$	3.43845i	0.208486i
$273$	− 37.3002i	− 2.25751i
$274$	−14.8078	−0.894570
$275$	0	0
$276$	−19.6847	−1.18488
$277$	0.876894i	0.0526875i	0.999653	+	0.0263437i	$0.00838644\pi$
$277$	0.876894i	0.0526875i	−0.999653	+	0.0263437i	$0.991614\pi$
$278$	16.4924i	0.989150i
$279$	18.2462	1.09237
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	16.0000i	0.952786i
$283$	21.1231i	1.25564i	0.778359	+	0.627819i	$0.216052\pi$
$283$	21.1231i	1.25564i	−0.778359	+	0.627819i	$0.783948\pi$
$284$	−10.2462	−0.608001
$285$	0	0
$286$	−22.7386	−1.34456
$287$	− 31.3693i	− 1.85167i
$288$	3.56155i	0.209867i
$289$	5.17708	0.304534
$290$	0	0
$291$	−15.3693	−0.900965
$292$	− 1.68466i	− 0.0985872i
$293$	22.1771i	1.29560i	0.761811	+	0.647799i	$0.224311\pi$
$293$	22.1771i	1.29560i	−0.761811	+	0.647799i	$0.775689\pi$
$294$	1.12311	0.0655009
$295$	0	0
$296$	6.00000	0.348743
$297$	− 5.75379i	− 0.333869i
$298$	13.3693i	0.774464i
$299$	−43.6847	−2.52635
$300$	0	0
$301$	7.36932	0.424760
$302$	− 5.12311i	− 0.294802i
$303$	− 44.4924i	− 2.55602i
$304$	1.00000	0.0573539
$305$	0	0
$306$	−12.2462	−0.700069
$307$	32.4924i	1.85444i	0.374516	+	0.927220i	$0.377809\pi$
$307$	32.4924i	1.85444i	−0.374516	+	0.927220i	$0.622191\pi$
$308$	10.2462i	0.583832i
$309$	5.75379	0.327322
$310$	0	0
$311$	−3.68466	−0.208938	−0.104469	−	0.994528i	$-0.533314\pi$
$311$	−3.68466	−0.208938	−0.104469	+	0.994528i	$0.533314\pi$
$312$	14.5616i	0.824386i
$313$	− 5.05398i	− 0.285668i	−0.989747	−	0.142834i	$-0.954379\pi$
$313$	− 5.05398i	− 0.285668i	0.989747	−	0.142834i	$-0.0456214\pi$
$314$	−20.2462	−1.14256
$315$	0	0
$316$	−5.12311	−0.288197
$317$	13.0540i	0.733184i	0.930382	+	0.366592i	$0.119476\pi$
$317$	13.0540i	0.733184i	−0.930382	+	0.366592i	$0.880524\pi$
$318$	11.6847i	0.655243i
$319$	22.7386	1.27312
$320$	0	0
$321$	13.9309	0.777545
$322$	19.6847i	1.09698i
$323$	3.43845i	0.191320i
$324$	7.00000	0.388889
$325$	0	0
$326$	15.3693	0.851228
$327$	1.43845i	0.0795463i
$328$	12.2462i	0.676184i
$329$	16.0000	0.882109
$330$	0	0
$331$	−2.56155	−0.140796	−0.0703978	−	0.997519i	$-0.522427\pi$
$331$	−2.56155	−0.140796	−0.0703978	+	0.997519i	$0.522427\pi$
$332$	2.87689i	0.157890i
$333$	21.3693i	1.17103i
$334$	−7.36932	−0.403231
$335$	0	0
$336$	6.56155	0.357962
$337$	− 26.0000i	− 1.41631i	−0.706057	−	0.708155i	$-0.749528\pi$
$337$	− 26.0000i	− 1.41631i	0.706057	−	0.708155i	$-0.250472\pi$
$338$	19.3153i	1.05062i
$339$	22.7386	1.23499
$340$	0	0
$341$	−20.4924	−1.10973
$342$	3.56155i	0.192587i
$343$	− 19.0540i	− 1.02882i
$344$	−2.87689	−0.155112
$345$	0	0
$346$	−20.2462	−1.08844
$347$	− 8.63068i	− 0.463319i	−0.972797	−	0.231660i	$-0.925584\pi$
$347$	− 8.63068i	− 0.463319i	0.972797	−	0.231660i	$-0.0744156\pi$
$348$	− 14.5616i	− 0.780581i
$349$	−3.75379	−0.200936	−0.100468	−	0.994940i	$-0.532034\pi$
$349$	−3.75379	−0.200936	−0.100468	+	0.994940i	$0.532034\pi$
$350$	0	0
$351$	−8.17708	−0.436460
$352$	− 4.00000i	− 0.213201i
$353$	3.93087i	0.209219i	0.994513	+	0.104610i	$0.0333593\pi$
$353$	3.93087i	0.209219i	−0.994513	+	0.104610i	$0.966641\pi$
$354$	−6.56155	−0.348743
$355$	0	0
$356$	2.00000	0.106000
$357$	22.5616i	1.19408i
$358$	− 22.2462i	− 1.17575i
$359$	−1.43845	−0.0759183	−0.0379592	−	0.999279i	$-0.512086\pi$
$359$	−1.43845	−0.0759183	−0.0379592	+	0.999279i	$0.512086\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	18.0000i	0.946059i
$363$	12.8078i	0.672233i
$364$	14.5616	0.763233
$365$	0	0
$366$	28.4924	1.48932
$367$	6.24621i	0.326050i	0.986622	+	0.163025i	$0.0521251\pi$
$367$	6.24621i	0.326050i	−0.986622	+	0.163025i	$0.947875\pi$
$368$	− 7.68466i	− 0.400591i
$369$	−43.6155	−2.27053
$370$	0	0
$371$	11.6847	0.606637
$372$	13.1231i	0.680401i
$373$	23.4384i	1.21360i	0.794856	+	0.606798i	$0.207546\pi$
$373$	23.4384i	1.21360i	−0.794856	+	0.606798i	$0.792454\pi$
$374$	13.7538	0.711191
$375$	0	0
$376$	−6.24621	−0.322124
$377$	− 32.3153i	− 1.66432i
$378$	3.68466i	0.189518i
$379$	−10.5616	−0.542511	−0.271255	−	0.962507i	$-0.587439\pi$
$379$	−10.5616	−0.542511	−0.271255	+	0.962507i	$0.587439\pi$
$380$	0	0
$381$	−33.6155	−1.72218
$382$	3.68466i	0.188524i
$383$	− 13.7538i	− 0.702786i	−0.936228	−	0.351393i	$-0.885708\pi$
$383$	− 13.7538i	− 0.702786i	0.936228	−	0.351393i	$-0.114292\pi$
$384$	−2.56155	−0.130719
$385$	0	0
$386$	14.4924	0.737645
$387$	− 10.2462i	− 0.520844i
$388$	− 6.00000i	− 0.304604i
$389$	7.12311	0.361156	0.180578	−	0.983561i	$-0.442203\pi$
$389$	7.12311	0.361156	0.180578	+	0.983561i	$0.442203\pi$
$390$	0	0
$391$	26.4233	1.33628
$392$	0.438447i	0.0221449i
$393$	− 42.2462i	− 2.13104i
$394$	−20.2462	−1.01999
$395$	0	0
$396$	14.2462	0.715899
$397$	− 7.12311i	− 0.357498i	−0.983895	−	0.178749i	$-0.942795\pi$
$397$	− 7.12311i	− 0.357498i	0.983895	−	0.178749i	$-0.0572051\pi$
$398$	16.8078i	0.842497i
$399$	6.56155	0.328489
$400$	0	0
$401$	−3.75379	−0.187455	−0.0937276	−	0.995598i	$-0.529878\pi$
$401$	−3.75379	−0.187455	−0.0937276	+	0.995598i	$0.529878\pi$
$402$	− 6.56155i	− 0.327261i
$403$	29.1231i	1.45073i
$404$	17.3693	0.864156
$405$	0	0
$406$	−14.5616	−0.722678
$407$	− 24.0000i	− 1.18964i
$408$	− 8.80776i	− 0.436049i
$409$	−24.7386	−1.22325	−0.611623	−	0.791149i	$-0.709483\pi$
$409$	−24.7386	−1.22325	−0.611623	+	0.791149i	$0.709483\pi$
$410$	0	0
$411$	37.9309	1.87099
$412$	2.24621i	0.110663i
$413$	6.56155i	0.322873i
$414$	27.3693	1.34513
$415$	0	0
$416$	−5.68466	−0.278713
$417$	− 42.2462i	− 2.06881i
$418$	− 4.00000i	− 0.195646i
$419$	−23.8617	−1.16572	−0.582861	−	0.812572i	$-0.698067\pi$
$419$	−23.8617	−1.16572	−0.582861	+	0.812572i	$0.698067\pi$
$420$	0	0
$421$	−23.9309	−1.16632	−0.583160	−	0.812358i	$-0.698184\pi$
$421$	−23.9309	−1.16632	−0.583160	+	0.812358i	$0.698184\pi$
$422$	− 8.31534i	− 0.404784i
$423$	− 22.2462i	− 1.08165i
$424$	−4.56155	−0.221529
$425$	0	0
$426$	26.2462	1.27163
$427$	− 28.4924i	− 1.37884i
$428$	5.43845i	0.262877i
$429$	58.2462	2.81215
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	− 1.43845i	− 0.0692073i
$433$	− 14.6307i	− 0.703106i	−0.936168	−	0.351553i	$-0.885654\pi$
$433$	− 14.6307i	− 0.703106i	0.936168	−	0.351553i	$-0.114346\pi$
$434$	13.1231	0.629929
$435$	0	0
$436$	−0.561553	−0.0268935
$437$	− 7.68466i	− 0.367607i
$438$	4.31534i	0.206195i
$439$	13.1231	0.626332	0.313166	−	0.949698i	$-0.398610\pi$
$439$	13.1231	0.626332	0.313166	+	0.949698i	$0.398610\pi$
$440$	0	0
$441$	−1.56155	−0.0743597
$442$	− 19.5464i	− 0.929727i
$443$	− 2.24621i	− 0.106721i	−0.998575	−	0.0533604i	$-0.983007\pi$
$443$	− 2.24621i	− 0.106721i	0.998575	−	0.0533604i	$-0.0169932\pi$
$444$	−15.3693	−0.729396
$445$	0	0
$446$	23.3693	1.10657
$447$	− 34.2462i	− 1.61979i
$448$	2.56155i	0.121022i
$449$	28.7386	1.35626	0.678130	−	0.734942i	$-0.262791\pi$
$449$	28.7386	1.35626	0.678130	+	0.734942i	$0.262791\pi$
$450$	0	0
$451$	48.9848	2.30661
$452$	8.87689i	0.417534i
$453$	13.1231i	0.616577i
$454$	25.9309	1.21700
$455$	0	0
$456$	−2.56155	−0.119956
$457$	6.31534i	0.295419i	0.989031	+	0.147710i	$0.0471901\pi$
$457$	6.31534i	0.295419i	−0.989031	+	0.147710i	$0.952810\pi$
$458$	− 14.4924i	− 0.677186i
$459$	4.94602	0.230861
$460$	0	0
$461$	3.75379	0.174831	0.0874157	−	0.996172i	$-0.472139\pi$
$461$	3.75379	0.174831	0.0874157	+	0.996172i	$0.472139\pi$
$462$	− 26.2462i	− 1.22108i
$463$	30.2462i	1.40566i	0.711358	+	0.702830i	$0.248081\pi$
$463$	30.2462i	1.40566i	−0.711358	+	0.702830i	$0.751919\pi$
$464$	5.68466	0.263904
$465$	0	0
$466$	−10.0000	−0.463241
$467$	18.2462i	0.844334i	0.906518	+	0.422167i	$0.138730\pi$
$467$	18.2462i	0.844334i	−0.906518	+	0.422167i	$0.861270\pi$
$468$	− 20.2462i	− 0.935881i
$469$	−6.56155	−0.302984
$470$	0	0
$471$	51.8617	2.38966
$472$	− 2.56155i	− 0.117905i
$473$	11.5076i	0.529119i
$474$	13.1231	0.602764
$475$	0	0
$476$	−8.80776	−0.403703
$477$	− 16.2462i	− 0.743863i
$478$	− 1.43845i	− 0.0657930i
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	−34.1080	−1.55519
$482$	− 23.1231i	− 1.05323i
$483$	− 50.4233i	− 2.29434i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−22.2462	−1.00911
$487$	− 17.6155i	− 0.798236i	−0.916900	−	0.399118i	$-0.869316\pi$
$487$	− 17.6155i	− 0.798236i	0.916900	−	0.399118i	$-0.130684\pi$
$488$	11.1231i	0.503519i
$489$	−39.3693	−1.78034
$490$	0	0
$491$	−1.12311	−0.0506850	−0.0253425	−	0.999679i	$-0.508068\pi$
$491$	−1.12311	−0.0506850	−0.0253425	+	0.999679i	$0.508068\pi$
$492$	− 31.3693i	− 1.41424i
$493$	19.5464i	0.880325i
$494$	−5.68466	−0.255765
$495$	0	0
$496$	−5.12311	−0.230034
$497$	− 26.2462i	− 1.17730i
$498$	− 7.36932i	− 0.330227i
$499$	−42.1080	−1.88501	−0.942505	−	0.334191i	$-0.891537\pi$
$499$	−42.1080	−1.88501	−0.942505	+	0.334191i	$0.891537\pi$
$500$	0	0
$501$	18.8769	0.843357
$502$	− 6.24621i	− 0.278782i
$503$	7.05398i	0.314521i	0.987557	+	0.157261i	$0.0502663\pi$
$503$	7.05398i	0.314521i	−0.987557	+	0.157261i	$0.949734\pi$
$504$	−9.12311	−0.406375
$505$	0	0
$506$	−30.7386	−1.36650
$507$	− 49.4773i	− 2.19736i
$508$	− 13.1231i	− 0.582244i
$509$	−2.49242	−0.110475	−0.0552373	−	0.998473i	$-0.517592\pi$
$509$	−2.49242	−0.110475	−0.0552373	+	0.998473i	$0.517592\pi$
$510$	0	0
$511$	4.31534	0.190899
$512$	− 1.00000i	− 0.0441942i
$513$	− 1.43845i	− 0.0635090i
$514$	14.0000	0.617514
$515$	0	0
$516$	7.36932	0.324416
$517$	24.9848i	1.09883i
$518$	15.3693i	0.675289i
$519$	51.8617	2.27648
$520$	0	0
$521$	−3.12311	−0.136826	−0.0684129	−	0.997657i	$-0.521794\pi$
$521$	−3.12311	−0.136826	−0.0684129	+	0.997657i	$0.521794\pi$
$522$	20.2462i	0.886153i
$523$	31.6847i	1.38547i	0.721191	+	0.692737i	$0.243595\pi$
$523$	31.6847i	1.38547i	−0.721191	+	0.692737i	$0.756405\pi$
$524$	16.4924	0.720475
$525$	0	0
$526$	22.2462	0.969981
$527$	− 17.6155i	− 0.767344i
$528$	10.2462i	0.445909i
$529$	−36.0540	−1.56756
$530$	0	0
$531$	9.12311	0.395909
$532$	2.56155i	0.111057i
$533$	− 69.6155i	− 3.01538i
$534$	−5.12311	−0.221698
$535$	0	0
$536$	2.56155	0.110642
$537$	56.9848i	2.45908i
$538$	− 26.0000i	− 1.12094i
$539$	1.75379	0.0755410
$540$	0	0
$541$	−0.384472	−0.0165297	−0.00826487	−	0.999966i	$-0.502631\pi$
$541$	−0.384472	−0.0165297	−0.00826487	+	0.999966i	$0.502631\pi$
$542$	21.9309i	0.942012i
$543$	− 46.1080i	− 1.97868i
$544$	3.43845	0.147422
$545$	0	0
$546$	−37.3002	−1.59630
$547$	16.4924i	0.705165i	0.935781	+	0.352583i	$0.114696\pi$
$547$	16.4924i	0.705165i	−0.935781	+	0.352583i	$0.885304\pi$
$548$	14.8078i	0.632556i
$549$	−39.6155	−1.69075
$550$	0	0
$551$	5.68466	0.242175
$552$	19.6847i	0.837835i
$553$	− 13.1231i	− 0.558051i
$554$	0.876894	0.0372557
$555$	0	0
$556$	16.4924	0.699435
$557$	39.6155i	1.67856i	0.543697	+	0.839282i	$0.317024\pi$
$557$	39.6155i	1.67856i	−0.543697	+	0.839282i	$0.682976\pi$
$558$	− 18.2462i	− 0.772424i
$559$	16.3542	0.691707
$560$	0	0
$561$	−35.2311	−1.48746
$562$	− 2.00000i	− 0.0843649i
$563$	8.49242i	0.357913i	0.983857	+	0.178956i	$0.0572721\pi$
$563$	8.49242i	0.357913i	−0.983857	+	0.178956i	$0.942728\pi$
$564$	16.0000	0.673722
$565$	0	0
$566$	21.1231	0.887870
$567$	17.9309i	0.753026i
$568$	10.2462i	0.429921i
$569$	3.12311	0.130927	0.0654637	−	0.997855i	$-0.479147\pi$
$569$	3.12311	0.130927	0.0654637	+	0.997855i	$0.479147\pi$
$570$	0	0
$571$	21.6155	0.904582	0.452291	−	0.891870i	$-0.350607\pi$
$571$	21.6155	0.904582	0.452291	+	0.891870i	$0.350607\pi$
$572$	22.7386i	0.950750i
$573$	− 9.43845i	− 0.394297i
$574$	−31.3693	−1.30933
$575$	0	0
$576$	3.56155	0.148398
$577$	− 10.3153i	− 0.429433i	−0.976676	−	0.214717i	$-0.931117\pi$
$577$	− 10.3153i	− 0.429433i	0.976676	−	0.214717i	$-0.0688828\pi$
$578$	− 5.17708i	− 0.215338i
$579$	−37.1231	−1.54278
$580$	0	0
$581$	−7.36932	−0.305731
$582$	15.3693i	0.637079i
$583$	18.2462i	0.755681i
$584$	−1.68466	−0.0697117
$585$	0	0
$586$	22.1771	0.916127
$587$	− 7.36932i	− 0.304164i	−0.988368	−	0.152082i	$-0.951402\pi$
$587$	− 7.36932i	− 0.304164i	0.988368	−	0.152082i	$-0.0485978\pi$
$588$	− 1.12311i	− 0.0463161i
$589$	−5.12311	−0.211094
$590$	0	0
$591$	51.8617	2.13331
$592$	− 6.00000i	− 0.246598i
$593$	− 7.75379i	− 0.318410i	−0.987246	−	0.159205i	$-0.949107\pi$
$593$	− 7.75379i	− 0.318410i	0.987246	−	0.159205i	$-0.0508931\pi$
$594$	−5.75379	−0.236081
$595$	0	0
$596$	13.3693	0.547629
$597$	− 43.0540i	− 1.76208i
$598$	43.6847i	1.78640i
$599$	11.8617	0.484658	0.242329	−	0.970194i	$-0.422089\pi$
$599$	11.8617	0.484658	0.242329	+	0.970194i	$0.422089\pi$
$600$	0	0
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	− 7.36932i	− 0.300351i
$603$	9.12311i	0.371522i
$604$	−5.12311	−0.208456
$605$	0	0
$606$	−44.4924	−1.80738
$607$	− 21.1231i	− 0.857360i	−0.903456	−	0.428680i	$-0.858979\pi$
$607$	− 21.1231i	− 0.857360i	0.903456	−	0.428680i	$-0.141021\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	37.3002	1.51148
$610$	0	0
$611$	35.5076	1.43648
$612$	12.2462i	0.495024i
$613$	− 5.36932i	− 0.216865i	−0.994104	−	0.108432i	$-0.965417\pi$
$613$	− 5.36932i	− 0.216865i	0.994104	−	0.108432i	$-0.0345831\pi$
$614$	32.4924	1.31129
$615$	0	0
$616$	10.2462	0.412832
$617$	12.2462i	0.493014i	0.969141	+	0.246507i	$0.0792828\pi$
$617$	12.2462i	0.493014i	−0.969141	+	0.246507i	$0.920717\pi$
$618$	− 5.75379i	− 0.231451i
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	−11.0540	−0.443581
$622$	3.68466i	0.147741i
$623$	5.12311i	0.205253i
$624$	14.5616	0.582929
$625$	0	0
$626$	−5.05398	−0.201997
$627$	10.2462i	0.409194i
$628$	20.2462i	0.807912i
$629$	20.6307	0.822599
$630$	0	0
$631$	−20.4924	−0.815790	−0.407895	−	0.913029i	$-0.633737\pi$
$631$	−20.4924	−0.815790	−0.407895	+	0.913029i	$0.633737\pi$
$632$	5.12311i	0.203786i
$633$	21.3002i	0.846606i
$634$	13.0540	0.518440
$635$	0	0
$636$	11.6847	0.463327
$637$	− 2.49242i	− 0.0987534i
$638$	− 22.7386i	− 0.900231i
$639$	−36.4924	−1.44362
$640$	0	0
$641$	21.8617	0.863487	0.431743	−	0.901996i	$-0.357899\pi$
$641$	21.8617	0.863487	0.431743	+	0.901996i	$0.357899\pi$
$642$	− 13.9309i	− 0.549808i
$643$	33.6155i	1.32567i	0.748767	+	0.662834i	$0.230646\pi$
$643$	33.6155i	1.32567i	−0.748767	+	0.662834i	$0.769354\pi$
$644$	19.6847	0.775684
$645$	0	0
$646$	3.43845	0.135284
$647$	5.43845i	0.213807i	0.994269	+	0.106904i	$0.0340937\pi$
$647$	5.43845i	0.213807i	−0.994269	+	0.106904i	$0.965906\pi$
$648$	− 7.00000i	− 0.274986i
$649$	−10.2462	−0.402199
$650$	0	0
$651$	−33.6155	−1.31750
$652$	− 15.3693i	− 0.601909i
$653$	7.12311i	0.278749i	0.990240	+	0.139374i	$0.0445091\pi$
$653$	7.12311i	0.278749i	−0.990240	+	0.139374i	$0.955491\pi$
$654$	1.43845	0.0562477
$655$	0	0
$656$	12.2462	0.478134
$657$	− 6.00000i	− 0.234082i
$658$	− 16.0000i	− 0.623745i
$659$	14.0691	0.548056	0.274028	−	0.961722i	$-0.411644\pi$
$659$	14.0691	0.548056	0.274028	+	0.961722i	$0.411644\pi$
$660$	0	0
$661$	−10.8078	−0.420373	−0.210187	−	0.977661i	$-0.567407\pi$
$661$	−10.8078	−0.420373	−0.210187	+	0.977661i	$0.567407\pi$
$662$	2.56155i	0.0995576i
$663$	50.0691i	1.94452i
$664$	2.87689	0.111645
$665$	0	0
$666$	21.3693	0.828044
$667$	− 43.6847i	− 1.69148i
$668$	7.36932i	0.285127i
$669$	−59.8617	−2.31439
$670$	0	0
$671$	44.4924	1.71761
$672$	− 6.56155i	− 0.253117i
$673$	− 21.3693i	− 0.823727i	−0.911246	−	0.411863i	$-0.864878\pi$
$673$	− 21.3693i	− 0.823727i	0.911246	−	0.411863i	$-0.135122\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	19.3153	0.742898
$677$	40.5616i	1.55891i	0.626460	+	0.779454i	$0.284503\pi$
$677$	40.5616i	1.55891i	−0.626460	+	0.779454i	$0.715497\pi$
$678$	− 22.7386i	− 0.873272i
$679$	15.3693	0.589820
$680$	0	0
$681$	−66.4233	−2.54535
$682$	20.4924i	0.784695i
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	3.56155	0.136179
$685$	0	0
$686$	−19.0540	−0.727484
$687$	37.1231i	1.41633i
$688$	2.87689i	0.109681i
$689$	25.9309	0.987887
$690$	0	0
$691$	−17.1231	−0.651394	−0.325697	−	0.945474i	$-0.605599\pi$
$691$	−17.1231	−0.651394	−0.325697	+	0.945474i	$0.605599\pi$
$692$	20.2462i	0.769645i
$693$	36.4924i	1.38623i
$694$	−8.63068	−0.327616
$695$	0	0
$696$	−14.5616	−0.551954
$697$	42.1080i	1.59495i
$698$	3.75379i	0.142083i
$699$	25.6155	0.968868
$700$	0	0
$701$	−20.8769	−0.788509	−0.394255	−	0.919001i	$-0.628997\pi$
$701$	−20.8769	−0.788509	−0.394255	+	0.919001i	$0.628997\pi$
$702$	8.17708i	0.308624i
$703$	− 6.00000i	− 0.226294i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	3.93087	0.147940
$707$	44.4924i	1.67331i
$708$	6.56155i	0.246598i
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	−18.2462	−0.684286
$712$	− 2.00000i	− 0.0749532i
$713$	39.3693i	1.47439i
$714$	22.5616	0.844345
$715$	0	0
$716$	−22.2462	−0.831380
$717$	3.68466i	0.137606i
$718$	1.43845i	0.0536824i
$719$	25.4384	0.948694	0.474347	−	0.880338i	$-0.342684\pi$
$719$	25.4384	0.948694	0.474347	+	0.880338i	$0.342684\pi$
$720$	0	0
$721$	−5.75379	−0.214282
$722$	− 1.00000i	− 0.0372161i
$723$	59.2311i	2.20283i
$724$	18.0000	0.668965
$725$	0	0
$726$	12.8078	0.475341
$727$	− 24.3153i	− 0.901806i	−0.892573	−	0.450903i	$-0.851102\pi$
$727$	− 24.3153i	− 0.901806i	0.892573	−	0.450903i	$-0.148898\pi$
$728$	− 14.5616i	− 0.539687i
$729$	35.9848	1.33277
$730$	0	0
$731$	−9.89205	−0.365871
$732$	− 28.4924i	− 1.05311i
$733$	36.8769i	1.36208i	0.732247	+	0.681040i	$0.238472\pi$
$733$	36.8769i	1.36208i	−0.732247	+	0.681040i	$0.761528\pi$
$734$	6.24621	0.230552
$735$	0	0
$736$	−7.68466	−0.283260
$737$	− 10.2462i	− 0.377424i
$738$	43.6155i	1.60551i
$739$	−25.1231	−0.924168	−0.462084	−	0.886836i	$-0.652898\pi$
$739$	−25.1231	−0.924168	−0.462084	+	0.886836i	$0.652898\pi$
$740$	0	0
$741$	14.5616	0.534932
$742$	− 11.6847i	− 0.428957i
$743$	− 18.8769i	− 0.692526i	−0.938137	−	0.346263i	$-0.887450\pi$
$743$	− 18.8769i	− 0.692526i	0.938137	−	0.346263i	$-0.112550\pi$
$744$	13.1231	0.481116
$745$	0	0
$746$	23.4384	0.858143
$747$	10.2462i	0.374889i
$748$	− 13.7538i	− 0.502888i
$749$	−13.9309	−0.509023
$750$	0	0
$751$	−34.8769	−1.27268	−0.636338	−	0.771410i	$-0.719552\pi$
$751$	−34.8769	−1.27268	−0.636338	+	0.771410i	$0.719552\pi$
$752$	6.24621i	0.227776i
$753$	16.0000i	0.583072i
$754$	−32.3153	−1.17686
$755$	0	0
$756$	3.68466	0.134010
$757$	42.4924i	1.54441i	0.635371	+	0.772207i	$0.280847\pi$
$757$	42.4924i	1.54441i	−0.635371	+	0.772207i	$0.719153\pi$
$758$	10.5616i	0.383613i
$759$	78.7386	2.85803
$760$	0	0
$761$	41.5464	1.50606	0.753028	−	0.657989i	$-0.228593\pi$
$761$	41.5464	1.50606	0.753028	+	0.657989i	$0.228593\pi$
$762$	33.6155i	1.21776i
$763$	− 1.43845i	− 0.0520753i
$764$	3.68466	0.133306
$765$	0	0
$766$	−13.7538	−0.496945
$767$	14.5616i	0.525787i
$768$	2.56155i	0.0924321i
$769$	−27.4384	−0.989456	−0.494728	−	0.869048i	$-0.664732\pi$
$769$	−27.4384	−0.989456	−0.494728	+	0.869048i	$0.664732\pi$
$770$	0	0
$771$	−35.8617	−1.29153
$772$	− 14.4924i	− 0.521594i
$773$	− 10.8078i	− 0.388728i	−0.980929	−	0.194364i	$-0.937736\pi$
$773$	− 10.8078i	− 0.388728i	0.980929	−	0.194364i	$-0.0622643\pi$
$774$	−10.2462	−0.368292
$775$	0	0
$776$	−6.00000	−0.215387
$777$	− 39.3693i	− 1.41237i
$778$	− 7.12311i	− 0.255376i
$779$	12.2462	0.438766
$780$	0	0
$781$	40.9848	1.46655
$782$	− 26.4233i	− 0.944895i
$783$	− 8.17708i	− 0.292225i
$784$	0.438447	0.0156588
$785$	0	0
$786$	−42.2462	−1.50687
$787$	11.1922i	0.398960i	0.979902	+	0.199480i	$0.0639253\pi$
$787$	11.1922i	0.398960i	−0.979902	+	0.199480i	$0.936075\pi$
$788$	20.2462i	0.721241i
$789$	−56.9848	−2.02871
$790$	0	0
$791$	−22.7386	−0.808493
$792$	− 14.2462i	− 0.506217i
$793$	− 63.2311i	− 2.24540i
$794$	−7.12311	−0.252790
$795$	0	0
$796$	16.8078	0.595735
$797$	− 11.3002i	− 0.400273i	−0.979768	−	0.200137i	$-0.935861\pi$
$797$	− 11.3002i	− 0.400273i	0.979768	−	0.200137i	$-0.0641386\pi$
$798$	− 6.56155i	− 0.232276i
$799$	−21.4773	−0.759811
$800$	0	0
$801$	7.12311	0.251683
$802$	3.75379i	0.132551i
$803$	6.73863i	0.237801i
$804$	−6.56155	−0.231408
$805$	0	0
$806$	29.1231	1.02582
$807$	66.6004i	2.34444i
$808$	− 17.3693i	− 0.611050i
$809$	12.5616	0.441641	0.220820	−	0.975315i	$-0.429127\pi$
$809$	12.5616	0.441641	0.220820	+	0.975315i	$0.429127\pi$
$810$	0	0
$811$	−20.8078	−0.730659	−0.365330	−	0.930878i	$-0.619044\pi$
$811$	−20.8078	−0.730659	−0.365330	+	0.930878i	$0.619044\pi$
$812$	14.5616i	0.511010i
$813$	− 56.1771i	− 1.97022i
$814$	−24.0000	−0.841200
$815$	0	0
$816$	−8.80776	−0.308333
$817$	2.87689i	0.100650i
$818$	24.7386i	0.864966i
$819$	51.8617	1.81220
$820$	0	0
$821$	−17.3693	−0.606193	−0.303097	−	0.952960i	$-0.598021\pi$
$821$	−17.3693	−0.606193	−0.303097	+	0.952960i	$0.598021\pi$
$822$	− 37.9309i	− 1.32299i
$823$	29.4384i	1.02616i	0.858341	+	0.513080i	$0.171496\pi$
$823$	29.4384i	1.02616i	−0.858341	+	0.513080i	$0.828504\pi$
$824$	2.24621	0.0782505
$825$	0	0
$826$	6.56155	0.228306
$827$	− 10.5616i	− 0.367261i	−0.982995	−	0.183631i	$-0.941215\pi$
$827$	− 10.5616i	− 0.367261i	0.982995	−	0.183631i	$-0.0587850\pi$
$828$	− 27.3693i	− 0.951150i
$829$	21.0540	0.731235	0.365617	−	0.930765i	$-0.380858\pi$
$829$	21.0540	0.731235	0.365617	+	0.930765i	$0.380858\pi$
$830$	0	0
$831$	−2.24621	−0.0779202
$832$	5.68466i	0.197080i
$833$	1.50758i	0.0522345i
$834$	−42.2462	−1.46287
$835$	0	0
$836$	−4.00000	−0.138343
$837$	7.36932i	0.254721i
$838$	23.8617i	0.824290i
$839$	−20.4924	−0.707477	−0.353738	−	0.935344i	$-0.615090\pi$
$839$	−20.4924	−0.707477	−0.353738	+	0.935344i	$0.615090\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	23.9309i	0.824712i
$843$	5.12311i	0.176449i
$844$	−8.31534	−0.286226
$845$	0	0
$846$	−22.2462	−0.764840
$847$	− 12.8078i	− 0.440080i
$848$	4.56155i	0.156644i
$849$	−54.1080	−1.85698
$850$	0	0
$851$	−46.1080	−1.58056
$852$	− 26.2462i	− 0.899180i
$853$	24.7386i	0.847035i	0.905888	+	0.423517i	$0.139205\pi$
$853$	24.7386i	0.847035i	−0.905888	+	0.423517i	$0.860795\pi$
$854$	−28.4924	−0.974991
$855$	0	0
$856$	5.43845	0.185882
$857$	14.6307i	0.499775i	0.968275	+	0.249887i	$0.0803936\pi$
$857$	14.6307i	0.499775i	−0.968275	+	0.249887i	$0.919606\pi$
$858$	− 58.2462i	− 1.98849i
$859$	52.9848	1.80782	0.903910	−	0.427723i	$-0.140684\pi$
$859$	52.9848	1.80782	0.903910	+	0.427723i	$0.140684\pi$
$860$	0	0
$861$	80.3542	2.73846
$862$	16.0000i	0.544962i
$863$	− 2.24621i	− 0.0764619i	−0.999269	−	0.0382310i	$-0.987828\pi$
$863$	− 2.24621i	− 0.0764619i	0.999269	−	0.0382310i	$-0.0121723\pi$
$864$	−1.43845	−0.0489370
$865$	0	0
$866$	−14.6307	−0.497171
$867$	13.2614i	0.450380i
$868$	− 13.1231i	− 0.445427i
$869$	20.4924	0.695158
$870$	0	0
$871$	−14.5616	−0.493399
$872$	0.561553i	0.0190166i
$873$	− 21.3693i	− 0.723242i
$874$	−7.68466	−0.259937
$875$	0	0
$876$	4.31534	0.145802
$877$	− 3.93087i	− 0.132736i	−0.997795	−	0.0663680i	$-0.978859\pi$
$877$	− 3.93087i	− 0.132736i	0.997795	−	0.0663680i	$-0.0211411\pi$
$878$	− 13.1231i	− 0.442883i
$879$	−56.8078	−1.91608
$880$	0	0
$881$	42.9848	1.44820	0.724098	−	0.689697i	$-0.242256\pi$
$881$	42.9848	1.44820	0.724098	+	0.689697i	$0.242256\pi$
$882$	1.56155i	0.0525802i
$883$	6.38447i	0.214855i	0.994213	+	0.107427i	$0.0342613\pi$
$883$	6.38447i	0.214855i	−0.994213	+	0.107427i	$0.965739\pi$
$884$	−19.5464	−0.657417
$885$	0	0
$886$	−2.24621	−0.0754629
$887$	28.4924i	0.956682i	0.878174	+	0.478341i	$0.158762\pi$
$887$	28.4924i	0.956682i	−0.878174	+	0.478341i	$0.841238\pi$
$888$	15.3693i	0.515761i
$889$	33.6155	1.12743
$890$	0	0
$891$	−28.0000	−0.938035
$892$	− 23.3693i	− 0.782463i
$893$	6.24621i	0.209021i
$894$	−34.2462	−1.14536
$895$	0	0
$896$	2.56155	0.0855755
$897$	− 111.901i	− 3.73625i
$898$	− 28.7386i	− 0.959021i
$899$	−29.1231	−0.971310
$900$	0	0
$901$	−15.6847	−0.522532
$902$	− 48.9848i	− 1.63102i
$903$	18.8769i	0.628184i
$904$	8.87689	0.295241
$905$	0	0
$906$	13.1231	0.435986
$907$	− 20.1771i	− 0.669969i	−0.942224	−	0.334984i	$-0.891269\pi$
$907$	− 20.1771i	− 0.669969i	0.942224	−	0.334984i	$-0.108731\pi$
$908$	− 25.9309i	− 0.860546i
$909$	61.8617	2.05182
$910$	0	0
$911$	4.49242	0.148841	0.0744203	−	0.997227i	$-0.476289\pi$
$911$	4.49242	0.148841	0.0744203	+	0.997227i	$0.476289\pi$
$912$	2.56155i	0.0848215i
$913$	− 11.5076i	− 0.380845i
$914$	6.31534	0.208893
$915$	0	0
$916$	−14.4924	−0.478843
$917$	42.2462i	1.39509i
$918$	− 4.94602i	− 0.163243i
$919$	2.06913	0.0682543	0.0341272	−	0.999417i	$-0.489135\pi$
$919$	2.06913	0.0682543	0.0341272	+	0.999417i	$0.489135\pi$
$920$	0	0
$921$	−83.2311	−2.74256
$922$	− 3.75379i	− 0.123624i
$923$	− 58.2462i	− 1.91720i
$924$	−26.2462	−0.863437
$925$	0	0
$926$	30.2462	0.993952
$927$	8.00000i	0.262754i
$928$	− 5.68466i	− 0.186608i
$929$	19.3002	0.633219	0.316609	−	0.948556i	$-0.397456\pi$
$929$	19.3002	0.633219	0.316609	+	0.948556i	$0.397456\pi$
$930$	0	0
$931$	0.438447	0.0143695
$932$	10.0000i	0.327561i
$933$	− 9.43845i	− 0.309001i
$934$	18.2462	0.597034
$935$	0	0
$936$	−20.2462	−0.661768
$937$	40.5616i	1.32509i	0.749023	+	0.662544i	$0.230523\pi$
$937$	40.5616i	1.32509i	−0.749023	+	0.662544i	$0.769477\pi$
$938$	6.56155i	0.214242i
$939$	12.9460	0.422478
$940$	0	0
$941$	54.8078	1.78668	0.893341	−	0.449379i	$-0.148355\pi$
$941$	54.8078	1.78668	0.893341	+	0.449379i	$0.148355\pi$
$942$	− 51.8617i	− 1.68975i
$943$	− 94.1080i	− 3.06458i
$944$	−2.56155	−0.0833714
$945$	0	0
$946$	11.5076	0.374144
$947$	− 34.2462i	− 1.11285i	−0.830897	−	0.556426i	$-0.812172\pi$
$947$	− 34.2462i	− 1.11285i	0.830897	−	0.556426i	$-0.187828\pi$
$948$	− 13.1231i	− 0.426219i
$949$	9.57671	0.310873
$950$	0	0
$951$	−33.4384	−1.08432
$952$	8.80776i	0.285461i
$953$	− 44.1080i	− 1.42880i	−0.699739	−	0.714398i	$-0.746700\pi$
$953$	− 44.1080i	− 1.42880i	0.699739	−	0.714398i	$-0.253300\pi$
$954$	−16.2462	−0.525991
$955$	0	0
$956$	−1.43845	−0.0465227
$957$	58.2462i	1.88283i
$958$	32.0000i	1.03387i
$959$	−37.9309	−1.22485
$960$	0	0
$961$	−4.75379	−0.153348
$962$	34.1080i	1.09968i
$963$	19.3693i	0.624168i
$964$	−23.1231	−0.744745
$965$	0	0
$966$	−50.4233	−1.62234
$967$	− 15.5076i	− 0.498690i	−0.968415	−	0.249345i	$-0.919785\pi$
$967$	− 15.5076i	− 0.498690i	0.968415	−	0.249345i	$-0.0802153\pi$
$968$	5.00000i	0.160706i
$969$	−8.80776	−0.282946
$970$	0	0
$971$	−56.4924	−1.81293	−0.906464	−	0.422283i	$-0.861229\pi$
$971$	−56.4924	−1.81293	−0.906464	+	0.422283i	$0.861229\pi$
$972$	22.2462i	0.713548i
$973$	42.2462i	1.35435i
$974$	−17.6155	−0.564438
$975$	0	0
$976$	11.1231	0.356042
$977$	28.7386i	0.919430i	0.888066	+	0.459715i	$0.152048\pi$
$977$	28.7386i	0.919430i	−0.888066	+	0.459715i	$0.847952\pi$
$978$	39.3693i	1.25889i
$979$	−8.00000	−0.255681
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	1.12311i	0.0358397i
$983$	− 18.8769i	− 0.602079i	−0.953612	−	0.301040i	$-0.902666\pi$
$983$	− 18.8769i	− 0.602079i	0.953612	−	0.301040i	$-0.0973337\pi$
$984$	−31.3693	−1.00002
$985$	0	0
$986$	19.5464	0.622484
$987$	40.9848i	1.30456i
$988$	5.68466i	0.180853i
$989$	22.1080	0.702992
$990$	0	0
$991$	2.87689	0.0913876	0.0456938	−	0.998955i	$-0.485450\pi$
$991$	2.87689	0.0913876	0.0456938	+	0.998955i	$0.485450\pi$
$992$	5.12311i	0.162659i
$993$	− 6.56155i	− 0.208225i
$994$	−26.2462	−0.832479
$995$	0	0
$996$	−7.36932	−0.233506
$997$	− 16.7386i	− 0.530118i	−0.964232	−	0.265059i	$-0.914609\pi$
$997$	− 16.7386i	− 0.530118i	0.964232	−	0.265059i	$-0.0853914\pi$
$998$	42.1080i	1.33290i
$999$	−8.63068	−0.273063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.b.f.799.2		4
5.2	odd	4		950.2.a.h.1.2		2
5.3	odd	4		190.2.a.d.1.1	✓	2
5.4	even	2	inner	950.2.b.f.799.3		4
15.2	even	4		8550.2.a.br.1.2		2
15.8	even	4		1710.2.a.w.1.1		2
20.3	even	4		1520.2.a.n.1.2		2
20.7	even	4		7600.2.a.y.1.1		2
35.13	even	4		9310.2.a.bc.1.2		2
40.3	even	4		6080.2.a.bb.1.1		2
40.13	odd	4		6080.2.a.bh.1.2		2
95.18	even	4		3610.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.a.d.1.1	✓	2	5.3	odd	4
950.2.a.h.1.2		2	5.2	odd	4
950.2.b.f.799.2		4	1.1	even	1	trivial
950.2.b.f.799.3		4	5.4	even	2	inner
1520.2.a.n.1.2		2	20.3	even	4
1710.2.a.w.1.1		2	15.8	even	4
3610.2.a.t.1.2		2	95.18	even	4
6080.2.a.bb.1.1		2	40.3	even	4
6080.2.a.bh.1.2		2	40.13	odd	4
7600.2.a.y.1.1		2	20.7	even	4
8550.2.a.br.1.2		2	15.2	even	4
9310.2.a.bc.1.2		2	35.13	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:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +2.56155i q^{3} -1.00000 q^{4} +2.56155 q^{6} -2.56155i q^{7} +1.00000i q^{8} -3.56155 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +2.56155i q^{3} -1.00000 q^{4} +2.56155 q^{6} -2.56155i q^{7} +1.00000i q^{8} -3.56155 q^{9} +4.00000 q^{11} -2.56155i q^{12} -5.68466i q^{13} -2.56155 q^{14} +1.00000 q^{16} +3.43845i q^{17} +3.56155i q^{18} +1.00000 q^{19} +6.56155 q^{21} -4.00000i q^{22} -7.68466i q^{23} -2.56155 q^{24} -5.68466 q^{26} -1.43845i q^{27} +2.56155i q^{28} +5.68466 q^{29} -5.12311 q^{31} -1.00000i q^{32} +10.2462i q^{33} +3.43845 q^{34} +3.56155 q^{36} -6.00000i q^{37} -1.00000i q^{38} +14.5616 q^{39} +12.2462 q^{41} -6.56155i q^{42} +2.87689i q^{43} -4.00000 q^{44} -7.68466 q^{46} +6.24621i q^{47} +2.56155i q^{48} +0.438447 q^{49} -8.80776 q^{51} +5.68466i q^{52} +4.56155i q^{53} -1.43845 q^{54} +2.56155 q^{56} +2.56155i q^{57} -5.68466i q^{58} -2.56155 q^{59} +11.1231 q^{61} +5.12311i q^{62} +9.12311i q^{63} -1.00000 q^{64} +10.2462 q^{66} -2.56155i q^{67} -3.43845i q^{68} +19.6847 q^{69} +10.2462 q^{71} -3.56155i q^{72} +1.68466i q^{73} -6.00000 q^{74} -1.00000 q^{76} -10.2462i q^{77} -14.5616i q^{78} +5.12311 q^{79} -7.00000 q^{81} -12.2462i q^{82} -2.87689i q^{83} -6.56155 q^{84} +2.87689 q^{86} +14.5616i q^{87} +4.00000i q^{88} -2.00000 q^{89} -14.5616 q^{91} +7.68466i q^{92} -13.1231i q^{93} +6.24621 q^{94} +2.56155 q^{96} +6.00000i q^{97} -0.438447i q^{98} -14.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 2 q^{6} - 6 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 2 * q^6 - 6 * q^9	\( 4 q - 4 q^{4} + 2 q^{6} - 6 q^{9} + 16 q^{11} - 2 q^{14} + 4 q^{16} + 4 q^{19} + 18 q^{21} - 2 q^{24} + 2 q^{26} - 2 q^{29} - 4 q^{31} + 22 q^{34} + 6 q^{36} + 50 q^{39} + 16 q^{41} - 16 q^{44} - 6 q^{46} + 10 q^{49} + 6 q^{51} - 14 q^{54} + 2 q^{56} - 2 q^{59} + 28 q^{61} - 4 q^{64} + 8 q^{66} + 54 q^{69} + 8 q^{71} - 24 q^{74} - 4 q^{76} + 4 q^{79} - 28 q^{81} - 18 q^{84} + 28 q^{86} - 8 q^{89} - 50 q^{91} - 8 q^{94} + 2 q^{96} - 24 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 2 * q^6 - 6 * q^9 + 16 * q^11 - 2 * q^14 + 4 * q^16 + 4 * q^19 + 18 * q^21 - 2 * q^24 + 2 * q^26 - 2 * q^29 - 4 * q^31 + 22 * q^34 + 6 * q^36 + 50 * q^39 + 16 * q^41 - 16 * q^44 - 6 * q^46 + 10 * q^49 + 6 * q^51 - 14 * q^54 + 2 * q^56 - 2 * q^59 + 28 * q^61 - 4 * q^64 + 8 * q^66 + 54 * q^69 + 8 * q^71 - 24 * q^74 - 4 * q^76 + 4 * q^79 - 28 * q^81 - 18 * q^84 + 28 * q^86 - 8 * q^89 - 50 * q^91 - 8 * q^94 + 2 * q^96 - 24 * q^99

Introduction

L-functions

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