LMFDB - Embedded newform 4009.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.a (trivial)

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:	yes
Analytic conductor:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.801938 q^{2} +0.801938 q^{3} -1.35690 q^{4} -2.69202 q^{5} -0.643104 q^{6} -2.04892 q^{7} +2.69202 q^{8} -2.35690 q^{9} +O(q^{10})$	\(q-0.801938 q^{2} +0.801938 q^{3} -1.35690 q^{4} -2.69202 q^{5} -0.643104 q^{6} -2.04892 q^{7} +2.69202 q^{8} -2.35690 q^{9} +2.15883 q^{10} +3.49396 q^{11} -1.08815 q^{12} +1.64310 q^{14} -2.15883 q^{15} +0.554958 q^{16} -2.24698 q^{17} +1.89008 q^{18} +1.00000 q^{19} +3.65279 q^{20} -1.64310 q^{21} -2.80194 q^{22} +0.554958 q^{23} +2.15883 q^{24} +2.24698 q^{25} -4.29590 q^{27} +2.78017 q^{28} +3.69202 q^{29} +1.73125 q^{30} +6.63102 q^{31} -5.82908 q^{32} +2.80194 q^{33} +1.80194 q^{34} +5.51573 q^{35} +3.19806 q^{36} +5.37867 q^{37} -0.801938 q^{38} -7.24698 q^{40} +2.63102 q^{41} +1.31767 q^{42} +2.60388 q^{43} -4.74094 q^{44} +6.34481 q^{45} -0.445042 q^{46} +6.11960 q^{47} +0.445042 q^{48} -2.80194 q^{49} -1.80194 q^{50} -1.80194 q^{51} -7.30798 q^{53} +3.44504 q^{54} -9.40581 q^{55} -5.51573 q^{56} +0.801938 q^{57} -2.96077 q^{58} -6.65279 q^{59} +2.92931 q^{60} -1.60388 q^{61} -5.31767 q^{62} +4.82908 q^{63} +3.56465 q^{64} -2.24698 q^{66} +9.65817 q^{67} +3.04892 q^{68} +0.445042 q^{69} -4.42327 q^{70} +10.3937 q^{71} -6.34481 q^{72} -0.405813 q^{73} -4.31336 q^{74} +1.80194 q^{75} -1.35690 q^{76} -7.15883 q^{77} -9.30798 q^{79} -1.49396 q^{80} +3.62565 q^{81} -2.10992 q^{82} -6.51573 q^{83} +2.22952 q^{84} +6.04892 q^{85} -2.08815 q^{86} +2.96077 q^{87} +9.40581 q^{88} +10.9433 q^{89} -5.08815 q^{90} -0.753020 q^{92} +5.31767 q^{93} -4.90754 q^{94} -2.69202 q^{95} -4.67456 q^{96} -5.15883 q^{97} +2.24698 q^{98} -8.23490 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} - 2 q^{3} - 3 q^{5} - 6 q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 - 2 * q^3 - 3 * q^5 - 6 * q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9	$ 3 q + 2 q^{2} - 2 q^{3} - 3 q^{5} - 6 q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9} - 2 q^{10} + q^{11} - 7 q^{12} + 9 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 5 q^{18} + 3 q^{19} - 7 q^{20} - 9 q^{21} - 4 q^{22} + 2 q^{23} - 2 q^{24} + 2 q^{25} + q^{27} + 7 q^{28} + 6 q^{29} + 13 q^{30} + 5 q^{31} - 7 q^{32} + 4 q^{33} + q^{34} + 4 q^{35} + 14 q^{36} + 9 q^{37} + 2 q^{38} - 17 q^{40} - 7 q^{41} - 13 q^{42} - q^{43} - 4 q^{45} - q^{46} - 3 q^{47} + q^{48} - 4 q^{49} - q^{50} - q^{51} - 27 q^{53} + 10 q^{54} - 15 q^{55} - 4 q^{56} - 2 q^{57} + 4 q^{58} - 2 q^{59} + 21 q^{60} + 4 q^{61} + q^{62} + 4 q^{63} - 11 q^{64} - 2 q^{66} + 8 q^{67} + q^{69} - 16 q^{70} - q^{71} + 4 q^{72} + 12 q^{73} - 15 q^{74} + q^{75} - 13 q^{77} - 33 q^{79} + 5 q^{80} - q^{81} - 7 q^{82} - 7 q^{83} - 14 q^{84} + 9 q^{85} - 10 q^{86} - 4 q^{87} + 15 q^{88} + 4 q^{89} - 19 q^{90} - 7 q^{92} - q^{93} - 30 q^{94} - 3 q^{95} + 7 q^{96} - 7 q^{97} + 2 q^{98} - q^{99}+O(q^{100}) $ 3 * q + 2 * q^2 - 2 * q^3 - 3 * q^5 - 6 * q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9 - 2 * q^10 + q^11 - 7 * q^12 + 9 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 + 5 * q^18 + 3 * q^19 - 7 * q^20 - 9 * q^21 - 4 * q^22 + 2 * q^23 - 2 * q^24 + 2 * q^25 + q^27 + 7 * q^28 + 6 * q^29 + 13 * q^30 + 5 * q^31 - 7 * q^32 + 4 * q^33 + q^34 + 4 * q^35 + 14 * q^36 + 9 * q^37 + 2 * q^38 - 17 * q^40 - 7 * q^41 - 13 * q^42 - q^43 - 4 * q^45 - q^46 - 3 * q^47 + q^48 - 4 * q^49 - q^50 - q^51 - 27 * q^53 + 10 * q^54 - 15 * q^55 - 4 * q^56 - 2 * q^57 + 4 * q^58 - 2 * q^59 + 21 * q^60 + 4 * q^61 + q^62 + 4 * q^63 - 11 * q^64 - 2 * q^66 + 8 * q^67 + q^69 - 16 * q^70 - q^71 + 4 * q^72 + 12 * q^73 - 15 * q^74 + q^75 - 13 * q^77 - 33 * q^79 + 5 * q^80 - q^81 - 7 * q^82 - 7 * q^83 - 14 * q^84 + 9 * q^85 - 10 * q^86 - 4 * q^87 + 15 * q^88 + 4 * q^89 - 19 * q^90 - 7 * q^92 - q^93 - 30 * q^94 - 3 * q^95 + 7 * q^96 - 7 * q^97 + 2 * q^98 - q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−0.801938	−0.567056	−0.283528	−	0.958964i	$-0.591505\pi$
$2$	−0.801938	−0.567056	−0.283528	+	0.958964i	$0.591505\pi$
$3$	0.801938	0.462999	0.231499	−	0.972835i	$-0.425637\pi$
$3$	0.801938	0.462999	0.231499	+	0.972835i	$0.425637\pi$
$4$	−1.35690	−0.678448
$5$	−2.69202	−1.20391	−0.601954	−	0.798531i	$-0.705611\pi$
$5$	−2.69202	−1.20391	−0.601954	+	0.798531i	$0.705611\pi$
$6$	−0.643104	−0.262546
$7$	−2.04892	−0.774418	−0.387209	−	0.921992i	$-0.626561\pi$
$7$	−2.04892	−0.774418	−0.387209	+	0.921992i	$0.626561\pi$
$8$	2.69202	0.951773
$9$	−2.35690	−0.785632
$10$	2.15883	0.682683
$11$	3.49396	1.05347	0.526734	−	0.850030i	$-0.323416\pi$
$11$	3.49396	1.05347	0.526734	+	0.850030i	$0.323416\pi$
$12$	−1.08815	−0.314121
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.64310	0.439138
$15$	−2.15883	−0.557408
$16$	0.554958	0.138740
$17$	−2.24698	−0.544973	−0.272486	−	0.962160i	$-0.587846\pi$
$17$	−2.24698	−0.544973	−0.272486	+	0.962160i	$0.587846\pi$
$18$	1.89008	0.445497
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.65279	0.816789
$21$	−1.64310	−0.358555
$22$	−2.80194	−0.597375
$23$	0.554958	0.115717	0.0578584	−	0.998325i	$-0.481573\pi$
$23$	0.554958	0.115717	0.0578584	+	0.998325i	$0.481573\pi$
$24$	2.15883	0.440670
$25$	2.24698	0.449396
$26$	0	0
$27$	−4.29590	−0.826746
$28$	2.78017	0.525402
$29$	3.69202	0.685591	0.342796	−	0.939410i	$-0.388626\pi$
$29$	3.69202	0.685591	0.342796	+	0.939410i	$0.388626\pi$
$30$	1.73125	0.316082
$31$	6.63102	1.19097	0.595483	−	0.803368i	$-0.296961\pi$
$31$	6.63102	1.19097	0.595483	+	0.803368i	$0.296961\pi$
$32$	−5.82908	−1.03045
$33$	2.80194	0.487755
$34$	1.80194	0.309030
$35$	5.51573	0.932328
$36$	3.19806	0.533010
$37$	5.37867	0.884247	0.442124	−	0.896954i	$-0.354225\pi$
$37$	5.37867	0.884247	0.442124	+	0.896954i	$0.354225\pi$
$38$	−0.801938	−0.130091
$39$	0	0
$40$	−7.24698	−1.14585
$41$	2.63102	0.410897	0.205448	−	0.978668i	$-0.434135\pi$
$41$	2.63102	0.410897	0.205448	+	0.978668i	$0.434135\pi$
$42$	1.31767	0.203320
$43$	2.60388	0.397087	0.198544	−	0.980092i	$-0.436379\pi$
$43$	2.60388	0.397087	0.198544	+	0.980092i	$0.436379\pi$
$44$	−4.74094	−0.714723
$45$	6.34481	0.945829
$46$	−0.445042	−0.0656178
$47$	6.11960	0.892636	0.446318	−	0.894874i	$-0.352735\pi$
$47$	6.11960	0.892636	0.446318	+	0.894874i	$0.352735\pi$
$48$	0.445042	0.0642363
$49$	−2.80194	−0.400277
$50$	−1.80194	−0.254832
$51$	−1.80194	−0.252322
$52$	0	0
$53$	−7.30798	−1.00383	−0.501914	−	0.864918i	$-0.667371\pi$
$53$	−7.30798	−1.00383	−0.501914	+	0.864918i	$0.667371\pi$
$54$	3.44504	0.468811
$55$	−9.40581	−1.26828
$56$	−5.51573	−0.737070
$57$	0.801938	0.106219
$58$	−2.96077	−0.388768
$59$	−6.65279	−0.866120	−0.433060	−	0.901365i	$-0.642566\pi$
$59$	−6.65279	−0.866120	−0.433060	+	0.901365i	$0.642566\pi$
$60$	2.92931	0.378173
$61$	−1.60388	−0.205355	−0.102678	−	0.994715i	$-0.532741\pi$
$61$	−1.60388	−0.205355	−0.102678	+	0.994715i	$0.532741\pi$
$62$	−5.31767	−0.675344
$63$	4.82908	0.608407
$64$	3.56465	0.445581
$65$	0	0
$66$	−2.24698	−0.276584
$67$	9.65817	1.17993	0.589967	−	0.807428i	$-0.299141\pi$
$67$	9.65817	1.17993	0.589967	+	0.807428i	$0.299141\pi$
$68$	3.04892	0.369736
$69$	0.445042	0.0535767
$70$	−4.42327	−0.528682
$71$	10.3937	1.23351	0.616755	−	0.787156i	$-0.288447\pi$
$71$	10.3937	1.23351	0.616755	+	0.787156i	$0.288447\pi$
$72$	−6.34481	−0.747744
$73$	−0.405813	−0.0474968	−0.0237484	−	0.999718i	$-0.507560\pi$
$73$	−0.405813	−0.0474968	−0.0237484	+	0.999718i	$0.507560\pi$
$74$	−4.31336	−0.501417
$75$	1.80194	0.208070
$76$	−1.35690	−0.155647
$77$	−7.15883	−0.815825
$78$	0	0
$79$	−9.30798	−1.04723	−0.523615	−	0.851955i	$-0.675417\pi$
$79$	−9.30798	−1.04723	−0.523615	+	0.851955i	$0.675417\pi$
$80$	−1.49396	−0.167030
$81$	3.62565	0.402850
$82$	−2.10992	−0.233001
$83$	−6.51573	−0.715194	−0.357597	−	0.933876i	$-0.616404\pi$
$83$	−6.51573	−0.715194	−0.357597	+	0.933876i	$0.616404\pi$
$84$	2.22952	0.243261
$85$	6.04892	0.656097
$86$	−2.08815	−0.225171
$87$	2.96077	0.317428
$88$	9.40581	1.00266
$89$	10.9433	1.15999	0.579994	−	0.814620i	$-0.303055\pi$
$89$	10.9433	1.15999	0.579994	+	0.814620i	$0.303055\pi$
$90$	−5.08815	−0.536338
$91$	0	0
$92$	−0.753020	−0.0785078
$93$	5.31767	0.551416
$94$	−4.90754	−0.506174
$95$	−2.69202	−0.276196
$96$	−4.67456	−0.477096
$97$	−5.15883	−0.523800	−0.261900	−	0.965095i	$-0.584349\pi$
$97$	−5.15883	−0.523800	−0.261900	+	0.965095i	$0.584349\pi$
$98$	2.24698	0.226979
$99$	−8.23490	−0.827638
$100$	−3.04892	−0.304892
$101$	−2.31336	−0.230187	−0.115094	−	0.993355i	$-0.536717\pi$
$101$	−2.31336	−0.230187	−0.115094	+	0.993355i	$0.536717\pi$
$102$	1.44504	0.143080
$103$	3.40581	0.335585	0.167792	−	0.985822i	$-0.446336\pi$
$103$	3.40581	0.335585	0.167792	+	0.985822i	$0.446336\pi$
$104$	0	0
$105$	4.42327	0.431667
$106$	5.86054	0.569226
$107$	17.9095	1.73137	0.865686	−	0.500587i	$-0.166882\pi$
$107$	17.9095	1.73137	0.865686	+	0.500587i	$0.166882\pi$
$108$	5.82908	0.560904
$109$	−10.6093	−1.01618	−0.508091	−	0.861303i	$-0.669649\pi$
$109$	−10.6093	−1.01618	−0.508091	+	0.861303i	$0.669649\pi$
$110$	7.54288	0.719185
$111$	4.31336	0.409406
$112$	−1.13706	−0.107442
$113$	−16.8901	−1.58889	−0.794443	−	0.607339i	$-0.792237\pi$
$113$	−16.8901	−1.58889	−0.794443	+	0.607339i	$0.792237\pi$
$114$	−0.643104	−0.0602322
$115$	−1.49396	−0.139312
$116$	−5.00969	−0.465138
$117$	0	0
$118$	5.33513	0.491138
$119$	4.60388	0.422037
$120$	−5.81163	−0.530526
$121$	1.20775	0.109796
$122$	1.28621	0.116448
$123$	2.10992	0.190245
$124$	−8.99761	−0.808009
$125$	7.41119	0.662877
$126$	−3.87263	−0.345001
$127$	−18.8877	−1.67601	−0.838006	−	0.545661i	$-0.816278\pi$
$127$	−18.8877	−1.67601	−0.838006	+	0.545661i	$0.816278\pi$
$128$	8.79954	0.777777
$129$	2.08815	0.183851
$130$	0	0
$131$	−9.96077	−0.870277	−0.435138	−	0.900364i	$-0.643301\pi$
$131$	−9.96077	−0.870277	−0.435138	+	0.900364i	$0.643301\pi$
$132$	−3.80194	−0.330916
$133$	−2.04892	−0.177664
$134$	−7.74525	−0.669088
$135$	11.5646	0.995326
$136$	−6.04892	−0.518690
$137$	2.70410	0.231027	0.115514	−	0.993306i	$-0.463149\pi$
$137$	2.70410	0.231027	0.115514	+	0.993306i	$0.463149\pi$
$138$	−0.356896	−0.0303810
$139$	−1.60925	−0.136495	−0.0682475	−	0.997668i	$-0.521741\pi$
$139$	−1.60925	−0.136495	−0.0682475	+	0.997668i	$0.521741\pi$
$140$	−7.48427	−0.632536
$141$	4.90754	0.413290
$142$	−8.33513	−0.699468
$143$	0	0
$144$	−1.30798	−0.108998
$145$	−9.93900	−0.825389
$146$	0.325437	0.0269334
$147$	−2.24698	−0.185328
$148$	−7.29829	−0.599916
$149$	−7.61596	−0.623924	−0.311962	−	0.950095i	$-0.600986\pi$
$149$	−7.61596	−0.623924	−0.311962	+	0.950095i	$0.600986\pi$
$150$	−1.44504	−0.117987
$151$	−5.16852	−0.420608	−0.210304	−	0.977636i	$-0.567445\pi$
$151$	−5.16852	−0.420608	−0.210304	+	0.977636i	$0.567445\pi$
$152$	2.69202	0.218352
$153$	5.29590	0.428148
$154$	5.74094	0.462618
$155$	−17.8509	−1.43382
$156$	0	0
$157$	−11.0218	−0.879633	−0.439817	−	0.898088i	$-0.644957\pi$
$157$	−11.0218	−0.879633	−0.439817	+	0.898088i	$0.644957\pi$
$158$	7.46442	0.593837
$159$	−5.86054	−0.464771
$160$	15.6920	1.24056
$161$	−1.13706	−0.0896131
$162$	−2.90754	−0.228438
$163$	24.9758	1.95626	0.978129	−	0.207998i	$-0.0666947\pi$
$163$	24.9758	1.95626	0.978129	+	0.207998i	$0.0666947\pi$
$164$	−3.57002	−0.278772
$165$	−7.54288	−0.587212
$166$	5.22521	0.405555
$167$	−0.215521	−0.0166775	−0.00833874	−	0.999965i	$-0.502654\pi$
$167$	−0.215521	−0.0166775	−0.00833874	+	0.999965i	$0.502654\pi$
$168$	−4.42327	−0.341263
$169$	−13.0000	−1.00000
$170$	−4.85086	−0.372044
$171$	−2.35690	−0.180236
$172$	−3.53319	−0.269403
$173$	−6.99462	−0.531791	−0.265896	−	0.964002i	$-0.585668\pi$
$173$	−6.99462	−0.531791	−0.265896	+	0.964002i	$0.585668\pi$
$174$	−2.37435	−0.179999
$175$	−4.60388	−0.348020
$176$	1.93900	0.146158
$177$	−5.33513	−0.401013
$178$	−8.77586	−0.657778
$179$	−20.2567	−1.51405	−0.757027	−	0.653383i	$-0.773349\pi$
$179$	−20.2567	−1.51405	−0.757027	+	0.653383i	$0.773349\pi$
$180$	−8.60925	−0.641696
$181$	−14.2392	−1.05839	−0.529196	−	0.848500i	$-0.677506\pi$
$181$	−14.2392	−1.05839	−0.529196	+	0.848500i	$0.677506\pi$
$182$	0	0
$183$	−1.28621	−0.0950793
$184$	1.49396	0.110136
$185$	−14.4795	−1.06455
$186$	−4.26444	−0.312684
$187$	−7.85086	−0.574111
$188$	−8.30367	−0.605607
$189$	8.80194	0.640247
$190$	2.15883	0.156618
$191$	5.02715	0.363752	0.181876	−	0.983322i	$-0.441783\pi$
$191$	5.02715	0.363752	0.181876	+	0.983322i	$0.441783\pi$
$192$	2.85862	0.206303
$193$	−11.4155	−0.821706	−0.410853	−	0.911702i	$-0.634769\pi$
$193$	−11.4155	−0.821706	−0.410853	+	0.911702i	$0.634769\pi$
$194$	4.13706	0.297024
$195$	0	0
$196$	3.80194	0.271567
$197$	−0.862937	−0.0614817	−0.0307408	−	0.999527i	$-0.509787\pi$
$197$	−0.862937	−0.0614817	−0.0307408	+	0.999527i	$0.509787\pi$
$198$	6.60388	0.469317
$199$	−14.8073	−1.04966	−0.524831	−	0.851206i	$-0.675872\pi$
$199$	−14.8073	−1.04966	−0.524831	+	0.851206i	$0.675872\pi$
$200$	6.04892	0.427723
$201$	7.74525	0.546308
$202$	1.85517	0.130529
$203$	−7.56465	−0.530934
$204$	2.44504	0.171187
$205$	−7.08277	−0.494682
$206$	−2.73125	−0.190295
$207$	−1.30798	−0.0909108
$208$	0	0
$209$	3.49396	0.241682
$210$	−3.54719	−0.244779
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	9.91617	0.681045
$213$	8.33513	0.571113
$214$	−14.3623	−0.981785
$215$	−7.00969	−0.478057
$216$	−11.5646	−0.786875
$217$	−13.5864	−0.922306
$218$	8.50796	0.576232
$219$	−0.325437	−0.0219910
$220$	12.7627	0.860462
$221$	0	0
$222$	−3.45904	−0.232156
$223$	27.0683	1.81263	0.906314	−	0.422606i	$-0.138885\pi$
$223$	27.0683	1.81263	0.906314	+	0.422606i	$0.138885\pi$
$224$	11.9433	0.797996
$225$	−5.29590	−0.353060
$226$	13.5448	0.900986
$227$	16.9323	1.12384	0.561918	−	0.827193i	$-0.310064\pi$
$227$	16.9323	1.12384	0.561918	+	0.827193i	$0.310064\pi$
$228$	−1.08815	−0.0720642
$229$	−13.7627	−0.909465	−0.454733	−	0.890628i	$-0.650265\pi$
$229$	−13.7627	−0.909465	−0.454733	+	0.890628i	$0.650265\pi$
$230$	1.19806	0.0789979
$231$	−5.74094	−0.377726
$232$	9.93900	0.652527
$233$	−28.8388	−1.88929	−0.944645	−	0.328093i	$-0.893594\pi$
$233$	−28.8388	−1.88929	−0.944645	+	0.328093i	$0.893594\pi$
$234$	0	0
$235$	−16.4741	−1.07465
$236$	9.02715	0.587617
$237$	−7.46442	−0.484866
$238$	−3.69202	−0.239318
$239$	−9.29350	−0.601147	−0.300573	−	0.953759i	$-0.597178\pi$
$239$	−9.29350	−0.601147	−0.300573	+	0.953759i	$0.597178\pi$
$240$	−1.19806	−0.0773346
$241$	9.12200	0.587600	0.293800	−	0.955867i	$-0.405080\pi$
$241$	9.12200	0.587600	0.293800	+	0.955867i	$0.405080\pi$
$242$	−0.968541	−0.0622602
$243$	15.7952	1.01326
$244$	2.17629	0.139323
$245$	7.54288	0.481897
$246$	−1.69202	−0.107879
$247$	0	0
$248$	17.8509	1.13353
$249$	−5.22521	−0.331134
$250$	−5.94331	−0.375888
$251$	−0.0459334	−0.00289929	−0.00144965	−	0.999999i	$-0.500461\pi$
$251$	−0.0459334	−0.00289929	−0.00144965	+	0.999999i	$0.500461\pi$
$252$	−6.55257	−0.412773
$253$	1.93900	0.121904
$254$	15.1468	0.950392
$255$	4.85086	0.303772
$256$	−14.1860	−0.886624
$257$	−10.1631	−0.633960	−0.316980	−	0.948432i	$-0.602669\pi$
$257$	−10.1631	−0.633960	−0.316980	+	0.948432i	$0.602669\pi$
$258$	−1.67456	−0.104254
$259$	−11.0204	−0.684777
$260$	0	0
$261$	−8.70171	−0.538622
$262$	7.98792	0.493495
$263$	−20.7778	−1.28121	−0.640606	−	0.767870i	$-0.721317\pi$
$263$	−20.7778	−1.28121	−0.640606	+	0.767870i	$0.721317\pi$
$264$	7.54288	0.464232
$265$	19.6732	1.20852
$266$	1.64310	0.100745
$267$	8.77586	0.537074
$268$	−13.1051	−0.800523
$269$	−3.58748	−0.218733	−0.109366	−	0.994002i	$-0.534882\pi$
$269$	−3.58748	−0.218733	−0.109366	+	0.994002i	$0.534882\pi$
$270$	−9.27413	−0.564405
$271$	−1.44743	−0.0879254	−0.0439627	−	0.999033i	$-0.513998\pi$
$271$	−1.44743	−0.0879254	−0.0439627	+	0.999033i	$0.513998\pi$
$272$	−1.24698	−0.0756092
$273$	0	0
$274$	−2.16852	−0.131005
$275$	7.85086	0.473424
$276$	−0.603875	−0.0363490
$277$	−5.38404	−0.323496	−0.161748	−	0.986832i	$-0.551713\pi$
$277$	−5.38404	−0.323496	−0.161748	+	0.986832i	$0.551713\pi$
$278$	1.29052	0.0774003
$279$	−15.6286	−0.935662
$280$	14.8485	0.887365
$281$	−5.40880	−0.322662	−0.161331	−	0.986900i	$-0.551579\pi$
$281$	−5.40880	−0.322662	−0.161331	+	0.986900i	$0.551579\pi$
$282$	−3.93554	−0.234358
$283$	−8.46442	−0.503157	−0.251579	−	0.967837i	$-0.580950\pi$
$283$	−8.46442	−0.503157	−0.251579	+	0.967837i	$0.580950\pi$
$284$	−14.1032	−0.836872
$285$	−2.15883	−0.127878
$286$	0	0
$287$	−5.39075	−0.318206
$288$	13.7385	0.809552
$289$	−11.9511	−0.703005
$290$	7.97046	0.468042
$291$	−4.13706	−0.242519
$292$	0.550646	0.0322241
$293$	−0.314683	−0.0183840	−0.00919200	−	0.999958i	$-0.502926\pi$
$293$	−0.314683	−0.0183840	−0.00919200	+	0.999958i	$0.502926\pi$
$294$	1.80194	0.105091
$295$	17.9095	1.04273
$296$	14.4795	0.841603
$297$	−15.0097	−0.870950
$298$	6.10752	0.353799
$299$	0	0
$300$	−2.44504	−0.141165
$301$	−5.33513	−0.307512
$302$	4.14483	0.238508
$303$	−1.85517	−0.106577
$304$	0.554958	0.0318290
$305$	4.31767	0.247229
$306$	−4.24698	−0.242784
$307$	0.819396	0.0467654	0.0233827	−	0.999727i	$-0.492556\pi$
$307$	0.819396	0.0467654	0.0233827	+	0.999727i	$0.492556\pi$
$308$	9.71379	0.553495
$309$	2.73125	0.155375
$310$	14.3153	0.813053
$311$	−13.5864	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$311$	−13.5864	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$312$	0	0
$313$	21.0586	1.19030	0.595151	−	0.803614i	$-0.297092\pi$
$313$	21.0586	1.19030	0.595151	+	0.803614i	$0.297092\pi$
$314$	8.83877	0.498801
$315$	−13.0000	−0.732467
$316$	12.6300	0.710491
$317$	−14.8224	−0.832508	−0.416254	−	0.909248i	$-0.636657\pi$
$317$	−14.8224	−0.832508	−0.416254	+	0.909248i	$0.636657\pi$
$318$	4.69979	0.263551
$319$	12.8998	0.722249
$320$	−9.59611	−0.536439
$321$	14.3623	0.801624
$322$	0.911854	0.0508156
$323$	−2.24698	−0.125025
$324$	−4.91962	−0.273312
$325$	0	0
$326$	−20.0291	−1.10931
$327$	−8.50796	−0.470491
$328$	7.08277	0.391081
$329$	−12.5386	−0.691273
$330$	6.04892	0.332982
$331$	34.7101	1.90784	0.953919	−	0.300064i	$-0.0970081\pi$
$331$	34.7101	1.90784	0.953919	+	0.300064i	$0.0970081\pi$
$332$	8.84117	0.485222
$333$	−12.6770	−0.694693
$334$	0.172834	0.00945706
$335$	−26.0000	−1.42053
$336$	−0.911854	−0.0497457
$337$	17.7362	0.966150	0.483075	−	0.875579i	$-0.339520\pi$
$337$	17.7362	0.966150	0.483075	+	0.875579i	$0.339520\pi$
$338$	10.4252	0.567056
$339$	−13.5448	−0.735652
$340$	−8.20775	−0.445128
$341$	23.1685	1.25465
$342$	1.89008	0.102204
$343$	20.0834	1.08440
$344$	7.00969	0.377937
$345$	−1.19806	−0.0645015
$346$	5.60925	0.301555
$347$	−7.76271	−0.416724	−0.208362	−	0.978052i	$-0.566813\pi$
$347$	−7.76271	−0.416724	−0.208362	+	0.978052i	$0.566813\pi$
$348$	−4.01746	−0.215358
$349$	−10.2185	−0.546984	−0.273492	−	0.961874i	$-0.588179\pi$
$349$	−10.2185	−0.546984	−0.273492	+	0.961874i	$0.588179\pi$
$350$	3.69202	0.197347
$351$	0	0
$352$	−20.3666	−1.08554
$353$	6.42865	0.342162	0.171081	−	0.985257i	$-0.445274\pi$
$353$	6.42865	0.342162	0.171081	+	0.985257i	$0.445274\pi$
$354$	4.27844	0.227396
$355$	−27.9801	−1.48503
$356$	−14.8489	−0.786992
$357$	3.69202	0.195402
$358$	16.2446	0.858553
$359$	−24.1183	−1.27291	−0.636457	−	0.771312i	$-0.719601\pi$
$359$	−24.1183	−1.27291	−0.636457	+	0.771312i	$0.719601\pi$
$360$	17.0804	0.900215
$361$	1.00000	0.0526316
$362$	11.4190	0.600167
$363$	0.968541	0.0508352
$364$	0	0
$365$	1.09246	0.0571819
$366$	1.03146	0.0539152
$367$	6.75973	0.352855	0.176427	−	0.984314i	$-0.443546\pi$
$367$	6.75973	0.352855	0.176427	+	0.984314i	$0.443546\pi$
$368$	0.307979	0.0160545
$369$	−6.20105	−0.322814
$370$	11.6116	0.603661
$371$	14.9734	0.777382
$372$	−7.21552	−0.374107
$373$	18.7289	0.969743	0.484872	−	0.874585i	$-0.338866\pi$
$373$	18.7289	0.969743	0.484872	+	0.874585i	$0.338866\pi$
$374$	6.29590	0.325553
$375$	5.94331	0.306911
$376$	16.4741	0.849587
$377$	0	0
$378$	−7.05861	−0.363056
$379$	13.8310	0.710451	0.355225	−	0.934781i	$-0.384404\pi$
$379$	13.8310	0.710451	0.355225	+	0.934781i	$0.384404\pi$
$380$	3.65279	0.187384
$381$	−15.1468	−0.775992
$382$	−4.03146	−0.206267
$383$	18.2416	0.932102	0.466051	−	0.884758i	$-0.345676\pi$
$383$	18.2416	0.932102	0.466051	+	0.884758i	$0.345676\pi$
$384$	7.05669	0.360110
$385$	19.2717	0.982178
$386$	9.15452	0.465953
$387$	−6.13706	−0.311964
$388$	7.00000	0.355371
$389$	−22.3370	−1.13253	−0.566267	−	0.824222i	$-0.691613\pi$
$389$	−22.3370	−1.13253	−0.566267	+	0.824222i	$0.691613\pi$
$390$	0	0
$391$	−1.24698	−0.0630625
$392$	−7.54288	−0.380973
$393$	−7.98792	−0.402937
$394$	0.692021	0.0348635
$395$	25.0573	1.26077
$396$	11.1739	0.561510
$397$	−30.6993	−1.54075	−0.770377	−	0.637588i	$-0.779932\pi$
$397$	−30.6993	−1.54075	−0.770377	+	0.637588i	$0.779932\pi$
$398$	11.8745	0.595217
$399$	−1.64310	−0.0822581
$400$	1.24698	0.0623490
$401$	26.7506	1.33586	0.667931	−	0.744223i	$-0.267180\pi$
$401$	26.7506	1.33586	0.667931	+	0.744223i	$0.267180\pi$
$402$	−6.21121	−0.309787
$403$	0	0
$404$	3.13898	0.156170
$405$	−9.76032	−0.484994
$406$	6.06638	0.301069
$407$	18.7928	0.931526
$408$	−4.85086	−0.240153
$409$	−7.62863	−0.377211	−0.188606	−	0.982053i	$-0.560397\pi$
$409$	−7.62863	−0.377211	−0.188606	+	0.982053i	$0.560397\pi$
$410$	5.67994	0.280512
$411$	2.16852	0.106965
$412$	−4.62133	−0.227677
$413$	13.6310	0.670739
$414$	1.04892	0.0515515
$415$	17.5405	0.861028
$416$	0	0
$417$	−1.29052	−0.0631970
$418$	−2.80194	−0.137047
$419$	2.42865	0.118647	0.0593236	−	0.998239i	$-0.481106\pi$
$419$	2.42865	0.118647	0.0593236	+	0.998239i	$0.481106\pi$
$420$	−6.00192	−0.292864
$421$	11.1395	0.542904	0.271452	−	0.962452i	$-0.412496\pi$
$421$	11.1395	0.542904	0.271452	+	0.962452i	$0.412496\pi$
$422$	−0.801938	−0.0390377
$423$	−14.4233	−0.701283
$424$	−19.6732	−0.955417
$425$	−5.04892	−0.244908
$426$	−6.68425	−0.323853
$427$	3.28621	0.159031
$428$	−24.3013	−1.17465
$429$	0	0
$430$	5.62133	0.271085
$431$	−5.61596	−0.270511	−0.135256	−	0.990811i	$-0.543186\pi$
$431$	−5.61596	−0.270511	−0.135256	+	0.990811i	$0.543186\pi$
$432$	−2.38404	−0.114702
$433$	−15.5133	−0.745523	−0.372762	−	0.927927i	$-0.621589\pi$
$433$	−15.5133	−0.745523	−0.372762	+	0.927927i	$0.621589\pi$
$434$	10.8955	0.522999
$435$	−7.97046	−0.382154
$436$	14.3957	0.689427
$437$	0.554958	0.0265472
$438$	0.260980	0.0124701
$439$	24.3967	1.16439	0.582196	−	0.813049i	$-0.302194\pi$
$439$	24.3967	1.16439	0.582196	+	0.813049i	$0.302194\pi$
$440$	−25.3207	−1.20711
$441$	6.60388	0.314470
$442$	0	0
$443$	31.4252	1.49306	0.746528	−	0.665354i	$-0.231719\pi$
$443$	31.4252	1.49306	0.746528	+	0.665354i	$0.231719\pi$
$444$	−5.85277	−0.277760
$445$	−29.4596	−1.39652
$446$	−21.7071	−1.02786
$447$	−6.10752	−0.288876
$448$	−7.30367	−0.345066
$449$	17.5429	0.827900	0.413950	−	0.910300i	$-0.364149\pi$
$449$	17.5429	0.827900	0.413950	+	0.910300i	$0.364149\pi$
$450$	4.24698	0.200205
$451$	9.19269	0.432867
$452$	22.9181	1.07798
$453$	−4.14483	−0.194741
$454$	−13.5786	−0.637277
$455$	0	0
$456$	2.15883	0.101097
$457$	−26.6165	−1.24507	−0.622535	−	0.782592i	$-0.713897\pi$
$457$	−26.6165	−1.24507	−0.622535	+	0.782592i	$0.713897\pi$
$458$	11.0368	0.515717
$459$	9.65279	0.450554
$460$	2.02715	0.0945162
$461$	−16.3623	−0.762067	−0.381034	−	0.924561i	$-0.624432\pi$
$461$	−16.3623	−0.762067	−0.381034	+	0.924561i	$0.624432\pi$
$462$	4.60388	0.214192
$463$	6.32304	0.293857	0.146928	−	0.989147i	$-0.453061\pi$
$463$	6.32304	0.293857	0.146928	+	0.989147i	$0.453061\pi$
$464$	2.04892	0.0951186
$465$	−14.3153	−0.663855
$466$	23.1269	1.07133
$467$	−12.6910	−0.587267	−0.293634	−	0.955918i	$-0.594865\pi$
$467$	−12.6910	−0.587267	−0.293634	+	0.955918i	$0.594865\pi$
$468$	0	0
$469$	−19.7888	−0.913761
$470$	13.2112	0.609388
$471$	−8.83877	−0.407269
$472$	−17.9095	−0.824350
$473$	9.09783	0.418319
$474$	5.98600	0.274946
$475$	2.24698	0.103098
$476$	−6.24698	−0.286330
$477$	17.2241	0.788639
$478$	7.45281	0.340884
$479$	−28.4959	−1.30201	−0.651005	−	0.759073i	$-0.725652\pi$
$479$	−28.4959	−1.30201	−0.651005	+	0.759073i	$0.725652\pi$
$480$	12.5840	0.574379
$481$	0	0
$482$	−7.31527	−0.333202
$483$	−0.911854	−0.0414908
$484$	−1.63879	−0.0744906
$485$	13.8877	0.630608
$486$	−12.6668	−0.574577
$487$	41.4174	1.87680	0.938401	−	0.345548i	$-0.112307\pi$
$487$	41.4174	1.87680	0.938401	+	0.345548i	$0.112307\pi$
$488$	−4.31767	−0.195452
$489$	20.0291	0.905746
$490$	−6.04892	−0.273262
$491$	−7.21313	−0.325524	−0.162762	−	0.986665i	$-0.552040\pi$
$491$	−7.21313	−0.325524	−0.162762	+	0.986665i	$0.552040\pi$
$492$	−2.86294	−0.129071
$493$	−8.29590	−0.373628
$494$	0	0
$495$	22.1685	0.996401
$496$	3.67994	0.165234
$497$	−21.2959	−0.955252
$498$	4.19029	0.187771
$499$	−1.86964	−0.0836966	−0.0418483	−	0.999124i	$-0.513325\pi$
$499$	−1.86964	−0.0836966	−0.0418483	+	0.999124i	$0.513325\pi$
$500$	−10.0562	−0.449728
$501$	−0.172834	−0.00772166
$502$	0.0368358	0.00164406
$503$	20.2838	0.904411	0.452205	−	0.891914i	$-0.350637\pi$
$503$	20.2838	0.904411	0.452205	+	0.891914i	$0.350637\pi$
$504$	13.0000	0.579066
$505$	6.22760	0.277125
$506$	−1.55496	−0.0691263
$507$	−10.4252	−0.462999
$508$	25.6286	1.13709
$509$	−1.74392	−0.0772980	−0.0386490	−	0.999253i	$-0.512305\pi$
$509$	−1.74392	−0.0772980	−0.0386490	+	0.999253i	$0.512305\pi$
$510$	−3.89008	−0.172256
$511$	0.831478	0.0367824
$512$	−6.22282	−0.275012
$513$	−4.29590	−0.189668
$514$	8.15021	0.359490
$515$	−9.16852	−0.404013
$516$	−2.83340	−0.124733
$517$	21.3817	0.940364
$518$	8.83771	0.388307
$519$	−5.60925	−0.246219
$520$	0	0
$521$	15.4886	0.678567	0.339284	−	0.940684i	$-0.389815\pi$
$521$	15.4886	0.678567	0.339284	+	0.940684i	$0.389815\pi$
$522$	6.97823	0.305429
$523$	13.5670	0.593245	0.296623	−	0.954995i	$-0.404140\pi$
$523$	13.5670	0.593245	0.296623	+	0.954995i	$0.404140\pi$
$524$	13.5157	0.590437
$525$	−3.69202	−0.161133
$526$	16.6625	0.726519
$527$	−14.8998	−0.649044
$528$	1.55496	0.0676709
$529$	−22.6920	−0.986610
$530$	−15.7767	−0.685296
$531$	15.6799	0.680451
$532$	2.78017	0.120536
$533$	0	0
$534$	−7.03769	−0.304551
$535$	−48.2127	−2.08441
$536$	26.0000	1.12303
$537$	−16.2446	−0.701006
$538$	2.87694	0.124034
$539$	−9.78986	−0.421679
$540$	−15.6920	−0.675277
$541$	−11.9463	−0.513611	−0.256806	−	0.966463i	$-0.582670\pi$
$541$	−11.9463	−0.513611	−0.256806	+	0.966463i	$0.582670\pi$
$542$	1.16075	0.0498586
$543$	−11.4190	−0.490034
$544$	13.0978	0.561565
$545$	28.5603	1.22339
$546$	0	0
$547$	38.0568	1.62719	0.813596	−	0.581431i	$-0.197507\pi$
$547$	38.0568	1.62719	0.813596	+	0.581431i	$0.197507\pi$
$548$	−3.66919	−0.156740
$549$	3.78017	0.161334
$550$	−6.29590	−0.268458
$551$	3.69202	0.157285
$552$	1.19806	0.0509929
$553$	19.0713	0.810993
$554$	4.31767	0.183440
$555$	−11.6116	−0.492887
$556$	2.18359	0.0926047
$557$	−45.1987	−1.91513	−0.957564	−	0.288220i	$-0.906937\pi$
$557$	−45.1987	−1.91513	−0.957564	+	0.288220i	$0.906937\pi$
$558$	12.5332	0.530572
$559$	0	0
$560$	3.06100	0.129351
$561$	−6.29590	−0.265813
$562$	4.33752	0.182967
$563$	−44.5894	−1.87922	−0.939610	−	0.342248i	$-0.888812\pi$
$563$	−44.5894	−1.87922	−0.939610	+	0.342248i	$0.888812\pi$
$564$	−6.65902	−0.280395
$565$	45.4685	1.91287
$566$	6.78794	0.285318
$567$	−7.42865	−0.311974
$568$	27.9801	1.17402
$569$	1.36121	0.0570648	0.0285324	−	0.999593i	$-0.490917\pi$
$569$	1.36121	0.0570648	0.0285324	+	0.999593i	$0.490917\pi$
$570$	1.73125	0.0725141
$571$	−17.5502	−0.734452	−0.367226	−	0.930132i	$-0.619692\pi$
$571$	−17.5502	−0.734452	−0.367226	+	0.930132i	$0.619692\pi$
$572$	0	0
$573$	4.03146	0.168417
$574$	4.32304	0.180440
$575$	1.24698	0.0520026
$576$	−8.40150	−0.350063
$577$	−10.6939	−0.445195	−0.222597	−	0.974910i	$-0.571453\pi$
$577$	−10.6939	−0.445195	−0.222597	+	0.974910i	$0.571453\pi$
$578$	9.58402	0.398643
$579$	−9.15452	−0.380449
$580$	13.4862	0.559984
$581$	13.3502	0.553859
$582$	3.31767	0.137522
$583$	−25.5338	−1.05750
$584$	−1.09246	−0.0452062
$585$	0	0
$586$	0.252356	0.0104247
$587$	−41.5894	−1.71658	−0.858289	−	0.513166i	$-0.828472\pi$
$587$	−41.5894	−1.71658	−0.858289	+	0.513166i	$0.828472\pi$
$588$	3.04892	0.125735
$589$	6.63102	0.273227
$590$	−14.3623	−0.591285
$591$	−0.692021	−0.0284660
$592$	2.98493	0.122680
$593$	−0.990902	−0.0406915	−0.0203457	−	0.999793i	$-0.506477\pi$
$593$	−0.990902	−0.0406915	−0.0203457	+	0.999793i	$0.506477\pi$
$594$	12.0368	0.493877
$595$	−12.3937	−0.508093
$596$	10.3341	0.423300
$597$	−11.8745	−0.485993
$598$	0	0
$599$	−22.6262	−0.924483	−0.462241	−	0.886754i	$-0.652955\pi$
$599$	−22.6262	−0.924483	−0.462241	+	0.886754i	$0.652955\pi$
$600$	4.85086	0.198035
$601$	4.88099	0.199100	0.0995498	−	0.995033i	$-0.468260\pi$
$601$	4.88099	0.199100	0.0995498	+	0.995033i	$0.468260\pi$
$602$	4.27844	0.174376
$603$	−22.7633	−0.926993
$604$	7.01315	0.285361
$605$	−3.25129	−0.132184
$606$	1.48773	0.0604348
$607$	42.4346	1.72237	0.861184	−	0.508293i	$-0.169723\pi$
$607$	42.4346	1.72237	0.861184	+	0.508293i	$0.169723\pi$
$608$	−5.82908	−0.236401
$609$	−6.06638	−0.245822
$610$	−3.46250	−0.140193
$611$	0	0
$612$	−7.18598	−0.290476
$613$	−9.37926	−0.378825	−0.189412	−	0.981898i	$-0.560658\pi$
$613$	−9.37926	−0.378825	−0.189412	+	0.981898i	$0.560658\pi$
$614$	−0.657105	−0.0265186
$615$	−5.67994	−0.229037
$616$	−19.2717	−0.776480
$617$	−18.7222	−0.753725	−0.376863	−	0.926269i	$-0.622997\pi$
$617$	−18.7222	−0.753725	−0.376863	+	0.926269i	$0.622997\pi$
$618$	−2.19029	−0.0881065
$619$	32.4547	1.30447	0.652233	−	0.758019i	$-0.273832\pi$
$619$	32.4547	1.30447	0.652233	+	0.758019i	$0.273832\pi$
$620$	24.2218	0.972769
$621$	−2.38404	−0.0956683
$622$	10.8955	0.436868
$623$	−22.4219	−0.898316
$624$	0	0
$625$	−31.1860	−1.24744
$626$	−16.8877	−0.674968
$627$	2.80194	0.111899
$628$	14.9554	0.596785
$629$	−12.0858	−0.481891
$630$	10.4252	0.415350
$631$	−29.0930	−1.15818	−0.579088	−	0.815265i	$-0.696591\pi$
$631$	−29.0930	−1.15818	−0.579088	+	0.815265i	$0.696591\pi$
$632$	−25.0573	−0.996725
$633$	0.801938	0.0318742
$634$	11.8866	0.472078
$635$	50.8461	2.01776
$636$	7.95215	0.315323
$637$	0	0
$638$	−10.3448	−0.409555
$639$	−24.4969	−0.969084
$640$	−23.6886	−0.936373
$641$	−9.28190	−0.366613	−0.183306	−	0.983056i	$-0.558680\pi$
$641$	−9.28190	−0.366613	−0.183306	+	0.983056i	$0.558680\pi$
$642$	−11.5176	−0.454565
$643$	−31.6437	−1.24791	−0.623953	−	0.781462i	$-0.714474\pi$
$643$	−31.6437	−1.24791	−0.623953	+	0.781462i	$0.714474\pi$
$644$	1.54288	0.0607979
$645$	−5.62133	−0.221340
$646$	1.80194	0.0708963
$647$	19.6698	0.773299	0.386649	−	0.922227i	$-0.373632\pi$
$647$	19.6698	0.773299	0.386649	+	0.922227i	$0.373632\pi$
$648$	9.76032	0.383421
$649$	−23.2446	−0.912430
$650$	0	0
$651$	−10.8955	−0.427027
$652$	−33.8896	−1.32722
$653$	−41.9939	−1.64335	−0.821674	−	0.569958i	$-0.806959\pi$
$653$	−41.9939	−1.64335	−0.821674	+	0.569958i	$0.806959\pi$
$654$	6.82285	0.266795
$655$	26.8146	1.04773
$656$	1.46011	0.0570076
$657$	0.956459	0.0373150
$658$	10.0551	0.391990
$659$	−7.42865	−0.289379	−0.144690	−	0.989477i	$-0.546218\pi$
$659$	−7.42865	−0.289379	−0.144690	+	0.989477i	$0.546218\pi$
$660$	10.2349	0.398393
$661$	3.80492	0.147994	0.0739971	−	0.997258i	$-0.476424\pi$
$661$	3.80492	0.147994	0.0739971	+	0.997258i	$0.476424\pi$
$662$	−27.8353	−1.08185
$663$	0	0
$664$	−17.5405	−0.680703
$665$	5.51573	0.213891
$666$	10.1661	0.393930
$667$	2.04892	0.0793344
$668$	0.292439	0.0113148
$669$	21.7071	0.839244
$670$	20.8504	0.805520
$671$	−5.60388	−0.216335
$672$	9.57779	0.369471
$673$	11.0435	0.425697	0.212849	−	0.977085i	$-0.431726\pi$
$673$	11.0435	0.425697	0.212849	+	0.977085i	$0.431726\pi$
$674$	−14.2233	−0.547861
$675$	−9.65279	−0.371536
$676$	17.6396	0.678448
$677$	−44.3682	−1.70521	−0.852605	−	0.522556i	$-0.824979\pi$
$677$	−44.3682	−1.70521	−0.852605	+	0.522556i	$0.824979\pi$
$678$	10.8621	0.417156
$679$	10.5700	0.405640
$680$	16.2838	0.624456
$681$	13.5786	0.520335
$682$	−18.5797	−0.711454
$683$	−40.7928	−1.56089	−0.780447	−	0.625222i	$-0.785009\pi$
$683$	−40.7928	−1.56089	−0.780447	+	0.625222i	$0.785009\pi$
$684$	3.19806	0.122281
$685$	−7.27950	−0.278136
$686$	−16.1056	−0.614915
$687$	−11.0368	−0.421082
$688$	1.44504	0.0550917
$689$	0	0
$690$	0.960771	0.0365759
$691$	−8.74392	−0.332634	−0.166317	−	0.986072i	$-0.553188\pi$
$691$	−8.74392	−0.332634	−0.166317	+	0.986072i	$0.553188\pi$
$692$	9.49098	0.360793
$693$	16.8726	0.640938
$694$	6.22521	0.236306
$695$	4.33214	0.164328
$696$	7.97046	0.302120
$697$	−5.91185	−0.223927
$698$	8.19460	0.310170
$699$	−23.1269	−0.874740
$700$	6.24698	0.236114
$701$	−41.4878	−1.56697	−0.783487	−	0.621408i	$-0.786561\pi$
$701$	−41.4878	−1.56697	−0.783487	+	0.621408i	$0.786561\pi$
$702$	0	0
$703$	5.37867	0.202860
$704$	12.4547	0.469405
$705$	−13.2112	−0.497563
$706$	−5.15538	−0.194025
$707$	4.73987	0.178261
$708$	7.23921	0.272066
$709$	10.3147	0.387376	0.193688	−	0.981063i	$-0.437955\pi$
$709$	10.3147	0.387376	0.193688	+	0.981063i	$0.437955\pi$
$710$	22.4383	0.842096
$711$	21.9379	0.822737
$712$	29.4596	1.10405
$713$	3.67994	0.137815
$714$	−2.96077	−0.110804
$715$	0	0
$716$	27.4862	1.02721
$717$	−7.45281	−0.278330
$718$	19.3414	0.721813
$719$	40.9482	1.52711	0.763555	−	0.645742i	$-0.223452\pi$
$719$	40.9482	1.52711	0.763555	+	0.645742i	$0.223452\pi$
$720$	3.52111	0.131224
$721$	−6.97823	−0.259883
$722$	−0.801938	−0.0298450
$723$	7.31527	0.272058
$724$	19.3211	0.718064
$725$	8.29590	0.308102
$726$	−0.776710	−0.0288264
$727$	44.5792	1.65335	0.826676	−	0.562678i	$-0.190229\pi$
$727$	44.5792	1.65335	0.826676	+	0.562678i	$0.190229\pi$
$728$	0	0
$729$	1.78986	0.0662910
$730$	−0.876083	−0.0324253
$731$	−5.85086	−0.216402
$732$	1.74525	0.0645063
$733$	23.4946	0.867791	0.433896	−	0.900963i	$-0.357139\pi$
$733$	23.4946	0.867791	0.433896	+	0.900963i	$0.357139\pi$
$734$	−5.42088	−0.200088
$735$	6.04892	0.223118
$736$	−3.23490	−0.119240
$737$	33.7453	1.24302
$738$	4.97285	0.183053
$739$	30.0043	1.10373	0.551863	−	0.833935i	$-0.313917\pi$
$739$	30.0043	1.10373	0.551863	+	0.833935i	$0.313917\pi$
$740$	19.6472	0.722244
$741$	0	0
$742$	−12.0078	−0.440819
$743$	2.02044	0.0741228	0.0370614	−	0.999313i	$-0.488200\pi$
$743$	2.02044	0.0741228	0.0370614	+	0.999313i	$0.488200\pi$
$744$	14.3153	0.524823
$745$	20.5023	0.751147
$746$	−15.0194	−0.549898
$747$	15.3569	0.561879
$748$	10.6528	0.389505
$749$	−36.6950	−1.34081
$750$	−4.76617	−0.174036
$751$	−3.86725	−0.141118	−0.0705590	−	0.997508i	$-0.522478\pi$
$751$	−3.86725	−0.141118	−0.0705590	+	0.997508i	$0.522478\pi$
$752$	3.39612	0.123844
$753$	−0.0368358	−0.00134237
$754$	0	0
$755$	13.9138	0.506374
$756$	−11.9433	−0.434374
$757$	19.2922	0.701186	0.350593	−	0.936528i	$-0.385980\pi$
$757$	19.2922	0.701186	0.350593	+	0.936528i	$0.385980\pi$
$758$	−11.0916	−0.402865
$759$	1.55496	0.0564414
$760$	−7.24698	−0.262876
$761$	15.4450	0.559882	0.279941	−	0.960017i	$-0.409685\pi$
$761$	15.4450	0.559882	0.279941	+	0.960017i	$0.409685\pi$
$762$	12.1468	0.440030
$763$	21.7375	0.786950
$764$	−6.82132	−0.246787
$765$	−14.2567	−0.515451
$766$	−14.6286	−0.528554
$767$	0	0
$768$	−11.3763	−0.410506
$769$	31.9420	1.15186	0.575929	−	0.817500i	$-0.304641\pi$
$769$	31.9420	1.15186	0.575929	+	0.817500i	$0.304641\pi$
$770$	−15.4547	−0.556950
$771$	−8.15021	−0.293523
$772$	15.4896	0.557485
$773$	25.3720	0.912566	0.456283	−	0.889835i	$-0.349180\pi$
$773$	25.3720	0.912566	0.456283	+	0.889835i	$0.349180\pi$
$774$	4.92154	0.176901
$775$	14.8998	0.535216
$776$	−13.8877	−0.498539
$777$	−8.83771	−0.317051
$778$	17.9129	0.642209
$779$	2.63102	0.0942662
$780$	0	0
$781$	36.3153	1.29946
$782$	1.00000	0.0357599
$783$	−15.8605	−0.566810
$784$	−1.55496	−0.0555342
$785$	29.6708	1.05900
$786$	6.40581	0.228488
$787$	−20.9584	−0.747086	−0.373543	−	0.927613i	$-0.621857\pi$
$787$	−20.9584	−0.747086	−0.373543	+	0.927613i	$0.621857\pi$
$788$	1.17092	0.0417121
$789$	−16.6625	−0.593200
$790$	−20.0944	−0.714926
$791$	34.6064	1.23046
$792$	−22.1685	−0.787724
$793$	0	0
$794$	24.6189	0.873694
$795$	15.7767	0.559542
$796$	20.0920	0.712141
$797$	−41.6426	−1.47506	−0.737529	−	0.675316i	$-0.764007\pi$
$797$	−41.6426	−1.47506	−0.737529	+	0.675316i	$0.764007\pi$
$798$	1.31767	0.0466449
$799$	−13.7506	−0.486462
$800$	−13.0978	−0.463078
$801$	−25.7922	−0.911324
$802$	−21.4523	−0.757508
$803$	−1.41789	−0.0500364
$804$	−10.5095	−0.370641
$805$	3.06100	0.107886
$806$	0	0
$807$	−2.87694	−0.101273
$808$	−6.22760	−0.219086
$809$	14.4819	0.509156	0.254578	−	0.967052i	$-0.418063\pi$
$809$	14.4819	0.509156	0.254578	+	0.967052i	$0.418063\pi$
$810$	7.82717	0.275019
$811$	−50.8189	−1.78449	−0.892247	−	0.451548i	$-0.850872\pi$
$811$	−50.8189	−1.78449	−0.892247	+	0.451548i	$0.850872\pi$
$812$	10.2644	0.360211
$813$	−1.16075	−0.0407094
$814$	−15.0707	−0.528227
$815$	−67.2355	−2.35516
$816$	−1.00000	−0.0350070
$817$	2.60388	0.0910981
$818$	6.11769	0.213900
$819$	0	0
$820$	9.61058	0.335616
$821$	−24.8286	−0.866524	−0.433262	−	0.901268i	$-0.642638\pi$
$821$	−24.8286	−0.866524	−0.433262	+	0.901268i	$0.642638\pi$
$822$	−1.73902	−0.0606553
$823$	−5.70278	−0.198786	−0.0993931	−	0.995048i	$-0.531690\pi$
$823$	−5.70278	−0.198786	−0.0993931	+	0.995048i	$0.531690\pi$
$824$	9.16852	0.319401
$825$	6.29590	0.219195
$826$	−10.9312	−0.380346
$827$	−1.49635	−0.0520333	−0.0260166	−	0.999662i	$-0.508282\pi$
$827$	−1.49635	−0.0520333	−0.0260166	+	0.999662i	$0.508282\pi$
$828$	1.77479	0.0616782
$829$	−8.26934	−0.287206	−0.143603	−	0.989635i	$-0.545869\pi$
$829$	−8.26934	−0.287206	−0.143603	+	0.989635i	$0.545869\pi$
$830$	−14.0664	−0.488251
$831$	−4.31767	−0.149778
$832$	0	0
$833$	6.29590	0.218140
$834$	1.03492	0.0358362
$835$	0.580186	0.0200782
$836$	−4.74094	−0.163969
$837$	−28.4862	−0.984627
$838$	−1.94762	−0.0672796
$839$	−34.9172	−1.20548	−0.602738	−	0.797939i	$-0.705924\pi$
$839$	−34.9172	−1.20548	−0.602738	+	0.797939i	$0.705924\pi$
$840$	11.9075	0.410849
$841$	−15.3690	−0.529965
$842$	−8.93315	−0.307857
$843$	−4.33752	−0.149392
$844$	−1.35690	−0.0467063
$845$	34.9963	1.20391
$846$	11.5666	0.397667
$847$	−2.47458	−0.0850276
$848$	−4.05562	−0.139271
$849$	−6.78794	−0.232961
$850$	4.04892	0.138877
$851$	2.98493	0.102322
$852$	−11.3099	−0.387471
$853$	15.5929	0.533889	0.266945	−	0.963712i	$-0.413986\pi$
$853$	15.5929	0.533889	0.266945	+	0.963712i	$0.413986\pi$
$854$	−2.63533	−0.0901793
$855$	6.34481	0.216988
$856$	48.2127	1.64787
$857$	8.93422	0.305187	0.152593	−	0.988289i	$-0.451237\pi$
$857$	8.93422	0.305187	0.152593	+	0.988289i	$0.451237\pi$
$858$	0	0
$859$	24.8219	0.846913	0.423456	−	0.905917i	$-0.360817\pi$
$859$	24.8219	0.846913	0.423456	+	0.905917i	$0.360817\pi$
$860$	9.51142	0.324337
$861$	−4.32304	−0.147329
$862$	4.50365	0.153395
$863$	39.1855	1.33389	0.666945	−	0.745107i	$-0.267602\pi$
$863$	39.1855	1.33389	0.666945	+	0.745107i	$0.267602\pi$
$864$	25.0411	0.851917
$865$	18.8297	0.640228
$866$	12.4407	0.422753
$867$	−9.58402	−0.325491
$868$	18.4354	0.625737
$869$	−32.5217	−1.10322
$870$	6.39181	0.216703
$871$	0	0
$872$	−28.5603	−0.967175
$873$	12.1588	0.411514
$874$	−0.445042	−0.0150538
$875$	−15.1849	−0.513344
$876$	0.441584	0.0149197
$877$	−9.06697	−0.306170	−0.153085	−	0.988213i	$-0.548921\pi$
$877$	−9.06697	−0.306170	−0.153085	+	0.988213i	$0.548921\pi$
$878$	−19.5646	−0.660275
$879$	−0.252356	−0.00851177
$880$	−5.21983	−0.175961
$881$	7.06638	0.238072	0.119036	−	0.992890i	$-0.462020\pi$
$881$	7.06638	0.238072	0.119036	+	0.992890i	$0.462020\pi$
$882$	−5.29590	−0.178322
$883$	39.3564	1.32445	0.662225	−	0.749305i	$-0.269613\pi$
$883$	39.3564	1.32445	0.662225	+	0.749305i	$0.269613\pi$
$884$	0	0
$885$	14.3623	0.482783
$886$	−25.2010	−0.846646
$887$	−19.3894	−0.651033	−0.325516	−	0.945536i	$-0.605538\pi$
$887$	−19.3894	−0.651033	−0.325516	+	0.945536i	$0.605538\pi$
$888$	11.6116	0.389661
$889$	38.6993	1.29793
$890$	23.6248	0.791905
$891$	12.6679	0.424389
$892$	−36.7289	−1.22977
$893$	6.11960	0.204785
$894$	4.89785	0.163809
$895$	54.5314	1.82278
$896$	−18.0295	−0.602325
$897$	0	0
$898$	−14.0683	−0.469465
$899$	24.4819	0.816516
$900$	7.18598	0.239533
$901$	16.4209	0.547059
$902$	−7.37196	−0.245459
$903$	−4.27844	−0.142378
$904$	−45.4685	−1.51226
$905$	38.3323	1.27421
$906$	3.32390	0.110429
$907$	−14.3357	−0.476010	−0.238005	−	0.971264i	$-0.576493\pi$
$907$	−14.3357	−0.476010	−0.238005	+	0.971264i	$0.576493\pi$
$908$	−22.9754	−0.762464
$909$	5.45234	0.180843
$910$	0	0
$911$	−16.5418	−0.548055	−0.274027	−	0.961722i	$-0.588356\pi$
$911$	−16.5418	−0.548055	−0.274027	+	0.961722i	$0.588356\pi$
$912$	0.445042	0.0147368
$913$	−22.7657	−0.753434
$914$	21.3448	0.706024
$915$	3.46250	0.114467
$916$	18.6746	0.617025
$917$	20.4088	0.673958
$918$	−7.74094	−0.255489
$919$	−34.7488	−1.14626	−0.573129	−	0.819465i	$-0.694271\pi$
$919$	−34.7488	−1.14626	−0.573129	+	0.819465i	$0.694271\pi$
$920$	−4.02177	−0.132594
$921$	0.657105	0.0216523
$922$	13.1215	0.432134
$923$	0	0
$924$	7.78986	0.256267
$925$	12.0858	0.397377
$926$	−5.07069	−0.166633
$927$	−8.02715	−0.263646
$928$	−21.5211	−0.706465
$929$	15.2131	0.499127	0.249563	−	0.968358i	$-0.419713\pi$
$929$	15.2131	0.499127	0.249563	+	0.968358i	$0.419713\pi$
$930$	11.4800	0.376443
$931$	−2.80194	−0.0918298
$932$	39.1312	1.28179
$933$	−10.8955	−0.356701
$934$	10.1774	0.333013
$935$	21.1347	0.691178
$936$	0	0
$937$	−24.9288	−0.814390	−0.407195	−	0.913341i	$-0.633493\pi$
$937$	−24.9288	−0.814390	−0.407195	+	0.913341i	$0.633493\pi$
$938$	15.8694	0.518154
$939$	16.8877	0.551109
$940$	22.3536	0.729096
$941$	46.4825	1.51529	0.757643	−	0.652670i	$-0.226351\pi$
$941$	46.4825	1.51529	0.757643	+	0.652670i	$0.226351\pi$
$942$	7.08815	0.230944
$943$	1.46011	0.0475476
$944$	−3.69202	−0.120165
$945$	−23.6950	−0.770799
$946$	−7.29590	−0.237210
$947$	−29.1403	−0.946933	−0.473466	−	0.880812i	$-0.656997\pi$
$947$	−29.1403	−0.946933	−0.473466	+	0.880812i	$0.656997\pi$
$948$	10.1284	0.328956
$949$	0	0
$950$	−1.80194	−0.0584626
$951$	−11.8866	−0.385450
$952$	12.3937	0.401683
$953$	14.5472	0.471230	0.235615	−	0.971847i	$-0.424290\pi$
$953$	14.5472	0.471230	0.235615	+	0.971847i	$0.424290\pi$
$954$	−13.8127	−0.447202
$955$	−13.5332	−0.437924
$956$	12.6103	0.407847
$957$	10.3448	0.334400
$958$	22.8519	0.738312
$959$	−5.54048	−0.178912
$960$	−7.69548	−0.248371
$961$	12.9705	0.418402
$962$	0	0
$963$	−42.2107	−1.36022
$964$	−12.3776	−0.398656
$965$	30.7308	0.989259
$966$	0.731250	0.0235276
$967$	30.3351	0.975512	0.487756	−	0.872980i	$-0.337816\pi$
$967$	30.3351	0.975512	0.487756	+	0.872980i	$0.337816\pi$
$968$	3.25129	0.104500
$969$	−1.80194	−0.0578866
$970$	−11.1371	−0.357590
$971$	38.5236	1.23628	0.618141	−	0.786067i	$-0.287886\pi$
$971$	38.5236	1.23628	0.618141	+	0.786067i	$0.287886\pi$
$972$	−21.4325	−0.687447
$973$	3.29722	0.105704
$974$	−33.2142	−1.06425
$975$	0	0
$976$	−0.890084	−0.0284909
$977$	−8.58940	−0.274799	−0.137400	−	0.990516i	$-0.543874\pi$
$977$	−8.58940	−0.274799	−0.137400	+	0.990516i	$0.543874\pi$
$978$	−16.0621	−0.513608
$979$	38.2355	1.22201
$980$	−10.2349	−0.326942
$981$	25.0049	0.798345
$982$	5.78448	0.184590
$983$	−35.9474	−1.14654	−0.573271	−	0.819366i	$-0.694326\pi$
$983$	−35.9474	−1.14654	−0.573271	+	0.819366i	$0.694326\pi$
$984$	5.67994	0.181070
$985$	2.32304	0.0740183
$986$	6.65279	0.211868
$987$	−10.0551	−0.320059
$988$	0	0
$989$	1.44504	0.0459497
$990$	−17.7778	−0.565015
$991$	1.18896	0.0377687	0.0188844	−	0.999822i	$-0.493989\pi$
$991$	1.18896	0.0377687	0.0188844	+	0.999822i	$0.493989\pi$
$992$	−38.6528	−1.22723
$993$	27.8353	0.883327
$994$	17.0780	0.541681
$995$	39.8616	1.26370
$996$	7.09006	0.224657
$997$	24.9409	0.789887	0.394943	−	0.918705i	$-0.370764\pi$
$997$	24.9409	0.789887	0.394943	+	0.918705i	$0.370764\pi$
$998$	1.49934	0.0474607
$999$	−23.1062	−0.731048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.b.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.b.1.1	✓	3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-0.801938 q^{2} +0.801938 q^{3} -1.35690 q^{4} -2.69202 q^{5} -0.643104 q^{6} -2.04892 q^{7} +2.69202 q^{8} -2.35690 q^{9} +O(q^{10})\)	\(q-0.801938 q^{2} +0.801938 q^{3} -1.35690 q^{4} -2.69202 q^{5} -0.643104 q^{6} -2.04892 q^{7} +2.69202 q^{8} -2.35690 q^{9} +2.15883 q^{10} +3.49396 q^{11} -1.08815 q^{12} +1.64310 q^{14} -2.15883 q^{15} +0.554958 q^{16} -2.24698 q^{17} +1.89008 q^{18} +1.00000 q^{19} +3.65279 q^{20} -1.64310 q^{21} -2.80194 q^{22} +0.554958 q^{23} +2.15883 q^{24} +2.24698 q^{25} -4.29590 q^{27} +2.78017 q^{28} +3.69202 q^{29} +1.73125 q^{30} +6.63102 q^{31} -5.82908 q^{32} +2.80194 q^{33} +1.80194 q^{34} +5.51573 q^{35} +3.19806 q^{36} +5.37867 q^{37} -0.801938 q^{38} -7.24698 q^{40} +2.63102 q^{41} +1.31767 q^{42} +2.60388 q^{43} -4.74094 q^{44} +6.34481 q^{45} -0.445042 q^{46} +6.11960 q^{47} +0.445042 q^{48} -2.80194 q^{49} -1.80194 q^{50} -1.80194 q^{51} -7.30798 q^{53} +3.44504 q^{54} -9.40581 q^{55} -5.51573 q^{56} +0.801938 q^{57} -2.96077 q^{58} -6.65279 q^{59} +2.92931 q^{60} -1.60388 q^{61} -5.31767 q^{62} +4.82908 q^{63} +3.56465 q^{64} -2.24698 q^{66} +9.65817 q^{67} +3.04892 q^{68} +0.445042 q^{69} -4.42327 q^{70} +10.3937 q^{71} -6.34481 q^{72} -0.405813 q^{73} -4.31336 q^{74} +1.80194 q^{75} -1.35690 q^{76} -7.15883 q^{77} -9.30798 q^{79} -1.49396 q^{80} +3.62565 q^{81} -2.10992 q^{82} -6.51573 q^{83} +2.22952 q^{84} +6.04892 q^{85} -2.08815 q^{86} +2.96077 q^{87} +9.40581 q^{88} +10.9433 q^{89} -5.08815 q^{90} -0.753020 q^{92} +5.31767 q^{93} -4.90754 q^{94} -2.69202 q^{95} -4.67456 q^{96} -5.15883 q^{97} +2.24698 q^{98} -8.23490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} - 2 q^{3} - 3 q^{5} - 6 q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 - 2 * q^3 - 3 * q^5 - 6 * q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9	\( 3 q + 2 q^{2} - 2 q^{3} - 3 q^{5} - 6 q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9} - 2 q^{10} + q^{11} - 7 q^{12} + 9 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 5 q^{18} + 3 q^{19} - 7 q^{20} - 9 q^{21} - 4 q^{22} + 2 q^{23} - 2 q^{24} + 2 q^{25} + q^{27} + 7 q^{28} + 6 q^{29} + 13 q^{30} + 5 q^{31} - 7 q^{32} + 4 q^{33} + q^{34} + 4 q^{35} + 14 q^{36} + 9 q^{37} + 2 q^{38} - 17 q^{40} - 7 q^{41} - 13 q^{42} - q^{43} - 4 q^{45} - q^{46} - 3 q^{47} + q^{48} - 4 q^{49} - q^{50} - q^{51} - 27 q^{53} + 10 q^{54} - 15 q^{55} - 4 q^{56} - 2 q^{57} + 4 q^{58} - 2 q^{59} + 21 q^{60} + 4 q^{61} + q^{62} + 4 q^{63} - 11 q^{64} - 2 q^{66} + 8 q^{67} + q^{69} - 16 q^{70} - q^{71} + 4 q^{72} + 12 q^{73} - 15 q^{74} + q^{75} - 13 q^{77} - 33 q^{79} + 5 q^{80} - q^{81} - 7 q^{82} - 7 q^{83} - 14 q^{84} + 9 q^{85} - 10 q^{86} - 4 q^{87} + 15 q^{88} + 4 q^{89} - 19 q^{90} - 7 q^{92} - q^{93} - 30 q^{94} - 3 q^{95} + 7 q^{96} - 7 q^{97} + 2 q^{98} - q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 - 2 * q^3 - 3 * q^5 - 6 * q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9 - 2 * q^10 + q^11 - 7 * q^12 + 9 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 + 5 * q^18 + 3 * q^19 - 7 * q^20 - 9 * q^21 - 4 * q^22 + 2 * q^23 - 2 * q^24 + 2 * q^25 + q^27 + 7 * q^28 + 6 * q^29 + 13 * q^30 + 5 * q^31 - 7 * q^32 + 4 * q^33 + q^34 + 4 * q^35 + 14 * q^36 + 9 * q^37 + 2 * q^38 - 17 * q^40 - 7 * q^41 - 13 * q^42 - q^43 - 4 * q^45 - q^46 - 3 * q^47 + q^48 - 4 * q^49 - q^50 - q^51 - 27 * q^53 + 10 * q^54 - 15 * q^55 - 4 * q^56 - 2 * q^57 + 4 * q^58 - 2 * q^59 + 21 * q^60 + 4 * q^61 + q^62 + 4 * q^63 - 11 * q^64 - 2 * q^66 + 8 * q^67 + q^69 - 16 * q^70 - q^71 + 4 * q^72 + 12 * q^73 - 15 * q^74 + q^75 - 13 * q^77 - 33 * q^79 + 5 * q^80 - q^81 - 7 * q^82 - 7 * q^83 - 14 * q^84 + 9 * q^85 - 10 * q^86 - 4 * q^87 + 15 * q^88 + 4 * q^89 - 19 * q^90 - 7 * q^92 - q^93 - 30 * q^94 - 3 * q^95 + 7 * q^96 - 7 * q^97 + 2 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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