LMFDB - Embedded newform 399.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [399,2,Mod(1,399)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(399, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("399.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	399.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:	$3.18603104065$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*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			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	399.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} +1.19394 q^{5} +1.48119 q^{6} -1.00000 q^{7} +2.67513 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} +1.19394 q^{5} +1.48119 q^{6} -1.00000 q^{7} +2.67513 q^{8} +1.00000 q^{9} -1.76845 q^{10} -3.35026 q^{11} -0.193937 q^{12} -1.35026 q^{13} +1.48119 q^{14} -1.19394 q^{15} -4.35026 q^{16} +2.80606 q^{17} -1.48119 q^{18} +1.00000 q^{19} +0.231548 q^{20} +1.00000 q^{21} +4.96239 q^{22} +9.27504 q^{23} -2.67513 q^{24} -3.57452 q^{25} +2.00000 q^{26} -1.00000 q^{27} -0.193937 q^{28} +10.1563 q^{29} +1.76845 q^{30} +5.73813 q^{31} +1.09332 q^{32} +3.35026 q^{33} -4.15633 q^{34} -1.19394 q^{35} +0.193937 q^{36} +3.92478 q^{37} -1.48119 q^{38} +1.35026 q^{39} +3.19394 q^{40} +8.57452 q^{41} -1.48119 q^{42} -5.73813 q^{43} -0.649738 q^{44} +1.19394 q^{45} -13.7381 q^{46} -0.932071 q^{47} +4.35026 q^{48} +1.00000 q^{49} +5.29455 q^{50} -2.80606 q^{51} -0.261865 q^{52} +4.54420 q^{53} +1.48119 q^{54} -4.00000 q^{55} -2.67513 q^{56} -1.00000 q^{57} -15.0435 q^{58} +4.00000 q^{59} -0.231548 q^{60} +11.9248 q^{61} -8.49929 q^{62} -1.00000 q^{63} +7.08110 q^{64} -1.61213 q^{65} -4.96239 q^{66} -8.31265 q^{67} +0.544198 q^{68} -9.27504 q^{69} +1.76845 q^{70} -5.89446 q^{71} +2.67513 q^{72} -3.61213 q^{73} -5.81336 q^{74} +3.57452 q^{75} +0.193937 q^{76} +3.35026 q^{77} -2.00000 q^{78} +8.62530 q^{79} -5.19394 q^{80} +1.00000 q^{81} -12.7005 q^{82} -15.6932 q^{83} +0.193937 q^{84} +3.35026 q^{85} +8.49929 q^{86} -10.1563 q^{87} -8.96239 q^{88} +8.57452 q^{89} -1.76845 q^{90} +1.35026 q^{91} +1.79877 q^{92} -5.73813 q^{93} +1.38058 q^{94} +1.19394 q^{95} -1.09332 q^{96} +11.1490 q^{97} -1.48119 q^{98} -3.35026 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 3 q^{3} + q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 - 3 * q^3 + q^4 + 4 * q^5 - q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} - 3 q^{3} + q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 6 q^{10} - q^{12} + 6 q^{13} - q^{14} - 4 q^{15} - 3 q^{16} + 8 q^{17} + q^{18} + 3 q^{19} + 12 q^{20} + 3 q^{21} + 4 q^{22} - 4 q^{23} - 3 q^{24} + q^{25} + 6 q^{26} - 3 q^{27} - q^{28} + 20 q^{29} - 6 q^{30} + 8 q^{31} - 3 q^{32} - 2 q^{34} - 4 q^{35} + q^{36} - 10 q^{37} + q^{38} - 6 q^{39} + 10 q^{40} + 14 q^{41} + q^{42} - 8 q^{43} - 12 q^{44} + 4 q^{45} - 32 q^{46} + 6 q^{47} + 3 q^{48} + 3 q^{49} + 23 q^{50} - 8 q^{51} - 10 q^{52} + 4 q^{53} - q^{54} - 12 q^{55} - 3 q^{56} - 3 q^{57} - 2 q^{58} + 12 q^{59} - 12 q^{60} + 14 q^{61} + 8 q^{62} - 3 q^{63} - 11 q^{64} - 4 q^{65} - 4 q^{66} - 4 q^{67} - 8 q^{68} + 4 q^{69} - 6 q^{70} + 2 q^{71} + 3 q^{72} - 10 q^{73} - 30 q^{74} - q^{75} + q^{76} - 6 q^{78} - 16 q^{79} - 16 q^{80} + 3 q^{81} - 18 q^{82} - 14 q^{83} + q^{84} - 8 q^{86} - 20 q^{87} - 16 q^{88} + 14 q^{89} + 6 q^{90} - 6 q^{91} - 8 q^{92} - 8 q^{93} - 8 q^{94} + 4 q^{95} + 3 q^{96} + 10 q^{97} + q^{98}+O(q^{100}) \) 3 * q + q^2 - 3 * q^3 + q^4 + 4 * q^5 - q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9 + 6 * q^10 - q^12 + 6 * q^13 - q^14 - 4 * q^15 - 3 * q^16 + 8 * q^17 + q^18 + 3 * q^19 + 12 * q^20 + 3 * q^21 + 4 * q^22 - 4 * q^23 - 3 * q^24 + q^25 + 6 * q^26 - 3 * q^27 - q^28 + 20 * q^29 - 6 * q^30 + 8 * q^31 - 3 * q^32 - 2 * q^34 - 4 * q^35 + q^36 - 10 * q^37 + q^38 - 6 * q^39 + 10 * q^40 + 14 * q^41 + q^42 - 8 * q^43 - 12 * q^44 + 4 * q^45 - 32 * q^46 + 6 * q^47 + 3 * q^48 + 3 * q^49 + 23 * q^50 - 8 * q^51 - 10 * q^52 + 4 * q^53 - q^54 - 12 * q^55 - 3 * q^56 - 3 * q^57 - 2 * q^58 + 12 * q^59 - 12 * q^60 + 14 * q^61 + 8 * q^62 - 3 * q^63 - 11 * q^64 - 4 * q^65 - 4 * q^66 - 4 * q^67 - 8 * q^68 + 4 * q^69 - 6 * q^70 + 2 * q^71 + 3 * q^72 - 10 * q^73 - 30 * q^74 - q^75 + q^76 - 6 * q^78 - 16 * q^79 - 16 * q^80 + 3 * q^81 - 18 * q^82 - 14 * q^83 + q^84 - 8 * q^86 - 20 * q^87 - 16 * q^88 + 14 * q^89 + 6 * q^90 - 6 * q^91 - 8 * q^92 - 8 * q^93 - 8 * q^94 + 4 * q^95 + 3 * q^96 + 10 * q^97 + q^98

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 $

$2$	−1.48119	−1.04736	−0.523681	−	0.851914i	$-0.675442\pi$
$2$	−1.48119	−1.04736	−0.523681	+	0.851914i	$0.675442\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.193937	0.0969683
$5$	1.19394	0.533945	0.266972	−	0.963704i	$-0.413977\pi$
$5$	1.19394	0.533945	0.266972	+	0.963704i	$0.413977\pi$
$6$	1.48119	0.604695
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.67513	0.945802
$9$	1.00000	0.333333
$10$	−1.76845	−0.559234
$11$	−3.35026	−1.01014	−0.505071	−	0.863078i	$-0.668534\pi$
$11$	−3.35026	−1.01014	−0.505071	+	0.863078i	$0.668534\pi$
$12$	−0.193937	−0.0559847
$13$	−1.35026	−0.374495	−0.187248	−	0.982313i	$-0.559957\pi$
$13$	−1.35026	−0.374495	−0.187248	+	0.982313i	$0.559957\pi$
$14$	1.48119	0.395866
$15$	−1.19394	−0.308273
$16$	−4.35026	−1.08757
$17$	2.80606	0.680570	0.340285	−	0.940322i	$-0.389476\pi$
$17$	2.80606	0.680570	0.340285	+	0.940322i	$0.389476\pi$
$18$	−1.48119	−0.349121
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.231548	0.0517757
$21$	1.00000	0.218218
$22$	4.96239	1.05798
$23$	9.27504	1.93398	0.966990	−	0.254816i	$-0.0820148\pi$
$23$	9.27504	1.93398	0.966990	+	0.254816i	$0.0820148\pi$
$24$	−2.67513	−0.546059
$25$	−3.57452	−0.714903
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−0.193937	−0.0366506
$29$	10.1563	1.88598	0.942991	−	0.332818i	$-0.107999\pi$
$29$	10.1563	1.88598	0.942991	+	0.332818i	$0.107999\pi$
$30$	1.76845	0.322874
$31$	5.73813	1.03060	0.515300	−	0.857010i	$-0.327681\pi$
$31$	5.73813	1.03060	0.515300	+	0.857010i	$0.327681\pi$
$32$	1.09332	0.193274
$33$	3.35026	0.583206
$34$	−4.15633	−0.712804
$35$	−1.19394	−0.201812
$36$	0.193937	0.0323228
$37$	3.92478	0.645229	0.322615	−	0.946530i	$-0.395438\pi$
$37$	3.92478	0.645229	0.322615	+	0.946530i	$0.395438\pi$
$38$	−1.48119	−0.240281
$39$	1.35026	0.216215
$40$	3.19394	0.505006
$41$	8.57452	1.33911	0.669557	−	0.742761i	$-0.266484\pi$
$41$	8.57452	1.33911	0.669557	+	0.742761i	$0.266484\pi$
$42$	−1.48119	−0.228553
$43$	−5.73813	−0.875057	−0.437529	−	0.899204i	$-0.644146\pi$
$43$	−5.73813	−0.875057	−0.437529	+	0.899204i	$0.644146\pi$
$44$	−0.649738	−0.0979517
$45$	1.19394	0.177982
$46$	−13.7381	−2.02558
$47$	−0.932071	−0.135957	−0.0679783	−	0.997687i	$-0.521655\pi$
$47$	−0.932071	−0.135957	−0.0679783	+	0.997687i	$0.521655\pi$
$48$	4.35026	0.627906
$49$	1.00000	0.142857
$50$	5.29455	0.748763
$51$	−2.80606	−0.392927
$52$	−0.261865	−0.0363142
$53$	4.54420	0.624194	0.312097	−	0.950050i	$-0.398969\pi$
$53$	4.54420	0.624194	0.312097	+	0.950050i	$0.398969\pi$
$54$	1.48119	0.201565
$55$	−4.00000	−0.539360
$56$	−2.67513	−0.357479
$57$	−1.00000	−0.132453
$58$	−15.0435	−1.97531
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−0.231548	−0.0298927
$61$	11.9248	1.52681	0.763406	−	0.645919i	$-0.223526\pi$
$61$	11.9248	1.52681	0.763406	+	0.645919i	$0.223526\pi$
$62$	−8.49929	−1.07941
$63$	−1.00000	−0.125988
$64$	7.08110	0.885138
$65$	−1.61213	−0.199960
$66$	−4.96239	−0.610828
$67$	−8.31265	−1.01555	−0.507776	−	0.861489i	$-0.669532\pi$
$67$	−8.31265	−1.01555	−0.507776	+	0.861489i	$0.669532\pi$
$68$	0.544198	0.0659937
$69$	−9.27504	−1.11658
$70$	1.76845	0.211370
$71$	−5.89446	−0.699544	−0.349772	−	0.936835i	$-0.613741\pi$
$71$	−5.89446	−0.699544	−0.349772	+	0.936835i	$0.613741\pi$
$72$	2.67513	0.315267
$73$	−3.61213	−0.422767	−0.211384	−	0.977403i	$-0.567797\pi$
$73$	−3.61213	−0.422767	−0.211384	+	0.977403i	$0.567797\pi$
$74$	−5.81336	−0.675789
$75$	3.57452	0.412749
$76$	0.193937	0.0222460
$77$	3.35026	0.381798
$78$	−2.00000	−0.226455
$79$	8.62530	0.970422	0.485211	−	0.874397i	$-0.338743\pi$
$79$	8.62530	0.970422	0.485211	+	0.874397i	$0.338743\pi$
$80$	−5.19394	−0.580700
$81$	1.00000	0.111111
$82$	−12.7005	−1.40254
$83$	−15.6932	−1.72256	−0.861278	−	0.508134i	$-0.830335\pi$
$83$	−15.6932	−1.72256	−0.861278	+	0.508134i	$0.830335\pi$
$84$	0.193937	0.0211602
$85$	3.35026	0.363387
$86$	8.49929	0.916502
$87$	−10.1563	−1.08887
$88$	−8.96239	−0.955394
$89$	8.57452	0.908897	0.454448	−	0.890773i	$-0.349836\pi$
$89$	8.57452	0.908897	0.454448	+	0.890773i	$0.349836\pi$
$90$	−1.76845	−0.186411
$91$	1.35026	0.141546
$92$	1.79877	0.187535
$93$	−5.73813	−0.595017
$94$	1.38058	0.142396
$95$	1.19394	0.122495
$96$	−1.09332	−0.111587
$97$	11.1490	1.13201	0.566006	−	0.824401i	$-0.308488\pi$
$97$	11.1490	1.13201	0.566006	+	0.824401i	$0.308488\pi$
$98$	−1.48119	−0.149623
$99$	−3.35026	−0.336714
$100$	−0.693229	−0.0693229
$101$	−9.50659	−0.945941	−0.472970	−	0.881078i	$-0.656818\pi$
$101$	−9.50659	−0.945941	−0.472970	+	0.881078i	$0.656818\pi$
$102$	4.15633	0.411538
$103$	−5.14903	−0.507349	−0.253675	−	0.967290i	$-0.581639\pi$
$103$	−5.14903	−0.507349	−0.253675	+	0.967290i	$0.581639\pi$
$104$	−3.61213	−0.354198
$105$	1.19394	0.116516
$106$	−6.73084	−0.653757
$107$	−11.0435	−1.06761	−0.533807	−	0.845606i	$-0.679239\pi$
$107$	−11.0435	−1.06761	−0.533807	+	0.845606i	$0.679239\pi$
$108$	−0.193937	−0.0186616
$109$	−5.53690	−0.530339	−0.265170	−	0.964202i	$-0.585428\pi$
$109$	−5.53690	−0.530339	−0.265170	+	0.964202i	$0.585428\pi$
$110$	5.92478	0.564905
$111$	−3.92478	−0.372523
$112$	4.35026	0.411061
$113$	−5.69323	−0.535574	−0.267787	−	0.963478i	$-0.586292\pi$
$113$	−5.69323	−0.535574	−0.267787	+	0.963478i	$0.586292\pi$
$114$	1.48119	0.138727
$115$	11.0738	1.03264
$116$	1.96968	0.182880
$117$	−1.35026	−0.124832
$118$	−5.92478	−0.545420
$119$	−2.80606	−0.257231
$120$	−3.19394	−0.291565
$121$	0.224254	0.0203867
$122$	−17.6629	−1.59912
$123$	−8.57452	−0.773138
$124$	1.11283	0.0999355
$125$	−10.2374	−0.915663
$126$	1.48119	0.131955
$127$	12.9380	1.14806	0.574029	−	0.818835i	$-0.305380\pi$
$127$	12.9380	1.14806	0.574029	+	0.818835i	$0.305380\pi$
$128$	−12.6751	−1.12033
$129$	5.73813	0.505215
$130$	2.38787	0.209430
$131$	20.4690	1.78838	0.894191	−	0.447685i	$-0.147751\pi$
$131$	20.4690	1.78838	0.894191	+	0.447685i	$0.147751\pi$
$132$	0.649738	0.0565525
$133$	−1.00000	−0.0867110
$134$	12.3127	1.06365
$135$	−1.19394	−0.102758
$136$	7.50659	0.643685
$137$	13.4763	1.15136	0.575678	−	0.817677i	$-0.304738\pi$
$137$	13.4763	1.15136	0.575678	+	0.817677i	$0.304738\pi$
$138$	13.7381	1.16947
$139$	15.0132	1.27340	0.636700	−	0.771111i	$-0.280299\pi$
$139$	15.0132	1.27340	0.636700	+	0.771111i	$0.280299\pi$
$140$	−0.231548	−0.0195694
$141$	0.932071	0.0784946
$142$	8.73084	0.732676
$143$	4.52373	0.378293
$144$	−4.35026	−0.362522
$145$	12.1260	1.00701
$146$	5.35026	0.442791
$147$	−1.00000	−0.0824786
$148$	0.761158	0.0625668
$149$	−8.76116	−0.717742	−0.358871	−	0.933387i	$-0.616838\pi$
$149$	−8.76116	−0.717742	−0.358871	+	0.933387i	$0.616838\pi$
$150$	−5.29455	−0.432298
$151$	−10.2374	−0.833110	−0.416555	−	0.909111i	$-0.636763\pi$
$151$	−10.2374	−0.833110	−0.416555	+	0.909111i	$0.636763\pi$
$152$	2.67513	0.216982
$153$	2.80606	0.226857
$154$	−4.96239	−0.399881
$155$	6.85097	0.550283
$156$	0.261865	0.0209660
$157$	−11.4010	−0.909903	−0.454951	−	0.890516i	$-0.650343\pi$
$157$	−11.4010	−0.909903	−0.454951	+	0.890516i	$0.650343\pi$
$158$	−12.7757	−1.01638
$159$	−4.54420	−0.360378
$160$	1.30536	0.103197
$161$	−9.27504	−0.730975
$162$	−1.48119	−0.116374
$163$	−24.9624	−1.95521	−0.977603	−	0.210459i	$-0.932504\pi$
$163$	−24.9624	−1.95521	−0.977603	+	0.210459i	$0.932504\pi$
$164$	1.66291	0.129852
$165$	4.00000	0.311400
$166$	23.2447	1.80414
$167$	21.7137	1.68026	0.840128	−	0.542388i	$-0.182480\pi$
$167$	21.7137	1.68026	0.840128	+	0.542388i	$0.182480\pi$
$168$	2.67513	0.206391
$169$	−11.1768	−0.859753
$170$	−4.96239	−0.380598
$171$	1.00000	0.0764719
$172$	−1.11283	−0.0848528
$173$	−7.27504	−0.553111	−0.276555	−	0.960998i	$-0.589193\pi$
$173$	−7.27504	−0.553111	−0.276555	+	0.960998i	$0.589193\pi$
$174$	15.0435	1.14044
$175$	3.57452	0.270208
$176$	14.5745	1.09860
$177$	−4.00000	−0.300658
$178$	−12.7005	−0.951944
$179$	15.8192	1.18239	0.591193	−	0.806530i	$-0.298657\pi$
$179$	15.8192	1.18239	0.591193	+	0.806530i	$0.298657\pi$
$180$	0.231548	0.0172586
$181$	24.7005	1.83598	0.917988	−	0.396609i	$-0.129813\pi$
$181$	24.7005	1.83598	0.917988	+	0.396609i	$0.129813\pi$
$182$	−2.00000	−0.148250
$183$	−11.9248	−0.881505
$184$	24.8119	1.82916
$185$	4.68594	0.344517
$186$	8.49929	0.623198
$187$	−9.40105	−0.687473
$188$	−0.180763	−0.0131835
$189$	1.00000	0.0727393
$190$	−1.76845	−0.128297
$191$	−15.9756	−1.15595	−0.577976	−	0.816054i	$-0.696157\pi$
$191$	−15.9756	−1.15595	−0.577976	+	0.816054i	$0.696157\pi$
$192$	−7.08110	−0.511035
$193$	−7.40105	−0.532739	−0.266370	−	0.963871i	$-0.585824\pi$
$193$	−7.40105	−0.532739	−0.266370	+	0.963871i	$0.585824\pi$
$194$	−16.5139	−1.18563
$195$	1.61213	0.115447
$196$	0.193937	0.0138526
$197$	13.3258	0.949426	0.474713	−	0.880141i	$-0.342552\pi$
$197$	13.3258	0.949426	0.474713	+	0.880141i	$0.342552\pi$
$198$	4.96239	0.352662
$199$	−2.91160	−0.206398	−0.103199	−	0.994661i	$-0.532908\pi$
$199$	−2.91160	−0.206398	−0.103199	+	0.994661i	$0.532908\pi$
$200$	−9.56230	−0.676156
$201$	8.31265	0.586329
$202$	14.0811	0.990743
$203$	−10.1563	−0.712834
$204$	−0.544198	−0.0381015
$205$	10.2374	0.715013
$206$	7.62672	0.531378
$207$	9.27504	0.644660
$208$	5.87399	0.407288
$209$	−3.35026	−0.231742
$210$	−1.76845	−0.122035
$211$	−5.14903	−0.354474	−0.177237	−	0.984168i	$-0.556716\pi$
$211$	−5.14903	−0.354474	−0.177237	+	0.984168i	$0.556716\pi$
$212$	0.881286	0.0605270
$213$	5.89446	0.403882
$214$	16.3576	1.11818
$215$	−6.85097	−0.467232
$216$	−2.67513	−0.182020
$217$	−5.73813	−0.389530
$218$	8.20123	0.555457
$219$	3.61213	0.244085
$220$	−0.775746	−0.0523008
$221$	−3.78892	−0.254870
$222$	5.81336	0.390167
$223$	−2.88717	−0.193339	−0.0966695	−	0.995317i	$-0.530819\pi$
$223$	−2.88717	−0.193339	−0.0966695	+	0.995317i	$0.530819\pi$
$224$	−1.09332	−0.0730506
$225$	−3.57452	−0.238301
$226$	8.43278	0.560940
$227$	15.7889	1.04795	0.523974	−	0.851734i	$-0.324449\pi$
$227$	15.7889	1.04795	0.523974	+	0.851734i	$0.324449\pi$
$228$	−0.193937	−0.0128438
$229$	21.3865	1.41326	0.706628	−	0.707585i	$-0.250215\pi$
$229$	21.3865	1.41326	0.706628	+	0.707585i	$0.250215\pi$
$230$	−16.4025	−1.08155
$231$	−3.35026	−0.220431
$232$	27.1695	1.78376
$233$	9.01317	0.590473	0.295236	−	0.955424i	$-0.404602\pi$
$233$	9.01317	0.590473	0.295236	+	0.955424i	$0.404602\pi$
$234$	2.00000	0.130744
$235$	−1.11283	−0.0725933
$236$	0.775746	0.0504968
$237$	−8.62530	−0.560273
$238$	4.15633	0.269415
$239$	−12.4387	−0.804590	−0.402295	−	0.915510i	$-0.631787\pi$
$239$	−12.4387	−0.804590	−0.402295	+	0.915510i	$0.631787\pi$
$240$	5.19394	0.335267
$241$	11.1490	0.718172	0.359086	−	0.933304i	$-0.383088\pi$
$241$	11.1490	0.718172	0.359086	+	0.933304i	$0.383088\pi$
$242$	−0.332163	−0.0213523
$243$	−1.00000	−0.0641500
$244$	2.31265	0.148052
$245$	1.19394	0.0762778
$246$	12.7005	0.809756
$247$	−1.35026	−0.0859151
$248$	15.3503	0.974743
$249$	15.6932	0.994518
$250$	15.1636	0.959031
$251$	−12.9321	−0.816265	−0.408133	−	0.912923i	$-0.633820\pi$
$251$	−12.9321	−0.816265	−0.408133	+	0.912923i	$0.633820\pi$
$252$	−0.193937	−0.0122169
$253$	−31.0738	−1.95359
$254$	−19.1636	−1.20243
$255$	−3.35026	−0.209802
$256$	4.61213	0.288258
$257$	5.03761	0.314238	0.157119	−	0.987580i	$-0.449779\pi$
$257$	5.03761	0.314238	0.157119	+	0.987580i	$0.449779\pi$
$258$	−8.49929	−0.529143
$259$	−3.92478	−0.243874
$260$	−0.312650	−0.0193898
$261$	10.1563	0.628661
$262$	−30.3185	−1.87309
$263$	10.2619	0.632774	0.316387	−	0.948630i	$-0.397530\pi$
$263$	10.2619	0.632774	0.316387	+	0.948630i	$0.397530\pi$
$264$	8.96239	0.551597
$265$	5.42548	0.333285
$266$	1.48119	0.0908178
$267$	−8.57452	−0.524752
$268$	−1.61213	−0.0984763
$269$	2.96239	0.180620	0.0903100	−	0.995914i	$-0.471214\pi$
$269$	2.96239	0.180620	0.0903100	+	0.995914i	$0.471214\pi$
$270$	1.76845	0.107625
$271$	−17.7743	−1.07971	−0.539857	−	0.841757i	$-0.681522\pi$
$271$	−17.7743	−1.07971	−0.539857	+	0.841757i	$0.681522\pi$
$272$	−12.2071	−0.740165
$273$	−1.35026	−0.0817216
$274$	−19.9610	−1.20589
$275$	11.9756	0.722154
$276$	−1.79877	−0.108273
$277$	−15.7988	−0.949256	−0.474628	−	0.880186i	$-0.657417\pi$
$277$	−15.7988	−0.949256	−0.474628	+	0.880186i	$0.657417\pi$
$278$	−22.2374	−1.33371
$279$	5.73813	0.343533
$280$	−3.19394	−0.190874
$281$	−1.31994	−0.0787413	−0.0393706	−	0.999225i	$-0.512535\pi$
$281$	−1.31994	−0.0787413	−0.0393706	+	0.999225i	$0.512535\pi$
$282$	−1.38058	−0.0822123
$283$	6.23743	0.370777	0.185388	−	0.982665i	$-0.440646\pi$
$283$	6.23743	0.370777	0.185388	+	0.982665i	$0.440646\pi$
$284$	−1.14315	−0.0678336
$285$	−1.19394	−0.0707227
$286$	−6.70052	−0.396210
$287$	−8.57452	−0.506138
$288$	1.09332	0.0644246
$289$	−9.12601	−0.536824
$290$	−17.9610	−1.05470
$291$	−11.1490	−0.653568
$292$	−0.700523	−0.0409950
$293$	−11.4255	−0.667484	−0.333742	−	0.942664i	$-0.608311\pi$
$293$	−11.4255	−0.667484	−0.333742	+	0.942664i	$0.608311\pi$
$294$	1.48119	0.0863850
$295$	4.77575	0.278055
$296$	10.4993	0.610259
$297$	3.35026	0.194402
$298$	12.9770	0.751736
$299$	−12.5237	−0.724266
$300$	0.693229	0.0400236
$301$	5.73813	0.330741
$302$	15.1636	0.872568
$303$	9.50659	0.546139
$304$	−4.35026	−0.249505
$305$	14.2374	0.815233
$306$	−4.15633	−0.237601
$307$	3.41090	0.194670	0.0973351	−	0.995252i	$-0.468968\pi$
$307$	3.41090	0.194670	0.0973351	+	0.995252i	$0.468968\pi$
$308$	0.649738	0.0370223
$309$	5.14903	0.292918
$310$	−10.1476	−0.576346
$311$	−16.5296	−0.937308	−0.468654	−	0.883382i	$-0.655261\pi$
$311$	−16.5296	−0.937308	−0.468654	+	0.883382i	$0.655261\pi$
$312$	3.61213	0.204496
$313$	−18.6253	−1.05276	−0.526382	−	0.850248i	$-0.676452\pi$
$313$	−18.6253	−1.05276	−0.526382	+	0.850248i	$0.676452\pi$
$314$	16.8872	0.952998
$315$	−1.19394	−0.0672707
$316$	1.67276	0.0941002
$317$	−25.9452	−1.45723	−0.728615	−	0.684923i	$-0.759836\pi$
$317$	−25.9452	−1.45723	−0.728615	+	0.684923i	$0.759836\pi$
$318$	6.73084	0.377447
$319$	−34.0263	−1.90511
$320$	8.45439	0.472615
$321$	11.0435	0.616388
$322$	13.7381	0.765596
$323$	2.80606	0.156134
$324$	0.193937	0.0107743
$325$	4.82653	0.267728
$326$	36.9741	2.04781
$327$	5.53690	0.306191
$328$	22.9380	1.26654
$329$	0.932071	0.0513868
$330$	−5.92478	−0.326148
$331$	−32.7875	−1.80216	−0.901082	−	0.433648i	$-0.857226\pi$
$331$	−32.7875	−1.80216	−0.901082	+	0.433648i	$0.857226\pi$
$332$	−3.04349	−0.167033
$333$	3.92478	0.215076
$334$	−32.1622	−1.75984
$335$	−9.92478	−0.542249
$336$	−4.35026	−0.237326
$337$	−11.0884	−0.604023	−0.302012	−	0.953304i	$-0.597658\pi$
$337$	−11.0884	−0.604023	−0.302012	+	0.953304i	$0.597658\pi$
$338$	16.5550	0.900473
$339$	5.69323	0.309214
$340$	0.649738	0.0352370
$341$	−19.2243	−1.04105
$342$	−1.48119	−0.0800938
$343$	−1.00000	−0.0539949
$344$	−15.3503	−0.827631
$345$	−11.0738	−0.596194
$346$	10.7757	0.579308
$347$	−4.58910	−0.246356	−0.123178	−	0.992385i	$-0.539309\pi$
$347$	−4.58910	−0.246356	−0.123178	+	0.992385i	$0.539309\pi$
$348$	−1.96968	−0.105586
$349$	21.3258	1.14155	0.570773	−	0.821108i	$-0.306644\pi$
$349$	21.3258	1.14155	0.570773	+	0.821108i	$0.306644\pi$
$350$	−5.29455	−0.283006
$351$	1.35026	0.0720716
$352$	−3.66291	−0.195234
$353$	16.9829	0.903906	0.451953	−	0.892042i	$-0.350727\pi$
$353$	16.9829	0.903906	0.451953	+	0.892042i	$0.350727\pi$
$354$	5.92478	0.314898
$355$	−7.03761	−0.373518
$356$	1.66291	0.0881342
$357$	2.80606	0.148513
$358$	−23.4314	−1.23839
$359$	−9.02302	−0.476217	−0.238108	−	0.971239i	$-0.576527\pi$
$359$	−9.02302	−0.476217	−0.238108	+	0.971239i	$0.576527\pi$
$360$	3.19394	0.168335
$361$	1.00000	0.0526316
$362$	−36.5863	−1.92293
$363$	−0.224254	−0.0117703
$364$	0.261865	0.0137255
$365$	−4.31265	−0.225734
$366$	17.6629	0.923255
$367$	22.0870	1.15293	0.576466	−	0.817121i	$-0.304432\pi$
$367$	22.0870	1.15293	0.576466	+	0.817121i	$0.304432\pi$
$368$	−40.3488	−2.10333
$369$	8.57452	0.446371
$370$	−6.94078	−0.360834
$371$	−4.54420	−0.235923
$372$	−1.11283	−0.0576978
$373$	−3.55149	−0.183889	−0.0919447	−	0.995764i	$-0.529308\pi$
$373$	−3.55149	−0.183889	−0.0919447	+	0.995764i	$0.529308\pi$
$374$	13.9248	0.720033
$375$	10.2374	0.528658
$376$	−2.49341	−0.128588
$377$	−13.7137	−0.706291
$378$	−1.48119	−0.0761844
$379$	−2.38787	−0.122657	−0.0613284	−	0.998118i	$-0.519534\pi$
$379$	−2.38787	−0.122657	−0.0613284	+	0.998118i	$0.519534\pi$
$380$	0.231548	0.0118782
$381$	−12.9380	−0.662831
$382$	23.6629	1.21070
$383$	7.01317	0.358356	0.179178	−	0.983817i	$-0.442656\pi$
$383$	7.01317	0.358356	0.179178	+	0.983817i	$0.442656\pi$
$384$	12.6751	0.646825
$385$	4.00000	0.203859
$386$	10.9624	0.557971
$387$	−5.73813	−0.291686
$388$	2.16220	0.109769
$389$	1.16362	0.0589978	0.0294989	−	0.999565i	$-0.490609\pi$
$389$	1.16362	0.0589978	0.0294989	+	0.999565i	$0.490609\pi$
$390$	−2.38787	−0.120915
$391$	26.0263	1.31621
$392$	2.67513	0.135115
$393$	−20.4690	−1.03252
$394$	−19.7381	−0.994393
$395$	10.2981	0.518152
$396$	−0.649738	−0.0326506
$397$	−3.67276	−0.184331	−0.0921653	−	0.995744i	$-0.529379\pi$
$397$	−3.67276	−0.184331	−0.0921653	+	0.995744i	$0.529379\pi$
$398$	4.31265	0.216174
$399$	1.00000	0.0500626
$400$	15.5501	0.777504
$401$	3.76845	0.188188	0.0940938	−	0.995563i	$-0.470005\pi$
$401$	3.76845	0.188188	0.0940938	+	0.995563i	$0.470005\pi$
$402$	−12.3127	−0.614099
$403$	−7.74798	−0.385955
$404$	−1.84367	−0.0917263
$405$	1.19394	0.0593272
$406$	15.0435	0.746596
$407$	−13.1490	−0.651773
$408$	−7.50659	−0.371631
$409$	−4.42407	−0.218756	−0.109378	−	0.994000i	$-0.534886\pi$
$409$	−4.42407	−0.218756	−0.109378	+	0.994000i	$0.534886\pi$
$410$	−15.1636	−0.748878
$411$	−13.4763	−0.664735
$412$	−0.998585	−0.0491968
$413$	−4.00000	−0.196827
$414$	−13.7381	−0.675192
$415$	−18.7367	−0.919749
$416$	−1.47627	−0.0723801
$417$	−15.0132	−0.735198
$418$	4.96239	0.242718
$419$	39.8554	1.94707	0.973533	−	0.228548i	$-0.0733977\pi$
$419$	39.8554	1.94707	0.973533	+	0.228548i	$0.0733977\pi$
$420$	0.231548	0.0112984
$421$	7.08840	0.345467	0.172734	−	0.984969i	$-0.444740\pi$
$421$	7.08840	0.345467	0.172734	+	0.984969i	$0.444740\pi$
$422$	7.62672	0.371263
$423$	−0.932071	−0.0453189
$424$	12.1563	0.590363
$425$	−10.0303	−0.486542
$426$	−8.73084	−0.423011
$427$	−11.9248	−0.577080
$428$	−2.14174	−0.103525
$429$	−4.52373	−0.218408
$430$	10.1476	0.489362
$431$	35.8799	1.72827	0.864136	−	0.503258i	$-0.167865\pi$
$431$	35.8799	1.72827	0.864136	+	0.503258i	$0.167865\pi$
$432$	4.35026	0.209302
$433$	−26.6253	−1.27953	−0.639765	−	0.768570i	$-0.720968\pi$
$433$	−26.6253	−1.27953	−0.639765	+	0.768570i	$0.720968\pi$
$434$	8.49929	0.407979
$435$	−12.1260	−0.581398
$436$	−1.07381	−0.0514261
$437$	9.27504	0.443685
$438$	−5.35026	−0.255645
$439$	−1.21440	−0.0579604	−0.0289802	−	0.999580i	$-0.509226\pi$
$439$	−1.21440	−0.0579604	−0.0289802	+	0.999580i	$0.509226\pi$
$440$	−10.7005	−0.510127
$441$	1.00000	0.0476190
$442$	5.61213	0.266942
$443$	−7.81336	−0.371224	−0.185612	−	0.982623i	$-0.559427\pi$
$443$	−7.81336	−0.371224	−0.185612	+	0.982623i	$0.559427\pi$
$444$	−0.761158	−0.0361230
$445$	10.2374	0.485301
$446$	4.27645	0.202496
$447$	8.76116	0.414389
$448$	−7.08110	−0.334551
$449$	12.1417	0.573004	0.286502	−	0.958080i	$-0.407507\pi$
$449$	12.1417	0.573004	0.286502	+	0.958080i	$0.407507\pi$
$450$	5.29455	0.249588
$451$	−28.7269	−1.35270
$452$	−1.10413	−0.0519337
$453$	10.2374	0.480996
$454$	−23.3865	−1.09758
$455$	1.61213	0.0755777
$456$	−2.67513	−0.125274
$457$	39.3766	1.84196	0.920980	−	0.389610i	$-0.127390\pi$
$457$	39.3766	1.84196	0.920980	+	0.389610i	$0.127390\pi$
$458$	−31.6775	−1.48019
$459$	−2.80606	−0.130976
$460$	2.14762	0.100133
$461$	0.670206	0.0312146	0.0156073	−	0.999878i	$-0.495032\pi$
$461$	0.670206	0.0312146	0.0156073	+	0.999878i	$0.495032\pi$
$462$	4.96239	0.230871
$463$	22.0263	1.02365	0.511826	−	0.859089i	$-0.328969\pi$
$463$	22.0263	1.02365	0.511826	+	0.859089i	$0.328969\pi$
$464$	−44.1827	−2.05113
$465$	−6.85097	−0.317706
$466$	−13.3503	−0.618439
$467$	−3.90431	−0.180670	−0.0903349	−	0.995911i	$-0.528794\pi$
$467$	−3.90431	−0.180670	−0.0903349	+	0.995911i	$0.528794\pi$
$468$	−0.261865	−0.0121047
$469$	8.31265	0.383843
$470$	1.64832	0.0760315
$471$	11.4010	0.525333
$472$	10.7005	0.492532
$473$	19.2243	0.883932
$474$	12.7757	0.586809
$475$	−3.57452	−0.164010
$476$	−0.544198	−0.0249433
$477$	4.54420	0.208065
$478$	18.4241	0.842697
$479$	31.3806	1.43382	0.716908	−	0.697168i	$-0.245557\pi$
$479$	31.3806	1.43382	0.716908	+	0.697168i	$0.245557\pi$
$480$	−1.30536	−0.0595811
$481$	−5.29948	−0.241635
$482$	−16.5139	−0.752187
$483$	9.27504	0.422029
$484$	0.0434910	0.00197686
$485$	13.3112	0.604432
$486$	1.48119	0.0671883
$487$	−5.55149	−0.251562	−0.125781	−	0.992058i	$-0.540144\pi$
$487$	−5.55149	−0.251562	−0.125781	+	0.992058i	$0.540144\pi$
$488$	31.9003	1.44406
$489$	24.9624	1.12884
$490$	−1.76845	−0.0798905
$491$	14.5139	0.655002	0.327501	−	0.944851i	$-0.393793\pi$
$491$	14.5139	0.655002	0.327501	+	0.944851i	$0.393793\pi$
$492$	−1.66291	−0.0749699
$493$	28.4993	1.28354
$494$	2.00000	0.0899843
$495$	−4.00000	−0.179787
$496$	−24.9624	−1.12084
$497$	5.89446	0.264403
$498$	−23.2447	−1.04162
$499$	33.7743	1.51195	0.755973	−	0.654602i	$-0.227164\pi$
$499$	33.7743	1.51195	0.755973	+	0.654602i	$0.227164\pi$
$500$	−1.98541	−0.0887903
$501$	−21.7137	−0.970096
$502$	19.1549	0.854925
$503$	−7.23013	−0.322376	−0.161188	−	0.986924i	$-0.551533\pi$
$503$	−7.23013	−0.322376	−0.161188	+	0.986924i	$0.551533\pi$
$504$	−2.67513	−0.119160
$505$	−11.3503	−0.505080
$506$	46.0263	2.04612
$507$	11.1768	0.496379
$508$	2.50914	0.111325
$509$	−35.0640	−1.55418	−0.777091	−	0.629388i	$-0.783306\pi$
$509$	−35.0640	−1.55418	−0.777091	+	0.629388i	$0.783306\pi$
$510$	4.96239	0.219738
$511$	3.61213	0.159791
$512$	18.5188	0.818423
$513$	−1.00000	−0.0441511
$514$	−7.46168	−0.329121
$515$	−6.14762	−0.270896
$516$	1.11283	0.0489898
$517$	3.12268	0.137335
$518$	5.81336	0.255424
$519$	7.27504	0.319339
$520$	−4.31265	−0.189122
$521$	5.28963	0.231743	0.115871	−	0.993264i	$-0.463034\pi$
$521$	5.28963	0.231743	0.115871	+	0.993264i	$0.463034\pi$
$522$	−15.0435	−0.658436
$523$	−20.3371	−0.889279	−0.444639	−	0.895710i	$-0.646668\pi$
$523$	−20.3371	−0.889279	−0.444639	+	0.895710i	$0.646668\pi$
$524$	3.96968	0.173416
$525$	−3.57452	−0.156005
$526$	−15.1998	−0.662743
$527$	16.1016	0.701395
$528$	−14.5745	−0.634274
$529$	63.0263	2.74028
$530$	−8.03620	−0.349070
$531$	4.00000	0.173585
$532$	−0.193937	−0.00840822
$533$	−11.5778	−0.501492
$534$	12.7005	0.549605
$535$	−13.1852	−0.570047
$536$	−22.2374	−0.960511
$537$	−15.8192	−0.682650
$538$	−4.38787	−0.189175
$539$	−3.35026	−0.144306
$540$	−0.231548	−0.00996424
$541$	8.29806	0.356762	0.178381	−	0.983962i	$-0.442914\pi$
$541$	8.29806	0.356762	0.178381	+	0.983962i	$0.442914\pi$
$542$	26.3272	1.13085
$543$	−24.7005	−1.06000
$544$	3.06793	0.131536
$545$	−6.61071	−0.283172
$546$	2.00000	0.0855921
$547$	−10.4485	−0.446746	−0.223373	−	0.974733i	$-0.571707\pi$
$547$	−10.4485	−0.446746	−0.223373	+	0.974733i	$0.571707\pi$
$548$	2.61354	0.111645
$549$	11.9248	0.508937
$550$	−17.7381	−0.756357
$551$	10.1563	0.432674
$552$	−24.8119	−1.05607
$553$	−8.62530	−0.366785
$554$	23.4010	0.994215
$555$	−4.68594	−0.198907
$556$	2.91160	0.123479
$557$	23.3112	0.987729	0.493864	−	0.869539i	$-0.335584\pi$
$557$	23.3112	0.987729	0.493864	+	0.869539i	$0.335584\pi$
$558$	−8.49929	−0.359804
$559$	7.74798	0.327705
$560$	5.19394	0.219484
$561$	9.40105	0.396913
$562$	1.95509	0.0824707
$563$	24.4631	1.03100	0.515498	−	0.856891i	$-0.327607\pi$
$563$	24.4631	1.03100	0.515498	+	0.856891i	$0.327607\pi$
$564$	0.180763	0.00761148
$565$	−6.79735	−0.285967
$566$	−9.23884	−0.388338
$567$	−1.00000	−0.0419961
$568$	−15.7685	−0.661630
$569$	−31.3463	−1.31410	−0.657052	−	0.753845i	$-0.728197\pi$
$569$	−31.3463	−1.31410	−0.657052	+	0.753845i	$0.728197\pi$
$570$	1.76845	0.0740723
$571$	−12.2520	−0.512731	−0.256365	−	0.966580i	$-0.582525\pi$
$571$	−12.2520	−0.512731	−0.256365	+	0.966580i	$0.582525\pi$
$572$	0.877317	0.0366825
$573$	15.9756	0.667389
$574$	12.7005	0.530110
$575$	−33.1538	−1.38261
$576$	7.08110	0.295046
$577$	5.07381	0.211225	0.105613	−	0.994407i	$-0.466320\pi$
$577$	5.07381	0.211225	0.105613	+	0.994407i	$0.466320\pi$
$578$	13.5174	0.562249
$579$	7.40105	0.307577
$580$	2.35168	0.0976480
$581$	15.6932	0.651065
$582$	16.5139	0.684522
$583$	−15.2243	−0.630524
$584$	−9.66291	−0.399854
$585$	−1.61213	−0.0666532
$586$	16.9234	0.699098
$587$	−1.82909	−0.0754945	−0.0377472	−	0.999287i	$-0.512018\pi$
$587$	−1.82909	−0.0754945	−0.0377472	+	0.999287i	$0.512018\pi$
$588$	−0.193937	−0.00799781
$589$	5.73813	0.236436
$590$	−7.07381	−0.291224
$591$	−13.3258	−0.548151
$592$	−17.0738	−0.701729
$593$	−16.5804	−0.680875	−0.340438	−	0.940267i	$-0.610575\pi$
$593$	−16.5804	−0.680875	−0.340438	+	0.940267i	$0.610575\pi$
$594$	−4.96239	−0.203609
$595$	−3.35026	−0.137347
$596$	−1.69911	−0.0695982
$597$	2.91160	0.119164
$598$	18.5501	0.758569
$599$	24.3430	0.994627	0.497313	−	0.867571i	$-0.334320\pi$
$599$	24.3430	0.994627	0.497313	+	0.867571i	$0.334320\pi$
$600$	9.56230	0.390379
$601$	20.9525	0.854672	0.427336	−	0.904093i	$-0.359452\pi$
$601$	20.9525	0.854672	0.427336	+	0.904093i	$0.359452\pi$
$602$	−8.49929	−0.346405
$603$	−8.31265	−0.338517
$604$	−1.98541	−0.0807853
$605$	0.267745	0.0108854
$606$	−14.0811	−0.572006
$607$	−2.85097	−0.115717	−0.0578586	−	0.998325i	$-0.518427\pi$
$607$	−2.85097	−0.115717	−0.0578586	+	0.998325i	$0.518427\pi$
$608$	1.09332	0.0443400
$609$	10.1563	0.411555
$610$	−21.0884	−0.853844
$611$	1.25854	0.0509151
$612$	0.544198	0.0219979
$613$	−15.7988	−0.638106	−0.319053	−	0.947737i	$-0.603365\pi$
$613$	−15.7988	−0.638106	−0.319053	+	0.947737i	$0.603365\pi$
$614$	−5.05220	−0.203890
$615$	−10.2374	−0.412813
$616$	8.96239	0.361105
$617$	−22.6253	−0.910860	−0.455430	−	0.890272i	$-0.650515\pi$
$617$	−22.6253	−0.910860	−0.455430	+	0.890272i	$0.650515\pi$
$618$	−7.62672	−0.306791
$619$	−35.4880	−1.42638	−0.713192	−	0.700969i	$-0.752751\pi$
$619$	−35.4880	−1.42638	−0.713192	+	0.700969i	$0.752751\pi$
$620$	1.32865	0.0533600
$621$	−9.27504	−0.372194
$622$	24.4836	0.981701
$623$	−8.57452	−0.343531
$624$	−5.87399	−0.235148
$625$	5.64974	0.225990
$626$	27.5877	1.10263
$627$	3.35026	0.133797
$628$	−2.21108	−0.0882317
$629$	11.0132	0.439124
$630$	1.76845	0.0704568
$631$	−3.03761	−0.120925	−0.0604627	−	0.998170i	$-0.519258\pi$
$631$	−3.03761	−0.120925	−0.0604627	+	0.998170i	$0.519258\pi$
$632$	23.0738	0.917827
$633$	5.14903	0.204656
$634$	38.4299	1.52625
$635$	15.4471	0.612999
$636$	−0.881286	−0.0349453
$637$	−1.35026	−0.0534993
$638$	50.3996	1.99534
$639$	−5.89446	−0.233181
$640$	−15.1333	−0.598196
$641$	−32.5560	−1.28588	−0.642942	−	0.765915i	$-0.722286\pi$
$641$	−32.5560	−1.28588	−0.642942	+	0.765915i	$0.722286\pi$
$642$	−16.3576	−0.645581
$643$	−30.8627	−1.21711	−0.608554	−	0.793513i	$-0.708250\pi$
$643$	−30.8627	−1.21711	−0.608554	+	0.793513i	$0.708250\pi$
$644$	−1.79877	−0.0708814
$645$	6.85097	0.269757
$646$	−4.15633	−0.163528
$647$	13.5164	0.531386	0.265693	−	0.964058i	$-0.414399\pi$
$647$	13.5164	0.531386	0.265693	+	0.964058i	$0.414399\pi$
$648$	2.67513	0.105089
$649$	−13.4010	−0.526037
$650$	−7.14903	−0.280408
$651$	5.73813	0.224895
$652$	−4.84112	−0.189593
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−8.20123	−0.320694
$655$	24.4387	0.954897
$656$	−37.3014	−1.45637
$657$	−3.61213	−0.140922
$658$	−1.38058	−0.0538206
$659$	−0.0303172	−0.00118099	−0.000590495	−	1.00000i	$-0.500188\pi$
$659$	−0.0303172	−0.00118099	−0.000590495		1.00000i	$0.500188\pi$
$660$	0.775746	0.0301959
$661$	−49.0494	−1.90780	−0.953900	−	0.300126i	$-0.902971\pi$
$661$	−49.0494	−1.90780	−0.953900	+	0.300126i	$0.902971\pi$
$662$	48.5647	1.88752
$663$	3.78892	0.147149
$664$	−41.9814	−1.62920
$665$	−1.19394	−0.0462989
$666$	−5.81336	−0.225263
$667$	94.2003	3.64745
$668$	4.21108	0.162932
$669$	2.88717	0.111624
$670$	14.7005	0.567931
$671$	−39.9511	−1.54230
$672$	1.09332	0.0421758
$673$	1.10299	0.0425169	0.0212585	−	0.999774i	$-0.493233\pi$
$673$	1.10299	0.0425169	0.0212585	+	0.999774i	$0.493233\pi$
$674$	16.4241	0.632632
$675$	3.57452	0.137583
$676$	−2.16759	−0.0833688
$677$	−1.81336	−0.0696930	−0.0348465	−	0.999393i	$-0.511094\pi$
$677$	−1.81336	−0.0696930	−0.0348465	+	0.999393i	$0.511094\pi$
$678$	−8.43278	−0.323859
$679$	−11.1490	−0.427861
$680$	8.96239	0.343692
$681$	−15.7889	−0.605033
$682$	28.4749	1.09036
$683$	−24.8061	−0.949178	−0.474589	−	0.880208i	$-0.657403\pi$
$683$	−24.8061	−0.949178	−0.474589	+	0.880208i	$0.657403\pi$
$684$	0.193937	0.00741535
$685$	16.0898	0.614760
$686$	1.48119	0.0565523
$687$	−21.3865	−0.815944
$688$	24.9624	0.951682
$689$	−6.13586	−0.233758
$690$	16.4025	0.624431
$691$	13.7743	0.524000	0.262000	−	0.965068i	$-0.415618\pi$
$691$	13.7743	0.524000	0.262000	+	0.965068i	$0.415618\pi$
$692$	−1.41090	−0.0536342
$693$	3.35026	0.127266
$694$	6.79735	0.258024
$695$	17.9248	0.679926
$696$	−27.1695	−1.02986
$697$	24.0606	0.911362
$698$	−31.5877	−1.19561
$699$	−9.01317	−0.340910
$700$	0.693229	0.0262016
$701$	32.5501	1.22940	0.614700	−	0.788761i	$-0.289277\pi$
$701$	32.5501	1.22940	0.614700	+	0.788761i	$0.289277\pi$
$702$	−2.00000	−0.0754851
$703$	3.92478	0.148026
$704$	−23.7235	−0.894115
$705$	1.11283	0.0419118
$706$	−25.1549	−0.946718
$707$	9.50659	0.357532
$708$	−0.775746	−0.0291543
$709$	−40.0508	−1.50414	−0.752069	−	0.659084i	$-0.770944\pi$
$709$	−40.0508	−1.50414	−0.752069	+	0.659084i	$0.770944\pi$
$710$	10.4241	0.391208
$711$	8.62530	0.323474
$712$	22.9380	0.859636
$713$	53.2214	1.99316
$714$	−4.15633	−0.155547
$715$	5.40105	0.201988
$716$	3.06793	0.114654
$717$	12.4387	0.464530
$718$	13.3649	0.498772
$719$	4.93207	0.183935	0.0919676	−	0.995762i	$-0.470684\pi$
$719$	4.93207	0.183935	0.0919676	+	0.995762i	$0.470684\pi$
$720$	−5.19394	−0.193567
$721$	5.14903	0.191760
$722$	−1.48119	−0.0551243
$723$	−11.1490	−0.414637
$724$	4.79033	0.178031
$725$	−36.3039	−1.34829
$726$	0.332163	0.0123277
$727$	−16.7757	−0.622178	−0.311089	−	0.950381i	$-0.600694\pi$
$727$	−16.7757	−0.622178	−0.311089	+	0.950381i	$0.600694\pi$
$728$	3.61213	0.133874
$729$	1.00000	0.0370370
$730$	6.38787	0.236426
$731$	−16.1016	−0.595538
$732$	−2.31265	−0.0854780
$733$	−40.8627	−1.50930	−0.754650	−	0.656128i	$-0.772193\pi$
$733$	−40.8627	−1.50930	−0.754650	+	0.656128i	$0.772193\pi$
$734$	−32.7151	−1.20754
$735$	−1.19394	−0.0440390
$736$	10.1406	0.373787
$737$	27.8496	1.02585
$738$	−12.7005	−0.467513
$739$	−25.5877	−0.941258	−0.470629	−	0.882331i	$-0.655973\pi$
$739$	−25.5877	−0.941258	−0.470629	+	0.882331i	$0.655973\pi$
$740$	0.908774	0.0334072
$741$	1.35026	0.0496031
$742$	6.73084	0.247097
$743$	33.9062	1.24390	0.621949	−	0.783058i	$-0.286341\pi$
$743$	33.9062	1.24390	0.621949	+	0.783058i	$0.286341\pi$
$744$	−15.3503	−0.562768
$745$	−10.4603	−0.383235
$746$	5.26045	0.192599
$747$	−15.6932	−0.574185
$748$	−1.82321	−0.0666630
$749$	11.0435	0.403520
$750$	−15.1636	−0.553697
$751$	38.7123	1.41263	0.706316	−	0.707897i	$-0.250356\pi$
$751$	38.7123	1.41263	0.706316	+	0.707897i	$0.250356\pi$
$752$	4.05475	0.147862
$753$	12.9321	0.471271
$754$	20.3127	0.739743
$755$	−12.2228	−0.444835
$756$	0.193937	0.00705340
$757$	−4.59895	−0.167152	−0.0835759	−	0.996501i	$-0.526634\pi$
$757$	−4.59895	−0.167152	−0.0835759	+	0.996501i	$0.526634\pi$
$758$	3.53690	0.128466
$759$	31.0738	1.12791
$760$	3.19394	0.115856
$761$	47.0698	1.70628	0.853140	−	0.521682i	$-0.174695\pi$
$761$	47.0698	1.70628	0.853140	+	0.521682i	$0.174695\pi$
$762$	19.1636	0.694225
$763$	5.53690	0.200449
$764$	−3.09825	−0.112091
$765$	3.35026	0.121129
$766$	−10.3879	−0.375329
$767$	−5.40105	−0.195021
$768$	−4.61213	−0.166426
$769$	11.4010	0.411132	0.205566	−	0.978643i	$-0.434096\pi$
$769$	11.4010	0.411132	0.205566	+	0.978643i	$0.434096\pi$
$770$	−5.92478	−0.213514
$771$	−5.03761	−0.181425
$772$	−1.43533	−0.0516588
$773$	−30.6497	−1.10239	−0.551197	−	0.834375i	$-0.685829\pi$
$773$	−30.6497	−1.10239	−0.551197	+	0.834375i	$0.685829\pi$
$774$	8.49929	0.305501
$775$	−20.5111	−0.736779
$776$	29.8251	1.07066
$777$	3.92478	0.140801
$778$	−1.72355	−0.0617921
$779$	8.57452	0.307214
$780$	0.312650	0.0111947
$781$	19.7480	0.706638
$782$	−38.5501	−1.37855
$783$	−10.1563	−0.362957
$784$	−4.35026	−0.155366
$785$	−13.6121	−0.485838
$786$	30.3185	1.08143
$787$	31.1392	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$787$	31.1392	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$788$	2.58436	0.0920642
$789$	−10.2619	−0.365332
$790$	−15.2534	−0.542693
$791$	5.69323	0.202428
$792$	−8.96239	−0.318465
$793$	−16.1016	−0.571784
$794$	5.44007	0.193061
$795$	−5.42548	−0.192422
$796$	−0.564666	−0.0200141
$797$	−19.7988	−0.701308	−0.350654	−	0.936505i	$-0.614041\pi$
$797$	−19.7988	−0.701308	−0.350654	+	0.936505i	$0.614041\pi$
$798$	−1.48119	−0.0524337
$799$	−2.61545	−0.0925280
$800$	−3.90809	−0.138172
$801$	8.57452	0.302966
$802$	−5.58181	−0.197101
$803$	12.1016	0.427055
$804$	1.61213	0.0568553
$805$	−11.0738	−0.390300
$806$	11.4763	0.404234
$807$	−2.96239	−0.104281
$808$	−25.4314	−0.894672
$809$	24.0752	0.846440	0.423220	−	0.906027i	$-0.360900\pi$
$809$	24.0752	0.846440	0.423220	+	0.906027i	$0.360900\pi$
$810$	−1.76845	−0.0621371
$811$	−39.4763	−1.38620	−0.693100	−	0.720842i	$-0.743755\pi$
$811$	−39.4763	−1.38620	−0.693100	+	0.720842i	$0.743755\pi$
$812$	−1.96968	−0.0691223
$813$	17.7743	0.623373
$814$	19.4763	0.682643
$815$	−29.8035	−1.04397
$816$	12.2071	0.427334
$817$	−5.73813	−0.200752
$818$	6.55291	0.229117
$819$	1.35026	0.0471820
$820$	1.98541	0.0693336
$821$	−35.9854	−1.25590	−0.627950	−	0.778254i	$-0.716106\pi$
$821$	−35.9854	−1.25590	−0.627950	+	0.778254i	$0.716106\pi$
$822$	19.9610	0.696219
$823$	−16.4387	−0.573016	−0.286508	−	0.958078i	$-0.592494\pi$
$823$	−16.4387	−0.573016	−0.286508	+	0.958078i	$0.592494\pi$
$824$	−13.7743	−0.479852
$825$	−11.9756	−0.416936
$826$	5.92478	0.206149
$827$	39.9814	1.39029	0.695145	−	0.718869i	$-0.255340\pi$
$827$	39.9814	1.39029	0.695145	+	0.718869i	$0.255340\pi$
$828$	1.79877	0.0625116
$829$	−43.3522	−1.50568	−0.752842	−	0.658202i	$-0.771317\pi$
$829$	−43.3522	−1.50568	−0.752842	+	0.658202i	$0.771317\pi$
$830$	27.7527	0.963311
$831$	15.7988	0.548053
$832$	−9.56134	−0.331480
$833$	2.80606	0.0972243
$834$	22.2374	0.770019
$835$	25.9248	0.897164
$836$	−0.649738	−0.0224717
$837$	−5.73813	−0.198339
$838$	−59.0336	−2.03928
$839$	−18.2981	−0.631719	−0.315860	−	0.948806i	$-0.602293\pi$
$839$	−18.2981	−0.631719	−0.315860	+	0.948806i	$0.602293\pi$
$840$	3.19394	0.110201
$841$	74.1509	2.55693
$842$	−10.4993	−0.361830
$843$	1.31994	0.0454613
$844$	−0.998585	−0.0343727
$845$	−13.3444	−0.459061
$846$	1.38058	0.0474653
$847$	−0.224254	−0.00770545
$848$	−19.7685	−0.678851
$849$	−6.23743	−0.214068
$850$	14.8568	0.509586
$851$	36.4025	1.24786
$852$	1.14315	0.0391637
$853$	−50.5764	−1.73170	−0.865852	−	0.500300i	$-0.833223\pi$
$853$	−50.5764	−1.73170	−0.865852	+	0.500300i	$0.833223\pi$
$854$	17.6629	0.604412
$855$	1.19394	0.0408318
$856$	−29.5428	−1.00975
$857$	−38.8726	−1.32786	−0.663931	−	0.747794i	$-0.731113\pi$
$857$	−38.8726	−1.32786	−0.663931	+	0.747794i	$0.731113\pi$
$858$	6.70052	0.228752
$859$	−53.5026	−1.82549	−0.912743	−	0.408535i	$-0.866040\pi$
$859$	−53.5026	−1.82549	−0.912743	+	0.408535i	$0.866040\pi$
$860$	−1.32865	−0.0453067
$861$	8.57452	0.292219
$862$	−53.1451	−1.81013
$863$	16.5950	0.564900	0.282450	−	0.959282i	$-0.408853\pi$
$863$	16.5950	0.564900	0.282450	+	0.959282i	$0.408853\pi$
$864$	−1.09332	−0.0371955
$865$	−8.68594	−0.295331
$866$	39.4372	1.34013
$867$	9.12601	0.309935
$868$	−1.11283	−0.0377721
$869$	−28.8970	−0.980264
$870$	17.9610	0.608934
$871$	11.2243	0.380319
$872$	−14.8119	−0.501596
$873$	11.1490	0.377338
$874$	−13.7381	−0.464699
$875$	10.2374	0.346088
$876$	0.700523	0.0236685
$877$	−22.7757	−0.769082	−0.384541	−	0.923108i	$-0.625640\pi$
$877$	−22.7757	−0.769082	−0.384541	+	0.923108i	$0.625640\pi$
$878$	1.79877	0.0607055
$879$	11.4255	0.385372
$880$	17.4010	0.586589
$881$	−13.3444	−0.449584	−0.224792	−	0.974407i	$-0.572170\pi$
$881$	−13.3444	−0.449584	−0.224792	+	0.974407i	$0.572170\pi$
$882$	−1.48119	−0.0498744
$883$	−19.4471	−0.654447	−0.327223	−	0.944947i	$-0.606113\pi$
$883$	−19.4471	−0.654447	−0.327223	+	0.944947i	$0.606113\pi$
$884$	−0.734810	−0.0247143
$885$	−4.77575	−0.160535
$886$	11.5731	0.388806
$887$	−26.8627	−0.901962	−0.450981	−	0.892534i	$-0.648926\pi$
$887$	−26.8627	−0.901962	−0.450981	+	0.892534i	$0.648926\pi$
$888$	−10.4993	−0.352333
$889$	−12.9380	−0.433925
$890$	−15.1636	−0.508286
$891$	−3.35026	−0.112238
$892$	−0.559927	−0.0187477
$893$	−0.932071	−0.0311906
$894$	−12.9770	−0.434015
$895$	18.8872	0.631328
$896$	12.6751	0.423446
$897$	12.5237	0.418155
$898$	−17.9843	−0.600143
$899$	58.2784	1.94369
$900$	−0.693229	−0.0231076
$901$	12.7513	0.424808
$902$	42.5501	1.41676
$903$	−5.73813	−0.190953
$904$	−15.2301	−0.506547
$905$	29.4909	0.980309
$906$	−15.1636	−0.503778
$907$	33.8759	1.12483	0.562415	−	0.826855i	$-0.309872\pi$
$907$	33.8759	1.12483	0.562415	+	0.826855i	$0.309872\pi$
$908$	3.06205	0.101618
$909$	−9.50659	−0.315314
$910$	−2.38787	−0.0791572
$911$	−23.2546	−0.770458	−0.385229	−	0.922821i	$-0.625878\pi$
$911$	−23.2546	−0.770458	−0.385229	+	0.922821i	$0.625878\pi$
$912$	4.35026	0.144052
$913$	52.5764	1.74003
$914$	−58.3244	−1.92920
$915$	−14.2374	−0.470675
$916$	4.14762	0.137041
$917$	−20.4690	−0.675945
$918$	4.15633	0.137179
$919$	33.5515	1.10676	0.553381	−	0.832928i	$-0.313337\pi$
$919$	33.5515	1.10676	0.553381	+	0.832928i	$0.313337\pi$
$920$	29.6239	0.976671
$921$	−3.41090	−0.112393
$922$	−0.992706	−0.0326930
$923$	7.95906	0.261976
$924$	−0.649738	−0.0213748
$925$	−14.0292	−0.461276
$926$	−32.6253	−1.07213
$927$	−5.14903	−0.169116
$928$	11.1041	0.364511
$929$	26.2532	0.861338	0.430669	−	0.902510i	$-0.358278\pi$
$929$	26.2532	0.861338	0.430669	+	0.902510i	$0.358278\pi$
$930$	10.1476	0.332753
$931$	1.00000	0.0327737
$932$	1.74798	0.0572571
$933$	16.5296	0.541155
$934$	5.78304	0.189227
$935$	−11.2243	−0.367072
$936$	−3.61213	−0.118066
$937$	37.8496	1.23649	0.618246	−	0.785985i	$-0.287844\pi$
$937$	37.8496	1.23649	0.618246	+	0.785985i	$0.287844\pi$
$938$	−12.3127	−0.402022
$939$	18.6253	0.607814
$940$	−0.215819	−0.00703925
$941$	−36.4847	−1.18937	−0.594684	−	0.803960i	$-0.702723\pi$
$941$	−36.4847	−1.18937	−0.594684	+	0.803960i	$0.702723\pi$
$942$	−16.8872	−0.550214
$943$	79.5290	2.58982
$944$	−17.4010	−0.566356
$945$	1.19394	0.0388388
$946$	−28.4749	−0.925797
$947$	−47.1509	−1.53220	−0.766100	−	0.642722i	$-0.777805\pi$
$947$	−47.1509	−1.53220	−0.766100	+	0.642722i	$0.777805\pi$
$948$	−1.67276	−0.0543288
$949$	4.87732	0.158324
$950$	5.29455	0.171778
$951$	25.9452	0.841332
$952$	−7.50659	−0.243290
$953$	−19.6786	−0.637454	−0.318727	−	0.947847i	$-0.603255\pi$
$953$	−19.6786	−0.637454	−0.318727	+	0.947847i	$0.603255\pi$
$954$	−6.73084	−0.217919
$955$	−19.0738	−0.617214
$956$	−2.41231	−0.0780197
$957$	34.0263	1.09992
$958$	−46.4807	−1.50172
$959$	−13.4763	−0.435171
$960$	−8.45439	−0.272864
$961$	1.92619	0.0621352
$962$	7.84955	0.253080
$963$	−11.0435	−0.355872
$964$	2.16220	0.0696399
$965$	−8.83638	−0.284453
$966$	−13.7381	−0.442017
$967$	−53.1655	−1.70969	−0.854844	−	0.518885i	$-0.826347\pi$
$967$	−53.1655	−1.70969	−0.854844	+	0.518885i	$0.826347\pi$
$968$	0.599908	0.0192818
$969$	−2.80606	−0.0901437
$970$	−19.7165	−0.633060
$971$	−16.7269	−0.536791	−0.268395	−	0.963309i	$-0.586493\pi$
$971$	−16.7269	−0.536791	−0.268395	+	0.963309i	$0.586493\pi$
$972$	−0.193937	−0.00622052
$973$	−15.0132	−0.481300
$974$	8.22284	0.263477
$975$	−4.82653	−0.154573
$976$	−51.8759	−1.66051
$977$	−34.2579	−1.09601	−0.548004	−	0.836476i	$-0.684612\pi$
$977$	−34.2579	−1.09601	−0.548004	+	0.836476i	$0.684612\pi$
$978$	−36.9741	−1.18230
$979$	−28.7269	−0.918115
$980$	0.231548	0.00739653
$981$	−5.53690	−0.176780
$982$	−21.4979	−0.686025
$983$	26.2981	0.838778	0.419389	−	0.907807i	$-0.362244\pi$
$983$	26.2981	0.838778	0.419389	+	0.907807i	$0.362244\pi$
$984$	−22.9380	−0.731235
$985$	15.9102	0.506941
$986$	−42.2130	−1.34434
$987$	−0.932071	−0.0296682
$988$	−0.261865	−0.00833104
$989$	−53.2214	−1.69234
$990$	5.92478	0.188302
$991$	−32.7757	−1.04116	−0.520578	−	0.853814i	$-0.674283\pi$
$991$	−32.7757	−1.04116	−0.520578	+	0.853814i	$0.674283\pi$
$992$	6.27362	0.199188
$993$	32.7875	1.04048
$994$	−8.73084	−0.276925
$995$	−3.47627	−0.110205
$996$	3.04349	0.0964367
$997$	−52.0263	−1.64769	−0.823845	−	0.566814i	$-0.808176\pi$
$997$	−52.0263	−1.64769	−0.823845	+	0.566814i	$0.808176\pi$
$998$	−50.0263	−1.58356
$999$	−3.92478	−0.124174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.2.a.d.1.1	✓	3
3.2	odd	2		1197.2.a.l.1.3		3
4.3	odd	2		6384.2.a.bx.1.2		3
5.4	even	2		9975.2.a.z.1.3		3
7.6	odd	2		2793.2.a.x.1.1		3
19.18	odd	2		7581.2.a.n.1.3		3
21.20	even	2		8379.2.a.bp.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.d.1.1	✓	3	1.1	even	1	trivial
1197.2.a.l.1.3		3	3.2	odd	2
2793.2.a.x.1.1		3	7.6	odd	2
6384.2.a.bx.1.2		3	4.3	odd	2
7581.2.a.n.1.3		3	19.18	odd	2
8379.2.a.bp.1.3		3	21.20	even	2
9975.2.a.z.1.3		3	5.4	even	2

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:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.a (trivial)

\(f(q)\)	\(=\)	\(q-1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} +1.19394 q^{5} +1.48119 q^{6} -1.00000 q^{7} +2.67513 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} +1.19394 q^{5} +1.48119 q^{6} -1.00000 q^{7} +2.67513 q^{8} +1.00000 q^{9} -1.76845 q^{10} -3.35026 q^{11} -0.193937 q^{12} -1.35026 q^{13} +1.48119 q^{14} -1.19394 q^{15} -4.35026 q^{16} +2.80606 q^{17} -1.48119 q^{18} +1.00000 q^{19} +0.231548 q^{20} +1.00000 q^{21} +4.96239 q^{22} +9.27504 q^{23} -2.67513 q^{24} -3.57452 q^{25} +2.00000 q^{26} -1.00000 q^{27} -0.193937 q^{28} +10.1563 q^{29} +1.76845 q^{30} +5.73813 q^{31} +1.09332 q^{32} +3.35026 q^{33} -4.15633 q^{34} -1.19394 q^{35} +0.193937 q^{36} +3.92478 q^{37} -1.48119 q^{38} +1.35026 q^{39} +3.19394 q^{40} +8.57452 q^{41} -1.48119 q^{42} -5.73813 q^{43} -0.649738 q^{44} +1.19394 q^{45} -13.7381 q^{46} -0.932071 q^{47} +4.35026 q^{48} +1.00000 q^{49} +5.29455 q^{50} -2.80606 q^{51} -0.261865 q^{52} +4.54420 q^{53} +1.48119 q^{54} -4.00000 q^{55} -2.67513 q^{56} -1.00000 q^{57} -15.0435 q^{58} +4.00000 q^{59} -0.231548 q^{60} +11.9248 q^{61} -8.49929 q^{62} -1.00000 q^{63} +7.08110 q^{64} -1.61213 q^{65} -4.96239 q^{66} -8.31265 q^{67} +0.544198 q^{68} -9.27504 q^{69} +1.76845 q^{70} -5.89446 q^{71} +2.67513 q^{72} -3.61213 q^{73} -5.81336 q^{74} +3.57452 q^{75} +0.193937 q^{76} +3.35026 q^{77} -2.00000 q^{78} +8.62530 q^{79} -5.19394 q^{80} +1.00000 q^{81} -12.7005 q^{82} -15.6932 q^{83} +0.193937 q^{84} +3.35026 q^{85} +8.49929 q^{86} -10.1563 q^{87} -8.96239 q^{88} +8.57452 q^{89} -1.76845 q^{90} +1.35026 q^{91} +1.79877 q^{92} -5.73813 q^{93} +1.38058 q^{94} +1.19394 q^{95} -1.09332 q^{96} +11.1490 q^{97} -1.48119 q^{98} -3.35026 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 3 q^{3} + q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 - 3 * q^3 + q^4 + 4 * q^5 - q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} - 3 q^{3} + q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 6 q^{10} - q^{12} + 6 q^{13} - q^{14} - 4 q^{15} - 3 q^{16} + 8 q^{17} + q^{18} + 3 q^{19} + 12 q^{20} + 3 q^{21} + 4 q^{22} - 4 q^{23} - 3 q^{24} + q^{25} + 6 q^{26} - 3 q^{27} - q^{28} + 20 q^{29} - 6 q^{30} + 8 q^{31} - 3 q^{32} - 2 q^{34} - 4 q^{35} + q^{36} - 10 q^{37} + q^{38} - 6 q^{39} + 10 q^{40} + 14 q^{41} + q^{42} - 8 q^{43} - 12 q^{44} + 4 q^{45} - 32 q^{46} + 6 q^{47} + 3 q^{48} + 3 q^{49} + 23 q^{50} - 8 q^{51} - 10 q^{52} + 4 q^{53} - q^{54} - 12 q^{55} - 3 q^{56} - 3 q^{57} - 2 q^{58} + 12 q^{59} - 12 q^{60} + 14 q^{61} + 8 q^{62} - 3 q^{63} - 11 q^{64} - 4 q^{65} - 4 q^{66} - 4 q^{67} - 8 q^{68} + 4 q^{69} - 6 q^{70} + 2 q^{71} + 3 q^{72} - 10 q^{73} - 30 q^{74} - q^{75} + q^{76} - 6 q^{78} - 16 q^{79} - 16 q^{80} + 3 q^{81} - 18 q^{82} - 14 q^{83} + q^{84} - 8 q^{86} - 20 q^{87} - 16 q^{88} + 14 q^{89} + 6 q^{90} - 6 q^{91} - 8 q^{92} - 8 q^{93} - 8 q^{94} + 4 q^{95} + 3 q^{96} + 10 q^{97} + q^{98}+O(q^{100}) \) 3 * q + q^2 - 3 * q^3 + q^4 + 4 * q^5 - q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9 + 6 * q^10 - q^12 + 6 * q^13 - q^14 - 4 * q^15 - 3 * q^16 + 8 * q^17 + q^18 + 3 * q^19 + 12 * q^20 + 3 * q^21 + 4 * q^22 - 4 * q^23 - 3 * q^24 + q^25 + 6 * q^26 - 3 * q^27 - q^28 + 20 * q^29 - 6 * q^30 + 8 * q^31 - 3 * q^32 - 2 * q^34 - 4 * q^35 + q^36 - 10 * q^37 + q^38 - 6 * q^39 + 10 * q^40 + 14 * q^41 + q^42 - 8 * q^43 - 12 * q^44 + 4 * q^45 - 32 * q^46 + 6 * q^47 + 3 * q^48 + 3 * q^49 + 23 * q^50 - 8 * q^51 - 10 * q^52 + 4 * q^53 - q^54 - 12 * q^55 - 3 * q^56 - 3 * q^57 - 2 * q^58 + 12 * q^59 - 12 * q^60 + 14 * q^61 + 8 * q^62 - 3 * q^63 - 11 * q^64 - 4 * q^65 - 4 * q^66 - 4 * q^67 - 8 * q^68 + 4 * q^69 - 6 * q^70 + 2 * q^71 + 3 * q^72 - 10 * q^73 - 30 * q^74 - q^75 + q^76 - 6 * q^78 - 16 * q^79 - 16 * q^80 + 3 * q^81 - 18 * q^82 - 14 * q^83 + q^84 - 8 * q^86 - 20 * q^87 - 16 * q^88 + 14 * q^89 + 6 * q^90 - 6 * q^91 - 8 * q^92 - 8 * q^93 - 8 * q^94 + 4 * q^95 + 3 * q^96 + 10 * q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more