LMFDB - Embedded newform 335.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [335,2,Mod(1,335)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(335, 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("335.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 335 = 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	335.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:	$2.67498846771$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	335.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +1.41421 q^{3} -1.00000 q^{5} -2.00000 q^{6} -2.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} +1.41421 q^{3} -1.00000 q^{5} -2.00000 q^{6} -2.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +1.41421 q^{10} -1.41421 q^{11} -2.00000 q^{13} +2.82843 q^{14} -1.41421 q^{15} -4.00000 q^{16} -4.41421 q^{17} +1.41421 q^{18} -1.00000 q^{19} -2.82843 q^{21} +2.00000 q^{22} -1.58579 q^{23} +4.00000 q^{24} +1.00000 q^{25} +2.82843 q^{26} -5.65685 q^{27} +5.82843 q^{29} +2.00000 q^{30} -2.24264 q^{31} -2.00000 q^{33} +6.24264 q^{34} +2.00000 q^{35} -9.24264 q^{37} +1.41421 q^{38} -2.82843 q^{39} -2.82843 q^{40} +1.41421 q^{41} +4.00000 q^{42} -6.00000 q^{43} +1.00000 q^{45} +2.24264 q^{46} +1.58579 q^{47} -5.65685 q^{48} -3.00000 q^{49} -1.41421 q^{50} -6.24264 q^{51} +4.24264 q^{53} +8.00000 q^{54} +1.41421 q^{55} -5.65685 q^{56} -1.41421 q^{57} -8.24264 q^{58} +3.00000 q^{59} +2.24264 q^{61} +3.17157 q^{62} +2.00000 q^{63} +8.00000 q^{64} +2.00000 q^{65} +2.82843 q^{66} -1.00000 q^{67} -2.24264 q^{69} -2.82843 q^{70} +11.6569 q^{71} -2.82843 q^{72} -11.2426 q^{73} +13.0711 q^{74} +1.41421 q^{75} +2.82843 q^{77} +4.00000 q^{78} +4.00000 q^{79} +4.00000 q^{80} -5.00000 q^{81} -2.00000 q^{82} +11.3137 q^{83} +4.41421 q^{85} +8.48528 q^{86} +8.24264 q^{87} -4.00000 q^{88} +14.6569 q^{89} -1.41421 q^{90} +4.00000 q^{91} -3.17157 q^{93} -2.24264 q^{94} +1.00000 q^{95} +10.7279 q^{97} +4.24264 q^{98} +1.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 4 q^{6} - 4 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 4 * q^6 - 4 * q^7 - 2 * q^9	$ 2 q - 2 q^{5} - 4 q^{6} - 4 q^{7} - 2 q^{9} - 4 q^{13} - 8 q^{16} - 6 q^{17} - 2 q^{19} + 4 q^{22} - 6 q^{23} + 8 q^{24} + 2 q^{25} + 6 q^{29} + 4 q^{30} + 4 q^{31} - 4 q^{33} + 4 q^{34} + 4 q^{35} - 10 q^{37} + 8 q^{42} - 12 q^{43} + 2 q^{45} - 4 q^{46} + 6 q^{47} - 6 q^{49} - 4 q^{51} + 16 q^{54} - 8 q^{58} + 6 q^{59} - 4 q^{61} + 12 q^{62} + 4 q^{63} + 16 q^{64} + 4 q^{65} - 2 q^{67} + 4 q^{69} + 12 q^{71} - 14 q^{73} + 12 q^{74} + 8 q^{78} + 8 q^{79} + 8 q^{80} - 10 q^{81} - 4 q^{82} + 6 q^{85} + 8 q^{87} - 8 q^{88} + 18 q^{89} + 8 q^{91} - 12 q^{93} + 4 q^{94} + 2 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 4 * q^6 - 4 * q^7 - 2 * q^9 - 4 * q^13 - 8 * q^16 - 6 * q^17 - 2 * q^19 + 4 * q^22 - 6 * q^23 + 8 * q^24 + 2 * q^25 + 6 * q^29 + 4 * q^30 + 4 * q^31 - 4 * q^33 + 4 * q^34 + 4 * q^35 - 10 * q^37 + 8 * q^42 - 12 * q^43 + 2 * q^45 - 4 * q^46 + 6 * q^47 - 6 * q^49 - 4 * q^51 + 16 * q^54 - 8 * q^58 + 6 * q^59 - 4 * q^61 + 12 * q^62 + 4 * q^63 + 16 * q^64 + 4 * q^65 - 2 * q^67 + 4 * q^69 + 12 * q^71 - 14 * q^73 + 12 * q^74 + 8 * q^78 + 8 * q^79 + 8 * q^80 - 10 * q^81 - 4 * q^82 + 6 * q^85 + 8 * q^87 - 8 * q^88 + 18 * q^89 + 8 * q^91 - 12 * q^93 + 4 * q^94 + 2 * q^95 - 4 * q^97

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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.00000	−0.816497
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	2.82843	1.00000
$9$	−1.00000	−0.333333
$10$	1.41421	0.447214
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.82843	0.755929
$15$	−1.41421	−0.365148
$16$	−4.00000	−1.00000
$17$	−4.41421	−1.07060	−0.535302	−	0.844661i	$-0.679802\pi$
$17$	−4.41421	−1.07060	−0.535302	+	0.844661i	$0.679802\pi$
$18$	1.41421	0.333333
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	2.00000	0.426401
$23$	−1.58579	−0.330659	−0.165330	−	0.986238i	$-0.552869\pi$
$23$	−1.58579	−0.330659	−0.165330	+	0.986238i	$0.552869\pi$
$24$	4.00000	0.816497
$25$	1.00000	0.200000
$26$	2.82843	0.554700
$27$	−5.65685	−1.08866
$28$	0	0
$29$	5.82843	1.08231	0.541156	−	0.840922i	$-0.317987\pi$
$29$	5.82843	1.08231	0.541156	+	0.840922i	$0.317987\pi$
$30$	2.00000	0.365148
$31$	−2.24264	−0.402790	−0.201395	−	0.979510i	$-0.564548\pi$
$31$	−2.24264	−0.402790	−0.201395	+	0.979510i	$0.564548\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	6.24264	1.07060
$35$	2.00000	0.338062
$36$	0	0
$37$	−9.24264	−1.51948	−0.759740	−	0.650227i	$-0.774674\pi$
$37$	−9.24264	−1.51948	−0.759740	+	0.650227i	$0.774674\pi$
$38$	1.41421	0.229416
$39$	−2.82843	−0.452911
$40$	−2.82843	−0.447214
$41$	1.41421	0.220863	0.110432	−	0.993884i	$-0.464777\pi$
$41$	1.41421	0.220863	0.110432	+	0.993884i	$0.464777\pi$
$42$	4.00000	0.617213
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	2.24264	0.330659
$47$	1.58579	0.231311	0.115655	−	0.993289i	$-0.463103\pi$
$47$	1.58579	0.231311	0.115655	+	0.993289i	$0.463103\pi$
$48$	−5.65685	−0.816497
$49$	−3.00000	−0.428571
$50$	−1.41421	−0.200000
$51$	−6.24264	−0.874145
$52$	0	0
$53$	4.24264	0.582772	0.291386	−	0.956606i	$-0.405884\pi$
$53$	4.24264	0.582772	0.291386	+	0.956606i	$0.405884\pi$
$54$	8.00000	1.08866
$55$	1.41421	0.190693
$56$	−5.65685	−0.755929
$57$	−1.41421	−0.187317
$58$	−8.24264	−1.08231
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	2.24264	0.287141	0.143570	−	0.989640i	$-0.454142\pi$
$61$	2.24264	0.287141	0.143570	+	0.989640i	$0.454142\pi$
$62$	3.17157	0.402790
$63$	2.00000	0.251976
$64$	8.00000	1.00000
$65$	2.00000	0.248069
$66$	2.82843	0.348155
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−2.24264	−0.269982
$70$	−2.82843	−0.338062
$71$	11.6569	1.38341	0.691707	−	0.722178i	$-0.256859\pi$
$71$	11.6569	1.38341	0.691707	+	0.722178i	$0.256859\pi$
$72$	−2.82843	−0.333333
$73$	−11.2426	−1.31585	−0.657926	−	0.753083i	$-0.728566\pi$
$73$	−11.2426	−1.31585	−0.657926	+	0.753083i	$0.728566\pi$
$74$	13.0711	1.51948
$75$	1.41421	0.163299
$76$	0	0
$77$	2.82843	0.322329
$78$	4.00000	0.452911
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	4.00000	0.447214
$81$	−5.00000	−0.555556
$82$	−2.00000	−0.220863
$83$	11.3137	1.24184	0.620920	−	0.783874i	$-0.286759\pi$
$83$	11.3137	1.24184	0.620920	+	0.783874i	$0.286759\pi$
$84$	0	0
$85$	4.41421	0.478789
$86$	8.48528	0.914991
$87$	8.24264	0.883704
$88$	−4.00000	−0.426401
$89$	14.6569	1.55362	0.776812	−	0.629733i	$-0.216836\pi$
$89$	14.6569	1.55362	0.776812	+	0.629733i	$0.216836\pi$
$90$	−1.41421	−0.149071
$91$	4.00000	0.419314
$92$	0	0
$93$	−3.17157	−0.328877
$94$	−2.24264	−0.231311
$95$	1.00000	0.102598
$96$	0	0
$97$	10.7279	1.08926	0.544628	−	0.838678i	$-0.316671\pi$
$97$	10.7279	1.08926	0.544628	+	0.838678i	$0.316671\pi$
$98$	4.24264	0.428571
$99$	1.41421	0.142134
$100$	0	0
$101$	7.41421	0.737742	0.368871	−	0.929481i	$-0.379744\pi$
$101$	7.41421	0.737742	0.368871	+	0.929481i	$0.379744\pi$
$102$	8.82843	0.874145
$103$	−0.485281	−0.0478162	−0.0239081	−	0.999714i	$-0.507611\pi$
$103$	−0.485281	−0.0478162	−0.0239081	+	0.999714i	$0.507611\pi$
$104$	−5.65685	−0.554700
$105$	2.82843	0.276026
$106$	−6.00000	−0.582772
$107$	−13.2426	−1.28021	−0.640107	−	0.768286i	$-0.721110\pi$
$107$	−13.2426	−1.28021	−0.640107	+	0.768286i	$0.721110\pi$
$108$	0	0
$109$	2.48528	0.238047	0.119023	−	0.992891i	$-0.462024\pi$
$109$	2.48528	0.238047	0.119023	+	0.992891i	$0.462024\pi$
$110$	−2.00000	−0.190693
$111$	−13.0711	−1.24065
$112$	8.00000	0.755929
$113$	−9.17157	−0.862789	−0.431394	−	0.902163i	$-0.641978\pi$
$113$	−9.17157	−0.862789	−0.431394	+	0.902163i	$0.641978\pi$
$114$	2.00000	0.187317
$115$	1.58579	0.147875
$116$	0	0
$117$	2.00000	0.184900
$118$	−4.24264	−0.390567
$119$	8.82843	0.809301
$120$	−4.00000	−0.365148
$121$	−9.00000	−0.818182
$122$	−3.17157	−0.287141
$123$	2.00000	0.180334
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−2.82843	−0.251976
$127$	5.72792	0.508271	0.254135	−	0.967169i	$-0.418209\pi$
$127$	5.72792	0.508271	0.254135	+	0.967169i	$0.418209\pi$
$128$	−11.3137	−1.00000
$129$	−8.48528	−0.747087
$130$	−2.82843	−0.248069
$131$	−2.48528	−0.217140	−0.108570	−	0.994089i	$-0.534627\pi$
$131$	−2.48528	−0.217140	−0.108570	+	0.994089i	$0.534627\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	1.41421	0.122169
$135$	5.65685	0.486864
$136$	−12.4853	−1.07060
$137$	−1.75736	−0.150141	−0.0750707	−	0.997178i	$-0.523918\pi$
$137$	−1.75736	−0.150141	−0.0750707	+	0.997178i	$0.523918\pi$
$138$	3.17157	0.269982
$139$	−18.7279	−1.58848	−0.794241	−	0.607603i	$-0.792131\pi$
$139$	−18.7279	−1.58848	−0.794241	+	0.607603i	$0.792131\pi$
$140$	0	0
$141$	2.24264	0.188864
$142$	−16.4853	−1.38341
$143$	2.82843	0.236525
$144$	4.00000	0.333333
$145$	−5.82843	−0.484025
$146$	15.8995	1.31585
$147$	−4.24264	−0.349927
$148$	0	0
$149$	−20.3137	−1.66416	−0.832082	−	0.554653i	$-0.812851\pi$
$149$	−20.3137	−1.66416	−0.832082	+	0.554653i	$0.812851\pi$
$150$	−2.00000	−0.163299
$151$	7.48528	0.609144	0.304572	−	0.952489i	$-0.401487\pi$
$151$	7.48528	0.609144	0.304572	+	0.952489i	$0.401487\pi$
$152$	−2.82843	−0.229416
$153$	4.41421	0.356868
$154$	−4.00000	−0.322329
$155$	2.24264	0.180133
$156$	0	0
$157$	3.24264	0.258791	0.129395	−	0.991593i	$-0.458696\pi$
$157$	3.24264	0.258791	0.129395	+	0.991593i	$0.458696\pi$
$158$	−5.65685	−0.450035
$159$	6.00000	0.475831
$160$	0	0
$161$	3.17157	0.249955
$162$	7.07107	0.555556
$163$	8.75736	0.685929	0.342965	−	0.939348i	$-0.388569\pi$
$163$	8.75736	0.685929	0.342965	+	0.939348i	$0.388569\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	−16.0000	−1.24184
$167$	−11.6569	−0.902034	−0.451017	−	0.892515i	$-0.648939\pi$
$167$	−11.6569	−0.902034	−0.451017	+	0.892515i	$0.648939\pi$
$168$	−8.00000	−0.617213
$169$	−9.00000	−0.692308
$170$	−6.24264	−0.478789
$171$	1.00000	0.0764719
$172$	0	0
$173$	−21.3848	−1.62585	−0.812927	−	0.582365i	$-0.802127\pi$
$173$	−21.3848	−1.62585	−0.812927	+	0.582365i	$0.802127\pi$
$174$	−11.6569	−0.883704
$175$	−2.00000	−0.151186
$176$	5.65685	0.426401
$177$	4.24264	0.318896
$178$	−20.7279	−1.55362
$179$	−5.65685	−0.422813	−0.211407	−	0.977398i	$-0.567804\pi$
$179$	−5.65685	−0.422813	−0.211407	+	0.977398i	$0.567804\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	−5.65685	−0.419314
$183$	3.17157	0.234449
$184$	−4.48528	−0.330659
$185$	9.24264	0.679532
$186$	4.48528	0.328877
$187$	6.24264	0.456507
$188$	0	0
$189$	11.3137	0.822951
$190$	−1.41421	−0.102598
$191$	−9.55635	−0.691473	−0.345737	−	0.938332i	$-0.612371\pi$
$191$	−9.55635	−0.691473	−0.345737	+	0.938332i	$0.612371\pi$
$192$	11.3137	0.816497
$193$	−25.2426	−1.81701	−0.908503	−	0.417879i	$-0.862773\pi$
$193$	−25.2426	−1.81701	−0.908503	+	0.417879i	$0.862773\pi$
$194$	−15.1716	−1.08926
$195$	2.82843	0.202548
$196$	0	0
$197$	13.4142	0.955723	0.477862	−	0.878435i	$-0.341412\pi$
$197$	13.4142	0.955723	0.477862	+	0.878435i	$0.341412\pi$
$198$	−2.00000	−0.142134
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	2.82843	0.200000
$201$	−1.41421	−0.0997509
$202$	−10.4853	−0.737742
$203$	−11.6569	−0.818151
$204$	0	0
$205$	−1.41421	−0.0987730
$206$	0.686292	0.0478162
$207$	1.58579	0.110220
$208$	8.00000	0.554700
$209$	1.41421	0.0978232
$210$	−4.00000	−0.276026
$211$	10.4853	0.721837	0.360918	−	0.932597i	$-0.382463\pi$
$211$	10.4853	0.721837	0.360918	+	0.932597i	$0.382463\pi$
$212$	0	0
$213$	16.4853	1.12955
$214$	18.7279	1.28021
$215$	6.00000	0.409197
$216$	−16.0000	−1.08866
$217$	4.48528	0.304481
$218$	−3.51472	−0.238047
$219$	−15.8995	−1.07439
$220$	0	0
$221$	8.82843	0.593864
$222$	18.4853	1.24065
$223$	24.2132	1.62144	0.810718	−	0.585437i	$-0.199077\pi$
$223$	24.2132	1.62144	0.810718	+	0.585437i	$0.199077\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	12.9706	0.862789
$227$	−5.10051	−0.338532	−0.169266	−	0.985570i	$-0.554140\pi$
$227$	−5.10051	−0.338532	−0.169266	+	0.985570i	$0.554140\pi$
$228$	0	0
$229$	−17.2132	−1.13748	−0.568740	−	0.822517i	$-0.692569\pi$
$229$	−17.2132	−1.13748	−0.568740	+	0.822517i	$0.692569\pi$
$230$	−2.24264	−0.147875
$231$	4.00000	0.263181
$232$	16.4853	1.08231
$233$	8.82843	0.578369	0.289185	−	0.957273i	$-0.406616\pi$
$233$	8.82843	0.578369	0.289185	+	0.957273i	$0.406616\pi$
$234$	−2.82843	−0.184900
$235$	−1.58579	−0.103445
$236$	0	0
$237$	5.65685	0.367452
$238$	−12.4853	−0.809301
$239$	−2.82843	−0.182956	−0.0914779	−	0.995807i	$-0.529159\pi$
$239$	−2.82843	−0.182956	−0.0914779	+	0.995807i	$0.529159\pi$
$240$	5.65685	0.365148
$241$	−21.0000	−1.35273	−0.676364	−	0.736567i	$-0.736446\pi$
$241$	−21.0000	−1.35273	−0.676364	+	0.736567i	$0.736446\pi$
$242$	12.7279	0.818182
$243$	9.89949	0.635053
$244$	0	0
$245$	3.00000	0.191663
$246$	−2.82843	−0.180334
$247$	2.00000	0.127257
$248$	−6.34315	−0.402790
$249$	16.0000	1.01396
$250$	1.41421	0.0894427
$251$	8.10051	0.511299	0.255650	−	0.966769i	$-0.417711\pi$
$251$	8.10051	0.511299	0.255650	+	0.966769i	$0.417711\pi$
$252$	0	0
$253$	2.24264	0.140994
$254$	−8.10051	−0.508271
$255$	6.24264	0.390929
$256$	0	0
$257$	−21.3848	−1.33395	−0.666973	−	0.745082i	$-0.732410\pi$
$257$	−21.3848	−1.33395	−0.666973	+	0.745082i	$0.732410\pi$
$258$	12.0000	0.747087
$259$	18.4853	1.14862
$260$	0	0
$261$	−5.82843	−0.360771
$262$	3.51472	0.217140
$263$	20.4853	1.26318	0.631588	−	0.775304i	$-0.282403\pi$
$263$	20.4853	1.26318	0.631588	+	0.775304i	$0.282403\pi$
$264$	−5.65685	−0.348155
$265$	−4.24264	−0.260623
$266$	−2.82843	−0.173422
$267$	20.7279	1.26853
$268$	0	0
$269$	−5.65685	−0.344904	−0.172452	−	0.985018i	$-0.555169\pi$
$269$	−5.65685	−0.344904	−0.172452	+	0.985018i	$0.555169\pi$
$270$	−8.00000	−0.486864
$271$	16.9706	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$272$	17.6569	1.07060
$273$	5.65685	0.342368
$274$	2.48528	0.150141
$275$	−1.41421	−0.0852803
$276$	0	0
$277$	−2.48528	−0.149326	−0.0746630	−	0.997209i	$-0.523788\pi$
$277$	−2.48528	−0.149326	−0.0746630	+	0.997209i	$0.523788\pi$
$278$	26.4853	1.58848
$279$	2.24264	0.134263
$280$	5.65685	0.338062
$281$	−21.5563	−1.28594	−0.642972	−	0.765890i	$-0.722299\pi$
$281$	−21.5563	−1.28594	−0.642972	+	0.765890i	$0.722299\pi$
$282$	−3.17157	−0.188864
$283$	19.2426	1.14386	0.571928	−	0.820304i	$-0.306196\pi$
$283$	19.2426	1.14386	0.571928	+	0.820304i	$0.306196\pi$
$284$	0	0
$285$	1.41421	0.0837708
$286$	−4.00000	−0.236525
$287$	−2.82843	−0.166957
$288$	0	0
$289$	2.48528	0.146193
$290$	8.24264	0.484025
$291$	15.1716	0.889373
$292$	0	0
$293$	−2.48528	−0.145192	−0.0725958	−	0.997361i	$-0.523128\pi$
$293$	−2.48528	−0.145192	−0.0725958	+	0.997361i	$0.523128\pi$
$294$	6.00000	0.349927
$295$	−3.00000	−0.174667
$296$	−26.1421	−1.51948
$297$	8.00000	0.464207
$298$	28.7279	1.66416
$299$	3.17157	0.183417
$300$	0	0
$301$	12.0000	0.691669
$302$	−10.5858	−0.609144
$303$	10.4853	0.602364
$304$	4.00000	0.229416
$305$	−2.24264	−0.128413
$306$	−6.24264	−0.356868
$307$	−26.2132	−1.49607	−0.748033	−	0.663661i	$-0.769002\pi$
$307$	−26.2132	−1.49607	−0.748033	+	0.663661i	$0.769002\pi$
$308$	0	0
$309$	−0.686292	−0.0390418
$310$	−3.17157	−0.180133
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	−8.00000	−0.452911
$313$	−18.7279	−1.05856	−0.529282	−	0.848446i	$-0.677539\pi$
$313$	−18.7279	−1.05856	−0.529282	+	0.848446i	$0.677539\pi$
$314$	−4.58579	−0.258791
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−31.7990	−1.78601	−0.893005	−	0.450048i	$-0.851407\pi$
$317$	−31.7990	−1.78601	−0.893005	+	0.450048i	$0.851407\pi$
$318$	−8.48528	−0.475831
$319$	−8.24264	−0.461499
$320$	−8.00000	−0.447214
$321$	−18.7279	−1.04529
$322$	−4.48528	−0.249955
$323$	4.41421	0.245613
$324$	0	0
$325$	−2.00000	−0.110940
$326$	−12.3848	−0.685929
$327$	3.51472	0.194364
$328$	4.00000	0.220863
$329$	−3.17157	−0.174854
$330$	−2.82843	−0.155700
$331$	−6.97056	−0.383137	−0.191568	−	0.981479i	$-0.561357\pi$
$331$	−6.97056	−0.383137	−0.191568	+	0.981479i	$0.561357\pi$
$332$	0	0
$333$	9.24264	0.506494
$334$	16.4853	0.902034
$335$	1.00000	0.0546358
$336$	11.3137	0.617213
$337$	−30.9706	−1.68707	−0.843537	−	0.537071i	$-0.819531\pi$
$337$	−30.9706	−1.68707	−0.843537	+	0.537071i	$0.819531\pi$
$338$	12.7279	0.692308
$339$	−12.9706	−0.704464
$340$	0	0
$341$	3.17157	0.171750
$342$	−1.41421	−0.0764719
$343$	20.0000	1.07990
$344$	−16.9706	−0.914991
$345$	2.24264	0.120740
$346$	30.2426	1.62585
$347$	2.10051	0.112761	0.0563805	−	0.998409i	$-0.482044\pi$
$347$	2.10051	0.112761	0.0563805	+	0.998409i	$0.482044\pi$
$348$	0	0
$349$	16.9706	0.908413	0.454207	−	0.890896i	$-0.349923\pi$
$349$	16.9706	0.908413	0.454207	+	0.890896i	$0.349923\pi$
$350$	2.82843	0.151186
$351$	11.3137	0.603881
$352$	0	0
$353$	19.4142	1.03331	0.516657	−	0.856192i	$-0.327176\pi$
$353$	19.4142	1.03331	0.516657	+	0.856192i	$0.327176\pi$
$354$	−6.00000	−0.318896
$355$	−11.6569	−0.618682
$356$	0	0
$357$	12.4853	0.660791
$358$	8.00000	0.422813
$359$	−6.51472	−0.343834	−0.171917	−	0.985111i	$-0.554996\pi$
$359$	−6.51472	−0.343834	−0.171917	+	0.985111i	$0.554996\pi$
$360$	2.82843	0.149071
$361$	−18.0000	−0.947368
$362$	−9.89949	−0.520306
$363$	−12.7279	−0.668043
$364$	0	0
$365$	11.2426	0.588467
$366$	−4.48528	−0.234449
$367$	10.7279	0.559993	0.279996	−	0.960001i	$-0.409667\pi$
$367$	10.7279	0.559993	0.279996	+	0.960001i	$0.409667\pi$
$368$	6.34315	0.330659
$369$	−1.41421	−0.0736210
$370$	−13.0711	−0.679532
$371$	−8.48528	−0.440534
$372$	0	0
$373$	−24.4853	−1.26780	−0.633900	−	0.773415i	$-0.718547\pi$
$373$	−24.4853	−1.26780	−0.633900	+	0.773415i	$0.718547\pi$
$374$	−8.82843	−0.456507
$375$	−1.41421	−0.0730297
$376$	4.48528	0.231311
$377$	−11.6569	−0.600359
$378$	−16.0000	−0.822951
$379$	10.7279	0.551056	0.275528	−	0.961293i	$-0.411147\pi$
$379$	10.7279	0.551056	0.275528	+	0.961293i	$0.411147\pi$
$380$	0	0
$381$	8.10051	0.415001
$382$	13.5147	0.691473
$383$	18.3431	0.937291	0.468645	−	0.883386i	$-0.344742\pi$
$383$	18.3431	0.937291	0.468645	+	0.883386i	$0.344742\pi$
$384$	−16.0000	−0.816497
$385$	−2.82843	−0.144150
$386$	35.6985	1.81701
$387$	6.00000	0.304997
$388$	0	0
$389$	−19.7990	−1.00385	−0.501924	−	0.864912i	$-0.667374\pi$
$389$	−19.7990	−1.00385	−0.501924	+	0.864912i	$0.667374\pi$
$390$	−4.00000	−0.202548
$391$	7.00000	0.354005
$392$	−8.48528	−0.428571
$393$	−3.51472	−0.177294
$394$	−18.9706	−0.955723
$395$	−4.00000	−0.201262
$396$	0	0
$397$	15.7279	0.789362	0.394681	−	0.918818i	$-0.370855\pi$
$397$	15.7279	0.789362	0.394681	+	0.918818i	$0.370855\pi$
$398$	15.5563	0.779769
$399$	2.82843	0.141598
$400$	−4.00000	−0.200000
$401$	21.2132	1.05934	0.529668	−	0.848205i	$-0.322316\pi$
$401$	21.2132	1.05934	0.529668	+	0.848205i	$0.322316\pi$
$402$	2.00000	0.0997509
$403$	4.48528	0.223428
$404$	0	0
$405$	5.00000	0.248452
$406$	16.4853	0.818151
$407$	13.0711	0.647909
$408$	−17.6569	−0.874145
$409$	−11.7574	−0.581364	−0.290682	−	0.956820i	$-0.593882\pi$
$409$	−11.7574	−0.581364	−0.290682	+	0.956820i	$0.593882\pi$
$410$	2.00000	0.0987730
$411$	−2.48528	−0.122590
$412$	0	0
$413$	−6.00000	−0.295241
$414$	−2.24264	−0.110220
$415$	−11.3137	−0.555368
$416$	0	0
$417$	−26.4853	−1.29699
$418$	−2.00000	−0.0978232
$419$	4.79899	0.234446	0.117223	−	0.993106i	$-0.462601\pi$
$419$	4.79899	0.234446	0.117223	+	0.993106i	$0.462601\pi$
$420$	0	0
$421$	28.9411	1.41050	0.705252	−	0.708957i	$-0.250834\pi$
$421$	28.9411	1.41050	0.705252	+	0.708957i	$0.250834\pi$
$422$	−14.8284	−0.721837
$423$	−1.58579	−0.0771036
$424$	12.0000	0.582772
$425$	−4.41421	−0.214121
$426$	−23.3137	−1.12955
$427$	−4.48528	−0.217058
$428$	0	0
$429$	4.00000	0.193122
$430$	−8.48528	−0.409197
$431$	−28.7990	−1.38720	−0.693599	−	0.720361i	$-0.743976\pi$
$431$	−28.7990	−1.38720	−0.693599	+	0.720361i	$0.743976\pi$
$432$	22.6274	1.08866
$433$	−25.6985	−1.23499	−0.617495	−	0.786575i	$-0.711852\pi$
$433$	−25.6985	−1.23499	−0.617495	+	0.786575i	$0.711852\pi$
$434$	−6.34315	−0.304481
$435$	−8.24264	−0.395204
$436$	0	0
$437$	1.58579	0.0758585
$438$	22.4853	1.07439
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	4.00000	0.190693
$441$	3.00000	0.142857
$442$	−12.4853	−0.593864
$443$	28.6274	1.36013	0.680065	−	0.733152i	$-0.261952\pi$
$443$	28.6274	1.36013	0.680065	+	0.733152i	$0.261952\pi$
$444$	0	0
$445$	−14.6569	−0.694802
$446$	−34.2426	−1.62144
$447$	−28.7279	−1.35878
$448$	−16.0000	−0.755929
$449$	28.7990	1.35911	0.679554	−	0.733625i	$-0.262173\pi$
$449$	28.7990	1.35911	0.679554	+	0.733625i	$0.262173\pi$
$450$	1.41421	0.0666667
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	10.5858	0.497364
$454$	7.21320	0.338532
$455$	−4.00000	−0.187523
$456$	−4.00000	−0.187317
$457$	22.2132	1.03909	0.519545	−	0.854443i	$-0.326101\pi$
$457$	22.2132	1.03909	0.519545	+	0.854443i	$0.326101\pi$
$458$	24.3431	1.13748
$459$	24.9706	1.16553
$460$	0	0
$461$	31.9706	1.48902	0.744509	−	0.667613i	$-0.232684\pi$
$461$	31.9706	1.48902	0.744509	+	0.667613i	$0.232684\pi$
$462$	−5.65685	−0.263181
$463$	21.7574	1.01115	0.505575	−	0.862783i	$-0.331280\pi$
$463$	21.7574	1.01115	0.505575	+	0.862783i	$0.331280\pi$
$464$	−23.3137	−1.08231
$465$	3.17157	0.147078
$466$	−12.4853	−0.578369
$467$	−40.2843	−1.86413	−0.932067	−	0.362286i	$-0.881996\pi$
$467$	−40.2843	−1.86413	−0.932067	+	0.362286i	$0.881996\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	2.24264	0.103445
$471$	4.58579	0.211302
$472$	8.48528	0.390567
$473$	8.48528	0.390154
$474$	−8.00000	−0.367452
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−4.24264	−0.194257
$478$	4.00000	0.182956
$479$	−9.68629	−0.442578	−0.221289	−	0.975208i	$-0.571026\pi$
$479$	−9.68629	−0.442578	−0.221289	+	0.975208i	$0.571026\pi$
$480$	0	0
$481$	18.4853	0.842856
$482$	29.6985	1.35273
$483$	4.48528	0.204087
$484$	0	0
$485$	−10.7279	−0.487130
$486$	−14.0000	−0.635053
$487$	16.7279	0.758015	0.379007	−	0.925394i	$-0.376266\pi$
$487$	16.7279	0.758015	0.379007	+	0.925394i	$0.376266\pi$
$488$	6.34315	0.287141
$489$	12.3848	0.560059
$490$	−4.24264	−0.191663
$491$	30.9411	1.39635	0.698177	−	0.715925i	$-0.253995\pi$
$491$	30.9411	1.39635	0.698177	+	0.715925i	$0.253995\pi$
$492$	0	0
$493$	−25.7279	−1.15873
$494$	−2.82843	−0.127257
$495$	−1.41421	−0.0635642
$496$	8.97056	0.402790
$497$	−23.3137	−1.04576
$498$	−22.6274	−1.01396
$499$	29.9411	1.34035	0.670174	−	0.742204i	$-0.266219\pi$
$499$	29.9411	1.34035	0.670174	+	0.742204i	$0.266219\pi$
$500$	0	0
$501$	−16.4853	−0.736508
$502$	−11.4558	−0.511299
$503$	−37.4142	−1.66822	−0.834109	−	0.551600i	$-0.814017\pi$
$503$	−37.4142	−1.66822	−0.834109	+	0.551600i	$0.814017\pi$
$504$	5.65685	0.251976
$505$	−7.41421	−0.329928
$506$	−3.17157	−0.140994
$507$	−12.7279	−0.565267
$508$	0	0
$509$	−11.4853	−0.509076	−0.254538	−	0.967063i	$-0.581923\pi$
$509$	−11.4853	−0.509076	−0.254538	+	0.967063i	$0.581923\pi$
$510$	−8.82843	−0.390929
$511$	22.4853	0.994690
$512$	22.6274	1.00000
$513$	5.65685	0.249756
$514$	30.2426	1.33395
$515$	0.485281	0.0213841
$516$	0	0
$517$	−2.24264	−0.0986312
$518$	−26.1421	−1.14862
$519$	−30.2426	−1.32750
$520$	5.65685	0.248069
$521$	36.7279	1.60908	0.804540	−	0.593899i	$-0.202412\pi$
$521$	36.7279	1.60908	0.804540	+	0.593899i	$0.202412\pi$
$522$	8.24264	0.360771
$523$	−15.2426	−0.666514	−0.333257	−	0.942836i	$-0.608148\pi$
$523$	−15.2426	−0.666514	−0.333257	+	0.942836i	$0.608148\pi$
$524$	0	0
$525$	−2.82843	−0.123443
$526$	−28.9706	−1.26318
$527$	9.89949	0.431229
$528$	8.00000	0.348155
$529$	−20.4853	−0.890664
$530$	6.00000	0.260623
$531$	−3.00000	−0.130189
$532$	0	0
$533$	−2.82843	−0.122513
$534$	−29.3137	−1.26853
$535$	13.2426	0.572529
$536$	−2.82843	−0.122169
$537$	−8.00000	−0.345225
$538$	8.00000	0.344904
$539$	4.24264	0.182743
$540$	0	0
$541$	14.4853	0.622771	0.311385	−	0.950284i	$-0.399207\pi$
$541$	14.4853	0.622771	0.311385	+	0.950284i	$0.399207\pi$
$542$	−24.0000	−1.03089
$543$	9.89949	0.424828
$544$	0	0
$545$	−2.48528	−0.106458
$546$	−8.00000	−0.342368
$547$	27.7574	1.18682	0.593409	−	0.804901i	$-0.297781\pi$
$547$	27.7574	1.18682	0.593409	+	0.804901i	$0.297781\pi$
$548$	0	0
$549$	−2.24264	−0.0957136
$550$	2.00000	0.0852803
$551$	−5.82843	−0.248299
$552$	−6.34315	−0.269982
$553$	−8.00000	−0.340195
$554$	3.51472	0.149326
$555$	13.0711	0.554836
$556$	0	0
$557$	−0.343146	−0.0145396	−0.00726978	−	0.999974i	$-0.502314\pi$
$557$	−0.343146	−0.0145396	−0.00726978	+	0.999974i	$0.502314\pi$
$558$	−3.17157	−0.134263
$559$	12.0000	0.507546
$560$	−8.00000	−0.338062
$561$	8.82843	0.372736
$562$	30.4853	1.28594
$563$	−22.2426	−0.937416	−0.468708	−	0.883353i	$-0.655280\pi$
$563$	−22.2426	−0.937416	−0.468708	+	0.883353i	$0.655280\pi$
$564$	0	0
$565$	9.17157	0.385851
$566$	−27.2132	−1.14386
$567$	10.0000	0.419961
$568$	32.9706	1.38341
$569$	−19.2843	−0.808439	−0.404219	−	0.914662i	$-0.632457\pi$
$569$	−19.2843	−0.808439	−0.404219	+	0.914662i	$0.632457\pi$
$570$	−2.00000	−0.0837708
$571$	−5.00000	−0.209243	−0.104622	−	0.994512i	$-0.533363\pi$
$571$	−5.00000	−0.209243	−0.104622	+	0.994512i	$0.533363\pi$
$572$	0	0
$573$	−13.5147	−0.564585
$574$	4.00000	0.166957
$575$	−1.58579	−0.0661319
$576$	−8.00000	−0.333333
$577$	22.7279	0.946176	0.473088	−	0.881015i	$-0.343139\pi$
$577$	22.7279	0.946176	0.473088	+	0.881015i	$0.343139\pi$
$578$	−3.51472	−0.146193
$579$	−35.6985	−1.48358
$580$	0	0
$581$	−22.6274	−0.938743
$582$	−21.4558	−0.889373
$583$	−6.00000	−0.248495
$584$	−31.7990	−1.31585
$585$	−2.00000	−0.0826898
$586$	3.51472	0.145192
$587$	−23.3137	−0.962260	−0.481130	−	0.876649i	$-0.659773\pi$
$587$	−23.3137	−0.962260	−0.481130	+	0.876649i	$0.659773\pi$
$588$	0	0
$589$	2.24264	0.0924064
$590$	4.24264	0.174667
$591$	18.9706	0.780345
$592$	36.9706	1.51948
$593$	−25.1127	−1.03125	−0.515627	−	0.856813i	$-0.672441\pi$
$593$	−25.1127	−1.03125	−0.515627	+	0.856813i	$0.672441\pi$
$594$	−11.3137	−0.464207
$595$	−8.82843	−0.361930
$596$	0	0
$597$	−15.5563	−0.636679
$598$	−4.48528	−0.183417
$599$	26.1421	1.06814	0.534069	−	0.845441i	$-0.320662\pi$
$599$	26.1421	1.06814	0.534069	+	0.845441i	$0.320662\pi$
$600$	4.00000	0.163299
$601$	−26.5147	−1.08156	−0.540779	−	0.841165i	$-0.681870\pi$
$601$	−26.5147	−1.08156	−0.540779	+	0.841165i	$0.681870\pi$
$602$	−16.9706	−0.691669
$603$	1.00000	0.0407231
$604$	0	0
$605$	9.00000	0.365902
$606$	−14.8284	−0.602364
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	−16.4853	−0.668017
$610$	3.17157	0.128413
$611$	−3.17157	−0.128308
$612$	0	0
$613$	−10.2132	−0.412507	−0.206254	−	0.978499i	$-0.566127\pi$
$613$	−10.2132	−0.412507	−0.206254	+	0.978499i	$0.566127\pi$
$614$	37.0711	1.49607
$615$	−2.00000	−0.0806478
$616$	8.00000	0.322329
$617$	−28.7574	−1.15773	−0.578864	−	0.815424i	$-0.696504\pi$
$617$	−28.7574	−1.15773	−0.578864	+	0.815424i	$0.696504\pi$
$618$	0.970563	0.0390418
$619$	18.4853	0.742986	0.371493	−	0.928436i	$-0.378846\pi$
$619$	18.4853	0.742986	0.371493	+	0.928436i	$0.378846\pi$
$620$	0	0
$621$	8.97056	0.359976
$622$	−25.4558	−1.02069
$623$	−29.3137	−1.17443
$624$	11.3137	0.452911
$625$	1.00000	0.0400000
$626$	26.4853	1.05856
$627$	2.00000	0.0798723
$628$	0	0
$629$	40.7990	1.62676
$630$	2.82843	0.112687
$631$	−35.4558	−1.41147	−0.705737	−	0.708473i	$-0.749384\pi$
$631$	−35.4558	−1.41147	−0.705737	+	0.708473i	$0.749384\pi$
$632$	11.3137	0.450035
$633$	14.8284	0.589377
$634$	44.9706	1.78601
$635$	−5.72792	−0.227306
$636$	0	0
$637$	6.00000	0.237729
$638$	11.6569	0.461499
$639$	−11.6569	−0.461138
$640$	11.3137	0.447214
$641$	−14.1005	−0.556936	−0.278468	−	0.960445i	$-0.589827\pi$
$641$	−14.1005	−0.556936	−0.278468	+	0.960445i	$0.589827\pi$
$642$	26.4853	1.04529
$643$	−22.9706	−0.905871	−0.452935	−	0.891543i	$-0.649623\pi$
$643$	−22.9706	−0.905871	−0.452935	+	0.891543i	$0.649623\pi$
$644$	0	0
$645$	8.48528	0.334108
$646$	−6.24264	−0.245613
$647$	15.5563	0.611583	0.305792	−	0.952098i	$-0.401079\pi$
$647$	15.5563	0.611583	0.305792	+	0.952098i	$0.401079\pi$
$648$	−14.1421	−0.555556
$649$	−4.24264	−0.166538
$650$	2.82843	0.110940
$651$	6.34315	0.248607
$652$	0	0
$653$	−24.3431	−0.952621	−0.476310	−	0.879277i	$-0.658026\pi$
$653$	−24.3431	−0.952621	−0.476310	+	0.879277i	$0.658026\pi$
$654$	−4.97056	−0.194364
$655$	2.48528	0.0971080
$656$	−5.65685	−0.220863
$657$	11.2426	0.438617
$658$	4.48528	0.174854
$659$	−15.3431	−0.597684	−0.298842	−	0.954303i	$-0.596600\pi$
$659$	−15.3431	−0.597684	−0.298842	+	0.954303i	$0.596600\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	9.85786	0.383137
$663$	12.4853	0.484888
$664$	32.0000	1.24184
$665$	−2.00000	−0.0775567
$666$	−13.0711	−0.506494
$667$	−9.24264	−0.357876
$668$	0	0
$669$	34.2426	1.32390
$670$	−1.41421	−0.0546358
$671$	−3.17157	−0.122437
$672$	0	0
$673$	−15.2721	−0.588695	−0.294348	−	0.955698i	$-0.595102\pi$
$673$	−15.2721	−0.588695	−0.294348	+	0.955698i	$0.595102\pi$
$674$	43.7990	1.68707
$675$	−5.65685	−0.217732
$676$	0	0
$677$	10.6274	0.408445	0.204222	−	0.978925i	$-0.434533\pi$
$677$	10.6274	0.408445	0.204222	+	0.978925i	$0.434533\pi$
$678$	18.3431	0.704464
$679$	−21.4558	−0.823400
$680$	12.4853	0.478789
$681$	−7.21320	−0.276411
$682$	−4.48528	−0.171750
$683$	49.7990	1.90551	0.952753	−	0.303747i	$-0.0982378\pi$
$683$	49.7990	1.90551	0.952753	+	0.303747i	$0.0982378\pi$
$684$	0	0
$685$	1.75736	0.0671452
$686$	−28.2843	−1.07990
$687$	−24.3431	−0.928749
$688$	24.0000	0.914991
$689$	−8.48528	−0.323263
$690$	−3.17157	−0.120740
$691$	33.9706	1.29230	0.646151	−	0.763210i	$-0.276378\pi$
$691$	33.9706	1.29230	0.646151	+	0.763210i	$0.276378\pi$
$692$	0	0
$693$	−2.82843	−0.107443
$694$	−2.97056	−0.112761
$695$	18.7279	0.710391
$696$	23.3137	0.883704
$697$	−6.24264	−0.236457
$698$	−24.0000	−0.908413
$699$	12.4853	0.472237
$700$	0	0
$701$	2.10051	0.0793350	0.0396675	−	0.999213i	$-0.487370\pi$
$701$	2.10051	0.0793350	0.0396675	+	0.999213i	$0.487370\pi$
$702$	−16.0000	−0.603881
$703$	9.24264	0.348593
$704$	−11.3137	−0.426401
$705$	−2.24264	−0.0844627
$706$	−27.4558	−1.03331
$707$	−14.8284	−0.557680
$708$	0	0
$709$	28.4853	1.06979	0.534894	−	0.844919i	$-0.320352\pi$
$709$	28.4853	1.06979	0.534894	+	0.844919i	$0.320352\pi$
$710$	16.4853	0.618682
$711$	−4.00000	−0.150012
$712$	41.4558	1.55362
$713$	3.55635	0.133186
$714$	−17.6569	−0.660791
$715$	−2.82843	−0.105777
$716$	0	0
$717$	−4.00000	−0.149383
$718$	9.21320	0.343834
$719$	−26.6569	−0.994133	−0.497066	−	0.867712i	$-0.665589\pi$
$719$	−26.6569	−0.994133	−0.497066	+	0.867712i	$0.665589\pi$
$720$	−4.00000	−0.149071
$721$	0.970563	0.0361456
$722$	25.4558	0.947368
$723$	−29.6985	−1.10450
$724$	0	0
$725$	5.82843	0.216462
$726$	18.0000	0.668043
$727$	23.4558	0.869929	0.434965	−	0.900448i	$-0.356761\pi$
$727$	23.4558	0.869929	0.434965	+	0.900448i	$0.356761\pi$
$728$	11.3137	0.419314
$729$	29.0000	1.07407
$730$	−15.8995	−0.588467
$731$	26.4853	0.979594
$732$	0	0
$733$	9.27208	0.342472	0.171236	−	0.985230i	$-0.445224\pi$
$733$	9.27208	0.342472	0.171236	+	0.985230i	$0.445224\pi$
$734$	−15.1716	−0.559993
$735$	4.24264	0.156492
$736$	0	0
$737$	1.41421	0.0520932
$738$	2.00000	0.0736210
$739$	−45.4558	−1.67212	−0.836060	−	0.548638i	$-0.815147\pi$
$739$	−45.4558	−1.67212	−0.836060	+	0.548638i	$0.815147\pi$
$740$	0	0
$741$	2.82843	0.103905
$742$	12.0000	0.440534
$743$	−28.2843	−1.03765	−0.518825	−	0.854881i	$-0.673630\pi$
$743$	−28.2843	−1.03765	−0.518825	+	0.854881i	$0.673630\pi$
$744$	−8.97056	−0.328877
$745$	20.3137	0.744237
$746$	34.6274	1.26780
$747$	−11.3137	−0.413947
$748$	0	0
$749$	26.4853	0.967751
$750$	2.00000	0.0730297
$751$	−49.9706	−1.82345	−0.911726	−	0.410799i	$-0.865250\pi$
$751$	−49.9706	−1.82345	−0.911726	+	0.410799i	$0.865250\pi$
$752$	−6.34315	−0.231311
$753$	11.4558	0.417474
$754$	16.4853	0.600359
$755$	−7.48528	−0.272417
$756$	0	0
$757$	20.9706	0.762188	0.381094	−	0.924536i	$-0.375547\pi$
$757$	20.9706	0.762188	0.381094	+	0.924536i	$0.375547\pi$
$758$	−15.1716	−0.551056
$759$	3.17157	0.115121
$760$	2.82843	0.102598
$761$	−11.4853	−0.416341	−0.208171	−	0.978093i	$-0.566751\pi$
$761$	−11.4853	−0.416341	−0.208171	+	0.978093i	$0.566751\pi$
$762$	−11.4558	−0.415001
$763$	−4.97056	−0.179946
$764$	0	0
$765$	−4.41421	−0.159596
$766$	−25.9411	−0.937291
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−24.1838	−0.872089	−0.436044	−	0.899925i	$-0.643621\pi$
$769$	−24.1838	−0.872089	−0.436044	+	0.899925i	$0.643621\pi$
$770$	4.00000	0.144150
$771$	−30.2426	−1.08916
$772$	0	0
$773$	6.89949	0.248158	0.124079	−	0.992272i	$-0.460402\pi$
$773$	6.89949	0.248158	0.124079	+	0.992272i	$0.460402\pi$
$774$	−8.48528	−0.304997
$775$	−2.24264	−0.0805580
$776$	30.3431	1.08926
$777$	26.1421	0.937844
$778$	28.0000	1.00385
$779$	−1.41421	−0.0506695
$780$	0	0
$781$	−16.4853	−0.589890
$782$	−9.89949	−0.354005
$783$	−32.9706	−1.17827
$784$	12.0000	0.428571
$785$	−3.24264	−0.115735
$786$	4.97056	0.177294
$787$	43.6985	1.55768	0.778841	−	0.627221i	$-0.215808\pi$
$787$	43.6985	1.55768	0.778841	+	0.627221i	$0.215808\pi$
$788$	0	0
$789$	28.9706	1.03138
$790$	5.65685	0.201262
$791$	18.3431	0.652207
$792$	4.00000	0.142134
$793$	−4.48528	−0.159277
$794$	−22.2426	−0.789362
$795$	−6.00000	−0.212798
$796$	0	0
$797$	19.1127	0.677007	0.338503	−	0.940965i	$-0.390079\pi$
$797$	19.1127	0.677007	0.338503	+	0.940965i	$0.390079\pi$
$798$	−4.00000	−0.141598
$799$	−7.00000	−0.247642
$800$	0	0
$801$	−14.6569	−0.517874
$802$	−30.0000	−1.05934
$803$	15.8995	0.561081
$804$	0	0
$805$	−3.17157	−0.111783
$806$	−6.34315	−0.223428
$807$	−8.00000	−0.281613
$808$	20.9706	0.737742
$809$	37.7574	1.32748	0.663739	−	0.747964i	$-0.268969\pi$
$809$	37.7574	1.32748	0.663739	+	0.747964i	$0.268969\pi$
$810$	−7.07107	−0.248452
$811$	−10.0000	−0.351147	−0.175574	−	0.984466i	$-0.556178\pi$
$811$	−10.0000	−0.351147	−0.175574	+	0.984466i	$0.556178\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	−18.4853	−0.647909
$815$	−8.75736	−0.306757
$816$	24.9706	0.874145
$817$	6.00000	0.209913
$818$	16.6274	0.581364
$819$	−4.00000	−0.139771
$820$	0	0
$821$	31.2843	1.09183	0.545914	−	0.837841i	$-0.316182\pi$
$821$	31.2843	1.09183	0.545914	+	0.837841i	$0.316182\pi$
$822$	3.51472	0.122590
$823$	32.7574	1.14185	0.570925	−	0.821002i	$-0.306585\pi$
$823$	32.7574	1.14185	0.570925	+	0.821002i	$0.306585\pi$
$824$	−1.37258	−0.0478162
$825$	−2.00000	−0.0696311
$826$	8.48528	0.295241
$827$	16.4142	0.570778	0.285389	−	0.958412i	$-0.407877\pi$
$827$	16.4142	0.570778	0.285389	+	0.958412i	$0.407877\pi$
$828$	0	0
$829$	−18.4558	−0.640998	−0.320499	−	0.947249i	$-0.603851\pi$
$829$	−18.4558	−0.640998	−0.320499	+	0.947249i	$0.603851\pi$
$830$	16.0000	0.555368
$831$	−3.51472	−0.121924
$832$	−16.0000	−0.554700
$833$	13.2426	0.458830
$834$	37.4558	1.29699
$835$	11.6569	0.403402
$836$	0	0
$837$	12.6863	0.438502
$838$	−6.78680	−0.234446
$839$	−43.6274	−1.50619	−0.753093	−	0.657914i	$-0.771439\pi$
$839$	−43.6274	−1.50619	−0.753093	+	0.657914i	$0.771439\pi$
$840$	8.00000	0.276026
$841$	4.97056	0.171399
$842$	−40.9289	−1.41050
$843$	−30.4853	−1.04997
$844$	0	0
$845$	9.00000	0.309609
$846$	2.24264	0.0771036
$847$	18.0000	0.618487
$848$	−16.9706	−0.582772
$849$	27.2132	0.933955
$850$	6.24264	0.214121
$851$	14.6569	0.502430
$852$	0	0
$853$	−48.6985	−1.66740	−0.833702	−	0.552214i	$-0.813783\pi$
$853$	−48.6985	−1.66740	−0.833702	+	0.552214i	$0.813783\pi$
$854$	6.34315	0.217058
$855$	−1.00000	−0.0341993
$856$	−37.4558	−1.28021
$857$	−4.54416	−0.155225	−0.0776127	−	0.996984i	$-0.524730\pi$
$857$	−4.54416	−0.155225	−0.0776127	+	0.996984i	$0.524730\pi$
$858$	−5.65685	−0.193122
$859$	−5.02944	−0.171602	−0.0858011	−	0.996312i	$-0.527345\pi$
$859$	−5.02944	−0.171602	−0.0858011	+	0.996312i	$0.527345\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	40.7279	1.38720
$863$	37.6690	1.28227	0.641135	−	0.767428i	$-0.278464\pi$
$863$	37.6690	1.28227	0.641135	+	0.767428i	$0.278464\pi$
$864$	0	0
$865$	21.3848	0.727104
$866$	36.3431	1.23499
$867$	3.51472	0.119366
$868$	0	0
$869$	−5.65685	−0.191896
$870$	11.6569	0.395204
$871$	2.00000	0.0677674
$872$	7.02944	0.238047
$873$	−10.7279	−0.363085
$874$	−2.24264	−0.0758585
$875$	2.00000	0.0676123
$876$	0	0
$877$	−30.6985	−1.03661	−0.518307	−	0.855195i	$-0.673438\pi$
$877$	−30.6985	−1.03661	−0.518307	+	0.855195i	$0.673438\pi$
$878$	18.3848	0.620456
$879$	−3.51472	−0.118549
$880$	−5.65685	−0.190693
$881$	−6.85786	−0.231047	−0.115524	−	0.993305i	$-0.536855\pi$
$881$	−6.85786	−0.231047	−0.115524	+	0.993305i	$0.536855\pi$
$882$	−4.24264	−0.142857
$883$	−51.6985	−1.73979	−0.869896	−	0.493235i	$-0.835814\pi$
$883$	−51.6985	−1.73979	−0.869896	+	0.493235i	$0.835814\pi$
$884$	0	0
$885$	−4.24264	−0.142615
$886$	−40.4853	−1.36013
$887$	40.0711	1.34545	0.672727	−	0.739890i	$-0.265123\pi$
$887$	40.0711	1.34545	0.672727	+	0.739890i	$0.265123\pi$
$888$	−36.9706	−1.24065
$889$	−11.4558	−0.384217
$890$	20.7279	0.694802
$891$	7.07107	0.236890
$892$	0	0
$893$	−1.58579	−0.0530663
$894$	40.6274	1.35878
$895$	5.65685	0.189088
$896$	22.6274	0.755929
$897$	4.48528	0.149759
$898$	−40.7279	−1.35911
$899$	−13.0711	−0.435945
$900$	0	0
$901$	−18.7279	−0.623918
$902$	2.82843	0.0941763
$903$	16.9706	0.564745
$904$	−25.9411	−0.862789
$905$	−7.00000	−0.232688
$906$	−14.9706	−0.497364
$907$	14.7574	0.490010	0.245005	−	0.969522i	$-0.421210\pi$
$907$	14.7574	0.490010	0.245005	+	0.969522i	$0.421210\pi$
$908$	0	0
$909$	−7.41421	−0.245914
$910$	5.65685	0.187523
$911$	−41.4853	−1.37447	−0.687234	−	0.726436i	$-0.741175\pi$
$911$	−41.4853	−1.37447	−0.687234	+	0.726436i	$0.741175\pi$
$912$	5.65685	0.187317
$913$	−16.0000	−0.529523
$914$	−31.4142	−1.03909
$915$	−3.17157	−0.104849
$916$	0	0
$917$	4.97056	0.164142
$918$	−35.3137	−1.16553
$919$	−8.97056	−0.295912	−0.147956	−	0.988994i	$-0.547269\pi$
$919$	−8.97056	−0.295912	−0.147956	+	0.988994i	$0.547269\pi$
$920$	4.48528	0.147875
$921$	−37.0711	−1.22153
$922$	−45.2132	−1.48902
$923$	−23.3137	−0.767380
$924$	0	0
$925$	−9.24264	−0.303896
$926$	−30.7696	−1.01115
$927$	0.485281	0.0159387
$928$	0	0
$929$	46.2843	1.51854	0.759269	−	0.650777i	$-0.225557\pi$
$929$	46.2843	1.51854	0.759269	+	0.650777i	$0.225557\pi$
$930$	−4.48528	−0.147078
$931$	3.00000	0.0983210
$932$	0	0
$933$	25.4558	0.833387
$934$	56.9706	1.86413
$935$	−6.24264	−0.204156
$936$	5.65685	0.184900
$937$	−2.48528	−0.0811906	−0.0405953	−	0.999176i	$-0.512925\pi$
$937$	−2.48528	−0.0811906	−0.0405953	+	0.999176i	$0.512925\pi$
$938$	−2.82843	−0.0923514
$939$	−26.4853	−0.864314
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−6.48528	−0.211302
$943$	−2.24264	−0.0730304
$944$	−12.0000	−0.390567
$945$	−11.3137	−0.368035
$946$	−12.0000	−0.390154
$947$	7.24264	0.235354	0.117677	−	0.993052i	$-0.462455\pi$
$947$	7.24264	0.235354	0.117677	+	0.993052i	$0.462455\pi$
$948$	0	0
$949$	22.4853	0.729903
$950$	1.41421	0.0458831
$951$	−44.9706	−1.45827
$952$	24.9706	0.809301
$953$	−9.38478	−0.304003	−0.152001	−	0.988380i	$-0.548572\pi$
$953$	−9.38478	−0.304003	−0.152001	+	0.988380i	$0.548572\pi$
$954$	6.00000	0.194257
$955$	9.55635	0.309236
$956$	0	0
$957$	−11.6569	−0.376813
$958$	13.6985	0.442578
$959$	3.51472	0.113496
$960$	−11.3137	−0.365148
$961$	−25.9706	−0.837760
$962$	−26.1421	−0.842856
$963$	13.2426	0.426738
$964$	0	0
$965$	25.2426	0.812589
$966$	−6.34315	−0.204087
$967$	29.9411	0.962842	0.481421	−	0.876490i	$-0.340121\pi$
$967$	29.9411	0.962842	0.481421	+	0.876490i	$0.340121\pi$
$968$	−25.4558	−0.818182
$969$	6.24264	0.200543
$970$	15.1716	0.487130
$971$	61.2843	1.96671	0.983353	−	0.181706i	$-0.0581619\pi$
$971$	61.2843	1.96671	0.983353	+	0.181706i	$0.0581619\pi$
$972$	0	0
$973$	37.4558	1.20078
$974$	−23.6569	−0.758015
$975$	−2.82843	−0.0905822
$976$	−8.97056	−0.287141
$977$	−5.87006	−0.187800	−0.0938999	−	0.995582i	$-0.529933\pi$
$977$	−5.87006	−0.187800	−0.0938999	+	0.995582i	$0.529933\pi$
$978$	−17.5147	−0.560059
$979$	−20.7279	−0.662467
$980$	0	0
$981$	−2.48528	−0.0793489
$982$	−43.7574	−1.39635
$983$	−28.9706	−0.924017	−0.462009	−	0.886875i	$-0.652871\pi$
$983$	−28.9706	−0.924017	−0.462009	+	0.886875i	$0.652871\pi$
$984$	5.65685	0.180334
$985$	−13.4142	−0.427412
$986$	36.3848	1.15873
$987$	−4.48528	−0.142768
$988$	0	0
$989$	9.51472	0.302550
$990$	2.00000	0.0635642
$991$	−53.6985	−1.70579	−0.852894	−	0.522084i	$-0.825155\pi$
$991$	−53.6985	−1.70579	−0.852894	+	0.522084i	$0.825155\pi$
$992$	0	0
$993$	−9.85786	−0.312830
$994$	32.9706	1.04576
$995$	11.0000	0.348723
$996$	0	0
$997$	−4.54416	−0.143915	−0.0719574	−	0.997408i	$-0.522925\pi$
$997$	−4.54416	−0.143915	−0.0719574	+	0.997408i	$0.522925\pi$
$998$	−42.3431	−1.34035
$999$	52.2843	1.65420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.2.a.b.1.1	✓	2
3.2	odd	2		3015.2.a.e.1.2		2
4.3	odd	2		5360.2.a.s.1.1		2
5.2	odd	4		1675.2.c.e.1274.1		4
5.3	odd	4		1675.2.c.e.1274.4		4
5.4	even	2		1675.2.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.2.a.b.1.1	✓	2	1.1	even	1	trivial
1675.2.a.f.1.2		2	5.4	even	2
1675.2.c.e.1274.1		4	5.2	odd	4
1675.2.c.e.1274.4		4	5.3	odd	4
3015.2.a.e.1.2		2	3.2	odd	2
5360.2.a.s.1.1		2	4.3	odd	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:	\( N \)	\(=\)	\( 335 = 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	335.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.41421 q^{3} -1.00000 q^{5} -2.00000 q^{6} -2.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} +1.41421 q^{3} -1.00000 q^{5} -2.00000 q^{6} -2.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +1.41421 q^{10} -1.41421 q^{11} -2.00000 q^{13} +2.82843 q^{14} -1.41421 q^{15} -4.00000 q^{16} -4.41421 q^{17} +1.41421 q^{18} -1.00000 q^{19} -2.82843 q^{21} +2.00000 q^{22} -1.58579 q^{23} +4.00000 q^{24} +1.00000 q^{25} +2.82843 q^{26} -5.65685 q^{27} +5.82843 q^{29} +2.00000 q^{30} -2.24264 q^{31} -2.00000 q^{33} +6.24264 q^{34} +2.00000 q^{35} -9.24264 q^{37} +1.41421 q^{38} -2.82843 q^{39} -2.82843 q^{40} +1.41421 q^{41} +4.00000 q^{42} -6.00000 q^{43} +1.00000 q^{45} +2.24264 q^{46} +1.58579 q^{47} -5.65685 q^{48} -3.00000 q^{49} -1.41421 q^{50} -6.24264 q^{51} +4.24264 q^{53} +8.00000 q^{54} +1.41421 q^{55} -5.65685 q^{56} -1.41421 q^{57} -8.24264 q^{58} +3.00000 q^{59} +2.24264 q^{61} +3.17157 q^{62} +2.00000 q^{63} +8.00000 q^{64} +2.00000 q^{65} +2.82843 q^{66} -1.00000 q^{67} -2.24264 q^{69} -2.82843 q^{70} +11.6569 q^{71} -2.82843 q^{72} -11.2426 q^{73} +13.0711 q^{74} +1.41421 q^{75} +2.82843 q^{77} +4.00000 q^{78} +4.00000 q^{79} +4.00000 q^{80} -5.00000 q^{81} -2.00000 q^{82} +11.3137 q^{83} +4.41421 q^{85} +8.48528 q^{86} +8.24264 q^{87} -4.00000 q^{88} +14.6569 q^{89} -1.41421 q^{90} +4.00000 q^{91} -3.17157 q^{93} -2.24264 q^{94} +1.00000 q^{95} +10.7279 q^{97} +4.24264 q^{98} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 4 q^{6} - 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 4 * q^6 - 4 * q^7 - 2 * q^9	\( 2 q - 2 q^{5} - 4 q^{6} - 4 q^{7} - 2 q^{9} - 4 q^{13} - 8 q^{16} - 6 q^{17} - 2 q^{19} + 4 q^{22} - 6 q^{23} + 8 q^{24} + 2 q^{25} + 6 q^{29} + 4 q^{30} + 4 q^{31} - 4 q^{33} + 4 q^{34} + 4 q^{35} - 10 q^{37} + 8 q^{42} - 12 q^{43} + 2 q^{45} - 4 q^{46} + 6 q^{47} - 6 q^{49} - 4 q^{51} + 16 q^{54} - 8 q^{58} + 6 q^{59} - 4 q^{61} + 12 q^{62} + 4 q^{63} + 16 q^{64} + 4 q^{65} - 2 q^{67} + 4 q^{69} + 12 q^{71} - 14 q^{73} + 12 q^{74} + 8 q^{78} + 8 q^{79} + 8 q^{80} - 10 q^{81} - 4 q^{82} + 6 q^{85} + 8 q^{87} - 8 q^{88} + 18 q^{89} + 8 q^{91} - 12 q^{93} + 4 q^{94} + 2 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 4 * q^6 - 4 * q^7 - 2 * q^9 - 4 * q^13 - 8 * q^16 - 6 * q^17 - 2 * q^19 + 4 * q^22 - 6 * q^23 + 8 * q^24 + 2 * q^25 + 6 * q^29 + 4 * q^30 + 4 * q^31 - 4 * q^33 + 4 * q^34 + 4 * q^35 - 10 * q^37 + 8 * q^42 - 12 * q^43 + 2 * q^45 - 4 * q^46 + 6 * q^47 - 6 * q^49 - 4 * q^51 + 16 * q^54 - 8 * q^58 + 6 * q^59 - 4 * q^61 + 12 * q^62 + 4 * q^63 + 16 * q^64 + 4 * q^65 - 2 * q^67 + 4 * q^69 + 12 * q^71 - 14 * q^73 + 12 * q^74 + 8 * q^78 + 8 * q^79 + 8 * q^80 - 10 * q^81 - 4 * q^82 + 6 * q^85 + 8 * q^87 - 8 * q^88 + 18 * q^89 + 8 * q^91 - 12 * q^93 + 4 * q^94 + 2 * q^95 - 4 * q^97

Introduction

L-functions

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Database

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