LMFDB - Embedded newform 666.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(1,666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, 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("666.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 666 = 2 \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	666.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:	$5.31803677462$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	666.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +2.56155 q^{7} -1.00000 q^{8} +2.00000 q^{10} -2.56155 q^{11} -5.68466 q^{13} -2.56155 q^{14} +1.00000 q^{16} +0.561553 q^{17} +0.561553 q^{19} -2.00000 q^{20} +2.56155 q^{22} +3.68466 q^{23} -1.00000 q^{25} +5.68466 q^{26} +2.56155 q^{28} -7.12311 q^{29} -0.876894 q^{31} -1.00000 q^{32} -0.561553 q^{34} -5.12311 q^{35} -1.00000 q^{37} -0.561553 q^{38} +2.00000 q^{40} -2.87689 q^{41} -7.12311 q^{43} -2.56155 q^{44} -3.68466 q^{46} -6.24621 q^{47} -0.438447 q^{49} +1.00000 q^{50} -5.68466 q^{52} -2.56155 q^{53} +5.12311 q^{55} -2.56155 q^{56} +7.12311 q^{58} -9.12311 q^{59} +4.24621 q^{61} +0.876894 q^{62} +1.00000 q^{64} +11.3693 q^{65} -14.2462 q^{67} +0.561553 q^{68} +5.12311 q^{70} +14.2462 q^{71} -0.561553 q^{73} +1.00000 q^{74} +0.561553 q^{76} -6.56155 q^{77} -6.00000 q^{79} -2.00000 q^{80} +2.87689 q^{82} +13.9309 q^{83} -1.12311 q^{85} +7.12311 q^{86} +2.56155 q^{88} -15.9309 q^{89} -14.5616 q^{91} +3.68466 q^{92} +6.24621 q^{94} -1.12311 q^{95} +11.1231 q^{97} +0.438447 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7} - 2 q^{8} + 4 q^{10} - q^{11} + q^{13} - q^{14} + 2 q^{16} - 3 q^{17} - 3 q^{19} - 4 q^{20} + q^{22} - 5 q^{23} - 2 q^{25} - q^{26} + q^{28} - 6 q^{29} - 10 q^{31}+ \cdots + 5 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + q^7 - 2 * q^8 + 4 * q^10 - q^11 + q^13 - q^14 + 2 * q^16 - 3 * q^17 - 3 * q^19 - 4 * q^20 + q^22 - 5 * q^23 - 2 * q^25 - q^26 + q^28 - 6 * q^29 - 10 * q^31 - 2 * q^32 + 3 * q^34 - 2 * q^35 - 2 * q^37 + 3 * q^38 + 4 * q^40 - 14 * q^41 - 6 * q^43 - q^44 + 5 * q^46 + 4 * q^47 - 5 * q^49 + 2 * q^50 + q^52 - q^53 + 2 * q^55 - q^56 + 6 * q^58 - 10 * q^59 - 8 * q^61 + 10 * q^62 + 2 * q^64 - 2 * q^65 - 12 * q^67 - 3 * q^68 + 2 * q^70 + 12 * q^71 + 3 * q^73 + 2 * q^74 - 3 * q^76 - 9 * q^77 - 12 * q^79 - 4 * q^80 + 14 * q^82 - q^83 + 6 * q^85 + 6 * q^86 + q^88 - 3 * q^89 - 25 * q^91 - 5 * q^92 - 4 * q^94 + 6 * q^95 + 14 * q^97 + 5 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	−5.68466	−1.57664	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	−5.68466	−1.57664	−0.788320	+	0.615265i	$0.789049\pi$
$14$	−2.56155	−0.684604
$15$	0	0
$16$	1.00000	0.250000
$17$	0.561553	0.136197	0.0680983	−	0.997679i	$-0.478307\pi$
$17$	0.561553	0.136197	0.0680983	+	0.997679i	$0.478307\pi$
$18$	0	0
$19$	0.561553	0.128829	0.0644145	−	0.997923i	$-0.479482\pi$
$19$	0.561553	0.128829	0.0644145	+	0.997923i	$0.479482\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	2.56155	0.546125
$23$	3.68466	0.768304	0.384152	−	0.923270i	$-0.374494\pi$
$23$	3.68466	0.768304	0.384152	+	0.923270i	$0.374494\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	5.68466	1.11485
$27$	0	0
$28$	2.56155	0.484088
$29$	−7.12311	−1.32273	−0.661364	−	0.750065i	$-0.730022\pi$
$29$	−7.12311	−1.32273	−0.661364	+	0.750065i	$0.730022\pi$
$30$	0	0
$31$	−0.876894	−0.157495	−0.0787474	−	0.996895i	$-0.525092\pi$
$31$	−0.876894	−0.157495	−0.0787474	+	0.996895i	$0.525092\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.561553	−0.0963055
$35$	−5.12311	−0.865963
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−0.561553	−0.0910959
$39$	0	0
$40$	2.00000	0.316228
$41$	−2.87689	−0.449295	−0.224648	−	0.974440i	$-0.572123\pi$
$41$	−2.87689	−0.449295	−0.224648	+	0.974440i	$0.572123\pi$
$42$	0	0
$43$	−7.12311	−1.08626	−0.543132	−	0.839648i	$-0.682762\pi$
$43$	−7.12311	−1.08626	−0.543132	+	0.839648i	$0.682762\pi$
$44$	−2.56155	−0.386169
$45$	0	0
$46$	−3.68466	−0.543273
$47$	−6.24621	−0.911104	−0.455552	−	0.890209i	$-0.650558\pi$
$47$	−6.24621	−0.911104	−0.455552	+	0.890209i	$0.650558\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	1.00000	0.141421
$51$	0	0
$52$	−5.68466	−0.788320
$53$	−2.56155	−0.351856	−0.175928	−	0.984403i	$-0.556293\pi$
$53$	−2.56155	−0.351856	−0.175928	+	0.984403i	$0.556293\pi$
$54$	0	0
$55$	5.12311	0.690799
$56$	−2.56155	−0.342302
$57$	0	0
$58$	7.12311	0.935310
$59$	−9.12311	−1.18773	−0.593864	−	0.804566i	$-0.702398\pi$
$59$	−9.12311	−1.18773	−0.593864	+	0.804566i	$0.702398\pi$
$60$	0	0
$61$	4.24621	0.543672	0.271836	−	0.962344i	$-0.412369\pi$
$61$	4.24621	0.543672	0.271836	+	0.962344i	$0.412369\pi$
$62$	0.876894	0.111366
$63$	0	0
$64$	1.00000	0.125000
$65$	11.3693	1.41019
$66$	0	0
$67$	−14.2462	−1.74045	−0.870226	−	0.492653i	$-0.836027\pi$
$67$	−14.2462	−1.74045	−0.870226	+	0.492653i	$0.836027\pi$
$68$	0.561553	0.0680983
$69$	0	0
$70$	5.12311	0.612328
$71$	14.2462	1.69071	0.845357	−	0.534202i	$-0.179388\pi$
$71$	14.2462	1.69071	0.845357	+	0.534202i	$0.179388\pi$
$72$	0	0
$73$	−0.561553	−0.0657248	−0.0328624	−	0.999460i	$-0.510462\pi$
$73$	−0.561553	−0.0657248	−0.0328624	+	0.999460i	$0.510462\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	0.561553	0.0644145
$77$	−6.56155	−0.747758
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	2.87689	0.317700
$83$	13.9309	1.52911	0.764556	−	0.644558i	$-0.222958\pi$
$83$	13.9309	1.52911	0.764556	+	0.644558i	$0.222958\pi$
$84$	0	0
$85$	−1.12311	−0.121818
$86$	7.12311	0.768104
$87$	0	0
$88$	2.56155	0.273062
$89$	−15.9309	−1.68867	−0.844334	−	0.535817i	$-0.820004\pi$
$89$	−15.9309	−1.68867	−0.844334	+	0.535817i	$0.820004\pi$
$90$	0	0
$91$	−14.5616	−1.52647
$92$	3.68466	0.384152
$93$	0	0
$94$	6.24621	0.644247
$95$	−1.12311	−0.115228
$96$	0	0
$97$	11.1231	1.12938	0.564690	−	0.825303i	$-0.308996\pi$
$97$	11.1231	1.12938	0.564690	+	0.825303i	$0.308996\pi$
$98$	0.438447	0.0442899
$99$	0	0
$100$	−1.00000	−0.100000
$101$	2.87689	0.286262	0.143131	−	0.989704i	$-0.454283\pi$
$101$	2.87689	0.286262	0.143131	+	0.989704i	$0.454283\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	5.68466	0.557427
$105$	0	0
$106$	2.56155	0.248800
$107$	10.5616	1.02102	0.510512	−	0.859871i	$-0.329456\pi$
$107$	10.5616	1.02102	0.510512	+	0.859871i	$0.329456\pi$
$108$	0	0
$109$	−0.561553	−0.0537870	−0.0268935	−	0.999638i	$-0.508561\pi$
$109$	−0.561553	−0.0537870	−0.0268935	+	0.999638i	$0.508561\pi$
$110$	−5.12311	−0.488469
$111$	0	0
$112$	2.56155	0.242044
$113$	−4.24621	−0.399450	−0.199725	−	0.979852i	$-0.564005\pi$
$113$	−4.24621	−0.399450	−0.199725	+	0.979852i	$0.564005\pi$
$114$	0	0
$115$	−7.36932	−0.687192
$116$	−7.12311	−0.661364
$117$	0	0
$118$	9.12311	0.839850
$119$	1.43845	0.131862
$120$	0	0
$121$	−4.43845	−0.403495
$122$	−4.24621	−0.384434
$123$	0	0
$124$	−0.876894	−0.0787474
$125$	12.0000	1.07331
$126$	0	0
$127$	1.43845	0.127642	0.0638208	−	0.997961i	$-0.479671\pi$
$127$	1.43845	0.127642	0.0638208	+	0.997961i	$0.479671\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−11.3693	−0.997155
$131$	−6.87689	−0.600837	−0.300419	−	0.953807i	$-0.597126\pi$
$131$	−6.87689	−0.600837	−0.300419	+	0.953807i	$0.597126\pi$
$132$	0	0
$133$	1.43845	0.124729
$134$	14.2462	1.23069
$135$	0	0
$136$	−0.561553	−0.0481528
$137$	22.2462	1.90062	0.950311	−	0.311302i	$-0.100765\pi$
$137$	22.2462	1.90062	0.950311	+	0.311302i	$0.100765\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	−5.12311	−0.432981
$141$	0	0
$142$	−14.2462	−1.19552
$143$	14.5616	1.21770
$144$	0	0
$145$	14.2462	1.18308
$146$	0.561553	0.0464744
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	5.12311	0.419701	0.209851	−	0.977733i	$-0.432702\pi$
$149$	5.12311	0.419701	0.209851	+	0.977733i	$0.432702\pi$
$150$	0	0
$151$	15.6847	1.27640	0.638200	−	0.769871i	$-0.279679\pi$
$151$	15.6847	1.27640	0.638200	+	0.769871i	$0.279679\pi$
$152$	−0.561553	−0.0455479
$153$	0	0
$154$	6.56155	0.528745
$155$	1.75379	0.140868
$156$	0	0
$157$	22.4924	1.79509	0.897545	−	0.440922i	$-0.145349\pi$
$157$	22.4924	1.79509	0.897545	+	0.440922i	$0.145349\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	2.00000	0.158114
$161$	9.43845	0.743854
$162$	0	0
$163$	7.93087	0.621194	0.310597	−	0.950542i	$-0.399471\pi$
$163$	7.93087	0.621194	0.310597	+	0.950542i	$0.399471\pi$
$164$	−2.87689	−0.224648
$165$	0	0
$166$	−13.9309	−1.08125
$167$	4.31534	0.333931	0.166966	−	0.985963i	$-0.446603\pi$
$167$	4.31534	0.333931	0.166966	+	0.985963i	$0.446603\pi$
$168$	0	0
$169$	19.3153	1.48580
$170$	1.12311	0.0861383
$171$	0	0
$172$	−7.12311	−0.543132
$173$	2.56155	0.194751	0.0973756	−	0.995248i	$-0.468955\pi$
$173$	2.56155	0.194751	0.0973756	+	0.995248i	$0.468955\pi$
$174$	0	0
$175$	−2.56155	−0.193635
$176$	−2.56155	−0.193084
$177$	0	0
$178$	15.9309	1.19407
$179$	1.12311	0.0839449	0.0419724	−	0.999119i	$-0.486636\pi$
$179$	1.12311	0.0839449	0.0419724	+	0.999119i	$0.486636\pi$
$180$	0	0
$181$	−18.4924	−1.37453	−0.687265	−	0.726406i	$-0.741189\pi$
$181$	−18.4924	−1.37453	−0.687265	+	0.726406i	$0.741189\pi$
$182$	14.5616	1.07937
$183$	0	0
$184$	−3.68466	−0.271637
$185$	2.00000	0.147043
$186$	0	0
$187$	−1.43845	−0.105190
$188$	−6.24621	−0.455552
$189$	0	0
$190$	1.12311	0.0814786
$191$	−6.56155	−0.474777	−0.237389	−	0.971415i	$-0.576291\pi$
$191$	−6.56155	−0.474777	−0.237389	+	0.971415i	$0.576291\pi$
$192$	0	0
$193$	25.3693	1.82612	0.913062	−	0.407821i	$-0.133711\pi$
$193$	25.3693	1.82612	0.913062	+	0.407821i	$0.133711\pi$
$194$	−11.1231	−0.798592
$195$	0	0
$196$	−0.438447	−0.0313177
$197$	8.80776	0.627527	0.313764	−	0.949501i	$-0.398410\pi$
$197$	8.80776	0.627527	0.313764	+	0.949501i	$0.398410\pi$
$198$	0	0
$199$	4.87689	0.345714	0.172857	−	0.984947i	$-0.444700\pi$
$199$	4.87689	0.345714	0.172857	+	0.984947i	$0.444700\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−2.87689	−0.202418
$203$	−18.2462	−1.28063
$204$	0	0
$205$	5.75379	0.401862
$206$	10.0000	0.696733
$207$	0	0
$208$	−5.68466	−0.394160
$209$	−1.43845	−0.0994995
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−2.56155	−0.175928
$213$	0	0
$214$	−10.5616	−0.721973
$215$	14.2462	0.971584
$216$	0	0
$217$	−2.24621	−0.152483
$218$	0.561553	0.0380332
$219$	0	0
$220$	5.12311	0.345400
$221$	−3.19224	−0.214733
$222$	0	0
$223$	−4.49242	−0.300835	−0.150417	−	0.988623i	$-0.548062\pi$
$223$	−4.49242	−0.300835	−0.150417	+	0.988623i	$0.548062\pi$
$224$	−2.56155	−0.171151
$225$	0	0
$226$	4.24621	0.282454
$227$	−19.3693	−1.28559	−0.642793	−	0.766040i	$-0.722225\pi$
$227$	−19.3693	−1.28559	−0.642793	+	0.766040i	$0.722225\pi$
$228$	0	0
$229$	2.63068	0.173840	0.0869202	−	0.996215i	$-0.472297\pi$
$229$	2.63068	0.173840	0.0869202	+	0.996215i	$0.472297\pi$
$230$	7.36932	0.485918
$231$	0	0
$232$	7.12311	0.467655
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	0	0
$235$	12.4924	0.814916
$236$	−9.12311	−0.593864
$237$	0	0
$238$	−1.43845	−0.0932407
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	8.24621	0.531185	0.265593	−	0.964085i	$-0.414432\pi$
$241$	8.24621	0.531185	0.265593	+	0.964085i	$0.414432\pi$
$242$	4.43845	0.285314
$243$	0	0
$244$	4.24621	0.271836
$245$	0.876894	0.0560227
$246$	0	0
$247$	−3.19224	−0.203117
$248$	0.876894	0.0556828
$249$	0	0
$250$	−12.0000	−0.758947
$251$	1.12311	0.0708898	0.0354449	−	0.999372i	$-0.488715\pi$
$251$	1.12311	0.0708898	0.0354449	+	0.999372i	$0.488715\pi$
$252$	0	0
$253$	−9.43845	−0.593390
$254$	−1.43845	−0.0902562
$255$	0	0
$256$	1.00000	0.0625000
$257$	−19.4384	−1.21254	−0.606269	−	0.795260i	$-0.707334\pi$
$257$	−19.4384	−1.21254	−0.606269	+	0.795260i	$0.707334\pi$
$258$	0	0
$259$	−2.56155	−0.159167
$260$	11.3693	0.705095
$261$	0	0
$262$	6.87689	0.424856
$263$	−30.2462	−1.86506	−0.932531	−	0.361091i	$-0.882404\pi$
$263$	−30.2462	−1.86506	−0.932531	+	0.361091i	$0.882404\pi$
$264$	0	0
$265$	5.12311	0.314710
$266$	−1.43845	−0.0881969
$267$	0	0
$268$	−14.2462	−0.870226
$269$	19.0540	1.16174	0.580871	−	0.813996i	$-0.302712\pi$
$269$	19.0540	1.16174	0.580871	+	0.813996i	$0.302712\pi$
$270$	0	0
$271$	−22.2462	−1.35136	−0.675681	−	0.737195i	$-0.736150\pi$
$271$	−22.2462	−1.35136	−0.675681	+	0.737195i	$0.736150\pi$
$272$	0.561553	0.0340491
$273$	0	0
$274$	−22.2462	−1.34394
$275$	2.56155	0.154467
$276$	0	0
$277$	23.4384	1.40828	0.704140	−	0.710061i	$-0.251333\pi$
$277$	23.4384	1.40828	0.704140	+	0.710061i	$0.251333\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	5.12311	0.306164
$281$	−15.9309	−0.950356	−0.475178	−	0.879890i	$-0.657616\pi$
$281$	−15.9309	−0.950356	−0.475178	+	0.879890i	$0.657616\pi$
$282$	0	0
$283$	−26.8078	−1.59356	−0.796778	−	0.604272i	$-0.793464\pi$
$283$	−26.8078	−1.59356	−0.796778	+	0.604272i	$0.793464\pi$
$284$	14.2462	0.845357
$285$	0	0
$286$	−14.5616	−0.861043
$287$	−7.36932	−0.434997
$288$	0	0
$289$	−16.6847	−0.981450
$290$	−14.2462	−0.836566
$291$	0	0
$292$	−0.561553	−0.0328624
$293$	−25.3002	−1.47805	−0.739026	−	0.673677i	$-0.764714\pi$
$293$	−25.3002	−1.47805	−0.739026	+	0.673677i	$0.764714\pi$
$294$	0	0
$295$	18.2462	1.06234
$296$	1.00000	0.0581238
$297$	0	0
$298$	−5.12311	−0.296774
$299$	−20.9460	−1.21134
$300$	0	0
$301$	−18.2462	−1.05169
$302$	−15.6847	−0.902551
$303$	0	0
$304$	0.561553	0.0322073
$305$	−8.49242	−0.486275
$306$	0	0
$307$	7.36932	0.420589	0.210295	−	0.977638i	$-0.432558\pi$
$307$	7.36932	0.420589	0.210295	+	0.977638i	$0.432558\pi$
$308$	−6.56155	−0.373879
$309$	0	0
$310$	−1.75379	−0.0996085
$311$	−13.7538	−0.779906	−0.389953	−	0.920835i	$-0.627509\pi$
$311$	−13.7538	−0.779906	−0.389953	+	0.920835i	$0.627509\pi$
$312$	0	0
$313$	−11.1231	−0.628715	−0.314358	−	0.949305i	$-0.601789\pi$
$313$	−11.1231	−0.628715	−0.314358	+	0.949305i	$0.601789\pi$
$314$	−22.4924	−1.26932
$315$	0	0
$316$	−6.00000	−0.337526
$317$	−1.12311	−0.0630799	−0.0315399	−	0.999502i	$-0.510041\pi$
$317$	−1.12311	−0.0630799	−0.0315399	+	0.999502i	$0.510041\pi$
$318$	0	0
$319$	18.2462	1.02159
$320$	−2.00000	−0.111803
$321$	0	0
$322$	−9.43845	−0.525984
$323$	0.315342	0.0175461
$324$	0	0
$325$	5.68466	0.315328
$326$	−7.93087	−0.439250
$327$	0	0
$328$	2.87689	0.158850
$329$	−16.0000	−0.882109
$330$	0	0
$331$	1.36932	0.0752645	0.0376322	−	0.999292i	$-0.488018\pi$
$331$	1.36932	0.0752645	0.0376322	+	0.999292i	$0.488018\pi$
$332$	13.9309	0.764556
$333$	0	0
$334$	−4.31534	−0.236125
$335$	28.4924	1.55671
$336$	0	0
$337$	23.9309	1.30360	0.651799	−	0.758392i	$-0.274015\pi$
$337$	23.9309	1.30360	0.651799	+	0.758392i	$0.274015\pi$
$338$	−19.3153	−1.05062
$339$	0	0
$340$	−1.12311	−0.0609090
$341$	2.24621	0.121639
$342$	0	0
$343$	−19.0540	−1.02882
$344$	7.12311	0.384052
$345$	0	0
$346$	−2.56155	−0.137710
$347$	29.6155	1.58984	0.794922	−	0.606711i	$-0.207512\pi$
$347$	29.6155	1.58984	0.794922	+	0.606711i	$0.207512\pi$
$348$	0	0
$349$	26.4924	1.41811	0.709053	−	0.705155i	$-0.249122\pi$
$349$	26.4924	1.41811	0.709053	+	0.705155i	$0.249122\pi$
$350$	2.56155	0.136921
$351$	0	0
$352$	2.56155	0.136531
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	−28.4924	−1.51222
$356$	−15.9309	−0.844334
$357$	0	0
$358$	−1.12311	−0.0593580
$359$	−4.49242	−0.237101	−0.118550	−	0.992948i	$-0.537825\pi$
$359$	−4.49242	−0.237101	−0.118550	+	0.992948i	$0.537825\pi$
$360$	0	0
$361$	−18.6847	−0.983403
$362$	18.4924	0.971940
$363$	0	0
$364$	−14.5616	−0.763233
$365$	1.12311	0.0587860
$366$	0	0
$367$	12.3153	0.642856	0.321428	−	0.946934i	$-0.395837\pi$
$367$	12.3153	0.642856	0.321428	+	0.946934i	$0.395837\pi$
$368$	3.68466	0.192076
$369$	0	0
$370$	−2.00000	−0.103975
$371$	−6.56155	−0.340659
$372$	0	0
$373$	2.63068	0.136212	0.0681058	−	0.997678i	$-0.478304\pi$
$373$	2.63068	0.136212	0.0681058	+	0.997678i	$0.478304\pi$
$374$	1.43845	0.0743803
$375$	0	0
$376$	6.24621	0.322124
$377$	40.4924	2.08547
$378$	0	0
$379$	−2.24621	−0.115380	−0.0576901	−	0.998335i	$-0.518374\pi$
$379$	−2.24621	−0.115380	−0.0576901	+	0.998335i	$0.518374\pi$
$380$	−1.12311	−0.0576141
$381$	0	0
$382$	6.56155	0.335718
$383$	8.80776	0.450056	0.225028	−	0.974352i	$-0.427753\pi$
$383$	8.80776	0.450056	0.225028	+	0.974352i	$0.427753\pi$
$384$	0	0
$385$	13.1231	0.668815
$386$	−25.3693	−1.29126
$387$	0	0
$388$	11.1231	0.564690
$389$	32.7386	1.65991	0.829957	−	0.557827i	$-0.188365\pi$
$389$	32.7386	1.65991	0.829957	+	0.557827i	$0.188365\pi$
$390$	0	0
$391$	2.06913	0.104640
$392$	0.438447	0.0221449
$393$	0	0
$394$	−8.80776	−0.443729
$395$	12.0000	0.603786
$396$	0	0
$397$	18.4924	0.928108	0.464054	−	0.885807i	$-0.346394\pi$
$397$	18.4924	0.928108	0.464054	+	0.885807i	$0.346394\pi$
$398$	−4.87689	−0.244457
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−7.43845	−0.371458	−0.185729	−	0.982601i	$-0.559465\pi$
$401$	−7.43845	−0.371458	−0.185729	+	0.982601i	$0.559465\pi$
$402$	0	0
$403$	4.98485	0.248313
$404$	2.87689	0.143131
$405$	0	0
$406$	18.2462	0.905544
$407$	2.56155	0.126971
$408$	0	0
$409$	3.75379	0.185613	0.0928065	−	0.995684i	$-0.470416\pi$
$409$	3.75379	0.185613	0.0928065	+	0.995684i	$0.470416\pi$
$410$	−5.75379	−0.284159
$411$	0	0
$412$	−10.0000	−0.492665
$413$	−23.3693	−1.14993
$414$	0	0
$415$	−27.8617	−1.36768
$416$	5.68466	0.278713
$417$	0	0
$418$	1.43845	0.0703568
$419$	−35.6847	−1.74331	−0.871655	−	0.490120i	$-0.836953\pi$
$419$	−35.6847	−1.74331	−0.871655	+	0.490120i	$0.836953\pi$
$420$	0	0
$421$	3.75379	0.182948	0.0914742	−	0.995807i	$-0.470842\pi$
$421$	3.75379	0.182948	0.0914742	+	0.995807i	$0.470842\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	2.56155	0.124400
$425$	−0.561553	−0.0272393
$426$	0	0
$427$	10.8769	0.526370
$428$	10.5616	0.510512
$429$	0	0
$430$	−14.2462	−0.687013
$431$	−8.17708	−0.393876	−0.196938	−	0.980416i	$-0.563100\pi$
$431$	−8.17708	−0.393876	−0.196938	+	0.980416i	$0.563100\pi$
$432$	0	0
$433$	24.5616	1.18035	0.590176	−	0.807274i	$-0.299058\pi$
$433$	24.5616	1.18035	0.590176	+	0.807274i	$0.299058\pi$
$434$	2.24621	0.107822
$435$	0	0
$436$	−0.561553	−0.0268935
$437$	2.06913	0.0989799
$438$	0	0
$439$	22.4924	1.07350	0.536752	−	0.843740i	$-0.319651\pi$
$439$	22.4924	1.07350	0.536752	+	0.843740i	$0.319651\pi$
$440$	−5.12311	−0.244234
$441$	0	0
$442$	3.19224	0.151839
$443$	−30.7386	−1.46044	−0.730218	−	0.683214i	$-0.760582\pi$
$443$	−30.7386	−1.46044	−0.730218	+	0.683214i	$0.760582\pi$
$444$	0	0
$445$	31.8617	1.51039
$446$	4.49242	0.212722
$447$	0	0
$448$	2.56155	0.121022
$449$	−15.7538	−0.743467	−0.371734	−	0.928339i	$-0.621237\pi$
$449$	−15.7538	−0.743467	−0.371734	+	0.928339i	$0.621237\pi$
$450$	0	0
$451$	7.36932	0.347008
$452$	−4.24621	−0.199725
$453$	0	0
$454$	19.3693	0.909047
$455$	29.1231	1.36531
$456$	0	0
$457$	−35.6155	−1.66602	−0.833012	−	0.553255i	$-0.813386\pi$
$457$	−35.6155	−1.66602	−0.833012	+	0.553255i	$0.813386\pi$
$458$	−2.63068	−0.122924
$459$	0	0
$460$	−7.36932	−0.343596
$461$	−38.9848	−1.81571	−0.907853	−	0.419289i	$-0.862279\pi$
$461$	−38.9848	−1.81571	−0.907853	+	0.419289i	$0.862279\pi$
$462$	0	0
$463$	13.3693	0.621325	0.310662	−	0.950520i	$-0.399449\pi$
$463$	13.3693	0.621325	0.310662	+	0.950520i	$0.399449\pi$
$464$	−7.12311	−0.330682
$465$	0	0
$466$	−16.0000	−0.741186
$467$	−21.6155	−1.00025	−0.500124	−	0.865954i	$-0.666712\pi$
$467$	−21.6155	−1.00025	−0.500124	+	0.865954i	$0.666712\pi$
$468$	0	0
$469$	−36.4924	−1.68506
$470$	−12.4924	−0.576232
$471$	0	0
$472$	9.12311	0.419925
$473$	18.2462	0.838962
$474$	0	0
$475$	−0.561553	−0.0257658
$476$	1.43845	0.0659311
$477$	0	0
$478$	8.00000	0.365911
$479$	−4.94602	−0.225990	−0.112995	−	0.993596i	$-0.536044\pi$
$479$	−4.94602	−0.225990	−0.112995	+	0.993596i	$0.536044\pi$
$480$	0	0
$481$	5.68466	0.259198
$482$	−8.24621	−0.375605
$483$	0	0
$484$	−4.43845	−0.201748
$485$	−22.2462	−1.01015
$486$	0	0
$487$	4.87689	0.220993	0.110497	−	0.993877i	$-0.464756\pi$
$487$	4.87689	0.220993	0.110497	+	0.993877i	$0.464756\pi$
$488$	−4.24621	−0.192217
$489$	0	0
$490$	−0.876894	−0.0396140
$491$	29.9309	1.35076	0.675381	−	0.737469i	$-0.263979\pi$
$491$	29.9309	1.35076	0.675381	+	0.737469i	$0.263979\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	3.19224	0.143625
$495$	0	0
$496$	−0.876894	−0.0393737
$497$	36.4924	1.63691
$498$	0	0
$499$	28.4233	1.27240	0.636201	−	0.771524i	$-0.280505\pi$
$499$	28.4233	1.27240	0.636201	+	0.771524i	$0.280505\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−1.12311	−0.0501267
$503$	32.9848	1.47072	0.735361	−	0.677676i	$-0.237013\pi$
$503$	32.9848	1.47072	0.735361	+	0.677676i	$0.237013\pi$
$504$	0	0
$505$	−5.75379	−0.256040
$506$	9.43845	0.419590
$507$	0	0
$508$	1.43845	0.0638208
$509$	−39.0540	−1.73104	−0.865519	−	0.500877i	$-0.833011\pi$
$509$	−39.0540	−1.73104	−0.865519	+	0.500877i	$0.833011\pi$
$510$	0	0
$511$	−1.43845	−0.0636332
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	19.4384	0.857393
$515$	20.0000	0.881305
$516$	0	0
$517$	16.0000	0.703679
$518$	2.56155	0.112548
$519$	0	0
$520$	−11.3693	−0.498578
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	−20.8769	−0.912883	−0.456441	−	0.889753i	$-0.650876\pi$
$523$	−20.8769	−0.912883	−0.456441	+	0.889753i	$0.650876\pi$
$524$	−6.87689	−0.300419
$525$	0	0
$526$	30.2462	1.31880
$527$	−0.492423	−0.0214503
$528$	0	0
$529$	−9.42329	−0.409708
$530$	−5.12311	−0.222533
$531$	0	0
$532$	1.43845	0.0623646
$533$	16.3542	0.708377
$534$	0	0
$535$	−21.1231	−0.913231
$536$	14.2462	0.615343
$537$	0	0
$538$	−19.0540	−0.821475
$539$	1.12311	0.0483756
$540$	0	0
$541$	−31.9309	−1.37282	−0.686408	−	0.727217i	$-0.740813\pi$
$541$	−31.9309	−1.37282	−0.686408	+	0.727217i	$0.740813\pi$
$542$	22.2462	0.955557
$543$	0	0
$544$	−0.561553	−0.0240764
$545$	1.12311	0.0481086
$546$	0	0
$547$	−22.8078	−0.975190	−0.487595	−	0.873070i	$-0.662126\pi$
$547$	−22.8078	−0.975190	−0.487595	+	0.873070i	$0.662126\pi$
$548$	22.2462	0.950311
$549$	0	0
$550$	−2.56155	−0.109225
$551$	−4.00000	−0.170406
$552$	0	0
$553$	−15.3693	−0.653570
$554$	−23.4384	−0.995804
$555$	0	0
$556$	−12.0000	−0.508913
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	0	0
$559$	40.4924	1.71265
$560$	−5.12311	−0.216491
$561$	0	0
$562$	15.9309	0.672003
$563$	17.1231	0.721653	0.360826	−	0.932633i	$-0.382495\pi$
$563$	17.1231	0.721653	0.360826	+	0.932633i	$0.382495\pi$
$564$	0	0
$565$	8.49242	0.357279
$566$	26.8078	1.12681
$567$	0	0
$568$	−14.2462	−0.597758
$569$	−15.3002	−0.641417	−0.320709	−	0.947178i	$-0.603921\pi$
$569$	−15.3002	−0.641417	−0.320709	+	0.947178i	$0.603921\pi$
$570$	0	0
$571$	−38.7386	−1.62116	−0.810581	−	0.585627i	$-0.800848\pi$
$571$	−38.7386	−1.62116	−0.810581	+	0.585627i	$0.800848\pi$
$572$	14.5616	0.608849
$573$	0	0
$574$	7.36932	0.307589
$575$	−3.68466	−0.153661
$576$	0	0
$577$	−0.876894	−0.0365056	−0.0182528	−	0.999833i	$-0.505810\pi$
$577$	−0.876894	−0.0365056	−0.0182528	+	0.999833i	$0.505810\pi$
$578$	16.6847	0.693990
$579$	0	0
$580$	14.2462	0.591542
$581$	35.6847	1.48045
$582$	0	0
$583$	6.56155	0.271752
$584$	0.561553	0.0232372
$585$	0	0
$586$	25.3002	1.04514
$587$	30.2462	1.24839	0.624197	−	0.781267i	$-0.285426\pi$
$587$	30.2462	1.24839	0.624197	+	0.781267i	$0.285426\pi$
$588$	0	0
$589$	−0.492423	−0.0202899
$590$	−18.2462	−0.751185
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	−8.63068	−0.354420	−0.177210	−	0.984173i	$-0.556707\pi$
$593$	−8.63068	−0.354420	−0.177210	+	0.984173i	$0.556707\pi$
$594$	0	0
$595$	−2.87689	−0.117941
$596$	5.12311	0.209851
$597$	0	0
$598$	20.9460	0.856547
$599$	17.7538	0.725400	0.362700	−	0.931906i	$-0.381855\pi$
$599$	17.7538	0.725400	0.362700	+	0.931906i	$0.381855\pi$
$600$	0	0
$601$	−21.6847	−0.884536	−0.442268	−	0.896883i	$-0.645826\pi$
$601$	−21.6847	−0.884536	−0.442268	+	0.896883i	$0.645826\pi$
$602$	18.2462	0.743660
$603$	0	0
$604$	15.6847	0.638200
$605$	8.87689	0.360897
$606$	0	0
$607$	−21.8617	−0.887341	−0.443670	−	0.896190i	$-0.646324\pi$
$607$	−21.8617	−0.887341	−0.443670	+	0.896190i	$0.646324\pi$
$608$	−0.561553	−0.0227740
$609$	0	0
$610$	8.49242	0.343848
$611$	35.5076	1.43648
$612$	0	0
$613$	34.9848	1.41302	0.706512	−	0.707701i	$-0.250268\pi$
$613$	34.9848	1.41302	0.706512	+	0.707701i	$0.250268\pi$
$614$	−7.36932	−0.297401
$615$	0	0
$616$	6.56155	0.264372
$617$	14.8769	0.598921	0.299461	−	0.954109i	$-0.403193\pi$
$617$	14.8769	0.598921	0.299461	+	0.954109i	$0.403193\pi$
$618$	0	0
$619$	−30.7386	−1.23549	−0.617745	−	0.786378i	$-0.711954\pi$
$619$	−30.7386	−1.23549	−0.617745	+	0.786378i	$0.711954\pi$
$620$	1.75379	0.0704339
$621$	0	0
$622$	13.7538	0.551477
$623$	−40.8078	−1.63493
$624$	0	0
$625$	−19.0000	−0.760000
$626$	11.1231	0.444569
$627$	0	0
$628$	22.4924	0.897545
$629$	−0.561553	−0.0223906
$630$	0	0
$631$	36.7386	1.46254	0.731271	−	0.682087i	$-0.238928\pi$
$631$	36.7386	1.46254	0.731271	+	0.682087i	$0.238928\pi$
$632$	6.00000	0.238667
$633$	0	0
$634$	1.12311	0.0446042
$635$	−2.87689	−0.114166
$636$	0	0
$637$	2.49242	0.0987534
$638$	−18.2462	−0.722374
$639$	0	0
$640$	2.00000	0.0790569
$641$	−35.8617	−1.41645	−0.708227	−	0.705985i	$-0.750505\pi$
$641$	−35.8617	−1.41645	−0.708227	+	0.705985i	$0.750505\pi$
$642$	0	0
$643$	22.8078	0.899450	0.449725	−	0.893167i	$-0.351522\pi$
$643$	22.8078	0.899450	0.449725	+	0.893167i	$0.351522\pi$
$644$	9.43845	0.371927
$645$	0	0
$646$	−0.315342	−0.0124069
$647$	13.9309	0.547679	0.273840	−	0.961775i	$-0.411706\pi$
$647$	13.9309	0.547679	0.273840	+	0.961775i	$0.411706\pi$
$648$	0	0
$649$	23.3693	0.917326
$650$	−5.68466	−0.222971
$651$	0	0
$652$	7.93087	0.310597
$653$	32.7386	1.28116	0.640581	−	0.767891i	$-0.278694\pi$
$653$	32.7386	1.28116	0.640581	+	0.767891i	$0.278694\pi$
$654$	0	0
$655$	13.7538	0.537405
$656$	−2.87689	−0.112324
$657$	0	0
$658$	16.0000	0.623745
$659$	1.75379	0.0683179	0.0341590	−	0.999416i	$-0.489125\pi$
$659$	1.75379	0.0683179	0.0341590	+	0.999416i	$0.489125\pi$
$660$	0	0
$661$	−15.4384	−0.600486	−0.300243	−	0.953863i	$-0.597068\pi$
$661$	−15.4384	−0.600486	−0.300243	+	0.953863i	$0.597068\pi$
$662$	−1.36932	−0.0532200
$663$	0	0
$664$	−13.9309	−0.540623
$665$	−2.87689	−0.111561
$666$	0	0
$667$	−26.2462	−1.01626
$668$	4.31534	0.166966
$669$	0	0
$670$	−28.4924	−1.10076
$671$	−10.8769	−0.419898
$672$	0	0
$673$	−37.0540	−1.42833	−0.714163	−	0.699980i	$-0.753192\pi$
$673$	−37.0540	−1.42833	−0.714163	+	0.699980i	$0.753192\pi$
$674$	−23.9309	−0.921783
$675$	0	0
$676$	19.3153	0.742898
$677$	26.5616	1.02084	0.510422	−	0.859924i	$-0.329489\pi$
$677$	26.5616	1.02084	0.510422	+	0.859924i	$0.329489\pi$
$678$	0	0
$679$	28.4924	1.09344
$680$	1.12311	0.0430691
$681$	0	0
$682$	−2.24621	−0.0860119
$683$	50.1080	1.91733	0.958664	−	0.284542i	$-0.0918414\pi$
$683$	50.1080	1.91733	0.958664	+	0.284542i	$0.0918414\pi$
$684$	0	0
$685$	−44.4924	−1.69997
$686$	19.0540	0.727484
$687$	0	0
$688$	−7.12311	−0.271566
$689$	14.5616	0.554751
$690$	0	0
$691$	−22.8769	−0.870278	−0.435139	−	0.900363i	$-0.643301\pi$
$691$	−22.8769	−0.870278	−0.435139	+	0.900363i	$0.643301\pi$
$692$	2.56155	0.0973756
$693$	0	0
$694$	−29.6155	−1.12419
$695$	24.0000	0.910372
$696$	0	0
$697$	−1.61553	−0.0611925
$698$	−26.4924	−1.00275
$699$	0	0
$700$	−2.56155	−0.0968176
$701$	−45.3693	−1.71358	−0.856788	−	0.515669i	$-0.827543\pi$
$701$	−45.3693	−1.71358	−0.856788	+	0.515669i	$0.827543\pi$
$702$	0	0
$703$	−0.561553	−0.0211794
$704$	−2.56155	−0.0965422
$705$	0	0
$706$	26.0000	0.978523
$707$	7.36932	0.277152
$708$	0	0
$709$	11.3002	0.424387	0.212194	−	0.977228i	$-0.431939\pi$
$709$	11.3002	0.424387	0.212194	+	0.977228i	$0.431939\pi$
$710$	28.4924	1.06930
$711$	0	0
$712$	15.9309	0.597035
$713$	−3.23106	−0.121004
$714$	0	0
$715$	−29.1231	−1.08914
$716$	1.12311	0.0419724
$717$	0	0
$718$	4.49242	0.167656
$719$	−47.3693	−1.76658	−0.883289	−	0.468829i	$-0.844676\pi$
$719$	−47.3693	−1.76658	−0.883289	+	0.468829i	$0.844676\pi$
$720$	0	0
$721$	−25.6155	−0.953972
$722$	18.6847	0.695371
$723$	0	0
$724$	−18.4924	−0.687265
$725$	7.12311	0.264546
$726$	0	0
$727$	−6.63068	−0.245918	−0.122959	−	0.992412i	$-0.539238\pi$
$727$	−6.63068	−0.245918	−0.122959	+	0.992412i	$0.539238\pi$
$728$	14.5616	0.539687
$729$	0	0
$730$	−1.12311	−0.0415680
$731$	−4.00000	−0.147945
$732$	0	0
$733$	9.86174	0.364252	0.182126	−	0.983275i	$-0.441702\pi$
$733$	9.86174	0.364252	0.182126	+	0.983275i	$0.441702\pi$
$734$	−12.3153	−0.454568
$735$	0	0
$736$	−3.68466	−0.135818
$737$	36.4924	1.34422
$738$	0	0
$739$	6.38447	0.234857	0.117428	−	0.993081i	$-0.462535\pi$
$739$	6.38447	0.234857	0.117428	+	0.993081i	$0.462535\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	6.56155	0.240882
$743$	−52.9848	−1.94383	−0.971913	−	0.235342i	$-0.924379\pi$
$743$	−52.9848	−1.94383	−0.971913	+	0.235342i	$0.924379\pi$
$744$	0	0
$745$	−10.2462	−0.375392
$746$	−2.63068	−0.0963162
$747$	0	0
$748$	−1.43845	−0.0525948
$749$	27.0540	0.988531
$750$	0	0
$751$	43.2311	1.57752	0.788762	−	0.614699i	$-0.210722\pi$
$751$	43.2311	1.57752	0.788762	+	0.614699i	$0.210722\pi$
$752$	−6.24621	−0.227776
$753$	0	0
$754$	−40.4924	−1.47465
$755$	−31.3693	−1.14165
$756$	0	0
$757$	1.68466	0.0612300	0.0306150	−	0.999531i	$-0.490253\pi$
$757$	1.68466	0.0612300	0.0306150	+	0.999531i	$0.490253\pi$
$758$	2.24621	0.0815861
$759$	0	0
$760$	1.12311	0.0407393
$761$	34.1080	1.23641	0.618206	−	0.786016i	$-0.287860\pi$
$761$	34.1080	1.23641	0.618206	+	0.786016i	$0.287860\pi$
$762$	0	0
$763$	−1.43845	−0.0520753
$764$	−6.56155	−0.237389
$765$	0	0
$766$	−8.80776	−0.318237
$767$	51.8617	1.87262
$768$	0	0
$769$	35.6155	1.28433	0.642164	−	0.766567i	$-0.278037\pi$
$769$	35.6155	1.28433	0.642164	+	0.766567i	$0.278037\pi$
$770$	−13.1231	−0.472924
$771$	0	0
$772$	25.3693	0.913062
$773$	4.31534	0.155212	0.0776060	−	0.996984i	$-0.475272\pi$
$773$	4.31534	0.155212	0.0776060	+	0.996984i	$0.475272\pi$
$774$	0	0
$775$	0.876894	0.0314990
$776$	−11.1231	−0.399296
$777$	0	0
$778$	−32.7386	−1.17374
$779$	−1.61553	−0.0578823
$780$	0	0
$781$	−36.4924	−1.30580
$782$	−2.06913	−0.0739919
$783$	0	0
$784$	−0.438447	−0.0156588
$785$	−44.9848	−1.60558
$786$	0	0
$787$	−47.2311	−1.68361	−0.841803	−	0.539785i	$-0.818505\pi$
$787$	−47.2311	−1.68361	−0.841803	+	0.539785i	$0.818505\pi$
$788$	8.80776	0.313764
$789$	0	0
$790$	−12.0000	−0.426941
$791$	−10.8769	−0.386738
$792$	0	0
$793$	−24.1383	−0.857175
$794$	−18.4924	−0.656272
$795$	0	0
$796$	4.87689	0.172857
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	−3.50758	−0.124089
$800$	1.00000	0.0353553
$801$	0	0
$802$	7.43845	0.262661
$803$	1.43845	0.0507617
$804$	0	0
$805$	−18.8769	−0.665323
$806$	−4.98485	−0.175584
$807$	0	0
$808$	−2.87689	−0.101209
$809$	−24.4233	−0.858677	−0.429339	−	0.903144i	$-0.641253\pi$
$809$	−24.4233	−0.858677	−0.429339	+	0.903144i	$0.641253\pi$
$810$	0	0
$811$	−7.36932	−0.258772	−0.129386	−	0.991594i	$-0.541301\pi$
$811$	−7.36932	−0.258772	−0.129386	+	0.991594i	$0.541301\pi$
$812$	−18.2462	−0.640316
$813$	0	0
$814$	−2.56155	−0.0897824
$815$	−15.8617	−0.555612
$816$	0	0
$817$	−4.00000	−0.139942
$818$	−3.75379	−0.131248
$819$	0	0
$820$	5.75379	0.200931
$821$	47.6847	1.66421	0.832103	−	0.554621i	$-0.187137\pi$
$821$	47.6847	1.66421	0.832103	+	0.554621i	$0.187137\pi$
$822$	0	0
$823$	−0.807764	−0.0281569	−0.0140784	−	0.999901i	$-0.504481\pi$
$823$	−0.807764	−0.0281569	−0.0140784	+	0.999901i	$0.504481\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	23.3693	0.813123
$827$	33.7538	1.17373	0.586867	−	0.809683i	$-0.300361\pi$
$827$	33.7538	1.17373	0.586867	+	0.809683i	$0.300361\pi$
$828$	0	0
$829$	−0.561553	−0.0195035	−0.00975177	−	0.999952i	$-0.503104\pi$
$829$	−0.561553	−0.0195035	−0.00975177	+	0.999952i	$0.503104\pi$
$830$	27.8617	0.967095
$831$	0	0
$832$	−5.68466	−0.197080
$833$	−0.246211	−0.00853071
$834$	0	0
$835$	−8.63068	−0.298677
$836$	−1.43845	−0.0497497
$837$	0	0
$838$	35.6847	1.23271
$839$	−22.8769	−0.789798	−0.394899	−	0.918725i	$-0.629220\pi$
$839$	−22.8769	−0.789798	−0.394899	+	0.918725i	$0.629220\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	−3.75379	−0.129364
$843$	0	0
$844$	−4.00000	−0.137686
$845$	−38.6307	−1.32894
$846$	0	0
$847$	−11.3693	−0.390654
$848$	−2.56155	−0.0879641
$849$	0	0
$850$	0.561553	0.0192611
$851$	−3.68466	−0.126308
$852$	0	0
$853$	40.5616	1.38880	0.694401	−	0.719589i	$-0.255670\pi$
$853$	40.5616	1.38880	0.694401	+	0.719589i	$0.255670\pi$
$854$	−10.8769	−0.372200
$855$	0	0
$856$	−10.5616	−0.360986
$857$	34.6695	1.18429	0.592144	−	0.805832i	$-0.298282\pi$
$857$	34.6695	1.18429	0.592144	+	0.805832i	$0.298282\pi$
$858$	0	0
$859$	−37.0540	−1.26427	−0.632133	−	0.774860i	$-0.717820\pi$
$859$	−37.0540	−1.26427	−0.632133	+	0.774860i	$0.717820\pi$
$860$	14.2462	0.485792
$861$	0	0
$862$	8.17708	0.278512
$863$	−32.6307	−1.11076	−0.555381	−	0.831596i	$-0.687427\pi$
$863$	−32.6307	−1.11076	−0.555381	+	0.831596i	$0.687427\pi$
$864$	0	0
$865$	−5.12311	−0.174191
$866$	−24.5616	−0.834636
$867$	0	0
$868$	−2.24621	−0.0762414
$869$	15.3693	0.521368
$870$	0	0
$871$	80.9848	2.74407
$872$	0.561553	0.0190166
$873$	0	0
$874$	−2.06913	−0.0699894
$875$	30.7386	1.03916
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−22.4924	−0.759082
$879$	0	0
$880$	5.12311	0.172700
$881$	16.4924	0.555644	0.277822	−	0.960633i	$-0.410387\pi$
$881$	16.4924	0.555644	0.277822	+	0.960633i	$0.410387\pi$
$882$	0	0
$883$	−32.5616	−1.09578	−0.547892	−	0.836549i	$-0.684569\pi$
$883$	−32.5616	−1.09578	−0.547892	+	0.836549i	$0.684569\pi$
$884$	−3.19224	−0.107367
$885$	0	0
$886$	30.7386	1.03268
$887$	−14.8769	−0.499517	−0.249759	−	0.968308i	$-0.580351\pi$
$887$	−14.8769	−0.499517	−0.249759	+	0.968308i	$0.580351\pi$
$888$	0	0
$889$	3.68466	0.123579
$890$	−31.8617	−1.06801
$891$	0	0
$892$	−4.49242	−0.150417
$893$	−3.50758	−0.117377
$894$	0	0
$895$	−2.24621	−0.0750826
$896$	−2.56155	−0.0855755
$897$	0	0
$898$	15.7538	0.525711
$899$	6.24621	0.208323
$900$	0	0
$901$	−1.43845	−0.0479216
$902$	−7.36932	−0.245371
$903$	0	0
$904$	4.24621	0.141227
$905$	36.9848	1.22942
$906$	0	0
$907$	−15.3002	−0.508034	−0.254017	−	0.967200i	$-0.581752\pi$
$907$	−15.3002	−0.508034	−0.254017	+	0.967200i	$0.581752\pi$
$908$	−19.3693	−0.642793
$909$	0	0
$910$	−29.1231	−0.965422
$911$	−3.50758	−0.116211	−0.0581056	−	0.998310i	$-0.518506\pi$
$911$	−3.50758	−0.116211	−0.0581056	+	0.998310i	$0.518506\pi$
$912$	0	0
$913$	−35.6847	−1.18099
$914$	35.6155	1.17806
$915$	0	0
$916$	2.63068	0.0869202
$917$	−17.6155	−0.581716
$918$	0	0
$919$	27.7538	0.915513	0.457757	−	0.889078i	$-0.348653\pi$
$919$	27.7538	0.915513	0.457757	+	0.889078i	$0.348653\pi$
$920$	7.36932	0.242959
$921$	0	0
$922$	38.9848	1.28390
$923$	−80.9848	−2.66565
$924$	0	0
$925$	1.00000	0.0328798
$926$	−13.3693	−0.439343
$927$	0	0
$928$	7.12311	0.233827
$929$	−29.7538	−0.976190	−0.488095	−	0.872790i	$-0.662308\pi$
$929$	−29.7538	−0.976190	−0.488095	+	0.872790i	$0.662308\pi$
$930$	0	0
$931$	−0.246211	−0.00806925
$932$	16.0000	0.524097
$933$	0	0
$934$	21.6155	0.707282
$935$	2.87689	0.0940845
$936$	0	0
$937$	−15.7538	−0.514654	−0.257327	−	0.966324i	$-0.582842\pi$
$937$	−15.7538	−0.514654	−0.257327	+	0.966324i	$0.582842\pi$
$938$	36.4924	1.19152
$939$	0	0
$940$	12.4924	0.407458
$941$	−7.86174	−0.256285	−0.128143	−	0.991756i	$-0.540902\pi$
$941$	−7.86174	−0.256285	−0.128143	+	0.991756i	$0.540902\pi$
$942$	0	0
$943$	−10.6004	−0.345196
$944$	−9.12311	−0.296932
$945$	0	0
$946$	−18.2462	−0.593235
$947$	−19.3693	−0.629418	−0.314709	−	0.949188i	$-0.601907\pi$
$947$	−19.3693	−0.629418	−0.314709	+	0.949188i	$0.601907\pi$
$948$	0	0
$949$	3.19224	0.103624
$950$	0.561553	0.0182192
$951$	0	0
$952$	−1.43845	−0.0466203
$953$	7.86174	0.254667	0.127333	−	0.991860i	$-0.459358\pi$
$953$	7.86174	0.254667	0.127333	+	0.991860i	$0.459358\pi$
$954$	0	0
$955$	13.1231	0.424654
$956$	−8.00000	−0.258738
$957$	0	0
$958$	4.94602	0.159799
$959$	56.9848	1.84014
$960$	0	0
$961$	−30.2311	−0.975195
$962$	−5.68466	−0.183281
$963$	0	0
$964$	8.24621	0.265593
$965$	−50.7386	−1.63333
$966$	0	0
$967$	21.3693	0.687191	0.343595	−	0.939118i	$-0.388355\pi$
$967$	21.3693	0.687191	0.343595	+	0.939118i	$0.388355\pi$
$968$	4.43845	0.142657
$969$	0	0
$970$	22.2462	0.714283
$971$	−26.2462	−0.842281	−0.421141	−	0.906995i	$-0.638370\pi$
$971$	−26.2462	−0.842281	−0.421141	+	0.906995i	$0.638370\pi$
$972$	0	0
$973$	−30.7386	−0.985435
$974$	−4.87689	−0.156266
$975$	0	0
$976$	4.24621	0.135918
$977$	−60.9157	−1.94887	−0.974433	−	0.224677i	$-0.927867\pi$
$977$	−60.9157	−1.94887	−0.974433	+	0.224677i	$0.927867\pi$
$978$	0	0
$979$	40.8078	1.30422
$980$	0.876894	0.0280114
$981$	0	0
$982$	−29.9309	−0.955132
$983$	−21.6155	−0.689428	−0.344714	−	0.938708i	$-0.612024\pi$
$983$	−21.6155	−0.689428	−0.344714	+	0.938708i	$0.612024\pi$
$984$	0	0
$985$	−17.6155	−0.561277
$986$	4.00000	0.127386
$987$	0	0
$988$	−3.19224	−0.101559
$989$	−26.2462	−0.834581
$990$	0	0
$991$	11.2614	0.357729	0.178865	−	0.983874i	$-0.442758\pi$
$991$	11.2614	0.357729	0.178865	+	0.983874i	$0.442758\pi$
$992$	0.876894	0.0278414
$993$	0	0
$994$	−36.4924	−1.15747
$995$	−9.75379	−0.309216
$996$	0	0
$997$	−8.06913	−0.255552	−0.127776	−	0.991803i	$-0.540784\pi$
$997$	−8.06913	−0.255552	−0.127776	+	0.991803i	$0.540784\pi$
$998$	−28.4233	−0.899724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	666.2.a.h.1.2	✓	2
3.2	odd	2		666.2.a.k.1.2	yes	2
4.3	odd	2		5328.2.a.y.1.1		2
12.11	even	2		5328.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
666.2.a.h.1.2	✓	2	1.1	even	1	trivial
666.2.a.k.1.2	yes	2	3.2	odd	2
5328.2.a.y.1.1		2	4.3	odd	2
5328.2.a.bh.1.1		2	12.11	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:	\( N \)	\(=\)	\( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	666.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +2.56155 q^{7} -1.00000 q^{8} +2.00000 q^{10} -2.56155 q^{11} -5.68466 q^{13} -2.56155 q^{14} +1.00000 q^{16} +0.561553 q^{17} +0.561553 q^{19} -2.00000 q^{20} +2.56155 q^{22} +3.68466 q^{23} -1.00000 q^{25} +5.68466 q^{26} +2.56155 q^{28} -7.12311 q^{29} -0.876894 q^{31} -1.00000 q^{32} -0.561553 q^{34} -5.12311 q^{35} -1.00000 q^{37} -0.561553 q^{38} +2.00000 q^{40} -2.87689 q^{41} -7.12311 q^{43} -2.56155 q^{44} -3.68466 q^{46} -6.24621 q^{47} -0.438447 q^{49} +1.00000 q^{50} -5.68466 q^{52} -2.56155 q^{53} +5.12311 q^{55} -2.56155 q^{56} +7.12311 q^{58} -9.12311 q^{59} +4.24621 q^{61} +0.876894 q^{62} +1.00000 q^{64} +11.3693 q^{65} -14.2462 q^{67} +0.561553 q^{68} +5.12311 q^{70} +14.2462 q^{71} -0.561553 q^{73} +1.00000 q^{74} +0.561553 q^{76} -6.56155 q^{77} -6.00000 q^{79} -2.00000 q^{80} +2.87689 q^{82} +13.9309 q^{83} -1.12311 q^{85} +7.12311 q^{86} +2.56155 q^{88} -15.9309 q^{89} -14.5616 q^{91} +3.68466 q^{92} +6.24621 q^{94} -1.12311 q^{95} +11.1231 q^{97} +0.438447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7} - 2 q^{8} + 4 q^{10} - q^{11} + q^{13} - q^{14} + 2 q^{16} - 3 q^{17} - 3 q^{19} - 4 q^{20} + q^{22} - 5 q^{23} - 2 q^{25} - q^{26} + q^{28} - 6 q^{29} - 10 q^{31}+ \cdots + 5 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + q^7 - 2 * q^8 + 4 * q^10 - q^11 + q^13 - q^14 + 2 * q^16 - 3 * q^17 - 3 * q^19 - 4 * q^20 + q^22 - 5 * q^23 - 2 * q^25 - q^26 + q^28 - 6 * q^29 - 10 * q^31 - 2 * q^32 + 3 * q^34 - 2 * q^35 - 2 * q^37 + 3 * q^38 + 4 * q^40 - 14 * q^41 - 6 * q^43 - q^44 + 5 * q^46 + 4 * q^47 - 5 * q^49 + 2 * q^50 + q^52 - q^53 + 2 * q^55 - q^56 + 6 * q^58 - 10 * q^59 - 8 * q^61 + 10 * q^62 + 2 * q^64 - 2 * q^65 - 12 * q^67 - 3 * q^68 + 2 * q^70 + 12 * q^71 + 3 * q^73 + 2 * q^74 - 3 * q^76 - 9 * q^77 - 12 * q^79 - 4 * q^80 + 14 * q^82 - q^83 + 6 * q^85 + 6 * q^86 + q^88 - 3 * q^89 - 25 * q^91 - 5 * q^92 - 4 * q^94 + 6 * q^95 + 14 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

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Database

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