LMFDB - Embedded newform 8664.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8664,2,Mod(1,8664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8664, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8664 = 2^{3} \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8664.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:	$69.1823883112$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	8664.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.75877 q^{5} +4.75877 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.75877 q^{5} +4.75877 q^{7} +1.00000 q^{9} -4.36959 q^{11} -1.00000 q^{13} -3.75877 q^{15} +6.12836 q^{17} +4.75877 q^{21} -3.75877 q^{23} +9.12836 q^{25} +1.00000 q^{27} -5.38919 q^{29} -7.36959 q^{31} -4.36959 q^{33} -17.8871 q^{35} +7.12836 q^{37} -1.00000 q^{39} +3.51754 q^{41} -0.758770 q^{43} -3.75877 q^{45} -6.00000 q^{47} +15.6459 q^{49} +6.12836 q^{51} -5.14796 q^{53} +16.4243 q^{55} -9.27631 q^{59} -8.51754 q^{61} +4.75877 q^{63} +3.75877 q^{65} -1.36959 q^{67} -3.75877 q^{69} +11.6459 q^{71} +4.51754 q^{73} +9.12836 q^{75} -20.7939 q^{77} -14.1480 q^{79} +1.00000 q^{81} +4.90673 q^{83} -23.0351 q^{85} -5.38919 q^{87} -8.36959 q^{89} -4.75877 q^{91} -7.36959 q^{93} +11.6459 q^{97} -4.36959 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 6 q^{11} - 3 q^{13} + 3 q^{21} + 9 q^{25} + 3 q^{27} - 12 q^{29} - 15 q^{31} - 6 q^{33} - 24 q^{35} + 3 q^{37} - 3 q^{39} - 12 q^{41} + 9 q^{43} - 18 q^{47} + 6 q^{49} + 6 q^{59} - 3 q^{61} + 3 q^{63} + 3 q^{67} - 6 q^{71} - 9 q^{73} + 9 q^{75} - 6 q^{77} - 27 q^{79} + 3 q^{81} - 12 q^{83} - 24 q^{85} - 12 q^{87} - 18 q^{89} - 3 q^{91} - 15 q^{93} - 6 q^{97} - 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 - 6 * q^11 - 3 * q^13 + 3 * q^21 + 9 * q^25 + 3 * q^27 - 12 * q^29 - 15 * q^31 - 6 * q^33 - 24 * q^35 + 3 * q^37 - 3 * q^39 - 12 * q^41 + 9 * q^43 - 18 * q^47 + 6 * q^49 + 6 * q^59 - 3 * q^61 + 3 * q^63 + 3 * q^67 - 6 * q^71 - 9 * q^73 + 9 * q^75 - 6 * q^77 - 27 * q^79 + 3 * q^81 - 12 * q^83 - 24 * q^85 - 12 * q^87 - 18 * q^89 - 3 * q^91 - 15 * q^93 - 6 * q^97 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.75877	−1.68097	−0.840487	−	0.541832i	$-0.817731\pi$
$5$	−3.75877	−1.68097	−0.840487	+	0.541832i	$0.817731\pi$
$6$	0	0
$7$	4.75877	1.79865	0.899323	−	0.437285i	$-0.144060\pi$
$7$	4.75877	1.79865	0.899323	+	0.437285i	$0.144060\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.36959	−1.31748	−0.658740	−	0.752371i	$-0.728910\pi$
$11$	−4.36959	−1.31748	−0.658740	+	0.752371i	$0.728910\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	−3.75877	−0.970510
$16$	0	0
$17$	6.12836	1.48634	0.743172	−	0.669100i	$-0.233320\pi$
$17$	6.12836	1.48634	0.743172	+	0.669100i	$0.233320\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	4.75877	1.03845
$22$	0	0
$23$	−3.75877	−0.783758	−0.391879	−	0.920017i	$-0.628175\pi$
$23$	−3.75877	−0.783758	−0.391879	+	0.920017i	$0.628175\pi$
$24$	0	0
$25$	9.12836	1.82567
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.38919	−1.00075	−0.500373	−	0.865810i	$-0.666804\pi$
$29$	−5.38919	−1.00075	−0.500373	+	0.865810i	$0.666804\pi$
$30$	0	0
$31$	−7.36959	−1.32362	−0.661808	−	0.749673i	$-0.730211\pi$
$31$	−7.36959	−1.32362	−0.661808	+	0.749673i	$0.730211\pi$
$32$	0	0
$33$	−4.36959	−0.760647
$34$	0	0
$35$	−17.8871	−3.02348
$36$	0	0
$37$	7.12836	1.17189	0.585947	−	0.810349i	$-0.300723\pi$
$37$	7.12836	1.17189	0.585947	+	0.810349i	$0.300723\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.51754	0.549348	0.274674	−	0.961537i	$-0.411430\pi$
$41$	3.51754	0.549348	0.274674	+	0.961537i	$0.411430\pi$
$42$	0	0
$43$	−0.758770	−0.115711	−0.0578557	−	0.998325i	$-0.518426\pi$
$43$	−0.758770	−0.115711	−0.0578557	+	0.998325i	$0.518426\pi$
$44$	0	0
$45$	−3.75877	−0.560324
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	15.6459	2.23513
$50$	0	0
$51$	6.12836	0.858141
$52$	0	0
$53$	−5.14796	−0.707126	−0.353563	−	0.935411i	$-0.615030\pi$
$53$	−5.14796	−0.707126	−0.353563	+	0.935411i	$0.615030\pi$
$54$	0	0
$55$	16.4243	2.21465
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.27631	−1.20767	−0.603836	−	0.797108i	$-0.706362\pi$
$59$	−9.27631	−1.20767	−0.603836	+	0.797108i	$0.706362\pi$
$60$	0	0
$61$	−8.51754	−1.09056	−0.545280	−	0.838254i	$-0.683577\pi$
$61$	−8.51754	−1.09056	−0.545280	+	0.838254i	$0.683577\pi$
$62$	0	0
$63$	4.75877	0.599549
$64$	0	0
$65$	3.75877	0.466218
$66$	0	0
$67$	−1.36959	−0.167321	−0.0836607	−	0.996494i	$-0.526661\pi$
$67$	−1.36959	−0.167321	−0.0836607	+	0.996494i	$0.526661\pi$
$68$	0	0
$69$	−3.75877	−0.452503
$70$	0	0
$71$	11.6459	1.38211	0.691057	−	0.722800i	$-0.257145\pi$
$71$	11.6459	1.38211	0.691057	+	0.722800i	$0.257145\pi$
$72$	0	0
$73$	4.51754	0.528738	0.264369	−	0.964422i	$-0.414836\pi$
$73$	4.51754	0.528738	0.264369	+	0.964422i	$0.414836\pi$
$74$	0	0
$75$	9.12836	1.05405
$76$	0	0
$77$	−20.7939	−2.36968
$78$	0	0
$79$	−14.1480	−1.59177	−0.795885	−	0.605448i	$-0.792994\pi$
$79$	−14.1480	−1.59177	−0.795885	+	0.605448i	$0.792994\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.90673	0.538583	0.269292	−	0.963059i	$-0.413210\pi$
$83$	4.90673	0.538583	0.269292	+	0.963059i	$0.413210\pi$
$84$	0	0
$85$	−23.0351	−2.49851
$86$	0	0
$87$	−5.38919	−0.577781
$88$	0	0
$89$	−8.36959	−0.887174	−0.443587	−	0.896231i	$-0.646294\pi$
$89$	−8.36959	−0.887174	−0.443587	+	0.896231i	$0.646294\pi$
$90$	0	0
$91$	−4.75877	−0.498855
$92$	0	0
$93$	−7.36959	−0.764190
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.6459	1.18246	0.591231	−	0.806502i	$-0.298642\pi$
$97$	11.6459	1.18246	0.591231	+	0.806502i	$0.298642\pi$
$98$	0	0
$99$	−4.36959	−0.439160
$100$	0	0
$101$	−4.12836	−0.410787	−0.205393	−	0.978680i	$-0.565847\pi$
$101$	−4.12836	−0.410787	−0.205393	+	0.978680i	$0.565847\pi$
$102$	0	0
$103$	6.75877	0.665961	0.332981	−	0.942934i	$-0.391946\pi$
$103$	6.75877	0.665961	0.332981	+	0.942934i	$0.391946\pi$
$104$	0	0
$105$	−17.8871	−1.74560
$106$	0	0
$107$	4.12836	0.399103	0.199552	−	0.979887i	$-0.436051\pi$
$107$	4.12836	0.399103	0.199552	+	0.979887i	$0.436051\pi$
$108$	0	0
$109$	0.128356	0.0122942	0.00614712	−	0.999981i	$-0.498043\pi$
$109$	0.128356	0.0122942	0.00614712	+	0.999981i	$0.498043\pi$
$110$	0	0
$111$	7.12836	0.676594
$112$	0	0
$113$	2.40879	0.226600	0.113300	−	0.993561i	$-0.463858\pi$
$113$	2.40879	0.226600	0.113300	+	0.993561i	$0.463858\pi$
$114$	0	0
$115$	14.1284	1.31748
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	29.1634	2.67341
$120$	0	0
$121$	8.09327	0.735752
$122$	0	0
$123$	3.51754	0.317166
$124$	0	0
$125$	−15.5175	−1.38793
$126$	0	0
$127$	−0.482459	−0.0428113	−0.0214057	−	0.999771i	$-0.506814\pi$
$127$	−0.482459	−0.0428113	−0.0214057	+	0.999771i	$0.506814\pi$
$128$	0	0
$129$	−0.758770	−0.0668060
$130$	0	0
$131$	−3.38919	−0.296115	−0.148057	−	0.988979i	$-0.547302\pi$
$131$	−3.38919	−0.296115	−0.148057	+	0.988979i	$0.547302\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.75877	−0.323503
$136$	0	0
$137$	−18.4243	−1.57409	−0.787046	−	0.616895i	$-0.788390\pi$
$137$	−18.4243	−1.57409	−0.787046	+	0.616895i	$0.788390\pi$
$138$	0	0
$139$	0.0196004	0.00166248	0.000831240	−	1.00000i	$-0.499735\pi$
$139$	0.0196004	0.00166248	0.000831240		1.00000i	$0.499735\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	4.36959	0.365403
$144$	0	0
$145$	20.2567	1.68223
$146$	0	0
$147$	15.6459	1.29045
$148$	0	0
$149$	6.24123	0.511301	0.255651	−	0.966769i	$-0.417710\pi$
$149$	6.24123	0.511301	0.255651	+	0.966769i	$0.417710\pi$
$150$	0	0
$151$	−0.739170	−0.0601528	−0.0300764	−	0.999548i	$-0.509575\pi$
$151$	−0.739170	−0.0601528	−0.0300764	+	0.999548i	$0.509575\pi$
$152$	0	0
$153$	6.12836	0.495448
$154$	0	0
$155$	27.7006	2.22496
$156$	0	0
$157$	−6.64590	−0.530400	−0.265200	−	0.964193i	$-0.585438\pi$
$157$	−6.64590	−0.530400	−0.265200	+	0.964193i	$0.585438\pi$
$158$	0	0
$159$	−5.14796	−0.408259
$160$	0	0
$161$	−17.8871	−1.40970
$162$	0	0
$163$	−11.3696	−0.890535	−0.445267	−	0.895398i	$-0.646891\pi$
$163$	−11.3696	−0.890535	−0.445267	+	0.895398i	$0.646891\pi$
$164$	0	0
$165$	16.4243	1.27863
$166$	0	0
$167$	−2.98040	−0.230630	−0.115315	−	0.993329i	$-0.536788\pi$
$167$	−2.98040	−0.230630	−0.115315	+	0.993329i	$0.536788\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.38919	0.409732	0.204866	−	0.978790i	$-0.434324\pi$
$173$	5.38919	0.409732	0.204866	+	0.978790i	$0.434324\pi$
$174$	0	0
$175$	43.4397	3.28374
$176$	0	0
$177$	−9.27631	−0.697250
$178$	0	0
$179$	−4.24123	−0.317004	−0.158502	−	0.987359i	$-0.550666\pi$
$179$	−4.24123	−0.317004	−0.158502	+	0.987359i	$0.550666\pi$
$180$	0	0
$181$	−12.6108	−0.937354	−0.468677	−	0.883369i	$-0.655269\pi$
$181$	−12.6108	−0.937354	−0.468677	+	0.883369i	$0.655269\pi$
$182$	0	0
$183$	−8.51754	−0.629635
$184$	0	0
$185$	−26.7939	−1.96992
$186$	0	0
$187$	−26.7784	−1.95823
$188$	0	0
$189$	4.75877	0.346150
$190$	0	0
$191$	−22.6263	−1.63718	−0.818591	−	0.574377i	$-0.805244\pi$
$191$	−22.6263	−1.63718	−0.818591	+	0.574377i	$0.805244\pi$
$192$	0	0
$193$	−24.6459	−1.77405	−0.887025	−	0.461721i	$-0.847232\pi$
$193$	−24.6459	−1.77405	−0.887025	+	0.461721i	$0.847232\pi$
$194$	0	0
$195$	3.75877	0.269171
$196$	0	0
$197$	22.6655	1.61485	0.807425	−	0.589970i	$-0.200861\pi$
$197$	22.6655	1.61485	0.807425	+	0.589970i	$0.200861\pi$
$198$	0	0
$199$	13.9222	0.986919	0.493460	−	0.869769i	$-0.335732\pi$
$199$	13.9222	0.986919	0.493460	+	0.869769i	$0.335732\pi$
$200$	0	0
$201$	−1.36959	−0.0966031
$202$	0	0
$203$	−25.6459	−1.79999
$204$	0	0
$205$	−13.2216	−0.923439
$206$	0	0
$207$	−3.75877	−0.261253
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2.01960	0.139035	0.0695175	−	0.997581i	$-0.477854\pi$
$211$	2.01960	0.139035	0.0695175	+	0.997581i	$0.477854\pi$
$212$	0	0
$213$	11.6459	0.797964
$214$	0	0
$215$	2.85204	0.194508
$216$	0	0
$217$	−35.0702	−2.38072
$218$	0	0
$219$	4.51754	0.305267
$220$	0	0
$221$	−6.12836	−0.412238
$222$	0	0
$223$	−28.3655	−1.89949	−0.949746	−	0.313021i	$-0.898659\pi$
$223$	−28.3655	−1.89949	−0.949746	+	0.313021i	$0.898659\pi$
$224$	0	0
$225$	9.12836	0.608557
$226$	0	0
$227$	−20.1830	−1.33960	−0.669798	−	0.742544i	$-0.733619\pi$
$227$	−20.1830	−1.33960	−0.669798	+	0.742544i	$0.733619\pi$
$228$	0	0
$229$	11.7784	0.778337	0.389168	−	0.921167i	$-0.372762\pi$
$229$	11.7784	0.778337	0.389168	+	0.921167i	$0.372762\pi$
$230$	0	0
$231$	−20.7939	−1.36814
$232$	0	0
$233$	−0.128356	−0.00840885	−0.00420443	−	0.999991i	$-0.501338\pi$
$233$	−0.128356	−0.00840885	−0.00420443	+	0.999991i	$0.501338\pi$
$234$	0	0
$235$	22.5526	1.47117
$236$	0	0
$237$	−14.1480	−0.919008
$238$	0	0
$239$	3.10876	0.201089	0.100544	−	0.994933i	$-0.467942\pi$
$239$	3.10876	0.201089	0.100544	+	0.994933i	$0.467942\pi$
$240$	0	0
$241$	−11.2959	−0.727634	−0.363817	−	0.931471i	$-0.618527\pi$
$241$	−11.2959	−0.727634	−0.363817	+	0.931471i	$0.618527\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−58.8093	−3.75719
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4.90673	0.310951
$250$	0	0
$251$	−1.51754	−0.0957863	−0.0478932	−	0.998852i	$-0.515251\pi$
$251$	−1.51754	−0.0957863	−0.0478932	+	0.998852i	$0.515251\pi$
$252$	0	0
$253$	16.4243	1.03258
$254$	0	0
$255$	−23.0351	−1.44251
$256$	0	0
$257$	12.1830	0.759957	0.379979	−	0.924995i	$-0.375931\pi$
$257$	12.1830	0.759957	0.379979	+	0.924995i	$0.375931\pi$
$258$	0	0
$259$	33.9222	2.10782
$260$	0	0
$261$	−5.38919	−0.333582
$262$	0	0
$263$	−19.3892	−1.19559	−0.597794	−	0.801650i	$-0.703956\pi$
$263$	−19.3892	−1.19559	−0.597794	+	0.801650i	$0.703956\pi$
$264$	0	0
$265$	19.3500	1.18866
$266$	0	0
$267$	−8.36959	−0.512210
$268$	0	0
$269$	−23.4047	−1.42701	−0.713504	−	0.700651i	$-0.752893\pi$
$269$	−23.4047	−1.42701	−0.713504	+	0.700651i	$0.752893\pi$
$270$	0	0
$271$	0.964918	0.0586146	0.0293073	−	0.999570i	$-0.490670\pi$
$271$	0.964918	0.0586146	0.0293073	+	0.999570i	$0.490670\pi$
$272$	0	0
$273$	−4.75877	−0.288014
$274$	0	0
$275$	−39.8871	−2.40528
$276$	0	0
$277$	15.3892	0.924647	0.462323	−	0.886711i	$-0.347016\pi$
$277$	15.3892	0.924647	0.462323	+	0.886711i	$0.347016\pi$
$278$	0	0
$279$	−7.36959	−0.441206
$280$	0	0
$281$	−8.49794	−0.506945	−0.253472	−	0.967343i	$-0.581573\pi$
$281$	−8.49794	−0.506945	−0.253472	+	0.967343i	$0.581573\pi$
$282$	0	0
$283$	−11.0351	−0.655968	−0.327984	−	0.944683i	$-0.606369\pi$
$283$	−11.0351	−0.655968	−0.327984	+	0.944683i	$0.606369\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.7392	0.988082
$288$	0	0
$289$	20.5567	1.20922
$290$	0	0
$291$	11.6459	0.682695
$292$	0	0
$293$	1.03508	0.0604701	0.0302351	−	0.999543i	$-0.490374\pi$
$293$	1.03508	0.0604701	0.0302351	+	0.999543i	$0.490374\pi$
$294$	0	0
$295$	34.8675	2.03007
$296$	0	0
$297$	−4.36959	−0.253549
$298$	0	0
$299$	3.75877	0.217375
$300$	0	0
$301$	−3.61081	−0.208124
$302$	0	0
$303$	−4.12836	−0.237168
$304$	0	0
$305$	32.0155	1.83320
$306$	0	0
$307$	−14.8675	−0.848535	−0.424267	−	0.905537i	$-0.639468\pi$
$307$	−14.8675	−0.848535	−0.424267	+	0.905537i	$0.639468\pi$
$308$	0	0
$309$	6.75877	0.384493
$310$	0	0
$311$	−16.0547	−0.910378	−0.455189	−	0.890395i	$-0.650428\pi$
$311$	−16.0547	−0.910378	−0.455189	+	0.890395i	$0.650428\pi$
$312$	0	0
$313$	35.1634	1.98755	0.993777	−	0.111383i	$-0.0355282\pi$
$313$	35.1634	1.98755	0.993777	+	0.111383i	$0.0355282\pi$
$314$	0	0
$315$	−17.8871	−1.00783
$316$	0	0
$317$	−18.5371	−1.04115	−0.520575	−	0.853816i	$-0.674282\pi$
$317$	−18.5371	−1.04115	−0.520575	+	0.853816i	$0.674282\pi$
$318$	0	0
$319$	23.5485	1.31846
$320$	0	0
$321$	4.12836	0.230422
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−9.12836	−0.506350
$326$	0	0
$327$	0.128356	0.00709808
$328$	0	0
$329$	−28.5526	−1.57416
$330$	0	0
$331$	8.75877	0.481426	0.240713	−	0.970596i	$-0.422619\pi$
$331$	8.75877	0.481426	0.240713	+	0.970596i	$0.422619\pi$
$332$	0	0
$333$	7.12836	0.390631
$334$	0	0
$335$	5.14796	0.281263
$336$	0	0
$337$	21.3851	1.16492	0.582459	−	0.812860i	$-0.302090\pi$
$337$	21.3851	1.16492	0.582459	+	0.812860i	$0.302090\pi$
$338$	0	0
$339$	2.40879	0.130827
$340$	0	0
$341$	32.2020	1.74384
$342$	0	0
$343$	41.1438	2.22156
$344$	0	0
$345$	14.1284	0.760645
$346$	0	0
$347$	23.9263	1.28443	0.642216	−	0.766524i	$-0.278015\pi$
$347$	23.9263	1.28443	0.642216	+	0.766524i	$0.278015\pi$
$348$	0	0
$349$	−23.0702	−1.23492	−0.617459	−	0.786603i	$-0.711838\pi$
$349$	−23.0702	−1.23492	−0.617459	+	0.786603i	$0.711838\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−28.4979	−1.51679	−0.758396	−	0.651794i	$-0.774017\pi$
$353$	−28.4979	−1.51679	−0.758396	+	0.651794i	$0.774017\pi$
$354$	0	0
$355$	−43.7743	−2.32330
$356$	0	0
$357$	29.1634	1.54349
$358$	0	0
$359$	4.96492	0.262038	0.131019	−	0.991380i	$-0.458175\pi$
$359$	4.96492	0.262038	0.131019	+	0.991380i	$0.458175\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	8.09327	0.424787
$364$	0	0
$365$	−16.9804	−0.888795
$366$	0	0
$367$	18.8479	0.983854	0.491927	−	0.870637i	$-0.336293\pi$
$367$	18.8479	0.983854	0.491927	+	0.870637i	$0.336293\pi$
$368$	0	0
$369$	3.51754	0.183116
$370$	0	0
$371$	−24.4979	−1.27187
$372$	0	0
$373$	−35.9026	−1.85897	−0.929483	−	0.368864i	$-0.879747\pi$
$373$	−35.9026	−1.85897	−0.929483	+	0.368864i	$0.879747\pi$
$374$	0	0
$375$	−15.5175	−0.801322
$376$	0	0
$377$	5.38919	0.277557
$378$	0	0
$379$	12.2763	0.630592	0.315296	−	0.948993i	$-0.397896\pi$
$379$	12.2763	0.630592	0.315296	+	0.948993i	$0.397896\pi$
$380$	0	0
$381$	−0.482459	−0.0247171
$382$	0	0
$383$	−14.4088	−0.736255	−0.368127	−	0.929775i	$-0.620001\pi$
$383$	−14.4088	−0.736255	−0.368127	+	0.929775i	$0.620001\pi$
$384$	0	0
$385$	78.1593	3.98337
$386$	0	0
$387$	−0.758770	−0.0385705
$388$	0	0
$389$	6.36959	0.322951	0.161475	−	0.986877i	$-0.448375\pi$
$389$	6.36959	0.322951	0.161475	+	0.986877i	$0.448375\pi$
$390$	0	0
$391$	−23.0351	−1.16493
$392$	0	0
$393$	−3.38919	−0.170962
$394$	0	0
$395$	53.1789	2.67572
$396$	0	0
$397$	−10.5567	−0.529828	−0.264914	−	0.964272i	$-0.585344\pi$
$397$	−10.5567	−0.529828	−0.264914	+	0.964272i	$0.585344\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.4439	−1.07086	−0.535428	−	0.844581i	$-0.679850\pi$
$401$	−21.4439	−1.07086	−0.535428	+	0.844581i	$0.679850\pi$
$402$	0	0
$403$	7.36959	0.367105
$404$	0	0
$405$	−3.75877	−0.186775
$406$	0	0
$407$	−31.1480	−1.54395
$408$	0	0
$409$	−6.65002	−0.328822	−0.164411	−	0.986392i	$-0.552572\pi$
$409$	−6.65002	−0.328822	−0.164411	+	0.986392i	$0.552572\pi$
$410$	0	0
$411$	−18.4243	−0.908802
$412$	0	0
$413$	−44.1438	−2.17218
$414$	0	0
$415$	−18.4433	−0.905344
$416$	0	0
$417$	0.0196004	0.000959834 0
$418$	0	0
$419$	5.84793	0.285690	0.142845	−	0.989745i	$-0.454375\pi$
$419$	5.84793	0.285690	0.142845	+	0.989745i	$0.454375\pi$
$420$	0	0
$421$	−1.42839	−0.0696153	−0.0348076	−	0.999394i	$-0.511082\pi$
$421$	−1.42839	−0.0696153	−0.0348076	+	0.999394i	$0.511082\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	55.9418	2.71358
$426$	0	0
$427$	−40.5330	−1.96153
$428$	0	0
$429$	4.36959	0.210966
$430$	0	0
$431$	−0.778371	−0.0374928	−0.0187464	−	0.999824i	$-0.505968\pi$
$431$	−0.778371	−0.0374928	−0.0187464	+	0.999824i	$0.505968\pi$
$432$	0	0
$433$	−26.6851	−1.28240	−0.641202	−	0.767372i	$-0.721564\pi$
$433$	−26.6851	−1.28240	−0.641202	+	0.767372i	$0.721564\pi$
$434$	0	0
$435$	20.2567	0.971235
$436$	0	0
$437$	0	0
$438$	0	0
$439$	28.5330	1.36181	0.680903	−	0.732373i	$-0.261587\pi$
$439$	28.5330	1.36181	0.680903	+	0.732373i	$0.261587\pi$
$440$	0	0
$441$	15.6459	0.745043
$442$	0	0
$443$	−17.6459	−0.838382	−0.419191	−	0.907898i	$-0.637686\pi$
$443$	−17.6459	−0.838382	−0.419191	+	0.907898i	$0.637686\pi$
$444$	0	0
$445$	31.4593	1.49132
$446$	0	0
$447$	6.24123	0.295200
$448$	0	0
$449$	1.38919	0.0655597	0.0327799	−	0.999463i	$-0.489564\pi$
$449$	1.38919	0.0655597	0.0327799	+	0.999463i	$0.489564\pi$
$450$	0	0
$451$	−15.3702	−0.723754
$452$	0	0
$453$	−0.739170	−0.0347292
$454$	0	0
$455$	17.8871	0.838561
$456$	0	0
$457$	24.6851	1.15472	0.577360	−	0.816490i	$-0.304083\pi$
$457$	24.6851	1.15472	0.577360	+	0.816490i	$0.304083\pi$
$458$	0	0
$459$	6.12836	0.286047
$460$	0	0
$461$	1.59121	0.0741102	0.0370551	−	0.999313i	$-0.488202\pi$
$461$	1.59121	0.0741102	0.0370551	+	0.999313i	$0.488202\pi$
$462$	0	0
$463$	−21.7547	−1.01102	−0.505512	−	0.862819i	$-0.668696\pi$
$463$	−21.7547	−1.01102	−0.505512	+	0.862819i	$0.668696\pi$
$464$	0	0
$465$	27.7006	1.28458
$466$	0	0
$467$	15.5175	0.718066	0.359033	−	0.933325i	$-0.383107\pi$
$467$	15.5175	0.718066	0.359033	+	0.933325i	$0.383107\pi$
$468$	0	0
$469$	−6.51754	−0.300952
$470$	0	0
$471$	−6.64590	−0.306227
$472$	0	0
$473$	3.31551	0.152447
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.14796	−0.235709
$478$	0	0
$479$	−11.6459	−0.532115	−0.266057	−	0.963957i	$-0.585721\pi$
$479$	−11.6459	−0.532115	−0.266057	+	0.963957i	$0.585721\pi$
$480$	0	0
$481$	−7.12836	−0.325025
$482$	0	0
$483$	−17.8871	−0.813892
$484$	0	0
$485$	−43.7743	−1.98769
$486$	0	0
$487$	−38.5526	−1.74699	−0.873493	−	0.486837i	$-0.838151\pi$
$487$	−38.5526	−1.74699	−0.873493	+	0.486837i	$0.838151\pi$
$488$	0	0
$489$	−11.3696	−0.514150
$490$	0	0
$491$	7.64590	0.345054	0.172527	−	0.985005i	$-0.444807\pi$
$491$	7.64590	0.345054	0.172527	+	0.985005i	$0.444807\pi$
$492$	0	0
$493$	−33.0268	−1.48745
$494$	0	0
$495$	16.4243	0.738216
$496$	0	0
$497$	55.4201	2.48593
$498$	0	0
$499$	30.6304	1.37121	0.685603	−	0.727976i	$-0.259539\pi$
$499$	30.6304	1.37121	0.685603	+	0.727976i	$0.259539\pi$
$500$	0	0
$501$	−2.98040	−0.133154
$502$	0	0
$503$	−5.38919	−0.240292	−0.120146	−	0.992756i	$-0.538336\pi$
$503$	−5.38919	−0.240292	−0.120146	+	0.992756i	$0.538336\pi$
$504$	0	0
$505$	15.5175	0.690522
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	2.25671	0.100027	0.0500135	−	0.998749i	$-0.484074\pi$
$509$	2.25671	0.100027	0.0500135	+	0.998749i	$0.484074\pi$
$510$	0	0
$511$	21.4979	0.951013
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−25.4047	−1.11946
$516$	0	0
$517$	26.2175	1.15304
$518$	0	0
$519$	5.38919	0.236559
$520$	0	0
$521$	−37.2763	−1.63310	−0.816552	−	0.577271i	$-0.804118\pi$
$521$	−37.2763	−1.63310	−0.816552	+	0.577271i	$0.804118\pi$
$522$	0	0
$523$	0.373704	0.0163409	0.00817046	−	0.999967i	$-0.497399\pi$
$523$	0.373704	0.0163409	0.00817046	+	0.999967i	$0.497399\pi$
$524$	0	0
$525$	43.4397	1.89587
$526$	0	0
$527$	−45.1634	−1.96735
$528$	0	0
$529$	−8.87164	−0.385724
$530$	0	0
$531$	−9.27631	−0.402558
$532$	0	0
$533$	−3.51754	−0.152362
$534$	0	0
$535$	−15.5175	−0.670882
$536$	0	0
$537$	−4.24123	−0.183023
$538$	0	0
$539$	−68.3661	−2.94474
$540$	0	0
$541$	−19.0892	−0.820707	−0.410353	−	0.911927i	$-0.634595\pi$
$541$	−19.0892	−0.820707	−0.410353	+	0.911927i	$0.634595\pi$
$542$	0	0
$543$	−12.6108	−0.541182
$544$	0	0
$545$	−0.482459	−0.0206663
$546$	0	0
$547$	−43.7547	−1.87081	−0.935407	−	0.353573i	$-0.884967\pi$
$547$	−43.7547	−1.87081	−0.935407	+	0.353573i	$0.884967\pi$
$548$	0	0
$549$	−8.51754	−0.363520
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−67.3269	−2.86303
$554$	0	0
$555$	−26.7939	−1.13734
$556$	0	0
$557$	31.9026	1.35176	0.675878	−	0.737013i	$-0.263764\pi$
$557$	31.9026	1.35176	0.675878	+	0.737013i	$0.263764\pi$
$558$	0	0
$559$	0.758770	0.0320926
$560$	0	0
$561$	−26.7784	−1.13058
$562$	0	0
$563$	3.87164	0.163170	0.0815852	−	0.996666i	$-0.474002\pi$
$563$	3.87164	0.163170	0.0815852	+	0.996666i	$0.474002\pi$
$564$	0	0
$565$	−9.05407	−0.380908
$566$	0	0
$567$	4.75877	0.199850
$568$	0	0
$569$	32.9959	1.38326	0.691630	−	0.722252i	$-0.256893\pi$
$569$	32.9959	1.38326	0.691630	+	0.722252i	$0.256893\pi$
$570$	0	0
$571$	−24.1789	−1.01186	−0.505928	−	0.862576i	$-0.668850\pi$
$571$	−24.1789	−1.01186	−0.505928	+	0.862576i	$0.668850\pi$
$572$	0	0
$573$	−22.6263	−0.945227
$574$	0	0
$575$	−34.3114	−1.43088
$576$	0	0
$577$	−3.16344	−0.131696	−0.0658478	−	0.997830i	$-0.520975\pi$
$577$	−3.16344	−0.131696	−0.0658478	+	0.997830i	$0.520975\pi$
$578$	0	0
$579$	−24.6459	−1.02425
$580$	0	0
$581$	23.3500	0.968721
$582$	0	0
$583$	22.4944	0.931624
$584$	0	0
$585$	3.75877	0.155406
$586$	0	0
$587$	−6.01548	−0.248285	−0.124143	−	0.992264i	$-0.539618\pi$
$587$	−6.01548	−0.248285	−0.124143	+	0.992264i	$0.539618\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	22.6655	0.932334
$592$	0	0
$593$	22.9222	0.941302	0.470651	−	0.882319i	$-0.344019\pi$
$593$	22.9222	0.941302	0.470651	+	0.882319i	$0.344019\pi$
$594$	0	0
$595$	−109.619	−4.49393
$596$	0	0
$597$	13.9222	0.569798
$598$	0	0
$599$	−34.4397	−1.40717	−0.703585	−	0.710611i	$-0.748419\pi$
$599$	−34.4397	−1.40717	−0.703585	+	0.710611i	$0.748419\pi$
$600$	0	0
$601$	−24.1634	−0.985647	−0.492824	−	0.870129i	$-0.664035\pi$
$601$	−24.1634	−0.985647	−0.492824	+	0.870129i	$0.664035\pi$
$602$	0	0
$603$	−1.36959	−0.0557738
$604$	0	0
$605$	−30.4208	−1.23678
$606$	0	0
$607$	6.84793	0.277949	0.138974	−	0.990296i	$-0.455619\pi$
$607$	6.84793	0.277949	0.138974	+	0.990296i	$0.455619\pi$
$608$	0	0
$609$	−25.6459	−1.03922
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	16.9459	0.684440	0.342220	−	0.939620i	$-0.388821\pi$
$613$	16.9459	0.684440	0.342220	+	0.939620i	$0.388821\pi$
$614$	0	0
$615$	−13.2216	−0.533148
$616$	0	0
$617$	42.2722	1.70181	0.850907	−	0.525316i	$-0.176053\pi$
$617$	42.2722	1.70181	0.850907	+	0.525316i	$0.176053\pi$
$618$	0	0
$619$	4.59121	0.184536	0.0922682	−	0.995734i	$-0.470588\pi$
$619$	4.59121	0.184536	0.0922682	+	0.995734i	$0.470588\pi$
$620$	0	0
$621$	−3.75877	−0.150834
$622$	0	0
$623$	−39.8289	−1.59571
$624$	0	0
$625$	12.6851	0.507404
$626$	0	0
$627$	0	0
$628$	0	0
$629$	43.6851	1.74184
$630$	0	0
$631$	−37.5681	−1.49556	−0.747781	−	0.663945i	$-0.768881\pi$
$631$	−37.5681	−1.49556	−0.747781	+	0.663945i	$0.768881\pi$
$632$	0	0
$633$	2.01960	0.0802719
$634$	0	0
$635$	1.81345	0.0719647
$636$	0	0
$637$	−15.6459	−0.619913
$638$	0	0
$639$	11.6459	0.460705
$640$	0	0
$641$	5.51754	0.217930	0.108965	−	0.994046i	$-0.465246\pi$
$641$	5.51754	0.217930	0.108965	+	0.994046i	$0.465246\pi$
$642$	0	0
$643$	−35.1830	−1.38748	−0.693742	−	0.720224i	$-0.744039\pi$
$643$	−35.1830	−1.38748	−0.693742	+	0.720224i	$0.744039\pi$
$644$	0	0
$645$	2.85204	0.112299
$646$	0	0
$647$	−3.57573	−0.140577	−0.0702883	−	0.997527i	$-0.522392\pi$
$647$	−3.57573	−0.140577	−0.0702883	+	0.997527i	$0.522392\pi$
$648$	0	0
$649$	40.5336	1.59108
$650$	0	0
$651$	−35.0702	−1.37451
$652$	0	0
$653$	32.6418	1.27737	0.638686	−	0.769468i	$-0.279478\pi$
$653$	32.6418	1.27737	0.638686	+	0.769468i	$0.279478\pi$
$654$	0	0
$655$	12.7392	0.497761
$656$	0	0
$657$	4.51754	0.176246
$658$	0	0
$659$	−48.9614	−1.90727	−0.953633	−	0.300972i	$-0.902689\pi$
$659$	−48.9614	−1.90727	−0.953633	+	0.300972i	$0.902689\pi$
$660$	0	0
$661$	−35.4783	−1.37995	−0.689974	−	0.723834i	$-0.742378\pi$
$661$	−35.4783	−1.37995	−0.689974	+	0.723834i	$0.742378\pi$
$662$	0	0
$663$	−6.12836	−0.238006
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.2567	0.784343
$668$	0	0
$669$	−28.3655	−1.09667
$670$	0	0
$671$	37.2181	1.43679
$672$	0	0
$673$	−2.68510	−0.103503	−0.0517514	−	0.998660i	$-0.516480\pi$
$673$	−2.68510	−0.103503	−0.0517514	+	0.998660i	$0.516480\pi$
$674$	0	0
$675$	9.12836	0.351351
$676$	0	0
$677$	−18.3351	−0.704676	−0.352338	−	0.935873i	$-0.614613\pi$
$677$	−18.3351	−0.704676	−0.352338	+	0.935873i	$0.614613\pi$
$678$	0	0
$679$	55.4201	2.12683
$680$	0	0
$681$	−20.1830	−0.773416
$682$	0	0
$683$	−4.07367	−0.155875	−0.0779374	−	0.996958i	$-0.524833\pi$
$683$	−4.07367	−0.155875	−0.0779374	+	0.996958i	$0.524833\pi$
$684$	0	0
$685$	69.2526	2.64601
$686$	0	0
$687$	11.7784	0.449373
$688$	0	0
$689$	5.14796	0.196122
$690$	0	0
$691$	43.5485	1.65666	0.828332	−	0.560238i	$-0.189290\pi$
$691$	43.5485	1.65666	0.828332	+	0.560238i	$0.189290\pi$
$692$	0	0
$693$	−20.7939	−0.789893
$694$	0	0
$695$	−0.0736733	−0.00279459
$696$	0	0
$697$	21.5567	0.816520
$698$	0	0
$699$	−0.128356	−0.00485485
$700$	0	0
$701$	31.3073	1.18246	0.591230	−	0.806503i	$-0.298643\pi$
$701$	31.3073	1.18246	0.591230	+	0.806503i	$0.298643\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	22.5526	0.849381
$706$	0	0
$707$	−19.6459	−0.738860
$708$	0	0
$709$	23.8093	0.894178	0.447089	−	0.894489i	$-0.352461\pi$
$709$	23.8093	0.894178	0.447089	+	0.894489i	$0.352461\pi$
$710$	0	0
$711$	−14.1480	−0.530590
$712$	0	0
$713$	27.7006	1.03739
$714$	0	0
$715$	−16.4243	−0.614233
$716$	0	0
$717$	3.10876	0.116099
$718$	0	0
$719$	31.7897	1.18556	0.592779	−	0.805366i	$-0.298031\pi$
$719$	31.7897	1.18556	0.592779	+	0.805366i	$0.298031\pi$
$720$	0	0
$721$	32.1634	1.19783
$722$	0	0
$723$	−11.2959	−0.420099
$724$	0	0
$725$	−49.1944	−1.82703
$726$	0	0
$727$	−6.44387	−0.238990	−0.119495	−	0.992835i	$-0.538128\pi$
$727$	−6.44387	−0.238990	−0.119495	+	0.992835i	$0.538128\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.65002	−0.171987
$732$	0	0
$733$	−33.9728	−1.25481	−0.627406	−	0.778692i	$-0.715884\pi$
$733$	−33.9728	−1.25481	−0.627406	+	0.778692i	$0.715884\pi$
$734$	0	0
$735$	−58.8093	−2.16921
$736$	0	0
$737$	5.98452	0.220443
$738$	0	0
$739$	22.9263	0.843359	0.421679	−	0.906745i	$-0.361441\pi$
$739$	22.9263	0.843359	0.421679	+	0.906745i	$0.361441\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.96553	0.292227	0.146113	−	0.989268i	$-0.453324\pi$
$743$	7.96553	0.292227	0.146113	+	0.989268i	$0.453324\pi$
$744$	0	0
$745$	−23.4593	−0.859484
$746$	0	0
$747$	4.90673	0.179528
$748$	0	0
$749$	19.6459	0.717845
$750$	0	0
$751$	22.7588	0.830479	0.415240	−	0.909712i	$-0.363698\pi$
$751$	22.7588	0.830479	0.415240	+	0.909712i	$0.363698\pi$
$752$	0	0
$753$	−1.51754	−0.0553023
$754$	0	0
$755$	2.77837	0.101115
$756$	0	0
$757$	4.77425	0.173523	0.0867616	−	0.996229i	$-0.472348\pi$
$757$	4.77425	0.173523	0.0867616	+	0.996229i	$0.472348\pi$
$758$	0	0
$759$	16.4243	0.596163
$760$	0	0
$761$	28.2222	1.02306	0.511528	−	0.859267i	$-0.329080\pi$
$761$	28.2222	1.02306	0.511528	+	0.859267i	$0.329080\pi$
$762$	0	0
$763$	0.610815	0.0221130
$764$	0	0
$765$	−23.0351	−0.832835
$766$	0	0
$767$	9.27631	0.334948
$768$	0	0
$769$	24.2216	0.873454	0.436727	−	0.899594i	$-0.356138\pi$
$769$	24.2216	0.873454	0.436727	+	0.899594i	$0.356138\pi$
$770$	0	0
$771$	12.1830	0.438761
$772$	0	0
$773$	−23.9608	−0.861810	−0.430905	−	0.902397i	$-0.641806\pi$
$773$	−23.9608	−0.861810	−0.430905	+	0.902397i	$0.641806\pi$
$774$	0	0
$775$	−67.2722	−2.41649
$776$	0	0
$777$	33.9222	1.21695
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−50.8877	−1.82091
$782$	0	0
$783$	−5.38919	−0.192594
$784$	0	0
$785$	24.9804	0.891589
$786$	0	0
$787$	28.8871	1.02971	0.514857	−	0.857276i	$-0.327845\pi$
$787$	28.8871	1.02971	0.514857	+	0.857276i	$0.327845\pi$
$788$	0	0
$789$	−19.3892	−0.690273
$790$	0	0
$791$	11.4629	0.407572
$792$	0	0
$793$	8.51754	0.302467
$794$	0	0
$795$	19.3500	0.686273
$796$	0	0
$797$	7.61493	0.269735	0.134867	−	0.990864i	$-0.456939\pi$
$797$	7.61493	0.269735	0.134867	+	0.990864i	$0.456939\pi$
$798$	0	0
$799$	−36.7701	−1.30083
$800$	0	0
$801$	−8.36959	−0.295725
$802$	0	0
$803$	−19.7398	−0.696602
$804$	0	0
$805$	67.2336	2.36967
$806$	0	0
$807$	−23.4047	−0.823883
$808$	0	0
$809$	22.5681	0.793452	0.396726	−	0.917937i	$-0.370146\pi$
$809$	22.5681	0.793452	0.396726	+	0.917937i	$0.370146\pi$
$810$	0	0
$811$	24.7392	0.868710	0.434355	−	0.900742i	$-0.356976\pi$
$811$	24.7392	0.868710	0.434355	+	0.900742i	$0.356976\pi$
$812$	0	0
$813$	0.964918	0.0338412
$814$	0	0
$815$	42.7357	1.49696
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−4.75877	−0.166285
$820$	0	0
$821$	−18.6108	−0.649522	−0.324761	−	0.945796i	$-0.605284\pi$
$821$	−18.6108	−0.649522	−0.324761	+	0.945796i	$0.605284\pi$
$822$	0	0
$823$	−21.9608	−0.765505	−0.382753	−	0.923851i	$-0.625024\pi$
$823$	−21.9608	−0.765505	−0.382753	+	0.923851i	$0.625024\pi$
$824$	0	0
$825$	−39.8871	−1.38869
$826$	0	0
$827$	25.9026	0.900722	0.450361	−	0.892847i	$-0.351295\pi$
$827$	25.9026	0.900722	0.450361	+	0.892847i	$0.351295\pi$
$828$	0	0
$829$	43.0310	1.49453	0.747264	−	0.664528i	$-0.231367\pi$
$829$	43.0310	1.49453	0.747264	+	0.664528i	$0.231367\pi$
$830$	0	0
$831$	15.3892	0.533845
$832$	0	0
$833$	95.8836	3.32217
$834$	0	0
$835$	11.2026	0.387683
$836$	0	0
$837$	−7.36959	−0.254730
$838$	0	0
$839$	−8.33038	−0.287597	−0.143798	−	0.989607i	$-0.545932\pi$
$839$	−8.33038	−0.287597	−0.143798	+	0.989607i	$0.545932\pi$
$840$	0	0
$841$	0.0433195	0.00149378
$842$	0	0
$843$	−8.49794	−0.292685
$844$	0	0
$845$	45.1052	1.55167
$846$	0	0
$847$	38.5140	1.32336
$848$	0	0
$849$	−11.0351	−0.378723
$850$	0	0
$851$	−26.7939	−0.918481
$852$	0	0
$853$	−43.3851	−1.48548	−0.742738	−	0.669582i	$-0.766473\pi$
$853$	−43.3851	−1.48548	−0.742738	+	0.669582i	$0.766473\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.5134	−1.17896	−0.589478	−	0.807784i	$-0.700667\pi$
$857$	−34.5134	−1.17896	−0.589478	+	0.807784i	$0.700667\pi$
$858$	0	0
$859$	−18.1789	−0.620257	−0.310128	−	0.950695i	$-0.600372\pi$
$859$	−18.1789	−0.620257	−0.310128	+	0.950695i	$0.600372\pi$
$860$	0	0
$861$	16.7392	0.570469
$862$	0	0
$863$	−44.3269	−1.50890	−0.754452	−	0.656355i	$-0.772097\pi$
$863$	−44.3269	−1.50890	−0.754452	+	0.656355i	$0.772097\pi$
$864$	0	0
$865$	−20.2567	−0.688749
$866$	0	0
$867$	20.5567	0.698144
$868$	0	0
$869$	61.8207	2.09712
$870$	0	0
$871$	1.36959	0.0464066
$872$	0	0
$873$	11.6459	0.394154
$874$	0	0
$875$	−73.8444	−2.49640
$876$	0	0
$877$	−11.9067	−0.402062	−0.201031	−	0.979585i	$-0.564429\pi$
$877$	−11.9067	−0.402062	−0.201031	+	0.979585i	$0.564429\pi$
$878$	0	0
$879$	1.03508	0.0349124
$880$	0	0
$881$	28.9804	0.976374	0.488187	−	0.872739i	$-0.337658\pi$
$881$	28.9804	0.976374	0.488187	+	0.872739i	$0.337658\pi$
$882$	0	0
$883$	4.46286	0.150187	0.0750936	−	0.997176i	$-0.476074\pi$
$883$	4.46286	0.150187	0.0750936	+	0.997176i	$0.476074\pi$
$884$	0	0
$885$	34.8675	1.17206
$886$	0	0
$887$	28.3114	0.950604	0.475302	−	0.879823i	$-0.342339\pi$
$887$	28.3114	0.950604	0.475302	+	0.879823i	$0.342339\pi$
$888$	0	0
$889$	−2.29591	−0.0770024
$890$	0	0
$891$	−4.36959	−0.146387
$892$	0	0
$893$	0	0
$894$	0	0
$895$	15.9418	0.532876
$896$	0	0
$897$	3.75877	0.125502
$898$	0	0
$899$	39.7161	1.32460
$900$	0	0
$901$	−31.5485	−1.05103
$902$	0	0
$903$	−3.61081	−0.120160
$904$	0	0
$905$	47.4012	1.57567
$906$	0	0
$907$	−20.5134	−0.681137	−0.340569	−	0.940220i	$-0.610620\pi$
$907$	−20.5134	−0.681137	−0.340569	+	0.940220i	$0.610620\pi$
$908$	0	0
$909$	−4.12836	−0.136929
$910$	0	0
$911$	−21.9608	−0.727594	−0.363797	−	0.931478i	$-0.618520\pi$
$911$	−21.9608	−0.727594	−0.363797	+	0.931478i	$0.618520\pi$
$912$	0	0
$913$	−21.4404	−0.709572
$914$	0	0
$915$	32.0155	1.05840
$916$	0	0
$917$	−16.1284	−0.532605
$918$	0	0
$919$	−22.0898	−0.728674	−0.364337	−	0.931267i	$-0.618704\pi$
$919$	−22.0898	−0.728674	−0.364337	+	0.931267i	$0.618704\pi$
$920$	0	0
$921$	−14.8675	−0.489902
$922$	0	0
$923$	−11.6459	−0.383329
$924$	0	0
$925$	65.0702	2.13949
$926$	0	0
$927$	6.75877	0.221987
$928$	0	0
$929$	52.5289	1.72342	0.861709	−	0.507403i	$-0.169395\pi$
$929$	52.5289	1.72342	0.861709	+	0.507403i	$0.169395\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−16.0547	−0.525607
$934$	0	0
$935$	100.654	3.29173
$936$	0	0
$937$	−25.9377	−0.847347	−0.423674	−	0.905815i	$-0.639260\pi$
$937$	−25.9377	−0.847347	−0.423674	+	0.905815i	$0.639260\pi$
$938$	0	0
$939$	35.1634	1.14752
$940$	0	0
$941$	−17.8135	−0.580702	−0.290351	−	0.956920i	$-0.593772\pi$
$941$	−17.8135	−0.580702	−0.290351	+	0.956920i	$0.593772\pi$
$942$	0	0
$943$	−13.2216	−0.430555
$944$	0	0
$945$	−17.8871	−0.581868
$946$	0	0
$947$	−34.0892	−1.10775	−0.553874	−	0.832600i	$-0.686851\pi$
$947$	−34.0892	−1.10775	−0.553874	+	0.832600i	$0.686851\pi$
$948$	0	0
$949$	−4.51754	−0.146646
$950$	0	0
$951$	−18.5371	−0.601108
$952$	0	0
$953$	12.6845	0.410891	0.205445	−	0.978669i	$-0.434136\pi$
$953$	12.6845	0.410891	0.205445	+	0.978669i	$0.434136\pi$
$954$	0	0
$955$	85.0471	2.75206
$956$	0	0
$957$	23.5485	0.761215
$958$	0	0
$959$	−87.6769	−2.83123
$960$	0	0
$961$	23.3108	0.751961
$962$	0	0
$963$	4.12836	0.133034
$964$	0	0
$965$	92.6383	2.98213
$966$	0	0
$967$	−47.2532	−1.51956	−0.759780	−	0.650180i	$-0.774694\pi$
$967$	−47.2532	−1.51956	−0.759780	+	0.650180i	$0.774694\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	35.3620	1.13482	0.567410	−	0.823436i	$-0.307946\pi$
$971$	35.3620	1.13482	0.567410	+	0.823436i	$0.307946\pi$
$972$	0	0
$973$	0.0932736	0.00299021
$974$	0	0
$975$	−9.12836	−0.292341
$976$	0	0
$977$	−45.0660	−1.44179	−0.720895	−	0.693044i	$-0.756269\pi$
$977$	−45.0660	−1.44179	−0.720895	+	0.693044i	$0.756269\pi$
$978$	0	0
$979$	36.5716	1.16883
$980$	0	0
$981$	0.128356	0.00409808
$982$	0	0
$983$	26.1830	0.835109	0.417555	−	0.908652i	$-0.362887\pi$
$983$	26.1830	0.835109	0.417555	+	0.908652i	$0.362887\pi$
$984$	0	0
$985$	−85.1944	−2.71452
$986$	0	0
$987$	−28.5526	−0.908840
$988$	0	0
$989$	2.85204	0.0906897
$990$	0	0
$991$	21.2140	0.673885	0.336942	−	0.941525i	$-0.390607\pi$
$991$	21.2140	0.673885	0.336942	+	0.941525i	$0.390607\pi$
$992$	0	0
$993$	8.75877	0.277951
$994$	0	0
$995$	−52.3304	−1.65898
$996$	0	0
$997$	−54.2728	−1.71884	−0.859418	−	0.511273i	$-0.829174\pi$
$997$	−54.2728	−1.71884	−0.859418	+	0.511273i	$0.829174\pi$
$998$	0	0
$999$	7.12836	0.225531

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8664.2.a.z.1.1		3
19.8	odd	6		456.2.q.f.121.3	yes	6
19.12	odd	6		456.2.q.f.49.3	✓	6
19.18	odd	2		8664.2.a.x.1.1		3
57.8	even	6		1368.2.s.j.577.1		6
57.50	even	6		1368.2.s.j.505.1		6
76.27	even	6		912.2.q.k.577.3		6
76.31	even	6		912.2.q.k.49.3		6
228.107	odd	6		2736.2.s.x.1873.1		6
228.179	odd	6		2736.2.s.x.577.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.q.f.49.3	✓	6	19.12	odd	6
456.2.q.f.121.3	yes	6	19.8	odd	6
912.2.q.k.49.3		6	76.31	even	6
912.2.q.k.577.3		6	76.27	even	6
1368.2.s.j.505.1		6	57.50	even	6
1368.2.s.j.577.1		6	57.8	even	6
2736.2.s.x.577.1		6	228.179	odd	6
2736.2.s.x.1873.1		6	228.107	odd	6
8664.2.a.x.1.1		3	19.18	odd	2
8664.2.a.z.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8664.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.75877 q^{5} +4.75877 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.75877 q^{5} +4.75877 q^{7} +1.00000 q^{9} -4.36959 q^{11} -1.00000 q^{13} -3.75877 q^{15} +6.12836 q^{17} +4.75877 q^{21} -3.75877 q^{23} +9.12836 q^{25} +1.00000 q^{27} -5.38919 q^{29} -7.36959 q^{31} -4.36959 q^{33} -17.8871 q^{35} +7.12836 q^{37} -1.00000 q^{39} +3.51754 q^{41} -0.758770 q^{43} -3.75877 q^{45} -6.00000 q^{47} +15.6459 q^{49} +6.12836 q^{51} -5.14796 q^{53} +16.4243 q^{55} -9.27631 q^{59} -8.51754 q^{61} +4.75877 q^{63} +3.75877 q^{65} -1.36959 q^{67} -3.75877 q^{69} +11.6459 q^{71} +4.51754 q^{73} +9.12836 q^{75} -20.7939 q^{77} -14.1480 q^{79} +1.00000 q^{81} +4.90673 q^{83} -23.0351 q^{85} -5.38919 q^{87} -8.36959 q^{89} -4.75877 q^{91} -7.36959 q^{93} +11.6459 q^{97} -4.36959 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 6 q^{11} - 3 q^{13} + 3 q^{21} + 9 q^{25} + 3 q^{27} - 12 q^{29} - 15 q^{31} - 6 q^{33} - 24 q^{35} + 3 q^{37} - 3 q^{39} - 12 q^{41} + 9 q^{43} - 18 q^{47} + 6 q^{49} + 6 q^{59} - 3 q^{61} + 3 q^{63} + 3 q^{67} - 6 q^{71} - 9 q^{73} + 9 q^{75} - 6 q^{77} - 27 q^{79} + 3 q^{81} - 12 q^{83} - 24 q^{85} - 12 q^{87} - 18 q^{89} - 3 q^{91} - 15 q^{93} - 6 q^{97} - 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 - 6 * q^11 - 3 * q^13 + 3 * q^21 + 9 * q^25 + 3 * q^27 - 12 * q^29 - 15 * q^31 - 6 * q^33 - 24 * q^35 + 3 * q^37 - 3 * q^39 - 12 * q^41 + 9 * q^43 - 18 * q^47 + 6 * q^49 + 6 * q^59 - 3 * q^61 + 3 * q^63 + 3 * q^67 - 6 * q^71 - 9 * q^73 + 9 * q^75 - 6 * q^77 - 27 * q^79 + 3 * q^81 - 12 * q^83 - 24 * q^85 - 12 * q^87 - 18 * q^89 - 3 * q^91 - 15 * q^93 - 6 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more