LMFDB - Embedded newform 6498.2.a.bn.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6498,2,Mod(1,6498)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6498, 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("6498.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6498 = 2 \cdot 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6498.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:	$51.8867912334$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	6498.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.41147 q^{5} -2.87939 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +3.41147 q^{5} -2.87939 q^{7} -1.00000 q^{8} -3.41147 q^{10} +0.347296 q^{11} +1.65270 q^{13} +2.87939 q^{14} +1.00000 q^{16} -6.94356 q^{17} +3.41147 q^{20} -0.347296 q^{22} -6.80066 q^{23} +6.63816 q^{25} -1.65270 q^{26} -2.87939 q^{28} +6.35504 q^{29} -1.59627 q^{31} -1.00000 q^{32} +6.94356 q^{34} -9.82295 q^{35} +11.2121 q^{37} -3.41147 q^{40} -3.49020 q^{41} +2.28312 q^{43} +0.347296 q^{44} +6.80066 q^{46} -5.59627 q^{47} +1.29086 q^{49} -6.63816 q^{50} +1.65270 q^{52} +1.98040 q^{53} +1.18479 q^{55} +2.87939 q^{56} -6.35504 q^{58} +0.445622 q^{59} -12.5321 q^{61} +1.59627 q^{62} +1.00000 q^{64} +5.63816 q^{65} +1.07873 q^{67} -6.94356 q^{68} +9.82295 q^{70} -16.6236 q^{71} +12.4192 q^{73} -11.2121 q^{74} -1.00000 q^{77} +10.9240 q^{79} +3.41147 q^{80} +3.49020 q^{82} -11.7169 q^{83} -23.6878 q^{85} -2.28312 q^{86} -0.347296 q^{88} +1.79292 q^{89} -4.75877 q^{91} -6.80066 q^{92} +5.59627 q^{94} -3.65270 q^{97} -1.29086 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 6 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 6 q^{23} + 3 q^{25} - 6 q^{26} - 3 q^{28} - 6 q^{29} + 9 q^{31} - 3 q^{32} + 6 q^{34} - 9 q^{35} + 9 q^{37} - 9 q^{41} + 15 q^{43} + 6 q^{46} - 3 q^{47} - 12 q^{49} - 3 q^{50} + 6 q^{52} + 3 q^{53} + 3 q^{56} + 6 q^{58} + 12 q^{59} - 33 q^{61} - 9 q^{62} + 3 q^{64} + 12 q^{67} - 6 q^{68} + 9 q^{70} - 15 q^{71} + 3 q^{73} - 9 q^{74} - 3 q^{77} + 15 q^{79} + 9 q^{82} - 27 q^{83} - 27 q^{85} - 15 q^{86} + 15 q^{89} - 3 q^{91} - 6 q^{92} + 3 q^{94} - 12 q^{97} + 12 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^7 - 3 * q^8 + 6 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^17 - 6 * q^23 + 3 * q^25 - 6 * q^26 - 3 * q^28 - 6 * q^29 + 9 * q^31 - 3 * q^32 + 6 * q^34 - 9 * q^35 + 9 * q^37 - 9 * q^41 + 15 * q^43 + 6 * q^46 - 3 * q^47 - 12 * q^49 - 3 * q^50 + 6 * q^52 + 3 * q^53 + 3 * q^56 + 6 * q^58 + 12 * q^59 - 33 * q^61 - 9 * q^62 + 3 * q^64 + 12 * q^67 - 6 * q^68 + 9 * q^70 - 15 * q^71 + 3 * q^73 - 9 * q^74 - 3 * q^77 + 15 * q^79 + 9 * q^82 - 27 * q^83 - 27 * q^85 - 15 * q^86 + 15 * q^89 - 3 * q^91 - 6 * q^92 + 3 * q^94 - 12 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.41147	1.52566	0.762829	−	0.646601i	$-0.223810\pi$
$5$	3.41147	1.52566	0.762829	+	0.646601i	$0.223810\pi$
$6$	0	0
$7$	−2.87939	−1.08831	−0.544153	−	0.838986i	$-0.683149\pi$
$7$	−2.87939	−1.08831	−0.544153	+	0.838986i	$0.683149\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.41147	−1.07880
$11$	0.347296	0.104714	0.0523569	−	0.998628i	$-0.483327\pi$
$11$	0.347296	0.104714	0.0523569	+	0.998628i	$0.483327\pi$
$12$	0	0
$13$	1.65270	0.458378	0.229189	−	0.973382i	$-0.426393\pi$
$13$	1.65270	0.458378	0.229189	+	0.973382i	$0.426393\pi$
$14$	2.87939	0.769548
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.94356	−1.68406	−0.842031	−	0.539430i	$-0.818640\pi$
$17$	−6.94356	−1.68406	−0.842031	+	0.539430i	$0.818640\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	3.41147	0.762829
$21$	0	0
$22$	−0.347296	−0.0740438
$23$	−6.80066	−1.41804	−0.709018	−	0.705191i	$-0.750861\pi$
$23$	−6.80066	−1.41804	−0.709018	+	0.705191i	$0.750861\pi$
$24$	0	0
$25$	6.63816	1.32763
$26$	−1.65270	−0.324122
$27$	0	0
$28$	−2.87939	−0.544153
$29$	6.35504	1.18010	0.590050	−	0.807366i	$-0.299108\pi$
$29$	6.35504	1.18010	0.590050	+	0.807366i	$0.299108\pi$
$30$	0	0
$31$	−1.59627	−0.286698	−0.143349	−	0.989672i	$-0.545787\pi$
$31$	−1.59627	−0.286698	−0.143349	+	0.989672i	$0.545787\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.94356	1.19081
$35$	−9.82295	−1.66038
$36$	0	0
$37$	11.2121	1.84326	0.921632	−	0.388066i	$-0.126857\pi$
$37$	11.2121	1.84326	0.921632	+	0.388066i	$0.126857\pi$
$38$	0	0
$39$	0	0
$40$	−3.41147	−0.539401
$41$	−3.49020	−0.545078	−0.272539	−	0.962145i	$-0.587863\pi$
$41$	−3.49020	−0.545078	−0.272539	+	0.962145i	$0.587863\pi$
$42$	0	0
$43$	2.28312	0.348172	0.174086	−	0.984730i	$-0.444303\pi$
$43$	2.28312	0.348172	0.174086	+	0.984730i	$0.444303\pi$
$44$	0.347296	0.0523569
$45$	0	0
$46$	6.80066	1.00270
$47$	−5.59627	−0.816299	−0.408150	−	0.912915i	$-0.633826\pi$
$47$	−5.59627	−0.816299	−0.408150	+	0.912915i	$0.633826\pi$
$48$	0	0
$49$	1.29086	0.184408
$50$	−6.63816	−0.938777
$51$	0	0
$52$	1.65270	0.229189
$53$	1.98040	0.272029	0.136014	−	0.990707i	$-0.456571\pi$
$53$	1.98040	0.272029	0.136014	+	0.990707i	$0.456571\pi$
$54$	0	0
$55$	1.18479	0.159757
$56$	2.87939	0.384774
$57$	0	0
$58$	−6.35504	−0.834457
$59$	0.445622	0.0580151	0.0290075	−	0.999579i	$-0.490765\pi$
$59$	0.445622	0.0580151	0.0290075	+	0.999579i	$0.490765\pi$
$60$	0	0
$61$	−12.5321	−1.60457	−0.802285	−	0.596941i	$-0.796382\pi$
$61$	−12.5321	−1.60457	−0.802285	+	0.596941i	$0.796382\pi$
$62$	1.59627	0.202726
$63$	0	0
$64$	1.00000	0.125000
$65$	5.63816	0.699327
$66$	0	0
$67$	1.07873	0.131787	0.0658937	−	0.997827i	$-0.479010\pi$
$67$	1.07873	0.131787	0.0658937	+	0.997827i	$0.479010\pi$
$68$	−6.94356	−0.842031
$69$	0	0
$70$	9.82295	1.17407
$71$	−16.6236	−1.97286	−0.986430	−	0.164185i	$-0.947501\pi$
$71$	−16.6236	−1.97286	−0.986430	+	0.164185i	$0.947501\pi$
$72$	0	0
$73$	12.4192	1.45356	0.726780	−	0.686871i	$-0.241016\pi$
$73$	12.4192	1.45356	0.726780	+	0.686871i	$0.241016\pi$
$74$	−11.2121	−1.30338
$75$	0	0
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	10.9240	1.22904	0.614521	−	0.788901i	$-0.289349\pi$
$79$	10.9240	1.22904	0.614521	+	0.788901i	$0.289349\pi$
$80$	3.41147	0.381414
$81$	0	0
$82$	3.49020	0.385428
$83$	−11.7169	−1.28609	−0.643047	−	0.765826i	$-0.722330\pi$
$83$	−11.7169	−1.28609	−0.643047	+	0.765826i	$0.722330\pi$
$84$	0	0
$85$	−23.6878	−2.56930
$86$	−2.28312	−0.246195
$87$	0	0
$88$	−0.347296	−0.0370219
$89$	1.79292	0.190049	0.0950245	−	0.995475i	$-0.469707\pi$
$89$	1.79292	0.190049	0.0950245	+	0.995475i	$0.469707\pi$
$90$	0	0
$91$	−4.75877	−0.498855
$92$	−6.80066	−0.709018
$93$	0	0
$94$	5.59627	0.577211
$95$	0	0
$96$	0	0
$97$	−3.65270	−0.370876	−0.185438	−	0.982656i	$-0.559370\pi$
$97$	−3.65270	−0.370876	−0.185438	+	0.982656i	$0.559370\pi$
$98$	−1.29086	−0.130396
$99$	0	0
$100$	6.63816	0.663816
$101$	9.90673	0.985756	0.492878	−	0.870098i	$-0.335945\pi$
$101$	9.90673	0.985756	0.492878	+	0.870098i	$0.335945\pi$
$102$	0	0
$103$	−7.84524	−0.773014	−0.386507	−	0.922286i	$-0.626319\pi$
$103$	−7.84524	−0.773014	−0.386507	+	0.922286i	$0.626319\pi$
$104$	−1.65270	−0.162061
$105$	0	0
$106$	−1.98040	−0.192353
$107$	−0.0273411	−0.00264317	−0.00132158	−	0.999999i	$-0.500421\pi$
$107$	−0.0273411	−0.00264317	−0.00132158	+	0.999999i	$0.500421\pi$
$108$	0	0
$109$	10.7733	1.03190	0.515948	−	0.856620i	$-0.327440\pi$
$109$	10.7733	1.03190	0.515948	+	0.856620i	$0.327440\pi$
$110$	−1.18479	−0.112966
$111$	0	0
$112$	−2.87939	−0.272076
$113$	−11.6604	−1.09692	−0.548461	−	0.836176i	$-0.684786\pi$
$113$	−11.6604	−1.09692	−0.548461	+	0.836176i	$0.684786\pi$
$114$	0	0
$115$	−23.2003	−2.16344
$116$	6.35504	0.590050
$117$	0	0
$118$	−0.445622	−0.0410229
$119$	19.9932	1.83277
$120$	0	0
$121$	−10.8794	−0.989035
$122$	12.5321	1.13460
$123$	0	0
$124$	−1.59627	−0.143349
$125$	5.58853	0.499853
$126$	0	0
$127$	−16.2121	−1.43859	−0.719297	−	0.694703i	$-0.755536\pi$
$127$	−16.2121	−1.43859	−0.719297	+	0.694703i	$0.755536\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−5.63816	−0.494499
$131$	2.80066	0.244695	0.122347	−	0.992487i	$-0.460958\pi$
$131$	2.80066	0.244695	0.122347	+	0.992487i	$0.460958\pi$
$132$	0	0
$133$	0	0
$134$	−1.07873	−0.0931877
$135$	0	0
$136$	6.94356	0.595406
$137$	−16.0351	−1.36997	−0.684985	−	0.728557i	$-0.740191\pi$
$137$	−16.0351	−1.36997	−0.684985	+	0.728557i	$0.740191\pi$
$138$	0	0
$139$	−2.07873	−0.176315	−0.0881576	−	0.996107i	$-0.528098\pi$
$139$	−2.07873	−0.176315	−0.0881576	+	0.996107i	$0.528098\pi$
$140$	−9.82295	−0.830191
$141$	0	0
$142$	16.6236	1.39502
$143$	0.573978	0.0479984
$144$	0	0
$145$	21.6800	1.80043
$146$	−12.4192	−1.02782
$147$	0	0
$148$	11.2121	0.921632
$149$	1.66725	0.136587	0.0682933	−	0.997665i	$-0.478245\pi$
$149$	1.66725	0.136587	0.0682933	+	0.997665i	$0.478245\pi$
$150$	0	0
$151$	−20.0523	−1.63183	−0.815917	−	0.578169i	$-0.803768\pi$
$151$	−20.0523	−1.63183	−0.815917	+	0.578169i	$0.803768\pi$
$152$	0	0
$153$	0	0
$154$	1.00000	0.0805823
$155$	−5.44562	−0.437403
$156$	0	0
$157$	−4.09833	−0.327082	−0.163541	−	0.986537i	$-0.552292\pi$
$157$	−4.09833	−0.327082	−0.163541	+	0.986537i	$0.552292\pi$
$158$	−10.9240	−0.869064
$159$	0	0
$160$	−3.41147	−0.269701
$161$	19.5817	1.54326
$162$	0	0
$163$	−8.13516	−0.637195	−0.318598	−	0.947890i	$-0.603212\pi$
$163$	−8.13516	−0.637195	−0.318598	+	0.947890i	$0.603212\pi$
$164$	−3.49020	−0.272539
$165$	0	0
$166$	11.7169	0.909406
$167$	−13.3969	−1.03669	−0.518343	−	0.855173i	$-0.673451\pi$
$167$	−13.3969	−1.03669	−0.518343	+	0.855173i	$0.673451\pi$
$168$	0	0
$169$	−10.2686	−0.789890
$170$	23.6878	1.81677
$171$	0	0
$172$	2.28312	0.174086
$173$	−1.02229	−0.0777232	−0.0388616	−	0.999245i	$-0.512373\pi$
$173$	−1.02229	−0.0777232	−0.0388616	+	0.999245i	$0.512373\pi$
$174$	0	0
$175$	−19.1138	−1.44487
$176$	0.347296	0.0261784
$177$	0	0
$178$	−1.79292	−0.134385
$179$	2.03508	0.152109	0.0760546	−	0.997104i	$-0.475768\pi$
$179$	2.03508	0.152109	0.0760546	+	0.997104i	$0.475768\pi$
$180$	0	0
$181$	−22.3037	−1.65782	−0.828909	−	0.559384i	$-0.811038\pi$
$181$	−22.3037	−1.65782	−0.828909	+	0.559384i	$0.811038\pi$
$182$	4.75877	0.352744
$183$	0	0
$184$	6.80066	0.501351
$185$	38.2499	2.81219
$186$	0	0
$187$	−2.41147	−0.176344
$188$	−5.59627	−0.408150
$189$	0	0
$190$	0	0
$191$	−10.7861	−0.780456	−0.390228	−	0.920718i	$-0.627604\pi$
$191$	−10.7861	−0.780456	−0.390228	+	0.920718i	$0.627604\pi$
$192$	0	0
$193$	2.86484	0.206216	0.103108	−	0.994670i	$-0.467121\pi$
$193$	2.86484	0.206216	0.103108	+	0.994670i	$0.467121\pi$
$194$	3.65270	0.262249
$195$	0	0
$196$	1.29086	0.0922042
$197$	8.33544	0.593875	0.296938	−	0.954897i	$-0.404035\pi$
$197$	8.33544	0.593875	0.296938	+	0.954897i	$0.404035\pi$
$198$	0	0
$199$	−3.03415	−0.215085	−0.107543	−	0.994200i	$-0.534298\pi$
$199$	−3.03415	−0.215085	−0.107543	+	0.994200i	$0.534298\pi$
$200$	−6.63816	−0.469388
$201$	0	0
$202$	−9.90673	−0.697035
$203$	−18.2986	−1.28431
$204$	0	0
$205$	−11.9067	−0.831602
$206$	7.84524	0.546604
$207$	0	0
$208$	1.65270	0.114594
$209$	0	0
$210$	0	0
$211$	16.6382	1.14542	0.572709	−	0.819759i	$-0.305893\pi$
$211$	16.6382	1.14542	0.572709	+	0.819759i	$0.305893\pi$
$212$	1.98040	0.136014
$213$	0	0
$214$	0.0273411	0.00186900
$215$	7.78880	0.531192
$216$	0	0
$217$	4.59627	0.312015
$218$	−10.7733	−0.729661
$219$	0	0
$220$	1.18479	0.0798787
$221$	−11.4757	−0.771936
$222$	0	0
$223$	8.17024	0.547120	0.273560	−	0.961855i	$-0.411799\pi$
$223$	8.17024	0.547120	0.273560	+	0.961855i	$0.411799\pi$
$224$	2.87939	0.192387
$225$	0	0
$226$	11.6604	0.775641
$227$	1.80066	0.119514	0.0597570	−	0.998213i	$-0.480967\pi$
$227$	1.80066	0.119514	0.0597570	+	0.998213i	$0.480967\pi$
$228$	0	0
$229$	−18.7392	−1.23832	−0.619160	−	0.785265i	$-0.712527\pi$
$229$	−18.7392	−1.23832	−0.619160	+	0.785265i	$0.712527\pi$
$230$	23.2003	1.52978
$231$	0	0
$232$	−6.35504	−0.417229
$233$	4.09833	0.268490	0.134245	−	0.990948i	$-0.457139\pi$
$233$	4.09833	0.268490	0.134245	+	0.990948i	$0.457139\pi$
$234$	0	0
$235$	−19.0915	−1.24539
$236$	0.445622	0.0290075
$237$	0	0
$238$	−19.9932	−1.29597
$239$	28.2276	1.82589	0.912946	−	0.408080i	$-0.133801\pi$
$239$	28.2276	1.82589	0.912946	+	0.408080i	$0.133801\pi$
$240$	0	0
$241$	−6.56212	−0.422703	−0.211352	−	0.977410i	$-0.567787\pi$
$241$	−6.56212	−0.422703	−0.211352	+	0.977410i	$0.567787\pi$
$242$	10.8794	0.699353
$243$	0	0
$244$	−12.5321	−0.802285
$245$	4.40373	0.281344
$246$	0	0
$247$	0	0
$248$	1.59627	0.101363
$249$	0	0
$250$	−5.58853	−0.353449
$251$	−8.89393	−0.561380	−0.280690	−	0.959798i	$-0.590563\pi$
$251$	−8.89393	−0.561380	−0.280690	+	0.959798i	$0.590563\pi$
$252$	0	0
$253$	−2.36184	−0.148488
$254$	16.2121	1.01724
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.4662	−0.839996	−0.419998	−	0.907525i	$-0.637969\pi$
$257$	−13.4662	−0.839996	−0.419998	+	0.907525i	$0.637969\pi$
$258$	0	0
$259$	−32.2841	−2.00603
$260$	5.63816	0.349664
$261$	0	0
$262$	−2.80066	−0.173025
$263$	16.7520	1.03297	0.516485	−	0.856296i	$-0.327240\pi$
$263$	16.7520	1.03297	0.516485	+	0.856296i	$0.327240\pi$
$264$	0	0
$265$	6.75608	0.415023
$266$	0	0
$267$	0	0
$268$	1.07873	0.0658937
$269$	3.58946	0.218853	0.109427	−	0.993995i	$-0.465099\pi$
$269$	3.58946	0.218853	0.109427	+	0.993995i	$0.465099\pi$
$270$	0	0
$271$	23.9590	1.45541	0.727704	−	0.685891i	$-0.240587\pi$
$271$	23.9590	1.45541	0.727704	+	0.685891i	$0.240587\pi$
$272$	−6.94356	−0.421015
$273$	0	0
$274$	16.0351	0.968715
$275$	2.30541	0.139021
$276$	0	0
$277$	−18.7219	−1.12489	−0.562446	−	0.826834i	$-0.690140\pi$
$277$	−18.7219	−1.12489	−0.562446	+	0.826834i	$0.690140\pi$
$278$	2.07873	0.124674
$279$	0	0
$280$	9.82295	0.587033
$281$	−10.6382	−0.634619	−0.317310	−	0.948322i	$-0.602779\pi$
$281$	−10.6382	−0.634619	−0.317310	+	0.948322i	$0.602779\pi$
$282$	0	0
$283$	−14.0128	−0.832974	−0.416487	−	0.909142i	$-0.636739\pi$
$283$	−14.0128	−0.832974	−0.416487	+	0.909142i	$0.636739\pi$
$284$	−16.6236	−0.986430
$285$	0	0
$286$	−0.573978	−0.0339400
$287$	10.0496	0.593211
$288$	0	0
$289$	31.2131	1.83606
$290$	−21.6800	−1.27310
$291$	0	0
$292$	12.4192	0.726780
$293$	−11.5202	−0.673019	−0.336509	−	0.941680i	$-0.609246\pi$
$293$	−11.5202	−0.673019	−0.336509	+	0.941680i	$0.609246\pi$
$294$	0	0
$295$	1.52023	0.0885112
$296$	−11.2121	−0.651692
$297$	0	0
$298$	−1.66725	−0.0965813
$299$	−11.2395	−0.649996
$300$	0	0
$301$	−6.57398	−0.378918
$302$	20.0523	1.15388
$303$	0	0
$304$	0	0
$305$	−42.7529	−2.44802
$306$	0	0
$307$	−4.29767	−0.245281	−0.122640	−	0.992451i	$-0.539136\pi$
$307$	−4.29767	−0.245281	−0.122640	+	0.992451i	$0.539136\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	5.44562	0.309291
$311$	22.9932	1.30382	0.651912	−	0.758294i	$-0.273967\pi$
$311$	22.9932	1.30382	0.651912	+	0.758294i	$0.273967\pi$
$312$	0	0
$313$	−9.20977	−0.520567	−0.260283	−	0.965532i	$-0.583816\pi$
$313$	−9.20977	−0.520567	−0.260283	+	0.965532i	$0.583816\pi$
$314$	4.09833	0.231282
$315$	0	0
$316$	10.9240	0.614521
$317$	3.35504	0.188438	0.0942188	−	0.995552i	$-0.469965\pi$
$317$	3.35504	0.188438	0.0942188	+	0.995552i	$0.469965\pi$
$318$	0	0
$319$	2.20708	0.123573
$320$	3.41147	0.190707
$321$	0	0
$322$	−19.5817	−1.09125
$323$	0	0
$324$	0	0
$325$	10.9709	0.608556
$326$	8.13516	0.450565
$327$	0	0
$328$	3.49020	0.192714
$329$	16.1138	0.888383
$330$	0	0
$331$	22.0743	1.21331	0.606656	−	0.794964i	$-0.292510\pi$
$331$	22.0743	1.21331	0.606656	+	0.794964i	$0.292510\pi$
$332$	−11.7169	−0.643047
$333$	0	0
$334$	13.3969	0.733047
$335$	3.68004	0.201062
$336$	0	0
$337$	13.1165	0.714501	0.357251	−	0.934009i	$-0.383714\pi$
$337$	13.1165	0.714501	0.357251	+	0.934009i	$0.383714\pi$
$338$	10.2686	0.558537
$339$	0	0
$340$	−23.6878	−1.28465
$341$	−0.554378	−0.0300212
$342$	0	0
$343$	16.4388	0.887613
$344$	−2.28312	−0.123098
$345$	0	0
$346$	1.02229	0.0549586
$347$	−8.75877	−0.470195	−0.235098	−	0.971972i	$-0.575541\pi$
$347$	−8.75877	−0.470195	−0.235098	+	0.971972i	$0.575541\pi$
$348$	0	0
$349$	3.26352	0.174692	0.0873461	−	0.996178i	$-0.472161\pi$
$349$	3.26352	0.174692	0.0873461	+	0.996178i	$0.472161\pi$
$350$	19.1138	1.02168
$351$	0	0
$352$	−0.347296	−0.0185110
$353$	−29.0838	−1.54797	−0.773987	−	0.633202i	$-0.781740\pi$
$353$	−29.0838	−1.54797	−0.773987	+	0.633202i	$0.781740\pi$
$354$	0	0
$355$	−56.7110	−3.00991
$356$	1.79292	0.0950245
$357$	0	0
$358$	−2.03508	−0.107557
$359$	−6.61081	−0.348905	−0.174453	−	0.984666i	$-0.555816\pi$
$359$	−6.61081	−0.348905	−0.174453	+	0.984666i	$0.555816\pi$
$360$	0	0
$361$	0	0
$362$	22.3037	1.17225
$363$	0	0
$364$	−4.75877	−0.249427
$365$	42.3678	2.21763
$366$	0	0
$367$	−5.79561	−0.302528	−0.151264	−	0.988493i	$-0.548334\pi$
$367$	−5.79561	−0.302528	−0.151264	+	0.988493i	$0.548334\pi$
$368$	−6.80066	−0.354509
$369$	0	0
$370$	−38.2499	−1.98852
$371$	−5.70233	−0.296050
$372$	0	0
$373$	25.8898	1.34052	0.670262	−	0.742125i	$-0.266182\pi$
$373$	25.8898	1.34052	0.670262	+	0.742125i	$0.266182\pi$
$374$	2.41147	0.124694
$375$	0	0
$376$	5.59627	0.288605
$377$	10.5030	0.540932
$378$	0	0
$379$	19.1557	0.983962	0.491981	−	0.870606i	$-0.336273\pi$
$379$	19.1557	0.983962	0.491981	+	0.870606i	$0.336273\pi$
$380$	0	0
$381$	0	0
$382$	10.7861	0.551865
$383$	−9.69459	−0.495371	−0.247685	−	0.968841i	$-0.579670\pi$
$383$	−9.69459	−0.495371	−0.247685	+	0.968841i	$0.579670\pi$
$384$	0	0
$385$	−3.41147	−0.173865
$386$	−2.86484	−0.145816
$387$	0	0
$388$	−3.65270	−0.185438
$389$	0.817896	0.0414690	0.0207345	−	0.999785i	$-0.493400\pi$
$389$	0.817896	0.0414690	0.0207345	+	0.999785i	$0.493400\pi$
$390$	0	0
$391$	47.2208	2.38806
$392$	−1.29086	−0.0651982
$393$	0	0
$394$	−8.33544	−0.419933
$395$	37.2668	1.87510
$396$	0	0
$397$	−3.69728	−0.185561	−0.0927806	−	0.995687i	$-0.529576\pi$
$397$	−3.69728	−0.185561	−0.0927806	+	0.995687i	$0.529576\pi$
$398$	3.03415	0.152088
$399$	0	0
$400$	6.63816	0.331908
$401$	−21.7401	−1.08565	−0.542824	−	0.839846i	$-0.682645\pi$
$401$	−21.7401	−1.08565	−0.542824	+	0.839846i	$0.682645\pi$
$402$	0	0
$403$	−2.63816	−0.131416
$404$	9.90673	0.492878
$405$	0	0
$406$	18.2986	0.908144
$407$	3.89393	0.193015
$408$	0	0
$409$	4.05468	0.200491	0.100246	−	0.994963i	$-0.468037\pi$
$409$	4.05468	0.200491	0.100246	+	0.994963i	$0.468037\pi$
$410$	11.9067	0.588031
$411$	0	0
$412$	−7.84524	−0.386507
$413$	−1.28312	−0.0631381
$414$	0	0
$415$	−39.9718	−1.96214
$416$	−1.65270	−0.0810305
$417$	0	0
$418$	0	0
$419$	−23.4989	−1.14800	−0.573998	−	0.818857i	$-0.694608\pi$
$419$	−23.4989	−1.14800	−0.573998	+	0.818857i	$0.694608\pi$
$420$	0	0
$421$	23.0729	1.12450	0.562251	−	0.826967i	$-0.309935\pi$
$421$	23.0729	1.12450	0.562251	+	0.826967i	$0.309935\pi$
$422$	−16.6382	−0.809933
$423$	0	0
$424$	−1.98040	−0.0961767
$425$	−46.0925	−2.23581
$426$	0	0
$427$	36.0847	1.74626
$428$	−0.0273411	−0.00132158
$429$	0	0
$430$	−7.78880	−0.375609
$431$	−3.51485	−0.169304	−0.0846522	−	0.996411i	$-0.526978\pi$
$431$	−3.51485	−0.169304	−0.0846522	+	0.996411i	$0.526978\pi$
$432$	0	0
$433$	34.1147	1.63945	0.819725	−	0.572757i	$-0.194126\pi$
$433$	34.1147	1.63945	0.819725	+	0.572757i	$0.194126\pi$
$434$	−4.59627	−0.220628
$435$	0	0
$436$	10.7733	0.515948
$437$	0	0
$438$	0	0
$439$	−9.70140	−0.463023	−0.231511	−	0.972832i	$-0.574367\pi$
$439$	−9.70140	−0.463023	−0.231511	+	0.972832i	$0.574367\pi$
$440$	−1.18479	−0.0564828
$441$	0	0
$442$	11.4757	0.545841
$443$	−15.1821	−0.721324	−0.360662	−	0.932697i	$-0.617449\pi$
$443$	−15.1821	−0.721324	−0.360662	+	0.932697i	$0.617449\pi$
$444$	0	0
$445$	6.11650	0.289950
$446$	−8.17024	−0.386872
$447$	0	0
$448$	−2.87939	−0.136038
$449$	34.8384	1.64413	0.822064	−	0.569396i	$-0.192823\pi$
$449$	34.8384	1.64413	0.822064	+	0.569396i	$0.192823\pi$
$450$	0	0
$451$	−1.21213	−0.0570771
$452$	−11.6604	−0.548461
$453$	0	0
$454$	−1.80066	−0.0845091
$455$	−16.2344	−0.761081
$456$	0	0
$457$	−31.8749	−1.49105	−0.745523	−	0.666479i	$-0.767800\pi$
$457$	−31.8749	−1.49105	−0.745523	+	0.666479i	$0.767800\pi$
$458$	18.7392	0.875624
$459$	0	0
$460$	−23.2003	−1.08172
$461$	10.6895	0.497862	0.248931	−	0.968521i	$-0.419921\pi$
$461$	10.6895	0.497862	0.248931	+	0.968521i	$0.419921\pi$
$462$	0	0
$463$	15.3182	0.711897	0.355949	−	0.934506i	$-0.384158\pi$
$463$	15.3182	0.711897	0.355949	+	0.934506i	$0.384158\pi$
$464$	6.35504	0.295025
$465$	0	0
$466$	−4.09833	−0.189851
$467$	−28.5125	−1.31940	−0.659700	−	0.751529i	$-0.729317\pi$
$467$	−28.5125	−1.31940	−0.659700	+	0.751529i	$0.729317\pi$
$468$	0	0
$469$	−3.10607	−0.143425
$470$	19.0915	0.880626
$471$	0	0
$472$	−0.445622	−0.0205114
$473$	0.792919	0.0364584
$474$	0	0
$475$	0	0
$476$	19.9932	0.916386
$477$	0	0
$478$	−28.2276	−1.29110
$479$	−8.05138	−0.367877	−0.183939	−	0.982938i	$-0.558885\pi$
$479$	−8.05138	−0.367877	−0.183939	+	0.982938i	$0.558885\pi$
$480$	0	0
$481$	18.5303	0.844911
$482$	6.56212	0.298896
$483$	0	0
$484$	−10.8794	−0.494518
$485$	−12.4611	−0.565830
$486$	0	0
$487$	−6.11556	−0.277123	−0.138561	−	0.990354i	$-0.544248\pi$
$487$	−6.11556	−0.277123	−0.138561	+	0.990354i	$0.544248\pi$
$488$	12.5321	0.567301
$489$	0	0
$490$	−4.40373	−0.198940
$491$	22.6851	1.02376	0.511882	−	0.859056i	$-0.328948\pi$
$491$	22.6851	1.02376	0.511882	+	0.859056i	$0.328948\pi$
$492$	0	0
$493$	−44.1266	−1.98736
$494$	0	0
$495$	0	0
$496$	−1.59627	−0.0716745
$497$	47.8658	2.14707
$498$	0	0
$499$	24.8557	1.11269	0.556346	−	0.830951i	$-0.312203\pi$
$499$	24.8557	1.11269	0.556346	+	0.830951i	$0.312203\pi$
$500$	5.58853	0.249926
$501$	0	0
$502$	8.89393	0.396956
$503$	−25.0232	−1.11573	−0.557865	−	0.829932i	$-0.688379\pi$
$503$	−25.0232	−1.11573	−0.557865	+	0.829932i	$0.688379\pi$
$504$	0	0
$505$	33.7965	1.50393
$506$	2.36184	0.104997
$507$	0	0
$508$	−16.2121	−0.719297
$509$	0.899310	0.0398612	0.0199306	−	0.999801i	$-0.493655\pi$
$509$	0.899310	0.0398612	0.0199306	+	0.999801i	$0.493655\pi$
$510$	0	0
$511$	−35.7597	−1.58192
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	13.4662	0.593967
$515$	−26.7638	−1.17935
$516$	0	0
$517$	−1.94356	−0.0854778
$518$	32.2841	1.41848
$519$	0	0
$520$	−5.63816	−0.247249
$521$	−6.63547	−0.290705	−0.145353	−	0.989380i	$-0.546432\pi$
$521$	−6.63547	−0.290705	−0.145353	+	0.989380i	$0.546432\pi$
$522$	0	0
$523$	−37.9522	−1.65954	−0.829768	−	0.558109i	$-0.811527\pi$
$523$	−37.9522	−1.65954	−0.829768	+	0.558109i	$0.811527\pi$
$524$	2.80066	0.122347
$525$	0	0
$526$	−16.7520	−0.730420
$527$	11.0838	0.482817
$528$	0	0
$529$	23.2490	1.01082
$530$	−6.75608	−0.293465
$531$	0	0
$532$	0	0
$533$	−5.76827	−0.249851
$534$	0	0
$535$	−0.0932736	−0.00403257
$536$	−1.07873	−0.0465939
$537$	0	0
$538$	−3.58946	−0.154753
$539$	0.448311	0.0193101
$540$	0	0
$541$	15.1307	0.650520	0.325260	−	0.945625i	$-0.394548\pi$
$541$	15.1307	0.650520	0.325260	+	0.945625i	$0.394548\pi$
$542$	−23.9590	−1.02913
$543$	0	0
$544$	6.94356	0.297703
$545$	36.7529	1.57432
$546$	0	0
$547$	−32.5868	−1.39331	−0.696655	−	0.717406i	$-0.745329\pi$
$547$	−32.5868	−1.39331	−0.696655	+	0.717406i	$0.745329\pi$
$548$	−16.0351	−0.684985
$549$	0	0
$550$	−2.30541	−0.0983029
$551$	0	0
$552$	0	0
$553$	−31.4543	−1.33757
$554$	18.7219	0.795419
$555$	0	0
$556$	−2.07873	−0.0881576
$557$	19.9436	0.845036	0.422518	−	0.906355i	$-0.361146\pi$
$557$	19.9436	0.845036	0.422518	+	0.906355i	$0.361146\pi$
$558$	0	0
$559$	3.77332	0.159594
$560$	−9.82295	−0.415095
$561$	0	0
$562$	10.6382	0.448744
$563$	3.96822	0.167240	0.0836202	−	0.996498i	$-0.473352\pi$
$563$	3.96822	0.167240	0.0836202	+	0.996498i	$0.473352\pi$
$564$	0	0
$565$	−39.7793	−1.67353
$566$	14.0128	0.589002
$567$	0	0
$568$	16.6236	0.697511
$569$	14.8135	0.621012	0.310506	−	0.950571i	$-0.399501\pi$
$569$	14.8135	0.621012	0.310506	+	0.950571i	$0.399501\pi$
$570$	0	0
$571$	21.0615	0.881396	0.440698	−	0.897655i	$-0.354731\pi$
$571$	21.0615	0.881396	0.440698	+	0.897655i	$0.354731\pi$
$572$	0.573978	0.0239992
$573$	0	0
$574$	−10.0496	−0.419463
$575$	−45.1438	−1.88263
$576$	0	0
$577$	−44.2422	−1.84183	−0.920913	−	0.389769i	$-0.872555\pi$
$577$	−44.2422	−1.84183	−0.920913	+	0.389769i	$0.872555\pi$
$578$	−31.2131	−1.29829
$579$	0	0
$580$	21.6800	0.900215
$581$	33.7374	1.39966
$582$	0	0
$583$	0.687786	0.0284852
$584$	−12.4192	−0.513911
$585$	0	0
$586$	11.5202	0.475896
$587$	−10.0024	−0.412842	−0.206421	−	0.978463i	$-0.566182\pi$
$587$	−10.0024	−0.412842	−0.206421	+	0.978463i	$0.566182\pi$
$588$	0	0
$589$	0	0
$590$	−1.52023	−0.0625869
$591$	0	0
$592$	11.2121	0.460816
$593$	11.4911	0.471884	0.235942	−	0.971767i	$-0.424182\pi$
$593$	11.4911	0.471884	0.235942	+	0.971767i	$0.424182\pi$
$594$	0	0
$595$	68.2063	2.79618
$596$	1.66725	0.0682933
$597$	0	0
$598$	11.2395	0.459616
$599$	6.41653	0.262172	0.131086	−	0.991371i	$-0.458154\pi$
$599$	6.41653	0.262172	0.131086	+	0.991371i	$0.458154\pi$
$600$	0	0
$601$	−31.5158	−1.28556	−0.642778	−	0.766053i	$-0.722218\pi$
$601$	−31.5158	−1.28556	−0.642778	+	0.766053i	$0.722218\pi$
$602$	6.57398	0.267935
$603$	0	0
$604$	−20.0523	−0.815917
$605$	−37.1147	−1.50893
$606$	0	0
$607$	−0.748341	−0.0303742	−0.0151871	−	0.999885i	$-0.504834\pi$
$607$	−0.748341	−0.0303742	−0.0151871	+	0.999885i	$0.504834\pi$
$608$	0	0
$609$	0	0
$610$	42.7529	1.73101
$611$	−9.24897	−0.374173
$612$	0	0
$613$	−30.6878	−1.23947	−0.619734	−	0.784812i	$-0.712760\pi$
$613$	−30.6878	−1.23947	−0.619734	+	0.784812i	$0.712760\pi$
$614$	4.29767	0.173440
$615$	0	0
$616$	1.00000	0.0402911
$617$	27.2918	1.09873	0.549363	−	0.835584i	$-0.314870\pi$
$617$	27.2918	1.09873	0.549363	+	0.835584i	$0.314870\pi$
$618$	0	0
$619$	−13.2371	−0.532044	−0.266022	−	0.963967i	$-0.585709\pi$
$619$	−13.2371	−0.532044	−0.266022	+	0.963967i	$0.585709\pi$
$620$	−5.44562	−0.218701
$621$	0	0
$622$	−22.9932	−0.921943
$623$	−5.16250	−0.206831
$624$	0	0
$625$	−14.1257	−0.565027
$626$	9.20977	0.368096
$627$	0	0
$628$	−4.09833	−0.163541
$629$	−77.8522	−3.10417
$630$	0	0
$631$	−12.2395	−0.487246	−0.243623	−	0.969870i	$-0.578336\pi$
$631$	−12.2395	−0.487246	−0.243623	+	0.969870i	$0.578336\pi$
$632$	−10.9240	−0.434532
$633$	0	0
$634$	−3.35504	−0.133246
$635$	−55.3073	−2.19480
$636$	0	0
$637$	2.13341	0.0845287
$638$	−2.20708	−0.0873792
$639$	0	0
$640$	−3.41147	−0.134850
$641$	8.95367	0.353649	0.176824	−	0.984242i	$-0.443418\pi$
$641$	8.95367	0.353649	0.176824	+	0.984242i	$0.443418\pi$
$642$	0	0
$643$	−31.3696	−1.23710	−0.618548	−	0.785747i	$-0.712279\pi$
$643$	−31.3696	−1.23710	−0.618548	+	0.785747i	$0.712279\pi$
$644$	19.5817	0.771628
$645$	0	0
$646$	0	0
$647$	5.54933	0.218166	0.109083	−	0.994033i	$-0.465208\pi$
$647$	5.54933	0.218166	0.109083	+	0.994033i	$0.465208\pi$
$648$	0	0
$649$	0.154763	0.00607498
$650$	−10.9709	−0.430314
$651$	0	0
$652$	−8.13516	−0.318598
$653$	−38.9103	−1.52268	−0.761340	−	0.648353i	$-0.775458\pi$
$653$	−38.9103	−1.52268	−0.761340	+	0.648353i	$0.775458\pi$
$654$	0	0
$655$	9.55438	0.373320
$656$	−3.49020	−0.136269
$657$	0	0
$658$	−16.1138	−0.628182
$659$	−39.5158	−1.53932	−0.769658	−	0.638456i	$-0.779573\pi$
$659$	−39.5158	−1.53932	−0.769658	+	0.638456i	$0.779573\pi$
$660$	0	0
$661$	0.0915189	0.00355967	0.00177984	−	0.999998i	$-0.499433\pi$
$661$	0.0915189	0.00355967	0.00177984	+	0.999998i	$0.499433\pi$
$662$	−22.0743	−0.857941
$663$	0	0
$664$	11.7169	0.454703
$665$	0	0
$666$	0	0
$667$	−43.2184	−1.67342
$668$	−13.3969	−0.518343
$669$	0	0
$670$	−3.68004	−0.142173
$671$	−4.35235	−0.168021
$672$	0	0
$673$	39.8563	1.53635	0.768173	−	0.640242i	$-0.221166\pi$
$673$	39.8563	1.53635	0.768173	+	0.640242i	$0.221166\pi$
$674$	−13.1165	−0.505229
$675$	0	0
$676$	−10.2686	−0.394945
$677$	29.3824	1.12926	0.564628	−	0.825345i	$-0.309020\pi$
$677$	29.3824	1.12926	0.564628	+	0.825345i	$0.309020\pi$
$678$	0	0
$679$	10.5175	0.403626
$680$	23.6878	0.908385
$681$	0	0
$682$	0.554378	0.0212282
$683$	−21.0933	−0.807112	−0.403556	−	0.914955i	$-0.632226\pi$
$683$	−21.0933	−0.807112	−0.403556	+	0.914955i	$0.632226\pi$
$684$	0	0
$685$	−54.7033	−2.09010
$686$	−16.4388	−0.627637
$687$	0	0
$688$	2.28312	0.0870431
$689$	3.27301	0.124692
$690$	0	0
$691$	−16.1634	−0.614886	−0.307443	−	0.951566i	$-0.599473\pi$
$691$	−16.1634	−0.614886	−0.307443	+	0.951566i	$0.599473\pi$
$692$	−1.02229	−0.0388616
$693$	0	0
$694$	8.75877	0.332478
$695$	−7.09152	−0.268997
$696$	0	0
$697$	24.2344	0.917944
$698$	−3.26352	−0.123526
$699$	0	0
$700$	−19.1138	−0.722434
$701$	20.0729	0.758141	0.379071	−	0.925368i	$-0.376244\pi$
$701$	20.0729	0.758141	0.379071	+	0.925368i	$0.376244\pi$
$702$	0	0
$703$	0	0
$704$	0.347296	0.0130892
$705$	0	0
$706$	29.0838	1.09458
$707$	−28.5253	−1.07280
$708$	0	0
$709$	−38.9009	−1.46095	−0.730476	−	0.682938i	$-0.760702\pi$
$709$	−38.9009	−1.46095	−0.730476	+	0.682938i	$0.760702\pi$
$710$	56.7110	2.12833
$711$	0	0
$712$	−1.79292	−0.0671925
$713$	10.8557	0.406548
$714$	0	0
$715$	1.95811	0.0732292
$716$	2.03508	0.0760546
$717$	0	0
$718$	6.61081	0.246713
$719$	−3.06181	−0.114186	−0.0570932	−	0.998369i	$-0.518183\pi$
$719$	−3.06181	−0.114186	−0.0570932	+	0.998369i	$0.518183\pi$
$720$	0	0
$721$	22.5895	0.841275
$722$	0	0
$723$	0	0
$724$	−22.3037	−0.828909
$725$	42.1857	1.56674
$726$	0	0
$727$	−17.4766	−0.648171	−0.324085	−	0.946028i	$-0.605056\pi$
$727$	−17.4766	−0.648171	−0.324085	+	0.946028i	$0.605056\pi$
$728$	4.75877	0.176372
$729$	0	0
$730$	−42.3678	−1.56810
$731$	−15.8530	−0.586344
$732$	0	0
$733$	−41.4175	−1.52979	−0.764894	−	0.644156i	$-0.777209\pi$
$733$	−41.4175	−1.52979	−0.764894	+	0.644156i	$0.777209\pi$
$734$	5.79561	0.213920
$735$	0	0
$736$	6.80066	0.250676
$737$	0.374638	0.0137999
$738$	0	0
$739$	26.4534	0.973103	0.486551	−	0.873652i	$-0.338255\pi$
$739$	26.4534	0.973103	0.486551	+	0.873652i	$0.338255\pi$
$740$	38.2499	1.40609
$741$	0	0
$742$	5.70233	0.209339
$743$	−15.0760	−0.553086	−0.276543	−	0.961002i	$-0.589189\pi$
$743$	−15.0760	−0.553086	−0.276543	+	0.961002i	$0.589189\pi$
$744$	0	0
$745$	5.68779	0.208384
$746$	−25.8898	−0.947893
$747$	0	0
$748$	−2.41147	−0.0881722
$749$	0.0787257	0.00287657
$750$	0	0
$751$	5.30541	0.193597	0.0967985	−	0.995304i	$-0.469140\pi$
$751$	5.30541	0.193597	0.0967985	+	0.995304i	$0.469140\pi$
$752$	−5.59627	−0.204075
$753$	0	0
$754$	−10.5030	−0.382496
$755$	−68.4080	−2.48962
$756$	0	0
$757$	8.13846	0.295797	0.147899	−	0.989003i	$-0.452749\pi$
$757$	8.13846	0.295797	0.147899	+	0.989003i	$0.452749\pi$
$758$	−19.1557	−0.695766
$759$	0	0
$760$	0	0
$761$	46.2113	1.67516	0.837579	−	0.546316i	$-0.183970\pi$
$761$	46.2113	1.67516	0.837579	+	0.546316i	$0.183970\pi$
$762$	0	0
$763$	−31.0205	−1.12302
$764$	−10.7861	−0.390228
$765$	0	0
$766$	9.69459	0.350280
$767$	0.736482	0.0265928
$768$	0	0
$769$	−26.8435	−0.968001	−0.484000	−	0.875068i	$-0.660817\pi$
$769$	−26.8435	−0.968001	−0.484000	+	0.875068i	$0.660817\pi$
$770$	3.41147	0.122941
$771$	0	0
$772$	2.86484	0.103108
$773$	−21.0256	−0.756238	−0.378119	−	0.925757i	$-0.623429\pi$
$773$	−21.0256	−0.756238	−0.378119	+	0.925757i	$0.623429\pi$
$774$	0	0
$775$	−10.5963	−0.380629
$776$	3.65270	0.131124
$777$	0	0
$778$	−0.817896	−0.0293230
$779$	0	0
$780$	0	0
$781$	−5.77332	−0.206586
$782$	−47.2208	−1.68861
$783$	0	0
$784$	1.29086	0.0461021
$785$	−13.9813	−0.499015
$786$	0	0
$787$	−45.5313	−1.62301	−0.811507	−	0.584342i	$-0.801353\pi$
$787$	−45.5313	−1.62301	−0.811507	+	0.584342i	$0.801353\pi$
$788$	8.33544	0.296938
$789$	0	0
$790$	−37.2668	−1.32589
$791$	33.5749	1.19379
$792$	0	0
$793$	−20.7118	−0.735499
$794$	3.69728	0.131212
$795$	0	0
$796$	−3.03415	−0.107543
$797$	36.1780	1.28149	0.640745	−	0.767754i	$-0.278626\pi$
$797$	36.1780	1.28149	0.640745	+	0.767754i	$0.278626\pi$
$798$	0	0
$799$	38.8580	1.37470
$800$	−6.63816	−0.234694
$801$	0	0
$802$	21.7401	0.767670
$803$	4.31315	0.152208
$804$	0	0
$805$	66.8025	2.35448
$806$	2.63816	0.0929251
$807$	0	0
$808$	−9.90673	−0.348517
$809$	−2.37876	−0.0836326	−0.0418163	−	0.999125i	$-0.513314\pi$
$809$	−2.37876	−0.0836326	−0.0418163	+	0.999125i	$0.513314\pi$
$810$	0	0
$811$	37.0610	1.30139	0.650694	−	0.759340i	$-0.274478\pi$
$811$	37.0610	1.30139	0.650694	+	0.759340i	$0.274478\pi$
$812$	−18.2986	−0.642155
$813$	0	0
$814$	−3.89393	−0.136482
$815$	−27.7529	−0.972142
$816$	0	0
$817$	0	0
$818$	−4.05468	−0.141769
$819$	0	0
$820$	−11.9067	−0.415801
$821$	25.9605	0.906027	0.453013	−	0.891504i	$-0.350349\pi$
$821$	25.9605	0.906027	0.453013	+	0.891504i	$0.350349\pi$
$822$	0	0
$823$	−45.0110	−1.56899	−0.784493	−	0.620138i	$-0.787077\pi$
$823$	−45.0110	−1.56899	−0.784493	+	0.620138i	$0.787077\pi$
$824$	7.84524	0.273302
$825$	0	0
$826$	1.28312	0.0446454
$827$	−16.8429	−0.585684	−0.292842	−	0.956161i	$-0.594601\pi$
$827$	−16.8429	−0.585684	−0.292842	+	0.956161i	$0.594601\pi$
$828$	0	0
$829$	13.0000	0.451509	0.225754	−	0.974184i	$-0.427515\pi$
$829$	13.0000	0.451509	0.225754	+	0.974184i	$0.427515\pi$
$830$	39.9718	1.38744
$831$	0	0
$832$	1.65270	0.0572972
$833$	−8.96316	−0.310555
$834$	0	0
$835$	−45.7033	−1.58163
$836$	0	0
$837$	0	0
$838$	23.4989	0.811755
$839$	1.34049	0.0462788	0.0231394	−	0.999732i	$-0.492634\pi$
$839$	1.34049	0.0462788	0.0231394	+	0.999732i	$0.492634\pi$
$840$	0	0
$841$	11.3865	0.392638
$842$	−23.0729	−0.795143
$843$	0	0
$844$	16.6382	0.572709
$845$	−35.0310	−1.20510
$846$	0	0
$847$	31.3259	1.07637
$848$	1.98040	0.0680072
$849$	0	0
$850$	46.0925	1.58096
$851$	−76.2499	−2.61381
$852$	0	0
$853$	−7.94532	−0.272042	−0.136021	−	0.990706i	$-0.543432\pi$
$853$	−7.94532	−0.272042	−0.136021	+	0.990706i	$0.543432\pi$
$854$	−36.0847	−1.23479
$855$	0	0
$856$	0.0273411	0.000934501 0
$857$	47.8759	1.63541	0.817704	−	0.575638i	$-0.195246\pi$
$857$	47.8759	1.63541	0.817704	+	0.575638i	$0.195246\pi$
$858$	0	0
$859$	1.63722	0.0558613	0.0279306	−	0.999610i	$-0.491108\pi$
$859$	1.63722	0.0558613	0.0279306	+	0.999610i	$0.491108\pi$
$860$	7.78880	0.265596
$861$	0	0
$862$	3.51485	0.119716
$863$	34.9231	1.18880	0.594399	−	0.804170i	$-0.297390\pi$
$863$	34.9231	1.18880	0.594399	+	0.804170i	$0.297390\pi$
$864$	0	0
$865$	−3.48751	−0.118579
$866$	−34.1147	−1.15927
$867$	0	0
$868$	4.59627	0.156007
$869$	3.79385	0.128698
$870$	0	0
$871$	1.78281	0.0604083
$872$	−10.7733	−0.364831
$873$	0	0
$874$	0	0
$875$	−16.0915	−0.543993
$876$	0	0
$877$	6.37195	0.215165	0.107583	−	0.994196i	$-0.465689\pi$
$877$	6.37195	0.215165	0.107583	+	0.994196i	$0.465689\pi$
$878$	9.70140	0.327406
$879$	0	0
$880$	1.18479	0.0399393
$881$	−5.27395	−0.177684	−0.0888419	−	0.996046i	$-0.528317\pi$
$881$	−5.27395	−0.177684	−0.0888419	+	0.996046i	$0.528317\pi$
$882$	0	0
$883$	−3.25133	−0.109416	−0.0547081	−	0.998502i	$-0.517423\pi$
$883$	−3.25133	−0.109416	−0.0547081	+	0.998502i	$0.517423\pi$
$884$	−11.4757	−0.385968
$885$	0	0
$886$	15.1821	0.510053
$887$	−22.3705	−0.751129	−0.375564	−	0.926796i	$-0.622551\pi$
$887$	−22.3705	−0.751129	−0.375564	+	0.926796i	$0.622551\pi$
$888$	0	0
$889$	46.6810	1.56563
$890$	−6.11650	−0.205025
$891$	0	0
$892$	8.17024	0.273560
$893$	0	0
$894$	0	0
$895$	6.94263	0.232067
$896$	2.87939	0.0961935
$897$	0	0
$898$	−34.8384	−1.16257
$899$	−10.1443	−0.338332
$900$	0	0
$901$	−13.7510	−0.458113
$902$	1.21213	0.0403596
$903$	0	0
$904$	11.6604	0.387821
$905$	−76.0883	−2.52926
$906$	0	0
$907$	−3.15745	−0.104841	−0.0524207	−	0.998625i	$-0.516694\pi$
$907$	−3.15745	−0.104841	−0.0524207	+	0.998625i	$0.516694\pi$
$908$	1.80066	0.0597570
$909$	0	0
$910$	16.2344	0.538166
$911$	−44.3387	−1.46901	−0.734504	−	0.678604i	$-0.762585\pi$
$911$	−44.3387	−1.46901	−0.734504	+	0.678604i	$0.762585\pi$
$912$	0	0
$913$	−4.06923	−0.134672
$914$	31.8749	1.05433
$915$	0	0
$916$	−18.7392	−0.619160
$917$	−8.06418	−0.266303
$918$	0	0
$919$	14.2044	0.468560	0.234280	−	0.972169i	$-0.424727\pi$
$919$	14.2044	0.468560	0.234280	+	0.972169i	$0.424727\pi$
$920$	23.2003	0.764890
$921$	0	0
$922$	−10.6895	−0.352041
$923$	−27.4739	−0.904314
$924$	0	0
$925$	74.4279	2.44717
$926$	−15.3182	−0.503387
$927$	0	0
$928$	−6.35504	−0.208614
$929$	15.4266	0.506131	0.253066	−	0.967449i	$-0.418561\pi$
$929$	15.4266	0.506131	0.253066	+	0.967449i	$0.418561\pi$
$930$	0	0
$931$	0	0
$932$	4.09833	0.134245
$933$	0	0
$934$	28.5125	0.932957
$935$	−8.22668	−0.269041
$936$	0	0
$937$	−9.69696	−0.316786	−0.158393	−	0.987376i	$-0.550631\pi$
$937$	−9.69696	−0.316786	−0.158393	+	0.987376i	$0.550631\pi$
$938$	3.10607	0.101417
$939$	0	0
$940$	−19.0915	−0.622697
$941$	−17.9855	−0.586309	−0.293154	−	0.956065i	$-0.594705\pi$
$941$	−17.9855	−0.586309	−0.293154	+	0.956065i	$0.594705\pi$
$942$	0	0
$943$	23.7357	0.772939
$944$	0.445622	0.0145038
$945$	0	0
$946$	−0.792919	−0.0257800
$947$	−2.19396	−0.0712942	−0.0356471	−	0.999364i	$-0.511349\pi$
$947$	−2.19396	−0.0712942	−0.0356471	+	0.999364i	$0.511349\pi$
$948$	0	0
$949$	20.5253	0.666279
$950$	0	0
$951$	0	0
$952$	−19.9932	−0.647983
$953$	−11.3108	−0.366392	−0.183196	−	0.983076i	$-0.558644\pi$
$953$	−11.3108	−0.366392	−0.183196	+	0.983076i	$0.558644\pi$
$954$	0	0
$955$	−36.7965	−1.19071
$956$	28.2276	0.912946
$957$	0	0
$958$	8.05138	0.260128
$959$	46.1712	1.49095
$960$	0	0
$961$	−28.4519	−0.917804
$962$	−18.5303	−0.597442
$963$	0	0
$964$	−6.56212	−0.211352
$965$	9.77332	0.314614
$966$	0	0
$967$	46.7844	1.50448	0.752242	−	0.658887i	$-0.228973\pi$
$967$	46.7844	1.50448	0.752242	+	0.658887i	$0.228973\pi$
$968$	10.8794	0.349677
$969$	0	0
$970$	12.4611	0.400102
$971$	60.3715	1.93741	0.968706	−	0.248211i	$-0.0798426\pi$
$971$	60.3715	1.93741	0.968706	+	0.248211i	$0.0798426\pi$
$972$	0	0
$973$	5.98545	0.191885
$974$	6.11556	0.195955
$975$	0	0
$976$	−12.5321	−0.401142
$977$	22.7956	0.729296	0.364648	−	0.931145i	$-0.381189\pi$
$977$	22.7956	0.729296	0.364648	+	0.931145i	$0.381189\pi$
$978$	0	0
$979$	0.622674	0.0199008
$980$	4.40373	0.140672
$981$	0	0
$982$	−22.6851	−0.723911
$983$	−30.7929	−0.982142	−0.491071	−	0.871120i	$-0.663394\pi$
$983$	−30.7929	−0.982142	−0.491071	+	0.871120i	$0.663394\pi$
$984$	0	0
$985$	28.4361	0.906050
$986$	44.1266	1.40528
$987$	0	0
$988$	0	0
$989$	−15.5267	−0.493721
$990$	0	0
$991$	38.1789	1.21279	0.606397	−	0.795162i	$-0.292614\pi$
$991$	38.1789	1.21279	0.606397	+	0.795162i	$0.292614\pi$
$992$	1.59627	0.0506815
$993$	0	0
$994$	−47.8658	−1.51821
$995$	−10.3509	−0.328146
$996$	0	0
$997$	30.9489	0.980163	0.490081	−	0.871677i	$-0.336967\pi$
$997$	30.9489	0.980163	0.490081	+	0.871677i	$0.336967\pi$
$998$	−24.8557	−0.786792
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bn.1.3		3
3.2	odd	2		2166.2.a.u.1.1		3
19.3	odd	18		342.2.u.a.199.1		6
19.13	odd	18		342.2.u.a.55.1		6
19.18	odd	2		6498.2.a.bs.1.3		3
57.32	even	18		114.2.i.d.55.1	✓	6
57.41	even	18		114.2.i.d.85.1	yes	6
57.56	even	2		2166.2.a.o.1.1		3
228.155	odd	18		912.2.bo.f.769.1		6
228.203	odd	18		912.2.bo.f.625.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.d.55.1	✓	6	57.32	even	18
114.2.i.d.85.1	yes	6	57.41	even	18
342.2.u.a.55.1		6	19.13	odd	18
342.2.u.a.199.1		6	19.3	odd	18
912.2.bo.f.625.1		6	228.203	odd	18
912.2.bo.f.769.1		6	228.155	odd	18
2166.2.a.o.1.1		3	57.56	even	2
2166.2.a.u.1.1		3	3.2	odd	2
6498.2.a.bn.1.3		3	1.1	even	1	trivial
6498.2.a.bs.1.3		3	19.18	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:	\( N \)	\(=\)	\( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6498.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.41147 q^{5} -2.87939 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.41147 q^{5} -2.87939 q^{7} -1.00000 q^{8} -3.41147 q^{10} +0.347296 q^{11} +1.65270 q^{13} +2.87939 q^{14} +1.00000 q^{16} -6.94356 q^{17} +3.41147 q^{20} -0.347296 q^{22} -6.80066 q^{23} +6.63816 q^{25} -1.65270 q^{26} -2.87939 q^{28} +6.35504 q^{29} -1.59627 q^{31} -1.00000 q^{32} +6.94356 q^{34} -9.82295 q^{35} +11.2121 q^{37} -3.41147 q^{40} -3.49020 q^{41} +2.28312 q^{43} +0.347296 q^{44} +6.80066 q^{46} -5.59627 q^{47} +1.29086 q^{49} -6.63816 q^{50} +1.65270 q^{52} +1.98040 q^{53} +1.18479 q^{55} +2.87939 q^{56} -6.35504 q^{58} +0.445622 q^{59} -12.5321 q^{61} +1.59627 q^{62} +1.00000 q^{64} +5.63816 q^{65} +1.07873 q^{67} -6.94356 q^{68} +9.82295 q^{70} -16.6236 q^{71} +12.4192 q^{73} -11.2121 q^{74} -1.00000 q^{77} +10.9240 q^{79} +3.41147 q^{80} +3.49020 q^{82} -11.7169 q^{83} -23.6878 q^{85} -2.28312 q^{86} -0.347296 q^{88} +1.79292 q^{89} -4.75877 q^{91} -6.80066 q^{92} +5.59627 q^{94} -3.65270 q^{97} -1.29086 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 6 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 6 q^{23} + 3 q^{25} - 6 q^{26} - 3 q^{28} - 6 q^{29} + 9 q^{31} - 3 q^{32} + 6 q^{34} - 9 q^{35} + 9 q^{37} - 9 q^{41} + 15 q^{43} + 6 q^{46} - 3 q^{47} - 12 q^{49} - 3 q^{50} + 6 q^{52} + 3 q^{53} + 3 q^{56} + 6 q^{58} + 12 q^{59} - 33 q^{61} - 9 q^{62} + 3 q^{64} + 12 q^{67} - 6 q^{68} + 9 q^{70} - 15 q^{71} + 3 q^{73} - 9 q^{74} - 3 q^{77} + 15 q^{79} + 9 q^{82} - 27 q^{83} - 27 q^{85} - 15 q^{86} + 15 q^{89} - 3 q^{91} - 6 q^{92} + 3 q^{94} - 12 q^{97} + 12 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^7 - 3 * q^8 + 6 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^17 - 6 * q^23 + 3 * q^25 - 6 * q^26 - 3 * q^28 - 6 * q^29 + 9 * q^31 - 3 * q^32 + 6 * q^34 - 9 * q^35 + 9 * q^37 - 9 * q^41 + 15 * q^43 + 6 * q^46 - 3 * q^47 - 12 * q^49 - 3 * q^50 + 6 * q^52 + 3 * q^53 + 3 * q^56 + 6 * q^58 + 12 * q^59 - 33 * q^61 - 9 * q^62 + 3 * q^64 + 12 * q^67 - 6 * q^68 + 9 * q^70 - 15 * q^71 + 3 * q^73 - 9 * q^74 - 3 * q^77 + 15 * q^79 + 9 * q^82 - 27 * q^83 - 27 * q^85 - 15 * q^86 + 15 * q^89 - 3 * q^91 - 6 * q^92 + 3 * q^94 - 12 * q^97 + 12 * q^98

Introduction

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