LMFDB - Embedded newform 6534.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6534,2,Mod(1,6534)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6534, 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("6534.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6534 = 2 \cdot 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6534.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:	$52.1742526807$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 594)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6534.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} +2.85410 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} +2.85410 q^{7} -1.00000 q^{8} -1.23607 q^{10} -6.23607 q^{13} -2.85410 q^{14} +1.00000 q^{16} +6.23607 q^{17} +5.00000 q^{19} +1.23607 q^{20} -7.61803 q^{23} -3.47214 q^{25} +6.23607 q^{26} +2.85410 q^{28} -7.09017 q^{29} -7.00000 q^{31} -1.00000 q^{32} -6.23607 q^{34} +3.52786 q^{35} +9.00000 q^{37} -5.00000 q^{38} -1.23607 q^{40} -3.38197 q^{41} -0.708204 q^{43} +7.61803 q^{46} -1.76393 q^{47} +1.14590 q^{49} +3.47214 q^{50} -6.23607 q^{52} -10.0902 q^{53} -2.85410 q^{56} +7.09017 q^{58} -9.32624 q^{59} +3.47214 q^{61} +7.00000 q^{62} +1.00000 q^{64} -7.70820 q^{65} -6.09017 q^{67} +6.23607 q^{68} -3.52786 q^{70} -4.52786 q^{71} -4.23607 q^{73} -9.00000 q^{74} +5.00000 q^{76} +15.0000 q^{79} +1.23607 q^{80} +3.38197 q^{82} +3.76393 q^{83} +7.70820 q^{85} +0.708204 q^{86} -4.23607 q^{89} -17.7984 q^{91} -7.61803 q^{92} +1.76393 q^{94} +6.18034 q^{95} +6.00000 q^{97} -1.14590 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - q^{7} - 2 q^{8} + 2 q^{10} - 8 q^{13} + q^{14} + 2 q^{16} + 8 q^{17} + 10 q^{19} - 2 q^{20} - 13 q^{23} + 2 q^{25} + 8 q^{26} - q^{28} - 3 q^{29} - 14 q^{31} - 2 q^{32} - 8 q^{34} + 16 q^{35} + 18 q^{37} - 10 q^{38} + 2 q^{40} - 9 q^{41} + 12 q^{43} + 13 q^{46} - 8 q^{47} + 9 q^{49} - 2 q^{50} - 8 q^{52} - 9 q^{53} + q^{56} + 3 q^{58} - 3 q^{59} - 2 q^{61} + 14 q^{62} + 2 q^{64} - 2 q^{65} - q^{67} + 8 q^{68} - 16 q^{70} - 18 q^{71} - 4 q^{73} - 18 q^{74} + 10 q^{76} + 30 q^{79} - 2 q^{80} + 9 q^{82} + 12 q^{83} + 2 q^{85} - 12 q^{86} - 4 q^{89} - 11 q^{91} - 13 q^{92} + 8 q^{94} - 10 q^{95} + 12 q^{97} - 9 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - q^7 - 2 * q^8 + 2 * q^10 - 8 * q^13 + q^14 + 2 * q^16 + 8 * q^17 + 10 * q^19 - 2 * q^20 - 13 * q^23 + 2 * q^25 + 8 * q^26 - q^28 - 3 * q^29 - 14 * q^31 - 2 * q^32 - 8 * q^34 + 16 * q^35 + 18 * q^37 - 10 * q^38 + 2 * q^40 - 9 * q^41 + 12 * q^43 + 13 * q^46 - 8 * q^47 + 9 * q^49 - 2 * q^50 - 8 * q^52 - 9 * q^53 + q^56 + 3 * q^58 - 3 * q^59 - 2 * q^61 + 14 * q^62 + 2 * q^64 - 2 * q^65 - q^67 + 8 * q^68 - 16 * q^70 - 18 * q^71 - 4 * q^73 - 18 * q^74 + 10 * q^76 + 30 * q^79 - 2 * q^80 + 9 * q^82 + 12 * q^83 + 2 * q^85 - 12 * q^86 - 4 * q^89 - 11 * q^91 - 13 * q^92 + 8 * q^94 - 10 * q^95 + 12 * q^97 - 9 * 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
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	2.85410	1.07875	0.539375	−	0.842066i	$-0.318661\pi$
$7$	2.85410	1.07875	0.539375	+	0.842066i	$0.318661\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.23607	−0.390879
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.23607	−1.72957	−0.864787	−	0.502139i	$-0.832547\pi$
$13$	−6.23607	−1.72957	−0.864787	+	0.502139i	$0.832547\pi$
$14$	−2.85410	−0.762791
$15$	0	0
$16$	1.00000	0.250000
$17$	6.23607	1.51247	0.756234	−	0.654301i	$-0.227037\pi$
$17$	6.23607	1.51247	0.756234	+	0.654301i	$0.227037\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	1.23607	0.276393
$21$	0	0
$22$	0	0
$23$	−7.61803	−1.58847	−0.794235	−	0.607611i	$-0.792128\pi$
$23$	−7.61803	−1.58847	−0.794235	+	0.607611i	$0.792128\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	6.23607	1.22299
$27$	0	0
$28$	2.85410	0.539375
$29$	−7.09017	−1.31661	−0.658306	−	0.752751i	$-0.728727\pi$
$29$	−7.09017	−1.31661	−0.658306	+	0.752751i	$0.728727\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.23607	−1.06948
$35$	3.52786	0.596318
$36$	0	0
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	−5.00000	−0.811107
$39$	0	0
$40$	−1.23607	−0.195440
$41$	−3.38197	−0.528174	−0.264087	−	0.964499i	$-0.585071\pi$
$41$	−3.38197	−0.528174	−0.264087	+	0.964499i	$0.585071\pi$
$42$	0	0
$43$	−0.708204	−0.108000	−0.0540000	−	0.998541i	$-0.517197\pi$
$43$	−0.708204	−0.108000	−0.0540000	+	0.998541i	$0.517197\pi$
$44$	0	0
$45$	0	0
$46$	7.61803	1.12322
$47$	−1.76393	−0.257296	−0.128648	−	0.991690i	$-0.541064\pi$
$47$	−1.76393	−0.257296	−0.128648	+	0.991690i	$0.541064\pi$
$48$	0	0
$49$	1.14590	0.163700
$50$	3.47214	0.491034
$51$	0	0
$52$	−6.23607	−0.864787
$53$	−10.0902	−1.38599	−0.692996	−	0.720942i	$-0.743710\pi$
$53$	−10.0902	−1.38599	−0.692996	+	0.720942i	$0.743710\pi$
$54$	0	0
$55$	0	0
$56$	−2.85410	−0.381395
$57$	0	0
$58$	7.09017	0.930985
$59$	−9.32624	−1.21417	−0.607086	−	0.794636i	$-0.707662\pi$
$59$	−9.32624	−1.21417	−0.607086	+	0.794636i	$0.707662\pi$
$60$	0	0
$61$	3.47214	0.444561	0.222281	−	0.974983i	$-0.428650\pi$
$61$	3.47214	0.444561	0.222281	+	0.974983i	$0.428650\pi$
$62$	7.00000	0.889001
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.70820	−0.956085
$66$	0	0
$67$	−6.09017	−0.744033	−0.372016	−	0.928226i	$-0.621333\pi$
$67$	−6.09017	−0.744033	−0.372016	+	0.928226i	$0.621333\pi$
$68$	6.23607	0.756234
$69$	0	0
$70$	−3.52786	−0.421660
$71$	−4.52786	−0.537359	−0.268679	−	0.963230i	$-0.586587\pi$
$71$	−4.52786	−0.537359	−0.268679	+	0.963230i	$0.586587\pi$
$72$	0	0
$73$	−4.23607	−0.495794	−0.247897	−	0.968786i	$-0.579739\pi$
$73$	−4.23607	−0.495794	−0.247897	+	0.968786i	$0.579739\pi$
$74$	−9.00000	−1.04623
$75$	0	0
$76$	5.00000	0.573539
$77$	0	0
$78$	0	0
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	1.23607	0.138197
$81$	0	0
$82$	3.38197	0.373476
$83$	3.76393	0.413145	0.206573	−	0.978431i	$-0.433769\pi$
$83$	3.76393	0.413145	0.206573	+	0.978431i	$0.433769\pi$
$84$	0	0
$85$	7.70820	0.836072
$86$	0.708204	0.0763676
$87$	0	0
$88$	0	0
$89$	−4.23607	−0.449022	−0.224511	−	0.974472i	$-0.572079\pi$
$89$	−4.23607	−0.449022	−0.224511	+	0.974472i	$0.572079\pi$
$90$	0	0
$91$	−17.7984	−1.86578
$92$	−7.61803	−0.794235
$93$	0	0
$94$	1.76393	0.181936
$95$	6.18034	0.634089
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−1.14590	−0.115753
$99$	0	0
$100$	−3.47214	−0.347214
$101$	−17.0344	−1.69499	−0.847495	−	0.530803i	$-0.821890\pi$
$101$	−17.0344	−1.69499	−0.847495	+	0.530803i	$0.821890\pi$
$102$	0	0
$103$	−5.56231	−0.548070	−0.274035	−	0.961720i	$-0.588358\pi$
$103$	−5.56231	−0.548070	−0.274035	+	0.961720i	$0.588358\pi$
$104$	6.23607	0.611497
$105$	0	0
$106$	10.0902	0.980044
$107$	−18.2705	−1.76628	−0.883138	−	0.469112i	$-0.844574\pi$
$107$	−18.2705	−1.76628	−0.883138	+	0.469112i	$0.844574\pi$
$108$	0	0
$109$	4.32624	0.414378	0.207189	−	0.978301i	$-0.433568\pi$
$109$	4.32624	0.414378	0.207189	+	0.978301i	$0.433568\pi$
$110$	0	0
$111$	0	0
$112$	2.85410	0.269687
$113$	2.56231	0.241041	0.120521	−	0.992711i	$-0.461544\pi$
$113$	2.56231	0.241041	0.120521	+	0.992711i	$0.461544\pi$
$114$	0	0
$115$	−9.41641	−0.878085
$116$	−7.09017	−0.658306
$117$	0	0
$118$	9.32624	0.858550
$119$	17.7984	1.63157
$120$	0	0
$121$	0	0
$122$	−3.47214	−0.314352
$123$	0	0
$124$	−7.00000	−0.628619
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−3.47214	−0.308102	−0.154051	−	0.988063i	$-0.549232\pi$
$127$	−3.47214	−0.308102	−0.154051	+	0.988063i	$0.549232\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	7.70820	0.676054
$131$	6.94427	0.606724	0.303362	−	0.952875i	$-0.401891\pi$
$131$	6.94427	0.606724	0.303362	+	0.952875i	$0.401891\pi$
$132$	0	0
$133$	14.2705	1.23741
$134$	6.09017	0.526111
$135$	0	0
$136$	−6.23607	−0.534738
$137$	−14.1803	−1.21151	−0.605754	−	0.795652i	$-0.707128\pi$
$137$	−14.1803	−1.21151	−0.605754	+	0.795652i	$0.707128\pi$
$138$	0	0
$139$	−8.56231	−0.726245	−0.363123	−	0.931741i	$-0.618289\pi$
$139$	−8.56231	−0.726245	−0.363123	+	0.931741i	$0.618289\pi$
$140$	3.52786	0.298159
$141$	0	0
$142$	4.52786	0.379970
$143$	0	0
$144$	0	0
$145$	−8.76393	−0.727805
$146$	4.23607	0.350579
$147$	0	0
$148$	9.00000	0.739795
$149$	7.65248	0.626915	0.313458	−	0.949602i	$-0.398513\pi$
$149$	7.65248	0.626915	0.313458	+	0.949602i	$0.398513\pi$
$150$	0	0
$151$	−3.85410	−0.313642	−0.156821	−	0.987627i	$-0.550125\pi$
$151$	−3.85410	−0.313642	−0.156821	+	0.987627i	$0.550125\pi$
$152$	−5.00000	−0.405554
$153$	0	0
$154$	0	0
$155$	−8.65248	−0.694984
$156$	0	0
$157$	−7.41641	−0.591894	−0.295947	−	0.955204i	$-0.595635\pi$
$157$	−7.41641	−0.591894	−0.295947	+	0.955204i	$0.595635\pi$
$158$	−15.0000	−1.19334
$159$	0	0
$160$	−1.23607	−0.0977198
$161$	−21.7426	−1.71356
$162$	0	0
$163$	−1.76393	−0.138162	−0.0690809	−	0.997611i	$-0.522007\pi$
$163$	−1.76393	−0.138162	−0.0690809	+	0.997611i	$0.522007\pi$
$164$	−3.38197	−0.264087
$165$	0	0
$166$	−3.76393	−0.292138
$167$	5.38197	0.416469	0.208235	−	0.978079i	$-0.433228\pi$
$167$	5.38197	0.416469	0.208235	+	0.978079i	$0.433228\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	−7.70820	−0.591192
$171$	0	0
$172$	−0.708204	−0.0540000
$173$	−3.94427	−0.299877	−0.149939	−	0.988695i	$-0.547908\pi$
$173$	−3.94427	−0.299877	−0.149939	+	0.988695i	$0.547908\pi$
$174$	0	0
$175$	−9.90983	−0.749113
$176$	0	0
$177$	0	0
$178$	4.23607	0.317507
$179$	1.79837	0.134417	0.0672084	−	0.997739i	$-0.478591\pi$
$179$	1.79837	0.134417	0.0672084	+	0.997739i	$0.478591\pi$
$180$	0	0
$181$	−11.3820	−0.846015	−0.423007	−	0.906126i	$-0.639026\pi$
$181$	−11.3820	−0.846015	−0.423007	+	0.906126i	$0.639026\pi$
$182$	17.7984	1.31930
$183$	0	0
$184$	7.61803	0.561609
$185$	11.1246	0.817898
$186$	0	0
$187$	0	0
$188$	−1.76393	−0.128648
$189$	0	0
$190$	−6.18034	−0.448369
$191$	−3.67376	−0.265824	−0.132912	−	0.991128i	$-0.542433\pi$
$191$	−3.67376	−0.265824	−0.132912	+	0.991128i	$0.542433\pi$
$192$	0	0
$193$	17.4164	1.25366	0.626830	−	0.779156i	$-0.284352\pi$
$193$	17.4164	1.25366	0.626830	+	0.779156i	$0.284352\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	1.14590	0.0818499
$197$	12.3262	0.878208	0.439104	−	0.898436i	$-0.355296\pi$
$197$	12.3262	0.878208	0.439104	+	0.898436i	$0.355296\pi$
$198$	0	0
$199$	1.94427	0.137826	0.0689129	−	0.997623i	$-0.478047\pi$
$199$	1.94427	0.137826	0.0689129	+	0.997623i	$0.478047\pi$
$200$	3.47214	0.245517
$201$	0	0
$202$	17.0344	1.19854
$203$	−20.2361	−1.42029
$204$	0	0
$205$	−4.18034	−0.291968
$206$	5.56231	0.387544
$207$	0	0
$208$	−6.23607	−0.432394
$209$	0	0
$210$	0	0
$211$	−11.3262	−0.779730	−0.389865	−	0.920872i	$-0.627478\pi$
$211$	−11.3262	−0.779730	−0.389865	+	0.920872i	$0.627478\pi$
$212$	−10.0902	−0.692996
$213$	0	0
$214$	18.2705	1.24895
$215$	−0.875388	−0.0597010
$216$	0	0
$217$	−19.9787	−1.35624
$218$	−4.32624	−0.293010
$219$	0	0
$220$	0	0
$221$	−38.8885	−2.61593
$222$	0	0
$223$	8.41641	0.563604	0.281802	−	0.959473i	$-0.409068\pi$
$223$	8.41641	0.563604	0.281802	+	0.959473i	$0.409068\pi$
$224$	−2.85410	−0.190698
$225$	0	0
$226$	−2.56231	−0.170442
$227$	16.7984	1.11495	0.557474	−	0.830195i	$-0.311771\pi$
$227$	16.7984	1.11495	0.557474	+	0.830195i	$0.311771\pi$
$228$	0	0
$229$	22.2705	1.47168	0.735838	−	0.677157i	$-0.236788\pi$
$229$	22.2705	1.47168	0.735838	+	0.677157i	$0.236788\pi$
$230$	9.41641	0.620900
$231$	0	0
$232$	7.09017	0.465492
$233$	19.6525	1.28748	0.643738	−	0.765246i	$-0.277383\pi$
$233$	19.6525	1.28748	0.643738	+	0.765246i	$0.277383\pi$
$234$	0	0
$235$	−2.18034	−0.142230
$236$	−9.32624	−0.607086
$237$	0	0
$238$	−17.7984	−1.15370
$239$	2.56231	0.165742	0.0828709	−	0.996560i	$-0.473591\pi$
$239$	2.56231	0.165742	0.0828709	+	0.996560i	$0.473591\pi$
$240$	0	0
$241$	−20.0344	−1.29053	−0.645266	−	0.763958i	$-0.723253\pi$
$241$	−20.0344	−1.29053	−0.645266	+	0.763958i	$0.723253\pi$
$242$	0	0
$243$	0	0
$244$	3.47214	0.222281
$245$	1.41641	0.0904910
$246$	0	0
$247$	−31.1803	−1.98396
$248$	7.00000	0.444500
$249$	0	0
$250$	10.4721	0.662316
$251$	26.2361	1.65601	0.828003	−	0.560724i	$-0.189477\pi$
$251$	26.2361	1.65601	0.828003	+	0.560724i	$0.189477\pi$
$252$	0	0
$253$	0	0
$254$	3.47214	0.217861
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.47214	−0.466099	−0.233050	−	0.972465i	$-0.574870\pi$
$257$	−7.47214	−0.466099	−0.233050	+	0.972465i	$0.574870\pi$
$258$	0	0
$259$	25.6869	1.59611
$260$	−7.70820	−0.478043
$261$	0	0
$262$	−6.94427	−0.429019
$263$	−17.9443	−1.10649	−0.553246	−	0.833018i	$-0.686611\pi$
$263$	−17.9443	−1.10649	−0.553246	+	0.833018i	$0.686611\pi$
$264$	0	0
$265$	−12.4721	−0.766157
$266$	−14.2705	−0.874981
$267$	0	0
$268$	−6.09017	−0.372016
$269$	−9.94427	−0.606313	−0.303156	−	0.952941i	$-0.598040\pi$
$269$	−9.94427	−0.606313	−0.303156	+	0.952941i	$0.598040\pi$
$270$	0	0
$271$	11.2705	0.684635	0.342317	−	0.939584i	$-0.388788\pi$
$271$	11.2705	0.684635	0.342317	+	0.939584i	$0.388788\pi$
$272$	6.23607	0.378117
$273$	0	0
$274$	14.1803	0.856666
$275$	0	0
$276$	0	0
$277$	−0.145898	−0.00876616	−0.00438308	−	0.999990i	$-0.501395\pi$
$277$	−0.145898	−0.00876616	−0.00438308	+	0.999990i	$0.501395\pi$
$278$	8.56231	0.513533
$279$	0	0
$280$	−3.52786	−0.210830
$281$	20.1803	1.20386	0.601929	−	0.798550i	$-0.294399\pi$
$281$	20.1803	1.20386	0.601929	+	0.798550i	$0.294399\pi$
$282$	0	0
$283$	6.70820	0.398761	0.199381	−	0.979922i	$-0.436107\pi$
$283$	6.70820	0.398761	0.199381	+	0.979922i	$0.436107\pi$
$284$	−4.52786	−0.268679
$285$	0	0
$286$	0	0
$287$	−9.65248	−0.569768
$288$	0	0
$289$	21.8885	1.28756
$290$	8.76393	0.514636
$291$	0	0
$292$	−4.23607	−0.247897
$293$	21.7639	1.27146	0.635731	−	0.771910i	$-0.280699\pi$
$293$	21.7639	1.27146	0.635731	+	0.771910i	$0.280699\pi$
$294$	0	0
$295$	−11.5279	−0.671178
$296$	−9.00000	−0.523114
$297$	0	0
$298$	−7.65248	−0.443296
$299$	47.5066	2.74738
$300$	0	0
$301$	−2.02129	−0.116505
$302$	3.85410	0.221779
$303$	0	0
$304$	5.00000	0.286770
$305$	4.29180	0.245748
$306$	0	0
$307$	−5.09017	−0.290511	−0.145256	−	0.989394i	$-0.546400\pi$
$307$	−5.09017	−0.290511	−0.145256	+	0.989394i	$0.546400\pi$
$308$	0	0
$309$	0	0
$310$	8.65248	0.491428
$311$	−13.4721	−0.763935	−0.381967	−	0.924176i	$-0.624753\pi$
$311$	−13.4721	−0.763935	−0.381967	+	0.924176i	$0.624753\pi$
$312$	0	0
$313$	33.0689	1.86916	0.934582	−	0.355748i	$-0.115774\pi$
$313$	33.0689	1.86916	0.934582	+	0.355748i	$0.115774\pi$
$314$	7.41641	0.418532
$315$	0	0
$316$	15.0000	0.843816
$317$	−16.1459	−0.906844	−0.453422	−	0.891296i	$-0.649797\pi$
$317$	−16.1459	−0.906844	−0.453422	+	0.891296i	$0.649797\pi$
$318$	0	0
$319$	0	0
$320$	1.23607	0.0690983
$321$	0	0
$322$	21.7426	1.21167
$323$	31.1803	1.73492
$324$	0	0
$325$	21.6525	1.20106
$326$	1.76393	0.0976952
$327$	0	0
$328$	3.38197	0.186738
$329$	−5.03444	−0.277558
$330$	0	0
$331$	−8.29180	−0.455758	−0.227879	−	0.973689i	$-0.573179\pi$
$331$	−8.29180	−0.455758	−0.227879	+	0.973689i	$0.573179\pi$
$332$	3.76393	0.206573
$333$	0	0
$334$	−5.38197	−0.294488
$335$	−7.52786	−0.411291
$336$	0	0
$337$	2.38197	0.129754	0.0648770	−	0.997893i	$-0.479335\pi$
$337$	2.38197	0.129754	0.0648770	+	0.997893i	$0.479335\pi$
$338$	−25.8885	−1.40815
$339$	0	0
$340$	7.70820	0.418036
$341$	0	0
$342$	0	0
$343$	−16.7082	−0.902158
$344$	0.708204	0.0381838
$345$	0	0
$346$	3.94427	0.212045
$347$	31.3262	1.68168	0.840840	−	0.541283i	$-0.182061\pi$
$347$	31.3262	1.68168	0.840840	+	0.541283i	$0.182061\pi$
$348$	0	0
$349$	17.3262	0.927452	0.463726	−	0.885979i	$-0.346512\pi$
$349$	17.3262	0.927452	0.463726	+	0.885979i	$0.346512\pi$
$350$	9.90983	0.529703
$351$	0	0
$352$	0	0
$353$	9.32624	0.496386	0.248193	−	0.968711i	$-0.420163\pi$
$353$	9.32624	0.496386	0.248193	+	0.968711i	$0.420163\pi$
$354$	0	0
$355$	−5.59675	−0.297045
$356$	−4.23607	−0.224511
$357$	0	0
$358$	−1.79837	−0.0950470
$359$	−32.1246	−1.69547	−0.847736	−	0.530418i	$-0.822035\pi$
$359$	−32.1246	−1.69547	−0.847736	+	0.530418i	$0.822035\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	11.3820	0.598223
$363$	0	0
$364$	−17.7984	−0.932888
$365$	−5.23607	−0.274068
$366$	0	0
$367$	−12.8541	−0.670979	−0.335489	−	0.942044i	$-0.608902\pi$
$367$	−12.8541	−0.670979	−0.335489	+	0.942044i	$0.608902\pi$
$368$	−7.61803	−0.397117
$369$	0	0
$370$	−11.1246	−0.578341
$371$	−28.7984	−1.49514
$372$	0	0
$373$	−23.7082	−1.22756	−0.613782	−	0.789475i	$-0.710353\pi$
$373$	−23.7082	−1.22756	−0.613782	+	0.789475i	$0.710353\pi$
$374$	0	0
$375$	0	0
$376$	1.76393	0.0909678
$377$	44.2148	2.27718
$378$	0	0
$379$	14.7082	0.755510	0.377755	−	0.925906i	$-0.376696\pi$
$379$	14.7082	0.755510	0.377755	+	0.925906i	$0.376696\pi$
$380$	6.18034	0.317045
$381$	0	0
$382$	3.67376	0.187966
$383$	−18.6525	−0.953097	−0.476548	−	0.879148i	$-0.658112\pi$
$383$	−18.6525	−0.953097	−0.476548	+	0.879148i	$0.658112\pi$
$384$	0	0
$385$	0	0
$386$	−17.4164	−0.886472
$387$	0	0
$388$	6.00000	0.304604
$389$	−29.9230	−1.51716	−0.758578	−	0.651582i	$-0.774105\pi$
$389$	−29.9230	−1.51716	−0.758578	+	0.651582i	$0.774105\pi$
$390$	0	0
$391$	−47.5066	−2.40251
$392$	−1.14590	−0.0578766
$393$	0	0
$394$	−12.3262	−0.620987
$395$	18.5410	0.932900
$396$	0	0
$397$	24.2705	1.21810	0.609051	−	0.793131i	$-0.291550\pi$
$397$	24.2705	1.21810	0.609051	+	0.793131i	$0.291550\pi$
$398$	−1.94427	−0.0974575
$399$	0	0
$400$	−3.47214	−0.173607
$401$	−19.6738	−0.982461	−0.491230	−	0.871030i	$-0.663453\pi$
$401$	−19.6738	−0.982461	−0.491230	+	0.871030i	$0.663453\pi$
$402$	0	0
$403$	43.6525	2.17448
$404$	−17.0344	−0.847495
$405$	0	0
$406$	20.2361	1.00430
$407$	0	0
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	4.18034	0.206452
$411$	0	0
$412$	−5.56231	−0.274035
$413$	−26.6180	−1.30979
$414$	0	0
$415$	4.65248	0.228381
$416$	6.23607	0.305748
$417$	0	0
$418$	0	0
$419$	−4.14590	−0.202540	−0.101270	−	0.994859i	$-0.532291\pi$
$419$	−4.14590	−0.202540	−0.101270	+	0.994859i	$0.532291\pi$
$420$	0	0
$421$	−34.4164	−1.67735	−0.838677	−	0.544630i	$-0.816670\pi$
$421$	−34.4164	−1.67735	−0.838677	+	0.544630i	$0.816670\pi$
$422$	11.3262	0.551353
$423$	0	0
$424$	10.0902	0.490022
$425$	−21.6525	−1.05030
$426$	0	0
$427$	9.90983	0.479570
$428$	−18.2705	−0.883138
$429$	0	0
$430$	0.875388	0.0422150
$431$	2.76393	0.133134	0.0665670	−	0.997782i	$-0.478795\pi$
$431$	2.76393	0.133134	0.0665670	+	0.997782i	$0.478795\pi$
$432$	0	0
$433$	0.527864	0.0253675	0.0126838	−	0.999920i	$-0.495963\pi$
$433$	0.527864	0.0253675	0.0126838	+	0.999920i	$0.495963\pi$
$434$	19.9787	0.959009
$435$	0	0
$436$	4.32624	0.207189
$437$	−38.0902	−1.82210
$438$	0	0
$439$	−5.27051	−0.251548	−0.125774	−	0.992059i	$-0.540141\pi$
$439$	−5.27051	−0.251548	−0.125774	+	0.992059i	$0.540141\pi$
$440$	0	0
$441$	0	0
$442$	38.8885	1.84974
$443$	−3.00000	−0.142534	−0.0712672	−	0.997457i	$-0.522704\pi$
$443$	−3.00000	−0.142534	−0.0712672	+	0.997457i	$0.522704\pi$
$444$	0	0
$445$	−5.23607	−0.248213
$446$	−8.41641	−0.398528
$447$	0	0
$448$	2.85410	0.134844
$449$	−25.4164	−1.19947	−0.599737	−	0.800197i	$-0.704728\pi$
$449$	−25.4164	−1.19947	−0.599737	+	0.800197i	$0.704728\pi$
$450$	0	0
$451$	0	0
$452$	2.56231	0.120521
$453$	0	0
$454$	−16.7984	−0.788387
$455$	−22.0000	−1.03138
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	−22.2705	−1.04063
$459$	0	0
$460$	−9.41641	−0.439042
$461$	5.03444	0.234477	0.117239	−	0.993104i	$-0.462596\pi$
$461$	5.03444	0.234477	0.117239	+	0.993104i	$0.462596\pi$
$462$	0	0
$463$	35.1246	1.63238	0.816190	−	0.577784i	$-0.196082\pi$
$463$	35.1246	1.63238	0.816190	+	0.577784i	$0.196082\pi$
$464$	−7.09017	−0.329153
$465$	0	0
$466$	−19.6525	−0.910383
$467$	−23.7426	−1.09868	−0.549339	−	0.835599i	$-0.685121\pi$
$467$	−23.7426	−1.09868	−0.549339	+	0.835599i	$0.685121\pi$
$468$	0	0
$469$	−17.3820	−0.802625
$470$	2.18034	0.100572
$471$	0	0
$472$	9.32624	0.429275
$473$	0	0
$474$	0	0
$475$	−17.3607	−0.796563
$476$	17.7984	0.815787
$477$	0	0
$478$	−2.56231	−0.117197
$479$	3.65248	0.166886	0.0834429	−	0.996513i	$-0.473408\pi$
$479$	3.65248	0.166886	0.0834429	+	0.996513i	$0.473408\pi$
$480$	0	0
$481$	−56.1246	−2.55906
$482$	20.0344	0.912544
$483$	0	0
$484$	0	0
$485$	7.41641	0.336762
$486$	0	0
$487$	16.3262	0.739812	0.369906	−	0.929069i	$-0.379390\pi$
$487$	16.3262	0.739812	0.369906	+	0.929069i	$0.379390\pi$
$488$	−3.47214	−0.157176
$489$	0	0
$490$	−1.41641	−0.0639868
$491$	−30.7082	−1.38584	−0.692921	−	0.721014i	$-0.743677\pi$
$491$	−30.7082	−1.38584	−0.692921	+	0.721014i	$0.743677\pi$
$492$	0	0
$493$	−44.2148	−1.99133
$494$	31.1803	1.40287
$495$	0	0
$496$	−7.00000	−0.314309
$497$	−12.9230	−0.579675
$498$	0	0
$499$	18.2918	0.818853	0.409427	−	0.912343i	$-0.365729\pi$
$499$	18.2918	0.818853	0.409427	+	0.912343i	$0.365729\pi$
$500$	−10.4721	−0.468328
$501$	0	0
$502$	−26.2361	−1.17097
$503$	−1.23607	−0.0551135	−0.0275568	−	0.999620i	$-0.508773\pi$
$503$	−1.23607	−0.0551135	−0.0275568	+	0.999620i	$0.508773\pi$
$504$	0	0
$505$	−21.0557	−0.936968
$506$	0	0
$507$	0	0
$508$	−3.47214	−0.154051
$509$	−24.5410	−1.08776	−0.543881	−	0.839162i	$-0.683046\pi$
$509$	−24.5410	−1.08776	−0.543881	+	0.839162i	$0.683046\pi$
$510$	0	0
$511$	−12.0902	−0.534838
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	7.47214	0.329582
$515$	−6.87539	−0.302966
$516$	0	0
$517$	0	0
$518$	−25.6869	−1.12862
$519$	0	0
$520$	7.70820	0.338027
$521$	0.236068	0.0103423	0.00517116	−	0.999987i	$-0.498354\pi$
$521$	0.236068	0.0103423	0.00517116	+	0.999987i	$0.498354\pi$
$522$	0	0
$523$	−27.8885	−1.21948	−0.609740	−	0.792601i	$-0.708726\pi$
$523$	−27.8885	−1.21948	−0.609740	+	0.792601i	$0.708726\pi$
$524$	6.94427	0.303362
$525$	0	0
$526$	17.9443	0.782407
$527$	−43.6525	−1.90153
$528$	0	0
$529$	35.0344	1.52324
$530$	12.4721	0.541755
$531$	0	0
$532$	14.2705	0.618705
$533$	21.0902	0.913517
$534$	0	0
$535$	−22.5836	−0.976374
$536$	6.09017	0.263055
$537$	0	0
$538$	9.94427	0.428728
$539$	0	0
$540$	0	0
$541$	−3.56231	−0.153155	−0.0765777	−	0.997064i	$-0.524399\pi$
$541$	−3.56231	−0.153155	−0.0765777	+	0.997064i	$0.524399\pi$
$542$	−11.2705	−0.484110
$543$	0	0
$544$	−6.23607	−0.267369
$545$	5.34752	0.229063
$546$	0	0
$547$	−26.4721	−1.13187	−0.565933	−	0.824451i	$-0.691484\pi$
$547$	−26.4721	−1.13187	−0.565933	+	0.824451i	$0.691484\pi$
$548$	−14.1803	−0.605754
$549$	0	0
$550$	0	0
$551$	−35.4508	−1.51026
$552$	0	0
$553$	42.8115	1.82053
$554$	0.145898	0.00619861
$555$	0	0
$556$	−8.56231	−0.363123
$557$	15.8197	0.670301	0.335150	−	0.942165i	$-0.391213\pi$
$557$	15.8197	0.670301	0.335150	+	0.942165i	$0.391213\pi$
$558$	0	0
$559$	4.41641	0.186794
$560$	3.52786	0.149079
$561$	0	0
$562$	−20.1803	−0.851256
$563$	−41.4508	−1.74695	−0.873473	−	0.486873i	$-0.838137\pi$
$563$	−41.4508	−1.74695	−0.873473	+	0.486873i	$0.838137\pi$
$564$	0	0
$565$	3.16718	0.133244
$566$	−6.70820	−0.281967
$567$	0	0
$568$	4.52786	0.189985
$569$	−14.3262	−0.600587	−0.300294	−	0.953847i	$-0.597085\pi$
$569$	−14.3262	−0.600587	−0.300294	+	0.953847i	$0.597085\pi$
$570$	0	0
$571$	31.4508	1.31618	0.658089	−	0.752941i	$-0.271365\pi$
$571$	31.4508	1.31618	0.658089	+	0.752941i	$0.271365\pi$
$572$	0	0
$573$	0	0
$574$	9.65248	0.402887
$575$	26.4508	1.10308
$576$	0	0
$577$	−32.1591	−1.33880	−0.669399	−	0.742903i	$-0.733449\pi$
$577$	−32.1591	−1.33880	−0.669399	+	0.742903i	$0.733449\pi$
$578$	−21.8885	−0.910443
$579$	0	0
$580$	−8.76393	−0.363902
$581$	10.7426	0.445680
$582$	0	0
$583$	0	0
$584$	4.23607	0.175290
$585$	0	0
$586$	−21.7639	−0.899060
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	0	0
$589$	−35.0000	−1.44215
$590$	11.5279	0.474595
$591$	0	0
$592$	9.00000	0.369898
$593$	−22.1459	−0.909423	−0.454712	−	0.890639i	$-0.650258\pi$
$593$	−22.1459	−0.909423	−0.454712	+	0.890639i	$0.650258\pi$
$594$	0	0
$595$	22.0000	0.901912
$596$	7.65248	0.313458
$597$	0	0
$598$	−47.5066	−1.94269
$599$	30.5410	1.24787	0.623936	−	0.781475i	$-0.285533\pi$
$599$	30.5410	1.24787	0.623936	+	0.781475i	$0.285533\pi$
$600$	0	0
$601$	9.12461	0.372201	0.186100	−	0.982531i	$-0.440415\pi$
$601$	9.12461	0.372201	0.186100	+	0.982531i	$0.440415\pi$
$602$	2.02129	0.0823815
$603$	0	0
$604$	−3.85410	−0.156821
$605$	0	0
$606$	0	0
$607$	−23.1246	−0.938599	−0.469300	−	0.883039i	$-0.655494\pi$
$607$	−23.1246	−0.938599	−0.469300	+	0.883039i	$0.655494\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	−4.29180	−0.173770
$611$	11.0000	0.445012
$612$	0	0
$613$	−44.4164	−1.79396	−0.896981	−	0.442069i	$-0.854245\pi$
$613$	−44.4164	−1.79396	−0.896981	+	0.442069i	$0.854245\pi$
$614$	5.09017	0.205423
$615$	0	0
$616$	0	0
$617$	22.4721	0.904694	0.452347	−	0.891842i	$-0.350587\pi$
$617$	22.4721	0.904694	0.452347	+	0.891842i	$0.350587\pi$
$618$	0	0
$619$	−27.7426	−1.11507	−0.557536	−	0.830153i	$-0.688253\pi$
$619$	−27.7426	−1.11507	−0.557536	+	0.830153i	$0.688253\pi$
$620$	−8.65248	−0.347492
$621$	0	0
$622$	13.4721	0.540183
$623$	−12.0902	−0.484382
$624$	0	0
$625$	4.41641	0.176656
$626$	−33.0689	−1.32170
$627$	0	0
$628$	−7.41641	−0.295947
$629$	56.1246	2.23783
$630$	0	0
$631$	0.854102	0.0340013	0.0170006	−	0.999855i	$-0.494588\pi$
$631$	0.854102	0.0340013	0.0170006	+	0.999855i	$0.494588\pi$
$632$	−15.0000	−0.596668
$633$	0	0
$634$	16.1459	0.641236
$635$	−4.29180	−0.170315
$636$	0	0
$637$	−7.14590	−0.283131
$638$	0	0
$639$	0	0
$640$	−1.23607	−0.0488599
$641$	−3.03444	−0.119853	−0.0599266	−	0.998203i	$-0.519087\pi$
$641$	−3.03444	−0.119853	−0.0599266	+	0.998203i	$0.519087\pi$
$642$	0	0
$643$	0.875388	0.0345219	0.0172610	−	0.999851i	$-0.494505\pi$
$643$	0.875388	0.0345219	0.0172610	+	0.999851i	$0.494505\pi$
$644$	−21.7426	−0.856780
$645$	0	0
$646$	−31.1803	−1.22677
$647$	−25.5066	−1.00277	−0.501384	−	0.865225i	$-0.667175\pi$
$647$	−25.5066	−1.00277	−0.501384	+	0.865225i	$0.667175\pi$
$648$	0	0
$649$	0	0
$650$	−21.6525	−0.849280
$651$	0	0
$652$	−1.76393	−0.0690809
$653$	9.50658	0.372021	0.186011	−	0.982548i	$-0.440444\pi$
$653$	9.50658	0.372021	0.186011	+	0.982548i	$0.440444\pi$
$654$	0	0
$655$	8.58359	0.335389
$656$	−3.38197	−0.132044
$657$	0	0
$658$	5.03444	0.196263
$659$	26.7771	1.04309	0.521544	−	0.853225i	$-0.325356\pi$
$659$	26.7771	1.04309	0.521544	+	0.853225i	$0.325356\pi$
$660$	0	0
$661$	−7.61803	−0.296307	−0.148154	−	0.988964i	$-0.547333\pi$
$661$	−7.61803	−0.296307	−0.148154	+	0.988964i	$0.547333\pi$
$662$	8.29180	0.322270
$663$	0	0
$664$	−3.76393	−0.146069
$665$	17.6393	0.684023
$666$	0	0
$667$	54.0132	2.09140
$668$	5.38197	0.208235
$669$	0	0
$670$	7.52786	0.290827
$671$	0	0
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−2.38197	−0.0917499
$675$	0	0
$676$	25.8885	0.995713
$677$	16.3262	0.627468	0.313734	−	0.949511i	$-0.398420\pi$
$677$	16.3262	0.627468	0.313734	+	0.949511i	$0.398420\pi$
$678$	0	0
$679$	17.1246	0.657182
$680$	−7.70820	−0.295596
$681$	0	0
$682$	0	0
$683$	33.8885	1.29671	0.648355	−	0.761339i	$-0.275457\pi$
$683$	33.8885	1.29671	0.648355	+	0.761339i	$0.275457\pi$
$684$	0	0
$685$	−17.5279	−0.669705
$686$	16.7082	0.637922
$687$	0	0
$688$	−0.708204	−0.0270000
$689$	62.9230	2.39717
$690$	0	0
$691$	33.2361	1.26436	0.632180	−	0.774822i	$-0.282160\pi$
$691$	33.2361	1.26436	0.632180	+	0.774822i	$0.282160\pi$
$692$	−3.94427	−0.149939
$693$	0	0
$694$	−31.3262	−1.18913
$695$	−10.5836	−0.401459
$696$	0	0
$697$	−21.0902	−0.798847
$698$	−17.3262	−0.655808
$699$	0	0
$700$	−9.90983	−0.374556
$701$	7.14590	0.269897	0.134948	−	0.990853i	$-0.456913\pi$
$701$	7.14590	0.269897	0.134948	+	0.990853i	$0.456913\pi$
$702$	0	0
$703$	45.0000	1.69721
$704$	0	0
$705$	0	0
$706$	−9.32624	−0.350998
$707$	−48.6180	−1.82847
$708$	0	0
$709$	−45.7214	−1.71710	−0.858551	−	0.512728i	$-0.828635\pi$
$709$	−45.7214	−1.71710	−0.858551	+	0.512728i	$0.828635\pi$
$710$	5.59675	0.210042
$711$	0	0
$712$	4.23607	0.158753
$713$	53.3262	1.99708
$714$	0	0
$715$	0	0
$716$	1.79837	0.0672084
$717$	0	0
$718$	32.1246	1.19888
$719$	48.7082	1.81651	0.908255	−	0.418418i	$-0.137415\pi$
$719$	48.7082	1.81651	0.908255	+	0.418418i	$0.137415\pi$
$720$	0	0
$721$	−15.8754	−0.591230
$722$	−6.00000	−0.223297
$723$	0	0
$724$	−11.3820	−0.423007
$725$	24.6180	0.914291
$726$	0	0
$727$	−5.14590	−0.190851	−0.0954254	−	0.995437i	$-0.530421\pi$
$727$	−5.14590	−0.190851	−0.0954254	+	0.995437i	$0.530421\pi$
$728$	17.7984	0.659652
$729$	0	0
$730$	5.23607	0.193796
$731$	−4.41641	−0.163347
$732$	0	0
$733$	−35.7082	−1.31891	−0.659456	−	0.751743i	$-0.729213\pi$
$733$	−35.7082	−1.31891	−0.659456	+	0.751743i	$0.729213\pi$
$734$	12.8541	0.474454
$735$	0	0
$736$	7.61803	0.280804
$737$	0	0
$738$	0	0
$739$	−40.3951	−1.48596	−0.742979	−	0.669314i	$-0.766588\pi$
$739$	−40.3951	−1.48596	−0.742979	+	0.669314i	$0.766588\pi$
$740$	11.1246	0.408949
$741$	0	0
$742$	28.7984	1.05722
$743$	46.2361	1.69624	0.848118	−	0.529807i	$-0.177736\pi$
$743$	46.2361	1.69624	0.848118	+	0.529807i	$0.177736\pi$
$744$	0	0
$745$	9.45898	0.346550
$746$	23.7082	0.868019
$747$	0	0
$748$	0	0
$749$	−52.1459	−1.90537
$750$	0	0
$751$	16.6738	0.608434	0.304217	−	0.952603i	$-0.401605\pi$
$751$	16.6738	0.608434	0.304217	+	0.952603i	$0.401605\pi$
$752$	−1.76393	−0.0643240
$753$	0	0
$754$	−44.2148	−1.61021
$755$	−4.76393	−0.173377
$756$	0	0
$757$	−36.0344	−1.30969	−0.654847	−	0.755761i	$-0.727267\pi$
$757$	−36.0344	−1.30969	−0.654847	+	0.755761i	$0.727267\pi$
$758$	−14.7082	−0.534226
$759$	0	0
$760$	−6.18034	−0.224184
$761$	−44.9443	−1.62923	−0.814614	−	0.580003i	$-0.803051\pi$
$761$	−44.9443	−1.62923	−0.814614	+	0.580003i	$0.803051\pi$
$762$	0	0
$763$	12.3475	0.447010
$764$	−3.67376	−0.132912
$765$	0	0
$766$	18.6525	0.673941
$767$	58.1591	2.10000
$768$	0	0
$769$	−22.8541	−0.824140	−0.412070	−	0.911152i	$-0.635194\pi$
$769$	−22.8541	−0.824140	−0.412070	+	0.911152i	$0.635194\pi$
$770$	0	0
$771$	0	0
$772$	17.4164	0.626830
$773$	4.81966	0.173351	0.0866756	−	0.996237i	$-0.472376\pi$
$773$	4.81966	0.173351	0.0866756	+	0.996237i	$0.472376\pi$
$774$	0	0
$775$	24.3050	0.873060
$776$	−6.00000	−0.215387
$777$	0	0
$778$	29.9230	1.07279
$779$	−16.9098	−0.605858
$780$	0	0
$781$	0	0
$782$	47.5066	1.69883
$783$	0	0
$784$	1.14590	0.0409249
$785$	−9.16718	−0.327191
$786$	0	0
$787$	9.47214	0.337645	0.168823	−	0.985646i	$-0.446004\pi$
$787$	9.47214	0.337645	0.168823	+	0.985646i	$0.446004\pi$
$788$	12.3262	0.439104
$789$	0	0
$790$	−18.5410	−0.659660
$791$	7.31308	0.260023
$792$	0	0
$793$	−21.6525	−0.768902
$794$	−24.2705	−0.861328
$795$	0	0
$796$	1.94427	0.0689129
$797$	10.7984	0.382498	0.191249	−	0.981542i	$-0.438746\pi$
$797$	10.7984	0.382498	0.191249	+	0.981542i	$0.438746\pi$
$798$	0	0
$799$	−11.0000	−0.389152
$800$	3.47214	0.122759
$801$	0	0
$802$	19.6738	0.694705
$803$	0	0
$804$	0	0
$805$	−26.8754	−0.947233
$806$	−43.6525	−1.53759
$807$	0	0
$808$	17.0344	0.599270
$809$	1.03444	0.0363690	0.0181845	−	0.999835i	$-0.494211\pi$
$809$	1.03444	0.0363690	0.0181845	+	0.999835i	$0.494211\pi$
$810$	0	0
$811$	4.18034	0.146792	0.0733958	−	0.997303i	$-0.476616\pi$
$811$	4.18034	0.146792	0.0733958	+	0.997303i	$0.476616\pi$
$812$	−20.2361	−0.710147
$813$	0	0
$814$	0	0
$815$	−2.18034	−0.0763740
$816$	0	0
$817$	−3.54102	−0.123885
$818$	0	0
$819$	0	0
$820$	−4.18034	−0.145984
$821$	5.83282	0.203567	0.101783	−	0.994807i	$-0.467545\pi$
$821$	5.83282	0.203567	0.101783	+	0.994807i	$0.467545\pi$
$822$	0	0
$823$	−23.2016	−0.808758	−0.404379	−	0.914592i	$-0.632512\pi$
$823$	−23.2016	−0.808758	−0.404379	+	0.914592i	$0.632512\pi$
$824$	5.56231	0.193772
$825$	0	0
$826$	26.6180	0.926160
$827$	35.0689	1.21946	0.609732	−	0.792607i	$-0.291277\pi$
$827$	35.0689	1.21946	0.609732	+	0.792607i	$0.291277\pi$
$828$	0	0
$829$	−12.2016	−0.423780	−0.211890	−	0.977294i	$-0.567962\pi$
$829$	−12.2016	−0.423780	−0.211890	+	0.977294i	$0.567962\pi$
$830$	−4.65248	−0.161490
$831$	0	0
$832$	−6.23607	−0.216197
$833$	7.14590	0.247591
$834$	0	0
$835$	6.65248	0.230218
$836$	0	0
$837$	0	0
$838$	4.14590	0.143218
$839$	−28.3050	−0.977195	−0.488598	−	0.872509i	$-0.662491\pi$
$839$	−28.3050	−0.977195	−0.488598	+	0.872509i	$0.662491\pi$
$840$	0	0
$841$	21.2705	0.733466
$842$	34.4164	1.18607
$843$	0	0
$844$	−11.3262	−0.389865
$845$	32.0000	1.10083
$846$	0	0
$847$	0	0
$848$	−10.0902	−0.346498
$849$	0	0
$850$	21.6525	0.742674
$851$	−68.5623	−2.35029
$852$	0	0
$853$	0.180340	0.00617472	0.00308736	−	0.999995i	$-0.499017\pi$
$853$	0.180340	0.00617472	0.00308736	+	0.999995i	$0.499017\pi$
$854$	−9.90983	−0.339107
$855$	0	0
$856$	18.2705	0.624473
$857$	9.11146	0.311241	0.155621	−	0.987817i	$-0.450262\pi$
$857$	9.11146	0.311241	0.155621	+	0.987817i	$0.450262\pi$
$858$	0	0
$859$	−6.50658	−0.222002	−0.111001	−	0.993820i	$-0.535406\pi$
$859$	−6.50658	−0.222002	−0.111001	+	0.993820i	$0.535406\pi$
$860$	−0.875388	−0.0298505
$861$	0	0
$862$	−2.76393	−0.0941399
$863$	10.7639	0.366409	0.183204	−	0.983075i	$-0.441353\pi$
$863$	10.7639	0.366409	0.183204	+	0.983075i	$0.441353\pi$
$864$	0	0
$865$	−4.87539	−0.165768
$866$	−0.527864	−0.0179376
$867$	0	0
$868$	−19.9787	−0.678122
$869$	0	0
$870$	0	0
$871$	37.9787	1.28686
$872$	−4.32624	−0.146505
$873$	0	0
$874$	38.0902	1.28842
$875$	−29.8885	−1.01042
$876$	0	0
$877$	7.59675	0.256524	0.128262	−	0.991740i	$-0.459060\pi$
$877$	7.59675	0.256524	0.128262	+	0.991740i	$0.459060\pi$
$878$	5.27051	0.177871
$879$	0	0
$880$	0	0
$881$	50.3262	1.69553	0.847767	−	0.530369i	$-0.177947\pi$
$881$	50.3262	1.69553	0.847767	+	0.530369i	$0.177947\pi$
$882$	0	0
$883$	25.2492	0.849704	0.424852	−	0.905263i	$-0.360326\pi$
$883$	25.2492	0.849704	0.424852	+	0.905263i	$0.360326\pi$
$884$	−38.8885	−1.30796
$885$	0	0
$886$	3.00000	0.100787
$887$	44.4296	1.49180	0.745899	−	0.666059i	$-0.232020\pi$
$887$	44.4296	1.49180	0.745899	+	0.666059i	$0.232020\pi$
$888$	0	0
$889$	−9.90983	−0.332365
$890$	5.23607	0.175513
$891$	0	0
$892$	8.41641	0.281802
$893$	−8.81966	−0.295139
$894$	0	0
$895$	2.22291	0.0743038
$896$	−2.85410	−0.0953489
$897$	0	0
$898$	25.4164	0.848157
$899$	49.6312	1.65529
$900$	0	0
$901$	−62.9230	−2.09627
$902$	0	0
$903$	0	0
$904$	−2.56231	−0.0852210
$905$	−14.0689	−0.467666
$906$	0	0
$907$	−5.87539	−0.195089	−0.0975445	−	0.995231i	$-0.531099\pi$
$907$	−5.87539	−0.195089	−0.0975445	+	0.995231i	$0.531099\pi$
$908$	16.7984	0.557474
$909$	0	0
$910$	22.0000	0.729293
$911$	−30.2361	−1.00177	−0.500883	−	0.865515i	$-0.666991\pi$
$911$	−30.2361	−1.00177	−0.500883	+	0.865515i	$0.666991\pi$
$912$	0	0
$913$	0	0
$914$	−6.00000	−0.198462
$915$	0	0
$916$	22.2705	0.735838
$917$	19.8197	0.654503
$918$	0	0
$919$	−33.5410	−1.10642	−0.553208	−	0.833043i	$-0.686597\pi$
$919$	−33.5410	−1.10642	−0.553208	+	0.833043i	$0.686597\pi$
$920$	9.41641	0.310450
$921$	0	0
$922$	−5.03444	−0.165801
$923$	28.2361	0.929401
$924$	0	0
$925$	−31.2492	−1.02747
$926$	−35.1246	−1.15427
$927$	0	0
$928$	7.09017	0.232746
$929$	20.4508	0.670971	0.335485	−	0.942045i	$-0.391100\pi$
$929$	20.4508	0.670971	0.335485	+	0.942045i	$0.391100\pi$
$930$	0	0
$931$	5.72949	0.187776
$932$	19.6525	0.643738
$933$	0	0
$934$	23.7426	0.776883
$935$	0	0
$936$	0	0
$937$	−9.70820	−0.317153	−0.158577	−	0.987347i	$-0.550690\pi$
$937$	−9.70820	−0.317153	−0.158577	+	0.987347i	$0.550690\pi$
$938$	17.3820	0.567541
$939$	0	0
$940$	−2.18034	−0.0711148
$941$	−27.2361	−0.887870	−0.443935	−	0.896059i	$-0.646418\pi$
$941$	−27.2361	−0.887870	−0.443935	+	0.896059i	$0.646418\pi$
$942$	0	0
$943$	25.7639	0.838989
$944$	−9.32624	−0.303543
$945$	0	0
$946$	0	0
$947$	30.5967	0.994261	0.497130	−	0.867676i	$-0.334387\pi$
$947$	30.5967	0.994261	0.497130	+	0.867676i	$0.334387\pi$
$948$	0	0
$949$	26.4164	0.857513
$950$	17.3607	0.563255
$951$	0	0
$952$	−17.7984	−0.576849
$953$	−37.8673	−1.22664	−0.613320	−	0.789834i	$-0.710167\pi$
$953$	−37.8673	−1.22664	−0.613320	+	0.789834i	$0.710167\pi$
$954$	0	0
$955$	−4.54102	−0.146944
$956$	2.56231	0.0828709
$957$	0	0
$958$	−3.65248	−0.118006
$959$	−40.4721	−1.30691
$960$	0	0
$961$	18.0000	0.580645
$962$	56.1246	1.80953
$963$	0	0
$964$	−20.0344	−0.645266
$965$	21.5279	0.693006
$966$	0	0
$967$	−6.29180	−0.202331	−0.101165	−	0.994870i	$-0.532257\pi$
$967$	−6.29180	−0.202331	−0.101165	+	0.994870i	$0.532257\pi$
$968$	0	0
$969$	0	0
$970$	−7.41641	−0.238127
$971$	27.8885	0.894986	0.447493	−	0.894287i	$-0.352317\pi$
$971$	27.8885	0.894986	0.447493	+	0.894287i	$0.352317\pi$
$972$	0	0
$973$	−24.4377	−0.783437
$974$	−16.3262	−0.523126
$975$	0	0
$976$	3.47214	0.111140
$977$	33.6525	1.07664	0.538319	−	0.842741i	$-0.319060\pi$
$977$	33.6525	1.07664	0.538319	+	0.842741i	$0.319060\pi$
$978$	0	0
$979$	0	0
$980$	1.41641	0.0452455
$981$	0	0
$982$	30.7082	0.979938
$983$	16.1459	0.514974	0.257487	−	0.966282i	$-0.417106\pi$
$983$	16.1459	0.514974	0.257487	+	0.966282i	$0.417106\pi$
$984$	0	0
$985$	15.2361	0.485461
$986$	44.2148	1.40809
$987$	0	0
$988$	−31.1803	−0.991979
$989$	5.39512	0.171555
$990$	0	0
$991$	39.7082	1.26137	0.630686	−	0.776038i	$-0.282773\pi$
$991$	39.7082	1.26137	0.630686	+	0.776038i	$0.282773\pi$
$992$	7.00000	0.222250
$993$	0	0
$994$	12.9230	0.409892
$995$	2.40325	0.0761882
$996$	0	0
$997$	−45.9443	−1.45507	−0.727535	−	0.686071i	$-0.759334\pi$
$997$	−45.9443	−1.45507	−0.727535	+	0.686071i	$0.759334\pi$
$998$	−18.2918	−0.579017
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6534.2.a.bf.1.2		2
3.2	odd	2		6534.2.a.cm.1.1		2
11.3	even	5		594.2.f.j.163.1	yes	4
11.4	even	5		594.2.f.j.379.1	yes	4
11.10	odd	2		6534.2.a.cb.1.2		2
33.14	odd	10		594.2.f.a.163.1	✓	4
33.26	odd	10		594.2.f.a.379.1	yes	4
33.32	even	2		6534.2.a.bv.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
594.2.f.a.163.1	✓	4	33.14	odd	10
594.2.f.a.379.1	yes	4	33.26	odd	10
594.2.f.j.163.1	yes	4	11.3	even	5
594.2.f.j.379.1	yes	4	11.4	even	5
6534.2.a.bf.1.2		2	1.1	even	1	trivial
6534.2.a.bv.1.1		2	33.32	even	2
6534.2.a.cb.1.2		2	11.10	odd	2
6534.2.a.cm.1.1		2	3.2	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:	\( N \)	\(=\)	\( 6534 = 2 \cdot 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6534.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} +2.85410 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} +2.85410 q^{7} -1.00000 q^{8} -1.23607 q^{10} -6.23607 q^{13} -2.85410 q^{14} +1.00000 q^{16} +6.23607 q^{17} +5.00000 q^{19} +1.23607 q^{20} -7.61803 q^{23} -3.47214 q^{25} +6.23607 q^{26} +2.85410 q^{28} -7.09017 q^{29} -7.00000 q^{31} -1.00000 q^{32} -6.23607 q^{34} +3.52786 q^{35} +9.00000 q^{37} -5.00000 q^{38} -1.23607 q^{40} -3.38197 q^{41} -0.708204 q^{43} +7.61803 q^{46} -1.76393 q^{47} +1.14590 q^{49} +3.47214 q^{50} -6.23607 q^{52} -10.0902 q^{53} -2.85410 q^{56} +7.09017 q^{58} -9.32624 q^{59} +3.47214 q^{61} +7.00000 q^{62} +1.00000 q^{64} -7.70820 q^{65} -6.09017 q^{67} +6.23607 q^{68} -3.52786 q^{70} -4.52786 q^{71} -4.23607 q^{73} -9.00000 q^{74} +5.00000 q^{76} +15.0000 q^{79} +1.23607 q^{80} +3.38197 q^{82} +3.76393 q^{83} +7.70820 q^{85} +0.708204 q^{86} -4.23607 q^{89} -17.7984 q^{91} -7.61803 q^{92} +1.76393 q^{94} +6.18034 q^{95} +6.00000 q^{97} -1.14590 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - q^{7} - 2 q^{8} + 2 q^{10} - 8 q^{13} + q^{14} + 2 q^{16} + 8 q^{17} + 10 q^{19} - 2 q^{20} - 13 q^{23} + 2 q^{25} + 8 q^{26} - q^{28} - 3 q^{29} - 14 q^{31} - 2 q^{32} - 8 q^{34} + 16 q^{35} + 18 q^{37} - 10 q^{38} + 2 q^{40} - 9 q^{41} + 12 q^{43} + 13 q^{46} - 8 q^{47} + 9 q^{49} - 2 q^{50} - 8 q^{52} - 9 q^{53} + q^{56} + 3 q^{58} - 3 q^{59} - 2 q^{61} + 14 q^{62} + 2 q^{64} - 2 q^{65} - q^{67} + 8 q^{68} - 16 q^{70} - 18 q^{71} - 4 q^{73} - 18 q^{74} + 10 q^{76} + 30 q^{79} - 2 q^{80} + 9 q^{82} + 12 q^{83} + 2 q^{85} - 12 q^{86} - 4 q^{89} - 11 q^{91} - 13 q^{92} + 8 q^{94} - 10 q^{95} + 12 q^{97} - 9 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - q^7 - 2 * q^8 + 2 * q^10 - 8 * q^13 + q^14 + 2 * q^16 + 8 * q^17 + 10 * q^19 - 2 * q^20 - 13 * q^23 + 2 * q^25 + 8 * q^26 - q^28 - 3 * q^29 - 14 * q^31 - 2 * q^32 - 8 * q^34 + 16 * q^35 + 18 * q^37 - 10 * q^38 + 2 * q^40 - 9 * q^41 + 12 * q^43 + 13 * q^46 - 8 * q^47 + 9 * q^49 - 2 * q^50 - 8 * q^52 - 9 * q^53 + q^56 + 3 * q^58 - 3 * q^59 - 2 * q^61 + 14 * q^62 + 2 * q^64 - 2 * q^65 - q^67 + 8 * q^68 - 16 * q^70 - 18 * q^71 - 4 * q^73 - 18 * q^74 + 10 * q^76 + 30 * q^79 - 2 * q^80 + 9 * q^82 + 12 * q^83 + 2 * q^85 - 12 * q^86 - 4 * q^89 - 11 * q^91 - 13 * q^92 + 8 * q^94 - 10 * q^95 + 12 * q^97 - 9 * q^98

Introduction

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