LMFDB - Embedded newform 3249.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3249,2,Mod(1,3249)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3249, 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("3249.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.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:	$25.9433956167$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 361)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} +1.23607 q^{5} +3.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	$q+0.618034 q^{2} -1.61803 q^{4} +1.23607 q^{5} +3.00000 q^{7} -2.23607 q^{8} +0.763932 q^{10} -0.618034 q^{11} -1.00000 q^{13} +1.85410 q^{14} +1.85410 q^{16} -5.23607 q^{17} -2.00000 q^{20} -0.381966 q^{22} -7.61803 q^{23} -3.47214 q^{25} -0.618034 q^{26} -4.85410 q^{28} +1.38197 q^{29} -2.14590 q^{31} +5.61803 q^{32} -3.23607 q^{34} +3.70820 q^{35} +2.14590 q^{37} -2.76393 q^{40} +3.00000 q^{41} -6.85410 q^{43} +1.00000 q^{44} -4.70820 q^{46} -3.00000 q^{47} +2.00000 q^{49} -2.14590 q^{50} +1.61803 q^{52} -9.32624 q^{53} -0.763932 q^{55} -6.70820 q^{56} +0.854102 q^{58} +15.3262 q^{59} -5.76393 q^{61} -1.32624 q^{62} -0.236068 q^{64} -1.23607 q^{65} -7.00000 q^{67} +8.47214 q^{68} +2.29180 q^{70} -1.47214 q^{71} +10.7082 q^{73} +1.32624 q^{74} -1.85410 q^{77} -13.4164 q^{79} +2.29180 q^{80} +1.85410 q^{82} +0.472136 q^{83} -6.47214 q^{85} -4.23607 q^{86} +1.38197 q^{88} +12.2361 q^{89} -3.00000 q^{91} +12.3262 q^{92} -1.85410 q^{94} +7.14590 q^{97} +1.23607 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 2 q^{5} + 6 q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 - 2 * q^5 + 6 * q^7	$ 2 q - q^{2} - q^{4} - 2 q^{5} + 6 q^{7} + 6 q^{10} + q^{11} - 2 q^{13} - 3 q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{20} - 3 q^{22} - 13 q^{23} + 2 q^{25} + q^{26} - 3 q^{28} + 5 q^{29} - 11 q^{31} + 9 q^{32} - 2 q^{34} - 6 q^{35} + 11 q^{37} - 10 q^{40} + 6 q^{41} - 7 q^{43} + 2 q^{44} + 4 q^{46} - 6 q^{47} + 4 q^{49} - 11 q^{50} + q^{52} - 3 q^{53} - 6 q^{55} - 5 q^{58} + 15 q^{59} - 16 q^{61} + 13 q^{62} + 4 q^{64} + 2 q^{65} - 14 q^{67} + 8 q^{68} + 18 q^{70} + 6 q^{71} + 8 q^{73} - 13 q^{74} + 3 q^{77} + 18 q^{80} - 3 q^{82} - 8 q^{83} - 4 q^{85} - 4 q^{86} + 5 q^{88} + 20 q^{89} - 6 q^{91} + 9 q^{92} + 3 q^{94} + 21 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 2 * q^5 + 6 * q^7 + 6 * q^10 + q^11 - 2 * q^13 - 3 * q^14 - 3 * q^16 - 6 * q^17 - 4 * q^20 - 3 * q^22 - 13 * q^23 + 2 * q^25 + q^26 - 3 * q^28 + 5 * q^29 - 11 * q^31 + 9 * q^32 - 2 * q^34 - 6 * q^35 + 11 * q^37 - 10 * q^40 + 6 * q^41 - 7 * q^43 + 2 * q^44 + 4 * q^46 - 6 * q^47 + 4 * q^49 - 11 * q^50 + q^52 - 3 * q^53 - 6 * q^55 - 5 * q^58 + 15 * q^59 - 16 * q^61 + 13 * q^62 + 4 * q^64 + 2 * q^65 - 14 * q^67 + 8 * q^68 + 18 * q^70 + 6 * q^71 + 8 * q^73 - 13 * q^74 + 3 * q^77 + 18 * q^80 - 3 * q^82 - 8 * q^83 - 4 * q^85 - 4 * q^86 + 5 * q^88 + 20 * q^89 - 6 * q^91 + 9 * q^92 + 3 * q^94 + 21 * q^97 - 2 * 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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	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
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0.763932	0.241577
$11$	−0.618034	−0.186344	−0.0931721	−	0.995650i	$-0.529701\pi$
$11$	−0.618034	−0.186344	−0.0931721	+	0.995650i	$0.529701\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	1.85410	0.495530
$15$	0	0
$16$	1.85410	0.463525
$17$	−5.23607	−1.26993	−0.634967	−	0.772540i	$-0.718986\pi$
$17$	−5.23607	−1.26993	−0.634967	+	0.772540i	$0.718986\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−0.381966	−0.0814354
$23$	−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
$26$	−0.618034	−0.121206
$27$	0	0
$28$	−4.85410	−0.917339
$29$	1.38197	0.256625	0.128312	−	0.991734i	$-0.459044\pi$
$29$	1.38197	0.256625	0.128312	+	0.991734i	$0.459044\pi$
$30$	0	0
$31$	−2.14590	−0.385415	−0.192707	−	0.981256i	$-0.561727\pi$
$31$	−2.14590	−0.385415	−0.192707	+	0.981256i	$0.561727\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	−3.23607	−0.554981
$35$	3.70820	0.626801
$36$	0	0
$37$	2.14590	0.352783	0.176392	−	0.984320i	$-0.443557\pi$
$37$	2.14590	0.352783	0.176392	+	0.984320i	$0.443557\pi$
$38$	0	0
$39$	0	0
$40$	−2.76393	−0.437016
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	−6.85410	−1.04524	−0.522620	−	0.852566i	$-0.675045\pi$
$43$	−6.85410	−1.04524	−0.522620	+	0.852566i	$0.675045\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.70820	−0.694187
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−2.14590	−0.303476
$51$	0	0
$52$	1.61803	0.224381
$53$	−9.32624	−1.28106	−0.640529	−	0.767934i	$-0.721285\pi$
$53$	−9.32624	−1.28106	−0.640529	+	0.767934i	$0.721285\pi$
$54$	0	0
$55$	−0.763932	−0.103009
$56$	−6.70820	−0.896421
$57$	0	0
$58$	0.854102	0.112149
$59$	15.3262	1.99531	0.997653	−	0.0684709i	$-0.0218120\pi$
$59$	15.3262	1.99531	0.997653	+	0.0684709i	$0.0218120\pi$
$60$	0	0
$61$	−5.76393	−0.737996	−0.368998	−	0.929430i	$-0.620299\pi$
$61$	−5.76393	−0.737996	−0.368998	+	0.929430i	$0.620299\pi$
$62$	−1.32624	−0.168432
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	−1.23607	−0.153315
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	8.47214	1.02740
$69$	0	0
$70$	2.29180	0.273922
$71$	−1.47214	−0.174710	−0.0873552	−	0.996177i	$-0.527842\pi$
$71$	−1.47214	−0.174710	−0.0873552	+	0.996177i	$0.527842\pi$
$72$	0	0
$73$	10.7082	1.25330	0.626650	−	0.779301i	$-0.284425\pi$
$73$	10.7082	1.25330	0.626650	+	0.779301i	$0.284425\pi$
$74$	1.32624	0.154172
$75$	0	0
$76$	0	0
$77$	−1.85410	−0.211295
$78$	0	0
$79$	−13.4164	−1.50946	−0.754732	−	0.656033i	$-0.772233\pi$
$79$	−13.4164	−1.50946	−0.754732	+	0.656033i	$0.772233\pi$
$80$	2.29180	0.256231
$81$	0	0
$82$	1.85410	0.204751
$83$	0.472136	0.0518237	0.0259118	−	0.999664i	$-0.491751\pi$
$83$	0.472136	0.0518237	0.0259118	+	0.999664i	$0.491751\pi$
$84$	0	0
$85$	−6.47214	−0.702002
$86$	−4.23607	−0.456787
$87$	0	0
$88$	1.38197	0.147318
$89$	12.2361	1.29702	0.648510	−	0.761206i	$-0.275392\pi$
$89$	12.2361	1.29702	0.648510	+	0.761206i	$0.275392\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	12.3262	1.28510
$93$	0	0
$94$	−1.85410	−0.191236
$95$	0	0
$96$	0	0
$97$	7.14590	0.725556	0.362778	−	0.931876i	$-0.381828\pi$
$97$	7.14590	0.725556	0.362778	+	0.931876i	$0.381828\pi$
$98$	1.23607	0.124862
$99$	0	0
$100$	5.61803	0.561803
$101$	−13.1803	−1.31149	−0.655746	−	0.754981i	$-0.727646\pi$
$101$	−13.1803	−1.31149	−0.655746	+	0.754981i	$0.727646\pi$
$102$	0	0
$103$	−1.32624	−0.130678	−0.0653391	−	0.997863i	$-0.520813\pi$
$103$	−1.32624	−0.130678	−0.0653391	+	0.997863i	$0.520813\pi$
$104$	2.23607	0.219265
$105$	0	0
$106$	−5.76393	−0.559843
$107$	10.4164	1.00699	0.503496	−	0.863998i	$-0.332047\pi$
$107$	10.4164	1.00699	0.503496	+	0.863998i	$0.332047\pi$
$108$	0	0
$109$	−16.7082	−1.60036	−0.800178	−	0.599763i	$-0.795262\pi$
$109$	−16.7082	−1.60036	−0.800178	+	0.599763i	$0.795262\pi$
$110$	−0.472136	−0.0450164
$111$	0	0
$112$	5.56231	0.525589
$113$	−11.2361	−1.05700	−0.528500	−	0.848933i	$-0.677245\pi$
$113$	−11.2361	−1.05700	−0.528500	+	0.848933i	$0.677245\pi$
$114$	0	0
$115$	−9.41641	−0.878085
$116$	−2.23607	−0.207614
$117$	0	0
$118$	9.47214	0.871981
$119$	−15.7082	−1.43997
$120$	0	0
$121$	−10.6180	−0.965276
$122$	−3.56231	−0.322516
$123$	0	0
$124$	3.47214	0.311807
$125$	−10.4721	−0.936656
$126$	0	0
$127$	10.2361	0.908304	0.454152	−	0.890924i	$-0.349942\pi$
$127$	10.2361	0.908304	0.454152	+	0.890924i	$0.349942\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	−0.763932	−0.0670013
$131$	−3.90983	−0.341603	−0.170802	−	0.985305i	$-0.554636\pi$
$131$	−3.90983	−0.341603	−0.170802	+	0.985305i	$0.554636\pi$
$132$	0	0
$133$	0	0
$134$	−4.32624	−0.373730
$135$	0	0
$136$	11.7082	1.00397
$137$	1.47214	0.125773	0.0628865	−	0.998021i	$-0.479969\pi$
$137$	1.47214	0.125773	0.0628865	+	0.998021i	$0.479969\pi$
$138$	0	0
$139$	−9.79837	−0.831087	−0.415544	−	0.909573i	$-0.636409\pi$
$139$	−9.79837	−0.831087	−0.415544	+	0.909573i	$0.636409\pi$
$140$	−6.00000	−0.507093
$141$	0	0
$142$	−0.909830	−0.0763512
$143$	0.618034	0.0516826
$144$	0	0
$145$	1.70820	0.141859
$146$	6.61803	0.547712
$147$	0	0
$148$	−3.47214	−0.285408
$149$	13.0902	1.07239	0.536194	−	0.844095i	$-0.319861\pi$
$149$	13.0902	1.07239	0.536194	+	0.844095i	$0.319861\pi$
$150$	0	0
$151$	−9.90983	−0.806451	−0.403225	−	0.915101i	$-0.632111\pi$
$151$	−9.90983	−0.806451	−0.403225	+	0.915101i	$0.632111\pi$
$152$	0	0
$153$	0	0
$154$	−1.14590	−0.0923391
$155$	−2.65248	−0.213052
$156$	0	0
$157$	−11.1459	−0.889540	−0.444770	−	0.895645i	$-0.646714\pi$
$157$	−11.1459	−0.889540	−0.444770	+	0.895645i	$0.646714\pi$
$158$	−8.29180	−0.659660
$159$	0	0
$160$	6.94427	0.548993
$161$	−22.8541	−1.80116
$162$	0	0
$163$	6.23607	0.488447	0.244223	−	0.969719i	$-0.421467\pi$
$163$	6.23607	0.488447	0.244223	+	0.969719i	$0.421467\pi$
$164$	−4.85410	−0.379042
$165$	0	0
$166$	0.291796	0.0226478
$167$	−15.7639	−1.21985	−0.609925	−	0.792459i	$-0.708800\pi$
$167$	−15.7639	−1.21985	−0.609925	+	0.792459i	$0.708800\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−4.00000	−0.306786
$171$	0	0
$172$	11.0902	0.845618
$173$	−8.47214	−0.644125	−0.322062	−	0.946718i	$-0.604376\pi$
$173$	−8.47214	−0.644125	−0.322062	+	0.946718i	$0.604376\pi$
$174$	0	0
$175$	−10.4164	−0.787406
$176$	−1.14590	−0.0863753
$177$	0	0
$178$	7.56231	0.566819
$179$	−7.76393	−0.580304	−0.290152	−	0.956981i	$-0.593706\pi$
$179$	−7.76393	−0.580304	−0.290152	+	0.956981i	$0.593706\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	−1.85410	−0.137435
$183$	0	0
$184$	17.0344	1.25580
$185$	2.65248	0.195014
$186$	0	0
$187$	3.23607	0.236645
$188$	4.85410	0.354022
$189$	0	0
$190$	0	0
$191$	−9.76393	−0.706493	−0.353247	−	0.935530i	$-0.614922\pi$
$191$	−9.76393	−0.706493	−0.353247	+	0.935530i	$0.614922\pi$
$192$	0	0
$193$	22.9443	1.65156	0.825782	−	0.563989i	$-0.190734\pi$
$193$	22.9443	1.65156	0.825782	+	0.563989i	$0.190734\pi$
$194$	4.41641	0.317080
$195$	0	0
$196$	−3.23607	−0.231148
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	13.4164	0.951064	0.475532	−	0.879698i	$-0.342256\pi$
$199$	13.4164	0.951064	0.475532	+	0.879698i	$0.342256\pi$
$200$	7.76393	0.548993
$201$	0	0
$202$	−8.14590	−0.573143
$203$	4.14590	0.290985
$204$	0	0
$205$	3.70820	0.258992
$206$	−0.819660	−0.0571084
$207$	0	0
$208$	−1.85410	−0.128559
$209$	0	0
$210$	0	0
$211$	2.85410	0.196484	0.0982422	−	0.995163i	$-0.468678\pi$
$211$	2.85410	0.196484	0.0982422	+	0.995163i	$0.468678\pi$
$212$	15.0902	1.03640
$213$	0	0
$214$	6.43769	0.440072
$215$	−8.47214	−0.577795
$216$	0	0
$217$	−6.43769	−0.437019
$218$	−10.3262	−0.699381
$219$	0	0
$220$	1.23607	0.0833357
$221$	5.23607	0.352216
$222$	0	0
$223$	19.6525	1.31603	0.658014	−	0.753006i	$-0.271397\pi$
$223$	19.6525	1.31603	0.658014	+	0.753006i	$0.271397\pi$
$224$	16.8541	1.12611
$225$	0	0
$226$	−6.94427	−0.461926
$227$	10.4164	0.691361	0.345681	−	0.938352i	$-0.387648\pi$
$227$	10.4164	0.691361	0.345681	+	0.938352i	$0.387648\pi$
$228$	0	0
$229$	−11.3820	−0.752141	−0.376071	−	0.926591i	$-0.622725\pi$
$229$	−11.3820	−0.752141	−0.376071	+	0.926591i	$0.622725\pi$
$230$	−5.81966	−0.383737
$231$	0	0
$232$	−3.09017	−0.202880
$233$	−13.4721	−0.882589	−0.441294	−	0.897362i	$-0.645481\pi$
$233$	−13.4721	−0.882589	−0.441294	+	0.897362i	$0.645481\pi$
$234$	0	0
$235$	−3.70820	−0.241897
$236$	−24.7984	−1.61424
$237$	0	0
$238$	−9.70820	−0.629289
$239$	−15.3262	−0.991372	−0.495686	−	0.868502i	$-0.665083\pi$
$239$	−15.3262	−0.991372	−0.495686	+	0.868502i	$0.665083\pi$
$240$	0	0
$241$	−19.1803	−1.23551	−0.617757	−	0.786369i	$-0.711959\pi$
$241$	−19.1803	−1.23551	−0.617757	+	0.786369i	$0.711959\pi$
$242$	−6.56231	−0.421841
$243$	0	0
$244$	9.32624	0.597051
$245$	2.47214	0.157939
$246$	0	0
$247$	0	0
$248$	4.79837	0.304697
$249$	0	0
$250$	−6.47214	−0.409334
$251$	−19.3607	−1.22204	−0.611018	−	0.791617i	$-0.709240\pi$
$251$	−19.3607	−1.22204	−0.611018	+	0.791617i	$0.709240\pi$
$252$	0	0
$253$	4.70820	0.296002
$254$	6.32624	0.396943
$255$	0	0
$256$	−6.56231	−0.410144
$257$	24.3607	1.51958	0.759789	−	0.650170i	$-0.225302\pi$
$257$	24.3607	1.51958	0.759789	+	0.650170i	$0.225302\pi$
$258$	0	0
$259$	6.43769	0.400019
$260$	2.00000	0.124035
$261$	0	0
$262$	−2.41641	−0.149286
$263$	−2.94427	−0.181552	−0.0907758	−	0.995871i	$-0.528935\pi$
$263$	−2.94427	−0.181552	−0.0907758	+	0.995871i	$0.528935\pi$
$264$	0	0
$265$	−11.5279	−0.708151
$266$	0	0
$267$	0	0
$268$	11.3262	0.691860
$269$	−14.6738	−0.894675	−0.447338	−	0.894365i	$-0.647628\pi$
$269$	−14.6738	−0.894675	−0.447338	+	0.894365i	$0.647628\pi$
$270$	0	0
$271$	7.85410	0.477103	0.238551	−	0.971130i	$-0.423327\pi$
$271$	7.85410	0.477103	0.238551	+	0.971130i	$0.423327\pi$
$272$	−9.70820	−0.588646
$273$	0	0
$274$	0.909830	0.0549648
$275$	2.14590	0.129403
$276$	0	0
$277$	−15.4164	−0.926282	−0.463141	−	0.886285i	$-0.653278\pi$
$277$	−15.4164	−0.926282	−0.463141	+	0.886285i	$0.653278\pi$
$278$	−6.05573	−0.363198
$279$	0	0
$280$	−8.29180	−0.495530
$281$	−8.50658	−0.507460	−0.253730	−	0.967275i	$-0.581657\pi$
$281$	−8.50658	−0.507460	−0.253730	+	0.967275i	$0.581657\pi$
$282$	0	0
$283$	−3.03444	−0.180379	−0.0901894	−	0.995925i	$-0.528747\pi$
$283$	−3.03444	−0.180379	−0.0901894	+	0.995925i	$0.528747\pi$
$284$	2.38197	0.141344
$285$	0	0
$286$	0.381966	0.0225861
$287$	9.00000	0.531253
$288$	0	0
$289$	10.4164	0.612730
$290$	1.05573	0.0619945
$291$	0	0
$292$	−17.3262	−1.01394
$293$	16.8541	0.984627	0.492314	−	0.870418i	$-0.336151\pi$
$293$	16.8541	0.984627	0.492314	+	0.870418i	$0.336151\pi$
$294$	0	0
$295$	18.9443	1.10298
$296$	−4.79837	−0.278900
$297$	0	0
$298$	8.09017	0.468651
$299$	7.61803	0.440562
$300$	0	0
$301$	−20.5623	−1.18519
$302$	−6.12461	−0.352432
$303$	0	0
$304$	0	0
$305$	−7.12461	−0.407954
$306$	0	0
$307$	−1.67376	−0.0955266	−0.0477633	−	0.998859i	$-0.515209\pi$
$307$	−1.67376	−0.0955266	−0.0477633	+	0.998859i	$0.515209\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	−1.63932	−0.0931071
$311$	18.6525	1.05768	0.528842	−	0.848720i	$-0.322626\pi$
$311$	18.6525	1.05768	0.528842	+	0.848720i	$0.322626\pi$
$312$	0	0
$313$	4.32624	0.244533	0.122267	−	0.992497i	$-0.460984\pi$
$313$	4.32624	0.244533	0.122267	+	0.992497i	$0.460984\pi$
$314$	−6.88854	−0.388743
$315$	0	0
$316$	21.7082	1.22118
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−0.854102	−0.0478205
$320$	−0.291796	−0.0163119
$321$	0	0
$322$	−14.1246	−0.787134
$323$	0	0
$324$	0	0
$325$	3.47214	0.192599
$326$	3.85410	0.213459
$327$	0	0
$328$	−6.70820	−0.370399
$329$	−9.00000	−0.496186
$330$	0	0
$331$	−6.94427	−0.381692	−0.190846	−	0.981620i	$-0.561123\pi$
$331$	−6.94427	−0.381692	−0.190846	+	0.981620i	$0.561123\pi$
$332$	−0.763932	−0.0419262
$333$	0	0
$334$	−9.74265	−0.533094
$335$	−8.65248	−0.472735
$336$	0	0
$337$	23.1246	1.25968	0.629839	−	0.776726i	$-0.283121\pi$
$337$	23.1246	1.25968	0.629839	+	0.776726i	$0.283121\pi$
$338$	−7.41641	−0.403399
$339$	0	0
$340$	10.4721	0.567931
$341$	1.32624	0.0718198
$342$	0	0
$343$	−15.0000	−0.809924
$344$	15.3262	0.826335
$345$	0	0
$346$	−5.23607	−0.281493
$347$	−1.41641	−0.0760368	−0.0380184	−	0.999277i	$-0.512105\pi$
$347$	−1.41641	−0.0760368	−0.0380184	+	0.999277i	$0.512105\pi$
$348$	0	0
$349$	25.9787	1.39061	0.695304	−	0.718715i	$-0.255270\pi$
$349$	25.9787	1.39061	0.695304	+	0.718715i	$0.255270\pi$
$350$	−6.43769	−0.344109
$351$	0	0
$352$	−3.47214	−0.185065
$353$	31.4508	1.67396	0.836980	−	0.547234i	$-0.184319\pi$
$353$	31.4508	1.67396	0.836980	+	0.547234i	$0.184319\pi$
$354$	0	0
$355$	−1.81966	−0.0965775
$356$	−19.7984	−1.04931
$357$	0	0
$358$	−4.79837	−0.253602
$359$	22.0344	1.16293	0.581467	−	0.813570i	$-0.302479\pi$
$359$	22.0344	1.16293	0.581467	+	0.813570i	$0.302479\pi$
$360$	0	0
$361$	0	0
$362$	7.41641	0.389798
$363$	0	0
$364$	4.85410	0.254424
$365$	13.2361	0.692807
$366$	0	0
$367$	1.94427	0.101490	0.0507451	−	0.998712i	$-0.483840\pi$
$367$	1.94427	0.101490	0.0507451	+	0.998712i	$0.483840\pi$
$368$	−14.1246	−0.736296
$369$	0	0
$370$	1.63932	0.0852242
$371$	−27.9787	−1.45258
$372$	0	0
$373$	3.47214	0.179780	0.0898902	−	0.995952i	$-0.471348\pi$
$373$	3.47214	0.179780	0.0898902	+	0.995952i	$0.471348\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	6.70820	0.345949
$377$	−1.38197	−0.0711749
$378$	0	0
$379$	25.1246	1.29056	0.645282	−	0.763944i	$-0.276740\pi$
$379$	25.1246	1.29056	0.645282	+	0.763944i	$0.276740\pi$
$380$	0	0
$381$	0	0
$382$	−6.03444	−0.308749
$383$	−2.61803	−0.133775	−0.0668876	−	0.997761i	$-0.521307\pi$
$383$	−2.61803	−0.133775	−0.0668876	+	0.997761i	$0.521307\pi$
$384$	0	0
$385$	−2.29180	−0.116801
$386$	14.1803	0.721760
$387$	0	0
$388$	−11.5623	−0.586987
$389$	24.2705	1.23056	0.615282	−	0.788307i	$-0.289042\pi$
$389$	24.2705	1.23056	0.615282	+	0.788307i	$0.289042\pi$
$390$	0	0
$391$	39.8885	2.01725
$392$	−4.47214	−0.225877
$393$	0	0
$394$	−1.85410	−0.0934083
$395$	−16.5836	−0.834411
$396$	0	0
$397$	−2.52786	−0.126870	−0.0634349	−	0.997986i	$-0.520206\pi$
$397$	−2.52786	−0.126870	−0.0634349	+	0.997986i	$0.520206\pi$
$398$	8.29180	0.415630
$399$	0	0
$400$	−6.43769	−0.321885
$401$	0.111456	0.00556586	0.00278293	−	0.999996i	$-0.499114\pi$
$401$	0.111456	0.00556586	0.00278293	+	0.999996i	$0.499114\pi$
$402$	0	0
$403$	2.14590	0.106895
$404$	21.3262	1.06102
$405$	0	0
$406$	2.56231	0.127165
$407$	−1.32624	−0.0657392
$408$	0	0
$409$	−21.7082	−1.07340	−0.536701	−	0.843773i	$-0.680330\pi$
$409$	−21.7082	−1.07340	−0.536701	+	0.843773i	$0.680330\pi$
$410$	2.29180	0.113184
$411$	0	0
$412$	2.14590	0.105721
$413$	45.9787	2.26246
$414$	0	0
$415$	0.583592	0.0286474
$416$	−5.61803	−0.275447
$417$	0	0
$418$	0	0
$419$	−8.94427	−0.436956	−0.218478	−	0.975842i	$-0.570109\pi$
$419$	−8.94427	−0.436956	−0.218478	+	0.975842i	$0.570109\pi$
$420$	0	0
$421$	−18.5279	−0.902993	−0.451496	−	0.892273i	$-0.649110\pi$
$421$	−18.5279	−0.902993	−0.451496	+	0.892273i	$0.649110\pi$
$422$	1.76393	0.0858669
$423$	0	0
$424$	20.8541	1.01276
$425$	18.1803	0.881876
$426$	0	0
$427$	−17.2918	−0.836809
$428$	−16.8541	−0.814674
$429$	0	0
$430$	−5.23607	−0.252506
$431$	3.65248	0.175934	0.0879668	−	0.996123i	$-0.471963\pi$
$431$	3.65248	0.175934	0.0879668	+	0.996123i	$0.471963\pi$
$432$	0	0
$433$	−23.5623	−1.13233	−0.566166	−	0.824291i	$-0.691574\pi$
$433$	−23.5623	−1.13233	−0.566166	+	0.824291i	$0.691574\pi$
$434$	−3.97871	−0.190984
$435$	0	0
$436$	27.0344	1.29471
$437$	0	0
$438$	0	0
$439$	14.5967	0.696665	0.348332	−	0.937371i	$-0.386748\pi$
$439$	14.5967	0.696665	0.348332	+	0.937371i	$0.386748\pi$
$440$	1.70820	0.0814354
$441$	0	0
$442$	3.23607	0.153924
$443$	34.4164	1.63517	0.817586	−	0.575806i	$-0.195312\pi$
$443$	34.4164	1.63517	0.817586	+	0.575806i	$0.195312\pi$
$444$	0	0
$445$	15.1246	0.716975
$446$	12.1459	0.575125
$447$	0	0
$448$	−0.708204	−0.0334595
$449$	32.8885	1.55211	0.776053	−	0.630667i	$-0.217219\pi$
$449$	32.8885	1.55211	0.776053	+	0.630667i	$0.217219\pi$
$450$	0	0
$451$	−1.85410	−0.0873063
$452$	18.1803	0.855131
$453$	0	0
$454$	6.43769	0.302136
$455$	−3.70820	−0.173843
$456$	0	0
$457$	6.29180	0.294318	0.147159	−	0.989113i	$-0.452987\pi$
$457$	6.29180	0.294318	0.147159	+	0.989113i	$0.452987\pi$
$458$	−7.03444	−0.328698
$459$	0	0
$460$	15.2361	0.710385
$461$	−3.05573	−0.142319	−0.0711597	−	0.997465i	$-0.522670\pi$
$461$	−3.05573	−0.142319	−0.0711597	+	0.997465i	$0.522670\pi$
$462$	0	0
$463$	−5.27051	−0.244941	−0.122471	−	0.992472i	$-0.539082\pi$
$463$	−5.27051	−0.244941	−0.122471	+	0.992472i	$0.539082\pi$
$464$	2.56231	0.118952
$465$	0	0
$466$	−8.32624	−0.385706
$467$	−1.94427	−0.0899702	−0.0449851	−	0.998988i	$-0.514324\pi$
$467$	−1.94427	−0.0899702	−0.0449851	+	0.998988i	$0.514324\pi$
$468$	0	0
$469$	−21.0000	−0.969690
$470$	−2.29180	−0.105713
$471$	0	0
$472$	−34.2705	−1.57743
$473$	4.23607	0.194775
$474$	0	0
$475$	0	0
$476$	25.4164	1.16496
$477$	0	0
$478$	−9.47214	−0.433245
$479$	−11.9098	−0.544174	−0.272087	−	0.962273i	$-0.587714\pi$
$479$	−11.9098	−0.544174	−0.272087	+	0.962273i	$0.587714\pi$
$480$	0	0
$481$	−2.14590	−0.0978445
$482$	−11.8541	−0.539940
$483$	0	0
$484$	17.1803	0.780925
$485$	8.83282	0.401078
$486$	0	0
$487$	−18.1803	−0.823830	−0.411915	−	0.911222i	$-0.635140\pi$
$487$	−18.1803	−0.823830	−0.411915	+	0.911222i	$0.635140\pi$
$488$	12.8885	0.583437
$489$	0	0
$490$	1.52786	0.0690219
$491$	−30.2148	−1.36357	−0.681787	−	0.731551i	$-0.738797\pi$
$491$	−30.2148	−1.36357	−0.681787	+	0.731551i	$0.738797\pi$
$492$	0	0
$493$	−7.23607	−0.325896
$494$	0	0
$495$	0	0
$496$	−3.97871	−0.178650
$497$	−4.41641	−0.198103
$498$	0	0
$499$	−15.1246	−0.677071	−0.338535	−	0.940954i	$-0.609931\pi$
$499$	−15.1246	−0.677071	−0.338535	+	0.940954i	$0.609931\pi$
$500$	16.9443	0.757771
$501$	0	0
$502$	−11.9656	−0.534049
$503$	17.8328	0.795126	0.397563	−	0.917575i	$-0.369856\pi$
$503$	17.8328	0.795126	0.397563	+	0.917575i	$0.369856\pi$
$504$	0	0
$505$	−16.2918	−0.724975
$506$	2.90983	0.129358
$507$	0	0
$508$	−16.5623	−0.734833
$509$	27.0344	1.19828	0.599140	−	0.800644i	$-0.295509\pi$
$509$	27.0344	1.19828	0.599140	+	0.800644i	$0.295509\pi$
$510$	0	0
$511$	32.1246	1.42111
$512$	18.7082	0.826794
$513$	0	0
$514$	15.0557	0.664080
$515$	−1.63932	−0.0722371
$516$	0	0
$517$	1.85410	0.0815433
$518$	3.97871	0.174815
$519$	0	0
$520$	2.76393	0.121206
$521$	27.2705	1.19474	0.597371	−	0.801965i	$-0.296212\pi$
$521$	27.2705	1.19474	0.597371	+	0.801965i	$0.296212\pi$
$522$	0	0
$523$	22.4164	0.980201	0.490101	−	0.871666i	$-0.336960\pi$
$523$	22.4164	0.980201	0.490101	+	0.871666i	$0.336960\pi$
$524$	6.32624	0.276363
$525$	0	0
$526$	−1.81966	−0.0793410
$527$	11.2361	0.489451
$528$	0	0
$529$	35.0344	1.52324
$530$	−7.12461	−0.309473
$531$	0	0
$532$	0	0
$533$	−3.00000	−0.129944
$534$	0	0
$535$	12.8754	0.556652
$536$	15.6525	0.676084
$537$	0	0
$538$	−9.06888	−0.390987
$539$	−1.23607	−0.0532412
$540$	0	0
$541$	23.8328	1.02465	0.512326	−	0.858791i	$-0.328784\pi$
$541$	23.8328	1.02465	0.512326	+	0.858791i	$0.328784\pi$
$542$	4.85410	0.208502
$543$	0	0
$544$	−29.4164	−1.26122
$545$	−20.6525	−0.884655
$546$	0	0
$547$	37.9230	1.62147	0.810735	−	0.585413i	$-0.199068\pi$
$547$	37.9230	1.62147	0.810735	+	0.585413i	$0.199068\pi$
$548$	−2.38197	−0.101753
$549$	0	0
$550$	1.32624	0.0565510
$551$	0	0
$552$	0	0
$553$	−40.2492	−1.71157
$554$	−9.52786	−0.404800
$555$	0	0
$556$	15.8541	0.672364
$557$	23.1803	0.982183	0.491091	−	0.871108i	$-0.336598\pi$
$557$	23.1803	0.982183	0.491091	+	0.871108i	$0.336598\pi$
$558$	0	0
$559$	6.85410	0.289898
$560$	6.87539	0.290538
$561$	0	0
$562$	−5.25735	−0.221768
$563$	−20.8328	−0.877999	−0.438999	−	0.898487i	$-0.644667\pi$
$563$	−20.8328	−0.877999	−0.438999	+	0.898487i	$0.644667\pi$
$564$	0	0
$565$	−13.8885	−0.584295
$566$	−1.87539	−0.0788284
$567$	0	0
$568$	3.29180	0.138121
$569$	−28.0902	−1.17760	−0.588801	−	0.808278i	$-0.700400\pi$
$569$	−28.0902	−1.17760	−0.588801	+	0.808278i	$0.700400\pi$
$570$	0	0
$571$	22.3262	0.934324	0.467162	−	0.884172i	$-0.345276\pi$
$571$	22.3262	0.934324	0.467162	+	0.884172i	$0.345276\pi$
$572$	−1.00000	−0.0418121
$573$	0	0
$574$	5.56231	0.232166
$575$	26.4508	1.10308
$576$	0	0
$577$	28.1246	1.17084	0.585421	−	0.810729i	$-0.300929\pi$
$577$	28.1246	1.17084	0.585421	+	0.810729i	$0.300929\pi$
$578$	6.43769	0.267773
$579$	0	0
$580$	−2.76393	−0.114766
$581$	1.41641	0.0587625
$582$	0	0
$583$	5.76393	0.238718
$584$	−23.9443	−0.990821
$585$	0	0
$586$	10.4164	0.430298
$587$	−38.1246	−1.57357	−0.786786	−	0.617226i	$-0.788256\pi$
$587$	−38.1246	−1.57357	−0.786786	+	0.617226i	$0.788256\pi$
$588$	0	0
$589$	0	0
$590$	11.7082	0.482019
$591$	0	0
$592$	3.97871	0.163524
$593$	12.7082	0.521863	0.260932	−	0.965357i	$-0.415970\pi$
$593$	12.7082	0.521863	0.260932	+	0.965357i	$0.415970\pi$
$594$	0	0
$595$	−19.4164	−0.795995
$596$	−21.1803	−0.867581
$597$	0	0
$598$	4.70820	0.192533
$599$	−1.58359	−0.0647038	−0.0323519	−	0.999477i	$-0.510300\pi$
$599$	−1.58359	−0.0647038	−0.0323519	+	0.999477i	$0.510300\pi$
$600$	0	0
$601$	33.7082	1.37499	0.687493	−	0.726191i	$-0.258711\pi$
$601$	33.7082	1.37499	0.687493	+	0.726191i	$0.258711\pi$
$602$	−12.7082	−0.517948
$603$	0	0
$604$	16.0344	0.652432
$605$	−13.1246	−0.533591
$606$	0	0
$607$	27.2705	1.10688	0.553438	−	0.832890i	$-0.313316\pi$
$607$	27.2705	1.10688	0.553438	+	0.832890i	$0.313316\pi$
$608$	0	0
$609$	0	0
$610$	−4.40325	−0.178282
$611$	3.00000	0.121367
$612$	0	0
$613$	−2.05573	−0.0830301	−0.0415150	−	0.999138i	$-0.513218\pi$
$613$	−2.05573	−0.0830301	−0.0415150	+	0.999138i	$0.513218\pi$
$614$	−1.03444	−0.0417467
$615$	0	0
$616$	4.14590	0.167043
$617$	−23.3262	−0.939079	−0.469539	−	0.882911i	$-0.655580\pi$
$617$	−23.3262	−0.939079	−0.469539	+	0.882911i	$0.655580\pi$
$618$	0	0
$619$	−30.1246	−1.21081	−0.605405	−	0.795917i	$-0.706989\pi$
$619$	−30.1246	−1.21081	−0.605405	+	0.795917i	$0.706989\pi$
$620$	4.29180	0.172363
$621$	0	0
$622$	11.5279	0.462225
$623$	36.7082	1.47068
$624$	0	0
$625$	4.41641	0.176656
$626$	2.67376	0.106865
$627$	0	0
$628$	18.0344	0.719653
$629$	−11.2361	−0.448011
$630$	0	0
$631$	−15.3607	−0.611499	−0.305750	−	0.952112i	$-0.598907\pi$
$631$	−15.3607	−0.611499	−0.305750	+	0.952112i	$0.598907\pi$
$632$	30.0000	1.19334
$633$	0	0
$634$	−11.1246	−0.441815
$635$	12.6525	0.502098
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	−0.527864	−0.0208983
$639$	0	0
$640$	−14.0689	−0.556121
$641$	1.49342	0.0589866	0.0294933	−	0.999565i	$-0.490611\pi$
$641$	1.49342	0.0589866	0.0294933	+	0.999565i	$0.490611\pi$
$642$	0	0
$643$	−37.7082	−1.48707	−0.743533	−	0.668699i	$-0.766851\pi$
$643$	−37.7082	−1.48707	−0.743533	+	0.668699i	$0.766851\pi$
$644$	36.9787	1.45717
$645$	0	0
$646$	0	0
$647$	1.47214	0.0578756	0.0289378	−	0.999581i	$-0.490788\pi$
$647$	1.47214	0.0578756	0.0289378	+	0.999581i	$0.490788\pi$
$648$	0	0
$649$	−9.47214	−0.371814
$650$	2.14590	0.0841690
$651$	0	0
$652$	−10.0902	−0.395162
$653$	3.43769	0.134527	0.0672637	−	0.997735i	$-0.478573\pi$
$653$	3.43769	0.134527	0.0672637	+	0.997735i	$0.478573\pi$
$654$	0	0
$655$	−4.83282	−0.188834
$656$	5.56231	0.217172
$657$	0	0
$658$	−5.56231	−0.216841
$659$	−45.7771	−1.78322	−0.891611	−	0.452802i	$-0.850424\pi$
$659$	−45.7771	−1.78322	−0.891611	+	0.452802i	$0.850424\pi$
$660$	0	0
$661$	−21.4164	−0.833002	−0.416501	−	0.909135i	$-0.636744\pi$
$661$	−21.4164	−0.833002	−0.416501	+	0.909135i	$0.636744\pi$
$662$	−4.29180	−0.166805
$663$	0	0
$664$	−1.05573	−0.0409702
$665$	0	0
$666$	0	0
$667$	−10.5279	−0.407641
$668$	25.5066	0.986879
$669$	0	0
$670$	−5.34752	−0.206593
$671$	3.56231	0.137521
$672$	0	0
$673$	−6.12461	−0.236086	−0.118043	−	0.993008i	$-0.537662\pi$
$673$	−6.12461	−0.236086	−0.118043	+	0.993008i	$0.537662\pi$
$674$	14.2918	0.550499
$675$	0	0
$676$	19.4164	0.746785
$677$	−11.7426	−0.451307	−0.225653	−	0.974208i	$-0.572452\pi$
$677$	−11.7426	−0.451307	−0.225653	+	0.974208i	$0.572452\pi$
$678$	0	0
$679$	21.4377	0.822703
$680$	14.4721	0.554981
$681$	0	0
$682$	0.819660	0.0313864
$683$	21.6525	0.828509	0.414254	−	0.910161i	$-0.364042\pi$
$683$	21.6525	0.828509	0.414254	+	0.910161i	$0.364042\pi$
$684$	0	0
$685$	1.81966	0.0695256
$686$	−9.27051	−0.353950
$687$	0	0
$688$	−12.7082	−0.484496
$689$	9.32624	0.355301
$690$	0	0
$691$	−16.8197	−0.639850	−0.319925	−	0.947443i	$-0.603658\pi$
$691$	−16.8197	−0.639850	−0.319925	+	0.947443i	$0.603658\pi$
$692$	13.7082	0.521108
$693$	0	0
$694$	−0.875388	−0.0332293
$695$	−12.1115	−0.459414
$696$	0	0
$697$	−15.7082	−0.594991
$698$	16.0557	0.607718
$699$	0	0
$700$	16.8541	0.637025
$701$	−38.6312	−1.45908	−0.729540	−	0.683938i	$-0.760266\pi$
$701$	−38.6312	−1.45908	−0.729540	+	0.683938i	$0.760266\pi$
$702$	0	0
$703$	0	0
$704$	0.145898	0.00549874
$705$	0	0
$706$	19.4377	0.731547
$707$	−39.5410	−1.48709
$708$	0	0
$709$	43.4164	1.63054	0.815269	−	0.579083i	$-0.196589\pi$
$709$	43.4164	1.63054	0.815269	+	0.579083i	$0.196589\pi$
$710$	−1.12461	−0.0422059
$711$	0	0
$712$	−27.3607	−1.02538
$713$	16.3475	0.612220
$714$	0	0
$715$	0.763932	0.0285694
$716$	12.5623	0.469475
$717$	0	0
$718$	13.6180	0.508221
$719$	−17.9656	−0.670002	−0.335001	−	0.942218i	$-0.608737\pi$
$719$	−17.9656	−0.670002	−0.335001	+	0.942218i	$0.608737\pi$
$720$	0	0
$721$	−3.97871	−0.148175
$722$	0	0
$723$	0	0
$724$	−19.4164	−0.721605
$725$	−4.79837	−0.178207
$726$	0	0
$727$	42.0689	1.56025	0.780124	−	0.625625i	$-0.215156\pi$
$727$	42.0689	1.56025	0.780124	+	0.625625i	$0.215156\pi$
$728$	6.70820	0.248623
$729$	0	0
$730$	8.18034	0.302768
$731$	35.8885	1.32739
$732$	0	0
$733$	40.9574	1.51280	0.756399	−	0.654111i	$-0.226957\pi$
$733$	40.9574	1.51280	0.756399	+	0.654111i	$0.226957\pi$
$734$	1.20163	0.0443528
$735$	0	0
$736$	−42.7984	−1.57757
$737$	4.32624	0.159359
$738$	0	0
$739$	−25.0000	−0.919640	−0.459820	−	0.888012i	$-0.652086\pi$
$739$	−25.0000	−0.919640	−0.459820	+	0.888012i	$0.652086\pi$
$740$	−4.29180	−0.157770
$741$	0	0
$742$	−17.2918	−0.634802
$743$	−41.3607	−1.51738	−0.758688	−	0.651454i	$-0.774159\pi$
$743$	−41.3607	−1.51738	−0.758688	+	0.651454i	$0.774159\pi$
$744$	0	0
$745$	16.1803	0.592802
$746$	2.14590	0.0785669
$747$	0	0
$748$	−5.23607	−0.191450
$749$	31.2492	1.14182
$750$	0	0
$751$	−23.8541	−0.870449	−0.435224	−	0.900322i	$-0.643331\pi$
$751$	−23.8541	−0.870449	−0.435224	+	0.900322i	$0.643331\pi$
$752$	−5.56231	−0.202836
$753$	0	0
$754$	−0.854102	−0.0311046
$755$	−12.2492	−0.445795
$756$	0	0
$757$	−15.7426	−0.572176	−0.286088	−	0.958203i	$-0.592355\pi$
$757$	−15.7426	−0.572176	−0.286088	+	0.958203i	$0.592355\pi$
$758$	15.5279	0.563997
$759$	0	0
$760$	0	0
$761$	30.8885	1.11971	0.559854	−	0.828591i	$-0.310857\pi$
$761$	30.8885	1.11971	0.559854	+	0.828591i	$0.310857\pi$
$762$	0	0
$763$	−50.1246	−1.81463
$764$	15.7984	0.571565
$765$	0	0
$766$	−1.61803	−0.0584619
$767$	−15.3262	−0.553398
$768$	0	0
$769$	−41.6312	−1.50126	−0.750630	−	0.660723i	$-0.770250\pi$
$769$	−41.6312	−1.50126	−0.750630	+	0.660723i	$0.770250\pi$
$770$	−1.41641	−0.0510438
$771$	0	0
$772$	−37.1246	−1.33614
$773$	−28.9230	−1.04029	−0.520144	−	0.854079i	$-0.674122\pi$
$773$	−28.9230	−1.04029	−0.520144	+	0.854079i	$0.674122\pi$
$774$	0	0
$775$	7.45085	0.267642
$776$	−15.9787	−0.573602
$777$	0	0
$778$	15.0000	0.537776
$779$	0	0
$780$	0	0
$781$	0.909830	0.0325563
$782$	24.6525	0.881571
$783$	0	0
$784$	3.70820	0.132436
$785$	−13.7771	−0.491725
$786$	0	0
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	4.85410	0.172920
$789$	0	0
$790$	−10.2492	−0.364651
$791$	−33.7082	−1.19853
$792$	0	0
$793$	5.76393	0.204683
$794$	−1.56231	−0.0554442
$795$	0	0
$796$	−21.7082	−0.769427
$797$	33.7082	1.19401	0.597003	−	0.802239i	$-0.296358\pi$
$797$	33.7082	1.19401	0.597003	+	0.802239i	$0.296358\pi$
$798$	0	0
$799$	15.7082	0.555716
$800$	−19.5066	−0.689662
$801$	0	0
$802$	0.0688837	0.00243237
$803$	−6.61803	−0.233545
$804$	0	0
$805$	−28.2492	−0.995654
$806$	1.32624	0.0467147
$807$	0	0
$808$	29.4721	1.03683
$809$	0.201626	0.00708880	0.00354440	−	0.999994i	$-0.498872\pi$
$809$	0.201626	0.00708880	0.00354440	+	0.999994i	$0.498872\pi$
$810$	0	0
$811$	−23.9787	−0.842007	−0.421003	−	0.907059i	$-0.638322\pi$
$811$	−23.9787	−0.842007	−0.421003	+	0.907059i	$0.638322\pi$
$812$	−6.70820	−0.235412
$813$	0	0
$814$	−0.819660	−0.0287291
$815$	7.70820	0.270007
$816$	0	0
$817$	0	0
$818$	−13.4164	−0.469094
$819$	0	0
$820$	−6.00000	−0.209529
$821$	53.0476	1.85137	0.925687	−	0.378290i	$-0.123488\pi$
$821$	53.0476	1.85137	0.925687	+	0.378290i	$0.123488\pi$
$822$	0	0
$823$	23.5967	0.822531	0.411265	−	0.911516i	$-0.365087\pi$
$823$	23.5967	0.822531	0.411265	+	0.911516i	$0.365087\pi$
$824$	2.96556	0.103310
$825$	0	0
$826$	28.4164	0.988733
$827$	16.5967	0.577125	0.288563	−	0.957461i	$-0.406823\pi$
$827$	16.5967	0.577125	0.288563	+	0.957461i	$0.406823\pi$
$828$	0	0
$829$	20.3262	0.705959	0.352980	−	0.935631i	$-0.385168\pi$
$829$	20.3262	0.705959	0.352980	+	0.935631i	$0.385168\pi$
$830$	0.360680	0.0125194
$831$	0	0
$832$	0.236068	0.00818418
$833$	−10.4721	−0.362838
$834$	0	0
$835$	−19.4853	−0.674316
$836$	0	0
$837$	0	0
$838$	−5.52786	−0.190957
$839$	−39.7984	−1.37399	−0.686996	−	0.726661i	$-0.741071\pi$
$839$	−39.7984	−1.37399	−0.686996	+	0.726661i	$0.741071\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	−11.4508	−0.394622
$843$	0	0
$844$	−4.61803	−0.158959
$845$	−14.8328	−0.510264
$846$	0	0
$847$	−31.8541	−1.09452
$848$	−17.2918	−0.593803
$849$	0	0
$850$	11.2361	0.385394
$851$	−16.3475	−0.560386
$852$	0	0
$853$	17.2918	0.592060	0.296030	−	0.955179i	$-0.404337\pi$
$853$	17.2918	0.592060	0.296030	+	0.955179i	$0.404337\pi$
$854$	−10.6869	−0.365699
$855$	0	0
$856$	−23.2918	−0.796097
$857$	8.38197	0.286323	0.143161	−	0.989699i	$-0.454273\pi$
$857$	8.38197	0.286323	0.143161	+	0.989699i	$0.454273\pi$
$858$	0	0
$859$	18.5410	0.632611	0.316306	−	0.948657i	$-0.397557\pi$
$859$	18.5410	0.632611	0.316306	+	0.948657i	$0.397557\pi$
$860$	13.7082	0.467446
$861$	0	0
$862$	2.25735	0.0768858
$863$	44.9443	1.52992	0.764960	−	0.644077i	$-0.222759\pi$
$863$	44.9443	1.52992	0.764960	+	0.644077i	$0.222759\pi$
$864$	0	0
$865$	−10.4721	−0.356063
$866$	−14.5623	−0.494847
$867$	0	0
$868$	10.4164	0.353556
$869$	8.29180	0.281280
$870$	0	0
$871$	7.00000	0.237186
$872$	37.3607	1.26519
$873$	0	0
$874$	0	0
$875$	−31.4164	−1.06207
$876$	0	0
$877$	34.1803	1.15419	0.577094	−	0.816678i	$-0.304187\pi$
$877$	34.1803	1.15419	0.577094	+	0.816678i	$0.304187\pi$
$878$	9.02129	0.304454
$879$	0	0
$880$	−1.41641	−0.0477471
$881$	23.4508	0.790079	0.395040	−	0.918664i	$-0.370731\pi$
$881$	23.4508	0.790079	0.395040	+	0.918664i	$0.370731\pi$
$882$	0	0
$883$	13.9230	0.468546	0.234273	−	0.972171i	$-0.424729\pi$
$883$	13.9230	0.468546	0.234273	+	0.972171i	$0.424729\pi$
$884$	−8.47214	−0.284949
$885$	0	0
$886$	21.2705	0.714597
$887$	−58.6525	−1.96936	−0.984679	−	0.174378i	$-0.944208\pi$
$887$	−58.6525	−1.96936	−0.984679	+	0.174378i	$0.944208\pi$
$888$	0	0
$889$	30.7082	1.02992
$890$	9.34752	0.313330
$891$	0	0
$892$	−31.7984	−1.06469
$893$	0	0
$894$	0	0
$895$	−9.59675	−0.320784
$896$	−34.1459	−1.14073
$897$	0	0
$898$	20.3262	0.678295
$899$	−2.96556	−0.0989069
$900$	0	0
$901$	48.8328	1.62686
$902$	−1.14590	−0.0381542
$903$	0	0
$904$	25.1246	0.835632
$905$	14.8328	0.493059
$906$	0	0
$907$	−17.5279	−0.582003	−0.291002	−	0.956723i	$-0.593988\pi$
$907$	−17.5279	−0.582003	−0.291002	+	0.956723i	$0.593988\pi$
$908$	−16.8541	−0.559323
$909$	0	0
$910$	−2.29180	−0.0759723
$911$	−5.61803	−0.186134	−0.0930669	−	0.995660i	$-0.529667\pi$
$911$	−5.61803	−0.186134	−0.0930669	+	0.995660i	$0.529667\pi$
$912$	0	0
$913$	−0.291796	−0.00965704
$914$	3.88854	0.128622
$915$	0	0
$916$	18.4164	0.608495
$917$	−11.7295	−0.387342
$918$	0	0
$919$	36.7082	1.21089	0.605446	−	0.795886i	$-0.292995\pi$
$919$	36.7082	1.21089	0.605446	+	0.795886i	$0.292995\pi$
$920$	21.0557	0.694187
$921$	0	0
$922$	−1.88854	−0.0621959
$923$	1.47214	0.0484559
$924$	0	0
$925$	−7.45085	−0.244982
$926$	−3.25735	−0.107043
$927$	0	0
$928$	7.76393	0.254864
$929$	−18.6180	−0.610838	−0.305419	−	0.952218i	$-0.598797\pi$
$929$	−18.6180	−0.610838	−0.305419	+	0.952218i	$0.598797\pi$
$930$	0	0
$931$	0	0
$932$	21.7984	0.714029
$933$	0	0
$934$	−1.20163	−0.0393184
$935$	4.00000	0.130814
$936$	0	0
$937$	20.4377	0.667670	0.333835	−	0.942631i	$-0.391657\pi$
$937$	20.4377	0.667670	0.333835	+	0.942631i	$0.391657\pi$
$938$	−12.9787	−0.423770
$939$	0	0
$940$	6.00000	0.195698
$941$	−49.6869	−1.61975	−0.809874	−	0.586604i	$-0.800464\pi$
$941$	−49.6869	−1.61975	−0.809874	+	0.586604i	$0.800464\pi$
$942$	0	0
$943$	−22.8541	−0.744232
$944$	28.4164	0.924875
$945$	0	0
$946$	2.61803	0.0851196
$947$	1.34752	0.0437887	0.0218943	−	0.999760i	$-0.493030\pi$
$947$	1.34752	0.0437887	0.0218943	+	0.999760i	$0.493030\pi$
$948$	0	0
$949$	−10.7082	−0.347603
$950$	0	0
$951$	0	0
$952$	35.1246	1.13840
$953$	−30.7082	−0.994736	−0.497368	−	0.867540i	$-0.665700\pi$
$953$	−30.7082	−0.994736	−0.497368	+	0.867540i	$0.665700\pi$
$954$	0	0
$955$	−12.0689	−0.390540
$956$	24.7984	0.802037
$957$	0	0
$958$	−7.36068	−0.237813
$959$	4.41641	0.142613
$960$	0	0
$961$	−26.3951	−0.851456
$962$	−1.32624	−0.0427596
$963$	0	0
$964$	31.0344	0.999552
$965$	28.3607	0.912963
$966$	0	0
$967$	−60.5410	−1.94687	−0.973434	−	0.228968i	$-0.926465\pi$
$967$	−60.5410	−1.94687	−0.973434	+	0.228968i	$0.926465\pi$
$968$	23.7426	0.763118
$969$	0	0
$970$	5.45898	0.175277
$971$	14.5066	0.465538	0.232769	−	0.972532i	$-0.425221\pi$
$971$	14.5066	0.465538	0.232769	+	0.972532i	$0.425221\pi$
$972$	0	0
$973$	−29.3951	−0.942364
$974$	−11.2361	−0.360027
$975$	0	0
$976$	−10.6869	−0.342080
$977$	−55.3607	−1.77115	−0.885573	−	0.464501i	$-0.846234\pi$
$977$	−55.3607	−1.77115	−0.885573	+	0.464501i	$0.846234\pi$
$978$	0	0
$979$	−7.56231	−0.241692
$980$	−4.00000	−0.127775
$981$	0	0
$982$	−18.6738	−0.595904
$983$	−30.3820	−0.969034	−0.484517	−	0.874782i	$-0.661005\pi$
$983$	−30.3820	−0.969034	−0.484517	+	0.874782i	$0.661005\pi$
$984$	0	0
$985$	−3.70820	−0.118153
$986$	−4.47214	−0.142422
$987$	0	0
$988$	0	0
$989$	52.2148	1.66033
$990$	0	0
$991$	−48.4508	−1.53909	−0.769546	−	0.638591i	$-0.779517\pi$
$991$	−48.4508	−1.53909	−0.769546	+	0.638591i	$0.779517\pi$
$992$	−12.0557	−0.382770
$993$	0	0
$994$	−2.72949	−0.0865742
$995$	16.5836	0.525735
$996$	0	0
$997$	26.2918	0.832670	0.416335	−	0.909211i	$-0.363314\pi$
$997$	26.2918	0.832670	0.416335	+	0.909211i	$0.363314\pi$
$998$	−9.34752	−0.295891
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.i.1.2		2
3.2	odd	2		361.2.a.f.1.1	yes	2
12.11	even	2		5776.2.a.s.1.1		2
15.14	odd	2		9025.2.a.n.1.2		2
19.18	odd	2		3249.2.a.o.1.1		2
57.2	even	18		361.2.e.i.99.1		12
57.5	odd	18		361.2.e.j.234.1		12
57.8	even	6		361.2.c.g.292.1		4
57.11	odd	6		361.2.c.d.292.2		4
57.14	even	18		361.2.e.i.234.2		12
57.17	odd	18		361.2.e.j.99.2		12
57.23	odd	18		361.2.e.j.54.1		12
57.26	odd	6		361.2.c.d.68.2		4
57.29	even	18		361.2.e.i.62.1		12
57.32	even	18		361.2.e.i.245.2		12
57.35	odd	18		361.2.e.j.28.1		12
57.41	even	18		361.2.e.i.28.2		12
57.44	odd	18		361.2.e.j.245.1		12
57.47	odd	18		361.2.e.j.62.2		12
57.50	even	6		361.2.c.g.68.1		4
57.53	even	18		361.2.e.i.54.2		12
57.56	even	2		361.2.a.c.1.2	✓	2
228.227	odd	2		5776.2.a.bg.1.2		2
285.284	even	2		9025.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
361.2.a.c.1.2	✓	2	57.56	even	2
361.2.a.f.1.1	yes	2	3.2	odd	2
361.2.c.d.68.2		4	57.26	odd	6
361.2.c.d.292.2		4	57.11	odd	6
361.2.c.g.68.1		4	57.50	even	6
361.2.c.g.292.1		4	57.8	even	6
361.2.e.i.28.2		12	57.41	even	18
361.2.e.i.54.2		12	57.53	even	18
361.2.e.i.62.1		12	57.29	even	18
361.2.e.i.99.1		12	57.2	even	18
361.2.e.i.234.2		12	57.14	even	18
361.2.e.i.245.2		12	57.32	even	18
361.2.e.j.28.1		12	57.35	odd	18
361.2.e.j.54.1		12	57.23	odd	18
361.2.e.j.62.2		12	57.47	odd	18
361.2.e.j.99.2		12	57.17	odd	18
361.2.e.j.234.1		12	57.5	odd	18
361.2.e.j.245.1		12	57.44	odd	18
3249.2.a.i.1.2		2	1.1	even	1	trivial
3249.2.a.o.1.1		2	19.18	odd	2
5776.2.a.s.1.1		2	12.11	even	2
5776.2.a.bg.1.2		2	228.227	odd	2
9025.2.a.n.1.2		2	15.14	odd	2
9025.2.a.s.1.1		2	285.284	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.23607 q^{5} +3.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.23607 q^{5} +3.00000 q^{7} -2.23607 q^{8} +0.763932 q^{10} -0.618034 q^{11} -1.00000 q^{13} +1.85410 q^{14} +1.85410 q^{16} -5.23607 q^{17} -2.00000 q^{20} -0.381966 q^{22} -7.61803 q^{23} -3.47214 q^{25} -0.618034 q^{26} -4.85410 q^{28} +1.38197 q^{29} -2.14590 q^{31} +5.61803 q^{32} -3.23607 q^{34} +3.70820 q^{35} +2.14590 q^{37} -2.76393 q^{40} +3.00000 q^{41} -6.85410 q^{43} +1.00000 q^{44} -4.70820 q^{46} -3.00000 q^{47} +2.00000 q^{49} -2.14590 q^{50} +1.61803 q^{52} -9.32624 q^{53} -0.763932 q^{55} -6.70820 q^{56} +0.854102 q^{58} +15.3262 q^{59} -5.76393 q^{61} -1.32624 q^{62} -0.236068 q^{64} -1.23607 q^{65} -7.00000 q^{67} +8.47214 q^{68} +2.29180 q^{70} -1.47214 q^{71} +10.7082 q^{73} +1.32624 q^{74} -1.85410 q^{77} -13.4164 q^{79} +2.29180 q^{80} +1.85410 q^{82} +0.472136 q^{83} -6.47214 q^{85} -4.23607 q^{86} +1.38197 q^{88} +12.2361 q^{89} -3.00000 q^{91} +12.3262 q^{92} -1.85410 q^{94} +7.14590 q^{97} +1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 2 q^{5} + 6 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 - 2 * q^5 + 6 * q^7	\( 2 q - q^{2} - q^{4} - 2 q^{5} + 6 q^{7} + 6 q^{10} + q^{11} - 2 q^{13} - 3 q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{20} - 3 q^{22} - 13 q^{23} + 2 q^{25} + q^{26} - 3 q^{28} + 5 q^{29} - 11 q^{31} + 9 q^{32} - 2 q^{34} - 6 q^{35} + 11 q^{37} - 10 q^{40} + 6 q^{41} - 7 q^{43} + 2 q^{44} + 4 q^{46} - 6 q^{47} + 4 q^{49} - 11 q^{50} + q^{52} - 3 q^{53} - 6 q^{55} - 5 q^{58} + 15 q^{59} - 16 q^{61} + 13 q^{62} + 4 q^{64} + 2 q^{65} - 14 q^{67} + 8 q^{68} + 18 q^{70} + 6 q^{71} + 8 q^{73} - 13 q^{74} + 3 q^{77} + 18 q^{80} - 3 q^{82} - 8 q^{83} - 4 q^{85} - 4 q^{86} + 5 q^{88} + 20 q^{89} - 6 q^{91} + 9 q^{92} + 3 q^{94} + 21 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 2 * q^5 + 6 * q^7 + 6 * q^10 + q^11 - 2 * q^13 - 3 * q^14 - 3 * q^16 - 6 * q^17 - 4 * q^20 - 3 * q^22 - 13 * q^23 + 2 * q^25 + q^26 - 3 * q^28 + 5 * q^29 - 11 * q^31 + 9 * q^32 - 2 * q^34 - 6 * q^35 + 11 * q^37 - 10 * q^40 + 6 * q^41 - 7 * q^43 + 2 * q^44 + 4 * q^46 - 6 * q^47 + 4 * q^49 - 11 * q^50 + q^52 - 3 * q^53 - 6 * q^55 - 5 * q^58 + 15 * q^59 - 16 * q^61 + 13 * q^62 + 4 * q^64 + 2 * q^65 - 14 * q^67 + 8 * q^68 + 18 * q^70 + 6 * q^71 + 8 * q^73 - 13 * q^74 + 3 * q^77 + 18 * q^80 - 3 * q^82 - 8 * q^83 - 4 * q^85 - 4 * q^86 + 5 * q^88 + 20 * q^89 - 6 * q^91 + 9 * q^92 + 3 * q^94 + 21 * q^97 - 2 * q^98

Introduction

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