LMFDB - Embedded newform 6825.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6825,2,Mod(1,6825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6825.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:	$54.4978993795$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1365)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.90211$ of defining polynomial
Character	$\chi$	$=$	6825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +2.80423 q^{11} +0.618034 q^{12} +1.00000 q^{13} -1.61803 q^{14} -4.85410 q^{16} -2.01062 q^{17} -1.61803 q^{18} -2.34458 q^{19} +1.00000 q^{21} -4.53733 q^{22} -0.939503 q^{23} +2.23607 q^{24} -1.61803 q^{26} +1.00000 q^{27} +0.618034 q^{28} -5.69572 q^{29} +0.895549 q^{31} +3.38197 q^{32} +2.80423 q^{33} +3.25325 q^{34} +0.618034 q^{36} -5.52015 q^{37} +3.79360 q^{38} +1.00000 q^{39} +3.07768 q^{41} -1.61803 q^{42} +5.12454 q^{43} +1.73311 q^{44} +1.52015 q^{46} -5.96261 q^{47} -4.85410 q^{48} +1.00000 q^{49} -2.01062 q^{51} +0.618034 q^{52} -1.30313 q^{53} -1.61803 q^{54} +2.23607 q^{56} -2.34458 q^{57} +9.21586 q^{58} -2.47027 q^{59} -4.15838 q^{61} -1.44903 q^{62} +1.00000 q^{63} +4.23607 q^{64} -4.53733 q^{66} -15.4797 q^{67} -1.24263 q^{68} -0.939503 q^{69} -12.3416 q^{71} +2.23607 q^{72} -2.77455 q^{73} +8.93179 q^{74} -1.44903 q^{76} +2.80423 q^{77} -1.61803 q^{78} -6.16495 q^{79} +1.00000 q^{81} -4.97980 q^{82} -15.5861 q^{83} +0.618034 q^{84} -8.29168 q^{86} -5.69572 q^{87} +6.27044 q^{88} -0.821412 q^{89} +1.00000 q^{91} -0.580644 q^{92} +0.895549 q^{93} +9.64771 q^{94} +3.38197 q^{96} -1.98281 q^{97} -1.61803 q^{98} +2.80423 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{14} - 6 q^{16} - 2 q^{17} - 2 q^{18} - 2 q^{19} + 4 q^{21} + 2 q^{22} - 8 q^{23} - 2 q^{26} + 4 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^3 - 2 * q^4 - 2 * q^6 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 2 * q^12 + 4 * q^13 - 2 * q^14 - 6 * q^16 - 2 * q^17 - 2 * q^18 - 2 * q^19 + 4 * q^21 + 2 * q^22 - 8 * q^23 - 2 * q^26 + 4 * q^27 - 2 * q^28 - 6 * q^29 - 2 * q^31 + 18 * q^32 - 4 * q^33 - 4 * q^34 - 2 * q^36 - 10 * q^37 + 6 * q^38 + 4 * q^39 - 2 * q^42 - 18 * q^43 + 2 * q^44 - 6 * q^46 - 12 * q^47 - 6 * q^48 + 4 * q^49 - 2 * q^51 - 2 * q^52 + 10 * q^53 - 2 * q^54 - 2 * q^57 + 8 * q^58 + 4 * q^59 - 20 * q^61 - 4 * q^62 + 4 * q^63 + 8 * q^64 + 2 * q^66 - 10 * q^67 + 6 * q^68 - 8 * q^69 - 14 * q^71 - 14 * q^73 + 10 * q^74 - 4 * q^76 - 4 * q^77 - 2 * q^78 - 26 * q^79 + 4 * q^81 - 12 * q^83 - 2 * q^84 + 4 * q^86 - 6 * q^87 + 20 * q^89 + 4 * q^91 + 14 * q^92 - 2 * q^93 + 16 * q^94 + 18 * q^96 - 16 * q^97 - 2 * q^98 - 4 * q^99

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 $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	2.80423	0.845506	0.422753	−	0.906245i	$-0.361064\pi$
$11$	2.80423	0.845506	0.422753	+	0.906245i	$0.361064\pi$
$12$	0.618034	0.178411
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.61803	−0.432438
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−2.01062	−0.487647	−0.243824	−	0.969820i	$-0.578402\pi$
$17$	−2.01062	−0.487647	−0.243824	+	0.969820i	$0.578402\pi$
$18$	−1.61803	−0.381374
$19$	−2.34458	−0.537883	−0.268941	−	0.963157i	$-0.586674\pi$
$19$	−2.34458	−0.537883	−0.268941	+	0.963157i	$0.586674\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−4.53733	−0.967363
$23$	−0.939503	−0.195900	−0.0979499	−	0.995191i	$-0.531228\pi$
$23$	−0.939503	−0.195900	−0.0979499	+	0.995191i	$0.531228\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−1.61803	−0.317323
$27$	1.00000	0.192450
$28$	0.618034	0.116797
$29$	−5.69572	−1.05767	−0.528834	−	0.848725i	$-0.677371\pi$
$29$	−5.69572	−1.05767	−0.528834	+	0.848725i	$0.677371\pi$
$30$	0	0
$31$	0.895549	0.160845	0.0804226	−	0.996761i	$-0.474373\pi$
$31$	0.895549	0.160845	0.0804226	+	0.996761i	$0.474373\pi$
$32$	3.38197	0.597853
$33$	2.80423	0.488153
$34$	3.25325	0.557928
$35$	0	0
$36$	0.618034	0.103006
$37$	−5.52015	−0.907507	−0.453753	−	0.891127i	$-0.649915\pi$
$37$	−5.52015	−0.907507	−0.453753	+	0.891127i	$0.649915\pi$
$38$	3.79360	0.615404
$39$	1.00000	0.160128
$40$	0	0
$41$	3.07768	0.480653	0.240327	−	0.970692i	$-0.422745\pi$
$41$	3.07768	0.480653	0.240327	+	0.970692i	$0.422745\pi$
$42$	−1.61803	−0.249668
$43$	5.12454	0.781485	0.390743	−	0.920500i	$-0.372218\pi$
$43$	5.12454	0.781485	0.390743	+	0.920500i	$0.372218\pi$
$44$	1.73311	0.261276
$45$	0	0
$46$	1.52015	0.224133
$47$	−5.96261	−0.869736	−0.434868	−	0.900494i	$-0.643205\pi$
$47$	−5.96261	−0.869736	−0.434868	+	0.900494i	$0.643205\pi$
$48$	−4.85410	−0.700629
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.01062	−0.281543
$52$	0.618034	0.0857059
$53$	−1.30313	−0.178999	−0.0894993	−	0.995987i	$-0.528527\pi$
$53$	−1.30313	−0.178999	−0.0894993	+	0.995987i	$0.528527\pi$
$54$	−1.61803	−0.220187
$55$	0	0
$56$	2.23607	0.298807
$57$	−2.34458	−0.310547
$58$	9.21586	1.21010
$59$	−2.47027	−0.321602	−0.160801	−	0.986987i	$-0.551408\pi$
$59$	−2.47027	−0.321602	−0.160801	+	0.986987i	$0.551408\pi$
$60$	0	0
$61$	−4.15838	−0.532427	−0.266213	−	0.963914i	$-0.585773\pi$
$61$	−4.15838	−0.532427	−0.266213	+	0.963914i	$0.585773\pi$
$62$	−1.44903	−0.184027
$63$	1.00000	0.125988
$64$	4.23607	0.529508
$65$	0	0
$66$	−4.53733	−0.558507
$67$	−15.4797	−1.89115	−0.945576	−	0.325402i	$-0.894500\pi$
$67$	−15.4797	−1.89115	−0.945576	+	0.325402i	$0.894500\pi$
$68$	−1.24263	−0.150691
$69$	−0.939503	−0.113103
$70$	0	0
$71$	−12.3416	−1.46467	−0.732337	−	0.680943i	$-0.761570\pi$
$71$	−12.3416	−1.46467	−0.732337	+	0.680943i	$0.761570\pi$
$72$	2.23607	0.263523
$73$	−2.77455	−0.324737	−0.162368	−	0.986730i	$-0.551913\pi$
$73$	−2.77455	−0.324737	−0.162368	+	0.986730i	$0.551913\pi$
$74$	8.93179	1.03830
$75$	0	0
$76$	−1.44903	−0.166215
$77$	2.80423	0.319571
$78$	−1.61803	−0.183206
$79$	−6.16495	−0.693611	−0.346805	−	0.937937i	$-0.612734\pi$
$79$	−6.16495	−0.693611	−0.346805	+	0.937937i	$0.612734\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−4.97980	−0.549927
$83$	−15.5861	−1.71079	−0.855396	−	0.517975i	$-0.826686\pi$
$83$	−15.5861	−1.71079	−0.855396	+	0.517975i	$0.826686\pi$
$84$	0.618034	0.0674330
$85$	0	0
$86$	−8.29168	−0.894115
$87$	−5.69572	−0.610645
$88$	6.27044	0.668431
$89$	−0.821412	−0.0870695	−0.0435348	−	0.999052i	$-0.513862\pi$
$89$	−0.821412	−0.0870695	−0.0435348	+	0.999052i	$0.513862\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−0.580644	−0.0605364
$93$	0.895549	0.0928641
$94$	9.64771	0.995085
$95$	0	0
$96$	3.38197	0.345170
$97$	−1.98281	−0.201324	−0.100662	−	0.994921i	$-0.532096\pi$
$97$	−1.98281	−0.201324	−0.100662	+	0.994921i	$0.532096\pi$
$98$	−1.61803	−0.163446
$99$	2.80423	0.281835
$100$	0	0
$101$	7.71883	0.768052	0.384026	−	0.923322i	$-0.374537\pi$
$101$	7.71883	0.768052	0.384026	+	0.923322i	$0.374537\pi$
$102$	3.25325	0.322120
$103$	−19.3309	−1.90473	−0.952367	−	0.304954i	$-0.901359\pi$
$103$	−19.3309	−1.90473	−0.952367	+	0.304954i	$0.901359\pi$
$104$	2.23607	0.219265
$105$	0	0
$106$	2.10851	0.204796
$107$	7.81736	0.755732	0.377866	−	0.925860i	$-0.376658\pi$
$107$	7.81736	0.755732	0.377866	+	0.925860i	$0.376658\pi$
$108$	0.618034	0.0594703
$109$	1.33686	0.128048	0.0640240	−	0.997948i	$-0.479607\pi$
$109$	1.33686	0.128048	0.0640240	+	0.997948i	$0.479607\pi$
$110$	0	0
$111$	−5.52015	−0.523949
$112$	−4.85410	−0.458670
$113$	−6.79766	−0.639470	−0.319735	−	0.947507i	$-0.603594\pi$
$113$	−6.79766	−0.639470	−0.319735	+	0.947507i	$0.603594\pi$
$114$	3.79360	0.355304
$115$	0	0
$116$	−3.52015	−0.326837
$117$	1.00000	0.0924500
$118$	3.99698	0.367952
$119$	−2.01062	−0.184313
$120$	0	0
$121$	−3.13632	−0.285120
$122$	6.72841	0.609161
$123$	3.07768	0.277505
$124$	0.553479	0.0497039
$125$	0	0
$126$	−1.61803	−0.144146
$127$	−10.4912	−0.930942	−0.465471	−	0.885063i	$-0.654115\pi$
$127$	−10.4912	−0.930942	−0.465471	+	0.885063i	$0.654115\pi$
$128$	−13.6180	−1.20368
$129$	5.12454	0.451191
$130$	0	0
$131$	−2.09319	−0.182883	−0.0914413	−	0.995810i	$-0.529147\pi$
$131$	−2.09319	−0.182883	−0.0914413	+	0.995810i	$0.529147\pi$
$132$	1.73311	0.150848
$133$	−2.34458	−0.203301
$134$	25.0467	2.16371
$135$	0	0
$136$	−4.49589	−0.385519
$137$	12.2704	1.04833	0.524167	−	0.851615i	$-0.324377\pi$
$137$	12.2704	1.04833	0.524167	+	0.851615i	$0.324377\pi$
$138$	1.52015	0.129404
$139$	13.4538	1.14114	0.570568	−	0.821250i	$-0.306723\pi$
$139$	13.4538	1.14114	0.570568	+	0.821250i	$0.306723\pi$
$140$	0	0
$141$	−5.96261	−0.502142
$142$	19.9691	1.67577
$143$	2.80423	0.234501
$144$	−4.85410	−0.404508
$145$	0	0
$146$	4.48932	0.371539
$147$	1.00000	0.0824786
$148$	−3.41164	−0.280435
$149$	12.6216	1.03400	0.517000	−	0.855985i	$-0.327049\pi$
$149$	12.6216	1.03400	0.517000	+	0.855985i	$0.327049\pi$
$150$	0	0
$151$	0.222429	0.0181010	0.00905051	−	0.999959i	$-0.497119\pi$
$151$	0.222429	0.0181010	0.00905051	+	0.999959i	$0.497119\pi$
$152$	−5.24263	−0.425234
$153$	−2.01062	−0.162549
$154$	−4.53733	−0.365629
$155$	0	0
$156$	0.618034	0.0494823
$157$	−19.5013	−1.55637	−0.778187	−	0.628033i	$-0.783860\pi$
$157$	−19.5013	−1.55637	−0.778187	+	0.628033i	$0.783860\pi$
$158$	9.97510	0.793576
$159$	−1.30313	−0.103345
$160$	0	0
$161$	−0.939503	−0.0740432
$162$	−1.61803	−0.127125
$163$	1.14712	0.0898494	0.0449247	−	0.998990i	$-0.485695\pi$
$163$	1.14712	0.0898494	0.0449247	+	0.998990i	$0.485695\pi$
$164$	1.90211	0.148530
$165$	0	0
$166$	25.2188	1.95736
$167$	17.8232	1.37920	0.689599	−	0.724191i	$-0.257787\pi$
$167$	17.8232	1.37920	0.689599	+	0.724191i	$0.257787\pi$
$168$	2.23607	0.172516
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.34458	−0.179294
$172$	3.16714	0.241492
$173$	−2.11443	−0.160757	−0.0803786	−	0.996764i	$-0.525613\pi$
$173$	−2.11443	−0.160757	−0.0803786	+	0.996764i	$0.525613\pi$
$174$	9.21586	0.698653
$175$	0	0
$176$	−13.6120	−1.02604
$177$	−2.47027	−0.185677
$178$	1.32907	0.0996182
$179$	18.6737	1.39574	0.697870	−	0.716225i	$-0.254131\pi$
$179$	18.6737	1.39574	0.697870	+	0.716225i	$0.254131\pi$
$180$	0	0
$181$	6.07872	0.451828	0.225914	−	0.974147i	$-0.427463\pi$
$181$	6.07872	0.451828	0.225914	+	0.974147i	$0.427463\pi$
$182$	−1.61803	−0.119937
$183$	−4.15838	−0.307397
$184$	−2.10079	−0.154872
$185$	0	0
$186$	−1.44903	−0.106248
$187$	−5.63824	−0.412309
$188$	−3.68510	−0.268763
$189$	1.00000	0.0727393
$190$	0	0
$191$	13.9329	1.00815	0.504076	−	0.863659i	$-0.331833\pi$
$191$	13.9329	1.00815	0.504076	+	0.863659i	$0.331833\pi$
$192$	4.23607	0.305712
$193$	−3.14339	−0.226266	−0.113133	−	0.993580i	$-0.536089\pi$
$193$	−3.14339	−0.226266	−0.113133	+	0.993580i	$0.536089\pi$
$194$	3.20826	0.230340
$195$	0	0
$196$	0.618034	0.0441453
$197$	5.11141	0.364173	0.182087	−	0.983283i	$-0.441715\pi$
$197$	5.11141	0.364173	0.182087	+	0.983283i	$0.441715\pi$
$198$	−4.53733	−0.322454
$199$	16.5135	1.17061	0.585304	−	0.810814i	$-0.300975\pi$
$199$	16.5135	1.17061	0.585304	+	0.810814i	$0.300975\pi$
$200$	0	0
$201$	−15.4797	−1.09186
$202$	−12.4893	−0.878746
$203$	−5.69572	−0.399761
$204$	−1.24263	−0.0870017
$205$	0	0
$206$	31.2781	2.17925
$207$	−0.939503	−0.0652999
$208$	−4.85410	−0.336571
$209$	−6.57472	−0.454783
$210$	0	0
$211$	−2.47078	−0.170096	−0.0850478	−	0.996377i	$-0.527104\pi$
$211$	−2.47078	−0.170096	−0.0850478	+	0.996377i	$0.527104\pi$
$212$	−0.805379	−0.0553136
$213$	−12.3416	−0.845630
$214$	−12.6487	−0.864650
$215$	0	0
$216$	2.23607	0.152145
$217$	0.895549	0.0607938
$218$	−2.16308	−0.146503
$219$	−2.77455	−0.187487
$220$	0	0
$221$	−2.01062	−0.135249
$222$	8.93179	0.599462
$223$	−9.10445	−0.609679	−0.304840	−	0.952404i	$-0.598603\pi$
$223$	−9.10445	−0.609679	−0.304840	+	0.952404i	$0.598603\pi$
$224$	3.38197	0.225967
$225$	0	0
$226$	10.9988	0.731632
$227$	7.00779	0.465123	0.232562	−	0.972582i	$-0.425289\pi$
$227$	7.00779	0.465123	0.232562	+	0.972582i	$0.425289\pi$
$228$	−1.44903	−0.0959642
$229$	−20.2385	−1.33740	−0.668698	−	0.743534i	$-0.733148\pi$
$229$	−20.2385	−1.33740	−0.668698	+	0.743534i	$0.733148\pi$
$230$	0	0
$231$	2.80423	0.184505
$232$	−12.7360	−0.836160
$233$	−14.6750	−0.961391	−0.480696	−	0.876887i	$-0.659616\pi$
$233$	−14.6750	−0.961391	−0.480696	+	0.876887i	$0.659616\pi$
$234$	−1.61803	−0.105774
$235$	0	0
$236$	−1.52671	−0.0993805
$237$	−6.16495	−0.400456
$238$	3.25325	0.210877
$239$	23.9963	1.55219	0.776097	−	0.630613i	$-0.217197\pi$
$239$	23.9963	1.55219	0.776097	+	0.630613i	$0.217197\pi$
$240$	0	0
$241$	25.2300	1.62521	0.812605	−	0.582815i	$-0.198049\pi$
$241$	25.2300	1.62521	0.812605	+	0.582815i	$0.198049\pi$
$242$	5.07467	0.326212
$243$	1.00000	0.0641500
$244$	−2.57002	−0.164529
$245$	0	0
$246$	−4.97980	−0.317500
$247$	−2.34458	−0.149182
$248$	2.00251	0.127159
$249$	−15.5861	−0.987726
$250$	0	0
$251$	11.8992	0.751069	0.375534	−	0.926808i	$-0.377459\pi$
$251$	11.8992	0.751069	0.375534	+	0.926808i	$0.377459\pi$
$252$	0.618034	0.0389325
$253$	−2.63458	−0.165634
$254$	16.9751	1.06511
$255$	0	0
$256$	13.5623	0.847644
$257$	5.49151	0.342551	0.171276	−	0.985223i	$-0.445211\pi$
$257$	5.49151	0.342551	0.171276	+	0.985223i	$0.445211\pi$
$258$	−8.29168	−0.516218
$259$	−5.52015	−0.343005
$260$	0	0
$261$	−5.69572	−0.352556
$262$	3.38685	0.209240
$263$	−2.38614	−0.147135	−0.0735677	−	0.997290i	$-0.523439\pi$
$263$	−2.38614	−0.147135	−0.0735677	+	0.997290i	$0.523439\pi$
$264$	6.27044	0.385919
$265$	0	0
$266$	3.79360	0.232601
$267$	−0.821412	−0.0502696
$268$	−9.56701	−0.584398
$269$	−18.6149	−1.13497	−0.567485	−	0.823383i	$-0.692084\pi$
$269$	−18.6149	−1.13497	−0.567485	+	0.823383i	$0.692084\pi$
$270$	0	0
$271$	−2.89400	−0.175798	−0.0878990	−	0.996129i	$-0.528015\pi$
$271$	−2.89400	−0.175798	−0.0878990	+	0.996129i	$0.528015\pi$
$272$	9.75976	0.591773
$273$	1.00000	0.0605228
$274$	−19.8540	−1.19942
$275$	0	0
$276$	−0.580644	−0.0349507
$277$	−14.3261	−0.860769	−0.430385	−	0.902646i	$-0.641622\pi$
$277$	−14.3261	−0.860769	−0.430385	+	0.902646i	$0.641622\pi$
$278$	−21.7687	−1.30560
$279$	0.895549	0.0536151
$280$	0	0
$281$	5.57543	0.332603	0.166301	−	0.986075i	$-0.446818\pi$
$281$	5.57543	0.332603	0.166301	+	0.986075i	$0.446818\pi$
$282$	9.64771	0.574513
$283$	18.6838	1.11064	0.555319	−	0.831637i	$-0.312596\pi$
$283$	18.6838	1.11064	0.555319	+	0.831637i	$0.312596\pi$
$284$	−7.62750	−0.452609
$285$	0	0
$286$	−4.53733	−0.268298
$287$	3.07768	0.181670
$288$	3.38197	0.199284
$289$	−12.9574	−0.762200
$290$	0	0
$291$	−1.98281	−0.116235
$292$	−1.71477	−0.100349
$293$	−31.0585	−1.81446	−0.907229	−	0.420637i	$-0.861807\pi$
$293$	−31.0585	−1.81446	−0.907229	+	0.420637i	$0.861807\pi$
$294$	−1.61803	−0.0943657
$295$	0	0
$296$	−12.3434	−0.717447
$297$	2.80423	0.162718
$298$	−20.4221	−1.18302
$299$	−0.939503	−0.0543328
$300$	0	0
$301$	5.12454	0.295374
$302$	−0.359898	−0.0207098
$303$	7.71883	0.443435
$304$	11.3808	0.652734
$305$	0	0
$306$	3.25325	0.185976
$307$	−1.72892	−0.0986745	−0.0493373	−	0.998782i	$-0.515711\pi$
$307$	−1.72892	−0.0986745	−0.0493373	+	0.998782i	$0.515711\pi$
$308$	1.73311	0.0987529
$309$	−19.3309	−1.09970
$310$	0	0
$311$	17.7385	1.00586	0.502930	−	0.864327i	$-0.332256\pi$
$311$	17.7385	1.00586	0.502930	+	0.864327i	$0.332256\pi$
$312$	2.23607	0.126592
$313$	−24.8522	−1.40473	−0.702365	−	0.711817i	$-0.747872\pi$
$313$	−24.8522	−1.40473	−0.702365	+	0.711817i	$0.747872\pi$
$314$	31.5538	1.78068
$315$	0	0
$316$	−3.81015	−0.214338
$317$	2.60626	0.146382	0.0731911	−	0.997318i	$-0.476682\pi$
$317$	2.60626	0.146382	0.0731911	+	0.997318i	$0.476682\pi$
$318$	2.10851	0.118239
$319$	−15.9721	−0.894265
$320$	0	0
$321$	7.81736	0.436322
$322$	1.52015	0.0847145
$323$	4.71406	0.262297
$324$	0.618034	0.0343352
$325$	0	0
$326$	−1.85608	−0.102799
$327$	1.33686	0.0739285
$328$	6.88191	0.379990
$329$	−5.96261	−0.328729
$330$	0	0
$331$	−30.7543	−1.69041	−0.845205	−	0.534443i	$-0.820522\pi$
$331$	−30.7543	−1.69041	−0.845205	+	0.534443i	$0.820522\pi$
$332$	−9.63271	−0.528664
$333$	−5.52015	−0.302502
$334$	−28.8385	−1.57797
$335$	0	0
$336$	−4.85410	−0.264813
$337$	14.3356	0.780912	0.390456	−	0.920622i	$-0.372317\pi$
$337$	14.3356	0.780912	0.390456	+	0.920622i	$0.372317\pi$
$338$	−1.61803	−0.0880094
$339$	−6.79766	−0.369198
$340$	0	0
$341$	2.51132	0.135996
$342$	3.79360	0.205135
$343$	1.00000	0.0539949
$344$	11.4588	0.617818
$345$	0	0
$346$	3.42122	0.183926
$347$	−28.2949	−1.51895	−0.759475	−	0.650537i	$-0.774544\pi$
$347$	−28.2949	−1.51895	−0.759475	+	0.650537i	$0.774544\pi$
$348$	−3.52015	−0.188700
$349$	−7.82174	−0.418688	−0.209344	−	0.977842i	$-0.567133\pi$
$349$	−7.82174	−0.418688	−0.209344	+	0.977842i	$0.567133\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	9.48380	0.505488
$353$	12.6656	0.674122	0.337061	−	0.941483i	$-0.390567\pi$
$353$	12.6656	0.674122	0.337061	+	0.941483i	$0.390567\pi$
$354$	3.99698	0.212437
$355$	0	0
$356$	−0.507661	−0.0269060
$357$	−2.01062	−0.106413
$358$	−30.2147	−1.59690
$359$	4.35926	0.230073	0.115036	−	0.993361i	$-0.463302\pi$
$359$	4.35926	0.230073	0.115036	+	0.993361i	$0.463302\pi$
$360$	0	0
$361$	−13.5030	−0.710682
$362$	−9.83558	−0.516947
$363$	−3.13632	−0.164614
$364$	0.618034	0.0323938
$365$	0	0
$366$	6.72841	0.351699
$367$	−23.7244	−1.23840	−0.619201	−	0.785233i	$-0.712543\pi$
$367$	−23.7244	−1.23840	−0.619201	+	0.785233i	$0.712543\pi$
$368$	4.56044	0.237729
$369$	3.07768	0.160218
$370$	0	0
$371$	−1.30313	−0.0676551
$372$	0.553479	0.0286966
$373$	−12.2595	−0.634773	−0.317387	−	0.948296i	$-0.602805\pi$
$373$	−12.2595	−0.634773	−0.317387	+	0.948296i	$0.602805\pi$
$374$	9.12286	0.471732
$375$	0	0
$376$	−13.3328	−0.687587
$377$	−5.69572	−0.293344
$378$	−1.61803	−0.0832227
$379$	−9.44164	−0.484984	−0.242492	−	0.970153i	$-0.577965\pi$
$379$	−9.44164	−0.484984	−0.242492	+	0.970153i	$0.577965\pi$
$380$	0	0
$381$	−10.4912	−0.537480
$382$	−22.5440	−1.15345
$383$	1.81922	0.0929578	0.0464789	−	0.998919i	$-0.485200\pi$
$383$	1.81922	0.0929578	0.0464789	+	0.998919i	$0.485200\pi$
$384$	−13.6180	−0.694942
$385$	0	0
$386$	5.08611	0.258876
$387$	5.12454	0.260495
$388$	−1.22545	−0.0622126
$389$	−11.2846	−0.572152	−0.286076	−	0.958207i	$-0.592351\pi$
$389$	−11.2846	−0.572152	−0.286076	+	0.958207i	$0.592351\pi$
$390$	0	0
$391$	1.88898	0.0955300
$392$	2.23607	0.112938
$393$	−2.09319	−0.105587
$394$	−8.27044	−0.416659
$395$	0	0
$396$	1.73311	0.0870919
$397$	21.6658	1.08738	0.543688	−	0.839287i	$-0.317027\pi$
$397$	21.6658	1.08738	0.543688	+	0.839287i	$0.317027\pi$
$398$	−26.7194	−1.33932
$399$	−2.34458	−0.117376
$400$	0	0
$401$	15.1128	0.754696	0.377348	−	0.926072i	$-0.376836\pi$
$401$	15.1128	0.754696	0.377348	+	0.926072i	$0.376836\pi$
$402$	25.0467	1.24922
$403$	0.895549	0.0446105
$404$	4.77050	0.237341
$405$	0	0
$406$	9.21586	0.457376
$407$	−15.4797	−0.767302
$408$	−4.49589	−0.222580
$409$	0.0919651	0.00454738	0.00227369	−	0.999997i	$-0.499276\pi$
$409$	0.0919651	0.00454738	0.00227369	+	0.999997i	$0.499276\pi$
$410$	0	0
$411$	12.2704	0.605256
$412$	−11.9472	−0.588595
$413$	−2.47027	−0.121554
$414$	1.52015	0.0747112
$415$	0	0
$416$	3.38197	0.165815
$417$	13.4538	0.658835
$418$	10.6381	0.520328
$419$	15.8705	0.775323	0.387661	−	0.921802i	$-0.373283\pi$
$419$	15.8705	0.775323	0.387661	+	0.921802i	$0.373283\pi$
$420$	0	0
$421$	−13.6257	−0.664076	−0.332038	−	0.943266i	$-0.607736\pi$
$421$	−13.6257	−0.664076	−0.332038	+	0.943266i	$0.607736\pi$
$422$	3.99781	0.194610
$423$	−5.96261	−0.289912
$424$	−2.91389	−0.141511
$425$	0	0
$426$	19.9691	0.967504
$427$	−4.15838	−0.201238
$428$	4.83139	0.233534
$429$	2.80423	0.135389
$430$	0	0
$431$	−25.4756	−1.22712	−0.613559	−	0.789649i	$-0.710263\pi$
$431$	−25.4756	−1.22712	−0.613559	+	0.789649i	$0.710263\pi$
$432$	−4.85410	−0.233543
$433$	−20.3956	−0.980151	−0.490076	−	0.871680i	$-0.663031\pi$
$433$	−20.3956	−0.980151	−0.490076	+	0.871680i	$0.663031\pi$
$434$	−1.44903	−0.0695556
$435$	0	0
$436$	0.826225	0.0395690
$437$	2.20274	0.105371
$438$	4.48932	0.214508
$439$	22.8539	1.09076	0.545379	−	0.838190i	$-0.316386\pi$
$439$	22.8539	1.09076	0.545379	+	0.838190i	$0.316386\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	3.25325	0.154742
$443$	15.9067	0.755750	0.377875	−	0.925857i	$-0.376655\pi$
$443$	15.9067	0.755750	0.377875	+	0.925857i	$0.376655\pi$
$444$	−3.41164	−0.161909
$445$	0	0
$446$	14.7313	0.697548
$447$	12.6216	0.596980
$448$	4.23607	0.200135
$449$	−23.3073	−1.09994	−0.549969	−	0.835185i	$-0.685360\pi$
$449$	−23.3073	−1.09994	−0.549969	+	0.835185i	$0.685360\pi$
$450$	0	0
$451$	8.63052	0.406395
$452$	−4.20119	−0.197607
$453$	0.222429	0.0104506
$454$	−11.3388	−0.532158
$455$	0	0
$456$	−5.24263	−0.245509
$457$	18.1531	0.849164	0.424582	−	0.905389i	$-0.360421\pi$
$457$	18.1531	0.849164	0.424582	+	0.905389i	$0.360421\pi$
$458$	32.7465	1.53014
$459$	−2.01062	−0.0938478
$460$	0	0
$461$	30.5416	1.42246	0.711232	−	0.702957i	$-0.248138\pi$
$461$	30.5416	1.42246	0.711232	+	0.702957i	$0.248138\pi$
$462$	−4.53733	−0.211096
$463$	−18.8199	−0.874636	−0.437318	−	0.899307i	$-0.644072\pi$
$463$	−18.8199	−0.874636	−0.437318	+	0.899307i	$0.644072\pi$
$464$	27.6476	1.28351
$465$	0	0
$466$	23.7447	1.09995
$467$	−14.3948	−0.666114	−0.333057	−	0.942907i	$-0.608080\pi$
$467$	−14.3948	−0.666114	−0.333057	+	0.942907i	$0.608080\pi$
$468$	0.618034	0.0285686
$469$	−15.4797	−0.714788
$470$	0	0
$471$	−19.5013	−0.898572
$472$	−5.52369	−0.254249
$473$	14.3704	0.660751
$474$	9.97510	0.458171
$475$	0	0
$476$	−1.24263	−0.0569560
$477$	−1.30313	−0.0596662
$478$	−38.8269	−1.77590
$479$	−25.3568	−1.15858	−0.579292	−	0.815120i	$-0.696671\pi$
$479$	−25.3568	−1.15858	−0.579292	+	0.815120i	$0.696671\pi$
$480$	0	0
$481$	−5.52015	−0.251697
$482$	−40.8231	−1.85944
$483$	−0.939503	−0.0427488
$484$	−1.93835	−0.0881068
$485$	0	0
$486$	−1.61803	−0.0733955
$487$	−6.55671	−0.297113	−0.148556	−	0.988904i	$-0.547463\pi$
$487$	−6.55671	−0.297113	−0.148556	+	0.988904i	$0.547463\pi$
$488$	−9.29843	−0.420920
$489$	1.14712	0.0518746
$490$	0	0
$491$	−36.7728	−1.65954	−0.829768	−	0.558109i	$-0.811527\pi$
$491$	−36.7728	−1.65954	−0.829768	+	0.558109i	$0.811527\pi$
$492$	1.90211	0.0857539
$493$	11.4519	0.515769
$494$	3.79360	0.170682
$495$	0	0
$496$	−4.34708	−0.195190
$497$	−12.3416	−0.553595
$498$	25.2188	1.13008
$499$	33.8073	1.51342	0.756712	−	0.653748i	$-0.226804\pi$
$499$	33.8073	1.51342	0.756712	+	0.653748i	$0.226804\pi$
$500$	0	0
$501$	17.8232	0.796280
$502$	−19.2533	−0.859315
$503$	2.57397	0.114768	0.0573838	−	0.998352i	$-0.481724\pi$
$503$	2.57397	0.114768	0.0573838	+	0.998352i	$0.481724\pi$
$504$	2.23607	0.0996024
$505$	0	0
$506$	4.26284	0.189506
$507$	1.00000	0.0444116
$508$	−6.48391	−0.287677
$509$	14.2906	0.633422	0.316711	−	0.948522i	$-0.397422\pi$
$509$	14.2906	0.633422	0.316711	+	0.948522i	$0.397422\pi$
$510$	0	0
$511$	−2.77455	−0.122739
$512$	5.29180	0.233867
$513$	−2.34458	−0.103516
$514$	−8.88546	−0.391921
$515$	0	0
$516$	3.16714	0.139426
$517$	−16.7205	−0.735367
$518$	8.93179	0.392440
$519$	−2.11443	−0.0928132
$520$	0	0
$521$	15.9346	0.698108	0.349054	−	0.937103i	$-0.386503\pi$
$521$	15.9346	0.698108	0.349054	+	0.937103i	$0.386503\pi$
$522$	9.21586	0.403367
$523$	−42.2459	−1.84729	−0.923643	−	0.383254i	$-0.874803\pi$
$523$	−42.2459	−1.84729	−0.923643	+	0.383254i	$0.874803\pi$
$524$	−1.29366	−0.0565138
$525$	0	0
$526$	3.86085	0.168341
$527$	−1.80061	−0.0784358
$528$	−13.6120	−0.592386
$529$	−22.1173	−0.961623
$530$	0	0
$531$	−2.47027	−0.107201
$532$	−1.44903	−0.0628233
$533$	3.07768	0.133309
$534$	1.32907	0.0575146
$535$	0	0
$536$	−34.6138	−1.49509
$537$	18.6737	0.805830
$538$	30.1195	1.29855
$539$	2.80423	0.120787
$540$	0	0
$541$	−32.0729	−1.37892	−0.689460	−	0.724324i	$-0.742152\pi$
$541$	−32.0729	−1.37892	−0.689460	+	0.724324i	$0.742152\pi$
$542$	4.68259	0.201134
$543$	6.07872	0.260863
$544$	−6.79985	−0.291541
$545$	0	0
$546$	−1.61803	−0.0692455
$547$	19.6201	0.838895	0.419448	−	0.907779i	$-0.362224\pi$
$547$	19.6201	0.838895	0.419448	+	0.907779i	$0.362224\pi$
$548$	7.58355	0.323953
$549$	−4.15838	−0.177476
$550$	0	0
$551$	13.3540	0.568901
$552$	−2.10079	−0.0894156
$553$	−6.16495	−0.262160
$554$	23.1800	0.984826
$555$	0	0
$556$	8.31490	0.352630
$557$	0.768499	0.0325624	0.0162812	−	0.999867i	$-0.494817\pi$
$557$	0.768499	0.0325624	0.0162812	+	0.999867i	$0.494817\pi$
$558$	−1.44903	−0.0613422
$559$	5.12454	0.216745
$560$	0	0
$561$	−5.63824	−0.238047
$562$	−9.02124	−0.380538
$563$	35.0266	1.47620	0.738098	−	0.674694i	$-0.235724\pi$
$563$	35.0266	1.47620	0.738098	+	0.674694i	$0.235724\pi$
$564$	−3.68510	−0.155171
$565$	0	0
$566$	−30.2311	−1.27071
$567$	1.00000	0.0419961
$568$	−27.5966	−1.15793
$569$	26.3124	1.10307	0.551537	−	0.834150i	$-0.314042\pi$
$569$	26.3124	1.10307	0.551537	+	0.834150i	$0.314042\pi$
$570$	0	0
$571$	−40.3772	−1.68973	−0.844866	−	0.534978i	$-0.820320\pi$
$571$	−40.3772	−1.68973	−0.844866	+	0.534978i	$0.820320\pi$
$572$	1.73311	0.0724648
$573$	13.9329	0.582057
$574$	−4.97980	−0.207853
$575$	0	0
$576$	4.23607	0.176503
$577$	33.7497	1.40502	0.702509	−	0.711675i	$-0.252063\pi$
$577$	33.7497	1.40502	0.702509	+	0.711675i	$0.252063\pi$
$578$	20.9655	0.872050
$579$	−3.14339	−0.130635
$580$	0	0
$581$	−15.5861	−0.646619
$582$	3.20826	0.132987
$583$	−3.65427	−0.151344
$584$	−6.20409	−0.256727
$585$	0	0
$586$	50.2537	2.07596
$587$	35.8502	1.47970	0.739848	−	0.672774i	$-0.234897\pi$
$587$	35.8502	1.47970	0.739848	+	0.672774i	$0.234897\pi$
$588$	0.618034	0.0254873
$589$	−2.09968	−0.0865159
$590$	0	0
$591$	5.11141	0.210255
$592$	26.7954	1.10128
$593$	−28.7642	−1.18120	−0.590602	−	0.806963i	$-0.701110\pi$
$593$	−28.7642	−1.18120	−0.590602	+	0.806963i	$0.701110\pi$
$594$	−4.53733	−0.186169
$595$	0	0
$596$	7.80057	0.319524
$597$	16.5135	0.675851
$598$	1.52015	0.0621634
$599$	−16.1612	−0.660327	−0.330163	−	0.943924i	$-0.607104\pi$
$599$	−16.1612	−0.660327	−0.330163	+	0.943924i	$0.607104\pi$
$600$	0	0
$601$	−40.3491	−1.64587	−0.822936	−	0.568134i	$-0.807666\pi$
$601$	−40.3491	−1.64587	−0.822936	+	0.568134i	$0.807666\pi$
$602$	−8.29168	−0.337944
$603$	−15.4797	−0.630384
$604$	0.137469	0.00559352
$605$	0	0
$606$	−12.4893	−0.507344
$607$	−34.9319	−1.41784	−0.708920	−	0.705289i	$-0.750817\pi$
$607$	−34.9319	−1.41784	−0.708920	+	0.705289i	$0.750817\pi$
$608$	−7.92928	−0.321575
$609$	−5.69572	−0.230802
$610$	0	0
$611$	−5.96261	−0.241221
$612$	−1.24263	−0.0502304
$613$	36.1928	1.46181	0.730907	−	0.682477i	$-0.239097\pi$
$613$	36.1928	1.46181	0.730907	+	0.682477i	$0.239097\pi$
$614$	2.79745	0.112896
$615$	0	0
$616$	6.27044	0.252643
$617$	27.8387	1.12074	0.560371	−	0.828242i	$-0.310658\pi$
$617$	27.8387	1.12074	0.560371	+	0.828242i	$0.310658\pi$
$618$	31.2781	1.25819
$619$	14.9807	0.602126	0.301063	−	0.953604i	$-0.402659\pi$
$619$	14.9807	0.602126	0.301063	+	0.953604i	$0.402659\pi$
$620$	0	0
$621$	−0.939503	−0.0377009
$622$	−28.7015	−1.15083
$623$	−0.821412	−0.0329092
$624$	−4.85410	−0.194320
$625$	0	0
$626$	40.2117	1.60718
$627$	−6.57472	−0.262569
$628$	−12.0525	−0.480946
$629$	11.0989	0.442543
$630$	0	0
$631$	10.7583	0.428281	0.214140	−	0.976803i	$-0.431305\pi$
$631$	10.7583	0.428281	0.214140	+	0.976803i	$0.431305\pi$
$632$	−13.7852	−0.548348
$633$	−2.47078	−0.0982047
$634$	−4.21702	−0.167479
$635$	0	0
$636$	−0.805379	−0.0319353
$637$	1.00000	0.0396214
$638$	25.8434	1.02315
$639$	−12.3416	−0.488225
$640$	0	0
$641$	−12.9112	−0.509961	−0.254981	−	0.966946i	$-0.582069\pi$
$641$	−12.9112	−0.509961	−0.254981	+	0.966946i	$0.582069\pi$
$642$	−12.6487	−0.499206
$643$	20.8057	0.820495	0.410248	−	0.911974i	$-0.365442\pi$
$643$	20.8057	0.820495	0.410248	+	0.911974i	$0.365442\pi$
$644$	−0.580644	−0.0228806
$645$	0	0
$646$	−7.62750	−0.300100
$647$	25.7610	1.01277	0.506384	−	0.862308i	$-0.330982\pi$
$647$	25.7610	1.01277	0.506384	+	0.862308i	$0.330982\pi$
$648$	2.23607	0.0878410
$649$	−6.92720	−0.271916
$650$	0	0
$651$	0.895549	0.0350993
$652$	0.708959	0.0277650
$653$	33.0209	1.29221	0.646103	−	0.763250i	$-0.276398\pi$
$653$	33.0209	1.29221	0.646103	+	0.763250i	$0.276398\pi$
$654$	−2.16308	−0.0845833
$655$	0	0
$656$	−14.9394	−0.583285
$657$	−2.77455	−0.108246
$658$	9.64771	0.376107
$659$	−36.4979	−1.42176	−0.710879	−	0.703315i	$-0.751702\pi$
$659$	−36.4979	−1.42176	−0.710879	+	0.703315i	$0.751702\pi$
$660$	0	0
$661$	35.1393	1.36676	0.683380	−	0.730063i	$-0.260509\pi$
$661$	35.1393	1.36676	0.683380	+	0.730063i	$0.260509\pi$
$662$	49.7615	1.93404
$663$	−2.01062	−0.0780861
$664$	−34.8515	−1.35250
$665$	0	0
$666$	8.93179	0.346100
$667$	5.35114	0.207197
$668$	11.0153	0.426196
$669$	−9.10445	−0.351998
$670$	0	0
$671$	−11.6611	−0.450170
$672$	3.38197	0.130462
$673$	10.0652	0.387985	0.193992	−	0.981003i	$-0.437856\pi$
$673$	10.0652	0.387985	0.193992	+	0.981003i	$0.437856\pi$
$674$	−23.1955	−0.893459
$675$	0	0
$676$	0.618034	0.0237705
$677$	−34.4229	−1.32298	−0.661489	−	0.749955i	$-0.730075\pi$
$677$	−34.4229	−1.32298	−0.661489	+	0.749955i	$0.730075\pi$
$678$	10.9988	0.422408
$679$	−1.98281	−0.0760934
$680$	0	0
$681$	7.00779	0.268539
$682$	−4.06340	−0.155596
$683$	7.19340	0.275248	0.137624	−	0.990485i	$-0.456053\pi$
$683$	7.19340	0.275248	0.137624	+	0.990485i	$0.456053\pi$
$684$	−1.44903	−0.0554050
$685$	0	0
$686$	−1.61803	−0.0617768
$687$	−20.2385	−0.772146
$688$	−24.8750	−0.948352
$689$	−1.30313	−0.0496453
$690$	0	0
$691$	2.19402	0.0834645	0.0417323	−	0.999129i	$-0.486712\pi$
$691$	2.19402	0.0834645	0.0417323	+	0.999129i	$0.486712\pi$
$692$	−1.30679	−0.0496767
$693$	2.80423	0.106524
$694$	45.7821	1.73787
$695$	0	0
$696$	−12.7360	−0.482757
$697$	−6.18806	−0.234389
$698$	12.6558	0.479031
$699$	−14.6750	−0.555060
$700$	0	0
$701$	9.08651	0.343193	0.171596	−	0.985167i	$-0.445108\pi$
$701$	9.08651	0.343193	0.171596	+	0.985167i	$0.445108\pi$
$702$	−1.61803	−0.0610688
$703$	12.9424	0.488132
$704$	11.8789	0.447703
$705$	0	0
$706$	−20.4934	−0.771278
$707$	7.71883	0.290296
$708$	−1.52671	−0.0573773
$709$	−17.0828	−0.641557	−0.320778	−	0.947154i	$-0.603944\pi$
$709$	−17.0828	−0.641557	−0.320778	+	0.947154i	$0.603944\pi$
$710$	0	0
$711$	−6.16495	−0.231204
$712$	−1.83673	−0.0688345
$713$	−0.841370	−0.0315096
$714$	3.25325	0.121750
$715$	0	0
$716$	11.5410	0.431307
$717$	23.9963	0.896160
$718$	−7.05342	−0.263231
$719$	−46.0179	−1.71618	−0.858088	−	0.513502i	$-0.828348\pi$
$719$	−46.0179	−1.71618	−0.858088	+	0.513502i	$0.828348\pi$
$720$	0	0
$721$	−19.3309	−0.719922
$722$	21.8482	0.813108
$723$	25.2300	0.938315
$724$	3.75686	0.139623
$725$	0	0
$726$	5.07467	0.188339
$727$	−41.4822	−1.53849	−0.769245	−	0.638954i	$-0.779368\pi$
$727$	−41.4822	−1.53849	−0.769245	+	0.638954i	$0.779368\pi$
$728$	2.23607	0.0828742
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.3035	−0.381089
$732$	−2.57002	−0.0949908
$733$	−8.95740	−0.330849	−0.165425	−	0.986222i	$-0.552899\pi$
$733$	−8.95740	−0.330849	−0.165425	+	0.986222i	$0.552899\pi$
$734$	38.3868	1.41688
$735$	0	0
$736$	−3.17737	−0.117119
$737$	−43.4087	−1.59898
$738$	−4.97980	−0.183309
$739$	41.1448	1.51354	0.756769	−	0.653683i	$-0.226777\pi$
$739$	41.1448	1.51354	0.756769	+	0.653683i	$0.226777\pi$
$740$	0	0
$741$	−2.34458	−0.0861302
$742$	2.10851	0.0774058
$743$	−51.2608	−1.88058	−0.940288	−	0.340379i	$-0.889445\pi$
$743$	−51.2608	−1.88058	−0.940288	+	0.340379i	$0.889445\pi$
$744$	2.00251	0.0734155
$745$	0	0
$746$	19.8363	0.726258
$747$	−15.5861	−0.570264
$748$	−3.48462	−0.127410
$749$	7.81736	0.285640
$750$	0	0
$751$	−9.69851	−0.353904	−0.176952	−	0.984220i	$-0.556624\pi$
$751$	−9.69851	−0.353904	−0.176952	+	0.984220i	$0.556624\pi$
$752$	28.9431	1.05545
$753$	11.8992	0.433630
$754$	9.21586	0.335622
$755$	0	0
$756$	0.618034	0.0224777
$757$	−42.0784	−1.52936	−0.764682	−	0.644408i	$-0.777104\pi$
$757$	−42.0784	−1.52936	−0.764682	+	0.644408i	$0.777104\pi$
$758$	15.2769	0.554882
$759$	−2.63458	−0.0956291
$760$	0	0
$761$	−20.1288	−0.729668	−0.364834	−	0.931072i	$-0.618874\pi$
$761$	−20.1288	−0.729668	−0.364834	+	0.931072i	$0.618874\pi$
$762$	16.9751	0.614943
$763$	1.33686	0.0483976
$764$	8.61103	0.311536
$765$	0	0
$766$	−2.94356	−0.106355
$767$	−2.47027	−0.0891963
$768$	13.5623	0.489388
$769$	23.5408	0.848902	0.424451	−	0.905451i	$-0.360467\pi$
$769$	23.5408	0.848902	0.424451	+	0.905451i	$0.360467\pi$
$770$	0	0
$771$	5.49151	0.197772
$772$	−1.94272	−0.0699201
$773$	9.93937	0.357494	0.178747	−	0.983895i	$-0.442796\pi$
$773$	9.93937	0.357494	0.178747	+	0.983895i	$0.442796\pi$
$774$	−8.29168	−0.298038
$775$	0	0
$776$	−4.43371	−0.159161
$777$	−5.52015	−0.198034
$778$	18.2589	0.654613
$779$	−7.21586	−0.258535
$780$	0	0
$781$	−34.6085	−1.23839
$782$	−3.05644	−0.109298
$783$	−5.69572	−0.203548
$784$	−4.85410	−0.173361
$785$	0	0
$786$	3.38685	0.120805
$787$	37.9659	1.35334	0.676670	−	0.736287i	$-0.263423\pi$
$787$	37.9659	1.35334	0.676670	+	0.736287i	$0.263423\pi$
$788$	3.15903	0.112536
$789$	−2.38614	−0.0849487
$790$	0	0
$791$	−6.79766	−0.241697
$792$	6.27044	0.222810
$793$	−4.15838	−0.147669
$794$	−35.0560	−1.24409
$795$	0	0
$796$	10.2059	0.361738
$797$	42.9372	1.52091	0.760456	−	0.649389i	$-0.224975\pi$
$797$	42.9372	1.52091	0.760456	+	0.649389i	$0.224975\pi$
$798$	3.79360	0.134292
$799$	11.9886	0.424125
$800$	0	0
$801$	−0.821412	−0.0290232
$802$	−24.4530	−0.863464
$803$	−7.78048	−0.274567
$804$	−9.56701	−0.337402
$805$	0	0
$806$	−1.44903	−0.0510398
$807$	−18.6149	−0.655276
$808$	17.2598	0.607198
$809$	47.5910	1.67321	0.836605	−	0.547807i	$-0.184537\pi$
$809$	47.5910	1.67321	0.836605	+	0.547807i	$0.184537\pi$
$810$	0	0
$811$	−40.0579	−1.40662	−0.703312	−	0.710882i	$-0.748296\pi$
$811$	−40.0579	−1.40662	−0.703312	+	0.710882i	$0.748296\pi$
$812$	−3.52015	−0.123533
$813$	−2.89400	−0.101497
$814$	25.0467	0.877888
$815$	0	0
$816$	9.75976	0.341660
$817$	−12.0149	−0.420347
$818$	−0.148803	−0.00520276
$819$	1.00000	0.0349428
$820$	0	0
$821$	−2.41652	−0.0843372	−0.0421686	−	0.999111i	$-0.513427\pi$
$821$	−2.41652	−0.0843372	−0.0421686	+	0.999111i	$0.513427\pi$
$822$	−19.8540	−0.692487
$823$	57.1896	1.99350	0.996752	−	0.0805350i	$-0.0256629\pi$
$823$	57.1896	1.99350	0.996752	+	0.0805350i	$0.0256629\pi$
$824$	−43.2253	−1.50582
$825$	0	0
$826$	3.99698	0.139073
$827$	−5.71058	−0.198576	−0.0992881	−	0.995059i	$-0.531657\pi$
$827$	−5.71058	−0.198576	−0.0992881	+	0.995059i	$0.531657\pi$
$828$	−0.580644	−0.0201788
$829$	−32.2179	−1.11898	−0.559488	−	0.828839i	$-0.689002\pi$
$829$	−32.2179	−1.11898	−0.559488	+	0.828839i	$0.689002\pi$
$830$	0	0
$831$	−14.3261	−0.496965
$832$	4.23607	0.146859
$833$	−2.01062	−0.0696639
$834$	−21.7687	−0.753788
$835$	0	0
$836$	−4.06340	−0.140536
$837$	0.895549	0.0309547
$838$	−25.6789	−0.887065
$839$	39.2649	1.35557	0.677787	−	0.735258i	$-0.262939\pi$
$839$	39.2649	1.35557	0.677787	+	0.735258i	$0.262939\pi$
$840$	0	0
$841$	3.44120	0.118662
$842$	22.0469	0.759785
$843$	5.57543	0.192028
$844$	−1.52703	−0.0525624
$845$	0	0
$846$	9.64771	0.331695
$847$	−3.13632	−0.107765
$848$	6.32553	0.217219
$849$	18.6838	0.641227
$850$	0	0
$851$	5.18619	0.177780
$852$	−7.62750	−0.261314
$853$	−13.4267	−0.459722	−0.229861	−	0.973224i	$-0.573827\pi$
$853$	−13.4267	−0.459722	−0.229861	+	0.973224i	$0.573827\pi$
$854$	6.72841	0.230241
$855$	0	0
$856$	17.4801	0.597459
$857$	−48.4140	−1.65379	−0.826895	−	0.562356i	$-0.809895\pi$
$857$	−48.4140	−1.65379	−0.826895	+	0.562356i	$0.809895\pi$
$858$	−4.53733	−0.154902
$859$	−16.8574	−0.575168	−0.287584	−	0.957755i	$-0.592852\pi$
$859$	−16.8574	−0.575168	−0.287584	+	0.957755i	$0.592852\pi$
$860$	0	0
$861$	3.07768	0.104887
$862$	41.2205	1.40397
$863$	−47.6860	−1.62325	−0.811625	−	0.584178i	$-0.801417\pi$
$863$	−47.6860	−1.62325	−0.811625	+	0.584178i	$0.801417\pi$
$864$	3.38197	0.115057
$865$	0	0
$866$	33.0008	1.12141
$867$	−12.9574	−0.440056
$868$	0.553479	0.0187863
$869$	−17.2879	−0.586452
$870$	0	0
$871$	−15.4797	−0.524511
$872$	2.98931	0.101231
$873$	−1.98281	−0.0671081
$874$	−3.56410	−0.120558
$875$	0	0
$876$	−1.71477	−0.0579367
$877$	5.74990	0.194160	0.0970801	−	0.995277i	$-0.469050\pi$
$877$	5.74990	0.194160	0.0970801	+	0.995277i	$0.469050\pi$
$878$	−36.9784	−1.24796
$879$	−31.0585	−1.04758
$880$	0	0
$881$	−16.9157	−0.569905	−0.284952	−	0.958542i	$-0.591978\pi$
$881$	−16.9157	−0.569905	−0.284952	+	0.958542i	$0.591978\pi$
$882$	−1.61803	−0.0544820
$883$	−8.09873	−0.272544	−0.136272	−	0.990671i	$-0.543512\pi$
$883$	−8.09873	−0.272544	−0.136272	+	0.990671i	$0.543512\pi$
$884$	−1.24263	−0.0417943
$885$	0	0
$886$	−25.7376	−0.864671
$887$	−44.5786	−1.49680	−0.748401	−	0.663247i	$-0.769178\pi$
$887$	−44.5786	−1.49680	−0.748401	+	0.663247i	$0.769178\pi$
$888$	−12.3434	−0.414218
$889$	−10.4912	−0.351863
$890$	0	0
$891$	2.80423	0.0939451
$892$	−5.62686	−0.188401
$893$	13.9798	0.467816
$894$	−20.4221	−0.683019
$895$	0	0
$896$	−13.6180	−0.454947
$897$	−0.939503	−0.0313691
$898$	37.7119	1.25846
$899$	−5.10079	−0.170121
$900$	0	0
$901$	2.62010	0.0872882
$902$	−13.9645	−0.464966
$903$	5.12454	0.170534
$904$	−15.2000	−0.505546
$905$	0	0
$906$	−0.359898	−0.0119568
$907$	−24.1868	−0.803110	−0.401555	−	0.915835i	$-0.631530\pi$
$907$	−24.1868	−0.803110	−0.401555	+	0.915835i	$0.631530\pi$
$908$	4.33105	0.143731
$909$	7.71883	0.256017
$910$	0	0
$911$	9.27522	0.307302	0.153651	−	0.988125i	$-0.450897\pi$
$911$	9.27522	0.307302	0.153651	+	0.988125i	$0.450897\pi$
$912$	11.3808	0.376856
$913$	−43.7068	−1.44648
$914$	−29.3723	−0.971548
$915$	0	0
$916$	−12.5081	−0.413278
$917$	−2.09319	−0.0691231
$918$	3.25325	0.107373
$919$	−2.91209	−0.0960611	−0.0480305	−	0.998846i	$-0.515294\pi$
$919$	−2.91209	−0.0960611	−0.0480305	+	0.998846i	$0.515294\pi$
$920$	0	0
$921$	−1.72892	−0.0569698
$922$	−49.4173	−1.62747
$923$	−12.3416	−0.406227
$924$	1.73311	0.0570150
$925$	0	0
$926$	30.4513	1.00069
$927$	−19.3309	−0.634911
$928$	−19.2627	−0.632330
$929$	35.9359	1.17902	0.589510	−	0.807761i	$-0.299321\pi$
$929$	35.9359	1.17902	0.589510	+	0.807761i	$0.299321\pi$
$930$	0	0
$931$	−2.34458	−0.0768404
$932$	−9.06965	−0.297086
$933$	17.7385	0.580733
$934$	23.2914	0.762116
$935$	0	0
$936$	2.23607	0.0730882
$937$	25.0249	0.817527	0.408763	−	0.912640i	$-0.365960\pi$
$937$	25.0249	0.817527	0.408763	+	0.912640i	$0.365960\pi$
$938$	25.0467	0.817805
$939$	−24.8522	−0.811021
$940$	0	0
$941$	12.7170	0.414562	0.207281	−	0.978281i	$-0.433538\pi$
$941$	12.7170	0.414562	0.207281	+	0.978281i	$0.433538\pi$
$942$	31.5538	1.02808
$943$	−2.89149	−0.0941599
$944$	11.9909	0.390272
$945$	0	0
$946$	−23.2518	−0.755980
$947$	−42.4461	−1.37931	−0.689656	−	0.724137i	$-0.742238\pi$
$947$	−42.4461	−1.37931	−0.689656	+	0.724137i	$0.742238\pi$
$948$	−3.81015	−0.123748
$949$	−2.77455	−0.0900658
$950$	0	0
$951$	2.60626	0.0845138
$952$	−4.49589	−0.145713
$953$	−44.1291	−1.42948	−0.714741	−	0.699389i	$-0.753456\pi$
$953$	−44.1291	−1.42948	−0.714741	+	0.699389i	$0.753456\pi$
$954$	2.10851	0.0682655
$955$	0	0
$956$	14.8306	0.479654
$957$	−15.9721	−0.516304
$958$	41.0282	1.32556
$959$	12.2704	0.396233
$960$	0	0
$961$	−30.1980	−0.974129
$962$	8.93179	0.287972
$963$	7.81736	0.251911
$964$	15.5930	0.502217
$965$	0	0
$966$	1.52015	0.0489099
$967$	−5.88712	−0.189317	−0.0946585	−	0.995510i	$-0.530176\pi$
$967$	−5.88712	−0.189317	−0.0946585	+	0.995510i	$0.530176\pi$
$968$	−7.01302	−0.225407
$969$	4.71406	0.151437
$970$	0	0
$971$	44.7978	1.43763	0.718814	−	0.695202i	$-0.244685\pi$
$971$	44.7978	1.43763	0.718814	+	0.695202i	$0.244685\pi$
$972$	0.618034	0.0198234
$973$	13.4538	0.431309
$974$	10.6090	0.339934
$975$	0	0
$976$	20.1852	0.646113
$977$	7.32375	0.234308	0.117154	−	0.993114i	$-0.462623\pi$
$977$	7.32375	0.234308	0.117154	+	0.993114i	$0.462623\pi$
$978$	−1.85608	−0.0593509
$979$	−2.30343	−0.0736178
$980$	0	0
$981$	1.33686	0.0426826
$982$	59.4997	1.89871
$983$	−15.2176	−0.485367	−0.242683	−	0.970106i	$-0.578028\pi$
$983$	−15.2176	−0.485367	−0.242683	+	0.970106i	$0.578028\pi$
$984$	6.88191	0.219387
$985$	0	0
$986$	−18.5296	−0.590103
$987$	−5.96261	−0.189792
$988$	−1.44903	−0.0460997
$989$	−4.81452	−0.153093
$990$	0	0
$991$	1.65833	0.0526785	0.0263393	−	0.999653i	$-0.491615\pi$
$991$	1.65833	0.0526785	0.0263393	+	0.999653i	$0.491615\pi$
$992$	3.02871	0.0961618
$993$	−30.7543	−0.975958
$994$	19.9691	0.633380
$995$	0	0
$996$	−9.63271	−0.305224
$997$	−2.49409	−0.0789887	−0.0394943	−	0.999220i	$-0.512575\pi$
$997$	−2.49409	−0.0789887	−0.0394943	+	0.999220i	$0.512575\pi$
$998$	−54.7014	−1.73154
$999$	−5.52015	−0.174650

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6825.2.a.be.1.2		4
5.2	odd	4		1365.2.f.c.274.2	✓	8
5.3	odd	4		1365.2.f.c.274.8	yes	8
5.4	even	2		6825.2.a.bm.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1365.2.f.c.274.2	✓	8	5.2	odd	4
1365.2.f.c.274.8	yes	8	5.3	odd	4
6825.2.a.be.1.2		4	1.1	even	1	trivial
6825.2.a.bm.1.4		4	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6825.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +2.80423 q^{11} +0.618034 q^{12} +1.00000 q^{13} -1.61803 q^{14} -4.85410 q^{16} -2.01062 q^{17} -1.61803 q^{18} -2.34458 q^{19} +1.00000 q^{21} -4.53733 q^{22} -0.939503 q^{23} +2.23607 q^{24} -1.61803 q^{26} +1.00000 q^{27} +0.618034 q^{28} -5.69572 q^{29} +0.895549 q^{31} +3.38197 q^{32} +2.80423 q^{33} +3.25325 q^{34} +0.618034 q^{36} -5.52015 q^{37} +3.79360 q^{38} +1.00000 q^{39} +3.07768 q^{41} -1.61803 q^{42} +5.12454 q^{43} +1.73311 q^{44} +1.52015 q^{46} -5.96261 q^{47} -4.85410 q^{48} +1.00000 q^{49} -2.01062 q^{51} +0.618034 q^{52} -1.30313 q^{53} -1.61803 q^{54} +2.23607 q^{56} -2.34458 q^{57} +9.21586 q^{58} -2.47027 q^{59} -4.15838 q^{61} -1.44903 q^{62} +1.00000 q^{63} +4.23607 q^{64} -4.53733 q^{66} -15.4797 q^{67} -1.24263 q^{68} -0.939503 q^{69} -12.3416 q^{71} +2.23607 q^{72} -2.77455 q^{73} +8.93179 q^{74} -1.44903 q^{76} +2.80423 q^{77} -1.61803 q^{78} -6.16495 q^{79} +1.00000 q^{81} -4.97980 q^{82} -15.5861 q^{83} +0.618034 q^{84} -8.29168 q^{86} -5.69572 q^{87} +6.27044 q^{88} -0.821412 q^{89} +1.00000 q^{91} -0.580644 q^{92} +0.895549 q^{93} +9.64771 q^{94} +3.38197 q^{96} -1.98281 q^{97} -1.61803 q^{98} +2.80423 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{14} - 6 q^{16} - 2 q^{17} - 2 q^{18} - 2 q^{19} + 4 q^{21} + 2 q^{22} - 8 q^{23} - 2 q^{26} + 4 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^3 - 2 * q^4 - 2 * q^6 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 2 * q^12 + 4 * q^13 - 2 * q^14 - 6 * q^16 - 2 * q^17 - 2 * q^18 - 2 * q^19 + 4 * q^21 + 2 * q^22 - 8 * q^23 - 2 * q^26 + 4 * q^27 - 2 * q^28 - 6 * q^29 - 2 * q^31 + 18 * q^32 - 4 * q^33 - 4 * q^34 - 2 * q^36 - 10 * q^37 + 6 * q^38 + 4 * q^39 - 2 * q^42 - 18 * q^43 + 2 * q^44 - 6 * q^46 - 12 * q^47 - 6 * q^48 + 4 * q^49 - 2 * q^51 - 2 * q^52 + 10 * q^53 - 2 * q^54 - 2 * q^57 + 8 * q^58 + 4 * q^59 - 20 * q^61 - 4 * q^62 + 4 * q^63 + 8 * q^64 + 2 * q^66 - 10 * q^67 + 6 * q^68 - 8 * q^69 - 14 * q^71 - 14 * q^73 + 10 * q^74 - 4 * q^76 - 4 * q^77 - 2 * q^78 - 26 * q^79 + 4 * q^81 - 12 * q^83 - 2 * q^84 + 4 * q^86 - 6 * q^87 + 20 * q^89 + 4 * q^91 + 14 * q^92 - 2 * q^93 + 16 * q^94 + 18 * q^96 - 16 * q^97 - 2 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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