LMFDB - Embedded newform 6825.2.a.bm.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:	$0$
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			$-1.17557$ 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)

$f(q)$	$=$	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +1.35114 q^{11} +1.61803 q^{12} -1.00000 q^{13} +0.618034 q^{14} +1.85410 q^{16} +5.87129 q^{17} -0.618034 q^{18} -2.45965 q^{19} +1.00000 q^{21} -0.835051 q^{22} +2.33395 q^{23} -2.23607 q^{24} +0.618034 q^{26} -1.00000 q^{27} +1.61803 q^{28} +0.344577 q^{29} +6.43945 q^{31} -5.61803 q^{32} -1.35114 q^{33} -3.62866 q^{34} -1.61803 q^{36} +2.55754 q^{37} +1.52015 q^{38} +1.00000 q^{39} -0.726543 q^{41} -0.618034 q^{42} +10.8753 q^{43} -2.18619 q^{44} -1.44246 q^{46} +3.84162 q^{47} -1.85410 q^{48} +1.00000 q^{49} -5.87129 q^{51} +1.61803 q^{52} -10.8339 q^{53} +0.618034 q^{54} -2.23607 q^{56} +2.45965 q^{57} -0.212960 q^{58} -4.76278 q^{59} -3.49047 q^{61} -3.97980 q^{62} -1.00000 q^{63} -0.236068 q^{64} +0.835051 q^{66} +3.45559 q^{67} -9.49994 q^{68} -2.33395 q^{69} -5.51609 q^{71} +2.23607 q^{72} +11.1074 q^{73} -1.58064 q^{74} +3.97980 q^{76} -1.35114 q^{77} -0.618034 q^{78} +0.773401 q^{79} +1.00000 q^{81} +0.449028 q^{82} -13.2188 q^{83} -1.61803 q^{84} -6.72133 q^{86} -0.344577 q^{87} +3.02124 q^{88} +3.04145 q^{89} +1.00000 q^{91} -3.77642 q^{92} -6.43945 q^{93} -2.37425 q^{94} +5.61803 q^{96} +4.39259 q^{97} -0.618034 q^{98} +1.35114 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	1.35114	0.407384	0.203692	−	0.979035i	$-0.434706\pi$
$11$	1.35114	0.407384	0.203692	+	0.979035i	$0.434706\pi$
$12$	1.61803	0.467086
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0.618034	0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	5.87129	1.42400	0.711998	−	0.702181i	$-0.247790\pi$
$17$	5.87129	1.42400	0.711998	+	0.702181i	$0.247790\pi$
$18$	−0.618034	−0.145672
$19$	−2.45965	−0.564282	−0.282141	−	0.959373i	$-0.591045\pi$
$19$	−2.45965	−0.564282	−0.282141	+	0.959373i	$0.591045\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−0.835051	−0.178033
$23$	2.33395	0.486663	0.243332	−	0.969943i	$-0.421760\pi$
$23$	2.33395	0.486663	0.243332	+	0.969943i	$0.421760\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	0.618034	0.121206
$27$	−1.00000	−0.192450
$28$	1.61803	0.305780
$29$	0.344577	0.0639863	0.0319931	−	0.999488i	$-0.489815\pi$
$29$	0.344577	0.0639863	0.0319931	+	0.999488i	$0.489815\pi$
$30$	0	0
$31$	6.43945	1.15656	0.578279	−	0.815839i	$-0.303724\pi$
$31$	6.43945	1.15656	0.578279	+	0.815839i	$0.303724\pi$
$32$	−5.61803	−0.993137
$33$	−1.35114	−0.235203
$34$	−3.62866	−0.622309
$35$	0	0
$36$	−1.61803	−0.269672
$37$	2.55754	0.420456	0.210228	−	0.977652i	$-0.432579\pi$
$37$	2.55754	0.420456	0.210228	+	0.977652i	$0.432579\pi$
$38$	1.52015	0.246600
$39$	1.00000	0.160128
$40$	0	0
$41$	−0.726543	−0.113467	−0.0567334	−	0.998389i	$-0.518069\pi$
$41$	−0.726543	−0.113467	−0.0567334	+	0.998389i	$0.518069\pi$
$42$	−0.618034	−0.0953647
$43$	10.8753	1.65847	0.829237	−	0.558897i	$-0.188775\pi$
$43$	10.8753	1.65847	0.829237	+	0.558897i	$0.188775\pi$
$44$	−2.18619	−0.329581
$45$	0	0
$46$	−1.44246	−0.212680
$47$	3.84162	0.560357	0.280179	−	0.959948i	$-0.409606\pi$
$47$	3.84162	0.560357	0.280179	+	0.959948i	$0.409606\pi$
$48$	−1.85410	−0.267617
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.87129	−0.822145
$52$	1.61803	0.224381
$53$	−10.8339	−1.48815	−0.744075	−	0.668096i	$-0.767110\pi$
$53$	−10.8339	−1.48815	−0.744075	+	0.668096i	$0.767110\pi$
$54$	0.618034	0.0841038
$55$	0	0
$56$	−2.23607	−0.298807
$57$	2.45965	0.325789
$58$	−0.212960	−0.0279630
$59$	−4.76278	−0.620061	−0.310031	−	0.950727i	$-0.600339\pi$
$59$	−4.76278	−0.620061	−0.310031	+	0.950727i	$0.600339\pi$
$60$	0	0
$61$	−3.49047	−0.446909	−0.223455	−	0.974714i	$-0.571733\pi$
$61$	−3.49047	−0.446909	−0.223455	+	0.974714i	$0.571733\pi$
$62$	−3.97980	−0.505435
$63$	−1.00000	−0.125988
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0.835051	0.102788
$67$	3.45559	0.422168	0.211084	−	0.977468i	$-0.432301\pi$
$67$	3.45559	0.422168	0.211084	+	0.977468i	$0.432301\pi$
$68$	−9.49994	−1.15204
$69$	−2.33395	−0.280975
$70$	0	0
$71$	−5.51609	−0.654639	−0.327320	−	0.944914i	$-0.606145\pi$
$71$	−5.51609	−0.654639	−0.327320	+	0.944914i	$0.606145\pi$
$72$	2.23607	0.263523
$73$	11.1074	1.30002	0.650009	−	0.759927i	$-0.274765\pi$
$73$	11.1074	1.30002	0.650009	+	0.759927i	$0.274765\pi$
$74$	−1.58064	−0.183746
$75$	0	0
$76$	3.97980	0.456514
$77$	−1.35114	−0.153977
$78$	−0.618034	−0.0699786
$79$	0.773401	0.0870144	0.0435072	−	0.999053i	$-0.486147\pi$
$79$	0.773401	0.0870144	0.0435072	+	0.999053i	$0.486147\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.449028	0.0495868
$83$	−13.2188	−1.45095	−0.725474	−	0.688249i	$-0.758380\pi$
$83$	−13.2188	−1.45095	−0.725474	+	0.688249i	$0.758380\pi$
$84$	−1.61803	−0.176542
$85$	0	0
$86$	−6.72133	−0.724780
$87$	−0.344577	−0.0369425
$88$	3.02124	0.322066
$89$	3.04145	0.322393	0.161196	−	0.986922i	$-0.448465\pi$
$89$	3.04145	0.322393	0.161196	+	0.986922i	$0.448465\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−3.77642	−0.393719
$93$	−6.43945	−0.667739
$94$	−2.37425	−0.244885
$95$	0	0
$96$	5.61803	0.573388
$97$	4.39259	0.446000	0.223000	−	0.974818i	$-0.428415\pi$
$97$	4.39259	0.446000	0.223000	+	0.974818i	$0.428415\pi$
$98$	−0.618034	−0.0624309
$99$	1.35114	0.135795
$100$	0	0
$101$	−1.83692	−0.182780	−0.0913900	−	0.995815i	$-0.529131\pi$
$101$	−1.83692	−0.182780	−0.0913900	+	0.995815i	$0.529131\pi$
$102$	3.62866	0.359290
$103$	8.64480	0.851798	0.425899	−	0.904771i	$-0.359958\pi$
$103$	8.64480	0.851798	0.425899	+	0.904771i	$0.359958\pi$
$104$	−2.23607	−0.219265
$105$	0	0
$106$	6.69572	0.650346
$107$	6.17661	0.597115	0.298558	−	0.954392i	$-0.403494\pi$
$107$	6.17661	0.597115	0.298558	+	0.954392i	$0.403494\pi$
$108$	1.61803	0.155695
$109$	−10.4549	−1.00140	−0.500701	−	0.865620i	$-0.666924\pi$
$109$	−10.4549	−1.00140	−0.500701	+	0.865620i	$0.666924\pi$
$110$	0	0
$111$	−2.55754	−0.242751
$112$	−1.85410	−0.175196
$113$	11.6150	1.09265	0.546324	−	0.837574i	$-0.316027\pi$
$113$	11.6150	1.09265	0.546324	+	0.837574i	$0.316027\pi$
$114$	−1.52015	−0.142375
$115$	0	0
$116$	−0.557537	−0.0517660
$117$	−1.00000	−0.0924500
$118$	2.94356	0.270977
$119$	−5.87129	−0.538220
$120$	0	0
$121$	−9.17442	−0.834038
$122$	2.15723	0.195307
$123$	0.726543	0.0655101
$124$	−10.4192	−0.935676
$125$	0	0
$126$	0.618034	0.0550588
$127$	−12.0996	−1.07367	−0.536835	−	0.843687i	$-0.680380\pi$
$127$	−12.0996	−1.07367	−0.536835	+	0.843687i	$0.680380\pi$
$128$	11.3820	1.00603
$129$	−10.8753	−0.957521
$130$	0	0
$131$	2.14661	0.187550	0.0937751	−	0.995593i	$-0.470107\pi$
$131$	2.14661	0.187550	0.0937751	+	0.995593i	$0.470107\pi$
$132$	2.18619	0.190284
$133$	2.45965	0.213279
$134$	−2.13567	−0.184494
$135$	0	0
$136$	13.1286	1.12577
$137$	−2.97876	−0.254492	−0.127246	−	0.991871i	$-0.540614\pi$
$137$	−2.97876	−0.254492	−0.127246	+	0.991871i	$0.540614\pi$
$138$	1.44246	0.122791
$139$	−11.2580	−0.954893	−0.477447	−	0.878661i	$-0.658438\pi$
$139$	−11.2580	−0.954893	−0.477447	+	0.878661i	$0.658438\pi$
$140$	0	0
$141$	−3.84162	−0.323522
$142$	3.40913	0.286088
$143$	−1.35114	−0.112988
$144$	1.85410	0.154508
$145$	0	0
$146$	−6.86472	−0.568129
$147$	−1.00000	−0.0824786
$148$	−4.13818	−0.340156
$149$	−2.82547	−0.231471	−0.115736	−	0.993280i	$-0.536923\pi$
$149$	−2.82547	−0.231471	−0.115736	+	0.993280i	$0.536923\pi$
$150$	0	0
$151$	−15.0509	−1.22483	−0.612413	−	0.790538i	$-0.709801\pi$
$151$	−15.0509	−1.22483	−0.612413	+	0.790538i	$0.709801\pi$
$152$	−5.49994	−0.446104
$153$	5.87129	0.474666
$154$	0.835051	0.0672903
$155$	0	0
$156$	−1.61803	−0.129546
$157$	−22.5709	−1.80136	−0.900679	−	0.434485i	$-0.856930\pi$
$157$	−22.5709	−1.80136	−0.900679	+	0.434485i	$0.856930\pi$
$158$	−0.477988	−0.0380267
$159$	10.8339	0.859184
$160$	0	0
$161$	−2.33395	−0.183941
$162$	−0.618034	−0.0485573
$163$	−1.78331	−0.139680	−0.0698398	−	0.997558i	$-0.522249\pi$
$163$	−1.78331	−0.139680	−0.0698398	+	0.997558i	$0.522249\pi$
$164$	1.17557	0.0917966
$165$	0	0
$166$	8.16965	0.634088
$167$	12.2632	0.948957	0.474479	−	0.880267i	$-0.342637\pi$
$167$	12.2632	0.948957	0.474479	+	0.880267i	$0.342637\pi$
$168$	2.23607	0.172516
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.45965	−0.188094
$172$	−17.5967	−1.34173
$173$	5.59597	0.425453	0.212727	−	0.977112i	$-0.431766\pi$
$173$	5.59597	0.425453	0.212727	+	0.977112i	$0.431766\pi$
$174$	0.212960	0.0161445
$175$	0	0
$176$	2.50515	0.188833
$177$	4.76278	0.357992
$178$	−1.87972	−0.140891
$179$	−4.90990	−0.366983	−0.183492	−	0.983021i	$-0.558740\pi$
$179$	−4.90990	−0.366983	−0.183492	+	0.983021i	$0.558740\pi$
$180$	0	0
$181$	5.46506	0.406215	0.203107	−	0.979156i	$-0.434896\pi$
$181$	5.46506	0.406215	0.203107	+	0.979156i	$0.434896\pi$
$182$	−0.618034	−0.0458117
$183$	3.49047	0.258023
$184$	5.21888	0.384741
$185$	0	0
$186$	3.97980	0.291813
$187$	7.93294	0.580114
$188$	−6.21586	−0.453339
$189$	1.00000	0.0727393
$190$	0	0
$191$	21.5978	1.56276	0.781382	−	0.624053i	$-0.214515\pi$
$191$	21.5978	1.56276	0.781382	+	0.624053i	$0.214515\pi$
$192$	0.236068	0.0170367
$193$	26.2531	1.88974	0.944871	−	0.327442i	$-0.106187\pi$
$193$	26.2531	1.88974	0.944871	+	0.327442i	$0.106187\pi$
$194$	−2.71477	−0.194909
$195$	0	0
$196$	−1.61803	−0.115574
$197$	−1.65241	−0.117729	−0.0588645	−	0.998266i	$-0.518748\pi$
$197$	−1.65241	−0.117729	−0.0588645	+	0.998266i	$0.518748\pi$
$198$	−0.835051	−0.0593445
$199$	4.83454	0.342712	0.171356	−	0.985209i	$-0.445185\pi$
$199$	4.83454	0.342712	0.171356	+	0.985209i	$0.445185\pi$
$200$	0	0
$201$	−3.45559	−0.243739
$202$	1.13528	0.0798778
$203$	−0.344577	−0.0241845
$204$	9.49994	0.665129
$205$	0	0
$206$	−5.34278	−0.372249
$207$	2.33395	0.162221
$208$	−1.85410	−0.128559
$209$	−3.32333	−0.229880
$210$	0	0
$211$	−28.1054	−1.93485	−0.967427	−	0.253150i	$-0.918533\pi$
$211$	−28.1054	−1.93485	−0.967427	+	0.253150i	$0.918533\pi$
$212$	17.5296	1.20394
$213$	5.51609	0.377956
$214$	−3.81736	−0.260949
$215$	0	0
$216$	−2.23607	−0.152145
$217$	−6.43945	−0.437138
$218$	6.46151	0.437629
$219$	−11.1074	−0.750566
$220$	0	0
$221$	−5.87129	−0.394946
$222$	1.58064	0.106086
$223$	3.56055	0.238432	0.119216	−	0.992868i	$-0.461962\pi$
$223$	3.56055	0.238432	0.119216	+	0.992868i	$0.461962\pi$
$224$	5.61803	0.375371
$225$	0	0
$226$	−7.17848	−0.477505
$227$	5.33467	0.354074	0.177037	−	0.984204i	$-0.443349\pi$
$227$	5.33467	0.354074	0.177037	+	0.984204i	$0.443349\pi$
$228$	−3.97980	−0.263568
$229$	15.6220	1.03233	0.516165	−	0.856489i	$-0.327359\pi$
$229$	15.6220	1.03233	0.516165	+	0.856489i	$0.327359\pi$
$230$	0	0
$231$	1.35114	0.0888986
$232$	0.770497	0.0505856
$233$	−19.2381	−1.26033	−0.630166	−	0.776460i	$-0.717013\pi$
$233$	−19.2381	−1.26033	−0.630166	+	0.776460i	$0.717013\pi$
$234$	0.618034	0.0404021
$235$	0	0
$236$	7.70634	0.501640
$237$	−0.773401	−0.0502378
$238$	3.62866	0.235211
$239$	22.2206	1.43733	0.718664	−	0.695357i	$-0.244754\pi$
$239$	22.2206	1.43733	0.718664	+	0.695357i	$0.244754\pi$
$240$	0	0
$241$	6.87681	0.442974	0.221487	−	0.975163i	$-0.428909\pi$
$241$	6.87681	0.442974	0.221487	+	0.975163i	$0.428909\pi$
$242$	5.67010	0.364488
$243$	−1.00000	−0.0641500
$244$	5.64771	0.361557
$245$	0	0
$246$	−0.449028	−0.0286290
$247$	2.45965	0.156504
$248$	14.3990	0.914340
$249$	13.2188	0.837705
$250$	0	0
$251$	−20.0173	−1.26348	−0.631739	−	0.775181i	$-0.717659\pi$
$251$	−20.0173	−1.26348	−0.631739	+	0.775181i	$0.717659\pi$
$252$	1.61803	0.101927
$253$	3.15350	0.198259
$254$	7.47799	0.469211
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−15.5054	−0.967198	−0.483599	−	0.875290i	$-0.660671\pi$
$257$	−15.5054	−0.967198	−0.483599	+	0.875290i	$0.660671\pi$
$258$	6.72133	0.418452
$259$	−2.55754	−0.158918
$260$	0	0
$261$	0.344577	0.0213288
$262$	−1.32668	−0.0819624
$263$	29.7401	1.83385	0.916926	−	0.399058i	$-0.130663\pi$
$263$	29.7401	1.83385	0.916926	+	0.399058i	$0.130663\pi$
$264$	−3.02124	−0.185945
$265$	0	0
$266$	−1.52015	−0.0932062
$267$	−3.04145	−0.186134
$268$	−5.59127	−0.341541
$269$	5.54846	0.338296	0.169148	−	0.985591i	$-0.445898\pi$
$269$	5.54846	0.338296	0.169148	+	0.985591i	$0.445898\pi$
$270$	0	0
$271$	18.0948	1.09918	0.549589	−	0.835435i	$-0.314784\pi$
$271$	18.0948	1.09918	0.549589	+	0.835435i	$0.314784\pi$
$272$	10.8860	0.660059
$273$	−1.00000	−0.0605228
$274$	1.84097	0.111217
$275$	0	0
$276$	3.77642	0.227314
$277$	7.93616	0.476838	0.238419	−	0.971162i	$-0.423371\pi$
$277$	7.93616	0.476838	0.238419	+	0.971162i	$0.423371\pi$
$278$	6.95784	0.417304
$279$	6.43945	0.385520
$280$	0	0
$281$	−27.0901	−1.61606	−0.808029	−	0.589143i	$-0.799465\pi$
$281$	−27.0901	−1.61606	−0.808029	+	0.589143i	$0.799465\pi$
$282$	2.37425	0.141384
$283$	24.3812	1.44931	0.724656	−	0.689111i	$-0.241999\pi$
$283$	24.3812	1.44931	0.724656	+	0.689111i	$0.241999\pi$
$284$	8.92522	0.529614
$285$	0	0
$286$	0.835051	0.0493776
$287$	0.726543	0.0428864
$288$	−5.61803	−0.331046
$289$	17.4720	1.02777
$290$	0	0
$291$	−4.39259	−0.257498
$292$	−17.9721	−1.05174
$293$	25.9141	1.51392	0.756959	−	0.653463i	$-0.226684\pi$
$293$	25.9141	1.51392	0.756959	+	0.653463i	$0.226684\pi$
$294$	0.618034	0.0360445
$295$	0	0
$296$	5.71883	0.332400
$297$	−1.35114	−0.0784012
$298$	1.74624	0.101157
$299$	−2.33395	−0.134976
$300$	0	0
$301$	−10.8753	−0.626844
$302$	9.30198	0.535269
$303$	1.83692	0.105528
$304$	−4.56044	−0.261559
$305$	0	0
$306$	−3.62866	−0.207436
$307$	16.1854	0.923748	0.461874	−	0.886946i	$-0.347177\pi$
$307$	16.1854	0.923748	0.461874	+	0.886946i	$0.347177\pi$
$308$	2.18619	0.124570
$309$	−8.64480	−0.491786
$310$	0	0
$311$	−10.6285	−0.602689	−0.301345	−	0.953515i	$-0.597435\pi$
$311$	−10.6285	−0.602689	−0.301345	+	0.953515i	$0.597435\pi$
$312$	2.23607	0.126592
$313$	−9.85532	−0.557056	−0.278528	−	0.960428i	$-0.589846\pi$
$313$	−9.85532	−0.557056	−0.278528	+	0.960428i	$0.589846\pi$
$314$	13.9496	0.787222
$315$	0	0
$316$	−1.25139	−0.0703961
$317$	21.6678	1.21698	0.608492	−	0.793560i	$-0.291775\pi$
$317$	21.6678	1.21698	0.608492	+	0.793560i	$0.291775\pi$
$318$	−6.69572	−0.375477
$319$	0.465571	0.0260670
$320$	0	0
$321$	−6.17661	−0.344745
$322$	1.44246	0.0803853
$323$	−14.4413	−0.803536
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	1.10215	0.0610422
$327$	10.4549	0.578160
$328$	−1.62460	−0.0897034
$329$	−3.84162	−0.211795
$330$	0	0
$331$	4.67975	0.257223	0.128611	−	0.991695i	$-0.458948\pi$
$331$	4.67975	0.257223	0.128611	+	0.991695i	$0.458948\pi$
$332$	21.3884	1.17384
$333$	2.55754	0.140152
$334$	−7.57909	−0.414710
$335$	0	0
$336$	1.85410	0.101150
$337$	−8.61584	−0.469335	−0.234667	−	0.972076i	$-0.575400\pi$
$337$	−8.61584	−0.469335	−0.234667	+	0.972076i	$0.575400\pi$
$338$	−0.618034	−0.0336166
$339$	−11.6150	−0.630841
$340$	0	0
$341$	8.70060	0.471164
$342$	1.52015	0.0822001
$343$	−1.00000	−0.0539949
$344$	24.3180	1.31114
$345$	0	0
$346$	−3.45850	−0.185930
$347$	−19.0911	−1.02486	−0.512432	−	0.858728i	$-0.671255\pi$
$347$	−19.0911	−1.02486	−0.512432	+	0.858728i	$0.671255\pi$
$348$	0.557537	0.0298871
$349$	−36.5635	−1.95720	−0.978601	−	0.205767i	$-0.934031\pi$
$349$	−36.5635	−1.95720	−0.978601	+	0.205767i	$0.934031\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−7.59076	−0.404589
$353$	31.1802	1.65956	0.829778	−	0.558094i	$-0.188467\pi$
$353$	31.1802	1.65956	0.829778	+	0.558094i	$0.188467\pi$
$354$	−2.94356	−0.156448
$355$	0	0
$356$	−4.92116	−0.260821
$357$	5.87129	0.310742
$358$	3.03448	0.160378
$359$	18.4661	0.974604	0.487302	−	0.873234i	$-0.337981\pi$
$359$	18.4661	0.974604	0.487302	+	0.873234i	$0.337981\pi$
$360$	0	0
$361$	−12.9501	−0.681585
$362$	−3.37759	−0.177522
$363$	9.17442	0.481532
$364$	−1.61803	−0.0848080
$365$	0	0
$366$	−2.15723	−0.112760
$367$	4.70760	0.245735	0.122867	−	0.992423i	$-0.460791\pi$
$367$	4.70760	0.245735	0.122867	+	0.992423i	$0.460791\pi$
$368$	4.32739	0.225581
$369$	−0.726543	−0.0378223
$370$	0	0
$371$	10.8339	0.562468
$372$	10.4192	0.540213
$373$	36.8766	1.90940	0.954699	−	0.297574i	$-0.0961776\pi$
$373$	36.8766	1.90940	0.954699	+	0.297574i	$0.0961776\pi$
$374$	−4.90283	−0.253519
$375$	0	0
$376$	8.59011	0.443001
$377$	−0.344577	−0.0177466
$378$	−0.618034	−0.0317882
$379$	−24.7106	−1.26930	−0.634649	−	0.772800i	$-0.718855\pi$
$379$	−24.7106	−1.26930	−0.634649	+	0.772800i	$0.718855\pi$
$380$	0	0
$381$	12.0996	0.619883
$382$	−13.3482	−0.682953
$383$	23.4115	1.19627	0.598137	−	0.801394i	$-0.295908\pi$
$383$	23.4115	1.19627	0.598137	+	0.801394i	$0.295908\pi$
$384$	−11.3820	−0.580834
$385$	0	0
$386$	−16.2253	−0.825848
$387$	10.8753	0.552825
$388$	−7.10736	−0.360821
$389$	7.35739	0.373034	0.186517	−	0.982452i	$-0.440280\pi$
$389$	7.35739	0.373034	0.186517	+	0.982452i	$0.440280\pi$
$390$	0	0
$391$	13.7033	0.693007
$392$	2.23607	0.112938
$393$	−2.14661	−0.108282
$394$	1.02124	0.0514495
$395$	0	0
$396$	−2.18619	−0.109860
$397$	4.56210	0.228965	0.114483	−	0.993425i	$-0.463479\pi$
$397$	4.56210	0.228965	0.114483	+	0.993425i	$0.463479\pi$
$398$	−2.98791	−0.149770
$399$	−2.45965	−0.123136
$400$	0	0
$401$	−22.9251	−1.14483	−0.572413	−	0.819966i	$-0.693992\pi$
$401$	−22.9251	−1.14483	−0.572413	+	0.819966i	$0.693992\pi$
$402$	2.13567	0.106518
$403$	−6.43945	−0.320772
$404$	2.97219	0.147872
$405$	0	0
$406$	0.212960	0.0105690
$407$	3.45559	0.171287
$408$	−13.1286	−0.649963
$409$	1.92418	0.0951446	0.0475723	−	0.998868i	$-0.484852\pi$
$409$	1.92418	0.0951446	0.0475723	+	0.998868i	$0.484852\pi$
$410$	0	0
$411$	2.97876	0.146931
$412$	−13.9876	−0.689119
$413$	4.76278	0.234361
$414$	−1.44246	−0.0708932
$415$	0	0
$416$	5.61803	0.275447
$417$	11.2580	0.551308
$418$	2.05393	0.100461
$419$	21.1798	1.03470	0.517351	−	0.855773i	$-0.326918\pi$
$419$	21.1798	1.03470	0.517351	+	0.855773i	$0.326918\pi$
$420$	0	0
$421$	15.9396	0.776847	0.388424	−	0.921481i	$-0.373020\pi$
$421$	15.9396	0.776847	0.388424	+	0.921481i	$0.373020\pi$
$422$	17.3701	0.845562
$423$	3.84162	0.186786
$424$	−24.2253	−1.17649
$425$	0	0
$426$	−3.40913	−0.165173
$427$	3.49047	0.168916
$428$	−9.99397	−0.483077
$429$	1.35114	0.0652337
$430$	0	0
$431$	35.9157	1.73000	0.864999	−	0.501773i	$-0.167319\pi$
$431$	35.9157	1.73000	0.864999	+	0.501773i	$0.167319\pi$
$432$	−1.85410	−0.0892055
$433$	−10.0607	−0.483487	−0.241743	−	0.970340i	$-0.577719\pi$
$433$	−10.0607	−0.483487	−0.241743	+	0.970340i	$0.577719\pi$
$434$	3.97980	0.191036
$435$	0	0
$436$	16.9165	0.810152
$437$	−5.74071	−0.274615
$438$	6.86472	0.328009
$439$	25.4083	1.21267	0.606336	−	0.795209i	$-0.292639\pi$
$439$	25.4083	1.21267	0.606336	+	0.795209i	$0.292639\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	3.62866	0.172598
$443$	−5.15974	−0.245147	−0.122573	−	0.992459i	$-0.539115\pi$
$443$	−5.15974	−0.245147	−0.122573	+	0.992459i	$0.539115\pi$
$444$	4.13818	0.196389
$445$	0	0
$446$	−2.20054	−0.104199
$447$	2.82547	0.133640
$448$	0.236068	0.0111532
$449$	2.94800	0.139125	0.0695624	−	0.997578i	$-0.477840\pi$
$449$	2.94800	0.139125	0.0695624	+	0.997578i	$0.477840\pi$
$450$	0	0
$451$	−0.981661	−0.0462246
$452$	−18.7935	−0.883971
$453$	15.0509	0.707154
$454$	−3.29701	−0.154736
$455$	0	0
$456$	5.49994	0.257558
$457$	25.8100	1.20734	0.603671	−	0.797233i	$-0.293704\pi$
$457$	25.8100	1.20734	0.603671	+	0.797233i	$0.293704\pi$
$458$	−9.65491	−0.451144
$459$	−5.87129	−0.274048
$460$	0	0
$461$	−17.4559	−0.813001	−0.406501	−	0.913651i	$-0.633251\pi$
$461$	−17.4559	−0.813001	−0.406501	+	0.913651i	$0.633251\pi$
$462$	−0.835051	−0.0388501
$463$	−35.8249	−1.66493	−0.832463	−	0.554081i	$-0.813070\pi$
$463$	−35.8249	−1.66493	−0.832463	+	0.554081i	$0.813070\pi$
$464$	0.638880	0.0296593
$465$	0	0
$466$	11.8898	0.550786
$467$	17.6101	0.814900	0.407450	−	0.913228i	$-0.366418\pi$
$467$	17.6101	0.814900	0.407450	+	0.913228i	$0.366418\pi$
$468$	1.61803	0.0747936
$469$	−3.45559	−0.159564
$470$	0	0
$471$	22.5709	1.04001
$472$	−10.6499	−0.490201
$473$	14.6941	0.675637
$474$	0.477988	0.0219547
$475$	0	0
$476$	9.49994	0.435429
$477$	−10.8339	−0.496050
$478$	−13.7331	−0.628136
$479$	11.8777	0.542707	0.271353	−	0.962480i	$-0.412529\pi$
$479$	11.8777	0.542707	0.271353	+	0.962480i	$0.412529\pi$
$480$	0	0
$481$	−2.55754	−0.116614
$482$	−4.25010	−0.193587
$483$	2.33395	0.106199
$484$	14.8445	0.674751
$485$	0	0
$486$	0.618034	0.0280346
$487$	20.1424	0.912741	0.456371	−	0.889790i	$-0.349149\pi$
$487$	20.1424	0.912741	0.456371	+	0.889790i	$0.349149\pi$
$488$	−7.80494	−0.353313
$489$	1.78331	0.0806441
$490$	0	0
$491$	35.6499	1.60886	0.804428	−	0.594051i	$-0.202472\pi$
$491$	35.6499	1.60886	0.804428	+	0.594051i	$0.202472\pi$
$492$	−1.17557	−0.0529988
$493$	2.02311	0.0911162
$494$	−1.52015	−0.0683946
$495$	0	0
$496$	11.9394	0.536094
$497$	5.51609	0.247430
$498$	−8.16965	−0.366091
$499$	41.9260	1.87686	0.938432	−	0.345465i	$-0.112279\pi$
$499$	41.9260	1.87686	0.938432	+	0.345465i	$0.112279\pi$
$500$	0	0
$501$	−12.2632	−0.547881
$502$	12.3713	0.552160
$503$	10.4994	0.468146	0.234073	−	0.972219i	$-0.424795\pi$
$503$	10.4994	0.468146	0.234073	+	0.972219i	$0.424795\pi$
$504$	−2.23607	−0.0996024
$505$	0	0
$506$	−1.94897	−0.0866424
$507$	−1.00000	−0.0444116
$508$	19.5776	0.868617
$509$	9.52973	0.422398	0.211199	−	0.977443i	$-0.432263\pi$
$509$	9.52973	0.422398	0.211199	+	0.977443i	$0.432263\pi$
$510$	0	0
$511$	−11.1074	−0.491361
$512$	−18.7082	−0.826794
$513$	2.45965	0.108596
$514$	9.58284	0.422681
$515$	0	0
$516$	17.5967	0.774651
$517$	5.19056	0.228281
$518$	1.58064	0.0694495
$519$	−5.59597	−0.245636
$520$	0	0
$521$	21.6253	0.947422	0.473711	−	0.880680i	$-0.342914\pi$
$521$	21.6253	0.947422	0.473711	+	0.880680i	$0.342914\pi$
$522$	−0.212960	−0.00932101
$523$	31.8125	1.39106	0.695531	−	0.718496i	$-0.255169\pi$
$523$	31.8125	1.39106	0.695531	+	0.718496i	$0.255169\pi$
$524$	−3.47329	−0.151731
$525$	0	0
$526$	−18.3804	−0.801422
$527$	37.8078	1.64694
$528$	−2.50515	−0.109023
$529$	−17.5527	−0.763159
$530$	0	0
$531$	−4.76278	−0.206687
$532$	−3.97980	−0.172546
$533$	0.726543	0.0314701
$534$	1.87972	0.0813433
$535$	0	0
$536$	7.72694	0.333753
$537$	4.90990	0.211878
$538$	−3.42914	−0.147841
$539$	1.35114	0.0581978
$540$	0	0
$541$	−8.31555	−0.357513	−0.178757	−	0.983893i	$-0.557207\pi$
$541$	−8.31555	−0.357513	−0.178757	+	0.983893i	$0.557207\pi$
$542$	−11.1832	−0.480358
$543$	−5.46506	−0.234528
$544$	−32.9851	−1.41422
$545$	0	0
$546$	0.618034	0.0264494
$547$	−23.7652	−1.01613	−0.508063	−	0.861320i	$-0.669638\pi$
$547$	−23.7652	−1.01613	−0.508063	+	0.861320i	$0.669638\pi$
$548$	4.81973	0.205889
$549$	−3.49047	−0.148970
$550$	0	0
$551$	−0.847537	−0.0361063
$552$	−5.21888	−0.222130
$553$	−0.773401	−0.0328884
$554$	−4.90482	−0.208386
$555$	0	0
$556$	18.2159	0.772525
$557$	−38.7138	−1.64036	−0.820179	−	0.572107i	$-0.806126\pi$
$557$	−38.7138	−1.64036	−0.820179	+	0.572107i	$0.806126\pi$
$558$	−3.97980	−0.168478
$559$	−10.8753	−0.459978
$560$	0	0
$561$	−7.93294	−0.334929
$562$	16.7426	0.706243
$563$	20.9359	0.882344	0.441172	−	0.897423i	$-0.354563\pi$
$563$	20.9359	0.882344	0.441172	+	0.897423i	$0.354563\pi$
$564$	6.21586	0.261735
$565$	0	0
$566$	−15.0684	−0.633373
$567$	−1.00000	−0.0419961
$568$	−12.3344	−0.517538
$569$	9.12131	0.382385	0.191193	−	0.981553i	$-0.438764\pi$
$569$	9.12131	0.382385	0.191193	+	0.981553i	$0.438764\pi$
$570$	0	0
$571$	20.8335	0.871854	0.435927	−	0.899982i	$-0.356421\pi$
$571$	20.8335	0.871854	0.435927	+	0.899982i	$0.356421\pi$
$572$	2.18619	0.0914093
$573$	−21.5978	−0.902262
$574$	−0.449028	−0.0187421
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	10.9082	0.454116	0.227058	−	0.973881i	$-0.427089\pi$
$577$	10.9082	0.454116	0.227058	+	0.973881i	$0.427089\pi$
$578$	−10.7983	−0.449150
$579$	−26.2531	−1.09104
$580$	0	0
$581$	13.2188	0.548407
$582$	2.71477	0.112531
$583$	−14.6381	−0.606249
$584$	24.8368	1.02775
$585$	0	0
$586$	−16.0158	−0.661606
$587$	2.60728	0.107614	0.0538070	−	0.998551i	$-0.482864\pi$
$587$	2.60728	0.107614	0.0538070	+	0.998551i	$0.482864\pi$
$588$	1.61803	0.0667266
$589$	−15.8388	−0.652626
$590$	0	0
$591$	1.65241	0.0679709
$592$	4.74193	0.194892
$593$	−43.7692	−1.79739	−0.898693	−	0.438579i	$-0.855482\pi$
$593$	−43.7692	−1.79739	−0.898693	+	0.438579i	$0.855482\pi$
$594$	0.835051	0.0342626
$595$	0	0
$596$	4.57171	0.187264
$597$	−4.83454	−0.197865
$598$	1.44246	0.0589867
$599$	−35.9539	−1.46904	−0.734518	−	0.678589i	$-0.762592\pi$
$599$	−35.9539	−1.46904	−0.734518	+	0.678589i	$0.762592\pi$
$600$	0	0
$601$	−15.4569	−0.630502	−0.315251	−	0.949008i	$-0.602089\pi$
$601$	−15.4569	−0.630502	−0.315251	+	0.949008i	$0.602089\pi$
$602$	6.72133	0.273941
$603$	3.45559	0.140723
$604$	24.3529	0.990905
$605$	0	0
$606$	−1.13528	−0.0461175
$607$	0.170089	0.00690369	0.00345185	−	0.999994i	$-0.498901\pi$
$607$	0.170089	0.00690369	0.00345185	+	0.999994i	$0.498901\pi$
$608$	13.8184	0.560410
$609$	0.344577	0.0139629
$610$	0	0
$611$	−3.84162	−0.155415
$612$	−9.49994	−0.384012
$613$	−6.45603	−0.260757	−0.130378	−	0.991464i	$-0.541619\pi$
$613$	−6.45603	−0.260757	−0.130378	+	0.991464i	$0.541619\pi$
$614$	−10.0031	−0.403693
$615$	0	0
$616$	−3.02124	−0.121729
$617$	2.68330	0.108026	0.0540128	−	0.998540i	$-0.482799\pi$
$617$	2.68330	0.108026	0.0540128	+	0.998540i	$0.482799\pi$
$618$	5.34278	0.214918
$619$	−1.34624	−0.0541099	−0.0270550	−	0.999634i	$-0.508613\pi$
$619$	−1.34624	−0.0541099	−0.0270550	+	0.999634i	$0.508613\pi$
$620$	0	0
$621$	−2.33395	−0.0936584
$622$	6.56880	0.263385
$623$	−3.04145	−0.121853
$624$	1.85410	0.0742235
$625$	0	0
$626$	6.09093	0.243442
$627$	3.32333	0.132721
$628$	36.5206	1.45733
$629$	15.0160	0.598728
$630$	0	0
$631$	9.70468	0.386337	0.193169	−	0.981166i	$-0.438124\pi$
$631$	9.70468	0.386337	0.193169	+	0.981166i	$0.438124\pi$
$632$	1.72938	0.0687909
$633$	28.1054	1.11709
$634$	−13.3914	−0.531842
$635$	0	0
$636$	−17.5296	−0.695094
$637$	−1.00000	−0.0396214
$638$	−0.287739	−0.0113917
$639$	−5.51609	−0.218213
$640$	0	0
$641$	10.4873	0.414225	0.207113	−	0.978317i	$-0.433593\pi$
$641$	10.4873	0.414225	0.207113	+	0.978317i	$0.433593\pi$
$642$	3.81736	0.150659
$643$	−30.8985	−1.21852	−0.609258	−	0.792972i	$-0.708533\pi$
$643$	−30.8985	−1.21852	−0.609258	+	0.792972i	$0.708533\pi$
$644$	3.77642	0.148812
$645$	0	0
$646$	8.92522	0.351158
$647$	0.956758	0.0376141	0.0188070	−	0.999823i	$-0.494013\pi$
$647$	0.956758	0.0376141	0.0188070	+	0.999823i	$0.494013\pi$
$648$	2.23607	0.0878410
$649$	−6.43519	−0.252603
$650$	0	0
$651$	6.43945	0.252382
$652$	2.88546	0.113003
$653$	31.0986	1.21698	0.608490	−	0.793561i	$-0.291775\pi$
$653$	31.0986	1.21698	0.608490	+	0.793561i	$0.291775\pi$
$654$	−6.46151	−0.252665
$655$	0	0
$656$	−1.34708	−0.0525948
$657$	11.1074	0.433339
$658$	2.37425	0.0925579
$659$	13.9815	0.544643	0.272322	−	0.962206i	$-0.412209\pi$
$659$	13.9815	0.544643	0.272322	+	0.962206i	$0.412209\pi$
$660$	0	0
$661$	8.88184	0.345464	0.172732	−	0.984969i	$-0.444741\pi$
$661$	8.88184	0.345464	0.172732	+	0.984969i	$0.444741\pi$
$662$	−2.89225	−0.112410
$663$	5.87129	0.228022
$664$	−29.5581	−1.14708
$665$	0	0
$666$	−1.58064	−0.0612487
$667$	0.804226	0.0311398
$668$	−19.8423	−0.767723
$669$	−3.56055	−0.137659
$670$	0	0
$671$	−4.71612	−0.182064
$672$	−5.61803	−0.216720
$673$	−13.6371	−0.525671	−0.262835	−	0.964841i	$-0.584658\pi$
$673$	−13.6371	−0.525671	−0.262835	+	0.964841i	$0.584658\pi$
$674$	5.32488	0.205107
$675$	0	0
$676$	−1.61803	−0.0622321
$677$	−13.6672	−0.525271	−0.262636	−	0.964895i	$-0.584592\pi$
$677$	−13.6672	−0.525271	−0.262636	+	0.964895i	$0.584592\pi$
$678$	7.17848	0.275688
$679$	−4.39259	−0.168572
$680$	0	0
$681$	−5.33467	−0.204425
$682$	−5.37727	−0.205906
$683$	3.45882	0.132348	0.0661741	−	0.997808i	$-0.478921\pi$
$683$	3.45882	0.132348	0.0661741	+	0.997808i	$0.478921\pi$
$684$	3.97980	0.152171
$685$	0	0
$686$	0.618034	0.0235966
$687$	−15.6220	−0.596016
$688$	20.1640	0.768745
$689$	10.8339	0.412739
$690$	0	0
$691$	29.8706	1.13633	0.568166	−	0.822914i	$-0.307653\pi$
$691$	29.8706	1.13633	0.568166	+	0.822914i	$0.307653\pi$
$692$	−9.05446	−0.344199
$693$	−1.35114	−0.0513256
$694$	11.7989	0.447882
$695$	0	0
$696$	−0.770497	−0.0292056
$697$	−4.26574	−0.161576
$698$	22.5975	0.855329
$699$	19.2381	0.727654
$700$	0	0
$701$	−3.86961	−0.146153	−0.0730765	−	0.997326i	$-0.523282\pi$
$701$	−3.86961	−0.146153	−0.0730765	+	0.997326i	$0.523282\pi$
$702$	−0.618034	−0.0233262
$703$	−6.29064	−0.237256
$704$	−0.318961	−0.0120213
$705$	0	0
$706$	−19.2704	−0.725253
$707$	1.83692	0.0690843
$708$	−7.70634	−0.289622
$709$	43.7739	1.64396	0.821981	−	0.569514i	$-0.192869\pi$
$709$	43.7739	1.64396	0.821981	+	0.569514i	$0.192869\pi$
$710$	0	0
$711$	0.773401	0.0290048
$712$	6.80088	0.254874
$713$	15.0294	0.562855
$714$	−3.62866	−0.135799
$715$	0	0
$716$	7.94438	0.296896
$717$	−22.2206	−0.829842
$718$	−11.4127	−0.425917
$719$	25.0421	0.933913	0.466957	−	0.884280i	$-0.345350\pi$
$719$	25.0421	0.933913	0.466957	+	0.884280i	$0.345350\pi$
$720$	0	0
$721$	−8.64480	−0.321949
$722$	8.00362	0.297864
$723$	−6.87681	−0.255751
$724$	−8.84266	−0.328635
$725$	0	0
$726$	−5.67010	−0.210437
$727$	44.1945	1.63908	0.819542	−	0.573019i	$-0.194228\pi$
$727$	44.1945	1.63908	0.819542	+	0.573019i	$0.194228\pi$
$728$	2.23607	0.0828742
$729$	1.00000	0.0370370
$730$	0	0
$731$	63.8523	2.36166
$732$	−5.64771	−0.208745
$733$	−21.4720	−0.793087	−0.396544	−	0.918016i	$-0.629790\pi$
$733$	−21.4720	−0.793087	−0.396544	+	0.918016i	$0.629790\pi$
$734$	−2.90946	−0.107390
$735$	0	0
$736$	−13.1122	−0.483324
$737$	4.66899	0.171985
$738$	0.449028	0.0165289
$739$	−38.0672	−1.40033	−0.700163	−	0.713983i	$-0.746889\pi$
$739$	−38.0672	−1.40033	−0.700163	+	0.713983i	$0.746889\pi$
$740$	0	0
$741$	−2.45965	−0.0903575
$742$	−6.69572	−0.245808
$743$	2.06294	0.0756818	0.0378409	−	0.999284i	$-0.487952\pi$
$743$	2.06294	0.0756818	0.0378409	+	0.999284i	$0.487952\pi$
$744$	−14.3990	−0.527894
$745$	0	0
$746$	−22.7910	−0.834437
$747$	−13.2188	−0.483649
$748$	−12.8358	−0.469322
$749$	−6.17661	−0.225688
$750$	0	0
$751$	19.3748	0.706995	0.353497	−	0.935435i	$-0.384992\pi$
$751$	19.3748	0.706995	0.353497	+	0.935435i	$0.384992\pi$
$752$	7.12275	0.259740
$753$	20.0173	0.729469
$754$	0.212960	0.00775555
$755$	0	0
$756$	−1.61803	−0.0588473
$757$	−30.3761	−1.10404	−0.552019	−	0.833832i	$-0.686142\pi$
$757$	−30.3761	−1.10404	−0.552019	+	0.833832i	$0.686142\pi$
$758$	15.2720	0.554704
$759$	−3.15350	−0.114465
$760$	0	0
$761$	38.4962	1.39549	0.697743	−	0.716349i	$-0.254188\pi$
$761$	38.4962	1.39549	0.697743	+	0.716349i	$0.254188\pi$
$762$	−7.47799	−0.270899
$763$	10.4549	0.378495
$764$	−34.9460	−1.26430
$765$	0	0
$766$	−14.4691	−0.522791
$767$	4.76278	0.171974
$768$	6.56231	0.236797
$769$	−10.0294	−0.361668	−0.180834	−	0.983514i	$-0.557880\pi$
$769$	−10.0294	−0.361668	−0.180834	+	0.983514i	$0.557880\pi$
$770$	0	0
$771$	15.5054	0.558412
$772$	−42.4785	−1.52883
$773$	−39.8407	−1.43297	−0.716485	−	0.697603i	$-0.754250\pi$
$773$	−39.8407	−1.43297	−0.716485	+	0.697603i	$0.754250\pi$
$774$	−6.72133	−0.241593
$775$	0	0
$776$	9.82212	0.352594
$777$	2.55754	0.0917511
$778$	−4.54712	−0.163022
$779$	1.78704	0.0640274
$780$	0	0
$781$	−7.45302	−0.266690
$782$	−8.46912	−0.302855
$783$	−0.344577	−0.0123142
$784$	1.85410	0.0662179
$785$	0	0
$786$	1.32668	0.0473210
$787$	7.33956	0.261627	0.130814	−	0.991407i	$-0.458241\pi$
$787$	7.33956	0.261627	0.130814	+	0.991407i	$0.458241\pi$
$788$	2.67365	0.0952448
$789$	−29.7401	−1.05877
$790$	0	0
$791$	−11.6150	−0.412982
$792$	3.02124	0.107355
$793$	3.49047	0.123950
$794$	−2.81954	−0.100062
$795$	0	0
$796$	−7.82245	−0.277259
$797$	31.2591	1.10726	0.553628	−	0.832764i	$-0.313243\pi$
$797$	31.2591	1.10726	0.553628	+	0.832764i	$0.313243\pi$
$798$	1.52015	0.0538126
$799$	22.5552	0.797947
$800$	0	0
$801$	3.04145	0.107464
$802$	14.1685	0.500307
$803$	15.0076	0.529607
$804$	5.59127	0.197189
$805$	0	0
$806$	3.97980	0.140182
$807$	−5.54846	−0.195315
$808$	−4.10747	−0.144500
$809$	41.3704	1.45451	0.727253	−	0.686370i	$-0.240797\pi$
$809$	41.3704	1.45451	0.727253	+	0.686370i	$0.240797\pi$
$810$	0	0
$811$	3.41538	0.119930	0.0599651	−	0.998200i	$-0.480901\pi$
$811$	3.41538	0.119930	0.0599651	+	0.998200i	$0.480901\pi$
$812$	0.557537	0.0195657
$813$	−18.0948	−0.634611
$814$	−2.13567	−0.0748553
$815$	0	0
$816$	−10.8860	−0.381085
$817$	−26.7495	−0.935848
$818$	−1.18921	−0.0415797
$819$	1.00000	0.0349428
$820$	0	0
$821$	9.42954	0.329093	0.164547	−	0.986369i	$-0.447384\pi$
$821$	9.42954	0.329093	0.164547	+	0.986369i	$0.447384\pi$
$822$	−1.84097	−0.0642113
$823$	9.46127	0.329799	0.164900	−	0.986310i	$-0.447270\pi$
$823$	9.46127	0.329799	0.164900	+	0.986310i	$0.447270\pi$
$824$	19.3304	0.673405
$825$	0	0
$826$	−2.94356	−0.102420
$827$	4.39948	0.152985	0.0764924	−	0.997070i	$-0.475628\pi$
$827$	4.39948	0.152985	0.0764924	+	0.997070i	$0.475628\pi$
$828$	−3.77642	−0.131240
$829$	−29.5962	−1.02792	−0.513959	−	0.857815i	$-0.671822\pi$
$829$	−29.5962	−1.02792	−0.513959	+	0.857815i	$0.671822\pi$
$830$	0	0
$831$	−7.93616	−0.275302
$832$	0.236068	0.00818418
$833$	5.87129	0.203428
$834$	−6.95784	−0.240930
$835$	0	0
$836$	5.37727	0.185977
$837$	−6.43945	−0.222580
$838$	−13.0899	−0.452182
$839$	−30.9399	−1.06816	−0.534082	−	0.845433i	$-0.679342\pi$
$839$	−30.9399	−1.06816	−0.534082	+	0.845433i	$0.679342\pi$
$840$	0	0
$841$	−28.8813	−0.995906
$842$	−9.85120	−0.339495
$843$	27.0901	0.933031
$844$	45.4755	1.56533
$845$	0	0
$846$	−2.37425	−0.0816284
$847$	9.17442	0.315237
$848$	−20.0872	−0.689796
$849$	−24.3812	−0.836761
$850$	0	0
$851$	5.96917	0.204621
$852$	−8.92522	−0.305773
$853$	−42.1501	−1.44319	−0.721596	−	0.692314i	$-0.756591\pi$
$853$	−42.1501	−1.44319	−0.721596	+	0.692314i	$0.756591\pi$
$854$	−2.15723	−0.0738190
$855$	0	0
$856$	13.8113	0.472061
$857$	−29.7602	−1.01659	−0.508295	−	0.861183i	$-0.669724\pi$
$857$	−29.7602	−1.01659	−0.508295	+	0.861183i	$0.669724\pi$
$858$	−0.835051	−0.0285082
$859$	20.6797	0.705581	0.352790	−	0.935702i	$-0.385233\pi$
$859$	20.6797	0.705581	0.352790	+	0.935702i	$0.385233\pi$
$860$	0	0
$861$	−0.726543	−0.0247605
$862$	−22.1971	−0.756037
$863$	−44.3852	−1.51089	−0.755446	−	0.655211i	$-0.772580\pi$
$863$	−44.3852	−1.51089	−0.755446	+	0.655211i	$0.772580\pi$
$864$	5.61803	0.191129
$865$	0	0
$866$	6.21786	0.211292
$867$	−17.4720	−0.593381
$868$	10.4192	0.353652
$869$	1.04497	0.0354483
$870$	0	0
$871$	−3.45559	−0.117088
$872$	−23.3780	−0.791678
$873$	4.39259	0.148667
$874$	3.54795	0.120011
$875$	0	0
$876$	17.9721	0.607220
$877$	8.93451	0.301697	0.150848	−	0.988557i	$-0.451799\pi$
$877$	8.93451	0.301697	0.150848	+	0.988557i	$0.451799\pi$
$878$	−15.7032	−0.529957
$879$	−25.9141	−0.874061
$880$	0	0
$881$	12.2457	0.412569	0.206284	−	0.978492i	$-0.433863\pi$
$881$	12.2457	0.412569	0.206284	+	0.978492i	$0.433863\pi$
$882$	−0.618034	−0.0208103
$883$	−49.0957	−1.65220	−0.826101	−	0.563523i	$-0.809446\pi$
$883$	−49.0957	−1.65220	−0.826101	+	0.563523i	$0.809446\pi$
$884$	9.49994	0.319518
$885$	0	0
$886$	3.18889	0.107133
$887$	21.4848	0.721387	0.360694	−	0.932684i	$-0.382540\pi$
$887$	21.4848	0.721387	0.360694	+	0.932684i	$0.382540\pi$
$888$	−5.71883	−0.191911
$889$	12.0996	0.405809
$890$	0	0
$891$	1.35114	0.0452649
$892$	−5.76110	−0.192896
$893$	−9.44903	−0.316200
$894$	−1.74624	−0.0584029
$895$	0	0
$896$	−11.3820	−0.380245
$897$	2.33395	0.0779285
$898$	−1.82197	−0.0607998
$899$	2.21888	0.0740039
$900$	0	0
$901$	−63.6089	−2.11912
$902$	0.606700	0.0202009
$903$	10.8753	0.361909
$904$	25.9720	0.863815
$905$	0	0
$906$	−9.30198	−0.309037
$907$	50.9430	1.69154	0.845768	−	0.533551i	$-0.179143\pi$
$907$	50.9430	1.69154	0.845768	+	0.533551i	$0.179143\pi$
$908$	−8.63167	−0.286452
$909$	−1.83692	−0.0609267
$910$	0	0
$911$	51.0746	1.69218	0.846089	−	0.533041i	$-0.178951\pi$
$911$	51.0746	1.69218	0.846089	+	0.533041i	$0.178951\pi$
$912$	4.56044	0.151011
$913$	−17.8604	−0.591094
$914$	−15.9515	−0.527628
$915$	0	0
$916$	−25.2769	−0.835172
$917$	−2.14661	−0.0708873
$918$	3.62866	0.119763
$919$	−11.2110	−0.369816	−0.184908	−	0.982756i	$-0.559199\pi$
$919$	−11.2110	−0.369816	−0.184908	+	0.982756i	$0.559199\pi$
$920$	0	0
$921$	−16.1854	−0.533326
$922$	10.7883	0.355295
$923$	5.51609	0.181564
$924$	−2.18619	−0.0719204
$925$	0	0
$926$	22.1410	0.727599
$927$	8.64480	0.283933
$928$	−1.93584	−0.0635471
$929$	11.3053	0.370913	0.185457	−	0.982652i	$-0.440624\pi$
$929$	11.3053	0.370913	0.185457	+	0.982652i	$0.440624\pi$
$930$	0	0
$931$	−2.45965	−0.0806118
$932$	31.1280	1.01963
$933$	10.6285	0.347963
$934$	−10.8837	−0.356124
$935$	0	0
$936$	−2.23607	−0.0730882
$937$	4.71413	0.154004	0.0770019	−	0.997031i	$-0.475465\pi$
$937$	4.71413	0.154004	0.0770019	+	0.997031i	$0.475465\pi$
$938$	2.13567	0.0697322
$939$	9.85532	0.321616
$940$	0	0
$941$	53.6341	1.74842	0.874211	−	0.485545i	$-0.161379\pi$
$941$	53.6341	1.74842	0.874211	+	0.485545i	$0.161379\pi$
$942$	−13.9496	−0.454503
$943$	−1.69572	−0.0552202
$944$	−8.83068	−0.287414
$945$	0	0
$946$	−9.08147	−0.295264
$947$	−37.6662	−1.22399	−0.611994	−	0.790863i	$-0.709632\pi$
$947$	−37.6662	−1.22399	−0.611994	+	0.790863i	$0.709632\pi$
$948$	1.25139	0.0406432
$949$	−11.1074	−0.360560
$950$	0	0
$951$	−21.6678	−0.702626
$952$	−13.1286	−0.425500
$953$	18.1088	0.586602	0.293301	−	0.956020i	$-0.405246\pi$
$953$	18.1088	0.586602	0.293301	+	0.956020i	$0.405246\pi$
$954$	6.69572	0.216782
$955$	0	0
$956$	−35.9536	−1.16282
$957$	−0.465571	−0.0150498
$958$	−7.34083	−0.237172
$959$	2.97876	0.0961891
$960$	0	0
$961$	10.4665	0.337628
$962$	1.58064	0.0509620
$963$	6.17661	0.199038
$964$	−11.1269	−0.358374
$965$	0	0
$966$	−1.44246	−0.0464105
$967$	28.9382	0.930591	0.465295	−	0.885155i	$-0.345948\pi$
$967$	28.9382	0.930591	0.465295	+	0.885155i	$0.345948\pi$
$968$	−20.5146	−0.659365
$969$	14.4413	0.463922
$970$	0	0
$971$	−18.1278	−0.581750	−0.290875	−	0.956761i	$-0.593946\pi$
$971$	−18.1278	−0.581750	−0.290875	+	0.956761i	$0.593946\pi$
$972$	1.61803	0.0518985
$973$	11.2580	0.360916
$974$	−12.4487	−0.398882
$975$	0	0
$976$	−6.47170	−0.207154
$977$	35.4208	1.13321	0.566606	−	0.823989i	$-0.308256\pi$
$977$	35.4208	1.13321	0.566606	+	0.823989i	$0.308256\pi$
$978$	−1.10215	−0.0352428
$979$	4.10942	0.131338
$980$	0	0
$981$	−10.4549	−0.333801
$982$	−22.0328	−0.703096
$983$	−20.0088	−0.638182	−0.319091	−	0.947724i	$-0.603378\pi$
$983$	−20.0088	−0.638182	−0.319091	+	0.947724i	$0.603378\pi$
$984$	1.62460	0.0517903
$985$	0	0
$986$	−1.25035	−0.0398192
$987$	3.84162	0.122280
$988$	−3.97980	−0.126614
$989$	25.3826	0.807119
$990$	0	0
$991$	−6.50296	−0.206573	−0.103287	−	0.994652i	$-0.532936\pi$
$991$	−6.50296	−0.206573	−0.103287	+	0.994652i	$0.532936\pi$
$992$	−36.1770	−1.14862
$993$	−4.67975	−0.148507
$994$	−3.40913	−0.108131
$995$	0	0
$996$	−21.3884	−0.677718
$997$	−28.1430	−0.891296	−0.445648	−	0.895208i	$-0.647027\pi$
$997$	−28.1430	−0.891296	−0.445648	+	0.895208i	$0.647027\pi$
$998$	−25.9117	−0.820219
$999$	−2.55754	−0.0809169

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6825.2.a.bm.1.2		4
5.2	odd	4		1365.2.f.c.274.4	✓	8
5.3	odd	4		1365.2.f.c.274.6	yes	8
5.4	even	2		6825.2.a.be.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1365.2.f.c.274.4	✓	8	5.2	odd	4
1365.2.f.c.274.6	yes	8	5.3	odd	4
6825.2.a.be.1.4		4	5.4	even	2
6825.2.a.bm.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +1.35114 q^{11} +1.61803 q^{12} -1.00000 q^{13} +0.618034 q^{14} +1.85410 q^{16} +5.87129 q^{17} -0.618034 q^{18} -2.45965 q^{19} +1.00000 q^{21} -0.835051 q^{22} +2.33395 q^{23} -2.23607 q^{24} +0.618034 q^{26} -1.00000 q^{27} +1.61803 q^{28} +0.344577 q^{29} +6.43945 q^{31} -5.61803 q^{32} -1.35114 q^{33} -3.62866 q^{34} -1.61803 q^{36} +2.55754 q^{37} +1.52015 q^{38} +1.00000 q^{39} -0.726543 q^{41} -0.618034 q^{42} +10.8753 q^{43} -2.18619 q^{44} -1.44246 q^{46} +3.84162 q^{47} -1.85410 q^{48} +1.00000 q^{49} -5.87129 q^{51} +1.61803 q^{52} -10.8339 q^{53} +0.618034 q^{54} -2.23607 q^{56} +2.45965 q^{57} -0.212960 q^{58} -4.76278 q^{59} -3.49047 q^{61} -3.97980 q^{62} -1.00000 q^{63} -0.236068 q^{64} +0.835051 q^{66} +3.45559 q^{67} -9.49994 q^{68} -2.33395 q^{69} -5.51609 q^{71} +2.23607 q^{72} +11.1074 q^{73} -1.58064 q^{74} +3.97980 q^{76} -1.35114 q^{77} -0.618034 q^{78} +0.773401 q^{79} +1.00000 q^{81} +0.449028 q^{82} -13.2188 q^{83} -1.61803 q^{84} -6.72133 q^{86} -0.344577 q^{87} +3.02124 q^{88} +3.04145 q^{89} +1.00000 q^{91} -3.77642 q^{92} -6.43945 q^{93} -2.37425 q^{94} +5.61803 q^{96} +4.39259 q^{97} -0.618034 q^{98} +1.35114 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} - 2 q^{4} - 2 q^{6} - 4 q^{7} + 4 q^{9} - 4 q^{11} + 2 q^{12} - 4 q^{13} - 2 q^{14} - 6 q^{16} + 2 q^{17} + 2 q^{18} - 2 q^{19} + 4 q^{21} - 2 q^{22} + 8 q^{23} - 2 q^{26} - 4 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) \) 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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