LMFDB - Embedded newform 2673.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2673,2,Mod(1,2673)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2673, 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("2673.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2673 = 3^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2673.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:	$21.3440124603$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.864654912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 14x^{2} - 16x - 9 $ x^6 - 2x^5 - 8x^4 + 14x^3 + 14x^2 - 16*x - 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.10820$ of defining polynomial
Character	$\chi$	$=$	2673.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.10820 q^{2} -0.771888 q^{4} -1.77189 q^{5} +4.26854 q^{7} +3.07181 q^{8} +O(q^{10})$	\(q-1.10820 q^{2} -0.771888 q^{4} -1.77189 q^{5} +4.26854 q^{7} +3.07181 q^{8} +1.96361 q^{10} +1.00000 q^{11} -6.83861 q^{13} -4.73041 q^{14} -1.86041 q^{16} +6.92714 q^{17} -1.55781 q^{19} +1.36770 q^{20} -1.10820 q^{22} +8.76519 q^{23} -1.86041 q^{25} +7.57856 q^{26} -3.29483 q^{28} +3.60995 q^{29} +0.545302 q^{31} -4.08191 q^{32} -7.67667 q^{34} -7.56338 q^{35} -5.43281 q^{37} +1.72637 q^{38} -5.44291 q^{40} +3.31719 q^{41} +7.12119 q^{43} -0.771888 q^{44} -9.71361 q^{46} -3.44856 q^{47} +11.2204 q^{49} +2.06172 q^{50} +5.27864 q^{52} -1.23215 q^{53} -1.77189 q^{55} +13.1122 q^{56} -4.00055 q^{58} -9.46081 q^{59} -5.78996 q^{61} -0.604305 q^{62} +8.24441 q^{64} +12.1172 q^{65} +3.08853 q^{67} -5.34697 q^{68} +8.38175 q^{70} -6.32123 q^{71} +0.633278 q^{73} +6.02065 q^{74} +1.20245 q^{76} +4.26854 q^{77} +2.87500 q^{79} +3.29644 q^{80} -3.67612 q^{82} -7.24610 q^{83} -12.2741 q^{85} -7.89171 q^{86} +3.07181 q^{88} +3.98829 q^{89} -29.1909 q^{91} -6.76575 q^{92} +3.82170 q^{94} +2.76026 q^{95} -9.55952 q^{97} -12.4345 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	$ 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 8 q^{10} + 6 q^{11} - 8 q^{13} - 4 q^{14} + 8 q^{16} + 2 q^{17} - 4 q^{19} + 40 q^{20} + 2 q^{22} + 10 q^{23} + 8 q^{25} + 2 q^{26} - 12 q^{28} + 6 q^{29} - 8 q^{31} + 4 q^{32} - 10 q^{34} - 10 q^{35} + 2 q^{37} + 30 q^{38} + 18 q^{40} - 4 q^{41} + 2 q^{43} + 8 q^{44} + 4 q^{46} + 28 q^{47} + 6 q^{49} + 16 q^{50} - 12 q^{52} + 24 q^{53} + 2 q^{55} - 6 q^{56} + 6 q^{58} - 8 q^{59} + 2 q^{61} - 10 q^{62} + 18 q^{64} - 4 q^{65} + 12 q^{67} - 14 q^{68} - 10 q^{70} + 30 q^{71} + 16 q^{73} + 78 q^{74} + 2 q^{76} - 2 q^{77} - 12 q^{79} + 58 q^{80} - 16 q^{82} - 16 q^{85} - 24 q^{86} + 6 q^{88} - 6 q^{89} + 6 q^{91} + 32 q^{92} + 10 q^{94} + 6 q^{95} - 18 q^{97} - 6 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 8 * q^10 + 6 * q^11 - 8 * q^13 - 4 * q^14 + 8 * q^16 + 2 * q^17 - 4 * q^19 + 40 * q^20 + 2 * q^22 + 10 * q^23 + 8 * q^25 + 2 * q^26 - 12 * q^28 + 6 * q^29 - 8 * q^31 + 4 * q^32 - 10 * q^34 - 10 * q^35 + 2 * q^37 + 30 * q^38 + 18 * q^40 - 4 * q^41 + 2 * q^43 + 8 * q^44 + 4 * q^46 + 28 * q^47 + 6 * q^49 + 16 * q^50 - 12 * q^52 + 24 * q^53 + 2 * q^55 - 6 * q^56 + 6 * q^58 - 8 * q^59 + 2 * q^61 - 10 * q^62 + 18 * q^64 - 4 * q^65 + 12 * q^67 - 14 * q^68 - 10 * q^70 + 30 * q^71 + 16 * q^73 + 78 * q^74 + 2 * q^76 - 2 * q^77 - 12 * q^79 + 58 * q^80 - 16 * q^82 - 16 * q^85 - 24 * q^86 + 6 * q^88 - 6 * q^89 + 6 * q^91 + 32 * q^92 + 10 * q^94 + 6 * q^95 - 18 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.10820	−0.783617	−0.391809	−	0.920047i	$-0.628150\pi$
$2$	−1.10820	−0.783617	−0.391809	+	0.920047i	$0.628150\pi$
$3$	0	0
$3$	0	0
$4$	−0.771888	−0.385944
$5$	−1.77189	−0.792412	−0.396206	−	0.918162i	$-0.629673\pi$
$5$	−1.77189	−0.792412	−0.396206	+	0.918162i	$0.629673\pi$
$6$	0	0
$7$	4.26854	1.61336	0.806678	−	0.590991i	$-0.201263\pi$
$7$	4.26854	1.61336	0.806678	+	0.590991i	$0.201263\pi$
$8$	3.07181	1.08605
$9$	0	0
$10$	1.96361	0.620948
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.83861	−1.89669	−0.948344	−	0.317242i	$-0.897243\pi$
$13$	−6.83861	−1.89669	−0.948344	+	0.317242i	$0.897243\pi$
$14$	−4.73041	−1.26425
$15$	0	0
$16$	−1.86041	−0.465104
$17$	6.92714	1.68008	0.840039	−	0.542527i	$-0.182532\pi$
$17$	6.92714	1.68008	0.840039	+	0.542527i	$0.182532\pi$
$18$	0	0
$19$	−1.55781	−0.357386	−0.178693	−	0.983905i	$-0.557187\pi$
$19$	−1.55781	−0.357386	−0.178693	+	0.983905i	$0.557187\pi$
$20$	1.36770	0.305827
$21$	0	0
$22$	−1.10820	−0.236270
$23$	8.76519	1.82767	0.913835	−	0.406086i	$-0.133107\pi$
$23$	8.76519	1.82767	0.913835	+	0.406086i	$0.133107\pi$
$24$	0	0
$25$	−1.86041	−0.372083
$26$	7.57856	1.48628
$27$	0	0
$28$	−3.29483	−0.622665
$29$	3.60995	0.670350	0.335175	−	0.942156i	$-0.391205\pi$
$29$	3.60995	0.670350	0.335175	+	0.942156i	$0.391205\pi$
$30$	0	0
$31$	0.545302	0.0979391	0.0489696	−	0.998800i	$-0.484406\pi$
$31$	0.545302	0.0979391	0.0489696	+	0.998800i	$0.484406\pi$
$32$	−4.08191	−0.721586
$33$	0	0
$34$	−7.67667	−1.31654
$35$	−7.56338	−1.27844
$36$	0	0
$37$	−5.43281	−0.893148	−0.446574	−	0.894747i	$-0.647356\pi$
$37$	−5.43281	−0.893148	−0.446574	+	0.894747i	$0.647356\pi$
$38$	1.72637	0.280054
$39$	0	0
$40$	−5.44291	−0.860599
$41$	3.31719	0.518058	0.259029	−	0.965870i	$-0.416597\pi$
$41$	3.31719	0.518058	0.259029	+	0.965870i	$0.416597\pi$
$42$	0	0
$43$	7.12119	1.08597	0.542985	−	0.839742i	$-0.317294\pi$
$43$	7.12119	1.08597	0.542985	+	0.839742i	$0.317294\pi$
$44$	−0.771888	−0.116366
$45$	0	0
$46$	−9.71361	−1.43219
$47$	−3.44856	−0.503024	−0.251512	−	0.967854i	$-0.580928\pi$
$47$	−3.44856	−0.503024	−0.251512	+	0.967854i	$0.580928\pi$
$48$	0	0
$49$	11.2204	1.60292
$50$	2.06172	0.291571
$51$	0	0
$52$	5.27864	0.732015
$53$	−1.23215	−0.169249	−0.0846245	−	0.996413i	$-0.526969\pi$
$53$	−1.23215	−0.169249	−0.0846245	+	0.996413i	$0.526969\pi$
$54$	0	0
$55$	−1.77189	−0.238921
$56$	13.1122	1.75219
$57$	0	0
$58$	−4.00055	−0.525298
$59$	−9.46081	−1.23169	−0.615847	−	0.787866i	$-0.711186\pi$
$59$	−9.46081	−1.23169	−0.615847	+	0.787866i	$0.711186\pi$
$60$	0	0
$61$	−5.78996	−0.741329	−0.370664	−	0.928767i	$-0.620870\pi$
$61$	−5.78996	−0.741329	−0.370664	+	0.928767i	$0.620870\pi$
$62$	−0.604305	−0.0767468
$63$	0	0
$64$	8.24441	1.03055
$65$	12.1172	1.50296
$66$	0	0
$67$	3.08853	0.377324	0.188662	−	0.982042i	$-0.439585\pi$
$67$	3.08853	0.377324	0.188662	+	0.982042i	$0.439585\pi$
$68$	−5.34697	−0.648415
$69$	0	0
$70$	8.38175	1.00181
$71$	−6.32123	−0.750192	−0.375096	−	0.926986i	$-0.622390\pi$
$71$	−6.32123	−0.750192	−0.375096	+	0.926986i	$0.622390\pi$
$72$	0	0
$73$	0.633278	0.0741196	0.0370598	−	0.999313i	$-0.488201\pi$
$73$	0.633278	0.0741196	0.0370598	+	0.999313i	$0.488201\pi$
$74$	6.02065	0.699887
$75$	0	0
$76$	1.20245	0.137931
$77$	4.26854	0.486445
$78$	0	0
$79$	2.87500	0.323463	0.161731	−	0.986835i	$-0.448292\pi$
$79$	2.87500	0.323463	0.161731	+	0.986835i	$0.448292\pi$
$80$	3.29644	0.368554
$81$	0	0
$82$	−3.67612	−0.405959
$83$	−7.24610	−0.795363	−0.397681	−	0.917524i	$-0.630185\pi$
$83$	−7.24610	−0.795363	−0.397681	+	0.917524i	$0.630185\pi$
$84$	0	0
$85$	−12.2741	−1.33131
$86$	−7.89171	−0.850985
$87$	0	0
$88$	3.07181	0.327456
$89$	3.98829	0.422758	0.211379	−	0.977404i	$-0.432205\pi$
$89$	3.98829	0.422758	0.211379	+	0.977404i	$0.432205\pi$
$90$	0	0
$91$	−29.1909	−3.06004
$92$	−6.76575	−0.705378
$93$	0	0
$94$	3.82170	0.394178
$95$	2.76026	0.283197
$96$	0	0
$97$	−9.55952	−0.970622	−0.485311	−	0.874341i	$-0.661294\pi$
$97$	−9.55952	−0.970622	−0.485311	+	0.874341i	$0.661294\pi$
$98$	−12.4345	−1.25608
$99$	0	0
$100$	1.43603	0.143603
$101$	12.3952	1.23337	0.616683	−	0.787212i	$-0.288476\pi$
$101$	12.3952	1.23337	0.616683	+	0.787212i	$0.288476\pi$
$102$	0	0
$103$	3.66761	0.361381	0.180690	−	0.983540i	$-0.442167\pi$
$103$	3.66761	0.361381	0.180690	+	0.983540i	$0.442167\pi$
$104$	−21.0069	−2.05990
$105$	0	0
$106$	1.36547	0.132626
$107$	16.1138	1.55778	0.778892	−	0.627158i	$-0.215782\pi$
$107$	16.1138	1.55778	0.778892	+	0.627158i	$0.215782\pi$
$108$	0	0
$109$	16.9559	1.62408	0.812038	−	0.583604i	$-0.198358\pi$
$109$	16.9559	1.62408	0.812038	+	0.583604i	$0.198358\pi$
$110$	1.96361	0.187223
$111$	0	0
$112$	−7.94125	−0.750378
$113$	−13.4587	−1.26609	−0.633046	−	0.774114i	$-0.718195\pi$
$113$	−13.4587	−1.26609	−0.633046	+	0.774114i	$0.718195\pi$
$114$	0	0
$115$	−15.5309	−1.44827
$116$	−2.78647	−0.258718
$117$	0	0
$118$	10.4845	0.965176
$119$	29.5688	2.71056
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.41645	0.580918
$123$	0	0
$124$	−0.420912	−0.0377990
$125$	12.1559	1.08726
$126$	0	0
$127$	−17.8683	−1.58556	−0.792778	−	0.609511i	$-0.791366\pi$
$127$	−17.8683	−1.58556	−0.792778	+	0.609511i	$0.791366\pi$
$128$	−0.972655	−0.0859714
$129$	0	0
$130$	−13.4284	−1.17775
$131$	−13.6408	−1.19180	−0.595902	−	0.803057i	$-0.703205\pi$
$131$	−13.6408	−1.19180	−0.595902	+	0.803057i	$0.703205\pi$
$132$	0	0
$133$	−6.64958	−0.576591
$134$	−3.42271	−0.295677
$135$	0	0
$136$	21.2789	1.82465
$137$	10.5764	0.903606	0.451803	−	0.892118i	$-0.350781\pi$
$137$	10.5764	0.903606	0.451803	+	0.892118i	$0.350781\pi$
$138$	0	0
$139$	20.4185	1.73188	0.865939	−	0.500150i	$-0.166722\pi$
$139$	20.4185	1.73188	0.865939	+	0.500150i	$0.166722\pi$
$140$	5.83808	0.493407
$141$	0	0
$142$	7.00520	0.587863
$143$	−6.83861	−0.571873
$144$	0	0
$145$	−6.39642	−0.531194
$146$	−0.701800	−0.0580814
$147$	0	0
$148$	4.19352	0.344705
$149$	23.2375	1.90369	0.951847	−	0.306574i	$-0.0991828\pi$
$149$	23.2375	1.90369	0.951847	+	0.306574i	$0.0991828\pi$
$150$	0	0
$151$	14.1005	1.14748	0.573739	−	0.819038i	$-0.305492\pi$
$151$	14.1005	1.14748	0.573739	+	0.819038i	$0.305492\pi$
$152$	−4.78530	−0.388139
$153$	0	0
$154$	−4.73041	−0.381187
$155$	−0.966214	−0.0776082
$156$	0	0
$157$	6.75000	0.538709	0.269354	−	0.963041i	$-0.413190\pi$
$157$	6.75000	0.538709	0.269354	+	0.963041i	$0.413190\pi$
$158$	−3.18608	−0.253471
$159$	0	0
$160$	7.23268	0.571794
$161$	37.4146	2.94868
$162$	0	0
$163$	2.16668	0.169707	0.0848537	−	0.996393i	$-0.472958\pi$
$163$	2.16668	0.169707	0.0848537	+	0.996393i	$0.472958\pi$
$164$	−2.56050	−0.199941
$165$	0	0
$166$	8.03015	0.623260
$167$	20.8869	1.61628	0.808140	−	0.588991i	$-0.200475\pi$
$167$	20.8869	1.61628	0.808140	+	0.588991i	$0.200475\pi$
$168$	0	0
$169$	33.7666	2.59743
$170$	13.6022	1.04324
$171$	0	0
$172$	−5.49675	−0.419124
$173$	−7.09883	−0.539714	−0.269857	−	0.962900i	$-0.586976\pi$
$173$	−7.09883	−0.539714	−0.269857	+	0.962900i	$0.586976\pi$
$174$	0	0
$175$	−7.94125	−0.600302
$176$	−1.86041	−0.140234
$177$	0	0
$178$	−4.41983	−0.331281
$179$	−4.56738	−0.341382	−0.170691	−	0.985325i	$-0.554600\pi$
$179$	−4.56738	−0.341382	−0.170691	+	0.985325i	$0.554600\pi$
$180$	0	0
$181$	7.15436	0.531779	0.265890	−	0.964003i	$-0.414334\pi$
$181$	7.15436	0.531779	0.265890	+	0.964003i	$0.414334\pi$
$182$	32.3494	2.39790
$183$	0	0
$184$	26.9250	1.98494
$185$	9.62633	0.707742
$186$	0	0
$187$	6.92714	0.506562
$188$	2.66190	0.194139
$189$	0	0
$190$	−3.05893	−0.221918
$191$	5.30701	0.384002	0.192001	−	0.981395i	$-0.438502\pi$
$191$	5.30701	0.384002	0.192001	+	0.981395i	$0.438502\pi$
$192$	0	0
$193$	15.4840	1.11456	0.557280	−	0.830324i	$-0.311845\pi$
$193$	15.4840	1.11456	0.557280	+	0.830324i	$0.311845\pi$
$194$	10.5939	0.760597
$195$	0	0
$196$	−8.66092	−0.618637
$197$	23.7002	1.68857	0.844285	−	0.535894i	$-0.180025\pi$
$197$	23.7002	1.68857	0.844285	+	0.535894i	$0.180025\pi$
$198$	0	0
$199$	−1.22561	−0.0868812	−0.0434406	−	0.999056i	$-0.513832\pi$
$199$	−1.22561	−0.0868812	−0.0434406	+	0.999056i	$0.513832\pi$
$200$	−5.71484	−0.404100
$201$	0	0
$202$	−13.7364	−0.966487
$203$	15.4092	1.08151
$204$	0	0
$205$	−5.87769	−0.410515
$206$	−4.06446	−0.283184
$207$	0	0
$208$	12.7226	0.882157
$209$	−1.55781	−0.107756
$210$	0	0
$211$	−12.6407	−0.870222	−0.435111	−	0.900377i	$-0.643291\pi$
$211$	−12.6407	−0.870222	−0.435111	+	0.900377i	$0.643291\pi$
$212$	0.951082	0.0653206
$213$	0	0
$214$	−17.8574	−1.22071
$215$	−12.6179	−0.860536
$216$	0	0
$217$	2.32764	0.158011
$218$	−18.7905	−1.27265
$219$	0	0
$220$	1.36770	0.0922102
$221$	−47.3720	−3.18658
$222$	0	0
$223$	−3.88848	−0.260392	−0.130196	−	0.991488i	$-0.541561\pi$
$223$	−3.88848	−0.260392	−0.130196	+	0.991488i	$0.541561\pi$
$224$	−17.4238	−1.16418
$225$	0	0
$226$	14.9150	0.992131
$227$	2.55989	0.169906	0.0849528	−	0.996385i	$-0.472926\pi$
$227$	2.55989	0.169906	0.0849528	+	0.996385i	$0.472926\pi$
$228$	0	0
$229$	−3.37655	−0.223128	−0.111564	−	0.993757i	$-0.535586\pi$
$229$	−3.37655	−0.223128	−0.111564	+	0.993757i	$0.535586\pi$
$230$	17.2114	1.13489
$231$	0	0
$232$	11.0891	0.728034
$233$	11.7472	0.769585	0.384793	−	0.923003i	$-0.374273\pi$
$233$	11.7472	0.769585	0.384793	+	0.923003i	$0.374273\pi$
$234$	0	0
$235$	6.11045	0.398602
$236$	7.30269	0.475364
$237$	0	0
$238$	−32.7682	−2.12405
$239$	3.08243	0.199386	0.0996931	−	0.995018i	$-0.468214\pi$
$239$	3.08243	0.199386	0.0996931	+	0.995018i	$0.468214\pi$
$240$	0	0
$241$	2.21069	0.142403	0.0712015	−	0.997462i	$-0.477317\pi$
$241$	2.21069	0.142403	0.0712015	+	0.997462i	$0.477317\pi$
$242$	−1.10820	−0.0712379
$243$	0	0
$244$	4.46920	0.286111
$245$	−19.8814	−1.27017
$246$	0	0
$247$	10.6533	0.677850
$248$	1.67507	0.106367
$249$	0	0
$250$	−13.4712	−0.851992
$251$	1.41073	0.0890445	0.0445223	−	0.999008i	$-0.485823\pi$
$251$	1.41073	0.0890445	0.0445223	+	0.999008i	$0.485823\pi$
$252$	0	0
$253$	8.76519	0.551063
$254$	19.8017	1.24247
$255$	0	0
$256$	−15.4109	−0.963183
$257$	−0.575883	−0.0359226	−0.0179613	−	0.999839i	$-0.505718\pi$
$257$	−0.575883	−0.0359226	−0.0179613	+	0.999839i	$0.505718\pi$
$258$	0	0
$259$	−23.1902	−1.44097
$260$	−9.35315	−0.580058
$261$	0	0
$262$	15.1168	0.933919
$263$	15.8359	0.976485	0.488242	−	0.872708i	$-0.337638\pi$
$263$	15.8359	0.976485	0.488242	+	0.872708i	$0.337638\pi$
$264$	0	0
$265$	2.18323	0.134115
$266$	7.36907	0.451827
$267$	0	0
$268$	−2.38400	−0.145626
$269$	19.9145	1.21421	0.607105	−	0.794621i	$-0.292331\pi$
$269$	19.9145	1.21421	0.607105	+	0.794621i	$0.292331\pi$
$270$	0	0
$271$	6.62946	0.402711	0.201356	−	0.979518i	$-0.435465\pi$
$271$	6.62946	0.402711	0.201356	+	0.979518i	$0.435465\pi$
$272$	−12.8873	−0.781410
$273$	0	0
$274$	−11.7208	−0.708081
$275$	−1.86041	−0.112187
$276$	0	0
$277$	−24.5688	−1.47619	−0.738097	−	0.674695i	$-0.764275\pi$
$277$	−24.5688	−1.47619	−0.738097	+	0.674695i	$0.764275\pi$
$278$	−22.6279	−1.35713
$279$	0	0
$280$	−23.2333	−1.38845
$281$	−5.21463	−0.311079	−0.155539	−	0.987830i	$-0.549712\pi$
$281$	−5.21463	−0.311079	−0.155539	+	0.987830i	$0.549712\pi$
$282$	0	0
$283$	−22.8831	−1.36026	−0.680129	−	0.733093i	$-0.738076\pi$
$283$	−22.8831	−1.36026	−0.680129	+	0.733093i	$0.738076\pi$
$284$	4.87928	0.289532
$285$	0	0
$286$	7.57856	0.448130
$287$	14.1596	0.835812
$288$	0	0
$289$	30.9852	1.82266
$290$	7.08853	0.416253
$291$	0	0
$292$	−0.488819	−0.0286060
$293$	−7.05753	−0.412305	−0.206153	−	0.978520i	$-0.566094\pi$
$293$	−7.05753	−0.412305	−0.206153	+	0.978520i	$0.566094\pi$
$294$	0	0
$295$	16.7635	0.976009
$296$	−16.6886	−0.970003
$297$	0	0
$298$	−25.7519	−1.49177
$299$	−59.9417	−3.46652
$300$	0	0
$301$	30.3971	1.75206
$302$	−15.6262	−0.899184
$303$	0	0
$304$	2.89817	0.166222
$305$	10.2592	0.587438
$306$	0	0
$307$	21.7980	1.24408	0.622040	−	0.782986i	$-0.286304\pi$
$307$	21.7980	1.24408	0.622040	+	0.782986i	$0.286304\pi$
$308$	−3.29483	−0.187741
$309$	0	0
$310$	1.07076	0.0608151
$311$	−14.5244	−0.823603	−0.411802	−	0.911274i	$-0.635100\pi$
$311$	−14.5244	−0.823603	−0.411802	+	0.911274i	$0.635100\pi$
$312$	0	0
$313$	13.8168	0.780972	0.390486	−	0.920609i	$-0.372307\pi$
$313$	13.8168	0.780972	0.390486	+	0.920609i	$0.372307\pi$
$314$	−7.48036	−0.422141
$315$	0	0
$316$	−2.21918	−0.124838
$317$	12.6573	0.710904	0.355452	−	0.934694i	$-0.384327\pi$
$317$	12.6573	0.710904	0.355452	+	0.934694i	$0.384327\pi$
$318$	0	0
$319$	3.60995	0.202118
$320$	−14.6082	−0.816621
$321$	0	0
$322$	−41.4629	−2.31064
$323$	−10.7912	−0.600436
$324$	0	0
$325$	12.7226	0.705725
$326$	−2.40112	−0.132986
$327$	0	0
$328$	10.1898	0.562637
$329$	−14.7203	−0.811557
$330$	0	0
$331$	−23.3466	−1.28325	−0.641624	−	0.767020i	$-0.721739\pi$
$331$	−23.3466	−1.28325	−0.641624	+	0.767020i	$0.721739\pi$
$332$	5.59318	0.306965
$333$	0	0
$334$	−23.1469	−1.26654
$335$	−5.47252	−0.298996
$336$	0	0
$337$	−3.99911	−0.217845	−0.108923	−	0.994050i	$-0.534740\pi$
$337$	−3.99911	−0.217845	−0.108923	+	0.994050i	$0.534740\pi$
$338$	−37.4202	−2.03539
$339$	0	0
$340$	9.47423	0.513812
$341$	0.545302	0.0295298
$342$	0	0
$343$	18.0151	0.972726
$344$	21.8749	1.17942
$345$	0	0
$346$	7.86694	0.422929
$347$	−5.56974	−0.298999	−0.149500	−	0.988762i	$-0.547766\pi$
$347$	−5.56974	−0.298999	−0.149500	+	0.988762i	$0.547766\pi$
$348$	0	0
$349$	−11.4451	−0.612640	−0.306320	−	0.951929i	$-0.599098\pi$
$349$	−11.4451	−0.612640	−0.306320	+	0.951929i	$0.599098\pi$
$350$	8.80052	0.470407
$351$	0	0
$352$	−4.08191	−0.217566
$353$	−2.26609	−0.120612	−0.0603060	−	0.998180i	$-0.519208\pi$
$353$	−2.26609	−0.120612	−0.0603060	+	0.998180i	$0.519208\pi$
$354$	0	0
$355$	11.2005	0.594461
$356$	−3.07851	−0.163161
$357$	0	0
$358$	5.06158	0.267513
$359$	−1.67255	−0.0882736	−0.0441368	−	0.999025i	$-0.514054\pi$
$359$	−1.67255	−0.0882736	−0.0441368	+	0.999025i	$0.514054\pi$
$360$	0	0
$361$	−16.5732	−0.872275
$362$	−7.92847	−0.416711
$363$	0	0
$364$	22.5321	1.18100
$365$	−1.12210	−0.0587332
$366$	0	0
$367$	7.71457	0.402697	0.201349	−	0.979520i	$-0.435468\pi$
$367$	7.71457	0.402697	0.201349	+	0.979520i	$0.435468\pi$
$368$	−16.3069	−0.850056
$369$	0	0
$370$	−10.6679	−0.554599
$371$	−5.25949	−0.273059
$372$	0	0
$373$	5.02159	0.260008	0.130004	−	0.991513i	$-0.458501\pi$
$373$	5.02159	0.260008	0.130004	+	0.991513i	$0.458501\pi$
$374$	−7.67667	−0.396951
$375$	0	0
$376$	−10.5933	−0.546309
$377$	−24.6870	−1.27145
$378$	0	0
$379$	18.6293	0.956922	0.478461	−	0.878109i	$-0.341195\pi$
$379$	18.6293	0.956922	0.478461	+	0.878109i	$0.341195\pi$
$380$	−2.13061	−0.109298
$381$	0	0
$382$	−5.88124	−0.300910
$383$	20.9673	1.07138	0.535690	−	0.844415i	$-0.320052\pi$
$383$	20.9673	1.07138	0.535690	+	0.844415i	$0.320052\pi$
$384$	0	0
$385$	−7.56338	−0.385465
$386$	−17.1594	−0.873389
$387$	0	0
$388$	7.37888	0.374606
$389$	−1.07850	−0.0546820	−0.0273410	−	0.999626i	$-0.508704\pi$
$389$	−1.07850	−0.0546820	−0.0273410	+	0.999626i	$0.508704\pi$
$390$	0	0
$391$	60.7177	3.07063
$392$	34.4671	1.74085
$393$	0	0
$394$	−26.2646	−1.32319
$395$	−5.09418	−0.256316
$396$	0	0
$397$	−21.7759	−1.09290	−0.546451	−	0.837491i	$-0.684021\pi$
$397$	−21.7759	−1.09290	−0.546451	+	0.837491i	$0.684021\pi$
$398$	1.35822	0.0680816
$399$	0	0
$400$	3.46114	0.173057
$401$	22.9316	1.14515	0.572576	−	0.819852i	$-0.305944\pi$
$401$	22.9316	1.14515	0.572576	+	0.819852i	$0.305944\pi$
$402$	0	0
$403$	−3.72911	−0.185760
$404$	−9.56768	−0.476010
$405$	0	0
$406$	−17.0765	−0.847493
$407$	−5.43281	−0.269294
$408$	0	0
$409$	−20.3245	−1.00498	−0.502492	−	0.864582i	$-0.667583\pi$
$409$	−20.3245	−1.00498	−0.502492	+	0.864582i	$0.667583\pi$
$410$	6.51367	0.321687
$411$	0	0
$412$	−2.83099	−0.139473
$413$	−40.3839	−1.98716
$414$	0	0
$415$	12.8393	0.630255
$416$	27.9146	1.36863
$417$	0	0
$418$	1.72637	0.0844394
$419$	8.47795	0.414175	0.207087	−	0.978322i	$-0.433602\pi$
$419$	8.47795	0.414175	0.207087	+	0.978322i	$0.433602\pi$
$420$	0	0
$421$	13.6883	0.667128	0.333564	−	0.942728i	$-0.391749\pi$
$421$	13.6883	0.667128	0.333564	+	0.942728i	$0.391749\pi$
$422$	14.0085	0.681921
$423$	0	0
$424$	−3.78494	−0.183813
$425$	−12.8873	−0.625128
$426$	0	0
$427$	−24.7147	−1.19603
$428$	−12.4381	−0.601217
$429$	0	0
$430$	13.9832	0.674331
$431$	26.1584	1.26001	0.630003	−	0.776593i	$-0.283054\pi$
$431$	26.1584	1.26001	0.630003	+	0.776593i	$0.283054\pi$
$432$	0	0
$433$	−21.2865	−1.02297	−0.511483	−	0.859294i	$-0.670904\pi$
$433$	−21.2865	−1.02297	−0.511483	+	0.859294i	$0.670904\pi$
$434$	−2.57950	−0.123820
$435$	0	0
$436$	−13.0880	−0.626802
$437$	−13.6545	−0.653184
$438$	0	0
$439$	34.3530	1.63958	0.819790	−	0.572664i	$-0.194090\pi$
$439$	34.3530	1.63958	0.819790	+	0.572664i	$0.194090\pi$
$440$	−5.44291	−0.259480
$441$	0	0
$442$	52.4977	2.49706
$443$	−31.0948	−1.47736	−0.738678	−	0.674058i	$-0.764550\pi$
$443$	−31.0948	−1.47736	−0.738678	+	0.674058i	$0.764550\pi$
$444$	0	0
$445$	−7.06681	−0.334999
$446$	4.30923	0.204048
$447$	0	0
$448$	35.1916	1.66265
$449$	−7.51821	−0.354806	−0.177403	−	0.984138i	$-0.556770\pi$
$449$	−7.51821	−0.354806	−0.177403	+	0.984138i	$0.556770\pi$
$450$	0	0
$451$	3.31719	0.156200
$452$	10.3886	0.488640
$453$	0	0
$454$	−2.83687	−0.133141
$455$	51.7230	2.42481
$456$	0	0
$457$	30.0560	1.40596	0.702981	−	0.711209i	$-0.251852\pi$
$457$	30.0560	1.40596	0.702981	+	0.711209i	$0.251852\pi$
$458$	3.74190	0.174847
$459$	0	0
$460$	11.9881	0.558950
$461$	28.8997	1.34599	0.672996	−	0.739646i	$-0.265007\pi$
$461$	28.8997	1.34599	0.672996	+	0.739646i	$0.265007\pi$
$462$	0	0
$463$	−36.8833	−1.71411	−0.857057	−	0.515222i	$-0.827710\pi$
$463$	−36.8833	−1.71411	−0.857057	+	0.515222i	$0.827710\pi$
$464$	−6.71600	−0.311782
$465$	0	0
$466$	−13.0183	−0.603060
$467$	4.15036	0.192056	0.0960278	−	0.995379i	$-0.469386\pi$
$467$	4.15036	0.192056	0.0960278	+	0.995379i	$0.469386\pi$
$468$	0	0
$469$	13.1835	0.608758
$470$	−6.77162	−0.312351
$471$	0	0
$472$	−29.0618	−1.33768
$473$	7.12119	0.327432
$474$	0	0
$475$	2.89817	0.132977
$476$	−22.8238	−1.04613
$477$	0	0
$478$	−3.41596	−0.156242
$479$	−43.2091	−1.97427	−0.987136	−	0.159882i	$-0.948889\pi$
$479$	−43.2091	−1.97427	−0.987136	+	0.159882i	$0.948889\pi$
$480$	0	0
$481$	37.1529	1.69402
$482$	−2.44989	−0.111589
$483$	0	0
$484$	−0.771888	−0.0350858
$485$	16.9384	0.769133
$486$	0	0
$487$	17.0924	0.774531	0.387266	−	0.921968i	$-0.373420\pi$
$487$	17.0924	0.774531	0.387266	+	0.921968i	$0.373420\pi$
$488$	−17.7857	−0.805120
$489$	0	0
$490$	22.0326	0.995330
$491$	−7.34035	−0.331265	−0.165633	−	0.986188i	$-0.552967\pi$
$491$	−7.34035	−0.331265	−0.165633	+	0.986188i	$0.552967\pi$
$492$	0	0
$493$	25.0066	1.12624
$494$	−11.8060	−0.531175
$495$	0	0
$496$	−1.01449	−0.0455518
$497$	−26.9824	−1.21033
$498$	0	0
$499$	37.8819	1.69583	0.847914	−	0.530134i	$-0.177858\pi$
$499$	37.8819	1.69583	0.847914	+	0.530134i	$0.177858\pi$
$500$	−9.38298	−0.419619
$501$	0	0
$502$	−1.56338	−0.0697769
$503$	−15.7894	−0.704013	−0.352007	−	0.935998i	$-0.614501\pi$
$503$	−15.7894	−0.704013	−0.352007	+	0.935998i	$0.614501\pi$
$504$	0	0
$505$	−21.9628	−0.977334
$506$	−9.71361	−0.431823
$507$	0	0
$508$	13.7923	0.611935
$509$	29.3167	1.29944	0.649720	−	0.760174i	$-0.274886\pi$
$509$	29.3167	1.29944	0.649720	+	0.760174i	$0.274886\pi$
$510$	0	0
$511$	2.70317	0.119581
$512$	19.0237	0.840738
$513$	0	0
$514$	0.638195	0.0281496
$515$	−6.49860	−0.286363
$516$	0	0
$517$	−3.44856	−0.151667
$518$	25.6994	1.12917
$519$	0	0
$520$	37.2219	1.63229
$521$	21.2065	0.929074	0.464537	−	0.885554i	$-0.346221\pi$
$521$	21.2065	0.929074	0.464537	+	0.885554i	$0.346221\pi$
$522$	0	0
$523$	−22.1005	−0.966386	−0.483193	−	0.875514i	$-0.660523\pi$
$523$	−22.1005	−0.966386	−0.483193	+	0.875514i	$0.660523\pi$
$524$	10.5292	0.459970
$525$	0	0
$526$	−17.5494	−0.765190
$527$	3.77738	0.164545
$528$	0	0
$529$	53.8286	2.34038
$530$	−2.41946	−0.105095
$531$	0	0
$532$	5.13272	0.222532
$533$	−22.6850	−0.982595
$534$	0	0
$535$	−28.5519	−1.23441
$536$	9.48737	0.409792
$537$	0	0
$538$	−22.0693	−0.951477
$539$	11.2204	0.483299
$540$	0	0
$541$	9.75292	0.419311	0.209655	−	0.977775i	$-0.432766\pi$
$541$	9.75292	0.419311	0.209655	+	0.977775i	$0.432766\pi$
$542$	−7.34679	−0.315571
$543$	0	0
$544$	−28.2759	−1.21232
$545$	−30.0439	−1.28694
$546$	0	0
$547$	−35.0219	−1.49743	−0.748715	−	0.662893i	$-0.769329\pi$
$547$	−35.0219	−1.49743	−0.748715	+	0.662893i	$0.769329\pi$
$548$	−8.16382	−0.348741
$549$	0	0
$550$	2.06172	0.0879118
$551$	−5.62361	−0.239574
$552$	0	0
$553$	12.2721	0.521861
$554$	27.2272	1.15677
$555$	0	0
$556$	−15.7608	−0.668407
$557$	−35.0988	−1.48718	−0.743592	−	0.668634i	$-0.766879\pi$
$557$	−35.0988	−1.48718	−0.743592	+	0.668634i	$0.766879\pi$
$558$	0	0
$559$	−48.6990	−2.05975
$560$	14.0710	0.594609
$561$	0	0
$562$	5.77886	0.243767
$563$	9.73226	0.410166	0.205083	−	0.978745i	$-0.434254\pi$
$563$	9.73226	0.410166	0.205083	+	0.978745i	$0.434254\pi$
$564$	0	0
$565$	23.8474	1.00327
$566$	25.3591	1.06592
$567$	0	0
$568$	−19.4176	−0.814746
$569$	−18.1970	−0.762858	−0.381429	−	0.924398i	$-0.624568\pi$
$569$	−18.1970	−0.762858	−0.381429	+	0.924398i	$0.624568\pi$
$570$	0	0
$571$	−39.2610	−1.64302	−0.821511	−	0.570193i	$-0.806868\pi$
$571$	−39.2610	−1.64302	−0.821511	+	0.570193i	$0.806868\pi$
$572$	5.27864	0.220711
$573$	0	0
$574$	−15.6917	−0.654957
$575$	−16.3069	−0.680044
$576$	0	0
$577$	−2.76079	−0.114933	−0.0574667	−	0.998347i	$-0.518302\pi$
$577$	−2.76079	−0.114933	−0.0574667	+	0.998347i	$0.518302\pi$
$578$	−34.3379	−1.42827
$579$	0	0
$580$	4.93732	0.205011
$581$	−30.9303	−1.28320
$582$	0	0
$583$	−1.23215	−0.0510305
$584$	1.94531	0.0804975
$585$	0	0
$586$	7.82117	0.323089
$587$	0.670127	0.0276591	0.0138295	−	0.999904i	$-0.495598\pi$
$587$	0.670127	0.0276591	0.0138295	+	0.999904i	$0.495598\pi$
$588$	0	0
$589$	−0.849477	−0.0350021
$590$	−18.5773	−0.764817
$591$	0	0
$592$	10.1073	0.415406
$593$	32.6571	1.34107	0.670534	−	0.741879i	$-0.266065\pi$
$593$	32.6571	1.34107	0.670534	+	0.741879i	$0.266065\pi$
$594$	0	0
$595$	−52.3925	−2.14788
$596$	−17.9368	−0.734719
$597$	0	0
$598$	66.4276	2.71643
$599$	27.5631	1.12620	0.563100	−	0.826389i	$-0.309609\pi$
$599$	27.5631	1.12620	0.563100	+	0.826389i	$0.309609\pi$
$600$	0	0
$601$	−38.8364	−1.58417	−0.792085	−	0.610411i	$-0.791004\pi$
$601$	−38.8364	−1.58417	−0.792085	+	0.610411i	$0.791004\pi$
$602$	−33.6861	−1.37294
$603$	0	0
$604$	−10.8840	−0.442862
$605$	−1.77189	−0.0720375
$606$	0	0
$607$	−6.11550	−0.248220	−0.124110	−	0.992268i	$-0.539608\pi$
$607$	−6.11550	−0.248220	−0.124110	+	0.992268i	$0.539608\pi$
$608$	6.35884	0.257885
$609$	0	0
$610$	−11.3692	−0.460326
$611$	23.5833	0.954079
$612$	0	0
$613$	−3.78531	−0.152887	−0.0764436	−	0.997074i	$-0.524357\pi$
$613$	−3.78531	−0.152887	−0.0764436	+	0.997074i	$0.524357\pi$
$614$	−24.1566	−0.974883
$615$	0	0
$616$	13.1122	0.528304
$617$	13.9260	0.560638	0.280319	−	0.959907i	$-0.409560\pi$
$617$	13.9260	0.560638	0.280319	+	0.959907i	$0.409560\pi$
$618$	0	0
$619$	7.80956	0.313893	0.156946	−	0.987607i	$-0.449835\pi$
$619$	7.80956	0.313893	0.156946	+	0.987607i	$0.449835\pi$
$620$	0.745809	0.0299524
$621$	0	0
$622$	16.0960	0.645390
$623$	17.0242	0.682060
$624$	0	0
$625$	−12.2368	−0.489472
$626$	−15.3118	−0.611983
$627$	0	0
$628$	−5.21024	−0.207911
$629$	−37.6338	−1.50056
$630$	0	0
$631$	−25.9524	−1.03315	−0.516574	−	0.856242i	$-0.672793\pi$
$631$	−25.9524	−1.03315	−0.516574	+	0.856242i	$0.672793\pi$
$632$	8.83146	0.351297
$633$	0	0
$634$	−14.0268	−0.557077
$635$	31.6606	1.25641
$636$	0	0
$637$	−76.7322	−3.04024
$638$	−4.00055	−0.158383
$639$	0	0
$640$	1.72344	0.0681248
$641$	−27.3469	−1.08014	−0.540069	−	0.841621i	$-0.681602\pi$
$641$	−27.3469	−1.08014	−0.540069	+	0.841621i	$0.681602\pi$
$642$	0	0
$643$	−7.19648	−0.283801	−0.141901	−	0.989881i	$-0.545321\pi$
$643$	−7.19648	−0.283801	−0.141901	+	0.989881i	$0.545321\pi$
$644$	−28.8799	−1.13803
$645$	0	0
$646$	11.9588	0.470512
$647$	39.9282	1.56974	0.784871	−	0.619660i	$-0.212729\pi$
$647$	39.9282	1.56974	0.784871	+	0.619660i	$0.212729\pi$
$648$	0	0
$649$	−9.46081	−0.371369
$650$	−14.0993	−0.553019
$651$	0	0
$652$	−1.67243	−0.0654975
$653$	−11.3655	−0.444765	−0.222382	−	0.974960i	$-0.571383\pi$
$653$	−11.3655	−0.444765	−0.222382	+	0.974960i	$0.571383\pi$
$654$	0	0
$655$	24.1700	0.944401
$656$	−6.17135	−0.240951
$657$	0	0
$658$	16.3131	0.635950
$659$	−26.9737	−1.05075	−0.525373	−	0.850872i	$-0.676074\pi$
$659$	−26.9737	−1.05075	−0.525373	+	0.850872i	$0.676074\pi$
$660$	0	0
$661$	−11.1490	−0.433647	−0.216823	−	0.976211i	$-0.569570\pi$
$661$	−11.1490	−0.433647	−0.216823	+	0.976211i	$0.569570\pi$
$662$	25.8728	1.00558
$663$	0	0
$664$	−22.2587	−0.863804
$665$	11.7823	0.456898
$666$	0	0
$667$	31.6419	1.22518
$668$	−16.1224	−0.623793
$669$	0	0
$670$	6.06466	0.234298
$671$	−5.78996	−0.223519
$672$	0	0
$673$	−1.54484	−0.0595490	−0.0297745	−	0.999557i	$-0.509479\pi$
$673$	−1.54484	−0.0595490	−0.0297745	+	0.999557i	$0.509479\pi$
$674$	4.43182	0.170707
$675$	0	0
$676$	−26.0640	−1.00246
$677$	−20.0946	−0.772300	−0.386150	−	0.922436i	$-0.626195\pi$
$677$	−20.0946	−0.772300	−0.386150	+	0.922436i	$0.626195\pi$
$678$	0	0
$679$	−40.8052	−1.56596
$680$	−37.7038	−1.44587
$681$	0	0
$682$	−0.604305	−0.0231400
$683$	34.9740	1.33824	0.669121	−	0.743154i	$-0.266671\pi$
$683$	34.9740	1.33824	0.669121	+	0.743154i	$0.266671\pi$
$684$	0	0
$685$	−18.7403	−0.716028
$686$	−19.9644	−0.762245
$687$	0	0
$688$	−13.2484	−0.505089
$689$	8.42620	0.321013
$690$	0	0
$691$	−13.8881	−0.528327	−0.264163	−	0.964478i	$-0.585096\pi$
$691$	−13.8881	−0.528327	−0.264163	+	0.964478i	$0.585096\pi$
$692$	5.47950	0.208299
$693$	0	0
$694$	6.17240	0.234301
$695$	−36.1793	−1.37236
$696$	0	0
$697$	22.9786	0.870377
$698$	12.6835	0.480076
$699$	0	0
$700$	6.12976	0.231683
$701$	20.3201	0.767478	0.383739	−	0.923442i	$-0.374636\pi$
$701$	20.3201	0.767478	0.383739	+	0.923442i	$0.374636\pi$
$702$	0	0
$703$	8.46328	0.319199
$704$	8.24441	0.310723
$705$	0	0
$706$	2.51129	0.0945136
$707$	52.9093	1.98986
$708$	0	0
$709$	46.5111	1.74676	0.873381	−	0.487038i	$-0.161923\pi$
$709$	46.5111	1.74676	0.873381	+	0.487038i	$0.161923\pi$
$710$	−12.4124	−0.465830
$711$	0	0
$712$	12.2513	0.459136
$713$	4.77968	0.179000
$714$	0	0
$715$	12.1172	0.453159
$716$	3.52550	0.131754
$717$	0	0
$718$	1.85352	0.0691727
$719$	10.0384	0.374368	0.187184	−	0.982325i	$-0.440064\pi$
$719$	10.0384	0.374368	0.187184	+	0.982325i	$0.440064\pi$
$720$	0	0
$721$	15.6554	0.583036
$722$	18.3665	0.683530
$723$	0	0
$724$	−5.52236	−0.205237
$725$	−6.71600	−0.249426
$726$	0	0
$727$	16.1178	0.597777	0.298888	−	0.954288i	$-0.403384\pi$
$727$	16.1178	0.597777	0.298888	+	0.954288i	$0.403384\pi$
$728$	−89.6689	−3.32335
$729$	0	0
$730$	1.24351	0.0460244
$731$	49.3294	1.82451
$732$	0	0
$733$	5.81522	0.214790	0.107395	−	0.994216i	$-0.465749\pi$
$733$	5.81522	0.214790	0.107395	+	0.994216i	$0.465749\pi$
$734$	−8.54931	−0.315561
$735$	0	0
$736$	−35.7787	−1.31882
$737$	3.08853	0.113767
$738$	0	0
$739$	−13.3238	−0.490122	−0.245061	−	0.969508i	$-0.578808\pi$
$739$	−13.3238	−0.490122	−0.245061	+	0.969508i	$0.578808\pi$
$740$	−7.43044	−0.273149
$741$	0	0
$742$	5.82858	0.213974
$743$	41.4085	1.51913	0.759564	−	0.650432i	$-0.225412\pi$
$743$	41.4085	1.51913	0.759564	+	0.650432i	$0.225412\pi$
$744$	0	0
$745$	−41.1743	−1.50851
$746$	−5.56493	−0.203747
$747$	0	0
$748$	−5.34697	−0.195505
$749$	68.7826	2.51326
$750$	0	0
$751$	−30.0065	−1.09495	−0.547477	−	0.836821i	$-0.684412\pi$
$751$	−30.0065	−1.09495	−0.547477	+	0.836821i	$0.684412\pi$
$752$	6.41574	0.233958
$753$	0	0
$754$	27.3582	0.996327
$755$	−24.9844	−0.909276
$756$	0	0
$757$	−6.04343	−0.219652	−0.109826	−	0.993951i	$-0.535029\pi$
$757$	−6.04343	−0.219652	−0.109826	+	0.993951i	$0.535029\pi$
$758$	−20.6450	−0.749861
$759$	0	0
$760$	8.47901	0.307566
$761$	−11.4583	−0.415362	−0.207681	−	0.978197i	$-0.566592\pi$
$761$	−11.4583	−0.415362	−0.207681	+	0.978197i	$0.566592\pi$
$762$	0	0
$763$	72.3768	2.62022
$764$	−4.09641	−0.148203
$765$	0	0
$766$	−23.2360	−0.839552
$767$	64.6988	2.33614
$768$	0	0
$769$	0.296199	0.0106812	0.00534060	−	0.999986i	$-0.498300\pi$
$769$	0.296199	0.0106812	0.00534060	+	0.999986i	$0.498300\pi$
$770$	8.38175	0.302057
$771$	0	0
$772$	−11.9519	−0.430158
$773$	−46.3039	−1.66544	−0.832718	−	0.553698i	$-0.813216\pi$
$773$	−46.3039	−1.66544	−0.832718	+	0.553698i	$0.813216\pi$
$774$	0	0
$775$	−1.01449	−0.0364415
$776$	−29.3651	−1.05414
$777$	0	0
$778$	1.19519	0.0428497
$779$	−5.16755	−0.185147
$780$	0	0
$781$	−6.32123	−0.226191
$782$	−67.2875	−2.40620
$783$	0	0
$784$	−20.8747	−0.745524
$785$	−11.9602	−0.426879
$786$	0	0
$787$	2.71145	0.0966529	0.0483264	−	0.998832i	$-0.484611\pi$
$787$	2.71145	0.0966529	0.0483264	+	0.998832i	$0.484611\pi$
$788$	−18.2939	−0.651694
$789$	0	0
$790$	5.64538	0.200854
$791$	−57.4492	−2.04266
$792$	0	0
$793$	39.5953	1.40607
$794$	24.1321	0.856416
$795$	0	0
$796$	0.946033	0.0335312
$797$	−37.0383	−1.31196	−0.655981	−	0.754777i	$-0.727745\pi$
$797$	−37.0383	−1.31196	−0.655981	+	0.754777i	$0.727745\pi$
$798$	0	0
$799$	−23.8886	−0.845118
$800$	7.59404	0.268490
$801$	0	0
$802$	−25.4129	−0.897361
$803$	0.633278	0.0223479
$804$	0	0
$805$	−66.2945	−2.33657
$806$	4.13261	0.145565
$807$	0	0
$808$	38.0756	1.33950
$809$	34.8322	1.22464	0.612318	−	0.790611i	$-0.290237\pi$
$809$	34.8322	1.22464	0.612318	+	0.790611i	$0.290237\pi$
$810$	0	0
$811$	−26.2526	−0.921853	−0.460927	−	0.887438i	$-0.652483\pi$
$811$	−26.2526	−0.921853	−0.460927	+	0.887438i	$0.652483\pi$
$812$	−11.8942	−0.417404
$813$	0	0
$814$	6.02065	0.211024
$815$	−3.83911	−0.134478
$816$	0	0
$817$	−11.0935	−0.388111
$818$	22.5237	0.787522
$819$	0	0
$820$	4.53691	0.158436
$821$	−44.4048	−1.54974	−0.774869	−	0.632122i	$-0.782184\pi$
$821$	−44.4048	−1.54974	−0.774869	+	0.632122i	$0.782184\pi$
$822$	0	0
$823$	3.58877	0.125097	0.0625483	−	0.998042i	$-0.480077\pi$
$823$	3.58877	0.125097	0.0625483	+	0.998042i	$0.480077\pi$
$824$	11.2662	0.392477
$825$	0	0
$826$	44.7535	1.55717
$827$	−28.2328	−0.981750	−0.490875	−	0.871230i	$-0.663323\pi$
$827$	−28.2328	−0.981750	−0.490875	+	0.871230i	$0.663323\pi$
$828$	0	0
$829$	−41.0926	−1.42721	−0.713603	−	0.700551i	$-0.752938\pi$
$829$	−41.0926	−1.42721	−0.713603	+	0.700551i	$0.752938\pi$
$830$	−14.2285	−0.493879
$831$	0	0
$832$	−56.3803	−1.95464
$833$	77.7255	2.69303
$834$	0	0
$835$	−37.0093	−1.28076
$836$	1.20245	0.0415877
$837$	0	0
$838$	−9.39528	−0.324555
$839$	19.0304	0.657003	0.328502	−	0.944503i	$-0.393456\pi$
$839$	19.0304	0.657003	0.328502	+	0.944503i	$0.393456\pi$
$840$	0	0
$841$	−15.9683	−0.550631
$842$	−15.1694	−0.522773
$843$	0	0
$844$	9.75721	0.335857
$845$	−59.8306	−2.05823
$846$	0	0
$847$	4.26854	0.146669
$848$	2.29231	0.0787183
$849$	0	0
$850$	14.2818	0.489861
$851$	−47.6196	−1.63238
$852$	0	0
$853$	24.4239	0.836259	0.418130	−	0.908387i	$-0.362686\pi$
$853$	24.4239	0.836259	0.418130	+	0.908387i	$0.362686\pi$
$854$	27.3889	0.937228
$855$	0	0
$856$	49.4987	1.69183
$857$	24.7991	0.847120	0.423560	−	0.905868i	$-0.360780\pi$
$857$	24.7991	0.847120	0.423560	+	0.905868i	$0.360780\pi$
$858$	0	0
$859$	37.8583	1.29171	0.645854	−	0.763461i	$-0.276501\pi$
$859$	37.8583	1.29171	0.645854	+	0.763461i	$0.276501\pi$
$860$	9.73963	0.332119
$861$	0	0
$862$	−28.9888	−0.987363
$863$	1.41766	0.0482576	0.0241288	−	0.999709i	$-0.492319\pi$
$863$	1.41766	0.0482576	0.0241288	+	0.999709i	$0.492319\pi$
$864$	0	0
$865$	12.5783	0.427676
$866$	23.5898	0.801613
$867$	0	0
$868$	−1.79668	−0.0609833
$869$	2.87500	0.0975277
$870$	0	0
$871$	−21.1212	−0.715665
$872$	52.0852	1.76383
$873$	0	0
$874$	15.1320	0.511846
$875$	51.8879	1.75413
$876$	0	0
$877$	41.2542	1.39306	0.696528	−	0.717530i	$-0.254727\pi$
$877$	41.2542	1.39306	0.696528	+	0.717530i	$0.254727\pi$
$878$	−38.0701	−1.28480
$879$	0	0
$880$	3.29644	0.111123
$881$	33.6154	1.13253	0.566266	−	0.824222i	$-0.308387\pi$
$881$	33.6154	1.13253	0.566266	+	0.824222i	$0.308387\pi$
$882$	0	0
$883$	41.9365	1.41128	0.705638	−	0.708573i	$-0.250661\pi$
$883$	41.9365	1.41128	0.705638	+	0.708573i	$0.250661\pi$
$884$	36.5658	1.22984
$885$	0	0
$886$	34.4593	1.15768
$887$	−53.0794	−1.78223	−0.891117	−	0.453775i	$-0.850077\pi$
$887$	−53.0794	−1.78223	−0.891117	+	0.453775i	$0.850077\pi$
$888$	0	0
$889$	−76.2716	−2.55807
$890$	7.83145	0.262511
$891$	0	0
$892$	3.00147	0.100497
$893$	5.37219	0.179774
$894$	0	0
$895$	8.09288	0.270515
$896$	−4.15182	−0.138703
$897$	0	0
$898$	8.33170	0.278032
$899$	1.96851	0.0656535
$900$	0	0
$901$	−8.53528	−0.284351
$902$	−3.67612	−0.122401
$903$	0	0
$904$	−41.3427	−1.37504
$905$	−12.6767	−0.421388
$906$	0	0
$907$	−11.2223	−0.372631	−0.186315	−	0.982490i	$-0.559655\pi$
$907$	−11.2223	−0.372631	−0.186315	+	0.982490i	$0.559655\pi$
$908$	−1.97595	−0.0655741
$909$	0	0
$910$	−57.3195	−1.90012
$911$	−2.18473	−0.0723834	−0.0361917	−	0.999345i	$-0.511523\pi$
$911$	−2.18473	−0.0723834	−0.0361917	+	0.999345i	$0.511523\pi$
$912$	0	0
$913$	−7.24610	−0.239811
$914$	−33.3082	−1.10174
$915$	0	0
$916$	2.60631	0.0861150
$917$	−58.2264	−1.92281
$918$	0	0
$919$	−10.6838	−0.352425	−0.176212	−	0.984352i	$-0.556385\pi$
$919$	−10.6838	−0.352425	−0.176212	+	0.984352i	$0.556385\pi$
$920$	−47.7081	−1.57289
$921$	0	0
$922$	−32.0267	−1.05474
$923$	43.2284	1.42288
$924$	0	0
$925$	10.1073	0.332325
$926$	40.8742	1.34321
$927$	0	0
$928$	−14.7355	−0.483716
$929$	3.85075	0.126339	0.0631694	−	0.998003i	$-0.479879\pi$
$929$	3.85075	0.126339	0.0631694	+	0.998003i	$0.479879\pi$
$930$	0	0
$931$	−17.4793	−0.572861
$932$	−9.06752	−0.297017
$933$	0	0
$934$	−4.59943	−0.150498
$935$	−12.2741	−0.401406
$936$	0	0
$937$	−12.7941	−0.417965	−0.208983	−	0.977919i	$-0.567015\pi$
$937$	−12.7941	−0.417965	−0.208983	+	0.977919i	$0.567015\pi$
$938$	−14.6100	−0.477033
$939$	0	0
$940$	−4.71658	−0.153838
$941$	−7.66012	−0.249713	−0.124856	−	0.992175i	$-0.539847\pi$
$941$	−7.66012	−0.249713	−0.124856	+	0.992175i	$0.539847\pi$
$942$	0	0
$943$	29.0758	0.946839
$944$	17.6010	0.572865
$945$	0	0
$946$	−7.89171	−0.256582
$947$	−13.9676	−0.453886	−0.226943	−	0.973908i	$-0.572873\pi$
$947$	−13.9676	−0.453886	−0.226943	+	0.973908i	$0.572873\pi$
$948$	0	0
$949$	−4.33074	−0.140582
$950$	−3.21176	−0.104203
$951$	0	0
$952$	90.8297	2.94381
$953$	49.1368	1.59170	0.795849	−	0.605495i	$-0.207025\pi$
$953$	49.1368	1.59170	0.795849	+	0.605495i	$0.207025\pi$
$954$	0	0
$955$	−9.40342	−0.304288
$956$	−2.37929	−0.0769518
$957$	0	0
$958$	47.8844	1.54707
$959$	45.1459	1.45784
$960$	0	0
$961$	−30.7026	−0.990408
$962$	−41.1729	−1.32747
$963$	0	0
$964$	−1.70640	−0.0549595
$965$	−27.4359	−0.883191
$966$	0	0
$967$	18.1888	0.584912	0.292456	−	0.956279i	$-0.405527\pi$
$967$	18.1888	0.584912	0.292456	+	0.956279i	$0.405527\pi$
$968$	3.07181	0.0987318
$969$	0	0
$970$	−18.7712	−0.602706
$971$	12.8069	0.410994	0.205497	−	0.978658i	$-0.434119\pi$
$971$	12.8069	0.410994	0.205497	+	0.978658i	$0.434119\pi$
$972$	0	0
$973$	87.1573	2.79414
$974$	−18.9419	−0.606936
$975$	0	0
$976$	10.7717	0.344795
$977$	−11.1097	−0.355432	−0.177716	−	0.984082i	$-0.556871\pi$
$977$	−11.1097	−0.355432	−0.177716	+	0.984082i	$0.556871\pi$
$978$	0	0
$979$	3.98829	0.127466
$980$	15.3462	0.490216
$981$	0	0
$982$	8.13459	0.259585
$983$	6.45645	0.205929	0.102964	−	0.994685i	$-0.467167\pi$
$983$	6.45645	0.205929	0.102964	+	0.994685i	$0.467167\pi$
$984$	0	0
$985$	−41.9941	−1.33804
$986$	−27.7124	−0.882541
$987$	0	0
$988$	−8.22311	−0.261612
$989$	62.4186	1.98480
$990$	0	0
$991$	16.0001	0.508261	0.254130	−	0.967170i	$-0.418211\pi$
$991$	16.0001	0.508261	0.254130	+	0.967170i	$0.418211\pi$
$992$	−2.22587	−0.0706716
$993$	0	0
$994$	29.9020	0.948433
$995$	2.17164	0.0688457
$996$	0	0
$997$	−34.7397	−1.10022	−0.550109	−	0.835093i	$-0.685414\pi$
$997$	−34.7397	−1.10022	−0.550109	+	0.835093i	$0.685414\pi$
$998$	−41.9808	−1.32888
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2673.2.a.p.1.2	yes	6
3.2	odd	2		2673.2.a.j.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2673.2.a.j.1.5	✓	6	3.2	odd	2
2673.2.a.p.1.2	yes	6	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:	\( N \)	\(=\)	\( 2673 = 3^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2673.a (trivial)

\(f(q)\)	\(=\)	\(q-1.10820 q^{2} -0.771888 q^{4} -1.77189 q^{5} +4.26854 q^{7} +3.07181 q^{8} +O(q^{10})\)	\(q-1.10820 q^{2} -0.771888 q^{4} -1.77189 q^{5} +4.26854 q^{7} +3.07181 q^{8} +1.96361 q^{10} +1.00000 q^{11} -6.83861 q^{13} -4.73041 q^{14} -1.86041 q^{16} +6.92714 q^{17} -1.55781 q^{19} +1.36770 q^{20} -1.10820 q^{22} +8.76519 q^{23} -1.86041 q^{25} +7.57856 q^{26} -3.29483 q^{28} +3.60995 q^{29} +0.545302 q^{31} -4.08191 q^{32} -7.67667 q^{34} -7.56338 q^{35} -5.43281 q^{37} +1.72637 q^{38} -5.44291 q^{40} +3.31719 q^{41} +7.12119 q^{43} -0.771888 q^{44} -9.71361 q^{46} -3.44856 q^{47} +11.2204 q^{49} +2.06172 q^{50} +5.27864 q^{52} -1.23215 q^{53} -1.77189 q^{55} +13.1122 q^{56} -4.00055 q^{58} -9.46081 q^{59} -5.78996 q^{61} -0.604305 q^{62} +8.24441 q^{64} +12.1172 q^{65} +3.08853 q^{67} -5.34697 q^{68} +8.38175 q^{70} -6.32123 q^{71} +0.633278 q^{73} +6.02065 q^{74} +1.20245 q^{76} +4.26854 q^{77} +2.87500 q^{79} +3.29644 q^{80} -3.67612 q^{82} -7.24610 q^{83} -12.2741 q^{85} -7.89171 q^{86} +3.07181 q^{88} +3.98829 q^{89} -29.1909 q^{91} -6.76575 q^{92} +3.82170 q^{94} +2.76026 q^{95} -9.55952 q^{97} -12.4345 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	\( 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 8 q^{10} + 6 q^{11} - 8 q^{13} - 4 q^{14} + 8 q^{16} + 2 q^{17} - 4 q^{19} + 40 q^{20} + 2 q^{22} + 10 q^{23} + 8 q^{25} + 2 q^{26} - 12 q^{28} + 6 q^{29} - 8 q^{31} + 4 q^{32} - 10 q^{34} - 10 q^{35} + 2 q^{37} + 30 q^{38} + 18 q^{40} - 4 q^{41} + 2 q^{43} + 8 q^{44} + 4 q^{46} + 28 q^{47} + 6 q^{49} + 16 q^{50} - 12 q^{52} + 24 q^{53} + 2 q^{55} - 6 q^{56} + 6 q^{58} - 8 q^{59} + 2 q^{61} - 10 q^{62} + 18 q^{64} - 4 q^{65} + 12 q^{67} - 14 q^{68} - 10 q^{70} + 30 q^{71} + 16 q^{73} + 78 q^{74} + 2 q^{76} - 2 q^{77} - 12 q^{79} + 58 q^{80} - 16 q^{82} - 16 q^{85} - 24 q^{86} + 6 q^{88} - 6 q^{89} + 6 q^{91} + 32 q^{92} + 10 q^{94} + 6 q^{95} - 18 q^{97} - 6 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 8 * q^10 + 6 * q^11 - 8 * q^13 - 4 * q^14 + 8 * q^16 + 2 * q^17 - 4 * q^19 + 40 * q^20 + 2 * q^22 + 10 * q^23 + 8 * q^25 + 2 * q^26 - 12 * q^28 + 6 * q^29 - 8 * q^31 + 4 * q^32 - 10 * q^34 - 10 * q^35 + 2 * q^37 + 30 * q^38 + 18 * q^40 - 4 * q^41 + 2 * q^43 + 8 * q^44 + 4 * q^46 + 28 * q^47 + 6 * q^49 + 16 * q^50 - 12 * q^52 + 24 * q^53 + 2 * q^55 - 6 * q^56 + 6 * q^58 - 8 * q^59 + 2 * q^61 - 10 * q^62 + 18 * q^64 - 4 * q^65 + 12 * q^67 - 14 * q^68 - 10 * q^70 + 30 * q^71 + 16 * q^73 + 78 * q^74 + 2 * q^76 - 2 * q^77 - 12 * q^79 + 58 * q^80 - 16 * q^82 - 16 * q^85 - 24 * q^86 + 6 * q^88 - 6 * q^89 + 6 * q^91 + 32 * q^92 + 10 * q^94 + 6 * q^95 - 18 * q^97 - 6 * q^98

Introduction

L-functions

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