LMFDB - Embedded newform 5103.2.a.i.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5103,2,Mod(1,5103)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5103, 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("5103.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5103 = 3^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5103.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:	$40.7476601515$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	5103.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.268830 q^{2} -1.92773 q^{4} +1.98449 q^{5} +1.00000 q^{7} -1.05589 q^{8} +O(q^{10})$	\(q+0.268830 q^{2} -1.92773 q^{4} +1.98449 q^{5} +1.00000 q^{7} -1.05589 q^{8} +0.533491 q^{10} -4.25324 q^{11} -0.109391 q^{13} +0.268830 q^{14} +3.57160 q^{16} +1.06748 q^{17} -6.42242 q^{19} -3.82556 q^{20} -1.14340 q^{22} +0.0715490 q^{23} -1.06180 q^{25} -0.0294076 q^{26} -1.92773 q^{28} +7.27006 q^{29} -2.69167 q^{31} +3.07194 q^{32} +0.286971 q^{34} +1.98449 q^{35} +9.47793 q^{37} -1.72654 q^{38} -2.09541 q^{40} -3.10599 q^{41} +6.55295 q^{43} +8.19910 q^{44} +0.0192345 q^{46} -11.1287 q^{47} +1.00000 q^{49} -0.285445 q^{50} +0.210876 q^{52} +6.12296 q^{53} -8.44050 q^{55} -1.05589 q^{56} +1.95441 q^{58} +5.37482 q^{59} +4.70539 q^{61} -0.723602 q^{62} -6.31738 q^{64} -0.217085 q^{65} +5.23154 q^{67} -2.05781 q^{68} +0.533491 q^{70} +11.5301 q^{71} +13.8468 q^{73} +2.54796 q^{74} +12.3807 q^{76} -4.25324 q^{77} -8.28311 q^{79} +7.08781 q^{80} -0.834985 q^{82} -16.2719 q^{83} +2.11840 q^{85} +1.76163 q^{86} +4.49097 q^{88} +10.5989 q^{89} -0.109391 q^{91} -0.137927 q^{92} -2.99174 q^{94} -12.7452 q^{95} +1.76383 q^{97} +0.268830 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 9 q^{2} + 27 q^{4} + 12 q^{5} + 27 q^{7} + 27 q^{8}+O(q^{10}) $ 27 * q + 9 * q^2 + 27 * q^4 + 12 * q^5 + 27 * q^7 + 27 * q^8	$ 27 q + 9 q^{2} + 27 q^{4} + 12 q^{5} + 27 q^{7} + 27 q^{8} + 24 q^{11} + 9 q^{14} + 27 q^{16} + 30 q^{17} + 30 q^{20} + 39 q^{23} + 27 q^{25} + 9 q^{26} + 27 q^{28} + 39 q^{29} + 63 q^{32} + 12 q^{35} + 9 q^{38} + 42 q^{41} + 42 q^{44} + 27 q^{47} + 27 q^{49} + 36 q^{50} + 66 q^{53} + 27 q^{56} + 18 q^{59} + 36 q^{62} + 27 q^{64} + 69 q^{65} + 21 q^{68} + 72 q^{71} + 54 q^{74} + 24 q^{77} + 21 q^{80} + 39 q^{83} + 27 q^{86} + 42 q^{89} + 75 q^{92} + 78 q^{95} + 9 q^{98}+O(q^{100}) $ 27 * q + 9 * q^2 + 27 * q^4 + 12 * q^5 + 27 * q^7 + 27 * q^8 + 24 * q^11 + 9 * q^14 + 27 * q^16 + 30 * q^17 + 30 * q^20 + 39 * q^23 + 27 * q^25 + 9 * q^26 + 27 * q^28 + 39 * q^29 + 63 * q^32 + 12 * q^35 + 9 * q^38 + 42 * q^41 + 42 * q^44 + 27 * q^47 + 27 * q^49 + 36 * q^50 + 66 * q^53 + 27 * q^56 + 18 * q^59 + 36 * q^62 + 27 * q^64 + 69 * q^65 + 21 * q^68 + 72 * q^71 + 54 * q^74 + 24 * q^77 + 21 * q^80 + 39 * q^83 + 27 * q^86 + 42 * q^89 + 75 * q^92 + 78 * q^95 + 9 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.268830	0.190092	0.0950459	−	0.995473i	$-0.469700\pi$
$2$	0.268830	0.190092	0.0950459	+	0.995473i	$0.469700\pi$
$3$	0	0
$3$	0	0
$4$	−1.92773	−0.963865
$5$	1.98449	0.887490	0.443745	−	0.896153i	$-0.353650\pi$
$5$	1.98449	0.887490	0.443745	+	0.896153i	$0.353650\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.05589	−0.373315
$9$	0	0
$10$	0.533491	0.168705
$11$	−4.25324	−1.28240	−0.641200	−	0.767374i	$-0.721563\pi$
$11$	−4.25324	−1.28240	−0.641200	+	0.767374i	$0.721563\pi$
$12$	0	0
$13$	−0.109391	−0.0303395	−0.0151698	−	0.999885i	$-0.504829\pi$
$13$	−0.109391	−0.0303395	−0.0151698	+	0.999885i	$0.504829\pi$
$14$	0.268830	0.0718479
$15$	0	0
$16$	3.57160	0.892901
$17$	1.06748	0.258902	0.129451	−	0.991586i	$-0.458678\pi$
$17$	1.06748	0.258902	0.129451	+	0.991586i	$0.458678\pi$
$18$	0	0
$19$	−6.42242	−1.47341	−0.736703	−	0.676217i	$-0.763618\pi$
$19$	−6.42242	−1.47341	−0.736703	+	0.676217i	$0.763618\pi$
$20$	−3.82556	−0.855421
$21$	0	0
$22$	−1.14340	−0.243774
$23$	0.0715490	0.0149190	0.00745950	−	0.999972i	$-0.497626\pi$
$23$	0.0715490	0.0149190	0.00745950	+	0.999972i	$0.497626\pi$
$24$	0	0
$25$	−1.06180	−0.212361
$26$	−0.0294076	−0.00576730
$27$	0	0
$28$	−1.92773	−0.364307
$29$	7.27006	1.35002	0.675008	−	0.737810i	$-0.264140\pi$
$29$	7.27006	1.35002	0.675008	+	0.737810i	$0.264140\pi$
$30$	0	0
$31$	−2.69167	−0.483438	−0.241719	−	0.970346i	$-0.577711\pi$
$31$	−2.69167	−0.483438	−0.241719	+	0.970346i	$0.577711\pi$
$32$	3.07194	0.543048
$33$	0	0
$34$	0.286971	0.0492152
$35$	1.98449	0.335440
$36$	0	0
$37$	9.47793	1.55816	0.779081	−	0.626923i	$-0.215686\pi$
$37$	9.47793	1.55816	0.779081	+	0.626923i	$0.215686\pi$
$38$	−1.72654	−0.280082
$39$	0	0
$40$	−2.09541	−0.331313
$41$	−3.10599	−0.485075	−0.242537	−	0.970142i	$-0.577980\pi$
$41$	−3.10599	−0.485075	−0.242537	+	0.970142i	$0.577980\pi$
$42$	0	0
$43$	6.55295	0.999316	0.499658	−	0.866223i	$-0.333459\pi$
$43$	6.55295	0.999316	0.499658	+	0.866223i	$0.333459\pi$
$44$	8.19910	1.23606
$45$	0	0
$46$	0.0192345	0.00283598
$47$	−11.1287	−1.62329	−0.811645	−	0.584151i	$-0.801428\pi$
$47$	−11.1287	−1.62329	−0.811645	+	0.584151i	$0.801428\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.285445	−0.0403680
$51$	0	0
$52$	0.210876	0.0292432
$53$	6.12296	0.841053	0.420526	−	0.907280i	$-0.361845\pi$
$53$	6.12296	0.841053	0.420526	+	0.907280i	$0.361845\pi$
$54$	0	0
$55$	−8.44050	−1.13812
$56$	−1.05589	−0.141100
$57$	0	0
$58$	1.95441	0.256627
$59$	5.37482	0.699742	0.349871	−	0.936798i	$-0.386225\pi$
$59$	5.37482	0.699742	0.349871	+	0.936798i	$0.386225\pi$
$60$	0	0
$61$	4.70539	0.602464	0.301232	−	0.953551i	$-0.402602\pi$
$61$	4.70539	0.602464	0.301232	+	0.953551i	$0.402602\pi$
$62$	−0.723602	−0.0918975
$63$	0	0
$64$	−6.31738	−0.789672
$65$	−0.217085	−0.0269260
$66$	0	0
$67$	5.23154	0.639134	0.319567	−	0.947564i	$-0.396463\pi$
$67$	5.23154	0.639134	0.319567	+	0.947564i	$0.396463\pi$
$68$	−2.05781	−0.249547
$69$	0	0
$70$	0.533491	0.0637644
$71$	11.5301	1.36837	0.684184	−	0.729310i	$-0.260159\pi$
$71$	11.5301	1.36837	0.684184	+	0.729310i	$0.260159\pi$
$72$	0	0
$73$	13.8468	1.62065	0.810325	−	0.585981i	$-0.199290\pi$
$73$	13.8468	1.62065	0.810325	+	0.585981i	$0.199290\pi$
$74$	2.54796	0.296194
$75$	0	0
$76$	12.3807	1.42016
$77$	−4.25324	−0.484701
$78$	0	0
$79$	−8.28311	−0.931923	−0.465961	−	0.884805i	$-0.654291\pi$
$79$	−8.28311	−0.931923	−0.465961	+	0.884805i	$0.654291\pi$
$80$	7.08781	0.792441
$81$	0	0
$82$	−0.834985	−0.0922087
$83$	−16.2719	−1.78608	−0.893038	−	0.449981i	$-0.851431\pi$
$83$	−16.2719	−1.78608	−0.893038	+	0.449981i	$0.851431\pi$
$84$	0	0
$85$	2.11840	0.229773
$86$	1.76163	0.189962
$87$	0	0
$88$	4.49097	0.478739
$89$	10.5989	1.12348	0.561741	−	0.827313i	$-0.310132\pi$
$89$	10.5989	1.12348	0.561741	+	0.827313i	$0.310132\pi$
$90$	0	0
$91$	−0.109391	−0.0114673
$92$	−0.137927	−0.0143799
$93$	0	0
$94$	−2.99174	−0.308574
$95$	−12.7452	−1.30763
$96$	0	0
$97$	1.76383	0.179089	0.0895447	−	0.995983i	$-0.471459\pi$
$97$	1.76383	0.179089	0.0895447	+	0.995983i	$0.471459\pi$
$98$	0.268830	0.0271560
$99$	0	0
$100$	2.04687	0.204687
$101$	7.25947	0.722344	0.361172	−	0.932499i	$-0.382377\pi$
$101$	7.25947	0.722344	0.361172	+	0.932499i	$0.382377\pi$
$102$	0	0
$103$	16.2067	1.59689	0.798446	−	0.602066i	$-0.205656\pi$
$103$	16.2067	1.59689	0.798446	+	0.602066i	$0.205656\pi$
$104$	0.115505	0.0113262
$105$	0	0
$106$	1.64604	0.159877
$107$	7.58438	0.733210	0.366605	−	0.930377i	$-0.380520\pi$
$107$	7.58438	0.733210	0.366605	+	0.930377i	$0.380520\pi$
$108$	0	0
$109$	−4.68262	−0.448513	−0.224257	−	0.974530i	$-0.571995\pi$
$109$	−4.68262	−0.448513	−0.224257	+	0.974530i	$0.571995\pi$
$110$	−2.26906	−0.216347
$111$	0	0
$112$	3.57160	0.337485
$113$	−3.37177	−0.317190	−0.158595	−	0.987344i	$-0.550696\pi$
$113$	−3.37177	−0.317190	−0.158595	+	0.987344i	$0.550696\pi$
$114$	0	0
$115$	0.141988	0.0132405
$116$	−14.0147	−1.30123
$117$	0	0
$118$	1.44491	0.133015
$119$	1.06748	0.0978558
$120$	0	0
$121$	7.09003	0.644548
$122$	1.26495	0.114523
$123$	0	0
$124$	5.18881	0.465969
$125$	−12.0296	−1.07596
$126$	0	0
$127$	−9.01219	−0.799702	−0.399851	−	0.916580i	$-0.630938\pi$
$127$	−9.01219	−0.799702	−0.399851	+	0.916580i	$0.630938\pi$
$128$	−7.84219	−0.693158
$129$	0	0
$130$	−0.0583590	−0.00511842
$131$	7.38926	0.645602	0.322801	−	0.946467i	$-0.395375\pi$
$131$	7.38926	0.645602	0.322801	+	0.946467i	$0.395375\pi$
$132$	0	0
$133$	−6.42242	−0.556895
$134$	1.40640	0.121494
$135$	0	0
$136$	−1.12715	−0.0966519
$137$	16.2358	1.38712	0.693558	−	0.720400i	$-0.256042\pi$
$137$	16.2358	1.38712	0.693558	+	0.720400i	$0.256042\pi$
$138$	0	0
$139$	−7.34736	−0.623195	−0.311597	−	0.950214i	$-0.600864\pi$
$139$	−7.34736	−0.623195	−0.311597	+	0.950214i	$0.600864\pi$
$140$	−3.82556	−0.323319
$141$	0	0
$142$	3.09963	0.260115
$143$	0.465265	0.0389074
$144$	0	0
$145$	14.4274	1.19813
$146$	3.72245	0.308072
$147$	0	0
$148$	−18.2709	−1.50186
$149$	16.4309	1.34607	0.673035	−	0.739611i	$-0.264990\pi$
$149$	16.4309	1.34607	0.673035	+	0.739611i	$0.264990\pi$
$150$	0	0
$151$	−21.2936	−1.73285	−0.866424	−	0.499309i	$-0.833587\pi$
$151$	−21.2936	−1.73285	−0.866424	+	0.499309i	$0.833587\pi$
$152$	6.78140	0.550044
$153$	0	0
$154$	−1.14340	−0.0921378
$155$	−5.34158	−0.429046
$156$	0	0
$157$	−4.84201	−0.386435	−0.193217	−	0.981156i	$-0.561892\pi$
$157$	−4.84201	−0.386435	−0.193217	+	0.981156i	$0.561892\pi$
$158$	−2.22675	−0.177151
$159$	0	0
$160$	6.09624	0.481950
$161$	0.0715490	0.00563885
$162$	0	0
$163$	17.1250	1.34133	0.670665	−	0.741760i	$-0.266009\pi$
$163$	17.1250	1.34133	0.670665	+	0.741760i	$0.266009\pi$
$164$	5.98752	0.467546
$165$	0	0
$166$	−4.37439	−0.339518
$167$	−2.54016	−0.196563	−0.0982817	−	0.995159i	$-0.531335\pi$
$167$	−2.54016	−0.196563	−0.0982817	+	0.995159i	$0.531335\pi$
$168$	0	0
$169$	−12.9880	−0.999080
$170$	0.569491	0.0436780
$171$	0	0
$172$	−12.6323	−0.963206
$173$	−11.7993	−0.897086	−0.448543	−	0.893761i	$-0.648057\pi$
$173$	−11.7993	−0.897086	−0.448543	+	0.893761i	$0.648057\pi$
$174$	0	0
$175$	−1.06180	−0.0802648
$176$	−15.1909	−1.14506
$177$	0	0
$178$	2.84931	0.213565
$179$	23.1128	1.72753	0.863766	−	0.503893i	$-0.168099\pi$
$179$	23.1128	1.72753	0.863766	+	0.503893i	$0.168099\pi$
$180$	0	0
$181$	13.2932	0.988076	0.494038	−	0.869440i	$-0.335520\pi$
$181$	13.2932	0.988076	0.494038	+	0.869440i	$0.335520\pi$
$182$	−0.0294076	−0.00217983
$183$	0	0
$184$	−0.0755481	−0.00556948
$185$	18.8088	1.38285
$186$	0	0
$187$	−4.54025	−0.332016
$188$	21.4532	1.56463
$189$	0	0
$190$	−3.42631	−0.248570
$191$	5.65034	0.408844	0.204422	−	0.978883i	$-0.434468\pi$
$191$	5.65034	0.408844	0.204422	+	0.978883i	$0.434468\pi$
$192$	0	0
$193$	−12.0026	−0.863964	−0.431982	−	0.901882i	$-0.642186\pi$
$193$	−12.0026	−0.863964	−0.431982	+	0.901882i	$0.642186\pi$
$194$	0.474170	0.0340434
$195$	0	0
$196$	−1.92773	−0.137695
$197$	21.1585	1.50748	0.753742	−	0.657171i	$-0.228247\pi$
$197$	21.1585	1.50748	0.753742	+	0.657171i	$0.228247\pi$
$198$	0	0
$199$	4.92519	0.349137	0.174569	−	0.984645i	$-0.444147\pi$
$199$	4.92519	0.349137	0.174569	+	0.984645i	$0.444147\pi$
$200$	1.12115	0.0792773
$201$	0	0
$202$	1.95157	0.137312
$203$	7.27006	0.510258
$204$	0	0
$205$	−6.16381	−0.430499
$206$	4.35685	0.303556
$207$	0	0
$208$	−0.390700	−0.0270902
$209$	27.3161	1.88949
$210$	0	0
$211$	−19.5738	−1.34752	−0.673758	−	0.738952i	$-0.735321\pi$
$211$	−19.5738	−1.34752	−0.673758	+	0.738952i	$0.735321\pi$
$212$	−11.8034	−0.810662
$213$	0	0
$214$	2.03891	0.139377
$215$	13.0043	0.886883
$216$	0	0
$217$	−2.69167	−0.182722
$218$	−1.25883	−0.0852587
$219$	0	0
$220$	16.2710	1.09699
$221$	−0.116772	−0.00785497
$222$	0	0
$223$	−2.57522	−0.172450	−0.0862248	−	0.996276i	$-0.527480\pi$
$223$	−2.57522	−0.172450	−0.0862248	+	0.996276i	$0.527480\pi$
$224$	3.07194	0.205253
$225$	0	0
$226$	−0.906435	−0.0602951
$227$	18.1727	1.20617	0.603084	−	0.797678i	$-0.293938\pi$
$227$	18.1727	1.20617	0.603084	+	0.797678i	$0.293938\pi$
$228$	0	0
$229$	23.7111	1.56687	0.783436	−	0.621473i	$-0.213465\pi$
$229$	23.7111	1.56687	0.783436	+	0.621473i	$0.213465\pi$
$230$	0.0381707	0.00251690
$231$	0	0
$232$	−7.67641	−0.503981
$233$	9.63100	0.630948	0.315474	−	0.948934i	$-0.397837\pi$
$233$	9.63100	0.630948	0.315474	+	0.948934i	$0.397837\pi$
$234$	0	0
$235$	−22.0848	−1.44065
$236$	−10.3612	−0.674456
$237$	0	0
$238$	0.286971	0.0186016
$239$	−10.0032	−0.647050	−0.323525	−	0.946220i	$-0.604868\pi$
$239$	−10.0032	−0.647050	−0.323525	+	0.946220i	$0.604868\pi$
$240$	0	0
$241$	1.69924	0.109458	0.0547289	−	0.998501i	$-0.482571\pi$
$241$	1.69924	0.109458	0.0547289	+	0.998501i	$0.482571\pi$
$242$	1.90602	0.122523
$243$	0	0
$244$	−9.07073	−0.580694
$245$	1.98449	0.126784
$246$	0	0
$247$	0.702554	0.0447024
$248$	2.84211	0.180474
$249$	0	0
$250$	−3.23392	−0.204531
$251$	21.2530	1.34148	0.670740	−	0.741692i	$-0.265977\pi$
$251$	21.2530	1.34148	0.670740	+	0.741692i	$0.265977\pi$
$252$	0	0
$253$	−0.304315	−0.0191321
$254$	−2.42275	−0.152017
$255$	0	0
$256$	10.5265	0.657908
$257$	4.75606	0.296675	0.148338	−	0.988937i	$-0.452608\pi$
$257$	4.75606	0.296675	0.148338	+	0.988937i	$0.452608\pi$
$258$	0	0
$259$	9.47793	0.588930
$260$	0.418481	0.0259531
$261$	0	0
$262$	1.98646	0.122724
$263$	0.0455692	0.00280992	0.00140496	−	0.999999i	$-0.499553\pi$
$263$	0.0455692	0.00280992	0.00140496	+	0.999999i	$0.499553\pi$
$264$	0	0
$265$	12.1509	0.746426
$266$	−1.72654	−0.105861
$267$	0	0
$268$	−10.0850	−0.616039
$269$	−3.67995	−0.224371	−0.112185	−	0.993687i	$-0.535785\pi$
$269$	−3.67995	−0.224371	−0.112185	+	0.993687i	$0.535785\pi$
$270$	0	0
$271$	21.3658	1.29788	0.648939	−	0.760841i	$-0.275213\pi$
$271$	21.3658	1.29788	0.648939	+	0.760841i	$0.275213\pi$
$272$	3.81262	0.231174
$273$	0	0
$274$	4.36467	0.263680
$275$	4.51610	0.272331
$276$	0	0
$277$	14.8421	0.891773	0.445887	−	0.895089i	$-0.352888\pi$
$277$	14.8421	0.891773	0.445887	+	0.895089i	$0.352888\pi$
$278$	−1.97519	−0.118464
$279$	0	0
$280$	−2.09541	−0.125225
$281$	−6.11526	−0.364806	−0.182403	−	0.983224i	$-0.558388\pi$
$281$	−6.11526	−0.364806	−0.182403	+	0.983224i	$0.558388\pi$
$282$	0	0
$283$	−16.1655	−0.960937	−0.480468	−	0.877012i	$-0.659533\pi$
$283$	−16.1655	−0.960937	−0.480468	+	0.877012i	$0.659533\pi$
$284$	−22.2269	−1.31892
$285$	0	0
$286$	0.125077	0.00739598
$287$	−3.10599	−0.183341
$288$	0	0
$289$	−15.8605	−0.932970
$290$	3.87851	0.227754
$291$	0	0
$292$	−26.6930	−1.56209
$293$	−8.28460	−0.483991	−0.241996	−	0.970277i	$-0.577802\pi$
$293$	−8.28460	−0.483991	−0.241996	+	0.970277i	$0.577802\pi$
$294$	0	0
$295$	10.6663	0.621014
$296$	−10.0077	−0.581685
$297$	0	0
$298$	4.41712	0.255877
$299$	−0.00782680	−0.000452635 0
$300$	0	0
$301$	6.55295	0.377706
$302$	−5.72436	−0.329400
$303$	0	0
$304$	−22.9384	−1.31561
$305$	9.33780	0.534681
$306$	0	0
$307$	−25.3127	−1.44467	−0.722336	−	0.691542i	$-0.756932\pi$
$307$	−25.3127	−1.44467	−0.722336	+	0.691542i	$0.756932\pi$
$308$	8.19910	0.467187
$309$	0	0
$310$	−1.43598	−0.0815582
$311$	33.2635	1.88620	0.943100	−	0.332509i	$-0.107895\pi$
$311$	33.2635	1.88620	0.943100	+	0.332509i	$0.107895\pi$
$312$	0	0
$313$	−13.4885	−0.762418	−0.381209	−	0.924489i	$-0.624492\pi$
$313$	−13.4885	−0.762418	−0.381209	+	0.924489i	$0.624492\pi$
$314$	−1.30168	−0.0734581
$315$	0	0
$316$	15.9676	0.898248
$317$	30.4736	1.71157	0.855785	−	0.517332i	$-0.173075\pi$
$317$	30.4736	1.71157	0.855785	+	0.517332i	$0.173075\pi$
$318$	0	0
$319$	−30.9213	−1.73126
$320$	−12.5368	−0.700826
$321$	0	0
$322$	0.0192345	0.00107190
$323$	−6.85581	−0.381468
$324$	0	0
$325$	0.116151	0.00644292
$326$	4.60371	0.254976
$327$	0	0
$328$	3.27960	0.181085
$329$	−11.1287	−0.613546
$330$	0	0
$331$	−4.01992	−0.220955	−0.110477	−	0.993879i	$-0.535238\pi$
$331$	−4.01992	−0.220955	−0.110477	+	0.993879i	$0.535238\pi$
$332$	31.3679	1.72154
$333$	0	0
$334$	−0.682872	−0.0373651
$335$	10.3819	0.567225
$336$	0	0
$337$	−11.8565	−0.645863	−0.322931	−	0.946422i	$-0.604668\pi$
$337$	−11.8565	−0.645863	−0.322931	+	0.946422i	$0.604668\pi$
$338$	−3.49158	−0.189917
$339$	0	0
$340$	−4.08371	−0.221470
$341$	11.4483	0.619960
$342$	0	0
$343$	1.00000	0.0539949
$344$	−6.91922	−0.373059
$345$	0	0
$346$	−3.17202	−0.170529
$347$	10.2831	0.552024	0.276012	−	0.961154i	$-0.410987\pi$
$347$	10.2831	0.552024	0.276012	+	0.961154i	$0.410987\pi$
$348$	0	0
$349$	4.27567	0.228871	0.114436	−	0.993431i	$-0.463494\pi$
$349$	4.27567	0.228871	0.114436	+	0.993431i	$0.463494\pi$
$350$	−0.285445	−0.0152577
$351$	0	0
$352$	−13.0657	−0.696404
$353$	−26.3971	−1.40498	−0.702489	−	0.711695i	$-0.747928\pi$
$353$	−26.3971	−1.40498	−0.702489	+	0.711695i	$0.747928\pi$
$354$	0	0
$355$	22.8813	1.21441
$356$	−20.4318	−1.08289
$357$	0	0
$358$	6.21343	0.328390
$359$	21.6348	1.14184	0.570921	−	0.821005i	$-0.306586\pi$
$359$	21.6348	1.14184	0.570921	+	0.821005i	$0.306586\pi$
$360$	0	0
$361$	22.2475	1.17092
$362$	3.57362	0.187825
$363$	0	0
$364$	0.210876	0.0110529
$365$	27.4789	1.43831
$366$	0	0
$367$	−2.25742	−0.117836	−0.0589182	−	0.998263i	$-0.518765\pi$
$367$	−2.25742	−0.117836	−0.0589182	+	0.998263i	$0.518765\pi$
$368$	0.255545	0.0133212
$369$	0	0
$370$	5.05639	0.262869
$371$	6.12296	0.317888
$372$	0	0
$373$	10.3806	0.537487	0.268743	−	0.963212i	$-0.413392\pi$
$373$	10.3806	0.537487	0.268743	+	0.963212i	$0.413392\pi$
$374$	−1.22056	−0.0631135
$375$	0	0
$376$	11.7507	0.605998
$377$	−0.795277	−0.0409589
$378$	0	0
$379$	10.1524	0.521496	0.260748	−	0.965407i	$-0.416031\pi$
$379$	10.1524	0.521496	0.260748	+	0.965407i	$0.416031\pi$
$380$	24.5694	1.26038
$381$	0	0
$382$	1.51898	0.0777180
$383$	7.61267	0.388989	0.194495	−	0.980904i	$-0.437693\pi$
$383$	7.61267	0.388989	0.194495	+	0.980904i	$0.437693\pi$
$384$	0	0
$385$	−8.44050	−0.430168
$386$	−3.22666	−0.164232
$387$	0	0
$388$	−3.40018	−0.172618
$389$	4.50304	0.228313	0.114157	−	0.993463i	$-0.463583\pi$
$389$	4.50304	0.228313	0.114157	+	0.993463i	$0.463583\pi$
$390$	0	0
$391$	0.0763771	0.00386256
$392$	−1.05589	−0.0533307
$393$	0	0
$394$	5.68806	0.286560
$395$	−16.4377	−0.827072
$396$	0	0
$397$	−1.41325	−0.0709289	−0.0354644	−	0.999371i	$-0.511291\pi$
$397$	−1.41325	−0.0709289	−0.0354644	+	0.999371i	$0.511291\pi$
$398$	1.32404	0.0663681
$399$	0	0
$400$	−3.79234	−0.189617
$401$	27.3096	1.36377	0.681887	−	0.731457i	$-0.261159\pi$
$401$	27.3096	1.36377	0.681887	+	0.731457i	$0.261159\pi$
$402$	0	0
$403$	0.294443	0.0146673
$404$	−13.9943	−0.696242
$405$	0	0
$406$	1.95441	0.0969959
$407$	−40.3119	−1.99819
$408$	0	0
$409$	−18.9077	−0.934924	−0.467462	−	0.884013i	$-0.654831\pi$
$409$	−18.9077	−0.934924	−0.467462	+	0.884013i	$0.654831\pi$
$410$	−1.65702	−0.0818343
$411$	0	0
$412$	−31.2421	−1.53919
$413$	5.37482	0.264477
$414$	0	0
$415$	−32.2915	−1.58513
$416$	−0.336042	−0.0164758
$417$	0	0
$418$	7.34340	0.359177
$419$	19.9807	0.976119	0.488060	−	0.872810i	$-0.337705\pi$
$419$	19.9807	0.976119	0.488060	+	0.872810i	$0.337705\pi$
$420$	0	0
$421$	25.3479	1.23538	0.617692	−	0.786420i	$-0.288068\pi$
$421$	25.3479	1.23538	0.617692	+	0.786420i	$0.288068\pi$
$422$	−5.26203	−0.256152
$423$	0	0
$424$	−6.46519	−0.313977
$425$	−1.13345	−0.0549806
$426$	0	0
$427$	4.70539	0.227710
$428$	−14.6206	−0.706715
$429$	0	0
$430$	3.49594	0.168589
$431$	9.73667	0.468999	0.234499	−	0.972116i	$-0.424655\pi$
$431$	9.73667	0.468999	0.234499	+	0.972116i	$0.424655\pi$
$432$	0	0
$433$	−10.2203	−0.491155	−0.245578	−	0.969377i	$-0.578978\pi$
$433$	−10.2203	−0.491155	−0.245578	+	0.969377i	$0.578978\pi$
$434$	−0.723602	−0.0347340
$435$	0	0
$436$	9.02682	0.432306
$437$	−0.459518	−0.0219817
$438$	0	0
$439$	33.7275	1.60973	0.804863	−	0.593461i	$-0.202239\pi$
$439$	33.7275	1.60973	0.804863	+	0.593461i	$0.202239\pi$
$440$	8.91227	0.424876
$441$	0	0
$442$	−0.0313920	−0.00149317
$443$	21.1342	1.00412	0.502059	−	0.864833i	$-0.332576\pi$
$443$	21.1342	1.00412	0.502059	+	0.864833i	$0.332576\pi$
$444$	0	0
$445$	21.0334	0.997080
$446$	−0.692298	−0.0327812
$447$	0	0
$448$	−6.31738	−0.298468
$449$	−18.4182	−0.869208	−0.434604	−	0.900622i	$-0.643112\pi$
$449$	−18.4182	−0.869208	−0.434604	+	0.900622i	$0.643112\pi$
$450$	0	0
$451$	13.2105	0.622059
$452$	6.49987	0.305728
$453$	0	0
$454$	4.88539	0.229283
$455$	−0.217085	−0.0101771
$456$	0	0
$457$	38.8038	1.81517	0.907584	−	0.419872i	$-0.137925\pi$
$457$	38.8038	1.81517	0.907584	+	0.419872i	$0.137925\pi$
$458$	6.37426	0.297849
$459$	0	0
$460$	−0.273715	−0.0127620
$461$	−36.5473	−1.70218	−0.851088	−	0.525024i	$-0.824056\pi$
$461$	−36.5473	−1.70218	−0.851088	+	0.525024i	$0.824056\pi$
$462$	0	0
$463$	−17.3513	−0.806382	−0.403191	−	0.915116i	$-0.632099\pi$
$463$	−17.3513	−0.806382	−0.403191	+	0.915116i	$0.632099\pi$
$464$	25.9658	1.20543
$465$	0	0
$466$	2.58911	0.119938
$467$	−23.2819	−1.07736	−0.538680	−	0.842511i	$-0.681077\pi$
$467$	−23.2819	−1.07736	−0.538680	+	0.842511i	$0.681077\pi$
$468$	0	0
$469$	5.23154	0.241570
$470$	−5.93707	−0.273857
$471$	0	0
$472$	−5.67523	−0.261224
$473$	−27.8713	−1.28152
$474$	0	0
$475$	6.81935	0.312893
$476$	−2.05781	−0.0943198
$477$	0	0
$478$	−2.68915	−0.122999
$479$	−0.779751	−0.0356278	−0.0178139	−	0.999841i	$-0.505671\pi$
$479$	−0.779751	−0.0356278	−0.0178139	+	0.999841i	$0.505671\pi$
$480$	0	0
$481$	−1.03680	−0.0472739
$482$	0.456808	0.0208070
$483$	0	0
$484$	−13.6677	−0.621258
$485$	3.50029	0.158940
$486$	0	0
$487$	35.3660	1.60259	0.801293	−	0.598272i	$-0.204146\pi$
$487$	35.3660	1.60259	0.801293	+	0.598272i	$0.204146\pi$
$488$	−4.96839	−0.224909
$489$	0	0
$490$	0.533491	0.0241007
$491$	−40.1437	−1.81166	−0.905830	−	0.423642i	$-0.860752\pi$
$491$	−40.1437	−1.81166	−0.905830	+	0.423642i	$0.860752\pi$
$492$	0	0
$493$	7.76065	0.349522
$494$	0.188868	0.00849757
$495$	0	0
$496$	−9.61357	−0.431662
$497$	11.5301	0.517194
$498$	0	0
$499$	5.85631	0.262164	0.131082	−	0.991372i	$-0.458155\pi$
$499$	5.85631	0.262164	0.131082	+	0.991372i	$0.458155\pi$
$500$	23.1898	1.03708
$501$	0	0
$502$	5.71346	0.255004
$503$	8.23611	0.367230	0.183615	−	0.982998i	$-0.441220\pi$
$503$	8.23611	0.367230	0.183615	+	0.982998i	$0.441220\pi$
$504$	0	0
$505$	14.4063	0.641073
$506$	−0.0818091	−0.00363686
$507$	0	0
$508$	17.3731	0.770805
$509$	21.7206	0.962748	0.481374	−	0.876515i	$-0.340138\pi$
$509$	21.7206	0.962748	0.481374	+	0.876515i	$0.340138\pi$
$510$	0	0
$511$	13.8468	0.612548
$512$	18.5142	0.818221
$513$	0	0
$514$	1.27857	0.0563955
$515$	32.1620	1.41723
$516$	0	0
$517$	47.3331	2.08171
$518$	2.54796	0.111951
$519$	0	0
$520$	0.229218	0.0100519
$521$	7.58951	0.332502	0.166251	−	0.986083i	$-0.446834\pi$
$521$	7.58951	0.332502	0.166251	+	0.986083i	$0.446834\pi$
$522$	0	0
$523$	−16.6859	−0.729623	−0.364811	−	0.931081i	$-0.618866\pi$
$523$	−16.6859	−0.729623	−0.364811	+	0.931081i	$0.618866\pi$
$524$	−14.2445	−0.622274
$525$	0	0
$526$	0.0122504	0.000534142 0
$527$	−2.87330	−0.125163
$528$	0	0
$529$	−22.9949	−0.999777
$530$	3.26654	0.141890
$531$	0	0
$532$	12.3807	0.536771
$533$	0.339767	0.0147169
$534$	0	0
$535$	15.0511	0.650717
$536$	−5.52394	−0.238598
$537$	0	0
$538$	−0.989284	−0.0426510
$539$	−4.25324	−0.183200
$540$	0	0
$541$	17.7745	0.764187	0.382093	−	0.924124i	$-0.375203\pi$
$541$	17.7745	0.764187	0.382093	+	0.924124i	$0.375203\pi$
$542$	5.74377	0.246716
$543$	0	0
$544$	3.27924	0.140596
$545$	−9.29260	−0.398051
$546$	0	0
$547$	2.60865	0.111538	0.0557690	−	0.998444i	$-0.482239\pi$
$547$	2.60865	0.111538	0.0557690	+	0.998444i	$0.482239\pi$
$548$	−31.2982	−1.33699
$549$	0	0
$550$	1.21407	0.0517679
$551$	−46.6914	−1.98912
$552$	0	0
$553$	−8.28311	−0.352234
$554$	3.99000	0.169519
$555$	0	0
$556$	14.1637	0.600676
$557$	20.4485	0.866429	0.433215	−	0.901291i	$-0.357379\pi$
$557$	20.4485	0.866429	0.433215	+	0.901291i	$0.357379\pi$
$558$	0	0
$559$	−0.716832	−0.0303188
$560$	7.08781	0.299515
$561$	0	0
$562$	−1.64397	−0.0693466
$563$	24.2667	1.02272	0.511359	−	0.859367i	$-0.329142\pi$
$563$	24.2667	1.02272	0.511359	+	0.859367i	$0.329142\pi$
$564$	0	0
$565$	−6.69124	−0.281503
$566$	−4.34577	−0.182666
$567$	0	0
$568$	−12.1745	−0.510832
$569$	3.27147	0.137147	0.0685735	−	0.997646i	$-0.478155\pi$
$569$	3.27147	0.137147	0.0685735	+	0.997646i	$0.478155\pi$
$570$	0	0
$571$	−39.9363	−1.67128	−0.835641	−	0.549276i	$-0.814904\pi$
$571$	−39.9363	−1.67128	−0.835641	+	0.549276i	$0.814904\pi$
$572$	−0.896905	−0.0375015
$573$	0	0
$574$	−0.834985	−0.0348516
$575$	−0.0759709	−0.00316821
$576$	0	0
$577$	−28.5036	−1.18662	−0.593311	−	0.804973i	$-0.702179\pi$
$577$	−28.5036	−1.18662	−0.593311	+	0.804973i	$0.702179\pi$
$578$	−4.26378	−0.177350
$579$	0	0
$580$	−27.8120	−1.15483
$581$	−16.2719	−0.675073
$582$	0	0
$583$	−26.0424	−1.07857
$584$	−14.6208	−0.605013
$585$	0	0
$586$	−2.22715	−0.0920028
$587$	−37.4354	−1.54512	−0.772562	−	0.634940i	$-0.781025\pi$
$587$	−37.4354	−1.54512	−0.772562	+	0.634940i	$0.781025\pi$
$588$	0	0
$589$	17.2870	0.712300
$590$	2.86742	0.118050
$591$	0	0
$592$	33.8514	1.39128
$593$	−1.95349	−0.0802204	−0.0401102	−	0.999195i	$-0.512771\pi$
$593$	−1.95349	−0.0802204	−0.0401102	+	0.999195i	$0.512771\pi$
$594$	0	0
$595$	2.11840	0.0868461
$596$	−31.6743	−1.29743
$597$	0	0
$598$	−0.00210408	−8.60423e−5 0
$599$	38.2229	1.56175	0.780873	−	0.624690i	$-0.214775\pi$
$599$	38.2229	1.56175	0.780873	+	0.624690i	$0.214775\pi$
$600$	0	0
$601$	16.0872	0.656210	0.328105	−	0.944641i	$-0.393590\pi$
$601$	16.0872	0.656210	0.328105	+	0.944641i	$0.393590\pi$
$602$	1.76163	0.0717988
$603$	0	0
$604$	41.0483	1.67023
$605$	14.0701	0.572031
$606$	0	0
$607$	2.80440	0.113827	0.0569135	−	0.998379i	$-0.481874\pi$
$607$	2.80440	0.113827	0.0569135	+	0.998379i	$0.481874\pi$
$608$	−19.7293	−0.800130
$609$	0	0
$610$	2.51029	0.101638
$611$	1.21738	0.0492499
$612$	0	0
$613$	−24.6455	−0.995424	−0.497712	−	0.867342i	$-0.665826\pi$
$613$	−24.6455	−0.995424	−0.497712	+	0.867342i	$0.665826\pi$
$614$	−6.80482	−0.274620
$615$	0	0
$616$	4.49097	0.180946
$617$	−12.9429	−0.521061	−0.260531	−	0.965466i	$-0.583897\pi$
$617$	−12.9429	−0.521061	−0.260531	+	0.965466i	$0.583897\pi$
$618$	0	0
$619$	−8.70764	−0.349990	−0.174995	−	0.984569i	$-0.555991\pi$
$619$	−8.70764	−0.349990	−0.174995	+	0.984569i	$0.555991\pi$
$620$	10.2971	0.413543
$621$	0	0
$622$	8.94224	0.358551
$623$	10.5989	0.424637
$624$	0	0
$625$	−18.5636	−0.742542
$626$	−3.62613	−0.144929
$627$	0	0
$628$	9.33410	0.372471
$629$	10.1175	0.403411
$630$	0	0
$631$	14.8276	0.590277	0.295139	−	0.955454i	$-0.404634\pi$
$631$	14.8276	0.590277	0.295139	+	0.955454i	$0.404634\pi$
$632$	8.74608	0.347900
$633$	0	0
$634$	8.19224	0.325355
$635$	−17.8846	−0.709728
$636$	0	0
$637$	−0.109391	−0.00433422
$638$	−8.31258	−0.329098
$639$	0	0
$640$	−15.5627	−0.615171
$641$	−13.2604	−0.523753	−0.261876	−	0.965101i	$-0.584341\pi$
$641$	−13.2604	−0.523753	−0.261876	+	0.965101i	$0.584341\pi$
$642$	0	0
$643$	−33.9031	−1.33701	−0.668504	−	0.743709i	$-0.733065\pi$
$643$	−33.9031	−1.33701	−0.668504	+	0.743709i	$0.733065\pi$
$644$	−0.137927	−0.00543509
$645$	0	0
$646$	−1.84305	−0.0725139
$647$	15.4959	0.609207	0.304604	−	0.952479i	$-0.401476\pi$
$647$	15.4959	0.609207	0.304604	+	0.952479i	$0.401476\pi$
$648$	0	0
$649$	−22.8604	−0.897348
$650$	0.0312250	0.00122475
$651$	0	0
$652$	−33.0123	−1.29286
$653$	−18.1551	−0.710463	−0.355232	−	0.934778i	$-0.615598\pi$
$653$	−18.1551	−0.710463	−0.355232	+	0.934778i	$0.615598\pi$
$654$	0	0
$655$	14.6639	0.572966
$656$	−11.0934	−0.433124
$657$	0	0
$658$	−2.99174	−0.116630
$659$	−12.5102	−0.487330	−0.243665	−	0.969859i	$-0.578350\pi$
$659$	−12.5102	−0.487330	−0.243665	+	0.969859i	$0.578350\pi$
$660$	0	0
$661$	−24.0089	−0.933837	−0.466918	−	0.884300i	$-0.654636\pi$
$661$	−24.0089	−0.933837	−0.466918	+	0.884300i	$0.654636\pi$
$662$	−1.08068	−0.0420017
$663$	0	0
$664$	17.1814	0.666768
$665$	−12.7452	−0.494239
$666$	0	0
$667$	0.520165	0.0201409
$668$	4.89674	0.189461
$669$	0	0
$670$	2.79098	0.107825
$671$	−20.0132	−0.772599
$672$	0	0
$673$	−33.4131	−1.28798	−0.643990	−	0.765034i	$-0.722722\pi$
$673$	−33.4131	−1.28798	−0.643990	+	0.765034i	$0.722722\pi$
$674$	−3.18738	−0.122773
$675$	0	0
$676$	25.0374	0.962978
$677$	−21.5096	−0.826680	−0.413340	−	0.910577i	$-0.635638\pi$
$677$	−21.5096	−0.826680	−0.413340	+	0.910577i	$0.635638\pi$
$678$	0	0
$679$	1.76383	0.0676894
$680$	−2.23681	−0.0857777
$681$	0	0
$682$	3.07765	0.117849
$683$	−42.4410	−1.62396	−0.811979	−	0.583687i	$-0.801610\pi$
$683$	−42.4410	−1.62396	−0.811979	+	0.583687i	$0.801610\pi$
$684$	0	0
$685$	32.2197	1.23105
$686$	0.268830	0.0102640
$687$	0	0
$688$	23.4046	0.892290
$689$	−0.669795	−0.0255172
$690$	0	0
$691$	−33.5025	−1.27449	−0.637247	−	0.770659i	$-0.719927\pi$
$691$	−33.5025	−1.27449	−0.637247	+	0.770659i	$0.719927\pi$
$692$	22.7459	0.864669
$693$	0	0
$694$	2.76440	0.104935
$695$	−14.5808	−0.553080
$696$	0	0
$697$	−3.31559	−0.125587
$698$	1.14943	0.0435066
$699$	0	0
$700$	2.04687	0.0773644
$701$	−4.77624	−0.180396	−0.0901980	−	0.995924i	$-0.528750\pi$
$701$	−4.77624	−0.180396	−0.0901980	+	0.995924i	$0.528750\pi$
$702$	0	0
$703$	−60.8713	−2.29580
$704$	26.8693	1.01268
$705$	0	0
$706$	−7.09635	−0.267075
$707$	7.25947	0.273020
$708$	0	0
$709$	3.75143	0.140888	0.0704439	−	0.997516i	$-0.477558\pi$
$709$	3.75143	0.140888	0.0704439	+	0.997516i	$0.477558\pi$
$710$	6.15119	0.230850
$711$	0	0
$712$	−11.1913	−0.419413
$713$	−0.192586	−0.00721240
$714$	0	0
$715$	0.923313	0.0345299
$716$	−44.5553	−1.66511
$717$	0	0
$718$	5.81609	0.217055
$719$	32.8054	1.22344	0.611718	−	0.791076i	$-0.290479\pi$
$719$	32.8054	1.22344	0.611718	+	0.791076i	$0.290479\pi$
$720$	0	0
$721$	16.2067	0.603569
$722$	5.98081	0.222583
$723$	0	0
$724$	−25.6257	−0.952372
$725$	−7.71937	−0.286690
$726$	0	0
$727$	−3.66946	−0.136093	−0.0680463	−	0.997682i	$-0.521677\pi$
$727$	−3.66946	−0.136093	−0.0680463	+	0.997682i	$0.521677\pi$
$728$	0.115505	0.00428090
$729$	0	0
$730$	7.38717	0.273411
$731$	6.99515	0.258725
$732$	0	0
$733$	−27.0739	−0.999997	−0.499998	−	0.866026i	$-0.666666\pi$
$733$	−27.0739	−0.999997	−0.499998	+	0.866026i	$0.666666\pi$
$734$	−0.606864	−0.0223998
$735$	0	0
$736$	0.219794	0.00810173
$737$	−22.2510	−0.819625
$738$	0	0
$739$	19.1679	0.705101	0.352551	−	0.935793i	$-0.385314\pi$
$739$	19.1679	0.705101	0.352551	+	0.935793i	$0.385314\pi$
$740$	−36.2584	−1.33288
$741$	0	0
$742$	1.64604	0.0604279
$743$	3.66732	0.134541	0.0672705	−	0.997735i	$-0.478571\pi$
$743$	3.66732	0.134541	0.0672705	+	0.997735i	$0.478571\pi$
$744$	0	0
$745$	32.6069	1.19462
$746$	2.79062	0.102172
$747$	0	0
$748$	8.75237	0.320018
$749$	7.58438	0.277127
$750$	0	0
$751$	−32.3105	−1.17903	−0.589514	−	0.807758i	$-0.700681\pi$
$751$	−32.3105	−1.17903	−0.589514	+	0.807758i	$0.700681\pi$
$752$	−39.7474	−1.44944
$753$	0	0
$754$	−0.213795	−0.00778594
$755$	−42.2569	−1.53789
$756$	0	0
$757$	−22.9786	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$757$	−22.9786	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$758$	2.72929	0.0991322
$759$	0	0
$760$	13.4576	0.488159
$761$	38.9742	1.41281	0.706407	−	0.707806i	$-0.250315\pi$
$761$	38.9742	1.41281	0.706407	+	0.707806i	$0.250315\pi$
$762$	0	0
$763$	−4.68262	−0.169522
$764$	−10.8923	−0.394071
$765$	0	0
$766$	2.04652	0.0739436
$767$	−0.587955	−0.0212298
$768$	0	0
$769$	−4.00488	−0.144420	−0.0722098	−	0.997389i	$-0.523005\pi$
$769$	−4.00488	−0.144420	−0.0722098	+	0.997389i	$0.523005\pi$
$770$	−2.26906	−0.0817714
$771$	0	0
$772$	23.1377	0.832745
$773$	−8.99583	−0.323558	−0.161779	−	0.986827i	$-0.551723\pi$
$773$	−8.99583	−0.323558	−0.161779	+	0.986827i	$0.551723\pi$
$774$	0	0
$775$	2.85802	0.102663
$776$	−1.86241	−0.0668567
$777$	0	0
$778$	1.21056	0.0434005
$779$	19.9480	0.714711
$780$	0	0
$781$	−49.0401	−1.75479
$782$	0.0205325	0.000734241 0
$783$	0	0
$784$	3.57160	0.127557
$785$	−9.60893	−0.342957
$786$	0	0
$787$	6.14098	0.218902	0.109451	−	0.993992i	$-0.465091\pi$
$787$	6.14098	0.218902	0.109451	+	0.993992i	$0.465091\pi$
$788$	−40.7880	−1.45301
$789$	0	0
$790$	−4.41896	−0.157220
$791$	−3.37177	−0.119886
$792$	0	0
$793$	−0.514727	−0.0182785
$794$	−0.379924	−0.0134830
$795$	0	0
$796$	−9.49443	−0.336521
$797$	42.8217	1.51682	0.758412	−	0.651776i	$-0.225976\pi$
$797$	42.8217	1.51682	0.758412	+	0.651776i	$0.225976\pi$
$798$	0	0
$799$	−11.8797	−0.420273
$800$	−3.26180	−0.115322
$801$	0	0
$802$	7.34164	0.259242
$803$	−58.8939	−2.07832
$804$	0	0
$805$	0.141988	0.00500442
$806$	0.0791554	0.00278813
$807$	0	0
$808$	−7.66522	−0.269662
$809$	−51.9664	−1.82704	−0.913521	−	0.406791i	$-0.866648\pi$
$809$	−51.9664	−1.82704	−0.913521	+	0.406791i	$0.866648\pi$
$810$	0	0
$811$	0.845745	0.0296981	0.0148491	−	0.999890i	$-0.495273\pi$
$811$	0.845745	0.0296981	0.0148491	+	0.999890i	$0.495273\pi$
$812$	−14.0147	−0.491820
$813$	0	0
$814$	−10.8371	−0.379839
$815$	33.9843	1.19042
$816$	0	0
$817$	−42.0858	−1.47240
$818$	−5.08295	−0.177721
$819$	0	0
$820$	11.8822	0.414943
$821$	−7.01109	−0.244689	−0.122344	−	0.992488i	$-0.539041\pi$
$821$	−7.01109	−0.244689	−0.122344	+	0.992488i	$0.539041\pi$
$822$	0	0
$823$	−4.80802	−0.167597	−0.0837984	−	0.996483i	$-0.526705\pi$
$823$	−4.80802	−0.167597	−0.0837984	+	0.996483i	$0.526705\pi$
$824$	−17.1125	−0.596144
$825$	0	0
$826$	1.44491	0.0502750
$827$	−41.2756	−1.43529	−0.717646	−	0.696408i	$-0.754780\pi$
$827$	−41.2756	−1.43529	−0.717646	+	0.696408i	$0.754780\pi$
$828$	0	0
$829$	−28.8000	−1.00027	−0.500133	−	0.865948i	$-0.666716\pi$
$829$	−28.8000	−1.00027	−0.500133	+	0.865948i	$0.666716\pi$
$830$	−8.68093	−0.301319
$831$	0	0
$832$	0.691063	0.0239583
$833$	1.06748	0.0369860
$834$	0	0
$835$	−5.04092	−0.174448
$836$	−52.6581	−1.82122
$837$	0	0
$838$	5.37141	0.185552
$839$	33.3431	1.15113	0.575566	−	0.817755i	$-0.304782\pi$
$839$	33.3431	1.15113	0.575566	+	0.817755i	$0.304782\pi$
$840$	0	0
$841$	23.8538	0.822543
$842$	6.81430	0.234836
$843$	0	0
$844$	37.7330	1.29882
$845$	−25.7746	−0.886674
$846$	0	0
$847$	7.09003	0.243616
$848$	21.8688	0.750977
$849$	0	0
$850$	−0.304707	−0.0104514
$851$	0.678136	0.0232462
$852$	0	0
$853$	42.1899	1.44455	0.722277	−	0.691604i	$-0.243096\pi$
$853$	42.1899	1.44455	0.722277	+	0.691604i	$0.243096\pi$
$854$	1.26495	0.0432858
$855$	0	0
$856$	−8.00829	−0.273718
$857$	29.7978	1.01787	0.508937	−	0.860804i	$-0.330039\pi$
$857$	29.7978	1.01787	0.508937	+	0.860804i	$0.330039\pi$
$858$	0	0
$859$	8.99932	0.307053	0.153526	−	0.988145i	$-0.450937\pi$
$859$	8.99932	0.307053	0.153526	+	0.988145i	$0.450937\pi$
$860$	−25.0687	−0.854836
$861$	0	0
$862$	2.61751	0.0891529
$863$	7.45910	0.253911	0.126955	−	0.991908i	$-0.459479\pi$
$863$	7.45910	0.253911	0.126955	+	0.991908i	$0.459479\pi$
$864$	0	0
$865$	−23.4156	−0.796155
$866$	−2.74752	−0.0933646
$867$	0	0
$868$	5.18881	0.176120
$869$	35.2300	1.19510
$870$	0	0
$871$	−0.572282	−0.0193910
$872$	4.94434	0.167437
$873$	0	0
$874$	−0.123532	−0.00417854
$875$	−12.0296	−0.406674
$876$	0	0
$877$	−28.5129	−0.962812	−0.481406	−	0.876498i	$-0.659874\pi$
$877$	−28.5129	−0.962812	−0.481406	+	0.876498i	$0.659874\pi$
$878$	9.06698	0.305996
$879$	0	0
$880$	−30.1461	−1.01623
$881$	−38.9446	−1.31208	−0.656038	−	0.754728i	$-0.727769\pi$
$881$	−38.9446	−1.31208	−0.656038	+	0.754728i	$0.727769\pi$
$882$	0	0
$883$	20.3440	0.684631	0.342315	−	0.939585i	$-0.388789\pi$
$883$	20.3440	0.684631	0.342315	+	0.939585i	$0.388789\pi$
$884$	0.225106	0.00757113
$885$	0	0
$886$	5.68153	0.190875
$887$	2.30968	0.0775514	0.0387757	−	0.999248i	$-0.487654\pi$
$887$	2.30968	0.0775514	0.0387757	+	0.999248i	$0.487654\pi$
$888$	0	0
$889$	−9.01219	−0.302259
$890$	5.65443	0.189537
$891$	0	0
$892$	4.96433	0.166218
$893$	71.4733	2.39176
$894$	0	0
$895$	45.8671	1.53317
$896$	−7.84219	−0.261989
$897$	0	0
$898$	−4.95137	−0.165229
$899$	−19.5686	−0.652649
$900$	0	0
$901$	6.53614	0.217750
$902$	3.55139	0.118248
$903$	0	0
$904$	3.56023	0.118412
$905$	26.3802	0.876908
$906$	0	0
$907$	16.2959	0.541095	0.270548	−	0.962707i	$-0.412795\pi$
$907$	16.2959	0.541095	0.270548	+	0.962707i	$0.412795\pi$
$908$	−35.0322	−1.16258
$909$	0	0
$910$	−0.0583590	−0.00193458
$911$	−14.8774	−0.492909	−0.246455	−	0.969154i	$-0.579266\pi$
$911$	−14.8774	−0.492909	−0.246455	+	0.969154i	$0.579266\pi$
$912$	0	0
$913$	69.2084	2.29046
$914$	10.4317	0.345048
$915$	0	0
$916$	−45.7085	−1.51025
$917$	7.38926	0.244015
$918$	0	0
$919$	−25.7942	−0.850871	−0.425435	−	0.904989i	$-0.639879\pi$
$919$	−25.7942	−0.850871	−0.425435	+	0.904989i	$0.639879\pi$
$920$	−0.149924	−0.00494286
$921$	0	0
$922$	−9.82501	−0.323570
$923$	−1.26128	−0.0415156
$924$	0	0
$925$	−10.0637	−0.330892
$926$	−4.66455	−0.153287
$927$	0	0
$928$	22.3332	0.733123
$929$	−57.1521	−1.87510	−0.937550	−	0.347851i	$-0.886911\pi$
$929$	−57.1521	−1.87510	−0.937550	+	0.347851i	$0.886911\pi$
$930$	0	0
$931$	−6.42242	−0.210486
$932$	−18.5660	−0.608149
$933$	0	0
$934$	−6.25889	−0.204797
$935$	−9.01007	−0.294661
$936$	0	0
$937$	−22.7694	−0.743845	−0.371922	−	0.928264i	$-0.621301\pi$
$937$	−22.7694	−0.743845	−0.371922	+	0.928264i	$0.621301\pi$
$938$	1.40640	0.0459205
$939$	0	0
$940$	42.5736	1.38860
$941$	5.93953	0.193623	0.0968115	−	0.995303i	$-0.469136\pi$
$941$	5.93953	0.193623	0.0968115	+	0.995303i	$0.469136\pi$
$942$	0	0
$943$	−0.222231	−0.00723682
$944$	19.1967	0.624800
$945$	0	0
$946$	−7.49264	−0.243607
$947$	8.11730	0.263777	0.131888	−	0.991265i	$-0.457896\pi$
$947$	8.11730	0.263777	0.131888	+	0.991265i	$0.457896\pi$
$948$	0	0
$949$	−1.51472	−0.0491698
$950$	1.83325	0.0594784
$951$	0	0
$952$	−1.12715	−0.0365310
$953$	14.4797	0.469044	0.234522	−	0.972111i	$-0.424648\pi$
$953$	14.4797	0.469044	0.234522	+	0.972111i	$0.424648\pi$
$954$	0	0
$955$	11.2130	0.362846
$956$	19.2834	0.623669
$957$	0	0
$958$	−0.209621	−0.00677254
$959$	16.2358	0.524281
$960$	0	0
$961$	−23.7549	−0.766288
$962$	−0.278723	−0.00898638
$963$	0	0
$964$	−3.27568	−0.105503
$965$	−23.8190	−0.766760
$966$	0	0
$967$	29.6967	0.954981	0.477491	−	0.878637i	$-0.341546\pi$
$967$	29.6967	0.954981	0.477491	+	0.878637i	$0.341546\pi$
$968$	−7.48632	−0.240619
$969$	0	0
$970$	0.940985	0.0302132
$971$	40.4986	1.29966	0.649831	−	0.760079i	$-0.274840\pi$
$971$	40.4986	1.29966	0.649831	+	0.760079i	$0.274840\pi$
$972$	0	0
$973$	−7.34736	−0.235546
$974$	9.50746	0.304639
$975$	0	0
$976$	16.8058	0.537941
$977$	24.0412	0.769145	0.384573	−	0.923095i	$-0.374349\pi$
$977$	24.0412	0.769145	0.384573	+	0.923095i	$0.374349\pi$
$978$	0	0
$979$	−45.0797	−1.44075
$980$	−3.82556	−0.122203
$981$	0	0
$982$	−10.7918	−0.344382
$983$	39.8861	1.27217	0.636085	−	0.771619i	$-0.280553\pi$
$983$	39.8861	1.27217	0.636085	+	0.771619i	$0.280553\pi$
$984$	0	0
$985$	41.9889	1.33788
$986$	2.08630	0.0664413
$987$	0	0
$988$	−1.35433	−0.0430871
$989$	0.468857	0.0149088
$990$	0	0
$991$	−41.0288	−1.30332	−0.651661	−	0.758510i	$-0.725928\pi$
$991$	−41.0288	−1.30332	−0.651661	+	0.758510i	$0.725928\pi$
$992$	−8.26865	−0.262530
$993$	0	0
$994$	3.09963	0.0983144
$995$	9.77398	0.309856
$996$	0	0
$997$	−18.0838	−0.572721	−0.286360	−	0.958122i	$-0.592445\pi$
$997$	−18.0838	−0.572721	−0.286360	+	0.958122i	$0.592445\pi$
$998$	1.57435	0.0498353
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5103.2.a.i.1.14		27
3.2	odd	2		5103.2.a.f.1.14		27
27.2	odd	18		567.2.v.b.64.5		54
27.13	even	9		189.2.v.a.169.5	yes	54
27.14	odd	18		567.2.v.b.505.5		54
27.25	even	9		189.2.v.a.85.5	✓	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.a.85.5	✓	54	27.25	even	9
189.2.v.a.169.5	yes	54	27.13	even	9
567.2.v.b.64.5		54	27.2	odd	18
567.2.v.b.505.5		54	27.14	odd	18
5103.2.a.f.1.14		27	3.2	odd	2
5103.2.a.i.1.14		27	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:	\( N \)	\(=\)	\( 5103 = 3^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5103.a (trivial)

\(f(q)\)	\(=\)	\(q+0.268830 q^{2} -1.92773 q^{4} +1.98449 q^{5} +1.00000 q^{7} -1.05589 q^{8} +O(q^{10})\)	\(q+0.268830 q^{2} -1.92773 q^{4} +1.98449 q^{5} +1.00000 q^{7} -1.05589 q^{8} +0.533491 q^{10} -4.25324 q^{11} -0.109391 q^{13} +0.268830 q^{14} +3.57160 q^{16} +1.06748 q^{17} -6.42242 q^{19} -3.82556 q^{20} -1.14340 q^{22} +0.0715490 q^{23} -1.06180 q^{25} -0.0294076 q^{26} -1.92773 q^{28} +7.27006 q^{29} -2.69167 q^{31} +3.07194 q^{32} +0.286971 q^{34} +1.98449 q^{35} +9.47793 q^{37} -1.72654 q^{38} -2.09541 q^{40} -3.10599 q^{41} +6.55295 q^{43} +8.19910 q^{44} +0.0192345 q^{46} -11.1287 q^{47} +1.00000 q^{49} -0.285445 q^{50} +0.210876 q^{52} +6.12296 q^{53} -8.44050 q^{55} -1.05589 q^{56} +1.95441 q^{58} +5.37482 q^{59} +4.70539 q^{61} -0.723602 q^{62} -6.31738 q^{64} -0.217085 q^{65} +5.23154 q^{67} -2.05781 q^{68} +0.533491 q^{70} +11.5301 q^{71} +13.8468 q^{73} +2.54796 q^{74} +12.3807 q^{76} -4.25324 q^{77} -8.28311 q^{79} +7.08781 q^{80} -0.834985 q^{82} -16.2719 q^{83} +2.11840 q^{85} +1.76163 q^{86} +4.49097 q^{88} +10.5989 q^{89} -0.109391 q^{91} -0.137927 q^{92} -2.99174 q^{94} -12.7452 q^{95} +1.76383 q^{97} +0.268830 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 9 q^{2} + 27 q^{4} + 12 q^{5} + 27 q^{7} + 27 q^{8}+O(q^{10}) \) 27 * q + 9 * q^2 + 27 * q^4 + 12 * q^5 + 27 * q^7 + 27 * q^8	\( 27 q + 9 q^{2} + 27 q^{4} + 12 q^{5} + 27 q^{7} + 27 q^{8} + 24 q^{11} + 9 q^{14} + 27 q^{16} + 30 q^{17} + 30 q^{20} + 39 q^{23} + 27 q^{25} + 9 q^{26} + 27 q^{28} + 39 q^{29} + 63 q^{32} + 12 q^{35} + 9 q^{38} + 42 q^{41} + 42 q^{44} + 27 q^{47} + 27 q^{49} + 36 q^{50} + 66 q^{53} + 27 q^{56} + 18 q^{59} + 36 q^{62} + 27 q^{64} + 69 q^{65} + 21 q^{68} + 72 q^{71} + 54 q^{74} + 24 q^{77} + 21 q^{80} + 39 q^{83} + 27 q^{86} + 42 q^{89} + 75 q^{92} + 78 q^{95} + 9 q^{98}+O(q^{100}) \) 27 * q + 9 * q^2 + 27 * q^4 + 12 * q^5 + 27 * q^7 + 27 * q^8 + 24 * q^11 + 9 * q^14 + 27 * q^16 + 30 * q^17 + 30 * q^20 + 39 * q^23 + 27 * q^25 + 9 * q^26 + 27 * q^28 + 39 * q^29 + 63 * q^32 + 12 * q^35 + 9 * q^38 + 42 * q^41 + 42 * q^44 + 27 * q^47 + 27 * q^49 + 36 * q^50 + 66 * q^53 + 27 * q^56 + 18 * q^59 + 36 * q^62 + 27 * q^64 + 69 * q^65 + 21 * q^68 + 72 * q^71 + 54 * q^74 + 24 * q^77 + 21 * q^80 + 39 * q^83 + 27 * q^86 + 42 * q^89 + 75 * q^92 + 78 * q^95 + 9 * q^98

Introduction

L-functions

Modular forms

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Representations

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Database

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