LMFDB - Embedded newform 1197.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1197,2,Mod(1,1197)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1197, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1197.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1197 = 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1197.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:	$9.55809312195$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	1197.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17009 q^{2} +2.70928 q^{4} -3.70928 q^{5} -1.00000 q^{7} -1.53919 q^{8} +O(q^{10})$	\(q-2.17009 q^{2} +2.70928 q^{4} -3.70928 q^{5} -1.00000 q^{7} -1.53919 q^{8} +8.04945 q^{10} +1.07838 q^{11} +0.921622 q^{13} +2.17009 q^{14} -2.07838 q^{16} -0.290725 q^{17} +1.00000 q^{19} -10.0494 q^{20} -2.34017 q^{22} +7.60197 q^{23} +8.75872 q^{25} -2.00000 q^{26} -2.70928 q^{28} -5.36910 q^{29} +8.49693 q^{31} +7.58864 q^{32} +0.630898 q^{34} +3.70928 q^{35} -10.6803 q^{37} -2.17009 q^{38} +5.70928 q^{40} +3.75872 q^{41} -8.49693 q^{43} +2.92162 q^{44} -16.4969 q^{46} +6.20620 q^{47} +1.00000 q^{49} -19.0072 q^{50} +2.49693 q^{52} -4.78765 q^{53} -4.00000 q^{55} +1.53919 q^{56} +11.6514 q^{58} -4.00000 q^{59} -2.68035 q^{61} -18.4391 q^{62} -12.3112 q^{64} -3.41855 q^{65} +1.26180 q^{67} -0.787653 q^{68} -8.04945 q^{70} +3.86603 q^{71} +1.41855 q^{73} +23.1773 q^{74} +2.70928 q^{76} -1.07838 q^{77} -10.5236 q^{79} +7.70928 q^{80} -8.15676 q^{82} -8.72979 q^{83} +1.07838 q^{85} +18.4391 q^{86} -1.65983 q^{88} +3.75872 q^{89} -0.921622 q^{91} +20.5958 q^{92} -13.4680 q^{94} -3.70928 q^{95} -13.5174 q^{97} -2.17009 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8	$ 3 q - q^{2} + q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} + 6 q^{10} + 6 q^{13} + q^{14} - 3 q^{16} - 8 q^{17} + 3 q^{19} - 12 q^{20} + 4 q^{22} + 4 q^{23} + q^{25} - 6 q^{26} - q^{28} - 20 q^{29} + 8 q^{31} + 3 q^{32} - 2 q^{34} + 4 q^{35} - 10 q^{37} - q^{38} + 10 q^{40} - 14 q^{41} - 8 q^{43} + 12 q^{44} - 32 q^{46} - 6 q^{47} + 3 q^{49} - 23 q^{50} - 10 q^{52} - 4 q^{53} - 12 q^{55} + 3 q^{56} - 2 q^{58} - 12 q^{59} + 14 q^{61} - 8 q^{62} - 11 q^{64} + 4 q^{65} - 4 q^{67} + 8 q^{68} - 6 q^{70} - 2 q^{71} - 10 q^{73} + 30 q^{74} + q^{76} - 16 q^{79} + 16 q^{80} - 18 q^{82} + 14 q^{83} + 8 q^{86} - 16 q^{88} - 14 q^{89} - 6 q^{91} + 8 q^{92} - 8 q^{94} - 4 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 + 6 * q^10 + 6 * q^13 + q^14 - 3 * q^16 - 8 * q^17 + 3 * q^19 - 12 * q^20 + 4 * q^22 + 4 * q^23 + q^25 - 6 * q^26 - q^28 - 20 * q^29 + 8 * q^31 + 3 * q^32 - 2 * q^34 + 4 * q^35 - 10 * q^37 - q^38 + 10 * q^40 - 14 * q^41 - 8 * q^43 + 12 * q^44 - 32 * q^46 - 6 * q^47 + 3 * q^49 - 23 * q^50 - 10 * q^52 - 4 * q^53 - 12 * q^55 + 3 * q^56 - 2 * q^58 - 12 * q^59 + 14 * q^61 - 8 * q^62 - 11 * q^64 + 4 * q^65 - 4 * q^67 + 8 * q^68 - 6 * q^70 - 2 * q^71 - 10 * q^73 + 30 * q^74 + q^76 - 16 * q^79 + 16 * q^80 - 18 * q^82 + 14 * q^83 + 8 * q^86 - 16 * q^88 - 14 * q^89 - 6 * q^91 + 8 * q^92 - 8 * q^94 - 4 * q^95 + 10 * q^97 - 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 $

$2$	−2.17009	−1.53448	−0.767241	−	0.641358i	$-0.778371\pi$
$2$	−2.17009	−1.53448	−0.767241	+	0.641358i	$0.778371\pi$
$3$	0	0
$3$	0	0
$4$	2.70928	1.35464
$5$	−3.70928	−1.65884	−0.829419	−	0.558627i	$-0.811328\pi$
$5$	−3.70928	−1.65884	−0.829419	+	0.558627i	$0.811328\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.53919	−0.544185
$9$	0	0
$10$	8.04945	2.54546
$11$	1.07838	0.325143	0.162572	−	0.986697i	$-0.448021\pi$
$11$	1.07838	0.325143	0.162572	+	0.986697i	$0.448021\pi$
$12$	0	0
$13$	0.921622	0.255612	0.127806	−	0.991799i	$-0.459207\pi$
$13$	0.921622	0.255612	0.127806	+	0.991799i	$0.459207\pi$
$14$	2.17009	0.579980
$15$	0	0
$16$	−2.07838	−0.519594
$17$	−0.290725	−0.0705111	−0.0352555	−	0.999378i	$-0.511225\pi$
$17$	−0.290725	−0.0705111	−0.0352555	+	0.999378i	$0.511225\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−10.0494	−2.24712
$21$	0	0
$22$	−2.34017	−0.498927
$23$	7.60197	1.58512	0.792560	−	0.609794i	$-0.208748\pi$
$23$	7.60197	1.58512	0.792560	+	0.609794i	$0.208748\pi$
$24$	0	0
$25$	8.75872	1.75174
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−2.70928	−0.512005
$29$	−5.36910	−0.997017	−0.498509	−	0.866885i	$-0.666119\pi$
$29$	−5.36910	−0.997017	−0.498509	+	0.866885i	$0.666119\pi$
$30$	0	0
$31$	8.49693	1.52609	0.763047	−	0.646343i	$-0.223703\pi$
$31$	8.49693	1.52609	0.763047	+	0.646343i	$0.223703\pi$
$32$	7.58864	1.34149
$33$	0	0
$34$	0.630898	0.108198
$35$	3.70928	0.626982
$36$	0	0
$37$	−10.6803	−1.75584	−0.877919	−	0.478809i	$-0.841069\pi$
$37$	−10.6803	−1.75584	−0.877919	+	0.478809i	$0.841069\pi$
$38$	−2.17009	−0.352035
$39$	0	0
$40$	5.70928	0.902716
$41$	3.75872	0.587014	0.293507	−	0.955957i	$-0.405178\pi$
$41$	3.75872	0.587014	0.293507	+	0.955957i	$0.405178\pi$
$42$	0	0
$43$	−8.49693	−1.29577	−0.647885	−	0.761738i	$-0.724346\pi$
$43$	−8.49693	−1.29577	−0.647885	+	0.761738i	$0.724346\pi$
$44$	2.92162	0.440451
$45$	0	0
$46$	−16.4969	−2.43234
$47$	6.20620	0.905268	0.452634	−	0.891696i	$-0.350484\pi$
$47$	6.20620	0.905268	0.452634	+	0.891696i	$0.350484\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−19.0072	−2.68802
$51$	0	0
$52$	2.49693	0.346262
$53$	−4.78765	−0.657635	−0.328817	−	0.944394i	$-0.606650\pi$
$53$	−4.78765	−0.657635	−0.328817	+	0.944394i	$0.606650\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	1.53919	0.205683
$57$	0	0
$58$	11.6514	1.52991
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−2.68035	−0.343183	−0.171592	−	0.985168i	$-0.554891\pi$
$61$	−2.68035	−0.343183	−0.171592	+	0.985168i	$0.554891\pi$
$62$	−18.4391	−2.34176
$63$	0	0
$64$	−12.3112	−1.53891
$65$	−3.41855	−0.424019
$66$	0	0
$67$	1.26180	0.154153	0.0770764	−	0.997025i	$-0.475441\pi$
$67$	1.26180	0.154153	0.0770764	+	0.997025i	$0.475441\pi$
$68$	−0.787653	−0.0955170
$69$	0	0
$70$	−8.04945	−0.962093
$71$	3.86603	0.458813	0.229407	−	0.973331i	$-0.426321\pi$
$71$	3.86603	0.458813	0.229407	+	0.973331i	$0.426321\pi$
$72$	0	0
$73$	1.41855	0.166029	0.0830144	−	0.996548i	$-0.473545\pi$
$73$	1.41855	0.166029	0.0830144	+	0.996548i	$0.473545\pi$
$74$	23.1773	2.69430
$75$	0	0
$76$	2.70928	0.310775
$77$	−1.07838	−0.122893
$78$	0	0
$79$	−10.5236	−1.18400	−0.591998	−	0.805939i	$-0.701661\pi$
$79$	−10.5236	−1.18400	−0.591998	+	0.805939i	$0.701661\pi$
$80$	7.70928	0.861923
$81$	0	0
$82$	−8.15676	−0.900763
$83$	−8.72979	−0.958219	−0.479110	−	0.877755i	$-0.659040\pi$
$83$	−8.72979	−0.958219	−0.479110	+	0.877755i	$0.659040\pi$
$84$	0	0
$85$	1.07838	0.116966
$86$	18.4391	1.98834
$87$	0	0
$88$	−1.65983	−0.176938
$89$	3.75872	0.398424	0.199212	−	0.979956i	$-0.436162\pi$
$89$	3.75872	0.398424	0.199212	+	0.979956i	$0.436162\pi$
$90$	0	0
$91$	−0.921622	−0.0966123
$92$	20.5958	2.14726
$93$	0	0
$94$	−13.4680	−1.38912
$95$	−3.70928	−0.380564
$96$	0	0
$97$	−13.5174	−1.37249	−0.686244	−	0.727371i	$-0.740742\pi$
$97$	−13.5174	−1.37249	−0.686244	+	0.727371i	$0.740742\pi$
$98$	−2.17009	−0.219212
$99$	0	0
$100$	23.7298	2.37298
$101$	2.44748	0.243533	0.121767	−	0.992559i	$-0.461144\pi$
$101$	2.44748	0.243533	0.121767	+	0.992559i	$0.461144\pi$
$102$	0	0
$103$	19.5174	1.92311	0.961556	−	0.274610i	$-0.0885488\pi$
$103$	19.5174	1.92311	0.961556	+	0.274610i	$0.0885488\pi$
$104$	−1.41855	−0.139100
$105$	0	0
$106$	10.3896	1.00913
$107$	−15.6514	−1.51308	−0.756540	−	0.653948i	$-0.773112\pi$
$107$	−15.6514	−1.51308	−0.756540	+	0.653948i	$0.773112\pi$
$108$	0	0
$109$	14.0989	1.35043	0.675215	−	0.737621i	$-0.264051\pi$
$109$	14.0989	1.35043	0.675215	+	0.737621i	$0.264051\pi$
$110$	8.68035	0.827639
$111$	0	0
$112$	2.07838	0.196388
$113$	−18.7298	−1.76195	−0.880975	−	0.473162i	$-0.843113\pi$
$113$	−18.7298	−1.76195	−0.880975	+	0.473162i	$0.843113\pi$
$114$	0	0
$115$	−28.1978	−2.62946
$116$	−14.5464	−1.35060
$117$	0	0
$118$	8.68035	0.799091
$119$	0.290725	0.0266507
$120$	0	0
$121$	−9.83710	−0.894282
$122$	5.81658	0.526609
$123$	0	0
$124$	23.0205	2.06730
$125$	−13.9421	−1.24702
$126$	0	0
$127$	−15.7854	−1.40073	−0.700363	−	0.713787i	$-0.746979\pi$
$127$	−15.7854	−1.40073	−0.700363	+	0.713787i	$0.746979\pi$
$128$	11.5392	1.01993
$129$	0	0
$130$	7.41855	0.650650
$131$	−6.10731	−0.533598	−0.266799	−	0.963752i	$-0.585966\pi$
$131$	−6.10731	−0.533598	−0.266799	+	0.963752i	$0.585966\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	−2.73820	−0.236545
$135$	0	0
$136$	0.447480	0.0383711
$137$	−18.9939	−1.62275	−0.811377	−	0.584523i	$-0.801282\pi$
$137$	−18.9939	−1.62275	−0.811377	+	0.584523i	$0.801282\pi$
$138$	0	0
$139$	0.894960	0.0759095	0.0379548	−	0.999279i	$-0.487916\pi$
$139$	0.894960	0.0759095	0.0379548	+	0.999279i	$0.487916\pi$
$140$	10.0494	0.849333
$141$	0	0
$142$	−8.38962	−0.704041
$143$	0.993857	0.0831105
$144$	0	0
$145$	19.9155	1.65389
$146$	−3.07838	−0.254768
$147$	0	0
$148$	−28.9360	−2.37852
$149$	−20.9360	−1.71514	−0.857572	−	0.514364i	$-0.828028\pi$
$149$	−20.9360	−1.71514	−0.857572	+	0.514364i	$0.828028\pi$
$150$	0	0
$151$	13.9421	1.13460	0.567298	−	0.823513i	$-0.307989\pi$
$151$	13.9421	1.13460	0.567298	+	0.823513i	$0.307989\pi$
$152$	−1.53919	−0.124845
$153$	0	0
$154$	2.34017	0.188577
$155$	−31.5174	−2.53154
$156$	0	0
$157$	−2.31351	−0.184638	−0.0923191	−	0.995729i	$-0.529428\pi$
$157$	−2.31351	−0.184638	−0.0923191	+	0.995729i	$0.529428\pi$
$158$	22.8371	1.81682
$159$	0	0
$160$	−28.1483	−2.22532
$161$	−7.60197	−0.599119
$162$	0	0
$163$	−17.6598	−1.38322	−0.691612	−	0.722269i	$-0.743099\pi$
$163$	−17.6598	−1.38322	−0.691612	+	0.722269i	$0.743099\pi$
$164$	10.1834	0.795191
$165$	0	0
$166$	18.9444	1.47037
$167$	−3.05172	−0.236149	−0.118074	−	0.993005i	$-0.537672\pi$
$167$	−3.05172	−0.236149	−0.118074	+	0.993005i	$0.537672\pi$
$168$	0	0
$169$	−12.1506	−0.934662
$170$	−2.34017	−0.179483
$171$	0	0
$172$	−23.0205	−1.75530
$173$	−9.60197	−0.730024	−0.365012	−	0.931003i	$-0.618935\pi$
$173$	−9.60197	−0.730024	−0.365012	+	0.931003i	$0.618935\pi$
$174$	0	0
$175$	−8.75872	−0.662097
$176$	−2.24128	−0.168943
$177$	0	0
$178$	−8.15676	−0.611375
$179$	0.814315	0.0608648	0.0304324	−	0.999537i	$-0.490312\pi$
$179$	0.814315	0.0608648	0.0304324	+	0.999537i	$0.490312\pi$
$180$	0	0
$181$	20.1568	1.49824	0.749120	−	0.662434i	$-0.230477\pi$
$181$	20.1568	1.49824	0.749120	+	0.662434i	$0.230477\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	−11.7009	−0.862599
$185$	39.6163	2.91265
$186$	0	0
$187$	−0.313511	−0.0229262
$188$	16.8143	1.22631
$189$	0	0
$190$	8.04945	0.583968
$191$	−5.44521	−0.394002	−0.197001	−	0.980403i	$-0.563120\pi$
$191$	−5.44521	−0.394002	−0.197001	+	0.980403i	$0.563120\pi$
$192$	0	0
$193$	1.68649	0.121396	0.0606981	−	0.998156i	$-0.480667\pi$
$193$	1.68649	0.121396	0.0606981	+	0.998156i	$0.480667\pi$
$194$	29.3340	2.10606
$195$	0	0
$196$	2.70928	0.193520
$197$	10.3668	0.738606	0.369303	−	0.929309i	$-0.379596\pi$
$197$	10.3668	0.738606	0.369303	+	0.929309i	$0.379596\pi$
$198$	0	0
$199$	−2.42469	−0.171882	−0.0859410	−	0.996300i	$-0.527390\pi$
$199$	−2.42469	−0.171882	−0.0859410	+	0.996300i	$0.527390\pi$
$200$	−13.4813	−0.953274
$201$	0	0
$202$	−5.31124	−0.373698
$203$	5.36910	0.376837
$204$	0	0
$205$	−13.9421	−0.973761
$206$	−42.3545	−2.95098
$207$	0	0
$208$	−1.91548	−0.132815
$209$	1.07838	0.0745929
$210$	0	0
$211$	19.5174	1.34364	0.671818	−	0.740716i	$-0.265514\pi$
$211$	19.5174	1.34364	0.671818	+	0.740716i	$0.265514\pi$
$212$	−12.9711	−0.890857
$213$	0	0
$214$	33.9649	2.32179
$215$	31.5174	2.14947
$216$	0	0
$217$	−8.49693	−0.576809
$218$	−30.5958	−2.07221
$219$	0	0
$220$	−10.8371	−0.730637
$221$	−0.267938	−0.0180235
$222$	0	0
$223$	19.0205	1.27371	0.636854	−	0.770984i	$-0.280235\pi$
$223$	19.0205	1.27371	0.636854	+	0.770984i	$0.280235\pi$
$224$	−7.58864	−0.507037
$225$	0	0
$226$	40.6453	2.70368
$227$	−11.7321	−0.778684	−0.389342	−	0.921093i	$-0.627298\pi$
$227$	−11.7321	−0.778684	−0.389342	+	0.921093i	$0.627298\pi$
$228$	0	0
$229$	−27.4596	−1.81458	−0.907290	−	0.420505i	$-0.861853\pi$
$229$	−27.4596	−1.81458	−0.907290	+	0.420505i	$0.861853\pi$
$230$	61.1917	4.03486
$231$	0	0
$232$	8.26406	0.542562
$233$	5.10504	0.334442	0.167221	−	0.985919i	$-0.446521\pi$
$233$	5.10504	0.334442	0.167221	+	0.985919i	$0.446521\pi$
$234$	0	0
$235$	−23.0205	−1.50169
$236$	−10.8371	−0.705435
$237$	0	0
$238$	−0.630898	−0.0408950
$239$	10.6537	0.689130	0.344565	−	0.938763i	$-0.388027\pi$
$239$	10.6537	0.689130	0.344565	+	0.938763i	$0.388027\pi$
$240$	0	0
$241$	−13.5174	−0.870735	−0.435368	−	0.900253i	$-0.643382\pi$
$241$	−13.5174	−0.870735	−0.435368	+	0.900253i	$0.643382\pi$
$242$	21.3474	1.37226
$243$	0	0
$244$	−7.26180	−0.464889
$245$	−3.70928	−0.236977
$246$	0	0
$247$	0.921622	0.0586414
$248$	−13.0784	−0.830478
$249$	0	0
$250$	30.2557	1.91354
$251$	18.2062	1.14917	0.574583	−	0.818447i	$-0.305164\pi$
$251$	18.2062	1.14917	0.574583	+	0.818447i	$0.305164\pi$
$252$	0	0
$253$	8.19779	0.515391
$254$	34.2557	2.14939
$255$	0	0
$256$	−0.418551	−0.0261594
$257$	−12.3402	−0.769759	−0.384879	−	0.922967i	$-0.625757\pi$
$257$	−12.3402	−0.769759	−0.384879	+	0.922967i	$0.625757\pi$
$258$	0	0
$259$	10.6803	0.663644
$260$	−9.26180	−0.574392
$261$	0	0
$262$	13.2534	0.818797
$263$	−7.50307	−0.462659	−0.231330	−	0.972875i	$-0.574308\pi$
$263$	−7.50307	−0.462659	−0.231330	+	0.972875i	$0.574308\pi$
$264$	0	0
$265$	17.7587	1.09091
$266$	2.17009	0.133057
$267$	0	0
$268$	3.41855	0.208821
$269$	4.34017	0.264625	0.132313	−	0.991208i	$-0.457760\pi$
$269$	4.34017	0.264625	0.132313	+	0.991208i	$0.457760\pi$
$270$	0	0
$271$	26.0410	1.58188	0.790940	−	0.611893i	$-0.209592\pi$
$271$	26.0410	1.58188	0.790940	+	0.611893i	$0.209592\pi$
$272$	0.604236	0.0366372
$273$	0	0
$274$	41.2183	2.49009
$275$	9.44521	0.569568
$276$	0	0
$277$	6.59583	0.396305	0.198152	−	0.980171i	$-0.436506\pi$
$277$	6.59583	0.396305	0.198152	+	0.980171i	$0.436506\pi$
$278$	−1.94214	−0.116482
$279$	0	0
$280$	−5.70928	−0.341194
$281$	11.6248	0.693475	0.346737	−	0.937962i	$-0.387290\pi$
$281$	11.6248	0.693475	0.346737	+	0.937962i	$0.387290\pi$
$282$	0	0
$283$	−17.9421	−1.06655	−0.533275	−	0.845942i	$-0.679039\pi$
$283$	−17.9421	−1.06655	−0.533275	+	0.845942i	$0.679039\pi$
$284$	10.4741	0.621526
$285$	0	0
$286$	−2.15676	−0.127532
$287$	−3.75872	−0.221870
$288$	0	0
$289$	−16.9155	−0.995028
$290$	−43.2183	−2.53787
$291$	0	0
$292$	3.84324	0.224909
$293$	23.7587	1.38800	0.694000	−	0.719975i	$-0.255847\pi$
$293$	23.7587	1.38800	0.694000	+	0.719975i	$0.255847\pi$
$294$	0	0
$295$	14.8371	0.863849
$296$	16.4391	0.955502
$297$	0	0
$298$	45.4329	2.63186
$299$	7.00614	0.405176
$300$	0	0
$301$	8.49693	0.489755
$302$	−30.2557	−1.74102
$303$	0	0
$304$	−2.07838	−0.119203
$305$	9.94214	0.569285
$306$	0	0
$307$	−24.0144	−1.37057	−0.685286	−	0.728274i	$-0.740323\pi$
$307$	−24.0144	−1.37057	−0.685286	+	0.728274i	$0.740323\pi$
$308$	−2.92162	−0.166475
$309$	0	0
$310$	68.3956	3.88461
$311$	−22.9854	−1.30339	−0.651693	−	0.758483i	$-0.725941\pi$
$311$	−22.9854	−1.30339	−0.651693	+	0.758483i	$0.725941\pi$
$312$	0	0
$313$	0.523590	0.0295951	0.0147975	−	0.999891i	$-0.495290\pi$
$313$	0.523590	0.0295951	0.0147975	+	0.999891i	$0.495290\pi$
$314$	5.02052	0.283324
$315$	0	0
$316$	−28.5113	−1.60389
$317$	17.1012	0.960497	0.480249	−	0.877132i	$-0.340546\pi$
$317$	17.1012	0.960497	0.480249	+	0.877132i	$0.340546\pi$
$318$	0	0
$319$	−5.78992	−0.324173
$320$	45.6658	2.55280
$321$	0	0
$322$	16.4969	0.919338
$323$	−0.290725	−0.0161764
$324$	0	0
$325$	8.07223	0.447767
$326$	38.3234	2.12253
$327$	0	0
$328$	−5.78539	−0.319444
$329$	−6.20620	−0.342159
$330$	0	0
$331$	25.1461	1.38215	0.691077	−	0.722781i	$-0.257137\pi$
$331$	25.1461	1.38215	0.691077	+	0.722781i	$0.257137\pi$
$332$	−23.6514	−1.29804
$333$	0	0
$334$	6.62249	0.362366
$335$	−4.68035	−0.255715
$336$	0	0
$337$	−11.5753	−0.630547	−0.315274	−	0.949001i	$-0.602096\pi$
$337$	−11.5753	−0.630547	−0.315274	+	0.949001i	$0.602096\pi$
$338$	26.3679	1.43422
$339$	0	0
$340$	2.92162	0.158447
$341$	9.16290	0.496199
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	13.0784	0.705139
$345$	0	0
$346$	20.8371	1.12021
$347$	32.0144	1.71862	0.859311	−	0.511454i	$-0.170893\pi$
$347$	32.0144	1.71862	0.859311	+	0.511454i	$0.170893\pi$
$348$	0	0
$349$	−2.36683	−0.126694	−0.0633469	−	0.997992i	$-0.520177\pi$
$349$	−2.36683	−0.126694	−0.0633469	+	0.997992i	$0.520177\pi$
$350$	19.0072	1.01598
$351$	0	0
$352$	8.18342	0.436178
$353$	−15.4413	−0.821859	−0.410930	−	0.911667i	$-0.634796\pi$
$353$	−15.4413	−0.821859	−0.410930	+	0.911667i	$0.634796\pi$
$354$	0	0
$355$	−14.3402	−0.761097
$356$	10.1834	0.539720
$357$	0	0
$358$	−1.76713	−0.0933959
$359$	−23.4329	−1.23674	−0.618371	−	0.785886i	$-0.712207\pi$
$359$	−23.4329	−1.23674	−0.618371	+	0.785886i	$0.712207\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−43.7419	−2.29902
$363$	0	0
$364$	−2.49693	−0.130875
$365$	−5.26180	−0.275415
$366$	0	0
$367$	−31.3028	−1.63399	−0.816997	−	0.576642i	$-0.804363\pi$
$367$	−31.3028	−1.63399	−0.816997	+	0.576642i	$0.804363\pi$
$368$	−15.7998	−0.823620
$369$	0	0
$370$	−85.9709	−4.46941
$371$	4.78765	0.248563
$372$	0	0
$373$	−23.6742	−1.22580	−0.612902	−	0.790159i	$-0.709998\pi$
$373$	−23.6742	−1.22580	−0.612902	+	0.790159i	$0.709998\pi$
$374$	0.680346	0.0351799
$375$	0	0
$376$	−9.55252	−0.492634
$377$	−4.94828	−0.254850
$378$	0	0
$379$	−7.41855	−0.381065	−0.190533	−	0.981681i	$-0.561022\pi$
$379$	−7.41855	−0.381065	−0.190533	+	0.981681i	$0.561022\pi$
$380$	−10.0494	−0.515526
$381$	0	0
$382$	11.8166	0.604589
$383$	7.10504	0.363051	0.181525	−	0.983386i	$-0.441897\pi$
$383$	7.10504	0.363051	0.181525	+	0.983386i	$0.441897\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	−3.65983	−0.186280
$387$	0	0
$388$	−36.6225	−1.85923
$389$	−16.2557	−0.824194	−0.412097	−	0.911140i	$-0.635204\pi$
$389$	−16.2557	−0.824194	−0.412097	+	0.911140i	$0.635204\pi$
$390$	0	0
$391$	−2.21008	−0.111769
$392$	−1.53919	−0.0777408
$393$	0	0
$394$	−22.4969	−1.13338
$395$	39.0349	1.96406
$396$	0	0
$397$	26.5113	1.33056	0.665282	−	0.746592i	$-0.268311\pi$
$397$	26.5113	1.33056	0.665282	+	0.746592i	$0.268311\pi$
$398$	5.26180	0.263750
$399$	0	0
$400$	−18.2039	−0.910197
$401$	6.04945	0.302095	0.151048	−	0.988527i	$-0.451735\pi$
$401$	6.04945	0.302095	0.151048	+	0.988527i	$0.451735\pi$
$402$	0	0
$403$	7.83096	0.390088
$404$	6.63090	0.329899
$405$	0	0
$406$	−11.6514	−0.578250
$407$	−11.5174	−0.570899
$408$	0	0
$409$	37.1194	1.83544	0.917718	−	0.397231i	$-0.130029\pi$
$409$	37.1194	1.83544	0.917718	+	0.397231i	$0.130029\pi$
$410$	30.2557	1.49422
$411$	0	0
$412$	52.8781	2.60512
$413$	4.00000	0.196827
$414$	0	0
$415$	32.3812	1.58953
$416$	6.99386	0.342902
$417$	0	0
$418$	−2.34017	−0.114462
$419$	23.3523	1.14083	0.570417	−	0.821355i	$-0.306782\pi$
$419$	23.3523	1.14083	0.570417	+	0.821355i	$0.306782\pi$
$420$	0	0
$421$	7.57531	0.369198	0.184599	−	0.982814i	$-0.440901\pi$
$421$	7.57531	0.369198	0.184599	+	0.982814i	$0.440901\pi$
$422$	−42.3545	−2.06179
$423$	0	0
$424$	7.36910	0.357875
$425$	−2.54638	−0.123517
$426$	0	0
$427$	2.68035	0.129711
$428$	−42.4040	−2.04967
$429$	0	0
$430$	−68.3956	−3.29833
$431$	5.90707	0.284533	0.142267	−	0.989828i	$-0.454561\pi$
$431$	5.90707	0.284533	0.142267	+	0.989828i	$0.454561\pi$
$432$	0	0
$433$	−7.47641	−0.359293	−0.179647	−	0.983731i	$-0.557495\pi$
$433$	−7.47641	−0.359293	−0.179647	+	0.983731i	$0.557495\pi$
$434$	18.4391	0.885104
$435$	0	0
$436$	38.1978	1.82934
$437$	7.60197	0.363651
$438$	0	0
$439$	−9.49079	−0.452970	−0.226485	−	0.974015i	$-0.572724\pi$
$439$	−9.49079	−0.452970	−0.226485	+	0.974015i	$0.572724\pi$
$440$	6.15676	0.293512
$441$	0	0
$442$	0.581449	0.0276567
$443$	25.1773	1.19621	0.598104	−	0.801418i	$-0.295921\pi$
$443$	25.1773	1.19621	0.598104	+	0.801418i	$0.295921\pi$
$444$	0	0
$445$	−13.9421	−0.660921
$446$	−41.2762	−1.95448
$447$	0	0
$448$	12.3112	0.581652
$449$	32.4040	1.52924	0.764620	−	0.644482i	$-0.222927\pi$
$449$	32.4040	1.52924	0.764620	+	0.644482i	$0.222927\pi$
$450$	0	0
$451$	4.05332	0.190864
$452$	−50.7442	−2.38680
$453$	0	0
$454$	25.4596	1.19488
$455$	3.41855	0.160264
$456$	0	0
$457$	8.86830	0.414841	0.207421	−	0.978252i	$-0.433493\pi$
$457$	8.86830	0.414841	0.207421	+	0.978252i	$0.433493\pi$
$458$	59.5897	2.78444
$459$	0	0
$460$	−76.3956	−3.56196
$461$	−8.70313	−0.405345	−0.202673	−	0.979247i	$-0.564963\pi$
$461$	−8.70313	−0.405345	−0.202673	+	0.979247i	$0.564963\pi$
$462$	0	0
$463$	−6.21008	−0.288607	−0.144303	−	0.989533i	$-0.546094\pi$
$463$	−6.21008	−0.288607	−0.144303	+	0.989533i	$0.546094\pi$
$464$	11.1590	0.518045
$465$	0	0
$466$	−11.0784	−0.513196
$467$	−16.4619	−0.761764	−0.380882	−	0.924624i	$-0.624380\pi$
$467$	−16.4619	−0.761764	−0.380882	+	0.924624i	$0.624380\pi$
$468$	0	0
$469$	−1.26180	−0.0582643
$470$	49.9565	2.30432
$471$	0	0
$472$	6.15676	0.283388
$473$	−9.16290	−0.421311
$474$	0	0
$475$	8.75872	0.401878
$476$	0.787653	0.0361020
$477$	0	0
$478$	−23.1194	−1.05746
$479$	−16.5320	−0.755366	−0.377683	−	0.925935i	$-0.623279\pi$
$479$	−16.5320	−0.755366	−0.377683	+	0.925935i	$0.623279\pi$
$480$	0	0
$481$	−9.84324	−0.448813
$482$	29.3340	1.33613
$483$	0	0
$484$	−26.6514	−1.21143
$485$	50.1399	2.27674
$486$	0	0
$487$	−25.6742	−1.16341	−0.581705	−	0.813400i	$-0.697614\pi$
$487$	−25.6742	−1.16341	−0.581705	+	0.813400i	$0.697614\pi$
$488$	4.12556	0.186755
$489$	0	0
$490$	8.04945	0.363637
$491$	−27.3340	−1.23357	−0.616784	−	0.787133i	$-0.711565\pi$
$491$	−27.3340	−1.23357	−0.616784	+	0.787133i	$0.711565\pi$
$492$	0	0
$493$	1.56093	0.0703008
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	−17.6598	−0.792950
$497$	−3.86603	−0.173415
$498$	0	0
$499$	−10.0410	−0.449499	−0.224749	−	0.974417i	$-0.572156\pi$
$499$	−10.0410	−0.449499	−0.224749	+	0.974417i	$0.572156\pi$
$500$	−37.7731	−1.68926
$501$	0	0
$502$	−39.5090	−1.76337
$503$	−36.8287	−1.64211	−0.821055	−	0.570849i	$-0.806614\pi$
$503$	−36.8287	−1.64211	−0.821055	+	0.570849i	$0.806614\pi$
$504$	0	0
$505$	−9.07838	−0.403983
$506$	−17.7899	−0.790858
$507$	0	0
$508$	−42.7670	−1.89748
$509$	14.1301	0.626305	0.313153	−	0.949703i	$-0.398615\pi$
$509$	14.1301	0.626305	0.313153	+	0.949703i	$0.398615\pi$
$510$	0	0
$511$	−1.41855	−0.0627530
$512$	−22.1701	−0.979789
$513$	0	0
$514$	26.7792	1.18118
$515$	−72.3956	−3.19013
$516$	0	0
$517$	6.69263	0.294342
$518$	−23.1773	−1.01835
$519$	0	0
$520$	5.26180	0.230745
$521$	−28.1711	−1.23420	−0.617100	−	0.786885i	$-0.711693\pi$
$521$	−28.1711	−1.23420	−0.617100	+	0.786885i	$0.711693\pi$
$522$	0	0
$523$	−32.1834	−1.40728	−0.703641	−	0.710555i	$-0.748444\pi$
$523$	−32.1834	−1.40728	−0.703641	+	0.710555i	$0.748444\pi$
$524$	−16.5464	−0.722832
$525$	0	0
$526$	16.2823	0.709943
$527$	−2.47027	−0.107606
$528$	0	0
$529$	34.7899	1.51261
$530$	−38.5380	−1.67398
$531$	0	0
$532$	−2.70928	−0.117462
$533$	3.46412	0.150048
$534$	0	0
$535$	58.0554	2.50995
$536$	−1.94214	−0.0838877
$537$	0	0
$538$	−9.41855	−0.406063
$539$	1.07838	0.0464490
$540$	0	0
$541$	−41.0349	−1.76423	−0.882114	−	0.471036i	$-0.843880\pi$
$541$	−41.0349	−1.76423	−0.882114	+	0.471036i	$0.843880\pi$
$542$	−56.5113	−2.42737
$543$	0	0
$544$	−2.20620	−0.0945902
$545$	−52.2967	−2.24014
$546$	0	0
$547$	9.67420	0.413639	0.206820	−	0.978379i	$-0.433689\pi$
$547$	9.67420	0.413639	0.206820	+	0.978379i	$0.433689\pi$
$548$	−51.4596	−2.19824
$549$	0	0
$550$	−20.4969	−0.873992
$551$	−5.36910	−0.228731
$552$	0	0
$553$	10.5236	0.447509
$554$	−14.3135	−0.608123
$555$	0	0
$556$	2.42469	0.102830
$557$	40.1399	1.70078	0.850392	−	0.526150i	$-0.176365\pi$
$557$	40.1399	1.70078	0.850392	+	0.526150i	$0.176365\pi$
$558$	0	0
$559$	−7.83096	−0.331214
$560$	−7.70928	−0.325776
$561$	0	0
$562$	−25.2267	−1.06413
$563$	−44.0989	−1.85855	−0.929273	−	0.369393i	$-0.879566\pi$
$563$	−44.0989	−1.85855	−0.929273	+	0.369393i	$0.879566\pi$
$564$	0	0
$565$	69.4740	2.92279
$566$	38.9360	1.63660
$567$	0	0
$568$	−5.95055	−0.249680
$569$	13.4147	0.562372	0.281186	−	0.959653i	$-0.409272\pi$
$569$	13.4147	0.562372	0.281186	+	0.959653i	$0.409272\pi$
$570$	0	0
$571$	−27.8310	−1.16469	−0.582345	−	0.812942i	$-0.697865\pi$
$571$	−27.8310	−1.16469	−0.582345	+	0.812942i	$0.697865\pi$
$572$	2.69263	0.112585
$573$	0	0
$574$	8.15676	0.340456
$575$	66.5835	2.77673
$576$	0	0
$577$	−34.1978	−1.42367	−0.711836	−	0.702345i	$-0.752136\pi$
$577$	−34.1978	−1.42367	−0.711836	+	0.702345i	$0.752136\pi$
$578$	36.7081	1.52685
$579$	0	0
$580$	53.9565	2.24042
$581$	8.72979	0.362173
$582$	0	0
$583$	−5.16290	−0.213825
$584$	−2.18342	−0.0903505
$585$	0	0
$586$	−51.5585	−2.12986
$587$	−33.1422	−1.36793	−0.683963	−	0.729517i	$-0.739745\pi$
$587$	−33.1422	−1.36793	−0.683963	+	0.729517i	$0.739745\pi$
$588$	0	0
$589$	8.49693	0.350110
$590$	−32.1978	−1.32556
$591$	0	0
$592$	22.1978	0.912324
$593$	−29.7503	−1.22170	−0.610849	−	0.791747i	$-0.709172\pi$
$593$	−29.7503	−1.22170	−0.610849	+	0.791747i	$0.709172\pi$
$594$	0	0
$595$	−1.07838	−0.0442092
$596$	−56.7214	−2.32340
$597$	0	0
$598$	−15.2039	−0.621735
$599$	−2.19183	−0.0895557	−0.0447778	−	0.998997i	$-0.514258\pi$
$599$	−2.19183	−0.0895557	−0.0447778	+	0.998997i	$0.514258\pi$
$600$	0	0
$601$	31.9877	1.30481	0.652403	−	0.757872i	$-0.273761\pi$
$601$	31.9877	1.30481	0.652403	+	0.757872i	$0.273761\pi$
$602$	−18.4391	−0.751520
$603$	0	0
$604$	37.7731	1.53697
$605$	36.4885	1.48347
$606$	0	0
$607$	−27.5174	−1.11690	−0.558449	−	0.829539i	$-0.688603\pi$
$607$	−27.5174	−1.11690	−0.558449	+	0.829539i	$0.688603\pi$
$608$	7.58864	0.307760
$609$	0	0
$610$	−21.5753	−0.873559
$611$	5.71978	0.231397
$612$	0	0
$613$	6.59583	0.266403	0.133201	−	0.991089i	$-0.457474\pi$
$613$	6.59583	0.266403	0.133201	+	0.991089i	$0.457474\pi$
$614$	52.1133	2.10312
$615$	0	0
$616$	1.65983	0.0668763
$617$	3.47641	0.139955	0.0699775	−	0.997549i	$-0.477707\pi$
$617$	3.47641	0.139955	0.0699775	+	0.997549i	$0.477707\pi$
$618$	0	0
$619$	26.9893	1.08479	0.542396	−	0.840123i	$-0.317517\pi$
$619$	26.9893	1.08479	0.542396	+	0.840123i	$0.317517\pi$
$620$	−85.3894	−3.42932
$621$	0	0
$622$	49.8804	2.00002
$623$	−3.75872	−0.150590
$624$	0	0
$625$	7.92162	0.316865
$626$	−1.13624	−0.0454131
$627$	0	0
$628$	−6.26794	−0.250118
$629$	3.10504	0.123806
$630$	0	0
$631$	−10.3402	−0.411636	−0.205818	−	0.978590i	$-0.565985\pi$
$631$	−10.3402	−0.411636	−0.205818	+	0.978590i	$0.565985\pi$
$632$	16.1978	0.644314
$633$	0	0
$634$	−37.1110	−1.47387
$635$	58.5523	2.32358
$636$	0	0
$637$	0.921622	0.0365160
$638$	12.5646	0.497438
$639$	0	0
$640$	−42.8020	−1.69190
$641$	−35.1955	−1.39014	−0.695070	−	0.718942i	$-0.744627\pi$
$641$	−35.1955	−1.39014	−0.695070	+	0.718942i	$0.744627\pi$
$642$	0	0
$643$	12.4657	0.491600	0.245800	−	0.969321i	$-0.420949\pi$
$643$	12.4657	0.491600	0.245800	+	0.969321i	$0.420949\pi$
$644$	−20.5958	−0.811589
$645$	0	0
$646$	0.630898	0.0248223
$647$	11.8804	0.467067	0.233533	−	0.972349i	$-0.424971\pi$
$647$	11.8804	0.467067	0.233533	+	0.972349i	$0.424971\pi$
$648$	0	0
$649$	−4.31351	−0.169320
$650$	−17.5174	−0.687091
$651$	0	0
$652$	−47.8453	−1.87377
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	22.6537	0.885153
$656$	−7.81205	−0.305009
$657$	0	0
$658$	13.4680	0.525037
$659$	−12.5464	−0.488737	−0.244369	−	0.969682i	$-0.578581\pi$
$659$	−12.5464	−0.488737	−0.244369	+	0.969682i	$0.578581\pi$
$660$	0	0
$661$	11.6430	0.452860	0.226430	−	0.974027i	$-0.427294\pi$
$661$	11.6430	0.452860	0.226430	+	0.974027i	$0.427294\pi$
$662$	−54.5692	−2.12089
$663$	0	0
$664$	13.4368	0.521449
$665$	3.70928	0.143840
$666$	0	0
$667$	−40.8157	−1.58039
$668$	−8.26794	−0.319896
$669$	0	0
$670$	10.1568	0.392390
$671$	−2.89043	−0.111584
$672$	0	0
$673$	41.3484	1.59386	0.796932	−	0.604069i	$-0.206455\pi$
$673$	41.3484	1.59386	0.796932	+	0.604069i	$0.206455\pi$
$674$	25.1194	0.967564
$675$	0	0
$676$	−32.9194	−1.26613
$677$	19.1773	0.737043	0.368521	−	0.929619i	$-0.379864\pi$
$677$	19.1773	0.737043	0.368521	+	0.929619i	$0.379864\pi$
$678$	0	0
$679$	13.5174	0.518752
$680$	−1.65983	−0.0636515
$681$	0	0
$682$	−19.8843	−0.761409
$683$	22.2907	0.852931	0.426465	−	0.904504i	$-0.359759\pi$
$683$	22.2907	0.852931	0.426465	+	0.904504i	$0.359759\pi$
$684$	0	0
$685$	70.4534	2.69189
$686$	2.17009	0.0828543
$687$	0	0
$688$	17.6598	0.673275
$689$	−4.41241	−0.168099
$690$	0	0
$691$	−30.0410	−1.14281	−0.571407	−	0.820666i	$-0.693602\pi$
$691$	−30.0410	−1.14281	−0.571407	+	0.820666i	$0.693602\pi$
$692$	−26.0144	−0.988918
$693$	0	0
$694$	−69.4740	−2.63720
$695$	−3.31965	−0.125922
$696$	0	0
$697$	−1.09275	−0.0413910
$698$	5.13624	0.194409
$699$	0	0
$700$	−23.7298	−0.896902
$701$	1.20394	0.0454721	0.0227360	−	0.999742i	$-0.492762\pi$
$701$	1.20394	0.0454721	0.0227360	+	0.999742i	$0.492762\pi$
$702$	0	0
$703$	−10.6803	−0.402817
$704$	−13.2762	−0.500365
$705$	0	0
$706$	33.5090	1.26113
$707$	−2.44748	−0.0920470
$708$	0	0
$709$	−33.2351	−1.24817	−0.624086	−	0.781356i	$-0.714528\pi$
$709$	−33.2351	−1.24817	−0.624086	+	0.781356i	$0.714528\pi$
$710$	31.1194	1.16789
$711$	0	0
$712$	−5.78539	−0.216816
$713$	64.5934	2.41904
$714$	0	0
$715$	−3.68649	−0.137867
$716$	2.20620	0.0824497
$717$	0	0
$718$	50.8515	1.89776
$719$	−10.2062	−0.380627	−0.190314	−	0.981723i	$-0.560950\pi$
$719$	−10.2062	−0.380627	−0.190314	+	0.981723i	$0.560950\pi$
$720$	0	0
$721$	−19.5174	−0.726868
$722$	−2.17009	−0.0807623
$723$	0	0
$724$	54.6102	2.02957
$725$	−47.0265	−1.74652
$726$	0	0
$727$	−26.8371	−0.995333	−0.497666	−	0.867368i	$-0.665810\pi$
$727$	−26.8371	−0.995333	−0.497666	+	0.867368i	$0.665810\pi$
$728$	1.41855	0.0525750
$729$	0	0
$730$	11.4186	0.422620
$731$	2.47027	0.0913661
$732$	0	0
$733$	2.46573	0.0910739	0.0455369	−	0.998963i	$-0.485500\pi$
$733$	2.46573	0.0910739	0.0455369	+	0.998963i	$0.485500\pi$
$734$	67.9299	2.50734
$735$	0	0
$736$	57.6886	2.12643
$737$	1.36069	0.0501217
$738$	0	0
$739$	0.863763	0.0317741	0.0158870	−	0.999874i	$-0.494943\pi$
$739$	0.863763	0.0317741	0.0158870	+	0.999874i	$0.494943\pi$
$740$	107.332	3.94559
$741$	0	0
$742$	−10.3896	−0.381415
$743$	36.1171	1.32501	0.662505	−	0.749058i	$-0.269493\pi$
$743$	36.1171	1.32501	0.662505	+	0.749058i	$0.269493\pi$
$744$	0	0
$745$	77.6574	2.84515
$746$	51.3751	1.88097
$747$	0	0
$748$	−0.849388	−0.0310567
$749$	15.6514	0.571890
$750$	0	0
$751$	−33.8264	−1.23434	−0.617172	−	0.786828i	$-0.711722\pi$
$751$	−33.8264	−1.23434	−0.617172	+	0.786828i	$0.711722\pi$
$752$	−12.8988	−0.470372
$753$	0	0
$754$	10.7382	0.391062
$755$	−51.7152	−1.88211
$756$	0	0
$757$	−13.6865	−0.497444	−0.248722	−	0.968575i	$-0.580011\pi$
$757$	−13.6865	−0.497444	−0.248722	+	0.968575i	$0.580011\pi$
$758$	16.0989	0.584738
$759$	0	0
$760$	5.70928	0.207097
$761$	7.86150	0.284979	0.142490	−	0.989796i	$-0.454489\pi$
$761$	7.86150	0.284979	0.142490	+	0.989796i	$0.454489\pi$
$762$	0	0
$763$	−14.0989	−0.510414
$764$	−14.7526	−0.533730
$765$	0	0
$766$	−15.4186	−0.557095
$767$	−3.68649	−0.133111
$768$	0	0
$769$	2.31351	0.0834273	0.0417137	−	0.999130i	$-0.486718\pi$
$769$	2.31351	0.0834273	0.0417137	+	0.999130i	$0.486718\pi$
$770$	−8.68035	−0.312818
$771$	0	0
$772$	4.56916	0.164448
$773$	32.9216	1.18411	0.592054	−	0.805898i	$-0.298317\pi$
$773$	32.9216	1.18411	0.592054	+	0.805898i	$0.298317\pi$
$774$	0	0
$775$	74.4222	2.67333
$776$	20.8059	0.746888
$777$	0	0
$778$	35.2762	1.26471
$779$	3.75872	0.134670
$780$	0	0
$781$	4.16904	0.149180
$782$	4.79606	0.171507
$783$	0	0
$784$	−2.07838	−0.0742278
$785$	8.58145	0.306285
$786$	0	0
$787$	24.8104	0.884397	0.442198	−	0.896917i	$-0.354199\pi$
$787$	24.8104	0.884397	0.442198	+	0.896917i	$0.354199\pi$
$788$	28.0866	1.00054
$789$	0	0
$790$	−84.7091	−3.01381
$791$	18.7298	0.665955
$792$	0	0
$793$	−2.47027	−0.0877217
$794$	−57.5318	−2.04173
$795$	0	0
$796$	−6.56916	−0.232838
$797$	−2.59583	−0.0919488	−0.0459744	−	0.998943i	$-0.514639\pi$
$797$	−2.59583	−0.0919488	−0.0459744	+	0.998943i	$0.514639\pi$
$798$	0	0
$799$	−1.80430	−0.0638314
$800$	66.4668	2.34996
$801$	0	0
$802$	−13.1278	−0.463560
$803$	1.52973	0.0539831
$804$	0	0
$805$	28.1978	0.993842
$806$	−16.9939	−0.598583
$807$	0	0
$808$	−3.76713	−0.132527
$809$	−38.6803	−1.35993	−0.679964	−	0.733245i	$-0.738005\pi$
$809$	−38.6803	−1.35993	−0.679964	+	0.733245i	$0.738005\pi$
$810$	0	0
$811$	−44.9939	−1.57995	−0.789974	−	0.613140i	$-0.789906\pi$
$811$	−44.9939	−1.57995	−0.789974	+	0.613140i	$0.789906\pi$
$812$	14.5464	0.510478
$813$	0	0
$814$	24.9939	0.876034
$815$	65.5052	2.29455
$816$	0	0
$817$	−8.49693	−0.297270
$818$	−80.5523	−2.81645
$819$	0	0
$820$	−37.7731	−1.31909
$821$	−3.77310	−0.131682	−0.0658410	−	0.997830i	$-0.520973\pi$
$821$	−3.77310	−0.131682	−0.0658410	+	0.997830i	$0.520973\pi$
$822$	0	0
$823$	−14.6537	−0.510795	−0.255398	−	0.966836i	$-0.582206\pi$
$823$	−14.6537	−0.510795	−0.255398	+	0.966836i	$0.582206\pi$
$824$	−30.0410	−1.04653
$825$	0	0
$826$	−8.68035	−0.302028
$827$	15.4368	0.536790	0.268395	−	0.963309i	$-0.413507\pi$
$827$	15.4368	0.536790	0.268395	+	0.963309i	$0.413507\pi$
$828$	0	0
$829$	8.57691	0.297889	0.148944	−	0.988846i	$-0.452412\pi$
$829$	8.57691	0.297889	0.148944	+	0.988846i	$0.452412\pi$
$830$	−70.2700	−2.43911
$831$	0	0
$832$	−11.3463	−0.393363
$833$	−0.290725	−0.0100730
$834$	0	0
$835$	11.3197	0.391733
$836$	2.92162	0.101046
$837$	0	0
$838$	−50.6765	−1.75059
$839$	−31.0349	−1.07144	−0.535722	−	0.844395i	$-0.679960\pi$
$839$	−31.0349	−1.07144	−0.535722	+	0.844395i	$0.679960\pi$
$840$	0	0
$841$	−0.172740	−0.00595654
$842$	−16.4391	−0.566528
$843$	0	0
$844$	52.8781	1.82014
$845$	45.0700	1.55045
$846$	0	0
$847$	9.83710	0.338007
$848$	9.95055	0.341703
$849$	0	0
$850$	5.52586	0.189535
$851$	−81.1917	−2.78321
$852$	0	0
$853$	11.4140	0.390808	0.195404	−	0.980723i	$-0.437398\pi$
$853$	11.4140	0.390808	0.195404	+	0.980723i	$0.437398\pi$
$854$	−5.81658	−0.199039
$855$	0	0
$856$	24.0905	0.823396
$857$	−22.7936	−0.778615	−0.389308	−	0.921108i	$-0.627286\pi$
$857$	−22.7936	−0.778615	−0.389308	+	0.921108i	$0.627286\pi$
$858$	0	0
$859$	−30.7838	−1.05033	−0.525164	−	0.851001i	$-0.675996\pi$
$859$	−30.7838	−1.05033	−0.525164	+	0.851001i	$0.675996\pi$
$860$	85.3894	2.91176
$861$	0	0
$862$	−12.8188	−0.436612
$863$	−10.0228	−0.341180	−0.170590	−	0.985342i	$-0.554567\pi$
$863$	−10.0228	−0.341180	−0.170590	+	0.985342i	$0.554567\pi$
$864$	0	0
$865$	35.6163	1.21099
$866$	16.2245	0.551329
$867$	0	0
$868$	−23.0205	−0.781367
$869$	−11.3484	−0.384968
$870$	0	0
$871$	1.16290	0.0394033
$872$	−21.7009	−0.734884
$873$	0	0
$874$	−16.4969	−0.558017
$875$	13.9421	0.471330
$876$	0	0
$877$	−32.8371	−1.10883	−0.554415	−	0.832240i	$-0.687058\pi$
$877$	−32.8371	−1.10883	−0.554415	+	0.832240i	$0.687058\pi$
$878$	20.5958	0.695075
$879$	0	0
$880$	8.31351	0.280248
$881$	45.0700	1.51845	0.759223	−	0.650831i	$-0.225579\pi$
$881$	45.0700	1.51845	0.759223	+	0.650831i	$0.225579\pi$
$882$	0	0
$883$	54.5523	1.83583	0.917916	−	0.396774i	$-0.129870\pi$
$883$	54.5523	1.83583	0.917916	+	0.396774i	$0.129870\pi$
$884$	−0.725919	−0.0244153
$885$	0	0
$886$	−54.6369	−1.83556
$887$	−16.4657	−0.552865	−0.276433	−	0.961033i	$-0.589152\pi$
$887$	−16.4657	−0.552865	−0.276433	+	0.961033i	$0.589152\pi$
$888$	0	0
$889$	15.7854	0.529425
$890$	30.2557	1.01417
$891$	0	0
$892$	51.5318	1.72541
$893$	6.20620	0.207683
$894$	0	0
$895$	−3.02052	−0.100965
$896$	−11.5392	−0.385497
$897$	0	0
$898$	−70.3195	−2.34659
$899$	−45.6209	−1.52154
$900$	0	0
$901$	1.39189	0.0463705
$902$	−8.79606	−0.292877
$903$	0	0
$904$	28.8287	0.958828
$905$	−74.7670	−2.48534
$906$	0	0
$907$	−23.5708	−0.782655	−0.391327	−	0.920252i	$-0.627984\pi$
$907$	−23.5708	−0.782655	−0.391327	+	0.920252i	$0.627984\pi$
$908$	−31.7854	−1.05484
$909$	0	0
$910$	−7.41855	−0.245923
$911$	0.616522	0.0204263	0.0102131	−	0.999948i	$-0.496749\pi$
$911$	0.616522	0.0204263	0.0102131	+	0.999948i	$0.496749\pi$
$912$	0	0
$913$	−9.41402	−0.311558
$914$	−19.2450	−0.636567
$915$	0	0
$916$	−74.3956	−2.45810
$917$	6.10731	0.201681
$918$	0	0
$919$	53.6742	1.77055	0.885274	−	0.465069i	$-0.153971\pi$
$919$	53.6742	1.77055	0.885274	+	0.465069i	$0.153971\pi$
$920$	43.4017	1.43091
$921$	0	0
$922$	18.8865	0.621995
$923$	3.56302	0.117278
$924$	0	0
$925$	−93.5462	−3.07578
$926$	13.4764	0.442862
$927$	0	0
$928$	−40.7442	−1.33749
$929$	50.2616	1.64903	0.824515	−	0.565840i	$-0.191448\pi$
$929$	50.2616	1.64903	0.824515	+	0.565840i	$0.191448\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	13.8310	0.453048
$933$	0	0
$934$	35.7237	1.16891
$935$	1.16290	0.0380308
$936$	0	0
$937$	8.63931	0.282234	0.141117	−	0.989993i	$-0.454931\pi$
$937$	8.63931	0.282234	0.141117	+	0.989993i	$0.454931\pi$
$938$	2.73820	0.0894056
$939$	0	0
$940$	−62.3689	−2.03425
$941$	−30.2122	−0.984889	−0.492444	−	0.870344i	$-0.663896\pi$
$941$	−30.2122	−0.984889	−0.492444	+	0.870344i	$0.663896\pi$
$942$	0	0
$943$	28.5737	0.930488
$944$	8.31351	0.270582
$945$	0	0
$946$	19.8843	0.646494
$947$	−27.1727	−0.882995	−0.441498	−	0.897262i	$-0.645553\pi$
$947$	−27.1727	−0.882995	−0.441498	+	0.897262i	$0.645553\pi$
$948$	0	0
$949$	1.30737	0.0424390
$950$	−19.0072	−0.616675
$951$	0	0
$952$	−0.447480	−0.0145029
$953$	−44.5029	−1.44159	−0.720795	−	0.693148i	$-0.756223\pi$
$953$	−44.5029	−1.44159	−0.720795	+	0.693148i	$0.756223\pi$
$954$	0	0
$955$	20.1978	0.653585
$956$	28.8638	0.933521
$957$	0	0
$958$	35.8759	1.15910
$959$	18.9939	0.613344
$960$	0	0
$961$	41.1978	1.32896
$962$	21.3607	0.688696
$963$	0	0
$964$	−36.6225	−1.17953
$965$	−6.25565	−0.201377
$966$	0	0
$967$	−18.6004	−0.598147	−0.299074	−	0.954230i	$-0.596678\pi$
$967$	−18.6004	−0.598147	−0.299074	+	0.954230i	$0.596678\pi$
$968$	15.1412	0.486655
$969$	0	0
$970$	−108.808	−3.49361
$971$	−16.0533	−0.515176	−0.257588	−	0.966255i	$-0.582928\pi$
$971$	−16.0533	−0.515176	−0.257588	+	0.966255i	$0.582928\pi$
$972$	0	0
$973$	−0.894960	−0.0286911
$974$	55.7152	1.78523
$975$	0	0
$976$	5.57077	0.178316
$977$	15.8394	0.506746	0.253373	−	0.967369i	$-0.418460\pi$
$977$	15.8394	0.506746	0.253373	+	0.967369i	$0.418460\pi$
$978$	0	0
$979$	4.05332	0.129545
$980$	−10.0494	−0.321018
$981$	0	0
$982$	59.3172	1.89289
$983$	23.0349	0.734699	0.367350	−	0.930083i	$-0.380265\pi$
$983$	23.0349	0.734699	0.367350	+	0.930083i	$0.380265\pi$
$984$	0	0
$985$	−38.4534	−1.22523
$986$	−3.38735	−0.107875
$987$	0	0
$988$	2.49693	0.0794379
$989$	−64.5934	−2.05395
$990$	0	0
$991$	−42.8371	−1.36077	−0.680383	−	0.732857i	$-0.738186\pi$
$991$	−42.8371	−1.36077	−0.680383	+	0.732857i	$0.738186\pi$
$992$	64.4801	2.04725
$993$	0	0
$994$	8.38962	0.266103
$995$	8.99386	0.285124
$996$	0	0
$997$	−23.7899	−0.753434	−0.376717	−	0.926328i	$-0.622947\pi$
$997$	−23.7899	−0.753434	−0.376717	+	0.926328i	$0.622947\pi$
$998$	21.7899	0.689748
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1197.2.a.l.1.1		3
3.2	odd	2		399.2.a.d.1.3	✓	3
7.6	odd	2		8379.2.a.bp.1.1		3
12.11	even	2		6384.2.a.bx.1.3		3
15.14	odd	2		9975.2.a.z.1.1		3
21.20	even	2		2793.2.a.x.1.3		3
57.56	even	2		7581.2.a.n.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.d.1.3	✓	3	3.2	odd	2
1197.2.a.l.1.1		3	1.1	even	1	trivial
2793.2.a.x.1.3		3	21.20	even	2
6384.2.a.bx.1.3		3	12.11	even	2
7581.2.a.n.1.1		3	57.56	even	2
8379.2.a.bp.1.1		3	7.6	odd	2
9975.2.a.z.1.1		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1197 = 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1197.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17009 q^{2} +2.70928 q^{4} -3.70928 q^{5} -1.00000 q^{7} -1.53919 q^{8} +O(q^{10})\)	\(q-2.17009 q^{2} +2.70928 q^{4} -3.70928 q^{5} -1.00000 q^{7} -1.53919 q^{8} +8.04945 q^{10} +1.07838 q^{11} +0.921622 q^{13} +2.17009 q^{14} -2.07838 q^{16} -0.290725 q^{17} +1.00000 q^{19} -10.0494 q^{20} -2.34017 q^{22} +7.60197 q^{23} +8.75872 q^{25} -2.00000 q^{26} -2.70928 q^{28} -5.36910 q^{29} +8.49693 q^{31} +7.58864 q^{32} +0.630898 q^{34} +3.70928 q^{35} -10.6803 q^{37} -2.17009 q^{38} +5.70928 q^{40} +3.75872 q^{41} -8.49693 q^{43} +2.92162 q^{44} -16.4969 q^{46} +6.20620 q^{47} +1.00000 q^{49} -19.0072 q^{50} +2.49693 q^{52} -4.78765 q^{53} -4.00000 q^{55} +1.53919 q^{56} +11.6514 q^{58} -4.00000 q^{59} -2.68035 q^{61} -18.4391 q^{62} -12.3112 q^{64} -3.41855 q^{65} +1.26180 q^{67} -0.787653 q^{68} -8.04945 q^{70} +3.86603 q^{71} +1.41855 q^{73} +23.1773 q^{74} +2.70928 q^{76} -1.07838 q^{77} -10.5236 q^{79} +7.70928 q^{80} -8.15676 q^{82} -8.72979 q^{83} +1.07838 q^{85} +18.4391 q^{86} -1.65983 q^{88} +3.75872 q^{89} -0.921622 q^{91} +20.5958 q^{92} -13.4680 q^{94} -3.70928 q^{95} -13.5174 q^{97} -2.17009 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - q^{2} + q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} + 6 q^{10} + 6 q^{13} + q^{14} - 3 q^{16} - 8 q^{17} + 3 q^{19} - 12 q^{20} + 4 q^{22} + 4 q^{23} + q^{25} - 6 q^{26} - q^{28} - 20 q^{29} + 8 q^{31} + 3 q^{32} - 2 q^{34} + 4 q^{35} - 10 q^{37} - q^{38} + 10 q^{40} - 14 q^{41} - 8 q^{43} + 12 q^{44} - 32 q^{46} - 6 q^{47} + 3 q^{49} - 23 q^{50} - 10 q^{52} - 4 q^{53} - 12 q^{55} + 3 q^{56} - 2 q^{58} - 12 q^{59} + 14 q^{61} - 8 q^{62} - 11 q^{64} + 4 q^{65} - 4 q^{67} + 8 q^{68} - 6 q^{70} - 2 q^{71} - 10 q^{73} + 30 q^{74} + q^{76} - 16 q^{79} + 16 q^{80} - 18 q^{82} + 14 q^{83} + 8 q^{86} - 16 q^{88} - 14 q^{89} - 6 q^{91} + 8 q^{92} - 8 q^{94} - 4 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 + 6 * q^10 + 6 * q^13 + q^14 - 3 * q^16 - 8 * q^17 + 3 * q^19 - 12 * q^20 + 4 * q^22 + 4 * q^23 + q^25 - 6 * q^26 - q^28 - 20 * q^29 + 8 * q^31 + 3 * q^32 - 2 * q^34 + 4 * q^35 - 10 * q^37 - q^38 + 10 * q^40 - 14 * q^41 - 8 * q^43 + 12 * q^44 - 32 * q^46 - 6 * q^47 + 3 * q^49 - 23 * q^50 - 10 * q^52 - 4 * q^53 - 12 * q^55 + 3 * q^56 - 2 * q^58 - 12 * q^59 + 14 * q^61 - 8 * q^62 - 11 * q^64 + 4 * q^65 - 4 * q^67 + 8 * q^68 - 6 * q^70 - 2 * q^71 - 10 * q^73 + 30 * q^74 + q^76 - 16 * q^79 + 16 * q^80 - 18 * q^82 + 14 * q^83 + 8 * q^86 - 16 * q^88 - 14 * q^89 - 6 * q^91 + 8 * q^92 - 8 * q^94 - 4 * q^95 + 10 * q^97 - q^98

Introduction

L-functions

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Database

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