LMFDB - Embedded newform 1197.2.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1197,2,Mod(1,1197)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [3,-1,0,9,0,0,-3,9,0,10,-4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$9.55809312195$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*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			1.3
Root			$-0.210756$ 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)

$f(q)$	$=$	\(q+2.74483 q^{2} +5.53407 q^{4} +2.53407 q^{5} -1.00000 q^{7} +9.70041 q^{8} +6.95558 q^{10} -2.42151 q^{11} -0.421512 q^{13} -2.74483 q^{14} +15.5578 q^{16} -6.53407 q^{17} -1.00000 q^{19} +14.0237 q^{20} -6.64663 q^{22} +1.57849 q^{23} +1.42151 q^{25} -1.15698 q^{26} -5.53407 q^{28} -6.02372 q^{29} +3.48965 q^{31} +23.3026 q^{32} -17.9349 q^{34} -2.53407 q^{35} -2.84302 q^{37} -2.74483 q^{38} +24.5815 q^{40} -8.42151 q^{41} +11.4897 q^{43} -13.4008 q^{44} +4.33268 q^{46} -8.02372 q^{47} +1.00000 q^{49} +3.90180 q^{50} -2.33268 q^{52} +10.8667 q^{53} -6.13628 q^{55} -9.70041 q^{56} -16.5341 q^{58} -14.1363 q^{59} +6.84302 q^{61} +9.57849 q^{62} +32.8461 q^{64} -1.06814 q^{65} -9.91116 q^{67} -36.1600 q^{68} -6.95558 q^{70} +13.3771 q^{71} +7.06814 q^{73} -7.80361 q^{74} -5.53407 q^{76} +2.42151 q^{77} +14.1363 q^{79} +39.4245 q^{80} -23.1156 q^{82} -2.11256 q^{83} -16.5578 q^{85} +31.5371 q^{86} -23.4897 q^{88} +9.71477 q^{89} +0.421512 q^{91} +8.73546 q^{92} -22.0237 q^{94} -2.53407 q^{95} -4.97930 q^{97} +2.74483 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 9 q^{4} - 3 q^{7} + 9 q^{8} + 10 q^{10} - 4 q^{11} + 2 q^{13} + q^{14} + 13 q^{16} - 12 q^{17} - 3 q^{19} + 16 q^{20} - 8 q^{22} + 8 q^{23} + q^{25} - 10 q^{26} - 9 q^{28} + 8 q^{29} - 8 q^{31}+ \cdots - q^{98}+O(q^{100}) $ 3 * q - q^2 + 9 * q^4 - 3 * q^7 + 9 * q^8 + 10 * q^10 - 4 * q^11 + 2 * q^13 + q^14 + 13 * q^16 - 12 * q^17 - 3 * q^19 + 16 * q^20 - 8 * q^22 + 8 * q^23 + q^25 - 10 * q^26 - 9 * q^28 + 8 * q^29 - 8 * q^31 + 27 * q^32 - 6 * q^34 - 2 * q^37 + q^38 + 14 * q^40 - 22 * q^41 + 16 * q^43 - 12 * q^46 + 2 * q^47 + 3 * q^49 + 9 * q^50 + 18 * q^52 + 12 * q^55 - 9 * q^56 - 42 * q^58 - 12 * q^59 + 14 * q^61 + 32 * q^62 + 29 * q^64 + 12 * q^65 - 8 * q^67 - 52 * q^68 - 10 * q^70 + 26 * q^71 + 6 * q^73 - 18 * q^74 - 9 * q^76 + 4 * q^77 + 12 * q^79 + 52 * q^80 - 2 * q^82 - 2 * q^83 - 16 * q^85 + 24 * q^86 - 52 * q^88 + 2 * q^89 - 2 * q^91 + 36 * q^92 - 40 * q^94 + 22 * q^97 - q^98

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 $

$2$	2.74483	1.94089	0.970443	−	0.241332i	$-0.0775843\pi$
$2$	2.74483	1.94089	0.970443	+	0.241332i	$0.0775843\pi$
$3$	0	0
$3$	0	0
$4$	5.53407	2.76704
$5$	2.53407	1.13327	0.566635	−	0.823969i	$-0.308245\pi$
$5$	2.53407	1.13327	0.566635	+	0.823969i	$0.308245\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	9.70041	3.42961
$9$	0	0
$10$	6.95558	2.19955
$11$	−2.42151	−0.730113	−0.365057	−	0.930985i	$-0.618950\pi$
$11$	−2.42151	−0.730113	−0.365057	+	0.930985i	$0.618950\pi$
$12$	0	0
$13$	−0.421512	−0.116906	−0.0584532	−	0.998290i	$-0.518617\pi$
$13$	−0.421512	−0.116906	−0.0584532	+	0.998290i	$0.518617\pi$
$14$	−2.74483	−0.733586
$15$	0	0
$16$	15.5578	3.88945
$17$	−6.53407	−1.58474	−0.792372	−	0.610038i	$-0.791154\pi$
$17$	−6.53407	−1.58474	−0.792372	+	0.610038i	$0.791154\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	14.0237	3.13580
$21$	0	0
$22$	−6.64663	−1.41707
$23$	1.57849	0.329138	0.164569	−	0.986366i	$-0.447377\pi$
$23$	1.57849	0.329138	0.164569	+	0.986366i	$0.447377\pi$
$24$	0	0
$25$	1.42151	0.284302
$26$	−1.15698	−0.226902
$27$	0	0
$28$	−5.53407	−1.04584
$29$	−6.02372	−1.11858	−0.559289	−	0.828973i	$-0.688926\pi$
$29$	−6.02372	−1.11858	−0.559289	+	0.828973i	$0.688926\pi$
$30$	0	0
$31$	3.48965	0.626760	0.313380	−	0.949628i	$-0.398539\pi$
$31$	3.48965	0.626760	0.313380	+	0.949628i	$0.398539\pi$
$32$	23.3026	4.11936
$33$	0	0
$34$	−17.9349	−3.07581
$35$	−2.53407	−0.428336
$36$	0	0
$37$	−2.84302	−0.467390	−0.233695	−	0.972310i	$-0.575082\pi$
$37$	−2.84302	−0.467390	−0.233695	+	0.972310i	$0.575082\pi$
$38$	−2.74483	−0.445270
$39$	0	0
$40$	24.5815	3.88668
$41$	−8.42151	−1.31522	−0.657610	−	0.753359i	$-0.728432\pi$
$41$	−8.42151	−1.31522	−0.657610	+	0.753359i	$0.728432\pi$
$42$	0	0
$43$	11.4897	1.75216	0.876078	−	0.482170i	$-0.160151\pi$
$43$	11.4897	1.75216	0.876078	+	0.482170i	$0.160151\pi$
$44$	−13.4008	−2.02025
$45$	0	0
$46$	4.33268	0.638818
$47$	−8.02372	−1.17038	−0.585190	−	0.810896i	$-0.698980\pi$
$47$	−8.02372	−1.17038	−0.585190	+	0.810896i	$0.698980\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.90180	0.551798
$51$	0	0
$52$	−2.33268	−0.323484
$53$	10.8667	1.49266	0.746331	−	0.665575i	$-0.231814\pi$
$53$	10.8667	1.49266	0.746331	+	0.665575i	$0.231814\pi$
$54$	0	0
$55$	−6.13628	−0.827416
$56$	−9.70041	−1.29627
$57$	0	0
$58$	−16.5341	−2.17103
$59$	−14.1363	−1.84039	−0.920194	−	0.391464i	$-0.871969\pi$
$59$	−14.1363	−1.84039	−0.920194	+	0.391464i	$0.871969\pi$
$60$	0	0
$61$	6.84302	0.876159	0.438080	−	0.898936i	$-0.355659\pi$
$61$	6.84302	0.876159	0.438080	+	0.898936i	$0.355659\pi$
$62$	9.57849	1.21647
$63$	0	0
$64$	32.8461	4.10576
$65$	−1.06814	−0.132487
$66$	0	0
$67$	−9.91116	−1.21084	−0.605421	−	0.795906i	$-0.706995\pi$
$67$	−9.91116	−1.21084	−0.605421	+	0.795906i	$0.706995\pi$
$68$	−36.1600	−4.38504
$69$	0	0
$70$	−6.95558	−0.831351
$71$	13.3771	1.58757	0.793784	−	0.608199i	$-0.208108\pi$
$71$	13.3771	1.58757	0.793784	+	0.608199i	$0.208108\pi$
$72$	0	0
$73$	7.06814	0.827263	0.413632	−	0.910444i	$-0.364260\pi$
$73$	7.06814	0.827263	0.413632	+	0.910444i	$0.364260\pi$
$74$	−7.80361	−0.907151
$75$	0	0
$76$	−5.53407	−0.634801
$77$	2.42151	0.275957
$78$	0	0
$79$	14.1363	1.59046	0.795228	−	0.606311i	$-0.207351\pi$
$79$	14.1363	1.59046	0.795228	+	0.606311i	$0.207351\pi$
$80$	39.4245	4.40780
$81$	0	0
$82$	−23.1156	−2.55269
$83$	−2.11256	−0.231883	−0.115942	−	0.993256i	$-0.536989\pi$
$83$	−2.11256	−0.231883	−0.115942	+	0.993256i	$0.536989\pi$
$84$	0	0
$85$	−16.5578	−1.79594
$86$	31.5371	3.40073
$87$	0	0
$88$	−23.4897	−2.50401
$89$	9.71477	1.02976	0.514882	−	0.857261i	$-0.327836\pi$
$89$	9.71477	1.02976	0.514882	+	0.857261i	$0.327836\pi$
$90$	0	0
$91$	0.421512	0.0441864
$92$	8.73546	0.910735
$93$	0	0
$94$	−22.0237	−2.27157
$95$	−2.53407	−0.259990
$96$	0	0
$97$	−4.97930	−0.505572	−0.252786	−	0.967522i	$-0.581347\pi$
$97$	−4.97930	−0.505572	−0.252786	+	0.967522i	$0.581347\pi$
$98$	2.74483	0.277269
$99$	0	0
$100$	7.86675	0.786675
$101$	1.69105	0.168265	0.0841327	−	0.996455i	$-0.473188\pi$
$101$	1.69105	0.168265	0.0841327	+	0.996455i	$0.473188\pi$
$102$	0	0
$103$	3.15698	0.311066	0.155533	−	0.987831i	$-0.450290\pi$
$103$	3.15698	0.311066	0.155533	+	0.987831i	$0.450290\pi$
$104$	−4.08884	−0.400943
$105$	0	0
$106$	29.8273	2.89709
$107$	16.5341	1.59841	0.799204	−	0.601059i	$-0.205254\pi$
$107$	16.5341	1.59841	0.799204	+	0.601059i	$0.205254\pi$
$108$	0	0
$109$	1.77488	0.170003	0.0850015	−	0.996381i	$-0.472910\pi$
$109$	1.77488	0.170003	0.0850015	+	0.996381i	$0.472910\pi$
$110$	−16.8430	−1.60592
$111$	0	0
$112$	−15.5578	−1.47007
$113$	−10.0237	−0.942952	−0.471476	−	0.881879i	$-0.656279\pi$
$113$	−10.0237	−0.942952	−0.471476	+	0.881879i	$0.656279\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	−33.3357	−3.09514
$117$	0	0
$118$	−38.8016	−3.57198
$119$	6.53407	0.598977
$120$	0	0
$121$	−5.13628	−0.466935
$122$	18.7829	1.70052
$123$	0	0
$124$	19.3120	1.73427
$125$	−9.06814	−0.811079
$126$	0	0
$127$	2.08884	0.185354	0.0926771	−	0.995696i	$-0.470458\pi$
$127$	2.08884	0.185354	0.0926771	+	0.995696i	$0.470458\pi$
$128$	43.5515	3.84944
$129$	0	0
$130$	−2.93186	−0.257141
$131$	13.9349	1.21750	0.608748	−	0.793363i	$-0.291672\pi$
$131$	13.9349	1.21750	0.608748	+	0.793363i	$0.291672\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	−27.2044	−2.35010
$135$	0	0
$136$	−63.3831	−5.43506
$137$	−16.9793	−1.45064	−0.725320	−	0.688412i	$-0.758308\pi$
$137$	−16.9793	−1.45064	−0.725320	+	0.688412i	$0.758308\pi$
$138$	0	0
$139$	−1.06814	−0.0905985	−0.0452992	−	0.998973i	$-0.514424\pi$
$139$	−1.06814	−0.0905985	−0.0452992	+	0.998973i	$0.514424\pi$
$140$	−14.0237	−1.18522
$141$	0	0
$142$	36.7178	3.08129
$143$	1.02070	0.0853549
$144$	0	0
$145$	−15.2645	−1.26765
$146$	19.4008	1.60562
$147$	0	0
$148$	−15.7335	−1.29329
$149$	−14.0474	−1.15081	−0.575406	−	0.817868i	$-0.695156\pi$
$149$	−14.0474	−1.15081	−0.575406	+	0.817868i	$0.695156\pi$
$150$	0	0
$151$	−14.7542	−1.20068	−0.600339	−	0.799745i	$-0.704968\pi$
$151$	−14.7542	−1.20068	−0.600339	+	0.799745i	$0.704968\pi$
$152$	−9.70041	−0.786807
$153$	0	0
$154$	6.64663	0.535601
$155$	8.84302	0.710289
$156$	0	0
$157$	−16.1363	−1.28782	−0.643908	−	0.765103i	$-0.722688\pi$
$157$	−16.1363	−1.28782	−0.643908	+	0.765103i	$0.722688\pi$
$158$	38.8016	3.08689
$159$	0	0
$160$	59.0505	4.66835
$161$	−1.57849	−0.124402
$162$	0	0
$163$	−6.64663	−0.520604	−0.260302	−	0.965527i	$-0.583822\pi$
$163$	−6.64663	−0.520604	−0.260302	+	0.965527i	$0.583822\pi$
$164$	−46.6052	−3.63926
$165$	0	0
$166$	−5.79861	−0.450059
$167$	14.3614	1.11132	0.555659	−	0.831410i	$-0.312466\pi$
$167$	14.3614	1.11132	0.555659	+	0.831410i	$0.312466\pi$
$168$	0	0
$169$	−12.8223	−0.986333
$170$	−45.4483	−3.48572
$171$	0	0
$172$	63.5845	4.84828
$173$	5.71477	0.434486	0.217243	−	0.976118i	$-0.430294\pi$
$173$	5.71477	0.434486	0.217243	+	0.976118i	$0.430294\pi$
$174$	0	0
$175$	−1.42151	−0.107456
$176$	−37.6734	−2.83974
$177$	0	0
$178$	26.6654	1.99865
$179$	9.60221	0.717703	0.358851	−	0.933395i	$-0.383168\pi$
$179$	9.60221	0.717703	0.358851	+	0.933395i	$0.383168\pi$
$180$	0	0
$181$	−0.979304	−0.0727911	−0.0363956	−	0.999337i	$-0.511588\pi$
$181$	−0.979304	−0.0727911	−0.0363956	+	0.999337i	$0.511588\pi$
$182$	1.15698	0.0857608
$183$	0	0
$184$	15.3120	1.12881
$185$	−7.20442	−0.529680
$186$	0	0
$187$	15.8223	1.15704
$188$	−44.4038	−3.23848
$189$	0	0
$190$	−6.95558	−0.504611
$191$	11.7148	0.847651	0.423825	−	0.905744i	$-0.360687\pi$
$191$	11.7148	0.847651	0.423825	+	0.905744i	$0.360687\pi$
$192$	0	0
$193$	17.8223	1.28288	0.641440	−	0.767174i	$-0.278337\pi$
$193$	17.8223	1.28288	0.641440	+	0.767174i	$0.278337\pi$
$194$	−13.6673	−0.981257
$195$	0	0
$196$	5.53407	0.395291
$197$	19.1156	1.36193	0.680965	−	0.732316i	$-0.261561\pi$
$197$	19.1156	1.36193	0.680965	+	0.732316i	$0.261561\pi$
$198$	0	0
$199$	16.0474	1.13757	0.568787	−	0.822485i	$-0.307413\pi$
$199$	16.0474	1.13757	0.568787	+	0.822485i	$0.307413\pi$
$200$	13.7892	0.975047
$201$	0	0
$202$	4.64163	0.326584
$203$	6.02372	0.422782
$204$	0	0
$205$	−21.3407	−1.49050
$206$	8.66535	0.603744
$207$	0	0
$208$	−6.55779	−0.454701
$209$	2.42151	0.167499
$210$	0	0
$211$	−9.29326	−0.639774	−0.319887	−	0.947456i	$-0.603645\pi$
$211$	−9.29326	−0.639774	−0.319887	+	0.947456i	$0.603645\pi$
$212$	60.1373	4.13025
$213$	0	0
$214$	45.3831	3.10233
$215$	29.1156	1.98567
$216$	0	0
$217$	−3.48965	−0.236893
$218$	4.87175	0.329956
$219$	0	0
$220$	−33.9586	−2.28949
$221$	2.75419	0.185267
$222$	0	0
$223$	−8.33268	−0.557997	−0.278999	−	0.960292i	$-0.590002\pi$
$223$	−8.33268	−0.557997	−0.278999	+	0.960292i	$0.590002\pi$
$224$	−23.3026	−1.55697
$225$	0	0
$226$	−27.5134	−1.83016
$227$	−16.8905	−1.12106	−0.560530	−	0.828134i	$-0.689403\pi$
$227$	−16.8905	−1.12106	−0.560530	+	0.828134i	$0.689403\pi$
$228$	0	0
$229$	6.04744	0.399626	0.199813	−	0.979834i	$-0.435966\pi$
$229$	6.04744	0.399626	0.199813	+	0.979834i	$0.435966\pi$
$230$	10.9793	0.723954
$231$	0	0
$232$	−58.4326	−3.83629
$233$	−21.2044	−1.38915	−0.694574	−	0.719421i	$-0.744407\pi$
$233$	−21.2044	−1.38915	−0.694574	+	0.719421i	$0.744407\pi$
$234$	0	0
$235$	−20.3327	−1.32636
$236$	−78.2312	−5.09242
$237$	0	0
$238$	17.9349	1.16255
$239$	5.80361	0.375404	0.187702	−	0.982226i	$-0.439896\pi$
$239$	5.80361	0.375404	0.187702	+	0.982226i	$0.439896\pi$
$240$	0	0
$241$	−12.9793	−0.836070	−0.418035	−	0.908431i	$-0.637281\pi$
$241$	−12.9793	−0.836070	−0.418035	+	0.908431i	$0.637281\pi$
$242$	−14.0982	−0.906266
$243$	0	0
$244$	37.8698	2.42436
$245$	2.53407	0.161896
$246$	0	0
$247$	0.421512	0.0268202
$248$	33.8510	2.14954
$249$	0	0
$250$	−24.8905	−1.57421
$251$	−10.3377	−0.652508	−0.326254	−	0.945282i	$-0.605787\pi$
$251$	−10.3377	−0.652508	−0.326254	+	0.945282i	$0.605787\pi$
$252$	0	0
$253$	−3.82233	−0.240308
$254$	5.73349	0.359751
$255$	0	0
$256$	53.8491	3.36557
$257$	15.1757	0.946634	0.473317	−	0.880892i	$-0.343056\pi$
$257$	15.1757	0.946634	0.473317	+	0.880892i	$0.343056\pi$
$258$	0	0
$259$	2.84302	0.176657
$260$	−5.91116	−0.366595
$261$	0	0
$262$	38.2488	2.36302
$263$	8.51035	0.524771	0.262385	−	0.964963i	$-0.415491\pi$
$263$	8.51035	0.524771	0.262385	+	0.964963i	$0.415491\pi$
$264$	0	0
$265$	27.5371	1.69159
$266$	2.74483	0.168296
$267$	0	0
$268$	−54.8491	−3.35044
$269$	−15.1757	−0.925279	−0.462639	−	0.886547i	$-0.653098\pi$
$269$	−15.1757	−0.925279	−0.462639	+	0.886547i	$0.653098\pi$
$270$	0	0
$271$	0.843024	0.0512100	0.0256050	−	0.999672i	$-0.491849\pi$
$271$	0.843024	0.0512100	0.0256050	+	0.999672i	$0.491849\pi$
$272$	−101.656	−6.16378
$273$	0	0
$274$	−46.6052	−2.81553
$275$	−3.44221	−0.207573
$276$	0	0
$277$	10.5578	0.634356	0.317178	−	0.948366i	$-0.397265\pi$
$277$	10.5578	0.634356	0.317178	+	0.948366i	$0.397265\pi$
$278$	−2.93186	−0.175841
$279$	0	0
$280$	−24.5815	−1.46903
$281$	−15.0444	−0.897475	−0.448737	−	0.893664i	$-0.648126\pi$
$281$	−15.0444	−0.897475	−0.448737	+	0.893664i	$0.648126\pi$
$282$	0	0
$283$	1.24581	0.0740559	0.0370279	−	0.999314i	$-0.488211\pi$
$283$	1.24581	0.0740559	0.0370279	+	0.999314i	$0.488211\pi$
$284$	74.0298	4.39286
$285$	0	0
$286$	2.80163	0.165664
$287$	8.42151	0.497106
$288$	0	0
$289$	25.6941	1.51142
$290$	−41.8985	−2.46036
$291$	0	0
$292$	39.1156	2.28907
$293$	−20.4215	−1.19304	−0.596519	−	0.802599i	$-0.703450\pi$
$293$	−20.4215	−1.19304	−0.596519	+	0.802599i	$0.703450\pi$
$294$	0	0
$295$	−35.8223	−2.08566
$296$	−27.5785	−1.60297
$297$	0	0
$298$	−38.5578	−2.23359
$299$	−0.665351	−0.0384783
$300$	0	0
$301$	−11.4897	−0.662253
$302$	−40.4977	−2.33038
$303$	0	0
$304$	−15.5578	−0.892301
$305$	17.3407	0.992926
$306$	0	0
$307$	−2.64663	−0.151051	−0.0755255	−	0.997144i	$-0.524063\pi$
$307$	−2.64663	−0.151051	−0.0755255	+	0.997144i	$0.524063\pi$
$308$	13.4008	0.763582
$309$	0	0
$310$	24.2726	1.37859
$311$	5.09186	0.288733	0.144367	−	0.989524i	$-0.453886\pi$
$311$	5.09186	0.288733	0.144367	+	0.989524i	$0.453886\pi$
$312$	0	0
$313$	19.9586	1.12813	0.564064	−	0.825731i	$-0.309237\pi$
$313$	19.9586	1.12813	0.564064	+	0.825731i	$0.309237\pi$
$314$	−44.2913	−2.49950
$315$	0	0
$316$	78.2312	4.40085
$317$	6.81930	0.383010	0.191505	−	0.981492i	$-0.438663\pi$
$317$	6.81930	0.383010	0.191505	+	0.981492i	$0.438663\pi$
$318$	0	0
$319$	14.5865	0.816688
$320$	83.2342	4.65293
$321$	0	0
$322$	−4.33268	−0.241451
$323$	6.53407	0.363565
$324$	0	0
$325$	−0.599184	−0.0332367
$326$	−18.2438	−1.01043
$327$	0	0
$328$	−81.6921	−4.51069
$329$	8.02372	0.442362
$330$	0	0
$331$	−5.91116	−0.324907	−0.162453	−	0.986716i	$-0.551941\pi$
$331$	−5.91116	−0.324907	−0.162453	+	0.986716i	$0.551941\pi$
$332$	−11.6910	−0.641630
$333$	0	0
$334$	39.4195	2.15694
$335$	−25.1156	−1.37221
$336$	0	0
$337$	4.08884	0.222733	0.111367	−	0.993779i	$-0.464477\pi$
$337$	4.08884	0.222733	0.111367	+	0.993779i	$0.464477\pi$
$338$	−35.1951	−1.91436
$339$	0	0
$340$	−91.6320	−4.96944
$341$	−8.45023	−0.457606
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	111.454	6.00921
$345$	0	0
$346$	15.6860	0.843287
$347$	17.3534	0.931578	0.465789	−	0.884896i	$-0.345771\pi$
$347$	17.3534	0.931578	0.465789	+	0.884896i	$0.345771\pi$
$348$	0	0
$349$	−4.70674	−0.251946	−0.125973	−	0.992034i	$-0.540205\pi$
$349$	−4.70674	−0.251946	−0.125973	+	0.992034i	$0.540205\pi$
$350$	−3.90180	−0.208560
$351$	0	0
$352$	−56.4276	−3.00760
$353$	−20.6704	−1.10017	−0.550086	−	0.835108i	$-0.685405\pi$
$353$	−20.6704	−1.10017	−0.550086	+	0.835108i	$0.685405\pi$
$354$	0	0
$355$	33.8985	1.79915
$356$	53.7622	2.84939
$357$	0	0
$358$	26.3564	1.39298
$359$	−8.10756	−0.427901	−0.213950	−	0.976845i	$-0.568633\pi$
$359$	−8.10756	−0.427901	−0.213950	+	0.976845i	$0.568633\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−2.68802	−0.141279
$363$	0	0
$364$	2.33268	0.122265
$365$	17.9112	0.937513
$366$	0	0
$367$	28.8905	1.50807	0.754035	−	0.656834i	$-0.228105\pi$
$367$	28.8905	1.50807	0.754035	+	0.656834i	$0.228105\pi$
$368$	24.5578	1.28016
$369$	0	0
$370$	−19.7749	−1.02805
$371$	−10.8667	−0.564173
$372$	0	0
$373$	11.9586	0.619193	0.309597	−	0.950868i	$-0.399806\pi$
$373$	11.9586	0.619193	0.309597	+	0.950868i	$0.399806\pi$
$374$	43.4295	2.24569
$375$	0	0
$376$	−77.8334	−4.01395
$377$	2.53907	0.130769
$378$	0	0
$379$	−15.3821	−0.790125	−0.395063	−	0.918654i	$-0.629277\pi$
$379$	−15.3821	−0.790125	−0.395063	+	0.918654i	$0.629277\pi$
$380$	−14.0237	−0.719402
$381$	0	0
$382$	32.1550	1.64519
$383$	0.617907	0.0315736	0.0157868	−	0.999875i	$-0.494975\pi$
$383$	0.617907	0.0315736	0.0157868	+	0.999875i	$0.494975\pi$
$384$	0	0
$385$	6.13628	0.312734
$386$	48.9192	2.48992
$387$	0	0
$388$	−27.5558	−1.39893
$389$	11.0681	0.561177	0.280588	−	0.959828i	$-0.409470\pi$
$389$	11.0681	0.561177	0.280588	+	0.959828i	$0.409470\pi$
$390$	0	0
$391$	−10.3140	−0.521599
$392$	9.70041	0.489945
$393$	0	0
$394$	52.4690	2.64335
$395$	35.8223	1.80242
$396$	0	0
$397$	−10.4502	−0.524482	−0.262241	−	0.965002i	$-0.584462\pi$
$397$	−10.4502	−0.524482	−0.262241	+	0.965002i	$0.584462\pi$
$398$	44.0474	2.20790
$399$	0	0
$400$	22.1156	1.10578
$401$	−3.26953	−0.163273	−0.0816364	−	0.996662i	$-0.526015\pi$
$401$	−3.26953	−0.163273	−0.0816364	+	0.996662i	$0.526015\pi$
$402$	0	0
$403$	−1.47093	−0.0732722
$404$	9.35837	0.465596
$405$	0	0
$406$	16.5341	0.820572
$407$	6.88441	0.341248
$408$	0	0
$409$	7.40082	0.365947	0.182973	−	0.983118i	$-0.441428\pi$
$409$	7.40082	0.365947	0.182973	+	0.983118i	$0.441428\pi$
$410$	−58.5765	−2.89289
$411$	0	0
$412$	17.4709	0.860731
$413$	14.1363	0.695601
$414$	0	0
$415$	−5.35337	−0.262787
$416$	−9.82233	−0.481579
$417$	0	0
$418$	6.64663	0.325097
$419$	31.8461	1.55578	0.777891	−	0.628400i	$-0.216290\pi$
$419$	31.8461	1.55578	0.777891	+	0.628400i	$0.216290\pi$
$420$	0	0
$421$	40.0061	1.94978	0.974888	−	0.222696i	$-0.0714858\pi$
$421$	40.0061	1.94978	0.974888	+	0.222696i	$0.0714858\pi$
$422$	−25.5084	−1.24173
$423$	0	0
$424$	105.412	5.11925
$425$	−9.28826	−0.450547
$426$	0	0
$427$	−6.84302	−0.331157
$428$	91.5007	4.42285
$429$	0	0
$430$	79.9172	3.85395
$431$	−3.91616	−0.188635	−0.0943175	−	0.995542i	$-0.530067\pi$
$431$	−3.91616	−0.188635	−0.0943175	+	0.995542i	$0.530067\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	−9.57849	−0.459782
$435$	0	0
$436$	9.82233	0.470404
$437$	−1.57849	−0.0755093
$438$	0	0
$439$	17.4483	0.832760	0.416380	−	0.909191i	$-0.363299\pi$
$439$	17.4483	0.832760	0.416380	+	0.909191i	$0.363299\pi$
$440$	−59.5244	−2.83772
$441$	0	0
$442$	7.55977	0.359581
$443$	3.03942	0.144407	0.0722036	−	0.997390i	$-0.476997\pi$
$443$	3.03942	0.144407	0.0722036	+	0.997390i	$0.476997\pi$
$444$	0	0
$445$	24.6179	1.16700
$446$	−22.8717	−1.08301
$447$	0	0
$448$	−32.8461	−1.55183
$449$	−35.9349	−1.69587	−0.847936	−	0.530099i	$-0.822155\pi$
$449$	−35.9349	−1.69587	−0.847936	+	0.530099i	$0.822155\pi$
$450$	0	0
$451$	20.3928	0.960259
$452$	−55.4720	−2.60918
$453$	0	0
$454$	−46.3614	−2.17585
$455$	1.06814	0.0500752
$456$	0	0
$457$	−24.4215	−1.14239	−0.571195	−	0.820814i	$-0.693520\pi$
$457$	−24.4215	−1.14239	−0.571195	+	0.820814i	$0.693520\pi$
$458$	16.5992	0.775629
$459$	0	0
$460$	22.1363	1.03211
$461$	−10.1313	−0.471861	−0.235930	−	0.971770i	$-0.575814\pi$
$461$	−10.1313	−0.471861	−0.235930	+	0.971770i	$0.575814\pi$
$462$	0	0
$463$	−5.86372	−0.272510	−0.136255	−	0.990674i	$-0.543507\pi$
$463$	−5.86372	−0.272510	−0.136255	+	0.990674i	$0.543507\pi$
$464$	−93.7158	−4.35065
$465$	0	0
$466$	−58.2024	−2.69618
$467$	14.9556	0.692062	0.346031	−	0.938223i	$-0.387529\pi$
$467$	14.9556	0.692062	0.346031	+	0.938223i	$0.387529\pi$
$468$	0	0
$469$	9.91116	0.457655
$470$	−55.8097	−2.57431
$471$	0	0
$472$	−137.128	−6.31181
$473$	−27.8223	−1.27927
$474$	0	0
$475$	−1.42151	−0.0652234
$476$	36.1600	1.65739
$477$	0	0
$478$	15.9299	0.728616
$479$	33.7098	1.54024	0.770119	−	0.637900i	$-0.220197\pi$
$479$	33.7098	1.54024	0.770119	+	0.637900i	$0.220197\pi$
$480$	0	0
$481$	1.19837	0.0546409
$482$	−35.6259	−1.62272
$483$	0	0
$484$	−28.4245	−1.29202
$485$	−12.6179	−0.572950
$486$	0	0
$487$	21.5084	0.974637	0.487319	−	0.873224i	$-0.337975\pi$
$487$	21.5084	0.974637	0.487319	+	0.873224i	$0.337975\pi$
$488$	66.3801	3.00489
$489$	0	0
$490$	6.95558	0.314221
$491$	17.3534	0.783147	0.391573	−	0.920147i	$-0.371931\pi$
$491$	17.3534	0.783147	0.391573	+	0.920147i	$0.371931\pi$
$492$	0	0
$493$	39.3594	1.77266
$494$	1.15698	0.0520548
$495$	0	0
$496$	54.2913	2.43775
$497$	−13.3771	−0.600045
$498$	0	0
$499$	−32.9379	−1.47450	−0.737252	−	0.675618i	$-0.763877\pi$
$499$	−32.9379	−1.47450	−0.737252	+	0.675618i	$0.763877\pi$
$500$	−50.1837	−2.24428
$501$	0	0
$502$	−28.3751	−1.26644
$503$	38.2074	1.70359	0.851793	−	0.523879i	$-0.175515\pi$
$503$	38.2074	1.70359	0.851793	+	0.523879i	$0.175515\pi$
$504$	0	0
$505$	4.28523	0.190690
$506$	−10.4916	−0.466410
$507$	0	0
$508$	11.5598	0.512882
$509$	8.82430	0.391130	0.195565	−	0.980691i	$-0.437346\pi$
$509$	8.82430	0.391130	0.195565	+	0.980691i	$0.437346\pi$
$510$	0	0
$511$	−7.06814	−0.312676
$512$	60.7034	2.68274
$513$	0	0
$514$	41.6547	1.83731
$515$	8.00000	0.352522
$516$	0	0
$517$	19.4295	0.854510
$518$	7.80361	0.342871
$519$	0	0
$520$	−10.3614	−0.454377
$521$	24.2913	1.06422	0.532110	−	0.846675i	$-0.321399\pi$
$521$	24.2913	1.06422	0.532110	+	0.846675i	$0.321399\pi$
$522$	0	0
$523$	19.0969	0.835047	0.417524	−	0.908666i	$-0.362898\pi$
$523$	19.0969	0.835047	0.417524	+	0.908666i	$0.362898\pi$
$524$	77.1166	3.36886
$525$	0	0
$526$	23.3594	1.01852
$527$	−22.8016	−0.993255
$528$	0	0
$529$	−20.5084	−0.891668
$530$	75.5845	3.28318
$531$	0	0
$532$	5.53407	0.239932
$533$	3.54977	0.153757
$534$	0	0
$535$	41.8985	1.81143
$536$	−96.1423	−4.15272
$537$	0	0
$538$	−41.6547	−1.79586
$539$	−2.42151	−0.104302
$540$	0	0
$541$	−36.2312	−1.55770	−0.778850	−	0.627210i	$-0.784197\pi$
$541$	−36.2312	−1.55770	−0.778850	+	0.627210i	$0.784197\pi$
$542$	2.31395	0.0993928
$543$	0	0
$544$	−152.261	−6.52813
$545$	4.49768	0.192659
$546$	0	0
$547$	−29.9586	−1.28094	−0.640469	−	0.767984i	$-0.721260\pi$
$547$	−29.9586	−1.28094	−0.640469	+	0.767984i	$0.721260\pi$
$548$	−93.9647	−4.01397
$549$	0	0
$550$	−9.44826	−0.402875
$551$	6.02372	0.256619
$552$	0	0
$553$	−14.1363	−0.601136
$554$	28.9793	1.23121
$555$	0	0
$556$	−5.91116	−0.250689
$557$	14.4402	0.611852	0.305926	−	0.952055i	$-0.401034\pi$
$557$	14.4402	0.611852	0.305926	+	0.952055i	$0.401034\pi$
$558$	0	0
$559$	−4.84302	−0.204838
$560$	−39.4245	−1.66599
$561$	0	0
$562$	−41.2943	−1.74190
$563$	17.9112	0.754866	0.377433	−	0.926037i	$-0.376807\pi$
$563$	17.9112	0.754866	0.377433	+	0.926037i	$0.376807\pi$
$564$	0	0
$565$	−25.4008	−1.06862
$566$	3.41954	0.143734
$567$	0	0
$568$	129.763	5.44475
$569$	8.50535	0.356563	0.178281	−	0.983980i	$-0.442946\pi$
$569$	8.50535	0.356563	0.178281	+	0.983980i	$0.442946\pi$
$570$	0	0
$571$	−6.80163	−0.284639	−0.142320	−	0.989821i	$-0.545456\pi$
$571$	−6.80163	−0.284639	−0.142320	+	0.989821i	$0.545456\pi$
$572$	5.64860	0.236180
$573$	0	0
$574$	23.1156	0.964826
$575$	2.24384	0.0935746
$576$	0	0
$577$	−14.2726	−0.594175	−0.297087	−	0.954850i	$-0.596015\pi$
$577$	−14.2726	−0.594175	−0.297087	+	0.954850i	$0.596015\pi$
$578$	70.5258	2.93348
$579$	0	0
$580$	−84.4750	−3.50763
$581$	2.11256	0.0876437
$582$	0	0
$583$	−26.3140	−1.08981
$584$	68.5638	2.83719
$585$	0	0
$586$	−56.0535	−2.31555
$587$	−40.2963	−1.66321	−0.831603	−	0.555371i	$-0.812576\pi$
$587$	−40.2963	−1.66321	−0.831603	+	0.555371i	$0.812576\pi$
$588$	0	0
$589$	−3.48965	−0.143789
$590$	−98.3261	−4.04802
$591$	0	0
$592$	−44.2312	−1.81789
$593$	−26.1313	−1.07308	−0.536542	−	0.843874i	$-0.680270\pi$
$593$	−26.1313	−1.07308	−0.536542	+	0.843874i	$0.680270\pi$
$594$	0	0
$595$	16.5578	0.678803
$596$	−77.7395	−3.18434
$597$	0	0
$598$	−1.82627	−0.0746819
$599$	−3.06314	−0.125157	−0.0625783	−	0.998040i	$-0.519932\pi$
$599$	−3.06314	−0.125157	−0.0625783	+	0.998040i	$0.519932\pi$
$600$	0	0
$601$	28.1363	1.14770	0.573851	−	0.818959i	$-0.305449\pi$
$601$	28.1363	1.14770	0.573851	+	0.818959i	$0.305449\pi$
$602$	−31.5371	−1.28536
$603$	0	0
$604$	−81.6507	−3.32232
$605$	−13.0157	−0.529163
$606$	0	0
$607$	28.8430	1.17070	0.585351	−	0.810780i	$-0.300957\pi$
$607$	28.8430	1.17070	0.585351	+	0.810780i	$0.300957\pi$
$608$	−23.3026	−0.945046
$609$	0	0
$610$	47.5972	1.92715
$611$	3.38209	0.136825
$612$	0	0
$613$	22.2852	0.900092	0.450046	−	0.893005i	$-0.351408\pi$
$613$	22.2852	0.900092	0.450046	+	0.893005i	$0.351408\pi$
$614$	−7.26454	−0.293173
$615$	0	0
$616$	23.4897	0.946425
$617$	−4.13628	−0.166520	−0.0832602	−	0.996528i	$-0.526533\pi$
$617$	−4.13628	−0.166520	−0.0832602	+	0.996528i	$0.526533\pi$
$618$	0	0
$619$	−9.06814	−0.364479	−0.182240	−	0.983254i	$-0.558335\pi$
$619$	−9.06814	−0.364479	−0.182240	+	0.983254i	$0.558335\pi$
$620$	48.9379	1.96539
$621$	0	0
$622$	13.9763	0.560398
$623$	−9.71477	−0.389214
$624$	0	0
$625$	−30.0869	−1.20347
$626$	54.7829	2.18957
$627$	0	0
$628$	−89.2993	−3.56343
$629$	18.5765	0.740694
$630$	0	0
$631$	−41.6259	−1.65710	−0.828551	−	0.559913i	$-0.810834\pi$
$631$	−41.6259	−1.65710	−0.828551	+	0.559913i	$0.810834\pi$
$632$	137.128	5.45465
$633$	0	0
$634$	18.7178	0.743379
$635$	5.29326	0.210057
$636$	0	0
$637$	−0.421512	−0.0167009
$638$	40.0374	1.58510
$639$	0	0
$640$	110.362	4.36246
$641$	−3.09186	−0.122121	−0.0610606	−	0.998134i	$-0.519448\pi$
$641$	−3.09186	−0.122121	−0.0610606	+	0.998134i	$0.519448\pi$
$642$	0	0
$643$	2.93186	0.115621	0.0578106	−	0.998328i	$-0.481588\pi$
$643$	2.93186	0.115621	0.0578106	+	0.998328i	$0.481588\pi$
$644$	−8.73546	−0.344226
$645$	0	0
$646$	17.9349	0.705639
$647$	5.93489	0.233324	0.116662	−	0.993172i	$-0.462781\pi$
$647$	5.93489	0.233324	0.116662	+	0.993172i	$0.462781\pi$
$648$	0	0
$649$	34.2312	1.34369
$650$	−1.64466	−0.0645087
$651$	0	0
$652$	−36.7829	−1.44053
$653$	−1.82233	−0.0713132	−0.0356566	−	0.999364i	$-0.511352\pi$
$653$	−1.82233	−0.0713132	−0.0356566	+	0.999364i	$0.511352\pi$
$654$	0	0
$655$	35.3120	1.37975
$656$	−131.020	−5.11548
$657$	0	0
$658$	22.0237	0.858574
$659$	−10.3978	−0.405040	−0.202520	−	0.979278i	$-0.564913\pi$
$659$	−10.3978	−0.405040	−0.202520	+	0.979278i	$0.564913\pi$
$660$	0	0
$661$	23.2231	0.903276	0.451638	−	0.892201i	$-0.350840\pi$
$661$	23.2231	0.903276	0.451638	+	0.892201i	$0.350840\pi$
$662$	−16.2251	−0.630607
$663$	0	0
$664$	−20.4927	−0.795270
$665$	2.53407	0.0982670
$666$	0	0
$667$	−9.50837	−0.368166
$668$	79.4770	3.07506
$669$	0	0
$670$	−68.9379	−2.66330
$671$	−16.5705	−0.639696
$672$	0	0
$673$	13.5498	0.522305	0.261153	−	0.965298i	$-0.415897\pi$
$673$	13.5498	0.522305	0.261153	+	0.965298i	$0.415897\pi$
$674$	11.2231	0.432299
$675$	0	0
$676$	−70.9597	−2.72922
$677$	12.6466	0.486049	0.243025	−	0.970020i	$-0.421860\pi$
$677$	12.6466	0.486049	0.243025	+	0.970020i	$0.421860\pi$
$678$	0	0
$679$	4.97930	0.191088
$680$	−160.617	−6.15939
$681$	0	0
$682$	−23.1944	−0.888160
$683$	13.7799	0.527273	0.263636	−	0.964622i	$-0.415078\pi$
$683$	13.7799	0.527273	0.263636	+	0.964622i	$0.415078\pi$
$684$	0	0
$685$	−43.0267	−1.64397
$686$	−2.74483	−0.104798
$687$	0	0
$688$	178.754	6.81492
$689$	−4.58046	−0.174502
$690$	0	0
$691$	1.11559	0.0424389	0.0212194	−	0.999775i	$-0.493245\pi$
$691$	1.11559	0.0424389	0.0212194	+	0.999775i	$0.493245\pi$
$692$	31.6259	1.20224
$693$	0	0
$694$	47.6320	1.80809
$695$	−2.70674	−0.102673
$696$	0	0
$697$	55.0267	2.08429
$698$	−12.9192	−0.488999
$699$	0	0
$700$	−7.86675	−0.297335
$701$	−16.7067	−0.631005	−0.315502	−	0.948925i	$-0.602173\pi$
$701$	−16.7067	−0.631005	−0.315502	+	0.948925i	$0.602173\pi$
$702$	0	0
$703$	2.84302	0.107227
$704$	−79.5371	−2.99767
$705$	0	0
$706$	−56.7365	−2.13531
$707$	−1.69105	−0.0635984
$708$	0	0
$709$	−25.1443	−0.944314	−0.472157	−	0.881514i	$-0.656525\pi$
$709$	−25.1443	−0.944314	−0.472157	+	0.881514i	$0.656525\pi$
$710$	93.0455	3.49193
$711$	0	0
$712$	94.2372	3.53169
$713$	5.50837	0.206290
$714$	0	0
$715$	2.58651	0.0967302
$716$	53.1393	1.98591
$717$	0	0
$718$	−22.2538	−0.830506
$719$	−32.2488	−1.20268	−0.601339	−	0.798994i	$-0.705366\pi$
$719$	−32.2488	−1.20268	−0.601339	+	0.798994i	$0.705366\pi$
$720$	0	0
$721$	−3.15698	−0.117572
$722$	2.74483	0.102152
$723$	0	0
$724$	−5.41954	−0.201416
$725$	−8.56279	−0.318014
$726$	0	0
$727$	−30.2312	−1.12121	−0.560606	−	0.828083i	$-0.689432\pi$
$727$	−30.2312	−1.12121	−0.560606	+	0.828083i	$0.689432\pi$
$728$	4.08884	0.151542
$729$	0	0
$730$	49.1630	1.81961
$731$	−75.0742	−2.77672
$732$	0	0
$733$	25.0267	0.924384	0.462192	−	0.886780i	$-0.347063\pi$
$733$	25.0267	0.924384	0.462192	+	0.886780i	$0.347063\pi$
$734$	79.2993	2.92699
$735$	0	0
$736$	36.7829	1.35584
$737$	24.0000	0.884051
$738$	0	0
$739$	5.53104	0.203463	0.101731	−	0.994812i	$-0.467562\pi$
$739$	5.53104	0.203463	0.101731	+	0.994812i	$0.467562\pi$
$740$	−39.8698	−1.46564
$741$	0	0
$742$	−29.8273	−1.09500
$743$	−21.6971	−0.795989	−0.397995	−	0.917388i	$-0.630294\pi$
$743$	−21.6971	−0.795989	−0.397995	+	0.917388i	$0.630294\pi$
$744$	0	0
$745$	−35.5972	−1.30418
$746$	32.8243	1.20178
$747$	0	0
$748$	87.5619	3.20158
$749$	−16.5341	−0.604142
$750$	0	0
$751$	−48.9854	−1.78750	−0.893751	−	0.448564i	$-0.851935\pi$
$751$	−48.9854	−1.78750	−0.893751	+	0.448564i	$0.851935\pi$
$752$	−124.831	−4.55213
$753$	0	0
$754$	6.96931	0.253807
$755$	−37.3881	−1.36069
$756$	0	0
$757$	−4.04139	−0.146887	−0.0734434	−	0.997299i	$-0.523399\pi$
$757$	−4.04139	−0.146887	−0.0734434	+	0.997299i	$0.523399\pi$
$758$	−42.2212	−1.53354
$759$	0	0
$760$	−24.5815	−0.891665
$761$	36.9429	1.33918	0.669590	−	0.742731i	$-0.266470\pi$
$761$	36.9429	1.33918	0.669590	+	0.742731i	$0.266470\pi$
$762$	0	0
$763$	−1.77488	−0.0642551
$764$	64.8304	2.34548
$765$	0	0
$766$	1.69605	0.0612806
$767$	5.95861	0.215153
$768$	0	0
$769$	24.1363	0.870377	0.435188	−	0.900339i	$-0.356682\pi$
$769$	24.1363	0.870377	0.435188	+	0.900339i	$0.356682\pi$
$770$	16.8430	0.606980
$771$	0	0
$772$	98.6300	3.54977
$773$	25.5371	0.918506	0.459253	−	0.888306i	$-0.348117\pi$
$773$	25.5371	0.918506	0.459253	+	0.888306i	$0.348117\pi$
$774$	0	0
$775$	4.96058	0.178189
$776$	−48.3013	−1.73392
$777$	0	0
$778$	30.3801	1.08918
$779$	8.42151	0.301732
$780$	0	0
$781$	−32.3928	−1.15911
$782$	−28.3100	−1.01236
$783$	0	0
$784$	15.5578	0.555635
$785$	−40.8905	−1.45944
$786$	0	0
$787$	−19.0969	−0.680730	−0.340365	−	0.940293i	$-0.610551\pi$
$787$	−19.0969	−0.680730	−0.340365	+	0.940293i	$0.610551\pi$
$788$	105.787	3.76851
$789$	0	0
$790$	98.3261	3.49828
$791$	10.0237	0.356403
$792$	0	0
$793$	−2.88441	−0.102429
$794$	−28.6841	−1.01796
$795$	0	0
$796$	88.8077	3.14770
$797$	−8.02872	−0.284392	−0.142196	−	0.989839i	$-0.545416\pi$
$797$	−8.02872	−0.284392	−0.142196	+	0.989839i	$0.545416\pi$
$798$	0	0
$799$	52.4276	1.85475
$800$	33.1249	1.17114
$801$	0	0
$802$	−8.97430	−0.316894
$803$	−17.1156	−0.603996
$804$	0	0
$805$	−4.00000	−0.140981
$806$	−4.03745	−0.142213
$807$	0	0
$808$	16.4038	0.577085
$809$	29.0742	1.02219	0.511097	−	0.859523i	$-0.329239\pi$
$809$	29.0742	1.02219	0.511097	+	0.859523i	$0.329239\pi$
$810$	0	0
$811$	4.66535	0.163823	0.0819113	−	0.996640i	$-0.473898\pi$
$811$	4.66535	0.163823	0.0819113	+	0.996640i	$0.473898\pi$
$812$	33.3357	1.16985
$813$	0	0
$814$	18.8965	0.662323
$815$	−16.8430	−0.589985
$816$	0	0
$817$	−11.4897	−0.401972
$818$	20.3140	0.710261
$819$	0	0
$820$	−118.101	−4.12426
$821$	17.5972	0.614147	0.307073	−	0.951686i	$-0.400650\pi$
$821$	17.5972	0.614147	0.307073	+	0.951686i	$0.400650\pi$
$822$	0	0
$823$	−7.93989	−0.276767	−0.138384	−	0.990379i	$-0.544191\pi$
$823$	−7.93989	−0.276767	−0.138384	+	0.990379i	$0.544191\pi$
$824$	30.6240	1.06684
$825$	0	0
$826$	38.8016	1.35008
$827$	31.7859	1.10531	0.552653	−	0.833412i	$-0.313616\pi$
$827$	31.7859	1.10531	0.552653	+	0.833412i	$0.313616\pi$
$828$	0	0
$829$	−48.9793	−1.70112	−0.850561	−	0.525877i	$-0.823737\pi$
$829$	−48.9793	−1.70112	−0.850561	+	0.525877i	$0.823737\pi$
$830$	−14.6941	−0.510039
$831$	0	0
$832$	−13.8450	−0.479989
$833$	−6.53407	−0.226392
$834$	0	0
$835$	36.3928	1.25942
$836$	13.4008	0.463477
$837$	0	0
$838$	87.4119	3.01959
$839$	−36.2726	−1.25227	−0.626134	−	0.779716i	$-0.715364\pi$
$839$	−36.2726	−1.25227	−0.626134	+	0.779716i	$0.715364\pi$
$840$	0	0
$841$	7.28523	0.251215
$842$	109.810	3.78429
$843$	0	0
$844$	−51.4295	−1.77028
$845$	−32.4927	−1.11778
$846$	0	0
$847$	5.13628	0.176485
$848$	169.063	5.80563
$849$	0	0
$850$	−25.4947	−0.874459
$851$	−4.48768	−0.153836
$852$	0	0
$853$	−34.6654	−1.18692	−0.593460	−	0.804864i	$-0.702238\pi$
$853$	−34.6654	−1.18692	−0.593460	+	0.804864i	$0.702238\pi$
$854$	−18.7829	−0.642738
$855$	0	0
$856$	160.387	5.48192
$857$	4.87175	0.166416	0.0832078	−	0.996532i	$-0.473483\pi$
$857$	4.87175	0.166416	0.0832078	+	0.996532i	$0.473483\pi$
$858$	0	0
$859$	−37.1156	−1.26637	−0.633184	−	0.774002i	$-0.718252\pi$
$859$	−37.1156	−1.26637	−0.633184	+	0.774002i	$0.718252\pi$
$860$	161.128	5.49441
$861$	0	0
$862$	−10.7492	−0.366119
$863$	29.1520	0.992345	0.496172	−	0.868224i	$-0.334738\pi$
$863$	29.1520	0.992345	0.496172	+	0.868224i	$0.334738\pi$
$864$	0	0
$865$	14.4816	0.492390
$866$	49.4069	1.67891
$867$	0	0
$868$	−19.3120	−0.655491
$869$	−34.2312	−1.16121
$870$	0	0
$871$	4.17767	0.141555
$872$	17.2171	0.583044
$873$	0	0
$874$	−4.33268	−0.146555
$875$	9.06814	0.306559
$876$	0	0
$877$	14.4502	0.487950	0.243975	−	0.969782i	$-0.421549\pi$
$877$	14.4502	0.487950	0.243975	+	0.969782i	$0.421549\pi$
$878$	47.8924	1.61629
$879$	0	0
$880$	−95.4670	−3.21819
$881$	−10.5341	−0.354902	−0.177451	−	0.984130i	$-0.556785\pi$
$881$	−10.5341	−0.354902	−0.177451	+	0.984130i	$0.556785\pi$
$882$	0	0
$883$	−5.11559	−0.172153	−0.0860766	−	0.996289i	$-0.527433\pi$
$883$	−5.11559	−0.172153	−0.0860766	+	0.996289i	$0.527433\pi$
$884$	15.2419	0.512639
$885$	0	0
$886$	8.34267	0.280278
$887$	0.617907	0.0207473	0.0103736	−	0.999946i	$-0.496698\pi$
$887$	0.617907	0.0207473	0.0103736	+	0.999946i	$0.496698\pi$
$888$	0	0
$889$	−2.08884	−0.0700573
$890$	67.5719	2.26501
$891$	0	0
$892$	−46.1136	−1.54400
$893$	8.02372	0.268504
$894$	0	0
$895$	24.3327	0.813352
$896$	−43.5515	−1.45495
$897$	0	0
$898$	−98.6350	−3.29149
$899$	−21.0207	−0.701079
$900$	0	0
$901$	−71.0041	−2.36549
$902$	55.9747	1.86375
$903$	0	0
$904$	−97.2342	−3.23396
$905$	−2.48163	−0.0824920
$906$	0	0
$907$	−8.27256	−0.274686	−0.137343	−	0.990524i	$-0.543856\pi$
$907$	−8.27256	−0.274686	−0.137343	+	0.990524i	$0.543856\pi$
$908$	−93.4730	−3.10201
$909$	0	0
$910$	2.93186	0.0971902
$911$	5.77988	0.191496	0.0957480	−	0.995406i	$-0.469476\pi$
$911$	5.77988	0.191496	0.0957480	+	0.995406i	$0.469476\pi$
$912$	0	0
$913$	5.11559	0.169301
$914$	−67.0328	−2.21725
$915$	0	0
$916$	33.4670	1.10578
$917$	−13.9349	−0.460170
$918$	0	0
$919$	4.62791	0.152661	0.0763303	−	0.997083i	$-0.475680\pi$
$919$	4.62791	0.152661	0.0763303	+	0.997083i	$0.475680\pi$
$920$	38.8016	1.27925
$921$	0	0
$922$	−27.8086	−0.915828
$923$	−5.63860	−0.185597
$924$	0	0
$925$	−4.04139	−0.132880
$926$	−16.0949	−0.528911
$927$	0	0
$928$	−140.369	−4.60782
$929$	29.9536	0.982746	0.491373	−	0.870949i	$-0.336495\pi$
$929$	29.9536	0.982746	0.491373	+	0.870949i	$0.336495\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	−117.347	−3.84382
$933$	0	0
$934$	41.0505	1.34321
$935$	40.0949	1.31124
$936$	0	0
$937$	4.58651	0.149835	0.0749174	−	0.997190i	$-0.476131\pi$
$937$	4.58651	0.149835	0.0749174	+	0.997190i	$0.476131\pi$
$938$	27.2044	0.888256
$939$	0	0
$940$	−112.522	−3.67008
$941$	−7.41082	−0.241586	−0.120793	−	0.992678i	$-0.538544\pi$
$941$	−7.41082	−0.241586	−0.120793	+	0.992678i	$0.538544\pi$
$942$	0	0
$943$	−13.2933	−0.432888
$944$	−219.929	−7.15809
$945$	0	0
$946$	−76.3675	−2.48292
$947$	−59.6320	−1.93778	−0.968890	−	0.247493i	$-0.920393\pi$
$947$	−59.6320	−1.93778	−0.968890	+	0.247493i	$0.920393\pi$
$948$	0	0
$949$	−2.97930	−0.0967123
$950$	−3.90180	−0.126591
$951$	0	0
$952$	63.3831	2.05426
$953$	−25.7986	−0.835699	−0.417849	−	0.908516i	$-0.637216\pi$
$953$	−25.7986	−0.835699	−0.417849	+	0.908516i	$0.637216\pi$
$954$	0	0
$955$	29.6860	0.960618
$956$	32.1176	1.03876
$957$	0	0
$958$	92.5275	2.98943
$959$	16.9793	0.548290
$960$	0	0
$961$	−18.8223	−0.607172
$962$	3.28931	0.106052
$963$	0	0
$964$	−71.8284	−2.31344
$965$	45.1630	1.45385
$966$	0	0
$967$	55.5845	1.78748	0.893739	−	0.448587i	$-0.148073\pi$
$967$	55.5845	1.78748	0.893739	+	0.448587i	$0.148073\pi$
$968$	−49.8240	−1.60140
$969$	0	0
$970$	−34.6340	−1.11203
$971$	48.4877	1.55604	0.778022	−	0.628237i	$-0.216223\pi$
$971$	48.4877	1.55604	0.778022	+	0.628237i	$0.216223\pi$
$972$	0	0
$973$	1.06814	0.0342430
$974$	59.0367	1.89166
$975$	0	0
$976$	106.462	3.40778
$977$	−53.0505	−1.69723	−0.848617	−	0.529007i	$-0.822565\pi$
$977$	−53.0505	−1.69723	−0.848617	+	0.529007i	$0.822565\pi$
$978$	0	0
$979$	−23.5244	−0.751844
$980$	14.0237	0.447971
$981$	0	0
$982$	47.6320	1.52000
$983$	−12.1777	−0.388407	−0.194204	−	0.980961i	$-0.562212\pi$
$983$	−12.1777	−0.388407	−0.194204	+	0.980961i	$0.562212\pi$
$984$	0	0
$985$	48.4402	1.54343
$986$	108.035	3.44053
$987$	0	0
$988$	2.33268	0.0742123
$989$	18.1363	0.576700
$990$	0	0
$991$	−2.04139	−0.0648469	−0.0324235	−	0.999474i	$-0.510323\pi$
$991$	−2.04139	−0.0648469	−0.0324235	+	0.999474i	$0.510323\pi$
$992$	81.3180	2.58185
$993$	0	0
$994$	−36.7178	−1.16462
$995$	40.6654	1.28918
$996$	0	0
$997$	−45.6447	−1.44558	−0.722790	−	0.691067i	$-0.757141\pi$
$997$	−45.6447	−1.44558	−0.722790	+	0.691067i	$0.757141\pi$
$998$	−90.4088	−2.86184
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1197.2.a.m.1.3		3
3.2	odd	2		399.2.a.e.1.1	✓	3
7.6	odd	2		8379.2.a.bq.1.3		3
12.11	even	2		6384.2.a.bu.1.1		3
15.14	odd	2		9975.2.a.x.1.3		3
21.20	even	2		2793.2.a.w.1.1		3
57.56	even	2		7581.2.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.e.1.1	✓	3	3.2	odd	2
1197.2.a.m.1.3		3	1.1	even	1	trivial
2793.2.a.w.1.1		3	21.20	even	2
6384.2.a.bu.1.1		3	12.11	even	2
7581.2.a.l.1.3		3	57.56	even	2
8379.2.a.bq.1.3		3	7.6	odd	2
9975.2.a.x.1.3		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.74483 q^{2} +5.53407 q^{4} +2.53407 q^{5} -1.00000 q^{7} +9.70041 q^{8} +6.95558 q^{10} -2.42151 q^{11} -0.421512 q^{13} -2.74483 q^{14} +15.5578 q^{16} -6.53407 q^{17} -1.00000 q^{19} +14.0237 q^{20} -6.64663 q^{22} +1.57849 q^{23} +1.42151 q^{25} -1.15698 q^{26} -5.53407 q^{28} -6.02372 q^{29} +3.48965 q^{31} +23.3026 q^{32} -17.9349 q^{34} -2.53407 q^{35} -2.84302 q^{37} -2.74483 q^{38} +24.5815 q^{40} -8.42151 q^{41} +11.4897 q^{43} -13.4008 q^{44} +4.33268 q^{46} -8.02372 q^{47} +1.00000 q^{49} +3.90180 q^{50} -2.33268 q^{52} +10.8667 q^{53} -6.13628 q^{55} -9.70041 q^{56} -16.5341 q^{58} -14.1363 q^{59} +6.84302 q^{61} +9.57849 q^{62} +32.8461 q^{64} -1.06814 q^{65} -9.91116 q^{67} -36.1600 q^{68} -6.95558 q^{70} +13.3771 q^{71} +7.06814 q^{73} -7.80361 q^{74} -5.53407 q^{76} +2.42151 q^{77} +14.1363 q^{79} +39.4245 q^{80} -23.1156 q^{82} -2.11256 q^{83} -16.5578 q^{85} +31.5371 q^{86} -23.4897 q^{88} +9.71477 q^{89} +0.421512 q^{91} +8.73546 q^{92} -22.0237 q^{94} -2.53407 q^{95} -4.97930 q^{97} +2.74483 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 9 q^{4} - 3 q^{7} + 9 q^{8} + 10 q^{10} - 4 q^{11} + 2 q^{13} + q^{14} + 13 q^{16} - 12 q^{17} - 3 q^{19} + 16 q^{20} - 8 q^{22} + 8 q^{23} + q^{25} - 10 q^{26} - 9 q^{28} + 8 q^{29} - 8 q^{31}+ \cdots - q^{98}+O(q^{100}) \) 3 * q - q^2 + 9 * q^4 - 3 * q^7 + 9 * q^8 + 10 * q^10 - 4 * q^11 + 2 * q^13 + q^14 + 13 * q^16 - 12 * q^17 - 3 * q^19 + 16 * q^20 - 8 * q^22 + 8 * q^23 + q^25 - 10 * q^26 - 9 * q^28 + 8 * q^29 - 8 * q^31 + 27 * q^32 - 6 * q^34 - 2 * q^37 + q^38 + 14 * q^40 - 22 * q^41 + 16 * q^43 - 12 * q^46 + 2 * q^47 + 3 * q^49 + 9 * q^50 + 18 * q^52 + 12 * q^55 - 9 * q^56 - 42 * q^58 - 12 * q^59 + 14 * q^61 + 32 * q^62 + 29 * q^64 + 12 * q^65 - 8 * q^67 - 52 * q^68 - 10 * q^70 + 26 * q^71 + 6 * q^73 - 18 * q^74 - 9 * q^76 + 4 * q^77 + 12 * q^79 + 52 * q^80 - 2 * q^82 - 2 * q^83 - 16 * q^85 + 24 * q^86 - 52 * q^88 + 2 * q^89 - 2 * q^91 + 36 * q^92 - 40 * q^94 + 22 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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