LMFDB - Embedded newform 1449.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,2,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, 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("1449.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1449.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:	$11.5703232529$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.61803 q^{2} +0.618034 q^{4} +0.618034 q^{5} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{10} -2.23607 q^{11} -2.38197 q^{13} -1.61803 q^{14} -4.85410 q^{16} -6.70820 q^{17} -3.47214 q^{19} +0.381966 q^{20} -3.61803 q^{22} +1.00000 q^{23} -4.61803 q^{25} -3.85410 q^{26} -0.618034 q^{28} +8.23607 q^{29} +6.70820 q^{31} -3.38197 q^{32} -10.8541 q^{34} -0.618034 q^{35} -11.0000 q^{37} -5.61803 q^{38} -1.38197 q^{40} +1.47214 q^{41} -1.61803 q^{43} -1.38197 q^{44} +1.61803 q^{46} +7.23607 q^{47} +1.00000 q^{49} -7.47214 q^{50} -1.47214 q^{52} +13.0902 q^{53} -1.38197 q^{55} +2.23607 q^{56} +13.3262 q^{58} -9.38197 q^{59} -4.85410 q^{61} +10.8541 q^{62} +4.23607 q^{64} -1.47214 q^{65} -5.09017 q^{67} -4.14590 q^{68} -1.00000 q^{70} -4.38197 q^{71} -12.7082 q^{73} -17.7984 q^{74} -2.14590 q^{76} +2.23607 q^{77} -9.47214 q^{79} -3.00000 q^{80} +2.38197 q^{82} +9.18034 q^{83} -4.14590 q^{85} -2.61803 q^{86} +5.00000 q^{88} -11.6180 q^{89} +2.38197 q^{91} +0.618034 q^{92} +11.7082 q^{94} -2.14590 q^{95} +10.4164 q^{97} +1.61803 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - q^{5} - 2 q^{7} + 2 q^{10} - 7 q^{13} - q^{14} - 3 q^{16} + 2 q^{19} + 3 q^{20} - 5 q^{22} + 2 q^{23} - 7 q^{25} - q^{26} + q^{28} + 12 q^{29} - 9 q^{32} - 15 q^{34} + q^{35}+ \cdots + q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - q^5 - 2 * q^7 + 2 * q^10 - 7 * q^13 - q^14 - 3 * q^16 + 2 * q^19 + 3 * q^20 - 5 * q^22 + 2 * q^23 - 7 * q^25 - q^26 + q^28 + 12 * q^29 - 9 * q^32 - 15 * q^34 + q^35 - 22 * q^37 - 9 * q^38 - 5 * q^40 - 6 * q^41 - q^43 - 5 * q^44 + q^46 + 10 * q^47 + 2 * q^49 - 6 * q^50 + 6 * q^52 + 15 * q^53 - 5 * q^55 + 11 * q^58 - 21 * q^59 - 3 * q^61 + 15 * q^62 + 4 * q^64 + 6 * q^65 + q^67 - 15 * q^68 - 2 * q^70 - 11 * q^71 - 12 * q^73 - 11 * q^74 - 11 * q^76 - 10 * q^79 - 6 * q^80 + 7 * q^82 - 4 * q^83 - 15 * q^85 - 3 * q^86 + 10 * q^88 - 21 * q^89 + 7 * q^91 - q^92 + 10 * q^94 - 11 * q^95 - 6 * 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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	0.618034	0.276393	0.138197	−	0.990405i	$-0.455869\pi$
$5$	0.618034	0.276393	0.138197	+	0.990405i	$0.455869\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	1.00000	0.316228
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	0	0
$13$	−2.38197	−0.660639	−0.330319	−	0.943869i	$-0.607156\pi$
$13$	−2.38197	−0.660639	−0.330319	+	0.943869i	$0.607156\pi$
$14$	−1.61803	−0.432438
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−6.70820	−1.62698	−0.813489	−	0.581580i	$-0.802435\pi$
$17$	−6.70820	−1.62698	−0.813489	+	0.581580i	$0.802435\pi$
$18$	0	0
$19$	−3.47214	−0.796563	−0.398281	−	0.917263i	$-0.630393\pi$
$19$	−3.47214	−0.796563	−0.398281	+	0.917263i	$0.630393\pi$
$20$	0.381966	0.0854102
$21$	0	0
$22$	−3.61803	−0.771367
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.61803	−0.923607
$26$	−3.85410	−0.755852
$27$	0	0
$28$	−0.618034	−0.116797
$29$	8.23607	1.52940	0.764700	−	0.644387i	$-0.222887\pi$
$29$	8.23607	1.52940	0.764700	+	0.644387i	$0.222887\pi$
$30$	0	0
$31$	6.70820	1.20483	0.602414	−	0.798183i	$-0.294205\pi$
$31$	6.70820	1.20483	0.602414	+	0.798183i	$0.294205\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	−10.8541	−1.86146
$35$	−0.618034	−0.104467
$36$	0	0
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	−5.61803	−0.911365
$39$	0	0
$40$	−1.38197	−0.218508
$41$	1.47214	0.229909	0.114955	−	0.993371i	$-0.463328\pi$
$41$	1.47214	0.229909	0.114955	+	0.993371i	$0.463328\pi$
$42$	0	0
$43$	−1.61803	−0.246748	−0.123374	−	0.992360i	$-0.539371\pi$
$43$	−1.61803	−0.246748	−0.123374	+	0.992360i	$0.539371\pi$
$44$	−1.38197	−0.208339
$45$	0	0
$46$	1.61803	0.238566
$47$	7.23607	1.05549	0.527744	−	0.849403i	$-0.323038\pi$
$47$	7.23607	1.05549	0.527744	+	0.849403i	$0.323038\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−7.47214	−1.05672
$51$	0	0
$52$	−1.47214	−0.204149
$53$	13.0902	1.79807	0.899037	−	0.437874i	$-0.144268\pi$
$53$	13.0902	1.79807	0.899037	+	0.437874i	$0.144268\pi$
$54$	0	0
$55$	−1.38197	−0.186344
$56$	2.23607	0.298807
$57$	0	0
$58$	13.3262	1.74982
$59$	−9.38197	−1.22143	−0.610714	−	0.791851i	$-0.709117\pi$
$59$	−9.38197	−1.22143	−0.610714	+	0.791851i	$0.709117\pi$
$60$	0	0
$61$	−4.85410	−0.621504	−0.310752	−	0.950491i	$-0.600581\pi$
$61$	−4.85410	−0.621504	−0.310752	+	0.950491i	$0.600581\pi$
$62$	10.8541	1.37847
$63$	0	0
$64$	4.23607	0.529508
$65$	−1.47214	−0.182596
$66$	0	0
$67$	−5.09017	−0.621863	−0.310932	−	0.950432i	$-0.600641\pi$
$67$	−5.09017	−0.621863	−0.310932	+	0.950432i	$0.600641\pi$
$68$	−4.14590	−0.502764
$69$	0	0
$70$	−1.00000	−0.119523
$71$	−4.38197	−0.520044	−0.260022	−	0.965603i	$-0.583730\pi$
$71$	−4.38197	−0.520044	−0.260022	+	0.965603i	$0.583730\pi$
$72$	0	0
$73$	−12.7082	−1.48738	−0.743691	−	0.668523i	$-0.766927\pi$
$73$	−12.7082	−1.48738	−0.743691	+	0.668523i	$0.766927\pi$
$74$	−17.7984	−2.06902
$75$	0	0
$76$	−2.14590	−0.246151
$77$	2.23607	0.254824
$78$	0	0
$79$	−9.47214	−1.06570	−0.532849	−	0.846210i	$-0.678879\pi$
$79$	−9.47214	−1.06570	−0.532849	+	0.846210i	$0.678879\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	2.38197	0.263044
$83$	9.18034	1.00767	0.503837	−	0.863799i	$-0.331921\pi$
$83$	9.18034	1.00767	0.503837	+	0.863799i	$0.331921\pi$
$84$	0	0
$85$	−4.14590	−0.449686
$86$	−2.61803	−0.282310
$87$	0	0
$88$	5.00000	0.533002
$89$	−11.6180	−1.23151	−0.615755	−	0.787938i	$-0.711149\pi$
$89$	−11.6180	−1.23151	−0.615755	+	0.787938i	$0.711149\pi$
$90$	0	0
$91$	2.38197	0.249698
$92$	0.618034	0.0644345
$93$	0	0
$94$	11.7082	1.20761
$95$	−2.14590	−0.220164
$96$	0	0
$97$	10.4164	1.05763	0.528813	−	0.848738i	$-0.322637\pi$
$97$	10.4164	1.05763	0.528813	+	0.848738i	$0.322637\pi$
$98$	1.61803	0.163446
$99$	0	0
$100$	−2.85410	−0.285410
$101$	−4.14590	−0.412532	−0.206266	−	0.978496i	$-0.566131\pi$
$101$	−4.14590	−0.412532	−0.206266	+	0.978496i	$0.566131\pi$
$102$	0	0
$103$	−7.41641	−0.730760	−0.365380	−	0.930858i	$-0.619061\pi$
$103$	−7.41641	−0.730760	−0.365380	+	0.930858i	$0.619061\pi$
$104$	5.32624	0.522281
$105$	0	0
$106$	21.1803	2.05722
$107$	−3.32624	−0.321560	−0.160780	−	0.986990i	$-0.551401\pi$
$107$	−3.32624	−0.321560	−0.160780	+	0.986990i	$0.551401\pi$
$108$	0	0
$109$	19.2705	1.84578	0.922890	−	0.385064i	$-0.125820\pi$
$109$	19.2705	1.84578	0.922890	+	0.385064i	$0.125820\pi$
$110$	−2.23607	−0.213201
$111$	0	0
$112$	4.85410	0.458670
$113$	0.909830	0.0855896	0.0427948	−	0.999084i	$-0.486374\pi$
$113$	0.909830	0.0855896	0.0427948	+	0.999084i	$0.486374\pi$
$114$	0	0
$115$	0.618034	0.0576320
$116$	5.09017	0.472610
$117$	0	0
$118$	−15.1803	−1.39746
$119$	6.70820	0.614940
$120$	0	0
$121$	−6.00000	−0.545455
$122$	−7.85410	−0.711077
$123$	0	0
$124$	4.14590	0.372313
$125$	−5.94427	−0.531672
$126$	0	0
$127$	22.2705	1.97619	0.988094	−	0.153851i	$-0.0491675\pi$
$127$	22.2705	1.97619	0.988094	+	0.153851i	$0.0491675\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	−2.38197	−0.208912
$131$	−15.1803	−1.32631	−0.663156	−	0.748481i	$-0.730784\pi$
$131$	−15.1803	−1.32631	−0.663156	+	0.748481i	$0.730784\pi$
$132$	0	0
$133$	3.47214	0.301072
$134$	−8.23607	−0.711488
$135$	0	0
$136$	15.0000	1.28624
$137$	16.4164	1.40255	0.701274	−	0.712892i	$-0.252615\pi$
$137$	16.4164	1.40255	0.701274	+	0.712892i	$0.252615\pi$
$138$	0	0
$139$	−11.3820	−0.965406	−0.482703	−	0.875784i	$-0.660345\pi$
$139$	−11.3820	−0.965406	−0.482703	+	0.875784i	$0.660345\pi$
$140$	−0.381966	−0.0322820
$141$	0	0
$142$	−7.09017	−0.594994
$143$	5.32624	0.445402
$144$	0	0
$145$	5.09017	0.422716
$146$	−20.5623	−1.70175
$147$	0	0
$148$	−6.79837	−0.558823
$149$	19.2361	1.57588	0.787940	−	0.615752i	$-0.211148\pi$
$149$	19.2361	1.57588	0.787940	+	0.615752i	$0.211148\pi$
$150$	0	0
$151$	−15.2361	−1.23989	−0.619947	−	0.784644i	$-0.712846\pi$
$151$	−15.2361	−1.23989	−0.619947	+	0.784644i	$0.712846\pi$
$152$	7.76393	0.629738
$153$	0	0
$154$	3.61803	0.291549
$155$	4.14590	0.333007
$156$	0	0
$157$	−0.291796	−0.0232879	−0.0116439	−	0.999932i	$-0.503706\pi$
$157$	−0.291796	−0.0232879	−0.0116439	+	0.999932i	$0.503706\pi$
$158$	−15.3262	−1.21929
$159$	0	0
$160$	−2.09017	−0.165242
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−0.618034	−0.0484082	−0.0242041	−	0.999707i	$-0.507705\pi$
$163$	−0.618034	−0.0484082	−0.0242041	+	0.999707i	$0.507705\pi$
$164$	0.909830	0.0710458
$165$	0	0
$166$	14.8541	1.15290
$167$	7.18034	0.555631	0.277816	−	0.960634i	$-0.410390\pi$
$167$	7.18034	0.555631	0.277816	+	0.960634i	$0.410390\pi$
$168$	0	0
$169$	−7.32624	−0.563557
$170$	−6.70820	−0.514496
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	3.47214	0.263982	0.131991	−	0.991251i	$-0.457863\pi$
$173$	3.47214	0.263982	0.131991	+	0.991251i	$0.457863\pi$
$174$	0	0
$175$	4.61803	0.349091
$176$	10.8541	0.818159
$177$	0	0
$178$	−18.7984	−1.40900
$179$	−1.85410	−0.138582	−0.0692910	−	0.997596i	$-0.522074\pi$
$179$	−1.85410	−0.138582	−0.0692910	+	0.997596i	$0.522074\pi$
$180$	0	0
$181$	−1.94427	−0.144517	−0.0722583	−	0.997386i	$-0.523021\pi$
$181$	−1.94427	−0.144517	−0.0722583	+	0.997386i	$0.523021\pi$
$182$	3.85410	0.285685
$183$	0	0
$184$	−2.23607	−0.164845
$185$	−6.79837	−0.499826
$186$	0	0
$187$	15.0000	1.09691
$188$	4.47214	0.326164
$189$	0	0
$190$	−3.47214	−0.251895
$191$	−16.1803	−1.17077	−0.585384	−	0.810756i	$-0.699056\pi$
$191$	−16.1803	−1.17077	−0.585384	+	0.810756i	$0.699056\pi$
$192$	0	0
$193$	8.29180	0.596857	0.298428	−	0.954432i	$-0.403538\pi$
$193$	8.29180	0.596857	0.298428	+	0.954432i	$0.403538\pi$
$194$	16.8541	1.21005
$195$	0	0
$196$	0.618034	0.0441453
$197$	25.5066	1.81727	0.908634	−	0.417593i	$-0.137126\pi$
$197$	25.5066	1.81727	0.908634	+	0.417593i	$0.137126\pi$
$198$	0	0
$199$	−6.90983	−0.489825	−0.244912	−	0.969545i	$-0.578759\pi$
$199$	−6.90983	−0.489825	−0.244912	+	0.969545i	$0.578759\pi$
$200$	10.3262	0.730175
$201$	0	0
$202$	−6.70820	−0.471988
$203$	−8.23607	−0.578059
$204$	0	0
$205$	0.909830	0.0635453
$206$	−12.0000	−0.836080
$207$	0	0
$208$	11.5623	0.801702
$209$	7.76393	0.537042
$210$	0	0
$211$	10.4164	0.717095	0.358548	−	0.933511i	$-0.383272\pi$
$211$	10.4164	0.717095	0.358548	+	0.933511i	$0.383272\pi$
$212$	8.09017	0.555635
$213$	0	0
$214$	−5.38197	−0.367904
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	−6.70820	−0.455383
$218$	31.1803	2.11180
$219$	0	0
$220$	−0.854102	−0.0575835
$221$	15.9787	1.07484
$222$	0	0
$223$	−17.8541	−1.19560	−0.597800	−	0.801646i	$-0.703958\pi$
$223$	−17.8541	−1.19560	−0.597800	+	0.801646i	$0.703958\pi$
$224$	3.38197	0.225967
$225$	0	0
$226$	1.47214	0.0979250
$227$	−24.3262	−1.61459	−0.807295	−	0.590149i	$-0.799069\pi$
$227$	−24.3262	−1.61459	−0.807295	+	0.590149i	$0.799069\pi$
$228$	0	0
$229$	−16.3262	−1.07887	−0.539434	−	0.842028i	$-0.681362\pi$
$229$	−16.3262	−1.07887	−0.539434	+	0.842028i	$0.681362\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	−18.4164	−1.20910
$233$	11.0902	0.726541	0.363271	−	0.931684i	$-0.381660\pi$
$233$	11.0902	0.726541	0.363271	+	0.931684i	$0.381660\pi$
$234$	0	0
$235$	4.47214	0.291730
$236$	−5.79837	−0.377442
$237$	0	0
$238$	10.8541	0.703567
$239$	4.79837	0.310381	0.155191	−	0.987885i	$-0.450401\pi$
$239$	4.79837	0.310381	0.155191	+	0.987885i	$0.450401\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	−9.70820	−0.624067
$243$	0	0
$244$	−3.00000	−0.192055
$245$	0.618034	0.0394847
$246$	0	0
$247$	8.27051	0.526240
$248$	−15.0000	−0.952501
$249$	0	0
$250$	−9.61803	−0.608298
$251$	−23.1246	−1.45961	−0.729806	−	0.683654i	$-0.760390\pi$
$251$	−23.1246	−1.45961	−0.729806	+	0.683654i	$0.760390\pi$
$252$	0	0
$253$	−2.23607	−0.140580
$254$	36.0344	2.26100
$255$	0	0
$256$	13.5623	0.847644
$257$	3.23607	0.201860	0.100930	−	0.994894i	$-0.467818\pi$
$257$	3.23607	0.201860	0.100930	+	0.994894i	$0.467818\pi$
$258$	0	0
$259$	11.0000	0.683507
$260$	−0.909830	−0.0564253
$261$	0	0
$262$	−24.5623	−1.51746
$263$	0.0557281	0.00343634	0.00171817	−	0.999999i	$-0.499453\pi$
$263$	0.0557281	0.00343634	0.00171817	+	0.999999i	$0.499453\pi$
$264$	0	0
$265$	8.09017	0.496975
$266$	5.61803	0.344464
$267$	0	0
$268$	−3.14590	−0.192166
$269$	32.0902	1.95657	0.978286	−	0.207259i	$-0.0664543\pi$
$269$	32.0902	1.95657	0.978286	+	0.207259i	$0.0664543\pi$
$270$	0	0
$271$	21.9443	1.33302	0.666510	−	0.745496i	$-0.267787\pi$
$271$	21.9443	1.33302	0.666510	+	0.745496i	$0.267787\pi$
$272$	32.5623	1.97438
$273$	0	0
$274$	26.5623	1.60469
$275$	10.3262	0.622696
$276$	0	0
$277$	4.27051	0.256590	0.128295	−	0.991736i	$-0.459050\pi$
$277$	4.27051	0.256590	0.128295	+	0.991736i	$0.459050\pi$
$278$	−18.4164	−1.10454
$279$	0	0
$280$	1.38197	0.0825883
$281$	3.70820	0.221213	0.110606	−	0.993864i	$-0.464721\pi$
$281$	3.70820	0.221213	0.110606	+	0.993864i	$0.464721\pi$
$282$	0	0
$283$	−26.2148	−1.55831	−0.779154	−	0.626833i	$-0.784351\pi$
$283$	−26.2148	−1.55831	−0.779154	+	0.626833i	$0.784351\pi$
$284$	−2.70820	−0.160702
$285$	0	0
$286$	8.61803	0.509595
$287$	−1.47214	−0.0868974
$288$	0	0
$289$	28.0000	1.64706
$290$	8.23607	0.483639
$291$	0	0
$292$	−7.85410	−0.459627
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	−5.79837	−0.337594
$296$	24.5967	1.42966
$297$	0	0
$298$	31.1246	1.80300
$299$	−2.38197	−0.137753
$300$	0	0
$301$	1.61803	0.0932619
$302$	−24.6525	−1.41859
$303$	0	0
$304$	16.8541	0.966649
$305$	−3.00000	−0.171780
$306$	0	0
$307$	16.1246	0.920280	0.460140	−	0.887846i	$-0.347799\pi$
$307$	16.1246	0.920280	0.460140	+	0.887846i	$0.347799\pi$
$308$	1.38197	0.0787448
$309$	0	0
$310$	6.70820	0.381000
$311$	−15.3262	−0.869071	−0.434536	−	0.900655i	$-0.643087\pi$
$311$	−15.3262	−0.869071	−0.434536	+	0.900655i	$0.643087\pi$
$312$	0	0
$313$	−2.47214	−0.139733	−0.0698667	−	0.997556i	$-0.522257\pi$
$313$	−2.47214	−0.139733	−0.0698667	+	0.997556i	$0.522257\pi$
$314$	−0.472136	−0.0266442
$315$	0	0
$316$	−5.85410	−0.329319
$317$	−14.5066	−0.814771	−0.407385	−	0.913256i	$-0.633559\pi$
$317$	−14.5066	−0.814771	−0.407385	+	0.913256i	$0.633559\pi$
$318$	0	0
$319$	−18.4164	−1.03112
$320$	2.61803	0.146353
$321$	0	0
$322$	−1.61803	−0.0901695
$323$	23.2918	1.29599
$324$	0	0
$325$	11.0000	0.610170
$326$	−1.00000	−0.0553849
$327$	0	0
$328$	−3.29180	−0.181759
$329$	−7.23607	−0.398937
$330$	0	0
$331$	−13.4164	−0.737432	−0.368716	−	0.929542i	$-0.620203\pi$
$331$	−13.4164	−0.737432	−0.368716	+	0.929542i	$0.620203\pi$
$332$	5.67376	0.311388
$333$	0	0
$334$	11.6180	0.635711
$335$	−3.14590	−0.171879
$336$	0	0
$337$	−7.50658	−0.408909	−0.204455	−	0.978876i	$-0.565542\pi$
$337$	−7.50658	−0.408909	−0.204455	+	0.978876i	$0.565542\pi$
$338$	−11.8541	−0.644778
$339$	0	0
$340$	−2.56231	−0.138961
$341$	−15.0000	−0.812296
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	3.61803	0.195071
$345$	0	0
$346$	5.61803	0.302027
$347$	−5.18034	−0.278095	−0.139048	−	0.990286i	$-0.544404\pi$
$347$	−5.18034	−0.278095	−0.139048	+	0.990286i	$0.544404\pi$
$348$	0	0
$349$	−11.8541	−0.634536	−0.317268	−	0.948336i	$-0.602765\pi$
$349$	−11.8541	−0.634536	−0.317268	+	0.948336i	$0.602765\pi$
$350$	7.47214	0.399402
$351$	0	0
$352$	7.56231	0.403072
$353$	−30.3050	−1.61297	−0.806485	−	0.591255i	$-0.798633\pi$
$353$	−30.3050	−1.61297	−0.806485	+	0.591255i	$0.798633\pi$
$354$	0	0
$355$	−2.70820	−0.143737
$356$	−7.18034	−0.380557
$357$	0	0
$358$	−3.00000	−0.158555
$359$	−23.0902	−1.21865	−0.609326	−	0.792920i	$-0.708560\pi$
$359$	−23.0902	−1.21865	−0.609326	+	0.792920i	$0.708560\pi$
$360$	0	0
$361$	−6.94427	−0.365488
$362$	−3.14590	−0.165345
$363$	0	0
$364$	1.47214	0.0771609
$365$	−7.85410	−0.411102
$366$	0	0
$367$	10.8541	0.566580	0.283290	−	0.959034i	$-0.408574\pi$
$367$	10.8541	0.566580	0.283290	+	0.959034i	$0.408574\pi$
$368$	−4.85410	−0.253038
$369$	0	0
$370$	−11.0000	−0.571863
$371$	−13.0902	−0.679608
$372$	0	0
$373$	24.4164	1.26423	0.632117	−	0.774873i	$-0.282186\pi$
$373$	24.4164	1.26423	0.632117	+	0.774873i	$0.282186\pi$
$374$	24.2705	1.25500
$375$	0	0
$376$	−16.1803	−0.834437
$377$	−19.6180	−1.01038
$378$	0	0
$379$	7.41641	0.380955	0.190478	−	0.981692i	$-0.438996\pi$
$379$	7.41641	0.380955	0.190478	+	0.981692i	$0.438996\pi$
$380$	−1.32624	−0.0680346
$381$	0	0
$382$	−26.1803	−1.33950
$383$	0.708204	0.0361875	0.0180938	−	0.999836i	$-0.494240\pi$
$383$	0.708204	0.0361875	0.0180938	+	0.999836i	$0.494240\pi$
$384$	0	0
$385$	1.38197	0.0704315
$386$	13.4164	0.682877
$387$	0	0
$388$	6.43769	0.326824
$389$	−5.29180	−0.268305	−0.134152	−	0.990961i	$-0.542831\pi$
$389$	−5.29180	−0.268305	−0.134152	+	0.990961i	$0.542831\pi$
$390$	0	0
$391$	−6.70820	−0.339248
$392$	−2.23607	−0.112938
$393$	0	0
$394$	41.2705	2.07918
$395$	−5.85410	−0.294552
$396$	0	0
$397$	22.0689	1.10761	0.553803	−	0.832648i	$-0.313176\pi$
$397$	22.0689	1.10761	0.553803	+	0.832648i	$0.313176\pi$
$398$	−11.1803	−0.560420
$399$	0	0
$400$	22.4164	1.12082
$401$	−33.1803	−1.65695	−0.828474	−	0.560028i	$-0.810790\pi$
$401$	−33.1803	−1.65695	−0.828474	+	0.560028i	$0.810790\pi$
$402$	0	0
$403$	−15.9787	−0.795956
$404$	−2.56231	−0.127479
$405$	0	0
$406$	−13.3262	−0.661370
$407$	24.5967	1.21922
$408$	0	0
$409$	6.41641	0.317271	0.158635	−	0.987337i	$-0.449291\pi$
$409$	6.41641	0.317271	0.158635	+	0.987337i	$0.449291\pi$
$410$	1.47214	0.0727036
$411$	0	0
$412$	−4.58359	−0.225817
$413$	9.38197	0.461656
$414$	0	0
$415$	5.67376	0.278514
$416$	8.05573	0.394965
$417$	0	0
$418$	12.5623	0.614442
$419$	−15.6738	−0.765713	−0.382857	−	0.923808i	$-0.625060\pi$
$419$	−15.6738	−0.765713	−0.382857	+	0.923808i	$0.625060\pi$
$420$	0	0
$421$	−10.2705	−0.500554	−0.250277	−	0.968174i	$-0.580522\pi$
$421$	−10.2705	−0.500554	−0.250277	+	0.968174i	$0.580522\pi$
$422$	16.8541	0.820445
$423$	0	0
$424$	−29.2705	−1.42150
$425$	30.9787	1.50269
$426$	0	0
$427$	4.85410	0.234906
$428$	−2.05573	−0.0993674
$429$	0	0
$430$	−1.61803	−0.0780285
$431$	−0.673762	−0.0324540	−0.0162270	−	0.999868i	$-0.505165\pi$
$431$	−0.673762	−0.0324540	−0.0162270	+	0.999868i	$0.505165\pi$
$432$	0	0
$433$	19.1246	0.919070	0.459535	−	0.888160i	$-0.348016\pi$
$433$	19.1246	0.919070	0.459535	+	0.888160i	$0.348016\pi$
$434$	−10.8541	−0.521014
$435$	0	0
$436$	11.9098	0.570377
$437$	−3.47214	−0.166095
$438$	0	0
$439$	23.6525	1.12887	0.564436	−	0.825477i	$-0.309094\pi$
$439$	23.6525	1.12887	0.564436	+	0.825477i	$0.309094\pi$
$440$	3.09017	0.147318
$441$	0	0
$442$	25.8541	1.22975
$443$	9.52786	0.452682	0.226341	−	0.974048i	$-0.427324\pi$
$443$	9.52786	0.452682	0.226341	+	0.974048i	$0.427324\pi$
$444$	0	0
$445$	−7.18034	−0.340381
$446$	−28.8885	−1.36791
$447$	0	0
$448$	−4.23607	−0.200135
$449$	18.4377	0.870129	0.435064	−	0.900399i	$-0.356726\pi$
$449$	18.4377	0.870129	0.435064	+	0.900399i	$0.356726\pi$
$450$	0	0
$451$	−3.29180	−0.155005
$452$	0.562306	0.0264486
$453$	0	0
$454$	−39.3607	−1.84729
$455$	1.47214	0.0690148
$456$	0	0
$457$	−14.3262	−0.670153	−0.335077	−	0.942191i	$-0.608762\pi$
$457$	−14.3262	−0.670153	−0.335077	+	0.942191i	$0.608762\pi$
$458$	−26.4164	−1.23436
$459$	0	0
$460$	0.381966	0.0178093
$461$	−25.4508	−1.18536	−0.592682	−	0.805436i	$-0.701931\pi$
$461$	−25.4508	−1.18536	−0.592682	+	0.805436i	$0.701931\pi$
$462$	0	0
$463$	17.2918	0.803618	0.401809	−	0.915724i	$-0.368382\pi$
$463$	17.2918	0.803618	0.401809	+	0.915724i	$0.368382\pi$
$464$	−39.9787	−1.85597
$465$	0	0
$466$	17.9443	0.831252
$467$	15.1803	0.702462	0.351231	−	0.936289i	$-0.385763\pi$
$467$	15.1803	0.702462	0.351231	+	0.936289i	$0.385763\pi$
$468$	0	0
$469$	5.09017	0.235042
$470$	7.23607	0.333775
$471$	0	0
$472$	20.9787	0.965624
$473$	3.61803	0.166357
$474$	0	0
$475$	16.0344	0.735711
$476$	4.14590	0.190027
$477$	0	0
$478$	7.76393	0.355114
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	0	0
$481$	26.2016	1.19469
$482$	−17.7984	−0.810694
$483$	0	0
$484$	−3.70820	−0.168555
$485$	6.43769	0.292321
$486$	0	0
$487$	−29.2918	−1.32734	−0.663669	−	0.748026i	$-0.731002\pi$
$487$	−29.2918	−1.32734	−0.663669	+	0.748026i	$0.731002\pi$
$488$	10.8541	0.491342
$489$	0	0
$490$	1.00000	0.0451754
$491$	−9.79837	−0.442194	−0.221097	−	0.975252i	$-0.570964\pi$
$491$	−9.79837	−0.442194	−0.221097	+	0.975252i	$0.570964\pi$
$492$	0	0
$493$	−55.2492	−2.48830
$494$	13.3820	0.602083
$495$	0	0
$496$	−32.5623	−1.46209
$497$	4.38197	0.196558
$498$	0	0
$499$	−38.3951	−1.71880	−0.859401	−	0.511302i	$-0.829163\pi$
$499$	−38.3951	−1.71880	−0.859401	+	0.511302i	$0.829163\pi$
$500$	−3.67376	−0.164296
$501$	0	0
$502$	−37.4164	−1.66998
$503$	−36.3262	−1.61971	−0.809853	−	0.586632i	$-0.800453\pi$
$503$	−36.3262	−1.61971	−0.809853	+	0.586632i	$0.800453\pi$
$504$	0	0
$505$	−2.56231	−0.114021
$506$	−3.61803	−0.160841
$507$	0	0
$508$	13.7639	0.610676
$509$	15.5967	0.691314	0.345657	−	0.938361i	$-0.387656\pi$
$509$	15.5967	0.691314	0.345657	+	0.938361i	$0.387656\pi$
$510$	0	0
$511$	12.7082	0.562178
$512$	−5.29180	−0.233867
$513$	0	0
$514$	5.23607	0.230953
$515$	−4.58359	−0.201977
$516$	0	0
$517$	−16.1803	−0.711611
$518$	17.7984	0.782016
$519$	0	0
$520$	3.29180	0.144355
$521$	−3.52786	−0.154559	−0.0772793	−	0.997009i	$-0.524623\pi$
$521$	−3.52786	−0.154559	−0.0772793	+	0.997009i	$0.524623\pi$
$522$	0	0
$523$	21.4164	0.936474	0.468237	−	0.883603i	$-0.344889\pi$
$523$	21.4164	0.936474	0.468237	+	0.883603i	$0.344889\pi$
$524$	−9.38197	−0.409853
$525$	0	0
$526$	0.0901699	0.00393160
$527$	−45.0000	−1.96023
$528$	0	0
$529$	1.00000	0.0434783
$530$	13.0902	0.568601
$531$	0	0
$532$	2.14590	0.0930365
$533$	−3.50658	−0.151887
$534$	0	0
$535$	−2.05573	−0.0888769
$536$	11.3820	0.491626
$537$	0	0
$538$	51.9230	2.23856
$539$	−2.23607	−0.0963143
$540$	0	0
$541$	−5.12461	−0.220324	−0.110162	−	0.993914i	$-0.535137\pi$
$541$	−5.12461	−0.220324	−0.110162	+	0.993914i	$0.535137\pi$
$542$	35.5066	1.52514
$543$	0	0
$544$	22.6869	0.972694
$545$	11.9098	0.510161
$546$	0	0
$547$	12.7984	0.547219	0.273609	−	0.961841i	$-0.411782\pi$
$547$	12.7984	0.547219	0.273609	+	0.961841i	$0.411782\pi$
$548$	10.1459	0.433411
$549$	0	0
$550$	16.7082	0.712440
$551$	−28.5967	−1.21826
$552$	0	0
$553$	9.47214	0.402796
$554$	6.90983	0.293571
$555$	0	0
$556$	−7.03444	−0.298327
$557$	5.81966	0.246587	0.123293	−	0.992370i	$-0.460654\pi$
$557$	5.81966	0.246587	0.123293	+	0.992370i	$0.460654\pi$
$558$	0	0
$559$	3.85410	0.163011
$560$	3.00000	0.126773
$561$	0	0
$562$	6.00000	0.253095
$563$	23.5066	0.990684	0.495342	−	0.868698i	$-0.335043\pi$
$563$	23.5066	0.990684	0.495342	+	0.868698i	$0.335043\pi$
$564$	0	0
$565$	0.562306	0.0236564
$566$	−42.4164	−1.78289
$567$	0	0
$568$	9.79837	0.411131
$569$	16.3607	0.685875	0.342938	−	0.939358i	$-0.388578\pi$
$569$	16.3607	0.685875	0.342938	+	0.939358i	$0.388578\pi$
$570$	0	0
$571$	−44.4164	−1.85877	−0.929384	−	0.369113i	$-0.879661\pi$
$571$	−44.4164	−1.85877	−0.929384	+	0.369113i	$0.879661\pi$
$572$	3.29180	0.137637
$573$	0	0
$574$	−2.38197	−0.0994213
$575$	−4.61803	−0.192585
$576$	0	0
$577$	−36.8328	−1.53337	−0.766685	−	0.642023i	$-0.778095\pi$
$577$	−36.8328	−1.53337	−0.766685	+	0.642023i	$0.778095\pi$
$578$	45.3050	1.88444
$579$	0	0
$580$	3.14590	0.130626
$581$	−9.18034	−0.380865
$582$	0	0
$583$	−29.2705	−1.21226
$584$	28.4164	1.17588
$585$	0	0
$586$	−19.4164	−0.802084
$587$	31.0344	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$587$	31.0344	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$588$	0	0
$589$	−23.2918	−0.959722
$590$	−9.38197	−0.386249
$591$	0	0
$592$	53.3951	2.19453
$593$	32.0689	1.31691	0.658456	−	0.752620i	$-0.271210\pi$
$593$	32.0689	1.31691	0.658456	+	0.752620i	$0.271210\pi$
$594$	0	0
$595$	4.14590	0.169965
$596$	11.8885	0.486974
$597$	0	0
$598$	−3.85410	−0.157606
$599$	15.5066	0.633582	0.316791	−	0.948495i	$-0.397395\pi$
$599$	15.5066	0.633582	0.316791	+	0.948495i	$0.397395\pi$
$600$	0	0
$601$	0.909830	0.0371127	0.0185564	−	0.999828i	$-0.494093\pi$
$601$	0.909830	0.0371127	0.0185564	+	0.999828i	$0.494093\pi$
$602$	2.61803	0.106703
$603$	0	0
$604$	−9.41641	−0.383148
$605$	−3.70820	−0.150760
$606$	0	0
$607$	−33.7984	−1.37183	−0.685917	−	0.727680i	$-0.740599\pi$
$607$	−33.7984	−1.37183	−0.685917	+	0.727680i	$0.740599\pi$
$608$	11.7426	0.476227
$609$	0	0
$610$	−4.85410	−0.196537
$611$	−17.2361	−0.697297
$612$	0	0
$613$	−11.6525	−0.470639	−0.235320	−	0.971918i	$-0.575614\pi$
$613$	−11.6525	−0.470639	−0.235320	+	0.971918i	$0.575614\pi$
$614$	26.0902	1.05291
$615$	0	0
$616$	−5.00000	−0.201456
$617$	2.67376	0.107642	0.0538208	−	0.998551i	$-0.482860\pi$
$617$	2.67376	0.107642	0.0538208	+	0.998551i	$0.482860\pi$
$618$	0	0
$619$	−19.6180	−0.788515	−0.394258	−	0.919000i	$-0.628998\pi$
$619$	−19.6180	−0.788515	−0.394258	+	0.919000i	$0.628998\pi$
$620$	2.56231	0.102905
$621$	0	0
$622$	−24.7984	−0.994324
$623$	11.6180	0.465467
$624$	0	0
$625$	19.4164	0.776656
$626$	−4.00000	−0.159872
$627$	0	0
$628$	−0.180340	−0.00719634
$629$	73.7902	2.94221
$630$	0	0
$631$	16.1246	0.641911	0.320955	−	0.947094i	$-0.395996\pi$
$631$	16.1246	0.641911	0.320955	+	0.947094i	$0.395996\pi$
$632$	21.1803	0.842509
$633$	0	0
$634$	−23.4721	−0.932198
$635$	13.7639	0.546205
$636$	0	0
$637$	−2.38197	−0.0943769
$638$	−29.7984	−1.17973
$639$	0	0
$640$	8.41641	0.332688
$641$	−17.7984	−0.702994	−0.351497	−	0.936189i	$-0.614327\pi$
$641$	−17.7984	−0.702994	−0.351497	+	0.936189i	$0.614327\pi$
$642$	0	0
$643$	−11.9787	−0.472394	−0.236197	−	0.971705i	$-0.575901\pi$
$643$	−11.9787	−0.472394	−0.236197	+	0.971705i	$0.575901\pi$
$644$	−0.618034	−0.0243540
$645$	0	0
$646$	37.6869	1.48277
$647$	24.0344	0.944891	0.472446	−	0.881360i	$-0.343371\pi$
$647$	24.0344	0.944891	0.472446	+	0.881360i	$0.343371\pi$
$648$	0	0
$649$	20.9787	0.823487
$650$	17.7984	0.698110
$651$	0	0
$652$	−0.381966	−0.0149589
$653$	−27.7426	−1.08565	−0.542827	−	0.839845i	$-0.682646\pi$
$653$	−27.7426	−1.08565	−0.542827	+	0.839845i	$0.682646\pi$
$654$	0	0
$655$	−9.38197	−0.366584
$656$	−7.14590	−0.279000
$657$	0	0
$658$	−11.7082	−0.456433
$659$	−9.65248	−0.376007	−0.188004	−	0.982168i	$-0.560202\pi$
$659$	−9.65248	−0.376007	−0.188004	+	0.982168i	$0.560202\pi$
$660$	0	0
$661$	14.4164	0.560733	0.280367	−	0.959893i	$-0.409544\pi$
$661$	14.4164	0.560733	0.280367	+	0.959893i	$0.409544\pi$
$662$	−21.7082	−0.843713
$663$	0	0
$664$	−20.5279	−0.796636
$665$	2.14590	0.0832144
$666$	0	0
$667$	8.23607	0.318902
$668$	4.43769	0.171700
$669$	0	0
$670$	−5.09017	−0.196650
$671$	10.8541	0.419018
$672$	0	0
$673$	12.4164	0.478617	0.239309	−	0.970944i	$-0.423079\pi$
$673$	12.4164	0.478617	0.239309	+	0.970944i	$0.423079\pi$
$674$	−12.1459	−0.467843
$675$	0	0
$676$	−4.52786	−0.174149
$677$	1.25735	0.0483240	0.0241620	−	0.999708i	$-0.492308\pi$
$677$	1.25735	0.0483240	0.0241620	+	0.999708i	$0.492308\pi$
$678$	0	0
$679$	−10.4164	−0.399745
$680$	9.27051	0.355508
$681$	0	0
$682$	−24.2705	−0.929366
$683$	−26.0132	−0.995366	−0.497683	−	0.867359i	$-0.665816\pi$
$683$	−26.0132	−0.995366	−0.497683	+	0.867359i	$0.665816\pi$
$684$	0	0
$685$	10.1459	0.387655
$686$	−1.61803	−0.0617768
$687$	0	0
$688$	7.85410	0.299435
$689$	−31.1803	−1.18788
$690$	0	0
$691$	−7.72949	−0.294044	−0.147022	−	0.989133i	$-0.546969\pi$
$691$	−7.72949	−0.294044	−0.147022	+	0.989133i	$0.546969\pi$
$692$	2.14590	0.0815748
$693$	0	0
$694$	−8.38197	−0.318175
$695$	−7.03444	−0.266832
$696$	0	0
$697$	−9.87539	−0.374057
$698$	−19.1803	−0.725987
$699$	0	0
$700$	2.85410	0.107875
$701$	−37.4508	−1.41450	−0.707250	−	0.706964i	$-0.750064\pi$
$701$	−37.4508	−1.41450	−0.707250	+	0.706964i	$0.750064\pi$
$702$	0	0
$703$	38.1935	1.44049
$704$	−9.47214	−0.356995
$705$	0	0
$706$	−49.0344	−1.84544
$707$	4.14590	0.155923
$708$	0	0
$709$	−2.56231	−0.0962294	−0.0481147	−	0.998842i	$-0.515321\pi$
$709$	−2.56231	−0.0962294	−0.0481147	+	0.998842i	$0.515321\pi$
$710$	−4.38197	−0.164452
$711$	0	0
$712$	25.9787	0.973593
$713$	6.70820	0.251224
$714$	0	0
$715$	3.29180	0.123106
$716$	−1.14590	−0.0428242
$717$	0	0
$718$	−37.3607	−1.39429
$719$	3.00000	0.111881	0.0559406	−	0.998434i	$-0.482184\pi$
$719$	3.00000	0.111881	0.0559406	+	0.998434i	$0.482184\pi$
$720$	0	0
$721$	7.41641	0.276201
$722$	−11.2361	−0.418163
$723$	0	0
$724$	−1.20163	−0.0446581
$725$	−38.0344	−1.41256
$726$	0	0
$727$	18.8885	0.700537	0.350269	−	0.936649i	$-0.386090\pi$
$727$	18.8885	0.700537	0.350269	+	0.936649i	$0.386090\pi$
$728$	−5.32624	−0.197404
$729$	0	0
$730$	−12.7082	−0.470352
$731$	10.8541	0.401453
$732$	0	0
$733$	25.5967	0.945437	0.472719	−	0.881213i	$-0.343273\pi$
$733$	25.5967	0.945437	0.472719	+	0.881213i	$0.343273\pi$
$734$	17.5623	0.648237
$735$	0	0
$736$	−3.38197	−0.124661
$737$	11.3820	0.419260
$738$	0	0
$739$	45.2492	1.66452	0.832260	−	0.554386i	$-0.187047\pi$
$739$	45.2492	1.66452	0.832260	+	0.554386i	$0.187047\pi$
$740$	−4.20163	−0.154455
$741$	0	0
$742$	−21.1803	−0.777555
$743$	−11.6738	−0.428269	−0.214134	−	0.976804i	$-0.568693\pi$
$743$	−11.6738	−0.428269	−0.214134	+	0.976804i	$0.568693\pi$
$744$	0	0
$745$	11.8885	0.435563
$746$	39.5066	1.44644
$747$	0	0
$748$	9.27051	0.338963
$749$	3.32624	0.121538
$750$	0	0
$751$	32.9787	1.20341	0.601705	−	0.798718i	$-0.294488\pi$
$751$	32.9787	1.20341	0.601705	+	0.798718i	$0.294488\pi$
$752$	−35.1246	−1.28086
$753$	0	0
$754$	−31.7426	−1.15600
$755$	−9.41641	−0.342698
$756$	0	0
$757$	−17.0000	−0.617876	−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000	−0.617876	−0.308938	+	0.951082i	$0.599973\pi$
$758$	12.0000	0.435860
$759$	0	0
$760$	4.79837	0.174055
$761$	14.4721	0.524615	0.262307	−	0.964984i	$-0.415517\pi$
$761$	14.4721	0.524615	0.262307	+	0.964984i	$0.415517\pi$
$762$	0	0
$763$	−19.2705	−0.697639
$764$	−10.0000	−0.361787
$765$	0	0
$766$	1.14590	0.0414030
$767$	22.3475	0.806922
$768$	0	0
$769$	33.2361	1.19852	0.599262	−	0.800553i	$-0.295461\pi$
$769$	33.2361	1.19852	0.599262	+	0.800553i	$0.295461\pi$
$770$	2.23607	0.0805823
$771$	0	0
$772$	5.12461	0.184439
$773$	19.9443	0.717346	0.358673	−	0.933463i	$-0.383229\pi$
$773$	19.9443	0.717346	0.358673	+	0.933463i	$0.383229\pi$
$774$	0	0
$775$	−30.9787	−1.11279
$776$	−23.2918	−0.836127
$777$	0	0
$778$	−8.56231	−0.306974
$779$	−5.11146	−0.183137
$780$	0	0
$781$	9.79837	0.350613
$782$	−10.8541	−0.388142
$783$	0	0
$784$	−4.85410	−0.173361
$785$	−0.180340	−0.00643661
$786$	0	0
$787$	−48.5755	−1.73153	−0.865764	−	0.500452i	$-0.833167\pi$
$787$	−48.5755	−1.73153	−0.865764	+	0.500452i	$0.833167\pi$
$788$	15.7639	0.561567
$789$	0	0
$790$	−9.47214	−0.337003
$791$	−0.909830	−0.0323498
$792$	0	0
$793$	11.5623	0.410590
$794$	35.7082	1.26724
$795$	0	0
$796$	−4.27051	−0.151364
$797$	−44.1803	−1.56495	−0.782474	−	0.622683i	$-0.786043\pi$
$797$	−44.1803	−1.56495	−0.782474	+	0.622683i	$0.786043\pi$
$798$	0	0
$799$	−48.5410	−1.71726
$800$	15.6180	0.552181
$801$	0	0
$802$	−53.6869	−1.89575
$803$	28.4164	1.00279
$804$	0	0
$805$	−0.618034	−0.0217828
$806$	−25.8541	−0.910672
$807$	0	0
$808$	9.27051	0.326135
$809$	9.43769	0.331812	0.165906	−	0.986142i	$-0.446945\pi$
$809$	9.43769	0.331812	0.165906	+	0.986142i	$0.446945\pi$
$810$	0	0
$811$	8.52786	0.299454	0.149727	−	0.988727i	$-0.452161\pi$
$811$	8.52786	0.299454	0.149727	+	0.988727i	$0.452161\pi$
$812$	−5.09017	−0.178630
$813$	0	0
$814$	39.7984	1.39493
$815$	−0.381966	−0.0133797
$816$	0	0
$817$	5.61803	0.196550
$818$	10.3820	0.362997
$819$	0	0
$820$	0.562306	0.0196366
$821$	29.1246	1.01646	0.508228	−	0.861223i	$-0.330301\pi$
$821$	29.1246	1.01646	0.508228	+	0.861223i	$0.330301\pi$
$822$	0	0
$823$	−39.8541	−1.38923	−0.694613	−	0.719383i	$-0.744425\pi$
$823$	−39.8541	−1.38923	−0.694613	+	0.719383i	$0.744425\pi$
$824$	16.5836	0.577717
$825$	0	0
$826$	15.1803	0.528192
$827$	28.2148	0.981124	0.490562	−	0.871406i	$-0.336792\pi$
$827$	28.2148	0.981124	0.490562	+	0.871406i	$0.336792\pi$
$828$	0	0
$829$	−30.4164	−1.05641	−0.528203	−	0.849118i	$-0.677134\pi$
$829$	−30.4164	−1.05641	−0.528203	+	0.849118i	$0.677134\pi$
$830$	9.18034	0.318654
$831$	0	0
$832$	−10.0902	−0.349814
$833$	−6.70820	−0.232425
$834$	0	0
$835$	4.43769	0.153573
$836$	4.79837	0.165955
$837$	0	0
$838$	−25.3607	−0.876070
$839$	−16.7426	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$839$	−16.7426	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$840$	0	0
$841$	38.8328	1.33906
$842$	−16.6180	−0.572695
$843$	0	0
$844$	6.43769	0.221595
$845$	−4.52786	−0.155763
$846$	0	0
$847$	6.00000	0.206162
$848$	−63.5410	−2.18201
$849$	0	0
$850$	50.1246	1.71926
$851$	−11.0000	−0.377075
$852$	0	0
$853$	−47.1246	−1.61352	−0.806758	−	0.590882i	$-0.798780\pi$
$853$	−47.1246	−1.61352	−0.806758	+	0.590882i	$0.798780\pi$
$854$	7.85410	0.268762
$855$	0	0
$856$	7.43769	0.254215
$857$	−39.7082	−1.35641	−0.678203	−	0.734874i	$-0.737241\pi$
$857$	−39.7082	−1.35641	−0.678203	+	0.734874i	$0.737241\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	−0.618034	−0.0210748
$861$	0	0
$862$	−1.09017	−0.0371313
$863$	8.06888	0.274668	0.137334	−	0.990525i	$-0.456147\pi$
$863$	8.06888	0.274668	0.137334	+	0.990525i	$0.456147\pi$
$864$	0	0
$865$	2.14590	0.0729627
$866$	30.9443	1.05153
$867$	0	0
$868$	−4.14590	−0.140721
$869$	21.1803	0.718494
$870$	0	0
$871$	12.1246	0.410827
$872$	−43.0902	−1.45922
$873$	0	0
$874$	−5.61803	−0.190033
$875$	5.94427	0.200953
$876$	0	0
$877$	16.5836	0.559988	0.279994	−	0.960002i	$-0.409667\pi$
$877$	16.5836	0.559988	0.279994	+	0.960002i	$0.409667\pi$
$878$	38.2705	1.29157
$879$	0	0
$880$	6.70820	0.226134
$881$	−1.81966	−0.0613059	−0.0306530	−	0.999530i	$-0.509759\pi$
$881$	−1.81966	−0.0613059	−0.0306530	+	0.999530i	$0.509759\pi$
$882$	0	0
$883$	−3.90983	−0.131576	−0.0657881	−	0.997834i	$-0.520956\pi$
$883$	−3.90983	−0.131576	−0.0657881	+	0.997834i	$0.520956\pi$
$884$	9.87539	0.332145
$885$	0	0
$886$	15.4164	0.517924
$887$	−9.79837	−0.328997	−0.164499	−	0.986377i	$-0.552601\pi$
$887$	−9.79837	−0.328997	−0.164499	+	0.986377i	$0.552601\pi$
$888$	0	0
$889$	−22.2705	−0.746929
$890$	−11.6180	−0.389437
$891$	0	0
$892$	−11.0344	−0.369460
$893$	−25.1246	−0.840763
$894$	0	0
$895$	−1.14590	−0.0383031
$896$	−13.6180	−0.454947
$897$	0	0
$898$	29.8328	0.995534
$899$	55.2492	1.84266
$900$	0	0
$901$	−87.8115	−2.92543
$902$	−5.32624	−0.177344
$903$	0	0
$904$	−2.03444	−0.0676645
$905$	−1.20163	−0.0399434
$906$	0	0
$907$	48.9787	1.62631	0.813156	−	0.582046i	$-0.197748\pi$
$907$	48.9787	1.62631	0.813156	+	0.582046i	$0.197748\pi$
$908$	−15.0344	−0.498935
$909$	0	0
$910$	2.38197	0.0789614
$911$	7.59675	0.251691	0.125846	−	0.992050i	$-0.459836\pi$
$911$	7.59675	0.251691	0.125846	+	0.992050i	$0.459836\pi$
$912$	0	0
$913$	−20.5279	−0.679373
$914$	−23.1803	−0.766737
$915$	0	0
$916$	−10.0902	−0.333389
$917$	15.1803	0.501299
$918$	0	0
$919$	−50.1246	−1.65346	−0.826729	−	0.562600i	$-0.809801\pi$
$919$	−50.1246	−1.65346	−0.826729	+	0.562600i	$0.809801\pi$
$920$	−1.38197	−0.0455621
$921$	0	0
$922$	−41.1803	−1.35620
$923$	10.4377	0.343561
$924$	0	0
$925$	50.7984	1.67024
$926$	27.9787	0.919438
$927$	0	0
$928$	−27.8541	−0.914356
$929$	−28.9098	−0.948501	−0.474250	−	0.880390i	$-0.657281\pi$
$929$	−28.9098	−0.948501	−0.474250	+	0.880390i	$0.657281\pi$
$930$	0	0
$931$	−3.47214	−0.113795
$932$	6.85410	0.224514
$933$	0	0
$934$	24.5623	0.803703
$935$	9.27051	0.303178
$936$	0	0
$937$	2.70820	0.0884732	0.0442366	−	0.999021i	$-0.485914\pi$
$937$	2.70820	0.0884732	0.0442366	+	0.999021i	$0.485914\pi$
$938$	8.23607	0.268917
$939$	0	0
$940$	2.76393	0.0901495
$941$	15.0557	0.490803	0.245401	−	0.969422i	$-0.421080\pi$
$941$	15.0557	0.490803	0.245401	+	0.969422i	$0.421080\pi$
$942$	0	0
$943$	1.47214	0.0479393
$944$	45.5410	1.48223
$945$	0	0
$946$	5.85410	0.190333
$947$	18.9443	0.615606	0.307803	−	0.951450i	$-0.400406\pi$
$947$	18.9443	0.615606	0.307803	+	0.951450i	$0.400406\pi$
$948$	0	0
$949$	30.2705	0.982622
$950$	25.9443	0.841743
$951$	0	0
$952$	−15.0000	−0.486153
$953$	−12.1591	−0.393870	−0.196935	−	0.980417i	$-0.563099\pi$
$953$	−12.1591	−0.393870	−0.196935	+	0.980417i	$0.563099\pi$
$954$	0	0
$955$	−10.0000	−0.323592
$956$	2.96556	0.0959130
$957$	0	0
$958$	−33.9787	−1.09780
$959$	−16.4164	−0.530113
$960$	0	0
$961$	14.0000	0.451613
$962$	42.3951	1.36687
$963$	0	0
$964$	−6.79837	−0.218961
$965$	5.12461	0.164967
$966$	0	0
$967$	50.3607	1.61949	0.809745	−	0.586782i	$-0.199605\pi$
$967$	50.3607	1.61949	0.809745	+	0.586782i	$0.199605\pi$
$968$	13.4164	0.431220
$969$	0	0
$970$	10.4164	0.334451
$971$	19.6869	0.631783	0.315892	−	0.948795i	$-0.397696\pi$
$971$	19.6869	0.631783	0.315892	+	0.948795i	$0.397696\pi$
$972$	0	0
$973$	11.3820	0.364889
$974$	−47.3951	−1.51864
$975$	0	0
$976$	23.5623	0.754211
$977$	−49.7984	−1.59319	−0.796596	−	0.604513i	$-0.793368\pi$
$977$	−49.7984	−1.59319	−0.796596	+	0.604513i	$0.793368\pi$
$978$	0	0
$979$	25.9787	0.830283
$980$	0.381966	0.0122015
$981$	0	0
$982$	−15.8541	−0.505925
$983$	7.40325	0.236127	0.118064	−	0.993006i	$-0.462331\pi$
$983$	7.40325	0.236127	0.118064	+	0.993006i	$0.462331\pi$
$984$	0	0
$985$	15.7639	0.502281
$986$	−89.3951	−2.84692
$987$	0	0
$988$	5.11146	0.162617
$989$	−1.61803	−0.0514505
$990$	0	0
$991$	−35.5755	−1.13009	−0.565046	−	0.825059i	$-0.691142\pi$
$991$	−35.5755	−1.13009	−0.565046	+	0.825059i	$0.691142\pi$
$992$	−22.6869	−0.720310
$993$	0	0
$994$	7.09017	0.224887
$995$	−4.27051	−0.135384
$996$	0	0
$997$	57.7214	1.82805	0.914027	−	0.405654i	$-0.132956\pi$
$997$	57.7214	1.82805	0.914027	+	0.405654i	$0.132956\pi$
$998$	−62.1246	−1.96652
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.h.1.2		2
3.2	odd	2		483.2.a.d.1.1	✓	2
12.11	even	2		7728.2.a.bn.1.1		2
21.20	even	2		3381.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.d.1.1	✓	2	3.2	odd	2
1449.2.a.h.1.2		2	1.1	even	1	trivial
3381.2.a.r.1.1		2	21.20	even	2
7728.2.a.bn.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} +0.618034 q^{5} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{10} -2.23607 q^{11} -2.38197 q^{13} -1.61803 q^{14} -4.85410 q^{16} -6.70820 q^{17} -3.47214 q^{19} +0.381966 q^{20} -3.61803 q^{22} +1.00000 q^{23} -4.61803 q^{25} -3.85410 q^{26} -0.618034 q^{28} +8.23607 q^{29} +6.70820 q^{31} -3.38197 q^{32} -10.8541 q^{34} -0.618034 q^{35} -11.0000 q^{37} -5.61803 q^{38} -1.38197 q^{40} +1.47214 q^{41} -1.61803 q^{43} -1.38197 q^{44} +1.61803 q^{46} +7.23607 q^{47} +1.00000 q^{49} -7.47214 q^{50} -1.47214 q^{52} +13.0902 q^{53} -1.38197 q^{55} +2.23607 q^{56} +13.3262 q^{58} -9.38197 q^{59} -4.85410 q^{61} +10.8541 q^{62} +4.23607 q^{64} -1.47214 q^{65} -5.09017 q^{67} -4.14590 q^{68} -1.00000 q^{70} -4.38197 q^{71} -12.7082 q^{73} -17.7984 q^{74} -2.14590 q^{76} +2.23607 q^{77} -9.47214 q^{79} -3.00000 q^{80} +2.38197 q^{82} +9.18034 q^{83} -4.14590 q^{85} -2.61803 q^{86} +5.00000 q^{88} -11.6180 q^{89} +2.38197 q^{91} +0.618034 q^{92} +11.7082 q^{94} -2.14590 q^{95} +10.4164 q^{97} +1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - q^{5} - 2 q^{7} + 2 q^{10} - 7 q^{13} - q^{14} - 3 q^{16} + 2 q^{19} + 3 q^{20} - 5 q^{22} + 2 q^{23} - 7 q^{25} - q^{26} + q^{28} + 12 q^{29} - 9 q^{32} - 15 q^{34} + q^{35}+ \cdots + q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - q^5 - 2 * q^7 + 2 * q^10 - 7 * q^13 - q^14 - 3 * q^16 + 2 * q^19 + 3 * q^20 - 5 * q^22 + 2 * q^23 - 7 * q^25 - q^26 + q^28 + 12 * q^29 - 9 * q^32 - 15 * q^34 + q^35 - 22 * q^37 - 9 * q^38 - 5 * q^40 - 6 * q^41 - q^43 - 5 * q^44 + q^46 + 10 * q^47 + 2 * q^49 - 6 * q^50 + 6 * q^52 + 15 * q^53 - 5 * q^55 + 11 * q^58 - 21 * q^59 - 3 * q^61 + 15 * q^62 + 4 * q^64 + 6 * q^65 + q^67 - 15 * q^68 - 2 * q^70 - 11 * q^71 - 12 * q^73 - 11 * q^74 - 11 * q^76 - 10 * q^79 - 6 * q^80 + 7 * q^82 - 4 * q^83 - 15 * q^85 - 3 * q^86 + 10 * q^88 - 21 * q^89 + 7 * q^91 - q^92 + 10 * q^94 - 11 * q^95 - 6 * q^97 + q^98

Introduction

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