LMFDB - Embedded newform 4598.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4598 = 2 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4598.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:	$36.7152148494$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4598.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} -0.618034 q^{7} -1.00000 q^{8} -3.00000 q^{9} +3.85410 q^{10} -2.47214 q^{13} +0.618034 q^{14} +1.00000 q^{16} -4.09017 q^{17} +3.00000 q^{18} -1.00000 q^{19} -3.85410 q^{20} +1.14590 q^{23} +9.85410 q^{25} +2.47214 q^{26} -0.618034 q^{28} -1.23607 q^{29} -5.70820 q^{31} -1.00000 q^{32} +4.09017 q^{34} +2.38197 q^{35} -3.00000 q^{36} -9.23607 q^{37} +1.00000 q^{38} +3.85410 q^{40} -10.4721 q^{41} +1.61803 q^{43} +11.5623 q^{45} -1.14590 q^{46} -11.6180 q^{47} -6.61803 q^{49} -9.85410 q^{50} -2.47214 q^{52} +2.00000 q^{53} +0.618034 q^{56} +1.23607 q^{58} -10.0000 q^{59} -10.0902 q^{61} +5.70820 q^{62} +1.85410 q^{63} +1.00000 q^{64} +9.52786 q^{65} -8.00000 q^{67} -4.09017 q^{68} -2.38197 q^{70} +8.47214 q^{71} +3.00000 q^{72} +8.47214 q^{73} +9.23607 q^{74} -1.00000 q^{76} +12.0000 q^{79} -3.85410 q^{80} +9.00000 q^{81} +10.4721 q^{82} -3.85410 q^{83} +15.7639 q^{85} -1.61803 q^{86} -14.1803 q^{89} -11.5623 q^{90} +1.52786 q^{91} +1.14590 q^{92} +11.6180 q^{94} +3.85410 q^{95} -7.52786 q^{97} +6.61803 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - q^{5} + q^{7} - 2 q^{8} - 6 q^{9} + q^{10} + 4 q^{13} - q^{14} + 2 q^{16} + 3 q^{17} + 6 q^{18} - 2 q^{19} - q^{20} + 9 q^{23} + 13 q^{25} - 4 q^{26} + q^{28} + 2 q^{29}+ \cdots + 11 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - q^5 + q^7 - 2 * q^8 - 6 * q^9 + q^10 + 4 * q^13 - q^14 + 2 * q^16 + 3 * q^17 + 6 * q^18 - 2 * q^19 - q^20 + 9 * q^23 + 13 * q^25 - 4 * q^26 + q^28 + 2 * q^29 + 2 * q^31 - 2 * q^32 - 3 * q^34 + 7 * q^35 - 6 * q^36 - 14 * q^37 + 2 * q^38 + q^40 - 12 * q^41 + q^43 + 3 * q^45 - 9 * q^46 - 21 * q^47 - 11 * q^49 - 13 * q^50 + 4 * q^52 + 4 * q^53 - q^56 - 2 * q^58 - 20 * q^59 - 9 * q^61 - 2 * q^62 - 3 * q^63 + 2 * q^64 + 28 * q^65 - 16 * q^67 + 3 * q^68 - 7 * q^70 + 8 * q^71 + 6 * q^72 + 8 * q^73 + 14 * q^74 - 2 * q^76 + 24 * q^79 - q^80 + 18 * q^81 + 12 * q^82 - q^83 + 36 * q^85 - q^86 - 6 * q^89 - 3 * q^90 + 12 * q^91 + 9 * q^92 + 21 * q^94 + q^95 - 24 * q^97 + 11 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	−3.85410	−1.72361	−0.861803	−	0.507242i	$-0.830665\pi$
$5$	−3.85410	−1.72361	−0.861803	+	0.507242i	$0.830665\pi$
$6$	0	0
$7$	−0.618034	−0.233595	−0.116797	−	0.993156i	$-0.537263\pi$
$7$	−0.618034	−0.233595	−0.116797	+	0.993156i	$0.537263\pi$
$8$	−1.00000	−0.353553
$9$	−3.00000	−1.00000
$10$	3.85410	1.21877
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.47214	−0.685647	−0.342824	−	0.939400i	$-0.611383\pi$
$13$	−2.47214	−0.685647	−0.342824	+	0.939400i	$0.611383\pi$
$14$	0.618034	0.165177
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.09017	−0.992012	−0.496006	−	0.868319i	$-0.665201\pi$
$17$	−4.09017	−0.992012	−0.496006	+	0.868319i	$0.665201\pi$
$18$	3.00000	0.707107
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−3.85410	−0.861803
$21$	0	0
$22$	0	0
$23$	1.14590	0.238936	0.119468	−	0.992838i	$-0.461881\pi$
$23$	1.14590	0.238936	0.119468	+	0.992838i	$0.461881\pi$
$24$	0	0
$25$	9.85410	1.97082
$26$	2.47214	0.484826
$27$	0	0
$28$	−0.618034	−0.116797
$29$	−1.23607	−0.229532	−0.114766	−	0.993393i	$-0.536612\pi$
$29$	−1.23607	−0.229532	−0.114766	+	0.993393i	$0.536612\pi$
$30$	0	0
$31$	−5.70820	−1.02522	−0.512612	−	0.858620i	$-0.671322\pi$
$31$	−5.70820	−1.02522	−0.512612	+	0.858620i	$0.671322\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.09017	0.701458
$35$	2.38197	0.402626
$36$	−3.00000	−0.500000
$37$	−9.23607	−1.51840	−0.759200	−	0.650857i	$-0.774410\pi$
$37$	−9.23607	−1.51840	−0.759200	+	0.650857i	$0.774410\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	3.85410	0.609387
$41$	−10.4721	−1.63547	−0.817736	−	0.575593i	$-0.804771\pi$
$41$	−10.4721	−1.63547	−0.817736	+	0.575593i	$0.804771\pi$
$42$	0	0
$43$	1.61803	0.246748	0.123374	−	0.992360i	$-0.460629\pi$
$43$	1.61803	0.246748	0.123374	+	0.992360i	$0.460629\pi$
$44$	0	0
$45$	11.5623	1.72361
$46$	−1.14590	−0.168953
$47$	−11.6180	−1.69466	−0.847332	−	0.531063i	$-0.821793\pi$
$47$	−11.6180	−1.69466	−0.847332	+	0.531063i	$0.821793\pi$
$48$	0	0
$49$	−6.61803	−0.945433
$50$	−9.85410	−1.39358
$51$	0	0
$52$	−2.47214	−0.342824
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	0	0
$56$	0.618034	0.0825883
$57$	0	0
$58$	1.23607	0.162304
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−10.0902	−1.29191	−0.645957	−	0.763374i	$-0.723541\pi$
$61$	−10.0902	−1.29191	−0.645957	+	0.763374i	$0.723541\pi$
$62$	5.70820	0.724943
$63$	1.85410	0.233595
$64$	1.00000	0.125000
$65$	9.52786	1.18179
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−4.09017	−0.496006
$69$	0	0
$70$	−2.38197	−0.284699
$71$	8.47214	1.00546	0.502729	−	0.864444i	$-0.332329\pi$
$71$	8.47214	1.00546	0.502729	+	0.864444i	$0.332329\pi$
$72$	3.00000	0.353553
$73$	8.47214	0.991589	0.495794	−	0.868440i	$-0.334877\pi$
$73$	8.47214	0.991589	0.495794	+	0.868440i	$0.334877\pi$
$74$	9.23607	1.07367
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	−3.85410	−0.430902
$81$	9.00000	1.00000
$82$	10.4721	1.15645
$83$	−3.85410	−0.423043	−0.211521	−	0.977373i	$-0.567842\pi$
$83$	−3.85410	−0.423043	−0.211521	+	0.977373i	$0.567842\pi$
$84$	0	0
$85$	15.7639	1.70984
$86$	−1.61803	−0.174477
$87$	0	0
$88$	0	0
$89$	−14.1803	−1.50311	−0.751557	−	0.659669i	$-0.770697\pi$
$89$	−14.1803	−1.50311	−0.751557	+	0.659669i	$0.770697\pi$
$90$	−11.5623	−1.21877
$91$	1.52786	0.160164
$92$	1.14590	0.119468
$93$	0	0
$94$	11.6180	1.19831
$95$	3.85410	0.395423
$96$	0	0
$97$	−7.52786	−0.764339	−0.382169	−	0.924092i	$-0.624823\pi$
$97$	−7.52786	−0.764339	−0.382169	+	0.924092i	$0.624823\pi$
$98$	6.61803	0.668522
$99$	0	0
$100$	9.85410	0.985410
$101$	−1.38197	−0.137511	−0.0687554	−	0.997634i	$-0.521903\pi$
$101$	−1.38197	−0.137511	−0.0687554	+	0.997634i	$0.521903\pi$
$102$	0	0
$103$	7.41641	0.730760	0.365380	−	0.930858i	$-0.380939\pi$
$103$	7.41641	0.730760	0.365380	+	0.930858i	$0.380939\pi$
$104$	2.47214	0.242413
$105$	0	0
$106$	−2.00000	−0.194257
$107$	9.23607	0.892884	0.446442	−	0.894812i	$-0.352691\pi$
$107$	9.23607	0.892884	0.446442	+	0.894812i	$0.352691\pi$
$108$	0	0
$109$	−1.70820	−0.163616	−0.0818081	−	0.996648i	$-0.526069\pi$
$109$	−1.70820	−0.163616	−0.0818081	+	0.996648i	$0.526069\pi$
$110$	0	0
$111$	0	0
$112$	−0.618034	−0.0583987
$113$	−13.7082	−1.28956	−0.644780	−	0.764368i	$-0.723051\pi$
$113$	−13.7082	−1.28956	−0.644780	+	0.764368i	$0.723051\pi$
$114$	0	0
$115$	−4.41641	−0.411832
$116$	−1.23607	−0.114766
$117$	7.41641	0.685647
$118$	10.0000	0.920575
$119$	2.52786	0.231729
$120$	0	0
$121$	0	0
$122$	10.0902	0.913521
$123$	0	0
$124$	−5.70820	−0.512612
$125$	−18.7082	−1.67331
$126$	−1.85410	−0.165177
$127$	5.23607	0.464626	0.232313	−	0.972641i	$-0.425371\pi$
$127$	5.23607	0.464626	0.232313	+	0.972641i	$0.425371\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−9.52786	−0.835649
$131$	6.32624	0.552726	0.276363	−	0.961053i	$-0.410871\pi$
$131$	6.32624	0.552726	0.276363	+	0.961053i	$0.410871\pi$
$132$	0	0
$133$	0.618034	0.0535903
$134$	8.00000	0.691095
$135$	0	0
$136$	4.09017	0.350729
$137$	−16.6180	−1.41977	−0.709887	−	0.704315i	$-0.751254\pi$
$137$	−16.6180	−1.41977	−0.709887	+	0.704315i	$0.751254\pi$
$138$	0	0
$139$	19.3262	1.63923	0.819615	−	0.572915i	$-0.194187\pi$
$139$	19.3262	1.63923	0.819615	+	0.572915i	$0.194187\pi$
$140$	2.38197	0.201313
$141$	0	0
$142$	−8.47214	−0.710966
$143$	0	0
$144$	−3.00000	−0.250000
$145$	4.76393	0.395623
$146$	−8.47214	−0.701159
$147$	0	0
$148$	−9.23607	−0.759200
$149$	−13.4164	−1.09911	−0.549557	−	0.835456i	$-0.685204\pi$
$149$	−13.4164	−1.09911	−0.549557	+	0.835456i	$0.685204\pi$
$150$	0	0
$151$	14.1803	1.15398	0.576990	−	0.816751i	$-0.304227\pi$
$151$	14.1803	1.15398	0.576990	+	0.816751i	$0.304227\pi$
$152$	1.00000	0.0811107
$153$	12.2705	0.992012
$154$	0	0
$155$	22.0000	1.76708
$156$	0	0
$157$	0.437694	0.0349318	0.0174659	−	0.999847i	$-0.494440\pi$
$157$	0.437694	0.0349318	0.0174659	+	0.999847i	$0.494440\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	3.85410	0.304694
$161$	−0.708204	−0.0558143
$162$	−9.00000	−0.707107
$163$	25.0902	1.96521	0.982607	−	0.185698i	$-0.0594546\pi$
$163$	25.0902	1.96521	0.982607	+	0.185698i	$0.0594546\pi$
$164$	−10.4721	−0.817736
$165$	0	0
$166$	3.85410	0.299136
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−6.88854	−0.529888
$170$	−15.7639	−1.20904
$171$	3.00000	0.229416
$172$	1.61803	0.123374
$173$	19.4164	1.47620	0.738101	−	0.674690i	$-0.235723\pi$
$173$	19.4164	1.47620	0.738101	+	0.674690i	$0.235723\pi$
$174$	0	0
$175$	−6.09017	−0.460374
$176$	0	0
$177$	0	0
$178$	14.1803	1.06286
$179$	7.23607	0.540849	0.270425	−	0.962741i	$-0.412836\pi$
$179$	7.23607	0.540849	0.270425	+	0.962741i	$0.412836\pi$
$180$	11.5623	0.861803
$181$	−12.6525	−0.940451	−0.470226	−	0.882546i	$-0.655827\pi$
$181$	−12.6525	−0.940451	−0.470226	+	0.882546i	$0.655827\pi$
$182$	−1.52786	−0.113253
$183$	0	0
$184$	−1.14590	−0.0844767
$185$	35.5967	2.61712
$186$	0	0
$187$	0	0
$188$	−11.6180	−0.847332
$189$	0	0
$190$	−3.85410	−0.279606
$191$	9.56231	0.691904	0.345952	−	0.938252i	$-0.387556\pi$
$191$	9.56231	0.691904	0.345952	+	0.938252i	$0.387556\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	7.52786	0.540469
$195$	0	0
$196$	−6.61803	−0.472717
$197$	−7.52786	−0.536338	−0.268169	−	0.963372i	$-0.586419\pi$
$197$	−7.52786	−0.536338	−0.268169	+	0.963372i	$0.586419\pi$
$198$	0	0
$199$	4.61803	0.327364	0.163682	−	0.986513i	$-0.447663\pi$
$199$	4.61803	0.327364	0.163682	+	0.986513i	$0.447663\pi$
$200$	−9.85410	−0.696790
$201$	0	0
$202$	1.38197	0.0972348
$203$	0.763932	0.0536175
$204$	0	0
$205$	40.3607	2.81891
$206$	−7.41641	−0.516726
$207$	−3.43769	−0.238936
$208$	−2.47214	−0.171412
$209$	0	0
$210$	0	0
$211$	17.1246	1.17891	0.589453	−	0.807802i	$-0.299343\pi$
$211$	17.1246	1.17891	0.589453	+	0.807802i	$0.299343\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−9.23607	−0.631365
$215$	−6.23607	−0.425296
$216$	0	0
$217$	3.52786	0.239487
$218$	1.70820	0.115694
$219$	0	0
$220$	0	0
$221$	10.1115	0.680170
$222$	0	0
$223$	4.47214	0.299476	0.149738	−	0.988726i	$-0.452157\pi$
$223$	4.47214	0.299476	0.149738	+	0.988726i	$0.452157\pi$
$224$	0.618034	0.0412941
$225$	−29.5623	−1.97082
$226$	13.7082	0.911856
$227$	−11.5279	−0.765131	−0.382566	−	0.923928i	$-0.624959\pi$
$227$	−11.5279	−0.765131	−0.382566	+	0.923928i	$0.624959\pi$
$228$	0	0
$229$	−0.562306	−0.0371582	−0.0185791	−	0.999827i	$-0.505914\pi$
$229$	−0.562306	−0.0371582	−0.0185791	+	0.999827i	$0.505914\pi$
$230$	4.41641	0.291209
$231$	0	0
$232$	1.23607	0.0811518
$233$	8.90983	0.583702	0.291851	−	0.956464i	$-0.405729\pi$
$233$	8.90983	0.583702	0.291851	+	0.956464i	$0.405729\pi$
$234$	−7.41641	−0.484826
$235$	44.7771	2.92094
$236$	−10.0000	−0.650945
$237$	0	0
$238$	−2.52786	−0.163857
$239$	−15.7984	−1.02191	−0.510956	−	0.859607i	$-0.670708\pi$
$239$	−15.7984	−1.02191	−0.510956	+	0.859607i	$0.670708\pi$
$240$	0	0
$241$	−9.05573	−0.583331	−0.291665	−	0.956520i	$-0.594209\pi$
$241$	−9.05573	−0.583331	−0.291665	+	0.956520i	$0.594209\pi$
$242$	0	0
$243$	0	0
$244$	−10.0902	−0.645957
$245$	25.5066	1.62956
$246$	0	0
$247$	2.47214	0.157298
$248$	5.70820	0.362471
$249$	0	0
$250$	18.7082	1.18321
$251$	−23.3262	−1.47234	−0.736170	−	0.676797i	$-0.763367\pi$
$251$	−23.3262	−1.47234	−0.736170	+	0.676797i	$0.763367\pi$
$252$	1.85410	0.116797
$253$	0	0
$254$	−5.23607	−0.328540
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.70820	−0.356068	−0.178034	−	0.984024i	$-0.556974\pi$
$257$	−5.70820	−0.356068	−0.178034	+	0.984024i	$0.556974\pi$
$258$	0	0
$259$	5.70820	0.354691
$260$	9.52786	0.590893
$261$	3.70820	0.229532
$262$	−6.32624	−0.390836
$263$	1.52786	0.0942121	0.0471061	−	0.998890i	$-0.485000\pi$
$263$	1.52786	0.0942121	0.0471061	+	0.998890i	$0.485000\pi$
$264$	0	0
$265$	−7.70820	−0.473511
$266$	−0.618034	−0.0378941
$267$	0	0
$268$	−8.00000	−0.488678
$269$	16.3607	0.997528	0.498764	−	0.866738i	$-0.333787\pi$
$269$	16.3607	0.997528	0.498764	+	0.866738i	$0.333787\pi$
$270$	0	0
$271$	14.0902	0.855917	0.427958	−	0.903798i	$-0.359233\pi$
$271$	14.0902	0.855917	0.427958	+	0.903798i	$0.359233\pi$
$272$	−4.09017	−0.248003
$273$	0	0
$274$	16.6180	1.00393
$275$	0	0
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−19.3262	−1.15911
$279$	17.1246	1.02522
$280$	−2.38197	−0.142350
$281$	−21.1246	−1.26019	−0.630094	−	0.776519i	$-0.716984\pi$
$281$	−21.1246	−1.26019	−0.630094	+	0.776519i	$0.716984\pi$
$282$	0	0
$283$	−27.2705	−1.62106	−0.810532	−	0.585695i	$-0.800822\pi$
$283$	−27.2705	−1.62106	−0.810532	+	0.585695i	$0.800822\pi$
$284$	8.47214	0.502729
$285$	0	0
$286$	0	0
$287$	6.47214	0.382038
$288$	3.00000	0.176777
$289$	−0.270510	−0.0159123
$290$	−4.76393	−0.279748
$291$	0	0
$292$	8.47214	0.495794
$293$	1.70820	0.0997943	0.0498972	−	0.998754i	$-0.484111\pi$
$293$	1.70820	0.0997943	0.0498972	+	0.998754i	$0.484111\pi$
$294$	0	0
$295$	38.5410	2.24394
$296$	9.23607	0.536836
$297$	0	0
$298$	13.4164	0.777192
$299$	−2.83282	−0.163826
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	−14.1803	−0.815987
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	38.8885	2.22675
$306$	−12.2705	−0.701458
$307$	19.4164	1.10815	0.554076	−	0.832466i	$-0.313072\pi$
$307$	19.4164	1.10815	0.554076	+	0.832466i	$0.313072\pi$
$308$	0	0
$309$	0	0
$310$	−22.0000	−1.24952
$311$	−33.3262	−1.88976	−0.944879	−	0.327420i	$-0.893821\pi$
$311$	−33.3262	−1.88976	−0.944879	+	0.327420i	$0.893821\pi$
$312$	0	0
$313$	−1.03444	−0.0584701	−0.0292351	−	0.999573i	$-0.509307\pi$
$313$	−1.03444	−0.0584701	−0.0292351	+	0.999573i	$0.509307\pi$
$314$	−0.437694	−0.0247005
$315$	−7.14590	−0.402626
$316$	12.0000	0.675053
$317$	15.4164	0.865872	0.432936	−	0.901425i	$-0.357478\pi$
$317$	15.4164	0.865872	0.432936	+	0.901425i	$0.357478\pi$
$318$	0	0
$319$	0	0
$320$	−3.85410	−0.215451
$321$	0	0
$322$	0.708204	0.0394667
$323$	4.09017	0.227583
$324$	9.00000	0.500000
$325$	−24.3607	−1.35129
$326$	−25.0902	−1.38962
$327$	0	0
$328$	10.4721	0.578227
$329$	7.18034	0.395865
$330$	0	0
$331$	−16.4721	−0.905390	−0.452695	−	0.891665i	$-0.649537\pi$
$331$	−16.4721	−0.905390	−0.452695	+	0.891665i	$0.649537\pi$
$332$	−3.85410	−0.211521
$333$	27.7082	1.51840
$334$	8.00000	0.437741
$335$	30.8328	1.68458
$336$	0	0
$337$	14.6525	0.798171	0.399086	−	0.916914i	$-0.369328\pi$
$337$	14.6525	0.798171	0.399086	+	0.916914i	$0.369328\pi$
$338$	6.88854	0.374687
$339$	0	0
$340$	15.7639	0.854919
$341$	0	0
$342$	−3.00000	−0.162221
$343$	8.41641	0.454443
$344$	−1.61803	−0.0872385
$345$	0	0
$346$	−19.4164	−1.04383
$347$	−17.2148	−0.924138	−0.462069	−	0.886844i	$-0.652893\pi$
$347$	−17.2148	−0.924138	−0.462069	+	0.886844i	$0.652893\pi$
$348$	0	0
$349$	−31.2705	−1.67387	−0.836936	−	0.547301i	$-0.815655\pi$
$349$	−31.2705	−1.67387	−0.836936	+	0.547301i	$0.815655\pi$
$350$	6.09017	0.325533
$351$	0	0
$352$	0	0
$353$	8.32624	0.443161	0.221580	−	0.975142i	$-0.428878\pi$
$353$	8.32624	0.443161	0.221580	+	0.975142i	$0.428878\pi$
$354$	0	0
$355$	−32.6525	−1.73301
$356$	−14.1803	−0.751557
$357$	0	0
$358$	−7.23607	−0.382438
$359$	−1.72949	−0.0912790	−0.0456395	−	0.998958i	$-0.514533\pi$
$359$	−1.72949	−0.0912790	−0.0456395	+	0.998958i	$0.514533\pi$
$360$	−11.5623	−0.609387
$361$	1.00000	0.0526316
$362$	12.6525	0.664999
$363$	0	0
$364$	1.52786	0.0800818
$365$	−32.6525	−1.70911
$366$	0	0
$367$	1.20163	0.0627244	0.0313622	−	0.999508i	$-0.490015\pi$
$367$	1.20163	0.0627244	0.0313622	+	0.999508i	$0.490015\pi$
$368$	1.14590	0.0597341
$369$	31.4164	1.63547
$370$	−35.5967	−1.85059
$371$	−1.23607	−0.0641735
$372$	0	0
$373$	−25.5967	−1.32535	−0.662675	−	0.748907i	$-0.730579\pi$
$373$	−25.5967	−1.32535	−0.662675	+	0.748907i	$0.730579\pi$
$374$	0	0
$375$	0	0
$376$	11.6180	0.599154
$377$	3.05573	0.157378
$378$	0	0
$379$	1.52786	0.0784811	0.0392406	−	0.999230i	$-0.487506\pi$
$379$	1.52786	0.0784811	0.0392406	+	0.999230i	$0.487506\pi$
$380$	3.85410	0.197711
$381$	0	0
$382$	−9.56231	−0.489250
$383$	18.4721	0.943882	0.471941	−	0.881630i	$-0.343554\pi$
$383$	18.4721	0.943882	0.471941	+	0.881630i	$0.343554\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	−4.85410	−0.246748
$388$	−7.52786	−0.382169
$389$	7.61803	0.386250	0.193125	−	0.981174i	$-0.438138\pi$
$389$	7.61803	0.386250	0.193125	+	0.981174i	$0.438138\pi$
$390$	0	0
$391$	−4.68692	−0.237028
$392$	6.61803	0.334261
$393$	0	0
$394$	7.52786	0.379248
$395$	−46.2492	−2.32705
$396$	0	0
$397$	7.79837	0.391389	0.195695	−	0.980665i	$-0.437304\pi$
$397$	7.79837	0.391389	0.195695	+	0.980665i	$0.437304\pi$
$398$	−4.61803	−0.231481
$399$	0	0
$400$	9.85410	0.492705
$401$	−3.41641	−0.170607	−0.0853036	−	0.996355i	$-0.527186\pi$
$401$	−3.41641	−0.170607	−0.0853036	+	0.996355i	$0.527186\pi$
$402$	0	0
$403$	14.1115	0.702942
$404$	−1.38197	−0.0687554
$405$	−34.6869	−1.72361
$406$	−0.763932	−0.0379133
$407$	0	0
$408$	0	0
$409$	−23.7082	−1.17230	−0.586148	−	0.810204i	$-0.699356\pi$
$409$	−23.7082	−1.17230	−0.586148	+	0.810204i	$0.699356\pi$
$410$	−40.3607	−1.99327
$411$	0	0
$412$	7.41641	0.365380
$413$	6.18034	0.304115
$414$	3.43769	0.168953
$415$	14.8541	0.729159
$416$	2.47214	0.121206
$417$	0	0
$418$	0	0
$419$	12.6738	0.619154	0.309577	−	0.950874i	$-0.399813\pi$
$419$	12.6738	0.619154	0.309577	+	0.950874i	$0.399813\pi$
$420$	0	0
$421$	−24.7639	−1.20692	−0.603460	−	0.797393i	$-0.706212\pi$
$421$	−24.7639	−1.20692	−0.603460	+	0.797393i	$0.706212\pi$
$422$	−17.1246	−0.833613
$423$	34.8541	1.69466
$424$	−2.00000	−0.0971286
$425$	−40.3050	−1.95508
$426$	0	0
$427$	6.23607	0.301784
$428$	9.23607	0.446442
$429$	0	0
$430$	6.23607	0.300730
$431$	27.8885	1.34334	0.671672	−	0.740849i	$-0.265576\pi$
$431$	27.8885	1.34334	0.671672	+	0.740849i	$0.265576\pi$
$432$	0	0
$433$	−36.5410	−1.75605	−0.878025	−	0.478615i	$-0.841139\pi$
$433$	−36.5410	−1.75605	−0.878025	+	0.478615i	$0.841139\pi$
$434$	−3.52786	−0.169343
$435$	0	0
$436$	−1.70820	−0.0818081
$437$	−1.14590	−0.0548157
$438$	0	0
$439$	36.3607	1.73540	0.867700	−	0.497088i	$-0.165597\pi$
$439$	36.3607	1.73540	0.867700	+	0.497088i	$0.165597\pi$
$440$	0	0
$441$	19.8541	0.945433
$442$	−10.1115	−0.480953
$443$	−2.61803	−0.124387	−0.0621933	−	0.998064i	$-0.519810\pi$
$443$	−2.61803	−0.124387	−0.0621933	+	0.998064i	$0.519810\pi$
$444$	0	0
$445$	54.6525	2.59078
$446$	−4.47214	−0.211762
$447$	0	0
$448$	−0.618034	−0.0291994
$449$	−32.9443	−1.55474	−0.777368	−	0.629046i	$-0.783446\pi$
$449$	−32.9443	−1.55474	−0.777368	+	0.629046i	$0.783446\pi$
$450$	29.5623	1.39358
$451$	0	0
$452$	−13.7082	−0.644780
$453$	0	0
$454$	11.5279	0.541029
$455$	−5.88854	−0.276059
$456$	0	0
$457$	30.7984	1.44069	0.720344	−	0.693617i	$-0.243984\pi$
$457$	30.7984	1.44069	0.720344	+	0.693617i	$0.243984\pi$
$458$	0.562306	0.0262748
$459$	0	0
$460$	−4.41641	−0.205916
$461$	−25.3820	−1.18216	−0.591078	−	0.806614i	$-0.701297\pi$
$461$	−25.3820	−1.18216	−0.591078	+	0.806614i	$0.701297\pi$
$462$	0	0
$463$	−17.8541	−0.829750	−0.414875	−	0.909878i	$-0.636175\pi$
$463$	−17.8541	−0.829750	−0.414875	+	0.909878i	$0.636175\pi$
$464$	−1.23607	−0.0573830
$465$	0	0
$466$	−8.90983	−0.412740
$467$	4.14590	0.191849	0.0959246	−	0.995389i	$-0.469419\pi$
$467$	4.14590	0.191849	0.0959246	+	0.995389i	$0.469419\pi$
$468$	7.41641	0.342824
$469$	4.94427	0.228305
$470$	−44.7771	−2.06541
$471$	0	0
$472$	10.0000	0.460287
$473$	0	0
$474$	0	0
$475$	−9.85410	−0.452137
$476$	2.52786	0.115864
$477$	−6.00000	−0.274721
$478$	15.7984	0.722601
$479$	−16.5623	−0.756751	−0.378376	−	0.925652i	$-0.623517\pi$
$479$	−16.5623	−0.756751	−0.378376	+	0.925652i	$0.623517\pi$
$480$	0	0
$481$	22.8328	1.04109
$482$	9.05573	0.412477
$483$	0	0
$484$	0	0
$485$	29.0132	1.31742
$486$	0	0
$487$	25.4164	1.15173	0.575864	−	0.817546i	$-0.304666\pi$
$487$	25.4164	1.15173	0.575864	+	0.817546i	$0.304666\pi$
$488$	10.0902	0.456761
$489$	0	0
$490$	−25.5066	−1.15227
$491$	35.5623	1.60490	0.802452	−	0.596716i	$-0.203528\pi$
$491$	35.5623	1.60490	0.802452	+	0.596716i	$0.203528\pi$
$492$	0	0
$493$	5.05573	0.227699
$494$	−2.47214	−0.111227
$495$	0	0
$496$	−5.70820	−0.256306
$497$	−5.23607	−0.234870
$498$	0	0
$499$	−23.2705	−1.04173	−0.520866	−	0.853639i	$-0.674391\pi$
$499$	−23.2705	−1.04173	−0.520866	+	0.853639i	$0.674391\pi$
$500$	−18.7082	−0.836656
$501$	0	0
$502$	23.3262	1.04110
$503$	−27.4164	−1.22244	−0.611219	−	0.791462i	$-0.709320\pi$
$503$	−27.4164	−1.22244	−0.611219	+	0.791462i	$0.709320\pi$
$504$	−1.85410	−0.0825883
$505$	5.32624	0.237014
$506$	0	0
$507$	0	0
$508$	5.23607	0.232313
$509$	22.9443	1.01699	0.508493	−	0.861066i	$-0.330203\pi$
$509$	22.9443	1.01699	0.508493	+	0.861066i	$0.330203\pi$
$510$	0	0
$511$	−5.23607	−0.231630
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	5.70820	0.251778
$515$	−28.5836	−1.25954
$516$	0	0
$517$	0	0
$518$	−5.70820	−0.250804
$519$	0	0
$520$	−9.52786	−0.417824
$521$	−5.70820	−0.250081	−0.125040	−	0.992152i	$-0.539906\pi$
$521$	−5.70820	−0.250081	−0.125040	+	0.992152i	$0.539906\pi$
$522$	−3.70820	−0.162304
$523$	23.7082	1.03669	0.518344	−	0.855172i	$-0.326549\pi$
$523$	23.7082	1.03669	0.518344	+	0.855172i	$0.326549\pi$
$524$	6.32624	0.276363
$525$	0	0
$526$	−1.52786	−0.0666180
$527$	23.3475	1.01703
$528$	0	0
$529$	−21.6869	−0.942909
$530$	7.70820	0.334823
$531$	30.0000	1.30189
$532$	0.618034	0.0267952
$533$	25.8885	1.12136
$534$	0	0
$535$	−35.5967	−1.53898
$536$	8.00000	0.345547
$537$	0	0
$538$	−16.3607	−0.705359
$539$	0	0
$540$	0	0
$541$	31.2148	1.34203	0.671014	−	0.741445i	$-0.265859\pi$
$541$	31.2148	1.34203	0.671014	+	0.741445i	$0.265859\pi$
$542$	−14.0902	−0.605225
$543$	0	0
$544$	4.09017	0.175365
$545$	6.58359	0.282010
$546$	0	0
$547$	−26.7639	−1.14434	−0.572172	−	0.820134i	$-0.693899\pi$
$547$	−26.7639	−1.14434	−0.572172	+	0.820134i	$0.693899\pi$
$548$	−16.6180	−0.709887
$549$	30.2705	1.29191
$550$	0	0
$551$	1.23607	0.0526583
$552$	0	0
$553$	−7.41641	−0.315378
$554$	−22.0000	−0.934690
$555$	0	0
$556$	19.3262	0.819615
$557$	32.1459	1.36207	0.681033	−	0.732253i	$-0.261531\pi$
$557$	32.1459	1.36207	0.681033	+	0.732253i	$0.261531\pi$
$558$	−17.1246	−0.724943
$559$	−4.00000	−0.169182
$560$	2.38197	0.100656
$561$	0	0
$562$	21.1246	0.891088
$563$	10.9443	0.461246	0.230623	−	0.973043i	$-0.425924\pi$
$563$	10.9443	0.461246	0.230623	+	0.973043i	$0.425924\pi$
$564$	0	0
$565$	52.8328	2.22269
$566$	27.2705	1.14627
$567$	−5.56231	−0.233595
$568$	−8.47214	−0.355483
$569$	−6.18034	−0.259093	−0.129547	−	0.991573i	$-0.541352\pi$
$569$	−6.18034	−0.259093	−0.129547	+	0.991573i	$0.541352\pi$
$570$	0	0
$571$	−24.5066	−1.02557	−0.512784	−	0.858518i	$-0.671386\pi$
$571$	−24.5066	−1.02557	−0.512784	+	0.858518i	$0.671386\pi$
$572$	0	0
$573$	0	0
$574$	−6.47214	−0.270142
$575$	11.2918	0.470900
$576$	−3.00000	−0.125000
$577$	32.4721	1.35183	0.675916	−	0.736978i	$-0.263748\pi$
$577$	32.4721	1.35183	0.675916	+	0.736978i	$0.263748\pi$
$578$	0.270510	0.0112517
$579$	0	0
$580$	4.76393	0.197812
$581$	2.38197	0.0988206
$582$	0	0
$583$	0	0
$584$	−8.47214	−0.350579
$585$	−28.5836	−1.18179
$586$	−1.70820	−0.0705653
$587$	1.88854	0.0779485	0.0389743	−	0.999240i	$-0.487591\pi$
$587$	1.88854	0.0779485	0.0389743	+	0.999240i	$0.487591\pi$
$588$	0	0
$589$	5.70820	0.235202
$590$	−38.5410	−1.58671
$591$	0	0
$592$	−9.23607	−0.379600
$593$	−31.9098	−1.31038	−0.655190	−	0.755464i	$-0.727411\pi$
$593$	−31.9098	−1.31038	−0.655190	+	0.755464i	$0.727411\pi$
$594$	0	0
$595$	−9.74265	−0.399410
$596$	−13.4164	−0.549557
$597$	0	0
$598$	2.83282	0.115842
$599$	15.1246	0.617975	0.308987	−	0.951066i	$-0.400010\pi$
$599$	15.1246	0.617975	0.308987	+	0.951066i	$0.400010\pi$
$600$	0	0
$601$	−22.0689	−0.900209	−0.450104	−	0.892976i	$-0.648613\pi$
$601$	−22.0689	−0.900209	−0.450104	+	0.892976i	$0.648613\pi$
$602$	1.00000	0.0407570
$603$	24.0000	0.977356
$604$	14.1803	0.576990
$605$	0	0
$606$	0	0
$607$	18.5836	0.754285	0.377142	−	0.926155i	$-0.376907\pi$
$607$	18.5836	0.754285	0.377142	+	0.926155i	$0.376907\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−38.8885	−1.57455
$611$	28.7214	1.16194
$612$	12.2705	0.496006
$613$	11.9787	0.483816	0.241908	−	0.970299i	$-0.422227\pi$
$613$	11.9787	0.483816	0.241908	+	0.970299i	$0.422227\pi$
$614$	−19.4164	−0.783582
$615$	0	0
$616$	0	0
$617$	30.9443	1.24577	0.622885	−	0.782314i	$-0.285961\pi$
$617$	30.9443	1.24577	0.622885	+	0.782314i	$0.285961\pi$
$618$	0	0
$619$	47.1033	1.89324	0.946621	−	0.322348i	$-0.104472\pi$
$619$	47.1033	1.89324	0.946621	+	0.322348i	$0.104472\pi$
$620$	22.0000	0.883541
$621$	0	0
$622$	33.3262	1.33626
$623$	8.76393	0.351120
$624$	0	0
$625$	22.8328	0.913313
$626$	1.03444	0.0413446
$627$	0	0
$628$	0.437694	0.0174659
$629$	37.7771	1.50627
$630$	7.14590	0.284699
$631$	17.5279	0.697773	0.348887	−	0.937165i	$-0.386560\pi$
$631$	17.5279	0.697773	0.348887	+	0.937165i	$0.386560\pi$
$632$	−12.0000	−0.477334
$633$	0	0
$634$	−15.4164	−0.612264
$635$	−20.1803	−0.800832
$636$	0	0
$637$	16.3607	0.648234
$638$	0	0
$639$	−25.4164	−1.00546
$640$	3.85410	0.152347
$641$	−34.4721	−1.36157	−0.680784	−	0.732484i	$-0.738361\pi$
$641$	−34.4721	−1.36157	−0.680784	+	0.732484i	$0.738361\pi$
$642$	0	0
$643$	−32.6738	−1.28853	−0.644264	−	0.764803i	$-0.722836\pi$
$643$	−32.6738	−1.28853	−0.644264	+	0.764803i	$0.722836\pi$
$644$	−0.708204	−0.0279071
$645$	0	0
$646$	−4.09017	−0.160926
$647$	20.3607	0.800461	0.400230	−	0.916415i	$-0.368930\pi$
$647$	20.3607	0.800461	0.400230	+	0.916415i	$0.368930\pi$
$648$	−9.00000	−0.353553
$649$	0	0
$650$	24.3607	0.955504
$651$	0	0
$652$	25.0902	0.982607
$653$	−11.6180	−0.454649	−0.227324	−	0.973819i	$-0.572998\pi$
$653$	−11.6180	−0.454649	−0.227324	+	0.973819i	$0.572998\pi$
$654$	0	0
$655$	−24.3820	−0.952682
$656$	−10.4721	−0.408868
$657$	−25.4164	−0.991589
$658$	−7.18034	−0.279919
$659$	−41.3050	−1.60901	−0.804506	−	0.593944i	$-0.797570\pi$
$659$	−41.3050	−1.60901	−0.804506	+	0.593944i	$0.797570\pi$
$660$	0	0
$661$	−6.94427	−0.270101	−0.135050	−	0.990839i	$-0.543120\pi$
$661$	−6.94427	−0.270101	−0.135050	+	0.990839i	$0.543120\pi$
$662$	16.4721	0.640208
$663$	0	0
$664$	3.85410	0.149568
$665$	−2.38197	−0.0923687
$666$	−27.7082	−1.07367
$667$	−1.41641	−0.0548435
$668$	−8.00000	−0.309529
$669$	0	0
$670$	−30.8328	−1.19118
$671$	0	0
$672$	0	0
$673$	−4.65248	−0.179340	−0.0896699	−	0.995972i	$-0.528581\pi$
$673$	−4.65248	−0.179340	−0.0896699	+	0.995972i	$0.528581\pi$
$674$	−14.6525	−0.564392
$675$	0	0
$676$	−6.88854	−0.264944
$677$	−44.0689	−1.69370	−0.846852	−	0.531828i	$-0.821505\pi$
$677$	−44.0689	−1.69370	−0.846852	+	0.531828i	$0.821505\pi$
$678$	0	0
$679$	4.65248	0.178546
$680$	−15.7639	−0.604519
$681$	0	0
$682$	0	0
$683$	16.8328	0.644090	0.322045	−	0.946724i	$-0.395630\pi$
$683$	16.8328	0.644090	0.322045	+	0.946724i	$0.395630\pi$
$684$	3.00000	0.114708
$685$	64.0476	2.44713
$686$	−8.41641	−0.321340
$687$	0	0
$688$	1.61803	0.0616870
$689$	−4.94427	−0.188362
$690$	0	0
$691$	27.5623	1.04852	0.524260	−	0.851558i	$-0.324342\pi$
$691$	27.5623	1.04852	0.524260	+	0.851558i	$0.324342\pi$
$692$	19.4164	0.738101
$693$	0	0
$694$	17.2148	0.653464
$695$	−74.4853	−2.82539
$696$	0	0
$697$	42.8328	1.62241
$698$	31.2705	1.18361
$699$	0	0
$700$	−6.09017	−0.230187
$701$	−7.43769	−0.280918	−0.140459	−	0.990087i	$-0.544858\pi$
$701$	−7.43769	−0.280918	−0.140459	+	0.990087i	$0.544858\pi$
$702$	0	0
$703$	9.23607	0.348345
$704$	0	0
$705$	0	0
$706$	−8.32624	−0.313362
$707$	0.854102	0.0321218
$708$	0	0
$709$	−14.9787	−0.562537	−0.281269	−	0.959629i	$-0.590755\pi$
$709$	−14.9787	−0.562537	−0.281269	+	0.959629i	$0.590755\pi$
$710$	32.6525	1.22543
$711$	−36.0000	−1.35011
$712$	14.1803	0.531431
$713$	−6.54102	−0.244963
$714$	0	0
$715$	0	0
$716$	7.23607	0.270425
$717$	0	0
$718$	1.72949	0.0645440
$719$	−51.8115	−1.93224	−0.966122	−	0.258086i	$-0.916908\pi$
$719$	−51.8115	−1.93224	−0.966122	+	0.258086i	$0.916908\pi$
$720$	11.5623	0.430902
$721$	−4.58359	−0.170702
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−12.6525	−0.470226
$725$	−12.1803	−0.452366
$726$	0	0
$727$	−25.9098	−0.960942	−0.480471	−	0.877011i	$-0.659534\pi$
$727$	−25.9098	−0.960942	−0.480471	+	0.877011i	$0.659534\pi$
$728$	−1.52786	−0.0566264
$729$	−27.0000	−1.00000
$730$	32.6525	1.20852
$731$	−6.61803	−0.244777
$732$	0	0
$733$	−37.4508	−1.38328	−0.691639	−	0.722243i	$-0.743111\pi$
$733$	−37.4508	−1.38328	−0.691639	+	0.722243i	$0.743111\pi$
$734$	−1.20163	−0.0443528
$735$	0	0
$736$	−1.14590	−0.0422384
$737$	0	0
$738$	−31.4164	−1.15645
$739$	22.0344	0.810550	0.405275	−	0.914195i	$-0.367176\pi$
$739$	22.0344	0.810550	0.405275	+	0.914195i	$0.367176\pi$
$740$	35.5967	1.30856
$741$	0	0
$742$	1.23607	0.0453775
$743$	−29.7082	−1.08989	−0.544944	−	0.838472i	$-0.683449\pi$
$743$	−29.7082	−1.08989	−0.544944	+	0.838472i	$0.683449\pi$
$744$	0	0
$745$	51.7082	1.89444
$746$	25.5967	0.937164
$747$	11.5623	0.423043
$748$	0	0
$749$	−5.70820	−0.208573
$750$	0	0
$751$	−33.2361	−1.21280	−0.606401	−	0.795159i	$-0.707387\pi$
$751$	−33.2361	−1.21280	−0.606401	+	0.795159i	$0.707387\pi$
$752$	−11.6180	−0.423666
$753$	0	0
$754$	−3.05573	−0.111283
$755$	−54.6525	−1.98901
$756$	0	0
$757$	−47.3050	−1.71933	−0.859664	−	0.510860i	$-0.829327\pi$
$757$	−47.3050	−1.71933	−0.859664	+	0.510860i	$0.829327\pi$
$758$	−1.52786	−0.0554945
$759$	0	0
$760$	−3.85410	−0.139803
$761$	−5.41641	−0.196345	−0.0981723	−	0.995169i	$-0.531300\pi$
$761$	−5.41641	−0.196345	−0.0981723	+	0.995169i	$0.531300\pi$
$762$	0	0
$763$	1.05573	0.0382199
$764$	9.56231	0.345952
$765$	−47.2918	−1.70984
$766$	−18.4721	−0.667425
$767$	24.7214	0.892637
$768$	0	0
$769$	31.7426	1.14467	0.572335	−	0.820020i	$-0.306038\pi$
$769$	31.7426	1.14467	0.572335	+	0.820020i	$0.306038\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	44.3607	1.59554	0.797771	−	0.602960i	$-0.206012\pi$
$773$	44.3607	1.59554	0.797771	+	0.602960i	$0.206012\pi$
$774$	4.85410	0.174477
$775$	−56.2492	−2.02053
$776$	7.52786	0.270235
$777$	0	0
$778$	−7.61803	−0.273120
$779$	10.4721	0.375203
$780$	0	0
$781$	0	0
$782$	4.68692	0.167604
$783$	0	0
$784$	−6.61803	−0.236358
$785$	−1.68692	−0.0602087
$786$	0	0
$787$	38.9443	1.38821	0.694107	−	0.719872i	$-0.255800\pi$
$787$	38.9443	1.38821	0.694107	+	0.719872i	$0.255800\pi$
$788$	−7.52786	−0.268169
$789$	0	0
$790$	46.2492	1.64547
$791$	8.47214	0.301234
$792$	0	0
$793$	24.9443	0.885797
$794$	−7.79837	−0.276754
$795$	0	0
$796$	4.61803	0.163682
$797$	−13.4164	−0.475234	−0.237617	−	0.971359i	$-0.576366\pi$
$797$	−13.4164	−0.475234	−0.237617	+	0.971359i	$0.576366\pi$
$798$	0	0
$799$	47.5197	1.68113
$800$	−9.85410	−0.348395
$801$	42.5410	1.50311
$802$	3.41641	0.120638
$803$	0	0
$804$	0	0
$805$	2.72949	0.0962019
$806$	−14.1115	−0.497055
$807$	0	0
$808$	1.38197	0.0486174
$809$	46.2148	1.62483	0.812413	−	0.583083i	$-0.198154\pi$
$809$	46.2148	1.62483	0.812413	+	0.583083i	$0.198154\pi$
$810$	34.6869	1.21877
$811$	−55.7771	−1.95860	−0.979299	−	0.202418i	$-0.935120\pi$
$811$	−55.7771	−1.95860	−0.979299	+	0.202418i	$0.935120\pi$
$812$	0.763932	0.0268088
$813$	0	0
$814$	0	0
$815$	−96.7001	−3.38726
$816$	0	0
$817$	−1.61803	−0.0566078
$818$	23.7082	0.828938
$819$	−4.58359	−0.160164
$820$	40.3607	1.40946
$821$	7.97871	0.278459	0.139230	−	0.990260i	$-0.455537\pi$
$821$	7.97871	0.278459	0.139230	+	0.990260i	$0.455537\pi$
$822$	0	0
$823$	−9.85410	−0.343492	−0.171746	−	0.985141i	$-0.554941\pi$
$823$	−9.85410	−0.343492	−0.171746	+	0.985141i	$0.554941\pi$
$824$	−7.41641	−0.258363
$825$	0	0
$826$	−6.18034	−0.215042
$827$	37.1246	1.29095	0.645475	−	0.763782i	$-0.276660\pi$
$827$	37.1246	1.29095	0.645475	+	0.763782i	$0.276660\pi$
$828$	−3.43769	−0.119468
$829$	14.0689	0.488633	0.244316	−	0.969696i	$-0.421436\pi$
$829$	14.0689	0.488633	0.244316	+	0.969696i	$0.421436\pi$
$830$	−14.8541	−0.515593
$831$	0	0
$832$	−2.47214	−0.0857059
$833$	27.0689	0.937881
$834$	0	0
$835$	30.8328	1.06701
$836$	0	0
$837$	0	0
$838$	−12.6738	−0.437808
$839$	25.1246	0.867398	0.433699	−	0.901058i	$-0.357208\pi$
$839$	25.1246	0.867398	0.433699	+	0.901058i	$0.357208\pi$
$840$	0	0
$841$	−27.4721	−0.947315
$842$	24.7639	0.853421
$843$	0	0
$844$	17.1246	0.589453
$845$	26.5492	0.913319
$846$	−34.8541	−1.19831
$847$	0	0
$848$	2.00000	0.0686803
$849$	0	0
$850$	40.3050	1.38245
$851$	−10.5836	−0.362801
$852$	0	0
$853$	16.3820	0.560908	0.280454	−	0.959867i	$-0.409515\pi$
$853$	16.3820	0.560908	0.280454	+	0.959867i	$0.409515\pi$
$854$	−6.23607	−0.213394
$855$	−11.5623	−0.395423
$856$	−9.23607	−0.315682
$857$	−32.8328	−1.12155	−0.560774	−	0.827969i	$-0.689496\pi$
$857$	−32.8328	−1.12155	−0.560774	+	0.827969i	$0.689496\pi$
$858$	0	0
$859$	−19.6869	−0.671709	−0.335854	−	0.941914i	$-0.609025\pi$
$859$	−19.6869	−0.671709	−0.335854	+	0.941914i	$0.609025\pi$
$860$	−6.23607	−0.212648
$861$	0	0
$862$	−27.8885	−0.949888
$863$	8.36068	0.284601	0.142300	−	0.989824i	$-0.454550\pi$
$863$	8.36068	0.284601	0.142300	+	0.989824i	$0.454550\pi$
$864$	0	0
$865$	−74.8328	−2.54439
$866$	36.5410	1.24171
$867$	0	0
$868$	3.52786	0.119744
$869$	0	0
$870$	0	0
$871$	19.7771	0.670121
$872$	1.70820	0.0578471
$873$	22.5836	0.764339
$874$	1.14590	0.0387606
$875$	11.5623	0.390877
$876$	0	0
$877$	17.3475	0.585784	0.292892	−	0.956145i	$-0.405382\pi$
$877$	17.3475	0.585784	0.292892	+	0.956145i	$0.405382\pi$
$878$	−36.3607	−1.22711
$879$	0	0
$880$	0	0
$881$	−10.9443	−0.368722	−0.184361	−	0.982859i	$-0.559022\pi$
$881$	−10.9443	−0.368722	−0.184361	+	0.982859i	$0.559022\pi$
$882$	−19.8541	−0.668522
$883$	−34.6180	−1.16499	−0.582495	−	0.812834i	$-0.697923\pi$
$883$	−34.6180	−1.16499	−0.582495	+	0.812834i	$0.697923\pi$
$884$	10.1115	0.340085
$885$	0	0
$886$	2.61803	0.0879546
$887$	−38.9443	−1.30762	−0.653810	−	0.756658i	$-0.726831\pi$
$887$	−38.9443	−1.30762	−0.653810	+	0.756658i	$0.726831\pi$
$888$	0	0
$889$	−3.23607	−0.108534
$890$	−54.6525	−1.83196
$891$	0	0
$892$	4.47214	0.149738
$893$	11.6180	0.388783
$894$	0	0
$895$	−27.8885	−0.932211
$896$	0.618034	0.0206471
$897$	0	0
$898$	32.9443	1.09936
$899$	7.05573	0.235322
$900$	−29.5623	−0.985410
$901$	−8.18034	−0.272527
$902$	0	0
$903$	0	0
$904$	13.7082	0.455928
$905$	48.7639	1.62097
$906$	0	0
$907$	39.7082	1.31849	0.659245	−	0.751929i	$-0.270876\pi$
$907$	39.7082	1.31849	0.659245	+	0.751929i	$0.270876\pi$
$908$	−11.5279	−0.382566
$909$	4.14590	0.137511
$910$	5.88854	0.195203
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	0	0
$914$	−30.7984	−1.01872
$915$	0	0
$916$	−0.562306	−0.0185791
$917$	−3.90983	−0.129114
$918$	0	0
$919$	9.56231	0.315431	0.157716	−	0.987485i	$-0.449587\pi$
$919$	9.56231	0.315431	0.157716	+	0.987485i	$0.449587\pi$
$920$	4.41641	0.145605
$921$	0	0
$922$	25.3820	0.835911
$923$	−20.9443	−0.689389
$924$	0	0
$925$	−91.0132	−2.99249
$926$	17.8541	0.586722
$927$	−22.2492	−0.730760
$928$	1.23607	0.0405759
$929$	34.7426	1.13987	0.569935	−	0.821690i	$-0.306969\pi$
$929$	34.7426	1.13987	0.569935	+	0.821690i	$0.306969\pi$
$930$	0	0
$931$	6.61803	0.216897
$932$	8.90983	0.291851
$933$	0	0
$934$	−4.14590	−0.135658
$935$	0	0
$936$	−7.41641	−0.242413
$937$	21.7984	0.712122	0.356061	−	0.934463i	$-0.384120\pi$
$937$	21.7984	0.712122	0.356061	+	0.934463i	$0.384120\pi$
$938$	−4.94427	−0.161436
$939$	0	0
$940$	44.7771	1.46047
$941$	8.06888	0.263038	0.131519	−	0.991314i	$-0.458015\pi$
$941$	8.06888	0.263038	0.131519	+	0.991314i	$0.458015\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	−10.0000	−0.325472
$945$	0	0
$946$	0	0
$947$	−50.0344	−1.62590	−0.812950	−	0.582333i	$-0.802140\pi$
$947$	−50.0344	−1.62590	−0.812950	+	0.582333i	$0.802140\pi$
$948$	0	0
$949$	−20.9443	−0.679880
$950$	9.85410	0.319709
$951$	0	0
$952$	−2.52786	−0.0819285
$953$	20.9443	0.678452	0.339226	−	0.940705i	$-0.389835\pi$
$953$	20.9443	0.678452	0.339226	+	0.940705i	$0.389835\pi$
$954$	6.00000	0.194257
$955$	−36.8541	−1.19257
$956$	−15.7984	−0.510956
$957$	0	0
$958$	16.5623	0.535104
$959$	10.2705	0.331652
$960$	0	0
$961$	1.58359	0.0510836
$962$	−22.8328	−0.736160
$963$	−27.7082	−0.892884
$964$	−9.05573	−0.291665
$965$	7.70820	0.248136
$966$	0	0
$967$	0.201626	0.00648386	0.00324193	−	0.999995i	$-0.498968\pi$
$967$	0.201626	0.00648386	0.00324193	+	0.999995i	$0.498968\pi$
$968$	0	0
$969$	0	0
$970$	−29.0132	−0.931556
$971$	−25.5967	−0.821439	−0.410719	−	0.911762i	$-0.634722\pi$
$971$	−25.5967	−0.821439	−0.410719	+	0.911762i	$0.634722\pi$
$972$	0	0
$973$	−11.9443	−0.382916
$974$	−25.4164	−0.814394
$975$	0	0
$976$	−10.0902	−0.322978
$977$	−8.65248	−0.276817	−0.138409	−	0.990375i	$-0.544199\pi$
$977$	−8.65248	−0.276817	−0.138409	+	0.990375i	$0.544199\pi$
$978$	0	0
$979$	0	0
$980$	25.5066	0.814778
$981$	5.12461	0.163616
$982$	−35.5623	−1.13484
$983$	18.5410	0.591367	0.295683	−	0.955286i	$-0.404453\pi$
$983$	18.5410	0.591367	0.295683	+	0.955286i	$0.404453\pi$
$984$	0	0
$985$	29.0132	0.924436
$986$	−5.05573	−0.161007
$987$	0	0
$988$	2.47214	0.0786491
$989$	1.85410	0.0589570
$990$	0	0
$991$	−39.7082	−1.26137	−0.630686	−	0.776038i	$-0.717227\pi$
$991$	−39.7082	−1.26137	−0.630686	+	0.776038i	$0.717227\pi$
$992$	5.70820	0.181236
$993$	0	0
$994$	5.23607	0.166078
$995$	−17.7984	−0.564246
$996$	0	0
$997$	−5.96556	−0.188931	−0.0944656	−	0.995528i	$-0.530114\pi$
$997$	−5.96556	−0.188931	−0.0944656	+	0.995528i	$0.530114\pi$
$998$	23.2705	0.736615
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.z.1.1		2
11.2	odd	10		418.2.f.b.191.1	✓	4
11.6	odd	10		418.2.f.b.267.1	yes	4
11.10	odd	2		4598.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.f.b.191.1	✓	4	11.2	odd	10
418.2.f.b.267.1	yes	4	11.6	odd	10
4598.2.a.z.1.1		2	1.1	even	1	trivial
4598.2.a.bh.1.1		2	11.10	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:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} -0.618034 q^{7} -1.00000 q^{8} -3.00000 q^{9} +3.85410 q^{10} -2.47214 q^{13} +0.618034 q^{14} +1.00000 q^{16} -4.09017 q^{17} +3.00000 q^{18} -1.00000 q^{19} -3.85410 q^{20} +1.14590 q^{23} +9.85410 q^{25} +2.47214 q^{26} -0.618034 q^{28} -1.23607 q^{29} -5.70820 q^{31} -1.00000 q^{32} +4.09017 q^{34} +2.38197 q^{35} -3.00000 q^{36} -9.23607 q^{37} +1.00000 q^{38} +3.85410 q^{40} -10.4721 q^{41} +1.61803 q^{43} +11.5623 q^{45} -1.14590 q^{46} -11.6180 q^{47} -6.61803 q^{49} -9.85410 q^{50} -2.47214 q^{52} +2.00000 q^{53} +0.618034 q^{56} +1.23607 q^{58} -10.0000 q^{59} -10.0902 q^{61} +5.70820 q^{62} +1.85410 q^{63} +1.00000 q^{64} +9.52786 q^{65} -8.00000 q^{67} -4.09017 q^{68} -2.38197 q^{70} +8.47214 q^{71} +3.00000 q^{72} +8.47214 q^{73} +9.23607 q^{74} -1.00000 q^{76} +12.0000 q^{79} -3.85410 q^{80} +9.00000 q^{81} +10.4721 q^{82} -3.85410 q^{83} +15.7639 q^{85} -1.61803 q^{86} -14.1803 q^{89} -11.5623 q^{90} +1.52786 q^{91} +1.14590 q^{92} +11.6180 q^{94} +3.85410 q^{95} -7.52786 q^{97} +6.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - q^{5} + q^{7} - 2 q^{8} - 6 q^{9} + q^{10} + 4 q^{13} - q^{14} + 2 q^{16} + 3 q^{17} + 6 q^{18} - 2 q^{19} - q^{20} + 9 q^{23} + 13 q^{25} - 4 q^{26} + q^{28} + 2 q^{29}+ \cdots + 11 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - q^5 + q^7 - 2 * q^8 - 6 * q^9 + q^10 + 4 * q^13 - q^14 + 2 * q^16 + 3 * q^17 + 6 * q^18 - 2 * q^19 - q^20 + 9 * q^23 + 13 * q^25 - 4 * q^26 + q^28 + 2 * q^29 + 2 * q^31 - 2 * q^32 - 3 * q^34 + 7 * q^35 - 6 * q^36 - 14 * q^37 + 2 * q^38 + q^40 - 12 * q^41 + q^43 + 3 * q^45 - 9 * q^46 - 21 * q^47 - 11 * q^49 - 13 * q^50 + 4 * q^52 + 4 * q^53 - q^56 - 2 * q^58 - 20 * q^59 - 9 * q^61 - 2 * q^62 - 3 * q^63 + 2 * q^64 + 28 * q^65 - 16 * q^67 + 3 * q^68 - 7 * q^70 + 8 * q^71 + 6 * q^72 + 8 * q^73 + 14 * q^74 - 2 * q^76 + 24 * q^79 - q^80 + 18 * q^81 + 12 * q^82 - q^83 + 36 * q^85 - q^86 - 6 * q^89 - 3 * q^90 + 12 * q^91 + 9 * q^92 + 21 * q^94 + q^95 - 24 * q^97 + 11 * q^98

Introduction

L-functions

Modular forms

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Database

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