LMFDB - Embedded newform 4842.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4842,2,Mod(1,4842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4842, 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("4842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4842 = 2 \cdot 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4842.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:	$38.6635646587$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +3.61803 q^{5} -4.23607 q^{7} -1.00000 q^{8} -3.61803 q^{10} -0.236068 q^{11} -1.00000 q^{13} +4.23607 q^{14} +1.00000 q^{16} -0.236068 q^{17} +2.00000 q^{19} +3.61803 q^{20} +0.236068 q^{22} +8.23607 q^{23} +8.09017 q^{25} +1.00000 q^{26} -4.23607 q^{28} -0.854102 q^{29} +7.47214 q^{31} -1.00000 q^{32} +0.236068 q^{34} -15.3262 q^{35} -8.70820 q^{37} -2.00000 q^{38} -3.61803 q^{40} -7.70820 q^{41} +2.23607 q^{43} -0.236068 q^{44} -8.23607 q^{46} +6.85410 q^{47} +10.9443 q^{49} -8.09017 q^{50} -1.00000 q^{52} -0.854102 q^{55} +4.23607 q^{56} +0.854102 q^{58} +6.56231 q^{59} -5.47214 q^{61} -7.47214 q^{62} +1.00000 q^{64} -3.61803 q^{65} +1.76393 q^{67} -0.236068 q^{68} +15.3262 q^{70} -9.70820 q^{71} -10.7082 q^{73} +8.70820 q^{74} +2.00000 q^{76} +1.00000 q^{77} -13.7984 q^{79} +3.61803 q^{80} +7.70820 q^{82} +5.23607 q^{83} -0.854102 q^{85} -2.23607 q^{86} +0.236068 q^{88} +9.79837 q^{89} +4.23607 q^{91} +8.23607 q^{92} -6.85410 q^{94} +7.23607 q^{95} +12.6525 q^{97} -10.9443 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 5 q^{5} - 4 q^{7} - 2 q^{8} - 5 q^{10} + 4 q^{11} - 2 q^{13} + 4 q^{14} + 2 q^{16} + 4 q^{17} + 4 q^{19} + 5 q^{20} - 4 q^{22} + 12 q^{23} + 5 q^{25} + 2 q^{26} - 4 q^{28} + 5 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 5 * q^5 - 4 * q^7 - 2 * q^8 - 5 * q^10 + 4 * q^11 - 2 * q^13 + 4 * q^14 + 2 * q^16 + 4 * q^17 + 4 * q^19 + 5 * q^20 - 4 * q^22 + 12 * q^23 + 5 * q^25 + 2 * q^26 - 4 * q^28 + 5 * q^29 + 6 * q^31 - 2 * q^32 - 4 * q^34 - 15 * q^35 - 4 * q^37 - 4 * q^38 - 5 * q^40 - 2 * q^41 + 4 * q^44 - 12 * q^46 + 7 * q^47 + 4 * q^49 - 5 * q^50 - 2 * q^52 + 5 * q^55 + 4 * q^56 - 5 * q^58 - 7 * q^59 - 2 * q^61 - 6 * q^62 + 2 * q^64 - 5 * q^65 + 8 * q^67 + 4 * q^68 + 15 * q^70 - 6 * q^71 - 8 * q^73 + 4 * q^74 + 4 * q^76 + 2 * q^77 - 3 * q^79 + 5 * q^80 + 2 * q^82 + 6 * q^83 + 5 * q^85 - 4 * q^88 - 5 * q^89 + 4 * q^91 + 12 * q^92 - 7 * q^94 + 10 * q^95 - 6 * q^97 - 4 * 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
$3$	0	0
$4$	1.00000	0.500000
$5$	3.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$5$	3.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$6$	0	0
$7$	−4.23607	−1.60108	−0.800542	−	0.599277i	$-0.795455\pi$
$7$	−4.23607	−1.60108	−0.800542	+	0.599277i	$0.795455\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.61803	−1.14412
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	4.23607	1.13214
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.236068	−0.0572549	−0.0286274	−	0.999590i	$-0.509114\pi$
$17$	−0.236068	−0.0572549	−0.0286274	+	0.999590i	$0.509114\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	3.61803	0.809017
$21$	0	0
$22$	0.236068	0.0503299
$23$	8.23607	1.71734	0.858669	−	0.512530i	$-0.171292\pi$
$23$	8.23607	1.71734	0.858669	+	0.512530i	$0.171292\pi$
$24$	0	0
$25$	8.09017	1.61803
$26$	1.00000	0.196116
$27$	0	0
$28$	−4.23607	−0.800542
$29$	−0.854102	−0.158603	−0.0793014	−	0.996851i	$-0.525269\pi$
$29$	−0.854102	−0.158603	−0.0793014	+	0.996851i	$0.525269\pi$
$30$	0	0
$31$	7.47214	1.34204	0.671018	−	0.741441i	$-0.265857\pi$
$31$	7.47214	1.34204	0.671018	+	0.741441i	$0.265857\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.236068	0.0404853
$35$	−15.3262	−2.59061
$36$	0	0
$37$	−8.70820	−1.43162	−0.715810	−	0.698295i	$-0.753942\pi$
$37$	−8.70820	−1.43162	−0.715810	+	0.698295i	$0.753942\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	−3.61803	−0.572061
$41$	−7.70820	−1.20382	−0.601910	−	0.798564i	$-0.705593\pi$
$41$	−7.70820	−1.20382	−0.601910	+	0.798564i	$0.705593\pi$
$42$	0	0
$43$	2.23607	0.340997	0.170499	−	0.985358i	$-0.445462\pi$
$43$	2.23607	0.340997	0.170499	+	0.985358i	$0.445462\pi$
$44$	−0.236068	−0.0355886
$45$	0	0
$46$	−8.23607	−1.21434
$47$	6.85410	0.999774	0.499887	−	0.866091i	$-0.333375\pi$
$47$	6.85410	0.999774	0.499887	+	0.866091i	$0.333375\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	−8.09017	−1.14412
$51$	0	0
$52$	−1.00000	−0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−0.854102	−0.115167
$56$	4.23607	0.566068
$57$	0	0
$58$	0.854102	0.112149
$59$	6.56231	0.854339	0.427170	−	0.904171i	$-0.359511\pi$
$59$	6.56231	0.854339	0.427170	+	0.904171i	$0.359511\pi$
$60$	0	0
$61$	−5.47214	−0.700635	−0.350318	−	0.936631i	$-0.613926\pi$
$61$	−5.47214	−0.700635	−0.350318	+	0.936631i	$0.613926\pi$
$62$	−7.47214	−0.948962
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.61803	−0.448762
$66$	0	0
$67$	1.76393	0.215499	0.107749	−	0.994178i	$-0.465636\pi$
$67$	1.76393	0.215499	0.107749	+	0.994178i	$0.465636\pi$
$68$	−0.236068	−0.0286274
$69$	0	0
$70$	15.3262	1.83184
$71$	−9.70820	−1.15215	−0.576076	−	0.817396i	$-0.695417\pi$
$71$	−9.70820	−1.15215	−0.576076	+	0.817396i	$0.695417\pi$
$72$	0	0
$73$	−10.7082	−1.25330	−0.626650	−	0.779301i	$-0.715575\pi$
$73$	−10.7082	−1.25330	−0.626650	+	0.779301i	$0.715575\pi$
$74$	8.70820	1.01231
$75$	0	0
$76$	2.00000	0.229416
$77$	1.00000	0.113961
$78$	0	0
$79$	−13.7984	−1.55244	−0.776219	−	0.630463i	$-0.782865\pi$
$79$	−13.7984	−1.55244	−0.776219	+	0.630463i	$0.782865\pi$
$80$	3.61803	0.404508
$81$	0	0
$82$	7.70820	0.851229
$83$	5.23607	0.574733	0.287367	−	0.957821i	$-0.407220\pi$
$83$	5.23607	0.574733	0.287367	+	0.957821i	$0.407220\pi$
$84$	0	0
$85$	−0.854102	−0.0926404
$86$	−2.23607	−0.241121
$87$	0	0
$88$	0.236068	0.0251649
$89$	9.79837	1.03863	0.519313	−	0.854584i	$-0.326188\pi$
$89$	9.79837	1.03863	0.519313	+	0.854584i	$0.326188\pi$
$90$	0	0
$91$	4.23607	0.444061
$92$	8.23607	0.858669
$93$	0	0
$94$	−6.85410	−0.706947
$95$	7.23607	0.742405
$96$	0	0
$97$	12.6525	1.28466	0.642332	−	0.766426i	$-0.277967\pi$
$97$	12.6525	1.28466	0.642332	+	0.766426i	$0.277967\pi$
$98$	−10.9443	−1.10554
$99$	0	0
$100$	8.09017	0.809017
$101$	5.09017	0.506491	0.253245	−	0.967402i	$-0.418502\pi$
$101$	5.09017	0.506491	0.253245	+	0.967402i	$0.418502\pi$
$102$	0	0
$103$	10.8541	1.06949	0.534743	−	0.845015i	$-0.320408\pi$
$103$	10.8541	1.06949	0.534743	+	0.845015i	$0.320408\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	0	0
$107$	12.2361	1.18291	0.591453	−	0.806340i	$-0.298555\pi$
$107$	12.2361	1.18291	0.591453	+	0.806340i	$0.298555\pi$
$108$	0	0
$109$	−4.70820	−0.450964	−0.225482	−	0.974247i	$-0.572396\pi$
$109$	−4.70820	−0.450964	−0.225482	+	0.974247i	$0.572396\pi$
$110$	0.854102	0.0814354
$111$	0	0
$112$	−4.23607	−0.400271
$113$	15.3820	1.44701	0.723507	−	0.690317i	$-0.242529\pi$
$113$	15.3820	1.44701	0.723507	+	0.690317i	$0.242529\pi$
$114$	0	0
$115$	29.7984	2.77871
$116$	−0.854102	−0.0793014
$117$	0	0
$118$	−6.56231	−0.604109
$119$	1.00000	0.0916698
$120$	0	0
$121$	−10.9443	−0.994934
$122$	5.47214	0.495424
$123$	0	0
$124$	7.47214	0.671018
$125$	11.1803	1.00000
$126$	0	0
$127$	20.7984	1.84556	0.922779	−	0.385331i	$-0.125913\pi$
$127$	20.7984	1.84556	0.922779	+	0.385331i	$0.125913\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	3.61803	0.317323
$131$	−2.76393	−0.241486	−0.120743	−	0.992684i	$-0.538528\pi$
$131$	−2.76393	−0.241486	−0.120743	+	0.992684i	$0.538528\pi$
$132$	0	0
$133$	−8.47214	−0.734627
$134$	−1.76393	−0.152381
$135$	0	0
$136$	0.236068	0.0202427
$137$	15.6180	1.33434	0.667169	−	0.744906i	$-0.267506\pi$
$137$	15.6180	1.33434	0.667169	+	0.744906i	$0.267506\pi$
$138$	0	0
$139$	10.5623	0.895883	0.447942	−	0.894063i	$-0.352157\pi$
$139$	10.5623	0.895883	0.447942	+	0.894063i	$0.352157\pi$
$140$	−15.3262	−1.29530
$141$	0	0
$142$	9.70820	0.814694
$143$	0.236068	0.0197410
$144$	0	0
$145$	−3.09017	−0.256625
$146$	10.7082	0.886217
$147$	0	0
$148$	−8.70820	−0.715810
$149$	−0.326238	−0.0267265	−0.0133632	−	0.999911i	$-0.504254\pi$
$149$	−0.326238	−0.0267265	−0.0133632	+	0.999911i	$0.504254\pi$
$150$	0	0
$151$	−9.94427	−0.809253	−0.404627	−	0.914482i	$-0.632599\pi$
$151$	−9.94427	−0.809253	−0.404627	+	0.914482i	$0.632599\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	27.0344	2.17146
$156$	0	0
$157$	22.6525	1.80786	0.903932	−	0.427676i	$-0.140668\pi$
$157$	22.6525	1.80786	0.903932	+	0.427676i	$0.140668\pi$
$158$	13.7984	1.09774
$159$	0	0
$160$	−3.61803	−0.286031
$161$	−34.8885	−2.74960
$162$	0	0
$163$	17.3262	1.35710	0.678548	−	0.734556i	$-0.262610\pi$
$163$	17.3262	1.35710	0.678548	+	0.734556i	$0.262610\pi$
$164$	−7.70820	−0.601910
$165$	0	0
$166$	−5.23607	−0.406398
$167$	−3.61803	−0.279972	−0.139986	−	0.990153i	$-0.544706\pi$
$167$	−3.61803	−0.279972	−0.139986	+	0.990153i	$0.544706\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0.854102	0.0655066
$171$	0	0
$172$	2.23607	0.170499
$173$	7.47214	0.568096	0.284048	−	0.958810i	$-0.408322\pi$
$173$	7.47214	0.568096	0.284048	+	0.958810i	$0.408322\pi$
$174$	0	0
$175$	−34.2705	−2.59061
$176$	−0.236068	−0.0177943
$177$	0	0
$178$	−9.79837	−0.734419
$179$	−18.7082	−1.39832	−0.699158	−	0.714967i	$-0.746442\pi$
$179$	−18.7082	−1.39832	−0.699158	+	0.714967i	$0.746442\pi$
$180$	0	0
$181$	19.4164	1.44321	0.721605	−	0.692305i	$-0.243405\pi$
$181$	19.4164	1.44321	0.721605	+	0.692305i	$0.243405\pi$
$182$	−4.23607	−0.313998
$183$	0	0
$184$	−8.23607	−0.607171
$185$	−31.5066	−2.31641
$186$	0	0
$187$	0.0557281	0.00407524
$188$	6.85410	0.499887
$189$	0	0
$190$	−7.23607	−0.524960
$191$	5.05573	0.365820	0.182910	−	0.983130i	$-0.441448\pi$
$191$	5.05573	0.365820	0.182910	+	0.983130i	$0.441448\pi$
$192$	0	0
$193$	−8.23607	−0.592845	−0.296423	−	0.955057i	$-0.595794\pi$
$193$	−8.23607	−0.592845	−0.296423	+	0.955057i	$0.595794\pi$
$194$	−12.6525	−0.908395
$195$	0	0
$196$	10.9443	0.781734
$197$	22.3820	1.59465	0.797325	−	0.603551i	$-0.206248\pi$
$197$	22.3820	1.59465	0.797325	+	0.603551i	$0.206248\pi$
$198$	0	0
$199$	−5.38197	−0.381517	−0.190759	−	0.981637i	$-0.561095\pi$
$199$	−5.38197	−0.381517	−0.190759	+	0.981637i	$0.561095\pi$
$200$	−8.09017	−0.572061
$201$	0	0
$202$	−5.09017	−0.358143
$203$	3.61803	0.253936
$204$	0	0
$205$	−27.8885	−1.94782
$206$	−10.8541	−0.756241
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−0.472136	−0.0326583
$210$	0	0
$211$	0.854102	0.0587988	0.0293994	−	0.999568i	$-0.490641\pi$
$211$	0.854102	0.0587988	0.0293994	+	0.999568i	$0.490641\pi$
$212$	0	0
$213$	0	0
$214$	−12.2361	−0.836440
$215$	8.09017	0.551745
$216$	0	0
$217$	−31.6525	−2.14871
$218$	4.70820	0.318880
$219$	0	0
$220$	−0.854102	−0.0575835
$221$	0.236068	0.0158797
$222$	0	0
$223$	−5.32624	−0.356671	−0.178336	−	0.983970i	$-0.557071\pi$
$223$	−5.32624	−0.356671	−0.178336	+	0.983970i	$0.557071\pi$
$224$	4.23607	0.283034
$225$	0	0
$226$	−15.3820	−1.02319
$227$	24.0344	1.59522	0.797611	−	0.603172i	$-0.206097\pi$
$227$	24.0344	1.59522	0.797611	+	0.603172i	$0.206097\pi$
$228$	0	0
$229$	−10.7082	−0.707618	−0.353809	−	0.935318i	$-0.615114\pi$
$229$	−10.7082	−0.707618	−0.353809	+	0.935318i	$0.615114\pi$
$230$	−29.7984	−1.96485
$231$	0	0
$232$	0.854102	0.0560745
$233$	−4.23607	−0.277514	−0.138757	−	0.990326i	$-0.544311\pi$
$233$	−4.23607	−0.277514	−0.138757	+	0.990326i	$0.544311\pi$
$234$	0	0
$235$	24.7984	1.61767
$236$	6.56231	0.427170
$237$	0	0
$238$	−1.00000	−0.0648204
$239$	19.2705	1.24651	0.623253	−	0.782020i	$-0.285811\pi$
$239$	19.2705	1.24651	0.623253	+	0.782020i	$0.285811\pi$
$240$	0	0
$241$	−4.70820	−0.303282	−0.151641	−	0.988436i	$-0.548456\pi$
$241$	−4.70820	−0.303282	−0.151641	+	0.988436i	$0.548456\pi$
$242$	10.9443	0.703524
$243$	0	0
$244$	−5.47214	−0.350318
$245$	39.5967	2.52974
$246$	0	0
$247$	−2.00000	−0.127257
$248$	−7.47214	−0.474481
$249$	0	0
$250$	−11.1803	−0.707107
$251$	−24.0902	−1.52056	−0.760279	−	0.649597i	$-0.774938\pi$
$251$	−24.0902	−1.52056	−0.760279	+	0.649597i	$0.774938\pi$
$252$	0	0
$253$	−1.94427	−0.122235
$254$	−20.7984	−1.30501
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.47214	0.341342	0.170671	−	0.985328i	$-0.445406\pi$
$257$	5.47214	0.341342	0.170671	+	0.985328i	$0.445406\pi$
$258$	0	0
$259$	36.8885	2.29214
$260$	−3.61803	−0.224381
$261$	0	0
$262$	2.76393	0.170756
$263$	−23.2705	−1.43492	−0.717461	−	0.696599i	$-0.754696\pi$
$263$	−23.2705	−1.43492	−0.717461	+	0.696599i	$0.754696\pi$
$264$	0	0
$265$	0	0
$266$	8.47214	0.519460
$267$	0	0
$268$	1.76393	0.107749
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	24.1246	1.46547	0.732733	−	0.680516i	$-0.238244\pi$
$271$	24.1246	1.46547	0.732733	+	0.680516i	$0.238244\pi$
$272$	−0.236068	−0.0143137
$273$	0	0
$274$	−15.6180	−0.943520
$275$	−1.90983	−0.115167
$276$	0	0
$277$	−21.3820	−1.28472	−0.642359	−	0.766404i	$-0.722044\pi$
$277$	−21.3820	−1.28472	−0.642359	+	0.766404i	$0.722044\pi$
$278$	−10.5623	−0.633485
$279$	0	0
$280$	15.3262	0.915918
$281$	29.1803	1.74075	0.870377	−	0.492387i	$-0.163875\pi$
$281$	29.1803	1.74075	0.870377	+	0.492387i	$0.163875\pi$
$282$	0	0
$283$	5.76393	0.342630	0.171315	−	0.985216i	$-0.445198\pi$
$283$	5.76393	0.342630	0.171315	+	0.985216i	$0.445198\pi$
$284$	−9.70820	−0.576076
$285$	0	0
$286$	−0.236068	−0.0139590
$287$	32.6525	1.92741
$288$	0	0
$289$	−16.9443	−0.996722
$290$	3.09017	0.181461
$291$	0	0
$292$	−10.7082	−0.626650
$293$	−7.52786	−0.439783	−0.219891	−	0.975524i	$-0.570570\pi$
$293$	−7.52786	−0.439783	−0.219891	+	0.975524i	$0.570570\pi$
$294$	0	0
$295$	23.7426	1.38235
$296$	8.70820	0.506154
$297$	0	0
$298$	0.326238	0.0188985
$299$	−8.23607	−0.476304
$300$	0	0
$301$	−9.47214	−0.545965
$302$	9.94427	0.572229
$303$	0	0
$304$	2.00000	0.114708
$305$	−19.7984	−1.13365
$306$	0	0
$307$	32.1246	1.83345	0.916724	−	0.399521i	$-0.130823\pi$
$307$	32.1246	1.83345	0.916724	+	0.399521i	$0.130823\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	−27.0344	−1.53545
$311$	17.2361	0.977368	0.488684	−	0.872461i	$-0.337477\pi$
$311$	17.2361	0.977368	0.488684	+	0.872461i	$0.337477\pi$
$312$	0	0
$313$	13.1459	0.743050	0.371525	−	0.928423i	$-0.378835\pi$
$313$	13.1459	0.743050	0.371525	+	0.928423i	$0.378835\pi$
$314$	−22.6525	−1.27835
$315$	0	0
$316$	−13.7984	−0.776219
$317$	30.2361	1.69823	0.849113	−	0.528211i	$-0.177137\pi$
$317$	30.2361	1.69823	0.849113	+	0.528211i	$0.177137\pi$
$318$	0	0
$319$	0.201626	0.0112889
$320$	3.61803	0.202254
$321$	0	0
$322$	34.8885	1.94426
$323$	−0.472136	−0.0262703
$324$	0	0
$325$	−8.09017	−0.448762
$326$	−17.3262	−0.959612
$327$	0	0
$328$	7.70820	0.425614
$329$	−29.0344	−1.60072
$330$	0	0
$331$	0.819660	0.0450526	0.0225263	−	0.999746i	$-0.492829\pi$
$331$	0.819660	0.0450526	0.0225263	+	0.999746i	$0.492829\pi$
$332$	5.23607	0.287367
$333$	0	0
$334$	3.61803	0.197970
$335$	6.38197	0.348684
$336$	0	0
$337$	−27.9787	−1.52410	−0.762049	−	0.647520i	$-0.775806\pi$
$337$	−27.9787	−1.52410	−0.762049	+	0.647520i	$0.775806\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	−0.854102	−0.0463202
$341$	−1.76393	−0.0955223
$342$	0	0
$343$	−16.7082	−0.902158
$344$	−2.23607	−0.120561
$345$	0	0
$346$	−7.47214	−0.401705
$347$	−13.3607	−0.717239	−0.358619	−	0.933484i	$-0.616752\pi$
$347$	−13.3607	−0.717239	−0.358619	+	0.933484i	$0.616752\pi$
$348$	0	0
$349$	21.2705	1.13858	0.569292	−	0.822135i	$-0.307217\pi$
$349$	21.2705	1.13858	0.569292	+	0.822135i	$0.307217\pi$
$350$	34.2705	1.83184
$351$	0	0
$352$	0.236068	0.0125825
$353$	9.85410	0.524481	0.262240	−	0.965003i	$-0.415539\pi$
$353$	9.85410	0.524481	0.262240	+	0.965003i	$0.415539\pi$
$354$	0	0
$355$	−35.1246	−1.86422
$356$	9.79837	0.519313
$357$	0	0
$358$	18.7082	0.988759
$359$	16.9443	0.894284	0.447142	−	0.894463i	$-0.352442\pi$
$359$	16.9443	0.894284	0.447142	+	0.894463i	$0.352442\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−19.4164	−1.02050
$363$	0	0
$364$	4.23607	0.222030
$365$	−38.7426	−2.02788
$366$	0	0
$367$	−9.20163	−0.480321	−0.240160	−	0.970733i	$-0.577200\pi$
$367$	−9.20163	−0.480321	−0.240160	+	0.970733i	$0.577200\pi$
$368$	8.23607	0.429335
$369$	0	0
$370$	31.5066	1.63795
$371$	0	0
$372$	0	0
$373$	6.94427	0.359561	0.179780	−	0.983707i	$-0.442461\pi$
$373$	6.94427	0.359561	0.179780	+	0.983707i	$0.442461\pi$
$374$	−0.0557281	−0.00288163
$375$	0	0
$376$	−6.85410	−0.353473
$377$	0.854102	0.0439885
$378$	0	0
$379$	6.14590	0.315694	0.157847	−	0.987464i	$-0.449545\pi$
$379$	6.14590	0.315694	0.157847	+	0.987464i	$0.449545\pi$
$380$	7.23607	0.371202
$381$	0	0
$382$	−5.05573	−0.258674
$383$	17.3820	0.888177	0.444088	−	0.895983i	$-0.353528\pi$
$383$	17.3820	0.888177	0.444088	+	0.895983i	$0.353528\pi$
$384$	0	0
$385$	3.61803	0.184392
$386$	8.23607	0.419205
$387$	0	0
$388$	12.6525	0.642332
$389$	−28.9443	−1.46753	−0.733766	−	0.679402i	$-0.762239\pi$
$389$	−28.9443	−1.46753	−0.733766	+	0.679402i	$0.762239\pi$
$390$	0	0
$391$	−1.94427	−0.0983261
$392$	−10.9443	−0.552769
$393$	0	0
$394$	−22.3820	−1.12759
$395$	−49.9230	−2.51190
$396$	0	0
$397$	−34.2705	−1.71999	−0.859994	−	0.510304i	$-0.829533\pi$
$397$	−34.2705	−1.71999	−0.859994	+	0.510304i	$0.829533\pi$
$398$	5.38197	0.269774
$399$	0	0
$400$	8.09017	0.404508
$401$	4.27051	0.213259	0.106630	−	0.994299i	$-0.465994\pi$
$401$	4.27051	0.213259	0.106630	+	0.994299i	$0.465994\pi$
$402$	0	0
$403$	−7.47214	−0.372214
$404$	5.09017	0.253245
$405$	0	0
$406$	−3.61803	−0.179560
$407$	2.05573	0.101899
$408$	0	0
$409$	−11.6525	−0.576178	−0.288089	−	0.957604i	$-0.593020\pi$
$409$	−11.6525	−0.576178	−0.288089	+	0.957604i	$0.593020\pi$
$410$	27.8885	1.37732
$411$	0	0
$412$	10.8541	0.534743
$413$	−27.7984	−1.36787
$414$	0	0
$415$	18.9443	0.929938
$416$	1.00000	0.0490290
$417$	0	0
$418$	0.472136	0.0230929
$419$	17.4377	0.851887	0.425944	−	0.904750i	$-0.359942\pi$
$419$	17.4377	0.851887	0.425944	+	0.904750i	$0.359942\pi$
$420$	0	0
$421$	11.4721	0.559118	0.279559	−	0.960129i	$-0.409812\pi$
$421$	11.4721	0.559118	0.279559	+	0.960129i	$0.409812\pi$
$422$	−0.854102	−0.0415770
$423$	0	0
$424$	0	0
$425$	−1.90983	−0.0926404
$426$	0	0
$427$	23.1803	1.12178
$428$	12.2361	0.591453
$429$	0	0
$430$	−8.09017	−0.390143
$431$	15.4721	0.745267	0.372633	−	0.927979i	$-0.378455\pi$
$431$	15.4721	0.745267	0.372633	+	0.927979i	$0.378455\pi$
$432$	0	0
$433$	12.2016	0.586373	0.293186	−	0.956055i	$-0.405284\pi$
$433$	12.2016	0.586373	0.293186	+	0.956055i	$0.405284\pi$
$434$	31.6525	1.51937
$435$	0	0
$436$	−4.70820	−0.225482
$437$	16.4721	0.787969
$438$	0	0
$439$	19.1459	0.913784	0.456892	−	0.889522i	$-0.348963\pi$
$439$	19.1459	0.913784	0.456892	+	0.889522i	$0.348963\pi$
$440$	0.854102	0.0407177
$441$	0	0
$442$	−0.236068	−0.0112286
$443$	−33.4164	−1.58766	−0.793831	−	0.608139i	$-0.791916\pi$
$443$	−33.4164	−1.58766	−0.793831	+	0.608139i	$0.791916\pi$
$444$	0	0
$445$	35.4508	1.68053
$446$	5.32624	0.252205
$447$	0	0
$448$	−4.23607	−0.200135
$449$	−2.90983	−0.137323	−0.0686617	−	0.997640i	$-0.521873\pi$
$449$	−2.90983	−0.137323	−0.0686617	+	0.997640i	$0.521873\pi$
$450$	0	0
$451$	1.81966	0.0856844
$452$	15.3820	0.723507
$453$	0	0
$454$	−24.0344	−1.12799
$455$	15.3262	0.718505
$456$	0	0
$457$	26.7984	1.25358	0.626788	−	0.779190i	$-0.284369\pi$
$457$	26.7984	1.25358	0.626788	+	0.779190i	$0.284369\pi$
$458$	10.7082	0.500362
$459$	0	0
$460$	29.7984	1.38936
$461$	−21.1246	−0.983871	−0.491936	−	0.870632i	$-0.663710\pi$
$461$	−21.1246	−0.983871	−0.491936	+	0.870632i	$0.663710\pi$
$462$	0	0
$463$	−20.0902	−0.933669	−0.466835	−	0.884345i	$-0.654606\pi$
$463$	−20.0902	−0.933669	−0.466835	+	0.884345i	$0.654606\pi$
$464$	−0.854102	−0.0396507
$465$	0	0
$466$	4.23607	0.196232
$467$	6.58359	0.304652	0.152326	−	0.988330i	$-0.451324\pi$
$467$	6.58359	0.304652	0.152326	+	0.988330i	$0.451324\pi$
$468$	0	0
$469$	−7.47214	−0.345031
$470$	−24.7984	−1.14386
$471$	0	0
$472$	−6.56231	−0.302055
$473$	−0.527864	−0.0242712
$474$	0	0
$475$	16.1803	0.742405
$476$	1.00000	0.0458349
$477$	0	0
$478$	−19.2705	−0.881413
$479$	36.5967	1.67215	0.836074	−	0.548617i	$-0.184845\pi$
$479$	36.5967	1.67215	0.836074	+	0.548617i	$0.184845\pi$
$480$	0	0
$481$	8.70820	0.397060
$482$	4.70820	0.214453
$483$	0	0
$484$	−10.9443	−0.497467
$485$	45.7771	2.07863
$486$	0	0
$487$	−14.8328	−0.672139	−0.336070	−	0.941837i	$-0.609098\pi$
$487$	−14.8328	−0.672139	−0.336070	+	0.941837i	$0.609098\pi$
$488$	5.47214	0.247712
$489$	0	0
$490$	−39.5967	−1.78880
$491$	−6.76393	−0.305252	−0.152626	−	0.988284i	$-0.548773\pi$
$491$	−6.76393	−0.305252	−0.152626	+	0.988284i	$0.548773\pi$
$492$	0	0
$493$	0.201626	0.00908078
$494$	2.00000	0.0899843
$495$	0	0
$496$	7.47214	0.335509
$497$	41.1246	1.84469
$498$	0	0
$499$	11.9443	0.534699	0.267350	−	0.963600i	$-0.413852\pi$
$499$	11.9443	0.534699	0.267350	+	0.963600i	$0.413852\pi$
$500$	11.1803	0.500000
$501$	0	0
$502$	24.0902	1.07520
$503$	−34.3607	−1.53207	−0.766033	−	0.642801i	$-0.777772\pi$
$503$	−34.3607	−1.53207	−0.766033	+	0.642801i	$0.777772\pi$
$504$	0	0
$505$	18.4164	0.819519
$506$	1.94427	0.0864334
$507$	0	0
$508$	20.7984	0.922779
$509$	−26.6738	−1.18229	−0.591147	−	0.806564i	$-0.701325\pi$
$509$	−26.6738	−1.18229	−0.591147	+	0.806564i	$0.701325\pi$
$510$	0	0
$511$	45.3607	2.00664
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−5.47214	−0.241366
$515$	39.2705	1.73047
$516$	0	0
$517$	−1.61803	−0.0711611
$518$	−36.8885	−1.62079
$519$	0	0
$520$	3.61803	0.158661
$521$	−30.4164	−1.33257	−0.666284	−	0.745699i	$-0.732116\pi$
$521$	−30.4164	−1.33257	−0.666284	+	0.745699i	$0.732116\pi$
$522$	0	0
$523$	27.6180	1.20765	0.603826	−	0.797116i	$-0.293642\pi$
$523$	27.6180	1.20765	0.603826	+	0.797116i	$0.293642\pi$
$524$	−2.76393	−0.120743
$525$	0	0
$526$	23.2705	1.01464
$527$	−1.76393	−0.0768381
$528$	0	0
$529$	44.8328	1.94925
$530$	0	0
$531$	0	0
$532$	−8.47214	−0.367314
$533$	7.70820	0.333879
$534$	0	0
$535$	44.2705	1.91398
$536$	−1.76393	−0.0761903
$537$	0	0
$538$	1.00000	0.0431131
$539$	−2.58359	−0.111283
$540$	0	0
$541$	20.4721	0.880166	0.440083	−	0.897957i	$-0.354949\pi$
$541$	20.4721	0.880166	0.440083	+	0.897957i	$0.354949\pi$
$542$	−24.1246	−1.03624
$543$	0	0
$544$	0.236068	0.0101213
$545$	−17.0344	−0.729675
$546$	0	0
$547$	3.20163	0.136892	0.0684458	−	0.997655i	$-0.478196\pi$
$547$	3.20163	0.136892	0.0684458	+	0.997655i	$0.478196\pi$
$548$	15.6180	0.667169
$549$	0	0
$550$	1.90983	0.0814354
$551$	−1.70820	−0.0727719
$552$	0	0
$553$	58.4508	2.48558
$554$	21.3820	0.908433
$555$	0	0
$556$	10.5623	0.447942
$557$	−28.2918	−1.19876	−0.599381	−	0.800464i	$-0.704587\pi$
$557$	−28.2918	−1.19876	−0.599381	+	0.800464i	$0.704587\pi$
$558$	0	0
$559$	−2.23607	−0.0945756
$560$	−15.3262	−0.647652
$561$	0	0
$562$	−29.1803	−1.23090
$563$	−7.29180	−0.307313	−0.153656	−	0.988124i	$-0.549105\pi$
$563$	−7.29180	−0.307313	−0.153656	+	0.988124i	$0.549105\pi$
$564$	0	0
$565$	55.6525	2.34132
$566$	−5.76393	−0.242276
$567$	0	0
$568$	9.70820	0.407347
$569$	5.90983	0.247753	0.123876	−	0.992298i	$-0.460467\pi$
$569$	5.90983	0.247753	0.123876	+	0.992298i	$0.460467\pi$
$570$	0	0
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	0.236068	0.00987050
$573$	0	0
$574$	−32.6525	−1.36289
$575$	66.6312	2.77871
$576$	0	0
$577$	3.00000	0.124892	0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000	0.124892	0.0624458	+	0.998048i	$0.480110\pi$
$578$	16.9443	0.704789
$579$	0	0
$580$	−3.09017	−0.128312
$581$	−22.1803	−0.920196
$582$	0	0
$583$	0	0
$584$	10.7082	0.443109
$585$	0	0
$586$	7.52786	0.310973
$587$	10.6738	0.440553	0.220277	−	0.975437i	$-0.429304\pi$
$587$	10.6738	0.440553	0.220277	+	0.975437i	$0.429304\pi$
$588$	0	0
$589$	14.9443	0.615768
$590$	−23.7426	−0.977469
$591$	0	0
$592$	−8.70820	−0.357905
$593$	0.583592	0.0239653	0.0119826	−	0.999928i	$-0.496186\pi$
$593$	0.583592	0.0239653	0.0119826	+	0.999928i	$0.496186\pi$
$594$	0	0
$595$	3.61803	0.148325
$596$	−0.326238	−0.0133632
$597$	0	0
$598$	8.23607	0.336798
$599$	15.5967	0.637266	0.318633	−	0.947878i	$-0.396776\pi$
$599$	15.5967	0.637266	0.318633	+	0.947878i	$0.396776\pi$
$600$	0	0
$601$	−25.6869	−1.04779	−0.523896	−	0.851782i	$-0.675522\pi$
$601$	−25.6869	−1.04779	−0.523896	+	0.851782i	$0.675522\pi$
$602$	9.47214	0.386055
$603$	0	0
$604$	−9.94427	−0.404627
$605$	−39.5967	−1.60984
$606$	0	0
$607$	47.5755	1.93103	0.965514	−	0.260350i	$-0.0838381\pi$
$607$	47.5755	1.93103	0.965514	+	0.260350i	$0.0838381\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	19.7984	0.801613
$611$	−6.85410	−0.277287
$612$	0	0
$613$	−26.5279	−1.07145	−0.535725	−	0.844392i	$-0.679962\pi$
$613$	−26.5279	−1.07145	−0.535725	+	0.844392i	$0.679962\pi$
$614$	−32.1246	−1.29644
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−36.1591	−1.45571	−0.727854	−	0.685732i	$-0.759482\pi$
$617$	−36.1591	−1.45571	−0.727854	+	0.685732i	$0.759482\pi$
$618$	0	0
$619$	−40.8328	−1.64121	−0.820605	−	0.571496i	$-0.806363\pi$
$619$	−40.8328	−1.64121	−0.820605	+	0.571496i	$0.806363\pi$
$620$	27.0344	1.08573
$621$	0	0
$622$	−17.2361	−0.691103
$623$	−41.5066	−1.66293
$624$	0	0
$625$	0	0
$626$	−13.1459	−0.525416
$627$	0	0
$628$	22.6525	0.903932
$629$	2.05573	0.0819672
$630$	0	0
$631$	5.20163	0.207073	0.103537	−	0.994626i	$-0.466984\pi$
$631$	5.20163	0.207073	0.103537	+	0.994626i	$0.466984\pi$
$632$	13.7984	0.548870
$633$	0	0
$634$	−30.2361	−1.20083
$635$	75.2492	2.98617
$636$	0	0
$637$	−10.9443	−0.433628
$638$	−0.201626	−0.00798245
$639$	0	0
$640$	−3.61803	−0.143015
$641$	−3.18034	−0.125616	−0.0628079	−	0.998026i	$-0.520006\pi$
$641$	−3.18034	−0.125616	−0.0628079	+	0.998026i	$0.520006\pi$
$642$	0	0
$643$	−4.14590	−0.163498	−0.0817491	−	0.996653i	$-0.526051\pi$
$643$	−4.14590	−0.163498	−0.0817491	+	0.996653i	$0.526051\pi$
$644$	−34.8885	−1.37480
$645$	0	0
$646$	0.472136	0.0185759
$647$	2.81966	0.110852	0.0554261	−	0.998463i	$-0.482348\pi$
$647$	2.81966	0.110852	0.0554261	+	0.998463i	$0.482348\pi$
$648$	0	0
$649$	−1.54915	−0.0608095
$650$	8.09017	0.317323
$651$	0	0
$652$	17.3262	0.678548
$653$	27.5410	1.07776	0.538882	−	0.842381i	$-0.318847\pi$
$653$	27.5410	1.07776	0.538882	+	0.842381i	$0.318847\pi$
$654$	0	0
$655$	−10.0000	−0.390732
$656$	−7.70820	−0.300955
$657$	0	0
$658$	29.0344	1.13188
$659$	31.3262	1.22030	0.610148	−	0.792287i	$-0.291110\pi$
$659$	31.3262	1.22030	0.610148	+	0.792287i	$0.291110\pi$
$660$	0	0
$661$	−20.8541	−0.811131	−0.405565	−	0.914066i	$-0.632925\pi$
$661$	−20.8541	−0.811131	−0.405565	+	0.914066i	$0.632925\pi$
$662$	−0.819660	−0.0318570
$663$	0	0
$664$	−5.23607	−0.203199
$665$	−30.6525	−1.18865
$666$	0	0
$667$	−7.03444	−0.272375
$668$	−3.61803	−0.139986
$669$	0	0
$670$	−6.38197	−0.246557
$671$	1.29180	0.0498692
$672$	0	0
$673$	10.7082	0.412771	0.206385	−	0.978471i	$-0.433830\pi$
$673$	10.7082	0.412771	0.206385	+	0.978471i	$0.433830\pi$
$674$	27.9787	1.07770
$675$	0	0
$676$	−12.0000	−0.461538
$677$	17.4377	0.670185	0.335093	−	0.942185i	$-0.391232\pi$
$677$	17.4377	0.670185	0.335093	+	0.942185i	$0.391232\pi$
$678$	0	0
$679$	−53.5967	−2.05685
$680$	0.854102	0.0327533
$681$	0	0
$682$	1.76393	0.0675444
$683$	21.2361	0.812576	0.406288	−	0.913745i	$-0.366823\pi$
$683$	21.2361	0.812576	0.406288	+	0.913745i	$0.366823\pi$
$684$	0	0
$685$	56.5066	2.15901
$686$	16.7082	0.637922
$687$	0	0
$688$	2.23607	0.0852493
$689$	0	0
$690$	0	0
$691$	−15.1246	−0.575367	−0.287684	−	0.957725i	$-0.592885\pi$
$691$	−15.1246	−0.575367	−0.287684	+	0.957725i	$0.592885\pi$
$692$	7.47214	0.284048
$693$	0	0
$694$	13.3607	0.507164
$695$	38.2148	1.44957
$696$	0	0
$697$	1.81966	0.0689245
$698$	−21.2705	−0.805101
$699$	0	0
$700$	−34.2705	−1.29530
$701$	10.9787	0.414660	0.207330	−	0.978271i	$-0.433523\pi$
$701$	10.9787	0.414660	0.207330	+	0.978271i	$0.433523\pi$
$702$	0	0
$703$	−17.4164	−0.656872
$704$	−0.236068	−0.00889715
$705$	0	0
$706$	−9.85410	−0.370864
$707$	−21.5623	−0.810934
$708$	0	0
$709$	−37.9443	−1.42503	−0.712514	−	0.701658i	$-0.752443\pi$
$709$	−37.9443	−1.42503	−0.712514	+	0.701658i	$0.752443\pi$
$710$	35.1246	1.31820
$711$	0	0
$712$	−9.79837	−0.367210
$713$	61.5410	2.30473
$714$	0	0
$715$	0.854102	0.0319416
$716$	−18.7082	−0.699158
$717$	0	0
$718$	−16.9443	−0.632355
$719$	−42.1803	−1.57306	−0.786531	−	0.617551i	$-0.788125\pi$
$719$	−42.1803	−1.57306	−0.786531	+	0.617551i	$0.788125\pi$
$720$	0	0
$721$	−45.9787	−1.71234
$722$	15.0000	0.558242
$723$	0	0
$724$	19.4164	0.721605
$725$	−6.90983	−0.256625
$726$	0	0
$727$	−44.7082	−1.65814	−0.829068	−	0.559148i	$-0.811128\pi$
$727$	−44.7082	−1.65814	−0.829068	+	0.559148i	$0.811128\pi$
$728$	−4.23607	−0.156999
$729$	0	0
$730$	38.7426	1.43393
$731$	−0.527864	−0.0195238
$732$	0	0
$733$	−33.3607	−1.23220	−0.616102	−	0.787666i	$-0.711289\pi$
$733$	−33.3607	−1.23220	−0.616102	+	0.787666i	$0.711289\pi$
$734$	9.20163	0.339638
$735$	0	0
$736$	−8.23607	−0.303585
$737$	−0.416408	−0.0153386
$738$	0	0
$739$	35.0344	1.28876	0.644381	−	0.764704i	$-0.277115\pi$
$739$	35.0344	1.28876	0.644381	+	0.764704i	$0.277115\pi$
$740$	−31.5066	−1.15820
$741$	0	0
$742$	0	0
$743$	9.20163	0.337575	0.168787	−	0.985652i	$-0.446015\pi$
$743$	9.20163	0.337575	0.168787	+	0.985652i	$0.446015\pi$
$744$	0	0
$745$	−1.18034	−0.0432443
$746$	−6.94427	−0.254248
$747$	0	0
$748$	0.0557281	0.00203762
$749$	−51.8328	−1.89393
$750$	0	0
$751$	39.8673	1.45478	0.727388	−	0.686226i	$-0.240734\pi$
$751$	39.8673	1.45478	0.727388	+	0.686226i	$0.240734\pi$
$752$	6.85410	0.249943
$753$	0	0
$754$	−0.854102	−0.0311046
$755$	−35.9787	−1.30940
$756$	0	0
$757$	−38.9443	−1.41545	−0.707727	−	0.706486i	$-0.750279\pi$
$757$	−38.9443	−1.41545	−0.707727	+	0.706486i	$0.750279\pi$
$758$	−6.14590	−0.223229
$759$	0	0
$760$	−7.23607	−0.262480
$761$	1.41641	0.0513447	0.0256724	−	0.999670i	$-0.491827\pi$
$761$	1.41641	0.0513447	0.0256724	+	0.999670i	$0.491827\pi$
$762$	0	0
$763$	19.9443	0.722031
$764$	5.05573	0.182910
$765$	0	0
$766$	−17.3820	−0.628036
$767$	−6.56231	−0.236951
$768$	0	0
$769$	−33.3262	−1.20177	−0.600887	−	0.799334i	$-0.705186\pi$
$769$	−33.3262	−1.20177	−0.600887	+	0.799334i	$0.705186\pi$
$770$	−3.61803	−0.130385
$771$	0	0
$772$	−8.23607	−0.296423
$773$	−47.5967	−1.71194	−0.855968	−	0.517029i	$-0.827038\pi$
$773$	−47.5967	−1.71194	−0.855968	+	0.517029i	$0.827038\pi$
$774$	0	0
$775$	60.4508	2.17146
$776$	−12.6525	−0.454197
$777$	0	0
$778$	28.9443	1.03770
$779$	−15.4164	−0.552350
$780$	0	0
$781$	2.29180	0.0820069
$782$	1.94427	0.0695270
$783$	0	0
$784$	10.9443	0.390867
$785$	81.9574	2.92519
$786$	0	0
$787$	−2.34752	−0.0836802	−0.0418401	−	0.999124i	$-0.513322\pi$
$787$	−2.34752	−0.0836802	−0.0418401	+	0.999124i	$0.513322\pi$
$788$	22.3820	0.797325
$789$	0	0
$790$	49.9230	1.77618
$791$	−65.1591	−2.31679
$792$	0	0
$793$	5.47214	0.194321
$794$	34.2705	1.21621
$795$	0	0
$796$	−5.38197	−0.190759
$797$	−13.2016	−0.467626	−0.233813	−	0.972282i	$-0.575120\pi$
$797$	−13.2016	−0.467626	−0.233813	+	0.972282i	$0.575120\pi$
$798$	0	0
$799$	−1.61803	−0.0572419
$800$	−8.09017	−0.286031
$801$	0	0
$802$	−4.27051	−0.150797
$803$	2.52786	0.0892064
$804$	0	0
$805$	−126.228	−4.44895
$806$	7.47214	0.263195
$807$	0	0
$808$	−5.09017	−0.179072
$809$	−15.5279	−0.545931	−0.272965	−	0.962024i	$-0.588004\pi$
$809$	−15.5279	−0.545931	−0.272965	+	0.962024i	$0.588004\pi$
$810$	0	0
$811$	−47.5967	−1.67135	−0.835674	−	0.549226i	$-0.814923\pi$
$811$	−47.5967	−1.67135	−0.835674	+	0.549226i	$0.814923\pi$
$812$	3.61803	0.126968
$813$	0	0
$814$	−2.05573	−0.0720532
$815$	62.6869	2.19583
$816$	0	0
$817$	4.47214	0.156460
$818$	11.6525	0.407419
$819$	0	0
$820$	−27.8885	−0.973910
$821$	−38.3607	−1.33880	−0.669398	−	0.742904i	$-0.733448\pi$
$821$	−38.3607	−1.33880	−0.669398	+	0.742904i	$0.733448\pi$
$822$	0	0
$823$	−50.8885	−1.77386	−0.886932	−	0.461901i	$-0.847168\pi$
$823$	−50.8885	−1.77386	−0.886932	+	0.461901i	$0.847168\pi$
$824$	−10.8541	−0.378121
$825$	0	0
$826$	27.7984	0.967229
$827$	37.7426	1.31244	0.656220	−	0.754569i	$-0.272154\pi$
$827$	37.7426	1.31244	0.656220	+	0.754569i	$0.272154\pi$
$828$	0	0
$829$	43.7984	1.52118	0.760590	−	0.649232i	$-0.224910\pi$
$829$	43.7984	1.52118	0.760590	+	0.649232i	$0.224910\pi$
$830$	−18.9443	−0.657565
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	−2.58359	−0.0895162
$834$	0	0
$835$	−13.0902	−0.453004
$836$	−0.472136	−0.0163292
$837$	0	0
$838$	−17.4377	−0.602375
$839$	−14.4508	−0.498899	−0.249449	−	0.968388i	$-0.580250\pi$
$839$	−14.4508	−0.498899	−0.249449	+	0.968388i	$0.580250\pi$
$840$	0	0
$841$	−28.2705	−0.974845
$842$	−11.4721	−0.395356
$843$	0	0
$844$	0.854102	0.0293994
$845$	−43.4164	−1.49357
$846$	0	0
$847$	46.3607	1.59297
$848$	0	0
$849$	0	0
$850$	1.90983	0.0655066
$851$	−71.7214	−2.45858
$852$	0	0
$853$	51.8115	1.77399	0.886996	−	0.461776i	$-0.152788\pi$
$853$	51.8115	1.77399	0.886996	+	0.461776i	$0.152788\pi$
$854$	−23.1803	−0.793215
$855$	0	0
$856$	−12.2361	−0.418220
$857$	42.0689	1.43705	0.718523	−	0.695503i	$-0.244819\pi$
$857$	42.0689	1.43705	0.718523	+	0.695503i	$0.244819\pi$
$858$	0	0
$859$	−12.7426	−0.434773	−0.217387	−	0.976086i	$-0.569753\pi$
$859$	−12.7426	−0.434773	−0.217387	+	0.976086i	$0.569753\pi$
$860$	8.09017	0.275873
$861$	0	0
$862$	−15.4721	−0.526983
$863$	−17.3262	−0.589792	−0.294896	−	0.955529i	$-0.595285\pi$
$863$	−17.3262	−0.589792	−0.294896	+	0.955529i	$0.595285\pi$
$864$	0	0
$865$	27.0344	0.919199
$866$	−12.2016	−0.414628
$867$	0	0
$868$	−31.6525	−1.07436
$869$	3.25735	0.110498
$870$	0	0
$871$	−1.76393	−0.0597686
$872$	4.70820	0.159440
$873$	0	0
$874$	−16.4721	−0.557178
$875$	−47.3607	−1.60108
$876$	0	0
$877$	42.5755	1.43767	0.718836	−	0.695180i	$-0.244675\pi$
$877$	42.5755	1.43767	0.718836	+	0.695180i	$0.244675\pi$
$878$	−19.1459	−0.646143
$879$	0	0
$880$	−0.854102	−0.0287918
$881$	44.5066	1.49946	0.749732	−	0.661741i	$-0.230182\pi$
$881$	44.5066	1.49946	0.749732	+	0.661741i	$0.230182\pi$
$882$	0	0
$883$	−50.1935	−1.68915	−0.844573	−	0.535441i	$-0.820146\pi$
$883$	−50.1935	−1.68915	−0.844573	+	0.535441i	$0.820146\pi$
$884$	0.236068	0.00793983
$885$	0	0
$886$	33.4164	1.12265
$887$	13.8328	0.464460	0.232230	−	0.972661i	$-0.425398\pi$
$887$	13.8328	0.464460	0.232230	+	0.972661i	$0.425398\pi$
$888$	0	0
$889$	−88.1033	−2.95489
$890$	−35.4508	−1.18832
$891$	0	0
$892$	−5.32624	−0.178336
$893$	13.7082	0.458728
$894$	0	0
$895$	−67.6869	−2.26252
$896$	4.23607	0.141517
$897$	0	0
$898$	2.90983	0.0971023
$899$	−6.38197	−0.212850
$900$	0	0
$901$	0	0
$902$	−1.81966	−0.0605881
$903$	0	0
$904$	−15.3820	−0.511597
$905$	70.2492	2.33516
$906$	0	0
$907$	−14.1246	−0.469000	−0.234500	−	0.972116i	$-0.575345\pi$
$907$	−14.1246	−0.469000	−0.234500	+	0.972116i	$0.575345\pi$
$908$	24.0344	0.797611
$909$	0	0
$910$	−15.3262	−0.508060
$911$	−47.3394	−1.56842	−0.784212	−	0.620493i	$-0.786933\pi$
$911$	−47.3394	−1.56842	−0.784212	+	0.620493i	$0.786933\pi$
$912$	0	0
$913$	−1.23607	−0.0409079
$914$	−26.7984	−0.886411
$915$	0	0
$916$	−10.7082	−0.353809
$917$	11.7082	0.386639
$918$	0	0
$919$	−5.34752	−0.176399	−0.0881993	−	0.996103i	$-0.528111\pi$
$919$	−5.34752	−0.176399	−0.0881993	+	0.996103i	$0.528111\pi$
$920$	−29.7984	−0.982423
$921$	0	0
$922$	21.1246	0.695702
$923$	9.70820	0.319549
$924$	0	0
$925$	−70.4508	−2.31641
$926$	20.0902	0.660204
$927$	0	0
$928$	0.854102	0.0280373
$929$	−17.8197	−0.584644	−0.292322	−	0.956320i	$-0.594428\pi$
$929$	−17.8197	−0.584644	−0.292322	+	0.956320i	$0.594428\pi$
$930$	0	0
$931$	21.8885	0.717368
$932$	−4.23607	−0.138757
$933$	0	0
$934$	−6.58359	−0.215422
$935$	0.201626	0.00659388
$936$	0	0
$937$	−33.0902	−1.08101	−0.540504	−	0.841341i	$-0.681767\pi$
$937$	−33.0902	−1.08101	−0.540504	+	0.841341i	$0.681767\pi$
$938$	7.47214	0.243974
$939$	0	0
$940$	24.7984	0.808834
$941$	46.9787	1.53146	0.765731	−	0.643161i	$-0.222377\pi$
$941$	46.9787	1.53146	0.765731	+	0.643161i	$0.222377\pi$
$942$	0	0
$943$	−63.4853	−2.06737
$944$	6.56231	0.213585
$945$	0	0
$946$	0.527864	0.0171623
$947$	50.2148	1.63176	0.815881	−	0.578220i	$-0.196253\pi$
$947$	50.2148	1.63176	0.815881	+	0.578220i	$0.196253\pi$
$948$	0	0
$949$	10.7082	0.347603
$950$	−16.1803	−0.524960
$951$	0	0
$952$	−1.00000	−0.0324102
$953$	−25.3050	−0.819708	−0.409854	−	0.912151i	$-0.634420\pi$
$953$	−25.3050	−0.819708	−0.409854	+	0.912151i	$0.634420\pi$
$954$	0	0
$955$	18.2918	0.591909
$956$	19.2705	0.623253
$957$	0	0
$958$	−36.5967	−1.18239
$959$	−66.1591	−2.13639
$960$	0	0
$961$	24.8328	0.801059
$962$	−8.70820	−0.280764
$963$	0	0
$964$	−4.70820	−0.151641
$965$	−29.7984	−0.959244
$966$	0	0
$967$	1.85410	0.0596239	0.0298119	−	0.999556i	$-0.490509\pi$
$967$	1.85410	0.0596239	0.0298119	+	0.999556i	$0.490509\pi$
$968$	10.9443	0.351762
$969$	0	0
$970$	−45.7771	−1.46981
$971$	0.652476	0.0209389	0.0104695	−	0.999945i	$-0.496667\pi$
$971$	0.652476	0.0209389	0.0104695	+	0.999945i	$0.496667\pi$
$972$	0	0
$973$	−44.7426	−1.43438
$974$	14.8328	0.475274
$975$	0	0
$976$	−5.47214	−0.175159
$977$	−31.1459	−0.996446	−0.498223	−	0.867049i	$-0.666014\pi$
$977$	−31.1459	−0.996446	−0.498223	+	0.867049i	$0.666014\pi$
$978$	0	0
$979$	−2.31308	−0.0739264
$980$	39.5967	1.26487
$981$	0	0
$982$	6.76393	0.215846
$983$	31.4377	1.00271	0.501353	−	0.865243i	$-0.332836\pi$
$983$	31.4377	1.00271	0.501353	+	0.865243i	$0.332836\pi$
$984$	0	0
$985$	80.9787	2.58020
$986$	−0.201626	−0.00642108
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	18.4164	0.585608
$990$	0	0
$991$	−16.3820	−0.520390	−0.260195	−	0.965556i	$-0.583787\pi$
$991$	−16.3820	−0.520390	−0.260195	+	0.965556i	$0.583787\pi$
$992$	−7.47214	−0.237241
$993$	0	0
$994$	−41.1246	−1.30439
$995$	−19.4721	−0.617308
$996$	0	0
$997$	−17.5836	−0.556878	−0.278439	−	0.960454i	$-0.589817\pi$
$997$	−17.5836	−0.556878	−0.278439	+	0.960454i	$0.589817\pi$
$998$	−11.9443	−0.378089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.f.1.2		2
3.2	odd	2		538.2.a.a.1.2	✓	2
12.11	even	2		4304.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.a.1.2	✓	2	3.2	odd	2
4304.2.a.d.1.1		2	12.11	even	2
4842.2.a.f.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.61803 q^{5} -4.23607 q^{7} -1.00000 q^{8} -3.61803 q^{10} -0.236068 q^{11} -1.00000 q^{13} +4.23607 q^{14} +1.00000 q^{16} -0.236068 q^{17} +2.00000 q^{19} +3.61803 q^{20} +0.236068 q^{22} +8.23607 q^{23} +8.09017 q^{25} +1.00000 q^{26} -4.23607 q^{28} -0.854102 q^{29} +7.47214 q^{31} -1.00000 q^{32} +0.236068 q^{34} -15.3262 q^{35} -8.70820 q^{37} -2.00000 q^{38} -3.61803 q^{40} -7.70820 q^{41} +2.23607 q^{43} -0.236068 q^{44} -8.23607 q^{46} +6.85410 q^{47} +10.9443 q^{49} -8.09017 q^{50} -1.00000 q^{52} -0.854102 q^{55} +4.23607 q^{56} +0.854102 q^{58} +6.56231 q^{59} -5.47214 q^{61} -7.47214 q^{62} +1.00000 q^{64} -3.61803 q^{65} +1.76393 q^{67} -0.236068 q^{68} +15.3262 q^{70} -9.70820 q^{71} -10.7082 q^{73} +8.70820 q^{74} +2.00000 q^{76} +1.00000 q^{77} -13.7984 q^{79} +3.61803 q^{80} +7.70820 q^{82} +5.23607 q^{83} -0.854102 q^{85} -2.23607 q^{86} +0.236068 q^{88} +9.79837 q^{89} +4.23607 q^{91} +8.23607 q^{92} -6.85410 q^{94} +7.23607 q^{95} +12.6525 q^{97} -10.9443 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 5 q^{5} - 4 q^{7} - 2 q^{8} - 5 q^{10} + 4 q^{11} - 2 q^{13} + 4 q^{14} + 2 q^{16} + 4 q^{17} + 4 q^{19} + 5 q^{20} - 4 q^{22} + 12 q^{23} + 5 q^{25} + 2 q^{26} - 4 q^{28} + 5 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 5 * q^5 - 4 * q^7 - 2 * q^8 - 5 * q^10 + 4 * q^11 - 2 * q^13 + 4 * q^14 + 2 * q^16 + 4 * q^17 + 4 * q^19 + 5 * q^20 - 4 * q^22 + 12 * q^23 + 5 * q^25 + 2 * q^26 - 4 * q^28 + 5 * q^29 + 6 * q^31 - 2 * q^32 - 4 * q^34 - 15 * q^35 - 4 * q^37 - 4 * q^38 - 5 * q^40 - 2 * q^41 + 4 * q^44 - 12 * q^46 + 7 * q^47 + 4 * q^49 - 5 * q^50 - 2 * q^52 + 5 * q^55 + 4 * q^56 - 5 * q^58 - 7 * q^59 - 2 * q^61 - 6 * q^62 + 2 * q^64 - 5 * q^65 + 8 * q^67 + 4 * q^68 + 15 * q^70 - 6 * q^71 - 8 * q^73 + 4 * q^74 + 4 * q^76 + 2 * q^77 - 3 * q^79 + 5 * q^80 + 2 * q^82 + 6 * q^83 + 5 * q^85 - 4 * q^88 - 5 * q^89 + 4 * q^91 + 12 * q^92 - 7 * q^94 + 10 * q^95 - 6 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more