LMFDB - Embedded newform 4014.2.a.t.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, 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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.103354048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 13x^{2} - 16x + 3 $ x^6 - 2x^5 - 8x^4 + 14x^3 + 13x^2 - 16*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.605879$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.63291 q^{5} -2.24656 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.63291 q^{5} -2.24656 q^{7} -1.00000 q^{8} -2.63291 q^{10} +4.09429 q^{11} +0.00776648 q^{13} +2.24656 q^{14} +1.00000 q^{16} -0.673658 q^{17} -3.57290 q^{19} +2.63291 q^{20} -4.09429 q^{22} -3.39412 q^{23} +1.93222 q^{25} -0.00776648 q^{26} -2.24656 q^{28} -8.00809 q^{29} -6.93425 q^{31} -1.00000 q^{32} +0.673658 q^{34} -5.91498 q^{35} +0.651332 q^{37} +3.57290 q^{38} -2.63291 q^{40} -4.43181 q^{41} +9.59164 q^{43} +4.09429 q^{44} +3.39412 q^{46} -10.6264 q^{47} -1.95299 q^{49} -1.93222 q^{50} +0.00776648 q^{52} -4.64357 q^{53} +10.7799 q^{55} +2.24656 q^{56} +8.00809 q^{58} -7.95351 q^{59} -11.0517 q^{61} +6.93425 q^{62} +1.00000 q^{64} +0.0204485 q^{65} +15.2010 q^{67} -0.673658 q^{68} +5.91498 q^{70} +9.64824 q^{71} +0.995761 q^{73} -0.651332 q^{74} -3.57290 q^{76} -9.19804 q^{77} -1.27648 q^{79} +2.63291 q^{80} +4.43181 q^{82} -5.75391 q^{83} -1.77368 q^{85} -9.59164 q^{86} -4.09429 q^{88} +2.07485 q^{89} -0.0174478 q^{91} -3.39412 q^{92} +10.6264 q^{94} -9.40712 q^{95} -17.5825 q^{97} +1.95299 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 6 * q^4 - 2 * q^5 + 8 * q^7 - 6 * q^8	$ 6 q - 6 q^{2} + 6 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} + 2 q^{10} - 2 q^{11} - 2 q^{13} - 8 q^{14} + 6 q^{16} - 2 q^{17} - 2 q^{19} - 2 q^{20} + 2 q^{22} - 22 q^{23} + 12 q^{25} + 2 q^{26} + 8 q^{28} - 8 q^{29} - 12 q^{31} - 6 q^{32} + 2 q^{34} - 20 q^{35} + 2 q^{38} + 2 q^{40} - 28 q^{41} + 14 q^{43} - 2 q^{44} + 22 q^{46} - 2 q^{47} - 12 q^{50} - 2 q^{52} - 26 q^{53} + 6 q^{55} - 8 q^{56} + 8 q^{58} - 4 q^{59} - 6 q^{61} + 12 q^{62} + 6 q^{64} - 16 q^{65} + 18 q^{67} - 2 q^{68} + 20 q^{70} - 12 q^{71} - 20 q^{73} - 2 q^{76} - 20 q^{77} + 12 q^{79} - 2 q^{80} + 28 q^{82} - 12 q^{83} - 10 q^{85} - 14 q^{86} + 2 q^{88} + 2 q^{89} - 22 q^{91} - 22 q^{92} + 2 q^{94} - 6 q^{95} - 6 q^{97}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^4 - 2 * q^5 + 8 * q^7 - 6 * q^8 + 2 * q^10 - 2 * q^11 - 2 * q^13 - 8 * q^14 + 6 * q^16 - 2 * q^17 - 2 * q^19 - 2 * q^20 + 2 * q^22 - 22 * q^23 + 12 * q^25 + 2 * q^26 + 8 * q^28 - 8 * q^29 - 12 * q^31 - 6 * q^32 + 2 * q^34 - 20 * q^35 + 2 * q^38 + 2 * q^40 - 28 * q^41 + 14 * q^43 - 2 * q^44 + 22 * q^46 - 2 * q^47 - 12 * q^50 - 2 * q^52 - 26 * q^53 + 6 * q^55 - 8 * q^56 + 8 * q^58 - 4 * q^59 - 6 * q^61 + 12 * q^62 + 6 * q^64 - 16 * q^65 + 18 * q^67 - 2 * q^68 + 20 * q^70 - 12 * q^71 - 20 * q^73 - 2 * q^76 - 20 * q^77 + 12 * q^79 - 2 * q^80 + 28 * q^82 - 12 * q^83 - 10 * q^85 - 14 * q^86 + 2 * q^88 + 2 * q^89 - 22 * q^91 - 22 * q^92 + 2 * q^94 - 6 * q^95 - 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.63291	1.17747	0.588737	−	0.808325i	$-0.299625\pi$
$5$	2.63291	1.17747	0.588737	+	0.808325i	$0.299625\pi$
$6$	0	0
$7$	−2.24656	−0.849118	−0.424559	−	0.905400i	$-0.639571\pi$
$7$	−2.24656	−0.849118	−0.424559	+	0.905400i	$0.639571\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.63291	−0.832600
$11$	4.09429	1.23447	0.617237	−	0.786777i	$-0.288252\pi$
$11$	4.09429	1.23447	0.617237	+	0.786777i	$0.288252\pi$
$12$	0	0
$13$	0.00776648	0.00215403	0.00107702	−	0.999999i	$-0.499657\pi$
$13$	0.00776648	0.00215403	0.00107702	+	0.999999i	$0.499657\pi$
$14$	2.24656	0.600417
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.673658	−0.163386	−0.0816931	−	0.996658i	$-0.526033\pi$
$17$	−0.673658	−0.163386	−0.0816931	+	0.996658i	$0.526033\pi$
$18$	0	0
$19$	−3.57290	−0.819679	−0.409839	−	0.912158i	$-0.634415\pi$
$19$	−3.57290	−0.819679	−0.409839	+	0.912158i	$0.634415\pi$
$20$	2.63291	0.588737
$21$	0	0
$22$	−4.09429	−0.872905
$23$	−3.39412	−0.707723	−0.353862	−	0.935298i	$-0.615132\pi$
$23$	−3.39412	−0.707723	−0.353862	+	0.935298i	$0.615132\pi$
$24$	0	0
$25$	1.93222	0.386444
$26$	−0.00776648	−0.00152313
$27$	0	0
$28$	−2.24656	−0.424559
$29$	−8.00809	−1.48706	−0.743532	−	0.668700i	$-0.766851\pi$
$29$	−8.00809	−1.48706	−0.743532	+	0.668700i	$0.766851\pi$
$30$	0	0
$31$	−6.93425	−1.24543	−0.622714	−	0.782450i	$-0.713970\pi$
$31$	−6.93425	−1.24543	−0.622714	+	0.782450i	$0.713970\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.673658	0.115531
$35$	−5.91498	−0.999815
$36$	0	0
$37$	0.651332	0.107078	0.0535391	−	0.998566i	$-0.482950\pi$
$37$	0.651332	0.107078	0.0535391	+	0.998566i	$0.482950\pi$
$38$	3.57290	0.579601
$39$	0	0
$40$	−2.63291	−0.416300
$41$	−4.43181	−0.692132	−0.346066	−	0.938210i	$-0.612483\pi$
$41$	−4.43181	−0.692132	−0.346066	+	0.938210i	$0.612483\pi$
$42$	0	0
$43$	9.59164	1.46271	0.731355	−	0.681996i	$-0.238888\pi$
$43$	9.59164	1.46271	0.731355	+	0.681996i	$0.238888\pi$
$44$	4.09429	0.617237
$45$	0	0
$46$	3.39412	0.500436
$47$	−10.6264	−1.55003	−0.775013	−	0.631946i	$-0.782256\pi$
$47$	−10.6264	−1.55003	−0.775013	+	0.631946i	$0.782256\pi$
$48$	0	0
$49$	−1.95299	−0.278998
$50$	−1.93222	−0.273257
$51$	0	0
$52$	0.00776648	0.00107702
$53$	−4.64357	−0.637843	−0.318921	−	0.947781i	$-0.603321\pi$
$53$	−4.64357	−0.637843	−0.318921	+	0.947781i	$0.603321\pi$
$54$	0	0
$55$	10.7799	1.45356
$56$	2.24656	0.300209
$57$	0	0
$58$	8.00809	1.05151
$59$	−7.95351	−1.03546	−0.517730	−	0.855544i	$-0.673223\pi$
$59$	−7.95351	−1.03546	−0.517730	+	0.855544i	$0.673223\pi$
$60$	0	0
$61$	−11.0517	−1.41502	−0.707510	−	0.706703i	$-0.750182\pi$
$61$	−11.0517	−1.41502	−0.707510	+	0.706703i	$0.750182\pi$
$62$	6.93425	0.880650
$63$	0	0
$64$	1.00000	0.125000
$65$	0.0204485	0.00253632
$66$	0	0
$67$	15.2010	1.85710	0.928551	−	0.371204i	$-0.121055\pi$
$67$	15.2010	1.85710	0.928551	+	0.371204i	$0.121055\pi$
$68$	−0.673658	−0.0816931
$69$	0	0
$70$	5.91498	0.706976
$71$	9.64824	1.14504	0.572518	−	0.819892i	$-0.305967\pi$
$71$	9.64824	1.14504	0.572518	+	0.819892i	$0.305967\pi$
$72$	0	0
$73$	0.995761	0.116545	0.0582725	−	0.998301i	$-0.481441\pi$
$73$	0.995761	0.116545	0.0582725	+	0.998301i	$0.481441\pi$
$74$	−0.651332	−0.0757157
$75$	0	0
$76$	−3.57290	−0.409839
$77$	−9.19804	−1.04821
$78$	0	0
$79$	−1.27648	−0.143615	−0.0718074	−	0.997419i	$-0.522877\pi$
$79$	−1.27648	−0.143615	−0.0718074	+	0.997419i	$0.522877\pi$
$80$	2.63291	0.294368
$81$	0	0
$82$	4.43181	0.489411
$83$	−5.75391	−0.631574	−0.315787	−	0.948830i	$-0.602268\pi$
$83$	−5.75391	−0.631574	−0.315787	+	0.948830i	$0.602268\pi$
$84$	0	0
$85$	−1.77368	−0.192383
$86$	−9.59164	−1.03429
$87$	0	0
$88$	−4.09429	−0.436452
$89$	2.07485	0.219933	0.109967	−	0.993935i	$-0.464926\pi$
$89$	2.07485	0.219933	0.109967	+	0.993935i	$0.464926\pi$
$90$	0	0
$91$	−0.0174478	−0.00182903
$92$	−3.39412	−0.353862
$93$	0	0
$94$	10.6264	1.09603
$95$	−9.40712	−0.965150
$96$	0	0
$97$	−17.5825	−1.78523	−0.892615	−	0.450819i	$-0.851132\pi$
$97$	−17.5825	−1.78523	−0.892615	+	0.450819i	$0.851132\pi$
$98$	1.95299	0.197281
$99$	0	0
$100$	1.93222	0.193222
$101$	8.56611	0.852359	0.426180	−	0.904639i	$-0.359859\pi$
$101$	8.56611	0.852359	0.426180	+	0.904639i	$0.359859\pi$
$102$	0	0
$103$	19.9320	1.96396	0.981978	−	0.188996i	$-0.0605234\pi$
$103$	19.9320	1.96396	0.981978	+	0.188996i	$0.0605234\pi$
$104$	−0.00776648	−0.000761566 0
$105$	0	0
$106$	4.64357	0.451023
$107$	−13.1728	−1.27347	−0.636733	−	0.771085i	$-0.719715\pi$
$107$	−13.1728	−1.27347	−0.636733	+	0.771085i	$0.719715\pi$
$108$	0	0
$109$	−5.10676	−0.489139	−0.244569	−	0.969632i	$-0.578647\pi$
$109$	−5.10676	−0.489139	−0.244569	+	0.969632i	$0.578647\pi$
$110$	−10.7799	−1.02782
$111$	0	0
$112$	−2.24656	−0.212280
$113$	−15.5553	−1.46332	−0.731659	−	0.681671i	$-0.761254\pi$
$113$	−15.5553	−1.46332	−0.731659	+	0.681671i	$0.761254\pi$
$114$	0	0
$115$	−8.93642	−0.833325
$116$	−8.00809	−0.743532
$117$	0	0
$118$	7.95351	0.732180
$119$	1.51341	0.138734
$120$	0	0
$121$	5.76317	0.523925
$122$	11.0517	1.00057
$123$	0	0
$124$	−6.93425	−0.622714
$125$	−8.07719	−0.722446
$126$	0	0
$127$	15.4106	1.36747	0.683737	−	0.729729i	$-0.260354\pi$
$127$	15.4106	1.36747	0.683737	+	0.729729i	$0.260354\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.0204485	−0.00179345
$131$	21.0190	1.83644	0.918219	−	0.396073i	$-0.129627\pi$
$131$	21.0190	1.83644	0.918219	+	0.396073i	$0.129627\pi$
$132$	0	0
$133$	8.02672	0.696005
$134$	−15.2010	−1.31317
$135$	0	0
$136$	0.673658	0.0577657
$137$	−12.5823	−1.07498	−0.537490	−	0.843270i	$-0.680627\pi$
$137$	−12.5823	−1.07498	−0.537490	+	0.843270i	$0.680627\pi$
$138$	0	0
$139$	−9.51306	−0.806887	−0.403444	−	0.915004i	$-0.632187\pi$
$139$	−9.51306	−0.806887	−0.403444	+	0.915004i	$0.632187\pi$
$140$	−5.91498	−0.499907
$141$	0	0
$142$	−9.64824	−0.809662
$143$	0.0317982	0.00265910
$144$	0	0
$145$	−21.0846	−1.75098
$146$	−0.995761	−0.0824097
$147$	0	0
$148$	0.651332	0.0535391
$149$	2.73070	0.223708	0.111854	−	0.993725i	$-0.464321\pi$
$149$	2.73070	0.223708	0.111854	+	0.993725i	$0.464321\pi$
$150$	0	0
$151$	7.96037	0.647806	0.323903	−	0.946090i	$-0.395005\pi$
$151$	7.96037	0.647806	0.323903	+	0.946090i	$0.395005\pi$
$152$	3.57290	0.289800
$153$	0	0
$154$	9.19804	0.741199
$155$	−18.2573	−1.46646
$156$	0	0
$157$	12.4903	0.996831	0.498416	−	0.866938i	$-0.333915\pi$
$157$	12.4903	0.996831	0.498416	+	0.866938i	$0.333915\pi$
$158$	1.27648	0.101551
$159$	0	0
$160$	−2.63291	−0.208150
$161$	7.62508	0.600941
$162$	0	0
$163$	−6.27007	−0.491109	−0.245555	−	0.969383i	$-0.578970\pi$
$163$	−6.27007	−0.491109	−0.245555	+	0.969383i	$0.578970\pi$
$164$	−4.43181	−0.346066
$165$	0	0
$166$	5.75391	0.446590
$167$	−2.90654	−0.224915	−0.112458	−	0.993657i	$-0.535872\pi$
$167$	−2.90654	−0.224915	−0.112458	+	0.993657i	$0.535872\pi$
$168$	0	0
$169$	−12.9999	−0.999995
$170$	1.77368	0.136035
$171$	0	0
$172$	9.59164	0.731355
$173$	9.28984	0.706293	0.353147	−	0.935568i	$-0.385112\pi$
$173$	9.28984	0.706293	0.353147	+	0.935568i	$0.385112\pi$
$174$	0	0
$175$	−4.34084	−0.328137
$176$	4.09429	0.308618
$177$	0	0
$178$	−2.07485	−0.155516
$179$	−21.1054	−1.57749	−0.788745	−	0.614721i	$-0.789269\pi$
$179$	−21.1054	−1.57749	−0.788745	+	0.614721i	$0.789269\pi$
$180$	0	0
$181$	0.543334	0.0403857	0.0201928	−	0.999796i	$-0.493572\pi$
$181$	0.543334	0.0403857	0.0201928	+	0.999796i	$0.493572\pi$
$182$	0.0174478	0.00129332
$183$	0	0
$184$	3.39412	0.250218
$185$	1.71490	0.126082
$186$	0	0
$187$	−2.75815	−0.201696
$188$	−10.6264	−0.775013
$189$	0	0
$190$	9.40712	0.682464
$191$	−14.0400	−1.01590	−0.507950	−	0.861387i	$-0.669596\pi$
$191$	−14.0400	−1.01590	−0.507950	+	0.861387i	$0.669596\pi$
$192$	0	0
$193$	−9.77786	−0.703826	−0.351913	−	0.936033i	$-0.614469\pi$
$193$	−9.77786	−0.703826	−0.351913	+	0.936033i	$0.614469\pi$
$194$	17.5825	1.26235
$195$	0	0
$196$	−1.95299	−0.139499
$197$	14.5046	1.03341	0.516705	−	0.856163i	$-0.327158\pi$
$197$	14.5046	1.03341	0.516705	+	0.856163i	$0.327158\pi$
$198$	0	0
$199$	22.6203	1.60351	0.801755	−	0.597653i	$-0.203900\pi$
$199$	22.6203	1.60351	0.801755	+	0.597653i	$0.203900\pi$
$200$	−1.93222	−0.136629
$201$	0	0
$202$	−8.56611	−0.602709
$203$	17.9906	1.26269
$204$	0	0
$205$	−11.6686	−0.814967
$206$	−19.9320	−1.38873
$207$	0	0
$208$	0.00776648	0.000538508 0
$209$	−14.6285	−1.01187
$210$	0	0
$211$	−10.0332	−0.690715	−0.345358	−	0.938471i	$-0.612242\pi$
$211$	−10.0332	−0.690715	−0.345358	+	0.938471i	$0.612242\pi$
$212$	−4.64357	−0.318921
$213$	0	0
$214$	13.1728	0.900476
$215$	25.2539	1.72230
$216$	0	0
$217$	15.5782	1.05752
$218$	5.10676	0.345873
$219$	0	0
$220$	10.7799	0.726780
$221$	−0.00523195	−0.000351939 0
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	2.24656	0.150104
$225$	0	0
$226$	15.5553	1.03472
$227$	20.8914	1.38661	0.693305	−	0.720645i	$-0.256154\pi$
$227$	20.8914	1.38661	0.693305	+	0.720645i	$0.256154\pi$
$228$	0	0
$229$	20.5829	1.36015	0.680077	−	0.733141i	$-0.261946\pi$
$229$	20.5829	1.36015	0.680077	+	0.733141i	$0.261946\pi$
$230$	8.93642	0.589250
$231$	0	0
$232$	8.00809	0.525757
$233$	−9.41577	−0.616848	−0.308424	−	0.951249i	$-0.599802\pi$
$233$	−9.41577	−0.616848	−0.308424	+	0.951249i	$0.599802\pi$
$234$	0	0
$235$	−27.9785	−1.82511
$236$	−7.95351	−0.517730
$237$	0	0
$238$	−1.51341	−0.0980999
$239$	14.6538	0.947877	0.473939	−	0.880558i	$-0.342832\pi$
$239$	14.6538	0.947877	0.473939	+	0.880558i	$0.342832\pi$
$240$	0	0
$241$	17.1216	1.10290	0.551448	−	0.834209i	$-0.314075\pi$
$241$	17.1216	1.10290	0.551448	+	0.834209i	$0.314075\pi$
$242$	−5.76317	−0.370471
$243$	0	0
$244$	−11.0517	−0.707510
$245$	−5.14204	−0.328513
$246$	0	0
$247$	−0.0277488	−0.00176562
$248$	6.93425	0.440325
$249$	0	0
$250$	8.07719	0.510846
$251$	−22.7295	−1.43467	−0.717337	−	0.696727i	$-0.754639\pi$
$251$	−22.7295	−1.43467	−0.717337	+	0.696727i	$0.754639\pi$
$252$	0	0
$253$	−13.8965	−0.873666
$254$	−15.4106	−0.966950
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.48452	0.466872	0.233436	−	0.972372i	$-0.425003\pi$
$257$	7.48452	0.466872	0.233436	+	0.972372i	$0.425003\pi$
$258$	0	0
$259$	−1.46325	−0.0909221
$260$	0.0204485	0.00126816
$261$	0	0
$262$	−21.0190	−1.29856
$263$	7.82448	0.482478	0.241239	−	0.970466i	$-0.422446\pi$
$263$	7.82448	0.482478	0.241239	+	0.970466i	$0.422446\pi$
$264$	0	0
$265$	−12.2261	−0.751043
$266$	−8.02672	−0.492150
$267$	0	0
$268$	15.2010	0.928551
$269$	−31.3794	−1.91324	−0.956619	−	0.291342i	$-0.905898\pi$
$269$	−31.3794	−1.91324	−0.956619	+	0.291342i	$0.905898\pi$
$270$	0	0
$271$	−1.63798	−0.0995000	−0.0497500	−	0.998762i	$-0.515842\pi$
$271$	−1.63798	−0.0995000	−0.0497500	+	0.998762i	$0.515842\pi$
$272$	−0.673658	−0.0408465
$273$	0	0
$274$	12.5823	0.760125
$275$	7.91106	0.477055
$276$	0	0
$277$	8.78445	0.527806	0.263903	−	0.964549i	$-0.414990\pi$
$277$	8.78445	0.527806	0.263903	+	0.964549i	$0.414990\pi$
$278$	9.51306	0.570555
$279$	0	0
$280$	5.91498	0.353488
$281$	13.1236	0.782887	0.391444	−	0.920202i	$-0.371976\pi$
$281$	13.1236	0.782887	0.391444	+	0.920202i	$0.371976\pi$
$282$	0	0
$283$	21.8961	1.30159	0.650795	−	0.759253i	$-0.274436\pi$
$283$	21.8961	1.30159	0.650795	+	0.759253i	$0.274436\pi$
$284$	9.64824	0.572518
$285$	0	0
$286$	−0.0317982	−0.00188027
$287$	9.95630	0.587702
$288$	0	0
$289$	−16.5462	−0.973305
$290$	21.0846	1.23813
$291$	0	0
$292$	0.995761	0.0582725
$293$	15.4295	0.901402	0.450701	−	0.892675i	$-0.351174\pi$
$293$	15.4295	0.901402	0.450701	+	0.892675i	$0.351174\pi$
$294$	0	0
$295$	−20.9409	−1.21923
$296$	−0.651332	−0.0378579
$297$	0	0
$298$	−2.73070	−0.158185
$299$	−0.0263604	−0.00152446
$300$	0	0
$301$	−21.5482	−1.24201
$302$	−7.96037	−0.458068
$303$	0	0
$304$	−3.57290	−0.204920
$305$	−29.0980	−1.66615
$306$	0	0
$307$	−18.6854	−1.06643	−0.533216	−	0.845979i	$-0.679017\pi$
$307$	−18.6854	−1.06643	−0.533216	+	0.845979i	$0.679017\pi$
$308$	−9.19804	−0.524107
$309$	0	0
$310$	18.2573	1.03694
$311$	−21.9347	−1.24380	−0.621902	−	0.783095i	$-0.713640\pi$
$311$	−21.9347	−1.24380	−0.621902	+	0.783095i	$0.713640\pi$
$312$	0	0
$313$	−5.51529	−0.311743	−0.155871	−	0.987777i	$-0.549819\pi$
$313$	−5.51529	−0.311743	−0.155871	+	0.987777i	$0.549819\pi$
$314$	−12.4903	−0.704866
$315$	0	0
$316$	−1.27648	−0.0718074
$317$	−10.2843	−0.577622	−0.288811	−	0.957386i	$-0.593260\pi$
$317$	−10.2843	−0.577622	−0.288811	+	0.957386i	$0.593260\pi$
$318$	0	0
$319$	−32.7874	−1.83574
$320$	2.63291	0.147184
$321$	0	0
$322$	−7.62508	−0.424929
$323$	2.40691	0.133924
$324$	0	0
$325$	0.0150066	0.000832414 0
$326$	6.27007	0.347267
$327$	0	0
$328$	4.43181	0.244706
$329$	23.8729	1.31615
$330$	0	0
$331$	−23.9394	−1.31583	−0.657913	−	0.753094i	$-0.728561\pi$
$331$	−23.9394	−1.31583	−0.657913	+	0.753094i	$0.728561\pi$
$332$	−5.75391	−0.315787
$333$	0	0
$334$	2.90654	0.159039
$335$	40.0230	2.18669
$336$	0	0
$337$	4.45595	0.242731	0.121366	−	0.992608i	$-0.461273\pi$
$337$	4.45595	0.242731	0.121366	+	0.992608i	$0.461273\pi$
$338$	12.9999	0.707104
$339$	0	0
$340$	−1.77368	−0.0961914
$341$	−28.3908	−1.53745
$342$	0	0
$343$	20.1134	1.08602
$344$	−9.59164	−0.517146
$345$	0	0
$346$	−9.28984	−0.499425
$347$	−30.7768	−1.65219	−0.826093	−	0.563534i	$-0.809442\pi$
$347$	−30.7768	−1.65219	−0.826093	+	0.563534i	$0.809442\pi$
$348$	0	0
$349$	10.6268	0.568837	0.284419	−	0.958700i	$-0.408199\pi$
$349$	10.6268	0.568837	0.284419	+	0.958700i	$0.408199\pi$
$350$	4.34084	0.232028
$351$	0	0
$352$	−4.09429	−0.218226
$353$	23.9565	1.27508	0.637538	−	0.770419i	$-0.279953\pi$
$353$	23.9565	1.27508	0.637538	+	0.770419i	$0.279953\pi$
$354$	0	0
$355$	25.4029	1.34825
$356$	2.07485	0.109967
$357$	0	0
$358$	21.1054	1.11545
$359$	17.1370	0.904457	0.452229	−	0.891902i	$-0.350629\pi$
$359$	17.1370	0.904457	0.452229	+	0.891902i	$0.350629\pi$
$360$	0	0
$361$	−6.23440	−0.328126
$362$	−0.543334	−0.0285570
$363$	0	0
$364$	−0.0174478	−0.000914515 0
$365$	2.62175	0.137229
$366$	0	0
$367$	−11.9256	−0.622511	−0.311255	−	0.950326i	$-0.600749\pi$
$367$	−11.9256	−0.622511	−0.311255	+	0.950326i	$0.600749\pi$
$368$	−3.39412	−0.176931
$369$	0	0
$370$	−1.71490	−0.0891533
$371$	10.4320	0.541604
$372$	0	0
$373$	−3.91021	−0.202463	−0.101232	−	0.994863i	$-0.532278\pi$
$373$	−3.91021	−0.202463	−0.101232	+	0.994863i	$0.532278\pi$
$374$	2.75815	0.142621
$375$	0	0
$376$	10.6264	0.548017
$377$	−0.0621946	−0.00320319
$378$	0	0
$379$	10.4984	0.539268	0.269634	−	0.962963i	$-0.413097\pi$
$379$	10.4984	0.539268	0.269634	+	0.962963i	$0.413097\pi$
$380$	−9.40712	−0.482575
$381$	0	0
$382$	14.0400	0.718350
$383$	−8.52206	−0.435457	−0.217729	−	0.976009i	$-0.569865\pi$
$383$	−8.52206	−0.435457	−0.217729	+	0.976009i	$0.569865\pi$
$384$	0	0
$385$	−24.2176	−1.23424
$386$	9.77786	0.497680
$387$	0	0
$388$	−17.5825	−0.892615
$389$	−31.7085	−1.60768	−0.803842	−	0.594843i	$-0.797214\pi$
$389$	−31.7085	−1.60768	−0.803842	+	0.594843i	$0.797214\pi$
$390$	0	0
$391$	2.28648	0.115632
$392$	1.95299	0.0986407
$393$	0	0
$394$	−14.5046	−0.730732
$395$	−3.36085	−0.169103
$396$	0	0
$397$	−5.92081	−0.297157	−0.148578	−	0.988901i	$-0.547470\pi$
$397$	−5.92081	−0.297157	−0.148578	+	0.988901i	$0.547470\pi$
$398$	−22.6203	−1.13385
$399$	0	0
$400$	1.93222	0.0966110
$401$	−8.12974	−0.405980	−0.202990	−	0.979181i	$-0.565066\pi$
$401$	−8.12974	−0.405980	−0.202990	+	0.979181i	$0.565066\pi$
$402$	0	0
$403$	−0.0538547	−0.00268269
$404$	8.56611	0.426180
$405$	0	0
$406$	−17.9906	−0.892859
$407$	2.66674	0.132185
$408$	0	0
$409$	12.9592	0.640791	0.320395	−	0.947284i	$-0.396184\pi$
$409$	12.9592	0.640791	0.320395	+	0.947284i	$0.396184\pi$
$410$	11.6686	0.576269
$411$	0	0
$412$	19.9320	0.981978
$413$	17.8680	0.879228
$414$	0	0
$415$	−15.1495	−0.743661
$416$	−0.00776648	−0.000380783 0
$417$	0	0
$418$	14.6285	0.715502
$419$	−30.4806	−1.48907	−0.744537	−	0.667581i	$-0.767330\pi$
$419$	−30.4806	−1.48907	−0.744537	+	0.667581i	$0.767330\pi$
$420$	0	0
$421$	−38.0654	−1.85520	−0.927598	−	0.373581i	$-0.878130\pi$
$421$	−38.0654	−1.85520	−0.927598	+	0.373581i	$0.878130\pi$
$422$	10.0332	0.488409
$423$	0	0
$424$	4.64357	0.225511
$425$	−1.30166	−0.0631396
$426$	0	0
$427$	24.8282	1.20152
$428$	−13.1728	−0.636733
$429$	0	0
$430$	−25.2539	−1.21785
$431$	−21.5664	−1.03881	−0.519407	−	0.854527i	$-0.673847\pi$
$431$	−21.5664	−1.03881	−0.519407	+	0.854527i	$0.673847\pi$
$432$	0	0
$433$	−8.92010	−0.428673	−0.214336	−	0.976760i	$-0.568759\pi$
$433$	−8.92010	−0.428673	−0.214336	+	0.976760i	$0.568759\pi$
$434$	−15.5782	−0.747776
$435$	0	0
$436$	−5.10676	−0.244569
$437$	12.1268	0.580106
$438$	0	0
$439$	−26.0102	−1.24140	−0.620700	−	0.784048i	$-0.713152\pi$
$439$	−26.0102	−1.24140	−0.620700	+	0.784048i	$0.713152\pi$
$440$	−10.7799	−0.513911
$441$	0	0
$442$	0.00523195	0.000248859 0
$443$	10.7806	0.512200	0.256100	−	0.966650i	$-0.417562\pi$
$443$	10.7806	0.512200	0.256100	+	0.966650i	$0.417562\pi$
$444$	0	0
$445$	5.46289	0.258966
$446$	1.00000	0.0473514
$447$	0	0
$448$	−2.24656	−0.106140
$449$	−20.0819	−0.947722	−0.473861	−	0.880600i	$-0.657140\pi$
$449$	−20.0819	−0.947722	−0.473861	+	0.880600i	$0.657140\pi$
$450$	0	0
$451$	−18.1451	−0.854419
$452$	−15.5553	−0.731659
$453$	0	0
$454$	−20.8914	−0.980481
$455$	−0.0459386	−0.00215363
$456$	0	0
$457$	−35.5020	−1.66071	−0.830357	−	0.557232i	$-0.811863\pi$
$457$	−35.5020	−1.66071	−0.830357	+	0.557232i	$0.811863\pi$
$458$	−20.5829	−0.961774
$459$	0	0
$460$	−8.93642	−0.416663
$461$	−3.26003	−0.151835	−0.0759173	−	0.997114i	$-0.524189\pi$
$461$	−3.26003	−0.151835	−0.0759173	+	0.997114i	$0.524189\pi$
$462$	0	0
$463$	5.67209	0.263604	0.131802	−	0.991276i	$-0.457924\pi$
$463$	5.67209	0.263604	0.131802	+	0.991276i	$0.457924\pi$
$464$	−8.00809	−0.371766
$465$	0	0
$466$	9.41577	0.436177
$467$	−10.7885	−0.499231	−0.249616	−	0.968345i	$-0.580304\pi$
$467$	−10.7885	−0.499231	−0.249616	+	0.968345i	$0.580304\pi$
$468$	0	0
$469$	−34.1500	−1.57690
$470$	27.9785	1.29055
$471$	0	0
$472$	7.95351	0.366090
$473$	39.2709	1.80568
$474$	0	0
$475$	−6.90363	−0.316760
$476$	1.51341	0.0693671
$477$	0	0
$478$	−14.6538	−0.670250
$479$	17.5354	0.801211	0.400605	−	0.916251i	$-0.368800\pi$
$479$	17.5354	0.801211	0.400605	+	0.916251i	$0.368800\pi$
$480$	0	0
$481$	0.00505855	0.000230650 0
$482$	−17.1216	−0.779866
$483$	0	0
$484$	5.76317	0.261962
$485$	−46.2931	−2.10206
$486$	0	0
$487$	21.6167	0.979547	0.489774	−	0.871850i	$-0.337079\pi$
$487$	21.6167	0.979547	0.489774	+	0.871850i	$0.337079\pi$
$488$	11.0517	0.500285
$489$	0	0
$490$	5.14204	0.232294
$491$	−11.0429	−0.498359	−0.249180	−	0.968457i	$-0.580161\pi$
$491$	−11.0429	−0.498359	−0.249180	+	0.968457i	$0.580161\pi$
$492$	0	0
$493$	5.39471	0.242966
$494$	0.0277488	0.00124848
$495$	0	0
$496$	−6.93425	−0.311357
$497$	−21.6753	−0.972270
$498$	0	0
$499$	−9.24468	−0.413849	−0.206924	−	0.978357i	$-0.566345\pi$
$499$	−9.24468	−0.413849	−0.206924	+	0.978357i	$0.566345\pi$
$500$	−8.07719	−0.361223
$501$	0	0
$502$	22.7295	1.01447
$503$	42.2295	1.88292	0.941461	−	0.337123i	$-0.109454\pi$
$503$	42.2295	1.88292	0.941461	+	0.337123i	$0.109454\pi$
$504$	0	0
$505$	22.5538	1.00363
$506$	13.8965	0.617775
$507$	0	0
$508$	15.4106	0.683737
$509$	−38.2695	−1.69627	−0.848134	−	0.529782i	$-0.822274\pi$
$509$	−38.2695	−1.69627	−0.848134	+	0.529782i	$0.822274\pi$
$510$	0	0
$511$	−2.23703	−0.0989605
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−7.48452	−0.330128
$515$	52.4791	2.31251
$516$	0	0
$517$	−43.5077	−1.91346
$518$	1.46325	0.0642916
$519$	0	0
$520$	−0.0204485	−0.000896724 0
$521$	20.1266	0.881760	0.440880	−	0.897566i	$-0.354666\pi$
$521$	20.1266	0.881760	0.440880	+	0.897566i	$0.354666\pi$
$522$	0	0
$523$	−3.72087	−0.162702	−0.0813512	−	0.996686i	$-0.525924\pi$
$523$	−3.72087	−0.162702	−0.0813512	+	0.996686i	$0.525924\pi$
$524$	21.0190	0.918219
$525$	0	0
$526$	−7.82448	−0.341163
$527$	4.67131	0.203486
$528$	0	0
$529$	−11.4799	−0.499128
$530$	12.2261	0.531068
$531$	0	0
$532$	8.02672	0.348002
$533$	−0.0344195	−0.00149088
$534$	0	0
$535$	−34.6829	−1.49947
$536$	−15.2010	−0.656585
$537$	0	0
$538$	31.3794	1.35286
$539$	−7.99608	−0.344416
$540$	0	0
$541$	−22.9511	−0.986745	−0.493373	−	0.869818i	$-0.664236\pi$
$541$	−22.9511	−0.986745	−0.493373	+	0.869818i	$0.664236\pi$
$542$	1.63798	0.0703571
$543$	0	0
$544$	0.673658	0.0288829
$545$	−13.4456	−0.575948
$546$	0	0
$547$	−2.69925	−0.115412	−0.0577058	−	0.998334i	$-0.518379\pi$
$547$	−2.69925	−0.115412	−0.0577058	+	0.998334i	$0.518379\pi$
$548$	−12.5823	−0.537490
$549$	0	0
$550$	−7.91106	−0.337329
$551$	28.6121	1.21892
$552$	0	0
$553$	2.86768	0.121946
$554$	−8.78445	−0.373215
$555$	0	0
$556$	−9.51306	−0.403444
$557$	−19.9531	−0.845440	−0.422720	−	0.906260i	$-0.638925\pi$
$557$	−19.9531	−0.845440	−0.422720	+	0.906260i	$0.638925\pi$
$558$	0	0
$559$	0.0744933	0.00315073
$560$	−5.91498	−0.249954
$561$	0	0
$562$	−13.1236	−0.553585
$563$	−1.38362	−0.0583125	−0.0291563	−	0.999575i	$-0.509282\pi$
$563$	−1.38362	−0.0583125	−0.0291563	+	0.999575i	$0.509282\pi$
$564$	0	0
$565$	−40.9557	−1.72302
$566$	−21.8961	−0.920364
$567$	0	0
$568$	−9.64824	−0.404831
$569$	−12.6605	−0.530757	−0.265379	−	0.964144i	$-0.585497\pi$
$569$	−12.6605	−0.530757	−0.265379	+	0.964144i	$0.585497\pi$
$570$	0	0
$571$	46.7306	1.95562	0.977808	−	0.209503i	$-0.0671846\pi$
$571$	46.7306	1.95562	0.977808	+	0.209503i	$0.0671846\pi$
$572$	0.0317982	0.00132955
$573$	0	0
$574$	−9.95630	−0.415568
$575$	−6.55819	−0.273495
$576$	0	0
$577$	−18.8790	−0.785944	−0.392972	−	0.919550i	$-0.628553\pi$
$577$	−18.8790	−0.785944	−0.392972	+	0.919550i	$0.628553\pi$
$578$	16.5462	0.688231
$579$	0	0
$580$	−21.0846	−0.875489
$581$	12.9265	0.536281
$582$	0	0
$583$	−19.0121	−0.787400
$584$	−0.995761	−0.0412049
$585$	0	0
$586$	−15.4295	−0.637387
$587$	7.31271	0.301828	0.150914	−	0.988547i	$-0.451778\pi$
$587$	7.31271	0.301828	0.150914	+	0.988547i	$0.451778\pi$
$588$	0	0
$589$	24.7754	1.02085
$590$	20.9409	0.862123
$591$	0	0
$592$	0.651332	0.0267696
$593$	−0.674078	−0.0276811	−0.0138405	−	0.999904i	$-0.504406\pi$
$593$	−0.674078	−0.0276811	−0.0138405	+	0.999904i	$0.504406\pi$
$594$	0	0
$595$	3.98468	0.163356
$596$	2.73070	0.111854
$597$	0	0
$598$	0.0263604	0.00107796
$599$	9.80274	0.400529	0.200265	−	0.979742i	$-0.435820\pi$
$599$	9.80274	0.400529	0.200265	+	0.979742i	$0.435820\pi$
$600$	0	0
$601$	22.8526	0.932178	0.466089	−	0.884738i	$-0.345663\pi$
$601$	22.8526	0.932178	0.466089	+	0.884738i	$0.345663\pi$
$602$	21.5482	0.878237
$603$	0	0
$604$	7.96037	0.323903
$605$	15.1739	0.616908
$606$	0	0
$607$	44.0179	1.78663	0.893316	−	0.449428i	$-0.148372\pi$
$607$	44.0179	1.78663	0.893316	+	0.449428i	$0.148372\pi$
$608$	3.57290	0.144900
$609$	0	0
$610$	29.0980	1.17815
$611$	−0.0825300	−0.00333881
$612$	0	0
$613$	23.4729	0.948063	0.474031	−	0.880508i	$-0.342798\pi$
$613$	23.4729	0.948063	0.474031	+	0.880508i	$0.342798\pi$
$614$	18.6854	0.754082
$615$	0	0
$616$	9.19804	0.370600
$617$	24.9763	1.00551	0.502753	−	0.864430i	$-0.332321\pi$
$617$	24.9763	1.00551	0.502753	+	0.864430i	$0.332321\pi$
$618$	0	0
$619$	−9.73956	−0.391466	−0.195733	−	0.980657i	$-0.562709\pi$
$619$	−9.73956	−0.391466	−0.195733	+	0.980657i	$0.562709\pi$
$620$	−18.2573	−0.733229
$621$	0	0
$622$	21.9347	0.879502
$623$	−4.66126	−0.186749
$624$	0	0
$625$	−30.9276	−1.23711
$626$	5.51529	0.220435
$627$	0	0
$628$	12.4903	0.498416
$629$	−0.438775	−0.0174951
$630$	0	0
$631$	−23.1790	−0.922740	−0.461370	−	0.887208i	$-0.652642\pi$
$631$	−23.1790	−0.922740	−0.461370	+	0.887208i	$0.652642\pi$
$632$	1.27648	0.0507755
$633$	0	0
$634$	10.2843	0.408440
$635$	40.5749	1.61016
$636$	0	0
$637$	−0.0151678	−0.000600971 0
$638$	32.7874	1.29807
$639$	0	0
$640$	−2.63291	−0.104075
$641$	11.3880	0.449799	0.224900	−	0.974382i	$-0.427795\pi$
$641$	11.3880	0.449799	0.224900	+	0.974382i	$0.427795\pi$
$642$	0	0
$643$	7.09359	0.279744	0.139872	−	0.990170i	$-0.455331\pi$
$643$	7.09359	0.279744	0.139872	+	0.990170i	$0.455331\pi$
$644$	7.62508	0.300470
$645$	0	0
$646$	−2.40691	−0.0946987
$647$	26.4990	1.04178	0.520892	−	0.853623i	$-0.325600\pi$
$647$	26.4990	1.04178	0.520892	+	0.853623i	$0.325600\pi$
$648$	0	0
$649$	−32.5640	−1.27825
$650$	−0.0150066	−0.000588605 0
$651$	0	0
$652$	−6.27007	−0.245555
$653$	11.3962	0.445966	0.222983	−	0.974822i	$-0.428421\pi$
$653$	11.3962	0.445966	0.222983	+	0.974822i	$0.428421\pi$
$654$	0	0
$655$	55.3412	2.16236
$656$	−4.43181	−0.173033
$657$	0	0
$658$	−23.8729	−0.930662
$659$	−41.0879	−1.60056	−0.800278	−	0.599629i	$-0.795315\pi$
$659$	−41.0879	−1.60056	−0.800278	+	0.599629i	$0.795315\pi$
$660$	0	0
$661$	43.6409	1.69743	0.848716	−	0.528849i	$-0.177376\pi$
$661$	43.6409	1.69743	0.848716	+	0.528849i	$0.177376\pi$
$662$	23.9394	0.930430
$663$	0	0
$664$	5.75391	0.223295
$665$	21.1336	0.819527
$666$	0	0
$667$	27.1804	1.05243
$668$	−2.90654	−0.112458
$669$	0	0
$670$	−40.0230	−1.54622
$671$	−45.2487	−1.74680
$672$	0	0
$673$	22.3619	0.861987	0.430994	−	0.902355i	$-0.358163\pi$
$673$	22.3619	0.861987	0.430994	+	0.902355i	$0.358163\pi$
$674$	−4.45595	−0.171637
$675$	0	0
$676$	−12.9999	−0.499998
$677$	34.4169	1.32275	0.661374	−	0.750056i	$-0.269974\pi$
$677$	34.4169	1.32275	0.661374	+	0.750056i	$0.269974\pi$
$678$	0	0
$679$	39.5000	1.51587
$680$	1.77368	0.0680176
$681$	0	0
$682$	28.3908	1.08714
$683$	−10.4429	−0.399588	−0.199794	−	0.979838i	$-0.564027\pi$
$683$	−10.4429	−0.399588	−0.199794	+	0.979838i	$0.564027\pi$
$684$	0	0
$685$	−33.1281	−1.26576
$686$	−20.1134	−0.767933
$687$	0	0
$688$	9.59164	0.365678
$689$	−0.0360642	−0.00137393
$690$	0	0
$691$	45.0269	1.71290	0.856452	−	0.516227i	$-0.172664\pi$
$691$	45.0269	1.71290	0.856452	+	0.516227i	$0.172664\pi$
$692$	9.28984	0.353147
$693$	0	0
$694$	30.7768	1.16827
$695$	−25.0470	−0.950088
$696$	0	0
$697$	2.98552	0.113085
$698$	−10.6268	−0.402229
$699$	0	0
$700$	−4.34084	−0.164068
$701$	19.7764	0.746944	0.373472	−	0.927641i	$-0.378167\pi$
$701$	19.7764	0.746944	0.373472	+	0.927641i	$0.378167\pi$
$702$	0	0
$703$	−2.32714	−0.0877698
$704$	4.09429	0.154309
$705$	0	0
$706$	−23.9565	−0.901615
$707$	−19.2442	−0.723754
$708$	0	0
$709$	11.3226	0.425230	0.212615	−	0.977136i	$-0.431802\pi$
$709$	11.3226	0.425230	0.212615	+	0.977136i	$0.431802\pi$
$710$	−25.4029	−0.953356
$711$	0	0
$712$	−2.07485	−0.0777582
$713$	23.5357	0.881418
$714$	0	0
$715$	0.0837218	0.00313102
$716$	−21.1054	−0.788745
$717$	0	0
$718$	−17.1370	−0.639548
$719$	42.4365	1.58261	0.791307	−	0.611419i	$-0.209401\pi$
$719$	42.4365	1.58261	0.791307	+	0.611419i	$0.209401\pi$
$720$	0	0
$721$	−44.7783	−1.66763
$722$	6.23440	0.232020
$723$	0	0
$724$	0.543334	0.0201928
$725$	−15.4734	−0.574667
$726$	0	0
$727$	16.3993	0.608218	0.304109	−	0.952637i	$-0.401641\pi$
$727$	16.3993	0.608218	0.304109	+	0.952637i	$0.401641\pi$
$728$	0.0174478	0.000646660 0
$729$	0	0
$730$	−2.62175	−0.0970353
$731$	−6.46149	−0.238987
$732$	0	0
$733$	−21.4011	−0.790468	−0.395234	−	0.918580i	$-0.629337\pi$
$733$	−21.4011	−0.790468	−0.395234	+	0.918580i	$0.629337\pi$
$734$	11.9256	0.440182
$735$	0	0
$736$	3.39412	0.125109
$737$	62.2374	2.29254
$738$	0	0
$739$	5.23739	0.192661	0.0963303	−	0.995349i	$-0.469289\pi$
$739$	5.23739	0.192661	0.0963303	+	0.995349i	$0.469289\pi$
$740$	1.71490	0.0630409
$741$	0	0
$742$	−10.4320	−0.382972
$743$	−37.6115	−1.37983	−0.689916	−	0.723889i	$-0.742353\pi$
$743$	−37.6115	−1.37983	−0.689916	+	0.723889i	$0.742353\pi$
$744$	0	0
$745$	7.18969	0.263410
$746$	3.91021	0.143163
$747$	0	0
$748$	−2.75815	−0.100848
$749$	29.5935	1.08132
$750$	0	0
$751$	15.3955	0.561790	0.280895	−	0.959738i	$-0.409369\pi$
$751$	15.3955	0.561790	0.280895	+	0.959738i	$0.409369\pi$
$752$	−10.6264	−0.387506
$753$	0	0
$754$	0.0621946	0.00226500
$755$	20.9589	0.762774
$756$	0	0
$757$	45.9995	1.67188	0.835941	−	0.548819i	$-0.184923\pi$
$757$	45.9995	1.67188	0.835941	+	0.548819i	$0.184923\pi$
$758$	−10.4984	−0.381320
$759$	0	0
$760$	9.40712	0.341232
$761$	−23.9738	−0.869050	−0.434525	−	0.900660i	$-0.643084\pi$
$761$	−23.9738	−0.869050	−0.434525	+	0.900660i	$0.643084\pi$
$762$	0	0
$763$	11.4726	0.415337
$764$	−14.0400	−0.507950
$765$	0	0
$766$	8.52206	0.307915
$767$	−0.0617708	−0.00223041
$768$	0	0
$769$	31.3460	1.13037	0.565183	−	0.824966i	$-0.308806\pi$
$769$	31.3460	1.13037	0.565183	+	0.824966i	$0.308806\pi$
$770$	24.2176	0.872743
$771$	0	0
$772$	−9.77786	−0.351913
$773$	−43.8102	−1.57574	−0.787871	−	0.615840i	$-0.788817\pi$
$773$	−43.8102	−1.57574	−0.787871	+	0.615840i	$0.788817\pi$
$774$	0	0
$775$	−13.3985	−0.481288
$776$	17.5825	0.631174
$777$	0	0
$778$	31.7085	1.13680
$779$	15.8344	0.567326
$780$	0	0
$781$	39.5026	1.41352
$782$	−2.28648	−0.0817643
$783$	0	0
$784$	−1.95299	−0.0697495
$785$	32.8857	1.17374
$786$	0	0
$787$	8.20309	0.292409	0.146204	−	0.989254i	$-0.453294\pi$
$787$	8.20309	0.292409	0.146204	+	0.989254i	$0.453294\pi$
$788$	14.5046	0.516705
$789$	0	0
$790$	3.36085	0.119574
$791$	34.9458	1.24253
$792$	0	0
$793$	−0.0858325	−0.00304800
$794$	5.92081	0.210122
$795$	0	0
$796$	22.6203	0.801755
$797$	22.7577	0.806118	0.403059	−	0.915174i	$-0.367947\pi$
$797$	22.7577	0.806118	0.403059	+	0.915174i	$0.367947\pi$
$798$	0	0
$799$	7.15859	0.253253
$800$	−1.93222	−0.0683143
$801$	0	0
$802$	8.12974	0.287071
$803$	4.07693	0.143872
$804$	0	0
$805$	20.0762	0.707592
$806$	0.0538547	0.00189695
$807$	0	0
$808$	−8.56611	−0.301355
$809$	−27.2179	−0.956930	−0.478465	−	0.878107i	$-0.658807\pi$
$809$	−27.2179	−0.956930	−0.478465	+	0.878107i	$0.658807\pi$
$810$	0	0
$811$	−9.81625	−0.344695	−0.172348	−	0.985036i	$-0.555135\pi$
$811$	−9.81625	−0.344695	−0.172348	+	0.985036i	$0.555135\pi$
$812$	17.9906	0.631347
$813$	0	0
$814$	−2.66674	−0.0934691
$815$	−16.5085	−0.578268
$816$	0	0
$817$	−34.2699	−1.19895
$818$	−12.9592	−0.453108
$819$	0	0
$820$	−11.6686	−0.407484
$821$	−17.8972	−0.624616	−0.312308	−	0.949981i	$-0.601102\pi$
$821$	−17.8972	−0.624616	−0.312308	+	0.949981i	$0.601102\pi$
$822$	0	0
$823$	16.0259	0.558627	0.279313	−	0.960200i	$-0.409893\pi$
$823$	16.0259	0.558627	0.279313	+	0.960200i	$0.409893\pi$
$824$	−19.9320	−0.694363
$825$	0	0
$826$	−17.8680	−0.621708
$827$	−18.7380	−0.651584	−0.325792	−	0.945441i	$-0.605631\pi$
$827$	−18.7380	−0.651584	−0.325792	+	0.945441i	$0.605631\pi$
$828$	0	0
$829$	−0.931833	−0.0323639	−0.0161820	−	0.999869i	$-0.505151\pi$
$829$	−0.931833	−0.0323639	−0.0161820	+	0.999869i	$0.505151\pi$
$830$	15.1495	0.525848
$831$	0	0
$832$	0.00776648	0.000269254 0
$833$	1.31564	0.0455844
$834$	0	0
$835$	−7.65267	−0.264832
$836$	−14.6285	−0.505936
$837$	0	0
$838$	30.4806	1.05293
$839$	10.4652	0.361300	0.180650	−	0.983547i	$-0.442180\pi$
$839$	10.4652	0.361300	0.180650	+	0.983547i	$0.442180\pi$
$840$	0	0
$841$	35.1294	1.21136
$842$	38.0654	1.31182
$843$	0	0
$844$	−10.0332	−0.345358
$845$	−34.2277	−1.17747
$846$	0	0
$847$	−12.9473	−0.444874
$848$	−4.64357	−0.159461
$849$	0	0
$850$	1.30166	0.0446464
$851$	−2.21070	−0.0757818
$852$	0	0
$853$	28.7039	0.982801	0.491401	−	0.870934i	$-0.336485\pi$
$853$	28.7039	0.982801	0.491401	+	0.870934i	$0.336485\pi$
$854$	−24.8282	−0.849603
$855$	0	0
$856$	13.1728	0.450238
$857$	−10.0329	−0.342717	−0.171359	−	0.985209i	$-0.554816\pi$
$857$	−10.0329	−0.342717	−0.171359	+	0.985209i	$0.554816\pi$
$858$	0	0
$859$	−32.0157	−1.09236	−0.546181	−	0.837667i	$-0.683919\pi$
$859$	−32.0157	−1.09236	−0.546181	+	0.837667i	$0.683919\pi$
$860$	25.2539	0.861152
$861$	0	0
$862$	21.5664	0.734553
$863$	4.53904	0.154511	0.0772553	−	0.997011i	$-0.475384\pi$
$863$	4.53904	0.154511	0.0772553	+	0.997011i	$0.475384\pi$
$864$	0	0
$865$	24.4593	0.831642
$866$	8.92010	0.303117
$867$	0	0
$868$	15.5782	0.528758
$869$	−5.22626	−0.177289
$870$	0	0
$871$	0.118059	0.00400026
$872$	5.10676	0.172937
$873$	0	0
$874$	−12.1268	−0.410197
$875$	18.1459	0.613442
$876$	0	0
$877$	40.9646	1.38328	0.691639	−	0.722243i	$-0.256889\pi$
$877$	40.9646	1.38328	0.691639	+	0.722243i	$0.256889\pi$
$878$	26.0102	0.877802
$879$	0	0
$880$	10.7799	0.363390
$881$	−0.505607	−0.0170343	−0.00851716	−	0.999964i	$-0.502711\pi$
$881$	−0.505607	−0.0170343	−0.00851716	+	0.999964i	$0.502711\pi$
$882$	0	0
$883$	27.7967	0.935434	0.467717	−	0.883878i	$-0.345077\pi$
$883$	27.7967	0.935434	0.467717	+	0.883878i	$0.345077\pi$
$884$	−0.00523195	−0.000175970 0
$885$	0	0
$886$	−10.7806	−0.362180
$887$	−6.73644	−0.226188	−0.113094	−	0.993584i	$-0.536076\pi$
$887$	−6.73644	−0.226188	−0.113094	+	0.993584i	$0.536076\pi$
$888$	0	0
$889$	−34.6209	−1.16115
$890$	−5.46289	−0.183116
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	37.9672	1.27052
$894$	0	0
$895$	−55.5686	−1.85745
$896$	2.24656	0.0750522
$897$	0	0
$898$	20.0819	0.670140
$899$	55.5301	1.85203
$900$	0	0
$901$	3.12818	0.104215
$902$	18.1451	0.604165
$903$	0	0
$904$	15.5553	0.517361
$905$	1.43055	0.0475531
$906$	0	0
$907$	−21.5625	−0.715972	−0.357986	−	0.933727i	$-0.616536\pi$
$907$	−21.5625	−0.715972	−0.357986	+	0.933727i	$0.616536\pi$
$908$	20.8914	0.693305
$909$	0	0
$910$	0.0459386	0.00152285
$911$	59.1815	1.96077	0.980386	−	0.197088i	$-0.0631485\pi$
$911$	59.1815	1.96077	0.980386	+	0.197088i	$0.0631485\pi$
$912$	0	0
$913$	−23.5581	−0.779661
$914$	35.5020	1.17430
$915$	0	0
$916$	20.5829	0.680077
$917$	−47.2204	−1.55935
$918$	0	0
$919$	−28.4787	−0.939426	−0.469713	−	0.882819i	$-0.655643\pi$
$919$	−28.4787	−0.939426	−0.469713	+	0.882819i	$0.655643\pi$
$920$	8.93642	0.294625
$921$	0	0
$922$	3.26003	0.107363
$923$	0.0749328	0.00246644
$924$	0	0
$925$	1.25852	0.0413798
$926$	−5.67209	−0.186397
$927$	0	0
$928$	8.00809	0.262878
$929$	32.6469	1.07111	0.535555	−	0.844500i	$-0.320102\pi$
$929$	32.6469	1.07111	0.535555	+	0.844500i	$0.320102\pi$
$930$	0	0
$931$	6.97782	0.228689
$932$	−9.41577	−0.308424
$933$	0	0
$934$	10.7885	0.353010
$935$	−7.26196	−0.237492
$936$	0	0
$937$	−26.4204	−0.863118	−0.431559	−	0.902085i	$-0.642036\pi$
$937$	−26.4204	−0.863118	−0.431559	+	0.902085i	$0.642036\pi$
$938$	34.1500	1.11504
$939$	0	0
$940$	−27.9785	−0.912557
$941$	27.0639	0.882259	0.441130	−	0.897443i	$-0.354578\pi$
$941$	27.0639	0.882259	0.441130	+	0.897443i	$0.354578\pi$
$942$	0	0
$943$	15.0421	0.489838
$944$	−7.95351	−0.258865
$945$	0	0
$946$	−39.2709	−1.27681
$947$	15.7697	0.512447	0.256223	−	0.966618i	$-0.417522\pi$
$947$	15.7697	0.512447	0.256223	+	0.966618i	$0.417522\pi$
$948$	0	0
$949$	0.00773355	0.000251042 0
$950$	6.90363	0.223983
$951$	0	0
$952$	−1.51341	−0.0490499
$953$	25.0914	0.812791	0.406395	−	0.913697i	$-0.366786\pi$
$953$	25.0914	0.812791	0.406395	+	0.913697i	$0.366786\pi$
$954$	0	0
$955$	−36.9661	−1.19620
$956$	14.6538	0.473939
$957$	0	0
$958$	−17.5354	−0.566542
$959$	28.2669	0.912785
$960$	0	0
$961$	17.0838	0.551090
$962$	−0.00505855	−0.000163094 0
$963$	0	0
$964$	17.1216	0.551448
$965$	−25.7442	−0.828737
$966$	0	0
$967$	40.5703	1.30465	0.652327	−	0.757938i	$-0.273793\pi$
$967$	40.5703	1.30465	0.652327	+	0.757938i	$0.273793\pi$
$968$	−5.76317	−0.185235
$969$	0	0
$970$	46.2931	1.48638
$971$	32.4433	1.04116	0.520578	−	0.853814i	$-0.325717\pi$
$971$	32.4433	1.04116	0.520578	+	0.853814i	$0.325717\pi$
$972$	0	0
$973$	21.3716	0.685143
$974$	−21.6167	−0.692645
$975$	0	0
$976$	−11.0517	−0.353755
$977$	−32.0199	−1.02441	−0.512204	−	0.858864i	$-0.671171\pi$
$977$	−32.0199	−1.02441	−0.512204	+	0.858864i	$0.671171\pi$
$978$	0	0
$979$	8.49501	0.271502
$980$	−5.14204	−0.164256
$981$	0	0
$982$	11.0429	0.352393
$983$	−11.3623	−0.362400	−0.181200	−	0.983446i	$-0.557998\pi$
$983$	−11.3623	−0.362400	−0.181200	+	0.983446i	$0.557998\pi$
$984$	0	0
$985$	38.1894	1.21681
$986$	−5.39471	−0.171803
$987$	0	0
$988$	−0.0277488	−0.000882808 0
$989$	−32.5552	−1.03519
$990$	0	0
$991$	57.7380	1.83411	0.917054	−	0.398764i	$-0.130561\pi$
$991$	57.7380	1.83411	0.917054	+	0.398764i	$0.130561\pi$
$992$	6.93425	0.220163
$993$	0	0
$994$	21.6753	0.687499
$995$	59.5572	1.88809
$996$	0	0
$997$	−48.5327	−1.53705	−0.768524	−	0.639821i	$-0.779008\pi$
$997$	−48.5327	−1.53705	−0.768524	+	0.639821i	$0.779008\pi$
$998$	9.24468	0.292635
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.t.1.5	✓	6
3.2	odd	2		4014.2.a.u.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.a.t.1.5	✓	6	1.1	even	1	trivial
4014.2.a.u.1.2	yes	6	3.2	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.63291 q^{5} -2.24656 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.63291 q^{5} -2.24656 q^{7} -1.00000 q^{8} -2.63291 q^{10} +4.09429 q^{11} +0.00776648 q^{13} +2.24656 q^{14} +1.00000 q^{16} -0.673658 q^{17} -3.57290 q^{19} +2.63291 q^{20} -4.09429 q^{22} -3.39412 q^{23} +1.93222 q^{25} -0.00776648 q^{26} -2.24656 q^{28} -8.00809 q^{29} -6.93425 q^{31} -1.00000 q^{32} +0.673658 q^{34} -5.91498 q^{35} +0.651332 q^{37} +3.57290 q^{38} -2.63291 q^{40} -4.43181 q^{41} +9.59164 q^{43} +4.09429 q^{44} +3.39412 q^{46} -10.6264 q^{47} -1.95299 q^{49} -1.93222 q^{50} +0.00776648 q^{52} -4.64357 q^{53} +10.7799 q^{55} +2.24656 q^{56} +8.00809 q^{58} -7.95351 q^{59} -11.0517 q^{61} +6.93425 q^{62} +1.00000 q^{64} +0.0204485 q^{65} +15.2010 q^{67} -0.673658 q^{68} +5.91498 q^{70} +9.64824 q^{71} +0.995761 q^{73} -0.651332 q^{74} -3.57290 q^{76} -9.19804 q^{77} -1.27648 q^{79} +2.63291 q^{80} +4.43181 q^{82} -5.75391 q^{83} -1.77368 q^{85} -9.59164 q^{86} -4.09429 q^{88} +2.07485 q^{89} -0.0174478 q^{91} -3.39412 q^{92} +10.6264 q^{94} -9.40712 q^{95} -17.5825 q^{97} +1.95299 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 6 * q^4 - 2 * q^5 + 8 * q^7 - 6 * q^8	\( 6 q - 6 q^{2} + 6 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} + 2 q^{10} - 2 q^{11} - 2 q^{13} - 8 q^{14} + 6 q^{16} - 2 q^{17} - 2 q^{19} - 2 q^{20} + 2 q^{22} - 22 q^{23} + 12 q^{25} + 2 q^{26} + 8 q^{28} - 8 q^{29} - 12 q^{31} - 6 q^{32} + 2 q^{34} - 20 q^{35} + 2 q^{38} + 2 q^{40} - 28 q^{41} + 14 q^{43} - 2 q^{44} + 22 q^{46} - 2 q^{47} - 12 q^{50} - 2 q^{52} - 26 q^{53} + 6 q^{55} - 8 q^{56} + 8 q^{58} - 4 q^{59} - 6 q^{61} + 12 q^{62} + 6 q^{64} - 16 q^{65} + 18 q^{67} - 2 q^{68} + 20 q^{70} - 12 q^{71} - 20 q^{73} - 2 q^{76} - 20 q^{77} + 12 q^{79} - 2 q^{80} + 28 q^{82} - 12 q^{83} - 10 q^{85} - 14 q^{86} + 2 q^{88} + 2 q^{89} - 22 q^{91} - 22 q^{92} + 2 q^{94} - 6 q^{95} - 6 q^{97}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^4 - 2 * q^5 + 8 * q^7 - 6 * q^8 + 2 * q^10 - 2 * q^11 - 2 * q^13 - 8 * q^14 + 6 * q^16 - 2 * q^17 - 2 * q^19 - 2 * q^20 + 2 * q^22 - 22 * q^23 + 12 * q^25 + 2 * q^26 + 8 * q^28 - 8 * q^29 - 12 * q^31 - 6 * q^32 + 2 * q^34 - 20 * q^35 + 2 * q^38 + 2 * q^40 - 28 * q^41 + 14 * q^43 - 2 * q^44 + 22 * q^46 - 2 * q^47 - 12 * q^50 - 2 * q^52 - 26 * q^53 + 6 * q^55 - 8 * q^56 + 8 * q^58 - 4 * q^59 - 6 * q^61 + 12 * q^62 + 6 * q^64 - 16 * q^65 + 18 * q^67 - 2 * q^68 + 20 * q^70 - 12 * q^71 - 20 * q^73 - 2 * q^76 - 20 * q^77 + 12 * q^79 - 2 * q^80 + 28 * q^82 - 12 * q^83 - 10 * q^85 - 14 * q^86 + 2 * q^88 + 2 * q^89 - 22 * q^91 - 22 * q^92 + 2 * q^94 - 6 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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