LMFDB - Embedded newform 133.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [133,2,Mod(1,133)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(133, 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("133.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$133 = 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	133.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:	$1.06201034688$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 3$ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	133.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278 q^{2} +0.302776 q^{3} +3.30278 q^{4} -3.00000 q^{5} -0.697224 q^{6} +1.00000 q^{7} -3.00000 q^{8} -2.90833 q^{9} +6.90833 q^{10} -0.697224 q^{11} +1.00000 q^{12} -5.60555 q^{13} -2.30278 q^{14} -0.908327 q^{15} +0.302776 q^{16} -5.30278 q^{17} +6.69722 q^{18} +1.00000 q^{19} -9.90833 q^{20} +0.302776 q^{21} +1.60555 q^{22} -3.00000 q^{23} -0.908327 q^{24} +4.00000 q^{25} +12.9083 q^{26} -1.78890 q^{27} +3.30278 q^{28} +9.90833 q^{29} +2.09167 q^{30} +1.30278 q^{31} +5.30278 q^{32} -0.211103 q^{33} +12.2111 q^{34} -3.00000 q^{35} -9.60555 q^{36} +3.60555 q^{37} -2.30278 q^{38} -1.69722 q^{39} +9.00000 q^{40} +0.697224 q^{41} -0.697224 q^{42} -10.0000 q^{43} -2.30278 q^{44} +8.72498 q^{45} +6.90833 q^{46} +6.21110 q^{47} +0.0916731 q^{48} +1.00000 q^{49} -9.21110 q^{50} -1.60555 q^{51} -18.5139 q^{52} -6.90833 q^{53} +4.11943 q^{54} +2.09167 q^{55} -3.00000 q^{56} +0.302776 q^{57} -22.8167 q^{58} -6.21110 q^{59} -3.00000 q^{60} -4.21110 q^{61} -3.00000 q^{62} -2.90833 q^{63} -12.8167 q^{64} +16.8167 q^{65} +0.486122 q^{66} -1.90833 q^{67} -17.5139 q^{68} -0.908327 q^{69} +6.90833 q^{70} +12.2111 q^{71} +8.72498 q^{72} +1.51388 q^{73} -8.30278 q^{74} +1.21110 q^{75} +3.30278 q^{76} -0.697224 q^{77} +3.90833 q^{78} +11.2111 q^{79} -0.908327 q^{80} +8.18335 q^{81} -1.60555 q^{82} -12.9083 q^{83} +1.00000 q^{84} +15.9083 q^{85} +23.0278 q^{86} +3.00000 q^{87} +2.09167 q^{88} -10.6056 q^{89} -20.0917 q^{90} -5.60555 q^{91} -9.90833 q^{92} +0.394449 q^{93} -14.3028 q^{94} -3.00000 q^{95} +1.60555 q^{96} +9.60555 q^{97} -2.30278 q^{98} +2.02776 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} - 5 q^{6} + 2 q^{7} - 6 q^{8} + 5 q^{9} + 3 q^{10} - 5 q^{11} + 2 q^{12} - 4 q^{13} - q^{14} + 9 q^{15} - 3 q^{16} - 7 q^{17} + 17 q^{18} + 2 q^{19} - 9 q^{20}+ \cdots - 32 q^{99}+O(q^{100})$ 2 * q - q^2 - 3 * q^3 + 3 * q^4 - 6 * q^5 - 5 * q^6 + 2 * q^7 - 6 * q^8 + 5 * q^9 + 3 * q^10 - 5 * q^11 + 2 * q^12 - 4 * q^13 - q^14 + 9 * q^15 - 3 * q^16 - 7 * q^17 + 17 * q^18 + 2 * q^19 - 9 * q^20 - 3 * q^21 - 4 * q^22 - 6 * q^23 + 9 * q^24 + 8 * q^25 + 15 * q^26 - 18 * q^27 + 3 * q^28 + 9 * q^29 + 15 * q^30 - q^31 + 7 * q^32 + 14 * q^33 + 10 * q^34 - 6 * q^35 - 12 * q^36 - q^38 - 7 * q^39 + 18 * q^40 + 5 * q^41 - 5 * q^42 - 20 * q^43 - q^44 - 15 * q^45 + 3 * q^46 - 2 * q^47 + 11 * q^48 + 2 * q^49 - 4 * q^50 + 4 * q^51 - 19 * q^52 - 3 * q^53 - 17 * q^54 + 15 * q^55 - 6 * q^56 - 3 * q^57 - 24 * q^58 + 2 * q^59 - 6 * q^60 + 6 * q^61 - 6 * q^62 + 5 * q^63 - 4 * q^64 + 12 * q^65 + 19 * q^66 + 7 * q^67 - 17 * q^68 + 9 * q^69 + 3 * q^70 + 10 * q^71 - 15 * q^72 - 15 * q^73 - 13 * q^74 - 12 * q^75 + 3 * q^76 - 5 * q^77 - 3 * q^78 + 8 * q^79 + 9 * q^80 + 38 * q^81 + 4 * q^82 - 15 * q^83 + 2 * q^84 + 21 * q^85 + 10 * q^86 + 6 * q^87 + 15 * q^88 - 14 * q^89 - 51 * q^90 - 4 * q^91 - 9 * q^92 + 8 * q^93 - 25 * q^94 - 6 * q^95 - 4 * q^96 + 12 * q^97 - q^98 - 32 * q^99

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	−2.30278	−1.62831	−0.814154	−	0.580649i	$-0.802799\pi$
$2$	−2.30278	−1.62831	−0.814154	+	0.580649i	$0.802799\pi$
$3$	0.302776	0.174808	0.0874038	−	0.996173i	$-0.472143\pi$
$3$	0.302776	0.174808	0.0874038	+	0.996173i	$0.472143\pi$
$4$	3.30278	1.65139
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−0.697224	−0.284641
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.00000	−1.06066
$9$	−2.90833	−0.969442
$10$	6.90833	2.18460
$11$	−0.697224	−0.210221	−0.105111	−	0.994461i	$-0.533520\pi$
$11$	−0.697224	−0.210221	−0.105111	+	0.994461i	$0.533520\pi$
$12$	1.00000	0.288675
$13$	−5.60555	−1.55470	−0.777350	−	0.629068i	$-0.783437\pi$
$13$	−5.60555	−1.55470	−0.777350	+	0.629068i	$0.783437\pi$
$14$	−2.30278	−0.615443
$15$	−0.908327	−0.234529
$16$	0.302776	0.0756939
$17$	−5.30278	−1.28611	−0.643056	−	0.765819i	$-0.722334\pi$
$17$	−5.30278	−1.28611	−0.643056	+	0.765819i	$0.722334\pi$
$18$	6.69722	1.57855
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−9.90833	−2.21557
$21$	0.302776	0.0660711
$22$	1.60555	0.342305
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	−0.908327	−0.185411
$25$	4.00000	0.800000
$26$	12.9083	2.53153
$27$	−1.78890	−0.344273
$28$	3.30278	0.624166
$29$	9.90833	1.83993	0.919965	−	0.392000i	$-0.128217\pi$
$29$	9.90833	1.83993	0.919965	+	0.392000i	$0.128217\pi$
$30$	2.09167	0.381886
$31$	1.30278	0.233985	0.116993	−	0.993133i	$-0.462675\pi$
$31$	1.30278	0.233985	0.116993	+	0.993133i	$0.462675\pi$
$32$	5.30278	0.937407
$33$	−0.211103	−0.0367482
$34$	12.2111	2.09419
$35$	−3.00000	−0.507093
$36$	−9.60555	−1.60093
$37$	3.60555	0.592749	0.296374	−	0.955072i	$-0.404222\pi$
$37$	3.60555	0.592749	0.296374	+	0.955072i	$0.404222\pi$
$38$	−2.30278	−0.373560
$39$	−1.69722	−0.271773
$40$	9.00000	1.42302
$41$	0.697224	0.108888	0.0544441	−	0.998517i	$-0.482661\pi$
$41$	0.697224	0.108888	0.0544441	+	0.998517i	$0.482661\pi$
$42$	−0.697224	−0.107584
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−2.30278	−0.347156
$45$	8.72498	1.30064
$46$	6.90833	1.01858
$47$	6.21110	0.905982	0.452991	−	0.891515i	$-0.350357\pi$
$47$	6.21110	0.905982	0.452991	+	0.891515i	$0.350357\pi$
$48$	0.0916731	0.0132319
$49$	1.00000	0.142857
$50$	−9.21110	−1.30265
$51$	−1.60555	−0.224822
$52$	−18.5139	−2.56741
$53$	−6.90833	−0.948932	−0.474466	−	0.880274i	$-0.657359\pi$
$53$	−6.90833	−0.948932	−0.474466	+	0.880274i	$0.657359\pi$
$54$	4.11943	0.560583
$55$	2.09167	0.282041
$56$	−3.00000	−0.400892
$57$	0.302776	0.0401036
$58$	−22.8167	−2.99597
$59$	−6.21110	−0.808617	−0.404308	−	0.914623i	$-0.632488\pi$
$59$	−6.21110	−0.808617	−0.404308	+	0.914623i	$0.632488\pi$
$60$	−3.00000	−0.387298
$61$	−4.21110	−0.539176	−0.269588	−	0.962976i	$-0.586888\pi$
$61$	−4.21110	−0.539176	−0.269588	+	0.962976i	$0.586888\pi$
$62$	−3.00000	−0.381000
$63$	−2.90833	−0.366415
$64$	−12.8167	−1.60208
$65$	16.8167	2.08585
$66$	0.486122	0.0598375
$67$	−1.90833	−0.233139	−0.116570	−	0.993183i	$-0.537190\pi$
$67$	−1.90833	−0.233139	−0.116570	+	0.993183i	$0.537190\pi$
$68$	−17.5139	−2.12387
$69$	−0.908327	−0.109350
$70$	6.90833	0.825703
$71$	12.2111	1.44919	0.724596	−	0.689174i	$-0.242027\pi$
$71$	12.2111	1.44919	0.724596	+	0.689174i	$0.242027\pi$
$72$	8.72498	1.02825
$73$	1.51388	0.177186	0.0885930	−	0.996068i	$-0.471763\pi$
$73$	1.51388	0.177186	0.0885930	+	0.996068i	$0.471763\pi$
$74$	−8.30278	−0.965178
$75$	1.21110	0.139846
$76$	3.30278	0.378854
$77$	−0.697224	−0.0794561
$78$	3.90833	0.442531
$79$	11.2111	1.26135	0.630674	−	0.776048i	$-0.282779\pi$
$79$	11.2111	1.26135	0.630674	+	0.776048i	$0.282779\pi$
$80$	−0.908327	−0.101554
$81$	8.18335	0.909261
$82$	−1.60555	−0.177303
$83$	−12.9083	−1.41687	−0.708436	−	0.705775i	$-0.750599\pi$
$83$	−12.9083	−1.41687	−0.708436	+	0.705775i	$0.750599\pi$
$84$	1.00000	0.109109
$85$	15.9083	1.72550
$86$	23.0278	2.48315
$87$	3.00000	0.321634
$88$	2.09167	0.222973
$89$	−10.6056	−1.12419	−0.562093	−	0.827074i	$-0.690004\pi$
$89$	−10.6056	−1.12419	−0.562093	+	0.827074i	$0.690004\pi$
$90$	−20.0917	−2.11785
$91$	−5.60555	−0.587621
$92$	−9.90833	−1.03301
$93$	0.394449	0.0409024
$94$	−14.3028	−1.47522
$95$	−3.00000	−0.307794
$96$	1.60555	0.163866
$97$	9.60555	0.975296	0.487648	−	0.873040i	$-0.337855\pi$
$97$	9.60555	0.975296	0.487648	+	0.873040i	$0.337855\pi$
$98$	−2.30278	−0.232615
$99$	2.02776	0.203797
$100$	13.2111	1.32111
$101$	−4.39445	−0.437264	−0.218632	−	0.975807i	$-0.570159\pi$
$101$	−4.39445	−0.437264	−0.218632	+	0.975807i	$0.570159\pi$
$102$	3.69722	0.366080
$103$	−16.2111	−1.59733	−0.798664	−	0.601778i	$-0.794459\pi$
$103$	−16.2111	−1.59733	−0.798664	+	0.601778i	$0.794459\pi$
$104$	16.8167	1.64901
$105$	−0.908327	−0.0886436
$106$	15.9083	1.54515
$107$	−5.78890	−0.559634	−0.279817	−	0.960053i	$-0.590274\pi$
$107$	−5.78890	−0.559634	−0.279817	+	0.960053i	$0.590274\pi$
$108$	−5.90833	−0.568529
$109$	−10.2111	−0.978046	−0.489023	−	0.872271i	$-0.662647\pi$
$109$	−10.2111	−0.978046	−0.489023	+	0.872271i	$0.662647\pi$
$110$	−4.81665	−0.459250
$111$	1.09167	0.103617
$112$	0.302776	0.0286096
$113$	9.69722	0.912238	0.456119	−	0.889919i	$-0.349239\pi$
$113$	9.69722	0.912238	0.456119	+	0.889919i	$0.349239\pi$
$114$	−0.697224	−0.0653010
$115$	9.00000	0.839254
$116$	32.7250	3.03844
$117$	16.3028	1.50719
$118$	14.3028	1.31668
$119$	−5.30278	−0.486105
$120$	2.72498	0.248756
$121$	−10.5139	−0.955807
$122$	9.69722	0.877945
$123$	0.211103	0.0190345
$124$	4.30278	0.386401
$125$	3.00000	0.268328
$126$	6.69722	0.596636
$127$	−11.8167	−1.04856	−0.524279	−	0.851546i	$-0.675665\pi$
$127$	−11.8167	−1.04856	−0.524279	+	0.851546i	$0.675665\pi$
$128$	18.9083	1.67128
$129$	−3.02776	−0.266579
$130$	−38.7250	−3.39641
$131$	−20.5139	−1.79231	−0.896153	−	0.443745i	$-0.853649\pi$
$131$	−20.5139	−1.79231	−0.896153	+	0.443745i	$0.853649\pi$
$132$	−0.697224	−0.0606856
$133$	1.00000	0.0867110
$134$	4.39445	0.379623
$135$	5.36669	0.461891
$136$	15.9083	1.36413
$137$	21.6333	1.84826	0.924129	−	0.382080i	$-0.124792\pi$
$137$	21.6333	1.84826	0.924129	+	0.382080i	$0.124792\pi$
$138$	2.09167	0.178055
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	−9.90833	−0.837406
$141$	1.88057	0.158373
$142$	−28.1194	−2.35973
$143$	3.90833	0.326831
$144$	−0.880571	−0.0733809
$145$	−29.7250	−2.46853
$146$	−3.48612	−0.288513
$147$	0.302776	0.0249725
$148$	11.9083	0.978858
$149$	−6.21110	−0.508833	−0.254417	−	0.967095i	$-0.581883\pi$
$149$	−6.21110	−0.508833	−0.254417	+	0.967095i	$0.581883\pi$
$150$	−2.78890	−0.227713
$151$	5.90833	0.480813	0.240406	−	0.970672i	$-0.422719\pi$
$151$	5.90833	0.480813	0.240406	+	0.970672i	$0.422719\pi$
$152$	−3.00000	−0.243332
$153$	15.4222	1.24681
$154$	1.60555	0.129379
$155$	−3.90833	−0.313924
$156$	−5.60555	−0.448803
$157$	4.51388	0.360247	0.180123	−	0.983644i	$-0.442350\pi$
$157$	4.51388	0.360247	0.180123	+	0.983644i	$0.442350\pi$
$158$	−25.8167	−2.05386
$159$	−2.09167	−0.165880
$160$	−15.9083	−1.25766
$161$	−3.00000	−0.236433
$162$	−18.8444	−1.48056
$163$	5.69722	0.446241	0.223121	−	0.974791i	$-0.428376\pi$
$163$	5.69722	0.446241	0.223121	+	0.974791i	$0.428376\pi$
$164$	2.30278	0.179817
$165$	0.633308	0.0493029
$166$	29.7250	2.30711
$167$	4.39445	0.340053	0.170026	−	0.985440i	$-0.445615\pi$
$167$	4.39445	0.340053	0.170026	+	0.985440i	$0.445615\pi$
$168$	−0.908327	−0.0700789
$169$	18.4222	1.41709
$170$	−36.6333	−2.80965
$171$	−2.90833	−0.222405
$172$	−33.0278	−2.51834
$173$	7.81665	0.594289	0.297145	−	0.954832i	$-0.403966\pi$
$173$	7.81665	0.594289	0.297145	+	0.954832i	$0.403966\pi$
$174$	−6.90833	−0.523719
$175$	4.00000	0.302372
$176$	−0.211103	−0.0159125
$177$	−1.88057	−0.141352
$178$	24.4222	1.83052
$179$	−11.7250	−0.876366	−0.438183	−	0.898886i	$-0.644378\pi$
$179$	−11.7250	−0.876366	−0.438183	+	0.898886i	$0.644378\pi$
$180$	28.8167	2.14787
$181$	−10.6972	−0.795118	−0.397559	−	0.917577i	$-0.630143\pi$
$181$	−10.6972	−0.795118	−0.397559	+	0.917577i	$0.630143\pi$
$182$	12.9083	0.956829
$183$	−1.27502	−0.0942521
$184$	9.00000	0.663489
$185$	−10.8167	−0.795256
$186$	−0.908327	−0.0666018
$187$	3.69722	0.270368
$188$	20.5139	1.49613
$189$	−1.78890	−0.130123
$190$	6.90833	0.501183
$191$	25.3305	1.83285	0.916426	−	0.400203i	$-0.131060\pi$
$191$	25.3305	1.83285	0.916426	+	0.400203i	$0.131060\pi$
$192$	−3.88057	−0.280056
$193$	−23.1194	−1.66417	−0.832086	−	0.554646i	$-0.812854\pi$
$193$	−23.1194	−1.66417	−0.832086	+	0.554646i	$0.812854\pi$
$194$	−22.1194	−1.58808
$195$	5.09167	0.364622
$196$	3.30278	0.235913
$197$	−6.90833	−0.492198	−0.246099	−	0.969245i	$-0.579149\pi$
$197$	−6.90833	−0.492198	−0.246099	+	0.969245i	$0.579149\pi$
$198$	−4.66947	−0.331845
$199$	−13.4222	−0.951475	−0.475737	−	0.879587i	$-0.657819\pi$
$199$	−13.4222	−0.951475	−0.475737	+	0.879587i	$0.657819\pi$
$200$	−12.0000	−0.848528
$201$	−0.577795	−0.0407545
$202$	10.1194	0.712001
$203$	9.90833	0.695428
$204$	−5.30278	−0.371269
$205$	−2.09167	−0.146089
$206$	37.3305	2.60094
$207$	8.72498	0.606428
$208$	−1.69722	−0.117681
$209$	−0.697224	−0.0482280
$210$	2.09167	0.144339
$211$	−7.69722	−0.529899	−0.264949	−	0.964262i	$-0.585355\pi$
$211$	−7.69722	−0.529899	−0.264949	+	0.964262i	$0.585355\pi$
$212$	−22.8167	−1.56705
$213$	3.69722	0.253330
$214$	13.3305	0.911256
$215$	30.0000	2.04598
$216$	5.36669	0.365157
$217$	1.30278	0.0884382
$218$	23.5139	1.59256
$219$	0.458365	0.0309735
$220$	6.90833	0.465759
$221$	29.7250	1.99952
$222$	−2.51388	−0.168720
$223$	12.8167	0.858267	0.429133	−	0.903241i	$-0.358819\pi$
$223$	12.8167	0.858267	0.429133	+	0.903241i	$0.358819\pi$
$224$	5.30278	0.354307
$225$	−11.6333	−0.775554
$226$	−22.3305	−1.48540
$227$	−25.1194	−1.66724	−0.833618	−	0.552342i	$-0.813734\pi$
$227$	−25.1194	−1.66724	−0.833618	+	0.552342i	$0.813734\pi$
$228$	1.00000	0.0662266
$229$	−0.788897	−0.0521318	−0.0260659	−	0.999660i	$-0.508298\pi$
$229$	−0.788897	−0.0521318	−0.0260659	+	0.999660i	$0.508298\pi$
$230$	−20.7250	−1.36656
$231$	−0.211103	−0.0138895
$232$	−29.7250	−1.95154
$233$	−2.09167	−0.137030	−0.0685150	−	0.997650i	$-0.521826\pi$
$233$	−2.09167	−0.137030	−0.0685150	+	0.997650i	$0.521826\pi$
$234$	−37.5416	−2.45417
$235$	−18.6333	−1.21550
$236$	−20.5139	−1.33534
$237$	3.39445	0.220493
$238$	12.2111	0.791528
$239$	−27.4222	−1.77379	−0.886897	−	0.461966i	$-0.847144\pi$
$239$	−27.4222	−1.77379	−0.886897	+	0.461966i	$0.847144\pi$
$240$	−0.275019	−0.0177524
$241$	−20.6056	−1.32732	−0.663660	−	0.748034i	$-0.730998\pi$
$241$	−20.6056	−1.32732	−0.663660	+	0.748034i	$0.730998\pi$
$242$	24.2111	1.55635
$243$	7.84441	0.503219
$244$	−13.9083	−0.890389
$245$	−3.00000	−0.191663
$246$	−0.486122	−0.0309940
$247$	−5.60555	−0.356673
$248$	−3.90833	−0.248179
$249$	−3.90833	−0.247680
$250$	−6.90833	−0.436921
$251$	−15.9083	−1.00412	−0.502062	−	0.864831i	$-0.667425\pi$
$251$	−15.9083	−1.00412	−0.502062	+	0.864831i	$0.667425\pi$
$252$	−9.60555	−0.605093
$253$	2.09167	0.131502
$254$	27.2111	1.70738
$255$	4.81665	0.301631
$256$	−17.9083	−1.11927
$257$	−0.908327	−0.0566599	−0.0283299	−	0.999599i	$-0.509019\pi$
$257$	−0.908327	−0.0566599	−0.0283299	+	0.999599i	$0.509019\pi$
$258$	6.97224	0.434073
$259$	3.60555	0.224038
$260$	55.5416	3.44455
$261$	−28.8167	−1.78371
$262$	47.2389	2.91843
$263$	−9.69722	−0.597956	−0.298978	−	0.954260i	$-0.596646\pi$
$263$	−9.69722	−0.597956	−0.298978	+	0.954260i	$0.596646\pi$
$264$	0.633308	0.0389774
$265$	20.7250	1.27313
$266$	−2.30278	−0.141192
$267$	−3.21110	−0.196516
$268$	−6.30278	−0.385003
$269$	20.7250	1.26362	0.631812	−	0.775122i	$-0.282311\pi$
$269$	20.7250	1.26362	0.631812	+	0.775122i	$0.282311\pi$
$270$	−12.3583	−0.752101
$271$	17.9083	1.08785	0.543927	−	0.839133i	$-0.316937\pi$
$271$	17.9083	1.08785	0.543927	+	0.839133i	$0.316937\pi$
$272$	−1.60555	−0.0973508
$273$	−1.69722	−0.102721
$274$	−49.8167	−3.00953
$275$	−2.78890	−0.168177
$276$	−3.00000	−0.180579
$277$	0.605551	0.0363840	0.0181920	−	0.999835i	$-0.494209\pi$
$277$	0.605551	0.0363840	0.0181920	+	0.999835i	$0.494209\pi$
$278$	−11.5139	−0.690557
$279$	−3.78890	−0.226835
$280$	9.00000	0.537853
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	−4.33053	−0.257879
$283$	27.3305	1.62463	0.812316	−	0.583218i	$-0.198207\pi$
$283$	27.3305	1.62463	0.812316	+	0.583218i	$0.198207\pi$
$284$	40.3305	2.39318
$285$	−0.908327	−0.0538046
$286$	−9.00000	−0.532181
$287$	0.697224	0.0411559
$288$	−15.4222	−0.908762
$289$	11.1194	0.654084
$290$	68.4500	4.01952
$291$	2.90833	0.170489
$292$	5.00000	0.292603
$293$	−12.6333	−0.738046	−0.369023	−	0.929420i	$-0.620308\pi$
$293$	−12.6333	−0.738046	−0.369023	+	0.929420i	$0.620308\pi$
$294$	−0.697224	−0.0406630
$295$	18.6333	1.08487
$296$	−10.8167	−0.628705
$297$	1.24726	0.0723735
$298$	14.3028	0.828538
$299$	16.8167	0.972532
$300$	4.00000	0.230940
$301$	−10.0000	−0.576390
$302$	−13.6056	−0.782911
$303$	−1.33053	−0.0764371
$304$	0.302776	0.0173654
$305$	12.6333	0.723381
$306$	−35.5139	−2.03019
$307$	14.6972	0.838815	0.419407	−	0.907798i	$-0.362238\pi$
$307$	14.6972	0.838815	0.419407	+	0.907798i	$0.362238\pi$
$308$	−2.30278	−0.131213
$309$	−4.90833	−0.279225
$310$	9.00000	0.511166
$311$	−2.51388	−0.142549	−0.0712745	−	0.997457i	$-0.522707\pi$
$311$	−2.51388	−0.142549	−0.0712745	+	0.997457i	$0.522707\pi$
$312$	5.09167	0.288259
$313$	−27.0278	−1.52770	−0.763850	−	0.645394i	$-0.776693\pi$
$313$	−27.0278	−1.52770	−0.763850	+	0.645394i	$0.776693\pi$
$314$	−10.3944	−0.586593
$315$	8.72498	0.491597
$316$	37.0278	2.08297
$317$	8.78890	0.493634	0.246817	−	0.969062i	$-0.420615\pi$
$317$	8.78890	0.493634	0.246817	+	0.969062i	$0.420615\pi$
$318$	4.81665	0.270105
$319$	−6.90833	−0.386792
$320$	38.4500	2.14942
$321$	−1.75274	−0.0978282
$322$	6.90833	0.384986
$323$	−5.30278	−0.295054
$324$	27.0278	1.50154
$325$	−22.4222	−1.24376
$326$	−13.1194	−0.726618
$327$	−3.09167	−0.170970
$328$	−2.09167	−0.115493
$329$	6.21110	0.342429
$330$	−1.45837	−0.0802804
$331$	11.6972	0.642938	0.321469	−	0.946920i	$-0.395823\pi$
$331$	11.6972	0.642938	0.321469	+	0.946920i	$0.395823\pi$
$332$	−42.6333	−2.33981
$333$	−10.4861	−0.574636
$334$	−10.1194	−0.553711
$335$	5.72498	0.312789
$336$	0.0916731	0.00500118
$337$	7.72498	0.420807	0.210403	−	0.977615i	$-0.432522\pi$
$337$	7.72498	0.420807	0.210403	+	0.977615i	$0.432522\pi$
$338$	−42.4222	−2.30746
$339$	2.93608	0.159466
$340$	52.5416	2.84947
$341$	−0.908327	−0.0491887
$342$	6.69722	0.362144
$343$	1.00000	0.0539949
$344$	30.0000	1.61749
$345$	2.72498	0.146708
$346$	−18.0000	−0.967686
$347$	−28.5416	−1.53220	−0.766098	−	0.642724i	$-0.777804\pi$
$347$	−28.5416	−1.53220	−0.766098	+	0.642724i	$0.777804\pi$
$348$	9.90833	0.531142
$349$	7.51388	0.402209	0.201104	−	0.979570i	$-0.435547\pi$
$349$	7.51388	0.402209	0.201104	+	0.979570i	$0.435547\pi$
$350$	−9.21110	−0.492354
$351$	10.0278	0.535242
$352$	−3.69722	−0.197063
$353$	9.90833	0.527367	0.263684	−	0.964609i	$-0.415063\pi$
$353$	9.90833	0.527367	0.263684	+	0.964609i	$0.415063\pi$
$354$	4.33053	0.230165
$355$	−36.6333	−1.94429
$356$	−35.0278	−1.85647
$357$	−1.60555	−0.0849748
$358$	27.0000	1.42699
$359$	−25.5416	−1.34804	−0.674018	−	0.738715i	$-0.735433\pi$
$359$	−25.5416	−1.34804	−0.674018	+	0.738715i	$0.735433\pi$
$360$	−26.1749	−1.37954
$361$	1.00000	0.0526316
$362$	24.6333	1.29470
$363$	−3.18335	−0.167082
$364$	−18.5139	−0.970391
$365$	−4.54163	−0.237720
$366$	2.93608	0.153472
$367$	−0.0277564	−0.00144887	−0.000724436	−	1.00000i	$-0.500231\pi$
$367$	−0.0277564	−0.00144887	−0.000724436		1.00000i	$0.500231\pi$
$368$	−0.908327	−0.0473498
$369$	−2.02776	−0.105561
$370$	24.9083	1.29492
$371$	−6.90833	−0.358662
$372$	1.30278	0.0675458
$373$	24.1194	1.24886	0.624428	−	0.781082i	$-0.285332\pi$
$373$	24.1194	1.24886	0.624428	+	0.781082i	$0.285332\pi$
$374$	−8.51388	−0.440242
$375$	0.908327	0.0469058
$376$	−18.6333	−0.960939
$377$	−55.5416	−2.86054
$378$	4.11943	0.211881
$379$	6.81665	0.350148	0.175074	−	0.984555i	$-0.443984\pi$
$379$	6.81665	0.350148	0.175074	+	0.984555i	$0.443984\pi$
$380$	−9.90833	−0.508286
$381$	−3.57779	−0.183296
$382$	−58.3305	−2.98445
$383$	6.63331	0.338946	0.169473	−	0.985535i	$-0.445793\pi$
$383$	6.63331	0.338946	0.169473	+	0.985535i	$0.445793\pi$
$384$	5.72498	0.292152
$385$	2.09167	0.106602
$386$	53.2389	2.70979
$387$	29.0833	1.47839
$388$	31.7250	1.61059
$389$	−24.1472	−1.22431	−0.612155	−	0.790737i	$-0.709697\pi$
$389$	−24.1472	−1.22431	−0.612155	+	0.790737i	$0.709697\pi$
$390$	−11.7250	−0.593717
$391$	15.9083	0.804519
$392$	−3.00000	−0.151523
$393$	−6.21110	−0.313309
$394$	15.9083	0.801450
$395$	−33.6333	−1.69228
$396$	6.69722	0.336548
$397$	37.0278	1.85837	0.929185	−	0.369615i	$-0.120510\pi$
$397$	37.0278	1.85837	0.929185	+	0.369615i	$0.120510\pi$
$398$	30.9083	1.54929
$399$	0.302776	0.0151577
$400$	1.21110	0.0605551
$401$	−3.48612	−0.174089	−0.0870443	−	0.996204i	$-0.527742\pi$
$401$	−3.48612	−0.174089	−0.0870443	+	0.996204i	$0.527742\pi$
$402$	1.33053	0.0663609
$403$	−7.30278	−0.363777
$404$	−14.5139	−0.722092
$405$	−24.5500	−1.21990
$406$	−22.8167	−1.13237
$407$	−2.51388	−0.124608
$408$	4.81665	0.238460
$409$	20.9083	1.03385	0.516925	−	0.856031i	$-0.327077\pi$
$409$	20.9083	1.03385	0.516925	+	0.856031i	$0.327077\pi$
$410$	4.81665	0.237878
$411$	6.55004	0.323090
$412$	−53.5416	−2.63781
$413$	−6.21110	−0.305628
$414$	−20.0917	−0.987452
$415$	38.7250	1.90093
$416$	−29.7250	−1.45739
$417$	1.51388	0.0741349
$418$	1.60555	0.0785301
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	−3.00000	−0.146385
$421$	−29.6056	−1.44289	−0.721443	−	0.692474i	$-0.756521\pi$
$421$	−29.6056	−1.44289	−0.721443	+	0.692474i	$0.756521\pi$
$422$	17.7250	0.862839
$423$	−18.0639	−0.878298
$424$	20.7250	1.00649
$425$	−21.2111	−1.02889
$426$	−8.51388	−0.412499
$427$	−4.21110	−0.203790
$428$	−19.1194	−0.924173
$429$	1.18335	0.0571325
$430$	−69.0833	−3.33149
$431$	−3.21110	−0.154673	−0.0773367	−	0.997005i	$-0.524642\pi$
$431$	−3.21110	−0.154673	−0.0773367	+	0.997005i	$0.524642\pi$
$432$	−0.541635	−0.0260594
$433$	−0.577795	−0.0277671	−0.0138835	−	0.999904i	$-0.504419\pi$
$433$	−0.577795	−0.0277671	−0.0138835	+	0.999904i	$0.504419\pi$
$434$	−3.00000	−0.144005
$435$	−9.00000	−0.431517
$436$	−33.7250	−1.61513
$437$	−3.00000	−0.143509
$438$	−1.05551	−0.0504344
$439$	22.7889	1.08765	0.543827	−	0.839197i	$-0.316975\pi$
$439$	22.7889	1.08765	0.543827	+	0.839197i	$0.316975\pi$
$440$	−6.27502	−0.299150
$441$	−2.90833	−0.138492
$442$	−68.4500	−3.25583
$443$	−16.1194	−0.765857	−0.382929	−	0.923778i	$-0.625084\pi$
$443$	−16.1194	−0.765857	−0.382929	+	0.923778i	$0.625084\pi$
$444$	3.60555	0.171112
$445$	31.8167	1.50825
$446$	−29.5139	−1.39752
$447$	−1.88057	−0.0889479
$448$	−12.8167	−0.605530
$449$	−0.486122	−0.0229415	−0.0114708	−	0.999934i	$-0.503651\pi$
$449$	−0.486122	−0.0229415	−0.0114708	+	0.999934i	$0.503651\pi$
$450$	26.7889	1.26284
$451$	−0.486122	−0.0228906
$452$	32.0278	1.50646
$453$	1.78890	0.0840497
$454$	57.8444	2.71477
$455$	16.8167	0.788377
$456$	−0.908327	−0.0425363
$457$	22.3028	1.04328	0.521640	−	0.853166i	$-0.325320\pi$
$457$	22.3028	1.04328	0.521640	+	0.853166i	$0.325320\pi$
$458$	1.81665	0.0848867
$459$	9.48612	0.442774
$460$	29.7250	1.38593
$461$	0.908327	0.0423050	0.0211525	−	0.999776i	$-0.493266\pi$
$461$	0.908327	0.0423050	0.0211525	+	0.999776i	$0.493266\pi$
$462$	0.486122	0.0226164
$463$	−36.4500	−1.69397	−0.846987	−	0.531614i	$-0.821586\pi$
$463$	−36.4500	−1.69397	−0.846987	+	0.531614i	$0.821586\pi$
$464$	3.00000	0.139272
$465$	−1.18335	−0.0548764
$466$	4.81665	0.223127
$467$	16.5416	0.765456	0.382728	−	0.923861i	$-0.374985\pi$
$467$	16.5416	0.765456	0.382728	+	0.923861i	$0.374985\pi$
$468$	53.8444	2.48896
$469$	−1.90833	−0.0881183
$470$	42.9083	1.97921
$471$	1.36669	0.0629739
$472$	18.6333	0.857668
$473$	6.97224	0.320584
$474$	−7.81665	−0.359031
$475$	4.00000	0.183533
$476$	−17.5139	−0.802747
$477$	20.0917	0.919935
$478$	63.1472	2.88829
$479$	−0.697224	−0.0318570	−0.0159285	−	0.999873i	$-0.505070\pi$
$479$	−0.697224	−0.0318570	−0.0159285	+	0.999873i	$0.505070\pi$
$480$	−4.81665	−0.219849
$481$	−20.2111	−0.921547
$482$	47.4500	2.16129
$483$	−0.908327	−0.0413303
$484$	−34.7250	−1.57841
$485$	−28.8167	−1.30850
$486$	−18.0639	−0.819396
$487$	4.57779	0.207440	0.103720	−	0.994607i	$-0.466925\pi$
$487$	4.57779	0.207440	0.103720	+	0.994607i	$0.466925\pi$
$488$	12.6333	0.571883
$489$	1.72498	0.0780063
$490$	6.90833	0.312086
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0.697224	0.0314333
$493$	−52.5416	−2.36636
$494$	12.9083	0.580773
$495$	−6.08327	−0.273423
$496$	0.394449	0.0177113
$497$	12.2111	0.547743
$498$	9.00000	0.403300
$499$	19.9361	0.892462	0.446231	−	0.894918i	$-0.352766\pi$
$499$	19.9361	0.892462	0.446231	+	0.894918i	$0.352766\pi$
$500$	9.90833	0.443114
$501$	1.33053	0.0594438
$502$	36.6333	1.63502
$503$	6.42221	0.286352	0.143176	−	0.989697i	$-0.454269\pi$
$503$	6.42221	0.286352	0.143176	+	0.989697i	$0.454269\pi$
$504$	8.72498	0.388642
$505$	13.1833	0.586651
$506$	−4.81665	−0.214126
$507$	5.57779	0.247719
$508$	−39.0278	−1.73158
$509$	33.6333	1.49077	0.745385	−	0.666634i	$-0.232266\pi$
$509$	33.6333	1.49077	0.745385	+	0.666634i	$0.232266\pi$
$510$	−11.0917	−0.491148
$511$	1.51388	0.0669700
$512$	3.42221	0.151242
$513$	−1.78890	−0.0789818
$514$	2.09167	0.0922597
$515$	48.6333	2.14304
$516$	−10.0000	−0.440225
$517$	−4.33053	−0.190457
$518$	−8.30278	−0.364803
$519$	2.36669	0.103886
$520$	−50.4500	−2.21238
$521$	36.6333	1.60493	0.802467	−	0.596696i	$-0.203520\pi$
$521$	36.6333	1.60493	0.802467	+	0.596696i	$0.203520\pi$
$522$	66.3583	2.90442
$523$	37.6611	1.64680	0.823402	−	0.567459i	$-0.192073\pi$
$523$	37.6611	1.64680	0.823402	+	0.567459i	$0.192073\pi$
$524$	−67.7527	−2.95979
$525$	1.21110	0.0528568
$526$	22.3305	0.973657
$527$	−6.90833	−0.300931
$528$	−0.0639167	−0.00278162
$529$	−14.0000	−0.608696
$530$	−47.7250	−2.07304
$531$	18.0639	0.783907
$532$	3.30278	0.143193
$533$	−3.90833	−0.169288
$534$	7.39445	0.319989
$535$	17.3667	0.750828
$536$	5.72498	0.247282
$537$	−3.55004	−0.153195
$538$	−47.7250	−2.05757
$539$	−0.697224	−0.0300316
$540$	17.7250	0.762762
$541$	16.7889	0.721811	0.360906	−	0.932602i	$-0.382468\pi$
$541$	16.7889	0.721811	0.360906	+	0.932602i	$0.382468\pi$
$542$	−41.2389	−1.77136
$543$	−3.23886	−0.138993
$544$	−28.1194	−1.20561
$545$	30.6333	1.31219
$546$	3.90833	0.167261
$547$	−10.4861	−0.448354	−0.224177	−	0.974548i	$-0.571969\pi$
$547$	−10.4861	−0.448354	−0.224177	+	0.974548i	$0.571969\pi$
$548$	71.4500	3.05219
$549$	12.2473	0.522700
$550$	6.42221	0.273844
$551$	9.90833	0.422109
$552$	2.72498	0.115983
$553$	11.2111	0.476745
$554$	−1.39445	−0.0592444
$555$	−3.27502	−0.139017
$556$	16.5139	0.700344
$557$	−1.11943	−0.0474317	−0.0237159	−	0.999719i	$-0.507550\pi$
$557$	−1.11943	−0.0474317	−0.0237159	+	0.999719i	$0.507550\pi$
$558$	8.72498	0.369358
$559$	56.0555	2.37090
$560$	−0.908327	−0.0383838
$561$	1.11943	0.0472623
$562$	−6.90833	−0.291410
$563$	−24.2111	−1.02038	−0.510188	−	0.860063i	$-0.670424\pi$
$563$	−24.2111	−1.02038	−0.510188	+	0.860063i	$0.670424\pi$
$564$	6.21110	0.261535
$565$	−29.0917	−1.22390
$566$	−62.9361	−2.64540
$567$	8.18335	0.343668
$568$	−36.6333	−1.53710
$569$	7.18335	0.301142	0.150571	−	0.988599i	$-0.451889\pi$
$569$	7.18335	0.301142	0.150571	+	0.988599i	$0.451889\pi$
$570$	2.09167	0.0876105
$571$	−2.39445	−0.100205	−0.0501023	−	0.998744i	$-0.515955\pi$
$571$	−2.39445	−0.100205	−0.0501023	+	0.998744i	$0.515955\pi$
$572$	12.9083	0.539724
$573$	7.66947	0.320397
$574$	−1.60555	−0.0670144
$575$	−12.0000	−0.500435
$576$	37.2750	1.55313
$577$	19.5139	0.812373	0.406187	−	0.913790i	$-0.366858\pi$
$577$	19.5139	0.812373	0.406187	+	0.913790i	$0.366858\pi$
$578$	−25.6056	−1.06505
$579$	−7.00000	−0.290910
$580$	−98.1749	−4.07649
$581$	−12.9083	−0.535528
$582$	−6.69722	−0.277609
$583$	4.81665	0.199485
$584$	−4.54163	−0.187934
$585$	−48.9083	−2.02211
$586$	29.0917	1.20177
$587$	0.422205	0.0174263	0.00871313	−	0.999962i	$-0.497226\pi$
$587$	0.422205	0.0174263	0.00871313	+	0.999962i	$0.497226\pi$
$588$	1.00000	0.0412393
$589$	1.30278	0.0536799
$590$	−42.9083	−1.76651
$591$	−2.09167	−0.0860399
$592$	1.09167	0.0448675
$593$	−4.81665	−0.197796	−0.0988981	−	0.995098i	$-0.531532\pi$
$593$	−4.81665	−0.197796	−0.0988981	+	0.995098i	$0.531532\pi$
$594$	−2.87217	−0.117846
$595$	15.9083	0.652178
$596$	−20.5139	−0.840281
$597$	−4.06392	−0.166325
$598$	−38.7250	−1.58358
$599$	−0.908327	−0.0371132	−0.0185566	−	0.999828i	$-0.505907\pi$
$599$	−0.908327	−0.0371132	−0.0185566	+	0.999828i	$0.505907\pi$
$600$	−3.63331	−0.148329
$601$	−27.5139	−1.12231	−0.561157	−	0.827709i	$-0.689644\pi$
$601$	−27.5139	−1.12231	−0.561157	+	0.827709i	$0.689644\pi$
$602$	23.0278	0.938541
$603$	5.55004	0.226015
$604$	19.5139	0.794008
$605$	31.5416	1.28235
$606$	3.06392	0.124463
$607$	−13.0000	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$608$	5.30278	0.215056
$609$	3.00000	0.121566
$610$	−29.0917	−1.17789
$611$	−34.8167	−1.40853
$612$	50.9361	2.05897
$613$	−0.724981	−0.0292817	−0.0146408	−	0.999893i	$-0.504660\pi$
$613$	−0.724981	−0.0292817	−0.0146408	+	0.999893i	$0.504660\pi$
$614$	−33.8444	−1.36585
$615$	−0.633308	−0.0255374
$616$	2.09167	0.0842759
$617$	−10.8806	−0.438035	−0.219018	−	0.975721i	$-0.570285\pi$
$617$	−10.8806	−0.438035	−0.219018	+	0.975721i	$0.570285\pi$
$618$	11.3028	0.454664
$619$	−41.3305	−1.66121	−0.830607	−	0.556859i	$-0.812006\pi$
$619$	−41.3305	−1.66121	−0.830607	+	0.556859i	$0.812006\pi$
$620$	−12.9083	−0.518411
$621$	5.36669	0.215358
$622$	5.78890	0.232114
$623$	−10.6056	−0.424902
$624$	−0.513878	−0.0205716
$625$	−29.0000	−1.16000
$626$	62.2389	2.48757
$627$	−0.211103	−0.00843062
$628$	14.9083	0.594907
$629$	−19.1194	−0.762342
$630$	−20.0917	−0.800471
$631$	13.2389	0.527031	0.263515	−	0.964655i	$-0.415118\pi$
$631$	13.2389	0.527031	0.263515	+	0.964655i	$0.415118\pi$
$632$	−33.6333	−1.33786
$633$	−2.33053	−0.0926303
$634$	−20.2389	−0.803788
$635$	35.4500	1.40679
$636$	−6.90833	−0.273933
$637$	−5.60555	−0.222100
$638$	15.9083	0.629817
$639$	−35.5139	−1.40491
$640$	−56.7250	−2.24225
$641$	9.90833	0.391355	0.195678	−	0.980668i	$-0.437309\pi$
$641$	9.90833	0.391355	0.195678	+	0.980668i	$0.437309\pi$
$642$	4.03616	0.159295
$643$	30.3944	1.19864	0.599320	−	0.800510i	$-0.295438\pi$
$643$	30.3944	1.19864	0.599320	+	0.800510i	$0.295438\pi$
$644$	−9.90833	−0.390443
$645$	9.08327	0.357653
$646$	12.2111	0.480439
$647$	15.4222	0.606309	0.303155	−	0.952941i	$-0.401960\pi$
$647$	15.4222	0.606309	0.303155	+	0.952941i	$0.401960\pi$
$648$	−24.5500	−0.964417
$649$	4.33053	0.169988
$650$	51.6333	2.02522
$651$	0.394449	0.0154597
$652$	18.8167	0.736917
$653$	−16.8167	−0.658087	−0.329043	−	0.944315i	$-0.606726\pi$
$653$	−16.8167	−0.658087	−0.329043	+	0.944315i	$0.606726\pi$
$654$	7.11943	0.278392
$655$	61.5416	2.40463
$656$	0.211103	0.00824217
$657$	−4.40285	−0.171772
$658$	−14.3028	−0.557580
$659$	16.3305	0.636147	0.318074	−	0.948066i	$-0.396964\pi$
$659$	16.3305	0.636147	0.318074	+	0.948066i	$0.396964\pi$
$660$	2.09167	0.0814183
$661$	−0.788897	−0.0306846	−0.0153423	−	0.999882i	$-0.504884\pi$
$661$	−0.788897	−0.0306846	−0.0153423	+	0.999882i	$0.504884\pi$
$662$	−26.9361	−1.04690
$663$	9.00000	0.349531
$664$	38.7250	1.50282
$665$	−3.00000	−0.116335
$666$	24.1472	0.935684
$667$	−29.7250	−1.15096
$668$	14.5139	0.561559
$669$	3.88057	0.150032
$670$	−13.1833	−0.509317
$671$	2.93608	0.113346
$672$	1.60555	0.0619355
$673$	14.9083	0.574674	0.287337	−	0.957830i	$-0.407230\pi$
$673$	14.9083	0.574674	0.287337	+	0.957830i	$0.407230\pi$
$674$	−17.7889	−0.685203
$675$	−7.15559	−0.275419
$676$	60.8444	2.34017
$677$	17.9361	0.689340	0.344670	−	0.938724i	$-0.387991\pi$
$677$	17.9361	0.689340	0.344670	+	0.938724i	$0.387991\pi$
$678$	−6.76114	−0.259660
$679$	9.60555	0.368627
$680$	−47.7250	−1.83017
$681$	−7.60555	−0.291445
$682$	2.09167	0.0800943
$683$	12.2111	0.467245	0.233622	−	0.972327i	$-0.424942\pi$
$683$	12.2111	0.467245	0.233622	+	0.972327i	$0.424942\pi$
$684$	−9.60555	−0.367277
$685$	−64.8999	−2.47970
$686$	−2.30278	−0.0879204
$687$	−0.238859	−0.00911304
$688$	−3.02776	−0.115432
$689$	38.7250	1.47530
$690$	−6.27502	−0.238886
$691$	−41.8167	−1.59078	−0.795390	−	0.606098i	$-0.792734\pi$
$691$	−41.8167	−1.59078	−0.795390	+	0.606098i	$0.792734\pi$
$692$	25.8167	0.981402
$693$	2.02776	0.0770281
$694$	65.7250	2.49489
$695$	−15.0000	−0.568982
$696$	−9.00000	−0.341144
$697$	−3.69722	−0.140042
$698$	−17.3028	−0.654920
$699$	−0.633308	−0.0239539
$700$	13.2111	0.499333
$701$	21.2111	0.801132	0.400566	−	0.916268i	$-0.368813\pi$
$701$	21.2111	0.801132	0.400566	+	0.916268i	$0.368813\pi$
$702$	−23.0917	−0.871539
$703$	3.60555	0.135986
$704$	8.93608	0.336791
$705$	−5.64171	−0.212479
$706$	−22.8167	−0.858716
$707$	−4.39445	−0.165270
$708$	−6.21110	−0.233428
$709$	−11.6056	−0.435856	−0.217928	−	0.975965i	$-0.569930\pi$
$709$	−11.6056	−0.435856	−0.217928	+	0.975965i	$0.569930\pi$
$710$	84.3583	3.16591
$711$	−32.6056	−1.22280
$712$	31.8167	1.19238
$713$	−3.90833	−0.146368
$714$	3.69722	0.138365
$715$	−11.7250	−0.438489
$716$	−38.7250	−1.44722
$717$	−8.30278	−0.310073
$718$	58.8167	2.19502
$719$	−27.6333	−1.03055	−0.515274	−	0.857025i	$-0.672310\pi$
$719$	−27.6333	−1.03055	−0.515274	+	0.857025i	$0.672310\pi$
$720$	2.64171	0.0984508
$721$	−16.2111	−0.603733
$722$	−2.30278	−0.0857004
$723$	−6.23886	−0.232026
$724$	−35.3305	−1.31305
$725$	39.6333	1.47194
$726$	7.33053	0.272062
$727$	−27.6611	−1.02589	−0.512946	−	0.858421i	$-0.671446\pi$
$727$	−27.6611	−1.02589	−0.512946	+	0.858421i	$0.671446\pi$
$728$	16.8167	0.623267
$729$	−22.1749	−0.821294
$730$	10.4584	0.387081
$731$	53.0278	1.96130
$732$	−4.21110	−0.155647
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	0.0639167	0.00235921
$735$	−0.908327	−0.0335041
$736$	−15.9083	−0.586389
$737$	1.33053	0.0490108
$738$	4.66947	0.171885
$739$	−18.5778	−0.683395	−0.341698	−	0.939810i	$-0.611002\pi$
$739$	−18.5778	−0.683395	−0.341698	+	0.939810i	$0.611002\pi$
$740$	−35.7250	−1.31328
$741$	−1.69722	−0.0623491
$742$	15.9083	0.584013
$743$	−12.6333	−0.463471	−0.231736	−	0.972779i	$-0.574440\pi$
$743$	−12.6333	−0.463471	−0.231736	+	0.972779i	$0.574440\pi$
$744$	−1.18335	−0.0433836
$745$	18.6333	0.682672
$746$	−55.5416	−2.03352
$747$	37.5416	1.37358
$748$	12.2111	0.446482
$749$	−5.78890	−0.211522
$750$	−2.09167	−0.0763771
$751$	26.3583	0.961828	0.480914	−	0.876768i	$-0.340305\pi$
$751$	26.3583	0.961828	0.480914	+	0.876768i	$0.340305\pi$
$752$	1.88057	0.0685774
$753$	−4.81665	−0.175529
$754$	127.900	4.65784
$755$	−17.7250	−0.645078
$756$	−5.90833	−0.214884
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−15.6972	−0.570149
$759$	0.633308	0.0229876
$760$	9.00000	0.326464
$761$	−24.2111	−0.877652	−0.438826	−	0.898572i	$-0.644606\pi$
$761$	−24.2111	−0.877652	−0.438826	+	0.898572i	$0.644606\pi$
$762$	8.23886	0.298462
$763$	−10.2111	−0.369667
$764$	83.6611	3.02675
$765$	−46.2666	−1.67277
$766$	−15.2750	−0.551909
$767$	34.8167	1.25716
$768$	−5.42221	−0.195657
$769$	−25.2111	−0.909136	−0.454568	−	0.890712i	$-0.650206\pi$
$769$	−25.2111	−0.909136	−0.454568	+	0.890712i	$0.650206\pi$
$770$	−4.81665	−0.173580
$771$	−0.275019	−0.00990458
$772$	−76.3583	−2.74819
$773$	−22.1833	−0.797880	−0.398940	−	0.916977i	$-0.630622\pi$
$773$	−22.1833	−0.797880	−0.398940	+	0.916977i	$0.630622\pi$
$774$	−66.9722	−2.40727
$775$	5.21110	0.187188
$776$	−28.8167	−1.03446
$777$	1.09167	0.0391636
$778$	55.6056	1.99356
$779$	0.697224	0.0249807
$780$	16.8167	0.602133
$781$	−8.51388	−0.304651
$782$	−36.6333	−1.31000
$783$	−17.7250	−0.633439
$784$	0.302776	0.0108134
$785$	−13.5416	−0.483322
$786$	14.3028	0.510163
$787$	−31.6333	−1.12761	−0.563803	−	0.825909i	$-0.690662\pi$
$787$	−31.6333	−1.12761	−0.563803	+	0.825909i	$0.690662\pi$
$788$	−22.8167	−0.812810
$789$	−2.93608	−0.104527
$790$	77.4500	2.75555
$791$	9.69722	0.344794
$792$	−6.08327	−0.216160
$793$	23.6056	0.838258
$794$	−85.2666	−3.02600
$795$	6.27502	0.222552
$796$	−44.3305	−1.57125
$797$	26.0917	0.924214	0.462107	−	0.886824i	$-0.347094\pi$
$797$	26.0917	0.924214	0.462107	+	0.886824i	$0.347094\pi$
$798$	−0.697224	−0.0246815
$799$	−32.9361	−1.16519
$800$	21.2111	0.749926
$801$	30.8444	1.08983
$802$	8.02776	0.283470
$803$	−1.05551	−0.0372482
$804$	−1.90833	−0.0673015
$805$	9.00000	0.317208
$806$	16.8167	0.592341
$807$	6.27502	0.220891
$808$	13.1833	0.463788
$809$	−26.0278	−0.915087	−0.457544	−	0.889187i	$-0.651271\pi$
$809$	−26.0278	−0.915087	−0.457544	+	0.889187i	$0.651271\pi$
$810$	56.5332	1.98638
$811$	−32.0555	−1.12562	−0.562811	−	0.826586i	$-0.690280\pi$
$811$	−32.0555	−1.12562	−0.562811	+	0.826586i	$0.690280\pi$
$812$	32.7250	1.14842
$813$	5.42221	0.190165
$814$	5.78890	0.202901
$815$	−17.0917	−0.598695
$816$	−0.486122	−0.0170177
$817$	−10.0000	−0.349856
$818$	−48.1472	−1.68343
$819$	16.3028	0.569665
$820$	−6.90833	−0.241249
$821$	−51.6333	−1.80201	−0.901007	−	0.433804i	$-0.857171\pi$
$821$	−51.6333	−1.80201	−0.901007	+	0.433804i	$0.857171\pi$
$822$	−15.0833	−0.526089
$823$	16.2389	0.566051	0.283026	−	0.959112i	$-0.408662\pi$
$823$	16.2389	0.566051	0.283026	+	0.959112i	$0.408662\pi$
$824$	48.6333	1.69422
$825$	−0.844410	−0.0293986
$826$	14.3028	0.497657
$827$	−45.0000	−1.56480	−0.782402	−	0.622774i	$-0.786006\pi$
$827$	−45.0000	−1.56480	−0.782402	+	0.622774i	$0.786006\pi$
$828$	28.8167	1.00145
$829$	54.0555	1.87743	0.938713	−	0.344700i	$-0.112019\pi$
$829$	54.0555	1.87743	0.938713	+	0.344700i	$0.112019\pi$
$830$	−89.1749	−3.09531
$831$	0.183346	0.00636021
$832$	71.8444	2.49076
$833$	−5.30278	−0.183730
$834$	−3.48612	−0.120715
$835$	−13.1833	−0.456229
$836$	−2.30278	−0.0796432
$837$	−2.33053	−0.0805550
$838$	−34.5416	−1.19322
$839$	−2.78890	−0.0962834	−0.0481417	−	0.998841i	$-0.515330\pi$
$839$	−2.78890	−0.0962834	−0.0481417	+	0.998841i	$0.515330\pi$
$840$	2.72498	0.0940208
$841$	69.1749	2.38534
$842$	68.1749	2.34946
$843$	0.908327	0.0312844
$844$	−25.4222	−0.875068
$845$	−55.2666	−1.90123
$846$	41.5971	1.43014
$847$	−10.5139	−0.361261
$848$	−2.09167	−0.0718283
$849$	8.27502	0.283998
$850$	48.8444	1.67535
$851$	−10.8167	−0.370790
$852$	12.2111	0.418345
$853$	3.33053	0.114035	0.0570176	−	0.998373i	$-0.481841\pi$
$853$	3.33053	0.114035	0.0570176	+	0.998373i	$0.481841\pi$
$854$	9.69722	0.331832
$855$	8.72498	0.298388
$856$	17.3667	0.593581
$857$	32.0917	1.09623	0.548115	−	0.836403i	$-0.315345\pi$
$857$	32.0917	1.09623	0.548115	+	0.836403i	$0.315345\pi$
$858$	−2.72498	−0.0930293
$859$	−27.3028	−0.931559	−0.465779	−	0.884901i	$-0.654226\pi$
$859$	−27.3028	−0.931559	−0.465779	+	0.884901i	$0.654226\pi$
$860$	99.0833	3.37871
$861$	0.211103	0.00719436
$862$	7.39445	0.251856
$863$	−16.5416	−0.563084	−0.281542	−	0.959549i	$-0.590846\pi$
$863$	−16.5416	−0.563084	−0.281542	+	0.959549i	$0.590846\pi$
$864$	−9.48612	−0.322724
$865$	−23.4500	−0.797323
$866$	1.33053	0.0452133
$867$	3.36669	0.114339
$868$	4.30278	0.146046
$869$	−7.81665	−0.265162
$870$	20.7250	0.702643
$871$	10.6972	0.362462
$872$	30.6333	1.03737
$873$	−27.9361	−0.945493
$874$	6.90833	0.233678
$875$	3.00000	0.101419
$876$	1.51388	0.0511492
$877$	−8.18335	−0.276332	−0.138166	−	0.990409i	$-0.544121\pi$
$877$	−8.18335	−0.276332	−0.138166	+	0.990409i	$0.544121\pi$
$878$	−52.4777	−1.77104
$879$	−3.82506	−0.129016
$880$	0.633308	0.0213488
$881$	−19.7527	−0.665487	−0.332743	−	0.943017i	$-0.607974\pi$
$881$	−19.7527	−0.665487	−0.332743	+	0.943017i	$0.607974\pi$
$882$	6.69722	0.225507
$883$	−46.2111	−1.55513	−0.777564	−	0.628804i	$-0.783545\pi$
$883$	−46.2111	−1.55513	−0.777564	+	0.628804i	$0.783545\pi$
$884$	98.1749	3.30198
$885$	5.64171	0.189644
$886$	37.1194	1.24705
$887$	19.1833	0.644114	0.322057	−	0.946720i	$-0.395626\pi$
$887$	19.1833	0.644114	0.322057	+	0.946720i	$0.395626\pi$
$888$	−3.27502	−0.109902
$889$	−11.8167	−0.396318
$890$	−73.2666	−2.45590
$891$	−5.70563	−0.191146
$892$	42.3305	1.41733
$893$	6.21110	0.207847
$894$	4.33053	0.144835
$895$	35.1749	1.17577
$896$	18.9083	0.631683
$897$	5.09167	0.170006
$898$	1.11943	0.0373558
$899$	12.9083	0.430517
$900$	−38.4222	−1.28074
$901$	36.6333	1.22043
$902$	1.11943	0.0372729
$903$	−3.02776	−0.100757
$904$	−29.0917	−0.967575
$905$	32.0917	1.06676
$906$	−4.11943	−0.136859
$907$	−5.18335	−0.172110	−0.0860551	−	0.996290i	$-0.527426\pi$
$907$	−5.18335	−0.172110	−0.0860551	+	0.996290i	$0.527426\pi$
$908$	−82.9638	−2.75325
$909$	12.7805	0.423902
$910$	−38.7250	−1.28372
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0.0916731	0.00303560
$913$	9.00000	0.297857
$914$	−51.3583	−1.69878
$915$	3.82506	0.126453
$916$	−2.60555	−0.0860898
$917$	−20.5139	−0.677428
$918$	−21.8444	−0.720973
$919$	−7.97224	−0.262980	−0.131490	−	0.991317i	$-0.541976\pi$
$919$	−7.97224	−0.262980	−0.131490	+	0.991317i	$0.541976\pi$
$920$	−27.0000	−0.890164
$921$	4.44996	0.146631
$922$	−2.09167	−0.0688856
$923$	−68.4500	−2.25306
$924$	−0.697224	−0.0229370
$925$	14.4222	0.474199
$926$	83.9361	2.75831
$927$	47.1472	1.54852
$928$	52.5416	1.72476
$929$	11.5139	0.377758	0.188879	−	0.982000i	$-0.439515\pi$
$929$	11.5139	0.377758	0.188879	+	0.982000i	$0.439515\pi$
$930$	2.72498	0.0893556
$931$	1.00000	0.0327737
$932$	−6.90833	−0.226290
$933$	−0.761141	−0.0249186
$934$	−38.0917	−1.24640
$935$	−11.0917	−0.362736
$936$	−48.9083	−1.59862
$937$	21.1194	0.689942	0.344971	−	0.938613i	$-0.387889\pi$
$937$	21.1194	0.689942	0.344971	+	0.938613i	$0.387889\pi$
$938$	4.39445	0.143484
$939$	−8.18335	−0.267053
$940$	−61.5416	−2.00727
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	−3.14719	−0.102541
$943$	−2.09167	−0.0681142
$944$	−1.88057	−0.0612074
$945$	5.36669	0.174579
$946$	−16.0555	−0.522010
$947$	36.9083	1.19936	0.599680	−	0.800240i	$-0.295295\pi$
$947$	36.9083	1.19936	0.599680	+	0.800240i	$0.295295\pi$
$948$	11.2111	0.364120
$949$	−8.48612	−0.275471
$950$	−9.21110	−0.298848
$951$	2.66106	0.0862909
$952$	15.9083	0.515592
$953$	4.33053	0.140280	0.0701398	−	0.997537i	$-0.477655\pi$
$953$	4.33053	0.140280	0.0701398	+	0.997537i	$0.477655\pi$
$954$	−46.2666	−1.49794
$955$	−75.9916	−2.45903
$956$	−90.5694	−2.92922
$957$	−2.09167	−0.0676142
$958$	1.60555	0.0518730
$959$	21.6333	0.698576
$960$	11.6417	0.375735
$961$	−29.3028	−0.945251
$962$	46.5416	1.50056
$963$	16.8360	0.542533
$964$	−68.0555	−2.19192
$965$	69.3583	2.23272
$966$	2.09167	0.0672985
$967$	26.6972	0.858525	0.429262	−	0.903180i	$-0.358774\pi$
$967$	26.6972	0.858525	0.429262	+	0.903180i	$0.358774\pi$
$968$	31.5416	1.01379
$969$	−1.60555	−0.0515777
$970$	66.3583	2.13064
$971$	−18.6333	−0.597971	−0.298986	−	0.954258i	$-0.596648\pi$
$971$	−18.6333	−0.597971	−0.298986	+	0.954258i	$0.596648\pi$
$972$	25.9083	0.831010
$973$	5.00000	0.160293
$974$	−10.5416	−0.337776
$975$	−6.78890	−0.217419
$976$	−1.27502	−0.0408124
$977$	−17.4500	−0.558274	−0.279137	−	0.960251i	$-0.590048\pi$
$977$	−17.4500	−0.558274	−0.279137	+	0.960251i	$0.590048\pi$
$978$	−3.97224	−0.127018
$979$	7.39445	0.236328
$980$	−9.90833	−0.316510
$981$	29.6972	0.948159
$982$	27.6333	0.881814
$983$	−49.0555	−1.56463	−0.782314	−	0.622884i	$-0.785961\pi$
$983$	−49.0555	−1.56463	−0.782314	+	0.622884i	$0.785961\pi$
$984$	−0.633308	−0.0201891
$985$	20.7250	0.660353
$986$	120.992	3.85316
$987$	1.88057	0.0598592
$988$	−18.5139	−0.589005
$989$	30.0000	0.953945
$990$	14.0084	0.445216
$991$	−16.9722	−0.539141	−0.269571	−	0.962981i	$-0.586882\pi$
$991$	−16.9722	−0.539141	−0.269571	+	0.962981i	$0.586882\pi$
$992$	6.90833	0.219340
$993$	3.54163	0.112390
$994$	−28.1194	−0.891894
$995$	40.2666	1.27654
$996$	−12.9083	−0.409016
$997$	5.27502	0.167062	0.0835308	−	0.996505i	$-0.473380\pi$
$997$	5.27502	0.167062	0.0835308	+	0.996505i	$0.473380\pi$
$998$	−45.9083	−1.45320
$999$	−6.44996	−0.204068

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	133.2.a.b.1.1	✓	2
3.2	odd	2		1197.2.a.h.1.2		2
4.3	odd	2		2128.2.a.l.1.1		2
5.4	even	2		3325.2.a.n.1.2		2
7.2	even	3		931.2.f.h.704.2		4
7.3	odd	6		931.2.f.g.324.2		4
7.4	even	3		931.2.f.h.324.2		4
7.5	odd	6		931.2.f.g.704.2		4
7.6	odd	2		931.2.a.g.1.1		2
8.3	odd	2		8512.2.a.l.1.2		2
8.5	even	2		8512.2.a.bh.1.1		2
19.18	odd	2		2527.2.a.d.1.2		2
21.20	even	2		8379.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.a.b.1.1	✓	2	1.1	even	1	trivial
931.2.a.g.1.1		2	7.6	odd	2
931.2.f.g.324.2		4	7.3	odd	6
931.2.f.g.704.2		4	7.5	odd	6
931.2.f.h.324.2		4	7.4	even	3
931.2.f.h.704.2		4	7.2	even	3
1197.2.a.h.1.2		2	3.2	odd	2
2128.2.a.l.1.1		2	4.3	odd	2
2527.2.a.d.1.2		2	19.18	odd	2
3325.2.a.n.1.2		2	5.4	even	2
8379.2.a.bf.1.2		2	21.20	even	2
8512.2.a.l.1.2		2	8.3	odd	2
8512.2.a.bh.1.1		2	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	133.2.a.b.1.1

Level	$133$
Weight	$2$
Character	133.1
Self dual	yes
Analytic conductor	$1.062$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$