LMFDB - Embedded newform 8037.2.a.o.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 28 x^{16} + 25 x^{15} + 322 x^{14} - 247 x^{13} - 1971 x^{12} + 1231 x^{11} + 6953 x^{10} + \cdots + 9 $ x^18 - x^17 - 28x^16 + 25x^15 + 322x^14 - 247x^13 - 1971x^12 + 1231x^11 + 6953x^10 - 3283x^9 - 14235x^8 + 4562x^7 + 15962x^6 - 2882x^5 - 8159x^4 + 606x^3 + 890x^2 - 179x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-0.494084$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.494084 q^{2} -1.75588 q^{4} +3.82270 q^{5} -3.40967 q^{7} -1.85572 q^{8} +O(q^{10})$	\(q+0.494084 q^{2} -1.75588 q^{4} +3.82270 q^{5} -3.40967 q^{7} -1.85572 q^{8} +1.88874 q^{10} +5.21862 q^{11} +3.85549 q^{13} -1.68466 q^{14} +2.59488 q^{16} +3.52637 q^{17} -1.00000 q^{19} -6.71220 q^{20} +2.57844 q^{22} +3.25122 q^{23} +9.61303 q^{25} +1.90494 q^{26} +5.98697 q^{28} -1.82253 q^{29} +5.41182 q^{31} +4.99353 q^{32} +1.74233 q^{34} -13.0341 q^{35} -8.34066 q^{37} -0.494084 q^{38} -7.09387 q^{40} +8.29688 q^{41} +3.89858 q^{43} -9.16328 q^{44} +1.60638 q^{46} -1.00000 q^{47} +4.62583 q^{49} +4.74965 q^{50} -6.76978 q^{52} -5.56376 q^{53} +19.9492 q^{55} +6.32739 q^{56} -0.900484 q^{58} -7.79504 q^{59} -13.1224 q^{61} +2.67390 q^{62} -2.72253 q^{64} +14.7384 q^{65} -3.15721 q^{67} -6.19189 q^{68} -6.43996 q^{70} +7.73085 q^{71} -10.8618 q^{73} -4.12099 q^{74} +1.75588 q^{76} -17.7938 q^{77} +16.9709 q^{79} +9.91944 q^{80} +4.09936 q^{82} +10.6109 q^{83} +13.4803 q^{85} +1.92623 q^{86} -9.68431 q^{88} +3.54943 q^{89} -13.1459 q^{91} -5.70876 q^{92} -0.494084 q^{94} -3.82270 q^{95} +12.4062 q^{97} +2.28555 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

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 $

$2$	0.494084	0.349370	0.174685	−	0.984624i	$-0.444109\pi$
$2$	0.494084	0.349370	0.174685	+	0.984624i	$0.444109\pi$
$3$	0	0
$3$	0	0
$4$	−1.75588	−0.877940
$5$	3.82270	1.70956	0.854782	−	0.518988i	$-0.173691\pi$
$5$	3.82270	1.70956	0.854782	+	0.518988i	$0.173691\pi$
$6$	0	0
$7$	−3.40967	−1.28873	−0.644366	−	0.764717i	$-0.722879\pi$
$7$	−3.40967	−1.28873	−0.644366	+	0.764717i	$0.722879\pi$
$8$	−1.85572	−0.656097
$9$	0	0
$10$	1.88874	0.597271
$11$	5.21862	1.57347	0.786737	−	0.617289i	$-0.211769\pi$
$11$	5.21862	1.57347	0.786737	+	0.617289i	$0.211769\pi$
$12$	0	0
$13$	3.85549	1.06932	0.534660	−	0.845067i	$-0.320440\pi$
$13$	3.85549	1.06932	0.534660	+	0.845067i	$0.320440\pi$
$14$	−1.68466	−0.450245
$15$	0	0
$16$	2.59488	0.648719
$17$	3.52637	0.855271	0.427636	−	0.903951i	$-0.359347\pi$
$17$	3.52637	0.855271	0.427636	+	0.903951i	$0.359347\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−6.71220	−1.50089
$21$	0	0
$22$	2.57844	0.549725
$23$	3.25122	0.677927	0.338964	−	0.940800i	$-0.389924\pi$
$23$	3.25122	0.677927	0.338964	+	0.940800i	$0.389924\pi$
$24$	0	0
$25$	9.61303	1.92261
$26$	1.90494	0.373589
$27$	0	0
$28$	5.98697	1.13143
$29$	−1.82253	−0.338435	−0.169218	−	0.985579i	$-0.554124\pi$
$29$	−1.82253	−0.338435	−0.169218	+	0.985579i	$0.554124\pi$
$30$	0	0
$31$	5.41182	0.971992	0.485996	−	0.873961i	$-0.338457\pi$
$31$	5.41182	0.971992	0.485996	+	0.873961i	$0.338457\pi$
$32$	4.99353	0.882740
$33$	0	0
$34$	1.74233	0.298807
$35$	−13.0341	−2.20317
$36$	0	0
$37$	−8.34066	−1.37120	−0.685598	−	0.727980i	$-0.740459\pi$
$37$	−8.34066	−1.37120	−0.685598	+	0.727980i	$0.740459\pi$
$38$	−0.494084	−0.0801511
$39$	0	0
$40$	−7.09387	−1.12164
$41$	8.29688	1.29576	0.647878	−	0.761744i	$-0.275657\pi$
$41$	8.29688	1.29576	0.647878	+	0.761744i	$0.275657\pi$
$42$	0	0
$43$	3.89858	0.594528	0.297264	−	0.954795i	$-0.403926\pi$
$43$	3.89858	0.594528	0.297264	+	0.954795i	$0.403926\pi$
$44$	−9.16328	−1.38142
$45$	0	0
$46$	1.60638	0.236848
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	4.62583	0.660832
$50$	4.74965	0.671702
$51$	0	0
$52$	−6.76978	−0.938799
$53$	−5.56376	−0.764241	−0.382121	−	0.924112i	$-0.624806\pi$
$53$	−5.56376	−0.764241	−0.382121	+	0.924112i	$0.624806\pi$
$54$	0	0
$55$	19.9492	2.68995
$56$	6.32739	0.845534
$57$	0	0
$58$	−0.900484	−0.118239
$59$	−7.79504	−1.01483	−0.507414	−	0.861702i	$-0.669398\pi$
$59$	−7.79504	−1.01483	−0.507414	+	0.861702i	$0.669398\pi$
$60$	0	0
$61$	−13.1224	−1.68015	−0.840075	−	0.542470i	$-0.817489\pi$
$61$	−13.1224	−1.68015	−0.840075	+	0.542470i	$0.817489\pi$
$62$	2.67390	0.339585
$63$	0	0
$64$	−2.72253	−0.340316
$65$	14.7384	1.82807
$66$	0	0
$67$	−3.15721	−0.385715	−0.192857	−	0.981227i	$-0.561775\pi$
$67$	−3.15721	−0.385715	−0.192857	+	0.981227i	$0.561775\pi$
$68$	−6.19189	−0.750877
$69$	0	0
$70$	−6.43996	−0.769723
$71$	7.73085	0.917483	0.458742	−	0.888570i	$-0.348300\pi$
$71$	7.73085	0.917483	0.458742	+	0.888570i	$0.348300\pi$
$72$	0	0
$73$	−10.8618	−1.27128	−0.635640	−	0.771985i	$-0.719264\pi$
$73$	−10.8618	−1.27128	−0.635640	+	0.771985i	$0.719264\pi$
$74$	−4.12099	−0.479056
$75$	0	0
$76$	1.75588	0.201413
$77$	−17.7938	−2.02779
$78$	0	0
$79$	16.9709	1.90937	0.954685	−	0.297618i	$-0.0961920\pi$
$79$	16.9709	1.90937	0.954685	+	0.297618i	$0.0961920\pi$
$80$	9.91944	1.10903
$81$	0	0
$82$	4.09936	0.452699
$83$	10.6109	1.16470	0.582349	−	0.812939i	$-0.302134\pi$
$83$	10.6109	1.16470	0.582349	+	0.812939i	$0.302134\pi$
$84$	0	0
$85$	13.4803	1.46214
$86$	1.92623	0.207710
$87$	0	0
$88$	−9.68431	−1.03235
$89$	3.54943	0.376239	0.188119	−	0.982146i	$-0.439761\pi$
$89$	3.54943	0.376239	0.188119	+	0.982146i	$0.439761\pi$
$90$	0	0
$91$	−13.1459	−1.37807
$92$	−5.70876	−0.595179
$93$	0	0
$94$	−0.494084	−0.0509609
$95$	−3.82270	−0.392201
$96$	0	0
$97$	12.4062	1.25966	0.629828	−	0.776735i	$-0.283125\pi$
$97$	12.4062	1.25966	0.629828	+	0.776735i	$0.283125\pi$
$98$	2.28555	0.230875
$99$	0	0
$100$	−16.8793	−1.68793
$101$	−2.70176	−0.268835	−0.134418	−	0.990925i	$-0.542916\pi$
$101$	−2.70176	−0.268835	−0.134418	+	0.990925i	$0.542916\pi$
$102$	0	0
$103$	−8.59124	−0.846520	−0.423260	−	0.906008i	$-0.639114\pi$
$103$	−8.59124	−0.846520	−0.423260	+	0.906008i	$0.639114\pi$
$104$	−7.15471	−0.701577
$105$	0	0
$106$	−2.74897	−0.267003
$107$	10.4852	1.01364	0.506819	−	0.862052i	$-0.330821\pi$
$107$	10.4852	1.01364	0.506819	+	0.862052i	$0.330821\pi$
$108$	0	0
$109$	−16.0924	−1.54137	−0.770686	−	0.637215i	$-0.780086\pi$
$109$	−16.0924	−1.54137	−0.770686	+	0.637215i	$0.780086\pi$
$110$	9.85660	0.939790
$111$	0	0
$112$	−8.84767	−0.836026
$113$	−10.7438	−1.01069	−0.505346	−	0.862917i	$-0.668635\pi$
$113$	−10.7438	−1.01069	−0.505346	+	0.862917i	$0.668635\pi$
$114$	0	0
$115$	12.4285	1.15896
$116$	3.20015	0.297126
$117$	0	0
$118$	−3.85141	−0.354551
$119$	−12.0238	−1.10222
$120$	0	0
$121$	16.2340	1.47582
$122$	−6.48357	−0.586995
$123$	0	0
$124$	−9.50251	−0.853351
$125$	17.6342	1.57725
$126$	0	0
$127$	−10.5391	−0.935191	−0.467595	−	0.883943i	$-0.654880\pi$
$127$	−10.5391	−0.935191	−0.467595	+	0.883943i	$0.654880\pi$
$128$	−11.3322	−1.00164
$129$	0	0
$130$	7.28200	0.638674
$131$	−3.77648	−0.329953	−0.164976	−	0.986298i	$-0.552755\pi$
$131$	−3.77648	−0.329953	−0.164976	+	0.986298i	$0.552755\pi$
$132$	0	0
$133$	3.40967	0.295656
$134$	−1.55993	−0.134757
$135$	0	0
$136$	−6.54397	−0.561141
$137$	−3.52993	−0.301582	−0.150791	−	0.988566i	$-0.548182\pi$
$137$	−3.52993	−0.301582	−0.150791	+	0.988566i	$0.548182\pi$
$138$	0	0
$139$	3.05961	0.259513	0.129756	−	0.991546i	$-0.458580\pi$
$139$	3.05961	0.259513	0.129756	+	0.991546i	$0.458580\pi$
$140$	22.8864	1.93425
$141$	0	0
$142$	3.81969	0.320542
$143$	20.1203	1.68255
$144$	0	0
$145$	−6.96699	−0.578577
$146$	−5.36666	−0.444148
$147$	0	0
$148$	14.6452	1.20383
$149$	−3.30241	−0.270544	−0.135272	−	0.990809i	$-0.543191\pi$
$149$	−3.30241	−0.270544	−0.135272	+	0.990809i	$0.543191\pi$
$150$	0	0
$151$	11.7253	0.954195	0.477097	−	0.878850i	$-0.341689\pi$
$151$	11.7253	0.954195	0.477097	+	0.878850i	$0.341689\pi$
$152$	1.85572	0.150519
$153$	0	0
$154$	−8.79162	−0.708449
$155$	20.6878	1.66168
$156$	0	0
$157$	22.6818	1.81021	0.905104	−	0.425191i	$-0.139793\pi$
$157$	22.6818	1.81021	0.905104	+	0.425191i	$0.139793\pi$
$158$	8.38504	0.667078
$159$	0	0
$160$	19.0888	1.50910
$161$	−11.0856	−0.873667
$162$	0	0
$163$	−2.12360	−0.166333	−0.0831666	−	0.996536i	$-0.526503\pi$
$163$	−2.12360	−0.166333	−0.0831666	+	0.996536i	$0.526503\pi$
$164$	−14.5683	−1.13760
$165$	0	0
$166$	5.24268	0.406911
$167$	11.3278	0.876569	0.438285	−	0.898836i	$-0.355586\pi$
$167$	11.3278	0.876569	0.438285	+	0.898836i	$0.355586\pi$
$168$	0	0
$169$	1.86479	0.143445
$170$	6.66039	0.510829
$171$	0	0
$172$	−6.84544	−0.521960
$173$	13.1604	1.00057	0.500283	−	0.865862i	$-0.333229\pi$
$173$	13.1604	1.00057	0.500283	+	0.865862i	$0.333229\pi$
$174$	0	0
$175$	−32.7772	−2.47773
$176$	13.5417	1.02074
$177$	0	0
$178$	1.75372	0.131447
$179$	−11.6564	−0.871237	−0.435619	−	0.900131i	$-0.643470\pi$
$179$	−11.6564	−0.871237	−0.435619	+	0.900131i	$0.643470\pi$
$180$	0	0
$181$	7.93263	0.589628	0.294814	−	0.955555i	$-0.404742\pi$
$181$	7.93263	0.589628	0.294814	+	0.955555i	$0.404742\pi$
$182$	−6.49520	−0.481456
$183$	0	0
$184$	−6.03337	−0.444786
$185$	−31.8838	−2.34415
$186$	0	0
$187$	18.4028	1.34575
$188$	1.75588	0.128061
$189$	0	0
$190$	−1.88874	−0.137023
$191$	10.9433	0.791828	0.395914	−	0.918288i	$-0.370428\pi$
$191$	10.9433	0.791828	0.395914	+	0.918288i	$0.370428\pi$
$192$	0	0
$193$	18.7710	1.35116	0.675582	−	0.737285i	$-0.263892\pi$
$193$	18.7710	1.35116	0.675582	+	0.737285i	$0.263892\pi$
$194$	6.12970	0.440087
$195$	0	0
$196$	−8.12240	−0.580171
$197$	13.2431	0.943531	0.471765	−	0.881724i	$-0.343617\pi$
$197$	13.2431	0.943531	0.471765	+	0.881724i	$0.343617\pi$
$198$	0	0
$199$	4.39461	0.311526	0.155763	−	0.987794i	$-0.450216\pi$
$199$	4.39461	0.311526	0.155763	+	0.987794i	$0.450216\pi$
$200$	−17.8391	−1.26142
$201$	0	0
$202$	−1.33490	−0.0939231
$203$	6.21422	0.436153
$204$	0	0
$205$	31.7165	2.21518
$206$	−4.24480	−0.295749
$207$	0	0
$208$	10.0045	0.693689
$209$	−5.21862	−0.360980
$210$	0	0
$211$	12.7821	0.879958	0.439979	−	0.898008i	$-0.354986\pi$
$211$	12.7821	0.879958	0.439979	+	0.898008i	$0.354986\pi$
$212$	9.76930	0.670958
$213$	0	0
$214$	5.18055	0.354135
$215$	14.9031	1.01638
$216$	0	0
$217$	−18.4525	−1.25264
$218$	−7.95100	−0.538510
$219$	0	0
$220$	−35.0284	−2.36162
$221$	13.5959	0.914559
$222$	0	0
$223$	7.71251	0.516468	0.258234	−	0.966082i	$-0.416860\pi$
$223$	7.71251	0.516468	0.258234	+	0.966082i	$0.416860\pi$
$224$	−17.0263	−1.13762
$225$	0	0
$226$	−5.30835	−0.353106
$227$	−4.92272	−0.326733	−0.163366	−	0.986565i	$-0.552235\pi$
$227$	−4.92272	−0.326733	−0.163366	+	0.986565i	$0.552235\pi$
$228$	0	0
$229$	−27.5190	−1.81851	−0.909255	−	0.416240i	$-0.863348\pi$
$229$	−27.5190	−1.81851	−0.909255	+	0.416240i	$0.863348\pi$
$230$	6.14070	0.404906
$231$	0	0
$232$	3.38211	0.222046
$233$	14.8606	0.973551	0.486775	−	0.873527i	$-0.338173\pi$
$233$	14.8606	0.973551	0.486775	+	0.873527i	$0.338173\pi$
$234$	0	0
$235$	−3.82270	−0.249365
$236$	13.6872	0.890958
$237$	0	0
$238$	−5.94075	−0.385082
$239$	3.95939	0.256112	0.128056	−	0.991767i	$-0.459126\pi$
$239$	3.95939	0.256112	0.128056	+	0.991767i	$0.459126\pi$
$240$	0	0
$241$	−23.9814	−1.54478	−0.772390	−	0.635149i	$-0.780939\pi$
$241$	−23.9814	−1.54478	−0.772390	+	0.635149i	$0.780939\pi$
$242$	8.02097	0.515608
$243$	0	0
$244$	23.0414	1.47507
$245$	17.6831	1.12973
$246$	0	0
$247$	−3.85549	−0.245319
$248$	−10.0428	−0.637721
$249$	0	0
$250$	8.71280	0.551046
$251$	−0.188664	−0.0119083	−0.00595417	−	0.999982i	$-0.501895\pi$
$251$	−0.188664	−0.0119083	−0.00595417	+	0.999982i	$0.501895\pi$
$252$	0	0
$253$	16.9669	1.06670
$254$	−5.20719	−0.326728
$255$	0	0
$256$	−0.154019	−0.00962617
$257$	10.3547	0.645909	0.322955	−	0.946414i	$-0.395324\pi$
$257$	10.3547	0.645909	0.322955	+	0.946414i	$0.395324\pi$
$258$	0	0
$259$	28.4389	1.76711
$260$	−25.8788	−1.60494
$261$	0	0
$262$	−1.86590	−0.115276
$263$	3.01901	0.186160	0.0930801	−	0.995659i	$-0.470329\pi$
$263$	3.01901	0.186160	0.0930801	+	0.995659i	$0.470329\pi$
$264$	0	0
$265$	−21.2686	−1.30652
$266$	1.68466	0.103293
$267$	0	0
$268$	5.54369	0.338635
$269$	11.3395	0.691381	0.345690	−	0.938349i	$-0.387645\pi$
$269$	11.3395	0.691381	0.345690	+	0.938349i	$0.387645\pi$
$270$	0	0
$271$	−13.2488	−0.804807	−0.402403	−	0.915462i	$-0.631825\pi$
$271$	−13.2488	−0.804807	−0.402403	+	0.915462i	$0.631825\pi$
$272$	9.15051	0.554831
$273$	0	0
$274$	−1.74408	−0.105364
$275$	50.1668	3.02517
$276$	0	0
$277$	−12.0833	−0.726014	−0.363007	−	0.931786i	$-0.618250\pi$
$277$	−12.0833	−0.726014	−0.363007	+	0.931786i	$0.618250\pi$
$278$	1.51171	0.0906661
$279$	0	0
$280$	24.1877	1.44549
$281$	19.3639	1.15516	0.577578	−	0.816336i	$-0.303998\pi$
$281$	19.3639	1.15516	0.577578	+	0.816336i	$0.303998\pi$
$282$	0	0
$283$	25.9084	1.54009	0.770046	−	0.637989i	$-0.220233\pi$
$283$	25.9084	1.54009	0.770046	+	0.637989i	$0.220233\pi$
$284$	−13.5745	−0.805495
$285$	0	0
$286$	9.94114	0.587832
$287$	−28.2896	−1.66988
$288$	0	0
$289$	−4.56469	−0.268511
$290$	−3.44228	−0.202138
$291$	0	0
$292$	19.0721	1.11611
$293$	−20.0273	−1.17001	−0.585003	−	0.811031i	$-0.698907\pi$
$293$	−20.0273	−1.17001	−0.585003	+	0.811031i	$0.698907\pi$
$294$	0	0
$295$	−29.7981	−1.73491
$296$	15.4780	0.899638
$297$	0	0
$298$	−1.63167	−0.0945201
$299$	12.5351	0.724921
$300$	0	0
$301$	−13.2929	−0.766188
$302$	5.79331	0.333368
$303$	0	0
$304$	−2.59488	−0.148826
$305$	−50.1630	−2.87232
$306$	0	0
$307$	31.5540	1.80088	0.900441	−	0.434977i	$-0.143244\pi$
$307$	31.5540	1.80088	0.900441	+	0.434977i	$0.143244\pi$
$308$	31.2437	1.78028
$309$	0	0
$310$	10.2215	0.580542
$311$	−8.07811	−0.458067	−0.229034	−	0.973419i	$-0.573557\pi$
$311$	−8.07811	−0.458067	−0.229034	+	0.973419i	$0.573557\pi$
$312$	0	0
$313$	−12.4930	−0.706148	−0.353074	−	0.935595i	$-0.614864\pi$
$313$	−12.4930	−0.706148	−0.353074	+	0.935595i	$0.614864\pi$
$314$	11.2067	0.632433
$315$	0	0
$316$	−29.7988	−1.67631
$317$	−8.60125	−0.483095	−0.241547	−	0.970389i	$-0.577655\pi$
$317$	−8.60125	−0.483095	−0.241547	+	0.970389i	$0.577655\pi$
$318$	0	0
$319$	−9.51110	−0.532519
$320$	−10.4074	−0.581792
$321$	0	0
$322$	−5.47722	−0.305233
$323$	−3.52637	−0.196213
$324$	0	0
$325$	37.0629	2.05588
$326$	−1.04924	−0.0581119
$327$	0	0
$328$	−15.3967	−0.850141
$329$	3.40967	0.187981
$330$	0	0
$331$	27.9884	1.53838	0.769190	−	0.639021i	$-0.220660\pi$
$331$	27.9884	1.53838	0.769190	+	0.639021i	$0.220660\pi$
$332$	−18.6315	−1.02253
$333$	0	0
$334$	5.59688	0.306247
$335$	−12.0691	−0.659404
$336$	0	0
$337$	12.2238	0.665875	0.332937	−	0.942949i	$-0.391960\pi$
$337$	12.2238	0.665875	0.332937	+	0.942949i	$0.391960\pi$
$338$	0.921362	0.0501155
$339$	0	0
$340$	−23.6697	−1.28367
$341$	28.2422	1.52940
$342$	0	0
$343$	8.09514	0.437097
$344$	−7.23468	−0.390068
$345$	0	0
$346$	6.50234	0.349568
$347$	−8.27020	−0.443968	−0.221984	−	0.975050i	$-0.571253\pi$
$347$	−8.27020	−0.443968	−0.221984	+	0.975050i	$0.571253\pi$
$348$	0	0
$349$	−1.85495	−0.0992934	−0.0496467	−	0.998767i	$-0.515810\pi$
$349$	−1.85495	−0.0992934	−0.0496467	+	0.998767i	$0.515810\pi$
$350$	−16.1947	−0.865644
$351$	0	0
$352$	26.0594	1.38897
$353$	1.97491	0.105114	0.0525569	−	0.998618i	$-0.483263\pi$
$353$	1.97491	0.105114	0.0525569	+	0.998618i	$0.483263\pi$
$354$	0	0
$355$	29.5527	1.56850
$356$	−6.23237	−0.330315
$357$	0	0
$358$	−5.75923	−0.304385
$359$	−36.7122	−1.93760	−0.968799	−	0.247850i	$-0.920276\pi$
$359$	−36.7122	−1.93760	−0.968799	+	0.247850i	$0.920276\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	3.91939	0.205999
$363$	0	0
$364$	23.0827	1.20986
$365$	−41.5215	−2.17333
$366$	0	0
$367$	7.13286	0.372332	0.186166	−	0.982518i	$-0.440394\pi$
$367$	7.13286	0.372332	0.186166	+	0.982518i	$0.440394\pi$
$368$	8.43653	0.439784
$369$	0	0
$370$	−15.7533	−0.818976
$371$	18.9706	0.984903
$372$	0	0
$373$	−23.9043	−1.23772	−0.618858	−	0.785503i	$-0.712404\pi$
$373$	−23.9043	−1.23772	−0.618858	+	0.785503i	$0.712404\pi$
$374$	9.09254	0.470164
$375$	0	0
$376$	1.85572	0.0957016
$377$	−7.02674	−0.361896
$378$	0	0
$379$	−17.7693	−0.912747	−0.456373	−	0.889788i	$-0.650852\pi$
$379$	−17.7693	−0.912747	−0.456373	+	0.889788i	$0.650852\pi$
$380$	6.71220	0.344329
$381$	0	0
$382$	5.40690	0.276641
$383$	16.0907	0.822195	0.411098	−	0.911591i	$-0.365146\pi$
$383$	16.0907	0.822195	0.411098	+	0.911591i	$0.365146\pi$
$384$	0	0
$385$	−68.0202	−3.46663
$386$	9.27445	0.472057
$387$	0	0
$388$	−21.7838	−1.10590
$389$	−0.777865	−0.0394393	−0.0197197	−	0.999806i	$-0.506277\pi$
$389$	−0.777865	−0.0394393	−0.0197197	+	0.999806i	$0.506277\pi$
$390$	0	0
$391$	11.4650	0.579812
$392$	−8.58425	−0.433570
$393$	0	0
$394$	6.54320	0.329642
$395$	64.8745	3.26419
$396$	0	0
$397$	38.0455	1.90945	0.954723	−	0.297495i	$-0.0961512\pi$
$397$	38.0455	1.90945	0.954723	+	0.297495i	$0.0961512\pi$
$398$	2.17131	0.108838
$399$	0	0
$400$	24.9446	1.24723
$401$	−22.0526	−1.10125	−0.550627	−	0.834751i	$-0.685612\pi$
$401$	−22.0526	−1.10125	−0.550627	+	0.834751i	$0.685612\pi$
$402$	0	0
$403$	20.8652	1.03937
$404$	4.74397	0.236021
$405$	0	0
$406$	3.07035	0.152379
$407$	−43.5268	−2.15754
$408$	0	0
$409$	−29.1909	−1.44340	−0.721699	−	0.692207i	$-0.756639\pi$
$409$	−29.1909	−1.44340	−0.721699	+	0.692207i	$0.756639\pi$
$410$	15.6706	0.773917
$411$	0	0
$412$	15.0852	0.743194
$413$	26.5785	1.30784
$414$	0	0
$415$	40.5623	1.99112
$416$	19.2525	0.943932
$417$	0	0
$418$	−2.57844	−0.126116
$419$	16.7780	0.819658	0.409829	−	0.912162i	$-0.365588\pi$
$419$	16.7780	0.819658	0.409829	+	0.912162i	$0.365588\pi$
$420$	0	0
$421$	22.2776	1.08574	0.542871	−	0.839816i	$-0.317337\pi$
$421$	22.2776	1.08574	0.542871	+	0.839816i	$0.317337\pi$
$422$	6.31545	0.307431
$423$	0	0
$424$	10.3248	0.501416
$425$	33.8991	1.64435
$426$	0	0
$427$	44.7430	2.16526
$428$	−18.4107	−0.889914
$429$	0	0
$430$	7.36339	0.355094
$431$	37.4125	1.80210	0.901048	−	0.433720i	$-0.142799\pi$
$431$	37.4125	1.80210	0.901048	+	0.433720i	$0.142799\pi$
$432$	0	0
$433$	17.6445	0.847939	0.423970	−	0.905676i	$-0.360636\pi$
$433$	17.6445	0.847939	0.423970	+	0.905676i	$0.360636\pi$
$434$	−9.11709	−0.437634
$435$	0	0
$436$	28.2563	1.35323
$437$	−3.25122	−0.155527
$438$	0	0
$439$	4.87483	0.232663	0.116331	−	0.993210i	$-0.462887\pi$
$439$	4.87483	0.232663	0.116331	+	0.993210i	$0.462887\pi$
$440$	−37.0202	−1.76487
$441$	0	0
$442$	6.71752	0.319520
$443$	−38.1271	−1.81147	−0.905736	−	0.423843i	$-0.860681\pi$
$443$	−38.1271	−1.81147	−0.905736	+	0.423843i	$0.860681\pi$
$444$	0	0
$445$	13.5684	0.643204
$446$	3.81063	0.180439
$447$	0	0
$448$	9.28291	0.438576
$449$	−14.2619	−0.673059	−0.336530	−	0.941673i	$-0.609253\pi$
$449$	−14.2619	−0.673059	−0.336530	+	0.941673i	$0.609253\pi$
$450$	0	0
$451$	43.2983	2.03884
$452$	18.8648	0.887328
$453$	0	0
$454$	−2.43224	−0.114151
$455$	−50.2529	−2.35589
$456$	0	0
$457$	19.8630	0.929150	0.464575	−	0.885534i	$-0.346207\pi$
$457$	19.8630	0.929150	0.464575	+	0.885534i	$0.346207\pi$
$458$	−13.5967	−0.635334
$459$	0	0
$460$	−21.8229	−1.01750
$461$	−31.5347	−1.46872	−0.734358	−	0.678762i	$-0.762517\pi$
$461$	−31.5347	−1.46872	−0.734358	+	0.678762i	$0.762517\pi$
$462$	0	0
$463$	−15.0687	−0.700303	−0.350152	−	0.936693i	$-0.613870\pi$
$463$	−15.0687	−0.700303	−0.350152	+	0.936693i	$0.613870\pi$
$464$	−4.72924	−0.219550
$465$	0	0
$466$	7.34240	0.340130
$467$	−22.7303	−1.05183	−0.525917	−	0.850536i	$-0.676278\pi$
$467$	−22.7303	−1.05183	−0.525917	+	0.850536i	$0.676278\pi$
$468$	0	0
$469$	10.7650	0.497083
$470$	−1.88874	−0.0871209
$471$	0	0
$472$	14.4654	0.665825
$473$	20.3452	0.935474
$474$	0	0
$475$	−9.61303	−0.441076
$476$	21.1123	0.967680
$477$	0	0
$478$	1.95627	0.0894779
$479$	−21.5766	−0.985861	−0.492931	−	0.870069i	$-0.664074\pi$
$479$	−21.5766	−0.985861	−0.492931	+	0.870069i	$0.664074\pi$
$480$	0	0
$481$	−32.1573	−1.46625
$482$	−11.8489	−0.539700
$483$	0	0
$484$	−28.5050	−1.29568
$485$	47.4251	2.15346
$486$	0	0
$487$	1.25162	0.0567163	0.0283581	−	0.999598i	$-0.490972\pi$
$487$	1.25162	0.0567163	0.0283581	+	0.999598i	$0.490972\pi$
$488$	24.3515	1.10234
$489$	0	0
$490$	8.73696	0.394696
$491$	21.7545	0.981769	0.490884	−	0.871225i	$-0.336674\pi$
$491$	21.7545	0.981769	0.490884	+	0.871225i	$0.336674\pi$
$492$	0	0
$493$	−6.42692	−0.289454
$494$	−1.90494	−0.0857071
$495$	0	0
$496$	14.0430	0.630550
$497$	−26.3596	−1.18239
$498$	0	0
$499$	32.0856	1.43635	0.718175	−	0.695863i	$-0.244978\pi$
$499$	32.0856	1.43635	0.718175	+	0.695863i	$0.244978\pi$
$500$	−30.9636	−1.38473
$501$	0	0
$502$	−0.0932157	−0.00416042
$503$	−23.6704	−1.05541	−0.527705	−	0.849428i	$-0.676947\pi$
$503$	−23.6704	−1.05541	−0.527705	+	0.849428i	$0.676947\pi$
$504$	0	0
$505$	−10.3280	−0.459591
$506$	8.38308	0.372674
$507$	0	0
$508$	18.5053	0.821042
$509$	−8.08921	−0.358548	−0.179274	−	0.983799i	$-0.557375\pi$
$509$	−8.08921	−0.358548	−0.179274	+	0.983799i	$0.557375\pi$
$510$	0	0
$511$	37.0352	1.63834
$512$	22.5884	0.998274
$513$	0	0
$514$	5.11610	0.225662
$515$	−32.8417	−1.44718
$516$	0	0
$517$	−5.21862	−0.229515
$518$	14.0512	0.617375
$519$	0	0
$520$	−27.3503	−1.19939
$521$	20.8531	0.913590	0.456795	−	0.889572i	$-0.348997\pi$
$521$	20.8531	0.913590	0.456795	+	0.889572i	$0.348997\pi$
$522$	0	0
$523$	23.7381	1.03800	0.518998	−	0.854775i	$-0.326305\pi$
$523$	23.7381	1.03800	0.518998	+	0.854775i	$0.326305\pi$
$524$	6.63105	0.289679
$525$	0	0
$526$	1.49165	0.0650389
$527$	19.0841	0.831316
$528$	0	0
$529$	−12.4295	−0.540415
$530$	−10.5085	−0.456459
$531$	0	0
$532$	−5.98697	−0.259568
$533$	31.9885	1.38558
$534$	0	0
$535$	40.0816	1.73288
$536$	5.85891	0.253066
$537$	0	0
$538$	5.60266	0.241548
$539$	24.1404	1.03980
$540$	0	0
$541$	−27.7942	−1.19496	−0.597482	−	0.801882i	$-0.703832\pi$
$541$	−27.7942	−1.19496	−0.597482	+	0.801882i	$0.703832\pi$
$542$	−6.54602	−0.281176
$543$	0	0
$544$	17.6091	0.754982
$545$	−61.5164	−2.63507
$546$	0	0
$547$	−4.71548	−0.201619	−0.100810	−	0.994906i	$-0.532143\pi$
$547$	−4.71548	−0.201619	−0.100810	+	0.994906i	$0.532143\pi$
$548$	6.19813	0.264771
$549$	0	0
$550$	24.7866	1.05690
$551$	1.82253	0.0776424
$552$	0	0
$553$	−57.8650	−2.46067
$554$	−5.97016	−0.253648
$555$	0	0
$556$	−5.37231	−0.227837
$557$	22.7817	0.965292	0.482646	−	0.875816i	$-0.339676\pi$
$557$	22.7817	0.965292	0.482646	+	0.875816i	$0.339676\pi$
$558$	0	0
$559$	15.0309	0.635741
$560$	−33.8220	−1.42924
$561$	0	0
$562$	9.56742	0.403577
$563$	−14.9339	−0.629390	−0.314695	−	0.949193i	$-0.601902\pi$
$563$	−14.9339	−0.629390	−0.314695	+	0.949193i	$0.601902\pi$
$564$	0	0
$565$	−41.0703	−1.72784
$566$	12.8009	0.538063
$567$	0	0
$568$	−14.3463	−0.601958
$569$	4.14992	0.173974	0.0869868	−	0.996209i	$-0.472276\pi$
$569$	4.14992	0.173974	0.0869868	+	0.996209i	$0.472276\pi$
$570$	0	0
$571$	−23.4903	−0.983038	−0.491519	−	0.870867i	$-0.663558\pi$
$571$	−23.4903	−0.983038	−0.491519	+	0.870867i	$0.663558\pi$
$572$	−35.3289	−1.47718
$573$	0	0
$574$	−13.9775	−0.583408
$575$	31.2541	1.30339
$576$	0	0
$577$	−34.5485	−1.43827	−0.719137	−	0.694868i	$-0.755463\pi$
$577$	−34.5485	−1.43827	−0.719137	+	0.694868i	$0.755463\pi$
$578$	−2.25534	−0.0938098
$579$	0	0
$580$	12.2332	0.507956
$581$	−36.1796	−1.50098
$582$	0	0
$583$	−29.0352	−1.20251
$584$	20.1565	0.834083
$585$	0	0
$586$	−9.89517	−0.408766
$587$	−18.8989	−0.780042	−0.390021	−	0.920806i	$-0.627532\pi$
$587$	−18.8989	−0.780042	−0.390021	+	0.920806i	$0.627532\pi$
$588$	0	0
$589$	−5.41182	−0.222990
$590$	−14.7228	−0.606127
$591$	0	0
$592$	−21.6430	−0.889522
$593$	−31.1479	−1.27909	−0.639545	−	0.768754i	$-0.720877\pi$
$593$	−31.1479	−1.27909	−0.639545	+	0.768754i	$0.720877\pi$
$594$	0	0
$595$	−45.9632	−1.88431
$596$	5.79864	0.237521
$597$	0	0
$598$	6.19338	0.253266
$599$	26.7030	1.09106	0.545528	−	0.838093i	$-0.316329\pi$
$599$	26.7030	1.09106	0.545528	+	0.838093i	$0.316329\pi$
$600$	0	0
$601$	16.4184	0.669719	0.334860	−	0.942268i	$-0.391311\pi$
$601$	16.4184	0.669719	0.334860	+	0.942268i	$0.391311\pi$
$602$	−6.56779	−0.267683
$603$	0	0
$604$	−20.5883	−0.837726
$605$	62.0577	2.52301
$606$	0	0
$607$	−42.2173	−1.71355	−0.856773	−	0.515694i	$-0.827534\pi$
$607$	−42.2173	−1.71355	−0.856773	+	0.515694i	$0.827534\pi$
$608$	−4.99353	−0.202515
$609$	0	0
$610$	−24.7847	−1.00350
$611$	−3.85549	−0.155976
$612$	0	0
$613$	−6.07060	−0.245189	−0.122595	−	0.992457i	$-0.539121\pi$
$613$	−6.07060	−0.245189	−0.122595	+	0.992457i	$0.539121\pi$
$614$	15.5903	0.629175
$615$	0	0
$616$	33.0203	1.33042
$617$	−19.6716	−0.791949	−0.395975	−	0.918261i	$-0.629593\pi$
$617$	−19.6716	−0.791949	−0.395975	+	0.918261i	$0.629593\pi$
$618$	0	0
$619$	−2.64487	−0.106306	−0.0531532	−	0.998586i	$-0.516927\pi$
$619$	−2.64487	−0.106306	−0.0531532	+	0.998586i	$0.516927\pi$
$620$	−36.3252	−1.45886
$621$	0	0
$622$	−3.99127	−0.160035
$623$	−12.1024	−0.484871
$624$	0	0
$625$	19.3452	0.773808
$626$	−6.17261	−0.246707
$627$	0	0
$628$	−39.8266	−1.58925
$629$	−29.4123	−1.17275
$630$	0	0
$631$	−40.7223	−1.62113	−0.810565	−	0.585648i	$-0.800840\pi$
$631$	−40.7223	−1.62113	−0.810565	+	0.585648i	$0.800840\pi$
$632$	−31.4932	−1.25273
$633$	0	0
$634$	−4.24975	−0.168779
$635$	−40.2877	−1.59877
$636$	0	0
$637$	17.8348	0.706641
$638$	−4.69928	−0.186046
$639$	0	0
$640$	−43.3197	−1.71236
$641$	46.3577	1.83102	0.915509	−	0.402298i	$-0.131789\pi$
$641$	46.3577	1.83102	0.915509	+	0.402298i	$0.131789\pi$
$642$	0	0
$643$	39.0561	1.54022	0.770110	−	0.637911i	$-0.220201\pi$
$643$	39.0561	1.54022	0.770110	+	0.637911i	$0.220201\pi$
$644$	19.4650	0.767027
$645$	0	0
$646$	−1.74233	−0.0685509
$647$	25.1982	0.990642	0.495321	−	0.868710i	$-0.335050\pi$
$647$	25.1982	0.990642	0.495321	+	0.868710i	$0.335050\pi$
$648$	0	0
$649$	−40.6794	−1.59680
$650$	18.3122	0.718264
$651$	0	0
$652$	3.72879	0.146031
$653$	−18.1812	−0.711487	−0.355744	−	0.934584i	$-0.615772\pi$
$653$	−18.1812	−0.711487	−0.355744	+	0.934584i	$0.615772\pi$
$654$	0	0
$655$	−14.4363	−0.564075
$656$	21.5294	0.840582
$657$	0	0
$658$	1.68466	0.0656750
$659$	−3.89585	−0.151761	−0.0758804	−	0.997117i	$-0.524177\pi$
$659$	−3.89585	−0.151761	−0.0758804	+	0.997117i	$0.524177\pi$
$660$	0	0
$661$	−14.1511	−0.550416	−0.275208	−	0.961385i	$-0.588747\pi$
$661$	−14.1511	−0.550416	−0.275208	+	0.961385i	$0.588747\pi$
$662$	13.8286	0.537464
$663$	0	0
$664$	−19.6909	−0.764154
$665$	13.0341	0.505442
$666$	0	0
$667$	−5.92545	−0.229435
$668$	−19.8902	−0.769575
$669$	0	0
$670$	−5.96314	−0.230376
$671$	−68.4808	−2.64367
$672$	0	0
$673$	25.3296	0.976383	0.488192	−	0.872736i	$-0.337657\pi$
$673$	25.3296	0.976383	0.488192	+	0.872736i	$0.337657\pi$
$674$	6.03961	0.232637
$675$	0	0
$676$	−3.27434	−0.125936
$677$	−16.4665	−0.632861	−0.316430	−	0.948616i	$-0.602484\pi$
$677$	−16.4665	−0.632861	−0.316430	+	0.948616i	$0.602484\pi$
$678$	0	0
$679$	−42.3009	−1.62336
$680$	−25.0156	−0.959306
$681$	0	0
$682$	13.9541	0.534328
$683$	40.3970	1.54575	0.772875	−	0.634559i	$-0.218818\pi$
$683$	40.3970	1.54575	0.772875	+	0.634559i	$0.218818\pi$
$684$	0	0
$685$	−13.4939	−0.515574
$686$	3.99968	0.152709
$687$	0	0
$688$	10.1163	0.385682
$689$	−21.4510	−0.817218
$690$	0	0
$691$	38.3359	1.45837	0.729183	−	0.684318i	$-0.239900\pi$
$691$	38.3359	1.45837	0.729183	+	0.684318i	$0.239900\pi$
$692$	−23.1081	−0.878437
$693$	0	0
$694$	−4.08618	−0.155109
$695$	11.6960	0.443654
$696$	0	0
$697$	29.2579	1.10822
$698$	−0.916503	−0.0346902
$699$	0	0
$700$	57.5529	2.17529
$701$	37.8001	1.42769	0.713845	−	0.700304i	$-0.246952\pi$
$701$	37.8001	1.42769	0.713845	+	0.700304i	$0.246952\pi$
$702$	0	0
$703$	8.34066	0.314574
$704$	−14.2078	−0.535478
$705$	0	0
$706$	0.975772	0.0367237
$707$	9.21211	0.346457
$708$	0	0
$709$	−19.2053	−0.721268	−0.360634	−	0.932707i	$-0.617440\pi$
$709$	−19.2053	−0.721268	−0.360634	+	0.932707i	$0.617440\pi$
$710$	14.6015	0.547986
$711$	0	0
$712$	−6.58675	−0.246849
$713$	17.5950	0.658939
$714$	0	0
$715$	76.9140	2.87642
$716$	20.4672	0.764894
$717$	0	0
$718$	−18.1389	−0.676939
$719$	−0.799565	−0.0298187	−0.0149094	−	0.999889i	$-0.504746\pi$
$719$	−0.799565	−0.0298187	−0.0149094	+	0.999889i	$0.504746\pi$
$720$	0	0
$721$	29.2933	1.09094
$722$	0.494084	0.0183879
$723$	0	0
$724$	−13.9288	−0.517658
$725$	−17.5200	−0.650678
$726$	0	0
$727$	−19.6334	−0.728161	−0.364081	−	0.931367i	$-0.618617\pi$
$727$	−19.6334	−0.728161	−0.364081	+	0.931367i	$0.618617\pi$
$728$	24.3952	0.904146
$729$	0	0
$730$	−20.5151	−0.759299
$731$	13.7479	0.508483
$732$	0	0
$733$	−5.81006	−0.214600	−0.107300	−	0.994227i	$-0.534220\pi$
$733$	−5.81006	−0.214600	−0.107300	+	0.994227i	$0.534220\pi$
$734$	3.52423	0.130082
$735$	0	0
$736$	16.2351	0.598433
$737$	−16.4763	−0.606912
$738$	0	0
$739$	−26.1259	−0.961058	−0.480529	−	0.876979i	$-0.659555\pi$
$739$	−26.1259	−0.961058	−0.480529	+	0.876979i	$0.659555\pi$
$740$	55.9842	2.05802
$741$	0	0
$742$	9.37306	0.344096
$743$	21.3873	0.784624	0.392312	−	0.919832i	$-0.371675\pi$
$743$	21.3873	0.784624	0.392312	+	0.919832i	$0.371675\pi$
$744$	0	0
$745$	−12.6241	−0.462512
$746$	−11.8107	−0.432422
$747$	0	0
$748$	−32.3131	−1.18149
$749$	−35.7509	−1.30631
$750$	0	0
$751$	19.5462	0.713249	0.356625	−	0.934248i	$-0.383928\pi$
$751$	19.5462	0.713249	0.356625	+	0.934248i	$0.383928\pi$
$752$	−2.59488	−0.0946255
$753$	0	0
$754$	−3.47180	−0.126436
$755$	44.8225	1.63126
$756$	0	0
$757$	−29.2634	−1.06359	−0.531797	−	0.846872i	$-0.678483\pi$
$757$	−29.2634	−1.06359	−0.531797	+	0.846872i	$0.678483\pi$
$758$	−8.77953	−0.318887
$759$	0	0
$760$	7.09387	0.257322
$761$	14.4051	0.522185	0.261092	−	0.965314i	$-0.415917\pi$
$761$	14.4051	0.522185	0.261092	+	0.965314i	$0.415917\pi$
$762$	0	0
$763$	54.8697	1.98642
$764$	−19.2151	−0.695178
$765$	0	0
$766$	7.95015	0.287251
$767$	−30.0537	−1.08518
$768$	0	0
$769$	19.2702	0.694902	0.347451	−	0.937698i	$-0.387047\pi$
$769$	19.2702	0.694902	0.347451	+	0.937698i	$0.387047\pi$
$770$	−33.6077	−1.21114
$771$	0	0
$772$	−32.9596	−1.18624
$773$	47.3598	1.70341	0.851706	−	0.524020i	$-0.175568\pi$
$773$	47.3598	1.70341	0.851706	+	0.524020i	$0.175568\pi$
$774$	0	0
$775$	52.0240	1.86876
$776$	−23.0224	−0.826456
$777$	0	0
$778$	−0.384331	−0.0137789
$779$	−8.29688	−0.297267
$780$	0	0
$781$	40.3444	1.44364
$782$	5.66469	0.202569
$783$	0	0
$784$	12.0035	0.428695
$785$	86.7058	3.09466
$786$	0	0
$787$	46.8581	1.67031	0.835155	−	0.550015i	$-0.185378\pi$
$787$	46.8581	1.67031	0.835155	+	0.550015i	$0.185378\pi$
$788$	−23.2533	−0.828364
$789$	0	0
$790$	32.0535	1.14041
$791$	36.6328	1.30251
$792$	0	0
$793$	−50.5932	−1.79662
$794$	18.7977	0.667104
$795$	0	0
$796$	−7.71641	−0.273501
$797$	−24.7561	−0.876906	−0.438453	−	0.898754i	$-0.644474\pi$
$797$	−24.7561	−0.876906	−0.438453	+	0.898754i	$0.644474\pi$
$798$	0	0
$799$	−3.52637	−0.124754
$800$	48.0030	1.69716
$801$	0	0
$802$	−10.8959	−0.384746
$803$	−56.6838	−2.00033
$804$	0	0
$805$	−42.3769	−1.49359
$806$	10.3092	0.363125
$807$	0	0
$808$	5.01372	0.176382
$809$	−56.2509	−1.97768	−0.988838	−	0.148993i	$-0.952397\pi$
$809$	−56.2509	−1.97768	−0.988838	+	0.148993i	$0.952397\pi$
$810$	0	0
$811$	−14.0546	−0.493523	−0.246761	−	0.969076i	$-0.579366\pi$
$811$	−14.0546	−0.493523	−0.246761	+	0.969076i	$0.579366\pi$
$812$	−10.9114	−0.382916
$813$	0	0
$814$	−21.5059	−0.753781
$815$	−8.11788	−0.284357
$816$	0	0
$817$	−3.89858	−0.136394
$818$	−14.4228	−0.504281
$819$	0	0
$820$	−55.6904	−1.94479
$821$	6.86603	0.239626	0.119813	−	0.992796i	$-0.461770\pi$
$821$	6.86603	0.239626	0.119813	+	0.992796i	$0.461770\pi$
$822$	0	0
$823$	−0.318229	−0.0110928	−0.00554639	−	0.999985i	$-0.501765\pi$
$823$	−0.318229	−0.0110928	−0.00554639	+	0.999985i	$0.501765\pi$
$824$	15.9430	0.555399
$825$	0	0
$826$	13.1320	0.456921
$827$	−33.6396	−1.16976	−0.584882	−	0.811118i	$-0.698859\pi$
$827$	−33.6396	−1.16976	−0.584882	+	0.811118i	$0.698859\pi$
$828$	0	0
$829$	−56.8908	−1.97590	−0.987949	−	0.154777i	$-0.950534\pi$
$829$	−56.8908	−1.97590	−0.987949	+	0.154777i	$0.950534\pi$
$830$	20.0412	0.695640
$831$	0	0
$832$	−10.4967	−0.363907
$833$	16.3124	0.565191
$834$	0	0
$835$	43.3027	1.49855
$836$	9.16328	0.316919
$837$	0	0
$838$	8.28974	0.286364
$839$	37.4682	1.29354	0.646772	−	0.762683i	$-0.276118\pi$
$839$	37.4682	1.29354	0.646772	+	0.762683i	$0.276118\pi$
$840$	0	0
$841$	−25.6784	−0.885461
$842$	11.0070	0.379326
$843$	0	0
$844$	−22.4439	−0.772550
$845$	7.12852	0.245229
$846$	0	0
$847$	−55.3526	−1.90194
$848$	−14.4373	−0.495778
$849$	0	0
$850$	16.7490	0.574487
$851$	−27.1174	−0.929571
$852$	0	0
$853$	21.9922	0.752997	0.376499	−	0.926417i	$-0.377128\pi$
$853$	21.9922	0.752997	0.376499	+	0.926417i	$0.377128\pi$
$854$	22.1068	0.756479
$855$	0	0
$856$	−19.4575	−0.665045
$857$	30.1374	1.02947	0.514737	−	0.857348i	$-0.327889\pi$
$857$	30.1374	1.02947	0.514737	+	0.857348i	$0.327889\pi$
$858$	0	0
$859$	36.1572	1.23367	0.616833	−	0.787094i	$-0.288415\pi$
$859$	36.1572	1.23367	0.616833	+	0.787094i	$0.288415\pi$
$860$	−26.1681	−0.892324
$861$	0	0
$862$	18.4849	0.629599
$863$	24.9411	0.849005	0.424503	−	0.905427i	$-0.360449\pi$
$863$	24.9411	0.849005	0.424503	+	0.905427i	$0.360449\pi$
$864$	0	0
$865$	50.3082	1.71053
$866$	8.71786	0.296245
$867$	0	0
$868$	32.4004	1.09974
$869$	88.5645	3.00434
$870$	0	0
$871$	−12.1726	−0.412453
$872$	29.8630	1.01129
$873$	0	0
$874$	−1.60638	−0.0543366
$875$	−60.1268	−2.03266
$876$	0	0
$877$	−16.0964	−0.543536	−0.271768	−	0.962363i	$-0.587608\pi$
$877$	−16.0964	−0.543536	−0.271768	+	0.962363i	$0.587608\pi$
$878$	2.40858	0.0812855
$879$	0	0
$880$	51.7658	1.74502
$881$	50.8865	1.71441	0.857206	−	0.514974i	$-0.172199\pi$
$881$	50.8865	1.71441	0.857206	+	0.514974i	$0.172199\pi$
$882$	0	0
$883$	37.1294	1.24950	0.624751	−	0.780824i	$-0.285200\pi$
$883$	37.1294	1.24950	0.624751	+	0.780824i	$0.285200\pi$
$884$	−23.8728	−0.802928
$885$	0	0
$886$	−18.8380	−0.632875
$887$	−4.38473	−0.147225	−0.0736124	−	0.997287i	$-0.523453\pi$
$887$	−4.38473	−0.147225	−0.0736124	+	0.997287i	$0.523453\pi$
$888$	0	0
$889$	35.9347	1.20521
$890$	6.70393	0.224716
$891$	0	0
$892$	−13.5422	−0.453428
$893$	1.00000	0.0334637
$894$	0	0
$895$	−44.5588	−1.48944
$896$	38.6391	1.29084
$897$	0	0
$898$	−7.04657	−0.235147
$899$	−9.86321	−0.328956
$900$	0	0
$901$	−19.6199	−0.653634
$902$	21.3930	0.712309
$903$	0	0
$904$	19.9375	0.663112
$905$	30.3241	1.00801
$906$	0	0
$907$	−34.3098	−1.13924	−0.569619	−	0.821909i	$-0.692909\pi$
$907$	−34.3098	−1.13924	−0.569619	+	0.821909i	$0.692909\pi$
$908$	8.64372	0.286852
$909$	0	0
$910$	−24.8292	−0.823080
$911$	−33.0856	−1.09618	−0.548088	−	0.836421i	$-0.684644\pi$
$911$	−33.0856	−1.09618	−0.548088	+	0.836421i	$0.684644\pi$
$912$	0	0
$913$	55.3743	1.83262
$914$	9.81397	0.324617
$915$	0	0
$916$	48.3202	1.59654
$917$	12.8765	0.425221
$918$	0	0
$919$	11.9301	0.393539	0.196770	−	0.980450i	$-0.436955\pi$
$919$	11.9301	0.393539	0.196770	+	0.980450i	$0.436955\pi$
$920$	−23.0638	−0.760389
$921$	0	0
$922$	−15.5808	−0.513126
$923$	29.8062	0.981083
$924$	0	0
$925$	−80.1790	−2.63627
$926$	−7.44522	−0.244665
$927$	0	0
$928$	−9.10087	−0.298751
$929$	−0.149981	−0.00492073	−0.00246037	−	0.999997i	$-0.500783\pi$
$929$	−0.149981	−0.00492073	−0.00246037	+	0.999997i	$0.500783\pi$
$930$	0	0
$931$	−4.62583	−0.151605
$932$	−26.0935	−0.854720
$933$	0	0
$934$	−11.2307	−0.367480
$935$	70.3484	2.30064
$936$	0	0
$937$	−11.8203	−0.386154	−0.193077	−	0.981184i	$-0.561847\pi$
$937$	−11.8203	−0.386154	−0.193077	+	0.981184i	$0.561847\pi$
$938$	5.31884	0.173666
$939$	0	0
$940$	6.71220	0.218928
$941$	29.9450	0.976178	0.488089	−	0.872794i	$-0.337694\pi$
$941$	29.9450	0.976178	0.488089	+	0.872794i	$0.337694\pi$
$942$	0	0
$943$	26.9750	0.878428
$944$	−20.2272	−0.658338
$945$	0	0
$946$	10.0523	0.326827
$947$	59.2191	1.92436	0.962182	−	0.272409i	$-0.0878202\pi$
$947$	59.2191	1.92436	0.962182	+	0.272409i	$0.0878202\pi$
$948$	0	0
$949$	−41.8776	−1.35941
$950$	−4.74965	−0.154099
$951$	0	0
$952$	22.3128	0.723161
$953$	−24.4368	−0.791587	−0.395794	−	0.918339i	$-0.629530\pi$
$953$	−24.4368	−0.791587	−0.395794	+	0.918339i	$0.629530\pi$
$954$	0	0
$955$	41.8329	1.35368
$956$	−6.95222	−0.224851
$957$	0	0
$958$	−10.6607	−0.344431
$959$	12.0359	0.388659
$960$	0	0
$961$	−1.71221	−0.0552324
$962$	−15.8884	−0.512264
$963$	0	0
$964$	42.1085	1.35622
$965$	71.7558	2.30990
$966$	0	0
$967$	16.7597	0.538956	0.269478	−	0.963007i	$-0.413149\pi$
$967$	16.7597	0.538956	0.269478	+	0.963007i	$0.413149\pi$
$968$	−30.1258	−0.968280
$969$	0	0
$970$	23.4320	0.752356
$971$	−35.2059	−1.12981	−0.564906	−	0.825155i	$-0.691088\pi$
$971$	−35.2059	−1.12981	−0.564906	+	0.825155i	$0.691088\pi$
$972$	0	0
$973$	−10.4323	−0.334443
$974$	0.618405	0.0198150
$975$	0	0
$976$	−34.0510	−1.08995
$977$	22.4826	0.719282	0.359641	−	0.933091i	$-0.382899\pi$
$977$	22.4826	0.719282	0.359641	+	0.933091i	$0.382899\pi$
$978$	0	0
$979$	18.5231	0.592001
$980$	−31.0495	−0.991839
$981$	0	0
$982$	10.7486	0.343001
$983$	−44.1166	−1.40710	−0.703550	−	0.710646i	$-0.748403\pi$
$983$	−44.1166	−1.40710	−0.703550	+	0.710646i	$0.748403\pi$
$984$	0	0
$985$	50.6243	1.61303
$986$	−3.17544	−0.101127
$987$	0	0
$988$	6.76978	0.215375
$989$	12.6752	0.403047
$990$	0	0
$991$	−12.6330	−0.401299	−0.200650	−	0.979663i	$-0.564305\pi$
$991$	−12.6330	−0.401299	−0.200650	+	0.979663i	$0.564305\pi$
$992$	27.0241	0.858016
$993$	0	0
$994$	−13.0239	−0.413092
$995$	16.7993	0.532573
$996$	0	0
$997$	54.6340	1.73028	0.865138	−	0.501534i	$-0.167231\pi$
$997$	54.6340	1.73028	0.865138	+	0.501534i	$0.167231\pi$
$998$	15.8530	0.501818
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.o.1.11		18
3.2	odd	2		893.2.a.c.1.8	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.c.1.8	✓	18	3.2	odd	2
8037.2.a.o.1.11		18	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.494084 q^{2} -1.75588 q^{4} +3.82270 q^{5} -3.40967 q^{7} -1.85572 q^{8} +O(q^{10})\)	\(q+0.494084 q^{2} -1.75588 q^{4} +3.82270 q^{5} -3.40967 q^{7} -1.85572 q^{8} +1.88874 q^{10} +5.21862 q^{11} +3.85549 q^{13} -1.68466 q^{14} +2.59488 q^{16} +3.52637 q^{17} -1.00000 q^{19} -6.71220 q^{20} +2.57844 q^{22} +3.25122 q^{23} +9.61303 q^{25} +1.90494 q^{26} +5.98697 q^{28} -1.82253 q^{29} +5.41182 q^{31} +4.99353 q^{32} +1.74233 q^{34} -13.0341 q^{35} -8.34066 q^{37} -0.494084 q^{38} -7.09387 q^{40} +8.29688 q^{41} +3.89858 q^{43} -9.16328 q^{44} +1.60638 q^{46} -1.00000 q^{47} +4.62583 q^{49} +4.74965 q^{50} -6.76978 q^{52} -5.56376 q^{53} +19.9492 q^{55} +6.32739 q^{56} -0.900484 q^{58} -7.79504 q^{59} -13.1224 q^{61} +2.67390 q^{62} -2.72253 q^{64} +14.7384 q^{65} -3.15721 q^{67} -6.19189 q^{68} -6.43996 q^{70} +7.73085 q^{71} -10.8618 q^{73} -4.12099 q^{74} +1.75588 q^{76} -17.7938 q^{77} +16.9709 q^{79} +9.91944 q^{80} +4.09936 q^{82} +10.6109 q^{83} +13.4803 q^{85} +1.92623 q^{86} -9.68431 q^{88} +3.54943 q^{89} -13.1459 q^{91} -5.70876 q^{92} -0.494084 q^{94} -3.82270 q^{95} +12.4062 q^{97} +2.28555 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more