LMFDB - Embedded newform 8001.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	7.7.118870813.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 9x^{5} - 3x^{4} + 20x^{3} + 7x^{2} - 13x - 4 $ x^7 - 9x^5 - 3x^4 + 20x^3 + 7x^2 - 13*x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.301070$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.09968 q^{2} +2.40865 q^{4} +3.29054 q^{5} +1.00000 q^{7} -0.858029 q^{8} +O(q^{10})$	\(q-2.09968 q^{2} +2.40865 q^{4} +3.29054 q^{5} +1.00000 q^{7} -0.858029 q^{8} -6.90907 q^{10} +3.62521 q^{11} -2.16007 q^{13} -2.09968 q^{14} -3.01571 q^{16} -2.19233 q^{17} -6.66601 q^{19} +7.92575 q^{20} -7.61177 q^{22} -2.36522 q^{23} +5.82765 q^{25} +4.53544 q^{26} +2.40865 q^{28} -2.06180 q^{29} -2.19207 q^{31} +8.04808 q^{32} +4.60319 q^{34} +3.29054 q^{35} -10.1454 q^{37} +13.9965 q^{38} -2.82338 q^{40} +0.647495 q^{41} +3.67859 q^{43} +8.73185 q^{44} +4.96620 q^{46} +8.59064 q^{47} +1.00000 q^{49} -12.2362 q^{50} -5.20284 q^{52} -2.84924 q^{53} +11.9289 q^{55} -0.858029 q^{56} +4.32912 q^{58} +0.592789 q^{59} -13.5269 q^{61} +4.60265 q^{62} -10.8670 q^{64} -7.10778 q^{65} +0.664083 q^{67} -5.28055 q^{68} -6.90907 q^{70} +5.65049 q^{71} -3.24624 q^{73} +21.3021 q^{74} -16.0561 q^{76} +3.62521 q^{77} +4.01101 q^{79} -9.92332 q^{80} -1.35953 q^{82} +3.40033 q^{83} -7.21395 q^{85} -7.72385 q^{86} -3.11054 q^{88} -6.31252 q^{89} -2.16007 q^{91} -5.69698 q^{92} -18.0376 q^{94} -21.9348 q^{95} -0.0689323 q^{97} -2.09968 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * q^97 + 2 * 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 $

$2$	−2.09968	−1.48470	−0.742348	−	0.670014i	$-0.766288\pi$
$2$	−2.09968	−1.48470	−0.742348	+	0.670014i	$0.766288\pi$
$3$	0	0
$3$	0	0
$4$	2.40865	1.20432
$5$	3.29054	1.47157	0.735787	−	0.677213i	$-0.236812\pi$
$5$	3.29054	1.47157	0.735787	+	0.677213i	$0.236812\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.858029	−0.303359
$9$	0	0
$10$	−6.90907	−2.18484
$11$	3.62521	1.09304	0.546521	−	0.837445i	$-0.315952\pi$
$11$	3.62521	1.09304	0.546521	+	0.837445i	$0.315952\pi$
$12$	0	0
$13$	−2.16007	−0.599095	−0.299547	−	0.954081i	$-0.596836\pi$
$13$	−2.16007	−0.599095	−0.299547	+	0.954081i	$0.596836\pi$
$14$	−2.09968	−0.561163
$15$	0	0
$16$	−3.01571	−0.753928
$17$	−2.19233	−0.531718	−0.265859	−	0.964012i	$-0.585656\pi$
$17$	−2.19233	−0.531718	−0.265859	+	0.964012i	$0.585656\pi$
$18$	0	0
$19$	−6.66601	−1.52929	−0.764643	−	0.644454i	$-0.777085\pi$
$19$	−6.66601	−1.52929	−0.764643	+	0.644454i	$0.777085\pi$
$20$	7.92575	1.77225
$21$	0	0
$22$	−7.61177	−1.62284
$23$	−2.36522	−0.493183	−0.246591	−	0.969120i	$-0.579311\pi$
$23$	−2.36522	−0.493183	−0.246591	+	0.969120i	$0.579311\pi$
$24$	0	0
$25$	5.82765	1.16553
$26$	4.53544	0.889474
$27$	0	0
$28$	2.40865	0.455192
$29$	−2.06180	−0.382867	−0.191433	−	0.981506i	$-0.561314\pi$
$29$	−2.06180	−0.382867	−0.191433	+	0.981506i	$0.561314\pi$
$30$	0	0
$31$	−2.19207	−0.393708	−0.196854	−	0.980433i	$-0.563072\pi$
$31$	−2.19207	−0.393708	−0.196854	+	0.980433i	$0.563072\pi$
$32$	8.04808	1.42271
$33$	0	0
$34$	4.60319	0.789440
$35$	3.29054	0.556203
$36$	0	0
$37$	−10.1454	−1.66789	−0.833947	−	0.551845i	$-0.813924\pi$
$37$	−10.1454	−1.66789	−0.833947	+	0.551845i	$0.813924\pi$
$38$	13.9965	2.27053
$39$	0	0
$40$	−2.82338	−0.446416
$41$	0.647495	0.101122	0.0505608	−	0.998721i	$-0.483899\pi$
$41$	0.647495	0.101122	0.0505608	+	0.998721i	$0.483899\pi$
$42$	0	0
$43$	3.67859	0.560979	0.280490	−	0.959857i	$-0.409503\pi$
$43$	3.67859	0.560979	0.280490	+	0.959857i	$0.409503\pi$
$44$	8.73185	1.31638
$45$	0	0
$46$	4.96620	0.732227
$47$	8.59064	1.25307	0.626537	−	0.779392i	$-0.284472\pi$
$47$	8.59064	1.25307	0.626537	+	0.779392i	$0.284472\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−12.2362	−1.73046
$51$	0	0
$52$	−5.20284	−0.721504
$53$	−2.84924	−0.391373	−0.195686	−	0.980667i	$-0.562693\pi$
$53$	−2.84924	−0.391373	−0.195686	+	0.980667i	$0.562693\pi$
$54$	0	0
$55$	11.9289	1.60849
$56$	−0.858029	−0.114659
$57$	0	0
$58$	4.32912	0.568441
$59$	0.592789	0.0771745	0.0385873	−	0.999255i	$-0.487714\pi$
$59$	0.592789	0.0771745	0.0385873	+	0.999255i	$0.487714\pi$
$60$	0	0
$61$	−13.5269	−1.73195	−0.865973	−	0.500091i	$-0.833300\pi$
$61$	−13.5269	−1.73195	−0.865973	+	0.500091i	$0.833300\pi$
$62$	4.60265	0.584537
$63$	0	0
$64$	−10.8670	−1.35837
$65$	−7.10778	−0.881612
$66$	0	0
$67$	0.664083	0.0811306	0.0405653	−	0.999177i	$-0.487084\pi$
$67$	0.664083	0.0811306	0.0405653	+	0.999177i	$0.487084\pi$
$68$	−5.28055	−0.640361
$69$	0	0
$70$	−6.90907	−0.825792
$71$	5.65049	0.670589	0.335295	−	0.942113i	$-0.391164\pi$
$71$	5.65049	0.670589	0.335295	+	0.942113i	$0.391164\pi$
$72$	0	0
$73$	−3.24624	−0.379943	−0.189972	−	0.981790i	$-0.560840\pi$
$73$	−3.24624	−0.379943	−0.189972	+	0.981790i	$0.560840\pi$
$74$	21.3021	2.47632
$75$	0	0
$76$	−16.0561	−1.84176
$77$	3.62521	0.413131
$78$	0	0
$79$	4.01101	0.451274	0.225637	−	0.974211i	$-0.427554\pi$
$79$	4.01101	0.451274	0.225637	+	0.974211i	$0.427554\pi$
$80$	−9.92332	−1.10946
$81$	0	0
$82$	−1.35953	−0.150135
$83$	3.40033	0.373235	0.186617	−	0.982433i	$-0.440248\pi$
$83$	3.40033	0.373235	0.186617	+	0.982433i	$0.440248\pi$
$84$	0	0
$85$	−7.21395	−0.782463
$86$	−7.72385	−0.832884
$87$	0	0
$88$	−3.11054	−0.331584
$89$	−6.31252	−0.669126	−0.334563	−	0.942373i	$-0.608589\pi$
$89$	−6.31252	−0.669126	−0.334563	+	0.942373i	$0.608589\pi$
$90$	0	0
$91$	−2.16007	−0.226436
$92$	−5.69698	−0.593952
$93$	0	0
$94$	−18.0376	−1.86043
$95$	−21.9348	−2.25046
$96$	0	0
$97$	−0.0689323	−0.00699902	−0.00349951	−	0.999994i	$-0.501114\pi$
$97$	−0.0689323	−0.00699902	−0.00349951	+	0.999994i	$0.501114\pi$
$98$	−2.09968	−0.212100
$99$	0	0
$100$	14.0368	1.40368
$101$	−6.53614	−0.650370	−0.325185	−	0.945650i	$-0.605427\pi$
$101$	−6.53614	−0.650370	−0.325185	+	0.945650i	$0.605427\pi$
$102$	0	0
$103$	1.42084	0.140000	0.0700000	−	0.997547i	$-0.477700\pi$
$103$	1.42084	0.140000	0.0700000	+	0.997547i	$0.477700\pi$
$104$	1.85340	0.181741
$105$	0	0
$106$	5.98248	0.581070
$107$	−17.1418	−1.65716	−0.828581	−	0.559869i	$-0.810851\pi$
$107$	−17.1418	−1.65716	−0.828581	+	0.559869i	$0.810851\pi$
$108$	0	0
$109$	14.3137	1.37100	0.685500	−	0.728073i	$-0.259584\pi$
$109$	14.3137	1.37100	0.685500	+	0.728073i	$0.259584\pi$
$110$	−25.0468	−2.38812
$111$	0	0
$112$	−3.01571	−0.284958
$113$	−9.27873	−0.872870	−0.436435	−	0.899736i	$-0.643759\pi$
$113$	−9.27873	−0.872870	−0.436435	+	0.899736i	$0.643759\pi$
$114$	0	0
$115$	−7.78285	−0.725755
$116$	−4.96615	−0.461095
$117$	0	0
$118$	−1.24467	−0.114581
$119$	−2.19233	−0.200971
$120$	0	0
$121$	2.14215	0.194741
$122$	28.4022	2.57141
$123$	0	0
$124$	−5.27993	−0.474152
$125$	2.72342	0.243590
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	6.72095	0.594054
$129$	0	0
$130$	14.9241	1.30893
$131$	−17.8118	−1.55623	−0.778113	−	0.628124i	$-0.783823\pi$
$131$	−17.8118	−1.55623	−0.778113	+	0.628124i	$0.783823\pi$
$132$	0	0
$133$	−6.66601	−0.578016
$134$	−1.39436	−0.120454
$135$	0	0
$136$	1.88108	0.161302
$137$	−7.51288	−0.641869	−0.320934	−	0.947101i	$-0.603997\pi$
$137$	−7.51288	−0.641869	−0.320934	+	0.947101i	$0.603997\pi$
$138$	0	0
$139$	−4.67226	−0.396296	−0.198148	−	0.980172i	$-0.563493\pi$
$139$	−4.67226	−0.396296	−0.198148	+	0.980172i	$0.563493\pi$
$140$	7.92575	0.669848
$141$	0	0
$142$	−11.8642	−0.995622
$143$	−7.83069	−0.654836
$144$	0	0
$145$	−6.78443	−0.563417
$146$	6.81605	0.564101
$147$	0	0
$148$	−24.4367	−2.00868
$149$	12.9632	1.06199	0.530994	−	0.847375i	$-0.321819\pi$
$149$	12.9632	1.06199	0.530994	+	0.847375i	$0.321819\pi$
$150$	0	0
$151$	10.9338	0.889780	0.444890	−	0.895585i	$-0.353243\pi$
$151$	10.9338	0.889780	0.444890	+	0.895585i	$0.353243\pi$
$152$	5.71963	0.463923
$153$	0	0
$154$	−7.61177	−0.613374
$155$	−7.21310	−0.579370
$156$	0	0
$157$	−20.5157	−1.63733	−0.818665	−	0.574271i	$-0.805285\pi$
$157$	−20.5157	−1.63733	−0.818665	+	0.574271i	$0.805285\pi$
$158$	−8.42183	−0.670005
$159$	0	0
$160$	26.4825	2.09363
$161$	−2.36522	−0.186406
$162$	0	0
$163$	1.33340	0.104440	0.0522199	−	0.998636i	$-0.483370\pi$
$163$	1.33340	0.104440	0.0522199	+	0.998636i	$0.483370\pi$
$164$	1.55959	0.121783
$165$	0	0
$166$	−7.13960	−0.554140
$167$	−0.822111	−0.0636169	−0.0318084	−	0.999494i	$-0.510127\pi$
$167$	−0.822111	−0.0636169	−0.0318084	+	0.999494i	$0.510127\pi$
$168$	0	0
$169$	−8.33411	−0.641086
$170$	15.1470	1.16172
$171$	0	0
$172$	8.86042	0.675601
$173$	22.8603	1.73804	0.869020	−	0.494777i	$-0.164750\pi$
$173$	22.8603	1.73804	0.869020	+	0.494777i	$0.164750\pi$
$174$	0	0
$175$	5.82765	0.440529
$176$	−10.9326	−0.824075
$177$	0	0
$178$	13.2543	0.993449
$179$	−1.52134	−0.113710	−0.0568552	−	0.998382i	$-0.518107\pi$
$179$	−1.52134	−0.113710	−0.0568552	+	0.998382i	$0.518107\pi$
$180$	0	0
$181$	14.4514	1.07417	0.537084	−	0.843529i	$-0.319526\pi$
$181$	14.4514	1.07417	0.537084	+	0.843529i	$0.319526\pi$
$182$	4.53544	0.336189
$183$	0	0
$184$	2.02943	0.149611
$185$	−33.3838	−2.45443
$186$	0	0
$187$	−7.94766	−0.581190
$188$	20.6918	1.50911
$189$	0	0
$190$	46.0559	3.34125
$191$	−6.46436	−0.467744	−0.233872	−	0.972267i	$-0.575140\pi$
$191$	−6.46436	−0.467744	−0.233872	+	0.972267i	$0.575140\pi$
$192$	0	0
$193$	−21.4158	−1.54154	−0.770771	−	0.637112i	$-0.780129\pi$
$193$	−21.4158	−1.54154	−0.770771	+	0.637112i	$0.780129\pi$
$194$	0.144736	0.0103914
$195$	0	0
$196$	2.40865	0.172046
$197$	−15.2826	−1.08884	−0.544419	−	0.838814i	$-0.683250\pi$
$197$	−15.2826	−1.08884	−0.544419	+	0.838814i	$0.683250\pi$
$198$	0	0
$199$	−9.68248	−0.686373	−0.343186	−	0.939267i	$-0.611506\pi$
$199$	−9.68248	−0.686373	−0.343186	+	0.939267i	$0.611506\pi$
$200$	−5.00030	−0.353574
$201$	0	0
$202$	13.7238	0.965603
$203$	−2.06180	−0.144710
$204$	0	0
$205$	2.13061	0.148808
$206$	−2.98332	−0.207857
$207$	0	0
$208$	6.51414	0.451674
$209$	−24.1657	−1.67157
$210$	0	0
$211$	−22.7122	−1.56357	−0.781785	−	0.623549i	$-0.785690\pi$
$211$	−22.7122	−1.56357	−0.781785	+	0.623549i	$0.785690\pi$
$212$	−6.86281	−0.471340
$213$	0	0
$214$	35.9923	2.46038
$215$	12.1045	0.825523
$216$	0	0
$217$	−2.19207	−0.148808
$218$	−30.0541	−2.03552
$219$	0	0
$220$	28.7325	1.93715
$221$	4.73558	0.318550
$222$	0	0
$223$	6.33989	0.424551	0.212275	−	0.977210i	$-0.431913\pi$
$223$	6.33989	0.424551	0.212275	+	0.977210i	$0.431913\pi$
$224$	8.04808	0.537735
$225$	0	0
$226$	19.4824	1.29595
$227$	15.7349	1.04436	0.522181	−	0.852835i	$-0.325119\pi$
$227$	15.7349	1.04436	0.522181	+	0.852835i	$0.325119\pi$
$228$	0	0
$229$	−0.365275	−0.0241380	−0.0120690	−	0.999927i	$-0.503842\pi$
$229$	−0.365275	−0.0241380	−0.0120690	+	0.999927i	$0.503842\pi$
$230$	16.3415	1.07753
$231$	0	0
$232$	1.76908	0.116146
$233$	4.21293	0.275998	0.137999	−	0.990432i	$-0.455933\pi$
$233$	4.21293	0.275998	0.137999	+	0.990432i	$0.455933\pi$
$234$	0	0
$235$	28.2678	1.84399
$236$	1.42782	0.0929431
$237$	0	0
$238$	4.60319	0.298380
$239$	3.38821	0.219165	0.109582	−	0.993978i	$-0.465049\pi$
$239$	3.38821	0.219165	0.109582	+	0.993978i	$0.465049\pi$
$240$	0	0
$241$	−10.0609	−0.648077	−0.324039	−	0.946044i	$-0.605041\pi$
$241$	−10.0609	−0.648077	−0.324039	+	0.946044i	$0.605041\pi$
$242$	−4.49782	−0.289131
$243$	0	0
$244$	−32.5816	−2.08582
$245$	3.29054	0.210225
$246$	0	0
$247$	14.3990	0.916188
$248$	1.88086	0.119435
$249$	0	0
$250$	−5.71830	−0.361657
$251$	−1.13195	−0.0714479	−0.0357240	−	0.999362i	$-0.511374\pi$
$251$	−1.13195	−0.0714479	−0.0357240	+	0.999362i	$0.511374\pi$
$252$	0	0
$253$	−8.57442	−0.539069
$254$	2.09968	0.131746
$255$	0	0
$256$	7.62208	0.476380
$257$	29.2462	1.82433	0.912165	−	0.409824i	$-0.134410\pi$
$257$	29.2462	1.82433	0.912165	+	0.409824i	$0.134410\pi$
$258$	0	0
$259$	−10.1454	−0.630404
$260$	−17.1202	−1.06175
$261$	0	0
$262$	37.3991	2.31052
$263$	−26.3656	−1.62577	−0.812887	−	0.582422i	$-0.802105\pi$
$263$	−26.3656	−1.62577	−0.812887	+	0.582422i	$0.802105\pi$
$264$	0	0
$265$	−9.37552	−0.575934
$266$	13.9965	0.858179
$267$	0	0
$268$	1.59954	0.0977076
$269$	22.2274	1.35523	0.677613	−	0.735419i	$-0.263014\pi$
$269$	22.2274	1.35523	0.677613	+	0.735419i	$0.263014\pi$
$270$	0	0
$271$	−26.8490	−1.63096	−0.815482	−	0.578783i	$-0.803528\pi$
$271$	−26.8490	−1.63096	−0.815482	+	0.578783i	$0.803528\pi$
$272$	6.61144	0.400877
$273$	0	0
$274$	15.7746	0.952981
$275$	21.1265	1.27397
$276$	0	0
$277$	2.10995	0.126775	0.0633873	−	0.997989i	$-0.479810\pi$
$277$	2.10995	0.126775	0.0633873	+	0.997989i	$0.479810\pi$
$278$	9.81024	0.588379
$279$	0	0
$280$	−2.82338	−0.168729
$281$	−19.6051	−1.16954	−0.584772	−	0.811198i	$-0.698816\pi$
$281$	−19.6051	−1.16954	−0.584772	+	0.811198i	$0.698816\pi$
$282$	0	0
$283$	−5.28771	−0.314322	−0.157161	−	0.987573i	$-0.550234\pi$
$283$	−5.28771	−0.314322	−0.157161	+	0.987573i	$0.550234\pi$
$284$	13.6100	0.807607
$285$	0	0
$286$	16.4419	0.972232
$287$	0.647495	0.0382204
$288$	0	0
$289$	−12.1937	−0.717276
$290$	14.2451	0.836503
$291$	0	0
$292$	−7.81904	−0.457575
$293$	−9.05623	−0.529071	−0.264535	−	0.964376i	$-0.585219\pi$
$293$	−9.05623	−0.529071	−0.264535	+	0.964376i	$0.585219\pi$
$294$	0	0
$295$	1.95059	0.113568
$296$	8.70505	0.505971
$297$	0	0
$298$	−27.2186	−1.57673
$299$	5.10903	0.295463
$300$	0	0
$301$	3.67859	0.212030
$302$	−22.9575	−1.32105
$303$	0	0
$304$	20.1027	1.15297
$305$	−44.5109	−2.54869
$306$	0	0
$307$	23.4922	1.34077	0.670386	−	0.742012i	$-0.266128\pi$
$307$	23.4922	1.34077	0.670386	+	0.742012i	$0.266128\pi$
$308$	8.73185	0.497544
$309$	0	0
$310$	15.1452	0.860189
$311$	0.106610	0.00604531	0.00302265	−	0.999995i	$-0.499038\pi$
$311$	0.106610	0.00604531	0.00302265	+	0.999995i	$0.499038\pi$
$312$	0	0
$313$	23.6312	1.33572	0.667858	−	0.744289i	$-0.267211\pi$
$313$	23.6312	1.33572	0.667858	+	0.744289i	$0.267211\pi$
$314$	43.0763	2.43094
$315$	0	0
$316$	9.66112	0.543480
$317$	−8.87083	−0.498236	−0.249118	−	0.968473i	$-0.580141\pi$
$317$	−8.87083	−0.498236	−0.249118	+	0.968473i	$0.580141\pi$
$318$	0	0
$319$	−7.47446	−0.418489
$320$	−35.7582	−1.99894
$321$	0	0
$322$	4.96620	0.276756
$323$	14.6141	0.813150
$324$	0	0
$325$	−12.5881	−0.698263
$326$	−2.79971	−0.155061
$327$	0	0
$328$	−0.555570	−0.0306762
$329$	8.59064	0.473617
$330$	0	0
$331$	−27.3847	−1.50520	−0.752599	−	0.658480i	$-0.771200\pi$
$331$	−27.3847	−1.50520	−0.752599	+	0.658480i	$0.771200\pi$
$332$	8.19020	0.449496
$333$	0	0
$334$	1.72617	0.0944518
$335$	2.18519	0.119390
$336$	0	0
$337$	18.4898	1.00720	0.503601	−	0.863936i	$-0.332008\pi$
$337$	18.4898	1.00720	0.503601	+	0.863936i	$0.332008\pi$
$338$	17.4990	0.951818
$339$	0	0
$340$	−17.3759	−0.942339
$341$	−7.94673	−0.430339
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.15634	−0.170178
$345$	0	0
$346$	−47.9994	−2.58046
$347$	−24.2408	−1.30132	−0.650658	−	0.759371i	$-0.725507\pi$
$347$	−24.2408	−1.30132	−0.650658	+	0.759371i	$0.725507\pi$
$348$	0	0
$349$	−7.74084	−0.414358	−0.207179	−	0.978303i	$-0.566428\pi$
$349$	−7.74084	−0.414358	−0.207179	+	0.978303i	$0.566428\pi$
$350$	−12.2362	−0.654052
$351$	0	0
$352$	29.1760	1.55508
$353$	10.6201	0.565253	0.282626	−	0.959230i	$-0.408794\pi$
$353$	10.6201	0.565253	0.282626	+	0.959230i	$0.408794\pi$
$354$	0	0
$355$	18.5932	0.986822
$356$	−15.2046	−0.805845
$357$	0	0
$358$	3.19433	0.168825
$359$	−11.5939	−0.611900	−0.305950	−	0.952048i	$-0.598974\pi$
$359$	−11.5939	−0.611900	−0.305950	+	0.952048i	$0.598974\pi$
$360$	0	0
$361$	25.4357	1.33872
$362$	−30.3434	−1.59481
$363$	0	0
$364$	−5.20284	−0.272703
$365$	−10.6819	−0.559115
$366$	0	0
$367$	−19.5505	−1.02053	−0.510263	−	0.860018i	$-0.670452\pi$
$367$	−19.5505	−1.02053	−0.510263	+	0.860018i	$0.670452\pi$
$368$	7.13282	0.371824
$369$	0	0
$370$	70.0953	3.64408
$371$	−2.84924	−0.147925
$372$	0	0
$373$	3.95621	0.204845	0.102423	−	0.994741i	$-0.467341\pi$
$373$	3.95621	0.204845	0.102423	+	0.994741i	$0.467341\pi$
$374$	16.6875	0.862891
$375$	0	0
$376$	−7.37102	−0.380131
$377$	4.45362	0.229373
$378$	0	0
$379$	−26.2433	−1.34803	−0.674014	−	0.738718i	$-0.735431\pi$
$379$	−26.2433	−1.34803	−0.674014	+	0.738718i	$0.735431\pi$
$380$	−52.8331	−2.71028
$381$	0	0
$382$	13.5731	0.694459
$383$	−2.25979	−0.115470	−0.0577349	−	0.998332i	$-0.518388\pi$
$383$	−2.25979	−0.115470	−0.0577349	+	0.998332i	$0.518388\pi$
$384$	0	0
$385$	11.9289	0.607953
$386$	44.9663	2.28872
$387$	0	0
$388$	−0.166034	−0.00842909
$389$	4.79264	0.242997	0.121498	−	0.992592i	$-0.461230\pi$
$389$	4.79264	0.242997	0.121498	+	0.992592i	$0.461230\pi$
$390$	0	0
$391$	5.18535	0.262234
$392$	−0.858029	−0.0433370
$393$	0	0
$394$	32.0885	1.61659
$395$	13.1984	0.664083
$396$	0	0
$397$	34.5066	1.73184	0.865919	−	0.500185i	$-0.166735\pi$
$397$	34.5066	1.73184	0.865919	+	0.500185i	$0.166735\pi$
$398$	20.3301	1.01906
$399$	0	0
$400$	−17.5745	−0.878725
$401$	2.10653	0.105195	0.0525975	−	0.998616i	$-0.483250\pi$
$401$	2.10653	0.105195	0.0525975	+	0.998616i	$0.483250\pi$
$402$	0	0
$403$	4.73502	0.235868
$404$	−15.7433	−0.783257
$405$	0	0
$406$	4.32912	0.214850
$407$	−36.7792	−1.82308
$408$	0	0
$409$	36.8469	1.82196	0.910981	−	0.412449i	$-0.135327\pi$
$409$	36.8469	1.82196	0.910981	+	0.412449i	$0.135327\pi$
$410$	−4.47359	−0.220935
$411$	0	0
$412$	3.42231	0.168605
$413$	0.592789	0.0291692
$414$	0	0
$415$	11.1889	0.549243
$416$	−17.3844	−0.852340
$417$	0	0
$418$	50.7401	2.48178
$419$	23.4163	1.14396	0.571981	−	0.820267i	$-0.306175\pi$
$419$	23.4163	1.14396	0.571981	+	0.820267i	$0.306175\pi$
$420$	0	0
$421$	7.19583	0.350703	0.175352	−	0.984506i	$-0.443894\pi$
$421$	7.19583	0.350703	0.175352	+	0.984506i	$0.443894\pi$
$422$	47.6882	2.32143
$423$	0	0
$424$	2.44473	0.118727
$425$	−12.7761	−0.619734
$426$	0	0
$427$	−13.5269	−0.654614
$428$	−41.2886	−1.99576
$429$	0	0
$430$	−25.4156	−1.22565
$431$	18.8941	0.910098	0.455049	−	0.890466i	$-0.349622\pi$
$431$	18.8941	0.910098	0.455049	+	0.890466i	$0.349622\pi$
$432$	0	0
$433$	18.2660	0.877810	0.438905	−	0.898534i	$-0.355367\pi$
$433$	18.2660	0.877810	0.438905	+	0.898534i	$0.355367\pi$
$434$	4.60265	0.220934
$435$	0	0
$436$	34.4766	1.65113
$437$	15.7666	0.754218
$438$	0	0
$439$	−31.3238	−1.49501	−0.747503	−	0.664259i	$-0.768747\pi$
$439$	−31.3238	−1.49501	−0.747503	+	0.664259i	$0.768747\pi$
$440$	−10.2353	−0.487951
$441$	0	0
$442$	−9.94319	−0.472950
$443$	7.54570	0.358507	0.179254	−	0.983803i	$-0.442632\pi$
$443$	7.54570	0.358507	0.179254	+	0.983803i	$0.442632\pi$
$444$	0	0
$445$	−20.7716	−0.984669
$446$	−13.3117	−0.630329
$447$	0	0
$448$	−10.8670	−0.513415
$449$	24.4439	1.15358	0.576790	−	0.816892i	$-0.304305\pi$
$449$	24.4439	1.15358	0.576790	+	0.816892i	$0.304305\pi$
$450$	0	0
$451$	2.34730	0.110530
$452$	−22.3492	−1.05122
$453$	0	0
$454$	−33.0382	−1.55056
$455$	−7.10778	−0.333218
$456$	0	0
$457$	10.3236	0.482915	0.241458	−	0.970411i	$-0.422375\pi$
$457$	10.3236	0.482915	0.241458	+	0.970411i	$0.422375\pi$
$458$	0.766959	0.0358377
$459$	0	0
$460$	−18.7462	−0.874044
$461$	0.224984	0.0104785	0.00523927	−	0.999986i	$-0.498332\pi$
$461$	0.224984	0.0104785	0.00523927	+	0.999986i	$0.498332\pi$
$462$	0	0
$463$	−21.4269	−0.995795	−0.497897	−	0.867236i	$-0.665894\pi$
$463$	−21.4269	−0.995795	−0.497897	+	0.867236i	$0.665894\pi$
$464$	6.21779	0.288654
$465$	0	0
$466$	−8.84580	−0.409774
$467$	−14.8070	−0.685186	−0.342593	−	0.939484i	$-0.611305\pi$
$467$	−14.8070	−0.685186	−0.342593	+	0.939484i	$0.611305\pi$
$468$	0	0
$469$	0.664083	0.0306645
$470$	−59.3534	−2.73777
$471$	0	0
$472$	−0.508630	−0.0234116
$473$	13.3357	0.613174
$474$	0	0
$475$	−38.8472	−1.78243
$476$	−5.28055	−0.242034
$477$	0	0
$478$	−7.11414	−0.325393
$479$	10.2074	0.466390	0.233195	−	0.972430i	$-0.425082\pi$
$479$	10.2074	0.466390	0.233195	+	0.972430i	$0.425082\pi$
$480$	0	0
$481$	21.9147	0.999226
$482$	21.1246	0.962198
$483$	0	0
$484$	5.15968	0.234531
$485$	−0.226825	−0.0102996
$486$	0	0
$487$	16.1543	0.732023	0.366011	−	0.930610i	$-0.380723\pi$
$487$	16.1543	0.732023	0.366011	+	0.930610i	$0.380723\pi$
$488$	11.6065	0.525402
$489$	0	0
$490$	−6.90907	−0.312120
$491$	12.4633	0.562459	0.281229	−	0.959641i	$-0.409258\pi$
$491$	12.4633	0.562459	0.281229	+	0.959641i	$0.409258\pi$
$492$	0	0
$493$	4.52015	0.203577
$494$	−30.2333	−1.36026
$495$	0	0
$496$	6.61066	0.296827
$497$	5.65049	0.253459
$498$	0	0
$499$	−13.4987	−0.604283	−0.302142	−	0.953263i	$-0.597702\pi$
$499$	−13.4987	−0.604283	−0.302142	+	0.953263i	$0.597702\pi$
$500$	6.55976	0.293361
$501$	0	0
$502$	2.37673	0.106078
$503$	−22.8347	−1.01815	−0.509074	−	0.860723i	$-0.670012\pi$
$503$	−22.8347	−1.01815	−0.509074	+	0.860723i	$0.670012\pi$
$504$	0	0
$505$	−21.5074	−0.957068
$506$	18.0035	0.800354
$507$	0	0
$508$	−2.40865	−0.106866
$509$	27.5341	1.22043	0.610214	−	0.792237i	$-0.291084\pi$
$509$	27.5341	1.22043	0.610214	+	0.792237i	$0.291084\pi$
$510$	0	0
$511$	−3.24624	−0.143605
$512$	−29.4458	−1.30133
$513$	0	0
$514$	−61.4077	−2.70858
$515$	4.67534	0.206020
$516$	0	0
$517$	31.1429	1.36966
$518$	21.3021	0.935959
$519$	0	0
$520$	6.09869	0.267445
$521$	−21.7185	−0.951504	−0.475752	−	0.879580i	$-0.657824\pi$
$521$	−21.7185	−0.951504	−0.475752	+	0.879580i	$0.657824\pi$
$522$	0	0
$523$	−25.3129	−1.10685	−0.553427	−	0.832898i	$-0.686680\pi$
$523$	−25.3129	−1.10685	−0.553427	+	0.832898i	$0.686680\pi$
$524$	−42.9024	−1.87420
$525$	0	0
$526$	55.3593	2.41378
$527$	4.80575	0.209342
$528$	0	0
$529$	−17.4057	−0.756771
$530$	19.6856	0.855087
$531$	0	0
$532$	−16.0561	−0.696119
$533$	−1.39863	−0.0605815
$534$	0	0
$535$	−56.4058	−2.43864
$536$	−0.569803	−0.0246117
$537$	0	0
$538$	−46.6703	−2.01210
$539$	3.62521	0.156149
$540$	0	0
$541$	−0.966528	−0.0415543	−0.0207771	−	0.999784i	$-0.506614\pi$
$541$	−0.966528	−0.0415543	−0.0207771	+	0.999784i	$0.506614\pi$
$542$	56.3744	2.42149
$543$	0	0
$544$	−17.6441	−0.756483
$545$	47.0997	2.01753
$546$	0	0
$547$	−6.92682	−0.296170	−0.148085	−	0.988975i	$-0.547311\pi$
$547$	−6.92682	−0.296170	−0.148085	+	0.988975i	$0.547311\pi$
$548$	−18.0959	−0.773018
$549$	0	0
$550$	−44.3588	−1.89146
$551$	13.7440	0.585513
$552$	0	0
$553$	4.01101	0.170566
$554$	−4.43021	−0.188222
$555$	0	0
$556$	−11.2538	−0.477268
$557$	41.5856	1.76204	0.881019	−	0.473080i	$-0.156858\pi$
$557$	41.5856	1.76204	0.881019	+	0.473080i	$0.156858\pi$
$558$	0	0
$559$	−7.94600	−0.336080
$560$	−9.92332	−0.419337
$561$	0	0
$562$	41.1645	1.73642
$563$	31.7414	1.33774	0.668871	−	0.743378i	$-0.266778\pi$
$563$	31.7414	1.33774	0.668871	+	0.743378i	$0.266778\pi$
$564$	0	0
$565$	−30.5320	−1.28449
$566$	11.1025	0.466673
$567$	0	0
$568$	−4.84828	−0.203429
$569$	0.333729	0.0139906	0.00699532	−	0.999976i	$-0.497773\pi$
$569$	0.333729	0.0139906	0.00699532	+	0.999976i	$0.497773\pi$
$570$	0	0
$571$	12.9921	0.543704	0.271852	−	0.962339i	$-0.412364\pi$
$571$	12.9921	0.543704	0.271852	+	0.962339i	$0.412364\pi$
$572$	−18.8614	−0.788634
$573$	0	0
$574$	−1.35953	−0.0567457
$575$	−13.7837	−0.574819
$576$	0	0
$577$	32.0896	1.33591	0.667954	−	0.744203i	$-0.267170\pi$
$577$	32.0896	1.33591	0.667954	+	0.744203i	$0.267170\pi$
$578$	25.6028	1.06494
$579$	0	0
$580$	−16.3413	−0.678536
$581$	3.40033	0.141070
$582$	0	0
$583$	−10.3291	−0.427787
$584$	2.78537	0.115259
$585$	0	0
$586$	19.0152	0.785509
$587$	40.9497	1.69017	0.845087	−	0.534629i	$-0.179549\pi$
$587$	40.9497	1.69017	0.845087	+	0.534629i	$0.179549\pi$
$588$	0	0
$589$	14.6124	0.602093
$590$	−4.09562	−0.168614
$591$	0	0
$592$	30.5956	1.25747
$593$	−42.4563	−1.74347	−0.871735	−	0.489978i	$-0.837005\pi$
$593$	−42.4563	−1.74347	−0.871735	+	0.489978i	$0.837005\pi$
$594$	0	0
$595$	−7.21395	−0.295743
$596$	31.2238	1.27898
$597$	0	0
$598$	−10.7273	−0.438673
$599$	−26.8144	−1.09560	−0.547802	−	0.836608i	$-0.684535\pi$
$599$	−26.8144	−1.09560	−0.547802	+	0.836608i	$0.684535\pi$
$600$	0	0
$601$	−32.3520	−1.31967	−0.659833	−	0.751412i	$-0.729373\pi$
$601$	−32.3520	−1.31967	−0.659833	+	0.751412i	$0.729373\pi$
$602$	−7.72385	−0.314801
$603$	0	0
$604$	26.3357	1.07158
$605$	7.04882	0.286575
$606$	0	0
$607$	−5.75375	−0.233538	−0.116769	−	0.993159i	$-0.537254\pi$
$607$	−5.75375	−0.233538	−0.116769	+	0.993159i	$0.537254\pi$
$608$	−53.6486	−2.17574
$609$	0	0
$610$	93.4585	3.78402
$611$	−18.5564	−0.750710
$612$	0	0
$613$	−28.0175	−1.13162	−0.565808	−	0.824537i	$-0.691436\pi$
$613$	−28.0175	−1.13162	−0.565808	+	0.824537i	$0.691436\pi$
$614$	−49.3261	−1.99064
$615$	0	0
$616$	−3.11054	−0.125327
$617$	1.13222	0.0455814	0.0227907	−	0.999740i	$-0.492745\pi$
$617$	1.13222	0.0455814	0.0227907	+	0.999740i	$0.492745\pi$
$618$	0	0
$619$	14.6756	0.589862	0.294931	−	0.955519i	$-0.404703\pi$
$619$	14.6756	0.589862	0.294931	+	0.955519i	$0.404703\pi$
$620$	−17.3738	−0.697750
$621$	0	0
$622$	−0.223847	−0.00897545
$623$	−6.31252	−0.252906
$624$	0	0
$625$	−20.1767	−0.807070
$626$	−49.6180	−1.98313
$627$	0	0
$628$	−49.4151	−1.97188
$629$	22.2421	0.886849
$630$	0	0
$631$	−33.4295	−1.33081	−0.665404	−	0.746483i	$-0.731741\pi$
$631$	−33.4295	−1.33081	−0.665404	+	0.746483i	$0.731741\pi$
$632$	−3.44157	−0.136898
$633$	0	0
$634$	18.6259	0.739729
$635$	−3.29054	−0.130581
$636$	0	0
$637$	−2.16007	−0.0855850
$638$	15.6940	0.621329
$639$	0	0
$640$	22.1156	0.874194
$641$	−13.4509	−0.531278	−0.265639	−	0.964073i	$-0.585583\pi$
$641$	−13.4509	−0.531278	−0.265639	+	0.964073i	$0.585583\pi$
$642$	0	0
$643$	11.7618	0.463838	0.231919	−	0.972735i	$-0.425499\pi$
$643$	11.7618	0.463838	0.231919	+	0.972735i	$0.425499\pi$
$644$	−5.69698	−0.224493
$645$	0	0
$646$	−30.6849	−1.20728
$647$	−8.12298	−0.319347	−0.159674	−	0.987170i	$-0.551044\pi$
$647$	−8.12298	−0.319347	−0.159674	+	0.987170i	$0.551044\pi$
$648$	0	0
$649$	2.14898	0.0843550
$650$	26.4310	1.03671
$651$	0	0
$652$	3.21169	0.125779
$653$	29.8203	1.16696	0.583479	−	0.812128i	$-0.301691\pi$
$653$	29.8203	1.16696	0.583479	+	0.812128i	$0.301691\pi$
$654$	0	0
$655$	−58.6105	−2.29010
$656$	−1.95266	−0.0762384
$657$	0	0
$658$	−18.0376	−0.703178
$659$	−7.08303	−0.275916	−0.137958	−	0.990438i	$-0.544054\pi$
$659$	−7.08303	−0.275916	−0.137958	+	0.990438i	$0.544054\pi$
$660$	0	0
$661$	36.2548	1.41015	0.705075	−	0.709133i	$-0.250913\pi$
$661$	36.2548	1.41015	0.705075	+	0.709133i	$0.250913\pi$
$662$	57.4990	2.23476
$663$	0	0
$664$	−2.91758	−0.113224
$665$	−21.9348	−0.850594
$666$	0	0
$667$	4.87661	0.188823
$668$	−1.98018	−0.0766153
$669$	0	0
$670$	−4.58820	−0.177258
$671$	−49.0379	−1.89309
$672$	0	0
$673$	22.9499	0.884656	0.442328	−	0.896853i	$-0.354153\pi$
$673$	22.9499	0.884656	0.442328	+	0.896853i	$0.354153\pi$
$674$	−38.8226	−1.49539
$675$	0	0
$676$	−20.0739	−0.772075
$677$	−3.57237	−0.137297	−0.0686486	−	0.997641i	$-0.521869\pi$
$677$	−3.57237	−0.137297	−0.0686486	+	0.997641i	$0.521869\pi$
$678$	0	0
$679$	−0.0689323	−0.00264538
$680$	6.18978	0.237367
$681$	0	0
$682$	16.6856	0.638923
$683$	22.1918	0.849145	0.424573	−	0.905394i	$-0.360424\pi$
$683$	22.1918	0.849145	0.424573	+	0.905394i	$0.360424\pi$
$684$	0	0
$685$	−24.7214	−0.944558
$686$	−2.09968	−0.0801661
$687$	0	0
$688$	−11.0936	−0.422938
$689$	6.15454	0.234469
$690$	0	0
$691$	−27.5328	−1.04740	−0.523698	−	0.851904i	$-0.675448\pi$
$691$	−27.5328	−1.04740	−0.523698	+	0.851904i	$0.675448\pi$
$692$	55.0625	2.09316
$693$	0	0
$694$	50.8979	1.93206
$695$	−15.3742	−0.583178
$696$	0	0
$697$	−1.41952	−0.0537683
$698$	16.2533	0.615195
$699$	0	0
$700$	14.0368	0.530540
$701$	33.7129	1.27332	0.636659	−	0.771146i	$-0.280316\pi$
$701$	33.7129	1.27332	0.636659	+	0.771146i	$0.280316\pi$
$702$	0	0
$703$	67.6293	2.55069
$704$	−39.3950	−1.48475
$705$	0	0
$706$	−22.2989	−0.839229
$707$	−6.53614	−0.245817
$708$	0	0
$709$	−43.1638	−1.62105	−0.810525	−	0.585704i	$-0.800818\pi$
$709$	−43.1638	−1.62105	−0.810525	+	0.585704i	$0.800818\pi$
$710$	−39.0396	−1.46513
$711$	0	0
$712$	5.41633	0.202986
$713$	5.18474	0.194170
$714$	0	0
$715$	−25.7672	−0.963639
$716$	−3.66438	−0.136944
$717$	0	0
$718$	24.3434	0.908486
$719$	−13.6734	−0.509931	−0.254965	−	0.966950i	$-0.582064\pi$
$719$	−13.6734	−0.509931	−0.254965	+	0.966950i	$0.582064\pi$
$720$	0	0
$721$	1.42084	0.0529150
$722$	−53.4067	−1.98759
$723$	0	0
$724$	34.8085	1.29365
$725$	−12.0154	−0.446243
$726$	0	0
$727$	−32.4355	−1.20297	−0.601483	−	0.798886i	$-0.705423\pi$
$727$	−32.4355	−1.20297	−0.601483	+	0.798886i	$0.705423\pi$
$728$	1.85340	0.0686916
$729$	0	0
$730$	22.4285	0.830116
$731$	−8.06468	−0.298283
$732$	0	0
$733$	−42.8281	−1.58189	−0.790946	−	0.611886i	$-0.790411\pi$
$733$	−42.8281	−1.58189	−0.790946	+	0.611886i	$0.790411\pi$
$734$	41.0497	1.51517
$735$	0	0
$736$	−19.0355	−0.701657
$737$	2.40744	0.0886792
$738$	0	0
$739$	38.7706	1.42620	0.713099	−	0.701063i	$-0.247291\pi$
$739$	38.7706	1.42620	0.713099	+	0.701063i	$0.247291\pi$
$740$	−80.4099	−2.95593
$741$	0	0
$742$	5.98248	0.219624
$743$	−13.9130	−0.510418	−0.255209	−	0.966886i	$-0.582144\pi$
$743$	−13.9130	−0.510418	−0.255209	+	0.966886i	$0.582144\pi$
$744$	0	0
$745$	42.6560	1.56280
$746$	−8.30678	−0.304133
$747$	0	0
$748$	−19.1431	−0.699942
$749$	−17.1418	−0.626348
$750$	0	0
$751$	−0.457421	−0.0166915	−0.00834576	−	0.999965i	$-0.502657\pi$
$751$	−0.457421	−0.0166915	−0.00834576	+	0.999965i	$0.502657\pi$
$752$	−25.9069	−0.944727
$753$	0	0
$754$	−9.35118	−0.340550
$755$	35.9781	1.30938
$756$	0	0
$757$	−20.9802	−0.762539	−0.381269	−	0.924464i	$-0.624513\pi$
$757$	−20.9802	−0.762539	−0.381269	+	0.924464i	$0.624513\pi$
$758$	55.1025	2.00141
$759$	0	0
$760$	18.8207	0.682697
$761$	30.4760	1.10476	0.552378	−	0.833594i	$-0.313721\pi$
$761$	30.4760	1.10476	0.552378	+	0.833594i	$0.313721\pi$
$762$	0	0
$763$	14.3137	0.518189
$764$	−15.5704	−0.563316
$765$	0	0
$766$	4.74483	0.171438
$767$	−1.28046	−0.0462348
$768$	0	0
$769$	−21.8499	−0.787927	−0.393963	−	0.919126i	$-0.628896\pi$
$769$	−21.8499	−0.787927	−0.393963	+	0.919126i	$0.628896\pi$
$770$	−25.0468	−0.902626
$771$	0	0
$772$	−51.5831	−1.85652
$773$	−19.4525	−0.699659	−0.349830	−	0.936813i	$-0.613761\pi$
$773$	−19.4525	−0.699659	−0.349830	+	0.936813i	$0.613761\pi$
$774$	0	0
$775$	−12.7746	−0.458879
$776$	0.0591460	0.00212322
$777$	0	0
$778$	−10.0630	−0.360776
$779$	−4.31620	−0.154644
$780$	0	0
$781$	20.4842	0.732982
$782$	−10.8876	−0.389338
$783$	0	0
$784$	−3.01571	−0.107704
$785$	−67.5077	−2.40945
$786$	0	0
$787$	41.2974	1.47209	0.736047	−	0.676930i	$-0.236690\pi$
$787$	41.2974	1.47209	0.736047	+	0.676930i	$0.236690\pi$
$788$	−36.8103	−1.31131
$789$	0	0
$790$	−27.7124	−0.985962
$791$	−9.27873	−0.329914
$792$	0	0
$793$	29.2191	1.03760
$794$	−72.4528	−2.57125
$795$	0	0
$796$	−23.3217	−0.826615
$797$	35.1784	1.24608	0.623042	−	0.782188i	$-0.285897\pi$
$797$	35.1784	1.24608	0.623042	+	0.782188i	$0.285897\pi$
$798$	0	0
$799$	−18.8335	−0.666282
$800$	46.9014	1.65822
$801$	0	0
$802$	−4.42303	−0.156183
$803$	−11.7683	−0.415294
$804$	0	0
$805$	−7.78285	−0.274310
$806$	−9.94203	−0.350193
$807$	0	0
$808$	5.60820	0.197296
$809$	−32.9258	−1.15761	−0.578804	−	0.815466i	$-0.696480\pi$
$809$	−32.9258	−1.15761	−0.578804	+	0.815466i	$0.696480\pi$
$810$	0	0
$811$	−46.5481	−1.63453	−0.817263	−	0.576265i	$-0.804510\pi$
$811$	−46.5481	−1.63453	−0.817263	+	0.576265i	$0.804510\pi$
$812$	−4.96615	−0.174278
$813$	0	0
$814$	77.2245	2.70672
$815$	4.38760	0.153691
$816$	0	0
$817$	−24.5215	−0.857899
$818$	−77.3666	−2.70506
$819$	0	0
$820$	5.13188	0.179213
$821$	−24.6736	−0.861114	−0.430557	−	0.902563i	$-0.641683\pi$
$821$	−24.6736	−0.861114	−0.430557	+	0.902563i	$0.641683\pi$
$822$	0	0
$823$	−35.8137	−1.24839	−0.624193	−	0.781270i	$-0.714572\pi$
$823$	−35.8137	−1.24839	−0.624193	+	0.781270i	$0.714572\pi$
$824$	−1.21913	−0.0424703
$825$	0	0
$826$	−1.24467	−0.0433075
$827$	43.4022	1.50924	0.754621	−	0.656161i	$-0.227821\pi$
$827$	43.4022	1.50924	0.754621	+	0.656161i	$0.227821\pi$
$828$	0	0
$829$	−12.6369	−0.438896	−0.219448	−	0.975624i	$-0.570426\pi$
$829$	−12.6369	−0.438896	−0.219448	+	0.975624i	$0.570426\pi$
$830$	−23.4931	−0.815459
$831$	0	0
$832$	23.4733	0.813792
$833$	−2.19233	−0.0759598
$834$	0	0
$835$	−2.70519	−0.0936170
$836$	−58.2066	−2.01312
$837$	0	0
$838$	−49.1668	−1.69844
$839$	−28.0483	−0.968336	−0.484168	−	0.874975i	$-0.660878\pi$
$839$	−28.0483	−0.968336	−0.484168	+	0.874975i	$0.660878\pi$
$840$	0	0
$841$	−24.7490	−0.853413
$842$	−15.1089	−0.520688
$843$	0	0
$844$	−54.7056	−1.88304
$845$	−27.4237	−0.943405
$846$	0	0
$847$	2.14215	0.0736050
$848$	8.59247	0.295067
$849$	0	0
$850$	26.8258	0.920117
$851$	23.9961	0.822576
$852$	0	0
$853$	19.5717	0.670121	0.335060	−	0.942197i	$-0.391243\pi$
$853$	19.5717	0.670121	0.335060	+	0.942197i	$0.391243\pi$
$854$	28.4022	0.971903
$855$	0	0
$856$	14.7082	0.502715
$857$	−5.60926	−0.191609	−0.0958043	−	0.995400i	$-0.530542\pi$
$857$	−5.60926	−0.191609	−0.0958043	+	0.995400i	$0.530542\pi$
$858$	0	0
$859$	9.59070	0.327231	0.163615	−	0.986524i	$-0.447684\pi$
$859$	9.59070	0.327231	0.163615	+	0.986524i	$0.447684\pi$
$860$	29.1556	0.994197
$861$	0	0
$862$	−39.6716	−1.35122
$863$	30.6668	1.04391	0.521955	−	0.852973i	$-0.325203\pi$
$863$	30.6668	1.04391	0.521955	+	0.852973i	$0.325203\pi$
$864$	0	0
$865$	75.2229	2.55765
$866$	−38.3528	−1.30328
$867$	0	0
$868$	−5.27993	−0.179213
$869$	14.5408	0.493262
$870$	0	0
$871$	−1.43446	−0.0486049
$872$	−12.2815	−0.415905
$873$	0	0
$874$	−33.1047	−1.11978
$875$	2.72342	0.0920684
$876$	0	0
$877$	−3.44789	−0.116427	−0.0582136	−	0.998304i	$-0.518540\pi$
$877$	−3.44789	−0.116427	−0.0582136	+	0.998304i	$0.518540\pi$
$878$	65.7700	2.21963
$879$	0	0
$880$	−35.9741	−1.21269
$881$	−37.1051	−1.25010	−0.625052	−	0.780583i	$-0.714922\pi$
$881$	−37.1051	−1.25010	−0.625052	+	0.780583i	$0.714922\pi$
$882$	0	0
$883$	−18.6304	−0.626964	−0.313482	−	0.949594i	$-0.601496\pi$
$883$	−18.6304	−0.626964	−0.313482	+	0.949594i	$0.601496\pi$
$884$	11.4063	0.383637
$885$	0	0
$886$	−15.8435	−0.532274
$887$	4.44415	0.149220	0.0746100	−	0.997213i	$-0.476229\pi$
$887$	4.44415	0.149220	0.0746100	+	0.997213i	$0.476229\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	43.6137	1.46193
$891$	0	0
$892$	15.2706	0.511297
$893$	−57.2653	−1.91631
$894$	0	0
$895$	−5.00604	−0.167333
$896$	6.72095	0.224531
$897$	0	0
$898$	−51.3244	−1.71272
$899$	4.51962	0.150738
$900$	0	0
$901$	6.24647	0.208100
$902$	−4.92858	−0.164104
$903$	0	0
$904$	7.96142	0.264793
$905$	47.5531	1.58072
$906$	0	0
$907$	48.7902	1.62005	0.810026	−	0.586393i	$-0.199453\pi$
$907$	48.7902	1.62005	0.810026	+	0.586393i	$0.199453\pi$
$908$	37.8998	1.25775
$909$	0	0
$910$	14.9241	0.494728
$911$	−33.0894	−1.09630	−0.548150	−	0.836380i	$-0.684668\pi$
$911$	−33.0894	−1.09630	−0.548150	+	0.836380i	$0.684668\pi$
$912$	0	0
$913$	12.3269	0.407961
$914$	−21.6761	−0.716983
$915$	0	0
$916$	−0.879818	−0.0290700
$917$	−17.8118	−0.588198
$918$	0	0
$919$	4.96601	0.163814	0.0819068	−	0.996640i	$-0.473899\pi$
$919$	4.96601	0.163814	0.0819068	+	0.996640i	$0.473899\pi$
$920$	6.67792	0.220164
$921$	0	0
$922$	−0.472394	−0.0155575
$923$	−12.2054	−0.401747
$924$	0	0
$925$	−59.1238	−1.94398
$926$	44.9897	1.47845
$927$	0	0
$928$	−16.5935	−0.544709
$929$	−2.64029	−0.0866251	−0.0433125	−	0.999062i	$-0.513791\pi$
$929$	−2.64029	−0.0866251	−0.0433125	+	0.999062i	$0.513791\pi$
$930$	0	0
$931$	−6.66601	−0.218470
$932$	10.1475	0.332391
$933$	0	0
$934$	31.0899	1.01729
$935$	−26.1521	−0.855265
$936$	0	0
$937$	−25.1677	−0.822194	−0.411097	−	0.911592i	$-0.634854\pi$
$937$	−25.1677	−0.822194	−0.411097	+	0.911592i	$0.634854\pi$
$938$	−1.39436	−0.0455275
$939$	0	0
$940$	68.0873	2.22076
$941$	−35.0361	−1.14215	−0.571073	−	0.820899i	$-0.693473\pi$
$941$	−35.0361	−1.14215	−0.571073	+	0.820899i	$0.693473\pi$
$942$	0	0
$943$	−1.53147	−0.0498715
$944$	−1.78768	−0.0581840
$945$	0	0
$946$	−28.0006	−0.910377
$947$	−34.0698	−1.10712	−0.553560	−	0.832809i	$-0.686731\pi$
$947$	−34.0698	−1.10712	−0.553560	+	0.832809i	$0.686731\pi$
$948$	0	0
$949$	7.01209	0.227622
$950$	81.5665	2.64637
$951$	0	0
$952$	1.88108	0.0609663
$953$	26.4808	0.857796	0.428898	−	0.903353i	$-0.358902\pi$
$953$	26.4808	0.857796	0.428898	+	0.903353i	$0.358902\pi$
$954$	0	0
$955$	−21.2712	−0.688321
$956$	8.16100	0.263945
$957$	0	0
$958$	−21.4323	−0.692447
$959$	−7.51288	−0.242604
$960$	0	0
$961$	−26.1948	−0.844994
$962$	−46.0139	−1.48355
$963$	0	0
$964$	−24.2331	−0.780495
$965$	−70.4695	−2.26849
$966$	0	0
$967$	−10.2513	−0.329659	−0.164829	−	0.986322i	$-0.552707\pi$
$967$	−10.2513	−0.329659	−0.164829	+	0.986322i	$0.552707\pi$
$968$	−1.83802	−0.0590763
$969$	0	0
$970$	0.476259	0.0152917
$971$	−10.1088	−0.324407	−0.162203	−	0.986757i	$-0.551860\pi$
$971$	−10.1088	−0.324407	−0.162203	+	0.986757i	$0.551860\pi$
$972$	0	0
$973$	−4.67226	−0.149786
$974$	−33.9189	−1.08683
$975$	0	0
$976$	40.7933	1.30576
$977$	−10.4900	−0.335604	−0.167802	−	0.985821i	$-0.553667\pi$
$977$	−10.4900	−0.335604	−0.167802	+	0.985821i	$0.553667\pi$
$978$	0	0
$979$	−22.8842	−0.731383
$980$	7.92575	0.253179
$981$	0	0
$982$	−26.1688	−0.835081
$983$	−15.4551	−0.492940	−0.246470	−	0.969150i	$-0.579271\pi$
$983$	−15.4551	−0.492940	−0.246470	+	0.969150i	$0.579271\pi$
$984$	0	0
$985$	−50.2879	−1.60230
$986$	−9.49085	−0.302250
$987$	0	0
$988$	34.6822	1.10339
$989$	−8.70067	−0.276665
$990$	0	0
$991$	58.7140	1.86511	0.932556	−	0.361026i	$-0.117574\pi$
$991$	58.7140	1.86511	0.932556	+	0.361026i	$0.117574\pi$
$992$	−17.6420	−0.560134
$993$	0	0
$994$	−11.8642	−0.376310
$995$	−31.8606	−1.01005
$996$	0	0
$997$	−9.14112	−0.289502	−0.144751	−	0.989468i	$-0.546238\pi$
$997$	−9.14112	−0.289502	−0.144751	+	0.989468i	$0.546238\pi$
$998$	28.3428	0.897177
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.l.1.1		7
3.2	odd	2		2667.2.a.j.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.j.1.7	✓	7	3.2	odd	2
8001.2.a.l.1.1		7	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.09968 q^{2} +2.40865 q^{4} +3.29054 q^{5} +1.00000 q^{7} -0.858029 q^{8} +O(q^{10})\)	\(q-2.09968 q^{2} +2.40865 q^{4} +3.29054 q^{5} +1.00000 q^{7} -0.858029 q^{8} -6.90907 q^{10} +3.62521 q^{11} -2.16007 q^{13} -2.09968 q^{14} -3.01571 q^{16} -2.19233 q^{17} -6.66601 q^{19} +7.92575 q^{20} -7.61177 q^{22} -2.36522 q^{23} +5.82765 q^{25} +4.53544 q^{26} +2.40865 q^{28} -2.06180 q^{29} -2.19207 q^{31} +8.04808 q^{32} +4.60319 q^{34} +3.29054 q^{35} -10.1454 q^{37} +13.9965 q^{38} -2.82338 q^{40} +0.647495 q^{41} +3.67859 q^{43} +8.73185 q^{44} +4.96620 q^{46} +8.59064 q^{47} +1.00000 q^{49} -12.2362 q^{50} -5.20284 q^{52} -2.84924 q^{53} +11.9289 q^{55} -0.858029 q^{56} +4.32912 q^{58} +0.592789 q^{59} -13.5269 q^{61} +4.60265 q^{62} -10.8670 q^{64} -7.10778 q^{65} +0.664083 q^{67} -5.28055 q^{68} -6.90907 q^{70} +5.65049 q^{71} -3.24624 q^{73} +21.3021 q^{74} -16.0561 q^{76} +3.62521 q^{77} +4.01101 q^{79} -9.92332 q^{80} -1.35953 q^{82} +3.40033 q^{83} -7.21395 q^{85} -7.72385 q^{86} -3.11054 q^{88} -6.31252 q^{89} -2.16007 q^{91} -5.69698 q^{92} -18.0376 q^{94} -21.9348 q^{95} -0.0689323 q^{97} -2.09968 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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