LMFDB - Embedded newform 8001.2.a.n.1.7

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} + \cdots - 1 $ x^12 - 5x^11 - 3x^10 + 41x^9 - 11x^8 - 123x^7 + 44x^6 + 159x^5 - 39x^4 - 71x^3 + 16x^2 + 7*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.127418$ 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)

$f(q)$	$=$	$q+0.872582 q^{2} -1.23860 q^{4} -3.32654 q^{5} +1.00000 q^{7} -2.82594 q^{8} +O(q^{10})$	\(q+0.872582 q^{2} -1.23860 q^{4} -3.32654 q^{5} +1.00000 q^{7} -2.82594 q^{8} -2.90267 q^{10} +6.60457 q^{11} -3.02878 q^{13} +0.872582 q^{14} +0.0113364 q^{16} -1.04478 q^{17} -2.10095 q^{19} +4.12025 q^{20} +5.76303 q^{22} +7.22640 q^{23} +6.06584 q^{25} -2.64286 q^{26} -1.23860 q^{28} -4.91361 q^{29} -7.63150 q^{31} +5.66178 q^{32} -0.911660 q^{34} -3.32654 q^{35} -2.71082 q^{37} -1.83325 q^{38} +9.40060 q^{40} -10.1822 q^{41} +6.25676 q^{43} -8.18043 q^{44} +6.30562 q^{46} +4.84267 q^{47} +1.00000 q^{49} +5.29294 q^{50} +3.75145 q^{52} -1.31492 q^{53} -21.9703 q^{55} -2.82594 q^{56} -4.28753 q^{58} +3.93161 q^{59} -13.7378 q^{61} -6.65911 q^{62} +4.91769 q^{64} +10.0753 q^{65} -13.5096 q^{67} +1.29407 q^{68} -2.90267 q^{70} +1.27486 q^{71} -4.47063 q^{73} -2.36542 q^{74} +2.60224 q^{76} +6.60457 q^{77} +10.5488 q^{79} -0.0377108 q^{80} -8.88484 q^{82} -0.980597 q^{83} +3.47551 q^{85} +5.45954 q^{86} -18.6642 q^{88} +10.5320 q^{89} -3.02878 q^{91} -8.95062 q^{92} +4.22563 q^{94} +6.98889 q^{95} -13.3696 q^{97} +0.872582 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * 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 $

$2$	0.872582	0.617008	0.308504	−	0.951223i	$-0.400172\pi$
$2$	0.872582	0.617008	0.308504	+	0.951223i	$0.400172\pi$
$3$	0	0
$3$	0	0
$4$	−1.23860	−0.619301
$5$	−3.32654	−1.48767	−0.743836	−	0.668362i	$-0.766996\pi$
$5$	−3.32654	−1.48767	−0.743836	+	0.668362i	$0.766996\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.82594	−0.999122
$9$	0	0
$10$	−2.90267	−0.917906
$11$	6.60457	1.99135	0.995677	−	0.0928853i	$-0.0296090\pi$
$11$	6.60457	1.99135	0.995677	+	0.0928853i	$0.0296090\pi$
$12$	0	0
$13$	−3.02878	−0.840032	−0.420016	−	0.907517i	$-0.637976\pi$
$13$	−3.02878	−0.840032	−0.420016	+	0.907517i	$0.637976\pi$
$14$	0.872582	0.233207
$15$	0	0
$16$	0.0113364	0.00283409
$17$	−1.04478	−0.253398	−0.126699	−	0.991941i	$-0.540438\pi$
$17$	−1.04478	−0.253398	−0.126699	+	0.991941i	$0.540438\pi$
$18$	0	0
$19$	−2.10095	−0.481992	−0.240996	−	0.970526i	$-0.577474\pi$
$19$	−2.10095	−0.481992	−0.240996	+	0.970526i	$0.577474\pi$
$20$	4.12025	0.921316
$21$	0	0
$22$	5.76303	1.22868
$23$	7.22640	1.50681	0.753404	−	0.657558i	$-0.228411\pi$
$23$	7.22640	1.50681	0.753404	+	0.657558i	$0.228411\pi$
$24$	0	0
$25$	6.06584	1.21317
$26$	−2.64286	−0.518307
$27$	0	0
$28$	−1.23860	−0.234074
$29$	−4.91361	−0.912434	−0.456217	−	0.889868i	$-0.650796\pi$
$29$	−4.91361	−0.912434	−0.456217	+	0.889868i	$0.650796\pi$
$30$	0	0
$31$	−7.63150	−1.37066	−0.685329	−	0.728234i	$-0.740342\pi$
$31$	−7.63150	−1.37066	−0.685329	+	0.728234i	$0.740342\pi$
$32$	5.66178	1.00087
$33$	0	0
$34$	−0.911660	−0.156348
$35$	−3.32654	−0.562287
$36$	0	0
$37$	−2.71082	−0.445657	−0.222828	−	0.974858i	$-0.571529\pi$
$37$	−2.71082	−0.445657	−0.222828	+	0.974858i	$0.571529\pi$
$38$	−1.83325	−0.297393
$39$	0	0
$40$	9.40060	1.48637
$41$	−10.1822	−1.59020	−0.795100	−	0.606478i	$-0.792582\pi$
$41$	−10.1822	−1.59020	−0.795100	+	0.606478i	$0.792582\pi$
$42$	0	0
$43$	6.25676	0.954147	0.477074	−	0.878863i	$-0.341697\pi$
$43$	6.25676	0.954147	0.477074	+	0.878863i	$0.341697\pi$
$44$	−8.18043	−1.23325
$45$	0	0
$46$	6.30562	0.929713
$47$	4.84267	0.706376	0.353188	−	0.935552i	$-0.385098\pi$
$47$	4.84267	0.706376	0.353188	+	0.935552i	$0.385098\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.29294	0.748535
$51$	0	0
$52$	3.75145	0.520233
$53$	−1.31492	−0.180618	−0.0903091	−	0.995914i	$-0.528785\pi$
$53$	−1.31492	−0.180618	−0.0903091	+	0.995914i	$0.528785\pi$
$54$	0	0
$55$	−21.9703	−2.96248
$56$	−2.82594	−0.377633
$57$	0	0
$58$	−4.28753	−0.562980
$59$	3.93161	0.511852	0.255926	−	0.966696i	$-0.417620\pi$
$59$	3.93161	0.511852	0.255926	+	0.966696i	$0.417620\pi$
$60$	0	0
$61$	−13.7378	−1.75895	−0.879476	−	0.475944i	$-0.842106\pi$
$61$	−13.7378	−1.75895	−0.879476	+	0.475944i	$0.842106\pi$
$62$	−6.65911	−0.845707
$63$	0	0
$64$	4.91769	0.614711
$65$	10.0753	1.24969
$66$	0	0
$67$	−13.5096	−1.65046	−0.825229	−	0.564799i	$-0.808954\pi$
$67$	−13.5096	−1.65046	−0.825229	+	0.564799i	$0.808954\pi$
$68$	1.29407	0.156929
$69$	0	0
$70$	−2.90267	−0.346936
$71$	1.27486	0.151298	0.0756488	−	0.997135i	$-0.475897\pi$
$71$	1.27486	0.151298	0.0756488	+	0.997135i	$0.475897\pi$
$72$	0	0
$73$	−4.47063	−0.523248	−0.261624	−	0.965170i	$-0.584258\pi$
$73$	−4.47063	−0.523248	−0.261624	+	0.965170i	$0.584258\pi$
$74$	−2.36542	−0.274974
$75$	0	0
$76$	2.60224	0.298498
$77$	6.60457	0.752661
$78$	0	0
$79$	10.5488	1.18683	0.593416	−	0.804896i	$-0.297779\pi$
$79$	10.5488	1.18683	0.593416	+	0.804896i	$0.297779\pi$
$80$	−0.0377108	−0.00421620
$81$	0	0
$82$	−8.88484	−0.981167
$83$	−0.980597	−0.107635	−0.0538173	−	0.998551i	$-0.517139\pi$
$83$	−0.980597	−0.107635	−0.0538173	+	0.998551i	$0.517139\pi$
$84$	0	0
$85$	3.47551	0.376972
$86$	5.45954	0.588717
$87$	0	0
$88$	−18.6642	−1.98961
$89$	10.5320	1.11639	0.558197	−	0.829709i	$-0.311493\pi$
$89$	10.5320	1.11639	0.558197	+	0.829709i	$0.311493\pi$
$90$	0	0
$91$	−3.02878	−0.317502
$92$	−8.95062	−0.933167
$93$	0	0
$94$	4.22563	0.435840
$95$	6.98889	0.717045
$96$	0	0
$97$	−13.3696	−1.35747	−0.678736	−	0.734382i	$-0.737472\pi$
$97$	−13.3696	−1.35747	−0.678736	+	0.734382i	$0.737472\pi$
$98$	0.872582	0.0881440
$99$	0	0
$100$	−7.51316	−0.751316
$101$	1.68723	0.167886	0.0839430	−	0.996471i	$-0.473249\pi$
$101$	1.68723	0.167886	0.0839430	+	0.996471i	$0.473249\pi$
$102$	0	0
$103$	11.8124	1.16391	0.581957	−	0.813220i	$-0.302287\pi$
$103$	11.8124	1.16391	0.581957	+	0.813220i	$0.302287\pi$
$104$	8.55916	0.839295
$105$	0	0
$106$	−1.14738	−0.111443
$107$	−2.72810	−0.263736	−0.131868	−	0.991267i	$-0.542097\pi$
$107$	−2.72810	−0.263736	−0.131868	+	0.991267i	$0.542097\pi$
$108$	0	0
$109$	3.86124	0.369840	0.184920	−	0.982754i	$-0.440797\pi$
$109$	3.86124	0.369840	0.184920	+	0.982754i	$0.440797\pi$
$110$	−19.1709	−1.82788
$111$	0	0
$112$	0.0113364	0.00107119
$113$	16.6002	1.56162	0.780810	−	0.624769i	$-0.214807\pi$
$113$	16.6002	1.56162	0.780810	+	0.624769i	$0.214807\pi$
$114$	0	0
$115$	−24.0389	−2.24164
$116$	6.08600	0.565071
$117$	0	0
$118$	3.43065	0.315817
$119$	−1.04478	−0.0957753
$120$	0	0
$121$	32.6204	2.96549
$122$	−11.9874	−1.08529
$123$	0	0
$124$	9.45239	0.848849
$125$	−3.54556	−0.317125
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−7.03247	−0.621589
$129$	0	0
$130$	8.79156	0.771071
$131$	−5.65694	−0.494249	−0.247125	−	0.968984i	$-0.579486\pi$
$131$	−5.65694	−0.494249	−0.247125	+	0.968984i	$0.579486\pi$
$132$	0	0
$133$	−2.10095	−0.182176
$134$	−11.7882	−1.01835
$135$	0	0
$136$	2.95250	0.253175
$137$	−5.93739	−0.507265	−0.253633	−	0.967301i	$-0.581625\pi$
$137$	−5.93739	−0.507265	−0.253633	+	0.967301i	$0.581625\pi$
$138$	0	0
$139$	1.99353	0.169089	0.0845443	−	0.996420i	$-0.473057\pi$
$139$	1.99353	0.169089	0.0845443	+	0.996420i	$0.473057\pi$
$140$	4.12025	0.348225
$141$	0	0
$142$	1.11242	0.0933518
$143$	−20.0038	−1.67280
$144$	0	0
$145$	16.3453	1.35740
$146$	−3.90099	−0.322848
$147$	0	0
$148$	3.35763	0.275996
$149$	11.7258	0.960612	0.480306	−	0.877101i	$-0.340526\pi$
$149$	11.7258	0.960612	0.480306	+	0.877101i	$0.340526\pi$
$150$	0	0
$151$	−1.22469	−0.0996640	−0.0498320	−	0.998758i	$-0.515869\pi$
$151$	−1.22469	−0.0996640	−0.0498320	+	0.998758i	$0.515869\pi$
$152$	5.93717	0.481568
$153$	0	0
$154$	5.76303	0.464398
$155$	25.3865	2.03909
$156$	0	0
$157$	17.6789	1.41093	0.705467	−	0.708743i	$-0.250737\pi$
$157$	17.6789	1.41093	0.705467	+	0.708743i	$0.250737\pi$
$158$	9.20468	0.732285
$159$	0	0
$160$	−18.8341	−1.48897
$161$	7.22640	0.569520
$162$	0	0
$163$	13.5753	1.06330	0.531650	−	0.846964i	$-0.321572\pi$
$163$	13.5753	1.06330	0.531650	+	0.846964i	$0.321572\pi$
$164$	12.6117	0.984812
$165$	0	0
$166$	−0.855651	−0.0664114
$167$	23.2440	1.79867	0.899336	−	0.437258i	$-0.144050\pi$
$167$	23.2440	1.79867	0.899336	+	0.437258i	$0.144050\pi$
$168$	0	0
$169$	−3.82649	−0.294345
$170$	3.03267	0.232595
$171$	0	0
$172$	−7.74964	−0.590904
$173$	−20.6386	−1.56913	−0.784563	−	0.620049i	$-0.787113\pi$
$173$	−20.6386	−1.56913	−0.784563	+	0.620049i	$0.787113\pi$
$174$	0	0
$175$	6.06584	0.458535
$176$	0.0748718	0.00564367
$177$	0	0
$178$	9.19006	0.688824
$179$	−10.1214	−0.756511	−0.378255	−	0.925701i	$-0.623476\pi$
$179$	−10.1214	−0.756511	−0.378255	+	0.925701i	$0.623476\pi$
$180$	0	0
$181$	22.9647	1.70696	0.853478	−	0.521130i	$-0.174489\pi$
$181$	22.9647	1.70696	0.853478	+	0.521130i	$0.174489\pi$
$182$	−2.64286	−0.195902
$183$	0	0
$184$	−20.4214	−1.50548
$185$	9.01765	0.662991
$186$	0	0
$187$	−6.90036	−0.504604
$188$	−5.99814	−0.437459
$189$	0	0
$190$	6.09838	0.442423
$191$	4.17969	0.302432	0.151216	−	0.988501i	$-0.451681\pi$
$191$	4.17969	0.302432	0.151216	+	0.988501i	$0.451681\pi$
$192$	0	0
$193$	−6.41287	−0.461608	−0.230804	−	0.973000i	$-0.574136\pi$
$193$	−6.41287	−0.461608	−0.230804	+	0.973000i	$0.574136\pi$
$194$	−11.6660	−0.837572
$195$	0	0
$196$	−1.23860	−0.0884715
$197$	18.1337	1.29197	0.645986	−	0.763349i	$-0.276446\pi$
$197$	18.1337	1.29197	0.645986	+	0.763349i	$0.276446\pi$
$198$	0	0
$199$	−5.80553	−0.411543	−0.205771	−	0.978600i	$-0.565970\pi$
$199$	−5.80553	−0.411543	−0.205771	+	0.978600i	$0.565970\pi$
$200$	−17.1417	−1.21210
$201$	0	0
$202$	1.47225	0.103587
$203$	−4.91361	−0.344868
$204$	0	0
$205$	33.8716	2.36570
$206$	10.3073	0.718144
$207$	0	0
$208$	−0.0343353	−0.00238073
$209$	−13.8759	−0.959816
$210$	0	0
$211$	13.9319	0.959111	0.479556	−	0.877511i	$-0.340798\pi$
$211$	13.9319	0.959111	0.479556	+	0.877511i	$0.340798\pi$
$212$	1.62866	0.111857
$213$	0	0
$214$	−2.38049	−0.162727
$215$	−20.8133	−1.41946
$216$	0	0
$217$	−7.63150	−0.518060
$218$	3.36925	0.228194
$219$	0	0
$220$	27.2125	1.83467
$221$	3.16442	0.212862
$222$	0	0
$223$	8.93412	0.598273	0.299136	−	0.954210i	$-0.403301\pi$
$223$	8.93412	0.598273	0.299136	+	0.954210i	$0.403301\pi$
$224$	5.66178	0.378294
$225$	0	0
$226$	14.4851	0.963532
$227$	26.1285	1.73421	0.867104	−	0.498126i	$-0.165978\pi$
$227$	26.1285	1.73421	0.867104	+	0.498126i	$0.165978\pi$
$228$	0	0
$229$	−3.03288	−0.200419	−0.100209	−	0.994966i	$-0.531951\pi$
$229$	−3.03288	−0.200419	−0.100209	+	0.994966i	$0.531951\pi$
$230$	−20.9759	−1.38311
$231$	0	0
$232$	13.8856	0.911633
$233$	−9.42978	−0.617766	−0.308883	−	0.951100i	$-0.599955\pi$
$233$	−9.42978	−0.617766	−0.308883	+	0.951100i	$0.599955\pi$
$234$	0	0
$235$	−16.1093	−1.05086
$236$	−4.86969	−0.316990
$237$	0	0
$238$	−0.911660	−0.0590941
$239$	21.0020	1.35851	0.679253	−	0.733904i	$-0.262304\pi$
$239$	21.0020	1.35851	0.679253	+	0.733904i	$0.262304\pi$
$240$	0	0
$241$	11.7376	0.756088	0.378044	−	0.925788i	$-0.376597\pi$
$241$	11.7376	0.756088	0.378044	+	0.925788i	$0.376597\pi$
$242$	28.4639	1.82973
$243$	0	0
$244$	17.0157	1.08932
$245$	−3.32654	−0.212525
$246$	0	0
$247$	6.36332	0.404889
$248$	21.5662	1.36945
$249$	0	0
$250$	−3.09379	−0.195668
$251$	−9.44194	−0.595970	−0.297985	−	0.954571i	$-0.596315\pi$
$251$	−9.44194	−0.595970	−0.297985	+	0.954571i	$0.596315\pi$
$252$	0	0
$253$	47.7273	3.00059
$254$	0.872582	0.0547506
$255$	0	0
$256$	−15.9718	−0.998237
$257$	24.0481	1.50008	0.750040	−	0.661392i	$-0.230034\pi$
$257$	24.0481	1.50008	0.750040	+	0.661392i	$0.230034\pi$
$258$	0	0
$259$	−2.71082	−0.168442
$260$	−12.4793	−0.773936
$261$	0	0
$262$	−4.93614	−0.304956
$263$	−17.0481	−1.05123	−0.525617	−	0.850721i	$-0.676165\pi$
$263$	−17.0481	−1.05123	−0.525617	+	0.850721i	$0.676165\pi$
$264$	0	0
$265$	4.37413	0.268701
$266$	−1.83325	−0.112404
$267$	0	0
$268$	16.7330	1.02213
$269$	3.11135	0.189702	0.0948512	−	0.995491i	$-0.469762\pi$
$269$	3.11135	0.189702	0.0948512	+	0.995491i	$0.469762\pi$
$270$	0	0
$271$	0.492934	0.0299436	0.0149718	−	0.999888i	$-0.495234\pi$
$271$	0.492934	0.0299436	0.0149718	+	0.999888i	$0.495234\pi$
$272$	−0.0118441	−0.000718151 0
$273$	0	0
$274$	−5.18085	−0.312987
$275$	40.0623	2.41585
$276$	0	0
$277$	27.7724	1.66868	0.834340	−	0.551250i	$-0.185849\pi$
$277$	27.7724	1.66868	0.834340	+	0.551250i	$0.185849\pi$
$278$	1.73951	0.104329
$279$	0	0
$280$	9.40060	0.561794
$281$	−17.6887	−1.05522	−0.527609	−	0.849487i	$-0.676911\pi$
$281$	−17.6887	−1.05522	−0.527609	+	0.849487i	$0.676911\pi$
$282$	0	0
$283$	−16.9144	−1.00546	−0.502728	−	0.864445i	$-0.667670\pi$
$283$	−16.9144	−1.00546	−0.502728	+	0.864445i	$0.667670\pi$
$284$	−1.57904	−0.0936987
$285$	0	0
$286$	−17.4549	−1.03213
$287$	−10.1822	−0.601039
$288$	0	0
$289$	−15.9084	−0.935790
$290$	14.2626	0.837529
$291$	0	0
$292$	5.53733	0.324048
$293$	0.0950654	0.00555378	0.00277689	−	0.999996i	$-0.499116\pi$
$293$	0.0950654	0.00555378	0.00277689	+	0.999996i	$0.499116\pi$
$294$	0	0
$295$	−13.0786	−0.761467
$296$	7.66064	0.445266
$297$	0	0
$298$	10.2317	0.592705
$299$	−21.8872	−1.26577
$300$	0	0
$301$	6.25676	0.360634
$302$	−1.06864	−0.0614935
$303$	0	0
$304$	−0.0238171	−0.00136601
$305$	45.6994	2.61674
$306$	0	0
$307$	−17.8559	−1.01909	−0.509545	−	0.860444i	$-0.670186\pi$
$307$	−17.8559	−1.01909	−0.509545	+	0.860444i	$0.670186\pi$
$308$	−8.18043	−0.466123
$309$	0	0
$310$	22.1518	1.25814
$311$	2.22365	0.126092	0.0630458	−	0.998011i	$-0.479919\pi$
$311$	2.22365	0.126092	0.0630458	+	0.998011i	$0.479919\pi$
$312$	0	0
$313$	20.2359	1.14380	0.571899	−	0.820324i	$-0.306207\pi$
$313$	20.2359	1.14380	0.571899	+	0.820324i	$0.306207\pi$
$314$	15.4263	0.870558
$315$	0	0
$316$	−13.0658	−0.735006
$317$	−9.48926	−0.532970	−0.266485	−	0.963839i	$-0.585862\pi$
$317$	−9.48926	−0.532970	−0.266485	+	0.963839i	$0.585862\pi$
$318$	0	0
$319$	−32.4523	−1.81698
$320$	−16.3589	−0.914489
$321$	0	0
$322$	6.30562	0.351398
$323$	2.19504	0.122135
$324$	0	0
$325$	−18.3721	−1.01910
$326$	11.8456	0.656065
$327$	0	0
$328$	28.7745	1.58880
$329$	4.84267	0.266985
$330$	0	0
$331$	−34.0368	−1.87083	−0.935417	−	0.353546i	$-0.884976\pi$
$331$	−34.0368	−1.87083	−0.935417	+	0.353546i	$0.884976\pi$
$332$	1.21457	0.0666581
$333$	0	0
$334$	20.2823	1.10980
$335$	44.9401	2.45534
$336$	0	0
$337$	−1.31149	−0.0714412	−0.0357206	−	0.999362i	$-0.511373\pi$
$337$	−1.31149	−0.0714412	−0.0357206	+	0.999362i	$0.511373\pi$
$338$	−3.33893	−0.181614
$339$	0	0
$340$	−4.30478	−0.233459
$341$	−50.4028	−2.72946
$342$	0	0
$343$	1.00000	0.0539949
$344$	−17.6813	−0.953310
$345$	0	0
$346$	−18.0089	−0.968164
$347$	5.30126	0.284586	0.142293	−	0.989825i	$-0.454552\pi$
$347$	5.30126	0.284586	0.142293	+	0.989825i	$0.454552\pi$
$348$	0	0
$349$	33.2250	1.77850	0.889248	−	0.457426i	$-0.151229\pi$
$349$	33.2250	1.77850	0.889248	+	0.457426i	$0.151229\pi$
$350$	5.29294	0.282920
$351$	0	0
$352$	37.3936	1.99309
$353$	−2.85830	−0.152132	−0.0760660	−	0.997103i	$-0.524236\pi$
$353$	−2.85830	−0.152132	−0.0760660	+	0.997103i	$0.524236\pi$
$354$	0	0
$355$	−4.24085	−0.225081
$356$	−13.0450	−0.691383
$357$	0	0
$358$	−8.83177	−0.466773
$359$	32.3399	1.70684	0.853418	−	0.521227i	$-0.174525\pi$
$359$	32.3399	1.70684	0.853418	+	0.521227i	$0.174525\pi$
$360$	0	0
$361$	−14.5860	−0.767684
$362$	20.0386	1.05321
$363$	0	0
$364$	3.75145	0.196629
$365$	14.8717	0.778421
$366$	0	0
$367$	31.7299	1.65629	0.828145	−	0.560514i	$-0.189397\pi$
$367$	31.7299	1.65629	0.828145	+	0.560514i	$0.189397\pi$
$368$	0.0819210	0.00427043
$369$	0	0
$370$	7.86864	0.409071
$371$	−1.31492	−0.0682673
$372$	0	0
$373$	−16.1393	−0.835659	−0.417830	−	0.908525i	$-0.637209\pi$
$373$	−16.1393	−0.835659	−0.417830	+	0.908525i	$0.637209\pi$
$374$	−6.02112	−0.311345
$375$	0	0
$376$	−13.6851	−0.705756
$377$	14.8822	0.766475
$378$	0	0
$379$	−36.0448	−1.85150	−0.925748	−	0.378141i	$-0.876563\pi$
$379$	−36.0448	−1.85150	−0.925748	+	0.378141i	$0.876563\pi$
$380$	−8.65645	−0.444067
$381$	0	0
$382$	3.64712	0.186603
$383$	−19.4364	−0.993154	−0.496577	−	0.867993i	$-0.665410\pi$
$383$	−19.4364	−0.993154	−0.496577	+	0.867993i	$0.665410\pi$
$384$	0	0
$385$	−21.9703	−1.11971
$386$	−5.59575	−0.284816
$387$	0	0
$388$	16.5596	0.840684
$389$	18.9145	0.959004	0.479502	−	0.877541i	$-0.340817\pi$
$389$	18.9145	0.959004	0.479502	+	0.877541i	$0.340817\pi$
$390$	0	0
$391$	−7.55003	−0.381821
$392$	−2.82594	−0.142732
$393$	0	0
$394$	15.8231	0.797158
$395$	−35.0909	−1.76562
$396$	0	0
$397$	3.51339	0.176332	0.0881660	−	0.996106i	$-0.471899\pi$
$397$	3.51339	0.176332	0.0881660	+	0.996106i	$0.471899\pi$
$398$	−5.06579	−0.253925
$399$	0	0
$400$	0.0687646	0.00343823
$401$	−20.7230	−1.03486	−0.517428	−	0.855727i	$-0.673110\pi$
$401$	−20.7230	−1.03486	−0.517428	+	0.855727i	$0.673110\pi$
$402$	0	0
$403$	23.1141	1.15140
$404$	−2.08981	−0.103972
$405$	0	0
$406$	−4.28753	−0.212786
$407$	−17.9038	−0.887460
$408$	0	0
$409$	6.51998	0.322392	0.161196	−	0.986922i	$-0.448465\pi$
$409$	6.51998	0.322392	0.161196	+	0.986922i	$0.448465\pi$
$410$	29.5557	1.45965
$411$	0	0
$412$	−14.6309	−0.720813
$413$	3.93161	0.193462
$414$	0	0
$415$	3.26199	0.160125
$416$	−17.1483	−0.840764
$417$	0	0
$418$	−12.1078	−0.592214
$419$	24.7810	1.21063	0.605317	−	0.795985i	$-0.293046\pi$
$419$	24.7810	1.21063	0.605317	+	0.795985i	$0.293046\pi$
$420$	0	0
$421$	23.6407	1.15218	0.576089	−	0.817387i	$-0.304578\pi$
$421$	23.6407	1.15218	0.576089	+	0.817387i	$0.304578\pi$
$422$	12.1567	0.591780
$423$	0	0
$424$	3.71589	0.180460
$425$	−6.33750	−0.307414
$426$	0	0
$427$	−13.7378	−0.664821
$428$	3.37903	0.163332
$429$	0	0
$430$	−18.1613	−0.875818
$431$	8.06794	0.388619	0.194310	−	0.980940i	$-0.437753\pi$
$431$	8.06794	0.388619	0.194310	+	0.980940i	$0.437753\pi$
$432$	0	0
$433$	27.0478	1.29983	0.649917	−	0.760005i	$-0.274804\pi$
$433$	27.0478	1.29983	0.649917	+	0.760005i	$0.274804\pi$
$434$	−6.65911	−0.319647
$435$	0	0
$436$	−4.78253	−0.229042
$437$	−15.1823	−0.726269
$438$	0	0
$439$	8.24982	0.393742	0.196871	−	0.980429i	$-0.436922\pi$
$439$	8.24982	0.393742	0.196871	+	0.980429i	$0.436922\pi$
$440$	62.0870	2.95988
$441$	0	0
$442$	2.76122	0.131338
$443$	−10.5434	−0.500933	−0.250467	−	0.968125i	$-0.580584\pi$
$443$	−10.5434	−0.500933	−0.250467	+	0.968125i	$0.580584\pi$
$444$	0	0
$445$	−35.0352	−1.66083
$446$	7.79575	0.369139
$447$	0	0
$448$	4.91769	0.232339
$449$	−10.4549	−0.493399	−0.246699	−	0.969092i	$-0.579346\pi$
$449$	−10.4549	−0.493399	−0.246699	+	0.969092i	$0.579346\pi$
$450$	0	0
$451$	−67.2494	−3.16665
$452$	−20.5611	−0.967112
$453$	0	0
$454$	22.7992	1.07002
$455$	10.0753	0.472340
$456$	0	0
$457$	−4.67165	−0.218531	−0.109265	−	0.994013i	$-0.534850\pi$
$457$	−4.67165	−0.218531	−0.109265	+	0.994013i	$0.534850\pi$
$458$	−2.64644	−0.123660
$459$	0	0
$460$	29.7746	1.38825
$461$	4.74068	0.220795	0.110398	−	0.993887i	$-0.464788\pi$
$461$	4.74068	0.220795	0.110398	+	0.993887i	$0.464788\pi$
$462$	0	0
$463$	36.0262	1.67428	0.837139	−	0.546990i	$-0.184226\pi$
$463$	36.0262	1.67428	0.837139	+	0.546990i	$0.184226\pi$
$464$	−0.0557024	−0.00258592
$465$	0	0
$466$	−8.22825	−0.381167
$467$	−11.2569	−0.520907	−0.260454	−	0.965486i	$-0.583872\pi$
$467$	−11.2569	−0.520907	−0.260454	+	0.965486i	$0.583872\pi$
$468$	0	0
$469$	−13.5096	−0.623814
$470$	−14.0567	−0.648387
$471$	0	0
$472$	−11.1105	−0.511402
$473$	41.3232	1.90005
$474$	0	0
$475$	−12.7440	−0.584737
$476$	1.29407	0.0593137
$477$	0	0
$478$	18.3259	0.838209
$479$	26.4912	1.21041	0.605206	−	0.796069i	$-0.293091\pi$
$479$	26.4912	1.21041	0.605206	+	0.796069i	$0.293091\pi$
$480$	0	0
$481$	8.21049	0.374366
$482$	10.2420	0.466512
$483$	0	0
$484$	−40.4037	−1.83653
$485$	44.4743	2.01947
$486$	0	0
$487$	2.91164	0.131939	0.0659694	−	0.997822i	$-0.478986\pi$
$487$	2.91164	0.131939	0.0659694	+	0.997822i	$0.478986\pi$
$488$	38.8224	1.75741
$489$	0	0
$490$	−2.90267	−0.131129
$491$	−6.54337	−0.295298	−0.147649	−	0.989040i	$-0.547171\pi$
$491$	−6.54337	−0.295298	−0.147649	+	0.989040i	$0.547171\pi$
$492$	0	0
$493$	5.13366	0.231209
$494$	5.55252	0.249820
$495$	0	0
$496$	−0.0865134	−0.00388457
$497$	1.27486	0.0571851
$498$	0	0
$499$	43.2433	1.93584	0.967918	−	0.251268i	$-0.0808476\pi$
$499$	43.2433	1.93584	0.967918	+	0.251268i	$0.0808476\pi$
$500$	4.39154	0.196395
$501$	0	0
$502$	−8.23886	−0.367719
$503$	18.1223	0.808034	0.404017	−	0.914752i	$-0.367614\pi$
$503$	18.1223	0.808034	0.404017	+	0.914752i	$0.367614\pi$
$504$	0	0
$505$	−5.61264	−0.249759
$506$	41.6459	1.85139
$507$	0	0
$508$	−1.23860	−0.0549541
$509$	34.2778	1.51934	0.759668	−	0.650311i	$-0.225362\pi$
$509$	34.2778	1.51934	0.759668	+	0.650311i	$0.225362\pi$
$510$	0	0
$511$	−4.47063	−0.197769
$512$	0.128256	0.00566816
$513$	0	0
$514$	20.9839	0.925562
$515$	−39.2945	−1.73152
$516$	0	0
$517$	31.9838	1.40665
$518$	−2.36542	−0.103930
$519$	0	0
$520$	−28.4724	−1.24860
$521$	31.2538	1.36926	0.684628	−	0.728893i	$-0.259965\pi$
$521$	31.2538	1.36926	0.684628	+	0.728893i	$0.259965\pi$
$522$	0	0
$523$	23.3748	1.02211	0.511055	−	0.859548i	$-0.329255\pi$
$523$	23.3748	1.02211	0.511055	+	0.859548i	$0.329255\pi$
$524$	7.00670	0.306089
$525$	0	0
$526$	−14.8759	−0.648620
$527$	7.97327	0.347321
$528$	0	0
$529$	29.2208	1.27047
$530$	3.81679	0.165791
$531$	0	0
$532$	2.60224	0.112822
$533$	30.8398	1.33582
$534$	0	0
$535$	9.07513	0.392352
$536$	38.1773	1.64901
$537$	0	0
$538$	2.71491	0.117048
$539$	6.60457	0.284479
$540$	0	0
$541$	−39.7725	−1.70995	−0.854977	−	0.518665i	$-0.826429\pi$
$541$	−39.7725	−1.70995	−0.854977	+	0.518665i	$0.826429\pi$
$542$	0.430125	0.0184754
$543$	0	0
$544$	−5.91534	−0.253618
$545$	−12.8445	−0.550200
$546$	0	0
$547$	−42.5481	−1.81922	−0.909612	−	0.415459i	$-0.863621\pi$
$547$	−42.5481	−1.81922	−0.909612	+	0.415459i	$0.863621\pi$
$548$	7.35405	0.314150
$549$	0	0
$550$	34.9576	1.49060
$551$	10.3233	0.439786
$552$	0	0
$553$	10.5488	0.448580
$554$	24.2337	1.02959
$555$	0	0
$556$	−2.46918	−0.104717
$557$	13.0751	0.554009	0.277005	−	0.960869i	$-0.410658\pi$
$557$	13.0751	0.554009	0.277005	+	0.960869i	$0.410658\pi$
$558$	0	0
$559$	−18.9504	−0.801515
$560$	−0.0377108	−0.00159357
$561$	0	0
$562$	−15.4348	−0.651078
$563$	15.3131	0.645371	0.322685	−	0.946506i	$-0.395414\pi$
$563$	15.3131	0.645371	0.322685	+	0.946506i	$0.395414\pi$
$564$	0	0
$565$	−55.2213	−2.32318
$566$	−14.7592	−0.620375
$567$	0	0
$568$	−3.60267	−0.151165
$569$	−20.5726	−0.862448	−0.431224	−	0.902245i	$-0.641918\pi$
$569$	−20.5726	−0.862448	−0.431224	+	0.902245i	$0.641918\pi$
$570$	0	0
$571$	6.30560	0.263881	0.131941	−	0.991258i	$-0.457879\pi$
$571$	6.30560	0.263881	0.131941	+	0.991258i	$0.457879\pi$
$572$	24.7767	1.03597
$573$	0	0
$574$	−8.88484	−0.370846
$575$	43.8342	1.82801
$576$	0	0
$577$	33.2579	1.38454	0.692272	−	0.721637i	$-0.256610\pi$
$577$	33.2579	1.38454	0.692272	+	0.721637i	$0.256610\pi$
$578$	−13.8814	−0.577390
$579$	0	0
$580$	−20.2453	−0.840641
$581$	−0.980597	−0.0406820
$582$	0	0
$583$	−8.68449	−0.359675
$584$	12.6338	0.522789
$585$	0	0
$586$	0.0829523	0.00342673
$587$	−17.1095	−0.706184	−0.353092	−	0.935589i	$-0.614870\pi$
$587$	−17.1095	−0.706184	−0.353092	+	0.935589i	$0.614870\pi$
$588$	0	0
$589$	16.0334	0.660645
$590$	−11.4122	−0.469832
$591$	0	0
$592$	−0.0307309	−0.00126303
$593$	6.45593	0.265113	0.132557	−	0.991175i	$-0.457681\pi$
$593$	6.45593	0.265113	0.132557	+	0.991175i	$0.457681\pi$
$594$	0	0
$595$	3.47551	0.142482
$596$	−14.5235	−0.594907
$597$	0	0
$598$	−19.0983	−0.780989
$599$	34.2077	1.39769	0.698844	−	0.715274i	$-0.253698\pi$
$599$	34.2077	1.39769	0.698844	+	0.715274i	$0.253698\pi$
$600$	0	0
$601$	−27.6074	−1.12613	−0.563065	−	0.826413i	$-0.690378\pi$
$601$	−27.6074	−1.12613	−0.563065	+	0.826413i	$0.690378\pi$
$602$	5.45954	0.222514
$603$	0	0
$604$	1.51691	0.0617220
$605$	−108.513	−4.41168
$606$	0	0
$607$	−9.78542	−0.397178	−0.198589	−	0.980083i	$-0.563636\pi$
$607$	−9.78542	−0.397178	−0.198589	+	0.980083i	$0.563636\pi$
$608$	−11.8951	−0.482411
$609$	0	0
$610$	39.8765	1.61455
$611$	−14.6674	−0.593379
$612$	0	0
$613$	23.7718	0.960135	0.480068	−	0.877232i	$-0.340612\pi$
$613$	23.7718	0.960135	0.480068	+	0.877232i	$0.340612\pi$
$614$	−15.5807	−0.628787
$615$	0	0
$616$	−18.6642	−0.752000
$617$	24.5701	0.989155	0.494577	−	0.869134i	$-0.335323\pi$
$617$	24.5701	0.989155	0.494577	+	0.869134i	$0.335323\pi$
$618$	0	0
$619$	28.0072	1.12571	0.562853	−	0.826557i	$-0.309704\pi$
$619$	28.0072	1.12571	0.562853	+	0.826557i	$0.309704\pi$
$620$	−31.4437	−1.26281
$621$	0	0
$622$	1.94031	0.0777995
$623$	10.5320	0.421957
$624$	0	0
$625$	−18.5348	−0.741391
$626$	17.6574	0.705733
$627$	0	0
$628$	−21.8972	−0.873792
$629$	2.83223	0.112928
$630$	0	0
$631$	−5.38443	−0.214351	−0.107175	−	0.994240i	$-0.534181\pi$
$631$	−5.38443	−0.214351	−0.107175	+	0.994240i	$0.534181\pi$
$632$	−29.8103	−1.18579
$633$	0	0
$634$	−8.28015	−0.328847
$635$	−3.32654	−0.132010
$636$	0	0
$637$	−3.02878	−0.120005
$638$	−28.3173	−1.12109
$639$	0	0
$640$	23.3938	0.924720
$641$	2.15591	0.0851532	0.0425766	−	0.999093i	$-0.486443\pi$
$641$	2.15591	0.0851532	0.0425766	+	0.999093i	$0.486443\pi$
$642$	0	0
$643$	−7.08832	−0.279536	−0.139768	−	0.990184i	$-0.544636\pi$
$643$	−7.08832	−0.279536	−0.139768	+	0.990184i	$0.544636\pi$
$644$	−8.95062	−0.352704
$645$	0	0
$646$	1.91535	0.0753586
$647$	−31.1591	−1.22499	−0.612495	−	0.790475i	$-0.709834\pi$
$647$	−31.1591	−1.22499	−0.612495	+	0.790475i	$0.709834\pi$
$648$	0	0
$649$	25.9666	1.01928
$650$	−16.0312	−0.628794
$651$	0	0
$652$	−16.8144	−0.658503
$653$	20.6765	0.809134	0.404567	−	0.914508i	$-0.367422\pi$
$653$	20.6765	0.809134	0.404567	+	0.914508i	$0.367422\pi$
$654$	0	0
$655$	18.8180	0.735281
$656$	−0.115430	−0.00450677
$657$	0	0
$658$	4.22563	0.164732
$659$	−22.2427	−0.866451	−0.433225	−	0.901286i	$-0.642625\pi$
$659$	−22.2427	−0.866451	−0.433225	+	0.901286i	$0.642625\pi$
$660$	0	0
$661$	−21.5285	−0.837363	−0.418681	−	0.908133i	$-0.637508\pi$
$661$	−21.5285	−0.837363	−0.418681	+	0.908133i	$0.637508\pi$
$662$	−29.6999	−1.15432
$663$	0	0
$664$	2.77111	0.107540
$665$	6.98889	0.271018
$666$	0	0
$667$	−35.5077	−1.37486
$668$	−28.7900	−1.11392
$669$	0	0
$670$	39.2139	1.51496
$671$	−90.7326	−3.50269
$672$	0	0
$673$	−8.41685	−0.324446	−0.162223	−	0.986754i	$-0.551866\pi$
$673$	−8.41685	−0.324446	−0.162223	+	0.986754i	$0.551866\pi$
$674$	−1.14438	−0.0440798
$675$	0	0
$676$	4.73950	0.182288
$677$	18.8247	0.723493	0.361746	−	0.932277i	$-0.382181\pi$
$677$	18.8247	0.723493	0.361746	+	0.932277i	$0.382181\pi$
$678$	0	0
$679$	−13.3696	−0.513076
$680$	−9.82161	−0.376641
$681$	0	0
$682$	−43.9806	−1.68410
$683$	−21.2533	−0.813234	−0.406617	−	0.913599i	$-0.633292\pi$
$683$	−21.2533	−0.813234	−0.406617	+	0.913599i	$0.633292\pi$
$684$	0	0
$685$	19.7509	0.754644
$686$	0.872582	0.0333153
$687$	0	0
$688$	0.0709289	0.00270414
$689$	3.98260	0.151725
$690$	0	0
$691$	−11.3622	−0.432239	−0.216119	−	0.976367i	$-0.569340\pi$
$691$	−11.3622	−0.432239	−0.216119	+	0.976367i	$0.569340\pi$
$692$	25.5630	0.971761
$693$	0	0
$694$	4.62578	0.175592
$695$	−6.63154	−0.251548
$696$	0	0
$697$	10.6383	0.402953
$698$	28.9915	1.09735
$699$	0	0
$700$	−7.51316	−0.283971
$701$	−21.8895	−0.826757	−0.413378	−	0.910559i	$-0.635651\pi$
$701$	−21.8895	−0.826757	−0.413378	+	0.910559i	$0.635651\pi$
$702$	0	0
$703$	5.69531	0.214803
$704$	32.4793	1.22411
$705$	0	0
$706$	−2.49410	−0.0938668
$707$	1.68723	0.0634549
$708$	0	0
$709$	16.2254	0.609356	0.304678	−	0.952455i	$-0.401451\pi$
$709$	16.2254	0.609356	0.304678	+	0.952455i	$0.401451\pi$
$710$	−3.70049	−0.138877
$711$	0	0
$712$	−29.7629	−1.11541
$713$	−55.1482	−2.06532
$714$	0	0
$715$	66.5434	2.48858
$716$	12.5364	0.468508
$717$	0	0
$718$	28.2192	1.05313
$719$	3.90181	0.145513	0.0727565	−	0.997350i	$-0.476820\pi$
$719$	3.90181	0.145513	0.0727565	+	0.997350i	$0.476820\pi$
$720$	0	0
$721$	11.8124	0.439918
$722$	−12.7275	−0.473668
$723$	0	0
$724$	−28.4442	−1.05712
$725$	−29.8052	−1.10694
$726$	0	0
$727$	9.56486	0.354741	0.177370	−	0.984144i	$-0.443241\pi$
$727$	9.56486	0.354741	0.177370	+	0.984144i	$0.443241\pi$
$728$	8.55916	0.317224
$729$	0	0
$730$	12.9768	0.480292
$731$	−6.53697	−0.241779
$732$	0	0
$733$	−34.9924	−1.29247	−0.646236	−	0.763137i	$-0.723658\pi$
$733$	−34.9924	−1.29247	−0.646236	+	0.763137i	$0.723658\pi$
$734$	27.6870	1.02194
$735$	0	0
$736$	40.9143	1.50812
$737$	−89.2250	−3.28664
$738$	0	0
$739$	11.9015	0.437802	0.218901	−	0.975747i	$-0.429753\pi$
$739$	11.9015	0.437802	0.218901	+	0.975747i	$0.429753\pi$
$740$	−11.1693	−0.410591
$741$	0	0
$742$	−1.14738	−0.0421215
$743$	21.4671	0.787550	0.393775	−	0.919207i	$-0.371169\pi$
$743$	21.4671	0.787550	0.393775	+	0.919207i	$0.371169\pi$
$744$	0	0
$745$	−39.0062	−1.42908
$746$	−14.0828	−0.515609
$747$	0	0
$748$	8.54679	0.312502
$749$	−2.72810	−0.0996827
$750$	0	0
$751$	−17.3470	−0.633001	−0.316500	−	0.948592i	$-0.602508\pi$
$751$	−17.3470	−0.633001	−0.316500	+	0.948592i	$0.602508\pi$
$752$	0.0548983	0.00200193
$753$	0	0
$754$	12.9860	0.472921
$755$	4.07398	0.148267
$756$	0	0
$757$	−44.2869	−1.60963	−0.804817	−	0.593523i	$-0.797737\pi$
$757$	−44.2869	−1.60963	−0.804817	+	0.593523i	$0.797737\pi$
$758$	−31.4520	−1.14239
$759$	0	0
$760$	−19.7502	−0.716416
$761$	6.94913	0.251906	0.125953	−	0.992036i	$-0.459801\pi$
$761$	6.94913	0.251906	0.125953	+	0.992036i	$0.459801\pi$
$762$	0	0
$763$	3.86124	0.139786
$764$	−5.17697	−0.187296
$765$	0	0
$766$	−16.9599	−0.612785
$767$	−11.9080	−0.429972
$768$	0	0
$769$	18.2806	0.659216	0.329608	−	0.944118i	$-0.393083\pi$
$769$	18.2806	0.659216	0.329608	+	0.944118i	$0.393083\pi$
$770$	−19.1709	−0.690872
$771$	0	0
$772$	7.94298	0.285874
$773$	16.5561	0.595480	0.297740	−	0.954647i	$-0.403767\pi$
$773$	16.5561	0.595480	0.297740	+	0.954647i	$0.403767\pi$
$774$	0	0
$775$	−46.2915	−1.66284
$776$	37.7816	1.35628
$777$	0	0
$778$	16.5045	0.591714
$779$	21.3924	0.766463
$780$	0	0
$781$	8.41988	0.301287
$782$	−6.58802	−0.235587
$783$	0	0
$784$	0.0113364	0.000404870 0
$785$	−58.8096	−2.09901
$786$	0	0
$787$	−15.7462	−0.561292	−0.280646	−	0.959811i	$-0.590549\pi$
$787$	−15.7462	−0.561292	−0.280646	+	0.959811i	$0.590549\pi$
$788$	−22.4604	−0.800120
$789$	0	0
$790$	−30.6197	−1.08940
$791$	16.6002	0.590237
$792$	0	0
$793$	41.6089	1.47758
$794$	3.06572	0.108798
$795$	0	0
$796$	7.19073	0.254869
$797$	−18.3944	−0.651563	−0.325782	−	0.945445i	$-0.605627\pi$
$797$	−18.3944	−0.651563	−0.325782	+	0.945445i	$0.605627\pi$
$798$	0	0
$799$	−5.05955	−0.178994
$800$	34.3435	1.21422
$801$	0	0
$802$	−18.0825	−0.638515
$803$	−29.5266	−1.04197
$804$	0	0
$805$	−24.0389	−0.847259
$806$	20.1690	0.710422
$807$	0	0
$808$	−4.76802	−0.167739
$809$	−27.0690	−0.951694	−0.475847	−	0.879528i	$-0.657858\pi$
$809$	−27.0690	−0.951694	−0.475847	+	0.879528i	$0.657858\pi$
$810$	0	0
$811$	2.80138	0.0983697	0.0491848	−	0.998790i	$-0.484338\pi$
$811$	2.80138	0.0983697	0.0491848	+	0.998790i	$0.484338\pi$
$812$	6.08600	0.213577
$813$	0	0
$814$	−15.6226	−0.547570
$815$	−45.1588	−1.58184
$816$	0	0
$817$	−13.1452	−0.459891
$818$	5.68921	0.198919
$819$	0	0
$820$	−41.9534	−1.46508
$821$	14.0701	0.491050	0.245525	−	0.969390i	$-0.421040\pi$
$821$	14.0701	0.491050	0.245525	+	0.969390i	$0.421040\pi$
$822$	0	0
$823$	−30.7025	−1.07022	−0.535110	−	0.844782i	$-0.679730\pi$
$823$	−30.7025	−1.07022	−0.535110	+	0.844782i	$0.679730\pi$
$824$	−33.3813	−1.16289
$825$	0	0
$826$	3.43065	0.119368
$827$	0.701213	0.0243836	0.0121918	−	0.999926i	$-0.496119\pi$
$827$	0.701213	0.0243836	0.0121918	+	0.999926i	$0.496119\pi$
$828$	0	0
$829$	35.3394	1.22739	0.613693	−	0.789544i	$-0.289683\pi$
$829$	35.3394	1.22739	0.613693	+	0.789544i	$0.289683\pi$
$830$	2.84635	0.0987984
$831$	0	0
$832$	−14.8946	−0.516378
$833$	−1.04478	−0.0361996
$834$	0	0
$835$	−77.3219	−2.67583
$836$	17.1867	0.594414
$837$	0	0
$838$	21.6235	0.746971
$839$	45.1445	1.55856	0.779281	−	0.626674i	$-0.215584\pi$
$839$	45.1445	1.55856	0.779281	+	0.626674i	$0.215584\pi$
$840$	0	0
$841$	−4.85644	−0.167463
$842$	20.6285	0.710904
$843$	0	0
$844$	−17.2561	−0.593978
$845$	12.7290	0.437890
$846$	0	0
$847$	32.6204	1.12085
$848$	−0.0149064	−0.000511888 0
$849$	0	0
$850$	−5.52998	−0.189677
$851$	−19.5895	−0.671519
$852$	0	0
$853$	10.6456	0.364497	0.182248	−	0.983253i	$-0.441662\pi$
$853$	10.6456	0.364497	0.182248	+	0.983253i	$0.441662\pi$
$854$	−11.9874	−0.410200
$855$	0	0
$856$	7.70946	0.263504
$857$	40.4813	1.38281	0.691407	−	0.722466i	$-0.256991\pi$
$857$	40.4813	1.38281	0.691407	+	0.722466i	$0.256991\pi$
$858$	0	0
$859$	−40.8061	−1.39229	−0.696143	−	0.717903i	$-0.745102\pi$
$859$	−40.8061	−1.39229	−0.696143	+	0.717903i	$0.745102\pi$
$860$	25.7794	0.879072
$861$	0	0
$862$	7.03994	0.239781
$863$	−30.5500	−1.03994	−0.519968	−	0.854186i	$-0.674056\pi$
$863$	−30.5500	−1.03994	−0.519968	+	0.854186i	$0.674056\pi$
$864$	0	0
$865$	68.6551	2.33434
$866$	23.6014	0.802008
$867$	0	0
$868$	9.45239	0.320835
$869$	69.6703	2.36340
$870$	0	0
$871$	40.9175	1.38644
$872$	−10.9116	−0.369515
$873$	0	0
$874$	−13.2478	−0.448114
$875$	−3.54556	−0.119862
$876$	0	0
$877$	26.1242	0.882152	0.441076	−	0.897470i	$-0.354597\pi$
$877$	26.1242	0.882152	0.441076	+	0.897470i	$0.354597\pi$
$878$	7.19864	0.242942
$879$	0	0
$880$	−0.249064	−0.00839594
$881$	34.3437	1.15707	0.578534	−	0.815658i	$-0.303625\pi$
$881$	34.3437	1.15707	0.578534	+	0.815658i	$0.303625\pi$
$882$	0	0
$883$	−0.923623	−0.0310824	−0.0155412	−	0.999879i	$-0.504947\pi$
$883$	−0.923623	−0.0310824	−0.0155412	+	0.999879i	$0.504947\pi$
$884$	−3.91946	−0.131826
$885$	0	0
$886$	−9.20000	−0.309080
$887$	26.5716	0.892185	0.446093	−	0.894987i	$-0.352815\pi$
$887$	26.5716	0.892185	0.446093	+	0.894987i	$0.352815\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−30.5711	−1.02474
$891$	0	0
$892$	−11.0658	−0.370511
$893$	−10.1742	−0.340467
$894$	0	0
$895$	33.6693	1.12544
$896$	−7.03247	−0.234938
$897$	0	0
$898$	−9.12278	−0.304431
$899$	37.4982	1.25064
$900$	0	0
$901$	1.37381	0.0457682
$902$	−58.6806	−1.95385
$903$	0	0
$904$	−46.9114	−1.56025
$905$	−76.3930	−2.53939
$906$	0	0
$907$	−15.7472	−0.522879	−0.261439	−	0.965220i	$-0.584197\pi$
$907$	−15.7472	−0.522879	−0.261439	+	0.965220i	$0.584197\pi$
$908$	−32.3628	−1.07400
$909$	0	0
$910$	8.79156	0.291437
$911$	−28.6067	−0.947781	−0.473891	−	0.880584i	$-0.657151\pi$
$911$	−28.6067	−0.947781	−0.473891	+	0.880584i	$0.657151\pi$
$912$	0	0
$913$	−6.47642	−0.214338
$914$	−4.07640	−0.134835
$915$	0	0
$916$	3.75654	0.124119
$917$	−5.65694	−0.186809
$918$	0	0
$919$	−43.3252	−1.42917	−0.714583	−	0.699550i	$-0.753384\pi$
$919$	−43.3252	−1.42917	−0.714583	+	0.699550i	$0.753384\pi$
$920$	67.9325	2.23967
$921$	0	0
$922$	4.13663	0.136233
$923$	−3.86126	−0.127095
$924$	0	0
$925$	−16.4434	−0.540657
$926$	31.4358	1.03304
$927$	0	0
$928$	−27.8198	−0.913229
$929$	35.6112	1.16837	0.584183	−	0.811622i	$-0.301415\pi$
$929$	35.6112	1.16837	0.584183	+	0.811622i	$0.301415\pi$
$930$	0	0
$931$	−2.10095	−0.0688559
$932$	11.6797	0.382583
$933$	0	0
$934$	−9.82256	−0.321404
$935$	22.9543	0.750685
$936$	0	0
$937$	−3.25543	−0.106350	−0.0531752	−	0.998585i	$-0.516934\pi$
$937$	−3.25543	−0.106350	−0.0531752	+	0.998585i	$0.516934\pi$
$938$	−11.7882	−0.384899
$939$	0	0
$940$	19.9530	0.650796
$941$	−10.9105	−0.355671	−0.177836	−	0.984060i	$-0.556910\pi$
$941$	−10.9105	−0.355671	−0.177836	+	0.984060i	$0.556910\pi$
$942$	0	0
$943$	−73.5810	−2.39613
$944$	0.0445701	0.00145063
$945$	0	0
$946$	36.0579	1.17234
$947$	29.1120	0.946013	0.473006	−	0.881059i	$-0.343169\pi$
$947$	29.1120	0.946013	0.473006	+	0.881059i	$0.343169\pi$
$948$	0	0
$949$	13.5406	0.439545
$950$	−11.1202	−0.360788
$951$	0	0
$952$	2.95250	0.0956912
$953$	27.4183	0.888166	0.444083	−	0.895986i	$-0.353529\pi$
$953$	27.4183	0.888166	0.444083	+	0.895986i	$0.353529\pi$
$954$	0	0
$955$	−13.9039	−0.449919
$956$	−26.0131	−0.841323
$957$	0	0
$958$	23.1157	0.746834
$959$	−5.93739	−0.191728
$960$	0	0
$961$	27.2398	0.878703
$962$	7.16432	0.230987
$963$	0	0
$964$	−14.5383	−0.468246
$965$	21.3326	0.686722
$966$	0	0
$967$	58.4120	1.87840	0.939202	−	0.343365i	$-0.111567\pi$
$967$	58.4120	1.87840	0.939202	+	0.343365i	$0.111567\pi$
$968$	−92.1834	−2.96289
$969$	0	0
$970$	38.8075	1.24603
$971$	−42.6688	−1.36931	−0.684654	−	0.728868i	$-0.740047\pi$
$971$	−42.6688	−1.36931	−0.684654	+	0.728868i	$0.740047\pi$
$972$	0	0
$973$	1.99353	0.0639095
$974$	2.54064	0.0814073
$975$	0	0
$976$	−0.155737	−0.00498502
$977$	42.0709	1.34597	0.672984	−	0.739657i	$-0.265012\pi$
$977$	42.0709	1.34597	0.672984	+	0.739657i	$0.265012\pi$
$978$	0	0
$979$	69.5596	2.22313
$980$	4.12025	0.131617
$981$	0	0
$982$	−5.70963	−0.182202
$983$	−41.0399	−1.30897	−0.654485	−	0.756075i	$-0.727114\pi$
$983$	−41.0399	−1.30897	−0.654485	+	0.756075i	$0.727114\pi$
$984$	0	0
$985$	−60.3224	−1.92203
$986$	4.47954	0.142658
$987$	0	0
$988$	−7.88162	−0.250748
$989$	45.2139	1.43772
$990$	0	0
$991$	−27.9222	−0.886979	−0.443490	−	0.896279i	$-0.646260\pi$
$991$	−27.9222	−0.886979	−0.443490	+	0.896279i	$0.646260\pi$
$992$	−43.2079	−1.37185
$993$	0	0
$994$	1.11242	0.0352837
$995$	19.3123	0.612241
$996$	0	0
$997$	−32.1660	−1.01871	−0.509354	−	0.860557i	$-0.670116\pi$
$997$	−32.1660	−1.01871	−0.509354	+	0.860557i	$0.670116\pi$
$998$	37.7333	1.19443
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.n.1.7		12
3.2	odd	2		889.2.a.a.1.6	✓	12
21.20	even	2		6223.2.a.i.1.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.a.1.6	✓	12	3.2	odd	2
6223.2.a.i.1.6		12	21.20	even	2
8001.2.a.n.1.7		12	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)

\(f(q)\)	\(=\)	\(q+0.872582 q^{2} -1.23860 q^{4} -3.32654 q^{5} +1.00000 q^{7} -2.82594 q^{8} +O(q^{10})\)	\(q+0.872582 q^{2} -1.23860 q^{4} -3.32654 q^{5} +1.00000 q^{7} -2.82594 q^{8} -2.90267 q^{10} +6.60457 q^{11} -3.02878 q^{13} +0.872582 q^{14} +0.0113364 q^{16} -1.04478 q^{17} -2.10095 q^{19} +4.12025 q^{20} +5.76303 q^{22} +7.22640 q^{23} +6.06584 q^{25} -2.64286 q^{26} -1.23860 q^{28} -4.91361 q^{29} -7.63150 q^{31} +5.66178 q^{32} -0.911660 q^{34} -3.32654 q^{35} -2.71082 q^{37} -1.83325 q^{38} +9.40060 q^{40} -10.1822 q^{41} +6.25676 q^{43} -8.18043 q^{44} +6.30562 q^{46} +4.84267 q^{47} +1.00000 q^{49} +5.29294 q^{50} +3.75145 q^{52} -1.31492 q^{53} -21.9703 q^{55} -2.82594 q^{56} -4.28753 q^{58} +3.93161 q^{59} -13.7378 q^{61} -6.65911 q^{62} +4.91769 q^{64} +10.0753 q^{65} -13.5096 q^{67} +1.29407 q^{68} -2.90267 q^{70} +1.27486 q^{71} -4.47063 q^{73} -2.36542 q^{74} +2.60224 q^{76} +6.60457 q^{77} +10.5488 q^{79} -0.0377108 q^{80} -8.88484 q^{82} -0.980597 q^{83} +3.47551 q^{85} +5.45954 q^{86} -18.6642 q^{88} +10.5320 q^{89} -3.02878 q^{91} -8.95062 q^{92} +4.22563 q^{94} +6.98889 q^{95} -13.3696 q^{97} +0.872582 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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