LMFDB - Embedded newform 8001.2.a.z.1.12

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:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-0.942656 q^{2} -1.11140 q^{4} -2.02754 q^{5} -1.00000 q^{7} +2.93298 q^{8} +O(q^{10})$	\(q-0.942656 q^{2} -1.11140 q^{4} -2.02754 q^{5} -1.00000 q^{7} +2.93298 q^{8} +1.91127 q^{10} -0.393608 q^{11} -1.52460 q^{13} +0.942656 q^{14} -0.541991 q^{16} +7.22498 q^{17} +5.33043 q^{19} +2.25341 q^{20} +0.371037 q^{22} -6.14596 q^{23} -0.889072 q^{25} +1.43717 q^{26} +1.11140 q^{28} -10.3543 q^{29} -5.51081 q^{31} -5.35505 q^{32} -6.81067 q^{34} +2.02754 q^{35} +1.11637 q^{37} -5.02476 q^{38} -5.94674 q^{40} +12.0128 q^{41} -0.215108 q^{43} +0.437456 q^{44} +5.79352 q^{46} +4.05988 q^{47} +1.00000 q^{49} +0.838089 q^{50} +1.69443 q^{52} -6.05109 q^{53} +0.798057 q^{55} -2.93298 q^{56} +9.76057 q^{58} -3.92402 q^{59} -3.48704 q^{61} +5.19480 q^{62} +6.13195 q^{64} +3.09118 q^{65} +9.31796 q^{67} -8.02984 q^{68} -1.91127 q^{70} +6.78717 q^{71} -11.2119 q^{73} -1.05235 q^{74} -5.92424 q^{76} +0.393608 q^{77} -5.16136 q^{79} +1.09891 q^{80} -11.3239 q^{82} +6.24052 q^{83} -14.6490 q^{85} +0.202772 q^{86} -1.15444 q^{88} +13.9448 q^{89} +1.52460 q^{91} +6.83062 q^{92} -3.82707 q^{94} -10.8077 q^{95} +10.5474 q^{97} -0.942656 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.942656	−0.666558	−0.333279	−	0.942828i	$-0.608155\pi$
$2$	−0.942656	−0.666558	−0.333279	+	0.942828i	$0.608155\pi$
$3$	0	0
$3$	0	0
$4$	−1.11140	−0.555700
$5$	−2.02754	−0.906745	−0.453372	−	0.891321i	$-0.649779\pi$
$5$	−2.02754	−0.906745	−0.453372	+	0.891321i	$0.649779\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.93298	1.03696
$9$	0	0
$10$	1.91127	0.604398
$11$	−0.393608	−0.118677	−0.0593387	−	0.998238i	$-0.518899\pi$
$11$	−0.393608	−0.118677	−0.0593387	+	0.998238i	$0.518899\pi$
$12$	0	0
$13$	−1.52460	−0.422847	−0.211423	−	0.977395i	$-0.567810\pi$
$13$	−1.52460	−0.422847	−0.211423	+	0.977395i	$0.567810\pi$
$14$	0.942656	0.251935
$15$	0	0
$16$	−0.541991	−0.135498
$17$	7.22498	1.75232	0.876158	−	0.482025i	$-0.160098\pi$
$17$	7.22498	1.75232	0.876158	+	0.482025i	$0.160098\pi$
$18$	0	0
$19$	5.33043	1.22288	0.611442	−	0.791289i	$-0.290590\pi$
$19$	5.33043	1.22288	0.611442	+	0.791289i	$0.290590\pi$
$20$	2.25341	0.503878
$21$	0	0
$22$	0.371037	0.0791054
$23$	−6.14596	−1.28152	−0.640760	−	0.767741i	$-0.721381\pi$
$23$	−6.14596	−1.28152	−0.640760	+	0.767741i	$0.721381\pi$
$24$	0	0
$25$	−0.889072	−0.177814
$26$	1.43717	0.281852
$27$	0	0
$28$	1.11140	0.210035
$29$	−10.3543	−1.92275	−0.961376	−	0.275239i	$-0.911243\pi$
$29$	−10.3543	−1.92275	−0.961376	+	0.275239i	$0.911243\pi$
$30$	0	0
$31$	−5.51081	−0.989771	−0.494885	−	0.868958i	$-0.664790\pi$
$31$	−5.51081	−0.989771	−0.494885	+	0.868958i	$0.664790\pi$
$32$	−5.35505	−0.946648
$33$	0	0
$34$	−6.81067	−1.16802
$35$	2.02754	0.342717
$36$	0	0
$37$	1.11637	0.183529	0.0917647	−	0.995781i	$-0.470749\pi$
$37$	1.11637	0.183529	0.0917647	+	0.995781i	$0.470749\pi$
$38$	−5.02476	−0.815124
$39$	0	0
$40$	−5.94674	−0.940262
$41$	12.0128	1.87609	0.938043	−	0.346519i	$-0.112636\pi$
$41$	12.0128	1.87609	0.938043	+	0.346519i	$0.112636\pi$
$42$	0	0
$43$	−0.215108	−0.0328036	−0.0164018	−	0.999865i	$-0.505221\pi$
$43$	−0.215108	−0.0328036	−0.0164018	+	0.999865i	$0.505221\pi$
$44$	0.437456	0.0659490
$45$	0	0
$46$	5.79352	0.854208
$47$	4.05988	0.592194	0.296097	−	0.955158i	$-0.404315\pi$
$47$	4.05988	0.592194	0.296097	+	0.955158i	$0.404315\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.838089	0.118524
$51$	0	0
$52$	1.69443	0.234976
$53$	−6.05109	−0.831181	−0.415591	−	0.909552i	$-0.636425\pi$
$53$	−6.05109	−0.831181	−0.415591	+	0.909552i	$0.636425\pi$
$54$	0	0
$55$	0.798057	0.107610
$56$	−2.93298	−0.391936
$57$	0	0
$58$	9.76057	1.28163
$59$	−3.92402	−0.510864	−0.255432	−	0.966827i	$-0.582218\pi$
$59$	−3.92402	−0.510864	−0.255432	+	0.966827i	$0.582218\pi$
$60$	0	0
$61$	−3.48704	−0.446469	−0.223235	−	0.974765i	$-0.571662\pi$
$61$	−3.48704	−0.446469	−0.223235	+	0.974765i	$0.571662\pi$
$62$	5.19480	0.659740
$63$	0	0
$64$	6.13195	0.766494
$65$	3.09118	0.383414
$66$	0	0
$67$	9.31796	1.13837	0.569185	−	0.822209i	$-0.307259\pi$
$67$	9.31796	1.13837	0.569185	+	0.822209i	$0.307259\pi$
$68$	−8.02984	−0.973761
$69$	0	0
$70$	−1.91127	−0.228441
$71$	6.78717	0.805489	0.402744	−	0.915313i	$-0.368056\pi$
$71$	6.78717	0.805489	0.402744	+	0.915313i	$0.368056\pi$
$72$	0	0
$73$	−11.2119	−1.31225	−0.656125	−	0.754652i	$-0.727806\pi$
$73$	−11.2119	−1.31225	−0.656125	+	0.754652i	$0.727806\pi$
$74$	−1.05235	−0.122333
$75$	0	0
$76$	−5.92424	−0.679557
$77$	0.393608	0.0448558
$78$	0	0
$79$	−5.16136	−0.580698	−0.290349	−	0.956921i	$-0.593771\pi$
$79$	−5.16136	−0.580698	−0.290349	+	0.956921i	$0.593771\pi$
$80$	1.09891	0.122862
$81$	0	0
$82$	−11.3239	−1.25052
$83$	6.24052	0.684986	0.342493	−	0.939520i	$-0.388729\pi$
$83$	6.24052	0.684986	0.342493	+	0.939520i	$0.388729\pi$
$84$	0	0
$85$	−14.6490	−1.58890
$86$	0.202772	0.0218655
$87$	0	0
$88$	−1.15444	−0.123064
$89$	13.9448	1.47815	0.739074	−	0.673624i	$-0.235263\pi$
$89$	13.9448	1.47815	0.739074	+	0.673624i	$0.235263\pi$
$90$	0	0
$91$	1.52460	0.159821
$92$	6.83062	0.712141
$93$	0	0
$94$	−3.82707	−0.394732
$95$	−10.8077	−1.10884
$96$	0	0
$97$	10.5474	1.07093	0.535465	−	0.844557i	$-0.320136\pi$
$97$	10.5474	1.07093	0.535465	+	0.844557i	$0.320136\pi$
$98$	−0.942656	−0.0952226
$99$	0	0
$100$	0.988114	0.0988114
$101$	15.7128	1.56348	0.781739	−	0.623605i	$-0.214333\pi$
$101$	15.7128	1.56348	0.781739	+	0.623605i	$0.214333\pi$
$102$	0	0
$103$	4.27528	0.421256	0.210628	−	0.977566i	$-0.432449\pi$
$103$	4.27528	0.421256	0.210628	+	0.977566i	$0.432449\pi$
$104$	−4.47161	−0.438477
$105$	0	0
$106$	5.70410	0.554031
$107$	8.30037	0.802427	0.401214	−	0.915985i	$-0.368588\pi$
$107$	8.30037	0.802427	0.401214	+	0.915985i	$0.368588\pi$
$108$	0	0
$109$	5.24166	0.502060	0.251030	−	0.967979i	$-0.419231\pi$
$109$	5.24166	0.502060	0.251030	+	0.967979i	$0.419231\pi$
$110$	−0.752293	−0.0717284
$111$	0	0
$112$	0.541991	0.0512133
$113$	1.08075	0.101669	0.0508344	−	0.998707i	$-0.483812\pi$
$113$	1.08075	0.101669	0.0508344	+	0.998707i	$0.483812\pi$
$114$	0	0
$115$	12.4612	1.16201
$116$	11.5078	1.06847
$117$	0	0
$118$	3.69900	0.340521
$119$	−7.22498	−0.662313
$120$	0	0
$121$	−10.8451	−0.985916
$122$	3.28708	0.297598
$123$	0	0
$124$	6.12472	0.550016
$125$	11.9403	1.06798
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	4.92978	0.435735
$129$	0	0
$130$	−2.91392	−0.255568
$131$	7.36931	0.643860	0.321930	−	0.946764i	$-0.395668\pi$
$131$	7.36931	0.643860	0.321930	+	0.946764i	$0.395668\pi$
$132$	0	0
$133$	−5.33043	−0.462207
$134$	−8.78363	−0.758790
$135$	0	0
$136$	21.1907	1.81709
$137$	2.37680	0.203063	0.101532	−	0.994832i	$-0.467626\pi$
$137$	2.37680	0.203063	0.101532	+	0.994832i	$0.467626\pi$
$138$	0	0
$139$	−14.1222	−1.19783	−0.598916	−	0.800812i	$-0.704402\pi$
$139$	−14.1222	−1.19783	−0.598916	+	0.800812i	$0.704402\pi$
$140$	−2.25341	−0.190448
$141$	0	0
$142$	−6.39796	−0.536905
$143$	0.600093	0.0501823
$144$	0	0
$145$	20.9939	1.74344
$146$	10.5689	0.874691
$147$	0	0
$148$	−1.24073	−0.101987
$149$	−13.5063	−1.10648	−0.553240	−	0.833022i	$-0.686609\pi$
$149$	−13.5063	−1.10648	−0.553240	+	0.833022i	$0.686609\pi$
$150$	0	0
$151$	21.6796	1.76426	0.882129	−	0.471007i	$-0.156109\pi$
$151$	21.6796	1.76426	0.882129	+	0.471007i	$0.156109\pi$
$152$	15.6340	1.26809
$153$	0	0
$154$	−0.371037	−0.0298990
$155$	11.1734	0.897469
$156$	0	0
$157$	−12.2097	−0.974438	−0.487219	−	0.873280i	$-0.661989\pi$
$157$	−12.2097	−0.974438	−0.487219	+	0.873280i	$0.661989\pi$
$158$	4.86538	0.387069
$159$	0	0
$160$	10.8576	0.858368
$161$	6.14596	0.484369
$162$	0	0
$163$	−11.4746	−0.898759	−0.449379	−	0.893341i	$-0.648355\pi$
$163$	−11.4746	−0.898759	−0.449379	+	0.893341i	$0.648355\pi$
$164$	−13.3510	−1.04254
$165$	0	0
$166$	−5.88266	−0.456583
$167$	9.48483	0.733958	0.366979	−	0.930229i	$-0.380392\pi$
$167$	9.48483	0.733958	0.366979	+	0.930229i	$0.380392\pi$
$168$	0	0
$169$	−10.6756	−0.821201
$170$	13.8089	1.05910
$171$	0	0
$172$	0.239071	0.0182290
$173$	21.7904	1.65669	0.828346	−	0.560216i	$-0.189282\pi$
$173$	21.7904	1.65669	0.828346	+	0.560216i	$0.189282\pi$
$174$	0	0
$175$	0.889072	0.0672075
$176$	0.213332	0.0160805
$177$	0	0
$178$	−13.1452	−0.985272
$179$	−4.32122	−0.322983	−0.161492	−	0.986874i	$-0.551630\pi$
$179$	−4.32122	−0.322983	−0.161492	+	0.986874i	$0.551630\pi$
$180$	0	0
$181$	9.17405	0.681902	0.340951	−	0.940081i	$-0.389251\pi$
$181$	9.17405	0.681902	0.340951	+	0.940081i	$0.389251\pi$
$182$	−1.43717	−0.106530
$183$	0	0
$184$	−18.0260	−1.32889
$185$	−2.26348	−0.166414
$186$	0	0
$187$	−2.84381	−0.207960
$188$	−4.51215	−0.329082
$189$	0	0
$190$	10.1879	0.739109
$191$	14.7115	1.06449	0.532244	−	0.846591i	$-0.321349\pi$
$191$	14.7115	1.06449	0.532244	+	0.846591i	$0.321349\pi$
$192$	0	0
$193$	−12.2579	−0.882345	−0.441173	−	0.897422i	$-0.645437\pi$
$193$	−12.2579	−0.882345	−0.441173	+	0.897422i	$0.645437\pi$
$194$	−9.94261	−0.713838
$195$	0	0
$196$	−1.11140	−0.0793857
$197$	3.36875	0.240013	0.120007	−	0.992773i	$-0.461708\pi$
$197$	3.36875	0.240013	0.120007	+	0.992773i	$0.461708\pi$
$198$	0	0
$199$	−20.0623	−1.42218	−0.711089	−	0.703102i	$-0.751798\pi$
$199$	−20.0623	−1.42218	−0.711089	+	0.703102i	$0.751798\pi$
$200$	−2.60763	−0.184387
$201$	0	0
$202$	−14.8117	−1.04215
$203$	10.3543	0.726732
$204$	0	0
$205$	−24.3565	−1.70113
$206$	−4.03012	−0.280792
$207$	0	0
$208$	0.826316	0.0572947
$209$	−2.09810	−0.145129
$210$	0	0
$211$	−26.1999	−1.80368	−0.901839	−	0.432071i	$-0.857783\pi$
$211$	−26.1999	−1.80368	−0.901839	+	0.432071i	$0.857783\pi$
$212$	6.72518	0.461887
$213$	0	0
$214$	−7.82440	−0.534865
$215$	0.436140	0.0297445
$216$	0	0
$217$	5.51081	0.374098
$218$	−4.94108	−0.334652
$219$	0	0
$220$	−0.886961	−0.0597989
$221$	−11.0152	−0.740961
$222$	0	0
$223$	−7.28885	−0.488098	−0.244049	−	0.969763i	$-0.578476\pi$
$223$	−7.28885	−0.488098	−0.244049	+	0.969763i	$0.578476\pi$
$224$	5.35505	0.357799
$225$	0	0
$226$	−1.01878	−0.0677681
$227$	−25.9000	−1.71904	−0.859522	−	0.511098i	$-0.829239\pi$
$227$	−25.9000	−1.71904	−0.859522	+	0.511098i	$0.829239\pi$
$228$	0	0
$229$	29.9555	1.97951	0.989757	−	0.142764i	$-0.0455990\pi$
$229$	29.9555	1.97951	0.989757	+	0.142764i	$0.0455990\pi$
$230$	−11.7466	−0.774549
$231$	0	0
$232$	−30.3691	−1.99383
$233$	−12.2443	−0.802152	−0.401076	−	0.916045i	$-0.631364\pi$
$233$	−12.2443	−0.802152	−0.401076	+	0.916045i	$0.631364\pi$
$234$	0	0
$235$	−8.23157	−0.536969
$236$	4.36116	0.283887
$237$	0	0
$238$	6.81067	0.441470
$239$	0.293187	0.0189647	0.00948234	−	0.999955i	$-0.496982\pi$
$239$	0.293187	0.0189647	0.00948234	+	0.999955i	$0.496982\pi$
$240$	0	0
$241$	−17.4415	−1.12351	−0.561753	−	0.827305i	$-0.689873\pi$
$241$	−17.4415	−1.12351	−0.561753	+	0.827305i	$0.689873\pi$
$242$	10.2232	0.657170
$243$	0	0
$244$	3.87549	0.248103
$245$	−2.02754	−0.129535
$246$	0	0
$247$	−8.12675	−0.517093
$248$	−16.1631	−1.02636
$249$	0	0
$250$	−11.2556	−0.711869
$251$	0.315392	0.0199074	0.00995368	−	0.999950i	$-0.496832\pi$
$251$	0.315392	0.0199074	0.00995368	+	0.999950i	$0.496832\pi$
$252$	0	0
$253$	2.41910	0.152087
$254$	0.942656	0.0591475
$255$	0	0
$256$	−16.9110	−1.05694
$257$	−6.88363	−0.429389	−0.214695	−	0.976681i	$-0.568876\pi$
$257$	−6.88363	−0.429389	−0.214695	+	0.976681i	$0.568876\pi$
$258$	0	0
$259$	−1.11637	−0.0693676
$260$	−3.43554	−0.213063
$261$	0	0
$262$	−6.94672	−0.429170
$263$	−17.9942	−1.10957	−0.554784	−	0.831994i	$-0.687199\pi$
$263$	−17.9942	−1.10957	−0.554784	+	0.831994i	$0.687199\pi$
$264$	0	0
$265$	12.2688	0.753669
$266$	5.02476	0.308088
$267$	0	0
$268$	−10.3560	−0.632592
$269$	−6.25866	−0.381597	−0.190799	−	0.981629i	$-0.561108\pi$
$269$	−6.25866	−0.381597	−0.190799	+	0.981629i	$0.561108\pi$
$270$	0	0
$271$	−15.0536	−0.914443	−0.457222	−	0.889353i	$-0.651155\pi$
$271$	−15.0536	−0.914443	−0.457222	+	0.889353i	$0.651155\pi$
$272$	−3.91587	−0.237435
$273$	0	0
$274$	−2.24050	−0.135354
$275$	0.349946	0.0211025
$276$	0	0
$277$	−14.9805	−0.900090	−0.450045	−	0.893006i	$-0.648592\pi$
$277$	−14.9805	−0.900090	−0.450045	+	0.893006i	$0.648592\pi$
$278$	13.3124	0.798425
$279$	0	0
$280$	5.94674	0.355386
$281$	−28.4947	−1.69985	−0.849925	−	0.526903i	$-0.823353\pi$
$281$	−28.4947	−1.69985	−0.849925	+	0.526903i	$0.823353\pi$
$282$	0	0
$283$	31.7506	1.88737	0.943687	−	0.330839i	$-0.107332\pi$
$283$	31.7506	1.88737	0.943687	+	0.330839i	$0.107332\pi$
$284$	−7.54326	−0.447610
$285$	0	0
$286$	−0.565681	−0.0334494
$287$	−12.0128	−0.709094
$288$	0	0
$289$	35.2003	2.07061
$290$	−19.7900	−1.16211
$291$	0	0
$292$	12.4609	0.729217
$293$	26.7569	1.56316	0.781578	−	0.623807i	$-0.214415\pi$
$293$	26.7569	1.56316	0.781578	+	0.623807i	$0.214415\pi$
$294$	0	0
$295$	7.95613	0.463224
$296$	3.27428	0.190314
$297$	0	0
$298$	12.7318	0.737533
$299$	9.37010	0.541887
$300$	0	0
$301$	0.215108	0.0123986
$302$	−20.4364	−1.17598
$303$	0	0
$304$	−2.88904	−0.165698
$305$	7.07011	0.404834
$306$	0	0
$307$	12.9334	0.738147	0.369073	−	0.929400i	$-0.379675\pi$
$307$	12.9334	0.738147	0.369073	+	0.929400i	$0.379675\pi$
$308$	−0.437456	−0.0249264
$309$	0	0
$310$	−10.5327	−0.598216
$311$	−33.8815	−1.92124	−0.960620	−	0.277864i	$-0.910374\pi$
$311$	−33.8815	−1.92124	−0.960620	+	0.277864i	$0.910374\pi$
$312$	0	0
$313$	−29.8784	−1.68883	−0.844413	−	0.535692i	$-0.820051\pi$
$313$	−29.8784	−1.68883	−0.844413	+	0.535692i	$0.820051\pi$
$314$	11.5095	0.649520
$315$	0	0
$316$	5.73633	0.322694
$317$	−12.1570	−0.682804	−0.341402	−	0.939917i	$-0.610902\pi$
$317$	−12.1570	−0.682804	−0.341402	+	0.939917i	$0.610902\pi$
$318$	0	0
$319$	4.07555	0.228187
$320$	−12.4328	−0.695014
$321$	0	0
$322$	−5.79352	−0.322860
$323$	38.5123	2.14288
$324$	0	0
$325$	1.35547	0.0751882
$326$	10.8166	0.599075
$327$	0	0
$328$	35.2333	1.94544
$329$	−4.05988	−0.223828
$330$	0	0
$331$	−3.00611	−0.165231	−0.0826155	−	0.996581i	$-0.526327\pi$
$331$	−3.00611	−0.165231	−0.0826155	+	0.996581i	$0.526327\pi$
$332$	−6.93571	−0.380647
$333$	0	0
$334$	−8.94093	−0.489226
$335$	−18.8926	−1.03221
$336$	0	0
$337$	−16.6283	−0.905799	−0.452899	−	0.891562i	$-0.649610\pi$
$337$	−16.6283	−0.905799	−0.452899	+	0.891562i	$0.649610\pi$
$338$	10.0634	0.547378
$339$	0	0
$340$	16.2808	0.882953
$341$	2.16910	0.117463
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−0.630906	−0.0340162
$345$	0	0
$346$	−20.5408	−1.10428
$347$	22.0246	1.18234	0.591172	−	0.806546i	$-0.298666\pi$
$347$	22.0246	1.18234	0.591172	+	0.806546i	$0.298666\pi$
$348$	0	0
$349$	5.78704	0.309773	0.154887	−	0.987932i	$-0.450499\pi$
$349$	5.78704	0.309773	0.154887	+	0.987932i	$0.450499\pi$
$350$	−0.838089	−0.0447977
$351$	0	0
$352$	2.10779	0.112346
$353$	−19.2731	−1.02580	−0.512902	−	0.858447i	$-0.671429\pi$
$353$	−19.2731	−1.02580	−0.512902	+	0.858447i	$0.671429\pi$
$354$	0	0
$355$	−13.7613	−0.730372
$356$	−15.4983	−0.821407
$357$	0	0
$358$	4.07343	0.215287
$359$	6.30280	0.332649	0.166325	−	0.986071i	$-0.446810\pi$
$359$	6.30280	0.332649	0.166325	+	0.986071i	$0.446810\pi$
$360$	0	0
$361$	9.41349	0.495447
$362$	−8.64798	−0.454528
$363$	0	0
$364$	−1.69443	−0.0888125
$365$	22.7325	1.18988
$366$	0	0
$367$	−15.8154	−0.825559	−0.412780	−	0.910831i	$-0.635442\pi$
$367$	−15.8154	−0.825559	−0.412780	+	0.910831i	$0.635442\pi$
$368$	3.33105	0.173643
$369$	0	0
$370$	2.13368	0.110925
$371$	6.05109	0.314157
$372$	0	0
$373$	−28.5380	−1.47764	−0.738821	−	0.673902i	$-0.764617\pi$
$373$	−28.5380	−1.47764	−0.738821	+	0.673902i	$0.764617\pi$
$374$	2.68074	0.138618
$375$	0	0
$376$	11.9075	0.614084
$377$	15.7862	0.813029
$378$	0	0
$379$	−33.4675	−1.71911	−0.859554	−	0.511045i	$-0.829258\pi$
$379$	−33.4675	−1.71911	−0.859554	+	0.511045i	$0.829258\pi$
$380$	12.0116	0.616185
$381$	0	0
$382$	−13.8679	−0.709543
$383$	−22.2341	−1.13611	−0.568055	−	0.822990i	$-0.692304\pi$
$383$	−22.2341	−1.13611	−0.568055	+	0.822990i	$0.692304\pi$
$384$	0	0
$385$	−0.798057	−0.0406728
$386$	11.5550	0.588135
$387$	0	0
$388$	−11.7224	−0.595116
$389$	19.0317	0.964943	0.482472	−	0.875912i	$-0.339739\pi$
$389$	19.0317	0.964943	0.482472	+	0.875912i	$0.339739\pi$
$390$	0	0
$391$	−44.4044	−2.24563
$392$	2.93298	0.148138
$393$	0	0
$394$	−3.17557	−0.159983
$395$	10.4649	0.526545
$396$	0	0
$397$	−6.12683	−0.307497	−0.153748	−	0.988110i	$-0.549135\pi$
$397$	−6.12683	−0.307497	−0.153748	+	0.988110i	$0.549135\pi$
$398$	18.9118	0.947964
$399$	0	0
$400$	0.481868	0.0240934
$401$	−7.88878	−0.393947	−0.196974	−	0.980409i	$-0.563111\pi$
$401$	−7.88878	−0.393947	−0.196974	+	0.980409i	$0.563111\pi$
$402$	0	0
$403$	8.40176	0.418521
$404$	−17.4632	−0.868825
$405$	0	0
$406$	−9.76057	−0.484409
$407$	−0.439411	−0.0217808
$408$	0	0
$409$	34.4698	1.70442	0.852211	−	0.523198i	$-0.175261\pi$
$409$	34.4698	1.70442	0.852211	+	0.523198i	$0.175261\pi$
$410$	22.9598	1.13390
$411$	0	0
$412$	−4.75155	−0.234092
$413$	3.92402	0.193089
$414$	0	0
$415$	−12.6529	−0.621107
$416$	8.16428	0.400287
$417$	0	0
$418$	1.97779	0.0967367
$419$	24.4461	1.19427	0.597135	−	0.802141i	$-0.296306\pi$
$419$	24.4461	1.19427	0.597135	+	0.802141i	$0.296306\pi$
$420$	0	0
$421$	29.1674	1.42153	0.710767	−	0.703428i	$-0.248348\pi$
$421$	29.1674	1.42153	0.710767	+	0.703428i	$0.248348\pi$
$422$	24.6975	1.20226
$423$	0	0
$424$	−17.7477	−0.861906
$425$	−6.42353	−0.311587
$426$	0	0
$427$	3.48704	0.168750
$428$	−9.22503	−0.445909
$429$	0	0
$430$	−0.411130	−0.0198264
$431$	−8.12744	−0.391485	−0.195742	−	0.980655i	$-0.562712\pi$
$431$	−8.12744	−0.391485	−0.195742	+	0.980655i	$0.562712\pi$
$432$	0	0
$433$	−12.8060	−0.615419	−0.307709	−	0.951480i	$-0.599562\pi$
$433$	−12.8060	−0.615419	−0.307709	+	0.951480i	$0.599562\pi$
$434$	−5.19480	−0.249358
$435$	0	0
$436$	−5.82558	−0.278995
$437$	−32.7606	−1.56715
$438$	0	0
$439$	−18.2637	−0.871679	−0.435840	−	0.900024i	$-0.643549\pi$
$439$	−18.2637	−0.871679	−0.435840	+	0.900024i	$0.643549\pi$
$440$	2.34069	0.111588
$441$	0	0
$442$	10.3835	0.493893
$443$	6.35839	0.302096	0.151048	−	0.988526i	$-0.451735\pi$
$443$	6.35839	0.302096	0.151048	+	0.988526i	$0.451735\pi$
$444$	0	0
$445$	−28.2737	−1.34030
$446$	6.87088	0.325346
$447$	0	0
$448$	−6.13195	−0.289707
$449$	37.4797	1.76878	0.884388	−	0.466753i	$-0.154576\pi$
$449$	37.4797	1.76878	0.884388	+	0.466753i	$0.154576\pi$
$450$	0	0
$451$	−4.72834	−0.222649
$452$	−1.20115	−0.0564973
$453$	0	0
$454$	24.4148	1.14584
$455$	−3.09118	−0.144917
$456$	0	0
$457$	38.1237	1.78335	0.891677	−	0.452673i	$-0.149529\pi$
$457$	38.1237	1.78335	0.891677	+	0.452673i	$0.149529\pi$
$458$	−28.2377	−1.31946
$459$	0	0
$460$	−13.8494	−0.645730
$461$	−4.34534	−0.202382	−0.101191	−	0.994867i	$-0.532265\pi$
$461$	−4.34534	−0.202382	−0.101191	+	0.994867i	$0.532265\pi$
$462$	0	0
$463$	32.5836	1.51429	0.757144	−	0.653248i	$-0.226594\pi$
$463$	32.5836	1.51429	0.757144	+	0.653248i	$0.226594\pi$
$464$	5.61195	0.260528
$465$	0	0
$466$	11.5422	0.534681
$467$	7.91842	0.366421	0.183210	−	0.983074i	$-0.441351\pi$
$467$	7.91842	0.366421	0.183210	+	0.983074i	$0.441351\pi$
$468$	0	0
$469$	−9.31796	−0.430263
$470$	7.75954	0.357921
$471$	0	0
$472$	−11.5091	−0.529748
$473$	0.0846681	0.00389304
$474$	0	0
$475$	−4.73914	−0.217446
$476$	8.02984	0.368047
$477$	0	0
$478$	−0.276374	−0.0126411
$479$	−28.9770	−1.32399	−0.661996	−	0.749508i	$-0.730290\pi$
$479$	−28.9770	−1.32399	−0.661996	+	0.749508i	$0.730290\pi$
$480$	0	0
$481$	−1.70201	−0.0776048
$482$	16.4413	0.748883
$483$	0	0
$484$	12.0532	0.547873
$485$	−21.3854	−0.971061
$486$	0	0
$487$	−9.20818	−0.417262	−0.208631	−	0.977994i	$-0.566901\pi$
$487$	−9.20818	−0.417262	−0.208631	+	0.977994i	$0.566901\pi$
$488$	−10.2274	−0.462973
$489$	0	0
$490$	1.91127	0.0863426
$491$	36.3971	1.64258	0.821288	−	0.570513i	$-0.193256\pi$
$491$	36.3971	1.64258	0.821288	+	0.570513i	$0.193256\pi$
$492$	0	0
$493$	−74.8099	−3.36927
$494$	7.66073	0.344672
$495$	0	0
$496$	2.98681	0.134112
$497$	−6.78717	−0.304446
$498$	0	0
$499$	8.80459	0.394148	0.197074	−	0.980389i	$-0.436856\pi$
$499$	8.80459	0.394148	0.197074	+	0.980389i	$0.436856\pi$
$500$	−13.2705	−0.593475
$501$	0	0
$502$	−0.297306	−0.0132694
$503$	−30.0843	−1.34139	−0.670696	−	0.741732i	$-0.734004\pi$
$503$	−30.0843	−1.34139	−0.670696	+	0.741732i	$0.734004\pi$
$504$	0	0
$505$	−31.8583	−1.41768
$506$	−2.28038	−0.101375
$507$	0	0
$508$	1.11140	0.0493104
$509$	−14.6613	−0.649849	−0.324924	−	0.945740i	$-0.605339\pi$
$509$	−14.6613	−0.649849	−0.324924	+	0.945740i	$0.605339\pi$
$510$	0	0
$511$	11.2119	0.495984
$512$	6.08168	0.268775
$513$	0	0
$514$	6.48890	0.286213
$515$	−8.66831	−0.381972
$516$	0	0
$517$	−1.59800	−0.0702800
$518$	1.05235	0.0462376
$519$	0	0
$520$	9.06637	0.397587
$521$	23.4040	1.02535	0.512674	−	0.858583i	$-0.328655\pi$
$521$	23.4040	1.02535	0.512674	+	0.858583i	$0.328655\pi$
$522$	0	0
$523$	−1.58294	−0.0692173	−0.0346087	−	0.999401i	$-0.511018\pi$
$523$	−1.58294	−0.0692173	−0.0346087	+	0.999401i	$0.511018\pi$
$524$	−8.19025	−0.357793
$525$	0	0
$526$	16.9623	0.739592
$527$	−39.8155	−1.73439
$528$	0	0
$529$	14.7728	0.642296
$530$	−11.5653	−0.502364
$531$	0	0
$532$	5.92424	0.256848
$533$	−18.3147	−0.793297
$534$	0	0
$535$	−16.8294	−0.727597
$536$	27.3294	1.18045
$537$	0	0
$538$	5.89976	0.254357
$539$	−0.393608	−0.0169539
$540$	0	0
$541$	−16.8950	−0.726372	−0.363186	−	0.931717i	$-0.618311\pi$
$541$	−16.8950	−0.726372	−0.363186	+	0.931717i	$0.618311\pi$
$542$	14.1904	0.609530
$543$	0	0
$544$	−38.6901	−1.65883
$545$	−10.6277	−0.455240
$546$	0	0
$547$	−28.0686	−1.20013	−0.600064	−	0.799952i	$-0.704858\pi$
$547$	−28.0686	−1.20013	−0.600064	+	0.799952i	$0.704858\pi$
$548$	−2.64157	−0.112842
$549$	0	0
$550$	−0.329879	−0.0140661
$551$	−55.1931	−2.35130
$552$	0	0
$553$	5.16136	0.219483
$554$	14.1214	0.599963
$555$	0	0
$556$	15.6954	0.665635
$557$	−30.4192	−1.28890	−0.644452	−	0.764645i	$-0.722914\pi$
$557$	−30.4192	−1.28890	−0.644452	+	0.764645i	$0.722914\pi$
$558$	0	0
$559$	0.327952	0.0138709
$560$	−1.09891	−0.0464374
$561$	0	0
$562$	26.8607	1.13305
$563$	38.0568	1.60390	0.801952	−	0.597388i	$-0.203795\pi$
$563$	38.0568	1.60390	0.801952	+	0.597388i	$0.203795\pi$
$564$	0	0
$565$	−2.19127	−0.0921876
$566$	−29.9298	−1.25805
$567$	0	0
$568$	19.9066	0.835263
$569$	8.39927	0.352116	0.176058	−	0.984380i	$-0.443665\pi$
$569$	8.39927	0.352116	0.176058	+	0.984380i	$0.443665\pi$
$570$	0	0
$571$	−33.2531	−1.39160	−0.695799	−	0.718236i	$-0.744950\pi$
$571$	−33.2531	−1.39160	−0.695799	+	0.718236i	$0.744950\pi$
$572$	−0.666943	−0.0278863
$573$	0	0
$574$	11.3239	0.472652
$575$	5.46420	0.227873
$576$	0	0
$577$	−18.8513	−0.784790	−0.392395	−	0.919797i	$-0.628353\pi$
$577$	−18.8513	−0.784790	−0.392395	+	0.919797i	$0.628353\pi$
$578$	−33.1818	−1.38018
$579$	0	0
$580$	−23.3326	−0.968832
$581$	−6.24052	−0.258900
$582$	0	0
$583$	2.38176	0.0986423
$584$	−32.8842	−1.36076
$585$	0	0
$586$	−25.2226	−1.04194
$587$	28.6206	1.18130	0.590648	−	0.806929i	$-0.298872\pi$
$587$	28.6206	1.18130	0.590648	+	0.806929i	$0.298872\pi$
$588$	0	0
$589$	−29.3750	−1.21038
$590$	−7.49989	−0.308766
$591$	0	0
$592$	−0.605060	−0.0248678
$593$	13.7191	0.563377	0.281689	−	0.959506i	$-0.409105\pi$
$593$	13.7191	0.563377	0.281689	+	0.959506i	$0.409105\pi$
$594$	0	0
$595$	14.6490	0.600549
$596$	15.0109	0.614870
$597$	0	0
$598$	−8.83278	−0.361199
$599$	−40.0016	−1.63442	−0.817211	−	0.576338i	$-0.804481\pi$
$599$	−40.0016	−1.63442	−0.817211	+	0.576338i	$0.804481\pi$
$600$	0	0
$601$	−10.8800	−0.443806	−0.221903	−	0.975069i	$-0.571227\pi$
$601$	−10.8800	−0.443806	−0.221903	+	0.975069i	$0.571227\pi$
$602$	−0.202772	−0.00826439
$603$	0	0
$604$	−24.0947	−0.980398
$605$	21.9888	0.893974
$606$	0	0
$607$	−20.1563	−0.818119	−0.409060	−	0.912508i	$-0.634143\pi$
$607$	−20.1563	−0.818119	−0.409060	+	0.912508i	$0.634143\pi$
$608$	−28.5447	−1.15764
$609$	0	0
$610$	−6.66468	−0.269845
$611$	−6.18967	−0.250407
$612$	0	0
$613$	24.4960	0.989382	0.494691	−	0.869069i	$-0.335281\pi$
$613$	24.4960	0.989382	0.494691	+	0.869069i	$0.335281\pi$
$614$	−12.1917	−0.492018
$615$	0	0
$616$	1.15444	0.0465139
$617$	33.8571	1.36304	0.681518	−	0.731801i	$-0.261320\pi$
$617$	33.8571	1.36304	0.681518	+	0.731801i	$0.261320\pi$
$618$	0	0
$619$	−47.1828	−1.89644	−0.948219	−	0.317619i	$-0.897117\pi$
$619$	−47.1828	−1.89644	−0.948219	+	0.317619i	$0.897117\pi$
$620$	−12.4181	−0.498724
$621$	0	0
$622$	31.9386	1.28062
$623$	−13.9448	−0.558688
$624$	0	0
$625$	−19.7642	−0.790568
$626$	28.1650	1.12570
$627$	0	0
$628$	13.5698	0.541495
$629$	8.06572	0.321601
$630$	0	0
$631$	−6.13033	−0.244045	−0.122022	−	0.992527i	$-0.538938\pi$
$631$	−6.13033	−0.244045	−0.122022	+	0.992527i	$0.538938\pi$
$632$	−15.1381	−0.602163
$633$	0	0
$634$	11.4599	0.455129
$635$	2.02754	0.0804606
$636$	0	0
$637$	−1.52460	−0.0604067
$638$	−3.84184	−0.152100
$639$	0	0
$640$	−9.99534	−0.395100
$641$	15.4511	0.610281	0.305140	−	0.952307i	$-0.401297\pi$
$641$	15.4511	0.610281	0.305140	+	0.952307i	$0.401297\pi$
$642$	0	0
$643$	28.9559	1.14191	0.570955	−	0.820982i	$-0.306573\pi$
$643$	28.9559	1.14191	0.570955	+	0.820982i	$0.306573\pi$
$644$	−6.83062	−0.269164
$645$	0	0
$646$	−36.3038	−1.42835
$647$	−17.7543	−0.697993	−0.348996	−	0.937124i	$-0.613477\pi$
$647$	−17.7543	−0.697993	−0.348996	+	0.937124i	$0.613477\pi$
$648$	0	0
$649$	1.54453	0.0606280
$650$	−1.27775	−0.0501173
$651$	0	0
$652$	12.7529	0.499440
$653$	−47.4601	−1.85726	−0.928628	−	0.371013i	$-0.879011\pi$
$653$	−47.4601	−1.85726	−0.928628	+	0.371013i	$0.879011\pi$
$654$	0	0
$655$	−14.9416	−0.583816
$656$	−6.51083	−0.254205
$657$	0	0
$658$	3.82707	0.149195
$659$	4.97190	0.193678	0.0968388	−	0.995300i	$-0.469127\pi$
$659$	4.97190	0.193678	0.0968388	+	0.995300i	$0.469127\pi$
$660$	0	0
$661$	−5.29780	−0.206060	−0.103030	−	0.994678i	$-0.532854\pi$
$661$	−5.29780	−0.206060	−0.103030	+	0.994678i	$0.532854\pi$
$662$	2.83373	0.110136
$663$	0	0
$664$	18.3033	0.710306
$665$	10.8077	0.419104
$666$	0	0
$667$	63.6373	2.46405
$668$	−10.5414	−0.407860
$669$	0	0
$670$	17.8092	0.688029
$671$	1.37253	0.0529858
$672$	0	0
$673$	−30.4834	−1.17505	−0.587524	−	0.809206i	$-0.699897\pi$
$673$	−30.4834	−1.17505	−0.587524	+	0.809206i	$0.699897\pi$
$674$	15.6747	0.603768
$675$	0	0
$676$	11.8649	0.456341
$677$	1.37014	0.0526588	0.0263294	−	0.999653i	$-0.491618\pi$
$677$	1.37014	0.0526588	0.0263294	+	0.999653i	$0.491618\pi$
$678$	0	0
$679$	−10.5474	−0.404774
$680$	−42.9651	−1.64764
$681$	0	0
$682$	−2.04472	−0.0782962
$683$	−39.0704	−1.49499	−0.747494	−	0.664268i	$-0.768743\pi$
$683$	−39.0704	−1.49499	−0.747494	+	0.664268i	$0.768743\pi$
$684$	0	0
$685$	−4.81905	−0.184127
$686$	0.942656	0.0359908
$687$	0	0
$688$	0.116586	0.00444481
$689$	9.22546	0.351462
$690$	0	0
$691$	39.0221	1.48447	0.742235	−	0.670140i	$-0.233766\pi$
$691$	39.0221	1.48447	0.742235	+	0.670140i	$0.233766\pi$
$692$	−24.2178	−0.920624
$693$	0	0
$694$	−20.7616	−0.788101
$695$	28.6334	1.08613
$696$	0	0
$697$	86.7923	3.28749
$698$	−5.45519	−0.206482
$699$	0	0
$700$	−0.988114	−0.0373472
$701$	5.33026	0.201321	0.100661	−	0.994921i	$-0.467904\pi$
$701$	5.33026	0.201321	0.100661	+	0.994921i	$0.467904\pi$
$702$	0	0
$703$	5.95071	0.224435
$704$	−2.41359	−0.0909654
$705$	0	0
$706$	18.1679	0.683758
$707$	−15.7128	−0.590939
$708$	0	0
$709$	8.31243	0.312180	0.156090	−	0.987743i	$-0.450111\pi$
$709$	8.31243	0.312180	0.156090	+	0.987743i	$0.450111\pi$
$710$	12.9721	0.486836
$711$	0	0
$712$	40.8999	1.53279
$713$	33.8692	1.26841
$714$	0	0
$715$	−1.21671	−0.0455025
$716$	4.80261	0.179482
$717$	0	0
$718$	−5.94137	−0.221730
$719$	15.6948	0.585319	0.292659	−	0.956217i	$-0.405460\pi$
$719$	15.6948	0.585319	0.292659	+	0.956217i	$0.405460\pi$
$720$	0	0
$721$	−4.27528	−0.159220
$722$	−8.87368	−0.330244
$723$	0	0
$724$	−10.1960	−0.378933
$725$	9.20575	0.341893
$726$	0	0
$727$	24.8055	0.919985	0.459993	−	0.887923i	$-0.347852\pi$
$727$	24.8055	0.919985	0.459993	+	0.887923i	$0.347852\pi$
$728$	4.47161	0.165729
$729$	0	0
$730$	−21.4290	−0.793121
$731$	−1.55415	−0.0574822
$732$	0	0
$733$	−40.1964	−1.48469	−0.742344	−	0.670018i	$-0.766286\pi$
$733$	−40.1964	−1.48469	−0.742344	+	0.670018i	$0.766286\pi$
$734$	14.9085	0.550283
$735$	0	0
$736$	32.9119	1.21315
$737$	−3.66763	−0.135099
$738$	0	0
$739$	−8.54978	−0.314509	−0.157254	−	0.987558i	$-0.550264\pi$
$739$	−8.54978	−0.314509	−0.157254	+	0.987558i	$0.550264\pi$
$740$	2.51563	0.0924764
$741$	0	0
$742$	−5.70410	−0.209404
$743$	16.6201	0.609732	0.304866	−	0.952395i	$-0.401388\pi$
$743$	16.6201	0.609732	0.304866	+	0.952395i	$0.401388\pi$
$744$	0	0
$745$	27.3846	1.00329
$746$	26.9015	0.984934
$747$	0	0
$748$	3.16061	0.115563
$749$	−8.30037	−0.303289
$750$	0	0
$751$	−24.5495	−0.895825	−0.447912	−	0.894077i	$-0.647832\pi$
$751$	−24.5495	−0.895825	−0.447912	+	0.894077i	$0.647832\pi$
$752$	−2.20041	−0.0802409
$753$	0	0
$754$	−14.8809	−0.541931
$755$	−43.9563	−1.59973
$756$	0	0
$757$	−11.8571	−0.430952	−0.215476	−	0.976509i	$-0.569130\pi$
$757$	−11.8571	−0.430952	−0.215476	+	0.976509i	$0.569130\pi$
$758$	31.5483	1.14589
$759$	0	0
$760$	−31.6987	−1.14983
$761$	−48.1062	−1.74385	−0.871924	−	0.489641i	$-0.837128\pi$
$761$	−48.1062	−1.74385	−0.871924	+	0.489641i	$0.837128\pi$
$762$	0	0
$763$	−5.24166	−0.189761
$764$	−16.3504	−0.591536
$765$	0	0
$766$	20.9591	0.757284
$767$	5.98255	0.216017
$768$	0	0
$769$	−33.7773	−1.21804	−0.609021	−	0.793154i	$-0.708437\pi$
$769$	−33.7773	−1.21804	−0.609021	+	0.793154i	$0.708437\pi$
$770$	0.752293	0.0271108
$771$	0	0
$772$	13.6235	0.490319
$773$	−22.2790	−0.801321	−0.400661	−	0.916227i	$-0.631219\pi$
$773$	−22.2790	−0.801321	−0.400661	+	0.916227i	$0.631219\pi$
$774$	0	0
$775$	4.89951	0.175995
$776$	30.9354	1.11052
$777$	0	0
$778$	−17.9403	−0.643191
$779$	64.0335	2.29424
$780$	0	0
$781$	−2.67148	−0.0955932
$782$	41.8581	1.49684
$783$	0	0
$784$	−0.541991	−0.0193568
$785$	24.7556	0.883566
$786$	0	0
$787$	−22.1520	−0.789634	−0.394817	−	0.918760i	$-0.629192\pi$
$787$	−22.1520	−0.789634	−0.394817	+	0.918760i	$0.629192\pi$
$788$	−3.74403	−0.133375
$789$	0	0
$790$	−9.86477	−0.350973
$791$	−1.08075	−0.0384272
$792$	0	0
$793$	5.31632	0.188788
$794$	5.77549	0.204965
$795$	0	0
$796$	22.2972	0.790304
$797$	6.03663	0.213828	0.106914	−	0.994268i	$-0.465903\pi$
$797$	6.03663	0.213828	0.106914	+	0.994268i	$0.465903\pi$
$798$	0	0
$799$	29.3325	1.03771
$800$	4.76102	0.168328
$801$	0	0
$802$	7.43641	0.262589
$803$	4.41308	0.155734
$804$	0	0
$805$	−12.4612	−0.439199
$806$	−7.91997	−0.278969
$807$	0	0
$808$	46.0852	1.62127
$809$	−1.64082	−0.0576880	−0.0288440	−	0.999584i	$-0.509183\pi$
$809$	−1.64082	−0.0576880	−0.0288440	+	0.999584i	$0.509183\pi$
$810$	0	0
$811$	24.1361	0.847532	0.423766	−	0.905772i	$-0.360708\pi$
$811$	24.1361	0.847532	0.423766	+	0.905772i	$0.360708\pi$
$812$	−11.5078	−0.403845
$813$	0	0
$814$	0.414213	0.0145182
$815$	23.2652	0.814945
$816$	0	0
$817$	−1.14662	−0.0401150
$818$	−32.4932	−1.13610
$819$	0	0
$820$	27.0698	0.945318
$821$	−8.59622	−0.300010	−0.150005	−	0.988685i	$-0.547929\pi$
$821$	−8.59622	−0.300010	−0.150005	+	0.988685i	$0.547929\pi$
$822$	0	0
$823$	11.4797	0.400156	0.200078	−	0.979780i	$-0.435880\pi$
$823$	11.4797	0.400156	0.200078	+	0.979780i	$0.435880\pi$
$824$	12.5393	0.436828
$825$	0	0
$826$	−3.69900	−0.128705
$827$	12.1662	0.423061	0.211531	−	0.977371i	$-0.432155\pi$
$827$	12.1662	0.423061	0.211531	+	0.977371i	$0.432155\pi$
$828$	0	0
$829$	−18.9606	−0.658530	−0.329265	−	0.944237i	$-0.606801\pi$
$829$	−18.9606	−0.658530	−0.329265	+	0.944237i	$0.606801\pi$
$830$	11.9273	0.414004
$831$	0	0
$832$	−9.34874	−0.324109
$833$	7.22498	0.250331
$834$	0	0
$835$	−19.2309	−0.665512
$836$	2.33183	0.0806480
$837$	0	0
$838$	−23.0443	−0.796051
$839$	37.6366	1.29936	0.649680	−	0.760208i	$-0.274903\pi$
$839$	37.6366	1.29936	0.649680	+	0.760208i	$0.274903\pi$
$840$	0	0
$841$	78.2122	2.69697
$842$	−27.4949	−0.947535
$843$	0	0
$844$	29.1186	1.00230
$845$	21.6453	0.744619
$846$	0	0
$847$	10.8451	0.372641
$848$	3.27963	0.112623
$849$	0	0
$850$	6.05517	0.207691
$851$	−6.86114	−0.235197
$852$	0	0
$853$	23.5742	0.807165	0.403583	−	0.914943i	$-0.367765\pi$
$853$	23.5742	0.807165	0.403583	+	0.914943i	$0.367765\pi$
$854$	−3.28708	−0.112481
$855$	0	0
$856$	24.3448	0.832089
$857$	−46.2533	−1.57998	−0.789991	−	0.613118i	$-0.789915\pi$
$857$	−46.2533	−1.57998	−0.789991	+	0.613118i	$0.789915\pi$
$858$	0	0
$859$	−38.4044	−1.31034	−0.655171	−	0.755480i	$-0.727404\pi$
$859$	−38.4044	−1.31034	−0.655171	+	0.755480i	$0.727404\pi$
$860$	−0.484726	−0.0165290
$861$	0	0
$862$	7.66138	0.260948
$863$	−40.9841	−1.39511	−0.697557	−	0.716529i	$-0.745730\pi$
$863$	−40.9841	−1.39511	−0.697557	+	0.716529i	$0.745730\pi$
$864$	0	0
$865$	−44.1809	−1.50220
$866$	12.0717	0.410212
$867$	0	0
$868$	−6.12472	−0.207886
$869$	2.03155	0.0689157
$870$	0	0
$871$	−14.2061	−0.481356
$872$	15.3737	0.520618
$873$	0	0
$874$	30.8820	1.04460
$875$	−11.9403	−0.403657
$876$	0	0
$877$	11.1446	0.376325	0.188163	−	0.982138i	$-0.439747\pi$
$877$	11.1446	0.376325	0.188163	+	0.982138i	$0.439747\pi$
$878$	17.2164	0.581025
$879$	0	0
$880$	−0.432539	−0.0145809
$881$	−50.5138	−1.70185	−0.850926	−	0.525285i	$-0.823959\pi$
$881$	−50.5138	−1.70185	−0.850926	+	0.525285i	$0.823959\pi$
$882$	0	0
$883$	26.0412	0.876357	0.438179	−	0.898888i	$-0.355624\pi$
$883$	26.0412	0.876357	0.438179	+	0.898888i	$0.355624\pi$
$884$	12.2423	0.411752
$885$	0	0
$886$	−5.99378	−0.201365
$887$	−31.8404	−1.06910	−0.534548	−	0.845138i	$-0.679518\pi$
$887$	−31.8404	−1.06910	−0.534548	+	0.845138i	$0.679518\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	26.6524	0.893390
$891$	0	0
$892$	8.10083	0.271236
$893$	21.6409	0.724185
$894$	0	0
$895$	8.76146	0.292863
$896$	−4.92978	−0.164692
$897$	0	0
$898$	−35.3304	−1.17899
$899$	57.0608	1.90308
$900$	0	0
$901$	−43.7190	−1.45649
$902$	4.45720	0.148408
$903$	0	0
$904$	3.16983	0.105427
$905$	−18.6008	−0.618311
$906$	0	0
$907$	−32.2737	−1.07163	−0.535815	−	0.844335i	$-0.679996\pi$
$907$	−32.2737	−1.07163	−0.535815	+	0.844335i	$0.679996\pi$
$908$	28.7853	0.955273
$909$	0	0
$910$	2.91392	0.0965955
$911$	−34.4345	−1.14087	−0.570433	−	0.821344i	$-0.693224\pi$
$911$	−34.4345	−1.14087	−0.570433	+	0.821344i	$0.693224\pi$
$912$	0	0
$913$	−2.45632	−0.0812923
$914$	−35.9376	−1.18871
$915$	0	0
$916$	−33.2925	−1.10002
$917$	−7.36931	−0.243356
$918$	0	0
$919$	5.27192	0.173904	0.0869522	−	0.996212i	$-0.472287\pi$
$919$	5.27192	0.173904	0.0869522	+	0.996212i	$0.472287\pi$
$920$	36.5484	1.20497
$921$	0	0
$922$	4.09616	0.134900
$923$	−10.3477	−0.340598
$924$	0	0
$925$	−0.992530	−0.0326342
$926$	−30.7151	−1.00936
$927$	0	0
$928$	55.4480	1.82017
$929$	7.44226	0.244173	0.122086	−	0.992519i	$-0.461042\pi$
$929$	7.44226	0.244173	0.122086	+	0.992519i	$0.461042\pi$
$930$	0	0
$931$	5.33043	0.174698
$932$	13.6083	0.445756
$933$	0	0
$934$	−7.46435	−0.244241
$935$	5.76595	0.188567
$936$	0	0
$937$	47.8635	1.56363	0.781816	−	0.623510i	$-0.214294\pi$
$937$	47.8635	1.56363	0.781816	+	0.623510i	$0.214294\pi$
$938$	8.78363	0.286796
$939$	0	0
$940$	9.14857	0.298393
$941$	34.0814	1.11102	0.555511	−	0.831509i	$-0.312523\pi$
$941$	34.0814	1.11102	0.555511	+	0.831509i	$0.312523\pi$
$942$	0	0
$943$	−73.8302	−2.40424
$944$	2.12678	0.0692209
$945$	0	0
$946$	−0.0798129	−0.00259494
$947$	−2.35915	−0.0766620	−0.0383310	−	0.999265i	$-0.512204\pi$
$947$	−2.35915	−0.0766620	−0.0383310	+	0.999265i	$0.512204\pi$
$948$	0	0
$949$	17.0936	0.554880
$950$	4.46737	0.144941
$951$	0	0
$952$	−21.1907	−0.686795
$953$	−38.2987	−1.24062	−0.620308	−	0.784358i	$-0.712993\pi$
$953$	−38.2987	−1.24062	−0.620308	+	0.784358i	$0.712993\pi$
$954$	0	0
$955$	−29.8282	−0.965218
$956$	−0.325848	−0.0105387
$957$	0	0
$958$	27.3153	0.882517
$959$	−2.37680	−0.0767507
$960$	0	0
$961$	−0.630958	−0.0203535
$962$	1.60441	0.0517281
$963$	0	0
$964$	19.3845	0.624333
$965$	24.8535	0.800062
$966$	0	0
$967$	26.3209	0.846423	0.423211	−	0.906031i	$-0.360903\pi$
$967$	26.3209	0.846423	0.423211	+	0.906031i	$0.360903\pi$
$968$	−31.8084	−1.02236
$969$	0	0
$970$	20.1591	0.647269
$971$	−28.2912	−0.907908	−0.453954	−	0.891025i	$-0.649987\pi$
$971$	−28.2912	−0.907908	−0.453954	+	0.891025i	$0.649987\pi$
$972$	0	0
$973$	14.1222	0.452738
$974$	8.68014	0.278130
$975$	0	0
$976$	1.88994	0.0604955
$977$	−12.7823	−0.408943	−0.204471	−	0.978873i	$-0.565548\pi$
$977$	−12.7823	−0.408943	−0.204471	+	0.978873i	$0.565548\pi$
$978$	0	0
$979$	−5.48880	−0.175423
$980$	2.25341	0.0719826
$981$	0	0
$982$	−34.3099	−1.09487
$983$	23.7874	0.758701	0.379350	−	0.925253i	$-0.376148\pi$
$983$	23.7874	0.758701	0.379350	+	0.925253i	$0.376148\pi$
$984$	0	0
$985$	−6.83028	−0.217631
$986$	70.5200	2.24581
$987$	0	0
$988$	9.03207	0.287348
$989$	1.32204	0.0420385
$990$	0	0
$991$	−13.9078	−0.441794	−0.220897	−	0.975297i	$-0.570899\pi$
$991$	−13.9078	−0.441794	−0.220897	+	0.975297i	$0.570899\pi$
$992$	29.5107	0.936964
$993$	0	0
$994$	6.39796	0.202931
$995$	40.6771	1.28955
$996$	0	0
$997$	−29.1698	−0.923816	−0.461908	−	0.886928i	$-0.652835\pi$
$997$	−29.1698	−0.923816	−0.461908	+	0.886928i	$0.652835\pi$
$998$	−8.29970	−0.262722
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.12	✓	32
3.2	odd	2	inner	8001.2.a.z.1.21	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.12	✓	32	1.1	even	1	trivial
8001.2.a.z.1.21	yes	32	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-0.942656 q^{2} -1.11140 q^{4} -2.02754 q^{5} -1.00000 q^{7} +2.93298 q^{8} +O(q^{10})\)	\(q-0.942656 q^{2} -1.11140 q^{4} -2.02754 q^{5} -1.00000 q^{7} +2.93298 q^{8} +1.91127 q^{10} -0.393608 q^{11} -1.52460 q^{13} +0.942656 q^{14} -0.541991 q^{16} +7.22498 q^{17} +5.33043 q^{19} +2.25341 q^{20} +0.371037 q^{22} -6.14596 q^{23} -0.889072 q^{25} +1.43717 q^{26} +1.11140 q^{28} -10.3543 q^{29} -5.51081 q^{31} -5.35505 q^{32} -6.81067 q^{34} +2.02754 q^{35} +1.11637 q^{37} -5.02476 q^{38} -5.94674 q^{40} +12.0128 q^{41} -0.215108 q^{43} +0.437456 q^{44} +5.79352 q^{46} +4.05988 q^{47} +1.00000 q^{49} +0.838089 q^{50} +1.69443 q^{52} -6.05109 q^{53} +0.798057 q^{55} -2.93298 q^{56} +9.76057 q^{58} -3.92402 q^{59} -3.48704 q^{61} +5.19480 q^{62} +6.13195 q^{64} +3.09118 q^{65} +9.31796 q^{67} -8.02984 q^{68} -1.91127 q^{70} +6.78717 q^{71} -11.2119 q^{73} -1.05235 q^{74} -5.92424 q^{76} +0.393608 q^{77} -5.16136 q^{79} +1.09891 q^{80} -11.3239 q^{82} +6.24052 q^{83} -14.6490 q^{85} +0.202772 q^{86} -1.15444 q^{88} +13.9448 q^{89} +1.52460 q^{91} +6.83062 q^{92} -3.82707 q^{94} -10.8077 q^{95} +10.5474 q^{97} -0.942656 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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