LMFDB - Embedded newform 8001.2.a.ba.1.2

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:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.68554 q^{2} +5.21214 q^{4} -3.72554 q^{5} +1.00000 q^{7} -8.62633 q^{8} +O(q^{10})$	\(q-2.68554 q^{2} +5.21214 q^{4} -3.72554 q^{5} +1.00000 q^{7} -8.62633 q^{8} +10.0051 q^{10} +4.70648 q^{11} -1.94350 q^{13} -2.68554 q^{14} +12.7421 q^{16} -6.80166 q^{17} -1.46364 q^{19} -19.4180 q^{20} -12.6394 q^{22} -1.68046 q^{23} +8.87962 q^{25} +5.21936 q^{26} +5.21214 q^{28} +5.63305 q^{29} +8.82330 q^{31} -16.9668 q^{32} +18.2662 q^{34} -3.72554 q^{35} -10.5716 q^{37} +3.93068 q^{38} +32.1377 q^{40} -10.3199 q^{41} -4.90580 q^{43} +24.5308 q^{44} +4.51294 q^{46} -1.59290 q^{47} +1.00000 q^{49} -23.8466 q^{50} -10.1298 q^{52} +3.28726 q^{53} -17.5341 q^{55} -8.62633 q^{56} -15.1278 q^{58} +14.6344 q^{59} -7.92208 q^{61} -23.6953 q^{62} +20.0808 q^{64} +7.24060 q^{65} +6.80143 q^{67} -35.4512 q^{68} +10.0051 q^{70} -8.27325 q^{71} -3.44600 q^{73} +28.3904 q^{74} -7.62872 q^{76} +4.70648 q^{77} +13.5315 q^{79} -47.4712 q^{80} +27.7145 q^{82} -0.628441 q^{83} +25.3398 q^{85} +13.1747 q^{86} -40.5996 q^{88} -5.11317 q^{89} -1.94350 q^{91} -8.75878 q^{92} +4.27780 q^{94} +5.45286 q^{95} -8.17544 q^{97} -2.68554 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * 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 $

$2$	−2.68554	−1.89897	−0.949483	−	0.313820i	$-0.898391\pi$
$2$	−2.68554	−1.89897	−0.949483	+	0.313820i	$0.898391\pi$
$3$	0	0
$3$	0	0
$4$	5.21214	2.60607
$5$	−3.72554	−1.66611	−0.833055	−	0.553190i	$-0.813410\pi$
$5$	−3.72554	−1.66611	−0.833055	+	0.553190i	$0.813410\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−8.62633	−3.04987
$9$	0	0
$10$	10.0051	3.16389
$11$	4.70648	1.41906	0.709528	−	0.704677i	$-0.248908\pi$
$11$	4.70648	1.41906	0.709528	+	0.704677i	$0.248908\pi$
$12$	0	0
$13$	−1.94350	−0.539031	−0.269516	−	0.962996i	$-0.586864\pi$
$13$	−1.94350	−0.539031	−0.269516	+	0.962996i	$0.586864\pi$
$14$	−2.68554	−0.717741
$15$	0	0
$16$	12.7421	3.18553
$17$	−6.80166	−1.64965	−0.824823	−	0.565391i	$-0.808725\pi$
$17$	−6.80166	−1.64965	−0.824823	+	0.565391i	$0.808725\pi$
$18$	0	0
$19$	−1.46364	−0.335783	−0.167892	−	0.985805i	$-0.553696\pi$
$19$	−1.46364	−0.335783	−0.167892	+	0.985805i	$0.553696\pi$
$20$	−19.4180	−4.34200
$21$	0	0
$22$	−12.6394	−2.69474
$23$	−1.68046	−0.350400	−0.175200	−	0.984533i	$-0.556057\pi$
$23$	−1.68046	−0.350400	−0.175200	+	0.984533i	$0.556057\pi$
$24$	0	0
$25$	8.87962	1.77592
$26$	5.21936	1.02360
$27$	0	0
$28$	5.21214	0.985001
$29$	5.63305	1.04603	0.523016	−	0.852323i	$-0.324807\pi$
$29$	5.63305	1.04603	0.523016	+	0.852323i	$0.324807\pi$
$30$	0	0
$31$	8.82330	1.58471	0.792355	−	0.610060i	$-0.208855\pi$
$31$	8.82330	1.58471	0.792355	+	0.610060i	$0.208855\pi$
$32$	−16.9668	−2.99933
$33$	0	0
$34$	18.2662	3.13262
$35$	−3.72554	−0.629730
$36$	0	0
$37$	−10.5716	−1.73795	−0.868976	−	0.494854i	$-0.835222\pi$
$37$	−10.5716	−1.73795	−0.868976	+	0.494854i	$0.835222\pi$
$38$	3.93068	0.637641
$39$	0	0
$40$	32.1377	5.08142
$41$	−10.3199	−1.61170	−0.805848	−	0.592123i	$-0.798290\pi$
$41$	−10.3199	−1.61170	−0.805848	+	0.592123i	$0.798290\pi$
$42$	0	0
$43$	−4.90580	−0.748127	−0.374064	−	0.927403i	$-0.622036\pi$
$43$	−4.90580	−0.748127	−0.374064	+	0.927403i	$0.622036\pi$
$44$	24.5308	3.69816
$45$	0	0
$46$	4.51294	0.665397
$47$	−1.59290	−0.232348	−0.116174	−	0.993229i	$-0.537063\pi$
$47$	−1.59290	−0.232348	−0.116174	+	0.993229i	$0.537063\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−23.8466	−3.37242
$51$	0	0
$52$	−10.1298	−1.40475
$53$	3.28726	0.451540	0.225770	−	0.974181i	$-0.427510\pi$
$53$	3.28726	0.451540	0.225770	+	0.974181i	$0.427510\pi$
$54$	0	0
$55$	−17.5341	−2.36430
$56$	−8.62633	−1.15274
$57$	0	0
$58$	−15.1278	−1.98638
$59$	14.6344	1.90524	0.952618	−	0.304169i	$-0.0983788\pi$
$59$	14.6344	1.90524	0.952618	+	0.304169i	$0.0983788\pi$
$60$	0	0
$61$	−7.92208	−1.01432	−0.507159	−	0.861853i	$-0.669304\pi$
$61$	−7.92208	−1.01432	−0.507159	+	0.861853i	$0.669304\pi$
$62$	−23.6953	−3.00931
$63$	0	0
$64$	20.0808	2.51011
$65$	7.24060	0.898086
$66$	0	0
$67$	6.80143	0.830927	0.415463	−	0.909610i	$-0.363619\pi$
$67$	6.80143	0.830927	0.415463	+	0.909610i	$0.363619\pi$
$68$	−35.4512	−4.29909
$69$	0	0
$70$	10.0051	1.19584
$71$	−8.27325	−0.981854	−0.490927	−	0.871201i	$-0.663342\pi$
$71$	−8.27325	−0.981854	−0.490927	+	0.871201i	$0.663342\pi$
$72$	0	0
$73$	−3.44600	−0.403323	−0.201662	−	0.979455i	$-0.564634\pi$
$73$	−3.44600	−0.403323	−0.201662	+	0.979455i	$0.564634\pi$
$74$	28.3904	3.30031
$75$	0	0
$76$	−7.62872	−0.875074
$77$	4.70648	0.536353
$78$	0	0
$79$	13.5315	1.52242	0.761209	−	0.648507i	$-0.224606\pi$
$79$	13.5315	1.52242	0.761209	+	0.648507i	$0.224606\pi$
$80$	−47.4712	−5.30744
$81$	0	0
$82$	27.7145	3.06055
$83$	−0.628441	−0.0689804	−0.0344902	−	0.999405i	$-0.510981\pi$
$83$	−0.628441	−0.0689804	−0.0344902	+	0.999405i	$0.510981\pi$
$84$	0	0
$85$	25.3398	2.74849
$86$	13.1747	1.42067
$87$	0	0
$88$	−40.5996	−4.32794
$89$	−5.11317	−0.541995	−0.270998	−	0.962580i	$-0.587354\pi$
$89$	−5.11317	−0.541995	−0.270998	+	0.962580i	$0.587354\pi$
$90$	0	0
$91$	−1.94350	−0.203735
$92$	−8.75878	−0.913166
$93$	0	0
$94$	4.27780	0.441221
$95$	5.45286	0.559452
$96$	0	0
$97$	−8.17544	−0.830090	−0.415045	−	0.909801i	$-0.636234\pi$
$97$	−8.17544	−0.830090	−0.415045	+	0.909801i	$0.636234\pi$
$98$	−2.68554	−0.271281
$99$	0	0
$100$	46.2818	4.62818
$101$	14.3720	1.43007	0.715035	−	0.699089i	$-0.246411\pi$
$101$	14.3720	1.43007	0.715035	+	0.699089i	$0.246411\pi$
$102$	0	0
$103$	−15.0744	−1.48533	−0.742663	−	0.669665i	$-0.766438\pi$
$103$	−15.0744	−1.48533	−0.742663	+	0.669665i	$0.766438\pi$
$104$	16.7653	1.64397
$105$	0	0
$106$	−8.82808	−0.857459
$107$	−1.24579	−0.120435	−0.0602177	−	0.998185i	$-0.519179\pi$
$107$	−1.24579	−0.120435	−0.0602177	+	0.998185i	$0.519179\pi$
$108$	0	0
$109$	6.78526	0.649910	0.324955	−	0.945730i	$-0.394651\pi$
$109$	6.78526	0.649910	0.324955	+	0.945730i	$0.394651\pi$
$110$	47.0887	4.48973
$111$	0	0
$112$	12.7421	1.20402
$113$	1.07357	0.100993	0.0504963	−	0.998724i	$-0.483920\pi$
$113$	1.07357	0.100993	0.0504963	+	0.998724i	$0.483920\pi$
$114$	0	0
$115$	6.26061	0.583805
$116$	29.3602	2.72603
$117$	0	0
$118$	−39.3013	−3.61798
$119$	−6.80166	−0.623507
$120$	0	0
$121$	11.1509	1.01372
$122$	21.2751	1.92615
$123$	0	0
$124$	45.9882	4.12987
$125$	−14.4536	−1.29277
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−19.9944	−1.76727
$129$	0	0
$130$	−19.4449	−1.70543
$131$	−20.9599	−1.83128	−0.915638	−	0.402003i	$-0.868314\pi$
$131$	−20.9599	−1.83128	−0.915638	+	0.402003i	$0.868314\pi$
$132$	0	0
$133$	−1.46364	−0.126914
$134$	−18.2655	−1.57790
$135$	0	0
$136$	58.6734	5.03120
$137$	14.4187	1.23188	0.615938	−	0.787795i	$-0.288777\pi$
$137$	14.4187	1.23188	0.615938	+	0.787795i	$0.288777\pi$
$138$	0	0
$139$	19.5521	1.65839	0.829193	−	0.558962i	$-0.188800\pi$
$139$	19.5521	1.65839	0.829193	+	0.558962i	$0.188800\pi$
$140$	−19.4180	−1.64112
$141$	0	0
$142$	22.2182	1.86451
$143$	−9.14706	−0.764916
$144$	0	0
$145$	−20.9861	−1.74280
$146$	9.25437	0.765897
$147$	0	0
$148$	−55.1004	−4.52922
$149$	−17.6661	−1.44727	−0.723633	−	0.690185i	$-0.757529\pi$
$149$	−17.6661	−1.44727	−0.723633	+	0.690185i	$0.757529\pi$
$150$	0	0
$151$	−2.22045	−0.180698	−0.0903489	−	0.995910i	$-0.528798\pi$
$151$	−2.22045	−0.180698	−0.0903489	+	0.995910i	$0.528798\pi$
$152$	12.6259	1.02409
$153$	0	0
$154$	−12.6394	−1.01852
$155$	−32.8715	−2.64030
$156$	0	0
$157$	−8.79803	−0.702159	−0.351080	−	0.936346i	$-0.614185\pi$
$157$	−8.79803	−0.702159	−0.351080	+	0.936346i	$0.614185\pi$
$158$	−36.3395	−2.89102
$159$	0	0
$160$	63.2104	4.99722
$161$	−1.68046	−0.132439
$162$	0	0
$163$	3.70643	0.290310	0.145155	−	0.989409i	$-0.453632\pi$
$163$	3.70643	0.290310	0.145155	+	0.989409i	$0.453632\pi$
$164$	−53.7887	−4.20019
$165$	0	0
$166$	1.68771	0.130991
$167$	18.0329	1.39543	0.697715	−	0.716375i	$-0.254200\pi$
$167$	18.0329	1.39543	0.697715	+	0.716375i	$0.254200\pi$
$168$	0	0
$169$	−9.22279	−0.709445
$170$	−68.0512	−5.21929
$171$	0	0
$172$	−25.5697	−1.94967
$173$	12.2737	0.933149	0.466574	−	0.884482i	$-0.345488\pi$
$173$	12.2737	0.933149	0.466574	+	0.884482i	$0.345488\pi$
$174$	0	0
$175$	8.87962	0.671236
$176$	59.9704	4.52044
$177$	0	0
$178$	13.7316	1.02923
$179$	−17.9631	−1.34263	−0.671313	−	0.741174i	$-0.734269\pi$
$179$	−17.9631	−1.34263	−0.671313	+	0.741174i	$0.734269\pi$
$180$	0	0
$181$	−13.1750	−0.979291	−0.489645	−	0.871922i	$-0.662874\pi$
$181$	−13.1750	−0.979291	−0.489645	+	0.871922i	$0.662874\pi$
$182$	5.21936	0.386885
$183$	0	0
$184$	14.4962	1.06867
$185$	39.3847	2.89562
$186$	0	0
$187$	−32.0119	−2.34094
$188$	−8.30241	−0.605516
$189$	0	0
$190$	−14.6439	−1.06238
$191$	−22.1546	−1.60305	−0.801527	−	0.597959i	$-0.795979\pi$
$191$	−22.1546	−1.60305	−0.801527	+	0.597959i	$0.795979\pi$
$192$	0	0
$193$	−18.3380	−1.32000	−0.659998	−	0.751268i	$-0.729443\pi$
$193$	−18.3380	−1.32000	−0.659998	+	0.751268i	$0.729443\pi$
$194$	21.9555	1.57631
$195$	0	0
$196$	5.21214	0.372296
$197$	22.7188	1.61865	0.809324	−	0.587363i	$-0.199834\pi$
$197$	22.7188	1.61865	0.809324	+	0.587363i	$0.199834\pi$
$198$	0	0
$199$	−19.9115	−1.41149	−0.705743	−	0.708468i	$-0.749387\pi$
$199$	−19.9115	−1.41149	−0.705743	+	0.708468i	$0.749387\pi$
$200$	−76.5985	−5.41633
$201$	0	0
$202$	−38.5967	−2.71565
$203$	5.63305	0.395363
$204$	0	0
$205$	38.4471	2.68526
$206$	40.4830	2.82058
$207$	0	0
$208$	−24.7643	−1.71710
$209$	−6.88861	−0.476495
$210$	0	0
$211$	11.2686	0.775759	0.387880	−	0.921710i	$-0.373208\pi$
$211$	11.2686	0.775759	0.387880	+	0.921710i	$0.373208\pi$
$212$	17.1337	1.17674
$213$	0	0
$214$	3.34563	0.228703
$215$	18.2767	1.24646
$216$	0	0
$217$	8.82330	0.598964
$218$	−18.2221	−1.23416
$219$	0	0
$220$	−91.3904	−6.16154
$221$	13.2191	0.889211
$222$	0	0
$223$	14.4122	0.965115	0.482558	−	0.875864i	$-0.339708\pi$
$223$	14.4122	0.965115	0.482558	+	0.875864i	$0.339708\pi$
$224$	−16.9668	−1.13364
$225$	0	0
$226$	−2.88311	−0.191781
$227$	−4.25396	−0.282345	−0.141173	−	0.989985i	$-0.545087\pi$
$227$	−4.25396	−0.282345	−0.141173	+	0.989985i	$0.545087\pi$
$228$	0	0
$229$	2.98232	0.197077	0.0985386	−	0.995133i	$-0.468583\pi$
$229$	2.98232	0.197077	0.0985386	+	0.995133i	$0.468583\pi$
$230$	−16.8131	−1.10862
$231$	0	0
$232$	−48.5926	−3.19026
$233$	25.4755	1.66896	0.834478	−	0.551042i	$-0.185769\pi$
$233$	25.4755	1.66896	0.834478	+	0.551042i	$0.185769\pi$
$234$	0	0
$235$	5.93440	0.387118
$236$	76.2765	4.96518
$237$	0	0
$238$	18.2662	1.18402
$239$	6.85932	0.443693	0.221846	−	0.975082i	$-0.428792\pi$
$239$	6.85932	0.443693	0.221846	+	0.975082i	$0.428792\pi$
$240$	0	0
$241$	0.702762	0.0452689	0.0226344	−	0.999744i	$-0.492795\pi$
$241$	0.702762	0.0452689	0.0226344	+	0.999744i	$0.492795\pi$
$242$	−29.9463	−1.92502
$243$	0	0
$244$	−41.2910	−2.64338
$245$	−3.72554	−0.238016
$246$	0	0
$247$	2.84460	0.180998
$248$	−76.1127	−4.83316
$249$	0	0
$250$	38.8159	2.45493
$251$	20.9515	1.32245	0.661224	−	0.750189i	$-0.270037\pi$
$251$	20.9515	1.32245	0.661224	+	0.750189i	$0.270037\pi$
$252$	0	0
$253$	−7.90904	−0.497237
$254$	2.68554	0.168506
$255$	0	0
$256$	13.5340	0.845877
$257$	−7.00299	−0.436834	−0.218417	−	0.975855i	$-0.570089\pi$
$257$	−7.00299	−0.436834	−0.218417	+	0.975855i	$0.570089\pi$
$258$	0	0
$259$	−10.5716	−0.656884
$260$	37.7390	2.34047
$261$	0	0
$262$	56.2888	3.47753
$263$	−11.8526	−0.730865	−0.365433	−	0.930838i	$-0.619079\pi$
$263$	−11.8526	−0.730865	−0.365433	+	0.930838i	$0.619079\pi$
$264$	0	0
$265$	−12.2468	−0.752315
$266$	3.93068	0.241005
$267$	0	0
$268$	35.4500	2.16545
$269$	16.7633	1.02208	0.511039	−	0.859558i	$-0.329261\pi$
$269$	16.7633	1.02208	0.511039	+	0.859558i	$0.329261\pi$
$270$	0	0
$271$	7.16652	0.435335	0.217668	−	0.976023i	$-0.430155\pi$
$271$	7.16652	0.435335	0.217668	+	0.976023i	$0.430155\pi$
$272$	−86.6675	−5.25499
$273$	0	0
$274$	−38.7221	−2.33929
$275$	41.7917	2.52013
$276$	0	0
$277$	−14.8325	−0.891200	−0.445600	−	0.895232i	$-0.647010\pi$
$277$	−14.8325	−0.891200	−0.445600	+	0.895232i	$0.647010\pi$
$278$	−52.5079	−3.14922
$279$	0	0
$280$	32.1377	1.92060
$281$	7.10840	0.424052	0.212026	−	0.977264i	$-0.431994\pi$
$281$	7.10840	0.424052	0.212026	+	0.977264i	$0.431994\pi$
$282$	0	0
$283$	−25.2287	−1.49969	−0.749844	−	0.661614i	$-0.769872\pi$
$283$	−25.2287	−1.49969	−0.749844	+	0.661614i	$0.769872\pi$
$284$	−43.1213	−2.55878
$285$	0	0
$286$	24.5648	1.45255
$287$	−10.3199	−0.609164
$288$	0	0
$289$	29.2626	1.72133
$290$	56.3592	3.30952
$291$	0	0
$292$	−17.9610	−1.05109
$293$	−22.7074	−1.32658	−0.663292	−	0.748361i	$-0.730841\pi$
$293$	−22.7074	−1.32658	−0.663292	+	0.748361i	$0.730841\pi$
$294$	0	0
$295$	−54.5210	−3.17433
$296$	91.1937	5.30053
$297$	0	0
$298$	47.4431	2.74831
$299$	3.26598	0.188876
$300$	0	0
$301$	−4.90580	−0.282765
$302$	5.96312	0.343139
$303$	0	0
$304$	−18.6499	−1.06965
$305$	29.5140	1.68997
$306$	0	0
$307$	1.60410	0.0915506	0.0457753	−	0.998952i	$-0.485424\pi$
$307$	1.60410	0.0915506	0.0457753	+	0.998952i	$0.485424\pi$
$308$	24.5308	1.39777
$309$	0	0
$310$	88.2778	5.01384
$311$	−12.3965	−0.702940	−0.351470	−	0.936199i	$-0.614318\pi$
$311$	−12.3965	−0.702940	−0.351470	+	0.936199i	$0.614318\pi$
$312$	0	0
$313$	−14.9901	−0.847293	−0.423647	−	0.905828i	$-0.639250\pi$
$313$	−14.9901	−0.847293	−0.423647	+	0.905828i	$0.639250\pi$
$314$	23.6275	1.33338
$315$	0	0
$316$	70.5283	3.96753
$317$	−5.23498	−0.294026	−0.147013	−	0.989135i	$-0.546966\pi$
$317$	−5.23498	−0.294026	−0.147013	+	0.989135i	$0.546966\pi$
$318$	0	0
$319$	26.5118	1.48438
$320$	−74.8119	−4.18211
$321$	0	0
$322$	4.51294	0.251496
$323$	9.95522	0.553923
$324$	0	0
$325$	−17.2576	−0.957278
$326$	−9.95378	−0.551289
$327$	0	0
$328$	89.0228	4.91546
$329$	−1.59290	−0.0878194
$330$	0	0
$331$	−3.26773	−0.179610	−0.0898052	−	0.995959i	$-0.528624\pi$
$331$	−3.26773	−0.179610	−0.0898052	+	0.995959i	$0.528624\pi$
$332$	−3.27552	−0.179768
$333$	0	0
$334$	−48.4282	−2.64987
$335$	−25.3390	−1.38442
$336$	0	0
$337$	12.4568	0.678564	0.339282	−	0.940685i	$-0.389816\pi$
$337$	12.4568	0.678564	0.339282	+	0.940685i	$0.389816\pi$
$338$	24.7682	1.34721
$339$	0	0
$340$	132.075	7.16276
$341$	41.5266	2.24879
$342$	0	0
$343$	1.00000	0.0539949
$344$	42.3190	2.28169
$345$	0	0
$346$	−32.9614	−1.77202
$347$	27.9350	1.49963	0.749814	−	0.661649i	$-0.230143\pi$
$347$	27.9350	1.49963	0.749814	+	0.661649i	$0.230143\pi$
$348$	0	0
$349$	−3.22642	−0.172707	−0.0863533	−	0.996265i	$-0.527521\pi$
$349$	−3.22642	−0.172707	−0.0863533	+	0.996265i	$0.527521\pi$
$350$	−23.8466	−1.27465
$351$	0	0
$352$	−79.8538	−4.25622
$353$	0.0678813	0.00361296	0.00180648	−	0.999998i	$-0.499425\pi$
$353$	0.0678813	0.00361296	0.00180648	+	0.999998i	$0.499425\pi$
$354$	0	0
$355$	30.8223	1.63588
$356$	−26.6506	−1.41248
$357$	0	0
$358$	48.2407	2.54960
$359$	−1.02053	−0.0538613	−0.0269307	−	0.999637i	$-0.508573\pi$
$359$	−1.02053	−0.0538613	−0.0269307	+	0.999637i	$0.508573\pi$
$360$	0	0
$361$	−16.8577	−0.887250
$362$	35.3820	1.85964
$363$	0	0
$364$	−10.1298	−0.530947
$365$	12.8382	0.671981
$366$	0	0
$367$	19.5442	1.02020	0.510100	−	0.860115i	$-0.329609\pi$
$367$	19.5442	1.02020	0.510100	+	0.860115i	$0.329609\pi$
$368$	−21.4126	−1.11621
$369$	0	0
$370$	−105.769	−5.49868
$371$	3.28726	0.170666
$372$	0	0
$373$	19.1990	0.994086	0.497043	−	0.867726i	$-0.334419\pi$
$373$	19.1990	0.994086	0.497043	+	0.867726i	$0.334419\pi$
$374$	85.9692	4.44536
$375$	0	0
$376$	13.7409	0.708632
$377$	−10.9479	−0.563844
$378$	0	0
$379$	−18.6260	−0.956755	−0.478378	−	0.878154i	$-0.658775\pi$
$379$	−18.6260	−0.956755	−0.478378	+	0.878154i	$0.658775\pi$
$380$	28.4211	1.45797
$381$	0	0
$382$	59.4972	3.04414
$383$	13.8872	0.709601	0.354800	−	0.934942i	$-0.384549\pi$
$383$	13.8872	0.709601	0.354800	+	0.934942i	$0.384549\pi$
$384$	0	0
$385$	−17.5341	−0.893623
$386$	49.2474	2.50662
$387$	0	0
$388$	−42.6115	−2.16327
$389$	−24.0078	−1.21724	−0.608622	−	0.793460i	$-0.708278\pi$
$389$	−24.0078	−1.21724	−0.608622	+	0.793460i	$0.708278\pi$
$390$	0	0
$391$	11.4299	0.578036
$392$	−8.62633	−0.435696
$393$	0	0
$394$	−61.0123	−3.07376
$395$	−50.4122	−2.53652
$396$	0	0
$397$	−19.7050	−0.988963	−0.494482	−	0.869188i	$-0.664642\pi$
$397$	−19.7050	−0.988963	−0.494482	+	0.869188i	$0.664642\pi$
$398$	53.4731	2.68036
$399$	0	0
$400$	113.145	5.65725
$401$	6.31211	0.315212	0.157606	−	0.987502i	$-0.449622\pi$
$401$	6.31211	0.315212	0.157606	+	0.987502i	$0.449622\pi$
$402$	0	0
$403$	−17.1481	−0.854209
$404$	74.9090	3.72686
$405$	0	0
$406$	−15.1278	−0.750780
$407$	−49.7548	−2.46625
$408$	0	0
$409$	−8.24099	−0.407491	−0.203745	−	0.979024i	$-0.565311\pi$
$409$	−8.24099	−0.407491	−0.203745	+	0.979024i	$0.565311\pi$
$410$	−103.251	−5.09922
$411$	0	0
$412$	−78.5699	−3.87086
$413$	14.6344	0.720112
$414$	0	0
$415$	2.34128	0.114929
$416$	32.9751	1.61674
$417$	0	0
$418$	18.4997	0.904848
$419$	−29.2409	−1.42851	−0.714255	−	0.699886i	$-0.753234\pi$
$419$	−29.2409	−1.42851	−0.714255	+	0.699886i	$0.753234\pi$
$420$	0	0
$421$	20.4488	0.996613	0.498306	−	0.867001i	$-0.333956\pi$
$421$	20.4488	0.996613	0.498306	+	0.867001i	$0.333956\pi$
$422$	−30.2622	−1.47314
$423$	0	0
$424$	−28.3570	−1.37714
$425$	−60.3962	−2.92964
$426$	0	0
$427$	−7.92208	−0.383376
$428$	−6.49325	−0.313863
$429$	0	0
$430$	−49.0829	−2.36699
$431$	−32.4710	−1.56407	−0.782036	−	0.623234i	$-0.785819\pi$
$431$	−32.4710	−1.56407	−0.782036	+	0.623234i	$0.785819\pi$
$432$	0	0
$433$	28.0765	1.34927	0.674634	−	0.738152i	$-0.264301\pi$
$433$	28.0765	1.34927	0.674634	+	0.738152i	$0.264301\pi$
$434$	−23.6953	−1.13741
$435$	0	0
$436$	35.3657	1.69371
$437$	2.45959	0.117658
$438$	0	0
$439$	−0.0674351	−0.00321850	−0.00160925	−	0.999999i	$-0.500512\pi$
$439$	−0.0674351	−0.00321850	−0.00160925	+	0.999999i	$0.500512\pi$
$440$	151.255	7.21082
$441$	0	0
$442$	−35.5004	−1.68858
$443$	6.46892	0.307348	0.153674	−	0.988122i	$-0.450889\pi$
$443$	6.46892	0.307348	0.153674	+	0.988122i	$0.450889\pi$
$444$	0	0
$445$	19.0493	0.903024
$446$	−38.7047	−1.83272
$447$	0	0
$448$	20.0808	0.948731
$449$	11.2504	0.530939	0.265470	−	0.964119i	$-0.414473\pi$
$449$	11.2504	0.530939	0.265470	+	0.964119i	$0.414473\pi$
$450$	0	0
$451$	−48.5703	−2.28709
$452$	5.59558	0.263194
$453$	0	0
$454$	11.4242	0.536164
$455$	7.24060	0.339444
$456$	0	0
$457$	28.3999	1.32849	0.664246	−	0.747514i	$-0.268753\pi$
$457$	28.3999	1.32849	0.664246	+	0.747514i	$0.268753\pi$
$458$	−8.00915	−0.374243
$459$	0	0
$460$	32.6312	1.52144
$461$	8.92186	0.415533	0.207766	−	0.978178i	$-0.433381\pi$
$461$	8.92186	0.415533	0.207766	+	0.978178i	$0.433381\pi$
$462$	0	0
$463$	31.2633	1.45293	0.726465	−	0.687204i	$-0.241162\pi$
$463$	31.2633	1.45293	0.726465	+	0.687204i	$0.241162\pi$
$464$	71.7770	3.33216
$465$	0	0
$466$	−68.4155	−3.16929
$467$	−2.23417	−0.103385	−0.0516925	−	0.998663i	$-0.516462\pi$
$467$	−2.23417	−0.103385	−0.0516925	+	0.998663i	$0.516462\pi$
$468$	0	0
$469$	6.80143	0.314061
$470$	−15.9371	−0.735123
$471$	0	0
$472$	−126.241	−5.81072
$473$	−23.0890	−1.06163
$474$	0	0
$475$	−12.9966	−0.596325
$476$	−35.4512	−1.62490
$477$	0	0
$478$	−18.4210	−0.842557
$479$	30.6455	1.40023	0.700115	−	0.714030i	$-0.253132\pi$
$479$	30.6455	1.40023	0.700115	+	0.714030i	$0.253132\pi$
$480$	0	0
$481$	20.5459	0.936811
$482$	−1.88730	−0.0859640
$483$	0	0
$484$	58.1201	2.64182
$485$	30.4579	1.38302
$486$	0	0
$487$	−1.68166	−0.0762034	−0.0381017	−	0.999274i	$-0.512131\pi$
$487$	−1.68166	−0.0762034	−0.0381017	+	0.999274i	$0.512131\pi$
$488$	68.3385	3.09354
$489$	0	0
$490$	10.0051	0.451984
$491$	−2.97369	−0.134201	−0.0671004	−	0.997746i	$-0.521375\pi$
$491$	−2.97369	−0.134201	−0.0671004	+	0.997746i	$0.521375\pi$
$492$	0	0
$493$	−38.3141	−1.72558
$494$	−7.63930	−0.343708
$495$	0	0
$496$	112.427	5.04814
$497$	−8.27325	−0.371106
$498$	0	0
$499$	−26.9053	−1.20445	−0.602223	−	0.798328i	$-0.705718\pi$
$499$	−26.9053	−1.20445	−0.602223	+	0.798328i	$0.705718\pi$
$500$	−75.3344	−3.36906
$501$	0	0
$502$	−56.2662	−2.51128
$503$	22.0087	0.981319	0.490659	−	0.871352i	$-0.336756\pi$
$503$	22.0087	0.981319	0.490659	+	0.871352i	$0.336756\pi$
$504$	0	0
$505$	−53.5435	−2.38265
$506$	21.2401	0.944236
$507$	0	0
$508$	−5.21214	−0.231251
$509$	−22.1431	−0.981477	−0.490738	−	0.871307i	$-0.663273\pi$
$509$	−22.1431	−0.981477	−0.490738	+	0.871307i	$0.663273\pi$
$510$	0	0
$511$	−3.44600	−0.152442
$512$	3.64252	0.160978
$513$	0	0
$514$	18.8068	0.829533
$515$	56.1603	2.47472
$516$	0	0
$517$	−7.49694	−0.329715
$518$	28.3904	1.24740
$519$	0	0
$520$	−62.4598	−2.73904
$521$	−34.8359	−1.52619	−0.763095	−	0.646287i	$-0.776321\pi$
$521$	−34.8359	−1.52619	−0.763095	+	0.646287i	$0.776321\pi$
$522$	0	0
$523$	−30.4063	−1.32957	−0.664786	−	0.747034i	$-0.731477\pi$
$523$	−30.4063	−1.32957	−0.664786	+	0.747034i	$0.731477\pi$
$524$	−109.246	−4.77243
$525$	0	0
$526$	31.8308	1.38789
$527$	−60.0131	−2.61421
$528$	0	0
$529$	−20.1761	−0.877220
$530$	32.8893	1.42862
$531$	0	0
$532$	−7.62872	−0.330747
$533$	20.0567	0.868754
$534$	0	0
$535$	4.64125	0.200659
$536$	−58.6714	−2.53422
$537$	0	0
$538$	−45.0186	−1.94089
$539$	4.70648	0.202722
$540$	0	0
$541$	27.6547	1.18897	0.594485	−	0.804107i	$-0.297356\pi$
$541$	27.6547	1.18897	0.594485	+	0.804107i	$0.297356\pi$
$542$	−19.2460	−0.826686
$543$	0	0
$544$	115.402	4.94784
$545$	−25.2787	−1.08282
$546$	0	0
$547$	43.8348	1.87424	0.937121	−	0.349005i	$-0.113481\pi$
$547$	43.8348	1.87424	0.937121	+	0.349005i	$0.113481\pi$
$548$	75.1524	3.21035
$549$	0	0
$550$	−112.233	−4.78565
$551$	−8.24479	−0.351240
$552$	0	0
$553$	13.5315	0.575420
$554$	39.8334	1.69236
$555$	0	0
$556$	101.908	4.32187
$557$	15.9394	0.675375	0.337687	−	0.941258i	$-0.390355\pi$
$557$	15.9394	0.675375	0.337687	+	0.941258i	$0.390355\pi$
$558$	0	0
$559$	9.53444	0.403264
$560$	−47.4712	−2.00602
$561$	0	0
$562$	−19.0899	−0.805260
$563$	29.9141	1.26073	0.630365	−	0.776299i	$-0.282905\pi$
$563$	29.9141	1.26073	0.630365	+	0.776299i	$0.282905\pi$
$564$	0	0
$565$	−3.99961	−0.168265
$566$	67.7527	2.84786
$567$	0	0
$568$	71.3678	2.99453
$569$	30.4151	1.27507	0.637533	−	0.770423i	$-0.279955\pi$
$569$	30.4151	1.27507	0.637533	+	0.770423i	$0.279955\pi$
$570$	0	0
$571$	−4.00791	−0.167726	−0.0838630	−	0.996477i	$-0.526726\pi$
$571$	−4.00791	−0.167726	−0.0838630	+	0.996477i	$0.526726\pi$
$572$	−47.6757	−1.99342
$573$	0	0
$574$	27.7145	1.15678
$575$	−14.9218	−0.622283
$576$	0	0
$577$	25.8417	1.07581	0.537903	−	0.843007i	$-0.319217\pi$
$577$	25.8417	1.07581	0.537903	+	0.843007i	$0.319217\pi$
$578$	−78.5860	−3.26875
$579$	0	0
$580$	−109.383	−4.54187
$581$	−0.628441	−0.0260721
$582$	0	0
$583$	15.4714	0.640761
$584$	29.7263	1.23008
$585$	0	0
$586$	60.9818	2.51914
$587$	−40.5893	−1.67530	−0.837650	−	0.546208i	$-0.816071\pi$
$587$	−40.5893	−1.67530	−0.837650	+	0.546208i	$0.816071\pi$
$588$	0	0
$589$	−12.9142	−0.532119
$590$	146.418	6.02795
$591$	0	0
$592$	−134.704	−5.53629
$593$	38.5249	1.58203	0.791014	−	0.611798i	$-0.209553\pi$
$593$	38.5249	1.58203	0.791014	+	0.611798i	$0.209553\pi$
$594$	0	0
$595$	25.3398	1.03883
$596$	−92.0783	−3.77167
$597$	0	0
$598$	−8.77093	−0.358670
$599$	−9.39269	−0.383775	−0.191888	−	0.981417i	$-0.561461\pi$
$599$	−9.39269	−0.383775	−0.191888	+	0.981417i	$0.561461\pi$
$600$	0	0
$601$	−8.40958	−0.343034	−0.171517	−	0.985181i	$-0.554867\pi$
$601$	−8.40958	−0.343034	−0.171517	+	0.985181i	$0.554867\pi$
$602$	13.1747	0.536962
$603$	0	0
$604$	−11.5733	−0.470911
$605$	−41.5432	−1.68897
$606$	0	0
$607$	18.2070	0.739001	0.369500	−	0.929231i	$-0.379529\pi$
$607$	18.2070	0.739001	0.369500	+	0.929231i	$0.379529\pi$
$608$	24.8334	1.00713
$609$	0	0
$610$	−79.2610	−3.20919
$611$	3.09581	0.125243
$612$	0	0
$613$	30.6088	1.23628	0.618138	−	0.786069i	$-0.287887\pi$
$613$	30.6088	1.23628	0.618138	+	0.786069i	$0.287887\pi$
$614$	−4.30787	−0.173851
$615$	0	0
$616$	−40.5996	−1.63581
$617$	34.4857	1.38834	0.694172	−	0.719810i	$-0.255771\pi$
$617$	34.4857	1.38834	0.694172	+	0.719810i	$0.255771\pi$
$618$	0	0
$619$	−21.3676	−0.858838	−0.429419	−	0.903105i	$-0.641282\pi$
$619$	−21.3676	−0.858838	−0.429419	+	0.903105i	$0.641282\pi$
$620$	−171.331	−6.88081
$621$	0	0
$622$	33.2913	1.33486
$623$	−5.11317	−0.204855
$624$	0	0
$625$	9.44950	0.377980
$626$	40.2567	1.60898
$627$	0	0
$628$	−45.8566	−1.82988
$629$	71.9041	2.86701
$630$	0	0
$631$	11.5746	0.460777	0.230389	−	0.973099i	$-0.426000\pi$
$631$	11.5746	0.460777	0.230389	+	0.973099i	$0.426000\pi$
$632$	−116.728	−4.64317
$633$	0	0
$634$	14.0588	0.558345
$635$	3.72554	0.147843
$636$	0	0
$637$	−1.94350	−0.0770045
$638$	−71.1986	−2.81878
$639$	0	0
$640$	74.4897	2.94447
$641$	−22.5477	−0.890579	−0.445290	−	0.895387i	$-0.646899\pi$
$641$	−22.5477	−0.890579	−0.445290	+	0.895387i	$0.646899\pi$
$642$	0	0
$643$	−5.01507	−0.197775	−0.0988875	−	0.995099i	$-0.531528\pi$
$643$	−5.01507	−0.197775	−0.0988875	+	0.995099i	$0.531528\pi$
$644$	−8.75878	−0.345144
$645$	0	0
$646$	−26.7352	−1.05188
$647$	−5.91012	−0.232351	−0.116175	−	0.993229i	$-0.537063\pi$
$647$	−5.91012	−0.232351	−0.116175	+	0.993229i	$0.537063\pi$
$648$	0	0
$649$	68.8764	2.70364
$650$	46.3460	1.81784
$651$	0	0
$652$	19.3184	0.756568
$653$	−15.8356	−0.619695	−0.309848	−	0.950786i	$-0.600278\pi$
$653$	−15.8356	−0.619695	−0.309848	+	0.950786i	$0.600278\pi$
$654$	0	0
$655$	78.0869	3.05111
$656$	−131.497	−5.13410
$657$	0	0
$658$	4.27780	0.166766
$659$	−9.25360	−0.360469	−0.180234	−	0.983624i	$-0.557686\pi$
$659$	−9.25360	−0.360469	−0.180234	+	0.983624i	$0.557686\pi$
$660$	0	0
$661$	17.4124	0.677262	0.338631	−	0.940919i	$-0.390036\pi$
$661$	17.4124	0.677262	0.338631	+	0.940919i	$0.390036\pi$
$662$	8.77561	0.341074
$663$	0	0
$664$	5.42114	0.210381
$665$	5.45286	0.211453
$666$	0	0
$667$	−9.46611	−0.366529
$668$	93.9902	3.63659
$669$	0	0
$670$	68.0489	2.62896
$671$	−37.2851	−1.43937
$672$	0	0
$673$	2.44074	0.0940837	0.0470419	−	0.998893i	$-0.485021\pi$
$673$	2.44074	0.0940837	0.0470419	+	0.998893i	$0.485021\pi$
$674$	−33.4532	−1.28857
$675$	0	0
$676$	−48.0704	−1.84886
$677$	37.8889	1.45619	0.728094	−	0.685477i	$-0.240406\pi$
$677$	37.8889	1.45619	0.728094	+	0.685477i	$0.240406\pi$
$678$	0	0
$679$	−8.17544	−0.313745
$680$	−218.590	−8.38254
$681$	0	0
$682$	−111.522	−4.27038
$683$	−26.5266	−1.01501	−0.507506	−	0.861648i	$-0.669432\pi$
$683$	−26.5266	−1.01501	−0.507506	+	0.861648i	$0.669432\pi$
$684$	0	0
$685$	−53.7175	−2.05244
$686$	−2.68554	−0.102534
$687$	0	0
$688$	−62.5102	−2.38318
$689$	−6.38881	−0.243394
$690$	0	0
$691$	41.6921	1.58604	0.793022	−	0.609193i	$-0.208507\pi$
$691$	41.6921	1.58604	0.793022	+	0.609193i	$0.208507\pi$
$692$	63.9720	2.43185
$693$	0	0
$694$	−75.0206	−2.84774
$695$	−72.8420	−2.76305
$696$	0	0
$697$	70.1924	2.65873
$698$	8.66470	0.327964
$699$	0	0
$700$	46.2818	1.74929
$701$	33.0125	1.24686	0.623432	−	0.781877i	$-0.285738\pi$
$701$	33.0125	1.24686	0.623432	+	0.781877i	$0.285738\pi$
$702$	0	0
$703$	15.4730	0.583575
$704$	94.5100	3.56198
$705$	0	0
$706$	−0.182298	−0.00686088
$707$	14.3720	0.540516
$708$	0	0
$709$	−36.9289	−1.38689	−0.693446	−	0.720508i	$-0.743909\pi$
$709$	−36.9289	−1.38689	−0.693446	+	0.720508i	$0.743909\pi$
$710$	−82.7746	−3.10647
$711$	0	0
$712$	44.1079	1.65301
$713$	−14.8272	−0.555282
$714$	0	0
$715$	34.0777	1.27443
$716$	−93.6262	−3.49898
$717$	0	0
$718$	2.74067	0.102281
$719$	−15.2876	−0.570132	−0.285066	−	0.958508i	$-0.592015\pi$
$719$	−15.2876	−0.570132	−0.285066	+	0.958508i	$0.592015\pi$
$720$	0	0
$721$	−15.0744	−0.561401
$722$	45.2722	1.68486
$723$	0	0
$724$	−68.6700	−2.55210
$725$	50.0193	1.85767
$726$	0	0
$727$	−24.5255	−0.909600	−0.454800	−	0.890594i	$-0.650289\pi$
$727$	−24.5255	−0.909600	−0.454800	+	0.890594i	$0.650289\pi$
$728$	16.7653	0.621364
$729$	0	0
$730$	−34.4775	−1.27607
$731$	33.3676	1.23414
$732$	0	0
$733$	17.1748	0.634366	0.317183	−	0.948364i	$-0.397263\pi$
$733$	17.1748	0.634366	0.317183	+	0.948364i	$0.397263\pi$
$734$	−52.4868	−1.93732
$735$	0	0
$736$	28.5120	1.05097
$737$	32.0108	1.17913
$738$	0	0
$739$	48.7451	1.79312	0.896559	−	0.442923i	$-0.146059\pi$
$739$	48.7451	1.79312	0.896559	+	0.442923i	$0.146059\pi$
$740$	205.278	7.54619
$741$	0	0
$742$	−8.82808	−0.324089
$743$	39.8236	1.46099	0.730494	−	0.682919i	$-0.239290\pi$
$743$	39.8236	1.46099	0.730494	+	0.682919i	$0.239290\pi$
$744$	0	0
$745$	65.8158	2.41130
$746$	−51.5597	−1.88773
$747$	0	0
$748$	−166.850	−6.10065
$749$	−1.24579	−0.0455203
$750$	0	0
$751$	18.7930	0.685767	0.342883	−	0.939378i	$-0.388596\pi$
$751$	18.7930	0.685767	0.342883	+	0.939378i	$0.388596\pi$
$752$	−20.2969	−0.740151
$753$	0	0
$754$	29.4010	1.07072
$755$	8.27237	0.301062
$756$	0	0
$757$	1.55235	0.0564210	0.0282105	−	0.999602i	$-0.491019\pi$
$757$	1.55235	0.0564210	0.0282105	+	0.999602i	$0.491019\pi$
$758$	50.0210	1.81685
$759$	0	0
$760$	−47.0382	−1.70625
$761$	−23.5314	−0.853013	−0.426506	−	0.904485i	$-0.640256\pi$
$761$	−23.5314	−0.853013	−0.426506	+	0.904485i	$0.640256\pi$
$762$	0	0
$763$	6.78526	0.245643
$764$	−115.473	−4.17767
$765$	0	0
$766$	−37.2945	−1.34751
$767$	−28.4420	−1.02698
$768$	0	0
$769$	−0.0257607	−0.000928956 0	−0.000464478	−	1.00000i	$-0.500148\pi$
$769$	−0.0257607	−0.000928956 0	−0.000464478		1.00000i	$0.500148\pi$
$770$	47.0887	1.69696
$771$	0	0
$772$	−95.5800	−3.44000
$773$	6.79173	0.244282	0.122141	−	0.992513i	$-0.461024\pi$
$773$	6.79173	0.244282	0.122141	+	0.992513i	$0.461024\pi$
$774$	0	0
$775$	78.3475	2.81432
$776$	70.5241	2.53167
$777$	0	0
$778$	64.4740	2.31151
$779$	15.1046	0.541180
$780$	0	0
$781$	−38.9379	−1.39331
$782$	−30.6955	−1.09767
$783$	0	0
$784$	12.7421	0.455075
$785$	32.7774	1.16988
$786$	0	0
$787$	11.1472	0.397354	0.198677	−	0.980065i	$-0.436336\pi$
$787$	11.1472	0.397354	0.198677	+	0.980065i	$0.436336\pi$
$788$	118.414	4.21831
$789$	0	0
$790$	135.384	4.81675
$791$	1.07357	0.0381716
$792$	0	0
$793$	15.3966	0.546749
$794$	52.9185	1.87801
$795$	0	0
$796$	−103.781	−3.67843
$797$	27.5886	0.977237	0.488618	−	0.872498i	$-0.337501\pi$
$797$	27.5886	0.977237	0.488618	+	0.872498i	$0.337501\pi$
$798$	0	0
$799$	10.8344	0.383292
$800$	−150.659	−5.32659
$801$	0	0
$802$	−16.9514	−0.598576
$803$	−16.2185	−0.572339
$804$	0	0
$805$	6.26061	0.220657
$806$	46.0520	1.62211
$807$	0	0
$808$	−123.978	−4.36153
$809$	21.2169	0.745946	0.372973	−	0.927842i	$-0.378338\pi$
$809$	21.2169	0.745946	0.372973	+	0.927842i	$0.378338\pi$
$810$	0	0
$811$	11.2280	0.394267	0.197134	−	0.980377i	$-0.436837\pi$
$811$	11.2280	0.394267	0.197134	+	0.980377i	$0.436837\pi$
$812$	29.3602	1.03034
$813$	0	0
$814$	133.619	4.68333
$815$	−13.8084	−0.483689
$816$	0	0
$817$	7.18035	0.251209
$818$	22.1315	0.773811
$819$	0	0
$820$	200.392	6.99798
$821$	7.73377	0.269911	0.134955	−	0.990852i	$-0.456911\pi$
$821$	7.73377	0.269911	0.134955	+	0.990852i	$0.456911\pi$
$822$	0	0
$823$	39.4351	1.37462	0.687311	−	0.726363i	$-0.258791\pi$
$823$	39.4351	1.37462	0.687311	+	0.726363i	$0.258791\pi$
$824$	130.037	4.53005
$825$	0	0
$826$	−39.3013	−1.36747
$827$	1.37339	0.0477575	0.0238788	−	0.999715i	$-0.492398\pi$
$827$	1.37339	0.0477575	0.0238788	+	0.999715i	$0.492398\pi$
$828$	0	0
$829$	36.2143	1.25777	0.628887	−	0.777497i	$-0.283511\pi$
$829$	36.2143	1.25777	0.628887	+	0.777497i	$0.283511\pi$
$830$	−6.28761	−0.218246
$831$	0	0
$832$	−39.0272	−1.35303
$833$	−6.80166	−0.235664
$834$	0	0
$835$	−67.1823	−2.32494
$836$	−35.9044	−1.24178
$837$	0	0
$838$	78.5276	2.71269
$839$	−19.1146	−0.659909	−0.329955	−	0.943997i	$-0.607033\pi$
$839$	−19.1146	−0.659909	−0.329955	+	0.943997i	$0.607033\pi$
$840$	0	0
$841$	2.73128	0.0941822
$842$	−54.9161	−1.89253
$843$	0	0
$844$	58.7333	2.02168
$845$	34.3598	1.18201
$846$	0	0
$847$	11.1509	0.383150
$848$	41.8866	1.43839
$849$	0	0
$850$	162.196	5.56329
$851$	17.7651	0.608978
$852$	0	0
$853$	0.564975	0.0193444	0.00967219	−	0.999953i	$-0.496921\pi$
$853$	0.564975	0.0193444	0.00967219	+	0.999953i	$0.496921\pi$
$854$	21.2751	0.728018
$855$	0	0
$856$	10.7466	0.367312
$857$	−30.9330	−1.05665	−0.528326	−	0.849042i	$-0.677180\pi$
$857$	−30.9330	−1.05665	−0.528326	+	0.849042i	$0.677180\pi$
$858$	0	0
$859$	−15.9714	−0.544936	−0.272468	−	0.962165i	$-0.587840\pi$
$859$	−15.9714	−0.544936	−0.272468	+	0.962165i	$0.587840\pi$
$860$	95.2608	3.24837
$861$	0	0
$862$	87.2022	2.97012
$863$	−21.0862	−0.717783	−0.358892	−	0.933379i	$-0.616845\pi$
$863$	−21.0862	−0.717783	−0.358892	+	0.933379i	$0.616845\pi$
$864$	0	0
$865$	−45.7259	−1.55473
$866$	−75.4005	−2.56221
$867$	0	0
$868$	45.9882	1.56094
$869$	63.6859	2.16040
$870$	0	0
$871$	−13.2186	−0.447896
$872$	−58.5319	−1.98214
$873$	0	0
$874$	−6.60535	−0.223429
$875$	−14.4536	−0.488622
$876$	0	0
$877$	25.6055	0.864635	0.432317	−	0.901721i	$-0.357696\pi$
$877$	25.6055	0.864635	0.432317	+	0.901721i	$0.357696\pi$
$878$	0.181100	0.00611182
$879$	0	0
$880$	−223.422	−7.53155
$881$	13.8775	0.467543	0.233772	−	0.972292i	$-0.424893\pi$
$881$	13.8775	0.467543	0.233772	+	0.972292i	$0.424893\pi$
$882$	0	0
$883$	34.7145	1.16823	0.584117	−	0.811669i	$-0.301441\pi$
$883$	34.7145	1.16823	0.584117	+	0.811669i	$0.301441\pi$
$884$	68.8996	2.31734
$885$	0	0
$886$	−17.3726	−0.583643
$887$	22.2057	0.745594	0.372797	−	0.927913i	$-0.378399\pi$
$887$	22.2057	0.745594	0.372797	+	0.927913i	$0.378399\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−51.1577	−1.71481
$891$	0	0
$892$	75.1186	2.51516
$893$	2.33144	0.0780186
$894$	0	0
$895$	66.9222	2.23696
$896$	−19.9944	−0.667965
$897$	0	0
$898$	−30.2134	−1.00824
$899$	49.7021	1.65766
$900$	0	0
$901$	−22.3588	−0.744881
$902$	130.438	4.34310
$903$	0	0
$904$	−9.26094	−0.308014
$905$	49.0840	1.63161
$906$	0	0
$907$	−4.71374	−0.156517	−0.0782587	−	0.996933i	$-0.524936\pi$
$907$	−4.71374	−0.156517	−0.0782587	+	0.996933i	$0.524936\pi$
$908$	−22.1722	−0.735811
$909$	0	0
$910$	−19.4449	−0.644593
$911$	35.4077	1.17311	0.586555	−	0.809909i	$-0.300484\pi$
$911$	35.4077	1.17311	0.586555	+	0.809909i	$0.300484\pi$
$912$	0	0
$913$	−2.95775	−0.0978871
$914$	−76.2692	−2.52276
$915$	0	0
$916$	15.5443	0.513597
$917$	−20.9599	−0.692158
$918$	0	0
$919$	−5.51318	−0.181863	−0.0909315	−	0.995857i	$-0.528984\pi$
$919$	−5.51318	−0.181863	−0.0909315	+	0.995857i	$0.528984\pi$
$920$	−54.0061	−1.78053
$921$	0	0
$922$	−23.9600	−0.789082
$923$	16.0791	0.529250
$924$	0	0
$925$	−93.8713	−3.08647
$926$	−83.9589	−2.75906
$927$	0	0
$928$	−95.5749	−3.13740
$929$	24.2188	0.794594	0.397297	−	0.917690i	$-0.369948\pi$
$929$	24.2188	0.794594	0.397297	+	0.917690i	$0.369948\pi$
$930$	0	0
$931$	−1.46364	−0.0479690
$932$	132.782	4.34941
$933$	0	0
$934$	5.99995	0.196324
$935$	119.261	3.90026
$936$	0	0
$937$	7.84414	0.256257	0.128128	−	0.991758i	$-0.459103\pi$
$937$	7.84414	0.256257	0.128128	+	0.991758i	$0.459103\pi$
$938$	−18.2655	−0.596391
$939$	0	0
$940$	30.9309	1.00886
$941$	7.51182	0.244878	0.122439	−	0.992476i	$-0.460928\pi$
$941$	7.51182	0.244878	0.122439	+	0.992476i	$0.460928\pi$
$942$	0	0
$943$	17.3421	0.564738
$944$	186.473	6.06918
$945$	0	0
$946$	62.0066	2.01601
$947$	10.7803	0.350314	0.175157	−	0.984540i	$-0.443957\pi$
$947$	10.7803	0.350314	0.175157	+	0.984540i	$0.443957\pi$
$948$	0	0
$949$	6.69731	0.217404
$950$	34.9029	1.13240
$951$	0	0
$952$	58.6734	1.90162
$953$	39.9858	1.29527	0.647634	−	0.761952i	$-0.275759\pi$
$953$	39.9858	1.29527	0.647634	+	0.761952i	$0.275759\pi$
$954$	0	0
$955$	82.5379	2.67086
$956$	35.7517	1.15629
$957$	0	0
$958$	−82.2999	−2.65899
$959$	14.4187	0.465605
$960$	0	0
$961$	46.8505	1.51131
$962$	−55.1768	−1.77897
$963$	0	0
$964$	3.66289	0.117974
$965$	68.3187	2.19926
$966$	0	0
$967$	23.3201	0.749923	0.374962	−	0.927040i	$-0.377656\pi$
$967$	23.3201	0.749923	0.374962	+	0.927040i	$0.377656\pi$
$968$	−96.1916	−3.09171
$969$	0	0
$970$	−81.7960	−2.62631
$971$	28.1326	0.902819	0.451410	−	0.892317i	$-0.350921\pi$
$971$	28.1326	0.902819	0.451410	+	0.892317i	$0.350921\pi$
$972$	0	0
$973$	19.5521	0.626811
$974$	4.51618	0.144708
$975$	0	0
$976$	−100.944	−3.23114
$977$	−10.6731	−0.341463	−0.170731	−	0.985318i	$-0.554613\pi$
$977$	−10.6731	−0.341463	−0.170731	+	0.985318i	$0.554613\pi$
$978$	0	0
$979$	−24.0650	−0.769121
$980$	−19.4180	−0.620285
$981$	0	0
$982$	7.98598	0.254843
$983$	35.8427	1.14321	0.571603	−	0.820531i	$-0.306322\pi$
$983$	35.8427	1.14321	0.571603	+	0.820531i	$0.306322\pi$
$984$	0	0
$985$	−84.6397	−2.69685
$986$	102.894	3.27682
$987$	0	0
$988$	14.8265	0.471692
$989$	8.24399	0.262144
$990$	0	0
$991$	50.1492	1.59304	0.796520	−	0.604612i	$-0.206672\pi$
$991$	50.1492	1.59304	0.796520	+	0.604612i	$0.206672\pi$
$992$	−149.703	−4.75308
$993$	0	0
$994$	22.2182	0.704718
$995$	74.1808	2.35169
$996$	0	0
$997$	21.3090	0.674863	0.337432	−	0.941350i	$-0.390442\pi$
$997$	21.3090	0.674863	0.337432	+	0.941350i	$0.390442\pi$
$998$	72.2553	2.28720
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.2	✓	40
3.2	odd	2	inner	8001.2.a.ba.1.39	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.2	✓	40	1.1	even	1	trivial
8001.2.a.ba.1.39	yes	40	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)

\(f(q)\)	\(=\)	\(q-2.68554 q^{2} +5.21214 q^{4} -3.72554 q^{5} +1.00000 q^{7} -8.62633 q^{8} +O(q^{10})\)	\(q-2.68554 q^{2} +5.21214 q^{4} -3.72554 q^{5} +1.00000 q^{7} -8.62633 q^{8} +10.0051 q^{10} +4.70648 q^{11} -1.94350 q^{13} -2.68554 q^{14} +12.7421 q^{16} -6.80166 q^{17} -1.46364 q^{19} -19.4180 q^{20} -12.6394 q^{22} -1.68046 q^{23} +8.87962 q^{25} +5.21936 q^{26} +5.21214 q^{28} +5.63305 q^{29} +8.82330 q^{31} -16.9668 q^{32} +18.2662 q^{34} -3.72554 q^{35} -10.5716 q^{37} +3.93068 q^{38} +32.1377 q^{40} -10.3199 q^{41} -4.90580 q^{43} +24.5308 q^{44} +4.51294 q^{46} -1.59290 q^{47} +1.00000 q^{49} -23.8466 q^{50} -10.1298 q^{52} +3.28726 q^{53} -17.5341 q^{55} -8.62633 q^{56} -15.1278 q^{58} +14.6344 q^{59} -7.92208 q^{61} -23.6953 q^{62} +20.0808 q^{64} +7.24060 q^{65} +6.80143 q^{67} -35.4512 q^{68} +10.0051 q^{70} -8.27325 q^{71} -3.44600 q^{73} +28.3904 q^{74} -7.62872 q^{76} +4.70648 q^{77} +13.5315 q^{79} -47.4712 q^{80} +27.7145 q^{82} -0.628441 q^{83} +25.3398 q^{85} +13.1747 q^{86} -40.5996 q^{88} -5.11317 q^{89} -1.94350 q^{91} -8.75878 q^{92} +4.27780 q^{94} +5.45286 q^{95} -8.17544 q^{97} -2.68554 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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