LMFDB - Embedded newform 8001.2.a.ba.1.6

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			1.6
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)

$f(q)$	$=$	$q-2.31568 q^{2} +3.36240 q^{4} +1.76797 q^{5} +1.00000 q^{7} -3.15488 q^{8} +O(q^{10})$	\(q-2.31568 q^{2} +3.36240 q^{4} +1.76797 q^{5} +1.00000 q^{7} -3.15488 q^{8} -4.09407 q^{10} -5.91269 q^{11} -6.94886 q^{13} -2.31568 q^{14} +0.580914 q^{16} +4.59921 q^{17} +2.11313 q^{19} +5.94463 q^{20} +13.6919 q^{22} -7.36361 q^{23} -1.87427 q^{25} +16.0914 q^{26} +3.36240 q^{28} +1.96837 q^{29} +8.41994 q^{31} +4.96455 q^{32} -10.6503 q^{34} +1.76797 q^{35} +10.4399 q^{37} -4.89334 q^{38} -5.57774 q^{40} -8.05785 q^{41} +2.89613 q^{43} -19.8808 q^{44} +17.0518 q^{46} -6.04059 q^{47} +1.00000 q^{49} +4.34022 q^{50} -23.3648 q^{52} -0.912285 q^{53} -10.4535 q^{55} -3.15488 q^{56} -4.55812 q^{58} -5.80425 q^{59} +0.182737 q^{61} -19.4979 q^{62} -12.6581 q^{64} -12.2854 q^{65} +6.40252 q^{67} +15.4644 q^{68} -4.09407 q^{70} -2.69489 q^{71} -3.40449 q^{73} -24.1756 q^{74} +7.10518 q^{76} -5.91269 q^{77} +1.85374 q^{79} +1.02704 q^{80} +18.6594 q^{82} +12.7542 q^{83} +8.13128 q^{85} -6.70653 q^{86} +18.6538 q^{88} -6.46307 q^{89} -6.94886 q^{91} -24.7594 q^{92} +13.9881 q^{94} +3.73596 q^{95} -11.2916 q^{97} -2.31568 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.31568	−1.63744	−0.818718	−	0.574196i	$-0.805315\pi$
$2$	−2.31568	−1.63744	−0.818718	+	0.574196i	$0.805315\pi$
$3$	0	0
$3$	0	0
$4$	3.36240	1.68120
$5$	1.76797	0.790662	0.395331	−	0.918539i	$-0.370630\pi$
$5$	1.76797	0.790662	0.395331	+	0.918539i	$0.370630\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.15488	−1.11542
$9$	0	0
$10$	−4.09407	−1.29466
$11$	−5.91269	−1.78274	−0.891371	−	0.453274i	$-0.850256\pi$
$11$	−5.91269	−1.78274	−0.891371	+	0.453274i	$0.850256\pi$
$12$	0	0
$13$	−6.94886	−1.92727	−0.963633	−	0.267227i	$-0.913893\pi$
$13$	−6.94886	−1.92727	−0.963633	+	0.267227i	$0.913893\pi$
$14$	−2.31568	−0.618893
$15$	0	0
$16$	0.580914	0.145228
$17$	4.59921	1.11547	0.557736	−	0.830018i	$-0.311670\pi$
$17$	4.59921	1.11547	0.557736	+	0.830018i	$0.311670\pi$
$18$	0	0
$19$	2.11313	0.484785	0.242393	−	0.970178i	$-0.422068\pi$
$19$	2.11313	0.484785	0.242393	+	0.970178i	$0.422068\pi$
$20$	5.94463	1.32926
$21$	0	0
$22$	13.6919	2.91913
$23$	−7.36361	−1.53542	−0.767710	−	0.640798i	$-0.778604\pi$
$23$	−7.36361	−1.53542	−0.767710	+	0.640798i	$0.778604\pi$
$24$	0	0
$25$	−1.87427	−0.374854
$26$	16.0914	3.15578
$27$	0	0
$28$	3.36240	0.635433
$29$	1.96837	0.365517	0.182759	−	0.983158i	$-0.441497\pi$
$29$	1.96837	0.365517	0.182759	+	0.983158i	$0.441497\pi$
$30$	0	0
$31$	8.41994	1.51227	0.756133	−	0.654418i	$-0.227086\pi$
$31$	8.41994	1.51227	0.756133	+	0.654418i	$0.227086\pi$
$32$	4.96455	0.877616
$33$	0	0
$34$	−10.6503	−1.82652
$35$	1.76797	0.298842
$36$	0	0
$37$	10.4399	1.71631	0.858157	−	0.513387i	$-0.171609\pi$
$37$	10.4399	1.71631	0.858157	+	0.513387i	$0.171609\pi$
$38$	−4.89334	−0.793805
$39$	0	0
$40$	−5.57774	−0.881919
$41$	−8.05785	−1.25842	−0.629212	−	0.777234i	$-0.716622\pi$
$41$	−8.05785	−1.25842	−0.629212	+	0.777234i	$0.716622\pi$
$42$	0	0
$43$	2.89613	0.441656	0.220828	−	0.975313i	$-0.429124\pi$
$43$	2.89613	0.441656	0.220828	+	0.975313i	$0.429124\pi$
$44$	−19.8808	−2.99714
$45$	0	0
$46$	17.0518	2.51415
$47$	−6.04059	−0.881111	−0.440555	−	0.897725i	$-0.645218\pi$
$47$	−6.04059	−0.881111	−0.440555	+	0.897725i	$0.645218\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.34022	0.613799
$51$	0	0
$52$	−23.3648	−3.24012
$53$	−0.912285	−0.125312	−0.0626560	−	0.998035i	$-0.519957\pi$
$53$	−0.912285	−0.125312	−0.0626560	+	0.998035i	$0.519957\pi$
$54$	0	0
$55$	−10.4535	−1.40955
$56$	−3.15488	−0.421588
$57$	0	0
$58$	−4.55812	−0.598511
$59$	−5.80425	−0.755649	−0.377824	−	0.925877i	$-0.623328\pi$
$59$	−5.80425	−0.755649	−0.377824	+	0.925877i	$0.623328\pi$
$60$	0	0
$61$	0.182737	0.0233971	0.0116986	−	0.999932i	$-0.496276\pi$
$61$	0.182737	0.0233971	0.0116986	+	0.999932i	$0.496276\pi$
$62$	−19.4979	−2.47624
$63$	0	0
$64$	−12.6581	−1.58227
$65$	−12.2854	−1.52382
$66$	0	0
$67$	6.40252	0.782192	0.391096	−	0.920350i	$-0.372096\pi$
$67$	6.40252	0.782192	0.391096	+	0.920350i	$0.372096\pi$
$68$	15.4644	1.87533
$69$	0	0
$70$	−4.09407	−0.489335
$71$	−2.69489	−0.319824	−0.159912	−	0.987131i	$-0.551121\pi$
$71$	−2.69489	−0.319824	−0.159912	+	0.987131i	$0.551121\pi$
$72$	0	0
$73$	−3.40449	−0.398465	−0.199232	−	0.979952i	$-0.563845\pi$
$73$	−3.40449	−0.398465	−0.199232	+	0.979952i	$0.563845\pi$
$74$	−24.1756	−2.81036
$75$	0	0
$76$	7.10518	0.815020
$77$	−5.91269	−0.673814
$78$	0	0
$79$	1.85374	0.208562	0.104281	−	0.994548i	$-0.466746\pi$
$79$	1.85374	0.208562	0.104281	+	0.994548i	$0.466746\pi$
$80$	1.02704	0.114827
$81$	0	0
$82$	18.6594	2.06059
$83$	12.7542	1.39996	0.699979	−	0.714163i	$-0.253193\pi$
$83$	12.7542	1.39996	0.699979	+	0.714163i	$0.253193\pi$
$84$	0	0
$85$	8.13128	0.881962
$86$	−6.70653	−0.723184
$87$	0	0
$88$	18.6538	1.98850
$89$	−6.46307	−0.685085	−0.342542	−	0.939502i	$-0.611288\pi$
$89$	−6.46307	−0.685085	−0.342542	+	0.939502i	$0.611288\pi$
$90$	0	0
$91$	−6.94886	−0.728438
$92$	−24.7594	−2.58134
$93$	0	0
$94$	13.9881	1.44276
$95$	3.73596	0.383301
$96$	0	0
$97$	−11.2916	−1.14649	−0.573243	−	0.819386i	$-0.694315\pi$
$97$	−11.2916	−1.14649	−0.573243	+	0.819386i	$0.694315\pi$
$98$	−2.31568	−0.233919
$99$	0	0
$100$	−6.30203	−0.630203
$101$	−5.28591	−0.525968	−0.262984	−	0.964800i	$-0.584707\pi$
$101$	−5.28591	−0.525968	−0.262984	+	0.964800i	$0.584707\pi$
$102$	0	0
$103$	−3.91726	−0.385979	−0.192989	−	0.981201i	$-0.561818\pi$
$103$	−3.91726	−0.385979	−0.192989	+	0.981201i	$0.561818\pi$
$104$	21.9228	2.14971
$105$	0	0
$106$	2.11256	0.205190
$107$	−15.3753	−1.48639	−0.743193	−	0.669077i	$-0.766690\pi$
$107$	−15.3753	−1.48639	−0.743193	+	0.669077i	$0.766690\pi$
$108$	0	0
$109$	5.05936	0.484599	0.242299	−	0.970202i	$-0.422098\pi$
$109$	5.05936	0.484599	0.242299	+	0.970202i	$0.422098\pi$
$110$	24.2070	2.30804
$111$	0	0
$112$	0.580914	0.0548912
$113$	11.7026	1.10089	0.550443	−	0.834873i	$-0.314459\pi$
$113$	11.7026	1.10089	0.550443	+	0.834873i	$0.314459\pi$
$114$	0	0
$115$	−13.0187	−1.21400
$116$	6.61844	0.614507
$117$	0	0
$118$	13.4408	1.23733
$119$	4.59921	0.421609
$120$	0	0
$121$	23.9599	2.17817
$122$	−0.423162	−0.0383113
$123$	0	0
$124$	28.3112	2.54242
$125$	−12.1535	−1.08704
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	19.3832	1.71325
$129$	0	0
$130$	28.4491	2.49515
$131$	−10.9100	−0.953210	−0.476605	−	0.879118i	$-0.658133\pi$
$131$	−10.9100	−0.953210	−0.476605	+	0.879118i	$0.658133\pi$
$132$	0	0
$133$	2.11313	0.183232
$134$	−14.8262	−1.28079
$135$	0	0
$136$	−14.5100	−1.24422
$137$	2.53474	0.216558	0.108279	−	0.994121i	$-0.465466\pi$
$137$	2.53474	0.216558	0.108279	+	0.994121i	$0.465466\pi$
$138$	0	0
$139$	19.5505	1.65825	0.829127	−	0.559060i	$-0.188838\pi$
$139$	19.5505	1.65825	0.829127	+	0.559060i	$0.188838\pi$
$140$	5.94463	0.502413
$141$	0	0
$142$	6.24051	0.523692
$143$	41.0865	3.43582
$144$	0	0
$145$	3.48003	0.289000
$146$	7.88372	0.652461
$147$	0	0
$148$	35.1032	2.88546
$149$	−15.6853	−1.28499	−0.642496	−	0.766289i	$-0.722101\pi$
$149$	−15.6853	−1.28499	−0.642496	+	0.766289i	$0.722101\pi$
$150$	0	0
$151$	10.9002	0.887049	0.443525	−	0.896262i	$-0.353728\pi$
$151$	10.9002	0.887049	0.443525	+	0.896262i	$0.353728\pi$
$152$	−6.66667	−0.540738
$153$	0	0
$154$	13.6919	1.10333
$155$	14.8862	1.19569
$156$	0	0
$157$	−11.9674	−0.955100	−0.477550	−	0.878605i	$-0.658475\pi$
$157$	−11.9674	−0.955100	−0.477550	+	0.878605i	$0.658475\pi$
$158$	−4.29267	−0.341506
$159$	0	0
$160$	8.77719	0.693897
$161$	−7.36361	−0.580334
$162$	0	0
$163$	4.38856	0.343738	0.171869	−	0.985120i	$-0.445019\pi$
$163$	4.38856	0.343738	0.171869	+	0.985120i	$0.445019\pi$
$164$	−27.0937	−2.11566
$165$	0	0
$166$	−29.5348	−2.29234
$167$	17.5758	1.36005	0.680027	−	0.733187i	$-0.261968\pi$
$167$	17.5758	1.36005	0.680027	+	0.733187i	$0.261968\pi$
$168$	0	0
$169$	35.2867	2.71436
$170$	−18.8295	−1.44416
$171$	0	0
$172$	9.73794	0.742511
$173$	8.55819	0.650667	0.325333	−	0.945599i	$-0.394523\pi$
$173$	8.55819	0.650667	0.325333	+	0.945599i	$0.394523\pi$
$174$	0	0
$175$	−1.87427	−0.141681
$176$	−3.43476	−0.258905
$177$	0	0
$178$	14.9664	1.12178
$179$	17.7524	1.32687	0.663437	−	0.748232i	$-0.269097\pi$
$179$	17.7524	1.32687	0.663437	+	0.748232i	$0.269097\pi$
$180$	0	0
$181$	8.05546	0.598757	0.299379	−	0.954134i	$-0.403221\pi$
$181$	8.05546	0.598757	0.299379	+	0.954134i	$0.403221\pi$
$182$	16.0914	1.19277
$183$	0	0
$184$	23.2313	1.71263
$185$	18.4575	1.35702
$186$	0	0
$187$	−27.1937	−1.98860
$188$	−20.3109	−1.48132
$189$	0	0
$190$	−8.65130	−0.627632
$191$	18.8813	1.36620	0.683102	−	0.730323i	$-0.260630\pi$
$191$	18.8813	1.36620	0.683102	+	0.730323i	$0.260630\pi$
$192$	0	0
$193$	−16.4053	−1.18088	−0.590440	−	0.807081i	$-0.701046\pi$
$193$	−16.4053	−1.18088	−0.590440	+	0.807081i	$0.701046\pi$
$194$	26.1477	1.87730
$195$	0	0
$196$	3.36240	0.240171
$197$	2.71976	0.193775	0.0968875	−	0.995295i	$-0.469111\pi$
$197$	2.71976	0.193775	0.0968875	+	0.995295i	$0.469111\pi$
$198$	0	0
$199$	8.80280	0.624014	0.312007	−	0.950080i	$-0.398999\pi$
$199$	8.80280	0.624014	0.312007	+	0.950080i	$0.398999\pi$
$200$	5.91309	0.418119
$201$	0	0
$202$	12.2405	0.861239
$203$	1.96837	0.138153
$204$	0	0
$205$	−14.2461	−0.994988
$206$	9.07113	0.632016
$207$	0	0
$208$	−4.03669	−0.279894
$209$	−12.4943	−0.864248
$210$	0	0
$211$	−15.7624	−1.08513	−0.542563	−	0.840015i	$-0.682546\pi$
$211$	−15.7624	−1.08513	−0.542563	+	0.840015i	$0.682546\pi$
$212$	−3.06746	−0.210674
$213$	0	0
$214$	35.6043	2.43386
$215$	5.12029	0.349201
$216$	0	0
$217$	8.41994	0.571583
$218$	−11.7159	−0.793500
$219$	0	0
$220$	−35.1487	−2.36973
$221$	−31.9593	−2.14981
$222$	0	0
$223$	−19.0840	−1.27796	−0.638980	−	0.769223i	$-0.720643\pi$
$223$	−19.0840	−1.27796	−0.638980	+	0.769223i	$0.720643\pi$
$224$	4.96455	0.331708
$225$	0	0
$226$	−27.0995	−1.80263
$227$	3.50438	0.232594	0.116297	−	0.993214i	$-0.462898\pi$
$227$	3.50438	0.232594	0.116297	+	0.993214i	$0.462898\pi$
$228$	0	0
$229$	16.4644	1.08800	0.544000	−	0.839085i	$-0.316909\pi$
$229$	16.4644	1.08800	0.544000	+	0.839085i	$0.316909\pi$
$230$	30.1471	1.98784
$231$	0	0
$232$	−6.20997	−0.407704
$233$	10.6030	0.694624	0.347312	−	0.937750i	$-0.387094\pi$
$233$	10.6030	0.694624	0.347312	+	0.937750i	$0.387094\pi$
$234$	0	0
$235$	−10.6796	−0.696661
$236$	−19.5162	−1.27040
$237$	0	0
$238$	−10.6503	−0.690358
$239$	−5.75241	−0.372092	−0.186046	−	0.982541i	$-0.559567\pi$
$239$	−5.75241	−0.372092	−0.186046	+	0.982541i	$0.559567\pi$
$240$	0	0
$241$	−5.20775	−0.335461	−0.167730	−	0.985833i	$-0.553644\pi$
$241$	−5.20775	−0.335461	−0.167730	+	0.985833i	$0.553644\pi$
$242$	−55.4836	−3.56662
$243$	0	0
$244$	0.614435	0.0393352
$245$	1.76797	0.112952
$246$	0	0
$247$	−14.6838	−0.934311
$248$	−26.5639	−1.68681
$249$	0	0
$250$	28.1437	1.77997
$251$	−4.79093	−0.302401	−0.151201	−	0.988503i	$-0.548314\pi$
$251$	−4.79093	−0.302401	−0.151201	+	0.988503i	$0.548314\pi$
$252$	0	0
$253$	43.5388	2.73726
$254$	2.31568	0.145299
$255$	0	0
$256$	−19.5691	−1.22307
$257$	0.460963	0.0287541	0.0143770	−	0.999897i	$-0.495423\pi$
$257$	0.460963	0.0287541	0.0143770	+	0.999897i	$0.495423\pi$
$258$	0	0
$259$	10.4399	0.648706
$260$	−41.3084	−2.56184
$261$	0	0
$262$	25.2641	1.56082
$263$	−18.1447	−1.11885	−0.559425	−	0.828881i	$-0.688978\pi$
$263$	−18.1447	−1.11885	−0.559425	+	0.828881i	$0.688978\pi$
$264$	0	0
$265$	−1.61290	−0.0990794
$266$	−4.89334	−0.300030
$267$	0	0
$268$	21.5278	1.31502
$269$	30.2879	1.84669	0.923343	−	0.383976i	$-0.125445\pi$
$269$	30.2879	1.84669	0.923343	+	0.383976i	$0.125445\pi$
$270$	0	0
$271$	32.2102	1.95663	0.978314	−	0.207127i	$-0.0664113\pi$
$271$	32.2102	1.95663	0.978314	+	0.207127i	$0.0664113\pi$
$272$	2.67174	0.161998
$273$	0	0
$274$	−5.86967	−0.354600
$275$	11.0820	0.668268
$276$	0	0
$277$	−30.9729	−1.86098	−0.930490	−	0.366316i	$-0.880619\pi$
$277$	−30.9729	−1.86098	−0.930490	+	0.366316i	$0.880619\pi$
$278$	−45.2729	−2.71529
$279$	0	0
$280$	−5.57774	−0.333334
$281$	−12.3546	−0.737014	−0.368507	−	0.929625i	$-0.620131\pi$
$281$	−12.3546	−0.737014	−0.368507	+	0.929625i	$0.620131\pi$
$282$	0	0
$283$	15.5250	0.922866	0.461433	−	0.887175i	$-0.347335\pi$
$283$	15.5250	0.922866	0.461433	+	0.887175i	$0.347335\pi$
$284$	−9.06128	−0.537688
$285$	0	0
$286$	−95.1433	−5.62594
$287$	−8.05785	−0.475640
$288$	0	0
$289$	4.15274	0.244279
$290$	−8.05864	−0.473220
$291$	0	0
$292$	−11.4472	−0.669898
$293$	18.7453	1.09511	0.547557	−	0.836769i	$-0.315558\pi$
$293$	18.7453	1.09511	0.547557	+	0.836769i	$0.315558\pi$
$294$	0	0
$295$	−10.2618	−0.597463
$296$	−32.9367	−1.91441
$297$	0	0
$298$	36.3222	2.10409
$299$	51.1687	2.95916
$300$	0	0
$301$	2.89613	0.166930
$302$	−25.2415	−1.45249
$303$	0	0
$304$	1.22755	0.0704046
$305$	0.323075	0.0184992
$306$	0	0
$307$	28.0269	1.59958	0.799789	−	0.600281i	$-0.204945\pi$
$307$	28.0269	1.59958	0.799789	+	0.600281i	$0.204945\pi$
$308$	−19.8808	−1.13281
$309$	0	0
$310$	−34.4718	−1.95787
$311$	10.6502	0.603917	0.301959	−	0.953321i	$-0.402360\pi$
$311$	10.6502	0.603917	0.301959	+	0.953321i	$0.402360\pi$
$312$	0	0
$313$	32.2611	1.82350	0.911751	−	0.410743i	$-0.134731\pi$
$313$	32.2611	1.82350	0.911751	+	0.410743i	$0.134731\pi$
$314$	27.7126	1.56391
$315$	0	0
$316$	6.23299	0.350633
$317$	−23.1759	−1.30169	−0.650845	−	0.759210i	$-0.725585\pi$
$317$	−23.1759	−1.30169	−0.650845	+	0.759210i	$0.725585\pi$
$318$	0	0
$319$	−11.6384	−0.651623
$320$	−22.3793	−1.25104
$321$	0	0
$322$	17.0518	0.950260
$323$	9.71873	0.540765
$324$	0	0
$325$	13.0240	0.722443
$326$	−10.1625	−0.562849
$327$	0	0
$328$	25.4215	1.40367
$329$	−6.04059	−0.333029
$330$	0	0
$331$	−5.81629	−0.319692	−0.159846	−	0.987142i	$-0.551100\pi$
$331$	−5.81629	−0.319692	−0.159846	+	0.987142i	$0.551100\pi$
$332$	42.8848	2.35361
$333$	0	0
$334$	−40.7000	−2.22700
$335$	11.3195	0.618449
$336$	0	0
$337$	2.62099	0.142775	0.0713873	−	0.997449i	$-0.477257\pi$
$337$	2.62099	0.142775	0.0713873	+	0.997449i	$0.477257\pi$
$338$	−81.7128	−4.44459
$339$	0	0
$340$	27.3406	1.48275
$341$	−49.7845	−2.69598
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.13695	−0.492631
$345$	0	0
$346$	−19.8181	−1.06543
$347$	30.3462	1.62907	0.814536	−	0.580113i	$-0.196992\pi$
$347$	30.3462	1.62907	0.814536	+	0.580113i	$0.196992\pi$
$348$	0	0
$349$	−9.87183	−0.528427	−0.264213	−	0.964464i	$-0.585112\pi$
$349$	−9.87183	−0.528427	−0.264213	+	0.964464i	$0.585112\pi$
$350$	4.34022	0.231994
$351$	0	0
$352$	−29.3538	−1.56456
$353$	−16.8509	−0.896881	−0.448441	−	0.893813i	$-0.648020\pi$
$353$	−16.8509	−0.896881	−0.448441	+	0.893813i	$0.648020\pi$
$354$	0	0
$355$	−4.76449	−0.252873
$356$	−21.7314	−1.15176
$357$	0	0
$358$	−41.1089	−2.17267
$359$	12.1102	0.639151	0.319575	−	0.947561i	$-0.396460\pi$
$359$	12.1102	0.639151	0.319575	+	0.947561i	$0.396460\pi$
$360$	0	0
$361$	−14.5347	−0.764983
$362$	−18.6539	−0.980427
$363$	0	0
$364$	−23.3648	−1.22465
$365$	−6.01904	−0.315051
$366$	0	0
$367$	−4.56992	−0.238548	−0.119274	−	0.992861i	$-0.538057\pi$
$367$	−4.56992	−0.238548	−0.119274	+	0.992861i	$0.538057\pi$
$368$	−4.27762	−0.222986
$369$	0	0
$370$	−42.7418	−2.22204
$371$	−0.912285	−0.0473635
$372$	0	0
$373$	−7.64372	−0.395777	−0.197888	−	0.980225i	$-0.563408\pi$
$373$	−7.64372	−0.395777	−0.197888	+	0.980225i	$0.563408\pi$
$374$	62.9721	3.25621
$375$	0	0
$376$	19.0573	0.982807
$377$	−13.6779	−0.704449
$378$	0	0
$379$	−16.3353	−0.839088	−0.419544	−	0.907735i	$-0.637810\pi$
$379$	−16.3353	−0.839088	−0.419544	+	0.907735i	$0.637810\pi$
$380$	12.5618	0.644405
$381$	0	0
$382$	−43.7232	−2.23707
$383$	34.8230	1.77937	0.889686	−	0.456572i	$-0.150923\pi$
$383$	34.8230	1.77937	0.889686	+	0.456572i	$0.150923\pi$
$384$	0	0
$385$	−10.4535	−0.532759
$386$	37.9895	1.93362
$387$	0	0
$388$	−37.9667	−1.92747
$389$	−9.43839	−0.478545	−0.239273	−	0.970952i	$-0.576909\pi$
$389$	−9.43839	−0.478545	−0.239273	+	0.970952i	$0.576909\pi$
$390$	0	0
$391$	−33.8668	−1.71272
$392$	−3.15488	−0.159345
$393$	0	0
$394$	−6.29811	−0.317294
$395$	3.27736	0.164902
$396$	0	0
$397$	1.13776	0.0571026	0.0285513	−	0.999592i	$-0.490911\pi$
$397$	1.13776	0.0571026	0.0285513	+	0.999592i	$0.490911\pi$
$398$	−20.3845	−1.02178
$399$	0	0
$400$	−1.08879	−0.0544394
$401$	29.1959	1.45798	0.728988	−	0.684527i	$-0.239991\pi$
$401$	29.1959	1.45798	0.728988	+	0.684527i	$0.239991\pi$
$402$	0	0
$403$	−58.5090	−2.91454
$404$	−17.7733	−0.884256
$405$	0	0
$406$	−4.55812	−0.226216
$407$	−61.7281	−3.05975
$408$	0	0
$409$	21.7186	1.07392	0.536958	−	0.843609i	$-0.319573\pi$
$409$	21.7186	1.07392	0.536958	+	0.843609i	$0.319573\pi$
$410$	32.9894	1.62923
$411$	0	0
$412$	−13.1714	−0.648907
$413$	−5.80425	−0.285608
$414$	0	0
$415$	22.5491	1.10689
$416$	−34.4979	−1.69140
$417$	0	0
$418$	28.9328	1.41515
$419$	−7.15554	−0.349571	−0.174785	−	0.984607i	$-0.555923\pi$
$419$	−7.15554	−0.349571	−0.174785	+	0.984607i	$0.555923\pi$
$420$	0	0
$421$	33.0533	1.61092	0.805459	−	0.592652i	$-0.201919\pi$
$421$	33.0533	1.61092	0.805459	+	0.592652i	$0.201919\pi$
$422$	36.5006	1.77682
$423$	0	0
$424$	2.87815	0.139775
$425$	−8.62016	−0.418139
$426$	0	0
$427$	0.182737	0.00884328
$428$	−51.6978	−2.49891
$429$	0	0
$430$	−11.8570	−0.571794
$431$	−16.2928	−0.784799	−0.392399	−	0.919795i	$-0.628355\pi$
$431$	−16.2928	−0.784799	−0.392399	+	0.919795i	$0.628355\pi$
$432$	0	0
$433$	−6.20459	−0.298173	−0.149087	−	0.988824i	$-0.547633\pi$
$433$	−6.20459	−0.298173	−0.149087	+	0.988824i	$0.547633\pi$
$434$	−19.4979	−0.935930
$435$	0	0
$436$	17.0116	0.814706
$437$	−15.5603	−0.744349
$438$	0	0
$439$	14.4802	0.691103	0.345551	−	0.938400i	$-0.387692\pi$
$439$	14.4802	0.691103	0.345551	+	0.938400i	$0.387692\pi$
$440$	32.9795	1.57223
$441$	0	0
$442$	74.0076	3.52018
$443$	8.33323	0.395924	0.197962	−	0.980210i	$-0.436568\pi$
$443$	8.33323	0.395924	0.197962	+	0.980210i	$0.436568\pi$
$444$	0	0
$445$	−11.4265	−0.541670
$446$	44.1926	2.09258
$447$	0	0
$448$	−12.6581	−0.598041
$449$	26.9278	1.27080	0.635400	−	0.772183i	$-0.280835\pi$
$449$	26.9278	1.27080	0.635400	+	0.772183i	$0.280835\pi$
$450$	0	0
$451$	47.6435	2.24345
$452$	39.3487	1.85081
$453$	0	0
$454$	−8.11504	−0.380858
$455$	−12.2854	−0.575949
$456$	0	0
$457$	22.3787	1.04683	0.523415	−	0.852078i	$-0.324658\pi$
$457$	22.3787	1.04683	0.523415	+	0.852078i	$0.324658\pi$
$458$	−38.1264	−1.78153
$459$	0	0
$460$	−43.7739	−2.04097
$461$	−5.96103	−0.277633	−0.138816	−	0.990318i	$-0.544330\pi$
$461$	−5.96103	−0.277633	−0.138816	+	0.990318i	$0.544330\pi$
$462$	0	0
$463$	−24.9899	−1.16138	−0.580689	−	0.814125i	$-0.697217\pi$
$463$	−24.9899	−1.16138	−0.580689	+	0.814125i	$0.697217\pi$
$464$	1.14345	0.0530835
$465$	0	0
$466$	−24.5531	−1.13740
$467$	35.2907	1.63306	0.816530	−	0.577303i	$-0.195895\pi$
$467$	35.2907	1.63306	0.816530	+	0.577303i	$0.195895\pi$
$468$	0	0
$469$	6.40252	0.295641
$470$	24.7306	1.14074
$471$	0	0
$472$	18.3117	0.842864
$473$	−17.1239	−0.787359
$474$	0	0
$475$	−3.96058	−0.181724
$476$	15.4644	0.708808
$477$	0	0
$478$	13.3208	0.609278
$479$	−15.1726	−0.693255	−0.346628	−	0.938003i	$-0.612673\pi$
$479$	−15.1726	−0.693255	−0.346628	+	0.938003i	$0.612673\pi$
$480$	0	0
$481$	−72.5456	−3.30780
$482$	12.0595	0.549296
$483$	0	0
$484$	80.5627	3.66194
$485$	−19.9632	−0.906482
$486$	0	0
$487$	6.54227	0.296459	0.148229	−	0.988953i	$-0.452643\pi$
$487$	6.54227	0.296459	0.148229	+	0.988953i	$0.452643\pi$
$488$	−0.576514	−0.0260976
$489$	0	0
$490$	−4.09407	−0.184951
$491$	−0.0806118	−0.00363796	−0.00181898	−	0.999998i	$-0.500579\pi$
$491$	−0.0806118	−0.00363796	−0.00181898	+	0.999998i	$0.500579\pi$
$492$	0	0
$493$	9.05295	0.407724
$494$	34.0032	1.52987
$495$	0	0
$496$	4.89125	0.219624
$497$	−2.69489	−0.120882
$498$	0	0
$499$	24.8849	1.11400	0.557001	−	0.830511i	$-0.311952\pi$
$499$	24.8849	1.11400	0.557001	+	0.830511i	$0.311952\pi$
$500$	−40.8650	−1.82754
$501$	0	0
$502$	11.0943	0.495162
$503$	22.6464	1.00976	0.504878	−	0.863191i	$-0.331538\pi$
$503$	22.6464	1.00976	0.504878	+	0.863191i	$0.331538\pi$
$504$	0	0
$505$	−9.34536	−0.415863
$506$	−100.822	−4.48209
$507$	0	0
$508$	−3.36240	−0.149182
$509$	17.6719	0.783294	0.391647	−	0.920115i	$-0.371905\pi$
$509$	17.6719	0.783294	0.391647	+	0.920115i	$0.371905\pi$
$510$	0	0
$511$	−3.40449	−0.150606
$512$	6.54940	0.289445
$513$	0	0
$514$	−1.06744	−0.0470830
$515$	−6.92561	−0.305179
$516$	0	0
$517$	35.7161	1.57079
$518$	−24.1756	−1.06221
$519$	0	0
$520$	38.7590	1.69969
$521$	16.8118	0.736537	0.368269	−	0.929719i	$-0.379951\pi$
$521$	16.8118	0.736537	0.368269	+	0.929719i	$0.379951\pi$
$522$	0	0
$523$	35.5721	1.55546	0.777730	−	0.628599i	$-0.216371\pi$
$523$	35.5721	1.55546	0.777730	+	0.628599i	$0.216371\pi$
$524$	−36.6837	−1.60253
$525$	0	0
$526$	42.0174	1.83205
$527$	38.7251	1.68689
$528$	0	0
$529$	31.2228	1.35751
$530$	3.73496	0.162236
$531$	0	0
$532$	7.10518	0.308049
$533$	55.9928	2.42532
$534$	0	0
$535$	−27.1831	−1.17523
$536$	−20.1992	−0.872471
$537$	0	0
$538$	−70.1372	−3.02383
$539$	−5.91269	−0.254678
$540$	0	0
$541$	7.21886	0.310363	0.155182	−	0.987886i	$-0.450404\pi$
$541$	7.21886	0.310363	0.155182	+	0.987886i	$0.450404\pi$
$542$	−74.5886	−3.20385
$543$	0	0
$544$	22.8330	0.978956
$545$	8.94481	0.383154
$546$	0	0
$547$	2.40675	0.102905	0.0514527	−	0.998675i	$-0.483615\pi$
$547$	2.40675	0.102905	0.0514527	+	0.998675i	$0.483615\pi$
$548$	8.52281	0.364077
$549$	0	0
$550$	−25.6624	−1.09425
$551$	4.15942	0.177197
$552$	0	0
$553$	1.85374	0.0788289
$554$	71.7235	3.04724
$555$	0	0
$556$	65.7366	2.78785
$557$	−16.5283	−0.700326	−0.350163	−	0.936689i	$-0.613874\pi$
$557$	−16.5283	−0.700326	−0.350163	+	0.936689i	$0.613874\pi$
$558$	0	0
$559$	−20.1248	−0.851189
$560$	1.02704	0.0434004
$561$	0	0
$562$	28.6094	1.20681
$563$	−1.36983	−0.0577316	−0.0288658	−	0.999583i	$-0.509190\pi$
$563$	−1.36983	−0.0577316	−0.0288658	+	0.999583i	$0.509190\pi$
$564$	0	0
$565$	20.6898	0.870428
$566$	−35.9510	−1.51113
$567$	0	0
$568$	8.50205	0.356738
$569$	10.4728	0.439042	0.219521	−	0.975608i	$-0.429551\pi$
$569$	10.4728	0.439042	0.219521	+	0.975608i	$0.429551\pi$
$570$	0	0
$571$	−6.37745	−0.266888	−0.133444	−	0.991056i	$-0.542604\pi$
$571$	−6.37745	−0.266888	−0.133444	+	0.991056i	$0.542604\pi$
$572$	138.149	5.77630
$573$	0	0
$574$	18.6594	0.778830
$575$	13.8014	0.575558
$576$	0	0
$577$	15.0856	0.628021	0.314010	−	0.949420i	$-0.398327\pi$
$577$	15.0856	0.628021	0.314010	+	0.949420i	$0.398327\pi$
$578$	−9.61645	−0.399991
$579$	0	0
$580$	11.7012	0.485867
$581$	12.7542	0.529134
$582$	0	0
$583$	5.39406	0.223399
$584$	10.7407	0.444455
$585$	0	0
$586$	−43.4083	−1.79318
$587$	−21.9901	−0.907629	−0.453815	−	0.891096i	$-0.649937\pi$
$587$	−21.9901	−0.907629	−0.453815	+	0.891096i	$0.649937\pi$
$588$	0	0
$589$	17.7924	0.733124
$590$	23.7630	0.978307
$591$	0	0
$592$	6.06470	0.249258
$593$	−32.9224	−1.35196	−0.675980	−	0.736920i	$-0.736279\pi$
$593$	−32.9224	−1.35196	−0.675980	+	0.736920i	$0.736279\pi$
$594$	0	0
$595$	8.13128	0.333350
$596$	−52.7402	−2.16032
$597$	0	0
$598$	−118.491	−4.84544
$599$	−14.2409	−0.581867	−0.290933	−	0.956743i	$-0.593966\pi$
$599$	−14.2409	−0.581867	−0.290933	+	0.956743i	$0.593966\pi$
$600$	0	0
$601$	31.1368	1.27009	0.635047	−	0.772473i	$-0.280981\pi$
$601$	31.1368	1.27009	0.635047	+	0.772473i	$0.280981\pi$
$602$	−6.70653	−0.273338
$603$	0	0
$604$	36.6509	1.49131
$605$	42.3605	1.72220
$606$	0	0
$607$	14.5720	0.591458	0.295729	−	0.955272i	$-0.404437\pi$
$607$	14.5720	0.591458	0.295729	+	0.955272i	$0.404437\pi$
$608$	10.4907	0.425455
$609$	0	0
$610$	−0.748140	−0.0302913
$611$	41.9752	1.69814
$612$	0	0
$613$	−17.1303	−0.691887	−0.345943	−	0.938255i	$-0.612441\pi$
$613$	−17.1303	−0.691887	−0.345943	+	0.938255i	$0.612441\pi$
$614$	−64.9014	−2.61921
$615$	0	0
$616$	18.6538	0.751584
$617$	−31.2009	−1.25610	−0.628051	−	0.778172i	$-0.716147\pi$
$617$	−31.2009	−1.25610	−0.628051	+	0.778172i	$0.716147\pi$
$618$	0	0
$619$	−4.53134	−0.182130	−0.0910650	−	0.995845i	$-0.529027\pi$
$619$	−4.53134	−0.182130	−0.0910650	+	0.995845i	$0.529027\pi$
$620$	50.0534	2.01019
$621$	0	0
$622$	−24.6625	−0.988876
$623$	−6.46307	−0.258938
$624$	0	0
$625$	−12.1158	−0.484631
$626$	−74.7064	−2.98587
$627$	0	0
$628$	−40.2390	−1.60571
$629$	48.0155	1.91450
$630$	0	0
$631$	34.7589	1.38373	0.691865	−	0.722027i	$-0.256790\pi$
$631$	34.7589	1.38373	0.691865	+	0.722027i	$0.256790\pi$
$632$	−5.84831	−0.232633
$633$	0	0
$634$	53.6682	2.13144
$635$	−1.76797	−0.0701599
$636$	0	0
$637$	−6.94886	−0.275324
$638$	26.9508	1.06699
$639$	0	0
$640$	34.2690	1.35460
$641$	−41.5851	−1.64251	−0.821257	−	0.570559i	$-0.806727\pi$
$641$	−41.5851	−1.64251	−0.821257	+	0.570559i	$0.806727\pi$
$642$	0	0
$643$	−28.7969	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$643$	−28.7969	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$644$	−24.7594	−0.975656
$645$	0	0
$646$	−22.5055	−0.885468
$647$	13.3570	0.525116	0.262558	−	0.964916i	$-0.415434\pi$
$647$	13.3570	0.525116	0.262558	+	0.964916i	$0.415434\pi$
$648$	0	0
$649$	34.3187	1.34713
$650$	−30.1596	−1.18296
$651$	0	0
$652$	14.7561	0.577892
$653$	4.70406	0.184084	0.0920420	−	0.995755i	$-0.470661\pi$
$653$	4.70406	0.184084	0.0920420	+	0.995755i	$0.470661\pi$
$654$	0	0
$655$	−19.2886	−0.753667
$656$	−4.68091	−0.182759
$657$	0	0
$658$	13.9881	0.545313
$659$	46.7318	1.82041	0.910206	−	0.414156i	$-0.135923\pi$
$659$	46.7318	1.82041	0.910206	+	0.414156i	$0.135923\pi$
$660$	0	0
$661$	−31.3013	−1.21748	−0.608741	−	0.793369i	$-0.708325\pi$
$661$	−31.3013	−1.21748	−0.608741	+	0.793369i	$0.708325\pi$
$662$	13.4687	0.523476
$663$	0	0
$664$	−40.2380	−1.56154
$665$	3.73596	0.144874
$666$	0	0
$667$	−14.4943	−0.561222
$668$	59.0967	2.28652
$669$	0	0
$670$	−26.2124	−1.01267
$671$	−1.08047	−0.0417111
$672$	0	0
$673$	18.3072	0.705690	0.352845	−	0.935682i	$-0.385214\pi$
$673$	18.3072	0.705690	0.352845	+	0.935682i	$0.385214\pi$
$674$	−6.06939	−0.233784
$675$	0	0
$676$	118.648	4.56337
$677$	−23.1634	−0.890241	−0.445120	−	0.895471i	$-0.646839\pi$
$677$	−23.1634	−0.890241	−0.445120	+	0.895471i	$0.646839\pi$
$678$	0	0
$679$	−11.2916	−0.433331
$680$	−25.6532	−0.983756
$681$	0	0
$682$	115.285	4.41450
$683$	20.2507	0.774871	0.387436	−	0.921897i	$-0.373361\pi$
$683$	20.2507	0.774871	0.387436	+	0.921897i	$0.373361\pi$
$684$	0	0
$685$	4.48136	0.171224
$686$	−2.31568	−0.0884133
$687$	0	0
$688$	1.68240	0.0641410
$689$	6.33934	0.241510
$690$	0	0
$691$	−9.08353	−0.345554	−0.172777	−	0.984961i	$-0.555274\pi$
$691$	−9.08353	−0.345554	−0.172777	+	0.984961i	$0.555274\pi$
$692$	28.7760	1.09390
$693$	0	0
$694$	−70.2723	−2.66750
$695$	34.5648	1.31112
$696$	0	0
$697$	−37.0597	−1.40374
$698$	22.8600	0.865265
$699$	0	0
$700$	−6.30203	−0.238195
$701$	34.5046	1.30322	0.651610	−	0.758554i	$-0.274094\pi$
$701$	34.5046	1.30322	0.651610	+	0.758554i	$0.274094\pi$
$702$	0	0
$703$	22.0609	0.832044
$704$	74.8437	2.82078
$705$	0	0
$706$	39.0213	1.46859
$707$	−5.28591	−0.198797
$708$	0	0
$709$	3.21751	0.120836	0.0604181	−	0.998173i	$-0.480757\pi$
$709$	3.21751	0.120836	0.0604181	+	0.998173i	$0.480757\pi$
$710$	11.0331	0.414063
$711$	0	0
$712$	20.3902	0.764156
$713$	−62.0012	−2.32196
$714$	0	0
$715$	72.6398	2.71657
$716$	59.6904	2.23074
$717$	0	0
$718$	−28.0434	−1.04657
$719$	28.3535	1.05741	0.528704	−	0.848806i	$-0.322678\pi$
$719$	28.3535	1.05741	0.528704	+	0.848806i	$0.322678\pi$
$720$	0	0
$721$	−3.91726	−0.145886
$722$	33.6577	1.25261
$723$	0	0
$724$	27.0856	1.00663
$725$	−3.68926	−0.137016
$726$	0	0
$727$	−2.17038	−0.0804948	−0.0402474	−	0.999190i	$-0.512815\pi$
$727$	−2.17038	−0.0804948	−0.0402474	+	0.999190i	$0.512815\pi$
$728$	21.9228	0.812513
$729$	0	0
$730$	13.9382	0.515876
$731$	13.3199	0.492655
$732$	0	0
$733$	10.1784	0.375947	0.187974	−	0.982174i	$-0.439808\pi$
$733$	10.1784	0.375947	0.187974	+	0.982174i	$0.439808\pi$
$734$	10.5825	0.390607
$735$	0	0
$736$	−36.5570	−1.34751
$737$	−37.8561	−1.39445
$738$	0	0
$739$	−34.1799	−1.25733	−0.628664	−	0.777677i	$-0.716398\pi$
$739$	−34.1799	−1.25733	−0.628664	+	0.777677i	$0.716398\pi$
$740$	62.0615	2.28143
$741$	0	0
$742$	2.11256	0.0775547
$743$	−5.72166	−0.209908	−0.104954	−	0.994477i	$-0.533469\pi$
$743$	−5.72166	−0.209908	−0.104954	+	0.994477i	$0.533469\pi$
$744$	0	0
$745$	−27.7312	−1.01599
$746$	17.7004	0.648059
$747$	0	0
$748$	−91.4360	−3.34323
$749$	−15.3753	−0.561801
$750$	0	0
$751$	−1.66395	−0.0607183	−0.0303592	−	0.999539i	$-0.509665\pi$
$751$	−1.66395	−0.0607183	−0.0303592	+	0.999539i	$0.509665\pi$
$752$	−3.50906	−0.127962
$753$	0	0
$754$	31.6738	1.15349
$755$	19.2713	0.701356
$756$	0	0
$757$	−3.21949	−0.117014	−0.0585071	−	0.998287i	$-0.518634\pi$
$757$	−3.21949	−0.117014	−0.0585071	+	0.998287i	$0.518634\pi$
$758$	37.8274	1.37395
$759$	0	0
$760$	−11.7865	−0.427541
$761$	26.8949	0.974939	0.487469	−	0.873140i	$-0.337920\pi$
$761$	26.8949	0.974939	0.487469	+	0.873140i	$0.337920\pi$
$762$	0	0
$763$	5.05936	0.183161
$764$	63.4864	2.29686
$765$	0	0
$766$	−80.6391	−2.91361
$767$	40.3329	1.45634
$768$	0	0
$769$	5.78089	0.208464	0.104232	−	0.994553i	$-0.466761\pi$
$769$	5.78089	0.208464	0.104232	+	0.994553i	$0.466761\pi$
$770$	24.2070	0.872358
$771$	0	0
$772$	−55.1612	−1.98529
$773$	8.24610	0.296592	0.148296	−	0.988943i	$-0.452621\pi$
$773$	8.24610	0.296592	0.148296	+	0.988943i	$0.452621\pi$
$774$	0	0
$775$	−15.7812	−0.566878
$776$	35.6235	1.27881
$777$	0	0
$778$	21.8563	0.783588
$779$	−17.0273	−0.610066
$780$	0	0
$781$	15.9340	0.570165
$782$	78.4249	2.80447
$783$	0	0
$784$	0.580914	0.0207469
$785$	−21.1580	−0.755161
$786$	0	0
$787$	5.97481	0.212979	0.106490	−	0.994314i	$-0.466039\pi$
$787$	5.97481	0.212979	0.106490	+	0.994314i	$0.466039\pi$
$788$	9.14492	0.325774
$789$	0	0
$790$	−7.58932	−0.270016
$791$	11.7026	0.416095
$792$	0	0
$793$	−1.26982	−0.0450925
$794$	−2.63470	−0.0935019
$795$	0	0
$796$	29.5985	1.04909
$797$	−18.7075	−0.662653	−0.331327	−	0.943516i	$-0.607496\pi$
$797$	−18.7075	−0.662653	−0.331327	+	0.943516i	$0.607496\pi$
$798$	0	0
$799$	−27.7819	−0.982855
$800$	−9.30489	−0.328978
$801$	0	0
$802$	−67.6086	−2.38734
$803$	20.1297	0.710360
$804$	0	0
$805$	−13.0187	−0.458848
$806$	135.488	4.77237
$807$	0	0
$808$	16.6764	0.586674
$809$	−12.7763	−0.449189	−0.224595	−	0.974452i	$-0.572106\pi$
$809$	−12.7763	−0.449189	−0.224595	+	0.974452i	$0.572106\pi$
$810$	0	0
$811$	−32.6347	−1.14596	−0.572980	−	0.819569i	$-0.694213\pi$
$811$	−32.6347	−1.14596	−0.572980	+	0.819569i	$0.694213\pi$
$812$	6.61844	0.232262
$813$	0	0
$814$	142.943	5.01014
$815$	7.75885	0.271781
$816$	0	0
$817$	6.11991	0.214108
$818$	−50.2934	−1.75847
$819$	0	0
$820$	−47.9009	−1.67277
$821$	39.0385	1.36245	0.681227	−	0.732073i	$-0.261447\pi$
$821$	39.0385	1.36245	0.681227	+	0.732073i	$0.261447\pi$
$822$	0	0
$823$	−3.63303	−0.126640	−0.0633198	−	0.997993i	$-0.520169\pi$
$823$	−3.63303	−0.126640	−0.0633198	+	0.997993i	$0.520169\pi$
$824$	12.3585	0.430528
$825$	0	0
$826$	13.4408	0.467666
$827$	40.9113	1.42262	0.711312	−	0.702876i	$-0.248101\pi$
$827$	40.9113	1.42262	0.711312	+	0.702876i	$0.248101\pi$
$828$	0	0
$829$	−36.1191	−1.25447	−0.627234	−	0.778831i	$-0.715813\pi$
$829$	−36.1191	−1.25447	−0.627234	+	0.778831i	$0.715813\pi$
$830$	−52.2167	−1.81247
$831$	0	0
$832$	87.9597	3.04945
$833$	4.59921	0.159353
$834$	0	0
$835$	31.0735	1.07534
$836$	−42.0107	−1.45297
$837$	0	0
$838$	16.5700	0.572400
$839$	34.1256	1.17815	0.589073	−	0.808080i	$-0.299493\pi$
$839$	34.1256	1.17815	0.589073	+	0.808080i	$0.299493\pi$
$840$	0	0
$841$	−25.1255	−0.866397
$842$	−76.5409	−2.63777
$843$	0	0
$844$	−52.9993	−1.82431
$845$	62.3859	2.14614
$846$	0	0
$847$	23.9599	0.823272
$848$	−0.529959	−0.0181989
$849$	0	0
$850$	19.9616	0.684676
$851$	−76.8756	−2.63526
$852$	0	0
$853$	14.9622	0.512296	0.256148	−	0.966638i	$-0.417547\pi$
$853$	14.9622	0.512296	0.256148	+	0.966638i	$0.417547\pi$
$854$	−0.423162	−0.0144803
$855$	0	0
$856$	48.5072	1.65794
$857$	−14.0821	−0.481035	−0.240517	−	0.970645i	$-0.577317\pi$
$857$	−14.0821	−0.481035	−0.240517	+	0.970645i	$0.577317\pi$
$858$	0	0
$859$	−37.8161	−1.29027	−0.645134	−	0.764069i	$-0.723199\pi$
$859$	−37.8161	−1.29027	−0.645134	+	0.764069i	$0.723199\pi$
$860$	17.2164	0.587075
$861$	0	0
$862$	37.7291	1.28506
$863$	−10.5696	−0.359795	−0.179897	−	0.983685i	$-0.557577\pi$
$863$	−10.5696	−0.359795	−0.179897	+	0.983685i	$0.557577\pi$
$864$	0	0
$865$	15.1306	0.514457
$866$	14.3679	0.488240
$867$	0	0
$868$	28.3112	0.960943
$869$	−10.9606	−0.371812
$870$	0	0
$871$	−44.4902	−1.50749
$872$	−15.9617	−0.540530
$873$	0	0
$874$	36.0327	1.21882
$875$	−12.1535	−0.410864
$876$	0	0
$877$	31.7486	1.07207	0.536036	−	0.844195i	$-0.319921\pi$
$877$	31.7486	1.07207	0.536036	+	0.844195i	$0.319921\pi$
$878$	−33.5316	−1.13164
$879$	0	0
$880$	−6.07257	−0.204706
$881$	−12.5955	−0.424355	−0.212177	−	0.977231i	$-0.568055\pi$
$881$	−12.5955	−0.424355	−0.212177	+	0.977231i	$0.568055\pi$
$882$	0	0
$883$	27.8755	0.938087	0.469043	−	0.883175i	$-0.344599\pi$
$883$	27.8755	0.938087	0.469043	+	0.883175i	$0.344599\pi$
$884$	−107.460	−3.61426
$885$	0	0
$886$	−19.2971	−0.648300
$887$	20.8105	0.698749	0.349374	−	0.936983i	$-0.386394\pi$
$887$	20.8105	0.698749	0.349374	+	0.936983i	$0.386394\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	26.4603	0.886951
$891$	0	0
$892$	−64.1680	−2.14850
$893$	−12.7646	−0.427150
$894$	0	0
$895$	31.3857	1.04911
$896$	19.3832	0.647547
$897$	0	0
$898$	−62.3562	−2.08085
$899$	16.5736	0.552759
$900$	0	0
$901$	−4.19579	−0.139782
$902$	−110.327	−3.67350
$903$	0	0
$904$	−36.9202	−1.22795
$905$	14.2418	0.473415
$906$	0	0
$907$	−47.7747	−1.58633	−0.793167	−	0.609004i	$-0.791569\pi$
$907$	−47.7747	−1.58633	−0.793167	+	0.609004i	$0.791569\pi$
$908$	11.7831	0.391036
$909$	0	0
$910$	28.4491	0.943079
$911$	−2.22205	−0.0736199	−0.0368099	−	0.999322i	$-0.511720\pi$
$911$	−2.22205	−0.0736199	−0.0368099	+	0.999322i	$0.511720\pi$
$912$	0	0
$913$	−75.4118	−2.49577
$914$	−51.8220	−1.71412
$915$	0	0
$916$	55.3600	1.82914
$917$	−10.9100	−0.360280
$918$	0	0
$919$	60.2641	1.98793	0.993965	−	0.109696i	$-0.0349876\pi$
$919$	60.2641	1.98793	0.993965	+	0.109696i	$0.0349876\pi$
$920$	41.0723	1.35411
$921$	0	0
$922$	13.8039	0.454606
$923$	18.7264	0.616387
$924$	0	0
$925$	−19.5672	−0.643367
$926$	57.8687	1.90168
$927$	0	0
$928$	9.77206	0.320784
$929$	−15.5165	−0.509078	−0.254539	−	0.967062i	$-0.581924\pi$
$929$	−15.5165	−0.509078	−0.254539	+	0.967062i	$0.581924\pi$
$930$	0	0
$931$	2.11313	0.0692551
$932$	35.6514	1.16780
$933$	0	0
$934$	−81.7222	−2.67403
$935$	−48.0778	−1.57231
$936$	0	0
$937$	35.9046	1.17295	0.586476	−	0.809966i	$-0.300515\pi$
$937$	35.9046	1.17295	0.586476	+	0.809966i	$0.300515\pi$
$938$	−14.8262	−0.484093
$939$	0	0
$940$	−35.9091	−1.17122
$941$	44.8431	1.46184	0.730922	−	0.682461i	$-0.239090\pi$
$941$	44.8431	1.46184	0.730922	+	0.682461i	$0.239090\pi$
$942$	0	0
$943$	59.3349	1.93221
$944$	−3.37177	−0.109742
$945$	0	0
$946$	39.6536	1.28925
$947$	5.65809	0.183863	0.0919317	−	0.995765i	$-0.470696\pi$
$947$	5.65809	0.183863	0.0919317	+	0.995765i	$0.470696\pi$
$948$	0	0
$949$	23.6573	0.767948
$950$	9.17144	0.297561
$951$	0	0
$952$	−14.5100	−0.470270
$953$	42.5761	1.37918	0.689588	−	0.724202i	$-0.257792\pi$
$953$	42.5761	1.37918	0.689588	+	0.724202i	$0.257792\pi$
$954$	0	0
$955$	33.3817	1.08021
$956$	−19.3419	−0.625561
$957$	0	0
$958$	35.1350	1.13516
$959$	2.53474	0.0818512
$960$	0	0
$961$	39.8953	1.28695
$962$	167.993	5.41631
$963$	0	0
$964$	−17.5105	−0.563976
$965$	−29.0042	−0.933677
$966$	0	0
$967$	−49.2627	−1.58418	−0.792091	−	0.610403i	$-0.791007\pi$
$967$	−49.2627	−1.58418	−0.792091	+	0.610403i	$0.791007\pi$
$968$	−75.5906	−2.42957
$969$	0	0
$970$	46.2285	1.48431
$971$	4.74041	0.152127	0.0760634	−	0.997103i	$-0.475765\pi$
$971$	4.74041	0.152127	0.0760634	+	0.997103i	$0.475765\pi$
$972$	0	0
$973$	19.5505	0.626761
$974$	−15.1498	−0.485432
$975$	0	0
$976$	0.106155	0.00339793
$977$	43.3923	1.38824	0.694121	−	0.719858i	$-0.255793\pi$
$977$	43.3923	1.38824	0.694121	+	0.719858i	$0.255793\pi$
$978$	0	0
$979$	38.2142	1.22133
$980$	5.94463	0.189894
$981$	0	0
$982$	0.186671	0.00595693
$983$	−37.4784	−1.19537	−0.597687	−	0.801729i	$-0.703913\pi$
$983$	−37.4784	−1.19537	−0.597687	+	0.801729i	$0.703913\pi$
$984$	0	0
$985$	4.80847	0.153211
$986$	−20.9638	−0.667623
$987$	0	0
$988$	−49.3729	−1.57076
$989$	−21.3260	−0.678127
$990$	0	0
$991$	33.4254	1.06179	0.530897	−	0.847437i	$-0.321855\pi$
$991$	33.4254	1.06179	0.530897	+	0.847437i	$0.321855\pi$
$992$	41.8012	1.32719
$993$	0	0
$994$	6.24051	0.197937
$995$	15.5631	0.493384
$996$	0	0
$997$	−38.4338	−1.21721	−0.608605	−	0.793473i	$-0.708271\pi$
$997$	−38.4338	−1.21721	−0.608605	+	0.793473i	$0.708271\pi$
$998$	−57.6257	−1.82411
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.6	✓	40
3.2	odd	2	inner	8001.2.a.ba.1.35	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.6	✓	40	1.1	even	1	trivial
8001.2.a.ba.1.35	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)

\(f(q)\)	\(=\)	\(q-2.31568 q^{2} +3.36240 q^{4} +1.76797 q^{5} +1.00000 q^{7} -3.15488 q^{8} +O(q^{10})\)	\(q-2.31568 q^{2} +3.36240 q^{4} +1.76797 q^{5} +1.00000 q^{7} -3.15488 q^{8} -4.09407 q^{10} -5.91269 q^{11} -6.94886 q^{13} -2.31568 q^{14} +0.580914 q^{16} +4.59921 q^{17} +2.11313 q^{19} +5.94463 q^{20} +13.6919 q^{22} -7.36361 q^{23} -1.87427 q^{25} +16.0914 q^{26} +3.36240 q^{28} +1.96837 q^{29} +8.41994 q^{31} +4.96455 q^{32} -10.6503 q^{34} +1.76797 q^{35} +10.4399 q^{37} -4.89334 q^{38} -5.57774 q^{40} -8.05785 q^{41} +2.89613 q^{43} -19.8808 q^{44} +17.0518 q^{46} -6.04059 q^{47} +1.00000 q^{49} +4.34022 q^{50} -23.3648 q^{52} -0.912285 q^{53} -10.4535 q^{55} -3.15488 q^{56} -4.55812 q^{58} -5.80425 q^{59} +0.182737 q^{61} -19.4979 q^{62} -12.6581 q^{64} -12.2854 q^{65} +6.40252 q^{67} +15.4644 q^{68} -4.09407 q^{70} -2.69489 q^{71} -3.40449 q^{73} -24.1756 q^{74} +7.10518 q^{76} -5.91269 q^{77} +1.85374 q^{79} +1.02704 q^{80} +18.6594 q^{82} +12.7542 q^{83} +8.13128 q^{85} -6.70653 q^{86} +18.6538 q^{88} -6.46307 q^{89} -6.94886 q^{91} -24.7594 q^{92} +13.9881 q^{94} +3.73596 q^{95} -11.2916 q^{97} -2.31568 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 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

Related objects

Downloads

Learn more