LMFDB - Embedded newform 8001.2.a.ba.1.33

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			1.33
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)

$f(q)$	$=$	$q+2.10140 q^{2} +2.41588 q^{4} +0.560790 q^{5} +1.00000 q^{7} +0.873928 q^{8} +O(q^{10})$	\(q+2.10140 q^{2} +2.41588 q^{4} +0.560790 q^{5} +1.00000 q^{7} +0.873928 q^{8} +1.17844 q^{10} -5.80921 q^{11} -0.507839 q^{13} +2.10140 q^{14} -2.99529 q^{16} +0.446823 q^{17} +5.43830 q^{19} +1.35480 q^{20} -12.2075 q^{22} +5.14284 q^{23} -4.68551 q^{25} -1.06717 q^{26} +2.41588 q^{28} +7.22466 q^{29} +9.14609 q^{31} -8.04215 q^{32} +0.938954 q^{34} +0.560790 q^{35} +3.98245 q^{37} +11.4280 q^{38} +0.490091 q^{40} -4.97303 q^{41} -1.04891 q^{43} -14.0343 q^{44} +10.8072 q^{46} +7.68734 q^{47} +1.00000 q^{49} -9.84614 q^{50} -1.22688 q^{52} +12.3221 q^{53} -3.25775 q^{55} +0.873928 q^{56} +15.1819 q^{58} -5.40822 q^{59} +7.99783 q^{61} +19.2196 q^{62} -10.9092 q^{64} -0.284791 q^{65} -5.65009 q^{67} +1.07947 q^{68} +1.17844 q^{70} -12.6636 q^{71} +12.0843 q^{73} +8.36871 q^{74} +13.1383 q^{76} -5.80921 q^{77} +14.9411 q^{79} -1.67973 q^{80} -10.4503 q^{82} +16.9898 q^{83} +0.250574 q^{85} -2.20419 q^{86} -5.07683 q^{88} -6.77722 q^{89} -0.507839 q^{91} +12.4245 q^{92} +16.1542 q^{94} +3.04974 q^{95} +5.81057 q^{97} +2.10140 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.10140	1.48591	0.742957	−	0.669339i	$-0.233423\pi$
$2$	2.10140	1.48591	0.742957	+	0.669339i	$0.233423\pi$
$3$	0	0
$3$	0	0
$4$	2.41588	1.20794
$5$	0.560790	0.250793	0.125397	−	0.992107i	$-0.459980\pi$
$5$	0.560790	0.250793	0.125397	+	0.992107i	$0.459980\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.873928	0.308980
$9$	0	0
$10$	1.17844	0.372657
$11$	−5.80921	−1.75154	−0.875771	−	0.482726i	$-0.839647\pi$
$11$	−5.80921	−1.75154	−0.875771	+	0.482726i	$0.839647\pi$
$12$	0	0
$13$	−0.507839	−0.140849	−0.0704246	−	0.997517i	$-0.522435\pi$
$13$	−0.507839	−0.140849	−0.0704246	+	0.997517i	$0.522435\pi$
$14$	2.10140	0.561623
$15$	0	0
$16$	−2.99529	−0.748822
$17$	0.446823	0.108371	0.0541853	−	0.998531i	$-0.482744\pi$
$17$	0.446823	0.108371	0.0541853	+	0.998531i	$0.482744\pi$
$18$	0	0
$19$	5.43830	1.24763	0.623815	−	0.781572i	$-0.285582\pi$
$19$	5.43830	1.24763	0.623815	+	0.781572i	$0.285582\pi$
$20$	1.35480	0.302943
$21$	0	0
$22$	−12.2075	−2.60264
$23$	5.14284	1.07236	0.536178	−	0.844105i	$-0.319868\pi$
$23$	5.14284	1.07236	0.536178	+	0.844105i	$0.319868\pi$
$24$	0	0
$25$	−4.68551	−0.937103
$26$	−1.06717	−0.209290
$27$	0	0
$28$	2.41588	0.456558
$29$	7.22466	1.34159	0.670793	−	0.741644i	$-0.265954\pi$
$29$	7.22466	1.34159	0.670793	+	0.741644i	$0.265954\pi$
$30$	0	0
$31$	9.14609	1.64269	0.821343	−	0.570434i	$-0.193225\pi$
$31$	9.14609	1.64269	0.821343	+	0.570434i	$0.193225\pi$
$32$	−8.04215	−1.42166
$33$	0	0
$34$	0.938954	0.161029
$35$	0.560790	0.0947909
$36$	0	0
$37$	3.98245	0.654710	0.327355	−	0.944901i	$-0.393843\pi$
$37$	3.98245	0.654710	0.327355	+	0.944901i	$0.393843\pi$
$38$	11.4280	1.85387
$39$	0	0
$40$	0.490091	0.0774901
$41$	−4.97303	−0.776657	−0.388328	−	0.921521i	$-0.626947\pi$
$41$	−4.97303	−0.776657	−0.388328	+	0.921521i	$0.626947\pi$
$42$	0	0
$43$	−1.04891	−0.159958	−0.0799789	−	0.996797i	$-0.525485\pi$
$43$	−1.04891	−0.159958	−0.0799789	+	0.996797i	$0.525485\pi$
$44$	−14.0343	−2.11576
$45$	0	0
$46$	10.8072	1.59343
$47$	7.68734	1.12131	0.560657	−	0.828048i	$-0.310549\pi$
$47$	7.68734	1.12131	0.560657	+	0.828048i	$0.310549\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−9.84614	−1.39245
$51$	0	0
$52$	−1.22688	−0.170137
$53$	12.3221	1.69257	0.846284	−	0.532732i	$-0.178835\pi$
$53$	12.3221	1.69257	0.846284	+	0.532732i	$0.178835\pi$
$54$	0	0
$55$	−3.25775	−0.439275
$56$	0.873928	0.116784
$57$	0	0
$58$	15.1819	1.99348
$59$	−5.40822	−0.704090	−0.352045	−	0.935983i	$-0.614514\pi$
$59$	−5.40822	−0.704090	−0.352045	+	0.935983i	$0.614514\pi$
$60$	0	0
$61$	7.99783	1.02402	0.512009	−	0.858980i	$-0.328902\pi$
$61$	7.99783	1.02402	0.512009	+	0.858980i	$0.328902\pi$
$62$	19.2196	2.44089
$63$	0	0
$64$	−10.9092	−1.36365
$65$	−0.284791	−0.0353240
$66$	0	0
$67$	−5.65009	−0.690269	−0.345134	−	0.938553i	$-0.612167\pi$
$67$	−5.65009	−0.690269	−0.345134	+	0.938553i	$0.612167\pi$
$68$	1.07947	0.130905
$69$	0	0
$70$	1.17844	0.140851
$71$	−12.6636	−1.50289	−0.751445	−	0.659795i	$-0.770643\pi$
$71$	−12.6636	−1.50289	−0.751445	+	0.659795i	$0.770643\pi$
$72$	0	0
$73$	12.0843	1.41436	0.707179	−	0.707035i	$-0.249968\pi$
$73$	12.0843	1.41436	0.707179	+	0.707035i	$0.249968\pi$
$74$	8.36871	0.972843
$75$	0	0
$76$	13.1383	1.50706
$77$	−5.80921	−0.662021
$78$	0	0
$79$	14.9411	1.68100	0.840501	−	0.541810i	$-0.182261\pi$
$79$	14.9411	1.68100	0.840501	+	0.541810i	$0.182261\pi$
$80$	−1.67973	−0.187799
$81$	0	0
$82$	−10.4503	−1.15404
$83$	16.9898	1.86487	0.932434	−	0.361339i	$-0.117680\pi$
$83$	16.9898	1.86487	0.932434	+	0.361339i	$0.117680\pi$
$84$	0	0
$85$	0.250574	0.0271786
$86$	−2.20419	−0.237683
$87$	0	0
$88$	−5.07683	−0.541192
$89$	−6.77722	−0.718384	−0.359192	−	0.933264i	$-0.616948\pi$
$89$	−6.77722	−0.718384	−0.359192	+	0.933264i	$0.616948\pi$
$90$	0	0
$91$	−0.507839	−0.0532360
$92$	12.4245	1.29534
$93$	0	0
$94$	16.1542	1.66617
$95$	3.04974	0.312897
$96$	0	0
$97$	5.81057	0.589974	0.294987	−	0.955501i	$-0.404685\pi$
$97$	5.81057	0.589974	0.294987	+	0.955501i	$0.404685\pi$
$98$	2.10140	0.212273
$99$	0	0
$100$	−11.3196	−1.13196
$101$	−6.94273	−0.690827	−0.345414	−	0.938451i	$-0.612261\pi$
$101$	−6.94273	−0.690827	−0.345414	+	0.938451i	$0.612261\pi$
$102$	0	0
$103$	14.2888	1.40792	0.703961	−	0.710239i	$-0.251413\pi$
$103$	14.2888	1.40792	0.703961	+	0.710239i	$0.251413\pi$
$104$	−0.443815	−0.0435196
$105$	0	0
$106$	25.8936	2.51501
$107$	6.01423	0.581418	0.290709	−	0.956812i	$-0.406109\pi$
$107$	6.01423	0.581418	0.290709	+	0.956812i	$0.406109\pi$
$108$	0	0
$109$	−1.90595	−0.182557	−0.0912785	−	0.995825i	$-0.529095\pi$
$109$	−1.90595	−0.182557	−0.0912785	+	0.995825i	$0.529095\pi$
$110$	−6.84583	−0.652724
$111$	0	0
$112$	−2.99529	−0.283028
$113$	−13.7760	−1.29594	−0.647968	−	0.761667i	$-0.724381\pi$
$113$	−13.7760	−1.29594	−0.647968	+	0.761667i	$0.724381\pi$
$114$	0	0
$115$	2.88405	0.268939
$116$	17.4539	1.62056
$117$	0	0
$118$	−11.3648	−1.04622
$119$	0.446823	0.0409602
$120$	0	0
$121$	22.7469	2.06790
$122$	16.8066	1.52160
$123$	0	0
$124$	22.0959	1.98427
$125$	−5.43154	−0.485812
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−6.84027	−0.604601
$129$	0	0
$130$	−0.598460	−0.0524884
$131$	−4.51284	−0.394289	−0.197144	−	0.980374i	$-0.563167\pi$
$131$	−4.51284	−0.394289	−0.197144	+	0.980374i	$0.563167\pi$
$132$	0	0
$133$	5.43830	0.471560
$134$	−11.8731	−1.02568
$135$	0	0
$136$	0.390492	0.0334844
$137$	22.0022	1.87977	0.939887	−	0.341485i	$-0.110930\pi$
$137$	22.0022	1.87977	0.939887	+	0.341485i	$0.110930\pi$
$138$	0	0
$139$	−13.7621	−1.16729	−0.583644	−	0.812009i	$-0.698374\pi$
$139$	−13.7621	−1.16729	−0.583644	+	0.812009i	$0.698374\pi$
$140$	1.35480	0.114502
$141$	0	0
$142$	−26.6112	−2.23317
$143$	2.95014	0.246703
$144$	0	0
$145$	4.05152	0.336461
$146$	25.3939	2.10161
$147$	0	0
$148$	9.62111	0.790850
$149$	8.51348	0.697451	0.348726	−	0.937225i	$-0.386614\pi$
$149$	8.51348	0.697451	0.348726	+	0.937225i	$0.386614\pi$
$150$	0	0
$151$	−8.17580	−0.665337	−0.332669	−	0.943044i	$-0.607949\pi$
$151$	−8.17580	−0.665337	−0.332669	+	0.943044i	$0.607949\pi$
$152$	4.75268	0.385493
$153$	0	0
$154$	−12.2075	−0.983706
$155$	5.12904	0.411974
$156$	0	0
$157$	3.71736	0.296678	0.148339	−	0.988937i	$-0.452607\pi$
$157$	3.71736	0.296678	0.148339	+	0.988937i	$0.452607\pi$
$158$	31.3972	2.49782
$159$	0	0
$160$	−4.50996	−0.356544
$161$	5.14284	0.405312
$162$	0	0
$163$	0.633075	0.0495863	0.0247931	−	0.999693i	$-0.492107\pi$
$163$	0.633075	0.0495863	0.0247931	+	0.999693i	$0.492107\pi$
$164$	−12.0142	−0.938154
$165$	0	0
$166$	35.7023	2.77103
$167$	14.9603	1.15766	0.578831	−	0.815448i	$-0.303509\pi$
$167$	14.9603	1.15766	0.578831	+	0.815448i	$0.303509\pi$
$168$	0	0
$169$	−12.7421	−0.980162
$170$	0.526557	0.0403851
$171$	0	0
$172$	−2.53405	−0.193219
$173$	−11.1657	−0.848916	−0.424458	−	0.905448i	$-0.639535\pi$
$173$	−11.1657	−0.848916	−0.424458	+	0.905448i	$0.639535\pi$
$174$	0	0
$175$	−4.68551	−0.354192
$176$	17.4002	1.31159
$177$	0	0
$178$	−14.2416	−1.06746
$179$	−4.45557	−0.333025	−0.166513	−	0.986039i	$-0.553251\pi$
$179$	−4.45557	−0.333025	−0.166513	+	0.986039i	$0.553251\pi$
$180$	0	0
$181$	−5.36462	−0.398749	−0.199375	−	0.979923i	$-0.563891\pi$
$181$	−5.36462	−0.398749	−0.199375	+	0.979923i	$0.563891\pi$
$182$	−1.06717	−0.0791041
$183$	0	0
$184$	4.49447	0.331337
$185$	2.23332	0.164197
$186$	0	0
$187$	−2.59569	−0.189816
$188$	18.5717	1.35448
$189$	0	0
$190$	6.40873	0.464938
$191$	−24.0005	−1.73661	−0.868306	−	0.496028i	$-0.834791\pi$
$191$	−24.0005	−1.73661	−0.868306	+	0.496028i	$0.834791\pi$
$192$	0	0
$193$	11.0050	0.792155	0.396078	−	0.918217i	$-0.370371\pi$
$193$	11.0050	0.792155	0.396078	+	0.918217i	$0.370371\pi$
$194$	12.2103	0.876651
$195$	0	0
$196$	2.41588	0.172563
$197$	−0.933248	−0.0664912	−0.0332456	−	0.999447i	$-0.510584\pi$
$197$	−0.933248	−0.0664912	−0.0332456	+	0.999447i	$0.510584\pi$
$198$	0	0
$199$	−7.73258	−0.548148	−0.274074	−	0.961709i	$-0.588371\pi$
$199$	−7.73258	−0.548148	−0.274074	+	0.961709i	$0.588371\pi$
$200$	−4.09480	−0.289546
$201$	0	0
$202$	−14.5894	−1.02651
$203$	7.22466	0.507072
$204$	0	0
$205$	−2.78883	−0.194780
$206$	30.0266	2.09205
$207$	0	0
$208$	1.52112	0.105471
$209$	−31.5922	−2.18528
$210$	0	0
$211$	−6.26746	−0.431470	−0.215735	−	0.976452i	$-0.569215\pi$
$211$	−6.26746	−0.431470	−0.215735	+	0.976452i	$0.569215\pi$
$212$	29.7687	2.04452
$213$	0	0
$214$	12.6383	0.863937
$215$	−0.588220	−0.0401163
$216$	0	0
$217$	9.14609	0.620877
$218$	−4.00517	−0.271264
$219$	0	0
$220$	−7.87033	−0.530617
$221$	−0.226914	−0.0152639
$222$	0	0
$223$	18.6747	1.25055	0.625275	−	0.780404i	$-0.284987\pi$
$223$	18.6747	1.25055	0.625275	+	0.780404i	$0.284987\pi$
$224$	−8.04215	−0.537339
$225$	0	0
$226$	−28.9489	−1.92565
$227$	−22.2469	−1.47658	−0.738289	−	0.674484i	$-0.764366\pi$
$227$	−22.2469	−1.47658	−0.738289	+	0.674484i	$0.764366\pi$
$228$	0	0
$229$	−1.06276	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$229$	−1.06276	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$230$	6.06055	0.399621
$231$	0	0
$232$	6.31384	0.414524
$233$	11.3686	0.744783	0.372391	−	0.928076i	$-0.378538\pi$
$233$	11.3686	0.744783	0.372391	+	0.928076i	$0.378538\pi$
$234$	0	0
$235$	4.31098	0.281218
$236$	−13.0656	−0.850499
$237$	0	0
$238$	0.938954	0.0608634
$239$	−15.5037	−1.00285	−0.501424	−	0.865202i	$-0.667190\pi$
$239$	−15.5037	−1.00285	−0.501424	+	0.865202i	$0.667190\pi$
$240$	0	0
$241$	12.2991	0.792257	0.396129	−	0.918195i	$-0.370353\pi$
$241$	12.2991	0.792257	0.396129	+	0.918195i	$0.370353\pi$
$242$	47.8003	3.07272
$243$	0	0
$244$	19.3218	1.23695
$245$	0.560790	0.0358276
$246$	0	0
$247$	−2.76178	−0.175728
$248$	7.99303	0.507558
$249$	0	0
$250$	−11.4138	−0.721875
$251$	27.6822	1.74729	0.873644	−	0.486566i	$-0.161751\pi$
$251$	27.6822	1.74729	0.873644	+	0.486566i	$0.161751\pi$
$252$	0	0
$253$	−29.8758	−1.87828
$254$	−2.10140	−0.131854
$255$	0	0
$256$	7.44424	0.465265
$257$	−21.1822	−1.32131	−0.660656	−	0.750689i	$-0.729722\pi$
$257$	−21.1822	−1.32131	−0.660656	+	0.750689i	$0.729722\pi$
$258$	0	0
$259$	3.98245	0.247457
$260$	−0.688021	−0.0426693
$261$	0	0
$262$	−9.48328	−0.585879
$263$	−0.729541	−0.0449854	−0.0224927	−	0.999747i	$-0.507160\pi$
$263$	−0.729541	−0.0449854	−0.0224927	+	0.999747i	$0.507160\pi$
$264$	0	0
$265$	6.91011	0.424484
$266$	11.4280	0.700697
$267$	0	0
$268$	−13.6499	−0.833803
$269$	−18.6944	−1.13981	−0.569907	−	0.821709i	$-0.693021\pi$
$269$	−18.6944	−1.13981	−0.569907	+	0.821709i	$0.693021\pi$
$270$	0	0
$271$	−24.8296	−1.50829	−0.754145	−	0.656708i	$-0.771948\pi$
$271$	−24.8296	−1.50829	−0.754145	+	0.656708i	$0.771948\pi$
$272$	−1.33836	−0.0811502
$273$	0	0
$274$	46.2354	2.79318
$275$	27.2191	1.64138
$276$	0	0
$277$	5.23567	0.314581	0.157291	−	0.987552i	$-0.449724\pi$
$277$	5.23567	0.314581	0.157291	+	0.987552i	$0.449724\pi$
$278$	−28.9197	−1.73449
$279$	0	0
$280$	0.490091	0.0292885
$281$	16.5734	0.988688	0.494344	−	0.869266i	$-0.335408\pi$
$281$	16.5734	0.988688	0.494344	+	0.869266i	$0.335408\pi$
$282$	0	0
$283$	−12.9465	−0.769587	−0.384794	−	0.923003i	$-0.625727\pi$
$283$	−12.9465	−0.769587	−0.384794	+	0.923003i	$0.625727\pi$
$284$	−30.5937	−1.81540
$285$	0	0
$286$	6.19943	0.366580
$287$	−4.97303	−0.293549
$288$	0	0
$289$	−16.8003	−0.988256
$290$	8.51387	0.499951
$291$	0	0
$292$	29.1942	1.70846
$293$	−22.8609	−1.33555	−0.667775	−	0.744363i	$-0.732753\pi$
$293$	−22.8609	−1.33555	−0.667775	+	0.744363i	$0.732753\pi$
$294$	0	0
$295$	−3.03288	−0.176581
$296$	3.48037	0.202293
$297$	0	0
$298$	17.8902	1.03635
$299$	−2.61173	−0.151040
$300$	0	0
$301$	−1.04891	−0.0604584
$302$	−17.1806	−0.988634
$303$	0	0
$304$	−16.2893	−0.934253
$305$	4.48511	0.256816
$306$	0	0
$307$	−13.6616	−0.779706	−0.389853	−	0.920877i	$-0.627474\pi$
$307$	−13.6616	−0.779706	−0.389853	+	0.920877i	$0.627474\pi$
$308$	−14.0343	−0.799681
$309$	0	0
$310$	10.7782	0.612158
$311$	−0.322143	−0.0182670	−0.00913351	−	0.999958i	$-0.502907\pi$
$311$	−0.322143	−0.0182670	−0.00913351	+	0.999958i	$0.502907\pi$
$312$	0	0
$313$	7.42475	0.419672	0.209836	−	0.977737i	$-0.432707\pi$
$313$	7.42475	0.419672	0.209836	+	0.977737i	$0.432707\pi$
$314$	7.81167	0.440838
$315$	0	0
$316$	36.0958	2.03055
$317$	−9.43372	−0.529850	−0.264925	−	0.964269i	$-0.585347\pi$
$317$	−9.43372	−0.529850	−0.264925	+	0.964269i	$0.585347\pi$
$318$	0	0
$319$	−41.9696	−2.34985
$320$	−6.11777	−0.341994
$321$	0	0
$322$	10.8072	0.602259
$323$	2.42996	0.135206
$324$	0	0
$325$	2.37949	0.131990
$326$	1.33034	0.0736809
$327$	0	0
$328$	−4.34607	−0.239972
$329$	7.68734	0.423817
$330$	0	0
$331$	21.4823	1.18078	0.590388	−	0.807120i	$-0.298975\pi$
$331$	21.4823	1.18078	0.590388	+	0.807120i	$0.298975\pi$
$332$	41.0452	2.25265
$333$	0	0
$334$	31.4375	1.72019
$335$	−3.16852	−0.173115
$336$	0	0
$337$	26.0480	1.41892	0.709462	−	0.704743i	$-0.248938\pi$
$337$	26.0480	1.41892	0.709462	+	0.704743i	$0.248938\pi$
$338$	−26.7762	−1.45644
$339$	0	0
$340$	0.605357	0.0328301
$341$	−53.1316	−2.87724
$342$	0	0
$343$	1.00000	0.0539949
$344$	−0.916675	−0.0494238
$345$	0	0
$346$	−23.4637	−1.26142
$347$	12.8632	0.690531	0.345265	−	0.938505i	$-0.387789\pi$
$347$	12.8632	0.690531	0.345265	+	0.938505i	$0.387789\pi$
$348$	0	0
$349$	7.04013	0.376850	0.188425	−	0.982088i	$-0.439662\pi$
$349$	7.04013	0.376850	0.188425	+	0.982088i	$0.439662\pi$
$350$	−9.84614	−0.526298
$351$	0	0
$352$	46.7185	2.49011
$353$	15.9182	0.847240	0.423620	−	0.905840i	$-0.360759\pi$
$353$	15.9182	0.847240	0.423620	+	0.905840i	$0.360759\pi$
$354$	0	0
$355$	−7.10161	−0.376915
$356$	−16.3729	−0.867764
$357$	0	0
$358$	−9.36294	−0.494847
$359$	1.38274	0.0729782	0.0364891	−	0.999334i	$-0.488383\pi$
$359$	1.38274	0.0729782	0.0364891	+	0.999334i	$0.488383\pi$
$360$	0	0
$361$	10.5751	0.556582
$362$	−11.2732	−0.592507
$363$	0	0
$364$	−1.22688	−0.0643058
$365$	6.77675	0.354711
$366$	0	0
$367$	−16.7609	−0.874912	−0.437456	−	0.899240i	$-0.644120\pi$
$367$	−16.7609	−0.874912	−0.437456	+	0.899240i	$0.644120\pi$
$368$	−15.4043	−0.803003
$369$	0	0
$370$	4.69309	0.243982
$371$	12.3221	0.639731
$372$	0	0
$373$	27.7753	1.43815	0.719074	−	0.694934i	$-0.244566\pi$
$373$	27.7753	1.43815	0.719074	+	0.694934i	$0.244566\pi$
$374$	−5.45458	−0.282050
$375$	0	0
$376$	6.71818	0.346464
$377$	−3.66897	−0.188961
$378$	0	0
$379$	0.772074	0.0396588	0.0198294	−	0.999803i	$-0.493688\pi$
$379$	0.772074	0.0396588	0.0198294	+	0.999803i	$0.493688\pi$
$380$	7.36781	0.377961
$381$	0	0
$382$	−50.4346	−2.58046
$383$	25.5845	1.30731	0.653653	−	0.756794i	$-0.273236\pi$
$383$	25.5845	1.30731	0.653653	+	0.756794i	$0.273236\pi$
$384$	0	0
$385$	−3.25775	−0.166030
$386$	23.1258	1.17707
$387$	0	0
$388$	14.0376	0.712653
$389$	16.9357	0.858676	0.429338	−	0.903144i	$-0.358747\pi$
$389$	16.9357	0.858676	0.429338	+	0.903144i	$0.358747\pi$
$390$	0	0
$391$	2.29794	0.116212
$392$	0.873928	0.0441400
$393$	0	0
$394$	−1.96113	−0.0988002
$395$	8.37881	0.421584
$396$	0	0
$397$	−20.2309	−1.01536	−0.507679	−	0.861546i	$-0.669496\pi$
$397$	−20.2309	−1.01536	−0.507679	+	0.861546i	$0.669496\pi$
$398$	−16.2492	−0.814500
$399$	0	0
$400$	14.0345	0.701723
$401$	4.97118	0.248249	0.124124	−	0.992267i	$-0.460388\pi$
$401$	4.97118	0.248249	0.124124	+	0.992267i	$0.460388\pi$
$402$	0	0
$403$	−4.64474	−0.231371
$404$	−16.7728	−0.834478
$405$	0	0
$406$	15.1819	0.753465
$407$	−23.1349	−1.14675
$408$	0	0
$409$	2.99618	0.148151	0.0740757	−	0.997253i	$-0.476399\pi$
$409$	2.99618	0.148151	0.0740757	+	0.997253i	$0.476399\pi$
$410$	−5.86044	−0.289426
$411$	0	0
$412$	34.5201	1.70068
$413$	−5.40822	−0.266121
$414$	0	0
$415$	9.52770	0.467696
$416$	4.08412	0.200240
$417$	0	0
$418$	−66.3878	−3.24713
$419$	11.5420	0.563865	0.281933	−	0.959434i	$-0.409025\pi$
$419$	11.5420	0.563865	0.281933	+	0.959434i	$0.409025\pi$
$420$	0	0
$421$	−19.1882	−0.935176	−0.467588	−	0.883946i	$-0.654877\pi$
$421$	−19.1882	−0.935176	−0.467588	+	0.883946i	$0.654877\pi$
$422$	−13.1704	−0.641127
$423$	0	0
$424$	10.7686	0.522970
$425$	−2.09360	−0.101554
$426$	0	0
$427$	7.99783	0.387042
$428$	14.5297	0.702318
$429$	0	0
$430$	−1.23609	−0.0596094
$431$	−10.8389	−0.522089	−0.261045	−	0.965327i	$-0.584067\pi$
$431$	−10.8389	−0.522089	−0.261045	+	0.965327i	$0.584067\pi$
$432$	0	0
$433$	16.1763	0.777381	0.388691	−	0.921368i	$-0.372928\pi$
$433$	16.1763	0.777381	0.388691	+	0.921368i	$0.372928\pi$
$434$	19.2196	0.922570
$435$	0	0
$436$	−4.60455	−0.220518
$437$	27.9683	1.33790
$438$	0	0
$439$	−37.2694	−1.77877	−0.889386	−	0.457157i	$-0.848868\pi$
$439$	−37.2694	−1.77877	−0.889386	+	0.457157i	$0.848868\pi$
$440$	−2.84704	−0.135727
$441$	0	0
$442$	−0.476838	−0.0226808
$443$	11.1356	0.529068	0.264534	−	0.964376i	$-0.414782\pi$
$443$	11.1356	0.529068	0.264534	+	0.964376i	$0.414782\pi$
$444$	0	0
$445$	−3.80060	−0.180166
$446$	39.2430	1.85821
$447$	0	0
$448$	−10.9092	−0.515411
$449$	−37.6261	−1.77569	−0.887843	−	0.460146i	$-0.847797\pi$
$449$	−37.6261	−1.77569	−0.887843	+	0.460146i	$0.847797\pi$
$450$	0	0
$451$	28.8894	1.36035
$452$	−33.2811	−1.56541
$453$	0	0
$454$	−46.7496	−2.19407
$455$	−0.284791	−0.0133512
$456$	0	0
$457$	8.15760	0.381596	0.190798	−	0.981629i	$-0.438892\pi$
$457$	8.15760	0.381596	0.190798	+	0.981629i	$0.438892\pi$
$458$	−2.23329	−0.104355
$459$	0	0
$460$	6.96752	0.324862
$461$	−14.2357	−0.663022	−0.331511	−	0.943451i	$-0.607558\pi$
$461$	−14.2357	−0.663022	−0.331511	+	0.943451i	$0.607558\pi$
$462$	0	0
$463$	14.9460	0.694601	0.347300	−	0.937754i	$-0.387098\pi$
$463$	14.9460	0.694601	0.347300	+	0.937754i	$0.387098\pi$
$464$	−21.6399	−1.00461
$465$	0	0
$466$	23.8900	1.10668
$467$	−12.3100	−0.569640	−0.284820	−	0.958581i	$-0.591934\pi$
$467$	−12.3100	−0.569640	−0.284820	+	0.958581i	$0.591934\pi$
$468$	0	0
$469$	−5.65009	−0.260897
$470$	9.05910	0.417865
$471$	0	0
$472$	−4.72640	−0.217550
$473$	6.09336	0.280173
$474$	0	0
$475$	−25.4812	−1.16916
$476$	1.07947	0.0494775
$477$	0	0
$478$	−32.5794	−1.49015
$479$	5.89044	0.269141	0.134571	−	0.990904i	$-0.457035\pi$
$479$	5.89044	0.269141	0.134571	+	0.990904i	$0.457035\pi$
$480$	0	0
$481$	−2.02244	−0.0922154
$482$	25.8454	1.17723
$483$	0	0
$484$	54.9538	2.49790
$485$	3.25851	0.147962
$486$	0	0
$487$	35.7636	1.62061	0.810303	−	0.586011i	$-0.199303\pi$
$487$	35.7636	1.62061	0.810303	+	0.586011i	$0.199303\pi$
$488$	6.98953	0.316401
$489$	0	0
$490$	1.17844	0.0532367
$491$	−8.22384	−0.371137	−0.185568	−	0.982631i	$-0.559413\pi$
$491$	−8.22384	−0.371137	−0.185568	+	0.982631i	$0.559413\pi$
$492$	0	0
$493$	3.22815	0.145389
$494$	−5.80360	−0.261116
$495$	0	0
$496$	−27.3952	−1.23008
$497$	−12.6636	−0.568039
$498$	0	0
$499$	1.21460	0.0543729	0.0271864	−	0.999630i	$-0.491345\pi$
$499$	1.21460	0.0543729	0.0271864	+	0.999630i	$0.491345\pi$
$500$	−13.1220	−0.586832
$501$	0	0
$502$	58.1715	2.59632
$503$	−35.4384	−1.58012	−0.790059	−	0.613031i	$-0.789950\pi$
$503$	−35.4384	−1.58012	−0.790059	+	0.613031i	$0.789950\pi$
$504$	0	0
$505$	−3.89342	−0.173255
$506$	−62.7810	−2.79096
$507$	0	0
$508$	−2.41588	−0.107187
$509$	−44.6263	−1.97803	−0.989014	−	0.147822i	$-0.952774\pi$
$509$	−44.6263	−1.97803	−0.989014	+	0.147822i	$0.952774\pi$
$510$	0	0
$511$	12.0843	0.534577
$512$	29.3239	1.29594
$513$	0	0
$514$	−44.5123	−1.96336
$515$	8.01305	0.353097
$516$	0	0
$517$	−44.6573	−1.96403
$518$	8.36871	0.367700
$519$	0	0
$520$	−0.248887	−0.0109144
$521$	31.5066	1.38033	0.690166	−	0.723651i	$-0.257538\pi$
$521$	31.5066	1.38033	0.690166	+	0.723651i	$0.257538\pi$
$522$	0	0
$523$	−11.7736	−0.514824	−0.257412	−	0.966302i	$-0.582870\pi$
$523$	−11.7736	−0.514824	−0.257412	+	0.966302i	$0.582870\pi$
$524$	−10.9025	−0.476277
$525$	0	0
$526$	−1.53306	−0.0668444
$527$	4.08669	0.178019
$528$	0	0
$529$	3.44876	0.149946
$530$	14.5209	0.630747
$531$	0	0
$532$	13.1383	0.569616
$533$	2.52550	0.109391
$534$	0	0
$535$	3.37272	0.145816
$536$	−4.93777	−0.213279
$537$	0	0
$538$	−39.2843	−1.69367
$539$	−5.80921	−0.250220
$540$	0	0
$541$	34.0266	1.46292	0.731459	−	0.681885i	$-0.238840\pi$
$541$	34.0266	1.46292	0.731459	+	0.681885i	$0.238840\pi$
$542$	−52.1768	−2.24119
$543$	0	0
$544$	−3.59342	−0.154067
$545$	−1.06884	−0.0457840
$546$	0	0
$547$	−13.6115	−0.581986	−0.290993	−	0.956725i	$-0.593986\pi$
$547$	−13.6115	−0.581986	−0.290993	+	0.956725i	$0.593986\pi$
$548$	53.1546	2.27065
$549$	0	0
$550$	57.1983	2.43894
$551$	39.2899	1.67380
$552$	0	0
$553$	14.9411	0.635359
$554$	11.0022	0.467440
$555$	0	0
$556$	−33.2476	−1.41001
$557$	−15.6048	−0.661195	−0.330598	−	0.943772i	$-0.607250\pi$
$557$	−15.6048	−0.661195	−0.330598	+	0.943772i	$0.607250\pi$
$558$	0	0
$559$	0.532679	0.0225299
$560$	−1.67973	−0.0709815
$561$	0	0
$562$	34.8274	1.46911
$563$	18.5157	0.780345	0.390173	−	0.920742i	$-0.372415\pi$
$563$	18.5157	0.780345	0.390173	+	0.920742i	$0.372415\pi$
$564$	0	0
$565$	−7.72545	−0.325012
$566$	−27.2057	−1.14354
$567$	0	0
$568$	−11.0671	−0.464363
$569$	13.5940	0.569891	0.284946	−	0.958544i	$-0.408024\pi$
$569$	13.5940	0.569891	0.284946	+	0.958544i	$0.408024\pi$
$570$	0	0
$571$	24.0122	1.00488	0.502439	−	0.864613i	$-0.332436\pi$
$571$	24.0122	1.00488	0.502439	+	0.864613i	$0.332436\pi$
$572$	7.12719	0.298003
$573$	0	0
$574$	−10.4503	−0.436188
$575$	−24.0968	−1.00491
$576$	0	0
$577$	−44.5452	−1.85444	−0.927220	−	0.374517i	$-0.877808\pi$
$577$	−44.5452	−1.85444	−0.927220	+	0.374517i	$0.877808\pi$
$578$	−35.3042	−1.46846
$579$	0	0
$580$	9.78799	0.406424
$581$	16.9898	0.704854
$582$	0	0
$583$	−71.5816	−2.96460
$584$	10.5608	0.437009
$585$	0	0
$586$	−48.0400	−1.98451
$587$	19.5144	0.805445	0.402723	−	0.915322i	$-0.368064\pi$
$587$	19.5144	0.805445	0.402723	+	0.915322i	$0.368064\pi$
$588$	0	0
$589$	49.7391	2.04947
$590$	−6.37329	−0.262384
$591$	0	0
$592$	−11.9286	−0.490261
$593$	18.9355	0.777587	0.388794	−	0.921325i	$-0.372892\pi$
$593$	18.9355	0.777587	0.388794	+	0.921325i	$0.372892\pi$
$594$	0	0
$595$	0.250574	0.0102725
$596$	20.5675	0.842479
$597$	0	0
$598$	−5.48829	−0.224433
$599$	−8.38945	−0.342783	−0.171392	−	0.985203i	$-0.554826\pi$
$599$	−8.38945	−0.342783	−0.171392	+	0.985203i	$0.554826\pi$
$600$	0	0
$601$	−26.4225	−1.07780	−0.538898	−	0.842371i	$-0.681159\pi$
$601$	−26.4225	−1.07780	−0.538898	+	0.842371i	$0.681159\pi$
$602$	−2.20419	−0.0898359
$603$	0	0
$604$	−19.7518	−0.803687
$605$	12.7562	0.518615
$606$	0	0
$607$	24.3695	0.989128	0.494564	−	0.869141i	$-0.335328\pi$
$607$	24.3695	0.989128	0.494564	+	0.869141i	$0.335328\pi$
$608$	−43.7356	−1.77371
$609$	0	0
$610$	9.42500	0.381607
$611$	−3.90393	−0.157936
$612$	0	0
$613$	−17.8924	−0.722667	−0.361333	−	0.932437i	$-0.617678\pi$
$613$	−17.8924	−0.722667	−0.361333	+	0.932437i	$0.617678\pi$
$614$	−28.7084	−1.15858
$615$	0	0
$616$	−5.07683	−0.204551
$617$	−12.6204	−0.508080	−0.254040	−	0.967194i	$-0.581759\pi$
$617$	−12.6204	−0.508080	−0.254040	+	0.967194i	$0.581759\pi$
$618$	0	0
$619$	28.6104	1.14995	0.574975	−	0.818171i	$-0.305012\pi$
$619$	28.6104	1.14995	0.574975	+	0.818171i	$0.305012\pi$
$620$	12.3911	0.497640
$621$	0	0
$622$	−0.676950	−0.0271432
$623$	−6.77722	−0.271524
$624$	0	0
$625$	20.3816	0.815265
$626$	15.6024	0.623596
$627$	0	0
$628$	8.98070	0.358369
$629$	1.77945	0.0709513
$630$	0	0
$631$	41.5735	1.65502	0.827508	−	0.561454i	$-0.189758\pi$
$631$	41.5735	1.65502	0.827508	+	0.561454i	$0.189758\pi$
$632$	13.0574	0.519396
$633$	0	0
$634$	−19.8240	−0.787312
$635$	−0.560790	−0.0222543
$636$	0	0
$637$	−0.507839	−0.0201213
$638$	−88.1949	−3.49167
$639$	0	0
$640$	−3.83596	−0.151630
$641$	−12.7729	−0.504498	−0.252249	−	0.967662i	$-0.581170\pi$
$641$	−12.7729	−0.504498	−0.252249	+	0.967662i	$0.581170\pi$
$642$	0	0
$643$	24.1097	0.950792	0.475396	−	0.879772i	$-0.342305\pi$
$643$	24.1097	0.950792	0.475396	+	0.879772i	$0.342305\pi$
$644$	12.4245	0.489593
$645$	0	0
$646$	5.10631	0.200905
$647$	−12.5791	−0.494537	−0.247268	−	0.968947i	$-0.579533\pi$
$647$	−12.5791	−0.494537	−0.247268	+	0.968947i	$0.579533\pi$
$648$	0	0
$649$	31.4175	1.23324
$650$	5.00025	0.196126
$651$	0	0
$652$	1.52943	0.0598972
$653$	8.39137	0.328380	0.164190	−	0.986429i	$-0.447499\pi$
$653$	8.39137	0.328380	0.164190	+	0.986429i	$0.447499\pi$
$654$	0	0
$655$	−2.53076	−0.0988849
$656$	14.8956	0.581577
$657$	0	0
$658$	16.1542	0.629755
$659$	27.9814	1.09000	0.545000	−	0.838436i	$-0.316530\pi$
$659$	27.9814	1.09000	0.545000	+	0.838436i	$0.316530\pi$
$660$	0	0
$661$	−19.1205	−0.743702	−0.371851	−	0.928292i	$-0.621277\pi$
$661$	−19.1205	−0.743702	−0.371851	+	0.928292i	$0.621277\pi$
$662$	45.1429	1.75453
$663$	0	0
$664$	14.8478	0.576208
$665$	3.04974	0.118264
$666$	0	0
$667$	37.1553	1.43866
$668$	36.1422	1.39839
$669$	0	0
$670$	−6.65832	−0.257233
$671$	−46.4611	−1.79361
$672$	0	0
$673$	24.0448	0.926860	0.463430	−	0.886134i	$-0.346619\pi$
$673$	24.0448	0.926860	0.463430	+	0.886134i	$0.346619\pi$
$674$	54.7372	2.10840
$675$	0	0
$676$	−30.7834	−1.18398
$677$	14.0232	0.538955	0.269478	−	0.963007i	$-0.413149\pi$
$677$	14.0232	0.538955	0.269478	+	0.963007i	$0.413149\pi$
$678$	0	0
$679$	5.81057	0.222989
$680$	0.218984	0.00839765
$681$	0	0
$682$	−111.651	−4.27532
$683$	5.98262	0.228918	0.114459	−	0.993428i	$-0.463486\pi$
$683$	5.98262	0.228918	0.114459	+	0.993428i	$0.463486\pi$
$684$	0	0
$685$	12.3386	0.471434
$686$	2.10140	0.0802318
$687$	0	0
$688$	3.14180	0.119780
$689$	−6.25763	−0.238397
$690$	0	0
$691$	24.4115	0.928658	0.464329	−	0.885663i	$-0.346296\pi$
$691$	24.4115	0.928658	0.464329	+	0.885663i	$0.346296\pi$
$692$	−26.9751	−1.02544
$693$	0	0
$694$	27.0306	1.02607
$695$	−7.71767	−0.292748
$696$	0	0
$697$	−2.22207	−0.0841667
$698$	14.7941	0.559966
$699$	0	0
$700$	−11.3196	−0.427842
$701$	−16.3369	−0.617037	−0.308518	−	0.951218i	$-0.599833\pi$
$701$	−16.3369	−0.617037	−0.308518	+	0.951218i	$0.599833\pi$
$702$	0	0
$703$	21.6577	0.816837
$704$	63.3738	2.38849
$705$	0	0
$706$	33.4505	1.25892
$707$	−6.94273	−0.261108
$708$	0	0
$709$	30.7342	1.15425	0.577124	−	0.816657i	$-0.304175\pi$
$709$	30.7342	1.15425	0.577124	+	0.816657i	$0.304175\pi$
$710$	−14.9233	−0.560062
$711$	0	0
$712$	−5.92280	−0.221966
$713$	47.0369	1.76154
$714$	0	0
$715$	1.65441	0.0618715
$716$	−10.7641	−0.402274
$717$	0	0
$718$	2.90569	0.108439
$719$	9.49665	0.354165	0.177083	−	0.984196i	$-0.443334\pi$
$719$	9.49665	0.354165	0.177083	+	0.984196i	$0.443334\pi$
$720$	0	0
$721$	14.2888	0.532144
$722$	22.2224	0.827032
$723$	0	0
$724$	−12.9603	−0.481665
$725$	−33.8513	−1.25720
$726$	0	0
$727$	−19.1297	−0.709480	−0.354740	−	0.934965i	$-0.615431\pi$
$727$	−19.1297	−0.709480	−0.354740	+	0.934965i	$0.615431\pi$
$728$	−0.443815	−0.0164489
$729$	0	0
$730$	14.2406	0.527070
$731$	−0.468679	−0.0173347
$732$	0	0
$733$	3.74313	0.138256	0.0691279	−	0.997608i	$-0.477978\pi$
$733$	3.74313	0.138256	0.0691279	+	0.997608i	$0.477978\pi$
$734$	−35.2213	−1.30004
$735$	0	0
$736$	−41.3595	−1.52453
$737$	32.8226	1.20903
$738$	0	0
$739$	8.67498	0.319114	0.159557	−	0.987189i	$-0.448993\pi$
$739$	8.67498	0.319114	0.159557	+	0.987189i	$0.448993\pi$
$740$	5.39543	0.198340
$741$	0	0
$742$	25.8936	0.950585
$743$	−36.0897	−1.32400	−0.662002	−	0.749502i	$-0.730293\pi$
$743$	−36.0897	−1.32400	−0.662002	+	0.749502i	$0.730293\pi$
$744$	0	0
$745$	4.77428	0.174916
$746$	58.3669	2.13696
$747$	0	0
$748$	−6.27087	−0.229286
$749$	6.01423	0.219755
$750$	0	0
$751$	31.1150	1.13540	0.567700	−	0.823235i	$-0.307833\pi$
$751$	31.1150	1.13540	0.567700	+	0.823235i	$0.307833\pi$
$752$	−23.0258	−0.839664
$753$	0	0
$754$	−7.70996	−0.280780
$755$	−4.58491	−0.166862
$756$	0	0
$757$	51.8888	1.88593	0.942966	−	0.332890i	$-0.108024\pi$
$757$	51.8888	1.88593	0.942966	+	0.332890i	$0.108024\pi$
$758$	1.62244	0.0589295
$759$	0	0
$760$	2.66526	0.0966790
$761$	−6.24052	−0.226219	−0.113109	−	0.993583i	$-0.536081\pi$
$761$	−6.24052	−0.226219	−0.113109	+	0.993583i	$0.536081\pi$
$762$	0	0
$763$	−1.90595	−0.0690001
$764$	−57.9822	−2.09772
$765$	0	0
$766$	53.7632	1.94254
$767$	2.74650	0.0991705
$768$	0	0
$769$	−13.6205	−0.491169	−0.245584	−	0.969375i	$-0.578980\pi$
$769$	−13.6205	−0.491169	−0.245584	+	0.969375i	$0.578980\pi$
$770$	−6.84583	−0.246707
$771$	0	0
$772$	26.5867	0.956876
$773$	−44.4719	−1.59954	−0.799771	−	0.600305i	$-0.795046\pi$
$773$	−44.4719	−1.59954	−0.799771	+	0.600305i	$0.795046\pi$
$774$	0	0
$775$	−42.8541	−1.53937
$776$	5.07802	0.182290
$777$	0	0
$778$	35.5888	1.27592
$779$	−27.0448	−0.968980
$780$	0	0
$781$	73.5654	2.63238
$782$	4.82889	0.172681
$783$	0	0
$784$	−2.99529	−0.106975
$785$	2.08466	0.0744048
$786$	0	0
$787$	−24.3894	−0.869388	−0.434694	−	0.900578i	$-0.643143\pi$
$787$	−24.3894	−0.869388	−0.434694	+	0.900578i	$0.643143\pi$
$788$	−2.25461	−0.0803173
$789$	0	0
$790$	17.6072	0.626437
$791$	−13.7760	−0.489818
$792$	0	0
$793$	−4.06161	−0.144232
$794$	−42.5131	−1.50873
$795$	0	0
$796$	−18.6810	−0.662130
$797$	−24.3222	−0.861538	−0.430769	−	0.902462i	$-0.641758\pi$
$797$	−24.3222	−0.861538	−0.430769	+	0.902462i	$0.641758\pi$
$798$	0	0
$799$	3.43488	0.121517
$800$	37.6816	1.33225
$801$	0	0
$802$	10.4464	0.368876
$803$	−70.2001	−2.47731
$804$	0	0
$805$	2.88405	0.101650
$806$	−9.76045	−0.343797
$807$	0	0
$808$	−6.06745	−0.213452
$809$	−9.21160	−0.323863	−0.161931	−	0.986802i	$-0.551772\pi$
$809$	−9.21160	−0.323863	−0.161931	+	0.986802i	$0.551772\pi$
$810$	0	0
$811$	−46.2910	−1.62550	−0.812748	−	0.582615i	$-0.802029\pi$
$811$	−46.2910	−1.62550	−0.812748	+	0.582615i	$0.802029\pi$
$812$	17.4539	0.612512
$813$	0	0
$814$	−48.6156	−1.70398
$815$	0.355022	0.0124359
$816$	0	0
$817$	−5.70430	−0.199568
$818$	6.29616	0.220140
$819$	0	0
$820$	−6.73747	−0.235283
$821$	−14.8527	−0.518362	−0.259181	−	0.965829i	$-0.583453\pi$
$821$	−14.8527	−0.518362	−0.259181	+	0.965829i	$0.583453\pi$
$822$	0	0
$823$	37.3700	1.30264	0.651318	−	0.758805i	$-0.274216\pi$
$823$	37.3700	1.30264	0.651318	+	0.758805i	$0.274216\pi$
$824$	12.4874	0.435020
$825$	0	0
$826$	−11.3648	−0.395433
$827$	31.0205	1.07869	0.539345	−	0.842085i	$-0.318672\pi$
$827$	31.0205	1.07869	0.539345	+	0.842085i	$0.318672\pi$
$828$	0	0
$829$	−50.2110	−1.74390	−0.871950	−	0.489595i	$-0.837145\pi$
$829$	−50.2110	−1.74390	−0.871950	+	0.489595i	$0.837145\pi$
$830$	20.0215	0.694956
$831$	0	0
$832$	5.54011	0.192069
$833$	0.446823	0.0154815
$834$	0	0
$835$	8.38958	0.290334
$836$	−76.3229	−2.63968
$837$	0	0
$838$	24.2544	0.837855
$839$	45.1203	1.55773	0.778863	−	0.627194i	$-0.215797\pi$
$839$	45.1203	1.55773	0.778863	+	0.627194i	$0.215797\pi$
$840$	0	0
$841$	23.1958	0.799855
$842$	−40.3221	−1.38959
$843$	0	0
$844$	−15.1414	−0.521190
$845$	−7.14565	−0.245818
$846$	0	0
$847$	22.7469	0.781593
$848$	−36.9082	−1.26743
$849$	0	0
$850$	−4.39948	−0.150901
$851$	20.4811	0.702082
$852$	0	0
$853$	−12.7022	−0.434916	−0.217458	−	0.976070i	$-0.569776\pi$
$853$	−12.7022	−0.434916	−0.217458	+	0.976070i	$0.569776\pi$
$854$	16.8066	0.575111
$855$	0	0
$856$	5.25601	0.179647
$857$	14.0200	0.478916	0.239458	−	0.970907i	$-0.423030\pi$
$857$	14.0200	0.478916	0.239458	+	0.970907i	$0.423030\pi$
$858$	0	0
$859$	41.9315	1.43068	0.715342	−	0.698775i	$-0.246271\pi$
$859$	41.9315	1.43068	0.715342	+	0.698775i	$0.246271\pi$
$860$	−1.42107	−0.0484581
$861$	0	0
$862$	−22.7768	−0.775780
$863$	−27.1712	−0.924919	−0.462460	−	0.886640i	$-0.653033\pi$
$863$	−27.1712	−0.924919	−0.462460	+	0.886640i	$0.653033\pi$
$864$	0	0
$865$	−6.26164	−0.212902
$866$	33.9928	1.15512
$867$	0	0
$868$	22.0959	0.749982
$869$	−86.7958	−2.94435
$870$	0	0
$871$	2.86934	0.0972238
$872$	−1.66566	−0.0564065
$873$	0	0
$874$	58.7725	1.98801
$875$	−5.43154	−0.183620
$876$	0	0
$877$	−29.7678	−1.00519	−0.502593	−	0.864523i	$-0.667620\pi$
$877$	−29.7678	−1.00519	−0.502593	+	0.864523i	$0.667620\pi$
$878$	−78.3179	−2.64310
$879$	0	0
$880$	9.75789	0.328938
$881$	−54.9329	−1.85074	−0.925368	−	0.379070i	$-0.876244\pi$
$881$	−54.9329	−1.85074	−0.925368	+	0.379070i	$0.876244\pi$
$882$	0	0
$883$	22.3984	0.753766	0.376883	−	0.926261i	$-0.376996\pi$
$883$	22.3984	0.753766	0.376883	+	0.926261i	$0.376996\pi$
$884$	−0.548197	−0.0184379
$885$	0	0
$886$	23.4003	0.786149
$887$	−11.4016	−0.382828	−0.191414	−	0.981509i	$-0.561307\pi$
$887$	−11.4016	−0.382828	−0.191414	+	0.981509i	$0.561307\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−7.98658	−0.267711
$891$	0	0
$892$	45.1158	1.51059
$893$	41.8060	1.39898
$894$	0	0
$895$	−2.49864	−0.0835204
$896$	−6.84027	−0.228518
$897$	0	0
$898$	−79.0675	−2.63852
$899$	66.0774	2.20381
$900$	0	0
$901$	5.50580	0.183425
$902$	60.7081	2.02136
$903$	0	0
$904$	−12.0392	−0.400419
$905$	−3.00843	−0.100004
$906$	0	0
$907$	9.40787	0.312383	0.156192	−	0.987727i	$-0.450078\pi$
$907$	9.40787	0.312383	0.156192	+	0.987727i	$0.450078\pi$
$908$	−53.7458	−1.78362
$909$	0	0
$910$	−0.598460	−0.0198388
$911$	31.7859	1.05311	0.526557	−	0.850140i	$-0.323483\pi$
$911$	31.7859	1.05311	0.526557	+	0.850140i	$0.323483\pi$
$912$	0	0
$913$	−98.6971	−3.26640
$914$	17.1424	0.567019
$915$	0	0
$916$	−2.56751	−0.0848329
$917$	−4.51284	−0.149027
$918$	0	0
$919$	11.2567	0.371326	0.185663	−	0.982614i	$-0.440557\pi$
$919$	11.2567	0.371326	0.185663	+	0.982614i	$0.440557\pi$
$920$	2.52046	0.0830969
$921$	0	0
$922$	−29.9149	−0.985193
$923$	6.43106	0.211681
$924$	0	0
$925$	−18.6598	−0.613531
$926$	31.4076	1.03212
$927$	0	0
$928$	−58.1018	−1.90729
$929$	31.3726	1.02930	0.514651	−	0.857400i	$-0.327921\pi$
$929$	31.3726	1.02930	0.514651	+	0.857400i	$0.327921\pi$
$930$	0	0
$931$	5.43830	0.178233
$932$	27.4652	0.899652
$933$	0	0
$934$	−25.8683	−0.846436
$935$	−1.45564	−0.0476045
$936$	0	0
$937$	7.76816	0.253775	0.126887	−	0.991917i	$-0.459501\pi$
$937$	7.76816	0.253775	0.126887	+	0.991917i	$0.459501\pi$
$938$	−11.8731	−0.387670
$939$	0	0
$940$	10.4148	0.339694
$941$	−12.0124	−0.391592	−0.195796	−	0.980645i	$-0.562729\pi$
$941$	−12.0124	−0.391592	−0.195796	+	0.980645i	$0.562729\pi$
$942$	0	0
$943$	−25.5755	−0.832852
$944$	16.1992	0.527238
$945$	0	0
$946$	12.8046	0.416313
$947$	−41.1666	−1.33773	−0.668867	−	0.743382i	$-0.733221\pi$
$947$	−41.1666	−1.33773	−0.668867	+	0.743382i	$0.733221\pi$
$948$	0	0
$949$	−6.13686	−0.199211
$950$	−53.5462	−1.73727
$951$	0	0
$952$	0.390492	0.0126559
$953$	−43.4646	−1.40796	−0.703978	−	0.710222i	$-0.748594\pi$
$953$	−43.4646	−1.40796	−0.703978	+	0.710222i	$0.748594\pi$
$954$	0	0
$955$	−13.4592	−0.435530
$956$	−37.4550	−1.21138
$957$	0	0
$958$	12.3782	0.399921
$959$	22.0022	0.710488
$960$	0	0
$961$	52.6510	1.69842
$962$	−4.24996	−0.137024
$963$	0	0
$964$	29.7132	0.956999
$965$	6.17148	0.198667
$966$	0	0
$967$	−33.1419	−1.06577	−0.532886	−	0.846187i	$-0.678893\pi$
$967$	−33.1419	−1.06577	−0.532886	+	0.846187i	$0.678893\pi$
$968$	19.8792	0.638940
$969$	0	0
$970$	6.84744	0.219858
$971$	−42.5237	−1.36465	−0.682326	−	0.731048i	$-0.739031\pi$
$971$	−42.5237	−1.36465	−0.682326	+	0.731048i	$0.739031\pi$
$972$	0	0
$973$	−13.7621	−0.441194
$974$	75.1537	2.40808
$975$	0	0
$976$	−23.9558	−0.766806
$977$	36.1740	1.15731	0.578655	−	0.815572i	$-0.303578\pi$
$977$	36.1740	1.15731	0.578655	+	0.815572i	$0.303578\pi$
$978$	0	0
$979$	39.3703	1.25828
$980$	1.35480	0.0432776
$981$	0	0
$982$	−17.2816	−0.551477
$983$	−24.7265	−0.788654	−0.394327	−	0.918970i	$-0.629022\pi$
$983$	−24.7265	−0.788654	−0.394327	+	0.918970i	$0.629022\pi$
$984$	0	0
$985$	−0.523357	−0.0166755
$986$	6.78363	0.216035
$987$	0	0
$988$	−6.67212	−0.212268
$989$	−5.39439	−0.171532
$990$	0	0
$991$	−60.9820	−1.93716	−0.968578	−	0.248709i	$-0.919994\pi$
$991$	−60.9820	−1.93716	−0.968578	+	0.248709i	$0.919994\pi$
$992$	−73.5542	−2.33535
$993$	0	0
$994$	−26.6112	−0.844057
$995$	−4.33636	−0.137472
$996$	0	0
$997$	29.5893	0.937101	0.468551	−	0.883437i	$-0.344776\pi$
$997$	29.5893	0.937101	0.468551	+	0.883437i	$0.344776\pi$
$998$	2.55236	0.0807934
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.33	yes	40
3.2	odd	2	inner	8001.2.a.ba.1.8	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.8	✓	40	3.2	odd	2	inner
8001.2.a.ba.1.33	yes	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+2.10140 q^{2} +2.41588 q^{4} +0.560790 q^{5} +1.00000 q^{7} +0.873928 q^{8} +O(q^{10})\)	\(q+2.10140 q^{2} +2.41588 q^{4} +0.560790 q^{5} +1.00000 q^{7} +0.873928 q^{8} +1.17844 q^{10} -5.80921 q^{11} -0.507839 q^{13} +2.10140 q^{14} -2.99529 q^{16} +0.446823 q^{17} +5.43830 q^{19} +1.35480 q^{20} -12.2075 q^{22} +5.14284 q^{23} -4.68551 q^{25} -1.06717 q^{26} +2.41588 q^{28} +7.22466 q^{29} +9.14609 q^{31} -8.04215 q^{32} +0.938954 q^{34} +0.560790 q^{35} +3.98245 q^{37} +11.4280 q^{38} +0.490091 q^{40} -4.97303 q^{41} -1.04891 q^{43} -14.0343 q^{44} +10.8072 q^{46} +7.68734 q^{47} +1.00000 q^{49} -9.84614 q^{50} -1.22688 q^{52} +12.3221 q^{53} -3.25775 q^{55} +0.873928 q^{56} +15.1819 q^{58} -5.40822 q^{59} +7.99783 q^{61} +19.2196 q^{62} -10.9092 q^{64} -0.284791 q^{65} -5.65009 q^{67} +1.07947 q^{68} +1.17844 q^{70} -12.6636 q^{71} +12.0843 q^{73} +8.36871 q^{74} +13.1383 q^{76} -5.80921 q^{77} +14.9411 q^{79} -1.67973 q^{80} -10.4503 q^{82} +16.9898 q^{83} +0.250574 q^{85} -2.20419 q^{86} -5.07683 q^{88} -6.77722 q^{89} -0.507839 q^{91} +12.4245 q^{92} +16.1542 q^{94} +3.04974 q^{95} +5.81057 q^{97} +2.10140 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 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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