LMFDB - Embedded newform 8046.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + \cdots - 7083 $ x^14 - 2x^13 - 42x^12 + 70x^11 + 680x^10 - 882x^9 - 5559x^8 + 5066x^7 + 24455x^6 - 12990x^5 - 55580x^4 + 9808x^3 + 53551x^2 + 6282*x - 7083
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.13855$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.13855 q^{5} -2.45592 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.13855 q^{5} -2.45592 q^{7} -1.00000 q^{8} +3.13855 q^{10} -1.65548 q^{11} +5.81470 q^{13} +2.45592 q^{14} +1.00000 q^{16} +6.74376 q^{17} +4.34108 q^{19} -3.13855 q^{20} +1.65548 q^{22} -1.27598 q^{23} +4.85047 q^{25} -5.81470 q^{26} -2.45592 q^{28} +10.5818 q^{29} -9.90340 q^{31} -1.00000 q^{32} -6.74376 q^{34} +7.70800 q^{35} +6.33116 q^{37} -4.34108 q^{38} +3.13855 q^{40} -8.45200 q^{41} +9.78035 q^{43} -1.65548 q^{44} +1.27598 q^{46} -7.30857 q^{47} -0.968479 q^{49} -4.85047 q^{50} +5.81470 q^{52} -10.0205 q^{53} +5.19581 q^{55} +2.45592 q^{56} -10.5818 q^{58} -4.28382 q^{59} +9.86429 q^{61} +9.90340 q^{62} +1.00000 q^{64} -18.2497 q^{65} +5.48095 q^{67} +6.74376 q^{68} -7.70800 q^{70} -12.5638 q^{71} -9.65654 q^{73} -6.33116 q^{74} +4.34108 q^{76} +4.06573 q^{77} +9.44894 q^{79} -3.13855 q^{80} +8.45200 q^{82} -2.53185 q^{83} -21.1656 q^{85} -9.78035 q^{86} +1.65548 q^{88} +7.55253 q^{89} -14.2804 q^{91} -1.27598 q^{92} +7.30857 q^{94} -13.6247 q^{95} +7.93248 q^{97} +0.968479 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) $ 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8	$ 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 14 q^{16} - 9 q^{17} + 14 q^{19} - 2 q^{20} + 2 q^{22} - 30 q^{23} + 18 q^{25} - 4 q^{26} + 4 q^{28} - 6 q^{29} + 11 q^{31} - 14 q^{32} + 9 q^{34} + 18 q^{35} + 13 q^{37} - 14 q^{38} + 2 q^{40} + 2 q^{41} + 12 q^{43} - 2 q^{44} + 30 q^{46} - 21 q^{47} + 32 q^{49} - 18 q^{50} + 4 q^{52} - 22 q^{53} - 7 q^{55} - 4 q^{56} + 6 q^{58} - 14 q^{59} + 31 q^{61} - 11 q^{62} + 14 q^{64} + 24 q^{67} - 9 q^{68} - 18 q^{70} - 28 q^{71} + 24 q^{73} - 13 q^{74} + 14 q^{76} - 16 q^{77} + 65 q^{79} - 2 q^{80} - 2 q^{82} + 15 q^{83} - 19 q^{85} - 12 q^{86} + 2 q^{88} + 11 q^{89} + 68 q^{91} - 30 q^{92} + 21 q^{94} - 8 q^{95} + 23 q^{97} - 32 q^{98}+O(q^{100}) $ 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 - 4 * q^14 + 14 * q^16 - 9 * q^17 + 14 * q^19 - 2 * q^20 + 2 * q^22 - 30 * q^23 + 18 * q^25 - 4 * q^26 + 4 * q^28 - 6 * q^29 + 11 * q^31 - 14 * q^32 + 9 * q^34 + 18 * q^35 + 13 * q^37 - 14 * q^38 + 2 * q^40 + 2 * q^41 + 12 * q^43 - 2 * q^44 + 30 * q^46 - 21 * q^47 + 32 * q^49 - 18 * q^50 + 4 * q^52 - 22 * q^53 - 7 * q^55 - 4 * q^56 + 6 * q^58 - 14 * q^59 + 31 * q^61 - 11 * q^62 + 14 * q^64 + 24 * q^67 - 9 * q^68 - 18 * q^70 - 28 * q^71 + 24 * q^73 - 13 * q^74 + 14 * q^76 - 16 * q^77 + 65 * q^79 - 2 * q^80 - 2 * q^82 + 15 * q^83 - 19 * q^85 - 12 * q^86 + 2 * q^88 + 11 * q^89 + 68 * q^91 - 30 * q^92 + 21 * q^94 - 8 * q^95 + 23 * q^97 - 32 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.13855	−1.40360	−0.701800	−	0.712374i	$-0.747620\pi$
$5$	−3.13855	−1.40360	−0.701800	+	0.712374i	$0.747620\pi$
$6$	0	0
$7$	−2.45592	−0.928249	−0.464124	−	0.885770i	$-0.653631\pi$
$7$	−2.45592	−0.928249	−0.464124	+	0.885770i	$0.653631\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.13855	0.992495
$11$	−1.65548	−0.499147	−0.249574	−	0.968356i	$-0.580290\pi$
$11$	−1.65548	−0.499147	−0.249574	+	0.968356i	$0.580290\pi$
$12$	0	0
$13$	5.81470	1.61271	0.806353	−	0.591434i	$-0.201438\pi$
$13$	5.81470	1.61271	0.806353	+	0.591434i	$0.201438\pi$
$14$	2.45592	0.656371
$15$	0	0
$16$	1.00000	0.250000
$17$	6.74376	1.63560	0.817801	−	0.575501i	$-0.195193\pi$
$17$	6.74376	1.63560	0.817801	+	0.575501i	$0.195193\pi$
$18$	0	0
$19$	4.34108	0.995912	0.497956	−	0.867202i	$-0.334084\pi$
$19$	4.34108	0.995912	0.497956	+	0.867202i	$0.334084\pi$
$20$	−3.13855	−0.701800
$21$	0	0
$22$	1.65548	0.352950
$23$	−1.27598	−0.266059	−0.133030	−	0.991112i	$-0.542471\pi$
$23$	−1.27598	−0.266059	−0.133030	+	0.991112i	$0.542471\pi$
$24$	0	0
$25$	4.85047	0.970093
$26$	−5.81470	−1.14036
$27$	0	0
$28$	−2.45592	−0.464124
$29$	10.5818	1.96500	0.982500	−	0.186263i	$-0.0596376\pi$
$29$	10.5818	1.96500	0.982500	+	0.186263i	$0.0596376\pi$
$30$	0	0
$31$	−9.90340	−1.77870	−0.889351	−	0.457224i	$-0.848844\pi$
$31$	−9.90340	−1.77870	−0.889351	+	0.457224i	$0.848844\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.74376	−1.15654
$35$	7.70800	1.30289
$36$	0	0
$37$	6.33116	1.04084	0.520418	−	0.853912i	$-0.325776\pi$
$37$	6.33116	1.04084	0.520418	+	0.853912i	$0.325776\pi$
$38$	−4.34108	−0.704216
$39$	0	0
$40$	3.13855	0.496248
$41$	−8.45200	−1.31998	−0.659990	−	0.751274i	$-0.729440\pi$
$41$	−8.45200	−1.31998	−0.659990	+	0.751274i	$0.729440\pi$
$42$	0	0
$43$	9.78035	1.49149	0.745744	−	0.666232i	$-0.232094\pi$
$43$	9.78035	1.49149	0.745744	+	0.666232i	$0.232094\pi$
$44$	−1.65548	−0.249574
$45$	0	0
$46$	1.27598	0.188132
$47$	−7.30857	−1.06606	−0.533032	−	0.846095i	$-0.678947\pi$
$47$	−7.30857	−1.06606	−0.533032	+	0.846095i	$0.678947\pi$
$48$	0	0
$49$	−0.968479	−0.138354
$50$	−4.85047	−0.685960
$51$	0	0
$52$	5.81470	0.806353
$53$	−10.0205	−1.37642	−0.688212	−	0.725510i	$-0.741604\pi$
$53$	−10.0205	−1.37642	−0.688212	+	0.725510i	$0.741604\pi$
$54$	0	0
$55$	5.19581	0.700603
$56$	2.45592	0.328186
$57$	0	0
$58$	−10.5818	−1.38946
$59$	−4.28382	−0.557705	−0.278853	−	0.960334i	$-0.589954\pi$
$59$	−4.28382	−0.557705	−0.278853	+	0.960334i	$0.589954\pi$
$60$	0	0
$61$	9.86429	1.26299	0.631496	−	0.775379i	$-0.282441\pi$
$61$	9.86429	1.26299	0.631496	+	0.775379i	$0.282441\pi$
$62$	9.90340	1.25773
$63$	0	0
$64$	1.00000	0.125000
$65$	−18.2497	−2.26360
$66$	0	0
$67$	5.48095	0.669605	0.334802	−	0.942288i	$-0.391330\pi$
$67$	5.48095	0.669605	0.334802	+	0.942288i	$0.391330\pi$
$68$	6.74376	0.817801
$69$	0	0
$70$	−7.70800	−0.921283
$71$	−12.5638	−1.49105	−0.745525	−	0.666477i	$-0.767801\pi$
$71$	−12.5638	−1.49105	−0.745525	+	0.666477i	$0.767801\pi$
$72$	0	0
$73$	−9.65654	−1.13021	−0.565106	−	0.825018i	$-0.691165\pi$
$73$	−9.65654	−1.13021	−0.565106	+	0.825018i	$0.691165\pi$
$74$	−6.33116	−0.735982
$75$	0	0
$76$	4.34108	0.497956
$77$	4.06573	0.463333
$78$	0	0
$79$	9.44894	1.06309	0.531545	−	0.847030i	$-0.321612\pi$
$79$	9.44894	1.06309	0.531545	+	0.847030i	$0.321612\pi$
$80$	−3.13855	−0.350900
$81$	0	0
$82$	8.45200	0.933367
$83$	−2.53185	−0.277906	−0.138953	−	0.990299i	$-0.544374\pi$
$83$	−2.53185	−0.277906	−0.138953	+	0.990299i	$0.544374\pi$
$84$	0	0
$85$	−21.1656	−2.29573
$86$	−9.78035	−1.05464
$87$	0	0
$88$	1.65548	0.176475
$89$	7.55253	0.800567	0.400283	−	0.916391i	$-0.368912\pi$
$89$	7.55253	0.800567	0.400283	+	0.916391i	$0.368912\pi$
$90$	0	0
$91$	−14.2804	−1.49699
$92$	−1.27598	−0.133030
$93$	0	0
$94$	7.30857	0.753821
$95$	−13.6247	−1.39786
$96$	0	0
$97$	7.93248	0.805421	0.402711	−	0.915327i	$-0.368068\pi$
$97$	7.93248	0.805421	0.402711	+	0.915327i	$0.368068\pi$
$98$	0.968479	0.0978311
$99$	0	0
$100$	4.85047	0.485047
$101$	7.84592	0.780698	0.390349	−	0.920667i	$-0.372354\pi$
$101$	7.84592	0.780698	0.390349	+	0.920667i	$0.372354\pi$
$102$	0	0
$103$	−16.0036	−1.57688	−0.788439	−	0.615113i	$-0.789110\pi$
$103$	−16.0036	−1.57688	−0.788439	+	0.615113i	$0.789110\pi$
$104$	−5.81470	−0.570178
$105$	0	0
$106$	10.0205	0.973278
$107$	−14.9000	−1.44044	−0.720218	−	0.693748i	$-0.755958\pi$
$107$	−14.9000	−1.44044	−0.720218	+	0.693748i	$0.755958\pi$
$108$	0	0
$109$	13.7347	1.31555	0.657774	−	0.753215i	$-0.271498\pi$
$109$	13.7347	1.31555	0.657774	+	0.753215i	$0.271498\pi$
$110$	−5.19581	−0.495401
$111$	0	0
$112$	−2.45592	−0.232062
$113$	10.2332	0.962656	0.481328	−	0.876541i	$-0.340155\pi$
$113$	10.2332	0.962656	0.481328	+	0.876541i	$0.340155\pi$
$114$	0	0
$115$	4.00471	0.373441
$116$	10.5818	0.982500
$117$	0	0
$118$	4.28382	0.394357
$119$	−16.5621	−1.51825
$120$	0	0
$121$	−8.25937	−0.750852
$122$	−9.86429	−0.893071
$123$	0	0
$124$	−9.90340	−0.889351
$125$	0.469315	0.0419768
$126$	0	0
$127$	−8.81207	−0.781944	−0.390972	−	0.920402i	$-0.627861\pi$
$127$	−8.81207	−0.781944	−0.390972	+	0.920402i	$0.627861\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	18.2497	1.60060
$131$	−4.20814	−0.367667	−0.183833	−	0.982957i	$-0.558851\pi$
$131$	−4.20814	−0.367667	−0.183833	+	0.982957i	$0.558851\pi$
$132$	0	0
$133$	−10.6613	−0.924454
$134$	−5.48095	−0.473482
$135$	0	0
$136$	−6.74376	−0.578272
$137$	13.3720	1.14245	0.571224	−	0.820795i	$-0.306469\pi$
$137$	13.3720	1.14245	0.571224	+	0.820795i	$0.306469\pi$
$138$	0	0
$139$	−11.8571	−1.00570	−0.502852	−	0.864373i	$-0.667716\pi$
$139$	−11.8571	−1.00570	−0.502852	+	0.864373i	$0.667716\pi$
$140$	7.70800	0.651445
$141$	0	0
$142$	12.5638	1.05433
$143$	−9.62614	−0.804978
$144$	0	0
$145$	−33.2116	−2.75807
$146$	9.65654	0.799181
$147$	0	0
$148$	6.33116	0.520418
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−4.94793	−0.402657	−0.201328	−	0.979524i	$-0.564526\pi$
$151$	−4.94793	−0.402657	−0.201328	+	0.979524i	$0.564526\pi$
$152$	−4.34108	−0.352108
$153$	0	0
$154$	−4.06573	−0.327626
$155$	31.0823	2.49659
$156$	0	0
$157$	−10.1868	−0.812994	−0.406497	−	0.913652i	$-0.633250\pi$
$157$	−10.1868	−0.812994	−0.406497	+	0.913652i	$0.633250\pi$
$158$	−9.44894	−0.751718
$159$	0	0
$160$	3.13855	0.248124
$161$	3.13369	0.246969
$162$	0	0
$163$	23.0384	1.80451	0.902253	−	0.431207i	$-0.141912\pi$
$163$	23.0384	1.80451	0.902253	+	0.431207i	$0.141912\pi$
$164$	−8.45200	−0.659990
$165$	0	0
$166$	2.53185	0.196510
$167$	9.27469	0.717697	0.358848	−	0.933396i	$-0.383169\pi$
$167$	9.27469	0.717697	0.358848	+	0.933396i	$0.383169\pi$
$168$	0	0
$169$	20.8107	1.60082
$170$	21.1656	1.62333
$171$	0	0
$172$	9.78035	0.745744
$173$	−10.3201	−0.784619	−0.392310	−	0.919833i	$-0.628324\pi$
$173$	−10.3201	−0.784619	−0.392310	+	0.919833i	$0.628324\pi$
$174$	0	0
$175$	−11.9123	−0.900488
$176$	−1.65548	−0.124787
$177$	0	0
$178$	−7.55253	−0.566086
$179$	2.62448	0.196163	0.0980813	−	0.995178i	$-0.468729\pi$
$179$	2.62448	0.196163	0.0980813	+	0.995178i	$0.468729\pi$
$180$	0	0
$181$	4.44211	0.330179	0.165090	−	0.986279i	$-0.447209\pi$
$181$	4.44211	0.330179	0.165090	+	0.986279i	$0.447209\pi$
$182$	14.2804	1.05853
$183$	0	0
$184$	1.27598	0.0940662
$185$	−19.8706	−1.46092
$186$	0	0
$187$	−11.1642	−0.816406
$188$	−7.30857	−0.533032
$189$	0	0
$190$	13.6247	0.988438
$191$	−8.43413	−0.610272	−0.305136	−	0.952309i	$-0.598702\pi$
$191$	−8.43413	−0.610272	−0.305136	+	0.952309i	$0.598702\pi$
$192$	0	0
$193$	−16.2065	−1.16657	−0.583284	−	0.812268i	$-0.698232\pi$
$193$	−16.2065	−1.16657	−0.583284	+	0.812268i	$0.698232\pi$
$194$	−7.93248	−0.569519
$195$	0	0
$196$	−0.968479	−0.0691770
$197$	21.0736	1.50143	0.750715	−	0.660626i	$-0.229709\pi$
$197$	21.0736	1.50143	0.750715	+	0.660626i	$0.229709\pi$
$198$	0	0
$199$	−2.72982	−0.193512	−0.0967558	−	0.995308i	$-0.530847\pi$
$199$	−2.72982	−0.193512	−0.0967558	+	0.995308i	$0.530847\pi$
$200$	−4.85047	−0.342980
$201$	0	0
$202$	−7.84592	−0.552037
$203$	−25.9881	−1.82401
$204$	0	0
$205$	26.5270	1.85272
$206$	16.0036	1.11502
$207$	0	0
$208$	5.81470	0.403177
$209$	−7.18659	−0.497107
$210$	0	0
$211$	24.5744	1.69177	0.845887	−	0.533362i	$-0.179072\pi$
$211$	24.5744	1.69177	0.845887	+	0.533362i	$0.179072\pi$
$212$	−10.0205	−0.688212
$213$	0	0
$214$	14.9000	1.01854
$215$	−30.6961	−2.09345
$216$	0	0
$217$	24.3219	1.65108
$218$	−13.7347	−0.930233
$219$	0	0
$220$	5.19581	0.350301
$221$	39.2129	2.63775
$222$	0	0
$223$	12.5158	0.838119	0.419059	−	0.907959i	$-0.362360\pi$
$223$	12.5158	0.838119	0.419059	+	0.907959i	$0.362360\pi$
$224$	2.45592	0.164093
$225$	0	0
$226$	−10.2332	−0.680701
$227$	12.7001	0.842933	0.421467	−	0.906844i	$-0.361515\pi$
$227$	12.7001	0.842933	0.421467	+	0.906844i	$0.361515\pi$
$228$	0	0
$229$	0.970181	0.0641114	0.0320557	−	0.999486i	$-0.489795\pi$
$229$	0.970181	0.0641114	0.0320557	+	0.999486i	$0.489795\pi$
$230$	−4.00471	−0.264063
$231$	0	0
$232$	−10.5818	−0.694732
$233$	10.5639	0.692063	0.346031	−	0.938223i	$-0.387529\pi$
$233$	10.5639	0.692063	0.346031	+	0.938223i	$0.387529\pi$
$234$	0	0
$235$	22.9383	1.49633
$236$	−4.28382	−0.278853
$237$	0	0
$238$	16.5621	1.07356
$239$	15.2653	0.987433	0.493717	−	0.869623i	$-0.335638\pi$
$239$	15.2653	0.987433	0.493717	+	0.869623i	$0.335638\pi$
$240$	0	0
$241$	18.1029	1.16611	0.583056	−	0.812432i	$-0.301857\pi$
$241$	18.1029	1.16611	0.583056	+	0.812432i	$0.301857\pi$
$242$	8.25937	0.530933
$243$	0	0
$244$	9.86429	0.631496
$245$	3.03961	0.194194
$246$	0	0
$247$	25.2421	1.60611
$248$	9.90340	0.628866
$249$	0	0
$250$	−0.469315	−0.0296821
$251$	−15.2081	−0.959929	−0.479964	−	0.877288i	$-0.659350\pi$
$251$	−15.2081	−0.959929	−0.479964	+	0.877288i	$0.659350\pi$
$252$	0	0
$253$	2.11236	0.132803
$254$	8.81207	0.552918
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.7569	1.04527	0.522635	−	0.852557i	$-0.324949\pi$
$257$	16.7569	1.04527	0.522635	+	0.852557i	$0.324949\pi$
$258$	0	0
$259$	−15.5488	−0.966155
$260$	−18.2497	−1.13180
$261$	0	0
$262$	4.20814	0.259980
$263$	−19.6426	−1.21121	−0.605607	−	0.795764i	$-0.707070\pi$
$263$	−19.6426	−1.21121	−0.605607	+	0.795764i	$0.707070\pi$
$264$	0	0
$265$	31.4498	1.93195
$266$	10.6613	0.653688
$267$	0	0
$268$	5.48095	0.334802
$269$	7.29567	0.444825	0.222412	−	0.974953i	$-0.428607\pi$
$269$	7.29567	0.444825	0.222412	+	0.974953i	$0.428607\pi$
$270$	0	0
$271$	−21.6197	−1.31330	−0.656652	−	0.754193i	$-0.728028\pi$
$271$	−21.6197	−1.31330	−0.656652	+	0.754193i	$0.728028\pi$
$272$	6.74376	0.408900
$273$	0	0
$274$	−13.3720	−0.807832
$275$	−8.02987	−0.484219
$276$	0	0
$277$	17.2265	1.03504	0.517519	−	0.855672i	$-0.326856\pi$
$277$	17.2265	1.03504	0.517519	+	0.855672i	$0.326856\pi$
$278$	11.8571	0.711140
$279$	0	0
$280$	−7.70800	−0.460641
$281$	−20.6398	−1.23127	−0.615635	−	0.788032i	$-0.711100\pi$
$281$	−20.6398	−1.23127	−0.615635	+	0.788032i	$0.711100\pi$
$282$	0	0
$283$	9.64089	0.573091	0.286546	−	0.958067i	$-0.407493\pi$
$283$	9.64089	0.573091	0.286546	+	0.958067i	$0.407493\pi$
$284$	−12.5638	−0.745525
$285$	0	0
$286$	9.62614	0.569205
$287$	20.7574	1.22527
$288$	0	0
$289$	28.4783	1.67519
$290$	33.2116	1.95025
$291$	0	0
$292$	−9.65654	−0.565106
$293$	−10.0379	−0.586418	−0.293209	−	0.956048i	$-0.594723\pi$
$293$	−10.0379	−0.586418	−0.293209	+	0.956048i	$0.594723\pi$
$294$	0	0
$295$	13.4449	0.782795
$296$	−6.33116	−0.367991
$297$	0	0
$298$	1.00000	0.0579284
$299$	−7.41941	−0.429076
$300$	0	0
$301$	−24.0197	−1.38447
$302$	4.94793	0.284721
$303$	0	0
$304$	4.34108	0.248978
$305$	−30.9595	−1.77274
$306$	0	0
$307$	4.47393	0.255341	0.127670	−	0.991817i	$-0.459250\pi$
$307$	4.47393	0.255341	0.127670	+	0.991817i	$0.459250\pi$
$308$	4.06573	0.231666
$309$	0	0
$310$	−31.0823	−1.76535
$311$	12.7688	0.724053	0.362027	−	0.932168i	$-0.382085\pi$
$311$	12.7688	0.724053	0.362027	+	0.932168i	$0.382085\pi$
$312$	0	0
$313$	20.6832	1.16908	0.584542	−	0.811364i	$-0.301274\pi$
$313$	20.6832	1.16908	0.584542	+	0.811364i	$0.301274\pi$
$314$	10.1868	0.574874
$315$	0	0
$316$	9.44894	0.531545
$317$	22.7877	1.27988	0.639941	−	0.768424i	$-0.278959\pi$
$317$	22.7877	1.27988	0.639941	+	0.768424i	$0.278959\pi$
$318$	0	0
$319$	−17.5181	−0.980824
$320$	−3.13855	−0.175450
$321$	0	0
$322$	−3.13369	−0.174634
$323$	29.2752	1.62892
$324$	0	0
$325$	28.2040	1.56448
$326$	−23.0384	−1.27598
$327$	0	0
$328$	8.45200	0.466683
$329$	17.9492	0.989573
$330$	0	0
$331$	−4.08945	−0.224777	−0.112388	−	0.993664i	$-0.535850\pi$
$331$	−4.08945	−0.224777	−0.112388	+	0.993664i	$0.535850\pi$
$332$	−2.53185	−0.138953
$333$	0	0
$334$	−9.27469	−0.507488
$335$	−17.2022	−0.939857
$336$	0	0
$337$	−4.62194	−0.251773	−0.125886	−	0.992045i	$-0.540178\pi$
$337$	−4.62194	−0.251773	−0.125886	+	0.992045i	$0.540178\pi$
$338$	−20.8107	−1.13195
$339$	0	0
$340$	−21.1656	−1.14787
$341$	16.3949	0.887834
$342$	0	0
$343$	19.5699	1.05668
$344$	−9.78035	−0.527321
$345$	0	0
$346$	10.3201	0.554810
$347$	13.2258	0.709998	0.354999	−	0.934867i	$-0.384481\pi$
$347$	13.2258	0.709998	0.354999	+	0.934867i	$0.384481\pi$
$348$	0	0
$349$	−22.2314	−1.19002	−0.595009	−	0.803719i	$-0.702851\pi$
$349$	−22.2314	−1.19002	−0.595009	+	0.803719i	$0.702851\pi$
$350$	11.9123	0.636741
$351$	0	0
$352$	1.65548	0.0882376
$353$	−7.40993	−0.394391	−0.197195	−	0.980364i	$-0.563183\pi$
$353$	−7.40993	−0.394391	−0.197195	+	0.980364i	$0.563183\pi$
$354$	0	0
$355$	39.4321	2.09284
$356$	7.55253	0.400283
$357$	0	0
$358$	−2.62448	−0.138708
$359$	10.1840	0.537489	0.268745	−	0.963211i	$-0.413391\pi$
$359$	10.1840	0.537489	0.268745	+	0.963211i	$0.413391\pi$
$360$	0	0
$361$	−0.155029	−0.00815943
$362$	−4.44211	−0.233472
$363$	0	0
$364$	−14.2804	−0.748497
$365$	30.3075	1.58637
$366$	0	0
$367$	4.57927	0.239036	0.119518	−	0.992832i	$-0.461865\pi$
$367$	4.57927	0.239036	0.119518	+	0.992832i	$0.461865\pi$
$368$	−1.27598	−0.0665148
$369$	0	0
$370$	19.8706	1.03302
$371$	24.6095	1.27766
$372$	0	0
$373$	26.4915	1.37168	0.685838	−	0.727754i	$-0.259436\pi$
$373$	26.4915	1.37168	0.685838	+	0.727754i	$0.259436\pi$
$374$	11.1642	0.577286
$375$	0	0
$376$	7.30857	0.376911
$377$	61.5302	3.16897
$378$	0	0
$379$	−2.14256	−0.110056	−0.0550279	−	0.998485i	$-0.517525\pi$
$379$	−2.14256	−0.110056	−0.0550279	+	0.998485i	$0.517525\pi$
$380$	−13.6247	−0.698931
$381$	0	0
$382$	8.43413	0.431528
$383$	−13.7583	−0.703015	−0.351508	−	0.936185i	$-0.614331\pi$
$383$	−13.7583	−0.703015	−0.351508	+	0.936185i	$0.614331\pi$
$384$	0	0
$385$	−12.7605	−0.650334
$386$	16.2065	0.824888
$387$	0	0
$388$	7.93248	0.402711
$389$	−13.9741	−0.708516	−0.354258	−	0.935148i	$-0.615267\pi$
$389$	−13.9741	−0.708516	−0.354258	+	0.935148i	$0.615267\pi$
$390$	0	0
$391$	−8.60487	−0.435167
$392$	0.968479	0.0489156
$393$	0	0
$394$	−21.0736	−1.06167
$395$	−29.6559	−1.49215
$396$	0	0
$397$	11.7975	0.592097	0.296049	−	0.955173i	$-0.404331\pi$
$397$	11.7975	0.592097	0.296049	+	0.955173i	$0.404331\pi$
$398$	2.72982	0.136833
$399$	0	0
$400$	4.85047	0.242523
$401$	7.48029	0.373548	0.186774	−	0.982403i	$-0.440197\pi$
$401$	7.48029	0.373548	0.186774	+	0.982403i	$0.440197\pi$
$402$	0	0
$403$	−57.5853	−2.86853
$404$	7.84592	0.390349
$405$	0	0
$406$	25.9881	1.28977
$407$	−10.4811	−0.519530
$408$	0	0
$409$	−23.3197	−1.15309	−0.576543	−	0.817067i	$-0.695599\pi$
$409$	−23.3197	−1.15309	−0.576543	+	0.817067i	$0.695599\pi$
$410$	−26.5270	−1.31007
$411$	0	0
$412$	−16.0036	−0.788439
$413$	10.5207	0.517689
$414$	0	0
$415$	7.94632	0.390070
$416$	−5.81470	−0.285089
$417$	0	0
$418$	7.18659	0.351507
$419$	39.9392	1.95116	0.975579	−	0.219650i	$-0.0704915\pi$
$419$	39.9392	1.95116	0.975579	+	0.219650i	$0.0704915\pi$
$420$	0	0
$421$	−18.9978	−0.925894	−0.462947	−	0.886386i	$-0.653208\pi$
$421$	−18.9978	−0.925894	−0.462947	+	0.886386i	$0.653208\pi$
$422$	−24.5744	−1.19626
$423$	0	0
$424$	10.0205	0.486639
$425$	32.7104	1.58669
$426$	0	0
$427$	−24.2259	−1.17237
$428$	−14.9000	−0.720218
$429$	0	0
$430$	30.6961	1.48030
$431$	−23.2157	−1.11826	−0.559132	−	0.829079i	$-0.688865\pi$
$431$	−23.2157	−1.11826	−0.559132	+	0.829079i	$0.688865\pi$
$432$	0	0
$433$	−26.9880	−1.29696	−0.648481	−	0.761231i	$-0.724595\pi$
$433$	−26.9880	−1.29696	−0.648481	+	0.761231i	$0.724595\pi$
$434$	−24.3219	−1.16749
$435$	0	0
$436$	13.7347	0.657774
$437$	−5.53911	−0.264972
$438$	0	0
$439$	0.671897	0.0320679	0.0160339	−	0.999871i	$-0.494896\pi$
$439$	0.671897	0.0320679	0.0160339	+	0.999871i	$0.494896\pi$
$440$	−5.19581	−0.247701
$441$	0	0
$442$	−39.2129	−1.86517
$443$	−17.4695	−0.830003	−0.415001	−	0.909821i	$-0.636219\pi$
$443$	−17.4695	−0.830003	−0.415001	+	0.909821i	$0.636219\pi$
$444$	0	0
$445$	−23.7040	−1.12368
$446$	−12.5158	−0.592640
$447$	0	0
$448$	−2.45592	−0.116031
$449$	4.17564	0.197061	0.0985303	−	0.995134i	$-0.468586\pi$
$449$	4.17564	0.197061	0.0985303	+	0.995134i	$0.468586\pi$
$450$	0	0
$451$	13.9921	0.658864
$452$	10.2332	0.481328
$453$	0	0
$454$	−12.7001	−0.596044
$455$	44.8197	2.10118
$456$	0	0
$457$	16.5057	0.772105	0.386052	−	0.922477i	$-0.373838\pi$
$457$	16.5057	0.772105	0.386052	+	0.922477i	$0.373838\pi$
$458$	−0.970181	−0.0453336
$459$	0	0
$460$	4.00471	0.186720
$461$	−12.3991	−0.577485	−0.288743	−	0.957407i	$-0.593237\pi$
$461$	−12.3991	−0.577485	−0.288743	+	0.957407i	$0.593237\pi$
$462$	0	0
$463$	−12.8673	−0.597992	−0.298996	−	0.954254i	$-0.596652\pi$
$463$	−12.8673	−0.597992	−0.298996	+	0.954254i	$0.596652\pi$
$464$	10.5818	0.491250
$465$	0	0
$466$	−10.5639	−0.489362
$467$	31.2259	1.44496	0.722480	−	0.691391i	$-0.243002\pi$
$467$	31.2259	1.44496	0.722480	+	0.691391i	$0.243002\pi$
$468$	0	0
$469$	−13.4608	−0.621560
$470$	−22.9383	−1.05806
$471$	0	0
$472$	4.28382	0.197179
$473$	−16.1912	−0.744472
$474$	0	0
$475$	21.0563	0.966128
$476$	−16.5621	−0.759123
$477$	0	0
$478$	−15.2653	−0.698221
$479$	10.0310	0.458330	0.229165	−	0.973388i	$-0.426400\pi$
$479$	10.0310	0.458330	0.229165	+	0.973388i	$0.426400\pi$
$480$	0	0
$481$	36.8138	1.67856
$482$	−18.1029	−0.824565
$483$	0	0
$484$	−8.25937	−0.375426
$485$	−24.8964	−1.13049
$486$	0	0
$487$	5.45193	0.247051	0.123525	−	0.992341i	$-0.460580\pi$
$487$	5.45193	0.247051	0.123525	+	0.992341i	$0.460580\pi$
$488$	−9.86429	−0.446535
$489$	0	0
$490$	−3.03961	−0.137316
$491$	17.2620	0.779022	0.389511	−	0.921022i	$-0.372644\pi$
$491$	17.2620	0.779022	0.389511	+	0.921022i	$0.372644\pi$
$492$	0	0
$493$	71.3614	3.21396
$494$	−25.2421	−1.13569
$495$	0	0
$496$	−9.90340	−0.444676
$497$	30.8557	1.38407
$498$	0	0
$499$	14.0141	0.627359	0.313679	−	0.949529i	$-0.398438\pi$
$499$	14.0141	0.627359	0.313679	+	0.949529i	$0.398438\pi$
$500$	0.469315	0.0209884
$501$	0	0
$502$	15.2081	0.678772
$503$	−35.1586	−1.56764	−0.783822	−	0.620985i	$-0.786733\pi$
$503$	−35.1586	−1.56764	−0.783822	+	0.620985i	$0.786733\pi$
$504$	0	0
$505$	−24.6248	−1.09579
$506$	−2.11236	−0.0939057
$507$	0	0
$508$	−8.81207	−0.390972
$509$	−3.74539	−0.166011	−0.0830057	−	0.996549i	$-0.526452\pi$
$509$	−3.74539	−0.166011	−0.0830057	+	0.996549i	$0.526452\pi$
$510$	0	0
$511$	23.7157	1.04912
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−16.7569	−0.739117
$515$	50.2279	2.21331
$516$	0	0
$517$	12.0992	0.532123
$518$	15.5488	0.683175
$519$	0	0
$520$	18.2497	0.800302
$521$	−22.9051	−1.00349	−0.501746	−	0.865015i	$-0.667309\pi$
$521$	−22.9051	−1.00349	−0.501746	+	0.865015i	$0.667309\pi$
$522$	0	0
$523$	29.2900	1.28076	0.640381	−	0.768057i	$-0.278776\pi$
$523$	29.2900	1.28076	0.640381	+	0.768057i	$0.278776\pi$
$524$	−4.20814	−0.183833
$525$	0	0
$526$	19.6426	0.856457
$527$	−66.7861	−2.90925
$528$	0	0
$529$	−21.3719	−0.929212
$530$	−31.4498	−1.36609
$531$	0	0
$532$	−10.6613	−0.462227
$533$	−49.1458	−2.12874
$534$	0	0
$535$	46.7643	2.02180
$536$	−5.48095	−0.236741
$537$	0	0
$538$	−7.29567	−0.314539
$539$	1.60330	0.0690590
$540$	0	0
$541$	−11.4261	−0.491247	−0.245623	−	0.969365i	$-0.578993\pi$
$541$	−11.4261	−0.491247	−0.245623	+	0.969365i	$0.578993\pi$
$542$	21.6197	0.928647
$543$	0	0
$544$	−6.74376	−0.289136
$545$	−43.1071	−1.84650
$546$	0	0
$547$	−27.6559	−1.18248	−0.591240	−	0.806495i	$-0.701362\pi$
$547$	−27.6559	−1.18248	−0.591240	+	0.806495i	$0.701362\pi$
$548$	13.3720	0.571224
$549$	0	0
$550$	8.02987	0.342395
$551$	45.9366	1.95697
$552$	0	0
$553$	−23.2058	−0.986811
$554$	−17.2265	−0.731882
$555$	0	0
$556$	−11.8571	−0.502852
$557$	−11.3963	−0.482877	−0.241438	−	0.970416i	$-0.577619\pi$
$557$	−11.3963	−0.482877	−0.241438	+	0.970416i	$0.577619\pi$
$558$	0	0
$559$	56.8697	2.40533
$560$	7.70800	0.325723
$561$	0	0
$562$	20.6398	0.870639
$563$	3.28777	0.138563	0.0692816	−	0.997597i	$-0.477929\pi$
$563$	3.28777	0.138563	0.0692816	+	0.997597i	$0.477929\pi$
$564$	0	0
$565$	−32.1173	−1.35118
$566$	−9.64089	−0.405237
$567$	0	0
$568$	12.5638	0.527166
$569$	13.0584	0.547438	0.273719	−	0.961810i	$-0.411746\pi$
$569$	13.0584	0.547438	0.273719	+	0.961810i	$0.411746\pi$
$570$	0	0
$571$	32.5893	1.36382	0.681909	−	0.731437i	$-0.261150\pi$
$571$	32.5893	1.36382	0.681909	+	0.731437i	$0.261150\pi$
$572$	−9.62614	−0.402489
$573$	0	0
$574$	−20.7574	−0.866397
$575$	−6.18908	−0.258102
$576$	0	0
$577$	−20.4382	−0.850854	−0.425427	−	0.904993i	$-0.639876\pi$
$577$	−20.4382	−0.850854	−0.425427	+	0.904993i	$0.639876\pi$
$578$	−28.4783	−1.18454
$579$	0	0
$580$	−33.2116	−1.37904
$581$	6.21801	0.257966
$582$	0	0
$583$	16.5888	0.687038
$584$	9.65654	0.399591
$585$	0	0
$586$	10.0379	0.414660
$587$	−9.44905	−0.390004	−0.195002	−	0.980803i	$-0.562471\pi$
$587$	−9.44905	−0.390004	−0.195002	+	0.980803i	$0.562471\pi$
$588$	0	0
$589$	−42.9914	−1.77143
$590$	−13.4449	−0.553520
$591$	0	0
$592$	6.33116	0.260209
$593$	21.8183	0.895971	0.447986	−	0.894041i	$-0.352142\pi$
$593$	21.8183	0.895971	0.447986	+	0.894041i	$0.352142\pi$
$594$	0	0
$595$	51.9809	2.13101
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	7.41941	0.303402
$599$	39.3662	1.60846	0.804229	−	0.594319i	$-0.202578\pi$
$599$	39.3662	1.60846	0.804229	+	0.594319i	$0.202578\pi$
$600$	0	0
$601$	−42.7828	−1.74515	−0.872573	−	0.488484i	$-0.837550\pi$
$601$	−42.7828	−1.74515	−0.872573	+	0.488484i	$0.837550\pi$
$602$	24.0197	0.978970
$603$	0	0
$604$	−4.94793	−0.201328
$605$	25.9224	1.05390
$606$	0	0
$607$	−4.96534	−0.201537	−0.100768	−	0.994910i	$-0.532130\pi$
$607$	−4.96534	−0.201537	−0.100768	+	0.994910i	$0.532130\pi$
$608$	−4.34108	−0.176054
$609$	0	0
$610$	30.9595	1.25351
$611$	−42.4971	−1.71925
$612$	0	0
$613$	−15.0641	−0.608435	−0.304217	−	0.952603i	$-0.598395\pi$
$613$	−15.0641	−0.608435	−0.304217	+	0.952603i	$0.598395\pi$
$614$	−4.47393	−0.180553
$615$	0	0
$616$	−4.06573	−0.163813
$617$	−8.67314	−0.349167	−0.174584	−	0.984642i	$-0.555858\pi$
$617$	−8.67314	−0.349167	−0.174584	+	0.984642i	$0.555858\pi$
$618$	0	0
$619$	6.39296	0.256955	0.128477	−	0.991712i	$-0.458991\pi$
$619$	6.39296	0.256955	0.128477	+	0.991712i	$0.458991\pi$
$620$	31.0823	1.24829
$621$	0	0
$622$	−12.7688	−0.511983
$623$	−18.5484	−0.743125
$624$	0	0
$625$	−25.7253	−1.02901
$626$	−20.6832	−0.826667
$627$	0	0
$628$	−10.1868	−0.406497
$629$	42.6958	1.70239
$630$	0	0
$631$	36.4507	1.45108	0.725539	−	0.688181i	$-0.241591\pi$
$631$	36.4507	1.45108	0.725539	+	0.688181i	$0.241591\pi$
$632$	−9.44894	−0.375859
$633$	0	0
$634$	−22.7877	−0.905014
$635$	27.6571	1.09754
$636$	0	0
$637$	−5.63141	−0.223125
$638$	17.5181	0.693547
$639$	0	0
$640$	3.13855	0.124062
$641$	22.2460	0.878664	0.439332	−	0.898325i	$-0.355215\pi$
$641$	22.2460	0.878664	0.439332	+	0.898325i	$0.355215\pi$
$642$	0	0
$643$	6.95106	0.274123	0.137061	−	0.990563i	$-0.456234\pi$
$643$	6.95106	0.274123	0.137061	+	0.990563i	$0.456234\pi$
$644$	3.13369	0.123485
$645$	0	0
$646$	−29.2752	−1.15182
$647$	−31.7627	−1.24872	−0.624360	−	0.781137i	$-0.714640\pi$
$647$	−31.7627	−1.24872	−0.624360	+	0.781137i	$0.714640\pi$
$648$	0	0
$649$	7.09179	0.278377
$650$	−28.2040	−1.10625
$651$	0	0
$652$	23.0384	0.902253
$653$	−48.7864	−1.90916	−0.954579	−	0.297958i	$-0.903694\pi$
$653$	−48.7864	−1.90916	−0.954579	+	0.297958i	$0.903694\pi$
$654$	0	0
$655$	13.2074	0.516057
$656$	−8.45200	−0.329995
$657$	0	0
$658$	−17.9492	−0.699734
$659$	40.5497	1.57959	0.789796	−	0.613370i	$-0.210187\pi$
$659$	40.5497	1.57959	0.789796	+	0.613370i	$0.210187\pi$
$660$	0	0
$661$	24.1970	0.941155	0.470578	−	0.882359i	$-0.344046\pi$
$661$	24.1970	0.941155	0.470578	+	0.882359i	$0.344046\pi$
$662$	4.08945	0.158941
$663$	0	0
$664$	2.53185	0.0982548
$665$	33.4611	1.29756
$666$	0	0
$667$	−13.5022	−0.522806
$668$	9.27469	0.358848
$669$	0	0
$670$	17.2022	0.664579
$671$	−16.3302	−0.630419
$672$	0	0
$673$	−18.4158	−0.709877	−0.354938	−	0.934890i	$-0.615498\pi$
$673$	−18.4158	−0.709877	−0.354938	+	0.934890i	$0.615498\pi$
$674$	4.62194	0.178030
$675$	0	0
$676$	20.8107	0.800411
$677$	38.8936	1.49480	0.747402	−	0.664372i	$-0.231301\pi$
$677$	38.8936	1.49480	0.747402	+	0.664372i	$0.231301\pi$
$678$	0	0
$679$	−19.4815	−0.747631
$680$	21.1656	0.811663
$681$	0	0
$682$	−16.3949	−0.627794
$683$	30.1305	1.15291	0.576456	−	0.817128i	$-0.304435\pi$
$683$	30.1305	1.15291	0.576456	+	0.817128i	$0.304435\pi$
$684$	0	0
$685$	−41.9686	−1.60354
$686$	−19.5699	−0.747183
$687$	0	0
$688$	9.78035	0.372872
$689$	−58.2662	−2.21977
$690$	0	0
$691$	−5.60838	−0.213353	−0.106676	−	0.994294i	$-0.534021\pi$
$691$	−5.60838	−0.213353	−0.106676	+	0.994294i	$0.534021\pi$
$692$	−10.3201	−0.392310
$693$	0	0
$694$	−13.2258	−0.502044
$695$	37.2139	1.41161
$696$	0	0
$697$	−56.9982	−2.15896
$698$	22.2314	0.841470
$699$	0	0
$700$	−11.9123	−0.450244
$701$	52.2163	1.97218	0.986091	−	0.166206i	$-0.0531518\pi$
$701$	52.2163	1.97218	0.986091	+	0.166206i	$0.0531518\pi$
$702$	0	0
$703$	27.4841	1.03658
$704$	−1.65548	−0.0623934
$705$	0	0
$706$	7.40993	0.278876
$707$	−19.2689	−0.724682
$708$	0	0
$709$	−16.7197	−0.627923	−0.313961	−	0.949436i	$-0.601656\pi$
$709$	−16.7197	−0.627923	−0.313961	+	0.949436i	$0.601656\pi$
$710$	−39.4321	−1.47986
$711$	0	0
$712$	−7.55253	−0.283043
$713$	12.6365	0.473240
$714$	0	0
$715$	30.2121	1.12987
$716$	2.62448	0.0980813
$717$	0	0
$718$	−10.1840	−0.380062
$719$	23.0237	0.858639	0.429320	−	0.903153i	$-0.358753\pi$
$719$	23.0237	0.858639	0.429320	+	0.903153i	$0.358753\pi$
$720$	0	0
$721$	39.3034	1.46374
$722$	0.155029	0.00576959
$723$	0	0
$724$	4.44211	0.165090
$725$	51.3269	1.90623
$726$	0	0
$727$	41.3092	1.53207	0.766036	−	0.642798i	$-0.222226\pi$
$727$	41.3092	1.53207	0.766036	+	0.642798i	$0.222226\pi$
$728$	14.2804	0.529267
$729$	0	0
$730$	−30.3075	−1.12173
$731$	65.9563	2.43948
$732$	0	0
$733$	−40.2265	−1.48580	−0.742901	−	0.669402i	$-0.766551\pi$
$733$	−40.2265	−1.48580	−0.742901	+	0.669402i	$0.766551\pi$
$734$	−4.57927	−0.169024
$735$	0	0
$736$	1.27598	0.0470331
$737$	−9.07362	−0.334231
$738$	0	0
$739$	50.0827	1.84232	0.921161	−	0.389182i	$-0.127242\pi$
$739$	50.0827	1.84232	0.921161	+	0.389182i	$0.127242\pi$
$740$	−19.8706	−0.730459
$741$	0	0
$742$	−24.6095	−0.903444
$743$	−2.30356	−0.0845094	−0.0422547	−	0.999107i	$-0.513454\pi$
$743$	−2.30356	−0.0845094	−0.0422547	+	0.999107i	$0.513454\pi$
$744$	0	0
$745$	3.13855	0.114987
$746$	−26.4915	−0.969922
$747$	0	0
$748$	−11.1642	−0.408203
$749$	36.5931	1.33708
$750$	0	0
$751$	−39.8018	−1.45239	−0.726195	−	0.687489i	$-0.758713\pi$
$751$	−39.8018	−1.45239	−0.726195	+	0.687489i	$0.758713\pi$
$752$	−7.30857	−0.266516
$753$	0	0
$754$	−61.5302	−2.24080
$755$	15.5293	0.565169
$756$	0	0
$757$	−49.7551	−1.80838	−0.904190	−	0.427130i	$-0.859525\pi$
$757$	−49.7551	−1.80838	−0.904190	+	0.427130i	$0.859525\pi$
$758$	2.14256	0.0778213
$759$	0	0
$760$	13.6247	0.494219
$761$	−11.6978	−0.424044	−0.212022	−	0.977265i	$-0.568005\pi$
$761$	−11.6978	−0.424044	−0.212022	+	0.977265i	$0.568005\pi$
$762$	0	0
$763$	−33.7313	−1.22116
$764$	−8.43413	−0.305136
$765$	0	0
$766$	13.7583	0.497107
$767$	−24.9091	−0.899415
$768$	0	0
$769$	30.9085	1.11459	0.557294	−	0.830315i	$-0.311840\pi$
$769$	30.9085	1.11459	0.557294	+	0.830315i	$0.311840\pi$
$770$	12.7605	0.459856
$771$	0	0
$772$	−16.2065	−0.583284
$773$	38.9509	1.40097	0.700483	−	0.713669i	$-0.252968\pi$
$773$	38.9509	1.40097	0.700483	+	0.713669i	$0.252968\pi$
$774$	0	0
$775$	−48.0361	−1.72551
$776$	−7.93248	−0.284759
$777$	0	0
$778$	13.9741	0.500997
$779$	−36.6908	−1.31458
$780$	0	0
$781$	20.7992	0.744253
$782$	8.60487	0.307709
$783$	0	0
$784$	−0.968479	−0.0345885
$785$	31.9717	1.14112
$786$	0	0
$787$	−32.9834	−1.17573	−0.587865	−	0.808959i	$-0.700031\pi$
$787$	−32.9834	−1.17573	−0.587865	+	0.808959i	$0.700031\pi$
$788$	21.0736	0.750715
$789$	0	0
$790$	29.6559	1.05511
$791$	−25.1318	−0.893584
$792$	0	0
$793$	57.3579	2.03684
$794$	−11.7975	−0.418676
$795$	0	0
$796$	−2.72982	−0.0967558
$797$	52.4425	1.85761	0.928805	−	0.370568i	$-0.120837\pi$
$797$	52.4425	1.85761	0.928805	+	0.370568i	$0.120837\pi$
$798$	0	0
$799$	−49.2872	−1.74366
$800$	−4.85047	−0.171490
$801$	0	0
$802$	−7.48029	−0.264138
$803$	15.9862	0.564142
$804$	0	0
$805$	−9.83522	−0.346646
$806$	57.5853	2.02835
$807$	0	0
$808$	−7.84592	−0.276019
$809$	−14.0324	−0.493353	−0.246676	−	0.969098i	$-0.579338\pi$
$809$	−14.0324	−0.493353	−0.246676	+	0.969098i	$0.579338\pi$
$810$	0	0
$811$	−18.8762	−0.662832	−0.331416	−	0.943485i	$-0.607526\pi$
$811$	−18.8762	−0.662832	−0.331416	+	0.943485i	$0.607526\pi$
$812$	−25.9881	−0.912004
$813$	0	0
$814$	10.4811	0.367363
$815$	−72.3070	−2.53280
$816$	0	0
$817$	42.4573	1.48539
$818$	23.3197	0.815355
$819$	0	0
$820$	26.5270	0.926362
$821$	−24.4806	−0.854380	−0.427190	−	0.904162i	$-0.640496\pi$
$821$	−24.4806	−0.854380	−0.427190	+	0.904162i	$0.640496\pi$
$822$	0	0
$823$	−14.7256	−0.513303	−0.256652	−	0.966504i	$-0.582619\pi$
$823$	−14.7256	−0.513303	−0.256652	+	0.966504i	$0.582619\pi$
$824$	16.0036	0.557510
$825$	0	0
$826$	−10.5207	−0.366062
$827$	35.7466	1.24303	0.621516	−	0.783402i	$-0.286517\pi$
$827$	35.7466	1.24303	0.621516	+	0.783402i	$0.286517\pi$
$828$	0	0
$829$	−20.8912	−0.725582	−0.362791	−	0.931870i	$-0.618176\pi$
$829$	−20.8912	−0.725582	−0.362791	+	0.931870i	$0.618176\pi$
$830$	−7.94632	−0.275821
$831$	0	0
$832$	5.81470	0.201588
$833$	−6.53118	−0.226292
$834$	0	0
$835$	−29.1090	−1.00736
$836$	−7.18659	−0.248553
$837$	0	0
$838$	−39.9392	−1.37968
$839$	5.13056	0.177127	0.0885634	−	0.996071i	$-0.471772\pi$
$839$	5.13056	0.177127	0.0885634	+	0.996071i	$0.471772\pi$
$840$	0	0
$841$	82.9755	2.86122
$842$	18.9978	0.654706
$843$	0	0
$844$	24.5744	0.845887
$845$	−65.3153	−2.24692
$846$	0	0
$847$	20.2843	0.696978
$848$	−10.0205	−0.344106
$849$	0	0
$850$	−32.7104	−1.12196
$851$	−8.07840	−0.276924
$852$	0	0
$853$	11.8595	0.406063	0.203032	−	0.979172i	$-0.434921\pi$
$853$	11.8595	0.406063	0.203032	+	0.979172i	$0.434921\pi$
$854$	24.2259	0.828992
$855$	0	0
$856$	14.9000	0.509271
$857$	−3.33149	−0.113801	−0.0569007	−	0.998380i	$-0.518122\pi$
$857$	−3.33149	−0.113801	−0.0569007	+	0.998380i	$0.518122\pi$
$858$	0	0
$859$	26.9189	0.918461	0.459231	−	0.888317i	$-0.348125\pi$
$859$	26.9189	0.918461	0.459231	+	0.888317i	$0.348125\pi$
$860$	−30.6961	−1.04673
$861$	0	0
$862$	23.2157	0.790731
$863$	34.3395	1.16893	0.584464	−	0.811419i	$-0.301305\pi$
$863$	34.3395	1.16893	0.584464	+	0.811419i	$0.301305\pi$
$864$	0	0
$865$	32.3900	1.10129
$866$	26.9880	0.917091
$867$	0	0
$868$	24.3219	0.825539
$869$	−15.6426	−0.530638
$870$	0	0
$871$	31.8701	1.07988
$872$	−13.7347	−0.465117
$873$	0	0
$874$	5.53911	0.187363
$875$	−1.15260	−0.0389649
$876$	0	0
$877$	−41.4384	−1.39928	−0.699638	−	0.714498i	$-0.746655\pi$
$877$	−41.4384	−1.39928	−0.699638	+	0.714498i	$0.746655\pi$
$878$	−0.671897	−0.0226754
$879$	0	0
$880$	5.19581	0.175151
$881$	20.3339	0.685065	0.342533	−	0.939506i	$-0.388715\pi$
$881$	20.3339	0.685065	0.342533	+	0.939506i	$0.388715\pi$
$882$	0	0
$883$	12.9192	0.434766	0.217383	−	0.976086i	$-0.430248\pi$
$883$	12.9192	0.434766	0.217383	+	0.976086i	$0.430248\pi$
$884$	39.2129	1.31887
$885$	0	0
$886$	17.4695	0.586900
$887$	−40.4182	−1.35711	−0.678554	−	0.734550i	$-0.737393\pi$
$887$	−40.4182	−1.35711	−0.678554	+	0.734550i	$0.737393\pi$
$888$	0	0
$889$	21.6417	0.725839
$890$	23.7040	0.794559
$891$	0	0
$892$	12.5158	0.419059
$893$	−31.7271	−1.06171
$894$	0	0
$895$	−8.23704	−0.275334
$896$	2.45592	0.0820464
$897$	0	0
$898$	−4.17564	−0.139343
$899$	−104.796	−3.49515
$900$	0	0
$901$	−67.5759	−2.25128
$902$	−13.9921	−0.465887
$903$	0	0
$904$	−10.2332	−0.340350
$905$	−13.9418	−0.463440
$906$	0	0
$907$	1.81640	0.0603124	0.0301562	−	0.999545i	$-0.490400\pi$
$907$	1.81640	0.0603124	0.0301562	+	0.999545i	$0.490400\pi$
$908$	12.7001	0.421467
$909$	0	0
$910$	−44.8197	−1.48576
$911$	22.0251	0.729724	0.364862	−	0.931062i	$-0.381116\pi$
$911$	22.0251	0.729724	0.364862	+	0.931062i	$0.381116\pi$
$912$	0	0
$913$	4.19143	0.138716
$914$	−16.5057	−0.545960
$915$	0	0
$916$	0.970181	0.0320557
$917$	10.3348	0.341286
$918$	0	0
$919$	18.0546	0.595566	0.297783	−	0.954634i	$-0.403753\pi$
$919$	18.0546	0.595566	0.297783	+	0.954634i	$0.403753\pi$
$920$	−4.00471	−0.132031
$921$	0	0
$922$	12.3991	0.408344
$923$	−73.0548	−2.40463
$924$	0	0
$925$	30.7091	1.00971
$926$	12.8673	0.422844
$927$	0	0
$928$	−10.5818	−0.347366
$929$	25.7663	0.845364	0.422682	−	0.906278i	$-0.361089\pi$
$929$	25.7663	0.845364	0.422682	+	0.906278i	$0.361089\pi$
$930$	0	0
$931$	−4.20424	−0.137788
$932$	10.5639	0.346031
$933$	0	0
$934$	−31.2259	−1.02174
$935$	35.0393	1.14591
$936$	0	0
$937$	32.9014	1.07484	0.537421	−	0.843314i	$-0.319399\pi$
$937$	32.9014	1.07484	0.537421	+	0.843314i	$0.319399\pi$
$938$	13.4608	0.439509
$939$	0	0
$940$	22.9383	0.748164
$941$	−18.7775	−0.612130	−0.306065	−	0.952011i	$-0.599012\pi$
$941$	−18.7775	−0.612130	−0.306065	+	0.952011i	$0.599012\pi$
$942$	0	0
$943$	10.7845	0.351193
$944$	−4.28382	−0.139426
$945$	0	0
$946$	16.1912	0.526421
$947$	7.70576	0.250404	0.125202	−	0.992131i	$-0.460042\pi$
$947$	7.70576	0.250404	0.125202	+	0.992131i	$0.460042\pi$
$948$	0	0
$949$	−56.1499	−1.82270
$950$	−21.0563	−0.683155
$951$	0	0
$952$	16.5621	0.536781
$953$	20.9431	0.678413	0.339207	−	0.940712i	$-0.389841\pi$
$953$	20.9431	0.678413	0.339207	+	0.940712i	$0.389841\pi$
$954$	0	0
$955$	26.4709	0.856578
$956$	15.2653	0.493717
$957$	0	0
$958$	−10.0310	−0.324088
$959$	−32.8405	−1.06048
$960$	0	0
$961$	67.0773	2.16378
$962$	−36.8138	−1.18692
$963$	0	0
$964$	18.1029	0.583056
$965$	50.8648	1.63739
$966$	0	0
$967$	−12.4181	−0.399339	−0.199669	−	0.979863i	$-0.563987\pi$
$967$	−12.4181	−0.399339	−0.199669	+	0.979863i	$0.563987\pi$
$968$	8.25937	0.265466
$969$	0	0
$970$	24.8964	0.799377
$971$	49.6265	1.59259	0.796295	−	0.604908i	$-0.206790\pi$
$971$	49.6265	1.59259	0.796295	+	0.604908i	$0.206790\pi$
$972$	0	0
$973$	29.1200	0.933543
$974$	−5.45193	−0.174691
$975$	0	0
$976$	9.86429	0.315748
$977$	18.1688	0.581271	0.290635	−	0.956834i	$-0.406133\pi$
$977$	18.1688	0.581271	0.290635	+	0.956834i	$0.406133\pi$
$978$	0	0
$979$	−12.5031	−0.399601
$980$	3.03961	0.0970969
$981$	0	0
$982$	−17.2620	−0.550852
$983$	−8.10825	−0.258613	−0.129307	−	0.991605i	$-0.541275\pi$
$983$	−8.10825	−0.258613	−0.129307	+	0.991605i	$0.541275\pi$
$984$	0	0
$985$	−66.1404	−2.10741
$986$	−71.3614	−2.27261
$987$	0	0
$988$	25.2421	0.803057
$989$	−12.4795	−0.396824
$990$	0	0
$991$	3.02887	0.0962151	0.0481076	−	0.998842i	$-0.484681\pi$
$991$	3.02887	0.0962151	0.0481076	+	0.998842i	$0.484681\pi$
$992$	9.90340	0.314433
$993$	0	0
$994$	−30.8557	−0.978682
$995$	8.56765	0.271613
$996$	0	0
$997$	16.2965	0.516115	0.258058	−	0.966130i	$-0.416918\pi$
$997$	16.2965	0.516115	0.258058	+	0.966130i	$0.416918\pi$
$998$	−14.0141	−0.443609
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.q.1.2	✓	14
3.2	odd	2		8046.2.a.r.1.13	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.q.1.2	✓	14	1.1	even	1	trivial
8046.2.a.r.1.13	yes	14	3.2	odd	2

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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.13855 q^{5} -2.45592 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.13855 q^{5} -2.45592 q^{7} -1.00000 q^{8} +3.13855 q^{10} -1.65548 q^{11} +5.81470 q^{13} +2.45592 q^{14} +1.00000 q^{16} +6.74376 q^{17} +4.34108 q^{19} -3.13855 q^{20} +1.65548 q^{22} -1.27598 q^{23} +4.85047 q^{25} -5.81470 q^{26} -2.45592 q^{28} +10.5818 q^{29} -9.90340 q^{31} -1.00000 q^{32} -6.74376 q^{34} +7.70800 q^{35} +6.33116 q^{37} -4.34108 q^{38} +3.13855 q^{40} -8.45200 q^{41} +9.78035 q^{43} -1.65548 q^{44} +1.27598 q^{46} -7.30857 q^{47} -0.968479 q^{49} -4.85047 q^{50} +5.81470 q^{52} -10.0205 q^{53} +5.19581 q^{55} +2.45592 q^{56} -10.5818 q^{58} -4.28382 q^{59} +9.86429 q^{61} +9.90340 q^{62} +1.00000 q^{64} -18.2497 q^{65} +5.48095 q^{67} +6.74376 q^{68} -7.70800 q^{70} -12.5638 q^{71} -9.65654 q^{73} -6.33116 q^{74} +4.34108 q^{76} +4.06573 q^{77} +9.44894 q^{79} -3.13855 q^{80} +8.45200 q^{82} -2.53185 q^{83} -21.1656 q^{85} -9.78035 q^{86} +1.65548 q^{88} +7.55253 q^{89} -14.2804 q^{91} -1.27598 q^{92} +7.30857 q^{94} -13.6247 q^{95} +7.93248 q^{97} +0.968479 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) \) 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8	\( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 14 q^{16} - 9 q^{17} + 14 q^{19} - 2 q^{20} + 2 q^{22} - 30 q^{23} + 18 q^{25} - 4 q^{26} + 4 q^{28} - 6 q^{29} + 11 q^{31} - 14 q^{32} + 9 q^{34} + 18 q^{35} + 13 q^{37} - 14 q^{38} + 2 q^{40} + 2 q^{41} + 12 q^{43} - 2 q^{44} + 30 q^{46} - 21 q^{47} + 32 q^{49} - 18 q^{50} + 4 q^{52} - 22 q^{53} - 7 q^{55} - 4 q^{56} + 6 q^{58} - 14 q^{59} + 31 q^{61} - 11 q^{62} + 14 q^{64} + 24 q^{67} - 9 q^{68} - 18 q^{70} - 28 q^{71} + 24 q^{73} - 13 q^{74} + 14 q^{76} - 16 q^{77} + 65 q^{79} - 2 q^{80} - 2 q^{82} + 15 q^{83} - 19 q^{85} - 12 q^{86} + 2 q^{88} + 11 q^{89} + 68 q^{91} - 30 q^{92} + 21 q^{94} - 8 q^{95} + 23 q^{97} - 32 q^{98}+O(q^{100}) \) 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 - 4 * q^14 + 14 * q^16 - 9 * q^17 + 14 * q^19 - 2 * q^20 + 2 * q^22 - 30 * q^23 + 18 * q^25 - 4 * q^26 + 4 * q^28 - 6 * q^29 + 11 * q^31 - 14 * q^32 + 9 * q^34 + 18 * q^35 + 13 * q^37 - 14 * q^38 + 2 * q^40 + 2 * q^41 + 12 * q^43 - 2 * q^44 + 30 * q^46 - 21 * q^47 + 32 * q^49 - 18 * q^50 + 4 * q^52 - 22 * q^53 - 7 * q^55 - 4 * q^56 + 6 * q^58 - 14 * q^59 + 31 * q^61 - 11 * q^62 + 14 * q^64 + 24 * q^67 - 9 * q^68 - 18 * q^70 - 28 * q^71 + 24 * q^73 - 13 * q^74 + 14 * q^76 - 16 * q^77 + 65 * q^79 - 2 * q^80 - 2 * q^82 + 15 * q^83 - 19 * q^85 - 12 * q^86 + 2 * q^88 + 11 * q^89 + 68 * q^91 - 30 * q^92 + 21 * q^94 - 8 * q^95 + 23 * q^97 - 32 * q^98

Introduction

L-functions

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