LMFDB - Embedded newform 8048.2.a.t.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.37429 q^{3} +2.26464 q^{5} +3.08988 q^{7} -1.11131 q^{9} +O(q^{10})$	$q-1.37429 q^{3} +2.26464 q^{5} +3.08988 q^{7} -1.11131 q^{9} +6.51225 q^{11} -1.74140 q^{13} -3.11229 q^{15} +0.835861 q^{17} -4.12347 q^{19} -4.24640 q^{21} -2.65631 q^{23} +0.128609 q^{25} +5.65016 q^{27} -6.05525 q^{29} -2.92404 q^{31} -8.94974 q^{33} +6.99747 q^{35} -5.45921 q^{37} +2.39319 q^{39} +11.7540 q^{41} +6.91557 q^{43} -2.51673 q^{45} +8.15938 q^{47} +2.54735 q^{49} -1.14872 q^{51} +1.38837 q^{53} +14.7479 q^{55} +5.66686 q^{57} +14.1812 q^{59} -10.9933 q^{61} -3.43383 q^{63} -3.94364 q^{65} +10.3683 q^{67} +3.65055 q^{69} +3.20096 q^{71} -4.08129 q^{73} -0.176747 q^{75} +20.1220 q^{77} -4.60634 q^{79} -4.43103 q^{81} +0.404366 q^{83} +1.89293 q^{85} +8.32170 q^{87} +13.3473 q^{89} -5.38070 q^{91} +4.01850 q^{93} -9.33818 q^{95} +8.09985 q^{97} -7.23716 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.37429	−0.793449	−0.396725	−	0.917938i	$-0.629853\pi$
$3$	−1.37429	−0.793449	−0.396725	+	0.917938i	$0.629853\pi$
$4$	0	0
$5$	2.26464	1.01278	0.506390	−	0.862305i	$-0.330980\pi$
$5$	2.26464	1.01278	0.506390	+	0.862305i	$0.330980\pi$
$6$	0	0
$7$	3.08988	1.16786	0.583932	−	0.811803i	$-0.301513\pi$
$7$	3.08988	1.16786	0.583932	+	0.811803i	$0.301513\pi$
$8$	0	0
$9$	−1.11131	−0.370438
$10$	0	0
$11$	6.51225	1.96352	0.981758	−	0.190134i	$-0.0608922\pi$
$11$	6.51225	1.96352	0.981758	+	0.190134i	$0.0608922\pi$
$12$	0	0
$13$	−1.74140	−0.482977	−0.241488	−	0.970404i	$-0.577636\pi$
$13$	−1.74140	−0.482977	−0.241488	+	0.970404i	$0.577636\pi$
$14$	0	0
$15$	−3.11229	−0.803589
$16$	0	0
$17$	0.835861	0.202726	0.101363	−	0.994850i	$-0.467680\pi$
$17$	0.835861	0.202726	0.101363	+	0.994850i	$0.467680\pi$
$18$	0	0
$19$	−4.12347	−0.945988	−0.472994	−	0.881066i	$-0.656827\pi$
$19$	−4.12347	−0.945988	−0.472994	+	0.881066i	$0.656827\pi$
$20$	0	0
$21$	−4.24640	−0.926641
$22$	0	0
$23$	−2.65631	−0.553879	−0.276940	−	0.960887i	$-0.589320\pi$
$23$	−2.65631	−0.553879	−0.276940	+	0.960887i	$0.589320\pi$
$24$	0	0
$25$	0.128609	0.0257218
$26$	0	0
$27$	5.65016	1.08737
$28$	0	0
$29$	−6.05525	−1.12443	−0.562216	−	0.826991i	$-0.690051\pi$
$29$	−6.05525	−1.12443	−0.562216	+	0.826991i	$0.690051\pi$
$30$	0	0
$31$	−2.92404	−0.525174	−0.262587	−	0.964908i	$-0.584576\pi$
$31$	−2.92404	−0.525174	−0.262587	+	0.964908i	$0.584576\pi$
$32$	0	0
$33$	−8.94974	−1.55795
$34$	0	0
$35$	6.99747	1.18279
$36$	0	0
$37$	−5.45921	−0.897489	−0.448744	−	0.893660i	$-0.648129\pi$
$37$	−5.45921	−0.897489	−0.448744	+	0.893660i	$0.648129\pi$
$38$	0	0
$39$	2.39319	0.383217
$40$	0	0
$41$	11.7540	1.83566	0.917830	−	0.396975i	$-0.129940\pi$
$41$	11.7540	1.83566	0.917830	+	0.396975i	$0.129940\pi$
$42$	0	0
$43$	6.91557	1.05461	0.527307	−	0.849675i	$-0.323202\pi$
$43$	6.91557	1.05461	0.527307	+	0.849675i	$0.323202\pi$
$44$	0	0
$45$	−2.51673	−0.375172
$46$	0	0
$47$	8.15938	1.19017	0.595084	−	0.803663i	$-0.297119\pi$
$47$	8.15938	1.19017	0.595084	+	0.803663i	$0.297119\pi$
$48$	0	0
$49$	2.54735	0.363906
$50$	0	0
$51$	−1.14872	−0.160853
$52$	0	0
$53$	1.38837	0.190707	0.0953535	−	0.995443i	$-0.469602\pi$
$53$	1.38837	0.190707	0.0953535	+	0.995443i	$0.469602\pi$
$54$	0	0
$55$	14.7479	1.98861
$56$	0	0
$57$	5.66686	0.750594
$58$	0	0
$59$	14.1812	1.84624	0.923119	−	0.384513i	$-0.125631\pi$
$59$	14.1812	1.84624	0.923119	+	0.384513i	$0.125631\pi$
$60$	0	0
$61$	−10.9933	−1.40755	−0.703775	−	0.710423i	$-0.748504\pi$
$61$	−10.9933	−1.40755	−0.703775	+	0.710423i	$0.748504\pi$
$62$	0	0
$63$	−3.43383	−0.432622
$64$	0	0
$65$	−3.94364	−0.489149
$66$	0	0
$67$	10.3683	1.26669	0.633346	−	0.773869i	$-0.281681\pi$
$67$	10.3683	1.26669	0.633346	+	0.773869i	$0.281681\pi$
$68$	0	0
$69$	3.65055	0.439475
$70$	0	0
$71$	3.20096	0.379884	0.189942	−	0.981795i	$-0.439170\pi$
$71$	3.20096	0.379884	0.189942	+	0.981795i	$0.439170\pi$
$72$	0	0
$73$	−4.08129	−0.477679	−0.238840	−	0.971059i	$-0.576767\pi$
$73$	−4.08129	−0.477679	−0.238840	+	0.971059i	$0.576767\pi$
$74$	0	0
$75$	−0.176747	−0.0204090
$76$	0	0
$77$	20.1220	2.29312
$78$	0	0
$79$	−4.60634	−0.518254	−0.259127	−	0.965843i	$-0.583435\pi$
$79$	−4.60634	−0.518254	−0.259127	+	0.965843i	$0.583435\pi$
$80$	0	0
$81$	−4.43103	−0.492337
$82$	0	0
$83$	0.404366	0.0443849	0.0221925	−	0.999754i	$-0.492935\pi$
$83$	0.404366	0.0443849	0.0221925	+	0.999754i	$0.492935\pi$
$84$	0	0
$85$	1.89293	0.205317
$86$	0	0
$87$	8.32170	0.892179
$88$	0	0
$89$	13.3473	1.41481	0.707407	−	0.706806i	$-0.249865\pi$
$89$	13.3473	1.41481	0.707407	+	0.706806i	$0.249865\pi$
$90$	0	0
$91$	−5.38070	−0.564051
$92$	0	0
$93$	4.01850	0.416699
$94$	0	0
$95$	−9.33818	−0.958077
$96$	0	0
$97$	8.09985	0.822415	0.411208	−	0.911542i	$-0.365107\pi$
$97$	8.09985	0.822415	0.411208	+	0.911542i	$0.365107\pi$
$98$	0	0
$99$	−7.23716	−0.727362
$100$	0	0
$101$	13.2417	1.31760	0.658801	−	0.752317i	$-0.271064\pi$
$101$	13.2417	1.31760	0.658801	+	0.752317i	$0.271064\pi$
$102$	0	0
$103$	−13.6280	−1.34280	−0.671401	−	0.741094i	$-0.734307\pi$
$103$	−13.6280	−1.34280	−0.671401	+	0.741094i	$0.734307\pi$
$104$	0	0
$105$	−9.61659	−0.938483
$106$	0	0
$107$	13.4808	1.30324	0.651618	−	0.758547i	$-0.274091\pi$
$107$	13.4808	1.30324	0.651618	+	0.758547i	$0.274091\pi$
$108$	0	0
$109$	6.39376	0.612411	0.306205	−	0.951965i	$-0.400941\pi$
$109$	6.39376	0.612411	0.306205	+	0.951965i	$0.400941\pi$
$110$	0	0
$111$	7.50256	0.712112
$112$	0	0
$113$	−3.85666	−0.362805	−0.181402	−	0.983409i	$-0.558064\pi$
$113$	−3.85666	−0.362805	−0.181402	+	0.983409i	$0.558064\pi$
$114$	0	0
$115$	−6.01560	−0.560957
$116$	0	0
$117$	1.93524	0.178913
$118$	0	0
$119$	2.58271	0.236756
$120$	0	0
$121$	31.4094	2.85540
$122$	0	0
$123$	−16.1534	−1.45650
$124$	0	0
$125$	−11.0320	−0.986729
$126$	0	0
$127$	8.65186	0.767728	0.383864	−	0.923390i	$-0.374593\pi$
$127$	8.65186	0.767728	0.383864	+	0.923390i	$0.374593\pi$
$128$	0	0
$129$	−9.50402	−0.836783
$130$	0	0
$131$	20.8848	1.82471	0.912357	−	0.409396i	$-0.134260\pi$
$131$	20.8848	1.82471	0.912357	+	0.409396i	$0.134260\pi$
$132$	0	0
$133$	−12.7410	−1.10479
$134$	0	0
$135$	12.7956	1.10127
$136$	0	0
$137$	4.65612	0.397799	0.198899	−	0.980020i	$-0.436263\pi$
$137$	4.65612	0.397799	0.198899	+	0.980020i	$0.436263\pi$
$138$	0	0
$139$	−23.1354	−1.96231	−0.981157	−	0.193210i	$-0.938110\pi$
$139$	−23.1354	−1.96231	−0.981157	+	0.193210i	$0.938110\pi$
$140$	0	0
$141$	−11.2134	−0.944338
$142$	0	0
$143$	−11.3404	−0.948332
$144$	0	0
$145$	−13.7130	−1.13880
$146$	0	0
$147$	−3.50080	−0.288741
$148$	0	0
$149$	−10.5902	−0.867581	−0.433790	−	0.901014i	$-0.642824\pi$
$149$	−10.5902	−0.867581	−0.433790	+	0.901014i	$0.642824\pi$
$150$	0	0
$151$	3.27812	0.266770	0.133385	−	0.991064i	$-0.457415\pi$
$151$	3.27812	0.266770	0.133385	+	0.991064i	$0.457415\pi$
$152$	0	0
$153$	−0.928904	−0.0750975
$154$	0	0
$155$	−6.62191	−0.531885
$156$	0	0
$157$	6.24105	0.498090	0.249045	−	0.968492i	$-0.419883\pi$
$157$	6.24105	0.498090	0.249045	+	0.968492i	$0.419883\pi$
$158$	0	0
$159$	−1.90803	−0.151316
$160$	0	0
$161$	−8.20768	−0.646856
$162$	0	0
$163$	−15.3480	−1.20215	−0.601075	−	0.799193i	$-0.705261\pi$
$163$	−15.3480	−1.20215	−0.601075	+	0.799193i	$0.705261\pi$
$164$	0	0
$165$	−20.2680	−1.57786
$166$	0	0
$167$	−21.0232	−1.62682	−0.813411	−	0.581689i	$-0.802392\pi$
$167$	−21.0232	−1.62682	−0.813411	+	0.581689i	$0.802392\pi$
$168$	0	0
$169$	−9.96754	−0.766734
$170$	0	0
$171$	4.58247	0.350430
$172$	0	0
$173$	5.56529	0.423121	0.211561	−	0.977365i	$-0.432145\pi$
$173$	5.56529	0.423121	0.211561	+	0.977365i	$0.432145\pi$
$174$	0	0
$175$	0.397386	0.0300396
$176$	0	0
$177$	−19.4892	−1.46490
$178$	0	0
$179$	15.0857	1.12756	0.563779	−	0.825926i	$-0.309347\pi$
$179$	15.0857	1.12756	0.563779	+	0.825926i	$0.309347\pi$
$180$	0	0
$181$	10.0687	0.748397	0.374199	−	0.927349i	$-0.377918\pi$
$181$	10.0687	0.748397	0.374199	+	0.927349i	$0.377918\pi$
$182$	0	0
$183$	15.1081	1.11682
$184$	0	0
$185$	−12.3632	−0.908958
$186$	0	0
$187$	5.44333	0.398056
$188$	0	0
$189$	17.4583	1.26990
$190$	0	0
$191$	−10.7526	−0.778030	−0.389015	−	0.921232i	$-0.627184\pi$
$191$	−10.7526	−0.778030	−0.389015	+	0.921232i	$0.627184\pi$
$192$	0	0
$193$	12.2947	0.884989	0.442494	−	0.896771i	$-0.354094\pi$
$193$	12.2947	0.884989	0.442494	+	0.896771i	$0.354094\pi$
$194$	0	0
$195$	5.41973	0.388115
$196$	0	0
$197$	2.96702	0.211391	0.105696	−	0.994399i	$-0.466293\pi$
$197$	2.96702	0.211391	0.105696	+	0.994399i	$0.466293\pi$
$198$	0	0
$199$	22.6307	1.60425	0.802123	−	0.597158i	$-0.203704\pi$
$199$	22.6307	1.60425	0.802123	+	0.597158i	$0.203704\pi$
$200$	0	0
$201$	−14.2491	−1.00506
$202$	0	0
$203$	−18.7100	−1.31318
$204$	0	0
$205$	26.6185	1.85912
$206$	0	0
$207$	2.95200	0.205178
$208$	0	0
$209$	−26.8530	−1.85746
$210$	0	0
$211$	−10.6096	−0.730395	−0.365198	−	0.930930i	$-0.618999\pi$
$211$	−10.6096	−0.730395	−0.365198	+	0.930930i	$0.618999\pi$
$212$	0	0
$213$	−4.39906	−0.301418
$214$	0	0
$215$	15.6613	1.06809
$216$	0	0
$217$	−9.03493	−0.613331
$218$	0	0
$219$	5.60890	0.379014
$220$	0	0
$221$	−1.45557	−0.0979119
$222$	0	0
$223$	2.13268	0.142815	0.0714074	−	0.997447i	$-0.477251\pi$
$223$	2.13268	0.142815	0.0714074	+	0.997447i	$0.477251\pi$
$224$	0	0
$225$	−0.142925	−0.00952835
$226$	0	0
$227$	19.2020	1.27448	0.637240	−	0.770666i	$-0.280076\pi$
$227$	19.2020	1.27448	0.637240	+	0.770666i	$0.280076\pi$
$228$	0	0
$229$	−1.96988	−0.130173	−0.0650865	−	0.997880i	$-0.520732\pi$
$229$	−1.96988	−0.130173	−0.0650865	+	0.997880i	$0.520732\pi$
$230$	0	0
$231$	−27.6536	−1.81947
$232$	0	0
$233$	19.7621	1.29466	0.647330	−	0.762210i	$-0.275886\pi$
$233$	19.7621	1.29466	0.647330	+	0.762210i	$0.275886\pi$
$234$	0	0
$235$	18.4781	1.20538
$236$	0	0
$237$	6.33047	0.411208
$238$	0	0
$239$	3.57916	0.231517	0.115758	−	0.993277i	$-0.463070\pi$
$239$	3.57916	0.231517	0.115758	+	0.993277i	$0.463070\pi$
$240$	0	0
$241$	−8.41343	−0.541956	−0.270978	−	0.962585i	$-0.587347\pi$
$241$	−8.41343	−0.541956	−0.270978	+	0.962585i	$0.587347\pi$
$242$	0	0
$243$	−10.8609	−0.696729
$244$	0	0
$245$	5.76883	0.368557
$246$	0	0
$247$	7.18059	0.456890
$248$	0	0
$249$	−0.555718	−0.0352172
$250$	0	0
$251$	−19.0734	−1.20390	−0.601950	−	0.798534i	$-0.705609\pi$
$251$	−19.0734	−1.20390	−0.601950	+	0.798534i	$0.705609\pi$
$252$	0	0
$253$	−17.2986	−1.08755
$254$	0	0
$255$	−2.60144	−0.162908
$256$	0	0
$257$	−12.5901	−0.785351	−0.392676	−	0.919677i	$-0.628450\pi$
$257$	−12.5901	−0.785351	−0.392676	+	0.919677i	$0.628450\pi$
$258$	0	0
$259$	−16.8683	−1.04814
$260$	0	0
$261$	6.72929	0.416533
$262$	0	0
$263$	−30.3038	−1.86861	−0.934306	−	0.356472i	$-0.883980\pi$
$263$	−30.3038	−1.86861	−0.934306	+	0.356472i	$0.883980\pi$
$264$	0	0
$265$	3.14416	0.193144
$266$	0	0
$267$	−18.3432	−1.12258
$268$	0	0
$269$	10.5782	0.644965	0.322483	−	0.946575i	$-0.395483\pi$
$269$	10.5782	0.644965	0.322483	+	0.946575i	$0.395483\pi$
$270$	0	0
$271$	18.3919	1.11723	0.558614	−	0.829428i	$-0.311333\pi$
$271$	18.3919	1.11723	0.558614	+	0.829428i	$0.311333\pi$
$272$	0	0
$273$	7.39467	0.447546
$274$	0	0
$275$	0.837534	0.0505052
$276$	0	0
$277$	−15.8542	−0.952586	−0.476293	−	0.879287i	$-0.658020\pi$
$277$	−15.8542	−0.952586	−0.476293	+	0.879287i	$0.658020\pi$
$278$	0	0
$279$	3.24953	0.194544
$280$	0	0
$281$	21.2898	1.27004	0.635022	−	0.772494i	$-0.280991\pi$
$281$	21.2898	1.27004	0.635022	+	0.772494i	$0.280991\pi$
$282$	0	0
$283$	26.3920	1.56884	0.784421	−	0.620229i	$-0.212960\pi$
$283$	26.3920	1.56884	0.784421	+	0.620229i	$0.212960\pi$
$284$	0	0
$285$	12.8334	0.760186
$286$	0	0
$287$	36.3183	2.14380
$288$	0	0
$289$	−16.3013	−0.958902
$290$	0	0
$291$	−11.1316	−0.652545
$292$	0	0
$293$	−4.42790	−0.258681	−0.129340	−	0.991600i	$-0.541286\pi$
$293$	−4.42790	−0.258681	−0.129340	+	0.991600i	$0.541286\pi$
$294$	0	0
$295$	32.1154	1.86983
$296$	0	0
$297$	36.7952	2.13508
$298$	0	0
$299$	4.62569	0.267511
$300$	0	0
$301$	21.3683	1.23165
$302$	0	0
$303$	−18.1980	−1.04545
$304$	0	0
$305$	−24.8959	−1.42554
$306$	0	0
$307$	−24.8357	−1.41745	−0.708724	−	0.705486i	$-0.750729\pi$
$307$	−24.8357	−1.41745	−0.708724	+	0.705486i	$0.750729\pi$
$308$	0	0
$309$	18.7288	1.06545
$310$	0	0
$311$	27.7257	1.57218	0.786089	−	0.618113i	$-0.212103\pi$
$311$	27.7257	1.57218	0.786089	+	0.618113i	$0.212103\pi$
$312$	0	0
$313$	10.5984	0.599058	0.299529	−	0.954087i	$-0.403170\pi$
$313$	10.5984	0.599058	0.299529	+	0.954087i	$0.403170\pi$
$314$	0	0
$315$	−7.77639	−0.438150
$316$	0	0
$317$	−0.261283	−0.0146751	−0.00733755	−	0.999973i	$-0.502336\pi$
$317$	−0.261283	−0.0146751	−0.00733755	+	0.999973i	$0.502336\pi$
$318$	0	0
$319$	−39.4333	−2.20784
$320$	0	0
$321$	−18.5266	−1.03405
$322$	0	0
$323$	−3.44664	−0.191776
$324$	0	0
$325$	−0.223959	−0.0124230
$326$	0	0
$327$	−8.78691	−0.485917
$328$	0	0
$329$	25.2115	1.38996
$330$	0	0
$331$	−0.802788	−0.0441252	−0.0220626	−	0.999757i	$-0.507023\pi$
$331$	−0.802788	−0.0441252	−0.0220626	+	0.999757i	$0.507023\pi$
$332$	0	0
$333$	6.06690	0.332464
$334$	0	0
$335$	23.4805	1.28288
$336$	0	0
$337$	11.3210	0.616692	0.308346	−	0.951274i	$-0.400225\pi$
$337$	11.3210	0.616692	0.308346	+	0.951274i	$0.400225\pi$
$338$	0	0
$339$	5.30019	0.287867
$340$	0	0
$341$	−19.0421	−1.03119
$342$	0	0
$343$	−13.7582	−0.742871
$344$	0	0
$345$	8.26720	0.445091
$346$	0	0
$347$	6.62662	0.355736	0.177868	−	0.984054i	$-0.443080\pi$
$347$	6.62662	0.355736	0.177868	+	0.984054i	$0.443080\pi$
$348$	0	0
$349$	−23.3099	−1.24775	−0.623876	−	0.781524i	$-0.714443\pi$
$349$	−23.3099	−1.24775	−0.623876	+	0.781524i	$0.714443\pi$
$350$	0	0
$351$	−9.83916	−0.525176
$352$	0	0
$353$	28.5477	1.51944	0.759721	−	0.650250i	$-0.225336\pi$
$353$	28.5477	1.51944	0.759721	+	0.650250i	$0.225336\pi$
$354$	0	0
$355$	7.24903	0.384738
$356$	0	0
$357$	−3.54940	−0.187854
$358$	0	0
$359$	16.7792	0.885574	0.442787	−	0.896627i	$-0.353990\pi$
$359$	16.7792	0.885574	0.442787	+	0.896627i	$0.353990\pi$
$360$	0	0
$361$	−1.99701	−0.105106
$362$	0	0
$363$	−43.1657	−2.26561
$364$	0	0
$365$	−9.24268	−0.483784
$366$	0	0
$367$	18.2497	0.952625	0.476313	−	0.879276i	$-0.341973\pi$
$367$	18.2497	0.952625	0.476313	+	0.879276i	$0.341973\pi$
$368$	0	0
$369$	−13.0623	−0.679998
$370$	0	0
$371$	4.28989	0.222720
$372$	0	0
$373$	−22.3673	−1.15814	−0.579068	−	0.815280i	$-0.696583\pi$
$373$	−22.3673	−1.15814	−0.579068	+	0.815280i	$0.696583\pi$
$374$	0	0
$375$	15.1612	0.782919
$376$	0	0
$377$	10.5446	0.543074
$378$	0	0
$379$	8.58954	0.441215	0.220607	−	0.975363i	$-0.429196\pi$
$379$	8.58954	0.441215	0.220607	+	0.975363i	$0.429196\pi$
$380$	0	0
$381$	−11.8902	−0.609153
$382$	0	0
$383$	34.5938	1.76766	0.883830	−	0.467807i	$-0.154956\pi$
$383$	34.5938	1.76766	0.883830	+	0.467807i	$0.154956\pi$
$384$	0	0
$385$	45.5693	2.32242
$386$	0	0
$387$	−7.68537	−0.390669
$388$	0	0
$389$	19.2484	0.975931	0.487965	−	0.872863i	$-0.337739\pi$
$389$	19.2484	0.975931	0.487965	+	0.872863i	$0.337739\pi$
$390$	0	0
$391$	−2.22031	−0.112286
$392$	0	0
$393$	−28.7019	−1.44782
$394$	0	0
$395$	−10.4317	−0.524877
$396$	0	0
$397$	−39.5085	−1.98287	−0.991437	−	0.130588i	$-0.958314\pi$
$397$	−39.5085	−1.98287	−0.991437	+	0.130588i	$0.958314\pi$
$398$	0	0
$399$	17.5099	0.876592
$400$	0	0
$401$	25.1799	1.25742	0.628712	−	0.777638i	$-0.283582\pi$
$401$	25.1799	1.25742	0.628712	+	0.777638i	$0.283582\pi$
$402$	0	0
$403$	5.09192	0.253647
$404$	0	0
$405$	−10.0347	−0.498629
$406$	0	0
$407$	−35.5517	−1.76223
$408$	0	0
$409$	3.78276	0.187045	0.0935227	−	0.995617i	$-0.470187\pi$
$409$	3.78276	0.187045	0.0935227	+	0.995617i	$0.470187\pi$
$410$	0	0
$411$	−6.39888	−0.315633
$412$	0	0
$413$	43.8183	2.15616
$414$	0	0
$415$	0.915744	0.0449521
$416$	0	0
$417$	31.7948	1.55700
$418$	0	0
$419$	−22.2755	−1.08823	−0.544116	−	0.839010i	$-0.683135\pi$
$419$	−22.2755	−1.08823	−0.544116	+	0.839010i	$0.683135\pi$
$420$	0	0
$421$	−3.95131	−0.192575	−0.0962876	−	0.995354i	$-0.530697\pi$
$421$	−3.95131	−0.192575	−0.0962876	+	0.995354i	$0.530697\pi$
$422$	0	0
$423$	−9.06764	−0.440884
$424$	0	0
$425$	0.107499	0.00521448
$426$	0	0
$427$	−33.9680	−1.64383
$428$	0	0
$429$	15.5851	0.752454
$430$	0	0
$431$	−9.78447	−0.471301	−0.235651	−	0.971838i	$-0.575722\pi$
$431$	−9.78447	−0.471301	−0.235651	+	0.971838i	$0.575722\pi$
$432$	0	0
$433$	4.43359	0.213065	0.106532	−	0.994309i	$-0.466025\pi$
$433$	4.43359	0.213065	0.106532	+	0.994309i	$0.466025\pi$
$434$	0	0
$435$	18.8457	0.903581
$436$	0	0
$437$	10.9532	0.523963
$438$	0	0
$439$	−2.15854	−0.103022	−0.0515108	−	0.998672i	$-0.516404\pi$
$439$	−2.15854	−0.103022	−0.0515108	+	0.998672i	$0.516404\pi$
$440$	0	0
$441$	−2.83090	−0.134805
$442$	0	0
$443$	−26.0296	−1.23670	−0.618352	−	0.785901i	$-0.712200\pi$
$443$	−26.0296	−1.23670	−0.618352	+	0.785901i	$0.712200\pi$
$444$	0	0
$445$	30.2269	1.43289
$446$	0	0
$447$	14.5540	0.688381
$448$	0	0
$449$	−22.4041	−1.05731	−0.528657	−	0.848836i	$-0.677304\pi$
$449$	−22.4041	−1.05731	−0.528657	+	0.848836i	$0.677304\pi$
$450$	0	0
$451$	76.5446	3.60435
$452$	0	0
$453$	−4.50510	−0.211668
$454$	0	0
$455$	−12.1854	−0.571259
$456$	0	0
$457$	−12.0889	−0.565495	−0.282748	−	0.959194i	$-0.591246\pi$
$457$	−12.0889	−0.565495	−0.282748	+	0.959194i	$0.591246\pi$
$458$	0	0
$459$	4.72274	0.220439
$460$	0	0
$461$	−4.12152	−0.191958	−0.0959792	−	0.995383i	$-0.530598\pi$
$461$	−4.12152	−0.191958	−0.0959792	+	0.995383i	$0.530598\pi$
$462$	0	0
$463$	24.6693	1.14648	0.573239	−	0.819388i	$-0.305687\pi$
$463$	24.6693	1.14648	0.573239	+	0.819388i	$0.305687\pi$
$464$	0	0
$465$	9.10046	0.422024
$466$	0	0
$467$	−10.5299	−0.487265	−0.243632	−	0.969868i	$-0.578339\pi$
$467$	−10.5299	−0.487265	−0.243632	+	0.969868i	$0.578339\pi$
$468$	0	0
$469$	32.0368	1.47932
$470$	0	0
$471$	−8.57704	−0.395209
$472$	0	0
$473$	45.0359	2.07075
$474$	0	0
$475$	−0.530316	−0.0243325
$476$	0	0
$477$	−1.54291	−0.0706452
$478$	0	0
$479$	15.1467	0.692069	0.346035	−	0.938222i	$-0.387528\pi$
$479$	15.1467	0.692069	0.346035	+	0.938222i	$0.387528\pi$
$480$	0	0
$481$	9.50665	0.433466
$482$	0	0
$483$	11.2798	0.513247
$484$	0	0
$485$	18.3433	0.832925
$486$	0	0
$487$	28.8706	1.30825	0.654125	−	0.756386i	$-0.273037\pi$
$487$	28.8706	1.30825	0.654125	+	0.756386i	$0.273037\pi$
$488$	0	0
$489$	21.0927	0.953844
$490$	0	0
$491$	−8.82286	−0.398170	−0.199085	−	0.979982i	$-0.563797\pi$
$491$	−8.82286	−0.398170	−0.199085	+	0.979982i	$0.563797\pi$
$492$	0	0
$493$	−5.06135	−0.227952
$494$	0	0
$495$	−16.3896	−0.736657
$496$	0	0
$497$	9.89057	0.443653
$498$	0	0
$499$	7.66999	0.343356	0.171678	−	0.985153i	$-0.445081\pi$
$499$	7.66999	0.343356	0.171678	+	0.985153i	$0.445081\pi$
$500$	0	0
$501$	28.8920	1.29080
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	29.9878	1.33444
$506$	0	0
$507$	13.6983	0.608364
$508$	0	0
$509$	13.8545	0.614088	0.307044	−	0.951695i	$-0.400660\pi$
$509$	13.8545	0.614088	0.307044	+	0.951695i	$0.400660\pi$
$510$	0	0
$511$	−12.6107	−0.557865
$512$	0	0
$513$	−23.2982	−1.02864
$514$	0	0
$515$	−30.8624	−1.35996
$516$	0	0
$517$	53.1359	2.33692
$518$	0	0
$519$	−7.64835	−0.335725
$520$	0	0
$521$	−35.5795	−1.55877	−0.779384	−	0.626547i	$-0.784468\pi$
$521$	−35.5795	−1.55877	−0.779384	+	0.626547i	$0.784468\pi$
$522$	0	0
$523$	−11.5529	−0.505174	−0.252587	−	0.967574i	$-0.581281\pi$
$523$	−11.5529	−0.505174	−0.252587	+	0.967574i	$0.581281\pi$
$524$	0	0
$525$	−0.546126	−0.0238349
$526$	0	0
$527$	−2.44409	−0.106466
$528$	0	0
$529$	−15.9440	−0.693218
$530$	0	0
$531$	−15.7598	−0.683918
$532$	0	0
$533$	−20.4683	−0.886580
$534$	0	0
$535$	30.5291	1.31989
$536$	0	0
$537$	−20.7322	−0.894660
$538$	0	0
$539$	16.5889	0.714536
$540$	0	0
$541$	5.89376	0.253392	0.126696	−	0.991942i	$-0.459563\pi$
$541$	5.89376	0.253392	0.126696	+	0.991942i	$0.459563\pi$
$542$	0	0
$543$	−13.8373	−0.593815
$544$	0	0
$545$	14.4796	0.620237
$546$	0	0
$547$	−7.07418	−0.302470	−0.151235	−	0.988498i	$-0.548325\pi$
$547$	−7.07418	−0.302470	−0.151235	+	0.988498i	$0.548325\pi$
$548$	0	0
$549$	12.2170	0.521411
$550$	0	0
$551$	24.9686	1.06370
$552$	0	0
$553$	−14.2330	−0.605250
$554$	0	0
$555$	16.9906	0.721212
$556$	0	0
$557$	−8.13168	−0.344550	−0.172275	−	0.985049i	$-0.555112\pi$
$557$	−8.13168	−0.344550	−0.172275	+	0.985049i	$0.555112\pi$
$558$	0	0
$559$	−12.0427	−0.509354
$560$	0	0
$561$	−7.48074	−0.315837
$562$	0	0
$563$	43.6049	1.83773	0.918864	−	0.394575i	$-0.129108\pi$
$563$	43.6049	1.83773	0.918864	+	0.394575i	$0.129108\pi$
$564$	0	0
$565$	−8.73397	−0.367441
$566$	0	0
$567$	−13.6914	−0.574983
$568$	0	0
$569$	34.9687	1.46596	0.732982	−	0.680248i	$-0.238128\pi$
$569$	34.9687	1.46596	0.732982	+	0.680248i	$0.238128\pi$
$570$	0	0
$571$	−5.39778	−0.225890	−0.112945	−	0.993601i	$-0.536028\pi$
$571$	−5.39778	−0.225890	−0.112945	+	0.993601i	$0.536028\pi$
$572$	0	0
$573$	14.7772	0.617327
$574$	0	0
$575$	−0.341626	−0.0142468
$576$	0	0
$577$	−32.6690	−1.36003	−0.680015	−	0.733198i	$-0.738027\pi$
$577$	−32.6690	−1.36003	−0.680015	+	0.733198i	$0.738027\pi$
$578$	0	0
$579$	−16.8965	−0.702194
$580$	0	0
$581$	1.24944	0.0518355
$582$	0	0
$583$	9.04140	0.374456
$584$	0	0
$585$	4.38263	0.181199
$586$	0	0
$587$	10.7902	0.445360	0.222680	−	0.974892i	$-0.428520\pi$
$587$	10.7902	0.445360	0.222680	+	0.974892i	$0.428520\pi$
$588$	0	0
$589$	12.0572	0.496808
$590$	0	0
$591$	−4.07756	−0.167728
$592$	0	0
$593$	17.5518	0.720766	0.360383	−	0.932804i	$-0.382646\pi$
$593$	17.5518	0.720766	0.360383	+	0.932804i	$0.382646\pi$
$594$	0	0
$595$	5.84891	0.239782
$596$	0	0
$597$	−31.1012	−1.27289
$598$	0	0
$599$	−40.2333	−1.64389	−0.821945	−	0.569567i	$-0.807111\pi$
$599$	−40.2333	−1.64389	−0.821945	+	0.569567i	$0.807111\pi$
$600$	0	0
$601$	−20.2293	−0.825170	−0.412585	−	0.910919i	$-0.635374\pi$
$601$	−20.2293	−0.825170	−0.412585	+	0.910919i	$0.635374\pi$
$602$	0	0
$603$	−11.5225	−0.469231
$604$	0	0
$605$	71.1310	2.89189
$606$	0	0
$607$	9.25572	0.375678	0.187839	−	0.982200i	$-0.439852\pi$
$607$	9.25572	0.375678	0.187839	+	0.982200i	$0.439852\pi$
$608$	0	0
$609$	25.7130	1.04194
$610$	0	0
$611$	−14.2087	−0.574823
$612$	0	0
$613$	−19.1929	−0.775194	−0.387597	−	0.921829i	$-0.626695\pi$
$613$	−19.1929	−0.775194	−0.387597	+	0.921829i	$0.626695\pi$
$614$	0	0
$615$	−36.5817	−1.47512
$616$	0	0
$617$	43.0807	1.73437	0.867183	−	0.497990i	$-0.165928\pi$
$617$	43.0807	1.73437	0.867183	+	0.497990i	$0.165928\pi$
$618$	0	0
$619$	−33.7919	−1.35821	−0.679106	−	0.734040i	$-0.737632\pi$
$619$	−33.7919	−1.35821	−0.679106	+	0.734040i	$0.737632\pi$
$620$	0	0
$621$	−15.0086	−0.602273
$622$	0	0
$623$	41.2416	1.65231
$624$	0	0
$625$	−25.6265	−1.02506
$626$	0	0
$627$	36.9040	1.47380
$628$	0	0
$629$	−4.56314	−0.181944
$630$	0	0
$631$	33.6234	1.33853	0.669264	−	0.743025i	$-0.266610\pi$
$631$	33.6234	1.33853	0.669264	+	0.743025i	$0.266610\pi$
$632$	0	0
$633$	14.5807	0.579531
$634$	0	0
$635$	19.5934	0.777539
$636$	0	0
$637$	−4.43594	−0.175758
$638$	0	0
$639$	−3.55727	−0.140723
$640$	0	0
$641$	−37.0731	−1.46430	−0.732150	−	0.681143i	$-0.761483\pi$
$641$	−37.0731	−1.46430	−0.732150	+	0.681143i	$0.761483\pi$
$642$	0	0
$643$	−1.95265	−0.0770051	−0.0385026	−	0.999259i	$-0.512259\pi$
$643$	−1.95265	−0.0770051	−0.0385026	+	0.999259i	$0.512259\pi$
$644$	0	0
$645$	−21.5232	−0.847476
$646$	0	0
$647$	44.3810	1.74480	0.872398	−	0.488795i	$-0.162564\pi$
$647$	44.3810	1.74480	0.872398	+	0.488795i	$0.162564\pi$
$648$	0	0
$649$	92.3517	3.62512
$650$	0	0
$651$	12.4167	0.486647
$652$	0	0
$653$	−42.7405	−1.67257	−0.836283	−	0.548298i	$-0.815276\pi$
$653$	−42.7405	−1.67257	−0.836283	+	0.548298i	$0.815276\pi$
$654$	0	0
$655$	47.2966	1.84803
$656$	0	0
$657$	4.53560	0.176951
$658$	0	0
$659$	−16.6005	−0.646665	−0.323332	−	0.946285i	$-0.604803\pi$
$659$	−16.6005	−0.646665	−0.323332	+	0.946285i	$0.604803\pi$
$660$	0	0
$661$	−45.8259	−1.78242	−0.891210	−	0.453591i	$-0.850143\pi$
$661$	−45.8259	−1.78242	−0.891210	+	0.453591i	$0.850143\pi$
$662$	0	0
$663$	2.00037	0.0776881
$664$	0	0
$665$	−28.8538	−1.11890
$666$	0	0
$667$	16.0846	0.622799
$668$	0	0
$669$	−2.93093	−0.113316
$670$	0	0
$671$	−71.5912	−2.76375
$672$	0	0
$673$	1.56534	0.0603394	0.0301697	−	0.999545i	$-0.490395\pi$
$673$	1.56534	0.0603394	0.0301697	+	0.999545i	$0.490395\pi$
$674$	0	0
$675$	0.726662	0.0279692
$676$	0	0
$677$	−5.14573	−0.197766	−0.0988832	−	0.995099i	$-0.531527\pi$
$677$	−5.14573	−0.197766	−0.0988832	+	0.995099i	$0.531527\pi$
$678$	0	0
$679$	25.0276	0.960469
$680$	0	0
$681$	−26.3892	−1.01123
$682$	0	0
$683$	13.2673	0.507660	0.253830	−	0.967249i	$-0.418310\pi$
$683$	13.2673	0.507660	0.253830	+	0.967249i	$0.418310\pi$
$684$	0	0
$685$	10.5444	0.402883
$686$	0	0
$687$	2.70719	0.103286
$688$	0	0
$689$	−2.41770	−0.0921070
$690$	0	0
$691$	−24.4580	−0.930426	−0.465213	−	0.885199i	$-0.654022\pi$
$691$	−24.4580	−0.930426	−0.465213	+	0.885199i	$0.654022\pi$
$692$	0	0
$693$	−22.3619	−0.849460
$694$	0	0
$695$	−52.3933	−1.98739
$696$	0	0
$697$	9.82467	0.372136
$698$	0	0
$699$	−27.1590	−1.02725
$700$	0	0
$701$	−37.2663	−1.40753	−0.703764	−	0.710434i	$-0.748499\pi$
$701$	−37.2663	−1.40753	−0.703764	+	0.710434i	$0.748499\pi$
$702$	0	0
$703$	22.5109	0.849014
$704$	0	0
$705$	−25.3943	−0.956406
$706$	0	0
$707$	40.9153	1.53878
$708$	0	0
$709$	23.3565	0.877170	0.438585	−	0.898690i	$-0.355480\pi$
$709$	23.3565	0.877170	0.438585	+	0.898690i	$0.355480\pi$
$710$	0	0
$711$	5.11909	0.191981
$712$	0	0
$713$	7.76717	0.290883
$714$	0	0
$715$	−25.6820	−0.960451
$716$	0	0
$717$	−4.91882	−0.183697
$718$	0	0
$719$	−6.11917	−0.228207	−0.114103	−	0.993469i	$-0.536400\pi$
$719$	−6.11917	−0.228207	−0.114103	+	0.993469i	$0.536400\pi$
$720$	0	0
$721$	−42.1087	−1.56821
$722$	0	0
$723$	11.5625	0.430015
$724$	0	0
$725$	−0.778760	−0.0289224
$726$	0	0
$727$	37.4642	1.38947	0.694736	−	0.719265i	$-0.255521\pi$
$727$	37.4642	1.38947	0.694736	+	0.719265i	$0.255521\pi$
$728$	0	0
$729$	28.2192	1.04516
$730$	0	0
$731$	5.78045	0.213798
$732$	0	0
$733$	−24.4668	−0.903703	−0.451852	−	0.892093i	$-0.649236\pi$
$733$	−24.4668	−0.903703	−0.451852	+	0.892093i	$0.649236\pi$
$734$	0	0
$735$	−7.92807	−0.292431
$736$	0	0
$737$	67.5210	2.48717
$738$	0	0
$739$	−4.26723	−0.156973	−0.0784864	−	0.996915i	$-0.525009\pi$
$739$	−4.26723	−0.156973	−0.0784864	+	0.996915i	$0.525009\pi$
$740$	0	0
$741$	−9.86825	−0.362519
$742$	0	0
$743$	4.79642	0.175963	0.0879817	−	0.996122i	$-0.471958\pi$
$743$	4.79642	0.175963	0.0879817	+	0.996122i	$0.471958\pi$
$744$	0	0
$745$	−23.9830	−0.878668
$746$	0	0
$747$	−0.449378	−0.0164419
$748$	0	0
$749$	41.6539	1.52200
$750$	0	0
$751$	11.5922	0.423006	0.211503	−	0.977377i	$-0.432164\pi$
$751$	11.5922	0.423006	0.211503	+	0.977377i	$0.432164\pi$
$752$	0	0
$753$	26.2124	0.955234
$754$	0	0
$755$	7.42377	0.270179
$756$	0	0
$757$	9.65563	0.350940	0.175470	−	0.984485i	$-0.443856\pi$
$757$	9.65563	0.350940	0.175470	+	0.984485i	$0.443856\pi$
$758$	0	0
$759$	23.7733	0.862917
$760$	0	0
$761$	12.2309	0.443371	0.221685	−	0.975118i	$-0.428844\pi$
$761$	12.2309	0.443371	0.221685	+	0.975118i	$0.428844\pi$
$762$	0	0
$763$	19.7559	0.715213
$764$	0	0
$765$	−2.10364	−0.0760572
$766$	0	0
$767$	−24.6951	−0.891690
$768$	0	0
$769$	2.39336	0.0863069	0.0431535	−	0.999068i	$-0.486260\pi$
$769$	2.39336	0.0863069	0.0431535	+	0.999068i	$0.486260\pi$
$770$	0	0
$771$	17.3026	0.623136
$772$	0	0
$773$	−12.0846	−0.434655	−0.217327	−	0.976099i	$-0.569734\pi$
$773$	−12.0846	−0.434655	−0.217327	+	0.976099i	$0.569734\pi$
$774$	0	0
$775$	−0.376059	−0.0135084
$776$	0	0
$777$	23.1820	0.831650
$778$	0	0
$779$	−48.4670	−1.73651
$780$	0	0
$781$	20.8454	0.745908
$782$	0	0
$783$	−34.2131	−1.22268
$784$	0	0
$785$	14.1338	0.504455
$786$	0	0
$787$	23.8100	0.848733	0.424367	−	0.905490i	$-0.360497\pi$
$787$	23.8100	0.848733	0.424367	+	0.905490i	$0.360497\pi$
$788$	0	0
$789$	41.6463	1.48265
$790$	0	0
$791$	−11.9166	−0.423706
$792$	0	0
$793$	19.1437	0.679814
$794$	0	0
$795$	−4.32100	−0.153250
$796$	0	0
$797$	3.46264	0.122653	0.0613265	−	0.998118i	$-0.480467\pi$
$797$	3.46264	0.122653	0.0613265	+	0.998118i	$0.480467\pi$
$798$	0	0
$799$	6.82011	0.241278
$800$	0	0
$801$	−14.8331	−0.524101
$802$	0	0
$803$	−26.5784	−0.937931
$804$	0	0
$805$	−18.5875	−0.655122
$806$	0	0
$807$	−14.5376	−0.511747
$808$	0	0
$809$	−7.70868	−0.271023	−0.135511	−	0.990776i	$-0.543268\pi$
$809$	−7.70868	−0.271023	−0.135511	+	0.990776i	$0.543268\pi$
$810$	0	0
$811$	−28.2030	−0.990343	−0.495171	−	0.868795i	$-0.664895\pi$
$811$	−28.2030	−0.990343	−0.495171	+	0.868795i	$0.664895\pi$
$812$	0	0
$813$	−25.2759	−0.886463
$814$	0	0
$815$	−34.7578	−1.21751
$816$	0	0
$817$	−28.5161	−0.997653
$818$	0	0
$819$	5.97966	0.208946
$820$	0	0
$821$	50.2320	1.75311	0.876555	−	0.481302i	$-0.159836\pi$
$821$	50.2320	1.75311	0.876555	+	0.481302i	$0.159836\pi$
$822$	0	0
$823$	11.3781	0.396617	0.198309	−	0.980140i	$-0.436455\pi$
$823$	11.3781	0.396617	0.198309	+	0.980140i	$0.436455\pi$
$824$	0	0
$825$	−1.15102	−0.0400733
$826$	0	0
$827$	−9.86810	−0.343148	−0.171574	−	0.985171i	$-0.554885\pi$
$827$	−9.86810	−0.343148	−0.171574	+	0.985171i	$0.554885\pi$
$828$	0	0
$829$	25.9000	0.899543	0.449772	−	0.893144i	$-0.351505\pi$
$829$	25.9000	0.899543	0.449772	+	0.893144i	$0.351505\pi$
$830$	0	0
$831$	21.7883	0.755828
$832$	0	0
$833$	2.12923	0.0737733
$834$	0	0
$835$	−47.6100	−1.64761
$836$	0	0
$837$	−16.5213	−0.571060
$838$	0	0
$839$	−3.04741	−0.105208	−0.0526042	−	0.998615i	$-0.516752\pi$
$839$	−3.04741	−0.105208	−0.0526042	+	0.998615i	$0.516752\pi$
$840$	0	0
$841$	7.66605	0.264346
$842$	0	0
$843$	−29.2585	−1.00771
$844$	0	0
$845$	−22.5729	−0.776532
$846$	0	0
$847$	97.0511	3.33471
$848$	0	0
$849$	−36.2704	−1.24480
$850$	0	0
$851$	14.5014	0.497100
$852$	0	0
$853$	−51.0981	−1.74957	−0.874783	−	0.484514i	$-0.838996\pi$
$853$	−51.0981	−1.74957	−0.874783	+	0.484514i	$0.838996\pi$
$854$	0	0
$855$	10.3777	0.354909
$856$	0	0
$857$	12.5990	0.430375	0.215188	−	0.976573i	$-0.430964\pi$
$857$	12.5990	0.430375	0.215188	+	0.976573i	$0.430964\pi$
$858$	0	0
$859$	−0.851857	−0.0290650	−0.0145325	−	0.999894i	$-0.504626\pi$
$859$	−0.851857	−0.0290650	−0.0145325	+	0.999894i	$0.504626\pi$
$860$	0	0
$861$	−49.9120	−1.70100
$862$	0	0
$863$	12.4587	0.424099	0.212050	−	0.977259i	$-0.431986\pi$
$863$	12.4587	0.424099	0.212050	+	0.977259i	$0.431986\pi$
$864$	0	0
$865$	12.6034	0.428528
$866$	0	0
$867$	22.4028	0.760840
$868$	0	0
$869$	−29.9976	−1.01760
$870$	0	0
$871$	−18.0554	−0.611782
$872$	0	0
$873$	−9.00149	−0.304654
$874$	0	0
$875$	−34.0874	−1.15237
$876$	0	0
$877$	23.0002	0.776663	0.388331	−	0.921520i	$-0.373052\pi$
$877$	23.0002	0.776663	0.388331	+	0.921520i	$0.373052\pi$
$878$	0	0
$879$	6.08524	0.205250
$880$	0	0
$881$	18.5223	0.624032	0.312016	−	0.950077i	$-0.398996\pi$
$881$	18.5223	0.624032	0.312016	+	0.950077i	$0.398996\pi$
$882$	0	0
$883$	51.6961	1.73971	0.869857	−	0.493304i	$-0.164211\pi$
$883$	51.6961	1.73971	0.869857	+	0.493304i	$0.164211\pi$
$884$	0	0
$885$	−44.1361	−1.48362
$886$	0	0
$887$	−34.9542	−1.17365	−0.586824	−	0.809715i	$-0.699622\pi$
$887$	−34.9542	−1.17365	−0.586824	+	0.809715i	$0.699622\pi$
$888$	0	0
$889$	26.7332	0.896602
$890$	0	0
$891$	−28.8560	−0.966712
$892$	0	0
$893$	−33.6450	−1.12589
$894$	0	0
$895$	34.1637	1.14197
$896$	0	0
$897$	−6.35706	−0.212256
$898$	0	0
$899$	17.7058	0.590522
$900$	0	0
$901$	1.16048	0.0386613
$902$	0	0
$903$	−29.3663	−0.977248
$904$	0	0
$905$	22.8019	0.757961
$906$	0	0
$907$	31.6724	1.05167	0.525833	−	0.850588i	$-0.323754\pi$
$907$	31.6724	1.05167	0.525833	+	0.850588i	$0.323754\pi$
$908$	0	0
$909$	−14.7157	−0.488090
$910$	0	0
$911$	34.7406	1.15101	0.575504	−	0.817799i	$-0.304806\pi$
$911$	34.7406	1.15101	0.575504	+	0.817799i	$0.304806\pi$
$912$	0	0
$913$	2.63333	0.0871505
$914$	0	0
$915$	34.2144	1.13109
$916$	0	0
$917$	64.5315	2.13102
$918$	0	0
$919$	23.6950	0.781626	0.390813	−	0.920470i	$-0.372194\pi$
$919$	23.6950	0.781626	0.390813	+	0.920470i	$0.372194\pi$
$920$	0	0
$921$	34.1316	1.12467
$922$	0	0
$923$	−5.57414	−0.183475
$924$	0	0
$925$	−0.702104	−0.0230850
$926$	0	0
$927$	15.1449	0.497425
$928$	0	0
$929$	32.0581	1.05179	0.525896	−	0.850549i	$-0.323730\pi$
$929$	32.0581	1.05179	0.525896	+	0.850549i	$0.323730\pi$
$930$	0	0
$931$	−10.5039	−0.344251
$932$	0	0
$933$	−38.1032	−1.24744
$934$	0	0
$935$	12.3272	0.403143
$936$	0	0
$937$	9.48141	0.309744	0.154872	−	0.987935i	$-0.450503\pi$
$937$	9.48141	0.309744	0.154872	+	0.987935i	$0.450503\pi$
$938$	0	0
$939$	−14.5653	−0.475322
$940$	0	0
$941$	6.10819	0.199121	0.0995607	−	0.995031i	$-0.468256\pi$
$941$	6.10819	0.199121	0.0995607	+	0.995031i	$0.468256\pi$
$942$	0	0
$943$	−31.2222	−1.01673
$944$	0	0
$945$	39.5368	1.28613
$946$	0	0
$947$	27.2564	0.885714	0.442857	−	0.896592i	$-0.353965\pi$
$947$	27.2564	0.885714	0.442857	+	0.896592i	$0.353965\pi$
$948$	0	0
$949$	7.10715	0.230708
$950$	0	0
$951$	0.359080	0.0116440
$952$	0	0
$953$	48.6365	1.57549	0.787745	−	0.616001i	$-0.211248\pi$
$953$	48.6365	1.57549	0.787745	+	0.616001i	$0.211248\pi$
$954$	0	0
$955$	−24.3508	−0.787972
$956$	0	0
$957$	54.1929	1.75181
$958$	0	0
$959$	14.3868	0.464575
$960$	0	0
$961$	−22.4500	−0.724193
$962$	0	0
$963$	−14.9814	−0.482768
$964$	0	0
$965$	27.8430	0.896298
$966$	0	0
$967$	−47.8999	−1.54036	−0.770178	−	0.637829i	$-0.779832\pi$
$967$	−47.8999	−1.54036	−0.770178	+	0.637829i	$0.779832\pi$
$968$	0	0
$969$	4.73670	0.152165
$970$	0	0
$971$	53.3839	1.71317	0.856586	−	0.516004i	$-0.172581\pi$
$971$	53.3839	1.71317	0.856586	+	0.516004i	$0.172581\pi$
$972$	0	0
$973$	−71.4854	−2.29172
$974$	0	0
$975$	0.307786	0.00985705
$976$	0	0
$977$	−15.5126	−0.496293	−0.248147	−	0.968722i	$-0.579821\pi$
$977$	−15.5126	−0.496293	−0.248147	+	0.968722i	$0.579821\pi$
$978$	0	0
$979$	86.9211	2.77801
$980$	0	0
$981$	−7.10548	−0.226860
$982$	0	0
$983$	7.38055	0.235403	0.117701	−	0.993049i	$-0.462447\pi$
$983$	7.38055	0.235403	0.117701	+	0.993049i	$0.462447\pi$
$984$	0	0
$985$	6.71924	0.214093
$986$	0	0
$987$	−34.6480	−1.10286
$988$	0	0
$989$	−18.3699	−0.584129
$990$	0	0
$991$	21.6653	0.688222	0.344111	−	0.938929i	$-0.388180\pi$
$991$	21.6653	0.688222	0.344111	+	0.938929i	$0.388180\pi$
$992$	0	0
$993$	1.10327	0.0350111
$994$	0	0
$995$	51.2504	1.62475
$996$	0	0
$997$	−18.9760	−0.600976	−0.300488	−	0.953786i	$-0.597150\pi$
$997$	−18.9760	−0.600976	−0.300488	+	0.953786i	$0.597150\pi$
$998$	0	0
$999$	−30.8454	−0.975905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.t.1.5		21
4.3	odd	2		2012.2.a.a.1.17	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.a.1.17	✓	21	4.3	odd	2
8048.2.a.t.1.5		21	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-1.37429 q^{3} +2.26464 q^{5} +3.08988 q^{7} -1.11131 q^{9} +O(q^{10})\)	\(q-1.37429 q^{3} +2.26464 q^{5} +3.08988 q^{7} -1.11131 q^{9} +6.51225 q^{11} -1.74140 q^{13} -3.11229 q^{15} +0.835861 q^{17} -4.12347 q^{19} -4.24640 q^{21} -2.65631 q^{23} +0.128609 q^{25} +5.65016 q^{27} -6.05525 q^{29} -2.92404 q^{31} -8.94974 q^{33} +6.99747 q^{35} -5.45921 q^{37} +2.39319 q^{39} +11.7540 q^{41} +6.91557 q^{43} -2.51673 q^{45} +8.15938 q^{47} +2.54735 q^{49} -1.14872 q^{51} +1.38837 q^{53} +14.7479 q^{55} +5.66686 q^{57} +14.1812 q^{59} -10.9933 q^{61} -3.43383 q^{63} -3.94364 q^{65} +10.3683 q^{67} +3.65055 q^{69} +3.20096 q^{71} -4.08129 q^{73} -0.176747 q^{75} +20.1220 q^{77} -4.60634 q^{79} -4.43103 q^{81} +0.404366 q^{83} +1.89293 q^{85} +8.32170 q^{87} +13.3473 q^{89} -5.38070 q^{91} +4.01850 q^{93} -9.33818 q^{95} +8.09985 q^{97} -7.23716 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * q^99

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