LMFDB - Embedded newform 8048.2.a.p.1.10

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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 $ x^10 - 2x^9 - 9x^8 + 14x^7 + 27x^6 - 27x^5 - 34x^4 + 14x^3 + 17x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.07227$ of defining polynomial
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)

$f(q)$	$=$	$q+3.07227 q^{3} +0.386144 q^{5} -0.194914 q^{7} +6.43884 q^{9} +O(q^{10})$	$q+3.07227 q^{3} +0.386144 q^{5} -0.194914 q^{7} +6.43884 q^{9} -2.36326 q^{11} -1.22636 q^{13} +1.18634 q^{15} -5.04830 q^{17} -4.24460 q^{19} -0.598827 q^{21} -1.53457 q^{23} -4.85089 q^{25} +10.5650 q^{27} -7.31602 q^{29} -3.33893 q^{31} -7.26057 q^{33} -0.0752647 q^{35} -2.17138 q^{37} -3.76771 q^{39} -1.04840 q^{41} +1.53067 q^{43} +2.48632 q^{45} +1.08422 q^{47} -6.96201 q^{49} -15.5098 q^{51} +1.55308 q^{53} -0.912558 q^{55} -13.0406 q^{57} -14.8451 q^{59} -5.45056 q^{61} -1.25502 q^{63} -0.473551 q^{65} +11.3202 q^{67} -4.71461 q^{69} +11.4236 q^{71} -0.902229 q^{73} -14.9032 q^{75} +0.460631 q^{77} +12.7431 q^{79} +13.1421 q^{81} -6.10241 q^{83} -1.94937 q^{85} -22.4768 q^{87} +6.44432 q^{89} +0.239034 q^{91} -10.2581 q^{93} -1.63903 q^{95} -12.9886 q^{97} -15.2166 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) $ 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9	$ 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9} + 3 q^{11} - 18 q^{13} + 2 q^{15} - 11 q^{17} + q^{21} + 2 q^{23} - 27 q^{25} + 2 q^{27} - 9 q^{29} + 22 q^{31} - 10 q^{33} + 6 q^{35} - 35 q^{37} - 8 q^{39} - 4 q^{41} + 20 q^{43} + 2 q^{45} - 7 q^{47} - 27 q^{49} - 9 q^{51} - 24 q^{53} + 11 q^{55} - 23 q^{57} - 17 q^{59} - 4 q^{61} - 10 q^{63} - 16 q^{65} + 6 q^{67} - 2 q^{69} + q^{71} - 31 q^{73} - 30 q^{75} + 3 q^{77} + 10 q^{79} - 6 q^{81} - 22 q^{83} - 6 q^{85} - 25 q^{87} + q^{89} - 10 q^{91} - 6 q^{93} - 39 q^{95} - 57 q^{97} - 35 q^{99}+O(q^{100}) $ 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9 + 3 * q^11 - 18 * q^13 + 2 * q^15 - 11 * q^17 + q^21 + 2 * q^23 - 27 * q^25 + 2 * q^27 - 9 * q^29 + 22 * q^31 - 10 * q^33 + 6 * q^35 - 35 * q^37 - 8 * q^39 - 4 * q^41 + 20 * q^43 + 2 * q^45 - 7 * q^47 - 27 * q^49 - 9 * q^51 - 24 * q^53 + 11 * q^55 - 23 * q^57 - 17 * q^59 - 4 * q^61 - 10 * q^63 - 16 * q^65 + 6 * q^67 - 2 * q^69 + q^71 - 31 * q^73 - 30 * q^75 + 3 * q^77 + 10 * q^79 - 6 * q^81 - 22 * q^83 - 6 * q^85 - 25 * q^87 + q^89 - 10 * q^91 - 6 * q^93 - 39 * q^95 - 57 * q^97 - 35 * 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
$3$	3.07227	1.77378	0.886888	−	0.461985i	$-0.152863\pi$
$3$	3.07227	1.77378	0.886888	+	0.461985i	$0.152863\pi$
$4$	0	0
$5$	0.386144	0.172689	0.0863445	−	0.996265i	$-0.472481\pi$
$5$	0.386144	0.172689	0.0863445	+	0.996265i	$0.472481\pi$
$6$	0	0
$7$	−0.194914	−0.0736704	−0.0368352	−	0.999321i	$-0.511728\pi$
$7$	−0.194914	−0.0736704	−0.0368352	+	0.999321i	$0.511728\pi$
$8$	0	0
$9$	6.43884	2.14628
$10$	0	0
$11$	−2.36326	−0.712549	−0.356275	−	0.934381i	$-0.615953\pi$
$11$	−2.36326	−0.712549	−0.356275	+	0.934381i	$0.615953\pi$
$12$	0	0
$13$	−1.22636	−0.340131	−0.170065	−	0.985433i	$-0.554398\pi$
$13$	−1.22636	−0.340131	−0.170065	+	0.985433i	$0.554398\pi$
$14$	0	0
$15$	1.18634	0.306311
$16$	0	0
$17$	−5.04830	−1.22439	−0.612197	−	0.790705i	$-0.709714\pi$
$17$	−5.04830	−1.22439	−0.612197	+	0.790705i	$0.709714\pi$
$18$	0	0
$19$	−4.24460	−0.973779	−0.486890	−	0.873464i	$-0.661869\pi$
$19$	−4.24460	−0.973779	−0.486890	+	0.873464i	$0.661869\pi$
$20$	0	0
$21$	−0.598827	−0.130675
$22$	0	0
$23$	−1.53457	−0.319980	−0.159990	−	0.987119i	$-0.551146\pi$
$23$	−1.53457	−0.319980	−0.159990	+	0.987119i	$0.551146\pi$
$24$	0	0
$25$	−4.85089	−0.970179
$26$	0	0
$27$	10.5650	2.03324
$28$	0	0
$29$	−7.31602	−1.35855	−0.679275	−	0.733883i	$-0.737706\pi$
$29$	−7.31602	−1.35855	−0.679275	+	0.733883i	$0.737706\pi$
$30$	0	0
$31$	−3.33893	−0.599690	−0.299845	−	0.953988i	$-0.596935\pi$
$31$	−3.33893	−0.599690	−0.299845	+	0.953988i	$0.596935\pi$
$32$	0	0
$33$	−7.26057	−1.26390
$34$	0	0
$35$	−0.0752647	−0.0127221
$36$	0	0
$37$	−2.17138	−0.356973	−0.178487	−	0.983942i	$-0.557120\pi$
$37$	−2.17138	−0.356973	−0.178487	+	0.983942i	$0.557120\pi$
$38$	0	0
$39$	−3.76771	−0.603316
$40$	0	0
$41$	−1.04840	−0.163733	−0.0818663	−	0.996643i	$-0.526088\pi$
$41$	−1.04840	−0.163733	−0.0818663	+	0.996643i	$0.526088\pi$
$42$	0	0
$43$	1.53067	0.233425	0.116712	−	0.993166i	$-0.462764\pi$
$43$	1.53067	0.233425	0.116712	+	0.993166i	$0.462764\pi$
$44$	0	0
$45$	2.48632	0.370639
$46$	0	0
$47$	1.08422	0.158150	0.0790748	−	0.996869i	$-0.474803\pi$
$47$	1.08422	0.158150	0.0790748	+	0.996869i	$0.474803\pi$
$48$	0	0
$49$	−6.96201	−0.994573
$50$	0	0
$51$	−15.5098	−2.17180
$52$	0	0
$53$	1.55308	0.213332	0.106666	−	0.994295i	$-0.465983\pi$
$53$	1.55308	0.213332	0.106666	+	0.994295i	$0.465983\pi$
$54$	0	0
$55$	−0.912558	−0.123049
$56$	0	0
$57$	−13.0406	−1.72727
$58$	0	0
$59$	−14.8451	−1.93267	−0.966334	−	0.257293i	$-0.917170\pi$
$59$	−14.8451	−1.93267	−0.966334	+	0.257293i	$0.917170\pi$
$60$	0	0
$61$	−5.45056	−0.697873	−0.348936	−	0.937146i	$-0.613457\pi$
$61$	−5.45056	−0.697873	−0.348936	+	0.937146i	$0.613457\pi$
$62$	0	0
$63$	−1.25502	−0.158117
$64$	0	0
$65$	−0.473551	−0.0587368
$66$	0	0
$67$	11.3202	1.38298	0.691490	−	0.722386i	$-0.256955\pi$
$67$	11.3202	1.38298	0.691490	+	0.722386i	$0.256955\pi$
$68$	0	0
$69$	−4.71461	−0.567572
$70$	0	0
$71$	11.4236	1.35573	0.677864	−	0.735187i	$-0.262906\pi$
$71$	11.4236	1.35573	0.677864	+	0.735187i	$0.262906\pi$
$72$	0	0
$73$	−0.902229	−0.105598	−0.0527990	−	0.998605i	$-0.516814\pi$
$73$	−0.902229	−0.105598	−0.0527990	+	0.998605i	$0.516814\pi$
$74$	0	0
$75$	−14.9032	−1.72088
$76$	0	0
$77$	0.460631	0.0524938
$78$	0	0
$79$	12.7431	1.43371	0.716853	−	0.697225i	$-0.245582\pi$
$79$	12.7431	1.43371	0.716853	+	0.697225i	$0.245582\pi$
$80$	0	0
$81$	13.1421	1.46024
$82$	0	0
$83$	−6.10241	−0.669826	−0.334913	−	0.942249i	$-0.608707\pi$
$83$	−6.10241	−0.669826	−0.334913	+	0.942249i	$0.608707\pi$
$84$	0	0
$85$	−1.94937	−0.211439
$86$	0	0
$87$	−22.4768	−2.40976
$88$	0	0
$89$	6.44432	0.683097	0.341548	−	0.939864i	$-0.389049\pi$
$89$	6.44432	0.683097	0.341548	+	0.939864i	$0.389049\pi$
$90$	0	0
$91$	0.239034	0.0250576
$92$	0	0
$93$	−10.2581	−1.06372
$94$	0	0
$95$	−1.63903	−0.168161
$96$	0	0
$97$	−12.9886	−1.31879	−0.659394	−	0.751797i	$-0.729187\pi$
$97$	−12.9886	−1.31879	−0.659394	+	0.751797i	$0.729187\pi$
$98$	0	0
$99$	−15.2166	−1.52933
$100$	0	0
$101$	6.18139	0.615072	0.307536	−	0.951537i	$-0.400496\pi$
$101$	6.18139	0.615072	0.307536	+	0.951537i	$0.400496\pi$
$102$	0	0
$103$	−7.78250	−0.766833	−0.383416	−	0.923576i	$-0.625253\pi$
$103$	−7.78250	−0.766833	−0.383416	+	0.923576i	$0.625253\pi$
$104$	0	0
$105$	−0.231233	−0.0225661
$106$	0	0
$107$	8.18217	0.791000	0.395500	−	0.918466i	$-0.370571\pi$
$107$	8.18217	0.791000	0.395500	+	0.918466i	$0.370571\pi$
$108$	0	0
$109$	−5.32486	−0.510029	−0.255015	−	0.966937i	$-0.582080\pi$
$109$	−5.32486	−0.510029	−0.255015	+	0.966937i	$0.582080\pi$
$110$	0	0
$111$	−6.67107	−0.633190
$112$	0	0
$113$	−3.60573	−0.339199	−0.169599	−	0.985513i	$-0.554247\pi$
$113$	−3.60573	−0.339199	−0.169599	+	0.985513i	$0.554247\pi$
$114$	0	0
$115$	−0.592565	−0.0552570
$116$	0	0
$117$	−7.89633	−0.730016
$118$	0	0
$119$	0.983983	0.0902016
$120$	0	0
$121$	−5.41501	−0.492274
$122$	0	0
$123$	−3.22097	−0.290425
$124$	0	0
$125$	−3.80386	−0.340228
$126$	0	0
$127$	17.8352	1.58262	0.791308	−	0.611418i	$-0.209401\pi$
$127$	17.8352	1.58262	0.791308	+	0.611418i	$0.209401\pi$
$128$	0	0
$129$	4.70262	0.414043
$130$	0	0
$131$	12.1841	1.06453	0.532267	−	0.846577i	$-0.321340\pi$
$131$	12.1841	1.06453	0.532267	+	0.846577i	$0.321340\pi$
$132$	0	0
$133$	0.827331	0.0717387
$134$	0	0
$135$	4.07963	0.351118
$136$	0	0
$137$	−13.9009	−1.18764	−0.593819	−	0.804599i	$-0.702380\pi$
$137$	−13.9009	−1.18764	−0.593819	+	0.804599i	$0.702380\pi$
$138$	0	0
$139$	−4.99920	−0.424027	−0.212013	−	0.977267i	$-0.568002\pi$
$139$	−4.99920	−0.424027	−0.212013	+	0.977267i	$0.568002\pi$
$140$	0	0
$141$	3.33101	0.280522
$142$	0	0
$143$	2.89820	0.242360
$144$	0	0
$145$	−2.82504	−0.234607
$146$	0	0
$147$	−21.3892	−1.76415
$148$	0	0
$149$	17.4937	1.43314	0.716570	−	0.697515i	$-0.245711\pi$
$149$	17.4937	1.43314	0.716570	+	0.697515i	$0.245711\pi$
$150$	0	0
$151$	−0.480306	−0.0390867	−0.0195434	−	0.999809i	$-0.506221\pi$
$151$	−0.480306	−0.0390867	−0.0195434	+	0.999809i	$0.506221\pi$
$152$	0	0
$153$	−32.5052	−2.62789
$154$	0	0
$155$	−1.28931	−0.103560
$156$	0	0
$157$	0.117389	0.00936869	0.00468434	−	0.999989i	$-0.498509\pi$
$157$	0.117389	0.00936869	0.00468434	+	0.999989i	$0.498509\pi$
$158$	0	0
$159$	4.77147	0.378402
$160$	0	0
$161$	0.299108	0.0235730
$162$	0	0
$163$	12.5273	0.981210	0.490605	−	0.871382i	$-0.336776\pi$
$163$	12.5273	0.981210	0.490605	+	0.871382i	$0.336776\pi$
$164$	0	0
$165$	−2.80362	−0.218262
$166$	0	0
$167$	0.744061	0.0575772	0.0287886	−	0.999586i	$-0.490835\pi$
$167$	0.744061	0.0575772	0.0287886	+	0.999586i	$0.490835\pi$
$168$	0	0
$169$	−11.4960	−0.884311
$170$	0	0
$171$	−27.3303	−2.09000
$172$	0	0
$173$	11.6073	0.882484	0.441242	−	0.897388i	$-0.354538\pi$
$173$	11.6073	0.882484	0.441242	+	0.897388i	$0.354538\pi$
$174$	0	0
$175$	0.945505	0.0714734
$176$	0	0
$177$	−45.6081	−3.42812
$178$	0	0
$179$	21.1974	1.58437	0.792183	−	0.610284i	$-0.208945\pi$
$179$	21.1974	1.58437	0.792183	+	0.610284i	$0.208945\pi$
$180$	0	0
$181$	11.9508	0.888299	0.444150	−	0.895953i	$-0.353506\pi$
$181$	11.9508	0.888299	0.444150	+	0.895953i	$0.353506\pi$
$182$	0	0
$183$	−16.7456	−1.23787
$184$	0	0
$185$	−0.838467	−0.0616453
$186$	0	0
$187$	11.9304	0.872441
$188$	0	0
$189$	−2.05927	−0.149790
$190$	0	0
$191$	−24.2243	−1.75281	−0.876406	−	0.481574i	$-0.840065\pi$
$191$	−24.2243	−1.75281	−0.876406	+	0.481574i	$0.840065\pi$
$192$	0	0
$193$	−5.78545	−0.416446	−0.208223	−	0.978081i	$-0.566768\pi$
$193$	−5.78545	−0.416446	−0.208223	+	0.978081i	$0.566768\pi$
$194$	0	0
$195$	−1.45488	−0.104186
$196$	0	0
$197$	4.06793	0.289828	0.144914	−	0.989444i	$-0.453709\pi$
$197$	4.06793	0.289828	0.144914	+	0.989444i	$0.453709\pi$
$198$	0	0
$199$	11.4204	0.809571	0.404785	−	0.914412i	$-0.367346\pi$
$199$	11.4204	0.809571	0.404785	+	0.914412i	$0.367346\pi$
$200$	0	0
$201$	34.7786	2.45310
$202$	0	0
$203$	1.42599	0.100085
$204$	0	0
$205$	−0.404834	−0.0282748
$206$	0	0
$207$	−9.88084	−0.686766
$208$	0	0
$209$	10.0311	0.693866
$210$	0	0
$211$	−18.0941	−1.24565	−0.622824	−	0.782362i	$-0.714015\pi$
$211$	−18.0941	−1.24565	−0.622824	+	0.782362i	$0.714015\pi$
$212$	0	0
$213$	35.0963	2.40476
$214$	0	0
$215$	0.591058	0.0403098
$216$	0	0
$217$	0.650804	0.0441794
$218$	0	0
$219$	−2.77189	−0.187307
$220$	0	0
$221$	6.19103	0.416454
$222$	0	0
$223$	21.1665	1.41741	0.708706	−	0.705504i	$-0.249279\pi$
$223$	21.1665	1.41741	0.708706	+	0.705504i	$0.249279\pi$
$224$	0	0
$225$	−31.2341	−2.08227
$226$	0	0
$227$	13.3828	0.888248	0.444124	−	0.895965i	$-0.353515\pi$
$227$	13.3828	0.888248	0.444124	+	0.895965i	$0.353515\pi$
$228$	0	0
$229$	−12.5484	−0.829225	−0.414612	−	0.909998i	$-0.636083\pi$
$229$	−12.5484	−0.829225	−0.414612	+	0.909998i	$0.636083\pi$
$230$	0	0
$231$	1.41518	0.0931122
$232$	0	0
$233$	15.6538	1.02551	0.512757	−	0.858534i	$-0.328624\pi$
$233$	15.6538	1.02551	0.512757	+	0.858534i	$0.328624\pi$
$234$	0	0
$235$	0.418665	0.0273107
$236$	0	0
$237$	39.1501	2.54307
$238$	0	0
$239$	−3.45659	−0.223588	−0.111794	−	0.993731i	$-0.535660\pi$
$239$	−3.45659	−0.223588	−0.111794	+	0.993731i	$0.535660\pi$
$240$	0	0
$241$	28.7843	1.85416	0.927080	−	0.374864i	$-0.122311\pi$
$241$	28.7843	1.85416	0.927080	+	0.374864i	$0.122311\pi$
$242$	0	0
$243$	8.68104	0.556889
$244$	0	0
$245$	−2.68834	−0.171752
$246$	0	0
$247$	5.20541	0.331212
$248$	0	0
$249$	−18.7482	−1.18812
$250$	0	0
$251$	1.82824	0.115397	0.0576986	−	0.998334i	$-0.481624\pi$
$251$	1.82824	0.115397	0.0576986	+	0.998334i	$0.481624\pi$
$252$	0	0
$253$	3.62658	0.228001
$254$	0	0
$255$	−5.98900	−0.375046
$256$	0	0
$257$	3.65882	0.228231	0.114116	−	0.993467i	$-0.463597\pi$
$257$	3.65882	0.228231	0.114116	+	0.993467i	$0.463597\pi$
$258$	0	0
$259$	0.423232	0.0262983
$260$	0	0
$261$	−47.1067	−2.91583
$262$	0	0
$263$	13.1581	0.811362	0.405681	−	0.914015i	$-0.367034\pi$
$263$	13.1581	0.811362	0.405681	+	0.914015i	$0.367034\pi$
$264$	0	0
$265$	0.599712	0.0368400
$266$	0	0
$267$	19.7987	1.21166
$268$	0	0
$269$	−30.0225	−1.83050	−0.915251	−	0.402885i	$-0.868008\pi$
$269$	−30.0225	−1.83050	−0.915251	+	0.402885i	$0.868008\pi$
$270$	0	0
$271$	9.04876	0.549673	0.274837	−	0.961491i	$-0.411376\pi$
$271$	9.04876	0.549673	0.274837	+	0.961491i	$0.411376\pi$
$272$	0	0
$273$	0.734377	0.0444465
$274$	0	0
$275$	11.4639	0.691300
$276$	0	0
$277$	10.3482	0.621762	0.310881	−	0.950449i	$-0.399376\pi$
$277$	10.3482	0.621762	0.310881	+	0.950449i	$0.399376\pi$
$278$	0	0
$279$	−21.4989	−1.28710
$280$	0	0
$281$	−19.3686	−1.15543	−0.577716	−	0.816238i	$-0.696056\pi$
$281$	−19.3686	−1.15543	−0.577716	+	0.816238i	$0.696056\pi$
$282$	0	0
$283$	−9.72949	−0.578358	−0.289179	−	0.957275i	$-0.593382\pi$
$283$	−9.72949	−0.578358	−0.289179	+	0.957275i	$0.593382\pi$
$284$	0	0
$285$	−5.03554	−0.298280
$286$	0	0
$287$	0.204347	0.0120622
$288$	0	0
$289$	8.48538	0.499140
$290$	0	0
$291$	−39.9044	−2.33924
$292$	0	0
$293$	26.2321	1.53249	0.766247	−	0.642546i	$-0.222122\pi$
$293$	26.2321	1.53249	0.766247	+	0.642546i	$0.222122\pi$
$294$	0	0
$295$	−5.73235	−0.333750
$296$	0	0
$297$	−24.9679	−1.44879
$298$	0	0
$299$	1.88193	0.108835
$300$	0	0
$301$	−0.298348	−0.0171965
$302$	0	0
$303$	18.9909	1.09100
$304$	0	0
$305$	−2.10470	−0.120515
$306$	0	0
$307$	13.7097	0.782455	0.391228	−	0.920294i	$-0.372050\pi$
$307$	13.7097	0.782455	0.391228	+	0.920294i	$0.372050\pi$
$308$	0	0
$309$	−23.9099	−1.36019
$310$	0	0
$311$	−30.9043	−1.75242	−0.876212	−	0.481926i	$-0.839937\pi$
$311$	−30.9043	−1.75242	−0.876212	+	0.481926i	$0.839937\pi$
$312$	0	0
$313$	−28.2287	−1.59558	−0.797790	−	0.602935i	$-0.793998\pi$
$313$	−28.2287	−1.59558	−0.797790	+	0.602935i	$0.793998\pi$
$314$	0	0
$315$	−0.484617	−0.0273051
$316$	0	0
$317$	−7.55793	−0.424496	−0.212248	−	0.977216i	$-0.568078\pi$
$317$	−7.55793	−0.424496	−0.212248	+	0.977216i	$0.568078\pi$
$318$	0	0
$319$	17.2896	0.968034
$320$	0	0
$321$	25.1378	1.40306
$322$	0	0
$323$	21.4281	1.19229
$324$	0	0
$325$	5.94894	0.329988
$326$	0	0
$327$	−16.3594	−0.904678
$328$	0	0
$329$	−0.211329	−0.0116509
$330$	0	0
$331$	−20.4225	−1.12252	−0.561260	−	0.827640i	$-0.689683\pi$
$331$	−20.4225	−1.12252	−0.561260	+	0.827640i	$0.689683\pi$
$332$	0	0
$333$	−13.9812	−0.766164
$334$	0	0
$335$	4.37122	0.238825
$336$	0	0
$337$	16.7603	0.912994	0.456497	−	0.889725i	$-0.349104\pi$
$337$	16.7603	0.912994	0.456497	+	0.889725i	$0.349104\pi$
$338$	0	0
$339$	−11.0778	−0.601662
$340$	0	0
$341$	7.89077	0.427309
$342$	0	0
$343$	2.72138	0.146941
$344$	0	0
$345$	−1.82052	−0.0980135
$346$	0	0
$347$	−14.1418	−0.759171	−0.379586	−	0.925157i	$-0.623933\pi$
$347$	−14.1418	−0.759171	−0.379586	+	0.925157i	$0.623933\pi$
$348$	0	0
$349$	−22.1523	−1.18579	−0.592893	−	0.805282i	$-0.702014\pi$
$349$	−22.1523	−1.18579	−0.592893	+	0.805282i	$0.702014\pi$
$350$	0	0
$351$	−12.9565	−0.691568
$352$	0	0
$353$	17.2209	0.916574	0.458287	−	0.888804i	$-0.348463\pi$
$353$	17.2209	0.916574	0.458287	+	0.888804i	$0.348463\pi$
$354$	0	0
$355$	4.41114	0.234119
$356$	0	0
$357$	3.02306	0.159997
$358$	0	0
$359$	−21.9672	−1.15938	−0.579692	−	0.814836i	$-0.696827\pi$
$359$	−21.9672	−1.15938	−0.579692	+	0.814836i	$0.696827\pi$
$360$	0	0
$361$	−0.983329	−0.0517541
$362$	0	0
$363$	−16.6364	−0.873183
$364$	0	0
$365$	−0.348390	−0.0182356
$366$	0	0
$367$	13.1841	0.688202	0.344101	−	0.938933i	$-0.388184\pi$
$367$	13.1841	0.688202	0.344101	+	0.938933i	$0.388184\pi$
$368$	0	0
$369$	−6.75048	−0.351416
$370$	0	0
$371$	−0.302716	−0.0157162
$372$	0	0
$373$	−35.2516	−1.82526	−0.912630	−	0.408787i	$-0.865952\pi$
$373$	−35.2516	−1.82526	−0.912630	+	0.408787i	$0.865952\pi$
$374$	0	0
$375$	−11.6865	−0.603488
$376$	0	0
$377$	8.97207	0.462085
$378$	0	0
$379$	−23.6003	−1.21227	−0.606134	−	0.795363i	$-0.707280\pi$
$379$	−23.6003	−1.21227	−0.606134	+	0.795363i	$0.707280\pi$
$380$	0	0
$381$	54.7945	2.80721
$382$	0	0
$383$	13.1311	0.670970	0.335485	−	0.942046i	$-0.391100\pi$
$383$	13.1311	0.670970	0.335485	+	0.942046i	$0.391100\pi$
$384$	0	0
$385$	0.177870	0.00906509
$386$	0	0
$387$	9.85572	0.500994
$388$	0	0
$389$	29.2216	1.48159	0.740797	−	0.671729i	$-0.234448\pi$
$389$	29.2216	1.48159	0.740797	+	0.671729i	$0.234448\pi$
$390$	0	0
$391$	7.74697	0.391781
$392$	0	0
$393$	37.4330	1.88824
$394$	0	0
$395$	4.92065	0.247585
$396$	0	0
$397$	26.2129	1.31559	0.657795	−	0.753197i	$-0.271489\pi$
$397$	26.2129	1.31559	0.657795	+	0.753197i	$0.271489\pi$
$398$	0	0
$399$	2.54178	0.127248
$400$	0	0
$401$	−24.3910	−1.21803	−0.609014	−	0.793160i	$-0.708435\pi$
$401$	−24.3910	−1.21803	−0.609014	+	0.793160i	$0.708435\pi$
$402$	0	0
$403$	4.09473	0.203973
$404$	0	0
$405$	5.07476	0.252167
$406$	0	0
$407$	5.13154	0.254361
$408$	0	0
$409$	13.2071	0.653051	0.326526	−	0.945188i	$-0.394122\pi$
$409$	13.2071	0.653051	0.326526	+	0.945188i	$0.394122\pi$
$410$	0	0
$411$	−42.7074	−2.10660
$412$	0	0
$413$	2.89351	0.142380
$414$	0	0
$415$	−2.35641	−0.115672
$416$	0	0
$417$	−15.3589	−0.752129
$418$	0	0
$419$	−15.8597	−0.774799	−0.387400	−	0.921912i	$-0.626627\pi$
$419$	−15.8597	−0.774799	−0.387400	+	0.921912i	$0.626627\pi$
$420$	0	0
$421$	−4.53957	−0.221245	−0.110622	−	0.993863i	$-0.535284\pi$
$421$	−4.53957	−0.221245	−0.110622	+	0.993863i	$0.535284\pi$
$422$	0	0
$423$	6.98111	0.339433
$424$	0	0
$425$	24.4888	1.18788
$426$	0	0
$427$	1.06239	0.0514126
$428$	0	0
$429$	8.90406	0.429892
$430$	0	0
$431$	24.0460	1.15825	0.579126	−	0.815238i	$-0.303394\pi$
$431$	24.0460	1.15825	0.579126	+	0.815238i	$0.303394\pi$
$432$	0	0
$433$	−20.8069	−0.999914	−0.499957	−	0.866050i	$-0.666651\pi$
$433$	−20.8069	−0.999914	−0.499957	+	0.866050i	$0.666651\pi$
$434$	0	0
$435$	−8.67928	−0.416140
$436$	0	0
$437$	6.51364	0.311590
$438$	0	0
$439$	27.0822	1.29256	0.646282	−	0.763098i	$-0.276323\pi$
$439$	27.0822	1.29256	0.646282	+	0.763098i	$0.276323\pi$
$440$	0	0
$441$	−44.8273	−2.13463
$442$	0	0
$443$	−26.1108	−1.24056	−0.620282	−	0.784379i	$-0.712982\pi$
$443$	−26.1108	−1.24056	−0.620282	+	0.784379i	$0.712982\pi$
$444$	0	0
$445$	2.48844	0.117963
$446$	0	0
$447$	53.7454	2.54207
$448$	0	0
$449$	18.4224	0.869407	0.434703	−	0.900574i	$-0.356853\pi$
$449$	18.4224	0.869407	0.434703	+	0.900574i	$0.356853\pi$
$450$	0	0
$451$	2.47764	0.116668
$452$	0	0
$453$	−1.47563	−0.0693311
$454$	0	0
$455$	0.0923016	0.00432716
$456$	0	0
$457$	−3.55147	−0.166131	−0.0830654	−	0.996544i	$-0.526471\pi$
$457$	−3.55147	−0.166131	−0.0830654	+	0.996544i	$0.526471\pi$
$458$	0	0
$459$	−53.3355	−2.48949
$460$	0	0
$461$	−2.86665	−0.133513	−0.0667565	−	0.997769i	$-0.521265\pi$
$461$	−2.86665	−0.133513	−0.0667565	+	0.997769i	$0.521265\pi$
$462$	0	0
$463$	−30.5287	−1.41879	−0.709395	−	0.704811i	$-0.751032\pi$
$463$	−30.5287	−1.41879	−0.709395	+	0.704811i	$0.751032\pi$
$464$	0	0
$465$	−3.96111	−0.183692
$466$	0	0
$467$	−18.7000	−0.865332	−0.432666	−	0.901554i	$-0.642427\pi$
$467$	−18.7000	−0.865332	−0.432666	+	0.901554i	$0.642427\pi$
$468$	0	0
$469$	−2.20646	−0.101885
$470$	0	0
$471$	0.360652	0.0166179
$472$	0	0
$473$	−3.61736	−0.166326
$474$	0	0
$475$	20.5901	0.944740
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	−7.50976	−0.343130	−0.171565	−	0.985173i	$-0.554882\pi$
$479$	−7.50976	−0.343130	−0.171565	+	0.985173i	$0.554882\pi$
$480$	0	0
$481$	2.66289	0.121418
$482$	0	0
$483$	0.918941	0.0418133
$484$	0	0
$485$	−5.01546	−0.227740
$486$	0	0
$487$	−32.6786	−1.48081	−0.740404	−	0.672162i	$-0.765366\pi$
$487$	−32.6786	−1.48081	−0.740404	+	0.672162i	$0.765366\pi$
$488$	0	0
$489$	38.4871	1.74045
$490$	0	0
$491$	−2.81716	−0.127137	−0.0635683	−	0.997977i	$-0.520248\pi$
$491$	−2.81716	−0.127137	−0.0635683	+	0.997977i	$0.520248\pi$
$492$	0	0
$493$	36.9335	1.66340
$494$	0	0
$495$	−5.87582	−0.264098
$496$	0	0
$497$	−2.22661	−0.0998770
$498$	0	0
$499$	−29.0591	−1.30086	−0.650432	−	0.759564i	$-0.725412\pi$
$499$	−29.0591	−1.30086	−0.650432	+	0.759564i	$0.725412\pi$
$500$	0	0
$501$	2.28596	0.102129
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	2.38691	0.106216
$506$	0	0
$507$	−35.3189	−1.56857
$508$	0	0
$509$	7.58042	0.335996	0.167998	−	0.985787i	$-0.446270\pi$
$509$	7.58042	0.335996	0.167998	+	0.985787i	$0.446270\pi$
$510$	0	0
$511$	0.175857	0.00777944
$512$	0	0
$513$	−44.8444	−1.97993
$514$	0	0
$515$	−3.00517	−0.132423
$516$	0	0
$517$	−2.56229	−0.112689
$518$	0	0
$519$	35.6607	1.56533
$520$	0	0
$521$	−8.90148	−0.389981	−0.194990	−	0.980805i	$-0.562468\pi$
$521$	−8.90148	−0.389981	−0.194990	+	0.980805i	$0.562468\pi$
$522$	0	0
$523$	−37.7311	−1.64986	−0.824932	−	0.565232i	$-0.808787\pi$
$523$	−37.7311	−1.64986	−0.824932	+	0.565232i	$0.808787\pi$
$524$	0	0
$525$	2.90484	0.126778
$526$	0	0
$527$	16.8560	0.734257
$528$	0	0
$529$	−20.6451	−0.897613
$530$	0	0
$531$	−95.5852	−4.14804
$532$	0	0
$533$	1.28572	0.0556905
$534$	0	0
$535$	3.15950	0.136597
$536$	0	0
$537$	65.1240	2.81031
$538$	0	0
$539$	16.4530	0.708682
$540$	0	0
$541$	11.6293	0.499981	0.249990	−	0.968248i	$-0.419573\pi$
$541$	11.6293	0.499981	0.249990	+	0.968248i	$0.419573\pi$
$542$	0	0
$543$	36.7162	1.57564
$544$	0	0
$545$	−2.05616	−0.0880764
$546$	0	0
$547$	13.4958	0.577040	0.288520	−	0.957474i	$-0.406837\pi$
$547$	13.4958	0.577040	0.288520	+	0.957474i	$0.406837\pi$
$548$	0	0
$549$	−35.0953	−1.49783
$550$	0	0
$551$	31.0536	1.32293
$552$	0	0
$553$	−2.48379	−0.105622
$554$	0	0
$555$	−2.57599	−0.109345
$556$	0	0
$557$	−34.2002	−1.44911	−0.724554	−	0.689218i	$-0.757954\pi$
$557$	−34.2002	−1.44911	−0.724554	+	0.689218i	$0.757954\pi$
$558$	0	0
$559$	−1.87715	−0.0793949
$560$	0	0
$561$	36.6536	1.54751
$562$	0	0
$563$	−25.6005	−1.07893	−0.539467	−	0.842007i	$-0.681374\pi$
$563$	−25.6005	−1.07893	−0.539467	+	0.842007i	$0.681374\pi$
$564$	0	0
$565$	−1.39233	−0.0585758
$566$	0	0
$567$	−2.56158	−0.107576
$568$	0	0
$569$	−17.2311	−0.722363	−0.361182	−	0.932495i	$-0.617627\pi$
$569$	−17.2311	−0.722363	−0.361182	+	0.932495i	$0.617627\pi$
$570$	0	0
$571$	−2.79325	−0.116894	−0.0584470	−	0.998291i	$-0.518615\pi$
$571$	−2.79325	−0.116894	−0.0584470	+	0.998291i	$0.518615\pi$
$572$	0	0
$573$	−74.4237	−3.10909
$574$	0	0
$575$	7.44403	0.310438
$576$	0	0
$577$	−40.1143	−1.66998	−0.834991	−	0.550264i	$-0.814527\pi$
$577$	−40.1143	−1.66998	−0.834991	+	0.550264i	$0.814527\pi$
$578$	0	0
$579$	−17.7745	−0.738682
$580$	0	0
$581$	1.18944	0.0493464
$582$	0	0
$583$	−3.67032	−0.152009
$584$	0	0
$585$	−3.04912	−0.126066
$586$	0	0
$587$	38.5445	1.59090	0.795451	−	0.606018i	$-0.207234\pi$
$587$	38.5445	1.59090	0.795451	+	0.606018i	$0.207234\pi$
$588$	0	0
$589$	14.1725	0.583966
$590$	0	0
$591$	12.4978	0.514090
$592$	0	0
$593$	−25.0372	−1.02816	−0.514078	−	0.857744i	$-0.671866\pi$
$593$	−25.0372	−1.02816	−0.514078	+	0.857744i	$0.671866\pi$
$594$	0	0
$595$	0.379959	0.0155768
$596$	0	0
$597$	35.0865	1.43600
$598$	0	0
$599$	23.0220	0.940655	0.470328	−	0.882492i	$-0.344136\pi$
$599$	23.0220	0.940655	0.470328	+	0.882492i	$0.344136\pi$
$600$	0	0
$601$	−33.9451	−1.38465	−0.692326	−	0.721585i	$-0.743414\pi$
$601$	−33.9451	−1.38465	−0.692326	+	0.721585i	$0.743414\pi$
$602$	0	0
$603$	72.8888	2.96826
$604$	0	0
$605$	−2.09097	−0.0850102
$606$	0	0
$607$	−24.3039	−0.986467	−0.493233	−	0.869897i	$-0.664185\pi$
$607$	−24.3039	−0.986467	−0.493233	+	0.869897i	$0.664185\pi$
$608$	0	0
$609$	4.38103	0.177528
$610$	0	0
$611$	−1.32964	−0.0537916
$612$	0	0
$613$	34.5604	1.39588	0.697940	−	0.716156i	$-0.254100\pi$
$613$	34.5604	1.39588	0.697940	+	0.716156i	$0.254100\pi$
$614$	0	0
$615$	−1.24376	−0.0501532
$616$	0	0
$617$	34.3607	1.38331	0.691655	−	0.722228i	$-0.256882\pi$
$617$	34.3607	1.38331	0.691655	+	0.722228i	$0.256882\pi$
$618$	0	0
$619$	−1.08451	−0.0435902	−0.0217951	−	0.999762i	$-0.506938\pi$
$619$	−1.08451	−0.0435902	−0.0217951	+	0.999762i	$0.506938\pi$
$620$	0	0
$621$	−16.2128	−0.650597
$622$	0	0
$623$	−1.25609	−0.0503240
$624$	0	0
$625$	22.7856	0.911425
$626$	0	0
$627$	30.8182	1.23076
$628$	0	0
$629$	10.9618	0.437076
$630$	0	0
$631$	6.03426	0.240220	0.120110	−	0.992761i	$-0.461675\pi$
$631$	6.03426	0.240220	0.120110	+	0.992761i	$0.461675\pi$
$632$	0	0
$633$	−55.5899	−2.20950
$634$	0	0
$635$	6.88695	0.273300
$636$	0	0
$637$	8.53792	0.338285
$638$	0	0
$639$	73.5545	2.90977
$640$	0	0
$641$	−22.0716	−0.871778	−0.435889	−	0.900001i	$-0.643566\pi$
$641$	−22.0716	−0.871778	−0.435889	+	0.900001i	$0.643566\pi$
$642$	0	0
$643$	40.6373	1.60258	0.801290	−	0.598277i	$-0.204148\pi$
$643$	40.6373	1.60258	0.801290	+	0.598277i	$0.204148\pi$
$644$	0	0
$645$	1.81589	0.0715006
$646$	0	0
$647$	−22.9719	−0.903118	−0.451559	−	0.892241i	$-0.649132\pi$
$647$	−22.9719	−0.903118	−0.451559	+	0.892241i	$0.649132\pi$
$648$	0	0
$649$	35.0828	1.37712
$650$	0	0
$651$	1.99944	0.0783644
$652$	0	0
$653$	−39.0459	−1.52798	−0.763992	−	0.645226i	$-0.776763\pi$
$653$	−39.0459	−1.52798	−0.763992	+	0.645226i	$0.776763\pi$
$654$	0	0
$655$	4.70483	0.183833
$656$	0	0
$657$	−5.80931	−0.226643
$658$	0	0
$659$	26.6054	1.03640	0.518199	−	0.855260i	$-0.326603\pi$
$659$	26.6054	1.03640	0.518199	+	0.855260i	$0.326603\pi$
$660$	0	0
$661$	−39.3661	−1.53116	−0.765582	−	0.643338i	$-0.777549\pi$
$661$	−39.3661	−1.53116	−0.765582	+	0.643338i	$0.777549\pi$
$662$	0	0
$663$	19.0205	0.738696
$664$	0	0
$665$	0.319469	0.0123885
$666$	0	0
$667$	11.2269	0.434709
$668$	0	0
$669$	65.0291	2.51417
$670$	0	0
$671$	12.8811	0.497269
$672$	0	0
$673$	−13.4145	−0.517090	−0.258545	−	0.965999i	$-0.583243\pi$
$673$	−13.4145	−0.517090	−0.258545	+	0.965999i	$0.583243\pi$
$674$	0	0
$675$	−51.2499	−1.97261
$676$	0	0
$677$	−2.13435	−0.0820299	−0.0410149	−	0.999159i	$-0.513059\pi$
$677$	−2.13435	−0.0820299	−0.0410149	+	0.999159i	$0.513059\pi$
$678$	0	0
$679$	2.53165	0.0971557
$680$	0	0
$681$	41.1156	1.57555
$682$	0	0
$683$	−21.3544	−0.817105	−0.408552	−	0.912735i	$-0.633966\pi$
$683$	−21.3544	−0.817105	−0.408552	+	0.912735i	$0.633966\pi$
$684$	0	0
$685$	−5.36777	−0.205092
$686$	0	0
$687$	−38.5522	−1.47086
$688$	0	0
$689$	−1.90463	−0.0725606
$690$	0	0
$691$	−13.8167	−0.525614	−0.262807	−	0.964848i	$-0.584648\pi$
$691$	−13.8167	−0.525614	−0.262807	+	0.964848i	$0.584648\pi$
$692$	0	0
$693$	2.96593	0.112666
$694$	0	0
$695$	−1.93041	−0.0732247
$696$	0	0
$697$	5.29265	0.200473
$698$	0	0
$699$	48.0927	1.81903
$700$	0	0
$701$	45.1209	1.70419	0.852097	−	0.523385i	$-0.175331\pi$
$701$	45.1209	1.70419	0.852097	+	0.523385i	$0.175331\pi$
$702$	0	0
$703$	9.21666	0.347613
$704$	0	0
$705$	1.28625	0.0484430
$706$	0	0
$707$	−1.20484	−0.0453126
$708$	0	0
$709$	−37.6350	−1.41341	−0.706706	−	0.707507i	$-0.749820\pi$
$709$	−37.6350	−1.41341	−0.706706	+	0.707507i	$0.749820\pi$
$710$	0	0
$711$	82.0505	3.07713
$712$	0	0
$713$	5.12383	0.191889
$714$	0	0
$715$	1.11912	0.0418529
$716$	0	0
$717$	−10.6196	−0.396596
$718$	0	0
$719$	34.9860	1.30476	0.652380	−	0.757892i	$-0.273771\pi$
$719$	34.9860	1.30476	0.652380	+	0.757892i	$0.273771\pi$
$720$	0	0
$721$	1.51691	0.0564929
$722$	0	0
$723$	88.4331	3.28886
$724$	0	0
$725$	35.4892	1.31804
$726$	0	0
$727$	−0.0594046	−0.00220319	−0.00110160	−	0.999999i	$-0.500351\pi$
$727$	−0.0594046	−0.00220319	−0.00110160	+	0.999999i	$0.500351\pi$
$728$	0	0
$729$	−12.7559	−0.472440
$730$	0	0
$731$	−7.72727	−0.285804
$732$	0	0
$733$	−23.6914	−0.875060	−0.437530	−	0.899204i	$-0.644147\pi$
$733$	−23.6914	−0.875060	−0.437530	+	0.899204i	$0.644147\pi$
$734$	0	0
$735$	−8.25930	−0.304649
$736$	0	0
$737$	−26.7525	−0.985441
$738$	0	0
$739$	13.2539	0.487551	0.243776	−	0.969832i	$-0.421614\pi$
$739$	13.2539	0.487551	0.243776	+	0.969832i	$0.421614\pi$
$740$	0	0
$741$	15.9924	0.587496
$742$	0	0
$743$	13.7983	0.506210	0.253105	−	0.967439i	$-0.418548\pi$
$743$	13.7983	0.506210	0.253105	+	0.967439i	$0.418548\pi$
$744$	0	0
$745$	6.75509	0.247487
$746$	0	0
$747$	−39.2924	−1.43763
$748$	0	0
$749$	−1.59481	−0.0582733
$750$	0	0
$751$	−3.82043	−0.139410	−0.0697048	−	0.997568i	$-0.522206\pi$
$751$	−3.82043	−0.139410	−0.0697048	+	0.997568i	$0.522206\pi$
$752$	0	0
$753$	5.61683	0.204689
$754$	0	0
$755$	−0.185467	−0.00674984
$756$	0	0
$757$	19.6215	0.713157	0.356578	−	0.934265i	$-0.383943\pi$
$757$	19.6215	0.713157	0.356578	+	0.934265i	$0.383943\pi$
$758$	0	0
$759$	11.1418	0.404423
$760$	0	0
$761$	47.5141	1.72238	0.861192	−	0.508280i	$-0.169719\pi$
$761$	47.5141	1.72238	0.861192	+	0.508280i	$0.169719\pi$
$762$	0	0
$763$	1.03789	0.0375741
$764$	0	0
$765$	−12.5517	−0.453808
$766$	0	0
$767$	18.2054	0.657360
$768$	0	0
$769$	−10.0139	−0.361111	−0.180556	−	0.983565i	$-0.557790\pi$
$769$	−10.0139	−0.361111	−0.180556	+	0.983565i	$0.557790\pi$
$770$	0	0
$771$	11.2409	0.404831
$772$	0	0
$773$	−35.8956	−1.29108	−0.645538	−	0.763728i	$-0.723367\pi$
$773$	−35.8956	−1.29108	−0.645538	+	0.763728i	$0.723367\pi$
$774$	0	0
$775$	16.1968	0.581807
$776$	0	0
$777$	1.30028	0.0466474
$778$	0	0
$779$	4.45005	0.159439
$780$	0	0
$781$	−26.9968	−0.966023
$782$	0	0
$783$	−77.2940	−2.76226
$784$	0	0
$785$	0.0453292	0.00161787
$786$	0	0
$787$	−1.89211	−0.0674464	−0.0337232	−	0.999431i	$-0.510736\pi$
$787$	−1.89211	−0.0674464	−0.0337232	+	0.999431i	$0.510736\pi$
$788$	0	0
$789$	40.4252	1.43917
$790$	0	0
$791$	0.702806	0.0249889
$792$	0	0
$793$	6.68434	0.237368
$794$	0	0
$795$	1.84248	0.0653459
$796$	0	0
$797$	24.4028	0.864393	0.432197	−	0.901779i	$-0.357739\pi$
$797$	24.4028	0.864393	0.432197	+	0.901779i	$0.357739\pi$
$798$	0	0
$799$	−5.47347	−0.193637
$800$	0	0
$801$	41.4940	1.46612
$802$	0	0
$803$	2.13220	0.0752437
$804$	0	0
$805$	0.115499	0.00407080
$806$	0	0
$807$	−92.2371	−3.24690
$808$	0	0
$809$	−32.9341	−1.15790	−0.578951	−	0.815363i	$-0.696538\pi$
$809$	−32.9341	−1.15790	−0.578951	+	0.815363i	$0.696538\pi$
$810$	0	0
$811$	42.8145	1.50342	0.751711	−	0.659493i	$-0.229229\pi$
$811$	42.8145	1.50342	0.751711	+	0.659493i	$0.229229\pi$
$812$	0	0
$813$	27.8002	0.974997
$814$	0	0
$815$	4.83732	0.169444
$816$	0	0
$817$	−6.49708	−0.227304
$818$	0	0
$819$	1.53910	0.0537805
$820$	0	0
$821$	−28.2632	−0.986393	−0.493196	−	0.869918i	$-0.664172\pi$
$821$	−28.2632	−0.986393	−0.493196	+	0.869918i	$0.664172\pi$
$822$	0	0
$823$	−10.4127	−0.362962	−0.181481	−	0.983394i	$-0.558089\pi$
$823$	−10.4127	−0.362962	−0.181481	+	0.983394i	$0.558089\pi$
$824$	0	0
$825$	35.2202	1.22621
$826$	0	0
$827$	0.0841446	0.00292600	0.00146300	−	0.999999i	$-0.499534\pi$
$827$	0.0841446	0.00292600	0.00146300	+	0.999999i	$0.499534\pi$
$828$	0	0
$829$	46.3470	1.60970	0.804850	−	0.593479i	$-0.202246\pi$
$829$	46.3470	1.60970	0.804850	+	0.593479i	$0.202246\pi$
$830$	0	0
$831$	31.7924	1.10287
$832$	0	0
$833$	35.1463	1.21775
$834$	0	0
$835$	0.287315	0.00994294
$836$	0	0
$837$	−35.2760	−1.21932
$838$	0	0
$839$	13.9611	0.481990	0.240995	−	0.970526i	$-0.422526\pi$
$839$	13.9611	0.481990	0.240995	+	0.970526i	$0.422526\pi$
$840$	0	0
$841$	24.5241	0.845660
$842$	0	0
$843$	−59.5054	−2.04948
$844$	0	0
$845$	−4.43913	−0.152711
$846$	0	0
$847$	1.05546	0.0362660
$848$	0	0
$849$	−29.8916	−1.02588
$850$	0	0
$851$	3.33214	0.114224
$852$	0	0
$853$	−54.5527	−1.86785	−0.933924	−	0.357472i	$-0.883639\pi$
$853$	−54.5527	−1.86785	−0.933924	+	0.357472i	$0.883639\pi$
$854$	0	0
$855$	−10.5534	−0.360920
$856$	0	0
$857$	27.6798	0.945525	0.472762	−	0.881190i	$-0.343257\pi$
$857$	27.6798	0.945525	0.472762	+	0.881190i	$0.343257\pi$
$858$	0	0
$859$	−27.1497	−0.926336	−0.463168	−	0.886270i	$-0.653287\pi$
$859$	−27.1497	−0.926336	−0.463168	+	0.886270i	$0.653287\pi$
$860$	0	0
$861$	0.627810	0.0213957
$862$	0	0
$863$	−25.9641	−0.883827	−0.441913	−	0.897058i	$-0.645700\pi$
$863$	−25.9641	−0.883827	−0.441913	+	0.897058i	$0.645700\pi$
$864$	0	0
$865$	4.48208	0.152395
$866$	0	0
$867$	26.0694	0.885363
$868$	0	0
$869$	−30.1151	−1.02159
$870$	0	0
$871$	−13.8826	−0.470394
$872$	0	0
$873$	−83.6313	−2.83049
$874$	0	0
$875$	0.741425	0.0250647
$876$	0	0
$877$	37.2176	1.25675	0.628374	−	0.777911i	$-0.283721\pi$
$877$	37.2176	1.25675	0.628374	+	0.777911i	$0.283721\pi$
$878$	0	0
$879$	80.5920	2.71830
$880$	0	0
$881$	−6.44185	−0.217031	−0.108516	−	0.994095i	$-0.534610\pi$
$881$	−6.44185	−0.217031	−0.108516	+	0.994095i	$0.534610\pi$
$882$	0	0
$883$	−45.9727	−1.54711	−0.773553	−	0.633732i	$-0.781522\pi$
$883$	−45.9727	−1.54711	−0.773553	+	0.633732i	$0.781522\pi$
$884$	0	0
$885$	−17.6113	−0.591998
$886$	0	0
$887$	28.3187	0.950849	0.475425	−	0.879756i	$-0.342294\pi$
$887$	28.3187	0.950849	0.475425	+	0.879756i	$0.342294\pi$
$888$	0	0
$889$	−3.47632	−0.116592
$890$	0	0
$891$	−31.0582	−1.04049
$892$	0	0
$893$	−4.60208	−0.154003
$894$	0	0
$895$	8.18524	0.273602
$896$	0	0
$897$	5.78181	0.193049
$898$	0	0
$899$	24.4277	0.814710
$900$	0	0
$901$	−7.84041	−0.261202
$902$	0	0
$903$	−0.916604	−0.0305027
$904$	0	0
$905$	4.61475	0.153399
$906$	0	0
$907$	12.5099	0.415384	0.207692	−	0.978194i	$-0.433405\pi$
$907$	12.5099	0.415384	0.207692	+	0.978194i	$0.433405\pi$
$908$	0	0
$909$	39.8010	1.32012
$910$	0	0
$911$	−2.41940	−0.0801584	−0.0400792	−	0.999197i	$-0.512761\pi$
$911$	−2.41940	−0.0801584	−0.0400792	+	0.999197i	$0.512761\pi$
$912$	0	0
$913$	14.4216	0.477284
$914$	0	0
$915$	−6.46621	−0.213766
$916$	0	0
$917$	−2.37485	−0.0784246
$918$	0	0
$919$	−50.3462	−1.66077	−0.830384	−	0.557192i	$-0.811879\pi$
$919$	−50.3462	−1.66077	−0.830384	+	0.557192i	$0.811879\pi$
$920$	0	0
$921$	42.1200	1.38790
$922$	0	0
$923$	−14.0094	−0.461125
$924$	0	0
$925$	10.5331	0.346328
$926$	0	0
$927$	−50.1103	−1.64584
$928$	0	0
$929$	−32.9858	−1.08223	−0.541115	−	0.840948i	$-0.681998\pi$
$929$	−32.9858	−1.08223	−0.541115	+	0.840948i	$0.681998\pi$
$930$	0	0
$931$	29.5510	0.968494
$932$	0	0
$933$	−94.9464	−3.10841
$934$	0	0
$935$	4.60687	0.150661
$936$	0	0
$937$	47.9023	1.56490	0.782450	−	0.622714i	$-0.213970\pi$
$937$	47.9023	1.56490	0.782450	+	0.622714i	$0.213970\pi$
$938$	0	0
$939$	−86.7262	−2.83020
$940$	0	0
$941$	−4.42362	−0.144206	−0.0721030	−	0.997397i	$-0.522971\pi$
$941$	−4.42362	−0.144206	−0.0721030	+	0.997397i	$0.522971\pi$
$942$	0	0
$943$	1.60884	0.0523912
$944$	0	0
$945$	−0.795175	−0.0258670
$946$	0	0
$947$	−19.3253	−0.627987	−0.313994	−	0.949425i	$-0.601667\pi$
$947$	−19.3253	−0.627987	−0.313994	+	0.949425i	$0.601667\pi$
$948$	0	0
$949$	1.10646	0.0359171
$950$	0	0
$951$	−23.2200	−0.752960
$952$	0	0
$953$	30.0968	0.974931	0.487465	−	0.873142i	$-0.337922\pi$
$953$	30.0968	0.974931	0.487465	+	0.873142i	$0.337922\pi$
$954$	0	0
$955$	−9.35408	−0.302691
$956$	0	0
$957$	53.1184	1.71708
$958$	0	0
$959$	2.70948	0.0874937
$960$	0	0
$961$	−19.8515	−0.640371
$962$	0	0
$963$	52.6836	1.69771
$964$	0	0
$965$	−2.23402	−0.0719156
$966$	0	0
$967$	42.6607	1.37187	0.685937	−	0.727661i	$-0.259392\pi$
$967$	42.6607	1.37187	0.685937	+	0.727661i	$0.259392\pi$
$968$	0	0
$969$	65.8328	2.11485
$970$	0	0
$971$	−33.0050	−1.05918	−0.529591	−	0.848253i	$-0.677655\pi$
$971$	−33.0050	−1.05918	−0.529591	+	0.848253i	$0.677655\pi$
$972$	0	0
$973$	0.974412	0.0312382
$974$	0	0
$975$	18.2767	0.585324
$976$	0	0
$977$	−5.79949	−0.185542	−0.0927710	−	0.995687i	$-0.529572\pi$
$977$	−5.79949	−0.185542	−0.0927710	+	0.995687i	$0.529572\pi$
$978$	0	0
$979$	−15.2296	−0.486740
$980$	0	0
$981$	−34.2859	−1.09467
$982$	0	0
$983$	35.2304	1.12367	0.561837	−	0.827248i	$-0.310095\pi$
$983$	35.2304	1.12367	0.561837	+	0.827248i	$0.310095\pi$
$984$	0	0
$985$	1.57081	0.0500501
$986$	0	0
$987$	−0.649260	−0.0206662
$988$	0	0
$989$	−2.34891	−0.0746911
$990$	0	0
$991$	−9.08023	−0.288443	−0.144221	−	0.989545i	$-0.546068\pi$
$991$	−9.08023	−0.288443	−0.144221	+	0.989545i	$0.546068\pi$
$992$	0	0
$993$	−62.7433	−1.99110
$994$	0	0
$995$	4.40992	0.139804
$996$	0	0
$997$	59.6440	1.88894	0.944472	−	0.328591i	$-0.106574\pi$
$997$	59.6440	1.88894	0.944472	+	0.328591i	$0.106574\pi$
$998$	0	0
$999$	−22.9407	−0.725813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.p.1.10		10
4.3	odd	2		503.2.a.e.1.5	✓	10
12.11	even	2		4527.2.a.k.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.5	✓	10	4.3	odd	2
4527.2.a.k.1.6		10	12.11	even	2
8048.2.a.p.1.10		10	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)

\(f(q)\)	\(=\)	\(q+3.07227 q^{3} +0.386144 q^{5} -0.194914 q^{7} +6.43884 q^{9} +O(q^{10})\)	\(q+3.07227 q^{3} +0.386144 q^{5} -0.194914 q^{7} +6.43884 q^{9} -2.36326 q^{11} -1.22636 q^{13} +1.18634 q^{15} -5.04830 q^{17} -4.24460 q^{19} -0.598827 q^{21} -1.53457 q^{23} -4.85089 q^{25} +10.5650 q^{27} -7.31602 q^{29} -3.33893 q^{31} -7.26057 q^{33} -0.0752647 q^{35} -2.17138 q^{37} -3.76771 q^{39} -1.04840 q^{41} +1.53067 q^{43} +2.48632 q^{45} +1.08422 q^{47} -6.96201 q^{49} -15.5098 q^{51} +1.55308 q^{53} -0.912558 q^{55} -13.0406 q^{57} -14.8451 q^{59} -5.45056 q^{61} -1.25502 q^{63} -0.473551 q^{65} +11.3202 q^{67} -4.71461 q^{69} +11.4236 q^{71} -0.902229 q^{73} -14.9032 q^{75} +0.460631 q^{77} +12.7431 q^{79} +13.1421 q^{81} -6.10241 q^{83} -1.94937 q^{85} -22.4768 q^{87} +6.44432 q^{89} +0.239034 q^{91} -10.2581 q^{93} -1.63903 q^{95} -12.9886 q^{97} -15.2166 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) \) 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9	\( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9} + 3 q^{11} - 18 q^{13} + 2 q^{15} - 11 q^{17} + q^{21} + 2 q^{23} - 27 q^{25} + 2 q^{27} - 9 q^{29} + 22 q^{31} - 10 q^{33} + 6 q^{35} - 35 q^{37} - 8 q^{39} - 4 q^{41} + 20 q^{43} + 2 q^{45} - 7 q^{47} - 27 q^{49} - 9 q^{51} - 24 q^{53} + 11 q^{55} - 23 q^{57} - 17 q^{59} - 4 q^{61} - 10 q^{63} - 16 q^{65} + 6 q^{67} - 2 q^{69} + q^{71} - 31 q^{73} - 30 q^{75} + 3 q^{77} + 10 q^{79} - 6 q^{81} - 22 q^{83} - 6 q^{85} - 25 q^{87} + q^{89} - 10 q^{91} - 6 q^{93} - 39 q^{95} - 57 q^{97} - 35 q^{99}+O(q^{100}) \) 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9 + 3 * q^11 - 18 * q^13 + 2 * q^15 - 11 * q^17 + q^21 + 2 * q^23 - 27 * q^25 + 2 * q^27 - 9 * q^29 + 22 * q^31 - 10 * q^33 + 6 * q^35 - 35 * q^37 - 8 * q^39 - 4 * q^41 + 20 * q^43 + 2 * q^45 - 7 * q^47 - 27 * q^49 - 9 * q^51 - 24 * q^53 + 11 * q^55 - 23 * q^57 - 17 * q^59 - 4 * q^61 - 10 * q^63 - 16 * q^65 + 6 * q^67 - 2 * q^69 + q^71 - 31 * q^73 - 30 * q^75 + 3 * q^77 + 10 * q^79 - 6 * q^81 - 22 * q^83 - 6 * q^85 - 25 * q^87 + q^89 - 10 * q^91 - 6 * q^93 - 39 * q^95 - 57 * q^97 - 35 * q^99

Introduction

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