LMFDB - Embedded newform 8048.2.a.p.1.7

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			1.7
Root			$-0.510671$ 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)

$f(q)$	$=$	$q+1.51067 q^{3} -2.23445 q^{5} +3.60329 q^{7} -0.717874 q^{9} +O(q^{10})$	$q+1.51067 q^{3} -2.23445 q^{5} +3.60329 q^{7} -0.717874 q^{9} +1.50306 q^{11} +0.00459548 q^{13} -3.37551 q^{15} +1.11927 q^{17} -2.49963 q^{19} +5.44339 q^{21} -1.72480 q^{23} -0.00724425 q^{25} -5.61648 q^{27} -0.572061 q^{29} -3.25375 q^{31} +2.27063 q^{33} -8.05137 q^{35} -6.13809 q^{37} +0.00694226 q^{39} -5.89549 q^{41} +7.66800 q^{43} +1.60405 q^{45} -6.44840 q^{47} +5.98373 q^{49} +1.69085 q^{51} -7.82500 q^{53} -3.35851 q^{55} -3.77612 q^{57} -0.253382 q^{59} +7.08244 q^{61} -2.58671 q^{63} -0.0102684 q^{65} -6.16046 q^{67} -2.60561 q^{69} +9.32311 q^{71} -15.2654 q^{73} -0.0109437 q^{75} +5.41596 q^{77} +1.99661 q^{79} -6.33104 q^{81} -6.91660 q^{83} -2.50096 q^{85} -0.864195 q^{87} +3.37422 q^{89} +0.0165589 q^{91} -4.91534 q^{93} +5.58529 q^{95} -14.8842 q^{97} -1.07901 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.51067	0.872186	0.436093	−	0.899902i	$-0.356362\pi$
$3$	1.51067	0.872186	0.436093	+	0.899902i	$0.356362\pi$
$4$	0	0
$5$	−2.23445	−0.999275	−0.499638	−	0.866235i	$-0.666534\pi$
$5$	−2.23445	−0.999275	−0.499638	+	0.866235i	$0.666534\pi$
$6$	0	0
$7$	3.60329	1.36192	0.680959	−	0.732322i	$-0.261563\pi$
$7$	3.60329	1.36192	0.680959	+	0.732322i	$0.261563\pi$
$8$	0	0
$9$	−0.717874	−0.239291
$10$	0	0
$11$	1.50306	0.453189	0.226595	−	0.973989i	$-0.427241\pi$
$11$	1.50306	0.453189	0.226595	+	0.973989i	$0.427241\pi$
$12$	0	0
$13$	0.00459548	0.00127456	0.000637279	−	1.00000i	$-0.499797\pi$
$13$	0.00459548	0.00127456	0.000637279		1.00000i	$0.499797\pi$
$14$	0	0
$15$	−3.37551	−0.871554
$16$	0	0
$17$	1.11927	0.271464	0.135732	−	0.990746i	$-0.456661\pi$
$17$	1.11927	0.271464	0.135732	+	0.990746i	$0.456661\pi$
$18$	0	0
$19$	−2.49963	−0.573454	−0.286727	−	0.958012i	$-0.592567\pi$
$19$	−2.49963	−0.573454	−0.286727	+	0.958012i	$0.592567\pi$
$20$	0	0
$21$	5.44339	1.18785
$22$	0	0
$23$	−1.72480	−0.359646	−0.179823	−	0.983699i	$-0.557553\pi$
$23$	−1.72480	−0.359646	−0.179823	+	0.983699i	$0.557553\pi$
$24$	0	0
$25$	−0.00724425	−0.00144885
$26$	0	0
$27$	−5.61648	−1.08089
$28$	0	0
$29$	−0.572061	−0.106229	−0.0531145	−	0.998588i	$-0.516915\pi$
$29$	−0.572061	−0.106229	−0.0531145	+	0.998588i	$0.516915\pi$
$30$	0	0
$31$	−3.25375	−0.584390	−0.292195	−	0.956359i	$-0.594386\pi$
$31$	−3.25375	−0.584390	−0.292195	+	0.956359i	$0.594386\pi$
$32$	0	0
$33$	2.27063	0.395266
$34$	0	0
$35$	−8.05137	−1.36093
$36$	0	0
$37$	−6.13809	−1.00910	−0.504548	−	0.863384i	$-0.668341\pi$
$37$	−6.13809	−1.00910	−0.504548	+	0.863384i	$0.668341\pi$
$38$	0	0
$39$	0.00694226	0.00111165
$40$	0	0
$41$	−5.89549	−0.920721	−0.460360	−	0.887732i	$-0.652280\pi$
$41$	−5.89549	−0.920721	−0.460360	+	0.887732i	$0.652280\pi$
$42$	0	0
$43$	7.66800	1.16936	0.584680	−	0.811264i	$-0.301220\pi$
$43$	7.66800	1.16936	0.584680	+	0.811264i	$0.301220\pi$
$44$	0	0
$45$	1.60405	0.239118
$46$	0	0
$47$	−6.44840	−0.940596	−0.470298	−	0.882508i	$-0.655854\pi$
$47$	−6.44840	−0.940596	−0.470298	+	0.882508i	$0.655854\pi$
$48$	0	0
$49$	5.98373	0.854819
$50$	0	0
$51$	1.69085	0.236767
$52$	0	0
$53$	−7.82500	−1.07485	−0.537423	−	0.843313i	$-0.680602\pi$
$53$	−7.82500	−1.07485	−0.537423	+	0.843313i	$0.680602\pi$
$54$	0	0
$55$	−3.35851	−0.452861
$56$	0	0
$57$	−3.77612	−0.500159
$58$	0	0
$59$	−0.253382	−0.0329875	−0.0164938	−	0.999864i	$-0.505250\pi$
$59$	−0.253382	−0.0329875	−0.0164938	+	0.999864i	$0.505250\pi$
$60$	0	0
$61$	7.08244	0.906814	0.453407	−	0.891304i	$-0.350208\pi$
$61$	7.08244	0.906814	0.453407	+	0.891304i	$0.350208\pi$
$62$	0	0
$63$	−2.58671	−0.325895
$64$	0	0
$65$	−0.0102684	−0.00127363
$66$	0	0
$67$	−6.16046	−0.752621	−0.376310	−	0.926494i	$-0.622807\pi$
$67$	−6.16046	−0.752621	−0.376310	+	0.926494i	$0.622807\pi$
$68$	0	0
$69$	−2.60561	−0.313678
$70$	0	0
$71$	9.32311	1.10645	0.553225	−	0.833032i	$-0.313397\pi$
$71$	9.32311	1.10645	0.553225	+	0.833032i	$0.313397\pi$
$72$	0	0
$73$	−15.2654	−1.78668	−0.893338	−	0.449385i	$-0.851643\pi$
$73$	−15.2654	−1.78668	−0.893338	+	0.449385i	$0.851643\pi$
$74$	0	0
$75$	−0.0109437	−0.00126367
$76$	0	0
$77$	5.41596	0.617206
$78$	0	0
$79$	1.99661	0.224636	0.112318	−	0.993672i	$-0.464172\pi$
$79$	1.99661	0.224636	0.112318	+	0.993672i	$0.464172\pi$
$80$	0	0
$81$	−6.33104	−0.703448
$82$	0	0
$83$	−6.91660	−0.759196	−0.379598	−	0.925152i	$-0.623938\pi$
$83$	−6.91660	−0.759196	−0.379598	+	0.925152i	$0.623938\pi$
$84$	0	0
$85$	−2.50096	−0.271267
$86$	0	0
$87$	−0.864195	−0.0926514
$88$	0	0
$89$	3.37422	0.357666	0.178833	−	0.983879i	$-0.442768\pi$
$89$	3.37422	0.357666	0.178833	+	0.983879i	$0.442768\pi$
$90$	0	0
$91$	0.0165589	0.00173584
$92$	0	0
$93$	−4.91534	−0.509697
$94$	0	0
$95$	5.58529	0.573039
$96$	0	0
$97$	−14.8842	−1.51126	−0.755631	−	0.654997i	$-0.772670\pi$
$97$	−14.8842	−1.51126	−0.755631	+	0.654997i	$0.772670\pi$
$98$	0	0
$99$	−1.07901	−0.108444
$100$	0	0
$101$	11.2591	1.12032	0.560159	−	0.828385i	$-0.310740\pi$
$101$	11.2591	1.12032	0.560159	+	0.828385i	$0.310740\pi$
$102$	0	0
$103$	−4.65004	−0.458182	−0.229091	−	0.973405i	$-0.573575\pi$
$103$	−4.65004	−0.458182	−0.229091	+	0.973405i	$0.573575\pi$
$104$	0	0
$105$	−12.1630	−1.18698
$106$	0	0
$107$	8.08323	0.781435	0.390718	−	0.920511i	$-0.372227\pi$
$107$	8.08323	0.781435	0.390718	+	0.920511i	$0.372227\pi$
$108$	0	0
$109$	−1.20991	−0.115889	−0.0579444	−	0.998320i	$-0.518455\pi$
$109$	−1.20991	−0.115889	−0.0579444	+	0.998320i	$0.518455\pi$
$110$	0	0
$111$	−9.27264	−0.880120
$112$	0	0
$113$	7.68412	0.722861	0.361431	−	0.932399i	$-0.382288\pi$
$113$	7.68412	0.722861	0.361431	+	0.932399i	$0.382288\pi$
$114$	0	0
$115$	3.85398	0.359386
$116$	0	0
$117$	−0.00329898	−0.000304991 0
$118$	0	0
$119$	4.03307	0.369711
$120$	0	0
$121$	−8.74081	−0.794619
$122$	0	0
$123$	−8.90614	−0.803040
$124$	0	0
$125$	11.1884	1.00072
$126$	0	0
$127$	−8.83928	−0.784360	−0.392180	−	0.919889i	$-0.628279\pi$
$127$	−8.83928	−0.784360	−0.392180	+	0.919889i	$0.628279\pi$
$128$	0	0
$129$	11.5838	1.01990
$130$	0	0
$131$	12.9102	1.12797	0.563984	−	0.825786i	$-0.309268\pi$
$131$	12.9102	1.12797	0.563984	+	0.825786i	$0.309268\pi$
$132$	0	0
$133$	−9.00690	−0.780998
$134$	0	0
$135$	12.5497	1.08011
$136$	0	0
$137$	3.83616	0.327745	0.163873	−	0.986482i	$-0.447601\pi$
$137$	3.83616	0.327745	0.163873	+	0.986482i	$0.447601\pi$
$138$	0	0
$139$	−9.22921	−0.782811	−0.391406	−	0.920218i	$-0.628011\pi$
$139$	−9.22921	−0.782811	−0.391406	+	0.920218i	$0.628011\pi$
$140$	0	0
$141$	−9.74142	−0.820375
$142$	0	0
$143$	0.00690728	0.000577616 0
$144$	0	0
$145$	1.27824	0.106152
$146$	0	0
$147$	9.03945	0.745561
$148$	0	0
$149$	22.6884	1.85871	0.929353	−	0.369191i	$-0.120365\pi$
$149$	22.6884	1.85871	0.929353	+	0.369191i	$0.120365\pi$
$150$	0	0
$151$	2.22521	0.181085	0.0905427	−	0.995893i	$-0.471140\pi$
$151$	2.22521	0.181085	0.0905427	+	0.995893i	$0.471140\pi$
$152$	0	0
$153$	−0.803497	−0.0649589
$154$	0	0
$155$	7.27033	0.583967
$156$	0	0
$157$	−22.3177	−1.78114	−0.890571	−	0.454844i	$-0.849695\pi$
$157$	−22.3177	−1.78114	−0.890571	+	0.454844i	$0.849695\pi$
$158$	0	0
$159$	−11.8210	−0.937466
$160$	0	0
$161$	−6.21497	−0.489808
$162$	0	0
$163$	−7.22321	−0.565765	−0.282883	−	0.959155i	$-0.591291\pi$
$163$	−7.22321	−0.565765	−0.282883	+	0.959155i	$0.591291\pi$
$164$	0	0
$165$	−5.07360	−0.394979
$166$	0	0
$167$	11.4485	0.885915	0.442958	−	0.896543i	$-0.353929\pi$
$167$	11.4485	0.885915	0.442958	+	0.896543i	$0.353929\pi$
$168$	0	0
$169$	−13.0000	−0.999998
$170$	0	0
$171$	1.79442	0.137223
$172$	0	0
$173$	−12.5821	−0.956599	−0.478300	−	0.878197i	$-0.658747\pi$
$173$	−12.5821	−0.956599	−0.478300	+	0.878197i	$0.658747\pi$
$174$	0	0
$175$	−0.0261032	−0.00197321
$176$	0	0
$177$	−0.382777	−0.0287713
$178$	0	0
$179$	−1.23407	−0.0922389	−0.0461194	−	0.998936i	$-0.514685\pi$
$179$	−1.23407	−0.0922389	−0.0461194	+	0.998936i	$0.514685\pi$
$180$	0	0
$181$	−17.0030	−1.26382	−0.631911	−	0.775041i	$-0.717729\pi$
$181$	−17.0030	−1.26382	−0.631911	+	0.775041i	$0.717729\pi$
$182$	0	0
$183$	10.6992	0.790910
$184$	0	0
$185$	13.7152	1.00836
$186$	0	0
$187$	1.68233	0.123024
$188$	0	0
$189$	−20.2378	−1.47209
$190$	0	0
$191$	−0.213652	−0.0154593	−0.00772966	−	0.999970i	$-0.502460\pi$
$191$	−0.213652	−0.0154593	−0.00772966	+	0.999970i	$0.502460\pi$
$192$	0	0
$193$	−3.51336	−0.252897	−0.126449	−	0.991973i	$-0.540358\pi$
$193$	−3.51336	−0.252897	−0.126449	+	0.991973i	$0.540358\pi$
$194$	0	0
$195$	−0.0155121	−0.00111085
$196$	0	0
$197$	−11.5456	−0.822587	−0.411294	−	0.911503i	$-0.634923\pi$
$197$	−11.5456	−0.822587	−0.411294	+	0.911503i	$0.634923\pi$
$198$	0	0
$199$	4.63698	0.328707	0.164353	−	0.986402i	$-0.447446\pi$
$199$	4.63698	0.328707	0.164353	+	0.986402i	$0.447446\pi$
$200$	0	0
$201$	−9.30643	−0.656425
$202$	0	0
$203$	−2.06130	−0.144675
$204$	0	0
$205$	13.1732	0.920054
$206$	0	0
$207$	1.23819	0.0860602
$208$	0	0
$209$	−3.75709	−0.259883
$210$	0	0
$211$	0.622411	0.0428485	0.0214243	−	0.999770i	$-0.493180\pi$
$211$	0.622411	0.0428485	0.0214243	+	0.999770i	$0.493180\pi$
$212$	0	0
$213$	14.0841	0.965030
$214$	0	0
$215$	−17.1338	−1.16851
$216$	0	0
$217$	−11.7242	−0.795891
$218$	0	0
$219$	−23.0609	−1.55831
$220$	0	0
$221$	0.00514360	0.000345996 0
$222$	0	0
$223$	−0.0281489	−0.00188499	−0.000942495	−	1.00000i	$-0.500300\pi$
$223$	−0.0281489	−0.00188499	−0.000942495		1.00000i	$0.500300\pi$
$224$	0	0
$225$	0.00520046	0.000346697 0
$226$	0	0
$227$	8.12837	0.539499	0.269749	−	0.962931i	$-0.413059\pi$
$227$	8.12837	0.539499	0.269749	+	0.962931i	$0.413059\pi$
$228$	0	0
$229$	23.9995	1.58593	0.792966	−	0.609266i	$-0.208536\pi$
$229$	23.9995	1.58593	0.792966	+	0.609266i	$0.208536\pi$
$230$	0	0
$231$	8.18174	0.538319
$232$	0	0
$233$	−12.5903	−0.824819	−0.412410	−	0.910999i	$-0.635313\pi$
$233$	−12.5903	−0.824819	−0.412410	+	0.910999i	$0.635313\pi$
$234$	0	0
$235$	14.4086	0.939915
$236$	0	0
$237$	3.01622	0.195925
$238$	0	0
$239$	−24.1351	−1.56117	−0.780586	−	0.625049i	$-0.785079\pi$
$239$	−24.1351	−1.56117	−0.780586	+	0.625049i	$0.785079\pi$
$240$	0	0
$241$	25.4713	1.64075	0.820374	−	0.571827i	$-0.193765\pi$
$241$	25.4713	1.64075	0.820374	+	0.571827i	$0.193765\pi$
$242$	0	0
$243$	7.28534	0.467355
$244$	0	0
$245$	−13.3703	−0.854199
$246$	0	0
$247$	−0.0114870	−0.000730901 0
$248$	0	0
$249$	−10.4487	−0.662160
$250$	0	0
$251$	−4.12389	−0.260298	−0.130149	−	0.991494i	$-0.541546\pi$
$251$	−4.12389	−0.260298	−0.130149	+	0.991494i	$0.541546\pi$
$252$	0	0
$253$	−2.59248	−0.162988
$254$	0	0
$255$	−3.77812	−0.236595
$256$	0	0
$257$	−16.7266	−1.04337	−0.521687	−	0.853137i	$-0.674697\pi$
$257$	−16.7266	−1.04337	−0.521687	+	0.853137i	$0.674697\pi$
$258$	0	0
$259$	−22.1174	−1.37431
$260$	0	0
$261$	0.410667	0.0254197
$262$	0	0
$263$	−28.7329	−1.77174	−0.885872	−	0.463930i	$-0.846439\pi$
$263$	−28.7329	−1.77174	−0.885872	+	0.463930i	$0.846439\pi$
$264$	0	0
$265$	17.4845	1.07407
$266$	0	0
$267$	5.09733	0.311951
$268$	0	0
$269$	28.6099	1.74438	0.872189	−	0.489169i	$-0.162700\pi$
$269$	28.6099	1.74438	0.872189	+	0.489169i	$0.162700\pi$
$270$	0	0
$271$	13.8852	0.843467	0.421733	−	0.906720i	$-0.361422\pi$
$271$	13.8852	0.843467	0.421733	+	0.906720i	$0.361422\pi$
$272$	0	0
$273$	0.0250150	0.00151398
$274$	0	0
$275$	−0.0108885	−0.000656603 0
$276$	0	0
$277$	−14.4664	−0.869200	−0.434600	−	0.900624i	$-0.643110\pi$
$277$	−14.4664	−0.869200	−0.434600	+	0.900624i	$0.643110\pi$
$278$	0	0
$279$	2.33578	0.139839
$280$	0	0
$281$	−15.4967	−0.924456	−0.462228	−	0.886761i	$-0.652950\pi$
$281$	−15.4967	−0.924456	−0.462228	+	0.886761i	$0.652950\pi$
$282$	0	0
$283$	−10.4295	−0.619970	−0.309985	−	0.950741i	$-0.600324\pi$
$283$	−10.4295	−0.619970	−0.309985	+	0.950741i	$0.600324\pi$
$284$	0	0
$285$	8.43754	0.499797
$286$	0	0
$287$	−21.2432	−1.25395
$288$	0	0
$289$	−15.7472	−0.926307
$290$	0	0
$291$	−22.4851	−1.31810
$292$	0	0
$293$	−1.90244	−0.111142	−0.0555708	−	0.998455i	$-0.517698\pi$
$293$	−1.90244	−0.111142	−0.0555708	+	0.998455i	$0.517698\pi$
$294$	0	0
$295$	0.566169	0.0329636
$296$	0	0
$297$	−8.44191	−0.489849
$298$	0	0
$299$	−0.00792630	−0.000458390 0
$300$	0	0
$301$	27.6301	1.59257
$302$	0	0
$303$	17.0087	0.977127
$304$	0	0
$305$	−15.8253	−0.906157
$306$	0	0
$307$	17.6876	1.00949	0.504744	−	0.863269i	$-0.331587\pi$
$307$	17.6876	1.00949	0.504744	+	0.863269i	$0.331587\pi$
$308$	0	0
$309$	−7.02468	−0.399620
$310$	0	0
$311$	28.0074	1.58815	0.794076	−	0.607818i	$-0.207955\pi$
$311$	28.0074	1.58815	0.794076	+	0.607818i	$0.207955\pi$
$312$	0	0
$313$	5.02123	0.283817	0.141908	−	0.989880i	$-0.454676\pi$
$313$	5.02123	0.283817	0.141908	+	0.989880i	$0.454676\pi$
$314$	0	0
$315$	5.77987	0.325659
$316$	0	0
$317$	0.356974	0.0200496	0.0100248	−	0.999950i	$-0.496809\pi$
$317$	0.356974	0.0200496	0.0100248	+	0.999950i	$0.496809\pi$
$318$	0	0
$319$	−0.859841	−0.0481418
$320$	0	0
$321$	12.2111	0.681557
$322$	0	0
$323$	−2.79777	−0.155672
$324$	0	0
$325$	−3.32908e−5 0	−1.84664e−6 0
$326$	0	0
$327$	−1.82778	−0.101077
$328$	0	0
$329$	−23.2355	−1.28101
$330$	0	0
$331$	−16.2490	−0.893123	−0.446562	−	0.894753i	$-0.647352\pi$
$331$	−16.2490	−0.893123	−0.446562	+	0.894753i	$0.647352\pi$
$332$	0	0
$333$	4.40637	0.241468
$334$	0	0
$335$	13.7652	0.752075
$336$	0	0
$337$	−24.0444	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$337$	−24.0444	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$338$	0	0
$339$	11.6082	0.630469
$340$	0	0
$341$	−4.89057	−0.264839
$342$	0	0
$343$	−3.66192	−0.197725
$344$	0	0
$345$	5.82210	0.313451
$346$	0	0
$347$	10.7861	0.579028	0.289514	−	0.957174i	$-0.406506\pi$
$347$	10.7861	0.579028	0.289514	+	0.957174i	$0.406506\pi$
$348$	0	0
$349$	17.2044	0.920933	0.460466	−	0.887677i	$-0.347682\pi$
$349$	17.2044	0.920933	0.460466	+	0.887677i	$0.347682\pi$
$350$	0	0
$351$	−0.0258105	−0.00137766
$352$	0	0
$353$	−2.83276	−0.150773	−0.0753863	−	0.997154i	$-0.524019\pi$
$353$	−2.83276	−0.150773	−0.0753863	+	0.997154i	$0.524019\pi$
$354$	0	0
$355$	−20.8320	−1.10565
$356$	0	0
$357$	6.09264	0.322457
$358$	0	0
$359$	−23.3759	−1.23373	−0.616867	−	0.787068i	$-0.711598\pi$
$359$	−23.3759	−1.23373	−0.616867	+	0.787068i	$0.711598\pi$
$360$	0	0
$361$	−12.7518	−0.671150
$362$	0	0
$363$	−13.2045	−0.693056
$364$	0	0
$365$	34.1097	1.78538
$366$	0	0
$367$	21.8406	1.14007	0.570034	−	0.821621i	$-0.306930\pi$
$367$	21.8406	1.14007	0.570034	+	0.821621i	$0.306930\pi$
$368$	0	0
$369$	4.23222	0.220320
$370$	0	0
$371$	−28.1958	−1.46385
$372$	0	0
$373$	0.594294	0.0307714	0.0153857	−	0.999882i	$-0.495102\pi$
$373$	0.594294	0.0307714	0.0153857	+	0.999882i	$0.495102\pi$
$374$	0	0
$375$	16.9020	0.872817
$376$	0	0
$377$	−0.00262890	−0.000135395 0
$378$	0	0
$379$	16.0631	0.825105	0.412553	−	0.910934i	$-0.364637\pi$
$379$	16.0631	0.825105	0.412553	+	0.910934i	$0.364637\pi$
$380$	0	0
$381$	−13.3532	−0.684108
$382$	0	0
$383$	−15.0745	−0.770271	−0.385135	−	0.922860i	$-0.625845\pi$
$383$	−15.0745	−0.770271	−0.385135	+	0.922860i	$0.625845\pi$
$384$	0	0
$385$	−12.1017	−0.616759
$386$	0	0
$387$	−5.50466	−0.279818
$388$	0	0
$389$	8.06793	0.409060	0.204530	−	0.978860i	$-0.434433\pi$
$389$	8.06793	0.409060	0.204530	+	0.978860i	$0.434433\pi$
$390$	0	0
$391$	−1.93053	−0.0976309
$392$	0	0
$393$	19.5030	0.983797
$394$	0	0
$395$	−4.46132	−0.224473
$396$	0	0
$397$	7.64285	0.383583	0.191792	−	0.981436i	$-0.438570\pi$
$397$	7.64285	0.383583	0.191792	+	0.981436i	$0.438570\pi$
$398$	0	0
$399$	−13.6065	−0.681175
$400$	0	0
$401$	37.0556	1.85047	0.925234	−	0.379397i	$-0.123868\pi$
$401$	37.0556	1.85047	0.925234	+	0.379397i	$0.123868\pi$
$402$	0	0
$403$	−0.0149525	−0.000744839 0
$404$	0	0
$405$	14.1464	0.702939
$406$	0	0
$407$	−9.22591	−0.457312
$408$	0	0
$409$	−29.1516	−1.44145	−0.720726	−	0.693220i	$-0.756191\pi$
$409$	−29.1516	−1.44145	−0.720726	+	0.693220i	$0.756191\pi$
$410$	0	0
$411$	5.79518	0.285855
$412$	0	0
$413$	−0.913010	−0.0449263
$414$	0	0
$415$	15.4548	0.758645
$416$	0	0
$417$	−13.9423	−0.682757
$418$	0	0
$419$	−31.8690	−1.55690	−0.778451	−	0.627705i	$-0.783994\pi$
$419$	−31.8690	−1.55690	−0.778451	+	0.627705i	$0.783994\pi$
$420$	0	0
$421$	−26.0459	−1.26940	−0.634699	−	0.772759i	$-0.718876\pi$
$421$	−26.0459	−1.26940	−0.634699	+	0.772759i	$0.718876\pi$
$422$	0	0
$423$	4.62914	0.225077
$424$	0	0
$425$	−0.00810829	−0.000393310 0
$426$	0	0
$427$	25.5201	1.23501
$428$	0	0
$429$	0.0104346	0.000503789 0
$430$	0	0
$431$	22.6196	1.08955	0.544774	−	0.838583i	$-0.316616\pi$
$431$	22.6196	1.08955	0.544774	+	0.838583i	$0.316616\pi$
$432$	0	0
$433$	−22.7272	−1.09220	−0.546099	−	0.837720i	$-0.683888\pi$
$433$	−22.7272	−1.09220	−0.546099	+	0.837720i	$0.683888\pi$
$434$	0	0
$435$	1.93100	0.0925843
$436$	0	0
$437$	4.31137	0.206241
$438$	0	0
$439$	−24.5936	−1.17379	−0.586895	−	0.809663i	$-0.699650\pi$
$439$	−24.5936	−1.17379	−0.586895	+	0.809663i	$0.699650\pi$
$440$	0	0
$441$	−4.29556	−0.204551
$442$	0	0
$443$	−23.4765	−1.11540	−0.557702	−	0.830041i	$-0.688317\pi$
$443$	−23.4765	−1.11540	−0.557702	+	0.830041i	$0.688317\pi$
$444$	0	0
$445$	−7.53951	−0.357407
$446$	0	0
$447$	34.2747	1.62114
$448$	0	0
$449$	−13.6452	−0.643958	−0.321979	−	0.946747i	$-0.604348\pi$
$449$	−13.6452	−0.643958	−0.321979	+	0.946747i	$0.604348\pi$
$450$	0	0
$451$	−8.86127	−0.417261
$452$	0	0
$453$	3.36157	0.157940
$454$	0	0
$455$	−0.0370000	−0.00173458
$456$	0	0
$457$	−2.17967	−0.101960	−0.0509802	−	0.998700i	$-0.516235\pi$
$457$	−2.17967	−0.101960	−0.0509802	+	0.998700i	$0.516235\pi$
$458$	0	0
$459$	−6.28638	−0.293423
$460$	0	0
$461$	12.2229	0.569276	0.284638	−	0.958635i	$-0.408127\pi$
$461$	12.2229	0.569276	0.284638	+	0.958635i	$0.408127\pi$
$462$	0	0
$463$	1.99436	0.0926860	0.0463430	−	0.998926i	$-0.485243\pi$
$463$	1.99436	0.0926860	0.0463430	+	0.998926i	$0.485243\pi$
$464$	0	0
$465$	10.9831	0.509328
$466$	0	0
$467$	−41.2601	−1.90929	−0.954645	−	0.297747i	$-0.903765\pi$
$467$	−41.2601	−1.90929	−0.954645	+	0.297747i	$0.903765\pi$
$468$	0	0
$469$	−22.1980	−1.02501
$470$	0	0
$471$	−33.7146	−1.55349
$472$	0	0
$473$	11.5255	0.529941
$474$	0	0
$475$	0.0181079	0.000830849 0
$476$	0	0
$477$	5.61736	0.257201
$478$	0	0
$479$	−23.4921	−1.07338	−0.536691	−	0.843779i	$-0.680326\pi$
$479$	−23.4921	−1.07338	−0.536691	+	0.843779i	$0.680326\pi$
$480$	0	0
$481$	−0.0282075	−0.00128615
$482$	0	0
$483$	−9.38878	−0.427204
$484$	0	0
$485$	33.2580	1.51017
$486$	0	0
$487$	−6.09387	−0.276139	−0.138070	−	0.990423i	$-0.544090\pi$
$487$	−6.09387	−0.276139	−0.138070	+	0.990423i	$0.544090\pi$
$488$	0	0
$489$	−10.9119	−0.493453
$490$	0	0
$491$	7.88983	0.356063	0.178032	−	0.984025i	$-0.443027\pi$
$491$	7.88983	0.356063	0.178032	+	0.984025i	$0.443027\pi$
$492$	0	0
$493$	−0.640292	−0.0288373
$494$	0	0
$495$	2.41098	0.108366
$496$	0	0
$497$	33.5939	1.50689
$498$	0	0
$499$	0.289120	0.0129428	0.00647139	−	0.999979i	$-0.497940\pi$
$499$	0.289120	0.0129428	0.00647139	+	0.999979i	$0.497940\pi$
$500$	0	0
$501$	17.2950	0.772683
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−25.1578	−1.11951
$506$	0	0
$507$	−19.6387	−0.872185
$508$	0	0
$509$	27.0889	1.20069	0.600346	−	0.799740i	$-0.295030\pi$
$509$	27.0889	1.20069	0.600346	+	0.799740i	$0.295030\pi$
$510$	0	0
$511$	−55.0056	−2.43331
$512$	0	0
$513$	14.0391	0.619843
$514$	0	0
$515$	10.3903	0.457850
$516$	0	0
$517$	−9.69233	−0.426268
$518$	0	0
$519$	−19.0074	−0.834333
$520$	0	0
$521$	21.1689	0.927424	0.463712	−	0.885986i	$-0.346517\pi$
$521$	21.1689	0.927424	0.463712	+	0.885986i	$0.346517\pi$
$522$	0	0
$523$	0.868369	0.0379711	0.0189856	−	0.999820i	$-0.493956\pi$
$523$	0.868369	0.0379711	0.0189856	+	0.999820i	$0.493956\pi$
$524$	0	0
$525$	−0.0394333	−0.00172101
$526$	0	0
$527$	−3.64183	−0.158641
$528$	0	0
$529$	−20.0251	−0.870655
$530$	0	0
$531$	0.181896	0.00789363
$532$	0	0
$533$	−0.0270926	−0.00117351
$534$	0	0
$535$	−18.0616	−0.780869
$536$	0	0
$537$	−1.86428	−0.0804495
$538$	0	0
$539$	8.99390	0.387395
$540$	0	0
$541$	14.8513	0.638508	0.319254	−	0.947669i	$-0.396568\pi$
$541$	14.8513	0.638508	0.319254	+	0.947669i	$0.396568\pi$
$542$	0	0
$543$	−25.6859	−1.10229
$544$	0	0
$545$	2.70349	0.115805
$546$	0	0
$547$	18.8256	0.804925	0.402463	−	0.915436i	$-0.368154\pi$
$547$	18.8256	0.804925	0.402463	+	0.915436i	$0.368154\pi$
$548$	0	0
$549$	−5.08430	−0.216993
$550$	0	0
$551$	1.42994	0.0609175
$552$	0	0
$553$	7.19437	0.305936
$554$	0	0
$555$	20.7192	0.879482
$556$	0	0
$557$	11.9201	0.505071	0.252535	−	0.967588i	$-0.418736\pi$
$557$	11.9201	0.505071	0.252535	+	0.967588i	$0.418736\pi$
$558$	0	0
$559$	0.0352382	0.00149042
$560$	0	0
$561$	2.54145	0.107300
$562$	0	0
$563$	22.7172	0.957414	0.478707	−	0.877975i	$-0.341106\pi$
$563$	22.7172	0.957414	0.478707	+	0.877975i	$0.341106\pi$
$564$	0	0
$565$	−17.1698	−0.722337
$566$	0	0
$567$	−22.8126	−0.958039
$568$	0	0
$569$	−16.9403	−0.710176	−0.355088	−	0.934833i	$-0.615549\pi$
$569$	−16.9403	−0.710176	−0.355088	+	0.934833i	$0.615549\pi$
$570$	0	0
$571$	−16.2940	−0.681883	−0.340941	−	0.940085i	$-0.610746\pi$
$571$	−16.2940	−0.681883	−0.340941	+	0.940085i	$0.610746\pi$
$572$	0	0
$573$	−0.322758	−0.0134834
$574$	0	0
$575$	0.0124949	0.000521073 0
$576$	0	0
$577$	−44.3876	−1.84788	−0.923941	−	0.382535i	$-0.875051\pi$
$577$	−44.3876	−1.84788	−0.923941	+	0.382535i	$0.875051\pi$
$578$	0	0
$579$	−5.30753	−0.220574
$580$	0	0
$581$	−24.9226	−1.03396
$582$	0	0
$583$	−11.7614	−0.487109
$584$	0	0
$585$	0.00737139	0.000304770 0
$586$	0	0
$587$	−33.6221	−1.38773	−0.693865	−	0.720105i	$-0.744094\pi$
$587$	−33.6221	−1.38773	−0.693865	+	0.720105i	$0.744094\pi$
$588$	0	0
$589$	8.13316	0.335121
$590$	0	0
$591$	−17.4415	−0.717449
$592$	0	0
$593$	14.3513	0.589336	0.294668	−	0.955600i	$-0.404791\pi$
$593$	14.3513	0.589336	0.294668	+	0.955600i	$0.404791\pi$
$594$	0	0
$595$	−9.01169	−0.369443
$596$	0	0
$597$	7.00495	0.286694
$598$	0	0
$599$	4.26954	0.174449	0.0872243	−	0.996189i	$-0.472200\pi$
$599$	4.26954	0.174449	0.0872243	+	0.996189i	$0.472200\pi$
$600$	0	0
$601$	18.2764	0.745510	0.372755	−	0.927930i	$-0.378413\pi$
$601$	18.2764	0.745510	0.372755	+	0.927930i	$0.378413\pi$
$602$	0	0
$603$	4.42244	0.180096
$604$	0	0
$605$	19.5309	0.794044
$606$	0	0
$607$	29.2048	1.18539	0.592694	−	0.805428i	$-0.298064\pi$
$607$	29.2048	1.18539	0.592694	+	0.805428i	$0.298064\pi$
$608$	0	0
$609$	−3.11395	−0.126184
$610$	0	0
$611$	−0.0296335	−0.00119884
$612$	0	0
$613$	−24.0622	−0.971861	−0.485931	−	0.873997i	$-0.661519\pi$
$613$	−24.0622	−0.971861	−0.485931	+	0.873997i	$0.661519\pi$
$614$	0	0
$615$	19.9003	0.802458
$616$	0	0
$617$	−20.7734	−0.836307	−0.418154	−	0.908376i	$-0.637323\pi$
$617$	−20.7734	−0.836307	−0.418154	+	0.908376i	$0.637323\pi$
$618$	0	0
$619$	6.43218	0.258531	0.129266	−	0.991610i	$-0.458738\pi$
$619$	6.43218	0.258531	0.129266	+	0.991610i	$0.458738\pi$
$620$	0	0
$621$	9.68732	0.388739
$622$	0	0
$623$	12.1583	0.487112
$624$	0	0
$625$	−24.9637	−0.998549
$626$	0	0
$627$	−5.67573	−0.226667
$628$	0	0
$629$	−6.87020	−0.273933
$630$	0	0
$631$	43.2199	1.72056	0.860279	−	0.509824i	$-0.170290\pi$
$631$	43.2199	1.72056	0.860279	+	0.509824i	$0.170290\pi$
$632$	0	0
$633$	0.940258	0.0373719
$634$	0	0
$635$	19.7509	0.783791
$636$	0	0
$637$	0.0274981	0.00108952
$638$	0	0
$639$	−6.69281	−0.264764
$640$	0	0
$641$	−49.6276	−1.96017	−0.980086	−	0.198574i	$-0.936369\pi$
$641$	−49.6276	−1.96017	−0.980086	+	0.198574i	$0.936369\pi$
$642$	0	0
$643$	45.1377	1.78006	0.890028	−	0.455906i	$-0.150685\pi$
$643$	45.1377	1.78006	0.890028	+	0.455906i	$0.150685\pi$
$644$	0	0
$645$	−25.8835	−1.01916
$646$	0	0
$647$	29.7930	1.17128	0.585642	−	0.810570i	$-0.300843\pi$
$647$	29.7930	1.17128	0.585642	+	0.810570i	$0.300843\pi$
$648$	0	0
$649$	−0.380848	−0.0149496
$650$	0	0
$651$	−17.7114	−0.694165
$652$	0	0
$653$	29.1127	1.13927	0.569635	−	0.821898i	$-0.307085\pi$
$653$	29.1127	1.13927	0.569635	+	0.821898i	$0.307085\pi$
$654$	0	0
$655$	−28.8471	−1.12715
$656$	0	0
$657$	10.9586	0.427536
$658$	0	0
$659$	−17.1327	−0.667395	−0.333697	−	0.942680i	$-0.608296\pi$
$659$	−17.1327	−0.667395	−0.333697	+	0.942680i	$0.608296\pi$
$660$	0	0
$661$	−0.266846	−0.0103791	−0.00518956	−	0.999987i	$-0.501652\pi$
$661$	−0.266846	−0.0103791	−0.00518956	+	0.999987i	$0.501652\pi$
$662$	0	0
$663$	0.00777029	0.000301773 0
$664$	0	0
$665$	20.1255	0.780432
$666$	0	0
$667$	0.986691	0.0382048
$668$	0	0
$669$	−0.0425237	−0.00164406
$670$	0	0
$671$	10.6453	0.410958
$672$	0	0
$673$	41.4814	1.59899	0.799495	−	0.600673i	$-0.205101\pi$
$673$	41.4814	1.59899	0.799495	+	0.600673i	$0.205101\pi$
$674$	0	0
$675$	0.0406872	0.00156605
$676$	0	0
$677$	33.0242	1.26922	0.634611	−	0.772832i	$-0.281160\pi$
$677$	33.0242	1.26922	0.634611	+	0.772832i	$0.281160\pi$
$678$	0	0
$679$	−53.6322	−2.05822
$680$	0	0
$681$	12.2793	0.470543
$682$	0	0
$683$	−0.471704	−0.0180492	−0.00902462	−	0.999959i	$-0.502873\pi$
$683$	−0.471704	−0.0180492	−0.00902462	+	0.999959i	$0.502873\pi$
$684$	0	0
$685$	−8.57170	−0.327508
$686$	0	0
$687$	36.2553	1.38323
$688$	0	0
$689$	−0.0359597	−0.00136995
$690$	0	0
$691$	12.1972	0.464003	0.232002	−	0.972715i	$-0.425473\pi$
$691$	12.1972	0.464003	0.232002	+	0.972715i	$0.425473\pi$
$692$	0	0
$693$	−3.88798	−0.147692
$694$	0	0
$695$	20.6222	0.782244
$696$	0	0
$697$	−6.59866	−0.249942
$698$	0	0
$699$	−19.0198	−0.719396
$700$	0	0
$701$	−47.0220	−1.77600	−0.887999	−	0.459846i	$-0.847905\pi$
$701$	−47.0220	−1.77600	−0.887999	+	0.459846i	$0.847905\pi$
$702$	0	0
$703$	15.3430	0.578671
$704$	0	0
$705$	21.7667	0.819781
$706$	0	0
$707$	40.5697	1.52578
$708$	0	0
$709$	26.6796	1.00197	0.500986	−	0.865455i	$-0.332971\pi$
$709$	26.6796	1.00197	0.500986	+	0.865455i	$0.332971\pi$
$710$	0	0
$711$	−1.43331	−0.0537535
$712$	0	0
$713$	5.61207	0.210174
$714$	0	0
$715$	−0.0154340	−0.000577198 0
$716$	0	0
$717$	−36.4602	−1.36163
$718$	0	0
$719$	−12.7457	−0.475336	−0.237668	−	0.971346i	$-0.576383\pi$
$719$	−12.7457	−0.475336	−0.237668	+	0.971346i	$0.576383\pi$
$720$	0	0
$721$	−16.7555	−0.624006
$722$	0	0
$723$	38.4787	1.43104
$724$	0	0
$725$	0.00414415	0.000153910 0
$726$	0	0
$727$	−1.14219	−0.0423614	−0.0211807	−	0.999776i	$-0.506743\pi$
$727$	−1.14219	−0.0423614	−0.0211807	+	0.999776i	$0.506743\pi$
$728$	0	0
$729$	29.9989	1.11107
$730$	0	0
$731$	8.58259	0.317439
$732$	0	0
$733$	−20.6111	−0.761289	−0.380644	−	0.924721i	$-0.624298\pi$
$733$	−20.6111	−0.761289	−0.380644	+	0.924721i	$0.624298\pi$
$734$	0	0
$735$	−20.1982	−0.745021
$736$	0	0
$737$	−9.25954	−0.341080
$738$	0	0
$739$	40.1054	1.47530	0.737651	−	0.675182i	$-0.235935\pi$
$739$	40.1054	1.47530	0.737651	+	0.675182i	$0.235935\pi$
$740$	0	0
$741$	−0.0173531	−0.000637482 0
$742$	0	0
$743$	31.7658	1.16538	0.582688	−	0.812696i	$-0.302001\pi$
$743$	31.7658	1.16538	0.582688	+	0.812696i	$0.302001\pi$
$744$	0	0
$745$	−50.6961	−1.85736
$746$	0	0
$747$	4.96525	0.181669
$748$	0	0
$749$	29.1263	1.06425
$750$	0	0
$751$	−3.19661	−0.116646	−0.0583230	−	0.998298i	$-0.518575\pi$
$751$	−3.19661	−0.116646	−0.0583230	+	0.998298i	$0.518575\pi$
$752$	0	0
$753$	−6.22984	−0.227028
$754$	0	0
$755$	−4.97212	−0.180954
$756$	0	0
$757$	45.6783	1.66021	0.830103	−	0.557610i	$-0.188281\pi$
$757$	45.6783	1.66021	0.830103	+	0.557610i	$0.188281\pi$
$758$	0	0
$759$	−3.91638	−0.142156
$760$	0	0
$761$	37.7902	1.36989	0.684946	−	0.728594i	$-0.259826\pi$
$761$	37.7902	1.36989	0.684946	+	0.728594i	$0.259826\pi$
$762$	0	0
$763$	−4.35968	−0.157831
$764$	0	0
$765$	1.79537	0.0649118
$766$	0	0
$767$	−0.00116441	−4.20445e−5 0
$768$	0	0
$769$	11.6770	0.421084	0.210542	−	0.977585i	$-0.432477\pi$
$769$	11.6770	0.421084	0.210542	+	0.977585i	$0.432477\pi$
$770$	0	0
$771$	−25.2683	−0.910017
$772$	0	0
$773$	−43.2250	−1.55470	−0.777348	−	0.629070i	$-0.783436\pi$
$773$	−43.2250	−1.55470	−0.777348	+	0.629070i	$0.783436\pi$
$774$	0	0
$775$	0.0235710	0.000846694 0
$776$	0	0
$777$	−33.4120	−1.19865
$778$	0	0
$779$	14.7365	0.527992
$780$	0	0
$781$	14.0132	0.501431
$782$	0	0
$783$	3.21297	0.114822
$784$	0	0
$785$	49.8676	1.77985
$786$	0	0
$787$	−41.6501	−1.48467	−0.742334	−	0.670030i	$-0.766281\pi$
$787$	−41.6501	−1.48467	−0.742334	+	0.670030i	$0.766281\pi$
$788$	0	0
$789$	−43.4059	−1.54529
$790$	0	0
$791$	27.6881	0.984477
$792$	0	0
$793$	0.0325473	0.00115579
$794$	0	0
$795$	26.4134	0.936787
$796$	0	0
$797$	20.0955	0.711819	0.355910	−	0.934520i	$-0.384171\pi$
$797$	20.0955	0.711819	0.355910	+	0.934520i	$0.384171\pi$
$798$	0	0
$799$	−7.21753	−0.255338
$800$	0	0
$801$	−2.42226	−0.0855864
$802$	0	0
$803$	−22.9448	−0.809703
$804$	0	0
$805$	13.8870	0.489453
$806$	0	0
$807$	43.2202	1.52142
$808$	0	0
$809$	−10.5903	−0.372334	−0.186167	−	0.982518i	$-0.559607\pi$
$809$	−10.5903	−0.372334	−0.186167	+	0.982518i	$0.559607\pi$
$810$	0	0
$811$	−29.9332	−1.05110	−0.525549	−	0.850764i	$-0.676140\pi$
$811$	−29.9332	−1.05110	−0.525549	+	0.850764i	$0.676140\pi$
$812$	0	0
$813$	20.9760	0.735660
$814$	0	0
$815$	16.1399	0.565355
$816$	0	0
$817$	−19.1672	−0.670575
$818$	0	0
$819$	−0.0118872	−0.000415372 0
$820$	0	0
$821$	1.47970	0.0516419	0.0258209	−	0.999667i	$-0.491780\pi$
$821$	1.47970	0.0516419	0.0258209	+	0.999667i	$0.491780\pi$
$822$	0	0
$823$	20.9277	0.729494	0.364747	−	0.931107i	$-0.381155\pi$
$823$	20.9277	0.729494	0.364747	+	0.931107i	$0.381155\pi$
$824$	0	0
$825$	−0.0164490	−0.000572680 0
$826$	0	0
$827$	−14.8863	−0.517646	−0.258823	−	0.965925i	$-0.583335\pi$
$827$	−14.8863	−0.517646	−0.258823	+	0.965925i	$0.583335\pi$
$828$	0	0
$829$	54.9732	1.90930	0.954649	−	0.297733i	$-0.0962304\pi$
$829$	54.9732	1.90930	0.954649	+	0.297733i	$0.0962304\pi$
$830$	0	0
$831$	−21.8539	−0.758104
$832$	0	0
$833$	6.69743	0.232052
$834$	0	0
$835$	−25.5812	−0.885273
$836$	0	0
$837$	18.2746	0.631663
$838$	0	0
$839$	40.8460	1.41016	0.705080	−	0.709128i	$-0.250911\pi$
$839$	40.8460	1.41016	0.705080	+	0.709128i	$0.250911\pi$
$840$	0	0
$841$	−28.6727	−0.988715
$842$	0	0
$843$	−23.4104	−0.806298
$844$	0	0
$845$	29.0478	0.999274
$846$	0	0
$847$	−31.4957	−1.08221
$848$	0	0
$849$	−15.7556	−0.540729
$850$	0	0
$851$	10.5870	0.362918
$852$	0	0
$853$	−8.90994	−0.305071	−0.152535	−	0.988298i	$-0.548744\pi$
$853$	−8.90994	−0.305071	−0.152535	+	0.988298i	$0.548744\pi$
$854$	0	0
$855$	−4.00953	−0.137123
$856$	0	0
$857$	32.1479	1.09815	0.549076	−	0.835773i	$-0.314980\pi$
$857$	32.1479	1.09815	0.549076	+	0.835773i	$0.314980\pi$
$858$	0	0
$859$	25.4577	0.868606	0.434303	−	0.900767i	$-0.356995\pi$
$859$	25.4577	0.868606	0.434303	+	0.900767i	$0.356995\pi$
$860$	0	0
$861$	−32.0915	−1.09367
$862$	0	0
$863$	42.3500	1.44161	0.720805	−	0.693138i	$-0.243772\pi$
$863$	42.3500	1.44161	0.720805	+	0.693138i	$0.243772\pi$
$864$	0	0
$865$	28.1140	0.955906
$866$	0	0
$867$	−23.7889	−0.807913
$868$	0	0
$869$	3.00102	0.101803
$870$	0	0
$871$	−0.0283103	−0.000959259 0
$872$	0	0
$873$	10.6850	0.361632
$874$	0	0
$875$	40.3152	1.36290
$876$	0	0
$877$	−54.7498	−1.84877	−0.924385	−	0.381461i	$-0.875421\pi$
$877$	−54.7498	−1.84877	−0.924385	+	0.381461i	$0.875421\pi$
$878$	0	0
$879$	−2.87396	−0.0969361
$880$	0	0
$881$	−42.7752	−1.44113	−0.720566	−	0.693386i	$-0.756118\pi$
$881$	−42.7752	−1.44113	−0.720566	+	0.693386i	$0.756118\pi$
$882$	0	0
$883$	51.0453	1.71781	0.858906	−	0.512133i	$-0.171144\pi$
$883$	51.0453	1.71781	0.858906	+	0.512133i	$0.171144\pi$
$884$	0	0
$885$	0.855295	0.0287504
$886$	0	0
$887$	15.6713	0.526191	0.263096	−	0.964770i	$-0.415256\pi$
$887$	15.6713	0.526191	0.263096	+	0.964770i	$0.415256\pi$
$888$	0	0
$889$	−31.8505	−1.06823
$890$	0	0
$891$	−9.51592	−0.318795
$892$	0	0
$893$	16.1186	0.539389
$894$	0	0
$895$	2.75747	0.0921720
$896$	0	0
$897$	−0.0119740	−0.000399801 0
$898$	0	0
$899$	1.86134	0.0620792
$900$	0	0
$901$	−8.75831	−0.291782
$902$	0	0
$903$	41.7400	1.38902
$904$	0	0
$905$	37.9923	1.26291
$906$	0	0
$907$	−50.8029	−1.68688	−0.843442	−	0.537221i	$-0.819474\pi$
$907$	−50.8029	−1.68688	−0.843442	+	0.537221i	$0.819474\pi$
$908$	0	0
$909$	−8.08259	−0.268083
$910$	0	0
$911$	47.4669	1.57265	0.786324	−	0.617815i	$-0.211982\pi$
$911$	47.4669	1.57265	0.786324	+	0.617815i	$0.211982\pi$
$912$	0	0
$913$	−10.3961	−0.344059
$914$	0	0
$915$	−23.9069	−0.790337
$916$	0	0
$917$	46.5192	1.53620
$918$	0	0
$919$	−18.1424	−0.598462	−0.299231	−	0.954181i	$-0.596730\pi$
$919$	−18.1424	−0.598462	−0.299231	+	0.954181i	$0.596730\pi$
$920$	0	0
$921$	26.7202	0.880461
$922$	0	0
$923$	0.0428442	0.00141023
$924$	0	0
$925$	0.0444659	0.00146203
$926$	0	0
$927$	3.33814	0.109639
$928$	0	0
$929$	−36.7632	−1.20616	−0.603081	−	0.797680i	$-0.706060\pi$
$929$	−36.7632	−1.20616	−0.603081	+	0.797680i	$0.706060\pi$
$930$	0	0
$931$	−14.9571	−0.490200
$932$	0	0
$933$	42.3099	1.38516
$934$	0	0
$935$	−3.75909	−0.122935
$936$	0	0
$937$	−29.1597	−0.952607	−0.476304	−	0.879281i	$-0.658024\pi$
$937$	−29.1597	−0.952607	−0.476304	+	0.879281i	$0.658024\pi$
$938$	0	0
$939$	7.58543	0.247541
$940$	0	0
$941$	−11.1041	−0.361983	−0.180991	−	0.983485i	$-0.557931\pi$
$941$	−11.1041	−0.361983	−0.180991	+	0.983485i	$0.557931\pi$
$942$	0	0
$943$	10.1686	0.331134
$944$	0	0
$945$	45.2204	1.47102
$946$	0	0
$947$	−7.44473	−0.241921	−0.120961	−	0.992657i	$-0.538597\pi$
$947$	−7.44473	−0.241921	−0.120961	+	0.992657i	$0.538597\pi$
$948$	0	0
$949$	−0.0701518	−0.00227722
$950$	0	0
$951$	0.539270	0.0174870
$952$	0	0
$953$	−40.8632	−1.32369	−0.661844	−	0.749641i	$-0.730226\pi$
$953$	−40.8632	−1.32369	−0.661844	+	0.749641i	$0.730226\pi$
$954$	0	0
$955$	0.477394	0.0154481
$956$	0	0
$957$	−1.29894	−0.0419886
$958$	0	0
$959$	13.8228	0.446362
$960$	0	0
$961$	−20.4131	−0.658488
$962$	0	0
$963$	−5.80274	−0.186991
$964$	0	0
$965$	7.85042	0.252714
$966$	0	0
$967$	−2.29088	−0.0736697	−0.0368349	−	0.999321i	$-0.511728\pi$
$967$	−2.29088	−0.0736697	−0.0368349	+	0.999321i	$0.511728\pi$
$968$	0	0
$969$	−4.22651	−0.135775
$970$	0	0
$971$	27.8481	0.893689	0.446845	−	0.894612i	$-0.352548\pi$
$971$	27.8481	0.893689	0.446845	+	0.894612i	$0.352548\pi$
$972$	0	0
$973$	−33.2556	−1.06612
$974$	0	0
$975$	−5.02915e−5 0	−1.61062e−6 0
$976$	0	0
$977$	2.65190	0.0848419	0.0424209	−	0.999100i	$-0.486493\pi$
$977$	2.65190	0.0848419	0.0424209	+	0.999100i	$0.486493\pi$
$978$	0	0
$979$	5.07165	0.162090
$980$	0	0
$981$	0.868565	0.0277312
$982$	0	0
$983$	−17.5940	−0.561162	−0.280581	−	0.959830i	$-0.590527\pi$
$983$	−17.5940	−0.561162	−0.280581	+	0.959830i	$0.590527\pi$
$984$	0	0
$985$	25.7980	0.821991
$986$	0	0
$987$	−35.1012	−1.11728
$988$	0	0
$989$	−13.2258	−0.420556
$990$	0	0
$991$	40.1464	1.27529	0.637646	−	0.770329i	$-0.279908\pi$
$991$	40.1464	1.27529	0.637646	+	0.770329i	$0.279908\pi$
$992$	0	0
$993$	−24.5468	−0.778970
$994$	0	0
$995$	−10.3611	−0.328469
$996$	0	0
$997$	−52.3474	−1.65786	−0.828930	−	0.559353i	$-0.811050\pi$
$997$	−52.3474	−1.65786	−0.828930	+	0.559353i	$0.811050\pi$
$998$	0	0
$999$	34.4745	1.09072

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.p.1.7		10
4.3	odd	2		503.2.a.e.1.10	✓	10
12.11	even	2		4527.2.a.k.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.10	✓	10	4.3	odd	2
4527.2.a.k.1.1		10	12.11	even	2
8048.2.a.p.1.7		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)

\(f(q)\)	\(=\)	\(q+1.51067 q^{3} -2.23445 q^{5} +3.60329 q^{7} -0.717874 q^{9} +O(q^{10})\)	\(q+1.51067 q^{3} -2.23445 q^{5} +3.60329 q^{7} -0.717874 q^{9} +1.50306 q^{11} +0.00459548 q^{13} -3.37551 q^{15} +1.11927 q^{17} -2.49963 q^{19} +5.44339 q^{21} -1.72480 q^{23} -0.00724425 q^{25} -5.61648 q^{27} -0.572061 q^{29} -3.25375 q^{31} +2.27063 q^{33} -8.05137 q^{35} -6.13809 q^{37} +0.00694226 q^{39} -5.89549 q^{41} +7.66800 q^{43} +1.60405 q^{45} -6.44840 q^{47} +5.98373 q^{49} +1.69085 q^{51} -7.82500 q^{53} -3.35851 q^{55} -3.77612 q^{57} -0.253382 q^{59} +7.08244 q^{61} -2.58671 q^{63} -0.0102684 q^{65} -6.16046 q^{67} -2.60561 q^{69} +9.32311 q^{71} -15.2654 q^{73} -0.0109437 q^{75} +5.41596 q^{77} +1.99661 q^{79} -6.33104 q^{81} -6.91660 q^{83} -2.50096 q^{85} -0.864195 q^{87} +3.37422 q^{89} +0.0165589 q^{91} -4.91534 q^{93} +5.58529 q^{95} -14.8842 q^{97} -1.07901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) \) 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

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