LMFDB - Embedded newform 855.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, 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("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.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:	$6.82720937282$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -1.41421 q^{7} -4.41421 q^{8} +O(q^{10})$	\(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -1.41421 q^{7} -4.41421 q^{8} -2.41421 q^{10} +2.24264 q^{11} -3.41421 q^{13} +3.41421 q^{14} +3.00000 q^{16} -1.17157 q^{17} -1.00000 q^{19} +3.82843 q^{20} -5.41421 q^{22} -7.65685 q^{23} +1.00000 q^{25} +8.24264 q^{26} -5.41421 q^{28} -1.41421 q^{29} -3.17157 q^{31} +1.58579 q^{32} +2.82843 q^{34} -1.41421 q^{35} -3.41421 q^{37} +2.41421 q^{38} -4.41421 q^{40} +0.242641 q^{41} +12.2426 q^{43} +8.58579 q^{44} +18.4853 q^{46} +7.65685 q^{47} -5.00000 q^{49} -2.41421 q^{50} -13.0711 q^{52} -8.00000 q^{53} +2.24264 q^{55} +6.24264 q^{56} +3.41421 q^{58} -12.4853 q^{59} +7.31371 q^{61} +7.65685 q^{62} -9.82843 q^{64} -3.41421 q^{65} -9.65685 q^{67} -4.48528 q^{68} +3.41421 q^{70} +10.8284 q^{71} -7.65685 q^{73} +8.24264 q^{74} -3.82843 q^{76} -3.17157 q^{77} +3.00000 q^{80} -0.585786 q^{82} -12.8284 q^{83} -1.17157 q^{85} -29.5563 q^{86} -9.89949 q^{88} -5.89949 q^{89} +4.82843 q^{91} -29.3137 q^{92} -18.4853 q^{94} -1.00000 q^{95} -9.75736 q^{97} +12.0711 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8} - 2 q^{10} - 4 q^{11} - 4 q^{13} + 4 q^{14} + 6 q^{16} - 8 q^{17} - 2 q^{19} + 2 q^{20} - 8 q^{22} - 4 q^{23} + 2 q^{25} + 8 q^{26} - 8 q^{28} - 12 q^{31} + 6 q^{32} - 4 q^{37} + 2 q^{38} - 6 q^{40} - 8 q^{41} + 16 q^{43} + 20 q^{44} + 20 q^{46} + 4 q^{47} - 10 q^{49} - 2 q^{50} - 12 q^{52} - 16 q^{53} - 4 q^{55} + 4 q^{56} + 4 q^{58} - 8 q^{59} - 8 q^{61} + 4 q^{62} - 14 q^{64} - 4 q^{65} - 8 q^{67} + 8 q^{68} + 4 q^{70} + 16 q^{71} - 4 q^{73} + 8 q^{74} - 2 q^{76} - 12 q^{77} + 6 q^{80} - 4 q^{82} - 20 q^{83} - 8 q^{85} - 28 q^{86} + 8 q^{89} + 4 q^{91} - 36 q^{92} - 20 q^{94} - 2 q^{95} - 28 q^{97} + 10 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8 - 2 * q^10 - 4 * q^11 - 4 * q^13 + 4 * q^14 + 6 * q^16 - 8 * q^17 - 2 * q^19 + 2 * q^20 - 8 * q^22 - 4 * q^23 + 2 * q^25 + 8 * q^26 - 8 * q^28 - 12 * q^31 + 6 * q^32 - 4 * q^37 + 2 * q^38 - 6 * q^40 - 8 * q^41 + 16 * q^43 + 20 * q^44 + 20 * q^46 + 4 * q^47 - 10 * q^49 - 2 * q^50 - 12 * q^52 - 16 * q^53 - 4 * q^55 + 4 * q^56 + 4 * q^58 - 8 * q^59 - 8 * q^61 + 4 * q^62 - 14 * q^64 - 4 * q^65 - 8 * q^67 + 8 * q^68 + 4 * q^70 + 16 * q^71 - 4 * q^73 + 8 * q^74 - 2 * q^76 - 12 * q^77 + 6 * q^80 - 4 * q^82 - 20 * q^83 - 8 * q^85 - 28 * q^86 + 8 * q^89 + 4 * q^91 - 36 * q^92 - 20 * q^94 - 2 * q^95 - 28 * q^97 + 10 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	−4.41421	−1.56066
$9$	0	0
$10$	−2.41421	−0.763441
$11$	2.24264	0.676182	0.338091	−	0.941113i	$-0.390219\pi$
$11$	2.24264	0.676182	0.338091	+	0.941113i	$0.390219\pi$
$12$	0	0
$13$	−3.41421	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$13$	−3.41421	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$14$	3.41421	0.912487
$15$	0	0
$16$	3.00000	0.750000
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	3.82843	0.856062
$21$	0	0
$22$	−5.41421	−1.15431
$23$	−7.65685	−1.59656	−0.798282	−	0.602284i	$-0.794258\pi$
$23$	−7.65685	−1.59656	−0.798282	+	0.602284i	$0.794258\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	8.24264	1.61651
$27$	0	0
$28$	−5.41421	−1.02319
$29$	−1.41421	−0.262613	−0.131306	−	0.991342i	$-0.541917\pi$
$29$	−1.41421	−0.262613	−0.131306	+	0.991342i	$0.541917\pi$
$30$	0	0
$31$	−3.17157	−0.569631	−0.284816	−	0.958582i	$-0.591932\pi$
$31$	−3.17157	−0.569631	−0.284816	+	0.958582i	$0.591932\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	2.82843	0.485071
$35$	−1.41421	−0.239046
$36$	0	0
$37$	−3.41421	−0.561293	−0.280647	−	0.959811i	$-0.590549\pi$
$37$	−3.41421	−0.561293	−0.280647	+	0.959811i	$0.590549\pi$
$38$	2.41421	0.391637
$39$	0	0
$40$	−4.41421	−0.697948
$41$	0.242641	0.0378941	0.0189471	−	0.999820i	$-0.493969\pi$
$41$	0.242641	0.0378941	0.0189471	+	0.999820i	$0.493969\pi$
$42$	0	0
$43$	12.2426	1.86699	0.933493	−	0.358597i	$-0.116745\pi$
$43$	12.2426	1.86699	0.933493	+	0.358597i	$0.116745\pi$
$44$	8.58579	1.29436
$45$	0	0
$46$	18.4853	2.72551
$47$	7.65685	1.11687	0.558433	−	0.829549i	$-0.311403\pi$
$47$	7.65685	1.11687	0.558433	+	0.829549i	$0.311403\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	−2.41421	−0.341421
$51$	0	0
$52$	−13.0711	−1.81263
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	0	0
$55$	2.24264	0.302398
$56$	6.24264	0.834208
$57$	0	0
$58$	3.41421	0.448308
$59$	−12.4853	−1.62545	−0.812723	−	0.582651i	$-0.802016\pi$
$59$	−12.4853	−1.62545	−0.812723	+	0.582651i	$0.802016\pi$
$60$	0	0
$61$	7.31371	0.936424	0.468212	−	0.883616i	$-0.344898\pi$
$61$	7.31371	0.936424	0.468212	+	0.883616i	$0.344898\pi$
$62$	7.65685	0.972421
$63$	0	0
$64$	−9.82843	−1.22855
$65$	−3.41421	−0.423481
$66$	0	0
$67$	−9.65685	−1.17977	−0.589886	−	0.807486i	$-0.700827\pi$
$67$	−9.65685	−1.17977	−0.589886	+	0.807486i	$0.700827\pi$
$68$	−4.48528	−0.543920
$69$	0	0
$70$	3.41421	0.408077
$71$	10.8284	1.28510	0.642549	−	0.766245i	$-0.277877\pi$
$71$	10.8284	1.28510	0.642549	+	0.766245i	$0.277877\pi$
$72$	0	0
$73$	−7.65685	−0.896167	−0.448084	−	0.893992i	$-0.647893\pi$
$73$	−7.65685	−0.896167	−0.448084	+	0.893992i	$0.647893\pi$
$74$	8.24264	0.958188
$75$	0	0
$76$	−3.82843	−0.439151
$77$	−3.17157	−0.361434
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−0.585786	−0.0646893
$83$	−12.8284	−1.40810	−0.704051	−	0.710149i	$-0.748628\pi$
$83$	−12.8284	−1.40810	−0.704051	+	0.710149i	$0.748628\pi$
$84$	0	0
$85$	−1.17157	−0.127075
$86$	−29.5563	−3.18714
$87$	0	0
$88$	−9.89949	−1.05529
$89$	−5.89949	−0.625345	−0.312673	−	0.949861i	$-0.601224\pi$
$89$	−5.89949	−0.625345	−0.312673	+	0.949861i	$0.601224\pi$
$90$	0	0
$91$	4.82843	0.506157
$92$	−29.3137	−3.05617
$93$	0	0
$94$	−18.4853	−1.90661
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−9.75736	−0.990710	−0.495355	−	0.868691i	$-0.664962\pi$
$97$	−9.75736	−0.990710	−0.495355	+	0.868691i	$0.664962\pi$
$98$	12.0711	1.21936
$99$	0	0
$100$	3.82843	0.382843
$101$	3.17157	0.315583	0.157792	−	0.987472i	$-0.449563\pi$
$101$	3.17157	0.315583	0.157792	+	0.987472i	$0.449563\pi$
$102$	0	0
$103$	−7.31371	−0.720641	−0.360321	−	0.932829i	$-0.617333\pi$
$103$	−7.31371	−0.720641	−0.360321	+	0.932829i	$0.617333\pi$
$104$	15.0711	1.47784
$105$	0	0
$106$	19.3137	1.87591
$107$	−19.3137	−1.86713	−0.933563	−	0.358412i	$-0.883318\pi$
$107$	−19.3137	−1.86713	−0.933563	+	0.358412i	$0.883318\pi$
$108$	0	0
$109$	−6.48528	−0.621177	−0.310589	−	0.950544i	$-0.600526\pi$
$109$	−6.48528	−0.621177	−0.310589	+	0.950544i	$0.600526\pi$
$110$	−5.41421	−0.516225
$111$	0	0
$112$	−4.24264	−0.400892
$113$	−10.1421	−0.954092	−0.477046	−	0.878878i	$-0.658292\pi$
$113$	−10.1421	−0.954092	−0.477046	+	0.878878i	$0.658292\pi$
$114$	0	0
$115$	−7.65685	−0.714005
$116$	−5.41421	−0.502697
$117$	0	0
$118$	30.1421	2.77481
$119$	1.65685	0.151884
$120$	0	0
$121$	−5.97056	−0.542778
$122$	−17.6569	−1.59858
$123$	0	0
$124$	−12.1421	−1.09040
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.3137	1.71381	0.856907	−	0.515471i	$-0.172383\pi$
$127$	19.3137	1.71381	0.856907	+	0.515471i	$0.172383\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	8.24264	0.722927
$131$	8.58579	0.750144	0.375072	−	0.926996i	$-0.377618\pi$
$131$	8.58579	0.750144	0.375072	+	0.926996i	$0.377618\pi$
$132$	0	0
$133$	1.41421	0.122628
$134$	23.3137	2.01400
$135$	0	0
$136$	5.17157	0.443459
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−14.1421	−1.19952	−0.599760	−	0.800180i	$-0.704737\pi$
$139$	−14.1421	−1.19952	−0.599760	+	0.800180i	$0.704737\pi$
$140$	−5.41421	−0.457585
$141$	0	0
$142$	−26.1421	−2.19380
$143$	−7.65685	−0.640298
$144$	0	0
$145$	−1.41421	−0.117444
$146$	18.4853	1.52985
$147$	0	0
$148$	−13.0711	−1.07444
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−6.48528	−0.527765	−0.263882	−	0.964555i	$-0.585003\pi$
$151$	−6.48528	−0.527765	−0.263882	+	0.964555i	$0.585003\pi$
$152$	4.41421	0.358040
$153$	0	0
$154$	7.65685	0.617007
$155$	−3.17157	−0.254747
$156$	0	0
$157$	10.4853	0.836817	0.418408	−	0.908259i	$-0.362588\pi$
$157$	10.4853	0.836817	0.418408	+	0.908259i	$0.362588\pi$
$158$	0	0
$159$	0	0
$160$	1.58579	0.125367
$161$	10.8284	0.853400
$162$	0	0
$163$	21.8995	1.71530	0.857650	−	0.514233i	$-0.171923\pi$
$163$	21.8995	1.71530	0.857650	+	0.514233i	$0.171923\pi$
$164$	0.928932	0.0725374
$165$	0	0
$166$	30.9706	2.40378
$167$	17.3137	1.33977	0.669887	−	0.742463i	$-0.266342\pi$
$167$	17.3137	1.33977	0.669887	+	0.742463i	$0.266342\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	2.82843	0.216930
$171$	0	0
$172$	46.8701	3.57381
$173$	19.7990	1.50529	0.752645	−	0.658427i	$-0.228778\pi$
$173$	19.7990	1.50529	0.752645	+	0.658427i	$0.228778\pi$
$174$	0	0
$175$	−1.41421	−0.106904
$176$	6.72792	0.507136
$177$	0	0
$178$	14.2426	1.06753
$179$	−16.4853	−1.23217	−0.616084	−	0.787681i	$-0.711282\pi$
$179$	−16.4853	−1.23217	−0.616084	+	0.787681i	$0.711282\pi$
$180$	0	0
$181$	−20.8284	−1.54816	−0.774082	−	0.633085i	$-0.781788\pi$
$181$	−20.8284	−1.54816	−0.774082	+	0.633085i	$0.781788\pi$
$182$	−11.6569	−0.864064
$183$	0	0
$184$	33.7990	2.49169
$185$	−3.41421	−0.251018
$186$	0	0
$187$	−2.62742	−0.192136
$188$	29.3137	2.13792
$189$	0	0
$190$	2.41421	0.175145
$191$	−10.2426	−0.741131	−0.370566	−	0.928806i	$-0.620836\pi$
$191$	−10.2426	−0.741131	−0.370566	+	0.928806i	$0.620836\pi$
$192$	0	0
$193$	5.07107	0.365023	0.182512	−	0.983204i	$-0.441577\pi$
$193$	5.07107	0.365023	0.182512	+	0.983204i	$0.441577\pi$
$194$	23.5563	1.69125
$195$	0	0
$196$	−19.1421	−1.36730
$197$	−6.82843	−0.486505	−0.243253	−	0.969963i	$-0.578214\pi$
$197$	−6.82843	−0.486505	−0.243253	+	0.969963i	$0.578214\pi$
$198$	0	0
$199$	18.1421	1.28606	0.643031	−	0.765840i	$-0.277677\pi$
$199$	18.1421	1.28606	0.643031	+	0.765840i	$0.277677\pi$
$200$	−4.41421	−0.312132
$201$	0	0
$202$	−7.65685	−0.538734
$203$	2.00000	0.140372
$204$	0	0
$205$	0.242641	0.0169468
$206$	17.6569	1.23021
$207$	0	0
$208$	−10.2426	−0.710199
$209$	−2.24264	−0.155127
$210$	0	0
$211$	7.31371	0.503496	0.251748	−	0.967793i	$-0.418995\pi$
$211$	7.31371	0.503496	0.251748	+	0.967793i	$0.418995\pi$
$212$	−30.6274	−2.10350
$213$	0	0
$214$	46.6274	3.18738
$215$	12.2426	0.834941
$216$	0	0
$217$	4.48528	0.304481
$218$	15.6569	1.06042
$219$	0	0
$220$	8.58579	0.578854
$221$	4.00000	0.269069
$222$	0	0
$223$	18.6274	1.24738	0.623692	−	0.781670i	$-0.285632\pi$
$223$	18.6274	1.24738	0.623692	+	0.781670i	$0.285632\pi$
$224$	−2.24264	−0.149843
$225$	0	0
$226$	24.4853	1.62874
$227$	−14.9706	−0.993631	−0.496816	−	0.867856i	$-0.665497\pi$
$227$	−14.9706	−0.993631	−0.496816	+	0.867856i	$0.665497\pi$
$228$	0	0
$229$	−22.6274	−1.49526	−0.747631	−	0.664114i	$-0.768809\pi$
$229$	−22.6274	−1.49526	−0.747631	+	0.664114i	$0.768809\pi$
$230$	18.4853	1.21888
$231$	0	0
$232$	6.24264	0.409849
$233$	0.343146	0.0224802	0.0112401	−	0.999937i	$-0.496422\pi$
$233$	0.343146	0.0224802	0.0112401	+	0.999937i	$0.496422\pi$
$234$	0	0
$235$	7.65685	0.499478
$236$	−47.7990	−3.11145
$237$	0	0
$238$	−4.00000	−0.259281
$239$	26.7279	1.72889	0.864443	−	0.502731i	$-0.167671\pi$
$239$	26.7279	1.72889	0.864443	+	0.502731i	$0.167671\pi$
$240$	0	0
$241$	−19.6569	−1.26621	−0.633105	−	0.774066i	$-0.718220\pi$
$241$	−19.6569	−1.26621	−0.633105	+	0.774066i	$0.718220\pi$
$242$	14.4142	0.926581
$243$	0	0
$244$	28.0000	1.79252
$245$	−5.00000	−0.319438
$246$	0	0
$247$	3.41421	0.217241
$248$	14.0000	0.889001
$249$	0	0
$250$	−2.41421	−0.152688
$251$	1.75736	0.110924	0.0554618	−	0.998461i	$-0.482337\pi$
$251$	1.75736	0.110924	0.0554618	+	0.998461i	$0.482337\pi$
$252$	0	0
$253$	−17.1716	−1.07957
$254$	−46.6274	−2.92566
$255$	0	0
$256$	−29.9706	−1.87316
$257$	4.48528	0.279784	0.139892	−	0.990167i	$-0.455324\pi$
$257$	4.48528	0.279784	0.139892	+	0.990167i	$0.455324\pi$
$258$	0	0
$259$	4.82843	0.300024
$260$	−13.0711	−0.810633
$261$	0	0
$262$	−20.7279	−1.28058
$263$	23.4558	1.44635	0.723175	−	0.690665i	$-0.242682\pi$
$263$	23.4558	1.44635	0.723175	+	0.690665i	$0.242682\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	−3.41421	−0.209339
$267$	0	0
$268$	−36.9706	−2.25834
$269$	3.07107	0.187246	0.0936232	−	0.995608i	$-0.470155\pi$
$269$	3.07107	0.187246	0.0936232	+	0.995608i	$0.470155\pi$
$270$	0	0
$271$	−10.8284	−0.657780	−0.328890	−	0.944368i	$-0.606675\pi$
$271$	−10.8284	−0.657780	−0.328890	+	0.944368i	$0.606675\pi$
$272$	−3.51472	−0.213111
$273$	0	0
$274$	24.1421	1.45848
$275$	2.24264	0.135236
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	34.1421	2.04771
$279$	0	0
$280$	6.24264	0.373069
$281$	14.5858	0.870115	0.435058	−	0.900403i	$-0.356728\pi$
$281$	14.5858	0.870115	0.435058	+	0.900403i	$0.356728\pi$
$282$	0	0
$283$	−10.3848	−0.617311	−0.308655	−	0.951174i	$-0.599879\pi$
$283$	−10.3848	−0.617311	−0.308655	+	0.951174i	$0.599879\pi$
$284$	41.4558	2.45995
$285$	0	0
$286$	18.4853	1.09306
$287$	−0.343146	−0.0202553
$288$	0	0
$289$	−15.6274	−0.919260
$290$	3.41421	0.200490
$291$	0	0
$292$	−29.3137	−1.71546
$293$	11.5147	0.672697	0.336349	−	0.941738i	$-0.390808\pi$
$293$	11.5147	0.672697	0.336349	+	0.941738i	$0.390808\pi$
$294$	0	0
$295$	−12.4853	−0.726921
$296$	15.0711	0.875988
$297$	0	0
$298$	−14.4853	−0.839110
$299$	26.1421	1.51184
$300$	0	0
$301$	−17.3137	−0.997946
$302$	15.6569	0.900951
$303$	0	0
$304$	−3.00000	−0.172062
$305$	7.31371	0.418782
$306$	0	0
$307$	21.1716	1.20833	0.604163	−	0.796861i	$-0.293508\pi$
$307$	21.1716	1.20833	0.604163	+	0.796861i	$0.293508\pi$
$308$	−12.1421	−0.691862
$309$	0	0
$310$	7.65685	0.434880
$311$	6.24264	0.353988	0.176994	−	0.984212i	$-0.443363\pi$
$311$	6.24264	0.353988	0.176994	+	0.984212i	$0.443363\pi$
$312$	0	0
$313$	5.79899	0.327778	0.163889	−	0.986479i	$-0.447596\pi$
$313$	5.79899	0.327778	0.163889	+	0.986479i	$0.447596\pi$
$314$	−25.3137	−1.42854
$315$	0	0
$316$	0	0
$317$	26.6274	1.49554	0.747772	−	0.663955i	$-0.231123\pi$
$317$	26.6274	1.49554	0.747772	+	0.663955i	$0.231123\pi$
$318$	0	0
$319$	−3.17157	−0.177574
$320$	−9.82843	−0.549426
$321$	0	0
$322$	−26.1421	−1.45684
$323$	1.17157	0.0651881
$324$	0	0
$325$	−3.41421	−0.189386
$326$	−52.8701	−2.92820
$327$	0	0
$328$	−1.07107	−0.0591398
$329$	−10.8284	−0.596991
$330$	0	0
$331$	28.1421	1.54683	0.773416	−	0.633899i	$-0.218547\pi$
$331$	28.1421	1.54683	0.773416	+	0.633899i	$0.218547\pi$
$332$	−49.1127	−2.69541
$333$	0	0
$334$	−41.7990	−2.28714
$335$	−9.65685	−0.527610
$336$	0	0
$337$	−24.5858	−1.33927	−0.669637	−	0.742689i	$-0.733550\pi$
$337$	−24.5858	−1.33927	−0.669637	+	0.742689i	$0.733550\pi$
$338$	3.24264	0.176376
$339$	0	0
$340$	−4.48528	−0.243249
$341$	−7.11270	−0.385174
$342$	0	0
$343$	16.9706	0.916324
$344$	−54.0416	−2.91373
$345$	0	0
$346$	−47.7990	−2.56969
$347$	27.4558	1.47391	0.736953	−	0.675943i	$-0.236264\pi$
$347$	27.4558	1.47391	0.736953	+	0.675943i	$0.236264\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	3.41421	0.182497
$351$	0	0
$352$	3.55635	0.189554
$353$	−3.65685	−0.194635	−0.0973174	−	0.995253i	$-0.531026\pi$
$353$	−3.65685	−0.194635	−0.0973174	+	0.995253i	$0.531026\pi$
$354$	0	0
$355$	10.8284	0.574713
$356$	−22.5858	−1.19704
$357$	0	0
$358$	39.7990	2.10344
$359$	−24.8701	−1.31259	−0.656296	−	0.754504i	$-0.727878\pi$
$359$	−24.8701	−1.31259	−0.656296	+	0.754504i	$0.727878\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	50.2843	2.64288
$363$	0	0
$364$	18.4853	0.968892
$365$	−7.65685	−0.400778
$366$	0	0
$367$	−14.3848	−0.750879	−0.375440	−	0.926847i	$-0.622508\pi$
$367$	−14.3848	−0.750879	−0.375440	+	0.926847i	$0.622508\pi$
$368$	−22.9706	−1.19742
$369$	0	0
$370$	8.24264	0.428514
$371$	11.3137	0.587378
$372$	0	0
$373$	0.585786	0.0303309	0.0151654	−	0.999885i	$-0.495173\pi$
$373$	0.585786	0.0303309	0.0151654	+	0.999885i	$0.495173\pi$
$374$	6.34315	0.327996
$375$	0	0
$376$	−33.7990	−1.74305
$377$	4.82843	0.248677
$378$	0	0
$379$	3.17157	0.162913	0.0814564	−	0.996677i	$-0.474043\pi$
$379$	3.17157	0.162913	0.0814564	+	0.996677i	$0.474043\pi$
$380$	−3.82843	−0.196394
$381$	0	0
$382$	24.7279	1.26519
$383$	−28.0000	−1.43073	−0.715367	−	0.698749i	$-0.753740\pi$
$383$	−28.0000	−1.43073	−0.715367	+	0.698749i	$0.753740\pi$
$384$	0	0
$385$	−3.17157	−0.161638
$386$	−12.2426	−0.623134
$387$	0	0
$388$	−37.3553	−1.89643
$389$	−30.9706	−1.57027	−0.785135	−	0.619325i	$-0.787406\pi$
$389$	−30.9706	−1.57027	−0.785135	+	0.619325i	$0.787406\pi$
$390$	0	0
$391$	8.97056	0.453661
$392$	22.0711	1.11476
$393$	0	0
$394$	16.4853	0.830516
$395$	0	0
$396$	0	0
$397$	28.6274	1.43677	0.718384	−	0.695646i	$-0.244882\pi$
$397$	28.6274	1.43677	0.718384	+	0.695646i	$0.244882\pi$
$398$	−43.7990	−2.19544
$399$	0	0
$400$	3.00000	0.150000
$401$	7.75736	0.387384	0.193692	−	0.981062i	$-0.437954\pi$
$401$	7.75736	0.387384	0.193692	+	0.981062i	$0.437954\pi$
$402$	0	0
$403$	10.8284	0.539402
$404$	12.1421	0.604094
$405$	0	0
$406$	−4.82843	−0.239631
$407$	−7.65685	−0.379536
$408$	0	0
$409$	−12.8284	−0.634325	−0.317162	−	0.948371i	$-0.602730\pi$
$409$	−12.8284	−0.634325	−0.317162	+	0.948371i	$0.602730\pi$
$410$	−0.585786	−0.0289299
$411$	0	0
$412$	−28.0000	−1.37946
$413$	17.6569	0.868837
$414$	0	0
$415$	−12.8284	−0.629723
$416$	−5.41421	−0.265454
$417$	0	0
$418$	5.41421	0.264818
$419$	−3.41421	−0.166795	−0.0833976	−	0.996516i	$-0.526577\pi$
$419$	−3.41421	−0.166795	−0.0833976	+	0.996516i	$0.526577\pi$
$420$	0	0
$421$	9.31371	0.453922	0.226961	−	0.973904i	$-0.427121\pi$
$421$	9.31371	0.453922	0.226961	+	0.973904i	$0.427121\pi$
$422$	−17.6569	−0.859522
$423$	0	0
$424$	35.3137	1.71499
$425$	−1.17157	−0.0568296
$426$	0	0
$427$	−10.3431	−0.500540
$428$	−73.9411	−3.57408
$429$	0	0
$430$	−29.5563	−1.42533
$431$	31.1127	1.49865	0.749323	−	0.662205i	$-0.230379\pi$
$431$	31.1127	1.49865	0.749323	+	0.662205i	$0.230379\pi$
$432$	0	0
$433$	−10.9289	−0.525211	−0.262605	−	0.964903i	$-0.584582\pi$
$433$	−10.9289	−0.525211	−0.262605	+	0.964903i	$0.584582\pi$
$434$	−10.8284	−0.519781
$435$	0	0
$436$	−24.8284	−1.18907
$437$	7.65685	0.366277
$438$	0	0
$439$	−21.6569	−1.03363	−0.516813	−	0.856099i	$-0.672882\pi$
$439$	−21.6569	−1.03363	−0.516813	+	0.856099i	$0.672882\pi$
$440$	−9.89949	−0.471940
$441$	0	0
$442$	−9.65685	−0.459330
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	−5.89949	−0.279663
$446$	−44.9706	−2.12942
$447$	0	0
$448$	13.8995	0.656689
$449$	26.8701	1.26808	0.634038	−	0.773302i	$-0.281396\pi$
$449$	26.8701	1.26808	0.634038	+	0.773302i	$0.281396\pi$
$450$	0	0
$451$	0.544156	0.0256233
$452$	−38.8284	−1.82634
$453$	0	0
$454$	36.1421	1.69623
$455$	4.82843	0.226360
$456$	0	0
$457$	32.8284	1.53565	0.767825	−	0.640660i	$-0.221339\pi$
$457$	32.8284	1.53565	0.767825	+	0.640660i	$0.221339\pi$
$458$	54.6274	2.55257
$459$	0	0
$460$	−29.3137	−1.36676
$461$	−19.6569	−0.915511	−0.457755	−	0.889078i	$-0.651346\pi$
$461$	−19.6569	−0.915511	−0.457755	+	0.889078i	$0.651346\pi$
$462$	0	0
$463$	24.2426	1.12665	0.563326	−	0.826235i	$-0.309522\pi$
$463$	24.2426	1.12665	0.563326	+	0.826235i	$0.309522\pi$
$464$	−4.24264	−0.196960
$465$	0	0
$466$	−0.828427	−0.0383761
$467$	−35.6569	−1.65000	−0.825001	−	0.565131i	$-0.808826\pi$
$467$	−35.6569	−1.65000	−0.825001	+	0.565131i	$0.808826\pi$
$468$	0	0
$469$	13.6569	0.630615
$470$	−18.4853	−0.852662
$471$	0	0
$472$	55.1127	2.53677
$473$	27.4558	1.26242
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	6.34315	0.290738
$477$	0	0
$478$	−64.5269	−2.95139
$479$	−17.0711	−0.779997	−0.389998	−	0.920815i	$-0.627524\pi$
$479$	−17.0711	−0.779997	−0.389998	+	0.920815i	$0.627524\pi$
$480$	0	0
$481$	11.6569	0.531507
$482$	47.4558	2.16155
$483$	0	0
$484$	−22.8579	−1.03899
$485$	−9.75736	−0.443059
$486$	0	0
$487$	20.4853	0.928277	0.464138	−	0.885763i	$-0.346364\pi$
$487$	20.4853	0.928277	0.464138	+	0.885763i	$0.346364\pi$
$488$	−32.2843	−1.46144
$489$	0	0
$490$	12.0711	0.545315
$491$	1.75736	0.0793085	0.0396543	−	0.999213i	$-0.487374\pi$
$491$	1.75736	0.0793085	0.0396543	+	0.999213i	$0.487374\pi$
$492$	0	0
$493$	1.65685	0.0746210
$494$	−8.24264	−0.370854
$495$	0	0
$496$	−9.51472	−0.427223
$497$	−15.3137	−0.686914
$498$	0	0
$499$	−5.17157	−0.231511	−0.115756	−	0.993278i	$-0.536929\pi$
$499$	−5.17157	−0.231511	−0.115756	+	0.993278i	$0.536929\pi$
$500$	3.82843	0.171212
$501$	0	0
$502$	−4.24264	−0.189358
$503$	−4.82843	−0.215289	−0.107644	−	0.994189i	$-0.534331\pi$
$503$	−4.82843	−0.215289	−0.107644	+	0.994189i	$0.534331\pi$
$504$	0	0
$505$	3.17157	0.141133
$506$	41.4558	1.84294
$507$	0	0
$508$	73.9411	3.28061
$509$	0.727922	0.0322646	0.0161323	−	0.999870i	$-0.494865\pi$
$509$	0.727922	0.0322646	0.0161323	+	0.999870i	$0.494865\pi$
$510$	0	0
$511$	10.8284	0.479021
$512$	31.2426	1.38074
$513$	0	0
$514$	−10.8284	−0.477621
$515$	−7.31371	−0.322281
$516$	0	0
$517$	17.1716	0.755205
$518$	−11.6569	−0.512173
$519$	0	0
$520$	15.0711	0.660910
$521$	−7.75736	−0.339856	−0.169928	−	0.985456i	$-0.554354\pi$
$521$	−7.75736	−0.339856	−0.169928	+	0.985456i	$0.554354\pi$
$522$	0	0
$523$	−15.5147	−0.678411	−0.339206	−	0.940712i	$-0.610158\pi$
$523$	−15.5147	−0.678411	−0.339206	+	0.940712i	$0.610158\pi$
$524$	32.8701	1.43594
$525$	0	0
$526$	−56.6274	−2.46907
$527$	3.71573	0.161860
$528$	0	0
$529$	35.6274	1.54902
$530$	19.3137	0.838934
$531$	0	0
$532$	5.41421	0.234736
$533$	−0.828427	−0.0358832
$534$	0	0
$535$	−19.3137	−0.835004
$536$	42.6274	1.84122
$537$	0	0
$538$	−7.41421	−0.319649
$539$	−11.2132	−0.482987
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	26.1421	1.12290
$543$	0	0
$544$	−1.85786	−0.0796553
$545$	−6.48528	−0.277799
$546$	0	0
$547$	24.4853	1.04692	0.523458	−	0.852052i	$-0.324642\pi$
$547$	24.4853	1.04692	0.523458	+	0.852052i	$0.324642\pi$
$548$	−38.2843	−1.63542
$549$	0	0
$550$	−5.41421	−0.230863
$551$	1.41421	0.0602475
$552$	0	0
$553$	0	0
$554$	53.1127	2.25654
$555$	0	0
$556$	−54.1421	−2.29614
$557$	25.3137	1.07258	0.536288	−	0.844035i	$-0.319826\pi$
$557$	25.3137	1.07258	0.536288	+	0.844035i	$0.319826\pi$
$558$	0	0
$559$	−41.7990	−1.76791
$560$	−4.24264	−0.179284
$561$	0	0
$562$	−35.2132	−1.48538
$563$	30.2843	1.27633	0.638165	−	0.769900i	$-0.279694\pi$
$563$	30.2843	1.27633	0.638165	+	0.769900i	$0.279694\pi$
$564$	0	0
$565$	−10.1421	−0.426683
$566$	25.0711	1.05382
$567$	0	0
$568$	−47.7990	−2.00560
$569$	−31.0711	−1.30257	−0.651283	−	0.758835i	$-0.725769\pi$
$569$	−31.0711	−1.30257	−0.651283	+	0.758835i	$0.725769\pi$
$570$	0	0
$571$	9.17157	0.383818	0.191909	−	0.981413i	$-0.438532\pi$
$571$	9.17157	0.383818	0.191909	+	0.981413i	$0.438532\pi$
$572$	−29.3137	−1.22567
$573$	0	0
$574$	0.828427	0.0345779
$575$	−7.65685	−0.319313
$576$	0	0
$577$	19.4558	0.809957	0.404979	−	0.914326i	$-0.367279\pi$
$577$	19.4558	0.809957	0.404979	+	0.914326i	$0.367279\pi$
$578$	37.7279	1.56927
$579$	0	0
$580$	−5.41421	−0.224813
$581$	18.1421	0.752663
$582$	0	0
$583$	−17.9411	−0.743045
$584$	33.7990	1.39861
$585$	0	0
$586$	−27.7990	−1.14837
$587$	12.3431	0.509456	0.254728	−	0.967013i	$-0.418014\pi$
$587$	12.3431	0.509456	0.254728	+	0.967013i	$0.418014\pi$
$588$	0	0
$589$	3.17157	0.130682
$590$	30.1421	1.24093
$591$	0	0
$592$	−10.2426	−0.420970
$593$	−36.6274	−1.50411	−0.752054	−	0.659102i	$-0.770937\pi$
$593$	−36.6274	−1.50411	−0.752054	+	0.659102i	$0.770937\pi$
$594$	0	0
$595$	1.65685	0.0679244
$596$	22.9706	0.940911
$597$	0	0
$598$	−63.1127	−2.58087
$599$	3.02944	0.123779	0.0618897	−	0.998083i	$-0.480287\pi$
$599$	3.02944	0.123779	0.0618897	+	0.998083i	$0.480287\pi$
$600$	0	0
$601$	8.14214	0.332125	0.166062	−	0.986115i	$-0.446895\pi$
$601$	8.14214	0.332125	0.166062	+	0.986115i	$0.446895\pi$
$602$	41.7990	1.70360
$603$	0	0
$604$	−24.8284	−1.01025
$605$	−5.97056	−0.242738
$606$	0	0
$607$	16.4853	0.669117	0.334558	−	0.942375i	$-0.391413\pi$
$607$	16.4853	0.669117	0.334558	+	0.942375i	$0.391413\pi$
$608$	−1.58579	−0.0643121
$609$	0	0
$610$	−17.6569	−0.714905
$611$	−26.1421	−1.05760
$612$	0	0
$613$	−10.4853	−0.423497	−0.211748	−	0.977324i	$-0.567916\pi$
$613$	−10.4853	−0.423497	−0.211748	+	0.977324i	$0.567916\pi$
$614$	−51.1127	−2.06274
$615$	0	0
$616$	14.0000	0.564076
$617$	−28.4853	−1.14677	−0.573387	−	0.819285i	$-0.694371\pi$
$617$	−28.4853	−1.14677	−0.573387	+	0.819285i	$0.694371\pi$
$618$	0	0
$619$	−20.4853	−0.823373	−0.411686	−	0.911326i	$-0.635060\pi$
$619$	−20.4853	−0.823373	−0.411686	+	0.911326i	$0.635060\pi$
$620$	−12.1421	−0.487640
$621$	0	0
$622$	−15.0711	−0.604295
$623$	8.34315	0.334261
$624$	0	0
$625$	1.00000	0.0400000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	40.1421	1.60185
$629$	4.00000	0.159490
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	−64.2843	−2.55305
$635$	19.3137	0.766441
$636$	0	0
$637$	17.0711	0.676380
$638$	7.65685	0.303138
$639$	0	0
$640$	20.5563	0.812561
$641$	31.3553	1.23846	0.619231	−	0.785209i	$-0.287445\pi$
$641$	31.3553	1.23846	0.619231	+	0.785209i	$0.287445\pi$
$642$	0	0
$643$	42.3848	1.67149	0.835746	−	0.549116i	$-0.185035\pi$
$643$	42.3848	1.67149	0.835746	+	0.549116i	$0.185035\pi$
$644$	41.4558	1.63359
$645$	0	0
$646$	−2.82843	−0.111283
$647$	36.1421	1.42089	0.710447	−	0.703751i	$-0.248493\pi$
$647$	36.1421	1.42089	0.710447	+	0.703751i	$0.248493\pi$
$648$	0	0
$649$	−28.0000	−1.09910
$650$	8.24264	0.323303
$651$	0	0
$652$	83.8406	3.28345
$653$	−42.8284	−1.67601	−0.838003	−	0.545666i	$-0.816277\pi$
$653$	−42.8284	−1.67601	−0.838003	+	0.545666i	$0.816277\pi$
$654$	0	0
$655$	8.58579	0.335474
$656$	0.727922	0.0284206
$657$	0	0
$658$	26.1421	1.01913
$659$	18.6274	0.725621	0.362811	−	0.931863i	$-0.381817\pi$
$659$	18.6274	0.725621	0.362811	+	0.931863i	$0.381817\pi$
$660$	0	0
$661$	7.45584	0.289999	0.144999	−	0.989432i	$-0.453682\pi$
$661$	7.45584	0.289999	0.144999	+	0.989432i	$0.453682\pi$
$662$	−67.9411	−2.64061
$663$	0	0
$664$	56.6274	2.19757
$665$	1.41421	0.0548408
$666$	0	0
$667$	10.8284	0.419278
$668$	66.2843	2.56462
$669$	0	0
$670$	23.3137	0.900687
$671$	16.4020	0.633193
$672$	0	0
$673$	−44.1838	−1.70316	−0.851580	−	0.524225i	$-0.824355\pi$
$673$	−44.1838	−1.70316	−0.851580	+	0.524225i	$0.824355\pi$
$674$	59.3553	2.28628
$675$	0	0
$676$	−5.14214	−0.197774
$677$	−16.9706	−0.652232	−0.326116	−	0.945330i	$-0.605740\pi$
$677$	−16.9706	−0.652232	−0.326116	+	0.945330i	$0.605740\pi$
$678$	0	0
$679$	13.7990	0.529557
$680$	5.17157	0.198321
$681$	0	0
$682$	17.1716	0.657534
$683$	−18.3431	−0.701881	−0.350940	−	0.936398i	$-0.614138\pi$
$683$	−18.3431	−0.701881	−0.350940	+	0.936398i	$0.614138\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	−40.9706	−1.56426
$687$	0	0
$688$	36.7279	1.40024
$689$	27.3137	1.04057
$690$	0	0
$691$	46.8284	1.78144	0.890719	−	0.454555i	$-0.150202\pi$
$691$	46.8284	1.78144	0.890719	+	0.454555i	$0.150202\pi$
$692$	75.7990	2.88145
$693$	0	0
$694$	−66.2843	−2.51612
$695$	−14.1421	−0.536442
$696$	0	0
$697$	−0.284271	−0.0107675
$698$	−43.4558	−1.64483
$699$	0	0
$700$	−5.41421	−0.204638
$701$	−6.68629	−0.252538	−0.126269	−	0.991996i	$-0.540300\pi$
$701$	−6.68629	−0.252538	−0.126269	+	0.991996i	$0.540300\pi$
$702$	0	0
$703$	3.41421	0.128770
$704$	−22.0416	−0.830725
$705$	0	0
$706$	8.82843	0.332262
$707$	−4.48528	−0.168686
$708$	0	0
$709$	28.9706	1.08801	0.544006	−	0.839081i	$-0.316907\pi$
$709$	28.9706	1.08801	0.544006	+	0.839081i	$0.316907\pi$
$710$	−26.1421	−0.981097
$711$	0	0
$712$	26.0416	0.975951
$713$	24.2843	0.909453
$714$	0	0
$715$	−7.65685	−0.286350
$716$	−63.1127	−2.35863
$717$	0	0
$718$	60.0416	2.24073
$719$	1.07107	0.0399441	0.0199720	−	0.999801i	$-0.493642\pi$
$719$	1.07107	0.0399441	0.0199720	+	0.999801i	$0.493642\pi$
$720$	0	0
$721$	10.3431	0.385199
$722$	−2.41421	−0.0898477
$723$	0	0
$724$	−79.7401	−2.96352
$725$	−1.41421	−0.0525226
$726$	0	0
$727$	47.3553	1.75631	0.878156	−	0.478374i	$-0.158774\pi$
$727$	47.3553	1.75631	0.878156	+	0.478374i	$0.158774\pi$
$728$	−21.3137	−0.789939
$729$	0	0
$730$	18.4853	0.684171
$731$	−14.3431	−0.530500
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	34.7279	1.28183
$735$	0	0
$736$	−12.1421	−0.447565
$737$	−21.6569	−0.797740
$738$	0	0
$739$	14.3431	0.527621	0.263811	−	0.964575i	$-0.415021\pi$
$739$	14.3431	0.527621	0.263811	+	0.964575i	$0.415021\pi$
$740$	−13.0711	−0.480502
$741$	0	0
$742$	−27.3137	−1.00272
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−1.41421	−0.0517780
$747$	0	0
$748$	−10.0589	−0.367789
$749$	27.3137	0.998021
$750$	0	0
$751$	−24.1421	−0.880959	−0.440480	−	0.897763i	$-0.645192\pi$
$751$	−24.1421	−0.880959	−0.440480	+	0.897763i	$0.645192\pi$
$752$	22.9706	0.837650
$753$	0	0
$754$	−11.6569	−0.424518
$755$	−6.48528	−0.236024
$756$	0	0
$757$	−32.4264	−1.17856	−0.589279	−	0.807930i	$-0.700588\pi$
$757$	−32.4264	−1.17856	−0.589279	+	0.807930i	$0.700588\pi$
$758$	−7.65685	−0.278109
$759$	0	0
$760$	4.41421	0.160120
$761$	−43.9411	−1.59286	−0.796432	−	0.604728i	$-0.793282\pi$
$761$	−43.9411	−1.59286	−0.796432	+	0.604728i	$0.793282\pi$
$762$	0	0
$763$	9.17157	0.332033
$764$	−39.2132	−1.41868
$765$	0	0
$766$	67.5980	2.44241
$767$	42.6274	1.53919
$768$	0	0
$769$	42.9706	1.54956	0.774779	−	0.632232i	$-0.217861\pi$
$769$	42.9706	1.54956	0.774779	+	0.632232i	$0.217861\pi$
$770$	7.65685	0.275934
$771$	0	0
$772$	19.4142	0.698733
$773$	25.6569	0.922813	0.461406	−	0.887189i	$-0.347345\pi$
$773$	25.6569	0.922813	0.461406	+	0.887189i	$0.347345\pi$
$774$	0	0
$775$	−3.17157	−0.113926
$776$	43.0711	1.54616
$777$	0	0
$778$	74.7696	2.68062
$779$	−0.242641	−0.00869350
$780$	0	0
$781$	24.2843	0.868960
$782$	−21.6569	−0.774448
$783$	0	0
$784$	−15.0000	−0.535714
$785$	10.4853	0.374236
$786$	0	0
$787$	−42.8284	−1.52667	−0.763334	−	0.646004i	$-0.776439\pi$
$787$	−42.8284	−1.52667	−0.763334	+	0.646004i	$0.776439\pi$
$788$	−26.1421	−0.931275
$789$	0	0
$790$	0	0
$791$	14.3431	0.509984
$792$	0	0
$793$	−24.9706	−0.886731
$794$	−69.1127	−2.45272
$795$	0	0
$796$	69.4558	2.46180
$797$	14.1421	0.500940	0.250470	−	0.968124i	$-0.419415\pi$
$797$	14.1421	0.500940	0.250470	+	0.968124i	$0.419415\pi$
$798$	0	0
$799$	−8.97056	−0.317356
$800$	1.58579	0.0560660
$801$	0	0
$802$	−18.7279	−0.661306
$803$	−17.1716	−0.605972
$804$	0	0
$805$	10.8284	0.381652
$806$	−26.1421	−0.920817
$807$	0	0
$808$	−14.0000	−0.492518
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	8.68629	0.305017	0.152508	−	0.988302i	$-0.451265\pi$
$811$	8.68629	0.305017	0.152508	+	0.988302i	$0.451265\pi$
$812$	7.65685	0.268703
$813$	0	0
$814$	18.4853	0.647909
$815$	21.8995	0.767106
$816$	0	0
$817$	−12.2426	−0.428316
$818$	30.9706	1.08286
$819$	0	0
$820$	0.928932	0.0324397
$821$	−14.4853	−0.505540	−0.252770	−	0.967526i	$-0.581342\pi$
$821$	−14.4853	−0.505540	−0.252770	+	0.967526i	$0.581342\pi$
$822$	0	0
$823$	−25.0122	−0.871870	−0.435935	−	0.899978i	$-0.643582\pi$
$823$	−25.0122	−0.871870	−0.435935	+	0.899978i	$0.643582\pi$
$824$	32.2843	1.12468
$825$	0	0
$826$	−42.6274	−1.48320
$827$	2.68629	0.0934115	0.0467058	−	0.998909i	$-0.485128\pi$
$827$	2.68629	0.0934115	0.0467058	+	0.998909i	$0.485128\pi$
$828$	0	0
$829$	46.4853	1.61450	0.807250	−	0.590209i	$-0.200955\pi$
$829$	46.4853	1.61450	0.807250	+	0.590209i	$0.200955\pi$
$830$	30.9706	1.07500
$831$	0	0
$832$	33.5563	1.16336
$833$	5.85786	0.202963
$834$	0	0
$835$	17.3137	0.599166
$836$	−8.58579	−0.296946
$837$	0	0
$838$	8.24264	0.284737
$839$	9.85786	0.340331	0.170166	−	0.985415i	$-0.445570\pi$
$839$	9.85786	0.340331	0.170166	+	0.985415i	$0.445570\pi$
$840$	0	0
$841$	−27.0000	−0.931034
$842$	−22.4853	−0.774894
$843$	0	0
$844$	28.0000	0.963800
$845$	−1.34315	−0.0462056
$846$	0	0
$847$	8.44365	0.290127
$848$	−24.0000	−0.824163
$849$	0	0
$850$	2.82843	0.0970143
$851$	26.1421	0.896141
$852$	0	0
$853$	−16.8284	−0.576194	−0.288097	−	0.957601i	$-0.593023\pi$
$853$	−16.8284	−0.576194	−0.288097	+	0.957601i	$0.593023\pi$
$854$	24.9706	0.854475
$855$	0	0
$856$	85.2548	2.91395
$857$	−12.6863	−0.433355	−0.216678	−	0.976243i	$-0.569522\pi$
$857$	−12.6863	−0.433355	−0.216678	+	0.976243i	$0.569522\pi$
$858$	0	0
$859$	−49.9411	−1.70397	−0.851985	−	0.523567i	$-0.824601\pi$
$859$	−49.9411	−1.70397	−0.851985	+	0.523567i	$0.824601\pi$
$860$	46.8701	1.59826
$861$	0	0
$862$	−75.1127	−2.55835
$863$	−8.68629	−0.295685	−0.147842	−	0.989011i	$-0.547233\pi$
$863$	−8.68629	−0.295685	−0.147842	+	0.989011i	$0.547233\pi$
$864$	0	0
$865$	19.7990	0.673186
$866$	26.3848	0.896591
$867$	0	0
$868$	17.1716	0.582841
$869$	0	0
$870$	0	0
$871$	32.9706	1.11716
$872$	28.6274	0.969447
$873$	0	0
$874$	−18.4853	−0.625274
$875$	−1.41421	−0.0478091
$876$	0	0
$877$	−34.9289	−1.17947	−0.589733	−	0.807598i	$-0.700767\pi$
$877$	−34.9289	−1.17947	−0.589733	+	0.807598i	$0.700767\pi$
$878$	52.2843	1.76451
$879$	0	0
$880$	6.72792	0.226798
$881$	5.79899	0.195373	0.0976865	−	0.995217i	$-0.468856\pi$
$881$	5.79899	0.195373	0.0976865	+	0.995217i	$0.468856\pi$
$882$	0	0
$883$	−12.0416	−0.405233	−0.202617	−	0.979258i	$-0.564945\pi$
$883$	−12.0416	−0.405233	−0.202617	+	0.979258i	$0.564945\pi$
$884$	15.3137	0.515056
$885$	0	0
$886$	43.4558	1.45993
$887$	−25.9411	−0.871018	−0.435509	−	0.900184i	$-0.643432\pi$
$887$	−25.9411	−0.871018	−0.435509	+	0.900184i	$0.643432\pi$
$888$	0	0
$889$	−27.3137	−0.916072
$890$	14.2426	0.477414
$891$	0	0
$892$	71.3137	2.38776
$893$	−7.65685	−0.256227
$894$	0	0
$895$	−16.4853	−0.551042
$896$	−29.0711	−0.971196
$897$	0	0
$898$	−64.8701	−2.16474
$899$	4.48528	0.149593
$900$	0	0
$901$	9.37258	0.312246
$902$	−1.31371	−0.0437417
$903$	0	0
$904$	44.7696	1.48901
$905$	−20.8284	−0.692360
$906$	0	0
$907$	−18.1421	−0.602400	−0.301200	−	0.953561i	$-0.597387\pi$
$907$	−18.1421	−0.602400	−0.301200	+	0.953561i	$0.597387\pi$
$908$	−57.3137	−1.90202
$909$	0	0
$910$	−11.6569	−0.386421
$911$	−8.68629	−0.287790	−0.143895	−	0.989593i	$-0.545963\pi$
$911$	−8.68629	−0.287790	−0.143895	+	0.989593i	$0.545963\pi$
$912$	0	0
$913$	−28.7696	−0.952133
$914$	−79.2548	−2.62152
$915$	0	0
$916$	−86.6274	−2.86225
$917$	−12.1421	−0.400969
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	33.7990	1.11432
$921$	0	0
$922$	47.4558	1.56287
$923$	−36.9706	−1.21690
$924$	0	0
$925$	−3.41421	−0.112259
$926$	−58.5269	−1.92331
$927$	0	0
$928$	−2.24264	−0.0736183
$929$	−33.1127	−1.08639	−0.543196	−	0.839606i	$-0.682786\pi$
$929$	−33.1127	−1.08639	−0.543196	+	0.839606i	$0.682786\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	1.31371	0.0430320
$933$	0	0
$934$	86.0833	2.81673
$935$	−2.62742	−0.0859257
$936$	0	0
$937$	−29.7990	−0.973491	−0.486745	−	0.873544i	$-0.661816\pi$
$937$	−29.7990	−0.973491	−0.486745	+	0.873544i	$0.661816\pi$
$938$	−32.9706	−1.07653
$939$	0	0
$940$	29.3137	0.956108
$941$	−22.8701	−0.745543	−0.372771	−	0.927923i	$-0.621592\pi$
$941$	−22.8701	−0.745543	−0.372771	+	0.927923i	$0.621592\pi$
$942$	0	0
$943$	−1.85786	−0.0605004
$944$	−37.4558	−1.21908
$945$	0	0
$946$	−66.2843	−2.15509
$947$	47.1716	1.53287	0.766435	−	0.642322i	$-0.222029\pi$
$947$	47.1716	1.53287	0.766435	+	0.642322i	$0.222029\pi$
$948$	0	0
$949$	26.1421	0.848610
$950$	2.41421	0.0783274
$951$	0	0
$952$	−7.31371	−0.237039
$953$	53.4558	1.73160	0.865802	−	0.500386i	$-0.166809\pi$
$953$	53.4558	1.73160	0.865802	+	0.500386i	$0.166809\pi$
$954$	0	0
$955$	−10.2426	−0.331444
$956$	102.326	3.30946
$957$	0	0
$958$	41.2132	1.33154
$959$	14.1421	0.456673
$960$	0	0
$961$	−20.9411	−0.675520
$962$	−28.1421	−0.907339
$963$	0	0
$964$	−75.2548	−2.42379
$965$	5.07107	0.163243
$966$	0	0
$967$	8.04163	0.258601	0.129301	−	0.991605i	$-0.458727\pi$
$967$	8.04163	0.258601	0.129301	+	0.991605i	$0.458727\pi$
$968$	26.3553	0.847093
$969$	0	0
$970$	23.5563	0.756349
$971$	−22.3431	−0.717026	−0.358513	−	0.933525i	$-0.616716\pi$
$971$	−22.3431	−0.717026	−0.358513	+	0.933525i	$0.616716\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	−49.4558	−1.58467
$975$	0	0
$976$	21.9411	0.702318
$977$	−32.2843	−1.03287	−0.516433	−	0.856328i	$-0.672740\pi$
$977$	−32.2843	−1.03287	−0.516433	+	0.856328i	$0.672740\pi$
$978$	0	0
$979$	−13.2304	−0.422847
$980$	−19.1421	−0.611473
$981$	0	0
$982$	−4.24264	−0.135388
$983$	24.6274	0.785493	0.392746	−	0.919647i	$-0.371525\pi$
$983$	24.6274	0.785493	0.392746	+	0.919647i	$0.371525\pi$
$984$	0	0
$985$	−6.82843	−0.217572
$986$	−4.00000	−0.127386
$987$	0	0
$988$	13.0711	0.415846
$989$	−93.7401	−2.98076
$990$	0	0
$991$	2.34315	0.0744325	0.0372162	−	0.999307i	$-0.488151\pi$
$991$	2.34315	0.0744325	0.0372162	+	0.999307i	$0.488151\pi$
$992$	−5.02944	−0.159685
$993$	0	0
$994$	36.9706	1.17264
$995$	18.1421	0.575144
$996$	0	0
$997$	−38.0833	−1.20611	−0.603054	−	0.797700i	$-0.706050\pi$
$997$	−38.0833	−1.20611	−0.603054	+	0.797700i	$0.706050\pi$
$998$	12.4853	0.395215
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.d.1.1		2
3.2	odd	2		285.2.a.g.1.2	✓	2
5.4	even	2		4275.2.a.y.1.2		2
12.11	even	2		4560.2.a.bf.1.2		2
15.2	even	4		1425.2.c.l.799.4		4
15.8	even	4		1425.2.c.l.799.1		4
15.14	odd	2		1425.2.a.k.1.1		2
57.56	even	2		5415.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.g.1.2	✓	2	3.2	odd	2
855.2.a.d.1.1		2	1.1	even	1	trivial
1425.2.a.k.1.1		2	15.14	odd	2
1425.2.c.l.799.1		4	15.8	even	4
1425.2.c.l.799.4		4	15.2	even	4
4275.2.a.y.1.2		2	5.4	even	2
4560.2.a.bf.1.2		2	12.11	even	2
5415.2.a.n.1.1		2	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -1.41421 q^{7} -4.41421 q^{8} +O(q^{10})\)	\(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -1.41421 q^{7} -4.41421 q^{8} -2.41421 q^{10} +2.24264 q^{11} -3.41421 q^{13} +3.41421 q^{14} +3.00000 q^{16} -1.17157 q^{17} -1.00000 q^{19} +3.82843 q^{20} -5.41421 q^{22} -7.65685 q^{23} +1.00000 q^{25} +8.24264 q^{26} -5.41421 q^{28} -1.41421 q^{29} -3.17157 q^{31} +1.58579 q^{32} +2.82843 q^{34} -1.41421 q^{35} -3.41421 q^{37} +2.41421 q^{38} -4.41421 q^{40} +0.242641 q^{41} +12.2426 q^{43} +8.58579 q^{44} +18.4853 q^{46} +7.65685 q^{47} -5.00000 q^{49} -2.41421 q^{50} -13.0711 q^{52} -8.00000 q^{53} +2.24264 q^{55} +6.24264 q^{56} +3.41421 q^{58} -12.4853 q^{59} +7.31371 q^{61} +7.65685 q^{62} -9.82843 q^{64} -3.41421 q^{65} -9.65685 q^{67} -4.48528 q^{68} +3.41421 q^{70} +10.8284 q^{71} -7.65685 q^{73} +8.24264 q^{74} -3.82843 q^{76} -3.17157 q^{77} +3.00000 q^{80} -0.585786 q^{82} -12.8284 q^{83} -1.17157 q^{85} -29.5563 q^{86} -9.89949 q^{88} -5.89949 q^{89} +4.82843 q^{91} -29.3137 q^{92} -18.4853 q^{94} -1.00000 q^{95} -9.75736 q^{97} +12.0711 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8} - 2 q^{10} - 4 q^{11} - 4 q^{13} + 4 q^{14} + 6 q^{16} - 8 q^{17} - 2 q^{19} + 2 q^{20} - 8 q^{22} - 4 q^{23} + 2 q^{25} + 8 q^{26} - 8 q^{28} - 12 q^{31} + 6 q^{32} - 4 q^{37} + 2 q^{38} - 6 q^{40} - 8 q^{41} + 16 q^{43} + 20 q^{44} + 20 q^{46} + 4 q^{47} - 10 q^{49} - 2 q^{50} - 12 q^{52} - 16 q^{53} - 4 q^{55} + 4 q^{56} + 4 q^{58} - 8 q^{59} - 8 q^{61} + 4 q^{62} - 14 q^{64} - 4 q^{65} - 8 q^{67} + 8 q^{68} + 4 q^{70} + 16 q^{71} - 4 q^{73} + 8 q^{74} - 2 q^{76} - 12 q^{77} + 6 q^{80} - 4 q^{82} - 20 q^{83} - 8 q^{85} - 28 q^{86} + 8 q^{89} + 4 q^{91} - 36 q^{92} - 20 q^{94} - 2 q^{95} - 28 q^{97} + 10 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8 - 2 * q^10 - 4 * q^11 - 4 * q^13 + 4 * q^14 + 6 * q^16 - 8 * q^17 - 2 * q^19 + 2 * q^20 - 8 * q^22 - 4 * q^23 + 2 * q^25 + 8 * q^26 - 8 * q^28 - 12 * q^31 + 6 * q^32 - 4 * q^37 + 2 * q^38 - 6 * q^40 - 8 * q^41 + 16 * q^43 + 20 * q^44 + 20 * q^46 + 4 * q^47 - 10 * q^49 - 2 * q^50 - 12 * q^52 - 16 * q^53 - 4 * q^55 + 4 * q^56 + 4 * q^58 - 8 * q^59 - 8 * q^61 + 4 * q^62 - 14 * q^64 - 4 * q^65 - 8 * q^67 + 8 * q^68 + 4 * q^70 + 16 * q^71 - 4 * q^73 + 8 * q^74 - 2 * q^76 - 12 * q^77 + 6 * q^80 - 4 * q^82 - 20 * q^83 - 8 * q^85 - 28 * q^86 + 8 * q^89 + 4 * q^91 - 36 * q^92 - 20 * q^94 - 2 * q^95 - 28 * q^97 + 10 * q^98

Introduction

L-functions

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Database

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