LMFDB - Embedded newform 1425.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, 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("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.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:	$11.3786822880$
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	$\chi$	$=$	1425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} +2.41421 q^{6} +1.41421 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} +2.41421 q^{6} +1.41421 q^{7} -4.41421 q^{8} +1.00000 q^{9} -2.24264 q^{11} -3.82843 q^{12} +3.41421 q^{13} -3.41421 q^{14} +3.00000 q^{16} -1.17157 q^{17} -2.41421 q^{18} -1.00000 q^{19} -1.41421 q^{21} +5.41421 q^{22} -7.65685 q^{23} +4.41421 q^{24} -8.24264 q^{26} -1.00000 q^{27} +5.41421 q^{28} +1.41421 q^{29} -3.17157 q^{31} +1.58579 q^{32} +2.24264 q^{33} +2.82843 q^{34} +3.82843 q^{36} +3.41421 q^{37} +2.41421 q^{38} -3.41421 q^{39} -0.242641 q^{41} +3.41421 q^{42} -12.2426 q^{43} -8.58579 q^{44} +18.4853 q^{46} +7.65685 q^{47} -3.00000 q^{48} -5.00000 q^{49} +1.17157 q^{51} +13.0711 q^{52} -8.00000 q^{53} +2.41421 q^{54} -6.24264 q^{56} +1.00000 q^{57} -3.41421 q^{58} +12.4853 q^{59} +7.31371 q^{61} +7.65685 q^{62} +1.41421 q^{63} -9.82843 q^{64} -5.41421 q^{66} +9.65685 q^{67} -4.48528 q^{68} +7.65685 q^{69} -10.8284 q^{71} -4.41421 q^{72} +7.65685 q^{73} -8.24264 q^{74} -3.82843 q^{76} -3.17157 q^{77} +8.24264 q^{78} +1.00000 q^{81} +0.585786 q^{82} -12.8284 q^{83} -5.41421 q^{84} +29.5563 q^{86} -1.41421 q^{87} +9.89949 q^{88} +5.89949 q^{89} +4.82843 q^{91} -29.3137 q^{92} +3.17157 q^{93} -18.4853 q^{94} -1.58579 q^{96} +9.75736 q^{97} +12.0711 q^{98} -2.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 4 q^{11} - 2 q^{12} + 4 q^{13} - 4 q^{14} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{19} + 8 q^{22} - 4 q^{23} + 6 q^{24} - 8 q^{26} - 2 q^{27} + 8 q^{28} - 12 q^{31} + 6 q^{32} - 4 q^{33} + 2 q^{36} + 4 q^{37} + 2 q^{38} - 4 q^{39} + 8 q^{41} + 4 q^{42} - 16 q^{43} - 20 q^{44} + 20 q^{46} + 4 q^{47} - 6 q^{48} - 10 q^{49} + 8 q^{51} + 12 q^{52} - 16 q^{53} + 2 q^{54} - 4 q^{56} + 2 q^{57} - 4 q^{58} + 8 q^{59} - 8 q^{61} + 4 q^{62} - 14 q^{64} - 8 q^{66} + 8 q^{67} + 8 q^{68} + 4 q^{69} - 16 q^{71} - 6 q^{72} + 4 q^{73} - 8 q^{74} - 2 q^{76} - 12 q^{77} + 8 q^{78} + 2 q^{81} + 4 q^{82} - 20 q^{83} - 8 q^{84} + 28 q^{86} - 8 q^{89} + 4 q^{91} - 36 q^{92} + 12 q^{93} - 20 q^{94} - 6 q^{96} + 28 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9 + 4 * q^11 - 2 * q^12 + 4 * q^13 - 4 * q^14 + 6 * q^16 - 8 * q^17 - 2 * q^18 - 2 * q^19 + 8 * q^22 - 4 * q^23 + 6 * q^24 - 8 * q^26 - 2 * q^27 + 8 * q^28 - 12 * q^31 + 6 * q^32 - 4 * q^33 + 2 * q^36 + 4 * q^37 + 2 * q^38 - 4 * q^39 + 8 * q^41 + 4 * q^42 - 16 * q^43 - 20 * q^44 + 20 * q^46 + 4 * q^47 - 6 * q^48 - 10 * q^49 + 8 * q^51 + 12 * q^52 - 16 * q^53 + 2 * q^54 - 4 * q^56 + 2 * q^57 - 4 * q^58 + 8 * q^59 - 8 * q^61 + 4 * q^62 - 14 * q^64 - 8 * q^66 + 8 * q^67 + 8 * q^68 + 4 * q^69 - 16 * q^71 - 6 * q^72 + 4 * q^73 - 8 * q^74 - 2 * q^76 - 12 * q^77 + 8 * q^78 + 2 * q^81 + 4 * q^82 - 20 * q^83 - 8 * q^84 + 28 * q^86 - 8 * q^89 + 4 * q^91 - 36 * q^92 + 12 * q^93 - 20 * q^94 - 6 * q^96 + 28 * q^97 + 10 * q^98 + 4 * 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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	2.41421	0.985599
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	−4.41421	−1.56066
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.24264	−0.676182	−0.338091	−	0.941113i	$-0.609781\pi$
$11$	−2.24264	−0.676182	−0.338091	+	0.941113i	$0.609781\pi$
$12$	−3.82843	−1.10517
$13$	3.41421	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$14$	−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	$0.545377\pi$
$18$	−2.41421	−0.569036
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.41421	−0.308607
$22$	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	$0.794258\pi$
$24$	4.41421	0.901048
$25$	0	0
$26$	−8.24264	−1.61651
$27$	−1.00000	−0.192450
$28$	5.41421	1.02319
$29$	1.41421	0.262613	0.131306	−	0.991342i	$-0.458083\pi$
$29$	1.41421	0.262613	0.131306	+	0.991342i	$0.458083\pi$
$30$	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
$33$	2.24264	0.390394
$34$	2.82843	0.485071
$35$	0	0
$36$	3.82843	0.638071
$37$	3.41421	0.561293	0.280647	−	0.959811i	$-0.409451\pi$
$37$	3.41421	0.561293	0.280647	+	0.959811i	$0.409451\pi$
$38$	2.41421	0.391637
$39$	−3.41421	−0.546712
$40$	0	0
$41$	−0.242641	−0.0378941	−0.0189471	−	0.999820i	$-0.506031\pi$
$41$	−0.242641	−0.0378941	−0.0189471	+	0.999820i	$0.506031\pi$
$42$	3.41421	0.526825
$43$	−12.2426	−1.86699	−0.933493	−	0.358597i	$-0.883255\pi$
$43$	−12.2426	−1.86699	−0.933493	+	0.358597i	$0.883255\pi$
$44$	−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	$0.311403\pi$
$48$	−3.00000	−0.433013
$49$	−5.00000	−0.714286
$50$	0	0
$51$	1.17157	0.164053
$52$	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	$0.685160\pi$
$54$	2.41421	0.328533
$55$	0	0
$56$	−6.24264	−0.834208
$57$	1.00000	0.132453
$58$	−3.41421	−0.448308
$59$	12.4853	1.62545	0.812723	−	0.582651i	$-0.197984\pi$
$59$	12.4853	1.62545	0.812723	+	0.582651i	$0.197984\pi$
$60$	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
$63$	1.41421	0.178174
$64$	−9.82843	−1.22855
$65$	0	0
$66$	−5.41421	−0.666444
$67$	9.65685	1.17977	0.589886	−	0.807486i	$-0.299173\pi$
$67$	9.65685	1.17977	0.589886	+	0.807486i	$0.299173\pi$
$68$	−4.48528	−0.543920
$69$	7.65685	0.921777
$70$	0	0
$71$	−10.8284	−1.28510	−0.642549	−	0.766245i	$-0.722123\pi$
$71$	−10.8284	−1.28510	−0.642549	+	0.766245i	$0.722123\pi$
$72$	−4.41421	−0.520220
$73$	7.65685	0.896167	0.448084	−	0.893992i	$-0.352107\pi$
$73$	7.65685	0.896167	0.448084	+	0.893992i	$0.352107\pi$
$74$	−8.24264	−0.958188
$75$	0	0
$76$	−3.82843	−0.439151
$77$	−3.17157	−0.361434
$78$	8.24264	0.933295
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	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	$0.748628\pi$
$84$	−5.41421	−0.590739
$85$	0	0
$86$	29.5563	3.18714
$87$	−1.41421	−0.151620
$88$	9.89949	1.05529
$89$	5.89949	0.625345	0.312673	−	0.949861i	$-0.398776\pi$
$89$	5.89949	0.625345	0.312673	+	0.949861i	$0.398776\pi$
$90$	0	0
$91$	4.82843	0.506157
$92$	−29.3137	−3.05617
$93$	3.17157	0.328877
$94$	−18.4853	−1.90661
$95$	0	0
$96$	−1.58579	−0.161849
$97$	9.75736	0.990710	0.495355	−	0.868691i	$-0.335038\pi$
$97$	9.75736	0.990710	0.495355	+	0.868691i	$0.335038\pi$
$98$	12.0711	1.21936
$99$	−2.24264	−0.225394
$100$	0	0
$101$	−3.17157	−0.315583	−0.157792	−	0.987472i	$-0.550437\pi$
$101$	−3.17157	−0.315583	−0.157792	+	0.987472i	$0.550437\pi$
$102$	−2.82843	−0.280056
$103$	7.31371	0.720641	0.360321	−	0.932829i	$-0.382667\pi$
$103$	7.31371	0.720641	0.360321	+	0.932829i	$0.382667\pi$
$104$	−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	$0.883318\pi$
$108$	−3.82843	−0.368391
$109$	−6.48528	−0.621177	−0.310589	−	0.950544i	$-0.600526\pi$
$109$	−6.48528	−0.621177	−0.310589	+	0.950544i	$0.600526\pi$
$110$	0	0
$111$	−3.41421	−0.324063
$112$	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	$0.658292\pi$
$114$	−2.41421	−0.226112
$115$	0	0
$116$	5.41421	0.502697
$117$	3.41421	0.315644
$118$	−30.1421	−2.77481
$119$	−1.65685	−0.151884
$120$	0	0
$121$	−5.97056	−0.542778
$122$	−17.6569	−1.59858
$123$	0.242641	0.0218782
$124$	−12.1421	−1.09040
$125$	0	0
$126$	−3.41421	−0.304162
$127$	−19.3137	−1.71381	−0.856907	−	0.515471i	$-0.827617\pi$
$127$	−19.3137	−1.71381	−0.856907	+	0.515471i	$0.827617\pi$
$128$	20.5563	1.81694
$129$	12.2426	1.07790
$130$	0	0
$131$	−8.58579	−0.750144	−0.375072	−	0.926996i	$-0.622382\pi$
$131$	−8.58579	−0.750144	−0.375072	+	0.926996i	$0.622382\pi$
$132$	8.58579	0.747297
$133$	−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	$0.640493\pi$
$138$	−18.4853	−1.57357
$139$	−14.1421	−1.19952	−0.599760	−	0.800180i	$-0.704737\pi$
$139$	−14.1421	−1.19952	−0.599760	+	0.800180i	$0.704737\pi$
$140$	0	0
$141$	−7.65685	−0.644823
$142$	26.1421	2.19380
$143$	−7.65685	−0.640298
$144$	3.00000	0.250000
$145$	0	0
$146$	−18.4853	−1.52985
$147$	5.00000	0.412393
$148$	13.0711	1.07444
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	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
$153$	−1.17157	−0.0947161
$154$	7.65685	0.617007
$155$	0	0
$156$	−13.0711	−1.04652
$157$	−10.4853	−0.836817	−0.418408	−	0.908259i	$-0.637412\pi$
$157$	−10.4853	−0.836817	−0.418408	+	0.908259i	$0.637412\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	−10.8284	−0.853400
$162$	−2.41421	−0.189679
$163$	−21.8995	−1.71530	−0.857650	−	0.514233i	$-0.828077\pi$
$163$	−21.8995	−1.71530	−0.857650	+	0.514233i	$0.828077\pi$
$164$	−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	$0.266342\pi$
$168$	6.24264	0.481630
$169$	−1.34315	−0.103319
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−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	$0.228778\pi$
$174$	3.41421	0.258831
$175$	0	0
$176$	−6.72792	−0.507136
$177$	−12.4853	−0.938451
$178$	−14.2426	−1.06753
$179$	16.4853	1.23217	0.616084	−	0.787681i	$-0.288718\pi$
$179$	16.4853	1.23217	0.616084	+	0.787681i	$0.288718\pi$
$180$	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
$183$	−7.31371	−0.540645
$184$	33.7990	2.49169
$185$	0	0
$186$	−7.65685	−0.561428
$187$	2.62742	0.192136
$188$	29.3137	2.13792
$189$	−1.41421	−0.102869
$190$	0	0
$191$	10.2426	0.741131	0.370566	−	0.928806i	$-0.379164\pi$
$191$	10.2426	0.741131	0.370566	+	0.928806i	$0.379164\pi$
$192$	9.82843	0.709306
$193$	−5.07107	−0.365023	−0.182512	−	0.983204i	$-0.558423\pi$
$193$	−5.07107	−0.365023	−0.182512	+	0.983204i	$0.558423\pi$
$194$	−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	$0.578214\pi$
$198$	5.41421	0.384771
$199$	18.1421	1.28606	0.643031	−	0.765840i	$-0.277677\pi$
$199$	18.1421	1.28606	0.643031	+	0.765840i	$0.277677\pi$
$200$	0	0
$201$	−9.65685	−0.681142
$202$	7.65685	0.538734
$203$	2.00000	0.140372
$204$	4.48528	0.314033
$205$	0	0
$206$	−17.6569	−1.23021
$207$	−7.65685	−0.532188
$208$	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
$213$	10.8284	0.741952
$214$	46.6274	3.18738
$215$	0	0
$216$	4.41421	0.300349
$217$	−4.48528	−0.304481
$218$	15.6569	1.06042
$219$	−7.65685	−0.517402
$220$	0	0
$221$	−4.00000	−0.269069
$222$	8.24264	0.553210
$223$	−18.6274	−1.24738	−0.623692	−	0.781670i	$-0.714368\pi$
$223$	−18.6274	−1.24738	−0.623692	+	0.781670i	$0.714368\pi$
$224$	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	$0.665497\pi$
$228$	3.82843	0.253544
$229$	−22.6274	−1.49526	−0.747631	−	0.664114i	$-0.768809\pi$
$229$	−22.6274	−1.49526	−0.747631	+	0.664114i	$0.768809\pi$
$230$	0	0
$231$	3.17157	0.208674
$232$	−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	$0.496422\pi$
$234$	−8.24264	−0.538838
$235$	0	0
$236$	47.7990	3.11145
$237$	0	0
$238$	4.00000	0.259281
$239$	−26.7279	−1.72889	−0.864443	−	0.502731i	$-0.832329\pi$
$239$	−26.7279	−1.72889	−0.864443	+	0.502731i	$0.832329\pi$
$240$	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
$243$	−1.00000	−0.0641500
$244$	28.0000	1.79252
$245$	0	0
$246$	−0.585786	−0.0373484
$247$	−3.41421	−0.217241
$248$	14.0000	0.889001
$249$	12.8284	0.812969
$250$	0	0
$251$	−1.75736	−0.110924	−0.0554618	−	0.998461i	$-0.517663\pi$
$251$	−1.75736	−0.110924	−0.0554618	+	0.998461i	$0.517663\pi$
$252$	5.41421	0.341063
$253$	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	$0.455324\pi$
$258$	−29.5563	−1.84010
$259$	4.82843	0.300024
$260$	0	0
$261$	1.41421	0.0875376
$262$	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	$0.242682\pi$
$264$	−9.89949	−0.609272
$265$	0	0
$266$	3.41421	0.209339
$267$	−5.89949	−0.361043
$268$	36.9706	2.25834
$269$	−3.07107	−0.187246	−0.0936232	−	0.995608i	$-0.529845\pi$
$269$	−3.07107	−0.187246	−0.0936232	+	0.995608i	$0.529845\pi$
$270$	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
$273$	−4.82843	−0.292230
$274$	24.1421	1.45848
$275$	0	0
$276$	29.3137	1.76448
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	34.1421	2.04771
$279$	−3.17157	−0.189877
$280$	0	0
$281$	−14.5858	−0.870115	−0.435058	−	0.900403i	$-0.643272\pi$
$281$	−14.5858	−0.870115	−0.435058	+	0.900403i	$0.643272\pi$
$282$	18.4853	1.10078
$283$	10.3848	0.617311	0.308655	−	0.951174i	$-0.400121\pi$
$283$	10.3848	0.617311	0.308655	+	0.951174i	$0.400121\pi$
$284$	−41.4558	−2.45995
$285$	0	0
$286$	18.4853	1.09306
$287$	−0.343146	−0.0202553
$288$	1.58579	0.0934434
$289$	−15.6274	−0.919260
$290$	0	0
$291$	−9.75736	−0.571987
$292$	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	$0.390808\pi$
$294$	−12.0711	−0.703999
$295$	0	0
$296$	−15.0711	−0.875988
$297$	2.24264	0.130131
$298$	14.4853	0.839110
$299$	−26.1421	−1.51184
$300$	0	0
$301$	−17.3137	−0.997946
$302$	15.6569	0.900951
$303$	3.17157	0.182202
$304$	−3.00000	−0.172062
$305$	0	0
$306$	2.82843	0.161690
$307$	−21.1716	−1.20833	−0.604163	−	0.796861i	$-0.706492\pi$
$307$	−21.1716	−1.20833	−0.604163	+	0.796861i	$0.706492\pi$
$308$	−12.1421	−0.691862
$309$	−7.31371	−0.416062
$310$	0	0
$311$	−6.24264	−0.353988	−0.176994	−	0.984212i	$-0.556637\pi$
$311$	−6.24264	−0.353988	−0.176994	+	0.984212i	$0.556637\pi$
$312$	15.0711	0.853231
$313$	−5.79899	−0.327778	−0.163889	−	0.986479i	$-0.552404\pi$
$313$	−5.79899	−0.327778	−0.163889	+	0.986479i	$0.552404\pi$
$314$	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	$0.231123\pi$
$318$	−19.3137	−1.08306
$319$	−3.17157	−0.177574
$320$	0	0
$321$	19.3137	1.07799
$322$	26.1421	1.45684
$323$	1.17157	0.0651881
$324$	3.82843	0.212690
$325$	0	0
$326$	52.8701	2.92820
$327$	6.48528	0.358637
$328$	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
$333$	3.41421	0.187098
$334$	−41.7990	−2.28714
$335$	0	0
$336$	−4.24264	−0.231455
$337$	24.5858	1.33927	0.669637	−	0.742689i	$-0.266450\pi$
$337$	24.5858	1.33927	0.669637	+	0.742689i	$0.266450\pi$
$338$	3.24264	0.176376
$339$	10.1421	0.550845
$340$	0	0
$341$	7.11270	0.385174
$342$	2.41421	0.130546
$343$	−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	$0.236264\pi$
$348$	−5.41421	−0.290232
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	−3.41421	−0.182237
$352$	−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	$0.531026\pi$
$354$	30.1421	1.60204
$355$	0	0
$356$	22.5858	1.19704
$357$	1.65685	0.0876900
$358$	−39.7990	−2.10344
$359$	24.8701	1.31259	0.656296	−	0.754504i	$-0.272122\pi$
$359$	24.8701	1.31259	0.656296	+	0.754504i	$0.272122\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	50.2843	2.64288
$363$	5.97056	0.313373
$364$	18.4853	0.968892
$365$	0	0
$366$	17.6569	0.922939
$367$	14.3848	0.750879	0.375440	−	0.926847i	$-0.377492\pi$
$367$	14.3848	0.750879	0.375440	+	0.926847i	$0.377492\pi$
$368$	−22.9706	−1.19742
$369$	−0.242641	−0.0126314
$370$	0	0
$371$	−11.3137	−0.587378
$372$	12.1421	0.629540
$373$	−0.585786	−0.0303309	−0.0151654	−	0.999885i	$-0.504827\pi$
$373$	−0.585786	−0.0303309	−0.0151654	+	0.999885i	$0.504827\pi$
$374$	−6.34315	−0.327996
$375$	0	0
$376$	−33.7990	−1.74305
$377$	4.82843	0.248677
$378$	3.41421	0.175608
$379$	3.17157	0.162913	0.0814564	−	0.996677i	$-0.474043\pi$
$379$	3.17157	0.162913	0.0814564	+	0.996677i	$0.474043\pi$
$380$	0	0
$381$	19.3137	0.989471
$382$	−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	$0.753740\pi$
$384$	−20.5563	−1.04901
$385$	0	0
$386$	12.2426	0.623134
$387$	−12.2426	−0.622328
$388$	37.3553	1.89643
$389$	30.9706	1.57027	0.785135	−	0.619325i	$-0.212594\pi$
$389$	30.9706	1.57027	0.785135	+	0.619325i	$0.212594\pi$
$390$	0	0
$391$	8.97056	0.453661
$392$	22.0711	1.11476
$393$	8.58579	0.433096
$394$	16.4853	0.830516
$395$	0	0
$396$	−8.58579	−0.431452
$397$	−28.6274	−1.43677	−0.718384	−	0.695646i	$-0.755118\pi$
$397$	−28.6274	−1.43677	−0.718384	+	0.695646i	$0.755118\pi$
$398$	−43.7990	−2.19544
$399$	1.41421	0.0707992
$400$	0	0
$401$	−7.75736	−0.387384	−0.193692	−	0.981062i	$-0.562046\pi$
$401$	−7.75736	−0.387384	−0.193692	+	0.981062i	$0.562046\pi$
$402$	23.3137	1.16278
$403$	−10.8284	−0.539402
$404$	−12.1421	−0.604094
$405$	0	0
$406$	−4.82843	−0.239631
$407$	−7.65685	−0.379536
$408$	−5.17157	−0.256031
$409$	−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	0
$411$	10.0000	0.493264
$412$	28.0000	1.37946
$413$	17.6569	0.868837
$414$	18.4853	0.908502
$415$	0	0
$416$	5.41421	0.265454
$417$	14.1421	0.692543
$418$	−5.41421	−0.264818
$419$	3.41421	0.166795	0.0833976	−	0.996516i	$-0.473423\pi$
$419$	3.41421	0.166795	0.0833976	+	0.996516i	$0.473423\pi$
$420$	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
$423$	7.65685	0.372289
$424$	35.3137	1.71499
$425$	0	0
$426$	−26.1421	−1.26659
$427$	10.3431	0.500540
$428$	−73.9411	−3.57408
$429$	7.65685	0.369676
$430$	0	0
$431$	−31.1127	−1.49865	−0.749323	−	0.662205i	$-0.769621\pi$
$431$	−31.1127	−1.49865	−0.749323	+	0.662205i	$0.769621\pi$
$432$	−3.00000	−0.144338
$433$	10.9289	0.525211	0.262605	−	0.964903i	$-0.415418\pi$
$433$	10.9289	0.525211	0.262605	+	0.964903i	$0.415418\pi$
$434$	10.8284	0.519781
$435$	0	0
$436$	−24.8284	−1.18907
$437$	7.65685	0.366277
$438$	18.4853	0.883261
$439$	−21.6569	−1.03363	−0.516813	−	0.856099i	$-0.672882\pi$
$439$	−21.6569	−1.03363	−0.516813	+	0.856099i	$0.672882\pi$
$440$	0	0
$441$	−5.00000	−0.238095
$442$	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	$0.640642\pi$
$444$	−13.0711	−0.620325
$445$	0	0
$446$	44.9706	2.12942
$447$	6.00000	0.283790
$448$	−13.8995	−0.656689
$449$	−26.8701	−1.26808	−0.634038	−	0.773302i	$-0.718604\pi$
$449$	−26.8701	−1.26808	−0.634038	+	0.773302i	$0.718604\pi$
$450$	0	0
$451$	0.544156	0.0256233
$452$	−38.8284	−1.82634
$453$	6.48528	0.304705
$454$	36.1421	1.69623
$455$	0	0
$456$	−4.41421	−0.206714
$457$	−32.8284	−1.53565	−0.767825	−	0.640660i	$-0.778661\pi$
$457$	−32.8284	−1.53565	−0.767825	+	0.640660i	$0.778661\pi$
$458$	54.6274	2.55257
$459$	1.17157	0.0546843
$460$	0	0
$461$	19.6569	0.915511	0.457755	−	0.889078i	$-0.348654\pi$
$461$	19.6569	0.915511	0.457755	+	0.889078i	$0.348654\pi$
$462$	−7.65685	−0.356229
$463$	−24.2426	−1.12665	−0.563326	−	0.826235i	$-0.690478\pi$
$463$	−24.2426	−1.12665	−0.563326	+	0.826235i	$0.690478\pi$
$464$	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	$0.808826\pi$
$468$	13.0711	0.604210
$469$	13.6569	0.630615
$470$	0	0
$471$	10.4853	0.483136
$472$	−55.1127	−2.53677
$473$	27.4558	1.26242
$474$	0	0
$475$	0	0
$476$	−6.34315	−0.290738
$477$	−8.00000	−0.366295
$478$	64.5269	2.95139
$479$	17.0711	0.779997	0.389998	−	0.920815i	$-0.372476\pi$
$479$	17.0711	0.779997	0.389998	+	0.920815i	$0.372476\pi$
$480$	0	0
$481$	11.6569	0.531507
$482$	47.4558	2.16155
$483$	10.8284	0.492710
$484$	−22.8579	−1.03899
$485$	0	0
$486$	2.41421	0.109511
$487$	−20.4853	−0.928277	−0.464138	−	0.885763i	$-0.653636\pi$
$487$	−20.4853	−0.928277	−0.464138	+	0.885763i	$0.653636\pi$
$488$	−32.2843	−1.46144
$489$	21.8995	0.990329
$490$	0	0
$491$	−1.75736	−0.0793085	−0.0396543	−	0.999213i	$-0.512626\pi$
$491$	−1.75736	−0.0793085	−0.0396543	+	0.999213i	$0.512626\pi$
$492$	0.928932	0.0418795
$493$	−1.65685	−0.0746210
$494$	8.24264	0.370854
$495$	0	0
$496$	−9.51472	−0.427223
$497$	−15.3137	−0.686914
$498$	−30.9706	−1.38782
$499$	−5.17157	−0.231511	−0.115756	−	0.993278i	$-0.536929\pi$
$499$	−5.17157	−0.231511	−0.115756	+	0.993278i	$0.536929\pi$
$500$	0	0
$501$	−17.3137	−0.773519
$502$	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	$0.534331\pi$
$504$	−6.24264	−0.278069
$505$	0	0
$506$	−41.4558	−1.84294
$507$	1.34315	0.0596512
$508$	−73.9411	−3.28061
$509$	−0.727922	−0.0322646	−0.0161323	−	0.999870i	$-0.505135\pi$
$509$	−0.727922	−0.0322646	−0.0161323	+	0.999870i	$0.505135\pi$
$510$	0	0
$511$	10.8284	0.479021
$512$	31.2426	1.38074
$513$	1.00000	0.0441511
$514$	−10.8284	−0.477621
$515$	0	0
$516$	46.8701	2.06334
$517$	−17.1716	−0.755205
$518$	−11.6569	−0.512173
$519$	−19.7990	−0.869079
$520$	0	0
$521$	7.75736	0.339856	0.169928	−	0.985456i	$-0.445646\pi$
$521$	7.75736	0.339856	0.169928	+	0.985456i	$0.445646\pi$
$522$	−3.41421	−0.149436
$523$	15.5147	0.678411	0.339206	−	0.940712i	$-0.389842\pi$
$523$	15.5147	0.678411	0.339206	+	0.940712i	$0.389842\pi$
$524$	−32.8701	−1.43594
$525$	0	0
$526$	−56.6274	−2.46907
$527$	3.71573	0.161860
$528$	6.72792	0.292795
$529$	35.6274	1.54902
$530$	0	0
$531$	12.4853	0.541815
$532$	−5.41421	−0.234736
$533$	−0.828427	−0.0358832
$534$	14.2426	0.616339
$535$	0	0
$536$	−42.6274	−1.84122
$537$	−16.4853	−0.711392
$538$	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
$543$	20.8284	0.893833
$544$	−1.85786	−0.0796553
$545$	0	0
$546$	11.6569	0.498867
$547$	−24.4853	−1.04692	−0.523458	−	0.852052i	$-0.675358\pi$
$547$	−24.4853	−1.04692	−0.523458	+	0.852052i	$0.675358\pi$
$548$	−38.2843	−1.63542
$549$	7.31371	0.312141
$550$	0	0
$551$	−1.41421	−0.0602475
$552$	−33.7990	−1.43858
$553$	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	$0.319826\pi$
$558$	7.65685	0.324140
$559$	−41.7990	−1.76791
$560$	0	0
$561$	−2.62742	−0.110930
$562$	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	$0.279694\pi$
$564$	−29.3137	−1.23433
$565$	0	0
$566$	−25.0711	−1.05382
$567$	1.41421	0.0593914
$568$	47.7990	2.00560
$569$	31.0711	1.30257	0.651283	−	0.758835i	$-0.274231\pi$
$569$	31.0711	1.30257	0.651283	+	0.758835i	$0.274231\pi$
$570$	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
$573$	−10.2426	−0.427892
$574$	0.828427	0.0345779
$575$	0	0
$576$	−9.82843	−0.409518
$577$	−19.4558	−0.809957	−0.404979	−	0.914326i	$-0.632721\pi$
$577$	−19.4558	−0.809957	−0.404979	+	0.914326i	$0.632721\pi$
$578$	37.7279	1.56927
$579$	5.07107	0.210746
$580$	0	0
$581$	−18.1421	−0.752663
$582$	23.5563	0.976442
$583$	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	$0.418014\pi$
$588$	19.1421	0.789408
$589$	3.17157	0.130682
$590$	0	0
$591$	6.82843	0.280884
$592$	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	$0.770937\pi$
$594$	−5.41421	−0.222148
$595$	0	0
$596$	−22.9706	−0.940911
$597$	−18.1421	−0.742508
$598$	63.1127	2.58087
$599$	−3.02944	−0.123779	−0.0618897	−	0.998083i	$-0.519713\pi$
$599$	−3.02944	−0.123779	−0.0618897	+	0.998083i	$0.519713\pi$
$600$	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
$603$	9.65685	0.393258
$604$	−24.8284	−1.01025
$605$	0	0
$606$	−7.65685	−0.311038
$607$	−16.4853	−0.669117	−0.334558	−	0.942375i	$-0.608587\pi$
$607$	−16.4853	−0.669117	−0.334558	+	0.942375i	$0.608587\pi$
$608$	−1.58579	−0.0643121
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	26.1421	1.05760
$612$	−4.48528	−0.181307
$613$	10.4853	0.423497	0.211748	−	0.977324i	$-0.432084\pi$
$613$	10.4853	0.423497	0.211748	+	0.977324i	$0.432084\pi$
$614$	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	$0.694371\pi$
$618$	17.6569	0.710263
$619$	−20.4853	−0.823373	−0.411686	−	0.911326i	$-0.635060\pi$
$619$	−20.4853	−0.823373	−0.411686	+	0.911326i	$0.635060\pi$
$620$	0	0
$621$	7.65685	0.307259
$622$	15.0711	0.604295
$623$	8.34315	0.334261
$624$	−10.2426	−0.410034
$625$	0	0
$626$	14.0000	0.559553
$627$	−2.24264	−0.0895624
$628$	−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
$633$	−7.31371	−0.290694
$634$	−64.2843	−2.55305
$635$	0	0
$636$	30.6274	1.21446
$637$	−17.0711	−0.676380
$638$	7.65685	0.303138
$639$	−10.8284	−0.428366
$640$	0	0
$641$	−31.3553	−1.23846	−0.619231	−	0.785209i	$-0.712555\pi$
$641$	−31.3553	−1.23846	−0.619231	+	0.785209i	$0.712555\pi$
$642$	−46.6274	−1.84024
$643$	−42.3848	−1.67149	−0.835746	−	0.549116i	$-0.814965\pi$
$643$	−42.3848	−1.67149	−0.835746	+	0.549116i	$0.814965\pi$
$644$	−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	$0.248493\pi$
$648$	−4.41421	−0.173407
$649$	−28.0000	−1.09910
$650$	0	0
$651$	4.48528	0.175792
$652$	−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	$0.816277\pi$
$654$	−15.6569	−0.612231
$655$	0	0
$656$	−0.727922	−0.0284206
$657$	7.65685	0.298722
$658$	−26.1421	−1.01913
$659$	−18.6274	−0.725621	−0.362811	−	0.931863i	$-0.618183\pi$
$659$	−18.6274	−0.725621	−0.362811	+	0.931863i	$0.618183\pi$
$660$	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
$663$	4.00000	0.155347
$664$	56.6274	2.19757
$665$	0	0
$666$	−8.24264	−0.319396
$667$	−10.8284	−0.419278
$668$	66.2843	2.56462
$669$	18.6274	0.720178
$670$	0	0
$671$	−16.4020	−0.633193
$672$	−2.24264	−0.0865117
$673$	44.1838	1.70316	0.851580	−	0.524225i	$-0.175645\pi$
$673$	44.1838	1.70316	0.851580	+	0.524225i	$0.175645\pi$
$674$	−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	$0.605740\pi$
$678$	−24.4853	−0.940352
$679$	13.7990	0.529557
$680$	0	0
$681$	14.9706	0.573673
$682$	−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	$0.614138\pi$
$684$	−3.82843	−0.146384
$685$	0	0
$686$	40.9706	1.56426
$687$	22.6274	0.863290
$688$	−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
$693$	−3.17157	−0.120478
$694$	−66.2843	−2.51612
$695$	0	0
$696$	6.24264	0.236627
$697$	0.284271	0.0107675
$698$	−43.4558	−1.64483
$699$	−0.343146	−0.0129790
$700$	0	0
$701$	6.68629	0.252538	0.126269	−	0.991996i	$-0.459700\pi$
$701$	6.68629	0.252538	0.126269	+	0.991996i	$0.459700\pi$
$702$	8.24264	0.311098
$703$	−3.41421	−0.128770
$704$	22.0416	0.830725
$705$	0	0
$706$	8.82843	0.332262
$707$	−4.48528	−0.168686
$708$	−47.7990	−1.79640
$709$	28.9706	1.08801	0.544006	−	0.839081i	$-0.316907\pi$
$709$	28.9706	1.08801	0.544006	+	0.839081i	$0.316907\pi$
$710$	0	0
$711$	0	0
$712$	−26.0416	−0.975951
$713$	24.2843	0.909453
$714$	−4.00000	−0.149696
$715$	0	0
$716$	63.1127	2.35863
$717$	26.7279	0.998173
$718$	−60.0416	−2.24073
$719$	−1.07107	−0.0399441	−0.0199720	−	0.999801i	$-0.506358\pi$
$719$	−1.07107	−0.0399441	−0.0199720	+	0.999801i	$0.506358\pi$
$720$	0	0
$721$	10.3431	0.385199
$722$	−2.41421	−0.0898477
$723$	19.6569	0.731046
$724$	−79.7401	−2.96352
$725$	0	0
$726$	−14.4142	−0.534962
$727$	−47.3553	−1.75631	−0.878156	−	0.478374i	$-0.841226\pi$
$727$	−47.3553	−1.75631	−0.878156	+	0.478374i	$0.841226\pi$
$728$	−21.3137	−0.789939
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.3431	0.530500
$732$	−28.0000	−1.03491
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−34.7279	−1.28183
$735$	0	0
$736$	−12.1421	−0.447565
$737$	−21.6569	−0.797740
$738$	0.585786	0.0215631
$739$	14.3431	0.527621	0.263811	−	0.964575i	$-0.415021\pi$
$739$	14.3431	0.527621	0.263811	+	0.964575i	$0.415021\pi$
$740$	0	0
$741$	3.41421	0.125424
$742$	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	$0.476624\pi$
$744$	−14.0000	−0.513265
$745$	0	0
$746$	1.41421	0.0517780
$747$	−12.8284	−0.469368
$748$	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
$753$	1.75736	0.0640417
$754$	−11.6569	−0.424518
$755$	0	0
$756$	−5.41421	−0.196913
$757$	32.4264	1.17856	0.589279	−	0.807930i	$-0.299412\pi$
$757$	32.4264	1.17856	0.589279	+	0.807930i	$0.299412\pi$
$758$	−7.65685	−0.278109
$759$	−17.1716	−0.623289
$760$	0	0
$761$	43.9411	1.59286	0.796432	−	0.604728i	$-0.206718\pi$
$761$	43.9411	1.59286	0.796432	+	0.604728i	$0.206718\pi$
$762$	−46.6274	−1.68913
$763$	−9.17157	−0.332033
$764$	39.2132	1.41868
$765$	0	0
$766$	67.5980	2.44241
$767$	42.6274	1.53919
$768$	29.9706	1.08147
$769$	42.9706	1.54956	0.774779	−	0.632232i	$-0.217861\pi$
$769$	42.9706	1.54956	0.774779	+	0.632232i	$0.217861\pi$
$770$	0	0
$771$	−4.48528	−0.161533
$772$	−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	$0.347345\pi$
$774$	29.5563	1.06238
$775$	0	0
$776$	−43.0711	−1.54616
$777$	−4.82843	−0.173219
$778$	−74.7696	−2.68062
$779$	0.242641	0.00869350
$780$	0	0
$781$	24.2843	0.868960
$782$	−21.6569	−0.774448
$783$	−1.41421	−0.0505399
$784$	−15.0000	−0.535714
$785$	0	0
$786$	−20.7279	−0.739340
$787$	42.8284	1.52667	0.763334	−	0.646004i	$-0.223561\pi$
$787$	42.8284	1.52667	0.763334	+	0.646004i	$0.223561\pi$
$788$	−26.1421	−0.931275
$789$	−23.4558	−0.835050
$790$	0	0
$791$	−14.3431	−0.509984
$792$	9.89949	0.351763
$793$	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	$0.419415\pi$
$798$	−3.41421	−0.120862
$799$	−8.97056	−0.317356
$800$	0	0
$801$	5.89949	0.208448
$802$	18.7279	0.661306
$803$	−17.1716	−0.605972
$804$	−36.9706	−1.30385
$805$	0	0
$806$	26.1421	0.920817
$807$	3.07107	0.108107
$808$	14.0000	0.492518
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	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
$813$	10.8284	0.379770
$814$	18.4853	0.647909
$815$	0	0
$816$	3.51472	0.123040
$817$	12.2426	0.428316
$818$	30.9706	1.08286
$819$	4.82843	0.168719
$820$	0	0
$821$	14.4853	0.505540	0.252770	−	0.967526i	$-0.418658\pi$
$821$	14.4853	0.505540	0.252770	+	0.967526i	$0.418658\pi$
$822$	−24.1421	−0.842054
$823$	25.0122	0.871870	0.435935	−	0.899978i	$-0.356418\pi$
$823$	25.0122	0.871870	0.435935	+	0.899978i	$0.356418\pi$
$824$	−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	$0.485128\pi$
$828$	−29.3137	−1.01872
$829$	46.4853	1.61450	0.807250	−	0.590209i	$-0.200955\pi$
$829$	46.4853	1.61450	0.807250	+	0.590209i	$0.200955\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	−33.5563	−1.16336
$833$	5.85786	0.202963
$834$	−34.1421	−1.18225
$835$	0	0
$836$	8.58579	0.296946
$837$	3.17157	0.109626
$838$	−8.24264	−0.284737
$839$	−9.85786	−0.340331	−0.170166	−	0.985415i	$-0.554430\pi$
$839$	−9.85786	−0.340331	−0.170166	+	0.985415i	$0.554430\pi$
$840$	0	0
$841$	−27.0000	−0.931034
$842$	−22.4853	−0.774894
$843$	14.5858	0.502361
$844$	28.0000	0.963800
$845$	0	0
$846$	−18.4853	−0.635537
$847$	−8.44365	−0.290127
$848$	−24.0000	−0.824163
$849$	−10.3848	−0.356405
$850$	0	0
$851$	−26.1421	−0.896141
$852$	41.4558	1.42025
$853$	16.8284	0.576194	0.288097	−	0.957601i	$-0.406977\pi$
$853$	16.8284	0.576194	0.288097	+	0.957601i	$0.406977\pi$
$854$	−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	$0.569522\pi$
$858$	−18.4853	−0.631077
$859$	−49.9411	−1.70397	−0.851985	−	0.523567i	$-0.824601\pi$
$859$	−49.9411	−1.70397	−0.851985	+	0.523567i	$0.824601\pi$
$860$	0	0
$861$	0.343146	0.0116944
$862$	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	$0.547233\pi$
$864$	−1.58579	−0.0539496
$865$	0	0
$866$	−26.3848	−0.896591
$867$	15.6274	0.530735
$868$	−17.1716	−0.582841
$869$	0	0
$870$	0	0
$871$	32.9706	1.11716
$872$	28.6274	0.969447
$873$	9.75736	0.330237
$874$	−18.4853	−0.625274
$875$	0	0
$876$	−29.3137	−0.990418
$877$	34.9289	1.17947	0.589733	−	0.807598i	$-0.299233\pi$
$877$	34.9289	1.17947	0.589733	+	0.807598i	$0.299233\pi$
$878$	52.2843	1.76451
$879$	−11.5147	−0.388382
$880$	0	0
$881$	−5.79899	−0.195373	−0.0976865	−	0.995217i	$-0.531144\pi$
$881$	−5.79899	−0.195373	−0.0976865	+	0.995217i	$0.531144\pi$
$882$	12.0711	0.406454
$883$	12.0416	0.405233	0.202617	−	0.979258i	$-0.435055\pi$
$883$	12.0416	0.405233	0.202617	+	0.979258i	$0.435055\pi$
$884$	−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	$0.643432\pi$
$888$	15.0711	0.505752
$889$	−27.3137	−0.916072
$890$	0	0
$891$	−2.24264	−0.0751313
$892$	−71.3137	−2.38776
$893$	−7.65685	−0.256227
$894$	−14.4853	−0.484460
$895$	0	0
$896$	29.0711	0.971196
$897$	26.1421	0.872861
$898$	64.8701	2.16474
$899$	−4.48528	−0.149593
$900$	0	0
$901$	9.37258	0.312246
$902$	−1.31371	−0.0437417
$903$	17.3137	0.576164
$904$	44.7696	1.48901
$905$	0	0
$906$	−15.6569	−0.520164
$907$	18.1421	0.602400	0.301200	−	0.953561i	$-0.402613\pi$
$907$	18.1421	0.602400	0.301200	+	0.953561i	$0.402613\pi$
$908$	−57.3137	−1.90202
$909$	−3.17157	−0.105194
$910$	0	0
$911$	8.68629	0.287790	0.143895	−	0.989593i	$-0.454037\pi$
$911$	8.68629	0.287790	0.143895	+	0.989593i	$0.454037\pi$
$912$	3.00000	0.0993399
$913$	28.7696	0.952133
$914$	79.2548	2.62152
$915$	0	0
$916$	−86.6274	−2.86225
$917$	−12.1421	−0.400969
$918$	−2.82843	−0.0933520
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	0	0
$921$	21.1716	0.697627
$922$	−47.4558	−1.56287
$923$	−36.9706	−1.21690
$924$	12.1421	0.399447
$925$	0	0
$926$	58.5269	1.92331
$927$	7.31371	0.240214
$928$	2.24264	0.0736183
$929$	33.1127	1.08639	0.543196	−	0.839606i	$-0.317214\pi$
$929$	33.1127	1.08639	0.543196	+	0.839606i	$0.317214\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	1.31371	0.0430320
$933$	6.24264	0.204375
$934$	86.0833	2.81673
$935$	0	0
$936$	−15.0711	−0.492613
$937$	29.7990	0.973491	0.486745	−	0.873544i	$-0.338184\pi$
$937$	29.7990	0.973491	0.486745	+	0.873544i	$0.338184\pi$
$938$	−32.9706	−1.07653
$939$	5.79899	0.189243
$940$	0	0
$941$	22.8701	0.745543	0.372771	−	0.927923i	$-0.378408\pi$
$941$	22.8701	0.745543	0.372771	+	0.927923i	$0.378408\pi$
$942$	−25.3137	−0.824765
$943$	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
$950$	0	0
$951$	−26.6274	−0.863453
$952$	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	$0.166809\pi$
$954$	19.3137	0.625304
$955$	0	0
$956$	−102.326	−3.30946
$957$	3.17157	0.102522
$958$	−41.2132	−1.33154
$959$	−14.1421	−0.456673
$960$	0	0
$961$	−20.9411	−0.675520
$962$	−28.1421	−0.907339
$963$	−19.3137	−0.622376
$964$	−75.2548	−2.42379
$965$	0	0
$966$	−26.1421	−0.841109
$967$	−8.04163	−0.258601	−0.129301	−	0.991605i	$-0.541273\pi$
$967$	−8.04163	−0.258601	−0.129301	+	0.991605i	$0.541273\pi$
$968$	26.3553	0.847093
$969$	−1.17157	−0.0376363
$970$	0	0
$971$	22.3431	0.717026	0.358513	−	0.933525i	$-0.383284\pi$
$971$	22.3431	0.717026	0.358513	+	0.933525i	$0.383284\pi$
$972$	−3.82843	−0.122797
$973$	−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	$0.672740\pi$
$978$	−52.8701	−1.69060
$979$	−13.2304	−0.422847
$980$	0	0
$981$	−6.48528	−0.207059
$982$	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	$0.371525\pi$
$984$	−1.07107	−0.0341444
$985$	0	0
$986$	4.00000	0.127386
$987$	−10.8284	−0.344673
$988$	−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
$993$	−28.1421	−0.893064
$994$	36.9706	1.17264
$995$	0	0
$996$	49.1127	1.55620
$997$	38.0833	1.20611	0.603054	−	0.797700i	$-0.293950\pi$
$997$	38.0833	1.20611	0.603054	+	0.797700i	$0.293950\pi$
$998$	12.4853	0.395215
$999$	−3.41421	−0.108021

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.a.k.1.1		2
3.2	odd	2		4275.2.a.y.1.2		2
5.2	odd	4		1425.2.c.l.799.1		4
5.3	odd	4		1425.2.c.l.799.4		4
5.4	even	2		285.2.a.g.1.2	✓	2
15.14	odd	2		855.2.a.d.1.1		2
20.19	odd	2		4560.2.a.bf.1.2		2
95.94	odd	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	5.4	even	2
855.2.a.d.1.1		2	15.14	odd	2
1425.2.a.k.1.1		2	1.1	even	1	trivial
1425.2.c.l.799.1		4	5.2	odd	4
1425.2.c.l.799.4		4	5.3	odd	4
4275.2.a.y.1.2		2	3.2	odd	2
4560.2.a.bf.1.2		2	20.19	odd	2
5415.2.a.n.1.1		2	95.94	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} +2.41421 q^{6} +1.41421 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} +2.41421 q^{6} +1.41421 q^{7} -4.41421 q^{8} +1.00000 q^{9} -2.24264 q^{11} -3.82843 q^{12} +3.41421 q^{13} -3.41421 q^{14} +3.00000 q^{16} -1.17157 q^{17} -2.41421 q^{18} -1.00000 q^{19} -1.41421 q^{21} +5.41421 q^{22} -7.65685 q^{23} +4.41421 q^{24} -8.24264 q^{26} -1.00000 q^{27} +5.41421 q^{28} +1.41421 q^{29} -3.17157 q^{31} +1.58579 q^{32} +2.24264 q^{33} +2.82843 q^{34} +3.82843 q^{36} +3.41421 q^{37} +2.41421 q^{38} -3.41421 q^{39} -0.242641 q^{41} +3.41421 q^{42} -12.2426 q^{43} -8.58579 q^{44} +18.4853 q^{46} +7.65685 q^{47} -3.00000 q^{48} -5.00000 q^{49} +1.17157 q^{51} +13.0711 q^{52} -8.00000 q^{53} +2.41421 q^{54} -6.24264 q^{56} +1.00000 q^{57} -3.41421 q^{58} +12.4853 q^{59} +7.31371 q^{61} +7.65685 q^{62} +1.41421 q^{63} -9.82843 q^{64} -5.41421 q^{66} +9.65685 q^{67} -4.48528 q^{68} +7.65685 q^{69} -10.8284 q^{71} -4.41421 q^{72} +7.65685 q^{73} -8.24264 q^{74} -3.82843 q^{76} -3.17157 q^{77} +8.24264 q^{78} +1.00000 q^{81} +0.585786 q^{82} -12.8284 q^{83} -5.41421 q^{84} +29.5563 q^{86} -1.41421 q^{87} +9.89949 q^{88} +5.89949 q^{89} +4.82843 q^{91} -29.3137 q^{92} +3.17157 q^{93} -18.4853 q^{94} -1.58579 q^{96} +9.75736 q^{97} +12.0711 q^{98} -2.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 4 q^{11} - 2 q^{12} + 4 q^{13} - 4 q^{14} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{19} + 8 q^{22} - 4 q^{23} + 6 q^{24} - 8 q^{26} - 2 q^{27} + 8 q^{28} - 12 q^{31} + 6 q^{32} - 4 q^{33} + 2 q^{36} + 4 q^{37} + 2 q^{38} - 4 q^{39} + 8 q^{41} + 4 q^{42} - 16 q^{43} - 20 q^{44} + 20 q^{46} + 4 q^{47} - 6 q^{48} - 10 q^{49} + 8 q^{51} + 12 q^{52} - 16 q^{53} + 2 q^{54} - 4 q^{56} + 2 q^{57} - 4 q^{58} + 8 q^{59} - 8 q^{61} + 4 q^{62} - 14 q^{64} - 8 q^{66} + 8 q^{67} + 8 q^{68} + 4 q^{69} - 16 q^{71} - 6 q^{72} + 4 q^{73} - 8 q^{74} - 2 q^{76} - 12 q^{77} + 8 q^{78} + 2 q^{81} + 4 q^{82} - 20 q^{83} - 8 q^{84} + 28 q^{86} - 8 q^{89} + 4 q^{91} - 36 q^{92} + 12 q^{93} - 20 q^{94} - 6 q^{96} + 28 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9 + 4 * q^11 - 2 * q^12 + 4 * q^13 - 4 * q^14 + 6 * q^16 - 8 * q^17 - 2 * q^18 - 2 * q^19 + 8 * q^22 - 4 * q^23 + 6 * q^24 - 8 * q^26 - 2 * q^27 + 8 * q^28 - 12 * q^31 + 6 * q^32 - 4 * q^33 + 2 * q^36 + 4 * q^37 + 2 * q^38 - 4 * q^39 + 8 * q^41 + 4 * q^42 - 16 * q^43 - 20 * q^44 + 20 * q^46 + 4 * q^47 - 6 * q^48 - 10 * q^49 + 8 * q^51 + 12 * q^52 - 16 * q^53 + 2 * q^54 - 4 * q^56 + 2 * q^57 - 4 * q^58 + 8 * q^59 - 8 * q^61 + 4 * q^62 - 14 * q^64 - 8 * q^66 + 8 * q^67 + 8 * q^68 + 4 * q^69 - 16 * q^71 - 6 * q^72 + 4 * q^73 - 8 * q^74 - 2 * q^76 - 12 * q^77 + 8 * q^78 + 2 * q^81 + 4 * q^82 - 20 * q^83 - 8 * q^84 + 28 * q^86 - 8 * q^89 + 4 * q^91 - 36 * q^92 + 12 * q^93 - 20 * q^94 - 6 * q^96 + 28 * q^97 + 10 * q^98 + 4 * q^99

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