LMFDB - Embedded newform 2280.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2280,2,Mod(1,2280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2280.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:	$18.2058916609$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	2280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{9} -6.19615 q^{11} -4.73205 q^{13} +1.00000 q^{15} +6.92820 q^{17} +1.00000 q^{19} -1.26795 q^{21} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.196152 q^{29} -2.00000 q^{31} -6.19615 q^{33} -1.26795 q^{35} -0.732051 q^{37} -4.73205 q^{39} -5.66025 q^{41} -1.26795 q^{43} +1.00000 q^{45} -10.0000 q^{47} -5.39230 q^{49} +6.92820 q^{51} -6.19615 q^{55} +1.00000 q^{57} -2.53590 q^{59} -1.46410 q^{61} -1.26795 q^{63} -4.73205 q^{65} -1.07180 q^{67} -6.00000 q^{69} -8.39230 q^{71} -12.9282 q^{73} +1.00000 q^{75} +7.85641 q^{77} -8.00000 q^{79} +1.00000 q^{81} +15.4641 q^{83} +6.92820 q^{85} +0.196152 q^{87} +9.66025 q^{89} +6.00000 q^{91} -2.00000 q^{93} +1.00000 q^{95} -15.6603 q^{97} -6.19615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} - 6 q^{13} + 2 q^{15} + 2 q^{19} - 6 q^{21} - 12 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 2 q^{33} - 6 q^{35} + 2 q^{37} - 6 q^{39} + 6 q^{41} - 6 q^{43} + 2 q^{45} - 20 q^{47} + 10 q^{49} - 2 q^{55} + 2 q^{57} - 12 q^{59} + 4 q^{61} - 6 q^{63} - 6 q^{65} - 16 q^{67} - 12 q^{69} + 4 q^{71} - 12 q^{73} + 2 q^{75} - 12 q^{77} - 16 q^{79} + 2 q^{81} + 24 q^{83} - 10 q^{87} + 2 q^{89} + 12 q^{91} - 4 q^{93} + 2 q^{95} - 14 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 - 2 * q^11 - 6 * q^13 + 2 * q^15 + 2 * q^19 - 6 * q^21 - 12 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 - 4 * q^31 - 2 * q^33 - 6 * q^35 + 2 * q^37 - 6 * q^39 + 6 * q^41 - 6 * q^43 + 2 * q^45 - 20 * q^47 + 10 * q^49 - 2 * q^55 + 2 * q^57 - 12 * q^59 + 4 * q^61 - 6 * q^63 - 6 * q^65 - 16 * q^67 - 12 * q^69 + 4 * q^71 - 12 * q^73 + 2 * q^75 - 12 * q^77 - 16 * q^79 + 2 * q^81 + 24 * q^83 - 10 * q^87 + 2 * q^89 + 12 * q^91 - 4 * q^93 + 2 * q^95 - 14 * q^97 - 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.26795	−0.479240	−0.239620	−	0.970867i	$-0.577023\pi$
$7$	−1.26795	−0.479240	−0.239620	+	0.970867i	$0.577023\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.19615	−1.86821	−0.934105	−	0.356998i	$-0.883800\pi$
$11$	−6.19615	−1.86821	−0.934105	+	0.356998i	$0.883800\pi$
$12$	0	0
$13$	−4.73205	−1.31243	−0.656217	−	0.754572i	$-0.727845\pi$
$13$	−4.73205	−1.31243	−0.656217	+	0.754572i	$0.727845\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	6.92820	1.68034	0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820	1.68034	0.840168	+	0.542326i	$0.182456\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.26795	−0.276689
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.196152	0.0364246	0.0182123	−	0.999834i	$-0.494203\pi$
$29$	0.196152	0.0364246	0.0182123	+	0.999834i	$0.494203\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	−6.19615	−1.07861
$34$	0	0
$35$	−1.26795	−0.214323
$36$	0	0
$37$	−0.732051	−0.120348	−0.0601742	−	0.998188i	$-0.519166\pi$
$37$	−0.732051	−0.120348	−0.0601742	+	0.998188i	$0.519166\pi$
$38$	0	0
$39$	−4.73205	−0.757735
$40$	0	0
$41$	−5.66025	−0.883983	−0.441992	−	0.897019i	$-0.645728\pi$
$41$	−5.66025	−0.883983	−0.441992	+	0.897019i	$0.645728\pi$
$42$	0	0
$43$	−1.26795	−0.193360	−0.0966802	−	0.995315i	$-0.530822\pi$
$43$	−1.26795	−0.193360	−0.0966802	+	0.995315i	$0.530822\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	0	0
$51$	6.92820	0.970143
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−6.19615	−0.835489
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−2.53590	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$59$	−2.53590	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$60$	0	0
$61$	−1.46410	−0.187459	−0.0937295	−	0.995598i	$-0.529879\pi$
$61$	−1.46410	−0.187459	−0.0937295	+	0.995598i	$0.529879\pi$
$62$	0	0
$63$	−1.26795	−0.159747
$64$	0	0
$65$	−4.73205	−0.586939
$66$	0	0
$67$	−1.07180	−0.130941	−0.0654704	−	0.997855i	$-0.520855\pi$
$67$	−1.07180	−0.130941	−0.0654704	+	0.997855i	$0.520855\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−8.39230	−0.995983	−0.497992	−	0.867182i	$-0.665929\pi$
$71$	−8.39230	−0.995983	−0.497992	+	0.867182i	$0.665929\pi$
$72$	0	0
$73$	−12.9282	−1.51313	−0.756566	−	0.653917i	$-0.773124\pi$
$73$	−12.9282	−1.51313	−0.756566	+	0.653917i	$0.773124\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	7.85641	0.895321
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.4641	1.69741	0.848703	−	0.528870i	$-0.177384\pi$
$83$	15.4641	1.69741	0.848703	+	0.528870i	$0.177384\pi$
$84$	0	0
$85$	6.92820	0.751469
$86$	0	0
$87$	0.196152	0.0210297
$88$	0	0
$89$	9.66025	1.02398	0.511992	−	0.858990i	$-0.328908\pi$
$89$	9.66025	1.02398	0.511992	+	0.858990i	$0.328908\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−15.6603	−1.59006	−0.795029	−	0.606571i	$-0.792544\pi$
$97$	−15.6603	−1.59006	−0.795029	+	0.606571i	$0.792544\pi$
$98$	0	0
$99$	−6.19615	−0.622737
$100$	0	0
$101$	3.46410	0.344691	0.172345	−	0.985037i	$-0.444865\pi$
$101$	3.46410	0.344691	0.172345	+	0.985037i	$0.444865\pi$
$102$	0	0
$103$	13.8564	1.36531	0.682656	−	0.730740i	$-0.260825\pi$
$103$	13.8564	1.36531	0.682656	+	0.730740i	$0.260825\pi$
$104$	0	0
$105$	−1.26795	−0.123739
$106$	0	0
$107$	2.92820	0.283080	0.141540	−	0.989933i	$-0.454795\pi$
$107$	2.92820	0.283080	0.141540	+	0.989933i	$0.454795\pi$
$108$	0	0
$109$	10.3923	0.995402	0.497701	−	0.867349i	$-0.334178\pi$
$109$	10.3923	0.995402	0.497701	+	0.867349i	$0.334178\pi$
$110$	0	0
$111$	−0.732051	−0.0694832
$112$	0	0
$113$	9.46410	0.890308	0.445154	−	0.895454i	$-0.353149\pi$
$113$	9.46410	0.890308	0.445154	+	0.895454i	$0.353149\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	0	0
$117$	−4.73205	−0.437478
$118$	0	0
$119$	−8.78461	−0.805284
$120$	0	0
$121$	27.3923	2.49021
$122$	0	0
$123$	−5.66025	−0.510368
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.9282	−1.32466	−0.662332	−	0.749211i	$-0.730433\pi$
$127$	−14.9282	−1.32466	−0.662332	+	0.749211i	$0.730433\pi$
$128$	0	0
$129$	−1.26795	−0.111637
$130$	0	0
$131$	10.5885	0.925118	0.462559	−	0.886589i	$-0.346931\pi$
$131$	10.5885	0.925118	0.462559	+	0.886589i	$0.346931\pi$
$132$	0	0
$133$	−1.26795	−0.109945
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−18.7846	−1.60488	−0.802439	−	0.596734i	$-0.796465\pi$
$137$	−18.7846	−1.60488	−0.802439	+	0.596734i	$0.796465\pi$
$138$	0	0
$139$	−10.5359	−0.893643	−0.446822	−	0.894623i	$-0.647444\pi$
$139$	−10.5359	−0.893643	−0.446822	+	0.894623i	$0.647444\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	0	0
$143$	29.3205	2.45190
$144$	0	0
$145$	0.196152	0.0162896
$146$	0	0
$147$	−5.39230	−0.444750
$148$	0	0
$149$	−4.92820	−0.403734	−0.201867	−	0.979413i	$-0.564701\pi$
$149$	−4.92820	−0.403734	−0.201867	+	0.979413i	$0.564701\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	6.92820	0.560112
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	3.46410	0.276465	0.138233	−	0.990400i	$-0.455858\pi$
$157$	3.46410	0.276465	0.138233	+	0.990400i	$0.455858\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.60770	0.599570
$162$	0	0
$163$	0.875644	0.0685858	0.0342929	−	0.999412i	$-0.489082\pi$
$163$	0.875644	0.0685858	0.0342929	+	0.999412i	$0.489082\pi$
$164$	0	0
$165$	−6.19615	−0.482370
$166$	0	0
$167$	−6.39230	−0.494651	−0.247326	−	0.968932i	$-0.579552\pi$
$167$	−6.39230	−0.494651	−0.247326	+	0.968932i	$0.579552\pi$
$168$	0	0
$169$	9.39230	0.722485
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−8.39230	−0.638055	−0.319028	−	0.947745i	$-0.603356\pi$
$173$	−8.39230	−0.638055	−0.319028	+	0.947745i	$0.603356\pi$
$174$	0	0
$175$	−1.26795	−0.0958479
$176$	0	0
$177$	−2.53590	−0.190610
$178$	0	0
$179$	14.5359	1.08646	0.543232	−	0.839583i	$-0.317200\pi$
$179$	14.5359	1.08646	0.543232	+	0.839583i	$0.317200\pi$
$180$	0	0
$181$	11.4641	0.852120	0.426060	−	0.904695i	$-0.359901\pi$
$181$	11.4641	0.852120	0.426060	+	0.904695i	$0.359901\pi$
$182$	0	0
$183$	−1.46410	−0.108230
$184$	0	0
$185$	−0.732051	−0.0538214
$186$	0	0
$187$	−42.9282	−3.13922
$188$	0	0
$189$	−1.26795	−0.0922297
$190$	0	0
$191$	−16.7321	−1.21069	−0.605344	−	0.795964i	$-0.706965\pi$
$191$	−16.7321	−1.21069	−0.605344	+	0.795964i	$0.706965\pi$
$192$	0	0
$193$	−22.1962	−1.59771	−0.798857	−	0.601521i	$-0.794562\pi$
$193$	−22.1962	−1.59771	−0.798857	+	0.601521i	$0.794562\pi$
$194$	0	0
$195$	−4.73205	−0.338869
$196$	0	0
$197$	14.9282	1.06359	0.531795	−	0.846873i	$-0.321518\pi$
$197$	14.9282	1.06359	0.531795	+	0.846873i	$0.321518\pi$
$198$	0	0
$199$	−9.46410	−0.670892	−0.335446	−	0.942059i	$-0.608887\pi$
$199$	−9.46410	−0.670892	−0.335446	+	0.942059i	$0.608887\pi$
$200$	0	0
$201$	−1.07180	−0.0755987
$202$	0	0
$203$	−0.248711	−0.0174561
$204$	0	0
$205$	−5.66025	−0.395329
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−6.19615	−0.428597
$210$	0	0
$211$	6.92820	0.476957	0.238479	−	0.971148i	$-0.423351\pi$
$211$	6.92820	0.476957	0.238479	+	0.971148i	$0.423351\pi$
$212$	0	0
$213$	−8.39230	−0.575031
$214$	0	0
$215$	−1.26795	−0.0864734
$216$	0	0
$217$	2.53590	0.172148
$218$	0	0
$219$	−12.9282	−0.873607
$220$	0	0
$221$	−32.7846	−2.20533
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	16.5359	1.09753	0.548763	−	0.835978i	$-0.315099\pi$
$227$	16.5359	1.09753	0.548763	+	0.835978i	$0.315099\pi$
$228$	0	0
$229$	10.5359	0.696232	0.348116	−	0.937452i	$-0.386822\pi$
$229$	10.5359	0.696232	0.348116	+	0.937452i	$0.386822\pi$
$230$	0	0
$231$	7.85641	0.516914
$232$	0	0
$233$	27.8564	1.82493	0.912467	−	0.409150i	$-0.134175\pi$
$233$	27.8564	1.82493	0.912467	+	0.409150i	$0.134175\pi$
$234$	0	0
$235$	−10.0000	−0.652328
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	16.0526	1.03835	0.519177	−	0.854667i	$-0.326239\pi$
$239$	16.0526	1.03835	0.519177	+	0.854667i	$0.326239\pi$
$240$	0	0
$241$	4.92820	0.317453	0.158727	−	0.987323i	$-0.449261\pi$
$241$	4.92820	0.317453	0.158727	+	0.987323i	$0.449261\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−5.39230	−0.344502
$246$	0	0
$247$	−4.73205	−0.301093
$248$	0	0
$249$	15.4641	0.979998
$250$	0	0
$251$	−17.8038	−1.12377	−0.561884	−	0.827216i	$-0.689923\pi$
$251$	−17.8038	−1.12377	−0.561884	+	0.827216i	$0.689923\pi$
$252$	0	0
$253$	37.1769	2.33729
$254$	0	0
$255$	6.92820	0.433861
$256$	0	0
$257$	−7.32051	−0.456641	−0.228320	−	0.973586i	$-0.573323\pi$
$257$	−7.32051	−0.456641	−0.228320	+	0.973586i	$0.573323\pi$
$258$	0	0
$259$	0.928203	0.0576757
$260$	0	0
$261$	0.196152	0.0121415
$262$	0	0
$263$	−3.46410	−0.213606	−0.106803	−	0.994280i	$-0.534061\pi$
$263$	−3.46410	−0.213606	−0.106803	+	0.994280i	$0.534061\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.66025	0.591198
$268$	0	0
$269$	19.5167	1.18995	0.594976	−	0.803744i	$-0.297162\pi$
$269$	19.5167	1.18995	0.594976	+	0.803744i	$0.297162\pi$
$270$	0	0
$271$	−11.3205	−0.687672	−0.343836	−	0.939030i	$-0.611726\pi$
$271$	−11.3205	−0.687672	−0.343836	+	0.939030i	$0.611726\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	−6.19615	−0.373642
$276$	0	0
$277$	−15.8564	−0.952719	−0.476360	−	0.879251i	$-0.658044\pi$
$277$	−15.8564	−0.952719	−0.476360	+	0.879251i	$0.658044\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	30.0526	1.79279	0.896393	−	0.443261i	$-0.146178\pi$
$281$	30.0526	1.79279	0.896393	+	0.443261i	$0.146178\pi$
$282$	0	0
$283$	−24.1962	−1.43831	−0.719156	−	0.694849i	$-0.755471\pi$
$283$	−24.1962	−1.43831	−0.719156	+	0.694849i	$0.755471\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	7.17691	0.423640
$288$	0	0
$289$	31.0000	1.82353
$290$	0	0
$291$	−15.6603	−0.918020
$292$	0	0
$293$	−3.60770	−0.210764	−0.105382	−	0.994432i	$-0.533606\pi$
$293$	−3.60770	−0.210764	−0.105382	+	0.994432i	$0.533606\pi$
$294$	0	0
$295$	−2.53590	−0.147646
$296$	0	0
$297$	−6.19615	−0.359537
$298$	0	0
$299$	28.3923	1.64197
$300$	0	0
$301$	1.60770	0.0926660
$302$	0	0
$303$	3.46410	0.199007
$304$	0	0
$305$	−1.46410	−0.0838342
$306$	0	0
$307$	−27.3205	−1.55926	−0.779632	−	0.626238i	$-0.784594\pi$
$307$	−27.3205	−1.55926	−0.779632	+	0.626238i	$0.784594\pi$
$308$	0	0
$309$	13.8564	0.788263
$310$	0	0
$311$	24.7321	1.40243	0.701213	−	0.712952i	$-0.252642\pi$
$311$	24.7321	1.40243	0.701213	+	0.712952i	$0.252642\pi$
$312$	0	0
$313$	21.3205	1.20511	0.602553	−	0.798079i	$-0.294150\pi$
$313$	21.3205	1.20511	0.602553	+	0.798079i	$0.294150\pi$
$314$	0	0
$315$	−1.26795	−0.0714408
$316$	0	0
$317$	14.9282	0.838451	0.419226	−	0.907882i	$-0.362302\pi$
$317$	14.9282	0.838451	0.419226	+	0.907882i	$0.362302\pi$
$318$	0	0
$319$	−1.21539	−0.0680488
$320$	0	0
$321$	2.92820	0.163436
$322$	0	0
$323$	6.92820	0.385496
$324$	0	0
$325$	−4.73205	−0.262487
$326$	0	0
$327$	10.3923	0.574696
$328$	0	0
$329$	12.6795	0.699043
$330$	0	0
$331$	−3.07180	−0.168841	−0.0844206	−	0.996430i	$-0.526904\pi$
$331$	−3.07180	−0.168841	−0.0844206	+	0.996430i	$0.526904\pi$
$332$	0	0
$333$	−0.732051	−0.0401161
$334$	0	0
$335$	−1.07180	−0.0585585
$336$	0	0
$337$	0.732051	0.0398773	0.0199387	−	0.999801i	$-0.493653\pi$
$337$	0.732051	0.0398773	0.0199387	+	0.999801i	$0.493653\pi$
$338$	0	0
$339$	9.46410	0.514019
$340$	0	0
$341$	12.3923	0.671081
$342$	0	0
$343$	15.7128	0.848412
$344$	0	0
$345$	−6.00000	−0.323029
$346$	0	0
$347$	3.46410	0.185963	0.0929814	−	0.995668i	$-0.470360\pi$
$347$	3.46410	0.185963	0.0929814	+	0.995668i	$0.470360\pi$
$348$	0	0
$349$	−25.7128	−1.37638	−0.688188	−	0.725533i	$-0.741593\pi$
$349$	−25.7128	−1.37638	−0.688188	+	0.725533i	$0.741593\pi$
$350$	0	0
$351$	−4.73205	−0.252578
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	−8.39230	−0.445417
$356$	0	0
$357$	−8.78461	−0.464931
$358$	0	0
$359$	−30.5885	−1.61440	−0.807199	−	0.590280i	$-0.799017\pi$
$359$	−30.5885	−1.61440	−0.807199	+	0.590280i	$0.799017\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	27.3923	1.43772
$364$	0	0
$365$	−12.9282	−0.676693
$366$	0	0
$367$	−9.26795	−0.483783	−0.241892	−	0.970303i	$-0.577768\pi$
$367$	−9.26795	−0.483783	−0.241892	+	0.970303i	$0.577768\pi$
$368$	0	0
$369$	−5.66025	−0.294661
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−14.5885	−0.755362	−0.377681	−	0.925936i	$-0.623278\pi$
$373$	−14.5885	−0.755362	−0.377681	+	0.925936i	$0.623278\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−0.928203	−0.0478049
$378$	0	0
$379$	−3.85641	−0.198090	−0.0990451	−	0.995083i	$-0.531579\pi$
$379$	−3.85641	−0.198090	−0.0990451	+	0.995083i	$0.531579\pi$
$380$	0	0
$381$	−14.9282	−0.764795
$382$	0	0
$383$	−34.6410	−1.77007	−0.885037	−	0.465521i	$-0.845867\pi$
$383$	−34.6410	−1.77007	−0.885037	+	0.465521i	$0.845867\pi$
$384$	0	0
$385$	7.85641	0.400400
$386$	0	0
$387$	−1.26795	−0.0644535
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−41.5692	−2.10225
$392$	0	0
$393$	10.5885	0.534117
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−3.07180	−0.154169	−0.0770845	−	0.997025i	$-0.524561\pi$
$397$	−3.07180	−0.154169	−0.0770845	+	0.997025i	$0.524561\pi$
$398$	0	0
$399$	−1.26795	−0.0634769
$400$	0	0
$401$	21.6603	1.08166	0.540831	−	0.841131i	$-0.318110\pi$
$401$	21.6603	1.08166	0.540831	+	0.841131i	$0.318110\pi$
$402$	0	0
$403$	9.46410	0.471440
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	4.53590	0.224836
$408$	0	0
$409$	−7.46410	−0.369076	−0.184538	−	0.982825i	$-0.559079\pi$
$409$	−7.46410	−0.369076	−0.184538	+	0.982825i	$0.559079\pi$
$410$	0	0
$411$	−18.7846	−0.926576
$412$	0	0
$413$	3.21539	0.158219
$414$	0	0
$415$	15.4641	0.759103
$416$	0	0
$417$	−10.5359	−0.515945
$418$	0	0
$419$	4.73205	0.231176	0.115588	−	0.993297i	$-0.463125\pi$
$419$	4.73205	0.231176	0.115588	+	0.993297i	$0.463125\pi$
$420$	0	0
$421$	−28.9282	−1.40987	−0.704937	−	0.709270i	$-0.749025\pi$
$421$	−28.9282	−1.40987	−0.704937	+	0.709270i	$0.749025\pi$
$422$	0	0
$423$	−10.0000	−0.486217
$424$	0	0
$425$	6.92820	0.336067
$426$	0	0
$427$	1.85641	0.0898378
$428$	0	0
$429$	29.3205	1.41561
$430$	0	0
$431$	14.2487	0.686336	0.343168	−	0.939274i	$-0.388500\pi$
$431$	14.2487	0.686336	0.343168	+	0.939274i	$0.388500\pi$
$432$	0	0
$433$	−28.0526	−1.34812	−0.674060	−	0.738677i	$-0.735451\pi$
$433$	−28.0526	−1.34812	−0.674060	+	0.738677i	$0.735451\pi$
$434$	0	0
$435$	0.196152	0.00940479
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−21.8564	−1.04315	−0.521575	−	0.853206i	$-0.674655\pi$
$439$	−21.8564	−1.04315	−0.521575	+	0.853206i	$0.674655\pi$
$440$	0	0
$441$	−5.39230	−0.256776
$442$	0	0
$443$	20.9282	0.994329	0.497164	−	0.867656i	$-0.334375\pi$
$443$	20.9282	0.994329	0.497164	+	0.867656i	$0.334375\pi$
$444$	0	0
$445$	9.66025	0.457940
$446$	0	0
$447$	−4.92820	−0.233096
$448$	0	0
$449$	20.1962	0.953115	0.476558	−	0.879143i	$-0.341884\pi$
$449$	20.1962	0.953115	0.476558	+	0.879143i	$0.341884\pi$
$450$	0	0
$451$	35.0718	1.65147
$452$	0	0
$453$	−2.00000	−0.0939682
$454$	0	0
$455$	6.00000	0.281284
$456$	0	0
$457$	−12.5359	−0.586405	−0.293202	−	0.956050i	$-0.594721\pi$
$457$	−12.5359	−0.586405	−0.293202	+	0.956050i	$0.594721\pi$
$458$	0	0
$459$	6.92820	0.323381
$460$	0	0
$461$	−30.7846	−1.43378	−0.716891	−	0.697185i	$-0.754436\pi$
$461$	−30.7846	−1.43378	−0.716891	+	0.697185i	$0.754436\pi$
$462$	0	0
$463$	−7.12436	−0.331097	−0.165548	−	0.986202i	$-0.552939\pi$
$463$	−7.12436	−0.331097	−0.165548	+	0.986202i	$0.552939\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	32.9282	1.52374	0.761868	−	0.647733i	$-0.224283\pi$
$467$	32.9282	1.52374	0.761868	+	0.647733i	$0.224283\pi$
$468$	0	0
$469$	1.35898	0.0627520
$470$	0	0
$471$	3.46410	0.159617
$472$	0	0
$473$	7.85641	0.361238
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	34.5885	1.58039	0.790193	−	0.612857i	$-0.209980\pi$
$479$	34.5885	1.58039	0.790193	+	0.612857i	$0.209980\pi$
$480$	0	0
$481$	3.46410	0.157949
$482$	0	0
$483$	7.60770	0.346162
$484$	0	0
$485$	−15.6603	−0.711096
$486$	0	0
$487$	−0.392305	−0.0177770	−0.00888851	−	0.999960i	$-0.502829\pi$
$487$	−0.392305	−0.0177770	−0.00888851	+	0.999960i	$0.502829\pi$
$488$	0	0
$489$	0.875644	0.0395980
$490$	0	0
$491$	2.19615	0.0991110	0.0495555	−	0.998771i	$-0.484220\pi$
$491$	2.19615	0.0991110	0.0495555	+	0.998771i	$0.484220\pi$
$492$	0	0
$493$	1.35898	0.0612056
$494$	0	0
$495$	−6.19615	−0.278496
$496$	0	0
$497$	10.6410	0.477315
$498$	0	0
$499$	27.3205	1.22303	0.611517	−	0.791231i	$-0.290560\pi$
$499$	27.3205	1.22303	0.611517	+	0.791231i	$0.290560\pi$
$500$	0	0
$501$	−6.39230	−0.285587
$502$	0	0
$503$	−23.1769	−1.03341	−0.516704	−	0.856164i	$-0.672841\pi$
$503$	−23.1769	−1.03341	−0.516704	+	0.856164i	$0.672841\pi$
$504$	0	0
$505$	3.46410	0.154150
$506$	0	0
$507$	9.39230	0.417127
$508$	0	0
$509$	−4.19615	−0.185991	−0.0929956	−	0.995667i	$-0.529644\pi$
$509$	−4.19615	−0.185991	−0.0929956	+	0.995667i	$0.529644\pi$
$510$	0	0
$511$	16.3923	0.725153
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	13.8564	0.610586
$516$	0	0
$517$	61.9615	2.72506
$518$	0	0
$519$	−8.39230	−0.368381
$520$	0	0
$521$	39.1244	1.71407	0.857035	−	0.515259i	$-0.172304\pi$
$521$	39.1244	1.71407	0.857035	+	0.515259i	$0.172304\pi$
$522$	0	0
$523$	−13.4641	−0.588744	−0.294372	−	0.955691i	$-0.595110\pi$
$523$	−13.4641	−0.588744	−0.294372	+	0.955691i	$0.595110\pi$
$524$	0	0
$525$	−1.26795	−0.0553378
$526$	0	0
$527$	−13.8564	−0.603595
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−2.53590	−0.110049
$532$	0	0
$533$	26.7846	1.16017
$534$	0	0
$535$	2.92820	0.126597
$536$	0	0
$537$	14.5359	0.627270
$538$	0	0
$539$	33.4115	1.43914
$540$	0	0
$541$	33.7128	1.44943	0.724714	−	0.689050i	$-0.241972\pi$
$541$	33.7128	1.44943	0.724714	+	0.689050i	$0.241972\pi$
$542$	0	0
$543$	11.4641	0.491972
$544$	0	0
$545$	10.3923	0.445157
$546$	0	0
$547$	30.2487	1.29334	0.646671	−	0.762769i	$-0.276161\pi$
$547$	30.2487	1.29334	0.646671	+	0.762769i	$0.276161\pi$
$548$	0	0
$549$	−1.46410	−0.0624863
$550$	0	0
$551$	0.196152	0.00835637
$552$	0	0
$553$	10.1436	0.431349
$554$	0	0
$555$	−0.732051	−0.0310738
$556$	0	0
$557$	−7.85641	−0.332887	−0.166443	−	0.986051i	$-0.553228\pi$
$557$	−7.85641	−0.332887	−0.166443	+	0.986051i	$0.553228\pi$
$558$	0	0
$559$	6.00000	0.253773
$560$	0	0
$561$	−42.9282	−1.81243
$562$	0	0
$563$	38.3923	1.61804	0.809021	−	0.587779i	$-0.199998\pi$
$563$	38.3923	1.61804	0.809021	+	0.587779i	$0.199998\pi$
$564$	0	0
$565$	9.46410	0.398158
$566$	0	0
$567$	−1.26795	−0.0532489
$568$	0	0
$569$	20.1962	0.846667	0.423333	−	0.905974i	$-0.360860\pi$
$569$	20.1962	0.846667	0.423333	+	0.905974i	$0.360860\pi$
$570$	0	0
$571$	28.3923	1.18818	0.594090	−	0.804398i	$-0.297512\pi$
$571$	28.3923	1.18818	0.594090	+	0.804398i	$0.297512\pi$
$572$	0	0
$573$	−16.7321	−0.698991
$574$	0	0
$575$	−6.00000	−0.250217
$576$	0	0
$577$	−9.32051	−0.388018	−0.194009	−	0.981000i	$-0.562149\pi$
$577$	−9.32051	−0.388018	−0.194009	+	0.981000i	$0.562149\pi$
$578$	0	0
$579$	−22.1962	−0.922441
$580$	0	0
$581$	−19.6077	−0.813464
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−4.73205	−0.195646
$586$	0	0
$587$	−18.7846	−0.775324	−0.387662	−	0.921802i	$-0.626717\pi$
$587$	−18.7846	−0.775324	−0.387662	+	0.921802i	$0.626717\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	14.9282	0.614064
$592$	0	0
$593$	−38.7846	−1.59269	−0.796347	−	0.604841i	$-0.793237\pi$
$593$	−38.7846	−1.59269	−0.796347	+	0.604841i	$0.793237\pi$
$594$	0	0
$595$	−8.78461	−0.360134
$596$	0	0
$597$	−9.46410	−0.387340
$598$	0	0
$599$	−28.0000	−1.14405	−0.572024	−	0.820237i	$-0.693842\pi$
$599$	−28.0000	−1.14405	−0.572024	+	0.820237i	$0.693842\pi$
$600$	0	0
$601$	−5.60770	−0.228743	−0.114371	−	0.993438i	$-0.536485\pi$
$601$	−5.60770	−0.228743	−0.114371	+	0.993438i	$0.536485\pi$
$602$	0	0
$603$	−1.07180	−0.0436469
$604$	0	0
$605$	27.3923	1.11366
$606$	0	0
$607$	−17.4641	−0.708846	−0.354423	−	0.935085i	$-0.615323\pi$
$607$	−17.4641	−0.708846	−0.354423	+	0.935085i	$0.615323\pi$
$608$	0	0
$609$	−0.248711	−0.0100783
$610$	0	0
$611$	47.3205	1.91438
$612$	0	0
$613$	35.1769	1.42078	0.710391	−	0.703807i	$-0.248518\pi$
$613$	35.1769	1.42078	0.710391	+	0.703807i	$0.248518\pi$
$614$	0	0
$615$	−5.66025	−0.228243
$616$	0	0
$617$	−36.7846	−1.48089	−0.740446	−	0.672116i	$-0.765386\pi$
$617$	−36.7846	−1.48089	−0.740446	+	0.672116i	$0.765386\pi$
$618$	0	0
$619$	21.1769	0.851172	0.425586	−	0.904918i	$-0.360068\pi$
$619$	21.1769	0.851172	0.425586	+	0.904918i	$0.360068\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	−12.2487	−0.490734
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.19615	−0.247450
$628$	0	0
$629$	−5.07180	−0.202226
$630$	0	0
$631$	35.7128	1.42170	0.710852	−	0.703341i	$-0.248309\pi$
$631$	35.7128	1.42170	0.710852	+	0.703341i	$0.248309\pi$
$632$	0	0
$633$	6.92820	0.275371
$634$	0	0
$635$	−14.9282	−0.592408
$636$	0	0
$637$	25.5167	1.01101
$638$	0	0
$639$	−8.39230	−0.331994
$640$	0	0
$641$	7.51666	0.296890	0.148445	−	0.988921i	$-0.452573\pi$
$641$	7.51666	0.296890	0.148445	+	0.988921i	$0.452573\pi$
$642$	0	0
$643$	20.9808	0.827400	0.413700	−	0.910413i	$-0.364236\pi$
$643$	20.9808	0.827400	0.413700	+	0.910413i	$0.364236\pi$
$644$	0	0
$645$	−1.26795	−0.0499255
$646$	0	0
$647$	−41.3205	−1.62448	−0.812238	−	0.583326i	$-0.801751\pi$
$647$	−41.3205	−1.62448	−0.812238	+	0.583326i	$0.801751\pi$
$648$	0	0
$649$	15.7128	0.616782
$650$	0	0
$651$	2.53590	0.0993897
$652$	0	0
$653$	−48.7846	−1.90909	−0.954545	−	0.298068i	$-0.903658\pi$
$653$	−48.7846	−1.90909	−0.954545	+	0.298068i	$0.903658\pi$
$654$	0	0
$655$	10.5885	0.413725
$656$	0	0
$657$	−12.9282	−0.504377
$658$	0	0
$659$	−11.7128	−0.456266	−0.228133	−	0.973630i	$-0.573262\pi$
$659$	−11.7128	−0.456266	−0.228133	+	0.973630i	$0.573262\pi$
$660$	0	0
$661$	13.6077	0.529278	0.264639	−	0.964348i	$-0.414747\pi$
$661$	13.6077	0.529278	0.264639	+	0.964348i	$0.414747\pi$
$662$	0	0
$663$	−32.7846	−1.27325
$664$	0	0
$665$	−1.26795	−0.0491690
$666$	0	0
$667$	−1.17691	−0.0455703
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	9.07180	0.350213
$672$	0	0
$673$	5.51666	0.212652	0.106326	−	0.994331i	$-0.466091\pi$
$673$	5.51666	0.212652	0.106326	+	0.994331i	$0.466091\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−40.0000	−1.53732	−0.768662	−	0.639655i	$-0.779077\pi$
$677$	−40.0000	−1.53732	−0.768662	+	0.639655i	$0.779077\pi$
$678$	0	0
$679$	19.8564	0.762019
$680$	0	0
$681$	16.5359	0.633657
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	−18.7846	−0.717723
$686$	0	0
$687$	10.5359	0.401970
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−12.3923	−0.471425	−0.235713	−	0.971823i	$-0.575742\pi$
$691$	−12.3923	−0.471425	−0.235713	+	0.971823i	$0.575742\pi$
$692$	0	0
$693$	7.85641	0.298440
$694$	0	0
$695$	−10.5359	−0.399649
$696$	0	0
$697$	−39.2154	−1.48539
$698$	0	0
$699$	27.8564	1.05363
$700$	0	0
$701$	11.0718	0.418176	0.209088	−	0.977897i	$-0.432950\pi$
$701$	11.0718	0.418176	0.209088	+	0.977897i	$0.432950\pi$
$702$	0	0
$703$	−0.732051	−0.0276098
$704$	0	0
$705$	−10.0000	−0.376622
$706$	0	0
$707$	−4.39230	−0.165190
$708$	0	0
$709$	−10.2487	−0.384898	−0.192449	−	0.981307i	$-0.561643\pi$
$709$	−10.2487	−0.384898	−0.192449	+	0.981307i	$0.561643\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	12.0000	0.449404
$714$	0	0
$715$	29.3205	1.09652
$716$	0	0
$717$	16.0526	0.599494
$718$	0	0
$719$	22.9808	0.857038	0.428519	−	0.903533i	$-0.359036\pi$
$719$	22.9808	0.857038	0.428519	+	0.903533i	$0.359036\pi$
$720$	0	0
$721$	−17.5692	−0.654312
$722$	0	0
$723$	4.92820	0.183282
$724$	0	0
$725$	0.196152	0.00728492
$726$	0	0
$727$	−32.5885	−1.20864	−0.604319	−	0.796742i	$-0.706555\pi$
$727$	−32.5885	−1.20864	−0.604319	+	0.796742i	$0.706555\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.78461	−0.324911
$732$	0	0
$733$	6.78461	0.250595	0.125298	−	0.992119i	$-0.460011\pi$
$733$	6.78461	0.250595	0.125298	+	0.992119i	$0.460011\pi$
$734$	0	0
$735$	−5.39230	−0.198898
$736$	0	0
$737$	6.64102	0.244625
$738$	0	0
$739$	−9.07180	−0.333711	−0.166856	−	0.985981i	$-0.553361\pi$
$739$	−9.07180	−0.333711	−0.166856	+	0.985981i	$0.553361\pi$
$740$	0	0
$741$	−4.73205	−0.173836
$742$	0	0
$743$	42.6410	1.56435	0.782174	−	0.623061i	$-0.214111\pi$
$743$	42.6410	1.56435	0.782174	+	0.623061i	$0.214111\pi$
$744$	0	0
$745$	−4.92820	−0.180555
$746$	0	0
$747$	15.4641	0.565802
$748$	0	0
$749$	−3.71281	−0.135663
$750$	0	0
$751$	−35.8564	−1.30842	−0.654209	−	0.756313i	$-0.726999\pi$
$751$	−35.8564	−1.30842	−0.654209	+	0.756313i	$0.726999\pi$
$752$	0	0
$753$	−17.8038	−0.648808
$754$	0	0
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	−4.53590	−0.164860	−0.0824300	−	0.996597i	$-0.526268\pi$
$757$	−4.53590	−0.164860	−0.0824300	+	0.996597i	$0.526268\pi$
$758$	0	0
$759$	37.1769	1.34944
$760$	0	0
$761$	−40.9282	−1.48365	−0.741823	−	0.670596i	$-0.766039\pi$
$761$	−40.9282	−1.48365	−0.741823	+	0.670596i	$0.766039\pi$
$762$	0	0
$763$	−13.1769	−0.477036
$764$	0	0
$765$	6.92820	0.250490
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	2.78461	0.100416	0.0502078	−	0.998739i	$-0.484012\pi$
$769$	2.78461	0.100416	0.0502078	+	0.998739i	$0.484012\pi$
$770$	0	0
$771$	−7.32051	−0.263642
$772$	0	0
$773$	−25.8564	−0.929990	−0.464995	−	0.885313i	$-0.653944\pi$
$773$	−25.8564	−0.929990	−0.464995	+	0.885313i	$0.653944\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0.928203	0.0332991
$778$	0	0
$779$	−5.66025	−0.202800
$780$	0	0
$781$	52.0000	1.86071
$782$	0	0
$783$	0.196152	0.00700992
$784$	0	0
$785$	3.46410	0.123639
$786$	0	0
$787$	−5.46410	−0.194774	−0.0973871	−	0.995247i	$-0.531048\pi$
$787$	−5.46410	−0.194774	−0.0973871	+	0.995247i	$0.531048\pi$
$788$	0	0
$789$	−3.46410	−0.123325
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	6.92820	0.246028
$794$	0	0
$795$	0	0
$796$	0	0
$797$	46.2487	1.63821	0.819107	−	0.573641i	$-0.194470\pi$
$797$	46.2487	1.63821	0.819107	+	0.573641i	$0.194470\pi$
$798$	0	0
$799$	−69.2820	−2.45102
$800$	0	0
$801$	9.66025	0.341328
$802$	0	0
$803$	80.1051	2.82685
$804$	0	0
$805$	7.60770	0.268136
$806$	0	0
$807$	19.5167	0.687019
$808$	0	0
$809$	−54.4974	−1.91603	−0.958014	−	0.286723i	$-0.907434\pi$
$809$	−54.4974	−1.91603	−0.958014	+	0.286723i	$0.907434\pi$
$810$	0	0
$811$	42.6410	1.49733	0.748664	−	0.662949i	$-0.230696\pi$
$811$	42.6410	1.49733	0.748664	+	0.662949i	$0.230696\pi$
$812$	0	0
$813$	−11.3205	−0.397028
$814$	0	0
$815$	0.875644	0.0306725
$816$	0	0
$817$	−1.26795	−0.0443599
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	13.3205	0.464889	0.232444	−	0.972610i	$-0.425328\pi$
$821$	13.3205	0.464889	0.232444	+	0.972610i	$0.425328\pi$
$822$	0	0
$823$	−40.9808	−1.42850	−0.714250	−	0.699891i	$-0.753232\pi$
$823$	−40.9808	−1.42850	−0.714250	+	0.699891i	$0.753232\pi$
$824$	0	0
$825$	−6.19615	−0.215722
$826$	0	0
$827$	−50.1051	−1.74233	−0.871163	−	0.490994i	$-0.836634\pi$
$827$	−50.1051	−1.74233	−0.871163	+	0.490994i	$0.836634\pi$
$828$	0	0
$829$	38.3923	1.33342	0.666710	−	0.745317i	$-0.267702\pi$
$829$	38.3923	1.33342	0.666710	+	0.745317i	$0.267702\pi$
$830$	0	0
$831$	−15.8564	−0.550053
$832$	0	0
$833$	−37.3590	−1.29441
$834$	0	0
$835$	−6.39230	−0.221215
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	18.5359	0.639930	0.319965	−	0.947429i	$-0.396329\pi$
$839$	18.5359	0.639930	0.319965	+	0.947429i	$0.396329\pi$
$840$	0	0
$841$	−28.9615	−0.998673
$842$	0	0
$843$	30.0526	1.03507
$844$	0	0
$845$	9.39230	0.323105
$846$	0	0
$847$	−34.7321	−1.19341
$848$	0	0
$849$	−24.1962	−0.830410
$850$	0	0
$851$	4.39230	0.150566
$852$	0	0
$853$	−31.4641	−1.07731	−0.538655	−	0.842526i	$-0.681067\pi$
$853$	−31.4641	−1.07731	−0.538655	+	0.842526i	$0.681067\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	−34.9282	−1.19312	−0.596562	−	0.802567i	$-0.703467\pi$
$857$	−34.9282	−1.19312	−0.596562	+	0.802567i	$0.703467\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	7.17691	0.244589
$862$	0	0
$863$	33.8564	1.15249	0.576243	−	0.817279i	$-0.304518\pi$
$863$	33.8564	1.15249	0.576243	+	0.817279i	$0.304518\pi$
$864$	0	0
$865$	−8.39230	−0.285347
$866$	0	0
$867$	31.0000	1.05282
$868$	0	0
$869$	49.5692	1.68152
$870$	0	0
$871$	5.07180	0.171851
$872$	0	0
$873$	−15.6603	−0.530019
$874$	0	0
$875$	−1.26795	−0.0428645
$876$	0	0
$877$	−46.1962	−1.55993	−0.779967	−	0.625821i	$-0.784764\pi$
$877$	−46.1962	−1.55993	−0.779967	+	0.625821i	$0.784764\pi$
$878$	0	0
$879$	−3.60770	−0.121685
$880$	0	0
$881$	44.2487	1.49078	0.745388	−	0.666630i	$-0.232264\pi$
$881$	44.2487	1.49078	0.745388	+	0.666630i	$0.232264\pi$
$882$	0	0
$883$	38.7321	1.30344	0.651719	−	0.758461i	$-0.274048\pi$
$883$	38.7321	1.30344	0.651719	+	0.758461i	$0.274048\pi$
$884$	0	0
$885$	−2.53590	−0.0852433
$886$	0	0
$887$	−18.9282	−0.635547	−0.317773	−	0.948167i	$-0.602935\pi$
$887$	−18.9282	−0.635547	−0.317773	+	0.948167i	$0.602935\pi$
$888$	0	0
$889$	18.9282	0.634832
$890$	0	0
$891$	−6.19615	−0.207579
$892$	0	0
$893$	−10.0000	−0.334637
$894$	0	0
$895$	14.5359	0.485881
$896$	0	0
$897$	28.3923	0.947991
$898$	0	0
$899$	−0.392305	−0.0130841
$900$	0	0
$901$	0	0
$902$	0	0
$903$	1.60770	0.0535007
$904$	0	0
$905$	11.4641	0.381080
$906$	0	0
$907$	43.0333	1.42890	0.714449	−	0.699688i	$-0.246677\pi$
$907$	43.0333	1.42890	0.714449	+	0.699688i	$0.246677\pi$
$908$	0	0
$909$	3.46410	0.114897
$910$	0	0
$911$	30.9282	1.02470	0.512349	−	0.858778i	$-0.328776\pi$
$911$	30.9282	1.02470	0.512349	+	0.858778i	$0.328776\pi$
$912$	0	0
$913$	−95.8179	−3.17111
$914$	0	0
$915$	−1.46410	−0.0484017
$916$	0	0
$917$	−13.4256	−0.443353
$918$	0	0
$919$	−9.07180	−0.299251	−0.149625	−	0.988743i	$-0.547807\pi$
$919$	−9.07180	−0.299251	−0.149625	+	0.988743i	$0.547807\pi$
$920$	0	0
$921$	−27.3205	−0.900241
$922$	0	0
$923$	39.7128	1.30716
$924$	0	0
$925$	−0.732051	−0.0240697
$926$	0	0
$927$	13.8564	0.455104
$928$	0	0
$929$	11.7513	0.385547	0.192774	−	0.981243i	$-0.438252\pi$
$929$	11.7513	0.385547	0.192774	+	0.981243i	$0.438252\pi$
$930$	0	0
$931$	−5.39230	−0.176726
$932$	0	0
$933$	24.7321	0.809691
$934$	0	0
$935$	−42.9282	−1.40390
$936$	0	0
$937$	−19.1769	−0.626482	−0.313241	−	0.949674i	$-0.601415\pi$
$937$	−19.1769	−0.626482	−0.313241	+	0.949674i	$0.601415\pi$
$938$	0	0
$939$	21.3205	0.695768
$940$	0	0
$941$	−54.0526	−1.76206	−0.881032	−	0.473058i	$-0.843150\pi$
$941$	−54.0526	−1.76206	−0.881032	+	0.473058i	$0.843150\pi$
$942$	0	0
$943$	33.9615	1.10594
$944$	0	0
$945$	−1.26795	−0.0412464
$946$	0	0
$947$	−53.3205	−1.73268	−0.866342	−	0.499451i	$-0.833535\pi$
$947$	−53.3205	−1.73268	−0.866342	+	0.499451i	$0.833535\pi$
$948$	0	0
$949$	61.1769	1.98589
$950$	0	0
$951$	14.9282	0.484080
$952$	0	0
$953$	−45.1769	−1.46342	−0.731712	−	0.681614i	$-0.761278\pi$
$953$	−45.1769	−1.46342	−0.731712	+	0.681614i	$0.761278\pi$
$954$	0	0
$955$	−16.7321	−0.541436
$956$	0	0
$957$	−1.21539	−0.0392880
$958$	0	0
$959$	23.8179	0.769121
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	2.92820	0.0943600
$964$	0	0
$965$	−22.1962	−0.714519
$966$	0	0
$967$	4.98076	0.160171	0.0800853	−	0.996788i	$-0.474481\pi$
$967$	4.98076	0.160171	0.0800853	+	0.996788i	$0.474481\pi$
$968$	0	0
$969$	6.92820	0.222566
$970$	0	0
$971$	−36.4974	−1.17126	−0.585629	−	0.810579i	$-0.699152\pi$
$971$	−36.4974	−1.17126	−0.585629	+	0.810579i	$0.699152\pi$
$972$	0	0
$973$	13.3590	0.428269
$974$	0	0
$975$	−4.73205	−0.151547
$976$	0	0
$977$	−9.85641	−0.315334	−0.157667	−	0.987492i	$-0.550397\pi$
$977$	−9.85641	−0.315334	−0.157667	+	0.987492i	$0.550397\pi$
$978$	0	0
$979$	−59.8564	−1.91302
$980$	0	0
$981$	10.3923	0.331801
$982$	0	0
$983$	54.3923	1.73485	0.867423	−	0.497572i	$-0.165775\pi$
$983$	54.3923	1.73485	0.867423	+	0.497572i	$0.165775\pi$
$984$	0	0
$985$	14.9282	0.475652
$986$	0	0
$987$	12.6795	0.403593
$988$	0	0
$989$	7.60770	0.241911
$990$	0	0
$991$	24.7846	0.787309	0.393655	−	0.919258i	$-0.371211\pi$
$991$	24.7846	0.787309	0.393655	+	0.919258i	$0.371211\pi$
$992$	0	0
$993$	−3.07180	−0.0974805
$994$	0	0
$995$	−9.46410	−0.300032
$996$	0	0
$997$	−58.3923	−1.84930	−0.924651	−	0.380815i	$-0.875644\pi$
$997$	−58.3923	−1.84930	−0.924651	+	0.380815i	$0.875644\pi$
$998$	0	0
$999$	−0.732051	−0.0231611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2280.2.a.q.1.2	✓	2
3.2	odd	2		6840.2.a.v.1.2		2
4.3	odd	2		4560.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.q.1.2	✓	2	1.1	even	1	trivial
4560.2.a.bi.1.1		2	4.3	odd	2
6840.2.a.v.1.2		2	3.2	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:	\( N \)	\(=\)	\( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{9} -6.19615 q^{11} -4.73205 q^{13} +1.00000 q^{15} +6.92820 q^{17} +1.00000 q^{19} -1.26795 q^{21} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.196152 q^{29} -2.00000 q^{31} -6.19615 q^{33} -1.26795 q^{35} -0.732051 q^{37} -4.73205 q^{39} -5.66025 q^{41} -1.26795 q^{43} +1.00000 q^{45} -10.0000 q^{47} -5.39230 q^{49} +6.92820 q^{51} -6.19615 q^{55} +1.00000 q^{57} -2.53590 q^{59} -1.46410 q^{61} -1.26795 q^{63} -4.73205 q^{65} -1.07180 q^{67} -6.00000 q^{69} -8.39230 q^{71} -12.9282 q^{73} +1.00000 q^{75} +7.85641 q^{77} -8.00000 q^{79} +1.00000 q^{81} +15.4641 q^{83} +6.92820 q^{85} +0.196152 q^{87} +9.66025 q^{89} +6.00000 q^{91} -2.00000 q^{93} +1.00000 q^{95} -15.6603 q^{97} -6.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} - 6 q^{13} + 2 q^{15} + 2 q^{19} - 6 q^{21} - 12 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 2 q^{33} - 6 q^{35} + 2 q^{37} - 6 q^{39} + 6 q^{41} - 6 q^{43} + 2 q^{45} - 20 q^{47} + 10 q^{49} - 2 q^{55} + 2 q^{57} - 12 q^{59} + 4 q^{61} - 6 q^{63} - 6 q^{65} - 16 q^{67} - 12 q^{69} + 4 q^{71} - 12 q^{73} + 2 q^{75} - 12 q^{77} - 16 q^{79} + 2 q^{81} + 24 q^{83} - 10 q^{87} + 2 q^{89} + 12 q^{91} - 4 q^{93} + 2 q^{95} - 14 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 - 2 * q^11 - 6 * q^13 + 2 * q^15 + 2 * q^19 - 6 * q^21 - 12 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 - 4 * q^31 - 2 * q^33 - 6 * q^35 + 2 * q^37 - 6 * q^39 + 6 * q^41 - 6 * q^43 + 2 * q^45 - 20 * q^47 + 10 * q^49 - 2 * q^55 + 2 * q^57 - 12 * q^59 + 4 * q^61 - 6 * q^63 - 6 * q^65 - 16 * q^67 - 12 * q^69 + 4 * q^71 - 12 * q^73 + 2 * q^75 - 12 * q^77 - 16 * q^79 + 2 * q^81 + 24 * q^83 - 10 * q^87 + 2 * q^89 + 12 * q^91 - 4 * q^93 + 2 * q^95 - 14 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

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Database

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