LMFDB - Embedded newform 2280.2.a.m.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:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +3.23607 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +3.23607 q^{7} +1.00000 q^{9} -3.23607 q^{11} +1.23607 q^{13} +1.00000 q^{15} -4.47214 q^{17} -1.00000 q^{19} -3.23607 q^{21} -2.47214 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.76393 q^{29} +4.00000 q^{31} +3.23607 q^{33} -3.23607 q^{35} -5.23607 q^{37} -1.23607 q^{39} +3.70820 q^{41} +3.23607 q^{43} -1.00000 q^{45} -2.47214 q^{47} +3.47214 q^{49} +4.47214 q^{51} -8.47214 q^{53} +3.23607 q^{55} +1.00000 q^{57} -1.52786 q^{59} -4.47214 q^{61} +3.23607 q^{63} -1.23607 q^{65} +2.47214 q^{67} +2.47214 q^{69} +4.94427 q^{71} +8.47214 q^{73} -1.00000 q^{75} -10.4721 q^{77} -12.9443 q^{79} +1.00000 q^{81} +4.47214 q^{85} +2.76393 q^{87} -4.29180 q^{89} +4.00000 q^{91} -4.00000 q^{93} +1.00000 q^{95} -18.1803 q^{97} -3.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{15} - 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} + 8 q^{31} + 2 q^{33} - 2 q^{35} - 6 q^{37} + 2 q^{39} - 6 q^{41} + 2 q^{43} - 2 q^{45} + 4 q^{47} - 2 q^{49} - 8 q^{53} + 2 q^{55} + 2 q^{57} - 12 q^{59} + 2 q^{63} + 2 q^{65} - 4 q^{67} - 4 q^{69} - 8 q^{71} + 8 q^{73} - 2 q^{75} - 12 q^{77} - 8 q^{79} + 2 q^{81} + 10 q^{87} - 22 q^{89} + 8 q^{91} - 8 q^{93} + 2 q^{95} - 14 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^15 - 2 * q^19 - 2 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 + 8 * q^31 + 2 * q^33 - 2 * q^35 - 6 * q^37 + 2 * q^39 - 6 * q^41 + 2 * q^43 - 2 * q^45 + 4 * q^47 - 2 * q^49 - 8 * q^53 + 2 * q^55 + 2 * q^57 - 12 * q^59 + 2 * q^63 + 2 * q^65 - 4 * q^67 - 4 * q^69 - 8 * q^71 + 8 * q^73 - 2 * q^75 - 12 * q^77 - 8 * q^79 + 2 * q^81 + 10 * q^87 - 22 * q^89 + 8 * q^91 - 8 * 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
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.23607	−0.975711	−0.487856	−	0.872924i	$-0.662221\pi$
$11$	−3.23607	−0.975711	−0.487856	+	0.872924i	$0.662221\pi$
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.23607	−0.706168
$22$	0	0
$23$	−2.47214	−0.515476	−0.257738	−	0.966215i	$-0.582977\pi$
$23$	−2.47214	−0.515476	−0.257738	+	0.966215i	$0.582977\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.76393	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$29$	−2.76393	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	3.23607	0.563327
$34$	0	0
$35$	−3.23607	−0.546995
$36$	0	0
$37$	−5.23607	−0.860804	−0.430402	−	0.902637i	$-0.641628\pi$
$37$	−5.23607	−0.860804	−0.430402	+	0.902637i	$0.641628\pi$
$38$	0	0
$39$	−1.23607	−0.197929
$40$	0	0
$41$	3.70820	0.579124	0.289562	−	0.957159i	$-0.406490\pi$
$41$	3.70820	0.579124	0.289562	+	0.957159i	$0.406490\pi$
$42$	0	0
$43$	3.23607	0.493496	0.246748	−	0.969080i	$-0.420638\pi$
$43$	3.23607	0.493496	0.246748	+	0.969080i	$0.420638\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−2.47214	−0.360598	−0.180299	−	0.983612i	$-0.557707\pi$
$47$	−2.47214	−0.360598	−0.180299	+	0.983612i	$0.557707\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	4.47214	0.626224
$52$	0	0
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0	0
$55$	3.23607	0.436351
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−1.52786	−0.198911	−0.0994555	−	0.995042i	$-0.531710\pi$
$59$	−1.52786	−0.198911	−0.0994555	+	0.995042i	$0.531710\pi$
$60$	0	0
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	0	0
$63$	3.23607	0.407706
$64$	0	0
$65$	−1.23607	−0.153315
$66$	0	0
$67$	2.47214	0.302019	0.151010	−	0.988532i	$-0.451748\pi$
$67$	2.47214	0.302019	0.151010	+	0.988532i	$0.451748\pi$
$68$	0	0
$69$	2.47214	0.297610
$70$	0	0
$71$	4.94427	0.586777	0.293389	−	0.955993i	$-0.405217\pi$
$71$	4.94427	0.586777	0.293389	+	0.955993i	$0.405217\pi$
$72$	0	0
$73$	8.47214	0.991589	0.495794	−	0.868440i	$-0.334877\pi$
$73$	8.47214	0.991589	0.495794	+	0.868440i	$0.334877\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−10.4721	−1.19341
$78$	0	0
$79$	−12.9443	−1.45634	−0.728172	−	0.685394i	$-0.759630\pi$
$79$	−12.9443	−1.45634	−0.728172	+	0.685394i	$0.759630\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	4.47214	0.485071
$86$	0	0
$87$	2.76393	0.296325
$88$	0	0
$89$	−4.29180	−0.454929	−0.227465	−	0.973786i	$-0.573044\pi$
$89$	−4.29180	−0.454929	−0.227465	+	0.973786i	$0.573044\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−18.1803	−1.84593	−0.922967	−	0.384879i	$-0.874243\pi$
$97$	−18.1803	−1.84593	−0.922967	+	0.384879i	$0.874243\pi$
$98$	0	0
$99$	−3.23607	−0.325237
$100$	0	0
$101$	−12.4721	−1.24102	−0.620512	−	0.784197i	$-0.713075\pi$
$101$	−12.4721	−1.24102	−0.620512	+	0.784197i	$0.713075\pi$
$102$	0	0
$103$	−7.41641	−0.730760	−0.365380	−	0.930858i	$-0.619061\pi$
$103$	−7.41641	−0.730760	−0.365380	+	0.930858i	$0.619061\pi$
$104$	0	0
$105$	3.23607	0.315808
$106$	0	0
$107$	8.94427	0.864675	0.432338	−	0.901712i	$-0.357689\pi$
$107$	8.94427	0.864675	0.432338	+	0.901712i	$0.357689\pi$
$108$	0	0
$109$	−8.47214	−0.811483	−0.405742	−	0.913988i	$-0.632987\pi$
$109$	−8.47214	−0.811483	−0.405742	+	0.913988i	$0.632987\pi$
$110$	0	0
$111$	5.23607	0.496986
$112$	0	0
$113$	−1.05573	−0.0993145	−0.0496573	−	0.998766i	$-0.515813\pi$
$113$	−1.05573	−0.0993145	−0.0496573	+	0.998766i	$0.515813\pi$
$114$	0	0
$115$	2.47214	0.230528
$116$	0	0
$117$	1.23607	0.114275
$118$	0	0
$119$	−14.4721	−1.32666
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	−3.70820	−0.334357
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	−3.23607	−0.284920
$130$	0	0
$131$	−9.70820	−0.848210	−0.424105	−	0.905613i	$-0.639411\pi$
$131$	−9.70820	−0.848210	−0.424105	+	0.905613i	$0.639411\pi$
$132$	0	0
$133$	−3.23607	−0.280603
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	2.94427	0.251546	0.125773	−	0.992059i	$-0.459859\pi$
$137$	2.94427	0.251546	0.125773	+	0.992059i	$0.459859\pi$
$138$	0	0
$139$	−18.4721	−1.56679	−0.783393	−	0.621527i	$-0.786513\pi$
$139$	−18.4721	−1.56679	−0.783393	+	0.621527i	$0.786513\pi$
$140$	0	0
$141$	2.47214	0.208191
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	2.76393	0.229532
$146$	0	0
$147$	−3.47214	−0.286377
$148$	0	0
$149$	−12.4721	−1.02176	−0.510879	−	0.859653i	$-0.670680\pi$
$149$	−12.4721	−1.02176	−0.510879	+	0.859653i	$0.670680\pi$
$150$	0	0
$151$	8.94427	0.727875	0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427	0.727875	0.363937	+	0.931423i	$0.381432\pi$
$152$	0	0
$153$	−4.47214	−0.361551
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−17.4164	−1.38998	−0.694990	−	0.719019i	$-0.744591\pi$
$157$	−17.4164	−1.38998	−0.694990	+	0.719019i	$0.744591\pi$
$158$	0	0
$159$	8.47214	0.671884
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	14.6525	1.14767	0.573835	−	0.818971i	$-0.305455\pi$
$163$	14.6525	1.14767	0.573835	+	0.818971i	$0.305455\pi$
$164$	0	0
$165$	−3.23607	−0.251928
$166$	0	0
$167$	24.9443	1.93025	0.965123	−	0.261797i	$-0.0843152\pi$
$167$	24.9443	1.93025	0.965123	+	0.261797i	$0.0843152\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−22.9443	−1.74442	−0.872210	−	0.489131i	$-0.837314\pi$
$173$	−22.9443	−1.74442	−0.872210	+	0.489131i	$0.837314\pi$
$174$	0	0
$175$	3.23607	0.244624
$176$	0	0
$177$	1.52786	0.114841
$178$	0	0
$179$	−14.4721	−1.08170	−0.540849	−	0.841120i	$-0.681897\pi$
$179$	−14.4721	−1.08170	−0.540849	+	0.841120i	$0.681897\pi$
$180$	0	0
$181$	−3.52786	−0.262224	−0.131112	−	0.991368i	$-0.541855\pi$
$181$	−3.52786	−0.262224	−0.131112	+	0.991368i	$0.541855\pi$
$182$	0	0
$183$	4.47214	0.330590
$184$	0	0
$185$	5.23607	0.384963
$186$	0	0
$187$	14.4721	1.05831
$188$	0	0
$189$	−3.23607	−0.235389
$190$	0	0
$191$	4.76393	0.344706	0.172353	−	0.985035i	$-0.444863\pi$
$191$	4.76393	0.344706	0.172353	+	0.985035i	$0.444863\pi$
$192$	0	0
$193$	−6.76393	−0.486878	−0.243439	−	0.969916i	$-0.578276\pi$
$193$	−6.76393	−0.486878	−0.243439	+	0.969916i	$0.578276\pi$
$194$	0	0
$195$	1.23607	0.0885167
$196$	0	0
$197$	9.41641	0.670891	0.335446	−	0.942060i	$-0.391113\pi$
$197$	9.41641	0.670891	0.335446	+	0.942060i	$0.391113\pi$
$198$	0	0
$199$	−1.52786	−0.108307	−0.0541537	−	0.998533i	$-0.517246\pi$
$199$	−1.52786	−0.108307	−0.0541537	+	0.998533i	$0.517246\pi$
$200$	0	0
$201$	−2.47214	−0.174371
$202$	0	0
$203$	−8.94427	−0.627765
$204$	0	0
$205$	−3.70820	−0.258992
$206$	0	0
$207$	−2.47214	−0.171825
$208$	0	0
$209$	3.23607	0.223844
$210$	0	0
$211$	−13.8885	−0.956127	−0.478063	−	0.878325i	$-0.658661\pi$
$211$	−13.8885	−0.956127	−0.478063	+	0.878325i	$0.658661\pi$
$212$	0	0
$213$	−4.94427	−0.338776
$214$	0	0
$215$	−3.23607	−0.220698
$216$	0	0
$217$	12.9443	0.878714
$218$	0	0
$219$	−8.47214	−0.572494
$220$	0	0
$221$	−5.52786	−0.371844
$222$	0	0
$223$	7.41641	0.496639	0.248320	−	0.968678i	$-0.420122\pi$
$223$	7.41641	0.496639	0.248320	+	0.968678i	$0.420122\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	11.0557	0.733794	0.366897	−	0.930261i	$-0.380420\pi$
$227$	11.0557	0.733794	0.366897	+	0.930261i	$0.380420\pi$
$228$	0	0
$229$	−4.47214	−0.295527	−0.147764	−	0.989023i	$-0.547207\pi$
$229$	−4.47214	−0.295527	−0.147764	+	0.989023i	$0.547207\pi$
$230$	0	0
$231$	10.4721	0.689016
$232$	0	0
$233$	−22.9443	−1.50313	−0.751565	−	0.659659i	$-0.770701\pi$
$233$	−22.9443	−1.50313	−0.751565	+	0.659659i	$0.770701\pi$
$234$	0	0
$235$	2.47214	0.161264
$236$	0	0
$237$	12.9443	0.840821
$238$	0	0
$239$	11.2361	0.726801	0.363400	−	0.931633i	$-0.381616\pi$
$239$	11.2361	0.726801	0.363400	+	0.931633i	$0.381616\pi$
$240$	0	0
$241$	−2.94427	−0.189657	−0.0948286	−	0.995494i	$-0.530230\pi$
$241$	−2.94427	−0.189657	−0.0948286	+	0.995494i	$0.530230\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.47214	−0.221827
$246$	0	0
$247$	−1.23607	−0.0786491
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.70820	0.107821	0.0539104	−	0.998546i	$-0.482831\pi$
$251$	1.70820	0.107821	0.0539104	+	0.998546i	$0.482831\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	−4.47214	−0.280056
$256$	0	0
$257$	−15.8885	−0.991100	−0.495550	−	0.868579i	$-0.665033\pi$
$257$	−15.8885	−0.991100	−0.495550	+	0.868579i	$0.665033\pi$
$258$	0	0
$259$	−16.9443	−1.05287
$260$	0	0
$261$	−2.76393	−0.171083
$262$	0	0
$263$	21.8885	1.34971	0.674853	−	0.737952i	$-0.264207\pi$
$263$	21.8885	1.34971	0.674853	+	0.737952i	$0.264207\pi$
$264$	0	0
$265$	8.47214	0.520439
$266$	0	0
$267$	4.29180	0.262654
$268$	0	0
$269$	−7.70820	−0.469977	−0.234989	−	0.971998i	$-0.575505\pi$
$269$	−7.70820	−0.469977	−0.234989	+	0.971998i	$0.575505\pi$
$270$	0	0
$271$	1.52786	0.0928111	0.0464056	−	0.998923i	$-0.485223\pi$
$271$	1.52786	0.0928111	0.0464056	+	0.998923i	$0.485223\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	−3.23607	−0.195142
$276$	0	0
$277$	24.4721	1.47039	0.735194	−	0.677857i	$-0.237091\pi$
$277$	24.4721	1.47039	0.735194	+	0.677857i	$0.237091\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	5.23607	0.312358	0.156179	−	0.987729i	$-0.450082\pi$
$281$	5.23607	0.312358	0.156179	+	0.987729i	$0.450082\pi$
$282$	0	0
$283$	1.70820	0.101542	0.0507711	−	0.998710i	$-0.483832\pi$
$283$	1.70820	0.101542	0.0507711	+	0.998710i	$0.483832\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	18.1803	1.06575
$292$	0	0
$293$	2.94427	0.172006	0.0860031	−	0.996295i	$-0.472591\pi$
$293$	2.94427	0.172006	0.0860031	+	0.996295i	$0.472591\pi$
$294$	0	0
$295$	1.52786	0.0889557
$296$	0	0
$297$	3.23607	0.187776
$298$	0	0
$299$	−3.05573	−0.176717
$300$	0	0
$301$	10.4721	0.603604
$302$	0	0
$303$	12.4721	0.716505
$304$	0	0
$305$	4.47214	0.256074
$306$	0	0
$307$	2.47214	0.141092	0.0705461	−	0.997509i	$-0.477526\pi$
$307$	2.47214	0.141092	0.0705461	+	0.997509i	$0.477526\pi$
$308$	0	0
$309$	7.41641	0.421905
$310$	0	0
$311$	12.7639	0.723776	0.361888	−	0.932222i	$-0.382132\pi$
$311$	12.7639	0.723776	0.361888	+	0.932222i	$0.382132\pi$
$312$	0	0
$313$	30.9443	1.74907	0.874537	−	0.484959i	$-0.161166\pi$
$313$	30.9443	1.74907	0.874537	+	0.484959i	$0.161166\pi$
$314$	0	0
$315$	−3.23607	−0.182332
$316$	0	0
$317$	14.3607	0.806576	0.403288	−	0.915073i	$-0.367867\pi$
$317$	14.3607	0.806576	0.403288	+	0.915073i	$0.367867\pi$
$318$	0	0
$319$	8.94427	0.500783
$320$	0	0
$321$	−8.94427	−0.499221
$322$	0	0
$323$	4.47214	0.248836
$324$	0	0
$325$	1.23607	0.0685647
$326$	0	0
$327$	8.47214	0.468510
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−9.88854	−0.543524	−0.271762	−	0.962365i	$-0.587606\pi$
$331$	−9.88854	−0.543524	−0.271762	+	0.962365i	$0.587606\pi$
$332$	0	0
$333$	−5.23607	−0.286935
$334$	0	0
$335$	−2.47214	−0.135067
$336$	0	0
$337$	15.7082	0.855680	0.427840	−	0.903854i	$-0.359275\pi$
$337$	15.7082	0.855680	0.427840	+	0.903854i	$0.359275\pi$
$338$	0	0
$339$	1.05573	0.0573393
$340$	0	0
$341$	−12.9443	−0.700972
$342$	0	0
$343$	−11.4164	−0.616428
$344$	0	0
$345$	−2.47214	−0.133095
$346$	0	0
$347$	9.88854	0.530845	0.265422	−	0.964132i	$-0.414489\pi$
$347$	9.88854	0.530845	0.265422	+	0.964132i	$0.414489\pi$
$348$	0	0
$349$	18.9443	1.01406	0.507032	−	0.861927i	$-0.330743\pi$
$349$	18.9443	1.01406	0.507032	+	0.861927i	$0.330743\pi$
$350$	0	0
$351$	−1.23607	−0.0659764
$352$	0	0
$353$	31.8885	1.69726	0.848628	−	0.528990i	$-0.177429\pi$
$353$	31.8885	1.69726	0.848628	+	0.528990i	$0.177429\pi$
$354$	0	0
$355$	−4.94427	−0.262415
$356$	0	0
$357$	14.4721	0.765947
$358$	0	0
$359$	−11.2361	−0.593017	−0.296508	−	0.955030i	$-0.595822\pi$
$359$	−11.2361	−0.593017	−0.296508	+	0.955030i	$0.595822\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.527864	0.0277057
$364$	0	0
$365$	−8.47214	−0.443452
$366$	0	0
$367$	12.7639	0.666272	0.333136	−	0.942879i	$-0.391893\pi$
$367$	12.7639	0.666272	0.333136	+	0.942879i	$0.391893\pi$
$368$	0	0
$369$	3.70820	0.193041
$370$	0	0
$371$	−27.4164	−1.42339
$372$	0	0
$373$	1.23607	0.0640012	0.0320006	−	0.999488i	$-0.489812\pi$
$373$	1.23607	0.0640012	0.0320006	+	0.999488i	$0.489812\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−3.41641	−0.175954
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	−25.8885	−1.32284	−0.661421	−	0.750014i	$-0.730046\pi$
$383$	−25.8885	−1.32284	−0.661421	+	0.750014i	$0.730046\pi$
$384$	0	0
$385$	10.4721	0.533709
$386$	0	0
$387$	3.23607	0.164499
$388$	0	0
$389$	5.05573	0.256336	0.128168	−	0.991752i	$-0.459090\pi$
$389$	5.05573	0.256336	0.128168	+	0.991752i	$0.459090\pi$
$390$	0	0
$391$	11.0557	0.559112
$392$	0	0
$393$	9.70820	0.489714
$394$	0	0
$395$	12.9443	0.651297
$396$	0	0
$397$	34.3607	1.72451	0.862257	−	0.506472i	$-0.169051\pi$
$397$	34.3607	1.72451	0.862257	+	0.506472i	$0.169051\pi$
$398$	0	0
$399$	3.23607	0.162006
$400$	0	0
$401$	−26.7639	−1.33653	−0.668263	−	0.743925i	$-0.732962\pi$
$401$	−26.7639	−1.33653	−0.668263	+	0.743925i	$0.732962\pi$
$402$	0	0
$403$	4.94427	0.246292
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	16.9443	0.839896
$408$	0	0
$409$	31.3050	1.54793	0.773965	−	0.633228i	$-0.218271\pi$
$409$	31.3050	1.54793	0.773965	+	0.633228i	$0.218271\pi$
$410$	0	0
$411$	−2.94427	−0.145230
$412$	0	0
$413$	−4.94427	−0.243292
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18.4721	0.904584
$418$	0	0
$419$	9.70820	0.474277	0.237138	−	0.971476i	$-0.423791\pi$
$419$	9.70820	0.474277	0.237138	+	0.971476i	$0.423791\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	0	0
$423$	−2.47214	−0.120199
$424$	0	0
$425$	−4.47214	−0.216930
$426$	0	0
$427$	−14.4721	−0.700356
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	−8.36068	−0.402720	−0.201360	−	0.979517i	$-0.564536\pi$
$431$	−8.36068	−0.402720	−0.201360	+	0.979517i	$0.564536\pi$
$432$	0	0
$433$	−13.5967	−0.653418	−0.326709	−	0.945125i	$-0.605940\pi$
$433$	−13.5967	−0.653418	−0.326709	+	0.945125i	$0.605940\pi$
$434$	0	0
$435$	−2.76393	−0.132520
$436$	0	0
$437$	2.47214	0.118258
$438$	0	0
$439$	4.94427	0.235977	0.117989	−	0.993015i	$-0.462355\pi$
$439$	4.94427	0.235977	0.117989	+	0.993015i	$0.462355\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	0	0
$443$	−24.3607	−1.15741	−0.578705	−	0.815537i	$-0.696442\pi$
$443$	−24.3607	−1.15741	−0.578705	+	0.815537i	$0.696442\pi$
$444$	0	0
$445$	4.29180	0.203451
$446$	0	0
$447$	12.4721	0.589912
$448$	0	0
$449$	−24.0689	−1.13588	−0.567940	−	0.823070i	$-0.692260\pi$
$449$	−24.0689	−1.13588	−0.567940	+	0.823070i	$0.692260\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	−8.94427	−0.420239
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−7.88854	−0.369011	−0.184505	−	0.982832i	$-0.559068\pi$
$457$	−7.88854	−0.369011	−0.184505	+	0.982832i	$0.559068\pi$
$458$	0	0
$459$	4.47214	0.208741
$460$	0	0
$461$	−2.58359	−0.120330	−0.0601649	−	0.998188i	$-0.519163\pi$
$461$	−2.58359	−0.120330	−0.0601649	+	0.998188i	$0.519163\pi$
$462$	0	0
$463$	−34.0689	−1.58332	−0.791658	−	0.610965i	$-0.790782\pi$
$463$	−34.0689	−1.58332	−0.791658	+	0.610965i	$0.790782\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	17.5279	0.811093	0.405546	−	0.914074i	$-0.367081\pi$
$467$	17.5279	0.811093	0.405546	+	0.914074i	$0.367081\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	17.4164	0.802506
$472$	0	0
$473$	−10.4721	−0.481509
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−8.47214	−0.387912
$478$	0	0
$479$	−9.34752	−0.427099	−0.213550	−	0.976932i	$-0.568503\pi$
$479$	−9.34752	−0.427099	−0.213550	+	0.976932i	$0.568503\pi$
$480$	0	0
$481$	−6.47214	−0.295104
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	18.1803	0.825527
$486$	0	0
$487$	−31.4164	−1.42361	−0.711807	−	0.702375i	$-0.752123\pi$
$487$	−31.4164	−1.42361	−0.711807	+	0.702375i	$0.752123\pi$
$488$	0	0
$489$	−14.6525	−0.662608
$490$	0	0
$491$	−12.7639	−0.576028	−0.288014	−	0.957626i	$-0.592995\pi$
$491$	−12.7639	−0.576028	−0.288014	+	0.957626i	$0.592995\pi$
$492$	0	0
$493$	12.3607	0.556697
$494$	0	0
$495$	3.23607	0.145450
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	12.3607	0.553340	0.276670	−	0.960965i	$-0.410769\pi$
$499$	12.3607	0.553340	0.276670	+	0.960965i	$0.410769\pi$
$500$	0	0
$501$	−24.9443	−1.11443
$502$	0	0
$503$	21.8885	0.975962	0.487981	−	0.872854i	$-0.337734\pi$
$503$	21.8885	0.975962	0.487981	+	0.872854i	$0.337734\pi$
$504$	0	0
$505$	12.4721	0.555003
$506$	0	0
$507$	11.4721	0.509495
$508$	0	0
$509$	29.5967	1.31185	0.655926	−	0.754825i	$-0.272278\pi$
$509$	29.5967	1.31185	0.655926	+	0.754825i	$0.272278\pi$
$510$	0	0
$511$	27.4164	1.21283
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	7.41641	0.326806
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	22.9443	1.00714
$520$	0	0
$521$	−9.59675	−0.420441	−0.210221	−	0.977654i	$-0.567418\pi$
$521$	−9.59675	−0.420441	−0.210221	+	0.977654i	$0.567418\pi$
$522$	0	0
$523$	−21.8885	−0.957119	−0.478560	−	0.878055i	$-0.658841\pi$
$523$	−21.8885	−0.957119	−0.478560	+	0.878055i	$0.658841\pi$
$524$	0	0
$525$	−3.23607	−0.141234
$526$	0	0
$527$	−17.8885	−0.779237
$528$	0	0
$529$	−16.8885	−0.734285
$530$	0	0
$531$	−1.52786	−0.0663037
$532$	0	0
$533$	4.58359	0.198537
$534$	0	0
$535$	−8.94427	−0.386695
$536$	0	0
$537$	14.4721	0.624519
$538$	0	0
$539$	−11.2361	−0.483972
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	3.52786	0.151395
$544$	0	0
$545$	8.47214	0.362906
$546$	0	0
$547$	8.58359	0.367008	0.183504	−	0.983019i	$-0.441256\pi$
$547$	8.58359	0.367008	0.183504	+	0.983019i	$0.441256\pi$
$548$	0	0
$549$	−4.47214	−0.190866
$550$	0	0
$551$	2.76393	0.117747
$552$	0	0
$553$	−41.8885	−1.78128
$554$	0	0
$555$	−5.23607	−0.222259
$556$	0	0
$557$	−28.8328	−1.22169	−0.610843	−	0.791752i	$-0.709169\pi$
$557$	−28.8328	−1.22169	−0.610843	+	0.791752i	$0.709169\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	−14.4721	−0.611014
$562$	0	0
$563$	30.8328	1.29945	0.649724	−	0.760170i	$-0.274884\pi$
$563$	30.8328	1.29945	0.649724	+	0.760170i	$0.274884\pi$
$564$	0	0
$565$	1.05573	0.0444148
$566$	0	0
$567$	3.23607	0.135902
$568$	0	0
$569$	−45.0132	−1.88705	−0.943525	−	0.331302i	$-0.892512\pi$
$569$	−45.0132	−1.88705	−0.943525	+	0.331302i	$0.892512\pi$
$570$	0	0
$571$	20.3607	0.852068	0.426034	−	0.904707i	$-0.359910\pi$
$571$	20.3607	0.852068	0.426034	+	0.904707i	$0.359910\pi$
$572$	0	0
$573$	−4.76393	−0.199016
$574$	0	0
$575$	−2.47214	−0.103095
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	6.76393	0.281099
$580$	0	0
$581$	0	0
$582$	0	0
$583$	27.4164	1.13547
$584$	0	0
$585$	−1.23607	−0.0511051
$586$	0	0
$587$	12.5836	0.519380	0.259690	−	0.965692i	$-0.416380\pi$
$587$	12.5836	0.519380	0.259690	+	0.965692i	$0.416380\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	−9.41641	−0.387339
$592$	0	0
$593$	12.8328	0.526981	0.263490	−	0.964662i	$-0.415126\pi$
$593$	12.8328	0.526981	0.263490	+	0.964662i	$0.415126\pi$
$594$	0	0
$595$	14.4721	0.593300
$596$	0	0
$597$	1.52786	0.0625313
$598$	0	0
$599$	16.3607	0.668479	0.334240	−	0.942488i	$-0.391521\pi$
$599$	16.3607	0.668479	0.334240	+	0.942488i	$0.391521\pi$
$600$	0	0
$601$	6.58359	0.268550	0.134275	−	0.990944i	$-0.457129\pi$
$601$	6.58359	0.268550	0.134275	+	0.990944i	$0.457129\pi$
$602$	0	0
$603$	2.47214	0.100673
$604$	0	0
$605$	0.527864	0.0214607
$606$	0	0
$607$	−15.4164	−0.625733	−0.312866	−	0.949797i	$-0.601289\pi$
$607$	−15.4164	−0.625733	−0.312866	+	0.949797i	$0.601289\pi$
$608$	0	0
$609$	8.94427	0.362440
$610$	0	0
$611$	−3.05573	−0.123622
$612$	0	0
$613$	10.3607	0.418464	0.209232	−	0.977866i	$-0.432904\pi$
$613$	10.3607	0.418464	0.209232	+	0.977866i	$0.432904\pi$
$614$	0	0
$615$	3.70820	0.149529
$616$	0	0
$617$	−6.36068	−0.256071	−0.128036	−	0.991770i	$-0.540867\pi$
$617$	−6.36068	−0.256071	−0.128036	+	0.991770i	$0.540867\pi$
$618$	0	0
$619$	0.583592	0.0234565	0.0117283	−	0.999931i	$-0.496267\pi$
$619$	0.583592	0.0234565	0.0117283	+	0.999931i	$0.496267\pi$
$620$	0	0
$621$	2.47214	0.0992034
$622$	0	0
$623$	−13.8885	−0.556433
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.23607	−0.129236
$628$	0	0
$629$	23.4164	0.933673
$630$	0	0
$631$	43.7771	1.74274	0.871369	−	0.490628i	$-0.163233\pi$
$631$	43.7771	1.74274	0.871369	+	0.490628i	$0.163233\pi$
$632$	0	0
$633$	13.8885	0.552020
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	4.29180	0.170047
$638$	0	0
$639$	4.94427	0.195592
$640$	0	0
$641$	34.5410	1.36429	0.682144	−	0.731218i	$-0.261048\pi$
$641$	34.5410	1.36429	0.682144	+	0.731218i	$0.261048\pi$
$642$	0	0
$643$	43.5967	1.71929	0.859644	−	0.510894i	$-0.170685\pi$
$643$	43.5967	1.71929	0.859644	+	0.510894i	$0.170685\pi$
$644$	0	0
$645$	3.23607	0.127420
$646$	0	0
$647$	37.8885	1.48955	0.744776	−	0.667314i	$-0.232556\pi$
$647$	37.8885	1.48955	0.744776	+	0.667314i	$0.232556\pi$
$648$	0	0
$649$	4.94427	0.194080
$650$	0	0
$651$	−12.9443	−0.507326
$652$	0	0
$653$	−19.5279	−0.764184	−0.382092	−	0.924124i	$-0.624796\pi$
$653$	−19.5279	−0.764184	−0.382092	+	0.924124i	$0.624796\pi$
$654$	0	0
$655$	9.70820	0.379331
$656$	0	0
$657$	8.47214	0.330530
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	39.5279	1.53746	0.768728	−	0.639576i	$-0.220890\pi$
$661$	39.5279	1.53746	0.768728	+	0.639576i	$0.220890\pi$
$662$	0	0
$663$	5.52786	0.214684
$664$	0	0
$665$	3.23607	0.125489
$666$	0	0
$667$	6.83282	0.264568
$668$	0	0
$669$	−7.41641	−0.286735
$670$	0	0
$671$	14.4721	0.558691
$672$	0	0
$673$	4.65248	0.179340	0.0896699	−	0.995972i	$-0.471419\pi$
$673$	4.65248	0.179340	0.0896699	+	0.995972i	$0.471419\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	17.4164	0.669367	0.334683	−	0.942331i	$-0.391371\pi$
$677$	17.4164	0.669367	0.334683	+	0.942331i	$0.391371\pi$
$678$	0	0
$679$	−58.8328	−2.25780
$680$	0	0
$681$	−11.0557	−0.423656
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	−2.94427	−0.112495
$686$	0	0
$687$	4.47214	0.170623
$688$	0	0
$689$	−10.4721	−0.398957
$690$	0	0
$691$	44.3607	1.68756	0.843780	−	0.536689i	$-0.180325\pi$
$691$	44.3607	1.68756	0.843780	+	0.536689i	$0.180325\pi$
$692$	0	0
$693$	−10.4721	−0.397804
$694$	0	0
$695$	18.4721	0.700688
$696$	0	0
$697$	−16.5836	−0.628148
$698$	0	0
$699$	22.9443	0.867832
$700$	0	0
$701$	−2.58359	−0.0975809	−0.0487905	−	0.998809i	$-0.515537\pi$
$701$	−2.58359	−0.0975809	−0.0487905	+	0.998809i	$0.515537\pi$
$702$	0	0
$703$	5.23607	0.197482
$704$	0	0
$705$	−2.47214	−0.0931060
$706$	0	0
$707$	−40.3607	−1.51792
$708$	0	0
$709$	−4.47214	−0.167955	−0.0839773	−	0.996468i	$-0.526762\pi$
$709$	−4.47214	−0.167955	−0.0839773	+	0.996468i	$0.526762\pi$
$710$	0	0
$711$	−12.9443	−0.485448
$712$	0	0
$713$	−9.88854	−0.370329
$714$	0	0
$715$	4.00000	0.149592
$716$	0	0
$717$	−11.2361	−0.419619
$718$	0	0
$719$	−17.7082	−0.660405	−0.330202	−	0.943910i	$-0.607117\pi$
$719$	−17.7082	−0.660405	−0.330202	+	0.943910i	$0.607117\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	0	0
$723$	2.94427	0.109499
$724$	0	0
$725$	−2.76393	−0.102650
$726$	0	0
$727$	−11.2361	−0.416723	−0.208361	−	0.978052i	$-0.566813\pi$
$727$	−11.2361	−0.416723	−0.208361	+	0.978052i	$0.566813\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.4721	−0.535271
$732$	0	0
$733$	−23.8885	−0.882343	−0.441172	−	0.897423i	$-0.645437\pi$
$733$	−23.8885	−0.882343	−0.441172	+	0.897423i	$0.645437\pi$
$734$	0	0
$735$	3.47214	0.128072
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−24.9443	−0.917590	−0.458795	−	0.888542i	$-0.651719\pi$
$739$	−24.9443	−0.917590	−0.458795	+	0.888542i	$0.651719\pi$
$740$	0	0
$741$	1.23607	0.0454081
$742$	0	0
$743$	30.8328	1.13115	0.565573	−	0.824698i	$-0.308655\pi$
$743$	30.8328	1.13115	0.565573	+	0.824698i	$0.308655\pi$
$744$	0	0
$745$	12.4721	0.456944
$746$	0	0
$747$	0	0
$748$	0	0
$749$	28.9443	1.05760
$750$	0	0
$751$	−21.8885	−0.798724	−0.399362	−	0.916793i	$-0.630768\pi$
$751$	−21.8885	−0.798724	−0.399362	+	0.916793i	$0.630768\pi$
$752$	0	0
$753$	−1.70820	−0.0622504
$754$	0	0
$755$	−8.94427	−0.325515
$756$	0	0
$757$	22.9443	0.833924	0.416962	−	0.908924i	$-0.363095\pi$
$757$	22.9443	0.833924	0.416962	+	0.908924i	$0.363095\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	15.3050	0.554804	0.277402	−	0.960754i	$-0.410527\pi$
$761$	15.3050	0.554804	0.277402	+	0.960754i	$0.410527\pi$
$762$	0	0
$763$	−27.4164	−0.992541
$764$	0	0
$765$	4.47214	0.161690
$766$	0	0
$767$	−1.88854	−0.0681914
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	15.8885	0.572212
$772$	0	0
$773$	−18.3607	−0.660388	−0.330194	−	0.943913i	$-0.607114\pi$
$773$	−18.3607	−0.660388	−0.330194	+	0.943913i	$0.607114\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	16.9443	0.607872
$778$	0	0
$779$	−3.70820	−0.132860
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	2.76393	0.0987749
$784$	0	0
$785$	17.4164	0.621618
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	0	0
$789$	−21.8885	−0.779253
$790$	0	0
$791$	−3.41641	−0.121473
$792$	0	0
$793$	−5.52786	−0.196300
$794$	0	0
$795$	−8.47214	−0.300476
$796$	0	0
$797$	−21.0557	−0.745832	−0.372916	−	0.927865i	$-0.621642\pi$
$797$	−21.0557	−0.745832	−0.372916	+	0.927865i	$0.621642\pi$
$798$	0	0
$799$	11.0557	0.391124
$800$	0	0
$801$	−4.29180	−0.151643
$802$	0	0
$803$	−27.4164	−0.967504
$804$	0	0
$805$	8.00000	0.281963
$806$	0	0
$807$	7.70820	0.271342
$808$	0	0
$809$	5.05573	0.177750	0.0888750	−	0.996043i	$-0.471673\pi$
$809$	5.05573	0.177750	0.0888750	+	0.996043i	$0.471673\pi$
$810$	0	0
$811$	−23.0557	−0.809596	−0.404798	−	0.914406i	$-0.632658\pi$
$811$	−23.0557	−0.809596	−0.404798	+	0.914406i	$0.632658\pi$
$812$	0	0
$813$	−1.52786	−0.0535845
$814$	0	0
$815$	−14.6525	−0.513254
$816$	0	0
$817$	−3.23607	−0.113216
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	8.47214	0.295680	0.147840	−	0.989011i	$-0.452768\pi$
$821$	8.47214	0.295680	0.147840	+	0.989011i	$0.452768\pi$
$822$	0	0
$823$	17.7082	0.617269	0.308635	−	0.951181i	$-0.400128\pi$
$823$	17.7082	0.617269	0.308635	+	0.951181i	$0.400128\pi$
$824$	0	0
$825$	3.23607	0.112665
$826$	0	0
$827$	27.0557	0.940820	0.470410	−	0.882448i	$-0.344106\pi$
$827$	27.0557	0.940820	0.470410	+	0.882448i	$0.344106\pi$
$828$	0	0
$829$	−8.47214	−0.294249	−0.147125	−	0.989118i	$-0.547002\pi$
$829$	−8.47214	−0.294249	−0.147125	+	0.989118i	$0.547002\pi$
$830$	0	0
$831$	−24.4721	−0.848929
$832$	0	0
$833$	−15.5279	−0.538009
$834$	0	0
$835$	−24.9443	−0.863232
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	−48.7214	−1.68205	−0.841024	−	0.540998i	$-0.818047\pi$
$839$	−48.7214	−1.68205	−0.841024	+	0.540998i	$0.818047\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	0	0
$843$	−5.23607	−0.180340
$844$	0	0
$845$	11.4721	0.394653
$846$	0	0
$847$	−1.70820	−0.0586946
$848$	0	0
$849$	−1.70820	−0.0586254
$850$	0	0
$851$	12.9443	0.443724
$852$	0	0
$853$	−2.94427	−0.100810	−0.0504050	−	0.998729i	$-0.516051\pi$
$853$	−2.94427	−0.100810	−0.0504050	+	0.998729i	$0.516051\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	18.3607	0.627189	0.313594	−	0.949557i	$-0.398467\pi$
$857$	18.3607	0.627189	0.313594	+	0.949557i	$0.398467\pi$
$858$	0	0
$859$	−24.9443	−0.851088	−0.425544	−	0.904938i	$-0.639917\pi$
$859$	−24.9443	−0.851088	−0.425544	+	0.904938i	$0.639917\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	38.8328	1.32188	0.660942	−	0.750437i	$-0.270157\pi$
$863$	38.8328	1.32188	0.660942	+	0.750437i	$0.270157\pi$
$864$	0	0
$865$	22.9443	0.780129
$866$	0	0
$867$	−3.00000	−0.101885
$868$	0	0
$869$	41.8885	1.42097
$870$	0	0
$871$	3.05573	0.103539
$872$	0	0
$873$	−18.1803	−0.615311
$874$	0	0
$875$	−3.23607	−0.109399
$876$	0	0
$877$	−39.4853	−1.33332	−0.666662	−	0.745360i	$-0.732277\pi$
$877$	−39.4853	−1.33332	−0.666662	+	0.745360i	$0.732277\pi$
$878$	0	0
$879$	−2.94427	−0.0993078
$880$	0	0
$881$	3.52786	0.118857	0.0594284	−	0.998233i	$-0.481072\pi$
$881$	3.52786	0.118857	0.0594284	+	0.998233i	$0.481072\pi$
$882$	0	0
$883$	−34.0689	−1.14651	−0.573255	−	0.819377i	$-0.694319\pi$
$883$	−34.0689	−1.14651	−0.573255	+	0.819377i	$0.694319\pi$
$884$	0	0
$885$	−1.52786	−0.0513586
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	12.9443	0.434137
$890$	0	0
$891$	−3.23607	−0.108412
$892$	0	0
$893$	2.47214	0.0827269
$894$	0	0
$895$	14.4721	0.483750
$896$	0	0
$897$	3.05573	0.102028
$898$	0	0
$899$	−11.0557	−0.368729
$900$	0	0
$901$	37.8885	1.26225
$902$	0	0
$903$	−10.4721	−0.348491
$904$	0	0
$905$	3.52786	0.117270
$906$	0	0
$907$	−56.9443	−1.89080	−0.945402	−	0.325907i	$-0.894330\pi$
$907$	−56.9443	−1.89080	−0.945402	+	0.325907i	$0.894330\pi$
$908$	0	0
$909$	−12.4721	−0.413675
$910$	0	0
$911$	17.8885	0.592674	0.296337	−	0.955083i	$-0.404235\pi$
$911$	17.8885	0.592674	0.296337	+	0.955083i	$0.404235\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−4.47214	−0.147844
$916$	0	0
$917$	−31.4164	−1.03746
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	−2.47214	−0.0814596
$922$	0	0
$923$	6.11146	0.201161
$924$	0	0
$925$	−5.23607	−0.172161
$926$	0	0
$927$	−7.41641	−0.243587
$928$	0	0
$929$	40.8328	1.33968	0.669841	−	0.742505i	$-0.266362\pi$
$929$	40.8328	1.33968	0.669841	+	0.742505i	$0.266362\pi$
$930$	0	0
$931$	−3.47214	−0.113795
$932$	0	0
$933$	−12.7639	−0.417872
$934$	0	0
$935$	−14.4721	−0.473289
$936$	0	0
$937$	23.3050	0.761340	0.380670	−	0.924711i	$-0.375693\pi$
$937$	23.3050	0.761340	0.380670	+	0.924711i	$0.375693\pi$
$938$	0	0
$939$	−30.9443	−1.00983
$940$	0	0
$941$	−12.6525	−0.412459	−0.206229	−	0.978504i	$-0.566119\pi$
$941$	−12.6525	−0.412459	−0.206229	+	0.978504i	$0.566119\pi$
$942$	0	0
$943$	−9.16718	−0.298525
$944$	0	0
$945$	3.23607	0.105269
$946$	0	0
$947$	27.7771	0.902634	0.451317	−	0.892364i	$-0.350954\pi$
$947$	27.7771	0.902634	0.451317	+	0.892364i	$0.350954\pi$
$948$	0	0
$949$	10.4721	0.339940
$950$	0	0
$951$	−14.3607	−0.465677
$952$	0	0
$953$	11.8885	0.385108	0.192554	−	0.981286i	$-0.438323\pi$
$953$	11.8885	0.385108	0.192554	+	0.981286i	$0.438323\pi$
$954$	0	0
$955$	−4.76393	−0.154157
$956$	0	0
$957$	−8.94427	−0.289127
$958$	0	0
$959$	9.52786	0.307671
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	8.94427	0.288225
$964$	0	0
$965$	6.76393	0.217739
$966$	0	0
$967$	1.70820	0.0549321	0.0274661	−	0.999623i	$-0.491256\pi$
$967$	1.70820	0.0549321	0.0274661	+	0.999623i	$0.491256\pi$
$968$	0	0
$969$	−4.47214	−0.143666
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	−59.7771	−1.91637
$974$	0	0
$975$	−1.23607	−0.0395859
$976$	0	0
$977$	−11.3050	−0.361677	−0.180839	−	0.983513i	$-0.557881\pi$
$977$	−11.3050	−0.361677	−0.180839	+	0.983513i	$0.557881\pi$
$978$	0	0
$979$	13.8885	0.443880
$980$	0	0
$981$	−8.47214	−0.270494
$982$	0	0
$983$	−26.8328	−0.855834	−0.427917	−	0.903818i	$-0.640752\pi$
$983$	−26.8328	−0.855834	−0.427917	+	0.903818i	$0.640752\pi$
$984$	0	0
$985$	−9.41641	−0.300032
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	41.8885	1.33063	0.665317	−	0.746561i	$-0.268297\pi$
$991$	41.8885	1.33063	0.665317	+	0.746561i	$0.268297\pi$
$992$	0	0
$993$	9.88854	0.313803
$994$	0	0
$995$	1.52786	0.0484365
$996$	0	0
$997$	45.4164	1.43835	0.719176	−	0.694828i	$-0.244519\pi$
$997$	45.4164	1.43835	0.719176	+	0.694828i	$0.244519\pi$
$998$	0	0
$999$	5.23607	0.165662

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.m.1.2	✓	2	1.1	even	1	trivial
4560.2.a.bk.1.1		2	4.3	odd	2
6840.2.a.bc.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +3.23607 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +3.23607 q^{7} +1.00000 q^{9} -3.23607 q^{11} +1.23607 q^{13} +1.00000 q^{15} -4.47214 q^{17} -1.00000 q^{19} -3.23607 q^{21} -2.47214 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.76393 q^{29} +4.00000 q^{31} +3.23607 q^{33} -3.23607 q^{35} -5.23607 q^{37} -1.23607 q^{39} +3.70820 q^{41} +3.23607 q^{43} -1.00000 q^{45} -2.47214 q^{47} +3.47214 q^{49} +4.47214 q^{51} -8.47214 q^{53} +3.23607 q^{55} +1.00000 q^{57} -1.52786 q^{59} -4.47214 q^{61} +3.23607 q^{63} -1.23607 q^{65} +2.47214 q^{67} +2.47214 q^{69} +4.94427 q^{71} +8.47214 q^{73} -1.00000 q^{75} -10.4721 q^{77} -12.9443 q^{79} +1.00000 q^{81} +4.47214 q^{85} +2.76393 q^{87} -4.29180 q^{89} +4.00000 q^{91} -4.00000 q^{93} +1.00000 q^{95} -18.1803 q^{97} -3.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{15} - 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} + 8 q^{31} + 2 q^{33} - 2 q^{35} - 6 q^{37} + 2 q^{39} - 6 q^{41} + 2 q^{43} - 2 q^{45} + 4 q^{47} - 2 q^{49} - 8 q^{53} + 2 q^{55} + 2 q^{57} - 12 q^{59} + 2 q^{63} + 2 q^{65} - 4 q^{67} - 4 q^{69} - 8 q^{71} + 8 q^{73} - 2 q^{75} - 12 q^{77} - 8 q^{79} + 2 q^{81} + 10 q^{87} - 22 q^{89} + 8 q^{91} - 8 q^{93} + 2 q^{95} - 14 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^15 - 2 * q^19 - 2 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 + 8 * q^31 + 2 * q^33 - 2 * q^35 - 6 * q^37 + 2 * q^39 - 6 * q^41 + 2 * q^43 - 2 * q^45 + 4 * q^47 - 2 * q^49 - 8 * q^53 + 2 * q^55 + 2 * q^57 - 12 * q^59 + 2 * q^63 + 2 * q^65 - 4 * q^67 - 4 * q^69 - 8 * q^71 + 8 * q^73 - 2 * q^75 - 12 * q^77 - 8 * q^79 + 2 * q^81 + 10 * q^87 - 22 * q^89 + 8 * q^91 - 8 * q^93 + 2 * q^95 - 14 * q^97 - 2 * q^99

Introduction

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