LMFDB - Embedded newform 4560.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, 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("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	4560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +3.64575 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +3.64575 q^{7} +1.00000 q^{9} -5.64575 q^{11} -5.64575 q^{13} +1.00000 q^{15} -4.00000 q^{17} +1.00000 q^{19} +3.64575 q^{21} +1.29150 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.93725 q^{29} -6.00000 q^{31} -5.64575 q^{33} +3.64575 q^{35} -1.64575 q^{37} -5.64575 q^{39} -4.35425 q^{41} -0.354249 q^{43} +1.00000 q^{45} -9.29150 q^{47} +6.29150 q^{49} -4.00000 q^{51} +0.708497 q^{53} -5.64575 q^{55} +1.00000 q^{57} -0.708497 q^{59} -0.708497 q^{61} +3.64575 q^{63} -5.64575 q^{65} -14.5830 q^{67} +1.29150 q^{69} +3.29150 q^{71} +10.0000 q^{73} +1.00000 q^{75} -20.5830 q^{77} +14.5830 q^{79} +1.00000 q^{81} -6.00000 q^{83} -4.00000 q^{85} -6.93725 q^{87} -1.06275 q^{89} -20.5830 q^{91} -6.00000 q^{93} +1.00000 q^{95} +12.9373 q^{97} -5.64575 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{17} + 2 q^{19} + 2 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} - 12 q^{31} - 6 q^{33} + 2 q^{35} + 2 q^{37} - 6 q^{39} - 14 q^{41} - 6 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 8 q^{51} + 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 12 q^{61} + 2 q^{63} - 6 q^{65} - 8 q^{67} - 8 q^{69} - 4 q^{71} + 20 q^{73} + 2 q^{75} - 20 q^{77} + 8 q^{79} + 2 q^{81} - 12 q^{83} - 8 q^{85} + 2 q^{87} - 18 q^{89} - 20 q^{91} - 12 q^{93} + 2 q^{95} + 10 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 6 * q^13 + 2 * q^15 - 8 * q^17 + 2 * q^19 + 2 * q^21 - 8 * q^23 + 2 * q^25 + 2 * q^27 + 2 * q^29 - 12 * q^31 - 6 * q^33 + 2 * q^35 + 2 * q^37 - 6 * q^39 - 14 * q^41 - 6 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 - 8 * q^51 + 12 * q^53 - 6 * q^55 + 2 * q^57 - 12 * q^59 - 12 * q^61 + 2 * q^63 - 6 * q^65 - 8 * q^67 - 8 * q^69 - 4 * q^71 + 20 * q^73 + 2 * q^75 - 20 * q^77 + 8 * q^79 + 2 * q^81 - 12 * q^83 - 8 * q^85 + 2 * q^87 - 18 * q^89 - 20 * q^91 - 12 * q^93 + 2 * q^95 + 10 * q^97 - 6 * 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
$7$	3.64575	1.37796	0.688982	−	0.724778i	$-0.258058\pi$
$7$	3.64575	1.37796	0.688982	+	0.724778i	$0.258058\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.64575	−1.70226	−0.851129	−	0.524957i	$-0.824082\pi$
$11$	−5.64575	−1.70226	−0.851129	+	0.524957i	$0.824082\pi$
$12$	0	0
$13$	−5.64575	−1.56585	−0.782925	−	0.622116i	$-0.786273\pi$
$13$	−5.64575	−1.56585	−0.782925	+	0.622116i	$0.786273\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.64575	0.795568
$22$	0	0
$23$	1.29150	0.269297	0.134648	−	0.990893i	$-0.457009\pi$
$23$	1.29150	0.269297	0.134648	+	0.990893i	$0.457009\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.93725	−1.28822	−0.644108	−	0.764935i	$-0.722771\pi$
$29$	−6.93725	−1.28822	−0.644108	+	0.764935i	$0.722771\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	−5.64575	−0.982799
$34$	0	0
$35$	3.64575	0.616244
$36$	0	0
$37$	−1.64575	−0.270560	−0.135280	−	0.990807i	$-0.543193\pi$
$37$	−1.64575	−0.270560	−0.135280	+	0.990807i	$0.543193\pi$
$38$	0	0
$39$	−5.64575	−0.904044
$40$	0	0
$41$	−4.35425	−0.680019	−0.340010	−	0.940422i	$-0.610430\pi$
$41$	−4.35425	−0.680019	−0.340010	+	0.940422i	$0.610430\pi$
$42$	0	0
$43$	−0.354249	−0.0540224	−0.0270112	−	0.999635i	$-0.508599\pi$
$43$	−0.354249	−0.0540224	−0.0270112	+	0.999635i	$0.508599\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−9.29150	−1.35530	−0.677652	−	0.735382i	$-0.737003\pi$
$47$	−9.29150	−1.35530	−0.677652	+	0.735382i	$0.737003\pi$
$48$	0	0
$49$	6.29150	0.898786
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	0.708497	0.0973196	0.0486598	−	0.998815i	$-0.484505\pi$
$53$	0.708497	0.0973196	0.0486598	+	0.998815i	$0.484505\pi$
$54$	0	0
$55$	−5.64575	−0.761273
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−0.708497	−0.0922385	−0.0461193	−	0.998936i	$-0.514685\pi$
$59$	−0.708497	−0.0922385	−0.0461193	+	0.998936i	$0.514685\pi$
$60$	0	0
$61$	−0.708497	−0.0907138	−0.0453569	−	0.998971i	$-0.514443\pi$
$61$	−0.708497	−0.0907138	−0.0453569	+	0.998971i	$0.514443\pi$
$62$	0	0
$63$	3.64575	0.459321
$64$	0	0
$65$	−5.64575	−0.700269
$66$	0	0
$67$	−14.5830	−1.78160	−0.890799	−	0.454398i	$-0.849854\pi$
$67$	−14.5830	−1.78160	−0.890799	+	0.454398i	$0.849854\pi$
$68$	0	0
$69$	1.29150	0.155479
$70$	0	0
$71$	3.29150	0.390629	0.195315	−	0.980741i	$-0.437427\pi$
$71$	3.29150	0.390629	0.195315	+	0.980741i	$0.437427\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−20.5830	−2.34565
$78$	0	0
$79$	14.5830	1.64072	0.820358	−	0.571850i	$-0.193774\pi$
$79$	14.5830	1.64072	0.820358	+	0.571850i	$0.193774\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	−6.93725	−0.743752
$88$	0	0
$89$	−1.06275	−0.112651	−0.0563254	−	0.998412i	$-0.517938\pi$
$89$	−1.06275	−0.112651	−0.0563254	+	0.998412i	$0.517938\pi$
$90$	0	0
$91$	−20.5830	−2.15769
$92$	0	0
$93$	−6.00000	−0.622171
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	12.9373	1.31358	0.656790	−	0.754074i	$-0.271914\pi$
$97$	12.9373	1.31358	0.656790	+	0.754074i	$0.271914\pi$
$98$	0	0
$99$	−5.64575	−0.567419
$100$	0	0
$101$	−1.29150	−0.128509	−0.0642547	−	0.997934i	$-0.520467\pi$
$101$	−1.29150	−0.128509	−0.0642547	+	0.997934i	$0.520467\pi$
$102$	0	0
$103$	−10.5830	−1.04277	−0.521387	−	0.853320i	$-0.674585\pi$
$103$	−10.5830	−1.04277	−0.521387	+	0.853320i	$0.674585\pi$
$104$	0	0
$105$	3.64575	0.355789
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−5.29150	−0.506834	−0.253417	−	0.967357i	$-0.581554\pi$
$109$	−5.29150	−0.506834	−0.253417	+	0.967357i	$0.581554\pi$
$110$	0	0
$111$	−1.64575	−0.156208
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	1.29150	0.120433
$116$	0	0
$117$	−5.64575	−0.521950
$118$	0	0
$119$	−14.5830	−1.33682
$120$	0	0
$121$	20.8745	1.89768
$122$	0	0
$123$	−4.35425	−0.392609
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	−0.354249	−0.0311899
$130$	0	0
$131$	−12.2288	−1.06843	−0.534216	−	0.845348i	$-0.679393\pi$
$131$	−12.2288	−1.06843	−0.534216	+	0.845348i	$0.679393\pi$
$132$	0	0
$133$	3.64575	0.316127
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	13.8745	1.17682	0.588410	−	0.808563i	$-0.299754\pi$
$139$	13.8745	1.17682	0.588410	+	0.808563i	$0.299754\pi$
$140$	0	0
$141$	−9.29150	−0.782486
$142$	0	0
$143$	31.8745	2.66548
$144$	0	0
$145$	−6.93725	−0.576108
$146$	0	0
$147$	6.29150	0.518914
$148$	0	0
$149$	0.583005	0.0477617	0.0238808	−	0.999715i	$-0.492398\pi$
$149$	0.583005	0.0477617	0.0238808	+	0.999715i	$0.492398\pi$
$150$	0	0
$151$	−12.5830	−1.02399	−0.511995	−	0.858988i	$-0.671093\pi$
$151$	−12.5830	−1.02399	−0.511995	+	0.858988i	$0.671093\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	−2.70850	−0.216162	−0.108081	−	0.994142i	$-0.534471\pi$
$157$	−2.70850	−0.216162	−0.108081	+	0.994142i	$0.534471\pi$
$158$	0	0
$159$	0.708497	0.0561875
$160$	0	0
$161$	4.70850	0.371082
$162$	0	0
$163$	7.64575	0.598861	0.299431	−	0.954118i	$-0.403203\pi$
$163$	7.64575	0.598861	0.299431	+	0.954118i	$0.403203\pi$
$164$	0	0
$165$	−5.64575	−0.439521
$166$	0	0
$167$	10.7085	0.828648	0.414324	−	0.910129i	$-0.364018\pi$
$167$	10.7085	0.828648	0.414324	+	0.910129i	$0.364018\pi$
$168$	0	0
$169$	18.8745	1.45189
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	3.64575	0.275593
$176$	0	0
$177$	−0.708497	−0.0532539
$178$	0	0
$179$	−3.29150	−0.246018	−0.123009	−	0.992406i	$-0.539254\pi$
$179$	−3.29150	−0.246018	−0.123009	+	0.992406i	$0.539254\pi$
$180$	0	0
$181$	6.70850	0.498639	0.249319	−	0.968421i	$-0.419793\pi$
$181$	6.70850	0.498639	0.249319	+	0.968421i	$0.419793\pi$
$182$	0	0
$183$	−0.708497	−0.0523736
$184$	0	0
$185$	−1.64575	−0.120998
$186$	0	0
$187$	22.5830	1.65143
$188$	0	0
$189$	3.64575	0.265189
$190$	0	0
$191$	20.2288	1.46370	0.731851	−	0.681465i	$-0.238657\pi$
$191$	20.2288	1.46370	0.731851	+	0.681465i	$0.238657\pi$
$192$	0	0
$193$	−6.35425	−0.457389	−0.228694	−	0.973498i	$-0.573446\pi$
$193$	−6.35425	−0.457389	−0.228694	+	0.973498i	$0.573446\pi$
$194$	0	0
$195$	−5.64575	−0.404301
$196$	0	0
$197$	17.1660	1.22303	0.611514	−	0.791234i	$-0.290561\pi$
$197$	17.1660	1.22303	0.611514	+	0.791234i	$0.290561\pi$
$198$	0	0
$199$	3.29150	0.233328	0.116664	−	0.993171i	$-0.462780\pi$
$199$	3.29150	0.233328	0.116664	+	0.993171i	$0.462780\pi$
$200$	0	0
$201$	−14.5830	−1.02861
$202$	0	0
$203$	−25.2915	−1.77512
$204$	0	0
$205$	−4.35425	−0.304114
$206$	0	0
$207$	1.29150	0.0897656
$208$	0	0
$209$	−5.64575	−0.390525
$210$	0	0
$211$	18.5830	1.27931	0.639653	−	0.768663i	$-0.279078\pi$
$211$	18.5830	1.27931	0.639653	+	0.768663i	$0.279078\pi$
$212$	0	0
$213$	3.29150	0.225530
$214$	0	0
$215$	−0.354249	−0.0241596
$216$	0	0
$217$	−21.8745	−1.48494
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	22.5830	1.51910
$222$	0	0
$223$	18.5830	1.24441	0.622205	−	0.782854i	$-0.286237\pi$
$223$	18.5830	1.24441	0.622205	+	0.782854i	$0.286237\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	21.2915	1.41317	0.706583	−	0.707630i	$-0.250236\pi$
$227$	21.2915	1.41317	0.706583	+	0.707630i	$0.250236\pi$
$228$	0	0
$229$	19.2915	1.27482	0.637409	−	0.770525i	$-0.280006\pi$
$229$	19.2915	1.27482	0.637409	+	0.770525i	$0.280006\pi$
$230$	0	0
$231$	−20.5830	−1.35426
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−9.29150	−0.606111
$236$	0	0
$237$	14.5830	0.947268
$238$	0	0
$239$	−10.3542	−0.669761	−0.334880	−	0.942261i	$-0.608696\pi$
$239$	−10.3542	−0.669761	−0.334880	+	0.942261i	$0.608696\pi$
$240$	0	0
$241$	−4.58301	−0.295217	−0.147609	−	0.989046i	$-0.547158\pi$
$241$	−4.58301	−0.295217	−0.147609	+	0.989046i	$0.547158\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.29150	0.401949
$246$	0	0
$247$	−5.64575	−0.359231
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	21.6458	1.36627	0.683134	−	0.730293i	$-0.260617\pi$
$251$	21.6458	1.36627	0.683134	+	0.730293i	$0.260617\pi$
$252$	0	0
$253$	−7.29150	−0.458413
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	−26.5830	−1.65820	−0.829101	−	0.559099i	$-0.811147\pi$
$257$	−26.5830	−1.65820	−0.829101	+	0.559099i	$0.811147\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−6.93725	−0.429405
$262$	0	0
$263$	16.5830	1.02255	0.511276	−	0.859417i	$-0.329173\pi$
$263$	16.5830	1.02255	0.511276	+	0.859417i	$0.329173\pi$
$264$	0	0
$265$	0.708497	0.0435226
$266$	0	0
$267$	−1.06275	−0.0650390
$268$	0	0
$269$	−22.2288	−1.35531	−0.677656	−	0.735379i	$-0.737004\pi$
$269$	−22.2288	−1.35531	−0.677656	+	0.735379i	$0.737004\pi$
$270$	0	0
$271$	15.2915	0.928893	0.464446	−	0.885601i	$-0.346253\pi$
$271$	15.2915	0.928893	0.464446	+	0.885601i	$0.346253\pi$
$272$	0	0
$273$	−20.5830	−1.24574
$274$	0	0
$275$	−5.64575	−0.340452
$276$	0	0
$277$	−20.5830	−1.23671	−0.618356	−	0.785898i	$-0.712201\pi$
$277$	−20.5830	−1.23671	−0.618356	+	0.785898i	$0.712201\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−5.77124	−0.344284	−0.172142	−	0.985072i	$-0.555069\pi$
$281$	−5.77124	−0.344284	−0.172142	+	0.985072i	$0.555069\pi$
$282$	0	0
$283$	−25.5203	−1.51702	−0.758511	−	0.651660i	$-0.774073\pi$
$283$	−25.5203	−1.51702	−0.758511	+	0.651660i	$0.774073\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	−15.8745	−0.937043
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	12.9373	0.758395
$292$	0	0
$293$	26.5830	1.55300	0.776498	−	0.630120i	$-0.216994\pi$
$293$	26.5830	1.55300	0.776498	+	0.630120i	$0.216994\pi$
$294$	0	0
$295$	−0.708497	−0.0412503
$296$	0	0
$297$	−5.64575	−0.327600
$298$	0	0
$299$	−7.29150	−0.421678
$300$	0	0
$301$	−1.29150	−0.0744410
$302$	0	0
$303$	−1.29150	−0.0741949
$304$	0	0
$305$	−0.708497	−0.0405684
$306$	0	0
$307$	−28.4575	−1.62416	−0.812078	−	0.583549i	$-0.801664\pi$
$307$	−28.4575	−1.62416	−0.812078	+	0.583549i	$0.801664\pi$
$308$	0	0
$309$	−10.5830	−0.602046
$310$	0	0
$311$	−7.06275	−0.400492	−0.200246	−	0.979746i	$-0.564174\pi$
$311$	−7.06275	−0.400492	−0.200246	+	0.979746i	$0.564174\pi$
$312$	0	0
$313$	6.70850	0.379187	0.189593	−	0.981863i	$-0.439283\pi$
$313$	6.70850	0.379187	0.189593	+	0.981863i	$0.439283\pi$
$314$	0	0
$315$	3.64575	0.205415
$316$	0	0
$317$	−32.4575	−1.82300	−0.911498	−	0.411305i	$-0.865073\pi$
$317$	−32.4575	−1.82300	−0.911498	+	0.411305i	$0.865073\pi$
$318$	0	0
$319$	39.1660	2.19288
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−5.64575	−0.313170
$326$	0	0
$327$	−5.29150	−0.292621
$328$	0	0
$329$	−33.8745	−1.86756
$330$	0	0
$331$	−32.5830	−1.79092	−0.895462	−	0.445138i	$-0.853155\pi$
$331$	−32.5830	−1.79092	−0.895462	+	0.445138i	$0.853155\pi$
$332$	0	0
$333$	−1.64575	−0.0901866
$334$	0	0
$335$	−14.5830	−0.796755
$336$	0	0
$337$	−0.937254	−0.0510555	−0.0255277	−	0.999674i	$-0.508127\pi$
$337$	−0.937254	−0.0510555	−0.0255277	+	0.999674i	$0.508127\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	33.8745	1.83441
$342$	0	0
$343$	−2.58301	−0.139469
$344$	0	0
$345$	1.29150	0.0695322
$346$	0	0
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	0	0
$349$	−19.1660	−1.02593	−0.512967	−	0.858409i	$-0.671454\pi$
$349$	−19.1660	−1.02593	−0.512967	+	0.858409i	$0.671454\pi$
$350$	0	0
$351$	−5.64575	−0.301348
$352$	0	0
$353$	−20.5830	−1.09552	−0.547761	−	0.836635i	$-0.684520\pi$
$353$	−20.5830	−1.09552	−0.547761	+	0.836635i	$0.684520\pi$
$354$	0	0
$355$	3.29150	0.174695
$356$	0	0
$357$	−14.5830	−0.771814
$358$	0	0
$359$	−22.1033	−1.16657	−0.583283	−	0.812269i	$-0.698232\pi$
$359$	−22.1033	−1.16657	−0.583283	+	0.812269i	$0.698232\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	20.8745	1.09563
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	3.64575	0.190307	0.0951533	−	0.995463i	$-0.469666\pi$
$367$	3.64575	0.190307	0.0951533	+	0.995463i	$0.469666\pi$
$368$	0	0
$369$	−4.35425	−0.226673
$370$	0	0
$371$	2.58301	0.134103
$372$	0	0
$373$	20.9373	1.08409	0.542045	−	0.840349i	$-0.317650\pi$
$373$	20.9373	1.08409	0.542045	+	0.840349i	$0.317650\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	39.1660	2.01715
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	−2.58301	−0.131985	−0.0659927	−	0.997820i	$-0.521021\pi$
$383$	−2.58301	−0.131985	−0.0659927	+	0.997820i	$0.521021\pi$
$384$	0	0
$385$	−20.5830	−1.04901
$386$	0	0
$387$	−0.354249	−0.0180075
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−5.16601	−0.261256
$392$	0	0
$393$	−12.2288	−0.616859
$394$	0	0
$395$	14.5830	0.733751
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	3.64575	0.182516
$400$	0	0
$401$	10.9373	0.546180	0.273090	−	0.961988i	$-0.411954\pi$
$401$	10.9373	0.546180	0.273090	+	0.961988i	$0.411954\pi$
$402$	0	0
$403$	33.8745	1.68741
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	9.29150	0.460563
$408$	0	0
$409$	−17.2915	−0.855010	−0.427505	−	0.904013i	$-0.640607\pi$
$409$	−17.2915	−0.855010	−0.427505	+	0.904013i	$0.640607\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	−2.58301	−0.127101
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	13.8745	0.679438
$418$	0	0
$419$	−23.0627	−1.12669	−0.563344	−	0.826222i	$-0.690486\pi$
$419$	−23.0627	−1.12669	−0.563344	+	0.826222i	$0.690486\pi$
$420$	0	0
$421$	7.41699	0.361482	0.180741	−	0.983531i	$-0.442150\pi$
$421$	7.41699	0.361482	0.180741	+	0.983531i	$0.442150\pi$
$422$	0	0
$423$	−9.29150	−0.451768
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−2.58301	−0.125000
$428$	0	0
$429$	31.8745	1.53892
$430$	0	0
$431$	7.29150	0.351219	0.175610	−	0.984460i	$-0.443810\pi$
$431$	7.29150	0.351219	0.175610	+	0.984460i	$0.443810\pi$
$432$	0	0
$433$	−22.3542	−1.07428	−0.537138	−	0.843494i	$-0.680495\pi$
$433$	−22.3542	−1.07428	−0.537138	+	0.843494i	$0.680495\pi$
$434$	0	0
$435$	−6.93725	−0.332616
$436$	0	0
$437$	1.29150	0.0617809
$438$	0	0
$439$	22.5830	1.07783	0.538914	−	0.842361i	$-0.318835\pi$
$439$	22.5830	1.07783	0.538914	+	0.842361i	$0.318835\pi$
$440$	0	0
$441$	6.29150	0.299595
$442$	0	0
$443$	−37.2915	−1.77177	−0.885886	−	0.463902i	$-0.846449\pi$
$443$	−37.2915	−1.77177	−0.885886	+	0.463902i	$0.846449\pi$
$444$	0	0
$445$	−1.06275	−0.0503790
$446$	0	0
$447$	0.583005	0.0275752
$448$	0	0
$449$	−9.77124	−0.461133	−0.230567	−	0.973057i	$-0.574058\pi$
$449$	−9.77124	−0.461133	−0.230567	+	0.973057i	$0.574058\pi$
$450$	0	0
$451$	24.5830	1.15757
$452$	0	0
$453$	−12.5830	−0.591201
$454$	0	0
$455$	−20.5830	−0.964946
$456$	0	0
$457$	31.8745	1.49103	0.745513	−	0.666491i	$-0.232204\pi$
$457$	31.8745	1.49103	0.745513	+	0.666491i	$0.232204\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	25.7490	1.19925	0.599626	−	0.800281i	$-0.295316\pi$
$461$	25.7490	1.19925	0.599626	+	0.800281i	$0.295316\pi$
$462$	0	0
$463$	−1.52026	−0.0706524	−0.0353262	−	0.999376i	$-0.511247\pi$
$463$	−1.52026	−0.0706524	−0.0353262	+	0.999376i	$0.511247\pi$
$464$	0	0
$465$	−6.00000	−0.278243
$466$	0	0
$467$	11.8745	0.549487	0.274743	−	0.961518i	$-0.411407\pi$
$467$	11.8745	0.549487	0.274743	+	0.961518i	$0.411407\pi$
$468$	0	0
$469$	−53.1660	−2.45498
$470$	0	0
$471$	−2.70850	−0.124801
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0.708497	0.0324399
$478$	0	0
$479$	−9.64575	−0.440726	−0.220363	−	0.975418i	$-0.570724\pi$
$479$	−9.64575	−0.440726	−0.220363	+	0.975418i	$0.570724\pi$
$480$	0	0
$481$	9.29150	0.423656
$482$	0	0
$483$	4.70850	0.214244
$484$	0	0
$485$	12.9373	0.587450
$486$	0	0
$487$	13.8745	0.628714	0.314357	−	0.949305i	$-0.398211\pi$
$487$	13.8745	0.628714	0.314357	+	0.949305i	$0.398211\pi$
$488$	0	0
$489$	7.64575	0.345753
$490$	0	0
$491$	32.2288	1.45446	0.727232	−	0.686392i	$-0.240807\pi$
$491$	32.2288	1.45446	0.727232	+	0.686392i	$0.240807\pi$
$492$	0	0
$493$	27.7490	1.24975
$494$	0	0
$495$	−5.64575	−0.253758
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	−43.0405	−1.92676	−0.963379	−	0.268143i	$-0.913590\pi$
$499$	−43.0405	−1.92676	−0.963379	+	0.268143i	$0.913590\pi$
$500$	0	0
$501$	10.7085	0.478420
$502$	0	0
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	−1.29150	−0.0574711
$506$	0	0
$507$	18.8745	0.838246
$508$	0	0
$509$	24.1033	1.06836	0.534179	−	0.845371i	$-0.320621\pi$
$509$	24.1033	1.06836	0.534179	+	0.845371i	$0.320621\pi$
$510$	0	0
$511$	36.4575	1.61279
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−10.5830	−0.466343
$516$	0	0
$517$	52.4575	2.30708
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−36.1033	−1.58171	−0.790856	−	0.612002i	$-0.790365\pi$
$521$	−36.1033	−1.58171	−0.790856	+	0.612002i	$0.790365\pi$
$522$	0	0
$523$	12.7085	0.555704	0.277852	−	0.960624i	$-0.410378\pi$
$523$	12.7085	0.555704	0.277852	+	0.960624i	$0.410378\pi$
$524$	0	0
$525$	3.64575	0.159114
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	−21.3320	−0.927479
$530$	0	0
$531$	−0.708497	−0.0307462
$532$	0	0
$533$	24.5830	1.06481
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3.29150	−0.142039
$538$	0	0
$539$	−35.5203	−1.52997
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	6.70850	0.287889
$544$	0	0
$545$	−5.29150	−0.226663
$546$	0	0
$547$	29.8745	1.27734	0.638671	−	0.769480i	$-0.279485\pi$
$547$	29.8745	1.27734	0.638671	+	0.769480i	$0.279485\pi$
$548$	0	0
$549$	−0.708497	−0.0302379
$550$	0	0
$551$	−6.93725	−0.295537
$552$	0	0
$553$	53.1660	2.26085
$554$	0	0
$555$	−1.64575	−0.0698583
$556$	0	0
$557$	−32.5830	−1.38059	−0.690293	−	0.723530i	$-0.742518\pi$
$557$	−32.5830	−1.38059	−0.690293	+	0.723530i	$0.742518\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	22.5830	0.953455
$562$	0	0
$563$	−39.8745	−1.68051	−0.840255	−	0.542191i	$-0.817595\pi$
$563$	−39.8745	−1.68051	−0.840255	+	0.542191i	$0.817595\pi$
$564$	0	0
$565$	−4.00000	−0.168281
$566$	0	0
$567$	3.64575	0.153107
$568$	0	0
$569$	−22.9373	−0.961580	−0.480790	−	0.876836i	$-0.659650\pi$
$569$	−22.9373	−0.961580	−0.480790	+	0.876836i	$0.659650\pi$
$570$	0	0
$571$	−27.2915	−1.14211	−0.571057	−	0.820910i	$-0.693466\pi$
$571$	−27.2915	−1.14211	−0.571057	+	0.820910i	$0.693466\pi$
$572$	0	0
$573$	20.2288	0.845068
$574$	0	0
$575$	1.29150	0.0538594
$576$	0	0
$577$	−2.70850	−0.112756	−0.0563781	−	0.998409i	$-0.517955\pi$
$577$	−2.70850	−0.112756	−0.0563781	+	0.998409i	$0.517955\pi$
$578$	0	0
$579$	−6.35425	−0.264074
$580$	0	0
$581$	−21.8745	−0.907508
$582$	0	0
$583$	−4.00000	−0.165663
$584$	0	0
$585$	−5.64575	−0.233423
$586$	0	0
$587$	22.7085	0.937280	0.468640	−	0.883389i	$-0.344744\pi$
$587$	22.7085	0.937280	0.468640	+	0.883389i	$0.344744\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	0	0
$591$	17.1660	0.706115
$592$	0	0
$593$	24.5830	1.00950	0.504752	−	0.863265i	$-0.331584\pi$
$593$	24.5830	1.00950	0.504752	+	0.863265i	$0.331584\pi$
$594$	0	0
$595$	−14.5830	−0.597845
$596$	0	0
$597$	3.29150	0.134712
$598$	0	0
$599$	−30.5830	−1.24959	−0.624794	−	0.780790i	$-0.714817\pi$
$599$	−30.5830	−1.24959	−0.624794	+	0.780790i	$0.714817\pi$
$600$	0	0
$601$	33.2915	1.35799	0.678994	−	0.734143i	$-0.262416\pi$
$601$	33.2915	1.35799	0.678994	+	0.734143i	$0.262416\pi$
$602$	0	0
$603$	−14.5830	−0.593866
$604$	0	0
$605$	20.8745	0.848669
$606$	0	0
$607$	−31.0405	−1.25990	−0.629948	−	0.776637i	$-0.716924\pi$
$607$	−31.0405	−1.25990	−0.629948	+	0.776637i	$0.716924\pi$
$608$	0	0
$609$	−25.2915	−1.02486
$610$	0	0
$611$	52.4575	2.12220
$612$	0	0
$613$	22.4575	0.907050	0.453525	−	0.891243i	$-0.350166\pi$
$613$	22.4575	0.907050	0.453525	+	0.891243i	$0.350166\pi$
$614$	0	0
$615$	−4.35425	−0.175580
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	−12.7085	−0.510798	−0.255399	−	0.966836i	$-0.582207\pi$
$619$	−12.7085	−0.510798	−0.255399	+	0.966836i	$0.582207\pi$
$620$	0	0
$621$	1.29150	0.0518262
$622$	0	0
$623$	−3.87451	−0.155229
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.64575	−0.225470
$628$	0	0
$629$	6.58301	0.262482
$630$	0	0
$631$	−6.58301	−0.262065	−0.131033	−	0.991378i	$-0.541829\pi$
$631$	−6.58301	−0.262065	−0.131033	+	0.991378i	$0.541829\pi$
$632$	0	0
$633$	18.5830	0.738608
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	−35.5203	−1.40736
$638$	0	0
$639$	3.29150	0.130210
$640$	0	0
$641$	−1.06275	−0.0419759	−0.0209880	−	0.999780i	$-0.506681\pi$
$641$	−1.06275	−0.0419759	−0.0209880	+	0.999780i	$0.506681\pi$
$642$	0	0
$643$	8.35425	0.329459	0.164730	−	0.986339i	$-0.447325\pi$
$643$	8.35425	0.329459	0.164730	+	0.986339i	$0.447325\pi$
$644$	0	0
$645$	−0.354249	−0.0139485
$646$	0	0
$647$	24.5830	0.966458	0.483229	−	0.875494i	$-0.339464\pi$
$647$	24.5830	0.966458	0.483229	+	0.875494i	$0.339464\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	−21.8745	−0.857330
$652$	0	0
$653$	30.5830	1.19681	0.598403	−	0.801195i	$-0.295802\pi$
$653$	30.5830	1.19681	0.598403	+	0.801195i	$0.295802\pi$
$654$	0	0
$655$	−12.2288	−0.477817
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	−46.5830	−1.81462	−0.907308	−	0.420466i	$-0.861866\pi$
$659$	−46.5830	−1.81462	−0.907308	+	0.420466i	$0.861866\pi$
$660$	0	0
$661$	−9.29150	−0.361398	−0.180699	−	0.983538i	$-0.557836\pi$
$661$	−9.29150	−0.361398	−0.180699	+	0.983538i	$0.557836\pi$
$662$	0	0
$663$	22.5830	0.877051
$664$	0	0
$665$	3.64575	0.141376
$666$	0	0
$667$	−8.95948	−0.346913
$668$	0	0
$669$	18.5830	0.718460
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	−28.9373	−1.11545	−0.557725	−	0.830026i	$-0.688325\pi$
$673$	−28.9373	−1.11545	−0.557725	+	0.830026i	$0.688325\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−12.4575	−0.478781	−0.239391	−	0.970923i	$-0.576948\pi$
$677$	−12.4575	−0.478781	−0.239391	+	0.970923i	$0.576948\pi$
$678$	0	0
$679$	47.1660	1.81007
$680$	0	0
$681$	21.2915	0.815892
$682$	0	0
$683$	43.7490	1.67401	0.837005	−	0.547196i	$-0.184305\pi$
$683$	43.7490	1.67401	0.837005	+	0.547196i	$0.184305\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	19.2915	0.736017
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−31.0405	−1.18084	−0.590418	−	0.807097i	$-0.701037\pi$
$691$	−31.0405	−1.18084	−0.590418	+	0.807097i	$0.701037\pi$
$692$	0	0
$693$	−20.5830	−0.781884
$694$	0	0
$695$	13.8745	0.526290
$696$	0	0
$697$	17.4170	0.659716
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−4.58301	−0.173098	−0.0865489	−	0.996248i	$-0.527584\pi$
$701$	−4.58301	−0.173098	−0.0865489	+	0.996248i	$0.527584\pi$
$702$	0	0
$703$	−1.64575	−0.0620707
$704$	0	0
$705$	−9.29150	−0.349938
$706$	0	0
$707$	−4.70850	−0.177081
$708$	0	0
$709$	−15.2915	−0.574284	−0.287142	−	0.957888i	$-0.592705\pi$
$709$	−15.2915	−0.574284	−0.287142	+	0.957888i	$0.592705\pi$
$710$	0	0
$711$	14.5830	0.546905
$712$	0	0
$713$	−7.74902	−0.290203
$714$	0	0
$715$	31.8745	1.19204
$716$	0	0
$717$	−10.3542	−0.386687
$718$	0	0
$719$	14.8118	0.552386	0.276193	−	0.961102i	$-0.410927\pi$
$719$	14.8118	0.552386	0.276193	+	0.961102i	$0.410927\pi$
$720$	0	0
$721$	−38.5830	−1.43691
$722$	0	0
$723$	−4.58301	−0.170444
$724$	0	0
$725$	−6.93725	−0.257643
$726$	0	0
$727$	44.1033	1.63570	0.817850	−	0.575432i	$-0.195166\pi$
$727$	44.1033	1.63570	0.817850	+	0.575432i	$0.195166\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.41699	0.0524094
$732$	0	0
$733$	28.5830	1.05574	0.527869	−	0.849326i	$-0.322991\pi$
$733$	28.5830	1.05574	0.527869	+	0.849326i	$0.322991\pi$
$734$	0	0
$735$	6.29150	0.232066
$736$	0	0
$737$	82.3320	3.03274
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−5.64575	−0.207402
$742$	0	0
$743$	−10.5830	−0.388253	−0.194126	−	0.980977i	$-0.562187\pi$
$743$	−10.5830	−0.388253	−0.194126	+	0.980977i	$0.562187\pi$
$744$	0	0
$745$	0.583005	0.0213597
$746$	0	0
$747$	−6.00000	−0.219529
$748$	0	0
$749$	0	0
$750$	0	0
$751$	23.4170	0.854498	0.427249	−	0.904134i	$-0.359483\pi$
$751$	23.4170	0.854498	0.427249	+	0.904134i	$0.359483\pi$
$752$	0	0
$753$	21.6458	0.788815
$754$	0	0
$755$	−12.5830	−0.457942
$756$	0	0
$757$	7.87451	0.286204	0.143102	−	0.989708i	$-0.454292\pi$
$757$	7.87451	0.286204	0.143102	+	0.989708i	$0.454292\pi$
$758$	0	0
$759$	−7.29150	−0.264665
$760$	0	0
$761$	−37.7490	−1.36840	−0.684200	−	0.729294i	$-0.739849\pi$
$761$	−37.7490	−1.36840	−0.684200	+	0.729294i	$0.739849\pi$
$762$	0	0
$763$	−19.2915	−0.698399
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	45.7490	1.64975	0.824876	−	0.565314i	$-0.191245\pi$
$769$	45.7490	1.64975	0.824876	+	0.565314i	$0.191245\pi$
$770$	0	0
$771$	−26.5830	−0.957364
$772$	0	0
$773$	−19.2915	−0.693867	−0.346934	−	0.937890i	$-0.612777\pi$
$773$	−19.2915	−0.693867	−0.346934	+	0.937890i	$0.612777\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	0	0
$777$	−6.00000	−0.215249
$778$	0	0
$779$	−4.35425	−0.156007
$780$	0	0
$781$	−18.5830	−0.664952
$782$	0	0
$783$	−6.93725	−0.247917
$784$	0	0
$785$	−2.70850	−0.0966704
$786$	0	0
$787$	−6.12549	−0.218350	−0.109175	−	0.994023i	$-0.534821\pi$
$787$	−6.12549	−0.218350	−0.109175	+	0.994023i	$0.534821\pi$
$788$	0	0
$789$	16.5830	0.590371
$790$	0	0
$791$	−14.5830	−0.518512
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	0.708497	0.0251278
$796$	0	0
$797$	40.0000	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$797$	40.0000	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$798$	0	0
$799$	37.1660	1.31484
$800$	0	0
$801$	−1.06275	−0.0375503
$802$	0	0
$803$	−56.4575	−1.99234
$804$	0	0
$805$	4.70850	0.165953
$806$	0	0
$807$	−22.2288	−0.782489
$808$	0	0
$809$	−40.5830	−1.42682	−0.713411	−	0.700746i	$-0.752851\pi$
$809$	−40.5830	−1.42682	−0.713411	+	0.700746i	$0.752851\pi$
$810$	0	0
$811$	46.3320	1.62694	0.813469	−	0.581609i	$-0.197577\pi$
$811$	46.3320	1.62694	0.813469	+	0.581609i	$0.197577\pi$
$812$	0	0
$813$	15.2915	0.536296
$814$	0	0
$815$	7.64575	0.267819
$816$	0	0
$817$	−0.354249	−0.0123936
$818$	0	0
$819$	−20.5830	−0.719228
$820$	0	0
$821$	−47.6235	−1.66207	−0.831036	−	0.556218i	$-0.812252\pi$
$821$	−47.6235	−1.66207	−0.831036	+	0.556218i	$0.812252\pi$
$822$	0	0
$823$	22.2288	0.774846	0.387423	−	0.921902i	$-0.373365\pi$
$823$	22.2288	0.774846	0.387423	+	0.921902i	$0.373365\pi$
$824$	0	0
$825$	−5.64575	−0.196560
$826$	0	0
$827$	22.4575	0.780924	0.390462	−	0.920619i	$-0.372315\pi$
$827$	22.4575	0.780924	0.390462	+	0.920619i	$0.372315\pi$
$828$	0	0
$829$	13.2915	0.461633	0.230816	−	0.972997i	$-0.425860\pi$
$829$	13.2915	0.461633	0.230816	+	0.972997i	$0.425860\pi$
$830$	0	0
$831$	−20.5830	−0.714017
$832$	0	0
$833$	−25.1660	−0.871951
$834$	0	0
$835$	10.7085	0.370583
$836$	0	0
$837$	−6.00000	−0.207390
$838$	0	0
$839$	−40.4575	−1.39675	−0.698374	−	0.715733i	$-0.746093\pi$
$839$	−40.4575	−1.39675	−0.698374	+	0.715733i	$0.746093\pi$
$840$	0	0
$841$	19.1255	0.659500
$842$	0	0
$843$	−5.77124	−0.198772
$844$	0	0
$845$	18.8745	0.649303
$846$	0	0
$847$	76.1033	2.61494
$848$	0	0
$849$	−25.5203	−0.875853
$850$	0	0
$851$	−2.12549	−0.0728609
$852$	0	0
$853$	25.2915	0.865965	0.432982	−	0.901402i	$-0.357461\pi$
$853$	25.2915	0.865965	0.432982	+	0.901402i	$0.357461\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	−43.0405	−1.47024	−0.735118	−	0.677939i	$-0.762873\pi$
$857$	−43.0405	−1.47024	−0.735118	+	0.677939i	$0.762873\pi$
$858$	0	0
$859$	−50.3320	−1.71731	−0.858653	−	0.512557i	$-0.828698\pi$
$859$	−50.3320	−1.71731	−0.858653	+	0.512557i	$0.828698\pi$
$860$	0	0
$861$	−15.8745	−0.541002
$862$	0	0
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−82.3320	−2.79292
$870$	0	0
$871$	82.3320	2.78971
$872$	0	0
$873$	12.9373	0.437860
$874$	0	0
$875$	3.64575	0.123249
$876$	0	0
$877$	−40.9373	−1.38235	−0.691176	−	0.722686i	$-0.742907\pi$
$877$	−40.9373	−1.38235	−0.691176	+	0.722686i	$0.742907\pi$
$878$	0	0
$879$	26.5830	0.896623
$880$	0	0
$881$	−31.8745	−1.07388	−0.536940	−	0.843621i	$-0.680420\pi$
$881$	−31.8745	−1.07388	−0.536940	+	0.843621i	$0.680420\pi$
$882$	0	0
$883$	36.8118	1.23881	0.619407	−	0.785070i	$-0.287373\pi$
$883$	36.8118	1.23881	0.619407	+	0.785070i	$0.287373\pi$
$884$	0	0
$885$	−0.708497	−0.0238159
$886$	0	0
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	−14.5830	−0.489098
$890$	0	0
$891$	−5.64575	−0.189140
$892$	0	0
$893$	−9.29150	−0.310928
$894$	0	0
$895$	−3.29150	−0.110023
$896$	0	0
$897$	−7.29150	−0.243456
$898$	0	0
$899$	41.6235	1.38822
$900$	0	0
$901$	−2.83399	−0.0944139
$902$	0	0
$903$	−1.29150	−0.0429785
$904$	0	0
$905$	6.70850	0.222998
$906$	0	0
$907$	−27.0405	−0.897866	−0.448933	−	0.893566i	$-0.648196\pi$
$907$	−27.0405	−0.897866	−0.448933	+	0.893566i	$0.648196\pi$
$908$	0	0
$909$	−1.29150	−0.0428364
$910$	0	0
$911$	−5.41699	−0.179473	−0.0897365	−	0.995966i	$-0.528602\pi$
$911$	−5.41699	−0.179473	−0.0897365	+	0.995966i	$0.528602\pi$
$912$	0	0
$913$	33.8745	1.12108
$914$	0	0
$915$	−0.708497	−0.0234222
$916$	0	0
$917$	−44.5830	−1.47226
$918$	0	0
$919$	−57.1660	−1.88573	−0.942866	−	0.333171i	$-0.891881\pi$
$919$	−57.1660	−1.88573	−0.942866	+	0.333171i	$0.891881\pi$
$920$	0	0
$921$	−28.4575	−0.937707
$922$	0	0
$923$	−18.5830	−0.611667
$924$	0	0
$925$	−1.64575	−0.0541120
$926$	0	0
$927$	−10.5830	−0.347591
$928$	0	0
$929$	19.8745	0.652061	0.326031	−	0.945359i	$-0.394289\pi$
$929$	19.8745	0.652061	0.326031	+	0.945359i	$0.394289\pi$
$930$	0	0
$931$	6.29150	0.206196
$932$	0	0
$933$	−7.06275	−0.231224
$934$	0	0
$935$	22.5830	0.738543
$936$	0	0
$937$	51.8745	1.69467	0.847333	−	0.531062i	$-0.178207\pi$
$937$	51.8745	1.69467	0.847333	+	0.531062i	$0.178207\pi$
$938$	0	0
$939$	6.70850	0.218924
$940$	0	0
$941$	−38.2288	−1.24622	−0.623111	−	0.782133i	$-0.714131\pi$
$941$	−38.2288	−1.24622	−0.623111	+	0.782133i	$0.714131\pi$
$942$	0	0
$943$	−5.62352	−0.183127
$944$	0	0
$945$	3.64575	0.118596
$946$	0	0
$947$	−32.5830	−1.05881	−0.529403	−	0.848371i	$-0.677584\pi$
$947$	−32.5830	−1.05881	−0.529403	+	0.848371i	$0.677584\pi$
$948$	0	0
$949$	−56.4575	−1.83269
$950$	0	0
$951$	−32.4575	−1.05251
$952$	0	0
$953$	10.5830	0.342817	0.171409	−	0.985200i	$-0.445168\pi$
$953$	10.5830	0.342817	0.171409	+	0.985200i	$0.445168\pi$
$954$	0	0
$955$	20.2288	0.654587
$956$	0	0
$957$	39.1660	1.26606
$958$	0	0
$959$	21.8745	0.706365
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.35425	−0.204551
$966$	0	0
$967$	36.3542	1.16907	0.584537	−	0.811367i	$-0.301276\pi$
$967$	36.3542	1.16907	0.584537	+	0.811367i	$0.301276\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	50.5830	1.62162
$974$	0	0
$975$	−5.64575	−0.180809
$976$	0	0
$977$	23.0405	0.737131	0.368566	−	0.929602i	$-0.379849\pi$
$977$	23.0405	0.737131	0.368566	+	0.929602i	$0.379849\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	−5.29150	−0.168945
$982$	0	0
$983$	18.4575	0.588703	0.294352	−	0.955697i	$-0.404896\pi$
$983$	18.4575	0.588703	0.294352	+	0.955697i	$0.404896\pi$
$984$	0	0
$985$	17.1660	0.546955
$986$	0	0
$987$	−33.8745	−1.07824
$988$	0	0
$989$	−0.457513	−0.0145481
$990$	0	0
$991$	−27.7490	−0.881477	−0.440738	−	0.897636i	$-0.645283\pi$
$991$	−27.7490	−0.881477	−0.440738	+	0.897636i	$0.645283\pi$
$992$	0	0
$993$	−32.5830	−1.03399
$994$	0	0
$995$	3.29150	0.104348
$996$	0	0
$997$	37.2915	1.18103	0.590517	−	0.807025i	$-0.298924\pi$
$997$	37.2915	1.18103	0.590517	+	0.807025i	$0.298924\pi$
$998$	0	0
$999$	−1.64575	−0.0520693

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bo.1.2		2
4.3	odd	2		285.2.a.d.1.2	✓	2
12.11	even	2		855.2.a.g.1.1		2
20.3	even	4		1425.2.c.i.799.1		4
20.7	even	4		1425.2.c.i.799.4		4
20.19	odd	2		1425.2.a.p.1.1		2
60.59	even	2		4275.2.a.u.1.2		2
76.75	even	2		5415.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.d.1.2	✓	2	4.3	odd	2
855.2.a.g.1.1		2	12.11	even	2
1425.2.a.p.1.1		2	20.19	odd	2
1425.2.c.i.799.1		4	20.3	even	4
1425.2.c.i.799.4		4	20.7	even	4
4275.2.a.u.1.2		2	60.59	even	2
4560.2.a.bo.1.2		2	1.1	even	1	trivial
5415.2.a.s.1.1		2	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +3.64575 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +3.64575 q^{7} +1.00000 q^{9} -5.64575 q^{11} -5.64575 q^{13} +1.00000 q^{15} -4.00000 q^{17} +1.00000 q^{19} +3.64575 q^{21} +1.29150 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.93725 q^{29} -6.00000 q^{31} -5.64575 q^{33} +3.64575 q^{35} -1.64575 q^{37} -5.64575 q^{39} -4.35425 q^{41} -0.354249 q^{43} +1.00000 q^{45} -9.29150 q^{47} +6.29150 q^{49} -4.00000 q^{51} +0.708497 q^{53} -5.64575 q^{55} +1.00000 q^{57} -0.708497 q^{59} -0.708497 q^{61} +3.64575 q^{63} -5.64575 q^{65} -14.5830 q^{67} +1.29150 q^{69} +3.29150 q^{71} +10.0000 q^{73} +1.00000 q^{75} -20.5830 q^{77} +14.5830 q^{79} +1.00000 q^{81} -6.00000 q^{83} -4.00000 q^{85} -6.93725 q^{87} -1.06275 q^{89} -20.5830 q^{91} -6.00000 q^{93} +1.00000 q^{95} +12.9373 q^{97} -5.64575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{17} + 2 q^{19} + 2 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} - 12 q^{31} - 6 q^{33} + 2 q^{35} + 2 q^{37} - 6 q^{39} - 14 q^{41} - 6 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 8 q^{51} + 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 12 q^{61} + 2 q^{63} - 6 q^{65} - 8 q^{67} - 8 q^{69} - 4 q^{71} + 20 q^{73} + 2 q^{75} - 20 q^{77} + 8 q^{79} + 2 q^{81} - 12 q^{83} - 8 q^{85} + 2 q^{87} - 18 q^{89} - 20 q^{91} - 12 q^{93} + 2 q^{95} + 10 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 6 * q^13 + 2 * q^15 - 8 * q^17 + 2 * q^19 + 2 * q^21 - 8 * q^23 + 2 * q^25 + 2 * q^27 + 2 * q^29 - 12 * q^31 - 6 * q^33 + 2 * q^35 + 2 * q^37 - 6 * q^39 - 14 * q^41 - 6 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 - 8 * q^51 + 12 * q^53 - 6 * q^55 + 2 * q^57 - 12 * q^59 - 12 * q^61 + 2 * q^63 - 6 * q^65 - 8 * q^67 - 8 * q^69 - 4 * q^71 + 20 * q^73 + 2 * q^75 - 20 * q^77 + 8 * q^79 + 2 * q^81 - 12 * q^83 - 8 * q^85 + 2 * q^87 - 18 * q^89 - 20 * q^91 - 12 * q^93 + 2 * q^95 + 10 * q^97 - 6 * q^99

Introduction

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