LMFDB - Embedded newform 4416.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4416,2,Mod(1,4416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4416, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4416.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4416 = 2^{6} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4416.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:	$35.2619375326$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.23607 q^{5} +3.23607 q^{7} +1.00000 q^{9} -4.00000 q^{11} +4.47214 q^{13} -1.23607 q^{15} -2.76393 q^{17} -7.23607 q^{19} +3.23607 q^{21} +1.00000 q^{23} -3.47214 q^{25} +1.00000 q^{27} -4.47214 q^{29} -6.47214 q^{31} -4.00000 q^{33} -4.00000 q^{35} -4.47214 q^{37} +4.47214 q^{39} -10.9443 q^{41} +5.70820 q^{43} -1.23607 q^{45} -4.00000 q^{47} +3.47214 q^{49} -2.76393 q^{51} +5.23607 q^{53} +4.94427 q^{55} -7.23607 q^{57} +4.94427 q^{59} -4.47214 q^{61} +3.23607 q^{63} -5.52786 q^{65} -0.763932 q^{67} +1.00000 q^{69} -8.00000 q^{71} +6.94427 q^{73} -3.47214 q^{75} -12.9443 q^{77} +9.70820 q^{79} +1.00000 q^{81} -4.00000 q^{83} +3.41641 q^{85} -4.47214 q^{87} -1.23607 q^{89} +14.4721 q^{91} -6.47214 q^{93} +8.94427 q^{95} +8.47214 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{11} + 2 q^{15} - 10 q^{17} - 10 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{31} - 8 q^{33} - 8 q^{35} - 4 q^{41} - 2 q^{43} + 2 q^{45} - 8 q^{47}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 8 * q^11 + 2 * q^15 - 10 * q^17 - 10 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^31 - 8 * q^33 - 8 * q^35 - 4 * q^41 - 2 * q^43 + 2 * q^45 - 8 * q^47 - 2 * q^49 - 10 * q^51 + 6 * q^53 - 8 * q^55 - 10 * q^57 - 8 * q^59 + 2 * q^63 - 20 * q^65 - 6 * q^67 + 2 * q^69 - 16 * q^71 - 4 * q^73 + 2 * q^75 - 8 * q^77 + 6 * q^79 + 2 * q^81 - 8 * q^83 - 20 * q^85 + 2 * q^89 + 20 * q^91 - 4 * q^93 + 8 * q^97 - 8 * q^99

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
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	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
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	0	0
$15$	−1.23607	−0.319151
$16$	0	0
$17$	−2.76393	−0.670352	−0.335176	−	0.942156i	$-0.608796\pi$
$17$	−2.76393	−0.670352	−0.335176	+	0.942156i	$0.608796\pi$
$18$	0	0
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	0	0
$21$	3.23607	0.706168
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	0	0
$39$	4.47214	0.716115
$40$	0	0
$41$	−10.9443	−1.70921	−0.854604	−	0.519280i	$-0.826200\pi$
$41$	−10.9443	−1.70921	−0.854604	+	0.519280i	$0.826200\pi$
$42$	0	0
$43$	5.70820	0.870493	0.435246	−	0.900311i	$-0.356661\pi$
$43$	5.70820	0.870493	0.435246	+	0.900311i	$0.356661\pi$
$44$	0	0
$45$	−1.23607	−0.184262
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	−2.76393	−0.387028
$52$	0	0
$53$	5.23607	0.719229	0.359615	−	0.933101i	$-0.382908\pi$
$53$	5.23607	0.719229	0.359615	+	0.933101i	$0.382908\pi$
$54$	0	0
$55$	4.94427	0.666685
$56$	0	0
$57$	−7.23607	−0.958441
$58$	0	0
$59$	4.94427	0.643689	0.321845	−	0.946792i	$-0.395697\pi$
$59$	4.94427	0.643689	0.321845	+	0.946792i	$0.395697\pi$
$60$	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
$65$	−5.52786	−0.685647
$66$	0	0
$67$	−0.763932	−0.0933292	−0.0466646	−	0.998911i	$-0.514859\pi$
$67$	−0.763932	−0.0933292	−0.0466646	+	0.998911i	$0.514859\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	6.94427	0.812766	0.406383	−	0.913703i	$-0.366790\pi$
$73$	6.94427	0.812766	0.406383	+	0.913703i	$0.366790\pi$
$74$	0	0
$75$	−3.47214	−0.400928
$76$	0	0
$77$	−12.9443	−1.47514
$78$	0	0
$79$	9.70820	1.09226	0.546129	−	0.837701i	$-0.316101\pi$
$79$	9.70820	1.09226	0.546129	+	0.837701i	$0.316101\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	3.41641	0.370561
$86$	0	0
$87$	−4.47214	−0.479463
$88$	0	0
$89$	−1.23607	−0.131023	−0.0655115	−	0.997852i	$-0.520868\pi$
$89$	−1.23607	−0.131023	−0.0655115	+	0.997852i	$0.520868\pi$
$90$	0	0
$91$	14.4721	1.51709
$92$	0	0
$93$	−6.47214	−0.671129
$94$	0	0
$95$	8.94427	0.917663
$96$	0	0
$97$	8.47214	0.860215	0.430108	−	0.902778i	$-0.358476\pi$
$97$	8.47214	0.860215	0.430108	+	0.902778i	$0.358476\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	6.94427	0.690981	0.345490	−	0.938422i	$-0.387713\pi$
$101$	6.94427	0.690981	0.345490	+	0.938422i	$0.387713\pi$
$102$	0	0
$103$	−11.2361	−1.10712	−0.553561	−	0.832808i	$-0.686732\pi$
$103$	−11.2361	−1.10712	−0.553561	+	0.832808i	$0.686732\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	−16.9443	−1.63806	−0.819032	−	0.573747i	$-0.805489\pi$
$107$	−16.9443	−1.63806	−0.819032	+	0.573747i	$0.805489\pi$
$108$	0	0
$109$	14.9443	1.43140	0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443	1.43140	0.715701	+	0.698407i	$0.246107\pi$
$110$	0	0
$111$	−4.47214	−0.424476
$112$	0	0
$113$	−6.18034	−0.581397	−0.290699	−	0.956815i	$-0.593888\pi$
$113$	−6.18034	−0.581397	−0.290699	+	0.956815i	$0.593888\pi$
$114$	0	0
$115$	−1.23607	−0.115264
$116$	0	0
$117$	4.47214	0.413449
$118$	0	0
$119$	−8.94427	−0.819920
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−10.9443	−0.986812
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	1.52786	0.135576	0.0677880	−	0.997700i	$-0.478406\pi$
$127$	1.52786	0.135576	0.0677880	+	0.997700i	$0.478406\pi$
$128$	0	0
$129$	5.70820	0.502579
$130$	0	0
$131$	−16.9443	−1.48043	−0.740214	−	0.672371i	$-0.765276\pi$
$131$	−16.9443	−1.48043	−0.740214	+	0.672371i	$0.765276\pi$
$132$	0	0
$133$	−23.4164	−2.03046
$134$	0	0
$135$	−1.23607	−0.106384
$136$	0	0
$137$	−1.23607	−0.105604	−0.0528022	−	0.998605i	$-0.516815\pi$
$137$	−1.23607	−0.105604	−0.0528022	+	0.998605i	$0.516815\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	−17.8885	−1.49592
$144$	0	0
$145$	5.52786	0.459064
$146$	0	0
$147$	3.47214	0.286377
$148$	0	0
$149$	11.7082	0.959173	0.479587	−	0.877494i	$-0.340787\pi$
$149$	11.7082	0.959173	0.479587	+	0.877494i	$0.340787\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	−2.76393	−0.223451
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	3.52786	0.281554	0.140777	−	0.990041i	$-0.455040\pi$
$157$	3.52786	0.281554	0.140777	+	0.990041i	$0.455040\pi$
$158$	0	0
$159$	5.23607	0.415247
$160$	0	0
$161$	3.23607	0.255038
$162$	0	0
$163$	−7.41641	−0.580898	−0.290449	−	0.956890i	$-0.593805\pi$
$163$	−7.41641	−0.580898	−0.290449	+	0.956890i	$0.593805\pi$
$164$	0	0
$165$	4.94427	0.384911
$166$	0	0
$167$	12.9443	1.00166	0.500829	−	0.865546i	$-0.333029\pi$
$167$	12.9443	1.00166	0.500829	+	0.865546i	$0.333029\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	−7.23607	−0.553356
$172$	0	0
$173$	−4.47214	−0.340010	−0.170005	−	0.985443i	$-0.554378\pi$
$173$	−4.47214	−0.340010	−0.170005	+	0.985443i	$0.554378\pi$
$174$	0	0
$175$	−11.2361	−0.849367
$176$	0	0
$177$	4.94427	0.371634
$178$	0	0
$179$	−20.9443	−1.56545	−0.782724	−	0.622369i	$-0.786170\pi$
$179$	−20.9443	−1.56545	−0.782724	+	0.622369i	$0.786170\pi$
$180$	0	0
$181$	11.8885	0.883669	0.441834	−	0.897097i	$-0.354328\pi$
$181$	11.8885	0.883669	0.441834	+	0.897097i	$0.354328\pi$
$182$	0	0
$183$	−4.47214	−0.330590
$184$	0	0
$185$	5.52786	0.406417
$186$	0	0
$187$	11.0557	0.808475
$188$	0	0
$189$	3.23607	0.235389
$190$	0	0
$191$	1.52786	0.110552	0.0552762	−	0.998471i	$-0.482396\pi$
$191$	1.52786	0.110552	0.0552762	+	0.998471i	$0.482396\pi$
$192$	0	0
$193$	17.4164	1.25366	0.626830	−	0.779156i	$-0.284352\pi$
$193$	17.4164	1.25366	0.626830	+	0.779156i	$0.284352\pi$
$194$	0	0
$195$	−5.52786	−0.395859
$196$	0	0
$197$	−17.4164	−1.24087	−0.620434	−	0.784259i	$-0.713043\pi$
$197$	−17.4164	−1.24087	−0.620434	+	0.784259i	$0.713043\pi$
$198$	0	0
$199$	9.70820	0.688196	0.344098	−	0.938934i	$-0.388185\pi$
$199$	9.70820	0.688196	0.344098	+	0.938934i	$0.388185\pi$
$200$	0	0
$201$	−0.763932	−0.0538836
$202$	0	0
$203$	−14.4721	−1.01574
$204$	0	0
$205$	13.5279	0.944827
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	28.9443	2.00212
$210$	0	0
$211$	−13.5279	−0.931297	−0.465648	−	0.884970i	$-0.654179\pi$
$211$	−13.5279	−0.931297	−0.465648	+	0.884970i	$0.654179\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	−7.05573	−0.481197
$216$	0	0
$217$	−20.9443	−1.42179
$218$	0	0
$219$	6.94427	0.469250
$220$	0	0
$221$	−12.3607	−0.831469
$222$	0	0
$223$	−25.8885	−1.73363	−0.866813	−	0.498634i	$-0.833835\pi$
$223$	−25.8885	−1.73363	−0.866813	+	0.498634i	$0.833835\pi$
$224$	0	0
$225$	−3.47214	−0.231476
$226$	0	0
$227$	13.5279	0.897876	0.448938	−	0.893563i	$-0.351802\pi$
$227$	13.5279	0.897876	0.448938	+	0.893563i	$0.351802\pi$
$228$	0	0
$229$	−2.94427	−0.194563	−0.0972815	−	0.995257i	$-0.531015\pi$
$229$	−2.94427	−0.194563	−0.0972815	+	0.995257i	$0.531015\pi$
$230$	0	0
$231$	−12.9443	−0.851671
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	4.94427	0.322529
$236$	0	0
$237$	9.70820	0.630616
$238$	0	0
$239$	4.94427	0.319818	0.159909	−	0.987132i	$-0.448880\pi$
$239$	4.94427	0.319818	0.159909	+	0.987132i	$0.448880\pi$
$240$	0	0
$241$	19.5279	1.25790	0.628950	−	0.777446i	$-0.283485\pi$
$241$	19.5279	1.25790	0.628950	+	0.777446i	$0.283485\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.29180	−0.274193
$246$	0	0
$247$	−32.3607	−2.05906
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	10.4721	0.660995	0.330498	−	0.943807i	$-0.392783\pi$
$251$	10.4721	0.660995	0.330498	+	0.943807i	$0.392783\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	3.41641	0.213944
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	−14.4721	−0.899255
$260$	0	0
$261$	−4.47214	−0.276818
$262$	0	0
$263$	6.47214	0.399089	0.199544	−	0.979889i	$-0.436054\pi$
$263$	6.47214	0.399089	0.199544	+	0.979889i	$0.436054\pi$
$264$	0	0
$265$	−6.47214	−0.397580
$266$	0	0
$267$	−1.23607	−0.0756461
$268$	0	0
$269$	0.472136	0.0287866	0.0143933	−	0.999896i	$-0.495418\pi$
$269$	0.472136	0.0287866	0.0143933	+	0.999896i	$0.495418\pi$
$270$	0	0
$271$	−17.5279	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$271$	−17.5279	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$272$	0	0
$273$	14.4721	0.875894
$274$	0	0
$275$	13.8885	0.837511
$276$	0	0
$277$	−15.8885	−0.954650	−0.477325	−	0.878727i	$-0.658394\pi$
$277$	−15.8885	−0.954650	−0.477325	+	0.878727i	$0.658394\pi$
$278$	0	0
$279$	−6.47214	−0.387477
$280$	0	0
$281$	24.6525	1.47064	0.735322	−	0.677718i	$-0.237031\pi$
$281$	24.6525	1.47064	0.735322	+	0.677718i	$0.237031\pi$
$282$	0	0
$283$	−5.70820	−0.339318	−0.169659	−	0.985503i	$-0.554267\pi$
$283$	−5.70820	−0.339318	−0.169659	+	0.985503i	$0.554267\pi$
$284$	0	0
$285$	8.94427	0.529813
$286$	0	0
$287$	−35.4164	−2.09056
$288$	0	0
$289$	−9.36068	−0.550628
$290$	0	0
$291$	8.47214	0.496645
$292$	0	0
$293$	−7.70820	−0.450318	−0.225159	−	0.974322i	$-0.572290\pi$
$293$	−7.70820	−0.450318	−0.225159	+	0.974322i	$0.572290\pi$
$294$	0	0
$295$	−6.11146	−0.355823
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	4.47214	0.258630
$300$	0	0
$301$	18.4721	1.06472
$302$	0	0
$303$	6.94427	0.398938
$304$	0	0
$305$	5.52786	0.316525
$306$	0	0
$307$	−2.47214	−0.141092	−0.0705461	−	0.997509i	$-0.522474\pi$
$307$	−2.47214	−0.141092	−0.0705461	+	0.997509i	$0.522474\pi$
$308$	0	0
$309$	−11.2361	−0.639198
$310$	0	0
$311$	7.05573	0.400094	0.200047	−	0.979786i	$-0.435891\pi$
$311$	7.05573	0.400094	0.200047	+	0.979786i	$0.435891\pi$
$312$	0	0
$313$	5.05573	0.285767	0.142883	−	0.989740i	$-0.454363\pi$
$313$	5.05573	0.285767	0.142883	+	0.989740i	$0.454363\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	−2.58359	−0.145109	−0.0725545	−	0.997364i	$-0.523115\pi$
$317$	−2.58359	−0.145109	−0.0725545	+	0.997364i	$0.523115\pi$
$318$	0	0
$319$	17.8885	1.00157
$320$	0	0
$321$	−16.9443	−0.945737
$322$	0	0
$323$	20.0000	1.11283
$324$	0	0
$325$	−15.5279	−0.861331
$326$	0	0
$327$	14.9443	0.826420
$328$	0	0
$329$	−12.9443	−0.713641
$330$	0	0
$331$	−23.4164	−1.28708	−0.643541	−	0.765412i	$-0.722535\pi$
$331$	−23.4164	−1.28708	−0.643541	+	0.765412i	$0.722535\pi$
$332$	0	0
$333$	−4.47214	−0.245072
$334$	0	0
$335$	0.944272	0.0515911
$336$	0	0
$337$	−14.3607	−0.782276	−0.391138	−	0.920332i	$-0.627918\pi$
$337$	−14.3607	−0.782276	−0.391138	+	0.920332i	$0.627918\pi$
$338$	0	0
$339$	−6.18034	−0.335670
$340$	0	0
$341$	25.8885	1.40194
$342$	0	0
$343$	−11.4164	−0.616428
$344$	0	0
$345$	−1.23607	−0.0665477
$346$	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
$349$	23.5279	1.25942	0.629709	−	0.776831i	$-0.283174\pi$
$349$	23.5279	1.25942	0.629709	+	0.776831i	$0.283174\pi$
$350$	0	0
$351$	4.47214	0.238705
$352$	0	0
$353$	−17.4164	−0.926982	−0.463491	−	0.886102i	$-0.653403\pi$
$353$	−17.4164	−0.926982	−0.463491	+	0.886102i	$0.653403\pi$
$354$	0	0
$355$	9.88854	0.524829
$356$	0	0
$357$	−8.94427	−0.473381
$358$	0	0
$359$	6.47214	0.341586	0.170793	−	0.985307i	$-0.445367\pi$
$359$	6.47214	0.341586	0.170793	+	0.985307i	$0.445367\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−8.58359	−0.449286
$366$	0	0
$367$	25.7082	1.34196	0.670979	−	0.741477i	$-0.265874\pi$
$367$	25.7082	1.34196	0.670979	+	0.741477i	$0.265874\pi$
$368$	0	0
$369$	−10.9443	−0.569736
$370$	0	0
$371$	16.9443	0.879703
$372$	0	0
$373$	−20.4721	−1.06001	−0.530004	−	0.847995i	$-0.677809\pi$
$373$	−20.4721	−1.06001	−0.530004	+	0.847995i	$0.677809\pi$
$374$	0	0
$375$	10.4721	0.540779
$376$	0	0
$377$	−20.0000	−1.03005
$378$	0	0
$379$	−30.0689	−1.54453	−0.772267	−	0.635298i	$-0.780877\pi$
$379$	−30.0689	−1.54453	−0.772267	+	0.635298i	$0.780877\pi$
$380$	0	0
$381$	1.52786	0.0782748
$382$	0	0
$383$	25.5279	1.30441	0.652206	−	0.758041i	$-0.273844\pi$
$383$	25.5279	1.30441	0.652206	+	0.758041i	$0.273844\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	5.70820	0.290164
$388$	0	0
$389$	−20.6525	−1.04712	−0.523561	−	0.851988i	$-0.675397\pi$
$389$	−20.6525	−1.04712	−0.523561	+	0.851988i	$0.675397\pi$
$390$	0	0
$391$	−2.76393	−0.139778
$392$	0	0
$393$	−16.9443	−0.854725
$394$	0	0
$395$	−12.0000	−0.603786
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	−23.4164	−1.17229
$400$	0	0
$401$	20.0689	1.00219	0.501096	−	0.865392i	$-0.332930\pi$
$401$	20.0689	1.00219	0.501096	+	0.865392i	$0.332930\pi$
$402$	0	0
$403$	−28.9443	−1.44182
$404$	0	0
$405$	−1.23607	−0.0614207
$406$	0	0
$407$	17.8885	0.886702
$408$	0	0
$409$	17.4164	0.861186	0.430593	−	0.902546i	$-0.358304\pi$
$409$	17.4164	0.861186	0.430593	+	0.902546i	$0.358304\pi$
$410$	0	0
$411$	−1.23607	−0.0609707
$412$	0	0
$413$	16.0000	0.787309
$414$	0	0
$415$	4.94427	0.242705
$416$	0	0
$417$	8.94427	0.438003
$418$	0	0
$419$	−13.8885	−0.678500	−0.339250	−	0.940696i	$-0.610173\pi$
$419$	−13.8885	−0.678500	−0.339250	+	0.940696i	$0.610173\pi$
$420$	0	0
$421$	30.9443	1.50813	0.754066	−	0.656799i	$-0.228090\pi$
$421$	30.9443	1.50813	0.754066	+	0.656799i	$0.228090\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	9.59675	0.465511
$426$	0	0
$427$	−14.4721	−0.700356
$428$	0	0
$429$	−17.8885	−0.863667
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	27.8885	1.34024	0.670119	−	0.742254i	$-0.266243\pi$
$433$	27.8885	1.34024	0.670119	+	0.742254i	$0.266243\pi$
$434$	0	0
$435$	5.52786	0.265041
$436$	0	0
$437$	−7.23607	−0.346148
$438$	0	0
$439$	−9.88854	−0.471954	−0.235977	−	0.971759i	$-0.575829\pi$
$439$	−9.88854	−0.471954	−0.235977	+	0.971759i	$0.575829\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	0	0
$443$	34.8328	1.65496	0.827479	−	0.561497i	$-0.189775\pi$
$443$	34.8328	1.65496	0.827479	+	0.561497i	$0.189775\pi$
$444$	0	0
$445$	1.52786	0.0724277
$446$	0	0
$447$	11.7082	0.553779
$448$	0	0
$449$	14.9443	0.705264	0.352632	−	0.935762i	$-0.385287\pi$
$449$	14.9443	0.705264	0.352632	+	0.935762i	$0.385287\pi$
$450$	0	0
$451$	43.7771	2.06138
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	−17.8885	−0.838628
$456$	0	0
$457$	8.47214	0.396310	0.198155	−	0.980171i	$-0.436505\pi$
$457$	8.47214	0.396310	0.198155	+	0.980171i	$0.436505\pi$
$458$	0	0
$459$	−2.76393	−0.129009
$460$	0	0
$461$	5.41641	0.252267	0.126134	−	0.992013i	$-0.459743\pi$
$461$	5.41641	0.252267	0.126134	+	0.992013i	$0.459743\pi$
$462$	0	0
$463$	−12.9443	−0.601571	−0.300786	−	0.953692i	$-0.597249\pi$
$463$	−12.9443	−0.601571	−0.300786	+	0.953692i	$0.597249\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	−20.3607	−0.942180	−0.471090	−	0.882085i	$-0.656139\pi$
$467$	−20.3607	−0.942180	−0.471090	+	0.882085i	$0.656139\pi$
$468$	0	0
$469$	−2.47214	−0.114153
$470$	0	0
$471$	3.52786	0.162555
$472$	0	0
$473$	−22.8328	−1.04985
$474$	0	0
$475$	25.1246	1.15280
$476$	0	0
$477$	5.23607	0.239743
$478$	0	0
$479$	−33.8885	−1.54841	−0.774204	−	0.632937i	$-0.781849\pi$
$479$	−33.8885	−1.54841	−0.774204	+	0.632937i	$0.781849\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	0	0
$483$	3.23607	0.147246
$484$	0	0
$485$	−10.4721	−0.475515
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	0	0
$489$	−7.41641	−0.335382
$490$	0	0
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	0	0
$493$	12.3607	0.556697
$494$	0	0
$495$	4.94427	0.222228
$496$	0	0
$497$	−25.8885	−1.16126
$498$	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
$501$	12.9443	0.578307
$502$	0	0
$503$	−19.0557	−0.849653	−0.424826	−	0.905275i	$-0.639665\pi$
$503$	−19.0557	−0.849653	−0.424826	+	0.905275i	$0.639665\pi$
$504$	0	0
$505$	−8.58359	−0.381965
$506$	0	0
$507$	7.00000	0.310881
$508$	0	0
$509$	16.4721	0.730115	0.365057	−	0.930985i	$-0.381049\pi$
$509$	16.4721	0.730115	0.365057	+	0.930985i	$0.381049\pi$
$510$	0	0
$511$	22.4721	0.994109
$512$	0	0
$513$	−7.23607	−0.319480
$514$	0	0
$515$	13.8885	0.612002
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	−4.47214	−0.196305
$520$	0	0
$521$	−33.2361	−1.45610	−0.728049	−	0.685525i	$-0.759573\pi$
$521$	−33.2361	−1.45610	−0.728049	+	0.685525i	$0.759573\pi$
$522$	0	0
$523$	−15.5967	−0.681998	−0.340999	−	0.940064i	$-0.610765\pi$
$523$	−15.5967	−0.681998	−0.340999	+	0.940064i	$0.610765\pi$
$524$	0	0
$525$	−11.2361	−0.490382
$526$	0	0
$527$	17.8885	0.779237
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.94427	0.214563
$532$	0	0
$533$	−48.9443	−2.12001
$534$	0	0
$535$	20.9443	0.905500
$536$	0	0
$537$	−20.9443	−0.903812
$538$	0	0
$539$	−13.8885	−0.598222
$540$	0	0
$541$	15.5279	0.667595	0.333798	−	0.942645i	$-0.391670\pi$
$541$	15.5279	0.667595	0.333798	+	0.942645i	$0.391670\pi$
$542$	0	0
$543$	11.8885	0.510186
$544$	0	0
$545$	−18.4721	−0.791259
$546$	0	0
$547$	−13.5279	−0.578410	−0.289205	−	0.957267i	$-0.593391\pi$
$547$	−13.5279	−0.578410	−0.289205	+	0.957267i	$0.593391\pi$
$548$	0	0
$549$	−4.47214	−0.190866
$550$	0	0
$551$	32.3607	1.37861
$552$	0	0
$553$	31.4164	1.33596
$554$	0	0
$555$	5.52786	0.234645
$556$	0	0
$557$	42.1803	1.78724	0.893619	−	0.448826i	$-0.148158\pi$
$557$	42.1803	1.78724	0.893619	+	0.448826i	$0.148158\pi$
$558$	0	0
$559$	25.5279	1.07971
$560$	0	0
$561$	11.0557	0.466773
$562$	0	0
$563$	7.41641	0.312564	0.156282	−	0.987712i	$-0.450049\pi$
$563$	7.41641	0.312564	0.156282	+	0.987712i	$0.450049\pi$
$564$	0	0
$565$	7.63932	0.321389
$566$	0	0
$567$	3.23607	0.135902
$568$	0	0
$569$	−10.7639	−0.451248	−0.225624	−	0.974215i	$-0.572442\pi$
$569$	−10.7639	−0.451248	−0.225624	+	0.974215i	$0.572442\pi$
$570$	0	0
$571$	−29.7082	−1.24325	−0.621625	−	0.783315i	$-0.713527\pi$
$571$	−29.7082	−1.24325	−0.621625	+	0.783315i	$0.713527\pi$
$572$	0	0
$573$	1.52786	0.0638274
$574$	0	0
$575$	−3.47214	−0.144798
$576$	0	0
$577$	7.52786	0.313389	0.156695	−	0.987647i	$-0.449916\pi$
$577$	7.52786	0.313389	0.156695	+	0.987647i	$0.449916\pi$
$578$	0	0
$579$	17.4164	0.723801
$580$	0	0
$581$	−12.9443	−0.537019
$582$	0	0
$583$	−20.9443	−0.867423
$584$	0	0
$585$	−5.52786	−0.228549
$586$	0	0
$587$	0.944272	0.0389743	0.0194871	−	0.999810i	$-0.493797\pi$
$587$	0.944272	0.0389743	0.0194871	+	0.999810i	$0.493797\pi$
$588$	0	0
$589$	46.8328	1.92971
$590$	0	0
$591$	−17.4164	−0.716415
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	11.0557	0.453241
$596$	0	0
$597$	9.70820	0.397330
$598$	0	0
$599$	−3.05573	−0.124854	−0.0624268	−	0.998050i	$-0.519884\pi$
$599$	−3.05573	−0.124854	−0.0624268	+	0.998050i	$0.519884\pi$
$600$	0	0
$601$	−42.3607	−1.72793	−0.863964	−	0.503553i	$-0.832026\pi$
$601$	−42.3607	−1.72793	−0.863964	+	0.503553i	$0.832026\pi$
$602$	0	0
$603$	−0.763932	−0.0311097
$604$	0	0
$605$	−6.18034	−0.251267
$606$	0	0
$607$	−14.8328	−0.602045	−0.301023	−	0.953617i	$-0.597328\pi$
$607$	−14.8328	−0.602045	−0.301023	+	0.953617i	$0.597328\pi$
$608$	0	0
$609$	−14.4721	−0.586441
$610$	0	0
$611$	−17.8885	−0.723693
$612$	0	0
$613$	40.4721	1.63465	0.817327	−	0.576174i	$-0.195455\pi$
$613$	40.4721	1.63465	0.817327	+	0.576174i	$0.195455\pi$
$614$	0	0
$615$	13.5279	0.545496
$616$	0	0
$617$	−20.2918	−0.816917	−0.408458	−	0.912777i	$-0.633934\pi$
$617$	−20.2918	−0.816917	−0.408458	+	0.912777i	$0.633934\pi$
$618$	0	0
$619$	9.12461	0.366749	0.183375	−	0.983043i	$-0.441298\pi$
$619$	9.12461	0.366749	0.183375	+	0.983043i	$0.441298\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−4.00000	−0.160257
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	28.9443	1.15592
$628$	0	0
$629$	12.3607	0.492853
$630$	0	0
$631$	−16.1803	−0.644129	−0.322065	−	0.946718i	$-0.604377\pi$
$631$	−16.1803	−0.644129	−0.322065	+	0.946718i	$0.604377\pi$
$632$	0	0
$633$	−13.5279	−0.537684
$634$	0	0
$635$	−1.88854	−0.0749446
$636$	0	0
$637$	15.5279	0.615236
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	49.0132	1.93590	0.967952	−	0.251137i	$-0.0808044\pi$
$641$	49.0132	1.93590	0.967952	+	0.251137i	$0.0808044\pi$
$642$	0	0
$643$	−38.0689	−1.50129	−0.750645	−	0.660706i	$-0.770257\pi$
$643$	−38.0689	−1.50129	−0.750645	+	0.660706i	$0.770257\pi$
$644$	0	0
$645$	−7.05573	−0.277819
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−19.7771	−0.776319
$650$	0	0
$651$	−20.9443	−0.820871
$652$	0	0
$653$	22.9443	0.897879	0.448939	−	0.893562i	$-0.351802\pi$
$653$	22.9443	0.897879	0.448939	+	0.893562i	$0.351802\pi$
$654$	0	0
$655$	20.9443	0.818360
$656$	0	0
$657$	6.94427	0.270922
$658$	0	0
$659$	−36.3607	−1.41641	−0.708205	−	0.706006i	$-0.750495\pi$
$659$	−36.3607	−1.41641	−0.708205	+	0.706006i	$0.750495\pi$
$660$	0	0
$661$	11.5279	0.448382	0.224191	−	0.974545i	$-0.428026\pi$
$661$	11.5279	0.448382	0.224191	+	0.974545i	$0.428026\pi$
$662$	0	0
$663$	−12.3607	−0.480049
$664$	0	0
$665$	28.9443	1.12241
$666$	0	0
$667$	−4.47214	−0.173162
$668$	0	0
$669$	−25.8885	−1.00091
$670$	0	0
$671$	17.8885	0.690580
$672$	0	0
$673$	2.58359	0.0995902	0.0497951	−	0.998759i	$-0.484143\pi$
$673$	2.58359	0.0995902	0.0497951	+	0.998759i	$0.484143\pi$
$674$	0	0
$675$	−3.47214	−0.133643
$676$	0	0
$677$	47.4853	1.82501	0.912504	−	0.409068i	$-0.134146\pi$
$677$	47.4853	1.82501	0.912504	+	0.409068i	$0.134146\pi$
$678$	0	0
$679$	27.4164	1.05215
$680$	0	0
$681$	13.5279	0.518389
$682$	0	0
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	1.52786	0.0583767
$686$	0	0
$687$	−2.94427	−0.112331
$688$	0	0
$689$	23.4164	0.892094
$690$	0	0
$691$	−4.36068	−0.165888	−0.0829440	−	0.996554i	$-0.526432\pi$
$691$	−4.36068	−0.165888	−0.0829440	+	0.996554i	$0.526432\pi$
$692$	0	0
$693$	−12.9443	−0.491712
$694$	0	0
$695$	−11.0557	−0.419368
$696$	0	0
$697$	30.2492	1.14577
$698$	0	0
$699$	−14.0000	−0.529529
$700$	0	0
$701$	−25.2361	−0.953153	−0.476577	−	0.879133i	$-0.658122\pi$
$701$	−25.2361	−0.953153	−0.476577	+	0.879133i	$0.658122\pi$
$702$	0	0
$703$	32.3607	1.22051
$704$	0	0
$705$	4.94427	0.186212
$706$	0	0
$707$	22.4721	0.845152
$708$	0	0
$709$	32.8328	1.23306	0.616531	−	0.787331i	$-0.288537\pi$
$709$	32.8328	1.23306	0.616531	+	0.787331i	$0.288537\pi$
$710$	0	0
$711$	9.70820	0.364086
$712$	0	0
$713$	−6.47214	−0.242383
$714$	0	0
$715$	22.1115	0.826922
$716$	0	0
$717$	4.94427	0.184647
$718$	0	0
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	0	0
$721$	−36.3607	−1.35414
$722$	0	0
$723$	19.5279	0.726249
$724$	0	0
$725$	15.5279	0.576690
$726$	0	0
$727$	1.34752	0.0499769	0.0249885	−	0.999688i	$-0.492045\pi$
$727$	1.34752	0.0499769	0.0249885	+	0.999688i	$0.492045\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−15.7771	−0.583537
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	−4.29180	−0.158305
$736$	0	0
$737$	3.05573	0.112559
$738$	0	0
$739$	26.8328	0.987061	0.493531	−	0.869728i	$-0.335706\pi$
$739$	26.8328	0.987061	0.493531	+	0.869728i	$0.335706\pi$
$740$	0	0
$741$	−32.3607	−1.18880
$742$	0	0
$743$	−24.3607	−0.893707	−0.446853	−	0.894607i	$-0.647455\pi$
$743$	−24.3607	−0.893707	−0.446853	+	0.894607i	$0.647455\pi$
$744$	0	0
$745$	−14.4721	−0.530218
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−54.8328	−2.00355
$750$	0	0
$751$	50.0689	1.82704	0.913520	−	0.406794i	$-0.133353\pi$
$751$	50.0689	1.82704	0.913520	+	0.406794i	$0.133353\pi$
$752$	0	0
$753$	10.4721	0.381626
$754$	0	0
$755$	19.7771	0.719762
$756$	0	0
$757$	−39.8885	−1.44977	−0.724887	−	0.688868i	$-0.758108\pi$
$757$	−39.8885	−1.44977	−0.724887	+	0.688868i	$0.758108\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	42.3607	1.53557	0.767787	−	0.640706i	$-0.221358\pi$
$761$	42.3607	1.53557	0.767787	+	0.640706i	$0.221358\pi$
$762$	0	0
$763$	48.3607	1.75077
$764$	0	0
$765$	3.41641	0.123520
$766$	0	0
$767$	22.1115	0.798398
$768$	0	0
$769$	16.8328	0.607007	0.303503	−	0.952830i	$-0.401844\pi$
$769$	16.8328	0.607007	0.303503	+	0.952830i	$0.401844\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	0	0
$773$	−36.2918	−1.30533	−0.652663	−	0.757649i	$-0.726348\pi$
$773$	−36.2918	−1.30533	−0.652663	+	0.757649i	$0.726348\pi$
$774$	0	0
$775$	22.4721	0.807223
$776$	0	0
$777$	−14.4721	−0.519185
$778$	0	0
$779$	79.1935	2.83740
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	−4.47214	−0.159821
$784$	0	0
$785$	−4.36068	−0.155639
$786$	0	0
$787$	13.7082	0.488645	0.244322	−	0.969694i	$-0.421434\pi$
$787$	13.7082	0.488645	0.244322	+	0.969694i	$0.421434\pi$
$788$	0	0
$789$	6.47214	0.230414
$790$	0	0
$791$	−20.0000	−0.711118
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	−6.47214	−0.229543
$796$	0	0
$797$	−12.6525	−0.448174	−0.224087	−	0.974569i	$-0.571940\pi$
$797$	−12.6525	−0.448174	−0.224087	+	0.974569i	$0.571940\pi$
$798$	0	0
$799$	11.0557	0.391124
$800$	0	0
$801$	−1.23607	−0.0436743
$802$	0	0
$803$	−27.7771	−0.980232
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0.472136	0.0166200
$808$	0	0
$809$	−43.3050	−1.52252	−0.761261	−	0.648446i	$-0.775419\pi$
$809$	−43.3050	−1.52252	−0.761261	+	0.648446i	$0.775419\pi$
$810$	0	0
$811$	23.4164	0.822261	0.411131	−	0.911576i	$-0.365134\pi$
$811$	23.4164	0.822261	0.411131	+	0.911576i	$0.365134\pi$
$812$	0	0
$813$	−17.5279	−0.614729
$814$	0	0
$815$	9.16718	0.321112
$816$	0	0
$817$	−41.3050	−1.44508
$818$	0	0
$819$	14.4721	0.505697
$820$	0	0
$821$	−12.1115	−0.422693	−0.211346	−	0.977411i	$-0.567785\pi$
$821$	−12.1115	−0.422693	−0.211346	+	0.977411i	$0.567785\pi$
$822$	0	0
$823$	−25.5279	−0.889845	−0.444923	−	0.895569i	$-0.646769\pi$
$823$	−25.5279	−0.889845	−0.444923	+	0.895569i	$0.646769\pi$
$824$	0	0
$825$	13.8885	0.483537
$826$	0	0
$827$	−8.58359	−0.298481	−0.149240	−	0.988801i	$-0.547683\pi$
$827$	−8.58359	−0.298481	−0.149240	+	0.988801i	$0.547683\pi$
$828$	0	0
$829$	−10.3607	−0.359841	−0.179921	−	0.983681i	$-0.557584\pi$
$829$	−10.3607	−0.359841	−0.179921	+	0.983681i	$0.557584\pi$
$830$	0	0
$831$	−15.8885	−0.551167
$832$	0	0
$833$	−9.59675	−0.332508
$834$	0	0
$835$	−16.0000	−0.553703
$836$	0	0
$837$	−6.47214	−0.223710
$838$	0	0
$839$	−12.5836	−0.434434	−0.217217	−	0.976123i	$-0.569698\pi$
$839$	−12.5836	−0.434434	−0.217217	+	0.976123i	$0.569698\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	24.6525	0.849076
$844$	0	0
$845$	−8.65248	−0.297654
$846$	0	0
$847$	16.1803	0.555963
$848$	0	0
$849$	−5.70820	−0.195905
$850$	0	0
$851$	−4.47214	−0.153303
$852$	0	0
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	0	0
$855$	8.94427	0.305888
$856$	0	0
$857$	−42.9443	−1.46695	−0.733474	−	0.679717i	$-0.762102\pi$
$857$	−42.9443	−1.46695	−0.733474	+	0.679717i	$0.762102\pi$
$858$	0	0
$859$	7.05573	0.240738	0.120369	−	0.992729i	$-0.461592\pi$
$859$	7.05573	0.240738	0.120369	+	0.992729i	$0.461592\pi$
$860$	0	0
$861$	−35.4164	−1.20699
$862$	0	0
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	5.52786	0.187953
$866$	0	0
$867$	−9.36068	−0.317905
$868$	0	0
$869$	−38.8328	−1.31731
$870$	0	0
$871$	−3.41641	−0.115761
$872$	0	0
$873$	8.47214	0.286738
$874$	0	0
$875$	33.8885	1.14564
$876$	0	0
$877$	−33.0557	−1.11621	−0.558106	−	0.829769i	$-0.688472\pi$
$877$	−33.0557	−1.11621	−0.558106	+	0.829769i	$0.688472\pi$
$878$	0	0
$879$	−7.70820	−0.259991
$880$	0	0
$881$	26.5410	0.894190	0.447095	−	0.894487i	$-0.352459\pi$
$881$	26.5410	0.894190	0.447095	+	0.894487i	$0.352459\pi$
$882$	0	0
$883$	7.41641	0.249582	0.124791	−	0.992183i	$-0.460174\pi$
$883$	7.41641	0.249582	0.124791	+	0.992183i	$0.460174\pi$
$884$	0	0
$885$	−6.11146	−0.205434
$886$	0	0
$887$	31.0557	1.04275	0.521375	−	0.853328i	$-0.325419\pi$
$887$	31.0557	1.04275	0.521375	+	0.853328i	$0.325419\pi$
$888$	0	0
$889$	4.94427	0.165826
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	28.9443	0.968583
$894$	0	0
$895$	25.8885	0.865359
$896$	0	0
$897$	4.47214	0.149320
$898$	0	0
$899$	28.9443	0.965346
$900$	0	0
$901$	−14.4721	−0.482137
$902$	0	0
$903$	18.4721	0.614714
$904$	0	0
$905$	−14.6950	−0.488480
$906$	0	0
$907$	1.12461	0.0373421	0.0186711	−	0.999826i	$-0.494056\pi$
$907$	1.12461	0.0373421	0.0186711	+	0.999826i	$0.494056\pi$
$908$	0	0
$909$	6.94427	0.230327
$910$	0	0
$911$	−40.7214	−1.34916	−0.674579	−	0.738202i	$-0.735675\pi$
$911$	−40.7214	−1.34916	−0.674579	+	0.738202i	$0.735675\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	0	0
$915$	5.52786	0.182746
$916$	0	0
$917$	−54.8328	−1.81074
$918$	0	0
$919$	−23.0132	−0.759134	−0.379567	−	0.925164i	$-0.623927\pi$
$919$	−23.0132	−0.759134	−0.379567	+	0.925164i	$0.623927\pi$
$920$	0	0
$921$	−2.47214	−0.0814596
$922$	0	0
$923$	−35.7771	−1.17762
$924$	0	0
$925$	15.5279	0.510553
$926$	0	0
$927$	−11.2361	−0.369041
$928$	0	0
$929$	−28.8328	−0.945974	−0.472987	−	0.881069i	$-0.656824\pi$
$929$	−28.8328	−0.945974	−0.472987	+	0.881069i	$0.656824\pi$
$930$	0	0
$931$	−25.1246	−0.823426
$932$	0	0
$933$	7.05573	0.230994
$934$	0	0
$935$	−13.6656	−0.446914
$936$	0	0
$937$	−12.8328	−0.419230	−0.209615	−	0.977784i	$-0.567221\pi$
$937$	−12.8328	−0.419230	−0.209615	+	0.977784i	$0.567221\pi$
$938$	0	0
$939$	5.05573	0.164987
$940$	0	0
$941$	42.5410	1.38680	0.693399	−	0.720554i	$-0.256112\pi$
$941$	42.5410	1.38680	0.693399	+	0.720554i	$0.256112\pi$
$942$	0	0
$943$	−10.9443	−0.356395
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	1.16718	0.0379284	0.0189642	−	0.999820i	$-0.493963\pi$
$947$	1.16718	0.0379284	0.0189642	+	0.999820i	$0.493963\pi$
$948$	0	0
$949$	31.0557	1.00811
$950$	0	0
$951$	−2.58359	−0.0837787
$952$	0	0
$953$	12.0689	0.390949	0.195475	−	0.980709i	$-0.437375\pi$
$953$	12.0689	0.390949	0.195475	+	0.980709i	$0.437375\pi$
$954$	0	0
$955$	−1.88854	−0.0611118
$956$	0	0
$957$	17.8885	0.578254
$958$	0	0
$959$	−4.00000	−0.129167
$960$	0	0
$961$	10.8885	0.351243
$962$	0	0
$963$	−16.9443	−0.546022
$964$	0	0
$965$	−21.5279	−0.693006
$966$	0	0
$967$	−7.63932	−0.245664	−0.122832	−	0.992427i	$-0.539198\pi$
$967$	−7.63932	−0.245664	−0.122832	+	0.992427i	$0.539198\pi$
$968$	0	0
$969$	20.0000	0.642493
$970$	0	0
$971$	−17.3050	−0.555342	−0.277671	−	0.960676i	$-0.589563\pi$
$971$	−17.3050	−0.555342	−0.277671	+	0.960676i	$0.589563\pi$
$972$	0	0
$973$	28.9443	0.927911
$974$	0	0
$975$	−15.5279	−0.497290
$976$	0	0
$977$	−32.0689	−1.02597	−0.512987	−	0.858396i	$-0.671461\pi$
$977$	−32.0689	−1.02597	−0.512987	+	0.858396i	$0.671461\pi$
$978$	0	0
$979$	4.94427	0.158020
$980$	0	0
$981$	14.9443	0.477134
$982$	0	0
$983$	22.8328	0.728254	0.364127	−	0.931349i	$-0.381367\pi$
$983$	22.8328	0.728254	0.364127	+	0.931349i	$0.381367\pi$
$984$	0	0
$985$	21.5279	0.685935
$986$	0	0
$987$	−12.9443	−0.412021
$988$	0	0
$989$	5.70820	0.181510
$990$	0	0
$991$	−1.52786	−0.0485342	−0.0242671	−	0.999706i	$-0.507725\pi$
$991$	−1.52786	−0.0485342	−0.0242671	+	0.999706i	$0.507725\pi$
$992$	0	0
$993$	−23.4164	−0.743097
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	−42.3607	−1.34158	−0.670788	−	0.741649i	$-0.734044\pi$
$997$	−42.3607	−1.34158	−0.670788	+	0.741649i	$0.734044\pi$
$998$	0	0
$999$	−4.47214	−0.141492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4416.2.a.bm.1.1		2
4.3	odd	2		4416.2.a.bg.1.1		2
8.3	odd	2		1104.2.a.m.1.2		2
8.5	even	2		69.2.a.b.1.1	✓	2
24.5	odd	2		207.2.a.c.1.2		2
24.11	even	2		3312.2.a.bb.1.1		2
40.13	odd	4		1725.2.b.o.1174.3		4
40.29	even	2		1725.2.a.ba.1.2		2
40.37	odd	4		1725.2.b.o.1174.2		4
56.13	odd	2		3381.2.a.t.1.1		2
88.21	odd	2		8349.2.a.i.1.2		2
120.29	odd	2		5175.2.a.bk.1.1		2
184.45	odd	2		1587.2.a.i.1.1		2
552.413	even	2		4761.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.a.b.1.1	✓	2	8.5	even	2
207.2.a.c.1.2		2	24.5	odd	2
1104.2.a.m.1.2		2	8.3	odd	2
1587.2.a.i.1.1		2	184.45	odd	2
1725.2.a.ba.1.2		2	40.29	even	2
1725.2.b.o.1174.2		4	40.37	odd	4
1725.2.b.o.1174.3		4	40.13	odd	4
3312.2.a.bb.1.1		2	24.11	even	2
3381.2.a.t.1.1		2	56.13	odd	2
4416.2.a.bg.1.1		2	4.3	odd	2
4416.2.a.bm.1.1		2	1.1	even	1	trivial
4761.2.a.v.1.2		2	552.413	even	2
5175.2.a.bk.1.1		2	120.29	odd	2
8349.2.a.i.1.2		2	88.21	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4416 = 2^{6} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4416.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.23607 q^{5} +3.23607 q^{7} +1.00000 q^{9} -4.00000 q^{11} +4.47214 q^{13} -1.23607 q^{15} -2.76393 q^{17} -7.23607 q^{19} +3.23607 q^{21} +1.00000 q^{23} -3.47214 q^{25} +1.00000 q^{27} -4.47214 q^{29} -6.47214 q^{31} -4.00000 q^{33} -4.00000 q^{35} -4.47214 q^{37} +4.47214 q^{39} -10.9443 q^{41} +5.70820 q^{43} -1.23607 q^{45} -4.00000 q^{47} +3.47214 q^{49} -2.76393 q^{51} +5.23607 q^{53} +4.94427 q^{55} -7.23607 q^{57} +4.94427 q^{59} -4.47214 q^{61} +3.23607 q^{63} -5.52786 q^{65} -0.763932 q^{67} +1.00000 q^{69} -8.00000 q^{71} +6.94427 q^{73} -3.47214 q^{75} -12.9443 q^{77} +9.70820 q^{79} +1.00000 q^{81} -4.00000 q^{83} +3.41641 q^{85} -4.47214 q^{87} -1.23607 q^{89} +14.4721 q^{91} -6.47214 q^{93} +8.94427 q^{95} +8.47214 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{11} + 2 q^{15} - 10 q^{17} - 10 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{31} - 8 q^{33} - 8 q^{35} - 4 q^{41} - 2 q^{43} + 2 q^{45} - 8 q^{47}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 8 * q^11 + 2 * q^15 - 10 * q^17 - 10 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^31 - 8 * q^33 - 8 * q^35 - 4 * q^41 - 2 * q^43 + 2 * q^45 - 8 * q^47 - 2 * q^49 - 10 * q^51 + 6 * q^53 - 8 * q^55 - 10 * q^57 - 8 * q^59 + 2 * q^63 - 20 * q^65 - 6 * q^67 + 2 * q^69 - 16 * q^71 - 4 * q^73 + 2 * q^75 - 8 * q^77 + 6 * q^79 + 2 * q^81 - 8 * q^83 - 20 * q^85 + 2 * q^89 + 20 * q^91 - 4 * q^93 + 8 * q^97 - 8 * q^99

Introduction

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Modular forms

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Database

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