LMFDB - Embedded newform 7728.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +0.618034 q^{13} -3.61803 q^{15} -5.47214 q^{17} -4.23607 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.09017 q^{25} -1.00000 q^{27} +1.76393 q^{29} +8.70820 q^{31} +1.00000 q^{33} -3.61803 q^{35} +0.236068 q^{37} -0.618034 q^{39} -3.47214 q^{41} +3.85410 q^{43} +3.61803 q^{45} -11.7082 q^{47} +1.00000 q^{49} +5.47214 q^{51} -0.0901699 q^{53} -3.61803 q^{55} +4.23607 q^{57} +3.61803 q^{59} -7.85410 q^{61} -1.00000 q^{63} +2.23607 q^{65} +8.09017 q^{67} +1.00000 q^{69} +10.3262 q^{71} +1.76393 q^{73} -8.09017 q^{75} +1.00000 q^{77} +14.2361 q^{79} +1.00000 q^{81} +17.9443 q^{83} -19.7984 q^{85} -1.76393 q^{87} +13.5623 q^{89} -0.618034 q^{91} -8.70820 q^{93} -15.3262 q^{95} +6.70820 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 5 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 5 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 5 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - q^{13} - 5 q^{15} - 2 q^{17} - 4 q^{19} + 2 q^{21} - 2 q^{23} + 5 q^{25} - 2 q^{27} + 8 q^{29} + 4 q^{31} + 2 q^{33} - 5 q^{35} - 4 q^{37} + q^{39} + 2 q^{41} + q^{43} + 5 q^{45} - 10 q^{47} + 2 q^{49} + 2 q^{51} + 11 q^{53} - 5 q^{55} + 4 q^{57} + 5 q^{59} - 9 q^{61} - 2 q^{63} + 5 q^{67} + 2 q^{69} + 5 q^{71} + 8 q^{73} - 5 q^{75} + 2 q^{77} + 24 q^{79} + 2 q^{81} + 18 q^{83} - 15 q^{85} - 8 q^{87} + 7 q^{89} + q^{91} - 4 q^{93} - 15 q^{95} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 5 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 - q^13 - 5 * q^15 - 2 * q^17 - 4 * q^19 + 2 * q^21 - 2 * q^23 + 5 * q^25 - 2 * q^27 + 8 * q^29 + 4 * q^31 + 2 * q^33 - 5 * q^35 - 4 * q^37 + q^39 + 2 * q^41 + q^43 + 5 * q^45 - 10 * q^47 + 2 * q^49 + 2 * q^51 + 11 * q^53 - 5 * q^55 + 4 * q^57 + 5 * q^59 - 9 * q^61 - 2 * q^63 + 5 * q^67 + 2 * q^69 + 5 * q^71 + 8 * q^73 - 5 * q^75 + 2 * q^77 + 24 * q^79 + 2 * q^81 + 18 * q^83 - 15 * q^85 - 8 * q^87 + 7 * q^89 + q^91 - 4 * q^93 - 15 * q^95 - 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
$5$	3.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$5$	3.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	0.618034	0.171412	0.0857059	−	0.996320i	$-0.472685\pi$
$13$	0.618034	0.171412	0.0857059	+	0.996320i	$0.472685\pi$
$14$	0	0
$15$	−3.61803	−0.934172
$16$	0	0
$17$	−5.47214	−1.32719	−0.663594	−	0.748093i	$-0.730970\pi$
$17$	−5.47214	−1.32719	−0.663594	+	0.748093i	$0.730970\pi$
$18$	0	0
$19$	−4.23607	−0.971821	−0.485910	−	0.874009i	$-0.661512\pi$
$19$	−4.23607	−0.971821	−0.485910	+	0.874009i	$0.661512\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	8.09017	1.61803
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.76393	0.327554	0.163777	−	0.986497i	$-0.447632\pi$
$29$	1.76393	0.327554	0.163777	+	0.986497i	$0.447632\pi$
$30$	0	0
$31$	8.70820	1.56404	0.782020	−	0.623254i	$-0.214190\pi$
$31$	8.70820	1.56404	0.782020	+	0.623254i	$0.214190\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−3.61803	−0.611559
$36$	0	0
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	0	0
$39$	−0.618034	−0.0989646
$40$	0	0
$41$	−3.47214	−0.542257	−0.271128	−	0.962543i	$-0.587397\pi$
$41$	−3.47214	−0.542257	−0.271128	+	0.962543i	$0.587397\pi$
$42$	0	0
$43$	3.85410	0.587745	0.293873	−	0.955845i	$-0.405056\pi$
$43$	3.85410	0.587745	0.293873	+	0.955845i	$0.405056\pi$
$44$	0	0
$45$	3.61803	0.539345
$46$	0	0
$47$	−11.7082	−1.70782	−0.853909	−	0.520423i	$-0.825774\pi$
$47$	−11.7082	−1.70782	−0.853909	+	0.520423i	$0.825774\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.47214	0.766252
$52$	0	0
$53$	−0.0901699	−0.0123858	−0.00619290	−	0.999981i	$-0.501971\pi$
$53$	−0.0901699	−0.0123858	−0.00619290	+	0.999981i	$0.501971\pi$
$54$	0	0
$55$	−3.61803	−0.487856
$56$	0	0
$57$	4.23607	0.561081
$58$	0	0
$59$	3.61803	0.471028	0.235514	−	0.971871i	$-0.424323\pi$
$59$	3.61803	0.471028	0.235514	+	0.971871i	$0.424323\pi$
$60$	0	0
$61$	−7.85410	−1.00561	−0.502807	−	0.864398i	$-0.667699\pi$
$61$	−7.85410	−1.00561	−0.502807	+	0.864398i	$0.667699\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	2.23607	0.277350
$66$	0	0
$67$	8.09017	0.988372	0.494186	−	0.869356i	$-0.335466\pi$
$67$	8.09017	0.988372	0.494186	+	0.869356i	$0.335466\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	10.3262	1.22550	0.612749	−	0.790277i	$-0.290063\pi$
$71$	10.3262	1.22550	0.612749	+	0.790277i	$0.290063\pi$
$72$	0	0
$73$	1.76393	0.206453	0.103226	−	0.994658i	$-0.467083\pi$
$73$	1.76393	0.206453	0.103226	+	0.994658i	$0.467083\pi$
$74$	0	0
$75$	−8.09017	−0.934172
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	14.2361	1.60168	0.800841	−	0.598877i	$-0.204386\pi$
$79$	14.2361	1.60168	0.800841	+	0.598877i	$0.204386\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	17.9443	1.96964	0.984820	−	0.173579i	$-0.0555334\pi$
$83$	17.9443	1.96964	0.984820	+	0.173579i	$0.0555334\pi$
$84$	0	0
$85$	−19.7984	−2.14744
$86$	0	0
$87$	−1.76393	−0.189113
$88$	0	0
$89$	13.5623	1.43760	0.718801	−	0.695216i	$-0.244691\pi$
$89$	13.5623	1.43760	0.718801	+	0.695216i	$0.244691\pi$
$90$	0	0
$91$	−0.618034	−0.0647876
$92$	0	0
$93$	−8.70820	−0.902999
$94$	0	0
$95$	−15.3262	−1.57244
$96$	0	0
$97$	6.70820	0.681115	0.340557	−	0.940224i	$-0.389384\pi$
$97$	6.70820	0.681115	0.340557	+	0.940224i	$0.389384\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	2.09017	0.207980	0.103990	−	0.994578i	$-0.466839\pi$
$101$	2.09017	0.207980	0.103990	+	0.994578i	$0.466839\pi$
$102$	0	0
$103$	18.4721	1.82011	0.910057	−	0.414484i	$-0.136038\pi$
$103$	18.4721	1.82011	0.910057	+	0.414484i	$0.136038\pi$
$104$	0	0
$105$	3.61803	0.353084
$106$	0	0
$107$	2.90983	0.281304	0.140652	−	0.990059i	$-0.455080\pi$
$107$	2.90983	0.281304	0.140652	+	0.990059i	$0.455080\pi$
$108$	0	0
$109$	2.38197	0.228151	0.114075	−	0.993472i	$-0.463609\pi$
$109$	2.38197	0.228151	0.114075	+	0.993472i	$0.463609\pi$
$110$	0	0
$111$	−0.236068	−0.0224066
$112$	0	0
$113$	14.0902	1.32549	0.662746	−	0.748844i	$-0.269391\pi$
$113$	14.0902	1.32549	0.662746	+	0.748844i	$0.269391\pi$
$114$	0	0
$115$	−3.61803	−0.337383
$116$	0	0
$117$	0.618034	0.0571373
$118$	0	0
$119$	5.47214	0.501630
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	3.47214	0.313072
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	7.85410	0.696939	0.348469	−	0.937320i	$-0.386702\pi$
$127$	7.85410	0.696939	0.348469	+	0.937320i	$0.386702\pi$
$128$	0	0
$129$	−3.85410	−0.339335
$130$	0	0
$131$	3.29180	0.287606	0.143803	−	0.989606i	$-0.454067\pi$
$131$	3.29180	0.287606	0.143803	+	0.989606i	$0.454067\pi$
$132$	0	0
$133$	4.23607	0.367314
$134$	0	0
$135$	−3.61803	−0.311391
$136$	0	0
$137$	15.1803	1.29694	0.648472	−	0.761239i	$-0.275408\pi$
$137$	15.1803	1.29694	0.648472	+	0.761239i	$0.275408\pi$
$138$	0	0
$139$	−21.0344	−1.78412	−0.892059	−	0.451919i	$-0.850740\pi$
$139$	−21.0344	−1.78412	−0.892059	+	0.451919i	$0.850740\pi$
$140$	0	0
$141$	11.7082	0.986009
$142$	0	0
$143$	−0.618034	−0.0516826
$144$	0	0
$145$	6.38197	0.529993
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	3.70820	0.303788	0.151894	−	0.988397i	$-0.451463\pi$
$149$	3.70820	0.303788	0.151894	+	0.988397i	$0.451463\pi$
$150$	0	0
$151$	−8.65248	−0.704128	−0.352064	−	0.935976i	$-0.614520\pi$
$151$	−8.65248	−0.704128	−0.352064	+	0.935976i	$0.614520\pi$
$152$	0	0
$153$	−5.47214	−0.442396
$154$	0	0
$155$	31.5066	2.53067
$156$	0	0
$157$	8.76393	0.699438	0.349719	−	0.936855i	$-0.386277\pi$
$157$	8.76393	0.699438	0.349719	+	0.936855i	$0.386277\pi$
$158$	0	0
$159$	0.0901699	0.00715094
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	20.2705	1.58771	0.793854	−	0.608108i	$-0.208071\pi$
$163$	20.2705	1.58771	0.793854	+	0.608108i	$0.208071\pi$
$164$	0	0
$165$	3.61803	0.281664
$166$	0	0
$167$	−10.7082	−0.828626	−0.414313	−	0.910135i	$-0.635978\pi$
$167$	−10.7082	−0.828626	−0.414313	+	0.910135i	$0.635978\pi$
$168$	0	0
$169$	−12.6180	−0.970618
$170$	0	0
$171$	−4.23607	−0.323940
$172$	0	0
$173$	−6.41641	−0.487830	−0.243915	−	0.969797i	$-0.578432\pi$
$173$	−6.41641	−0.487830	−0.243915	+	0.969797i	$0.578432\pi$
$174$	0	0
$175$	−8.09017	−0.611559
$176$	0	0
$177$	−3.61803	−0.271948
$178$	0	0
$179$	13.7984	1.03134	0.515669	−	0.856788i	$-0.327543\pi$
$179$	13.7984	1.03134	0.515669	+	0.856788i	$0.327543\pi$
$180$	0	0
$181$	−3.18034	−0.236393	−0.118196	−	0.992990i	$-0.537711\pi$
$181$	−3.18034	−0.236393	−0.118196	+	0.992990i	$0.537711\pi$
$182$	0	0
$183$	7.85410	0.580592
$184$	0	0
$185$	0.854102	0.0627948
$186$	0	0
$187$	5.47214	0.400162
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−14.1803	−1.02605	−0.513027	−	0.858373i	$-0.671476\pi$
$191$	−14.1803	−1.02605	−0.513027	+	0.858373i	$0.671476\pi$
$192$	0	0
$193$	0.763932	0.0549890	0.0274945	−	0.999622i	$-0.491247\pi$
$193$	0.763932	0.0549890	0.0274945	+	0.999622i	$0.491247\pi$
$194$	0	0
$195$	−2.23607	−0.160128
$196$	0	0
$197$	11.5623	0.823780	0.411890	−	0.911234i	$-0.364869\pi$
$197$	11.5623	0.823780	0.411890	+	0.911234i	$0.364869\pi$
$198$	0	0
$199$	−1.43769	−0.101915	−0.0509577	−	0.998701i	$-0.516227\pi$
$199$	−1.43769	−0.101915	−0.0509577	+	0.998701i	$0.516227\pi$
$200$	0	0
$201$	−8.09017	−0.570637
$202$	0	0
$203$	−1.76393	−0.123804
$204$	0	0
$205$	−12.5623	−0.877390
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	4.23607	0.293015
$210$	0	0
$211$	−8.88854	−0.611913	−0.305956	−	0.952046i	$-0.598976\pi$
$211$	−8.88854	−0.611913	−0.305956	+	0.952046i	$0.598976\pi$
$212$	0	0
$213$	−10.3262	−0.707542
$214$	0	0
$215$	13.9443	0.950991
$216$	0	0
$217$	−8.70820	−0.591151
$218$	0	0
$219$	−1.76393	−0.119195
$220$	0	0
$221$	−3.38197	−0.227496
$222$	0	0
$223$	−1.61803	−0.108352	−0.0541758	−	0.998531i	$-0.517253\pi$
$223$	−1.61803	−0.108352	−0.0541758	+	0.998531i	$0.517253\pi$
$224$	0	0
$225$	8.09017	0.539345
$226$	0	0
$227$	−23.9787	−1.59152	−0.795762	−	0.605610i	$-0.792929\pi$
$227$	−23.9787	−1.59152	−0.795762	+	0.605610i	$0.792929\pi$
$228$	0	0
$229$	27.7984	1.83697	0.918484	−	0.395458i	$-0.129414\pi$
$229$	27.7984	1.83697	0.918484	+	0.395458i	$0.129414\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	8.67376	0.568237	0.284119	−	0.958789i	$-0.408299\pi$
$233$	8.67376	0.568237	0.284119	+	0.958789i	$0.408299\pi$
$234$	0	0
$235$	−42.3607	−2.76331
$236$	0	0
$237$	−14.2361	−0.924732
$238$	0	0
$239$	−15.7984	−1.02191	−0.510956	−	0.859607i	$-0.670708\pi$
$239$	−15.7984	−1.02191	−0.510956	+	0.859607i	$0.670708\pi$
$240$	0	0
$241$	−3.29180	−0.212043	−0.106022	−	0.994364i	$-0.533811\pi$
$241$	−3.29180	−0.212043	−0.106022	+	0.994364i	$0.533811\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.61803	0.231148
$246$	0	0
$247$	−2.61803	−0.166582
$248$	0	0
$249$	−17.9443	−1.13717
$250$	0	0
$251$	−17.7082	−1.11773	−0.558866	−	0.829258i	$-0.688763\pi$
$251$	−17.7082	−1.11773	−0.558866	+	0.829258i	$0.688763\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	0	0
$255$	19.7984	1.23982
$256$	0	0
$257$	23.7082	1.47888	0.739439	−	0.673224i	$-0.235091\pi$
$257$	23.7082	1.47888	0.739439	+	0.673224i	$0.235091\pi$
$258$	0	0
$259$	−0.236068	−0.0146686
$260$	0	0
$261$	1.76393	0.109185
$262$	0	0
$263$	−21.6525	−1.33515	−0.667574	−	0.744543i	$-0.732667\pi$
$263$	−21.6525	−1.33515	−0.667574	+	0.744543i	$0.732667\pi$
$264$	0	0
$265$	−0.326238	−0.0200406
$266$	0	0
$267$	−13.5623	−0.830000
$268$	0	0
$269$	15.7426	0.959846	0.479923	−	0.877311i	$-0.340665\pi$
$269$	15.7426	0.959846	0.479923	+	0.877311i	$0.340665\pi$
$270$	0	0
$271$	15.4721	0.939865	0.469933	−	0.882702i	$-0.344278\pi$
$271$	15.4721	0.939865	0.469933	+	0.882702i	$0.344278\pi$
$272$	0	0
$273$	0.618034	0.0374051
$274$	0	0
$275$	−8.09017	−0.487856
$276$	0	0
$277$	−24.3262	−1.46162	−0.730811	−	0.682580i	$-0.760858\pi$
$277$	−24.3262	−1.46162	−0.730811	+	0.682580i	$0.760858\pi$
$278$	0	0
$279$	8.70820	0.521347
$280$	0	0
$281$	−14.6525	−0.874093	−0.437047	−	0.899439i	$-0.643976\pi$
$281$	−14.6525	−0.874093	−0.437047	+	0.899439i	$0.643976\pi$
$282$	0	0
$283$	−25.3262	−1.50549	−0.752744	−	0.658313i	$-0.771270\pi$
$283$	−25.3262	−1.50549	−0.752744	+	0.658313i	$0.771270\pi$
$284$	0	0
$285$	15.3262	0.907848
$286$	0	0
$287$	3.47214	0.204954
$288$	0	0
$289$	12.9443	0.761428
$290$	0	0
$291$	−6.70820	−0.393242
$292$	0	0
$293$	−22.8328	−1.33391	−0.666954	−	0.745099i	$-0.732402\pi$
$293$	−22.8328	−1.33391	−0.666954	+	0.745099i	$0.732402\pi$
$294$	0	0
$295$	13.0902	0.762139
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−0.618034	−0.0357418
$300$	0	0
$301$	−3.85410	−0.222147
$302$	0	0
$303$	−2.09017	−0.120077
$304$	0	0
$305$	−28.4164	−1.62712
$306$	0	0
$307$	34.1246	1.94759	0.973797	−	0.227418i	$-0.0730284\pi$
$307$	34.1246	1.94759	0.973797	+	0.227418i	$0.0730284\pi$
$308$	0	0
$309$	−18.4721	−1.05084
$310$	0	0
$311$	10.5066	0.595773	0.297887	−	0.954601i	$-0.403718\pi$
$311$	10.5066	0.595773	0.297887	+	0.954601i	$0.403718\pi$
$312$	0	0
$313$	33.8885	1.91549	0.957747	−	0.287612i	$-0.0928615\pi$
$313$	33.8885	1.91549	0.957747	+	0.287612i	$0.0928615\pi$
$314$	0	0
$315$	−3.61803	−0.203853
$316$	0	0
$317$	−0.0901699	−0.00506445	−0.00253222	−	0.999997i	$-0.500806\pi$
$317$	−0.0901699	−0.00506445	−0.00253222	+	0.999997i	$0.500806\pi$
$318$	0	0
$319$	−1.76393	−0.0987612
$320$	0	0
$321$	−2.90983	−0.162411
$322$	0	0
$323$	23.1803	1.28979
$324$	0	0
$325$	5.00000	0.277350
$326$	0	0
$327$	−2.38197	−0.131723
$328$	0	0
$329$	11.7082	0.645494
$330$	0	0
$331$	−7.88854	−0.433594	−0.216797	−	0.976217i	$-0.569561\pi$
$331$	−7.88854	−0.433594	−0.216797	+	0.976217i	$0.569561\pi$
$332$	0	0
$333$	0.236068	0.0129364
$334$	0	0
$335$	29.2705	1.59922
$336$	0	0
$337$	10.0344	0.546611	0.273305	−	0.961927i	$-0.411883\pi$
$337$	10.0344	0.546611	0.273305	+	0.961927i	$0.411883\pi$
$338$	0	0
$339$	−14.0902	−0.765273
$340$	0	0
$341$	−8.70820	−0.471576
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.61803	0.194788
$346$	0	0
$347$	7.29180	0.391444	0.195722	−	0.980659i	$-0.437295\pi$
$347$	7.29180	0.391444	0.195722	+	0.980659i	$0.437295\pi$
$348$	0	0
$349$	−19.7984	−1.05978	−0.529891	−	0.848066i	$-0.677767\pi$
$349$	−19.7984	−1.05978	−0.529891	+	0.848066i	$0.677767\pi$
$350$	0	0
$351$	−0.618034	−0.0329882
$352$	0	0
$353$	6.41641	0.341511	0.170755	−	0.985313i	$-0.445379\pi$
$353$	6.41641	0.341511	0.170755	+	0.985313i	$0.445379\pi$
$354$	0	0
$355$	37.3607	1.98290
$356$	0	0
$357$	−5.47214	−0.289616
$358$	0	0
$359$	−8.56231	−0.451901	−0.225951	−	0.974139i	$-0.572549\pi$
$359$	−8.56231	−0.451901	−0.225951	+	0.974139i	$0.572549\pi$
$360$	0	0
$361$	−1.05573	−0.0555646
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	6.38197	0.334047
$366$	0	0
$367$	−13.2705	−0.692715	−0.346357	−	0.938103i	$-0.612582\pi$
$367$	−13.2705	−0.692715	−0.346357	+	0.938103i	$0.612582\pi$
$368$	0	0
$369$	−3.47214	−0.180752
$370$	0	0
$371$	0.0901699	0.00468139
$372$	0	0
$373$	34.1246	1.76691	0.883453	−	0.468520i	$-0.155213\pi$
$373$	34.1246	1.76691	0.883453	+	0.468520i	$0.155213\pi$
$374$	0	0
$375$	−11.1803	−0.577350
$376$	0	0
$377$	1.09017	0.0561466
$378$	0	0
$379$	25.5279	1.31128	0.655639	−	0.755074i	$-0.272399\pi$
$379$	25.5279	1.31128	0.655639	+	0.755074i	$0.272399\pi$
$380$	0	0
$381$	−7.85410	−0.402378
$382$	0	0
$383$	3.94427	0.201543	0.100771	−	0.994910i	$-0.467869\pi$
$383$	3.94427	0.201543	0.100771	+	0.994910i	$0.467869\pi$
$384$	0	0
$385$	3.61803	0.184392
$386$	0	0
$387$	3.85410	0.195915
$388$	0	0
$389$	−19.9443	−1.01121	−0.505607	−	0.862764i	$-0.668732\pi$
$389$	−19.9443	−1.01121	−0.505607	+	0.862764i	$0.668732\pi$
$390$	0	0
$391$	5.47214	0.276738
$392$	0	0
$393$	−3.29180	−0.166049
$394$	0	0
$395$	51.5066	2.59158
$396$	0	0
$397$	22.1803	1.11320	0.556600	−	0.830781i	$-0.312106\pi$
$397$	22.1803	1.11320	0.556600	+	0.830781i	$0.312106\pi$
$398$	0	0
$399$	−4.23607	−0.212069
$400$	0	0
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	0	0
$403$	5.38197	0.268095
$404$	0	0
$405$	3.61803	0.179782
$406$	0	0
$407$	−0.236068	−0.0117015
$408$	0	0
$409$	−24.8885	−1.23066	−0.615330	−	0.788270i	$-0.710977\pi$
$409$	−24.8885	−1.23066	−0.615330	+	0.788270i	$0.710977\pi$
$410$	0	0
$411$	−15.1803	−0.748791
$412$	0	0
$413$	−3.61803	−0.178032
$414$	0	0
$415$	64.9230	3.18694
$416$	0	0
$417$	21.0344	1.03006
$418$	0	0
$419$	−13.9098	−0.679540	−0.339770	−	0.940509i	$-0.610349\pi$
$419$	−13.9098	−0.679540	−0.339770	+	0.940509i	$0.610349\pi$
$420$	0	0
$421$	−29.5623	−1.44078	−0.720389	−	0.693570i	$-0.756037\pi$
$421$	−29.5623	−1.44078	−0.720389	+	0.693570i	$0.756037\pi$
$422$	0	0
$423$	−11.7082	−0.569272
$424$	0	0
$425$	−44.2705	−2.14744
$426$	0	0
$427$	7.85410	0.380087
$428$	0	0
$429$	0.618034	0.0298390
$430$	0	0
$431$	−11.2705	−0.542881	−0.271441	−	0.962455i	$-0.587500\pi$
$431$	−11.2705	−0.542881	−0.271441	+	0.962455i	$0.587500\pi$
$432$	0	0
$433$	20.7639	0.997851	0.498925	−	0.866645i	$-0.333728\pi$
$433$	20.7639	0.997851	0.498925	+	0.866645i	$0.333728\pi$
$434$	0	0
$435$	−6.38197	−0.305992
$436$	0	0
$437$	4.23607	0.202639
$438$	0	0
$439$	−9.18034	−0.438154	−0.219077	−	0.975708i	$-0.570305\pi$
$439$	−9.18034	−0.438154	−0.219077	+	0.975708i	$0.570305\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	26.8328	1.27487	0.637433	−	0.770506i	$-0.279996\pi$
$443$	26.8328	1.27487	0.637433	+	0.770506i	$0.279996\pi$
$444$	0	0
$445$	49.0689	2.32609
$446$	0	0
$447$	−3.70820	−0.175392
$448$	0	0
$449$	40.2705	1.90048	0.950241	−	0.311514i	$-0.100836\pi$
$449$	40.2705	1.90048	0.950241	+	0.311514i	$0.100836\pi$
$450$	0	0
$451$	3.47214	0.163496
$452$	0	0
$453$	8.65248	0.406529
$454$	0	0
$455$	−2.23607	−0.104828
$456$	0	0
$457$	−2.74265	−0.128296	−0.0641478	−	0.997940i	$-0.520433\pi$
$457$	−2.74265	−0.128296	−0.0641478	+	0.997940i	$0.520433\pi$
$458$	0	0
$459$	5.47214	0.255417
$460$	0	0
$461$	−5.32624	−0.248068	−0.124034	−	0.992278i	$-0.539583\pi$
$461$	−5.32624	−0.248068	−0.124034	+	0.992278i	$0.539583\pi$
$462$	0	0
$463$	−14.1246	−0.656426	−0.328213	−	0.944604i	$-0.606446\pi$
$463$	−14.1246	−0.656426	−0.328213	+	0.944604i	$0.606446\pi$
$464$	0	0
$465$	−31.5066	−1.46108
$466$	0	0
$467$	32.8885	1.52190	0.760950	−	0.648810i	$-0.224733\pi$
$467$	32.8885	1.52190	0.760950	+	0.648810i	$0.224733\pi$
$468$	0	0
$469$	−8.09017	−0.373569
$470$	0	0
$471$	−8.76393	−0.403821
$472$	0	0
$473$	−3.85410	−0.177212
$474$	0	0
$475$	−34.2705	−1.57244
$476$	0	0
$477$	−0.0901699	−0.00412860
$478$	0	0
$479$	22.7082	1.03756	0.518782	−	0.854906i	$-0.326386\pi$
$479$	22.7082	1.03756	0.518782	+	0.854906i	$0.326386\pi$
$480$	0	0
$481$	0.145898	0.00665238
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	24.2705	1.10207
$486$	0	0
$487$	30.1246	1.36508	0.682538	−	0.730850i	$-0.260876\pi$
$487$	30.1246	1.36508	0.682538	+	0.730850i	$0.260876\pi$
$488$	0	0
$489$	−20.2705	−0.916664
$490$	0	0
$491$	2.43769	0.110012	0.0550058	−	0.998486i	$-0.482482\pi$
$491$	2.43769	0.110012	0.0550058	+	0.998486i	$0.482482\pi$
$492$	0	0
$493$	−9.65248	−0.434726
$494$	0	0
$495$	−3.61803	−0.162619
$496$	0	0
$497$	−10.3262	−0.463195
$498$	0	0
$499$	−16.5623	−0.741431	−0.370715	−	0.928747i	$-0.620887\pi$
$499$	−16.5623	−0.741431	−0.370715	+	0.928747i	$0.620887\pi$
$500$	0	0
$501$	10.7082	0.478407
$502$	0	0
$503$	25.6869	1.14532	0.572662	−	0.819792i	$-0.305911\pi$
$503$	25.6869	1.14532	0.572662	+	0.819792i	$0.305911\pi$
$504$	0	0
$505$	7.56231	0.336518
$506$	0	0
$507$	12.6180	0.560387
$508$	0	0
$509$	30.7639	1.36359	0.681794	−	0.731545i	$-0.261200\pi$
$509$	30.7639	1.36359	0.681794	+	0.731545i	$0.261200\pi$
$510$	0	0
$511$	−1.76393	−0.0780318
$512$	0	0
$513$	4.23607	0.187027
$514$	0	0
$515$	66.8328	2.94501
$516$	0	0
$517$	11.7082	0.514926
$518$	0	0
$519$	6.41641	0.281649
$520$	0	0
$521$	−21.4164	−0.938270	−0.469135	−	0.883127i	$-0.655434\pi$
$521$	−21.4164	−0.938270	−0.469135	+	0.883127i	$0.655434\pi$
$522$	0	0
$523$	10.5836	0.462788	0.231394	−	0.972860i	$-0.425671\pi$
$523$	10.5836	0.462788	0.231394	+	0.972860i	$0.425671\pi$
$524$	0	0
$525$	8.09017	0.353084
$526$	0	0
$527$	−47.6525	−2.07577
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.61803	0.157009
$532$	0	0
$533$	−2.14590	−0.0929492
$534$	0	0
$535$	10.5279	0.455159
$536$	0	0
$537$	−13.7984	−0.595444
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−15.7082	−0.675348	−0.337674	−	0.941263i	$-0.609640\pi$
$541$	−15.7082	−0.675348	−0.337674	+	0.941263i	$0.609640\pi$
$542$	0	0
$543$	3.18034	0.136481
$544$	0	0
$545$	8.61803	0.369156
$546$	0	0
$547$	38.8541	1.66128	0.830641	−	0.556809i	$-0.187974\pi$
$547$	38.8541	1.66128	0.830641	+	0.556809i	$0.187974\pi$
$548$	0	0
$549$	−7.85410	−0.335205
$550$	0	0
$551$	−7.47214	−0.318324
$552$	0	0
$553$	−14.2361	−0.605379
$554$	0	0
$555$	−0.854102	−0.0362546
$556$	0	0
$557$	12.7639	0.540825	0.270413	−	0.962745i	$-0.412840\pi$
$557$	12.7639	0.540825	0.270413	+	0.962745i	$0.412840\pi$
$558$	0	0
$559$	2.38197	0.100746
$560$	0	0
$561$	−5.47214	−0.231034
$562$	0	0
$563$	−36.2148	−1.52627	−0.763136	−	0.646238i	$-0.776341\pi$
$563$	−36.2148	−1.52627	−0.763136	+	0.646238i	$0.776341\pi$
$564$	0	0
$565$	50.9787	2.14469
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	25.8885	1.08530	0.542652	−	0.839958i	$-0.317420\pi$
$569$	25.8885	1.08530	0.542652	+	0.839958i	$0.317420\pi$
$570$	0	0
$571$	24.5967	1.02934	0.514671	−	0.857388i	$-0.327914\pi$
$571$	24.5967	1.02934	0.514671	+	0.857388i	$0.327914\pi$
$572$	0	0
$573$	14.1803	0.592392
$574$	0	0
$575$	−8.09017	−0.337383
$576$	0	0
$577$	−40.2492	−1.67560	−0.837799	−	0.545979i	$-0.816158\pi$
$577$	−40.2492	−1.67560	−0.837799	+	0.545979i	$0.816158\pi$
$578$	0	0
$579$	−0.763932	−0.0317479
$580$	0	0
$581$	−17.9443	−0.744454
$582$	0	0
$583$	0.0901699	0.00373446
$584$	0	0
$585$	2.23607	0.0924500
$586$	0	0
$587$	−44.7984	−1.84903	−0.924513	−	0.381150i	$-0.875528\pi$
$587$	−44.7984	−1.84903	−0.924513	+	0.381150i	$0.875528\pi$
$588$	0	0
$589$	−36.8885	−1.51997
$590$	0	0
$591$	−11.5623	−0.475610
$592$	0	0
$593$	−45.1246	−1.85305	−0.926523	−	0.376238i	$-0.877217\pi$
$593$	−45.1246	−1.85305	−0.926523	+	0.376238i	$0.877217\pi$
$594$	0	0
$595$	19.7984	0.811654
$596$	0	0
$597$	1.43769	0.0588409
$598$	0	0
$599$	7.38197	0.301619	0.150809	−	0.988563i	$-0.451812\pi$
$599$	7.38197	0.301619	0.150809	+	0.988563i	$0.451812\pi$
$600$	0	0
$601$	31.6869	1.29254	0.646268	−	0.763110i	$-0.276329\pi$
$601$	31.6869	1.29254	0.646268	+	0.763110i	$0.276329\pi$
$602$	0	0
$603$	8.09017	0.329457
$604$	0	0
$605$	−36.1803	−1.47094
$606$	0	0
$607$	−6.50658	−0.264094	−0.132047	−	0.991243i	$-0.542155\pi$
$607$	−6.50658	−0.264094	−0.132047	+	0.991243i	$0.542155\pi$
$608$	0	0
$609$	1.76393	0.0714781
$610$	0	0
$611$	−7.23607	−0.292740
$612$	0	0
$613$	−21.8328	−0.881819	−0.440910	−	0.897552i	$-0.645344\pi$
$613$	−21.8328	−0.881819	−0.440910	+	0.897552i	$0.645344\pi$
$614$	0	0
$615$	12.5623	0.506561
$616$	0	0
$617$	−7.74265	−0.311707	−0.155854	−	0.987780i	$-0.549813\pi$
$617$	−7.74265	−0.311707	−0.155854	+	0.987780i	$0.549813\pi$
$618$	0	0
$619$	−4.90983	−0.197343	−0.0986714	−	0.995120i	$-0.531459\pi$
$619$	−4.90983	−0.197343	−0.0986714	+	0.995120i	$0.531459\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−13.5623	−0.543362
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.23607	−0.169172
$628$	0	0
$629$	−1.29180	−0.0515073
$630$	0	0
$631$	−5.00000	−0.199047	−0.0995234	−	0.995035i	$-0.531732\pi$
$631$	−5.00000	−0.199047	−0.0995234	+	0.995035i	$0.531732\pi$
$632$	0	0
$633$	8.88854	0.353288
$634$	0	0
$635$	28.4164	1.12767
$636$	0	0
$637$	0.618034	0.0244874
$638$	0	0
$639$	10.3262	0.408500
$640$	0	0
$641$	−24.7984	−0.979477	−0.489738	−	0.871869i	$-0.662908\pi$
$641$	−24.7984	−0.979477	−0.489738	+	0.871869i	$0.662908\pi$
$642$	0	0
$643$	−3.96556	−0.156386	−0.0781932	−	0.996938i	$-0.524915\pi$
$643$	−3.96556	−0.156386	−0.0781932	+	0.996938i	$0.524915\pi$
$644$	0	0
$645$	−13.9443	−0.549055
$646$	0	0
$647$	−40.2705	−1.58320	−0.791599	−	0.611042i	$-0.790751\pi$
$647$	−40.2705	−1.58320	−0.791599	+	0.611042i	$0.790751\pi$
$648$	0	0
$649$	−3.61803	−0.142020
$650$	0	0
$651$	8.70820	0.341301
$652$	0	0
$653$	4.09017	0.160061	0.0800304	−	0.996792i	$-0.474498\pi$
$653$	4.09017	0.160061	0.0800304	+	0.996792i	$0.474498\pi$
$654$	0	0
$655$	11.9098	0.465356
$656$	0	0
$657$	1.76393	0.0688175
$658$	0	0
$659$	35.9443	1.40019	0.700095	−	0.714050i	$-0.253141\pi$
$659$	35.9443	1.40019	0.700095	+	0.714050i	$0.253141\pi$
$660$	0	0
$661$	23.6525	0.919975	0.459987	−	0.887925i	$-0.347854\pi$
$661$	23.6525	0.919975	0.459987	+	0.887925i	$0.347854\pi$
$662$	0	0
$663$	3.38197	0.131345
$664$	0	0
$665$	15.3262	0.594326
$666$	0	0
$667$	−1.76393	−0.0682997
$668$	0	0
$669$	1.61803	0.0625568
$670$	0	0
$671$	7.85410	0.303204
$672$	0	0
$673$	−34.4164	−1.32666	−0.663328	−	0.748329i	$-0.730856\pi$
$673$	−34.4164	−1.32666	−0.663328	+	0.748329i	$0.730856\pi$
$674$	0	0
$675$	−8.09017	−0.311391
$676$	0	0
$677$	34.3262	1.31926	0.659632	−	0.751589i	$-0.270712\pi$
$677$	34.3262	1.31926	0.659632	+	0.751589i	$0.270712\pi$
$678$	0	0
$679$	−6.70820	−0.257437
$680$	0	0
$681$	23.9787	0.918866
$682$	0	0
$683$	−14.1246	−0.540463	−0.270232	−	0.962795i	$-0.587100\pi$
$683$	−14.1246	−0.540463	−0.270232	+	0.962795i	$0.587100\pi$
$684$	0	0
$685$	54.9230	2.09850
$686$	0	0
$687$	−27.7984	−1.06057
$688$	0	0
$689$	−0.0557281	−0.00212307
$690$	0	0
$691$	20.6180	0.784347	0.392173	−	0.919891i	$-0.371723\pi$
$691$	20.6180	0.784347	0.392173	+	0.919891i	$0.371723\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	−76.1033	−2.88676
$696$	0	0
$697$	19.0000	0.719676
$698$	0	0
$699$	−8.67376	−0.328072
$700$	0	0
$701$	8.09017	0.305562	0.152781	−	0.988260i	$-0.451177\pi$
$701$	8.09017	0.305562	0.152781	+	0.988260i	$0.451177\pi$
$702$	0	0
$703$	−1.00000	−0.0377157
$704$	0	0
$705$	42.3607	1.59540
$706$	0	0
$707$	−2.09017	−0.0786089
$708$	0	0
$709$	8.72949	0.327843	0.163921	−	0.986473i	$-0.447586\pi$
$709$	8.72949	0.327843	0.163921	+	0.986473i	$0.447586\pi$
$710$	0	0
$711$	14.2361	0.533894
$712$	0	0
$713$	−8.70820	−0.326125
$714$	0	0
$715$	−2.23607	−0.0836242
$716$	0	0
$717$	15.7984	0.590001
$718$	0	0
$719$	−32.8885	−1.22654	−0.613268	−	0.789875i	$-0.710145\pi$
$719$	−32.8885	−1.22654	−0.613268	+	0.789875i	$0.710145\pi$
$720$	0	0
$721$	−18.4721	−0.687938
$722$	0	0
$723$	3.29180	0.122423
$724$	0	0
$725$	14.2705	0.529993
$726$	0	0
$727$	−51.5410	−1.91155	−0.955775	−	0.294098i	$-0.904981\pi$
$727$	−51.5410	−1.91155	−0.955775	+	0.294098i	$0.904981\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−21.0902	−0.780048
$732$	0	0
$733$	47.2361	1.74470	0.872352	−	0.488878i	$-0.162594\pi$
$733$	47.2361	1.74470	0.872352	+	0.488878i	$0.162594\pi$
$734$	0	0
$735$	−3.61803	−0.133453
$736$	0	0
$737$	−8.09017	−0.298005
$738$	0	0
$739$	31.4721	1.15772	0.578861	−	0.815427i	$-0.303498\pi$
$739$	31.4721	1.15772	0.578861	+	0.815427i	$0.303498\pi$
$740$	0	0
$741$	2.61803	0.0961759
$742$	0	0
$743$	−15.6180	−0.572970	−0.286485	−	0.958085i	$-0.592487\pi$
$743$	−15.6180	−0.572970	−0.286485	+	0.958085i	$0.592487\pi$
$744$	0	0
$745$	13.4164	0.491539
$746$	0	0
$747$	17.9443	0.656547
$748$	0	0
$749$	−2.90983	−0.106323
$750$	0	0
$751$	−10.4508	−0.381357	−0.190678	−	0.981653i	$-0.561069\pi$
$751$	−10.4508	−0.381357	−0.190678	+	0.981653i	$0.561069\pi$
$752$	0	0
$753$	17.7082	0.645323
$754$	0	0
$755$	−31.3050	−1.13930
$756$	0	0
$757$	53.6525	1.95003	0.975016	−	0.222134i	$-0.0713022\pi$
$757$	53.6525	1.95003	0.975016	+	0.222134i	$0.0713022\pi$
$758$	0	0
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	41.8885	1.51846	0.759229	−	0.650823i	$-0.225576\pi$
$761$	41.8885	1.51846	0.759229	+	0.650823i	$0.225576\pi$
$762$	0	0
$763$	−2.38197	−0.0862330
$764$	0	0
$765$	−19.7984	−0.715812
$766$	0	0
$767$	2.23607	0.0807397
$768$	0	0
$769$	−10.6525	−0.384138	−0.192069	−	0.981381i	$-0.561520\pi$
$769$	−10.6525	−0.384138	−0.192069	+	0.981381i	$0.561520\pi$
$770$	0	0
$771$	−23.7082	−0.853830
$772$	0	0
$773$	−20.5967	−0.740814	−0.370407	−	0.928870i	$-0.620782\pi$
$773$	−20.5967	−0.740814	−0.370407	+	0.928870i	$0.620782\pi$
$774$	0	0
$775$	70.4508	2.53067
$776$	0	0
$777$	0.236068	0.00846889
$778$	0	0
$779$	14.7082	0.526976
$780$	0	0
$781$	−10.3262	−0.369502
$782$	0	0
$783$	−1.76393	−0.0630378
$784$	0	0
$785$	31.7082	1.13171
$786$	0	0
$787$	−13.7984	−0.491859	−0.245929	−	0.969288i	$-0.579093\pi$
$787$	−13.7984	−0.491859	−0.245929	+	0.969288i	$0.579093\pi$
$788$	0	0
$789$	21.6525	0.770849
$790$	0	0
$791$	−14.0902	−0.500989
$792$	0	0
$793$	−4.85410	−0.172374
$794$	0	0
$795$	0.326238	0.0115705
$796$	0	0
$797$	7.12461	0.252367	0.126183	−	0.992007i	$-0.459727\pi$
$797$	7.12461	0.252367	0.126183	+	0.992007i	$0.459727\pi$
$798$	0	0
$799$	64.0689	2.26659
$800$	0	0
$801$	13.5623	0.479201
$802$	0	0
$803$	−1.76393	−0.0622478
$804$	0	0
$805$	3.61803	0.127519
$806$	0	0
$807$	−15.7426	−0.554167
$808$	0	0
$809$	−6.72949	−0.236596	−0.118298	−	0.992978i	$-0.537744\pi$
$809$	−6.72949	−0.236596	−0.118298	+	0.992978i	$0.537744\pi$
$810$	0	0
$811$	12.3050	0.432085	0.216043	−	0.976384i	$-0.430685\pi$
$811$	12.3050	0.432085	0.216043	+	0.976384i	$0.430685\pi$
$812$	0	0
$813$	−15.4721	−0.542631
$814$	0	0
$815$	73.3394	2.56897
$816$	0	0
$817$	−16.3262	−0.571183
$818$	0	0
$819$	−0.618034	−0.0215959
$820$	0	0
$821$	−30.5410	−1.06589	−0.532944	−	0.846150i	$-0.678915\pi$
$821$	−30.5410	−1.06589	−0.532944	+	0.846150i	$0.678915\pi$
$822$	0	0
$823$	−43.2148	−1.50637	−0.753186	−	0.657807i	$-0.771484\pi$
$823$	−43.2148	−1.50637	−0.753186	+	0.657807i	$0.771484\pi$
$824$	0	0
$825$	8.09017	0.281664
$826$	0	0
$827$	0.854102	0.0297000	0.0148500	−	0.999890i	$-0.495273\pi$
$827$	0.854102	0.0297000	0.0148500	+	0.999890i	$0.495273\pi$
$828$	0	0
$829$	38.7771	1.34678	0.673392	−	0.739286i	$-0.264837\pi$
$829$	38.7771	1.34678	0.673392	+	0.739286i	$0.264837\pi$
$830$	0	0
$831$	24.3262	0.843868
$832$	0	0
$833$	−5.47214	−0.189598
$834$	0	0
$835$	−38.7426	−1.34074
$836$	0	0
$837$	−8.70820	−0.301000
$838$	0	0
$839$	0.978714	0.0337890	0.0168945	−	0.999857i	$-0.494622\pi$
$839$	0.978714	0.0337890	0.0168945	+	0.999857i	$0.494622\pi$
$840$	0	0
$841$	−25.8885	−0.892708
$842$	0	0
$843$	14.6525	0.504658
$844$	0	0
$845$	−45.6525	−1.57049
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	25.3262	0.869194
$850$	0	0
$851$	−0.236068	−0.00809231
$852$	0	0
$853$	22.0689	0.755624	0.377812	−	0.925882i	$-0.376677\pi$
$853$	22.0689	0.755624	0.377812	+	0.925882i	$0.376677\pi$
$854$	0	0
$855$	−15.3262	−0.524146
$856$	0	0
$857$	1.12461	0.0384160	0.0192080	−	0.999816i	$-0.493886\pi$
$857$	1.12461	0.0384160	0.0192080	+	0.999816i	$0.493886\pi$
$858$	0	0
$859$	1.63932	0.0559329	0.0279664	−	0.999609i	$-0.491097\pi$
$859$	1.63932	0.0559329	0.0279664	+	0.999609i	$0.491097\pi$
$860$	0	0
$861$	−3.47214	−0.118330
$862$	0	0
$863$	−7.81966	−0.266184	−0.133092	−	0.991104i	$-0.542491\pi$
$863$	−7.81966	−0.266184	−0.133092	+	0.991104i	$0.542491\pi$
$864$	0	0
$865$	−23.2148	−0.789326
$866$	0	0
$867$	−12.9443	−0.439611
$868$	0	0
$869$	−14.2361	−0.482926
$870$	0	0
$871$	5.00000	0.169419
$872$	0	0
$873$	6.70820	0.227038
$874$	0	0
$875$	−11.1803	−0.377964
$876$	0	0
$877$	20.0000	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$877$	20.0000	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$878$	0	0
$879$	22.8328	0.770132
$880$	0	0
$881$	−34.2918	−1.15532	−0.577660	−	0.816277i	$-0.696034\pi$
$881$	−34.2918	−1.15532	−0.577660	+	0.816277i	$0.696034\pi$
$882$	0	0
$883$	−4.90983	−0.165229	−0.0826145	−	0.996582i	$-0.526327\pi$
$883$	−4.90983	−0.165229	−0.0826145	+	0.996582i	$0.526327\pi$
$884$	0	0
$885$	−13.0902	−0.440021
$886$	0	0
$887$	−3.02129	−0.101445	−0.0507224	−	0.998713i	$-0.516152\pi$
$887$	−3.02129	−0.101445	−0.0507224	+	0.998713i	$0.516152\pi$
$888$	0	0
$889$	−7.85410	−0.263418
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	49.5967	1.65969
$894$	0	0
$895$	49.9230	1.66874
$896$	0	0
$897$	0.618034	0.0206356
$898$	0	0
$899$	15.3607	0.512307
$900$	0	0
$901$	0.493422	0.0164383
$902$	0	0
$903$	3.85410	0.128256
$904$	0	0
$905$	−11.5066	−0.382492
$906$	0	0
$907$	2.27051	0.0753910	0.0376955	−	0.999289i	$-0.487998\pi$
$907$	2.27051	0.0753910	0.0376955	+	0.999289i	$0.487998\pi$
$908$	0	0
$909$	2.09017	0.0693266
$910$	0	0
$911$	41.0132	1.35883	0.679413	−	0.733756i	$-0.262234\pi$
$911$	41.0132	1.35883	0.679413	+	0.733756i	$0.262234\pi$
$912$	0	0
$913$	−17.9443	−0.593869
$914$	0	0
$915$	28.4164	0.939417
$916$	0	0
$917$	−3.29180	−0.108705
$918$	0	0
$919$	42.1935	1.39183	0.695917	−	0.718122i	$-0.254998\pi$
$919$	42.1935	1.39183	0.695917	+	0.718122i	$0.254998\pi$
$920$	0	0
$921$	−34.1246	−1.12444
$922$	0	0
$923$	6.38197	0.210065
$924$	0	0
$925$	1.90983	0.0627948
$926$	0	0
$927$	18.4721	0.606705
$928$	0	0
$929$	−34.9230	−1.14579	−0.572893	−	0.819630i	$-0.694179\pi$
$929$	−34.9230	−1.14579	−0.572893	+	0.819630i	$0.694179\pi$
$930$	0	0
$931$	−4.23607	−0.138832
$932$	0	0
$933$	−10.5066	−0.343970
$934$	0	0
$935$	19.7984	0.647476
$936$	0	0
$937$	21.3607	0.697823	0.348911	−	0.937156i	$-0.386551\pi$
$937$	21.3607	0.697823	0.348911	+	0.937156i	$0.386551\pi$
$938$	0	0
$939$	−33.8885	−1.10591
$940$	0	0
$941$	−0.360680	−0.0117578	−0.00587891	−	0.999983i	$-0.501871\pi$
$941$	−0.360680	−0.0117578	−0.00587891	+	0.999983i	$0.501871\pi$
$942$	0	0
$943$	3.47214	0.113068
$944$	0	0
$945$	3.61803	0.117695
$946$	0	0
$947$	26.5836	0.863851	0.431925	−	0.901909i	$-0.357834\pi$
$947$	26.5836	0.863851	0.431925	+	0.901909i	$0.357834\pi$
$948$	0	0
$949$	1.09017	0.0353884
$950$	0	0
$951$	0.0901699	0.00292396
$952$	0	0
$953$	12.9787	0.420422	0.210211	−	0.977656i	$-0.432585\pi$
$953$	12.9787	0.420422	0.210211	+	0.977656i	$0.432585\pi$
$954$	0	0
$955$	−51.3050	−1.66019
$956$	0	0
$957$	1.76393	0.0570198
$958$	0	0
$959$	−15.1803	−0.490199
$960$	0	0
$961$	44.8328	1.44622
$962$	0	0
$963$	2.90983	0.0937680
$964$	0	0
$965$	2.76393	0.0889741
$966$	0	0
$967$	−27.8885	−0.896835	−0.448418	−	0.893824i	$-0.648012\pi$
$967$	−27.8885	−0.896835	−0.448418	+	0.893824i	$0.648012\pi$
$968$	0	0
$969$	−23.1803	−0.744660
$970$	0	0
$971$	−50.0344	−1.60568	−0.802841	−	0.596193i	$-0.796679\pi$
$971$	−50.0344	−1.60568	−0.802841	+	0.596193i	$0.796679\pi$
$972$	0	0
$973$	21.0344	0.674333
$974$	0	0
$975$	−5.00000	−0.160128
$976$	0	0
$977$	−12.2148	−0.390785	−0.195393	−	0.980725i	$-0.562598\pi$
$977$	−12.2148	−0.390785	−0.195393	+	0.980725i	$0.562598\pi$
$978$	0	0
$979$	−13.5623	−0.433453
$980$	0	0
$981$	2.38197	0.0760503
$982$	0	0
$983$	−5.00000	−0.159475	−0.0797376	−	0.996816i	$-0.525408\pi$
$983$	−5.00000	−0.159475	−0.0797376	+	0.996816i	$0.525408\pi$
$984$	0	0
$985$	41.8328	1.33290
$986$	0	0
$987$	−11.7082	−0.372676
$988$	0	0
$989$	−3.85410	−0.122553
$990$	0	0
$991$	50.6180	1.60793	0.803967	−	0.594673i	$-0.202719\pi$
$991$	50.6180	1.60793	0.803967	+	0.594673i	$0.202719\pi$
$992$	0	0
$993$	7.88854	0.250335
$994$	0	0
$995$	−5.20163	−0.164903
$996$	0	0
$997$	40.3050	1.27647	0.638235	−	0.769841i	$-0.279665\pi$
$997$	40.3050	1.27647	0.638235	+	0.769841i	$0.279665\pi$
$998$	0	0
$999$	−0.236068	−0.00746886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.be.1.2		2
4.3	odd	2		483.2.a.e.1.2	✓	2
12.11	even	2		1449.2.a.g.1.1		2
28.27	even	2		3381.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.e.1.2	✓	2	4.3	odd	2
1449.2.a.g.1.1		2	12.11	even	2
3381.2.a.o.1.2		2	28.27	even	2
7728.2.a.be.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +0.618034 q^{13} -3.61803 q^{15} -5.47214 q^{17} -4.23607 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.09017 q^{25} -1.00000 q^{27} +1.76393 q^{29} +8.70820 q^{31} +1.00000 q^{33} -3.61803 q^{35} +0.236068 q^{37} -0.618034 q^{39} -3.47214 q^{41} +3.85410 q^{43} +3.61803 q^{45} -11.7082 q^{47} +1.00000 q^{49} +5.47214 q^{51} -0.0901699 q^{53} -3.61803 q^{55} +4.23607 q^{57} +3.61803 q^{59} -7.85410 q^{61} -1.00000 q^{63} +2.23607 q^{65} +8.09017 q^{67} +1.00000 q^{69} +10.3262 q^{71} +1.76393 q^{73} -8.09017 q^{75} +1.00000 q^{77} +14.2361 q^{79} +1.00000 q^{81} +17.9443 q^{83} -19.7984 q^{85} -1.76393 q^{87} +13.5623 q^{89} -0.618034 q^{91} -8.70820 q^{93} -15.3262 q^{95} +6.70820 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 5 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 5 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 5 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - q^{13} - 5 q^{15} - 2 q^{17} - 4 q^{19} + 2 q^{21} - 2 q^{23} + 5 q^{25} - 2 q^{27} + 8 q^{29} + 4 q^{31} + 2 q^{33} - 5 q^{35} - 4 q^{37} + q^{39} + 2 q^{41} + q^{43} + 5 q^{45} - 10 q^{47} + 2 q^{49} + 2 q^{51} + 11 q^{53} - 5 q^{55} + 4 q^{57} + 5 q^{59} - 9 q^{61} - 2 q^{63} + 5 q^{67} + 2 q^{69} + 5 q^{71} + 8 q^{73} - 5 q^{75} + 2 q^{77} + 24 q^{79} + 2 q^{81} + 18 q^{83} - 15 q^{85} - 8 q^{87} + 7 q^{89} + q^{91} - 4 q^{93} - 15 q^{95} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 5 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 - q^13 - 5 * q^15 - 2 * q^17 - 4 * q^19 + 2 * q^21 - 2 * q^23 + 5 * q^25 - 2 * q^27 + 8 * q^29 + 4 * q^31 + 2 * q^33 - 5 * q^35 - 4 * q^37 + q^39 + 2 * q^41 + q^43 + 5 * q^45 - 10 * q^47 + 2 * q^49 + 2 * q^51 + 11 * q^53 - 5 * q^55 + 4 * q^57 + 5 * q^59 - 9 * q^61 - 2 * q^63 + 5 * q^67 + 2 * q^69 + 5 * q^71 + 8 * q^73 - 5 * q^75 + 2 * q^77 + 24 * q^79 + 2 * q^81 + 18 * q^83 - 15 * q^85 - 8 * q^87 + 7 * q^89 + q^91 - 4 * q^93 - 15 * q^95 - 2 * q^99

Introduction

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