LMFDB - Embedded newform 945.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(1,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("945.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.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:	$7.54586299101$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	\(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} -1.61803 q^{10} -1.23607 q^{11} -6.47214 q^{13} -1.61803 q^{14} -4.85410 q^{16} +0.236068 q^{17} +1.47214 q^{19} -0.618034 q^{20} -2.00000 q^{22} +1.76393 q^{23} +1.00000 q^{25} -10.4721 q^{26} -0.618034 q^{28} -5.70820 q^{29} -9.00000 q^{31} -3.38197 q^{32} +0.381966 q^{34} +1.00000 q^{35} +3.70820 q^{37} +2.38197 q^{38} +2.23607 q^{40} +0.472136 q^{41} +0.472136 q^{43} -0.763932 q^{44} +2.85410 q^{46} +4.00000 q^{47} +1.00000 q^{49} +1.61803 q^{50} -4.00000 q^{52} +9.94427 q^{53} +1.23607 q^{55} +2.23607 q^{56} -9.23607 q^{58} +1.70820 q^{59} -12.7082 q^{61} -14.5623 q^{62} +4.23607 q^{64} +6.47214 q^{65} -6.94427 q^{67} +0.145898 q^{68} +1.61803 q^{70} -2.00000 q^{71} -8.47214 q^{73} +6.00000 q^{74} +0.909830 q^{76} +1.23607 q^{77} +5.18034 q^{79} +4.85410 q^{80} +0.763932 q^{82} +11.0000 q^{83} -0.236068 q^{85} +0.763932 q^{86} +2.76393 q^{88} -14.1803 q^{89} +6.47214 q^{91} +1.09017 q^{92} +6.47214 q^{94} -1.47214 q^{95} -0.763932 q^{97} +1.61803 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7	$ 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7} - q^{10} + 2 q^{11} - 4 q^{13} - q^{14} - 3 q^{16} - 4 q^{17} - 6 q^{19} + q^{20} - 4 q^{22} + 8 q^{23} + 2 q^{25} - 12 q^{26} + q^{28} + 2 q^{29} - 18 q^{31} - 9 q^{32} + 3 q^{34} + 2 q^{35} - 6 q^{37} + 7 q^{38} - 8 q^{41} - 8 q^{43} - 6 q^{44} - q^{46} + 8 q^{47} + 2 q^{49} + q^{50} - 8 q^{52} + 2 q^{53} - 2 q^{55} - 14 q^{58} - 10 q^{59} - 12 q^{61} - 9 q^{62} + 4 q^{64} + 4 q^{65} + 4 q^{67} + 7 q^{68} + q^{70} - 4 q^{71} - 8 q^{73} + 12 q^{74} + 13 q^{76} - 2 q^{77} - 12 q^{79} + 3 q^{80} + 6 q^{82} + 22 q^{83} + 4 q^{85} + 6 q^{86} + 10 q^{88} - 6 q^{89} + 4 q^{91} - 9 q^{92} + 4 q^{94} + 6 q^{95} - 6 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7 - q^10 + 2 * q^11 - 4 * q^13 - q^14 - 3 * q^16 - 4 * q^17 - 6 * q^19 + q^20 - 4 * q^22 + 8 * q^23 + 2 * q^25 - 12 * q^26 + q^28 + 2 * q^29 - 18 * q^31 - 9 * q^32 + 3 * q^34 + 2 * q^35 - 6 * q^37 + 7 * q^38 - 8 * q^41 - 8 * q^43 - 6 * q^44 - q^46 + 8 * q^47 + 2 * q^49 + q^50 - 8 * q^52 + 2 * q^53 - 2 * q^55 - 14 * q^58 - 10 * q^59 - 12 * q^61 - 9 * q^62 + 4 * q^64 + 4 * q^65 + 4 * q^67 + 7 * q^68 + q^70 - 4 * q^71 - 8 * q^73 + 12 * q^74 + 13 * q^76 - 2 * q^77 - 12 * q^79 + 3 * q^80 + 6 * q^82 + 22 * q^83 + 4 * q^85 + 6 * q^86 + 10 * q^88 - 6 * q^89 + 4 * q^91 - 9 * q^92 + 4 * q^94 + 6 * q^95 - 6 * q^97 + q^98

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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	−1.61803	−0.511667
$11$	−1.23607	−0.372689	−0.186344	−	0.982485i	$-0.559664\pi$
$11$	−1.23607	−0.372689	−0.186344	+	0.982485i	$0.559664\pi$
$12$	0	0
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	−1.61803	−0.432438
$15$	0	0
$16$	−4.85410	−1.21353
$17$	0.236068	0.0572549	0.0286274	−	0.999590i	$-0.490886\pi$
$17$	0.236068	0.0572549	0.0286274	+	0.999590i	$0.490886\pi$
$18$	0	0
$19$	1.47214	0.337731	0.168866	−	0.985639i	$-0.445990\pi$
$19$	1.47214	0.337731	0.168866	+	0.985639i	$0.445990\pi$
$20$	−0.618034	−0.138197
$21$	0	0
$22$	−2.00000	−0.426401
$23$	1.76393	0.367805	0.183903	−	0.982944i	$-0.441127\pi$
$23$	1.76393	0.367805	0.183903	+	0.982944i	$0.441127\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−10.4721	−2.05375
$27$	0	0
$28$	−0.618034	−0.116797
$29$	−5.70820	−1.05999	−0.529993	−	0.848002i	$-0.677806\pi$
$29$	−5.70820	−1.05999	−0.529993	+	0.848002i	$0.677806\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	0.381966	0.0655066
$35$	1.00000	0.169031
$36$	0	0
$37$	3.70820	0.609625	0.304812	−	0.952412i	$-0.401406\pi$
$37$	3.70820	0.609625	0.304812	+	0.952412i	$0.401406\pi$
$38$	2.38197	0.386406
$39$	0	0
$40$	2.23607	0.353553
$41$	0.472136	0.0737352	0.0368676	−	0.999320i	$-0.488262\pi$
$41$	0.472136	0.0737352	0.0368676	+	0.999320i	$0.488262\pi$
$42$	0	0
$43$	0.472136	0.0720001	0.0360000	−	0.999352i	$-0.488538\pi$
$43$	0.472136	0.0720001	0.0360000	+	0.999352i	$0.488538\pi$
$44$	−0.763932	−0.115167
$45$	0	0
$46$	2.85410	0.420814
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.61803	0.228825
$51$	0	0
$52$	−4.00000	−0.554700
$53$	9.94427	1.36595	0.682975	−	0.730441i	$-0.260686\pi$
$53$	9.94427	1.36595	0.682975	+	0.730441i	$0.260686\pi$
$54$	0	0
$55$	1.23607	0.166671
$56$	2.23607	0.298807
$57$	0	0
$58$	−9.23607	−1.21276
$59$	1.70820	0.222389	0.111195	−	0.993799i	$-0.464532\pi$
$59$	1.70820	0.222389	0.111195	+	0.993799i	$0.464532\pi$
$60$	0	0
$61$	−12.7082	−1.62712	−0.813559	−	0.581482i	$-0.802473\pi$
$61$	−12.7082	−1.62712	−0.813559	+	0.581482i	$0.802473\pi$
$62$	−14.5623	−1.84941
$63$	0	0
$64$	4.23607	0.529508
$65$	6.47214	0.802770
$66$	0	0
$67$	−6.94427	−0.848378	−0.424189	−	0.905574i	$-0.639441\pi$
$67$	−6.94427	−0.848378	−0.424189	+	0.905574i	$0.639441\pi$
$68$	0.145898	0.0176927
$69$	0	0
$70$	1.61803	0.193392
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	−8.47214	−0.991589	−0.495794	−	0.868440i	$-0.665123\pi$
$73$	−8.47214	−0.991589	−0.495794	+	0.868440i	$0.665123\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	0.909830	0.104365
$77$	1.23607	0.140863
$78$	0	0
$79$	5.18034	0.582834	0.291417	−	0.956596i	$-0.405873\pi$
$79$	5.18034	0.582834	0.291417	+	0.956596i	$0.405873\pi$
$80$	4.85410	0.542705
$81$	0	0
$82$	0.763932	0.0843622
$83$	11.0000	1.20741	0.603703	−	0.797209i	$-0.293691\pi$
$83$	11.0000	1.20741	0.603703	+	0.797209i	$0.293691\pi$
$84$	0	0
$85$	−0.236068	−0.0256052
$86$	0.763932	0.0823769
$87$	0	0
$88$	2.76393	0.294636
$89$	−14.1803	−1.50311	−0.751557	−	0.659669i	$-0.770697\pi$
$89$	−14.1803	−1.50311	−0.751557	+	0.659669i	$0.770697\pi$
$90$	0	0
$91$	6.47214	0.678464
$92$	1.09017	0.113658
$93$	0	0
$94$	6.47214	0.667550
$95$	−1.47214	−0.151038
$96$	0	0
$97$	−0.763932	−0.0775655	−0.0387828	−	0.999248i	$-0.512348\pi$
$97$	−0.763932	−0.0775655	−0.0387828	+	0.999248i	$0.512348\pi$
$98$	1.61803	0.163446
$99$	0	0
$100$	0.618034	0.0618034
$101$	16.1803	1.61000	0.805002	−	0.593272i	$-0.202164\pi$
$101$	16.1803	1.61000	0.805002	+	0.593272i	$0.202164\pi$
$102$	0	0
$103$	−15.2361	−1.50125	−0.750627	−	0.660726i	$-0.770249\pi$
$103$	−15.2361	−1.50125	−0.750627	+	0.660726i	$0.770249\pi$
$104$	14.4721	1.41911
$105$	0	0
$106$	16.0902	1.56282
$107$	15.4164	1.49036	0.745180	−	0.666863i	$-0.232363\pi$
$107$	15.4164	1.49036	0.745180	+	0.666863i	$0.232363\pi$
$108$	0	0
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	4.85410	0.458670
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−1.76393	−0.164488
$116$	−3.52786	−0.327554
$117$	0	0
$118$	2.76393	0.254441
$119$	−0.236068	−0.0216403
$120$	0	0
$121$	−9.47214	−0.861103
$122$	−20.5623	−1.86162
$123$	0	0
$124$	−5.56231	−0.499510
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.18034	−0.548416	−0.274208	−	0.961670i	$-0.588416\pi$
$127$	−6.18034	−0.548416	−0.274208	+	0.961670i	$0.588416\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	10.4721	0.918467
$131$	−21.2361	−1.85540	−0.927702	−	0.373322i	$-0.878219\pi$
$131$	−21.2361	−1.85540	−0.927702	+	0.373322i	$0.878219\pi$
$132$	0	0
$133$	−1.47214	−0.127650
$134$	−11.2361	−0.970648
$135$	0	0
$136$	−0.527864	−0.0452640
$137$	15.9443	1.36221	0.681106	−	0.732185i	$-0.261499\pi$
$137$	15.9443	1.36221	0.681106	+	0.732185i	$0.261499\pi$
$138$	0	0
$139$	14.4721	1.22751	0.613755	−	0.789496i	$-0.289658\pi$
$139$	14.4721	1.22751	0.613755	+	0.789496i	$0.289658\pi$
$140$	0.618034	0.0522334
$141$	0	0
$142$	−3.23607	−0.271565
$143$	8.00000	0.668994
$144$	0	0
$145$	5.70820	0.474041
$146$	−13.7082	−1.13450
$147$	0	0
$148$	2.29180	0.188384
$149$	10.9443	0.896590	0.448295	−	0.893886i	$-0.352031\pi$
$149$	10.9443	0.896590	0.448295	+	0.893886i	$0.352031\pi$
$150$	0	0
$151$	−17.8885	−1.45575	−0.727875	−	0.685710i	$-0.759492\pi$
$151$	−17.8885	−1.45575	−0.727875	+	0.685710i	$0.759492\pi$
$152$	−3.29180	−0.267000
$153$	0	0
$154$	2.00000	0.161165
$155$	9.00000	0.722897
$156$	0	0
$157$	−5.23607	−0.417884	−0.208942	−	0.977928i	$-0.567002\pi$
$157$	−5.23607	−0.417884	−0.208942	+	0.977928i	$0.567002\pi$
$158$	8.38197	0.666833
$159$	0	0
$160$	3.38197	0.267368
$161$	−1.76393	−0.139017
$162$	0	0
$163$	6.76393	0.529792	0.264896	−	0.964277i	$-0.414662\pi$
$163$	6.76393	0.529792	0.264896	+	0.964277i	$0.414662\pi$
$164$	0.291796	0.0227854
$165$	0	0
$166$	17.7984	1.38142
$167$	−11.9443	−0.924276	−0.462138	−	0.886808i	$-0.652917\pi$
$167$	−11.9443	−0.924276	−0.462138	+	0.886808i	$0.652917\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	−0.381966	−0.0292955
$171$	0	0
$172$	0.291796	0.0222492
$173$	−9.18034	−0.697968	−0.348984	−	0.937129i	$-0.613473\pi$
$173$	−9.18034	−0.697968	−0.348984	+	0.937129i	$0.613473\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	6.00000	0.452267
$177$	0	0
$178$	−22.9443	−1.71975
$179$	11.4164	0.853302	0.426651	−	0.904416i	$-0.359693\pi$
$179$	11.4164	0.853302	0.426651	+	0.904416i	$0.359693\pi$
$180$	0	0
$181$	8.23607	0.612182	0.306091	−	0.952002i	$-0.400979\pi$
$181$	8.23607	0.612182	0.306091	+	0.952002i	$0.400979\pi$
$182$	10.4721	0.776246
$183$	0	0
$184$	−3.94427	−0.290776
$185$	−3.70820	−0.272633
$186$	0	0
$187$	−0.291796	−0.0213382
$188$	2.47214	0.180299
$189$	0	0
$190$	−2.38197	−0.172806
$191$	−25.2361	−1.82602	−0.913009	−	0.407940i	$-0.866247\pi$
$191$	−25.2361	−1.82602	−0.913009	+	0.407940i	$0.866247\pi$
$192$	0	0
$193$	23.4164	1.68555	0.842775	−	0.538266i	$-0.180920\pi$
$193$	23.4164	1.68555	0.842775	+	0.538266i	$0.180920\pi$
$194$	−1.23607	−0.0887445
$195$	0	0
$196$	0.618034	0.0441453
$197$	−17.9443	−1.27848	−0.639238	−	0.769009i	$-0.720750\pi$
$197$	−17.9443	−1.27848	−0.639238	+	0.769009i	$0.720750\pi$
$198$	0	0
$199$	−17.5279	−1.24252	−0.621259	−	0.783606i	$-0.713378\pi$
$199$	−17.5279	−1.24252	−0.621259	+	0.783606i	$0.713378\pi$
$200$	−2.23607	−0.158114
$201$	0	0
$202$	26.1803	1.84204
$203$	5.70820	0.400637
$204$	0	0
$205$	−0.472136	−0.0329754
$206$	−24.6525	−1.71762
$207$	0	0
$208$	31.4164	2.17834
$209$	−1.81966	−0.125869
$210$	0	0
$211$	−12.7082	−0.874869	−0.437434	−	0.899250i	$-0.644113\pi$
$211$	−12.7082	−0.874869	−0.437434	+	0.899250i	$0.644113\pi$
$212$	6.14590	0.422102
$213$	0	0
$214$	24.9443	1.70516
$215$	−0.472136	−0.0321994
$216$	0	0
$217$	9.00000	0.610960
$218$	1.61803	0.109587
$219$	0	0
$220$	0.763932	0.0515043
$221$	−1.52786	−0.102775
$222$	0	0
$223$	−12.6525	−0.847272	−0.423636	−	0.905832i	$-0.639247\pi$
$223$	−12.6525	−0.847272	−0.423636	+	0.905832i	$0.639247\pi$
$224$	3.38197	0.225967
$225$	0	0
$226$	3.23607	0.215260
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	0	0
$229$	3.76393	0.248728	0.124364	−	0.992237i	$-0.460311\pi$
$229$	3.76393	0.248728	0.124364	+	0.992237i	$0.460311\pi$
$230$	−2.85410	−0.188194
$231$	0	0
$232$	12.7639	0.837993
$233$	−26.3607	−1.72695	−0.863473	−	0.504395i	$-0.831715\pi$
$233$	−26.3607	−1.72695	−0.863473	+	0.504395i	$0.831715\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	1.05573	0.0687220
$237$	0	0
$238$	−0.381966	−0.0247592
$239$	−3.52786	−0.228199	−0.114099	−	0.993469i	$-0.536398\pi$
$239$	−3.52786	−0.228199	−0.114099	+	0.993469i	$0.536398\pi$
$240$	0	0
$241$	25.6525	1.65242	0.826211	−	0.563361i	$-0.190492\pi$
$241$	25.6525	1.65242	0.826211	+	0.563361i	$0.190492\pi$
$242$	−15.3262	−0.985208
$243$	0	0
$244$	−7.85410	−0.502807
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−9.52786	−0.606243
$248$	20.1246	1.27791
$249$	0	0
$250$	−1.61803	−0.102333
$251$	−7.23607	−0.456737	−0.228368	−	0.973575i	$-0.573339\pi$
$251$	−7.23607	−0.456737	−0.228368	+	0.973575i	$0.573339\pi$
$252$	0	0
$253$	−2.18034	−0.137077
$254$	−10.0000	−0.627456
$255$	0	0
$256$	13.5623	0.847644
$257$	−24.5967	−1.53430	−0.767151	−	0.641466i	$-0.778327\pi$
$257$	−24.5967	−1.53430	−0.767151	+	0.641466i	$0.778327\pi$
$258$	0	0
$259$	−3.70820	−0.230417
$260$	4.00000	0.248069
$261$	0	0
$262$	−34.3607	−2.12281
$263$	18.4721	1.13904	0.569520	−	0.821977i	$-0.307129\pi$
$263$	18.4721	1.13904	0.569520	+	0.821977i	$0.307129\pi$
$264$	0	0
$265$	−9.94427	−0.610872
$266$	−2.38197	−0.146048
$267$	0	0
$268$	−4.29180	−0.262163
$269$	−7.70820	−0.469977	−0.234989	−	0.971998i	$-0.575505\pi$
$269$	−7.70820	−0.469977	−0.234989	+	0.971998i	$0.575505\pi$
$270$	0	0
$271$	−20.4164	−1.24021	−0.620104	−	0.784519i	$-0.712910\pi$
$271$	−20.4164	−1.24021	−0.620104	+	0.784519i	$0.712910\pi$
$272$	−1.14590	−0.0694803
$273$	0	0
$274$	25.7984	1.55854
$275$	−1.23607	−0.0745377
$276$	0	0
$277$	30.3607	1.82420	0.912098	−	0.409972i	$-0.134461\pi$
$277$	30.3607	1.82420	0.912098	+	0.409972i	$0.134461\pi$
$278$	23.4164	1.40442
$279$	0	0
$280$	−2.23607	−0.133631
$281$	−10.6525	−0.635473	−0.317737	−	0.948179i	$-0.602923\pi$
$281$	−10.6525	−0.635473	−0.317737	+	0.948179i	$0.602923\pi$
$282$	0	0
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	−1.23607	−0.0733471
$285$	0	0
$286$	12.9443	0.765411
$287$	−0.472136	−0.0278693
$288$	0	0
$289$	−16.9443	−0.996722
$290$	9.23607	0.542361
$291$	0	0
$292$	−5.23607	−0.306418
$293$	9.29180	0.542833	0.271416	−	0.962462i	$-0.412508\pi$
$293$	9.29180	0.542833	0.271416	+	0.962462i	$0.412508\pi$
$294$	0	0
$295$	−1.70820	−0.0994555
$296$	−8.29180	−0.481951
$297$	0	0
$298$	17.7082	1.02581
$299$	−11.4164	−0.660228
$300$	0	0
$301$	−0.472136	−0.0272135
$302$	−28.9443	−1.66556
$303$	0	0
$304$	−7.14590	−0.409845
$305$	12.7082	0.727670
$306$	0	0
$307$	23.1246	1.31979	0.659896	−	0.751357i	$-0.270600\pi$
$307$	23.1246	1.31979	0.659896	+	0.751357i	$0.270600\pi$
$308$	0.763932	0.0435291
$309$	0	0
$310$	14.5623	0.827083
$311$	0.944272	0.0535447	0.0267724	−	0.999642i	$-0.491477\pi$
$311$	0.944272	0.0535447	0.0267724	+	0.999642i	$0.491477\pi$
$312$	0	0
$313$	−10.4721	−0.591920	−0.295960	−	0.955200i	$-0.595640\pi$
$313$	−10.4721	−0.591920	−0.295960	+	0.955200i	$0.595640\pi$
$314$	−8.47214	−0.478110
$315$	0	0
$316$	3.20163	0.180106
$317$	−11.9443	−0.670857	−0.335429	−	0.942066i	$-0.608881\pi$
$317$	−11.9443	−0.670857	−0.335429	+	0.942066i	$0.608881\pi$
$318$	0	0
$319$	7.05573	0.395045
$320$	−4.23607	−0.236803
$321$	0	0
$322$	−2.85410	−0.159053
$323$	0.347524	0.0193368
$324$	0	0
$325$	−6.47214	−0.359010
$326$	10.9443	0.606147
$327$	0	0
$328$	−1.05573	−0.0582928
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−19.4164	−1.06722	−0.533611	−	0.845730i	$-0.679165\pi$
$331$	−19.4164	−1.06722	−0.533611	+	0.845730i	$0.679165\pi$
$332$	6.79837	0.373109
$333$	0	0
$334$	−19.3262	−1.05748
$335$	6.94427	0.379406
$336$	0	0
$337$	20.9443	1.14091	0.570454	−	0.821330i	$-0.306767\pi$
$337$	20.9443	1.14091	0.570454	+	0.821330i	$0.306767\pi$
$338$	46.7426	2.54246
$339$	0	0
$340$	−0.145898	−0.00791243
$341$	11.1246	0.602432
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−1.05573	−0.0569210
$345$	0	0
$346$	−14.8541	−0.798561
$347$	−20.9443	−1.12435	−0.562174	−	0.827019i	$-0.690035\pi$
$347$	−20.9443	−1.12435	−0.562174	+	0.827019i	$0.690035\pi$
$348$	0	0
$349$	−11.1803	−0.598470	−0.299235	−	0.954179i	$-0.596731\pi$
$349$	−11.1803	−0.598470	−0.299235	+	0.954179i	$0.596731\pi$
$350$	−1.61803	−0.0864876
$351$	0	0
$352$	4.18034	0.222813
$353$	4.47214	0.238028	0.119014	−	0.992893i	$-0.462027\pi$
$353$	4.47214	0.238028	0.119014	+	0.992893i	$0.462027\pi$
$354$	0	0
$355$	2.00000	0.106149
$356$	−8.76393	−0.464487
$357$	0	0
$358$	18.4721	0.976283
$359$	17.4164	0.919203	0.459601	−	0.888125i	$-0.347992\pi$
$359$	17.4164	0.919203	0.459601	+	0.888125i	$0.347992\pi$
$360$	0	0
$361$	−16.8328	−0.885938
$362$	13.3262	0.700412
$363$	0	0
$364$	4.00000	0.209657
$365$	8.47214	0.443452
$366$	0	0
$367$	3.41641	0.178335	0.0891675	−	0.996017i	$-0.471579\pi$
$367$	3.41641	0.178335	0.0891675	+	0.996017i	$0.471579\pi$
$368$	−8.56231	−0.446341
$369$	0	0
$370$	−6.00000	−0.311925
$371$	−9.94427	−0.516281
$372$	0	0
$373$	15.4164	0.798231	0.399116	−	0.916901i	$-0.369317\pi$
$373$	15.4164	0.798231	0.399116	+	0.916901i	$0.369317\pi$
$374$	−0.472136	−0.0244136
$375$	0	0
$376$	−8.94427	−0.461266
$377$	36.9443	1.90273
$378$	0	0
$379$	−33.6525	−1.72861	−0.864306	−	0.502967i	$-0.832242\pi$
$379$	−33.6525	−1.72861	−0.864306	+	0.502967i	$0.832242\pi$
$380$	−0.909830	−0.0466733
$381$	0	0
$382$	−40.8328	−2.08919
$383$	−10.0557	−0.513824	−0.256912	−	0.966435i	$-0.582705\pi$
$383$	−10.0557	−0.513824	−0.256912	+	0.966435i	$0.582705\pi$
$384$	0	0
$385$	−1.23607	−0.0629959
$386$	37.8885	1.92848
$387$	0	0
$388$	−0.472136	−0.0239691
$389$	−24.6525	−1.24993	−0.624965	−	0.780653i	$-0.714887\pi$
$389$	−24.6525	−1.24993	−0.624965	+	0.780653i	$0.714887\pi$
$390$	0	0
$391$	0.416408	0.0210587
$392$	−2.23607	−0.112938
$393$	0	0
$394$	−29.0344	−1.46273
$395$	−5.18034	−0.260651
$396$	0	0
$397$	−11.2361	−0.563922	−0.281961	−	0.959426i	$-0.590985\pi$
$397$	−11.2361	−0.563922	−0.281961	+	0.959426i	$0.590985\pi$
$398$	−28.3607	−1.42159
$399$	0	0
$400$	−4.85410	−0.242705
$401$	15.5279	0.775425	0.387712	−	0.921780i	$-0.373265\pi$
$401$	15.5279	0.775425	0.387712	+	0.921780i	$0.373265\pi$
$402$	0	0
$403$	58.2492	2.90160
$404$	10.0000	0.497519
$405$	0	0
$406$	9.23607	0.458378
$407$	−4.58359	−0.227200
$408$	0	0
$409$	4.23607	0.209460	0.104730	−	0.994501i	$-0.466602\pi$
$409$	4.23607	0.209460	0.104730	+	0.994501i	$0.466602\pi$
$410$	−0.763932	−0.0377279
$411$	0	0
$412$	−9.41641	−0.463913
$413$	−1.70820	−0.0840552
$414$	0	0
$415$	−11.0000	−0.539969
$416$	21.8885	1.07317
$417$	0	0
$418$	−2.94427	−0.144009
$419$	12.6525	0.618114	0.309057	−	0.951044i	$-0.399987\pi$
$419$	12.6525	0.618114	0.309057	+	0.951044i	$0.399987\pi$
$420$	0	0
$421$	25.8328	1.25901	0.629507	−	0.776995i	$-0.283257\pi$
$421$	25.8328	1.25901	0.629507	+	0.776995i	$0.283257\pi$
$422$	−20.5623	−1.00096
$423$	0	0
$424$	−22.2361	−1.07988
$425$	0.236068	0.0114510
$426$	0	0
$427$	12.7082	0.614993
$428$	9.52786	0.460547
$429$	0	0
$430$	−0.763932	−0.0368401
$431$	9.52786	0.458941	0.229471	−	0.973316i	$-0.426301\pi$
$431$	9.52786	0.458941	0.229471	+	0.973316i	$0.426301\pi$
$432$	0	0
$433$	0.763932	0.0367122	0.0183561	−	0.999832i	$-0.494157\pi$
$433$	0.763932	0.0367122	0.0183561	+	0.999832i	$0.494157\pi$
$434$	14.5623	0.699013
$435$	0	0
$436$	0.618034	0.0295985
$437$	2.59675	0.124219
$438$	0	0
$439$	−21.0000	−1.00228	−0.501138	−	0.865368i	$-0.667085\pi$
$439$	−21.0000	−1.00228	−0.501138	+	0.865368i	$0.667085\pi$
$440$	−2.76393	−0.131765
$441$	0	0
$442$	−2.47214	−0.117588
$443$	−30.7082	−1.45899	−0.729495	−	0.683986i	$-0.760245\pi$
$443$	−30.7082	−1.45899	−0.729495	+	0.683986i	$0.760245\pi$
$444$	0	0
$445$	14.1803	0.672213
$446$	−20.4721	−0.969384
$447$	0	0
$448$	−4.23607	−0.200135
$449$	−7.05573	−0.332980	−0.166490	−	0.986043i	$-0.553243\pi$
$449$	−7.05573	−0.332980	−0.166490	+	0.986043i	$0.553243\pi$
$450$	0	0
$451$	−0.583592	−0.0274803
$452$	1.23607	0.0581397
$453$	0	0
$454$	17.7984	0.835319
$455$	−6.47214	−0.303418
$456$	0	0
$457$	−18.0689	−0.845227	−0.422613	−	0.906310i	$-0.638887\pi$
$457$	−18.0689	−0.845227	−0.422613	+	0.906310i	$0.638887\pi$
$458$	6.09017	0.284575
$459$	0	0
$460$	−1.09017	−0.0508294
$461$	−28.4721	−1.32608	−0.663040	−	0.748584i	$-0.730734\pi$
$461$	−28.4721	−1.32608	−0.663040	+	0.748584i	$0.730734\pi$
$462$	0	0
$463$	−32.4721	−1.50911	−0.754554	−	0.656238i	$-0.772147\pi$
$463$	−32.4721	−1.50911	−0.754554	+	0.656238i	$0.772147\pi$
$464$	27.7082	1.28632
$465$	0	0
$466$	−42.6525	−1.97584
$467$	41.4721	1.91910	0.959551	−	0.281536i	$-0.0908437\pi$
$467$	41.4721	1.91910	0.959551	+	0.281536i	$0.0908437\pi$
$468$	0	0
$469$	6.94427	0.320657
$470$	−6.47214	−0.298537
$471$	0	0
$472$	−3.81966	−0.175814
$473$	−0.583592	−0.0268336
$474$	0	0
$475$	1.47214	0.0675462
$476$	−0.145898	−0.00668723
$477$	0	0
$478$	−5.70820	−0.261087
$479$	−21.5967	−0.986781	−0.493390	−	0.869808i	$-0.664243\pi$
$479$	−21.5967	−0.986781	−0.493390	+	0.869808i	$0.664243\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	41.5066	1.89057
$483$	0	0
$484$	−5.85410	−0.266096
$485$	0.763932	0.0346884
$486$	0	0
$487$	−1.23607	−0.0560116	−0.0280058	−	0.999608i	$-0.508916\pi$
$487$	−1.23607	−0.0560116	−0.0280058	+	0.999608i	$0.508916\pi$
$488$	28.4164	1.28635
$489$	0	0
$490$	−1.61803	−0.0730953
$491$	−3.88854	−0.175488	−0.0877438	−	0.996143i	$-0.527966\pi$
$491$	−3.88854	−0.175488	−0.0877438	+	0.996143i	$0.527966\pi$
$492$	0	0
$493$	−1.34752	−0.0606894
$494$	−15.4164	−0.693617
$495$	0	0
$496$	43.6869	1.96160
$497$	2.00000	0.0897123
$498$	0	0
$499$	10.2361	0.458229	0.229115	−	0.973399i	$-0.426417\pi$
$499$	10.2361	0.458229	0.229115	+	0.973399i	$0.426417\pi$
$500$	−0.618034	−0.0276393
$501$	0	0
$502$	−11.7082	−0.522563
$503$	9.94427	0.443393	0.221697	−	0.975116i	$-0.428841\pi$
$503$	9.94427	0.443393	0.221697	+	0.975116i	$0.428841\pi$
$504$	0	0
$505$	−16.1803	−0.720016
$506$	−3.52786	−0.156833
$507$	0	0
$508$	−3.81966	−0.169470
$509$	36.3607	1.61166	0.805829	−	0.592148i	$-0.201720\pi$
$509$	36.3607	1.61166	0.805829	+	0.592148i	$0.201720\pi$
$510$	0	0
$511$	8.47214	0.374785
$512$	−5.29180	−0.233867
$513$	0	0
$514$	−39.7984	−1.75543
$515$	15.2361	0.671381
$516$	0	0
$517$	−4.94427	−0.217449
$518$	−6.00000	−0.263625
$519$	0	0
$520$	−14.4721	−0.634645
$521$	28.2918	1.23949	0.619743	−	0.784805i	$-0.287237\pi$
$521$	28.2918	1.23949	0.619743	+	0.784805i	$0.287237\pi$
$522$	0	0
$523$	38.0689	1.66464	0.832318	−	0.554298i	$-0.187013\pi$
$523$	38.0689	1.66464	0.832318	+	0.554298i	$0.187013\pi$
$524$	−13.1246	−0.573351
$525$	0	0
$526$	29.8885	1.30320
$527$	−2.12461	−0.0925495
$528$	0	0
$529$	−19.8885	−0.864719
$530$	−16.0902	−0.698912
$531$	0	0
$532$	−0.909830	−0.0394461
$533$	−3.05573	−0.132358
$534$	0	0
$535$	−15.4164	−0.666509
$536$	15.5279	0.670702
$537$	0	0
$538$	−12.4721	−0.537712
$539$	−1.23607	−0.0532412
$540$	0	0
$541$	8.47214	0.364246	0.182123	−	0.983276i	$-0.441703\pi$
$541$	8.47214	0.364246	0.182123	+	0.983276i	$0.441703\pi$
$542$	−33.0344	−1.41895
$543$	0	0
$544$	−0.798374	−0.0342300
$545$	−1.00000	−0.0428353
$546$	0	0
$547$	−6.76393	−0.289205	−0.144602	−	0.989490i	$-0.546190\pi$
$547$	−6.76393	−0.289205	−0.144602	+	0.989490i	$0.546190\pi$
$548$	9.85410	0.420946
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	−8.40325	−0.357991
$552$	0	0
$553$	−5.18034	−0.220290
$554$	49.1246	2.08710
$555$	0	0
$556$	8.94427	0.379322
$557$	14.9443	0.633209	0.316605	−	0.948558i	$-0.397457\pi$
$557$	14.9443	0.633209	0.316605	+	0.948558i	$0.397457\pi$
$558$	0	0
$559$	−3.05573	−0.129244
$560$	−4.85410	−0.205123
$561$	0	0
$562$	−17.2361	−0.727060
$563$	−27.4164	−1.15546	−0.577732	−	0.816227i	$-0.696062\pi$
$563$	−27.4164	−1.15546	−0.577732	+	0.816227i	$0.696062\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	−35.5967	−1.49624
$567$	0	0
$568$	4.47214	0.187647
$569$	−33.5279	−1.40556	−0.702781	−	0.711407i	$-0.748058\pi$
$569$	−33.5279	−1.40556	−0.702781	+	0.711407i	$0.748058\pi$
$570$	0	0
$571$	−5.18034	−0.216790	−0.108395	−	0.994108i	$-0.534571\pi$
$571$	−5.18034	−0.216790	−0.108395	+	0.994108i	$0.534571\pi$
$572$	4.94427	0.206730
$573$	0	0
$574$	−0.763932	−0.0318859
$575$	1.76393	0.0735611
$576$	0	0
$577$	38.5410	1.60448	0.802242	−	0.596999i	$-0.203640\pi$
$577$	38.5410	1.60448	0.802242	+	0.596999i	$0.203640\pi$
$578$	−27.4164	−1.14037
$579$	0	0
$580$	3.52786	0.146487
$581$	−11.0000	−0.456357
$582$	0	0
$583$	−12.2918	−0.509074
$584$	18.9443	0.783920
$585$	0	0
$586$	15.0344	0.621067
$587$	46.3050	1.91121	0.955605	−	0.294651i	$-0.0952034\pi$
$587$	46.3050	1.91121	0.955605	+	0.294651i	$0.0952034\pi$
$588$	0	0
$589$	−13.2492	−0.545925
$590$	−2.76393	−0.113789
$591$	0	0
$592$	−18.0000	−0.739795
$593$	23.1803	0.951902	0.475951	−	0.879472i	$-0.342104\pi$
$593$	23.1803	0.951902	0.475951	+	0.879472i	$0.342104\pi$
$594$	0	0
$595$	0.236068	0.00967784
$596$	6.76393	0.277061
$597$	0	0
$598$	−18.4721	−0.755382
$599$	−41.8885	−1.71152	−0.855760	−	0.517373i	$-0.826910\pi$
$599$	−41.8885	−1.71152	−0.855760	+	0.517373i	$0.826910\pi$
$600$	0	0
$601$	4.23607	0.172793	0.0863964	−	0.996261i	$-0.472465\pi$
$601$	4.23607	0.172793	0.0863964	+	0.996261i	$0.472465\pi$
$602$	−0.763932	−0.0311355
$603$	0	0
$604$	−11.0557	−0.449851
$605$	9.47214	0.385097
$606$	0	0
$607$	−2.58359	−0.104865	−0.0524324	−	0.998624i	$-0.516697\pi$
$607$	−2.58359	−0.104865	−0.0524324	+	0.998624i	$0.516697\pi$
$608$	−4.97871	−0.201914
$609$	0	0
$610$	20.5623	0.832543
$611$	−25.8885	−1.04734
$612$	0	0
$613$	25.7082	1.03834	0.519172	−	0.854670i	$-0.326240\pi$
$613$	25.7082	1.03834	0.519172	+	0.854670i	$0.326240\pi$
$614$	37.4164	1.51000
$615$	0	0
$616$	−2.76393	−0.111362
$617$	−8.88854	−0.357839	−0.178920	−	0.983864i	$-0.557260\pi$
$617$	−8.88854	−0.357839	−0.178920	+	0.983864i	$0.557260\pi$
$618$	0	0
$619$	−10.8328	−0.435408	−0.217704	−	0.976015i	$-0.569857\pi$
$619$	−10.8328	−0.435408	−0.217704	+	0.976015i	$0.569857\pi$
$620$	5.56231	0.223388
$621$	0	0
$622$	1.52786	0.0612618
$623$	14.1803	0.568123
$624$	0	0
$625$	1.00000	0.0400000
$626$	−16.9443	−0.677229
$627$	0	0
$628$	−3.23607	−0.129133
$629$	0.875388	0.0349040
$630$	0	0
$631$	17.1803	0.683939	0.341969	−	0.939711i	$-0.388906\pi$
$631$	17.1803	0.683939	0.341969	+	0.939711i	$0.388906\pi$
$632$	−11.5836	−0.460771
$633$	0	0
$634$	−19.3262	−0.767543
$635$	6.18034	0.245259
$636$	0	0
$637$	−6.47214	−0.256435
$638$	11.4164	0.451980
$639$	0	0
$640$	−13.6180	−0.538300
$641$	−13.1246	−0.518391	−0.259195	−	0.965825i	$-0.583457\pi$
$641$	−13.1246	−0.518391	−0.259195	+	0.965825i	$0.583457\pi$
$642$	0	0
$643$	−14.6525	−0.577837	−0.288919	−	0.957354i	$-0.593296\pi$
$643$	−14.6525	−0.577837	−0.288919	+	0.957354i	$0.593296\pi$
$644$	−1.09017	−0.0429587
$645$	0	0
$646$	0.562306	0.0221236
$647$	11.4721	0.451016	0.225508	−	0.974241i	$-0.427596\pi$
$647$	11.4721	0.451016	0.225508	+	0.974241i	$0.427596\pi$
$648$	0	0
$649$	−2.11146	−0.0828819
$650$	−10.4721	−0.410751
$651$	0	0
$652$	4.18034	0.163715
$653$	−4.88854	−0.191304	−0.0956518	−	0.995415i	$-0.530494\pi$
$653$	−4.88854	−0.191304	−0.0956518	+	0.995415i	$0.530494\pi$
$654$	0	0
$655$	21.2361	0.829762
$656$	−2.29180	−0.0894796
$657$	0	0
$658$	−6.47214	−0.252310
$659$	−13.5279	−0.526971	−0.263485	−	0.964663i	$-0.584872\pi$
$659$	−13.5279	−0.526971	−0.263485	+	0.964663i	$0.584872\pi$
$660$	0	0
$661$	−26.3607	−1.02531	−0.512656	−	0.858594i	$-0.671338\pi$
$661$	−26.3607	−1.02531	−0.512656	+	0.858594i	$0.671338\pi$
$662$	−31.4164	−1.22103
$663$	0	0
$664$	−24.5967	−0.954539
$665$	1.47214	0.0570870
$666$	0	0
$667$	−10.0689	−0.389869
$668$	−7.38197	−0.285617
$669$	0	0
$670$	11.2361	0.434087
$671$	15.7082	0.606408
$672$	0	0
$673$	35.2361	1.35825	0.679125	−	0.734022i	$-0.262359\pi$
$673$	35.2361	1.35825	0.679125	+	0.734022i	$0.262359\pi$
$674$	33.8885	1.30534
$675$	0	0
$676$	17.8541	0.686696
$677$	15.5279	0.596784	0.298392	−	0.954443i	$-0.403550\pi$
$677$	15.5279	0.596784	0.298392	+	0.954443i	$0.403550\pi$
$678$	0	0
$679$	0.763932	0.0293170
$680$	0.527864	0.0202427
$681$	0	0
$682$	18.0000	0.689256
$683$	−38.1246	−1.45880	−0.729399	−	0.684089i	$-0.760200\pi$
$683$	−38.1246	−1.45880	−0.729399	+	0.684089i	$0.760200\pi$
$684$	0	0
$685$	−15.9443	−0.609199
$686$	−1.61803	−0.0617768
$687$	0	0
$688$	−2.29180	−0.0873739
$689$	−64.3607	−2.45195
$690$	0	0
$691$	−27.3607	−1.04085	−0.520425	−	0.853908i	$-0.674226\pi$
$691$	−27.3607	−1.04085	−0.520425	+	0.853908i	$0.674226\pi$
$692$	−5.67376	−0.215684
$693$	0	0
$694$	−33.8885	−1.28639
$695$	−14.4721	−0.548959
$696$	0	0
$697$	0.111456	0.00422170
$698$	−18.0902	−0.684723
$699$	0	0
$700$	−0.618034	−0.0233595
$701$	46.9443	1.77306	0.886530	−	0.462670i	$-0.153109\pi$
$701$	46.9443	1.77306	0.886530	+	0.462670i	$0.153109\pi$
$702$	0	0
$703$	5.45898	0.205889
$704$	−5.23607	−0.197342
$705$	0	0
$706$	7.23607	0.272333
$707$	−16.1803	−0.608524
$708$	0	0
$709$	38.9443	1.46258	0.731291	−	0.682065i	$-0.238918\pi$
$709$	38.9443	1.46258	0.731291	+	0.682065i	$0.238918\pi$
$710$	3.23607	0.121447
$711$	0	0
$712$	31.7082	1.18832
$713$	−15.8754	−0.594538
$714$	0	0
$715$	−8.00000	−0.299183
$716$	7.05573	0.263685
$717$	0	0
$718$	28.1803	1.05168
$719$	−37.7771	−1.40885	−0.704424	−	0.709780i	$-0.748795\pi$
$719$	−37.7771	−1.40885	−0.704424	+	0.709780i	$0.748795\pi$
$720$	0	0
$721$	15.2361	0.567421
$722$	−27.2361	−1.01362
$723$	0	0
$724$	5.09017	0.189175
$725$	−5.70820	−0.211997
$726$	0	0
$727$	−15.0557	−0.558386	−0.279193	−	0.960235i	$-0.590067\pi$
$727$	−15.0557	−0.558386	−0.279193	+	0.960235i	$0.590067\pi$
$728$	−14.4721	−0.536373
$729$	0	0
$730$	13.7082	0.507363
$731$	0.111456	0.00412236
$732$	0	0
$733$	25.1246	0.927999	0.463999	−	0.885836i	$-0.346414\pi$
$733$	25.1246	0.927999	0.463999	+	0.885836i	$0.346414\pi$
$734$	5.52786	0.204037
$735$	0	0
$736$	−5.96556	−0.219893
$737$	8.58359	0.316181
$738$	0	0
$739$	−10.2361	−0.376540	−0.188270	−	0.982117i	$-0.560288\pi$
$739$	−10.2361	−0.376540	−0.188270	+	0.982117i	$0.560288\pi$
$740$	−2.29180	−0.0842481
$741$	0	0
$742$	−16.0902	−0.590689
$743$	−6.11146	−0.224208	−0.112104	−	0.993697i	$-0.535759\pi$
$743$	−6.11146	−0.224208	−0.112104	+	0.993697i	$0.535759\pi$
$744$	0	0
$745$	−10.9443	−0.400967
$746$	24.9443	0.913275
$747$	0	0
$748$	−0.180340	−0.00659388
$749$	−15.4164	−0.563303
$750$	0	0
$751$	−8.34752	−0.304605	−0.152303	−	0.988334i	$-0.548669\pi$
$751$	−8.34752	−0.304605	−0.152303	+	0.988334i	$0.548669\pi$
$752$	−19.4164	−0.708044
$753$	0	0
$754$	59.7771	2.17695
$755$	17.8885	0.651031
$756$	0	0
$757$	1.41641	0.0514802	0.0257401	−	0.999669i	$-0.491806\pi$
$757$	1.41641	0.0514802	0.0257401	+	0.999669i	$0.491806\pi$
$758$	−54.4508	−1.97774
$759$	0	0
$760$	3.29180	0.119406
$761$	−2.47214	−0.0896149	−0.0448074	−	0.998996i	$-0.514267\pi$
$761$	−2.47214	−0.0896149	−0.0448074	+	0.998996i	$0.514267\pi$
$762$	0	0
$763$	−1.00000	−0.0362024
$764$	−15.5967	−0.564271
$765$	0	0
$766$	−16.2705	−0.587877
$767$	−11.0557	−0.399199
$768$	0	0
$769$	28.5967	1.03123	0.515613	−	0.856822i	$-0.327564\pi$
$769$	28.5967	1.03123	0.515613	+	0.856822i	$0.327564\pi$
$770$	−2.00000	−0.0720750
$771$	0	0
$772$	14.4721	0.520864
$773$	−47.1803	−1.69696	−0.848479	−	0.529228i	$-0.822481\pi$
$773$	−47.1803	−1.69696	−0.848479	+	0.529228i	$0.822481\pi$
$774$	0	0
$775$	−9.00000	−0.323290
$776$	1.70820	0.0613209
$777$	0	0
$778$	−39.8885	−1.43007
$779$	0.695048	0.0249027
$780$	0	0
$781$	2.47214	0.0884600
$782$	0.673762	0.0240937
$783$	0	0
$784$	−4.85410	−0.173361
$785$	5.23607	0.186883
$786$	0	0
$787$	34.1803	1.21840	0.609199	−	0.793018i	$-0.291491\pi$
$787$	34.1803	1.21840	0.609199	+	0.793018i	$0.291491\pi$
$788$	−11.0902	−0.395071
$789$	0	0
$790$	−8.38197	−0.298217
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	82.2492	2.92076
$794$	−18.1803	−0.645196
$795$	0	0
$796$	−10.8328	−0.383959
$797$	25.2918	0.895881	0.447941	−	0.894063i	$-0.352158\pi$
$797$	25.2918	0.895881	0.447941	+	0.894063i	$0.352158\pi$
$798$	0	0
$799$	0.944272	0.0334059
$800$	−3.38197	−0.119571
$801$	0	0
$802$	25.1246	0.887181
$803$	10.4721	0.369554
$804$	0	0
$805$	1.76393	0.0621704
$806$	94.2492	3.31979
$807$	0	0
$808$	−36.1803	−1.27282
$809$	29.0557	1.02154	0.510772	−	0.859716i	$-0.329359\pi$
$809$	29.0557	1.02154	0.510772	+	0.859716i	$0.329359\pi$
$810$	0	0
$811$	35.0557	1.23097	0.615487	−	0.788147i	$-0.288960\pi$
$811$	35.0557	1.23097	0.615487	+	0.788147i	$0.288960\pi$
$812$	3.52786	0.123804
$813$	0	0
$814$	−7.41641	−0.259945
$815$	−6.76393	−0.236930
$816$	0	0
$817$	0.695048	0.0243167
$818$	6.85410	0.239648
$819$	0	0
$820$	−0.291796	−0.0101900
$821$	5.41641	0.189034	0.0945170	−	0.995523i	$-0.469869\pi$
$821$	5.41641	0.189034	0.0945170	+	0.995523i	$0.469869\pi$
$822$	0	0
$823$	23.3050	0.812360	0.406180	−	0.913793i	$-0.366861\pi$
$823$	23.3050	0.812360	0.406180	+	0.913793i	$0.366861\pi$
$824$	34.0689	1.18685
$825$	0	0
$826$	−2.76393	−0.0961695
$827$	−21.2918	−0.740388	−0.370194	−	0.928954i	$-0.620709\pi$
$827$	−21.2918	−0.740388	−0.370194	+	0.928954i	$0.620709\pi$
$828$	0	0
$829$	28.4721	0.988878	0.494439	−	0.869212i	$-0.335374\pi$
$829$	28.4721	0.988878	0.494439	+	0.869212i	$0.335374\pi$
$830$	−17.7984	−0.617791
$831$	0	0
$832$	−27.4164	−0.950493
$833$	0.236068	0.00817927
$834$	0	0
$835$	11.9443	0.413349
$836$	−1.12461	−0.0388955
$837$	0	0
$838$	20.4721	0.707198
$839$	−3.81966	−0.131869	−0.0659347	−	0.997824i	$-0.521003\pi$
$839$	−3.81966	−0.131869	−0.0659347	+	0.997824i	$0.521003\pi$
$840$	0	0
$841$	3.58359	0.123572
$842$	41.7984	1.44047
$843$	0	0
$844$	−7.85410	−0.270349
$845$	−28.8885	−0.993796
$846$	0	0
$847$	9.47214	0.325466
$848$	−48.2705	−1.65762
$849$	0	0
$850$	0.381966	0.0131013
$851$	6.54102	0.224223
$852$	0	0
$853$	−42.2492	−1.44659	−0.723293	−	0.690541i	$-0.757372\pi$
$853$	−42.2492	−1.44659	−0.723293	+	0.690541i	$0.757372\pi$
$854$	20.5623	0.703628
$855$	0	0
$856$	−34.4721	−1.17823
$857$	0.819660	0.0279991	0.0139995	−	0.999902i	$-0.495544\pi$
$857$	0.819660	0.0279991	0.0139995	+	0.999902i	$0.495544\pi$
$858$	0	0
$859$	−11.9443	−0.407533	−0.203767	−	0.979019i	$-0.565318\pi$
$859$	−11.9443	−0.407533	−0.203767	+	0.979019i	$0.565318\pi$
$860$	−0.291796	−0.00995016
$861$	0	0
$862$	15.4164	0.525085
$863$	4.81966	0.164063	0.0820316	−	0.996630i	$-0.473859\pi$
$863$	4.81966	0.164063	0.0820316	+	0.996630i	$0.473859\pi$
$864$	0	0
$865$	9.18034	0.312141
$866$	1.23607	0.0420033
$867$	0	0
$868$	5.56231	0.188797
$869$	−6.40325	−0.217215
$870$	0	0
$871$	44.9443	1.52288
$872$	−2.23607	−0.0757228
$873$	0	0
$874$	4.20163	0.142122
$875$	1.00000	0.0338062
$876$	0	0
$877$	52.6525	1.77795	0.888974	−	0.457958i	$-0.151419\pi$
$877$	52.6525	1.77795	0.888974	+	0.457958i	$0.151419\pi$
$878$	−33.9787	−1.14673
$879$	0	0
$880$	−6.00000	−0.202260
$881$	37.4853	1.26291	0.631456	−	0.775412i	$-0.282458\pi$
$881$	37.4853	1.26291	0.631456	+	0.775412i	$0.282458\pi$
$882$	0	0
$883$	−34.6525	−1.16615	−0.583074	−	0.812419i	$-0.698150\pi$
$883$	−34.6525	−1.16615	−0.583074	+	0.812419i	$0.698150\pi$
$884$	−0.944272	−0.0317593
$885$	0	0
$886$	−49.6869	−1.66926
$887$	37.7214	1.26656	0.633280	−	0.773923i	$-0.281708\pi$
$887$	37.7214	1.26656	0.633280	+	0.773923i	$0.281708\pi$
$888$	0	0
$889$	6.18034	0.207282
$890$	22.9443	0.769094
$891$	0	0
$892$	−7.81966	−0.261822
$893$	5.88854	0.197053
$894$	0	0
$895$	−11.4164	−0.381608
$896$	−13.6180	−0.454947
$897$	0	0
$898$	−11.4164	−0.380970
$899$	51.3738	1.71341
$900$	0	0
$901$	2.34752	0.0782074
$902$	−0.944272	−0.0314408
$903$	0	0
$904$	−4.47214	−0.148741
$905$	−8.23607	−0.273776
$906$	0	0
$907$	−13.1246	−0.435796	−0.217898	−	0.975972i	$-0.569920\pi$
$907$	−13.1246	−0.435796	−0.217898	+	0.975972i	$0.569920\pi$
$908$	6.79837	0.225612
$909$	0	0
$910$	−10.4721	−0.347148
$911$	59.2361	1.96258	0.981289	−	0.192539i	$-0.0616723\pi$
$911$	59.2361	1.96258	0.981289	+	0.192539i	$0.0616723\pi$
$912$	0	0
$913$	−13.5967	−0.449987
$914$	−29.2361	−0.967043
$915$	0	0
$916$	2.32624	0.0768611
$917$	21.2361	0.701277
$918$	0	0
$919$	13.3050	0.438890	0.219445	−	0.975625i	$-0.429575\pi$
$919$	13.3050	0.438890	0.219445	+	0.975625i	$0.429575\pi$
$920$	3.94427	0.130039
$921$	0	0
$922$	−46.0689	−1.51720
$923$	12.9443	0.426066
$924$	0	0
$925$	3.70820	0.121925
$926$	−52.5410	−1.72661
$927$	0	0
$928$	19.3050	0.633716
$929$	35.1246	1.15240	0.576201	−	0.817308i	$-0.304535\pi$
$929$	35.1246	1.15240	0.576201	+	0.817308i	$0.304535\pi$
$930$	0	0
$931$	1.47214	0.0482473
$932$	−16.2918	−0.533656
$933$	0	0
$934$	67.1033	2.19569
$935$	0.291796	0.00954275
$936$	0	0
$937$	13.5279	0.441936	0.220968	−	0.975281i	$-0.429078\pi$
$937$	13.5279	0.441936	0.220968	+	0.975281i	$0.429078\pi$
$938$	11.2361	0.366871
$939$	0	0
$940$	−2.47214	−0.0806322
$941$	30.7639	1.00288	0.501438	−	0.865194i	$-0.332805\pi$
$941$	30.7639	1.00288	0.501438	+	0.865194i	$0.332805\pi$
$942$	0	0
$943$	0.832816	0.0271202
$944$	−8.29180	−0.269875
$945$	0	0
$946$	−0.944272	−0.0307009
$947$	−25.5410	−0.829972	−0.414986	−	0.909828i	$-0.636213\pi$
$947$	−25.5410	−0.829972	−0.414986	+	0.909828i	$0.636213\pi$
$948$	0	0
$949$	54.8328	1.77995
$950$	2.38197	0.0772812
$951$	0	0
$952$	0.527864	0.0171082
$953$	13.7771	0.446284	0.223142	−	0.974786i	$-0.428369\pi$
$953$	13.7771	0.446284	0.223142	+	0.974786i	$0.428369\pi$
$954$	0	0
$955$	25.2361	0.816620
$956$	−2.18034	−0.0705172
$957$	0	0
$958$	−34.9443	−1.12900
$959$	−15.9443	−0.514867
$960$	0	0
$961$	50.0000	1.61290
$962$	−38.8328	−1.25202
$963$	0	0
$964$	15.8541	0.510626
$965$	−23.4164	−0.753801
$966$	0	0
$967$	29.5967	0.951767	0.475884	−	0.879508i	$-0.342128\pi$
$967$	29.5967	0.951767	0.475884	+	0.879508i	$0.342128\pi$
$968$	21.1803	0.680762
$969$	0	0
$970$	1.23607	0.0396878
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	0	0
$973$	−14.4721	−0.463955
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	61.6869	1.97455
$977$	55.8885	1.78803	0.894016	−	0.448034i	$-0.147876\pi$
$977$	55.8885	1.78803	0.894016	+	0.448034i	$0.147876\pi$
$978$	0	0
$979$	17.5279	0.560193
$980$	−0.618034	−0.0197424
$981$	0	0
$982$	−6.29180	−0.200779
$983$	41.0000	1.30770	0.653848	−	0.756626i	$-0.273153\pi$
$983$	41.0000	1.30770	0.653848	+	0.756626i	$0.273153\pi$
$984$	0	0
$985$	17.9443	0.571752
$986$	−2.18034	−0.0694362
$987$	0	0
$988$	−5.88854	−0.187340
$989$	0.832816	0.0264820
$990$	0	0
$991$	−42.0132	−1.33459	−0.667296	−	0.744793i	$-0.732548\pi$
$991$	−42.0132	−1.33459	−0.667296	+	0.744793i	$0.732548\pi$
$992$	30.4377	0.966398
$993$	0	0
$994$	3.23607	0.102642
$995$	17.5279	0.555671
$996$	0	0
$997$	33.4164	1.05831	0.529154	−	0.848526i	$-0.322509\pi$
$997$	33.4164	1.05831	0.529154	+	0.848526i	$0.322509\pi$
$998$	16.5623	0.524271
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.h.1.2	yes	2
3.2	odd	2		945.2.a.c.1.1	✓	2
5.4	even	2		4725.2.a.y.1.1		2
7.6	odd	2		6615.2.a.t.1.2		2
15.14	odd	2		4725.2.a.bd.1.2		2
21.20	even	2		6615.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.c.1.1	✓	2	3.2	odd	2
945.2.a.h.1.2	yes	2	1.1	even	1	trivial
4725.2.a.y.1.1		2	5.4	even	2
4725.2.a.bd.1.2		2	15.14	odd	2
6615.2.a.n.1.1		2	21.20	even	2
6615.2.a.t.1.2		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} -1.61803 q^{10} -1.23607 q^{11} -6.47214 q^{13} -1.61803 q^{14} -4.85410 q^{16} +0.236068 q^{17} +1.47214 q^{19} -0.618034 q^{20} -2.00000 q^{22} +1.76393 q^{23} +1.00000 q^{25} -10.4721 q^{26} -0.618034 q^{28} -5.70820 q^{29} -9.00000 q^{31} -3.38197 q^{32} +0.381966 q^{34} +1.00000 q^{35} +3.70820 q^{37} +2.38197 q^{38} +2.23607 q^{40} +0.472136 q^{41} +0.472136 q^{43} -0.763932 q^{44} +2.85410 q^{46} +4.00000 q^{47} +1.00000 q^{49} +1.61803 q^{50} -4.00000 q^{52} +9.94427 q^{53} +1.23607 q^{55} +2.23607 q^{56} -9.23607 q^{58} +1.70820 q^{59} -12.7082 q^{61} -14.5623 q^{62} +4.23607 q^{64} +6.47214 q^{65} -6.94427 q^{67} +0.145898 q^{68} +1.61803 q^{70} -2.00000 q^{71} -8.47214 q^{73} +6.00000 q^{74} +0.909830 q^{76} +1.23607 q^{77} +5.18034 q^{79} +4.85410 q^{80} +0.763932 q^{82} +11.0000 q^{83} -0.236068 q^{85} +0.763932 q^{86} +2.76393 q^{88} -14.1803 q^{89} +6.47214 q^{91} +1.09017 q^{92} +6.47214 q^{94} -1.47214 q^{95} -0.763932 q^{97} +1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7	\( 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7} - q^{10} + 2 q^{11} - 4 q^{13} - q^{14} - 3 q^{16} - 4 q^{17} - 6 q^{19} + q^{20} - 4 q^{22} + 8 q^{23} + 2 q^{25} - 12 q^{26} + q^{28} + 2 q^{29} - 18 q^{31} - 9 q^{32} + 3 q^{34} + 2 q^{35} - 6 q^{37} + 7 q^{38} - 8 q^{41} - 8 q^{43} - 6 q^{44} - q^{46} + 8 q^{47} + 2 q^{49} + q^{50} - 8 q^{52} + 2 q^{53} - 2 q^{55} - 14 q^{58} - 10 q^{59} - 12 q^{61} - 9 q^{62} + 4 q^{64} + 4 q^{65} + 4 q^{67} + 7 q^{68} + q^{70} - 4 q^{71} - 8 q^{73} + 12 q^{74} + 13 q^{76} - 2 q^{77} - 12 q^{79} + 3 q^{80} + 6 q^{82} + 22 q^{83} + 4 q^{85} + 6 q^{86} + 10 q^{88} - 6 q^{89} + 4 q^{91} - 9 q^{92} + 4 q^{94} + 6 q^{95} - 6 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7 - q^10 + 2 * q^11 - 4 * q^13 - q^14 - 3 * q^16 - 4 * q^17 - 6 * q^19 + q^20 - 4 * q^22 + 8 * q^23 + 2 * q^25 - 12 * q^26 + q^28 + 2 * q^29 - 18 * q^31 - 9 * q^32 + 3 * q^34 + 2 * q^35 - 6 * q^37 + 7 * q^38 - 8 * q^41 - 8 * q^43 - 6 * q^44 - q^46 + 8 * q^47 + 2 * q^49 + q^50 - 8 * q^52 + 2 * q^53 - 2 * q^55 - 14 * q^58 - 10 * q^59 - 12 * q^61 - 9 * q^62 + 4 * q^64 + 4 * q^65 + 4 * q^67 + 7 * q^68 + q^70 - 4 * q^71 - 8 * q^73 + 12 * q^74 + 13 * q^76 - 2 * q^77 - 12 * q^79 + 3 * q^80 + 6 * q^82 + 22 * q^83 + 4 * q^85 + 6 * q^86 + 10 * q^88 - 6 * q^89 + 4 * q^91 - 9 * q^92 + 4 * q^94 + 6 * q^95 - 6 * q^97 + q^98

Introduction

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