LMFDB - Embedded newform 945.2.a.i.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:	$0$
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:	$\Z[a_1, a_2]$
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)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	\(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} -1.61803 q^{10} +5.47214 q^{11} +5.09017 q^{13} -1.61803 q^{14} -4.85410 q^{16} +4.38197 q^{17} +2.61803 q^{19} -0.618034 q^{20} +8.85410 q^{22} +6.61803 q^{23} +1.00000 q^{25} +8.23607 q^{26} -0.618034 q^{28} -3.85410 q^{29} +3.00000 q^{31} -3.38197 q^{32} +7.09017 q^{34} +1.00000 q^{35} -3.00000 q^{37} +4.23607 q^{38} +2.23607 q^{40} +1.61803 q^{41} -12.2361 q^{43} +3.38197 q^{44} +10.7082 q^{46} -2.70820 q^{47} +1.00000 q^{49} +1.61803 q^{50} +3.14590 q^{52} +4.38197 q^{53} -5.47214 q^{55} +2.23607 q^{56} -6.23607 q^{58} -2.70820 q^{59} +7.85410 q^{61} +4.85410 q^{62} +4.23607 q^{64} -5.09017 q^{65} -11.7984 q^{67} +2.70820 q^{68} +1.61803 q^{70} +2.14590 q^{71} -11.4721 q^{73} -4.85410 q^{74} +1.61803 q^{76} -5.47214 q^{77} -13.7984 q^{79} +4.85410 q^{80} +2.61803 q^{82} +12.4164 q^{83} -4.38197 q^{85} -19.7984 q^{86} -12.2361 q^{88} +4.52786 q^{89} -5.09017 q^{91} +4.09017 q^{92} -4.38197 q^{94} -2.61803 q^{95} -13.0344 q^{97} +1.61803 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - q^{13} - q^{14} - 3 q^{16} + 11 q^{17} + 3 q^{19} + q^{20} + 11 q^{22} + 11 q^{23} + 2 q^{25} + 12 q^{26} + q^{28} - q^{29} + 6 q^{31} - 9 q^{32} + 3 q^{34} + 2 q^{35} - 6 q^{37} + 4 q^{38} + q^{41} - 20 q^{43} + 9 q^{44} + 8 q^{46} + 8 q^{47} + 2 q^{49} + q^{50} + 13 q^{52} + 11 q^{53} - 2 q^{55} - 8 q^{58} + 8 q^{59} + 9 q^{61} + 3 q^{62} + 4 q^{64} + q^{65} + q^{67} - 8 q^{68} + q^{70} + 11 q^{71} - 14 q^{73} - 3 q^{74} + q^{76} - 2 q^{77} - 3 q^{79} + 3 q^{80} + 3 q^{82} - 2 q^{83} - 11 q^{85} - 15 q^{86} - 20 q^{88} + 18 q^{89} + q^{91} - 3 q^{92} - 11 q^{94} - 3 q^{95} + 3 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7 - q^10 + 2 * q^11 - q^13 - q^14 - 3 * q^16 + 11 * q^17 + 3 * q^19 + q^20 + 11 * q^22 + 11 * q^23 + 2 * q^25 + 12 * q^26 + q^28 - q^29 + 6 * q^31 - 9 * q^32 + 3 * q^34 + 2 * q^35 - 6 * q^37 + 4 * q^38 + q^41 - 20 * q^43 + 9 * q^44 + 8 * q^46 + 8 * q^47 + 2 * q^49 + q^50 + 13 * q^52 + 11 * q^53 - 2 * q^55 - 8 * q^58 + 8 * q^59 + 9 * q^61 + 3 * q^62 + 4 * q^64 + q^65 + q^67 - 8 * q^68 + q^70 + 11 * q^71 - 14 * q^73 - 3 * q^74 + q^76 - 2 * q^77 - 3 * q^79 + 3 * q^80 + 3 * q^82 - 2 * q^83 - 11 * q^85 - 15 * q^86 - 20 * q^88 + 18 * q^89 + q^91 - 3 * q^92 - 11 * q^94 - 3 * q^95 + 3 * 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
$11$	5.47214	1.64991	0.824956	−	0.565198i	$-0.191200\pi$
$11$	5.47214	1.64991	0.824956	+	0.565198i	$0.191200\pi$
$12$	0	0
$13$	5.09017	1.41176	0.705880	−	0.708332i	$-0.250552\pi$
$13$	5.09017	1.41176	0.705880	+	0.708332i	$0.250552\pi$
$14$	−1.61803	−0.432438
$15$	0	0
$16$	−4.85410	−1.21353
$17$	4.38197	1.06278	0.531391	−	0.847126i	$-0.321669\pi$
$17$	4.38197	1.06278	0.531391	+	0.847126i	$0.321669\pi$
$18$	0	0
$19$	2.61803	0.600618	0.300309	−	0.953842i	$-0.402910\pi$
$19$	2.61803	0.600618	0.300309	+	0.953842i	$0.402910\pi$
$20$	−0.618034	−0.138197
$21$	0	0
$22$	8.85410	1.88770
$23$	6.61803	1.37996	0.689978	−	0.723831i	$-0.257620\pi$
$23$	6.61803	1.37996	0.689978	+	0.723831i	$0.257620\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	8.23607	1.61523
$27$	0	0
$28$	−0.618034	−0.116797
$29$	−3.85410	−0.715689	−0.357844	−	0.933781i	$-0.616488\pi$
$29$	−3.85410	−0.715689	−0.357844	+	0.933781i	$0.616488\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	7.09017	1.21595
$35$	1.00000	0.169031
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	4.23607	0.687181
$39$	0	0
$40$	2.23607	0.353553
$41$	1.61803	0.252694	0.126347	−	0.991986i	$-0.459675\pi$
$41$	1.61803	0.252694	0.126347	+	0.991986i	$0.459675\pi$
$42$	0	0
$43$	−12.2361	−1.86598	−0.932991	−	0.359899i	$-0.882811\pi$
$43$	−12.2361	−1.86598	−0.932991	+	0.359899i	$0.882811\pi$
$44$	3.38197	0.509851
$45$	0	0
$46$	10.7082	1.57884
$47$	−2.70820	−0.395032	−0.197516	−	0.980300i	$-0.563287\pi$
$47$	−2.70820	−0.395032	−0.197516	+	0.980300i	$0.563287\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.61803	0.228825
$51$	0	0
$52$	3.14590	0.436258
$53$	4.38197	0.601909	0.300955	−	0.953638i	$-0.402695\pi$
$53$	4.38197	0.601909	0.300955	+	0.953638i	$0.402695\pi$
$54$	0	0
$55$	−5.47214	−0.737863
$56$	2.23607	0.298807
$57$	0	0
$58$	−6.23607	−0.818836
$59$	−2.70820	−0.352578	−0.176289	−	0.984338i	$-0.556409\pi$
$59$	−2.70820	−0.352578	−0.176289	+	0.984338i	$0.556409\pi$
$60$	0	0
$61$	7.85410	1.00561	0.502807	−	0.864398i	$-0.332301\pi$
$61$	7.85410	1.00561	0.502807	+	0.864398i	$0.332301\pi$
$62$	4.85410	0.616472
$63$	0	0
$64$	4.23607	0.529508
$65$	−5.09017	−0.631358
$66$	0	0
$67$	−11.7984	−1.44140	−0.720700	−	0.693247i	$-0.756180\pi$
$67$	−11.7984	−1.44140	−0.720700	+	0.693247i	$0.756180\pi$
$68$	2.70820	0.328418
$69$	0	0
$70$	1.61803	0.193392
$71$	2.14590	0.254671	0.127336	−	0.991860i	$-0.459357\pi$
$71$	2.14590	0.254671	0.127336	+	0.991860i	$0.459357\pi$
$72$	0	0
$73$	−11.4721	−1.34271	−0.671356	−	0.741135i	$-0.734288\pi$
$73$	−11.4721	−1.34271	−0.671356	+	0.741135i	$0.734288\pi$
$74$	−4.85410	−0.564278
$75$	0	0
$76$	1.61803	0.185601
$77$	−5.47214	−0.623608
$78$	0	0
$79$	−13.7984	−1.55244	−0.776219	−	0.630463i	$-0.782865\pi$
$79$	−13.7984	−1.55244	−0.776219	+	0.630463i	$0.782865\pi$
$80$	4.85410	0.542705
$81$	0	0
$82$	2.61803	0.289113
$83$	12.4164	1.36288	0.681439	−	0.731875i	$-0.261355\pi$
$83$	12.4164	1.36288	0.681439	+	0.731875i	$0.261355\pi$
$84$	0	0
$85$	−4.38197	−0.475291
$86$	−19.7984	−2.13491
$87$	0	0
$88$	−12.2361	−1.30437
$89$	4.52786	0.479953	0.239976	−	0.970779i	$-0.422860\pi$
$89$	4.52786	0.479953	0.239976	+	0.970779i	$0.422860\pi$
$90$	0	0
$91$	−5.09017	−0.533595
$92$	4.09017	0.426430
$93$	0	0
$94$	−4.38197	−0.451965
$95$	−2.61803	−0.268605
$96$	0	0
$97$	−13.0344	−1.32345	−0.661724	−	0.749748i	$-0.730175\pi$
$97$	−13.0344	−1.32345	−0.661724	+	0.749748i	$0.730175\pi$
$98$	1.61803	0.163446
$99$	0	0
$100$	0.618034	0.0618034
$101$	6.47214	0.644002	0.322001	−	0.946739i	$-0.395645\pi$
$101$	6.47214	0.644002	0.322001	+	0.946739i	$0.395645\pi$
$102$	0	0
$103$	14.3262	1.41161	0.705803	−	0.708408i	$-0.250586\pi$
$103$	14.3262	1.41161	0.705803	+	0.708408i	$0.250586\pi$
$104$	−11.3820	−1.11609
$105$	0	0
$106$	7.09017	0.688658
$107$	14.7082	1.42190	0.710948	−	0.703245i	$-0.248266\pi$
$107$	14.7082	1.42190	0.710948	+	0.703245i	$0.248266\pi$
$108$	0	0
$109$	−6.85410	−0.656504	−0.328252	−	0.944590i	$-0.606460\pi$
$109$	−6.85410	−0.656504	−0.328252	+	0.944590i	$0.606460\pi$
$110$	−8.85410	−0.844205
$111$	0	0
$112$	4.85410	0.458670
$113$	1.56231	0.146969	0.0734847	−	0.997296i	$-0.476588\pi$
$113$	1.56231	0.146969	0.0734847	+	0.997296i	$0.476588\pi$
$114$	0	0
$115$	−6.61803	−0.617135
$116$	−2.38197	−0.221160
$117$	0	0
$118$	−4.38197	−0.403393
$119$	−4.38197	−0.401694
$120$	0	0
$121$	18.9443	1.72221
$122$	12.7082	1.15055
$123$	0	0
$124$	1.85410	0.166503
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.7984	1.40188	0.700939	−	0.713221i	$-0.252764\pi$
$127$	15.7984	1.40188	0.700939	+	0.713221i	$0.252764\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	−8.23607	−0.722351
$131$	17.3262	1.51380	0.756900	−	0.653530i	$-0.226713\pi$
$131$	17.3262	1.51380	0.756900	+	0.653530i	$0.226713\pi$
$132$	0	0
$133$	−2.61803	−0.227012
$134$	−19.0902	−1.64914
$135$	0	0
$136$	−9.79837	−0.840204
$137$	−18.4721	−1.57818	−0.789091	−	0.614277i	$-0.789448\pi$
$137$	−18.4721	−1.57818	−0.789091	+	0.614277i	$0.789448\pi$
$138$	0	0
$139$	−0.527864	−0.0447728	−0.0223864	−	0.999749i	$-0.507126\pi$
$139$	−0.527864	−0.0447728	−0.0223864	+	0.999749i	$0.507126\pi$
$140$	0.618034	0.0522334
$141$	0	0
$142$	3.47214	0.291375
$143$	27.8541	2.32928
$144$	0	0
$145$	3.85410	0.320066
$146$	−18.5623	−1.53623
$147$	0	0
$148$	−1.85410	−0.152406
$149$	0.0901699	0.00738701	0.00369350	−	0.999993i	$-0.498824\pi$
$149$	0.0901699	0.00738701	0.00369350	+	0.999993i	$0.498824\pi$
$150$	0	0
$151$	−14.8885	−1.21161	−0.605806	−	0.795612i	$-0.707149\pi$
$151$	−14.8885	−1.21161	−0.605806	+	0.795612i	$0.707149\pi$
$152$	−5.85410	−0.474830
$153$	0	0
$154$	−8.85410	−0.713484
$155$	−3.00000	−0.240966
$156$	0	0
$157$	−5.23607	−0.417884	−0.208942	−	0.977928i	$-0.567002\pi$
$157$	−5.23607	−0.417884	−0.208942	+	0.977928i	$0.567002\pi$
$158$	−22.3262	−1.77618
$159$	0	0
$160$	3.38197	0.267368
$161$	−6.61803	−0.521574
$162$	0	0
$163$	−7.52786	−0.589628	−0.294814	−	0.955555i	$-0.595258\pi$
$163$	−7.52786	−0.589628	−0.294814	+	0.955555i	$0.595258\pi$
$164$	1.00000	0.0780869
$165$	0	0
$166$	20.0902	1.55930
$167$	−8.23607	−0.637326	−0.318663	−	0.947868i	$-0.603234\pi$
$167$	−8.23607	−0.637326	−0.318663	+	0.947868i	$0.603234\pi$
$168$	0	0
$169$	12.9098	0.993064
$170$	−7.09017	−0.543791
$171$	0	0
$172$	−7.56231	−0.576620
$173$	−15.1803	−1.15414	−0.577070	−	0.816695i	$-0.695804\pi$
$173$	−15.1803	−1.15414	−0.577070	+	0.816695i	$0.695804\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−26.5623	−2.00221
$177$	0	0
$178$	7.32624	0.549125
$179$	−14.7082	−1.09934	−0.549671	−	0.835381i	$-0.685247\pi$
$179$	−14.7082	−1.09934	−0.549671	+	0.835381i	$0.685247\pi$
$180$	0	0
$181$	0.819660	0.0609249	0.0304624	−	0.999536i	$-0.490302\pi$
$181$	0.819660	0.0609249	0.0304624	+	0.999536i	$0.490302\pi$
$182$	−8.23607	−0.610498
$183$	0	0
$184$	−14.7984	−1.09095
$185$	3.00000	0.220564
$186$	0	0
$187$	23.9787	1.75350
$188$	−1.67376	−0.122072
$189$	0	0
$190$	−4.23607	−0.307317
$191$	18.6180	1.34715	0.673577	−	0.739117i	$-0.264757\pi$
$191$	18.6180	1.34715	0.673577	+	0.739117i	$0.264757\pi$
$192$	0	0
$193$	−16.5623	−1.19218	−0.596090	−	0.802917i	$-0.703280\pi$
$193$	−16.5623	−1.19218	−0.596090	+	0.802917i	$0.703280\pi$
$194$	−21.0902	−1.51419
$195$	0	0
$196$	0.618034	0.0441453
$197$	14.8885	1.06076	0.530382	−	0.847759i	$-0.322048\pi$
$197$	14.8885	1.06076	0.530382	+	0.847759i	$0.322048\pi$
$198$	0	0
$199$	27.0344	1.91642	0.958210	−	0.286064i	$-0.0923471\pi$
$199$	27.0344	1.91642	0.958210	+	0.286064i	$0.0923471\pi$
$200$	−2.23607	−0.158114
$201$	0	0
$202$	10.4721	0.736817
$203$	3.85410	0.270505
$204$	0	0
$205$	−1.61803	−0.113008
$206$	23.1803	1.61505
$207$	0	0
$208$	−24.7082	−1.71321
$209$	14.3262	0.990967
$210$	0	0
$211$	−9.70820	−0.668340	−0.334170	−	0.942513i	$-0.608456\pi$
$211$	−9.70820	−0.668340	−0.334170	+	0.942513i	$0.608456\pi$
$212$	2.70820	0.186000
$213$	0	0
$214$	23.7984	1.62682
$215$	12.2361	0.834493
$216$	0	0
$217$	−3.00000	−0.203653
$218$	−11.0902	−0.751121
$219$	0	0
$220$	−3.38197	−0.228012
$221$	22.3050	1.50039
$222$	0	0
$223$	−10.5279	−0.704998	−0.352499	−	0.935812i	$-0.614668\pi$
$223$	−10.5279	−0.704998	−0.352499	+	0.935812i	$0.614668\pi$
$224$	3.38197	0.225967
$225$	0	0
$226$	2.52786	0.168151
$227$	−28.2705	−1.87638	−0.938190	−	0.346121i	$-0.887499\pi$
$227$	−28.2705	−1.87638	−0.938190	+	0.346121i	$0.887499\pi$
$228$	0	0
$229$	5.18034	0.342326	0.171163	−	0.985243i	$-0.445247\pi$
$229$	5.18034	0.342326	0.171163	+	0.985243i	$0.445247\pi$
$230$	−10.7082	−0.706078
$231$	0	0
$232$	8.61803	0.565802
$233$	−29.0902	−1.90576	−0.952880	−	0.303347i	$-0.901896\pi$
$233$	−29.0902	−1.90576	−0.952880	+	0.303347i	$0.901896\pi$
$234$	0	0
$235$	2.70820	0.176664
$236$	−1.67376	−0.108953
$237$	0	0
$238$	−7.09017	−0.459587
$239$	−21.3607	−1.38171	−0.690854	−	0.722995i	$-0.742765\pi$
$239$	−21.3607	−1.38171	−0.690854	+	0.722995i	$0.742765\pi$
$240$	0	0
$241$	5.79837	0.373506	0.186753	−	0.982407i	$-0.440204\pi$
$241$	5.79837	0.373506	0.186753	+	0.982407i	$0.440204\pi$
$242$	30.6525	1.97042
$243$	0	0
$244$	4.85410	0.310752
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	13.3262	0.847928
$248$	−6.70820	−0.425971
$249$	0	0
$250$	−1.61803	−0.102333
$251$	−20.6525	−1.30357	−0.651786	−	0.758403i	$-0.725980\pi$
$251$	−20.6525	−1.30357	−0.651786	+	0.758403i	$0.725980\pi$
$252$	0	0
$253$	36.2148	2.27680
$254$	25.5623	1.60392
$255$	0	0
$256$	13.5623	0.847644
$257$	1.52786	0.0953055	0.0476528	−	0.998864i	$-0.484826\pi$
$257$	1.52786	0.0953055	0.0476528	+	0.998864i	$0.484826\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	−3.14590	−0.195100
$261$	0	0
$262$	28.0344	1.73197
$263$	−20.0902	−1.23881	−0.619406	−	0.785070i	$-0.712627\pi$
$263$	−20.0902	−1.23881	−0.619406	+	0.785070i	$0.712627\pi$
$264$	0	0
$265$	−4.38197	−0.269182
$266$	−4.23607	−0.259730
$267$	0	0
$268$	−7.29180	−0.445417
$269$	−12.1246	−0.739251	−0.369625	−	0.929181i	$-0.620514\pi$
$269$	−12.1246	−0.739251	−0.369625	+	0.929181i	$0.620514\pi$
$270$	0	0
$271$	24.8541	1.50978	0.754890	−	0.655852i	$-0.227690\pi$
$271$	24.8541	1.50978	0.754890	+	0.655852i	$0.227690\pi$
$272$	−21.2705	−1.28971
$273$	0	0
$274$	−29.8885	−1.80563
$275$	5.47214	0.329982
$276$	0	0
$277$	15.7984	0.949232	0.474616	−	0.880193i	$-0.342587\pi$
$277$	15.7984	0.949232	0.474616	+	0.880193i	$0.342587\pi$
$278$	−0.854102	−0.0512256
$279$	0	0
$280$	−2.23607	−0.133631
$281$	−6.67376	−0.398123	−0.199062	−	0.979987i	$-0.563789\pi$
$281$	−6.67376	−0.398123	−0.199062	+	0.979987i	$0.563789\pi$
$282$	0	0
$283$	−19.9787	−1.18761	−0.593806	−	0.804609i	$-0.702375\pi$
$283$	−19.9787	−1.18761	−0.593806	+	0.804609i	$0.702375\pi$
$284$	1.32624	0.0786977
$285$	0	0
$286$	45.0689	2.66498
$287$	−1.61803	−0.0955095
$288$	0	0
$289$	2.20163	0.129507
$290$	6.23607	0.366195
$291$	0	0
$292$	−7.09017	−0.414921
$293$	−13.1246	−0.766748	−0.383374	−	0.923593i	$-0.625238\pi$
$293$	−13.1246	−0.766748	−0.383374	+	0.923593i	$0.625238\pi$
$294$	0	0
$295$	2.70820	0.157678
$296$	6.70820	0.389906
$297$	0	0
$298$	0.145898	0.00845165
$299$	33.6869	1.94816
$300$	0	0
$301$	12.2361	0.705275
$302$	−24.0902	−1.38623
$303$	0	0
$304$	−12.7082	−0.728865
$305$	−7.85410	−0.449725
$306$	0	0
$307$	−4.58359	−0.261599	−0.130800	−	0.991409i	$-0.541754\pi$
$307$	−4.58359	−0.261599	−0.130800	+	0.991409i	$0.541754\pi$
$308$	−3.38197	−0.192705
$309$	0	0
$310$	−4.85410	−0.275694
$311$	5.96556	0.338276	0.169138	−	0.985592i	$-0.445902\pi$
$311$	5.96556	0.338276	0.169138	+	0.985592i	$0.445902\pi$
$312$	0	0
$313$	−29.1803	−1.64937	−0.824685	−	0.565592i	$-0.808648\pi$
$313$	−29.1803	−1.64937	−0.824685	+	0.565592i	$0.808648\pi$
$314$	−8.47214	−0.478110
$315$	0	0
$316$	−8.52786	−0.479730
$317$	6.05573	0.340124	0.170062	−	0.985433i	$-0.445603\pi$
$317$	6.05573	0.340124	0.170062	+	0.985433i	$0.445603\pi$
$318$	0	0
$319$	−21.0902	−1.18082
$320$	−4.23607	−0.236803
$321$	0	0
$322$	−10.7082	−0.596745
$323$	11.4721	0.638327
$324$	0	0
$325$	5.09017	0.282352
$326$	−12.1803	−0.674607
$327$	0	0
$328$	−3.61803	−0.199773
$329$	2.70820	0.149308
$330$	0	0
$331$	4.14590	0.227879	0.113940	−	0.993488i	$-0.463653\pi$
$331$	4.14590	0.227879	0.113940	+	0.993488i	$0.463653\pi$
$332$	7.67376	0.421152
$333$	0	0
$334$	−13.3262	−0.729179
$335$	11.7984	0.644614
$336$	0	0
$337$	−13.7426	−0.748610	−0.374305	−	0.927306i	$-0.622119\pi$
$337$	−13.7426	−0.748610	−0.374305	+	0.927306i	$0.622119\pi$
$338$	20.8885	1.13619
$339$	0	0
$340$	−2.70820	−0.146873
$341$	16.4164	0.888998
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	27.3607	1.47519
$345$	0	0
$346$	−24.5623	−1.32048
$347$	20.1803	1.08334	0.541669	−	0.840592i	$-0.317793\pi$
$347$	20.1803	1.08334	0.541669	+	0.840592i	$0.317793\pi$
$348$	0	0
$349$	12.8197	0.686221	0.343110	−	0.939295i	$-0.388520\pi$
$349$	12.8197	0.686221	0.343110	+	0.939295i	$0.388520\pi$
$350$	−1.61803	−0.0864876
$351$	0	0
$352$	−18.5066	−0.986404
$353$	29.4508	1.56751	0.783755	−	0.621070i	$-0.213302\pi$
$353$	29.4508	1.56751	0.783755	+	0.621070i	$0.213302\pi$
$354$	0	0
$355$	−2.14590	−0.113892
$356$	2.79837	0.148314
$357$	0	0
$358$	−23.7984	−1.25778
$359$	−14.7082	−0.776269	−0.388135	−	0.921603i	$-0.626880\pi$
$359$	−14.7082	−0.776269	−0.388135	+	0.921603i	$0.626880\pi$
$360$	0	0
$361$	−12.1459	−0.639258
$362$	1.32624	0.0697055
$363$	0	0
$364$	−3.14590	−0.164890
$365$	11.4721	0.600479
$366$	0	0
$367$	−31.9787	−1.66928	−0.834638	−	0.550799i	$-0.814323\pi$
$367$	−31.9787	−1.66928	−0.834638	+	0.550799i	$0.814323\pi$
$368$	−32.1246	−1.67461
$369$	0	0
$370$	4.85410	0.252353
$371$	−4.38197	−0.227500
$372$	0	0
$373$	15.8541	0.820894	0.410447	−	0.911884i	$-0.365373\pi$
$373$	15.8541	0.820894	0.410447	+	0.911884i	$0.365373\pi$
$374$	38.7984	2.00622
$375$	0	0
$376$	6.05573	0.312300
$377$	−19.6180	−1.01038
$378$	0	0
$379$	23.8885	1.22707	0.613536	−	0.789667i	$-0.289747\pi$
$379$	23.8885	1.22707	0.613536	+	0.789667i	$0.289747\pi$
$380$	−1.61803	−0.0830034
$381$	0	0
$382$	30.1246	1.54131
$383$	24.5279	1.25332	0.626658	−	0.779295i	$-0.284422\pi$
$383$	24.5279	1.25332	0.626658	+	0.779295i	$0.284422\pi$
$384$	0	0
$385$	5.47214	0.278886
$386$	−26.7984	−1.36400
$387$	0	0
$388$	−8.05573	−0.408968
$389$	−22.0902	−1.12002	−0.560008	−	0.828487i	$-0.689202\pi$
$389$	−22.0902	−1.12002	−0.560008	+	0.828487i	$0.689202\pi$
$390$	0	0
$391$	29.0000	1.46659
$392$	−2.23607	−0.112938
$393$	0	0
$394$	24.0902	1.21365
$395$	13.7984	0.694272
$396$	0	0
$397$	−34.3607	−1.72451	−0.862257	−	0.506472i	$-0.830949\pi$
$397$	−34.3607	−1.72451	−0.862257	+	0.506472i	$0.830949\pi$
$398$	43.7426	2.19262
$399$	0	0
$400$	−4.85410	−0.242705
$401$	11.3820	0.568388	0.284194	−	0.958767i	$-0.408274\pi$
$401$	11.3820	0.568388	0.284194	+	0.958767i	$0.408274\pi$
$402$	0	0
$403$	15.2705	0.760678
$404$	4.00000	0.199007
$405$	0	0
$406$	6.23607	0.309491
$407$	−16.4164	−0.813731
$408$	0	0
$409$	−8.47214	−0.418920	−0.209460	−	0.977817i	$-0.567171\pi$
$409$	−8.47214	−0.418920	−0.209460	+	0.977817i	$0.567171\pi$
$410$	−2.61803	−0.129295
$411$	0	0
$412$	8.85410	0.436210
$413$	2.70820	0.133262
$414$	0	0
$415$	−12.4164	−0.609497
$416$	−17.2148	−0.844024
$417$	0	0
$418$	23.1803	1.13379
$419$	10.5279	0.514320	0.257160	−	0.966369i	$-0.417213\pi$
$419$	10.5279	0.514320	0.257160	+	0.966369i	$0.417213\pi$
$420$	0	0
$421$	−32.1459	−1.56670	−0.783348	−	0.621584i	$-0.786489\pi$
$421$	−32.1459	−1.56670	−0.783348	+	0.621584i	$0.786489\pi$
$422$	−15.7082	−0.764663
$423$	0	0
$424$	−9.79837	−0.475851
$425$	4.38197	0.212557
$426$	0	0
$427$	−7.85410	−0.380087
$428$	9.09017	0.439390
$429$	0	0
$430$	19.7984	0.954762
$431$	4.67376	0.225127	0.112564	−	0.993645i	$-0.464094\pi$
$431$	4.67376	0.225127	0.112564	+	0.993645i	$0.464094\pi$
$432$	0	0
$433$	−4.09017	−0.196561	−0.0982805	−	0.995159i	$-0.531334\pi$
$433$	−4.09017	−0.196561	−0.0982805	+	0.995159i	$0.531334\pi$
$434$	−4.85410	−0.233004
$435$	0	0
$436$	−4.23607	−0.202871
$437$	17.3262	0.828826
$438$	0	0
$439$	34.4164	1.64261	0.821303	−	0.570493i	$-0.193248\pi$
$439$	34.4164	1.64261	0.821303	+	0.570493i	$0.193248\pi$
$440$	12.2361	0.583332
$441$	0	0
$442$	36.0902	1.71663
$443$	−12.9787	−0.616637	−0.308319	−	0.951283i	$-0.599766\pi$
$443$	−12.9787	−0.616637	−0.308319	+	0.951283i	$0.599766\pi$
$444$	0	0
$445$	−4.52786	−0.214641
$446$	−17.0344	−0.806604
$447$	0	0
$448$	−4.23607	−0.200135
$449$	−13.0557	−0.616138	−0.308069	−	0.951364i	$-0.599683\pi$
$449$	−13.0557	−0.616138	−0.308069	+	0.951364i	$0.599683\pi$
$450$	0	0
$451$	8.85410	0.416923
$452$	0.965558	0.0454160
$453$	0	0
$454$	−45.7426	−2.14681
$455$	5.09017	0.238631
$456$	0	0
$457$	−2.96556	−0.138723	−0.0693615	−	0.997592i	$-0.522096\pi$
$457$	−2.96556	−0.138723	−0.0693615	+	0.997592i	$0.522096\pi$
$458$	8.38197	0.391664
$459$	0	0
$460$	−4.09017	−0.190705
$461$	−19.7426	−0.919507	−0.459753	−	0.888047i	$-0.652062\pi$
$461$	−19.7426	−0.919507	−0.459753	+	0.888047i	$0.652062\pi$
$462$	0	0
$463$	12.7984	0.594791	0.297395	−	0.954754i	$-0.403882\pi$
$463$	12.7984	0.594791	0.297395	+	0.954754i	$0.403882\pi$
$464$	18.7082	0.868507
$465$	0	0
$466$	−47.0689	−2.18042
$467$	26.4721	1.22498	0.612492	−	0.790477i	$-0.290167\pi$
$467$	26.4721	1.22498	0.612492	+	0.790477i	$0.290167\pi$
$468$	0	0
$469$	11.7984	0.544798
$470$	4.38197	0.202125
$471$	0	0
$472$	6.05573	0.278737
$473$	−66.9574	−3.07871
$474$	0	0
$475$	2.61803	0.120124
$476$	−2.70820	−0.124130
$477$	0	0
$478$	−34.5623	−1.58084
$479$	19.7984	0.904611	0.452305	−	0.891863i	$-0.350602\pi$
$479$	19.7984	0.904611	0.452305	+	0.891863i	$0.350602\pi$
$480$	0	0
$481$	−15.2705	−0.696275
$482$	9.38197	0.427337
$483$	0	0
$484$	11.7082	0.532191
$485$	13.0344	0.591864
$486$	0	0
$487$	8.47214	0.383909	0.191955	−	0.981404i	$-0.438517\pi$
$487$	8.47214	0.383909	0.191955	+	0.981404i	$0.438517\pi$
$488$	−17.5623	−0.795008
$489$	0	0
$490$	−1.61803	−0.0730953
$491$	15.7984	0.712971	0.356485	−	0.934301i	$-0.383975\pi$
$491$	15.7984	0.712971	0.356485	+	0.934301i	$0.383975\pi$
$492$	0	0
$493$	−16.8885	−0.760622
$494$	21.5623	0.970134
$495$	0	0
$496$	−14.5623	−0.653867
$497$	−2.14590	−0.0962567
$498$	0	0
$499$	3.36068	0.150445	0.0752223	−	0.997167i	$-0.476033\pi$
$499$	3.36068	0.150445	0.0752223	+	0.997167i	$0.476033\pi$
$500$	−0.618034	−0.0276393
$501$	0	0
$502$	−33.4164	−1.49145
$503$	2.79837	0.124773	0.0623867	−	0.998052i	$-0.480129\pi$
$503$	2.79837	0.124773	0.0623867	+	0.998052i	$0.480129\pi$
$504$	0	0
$505$	−6.47214	−0.288006
$506$	58.5967	2.60494
$507$	0	0
$508$	9.76393	0.433204
$509$	12.5279	0.555288	0.277644	−	0.960684i	$-0.410446\pi$
$509$	12.5279	0.555288	0.277644	+	0.960684i	$0.410446\pi$
$510$	0	0
$511$	11.4721	0.507497
$512$	−5.29180	−0.233867
$513$	0	0
$514$	2.47214	0.109041
$515$	−14.3262	−0.631289
$516$	0	0
$517$	−14.8197	−0.651768
$518$	4.85410	0.213277
$519$	0	0
$520$	11.3820	0.499132
$521$	−8.41641	−0.368730	−0.184365	−	0.982858i	$-0.559023\pi$
$521$	−8.41641	−0.368730	−0.184365	+	0.982858i	$0.559023\pi$
$522$	0	0
$523$	−19.7426	−0.863286	−0.431643	−	0.902045i	$-0.642066\pi$
$523$	−19.7426	−0.863286	−0.431643	+	0.902045i	$0.642066\pi$
$524$	10.7082	0.467790
$525$	0	0
$526$	−32.5066	−1.41735
$527$	13.1459	0.572644
$528$	0	0
$529$	20.7984	0.904277
$530$	−7.09017	−0.307977
$531$	0	0
$532$	−1.61803	−0.0701507
$533$	8.23607	0.356744
$534$	0	0
$535$	−14.7082	−0.635891
$536$	26.3820	1.13953
$537$	0	0
$538$	−19.6180	−0.845794
$539$	5.47214	0.235702
$540$	0	0
$541$	32.7426	1.40772	0.703858	−	0.710341i	$-0.251459\pi$
$541$	32.7426	1.40772	0.703858	+	0.710341i	$0.251459\pi$
$542$	40.2148	1.72737
$543$	0	0
$544$	−14.8197	−0.635388
$545$	6.85410	0.293597
$546$	0	0
$547$	6.81966	0.291588	0.145794	−	0.989315i	$-0.453426\pi$
$547$	6.81966	0.291588	0.145794	+	0.989315i	$0.453426\pi$
$548$	−11.4164	−0.487685
$549$	0	0
$550$	8.85410	0.377540
$551$	−10.0902	−0.429856
$552$	0	0
$553$	13.7984	0.586767
$554$	25.5623	1.08604
$555$	0	0
$556$	−0.326238	−0.0138356
$557$	−7.20163	−0.305143	−0.152571	−	0.988292i	$-0.548755\pi$
$557$	−7.20163	−0.305143	−0.152571	+	0.988292i	$0.548755\pi$
$558$	0	0
$559$	−62.2837	−2.63432
$560$	−4.85410	−0.205123
$561$	0	0
$562$	−10.7984	−0.455502
$563$	11.1459	0.469744	0.234872	−	0.972026i	$-0.424533\pi$
$563$	11.1459	0.469744	0.234872	+	0.972026i	$0.424533\pi$
$564$	0	0
$565$	−1.56231	−0.0657267
$566$	−32.3262	−1.35877
$567$	0	0
$568$	−4.79837	−0.201335
$569$	4.05573	0.170025	0.0850125	−	0.996380i	$-0.472907\pi$
$569$	4.05573	0.170025	0.0850125	+	0.996380i	$0.472907\pi$
$570$	0	0
$571$	20.6738	0.865170	0.432585	−	0.901593i	$-0.357602\pi$
$571$	20.6738	0.865170	0.432585	+	0.901593i	$0.357602\pi$
$572$	17.2148	0.719786
$573$	0	0
$574$	−2.61803	−0.109275
$575$	6.61803	0.275991
$576$	0	0
$577$	28.1246	1.17084	0.585421	−	0.810729i	$-0.300929\pi$
$577$	28.1246	1.17084	0.585421	+	0.810729i	$0.300929\pi$
$578$	3.56231	0.148172
$579$	0	0
$580$	2.38197	0.0989058
$581$	−12.4164	−0.515119
$582$	0	0
$583$	23.9787	0.993097
$584$	25.6525	1.06151
$585$	0	0
$586$	−21.2361	−0.877254
$587$	−14.5066	−0.598751	−0.299375	−	0.954135i	$-0.596778\pi$
$587$	−14.5066	−0.598751	−0.299375	+	0.954135i	$0.596778\pi$
$588$	0	0
$589$	7.85410	0.323623
$590$	4.38197	0.180403
$591$	0	0
$592$	14.5623	0.598507
$593$	−35.0689	−1.44011	−0.720053	−	0.693919i	$-0.755883\pi$
$593$	−35.0689	−1.44011	−0.720053	+	0.693919i	$0.755883\pi$
$594$	0	0
$595$	4.38197	0.179643
$596$	0.0557281	0.00228271
$597$	0	0
$598$	54.5066	2.22894
$599$	46.3607	1.89425	0.947123	−	0.320871i	$-0.103975\pi$
$599$	46.3607	1.89425	0.947123	+	0.320871i	$0.103975\pi$
$600$	0	0
$601$	−28.3262	−1.15545	−0.577726	−	0.816231i	$-0.696060\pi$
$601$	−28.3262	−1.15545	−0.577726	+	0.816231i	$0.696060\pi$
$602$	19.7984	0.806921
$603$	0	0
$604$	−9.20163	−0.374409
$605$	−18.9443	−0.770194
$606$	0	0
$607$	6.41641	0.260434	0.130217	−	0.991486i	$-0.458433\pi$
$607$	6.41641	0.260434	0.130217	+	0.991486i	$0.458433\pi$
$608$	−8.85410	−0.359081
$609$	0	0
$610$	−12.7082	−0.514540
$611$	−13.7852	−0.557690
$612$	0	0
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	−7.41641	−0.299302
$615$	0	0
$616$	12.2361	0.493005
$617$	−1.03444	−0.0416451	−0.0208225	−	0.999783i	$-0.506628\pi$
$617$	−1.03444	−0.0416451	−0.0208225	+	0.999783i	$0.506628\pi$
$618$	0	0
$619$	31.0000	1.24600	0.622998	−	0.782224i	$-0.285915\pi$
$619$	31.0000	1.24600	0.622998	+	0.782224i	$0.285915\pi$
$620$	−1.85410	−0.0744625
$621$	0	0
$622$	9.65248	0.387029
$623$	−4.52786	−0.181405
$624$	0	0
$625$	1.00000	0.0400000
$626$	−47.2148	−1.88708
$627$	0	0
$628$	−3.23607	−0.129133
$629$	−13.1459	−0.524161
$630$	0	0
$631$	13.3050	0.529662	0.264831	−	0.964295i	$-0.414684\pi$
$631$	13.3050	0.529662	0.264831	+	0.964295i	$0.414684\pi$
$632$	30.8541	1.22731
$633$	0	0
$634$	9.79837	0.389143
$635$	−15.7984	−0.626939
$636$	0	0
$637$	5.09017	0.201680
$638$	−34.1246	−1.35101
$639$	0	0
$640$	−13.6180	−0.538300
$641$	36.3951	1.43752	0.718760	−	0.695258i	$-0.244710\pi$
$641$	36.3951	1.43752	0.718760	+	0.695258i	$0.244710\pi$
$642$	0	0
$643$	18.6180	0.734224	0.367112	−	0.930177i	$-0.380347\pi$
$643$	18.6180	0.734224	0.367112	+	0.930177i	$0.380347\pi$
$644$	−4.09017	−0.161175
$645$	0	0
$646$	18.5623	0.730324
$647$	1.05573	0.0415050	0.0207525	−	0.999785i	$-0.493394\pi$
$647$	1.05573	0.0415050	0.0207525	+	0.999785i	$0.493394\pi$
$648$	0	0
$649$	−14.8197	−0.581723
$650$	8.23607	0.323045
$651$	0	0
$652$	−4.65248	−0.182205
$653$	−49.4508	−1.93516	−0.967581	−	0.252562i	$-0.918727\pi$
$653$	−49.4508	−1.93516	−0.967581	+	0.252562i	$0.918727\pi$
$654$	0	0
$655$	−17.3262	−0.676992
$656$	−7.85410	−0.306651
$657$	0	0
$658$	4.38197	0.170827
$659$	12.7639	0.497212	0.248606	−	0.968605i	$-0.420028\pi$
$659$	12.7639	0.497212	0.248606	+	0.968605i	$0.420028\pi$
$660$	0	0
$661$	−15.6738	−0.609639	−0.304819	−	0.952410i	$-0.598596\pi$
$661$	−15.6738	−0.609639	−0.304819	+	0.952410i	$0.598596\pi$
$662$	6.70820	0.260722
$663$	0	0
$664$	−27.7639	−1.07745
$665$	2.61803	0.101523
$666$	0	0
$667$	−25.5066	−0.987619
$668$	−5.09017	−0.196945
$669$	0	0
$670$	19.0902	0.737518
$671$	42.9787	1.65917
$672$	0	0
$673$	20.9443	0.807342	0.403671	−	0.914904i	$-0.367734\pi$
$673$	20.9443	0.807342	0.403671	+	0.914904i	$0.367734\pi$
$674$	−22.2361	−0.856501
$675$	0	0
$676$	7.97871	0.306874
$677$	−29.4721	−1.13271	−0.566353	−	0.824163i	$-0.691646\pi$
$677$	−29.4721	−1.13271	−0.566353	+	0.824163i	$0.691646\pi$
$678$	0	0
$679$	13.0344	0.500216
$680$	9.79837	0.375750
$681$	0	0
$682$	26.5623	1.01712
$683$	−19.8541	−0.759696	−0.379848	−	0.925049i	$-0.624024\pi$
$683$	−19.8541	−0.759696	−0.379848	+	0.925049i	$0.624024\pi$
$684$	0	0
$685$	18.4721	0.705784
$686$	−1.61803	−0.0617768
$687$	0	0
$688$	59.3951	2.26442
$689$	22.3050	0.849751
$690$	0	0
$691$	−17.6525	−0.671532	−0.335766	−	0.941945i	$-0.608995\pi$
$691$	−17.6525	−0.671532	−0.335766	+	0.941945i	$0.608995\pi$
$692$	−9.38197	−0.356649
$693$	0	0
$694$	32.6525	1.23947
$695$	0.527864	0.0200230
$696$	0	0
$697$	7.09017	0.268559
$698$	20.7426	0.785121
$699$	0	0
$700$	−0.618034	−0.0233595
$701$	−21.1803	−0.799970	−0.399985	−	0.916522i	$-0.630985\pi$
$701$	−21.1803	−0.799970	−0.399985	+	0.916522i	$0.630985\pi$
$702$	0	0
$703$	−7.85410	−0.296223
$704$	23.1803	0.873642
$705$	0	0
$706$	47.6525	1.79342
$707$	−6.47214	−0.243410
$708$	0	0
$709$	17.2361	0.647314	0.323657	−	0.946174i	$-0.395088\pi$
$709$	17.2361	0.647314	0.323657	+	0.946174i	$0.395088\pi$
$710$	−3.47214	−0.130307
$711$	0	0
$712$	−10.1246	−0.379436
$713$	19.8541	0.743542
$714$	0	0
$715$	−27.8541	−1.04168
$716$	−9.09017	−0.339716
$717$	0	0
$718$	−23.7984	−0.888147
$719$	32.7426	1.22109	0.610547	−	0.791980i	$-0.290950\pi$
$719$	32.7426	1.22109	0.610547	+	0.791980i	$0.290950\pi$
$720$	0	0
$721$	−14.3262	−0.533537
$722$	−19.6525	−0.731389
$723$	0	0
$724$	0.506578	0.0188268
$725$	−3.85410	−0.143138
$726$	0	0
$727$	−43.7426	−1.62232	−0.811162	−	0.584821i	$-0.801165\pi$
$727$	−43.7426	−1.62232	−0.811162	+	0.584821i	$0.801165\pi$
$728$	11.3820	0.421844
$729$	0	0
$730$	18.5623	0.687022
$731$	−53.6180	−1.98313
$732$	0	0
$733$	17.4377	0.644076	0.322038	−	0.946727i	$-0.395632\pi$
$733$	17.4377	0.644076	0.322038	+	0.946727i	$0.395632\pi$
$734$	−51.7426	−1.90986
$735$	0	0
$736$	−22.3820	−0.825010
$737$	−64.5623	−2.37818
$738$	0	0
$739$	42.1803	1.55163	0.775814	−	0.630961i	$-0.217339\pi$
$739$	42.1803	1.55163	0.775814	+	0.630961i	$0.217339\pi$
$740$	1.85410	0.0681581
$741$	0	0
$742$	−7.09017	−0.260288
$743$	42.1591	1.54667	0.773333	−	0.634000i	$-0.218588\pi$
$743$	42.1591	1.54667	0.773333	+	0.634000i	$0.218588\pi$
$744$	0	0
$745$	−0.0901699	−0.00330357
$746$	25.6525	0.939204
$747$	0	0
$748$	14.8197	0.541860
$749$	−14.7082	−0.537426
$750$	0	0
$751$	14.9443	0.545324	0.272662	−	0.962110i	$-0.412096\pi$
$751$	14.9443	0.545324	0.272662	+	0.962110i	$0.412096\pi$
$752$	13.1459	0.479382
$753$	0	0
$754$	−31.7426	−1.15600
$755$	14.8885	0.541850
$756$	0	0
$757$	−31.6869	−1.15168	−0.575840	−	0.817562i	$-0.695325\pi$
$757$	−31.6869	−1.15168	−0.575840	+	0.817562i	$0.695325\pi$
$758$	38.6525	1.40392
$759$	0	0
$760$	5.85410	0.212351
$761$	21.0902	0.764518	0.382259	−	0.924055i	$-0.375146\pi$
$761$	21.0902	0.764518	0.382259	+	0.924055i	$0.375146\pi$
$762$	0	0
$763$	6.85410	0.248135
$764$	11.5066	0.416293
$765$	0	0
$766$	39.6869	1.43395
$767$	−13.7852	−0.497755
$768$	0	0
$769$	30.8885	1.11387	0.556935	−	0.830556i	$-0.311977\pi$
$769$	30.8885	1.11387	0.556935	+	0.830556i	$0.311977\pi$
$770$	8.85410	0.319080
$771$	0	0
$772$	−10.2361	−0.368404
$773$	−13.2016	−0.474829	−0.237415	−	0.971408i	$-0.576300\pi$
$773$	−13.2016	−0.474829	−0.237415	+	0.971408i	$0.576300\pi$
$774$	0	0
$775$	3.00000	0.107763
$776$	29.1459	1.04628
$777$	0	0
$778$	−35.7426	−1.28144
$779$	4.23607	0.151773
$780$	0	0
$781$	11.7426	0.420185
$782$	46.9230	1.67796
$783$	0	0
$784$	−4.85410	−0.173361
$785$	5.23607	0.186883
$786$	0	0
$787$	23.5967	0.841133	0.420567	−	0.907262i	$-0.361831\pi$
$787$	23.5967	0.841133	0.420567	+	0.907262i	$0.361831\pi$
$788$	9.20163	0.327794
$789$	0	0
$790$	22.3262	0.794332
$791$	−1.56231	−0.0555492
$792$	0	0
$793$	39.9787	1.41969
$794$	−55.5967	−1.97305
$795$	0	0
$796$	16.7082	0.592207
$797$	4.56231	0.161605	0.0808026	−	0.996730i	$-0.474252\pi$
$797$	4.56231	0.161605	0.0808026	+	0.996730i	$0.474252\pi$
$798$	0	0
$799$	−11.8673	−0.419833
$800$	−3.38197	−0.119571
$801$	0	0
$802$	18.4164	0.650306
$803$	−62.7771	−2.21536
$804$	0	0
$805$	6.61803	0.233255
$806$	24.7082	0.870309
$807$	0	0
$808$	−14.4721	−0.509128
$809$	7.18034	0.252447	0.126224	−	0.992002i	$-0.459714\pi$
$809$	7.18034	0.252447	0.126224	+	0.992002i	$0.459714\pi$
$810$	0	0
$811$	−32.7984	−1.15171	−0.575853	−	0.817553i	$-0.695330\pi$
$811$	−32.7984	−1.15171	−0.575853	+	0.817553i	$0.695330\pi$
$812$	2.38197	0.0835906
$813$	0	0
$814$	−26.5623	−0.931008
$815$	7.52786	0.263690
$816$	0	0
$817$	−32.0344	−1.12074
$818$	−13.7082	−0.479296
$819$	0	0
$820$	−1.00000	−0.0349215
$821$	18.1246	0.632553	0.316277	−	0.948667i	$-0.397567\pi$
$821$	18.1246	0.632553	0.316277	+	0.948667i	$0.397567\pi$
$822$	0	0
$823$	7.76393	0.270634	0.135317	−	0.990802i	$-0.456795\pi$
$823$	7.76393	0.270634	0.135317	+	0.990802i	$0.456795\pi$
$824$	−32.0344	−1.11597
$825$	0	0
$826$	4.38197	0.152468
$827$	−44.9574	−1.56332	−0.781661	−	0.623703i	$-0.785628\pi$
$827$	−44.9574	−1.56332	−0.781661	+	0.623703i	$0.785628\pi$
$828$	0	0
$829$	9.76393	0.339115	0.169558	−	0.985520i	$-0.445766\pi$
$829$	9.76393	0.339115	0.169558	+	0.985520i	$0.445766\pi$
$830$	−20.0902	−0.697340
$831$	0	0
$832$	21.5623	0.747538
$833$	4.38197	0.151826
$834$	0	0
$835$	8.23607	0.285021
$836$	8.85410	0.306226
$837$	0	0
$838$	17.0344	0.588445
$839$	−29.5066	−1.01868	−0.509340	−	0.860565i	$-0.670110\pi$
$839$	−29.5066	−1.01868	−0.509340	+	0.860565i	$0.670110\pi$
$840$	0	0
$841$	−14.1459	−0.487790
$842$	−52.0132	−1.79249
$843$	0	0
$844$	−6.00000	−0.206529
$845$	−12.9098	−0.444112
$846$	0	0
$847$	−18.9443	−0.650933
$848$	−21.2705	−0.730432
$849$	0	0
$850$	7.09017	0.243191
$851$	−19.8541	−0.680590
$852$	0	0
$853$	1.00000	0.0342393	0.0171197	−	0.999853i	$-0.494550\pi$
$853$	1.00000	0.0342393	0.0171197	+	0.999853i	$0.494550\pi$
$854$	−12.7082	−0.434866
$855$	0	0
$856$	−32.8885	−1.12411
$857$	−44.8885	−1.53336	−0.766682	−	0.642027i	$-0.778094\pi$
$857$	−44.8885	−1.53336	−0.766682	+	0.642027i	$0.778094\pi$
$858$	0	0
$859$	−44.0689	−1.50361	−0.751805	−	0.659385i	$-0.770817\pi$
$859$	−44.0689	−1.50361	−0.751805	+	0.659385i	$0.770817\pi$
$860$	7.56231	0.257872
$861$	0	0
$862$	7.56231	0.257573
$863$	0.944272	0.0321434	0.0160717	−	0.999871i	$-0.494884\pi$
$863$	0.944272	0.0321434	0.0160717	+	0.999871i	$0.494884\pi$
$864$	0	0
$865$	15.1803	0.516147
$866$	−6.61803	−0.224890
$867$	0	0
$868$	−1.85410	−0.0629323
$869$	−75.5066	−2.56139
$870$	0	0
$871$	−60.0557	−2.03491
$872$	15.3262	0.519012
$873$	0	0
$874$	28.0344	0.948279
$875$	1.00000	0.0338062
$876$	0	0
$877$	−22.3475	−0.754622	−0.377311	−	0.926087i	$-0.623151\pi$
$877$	−22.3475	−0.754622	−0.377311	+	0.926087i	$0.623151\pi$
$878$	55.6869	1.87934
$879$	0	0
$880$	26.5623	0.895415
$881$	−36.4721	−1.22878	−0.614389	−	0.789003i	$-0.710597\pi$
$881$	−36.4721	−1.22878	−0.614389	+	0.789003i	$0.710597\pi$
$882$	0	0
$883$	−9.94427	−0.334651	−0.167326	−	0.985902i	$-0.553513\pi$
$883$	−9.94427	−0.334651	−0.167326	+	0.985902i	$0.553513\pi$
$884$	13.7852	0.463647
$885$	0	0
$886$	−21.0000	−0.705509
$887$	46.1803	1.55058	0.775292	−	0.631603i	$-0.217603\pi$
$887$	46.1803	1.55058	0.775292	+	0.631603i	$0.217603\pi$
$888$	0	0
$889$	−15.7984	−0.529860
$890$	−7.32624	−0.245576
$891$	0	0
$892$	−6.50658	−0.217856
$893$	−7.09017	−0.237263
$894$	0	0
$895$	14.7082	0.491641
$896$	−13.6180	−0.454947
$897$	0	0
$898$	−21.1246	−0.704937
$899$	−11.5623	−0.385624
$900$	0	0
$901$	19.2016	0.639699
$902$	14.3262	0.477012
$903$	0	0
$904$	−3.49342	−0.116189
$905$	−0.819660	−0.0272464
$906$	0	0
$907$	−24.6869	−0.819716	−0.409858	−	0.912149i	$-0.634422\pi$
$907$	−24.6869	−0.819716	−0.409858	+	0.912149i	$0.634422\pi$
$908$	−17.4721	−0.579833
$909$	0	0
$910$	8.23607	0.273023
$911$	25.5279	0.845776	0.422888	−	0.906182i	$-0.361016\pi$
$911$	25.5279	0.845776	0.422888	+	0.906182i	$0.361016\pi$
$912$	0	0
$913$	67.9443	2.24863
$914$	−4.79837	−0.158716
$915$	0	0
$916$	3.20163	0.105785
$917$	−17.3262	−0.572163
$918$	0	0
$919$	−4.09017	−0.134922	−0.0674611	−	0.997722i	$-0.521490\pi$
$919$	−4.09017	−0.134922	−0.0674611	+	0.997722i	$0.521490\pi$
$920$	14.7984	0.487888
$921$	0	0
$922$	−31.9443	−1.05203
$923$	10.9230	0.359534
$924$	0	0
$925$	−3.00000	−0.0986394
$926$	20.7082	0.680514
$927$	0	0
$928$	13.0344	0.427877
$929$	3.97871	0.130537	0.0652687	−	0.997868i	$-0.479210\pi$
$929$	3.97871	0.130537	0.0652687	+	0.997868i	$0.479210\pi$
$930$	0	0
$931$	2.61803	0.0858026
$932$	−17.9787	−0.588912
$933$	0	0
$934$	42.8328	1.40153
$935$	−23.9787	−0.784188
$936$	0	0
$937$	−52.4721	−1.71419	−0.857095	−	0.515158i	$-0.827733\pi$
$937$	−52.4721	−1.71419	−0.857095	+	0.515158i	$0.827733\pi$
$938$	19.0902	0.623316
$939$	0	0
$940$	1.67376	0.0545921
$941$	−27.6525	−0.901445	−0.450722	−	0.892664i	$-0.648834\pi$
$941$	−27.6525	−0.901445	−0.450722	+	0.892664i	$0.648834\pi$
$942$	0	0
$943$	10.7082	0.348707
$944$	13.1459	0.427863
$945$	0	0
$946$	−108.339	−3.52242
$947$	39.6869	1.28965	0.644826	−	0.764330i	$-0.276930\pi$
$947$	39.6869	1.28965	0.644826	+	0.764330i	$0.276930\pi$
$948$	0	0
$949$	−58.3951	−1.89559
$950$	4.23607	0.137436
$951$	0	0
$952$	9.79837	0.317567
$953$	38.4853	1.24666	0.623330	−	0.781959i	$-0.285779\pi$
$953$	38.4853	1.24666	0.623330	+	0.781959i	$0.285779\pi$
$954$	0	0
$955$	−18.6180	−0.602465
$956$	−13.2016	−0.426971
$957$	0	0
$958$	32.0344	1.03499
$959$	18.4721	0.596496
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−24.7082	−0.796624
$963$	0	0
$964$	3.58359	0.115420
$965$	16.5623	0.533159
$966$	0	0
$967$	56.5967	1.82003	0.910014	−	0.414577i	$-0.136070\pi$
$967$	56.5967	1.82003	0.910014	+	0.414577i	$0.136070\pi$
$968$	−42.3607	−1.36152
$969$	0	0
$970$	21.0902	0.677165
$971$	16.5836	0.532193	0.266096	−	0.963946i	$-0.414266\pi$
$971$	16.5836	0.532193	0.266096	+	0.963946i	$0.414266\pi$
$972$	0	0
$973$	0.527864	0.0169225
$974$	13.7082	0.439239
$975$	0	0
$976$	−38.1246	−1.22034
$977$	−33.7771	−1.08062	−0.540312	−	0.841465i	$-0.681694\pi$
$977$	−33.7771	−1.08062	−0.540312	+	0.841465i	$0.681694\pi$
$978$	0	0
$979$	24.7771	0.791879
$980$	−0.618034	−0.0197424
$981$	0	0
$982$	25.5623	0.815726
$983$	8.97871	0.286376	0.143188	−	0.989695i	$-0.454265\pi$
$983$	8.97871	0.286376	0.143188	+	0.989695i	$0.454265\pi$
$984$	0	0
$985$	−14.8885	−0.474388
$986$	−27.3262	−0.870245
$987$	0	0
$988$	8.23607	0.262024
$989$	−80.9787	−2.57497
$990$	0	0
$991$	−4.05573	−0.128834	−0.0644172	−	0.997923i	$-0.520519\pi$
$991$	−4.05573	−0.128834	−0.0644172	+	0.997923i	$0.520519\pi$
$992$	−10.1459	−0.322133
$993$	0	0
$994$	−3.47214	−0.110129
$995$	−27.0344	−0.857049
$996$	0	0
$997$	28.1246	0.890715	0.445358	−	0.895353i	$-0.353077\pi$
$997$	28.1246	0.890715	0.445358	+	0.895353i	$0.353077\pi$
$998$	5.43769	0.172127
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.i.1.2	yes	2
3.2	odd	2		945.2.a.d.1.1	✓	2
5.4	even	2		4725.2.a.x.1.1		2
7.6	odd	2		6615.2.a.s.1.2		2
15.14	odd	2		4725.2.a.bc.1.2		2
21.20	even	2		6615.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.d.1.1	✓	2	3.2	odd	2
945.2.a.i.1.2	yes	2	1.1	even	1	trivial
4725.2.a.x.1.1		2	5.4	even	2
4725.2.a.bc.1.2		2	15.14	odd	2
6615.2.a.m.1.1		2	21.20	even	2
6615.2.a.s.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)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} -1.61803 q^{10} +5.47214 q^{11} +5.09017 q^{13} -1.61803 q^{14} -4.85410 q^{16} +4.38197 q^{17} +2.61803 q^{19} -0.618034 q^{20} +8.85410 q^{22} +6.61803 q^{23} +1.00000 q^{25} +8.23607 q^{26} -0.618034 q^{28} -3.85410 q^{29} +3.00000 q^{31} -3.38197 q^{32} +7.09017 q^{34} +1.00000 q^{35} -3.00000 q^{37} +4.23607 q^{38} +2.23607 q^{40} +1.61803 q^{41} -12.2361 q^{43} +3.38197 q^{44} +10.7082 q^{46} -2.70820 q^{47} +1.00000 q^{49} +1.61803 q^{50} +3.14590 q^{52} +4.38197 q^{53} -5.47214 q^{55} +2.23607 q^{56} -6.23607 q^{58} -2.70820 q^{59} +7.85410 q^{61} +4.85410 q^{62} +4.23607 q^{64} -5.09017 q^{65} -11.7984 q^{67} +2.70820 q^{68} +1.61803 q^{70} +2.14590 q^{71} -11.4721 q^{73} -4.85410 q^{74} +1.61803 q^{76} -5.47214 q^{77} -13.7984 q^{79} +4.85410 q^{80} +2.61803 q^{82} +12.4164 q^{83} -4.38197 q^{85} -19.7984 q^{86} -12.2361 q^{88} +4.52786 q^{89} -5.09017 q^{91} +4.09017 q^{92} -4.38197 q^{94} -2.61803 q^{95} -13.0344 q^{97} +1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7	\( 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7} - q^{10} + 2 q^{11} - q^{13} - q^{14} - 3 q^{16} + 11 q^{17} + 3 q^{19} + q^{20} + 11 q^{22} + 11 q^{23} + 2 q^{25} + 12 q^{26} + q^{28} - q^{29} + 6 q^{31} - 9 q^{32} + 3 q^{34} + 2 q^{35} - 6 q^{37} + 4 q^{38} + q^{41} - 20 q^{43} + 9 q^{44} + 8 q^{46} + 8 q^{47} + 2 q^{49} + q^{50} + 13 q^{52} + 11 q^{53} - 2 q^{55} - 8 q^{58} + 8 q^{59} + 9 q^{61} + 3 q^{62} + 4 q^{64} + q^{65} + q^{67} - 8 q^{68} + q^{70} + 11 q^{71} - 14 q^{73} - 3 q^{74} + q^{76} - 2 q^{77} - 3 q^{79} + 3 q^{80} + 3 q^{82} - 2 q^{83} - 11 q^{85} - 15 q^{86} - 20 q^{88} + 18 q^{89} + q^{91} - 3 q^{92} - 11 q^{94} - 3 q^{95} + 3 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7 - q^10 + 2 * q^11 - q^13 - q^14 - 3 * q^16 + 11 * q^17 + 3 * q^19 + q^20 + 11 * q^22 + 11 * q^23 + 2 * q^25 + 12 * q^26 + q^28 - q^29 + 6 * q^31 - 9 * q^32 + 3 * q^34 + 2 * q^35 - 6 * q^37 + 4 * q^38 + q^41 - 20 * q^43 + 9 * q^44 + 8 * q^46 + 8 * q^47 + 2 * q^49 + q^50 + 13 * q^52 + 11 * q^53 - 2 * q^55 - 8 * q^58 + 8 * q^59 + 9 * q^61 + 3 * q^62 + 4 * q^64 + q^65 + q^67 - 8 * q^68 + q^70 + 11 * q^71 - 14 * q^73 - 3 * q^74 + q^76 - 2 * q^77 - 3 * q^79 + 3 * q^80 + 3 * q^82 - 2 * q^83 - 11 * q^85 - 15 * q^86 - 20 * q^88 + 18 * q^89 + q^91 - 3 * q^92 - 11 * q^94 - 3 * q^95 + 3 * q^97 + q^98

Introduction

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