LMFDB - Embedded newform 945.2.a.d.1.1

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			1.1
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	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	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	$-0.808800\pi$
$11$	−5.47214	−1.64991	−0.824956	+	0.565198i	$0.808800\pi$
$12$	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	$-0.678331\pi$
$17$	−4.38197	−1.06278	−0.531391	+	0.847126i	$0.678331\pi$
$18$	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	$-0.742380\pi$
$23$	−6.61803	−1.37996	−0.689978	+	0.723831i	$0.742380\pi$
$24$	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	$-0.383512\pi$
$29$	3.85410	0.715689	0.357844	+	0.933781i	$0.383512\pi$
$30$	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	$-0.540325\pi$
$41$	−1.61803	−0.252694	−0.126347	+	0.991986i	$0.540325\pi$
$42$	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	$-0.436713\pi$
$47$	2.70820	0.395032	0.197516	+	0.980300i	$0.436713\pi$
$48$	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	$-0.597305\pi$
$53$	−4.38197	−0.601909	−0.300955	+	0.953638i	$0.597305\pi$
$54$	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	$-0.443591\pi$
$59$	2.70820	0.352578	0.176289	+	0.984338i	$0.443591\pi$
$60$	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	$-0.540643\pi$
$71$	−2.14590	−0.254671	−0.127336	+	0.991860i	$0.540643\pi$
$72$	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	$-0.738645\pi$
$83$	−12.4164	−1.36288	−0.681439	+	0.731875i	$0.738645\pi$
$84$	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	$-0.577140\pi$
$89$	−4.52786	−0.479953	−0.239976	+	0.970779i	$0.577140\pi$
$90$	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	$-0.604355\pi$
$101$	−6.47214	−0.644002	−0.322001	+	0.946739i	$0.604355\pi$
$102$	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	$-0.751734\pi$
$107$	−14.7082	−1.42190	−0.710948	+	0.703245i	$0.751734\pi$
$108$	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	$-0.523412\pi$
$113$	−1.56231	−0.146969	−0.0734847	+	0.997296i	$0.523412\pi$
$114$	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	$-0.773287\pi$
$131$	−17.3262	−1.51380	−0.756900	+	0.653530i	$0.773287\pi$
$132$	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	$-0.210552\pi$
$137$	18.4721	1.57818	0.789091	+	0.614277i	$0.210552\pi$
$138$	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	$-0.501176\pi$
$149$	−0.0901699	−0.00738701	−0.00369350	+	0.999993i	$0.501176\pi$
$150$	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	$-0.396766\pi$
$167$	8.23607	0.637326	0.318663	+	0.947868i	$0.396766\pi$
$168$	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	$-0.304196\pi$
$173$	15.1803	1.15414	0.577070	+	0.816695i	$0.304196\pi$
$174$	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	$-0.314753\pi$
$179$	14.7082	1.09934	0.549671	+	0.835381i	$0.314753\pi$
$180$	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	$-0.735243\pi$
$191$	−18.6180	−1.34715	−0.673577	+	0.739117i	$0.735243\pi$
$192$	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	$-0.677952\pi$
$197$	−14.8885	−1.06076	−0.530382	+	0.847759i	$0.677952\pi$
$198$	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	$-0.112501\pi$
$227$	28.2705	1.87638	0.938190	+	0.346121i	$0.112501\pi$
$228$	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	$-0.0981041\pi$
$233$	29.0902	1.90576	0.952880	+	0.303347i	$0.0981041\pi$
$234$	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	$-0.257235\pi$
$239$	21.3607	1.38171	0.690854	+	0.722995i	$0.257235\pi$
$240$	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	$-0.274020\pi$
$251$	20.6525	1.30357	0.651786	+	0.758403i	$0.274020\pi$
$252$	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	$-0.515174\pi$
$257$	−1.52786	−0.0953055	−0.0476528	+	0.998864i	$0.515174\pi$
$258$	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	$-0.287373\pi$
$263$	20.0902	1.23881	0.619406	+	0.785070i	$0.287373\pi$
$264$	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	$-0.379486\pi$
$269$	12.1246	0.739251	0.369625	+	0.929181i	$0.379486\pi$
$270$	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	$-0.436211\pi$
$281$	6.67376	0.398123	0.199062	+	0.979987i	$0.436211\pi$
$282$	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	$-0.374762\pi$
$293$	13.1246	0.766748	0.383374	+	0.923593i	$0.374762\pi$
$294$	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	$-0.554098\pi$
$311$	−5.96556	−0.338276	−0.169138	+	0.985592i	$0.554098\pi$
$312$	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	$-0.554397\pi$
$317$	−6.05573	−0.340124	−0.170062	+	0.985433i	$0.554397\pi$
$318$	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	$-0.682207\pi$
$347$	−20.1803	−1.08334	−0.541669	+	0.840592i	$0.682207\pi$
$348$	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	$-0.786698\pi$
$353$	−29.4508	−1.56751	−0.783755	+	0.621070i	$0.786698\pi$
$354$	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	$-0.373120\pi$
$359$	14.7082	0.776269	0.388135	+	0.921603i	$0.373120\pi$
$360$	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	$-0.715578\pi$
$383$	−24.5279	−1.25332	−0.626658	+	0.779295i	$0.715578\pi$
$384$	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	$-0.310798\pi$
$389$	22.0902	1.12002	0.560008	+	0.828487i	$0.310798\pi$
$390$	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	$-0.591726\pi$
$401$	−11.3820	−0.568388	−0.284194	+	0.958767i	$0.591726\pi$
$402$	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	$-0.582787\pi$
$419$	−10.5279	−0.514320	−0.257160	+	0.966369i	$0.582787\pi$
$420$	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	$-0.535906\pi$
$431$	−4.67376	−0.225127	−0.112564	+	0.993645i	$0.535906\pi$
$432$	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	$-0.400234\pi$
$443$	12.9787	0.616637	0.308319	+	0.951283i	$0.400234\pi$
$444$	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	$-0.400317\pi$
$449$	13.0557	0.616138	0.308069	+	0.951364i	$0.400317\pi$
$450$	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	$-0.347938\pi$
$461$	19.7426	0.919507	0.459753	+	0.888047i	$0.347938\pi$
$462$	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	$-0.709833\pi$
$467$	−26.4721	−1.22498	−0.612492	+	0.790477i	$0.709833\pi$
$468$	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	$-0.649398\pi$
$479$	−19.7984	−0.904611	−0.452305	+	0.891863i	$0.649398\pi$
$480$	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	$-0.616025\pi$
$491$	−15.7984	−0.712971	−0.356485	+	0.934301i	$0.616025\pi$
$492$	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	$-0.519871\pi$
$503$	−2.79837	−0.124773	−0.0623867	+	0.998052i	$0.519871\pi$
$504$	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	$-0.589554\pi$
$509$	−12.5279	−0.555288	−0.277644	+	0.960684i	$0.589554\pi$
$510$	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	$-0.440977\pi$
$521$	8.41641	0.368730	0.184365	+	0.982858i	$0.440977\pi$
$522$	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	$-0.451245\pi$
$557$	7.20163	0.305143	0.152571	+	0.988292i	$0.451245\pi$
$558$	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	$-0.575467\pi$
$563$	−11.1459	−0.469744	−0.234872	+	0.972026i	$0.575467\pi$
$564$	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	$-0.527093\pi$
$569$	−4.05573	−0.170025	−0.0850125	+	0.996380i	$0.527093\pi$
$570$	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	$-0.403222\pi$
$587$	14.5066	0.598751	0.299375	+	0.954135i	$0.403222\pi$
$588$	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	$-0.244117\pi$
$593$	35.0689	1.44011	0.720053	+	0.693919i	$0.244117\pi$
$594$	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	$-0.896025\pi$
$599$	−46.3607	−1.89425	−0.947123	+	0.320871i	$0.896025\pi$
$600$	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	$-0.493372\pi$
$617$	1.03444	0.0416451	0.0208225	+	0.999783i	$0.493372\pi$
$618$	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	$-0.755290\pi$
$641$	−36.3951	−1.43752	−0.718760	+	0.695258i	$0.755290\pi$
$642$	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	$-0.506606\pi$
$647$	−1.05573	−0.0415050	−0.0207525	+	0.999785i	$0.506606\pi$
$648$	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	$-0.0812733\pi$
$653$	49.4508	1.93516	0.967581	+	0.252562i	$0.0812733\pi$
$654$	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	$-0.579972\pi$
$659$	−12.7639	−0.497212	−0.248606	+	0.968605i	$0.579972\pi$
$660$	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	$-0.308354\pi$
$677$	29.4721	1.13271	0.566353	+	0.824163i	$0.308354\pi$
$678$	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	$-0.375976\pi$
$683$	19.8541	0.759696	0.379848	+	0.925049i	$0.375976\pi$
$684$	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	$-0.369015\pi$
$701$	21.1803	0.799970	0.399985	+	0.916522i	$0.369015\pi$
$702$	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	$-0.709050\pi$
$719$	−32.7426	−1.22109	−0.610547	+	0.791980i	$0.709050\pi$
$720$	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	$-0.781412\pi$
$743$	−42.1591	−1.54667	−0.773333	+	0.634000i	$0.781412\pi$
$744$	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	$-0.624854\pi$
$761$	−21.0902	−0.764518	−0.382259	+	0.924055i	$0.624854\pi$
$762$	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	$-0.423700\pi$
$773$	13.2016	0.474829	0.237415	+	0.971408i	$0.423700\pi$
$774$	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	$-0.525748\pi$
$797$	−4.56231	−0.161605	−0.0808026	+	0.996730i	$0.525748\pi$
$798$	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	$-0.540286\pi$
$809$	−7.18034	−0.252447	−0.126224	+	0.992002i	$0.540286\pi$
$810$	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	$-0.602433\pi$
$821$	−18.1246	−0.632553	−0.316277	+	0.948667i	$0.602433\pi$
$822$	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	$-0.214372\pi$
$827$	44.9574	1.56332	0.781661	+	0.623703i	$0.214372\pi$
$828$	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	$-0.329890\pi$
$839$	29.5066	1.01868	0.509340	+	0.860565i	$0.329890\pi$
$840$	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	$-0.221906\pi$
$857$	44.8885	1.53336	0.766682	+	0.642027i	$0.221906\pi$
$858$	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	$-0.505116\pi$
$863$	−0.944272	−0.0321434	−0.0160717	+	0.999871i	$0.505116\pi$
$864$	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	$-0.289403\pi$
$881$	36.4721	1.22878	0.614389	+	0.789003i	$0.289403\pi$
$882$	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	$-0.782397\pi$
$887$	−46.1803	−1.55058	−0.775292	+	0.631603i	$0.782397\pi$
$888$	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	$-0.638984\pi$
$911$	−25.5279	−0.845776	−0.422888	+	0.906182i	$0.638984\pi$
$912$	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	$-0.520790\pi$
$929$	−3.97871	−0.130537	−0.0652687	+	0.997868i	$0.520790\pi$
$930$	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	$-0.351166\pi$
$941$	27.6525	0.901445	0.450722	+	0.892664i	$0.351166\pi$
$942$	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	$-0.723070\pi$
$947$	−39.6869	−1.28965	−0.644826	+	0.764330i	$0.723070\pi$
$948$	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	$-0.714221\pi$
$953$	−38.4853	−1.24666	−0.623330	+	0.781959i	$0.714221\pi$
$954$	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	$-0.585734\pi$
$971$	−16.5836	−0.532193	−0.266096	+	0.963946i	$0.585734\pi$
$972$	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	$-0.318306\pi$
$977$	33.7771	1.08062	0.540312	+	0.841465i	$0.318306\pi$
$978$	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	$-0.545735\pi$
$983$	−8.97871	−0.286376	−0.143188	+	0.989695i	$0.545735\pi$
$984$	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.d.1.1	✓	2
3.2	odd	2		945.2.a.i.1.2	yes	2
5.4	even	2		4725.2.a.bc.1.2		2
7.6	odd	2		6615.2.a.m.1.1		2
15.14	odd	2		4725.2.a.x.1.1		2
21.20	even	2		6615.2.a.s.1.2		2

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
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

L-functions

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