LMFDB - Embedded newform 1755.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,2,Mod(1,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, 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("1755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1755.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:	$14.0137455547$
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	$\chi$	$=$	1755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -3.23607 q^{7} +2.23607 q^{8} +O(q^{10})$	\(q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -3.23607 q^{7} +2.23607 q^{8} +1.61803 q^{10} -1.23607 q^{11} -1.00000 q^{13} +5.23607 q^{14} -4.85410 q^{16} +4.23607 q^{17} +3.47214 q^{19} -0.618034 q^{20} +2.00000 q^{22} +7.47214 q^{23} +1.00000 q^{25} +1.61803 q^{26} -2.00000 q^{28} -0.763932 q^{29} -3.00000 q^{31} +3.38197 q^{32} -6.85410 q^{34} +3.23607 q^{35} -6.00000 q^{37} -5.61803 q^{38} -2.23607 q^{40} +1.52786 q^{41} -2.47214 q^{43} -0.763932 q^{44} -12.0902 q^{46} +6.47214 q^{47} +3.47214 q^{49} -1.61803 q^{50} -0.618034 q^{52} +6.70820 q^{53} +1.23607 q^{55} -7.23607 q^{56} +1.23607 q^{58} -6.00000 q^{59} -11.0000 q^{61} +4.85410 q^{62} +4.23607 q^{64} +1.00000 q^{65} +1.70820 q^{67} +2.61803 q^{68} -5.23607 q^{70} -13.2361 q^{71} +9.70820 q^{74} +2.14590 q^{76} +4.00000 q^{77} +6.70820 q^{79} +4.85410 q^{80} -2.47214 q^{82} -3.76393 q^{83} -4.23607 q^{85} +4.00000 q^{86} -2.76393 q^{88} -3.70820 q^{89} +3.23607 q^{91} +4.61803 q^{92} -10.4721 q^{94} -3.47214 q^{95} +2.47214 q^{97} -5.61803 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{13} + 6 q^{14} - 3 q^{16} + 4 q^{17} - 2 q^{19} + q^{20} + 4 q^{22} + 6 q^{23} + 2 q^{25} + q^{26} - 4 q^{28} - 6 q^{29} - 6 q^{31} + 9 q^{32} - 7 q^{34} + 2 q^{35} - 12 q^{37} - 9 q^{38} + 12 q^{41} + 4 q^{43} - 6 q^{44} - 13 q^{46} + 4 q^{47} - 2 q^{49} - q^{50} + q^{52} - 2 q^{55} - 10 q^{56} - 2 q^{58} - 12 q^{59} - 22 q^{61} + 3 q^{62} + 4 q^{64} + 2 q^{65} - 10 q^{67} + 3 q^{68} - 6 q^{70} - 22 q^{71} + 6 q^{74} + 11 q^{76} + 8 q^{77} + 3 q^{80} + 4 q^{82} - 12 q^{83} - 4 q^{85} + 8 q^{86} - 10 q^{88} + 6 q^{89} + 2 q^{91} + 7 q^{92} - 12 q^{94} + 2 q^{95} - 4 q^{97} - 9 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 2 * q^5 - 2 * q^7 + q^10 + 2 * q^11 - 2 * q^13 + 6 * q^14 - 3 * q^16 + 4 * q^17 - 2 * q^19 + q^20 + 4 * q^22 + 6 * q^23 + 2 * q^25 + q^26 - 4 * q^28 - 6 * q^29 - 6 * q^31 + 9 * q^32 - 7 * q^34 + 2 * q^35 - 12 * q^37 - 9 * q^38 + 12 * q^41 + 4 * q^43 - 6 * q^44 - 13 * q^46 + 4 * q^47 - 2 * q^49 - q^50 + q^52 - 2 * q^55 - 10 * q^56 - 2 * q^58 - 12 * q^59 - 22 * q^61 + 3 * q^62 + 4 * q^64 + 2 * q^65 - 10 * q^67 + 3 * q^68 - 6 * q^70 - 22 * q^71 + 6 * q^74 + 11 * q^76 + 8 * q^77 + 3 * q^80 + 4 * q^82 - 12 * q^83 - 4 * q^85 + 8 * q^86 - 10 * q^88 + 6 * q^89 + 2 * q^91 + 7 * q^92 - 12 * q^94 + 2 * q^95 - 4 * q^97 - 9 * 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
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	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
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	5.23607	1.39940
$15$	0	0
$16$	−4.85410	−1.21353
$17$	4.23607	1.02740	0.513699	−	0.857971i	$-0.328275\pi$
$17$	4.23607	1.02740	0.513699	+	0.857971i	$0.328275\pi$
$18$	0	0
$19$	3.47214	0.796563	0.398281	−	0.917263i	$-0.369607\pi$
$19$	3.47214	0.796563	0.398281	+	0.917263i	$0.369607\pi$
$20$	−0.618034	−0.138197
$21$	0	0
$22$	2.00000	0.426401
$23$	7.47214	1.55805	0.779024	−	0.626994i	$-0.215715\pi$
$23$	7.47214	1.55805	0.779024	+	0.626994i	$0.215715\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.61803	0.317323
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−0.763932	−0.141859	−0.0709293	−	0.997481i	$-0.522596\pi$
$29$	−0.763932	−0.141859	−0.0709293	+	0.997481i	$0.522596\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	3.38197	0.597853
$33$	0	0
$34$	−6.85410	−1.17547
$35$	3.23607	0.546995
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−5.61803	−0.911365
$39$	0	0
$40$	−2.23607	−0.353553
$41$	1.52786	0.238612	0.119306	−	0.992858i	$-0.461933\pi$
$41$	1.52786	0.238612	0.119306	+	0.992858i	$0.461933\pi$
$42$	0	0
$43$	−2.47214	−0.376997	−0.188499	−	0.982073i	$-0.560362\pi$
$43$	−2.47214	−0.376997	−0.188499	+	0.982073i	$0.560362\pi$
$44$	−0.763932	−0.115167
$45$	0	0
$46$	−12.0902	−1.78260
$47$	6.47214	0.944058	0.472029	−	0.881583i	$-0.343522\pi$
$47$	6.47214	0.944058	0.472029	+	0.881583i	$0.343522\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	−1.61803	−0.228825
$51$	0	0
$52$	−0.618034	−0.0857059
$53$	6.70820	0.921443	0.460721	−	0.887545i	$-0.347591\pi$
$53$	6.70820	0.921443	0.460721	+	0.887545i	$0.347591\pi$
$54$	0	0
$55$	1.23607	0.166671
$56$	−7.23607	−0.966960
$57$	0	0
$58$	1.23607	0.162304
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	4.85410	0.616472
$63$	0	0
$64$	4.23607	0.529508
$65$	1.00000	0.124035
$66$	0	0
$67$	1.70820	0.208690	0.104345	−	0.994541i	$-0.466725\pi$
$67$	1.70820	0.208690	0.104345	+	0.994541i	$0.466725\pi$
$68$	2.61803	0.317483
$69$	0	0
$70$	−5.23607	−0.625830
$71$	−13.2361	−1.57083	−0.785416	−	0.618968i	$-0.787551\pi$
$71$	−13.2361	−1.57083	−0.785416	+	0.618968i	$0.787551\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	9.70820	1.12856
$75$	0	0
$76$	2.14590	0.246151
$77$	4.00000	0.455842
$78$	0	0
$79$	6.70820	0.754732	0.377366	−	0.926064i	$-0.376830\pi$
$79$	6.70820	0.754732	0.377366	+	0.926064i	$0.376830\pi$
$80$	4.85410	0.542705
$81$	0	0
$82$	−2.47214	−0.273002
$83$	−3.76393	−0.413145	−0.206573	−	0.978431i	$-0.566231\pi$
$83$	−3.76393	−0.413145	−0.206573	+	0.978431i	$0.566231\pi$
$84$	0	0
$85$	−4.23607	−0.459466
$86$	4.00000	0.431331
$87$	0	0
$88$	−2.76393	−0.294636
$89$	−3.70820	−0.393069	−0.196534	−	0.980497i	$-0.562969\pi$
$89$	−3.70820	−0.393069	−0.196534	+	0.980497i	$0.562969\pi$
$90$	0	0
$91$	3.23607	0.339232
$92$	4.61803	0.481463
$93$	0	0
$94$	−10.4721	−1.08012
$95$	−3.47214	−0.356234
$96$	0	0
$97$	2.47214	0.251007	0.125504	−	0.992093i	$-0.459945\pi$
$97$	2.47214	0.251007	0.125504	+	0.992093i	$0.459945\pi$
$98$	−5.61803	−0.567507
$99$	0	0
$100$	0.618034	0.0618034
$101$	−2.76393	−0.275022	−0.137511	−	0.990500i	$-0.543910\pi$
$101$	−2.76393	−0.275022	−0.137511	+	0.990500i	$0.543910\pi$
$102$	0	0
$103$	−13.2361	−1.30419	−0.652094	−	0.758138i	$-0.726109\pi$
$103$	−13.2361	−1.30419	−0.652094	+	0.758138i	$0.726109\pi$
$104$	−2.23607	−0.219265
$105$	0	0
$106$	−10.8541	−1.05424
$107$	−2.47214	−0.238990	−0.119495	−	0.992835i	$-0.538128\pi$
$107$	−2.47214	−0.238990	−0.119495	+	0.992835i	$0.538128\pi$
$108$	0	0
$109$	−7.29180	−0.698427	−0.349214	−	0.937043i	$-0.613551\pi$
$109$	−7.29180	−0.698427	−0.349214	+	0.937043i	$0.613551\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	15.7082	1.48429
$113$	−2.94427	−0.276974	−0.138487	−	0.990364i	$-0.544224\pi$
$113$	−2.94427	−0.276974	−0.138487	+	0.990364i	$0.544224\pi$
$114$	0	0
$115$	−7.47214	−0.696780
$116$	−0.472136	−0.0438367
$117$	0	0
$118$	9.70820	0.893713
$119$	−13.7082	−1.25663
$120$	0	0
$121$	−9.47214	−0.861103
$122$	17.7984	1.61139
$123$	0	0
$124$	−1.85410	−0.166503
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.94427	0.616204	0.308102	−	0.951353i	$-0.400306\pi$
$127$	6.94427	0.616204	0.308102	+	0.951353i	$0.400306\pi$
$128$	−13.6180	−1.20368
$129$	0	0
$130$	−1.61803	−0.141911
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	−11.2361	−0.974291
$134$	−2.76393	−0.238767
$135$	0	0
$136$	9.47214	0.812229
$137$	0.0557281	0.00476117	0.00238059	−	0.999997i	$-0.499242\pi$
$137$	0.0557281	0.00476117	0.00238059	+	0.999997i	$0.499242\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	21.4164	1.79723
$143$	1.23607	0.103365
$144$	0	0
$145$	0.763932	0.0634411
$146$	0	0
$147$	0	0
$148$	−3.70820	−0.304812
$149$	−14.1803	−1.16170	−0.580849	−	0.814011i	$-0.697279\pi$
$149$	−14.1803	−1.16170	−0.580849	+	0.814011i	$0.697279\pi$
$150$	0	0
$151$	−11.4164	−0.929054	−0.464527	−	0.885559i	$-0.653776\pi$
$151$	−11.4164	−0.929054	−0.464527	+	0.885559i	$0.653776\pi$
$152$	7.76393	0.629738
$153$	0	0
$154$	−6.47214	−0.521540
$155$	3.00000	0.240966
$156$	0	0
$157$	3.70820	0.295947	0.147973	−	0.988991i	$-0.452725\pi$
$157$	3.70820	0.295947	0.147973	+	0.988991i	$0.452725\pi$
$158$	−10.8541	−0.863506
$159$	0	0
$160$	−3.38197	−0.267368
$161$	−24.1803	−1.90568
$162$	0	0
$163$	21.4164	1.67746	0.838731	−	0.544546i	$-0.183298\pi$
$163$	21.4164	1.67746	0.838731	+	0.544546i	$0.183298\pi$
$164$	0.944272	0.0737352
$165$	0	0
$166$	6.09017	0.472689
$167$	14.2361	1.10162	0.550810	−	0.834631i	$-0.314319\pi$
$167$	14.2361	1.10162	0.550810	+	0.834631i	$0.314319\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	6.85410	0.525686
$171$	0	0
$172$	−1.52786	−0.116499
$173$	−13.1803	−1.00208	−0.501041	−	0.865423i	$-0.667050\pi$
$173$	−13.1803	−1.00208	−0.501041	+	0.865423i	$0.667050\pi$
$174$	0	0
$175$	−3.23607	−0.244624
$176$	6.00000	0.452267
$177$	0	0
$178$	6.00000	0.449719
$179$	−24.6525	−1.84261	−0.921306	−	0.388838i	$-0.872877\pi$
$179$	−24.6525	−1.84261	−0.921306	+	0.388838i	$0.872877\pi$
$180$	0	0
$181$	−10.4164	−0.774245	−0.387123	−	0.922028i	$-0.626531\pi$
$181$	−10.4164	−0.774245	−0.387123	+	0.922028i	$0.626531\pi$
$182$	−5.23607	−0.388123
$183$	0	0
$184$	16.7082	1.23175
$185$	6.00000	0.441129
$186$	0	0
$187$	−5.23607	−0.382899
$188$	4.00000	0.291730
$189$	0	0
$190$	5.61803	0.407575
$191$	3.23607	0.234154	0.117077	−	0.993123i	$-0.462648\pi$
$191$	3.23607	0.234154	0.117077	+	0.993123i	$0.462648\pi$
$192$	0	0
$193$	−22.9443	−1.65156	−0.825782	−	0.563989i	$-0.809266\pi$
$193$	−22.9443	−1.65156	−0.825782	+	0.563989i	$0.809266\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	2.14590	0.153278
$197$	−22.4164	−1.59710	−0.798551	−	0.601927i	$-0.794400\pi$
$197$	−22.4164	−1.59710	−0.798551	+	0.601927i	$0.794400\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	2.23607	0.158114
$201$	0	0
$202$	4.47214	0.314658
$203$	2.47214	0.173510
$204$	0	0
$205$	−1.52786	−0.106711
$206$	21.4164	1.49215
$207$	0	0
$208$	4.85410	0.336571
$209$	−4.29180	−0.296870
$210$	0	0
$211$	−22.2361	−1.53079	−0.765397	−	0.643558i	$-0.777457\pi$
$211$	−22.2361	−1.53079	−0.765397	+	0.643558i	$0.777457\pi$
$212$	4.14590	0.284741
$213$	0	0
$214$	4.00000	0.273434
$215$	2.47214	0.168598
$216$	0	0
$217$	9.70820	0.659036
$218$	11.7984	0.799087
$219$	0	0
$220$	0.763932	0.0515043
$221$	−4.23607	−0.284949
$222$	0	0
$223$	−29.1246	−1.95033	−0.975164	−	0.221483i	$-0.928910\pi$
$223$	−29.1246	−1.95033	−0.975164	+	0.221483i	$0.928910\pi$
$224$	−10.9443	−0.731245
$225$	0	0
$226$	4.76393	0.316892
$227$	−29.6525	−1.96810	−0.984052	−	0.177881i	$-0.943076\pi$
$227$	−29.6525	−1.96810	−0.984052	+	0.177881i	$0.943076\pi$
$228$	0	0
$229$	−26.5967	−1.75756	−0.878781	−	0.477225i	$-0.841643\pi$
$229$	−26.5967	−1.75756	−0.878781	+	0.477225i	$0.841643\pi$
$230$	12.0902	0.797202
$231$	0	0
$232$	−1.70820	−0.112149
$233$	6.94427	0.454934	0.227467	−	0.973786i	$-0.426956\pi$
$233$	6.94427	0.454934	0.227467	+	0.973786i	$0.426956\pi$
$234$	0	0
$235$	−6.47214	−0.422196
$236$	−3.70820	−0.241384
$237$	0	0
$238$	22.1803	1.43774
$239$	26.6525	1.72401	0.862003	−	0.506904i	$-0.169210\pi$
$239$	26.6525	1.72401	0.862003	+	0.506904i	$0.169210\pi$
$240$	0	0
$241$	−12.7082	−0.818607	−0.409304	−	0.912398i	$-0.634228\pi$
$241$	−12.7082	−0.818607	−0.409304	+	0.912398i	$0.634228\pi$
$242$	15.3262	0.985208
$243$	0	0
$244$	−6.79837	−0.435221
$245$	−3.47214	−0.221827
$246$	0	0
$247$	−3.47214	−0.220927
$248$	−6.70820	−0.425971
$249$	0	0
$250$	1.61803	0.102333
$251$	−17.8885	−1.12911	−0.564557	−	0.825394i	$-0.690953\pi$
$251$	−17.8885	−1.12911	−0.564557	+	0.825394i	$0.690953\pi$
$252$	0	0
$253$	−9.23607	−0.580667
$254$	−11.2361	−0.705014
$255$	0	0
$256$	13.5623	0.847644
$257$	11.7639	0.733814	0.366907	−	0.930258i	$-0.380417\pi$
$257$	11.7639	0.733814	0.366907	+	0.930258i	$0.380417\pi$
$258$	0	0
$259$	19.4164	1.20648
$260$	0.618034	0.0383288
$261$	0	0
$262$	−9.70820	−0.599775
$263$	5.88854	0.363103	0.181552	−	0.983381i	$-0.441888\pi$
$263$	5.88854	0.363103	0.181552	+	0.983381i	$0.441888\pi$
$264$	0	0
$265$	−6.70820	−0.412082
$266$	18.1803	1.11471
$267$	0	0
$268$	1.05573	0.0644889
$269$	−7.81966	−0.476773	−0.238387	−	0.971170i	$-0.576619\pi$
$269$	−7.81966	−0.476773	−0.238387	+	0.971170i	$0.576619\pi$
$270$	0	0
$271$	−14.4164	−0.875734	−0.437867	−	0.899040i	$-0.644266\pi$
$271$	−14.4164	−0.875734	−0.437867	+	0.899040i	$0.644266\pi$
$272$	−20.5623	−1.24677
$273$	0	0
$274$	−0.0901699	−0.00544737
$275$	−1.23607	−0.0745377
$276$	0	0
$277$	−9.88854	−0.594145	−0.297073	−	0.954855i	$-0.596010\pi$
$277$	−9.88854	−0.594145	−0.297073	+	0.954855i	$0.596010\pi$
$278$	−12.9443	−0.776346
$279$	0	0
$280$	7.23607	0.432438
$281$	30.0689	1.79376	0.896880	−	0.442275i	$-0.145828\pi$
$281$	30.0689	1.79376	0.896880	+	0.442275i	$0.145828\pi$
$282$	0	0
$283$	23.7082	1.40931	0.704653	−	0.709552i	$-0.251103\pi$
$283$	23.7082	1.40931	0.704653	+	0.709552i	$0.251103\pi$
$284$	−8.18034	−0.485414
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−4.94427	−0.291851
$288$	0	0
$289$	0.944272	0.0555454
$290$	−1.23607	−0.0725844
$291$	0	0
$292$	0	0
$293$	−11.9443	−0.697792	−0.348896	−	0.937161i	$-0.613443\pi$
$293$	−11.9443	−0.697792	−0.348896	+	0.937161i	$0.613443\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	−13.4164	−0.779813
$297$	0	0
$298$	22.9443	1.32913
$299$	−7.47214	−0.432125
$300$	0	0
$301$	8.00000	0.461112
$302$	18.4721	1.06295
$303$	0	0
$304$	−16.8541	−0.966649
$305$	11.0000	0.629858
$306$	0	0
$307$	27.8885	1.59168	0.795842	−	0.605505i	$-0.207029\pi$
$307$	27.8885	1.59168	0.795842	+	0.605505i	$0.207029\pi$
$308$	2.47214	0.140863
$309$	0	0
$310$	−4.85410	−0.275694
$311$	13.4164	0.760775	0.380387	−	0.924827i	$-0.375791\pi$
$311$	13.4164	0.760775	0.380387	+	0.924827i	$0.375791\pi$
$312$	0	0
$313$	21.2361	1.20033	0.600167	−	0.799875i	$-0.295101\pi$
$313$	21.2361	1.20033	0.600167	+	0.799875i	$0.295101\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	4.14590	0.233225
$317$	16.5279	0.928297	0.464149	−	0.885757i	$-0.346360\pi$
$317$	16.5279	0.928297	0.464149	+	0.885757i	$0.346360\pi$
$318$	0	0
$319$	0.944272	0.0528691
$320$	−4.23607	−0.236803
$321$	0	0
$322$	39.1246	2.18033
$323$	14.7082	0.818386
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−34.6525	−1.91922
$327$	0	0
$328$	3.41641	0.188640
$329$	−20.9443	−1.15470
$330$	0	0
$331$	−11.0557	−0.607678	−0.303839	−	0.952723i	$-0.598268\pi$
$331$	−11.0557	−0.607678	−0.303839	+	0.952723i	$0.598268\pi$
$332$	−2.32624	−0.127669
$333$	0	0
$334$	−23.0344	−1.26039
$335$	−1.70820	−0.0933292
$336$	0	0
$337$	−10.9443	−0.596172	−0.298086	−	0.954539i	$-0.596348\pi$
$337$	−10.9443	−0.596172	−0.298086	+	0.954539i	$0.596348\pi$
$338$	−1.61803	−0.0880094
$339$	0	0
$340$	−2.61803	−0.141983
$341$	3.70820	0.200811
$342$	0	0
$343$	11.4164	0.616428
$344$	−5.52786	−0.298042
$345$	0	0
$346$	21.3262	1.14651
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	16.7082	0.894370	0.447185	−	0.894442i	$-0.352427\pi$
$349$	16.7082	0.894370	0.447185	+	0.894442i	$0.352427\pi$
$350$	5.23607	0.279880
$351$	0	0
$352$	−4.18034	−0.222813
$353$	7.52786	0.400668	0.200334	−	0.979728i	$-0.435797\pi$
$353$	7.52786	0.400668	0.200334	+	0.979728i	$0.435797\pi$
$354$	0	0
$355$	13.2361	0.702498
$356$	−2.29180	−0.121465
$357$	0	0
$358$	39.8885	2.10818
$359$	4.47214	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$359$	4.47214	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$360$	0	0
$361$	−6.94427	−0.365488
$362$	16.8541	0.885832
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	8.76393	0.457474	0.228737	−	0.973488i	$-0.426540\pi$
$367$	8.76393	0.457474	0.228737	+	0.973488i	$0.426540\pi$
$368$	−36.2705	−1.89073
$369$	0	0
$370$	−9.70820	−0.504705
$371$	−21.7082	−1.12703
$372$	0	0
$373$	25.5967	1.32535	0.662675	−	0.748907i	$-0.269421\pi$
$373$	25.5967	1.32535	0.662675	+	0.748907i	$0.269421\pi$
$374$	8.47214	0.438084
$375$	0	0
$376$	14.4721	0.746343
$377$	0.763932	0.0393445
$378$	0	0
$379$	24.8885	1.27844	0.639219	−	0.769024i	$-0.279258\pi$
$379$	24.8885	1.27844	0.639219	+	0.769024i	$0.279258\pi$
$380$	−2.14590	−0.110082
$381$	0	0
$382$	−5.23607	−0.267901
$383$	−32.5967	−1.66562	−0.832808	−	0.553562i	$-0.813268\pi$
$383$	−32.5967	−1.66562	−0.832808	+	0.553562i	$0.813268\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	37.1246	1.88959
$387$	0	0
$388$	1.52786	0.0775655
$389$	−20.9443	−1.06192	−0.530958	−	0.847398i	$-0.678168\pi$
$389$	−20.9443	−1.06192	−0.530958	+	0.847398i	$0.678168\pi$
$390$	0	0
$391$	31.6525	1.60073
$392$	7.76393	0.392138
$393$	0	0
$394$	36.2705	1.82728
$395$	−6.70820	−0.337526
$396$	0	0
$397$	−11.8197	−0.593212	−0.296606	−	0.955000i	$-0.595855\pi$
$397$	−11.8197	−0.593212	−0.296606	+	0.955000i	$0.595855\pi$
$398$	−25.8885	−1.29768
$399$	0	0
$400$	−4.85410	−0.242705
$401$	31.3050	1.56329	0.781647	−	0.623721i	$-0.214380\pi$
$401$	31.3050	1.56329	0.781647	+	0.623721i	$0.214380\pi$
$402$	0	0
$403$	3.00000	0.149441
$404$	−1.70820	−0.0849863
$405$	0	0
$406$	−4.00000	−0.198517
$407$	7.41641	0.367618
$408$	0	0
$409$	−32.5967	−1.61181	−0.805903	−	0.592048i	$-0.798320\pi$
$409$	−32.5967	−1.61181	−0.805903	+	0.592048i	$0.798320\pi$
$410$	2.47214	0.122090
$411$	0	0
$412$	−8.18034	−0.403016
$413$	19.4164	0.955419
$414$	0	0
$415$	3.76393	0.184764
$416$	−3.38197	−0.165815
$417$	0	0
$418$	6.94427	0.339655
$419$	36.0689	1.76208	0.881040	−	0.473042i	$-0.156844\pi$
$419$	36.0689	1.76208	0.881040	+	0.473042i	$0.156844\pi$
$420$	0	0
$421$	−24.7082	−1.20420	−0.602102	−	0.798419i	$-0.705670\pi$
$421$	−24.7082	−1.20420	−0.602102	+	0.798419i	$0.705670\pi$
$422$	35.9787	1.75142
$423$	0	0
$424$	15.0000	0.728464
$425$	4.23607	0.205479
$426$	0	0
$427$	35.5967	1.72265
$428$	−1.52786	−0.0738521
$429$	0	0
$430$	−4.00000	−0.192897
$431$	−15.2361	−0.733896	−0.366948	−	0.930242i	$-0.619597\pi$
$431$	−15.2361	−0.733896	−0.366948	+	0.930242i	$0.619597\pi$
$432$	0	0
$433$	−12.1803	−0.585350	−0.292675	−	0.956212i	$-0.594545\pi$
$433$	−12.1803	−0.585350	−0.292675	+	0.956212i	$0.594545\pi$
$434$	−15.7082	−0.754018
$435$	0	0
$436$	−4.50658	−0.215826
$437$	25.9443	1.24108
$438$	0	0
$439$	−3.29180	−0.157109	−0.0785544	−	0.996910i	$-0.525030\pi$
$439$	−3.29180	−0.157109	−0.0785544	+	0.996910i	$0.525030\pi$
$440$	2.76393	0.131765
$441$	0	0
$442$	6.85410	0.326016
$443$	12.5279	0.595217	0.297608	−	0.954688i	$-0.403811\pi$
$443$	12.5279	0.595217	0.297608	+	0.954688i	$0.403811\pi$
$444$	0	0
$445$	3.70820	0.175786
$446$	47.1246	2.23142
$447$	0	0
$448$	−13.7082	−0.647652
$449$	2.65248	0.125178	0.0625890	−	0.998039i	$-0.480064\pi$
$449$	2.65248	0.125178	0.0625890	+	0.998039i	$0.480064\pi$
$450$	0	0
$451$	−1.88854	−0.0889281
$452$	−1.81966	−0.0855896
$453$	0	0
$454$	47.9787	2.25175
$455$	−3.23607	−0.151709
$456$	0	0
$457$	19.4164	0.908261	0.454131	−	0.890935i	$-0.349950\pi$
$457$	19.4164	0.908261	0.454131	+	0.890935i	$0.349950\pi$
$458$	43.0344	2.01087
$459$	0	0
$460$	−4.61803	−0.215317
$461$	29.1246	1.35647	0.678234	−	0.734846i	$-0.262745\pi$
$461$	29.1246	1.35647	0.678234	+	0.734846i	$0.262745\pi$
$462$	0	0
$463$	−23.3050	−1.08307	−0.541536	−	0.840677i	$-0.682157\pi$
$463$	−23.3050	−1.08307	−0.541536	+	0.840677i	$0.682157\pi$
$464$	3.70820	0.172149
$465$	0	0
$466$	−11.2361	−0.520501
$467$	31.3607	1.45120	0.725600	−	0.688117i	$-0.241563\pi$
$467$	31.3607	1.45120	0.725600	+	0.688117i	$0.241563\pi$
$468$	0	0
$469$	−5.52786	−0.255253
$470$	10.4721	0.483044
$471$	0	0
$472$	−13.4164	−0.617540
$473$	3.05573	0.140503
$474$	0	0
$475$	3.47214	0.159313
$476$	−8.47214	−0.388320
$477$	0	0
$478$	−43.1246	−1.97247
$479$	−31.4164	−1.43545	−0.717726	−	0.696325i	$-0.754817\pi$
$479$	−31.4164	−1.43545	−0.717726	+	0.696325i	$0.754817\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	20.5623	0.936587
$483$	0	0
$484$	−5.85410	−0.266096
$485$	−2.47214	−0.112254
$486$	0	0
$487$	−26.8328	−1.21591	−0.607955	−	0.793971i	$-0.708010\pi$
$487$	−26.8328	−1.21591	−0.607955	+	0.793971i	$0.708010\pi$
$488$	−24.5967	−1.11344
$489$	0	0
$490$	5.61803	0.253797
$491$	−26.6525	−1.20281	−0.601405	−	0.798945i	$-0.705392\pi$
$491$	−26.6525	−1.20281	−0.601405	+	0.798945i	$0.705392\pi$
$492$	0	0
$493$	−3.23607	−0.145745
$494$	5.61803	0.252767
$495$	0	0
$496$	14.5623	0.653867
$497$	42.8328	1.92131
$498$	0	0
$499$	3.47214	0.155434	0.0777171	−	0.996975i	$-0.475237\pi$
$499$	3.47214	0.155434	0.0777171	+	0.996975i	$0.475237\pi$
$500$	−0.618034	−0.0276393
$501$	0	0
$502$	28.9443	1.29185
$503$	30.3050	1.35123	0.675616	−	0.737254i	$-0.263878\pi$
$503$	30.3050	1.35123	0.675616	+	0.737254i	$0.263878\pi$
$504$	0	0
$505$	2.76393	0.122993
$506$	14.9443	0.664354
$507$	0	0
$508$	4.29180	0.190418
$509$	−1.81966	−0.0806550	−0.0403275	−	0.999187i	$-0.512840\pi$
$509$	−1.81966	−0.0806550	−0.0403275	+	0.999187i	$0.512840\pi$
$510$	0	0
$511$	0	0
$512$	5.29180	0.233867
$513$	0	0
$514$	−19.0344	−0.839573
$515$	13.2361	0.583251
$516$	0	0
$517$	−8.00000	−0.351840
$518$	−31.4164	−1.38036
$519$	0	0
$520$	2.23607	0.0980581
$521$	−26.1803	−1.14698	−0.573491	−	0.819212i	$-0.694411\pi$
$521$	−26.1803	−1.14698	−0.573491	+	0.819212i	$0.694411\pi$
$522$	0	0
$523$	−9.70820	−0.424510	−0.212255	−	0.977214i	$-0.568081\pi$
$523$	−9.70820	−0.424510	−0.212255	+	0.977214i	$0.568081\pi$
$524$	3.70820	0.161994
$525$	0	0
$526$	−9.52786	−0.415435
$527$	−12.7082	−0.553578
$528$	0	0
$529$	32.8328	1.42751
$530$	10.8541	0.471472
$531$	0	0
$532$	−6.94427	−0.301072
$533$	−1.52786	−0.0661791
$534$	0	0
$535$	2.47214	0.106880
$536$	3.81966	0.164984
$537$	0	0
$538$	12.6525	0.545487
$539$	−4.29180	−0.184861
$540$	0	0
$541$	31.8885	1.37100	0.685498	−	0.728075i	$-0.259585\pi$
$541$	31.8885	1.37100	0.685498	+	0.728075i	$0.259585\pi$
$542$	23.3262	1.00195
$543$	0	0
$544$	14.3262	0.614232
$545$	7.29180	0.312346
$546$	0	0
$547$	40.0000	1.71028	0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000	1.71028	0.855138	+	0.518400i	$0.173472\pi$
$548$	0.0344419	0.00147128
$549$	0	0
$550$	2.00000	0.0852803
$551$	−2.65248	−0.112999
$552$	0	0
$553$	−21.7082	−0.923127
$554$	16.0000	0.679775
$555$	0	0
$556$	4.94427	0.209684
$557$	−22.9443	−0.972180	−0.486090	−	0.873909i	$-0.661577\pi$
$557$	−22.9443	−0.972180	−0.486090	+	0.873909i	$0.661577\pi$
$558$	0	0
$559$	2.47214	0.104560
$560$	−15.7082	−0.663793
$561$	0	0
$562$	−48.6525	−2.05228
$563$	−26.8328	−1.13087	−0.565434	−	0.824793i	$-0.691291\pi$
$563$	−26.8328	−1.13087	−0.565434	+	0.824793i	$0.691291\pi$
$564$	0	0
$565$	2.94427	0.123866
$566$	−38.3607	−1.61242
$567$	0	0
$568$	−29.5967	−1.24185
$569$	4.76393	0.199714	0.0998572	−	0.995002i	$-0.468161\pi$
$569$	4.76393	0.199714	0.0998572	+	0.995002i	$0.468161\pi$
$570$	0	0
$571$	−41.5410	−1.73844	−0.869219	−	0.494428i	$-0.835378\pi$
$571$	−41.5410	−1.73844	−0.869219	+	0.494428i	$0.835378\pi$
$572$	0.763932	0.0319416
$573$	0	0
$574$	8.00000	0.333914
$575$	7.47214	0.311610
$576$	0	0
$577$	−26.7639	−1.11420	−0.557099	−	0.830446i	$-0.688085\pi$
$577$	−26.7639	−1.11420	−0.557099	+	0.830446i	$0.688085\pi$
$578$	−1.52786	−0.0635508
$579$	0	0
$580$	0.472136	0.0196044
$581$	12.1803	0.505326
$582$	0	0
$583$	−8.29180	−0.343411
$584$	0	0
$585$	0	0
$586$	19.3262	0.798360
$587$	25.6525	1.05879	0.529395	−	0.848375i	$-0.322419\pi$
$587$	25.6525	1.05879	0.529395	+	0.848375i	$0.322419\pi$
$588$	0	0
$589$	−10.4164	−0.429201
$590$	−9.70820	−0.399680
$591$	0	0
$592$	29.1246	1.19701
$593$	21.3607	0.877178	0.438589	−	0.898688i	$-0.355478\pi$
$593$	21.3607	0.877178	0.438589	+	0.898688i	$0.355478\pi$
$594$	0	0
$595$	13.7082	0.561982
$596$	−8.76393	−0.358985
$597$	0	0
$598$	12.0902	0.494404
$599$	−9.70820	−0.396666	−0.198333	−	0.980135i	$-0.563553\pi$
$599$	−9.70820	−0.396666	−0.198333	+	0.980135i	$0.563553\pi$
$600$	0	0
$601$	−20.5279	−0.837349	−0.418675	−	0.908136i	$-0.637505\pi$
$601$	−20.5279	−0.837349	−0.418675	+	0.908136i	$0.637505\pi$
$602$	−12.9443	−0.527569
$603$	0	0
$604$	−7.05573	−0.287094
$605$	9.47214	0.385097
$606$	0	0
$607$	−14.8328	−0.602045	−0.301023	−	0.953617i	$-0.597328\pi$
$607$	−14.8328	−0.602045	−0.301023	+	0.953617i	$0.597328\pi$
$608$	11.7426	0.476227
$609$	0	0
$610$	−17.7984	−0.720635
$611$	−6.47214	−0.261835
$612$	0	0
$613$	−32.6525	−1.31882	−0.659411	−	0.751783i	$-0.729194\pi$
$613$	−32.6525	−1.31882	−0.659411	+	0.751783i	$0.729194\pi$
$614$	−45.1246	−1.82108
$615$	0	0
$616$	8.94427	0.360375
$617$	−23.8328	−0.959473	−0.479737	−	0.877413i	$-0.659268\pi$
$617$	−23.8328	−0.959473	−0.479737	+	0.877413i	$0.659268\pi$
$618$	0	0
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	1.85410	0.0744625
$621$	0	0
$622$	−21.7082	−0.870420
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	−34.3607	−1.37333
$627$	0	0
$628$	2.29180	0.0914526
$629$	−25.4164	−1.01342
$630$	0	0
$631$	11.5836	0.461136	0.230568	−	0.973056i	$-0.425942\pi$
$631$	11.5836	0.461136	0.230568	+	0.973056i	$0.425942\pi$
$632$	15.0000	0.596668
$633$	0	0
$634$	−26.7426	−1.06209
$635$	−6.94427	−0.275575
$636$	0	0
$637$	−3.47214	−0.137571
$638$	−1.52786	−0.0604887
$639$	0	0
$640$	13.6180	0.538300
$641$	14.1803	0.560090	0.280045	−	0.959987i	$-0.409651\pi$
$641$	14.1803	0.560090	0.280045	+	0.959987i	$0.409651\pi$
$642$	0	0
$643$	−32.8328	−1.29480	−0.647400	−	0.762150i	$-0.724144\pi$
$643$	−32.8328	−1.29480	−0.647400	+	0.762150i	$0.724144\pi$
$644$	−14.9443	−0.588887
$645$	0	0
$646$	−23.7984	−0.936334
$647$	−3.36068	−0.132122	−0.0660610	−	0.997816i	$-0.521043\pi$
$647$	−3.36068	−0.132122	−0.0660610	+	0.997816i	$0.521043\pi$
$648$	0	0
$649$	7.41641	0.291119
$650$	1.61803	0.0634645
$651$	0	0
$652$	13.2361	0.518364
$653$	−20.1246	−0.787537	−0.393768	−	0.919210i	$-0.628829\pi$
$653$	−20.1246	−0.787537	−0.393768	+	0.919210i	$0.628829\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	−7.41641	−0.289562
$657$	0	0
$658$	33.8885	1.32111
$659$	42.9443	1.67287	0.836436	−	0.548065i	$-0.184635\pi$
$659$	42.9443	1.67287	0.836436	+	0.548065i	$0.184635\pi$
$660$	0	0
$661$	30.3607	1.18089	0.590447	−	0.807077i	$-0.298952\pi$
$661$	30.3607	1.18089	0.590447	+	0.807077i	$0.298952\pi$
$662$	17.8885	0.695258
$663$	0	0
$664$	−8.41641	−0.326620
$665$	11.2361	0.435716
$666$	0	0
$667$	−5.70820	−0.221023
$668$	8.79837	0.340419
$669$	0	0
$670$	2.76393	0.106780
$671$	13.5967	0.524897
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	17.7082	0.682095
$675$	0	0
$676$	0.618034	0.0237705
$677$	38.9443	1.49675	0.748375	−	0.663276i	$-0.230834\pi$
$677$	38.9443	1.49675	0.748375	+	0.663276i	$0.230834\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	−9.47214	−0.363240
$681$	0	0
$682$	−6.00000	−0.229752
$683$	−9.06888	−0.347011	−0.173506	−	0.984833i	$-0.555509\pi$
$683$	−9.06888	−0.347011	−0.173506	+	0.984833i	$0.555509\pi$
$684$	0	0
$685$	−0.0557281	−0.00212926
$686$	−18.4721	−0.705269
$687$	0	0
$688$	12.0000	0.457496
$689$	−6.70820	−0.255562
$690$	0	0
$691$	34.7771	1.32298	0.661491	−	0.749953i	$-0.269924\pi$
$691$	34.7771	1.32298	0.661491	+	0.749953i	$0.269924\pi$
$692$	−8.14590	−0.309661
$693$	0	0
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	6.47214	0.245150
$698$	−27.0344	−1.02327
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	−18.3607	−0.693473	−0.346737	−	0.937963i	$-0.612710\pi$
$701$	−18.3607	−0.693473	−0.346737	+	0.937963i	$0.612710\pi$
$702$	0	0
$703$	−20.8328	−0.785725
$704$	−5.23607	−0.197342
$705$	0	0
$706$	−12.1803	−0.458413
$707$	8.94427	0.336384
$708$	0	0
$709$	−35.3050	−1.32591	−0.662953	−	0.748661i	$-0.730697\pi$
$709$	−35.3050	−1.32591	−0.662953	+	0.748661i	$0.730697\pi$
$710$	−21.4164	−0.803743
$711$	0	0
$712$	−8.29180	−0.310748
$713$	−22.4164	−0.839501
$714$	0	0
$715$	−1.23607	−0.0462263
$716$	−15.2361	−0.569399
$717$	0	0
$718$	−7.23607	−0.270048
$719$	−2.29180	−0.0854696	−0.0427348	−	0.999086i	$-0.513607\pi$
$719$	−2.29180	−0.0854696	−0.0427348	+	0.999086i	$0.513607\pi$
$720$	0	0
$721$	42.8328	1.59518
$722$	11.2361	0.418163
$723$	0	0
$724$	−6.43769	−0.239255
$725$	−0.763932	−0.0283717
$726$	0	0
$727$	−20.3607	−0.755136	−0.377568	−	0.925982i	$-0.623240\pi$
$727$	−20.3607	−0.755136	−0.377568	+	0.925982i	$0.623240\pi$
$728$	7.23607	0.268187
$729$	0	0
$730$	0	0
$731$	−10.4721	−0.387326
$732$	0	0
$733$	−49.2361	−1.81858	−0.909288	−	0.416168i	$-0.863373\pi$
$733$	−49.2361	−1.81858	−0.909288	+	0.416168i	$0.863373\pi$
$734$	−14.1803	−0.523406
$735$	0	0
$736$	25.2705	0.931483
$737$	−2.11146	−0.0777765
$738$	0	0
$739$	−45.0000	−1.65535	−0.827676	−	0.561206i	$-0.810337\pi$
$739$	−45.0000	−1.65535	−0.827676	+	0.561206i	$0.810337\pi$
$740$	3.70820	0.136316
$741$	0	0
$742$	35.1246	1.28947
$743$	26.8328	0.984401	0.492200	−	0.870482i	$-0.336193\pi$
$743$	26.8328	0.984401	0.492200	+	0.870482i	$0.336193\pi$
$744$	0	0
$745$	14.1803	0.519527
$746$	−41.4164	−1.51636
$747$	0	0
$748$	−3.23607	−0.118322
$749$	8.00000	0.292314
$750$	0	0
$751$	−20.1246	−0.734358	−0.367179	−	0.930150i	$-0.619676\pi$
$751$	−20.1246	−0.734358	−0.367179	+	0.930150i	$0.619676\pi$
$752$	−31.4164	−1.14564
$753$	0	0
$754$	−1.23607	−0.0450149
$755$	11.4164	0.415486
$756$	0	0
$757$	−35.5279	−1.29128	−0.645641	−	0.763641i	$-0.723410\pi$
$757$	−35.5279	−1.29128	−0.645641	+	0.763641i	$0.723410\pi$
$758$	−40.2705	−1.46269
$759$	0	0
$760$	−7.76393	−0.281627
$761$	−18.9443	−0.686729	−0.343365	−	0.939202i	$-0.611567\pi$
$761$	−18.9443	−0.686729	−0.343365	+	0.939202i	$0.611567\pi$
$762$	0	0
$763$	23.5967	0.854260
$764$	2.00000	0.0723575
$765$	0	0
$766$	52.7426	1.90567
$767$	6.00000	0.216647
$768$	0	0
$769$	−6.70820	−0.241904	−0.120952	−	0.992658i	$-0.538595\pi$
$769$	−6.70820	−0.241904	−0.120952	+	0.992658i	$0.538595\pi$
$770$	6.47214	0.233240
$771$	0	0
$772$	−14.1803	−0.510362
$773$	−35.4721	−1.27584	−0.637922	−	0.770101i	$-0.720206\pi$
$773$	−35.4721	−1.27584	−0.637922	+	0.770101i	$0.720206\pi$
$774$	0	0
$775$	−3.00000	−0.107763
$776$	5.52786	0.198439
$777$	0	0
$778$	33.8885	1.21496
$779$	5.30495	0.190070
$780$	0	0
$781$	16.3607	0.585431
$782$	−51.2148	−1.83144
$783$	0	0
$784$	−16.8541	−0.601932
$785$	−3.70820	−0.132351
$786$	0	0
$787$	−35.4164	−1.26246	−0.631229	−	0.775596i	$-0.717449\pi$
$787$	−35.4164	−1.26246	−0.631229	+	0.775596i	$0.717449\pi$
$788$	−13.8541	−0.493532
$789$	0	0
$790$	10.8541	0.386172
$791$	9.52786	0.338772
$792$	0	0
$793$	11.0000	0.390621
$794$	19.1246	0.678707
$795$	0	0
$796$	9.88854	0.350490
$797$	5.29180	0.187445	0.0937225	−	0.995598i	$-0.470123\pi$
$797$	5.29180	0.187445	0.0937225	+	0.995598i	$0.470123\pi$
$798$	0	0
$799$	27.4164	0.969923
$800$	3.38197	0.119571
$801$	0	0
$802$	−50.6525	−1.78860
$803$	0	0
$804$	0	0
$805$	24.1803	0.852245
$806$	−4.85410	−0.170978
$807$	0	0
$808$	−6.18034	−0.217424
$809$	47.7771	1.67975	0.839876	−	0.542778i	$-0.182627\pi$
$809$	47.7771	1.67975	0.839876	+	0.542778i	$0.182627\pi$
$810$	0	0
$811$	13.8885	0.487693	0.243846	−	0.969814i	$-0.421591\pi$
$811$	13.8885	0.487693	0.243846	+	0.969814i	$0.421591\pi$
$812$	1.52786	0.0536175
$813$	0	0
$814$	−12.0000	−0.420600
$815$	−21.4164	−0.750184
$816$	0	0
$817$	−8.58359	−0.300302
$818$	52.7426	1.84410
$819$	0	0
$820$	−0.944272	−0.0329754
$821$	36.7639	1.28307	0.641535	−	0.767094i	$-0.278298\pi$
$821$	36.7639	1.28307	0.641535	+	0.767094i	$0.278298\pi$
$822$	0	0
$823$	45.0132	1.56906	0.784530	−	0.620091i	$-0.212904\pi$
$823$	45.0132	1.56906	0.784530	+	0.620091i	$0.212904\pi$
$824$	−29.5967	−1.03105
$825$	0	0
$826$	−31.4164	−1.09312
$827$	23.1803	0.806059	0.403030	−	0.915187i	$-0.367957\pi$
$827$	23.1803	0.806059	0.403030	+	0.915187i	$0.367957\pi$
$828$	0	0
$829$	−20.8328	−0.723554	−0.361777	−	0.932265i	$-0.617830\pi$
$829$	−20.8328	−0.723554	−0.361777	+	0.932265i	$0.617830\pi$
$830$	−6.09017	−0.211393
$831$	0	0
$832$	−4.23607	−0.146859
$833$	14.7082	0.509609
$834$	0	0
$835$	−14.2361	−0.492659
$836$	−2.65248	−0.0917378
$837$	0	0
$838$	−58.3607	−2.01604
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−28.4164	−0.979876
$842$	39.9787	1.37776
$843$	0	0
$844$	−13.7426	−0.473041
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	30.6525	1.05323
$848$	−32.5623	−1.11819
$849$	0	0
$850$	−6.85410	−0.235094
$851$	−44.8328	−1.53685
$852$	0	0
$853$	19.4164	0.664805	0.332403	−	0.943138i	$-0.392141\pi$
$853$	19.4164	0.664805	0.332403	+	0.943138i	$0.392141\pi$
$854$	−57.5967	−1.97092
$855$	0	0
$856$	−5.52786	−0.188939
$857$	−45.6525	−1.55946	−0.779729	−	0.626117i	$-0.784643\pi$
$857$	−45.6525	−1.55946	−0.779729	+	0.626117i	$0.784643\pi$
$858$	0	0
$859$	43.6525	1.48940	0.744702	−	0.667398i	$-0.232592\pi$
$859$	43.6525	1.48940	0.744702	+	0.667398i	$0.232592\pi$
$860$	1.52786	0.0520997
$861$	0	0
$862$	24.6525	0.839667
$863$	39.5410	1.34599	0.672996	−	0.739646i	$-0.265007\pi$
$863$	39.5410	1.34599	0.672996	+	0.739646i	$0.265007\pi$
$864$	0	0
$865$	13.1803	0.448145
$866$	19.7082	0.669712
$867$	0	0
$868$	6.00000	0.203653
$869$	−8.29180	−0.281280
$870$	0	0
$871$	−1.70820	−0.0578803
$872$	−16.3050	−0.552155
$873$	0	0
$874$	−41.9787	−1.41995
$875$	3.23607	0.109399
$876$	0	0
$877$	−6.40325	−0.216222	−0.108111	−	0.994139i	$-0.534480\pi$
$877$	−6.40325	−0.216222	−0.108111	+	0.994139i	$0.534480\pi$
$878$	5.32624	0.179752
$879$	0	0
$880$	−6.00000	−0.202260
$881$	5.12461	0.172653	0.0863263	−	0.996267i	$-0.472487\pi$
$881$	5.12461	0.172653	0.0863263	+	0.996267i	$0.472487\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	−2.61803	−0.0880540
$885$	0	0
$886$	−20.2705	−0.681001
$887$	−52.8885	−1.77582	−0.887912	−	0.460014i	$-0.847844\pi$
$887$	−52.8885	−1.77582	−0.887912	+	0.460014i	$0.847844\pi$
$888$	0	0
$889$	−22.4721	−0.753691
$890$	−6.00000	−0.201120
$891$	0	0
$892$	−18.0000	−0.602685
$893$	22.4721	0.752001
$894$	0	0
$895$	24.6525	0.824041
$896$	44.0689	1.47224
$897$	0	0
$898$	−4.29180	−0.143219
$899$	2.29180	0.0764357
$900$	0	0
$901$	28.4164	0.946688
$902$	3.05573	0.101745
$903$	0	0
$904$	−6.58359	−0.218967
$905$	10.4164	0.346253
$906$	0	0
$907$	−47.1246	−1.56475	−0.782374	−	0.622809i	$-0.785991\pi$
$907$	−47.1246	−1.56475	−0.782374	+	0.622809i	$0.785991\pi$
$908$	−18.3262	−0.608178
$909$	0	0
$910$	5.23607	0.173574
$911$	−44.8328	−1.48538	−0.742689	−	0.669637i	$-0.766450\pi$
$911$	−44.8328	−1.48538	−0.742689	+	0.669637i	$0.766450\pi$
$912$	0	0
$913$	4.65248	0.153974
$914$	−31.4164	−1.03916
$915$	0	0
$916$	−16.4377	−0.543117
$917$	−19.4164	−0.641186
$918$	0	0
$919$	−22.8328	−0.753185	−0.376593	−	0.926379i	$-0.622904\pi$
$919$	−22.8328	−0.753185	−0.376593	+	0.926379i	$0.622904\pi$
$920$	−16.7082	−0.550853
$921$	0	0
$922$	−47.1246	−1.55197
$923$	13.2361	0.435670
$924$	0	0
$925$	−6.00000	−0.197279
$926$	37.7082	1.23917
$927$	0	0
$928$	−2.58359	−0.0848106
$929$	−13.8885	−0.455668	−0.227834	−	0.973700i	$-0.573164\pi$
$929$	−13.8885	−0.455668	−0.227834	+	0.973700i	$0.573164\pi$
$930$	0	0
$931$	12.0557	0.395111
$932$	4.29180	0.140582
$933$	0	0
$934$	−50.7426	−1.66035
$935$	5.23607	0.171238
$936$	0	0
$937$	−53.6656	−1.75318	−0.876590	−	0.481238i	$-0.840187\pi$
$937$	−53.6656	−1.75318	−0.876590	+	0.481238i	$0.840187\pi$
$938$	8.94427	0.292041
$939$	0	0
$940$	−4.00000	−0.130466
$941$	2.06888	0.0674437	0.0337218	−	0.999431i	$-0.489264\pi$
$941$	2.06888	0.0674437	0.0337218	+	0.999431i	$0.489264\pi$
$942$	0	0
$943$	11.4164	0.371769
$944$	29.1246	0.947925
$945$	0	0
$946$	−4.94427	−0.160752
$947$	20.5967	0.669304	0.334652	−	0.942342i	$-0.391381\pi$
$947$	20.5967	0.669304	0.334652	+	0.942342i	$0.391381\pi$
$948$	0	0
$949$	0	0
$950$	−5.61803	−0.182273
$951$	0	0
$952$	−30.6525	−0.993452
$953$	−7.88854	−0.255535	−0.127767	−	0.991804i	$-0.540781\pi$
$953$	−7.88854	−0.255535	−0.127767	+	0.991804i	$0.540781\pi$
$954$	0	0
$955$	−3.23607	−0.104717
$956$	16.4721	0.532747
$957$	0	0
$958$	50.8328	1.64233
$959$	−0.180340	−0.00582348
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−9.70820	−0.313005
$963$	0	0
$964$	−7.85410	−0.252964
$965$	22.9443	0.738602
$966$	0	0
$967$	−33.1246	−1.06522	−0.532608	−	0.846362i	$-0.678788\pi$
$967$	−33.1246	−1.06522	−0.532608	+	0.846362i	$0.678788\pi$
$968$	−21.1803	−0.680762
$969$	0	0
$970$	4.00000	0.128432
$971$	−44.4721	−1.42718	−0.713589	−	0.700564i	$-0.752932\pi$
$971$	−44.4721	−1.42718	−0.713589	+	0.700564i	$0.752932\pi$
$972$	0	0
$973$	−25.8885	−0.829949
$974$	43.4164	1.39115
$975$	0	0
$976$	53.3951	1.70914
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	4.58359	0.146492
$980$	−2.14590	−0.0685482
$981$	0	0
$982$	43.1246	1.37616
$983$	−9.54102	−0.304311	−0.152156	−	0.988357i	$-0.548621\pi$
$983$	−9.54102	−0.304311	−0.152156	+	0.988357i	$0.548621\pi$
$984$	0	0
$985$	22.4164	0.714246
$986$	5.23607	0.166750
$987$	0	0
$988$	−2.14590	−0.0682701
$989$	−18.4721	−0.587380
$990$	0	0
$991$	10.1246	0.321619	0.160809	−	0.986985i	$-0.448590\pi$
$991$	10.1246	0.321619	0.160809	+	0.986985i	$0.448590\pi$
$992$	−10.1459	−0.322133
$993$	0	0
$994$	−69.3050	−2.19822
$995$	−16.0000	−0.507234
$996$	0	0
$997$	−0.0688837	−0.00218157	−0.00109078	−	0.999999i	$-0.500347\pi$
$997$	−0.0688837	−0.00218157	−0.00109078	+	0.999999i	$0.500347\pi$
$998$	−5.61803	−0.177836
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.i.1.1	✓	2
3.2	odd	2		1755.2.a.j.1.2	yes	2
5.4	even	2		8775.2.a.bc.1.2		2
15.14	odd	2		8775.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.i.1.1	✓	2	1.1	even	1	trivial
1755.2.a.j.1.2	yes	2	3.2	odd	2
8775.2.a.v.1.1		2	15.14	odd	2
8775.2.a.bc.1.2		2	5.4	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:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -3.23607 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -3.23607 q^{7} +2.23607 q^{8} +1.61803 q^{10} -1.23607 q^{11} -1.00000 q^{13} +5.23607 q^{14} -4.85410 q^{16} +4.23607 q^{17} +3.47214 q^{19} -0.618034 q^{20} +2.00000 q^{22} +7.47214 q^{23} +1.00000 q^{25} +1.61803 q^{26} -2.00000 q^{28} -0.763932 q^{29} -3.00000 q^{31} +3.38197 q^{32} -6.85410 q^{34} +3.23607 q^{35} -6.00000 q^{37} -5.61803 q^{38} -2.23607 q^{40} +1.52786 q^{41} -2.47214 q^{43} -0.763932 q^{44} -12.0902 q^{46} +6.47214 q^{47} +3.47214 q^{49} -1.61803 q^{50} -0.618034 q^{52} +6.70820 q^{53} +1.23607 q^{55} -7.23607 q^{56} +1.23607 q^{58} -6.00000 q^{59} -11.0000 q^{61} +4.85410 q^{62} +4.23607 q^{64} +1.00000 q^{65} +1.70820 q^{67} +2.61803 q^{68} -5.23607 q^{70} -13.2361 q^{71} +9.70820 q^{74} +2.14590 q^{76} +4.00000 q^{77} +6.70820 q^{79} +4.85410 q^{80} -2.47214 q^{82} -3.76393 q^{83} -4.23607 q^{85} +4.00000 q^{86} -2.76393 q^{88} -3.70820 q^{89} +3.23607 q^{91} +4.61803 q^{92} -10.4721 q^{94} -3.47214 q^{95} +2.47214 q^{97} -5.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 - 2 * q^5 - 2 * q^7	\( 2 q - q^{2} - q^{4} - 2 q^{5} - 2 q^{7} + q^{10} + 2 q^{11} - 2 q^{13} + 6 q^{14} - 3 q^{16} + 4 q^{17} - 2 q^{19} + q^{20} + 4 q^{22} + 6 q^{23} + 2 q^{25} + q^{26} - 4 q^{28} - 6 q^{29} - 6 q^{31} + 9 q^{32} - 7 q^{34} + 2 q^{35} - 12 q^{37} - 9 q^{38} + 12 q^{41} + 4 q^{43} - 6 q^{44} - 13 q^{46} + 4 q^{47} - 2 q^{49} - q^{50} + q^{52} - 2 q^{55} - 10 q^{56} - 2 q^{58} - 12 q^{59} - 22 q^{61} + 3 q^{62} + 4 q^{64} + 2 q^{65} - 10 q^{67} + 3 q^{68} - 6 q^{70} - 22 q^{71} + 6 q^{74} + 11 q^{76} + 8 q^{77} + 3 q^{80} + 4 q^{82} - 12 q^{83} - 4 q^{85} + 8 q^{86} - 10 q^{88} + 6 q^{89} + 2 q^{91} + 7 q^{92} - 12 q^{94} + 2 q^{95} - 4 q^{97} - 9 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 2 * q^5 - 2 * q^7 + q^10 + 2 * q^11 - 2 * q^13 + 6 * q^14 - 3 * q^16 + 4 * q^17 - 2 * q^19 + q^20 + 4 * q^22 + 6 * q^23 + 2 * q^25 + q^26 - 4 * q^28 - 6 * q^29 - 6 * q^31 + 9 * q^32 - 7 * q^34 + 2 * q^35 - 12 * q^37 - 9 * q^38 + 12 * q^41 + 4 * q^43 - 6 * q^44 - 13 * q^46 + 4 * q^47 - 2 * q^49 - q^50 + q^52 - 2 * q^55 - 10 * q^56 - 2 * q^58 - 12 * q^59 - 22 * q^61 + 3 * q^62 + 4 * q^64 + 2 * q^65 - 10 * q^67 + 3 * q^68 - 6 * q^70 - 22 * q^71 + 6 * q^74 + 11 * q^76 + 8 * q^77 + 3 * q^80 + 4 * q^82 - 12 * q^83 - 4 * q^85 + 8 * q^86 - 10 * q^88 + 6 * q^89 + 2 * q^91 + 7 * q^92 - 12 * q^94 + 2 * q^95 - 4 * q^97 - 9 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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