LMFDB - Embedded newform 1755.2.a.o.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:	$4$
Coefficient field:	4.4.12357.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 3 $ x^4 - x^3 - 5x^2 + 3x + 3
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			$2.35284$ 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)

$f(q)$	$=$	$q-2.35284 q^{2} +3.53587 q^{4} -1.00000 q^{5} +2.43063 q^{7} -3.61366 q^{8} +O(q^{10})$	\(q-2.35284 q^{2} +3.53587 q^{4} -1.00000 q^{5} +2.43063 q^{7} -3.61366 q^{8} +2.35284 q^{10} +0.183026 q^{11} +1.00000 q^{13} -5.71890 q^{14} +1.43063 q^{16} -1.07779 q^{17} -2.44384 q^{19} -3.53587 q^{20} -0.430632 q^{22} -4.45808 q^{23} +1.00000 q^{25} -2.35284 q^{26} +8.59440 q^{28} -4.26082 q^{29} -3.53587 q^{31} +3.86126 q^{32} +2.53587 q^{34} -2.43063 q^{35} +0.0920270 q^{37} +5.74998 q^{38} +3.61366 q^{40} -7.88871 q^{41} -1.23337 q^{43} +0.647157 q^{44} +10.4892 q^{46} +10.5741 q^{47} -1.09203 q^{49} -2.35284 q^{50} +3.53587 q^{52} +3.39713 q^{53} -0.183026 q^{55} -8.78348 q^{56} +10.0250 q^{58} -5.95226 q^{59} -5.64111 q^{61} +8.31934 q^{62} -11.9462 q^{64} -1.00000 q^{65} -12.8995 q^{67} -3.81092 q^{68} +5.71890 q^{70} +0.168789 q^{71} +8.08495 q^{73} -0.216525 q^{74} -8.64111 q^{76} +0.444870 q^{77} +9.57411 q^{79} -1.43063 q^{80} +18.5609 q^{82} +3.74634 q^{83} +1.07779 q^{85} +2.90192 q^{86} -0.661395 q^{88} +12.7439 q^{89} +2.43063 q^{91} -15.7632 q^{92} -24.8792 q^{94} +2.44384 q^{95} -8.57411 q^{97} +2.56937 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 3 q^{4} - 4 q^{5} + q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q - q^2 + 3 * q^4 - 4 * q^5 + q^7 - 3 * q^8	$ 4 q - q^{2} + 3 q^{4} - 4 q^{5} + q^{7} - 3 q^{8} + q^{10} - 2 q^{11} + 4 q^{13} - 9 q^{14} - 3 q^{16} - 4 q^{17} - 4 q^{19} - 3 q^{20} + 7 q^{22} - 7 q^{23} + 4 q^{25} - q^{26} - 2 q^{28} - 14 q^{29} - 3 q^{31} - 2 q^{32} - q^{34} - q^{35} - 5 q^{37} - 14 q^{38} + 3 q^{40} - 12 q^{41} - 4 q^{43} + 11 q^{44} + 8 q^{46} - 11 q^{47} + q^{49} - q^{50} + 3 q^{52} - 15 q^{53} + 2 q^{55} - 18 q^{56} - 5 q^{58} - 9 q^{59} - 9 q^{61} + 5 q^{62} - 11 q^{64} - 4 q^{65} + 8 q^{67} + 4 q^{68} + 9 q^{70} + 3 q^{71} + 13 q^{73} - 18 q^{74} - 21 q^{76} - 12 q^{77} - 15 q^{79} + 3 q^{80} + 18 q^{82} - q^{83} + 4 q^{85} - 5 q^{86} - 6 q^{88} - 8 q^{89} + q^{91} - 29 q^{92} - 23 q^{94} + 4 q^{95} + 19 q^{97} + 19 q^{98}+O(q^{100}) $ 4 * q - q^2 + 3 * q^4 - 4 * q^5 + q^7 - 3 * q^8 + q^10 - 2 * q^11 + 4 * q^13 - 9 * q^14 - 3 * q^16 - 4 * q^17 - 4 * q^19 - 3 * q^20 + 7 * q^22 - 7 * q^23 + 4 * q^25 - q^26 - 2 * q^28 - 14 * q^29 - 3 * q^31 - 2 * q^32 - q^34 - q^35 - 5 * q^37 - 14 * q^38 + 3 * q^40 - 12 * q^41 - 4 * q^43 + 11 * q^44 + 8 * q^46 - 11 * q^47 + q^49 - q^50 + 3 * q^52 - 15 * q^53 + 2 * q^55 - 18 * q^56 - 5 * q^58 - 9 * q^59 - 9 * q^61 + 5 * q^62 - 11 * q^64 - 4 * q^65 + 8 * q^67 + 4 * q^68 + 9 * q^70 + 3 * q^71 + 13 * q^73 - 18 * q^74 - 21 * q^76 - 12 * q^77 - 15 * q^79 + 3 * q^80 + 18 * q^82 - q^83 + 4 * q^85 - 5 * q^86 - 6 * q^88 - 8 * q^89 + q^91 - 29 * q^92 - 23 * q^94 + 4 * q^95 + 19 * q^97 + 19 * 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 $

$2$	−2.35284	−1.66371	−0.831856	−	0.554992i	$-0.812721\pi$
$2$	−2.35284	−1.66371	−0.831856	+	0.554992i	$0.812721\pi$
$3$	0	0
$3$	0	0
$4$	3.53587	1.76793
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.43063	0.918693	0.459346	−	0.888257i	$-0.348084\pi$
$7$	2.43063	0.918693	0.459346	+	0.888257i	$0.348084\pi$
$8$	−3.61366	−1.27762
$9$	0	0
$10$	2.35284	0.744034
$11$	0.183026	0.0551845	0.0275923	−	0.999619i	$-0.491216\pi$
$11$	0.183026	0.0551845	0.0275923	+	0.999619i	$0.491216\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−5.71890	−1.52844
$15$	0	0
$16$	1.43063	0.357658
$17$	−1.07779	−0.261402	−0.130701	−	0.991422i	$-0.541723\pi$
$17$	−1.07779	−0.261402	−0.130701	+	0.991422i	$0.541723\pi$
$18$	0	0
$19$	−2.44384	−0.560656	−0.280328	−	0.959904i	$-0.590443\pi$
$19$	−2.44384	−0.560656	−0.280328	+	0.959904i	$0.590443\pi$
$20$	−3.53587	−0.790644
$21$	0	0
$22$	−0.430632	−0.0918111
$23$	−4.45808	−0.929574	−0.464787	−	0.885423i	$-0.653869\pi$
$23$	−4.45808	−0.929574	−0.464787	+	0.885423i	$0.653869\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.35284	−0.461430
$27$	0	0
$28$	8.59440	1.62419
$29$	−4.26082	−0.791214	−0.395607	−	0.918420i	$-0.629466\pi$
$29$	−4.26082	−0.791214	−0.395607	+	0.918420i	$0.629466\pi$
$30$	0	0
$31$	−3.53587	−0.635061	−0.317530	−	0.948248i	$-0.602854\pi$
$31$	−3.53587	−0.635061	−0.317530	+	0.948248i	$0.602854\pi$
$32$	3.86126	0.682582
$33$	0	0
$34$	2.53587	0.434898
$35$	−2.43063	−0.410852
$36$	0	0
$37$	0.0920270	0.0151291	0.00756457	−	0.999971i	$-0.497592\pi$
$37$	0.0920270	0.0151291	0.00756457	+	0.999971i	$0.497592\pi$
$38$	5.74998	0.932769
$39$	0	0
$40$	3.61366	0.571370
$41$	−7.88871	−1.23201	−0.616005	−	0.787742i	$-0.711250\pi$
$41$	−7.88871	−1.23201	−0.616005	+	0.787742i	$0.711250\pi$
$42$	0	0
$43$	−1.23337	−0.188087	−0.0940435	−	0.995568i	$-0.529979\pi$
$43$	−1.23337	−0.188087	−0.0940435	+	0.995568i	$0.529979\pi$
$44$	0.647157	0.0975626
$45$	0	0
$46$	10.4892	1.54654
$47$	10.5741	1.54239	0.771196	−	0.636598i	$-0.219659\pi$
$47$	10.5741	1.54239	0.771196	+	0.636598i	$0.219659\pi$
$48$	0	0
$49$	−1.09203	−0.156004
$50$	−2.35284	−0.332742
$51$	0	0
$52$	3.53587	0.490337
$53$	3.39713	0.466632	0.233316	−	0.972401i	$-0.425042\pi$
$53$	3.39713	0.466632	0.233316	+	0.972401i	$0.425042\pi$
$54$	0	0
$55$	−0.183026	−0.0246793
$56$	−8.78348	−1.17374
$57$	0	0
$58$	10.0250	1.31635
$59$	−5.95226	−0.774919	−0.387459	−	0.921887i	$-0.626647\pi$
$59$	−5.95226	−0.774919	−0.387459	+	0.921887i	$0.626647\pi$
$60$	0	0
$61$	−5.64111	−0.722270	−0.361135	−	0.932514i	$-0.617611\pi$
$61$	−5.64111	−0.722270	−0.361135	+	0.932514i	$0.617611\pi$
$62$	8.31934	1.05656
$63$	0	0
$64$	−11.9462	−1.49328
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−12.8995	−1.57593	−0.787963	−	0.615723i	$-0.788864\pi$
$67$	−12.8995	−1.57593	−0.787963	+	0.615723i	$0.788864\pi$
$68$	−3.81092	−0.462142
$69$	0	0
$70$	5.71890	0.683539
$71$	0.168789	0.0200315	0.0100158	−	0.999950i	$-0.496812\pi$
$71$	0.168789	0.0200315	0.0100158	+	0.999950i	$0.496812\pi$
$72$	0	0
$73$	8.08495	0.946272	0.473136	−	0.880990i	$-0.343122\pi$
$73$	8.08495	0.946272	0.473136	+	0.880990i	$0.343122\pi$
$74$	−0.216525	−0.0251705
$75$	0	0
$76$	−8.64111	−0.991203
$77$	0.444870	0.0506976
$78$	0	0
$79$	9.57411	1.07717	0.538586	−	0.842571i	$-0.318959\pi$
$79$	9.57411	1.07717	0.538586	+	0.842571i	$0.318959\pi$
$80$	−1.43063	−0.159950
$81$	0	0
$82$	18.5609	2.04971
$83$	3.74634	0.411215	0.205607	−	0.978635i	$-0.434083\pi$
$83$	3.74634	0.411215	0.205607	+	0.978635i	$0.434083\pi$
$84$	0	0
$85$	1.07779	0.116903
$86$	2.90192	0.312922
$87$	0	0
$88$	−0.661395	−0.0705049
$89$	12.7439	1.35085	0.675427	−	0.737427i	$-0.263959\pi$
$89$	12.7439	1.35085	0.675427	+	0.737427i	$0.263959\pi$
$90$	0	0
$91$	2.43063	0.254799
$92$	−15.7632	−1.64343
$93$	0	0
$94$	−24.8792	−2.56610
$95$	2.44384	0.250733
$96$	0	0
$97$	−8.57411	−0.870569	−0.435284	−	0.900293i	$-0.643352\pi$
$97$	−8.57411	−0.870569	−0.435284	+	0.900293i	$0.643352\pi$
$98$	2.56937	0.259545
$99$	0	0
$100$	3.53587	0.353587
$101$	−11.5501	−1.14928	−0.574639	−	0.818407i	$-0.694858\pi$
$101$	−11.5501	−1.14928	−0.574639	+	0.818407i	$0.694858\pi$
$102$	0	0
$103$	2.35182	0.231731	0.115866	−	0.993265i	$-0.463036\pi$
$103$	2.35182	0.231731	0.115866	+	0.993265i	$0.463036\pi$
$104$	−3.61366	−0.354348
$105$	0	0
$106$	−7.99292	−0.776341
$107$	−6.56592	−0.634752	−0.317376	−	0.948300i	$-0.602802\pi$
$107$	−6.56592	−0.634752	−0.317376	+	0.948300i	$0.602802\pi$
$108$	0	0
$109$	9.80748	0.939386	0.469693	−	0.882830i	$-0.344365\pi$
$109$	9.80748	0.939386	0.469693	+	0.882830i	$0.344365\pi$
$110$	0.430632	0.0410592
$111$	0	0
$112$	3.47734	0.328578
$113$	−6.69248	−0.629575	−0.314788	−	0.949162i	$-0.601933\pi$
$113$	−6.69248	−0.629575	−0.314788	+	0.949162i	$0.601933\pi$
$114$	0	0
$115$	4.45808	0.415718
$116$	−15.0657	−1.39881
$117$	0	0
$118$	14.0047	1.28924
$119$	−2.61971	−0.240148
$120$	0	0
$121$	−10.9665	−0.996955
$122$	13.2726	1.20165
$123$	0	0
$124$	−12.5024	−1.12275
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.33861	−0.296253	−0.148127	−	0.988968i	$-0.547324\pi$
$127$	−3.33861	−0.296253	−0.148127	+	0.988968i	$0.547324\pi$
$128$	20.3850	1.80180
$129$	0	0
$130$	2.35284	0.206358
$131$	−10.0585	−0.878818	−0.439409	−	0.898287i	$-0.644812\pi$
$131$	−10.0585	−0.878818	−0.439409	+	0.898287i	$0.644812\pi$
$132$	0	0
$133$	−5.94008	−0.515070
$134$	30.3505	2.62188
$135$	0	0
$136$	3.89476	0.333973
$137$	13.7882	1.17801	0.589003	−	0.808131i	$-0.299520\pi$
$137$	13.7882	1.17801	0.589003	+	0.808131i	$0.299520\pi$
$138$	0	0
$139$	−10.7067	−0.908132	−0.454066	−	0.890968i	$-0.650027\pi$
$139$	−10.7067	−0.908132	−0.454066	+	0.890968i	$0.650027\pi$
$140$	−8.59440	−0.726359
$141$	0	0
$142$	−0.397134	−0.0333267
$143$	0.183026	0.0153054
$144$	0	0
$145$	4.26082	0.353841
$146$	−19.0226	−1.57432
$147$	0	0
$148$	0.325395	0.0267473
$149$	5.98074	0.489961	0.244981	−	0.969528i	$-0.421218\pi$
$149$	5.98074	0.489961	0.244981	+	0.969528i	$0.421218\pi$
$150$	0	0
$151$	−15.5116	−1.26231	−0.631157	−	0.775655i	$-0.717420\pi$
$151$	−15.5116	−1.26231	−0.631157	+	0.775655i	$0.717420\pi$
$152$	8.83121	0.716306
$153$	0	0
$154$	−1.04671	−0.0843462
$155$	3.53587	0.284008
$156$	0	0
$157$	−7.87048	−0.628133	−0.314066	−	0.949401i	$-0.601691\pi$
$157$	−7.87048	−0.628133	−0.314066	+	0.949401i	$0.601691\pi$
$158$	−22.5264	−1.79210
$159$	0	0
$160$	−3.86126	−0.305260
$161$	−10.8360	−0.853993
$162$	0	0
$163$	18.3407	1.43656	0.718279	−	0.695755i	$-0.244930\pi$
$163$	18.3407	1.43656	0.718279	+	0.695755i	$0.244930\pi$
$164$	−27.8935	−2.17811
$165$	0	0
$166$	−8.81456	−0.684142
$167$	11.0549	0.855453	0.427727	−	0.903908i	$-0.359315\pi$
$167$	11.0549	0.855453	0.427727	+	0.903908i	$0.359315\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−2.53587	−0.194492
$171$	0	0
$172$	−4.36103	−0.332525
$173$	−3.48339	−0.264837	−0.132419	−	0.991194i	$-0.542274\pi$
$173$	−3.48339	−0.264837	−0.132419	+	0.991194i	$0.542274\pi$
$174$	0	0
$175$	2.43063	0.183739
$176$	0.261843	0.0197372
$177$	0	0
$178$	−29.9845	−2.24743
$179$	−11.9869	−0.895941	−0.447970	−	0.894048i	$-0.647853\pi$
$179$	−11.9869	−0.895941	−0.447970	+	0.894048i	$0.647853\pi$
$180$	0	0
$181$	−10.1664	−0.755660	−0.377830	−	0.925875i	$-0.623330\pi$
$181$	−10.1664	−0.755660	−0.377830	+	0.925875i	$0.623330\pi$
$182$	−5.71890	−0.423913
$183$	0	0
$184$	16.1100	1.18764
$185$	−0.0920270	−0.00676596
$186$	0	0
$187$	−0.197264	−0.0144254
$188$	37.3887	2.72685
$189$	0	0
$190$	−5.74998	−0.417147
$191$	−14.6828	−1.06241	−0.531205	−	0.847243i	$-0.678261\pi$
$191$	−14.6828	−1.06241	−0.531205	+	0.847243i	$0.678261\pi$
$192$	0	0
$193$	−16.4425	−1.18355	−0.591777	−	0.806102i	$-0.701573\pi$
$193$	−16.4425	−1.18355	−0.591777	+	0.806102i	$0.701573\pi$
$194$	20.1735	1.44838
$195$	0	0
$196$	−3.86126	−0.275805
$197$	−12.6483	−0.901152	−0.450576	−	0.892738i	$-0.648781\pi$
$197$	−12.6483	−0.901152	−0.450576	+	0.892738i	$0.648781\pi$
$198$	0	0
$199$	−21.5873	−1.53028	−0.765142	−	0.643861i	$-0.777331\pi$
$199$	−21.5873	−1.53028	−0.765142	+	0.643861i	$0.777331\pi$
$200$	−3.61366	−0.255524
$201$	0	0
$202$	27.1756	1.91207
$203$	−10.3565	−0.726882
$204$	0	0
$205$	7.88871	0.550972
$206$	−5.53345	−0.385534
$207$	0	0
$208$	1.43063	0.0991965
$209$	−0.447288	−0.0309395
$210$	0	0
$211$	−23.9651	−1.64983	−0.824913	−	0.565259i	$-0.808776\pi$
$211$	−23.9651	−1.64983	−0.824913	+	0.565259i	$0.808776\pi$
$212$	12.0118	0.824975
$213$	0	0
$214$	15.4486	1.05604
$215$	1.23337	0.0841150
$216$	0	0
$217$	−8.59440	−0.583426
$218$	−23.0755	−1.56287
$219$	0	0
$220$	−0.647157	−0.0436313
$221$	−1.07779	−0.0725000
$222$	0	0
$223$	−16.5670	−1.10941	−0.554705	−	0.832047i	$-0.687169\pi$
$223$	−16.5670	−1.10941	−0.554705	+	0.832047i	$0.687169\pi$
$224$	9.38531	0.627083
$225$	0	0
$226$	15.7463	1.04743
$227$	−3.92937	−0.260801	−0.130401	−	0.991461i	$-0.541626\pi$
$227$	−3.92937	−0.260801	−0.130401	+	0.991461i	$0.541626\pi$
$228$	0	0
$229$	−19.7582	−1.30566	−0.652828	−	0.757506i	$-0.726418\pi$
$229$	−19.7582	−1.30566	−0.652828	+	0.757506i	$0.726418\pi$
$230$	−10.4892	−0.691635
$231$	0	0
$232$	15.3971	1.01087
$233$	−20.7846	−1.36164	−0.680822	−	0.732449i	$-0.738377\pi$
$233$	−20.7846	−1.36164	−0.680822	+	0.732449i	$0.738377\pi$
$234$	0	0
$235$	−10.5741	−0.689779
$236$	−21.0464	−1.37001
$237$	0	0
$238$	6.16377	0.399538
$239$	12.0814	0.781482	0.390741	−	0.920501i	$-0.372219\pi$
$239$	12.0814	0.781482	0.390741	+	0.920501i	$0.372219\pi$
$240$	0	0
$241$	1.73313	0.111641	0.0558205	−	0.998441i	$-0.482223\pi$
$241$	1.73313	0.111641	0.0558205	+	0.998441i	$0.482223\pi$
$242$	25.8025	1.65864
$243$	0	0
$244$	−19.9462	−1.27693
$245$	1.09203	0.0697670
$246$	0	0
$247$	−2.44384	−0.155498
$248$	12.7774	0.811367
$249$	0	0
$250$	2.35284	0.148807
$251$	9.18413	0.579697	0.289849	−	0.957072i	$-0.406395\pi$
$251$	9.18413	0.579697	0.289849	+	0.957072i	$0.406395\pi$
$252$	0	0
$253$	−0.815946	−0.0512981
$254$	7.85521	0.492880
$255$	0	0
$256$	−24.0703	−1.50440
$257$	19.0922	1.19094	0.595470	−	0.803378i	$-0.296966\pi$
$257$	19.0922	1.19094	0.595470	+	0.803378i	$0.296966\pi$
$258$	0	0
$259$	0.223684	0.0138990
$260$	−3.53587	−0.219285
$261$	0	0
$262$	23.6661	1.46210
$263$	−9.81203	−0.605036	−0.302518	−	0.953144i	$-0.597827\pi$
$263$	−9.81203	−0.605036	−0.302518	+	0.953144i	$0.597827\pi$
$264$	0	0
$265$	−3.39713	−0.208684
$266$	13.9761	0.856928
$267$	0	0
$268$	−45.6110	−2.78613
$269$	14.4008	0.878030	0.439015	−	0.898480i	$-0.355327\pi$
$269$	14.4008	0.878030	0.439015	+	0.898480i	$0.355327\pi$
$270$	0	0
$271$	11.5970	0.704468	0.352234	−	0.935912i	$-0.385422\pi$
$271$	11.5970	0.704468	0.352234	+	0.935912i	$0.385422\pi$
$272$	−1.54192	−0.0934927
$273$	0	0
$274$	−32.4415	−1.95986
$275$	0.183026	0.0110369
$276$	0	0
$277$	23.0932	1.38754	0.693769	−	0.720197i	$-0.255949\pi$
$277$	23.0932	1.38754	0.693769	+	0.720197i	$0.255949\pi$
$278$	25.1912	1.51087
$279$	0	0
$280$	8.78348	0.524913
$281$	−19.3707	−1.15556	−0.577780	−	0.816193i	$-0.696081\pi$
$281$	−19.3707	−1.15556	−0.577780	+	0.816193i	$0.696081\pi$
$282$	0	0
$283$	1.38653	0.0824206	0.0412103	−	0.999150i	$-0.486879\pi$
$283$	1.38653	0.0824206	0.0412103	+	0.999150i	$0.486879\pi$
$284$	0.596815	0.0354145
$285$	0	0
$286$	−0.430632	−0.0254638
$287$	−19.1746	−1.13184
$288$	0	0
$289$	−15.8384	−0.931669
$290$	−10.0250	−0.588690
$291$	0	0
$292$	28.5873	1.67295
$293$	−12.6682	−0.740084	−0.370042	−	0.929015i	$-0.620657\pi$
$293$	−12.6682	−0.740084	−0.370042	+	0.929015i	$0.620657\pi$
$294$	0	0
$295$	5.95226	0.346554
$296$	−0.332554	−0.0193293
$297$	0	0
$298$	−14.0717	−0.815154
$299$	−4.45808	−0.257817
$300$	0	0
$301$	−2.99786	−0.172794
$302$	36.4963	2.10013
$303$	0	0
$304$	−3.49624	−0.200523
$305$	5.64111	0.323009
$306$	0	0
$307$	−16.7556	−0.956290	−0.478145	−	0.878281i	$-0.658691\pi$
$307$	−16.7556	−0.956290	−0.478145	+	0.878281i	$0.658691\pi$
$308$	1.57300	0.0896301
$309$	0	0
$310$	−8.31934	−0.472507
$311$	−8.39005	−0.475756	−0.237878	−	0.971295i	$-0.576452\pi$
$311$	−8.39005	−0.475756	−0.237878	+	0.971295i	$0.576452\pi$
$312$	0	0
$313$	17.9580	1.01505	0.507524	−	0.861638i	$-0.330561\pi$
$313$	17.9580	1.01505	0.507524	+	0.861638i	$0.330561\pi$
$314$	18.5180	1.04503
$315$	0	0
$316$	33.8528	1.90437
$317$	−16.4248	−0.922507	−0.461253	−	0.887268i	$-0.652600\pi$
$317$	−16.4248	−0.922507	−0.461253	+	0.887268i	$0.652600\pi$
$318$	0	0
$319$	−0.779842	−0.0436627
$320$	11.9462	0.667814
$321$	0	0
$322$	25.4953	1.42080
$323$	2.63395	0.146557
$324$	0	0
$325$	1.00000	0.0554700
$326$	−43.1529	−2.39002
$327$	0	0
$328$	28.5071	1.57404
$329$	25.7018	1.41698
$330$	0	0
$331$	29.2270	1.60646	0.803232	−	0.595667i	$-0.203112\pi$
$331$	29.2270	1.60646	0.803232	+	0.595667i	$0.203112\pi$
$332$	13.2466	0.727000
$333$	0	0
$334$	−26.0104	−1.42323
$335$	12.8995	0.704775
$336$	0	0
$337$	22.9665	1.25107	0.625533	−	0.780198i	$-0.284882\pi$
$337$	22.9665	1.25107	0.625533	+	0.780198i	$0.284882\pi$
$338$	−2.35284	−0.127978
$339$	0	0
$340$	3.81092	0.206676
$341$	−0.647157	−0.0350455
$342$	0	0
$343$	−19.6687	−1.06201
$344$	4.45697	0.240304
$345$	0	0
$346$	8.19587	0.440613
$347$	10.8644	0.583233	0.291617	−	0.956535i	$-0.405807\pi$
$347$	10.8644	0.583233	0.291617	+	0.956535i	$0.405807\pi$
$348$	0	0
$349$	22.5965	1.20956	0.604782	−	0.796391i	$-0.293260\pi$
$349$	22.5965	1.20956	0.604782	+	0.796391i	$0.293260\pi$
$350$	−5.71890	−0.305688
$351$	0	0
$352$	0.706713	0.0376679
$353$	−27.5386	−1.46573	−0.732865	−	0.680374i	$-0.761817\pi$
$353$	−27.5386	−1.46573	−0.732865	+	0.680374i	$0.761817\pi$
$354$	0	0
$355$	−0.168789	−0.00895838
$356$	45.0609	2.38822
$357$	0	0
$358$	28.2032	1.49059
$359$	10.6168	0.560334	0.280167	−	0.959951i	$-0.409610\pi$
$359$	10.6168	0.560334	0.280167	+	0.959951i	$0.409610\pi$
$360$	0	0
$361$	−13.0276	−0.685665
$362$	23.9199	1.25720
$363$	0	0
$364$	8.59440	0.450469
$365$	−8.08495	−0.423186
$366$	0	0
$367$	−14.3037	−0.746648	−0.373324	−	0.927701i	$-0.621782\pi$
$367$	−14.3037	−0.746648	−0.373324	+	0.927701i	$0.621782\pi$
$368$	−6.37787	−0.332470
$369$	0	0
$370$	0.216525	0.0112566
$371$	8.25718	0.428692
$372$	0	0
$373$	−15.3143	−0.792945	−0.396473	−	0.918046i	$-0.629766\pi$
$373$	−15.3143	−0.792945	−0.396473	+	0.918046i	$0.629766\pi$
$374$	0.464131	0.0239996
$375$	0	0
$376$	−38.2112	−1.97059
$377$	−4.26082	−0.219443
$378$	0	0
$379$	14.6458	0.752307	0.376153	−	0.926557i	$-0.377247\pi$
$379$	14.6458	0.752307	0.376153	+	0.926557i	$0.377247\pi$
$380$	8.64111	0.443279
$381$	0	0
$382$	34.5463	1.76754
$383$	2.30492	0.117776	0.0588879	−	0.998265i	$-0.481245\pi$
$383$	2.30492	0.117776	0.0588879	+	0.998265i	$0.481245\pi$
$384$	0	0
$385$	−0.444870	−0.0226727
$386$	38.6865	1.96909
$387$	0	0
$388$	−30.3169	−1.53911
$389$	27.8157	1.41031	0.705157	−	0.709051i	$-0.250876\pi$
$389$	27.8157	1.41031	0.705157	+	0.709051i	$0.250876\pi$
$390$	0	0
$391$	4.80487	0.242993
$392$	3.94621	0.199314
$393$	0	0
$394$	29.7594	1.49926
$395$	−9.57411	−0.481726
$396$	0	0
$397$	20.6432	1.03605	0.518027	−	0.855364i	$-0.326667\pi$
$397$	20.6432	1.03605	0.518027	+	0.855364i	$0.326667\pi$
$398$	50.7916	2.54595
$399$	0	0
$400$	1.43063	0.0715316
$401$	30.1975	1.50799	0.753994	−	0.656881i	$-0.228125\pi$
$401$	30.1975	1.50799	0.753994	+	0.656881i	$0.228125\pi$
$402$	0	0
$403$	−3.53587	−0.176134
$404$	−40.8397	−2.03185
$405$	0	0
$406$	24.3672	1.20932
$407$	0.0168434	0.000834894 0
$408$	0	0
$409$	30.8620	1.52603	0.763014	−	0.646382i	$-0.223719\pi$
$409$	30.8620	1.52603	0.763014	+	0.646382i	$0.223719\pi$
$410$	−18.5609	−0.916657
$411$	0	0
$412$	8.31571	0.409686
$413$	−14.4678	−0.711912
$414$	0	0
$415$	−3.74634	−0.183901
$416$	3.86126	0.189314
$417$	0	0
$418$	1.05240	0.0514744
$419$	10.8876	0.531894	0.265947	−	0.963988i	$-0.414315\pi$
$419$	10.8876	0.531894	0.265947	+	0.963988i	$0.414315\pi$
$420$	0	0
$421$	−10.2848	−0.501251	−0.250626	−	0.968084i	$-0.580636\pi$
$421$	−10.2848	−0.501251	−0.250626	+	0.968084i	$0.580636\pi$
$422$	56.3861	2.74483
$423$	0	0
$424$	−12.2761	−0.596179
$425$	−1.07779	−0.0522805
$426$	0	0
$427$	−13.7115	−0.663544
$428$	−23.2162	−1.12220
$429$	0	0
$430$	−2.90192	−0.139943
$431$	−15.0775	−0.726258	−0.363129	−	0.931739i	$-0.618292\pi$
$431$	−15.0775	−0.726258	−0.363129	+	0.931739i	$0.618292\pi$
$432$	0	0
$433$	9.25579	0.444805	0.222402	−	0.974955i	$-0.428610\pi$
$433$	9.25579	0.444805	0.222402	+	0.974955i	$0.428610\pi$
$434$	20.2213	0.970652
$435$	0	0
$436$	34.6780	1.66077
$437$	10.8948	0.521171
$438$	0	0
$439$	28.6419	1.36700	0.683500	−	0.729950i	$-0.260457\pi$
$439$	28.6419	1.36700	0.683500	+	0.729950i	$0.260457\pi$
$440$	0.661395	0.0315308
$441$	0	0
$442$	2.53587	0.120619
$443$	29.2138	1.38799	0.693995	−	0.719979i	$-0.255849\pi$
$443$	29.2138	1.38799	0.693995	+	0.719979i	$0.255849\pi$
$444$	0	0
$445$	−12.7439	−0.604120
$446$	38.9796	1.84574
$447$	0	0
$448$	−29.0368	−1.37186
$449$	−26.1604	−1.23459	−0.617293	−	0.786734i	$-0.711771\pi$
$449$	−26.1604	−1.23459	−0.617293	+	0.786734i	$0.711771\pi$
$450$	0	0
$451$	−1.44384	−0.0679879
$452$	−23.6637	−1.11305
$453$	0	0
$454$	9.24519	0.433898
$455$	−2.43063	−0.113950
$456$	0	0
$457$	26.3575	1.23295	0.616476	−	0.787374i	$-0.288560\pi$
$457$	26.3575	1.23295	0.616476	+	0.787374i	$0.288560\pi$
$458$	46.4879	2.17223
$459$	0	0
$460$	15.7632	0.734962
$461$	−26.1120	−1.21616	−0.608079	−	0.793876i	$-0.708060\pi$
$461$	−26.1120	−1.21616	−0.608079	+	0.793876i	$0.708060\pi$
$462$	0	0
$463$	−36.0253	−1.67424	−0.837119	−	0.547021i	$-0.815762\pi$
$463$	−36.0253	−1.67424	−0.837119	+	0.547021i	$0.815762\pi$
$464$	−6.09566	−0.282984
$465$	0	0
$466$	48.9029	2.26538
$467$	−25.5706	−1.18327	−0.591633	−	0.806208i	$-0.701516\pi$
$467$	−25.5706	−1.18327	−0.591633	+	0.806208i	$0.701516\pi$
$468$	0	0
$469$	−31.3540	−1.44779
$470$	24.8792	1.14759
$471$	0	0
$472$	21.5094	0.990053
$473$	−0.225739	−0.0103795
$474$	0	0
$475$	−2.44384	−0.112131
$476$	−9.26295	−0.424567
$477$	0	0
$478$	−28.4257	−1.30016
$479$	35.6519	1.62898	0.814488	−	0.580180i	$-0.197018\pi$
$479$	35.6519	1.62898	0.814488	+	0.580180i	$0.197018\pi$
$480$	0	0
$481$	0.0920270	0.00419607
$482$	−4.07779	−0.185738
$483$	0	0
$484$	−38.7761	−1.76255
$485$	8.57411	0.389330
$486$	0	0
$487$	8.55829	0.387813	0.193907	−	0.981020i	$-0.437884\pi$
$487$	8.55829	0.387813	0.193907	+	0.981020i	$0.437884\pi$
$488$	20.3850	0.922787
$489$	0	0
$490$	−2.56937	−0.116072
$491$	−6.30519	−0.284549	−0.142275	−	0.989827i	$-0.545442\pi$
$491$	−6.30519	−0.284549	−0.142275	+	0.989827i	$0.545442\pi$
$492$	0	0
$493$	4.59226	0.206825
$494$	5.74998	0.258704
$495$	0	0
$496$	−5.05853	−0.227135
$497$	0.410264	0.0184028
$498$	0	0
$499$	31.5630	1.41296	0.706478	−	0.707735i	$-0.250283\pi$
$499$	31.5630	1.41296	0.706478	+	0.707735i	$0.250283\pi$
$500$	−3.53587	−0.158129
$501$	0	0
$502$	−21.6088	−0.964449
$503$	−16.8478	−0.751205	−0.375603	−	0.926781i	$-0.622564\pi$
$503$	−16.8478	−0.751205	−0.375603	+	0.926781i	$0.622564\pi$
$504$	0	0
$505$	11.5501	0.513973
$506$	1.91979	0.0853452
$507$	0	0
$508$	−11.8049	−0.523756
$509$	−5.78348	−0.256348	−0.128174	−	0.991752i	$-0.540912\pi$
$509$	−5.78348	−0.256348	−0.128174	+	0.991752i	$0.540912\pi$
$510$	0	0
$511$	19.6515	0.869333
$512$	15.8637	0.701082
$513$	0	0
$514$	−44.9210	−1.98138
$515$	−2.35182	−0.103633
$516$	0	0
$517$	1.93534	0.0851162
$518$	−0.526293	−0.0231240
$519$	0	0
$520$	3.61366	0.158469
$521$	−2.47829	−0.108576	−0.0542879	−	0.998525i	$-0.517289\pi$
$521$	−2.47829	−0.108576	−0.0542879	+	0.998525i	$0.517289\pi$
$522$	0	0
$523$	22.3138	0.975716	0.487858	−	0.872923i	$-0.337778\pi$
$523$	22.3138	0.975716	0.487858	+	0.872923i	$0.337778\pi$
$524$	−35.5656	−1.55369
$525$	0	0
$526$	23.0862	1.00660
$527$	3.81092	0.166006
$528$	0	0
$529$	−3.12553	−0.135892
$530$	7.99292	0.347190
$531$	0	0
$532$	−21.0034	−0.910611
$533$	−7.88871	−0.341698
$534$	0	0
$535$	6.56592	0.283870
$536$	46.6144	2.01344
$537$	0	0
$538$	−33.8827	−1.46079
$539$	−0.199870	−0.00860900
$540$	0	0
$541$	6.00400	0.258132	0.129066	−	0.991636i	$-0.458802\pi$
$541$	6.00400	0.258132	0.129066	+	0.991636i	$0.458802\pi$
$542$	−27.2859	−1.17203
$543$	0	0
$544$	−4.16163	−0.178428
$545$	−9.80748	−0.420106
$546$	0	0
$547$	18.3103	0.782893	0.391446	−	0.920201i	$-0.371975\pi$
$547$	18.3103	0.782893	0.391446	+	0.920201i	$0.371975\pi$
$548$	48.7533	2.08264
$549$	0	0
$550$	−0.430632	−0.0183622
$551$	10.4128	0.443599
$552$	0	0
$553$	23.2711	0.989589
$554$	−54.3348	−2.30846
$555$	0	0
$556$	−37.8575	−1.60552
$557$	−25.5043	−1.08065	−0.540326	−	0.841456i	$-0.681699\pi$
$557$	−25.5043	−1.08065	−0.540326	+	0.841456i	$0.681699\pi$
$558$	0	0
$559$	−1.23337	−0.0521659
$560$	−3.47734	−0.146944
$561$	0	0
$562$	45.5762	1.92252
$563$	−11.7255	−0.494171	−0.247086	−	0.968994i	$-0.579473\pi$
$563$	−11.7255	−0.494171	−0.247086	+	0.968994i	$0.579473\pi$
$564$	0	0
$565$	6.69248	0.281555
$566$	−3.26229	−0.137124
$567$	0	0
$568$	−0.609945	−0.0255927
$569$	−46.1412	−1.93434	−0.967171	−	0.254127i	$-0.918212\pi$
$569$	−46.1412	−1.93434	−0.967171	+	0.254127i	$0.918212\pi$
$570$	0	0
$571$	38.2903	1.60240	0.801200	−	0.598397i	$-0.204195\pi$
$571$	38.2903	1.60240	0.801200	+	0.598397i	$0.204195\pi$
$572$	0.647157	0.0270590
$573$	0	0
$574$	45.1147	1.88305
$575$	−4.45808	−0.185915
$576$	0	0
$577$	−15.8454	−0.659655	−0.329827	−	0.944041i	$-0.606991\pi$
$577$	−15.8454	−0.659655	−0.329827	+	0.944041i	$0.606991\pi$
$578$	37.2652	1.55003
$579$	0	0
$580$	15.0657	0.625569
$581$	9.10598	0.377780
$582$	0	0
$583$	0.621765	0.0257509
$584$	−29.2162	−1.20898
$585$	0	0
$586$	29.8063	1.23129
$587$	3.17381	0.130997	0.0654986	−	0.997853i	$-0.479136\pi$
$587$	3.17381	0.130997	0.0654986	+	0.997853i	$0.479136\pi$
$588$	0	0
$589$	8.64111	0.356051
$590$	−14.0047	−0.576566
$591$	0	0
$592$	0.131657	0.00541106
$593$	6.10290	0.250616	0.125308	−	0.992118i	$-0.460008\pi$
$593$	6.10290	0.250616	0.125308	+	0.992118i	$0.460008\pi$
$594$	0	0
$595$	2.61971	0.107398
$596$	21.1471	0.866219
$597$	0	0
$598$	10.4892	0.428934
$599$	23.4498	0.958133	0.479066	−	0.877779i	$-0.340975\pi$
$599$	23.4498	0.958133	0.479066	+	0.877779i	$0.340975\pi$
$600$	0	0
$601$	3.22842	0.131690	0.0658451	−	0.997830i	$-0.479026\pi$
$601$	3.22842	0.131690	0.0658451	+	0.997830i	$0.479026\pi$
$602$	7.05350	0.287479
$603$	0	0
$604$	−54.8469	−2.23169
$605$	10.9665	0.445852
$606$	0	0
$607$	−0.626506	−0.0254291	−0.0127145	−	0.999919i	$-0.504047\pi$
$607$	−0.626506	−0.0254291	−0.0127145	+	0.999919i	$0.504047\pi$
$608$	−9.43632	−0.382693
$609$	0	0
$610$	−13.2726	−0.537393
$611$	10.5741	0.427783
$612$	0	0
$613$	22.3221	0.901583	0.450791	−	0.892629i	$-0.351142\pi$
$613$	22.3221	0.901583	0.450791	+	0.892629i	$0.351142\pi$
$614$	39.4232	1.59099
$615$	0	0
$616$	−1.60761	−0.0647723
$617$	−46.2105	−1.86036	−0.930182	−	0.367100i	$-0.880351\pi$
$617$	−46.2105	−1.86036	−0.930182	+	0.367100i	$0.880351\pi$
$618$	0	0
$619$	−37.8000	−1.51931	−0.759654	−	0.650327i	$-0.774632\pi$
$619$	−37.8000	−1.51931	−0.759654	+	0.650327i	$0.774632\pi$
$620$	12.5024	0.502107
$621$	0	0
$622$	19.7405	0.791521
$623$	30.9758	1.24102
$624$	0	0
$625$	1.00000	0.0400000
$626$	−42.2524	−1.68875
$627$	0	0
$628$	−27.8290	−1.11050
$629$	−0.0991857	−0.00395479
$630$	0	0
$631$	35.7155	1.42181	0.710905	−	0.703288i	$-0.248286\pi$
$631$	35.7155	1.42181	0.710905	+	0.703288i	$0.248286\pi$
$632$	−34.5976	−1.37622
$633$	0	0
$634$	38.6449	1.53479
$635$	3.33861	0.132489
$636$	0	0
$637$	−1.09203	−0.0432677
$638$	1.83484	0.0726422
$639$	0	0
$640$	−20.3850	−0.805789
$641$	−40.9261	−1.61648	−0.808242	−	0.588850i	$-0.799581\pi$
$641$	−40.9261	−1.61648	−0.808242	+	0.588850i	$0.799581\pi$
$642$	0	0
$643$	28.9087	1.14005	0.570024	−	0.821628i	$-0.306934\pi$
$643$	28.9087	1.14005	0.570024	+	0.821628i	$0.306934\pi$
$644$	−38.3145	−1.50980
$645$	0	0
$646$	−6.19726	−0.243828
$647$	46.9080	1.84414	0.922071	−	0.387021i	$-0.126496\pi$
$647$	46.9080	1.84414	0.922071	+	0.387021i	$0.126496\pi$
$648$	0	0
$649$	−1.08942	−0.0427635
$650$	−2.35284	−0.0922861
$651$	0	0
$652$	64.8505	2.53974
$653$	−27.5596	−1.07849	−0.539245	−	0.842149i	$-0.681290\pi$
$653$	−27.5596	−1.07849	−0.539245	+	0.842149i	$0.681290\pi$
$654$	0	0
$655$	10.0585	0.393019
$656$	−11.2858	−0.440638
$657$	0	0
$658$	−60.4722	−2.35745
$659$	−10.0108	−0.389965	−0.194983	−	0.980807i	$-0.562465\pi$
$659$	−10.0108	−0.389965	−0.194983	+	0.980807i	$0.562465\pi$
$660$	0	0
$661$	−38.3165	−1.49034	−0.745169	−	0.666876i	$-0.767631\pi$
$661$	−38.3165	−1.49034	−0.745169	+	0.666876i	$0.767631\pi$
$662$	−68.7666	−2.67269
$663$	0	0
$664$	−13.5380	−0.525376
$665$	5.94008	0.230346
$666$	0	0
$667$	18.9951	0.735492
$668$	39.0887	1.51239
$669$	0	0
$670$	−30.3505	−1.17254
$671$	−1.03247	−0.0398581
$672$	0	0
$673$	3.50118	0.134961	0.0674803	−	0.997721i	$-0.478504\pi$
$673$	3.50118	0.134961	0.0674803	+	0.997721i	$0.478504\pi$
$674$	−54.0366	−2.08141
$675$	0	0
$676$	3.53587	0.135995
$677$	−24.5852	−0.944885	−0.472443	−	0.881361i	$-0.656628\pi$
$677$	−24.5852	−0.944885	−0.472443	+	0.881361i	$0.656628\pi$
$678$	0	0
$679$	−20.8405	−0.799785
$680$	−3.89476	−0.149357
$681$	0	0
$682$	1.52266	0.0583056
$683$	−40.6170	−1.55417	−0.777083	−	0.629398i	$-0.783302\pi$
$683$	−40.6170	−1.55417	−0.777083	+	0.629398i	$0.783302\pi$
$684$	0	0
$685$	−13.7882	−0.526821
$686$	46.2775	1.76688
$687$	0	0
$688$	−1.76450	−0.0672708
$689$	3.39713	0.129420
$690$	0	0
$691$	−1.92213	−0.0731213	−0.0365606	−	0.999331i	$-0.511640\pi$
$691$	−1.92213	−0.0731213	−0.0365606	+	0.999331i	$0.511640\pi$
$692$	−12.3168	−0.468215
$693$	0	0
$694$	−25.5623	−0.970331
$695$	10.7067	0.406129
$696$	0	0
$697$	8.50237	0.322050
$698$	−53.1661	−2.01237
$699$	0	0
$700$	8.59440	0.324838
$701$	−30.9436	−1.16872	−0.584362	−	0.811493i	$-0.698655\pi$
$701$	−30.9436	−1.16872	−0.584362	+	0.811493i	$0.698655\pi$
$702$	0	0
$703$	−0.224899	−0.00848224
$704$	−2.18647	−0.0824058
$705$	0	0
$706$	64.7939	2.43855
$707$	−28.0741	−1.05583
$708$	0	0
$709$	−18.8450	−0.707738	−0.353869	−	0.935295i	$-0.615134\pi$
$709$	−18.8450	−0.707738	−0.353869	+	0.935295i	$0.615134\pi$
$710$	0.397134	0.0149042
$711$	0	0
$712$	−46.0522	−1.72588
$713$	15.7632	0.590336
$714$	0	0
$715$	−0.183026	−0.00684480
$716$	−42.3840	−1.58396
$717$	0	0
$718$	−24.9797	−0.932235
$719$	21.5200	0.802559	0.401280	−	0.915956i	$-0.368566\pi$
$719$	21.5200	0.802559	0.401280	+	0.915956i	$0.368566\pi$
$720$	0	0
$721$	5.71640	0.212890
$722$	30.6520	1.14075
$723$	0	0
$724$	−35.9470	−1.33596
$725$	−4.26082	−0.158243
$726$	0	0
$727$	13.3927	0.496706	0.248353	−	0.968670i	$-0.420111\pi$
$727$	13.3927	0.496706	0.248353	+	0.968670i	$0.420111\pi$
$728$	−8.78348	−0.325537
$729$	0	0
$730$	19.0226	0.704058
$731$	1.32931	0.0491664
$732$	0	0
$733$	−20.2980	−0.749725	−0.374862	−	0.927080i	$-0.622310\pi$
$733$	−20.2980	−0.749725	−0.374862	+	0.927080i	$0.622310\pi$
$734$	33.6544	1.24221
$735$	0	0
$736$	−17.2138	−0.634510
$737$	−2.36095	−0.0869667
$738$	0	0
$739$	22.0444	0.810915	0.405458	−	0.914114i	$-0.367112\pi$
$739$	22.0444	0.810915	0.405458	+	0.914114i	$0.367112\pi$
$740$	−0.325395	−0.0119618
$741$	0	0
$742$	−19.4279	−0.713219
$743$	12.8676	0.472066	0.236033	−	0.971745i	$-0.424153\pi$
$743$	12.8676	0.472066	0.236033	+	0.971745i	$0.424153\pi$
$744$	0	0
$745$	−5.98074	−0.219117
$746$	36.0322	1.31923
$747$	0	0
$748$	−0.697499	−0.0255031
$749$	−15.9593	−0.583142
$750$	0	0
$751$	−24.4998	−0.894009	−0.447005	−	0.894532i	$-0.647509\pi$
$751$	−24.4998	−0.894009	−0.447005	+	0.894532i	$0.647509\pi$
$752$	15.1277	0.551649
$753$	0	0
$754$	10.0250	0.365090
$755$	15.5116	0.564524
$756$	0	0
$757$	−32.8193	−1.19284	−0.596419	−	0.802673i	$-0.703410\pi$
$757$	−32.8193	−1.19284	−0.596419	+	0.802673i	$0.703410\pi$
$758$	−34.4594	−1.25162
$759$	0	0
$760$	−8.83121	−0.320342
$761$	47.3069	1.71487	0.857437	−	0.514590i	$-0.172056\pi$
$761$	47.3069	1.71487	0.857437	+	0.514590i	$0.172056\pi$
$762$	0	0
$763$	23.8384	0.863007
$764$	−51.9164	−1.87827
$765$	0	0
$766$	−5.42311	−0.195945
$767$	−5.95226	−0.214924
$768$	0	0
$769$	−0.823293	−0.0296887	−0.0148443	−	0.999890i	$-0.504725\pi$
$769$	−0.823293	−0.0296887	−0.0148443	+	0.999890i	$0.504725\pi$
$770$	1.04671	0.0377208
$771$	0	0
$772$	−58.1384	−2.09245
$773$	11.2193	0.403531	0.201766	−	0.979434i	$-0.435332\pi$
$773$	11.2193	0.403531	0.201766	+	0.979434i	$0.435332\pi$
$774$	0	0
$775$	−3.53587	−0.127012
$776$	30.9839	1.11226
$777$	0	0
$778$	−65.4461	−2.34636
$779$	19.2788	0.690733
$780$	0	0
$781$	0.0308928	0.00110543
$782$	−11.3051	−0.404270
$783$	0	0
$784$	−1.56229	−0.0557960
$785$	7.87048	0.280909
$786$	0	0
$787$	−43.2754	−1.54260	−0.771301	−	0.636471i	$-0.780394\pi$
$787$	−43.2754	−1.54260	−0.771301	+	0.636471i	$0.780394\pi$
$788$	−44.7226	−1.59318
$789$	0	0
$790$	22.5264	0.801452
$791$	−16.2669	−0.578386
$792$	0	0
$793$	−5.64111	−0.200322
$794$	−48.5703	−1.72370
$795$	0	0
$796$	−76.3299	−2.70544
$797$	−51.3338	−1.81834	−0.909169	−	0.416427i	$-0.863282\pi$
$797$	−51.3338	−1.81834	−0.909169	+	0.416427i	$0.863282\pi$
$798$	0	0
$799$	−11.3967	−0.403185
$800$	3.86126	0.136516
$801$	0	0
$802$	−71.0499	−2.50886
$803$	1.47976	0.0522195
$804$	0	0
$805$	10.8360	0.381917
$806$	8.31934	0.293036
$807$	0	0
$808$	41.7381	1.46834
$809$	7.68184	0.270079	0.135040	−	0.990840i	$-0.456884\pi$
$809$	7.68184	0.270079	0.135040	+	0.990840i	$0.456884\pi$
$810$	0	0
$811$	−34.0847	−1.19688	−0.598439	−	0.801169i	$-0.704212\pi$
$811$	−34.0847	−1.19688	−0.598439	+	0.801169i	$0.704212\pi$
$812$	−36.6191	−1.28508
$813$	0	0
$814$	−0.0396298	−0.00138902
$815$	−18.3407	−0.642448
$816$	0	0
$817$	3.01416	0.105452
$818$	−72.6135	−2.53887
$819$	0	0
$820$	27.8935	0.974082
$821$	21.3374	0.744680	0.372340	−	0.928096i	$-0.378556\pi$
$821$	21.3374	0.744680	0.372340	+	0.928096i	$0.378556\pi$
$822$	0	0
$823$	36.9078	1.28652	0.643262	−	0.765646i	$-0.277581\pi$
$823$	36.9078	1.28652	0.643262	+	0.765646i	$0.277581\pi$
$824$	−8.49866	−0.296065
$825$	0	0
$826$	34.0404	1.18442
$827$	−25.2650	−0.878550	−0.439275	−	0.898353i	$-0.644765\pi$
$827$	−25.2650	−0.878550	−0.439275	+	0.898353i	$0.644765\pi$
$828$	0	0
$829$	46.5989	1.61845	0.809224	−	0.587500i	$-0.199888\pi$
$829$	46.5989	1.61845	0.809224	+	0.587500i	$0.199888\pi$
$830$	8.81456	0.305958
$831$	0	0
$832$	−11.9462	−0.414160
$833$	1.17698	0.0407798
$834$	0	0
$835$	−11.0549	−0.382570
$836$	−1.58155	−0.0546991
$837$	0	0
$838$	−25.6168	−0.884918
$839$	24.5324	0.846953	0.423477	−	0.905907i	$-0.360810\pi$
$839$	24.5324	0.846953	0.423477	+	0.905907i	$0.360810\pi$
$840$	0	0
$841$	−10.8454	−0.373981
$842$	24.1986	0.833937
$843$	0	0
$844$	−84.7375	−2.91678
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−26.6555	−0.915895
$848$	4.86005	0.166895
$849$	0	0
$850$	2.53587	0.0869796
$851$	−0.410264	−0.0140637
$852$	0	0
$853$	16.7480	0.573441	0.286720	−	0.958014i	$-0.407435\pi$
$853$	16.7480	0.573441	0.286720	+	0.958014i	$0.407435\pi$
$854$	32.2609	1.10395
$855$	0	0
$856$	23.7270	0.810972
$857$	10.4809	0.358020	0.179010	−	0.983847i	$-0.442711\pi$
$857$	10.4809	0.358020	0.179010	+	0.983847i	$0.442711\pi$
$858$	0	0
$859$	−48.3559	−1.64988	−0.824941	−	0.565219i	$-0.808792\pi$
$859$	−48.3559	−1.64988	−0.824941	+	0.565219i	$0.808792\pi$
$860$	4.36103	0.148710
$861$	0	0
$862$	35.4750	1.20828
$863$	23.8110	0.810537	0.405268	−	0.914198i	$-0.367178\pi$
$863$	23.8110	0.810537	0.405268	+	0.914198i	$0.367178\pi$
$864$	0	0
$865$	3.48339	0.118439
$866$	−21.7774	−0.740027
$867$	0	0
$868$	−30.3887	−1.03146
$869$	1.75231	0.0594432
$870$	0	0
$871$	−12.8995	−0.437083
$872$	−35.4409	−1.20018
$873$	0	0
$874$	−25.6339	−0.867078
$875$	−2.43063	−0.0821704
$876$	0	0
$877$	44.2353	1.49372	0.746860	−	0.664981i	$-0.231560\pi$
$877$	44.2353	1.49372	0.746860	+	0.664981i	$0.231560\pi$
$878$	−67.3898	−2.27429
$879$	0	0
$880$	−0.261843	−0.00882674
$881$	2.89187	0.0974297	0.0487149	−	0.998813i	$-0.484487\pi$
$881$	2.89187	0.0974297	0.0487149	+	0.998813i	$0.484487\pi$
$882$	0	0
$883$	24.4592	0.823117	0.411559	−	0.911383i	$-0.364985\pi$
$883$	24.4592	0.823117	0.411559	+	0.911383i	$0.364985\pi$
$884$	−3.81092	−0.128175
$885$	0	0
$886$	−68.7355	−2.30922
$887$	43.5599	1.46260	0.731299	−	0.682057i	$-0.238915\pi$
$887$	43.5599	1.46260	0.731299	+	0.682057i	$0.238915\pi$
$888$	0	0
$889$	−8.11492	−0.272166
$890$	29.9845	1.00508
$891$	0	0
$892$	−58.5788	−1.96136
$893$	−25.8415	−0.864751
$894$	0	0
$895$	11.9869	0.400677
$896$	49.5485	1.65530
$897$	0	0
$898$	61.5513	2.05399
$899$	15.0657	0.502469
$900$	0	0
$901$	−3.66139	−0.121979
$902$	3.39713	0.113112
$903$	0	0
$904$	24.1843	0.804359
$905$	10.1664	0.337942
$906$	0	0
$907$	−13.9344	−0.462684	−0.231342	−	0.972872i	$-0.574312\pi$
$907$	−13.9344	−0.462684	−0.231342	+	0.972872i	$0.574312\pi$
$908$	−13.8937	−0.461080
$909$	0	0
$910$	5.71890	0.189580
$911$	21.5208	0.713016	0.356508	−	0.934292i	$-0.383967\pi$
$911$	21.5208	0.713016	0.356508	+	0.934292i	$0.383967\pi$
$912$	0	0
$913$	0.685679	0.0226927
$914$	−62.0151	−2.05128
$915$	0	0
$916$	−69.8623	−2.30831
$917$	−24.4486	−0.807363
$918$	0	0
$919$	36.9897	1.22018	0.610088	−	0.792333i	$-0.291134\pi$
$919$	36.9897	1.22018	0.610088	+	0.792333i	$0.291134\pi$
$920$	−16.1100	−0.531130
$921$	0	0
$922$	61.4375	2.02334
$923$	0.168789	0.00555575
$924$	0	0
$925$	0.0920270	0.00302583
$926$	84.7619	2.78545
$927$	0	0
$928$	−16.4521	−0.540068
$929$	−4.50934	−0.147947	−0.0739733	−	0.997260i	$-0.523568\pi$
$929$	−4.50934	−0.147947	−0.0739733	+	0.997260i	$0.523568\pi$
$930$	0	0
$931$	2.66874	0.0874645
$932$	−73.4916	−2.40730
$933$	0	0
$934$	60.1636	1.96861
$935$	0.197264	0.00645122
$936$	0	0
$937$	25.2200	0.823900	0.411950	−	0.911206i	$-0.364848\pi$
$937$	25.2200	0.823900	0.411950	+	0.911206i	$0.364848\pi$
$938$	73.7709	2.40871
$939$	0	0
$940$	−37.3887	−1.21948
$941$	6.50684	0.212117	0.106059	−	0.994360i	$-0.466177\pi$
$941$	6.50684	0.212117	0.106059	+	0.994360i	$0.466177\pi$
$942$	0	0
$943$	35.1685	1.14524
$944$	−8.51550	−0.277156
$945$	0	0
$946$	0.531128	0.0172685
$947$	37.5599	1.22053	0.610266	−	0.792197i	$-0.291063\pi$
$947$	37.5599	1.22053	0.610266	+	0.792197i	$0.291063\pi$
$948$	0	0
$949$	8.08495	0.262449
$950$	5.74998	0.186554
$951$	0	0
$952$	9.46674	0.306819
$953$	−48.5070	−1.57130	−0.785648	−	0.618673i	$-0.787671\pi$
$953$	−48.5070	−1.57130	−0.785648	+	0.618673i	$0.787671\pi$
$954$	0	0
$955$	14.6828	0.475124
$956$	42.7183	1.38161
$957$	0	0
$958$	−83.8833	−2.71015
$959$	33.5141	1.08223
$960$	0	0
$961$	−18.4976	−0.596698
$962$	−0.216525	−0.00698105
$963$	0	0
$964$	6.12813	0.197374
$965$	16.4425	0.529301
$966$	0	0
$967$	−3.86621	−0.124329	−0.0621644	−	0.998066i	$-0.519800\pi$
$967$	−3.86621	−0.124329	−0.0621644	+	0.998066i	$0.519800\pi$
$968$	39.6292	1.27373
$969$	0	0
$970$	−20.1735	−0.647733
$971$	13.9432	0.447460	0.223730	−	0.974651i	$-0.428177\pi$
$971$	13.9432	0.447460	0.223730	+	0.974651i	$0.428177\pi$
$972$	0	0
$973$	−26.0241	−0.834294
$974$	−20.1363	−0.645209
$975$	0	0
$976$	−8.07035	−0.258326
$977$	46.9476	1.50199	0.750993	−	0.660310i	$-0.229575\pi$
$977$	46.9476	1.50199	0.750993	+	0.660310i	$0.229575\pi$
$978$	0	0
$979$	2.33247	0.0745462
$980$	3.86126	0.123344
$981$	0	0
$982$	14.8351	0.473407
$983$	−44.2683	−1.41194	−0.705970	−	0.708242i	$-0.749489\pi$
$983$	−44.2683	−1.41194	−0.705970	+	0.708242i	$0.749489\pi$
$984$	0	0
$985$	12.6483	0.403007
$986$	−10.8049	−0.344097
$987$	0	0
$988$	−8.64111	−0.274910
$989$	5.49845	0.174841
$990$	0	0
$991$	−1.13027	−0.0359041	−0.0179521	−	0.999839i	$-0.505715\pi$
$991$	−1.13027	−0.0359041	−0.0179521	+	0.999839i	$0.505715\pi$
$992$	−13.6529	−0.433481
$993$	0	0
$994$	−0.965286	−0.0306170
$995$	21.5873	0.684364
$996$	0	0
$997$	32.7216	1.03630	0.518151	−	0.855289i	$-0.326620\pi$
$997$	32.7216	1.03630	0.518151	+	0.855289i	$0.326620\pi$
$998$	−74.2629	−2.35075
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.o.1.1	✓	4
3.2	odd	2		1755.2.a.q.1.4	yes	4
5.4	even	2		8775.2.a.br.1.4		4
15.14	odd	2		8775.2.a.bj.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.o.1.1	✓	4	1.1	even	1	trivial
1755.2.a.q.1.4	yes	4	3.2	odd	2
8775.2.a.bj.1.1		4	15.14	odd	2
8775.2.a.br.1.4		4	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)

\(f(q)\)	\(=\)	\(q-2.35284 q^{2} +3.53587 q^{4} -1.00000 q^{5} +2.43063 q^{7} -3.61366 q^{8} +O(q^{10})\)	\(q-2.35284 q^{2} +3.53587 q^{4} -1.00000 q^{5} +2.43063 q^{7} -3.61366 q^{8} +2.35284 q^{10} +0.183026 q^{11} +1.00000 q^{13} -5.71890 q^{14} +1.43063 q^{16} -1.07779 q^{17} -2.44384 q^{19} -3.53587 q^{20} -0.430632 q^{22} -4.45808 q^{23} +1.00000 q^{25} -2.35284 q^{26} +8.59440 q^{28} -4.26082 q^{29} -3.53587 q^{31} +3.86126 q^{32} +2.53587 q^{34} -2.43063 q^{35} +0.0920270 q^{37} +5.74998 q^{38} +3.61366 q^{40} -7.88871 q^{41} -1.23337 q^{43} +0.647157 q^{44} +10.4892 q^{46} +10.5741 q^{47} -1.09203 q^{49} -2.35284 q^{50} +3.53587 q^{52} +3.39713 q^{53} -0.183026 q^{55} -8.78348 q^{56} +10.0250 q^{58} -5.95226 q^{59} -5.64111 q^{61} +8.31934 q^{62} -11.9462 q^{64} -1.00000 q^{65} -12.8995 q^{67} -3.81092 q^{68} +5.71890 q^{70} +0.168789 q^{71} +8.08495 q^{73} -0.216525 q^{74} -8.64111 q^{76} +0.444870 q^{77} +9.57411 q^{79} -1.43063 q^{80} +18.5609 q^{82} +3.74634 q^{83} +1.07779 q^{85} +2.90192 q^{86} -0.661395 q^{88} +12.7439 q^{89} +2.43063 q^{91} -15.7632 q^{92} -24.8792 q^{94} +2.44384 q^{95} -8.57411 q^{97} +2.56937 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 3 q^{4} - 4 q^{5} + q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q - q^2 + 3 * q^4 - 4 * q^5 + q^7 - 3 * q^8	\( 4 q - q^{2} + 3 q^{4} - 4 q^{5} + q^{7} - 3 q^{8} + q^{10} - 2 q^{11} + 4 q^{13} - 9 q^{14} - 3 q^{16} - 4 q^{17} - 4 q^{19} - 3 q^{20} + 7 q^{22} - 7 q^{23} + 4 q^{25} - q^{26} - 2 q^{28} - 14 q^{29} - 3 q^{31} - 2 q^{32} - q^{34} - q^{35} - 5 q^{37} - 14 q^{38} + 3 q^{40} - 12 q^{41} - 4 q^{43} + 11 q^{44} + 8 q^{46} - 11 q^{47} + q^{49} - q^{50} + 3 q^{52} - 15 q^{53} + 2 q^{55} - 18 q^{56} - 5 q^{58} - 9 q^{59} - 9 q^{61} + 5 q^{62} - 11 q^{64} - 4 q^{65} + 8 q^{67} + 4 q^{68} + 9 q^{70} + 3 q^{71} + 13 q^{73} - 18 q^{74} - 21 q^{76} - 12 q^{77} - 15 q^{79} + 3 q^{80} + 18 q^{82} - q^{83} + 4 q^{85} - 5 q^{86} - 6 q^{88} - 8 q^{89} + q^{91} - 29 q^{92} - 23 q^{94} + 4 q^{95} + 19 q^{97} + 19 q^{98}+O(q^{100}) \) 4 * q - q^2 + 3 * q^4 - 4 * q^5 + q^7 - 3 * q^8 + q^10 - 2 * q^11 + 4 * q^13 - 9 * q^14 - 3 * q^16 - 4 * q^17 - 4 * q^19 - 3 * q^20 + 7 * q^22 - 7 * q^23 + 4 * q^25 - q^26 - 2 * q^28 - 14 * q^29 - 3 * q^31 - 2 * q^32 - q^34 - q^35 - 5 * q^37 - 14 * q^38 + 3 * q^40 - 12 * q^41 - 4 * q^43 + 11 * q^44 + 8 * q^46 - 11 * q^47 + q^49 - q^50 + 3 * q^52 - 15 * q^53 + 2 * q^55 - 18 * q^56 - 5 * q^58 - 9 * q^59 - 9 * q^61 + 5 * q^62 - 11 * q^64 - 4 * q^65 + 8 * q^67 + 4 * q^68 + 9 * q^70 + 3 * q^71 + 13 * q^73 - 18 * q^74 - 21 * q^76 - 12 * q^77 - 15 * q^79 + 3 * q^80 + 18 * q^82 - q^83 + 4 * q^85 - 5 * q^86 - 6 * q^88 - 8 * q^89 + q^91 - 29 * q^92 - 23 * q^94 + 4 * q^95 + 19 * q^97 + 19 * q^98

Introduction

L-functions

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