LMFDB - Embedded newform 5775.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5775,2,Mod(1,5775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5775.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:	$46.1136071673$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	5775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} +0.381966 q^{6} -1.00000 q^{7} +1.47214 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} +0.381966 q^{6} -1.00000 q^{7} +1.47214 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.85410 q^{12} -1.38197 q^{13} +0.381966 q^{14} +3.14590 q^{16} +3.23607 q^{17} -0.381966 q^{18} -3.61803 q^{19} +1.00000 q^{21} -0.381966 q^{22} -5.47214 q^{23} -1.47214 q^{24} +0.527864 q^{26} -1.00000 q^{27} +1.85410 q^{28} +3.47214 q^{31} -4.14590 q^{32} -1.00000 q^{33} -1.23607 q^{34} -1.85410 q^{36} -6.85410 q^{37} +1.38197 q^{38} +1.38197 q^{39} +1.76393 q^{41} -0.381966 q^{42} +8.47214 q^{43} -1.85410 q^{44} +2.09017 q^{46} +3.00000 q^{47} -3.14590 q^{48} +1.00000 q^{49} -3.23607 q^{51} +2.56231 q^{52} -1.47214 q^{53} +0.381966 q^{54} -1.47214 q^{56} +3.61803 q^{57} -5.32624 q^{59} +4.23607 q^{61} -1.32624 q^{62} -1.00000 q^{63} -4.70820 q^{64} +0.381966 q^{66} +15.7082 q^{67} -6.00000 q^{68} +5.47214 q^{69} -2.52786 q^{71} +1.47214 q^{72} -9.00000 q^{73} +2.61803 q^{74} +6.70820 q^{76} -1.00000 q^{77} -0.527864 q^{78} +8.70820 q^{79} +1.00000 q^{81} -0.673762 q^{82} -9.32624 q^{83} -1.85410 q^{84} -3.23607 q^{86} +1.47214 q^{88} +8.52786 q^{89} +1.38197 q^{91} +10.1459 q^{92} -3.47214 q^{93} -1.14590 q^{94} +4.14590 q^{96} -8.56231 q^{97} -0.381966 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 3 q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 3 * q^6 - 2 * q^7 - 6 * q^8 + 2 * q^9	$ 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 3 q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{11} - 3 q^{12} - 5 q^{13} + 3 q^{14} + 13 q^{16} + 2 q^{17} - 3 q^{18} - 5 q^{19} + 2 q^{21} - 3 q^{22} - 2 q^{23} + 6 q^{24} + 10 q^{26} - 2 q^{27} - 3 q^{28} - 2 q^{31} - 15 q^{32} - 2 q^{33} + 2 q^{34} + 3 q^{36} - 7 q^{37} + 5 q^{38} + 5 q^{39} + 8 q^{41} - 3 q^{42} + 8 q^{43} + 3 q^{44} - 7 q^{46} + 6 q^{47} - 13 q^{48} + 2 q^{49} - 2 q^{51} - 15 q^{52} + 6 q^{53} + 3 q^{54} + 6 q^{56} + 5 q^{57} + 5 q^{59} + 4 q^{61} + 13 q^{62} - 2 q^{63} + 4 q^{64} + 3 q^{66} + 18 q^{67} - 12 q^{68} + 2 q^{69} - 14 q^{71} - 6 q^{72} - 18 q^{73} + 3 q^{74} - 2 q^{77} - 10 q^{78} + 4 q^{79} + 2 q^{81} - 17 q^{82} - 3 q^{83} + 3 q^{84} - 2 q^{86} - 6 q^{88} + 26 q^{89} + 5 q^{91} + 27 q^{92} + 2 q^{93} - 9 q^{94} + 15 q^{96} + 3 q^{97} - 3 q^{98} + 2 q^{99}+O(q^{100}) $ 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 3 * q^6 - 2 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^11 - 3 * q^12 - 5 * q^13 + 3 * q^14 + 13 * q^16 + 2 * q^17 - 3 * q^18 - 5 * q^19 + 2 * q^21 - 3 * q^22 - 2 * q^23 + 6 * q^24 + 10 * q^26 - 2 * q^27 - 3 * q^28 - 2 * q^31 - 15 * q^32 - 2 * q^33 + 2 * q^34 + 3 * q^36 - 7 * q^37 + 5 * q^38 + 5 * q^39 + 8 * q^41 - 3 * q^42 + 8 * q^43 + 3 * q^44 - 7 * q^46 + 6 * q^47 - 13 * q^48 + 2 * q^49 - 2 * q^51 - 15 * q^52 + 6 * q^53 + 3 * q^54 + 6 * q^56 + 5 * q^57 + 5 * q^59 + 4 * q^61 + 13 * q^62 - 2 * q^63 + 4 * q^64 + 3 * q^66 + 18 * q^67 - 12 * q^68 + 2 * q^69 - 14 * q^71 - 6 * q^72 - 18 * q^73 + 3 * q^74 - 2 * q^77 - 10 * q^78 + 4 * q^79 + 2 * q^81 - 17 * q^82 - 3 * q^83 + 3 * q^84 - 2 * q^86 - 6 * q^88 + 26 * q^89 + 5 * q^91 + 27 * q^92 + 2 * q^93 - 9 * q^94 + 15 * q^96 + 3 * q^97 - 3 * q^98 + 2 * q^99

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 $

$2$	−0.381966	−0.270091	−0.135045	−	0.990839i	$-0.543118\pi$
$2$	−0.381966	−0.270091	−0.135045	+	0.990839i	$0.543118\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.85410	−0.927051
$5$	0	0
$5$	0	0
$6$	0.381966	0.155937
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.47214	0.520479
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.85410	0.535233
$13$	−1.38197	−0.383288	−0.191644	−	0.981464i	$-0.561382\pi$
$13$	−1.38197	−0.383288	−0.191644	+	0.981464i	$0.561382\pi$
$14$	0.381966	0.102085
$15$	0	0
$16$	3.14590	0.786475
$17$	3.23607	0.784862	0.392431	−	0.919781i	$-0.371634\pi$
$17$	3.23607	0.784862	0.392431	+	0.919781i	$0.371634\pi$
$18$	−0.381966	−0.0900303
$19$	−3.61803	−0.830034	−0.415017	−	0.909814i	$-0.636224\pi$
$19$	−3.61803	−0.830034	−0.415017	+	0.909814i	$0.636224\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−0.381966	−0.0814354
$23$	−5.47214	−1.14102	−0.570510	−	0.821291i	$-0.693254\pi$
$23$	−5.47214	−1.14102	−0.570510	+	0.821291i	$0.693254\pi$
$24$	−1.47214	−0.300498
$25$	0	0
$26$	0.527864	0.103523
$27$	−1.00000	−0.192450
$28$	1.85410	0.350392
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	3.47214	0.623614	0.311807	−	0.950145i	$-0.399066\pi$
$31$	3.47214	0.623614	0.311807	+	0.950145i	$0.399066\pi$
$32$	−4.14590	−0.732898
$33$	−1.00000	−0.174078
$34$	−1.23607	−0.211984
$35$	0	0
$36$	−1.85410	−0.309017
$37$	−6.85410	−1.12681	−0.563404	−	0.826182i	$-0.690508\pi$
$37$	−6.85410	−1.12681	−0.563404	+	0.826182i	$0.690508\pi$
$38$	1.38197	0.224184
$39$	1.38197	0.221292
$40$	0	0
$41$	1.76393	0.275480	0.137740	−	0.990468i	$-0.456016\pi$
$41$	1.76393	0.275480	0.137740	+	0.990468i	$0.456016\pi$
$42$	−0.381966	−0.0589386
$43$	8.47214	1.29199	0.645994	−	0.763342i	$-0.276443\pi$
$43$	8.47214	1.29199	0.645994	+	0.763342i	$0.276443\pi$
$44$	−1.85410	−0.279516
$45$	0	0
$46$	2.09017	0.308179
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−3.14590	−0.454071
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.23607	−0.453140
$52$	2.56231	0.355328
$53$	−1.47214	−0.202213	−0.101107	−	0.994876i	$-0.532238\pi$
$53$	−1.47214	−0.202213	−0.101107	+	0.994876i	$0.532238\pi$
$54$	0.381966	0.0519790
$55$	0	0
$56$	−1.47214	−0.196722
$57$	3.61803	0.479220
$58$	0	0
$59$	−5.32624	−0.693417	−0.346709	−	0.937973i	$-0.612701\pi$
$59$	−5.32624	−0.693417	−0.346709	+	0.937973i	$0.612701\pi$
$60$	0	0
$61$	4.23607	0.542373	0.271186	−	0.962527i	$-0.412584\pi$
$61$	4.23607	0.542373	0.271186	+	0.962527i	$0.412584\pi$
$62$	−1.32624	−0.168432
$63$	−1.00000	−0.125988
$64$	−4.70820	−0.588525
$65$	0	0
$66$	0.381966	0.0470168
$67$	15.7082	1.91906	0.959531	−	0.281602i	$-0.0908658\pi$
$67$	15.7082	1.91906	0.959531	+	0.281602i	$0.0908658\pi$
$68$	−6.00000	−0.727607
$69$	5.47214	0.658768
$70$	0	0
$71$	−2.52786	−0.300002	−0.150001	−	0.988686i	$-0.547928\pi$
$71$	−2.52786	−0.300002	−0.150001	+	0.988686i	$0.547928\pi$
$72$	1.47214	0.173493
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	2.61803	0.304340
$75$	0	0
$76$	6.70820	0.769484
$77$	−1.00000	−0.113961
$78$	−0.527864	−0.0597688
$79$	8.70820	0.979749	0.489875	−	0.871793i	$-0.337043\pi$
$79$	8.70820	0.979749	0.489875	+	0.871793i	$0.337043\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.673762	−0.0744046
$83$	−9.32624	−1.02369	−0.511844	−	0.859079i	$-0.671037\pi$
$83$	−9.32624	−1.02369	−0.511844	+	0.859079i	$0.671037\pi$
$84$	−1.85410	−0.202299
$85$	0	0
$86$	−3.23607	−0.348954
$87$	0	0
$88$	1.47214	0.156930
$89$	8.52786	0.903952	0.451976	−	0.892030i	$-0.350719\pi$
$89$	8.52786	0.903952	0.451976	+	0.892030i	$0.350719\pi$
$90$	0	0
$91$	1.38197	0.144869
$92$	10.1459	1.05778
$93$	−3.47214	−0.360044
$94$	−1.14590	−0.118190
$95$	0	0
$96$	4.14590	0.423139
$97$	−8.56231	−0.869370	−0.434685	−	0.900582i	$-0.643140\pi$
$97$	−8.56231	−0.869370	−0.434685	+	0.900582i	$0.643140\pi$
$98$	−0.381966	−0.0385844
$99$	1.00000	0.100504
$100$	0	0
$101$	17.1803	1.70951	0.854754	−	0.519034i	$-0.173708\pi$
$101$	17.1803	1.70951	0.854754	+	0.519034i	$0.173708\pi$
$102$	1.23607	0.122389
$103$	8.85410	0.872421	0.436210	−	0.899845i	$-0.356320\pi$
$103$	8.85410	0.872421	0.436210	+	0.899845i	$0.356320\pi$
$104$	−2.03444	−0.199493
$105$	0	0
$106$	0.562306	0.0546160
$107$	−2.23607	−0.216169	−0.108084	−	0.994142i	$-0.534472\pi$
$107$	−2.23607	−0.216169	−0.108084	+	0.994142i	$0.534472\pi$
$108$	1.85410	0.178411
$109$	7.32624	0.701726	0.350863	−	0.936427i	$-0.385888\pi$
$109$	7.32624	0.701726	0.350863	+	0.936427i	$0.385888\pi$
$110$	0	0
$111$	6.85410	0.650563
$112$	−3.14590	−0.297259
$113$	−12.3820	−1.16480	−0.582399	−	0.812903i	$-0.697886\pi$
$113$	−12.3820	−1.16480	−0.582399	+	0.812903i	$0.697886\pi$
$114$	−1.38197	−0.129433
$115$	0	0
$116$	0	0
$117$	−1.38197	−0.127763
$118$	2.03444	0.187286
$119$	−3.23607	−0.296650
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.61803	−0.146490
$123$	−1.76393	−0.159048
$124$	−6.43769	−0.578122
$125$	0	0
$126$	0.381966	0.0340282
$127$	−7.56231	−0.671046	−0.335523	−	0.942032i	$-0.608913\pi$
$127$	−7.56231	−0.671046	−0.335523	+	0.942032i	$0.608913\pi$
$128$	10.0902	0.891853
$129$	−8.47214	−0.745930
$130$	0	0
$131$	16.4164	1.43431	0.717154	−	0.696915i	$-0.245444\pi$
$131$	16.4164	1.43431	0.717154	+	0.696915i	$0.245444\pi$
$132$	1.85410	0.161379
$133$	3.61803	0.313723
$134$	−6.00000	−0.518321
$135$	0	0
$136$	4.76393	0.408504
$137$	7.18034	0.613458	0.306729	−	0.951797i	$-0.400765\pi$
$137$	7.18034	0.613458	0.306729	+	0.951797i	$0.400765\pi$
$138$	−2.09017	−0.177927
$139$	−1.70820	−0.144888	−0.0724440	−	0.997372i	$-0.523080\pi$
$139$	−1.70820	−0.144888	−0.0724440	+	0.997372i	$0.523080\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0.965558	0.0810278
$143$	−1.38197	−0.115566
$144$	3.14590	0.262158
$145$	0	0
$146$	3.43769	0.284506
$147$	−1.00000	−0.0824786
$148$	12.7082	1.04461
$149$	−13.3820	−1.09629	−0.548147	−	0.836382i	$-0.684666\pi$
$149$	−13.3820	−1.09629	−0.548147	+	0.836382i	$0.684666\pi$
$150$	0	0
$151$	9.79837	0.797380	0.398690	−	0.917086i	$-0.369465\pi$
$151$	9.79837	0.797380	0.398690	+	0.917086i	$0.369465\pi$
$152$	−5.32624	−0.432015
$153$	3.23607	0.261621
$154$	0.381966	0.0307797
$155$	0	0
$156$	−2.56231	−0.205149
$157$	−6.23607	−0.497692	−0.248846	−	0.968543i	$-0.580051\pi$
$157$	−6.23607	−0.497692	−0.248846	+	0.968543i	$0.580051\pi$
$158$	−3.32624	−0.264621
$159$	1.47214	0.116748
$160$	0	0
$161$	5.47214	0.431265
$162$	−0.381966	−0.0300101
$163$	−7.29180	−0.571138	−0.285569	−	0.958358i	$-0.592182\pi$
$163$	−7.29180	−0.571138	−0.285569	+	0.958358i	$0.592182\pi$
$164$	−3.27051	−0.255384
$165$	0	0
$166$	3.56231	0.276489
$167$	−1.09017	−0.0843599	−0.0421799	−	0.999110i	$-0.513430\pi$
$167$	−1.09017	−0.0843599	−0.0421799	+	0.999110i	$0.513430\pi$
$168$	1.47214	0.113578
$169$	−11.0902	−0.853090
$170$	0	0
$171$	−3.61803	−0.276678
$172$	−15.7082	−1.19774
$173$	−7.38197	−0.561240	−0.280620	−	0.959819i	$-0.590540\pi$
$173$	−7.38197	−0.561240	−0.280620	+	0.959819i	$0.590540\pi$
$174$	0	0
$175$	0	0
$176$	3.14590	0.237131
$177$	5.32624	0.400345
$178$	−3.25735	−0.244149
$179$	−1.14590	−0.0856484	−0.0428242	−	0.999083i	$-0.513636\pi$
$179$	−1.14590	−0.0856484	−0.0428242	+	0.999083i	$0.513636\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	−0.527864	−0.0391279
$183$	−4.23607	−0.313139
$184$	−8.05573	−0.593876
$185$	0	0
$186$	1.32624	0.0972445
$187$	3.23607	0.236645
$188$	−5.56231	−0.405673
$189$	1.00000	0.0727393
$190$	0	0
$191$	8.67376	0.627611	0.313806	−	0.949487i	$-0.398396\pi$
$191$	8.67376	0.627611	0.313806	+	0.949487i	$0.398396\pi$
$192$	4.70820	0.339785
$193$	−7.85410	−0.565351	−0.282675	−	0.959216i	$-0.591222\pi$
$193$	−7.85410	−0.565351	−0.282675	+	0.959216i	$0.591222\pi$
$194$	3.27051	0.234809
$195$	0	0
$196$	−1.85410	−0.132436
$197$	−10.0344	−0.714924	−0.357462	−	0.933928i	$-0.616358\pi$
$197$	−10.0344	−0.714924	−0.357462	+	0.933928i	$0.616358\pi$
$198$	−0.381966	−0.0271451
$199$	−10.1803	−0.721665	−0.360833	−	0.932631i	$-0.617507\pi$
$199$	−10.1803	−0.721665	−0.360833	+	0.932631i	$0.617507\pi$
$200$	0	0
$201$	−15.7082	−1.10797
$202$	−6.56231	−0.461722
$203$	0	0
$204$	6.00000	0.420084
$205$	0	0
$206$	−3.38197	−0.235633
$207$	−5.47214	−0.380340
$208$	−4.34752	−0.301447
$209$	−3.61803	−0.250265
$210$	0	0
$211$	−1.29180	−0.0889309	−0.0444655	−	0.999011i	$-0.514158\pi$
$211$	−1.29180	−0.0889309	−0.0444655	+	0.999011i	$0.514158\pi$
$212$	2.72949	0.187462
$213$	2.52786	0.173206
$214$	0.854102	0.0583852
$215$	0	0
$216$	−1.47214	−0.100166
$217$	−3.47214	−0.235704
$218$	−2.79837	−0.189530
$219$	9.00000	0.608164
$220$	0	0
$221$	−4.47214	−0.300828
$222$	−2.61803	−0.175711
$223$	−10.4164	−0.697534	−0.348767	−	0.937209i	$-0.613400\pi$
$223$	−10.4164	−0.697534	−0.348767	+	0.937209i	$0.613400\pi$
$224$	4.14590	0.277009
$225$	0	0
$226$	4.72949	0.314601
$227$	−13.6525	−0.906147	−0.453073	−	0.891473i	$-0.649672\pi$
$227$	−13.6525	−0.906147	−0.453073	+	0.891473i	$0.649672\pi$
$228$	−6.70820	−0.444262
$229$	0.437694	0.0289236	0.0144618	−	0.999895i	$-0.495396\pi$
$229$	0.437694	0.0289236	0.0144618	+	0.999895i	$0.495396\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−3.43769	−0.225211	−0.112605	−	0.993640i	$-0.535920\pi$
$233$	−3.43769	−0.225211	−0.112605	+	0.993640i	$0.535920\pi$
$234$	0.527864	0.0345076
$235$	0	0
$236$	9.87539	0.642833
$237$	−8.70820	−0.565659
$238$	1.23607	0.0801224
$239$	−12.0902	−0.782048	−0.391024	−	0.920380i	$-0.627879\pi$
$239$	−12.0902	−0.782048	−0.391024	+	0.920380i	$0.627879\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	−0.381966	−0.0245537
$243$	−1.00000	−0.0641500
$244$	−7.85410	−0.502807
$245$	0	0
$246$	0.673762	0.0429575
$247$	5.00000	0.318142
$248$	5.11146	0.324578
$249$	9.32624	0.591026
$250$	0	0
$251$	−11.9443	−0.753916	−0.376958	−	0.926230i	$-0.623030\pi$
$251$	−11.9443	−0.753916	−0.376958	+	0.926230i	$0.623030\pi$
$252$	1.85410	0.116797
$253$	−5.47214	−0.344030
$254$	2.88854	0.181243
$255$	0	0
$256$	5.56231	0.347644
$257$	4.85410	0.302791	0.151395	−	0.988473i	$-0.451623\pi$
$257$	4.85410	0.302791	0.151395	+	0.988473i	$0.451623\pi$
$258$	3.23607	0.201469
$259$	6.85410	0.425893
$260$	0	0
$261$	0	0
$262$	−6.27051	−0.387393
$263$	−31.7984	−1.96077	−0.980386	−	0.197088i	$-0.936852\pi$
$263$	−31.7984	−1.96077	−0.980386	+	0.197088i	$0.936852\pi$
$264$	−1.47214	−0.0906037
$265$	0	0
$266$	−1.38197	−0.0847338
$267$	−8.52786	−0.521897
$268$	−29.1246	−1.77907
$269$	−1.03444	−0.0630710	−0.0315355	−	0.999503i	$-0.510040\pi$
$269$	−1.03444	−0.0630710	−0.0315355	+	0.999503i	$0.510040\pi$
$270$	0	0
$271$	16.6525	1.01157	0.505783	−	0.862661i	$-0.331204\pi$
$271$	16.6525	1.01157	0.505783	+	0.862661i	$0.331204\pi$
$272$	10.1803	0.617274
$273$	−1.38197	−0.0836404
$274$	−2.74265	−0.165689
$275$	0	0
$276$	−10.1459	−0.610711
$277$	−24.7984	−1.48999	−0.744995	−	0.667070i	$-0.767548\pi$
$277$	−24.7984	−1.48999	−0.744995	+	0.667070i	$0.767548\pi$
$278$	0.652476	0.0391329
$279$	3.47214	0.207871
$280$	0	0
$281$	−26.8885	−1.60404	−0.802018	−	0.597300i	$-0.796240\pi$
$281$	−26.8885	−1.60404	−0.802018	+	0.597300i	$0.796240\pi$
$282$	1.14590	0.0682372
$283$	15.2705	0.907738	0.453869	−	0.891069i	$-0.350043\pi$
$283$	15.2705	0.907738	0.453869	+	0.891069i	$0.350043\pi$
$284$	4.68692	0.278117
$285$	0	0
$286$	0.527864	0.0312133
$287$	−1.76393	−0.104122
$288$	−4.14590	−0.244299
$289$	−6.52786	−0.383992
$290$	0	0
$291$	8.56231	0.501931
$292$	16.6869	0.976528
$293$	0.673762	0.0393616	0.0196808	−	0.999806i	$-0.493735\pi$
$293$	0.673762	0.0393616	0.0196808	+	0.999806i	$0.493735\pi$
$294$	0.381966	0.0222767
$295$	0	0
$296$	−10.0902	−0.586479
$297$	−1.00000	−0.0580259
$298$	5.11146	0.296099
$299$	7.56231	0.437339
$300$	0	0
$301$	−8.47214	−0.488326
$302$	−3.74265	−0.215365
$303$	−17.1803	−0.986985
$304$	−11.3820	−0.652801
$305$	0	0
$306$	−1.23607	−0.0706613
$307$	−8.41641	−0.480350	−0.240175	−	0.970730i	$-0.577205\pi$
$307$	−8.41641	−0.480350	−0.240175	+	0.970730i	$0.577205\pi$
$308$	1.85410	0.105647
$309$	−8.85410	−0.503692
$310$	0	0
$311$	25.4721	1.44439	0.722196	−	0.691688i	$-0.243133\pi$
$311$	25.4721	1.44439	0.722196	+	0.691688i	$0.243133\pi$
$312$	2.03444	0.115178
$313$	−5.43769	−0.307357	−0.153678	−	0.988121i	$-0.549112\pi$
$313$	−5.43769	−0.307357	−0.153678	+	0.988121i	$0.549112\pi$
$314$	2.38197	0.134422
$315$	0	0
$316$	−16.1459	−0.908278
$317$	2.90983	0.163432	0.0817162	−	0.996656i	$-0.473960\pi$
$317$	2.90983	0.163432	0.0817162	+	0.996656i	$0.473960\pi$
$318$	−0.562306	−0.0315325
$319$	0	0
$320$	0	0
$321$	2.23607	0.124805
$322$	−2.09017	−0.116481
$323$	−11.7082	−0.651462
$324$	−1.85410	−0.103006
$325$	0	0
$326$	2.78522	0.154259
$327$	−7.32624	−0.405142
$328$	2.59675	0.143381
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−14.3820	−0.790504	−0.395252	−	0.918573i	$-0.629343\pi$
$331$	−14.3820	−0.790504	−0.395252	+	0.918573i	$0.629343\pi$
$332$	17.2918	0.949011
$333$	−6.85410	−0.375602
$334$	0.416408	0.0227848
$335$	0	0
$336$	3.14590	0.171623
$337$	−32.1246	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$337$	−32.1246	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$338$	4.23607	0.230412
$339$	12.3820	0.672496
$340$	0	0
$341$	3.47214	0.188027
$342$	1.38197	0.0747282
$343$	−1.00000	−0.0539949
$344$	12.4721	0.672453
$345$	0	0
$346$	2.81966	0.151586
$347$	15.5967	0.837277	0.418639	−	0.908153i	$-0.362507\pi$
$347$	15.5967	0.837277	0.418639	+	0.908153i	$0.362507\pi$
$348$	0	0
$349$	−24.4164	−1.30698	−0.653490	−	0.756935i	$-0.726696\pi$
$349$	−24.4164	−1.30698	−0.653490	+	0.756935i	$0.726696\pi$
$350$	0	0
$351$	1.38197	0.0737639
$352$	−4.14590	−0.220977
$353$	12.7639	0.679356	0.339678	−	0.940542i	$-0.389682\pi$
$353$	12.7639	0.679356	0.339678	+	0.940542i	$0.389682\pi$
$354$	−2.03444	−0.108129
$355$	0	0
$356$	−15.8115	−0.838009
$357$	3.23607	0.171271
$358$	0.437694	0.0231329
$359$	22.5967	1.19261	0.596305	−	0.802758i	$-0.296635\pi$
$359$	22.5967	1.19261	0.596305	+	0.802758i	$0.296635\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	2.67376	0.140530
$363$	−1.00000	−0.0524864
$364$	−2.56231	−0.134301
$365$	0	0
$366$	1.61803	0.0845760
$367$	−24.9787	−1.30388	−0.651939	−	0.758271i	$-0.726044\pi$
$367$	−24.9787	−1.30388	−0.651939	+	0.758271i	$0.726044\pi$
$368$	−17.2148	−0.897383
$369$	1.76393	0.0918266
$370$	0	0
$371$	1.47214	0.0764295
$372$	6.43769	0.333779
$373$	−6.94427	−0.359561	−0.179780	−	0.983707i	$-0.557539\pi$
$373$	−6.94427	−0.359561	−0.179780	+	0.983707i	$0.557539\pi$
$374$	−1.23607	−0.0639156
$375$	0	0
$376$	4.41641	0.227759
$377$	0	0
$378$	−0.381966	−0.0196462
$379$	30.7984	1.58201	0.791003	−	0.611812i	$-0.209559\pi$
$379$	30.7984	1.58201	0.791003	+	0.611812i	$0.209559\pi$
$380$	0	0
$381$	7.56231	0.387429
$382$	−3.31308	−0.169512
$383$	−20.4721	−1.04608	−0.523039	−	0.852309i	$-0.675202\pi$
$383$	−20.4721	−1.04608	−0.523039	+	0.852309i	$0.675202\pi$
$384$	−10.0902	−0.514912
$385$	0	0
$386$	3.00000	0.152696
$387$	8.47214	0.430663
$388$	15.8754	0.805951
$389$	15.7082	0.796438	0.398219	−	0.917290i	$-0.369628\pi$
$389$	15.7082	0.796438	0.398219	+	0.917290i	$0.369628\pi$
$390$	0	0
$391$	−17.7082	−0.895542
$392$	1.47214	0.0743541
$393$	−16.4164	−0.828098
$394$	3.83282	0.193094
$395$	0	0
$396$	−1.85410	−0.0931721
$397$	0.708204	0.0355437	0.0177719	−	0.999842i	$-0.494343\pi$
$397$	0.708204	0.0355437	0.0177719	+	0.999842i	$0.494343\pi$
$398$	3.88854	0.194915
$399$	−3.61803	−0.181128
$400$	0	0
$401$	−2.90983	−0.145310	−0.0726550	−	0.997357i	$-0.523147\pi$
$401$	−2.90983	−0.145310	−0.0726550	+	0.997357i	$0.523147\pi$
$402$	6.00000	0.299253
$403$	−4.79837	−0.239024
$404$	−31.8541	−1.58480
$405$	0	0
$406$	0	0
$407$	−6.85410	−0.339745
$408$	−4.76393	−0.235850
$409$	−16.5623	−0.818953	−0.409477	−	0.912321i	$-0.634289\pi$
$409$	−16.5623	−0.818953	−0.409477	+	0.912321i	$0.634289\pi$
$410$	0	0
$411$	−7.18034	−0.354180
$412$	−16.4164	−0.808778
$413$	5.32624	0.262087
$414$	2.09017	0.102726
$415$	0	0
$416$	5.72949	0.280911
$417$	1.70820	0.0836511
$418$	1.38197	0.0675942
$419$	−15.4721	−0.755863	−0.377932	−	0.925833i	$-0.623365\pi$
$419$	−15.4721	−0.755863	−0.377932	+	0.925833i	$0.623365\pi$
$420$	0	0
$421$	−0.618034	−0.0301211	−0.0150606	−	0.999887i	$-0.504794\pi$
$421$	−0.618034	−0.0301211	−0.0150606	+	0.999887i	$0.504794\pi$
$422$	0.493422	0.0240194
$423$	3.00000	0.145865
$424$	−2.16718	−0.105248
$425$	0	0
$426$	−0.965558	−0.0467814
$427$	−4.23607	−0.204998
$428$	4.14590	0.200400
$429$	1.38197	0.0667219
$430$	0	0
$431$	4.09017	0.197017	0.0985083	−	0.995136i	$-0.468593\pi$
$431$	4.09017	0.197017	0.0985083	+	0.995136i	$0.468593\pi$
$432$	−3.14590	−0.151357
$433$	−36.0344	−1.73170	−0.865852	−	0.500300i	$-0.833223\pi$
$433$	−36.0344	−1.73170	−0.865852	+	0.500300i	$0.833223\pi$
$434$	1.32624	0.0636615
$435$	0	0
$436$	−13.5836	−0.650536
$437$	19.7984	0.947085
$438$	−3.43769	−0.164259
$439$	−16.1459	−0.770602	−0.385301	−	0.922791i	$-0.625902\pi$
$439$	−16.1459	−0.770602	−0.385301	+	0.922791i	$0.625902\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	1.70820	0.0812510
$443$	−11.0902	−0.526910	−0.263455	−	0.964672i	$-0.584862\pi$
$443$	−11.0902	−0.526910	−0.263455	+	0.964672i	$0.584862\pi$
$444$	−12.7082	−0.603105
$445$	0	0
$446$	3.97871	0.188398
$447$	13.3820	0.632945
$448$	4.70820	0.222442
$449$	−3.11146	−0.146839	−0.0734193	−	0.997301i	$-0.523391\pi$
$449$	−3.11146	−0.146839	−0.0734193	+	0.997301i	$0.523391\pi$
$450$	0	0
$451$	1.76393	0.0830603
$452$	22.9574	1.07983
$453$	−9.79837	−0.460368
$454$	5.21478	0.244742
$455$	0	0
$456$	5.32624	0.249424
$457$	8.09017	0.378442	0.189221	−	0.981935i	$-0.439404\pi$
$457$	8.09017	0.378442	0.189221	+	0.981935i	$0.439404\pi$
$458$	−0.167184	−0.00781201
$459$	−3.23607	−0.151047
$460$	0	0
$461$	−12.6525	−0.589285	−0.294642	−	0.955608i	$-0.595200\pi$
$461$	−12.6525	−0.589285	−0.294642	+	0.955608i	$0.595200\pi$
$462$	−0.381966	−0.0177707
$463$	7.29180	0.338879	0.169439	−	0.985541i	$-0.445804\pi$
$463$	7.29180	0.338879	0.169439	+	0.985541i	$0.445804\pi$
$464$	0	0
$465$	0	0
$466$	1.31308	0.0608274
$467$	10.1459	0.469496	0.234748	−	0.972056i	$-0.424573\pi$
$467$	10.1459	0.469496	0.234748	+	0.972056i	$0.424573\pi$
$468$	2.56231	0.118443
$469$	−15.7082	−0.725337
$470$	0	0
$471$	6.23607	0.287343
$472$	−7.84095	−0.360909
$473$	8.47214	0.389549
$474$	3.32624	0.152779
$475$	0	0
$476$	6.00000	0.275010
$477$	−1.47214	−0.0674045
$478$	4.61803	0.211224
$479$	33.7984	1.54429	0.772144	−	0.635448i	$-0.219185\pi$
$479$	33.7984	1.54429	0.772144	+	0.635448i	$0.219185\pi$
$480$	0	0
$481$	9.47214	0.431892
$482$	−2.29180	−0.104388
$483$	−5.47214	−0.248991
$484$	−1.85410	−0.0842774
$485$	0	0
$486$	0.381966	0.0173263
$487$	4.49342	0.203616	0.101808	−	0.994804i	$-0.467537\pi$
$487$	4.49342	0.203616	0.101808	+	0.994804i	$0.467537\pi$
$488$	6.23607	0.282294
$489$	7.29180	0.329746
$490$	0	0
$491$	−16.3262	−0.736793	−0.368396	−	0.929669i	$-0.620093\pi$
$491$	−16.3262	−0.736793	−0.368396	+	0.929669i	$0.620093\pi$
$492$	3.27051	0.147446
$493$	0	0
$494$	−1.90983	−0.0859273
$495$	0	0
$496$	10.9230	0.490457
$497$	2.52786	0.113390
$498$	−3.56231	−0.159631
$499$	−33.7082	−1.50899	−0.754493	−	0.656308i	$-0.772117\pi$
$499$	−33.7082	−1.50899	−0.754493	+	0.656308i	$0.772117\pi$
$500$	0	0
$501$	1.09017	0.0487052
$502$	4.56231	0.203626
$503$	−16.5836	−0.739426	−0.369713	−	0.929146i	$-0.620544\pi$
$503$	−16.5836	−0.739426	−0.369713	+	0.929146i	$0.620544\pi$
$504$	−1.47214	−0.0655741
$505$	0	0
$506$	2.09017	0.0929194
$507$	11.0902	0.492532
$508$	14.0213	0.622094
$509$	44.7214	1.98224	0.991120	−	0.132973i	$-0.0424523\pi$
$509$	44.7214	1.98224	0.991120	+	0.132973i	$0.0424523\pi$
$510$	0	0
$511$	9.00000	0.398137
$512$	−22.3050	−0.985749
$513$	3.61803	0.159740
$514$	−1.85410	−0.0817809
$515$	0	0
$516$	15.7082	0.691515
$517$	3.00000	0.131940
$518$	−2.61803	−0.115030
$519$	7.38197	0.324032
$520$	0	0
$521$	−27.1803	−1.19079	−0.595396	−	0.803432i	$-0.703005\pi$
$521$	−27.1803	−1.19079	−0.595396	+	0.803432i	$0.703005\pi$
$522$	0	0
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	−30.4377	−1.32968
$525$	0	0
$526$	12.1459	0.529586
$527$	11.2361	0.489451
$528$	−3.14590	−0.136908
$529$	6.94427	0.301925
$530$	0	0
$531$	−5.32624	−0.231139
$532$	−6.70820	−0.290838
$533$	−2.43769	−0.105588
$534$	3.25735	0.140960
$535$	0	0
$536$	23.1246	0.998831
$537$	1.14590	0.0494492
$538$	0.395122	0.0170349
$539$	1.00000	0.0430730
$540$	0	0
$541$	−37.0689	−1.59372	−0.796858	−	0.604167i	$-0.793506\pi$
$541$	−37.0689	−1.59372	−0.796858	+	0.604167i	$0.793506\pi$
$542$	−6.36068	−0.273215
$543$	7.00000	0.300399
$544$	−13.4164	−0.575224
$545$	0	0
$546$	0.527864	0.0225905
$547$	21.8328	0.933504	0.466752	−	0.884388i	$-0.345424\pi$
$547$	21.8328	0.933504	0.466752	+	0.884388i	$0.345424\pi$
$548$	−13.3131	−0.568707
$549$	4.23607	0.180791
$550$	0	0
$551$	0	0
$552$	8.05573	0.342875
$553$	−8.70820	−0.370310
$554$	9.47214	0.402432
$555$	0	0
$556$	3.16718	0.134319
$557$	−26.6738	−1.13020	−0.565102	−	0.825021i	$-0.691163\pi$
$557$	−26.6738	−1.13020	−0.565102	+	0.825021i	$0.691163\pi$
$558$	−1.32624	−0.0561441
$559$	−11.7082	−0.495204
$560$	0	0
$561$	−3.23607	−0.136627
$562$	10.2705	0.433235
$563$	26.1246	1.10102	0.550511	−	0.834828i	$-0.314433\pi$
$563$	26.1246	1.10102	0.550511	+	0.834828i	$0.314433\pi$
$564$	5.56231	0.234215
$565$	0	0
$566$	−5.83282	−0.245172
$567$	−1.00000	−0.0419961
$568$	−3.72136	−0.156145
$569$	−30.5410	−1.28035	−0.640173	−	0.768231i	$-0.721137\pi$
$569$	−30.5410	−1.28035	−0.640173	+	0.768231i	$0.721137\pi$
$570$	0	0
$571$	3.50658	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$571$	3.50658	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$572$	2.56231	0.107135
$573$	−8.67376	−0.362352
$574$	0.673762	0.0281223
$575$	0	0
$576$	−4.70820	−0.196175
$577$	−29.5623	−1.23069	−0.615347	−	0.788256i	$-0.710984\pi$
$577$	−29.5623	−1.23069	−0.615347	+	0.788256i	$0.710984\pi$
$578$	2.49342	0.103713
$579$	7.85410	0.326405
$580$	0	0
$581$	9.32624	0.386918
$582$	−3.27051	−0.135567
$583$	−1.47214	−0.0609696
$584$	−13.2492	−0.548257
$585$	0	0
$586$	−0.257354	−0.0106312
$587$	−32.2148	−1.32965	−0.664823	−	0.747001i	$-0.731493\pi$
$587$	−32.2148	−1.32965	−0.664823	+	0.747001i	$0.731493\pi$
$588$	1.85410	0.0764619
$589$	−12.5623	−0.517621
$590$	0	0
$591$	10.0344	0.412762
$592$	−21.5623	−0.886205
$593$	−24.2705	−0.996670	−0.498335	−	0.866984i	$-0.666055\pi$
$593$	−24.2705	−0.996670	−0.498335	+	0.866984i	$0.666055\pi$
$594$	0.381966	0.0156723
$595$	0	0
$596$	24.8115	1.01632
$597$	10.1803	0.416654
$598$	−2.88854	−0.118121
$599$	40.5066	1.65505	0.827527	−	0.561426i	$-0.189747\pi$
$599$	40.5066	1.65505	0.827527	+	0.561426i	$0.189747\pi$
$600$	0	0
$601$	4.59675	0.187505	0.0937526	−	0.995596i	$-0.470114\pi$
$601$	4.59675	0.187505	0.0937526	+	0.995596i	$0.470114\pi$
$602$	3.23607	0.131892
$603$	15.7082	0.639688
$604$	−18.1672	−0.739212
$605$	0	0
$606$	6.56231	0.266575
$607$	14.5967	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$607$	14.5967	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$608$	15.0000	0.608330
$609$	0	0
$610$	0	0
$611$	−4.14590	−0.167725
$612$	−6.00000	−0.242536
$613$	14.7082	0.594059	0.297029	−	0.954868i	$-0.404004\pi$
$613$	14.7082	0.594059	0.297029	+	0.954868i	$0.404004\pi$
$614$	3.21478	0.129738
$615$	0	0
$616$	−1.47214	−0.0593140
$617$	8.83282	0.355596	0.177798	−	0.984067i	$-0.443103\pi$
$617$	8.83282	0.355596	0.177798	+	0.984067i	$0.443103\pi$
$618$	3.38197	0.136043
$619$	−47.1246	−1.89410	−0.947049	−	0.321089i	$-0.895951\pi$
$619$	−47.1246	−1.89410	−0.947049	+	0.321089i	$0.895951\pi$
$620$	0	0
$621$	5.47214	0.219589
$622$	−9.72949	−0.390117
$623$	−8.52786	−0.341662
$624$	4.34752	0.174040
$625$	0	0
$626$	2.07701	0.0830142
$627$	3.61803	0.144490
$628$	11.5623	0.461386
$629$	−22.1803	−0.884388
$630$	0	0
$631$	−42.2705	−1.68276	−0.841381	−	0.540442i	$-0.818257\pi$
$631$	−42.2705	−1.68276	−0.841381	+	0.540442i	$0.818257\pi$
$632$	12.8197	0.509939
$633$	1.29180	0.0513443
$634$	−1.11146	−0.0441416
$635$	0	0
$636$	−2.72949	−0.108231
$637$	−1.38197	−0.0547555
$638$	0	0
$639$	−2.52786	−0.100001
$640$	0	0
$641$	−25.5279	−1.00829	−0.504145	−	0.863619i	$-0.668192\pi$
$641$	−25.5279	−1.00829	−0.504145	+	0.863619i	$0.668192\pi$
$642$	−0.854102	−0.0337087
$643$	26.2148	1.03381	0.516905	−	0.856043i	$-0.327084\pi$
$643$	26.2148	1.03381	0.516905	+	0.856043i	$0.327084\pi$
$644$	−10.1459	−0.399804
$645$	0	0
$646$	4.47214	0.175954
$647$	38.1246	1.49883	0.749417	−	0.662099i	$-0.230334\pi$
$647$	38.1246	1.49883	0.749417	+	0.662099i	$0.230334\pi$
$648$	1.47214	0.0578310
$649$	−5.32624	−0.209073
$650$	0	0
$651$	3.47214	0.136084
$652$	13.5197	0.529474
$653$	−39.3262	−1.53895	−0.769477	−	0.638674i	$-0.779483\pi$
$653$	−39.3262	−1.53895	−0.769477	+	0.638674i	$0.779483\pi$
$654$	2.79837	0.109425
$655$	0	0
$656$	5.54915	0.216658
$657$	−9.00000	−0.351123
$658$	1.14590	0.0446718
$659$	−8.32624	−0.324344	−0.162172	−	0.986762i	$-0.551850\pi$
$659$	−8.32624	−0.324344	−0.162172	+	0.986762i	$0.551850\pi$
$660$	0	0
$661$	8.36068	0.325193	0.162596	−	0.986693i	$-0.448013\pi$
$661$	8.36068	0.325193	0.162596	+	0.986693i	$0.448013\pi$
$662$	5.49342	0.213508
$663$	4.47214	0.173683
$664$	−13.7295	−0.532808
$665$	0	0
$666$	2.61803	0.101447
$667$	0	0
$668$	2.02129	0.0782059
$669$	10.4164	0.402722
$670$	0	0
$671$	4.23607	0.163532
$672$	−4.14590	−0.159931
$673$	14.5279	0.560008	0.280004	−	0.959999i	$-0.409664\pi$
$673$	14.5279	0.560008	0.280004	+	0.959999i	$0.409664\pi$
$674$	12.2705	0.472642
$675$	0	0
$676$	20.5623	0.790858
$677$	−35.6525	−1.37024	−0.685118	−	0.728432i	$-0.740249\pi$
$677$	−35.6525	−1.37024	−0.685118	+	0.728432i	$0.740249\pi$
$678$	−4.72949	−0.181635
$679$	8.56231	0.328591
$680$	0	0
$681$	13.6525	0.523164
$682$	−1.32624	−0.0507843
$683$	43.0132	1.64585	0.822926	−	0.568148i	$-0.192340\pi$
$683$	43.0132	1.64585	0.822926	+	0.568148i	$0.192340\pi$
$684$	6.70820	0.256495
$685$	0	0
$686$	0.381966	0.0145835
$687$	−0.437694	−0.0166991
$688$	26.6525	1.01612
$689$	2.03444	0.0775061
$690$	0	0
$691$	−41.0000	−1.55971	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	−41.0000	−1.55971	−0.779857	+	0.625958i	$0.784708\pi$
$692$	13.6869	0.520299
$693$	−1.00000	−0.0379869
$694$	−5.95743	−0.226141
$695$	0	0
$696$	0	0
$697$	5.70820	0.216214
$698$	9.32624	0.353003
$699$	3.43769	0.130026
$700$	0	0
$701$	34.7771	1.31351	0.656756	−	0.754103i	$-0.271928\pi$
$701$	34.7771	1.31351	0.656756	+	0.754103i	$0.271928\pi$
$702$	−0.527864	−0.0199229
$703$	24.7984	0.935288
$704$	−4.70820	−0.177447
$705$	0	0
$706$	−4.87539	−0.183488
$707$	−17.1803	−0.646133
$708$	−9.87539	−0.371140
$709$	3.90983	0.146837	0.0734184	−	0.997301i	$-0.476609\pi$
$709$	3.90983	0.146837	0.0734184	+	0.997301i	$0.476609\pi$
$710$	0	0
$711$	8.70820	0.326583
$712$	12.5542	0.470488
$713$	−19.0000	−0.711556
$714$	−1.23607	−0.0462587
$715$	0	0
$716$	2.12461	0.0794005
$717$	12.0902	0.451516
$718$	−8.63119	−0.322113
$719$	6.76393	0.252252	0.126126	−	0.992014i	$-0.459746\pi$
$719$	6.76393	0.252252	0.126126	+	0.992014i	$0.459746\pi$
$720$	0	0
$721$	−8.85410	−0.329744
$722$	2.25735	0.0840100
$723$	−6.00000	−0.223142
$724$	12.9787	0.482350
$725$	0	0
$726$	0.381966	0.0141761
$727$	−17.9098	−0.664239	−0.332119	−	0.943237i	$-0.607764\pi$
$727$	−17.9098	−0.664239	−0.332119	+	0.943237i	$0.607764\pi$
$728$	2.03444	0.0754014
$729$	1.00000	0.0370370
$730$	0	0
$731$	27.4164	1.01403
$732$	7.85410	0.290296
$733$	34.0689	1.25836	0.629181	−	0.777258i	$-0.283390\pi$
$733$	34.0689	1.25836	0.629181	+	0.777258i	$0.283390\pi$
$734$	9.54102	0.352165
$735$	0	0
$736$	22.6869	0.836251
$737$	15.7082	0.578619
$738$	−0.673762	−0.0248015
$739$	22.5836	0.830751	0.415375	−	0.909650i	$-0.363650\pi$
$739$	22.5836	0.830751	0.415375	+	0.909650i	$0.363650\pi$
$740$	0	0
$741$	−5.00000	−0.183680
$742$	−0.562306	−0.0206429
$743$	22.9098	0.840480	0.420240	−	0.907413i	$-0.361946\pi$
$743$	22.9098	0.840480	0.420240	+	0.907413i	$0.361946\pi$
$744$	−5.11146	−0.187395
$745$	0	0
$746$	2.65248	0.0971140
$747$	−9.32624	−0.341229
$748$	−6.00000	−0.219382
$749$	2.23607	0.0817041
$750$	0	0
$751$	−53.4164	−1.94919	−0.974596	−	0.223969i	$-0.928098\pi$
$751$	−53.4164	−1.94919	−0.974596	+	0.223969i	$0.928098\pi$
$752$	9.43769	0.344157
$753$	11.9443	0.435273
$754$	0	0
$755$	0	0
$756$	−1.85410	−0.0674330
$757$	3.25735	0.118391	0.0591953	−	0.998246i	$-0.481147\pi$
$757$	3.25735	0.118391	0.0591953	+	0.998246i	$0.481147\pi$
$758$	−11.7639	−0.427285
$759$	5.47214	0.198626
$760$	0	0
$761$	21.3050	0.772304	0.386152	−	0.922435i	$-0.373804\pi$
$761$	21.3050	0.772304	0.386152	+	0.922435i	$0.373804\pi$
$762$	−2.88854	−0.104641
$763$	−7.32624	−0.265228
$764$	−16.0820	−0.581828
$765$	0	0
$766$	7.81966	0.282536
$767$	7.36068	0.265779
$768$	−5.56231	−0.200712
$769$	−15.5410	−0.560424	−0.280212	−	0.959938i	$-0.590405\pi$
$769$	−15.5410	−0.560424	−0.280212	+	0.959938i	$0.590405\pi$
$770$	0	0
$771$	−4.85410	−0.174816
$772$	14.5623	0.524109
$773$	40.4853	1.45615	0.728077	−	0.685495i	$-0.240414\pi$
$773$	40.4853	1.45615	0.728077	+	0.685495i	$0.240414\pi$
$774$	−3.23607	−0.116318
$775$	0	0
$776$	−12.6049	−0.452489
$777$	−6.85410	−0.245890
$778$	−6.00000	−0.215110
$779$	−6.38197	−0.228658
$780$	0	0
$781$	−2.52786	−0.0904541
$782$	6.76393	0.241878
$783$	0	0
$784$	3.14590	0.112354
$785$	0	0
$786$	6.27051	0.223662
$787$	−10.0000	−0.356462	−0.178231	−	0.983989i	$-0.557037\pi$
$787$	−10.0000	−0.356462	−0.178231	+	0.983989i	$0.557037\pi$
$788$	18.6049	0.662771
$789$	31.7984	1.13205
$790$	0	0
$791$	12.3820	0.440252
$792$	1.47214	0.0523101
$793$	−5.85410	−0.207885
$794$	−0.270510	−0.00960003
$795$	0	0
$796$	18.8754	0.669020
$797$	46.7984	1.65768	0.828842	−	0.559483i	$-0.189000\pi$
$797$	46.7984	1.65768	0.828842	+	0.559483i	$0.189000\pi$
$798$	1.38197	0.0489211
$799$	9.70820	0.343452
$800$	0	0
$801$	8.52786	0.301317
$802$	1.11146	0.0392469
$803$	−9.00000	−0.317603
$804$	29.1246	1.02715
$805$	0	0
$806$	1.83282	0.0645582
$807$	1.03444	0.0364141
$808$	25.2918	0.889762
$809$	−7.76393	−0.272965	−0.136483	−	0.990642i	$-0.543580\pi$
$809$	−7.76393	−0.272965	−0.136483	+	0.990642i	$0.543580\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	−16.6525	−0.584028
$814$	2.61803	0.0917620
$815$	0	0
$816$	−10.1803	−0.356383
$817$	−30.6525	−1.07239
$818$	6.32624	0.221192
$819$	1.38197	0.0482898
$820$	0	0
$821$	41.0689	1.43331	0.716657	−	0.697426i	$-0.245671\pi$
$821$	41.0689	1.43331	0.716657	+	0.697426i	$0.245671\pi$
$822$	2.74265	0.0956608
$823$	−31.8541	−1.11036	−0.555182	−	0.831729i	$-0.687351\pi$
$823$	−31.8541	−1.11036	−0.555182	+	0.831729i	$0.687351\pi$
$824$	13.0344	0.454076
$825$	0	0
$826$	−2.03444	−0.0707873
$827$	−20.3050	−0.706072	−0.353036	−	0.935610i	$-0.614851\pi$
$827$	−20.3050	−0.706072	−0.353036	+	0.935610i	$0.614851\pi$
$828$	10.1459	0.352594
$829$	−7.14590	−0.248187	−0.124094	−	0.992271i	$-0.539602\pi$
$829$	−7.14590	−0.248187	−0.124094	+	0.992271i	$0.539602\pi$
$830$	0	0
$831$	24.7984	0.860246
$832$	6.50658	0.225575
$833$	3.23607	0.112123
$834$	−0.652476	−0.0225934
$835$	0	0
$836$	6.70820	0.232008
$837$	−3.47214	−0.120015
$838$	5.90983	0.204152
$839$	16.5279	0.570605	0.285303	−	0.958437i	$-0.407906\pi$
$839$	16.5279	0.570605	0.285303	+	0.958437i	$0.407906\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0.236068	0.00813544
$843$	26.8885	0.926091
$844$	2.39512	0.0824435
$845$	0	0
$846$	−1.14590	−0.0393968
$847$	−1.00000	−0.0343604
$848$	−4.63119	−0.159036
$849$	−15.2705	−0.524083
$850$	0	0
$851$	37.5066	1.28571
$852$	−4.68692	−0.160571
$853$	56.0476	1.91903	0.959517	−	0.281652i	$-0.0908824\pi$
$853$	56.0476	1.91903	0.959517	+	0.281652i	$0.0908824\pi$
$854$	1.61803	0.0553680
$855$	0	0
$856$	−3.29180	−0.112511
$857$	5.32624	0.181941	0.0909704	−	0.995854i	$-0.471003\pi$
$857$	5.32624	0.181941	0.0909704	+	0.995854i	$0.471003\pi$
$858$	−0.527864	−0.0180210
$859$	−30.6312	−1.04512	−0.522561	−	0.852602i	$-0.675023\pi$
$859$	−30.6312	−1.04512	−0.522561	+	0.852602i	$0.675023\pi$
$860$	0	0
$861$	1.76393	0.0601146
$862$	−1.56231	−0.0532124
$863$	35.8885	1.22166	0.610830	−	0.791762i	$-0.290836\pi$
$863$	35.8885	1.22166	0.610830	+	0.791762i	$0.290836\pi$
$864$	4.14590	0.141046
$865$	0	0
$866$	13.7639	0.467717
$867$	6.52786	0.221698
$868$	6.43769	0.218510
$869$	8.70820	0.295406
$870$	0	0
$871$	−21.7082	−0.735554
$872$	10.7852	0.365234
$873$	−8.56231	−0.289790
$874$	−7.56231	−0.255799
$875$	0	0
$876$	−16.6869	−0.563799
$877$	13.5967	0.459130	0.229565	−	0.973293i	$-0.426270\pi$
$877$	13.5967	0.459130	0.229565	+	0.973293i	$0.426270\pi$
$878$	6.16718	0.208132
$879$	−0.673762	−0.0227254
$880$	0	0
$881$	−50.4853	−1.70089	−0.850446	−	0.526062i	$-0.823668\pi$
$881$	−50.4853	−1.70089	−0.850446	+	0.526062i	$0.823668\pi$
$882$	−0.381966	−0.0128615
$883$	41.8541	1.40850	0.704251	−	0.709951i	$-0.251283\pi$
$883$	41.8541	1.40850	0.704251	+	0.709951i	$0.251283\pi$
$884$	8.29180	0.278883
$885$	0	0
$886$	4.23607	0.142313
$887$	−42.9443	−1.44193	−0.720964	−	0.692973i	$-0.756301\pi$
$887$	−42.9443	−1.44193	−0.720964	+	0.692973i	$0.756301\pi$
$888$	10.0902	0.338604
$889$	7.56231	0.253632
$890$	0	0
$891$	1.00000	0.0335013
$892$	19.3131	0.646650
$893$	−10.8541	−0.363219
$894$	−5.11146	−0.170953
$895$	0	0
$896$	−10.0902	−0.337089
$897$	−7.56231	−0.252498
$898$	1.18847	0.0396598
$899$	0	0
$900$	0	0
$901$	−4.76393	−0.158710
$902$	−0.673762	−0.0224338
$903$	8.47214	0.281935
$904$	−18.2279	−0.606252
$905$	0	0
$906$	3.74265	0.124341
$907$	−46.7082	−1.55092	−0.775460	−	0.631396i	$-0.782482\pi$
$907$	−46.7082	−1.55092	−0.775460	+	0.631396i	$0.782482\pi$
$908$	25.3131	0.840044
$909$	17.1803	0.569836
$910$	0	0
$911$	−27.3607	−0.906500	−0.453250	−	0.891384i	$-0.649735\pi$
$911$	−27.3607	−0.906500	−0.453250	+	0.891384i	$0.649735\pi$
$912$	11.3820	0.376895
$913$	−9.32624	−0.308653
$914$	−3.09017	−0.102214
$915$	0	0
$916$	−0.811529	−0.0268137
$917$	−16.4164	−0.542118
$918$	1.23607	0.0407963
$919$	−2.74265	−0.0904715	−0.0452358	−	0.998976i	$-0.514404\pi$
$919$	−2.74265	−0.0904715	−0.0452358	+	0.998976i	$0.514404\pi$
$920$	0	0
$921$	8.41641	0.277330
$922$	4.83282	0.159160
$923$	3.49342	0.114987
$924$	−1.85410	−0.0609955
$925$	0	0
$926$	−2.78522	−0.0915280
$927$	8.85410	0.290807
$928$	0	0
$929$	−1.63932	−0.0537844	−0.0268922	−	0.999638i	$-0.508561\pi$
$929$	−1.63932	−0.0537844	−0.0268922	+	0.999638i	$0.508561\pi$
$930$	0	0
$931$	−3.61803	−0.118576
$932$	6.37384	0.208782
$933$	−25.4721	−0.833920
$934$	−3.87539	−0.126807
$935$	0	0
$936$	−2.03444	−0.0664978
$937$	59.4853	1.94330	0.971650	−	0.236424i	$-0.0759756\pi$
$937$	59.4853	1.94330	0.971650	+	0.236424i	$0.0759756\pi$
$938$	6.00000	0.195907
$939$	5.43769	0.177452
$940$	0	0
$941$	10.2361	0.333686	0.166843	−	0.985983i	$-0.446643\pi$
$941$	10.2361	0.333686	0.166843	+	0.985983i	$0.446643\pi$
$942$	−2.38197	−0.0776086
$943$	−9.65248	−0.314328
$944$	−16.7558	−0.545355
$945$	0	0
$946$	−3.23607	−0.105214
$947$	46.5755	1.51350	0.756750	−	0.653705i	$-0.226786\pi$
$947$	46.5755	1.51350	0.756750	+	0.653705i	$0.226786\pi$
$948$	16.1459	0.524394
$949$	12.4377	0.403745
$950$	0	0
$951$	−2.90983	−0.0943577
$952$	−4.76393	−0.154400
$953$	11.9443	0.386913	0.193456	−	0.981109i	$-0.438030\pi$
$953$	11.9443	0.386913	0.193456	+	0.981109i	$0.438030\pi$
$954$	0.562306	0.0182053
$955$	0	0
$956$	22.4164	0.724998
$957$	0	0
$958$	−12.9098	−0.417098
$959$	−7.18034	−0.231865
$960$	0	0
$961$	−18.9443	−0.611106
$962$	−3.61803	−0.116650
$963$	−2.23607	−0.0720563
$964$	−11.1246	−0.358300
$965$	0	0
$966$	2.09017	0.0672501
$967$	9.68692	0.311510	0.155755	−	0.987796i	$-0.450219\pi$
$967$	9.68692	0.311510	0.155755	+	0.987796i	$0.450219\pi$
$968$	1.47214	0.0473162
$969$	11.7082	0.376122
$970$	0	0
$971$	−7.52786	−0.241581	−0.120790	−	0.992678i	$-0.538543\pi$
$971$	−7.52786	−0.241581	−0.120790	+	0.992678i	$0.538543\pi$
$972$	1.85410	0.0594703
$973$	1.70820	0.0547625
$974$	−1.71633	−0.0549949
$975$	0	0
$976$	13.3262	0.426562
$977$	−11.2148	−0.358793	−0.179396	−	0.983777i	$-0.557414\pi$
$977$	−11.2148	−0.358793	−0.179396	+	0.983777i	$0.557414\pi$
$978$	−2.78522	−0.0890615
$979$	8.52786	0.272552
$980$	0	0
$981$	7.32624	0.233909
$982$	6.23607	0.199001
$983$	−59.9230	−1.91125	−0.955623	−	0.294592i	$-0.904816\pi$
$983$	−59.9230	−1.91125	−0.955623	+	0.294592i	$0.904816\pi$
$984$	−2.59675	−0.0827813
$985$	0	0
$986$	0	0
$987$	3.00000	0.0954911
$988$	−9.27051	−0.294934
$989$	−46.3607	−1.47418
$990$	0	0
$991$	−31.1246	−0.988706	−0.494353	−	0.869261i	$-0.664595\pi$
$991$	−31.1246	−0.988706	−0.494353	+	0.869261i	$0.664595\pi$
$992$	−14.3951	−0.457046
$993$	14.3820	0.456398
$994$	−0.965558	−0.0306256
$995$	0	0
$996$	−17.2918	−0.547912
$997$	−2.83282	−0.0897162	−0.0448581	−	0.998993i	$-0.514284\pi$
$997$	−2.83282	−0.0897162	−0.0448581	+	0.998993i	$0.514284\pi$
$998$	12.8754	0.407563
$999$	6.85410	0.216854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.bb.1.2	✓	2
5.4	even	2		5775.2.a.bo.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5775.2.a.bb.1.2	✓	2	1.1	even	1	trivial
5775.2.a.bo.1.1	yes	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:	\( N \)	\(=\)	\( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5775.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} +0.381966 q^{6} -1.00000 q^{7} +1.47214 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} +0.381966 q^{6} -1.00000 q^{7} +1.47214 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.85410 q^{12} -1.38197 q^{13} +0.381966 q^{14} +3.14590 q^{16} +3.23607 q^{17} -0.381966 q^{18} -3.61803 q^{19} +1.00000 q^{21} -0.381966 q^{22} -5.47214 q^{23} -1.47214 q^{24} +0.527864 q^{26} -1.00000 q^{27} +1.85410 q^{28} +3.47214 q^{31} -4.14590 q^{32} -1.00000 q^{33} -1.23607 q^{34} -1.85410 q^{36} -6.85410 q^{37} +1.38197 q^{38} +1.38197 q^{39} +1.76393 q^{41} -0.381966 q^{42} +8.47214 q^{43} -1.85410 q^{44} +2.09017 q^{46} +3.00000 q^{47} -3.14590 q^{48} +1.00000 q^{49} -3.23607 q^{51} +2.56231 q^{52} -1.47214 q^{53} +0.381966 q^{54} -1.47214 q^{56} +3.61803 q^{57} -5.32624 q^{59} +4.23607 q^{61} -1.32624 q^{62} -1.00000 q^{63} -4.70820 q^{64} +0.381966 q^{66} +15.7082 q^{67} -6.00000 q^{68} +5.47214 q^{69} -2.52786 q^{71} +1.47214 q^{72} -9.00000 q^{73} +2.61803 q^{74} +6.70820 q^{76} -1.00000 q^{77} -0.527864 q^{78} +8.70820 q^{79} +1.00000 q^{81} -0.673762 q^{82} -9.32624 q^{83} -1.85410 q^{84} -3.23607 q^{86} +1.47214 q^{88} +8.52786 q^{89} +1.38197 q^{91} +10.1459 q^{92} -3.47214 q^{93} -1.14590 q^{94} +4.14590 q^{96} -8.56231 q^{97} -0.381966 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 3 q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 3 * q^6 - 2 * q^7 - 6 * q^8 + 2 * q^9	\( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 3 q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{11} - 3 q^{12} - 5 q^{13} + 3 q^{14} + 13 q^{16} + 2 q^{17} - 3 q^{18} - 5 q^{19} + 2 q^{21} - 3 q^{22} - 2 q^{23} + 6 q^{24} + 10 q^{26} - 2 q^{27} - 3 q^{28} - 2 q^{31} - 15 q^{32} - 2 q^{33} + 2 q^{34} + 3 q^{36} - 7 q^{37} + 5 q^{38} + 5 q^{39} + 8 q^{41} - 3 q^{42} + 8 q^{43} + 3 q^{44} - 7 q^{46} + 6 q^{47} - 13 q^{48} + 2 q^{49} - 2 q^{51} - 15 q^{52} + 6 q^{53} + 3 q^{54} + 6 q^{56} + 5 q^{57} + 5 q^{59} + 4 q^{61} + 13 q^{62} - 2 q^{63} + 4 q^{64} + 3 q^{66} + 18 q^{67} - 12 q^{68} + 2 q^{69} - 14 q^{71} - 6 q^{72} - 18 q^{73} + 3 q^{74} - 2 q^{77} - 10 q^{78} + 4 q^{79} + 2 q^{81} - 17 q^{82} - 3 q^{83} + 3 q^{84} - 2 q^{86} - 6 q^{88} + 26 q^{89} + 5 q^{91} + 27 q^{92} + 2 q^{93} - 9 q^{94} + 15 q^{96} + 3 q^{97} - 3 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 3 * q^6 - 2 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^11 - 3 * q^12 - 5 * q^13 + 3 * q^14 + 13 * q^16 + 2 * q^17 - 3 * q^18 - 5 * q^19 + 2 * q^21 - 3 * q^22 - 2 * q^23 + 6 * q^24 + 10 * q^26 - 2 * q^27 - 3 * q^28 - 2 * q^31 - 15 * q^32 - 2 * q^33 + 2 * q^34 + 3 * q^36 - 7 * q^37 + 5 * q^38 + 5 * q^39 + 8 * q^41 - 3 * q^42 + 8 * q^43 + 3 * q^44 - 7 * q^46 + 6 * q^47 - 13 * q^48 + 2 * q^49 - 2 * q^51 - 15 * q^52 + 6 * q^53 + 3 * q^54 + 6 * q^56 + 5 * q^57 + 5 * q^59 + 4 * q^61 + 13 * q^62 - 2 * q^63 + 4 * q^64 + 3 * q^66 + 18 * q^67 - 12 * q^68 + 2 * q^69 - 14 * q^71 - 6 * q^72 - 18 * q^73 + 3 * q^74 - 2 * q^77 - 10 * q^78 + 4 * q^79 + 2 * q^81 - 17 * q^82 - 3 * q^83 + 3 * q^84 - 2 * q^86 - 6 * q^88 + 26 * q^89 + 5 * q^91 + 27 * q^92 + 2 * q^93 - 9 * q^94 + 15 * q^96 + 3 * q^97 - 3 * q^98 + 2 * q^99

Introduction

L-functions

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