LMFDB - Embedded newform 5775.2.a.bz.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.546295$ 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)

$f(q)$	$=$	$q-0.546295 q^{2} -1.00000 q^{3} -1.70156 q^{4} +0.546295 q^{6} +1.00000 q^{7} +2.02214 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.546295 q^{2} -1.00000 q^{3} -1.70156 q^{4} +0.546295 q^{6} +1.00000 q^{7} +2.02214 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.70156 q^{12} +0.568438 q^{13} -0.546295 q^{14} +2.29844 q^{16} -2.70156 q^{17} -0.546295 q^{18} +6.62049 q^{19} -1.00000 q^{21} +0.546295 q^{22} -8.93103 q^{23} -2.02214 q^{24} -0.310535 q^{26} -1.00000 q^{27} -1.70156 q^{28} +5.27000 q^{29} +9.66103 q^{31} -5.29991 q^{32} +1.00000 q^{33} +1.47585 q^{34} -1.70156 q^{36} +2.75362 q^{37} -3.61674 q^{38} -0.568438 q^{39} -10.9831 q^{41} +0.546295 q^{42} +0.0520550 q^{43} +1.70156 q^{44} +4.87897 q^{46} +10.1567 q^{47} -2.29844 q^{48} +1.00000 q^{49} +2.70156 q^{51} -0.967233 q^{52} -1.56469 q^{53} +0.546295 q^{54} +2.02214 q^{56} -6.62049 q^{57} -2.87897 q^{58} +6.36259 q^{59} -3.71308 q^{61} -5.27777 q^{62} +1.00000 q^{63} -1.70156 q^{64} -0.546295 q^{66} +10.4146 q^{67} +4.59688 q^{68} +8.93103 q^{69} +2.56844 q^{71} +2.02214 q^{72} -2.26625 q^{73} -1.50429 q^{74} -11.2652 q^{76} -1.00000 q^{77} +0.310535 q^{78} -2.56844 q^{79} +1.00000 q^{81} +6.00000 q^{82} -17.1605 q^{83} +1.70156 q^{84} -0.0284374 q^{86} -5.27000 q^{87} -2.02214 q^{88} -10.7772 q^{89} +0.568438 q^{91} +15.1967 q^{92} -9.66103 q^{93} -5.54857 q^{94} +5.29991 q^{96} -17.9952 q^{97} -0.546295 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 6 q^{4} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 6 * q^4 + 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 6 q^{4} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 6 q^{12} - 8 q^{13} + 22 q^{16} + 2 q^{17} + 10 q^{19} - 4 q^{21} + 2 q^{23} + 20 q^{26} - 4 q^{27} + 6 q^{28} - 2 q^{29} + 24 q^{31} + 4 q^{33} + 6 q^{36} - 8 q^{37} - 16 q^{38} + 8 q^{39} - 6 q^{43} - 6 q^{44} - 12 q^{46} - 4 q^{47} - 22 q^{48} + 4 q^{49} - 2 q^{51} - 12 q^{52} - 14 q^{53} - 10 q^{57} + 20 q^{58} - 2 q^{59} + 6 q^{61} - 8 q^{62} + 4 q^{63} + 6 q^{64} + 8 q^{67} + 44 q^{68} - 2 q^{69} - 4 q^{73} - 36 q^{74} + 56 q^{76} - 4 q^{77} - 20 q^{78} + 4 q^{81} + 24 q^{82} - 6 q^{83} - 6 q^{84} - 36 q^{86} + 2 q^{87} + 18 q^{89} - 8 q^{91} + 44 q^{92} - 24 q^{93} - 36 q^{94} + 6 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 6 * q^4 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 6 * q^12 - 8 * q^13 + 22 * q^16 + 2 * q^17 + 10 * q^19 - 4 * q^21 + 2 * q^23 + 20 * q^26 - 4 * q^27 + 6 * q^28 - 2 * q^29 + 24 * q^31 + 4 * q^33 + 6 * q^36 - 8 * q^37 - 16 * q^38 + 8 * q^39 - 6 * q^43 - 6 * q^44 - 12 * q^46 - 4 * q^47 - 22 * q^48 + 4 * q^49 - 2 * q^51 - 12 * q^52 - 14 * q^53 - 10 * q^57 + 20 * q^58 - 2 * q^59 + 6 * q^61 - 8 * q^62 + 4 * q^63 + 6 * q^64 + 8 * q^67 + 44 * q^68 - 2 * q^69 - 4 * q^73 - 36 * q^74 + 56 * q^76 - 4 * q^77 - 20 * q^78 + 4 * q^81 + 24 * q^82 - 6 * q^83 - 6 * q^84 - 36 * q^86 + 2 * q^87 + 18 * q^89 - 8 * q^91 + 44 * q^92 - 24 * q^93 - 36 * q^94 + 6 * q^97 - 4 * 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 $

$2$	−0.546295	−0.386289	−0.193144	−	0.981170i	$-0.561869\pi$
$2$	−0.546295	−0.386289	−0.193144	+	0.981170i	$0.561869\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.70156	−0.850781
$5$	0	0
$5$	0	0
$6$	0.546295	0.223024
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.02214	0.714936
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.70156	0.491199
$13$	0.568438	0.157656	0.0788282	−	0.996888i	$-0.474882\pi$
$13$	0.568438	0.157656	0.0788282	+	0.996888i	$0.474882\pi$
$14$	−0.546295	−0.146003
$15$	0	0
$16$	2.29844	0.574609
$17$	−2.70156	−0.655225	−0.327613	−	0.944812i	$-0.606244\pi$
$17$	−2.70156	−0.655225	−0.327613	+	0.944812i	$0.606244\pi$
$18$	−0.546295	−0.128763
$19$	6.62049	1.51885	0.759423	−	0.650598i	$-0.225482\pi$
$19$	6.62049	1.51885	0.759423	+	0.650598i	$0.225482\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0.546295	0.116470
$23$	−8.93103	−1.86225	−0.931124	−	0.364703i	$-0.881171\pi$
$23$	−8.93103	−1.86225	−0.931124	+	0.364703i	$0.881171\pi$
$24$	−2.02214	−0.412768
$25$	0	0
$26$	−0.310535	−0.0609009
$27$	−1.00000	−0.192450
$28$	−1.70156	−0.321565
$29$	5.27000	0.978615	0.489307	−	0.872111i	$-0.337250\pi$
$29$	5.27000	0.978615	0.489307	+	0.872111i	$0.337250\pi$
$30$	0	0
$31$	9.66103	1.73517	0.867586	−	0.497287i	$-0.165671\pi$
$31$	9.66103	1.73517	0.867586	+	0.497287i	$0.165671\pi$
$32$	−5.29991	−0.936901
$33$	1.00000	0.174078
$34$	1.47585	0.253106
$35$	0	0
$36$	−1.70156	−0.283594
$37$	2.75362	0.452692	0.226346	−	0.974047i	$-0.427322\pi$
$37$	2.75362	0.452692	0.226346	+	0.974047i	$0.427322\pi$
$38$	−3.61674	−0.586713
$39$	−0.568438	−0.0910230
$40$	0	0
$41$	−10.9831	−1.71527	−0.857635	−	0.514259i	$-0.828067\pi$
$41$	−10.9831	−1.71527	−0.857635	+	0.514259i	$0.828067\pi$
$42$	0.546295	0.0842951
$43$	0.0520550	0.00793831	0.00396916	−	0.999992i	$-0.498737\pi$
$43$	0.0520550	0.00793831	0.00396916	+	0.999992i	$0.498737\pi$
$44$	1.70156	0.256520
$45$	0	0
$46$	4.87897	0.719365
$47$	10.1567	1.48151	0.740756	−	0.671774i	$-0.234467\pi$
$47$	10.1567	1.48151	0.740756	+	0.671774i	$0.234467\pi$
$48$	−2.29844	−0.331751
$49$	1.00000	0.142857
$50$	0	0
$51$	2.70156	0.378294
$52$	−0.967233	−0.134131
$53$	−1.56469	−0.214926	−0.107463	−	0.994209i	$-0.534273\pi$
$53$	−1.56469	−0.214926	−0.107463	+	0.994209i	$0.534273\pi$
$54$	0.546295	0.0743413
$55$	0	0
$56$	2.02214	0.270220
$57$	−6.62049	−0.876906
$58$	−2.87897	−0.378028
$59$	6.36259	0.828339	0.414169	−	0.910200i	$-0.364072\pi$
$59$	6.36259	0.828339	0.414169	+	0.910200i	$0.364072\pi$
$60$	0	0
$61$	−3.71308	−0.475412	−0.237706	−	0.971337i	$-0.576395\pi$
$61$	−3.71308	−0.475412	−0.237706	+	0.971337i	$0.576395\pi$
$62$	−5.27777	−0.670277
$63$	1.00000	0.125988
$64$	−1.70156	−0.212695
$65$	0	0
$66$	−0.546295	−0.0672442
$67$	10.4146	1.27235	0.636176	−	0.771544i	$-0.280515\pi$
$67$	10.4146	1.27235	0.636176	+	0.771544i	$0.280515\pi$
$68$	4.59688	0.557453
$69$	8.93103	1.07517
$70$	0	0
$71$	2.56844	0.304818	0.152409	−	0.988318i	$-0.451297\pi$
$71$	2.56844	0.304818	0.152409	+	0.988318i	$0.451297\pi$
$72$	2.02214	0.238312
$73$	−2.26625	−0.265244	−0.132622	−	0.991167i	$-0.542340\pi$
$73$	−2.26625	−0.265244	−0.132622	+	0.991167i	$0.542340\pi$
$74$	−1.50429	−0.174870
$75$	0	0
$76$	−11.2652	−1.29220
$77$	−1.00000	−0.113961
$78$	0.310535	0.0351612
$79$	−2.56844	−0.288972	−0.144486	−	0.989507i	$-0.546153\pi$
$79$	−2.56844	−0.288972	−0.144486	+	0.989507i	$0.546153\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−17.1605	−1.88361	−0.941804	−	0.336161i	$-0.890871\pi$
$83$	−17.1605	−1.88361	−0.941804	+	0.336161i	$0.890871\pi$
$84$	1.70156	0.185656
$85$	0	0
$86$	−0.0284374	−0.00306648
$87$	−5.27000	−0.565003
$88$	−2.02214	−0.215561
$89$	−10.7772	−1.14238	−0.571192	−	0.820816i	$-0.693519\pi$
$89$	−10.7772	−1.14238	−0.571192	+	0.820816i	$0.693519\pi$
$90$	0	0
$91$	0.568438	0.0595885
$92$	15.1967	1.58437
$93$	−9.66103	−1.00180
$94$	−5.54857	−0.572292
$95$	0	0
$96$	5.29991	0.540920
$97$	−17.9952	−1.82713	−0.913567	−	0.406689i	$-0.866683\pi$
$97$	−17.9952	−1.82713	−0.913567	+	0.406689i	$0.866683\pi$
$98$	−0.546295	−0.0551841
$99$	−1.00000	−0.100504
$100$	0	0
$101$	0.826342	0.0822241	0.0411120	−	0.999155i	$-0.486910\pi$
$101$	0.826342	0.0822241	0.0411120	+	0.999155i	$0.486910\pi$
$102$	−1.47585	−0.146131
$103$	−10.5921	−1.04367	−0.521833	−	0.853048i	$-0.674752\pi$
$103$	−10.5921	−1.04367	−0.521833	+	0.853048i	$0.674752\pi$
$104$	1.14946	0.112714
$105$	0	0
$106$	0.854779	0.0830235
$107$	13.7864	1.33278	0.666390	−	0.745603i	$-0.267839\pi$
$107$	13.7864	1.33278	0.666390	+	0.745603i	$0.267839\pi$
$108$	1.70156	0.163733
$109$	19.8905	1.90516	0.952582	−	0.304282i	$-0.0984166\pi$
$109$	19.8905	1.90516	0.952582	+	0.304282i	$0.0984166\pi$
$110$	0	0
$111$	−2.75362	−0.261362
$112$	2.29844	0.217182
$113$	−6.43531	−0.605383	−0.302692	−	0.953089i	$-0.597885\pi$
$113$	−6.43531	−0.605383	−0.302692	+	0.953089i	$0.597885\pi$
$114$	3.61674	0.338739
$115$	0	0
$116$	−8.96723	−0.832587
$117$	0.568438	0.0525521
$118$	−3.47585	−0.319978
$119$	−2.70156	−0.247652
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.02844	0.183646
$123$	10.9831	0.990311
$124$	−16.4388	−1.47625
$125$	0	0
$126$	−0.546295	−0.0486678
$127$	−4.05206	−0.359562	−0.179781	−	0.983707i	$-0.557539\pi$
$127$	−4.05206	−0.359562	−0.179781	+	0.983707i	$0.557539\pi$
$128$	11.5294	1.01906
$129$	−0.0520550	−0.00458319
$130$	0	0
$131$	−5.69781	−0.497820	−0.248910	−	0.968527i	$-0.580072\pi$
$131$	−5.69781	−0.497820	−0.248910	+	0.968527i	$0.580072\pi$
$132$	−1.70156	−0.148102
$133$	6.62049	0.574070
$134$	−5.68947	−0.491495
$135$	0	0
$136$	−5.46295	−0.468444
$137$	3.40312	0.290749	0.145374	−	0.989377i	$-0.453561\pi$
$137$	3.40312	0.290749	0.145374	+	0.989377i	$0.453561\pi$
$138$	−4.87897	−0.415326
$139$	−8.04724	−0.682558	−0.341279	−	0.939962i	$-0.610860\pi$
$139$	−8.04724	−0.682558	−0.341279	+	0.939962i	$0.610860\pi$
$140$	0	0
$141$	−10.1567	−0.855352
$142$	−1.40312	−0.117748
$143$	−0.568438	−0.0475352
$144$	2.29844	0.191536
$145$	0	0
$146$	1.23804	0.102461
$147$	−1.00000	−0.0824786
$148$	−4.68545	−0.385142
$149$	7.50723	0.615017	0.307508	−	0.951545i	$-0.400505\pi$
$149$	7.50723	0.615017	0.307508	+	0.951545i	$0.400505\pi$
$150$	0	0
$151$	3.21795	0.261873	0.130936	−	0.991391i	$-0.458202\pi$
$151$	3.21795	0.261873	0.130936	+	0.991391i	$0.458202\pi$
$152$	13.3876	1.08588
$153$	−2.70156	−0.218408
$154$	0.546295	0.0440217
$155$	0	0
$156$	0.967233	0.0774406
$157$	3.35107	0.267444	0.133722	−	0.991019i	$-0.457307\pi$
$157$	3.35107	0.267444	0.133722	+	0.991019i	$0.457307\pi$
$158$	1.40312	0.111627
$159$	1.56469	0.124088
$160$	0	0
$161$	−8.93103	−0.703864
$162$	−0.546295	−0.0429210
$163$	−10.9600	−0.858457	−0.429228	−	0.903196i	$-0.641214\pi$
$163$	−10.9600	−0.858457	−0.429228	+	0.903196i	$0.641214\pi$
$164$	18.6884	1.45932
$165$	0	0
$166$	9.37469	0.727617
$167$	7.58830	0.587201	0.293600	−	0.955928i	$-0.405147\pi$
$167$	7.58830	0.587201	0.293600	+	0.955928i	$0.405147\pi$
$168$	−2.02214	−0.156012
$169$	−12.6769	−0.975144
$170$	0	0
$171$	6.62049	0.506282
$172$	−0.0885748	−0.00675377
$173$	18.2094	1.38443	0.692216	−	0.721690i	$-0.256634\pi$
$173$	18.2094	1.38443	0.692216	+	0.721690i	$0.256634\pi$
$174$	2.87897	0.218254
$175$	0	0
$176$	−2.29844	−0.173251
$177$	−6.36259	−0.478242
$178$	5.88755	0.441290
$179$	4.75362	0.355302	0.177651	−	0.984094i	$-0.443150\pi$
$179$	4.75362	0.355302	0.177651	+	0.984094i	$0.443150\pi$
$180$	0	0
$181$	25.2652	1.87795	0.938973	−	0.343991i	$-0.111779\pi$
$181$	25.2652	1.87795	0.938973	+	0.343991i	$0.111779\pi$
$182$	−0.310535	−0.0230184
$183$	3.71308	0.274479
$184$	−18.0598	−1.33139
$185$	0	0
$186$	5.27777	0.386985
$187$	2.70156	0.197558
$188$	−17.2823	−1.26044
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−17.4504	−1.26266	−0.631332	−	0.775513i	$-0.717491\pi$
$191$	−17.4504	−1.26266	−0.631332	+	0.775513i	$0.717491\pi$
$192$	1.70156	0.122800
$193$	−22.9546	−1.65231	−0.826156	−	0.563442i	$-0.809477\pi$
$193$	−22.9546	−1.65231	−0.826156	+	0.563442i	$0.809477\pi$
$194$	9.83067	0.705801
$195$	0	0
$196$	−1.70156	−0.121540
$197$	3.36634	0.239842	0.119921	−	0.992783i	$-0.461736\pi$
$197$	3.36634	0.239842	0.119921	+	0.992783i	$0.461736\pi$
$198$	0.546295	0.0388235
$199$	9.54402	0.676557	0.338279	−	0.941046i	$-0.390155\pi$
$199$	9.54402	0.676557	0.338279	+	0.941046i	$0.390155\pi$
$200$	0	0
$201$	−10.4146	−0.734592
$202$	−0.451426	−0.0317622
$203$	5.27000	0.369882
$204$	−4.59688	−0.321846
$205$	0	0
$206$	5.78638	0.403156
$207$	−8.93103	−0.620749
$208$	1.30652	0.0905909
$209$	−6.62049	−0.457949
$210$	0	0
$211$	−7.73375	−0.532413	−0.266207	−	0.963916i	$-0.585770\pi$
$211$	−7.73375	−0.532413	−0.266207	+	0.963916i	$0.585770\pi$
$212$	2.66241	0.182855
$213$	−2.56844	−0.175986
$214$	−7.53143	−0.514838
$215$	0	0
$216$	−2.02214	−0.137589
$217$	9.66103	0.655833
$218$	−10.8661	−0.735943
$219$	2.26625	0.153139
$220$	0	0
$221$	−1.53567	−0.103300
$222$	1.50429	0.100961
$223$	13.9141	0.931758	0.465879	−	0.884848i	$-0.345738\pi$
$223$	13.9141	0.931758	0.465879	+	0.884848i	$0.345738\pi$
$224$	−5.29991	−0.354115
$225$	0	0
$226$	3.51558	0.233853
$227$	−0.597452	−0.0396543	−0.0198271	−	0.999803i	$-0.506312\pi$
$227$	−0.597452	−0.0396543	−0.0198271	+	0.999803i	$0.506312\pi$
$228$	11.2652	0.746055
$229$	21.1895	1.40024	0.700121	−	0.714024i	$-0.253129\pi$
$229$	21.1895	1.40024	0.700121	+	0.714024i	$0.253129\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	10.6567	0.699647
$233$	20.2821	1.32872	0.664362	−	0.747411i	$-0.268703\pi$
$233$	20.2821	1.32872	0.664362	+	0.747411i	$0.268703\pi$
$234$	−0.310535	−0.0203003
$235$	0	0
$236$	−10.8263	−0.704735
$237$	2.56844	0.166838
$238$	1.47585	0.0956651
$239$	2.23723	0.144715	0.0723573	−	0.997379i	$-0.476948\pi$
$239$	2.23723	0.144715	0.0723573	+	0.997379i	$0.476948\pi$
$240$	0	0
$241$	8.93045	0.575261	0.287630	−	0.957741i	$-0.407133\pi$
$241$	8.93045	0.575261	0.287630	+	0.957741i	$0.407133\pi$
$242$	−0.546295	−0.0351172
$243$	−1.00000	−0.0641500
$244$	6.31804	0.404471
$245$	0	0
$246$	−6.00000	−0.382546
$247$	3.76334	0.239456
$248$	19.5360	1.24054
$249$	17.1605	1.08750
$250$	0	0
$251$	3.35884	0.212008	0.106004	−	0.994366i	$-0.466194\pi$
$251$	3.35884	0.212008	0.106004	+	0.994366i	$0.466194\pi$
$252$	−1.70156	−0.107188
$253$	8.93103	0.561489
$254$	2.21362	0.138895
$255$	0	0
$256$	−2.89531	−0.180957
$257$	−5.80943	−0.362382	−0.181191	−	0.983448i	$-0.557995\pi$
$257$	−5.80943	−0.362382	−0.181191	+	0.983448i	$0.557995\pi$
$258$	0.0284374	0.00177043
$259$	2.75362	0.171101
$260$	0	0
$261$	5.27000	0.326205
$262$	3.11268	0.192302
$263$	23.3221	1.43810	0.719050	−	0.694959i	$-0.244577\pi$
$263$	23.3221	1.43810	0.719050	+	0.694959i	$0.244577\pi$
$264$	2.02214	0.124454
$265$	0	0
$266$	−3.61674	−0.221757
$267$	10.7772	0.659556
$268$	−17.7212	−1.08249
$269$	29.1246	1.77576	0.887878	−	0.460080i	$-0.152179\pi$
$269$	29.1246	1.77576	0.887878	+	0.460080i	$0.152179\pi$
$270$	0	0
$271$	31.2416	1.89779	0.948895	−	0.315592i	$-0.102203\pi$
$271$	31.2416	1.89779	0.948895	+	0.315592i	$0.102203\pi$
$272$	−6.20937	−0.376499
$273$	−0.568438	−0.0344035
$274$	−1.85911	−0.112313
$275$	0	0
$276$	−15.1967	−0.914734
$277$	15.8988	0.955269	0.477634	−	0.878559i	$-0.341494\pi$
$277$	15.8988	0.955269	0.477634	+	0.878559i	$0.341494\pi$
$278$	4.39616	0.263664
$279$	9.66103	0.578391
$280$	0	0
$281$	−14.1041	−0.841381	−0.420690	−	0.907204i	$-0.638212\pi$
$281$	−14.1041	−0.841381	−0.420690	+	0.907204i	$0.638212\pi$
$282$	5.54857	0.330413
$283$	1.32745	0.0789088	0.0394544	−	0.999221i	$-0.487438\pi$
$283$	1.32745	0.0789088	0.0394544	+	0.999221i	$0.487438\pi$
$284$	−4.37036	−0.259333
$285$	0	0
$286$	0.310535	0.0183623
$287$	−10.9831	−0.648311
$288$	−5.29991	−0.312300
$289$	−9.70156	−0.570680
$290$	0	0
$291$	17.9952	1.05490
$292$	3.85616	0.225665
$293$	−18.3784	−1.07368	−0.536840	−	0.843684i	$-0.680382\pi$
$293$	−18.3784	−1.07368	−0.536840	+	0.843684i	$0.680382\pi$
$294$	0.546295	0.0318606
$295$	0	0
$296$	5.56821	0.323646
$297$	1.00000	0.0580259
$298$	−4.10116	−0.237574
$299$	−5.07674	−0.293595
$300$	0	0
$301$	0.0520550	0.00300040
$302$	−1.75795	−0.101158
$303$	−0.826342	−0.0474721
$304$	15.2168	0.872743
$305$	0	0
$306$	1.47585	0.0843687
$307$	−6.45143	−0.368202	−0.184101	−	0.982907i	$-0.558937\pi$
$307$	−6.45143	−0.368202	−0.184101	+	0.982907i	$0.558937\pi$
$308$	1.70156	0.0969555
$309$	10.5921	0.602561
$310$	0	0
$311$	−29.4136	−1.66789	−0.833946	−	0.551847i	$-0.813923\pi$
$311$	−29.4136	−1.66789	−0.833946	+	0.551847i	$0.813923\pi$
$312$	−1.14946	−0.0650756
$313$	24.7542	1.39919	0.699595	−	0.714540i	$-0.253364\pi$
$313$	24.7542	1.39919	0.699595	+	0.714540i	$0.253364\pi$
$314$	−1.83067	−0.103311
$315$	0	0
$316$	4.37036	0.245852
$317$	15.0955	0.847850	0.423925	−	0.905697i	$-0.360652\pi$
$317$	15.0955	0.847850	0.423925	+	0.905697i	$0.360652\pi$
$318$	−0.854779	−0.0479336
$319$	−5.27000	−0.295063
$320$	0	0
$321$	−13.7864	−0.769481
$322$	4.87897	0.271895
$323$	−17.8857	−0.995186
$324$	−1.70156	−0.0945312
$325$	0	0
$326$	5.98741	0.331612
$327$	−19.8905	−1.09995
$328$	−22.2094	−1.22631
$329$	10.1567	0.559959
$330$	0	0
$331$	6.78263	0.372807	0.186404	−	0.982473i	$-0.440317\pi$
$331$	6.78263	0.372807	0.186404	+	0.982473i	$0.440317\pi$
$332$	29.1996	1.60254
$333$	2.75362	0.150897
$334$	−4.14545	−0.226829
$335$	0	0
$336$	−2.29844	−0.125390
$337$	31.9509	1.74048	0.870238	−	0.492631i	$-0.163965\pi$
$337$	31.9509	1.74048	0.870238	+	0.492631i	$0.163965\pi$
$338$	6.92531	0.376687
$339$	6.43531	0.349518
$340$	0	0
$341$	−9.66103	−0.523174
$342$	−3.61674	−0.195571
$343$	1.00000	0.0539949
$344$	0.105263	0.00567538
$345$	0	0
$346$	−9.94768	−0.534791
$347$	13.6694	0.733810	0.366905	−	0.930258i	$-0.380417\pi$
$347$	13.6694	0.733810	0.366905	+	0.930258i	$0.380417\pi$
$348$	8.96723	0.480694
$349$	16.1720	0.865668	0.432834	−	0.901474i	$-0.357514\pi$
$349$	16.1720	0.865668	0.432834	+	0.901474i	$0.357514\pi$
$350$	0	0
$351$	−0.568438	−0.0303410
$352$	5.29991	0.282486
$353$	9.94313	0.529219	0.264610	−	0.964356i	$-0.414757\pi$
$353$	9.94313	0.529219	0.264610	+	0.964356i	$0.414757\pi$
$354$	3.47585	0.184739
$355$	0	0
$356$	18.3381	0.971919
$357$	2.70156	0.142982
$358$	−2.59688	−0.137249
$359$	−12.5845	−0.664187	−0.332094	−	0.943246i	$-0.607755\pi$
$359$	−12.5845	−0.664187	−0.332094	+	0.943246i	$0.607755\pi$
$360$	0	0
$361$	24.8309	1.30689
$362$	−13.8022	−0.725429
$363$	−1.00000	−0.0524864
$364$	−0.967233	−0.0506968
$365$	0	0
$366$	−2.02844	−0.106028
$367$	−36.9699	−1.92981	−0.964907	−	0.262592i	$-0.915423\pi$
$367$	−36.9699	−1.92981	−0.964907	+	0.262592i	$0.915423\pi$
$368$	−20.5274	−1.07007
$369$	−10.9831	−0.571756
$370$	0	0
$371$	−1.56469	−0.0812344
$372$	16.4388	0.852314
$373$	13.4109	0.694390	0.347195	−	0.937793i	$-0.387134\pi$
$373$	13.4109	0.694390	0.347195	+	0.937793i	$0.387134\pi$
$374$	−1.47585	−0.0763143
$375$	0	0
$376$	20.5384	1.05919
$377$	2.99567	0.154285
$378$	0.546295	0.0280984
$379$	−0.805672	−0.0413846	−0.0206923	−	0.999786i	$-0.506587\pi$
$379$	−0.805672	−0.0413846	−0.0206923	+	0.999786i	$0.506587\pi$
$380$	0	0
$381$	4.05206	0.207593
$382$	9.53304	0.487753
$383$	22.3494	1.14200	0.571001	−	0.820949i	$-0.306555\pi$
$383$	22.3494	1.14200	0.571001	+	0.820949i	$0.306555\pi$
$384$	−11.5294	−0.588356
$385$	0	0
$386$	12.5400	0.638269
$387$	0.0520550	0.00264610
$388$	30.6199	1.55449
$389$	−29.4863	−1.49501	−0.747507	−	0.664253i	$-0.768750\pi$
$389$	−29.4863	−1.49501	−0.747507	+	0.664253i	$0.768750\pi$
$390$	0	0
$391$	24.1277	1.22019
$392$	2.02214	0.102134
$393$	5.69781	0.287416
$394$	−1.83902	−0.0926482
$395$	0	0
$396$	1.70156	0.0855067
$397$	−4.45143	−0.223411	−0.111705	−	0.993741i	$-0.535631\pi$
$397$	−4.45143	−0.223411	−0.111705	+	0.993741i	$0.535631\pi$
$398$	−5.21384	−0.261346
$399$	−6.62049	−0.331439
$400$	0	0
$401$	22.7123	1.13420	0.567099	−	0.823650i	$-0.308066\pi$
$401$	22.7123	1.13420	0.567099	+	0.823650i	$0.308066\pi$
$402$	5.68947	0.283765
$403$	5.49170	0.273561
$404$	−1.40607	−0.0699547
$405$	0	0
$406$	−2.87897	−0.142881
$407$	−2.75362	−0.136492
$408$	5.46295	0.270456
$409$	5.09259	0.251812	0.125906	−	0.992042i	$-0.459816\pi$
$409$	5.09259	0.251812	0.125906	+	0.992042i	$0.459816\pi$
$410$	0	0
$411$	−3.40312	−0.167864
$412$	18.0230	0.887932
$413$	6.36259	0.313083
$414$	4.87897	0.239788
$415$	0	0
$416$	−3.01267	−0.147708
$417$	8.04724	0.394075
$418$	3.61674	0.176901
$419$	−2.86286	−0.139860	−0.0699300	−	0.997552i	$-0.522278\pi$
$419$	−2.86286	−0.139860	−0.0699300	+	0.997552i	$0.522278\pi$
$420$	0	0
$421$	−16.1047	−0.784894	−0.392447	−	0.919775i	$-0.628371\pi$
$421$	−16.1047	−0.784894	−0.392447	+	0.919775i	$0.628371\pi$
$422$	4.22491	0.205665
$423$	10.1567	0.493838
$424$	−3.16402	−0.153658
$425$	0	0
$426$	1.40312	0.0679816
$427$	−3.71308	−0.179689
$428$	−23.4584	−1.13390
$429$	0.568438	0.0274445
$430$	0	0
$431$	−4.67794	−0.225329	−0.112664	−	0.993633i	$-0.535938\pi$
$431$	−4.67794	−0.225329	−0.112664	+	0.993633i	$0.535938\pi$
$432$	−2.29844	−0.110584
$433$	27.4337	1.31838	0.659189	−	0.751977i	$-0.270900\pi$
$433$	27.4337	1.31838	0.659189	+	0.751977i	$0.270900\pi$
$434$	−5.27777	−0.253341
$435$	0	0
$436$	−33.8449	−1.62088
$437$	−59.1278	−2.82847
$438$	−1.23804	−0.0591558
$439$	5.73433	0.273685	0.136842	−	0.990593i	$-0.456305\pi$
$439$	5.73433	0.273685	0.136842	+	0.990593i	$0.456305\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0.838929	0.0399038
$443$	9.58536	0.455414	0.227707	−	0.973730i	$-0.426877\pi$
$443$	9.58536	0.455414	0.227707	+	0.973730i	$0.426877\pi$
$444$	4.68545	0.222362
$445$	0	0
$446$	−7.60121	−0.359927
$447$	−7.50723	−0.355080
$448$	−1.70156	−0.0803913
$449$	30.9463	1.46045	0.730223	−	0.683209i	$-0.239416\pi$
$449$	30.9463	1.46045	0.730223	+	0.683209i	$0.239416\pi$
$450$	0	0
$451$	10.9831	0.517173
$452$	10.9501	0.515049
$453$	−3.21795	−0.151192
$454$	0.326385	0.0153180
$455$	0	0
$456$	−13.3876	−0.626931
$457$	42.2246	1.97519	0.987593	−	0.157036i	$-0.0501940\pi$
$457$	42.2246	1.97519	0.987593	+	0.157036i	$0.0501940\pi$
$458$	−11.5757	−0.540898
$459$	2.70156	0.126098
$460$	0	0
$461$	9.87464	0.459908	0.229954	−	0.973201i	$-0.426142\pi$
$461$	9.87464	0.459908	0.229954	+	0.973201i	$0.426142\pi$
$462$	−0.546295	−0.0254159
$463$	−2.56009	−0.118978	−0.0594888	−	0.998229i	$-0.518947\pi$
$463$	−2.56009	−0.118978	−0.0594888	+	0.998229i	$0.518947\pi$
$464$	12.1128	0.562321
$465$	0	0
$466$	−11.0800	−0.513271
$467$	19.6924	0.911256	0.455628	−	0.890170i	$-0.349415\pi$
$467$	19.6924	0.911256	0.455628	+	0.890170i	$0.349415\pi$
$468$	−0.967233	−0.0447104
$469$	10.4146	0.480904
$470$	0	0
$471$	−3.35107	−0.154409
$472$	12.8661	0.592209
$473$	−0.0520550	−0.00239349
$474$	−1.40312	−0.0644476
$475$	0	0
$476$	4.59688	0.210697
$477$	−1.56469	−0.0716420
$478$	−1.22219	−0.0559016
$479$	15.3672	0.702144	0.351072	−	0.936348i	$-0.385817\pi$
$479$	15.3672	0.702144	0.351072	+	0.936348i	$0.385817\pi$
$480$	0	0
$481$	1.56526	0.0713698
$482$	−4.87866	−0.222217
$483$	8.93103	0.406376
$484$	−1.70156	−0.0773437
$485$	0	0
$486$	0.546295	0.0247804
$487$	1.52848	0.0692621	0.0346310	−	0.999400i	$-0.488974\pi$
$487$	1.52848	0.0692621	0.0346310	+	0.999400i	$0.488974\pi$
$488$	−7.50839	−0.339889
$489$	10.9600	0.495630
$490$	0	0
$491$	−40.7359	−1.83839	−0.919193	−	0.393808i	$-0.871157\pi$
$491$	−40.7359	−1.83839	−0.919193	+	0.393808i	$0.871157\pi$
$492$	−18.6884	−0.842538
$493$	−14.2372	−0.641213
$494$	−2.05589	−0.0924990
$495$	0	0
$496$	22.2053	0.997046
$497$	2.56844	0.115210
$498$	−9.37469	−0.420090
$499$	−20.3312	−0.910150	−0.455075	−	0.890453i	$-0.650388\pi$
$499$	−20.3312	−0.910150	−0.455075	+	0.890453i	$0.650388\pi$
$500$	0	0
$501$	−7.58830	−0.339020
$502$	−1.83491	−0.0818963
$503$	23.5952	1.05206	0.526030	−	0.850466i	$-0.323680\pi$
$503$	23.5952	1.05206	0.526030	+	0.850466i	$0.323680\pi$
$504$	2.02214	0.0900734
$505$	0	0
$506$	−4.87897	−0.216897
$507$	12.6769	0.563000
$508$	6.89482	0.305908
$509$	38.0993	1.68872	0.844361	−	0.535775i	$-0.179981\pi$
$509$	38.0993	1.68872	0.844361	+	0.535775i	$0.179981\pi$
$510$	0	0
$511$	−2.26625	−0.100253
$512$	−21.4771	−0.949161
$513$	−6.62049	−0.292302
$514$	3.17366	0.139984
$515$	0	0
$516$	0.0885748	0.00389929
$517$	−10.1567	−0.446693
$518$	−1.50429	−0.0660945
$519$	−18.2094	−0.799303
$520$	0	0
$521$	33.9952	1.48936	0.744678	−	0.667424i	$-0.232603\pi$
$521$	33.9952	1.48936	0.744678	+	0.667424i	$0.232603\pi$
$522$	−2.87897	−0.126009
$523$	−10.9517	−0.478884	−0.239442	−	0.970911i	$-0.576965\pi$
$523$	−10.9517	−0.478884	−0.239442	+	0.970911i	$0.576965\pi$
$524$	9.69518	0.423536
$525$	0	0
$526$	−12.7407	−0.555522
$527$	−26.0999	−1.13693
$528$	2.29844	0.100027
$529$	56.7633	2.46797
$530$	0	0
$531$	6.36259	0.276113
$532$	−11.2652	−0.488408
$533$	−6.24321	−0.270423
$534$	−5.88755	−0.254779
$535$	0	0
$536$	21.0599	0.909650
$537$	−4.75362	−0.205134
$538$	−15.9106	−0.685954
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	13.4971	0.580285	0.290143	−	0.956983i	$-0.406297\pi$
$541$	13.4971	0.580285	0.290143	+	0.956983i	$0.406297\pi$
$542$	−17.0671	−0.733095
$543$	−25.2652	−1.08423
$544$	14.3180	0.613881
$545$	0	0
$546$	0.310535	0.0132897
$547$	42.0182	1.79657	0.898285	−	0.439414i	$-0.144814\pi$
$547$	42.0182	1.79657	0.898285	+	0.439414i	$0.144814\pi$
$548$	−5.79063	−0.247363
$549$	−3.71308	−0.158471
$550$	0	0
$551$	34.8900	1.48636
$552$	18.0598	0.768677
$553$	−2.56844	−0.109221
$554$	−8.68545	−0.369009
$555$	0	0
$556$	13.6929	0.580707
$557$	27.9461	1.18411	0.592057	−	0.805896i	$-0.298316\pi$
$557$	27.9461	1.18411	0.592057	+	0.805896i	$0.298316\pi$
$558$	−5.27777	−0.223426
$559$	0.0295901	0.00125153
$560$	0	0
$561$	−2.70156	−0.114060
$562$	7.70500	0.325016
$563$	0.266247	0.0112210	0.00561050	−	0.999984i	$-0.498214\pi$
$563$	0.266247	0.0112210	0.00561050	+	0.999984i	$0.498214\pi$
$564$	17.2823	0.727717
$565$	0	0
$566$	−0.725180	−0.0304816
$567$	1.00000	0.0419961
$568$	5.19375	0.217925
$569$	−35.3983	−1.48397	−0.741987	−	0.670414i	$-0.766116\pi$
$569$	−35.3983	−1.48397	−0.741987	+	0.670414i	$0.766116\pi$
$570$	0	0
$571$	−0.440134	−0.0184191	−0.00920953	−	0.999958i	$-0.502932\pi$
$571$	−0.440134	−0.0184191	−0.00920953	+	0.999958i	$0.502932\pi$
$572$	0.967233	0.0404421
$573$	17.4504	0.728999
$574$	6.00000	0.250435
$575$	0	0
$576$	−1.70156	−0.0708984
$577$	25.1030	1.04505	0.522527	−	0.852623i	$-0.324990\pi$
$577$	25.1030	1.04505	0.522527	+	0.852623i	$0.324990\pi$
$578$	5.29991	0.220447
$579$	22.9546	0.953963
$580$	0	0
$581$	−17.1605	−0.711937
$582$	−9.83067	−0.407494
$583$	1.56469	0.0648026
$584$	−4.58268	−0.189633
$585$	0	0
$586$	10.0400	0.414750
$587$	−3.24638	−0.133993	−0.0669963	−	0.997753i	$-0.521342\pi$
$587$	−3.24638	−0.133993	−0.0669963	+	0.997753i	$0.521342\pi$
$588$	1.70156	0.0701712
$589$	63.9608	2.63546
$590$	0	0
$591$	−3.36634	−0.138473
$592$	6.32902	0.260121
$593$	−10.7252	−0.440430	−0.220215	−	0.975451i	$-0.570676\pi$
$593$	−10.7252	−0.440430	−0.220215	+	0.975451i	$0.570676\pi$
$594$	−0.546295	−0.0224147
$595$	0	0
$596$	−12.7740	−0.523244
$597$	−9.54402	−0.390611
$598$	2.77340	0.113413
$599$	36.3607	1.48566	0.742829	−	0.669481i	$-0.233483\pi$
$599$	36.3607	1.48566	0.742829	+	0.669481i	$0.233483\pi$
$600$	0	0
$601$	−39.7034	−1.61954	−0.809769	−	0.586749i	$-0.800407\pi$
$601$	−39.7034	−1.61954	−0.809769	+	0.586749i	$0.800407\pi$
$602$	−0.0284374	−0.00115902
$603$	10.4146	0.424117
$604$	−5.47553	−0.222796
$605$	0	0
$606$	0.451426	0.0183379
$607$	9.91893	0.402597	0.201299	−	0.979530i	$-0.435484\pi$
$607$	9.91893	0.402597	0.201299	+	0.979530i	$0.435484\pi$
$608$	−35.0880	−1.42301
$609$	−5.27000	−0.213551
$610$	0	0
$611$	5.77348	0.233570
$612$	4.59688	0.185818
$613$	−39.2681	−1.58602	−0.793012	−	0.609206i	$-0.791488\pi$
$613$	−39.2681	−1.58602	−0.793012	+	0.609206i	$0.791488\pi$
$614$	3.52438	0.142232
$615$	0	0
$616$	−2.02214	−0.0814745
$617$	−31.3462	−1.26195	−0.630976	−	0.775802i	$-0.717345\pi$
$617$	−31.3462	−1.26195	−0.630976	+	0.775802i	$0.717345\pi$
$618$	−5.78638	−0.232762
$619$	−12.3916	−0.498061	−0.249030	−	0.968496i	$-0.580112\pi$
$619$	−12.3916	−0.498061	−0.249030	+	0.968496i	$0.580112\pi$
$620$	0	0
$621$	8.93103	0.358390
$622$	16.0685	0.644287
$623$	−10.7772	−0.431781
$624$	−1.30652	−0.0523027
$625$	0	0
$626$	−13.5231	−0.540491
$627$	6.62049	0.264397
$628$	−5.70205	−0.227537
$629$	−7.43907	−0.296615
$630$	0	0
$631$	33.2233	1.32260	0.661299	−	0.750123i	$-0.270006\pi$
$631$	33.2233	1.32260	0.661299	+	0.750123i	$0.270006\pi$
$632$	−5.19375	−0.206596
$633$	7.73375	0.307389
$634$	−8.24661	−0.327515
$635$	0	0
$636$	−2.66241	−0.105571
$637$	0.568438	0.0225223
$638$	2.87897	0.113980
$639$	2.56844	0.101606
$640$	0	0
$641$	18.9179	0.747211	0.373605	−	0.927588i	$-0.378121\pi$
$641$	18.9179	0.747211	0.373605	+	0.927588i	$0.378121\pi$
$642$	7.53143	0.297242
$643$	5.82554	0.229737	0.114868	−	0.993381i	$-0.463355\pi$
$643$	5.82554	0.229737	0.114868	+	0.993381i	$0.463355\pi$
$644$	15.1967	0.598834
$645$	0	0
$646$	9.77085	0.384429
$647$	−8.30759	−0.326605	−0.163302	−	0.986576i	$-0.552215\pi$
$647$	−8.30759	−0.326605	−0.163302	+	0.986576i	$0.552215\pi$
$648$	2.02214	0.0794373
$649$	−6.36259	−0.249753
$650$	0	0
$651$	−9.66103	−0.378646
$652$	18.6492	0.730359
$653$	−44.4182	−1.73822	−0.869109	−	0.494621i	$-0.835307\pi$
$653$	−44.4182	−1.73822	−0.869109	+	0.494621i	$0.835307\pi$
$654$	10.8661	0.424897
$655$	0	0
$656$	−25.2439	−0.985610
$657$	−2.26625	−0.0884147
$658$	−5.54857	−0.216306
$659$	−13.4965	−0.525750	−0.262875	−	0.964830i	$-0.584671\pi$
$659$	−13.4965	−0.525750	−0.262875	+	0.964830i	$0.584671\pi$
$660$	0	0
$661$	31.7450	1.23474	0.617370	−	0.786673i	$-0.288198\pi$
$661$	31.7450	1.23474	0.617370	+	0.786673i	$0.288198\pi$
$662$	−3.70532	−0.144011
$663$	1.53567	0.0596405
$664$	−34.7010	−1.34666
$665$	0	0
$666$	−1.50429	−0.0582899
$667$	−47.0665	−1.82242
$668$	−12.9120	−0.499579
$669$	−13.9141	−0.537951
$670$	0	0
$671$	3.71308	0.143342
$672$	5.29991	0.204449
$673$	−5.81295	−0.224073	−0.112036	−	0.993704i	$-0.535737\pi$
$673$	−5.81295	−0.224073	−0.112036	+	0.993704i	$0.535737\pi$
$674$	−17.4546	−0.672326
$675$	0	0
$676$	21.5705	0.829634
$677$	23.3699	0.898177	0.449088	−	0.893487i	$-0.351749\pi$
$677$	23.3699	0.898177	0.449088	+	0.893487i	$0.351749\pi$
$678$	−3.51558	−0.135015
$679$	−17.9952	−0.690592
$680$	0	0
$681$	0.597452	0.0228944
$682$	5.27777	0.202096
$683$	−13.4705	−0.515433	−0.257716	−	0.966221i	$-0.582970\pi$
$683$	−13.4705	−0.515433	−0.257716	+	0.966221i	$0.582970\pi$
$684$	−11.2652	−0.430735
$685$	0	0
$686$	−0.546295	−0.0208576
$687$	−21.1895	−0.808430
$688$	0.119645	0.00456143
$689$	−0.889427	−0.0338845
$690$	0	0
$691$	9.79358	0.372565	0.186283	−	0.982496i	$-0.440356\pi$
$691$	9.79358	0.372565	0.186283	+	0.982496i	$0.440356\pi$
$692$	−30.9844	−1.17785
$693$	−1.00000	−0.0379869
$694$	−7.46751	−0.283463
$695$	0	0
$696$	−10.6567	−0.403941
$697$	29.6715	1.12389
$698$	−8.83469	−0.334398
$699$	−20.2821	−0.767139
$700$	0	0
$701$	−9.27000	−0.350123	−0.175062	−	0.984557i	$-0.556012\pi$
$701$	−9.27000	−0.350123	−0.175062	+	0.984557i	$0.556012\pi$
$702$	0.310535	0.0117204
$703$	18.2303	0.687569
$704$	1.70156	0.0641300
$705$	0	0
$706$	−5.43188	−0.204431
$707$	0.826342	0.0310778
$708$	10.8263	0.406879
$709$	−19.9898	−0.750732	−0.375366	−	0.926877i	$-0.622483\pi$
$709$	−19.9898	−0.750732	−0.375366	+	0.926877i	$0.622483\pi$
$710$	0	0
$711$	−2.56844	−0.0963240
$712$	−21.7931	−0.816732
$713$	−86.2829	−3.23132
$714$	−1.47585	−0.0552323
$715$	0	0
$716$	−8.08857	−0.302284
$717$	−2.23723	−0.0835510
$718$	6.87487	0.256568
$719$	−26.7174	−0.996391	−0.498196	−	0.867065i	$-0.666004\pi$
$719$	−26.7174	−0.996391	−0.498196	+	0.867065i	$0.666004\pi$
$720$	0	0
$721$	−10.5921	−0.394469
$722$	−13.5650	−0.504837
$723$	−8.93045	−0.332127
$724$	−42.9903	−1.59772
$725$	0	0
$726$	0.546295	0.0202749
$727$	−16.4955	−0.611782	−0.305891	−	0.952066i	$-0.598954\pi$
$727$	−16.4955	−0.611782	−0.305891	+	0.952066i	$0.598954\pi$
$728$	1.14946	0.0426020
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.140630	−0.00520138
$732$	−6.31804	−0.233522
$733$	22.6838	0.837847	0.418923	−	0.908022i	$-0.362408\pi$
$733$	22.6838	0.837847	0.418923	+	0.908022i	$0.362408\pi$
$734$	20.1965	0.745465
$735$	0	0
$736$	47.3337	1.74474
$737$	−10.4146	−0.383628
$738$	6.00000	0.220863
$739$	−41.7166	−1.53457	−0.767285	−	0.641306i	$-0.778393\pi$
$739$	−41.7166	−1.53457	−0.767285	+	0.641306i	$0.778393\pi$
$740$	0	0
$741$	−3.76334	−0.138250
$742$	0.854779	0.0313799
$743$	26.4664	0.970959	0.485480	−	0.874248i	$-0.338645\pi$
$743$	26.4664	0.970959	0.485480	+	0.874248i	$0.338645\pi$
$744$	−19.5360	−0.716224
$745$	0	0
$746$	−7.32630	−0.268235
$747$	−17.1605	−0.627870
$748$	−4.59688	−0.168078
$749$	13.7864	0.503744
$750$	0	0
$751$	−36.2802	−1.32388	−0.661942	−	0.749555i	$-0.730268\pi$
$751$	−36.2802	−1.32388	−0.661942	+	0.749555i	$0.730268\pi$
$752$	23.3446	0.851291
$753$	−3.35884	−0.122403
$754$	−1.63652	−0.0595985
$755$	0	0
$756$	1.70156	0.0618852
$757$	−27.3451	−0.993874	−0.496937	−	0.867786i	$-0.665542\pi$
$757$	−27.3451	−0.993874	−0.496937	+	0.867786i	$0.665542\pi$
$758$	0.440134	0.0159864
$759$	−8.93103	−0.324176
$760$	0	0
$761$	−16.2537	−0.589195	−0.294597	−	0.955621i	$-0.595186\pi$
$761$	−16.2537	−0.589195	−0.294597	+	0.955621i	$0.595186\pi$
$762$	−2.21362	−0.0801909
$763$	19.8905	0.720084
$764$	29.6929	1.07425
$765$	0	0
$766$	−12.2094	−0.441143
$767$	3.61674	0.130593
$768$	2.89531	0.104476
$769$	−38.2698	−1.38004	−0.690022	−	0.723789i	$-0.742399\pi$
$769$	−38.2698	−1.38004	−0.690022	+	0.723789i	$0.742399\pi$
$770$	0	0
$771$	5.80943	0.209221
$772$	39.0588	1.40576
$773$	−6.60438	−0.237543	−0.118772	−	0.992922i	$-0.537896\pi$
$773$	−6.60438	−0.237543	−0.118772	+	0.992922i	$0.537896\pi$
$774$	−0.0284374	−0.00102216
$775$	0	0
$776$	−36.3888	−1.30628
$777$	−2.75362	−0.0987855
$778$	16.1082	0.577507
$779$	−72.7134	−2.60523
$780$	0	0
$781$	−2.56844	−0.0919060
$782$	−13.1808	−0.471346
$783$	−5.27000	−0.188334
$784$	2.29844	0.0820871
$785$	0	0
$786$	−3.11268	−0.111026
$787$	23.1916	0.826692	0.413346	−	0.910574i	$-0.364360\pi$
$787$	23.1916	0.826692	0.413346	+	0.910574i	$0.364360\pi$
$788$	−5.72804	−0.204053
$789$	−23.3221	−0.830287
$790$	0	0
$791$	−6.43531	−0.228813
$792$	−2.02214	−0.0718537
$793$	−2.11066	−0.0749517
$794$	2.43179	0.0863010
$795$	0	0
$796$	−16.2397	−0.575602
$797$	−32.5872	−1.15430	−0.577150	−	0.816638i	$-0.695835\pi$
$797$	−32.5872	−1.15430	−0.577150	+	0.816638i	$0.695835\pi$
$798$	3.61674	0.128031
$799$	−27.4391	−0.970724
$800$	0	0
$801$	−10.7772	−0.380795
$802$	−12.4076	−0.438127
$803$	2.26625	0.0799741
$804$	17.7212	0.624977
$805$	0	0
$806$	−3.00009	−0.105674
$807$	−29.1246	−1.02523
$808$	1.67098	0.0587849
$809$	13.5303	0.475699	0.237850	−	0.971302i	$-0.423557\pi$
$809$	13.5303	0.475699	0.237850	+	0.971302i	$0.423557\pi$
$810$	0	0
$811$	27.7241	0.973525	0.486763	−	0.873534i	$-0.338178\pi$
$811$	27.7241	0.973525	0.486763	+	0.873534i	$0.338178\pi$
$812$	−8.96723	−0.314688
$813$	−31.2416	−1.09569
$814$	1.50429	0.0527252
$815$	0	0
$816$	6.20937	0.217372
$817$	0.344630	0.0120571
$818$	−2.78205	−0.0972723
$819$	0.568438	0.0198628
$820$	0	0
$821$	40.9774	1.43012	0.715061	−	0.699062i	$-0.246399\pi$
$821$	40.9774	1.43012	0.715061	+	0.699062i	$0.246399\pi$
$822$	1.85911	0.0648439
$823$	−3.27352	−0.114108	−0.0570539	−	0.998371i	$-0.518171\pi$
$823$	−3.27352	−0.114108	−0.0570539	+	0.998371i	$0.518171\pi$
$824$	−21.4187	−0.746154
$825$	0	0
$826$	−3.47585	−0.120940
$827$	36.5217	1.26998	0.634992	−	0.772519i	$-0.281003\pi$
$827$	36.5217	1.26998	0.634992	+	0.772519i	$0.281003\pi$
$828$	15.1967	0.528122
$829$	21.2335	0.737469	0.368735	−	0.929535i	$-0.379791\pi$
$829$	21.2335	0.737469	0.368735	+	0.929535i	$0.379791\pi$
$830$	0	0
$831$	−15.8988	−0.551525
$832$	−0.967233	−0.0335328
$833$	−2.70156	−0.0936036
$834$	−4.39616	−0.152227
$835$	0	0
$836$	11.2652	0.389614
$837$	−9.66103	−0.333934
$838$	1.56397	0.0540263
$839$	20.7571	0.716616	0.358308	−	0.933603i	$-0.383354\pi$
$839$	20.7571	0.716616	0.358308	+	0.933603i	$0.383354\pi$
$840$	0	0
$841$	−1.22709	−0.0423136
$842$	8.79790	0.303196
$843$	14.1041	0.485771
$844$	13.1595	0.452967
$845$	0	0
$846$	−5.54857	−0.190764
$847$	1.00000	0.0343604
$848$	−3.59633	−0.123499
$849$	−1.32745	−0.0455580
$850$	0	0
$851$	−24.5926	−0.843025
$852$	4.37036	0.149726
$853$	31.3333	1.07283	0.536417	−	0.843953i	$-0.319778\pi$
$853$	31.3333	1.07283	0.536417	+	0.843953i	$0.319778\pi$
$854$	2.02844	0.0694117
$855$	0	0
$856$	27.8780	0.952852
$857$	−13.5411	−0.462554	−0.231277	−	0.972888i	$-0.574290\pi$
$857$	−13.5411	−0.462554	−0.231277	+	0.972888i	$0.574290\pi$
$858$	−0.310535	−0.0106015
$859$	−19.5644	−0.667530	−0.333765	−	0.942656i	$-0.608319\pi$
$859$	−19.5644	−0.667530	−0.333765	+	0.942656i	$0.608319\pi$
$860$	0	0
$861$	10.9831	0.374302
$862$	2.55554	0.0870419
$863$	8.91434	0.303448	0.151724	−	0.988423i	$-0.451518\pi$
$863$	8.91434	0.303448	0.151724	+	0.988423i	$0.451518\pi$
$864$	5.29991	0.180307
$865$	0	0
$866$	−14.9869	−0.509275
$867$	9.70156	0.329482
$868$	−16.4388	−0.557971
$869$	2.56844	0.0871283
$870$	0	0
$871$	5.92008	0.200594
$872$	40.2214	1.36207
$873$	−17.9952	−0.609045
$874$	32.3012	1.09260
$875$	0	0
$876$	−3.85616	−0.130288
$877$	−6.17626	−0.208557	−0.104279	−	0.994548i	$-0.533253\pi$
$877$	−6.17626	−0.208557	−0.104279	+	0.994548i	$0.533253\pi$
$878$	−3.13263	−0.105721
$879$	18.3784	0.619889
$880$	0	0
$881$	19.3511	0.651954	0.325977	−	0.945378i	$-0.394307\pi$
$881$	19.3511	0.651954	0.325977	+	0.945378i	$0.394307\pi$
$882$	−0.546295	−0.0183947
$883$	−28.7925	−0.968945	−0.484473	−	0.874806i	$-0.660988\pi$
$883$	−28.7925	−0.968945	−0.484473	+	0.874806i	$0.660988\pi$
$884$	2.61304	0.0878861
$885$	0	0
$886$	−5.23643	−0.175921
$887$	43.4970	1.46049	0.730243	−	0.683187i	$-0.239407\pi$
$887$	43.4970	1.46049	0.730243	+	0.683187i	$0.239407\pi$
$888$	−5.56821	−0.186857
$889$	−4.05206	−0.135902
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−23.6757	−0.792722
$893$	67.2426	2.25019
$894$	4.10116	0.137163
$895$	0	0
$896$	11.5294	0.385169
$897$	5.07674	0.169507
$898$	−16.9058	−0.564154
$899$	50.9136	1.69806
$900$	0	0
$901$	4.22709	0.140825
$902$	−6.00000	−0.199778
$903$	−0.0520550	−0.00173228
$904$	−13.0131	−0.432810
$905$	0	0
$906$	1.75795	0.0584039
$907$	9.20210	0.305551	0.152775	−	0.988261i	$-0.451179\pi$
$907$	9.20210	0.305551	0.152775	+	0.988261i	$0.451179\pi$
$908$	1.01660	0.0337371
$909$	0.826342	0.0274080
$910$	0	0
$911$	7.98331	0.264499	0.132249	−	0.991216i	$-0.457780\pi$
$911$	7.98331	0.264499	0.132249	+	0.991216i	$0.457780\pi$
$912$	−15.2168	−0.503878
$913$	17.1605	0.567929
$914$	−23.0671	−0.762992
$915$	0	0
$916$	−36.0553	−1.19130
$917$	−5.69781	−0.188158
$918$	−1.47585	−0.0487103
$919$	−36.4176	−1.20131	−0.600653	−	0.799510i	$-0.705093\pi$
$919$	−36.4176	−1.20131	−0.600653	+	0.799510i	$0.705093\pi$
$920$	0	0
$921$	6.45143	0.212582
$922$	−5.39447	−0.177657
$923$	1.46000	0.0480565
$924$	−1.70156	−0.0559773
$925$	0	0
$926$	1.39857	0.0459597
$927$	−10.5921	−0.347889
$928$	−27.9305	−0.916865
$929$	1.06660	0.0349940	0.0174970	−	0.999847i	$-0.494430\pi$
$929$	1.06660	0.0349940	0.0174970	+	0.999847i	$0.494430\pi$
$930$	0	0
$931$	6.62049	0.216978
$932$	−34.5112	−1.13045
$933$	29.4136	0.962957
$934$	−10.7579	−0.352008
$935$	0	0
$936$	1.14946	0.0375714
$937$	4.74611	0.155049	0.0775243	−	0.996990i	$-0.475298\pi$
$937$	4.74611	0.155049	0.0775243	+	0.996990i	$0.475298\pi$
$938$	−5.68947	−0.185768
$939$	−24.7542	−0.807823
$940$	0	0
$941$	33.1988	1.08225	0.541125	−	0.840942i	$-0.317999\pi$
$941$	33.1988	1.08225	0.541125	+	0.840942i	$0.317999\pi$
$942$	1.83067	0.0596465
$943$	98.0902	3.19426
$944$	14.6240	0.475971
$945$	0	0
$946$	0.0284374	0.000924579 0
$947$	−6.30361	−0.204840	−0.102420	−	0.994741i	$-0.532659\pi$
$947$	−6.30361	−0.204840	−0.102420	+	0.994741i	$0.532659\pi$
$948$	−4.37036	−0.141943
$949$	−1.28822	−0.0418175
$950$	0	0
$951$	−15.0955	−0.489506
$952$	−5.46295	−0.177055
$953$	39.4705	1.27857	0.639287	−	0.768968i	$-0.279230\pi$
$953$	39.4705	1.27857	0.639287	+	0.768968i	$0.279230\pi$
$954$	0.854779	0.0276745
$955$	0	0
$956$	−3.80679	−0.123120
$957$	5.27000	0.170355
$958$	−8.39501	−0.271230
$959$	3.40312	0.109893
$960$	0	0
$961$	62.3355	2.01082
$962$	−0.855094	−0.0275693
$963$	13.7864	0.444260
$964$	−15.1957	−0.489421
$965$	0	0
$966$	−4.87897	−0.156978
$967$	20.7192	0.666285	0.333142	−	0.942877i	$-0.391891\pi$
$967$	20.7192	0.666285	0.333142	+	0.942877i	$0.391891\pi$
$968$	2.02214	0.0649942
$969$	17.8857	0.574571
$970$	0	0
$971$	0.281521	0.00903444	0.00451722	−	0.999990i	$-0.498562\pi$
$971$	0.281521	0.00903444	0.00451722	+	0.999990i	$0.498562\pi$
$972$	1.70156	0.0545776
$973$	−8.04724	−0.257983
$974$	−0.835001	−0.0267551
$975$	0	0
$976$	−8.53429	−0.273176
$977$	−53.1267	−1.69967	−0.849836	−	0.527047i	$-0.823299\pi$
$977$	−53.1267	−1.69967	−0.849836	+	0.527047i	$0.823299\pi$
$978$	−5.98741	−0.191456
$979$	10.7772	0.344442
$980$	0	0
$981$	19.8905	0.635055
$982$	22.2538	0.710147
$983$	47.2072	1.50567	0.752837	−	0.658207i	$-0.228685\pi$
$983$	47.2072	1.50567	0.752837	+	0.658207i	$0.228685\pi$
$984$	22.2094	0.708009
$985$	0	0
$986$	7.77773	0.247693
$987$	−10.1567	−0.323293
$988$	−6.40356	−0.203724
$989$	−0.464905	−0.0147831
$990$	0	0
$991$	−30.4901	−0.968549	−0.484274	−	0.874916i	$-0.660916\pi$
$991$	−30.4901	−0.968549	−0.484274	+	0.874916i	$0.660916\pi$
$992$	−51.2026	−1.62568
$993$	−6.78263	−0.215240
$994$	−1.40312	−0.0445044
$995$	0	0
$996$	−29.1996	−0.925226
$997$	−6.62107	−0.209691	−0.104846	−	0.994489i	$-0.533435\pi$
$997$	−6.62107	−0.209691	−0.104846	+	0.994489i	$0.533435\pi$
$998$	11.1068	0.351581
$999$	−2.75362	−0.0871206

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.bz.1.2		4
5.4	even	2		1155.2.a.u.1.3	✓	4
15.14	odd	2		3465.2.a.bl.1.2		4
35.34	odd	2		8085.2.a.bn.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.u.1.3	✓	4	5.4	even	2
3465.2.a.bl.1.2		4	15.14	odd	2
5775.2.a.bz.1.2		4	1.1	even	1	trivial
8085.2.a.bn.1.3		4	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5775.a (trivial)

\(f(q)\)	\(=\)	\(q-0.546295 q^{2} -1.00000 q^{3} -1.70156 q^{4} +0.546295 q^{6} +1.00000 q^{7} +2.02214 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.546295 q^{2} -1.00000 q^{3} -1.70156 q^{4} +0.546295 q^{6} +1.00000 q^{7} +2.02214 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.70156 q^{12} +0.568438 q^{13} -0.546295 q^{14} +2.29844 q^{16} -2.70156 q^{17} -0.546295 q^{18} +6.62049 q^{19} -1.00000 q^{21} +0.546295 q^{22} -8.93103 q^{23} -2.02214 q^{24} -0.310535 q^{26} -1.00000 q^{27} -1.70156 q^{28} +5.27000 q^{29} +9.66103 q^{31} -5.29991 q^{32} +1.00000 q^{33} +1.47585 q^{34} -1.70156 q^{36} +2.75362 q^{37} -3.61674 q^{38} -0.568438 q^{39} -10.9831 q^{41} +0.546295 q^{42} +0.0520550 q^{43} +1.70156 q^{44} +4.87897 q^{46} +10.1567 q^{47} -2.29844 q^{48} +1.00000 q^{49} +2.70156 q^{51} -0.967233 q^{52} -1.56469 q^{53} +0.546295 q^{54} +2.02214 q^{56} -6.62049 q^{57} -2.87897 q^{58} +6.36259 q^{59} -3.71308 q^{61} -5.27777 q^{62} +1.00000 q^{63} -1.70156 q^{64} -0.546295 q^{66} +10.4146 q^{67} +4.59688 q^{68} +8.93103 q^{69} +2.56844 q^{71} +2.02214 q^{72} -2.26625 q^{73} -1.50429 q^{74} -11.2652 q^{76} -1.00000 q^{77} +0.310535 q^{78} -2.56844 q^{79} +1.00000 q^{81} +6.00000 q^{82} -17.1605 q^{83} +1.70156 q^{84} -0.0284374 q^{86} -5.27000 q^{87} -2.02214 q^{88} -10.7772 q^{89} +0.568438 q^{91} +15.1967 q^{92} -9.66103 q^{93} -5.54857 q^{94} +5.29991 q^{96} -17.9952 q^{97} -0.546295 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 6 q^{4} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 6 * q^4 + 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 6 q^{4} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 6 q^{12} - 8 q^{13} + 22 q^{16} + 2 q^{17} + 10 q^{19} - 4 q^{21} + 2 q^{23} + 20 q^{26} - 4 q^{27} + 6 q^{28} - 2 q^{29} + 24 q^{31} + 4 q^{33} + 6 q^{36} - 8 q^{37} - 16 q^{38} + 8 q^{39} - 6 q^{43} - 6 q^{44} - 12 q^{46} - 4 q^{47} - 22 q^{48} + 4 q^{49} - 2 q^{51} - 12 q^{52} - 14 q^{53} - 10 q^{57} + 20 q^{58} - 2 q^{59} + 6 q^{61} - 8 q^{62} + 4 q^{63} + 6 q^{64} + 8 q^{67} + 44 q^{68} - 2 q^{69} - 4 q^{73} - 36 q^{74} + 56 q^{76} - 4 q^{77} - 20 q^{78} + 4 q^{81} + 24 q^{82} - 6 q^{83} - 6 q^{84} - 36 q^{86} + 2 q^{87} + 18 q^{89} - 8 q^{91} + 44 q^{92} - 24 q^{93} - 36 q^{94} + 6 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 6 * q^4 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 6 * q^12 - 8 * q^13 + 22 * q^16 + 2 * q^17 + 10 * q^19 - 4 * q^21 + 2 * q^23 + 20 * q^26 - 4 * q^27 + 6 * q^28 - 2 * q^29 + 24 * q^31 + 4 * q^33 + 6 * q^36 - 8 * q^37 - 16 * q^38 + 8 * q^39 - 6 * q^43 - 6 * q^44 - 12 * q^46 - 4 * q^47 - 22 * q^48 + 4 * q^49 - 2 * q^51 - 12 * q^52 - 14 * q^53 - 10 * q^57 + 20 * q^58 - 2 * q^59 + 6 * q^61 - 8 * q^62 + 4 * q^63 + 6 * q^64 + 8 * q^67 + 44 * q^68 - 2 * q^69 - 4 * q^73 - 36 * q^74 + 56 * q^76 - 4 * q^77 - 20 * q^78 + 4 * q^81 + 24 * q^82 - 6 * q^83 - 6 * q^84 - 36 * q^86 + 2 * q^87 + 18 * q^89 - 8 * q^91 + 44 * q^92 - 24 * q^93 - 36 * q^94 + 6 * q^97 - 4 * q^99

Introduction

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Database

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