LMFDB - Embedded newform 5775.2.a.bw.1.1

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:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.52892$ 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)

$f(q)$	$=$	$q-2.52892 q^{2} +1.00000 q^{3} +4.39543 q^{4} -2.52892 q^{6} +1.00000 q^{7} -6.05784 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.52892 q^{2} +1.00000 q^{3} +4.39543 q^{4} -2.52892 q^{6} +1.00000 q^{7} -6.05784 q^{8} +1.00000 q^{9} +1.00000 q^{11} +4.39543 q^{12} -0.133492 q^{13} -2.52892 q^{14} +6.52892 q^{16} +5.05784 q^{17} -2.52892 q^{18} -0.924344 q^{19} +1.00000 q^{21} -2.52892 q^{22} +7.05784 q^{23} -6.05784 q^{24} +0.337590 q^{26} +1.00000 q^{27} +4.39543 q^{28} +3.86651 q^{29} +2.79085 q^{31} -4.39543 q^{32} +1.00000 q^{33} -12.7909 q^{34} +4.39543 q^{36} -9.98218 q^{37} +2.33759 q^{38} -0.133492 q^{39} +11.8487 q^{41} -2.52892 q^{42} +3.05784 q^{43} +4.39543 q^{44} -17.8487 q^{46} +3.07566 q^{47} +6.52892 q^{48} +1.00000 q^{49} +5.05784 q^{51} -0.586754 q^{52} +4.79085 q^{53} -2.52892 q^{54} -6.05784 q^{56} -0.924344 q^{57} -9.77808 q^{58} -12.6574 q^{59} +6.00000 q^{61} -7.05784 q^{62} +1.00000 q^{63} -1.94216 q^{64} -2.52892 q^{66} -8.92434 q^{67} +22.2313 q^{68} +7.05784 q^{69} -6.11567 q^{71} -6.05784 q^{72} -7.86651 q^{73} +25.2441 q^{74} -4.06289 q^{76} +1.00000 q^{77} +0.337590 q^{78} +14.1157 q^{79} +1.00000 q^{81} -29.9644 q^{82} +1.20915 q^{83} +4.39543 q^{84} -7.73302 q^{86} +3.86651 q^{87} -6.05784 q^{88} +15.5817 q^{89} -0.133492 q^{91} +31.0222 q^{92} +2.79085 q^{93} -7.77808 q^{94} -4.39543 q^{96} -12.7909 q^{97} -2.52892 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{12} + 12 q^{16} + 12 q^{19} + 3 q^{21} + 6 q^{23} - 3 q^{24} + 9 q^{26} + 3 q^{27} + 6 q^{28} + 12 q^{29} - 6 q^{31} - 6 q^{32} + 3 q^{33} - 24 q^{34} + 6 q^{36} + 15 q^{38} + 6 q^{41} - 6 q^{43} + 6 q^{44} - 24 q^{46} + 24 q^{47} + 12 q^{48} + 3 q^{49} + 21 q^{52} - 3 q^{56} + 12 q^{57} + 9 q^{58} - 24 q^{59} + 18 q^{61} - 6 q^{62} + 3 q^{63} - 21 q^{64} - 12 q^{67} + 6 q^{68} + 6 q^{69} + 12 q^{71} - 3 q^{72} - 24 q^{73} + 39 q^{74} - 3 q^{76} + 3 q^{77} + 9 q^{78} + 12 q^{79} + 3 q^{81} - 30 q^{82} + 18 q^{83} + 6 q^{84} - 24 q^{86} + 12 q^{87} - 3 q^{88} + 18 q^{89} + 18 q^{92} - 6 q^{93} + 15 q^{94} - 6 q^{96} - 24 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^12 + 12 * q^16 + 12 * q^19 + 3 * q^21 + 6 * q^23 - 3 * q^24 + 9 * q^26 + 3 * q^27 + 6 * q^28 + 12 * q^29 - 6 * q^31 - 6 * q^32 + 3 * q^33 - 24 * q^34 + 6 * q^36 + 15 * q^38 + 6 * q^41 - 6 * q^43 + 6 * q^44 - 24 * q^46 + 24 * q^47 + 12 * q^48 + 3 * q^49 + 21 * q^52 - 3 * q^56 + 12 * q^57 + 9 * q^58 - 24 * q^59 + 18 * q^61 - 6 * q^62 + 3 * q^63 - 21 * q^64 - 12 * q^67 + 6 * q^68 + 6 * q^69 + 12 * q^71 - 3 * q^72 - 24 * q^73 + 39 * q^74 - 3 * q^76 + 3 * q^77 + 9 * q^78 + 12 * q^79 + 3 * q^81 - 30 * q^82 + 18 * q^83 + 6 * q^84 - 24 * q^86 + 12 * q^87 - 3 * q^88 + 18 * q^89 + 18 * q^92 - 6 * q^93 + 15 * q^94 - 6 * q^96 - 24 * q^97 + 3 * 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 $

$2$	−2.52892	−1.78822	−0.894108	−	0.447852i	$-0.852189\pi$
$2$	−2.52892	−1.78822	−0.894108	+	0.447852i	$0.852189\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.39543	2.19771
$5$	0	0
$5$	0	0
$6$	−2.52892	−1.03243
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−6.05784	−2.14177
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	4.39543	1.26885
$13$	−0.133492	−0.0370240	−0.0185120	−	0.999829i	$-0.505893\pi$
$13$	−0.133492	−0.0370240	−0.0185120	+	0.999829i	$0.505893\pi$
$14$	−2.52892	−0.675882
$15$	0	0
$16$	6.52892	1.63223
$17$	5.05784	1.22671	0.613353	−	0.789809i	$-0.289820\pi$
$17$	5.05784	1.22671	0.613353	+	0.789809i	$0.289820\pi$
$18$	−2.52892	−0.596072
$19$	−0.924344	−0.212059	−0.106030	−	0.994363i	$-0.533814\pi$
$19$	−0.924344	−0.212059	−0.106030	+	0.994363i	$0.533814\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−2.52892	−0.539167
$23$	7.05784	1.47166	0.735830	−	0.677166i	$-0.236792\pi$
$23$	7.05784	1.47166	0.735830	+	0.677166i	$0.236792\pi$
$24$	−6.05784	−1.23655
$25$	0	0
$26$	0.337590	0.0662069
$27$	1.00000	0.192450
$28$	4.39543	0.830657
$29$	3.86651	0.717993	0.358996	−	0.933339i	$-0.383119\pi$
$29$	3.86651	0.717993	0.358996	+	0.933339i	$0.383119\pi$
$30$	0	0
$31$	2.79085	0.501252	0.250626	−	0.968084i	$-0.419364\pi$
$31$	2.79085	0.501252	0.250626	+	0.968084i	$0.419364\pi$
$32$	−4.39543	−0.777009
$33$	1.00000	0.174078
$34$	−12.7909	−2.19361
$35$	0	0
$36$	4.39543	0.732571
$37$	−9.98218	−1.64106	−0.820530	−	0.571603i	$-0.806322\pi$
$37$	−9.98218	−1.64106	−0.820530	+	0.571603i	$0.806322\pi$
$38$	2.33759	0.379207
$39$	−0.133492	−0.0213758
$40$	0	0
$41$	11.8487	1.85045	0.925227	−	0.379414i	$-0.123874\pi$
$41$	11.8487	1.85045	0.925227	+	0.379414i	$0.123874\pi$
$42$	−2.52892	−0.390221
$43$	3.05784	0.466316	0.233158	−	0.972439i	$-0.425094\pi$
$43$	3.05784	0.466316	0.233158	+	0.972439i	$0.425094\pi$
$44$	4.39543	0.662635
$45$	0	0
$46$	−17.8487	−2.63165
$47$	3.07566	0.448631	0.224315	−	0.974517i	$-0.427985\pi$
$47$	3.07566	0.448631	0.224315	+	0.974517i	$0.427985\pi$
$48$	6.52892	0.942368
$49$	1.00000	0.142857
$50$	0	0
$51$	5.05784	0.708239
$52$	−0.586754	−0.0813681
$53$	4.79085	0.658074	0.329037	−	0.944317i	$-0.393276\pi$
$53$	4.79085	0.658074	0.329037	+	0.944317i	$0.393276\pi$
$54$	−2.52892	−0.344142
$55$	0	0
$56$	−6.05784	−0.809512
$57$	−0.924344	−0.122432
$58$	−9.77808	−1.28393
$59$	−12.6574	−1.64785	−0.823924	−	0.566700i	$-0.808220\pi$
$59$	−12.6574	−1.64785	−0.823924	+	0.566700i	$0.808220\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−7.05784	−0.896346
$63$	1.00000	0.125988
$64$	−1.94216	−0.242771
$65$	0	0
$66$	−2.52892	−0.311288
$67$	−8.92434	−1.09028	−0.545141	−	0.838344i	$-0.683524\pi$
$67$	−8.92434	−1.09028	−0.545141	+	0.838344i	$0.683524\pi$
$68$	22.2313	2.69595
$69$	7.05784	0.849664
$70$	0	0
$71$	−6.11567	−0.725797	−0.362898	−	0.931829i	$-0.618213\pi$
$71$	−6.11567	−0.725797	−0.362898	+	0.931829i	$0.618213\pi$
$72$	−6.05784	−0.713923
$73$	−7.86651	−0.920705	−0.460353	−	0.887736i	$-0.652277\pi$
$73$	−7.86651	−0.920705	−0.460353	+	0.887736i	$0.652277\pi$
$74$	25.2441	2.93457
$75$	0	0
$76$	−4.06289	−0.466045
$77$	1.00000	0.113961
$78$	0.337590	0.0382246
$79$	14.1157	1.58814	0.794069	−	0.607828i	$-0.207959\pi$
$79$	14.1157	1.58814	0.794069	+	0.607828i	$0.207959\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−29.9644	−3.30901
$83$	1.20915	0.132721	0.0663606	−	0.997796i	$-0.478861\pi$
$83$	1.20915	0.132721	0.0663606	+	0.997796i	$0.478861\pi$
$84$	4.39543	0.479580
$85$	0	0
$86$	−7.73302	−0.833873
$87$	3.86651	0.414533
$88$	−6.05784	−0.645767
$89$	15.5817	1.65166	0.825829	−	0.563921i	$-0.190708\pi$
$89$	15.5817	1.65166	0.825829	+	0.563921i	$0.190708\pi$
$90$	0	0
$91$	−0.133492	−0.0139938
$92$	31.0222	3.23429
$93$	2.79085	0.289398
$94$	−7.77808	−0.802248
$95$	0	0
$96$	−4.39543	−0.448606
$97$	−12.7909	−1.29871	−0.649357	−	0.760484i	$-0.724962\pi$
$97$	−12.7909	−1.29871	−0.649357	+	0.760484i	$0.724962\pi$
$98$	−2.52892	−0.255459
$99$	1.00000	0.100504
$100$	0	0
$101$	−9.59180	−0.954420	−0.477210	−	0.878789i	$-0.658352\pi$
$101$	−9.59180	−0.954420	−0.477210	+	0.878789i	$0.658352\pi$
$102$	−12.7909	−1.26648
$103$	−9.84869	−0.970420	−0.485210	−	0.874398i	$-0.661257\pi$
$103$	−9.84869	−0.970420	−0.485210	+	0.874398i	$0.661257\pi$
$104$	0.808672	0.0792968
$105$	0	0
$106$	−12.1157	−1.17678
$107$	−0.924344	−0.0893597	−0.0446799	−	0.999001i	$-0.514227\pi$
$107$	−0.924344	−0.0893597	−0.0446799	+	0.999001i	$0.514227\pi$
$108$	4.39543	0.422950
$109$	−8.52387	−0.816439	−0.408219	−	0.912884i	$-0.633850\pi$
$109$	−8.52387	−0.816439	−0.408219	+	0.912884i	$0.633850\pi$
$110$	0	0
$111$	−9.98218	−0.947467
$112$	6.52892	0.616925
$113$	12.1157	1.13975	0.569873	−	0.821733i	$-0.306992\pi$
$113$	12.1157	1.13975	0.569873	+	0.821733i	$0.306992\pi$
$114$	2.33759	0.218935
$115$	0	0
$116$	16.9950	1.57794
$117$	−0.133492	−0.0123413
$118$	32.0094	2.94671
$119$	5.05784	0.463651
$120$	0	0
$121$	1.00000	0.0909091
$122$	−15.1735	−1.37374
$123$	11.8487	1.06836
$124$	12.2670	1.10161
$125$	0	0
$126$	−2.52892	−0.225294
$127$	8.90652	0.790326	0.395163	−	0.918611i	$-0.370688\pi$
$127$	8.90652	0.790326	0.395163	+	0.918611i	$0.370688\pi$
$128$	13.7024	1.21113
$129$	3.05784	0.269227
$130$	0	0
$131$	−15.6974	−1.37149	−0.685743	−	0.727844i	$-0.740523\pi$
$131$	−15.6974	−1.37149	−0.685743	+	0.727844i	$0.740523\pi$
$132$	4.39543	0.382573
$133$	−0.924344	−0.0801508
$134$	22.5689	1.94966
$135$	0	0
$136$	−30.6395	−2.62732
$137$	−14.6395	−1.25074	−0.625370	−	0.780328i	$-0.715052\pi$
$137$	−14.6395	−1.25074	−0.625370	+	0.780328i	$0.715052\pi$
$138$	−17.8487	−1.51938
$139$	18.6496	1.58184	0.790921	−	0.611918i	$-0.209602\pi$
$139$	18.6496	1.58184	0.790921	+	0.611918i	$0.209602\pi$
$140$	0	0
$141$	3.07566	0.259017
$142$	15.4660	1.29788
$143$	−0.133492	−0.0111632
$144$	6.52892	0.544076
$145$	0	0
$146$	19.8938	1.64642
$147$	1.00000	0.0824786
$148$	−43.8759	−3.60658
$149$	11.8665	0.972142	0.486071	−	0.873919i	$-0.338430\pi$
$149$	11.8665	0.972142	0.486071	+	0.873919i	$0.338430\pi$
$150$	0	0
$151$	−11.3248	−0.921601	−0.460800	−	0.887504i	$-0.652438\pi$
$151$	−11.3248	−0.921601	−0.460800	+	0.887504i	$0.652438\pi$
$152$	5.59952	0.454181
$153$	5.05784	0.408902
$154$	−2.52892	−0.203786
$155$	0	0
$156$	−0.586754	−0.0469779
$157$	20.3827	1.62671	0.813357	−	0.581766i	$-0.197638\pi$
$157$	20.3827	1.62671	0.813357	+	0.581766i	$0.197638\pi$
$158$	−35.6974	−2.83993
$159$	4.79085	0.379939
$160$	0	0
$161$	7.05784	0.556235
$162$	−2.52892	−0.198691
$163$	19.3070	1.51224	0.756120	−	0.654432i	$-0.227092\pi$
$163$	19.3070	1.51224	0.756120	+	0.654432i	$0.227092\pi$
$164$	52.0800	4.06677
$165$	0	0
$166$	−3.05784	−0.237334
$167$	−12.6395	−0.978077	−0.489038	−	0.872262i	$-0.662652\pi$
$167$	−12.6395	−0.978077	−0.489038	+	0.872262i	$0.662652\pi$
$168$	−6.05784	−0.467372
$169$	−12.9822	−0.998629
$170$	0	0
$171$	−0.924344	−0.0706864
$172$	13.4405	1.02483
$173$	19.8487	1.50907	0.754534	−	0.656261i	$-0.227863\pi$
$173$	19.8487	1.50907	0.754534	+	0.656261i	$0.227863\pi$
$174$	−9.77808	−0.741274
$175$	0	0
$176$	6.52892	0.492136
$177$	−12.6574	−0.951385
$178$	−39.4049	−2.95352
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	2.67518	0.198845	0.0994223	−	0.995045i	$-0.468301\pi$
$181$	2.67518	0.198845	0.0994223	+	0.995045i	$0.468301\pi$
$182$	0.337590	0.0250238
$183$	6.00000	0.443533
$184$	−42.7552	−3.15196
$185$	0	0
$186$	−7.05784	−0.517506
$187$	5.05784	0.369866
$188$	13.5188	0.985961
$189$	1.00000	0.0727393
$190$	0	0
$191$	3.32482	0.240576	0.120288	−	0.992739i	$-0.461618\pi$
$191$	3.32482	0.240576	0.120288	+	0.992739i	$0.461618\pi$
$192$	−1.94216	−0.140164
$193$	−18.3726	−1.32249	−0.661243	−	0.750172i	$-0.729971\pi$
$193$	−18.3726	−1.32249	−0.661243	+	0.750172i	$0.729971\pi$
$194$	32.3470	2.32238
$195$	0	0
$196$	4.39543	0.313959
$197$	8.11567	0.578218	0.289109	−	0.957296i	$-0.406641\pi$
$197$	8.11567	0.578218	0.289109	+	0.957296i	$0.406641\pi$
$198$	−2.52892	−0.179722
$199$	−6.52387	−0.462465	−0.231232	−	0.972899i	$-0.574276\pi$
$199$	−6.52387	−0.462465	−0.231232	+	0.972899i	$0.574276\pi$
$200$	0	0
$201$	−8.92434	−0.629475
$202$	24.2569	1.70671
$203$	3.86651	0.271376
$204$	22.2313	1.55651
$205$	0	0
$206$	24.9065	1.73532
$207$	7.05784	0.490554
$208$	−0.871558	−0.0604317
$209$	−0.924344	−0.0639382
$210$	0	0
$211$	13.8487	0.953383	0.476691	−	0.879071i	$-0.341836\pi$
$211$	13.8487	0.953383	0.476691	+	0.879071i	$0.341836\pi$
$212$	21.0578	1.44626
$213$	−6.11567	−0.419039
$214$	2.33759	0.159794
$215$	0	0
$216$	−6.05784	−0.412184
$217$	2.79085	0.189455
$218$	21.5562	1.45997
$219$	−7.86651	−0.531569
$220$	0	0
$221$	−0.675180	−0.0454175
$222$	25.2441	1.69427
$223$	24.4983	1.64053	0.820265	−	0.571984i	$-0.193826\pi$
$223$	24.4983	1.64053	0.820265	+	0.571984i	$0.193826\pi$
$224$	−4.39543	−0.293682
$225$	0	0
$226$	−30.6395	−2.03811
$227$	7.73302	0.513258	0.256629	−	0.966510i	$-0.417388\pi$
$227$	7.73302	0.513258	0.256629	+	0.966510i	$0.417388\pi$
$228$	−4.06289	−0.269071
$229$	−4.79085	−0.316588	−0.158294	−	0.987392i	$-0.550599\pi$
$229$	−4.79085	−0.316588	−0.158294	+	0.987392i	$0.550599\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	−23.4227	−1.53777
$233$	9.69738	0.635296	0.317648	−	0.948209i	$-0.397107\pi$
$233$	9.69738	0.635296	0.317648	+	0.948209i	$0.397107\pi$
$234$	0.337590	0.0220690
$235$	0	0
$236$	−55.6345	−3.62150
$237$	14.1157	0.916911
$238$	−12.7909	−0.829108
$239$	23.3070	1.50760	0.753802	−	0.657101i	$-0.228218\pi$
$239$	23.3070	1.50760	0.753802	+	0.657101i	$0.228218\pi$
$240$	0	0
$241$	9.71520	0.625811	0.312905	−	0.949784i	$-0.398698\pi$
$241$	9.71520	0.625811	0.312905	+	0.949784i	$0.398698\pi$
$242$	−2.52892	−0.162565
$243$	1.00000	0.0641500
$244$	26.3726	1.68833
$245$	0	0
$246$	−29.9644	−1.91046
$247$	0.123392	0.00785127
$248$	−16.9065	−1.07357
$249$	1.20915	0.0766266
$250$	0	0
$251$	−15.0400	−0.949317	−0.474659	−	0.880170i	$-0.657428\pi$
$251$	−15.0400	−0.949317	−0.474659	+	0.880170i	$0.657428\pi$
$252$	4.39543	0.276886
$253$	7.05784	0.443722
$254$	−22.5239	−1.41327
$255$	0	0
$256$	−30.7680	−1.92300
$257$	−15.8309	−0.987502	−0.493751	−	0.869603i	$-0.664375\pi$
$257$	−15.8309	−0.987502	−0.493751	+	0.869603i	$0.664375\pi$
$258$	−7.73302	−0.481437
$259$	−9.98218	−0.620262
$260$	0	0
$261$	3.86651	0.239331
$262$	39.6974	2.45251
$263$	3.07566	0.189653	0.0948265	−	0.995494i	$-0.469770\pi$
$263$	3.07566	0.189653	0.0948265	+	0.995494i	$0.469770\pi$
$264$	−6.05784	−0.372834
$265$	0	0
$266$	2.33759	0.143327
$267$	15.5817	0.953585
$268$	−39.2263	−2.39613
$269$	−10.5340	−0.642267	−0.321134	−	0.947034i	$-0.604064\pi$
$269$	−10.5340	−0.642267	−0.321134	+	0.947034i	$0.604064\pi$
$270$	0	0
$271$	−4.92434	−0.299133	−0.149566	−	0.988752i	$-0.547788\pi$
$271$	−4.92434	−0.299133	−0.149566	+	0.988752i	$0.547788\pi$
$272$	33.0222	2.00226
$273$	−0.133492	−0.00807930
$274$	37.0222	2.23659
$275$	0	0
$276$	31.0222	1.86732
$277$	0.151312	0.00909146	0.00454573	−	0.999990i	$-0.498553\pi$
$277$	0.151312	0.00909146	0.00454573	+	0.999990i	$0.498553\pi$
$278$	−47.1634	−2.82867
$279$	2.79085	0.167084
$280$	0	0
$281$	6.51615	0.388721	0.194360	−	0.980930i	$-0.437737\pi$
$281$	6.51615	0.388721	0.194360	+	0.980930i	$0.437737\pi$
$282$	−7.77808	−0.463178
$283$	0.390376	0.0232055	0.0116027	−	0.999933i	$-0.496307\pi$
$283$	0.390376	0.0232055	0.0116027	+	0.999933i	$0.496307\pi$
$284$	−26.8810	−1.59509
$285$	0	0
$286$	0.337590	0.0199621
$287$	11.8487	0.699406
$288$	−4.39543	−0.259003
$289$	8.58170	0.504806
$290$	0	0
$291$	−12.7909	−0.749813
$292$	−34.5767	−2.02345
$293$	−29.4304	−1.71934	−0.859671	−	0.510848i	$-0.829331\pi$
$293$	−29.4304	−1.71934	−0.859671	+	0.510848i	$0.829331\pi$
$294$	−2.52892	−0.147489
$295$	0	0
$296$	60.4704	3.51477
$297$	1.00000	0.0580259
$298$	−30.0094	−1.73840
$299$	−0.942164	−0.0544868
$300$	0	0
$301$	3.05784	0.176251
$302$	28.6395	1.64802
$303$	−9.59180	−0.551035
$304$	−6.03497	−0.346129
$305$	0	0
$306$	−12.7909	−0.731204
$307$	−23.6974	−1.35248	−0.676240	−	0.736681i	$-0.736392\pi$
$307$	−23.6974	−1.35248	−0.676240	+	0.736681i	$0.736392\pi$
$308$	4.39543	0.250453
$309$	−9.84869	−0.560272
$310$	0	0
$311$	−12.2313	−0.693576	−0.346788	−	0.937944i	$-0.612728\pi$
$311$	−12.2313	−0.693576	−0.346788	+	0.937944i	$0.612728\pi$
$312$	0.808672	0.0457820
$313$	19.1735	1.08375	0.541875	−	0.840459i	$-0.317714\pi$
$313$	19.1735	1.08375	0.541875	+	0.840459i	$0.317714\pi$
$314$	−51.5461	−2.90891
$315$	0	0
$316$	62.0444	3.49027
$317$	−29.5562	−1.66004	−0.830020	−	0.557734i	$-0.811671\pi$
$317$	−29.5562	−1.66004	−0.830020	+	0.557734i	$0.811671\pi$
$318$	−12.1157	−0.679413
$319$	3.86651	0.216483
$320$	0	0
$321$	−0.924344	−0.0515919
$322$	−17.8487	−0.994668
$323$	−4.67518	−0.260134
$324$	4.39543	0.244190
$325$	0	0
$326$	−48.8258	−2.70421
$327$	−8.52387	−0.471371
$328$	−71.7774	−3.96324
$329$	3.07566	0.169566
$330$	0	0
$331$	18.6496	1.02508	0.512538	−	0.858664i	$-0.328705\pi$
$331$	18.6496	1.02508	0.512538	+	0.858664i	$0.328705\pi$
$332$	5.31472	0.291683
$333$	−9.98218	−0.547020
$334$	31.9644	1.74901
$335$	0	0
$336$	6.52892	0.356182
$337$	21.0222	1.14515	0.572576	−	0.819852i	$-0.305944\pi$
$337$	21.0222	1.14515	0.572576	+	0.819852i	$0.305944\pi$
$338$	32.8309	1.78576
$339$	12.1157	0.658033
$340$	0	0
$341$	2.79085	0.151133
$342$	2.33759	0.126402
$343$	1.00000	0.0539949
$344$	−18.5239	−0.998740
$345$	0	0
$346$	−50.1957	−2.69854
$347$	−23.1634	−1.24348	−0.621738	−	0.783225i	$-0.713573\pi$
$347$	−23.1634	−1.24348	−0.621738	+	0.783225i	$0.713573\pi$
$348$	16.9950	0.911025
$349$	−28.0979	−1.50404	−0.752022	−	0.659138i	$-0.770921\pi$
$349$	−28.0979	−1.50404	−0.752022	+	0.659138i	$0.770921\pi$
$350$	0	0
$351$	−0.133492	−0.00712527
$352$	−4.39543	−0.234277
$353$	33.4126	1.77837	0.889186	−	0.457546i	$-0.151272\pi$
$353$	33.4126	1.77837	0.889186	+	0.457546i	$0.151272\pi$
$354$	32.0094	1.70128
$355$	0	0
$356$	68.4882	3.62987
$357$	5.05784	0.267689
$358$	30.3470	1.60389
$359$	13.5817	0.716815	0.358407	−	0.933565i	$-0.383320\pi$
$359$	13.5817	0.716815	0.358407	+	0.933565i	$0.383320\pi$
$360$	0	0
$361$	−18.1456	−0.955031
$362$	−6.76531	−0.355577
$363$	1.00000	0.0524864
$364$	−0.586754	−0.0307543
$365$	0	0
$366$	−15.1735	−0.793132
$367$	3.73302	0.194862	0.0974309	−	0.995242i	$-0.468937\pi$
$367$	3.73302	0.194862	0.0974309	+	0.995242i	$0.468937\pi$
$368$	46.0800	2.40209
$369$	11.8487	0.616818
$370$	0	0
$371$	4.79085	0.248729
$372$	12.2670	0.636013
$373$	−6.94216	−0.359452	−0.179726	−	0.983717i	$-0.557521\pi$
$373$	−6.94216	−0.359452	−0.179726	+	0.983717i	$0.557521\pi$
$374$	−12.7909	−0.661399
$375$	0	0
$376$	−18.6318	−0.960863
$377$	−0.516148	−0.0265830
$378$	−2.52892	−0.130074
$379$	−0.390376	−0.0200523	−0.0100261	−	0.999950i	$-0.503191\pi$
$379$	−0.390376	−0.0200523	−0.0100261	+	0.999950i	$0.503191\pi$
$380$	0	0
$381$	8.90652	0.456295
$382$	−8.40820	−0.430201
$383$	0.533968	0.0272845	0.0136422	−	0.999907i	$-0.495657\pi$
$383$	0.533968	0.0272845	0.0136422	+	0.999907i	$0.495657\pi$
$384$	13.7024	0.699249
$385$	0	0
$386$	46.4627	2.36489
$387$	3.05784	0.155439
$388$	−56.2212	−2.85420
$389$	−10.4983	−0.532286	−0.266143	−	0.963934i	$-0.585749\pi$
$389$	−10.4983	−0.532286	−0.266143	+	0.963934i	$0.585749\pi$
$390$	0	0
$391$	35.6974	1.80529
$392$	−6.05784	−0.305967
$393$	−15.6974	−0.791828
$394$	−20.5239	−1.03398
$395$	0	0
$396$	4.39543	0.220878
$397$	20.7552	1.04167	0.520837	−	0.853656i	$-0.325620\pi$
$397$	20.7552	1.04167	0.520837	+	0.853656i	$0.325620\pi$
$398$	16.4983	0.826986
$399$	−0.924344	−0.0462751
$400$	0	0
$401$	21.3248	1.06491	0.532455	−	0.846458i	$-0.321269\pi$
$401$	21.3248	1.06491	0.532455	+	0.846458i	$0.321269\pi$
$402$	22.5689	1.12564
$403$	−0.372556	−0.0185583
$404$	−42.1601	−2.09754
$405$	0	0
$406$	−9.77808	−0.485278
$407$	−9.98218	−0.494798
$408$	−30.6395	−1.51688
$409$	33.9287	1.67767	0.838834	−	0.544388i	$-0.183238\pi$
$409$	33.9287	1.67767	0.838834	+	0.544388i	$0.183238\pi$
$410$	0	0
$411$	−14.6395	−0.722115
$412$	−43.2892	−2.13270
$413$	−12.6574	−0.622828
$414$	−17.8487	−0.877215
$415$	0	0
$416$	0.586754	0.0287680
$417$	18.6496	0.913277
$418$	2.33759	0.114335
$419$	−9.99228	−0.488155	−0.244077	−	0.969756i	$-0.578485\pi$
$419$	−9.99228	−0.488155	−0.244077	+	0.969756i	$0.578485\pi$
$420$	0	0
$421$	−29.9822	−1.46124	−0.730621	−	0.682783i	$-0.760769\pi$
$421$	−29.9822	−1.46124	−0.730621	+	0.682783i	$0.760769\pi$
$422$	−35.0222	−1.70485
$423$	3.07566	0.149544
$424$	−29.0222	−1.40944
$425$	0	0
$426$	15.4660	0.749332
$427$	6.00000	0.290360
$428$	−4.06289	−0.196387
$429$	−0.133492	−0.00644505
$430$	0	0
$431$	−2.54169	−0.122429	−0.0612144	−	0.998125i	$-0.519497\pi$
$431$	−2.54169	−0.122429	−0.0612144	+	0.998125i	$0.519497\pi$
$432$	6.52892	0.314123
$433$	−20.3827	−0.979528	−0.489764	−	0.871855i	$-0.662917\pi$
$433$	−20.3827	−0.979528	−0.489764	+	0.871855i	$0.662917\pi$
$434$	−7.05784	−0.338787
$435$	0	0
$436$	−37.4660	−1.79430
$437$	−6.52387	−0.312079
$438$	19.8938	0.950560
$439$	−5.45831	−0.260511	−0.130256	−	0.991480i	$-0.541580\pi$
$439$	−5.45831	−0.260511	−0.130256	+	0.991480i	$0.541580\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	1.70748	0.0812163
$443$	10.6496	0.505980	0.252990	−	0.967469i	$-0.418586\pi$
$443$	10.6496	0.505980	0.252990	+	0.967469i	$0.418586\pi$
$444$	−43.8759	−2.08226
$445$	0	0
$446$	−61.9543	−2.93362
$447$	11.8665	0.561267
$448$	−1.94216	−0.0917586
$449$	27.0323	1.27573	0.637866	−	0.770147i	$-0.279817\pi$
$449$	27.0323	1.27573	0.637866	+	0.770147i	$0.279817\pi$
$450$	0	0
$451$	11.8487	0.557933
$452$	53.2535	2.50484
$453$	−11.3248	−0.532086
$454$	−19.5562	−0.917816
$455$	0	0
$456$	5.59952	0.262222
$457$	−4.79085	−0.224107	−0.112053	−	0.993702i	$-0.535743\pi$
$457$	−4.79085	−0.224107	−0.112053	+	0.993702i	$0.535743\pi$
$458$	12.1157	0.566128
$459$	5.05784	0.236080
$460$	0	0
$461$	22.4983	1.04785	0.523926	−	0.851764i	$-0.324467\pi$
$461$	22.4983	1.04785	0.523926	+	0.851764i	$0.324467\pi$
$462$	−2.52892	−0.117656
$463$	−25.6897	−1.19390	−0.596950	−	0.802279i	$-0.703621\pi$
$463$	−25.6897	−1.19390	−0.596950	+	0.802279i	$0.703621\pi$
$464$	25.2441	1.17193
$465$	0	0
$466$	−24.5239	−1.13605
$467$	−4.62172	−0.213868	−0.106934	−	0.994266i	$-0.534103\pi$
$467$	−4.62172	−0.213868	−0.106934	+	0.994266i	$0.534103\pi$
$468$	−0.586754	−0.0271227
$469$	−8.92434	−0.412088
$470$	0	0
$471$	20.3827	0.939183
$472$	76.6762	3.52931
$473$	3.05784	0.140599
$474$	−35.6974	−1.63963
$475$	0	0
$476$	22.2313	1.01897
$477$	4.79085	0.219358
$478$	−58.9415	−2.69592
$479$	−0.372556	−0.0170225	−0.00851126	−	0.999964i	$-0.502709\pi$
$479$	−0.372556	−0.0170225	−0.00851126	+	0.999964i	$0.502709\pi$
$480$	0	0
$481$	1.33254	0.0607586
$482$	−24.5689	−1.11908
$483$	7.05784	0.321143
$484$	4.39543	0.199792
$485$	0	0
$486$	−2.52892	−0.114714
$487$	−4.30262	−0.194971	−0.0974853	−	0.995237i	$-0.531080\pi$
$487$	−4.30262	−0.194971	−0.0974853	+	0.995237i	$0.531080\pi$
$488$	−36.3470	−1.64535
$489$	19.3070	0.873093
$490$	0	0
$491$	25.4583	1.14892	0.574459	−	0.818534i	$-0.305213\pi$
$491$	25.4583	1.14892	0.574459	+	0.818534i	$0.305213\pi$
$492$	52.0800	2.34795
$493$	19.5562	0.880765
$494$	−0.312049	−0.0140398
$495$	0	0
$496$	18.2212	0.818158
$497$	−6.11567	−0.274325
$498$	−3.05784	−0.137025
$499$	−8.39038	−0.375605	−0.187802	−	0.982207i	$-0.560136\pi$
$499$	−8.39038	−0.375605	−0.187802	+	0.982207i	$0.560136\pi$
$500$	0	0
$501$	−12.6395	−0.564693
$502$	38.0350	1.69758
$503$	26.7552	1.19296	0.596478	−	0.802629i	$-0.296566\pi$
$503$	26.7552	1.19296	0.596478	+	0.802629i	$0.296566\pi$
$504$	−6.05784	−0.269837
$505$	0	0
$506$	−17.8487	−0.793471
$507$	−12.9822	−0.576559
$508$	39.1480	1.73691
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−7.86651	−0.347994
$512$	50.4049	2.22760
$513$	−0.924344	−0.0408108
$514$	40.0350	1.76587
$515$	0	0
$516$	13.4405	0.591685
$517$	3.07566	0.135267
$518$	25.2441	1.10916
$519$	19.8487	0.871261
$520$	0	0
$521$	−1.44821	−0.0634473	−0.0317237	−	0.999497i	$-0.510100\pi$
$521$	−1.44821	−0.0634473	−0.0317237	+	0.999497i	$0.510100\pi$
$522$	−9.77808	−0.427975
$523$	−23.5740	−1.03082	−0.515409	−	0.856944i	$-0.672360\pi$
$523$	−23.5740	−1.03082	−0.515409	+	0.856944i	$0.672360\pi$
$524$	−68.9967	−3.01413
$525$	0	0
$526$	−7.77808	−0.339140
$527$	14.1157	0.614888
$528$	6.52892	0.284135
$529$	26.8130	1.16578
$530$	0	0
$531$	−12.6574	−0.549283
$532$	−4.06289	−0.176148
$533$	−1.58170	−0.0685112
$534$	−39.4049	−1.70521
$535$	0	0
$536$	54.0622	2.33513
$537$	−12.0000	−0.517838
$538$	26.6395	1.14851
$539$	1.00000	0.0430730
$540$	0	0
$541$	−18.4983	−0.795305	−0.397653	−	0.917536i	$-0.630175\pi$
$541$	−18.4983	−0.795305	−0.397653	+	0.917536i	$0.630175\pi$
$542$	12.4533	0.534913
$543$	2.67518	0.114803
$544$	−22.2313	−0.953161
$545$	0	0
$546$	0.337590	0.0144475
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−64.3470	−2.74877
$549$	6.00000	0.256074
$550$	0	0
$551$	−3.57398	−0.152257
$552$	−42.7552	−1.81978
$553$	14.1157	0.600259
$554$	−0.382656	−0.0162575
$555$	0	0
$556$	81.9731	3.47643
$557$	−1.98218	−0.0839877	−0.0419938	−	0.999118i	$-0.513371\pi$
$557$	−1.98218	−0.0839877	−0.0419938	+	0.999118i	$0.513371\pi$
$558$	−7.05784	−0.298782
$559$	−0.408196	−0.0172649
$560$	0	0
$561$	5.05784	0.213542
$562$	−16.4788	−0.695116
$563$	−1.98990	−0.0838643	−0.0419322	−	0.999120i	$-0.513351\pi$
$563$	−1.98990	−0.0838643	−0.0419322	+	0.999120i	$0.513351\pi$
$564$	13.5188	0.569245
$565$	0	0
$566$	−0.987230	−0.0414964
$567$	1.00000	0.0419961
$568$	37.0477	1.55449
$569$	−14.5340	−0.609296	−0.304648	−	0.952465i	$-0.598539\pi$
$569$	−14.5340	−0.609296	−0.304648	+	0.952465i	$0.598539\pi$
$570$	0	0
$571$	−29.2791	−1.22529	−0.612646	−	0.790358i	$-0.709895\pi$
$571$	−29.2791	−1.22529	−0.612646	+	0.790358i	$0.709895\pi$
$572$	−0.586754	−0.0245334
$573$	3.32482	0.138896
$574$	−29.9644	−1.25069
$575$	0	0
$576$	−1.94216	−0.0809235
$577$	5.59180	0.232790	0.116395	−	0.993203i	$-0.462866\pi$
$577$	5.59180	0.232790	0.116395	+	0.993203i	$0.462866\pi$
$578$	−21.7024	−0.902702
$579$	−18.3726	−0.763537
$580$	0	0
$581$	1.20915	0.0501639
$582$	32.3470	1.34083
$583$	4.79085	0.198417
$584$	47.6540	1.97194
$585$	0	0
$586$	74.4270	3.07455
$587$	2.80867	0.115926	0.0579632	−	0.998319i	$-0.481539\pi$
$587$	2.80867	0.115926	0.0579632	+	0.998319i	$0.481539\pi$
$588$	4.39543	0.181264
$589$	−2.57971	−0.106295
$590$	0	0
$591$	8.11567	0.333834
$592$	−65.1728	−2.67859
$593$	42.1957	1.73277	0.866385	−	0.499377i	$-0.166438\pi$
$593$	42.1957	1.73277	0.866385	+	0.499377i	$0.166438\pi$
$594$	−2.52892	−0.103763
$595$	0	0
$596$	52.1584	2.13649
$597$	−6.52387	−0.267004
$598$	2.38266	0.0974340
$599$	34.3470	1.40338	0.701691	−	0.712482i	$-0.252429\pi$
$599$	34.3470	1.40338	0.701691	+	0.712482i	$0.252429\pi$
$600$	0	0
$601$	5.98218	0.244018	0.122009	−	0.992529i	$-0.461066\pi$
$601$	5.98218	0.244018	0.122009	+	0.992529i	$0.461066\pi$
$602$	−7.73302	−0.315174
$603$	−8.92434	−0.363427
$604$	−49.7774	−2.02541
$605$	0	0
$606$	24.2569	0.985369
$607$	−8.12339	−0.329718	−0.164859	−	0.986317i	$-0.552717\pi$
$607$	−8.12339	−0.329718	−0.164859	+	0.986317i	$0.552717\pi$
$608$	4.06289	0.164772
$609$	3.86651	0.156679
$610$	0	0
$611$	−0.410575	−0.0166101
$612$	22.2313	0.898649
$613$	16.5239	0.667393	0.333696	−	0.942681i	$-0.391704\pi$
$613$	16.5239	0.667393	0.333696	+	0.942681i	$0.391704\pi$
$614$	59.9287	2.41853
$615$	0	0
$616$	−6.05784	−0.244077
$617$	−11.8843	−0.478445	−0.239223	−	0.970965i	$-0.576893\pi$
$617$	−11.8843	−0.478445	−0.239223	+	0.970965i	$0.576893\pi$
$618$	24.9065	1.00189
$619$	43.0222	1.72921	0.864604	−	0.502454i	$-0.167569\pi$
$619$	43.0222	1.72921	0.864604	+	0.502454i	$0.167569\pi$
$620$	0	0
$621$	7.05784	0.283221
$622$	30.9321	1.24026
$623$	15.5817	0.624268
$624$	−0.871558	−0.0348902
$625$	0	0
$626$	−48.4882	−1.93798
$627$	−0.924344	−0.0369147
$628$	89.5905	3.57505
$629$	−50.4882	−2.01310
$630$	0	0
$631$	13.5817	0.540679	0.270340	−	0.962765i	$-0.412864\pi$
$631$	13.5817	0.540679	0.270340	+	0.962765i	$0.412864\pi$
$632$	−85.5104	−3.40142
$633$	13.8487	0.550436
$634$	74.7451	2.96851
$635$	0	0
$636$	21.0578	0.834998
$637$	−0.133492	−0.00528914
$638$	−9.77808	−0.387118
$639$	−6.11567	−0.241932
$640$	0	0
$641$	36.7552	1.45174	0.725872	−	0.687830i	$-0.241437\pi$
$641$	36.7552	1.45174	0.725872	+	0.687830i	$0.241437\pi$
$642$	2.33759	0.0922573
$643$	44.3369	1.74848	0.874239	−	0.485496i	$-0.161361\pi$
$643$	44.3369	1.74848	0.874239	+	0.485496i	$0.161361\pi$
$644$	31.0222	1.22245
$645$	0	0
$646$	11.8231	0.465176
$647$	17.9721	0.706555	0.353278	−	0.935519i	$-0.385067\pi$
$647$	17.9721	0.706555	0.353278	+	0.935519i	$0.385067\pi$
$648$	−6.05784	−0.237974
$649$	−12.6574	−0.496845
$650$	0	0
$651$	2.79085	0.109382
$652$	84.8625	3.32347
$653$	22.1957	0.868585	0.434292	−	0.900772i	$-0.356998\pi$
$653$	22.1957	0.868585	0.434292	+	0.900772i	$0.356998\pi$
$654$	21.5562	0.842913
$655$	0	0
$656$	77.3591	3.02037
$657$	−7.86651	−0.306902
$658$	−7.77808	−0.303221
$659$	9.99228	0.389244	0.194622	−	0.980878i	$-0.437652\pi$
$659$	9.99228	0.389244	0.194622	+	0.980878i	$0.437652\pi$
$660$	0	0
$661$	27.9543	1.08729	0.543647	−	0.839314i	$-0.317043\pi$
$661$	27.9543	1.08729	0.543647	+	0.839314i	$0.317043\pi$
$662$	−47.1634	−1.83306
$663$	−0.675180	−0.0262218
$664$	−7.32482	−0.284258
$665$	0	0
$666$	25.2441	0.978190
$667$	27.2892	1.05664
$668$	−55.5562	−2.14953
$669$	24.4983	0.947160
$670$	0	0
$671$	6.00000	0.231627
$672$	−4.39543	−0.169557
$673$	43.4203	1.67373	0.836865	−	0.547410i	$-0.184386\pi$
$673$	43.4203	1.67373	0.836865	+	0.547410i	$0.184386\pi$
$674$	−53.1634	−2.04778
$675$	0	0
$676$	−57.0622	−2.19470
$677$	12.3470	0.474534	0.237267	−	0.971444i	$-0.423748\pi$
$677$	12.3470	0.474534	0.237267	+	0.971444i	$0.423748\pi$
$678$	−30.6395	−1.17670
$679$	−12.7909	−0.490868
$680$	0	0
$681$	7.73302	0.296330
$682$	−7.05784	−0.270259
$683$	27.8386	1.06521	0.532607	−	0.846363i	$-0.321212\pi$
$683$	27.8386	1.06521	0.532607	+	0.846363i	$0.321212\pi$
$684$	−4.06289	−0.155348
$685$	0	0
$686$	−2.52892	−0.0965545
$687$	−4.79085	−0.182782
$688$	19.9644	0.761134
$689$	−0.639540	−0.0243645
$690$	0	0
$691$	−16.8010	−0.639138	−0.319569	−	0.947563i	$-0.603538\pi$
$691$	−16.8010	−0.639138	−0.319569	+	0.947563i	$0.603538\pi$
$692$	87.2434	3.31650
$693$	1.00000	0.0379869
$694$	58.5784	2.22360
$695$	0	0
$696$	−23.4227	−0.887834
$697$	59.9287	2.26996
$698$	71.0572	2.68955
$699$	9.69738	0.366788
$700$	0	0
$701$	9.16341	0.346097	0.173049	−	0.984913i	$-0.444638\pi$
$701$	9.16341	0.346097	0.173049	+	0.984913i	$0.444638\pi$
$702$	0.337590	0.0127415
$703$	9.22697	0.348002
$704$	−1.94216	−0.0731981
$705$	0	0
$706$	−84.4977	−3.18011
$707$	−9.59180	−0.360737
$708$	−55.6345	−2.09087
$709$	−34.7475	−1.30497	−0.652485	−	0.757802i	$-0.726273\pi$
$709$	−34.7475	−1.30497	−0.652485	+	0.757802i	$0.726273\pi$
$710$	0	0
$711$	14.1157	0.529379
$712$	−94.3914	−3.53747
$713$	19.6974	0.737673
$714$	−12.7909	−0.478686
$715$	0	0
$716$	−52.7451	−1.97118
$717$	23.3070	0.870416
$718$	−34.3470	−1.28182
$719$	−25.1557	−0.938149	−0.469074	−	0.883159i	$-0.655412\pi$
$719$	−25.1557	−0.938149	−0.469074	+	0.883159i	$0.655412\pi$
$720$	0	0
$721$	−9.84869	−0.366784
$722$	45.8887	1.70780
$723$	9.71520	0.361312
$724$	11.7586	0.437003
$725$	0	0
$726$	−2.52892	−0.0938569
$727$	23.5918	0.874972	0.437486	−	0.899225i	$-0.355869\pi$
$727$	23.5918	0.874972	0.437486	+	0.899225i	$0.355869\pi$
$728$	0.808672	0.0299714
$729$	1.00000	0.0370370
$730$	0	0
$731$	15.4660	0.572032
$732$	26.3726	0.974758
$733$	−21.9287	−0.809956	−0.404978	−	0.914326i	$-0.632721\pi$
$733$	−21.9287	−0.809956	−0.404978	+	0.914326i	$0.632721\pi$
$734$	−9.44049	−0.348455
$735$	0	0
$736$	−31.0222	−1.14349
$737$	−8.92434	−0.328732
$738$	−29.9644	−1.10300
$739$	35.0578	1.28962	0.644812	−	0.764341i	$-0.276936\pi$
$739$	35.0578	1.28962	0.644812	+	0.764341i	$0.276936\pi$
$740$	0	0
$741$	0.123392	0.00453294
$742$	−12.1157	−0.444780
$743$	31.3070	1.14854	0.574271	−	0.818665i	$-0.305285\pi$
$743$	31.3070	1.14854	0.574271	+	0.818665i	$0.305285\pi$
$744$	−16.9065	−0.619823
$745$	0	0
$746$	17.5562	0.642777
$747$	1.20915	0.0442404
$748$	22.2313	0.812858
$749$	−0.924344	−0.0337748
$750$	0	0
$751$	−4.42602	−0.161508	−0.0807538	−	0.996734i	$-0.525733\pi$
$751$	−4.42602	−0.161508	−0.0807538	+	0.996734i	$0.525733\pi$
$752$	20.0807	0.732268
$753$	−15.0400	−0.548089
$754$	1.30529	0.0475360
$755$	0	0
$756$	4.39543	0.159860
$757$	−2.51615	−0.0914509	−0.0457255	−	0.998954i	$-0.514560\pi$
$757$	−2.51615	−0.0914509	−0.0457255	+	0.998954i	$0.514560\pi$
$758$	0.987230	0.0358578
$759$	7.05784	0.256183
$760$	0	0
$761$	13.7330	0.497821	0.248911	−	0.968526i	$-0.419927\pi$
$761$	13.7330	0.497821	0.248911	+	0.968526i	$0.419927\pi$
$762$	−22.5239	−0.815954
$763$	−8.52387	−0.308585
$764$	14.6140	0.528716
$765$	0	0
$766$	−1.35036	−0.0487905
$767$	1.68966	0.0610099
$768$	−30.7680	−1.11024
$769$	46.4805	1.67613	0.838065	−	0.545570i	$-0.183687\pi$
$769$	46.4805	1.67613	0.838065	+	0.545570i	$0.183687\pi$
$770$	0	0
$771$	−15.8309	−0.570135
$772$	−80.7552	−2.90644
$773$	45.9822	1.65386	0.826932	−	0.562302i	$-0.190084\pi$
$773$	45.9822	1.65386	0.826932	+	0.562302i	$0.190084\pi$
$774$	−7.73302	−0.277958
$775$	0	0
$776$	77.4849	2.78155
$777$	−9.98218	−0.358109
$778$	26.5494	0.951842
$779$	−10.9523	−0.392406
$780$	0	0
$781$	−6.11567	−0.218836
$782$	−90.2757	−3.22825
$783$	3.86651	0.138178
$784$	6.52892	0.233176
$785$	0	0
$786$	39.6974	1.41596
$787$	−40.9243	−1.45880	−0.729398	−	0.684090i	$-0.760200\pi$
$787$	−40.9243	−1.45880	−0.729398	+	0.684090i	$0.760200\pi$
$788$	35.6718	1.27076
$789$	3.07566	0.109496
$790$	0	0
$791$	12.1157	0.430784
$792$	−6.05784	−0.215256
$793$	−0.800952	−0.0284426
$794$	−52.4882	−1.86274
$795$	0	0
$796$	−28.6752	−1.01636
$797$	28.8786	1.02293	0.511466	−	0.859303i	$-0.329102\pi$
$797$	28.8786	1.02293	0.511466	+	0.859303i	$0.329102\pi$
$798$	2.33759	0.0827498
$799$	15.5562	0.550338
$800$	0	0
$801$	15.5817	0.550552
$802$	−53.9287	−1.90429
$803$	−7.86651	−0.277603
$804$	−39.2263	−1.38340
$805$	0	0
$806$	0.942164	0.0331863
$807$	−10.5340	−0.370813
$808$	58.1056	2.04415
$809$	6.51615	0.229096	0.114548	−	0.993418i	$-0.463458\pi$
$809$	6.51615	0.229096	0.114548	+	0.993418i	$0.463458\pi$
$810$	0	0
$811$	38.5060	1.35213	0.676065	−	0.736842i	$-0.263684\pi$
$811$	38.5060	1.35213	0.676065	+	0.736842i	$0.263684\pi$
$812$	16.9950	0.596406
$813$	−4.92434	−0.172704
$814$	25.2441	0.884806
$815$	0	0
$816$	33.0222	1.15601
$817$	−2.82649	−0.0988864
$818$	−85.8029	−3.00003
$819$	−0.133492	−0.00466459
$820$	0	0
$821$	−37.3769	−1.30446	−0.652232	−	0.758019i	$-0.726167\pi$
$821$	−37.3769	−1.30446	−0.652232	+	0.758019i	$0.726167\pi$
$822$	37.0222	1.29130
$823$	−43.0400	−1.50028	−0.750140	−	0.661279i	$-0.770014\pi$
$823$	−43.0400	−1.50028	−0.750140	+	0.661279i	$0.770014\pi$
$824$	59.6617	2.07842
$825$	0	0
$826$	32.0094	1.11375
$827$	−11.3426	−0.394422	−0.197211	−	0.980361i	$-0.563188\pi$
$827$	−11.3426	−0.394422	−0.197211	+	0.980361i	$0.563188\pi$
$828$	31.0222	1.07810
$829$	14.4983	0.503548	0.251774	−	0.967786i	$-0.418986\pi$
$829$	14.4983	0.503548	0.251774	+	0.967786i	$0.418986\pi$
$830$	0	0
$831$	0.151312	0.00524896
$832$	0.259263	0.00898833
$833$	5.05784	0.175244
$834$	−47.1634	−1.63314
$835$	0	0
$836$	−4.06289	−0.140518
$837$	2.79085	0.0964660
$838$	25.2697	0.872926
$839$	−3.60962	−0.124618	−0.0623090	−	0.998057i	$-0.519846\pi$
$839$	−3.60962	−0.124618	−0.0623090	+	0.998057i	$0.519846\pi$
$840$	0	0
$841$	−14.0501	−0.484487
$842$	75.8225	2.61301
$843$	6.51615	0.224428
$844$	60.8709	2.09526
$845$	0	0
$846$	−7.77808	−0.267416
$847$	1.00000	0.0343604
$848$	31.2791	1.07413
$849$	0.390376	0.0133977
$850$	0	0
$851$	−70.4526	−2.41508
$852$	−26.8810	−0.920927
$853$	24.6496	0.843988	0.421994	−	0.906599i	$-0.361330\pi$
$853$	24.6496	0.843988	0.421994	+	0.906599i	$0.361330\pi$
$854$	−15.1735	−0.519227
$855$	0	0
$856$	5.59952	0.191388
$857$	−17.9287	−0.612433	−0.306217	−	0.951962i	$-0.599063\pi$
$857$	−17.9287	−0.612433	−0.306217	+	0.951962i	$0.599063\pi$
$858$	0.337590	0.0115251
$859$	−15.6974	−0.535588	−0.267794	−	0.963476i	$-0.586295\pi$
$859$	−15.6974	−0.535588	−0.267794	+	0.963476i	$0.586295\pi$
$860$	0	0
$861$	11.8487	0.403802
$862$	6.42772	0.218929
$863$	15.0578	0.512575	0.256287	−	0.966601i	$-0.417501\pi$
$863$	15.0578	0.512575	0.256287	+	0.966601i	$0.417501\pi$
$864$	−4.39543	−0.149535
$865$	0	0
$866$	51.5461	1.75161
$867$	8.58170	0.291450
$868$	12.2670	0.416369
$869$	14.1157	0.478841
$870$	0	0
$871$	1.19133	0.0403666
$872$	51.6362	1.74862
$873$	−12.7909	−0.432905
$874$	16.4983	0.558064
$875$	0	0
$876$	−34.5767	−1.16824
$877$	43.8487	1.48066	0.740332	−	0.672241i	$-0.234668\pi$
$877$	43.8487	1.48066	0.740332	+	0.672241i	$0.234668\pi$
$878$	13.8036	0.465850
$879$	−29.4304	−0.992662
$880$	0	0
$881$	11.0656	0.372808	0.186404	−	0.982473i	$-0.440317\pi$
$881$	11.0656	0.372808	0.186404	+	0.982473i	$0.440317\pi$
$882$	−2.52892	−0.0851531
$883$	−5.72530	−0.192672	−0.0963358	−	0.995349i	$-0.530712\pi$
$883$	−5.72530	−0.192672	−0.0963358	+	0.995349i	$0.530712\pi$
$884$	−2.96770	−0.0998147
$885$	0	0
$886$	−26.9321	−0.904800
$887$	−15.0935	−0.506789	−0.253395	−	0.967363i	$-0.581547\pi$
$887$	−15.0935	−0.506789	−0.253395	+	0.967363i	$0.581547\pi$
$888$	60.4704	2.02925
$889$	8.90652	0.298715
$890$	0	0
$891$	1.00000	0.0335013
$892$	107.681	3.60541
$893$	−2.84296	−0.0951362
$894$	−30.0094	−1.00367
$895$	0	0
$896$	13.7024	0.457766
$897$	−0.942164	−0.0314579
$898$	−68.3625	−2.28128
$899$	10.7909	0.359895
$900$	0	0
$901$	24.2313	0.807263
$902$	−29.9644	−0.997704
$903$	3.05784	0.101758
$904$	−73.3948	−2.44107
$905$	0	0
$906$	28.6395	0.951485
$907$	−41.2993	−1.37132	−0.685660	−	0.727922i	$-0.740486\pi$
$907$	−41.2993	−1.37132	−0.685660	+	0.727922i	$0.740486\pi$
$908$	33.9899	1.12799
$909$	−9.59180	−0.318140
$910$	0	0
$911$	3.45059	0.114323	0.0571616	−	0.998365i	$-0.481795\pi$
$911$	3.45059	0.114323	0.0571616	+	0.998365i	$0.481795\pi$
$912$	−6.03497	−0.199838
$913$	1.20915	0.0400170
$914$	12.1157	0.400751
$915$	0	0
$916$	−21.0578	−0.695770
$917$	−15.6974	−0.518373
$918$	−12.7909	−0.422161
$919$	−10.8265	−0.357133	−0.178567	−	0.983928i	$-0.557146\pi$
$919$	−10.8265	−0.357133	−0.178567	+	0.983928i	$0.557146\pi$
$920$	0	0
$921$	−23.6974	−0.780855
$922$	−56.8964	−1.87378
$923$	0.816393	0.0268719
$924$	4.39543	0.144599
$925$	0	0
$926$	64.9670	2.13495
$927$	−9.84869	−0.323473
$928$	−16.9950	−0.557887
$929$	17.4684	0.573120	0.286560	−	0.958062i	$-0.407488\pi$
$929$	17.4684	0.573120	0.286560	+	0.958062i	$0.407488\pi$
$930$	0	0
$931$	−0.924344	−0.0302942
$932$	42.6241	1.39620
$933$	−12.2313	−0.400436
$934$	11.6880	0.382441
$935$	0	0
$936$	0.808672	0.0264323
$937$	−10.5340	−0.344130	−0.172065	−	0.985086i	$-0.555044\pi$
$937$	−10.5340	−0.344130	−0.172065	+	0.985086i	$0.555044\pi$
$938$	22.5689	0.736902
$939$	19.1735	0.625704
$940$	0	0
$941$	−16.1513	−0.526518	−0.263259	−	0.964725i	$-0.584797\pi$
$941$	−16.1513	−0.526518	−0.263259	+	0.964725i	$0.584797\pi$
$942$	−51.5461	−1.67946
$943$	83.6261	2.72324
$944$	−82.6389	−2.68967
$945$	0	0
$946$	−7.73302	−0.251422
$947$	6.91662	0.224760	0.112380	−	0.993665i	$-0.464153\pi$
$947$	6.91662	0.224760	0.112380	+	0.993665i	$0.464153\pi$
$948$	62.0444	2.01511
$949$	1.05012	0.0340882
$950$	0	0
$951$	−29.5562	−0.958424
$952$	−30.6395	−0.993033
$953$	16.1335	0.522615	0.261308	−	0.965256i	$-0.415846\pi$
$953$	16.1335	0.522615	0.261308	+	0.965256i	$0.415846\pi$
$954$	−12.1157	−0.392259
$955$	0	0
$956$	102.444	3.31328
$957$	3.86651	0.124986
$958$	0.942164	0.0304399
$959$	−14.6395	−0.472735
$960$	0	0
$961$	−23.2111	−0.748747
$962$	−3.36989	−0.108649
$963$	−0.924344	−0.0297866
$964$	42.7024	1.37535
$965$	0	0
$966$	−17.8487	−0.574272
$967$	−17.8487	−0.573975	−0.286988	−	0.957934i	$-0.592654\pi$
$967$	−17.8487	−0.573975	−0.286988	+	0.957934i	$0.592654\pi$
$968$	−6.05784	−0.194706
$969$	−4.67518	−0.150188
$970$	0	0
$971$	33.9566	1.08972	0.544860	−	0.838527i	$-0.316583\pi$
$971$	33.9566	1.08972	0.544860	+	0.838527i	$0.316583\pi$
$972$	4.39543	0.140983
$973$	18.6496	0.597880
$974$	10.8810	0.348649
$975$	0	0
$976$	39.1735	1.25391
$977$	−41.0222	−1.31242	−0.656208	−	0.754580i	$-0.727841\pi$
$977$	−41.0222	−1.31242	−0.656208	+	0.754580i	$0.727841\pi$
$978$	−48.8258	−1.56128
$979$	15.5817	0.497993
$980$	0	0
$981$	−8.52387	−0.272146
$982$	−64.3820	−2.05451
$983$	−42.3470	−1.35066	−0.675330	−	0.737516i	$-0.735999\pi$
$983$	−42.3470	−1.35066	−0.675330	+	0.737516i	$0.735999\pi$
$984$	−71.7774	−2.28818
$985$	0	0
$986$	−49.4559	−1.57500
$987$	3.07566	0.0978992
$988$	0.542362	0.0172548
$989$	21.5817	0.686258
$990$	0	0
$991$	50.9687	1.61908	0.809538	−	0.587068i	$-0.199718\pi$
$991$	50.9687	1.61908	0.809538	+	0.587068i	$0.199718\pi$
$992$	−12.2670	−0.389477
$993$	18.6496	0.591828
$994$	15.4660	0.490553
$995$	0	0
$996$	5.31472	0.168403
$997$	32.8608	1.04071	0.520356	−	0.853950i	$-0.325799\pi$
$997$	32.8608	1.04071	0.520356	+	0.853950i	$0.325799\pi$
$998$	21.2186	0.671662
$999$	−9.98218	−0.315822

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.bw.1.1		3
5.4	even	2		231.2.a.d.1.3	✓	3
15.14	odd	2		693.2.a.m.1.1		3
20.19	odd	2		3696.2.a.bp.1.2		3
35.34	odd	2		1617.2.a.s.1.3		3
55.54	odd	2		2541.2.a.bi.1.1		3
105.104	even	2		4851.2.a.bp.1.1		3
165.164	even	2		7623.2.a.cb.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.d.1.3	✓	3	5.4	even	2
693.2.a.m.1.1		3	15.14	odd	2
1617.2.a.s.1.3		3	35.34	odd	2
2541.2.a.bi.1.1		3	55.54	odd	2
3696.2.a.bp.1.2		3	20.19	odd	2
4851.2.a.bp.1.1		3	105.104	even	2
5775.2.a.bw.1.1		3	1.1	even	1	trivial
7623.2.a.cb.1.3		3	165.164	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)

\(f(q)\)	\(=\)	\(q-2.52892 q^{2} +1.00000 q^{3} +4.39543 q^{4} -2.52892 q^{6} +1.00000 q^{7} -6.05784 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.52892 q^{2} +1.00000 q^{3} +4.39543 q^{4} -2.52892 q^{6} +1.00000 q^{7} -6.05784 q^{8} +1.00000 q^{9} +1.00000 q^{11} +4.39543 q^{12} -0.133492 q^{13} -2.52892 q^{14} +6.52892 q^{16} +5.05784 q^{17} -2.52892 q^{18} -0.924344 q^{19} +1.00000 q^{21} -2.52892 q^{22} +7.05784 q^{23} -6.05784 q^{24} +0.337590 q^{26} +1.00000 q^{27} +4.39543 q^{28} +3.86651 q^{29} +2.79085 q^{31} -4.39543 q^{32} +1.00000 q^{33} -12.7909 q^{34} +4.39543 q^{36} -9.98218 q^{37} +2.33759 q^{38} -0.133492 q^{39} +11.8487 q^{41} -2.52892 q^{42} +3.05784 q^{43} +4.39543 q^{44} -17.8487 q^{46} +3.07566 q^{47} +6.52892 q^{48} +1.00000 q^{49} +5.05784 q^{51} -0.586754 q^{52} +4.79085 q^{53} -2.52892 q^{54} -6.05784 q^{56} -0.924344 q^{57} -9.77808 q^{58} -12.6574 q^{59} +6.00000 q^{61} -7.05784 q^{62} +1.00000 q^{63} -1.94216 q^{64} -2.52892 q^{66} -8.92434 q^{67} +22.2313 q^{68} +7.05784 q^{69} -6.11567 q^{71} -6.05784 q^{72} -7.86651 q^{73} +25.2441 q^{74} -4.06289 q^{76} +1.00000 q^{77} +0.337590 q^{78} +14.1157 q^{79} +1.00000 q^{81} -29.9644 q^{82} +1.20915 q^{83} +4.39543 q^{84} -7.73302 q^{86} +3.86651 q^{87} -6.05784 q^{88} +15.5817 q^{89} -0.133492 q^{91} +31.0222 q^{92} +2.79085 q^{93} -7.77808 q^{94} -4.39543 q^{96} -12.7909 q^{97} -2.52892 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{12} + 12 q^{16} + 12 q^{19} + 3 q^{21} + 6 q^{23} - 3 q^{24} + 9 q^{26} + 3 q^{27} + 6 q^{28} + 12 q^{29} - 6 q^{31} - 6 q^{32} + 3 q^{33} - 24 q^{34} + 6 q^{36} + 15 q^{38} + 6 q^{41} - 6 q^{43} + 6 q^{44} - 24 q^{46} + 24 q^{47} + 12 q^{48} + 3 q^{49} + 21 q^{52} - 3 q^{56} + 12 q^{57} + 9 q^{58} - 24 q^{59} + 18 q^{61} - 6 q^{62} + 3 q^{63} - 21 q^{64} - 12 q^{67} + 6 q^{68} + 6 q^{69} + 12 q^{71} - 3 q^{72} - 24 q^{73} + 39 q^{74} - 3 q^{76} + 3 q^{77} + 9 q^{78} + 12 q^{79} + 3 q^{81} - 30 q^{82} + 18 q^{83} + 6 q^{84} - 24 q^{86} + 12 q^{87} - 3 q^{88} + 18 q^{89} + 18 q^{92} - 6 q^{93} + 15 q^{94} - 6 q^{96} - 24 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^12 + 12 * q^16 + 12 * q^19 + 3 * q^21 + 6 * q^23 - 3 * q^24 + 9 * q^26 + 3 * q^27 + 6 * q^28 + 12 * q^29 - 6 * q^31 - 6 * q^32 + 3 * q^33 - 24 * q^34 + 6 * q^36 + 15 * q^38 + 6 * q^41 - 6 * q^43 + 6 * q^44 - 24 * q^46 + 24 * q^47 + 12 * q^48 + 3 * q^49 + 21 * q^52 - 3 * q^56 + 12 * q^57 + 9 * q^58 - 24 * q^59 + 18 * q^61 - 6 * q^62 + 3 * q^63 - 21 * q^64 - 12 * q^67 + 6 * q^68 + 6 * q^69 + 12 * q^71 - 3 * q^72 - 24 * q^73 + 39 * q^74 - 3 * q^76 + 3 * q^77 + 9 * q^78 + 12 * q^79 + 3 * q^81 - 30 * q^82 + 18 * q^83 + 6 * q^84 - 24 * q^86 + 12 * q^87 - 3 * q^88 + 18 * q^89 + 18 * q^92 - 6 * q^93 + 15 * q^94 - 6 * q^96 - 24 * q^97 + 3 * q^99

Introduction

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