LMFDB - Embedded newform 5775.2.a.bq.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.34292$ 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)

$f(q)$	$=$	$q-2.34292 q^{2} -1.00000 q^{3} +3.48929 q^{4} +2.34292 q^{6} +1.00000 q^{7} -3.48929 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.34292 q^{2} -1.00000 q^{3} +3.48929 q^{4} +2.34292 q^{6} +1.00000 q^{7} -3.48929 q^{8} +1.00000 q^{9} +1.00000 q^{11} -3.48929 q^{12} -3.14637 q^{13} -2.34292 q^{14} +1.19656 q^{16} -4.19656 q^{17} -2.34292 q^{18} -7.17513 q^{19} -1.00000 q^{21} -2.34292 q^{22} +4.48929 q^{23} +3.48929 q^{24} +7.37169 q^{26} -1.00000 q^{27} +3.48929 q^{28} +5.34292 q^{29} +3.83221 q^{31} +4.17513 q^{32} -1.00000 q^{33} +9.83221 q^{34} +3.48929 q^{36} +6.81079 q^{37} +16.8108 q^{38} +3.14637 q^{39} -4.12494 q^{41} +2.34292 q^{42} +1.05019 q^{43} +3.48929 q^{44} -10.5181 q^{46} +3.83221 q^{47} -1.19656 q^{48} +1.00000 q^{49} +4.19656 q^{51} -10.9786 q^{52} -2.48929 q^{53} +2.34292 q^{54} -3.48929 q^{56} +7.17513 q^{57} -12.5181 q^{58} +2.02877 q^{59} +4.78202 q^{61} -8.97858 q^{62} +1.00000 q^{63} -12.1751 q^{64} +2.34292 q^{66} +4.97858 q^{67} -14.6430 q^{68} -4.48929 q^{69} -7.43910 q^{71} -3.48929 q^{72} -8.68585 q^{73} -15.9572 q^{74} -25.0361 q^{76} +1.00000 q^{77} -7.37169 q^{78} -15.8898 q^{79} +1.00000 q^{81} +9.66442 q^{82} +2.58967 q^{83} -3.48929 q^{84} -2.46052 q^{86} -5.34292 q^{87} -3.48929 q^{88} +4.61423 q^{89} -3.14637 q^{91} +15.6644 q^{92} -3.83221 q^{93} -8.97858 q^{94} -4.17513 q^{96} -8.22112 q^{97} -2.34292 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 3 q^{3} + 3 q^{4} + q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - q^2 - 3 * q^3 + 3 * q^4 + q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - q^{2} - 3 q^{3} + 3 q^{4} + q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} - 3 q^{12} - 8 q^{13} - q^{14} - q^{16} - 8 q^{17} - q^{18} - 2 q^{19} - 3 q^{21} - q^{22} + 6 q^{23} + 3 q^{24} - 2 q^{26} - 3 q^{27} + 3 q^{28} + 10 q^{29} - 2 q^{31} - 7 q^{32} - 3 q^{33} + 16 q^{34} + 3 q^{36} - 8 q^{37} + 22 q^{38} + 8 q^{39} + 4 q^{41} + q^{42} + 3 q^{44} - 6 q^{46} - 2 q^{47} + q^{48} + 3 q^{49} + 8 q^{51} - 18 q^{52} + q^{54} - 3 q^{56} + 2 q^{57} - 12 q^{58} - 12 q^{59} + 4 q^{61} - 12 q^{62} + 3 q^{63} - 17 q^{64} + q^{66} - 2 q^{68} - 6 q^{69} - 18 q^{71} - 3 q^{72} - 14 q^{73} - 18 q^{74} - 24 q^{76} + 3 q^{77} + 2 q^{78} + 2 q^{79} + 3 q^{81} + 2 q^{82} - 6 q^{83} - 3 q^{84} - 18 q^{86} - 10 q^{87} - 3 q^{88} - 10 q^{89} - 8 q^{91} + 20 q^{92} + 2 q^{93} - 12 q^{94} + 7 q^{96} - 10 q^{97} - q^{98} + 3 q^{99}+O(q^{100}) $ 3 * q - q^2 - 3 * q^3 + 3 * q^4 + q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 - 3 * q^12 - 8 * q^13 - q^14 - q^16 - 8 * q^17 - q^18 - 2 * q^19 - 3 * q^21 - q^22 + 6 * q^23 + 3 * q^24 - 2 * q^26 - 3 * q^27 + 3 * q^28 + 10 * q^29 - 2 * q^31 - 7 * q^32 - 3 * q^33 + 16 * q^34 + 3 * q^36 - 8 * q^37 + 22 * q^38 + 8 * q^39 + 4 * q^41 + q^42 + 3 * q^44 - 6 * q^46 - 2 * q^47 + q^48 + 3 * q^49 + 8 * q^51 - 18 * q^52 + q^54 - 3 * q^56 + 2 * q^57 - 12 * q^58 - 12 * q^59 + 4 * q^61 - 12 * q^62 + 3 * q^63 - 17 * q^64 + q^66 - 2 * q^68 - 6 * q^69 - 18 * q^71 - 3 * q^72 - 14 * q^73 - 18 * q^74 - 24 * q^76 + 3 * q^77 + 2 * q^78 + 2 * q^79 + 3 * q^81 + 2 * q^82 - 6 * q^83 - 3 * q^84 - 18 * q^86 - 10 * q^87 - 3 * q^88 - 10 * q^89 - 8 * q^91 + 20 * q^92 + 2 * q^93 - 12 * q^94 + 7 * q^96 - 10 * q^97 - q^98 + 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 $

$2$	−2.34292	−1.65670	−0.828348	−	0.560213i	$-0.810719\pi$
$2$	−2.34292	−1.65670	−0.828348	+	0.560213i	$0.810719\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.48929	1.74464
$5$	0	0
$5$	0	0
$6$	2.34292	0.956494
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.48929	−1.23365
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−3.48929	−1.00727
$13$	−3.14637	−0.872645	−0.436322	−	0.899790i	$-0.643719\pi$
$13$	−3.14637	−0.872645	−0.436322	+	0.899790i	$0.643719\pi$
$14$	−2.34292	−0.626173
$15$	0	0
$16$	1.19656	0.299139
$17$	−4.19656	−1.01781	−0.508907	−	0.860821i	$-0.669950\pi$
$17$	−4.19656	−1.01781	−0.508907	+	0.860821i	$0.669950\pi$
$18$	−2.34292	−0.552232
$19$	−7.17513	−1.64609	−0.823044	−	0.567977i	$-0.807726\pi$
$19$	−7.17513	−1.64609	−0.823044	+	0.567977i	$0.807726\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−2.34292	−0.499513
$23$	4.48929	0.936081	0.468041	−	0.883707i	$-0.344960\pi$
$23$	4.48929	0.936081	0.468041	+	0.883707i	$0.344960\pi$
$24$	3.48929	0.712248
$25$	0	0
$26$	7.37169	1.44571
$27$	−1.00000	−0.192450
$28$	3.48929	0.659414
$29$	5.34292	0.992156	0.496078	−	0.868278i	$-0.334773\pi$
$29$	5.34292	0.992156	0.496078	+	0.868278i	$0.334773\pi$
$30$	0	0
$31$	3.83221	0.688286	0.344143	−	0.938917i	$-0.388170\pi$
$31$	3.83221	0.688286	0.344143	+	0.938917i	$0.388170\pi$
$32$	4.17513	0.738067
$33$	−1.00000	−0.174078
$34$	9.83221	1.68621
$35$	0	0
$36$	3.48929	0.581548
$37$	6.81079	1.11969	0.559843	−	0.828598i	$-0.310861\pi$
$37$	6.81079	1.11969	0.559843	+	0.828598i	$0.310861\pi$
$38$	16.8108	2.72707
$39$	3.14637	0.503822
$40$	0	0
$41$	−4.12494	−0.644208	−0.322104	−	0.946704i	$-0.604390\pi$
$41$	−4.12494	−0.644208	−0.322104	+	0.946704i	$0.604390\pi$
$42$	2.34292	0.361521
$43$	1.05019	0.160153	0.0800764	−	0.996789i	$-0.474484\pi$
$43$	1.05019	0.160153	0.0800764	+	0.996789i	$0.474484\pi$
$44$	3.48929	0.526030
$45$	0	0
$46$	−10.5181	−1.55080
$47$	3.83221	0.558986	0.279493	−	0.960148i	$-0.409834\pi$
$47$	3.83221	0.558986	0.279493	+	0.960148i	$0.409834\pi$
$48$	−1.19656	−0.172708
$49$	1.00000	0.142857
$50$	0	0
$51$	4.19656	0.587636
$52$	−10.9786	−1.52245
$53$	−2.48929	−0.341930	−0.170965	−	0.985277i	$-0.554689\pi$
$53$	−2.48929	−0.341930	−0.170965	+	0.985277i	$0.554689\pi$
$54$	2.34292	0.318831
$55$	0	0
$56$	−3.48929	−0.466276
$57$	7.17513	0.950370
$58$	−12.5181	−1.64370
$59$	2.02877	0.264123	0.132062	−	0.991242i	$-0.457840\pi$
$59$	2.02877	0.264123	0.132062	+	0.991242i	$0.457840\pi$
$60$	0	0
$61$	4.78202	0.612275	0.306137	−	0.951987i	$-0.400963\pi$
$61$	4.78202	0.612275	0.306137	+	0.951987i	$0.400963\pi$
$62$	−8.97858	−1.14028
$63$	1.00000	0.125988
$64$	−12.1751	−1.52189
$65$	0	0
$66$	2.34292	0.288394
$67$	4.97858	0.608230	0.304115	−	0.952635i	$-0.401639\pi$
$67$	4.97858	0.608230	0.304115	+	0.952635i	$0.401639\pi$
$68$	−14.6430	−1.77572
$69$	−4.48929	−0.540447
$70$	0	0
$71$	−7.43910	−0.882858	−0.441429	−	0.897296i	$-0.645528\pi$
$71$	−7.43910	−0.882858	−0.441429	+	0.897296i	$0.645528\pi$
$72$	−3.48929	−0.411217
$73$	−8.68585	−1.01660	−0.508301	−	0.861180i	$-0.669726\pi$
$73$	−8.68585	−1.01660	−0.508301	+	0.861180i	$0.669726\pi$
$74$	−15.9572	−1.85498
$75$	0	0
$76$	−25.0361	−2.87184
$77$	1.00000	0.113961
$78$	−7.37169	−0.834680
$79$	−15.8898	−1.78774	−0.893868	−	0.448330i	$-0.852019\pi$
$79$	−15.8898	−1.78774	−0.893868	+	0.448330i	$0.852019\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	9.66442	1.06726
$83$	2.58967	0.284254	0.142127	−	0.989848i	$-0.454606\pi$
$83$	2.58967	0.284254	0.142127	+	0.989848i	$0.454606\pi$
$84$	−3.48929	−0.380713
$85$	0	0
$86$	−2.46052	−0.265325
$87$	−5.34292	−0.572821
$88$	−3.48929	−0.371959
$89$	4.61423	0.489108	0.244554	−	0.969636i	$-0.421359\pi$
$89$	4.61423	0.489108	0.244554	+	0.969636i	$0.421359\pi$
$90$	0	0
$91$	−3.14637	−0.329829
$92$	15.6644	1.63313
$93$	−3.83221	−0.397382
$94$	−8.97858	−0.926070
$95$	0	0
$96$	−4.17513	−0.426123
$97$	−8.22112	−0.834728	−0.417364	−	0.908739i	$-0.637046\pi$
$97$	−8.22112	−0.834728	−0.417364	+	0.908739i	$0.637046\pi$
$98$	−2.34292	−0.236671
$99$	1.00000	0.100504
$100$	0	0
$101$	1.41454	0.140752	0.0703759	−	0.997521i	$-0.477580\pi$
$101$	1.41454	0.140752	0.0703759	+	0.997521i	$0.477580\pi$
$102$	−9.83221	−0.973534
$103$	−0.657077	−0.0647437	−0.0323719	−	0.999476i	$-0.510306\pi$
$103$	−0.657077	−0.0647437	−0.0323719	+	0.999476i	$0.510306\pi$
$104$	10.9786	1.07654
$105$	0	0
$106$	5.83221	0.566474
$107$	12.4177	1.20046	0.600231	−	0.799827i	$-0.295075\pi$
$107$	12.4177	1.20046	0.600231	+	0.799827i	$0.295075\pi$
$108$	−3.48929	−0.335757
$109$	−9.10352	−0.871959	−0.435980	−	0.899957i	$-0.643598\pi$
$109$	−9.10352	−0.871959	−0.435980	+	0.899957i	$0.643598\pi$
$110$	0	0
$111$	−6.81079	−0.646451
$112$	1.19656	0.113064
$113$	13.5682	1.27639	0.638197	−	0.769873i	$-0.279681\pi$
$113$	13.5682	1.27639	0.638197	+	0.769873i	$0.279681\pi$
$114$	−16.8108	−1.57447
$115$	0	0
$116$	18.6430	1.73096
$117$	−3.14637	−0.290882
$118$	−4.75325	−0.437572
$119$	−4.19656	−0.384698
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.2039	−1.01435
$123$	4.12494	0.371934
$124$	13.3717	1.20081
$125$	0	0
$126$	−2.34292	−0.208724
$127$	−11.9284	−1.05847	−0.529237	−	0.848474i	$-0.677522\pi$
$127$	−11.9284	−1.05847	−0.529237	+	0.848474i	$0.677522\pi$
$128$	20.1751	1.78325
$129$	−1.05019	−0.0924643
$130$	0	0
$131$	9.78937	0.855301	0.427650	−	0.903944i	$-0.359341\pi$
$131$	9.78937	0.855301	0.427650	+	0.903944i	$0.359341\pi$
$132$	−3.48929	−0.303704
$133$	−7.17513	−0.622163
$134$	−11.6644	−1.00765
$135$	0	0
$136$	14.6430	1.25563
$137$	13.4721	1.15100	0.575499	−	0.817803i	$-0.304808\pi$
$137$	13.4721	1.15100	0.575499	+	0.817803i	$0.304808\pi$
$138$	10.5181	0.895357
$139$	−20.6430	−1.75092	−0.875458	−	0.483294i	$-0.839440\pi$
$139$	−20.6430	−1.75092	−0.875458	+	0.483294i	$0.839440\pi$
$140$	0	0
$141$	−3.83221	−0.322730
$142$	17.4292	1.46263
$143$	−3.14637	−0.263112
$144$	1.19656	0.0997131
$145$	0	0
$146$	20.3503	1.68420
$147$	−1.00000	−0.0824786
$148$	23.7648	1.95346
$149$	18.2499	1.49509	0.747544	−	0.664212i	$-0.231233\pi$
$149$	18.2499	1.49509	0.747544	+	0.664212i	$0.231233\pi$
$150$	0	0
$151$	16.9786	1.38170	0.690849	−	0.723000i	$-0.257237\pi$
$151$	16.9786	1.38170	0.690849	+	0.723000i	$0.257237\pi$
$152$	25.0361	2.03070
$153$	−4.19656	−0.339272
$154$	−2.34292	−0.188798
$155$	0	0
$156$	10.9786	0.878990
$157$	−2.07161	−0.165333	−0.0826664	−	0.996577i	$-0.526344\pi$
$157$	−2.07161	−0.165333	−0.0826664	+	0.996577i	$0.526344\pi$
$158$	37.2285	2.96174
$159$	2.48929	0.197413
$160$	0	0
$161$	4.48929	0.353806
$162$	−2.34292	−0.184077
$163$	−23.1611	−1.81411	−0.907057	−	0.421008i	$-0.861677\pi$
$163$	−23.1611	−1.81411	−0.907057	+	0.421008i	$0.861677\pi$
$164$	−14.3931	−1.12391
$165$	0	0
$166$	−6.06740	−0.470922
$167$	−9.31415	−0.720751	−0.360375	−	0.932807i	$-0.617351\pi$
$167$	−9.31415	−0.720751	−0.360375	+	0.932807i	$0.617351\pi$
$168$	3.48929	0.269204
$169$	−3.10038	−0.238491
$170$	0	0
$171$	−7.17513	−0.548696
$172$	3.66442	0.279410
$173$	−3.89962	−0.296482	−0.148241	−	0.988951i	$-0.547361\pi$
$173$	−3.89962	−0.296482	−0.148241	+	0.988951i	$0.547361\pi$
$174$	12.5181	0.948992
$175$	0	0
$176$	1.19656	0.0901939
$177$	−2.02877	−0.152492
$178$	−10.8108	−0.810303
$179$	−6.85363	−0.512265	−0.256132	−	0.966642i	$-0.582448\pi$
$179$	−6.85363	−0.512265	−0.256132	+	0.966642i	$0.582448\pi$
$180$	0	0
$181$	1.41454	0.105142	0.0525709	−	0.998617i	$-0.483258\pi$
$181$	1.41454	0.105142	0.0525709	+	0.998617i	$0.483258\pi$
$182$	7.37169	0.546426
$183$	−4.78202	−0.353497
$184$	−15.6644	−1.15480
$185$	0	0
$186$	8.97858	0.658341
$187$	−4.19656	−0.306883
$188$	13.3717	0.975231
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	15.3288	1.10916	0.554578	−	0.832132i	$-0.312880\pi$
$191$	15.3288	1.10916	0.554578	+	0.832132i	$0.312880\pi$
$192$	12.1751	0.878665
$193$	5.66442	0.407734	0.203867	−	0.978999i	$-0.434649\pi$
$193$	5.66442	0.407734	0.203867	+	0.978999i	$0.434649\pi$
$194$	19.2614	1.38289
$195$	0	0
$196$	3.48929	0.249235
$197$	8.68585	0.618841	0.309420	−	0.950925i	$-0.399865\pi$
$197$	8.68585	0.618841	0.309420	+	0.950925i	$0.399865\pi$
$198$	−2.34292	−0.166504
$199$	0.786230	0.0557344	0.0278672	−	0.999612i	$-0.491128\pi$
$199$	0.786230	0.0557344	0.0278672	+	0.999612i	$0.491128\pi$
$200$	0	0
$201$	−4.97858	−0.351162
$202$	−3.31415	−0.233183
$203$	5.34292	0.375000
$204$	14.6430	1.02522
$205$	0	0
$206$	1.53948	0.107261
$207$	4.48929	0.312027
$208$	−3.76481	−0.261042
$209$	−7.17513	−0.496314
$210$	0	0
$211$	20.6430	1.42112	0.710561	−	0.703635i	$-0.248441\pi$
$211$	20.6430	1.42112	0.710561	+	0.703635i	$0.248441\pi$
$212$	−8.68585	−0.596546
$213$	7.43910	0.509718
$214$	−29.0937	−1.98880
$215$	0	0
$216$	3.48929	0.237416
$217$	3.83221	0.260147
$218$	21.3288	1.44457
$219$	8.68585	0.586935
$220$	0	0
$221$	13.2039	0.888191
$222$	15.9572	1.07097
$223$	−5.83642	−0.390836	−0.195418	−	0.980720i	$-0.562606\pi$
$223$	−5.83642	−0.390836	−0.195418	+	0.980720i	$0.562606\pi$
$224$	4.17513	0.278963
$225$	0	0
$226$	−31.7894	−2.11460
$227$	−27.8757	−1.85017	−0.925087	−	0.379756i	$-0.876008\pi$
$227$	−27.8757	−1.85017	−0.925087	+	0.379756i	$0.876008\pi$
$228$	25.0361	1.65806
$229$	−20.8536	−1.37805	−0.689023	−	0.724739i	$-0.741960\pi$
$229$	−20.8536	−1.37805	−0.689023	+	0.724739i	$0.741960\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	−18.6430	−1.22397
$233$	−24.5756	−1.61000	−0.805000	−	0.593274i	$-0.797835\pi$
$233$	−24.5756	−1.61000	−0.805000	+	0.593274i	$0.797835\pi$
$234$	7.37169	0.481903
$235$	0	0
$236$	7.07896	0.460801
$237$	15.8898	1.03215
$238$	9.83221	0.637328
$239$	−20.7146	−1.33992	−0.669959	−	0.742399i	$-0.733688\pi$
$239$	−20.7146	−1.33992	−0.669959	+	0.742399i	$0.733688\pi$
$240$	0	0
$241$	21.7220	1.39923	0.699617	−	0.714518i	$-0.253354\pi$
$241$	21.7220	1.39923	0.699617	+	0.714518i	$0.253354\pi$
$242$	−2.34292	−0.150609
$243$	−1.00000	−0.0641500
$244$	16.6858	1.06820
$245$	0	0
$246$	−9.66442	−0.616181
$247$	22.5756	1.43645
$248$	−13.3717	−0.849103
$249$	−2.58967	−0.164114
$250$	0	0
$251$	0.585462	0.0369540	0.0184770	−	0.999829i	$-0.494118\pi$
$251$	0.585462	0.0369540	0.0184770	+	0.999829i	$0.494118\pi$
$252$	3.48929	0.219805
$253$	4.48929	0.282239
$254$	27.9473	1.75357
$255$	0	0
$256$	−22.9185	−1.43241
$257$	7.53948	0.470300	0.235150	−	0.971959i	$-0.424442\pi$
$257$	7.53948	0.470300	0.235150	+	0.971959i	$0.424442\pi$
$258$	2.46052	0.153185
$259$	6.81079	0.423202
$260$	0	0
$261$	5.34292	0.330719
$262$	−22.9357	−1.41697
$263$	24.4507	1.50769	0.753846	−	0.657051i	$-0.228196\pi$
$263$	24.4507	1.50769	0.753846	+	0.657051i	$0.228196\pi$
$264$	3.48929	0.214751
$265$	0	0
$266$	16.8108	1.03074
$267$	−4.61423	−0.282386
$268$	17.3717	1.06114
$269$	−24.4794	−1.49254	−0.746268	−	0.665645i	$-0.768156\pi$
$269$	−24.4794	−1.49254	−0.746268	+	0.665645i	$0.768156\pi$
$270$	0	0
$271$	11.8181	0.717901	0.358950	−	0.933357i	$-0.383135\pi$
$271$	11.8181	0.717901	0.358950	+	0.933357i	$0.383135\pi$
$272$	−5.02142	−0.304468
$273$	3.14637	0.190427
$274$	−31.5640	−1.90685
$275$	0	0
$276$	−15.6644	−0.942887
$277$	−5.80765	−0.348948	−0.174474	−	0.984662i	$-0.555823\pi$
$277$	−5.80765	−0.348948	−0.174474	+	0.984662i	$0.555823\pi$
$278$	48.3650	2.90074
$279$	3.83221	0.229429
$280$	0	0
$281$	−25.7220	−1.53444	−0.767222	−	0.641382i	$-0.778361\pi$
$281$	−25.7220	−1.53444	−0.767222	+	0.641382i	$0.778361\pi$
$282$	8.97858	0.534666
$283$	−18.9112	−1.12415	−0.562076	−	0.827085i	$-0.689997\pi$
$283$	−18.9112	−1.12415	−0.562076	+	0.827085i	$0.689997\pi$
$284$	−25.9572	−1.54027
$285$	0	0
$286$	7.37169	0.435897
$287$	−4.12494	−0.243488
$288$	4.17513	0.246022
$289$	0.611096	0.0359468
$290$	0	0
$291$	8.22112	0.481930
$292$	−30.3074	−1.77361
$293$	−30.9975	−1.81089	−0.905446	−	0.424461i	$-0.860463\pi$
$293$	−30.9975	−1.81089	−0.905446	+	0.424461i	$0.860463\pi$
$294$	2.34292	0.136642
$295$	0	0
$296$	−23.7648	−1.38130
$297$	−1.00000	−0.0580259
$298$	−42.7581	−2.47691
$299$	−14.1249	−0.816867
$300$	0	0
$301$	1.05019	0.0605321
$302$	−39.7795	−2.28905
$303$	−1.41454	−0.0812631
$304$	−8.58546	−0.492410
$305$	0	0
$306$	9.83221	0.562070
$307$	−3.35700	−0.191594	−0.0957970	−	0.995401i	$-0.530540\pi$
$307$	−3.35700	−0.191594	−0.0957970	+	0.995401i	$0.530540\pi$
$308$	3.48929	0.198821
$309$	0.657077	0.0373798
$310$	0	0
$311$	−17.6216	−0.999228	−0.499614	−	0.866248i	$-0.666525\pi$
$311$	−17.6216	−0.999228	−0.499614	+	0.866248i	$0.666525\pi$
$312$	−10.9786	−0.621540
$313$	−2.46473	−0.139315	−0.0696574	−	0.997571i	$-0.522191\pi$
$313$	−2.46473	−0.139315	−0.0696574	+	0.997571i	$0.522191\pi$
$314$	4.85363	0.273906
$315$	0	0
$316$	−55.4439	−3.11896
$317$	0.235192	0.0132097	0.00660486	−	0.999978i	$-0.497898\pi$
$317$	0.235192	0.0132097	0.00660486	+	0.999978i	$0.497898\pi$
$318$	−5.83221	−0.327054
$319$	5.34292	0.299146
$320$	0	0
$321$	−12.4177	−0.693087
$322$	−10.5181	−0.586148
$323$	30.1109	1.67541
$324$	3.48929	0.193849
$325$	0	0
$326$	54.2646	3.00544
$327$	9.10352	0.503426
$328$	14.3931	0.794727
$329$	3.83221	0.211277
$330$	0	0
$331$	7.81814	0.429724	0.214862	−	0.976644i	$-0.431070\pi$
$331$	7.81814	0.429724	0.214862	+	0.976644i	$0.431070\pi$
$332$	9.03612	0.495921
$333$	6.81079	0.373229
$334$	21.8223	1.19407
$335$	0	0
$336$	−1.19656	−0.0652776
$337$	5.53527	0.301525	0.150763	−	0.988570i	$-0.451827\pi$
$337$	5.53527	0.301525	0.150763	+	0.988570i	$0.451827\pi$
$338$	7.26396	0.395107
$339$	−13.5682	−0.736926
$340$	0	0
$341$	3.83221	0.207526
$342$	16.8108	0.909023
$343$	1.00000	0.0539949
$344$	−3.66442	−0.197572
$345$	0	0
$346$	9.13650	0.491181
$347$	18.8782	1.01343	0.506717	−	0.862112i	$-0.330859\pi$
$347$	18.8782	1.01343	0.506717	+	0.862112i	$0.330859\pi$
$348$	−18.6430	−0.999370
$349$	16.8396	0.901401	0.450700	−	0.892675i	$-0.351174\pi$
$349$	16.8396	0.901401	0.450700	+	0.892675i	$0.351174\pi$
$350$	0	0
$351$	3.14637	0.167941
$352$	4.17513	0.222535
$353$	−5.75011	−0.306048	−0.153024	−	0.988222i	$-0.548901\pi$
$353$	−5.75011	−0.306048	−0.153024	+	0.988222i	$0.548901\pi$
$354$	4.75325	0.252632
$355$	0	0
$356$	16.1004	0.853319
$357$	4.19656	0.222105
$358$	16.0575	0.848667
$359$	17.7507	0.936848	0.468424	−	0.883504i	$-0.344822\pi$
$359$	17.7507	0.936848	0.468424	+	0.883504i	$0.344822\pi$
$360$	0	0
$361$	32.4826	1.70961
$362$	−3.31415	−0.174188
$363$	−1.00000	−0.0524864
$364$	−10.9786	−0.575434
$365$	0	0
$366$	11.2039	0.585637
$367$	−37.5500	−1.96009	−0.980046	−	0.198771i	$-0.936305\pi$
$367$	−37.5500	−1.96009	−0.980046	+	0.198771i	$0.936305\pi$
$368$	5.37169	0.280019
$369$	−4.12494	−0.214736
$370$	0	0
$371$	−2.48929	−0.129237
$372$	−13.3717	−0.693290
$373$	−18.5714	−0.961590	−0.480795	−	0.876833i	$-0.659652\pi$
$373$	−18.5714	−0.961590	−0.480795	+	0.876833i	$0.659652\pi$
$374$	9.83221	0.508412
$375$	0	0
$376$	−13.3717	−0.689592
$377$	−16.8108	−0.865800
$378$	2.34292	0.120507
$379$	11.7606	0.604101	0.302051	−	0.953292i	$-0.402329\pi$
$379$	11.7606	0.604101	0.302051	+	0.953292i	$0.402329\pi$
$380$	0	0
$381$	11.9284	0.611110
$382$	−35.9143	−1.83754
$383$	38.4324	1.96380	0.981901	−	0.189394i	$-0.0606525\pi$
$383$	38.4324	1.96380	0.981901	+	0.189394i	$0.0606525\pi$
$384$	−20.1751	−1.02956
$385$	0	0
$386$	−13.2713	−0.675492
$387$	1.05019	0.0533843
$388$	−28.6858	−1.45630
$389$	−19.3471	−0.980939	−0.490469	−	0.871458i	$-0.663175\pi$
$389$	−19.3471	−0.980939	−0.490469	+	0.871458i	$0.663175\pi$
$390$	0	0
$391$	−18.8396	−0.952757
$392$	−3.48929	−0.176236
$393$	−9.78937	−0.493808
$394$	−20.3503	−1.02523
$395$	0	0
$396$	3.48929	0.175343
$397$	5.72196	0.287177	0.143589	−	0.989637i	$-0.454136\pi$
$397$	5.72196	0.287177	0.143589	+	0.989637i	$0.454136\pi$
$398$	−1.84208	−0.0923350
$399$	7.17513	0.359206
$400$	0	0
$401$	−4.46052	−0.222748	−0.111374	−	0.993779i	$-0.535525\pi$
$401$	−4.46052	−0.222748	−0.111374	+	0.993779i	$0.535525\pi$
$402$	11.6644	0.581769
$403$	−12.0575	−0.600629
$404$	4.93573	0.245562
$405$	0	0
$406$	−12.5181	−0.621261
$407$	6.81079	0.337598
$408$	−14.6430	−0.724937
$409$	36.2646	1.79317	0.896584	−	0.442874i	$-0.146041\pi$
$409$	36.2646	1.79317	0.896584	+	0.442874i	$0.146041\pi$
$410$	0	0
$411$	−13.4721	−0.664529
$412$	−2.29273	−0.112955
$413$	2.02877	0.0998292
$414$	−10.5181	−0.516934
$415$	0	0
$416$	−13.1365	−0.644070
$417$	20.6430	1.01089
$418$	16.8108	0.822243
$419$	−7.92839	−0.387327	−0.193663	−	0.981068i	$-0.562037\pi$
$419$	−7.92839	−0.387327	−0.193663	+	0.981068i	$0.562037\pi$
$420$	0	0
$421$	10.3461	0.504236	0.252118	−	0.967696i	$-0.418873\pi$
$421$	10.3461	0.504236	0.252118	+	0.967696i	$0.418873\pi$
$422$	−48.3650	−2.35437
$423$	3.83221	0.186329
$424$	8.68585	0.421822
$425$	0	0
$426$	−17.4292	−0.844449
$427$	4.78202	0.231418
$428$	43.3288	2.09438
$429$	3.14637	0.151908
$430$	0	0
$431$	29.8223	1.43649	0.718246	−	0.695789i	$-0.244945\pi$
$431$	29.8223	1.43649	0.718246	+	0.695789i	$0.244945\pi$
$432$	−1.19656	−0.0575694
$433$	25.5787	1.22924	0.614618	−	0.788825i	$-0.289310\pi$
$433$	25.5787	1.22924	0.614618	+	0.788825i	$0.289310\pi$
$434$	−8.97858	−0.430985
$435$	0	0
$436$	−31.7648	−1.52126
$437$	−32.2113	−1.54087
$438$	−20.3503	−0.972373
$439$	13.2755	0.633606	0.316803	−	0.948491i	$-0.397391\pi$
$439$	13.2755	0.633606	0.316803	+	0.948491i	$0.397391\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−30.9357	−1.47146
$443$	−3.85677	−0.183241	−0.0916203	−	0.995794i	$-0.529205\pi$
$443$	−3.85677	−0.183241	−0.0916203	+	0.995794i	$0.529205\pi$
$444$	−23.7648	−1.12783
$445$	0	0
$446$	13.6743	0.647496
$447$	−18.2499	−0.863190
$448$	−12.1751	−0.575221
$449$	−34.2829	−1.61791	−0.808954	−	0.587872i	$-0.799966\pi$
$449$	−34.2829	−1.61791	−0.808954	+	0.587872i	$0.799966\pi$
$450$	0	0
$451$	−4.12494	−0.194236
$452$	47.3435	2.22685
$453$	−16.9786	−0.797723
$454$	65.3106	3.06518
$455$	0	0
$456$	−25.0361	−1.17242
$457$	−31.5500	−1.47585	−0.737923	−	0.674885i	$-0.764193\pi$
$457$	−31.5500	−1.47585	−0.737923	+	0.674885i	$0.764193\pi$
$458$	48.8585	2.28301
$459$	4.19656	0.195879
$460$	0	0
$461$	−0.878193	−0.0409015	−0.0204508	−	0.999791i	$-0.506510\pi$
$461$	−0.878193	−0.0409015	−0.0204508	+	0.999791i	$0.506510\pi$
$462$	2.34292	0.109003
$463$	−39.1856	−1.82111	−0.910555	−	0.413388i	$-0.864345\pi$
$463$	−39.1856	−1.82111	−0.910555	+	0.413388i	$0.864345\pi$
$464$	6.39312	0.296793
$465$	0	0
$466$	57.5787	2.66728
$467$	33.7073	1.55979	0.779893	−	0.625913i	$-0.215273\pi$
$467$	33.7073	1.55979	0.779893	+	0.625913i	$0.215273\pi$
$468$	−10.9786	−0.507485
$469$	4.97858	0.229889
$470$	0	0
$471$	2.07161	0.0954550
$472$	−7.07896	−0.325836
$473$	1.05019	0.0482879
$474$	−37.2285	−1.70996
$475$	0	0
$476$	−14.6430	−0.671161
$477$	−2.48929	−0.113977
$478$	48.5328	2.21984
$479$	−22.2400	−1.01617	−0.508086	−	0.861306i	$-0.669647\pi$
$479$	−22.2400	−1.01617	−0.508086	+	0.861306i	$0.669647\pi$
$480$	0	0
$481$	−21.4292	−0.977089
$482$	−50.8929	−2.31811
$483$	−4.48929	−0.204270
$484$	3.48929	0.158604
$485$	0	0
$486$	2.34292	0.106277
$487$	−42.2155	−1.91297	−0.956483	−	0.291789i	$-0.905749\pi$
$487$	−42.2155	−1.91297	−0.956483	+	0.291789i	$0.905749\pi$
$488$	−16.6858	−0.755333
$489$	23.1611	1.04738
$490$	0	0
$491$	−18.0779	−0.815844	−0.407922	−	0.913017i	$-0.633746\pi$
$491$	−18.0779	−0.815844	−0.407922	+	0.913017i	$0.633746\pi$
$492$	14.3931	0.648892
$493$	−22.4219	−1.00983
$494$	−52.8929	−2.37976
$495$	0	0
$496$	4.58546	0.205893
$497$	−7.43910	−0.333689
$498$	6.06740	0.271887
$499$	18.4464	0.825776	0.412888	−	0.910782i	$-0.364520\pi$
$499$	18.4464	0.825776	0.412888	+	0.910782i	$0.364520\pi$
$500$	0	0
$501$	9.31415	0.416126
$502$	−1.37169	−0.0612216
$503$	−42.3608	−1.88877	−0.944386	−	0.328838i	$-0.893343\pi$
$503$	−42.3608	−1.88877	−0.944386	+	0.328838i	$0.893343\pi$
$504$	−3.48929	−0.155425
$505$	0	0
$506$	−10.5181	−0.467585
$507$	3.10038	0.137693
$508$	−41.6216	−1.84666
$509$	6.46473	0.286544	0.143272	−	0.989683i	$-0.454238\pi$
$509$	6.46473	0.286544	0.143272	+	0.989683i	$0.454238\pi$
$510$	0	0
$511$	−8.68585	−0.384239
$512$	13.3461	0.589818
$513$	7.17513	0.316790
$514$	−17.6644	−0.779144
$515$	0	0
$516$	−3.66442	−0.161317
$517$	3.83221	0.168540
$518$	−15.9572	−0.701117
$519$	3.89962	0.171174
$520$	0	0
$521$	20.6142	0.903126	0.451563	−	0.892239i	$-0.350867\pi$
$521$	20.6142	0.903126	0.451563	+	0.892239i	$0.350867\pi$
$522$	−12.5181	−0.547901
$523$	36.1642	1.58135	0.790675	−	0.612236i	$-0.209730\pi$
$523$	36.1642	1.58135	0.790675	+	0.612236i	$0.209730\pi$
$524$	34.1579	1.49220
$525$	0	0
$526$	−57.2860	−2.49779
$527$	−16.0821	−0.700547
$528$	−1.19656	−0.0520735
$529$	−2.84629	−0.123752
$530$	0	0
$531$	2.02877	0.0880411
$532$	−25.0361	−1.08545
$533$	12.9786	0.562165
$534$	10.8108	0.467829
$535$	0	0
$536$	−17.3717	−0.750343
$537$	6.85363	0.295756
$538$	57.3534	2.47268
$539$	1.00000	0.0430730
$540$	0	0
$541$	6.89648	0.296503	0.148251	−	0.988950i	$-0.452636\pi$
$541$	6.89648	0.296503	0.148251	+	0.988950i	$0.452636\pi$
$542$	−27.6890	−1.18934
$543$	−1.41454	−0.0607036
$544$	−17.5212	−0.751215
$545$	0	0
$546$	−7.37169	−0.315479
$547$	−1.97965	−0.0846438	−0.0423219	−	0.999104i	$-0.513476\pi$
$547$	−1.97965	−0.0846438	−0.0423219	+	0.999104i	$0.513476\pi$
$548$	47.0080	2.00808
$549$	4.78202	0.204092
$550$	0	0
$551$	−38.3362	−1.63318
$552$	15.6644	0.666722
$553$	−15.8898	−0.675701
$554$	13.6069	0.578101
$555$	0	0
$556$	−72.0294	−3.05473
$557$	−39.2369	−1.66252	−0.831260	−	0.555884i	$-0.812380\pi$
$557$	−39.2369	−1.66252	−0.831260	+	0.555884i	$0.812380\pi$
$558$	−8.97858	−0.380093
$559$	−3.30429	−0.139756
$560$	0	0
$561$	4.19656	0.177179
$562$	60.2646	2.54211
$563$	−28.9786	−1.22130	−0.610651	−	0.791900i	$-0.709092\pi$
$563$	−28.9786	−1.22130	−0.610651	+	0.791900i	$0.709092\pi$
$564$	−13.3717	−0.563050
$565$	0	0
$566$	44.3074	1.86238
$567$	1.00000	0.0419961
$568$	25.9572	1.08914
$569$	−32.5861	−1.36608	−0.683040	−	0.730381i	$-0.739342\pi$
$569$	−32.5861	−1.36608	−0.683040	+	0.730381i	$0.739342\pi$
$570$	0	0
$571$	−46.1825	−1.93268	−0.966338	−	0.257275i	$-0.917176\pi$
$571$	−46.1825	−1.93268	−0.966338	+	0.257275i	$0.917176\pi$
$572$	−10.9786	−0.459037
$573$	−15.3288	−0.640372
$574$	9.66442	0.403385
$575$	0	0
$576$	−12.1751	−0.507297
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−1.43175	−0.0595530
$579$	−5.66442	−0.235405
$580$	0	0
$581$	2.58967	0.107438
$582$	−19.2614	−0.798412
$583$	−2.48929	−0.103096
$584$	30.3074	1.25413
$585$	0	0
$586$	72.6247	3.00010
$587$	30.1825	1.24576	0.622882	−	0.782316i	$-0.285962\pi$
$587$	30.1825	1.24576	0.622882	+	0.782316i	$0.285962\pi$
$588$	−3.48929	−0.143896
$589$	−27.4966	−1.13298
$590$	0	0
$591$	−8.68585	−0.357288
$592$	8.14950	0.334942
$593$	11.1793	0.459081	0.229540	−	0.973299i	$-0.426278\pi$
$593$	11.1793	0.459081	0.229540	+	0.973299i	$0.426278\pi$
$594$	2.34292	0.0961313
$595$	0	0
$596$	63.6791	2.60840
$597$	−0.786230	−0.0321783
$598$	33.0937	1.35330
$599$	15.8568	0.647890	0.323945	−	0.946076i	$-0.394991\pi$
$599$	15.8568	0.647890	0.323945	+	0.946076i	$0.394991\pi$
$600$	0	0
$601$	−40.6472	−1.65803	−0.829017	−	0.559223i	$-0.811099\pi$
$601$	−40.6472	−1.65803	−0.829017	+	0.559223i	$0.811099\pi$
$602$	−2.46052	−0.100283
$603$	4.97858	0.202743
$604$	59.2432	2.41057
$605$	0	0
$606$	3.31415	0.134628
$607$	38.1298	1.54764	0.773820	−	0.633406i	$-0.218344\pi$
$607$	38.1298	1.54764	0.773820	+	0.633406i	$0.218344\pi$
$608$	−29.9572	−1.21492
$609$	−5.34292	−0.216506
$610$	0	0
$611$	−12.0575	−0.487796
$612$	−14.6430	−0.591908
$613$	−26.5855	−1.07378	−0.536888	−	0.843653i	$-0.680400\pi$
$613$	−26.5855	−1.07378	−0.536888	+	0.843653i	$0.680400\pi$
$614$	7.86519	0.317413
$615$	0	0
$616$	−3.48929	−0.140587
$617$	−35.8799	−1.44447	−0.722235	−	0.691648i	$-0.756885\pi$
$617$	−35.8799	−1.44447	−0.722235	+	0.691648i	$0.756885\pi$
$618$	−1.53948	−0.0619270
$619$	5.64973	0.227082	0.113541	−	0.993533i	$-0.463781\pi$
$619$	5.64973	0.227082	0.113541	+	0.993533i	$0.463781\pi$
$620$	0	0
$621$	−4.48929	−0.180149
$622$	41.2860	1.65542
$623$	4.61423	0.184865
$624$	3.76481	0.150713
$625$	0	0
$626$	5.77467	0.230802
$627$	7.17513	0.286547
$628$	−7.22846	−0.288447
$629$	−28.5819	−1.13963
$630$	0	0
$631$	−7.56825	−0.301287	−0.150644	−	0.988588i	$-0.548135\pi$
$631$	−7.56825	−0.301287	−0.150644	+	0.988588i	$0.548135\pi$
$632$	55.4439	2.20544
$633$	−20.6430	−0.820486
$634$	−0.551038	−0.0218845
$635$	0	0
$636$	8.68585	0.344416
$637$	−3.14637	−0.124664
$638$	−12.5181	−0.495595
$639$	−7.43910	−0.294286
$640$	0	0
$641$	−6.78623	−0.268040	−0.134020	−	0.990979i	$-0.542789\pi$
$641$	−6.78623	−0.268040	−0.134020	+	0.990979i	$0.542789\pi$
$642$	29.0937	1.14823
$643$	−35.8427	−1.41350	−0.706749	−	0.707464i	$-0.749839\pi$
$643$	−35.8427	−1.41350	−0.706749	+	0.707464i	$0.749839\pi$
$644$	15.6644	0.617265
$645$	0	0
$646$	−70.5474	−2.77565
$647$	−42.0231	−1.65210	−0.826050	−	0.563597i	$-0.809417\pi$
$647$	−42.0231	−1.65210	−0.826050	+	0.563597i	$0.809417\pi$
$648$	−3.48929	−0.137072
$649$	2.02877	0.0796362
$650$	0	0
$651$	−3.83221	−0.150196
$652$	−80.8156	−3.16498
$653$	19.0748	0.746453	0.373226	−	0.927740i	$-0.378251\pi$
$653$	19.0748	0.746453	0.373226	+	0.927740i	$0.378251\pi$
$654$	−21.3288	−0.834024
$655$	0	0
$656$	−4.93573	−0.192708
$657$	−8.68585	−0.338867
$658$	−8.97858	−0.350021
$659$	−4.27238	−0.166428	−0.0832142	−	0.996532i	$-0.526519\pi$
$659$	−4.27238	−0.166428	−0.0832142	+	0.996532i	$0.526519\pi$
$660$	0	0
$661$	−26.7251	−1.03949	−0.519743	−	0.854323i	$-0.673972\pi$
$661$	−26.7251	−1.03949	−0.519743	+	0.854323i	$0.673972\pi$
$662$	−18.3173	−0.711922
$663$	−13.2039	−0.512797
$664$	−9.03612	−0.350669
$665$	0	0
$666$	−15.9572	−0.618327
$667$	23.9859	0.928739
$668$	−32.4998	−1.25745
$669$	5.83642	0.225649
$670$	0	0
$671$	4.78202	0.184608
$672$	−4.17513	−0.161059
$673$	−25.7936	−0.994269	−0.497135	−	0.867673i	$-0.665615\pi$
$673$	−25.7936	−0.994269	−0.497135	+	0.867673i	$0.665615\pi$
$674$	−12.9687	−0.499536
$675$	0	0
$676$	−10.8181	−0.416082
$677$	5.17513	0.198897	0.0994483	−	0.995043i	$-0.468292\pi$
$677$	5.17513	0.198897	0.0994483	+	0.995043i	$0.468292\pi$
$678$	31.7894	1.22086
$679$	−8.22112	−0.315497
$680$	0	0
$681$	27.8757	1.06820
$682$	−8.97858	−0.343807
$683$	24.5855	0.940737	0.470368	−	0.882470i	$-0.344121\pi$
$683$	24.5855	0.940737	0.470368	+	0.882470i	$0.344121\pi$
$684$	−25.0361	−0.957280
$685$	0	0
$686$	−2.34292	−0.0894532
$687$	20.8536	0.795616
$688$	1.25662	0.0479080
$689$	7.83221	0.298384
$690$	0	0
$691$	14.9357	0.568182	0.284091	−	0.958797i	$-0.408308\pi$
$691$	14.9357	0.568182	0.284091	+	0.958797i	$0.408308\pi$
$692$	−13.6069	−0.517256
$693$	1.00000	0.0379869
$694$	−44.2302	−1.67895
$695$	0	0
$696$	18.6430	0.706661
$697$	17.3106	0.655684
$698$	−39.4538	−1.49335
$699$	24.5756	0.929534
$700$	0	0
$701$	2.07161	0.0782438	0.0391219	−	0.999234i	$-0.487544\pi$
$701$	2.07161	0.0782438	0.0391219	+	0.999234i	$0.487544\pi$
$702$	−7.37169	−0.278227
$703$	−48.8683	−1.84310
$704$	−12.1751	−0.458868
$705$	0	0
$706$	13.4721	0.507028
$707$	1.41454	0.0531992
$708$	−7.07896	−0.266044
$709$	−30.1256	−1.13139	−0.565695	−	0.824615i	$-0.691392\pi$
$709$	−30.1256	−1.13139	−0.565695	+	0.824615i	$0.691392\pi$
$710$	0	0
$711$	−15.8898	−0.595912
$712$	−16.1004	−0.603387
$713$	17.2039	0.644291
$714$	−9.83221	−0.367961
$715$	0	0
$716$	−23.9143	−0.893720
$717$	20.7146	0.773601
$718$	−41.5886	−1.55207
$719$	−34.0863	−1.27120	−0.635602	−	0.772017i	$-0.719248\pi$
$719$	−34.0863	−1.27120	−0.635602	+	0.772017i	$0.719248\pi$
$720$	0	0
$721$	−0.657077	−0.0244708
$722$	−76.1041	−2.83230
$723$	−21.7220	−0.807848
$724$	4.93573	0.183435
$725$	0	0
$726$	2.34292	0.0869540
$727$	10.3068	0.382258	0.191129	−	0.981565i	$-0.438785\pi$
$727$	10.3068	0.382258	0.191129	+	0.981565i	$0.438785\pi$
$728$	10.9786	0.406893
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.40719	−0.163006
$732$	−16.6858	−0.616727
$733$	−36.6086	−1.35217	−0.676084	−	0.736824i	$-0.736324\pi$
$733$	−36.6086	−1.35217	−0.676084	+	0.736824i	$0.736324\pi$
$734$	87.9767	3.24728
$735$	0	0
$736$	18.7434	0.690890
$737$	4.97858	0.183388
$738$	9.66442	0.355752
$739$	6.88661	0.253328	0.126664	−	0.991946i	$-0.459573\pi$
$739$	6.88661	0.253328	0.126664	+	0.991946i	$0.459573\pi$
$740$	0	0
$741$	−22.5756	−0.829335
$742$	5.83221	0.214107
$743$	−28.5573	−1.04767	−0.523833	−	0.851821i	$-0.675498\pi$
$743$	−28.5573	−1.04767	−0.523833	+	0.851821i	$0.675498\pi$
$744$	13.3717	0.490230
$745$	0	0
$746$	43.5113	1.59306
$747$	2.58967	0.0947512
$748$	−14.6430	−0.535401
$749$	12.4177	0.453732
$750$	0	0
$751$	−12.2969	−0.448722	−0.224361	−	0.974506i	$-0.572029\pi$
$751$	−12.2969	−0.448722	−0.224361	+	0.974506i	$0.572029\pi$
$752$	4.58546	0.167215
$753$	−0.585462	−0.0213354
$754$	39.3864	1.43437
$755$	0	0
$756$	−3.48929	−0.126904
$757$	−12.9933	−0.472248	−0.236124	−	0.971723i	$-0.575877\pi$
$757$	−12.9933	−0.472248	−0.236124	+	0.971723i	$0.575877\pi$
$758$	−27.5542	−1.00081
$759$	−4.48929	−0.162951
$760$	0	0
$761$	7.37169	0.267224	0.133612	−	0.991034i	$-0.457342\pi$
$761$	7.37169	0.267224	0.133612	+	0.991034i	$0.457342\pi$
$762$	−27.9473	−1.01242
$763$	−9.10352	−0.329570
$764$	53.4868	1.93508
$765$	0	0
$766$	−90.0441	−3.25342
$767$	−6.38325	−0.230486
$768$	22.9185	0.827001
$769$	29.6258	1.06833	0.534167	−	0.845379i	$-0.320625\pi$
$769$	29.6258	1.06833	0.534167	+	0.845379i	$0.320625\pi$
$770$	0	0
$771$	−7.53948	−0.271528
$772$	19.7648	0.711351
$773$	−12.3784	−0.445221	−0.222610	−	0.974907i	$-0.571458\pi$
$773$	−12.3784	−0.445221	−0.222610	+	0.974907i	$0.571458\pi$
$774$	−2.46052	−0.0884415
$775$	0	0
$776$	28.6858	1.02976
$777$	−6.81079	−0.244336
$778$	45.3288	1.62512
$779$	29.5970	1.06042
$780$	0	0
$781$	−7.43910	−0.266192
$782$	44.1396	1.57843
$783$	−5.34292	−0.190940
$784$	1.19656	0.0427342
$785$	0	0
$786$	22.9357	0.818090
$787$	45.9572	1.63820	0.819098	−	0.573654i	$-0.194475\pi$
$787$	45.9572	1.63820	0.819098	+	0.573654i	$0.194475\pi$
$788$	30.3074	1.07966
$789$	−24.4507	−0.870466
$790$	0	0
$791$	13.5682	0.482431
$792$	−3.48929	−0.123986
$793$	−15.0460	−0.534298
$794$	−13.4061	−0.475765
$795$	0	0
$796$	2.74338	0.0972367
$797$	18.4507	0.653556	0.326778	−	0.945101i	$-0.394037\pi$
$797$	18.4507	0.653556	0.326778	+	0.945101i	$0.394037\pi$
$798$	−16.8108	−0.595095
$799$	−16.0821	−0.568944
$800$	0	0
$801$	4.61423	0.163036
$802$	10.4507	0.369025
$803$	−8.68585	−0.306517
$804$	−17.3717	−0.612652
$805$	0	0
$806$	28.2499	0.995060
$807$	24.4794	0.861716
$808$	−4.93573	−0.173638
$809$	−16.0063	−0.562751	−0.281375	−	0.959598i	$-0.590791\pi$
$809$	−16.0063	−0.562751	−0.281375	+	0.959598i	$0.590791\pi$
$810$	0	0
$811$	−41.4868	−1.45680	−0.728399	−	0.685153i	$-0.759735\pi$
$811$	−41.4868	−1.45680	−0.728399	+	0.685153i	$0.759735\pi$
$812$	18.6430	0.654241
$813$	−11.8181	−0.414480
$814$	−15.9572	−0.559298
$815$	0	0
$816$	5.02142	0.175785
$817$	−7.53527	−0.263626
$818$	−84.9651	−2.97074
$819$	−3.14637	−0.109943
$820$	0	0
$821$	31.0136	1.08238	0.541191	−	0.840899i	$-0.317973\pi$
$821$	31.0136	1.08238	0.541191	+	0.840899i	$0.317973\pi$
$822$	31.5640	1.10092
$823$	−27.6117	−0.962484	−0.481242	−	0.876588i	$-0.659814\pi$
$823$	−27.6117	−0.962484	−0.481242	+	0.876588i	$0.659814\pi$
$824$	2.29273	0.0798711
$825$	0	0
$826$	−4.75325	−0.165387
$827$	25.0361	0.870591	0.435296	−	0.900288i	$-0.356644\pi$
$827$	25.0361	0.870591	0.435296	+	0.900288i	$0.356644\pi$
$828$	15.6644	0.544376
$829$	−5.75011	−0.199710	−0.0998549	−	0.995002i	$-0.531838\pi$
$829$	−5.75011	−0.199710	−0.0998549	+	0.995002i	$0.531838\pi$
$830$	0	0
$831$	5.80765	0.201465
$832$	38.3074	1.32807
$833$	−4.19656	−0.145402
$834$	−48.3650	−1.67474
$835$	0	0
$836$	−25.0361	−0.865892
$837$	−3.83221	−0.132461
$838$	18.5756	0.641683
$839$	−18.6142	−0.642635	−0.321317	−	0.946972i	$-0.604126\pi$
$839$	−18.6142	−0.642635	−0.321317	+	0.946972i	$0.604126\pi$
$840$	0	0
$841$	−0.453173	−0.0156267
$842$	−24.2400	−0.835366
$843$	25.7220	0.885911
$844$	72.0294	2.47935
$845$	0	0
$846$	−8.97858	−0.308690
$847$	1.00000	0.0343604
$848$	−2.97858	−0.102285
$849$	18.9112	0.649030
$850$	0	0
$851$	30.5756	1.04812
$852$	25.9572	0.889277
$853$	29.1611	0.998456	0.499228	−	0.866471i	$-0.333617\pi$
$853$	29.1611	0.998456	0.499228	+	0.866471i	$0.333617\pi$
$854$	−11.2039	−0.383390
$855$	0	0
$856$	−43.3288	−1.48095
$857$	29.4439	1.00579	0.502893	−	0.864349i	$-0.332269\pi$
$857$	29.4439	1.00579	0.502893	+	0.864349i	$0.332269\pi$
$858$	−7.37169	−0.251665
$859$	49.9536	1.70439	0.852197	−	0.523221i	$-0.175270\pi$
$859$	49.9536	1.70439	0.852197	+	0.523221i	$0.175270\pi$
$860$	0	0
$861$	4.12494	0.140578
$862$	−69.8715	−2.37983
$863$	−11.8181	−0.402294	−0.201147	−	0.979561i	$-0.564467\pi$
$863$	−11.8181	−0.402294	−0.201147	+	0.979561i	$0.564467\pi$
$864$	−4.17513	−0.142041
$865$	0	0
$866$	−59.9290	−2.03647
$867$	−0.611096	−0.0207539
$868$	13.3717	0.453865
$869$	−15.8898	−0.539023
$870$	0	0
$871$	−15.6644	−0.530769
$872$	31.7648	1.07569
$873$	−8.22112	−0.278243
$874$	75.4685	2.55276
$875$	0	0
$876$	30.3074	1.02399
$877$	−38.8213	−1.31090	−0.655451	−	0.755238i	$-0.727521\pi$
$877$	−38.8213	−1.31090	−0.655451	+	0.755238i	$0.727521\pi$
$878$	−31.1035	−1.04969
$879$	30.9975	1.04552
$880$	0	0
$881$	−44.3362	−1.49372	−0.746862	−	0.664979i	$-0.768441\pi$
$881$	−44.3362	−1.49372	−0.746862	+	0.664979i	$0.768441\pi$
$882$	−2.34292	−0.0788903
$883$	39.9634	1.34488	0.672438	−	0.740153i	$-0.265247\pi$
$883$	39.9634	1.34488	0.672438	+	0.740153i	$0.265247\pi$
$884$	46.0722	1.54958
$885$	0	0
$886$	9.03612	0.303574
$887$	−13.2474	−0.444803	−0.222402	−	0.974955i	$-0.571390\pi$
$887$	−13.2474	−0.444803	−0.222402	+	0.974955i	$0.571390\pi$
$888$	23.7648	0.797495
$889$	−11.9284	−0.400065
$890$	0	0
$891$	1.00000	0.0335013
$892$	−20.3650	−0.681869
$893$	−27.4966	−0.920140
$894$	42.7581	1.43004
$895$	0	0
$896$	20.1751	0.674004
$897$	14.1249	0.471618
$898$	80.3221	2.68038
$899$	20.4752	0.682887
$900$	0	0
$901$	10.4464	0.348021
$902$	9.66442	0.321790
$903$	−1.05019	−0.0349482
$904$	−47.3435	−1.57462
$905$	0	0
$906$	39.7795	1.32159
$907$	18.7961	0.624114	0.312057	−	0.950063i	$-0.398982\pi$
$907$	18.7961	0.624114	0.312057	+	0.950063i	$0.398982\pi$
$908$	−97.2663	−3.22789
$909$	1.41454	0.0469173
$910$	0	0
$911$	14.8782	0.492937	0.246468	−	0.969151i	$-0.420730\pi$
$911$	14.8782	0.492937	0.246468	+	0.969151i	$0.420730\pi$
$912$	8.58546	0.284293
$913$	2.58967	0.0857057
$914$	73.9191	2.44503
$915$	0	0
$916$	−72.7643	−2.40420
$917$	9.78937	0.323273
$918$	−9.83221	−0.324511
$919$	−13.6791	−0.451232	−0.225616	−	0.974216i	$-0.572440\pi$
$919$	−13.6791	−0.451232	−0.225616	+	0.974216i	$0.572440\pi$
$920$	0	0
$921$	3.35700	0.110617
$922$	2.05754	0.0677614
$923$	23.4061	0.770422
$924$	−3.48929	−0.114789
$925$	0	0
$926$	91.8089	3.01703
$927$	−0.657077	−0.0215812
$928$	22.3074	0.732277
$929$	12.9357	0.424408	0.212204	−	0.977225i	$-0.431936\pi$
$929$	12.9357	0.424408	0.212204	+	0.977225i	$0.431936\pi$
$930$	0	0
$931$	−7.17513	−0.235156
$932$	−85.7513	−2.80888
$933$	17.6216	0.576905
$934$	−78.9735	−2.58409
$935$	0	0
$936$	10.9786	0.358846
$937$	33.1119	1.08172	0.540860	−	0.841113i	$-0.318099\pi$
$937$	33.1119	1.08172	0.540860	+	0.841113i	$0.318099\pi$
$938$	−11.6644	−0.380857
$939$	2.46473	0.0804334
$940$	0	0
$941$	−35.1758	−1.14670	−0.573348	−	0.819312i	$-0.694356\pi$
$941$	−35.1758	−1.14670	−0.573348	+	0.819312i	$0.694356\pi$
$942$	−4.85363	−0.158140
$943$	−18.5181	−0.603031
$944$	2.42754	0.0790097
$945$	0	0
$946$	−2.46052	−0.0799984
$947$	32.8249	1.06666	0.533332	−	0.845906i	$-0.320940\pi$
$947$	32.8249	1.06666	0.533332	+	0.845906i	$0.320940\pi$
$948$	55.4439	1.80073
$949$	27.3288	0.887132
$950$	0	0
$951$	−0.235192	−0.00762664
$952$	14.6430	0.474582
$953$	−41.3288	−1.33877	−0.669386	−	0.742914i	$-0.733443\pi$
$953$	−41.3288	−1.33877	−0.669386	+	0.742914i	$0.733443\pi$
$954$	5.83221	0.188825
$955$	0	0
$956$	−72.2793	−2.33768
$957$	−5.34292	−0.172712
$958$	52.1067	1.68349
$959$	13.4721	0.435036
$960$	0	0
$961$	−16.3142	−0.526263
$962$	50.2070	1.61874
$963$	12.4177	0.400154
$964$	75.7942	2.44117
$965$	0	0
$966$	10.5181	0.338413
$967$	−52.3509	−1.68349	−0.841745	−	0.539875i	$-0.818472\pi$
$967$	−52.3509	−1.68349	−0.841745	+	0.539875i	$0.818472\pi$
$968$	−3.48929	−0.112150
$969$	−30.1109	−0.967300
$970$	0	0
$971$	20.1783	0.647552	0.323776	−	0.946134i	$-0.395048\pi$
$971$	20.1783	0.647552	0.323776	+	0.946134i	$0.395048\pi$
$972$	−3.48929	−0.111919
$973$	−20.6430	−0.661784
$974$	98.9076	3.16920
$975$	0	0
$976$	5.72196	0.183156
$977$	−29.1470	−0.932495	−0.466247	−	0.884654i	$-0.654394\pi$
$977$	−29.1470	−0.932495	−0.466247	+	0.884654i	$0.654394\pi$
$978$	−54.2646	−1.73519
$979$	4.61423	0.147471
$980$	0	0
$981$	−9.10352	−0.290653
$982$	42.3551	1.35161
$983$	−6.68585	−0.213245	−0.106623	−	0.994300i	$-0.534004\pi$
$983$	−6.68585	−0.213245	−0.106623	+	0.994300i	$0.534004\pi$
$984$	−14.3931	−0.458836
$985$	0	0
$986$	52.5328	1.67298
$987$	−3.83221	−0.121981
$988$	78.7728	2.50610
$989$	4.71462	0.149916
$990$	0	0
$991$	−11.3675	−0.361100	−0.180550	−	0.983566i	$-0.557788\pi$
$991$	−11.3675	−0.361100	−0.180550	+	0.983566i	$0.557788\pi$
$992$	16.0000	0.508001
$993$	−7.81814	−0.248101
$994$	17.4292	0.552822
$995$	0	0
$996$	−9.03612	−0.286320
$997$	25.3288	0.802173	0.401086	−	0.916040i	$-0.368633\pi$
$997$	25.3288	0.802173	0.401086	+	0.916040i	$0.368633\pi$
$998$	−43.2186	−1.36806
$999$	−6.81079	−0.215484

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.bq.1.1		3
5.4	even	2		1155.2.a.t.1.3	✓	3
15.14	odd	2		3465.2.a.bb.1.1		3
35.34	odd	2		8085.2.a.bl.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.t.1.3	✓	3	5.4	even	2
3465.2.a.bb.1.1		3	15.14	odd	2
5775.2.a.bq.1.1		3	1.1	even	1	trivial
8085.2.a.bl.1.3		3	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)

\(f(q)\)	\(=\)	\(q-2.34292 q^{2} -1.00000 q^{3} +3.48929 q^{4} +2.34292 q^{6} +1.00000 q^{7} -3.48929 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.34292 q^{2} -1.00000 q^{3} +3.48929 q^{4} +2.34292 q^{6} +1.00000 q^{7} -3.48929 q^{8} +1.00000 q^{9} +1.00000 q^{11} -3.48929 q^{12} -3.14637 q^{13} -2.34292 q^{14} +1.19656 q^{16} -4.19656 q^{17} -2.34292 q^{18} -7.17513 q^{19} -1.00000 q^{21} -2.34292 q^{22} +4.48929 q^{23} +3.48929 q^{24} +7.37169 q^{26} -1.00000 q^{27} +3.48929 q^{28} +5.34292 q^{29} +3.83221 q^{31} +4.17513 q^{32} -1.00000 q^{33} +9.83221 q^{34} +3.48929 q^{36} +6.81079 q^{37} +16.8108 q^{38} +3.14637 q^{39} -4.12494 q^{41} +2.34292 q^{42} +1.05019 q^{43} +3.48929 q^{44} -10.5181 q^{46} +3.83221 q^{47} -1.19656 q^{48} +1.00000 q^{49} +4.19656 q^{51} -10.9786 q^{52} -2.48929 q^{53} +2.34292 q^{54} -3.48929 q^{56} +7.17513 q^{57} -12.5181 q^{58} +2.02877 q^{59} +4.78202 q^{61} -8.97858 q^{62} +1.00000 q^{63} -12.1751 q^{64} +2.34292 q^{66} +4.97858 q^{67} -14.6430 q^{68} -4.48929 q^{69} -7.43910 q^{71} -3.48929 q^{72} -8.68585 q^{73} -15.9572 q^{74} -25.0361 q^{76} +1.00000 q^{77} -7.37169 q^{78} -15.8898 q^{79} +1.00000 q^{81} +9.66442 q^{82} +2.58967 q^{83} -3.48929 q^{84} -2.46052 q^{86} -5.34292 q^{87} -3.48929 q^{88} +4.61423 q^{89} -3.14637 q^{91} +15.6644 q^{92} -3.83221 q^{93} -8.97858 q^{94} -4.17513 q^{96} -8.22112 q^{97} -2.34292 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 3 q^{3} + 3 q^{4} + q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - q^2 - 3 * q^3 + 3 * q^4 + q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - q^{2} - 3 q^{3} + 3 q^{4} + q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} - 3 q^{12} - 8 q^{13} - q^{14} - q^{16} - 8 q^{17} - q^{18} - 2 q^{19} - 3 q^{21} - q^{22} + 6 q^{23} + 3 q^{24} - 2 q^{26} - 3 q^{27} + 3 q^{28} + 10 q^{29} - 2 q^{31} - 7 q^{32} - 3 q^{33} + 16 q^{34} + 3 q^{36} - 8 q^{37} + 22 q^{38} + 8 q^{39} + 4 q^{41} + q^{42} + 3 q^{44} - 6 q^{46} - 2 q^{47} + q^{48} + 3 q^{49} + 8 q^{51} - 18 q^{52} + q^{54} - 3 q^{56} + 2 q^{57} - 12 q^{58} - 12 q^{59} + 4 q^{61} - 12 q^{62} + 3 q^{63} - 17 q^{64} + q^{66} - 2 q^{68} - 6 q^{69} - 18 q^{71} - 3 q^{72} - 14 q^{73} - 18 q^{74} - 24 q^{76} + 3 q^{77} + 2 q^{78} + 2 q^{79} + 3 q^{81} + 2 q^{82} - 6 q^{83} - 3 q^{84} - 18 q^{86} - 10 q^{87} - 3 q^{88} - 10 q^{89} - 8 q^{91} + 20 q^{92} + 2 q^{93} - 12 q^{94} + 7 q^{96} - 10 q^{97} - q^{98} + 3 q^{99}+O(q^{100}) \) 3 * q - q^2 - 3 * q^3 + 3 * q^4 + q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 - 3 * q^12 - 8 * q^13 - q^14 - q^16 - 8 * q^17 - q^18 - 2 * q^19 - 3 * q^21 - q^22 + 6 * q^23 + 3 * q^24 - 2 * q^26 - 3 * q^27 + 3 * q^28 + 10 * q^29 - 2 * q^31 - 7 * q^32 - 3 * q^33 + 16 * q^34 + 3 * q^36 - 8 * q^37 + 22 * q^38 + 8 * q^39 + 4 * q^41 + q^42 + 3 * q^44 - 6 * q^46 - 2 * q^47 + q^48 + 3 * q^49 + 8 * q^51 - 18 * q^52 + q^54 - 3 * q^56 + 2 * q^57 - 12 * q^58 - 12 * q^59 + 4 * q^61 - 12 * q^62 + 3 * q^63 - 17 * q^64 + q^66 - 2 * q^68 - 6 * q^69 - 18 * q^71 - 3 * q^72 - 14 * q^73 - 18 * q^74 - 24 * q^76 + 3 * q^77 + 2 * q^78 + 2 * q^79 + 3 * q^81 + 2 * q^82 - 6 * q^83 - 3 * q^84 - 18 * q^86 - 10 * q^87 - 3 * q^88 - 10 * q^89 - 8 * q^91 + 20 * q^92 + 2 * q^93 - 12 * q^94 + 7 * q^96 - 10 * q^97 - q^98 + 3 * q^99

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