LMFDB - Embedded newform 275.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,2,Mod(1,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	275.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:	$2.19588605559$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.30278 q^{2} -2.30278 q^{3} -0.302776 q^{4} -3.00000 q^{6} -0.697224 q^{7} -3.00000 q^{8} +2.30278 q^{9} -1.00000 q^{11} +0.697224 q^{12} -5.00000 q^{13} -0.908327 q^{14} -3.30278 q^{16} -6.90833 q^{17} +3.00000 q^{18} -1.00000 q^{19} +1.60555 q^{21} -1.30278 q^{22} +7.30278 q^{23} +6.90833 q^{24} -6.51388 q^{26} +1.60555 q^{27} +0.211103 q^{28} +0.908327 q^{29} +10.2111 q^{31} +1.69722 q^{32} +2.30278 q^{33} -9.00000 q^{34} -0.697224 q^{36} -2.39445 q^{37} -1.30278 q^{38} +11.5139 q^{39} -5.60555 q^{41} +2.09167 q^{42} -7.21110 q^{43} +0.302776 q^{44} +9.51388 q^{46} +3.00000 q^{47} +7.60555 q^{48} -6.51388 q^{49} +15.9083 q^{51} +1.51388 q^{52} +1.30278 q^{53} +2.09167 q^{54} +2.09167 q^{56} +2.30278 q^{57} +1.18335 q^{58} -14.2111 q^{59} -7.90833 q^{61} +13.3028 q^{62} -1.60555 q^{63} +8.81665 q^{64} +3.00000 q^{66} +4.00000 q^{67} +2.09167 q^{68} -16.8167 q^{69} -2.60555 q^{71} -6.90833 q^{72} +7.90833 q^{73} -3.11943 q^{74} +0.302776 q^{76} +0.697224 q^{77} +15.0000 q^{78} -10.9083 q^{79} -10.6056 q^{81} -7.30278 q^{82} +3.51388 q^{83} -0.486122 q^{84} -9.39445 q^{86} -2.09167 q^{87} +3.00000 q^{88} +1.69722 q^{89} +3.48612 q^{91} -2.21110 q^{92} -23.5139 q^{93} +3.90833 q^{94} -3.90833 q^{96} -15.3028 q^{97} -8.48612 q^{98} -2.30278 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{3} + 3 q^{4} - 6 q^{6} - 5 q^{7} - 6 q^{8} + q^{9} - 2 q^{11} + 5 q^{12} - 10 q^{13} + 9 q^{14} - 3 q^{16} - 3 q^{17} + 6 q^{18} - 2 q^{19} - 4 q^{21} + q^{22} + 11 q^{23} + 3 q^{24}+ \cdots - q^{99}+O(q^{100}) $ 2 * q - q^2 - q^3 + 3 * q^4 - 6 * q^6 - 5 * q^7 - 6 * q^8 + q^9 - 2 * q^11 + 5 * q^12 - 10 * q^13 + 9 * q^14 - 3 * q^16 - 3 * q^17 + 6 * q^18 - 2 * q^19 - 4 * q^21 + q^22 + 11 * q^23 + 3 * q^24 + 5 * q^26 - 4 * q^27 - 14 * q^28 - 9 * q^29 + 6 * q^31 + 7 * q^32 + q^33 - 18 * q^34 - 5 * q^36 - 12 * q^37 + q^38 + 5 * q^39 - 4 * q^41 + 15 * q^42 - 3 * q^44 + q^46 + 6 * q^47 + 8 * q^48 + 5 * q^49 + 21 * q^51 - 15 * q^52 - q^53 + 15 * q^54 + 15 * q^56 + q^57 + 24 * q^58 - 14 * q^59 - 5 * q^61 + 23 * q^62 + 4 * q^63 - 4 * q^64 + 6 * q^66 + 8 * q^67 + 15 * q^68 - 12 * q^69 + 2 * q^71 - 3 * q^72 + 5 * q^73 + 19 * q^74 - 3 * q^76 + 5 * q^77 + 30 * q^78 - 11 * q^79 - 14 * q^81 - 11 * q^82 - 11 * q^83 - 19 * q^84 - 26 * q^86 - 15 * q^87 + 6 * q^88 + 7 * q^89 + 25 * q^91 + 10 * q^92 - 29 * q^93 - 3 * q^94 + 3 * q^96 - 27 * q^97 - 35 * q^98 - q^99

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 $

$2$	1.30278	0.921201	0.460601	−	0.887607i	$-0.347634\pi$
$2$	1.30278	0.921201	0.460601	+	0.887607i	$0.347634\pi$
$3$	−2.30278	−1.32951	−0.664754	−	0.747062i	$-0.731464\pi$
$3$	−2.30278	−1.32951	−0.664754	+	0.747062i	$0.731464\pi$
$4$	−0.302776	−0.151388
$5$	0	0
$5$	0	0
$6$	−3.00000	−1.22474
$7$	−0.697224	−0.263526	−0.131763	−	0.991281i	$-0.542064\pi$
$7$	−0.697224	−0.263526	−0.131763	+	0.991281i	$0.542064\pi$
$8$	−3.00000	−1.06066
$9$	2.30278	0.767592
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.697224	0.201271
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	−0.908327	−0.242761
$15$	0	0
$16$	−3.30278	−0.825694
$17$	−6.90833	−1.67552	−0.837758	−	0.546042i	$-0.816134\pi$
$17$	−6.90833	−1.67552	−0.837758	+	0.546042i	$0.816134\pi$
$18$	3.00000	0.707107
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	1.60555	0.350360
$22$	−1.30278	−0.277753
$23$	7.30278	1.52273	0.761367	−	0.648321i	$-0.224529\pi$
$23$	7.30278	1.52273	0.761367	+	0.648321i	$0.224529\pi$
$24$	6.90833	1.41016
$25$	0	0
$26$	−6.51388	−1.27748
$27$	1.60555	0.308988
$28$	0.211103	0.0398946
$29$	0.908327	0.168672	0.0843360	−	0.996437i	$-0.473123\pi$
$29$	0.908327	0.168672	0.0843360	+	0.996437i	$0.473123\pi$
$30$	0	0
$31$	10.2111	1.83397	0.916984	−	0.398924i	$-0.130616\pi$
$31$	10.2111	1.83397	0.916984	+	0.398924i	$0.130616\pi$
$32$	1.69722	0.300030
$33$	2.30278	0.400862
$34$	−9.00000	−1.54349
$35$	0	0
$36$	−0.697224	−0.116204
$37$	−2.39445	−0.393645	−0.196822	−	0.980439i	$-0.563062\pi$
$37$	−2.39445	−0.393645	−0.196822	+	0.980439i	$0.563062\pi$
$38$	−1.30278	−0.211338
$39$	11.5139	1.84370
$40$	0	0
$41$	−5.60555	−0.875440	−0.437720	−	0.899111i	$-0.644214\pi$
$41$	−5.60555	−0.875440	−0.437720	+	0.899111i	$0.644214\pi$
$42$	2.09167	0.322752
$43$	−7.21110	−1.09968	−0.549841	−	0.835269i	$-0.685312\pi$
$43$	−7.21110	−1.09968	−0.549841	+	0.835269i	$0.685312\pi$
$44$	0.302776	0.0456451
$45$	0	0
$46$	9.51388	1.40274
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	7.60555	1.09777
$49$	−6.51388	−0.930554
$50$	0	0
$51$	15.9083	2.22761
$52$	1.51388	0.209937
$53$	1.30278	0.178950	0.0894750	−	0.995989i	$-0.471481\pi$
$53$	1.30278	0.178950	0.0894750	+	0.995989i	$0.471481\pi$
$54$	2.09167	0.284641
$55$	0	0
$56$	2.09167	0.279512
$57$	2.30278	0.305010
$58$	1.18335	0.155381
$59$	−14.2111	−1.85013	−0.925064	−	0.379811i	$-0.875989\pi$
$59$	−14.2111	−1.85013	−0.925064	+	0.379811i	$0.875989\pi$
$60$	0	0
$61$	−7.90833	−1.01256	−0.506279	−	0.862370i	$-0.668979\pi$
$61$	−7.90833	−1.01256	−0.506279	+	0.862370i	$0.668979\pi$
$62$	13.3028	1.68945
$63$	−1.60555	−0.202280
$64$	8.81665	1.10208
$65$	0	0
$66$	3.00000	0.369274
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	2.09167	0.253653
$69$	−16.8167	−2.02449
$70$	0	0
$71$	−2.60555	−0.309222	−0.154611	−	0.987975i	$-0.549412\pi$
$71$	−2.60555	−0.309222	−0.154611	+	0.987975i	$0.549412\pi$
$72$	−6.90833	−0.814154
$73$	7.90833	0.925600	0.462800	−	0.886463i	$-0.346845\pi$
$73$	7.90833	0.925600	0.462800	+	0.886463i	$0.346845\pi$
$74$	−3.11943	−0.362626
$75$	0	0
$76$	0.302776	0.0347307
$77$	0.697224	0.0794561
$78$	15.0000	1.69842
$79$	−10.9083	−1.22728	−0.613641	−	0.789585i	$-0.710296\pi$
$79$	−10.9083	−1.22728	−0.613641	+	0.789585i	$0.710296\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	−7.30278	−0.806457
$83$	3.51388	0.385698	0.192849	−	0.981228i	$-0.438227\pi$
$83$	3.51388	0.385698	0.192849	+	0.981228i	$0.438227\pi$
$84$	−0.486122	−0.0530402
$85$	0	0
$86$	−9.39445	−1.01303
$87$	−2.09167	−0.224251
$88$	3.00000	0.319801
$89$	1.69722	0.179905	0.0899527	−	0.995946i	$-0.471328\pi$
$89$	1.69722	0.179905	0.0899527	+	0.995946i	$0.471328\pi$
$90$	0	0
$91$	3.48612	0.365445
$92$	−2.21110	−0.230523
$93$	−23.5139	−2.43828
$94$	3.90833	0.403113
$95$	0	0
$96$	−3.90833	−0.398892
$97$	−15.3028	−1.55376	−0.776881	−	0.629648i	$-0.783199\pi$
$97$	−15.3028	−1.55376	−0.776881	+	0.629648i	$0.783199\pi$
$98$	−8.48612	−0.857228
$99$	−2.30278	−0.231438
$100$	0	0
$101$	−0.513878	−0.0511328	−0.0255664	−	0.999673i	$-0.508139\pi$
$101$	−0.513878	−0.0511328	−0.0255664	+	0.999673i	$0.508139\pi$
$102$	20.7250	2.05208
$103$	−2.90833	−0.286566	−0.143283	−	0.989682i	$-0.545766\pi$
$103$	−2.90833	−0.286566	−0.143283	+	0.989682i	$0.545766\pi$
$104$	15.0000	1.47087
$105$	0	0
$106$	1.69722	0.164849
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	−0.486122	−0.0467771
$109$	11.5139	1.10283	0.551415	−	0.834231i	$-0.314088\pi$
$109$	11.5139	1.10283	0.551415	+	0.834231i	$0.314088\pi$
$110$	0	0
$111$	5.51388	0.523354
$112$	2.30278	0.217592
$113$	10.8167	1.01755	0.508773	−	0.860901i	$-0.330099\pi$
$113$	10.8167	1.01755	0.508773	+	0.860901i	$0.330099\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	−0.275019	−0.0255349
$117$	−11.5139	−1.06446
$118$	−18.5139	−1.70434
$119$	4.81665	0.441542
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.3028	−0.932769
$123$	12.9083	1.16390
$124$	−3.09167	−0.277640
$125$	0	0
$126$	−2.09167	−0.186341
$127$	−8.11943	−0.720483	−0.360241	−	0.932859i	$-0.617306\pi$
$127$	−8.11943	−0.720483	−0.360241	+	0.932859i	$0.617306\pi$
$128$	8.09167	0.715210
$129$	16.6056	1.46204
$130$	0	0
$131$	−9.90833	−0.865695	−0.432847	−	0.901467i	$-0.642491\pi$
$131$	−9.90833	−0.865695	−0.432847	+	0.901467i	$0.642491\pi$
$132$	−0.697224	−0.0606856
$133$	0.697224	0.0604570
$134$	5.21110	0.450171
$135$	0	0
$136$	20.7250	1.77715
$137$	12.9083	1.10283	0.551416	−	0.834230i	$-0.314088\pi$
$137$	12.9083	1.10283	0.551416	+	0.834230i	$0.314088\pi$
$138$	−21.9083	−1.86496
$139$	−6.21110	−0.526819	−0.263409	−	0.964684i	$-0.584847\pi$
$139$	−6.21110	−0.526819	−0.263409	+	0.964684i	$0.584847\pi$
$140$	0	0
$141$	−6.90833	−0.581786
$142$	−3.39445	−0.284856
$143$	5.00000	0.418121
$144$	−7.60555	−0.633796
$145$	0	0
$146$	10.3028	0.852664
$147$	15.0000	1.23718
$148$	0.724981	0.0595930
$149$	17.2111	1.40999	0.704994	−	0.709213i	$-0.250950\pi$
$149$	17.2111	1.40999	0.704994	+	0.709213i	$0.250950\pi$
$150$	0	0
$151$	0.816654	0.0664583	0.0332292	−	0.999448i	$-0.489421\pi$
$151$	0.816654	0.0664583	0.0332292	+	0.999448i	$0.489421\pi$
$152$	3.00000	0.243332
$153$	−15.9083	−1.28611
$154$	0.908327	0.0731951
$155$	0	0
$156$	−3.48612	−0.279113
$157$	−19.2111	−1.53321	−0.766606	−	0.642117i	$-0.778056\pi$
$157$	−19.2111	−1.53321	−0.766606	+	0.642117i	$0.778056\pi$
$158$	−14.2111	−1.13057
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−5.09167	−0.401280
$162$	−13.8167	−1.08554
$163$	−9.30278	−0.728650	−0.364325	−	0.931272i	$-0.618700\pi$
$163$	−9.30278	−0.728650	−0.364325	+	0.931272i	$0.618700\pi$
$164$	1.69722	0.132531
$165$	0	0
$166$	4.57779	0.355306
$167$	−13.4222	−1.03864	−0.519321	−	0.854579i	$-0.673815\pi$
$167$	−13.4222	−1.03864	−0.519321	+	0.854579i	$0.673815\pi$
$168$	−4.81665	−0.371613
$169$	12.0000	0.923077
$170$	0	0
$171$	−2.30278	−0.176098
$172$	2.18335	0.166479
$173$	4.81665	0.366203	0.183102	−	0.983094i	$-0.441386\pi$
$173$	4.81665	0.366203	0.183102	+	0.983094i	$0.441386\pi$
$174$	−2.72498	−0.206580
$175$	0	0
$176$	3.30278	0.248956
$177$	32.7250	2.45976
$178$	2.21110	0.165729
$179$	12.5139	0.935331	0.467666	−	0.883905i	$-0.345095\pi$
$179$	12.5139	0.935331	0.467666	+	0.883905i	$0.345095\pi$
$180$	0	0
$181$	−19.9083	−1.47977	−0.739887	−	0.672731i	$-0.765121\pi$
$181$	−19.9083	−1.47977	−0.739887	+	0.672731i	$0.765121\pi$
$182$	4.54163	0.336648
$183$	18.2111	1.34620
$184$	−21.9083	−1.61510
$185$	0	0
$186$	−30.6333	−2.24614
$187$	6.90833	0.505187
$188$	−0.908327	−0.0662465
$189$	−1.11943	−0.0814265
$190$	0	0
$191$	10.3028	0.745483	0.372741	−	0.927935i	$-0.378418\pi$
$191$	10.3028	0.745483	0.372741	+	0.927935i	$0.378418\pi$
$192$	−20.3028	−1.46523
$193$	−13.2111	−0.950956	−0.475478	−	0.879728i	$-0.657725\pi$
$193$	−13.2111	−0.950956	−0.475478	+	0.879728i	$0.657725\pi$
$194$	−19.9361	−1.43133
$195$	0	0
$196$	1.97224	0.140875
$197$	−13.3028	−0.947784	−0.473892	−	0.880583i	$-0.657151\pi$
$197$	−13.3028	−0.947784	−0.473892	+	0.880583i	$0.657151\pi$
$198$	−3.00000	−0.213201
$199$	−6.48612	−0.459789	−0.229894	−	0.973216i	$-0.573838\pi$
$199$	−6.48612	−0.459789	−0.229894	+	0.973216i	$0.573838\pi$
$200$	0	0
$201$	−9.21110	−0.649701
$202$	−0.669468	−0.0471036
$203$	−0.633308	−0.0444495
$204$	−4.81665	−0.337233
$205$	0	0
$206$	−3.78890	−0.263985
$207$	16.8167	1.16884
$208$	16.5139	1.14503
$209$	1.00000	0.0691714
$210$	0	0
$211$	−25.2389	−1.73751	−0.868757	−	0.495238i	$-0.835081\pi$
$211$	−25.2389	−1.73751	−0.868757	+	0.495238i	$0.835081\pi$
$212$	−0.394449	−0.0270908
$213$	6.00000	0.411113
$214$	3.90833	0.267168
$215$	0	0
$216$	−4.81665	−0.327732
$217$	−7.11943	−0.483298
$218$	15.0000	1.01593
$219$	−18.2111	−1.23059
$220$	0	0
$221$	34.5416	2.32352
$222$	7.18335	0.482115
$223$	22.6333	1.51564	0.757819	−	0.652465i	$-0.226265\pi$
$223$	22.6333	1.51564	0.757819	+	0.652465i	$0.226265\pi$
$224$	−1.18335	−0.0790656
$225$	0	0
$226$	14.0917	0.937364
$227$	−1.69722	−0.112649	−0.0563244	−	0.998413i	$-0.517938\pi$
$227$	−1.69722	−0.112649	−0.0563244	+	0.998413i	$0.517938\pi$
$228$	−0.697224	−0.0461748
$229$	−18.7250	−1.23738	−0.618691	−	0.785635i	$-0.712337\pi$
$229$	−18.7250	−1.23738	−0.618691	+	0.785635i	$0.712337\pi$
$230$	0	0
$231$	−1.60555	−0.105638
$232$	−2.72498	−0.178904
$233$	−15.9083	−1.04219	−0.521095	−	0.853499i	$-0.674476\pi$
$233$	−15.9083	−1.04219	−0.521095	+	0.853499i	$0.674476\pi$
$234$	−15.0000	−0.980581
$235$	0	0
$236$	4.30278	0.280087
$237$	25.1194	1.63168
$238$	6.27502	0.406749
$239$	21.1194	1.36610	0.683051	−	0.730371i	$-0.260653\pi$
$239$	21.1194	1.36610	0.683051	+	0.730371i	$0.260653\pi$
$240$	0	0
$241$	21.9361	1.41303	0.706514	−	0.707699i	$-0.250267\pi$
$241$	21.9361	1.41303	0.706514	+	0.707699i	$0.250267\pi$
$242$	1.30278	0.0837456
$243$	19.6056	1.25770
$244$	2.39445	0.153289
$245$	0	0
$246$	16.8167	1.07219
$247$	5.00000	0.318142
$248$	−30.6333	−1.94522
$249$	−8.09167	−0.512789
$250$	0	0
$251$	6.90833	0.436050	0.218025	−	0.975943i	$-0.430039\pi$
$251$	6.90833	0.436050	0.218025	+	0.975943i	$0.430039\pi$
$252$	0.486122	0.0306228
$253$	−7.30278	−0.459122
$254$	−10.5778	−0.663710
$255$	0	0
$256$	−7.09167	−0.443230
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	21.6333	1.34683
$259$	1.66947	0.103736
$260$	0	0
$261$	2.09167	0.129471
$262$	−12.9083	−0.797479
$263$	22.8167	1.40694	0.703468	−	0.710727i	$-0.251634\pi$
$263$	22.8167	1.40694	0.703468	+	0.710727i	$0.251634\pi$
$264$	−6.90833	−0.425178
$265$	0	0
$266$	0.908327	0.0556931
$267$	−3.90833	−0.239186
$268$	−1.21110	−0.0739799
$269$	8.72498	0.531971	0.265986	−	0.963977i	$-0.414303\pi$
$269$	8.72498	0.531971	0.265986	+	0.963977i	$0.414303\pi$
$270$	0	0
$271$	−0.211103	−0.0128236	−0.00641178	−	0.999979i	$-0.502041\pi$
$271$	−0.211103	−0.0128236	−0.00641178	+	0.999979i	$0.502041\pi$
$272$	22.8167	1.38346
$273$	−8.02776	−0.485862
$274$	16.8167	1.01593
$275$	0	0
$276$	5.09167	0.306483
$277$	−14.3944	−0.864879	−0.432439	−	0.901663i	$-0.642347\pi$
$277$	−14.3944	−0.864879	−0.432439	+	0.901663i	$0.642347\pi$
$278$	−8.09167	−0.485306
$279$	23.5139	1.40774
$280$	0	0
$281$	−1.18335	−0.0705925	−0.0352963	−	0.999377i	$-0.511237\pi$
$281$	−1.18335	−0.0705925	−0.0352963	+	0.999377i	$0.511237\pi$
$282$	−9.00000	−0.535942
$283$	−6.30278	−0.374661	−0.187331	−	0.982297i	$-0.559984\pi$
$283$	−6.30278	−0.374661	−0.187331	+	0.982297i	$0.559984\pi$
$284$	0.788897	0.0468125
$285$	0	0
$286$	6.51388	0.385174
$287$	3.90833	0.230701
$288$	3.90833	0.230300
$289$	30.7250	1.80735
$290$	0	0
$291$	35.2389	2.06574
$292$	−2.39445	−0.140125
$293$	−0.788897	−0.0460879	−0.0230439	−	0.999734i	$-0.507336\pi$
$293$	−0.788897	−0.0460879	−0.0230439	+	0.999734i	$0.507336\pi$
$294$	19.5416	1.13969
$295$	0	0
$296$	7.18335	0.417524
$297$	−1.60555	−0.0931635
$298$	22.4222	1.29888
$299$	−36.5139	−2.11165
$300$	0	0
$301$	5.02776	0.289795
$302$	1.06392	0.0612215
$303$	1.18335	0.0679815
$304$	3.30278	0.189427
$305$	0	0
$306$	−20.7250	−1.18477
$307$	16.9083	0.965009	0.482505	−	0.875893i	$-0.339727\pi$
$307$	16.9083	0.965009	0.482505	+	0.875893i	$0.339727\pi$
$308$	−0.211103	−0.0120287
$309$	6.69722	0.380992
$310$	0	0
$311$	4.81665	0.273127	0.136564	−	0.990631i	$-0.456394\pi$
$311$	4.81665	0.273127	0.136564	+	0.990631i	$0.456394\pi$
$312$	−34.5416	−1.95553
$313$	−0.183346	−0.0103633	−0.00518167	−	0.999987i	$-0.501649\pi$
$313$	−0.183346	−0.0103633	−0.00518167	+	0.999987i	$0.501649\pi$
$314$	−25.0278	−1.41240
$315$	0	0
$316$	3.30278	0.185796
$317$	0.908327	0.0510167	0.0255084	−	0.999675i	$-0.491880\pi$
$317$	0.908327	0.0510167	0.0255084	+	0.999675i	$0.491880\pi$
$318$	−3.90833	−0.219168
$319$	−0.908327	−0.0508565
$320$	0	0
$321$	−6.90833	−0.385585
$322$	−6.63331	−0.369660
$323$	6.90833	0.384390
$324$	3.21110	0.178395
$325$	0	0
$326$	−12.1194	−0.671233
$327$	−26.5139	−1.46622
$328$	16.8167	0.928544
$329$	−2.09167	−0.115318
$330$	0	0
$331$	−21.6056	−1.18755	−0.593774	−	0.804632i	$-0.702363\pi$
$331$	−21.6056	−1.18755	−0.593774	+	0.804632i	$0.702363\pi$
$332$	−1.06392	−0.0583900
$333$	−5.51388	−0.302159
$334$	−17.4861	−0.956798
$335$	0	0
$336$	−5.30278	−0.289290
$337$	30.8444	1.68020	0.840101	−	0.542430i	$-0.182496\pi$
$337$	30.8444	1.68020	0.840101	+	0.542430i	$0.182496\pi$
$338$	15.6333	0.850340
$339$	−24.9083	−1.35283
$340$	0	0
$341$	−10.2111	−0.552962
$342$	−3.00000	−0.162221
$343$	9.42221	0.508751
$344$	21.6333	1.16639
$345$	0	0
$346$	6.27502	0.337347
$347$	12.5139	0.671780	0.335890	−	0.941901i	$-0.390963\pi$
$347$	12.5139	0.671780	0.335890	+	0.941901i	$0.390963\pi$
$348$	0.633308	0.0339489
$349$	−5.18335	−0.277458	−0.138729	−	0.990330i	$-0.544302\pi$
$349$	−5.18335	−0.277458	−0.138729	+	0.990330i	$0.544302\pi$
$350$	0	0
$351$	−8.02776	−0.428490
$352$	−1.69722	−0.0904624
$353$	−18.6333	−0.991751	−0.495875	−	0.868394i	$-0.665153\pi$
$353$	−18.6333	−0.991751	−0.495875	+	0.868394i	$0.665153\pi$
$354$	42.6333	2.26593
$355$	0	0
$356$	−0.513878	−0.0272355
$357$	−11.0917	−0.587034
$358$	16.3028	0.861628
$359$	0.788897	0.0416364	0.0208182	−	0.999783i	$-0.493373\pi$
$359$	0.788897	0.0416364	0.0208182	+	0.999783i	$0.493373\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−25.9361	−1.36317
$363$	−2.30278	−0.120864
$364$	−1.05551	−0.0553239
$365$	0	0
$366$	23.7250	1.24012
$367$	20.6972	1.08039	0.540193	−	0.841541i	$-0.318351\pi$
$367$	20.6972	1.08039	0.540193	+	0.841541i	$0.318351\pi$
$368$	−24.1194	−1.25731
$369$	−12.9083	−0.671981
$370$	0	0
$371$	−0.908327	−0.0471580
$372$	7.11943	0.369125
$373$	−27.4222	−1.41987	−0.709934	−	0.704268i	$-0.751275\pi$
$373$	−27.4222	−1.41987	−0.709934	+	0.704268i	$0.751275\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−9.00000	−0.464140
$377$	−4.54163	−0.233906
$378$	−1.45837	−0.0750102
$379$	3.18335	0.163518	0.0817588	−	0.996652i	$-0.473946\pi$
$379$	3.18335	0.163518	0.0817588	+	0.996652i	$0.473946\pi$
$380$	0	0
$381$	18.6972	0.957888
$382$	13.4222	0.686740
$383$	−21.6333	−1.10541	−0.552705	−	0.833377i	$-0.686404\pi$
$383$	−21.6333	−1.10541	−0.552705	+	0.833377i	$0.686404\pi$
$384$	−18.6333	−0.950877
$385$	0	0
$386$	−17.2111	−0.876022
$387$	−16.6056	−0.844108
$388$	4.63331	0.235221
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	−50.4500	−2.55136
$392$	19.5416	0.987002
$393$	22.8167	1.15095
$394$	−17.3305	−0.873100
$395$	0	0
$396$	0.697224	0.0350368
$397$	−21.6972	−1.08895	−0.544476	−	0.838776i	$-0.683272\pi$
$397$	−21.6972	−1.08895	−0.544476	+	0.838776i	$0.683272\pi$
$398$	−8.44996	−0.423558
$399$	−1.60555	−0.0803781
$400$	0	0
$401$	−12.7889	−0.638647	−0.319324	−	0.947646i	$-0.603456\pi$
$401$	−12.7889	−0.638647	−0.319324	+	0.947646i	$0.603456\pi$
$402$	−12.0000	−0.598506
$403$	−51.0555	−2.54326
$404$	0.155590	0.00774088
$405$	0	0
$406$	−0.825058	−0.0409469
$407$	2.39445	0.118688
$408$	−47.7250	−2.36274
$409$	−6.21110	−0.307119	−0.153560	−	0.988139i	$-0.549074\pi$
$409$	−6.21110	−0.307119	−0.153560	+	0.988139i	$0.549074\pi$
$410$	0	0
$411$	−29.7250	−1.46623
$412$	0.880571	0.0433826
$413$	9.90833	0.487557
$414$	21.9083	1.07674
$415$	0	0
$416$	−8.48612	−0.416066
$417$	14.3028	0.700410
$418$	1.30278	0.0637208
$419$	6.39445	0.312389	0.156195	−	0.987726i	$-0.450077\pi$
$419$	6.39445	0.312389	0.156195	+	0.987726i	$0.450077\pi$
$420$	0	0
$421$	0.697224	0.0339806	0.0169903	−	0.999856i	$-0.494592\pi$
$421$	0.697224	0.0339806	0.0169903	+	0.999856i	$0.494592\pi$
$422$	−32.8806	−1.60060
$423$	6.90833	0.335894
$424$	−3.90833	−0.189805
$425$	0	0
$426$	7.81665	0.378718
$427$	5.51388	0.266835
$428$	−0.908327	−0.0439056
$429$	−11.5139	−0.555895
$430$	0	0
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	−5.30278	−0.255130
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	−9.27502	−0.445215
$435$	0	0
$436$	−3.48612	−0.166955
$437$	−7.30278	−0.349339
$438$	−23.7250	−1.13362
$439$	24.3028	1.15991	0.579954	−	0.814649i	$-0.303070\pi$
$439$	24.3028	1.15991	0.579954	+	0.814649i	$0.303070\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	45.0000	2.14043
$443$	8.60555	0.408862	0.204431	−	0.978881i	$-0.434466\pi$
$443$	8.60555	0.408862	0.204431	+	0.978881i	$0.434466\pi$
$444$	−1.66947	−0.0792294
$445$	0	0
$446$	29.4861	1.39621
$447$	−39.6333	−1.87459
$448$	−6.14719	−0.290427
$449$	23.4861	1.10838	0.554189	−	0.832391i	$-0.313028\pi$
$449$	23.4861	1.10838	0.554189	+	0.832391i	$0.313028\pi$
$450$	0	0
$451$	5.60555	0.263955
$452$	−3.27502	−0.154044
$453$	−1.88057	−0.0883569
$454$	−2.21110	−0.103772
$455$	0	0
$456$	−6.90833	−0.323512
$457$	20.6972	0.968175	0.484088	−	0.875020i	$-0.339152\pi$
$457$	20.6972	0.968175	0.484088	+	0.875020i	$0.339152\pi$
$458$	−24.3944	−1.13988
$459$	−11.0917	−0.517715
$460$	0	0
$461$	32.2111	1.50022	0.750110	−	0.661313i	$-0.230000\pi$
$461$	32.2111	1.50022	0.750110	+	0.661313i	$0.230000\pi$
$462$	−2.09167	−0.0973134
$463$	−11.7889	−0.547877	−0.273938	−	0.961747i	$-0.588326\pi$
$463$	−11.7889	−0.547877	−0.273938	+	0.961747i	$0.588326\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−20.7250	−0.960066
$467$	18.6333	0.862247	0.431123	−	0.902293i	$-0.358117\pi$
$467$	18.6333	0.862247	0.431123	+	0.902293i	$0.358117\pi$
$468$	3.48612	0.161146
$469$	−2.78890	−0.128779
$470$	0	0
$471$	44.2389	2.03842
$472$	42.6333	1.96236
$473$	7.21110	0.331567
$474$	32.7250	1.50311
$475$	0	0
$476$	−1.45837	−0.0668441
$477$	3.00000	0.137361
$478$	27.5139	1.25846
$479$	−34.8167	−1.59081	−0.795407	−	0.606076i	$-0.792743\pi$
$479$	−34.8167	−1.59081	−0.795407	+	0.606076i	$0.792743\pi$
$480$	0	0
$481$	11.9722	0.545887
$482$	28.5778	1.30168
$483$	11.7250	0.533505
$484$	−0.302776	−0.0137625
$485$	0	0
$486$	25.5416	1.15859
$487$	−4.21110	−0.190823	−0.0954116	−	0.995438i	$-0.530417\pi$
$487$	−4.21110	−0.190823	−0.0954116	+	0.995438i	$0.530417\pi$
$488$	23.7250	1.07398
$489$	21.4222	0.968746
$490$	0	0
$491$	−9.78890	−0.441767	−0.220883	−	0.975300i	$-0.570894\pi$
$491$	−9.78890	−0.441767	−0.220883	+	0.975300i	$0.570894\pi$
$492$	−3.90833	−0.176201
$493$	−6.27502	−0.282613
$494$	6.51388	0.293073
$495$	0	0
$496$	−33.7250	−1.51430
$497$	1.81665	0.0814881
$498$	−10.5416	−0.472382
$499$	−3.48612	−0.156060	−0.0780301	−	0.996951i	$-0.524863\pi$
$499$	−3.48612	−0.156060	−0.0780301	+	0.996951i	$0.524863\pi$
$500$	0	0
$501$	30.9083	1.38088
$502$	9.00000	0.401690
$503$	−9.39445	−0.418878	−0.209439	−	0.977822i	$-0.567164\pi$
$503$	−9.39445	−0.418878	−0.209439	+	0.977822i	$0.567164\pi$
$504$	4.81665	0.214551
$505$	0	0
$506$	−9.51388	−0.422943
$507$	−27.6333	−1.22724
$508$	2.45837	0.109072
$509$	−22.6972	−1.00604	−0.503018	−	0.864276i	$-0.667777\pi$
$509$	−22.6972	−1.00604	−0.503018	+	0.864276i	$0.667777\pi$
$510$	0	0
$511$	−5.51388	−0.243920
$512$	−25.4222	−1.12351
$513$	−1.60555	−0.0708868
$514$	23.4500	1.03433
$515$	0	0
$516$	−5.02776	−0.221335
$517$	−3.00000	−0.131940
$518$	2.17494	0.0955615
$519$	−11.0917	−0.486870
$520$	0	0
$521$	−41.4500	−1.81596	−0.907978	−	0.419018i	$-0.862374\pi$
$521$	−41.4500	−1.81596	−0.907978	+	0.419018i	$0.862374\pi$
$522$	2.72498	0.119269
$523$	32.4222	1.41772	0.708862	−	0.705347i	$-0.249209\pi$
$523$	32.4222	1.41772	0.708862	+	0.705347i	$0.249209\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	29.7250	1.29607
$527$	−70.5416	−3.07284
$528$	−7.60555	−0.330989
$529$	30.3305	1.31872
$530$	0	0
$531$	−32.7250	−1.42014
$532$	−0.211103	−0.00915246
$533$	28.0278	1.21402
$534$	−5.09167	−0.220338
$535$	0	0
$536$	−12.0000	−0.518321
$537$	−28.8167	−1.24353
$538$	11.3667	0.490053
$539$	6.51388	0.280573
$540$	0	0
$541$	25.7250	1.10600	0.553002	−	0.833180i	$-0.313482\pi$
$541$	25.7250	1.10600	0.553002	+	0.833180i	$0.313482\pi$
$542$	−0.275019	−0.0118131
$543$	45.8444	1.96737
$544$	−11.7250	−0.502704
$545$	0	0
$546$	−10.4584	−0.447577
$547$	7.11943	0.304405	0.152202	−	0.988349i	$-0.451363\pi$
$547$	7.11943	0.304405	0.152202	+	0.988349i	$0.451363\pi$
$548$	−3.90833	−0.166955
$549$	−18.2111	−0.777231
$550$	0	0
$551$	−0.908327	−0.0386960
$552$	50.4500	2.14729
$553$	7.60555	0.323421
$554$	−18.7527	−0.796727
$555$	0	0
$556$	1.88057	0.0797540
$557$	−19.4222	−0.822945	−0.411473	−	0.911422i	$-0.634985\pi$
$557$	−19.4222	−0.822945	−0.411473	+	0.911422i	$0.634985\pi$
$558$	30.6333	1.29681
$559$	36.0555	1.52499
$560$	0	0
$561$	−15.9083	−0.671650
$562$	−1.54163	−0.0650299
$563$	−8.09167	−0.341023	−0.170512	−	0.985356i	$-0.554542\pi$
$563$	−8.09167	−0.341023	−0.170512	+	0.985356i	$0.554542\pi$
$564$	2.09167	0.0880753
$565$	0	0
$566$	−8.21110	−0.345138
$567$	7.39445	0.310538
$568$	7.81665	0.327980
$569$	−46.1472	−1.93459	−0.967295	−	0.253653i	$-0.918368\pi$
$569$	−46.1472	−1.93459	−0.967295	+	0.253653i	$0.918368\pi$
$570$	0	0
$571$	22.3305	0.934504	0.467252	−	0.884124i	$-0.345244\pi$
$571$	22.3305	0.934504	0.467252	+	0.884124i	$0.345244\pi$
$572$	−1.51388	−0.0632984
$573$	−23.7250	−0.991125
$574$	5.09167	0.212522
$575$	0	0
$576$	20.3028	0.845949
$577$	−44.3583	−1.84666	−0.923330	−	0.384008i	$-0.874544\pi$
$577$	−44.3583	−1.84666	−0.923330	+	0.384008i	$0.874544\pi$
$578$	40.0278	1.66494
$579$	30.4222	1.26430
$580$	0	0
$581$	−2.44996	−0.101642
$582$	45.9083	1.90296
$583$	−1.30278	−0.0539555
$584$	−23.7250	−0.981747
$585$	0	0
$586$	−1.02776	−0.0424562
$587$	−16.5416	−0.682746	−0.341373	−	0.939928i	$-0.610892\pi$
$587$	−16.5416	−0.682746	−0.341373	+	0.939928i	$0.610892\pi$
$588$	−4.54163	−0.187294
$589$	−10.2111	−0.420741
$590$	0	0
$591$	30.6333	1.26009
$592$	7.90833	0.325030
$593$	6.39445	0.262589	0.131294	−	0.991343i	$-0.458087\pi$
$593$	6.39445	0.262589	0.131294	+	0.991343i	$0.458087\pi$
$594$	−2.09167	−0.0858224
$595$	0	0
$596$	−5.21110	−0.213455
$597$	14.9361	0.611293
$598$	−47.5694	−1.94526
$599$	−24.9083	−1.01773	−0.508863	−	0.860847i	$-0.669934\pi$
$599$	−24.9083	−1.01773	−0.508863	+	0.860847i	$0.669934\pi$
$600$	0	0
$601$	−1.90833	−0.0778423	−0.0389211	−	0.999242i	$-0.512392\pi$
$601$	−1.90833	−0.0778423	−0.0389211	+	0.999242i	$0.512392\pi$
$602$	6.55004	0.266960
$603$	9.21110	0.375105
$604$	−0.247263	−0.0100610
$605$	0	0
$606$	1.54163	0.0626246
$607$	−7.21110	−0.292690	−0.146345	−	0.989234i	$-0.546751\pi$
$607$	−7.21110	−0.292690	−0.146345	+	0.989234i	$0.546751\pi$
$608$	−1.69722	−0.0688315
$609$	1.45837	0.0590959
$610$	0	0
$611$	−15.0000	−0.606835
$612$	4.81665	0.194702
$613$	15.8806	0.641410	0.320705	−	0.947179i	$-0.396080\pi$
$613$	15.8806	0.641410	0.320705	+	0.947179i	$0.396080\pi$
$614$	22.0278	0.888968
$615$	0	0
$616$	−2.09167	−0.0842759
$617$	−3.39445	−0.136655	−0.0683277	−	0.997663i	$-0.521766\pi$
$617$	−3.39445	−0.136655	−0.0683277	+	0.997663i	$0.521766\pi$
$618$	8.72498	0.350970
$619$	−11.4222	−0.459097	−0.229549	−	0.973297i	$-0.573725\pi$
$619$	−11.4222	−0.459097	−0.229549	+	0.973297i	$0.573725\pi$
$620$	0	0
$621$	11.7250	0.470507
$622$	6.27502	0.251605
$623$	−1.18335	−0.0474098
$624$	−38.0278	−1.52233
$625$	0	0
$626$	−0.238859	−0.00954672
$627$	−2.30278	−0.0919640
$628$	5.81665	0.232110
$629$	16.5416	0.659558
$630$	0	0
$631$	6.93608	0.276121	0.138061	−	0.990424i	$-0.455913\pi$
$631$	6.93608	0.276121	0.138061	+	0.990424i	$0.455913\pi$
$632$	32.7250	1.30173
$633$	58.1194	2.31004
$634$	1.18335	0.0469967
$635$	0	0
$636$	0.908327	0.0360175
$637$	32.5694	1.29045
$638$	−1.18335	−0.0468491
$639$	−6.00000	−0.237356
$640$	0	0
$641$	27.7889	1.09760	0.548798	−	0.835955i	$-0.315086\pi$
$641$	27.7889	1.09760	0.548798	+	0.835955i	$0.315086\pi$
$642$	−9.00000	−0.355202
$643$	22.0000	0.867595	0.433798	−	0.901010i	$-0.357173\pi$
$643$	22.0000	0.867595	0.433798	+	0.901010i	$0.357173\pi$
$644$	1.54163	0.0607489
$645$	0	0
$646$	9.00000	0.354100
$647$	33.2389	1.30675	0.653377	−	0.757033i	$-0.273352\pi$
$647$	33.2389	1.30675	0.653377	+	0.757033i	$0.273352\pi$
$648$	31.8167	1.24988
$649$	14.2111	0.557835
$650$	0	0
$651$	16.3944	0.642549
$652$	2.81665	0.110309
$653$	−6.11943	−0.239472	−0.119736	−	0.992806i	$-0.538205\pi$
$653$	−6.11943	−0.239472	−0.119736	+	0.992806i	$0.538205\pi$
$654$	−34.5416	−1.35068
$655$	0	0
$656$	18.5139	0.722846
$657$	18.2111	0.710483
$658$	−2.72498	−0.106231
$659$	−30.9083	−1.20402	−0.602009	−	0.798489i	$-0.705633\pi$
$659$	−30.9083	−1.20402	−0.602009	+	0.798489i	$0.705633\pi$
$660$	0	0
$661$	−8.81665	−0.342928	−0.171464	−	0.985190i	$-0.554850\pi$
$661$	−8.81665	−0.342928	−0.171464	+	0.985190i	$0.554850\pi$
$662$	−28.1472	−1.09397
$663$	−79.5416	−3.08914
$664$	−10.5416	−0.409095
$665$	0	0
$666$	−7.18335	−0.278349
$667$	6.63331	0.256843
$668$	4.06392	0.157238
$669$	−52.1194	−2.01505
$670$	0	0
$671$	7.90833	0.305298
$672$	2.72498	0.105118
$673$	−30.0278	−1.15748	−0.578742	−	0.815510i	$-0.696456\pi$
$673$	−30.0278	−1.15748	−0.578742	+	0.815510i	$0.696456\pi$
$674$	40.1833	1.54780
$675$	0	0
$676$	−3.63331	−0.139743
$677$	−24.2389	−0.931575	−0.465788	−	0.884897i	$-0.654229\pi$
$677$	−24.2389	−0.931575	−0.465788	+	0.884897i	$0.654229\pi$
$678$	−32.4500	−1.24623
$679$	10.6695	0.409457
$680$	0	0
$681$	3.90833	0.149767
$682$	−13.3028	−0.509390
$683$	−47.8444	−1.83072	−0.915358	−	0.402642i	$-0.868092\pi$
$683$	−47.8444	−1.83072	−0.915358	+	0.402642i	$0.868092\pi$
$684$	0.697224	0.0266590
$685$	0	0
$686$	12.2750	0.468662
$687$	43.1194	1.64511
$688$	23.8167	0.908001
$689$	−6.51388	−0.248159
$690$	0	0
$691$	27.5416	1.04773	0.523867	−	0.851800i	$-0.324489\pi$
$691$	27.5416	1.04773	0.523867	+	0.851800i	$0.324489\pi$
$692$	−1.45837	−0.0554387
$693$	1.60555	0.0609898
$694$	16.3028	0.618845
$695$	0	0
$696$	6.27502	0.237854
$697$	38.7250	1.46681
$698$	−6.75274	−0.255595
$699$	36.6333	1.38560
$700$	0	0
$701$	−41.2111	−1.55652	−0.778261	−	0.627941i	$-0.783898\pi$
$701$	−41.2111	−1.55652	−0.778261	+	0.627941i	$0.783898\pi$
$702$	−10.4584	−0.394726
$703$	2.39445	0.0903083
$704$	−8.81665	−0.332290
$705$	0	0
$706$	−24.2750	−0.913602
$707$	0.358288	0.0134748
$708$	−9.90833	−0.372378
$709$	−31.6333	−1.18801	−0.594007	−	0.804460i	$-0.702455\pi$
$709$	−31.6333	−1.18801	−0.594007	+	0.804460i	$0.702455\pi$
$710$	0	0
$711$	−25.1194	−0.942052
$712$	−5.09167	−0.190819
$713$	74.5694	2.79265
$714$	−14.4500	−0.540776
$715$	0	0
$716$	−3.78890	−0.141598
$717$	−48.6333	−1.81624
$718$	1.02776	0.0383555
$719$	7.18335	0.267894	0.133947	−	0.990989i	$-0.457235\pi$
$719$	7.18335	0.267894	0.133947	+	0.990989i	$0.457235\pi$
$720$	0	0
$721$	2.02776	0.0755176
$722$	−23.4500	−0.872717
$723$	−50.5139	−1.87863
$724$	6.02776	0.224020
$725$	0	0
$726$	−3.00000	−0.111340
$727$	39.3305	1.45869	0.729344	−	0.684147i	$-0.239825\pi$
$727$	39.3305	1.45869	0.729344	+	0.684147i	$0.239825\pi$
$728$	−10.4584	−0.387613
$729$	−13.3305	−0.493723
$730$	0	0
$731$	49.8167	1.84254
$732$	−5.51388	−0.203799
$733$	−19.6056	−0.724148	−0.362074	−	0.932149i	$-0.617931\pi$
$733$	−19.6056	−0.724148	−0.362074	+	0.932149i	$0.617931\pi$
$734$	26.9638	0.995253
$735$	0	0
$736$	12.3944	0.456865
$737$	−4.00000	−0.147342
$738$	−16.8167	−0.619030
$739$	35.1194	1.29189	0.645945	−	0.763384i	$-0.276464\pi$
$739$	35.1194	1.29189	0.645945	+	0.763384i	$0.276464\pi$
$740$	0	0
$741$	−11.5139	−0.422973
$742$	−1.18335	−0.0434420
$743$	−40.6972	−1.49304	−0.746518	−	0.665365i	$-0.768276\pi$
$743$	−40.6972	−1.49304	−0.746518	+	0.665365i	$0.768276\pi$
$744$	70.5416	2.58618
$745$	0	0
$746$	−35.7250	−1.30798
$747$	8.09167	0.296059
$748$	−2.09167	−0.0764791
$749$	−2.09167	−0.0764281
$750$	0	0
$751$	−45.3305	−1.65413	−0.827067	−	0.562103i	$-0.809992\pi$
$751$	−45.3305	−1.65413	−0.827067	+	0.562103i	$0.809992\pi$
$752$	−9.90833	−0.361320
$753$	−15.9083	−0.579732
$754$	−5.91673	−0.215475
$755$	0	0
$756$	0.338936	0.0123270
$757$	−49.0555	−1.78295	−0.891476	−	0.453067i	$-0.850330\pi$
$757$	−49.0555	−1.78295	−0.891476	+	0.453067i	$0.850330\pi$
$758$	4.14719	0.150633
$759$	16.8167	0.610406
$760$	0	0
$761$	−13.5778	−0.492195	−0.246097	−	0.969245i	$-0.579148\pi$
$761$	−13.5778	−0.492195	−0.246097	+	0.969245i	$0.579148\pi$
$762$	24.3583	0.882408
$763$	−8.02776	−0.290624
$764$	−3.11943	−0.112857
$765$	0	0
$766$	−28.1833	−1.01831
$767$	71.0555	2.56567
$768$	16.3305	0.589277
$769$	−5.18335	−0.186916	−0.0934581	−	0.995623i	$-0.529792\pi$
$769$	−5.18335	−0.186916	−0.0934581	+	0.995623i	$0.529792\pi$
$770$	0	0
$771$	−41.4500	−1.49278
$772$	4.00000	0.143963
$773$	−3.11943	−0.112198	−0.0560990	−	0.998425i	$-0.517866\pi$
$773$	−3.11943	−0.112198	−0.0560990	+	0.998425i	$0.517866\pi$
$774$	−21.6333	−0.777593
$775$	0	0
$776$	45.9083	1.64801
$777$	−3.84441	−0.137917
$778$	15.6333	0.560481
$779$	5.60555	0.200840
$780$	0	0
$781$	2.60555	0.0932340
$782$	−65.7250	−2.35032
$783$	1.45837	0.0521177
$784$	21.5139	0.768353
$785$	0	0
$786$	29.7250	1.06025
$787$	−10.2111	−0.363986	−0.181993	−	0.983300i	$-0.558255\pi$
$787$	−10.2111	−0.363986	−0.181993	+	0.983300i	$0.558255\pi$
$788$	4.02776	0.143483
$789$	−52.5416	−1.87053
$790$	0	0
$791$	−7.54163	−0.268150
$792$	6.90833	0.245477
$793$	39.5416	1.40416
$794$	−28.2666	−1.00314
$795$	0	0
$796$	1.96384	0.0696065
$797$	−3.51388	−0.124468	−0.0622340	−	0.998062i	$-0.519822\pi$
$797$	−3.51388	−0.124468	−0.0622340	+	0.998062i	$0.519822\pi$
$798$	−2.09167	−0.0740444
$799$	−20.7250	−0.733197
$800$	0	0
$801$	3.90833	0.138094
$802$	−16.6611	−0.588323
$803$	−7.90833	−0.279079
$804$	2.78890	0.0983568
$805$	0	0
$806$	−66.5139	−2.34285
$807$	−20.0917	−0.707260
$808$	1.54163	0.0542345
$809$	39.6333	1.39343	0.696716	−	0.717347i	$-0.254644\pi$
$809$	39.6333	1.39343	0.696716	+	0.717347i	$0.254644\pi$
$810$	0	0
$811$	38.8722	1.36499	0.682493	−	0.730892i	$-0.260896\pi$
$811$	38.8722	1.36499	0.682493	+	0.730892i	$0.260896\pi$
$812$	0.191750	0.00672911
$813$	0.486122	0.0170490
$814$	3.11943	0.109336
$815$	0	0
$816$	−52.5416	−1.83933
$817$	7.21110	0.252285
$818$	−8.09167	−0.282919
$819$	8.02776	0.280513
$820$	0	0
$821$	12.0000	0.418803	0.209401	−	0.977830i	$-0.432848\pi$
$821$	12.0000	0.418803	0.209401	+	0.977830i	$0.432848\pi$
$822$	−38.7250	−1.35069
$823$	−18.4222	−0.642158	−0.321079	−	0.947052i	$-0.604045\pi$
$823$	−18.4222	−0.642158	−0.321079	+	0.947052i	$0.604045\pi$
$824$	8.72498	0.303949
$825$	0	0
$826$	12.9083	0.449138
$827$	−13.8167	−0.480452	−0.240226	−	0.970717i	$-0.577222\pi$
$827$	−13.8167	−0.480452	−0.240226	+	0.970717i	$0.577222\pi$
$828$	−5.09167	−0.176948
$829$	29.7527	1.03336	0.516678	−	0.856180i	$-0.327169\pi$
$829$	29.7527	1.03336	0.516678	+	0.856180i	$0.327169\pi$
$830$	0	0
$831$	33.1472	1.14986
$832$	−44.0833	−1.52831
$833$	45.0000	1.55916
$834$	18.6333	0.645219
$835$	0	0
$836$	−0.302776	−0.0104717
$837$	16.3944	0.566675
$838$	8.33053	0.287773
$839$	−9.11943	−0.314838	−0.157419	−	0.987532i	$-0.550317\pi$
$839$	−9.11943	−0.314838	−0.157419	+	0.987532i	$0.550317\pi$
$840$	0	0
$841$	−28.1749	−0.971550
$842$	0.908327	0.0313030
$843$	2.72498	0.0938533
$844$	7.64171	0.263039
$845$	0	0
$846$	9.00000	0.309426
$847$	−0.697224	−0.0239569
$848$	−4.30278	−0.147758
$849$	14.5139	0.498115
$850$	0	0
$851$	−17.4861	−0.599417
$852$	−1.81665	−0.0622375
$853$	12.7250	0.435695	0.217848	−	0.975983i	$-0.430096\pi$
$853$	12.7250	0.435695	0.217848	+	0.975983i	$0.430096\pi$
$854$	7.18335	0.245809
$855$	0	0
$856$	−9.00000	−0.307614
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	−15.0000	−0.512092
$859$	41.3944	1.41236	0.706180	−	0.708032i	$-0.250417\pi$
$859$	41.3944	1.41236	0.706180	+	0.708032i	$0.250417\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	42.9916	1.46430
$863$	12.3944	0.421912	0.210956	−	0.977496i	$-0.432342\pi$
$863$	12.3944	0.421912	0.210956	+	0.977496i	$0.432342\pi$
$864$	2.72498	0.0927057
$865$	0	0
$866$	−6.51388	−0.221351
$867$	−70.7527	−2.40289
$868$	2.15559	0.0731655
$869$	10.9083	0.370040
$870$	0	0
$871$	−20.0000	−0.677674
$872$	−34.5416	−1.16973
$873$	−35.2389	−1.19265
$874$	−9.51388	−0.321812
$875$	0	0
$876$	5.51388	0.186297
$877$	−20.0000	−0.675352	−0.337676	−	0.941262i	$-0.609641\pi$
$877$	−20.0000	−0.675352	−0.337676	+	0.941262i	$0.609641\pi$
$878$	31.6611	1.06851
$879$	1.81665	0.0612742
$880$	0	0
$881$	19.5416	0.658374	0.329187	−	0.944265i	$-0.393225\pi$
$881$	19.5416	0.658374	0.329187	+	0.944265i	$0.393225\pi$
$882$	−19.5416	−0.658001
$883$	−52.4500	−1.76508	−0.882541	−	0.470236i	$-0.844169\pi$
$883$	−52.4500	−1.76508	−0.882541	+	0.470236i	$0.844169\pi$
$884$	−10.4584	−0.351753
$885$	0	0
$886$	11.2111	0.376644
$887$	3.23886	0.108750	0.0543751	−	0.998521i	$-0.482683\pi$
$887$	3.23886	0.108750	0.0543751	+	0.998521i	$0.482683\pi$
$888$	−16.5416	−0.555101
$889$	5.66106	0.189866
$890$	0	0
$891$	10.6056	0.355299
$892$	−6.85281	−0.229449
$893$	−3.00000	−0.100391
$894$	−51.6333	−1.72688
$895$	0	0
$896$	−5.64171	−0.188476
$897$	84.0833	2.80746
$898$	30.5971	1.02104
$899$	9.27502	0.309339
$900$	0	0
$901$	−9.00000	−0.299833
$902$	7.30278	0.243156
$903$	−11.5778	−0.385285
$904$	−32.4500	−1.07927
$905$	0	0
$906$	−2.44996	−0.0813945
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	0.513878	0.0170536
$909$	−1.18335	−0.0392491
$910$	0	0
$911$	−24.7889	−0.821293	−0.410646	−	0.911795i	$-0.634697\pi$
$911$	−24.7889	−0.821293	−0.410646	+	0.911795i	$0.634697\pi$
$912$	−7.60555	−0.251845
$913$	−3.51388	−0.116292
$914$	26.9638	0.891885
$915$	0	0
$916$	5.66947	0.187324
$917$	6.90833	0.228133
$918$	−14.4500	−0.476920
$919$	26.7889	0.883684	0.441842	−	0.897093i	$-0.354325\pi$
$919$	26.7889	0.883684	0.441842	+	0.897093i	$0.354325\pi$
$920$	0	0
$921$	−38.9361	−1.28299
$922$	41.9638	1.38201
$923$	13.0278	0.428814
$924$	0.486122	0.0159922
$925$	0	0
$926$	−15.3583	−0.504705
$927$	−6.69722	−0.219966
$928$	1.54163	0.0506066
$929$	53.6056	1.75874	0.879371	−	0.476138i	$-0.157964\pi$
$929$	53.6056	1.75874	0.879371	+	0.476138i	$0.157964\pi$
$930$	0	0
$931$	6.51388	0.213484
$932$	4.81665	0.157775
$933$	−11.0917	−0.363125
$934$	24.2750	0.794303
$935$	0	0
$936$	34.5416	1.12903
$937$	9.21110	0.300914	0.150457	−	0.988617i	$-0.451926\pi$
$937$	9.21110	0.300914	0.150457	+	0.988617i	$0.451926\pi$
$938$	−3.63331	−0.118632
$939$	0.422205	0.0137781
$940$	0	0
$941$	59.6056	1.94309	0.971543	−	0.236864i	$-0.0761197\pi$
$941$	59.6056	1.94309	0.971543	+	0.236864i	$0.0761197\pi$
$942$	57.6333	1.87779
$943$	−40.9361	−1.33306
$944$	46.9361	1.52764
$945$	0	0
$946$	9.39445	0.305440
$947$	−6.63331	−0.215554	−0.107777	−	0.994175i	$-0.534373\pi$
$947$	−6.63331	−0.215554	−0.107777	+	0.994175i	$0.534373\pi$
$948$	−7.60555	−0.247017
$949$	−39.5416	−1.28358
$950$	0	0
$951$	−2.09167	−0.0678271
$952$	−14.4500	−0.468326
$953$	−37.2666	−1.20718	−0.603592	−	0.797293i	$-0.706264\pi$
$953$	−37.2666	−1.20718	−0.603592	+	0.797293i	$0.706264\pi$
$954$	3.90833	0.126537
$955$	0	0
$956$	−6.39445	−0.206811
$957$	2.09167	0.0676142
$958$	−45.3583	−1.46546
$959$	−9.00000	−0.290625
$960$	0	0
$961$	73.2666	2.36344
$962$	15.5971	0.502872
$963$	6.90833	0.222618
$964$	−6.64171	−0.213915
$965$	0	0
$966$	15.2750	0.491466
$967$	−14.9083	−0.479419	−0.239710	−	0.970845i	$-0.577052\pi$
$967$	−14.9083	−0.479419	−0.239710	+	0.970845i	$0.577052\pi$
$968$	−3.00000	−0.0964237
$969$	−15.9083	−0.511049
$970$	0	0
$971$	45.3583	1.45562	0.727808	−	0.685781i	$-0.240539\pi$
$971$	45.3583	1.45562	0.727808	+	0.685781i	$0.240539\pi$
$972$	−5.93608	−0.190400
$973$	4.33053	0.138830
$974$	−5.48612	−0.175787
$975$	0	0
$976$	26.1194	0.836063
$977$	52.0278	1.66452	0.832258	−	0.554389i	$-0.187048\pi$
$977$	52.0278	1.66452	0.832258	+	0.554389i	$0.187048\pi$
$978$	27.9083	0.892410
$979$	−1.69722	−0.0542435
$980$	0	0
$981$	26.5139	0.846523
$982$	−12.7527	−0.406956
$983$	−8.84441	−0.282093	−0.141046	−	0.990003i	$-0.545047\pi$
$983$	−8.84441	−0.282093	−0.141046	+	0.990003i	$0.545047\pi$
$984$	−38.7250	−1.23451
$985$	0	0
$986$	−8.17494	−0.260343
$987$	4.81665	0.153316
$988$	−1.51388	−0.0481629
$989$	−52.6611	−1.67452
$990$	0	0
$991$	−16.9083	−0.537111	−0.268555	−	0.963264i	$-0.586546\pi$
$991$	−16.9083	−0.537111	−0.268555	+	0.963264i	$0.586546\pi$
$992$	17.3305	0.550245
$993$	49.7527	1.57886
$994$	2.36669	0.0750669
$995$	0	0
$996$	2.44996	0.0776300
$997$	−46.7250	−1.47979	−0.739897	−	0.672720i	$-0.765126\pi$
$997$	−46.7250	−1.47979	−0.739897	+	0.672720i	$0.765126\pi$
$998$	−4.54163	−0.143763
$999$	−3.84441	−0.121632

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.2.a.e.1.2	✓	2
3.2	odd	2		2475.2.a.t.1.1		2
4.3	odd	2		4400.2.a.bs.1.2		2
5.2	odd	4		275.2.b.c.199.3		4
5.3	odd	4		275.2.b.c.199.2		4
5.4	even	2		275.2.a.f.1.1	yes	2
11.10	odd	2		3025.2.a.n.1.1		2
15.2	even	4		2475.2.c.k.199.2		4
15.8	even	4		2475.2.c.k.199.3		4
15.14	odd	2		2475.2.a.o.1.2		2
20.3	even	4		4400.2.b.y.4049.4		4
20.7	even	4		4400.2.b.y.4049.1		4
20.19	odd	2		4400.2.a.bh.1.1		2
55.54	odd	2		3025.2.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.2	✓	2	1.1	even	1	trivial
275.2.a.f.1.1	yes	2	5.4	even	2
275.2.b.c.199.2		4	5.3	odd	4
275.2.b.c.199.3		4	5.2	odd	4
2475.2.a.o.1.2		2	15.14	odd	2
2475.2.a.t.1.1		2	3.2	odd	2
2475.2.c.k.199.2		4	15.2	even	4
2475.2.c.k.199.3		4	15.8	even	4
3025.2.a.h.1.2		2	55.54	odd	2
3025.2.a.n.1.1		2	11.10	odd	2
4400.2.a.bh.1.1		2	20.19	odd	2
4400.2.a.bs.1.2		2	4.3	odd	2
4400.2.b.y.4049.1		4	20.7	even	4
4400.2.b.y.4049.4		4	20.3	even	4

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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30278 q^{2} -2.30278 q^{3} -0.302776 q^{4} -3.00000 q^{6} -0.697224 q^{7} -3.00000 q^{8} +2.30278 q^{9} -1.00000 q^{11} +0.697224 q^{12} -5.00000 q^{13} -0.908327 q^{14} -3.30278 q^{16} -6.90833 q^{17} +3.00000 q^{18} -1.00000 q^{19} +1.60555 q^{21} -1.30278 q^{22} +7.30278 q^{23} +6.90833 q^{24} -6.51388 q^{26} +1.60555 q^{27} +0.211103 q^{28} +0.908327 q^{29} +10.2111 q^{31} +1.69722 q^{32} +2.30278 q^{33} -9.00000 q^{34} -0.697224 q^{36} -2.39445 q^{37} -1.30278 q^{38} +11.5139 q^{39} -5.60555 q^{41} +2.09167 q^{42} -7.21110 q^{43} +0.302776 q^{44} +9.51388 q^{46} +3.00000 q^{47} +7.60555 q^{48} -6.51388 q^{49} +15.9083 q^{51} +1.51388 q^{52} +1.30278 q^{53} +2.09167 q^{54} +2.09167 q^{56} +2.30278 q^{57} +1.18335 q^{58} -14.2111 q^{59} -7.90833 q^{61} +13.3028 q^{62} -1.60555 q^{63} +8.81665 q^{64} +3.00000 q^{66} +4.00000 q^{67} +2.09167 q^{68} -16.8167 q^{69} -2.60555 q^{71} -6.90833 q^{72} +7.90833 q^{73} -3.11943 q^{74} +0.302776 q^{76} +0.697224 q^{77} +15.0000 q^{78} -10.9083 q^{79} -10.6056 q^{81} -7.30278 q^{82} +3.51388 q^{83} -0.486122 q^{84} -9.39445 q^{86} -2.09167 q^{87} +3.00000 q^{88} +1.69722 q^{89} +3.48612 q^{91} -2.21110 q^{92} -23.5139 q^{93} +3.90833 q^{94} -3.90833 q^{96} -15.3028 q^{97} -8.48612 q^{98} -2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{3} + 3 q^{4} - 6 q^{6} - 5 q^{7} - 6 q^{8} + q^{9} - 2 q^{11} + 5 q^{12} - 10 q^{13} + 9 q^{14} - 3 q^{16} - 3 q^{17} + 6 q^{18} - 2 q^{19} - 4 q^{21} + q^{22} + 11 q^{23} + 3 q^{24}+ \cdots - q^{99}+O(q^{100}) \) 2 * q - q^2 - q^3 + 3 * q^4 - 6 * q^6 - 5 * q^7 - 6 * q^8 + q^9 - 2 * q^11 + 5 * q^12 - 10 * q^13 + 9 * q^14 - 3 * q^16 - 3 * q^17 + 6 * q^18 - 2 * q^19 - 4 * q^21 + q^22 + 11 * q^23 + 3 * q^24 + 5 * q^26 - 4 * q^27 - 14 * q^28 - 9 * q^29 + 6 * q^31 + 7 * q^32 + q^33 - 18 * q^34 - 5 * q^36 - 12 * q^37 + q^38 + 5 * q^39 - 4 * q^41 + 15 * q^42 - 3 * q^44 + q^46 + 6 * q^47 + 8 * q^48 + 5 * q^49 + 21 * q^51 - 15 * q^52 - q^53 + 15 * q^54 + 15 * q^56 + q^57 + 24 * q^58 - 14 * q^59 - 5 * q^61 + 23 * q^62 + 4 * q^63 - 4 * q^64 + 6 * q^66 + 8 * q^67 + 15 * q^68 - 12 * q^69 + 2 * q^71 - 3 * q^72 + 5 * q^73 + 19 * q^74 - 3 * q^76 + 5 * q^77 + 30 * q^78 - 11 * q^79 - 14 * q^81 - 11 * q^82 - 11 * q^83 - 19 * q^84 - 26 * q^86 - 15 * q^87 + 6 * q^88 + 7 * q^89 + 25 * q^91 + 10 * q^92 - 29 * q^93 - 3 * q^94 + 3 * q^96 - 27 * q^97 - 35 * q^98 - q^99

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