LMFDB - Embedded newform 1127.2.c.c.1126.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1127,2,Mod(1126,1127)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1127, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1127.1126");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1127 = 7^{2} \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1127.c (of order $2$ , degree $1$ , minimal)

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:	no
Analytic conductor:	$8.99914030780$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 161)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1126.28
Character	$\chi$	$=$	1127.1126
Dual form			1127.2.c.c.1126.25

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.46597 q^{2} -0.986579i q^{3} +4.08101 q^{4} +2.48691 q^{5} -2.43287i q^{6} +5.13171 q^{8} +2.02666 q^{9} +6.13265 q^{10} +4.49657i q^{11} -4.02624i q^{12} -2.10932i q^{13} -2.45353i q^{15} +4.49263 q^{16} -4.01646 q^{17} +4.99769 q^{18} -5.83898 q^{19} +10.1491 q^{20} +11.0884i q^{22} +(-4.49263 + 1.67817i) q^{23} -5.06284i q^{24} +1.18473 q^{25} -5.20153i q^{26} -4.95920i q^{27} +2.64981 q^{29} -6.05034i q^{30} +5.60378i q^{31} +0.815274 q^{32} +4.43622 q^{33} -9.90448 q^{34} +8.27083 q^{36} +0.778402i q^{37} -14.3988 q^{38} -2.08101 q^{39} +12.7621 q^{40} -1.68118i q^{41} -8.94389i q^{43} +18.3506i q^{44} +5.04013 q^{45} +(-11.0787 + 4.13831i) q^{46} -8.06722i q^{47} -4.43234i q^{48} +2.92150 q^{50} +3.96256i q^{51} -8.60817i q^{52} -6.34746i q^{53} -12.2292i q^{54} +11.1826i q^{55} +5.76061i q^{57} +6.53434 q^{58} +12.1792i q^{59} -10.0129i q^{60} +5.87140 q^{61} +13.8188i q^{62} -6.97482 q^{64} -5.24569i q^{65} +10.9396 q^{66} -15.7279i q^{67} -16.3912 q^{68} +(1.65564 + 4.43234i) q^{69} +5.33546 q^{71} +10.4003 q^{72} -3.64596i q^{73} +1.91952i q^{74} -1.16883i q^{75} -23.8290 q^{76} -5.13171 q^{78} -5.04997i q^{79} +11.1728 q^{80} +1.18735 q^{81} -4.14575i q^{82} +0.851415 q^{83} -9.98859 q^{85} -22.0554i q^{86} -2.61424i q^{87} +23.0751i q^{88} +7.95517 q^{89} +12.4288 q^{90} +(-18.3345 + 6.84862i) q^{92} +5.52857 q^{93} -19.8935i q^{94} -14.5210 q^{95} -0.804332i q^{96} +2.72009 q^{97} +9.11304i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$28 q + 8 q^{2} + 24 q^{4} + 12 q^{8} - 8 q^{9} - 44 q^{18} + 44 q^{25} - 44 q^{29} + 12 q^{32} + 32 q^{39} - 36 q^{46} + 84 q^{50} - 28 q^{58} - 68 q^{64} + 16 q^{71} + 8 q^{72} - 12 q^{78} - 44 q^{81}+ \cdots - 112 q^{95}+O(q^{100})$ 28 * q + 8 * q^2 + 24 * q^4 + 12 * q^8 - 8 * q^9 - 44 * q^18 + 44 * q^25 - 44 * q^29 + 12 * q^32 + 32 * q^39 - 36 * q^46 + 84 * q^50 - 28 * q^58 - 68 * q^64 + 16 * q^71 + 8 * q^72 - 12 * q^78 - 44 * q^81 - 52 * q^85 - 28 * q^92 + 20 * q^93 - 112 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1127\mathbb{Z}\right)^\times$ .

$n$	$346$	$442$
$\chi(n)$	$-1$	$-1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	2.46597			1.74370			0.871852	−	0.489769i	$-0.162919\pi$
$2$	2.46597			1.74370			0.871852	+	0.489769i	$0.162919\pi$
$3$		−	0.986579i		−	0.569601i	−0.958587	−	0.284801i	$-0.908073\pi$
$3$		−	0.986579i		−	0.569601i	0.958587	−	0.284801i	$-0.0919274\pi$
$4$	4.08101			2.04051
$5$	2.48691			1.11218			0.556090	−	0.831122i	$-0.312301\pi$
$5$	2.48691			1.11218			0.556090	+	0.831122i	$0.312301\pi$
$6$		−	2.43287i		−	0.993217i
$7$		0			0
$7$		0			0
$8$	5.13171			1.81433
$9$	2.02666			0.675554
$10$	6.13265			1.93931
$11$			4.49657i			1.35577i	0.735169	+	0.677884i	$0.237103\pi$
$11$			4.49657i			1.35577i	−0.735169	+	0.677884i	$0.762897\pi$
$12$		−	4.02624i		−	1.16228i
$13$		−	2.10932i		−	0.585021i	−0.956262	−	0.292510i	$-0.905509\pi$
$13$		−	2.10932i		−	0.585021i	0.956262	−	0.292510i	$-0.0944906\pi$
$14$		0			0
$15$		−	2.45353i		−	0.633500i
$16$	4.49263			1.12316
$17$	−4.01646			−0.974135			−0.487068	−	0.873364i	$-0.661934\pi$
$17$	−4.01646			−0.974135			−0.487068	+	0.873364i	$0.661934\pi$
$18$	4.99769			1.17797
$19$	−5.83898			−1.33955			−0.669777	−	0.742562i	$-0.733610\pi$
$19$	−5.83898			−1.33955			−0.669777	+	0.742562i	$0.733610\pi$
$20$	10.1491			2.26941
$21$		0			0
$22$			11.0884i			2.36406i
$23$	−4.49263	+	1.67817i	−0.936779	+	0.349922i
$23$	−4.49263	+	1.67817i	−0.936779	+	0.349922i
$24$		−	5.06284i		−	1.03345i
$25$	1.18473			0.236945
$26$		−	5.20153i		−	1.02010i
$27$		−	4.95920i		−	0.954398i
$28$		0			0
$29$	2.64981			0.492057			0.246028	−	0.969263i	$-0.420874\pi$
$29$	2.64981			0.492057			0.246028	+	0.969263i	$0.420874\pi$
$30$		−	6.05034i		−	1.10464i
$31$			5.60378i			1.00647i	0.864150	+	0.503234i	$0.167857\pi$
$31$			5.60378i			1.00647i	−0.864150	+	0.503234i	$0.832143\pi$
$32$	0.815274			0.144121
$33$	4.43622			0.772247
$34$	−9.90448			−1.69860
$35$		0			0
$36$	8.27083			1.37847
$37$			0.778402i			0.127968i	0.997951	+	0.0639842i	$0.0203807\pi$
$37$			0.778402i			0.127968i	−0.997951	+	0.0639842i	$0.979619\pi$
$38$	−14.3988			−2.33579
$39$	−2.08101			−0.333229
$40$	12.7621			2.01787
$41$		−	1.68118i		−	0.262557i	−0.991346	−	0.131278i	$-0.958092\pi$
$41$		−	1.68118i		−	0.262557i	0.991346	−	0.131278i	$-0.0419082\pi$
$42$		0			0
$43$		−	8.94389i		−	1.36393i	−0.731384	−	0.681965i	$-0.761125\pi$
$43$		−	8.94389i		−	1.36393i	0.731384	−	0.681965i	$-0.238875\pi$
$44$			18.3506i			2.76645i
$45$	5.04013			0.751338
$46$	−11.0787	+	4.13831i	−1.63347	+	0.610161i
$47$		−	8.06722i		−	1.17673i	−0.808597	−	0.588363i	$-0.799773\pi$
$47$		−	8.06722i		−	1.17673i	0.808597	−	0.588363i	$-0.200227\pi$
$48$		−	4.43234i		−	0.639753i
$49$		0			0
$50$	2.92150			0.413162
$51$			3.96256i			0.554869i
$52$		−	8.60817i		−	1.19374i
$53$		−	6.34746i		−	0.871891i	−0.899973	−	0.435945i	$-0.856414\pi$
$53$		−	6.34746i		−	0.871891i	0.899973	−	0.435945i	$-0.143586\pi$
$54$		−	12.2292i		−	1.66419i
$55$			11.1826i			1.50786i
$56$		0			0
$57$			5.76061i			0.763012i
$58$	6.53434			0.858001
$59$			12.1792i			1.58560i	0.609483	+	0.792799i	$0.291377\pi$
$59$			12.1792i			1.58560i	−0.609483	+	0.792799i	$0.708623\pi$
$60$		−	10.0129i		−	1.29266i
$61$	5.87140			0.751755			0.375878	−	0.926669i	$-0.377341\pi$
$61$	5.87140			0.751755			0.375878	+	0.926669i	$0.377341\pi$
$62$			13.8188i			1.75498i
$63$		0			0
$64$	−6.97482			−0.871853
$65$		−	5.24569i		−	0.650648i
$66$	10.9396			1.34657
$67$		−	15.7279i		−	1.92147i	−0.277471	−	0.960734i	$-0.589496\pi$
$67$		−	15.7279i		−	1.92147i	0.277471	−	0.960734i	$-0.410504\pi$
$68$	−16.3912			−1.98773
$69$	1.65564	+	4.43234i	0.199316	+	0.533591i
$70$		0			0
$71$	5.33546			0.633203			0.316601	−	0.948559i	$-0.397458\pi$
$71$	5.33546			0.633203			0.316601	+	0.948559i	$0.397458\pi$
$72$	10.4003			1.22568
$73$		−	3.64596i		−	0.426727i	−0.976973	−	0.213364i	$-0.931558\pi$
$73$		−	3.64596i		−	0.426727i	0.976973	−	0.213364i	$-0.0684419\pi$
$74$			1.91952i			0.223139i
$75$		−	1.16883i		−	0.134964i
$76$	−23.8290			−2.73337
$77$		0			0
$78$	−5.13171			−0.581052
$79$		−	5.04997i		−	0.568166i	−0.958800	−	0.284083i	$-0.908311\pi$
$79$		−	5.04997i		−	0.568166i	0.958800	−	0.284083i	$-0.0916891\pi$
$80$	11.1728			1.24915
$81$	1.18735			0.131928
$82$		−	4.14575i		−	0.457822i
$83$	0.851415			0.0934550			0.0467275	−	0.998908i	$-0.485121\pi$
$83$	0.851415			0.0934550			0.0467275	+	0.998908i	$0.485121\pi$
$84$		0			0
$85$	−9.98859			−1.08341
$86$		−	22.0554i		−	2.37829i
$87$		−	2.61424i		−	0.280276i
$88$			23.0751i			2.45982i
$89$	7.95517			0.843246			0.421623	−	0.906771i	$-0.361461\pi$
$89$	7.95517			0.843246			0.421623	+	0.906771i	$0.361461\pi$
$90$	12.4288			1.31011
$91$		0			0
$92$	−18.3345	+	6.84862i	−1.91150	+	0.714018i
$93$	5.52857			0.573286
$94$		−	19.8935i		−	2.05186i
$95$	−14.5210			−1.48983
$96$		−	0.804332i		−	0.0820918i
$97$	2.72009			0.276184			0.138092	−	0.990419i	$-0.455903\pi$
$97$	2.72009			0.276184			0.138092	+	0.990419i	$0.455903\pi$
$98$		0			0
$99$			9.11304i			0.915895i
$100$	4.83488			0.483488
$101$			14.2377i			1.41670i	0.705861	+	0.708351i	$0.250560\pi$
$101$			14.2377i			1.41670i	−0.705861	+	0.708351i	$0.749440\pi$
$102$			9.77155i			0.967528i
$103$	−9.42494			−0.928666			−0.464333	−	0.885661i	$-0.653706\pi$
$103$	−9.42494			−0.928666			−0.464333	+	0.885661i	$0.653706\pi$
$104$		−	10.8244i		−	1.06142i
$105$		0			0
$106$		−	15.6527i		−	1.52032i
$107$			17.7613i			1.71705i	0.512773	+	0.858524i	$0.328618\pi$
$107$			17.7613i			1.71705i	−0.512773	+	0.858524i	$0.671382\pi$
$108$		−	20.2385i		−	1.94745i
$109$			6.50372i			0.622944i	0.950255	+	0.311472i	$0.100822\pi$
$109$			6.50372i			0.622944i	−0.950255	+	0.311472i	$0.899178\pi$
$110$			27.5759i			2.62926i
$111$	0.767955			0.0728910
$112$		0			0
$113$			3.54545i			0.333528i	0.985997	+	0.166764i	$0.0533318\pi$
$113$			3.54545i			0.333528i	−0.985997	+	0.166764i	$0.946668\pi$
$114$			14.2055i			1.33047i
$115$	−11.1728	+	4.17345i	−1.04187	+	0.389177i
$116$	10.8139			1.00404
$117$		−	4.27488i		−	0.395213i
$118$			30.0336i			2.76482i
$119$		0			0
$120$		−	12.5908i		−	1.14938i
$121$	−9.21917			−0.838107
$122$	14.4787			1.31084
$123$	−1.65862			−0.149553
$124$			22.8691i			2.05371i
$125$	−9.48825			−0.848655
$126$		0			0
$127$	0.227609			0.0201970			0.0100985	−	0.999949i	$-0.496785\pi$
$127$	0.227609			0.0201970			0.0100985	+	0.999949i	$0.496785\pi$
$128$	−18.8303			−1.66438
$129$	−8.82385			−0.776897
$130$		−	12.9357i		−	1.13454i
$131$			10.2755i			0.897779i	0.893587	+	0.448889i	$0.148180\pi$
$131$			10.2755i			0.897779i	−0.893587	+	0.448889i	$0.851820\pi$
$132$	18.1043			1.57578
$133$		0			0
$134$		−	38.7845i		−	3.35047i
$135$		−	12.3331i		−	1.06146i
$136$	−20.6113			−1.76741
$137$			10.4070i			0.889126i	0.895748	+	0.444563i	$0.146641\pi$
$137$			10.4070i			0.889126i	−0.895748	+	0.444563i	$0.853359\pi$
$138$	4.08277	+	10.9300i	0.347548	+	0.930424i
$139$			6.21923i			0.527508i	0.964590	+	0.263754i	$0.0849607\pi$
$139$			6.21923i			0.527508i	−0.964590	+	0.263754i	$0.915039\pi$
$140$		0			0
$141$	−7.95895			−0.670265
$142$	13.1571			1.10412
$143$	9.48472			0.793152
$144$	9.10505			0.758754
$145$	6.58983			0.547256
$146$		−	8.99082i		−	0.744086i
$147$		0			0
$148$			3.17667i			0.261120i
$149$		−	17.7095i		−	1.45082i	−0.688316	−	0.725411i	$-0.741650\pi$
$149$		−	17.7095i		−	1.45082i	0.688316	−	0.725411i	$-0.258350\pi$
$150$		−	2.88229i		−	0.235338i
$151$	−3.58392			−0.291656			−0.145828	−	0.989310i	$-0.546585\pi$
$151$	−3.58392			−0.291656			−0.145828	+	0.989310i	$0.546585\pi$
$152$	−29.9640			−2.43040
$153$	−8.14002			−0.658081
$154$		0			0
$155$			13.9361i			1.11937i
$156$	−8.49263			−0.679955
$157$	8.71438			0.695484			0.347742	−	0.937590i	$-0.386949\pi$
$157$	8.71438			0.695484			0.347742	+	0.937590i	$0.386949\pi$
$158$		−	12.4531i		−	0.990713i
$159$	−6.26227			−0.496630
$160$	2.02751			0.160289
$161$		0			0
$162$	2.92797			0.230043
$163$	6.06005			0.474660			0.237330	−	0.971429i	$-0.423728\pi$
$163$	6.06005			0.474660			0.237330	+	0.971429i	$0.423728\pi$
$164$		−	6.86093i		−	0.535749i
$165$	11.0325			0.858878
$166$	2.09956			0.162958
$167$		−	4.39200i		−	0.339863i	−0.985456	−	0.169931i	$-0.945645\pi$
$167$		−	4.39200i		−	0.339863i	0.985456	−	0.169931i	$-0.0543546\pi$
$168$		0			0
$169$	8.55076			0.657751
$170$	−24.6316			−1.88915
$171$	−11.8336			−0.904941
$172$		−	36.5001i		−	2.78311i
$173$			8.42394i			0.640460i	0.947340	+	0.320230i	$0.103760\pi$
$173$			8.42394i			0.640460i	−0.947340	+	0.320230i	$0.896240\pi$
$174$		−	6.44664i		−	0.488719i
$175$		0			0
$176$			20.2015i			1.52274i
$177$	12.0158			0.903159
$178$	19.6172			1.47037
$179$	7.62608			0.570000			0.285000	−	0.958528i	$-0.408006\pi$
$179$	7.62608			0.570000			0.285000	+	0.958528i	$0.408006\pi$
$180$	20.5688			1.53311
$181$	13.6135			1.01189			0.505943	−	0.862567i	$-0.331145\pi$
$181$	13.6135			1.01189			0.505943	+	0.862567i	$0.331145\pi$
$182$		0			0
$183$		−	5.79259i		−	0.428201i
$184$	−23.0549	+	8.61188i	−1.69963	+	0.634876i
$185$			1.93582i			0.142324i
$186$	13.6333			0.999642
$187$		−	18.0603i		−	1.32070i
$188$		−	32.9224i		−	2.40112i
$189$		0			0
$190$	−35.8084			−2.59782
$191$			10.0258i			0.725440i	0.931898	+	0.362720i	$0.118152\pi$
$191$			10.0258i			0.725440i	−0.931898	+	0.362720i	$0.881848\pi$
$192$			6.88121i			0.496609i
$193$	−13.5576			−0.975899			−0.487950	−	0.872872i	$-0.662255\pi$
$193$	−13.5576			−0.975899			−0.487950	+	0.872872i	$0.662255\pi$
$194$	6.70767			0.481582
$195$	−5.17529			−0.370610
$196$		0			0
$197$	24.0652			1.71458			0.857288	−	0.514837i	$-0.172148\pi$
$197$	24.0652			1.71458			0.857288	+	0.514837i	$0.172148\pi$
$198$			22.4725i			1.59705i
$199$	24.1281			1.71040			0.855198	−	0.518301i	$-0.173435\pi$
$199$	24.1281			1.71040			0.855198	+	0.518301i	$0.173435\pi$
$200$	6.07967			0.429898
$201$	−15.5168			−1.09447
$202$			35.1097i			2.47031i
$203$		0			0
$204$			16.1712i			1.13221i
$205$		−	4.18095i		−	0.292011i
$206$	−23.2416			−1.61932
$207$	−9.10505	+	3.40108i	−0.632845	+	0.236391i
$208$		−	9.47641i		−	0.657071i
$209$		−	26.2554i		−	1.81612i
$210$		0			0
$211$	22.0361			1.51703			0.758515	−	0.651656i	$-0.225925\pi$
$211$	22.0361			1.51703			0.758515	+	0.651656i	$0.225925\pi$
$212$		−	25.9041i		−	1.77910i
$213$		−	5.26385i		−	0.360673i
$214$			43.7988i			2.99402i
$215$		−	22.2427i		−	1.51694i
$216$		−	25.4492i		−	1.73160i
$217$		0			0
$218$			16.0380i			1.08623i
$219$	−3.59702			−0.243064
$220$			45.6362i			3.07679i
$221$			8.47201i			0.569889i
$222$	1.89375			0.127100
$223$			16.7338i			1.12058i	0.828297	+	0.560290i	$0.189310\pi$
$223$			16.7338i			1.12058i	−0.828297	+	0.560290i	$0.810690\pi$
$224$		0			0
$225$	2.40104			0.160069
$226$			8.74298i			0.581575i
$227$	−29.2679			−1.94258			−0.971289	−	0.237904i	$-0.923540\pi$
$227$	−29.2679			−1.94258			−0.971289	+	0.237904i	$0.923540\pi$
$228$			23.5091i			1.55693i
$229$	5.06294			0.334569			0.167284	−	0.985909i	$-0.446500\pi$
$229$	5.06294			0.334569			0.167284	+	0.985909i	$0.446500\pi$
$230$	−27.5517	+	10.2916i	−1.81671	+	0.678609i
$231$		0			0
$232$	13.5980			0.892755
$233$	−2.11926			−0.138837			−0.0694186	−	0.997588i	$-0.522114\pi$
$233$	−2.11926			−0.138837			−0.0694186	+	0.997588i	$0.522114\pi$
$234$		−	10.5417i		−	0.689135i
$235$		−	20.0625i		−	1.30873i
$236$			49.7035i			3.23542i
$237$	−4.98219			−0.323628
$238$		0			0
$239$	−11.1996			−0.724444			−0.362222	−	0.932092i	$-0.617982\pi$
$239$	−11.1996			−0.724444			−0.362222	+	0.932092i	$0.617982\pi$
$240$		−	11.0228i		−	0.711520i
$241$	−6.65950			−0.428976			−0.214488	−	0.976727i	$-0.568808\pi$
$241$	−6.65950			−0.428976			−0.214488	+	0.976727i	$0.568808\pi$
$242$	−22.7342			−1.46141
$243$		−	16.0490i		−	1.02954i
$244$	23.9612			1.53396
$245$		0			0
$246$	−4.09011			−0.260776
$247$			12.3163i			0.783667i
$248$			28.7570i			1.82607i
$249$		−	0.839988i		−	0.0532321i
$250$	−23.3977			−1.47980
$251$	−2.15008			−0.135712			−0.0678559	−	0.997695i	$-0.521616\pi$
$251$	−2.15008			−0.135712			−0.0678559	+	0.997695i	$0.521616\pi$
$252$		0			0
$253$	−7.54600	−	20.2015i	−0.474413	−	1.27005i
$254$	0.561276			0.0352176
$255$			9.85453i			0.617114i
$256$	−32.4852			−2.03033
$257$		−	28.4754i		−	1.77624i	−0.459608	−	0.888122i	$-0.652010\pi$
$257$		−	28.4754i		−	1.77624i	0.459608	−	0.888122i	$-0.347990\pi$
$258$	−21.7594			−1.35468
$259$		0			0
$260$		−	21.4077i		−	1.32765i
$261$	5.37026			0.332411
$262$			25.3392i			1.56546i
$263$			12.5676i			0.774949i	0.921880	+	0.387475i	$0.126653\pi$
$263$			12.5676i			0.774949i	−0.921880	+	0.387475i	$0.873347\pi$
$264$	22.7654			1.40112
$265$		−	15.7856i		−	0.969700i
$266$		0			0
$267$		−	7.84840i		−	0.480314i
$268$		−	64.1857i		−	3.92077i
$269$		−	15.6214i		−	0.952453i	−0.879323	−	0.476226i	$-0.842004\pi$
$269$		−	15.6214i		−	0.952453i	0.879323	−	0.476226i	$-0.157996\pi$
$270$		−	30.4130i		−	1.85088i
$271$			15.5059i			0.941915i	0.882156	+	0.470957i	$0.156091\pi$
$271$			15.5059i			0.941915i	−0.882156	+	0.470957i	$0.843909\pi$
$272$	−18.0445			−1.09411
$273$		0			0
$274$			25.6632i			1.55037i
$275$			5.32721i			0.321243i
$276$	6.75670	+	18.0884i	0.406706	+	1.08879i
$277$	1.81998			0.109352			0.0546759	−	0.998504i	$-0.482587\pi$
$277$	1.81998			0.109352			0.0546759	+	0.998504i	$0.482587\pi$
$278$			15.3364i			0.919818i
$279$			11.3570i			0.679924i
$280$		0			0
$281$			16.5334i			0.986301i	0.869944	+	0.493150i	$0.164155\pi$
$281$			16.5334i			0.986301i	−0.869944	+	0.493150i	$0.835845\pi$
$282$	−19.6265			−1.16874
$283$	−29.2823			−1.74065			−0.870327	−	0.492475i	$-0.836092\pi$
$283$	−29.2823			−1.74065			−0.870327	+	0.492475i	$0.836092\pi$
$284$	21.7741			1.29205
$285$			14.3261i			0.848607i
$286$	23.3890			1.38302
$287$		0			0
$288$	1.65229			0.0973619
$289$	−0.868025			−0.0510603
$290$	16.2503			0.954252
$291$		−	2.68358i		−	0.157315i
$292$		−	14.8792i		−	0.870739i
$293$	19.5508			1.14217			0.571085	−	0.820891i	$-0.306523\pi$
$293$	19.5508			1.14217			0.571085	+	0.820891i	$0.306523\pi$
$294$		0			0
$295$			30.2886i			1.76347i
$296$			3.99454i			0.232178i
$297$	22.2994			1.29394
$298$		−	43.6712i		−	2.52981i
$299$	3.53980	+	9.47641i	0.204712	+	0.548035i
$300$		−	4.76999i		−	0.275395i
$301$		0			0
$302$	−8.83785			−0.508561
$303$	14.0466			0.806955
$304$	−26.2324			−1.50453
$305$	14.6016			0.836087
$306$	−20.0730			−1.14750
$307$			0.893505i			0.0509950i	0.999675	+	0.0254975i	$0.00811699\pi$
$307$			0.893505i			0.0509950i	−0.999675	+	0.0254975i	$0.991883\pi$
$308$		0			0
$309$			9.29844i			0.528970i
$310$			34.3660i			1.95186i
$311$			27.2117i			1.54303i	0.636210	+	0.771516i	$0.280501\pi$
$311$			27.2117i			1.54303i	−0.636210	+	0.771516i	$0.719499\pi$
$312$	−10.6792			−0.604588
$313$	17.9576			1.01502			0.507511	−	0.861646i	$-0.330566\pi$
$313$	17.9576			1.01502			0.507511	+	0.861646i	$0.330566\pi$
$314$	21.4894			1.21272
$315$		0			0
$316$		−	20.6090i		−	1.15935i
$317$	−11.3066			−0.635043			−0.317522	−	0.948251i	$-0.602851\pi$
$317$	−11.3066			−0.635043			−0.317522	+	0.948251i	$0.602851\pi$
$318$	−15.4426			−0.865976
$319$			11.9150i			0.667114i
$320$	−17.3458			−0.969658
$321$	17.5229			0.978033
$322$		0			0
$323$	23.4521			1.30491
$324$	4.84558			0.269199
$325$		−	2.49897i		−	0.138618i
$326$	14.9439			0.827667
$327$	6.41643			0.354830
$328$		−	8.62736i		−	0.476366i
$329$		0			0
$330$	27.2058			1.49763
$331$	−35.6885			−1.96162			−0.980808	−	0.194975i	$-0.937537\pi$
$331$	−35.6885			−1.96162			−0.980808	+	0.194975i	$0.937537\pi$
$332$	3.47464			0.190695
$333$			1.57756i			0.0864497i
$334$		−	10.8305i		−	0.592621i
$335$		−	39.1139i		−	2.13702i
$336$		0			0
$337$		−	17.0125i		−	0.926730i	−0.886167	−	0.463365i	$-0.846642\pi$
$337$		−	17.0125i		−	0.926730i	0.886167	−	0.463365i	$-0.153358\pi$
$338$	21.0859			1.14692
$339$	3.49787			0.189978
$340$	−40.7635			−2.21071
$341$	−25.1978			−1.36454
$342$	−29.1814			−1.57795
$343$		0			0
$344$		−	45.8975i		−	2.47463i
$345$	4.11744	+	11.0228i	0.221676	+	0.593449i
$346$			20.7732i			1.11677i
$347$	1.88896			0.101405			0.0507024	−	0.998714i	$-0.483854\pi$
$347$	1.88896			0.101405			0.0507024	+	0.998714i	$0.483854\pi$
$348$		−	10.6687i		−	0.571905i
$349$		−	29.0312i		−	1.55401i	−0.629497	−	0.777003i	$-0.716739\pi$
$349$		−	29.0312i		−	1.55401i	0.629497	−	0.777003i	$-0.283261\pi$
$350$		0			0
$351$	−10.4605			−0.558343
$352$			3.66594i			0.195395i
$353$		−	18.2642i		−	0.972103i	−0.873930	−	0.486052i	$-0.838437\pi$
$353$		−	18.2642i		−	0.972103i	0.873930	−	0.486052i	$-0.161563\pi$
$354$	29.6305			1.57484
$355$	13.2688			0.704236
$356$	32.4651			1.72065
$357$		0			0
$358$	18.8057			0.993911
$359$		−	5.13956i		−	0.271256i	−0.990760	−	0.135628i	$-0.956695\pi$
$359$		−	5.13956i		−	0.271256i	0.990760	−	0.135628i	$-0.0433052\pi$
$360$	25.8645			1.36318
$361$	15.0937			0.794405
$362$	33.5706			1.76443
$363$			9.09544i			0.477387i
$364$		0			0
$365$		−	9.06717i		−	0.474597i
$366$		−	14.2844i		−	0.746656i
$367$	9.65616			0.504047			0.252024	−	0.967721i	$-0.418904\pi$
$367$	9.65616			0.504047			0.252024	+	0.967721i	$0.418904\pi$
$368$	−20.1838	+	7.53939i	−1.05215	+	0.393018i
$369$		−	3.40719i		−	0.177371i
$370$			4.77367i			0.248171i
$371$		0			0
$372$	22.5622			1.16979
$373$			6.18433i			0.320212i	0.987100	+	0.160106i	$0.0511837\pi$
$373$			6.18433i			0.320212i	−0.987100	+	0.160106i	$0.948816\pi$
$374$		−	44.5362i		−	2.30291i
$375$			9.36090i			0.483395i
$376$		−	41.3987i		−	2.13497i
$377$		−	5.58929i		−	0.287863i
$378$		0			0
$379$		−	13.0402i		−	0.669830i	−0.942248	−	0.334915i	$-0.891292\pi$
$379$		−	13.0402i		−	0.669830i	0.942248	−	0.334915i	$-0.108708\pi$
$380$	−59.2605			−3.04000
$381$		−	0.224554i		−	0.0115042i
$382$			24.7233i			1.26495i
$383$	24.4317			1.24840			0.624201	−	0.781264i	$-0.285425\pi$
$383$	24.4317			1.24840			0.624201	+	0.781264i	$0.285425\pi$
$384$			18.5775i			0.948031i
$385$		0			0
$386$	−33.4327			−1.70168
$387$		−	18.1263i		−	0.921409i
$388$	11.1007			0.563554
$389$		−	15.9172i		−	0.807034i	−0.914972	−	0.403517i	$-0.867788\pi$
$389$		−	15.9172i		−	0.807034i	0.914972	−	0.403517i	$-0.132212\pi$
$390$	−12.7621			−0.646235
$391$	18.0445	−	6.74030i	0.912549	−	0.340872i
$392$		0			0
$393$	10.1376			0.511376
$394$	59.3441			2.98971
$395$		−	12.5588i		−	0.631903i
$396$			37.1904i			1.86889i
$397$			15.8323i			0.794599i	0.917689	+	0.397299i	$0.130053\pi$
$397$			15.8323i			0.794599i	−0.917689	+	0.397299i	$0.869947\pi$
$398$	59.4992			2.98243
$399$		0			0
$400$	5.32254			0.266127
$401$		−	0.706831i		−	0.0352975i	−0.999844	−	0.0176487i	$-0.994382\pi$
$401$		−	0.706831i		−	0.0352975i	0.999844	−	0.0176487i	$-0.00561806\pi$
$402$	−38.2640			−1.90843
$403$	11.8202			0.588805
$404$			58.1041i			2.89079i
$405$	2.95283			0.146727
$406$		0			0
$407$	−3.50014			−0.173496
$408$			20.3347i			1.00672i
$409$		−	22.9311i		−	1.13387i	−0.823763	−	0.566935i	$-0.808129\pi$
$409$		−	22.9311i		−	1.13387i	0.823763	−	0.566935i	$-0.191871\pi$
$410$		−	10.3101i		−	0.509180i
$411$	10.2673			0.506448
$412$	−38.4633			−1.89495
$413$		0			0
$414$	−22.4528	+	8.38696i	−1.10349	+	0.412197i
$415$	2.11739			0.103939
$416$		−	1.71968i		−	0.0843140i
$417$	6.13576			0.300469
$418$		−	64.7451i		−	3.16678i
$419$	8.27756			0.404385			0.202193	−	0.979346i	$-0.435193\pi$
$419$	8.27756			0.404385			0.202193	+	0.979346i	$0.435193\pi$
$420$		0			0
$421$		−	2.27003i		−	0.110635i	−0.998469	−	0.0553173i	$-0.982383\pi$
$421$		−	2.27003i		−	0.110635i	0.998469	−	0.0553173i	$-0.0176170\pi$
$422$	54.3404			2.64525
$423$		−	16.3495i		−	0.794942i
$424$		−	32.5734i		−	1.58190i
$425$	−4.75841			−0.230817
$426$		−	12.9805i		−	0.628908i
$427$		0			0
$428$			72.4840i			3.50365i
$429$		−	9.35742i		−	0.451781i
$430$		−	54.8498i		−	2.64509i
$431$		−	0.603523i		−	0.0290707i	−0.999894	−	0.0145353i	$-0.995373\pi$
$431$		−	0.603523i		−	0.0290707i	0.999894	−	0.0145353i	$-0.00462690\pi$
$432$		−	22.2799i		−	1.07194i
$433$	−27.0009			−1.29758			−0.648790	−	0.760968i	$-0.724725\pi$
$433$	−27.0009			−1.29758			−0.648790	+	0.760968i	$0.724725\pi$
$434$		0			0
$435$		−	6.50139i		−	0.311718i
$436$			26.5418i			1.27112i
$437$	26.2324	−	9.79879i	1.25487	−	0.468740i
$438$	−8.87016			−0.423832
$439$		−	7.81671i		−	0.373071i	−0.982448	−	0.186536i	$-0.940274\pi$
$439$		−	7.81671i		−	0.373071i	0.982448	−	0.186536i	$-0.0597260\pi$
$440$			57.3858i			2.73576i
$441$		0			0
$442$			20.8917i			0.993718i
$443$	6.19844			0.294497			0.147248	−	0.989100i	$-0.452958\pi$
$443$	6.19844			0.294497			0.147248	+	0.989100i	$0.452958\pi$
$444$	3.13403			0.148735
$445$	19.7838			0.937842
$446$			41.2651i			1.95396i
$447$	−17.4719			−0.826391
$448$		0			0
$449$	−12.0380			−0.568109			−0.284055	−	0.958808i	$-0.591680\pi$
$449$	−12.0380			−0.568109			−0.284055	+	0.958808i	$0.591680\pi$
$450$	5.92089			0.279114
$451$	7.55957			0.355966
$452$			14.4690i			0.680566i
$453$			3.53582i			0.166127i
$454$	−72.1737			−3.38728
$455$		0			0
$456$			29.5618i			1.38436i
$457$			17.9498i			0.839656i	0.907604	+	0.419828i	$0.137910\pi$
$457$			17.9498i			0.839656i	−0.907604	+	0.419828i	$0.862090\pi$
$458$	12.4851			0.583389
$459$			19.9184i			0.929713i
$460$	−45.5962	+	17.0319i	−2.12594	+	0.794117i
$461$		−	12.3887i		−	0.577001i	−0.957480	−	0.288501i	$-0.906843\pi$
$461$		−	12.3887i		−	0.577001i	0.957480	−	0.288501i	$-0.0931567\pi$
$462$		0			0
$463$	12.7244			0.591354			0.295677	−	0.955288i	$-0.404455\pi$
$463$	12.7244			0.591354			0.295677	+	0.955288i	$0.404455\pi$
$464$	11.9046			0.552657
$465$	13.7491			0.637598
$466$	−5.22603			−0.242091
$467$	−4.10530			−0.189971			−0.0949853	−	0.995479i	$-0.530280\pi$
$467$	−4.10530			−0.189971			−0.0949853	+	0.995479i	$0.530280\pi$
$468$		−	17.4458i		−	0.806435i
$469$		0			0
$470$		−	49.4735i		−	2.28204i
$471$		−	8.59742i		−	0.396148i
$472$			62.5003i			2.87681i
$473$	40.2169			1.84917
$474$	−12.2859			−0.564312
$475$	−6.91759			−0.317401
$476$		0			0
$477$		−	12.8642i		−	0.589009i
$478$	−27.6180			−1.26322
$479$	20.0488			0.916051			0.458025	−	0.888939i	$-0.348557\pi$
$479$	20.0488			0.916051			0.458025	+	0.888939i	$0.348557\pi$
$480$		−	2.00030i		−	0.0913009i
$481$	1.64190			0.0748642
$482$	−16.4221			−0.748008
$483$		0			0
$484$	−37.6236			−1.71016
$485$	6.76463			0.307166
$486$		−	39.5764i		−	1.79522i
$487$	−27.0461			−1.22557			−0.612787	−	0.790248i	$-0.709952\pi$
$487$	−27.0461			−1.22557			−0.612787	+	0.790248i	$0.709952\pi$
$488$	30.1303			1.36394
$489$		−	5.97872i		−	0.270367i
$490$		0			0
$491$	10.5974			0.478256			0.239128	−	0.970988i	$-0.423138\pi$
$491$	10.5974			0.478256			0.239128	+	0.970988i	$0.423138\pi$
$492$	−6.76885			−0.305163
$493$	−10.6428			−0.479330
$494$			30.3716i			1.36648i
$495$			22.6633i			1.01864i
$496$			25.1757i			1.13042i
$497$		0			0
$498$		−	2.07139i		−	0.0928210i
$499$	27.7025			1.24013			0.620067	−	0.784549i	$-0.287105\pi$
$499$	27.7025			1.24013			0.620067	+	0.784549i	$0.287105\pi$
$500$	−38.7216			−1.73168
$501$	−4.33305			−0.193586
$502$	−5.30203			−0.236641
$503$	−14.7700			−0.658561			−0.329280	−	0.944232i	$-0.606806\pi$
$503$	−14.7700			−0.658561			−0.329280	+	0.944232i	$0.606806\pi$
$504$		0			0
$505$			35.4078i			1.57563i
$506$	−18.6082	−	49.8162i	−0.827236	−	2.21460i
$507$		−	8.43600i		−	0.374656i
$508$	0.928874			0.0412121
$509$			12.6691i			0.561550i	0.959774	+	0.280775i	$0.0905914\pi$
$509$			12.6691i			0.561550i	−0.959774	+	0.280775i	$0.909409\pi$
$510$			24.3010i			1.07607i
$511$		0			0
$512$	−42.4471			−1.87591
$513$			28.9567i			1.27847i
$514$		−	70.2194i		−	3.09724i
$515$	−23.4390			−1.03284
$516$	−36.0103			−1.58526
$517$	36.2749			1.59537
$518$		0			0
$519$	8.31088			0.364807
$520$		−	26.9194i		−	1.18049i
$521$	−7.79378			−0.341452			−0.170726	−	0.985319i	$-0.554611\pi$
$521$	−7.79378			−0.341452			−0.170726	+	0.985319i	$0.554611\pi$
$522$	13.2429			0.579626
$523$	−3.76428			−0.164600			−0.0823002	−	0.996608i	$-0.526227\pi$
$523$	−3.76428			−0.164600			−0.0823002	+	0.996608i	$0.526227\pi$
$524$			41.9346i			1.83192i
$525$		0			0
$526$			30.9912i			1.35128i
$527$		−	22.5074i		−	0.980437i
$528$	19.9303			0.867356
$529$	17.3675	−	15.0788i	0.755109	−	0.655599i
$530$		−	38.9267i		−	1.69087i
$531$			24.6832i			1.07116i
$532$		0			0
$533$	−3.54616			−0.153601
$534$		−	19.3539i		−	0.837526i
$535$			44.1707i			1.90967i
$536$		−	80.7110i		−	3.48619i
$537$		−	7.52373i		−	0.324673i
$538$		−	38.5219i		−	1.66080i
$539$		0			0
$540$		−	50.3315i		−	2.16592i
$541$	−13.2377			−0.569133			−0.284567	−	0.958656i	$-0.591850\pi$
$541$	−13.2377			−0.569133			−0.284567	+	0.958656i	$0.591850\pi$
$542$			38.2370i			1.64242i
$543$		−	13.4308i		−	0.576372i
$544$	−3.27452			−0.140394
$545$			16.1742i			0.692826i
$546$		0			0
$547$	−40.0668			−1.71313			−0.856567	−	0.516036i	$-0.827407\pi$
$547$	−40.0668			−1.71313			−0.856567	+	0.516036i	$0.827407\pi$
$548$			42.4709i			1.81427i
$549$	11.8993			0.507851
$550$			13.1367i			0.560152i
$551$	−15.4722			−0.659136
$552$	8.49629	+	22.7455i	0.361626	+	0.968112i
$553$		0			0
$554$	4.48801			0.190677
$555$	1.90984			0.0810680
$556$			25.3807i			1.07638i
$557$		−	10.4022i		−	0.440755i	−0.975415	−	0.220377i	$-0.929271\pi$
$557$		−	10.4022i		−	0.440755i	0.975415	−	0.220377i	$-0.0707289\pi$
$558$			28.0060i			1.18559i
$559$	−18.8655			−0.797928
$560$		0			0
$561$	−17.8179			−0.752274
$562$			40.7709i			1.71982i
$563$	1.95284			0.0823024			0.0411512	−	0.999153i	$-0.486897\pi$
$563$	1.95284			0.0823024			0.0411512	+	0.999153i	$0.486897\pi$
$564$	−32.4806			−1.36768
$565$			8.81722i			0.370943i
$566$	−72.2093			−3.03518
$567$		0			0
$568$	27.3801			1.14884
$569$		−	13.0841i		−	0.548513i	−0.961657	−	0.274256i	$-0.911568\pi$
$569$		−	13.0841i		−	0.548513i	0.961657	−	0.274256i	$-0.0884317\pi$
$570$			35.3278i			1.47972i
$571$		−	17.6990i		−	0.740680i	−0.928896	−	0.370340i	$-0.879241\pi$
$571$		−	17.6990i		−	0.740680i	0.928896	−	0.370340i	$-0.120759\pi$
$572$	38.7073			1.61843
$573$	9.89122			0.413212
$574$		0			0
$575$	−5.32254	+	1.98817i	−0.221965	+	0.0829124i
$576$	−14.1356			−0.588984
$577$			6.79439i			0.282854i	0.989949	+	0.141427i	$0.0451691\pi$
$577$			6.79439i			0.282854i	−0.989949	+	0.141427i	$0.954831\pi$
$578$	−2.14052			−0.0890340
$579$			13.3757i			0.555874i
$580$	26.8932			1.11668
$581$		0			0
$582$		−	6.61764i		−	0.274310i
$583$	28.5418			1.18208
$584$		−	18.7100i		−	0.774226i
$585$		−	10.6313i		−	0.439548i
$586$	48.2117			1.99161
$587$			28.6736i			1.18349i	0.806126	+	0.591744i	$0.201560\pi$
$587$			28.6736i			1.18349i	−0.806126	+	0.591744i	$0.798440\pi$
$588$		0			0
$589$		−	32.7204i		−	1.34822i
$590$			74.6909i			3.07497i
$591$		−	23.7422i		−	0.976625i
$592$			3.49707i			0.143729i
$593$		−	15.5881i		−	0.640126i	−0.947396	−	0.320063i	$-0.896296\pi$
$593$		−	15.5881i		−	0.640126i	0.947396	−	0.320063i	$-0.103704\pi$
$594$	54.9897			2.25625
$595$		0			0
$596$		−	72.2729i		−	2.96041i
$597$		−	23.8043i		−	0.974244i
$598$	8.72903	+	23.3685i	0.356957	+	0.955611i
$599$	1.11974			0.0457515			0.0228757	−	0.999738i	$-0.492718\pi$
$599$	1.11974			0.0457515			0.0228757	+	0.999738i	$0.492718\pi$
$600$		−	5.99808i		−	0.244870i
$601$		−	3.71483i		−	0.151531i	−0.997126	−	0.0757656i	$-0.975860\pi$
$601$		−	3.71483i		−	0.151531i	0.997126	−	0.0757656i	$-0.0241401\pi$
$602$		0			0
$603$		−	31.8751i		−	1.29806i
$604$	−14.6260			−0.595125
$605$	−22.9273			−0.932126
$606$	34.6385			1.40709
$607$			10.2553i			0.416251i	0.978102	+	0.208126i	$0.0667363\pi$
$607$			10.2553i			0.416251i	−0.978102	+	0.208126i	$0.933264\pi$
$608$	−4.76037			−0.193059
$609$		0			0
$610$	36.0072			1.45789
$611$	−17.0164			−0.688409
$612$	−33.2195			−1.34282
$613$			39.4514i			1.59343i	0.604357	+	0.796714i	$0.293430\pi$
$613$			39.4514i			1.59343i	−0.604357	+	0.796714i	$0.706570\pi$
$614$			2.20336i			0.0889202i
$615$	−4.12484			−0.166330
$616$		0			0
$617$			32.7292i			1.31763i	0.752305	+	0.658815i	$0.228942\pi$
$617$			32.7292i			1.31763i	−0.752305	+	0.658815i	$0.771058\pi$
$618$			22.9297i			0.922367i
$619$	26.7689			1.07593			0.537965	−	0.842967i	$-0.319193\pi$
$619$	26.7689			1.07593			0.537965	+	0.842967i	$0.319193\pi$
$620$			56.8734i			2.28409i
$621$	8.32237	+	22.2799i	0.333965	+	0.894060i
$622$			67.1032i			2.69059i
$623$		0			0
$624$	−9.34922			−0.374268
$625$	−29.5201			−1.18080
$626$	44.2828			1.76990
$627$	−25.9030			−1.03447
$628$	35.5635			1.41914
$629$		−	3.12642i		−	0.124659i
$630$		0			0
$631$			3.62895i			0.144466i	0.997388	+	0.0722331i	$0.0230125\pi$
$631$			3.62895i			0.144466i	−0.997388	+	0.0722331i	$0.976987\pi$
$632$		−	25.9150i		−	1.03084i
$633$		−	21.7404i		−	0.864102i
$634$	−27.8818			−1.10733
$635$	0.566043			0.0224627
$636$	−25.5564			−1.01338
$637$		0			0
$638$			29.3822i			1.16325i
$639$	10.8132			0.427763
$640$	−46.8292			−1.85109
$641$		−	18.7443i		−	0.740355i	−0.928961	−	0.370177i	$-0.879297\pi$
$641$		−	18.7443i		−	0.740355i	0.928961	−	0.370177i	$-0.120703\pi$
$642$	43.2110			1.70540
$643$	21.3880			0.843460			0.421730	−	0.906722i	$-0.361423\pi$
$643$	21.3880			0.843460			0.421730	+	0.906722i	$0.361423\pi$
$644$		0			0
$645$	−21.9441			−0.864050
$646$	57.8321			2.27537
$647$		−	11.1153i		−	0.436988i	−0.975838	−	0.218494i	$-0.929886\pi$
$647$		−	11.1153i		−	0.436988i	0.975838	−	0.218494i	$-0.0701145\pi$
$648$	6.09313			0.239361
$649$	−54.7647			−2.14970
$650$		−	6.16238i		−	0.241708i
$651$		0			0
$652$	24.7312			0.968547
$653$	2.94991			0.115439			0.0577194	−	0.998333i	$-0.481617\pi$
$653$	2.94991			0.115439			0.0577194	+	0.998333i	$0.481617\pi$
$654$	15.8227			0.618718
$655$			25.5544i			0.998492i
$656$		−	7.55294i		−	0.294893i
$657$		−	7.38913i		−	0.288277i
$658$		0			0
$659$			13.6891i			0.533252i	0.963800	+	0.266626i	$0.0859089\pi$
$659$			13.6891i			0.533252i	−0.963800	+	0.266626i	$0.914091\pi$
$660$	45.0237			1.75255
$661$	−20.2120			−0.786155			−0.393078	−	0.919505i	$-0.628590\pi$
$661$	−20.2120			−0.786155			−0.393078	+	0.919505i	$0.628590\pi$
$662$	−88.0068			−3.42048
$663$	8.35831			0.324610
$664$	4.36922			0.169559
$665$		0			0
$666$			3.89021i			0.150743i
$667$	−11.9046	+	4.44682i	−0.460948	+	0.172181i
$668$		−	17.9238i		−	0.693492i
$669$	16.5092			0.638283
$670$		−	96.4537i		−	3.72633i
$671$			26.4012i			1.01921i
$672$		0			0
$673$	35.7251			1.37710			0.688551	−	0.725188i	$-0.258247\pi$
$673$	35.7251			1.37710			0.688551	+	0.725188i	$0.258247\pi$
$674$		−	41.9523i		−	1.61594i
$675$		−	5.87529i		−	0.226140i
$676$	34.8958			1.34214
$677$	−32.2210			−1.23835			−0.619177	−	0.785252i	$-0.712533\pi$
$677$	−32.2210			−1.23835			−0.619177	+	0.785252i	$0.712533\pi$
$678$	8.62564			0.331266
$679$		0			0
$680$	−51.2586			−1.96568
$681$			28.8751i			1.10649i
$682$	−62.1371			−2.37935
$683$	5.63137			0.215478			0.107739	−	0.994179i	$-0.465639\pi$
$683$	5.63137			0.215478			0.107739	+	0.994179i	$0.465639\pi$
$684$	−48.2932			−1.84654
$685$			25.8812i			0.988869i
$686$		0			0
$687$		−	4.99499i		−	0.190571i
$688$		−	40.1816i		−	1.53191i
$689$	−13.3888			−0.510074
$690$	10.1535	+	27.1820i	0.386537	+	1.03480i
$691$		−	10.1159i		−	0.384827i	−0.981314	−	0.192414i	$-0.938368\pi$
$691$		−	10.1159i		−	0.384827i	0.981314	−	0.192414i	$-0.0616315\pi$
$692$			34.3782i			1.30686i
$693$		0			0
$694$	4.65812			0.176820
$695$			15.4667i			0.586684i
$696$		−	13.4155i		−	0.508515i
$697$			6.75241i			0.255766i
$698$		−	71.5901i		−	2.70973i
$699$			2.09081i			0.0790819i
$700$		0			0
$701$			33.1633i			1.25256i	0.779598	+	0.626280i	$0.215423\pi$
$701$			33.1633i			1.25256i	−0.779598	+	0.626280i	$0.784577\pi$
$702$	−25.7954			−0.973584
$703$		−	4.54507i		−	0.171421i
$704$		−	31.3628i		−	1.18203i
$705$	−19.7932			−0.745455
$706$		−	45.0389i		−	1.69506i
$707$		0			0
$708$	49.0364			1.84290
$709$			29.7999i			1.11916i	0.828776	+	0.559580i	$0.189038\pi$
$709$			29.7999i			1.11916i	−0.828776	+	0.559580i	$0.810962\pi$
$710$	32.7205			1.22798
$711$		−	10.2346i		−	0.383827i
$712$	40.8236			1.52993
$713$	−9.40408	−	25.1757i	−0.352186	−	0.942839i
$714$		0			0
$715$	23.5877			0.882128
$716$	31.1221			1.16309
$717$			11.0493i			0.412644i
$718$		−	12.6740i		−	0.472990i
$719$		−	46.4035i		−	1.73056i	−0.501291	−	0.865279i	$-0.667141\pi$
$719$		−	46.4035i		−	1.73056i	0.501291	−	0.865279i	$-0.332859\pi$
$720$	22.6435			0.843872
$721$		0			0
$722$	37.2206			1.38521
$723$			6.57012i			0.244345i
$724$	55.5570			2.06476
$725$	3.13929			0.116590
$726$			22.4291i			0.832422i
$727$	2.32964			0.0864016			0.0432008	−	0.999066i	$-0.486244\pi$
$727$	2.32964			0.0864016			0.0432008	+	0.999066i	$0.486244\pi$
$728$		0			0
$729$	−12.2716			−0.454502
$730$		−	22.3594i		−	0.827558i
$731$			35.9228i			1.32865i
$732$		−	23.6396i		−	0.873746i
$733$	−32.3300			−1.19413			−0.597067	−	0.802191i	$-0.703668\pi$
$733$	−32.3300			−1.19413			−0.597067	+	0.802191i	$0.703668\pi$
$734$	23.8118			0.878909
$735$		0			0
$736$	−3.66273	+	1.36817i	−0.135010	+	0.0504313i
$737$	70.7216			2.60506
$738$		−	8.40204i		−	0.309283i
$739$	20.5438			0.755716			0.377858	−	0.925864i	$-0.376661\pi$
$739$	20.5438			0.755716			0.377858	+	0.925864i	$0.376661\pi$
$740$			7.90009i			0.290413i
$741$	12.1510			0.446378
$742$		0			0
$743$			7.41754i			0.272123i	0.990700	+	0.136062i	$0.0434445\pi$
$743$			7.41754i			0.272123i	−0.990700	+	0.136062i	$0.956556\pi$
$744$	28.3710			1.04013
$745$		−	44.0421i		−	1.61358i
$746$			15.2504i			0.558356i
$747$	1.72553			0.0631339
$748$		−	73.7044i		−	2.69490i
$749$		0			0
$750$			23.0837i			0.842898i
$751$		−	31.5847i		−	1.15254i	−0.817258	−	0.576272i	$-0.804507\pi$
$751$		−	31.5847i		−	1.15254i	0.817258	−	0.576272i	$-0.195493\pi$
$752$		−	36.2431i		−	1.32165i
$753$			2.12122i			0.0773016i
$754$		−	13.7830i		−	0.501948i
$755$	−8.91290			−0.324374
$756$		0			0
$757$		−	31.9227i		−	1.16025i	−0.814528	−	0.580125i	$-0.803004\pi$
$757$		−	31.9227i		−	1.16025i	0.814528	−	0.580125i	$-0.196996\pi$
$758$		−	32.1567i		−	1.16799i
$759$	−19.9303	+	7.44473i	−0.723425	+	0.270226i
$760$	−74.5178			−2.70304
$761$		−	4.34317i		−	0.157440i	−0.996897	−	0.0787199i	$-0.974917\pi$
$761$		−	4.34317i		−	0.157440i	0.996897	−	0.0787199i	$-0.0250833\pi$
$762$		−	0.553743i		−	0.0200600i
$763$		0			0
$764$			40.9153i			1.48026i
$765$	−20.2435			−0.731905
$766$	60.2479			2.17684
$767$	25.6899			0.927608
$768$			32.0492i			1.15648i
$769$	−30.9232			−1.11512			−0.557559	−	0.830137i	$-0.688262\pi$
$769$	−30.9232			−1.11512			−0.557559	+	0.830137i	$0.688262\pi$
$770$		0			0
$771$	−28.0932			−1.01175
$772$	−55.3288			−1.99133
$773$	9.67418			0.347956			0.173978	−	0.984750i	$-0.444338\pi$
$773$	9.67418			0.347956			0.173978	+	0.984750i	$0.444338\pi$
$774$		−	44.6988i		−	1.60667i
$775$			6.63894i			0.238478i
$776$	13.9587			0.501089
$777$		0			0
$778$		−	39.2514i		−	1.40723i
$779$			9.81640i			0.351709i
$780$	−21.1204			−0.756232
$781$			23.9913i			0.858476i
$782$	44.4972	−	16.6214i	1.59122	−	0.594379i
$783$		−	13.1409i		−	0.469618i
$784$		0			0
$785$	21.6719			0.773503
$786$	24.9991			0.891689
$787$	−21.8488			−0.778827			−0.389413	−	0.921063i	$-0.627322\pi$
$787$	−21.8488			−0.778827			−0.389413	+	0.921063i	$0.627322\pi$
$788$	98.2104			3.49860
$789$	12.3989			0.441412
$790$		−	30.9697i		−	1.10185i
$791$		0			0
$792$			46.7655i			1.66174i
$793$		−	12.3847i		−	0.439792i
$794$			39.0419i			1.38555i
$795$	−15.5737			−0.552342
$796$	98.4671			3.49007
$797$	21.0566			0.745864			0.372932	−	0.927859i	$-0.378352\pi$
$797$	21.0566			0.745864			0.372932	+	0.927859i	$0.378352\pi$
$798$		0			0
$799$			32.4017i			1.14629i
$800$	0.965876			0.0341489
$801$	16.1224			0.569658
$802$		−	1.74302i		−	0.0615483i
$803$	16.3943			0.578543
$804$	−63.3243			−2.23327
$805$		0			0
$806$	29.1482			1.02670
$807$	−15.4117			−0.542519
$808$			73.0637i			2.57037i
$809$	−21.4849			−0.755369			−0.377684	−	0.925934i	$-0.623280\pi$
$809$	−21.4849			−0.755369			−0.377684	+	0.925934i	$0.623280\pi$
$810$	7.28159			0.255849
$811$		−	20.3429i		−	0.714337i	−0.934040	−	0.357169i	$-0.883742\pi$
$811$		−	20.3429i		−	0.714337i	0.934040	−	0.357169i	$-0.116258\pi$
$812$		0			0
$813$	15.2978			0.536516
$814$	−8.63125			−0.302525
$815$	15.0708			0.527908
$816$			17.8023i			0.623206i
$817$			52.2232i			1.82706i
$818$		−	56.5474i		−	1.97713i
$819$		0			0
$820$		−	17.0625i		−	0.595849i
$821$	−28.1375			−0.982005			−0.491003	−	0.871158i	$-0.663369\pi$
$821$	−28.1375			−0.982005			−0.491003	+	0.871158i	$0.663369\pi$
$822$	25.3188			0.883095
$823$	−33.5526			−1.16957			−0.584786	−	0.811188i	$-0.698821\pi$
$823$	−33.5526			−1.16957			−0.584786	+	0.811188i	$0.698821\pi$
$824$	−48.3661			−1.68491
$825$	5.25571			0.182980
$826$		0			0
$827$		−	27.0254i		−	0.939764i	−0.882729	−	0.469882i	$-0.844297\pi$
$827$		−	27.0254i		−	0.939764i	0.882729	−	0.469882i	$-0.155703\pi$
$828$	−37.1578	+	13.8798i	−1.29132	+	0.482358i
$829$		−	16.1942i		−	0.562448i	−0.959642	−	0.281224i	$-0.909260\pi$
$829$		−	16.1942i		−	0.562448i	0.959642	−	0.281224i	$-0.0907403\pi$
$830$	5.22143			0.181239
$831$		−	1.79555i		−	0.0622869i
$832$			14.7121i			0.510052i
$833$		0			0
$834$	15.1306			0.523930
$835$		−	10.9225i		−	0.377989i
$836$		−	107.149i		−	3.70581i
$837$	27.7903			0.960572
$838$	20.4122			0.705128
$839$	−43.1817			−1.49080			−0.745399	−	0.666618i	$-0.767741\pi$
$839$	−43.1817			−1.49080			−0.745399	+	0.666618i	$0.767741\pi$
$840$		0			0
$841$	−21.9785			−0.757880
$842$		−	5.59784i		−	0.192914i
$843$	16.3115			0.561798
$844$	89.9297			3.09551
$845$	21.2650			0.731538
$846$		−	40.3175i		−	1.38614i
$847$		0			0
$848$		−	28.5168i		−	0.979271i
$849$			28.8893i			0.991479i
$850$	−11.7341			−0.402476
$851$	−1.30629	−	3.49707i	−0.0447790	−	0.119878i
$852$		−	21.4818i		−	0.735956i
$853$		−	18.9128i		−	0.647562i	−0.946132	−	0.323781i	$-0.895046\pi$
$853$		−	18.9128i		−	0.647562i	0.946132	−	0.323781i	$-0.104954\pi$
$854$		0			0
$855$	−29.4292			−1.00646
$856$			91.1458i			3.11530i
$857$			0.263693i			0.00900758i	0.999990	+	0.00450379i	$0.00143361\pi$
$857$			0.263693i			0.00900758i	−0.999990	+	0.00450379i	$0.998566\pi$
$858$		−	23.0751i		−	0.787772i
$859$			40.7829i			1.39150i	0.718286	+	0.695748i	$0.244927\pi$
$859$			40.7829i			1.39150i	−0.718286	+	0.695748i	$0.755073\pi$
$860$		−	90.7726i		−	3.09532i
$861$		0			0
$862$		−	1.48827i		−	0.0506907i
$863$	12.6099			0.429248			0.214624	−	0.976697i	$-0.431147\pi$
$863$	12.6099			0.429248			0.214624	+	0.976697i	$0.431147\pi$
$864$		−	4.04311i		−	0.137549i
$865$			20.9496i			0.712307i
$866$	−66.5834			−2.26260
$867$			0.856375i			0.0290840i
$868$		0			0
$869$	22.7075			0.770301
$870$		−	16.0322i		−	0.543543i
$871$	−33.1752			−1.12410
$872$			33.3752i			1.13023i
$873$	5.51271			0.186577
$874$	64.6883	−	24.1635i	2.18812	−	0.817343i
$875$		0			0
$876$	−14.6795			−0.495974
$877$	−16.8956			−0.570524			−0.285262	−	0.958450i	$-0.592081\pi$
$877$	−16.8956			−0.570524			−0.285262	+	0.958450i	$0.592081\pi$
$878$		−	19.2758i		−	0.650526i
$879$		−	19.2884i		−	0.650581i
$880$			50.2392i			1.69356i
$881$	12.0628			0.406406			0.203203	−	0.979137i	$-0.434865\pi$
$881$	12.0628			0.406406			0.203203	+	0.979137i	$0.434865\pi$
$882$		0			0
$883$	−49.2906			−1.65876			−0.829381	−	0.558684i	$-0.811307\pi$
$883$	−49.2906			−1.65876			−0.829381	+	0.558684i	$0.811307\pi$
$884$			34.5744i			1.16286i
$885$	29.8821			1.00448
$886$	15.2852			0.513516
$887$		−	6.00364i		−	0.201583i	−0.994908	−	0.100791i	$-0.967863\pi$
$887$		−	6.00364i		−	0.201583i	0.994908	−	0.100791i	$-0.0321374\pi$
$888$	3.94092			0.132249
$889$		0			0
$890$	48.7863			1.63532
$891$			5.33900i			0.178863i
$892$			68.2909i			2.28655i
$893$			47.1044i			1.57629i
$894$	−43.0851			−1.44098
$895$	18.9654			0.633943
$896$		0			0
$897$	9.34922	−	3.49229i	0.312161	−	0.116604i
$898$	−29.6854			−0.990615
$899$			14.8489i			0.495240i
$900$	9.79867			0.326622
$901$			25.4943i			0.849340i
$902$	18.6417			0.620700
$903$		0			0
$904$			18.1942i			0.605132i
$905$	33.8556			1.12540
$906$			8.71923i			0.289677i
$907$		−	19.4967i		−	0.647377i	−0.946164	−	0.323688i	$-0.895077\pi$
$907$		−	19.4967i		−	0.647377i	0.946164	−	0.323688i	$-0.104923\pi$
$908$	−119.443			−3.96384
$909$			28.8550i			0.957058i
$910$		0			0
$911$		−	39.2121i		−	1.29916i	−0.760295	−	0.649578i	$-0.774946\pi$
$911$		−	39.2121i		−	1.29916i	0.760295	−	0.649578i	$-0.225054\pi$
$912$			25.8803i			0.856983i
$913$			3.82845i			0.126703i
$914$			44.2637i			1.46411i
$915$		−	14.4057i		−	0.476237i
$916$	20.6619			0.682689
$917$		0			0
$918$			49.1183i			1.62114i
$919$		−	40.4966i		−	1.33586i	−0.744225	−	0.667929i	$-0.767181\pi$
$919$		−	40.4966i		−	1.33586i	0.744225	−	0.667929i	$-0.232819\pi$
$920$	−57.3355	+	21.4170i	−1.89030	+	0.706097i
$921$	0.881513			0.0290468
$922$		−	30.5503i		−	1.00612i
$923$		−	11.2542i		−	0.370437i
$924$		0			0
$925$			0.922193i			0.0303215i
$926$	31.3781			1.03115
$927$	−19.1012			−0.627364
$928$	2.16032			0.0709159
$929$			22.1070i			0.725306i	0.931924	+	0.362653i	$0.118129\pi$
$929$			22.1070i			0.725306i	−0.931924	+	0.362653i	$0.881871\pi$
$930$	33.9048			1.11178
$931$		0			0
$932$	−8.64872			−0.283298
$933$	26.8464			0.878913
$934$	−10.1236			−0.331253
$935$		−	44.9144i		−	1.46886i
$936$		−	21.9375i		−	0.717049i
$937$	−18.7417			−0.612266			−0.306133	−	0.951989i	$-0.599035\pi$
$937$	−18.7417			−0.612266			−0.306133	+	0.951989i	$0.599035\pi$
$938$		0			0
$939$		−	17.7165i		−	0.578157i
$940$		−	81.8752i		−	2.67047i
$941$	−22.5284			−0.734406			−0.367203	−	0.930141i	$-0.619685\pi$
$941$	−22.5284			−0.734406			−0.367203	+	0.930141i	$0.619685\pi$
$942$		−	21.2010i		−	0.690766i
$943$	2.82131	+	7.55294i	0.0918744	+	0.245958i
$944$			54.7167i			1.78088i
$945$		0			0
$946$	99.1736			3.22441
$947$	−21.2167			−0.689449			−0.344724	−	0.938704i	$-0.612028\pi$
$947$	−21.2167			−0.689449			−0.344724	+	0.938704i	$0.612028\pi$
$948$	−20.3324			−0.660365
$949$	−7.69050			−0.249644
$950$	−17.0586			−0.553453
$951$			11.1549i			0.361722i
$952$		0			0
$953$			37.6405i			1.21930i	0.792673	+	0.609648i	$0.208689\pi$
$953$			37.6405i			1.21930i	−0.792673	+	0.609648i	$0.791311\pi$
$954$		−	31.7226i		−	1.02706i
$955$			24.9332i			0.806820i
$956$	−45.7058			−1.47823
$957$	11.7551			0.379989
$958$	49.4396			1.59732
$959$		0			0
$960$			17.1130i			0.552319i
$961$	−0.402367			−0.0129796
$962$	4.04888			0.130541
$963$			35.9961i			1.15996i
$964$	−27.1775			−0.875328
$965$	−33.7166			−1.08538
$966$		0			0
$967$	−22.5027			−0.723636			−0.361818	−	0.932249i	$-0.617844\pi$
$967$	−22.5027			−0.723636			−0.361818	+	0.932249i	$0.617844\pi$
$968$	−47.3102			−1.52061
$969$		−	23.1373i		−	0.743277i
$970$	16.6814			0.535607
$971$	47.1721			1.51383			0.756913	−	0.653516i	$-0.226707\pi$
$971$	47.1721			1.51383			0.756913	+	0.653516i	$0.226707\pi$
$972$		−	65.4962i		−	2.10079i
$973$		0			0
$974$	−66.6948			−2.13704
$975$	−2.46543			−0.0789569
$976$	26.3780			0.844340
$977$		−	2.43944i		−	0.0780445i	−0.999238	−	0.0390223i	$-0.987576\pi$
$977$		−	2.43944i		−	0.0780445i	0.999238	−	0.0390223i	$-0.0124243\pi$
$978$		−	14.7433i		−	0.471440i
$979$			35.7710i			1.14325i
$980$		0			0
$981$			13.1808i			0.420832i
$982$	26.1330			0.833937
$983$	24.8899			0.793864			0.396932	−	0.917848i	$-0.370075\pi$
$983$	24.8899			0.793864			0.396932	+	0.917848i	$0.370075\pi$
$984$	−8.51156			−0.271339
$985$	59.8481			1.90692
$986$	−26.2449			−0.835809
$987$		0			0
$988$			50.2629i			1.59908i
$989$	15.0094	+	40.1816i	0.477270	+	1.27770i
$990$			55.8871i			1.77621i
$991$	−24.2765			−0.771170			−0.385585	−	0.922672i	$-0.626000\pi$
$991$	−24.2765			−0.771170			−0.385585	+	0.922672i	$0.626000\pi$
$992$			4.56862i			0.145054i
$993$			35.2095i			1.11734i
$994$		0			0
$995$	60.0045			1.90227
$996$		−	3.42800i		−	0.108620i
$997$		−	3.32387i		−	0.105268i	−0.998614	−	0.0526340i	$-0.983238\pi$
$997$		−	3.32387i		−	0.105268i	0.998614	−	0.0526340i	$-0.0167617\pi$
$998$	68.3135			2.16243
$999$	3.86025			0.122133

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a_n

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a_n

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a_n

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up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.2.c.c.1126.28		28
7.2	even	3		161.2.g.a.45.1	✓	28
7.3	odd	6		161.2.g.a.68.2	yes	28
7.6	odd	2	inner	1127.2.c.c.1126.27		28
23.22	odd	2	inner	1127.2.c.c.1126.26		28
161.45	even	6		161.2.g.a.68.1	yes	28
161.114	odd	6		161.2.g.a.45.2	yes	28
161.160	even	2	inner	1127.2.c.c.1126.25		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.g.a.45.1	✓	28	7.2	even	3
161.2.g.a.45.2	yes	28	161.114	odd	6
161.2.g.a.68.1	yes	28	161.45	even	6
161.2.g.a.68.2	yes	28	7.3	odd	6
1127.2.c.c.1126.25		28	161.160	even	2	inner
1127.2.c.c.1126.26		28	23.22	odd	2	inner
1127.2.c.c.1126.27		28	7.6	odd	2	inner
1127.2.c.c.1126.28		28	1.1	even	1	trivial

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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1127.2.c.c.1126.28

Level	$1127$
Weight	$2$
Character	1127.1126
Analytic conductor	$8.999$
Analytic rank	$0$
Dimension	$28$
Inner twists	$4$