LMFDB - Embedded newform 1755.2.b.e.1351.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,2,Mod(1351,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("1755.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1755.b (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:	$14.0137455547$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 32 x^{16} + 428 x^{14} + 3114 x^{12} + 13440 x^{10} + 35180 x^{8} + 54641 x^{6} + 46624 x^{4} + \cdots + 1764 $ x^18 + 32x^16 + 428x^14 + 3114x^12 + 13440x^10 + 35180x^8 + 54641x^6 + 46624x^4 + 18060x^2 + 1764
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.9
Root			$-0.376265i$ of defining polynomial
Character	$\chi$	$=$	1755.1351
Dual form			1755.2.b.e.1351.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.376265i q^{2} +1.85842 q^{4} +1.00000i q^{5} -4.44363i q^{7} -1.45179i q^{8} +O(q^{10})$	\(q-0.376265i q^{2} +1.85842 q^{4} +1.00000i q^{5} -4.44363i q^{7} -1.45179i q^{8} +0.376265 q^{10} +0.203271i q^{11} +(-3.56791 - 0.519657i) q^{13} -1.67198 q^{14} +3.17059 q^{16} +3.41326 q^{17} -2.56759i q^{19} +1.85842i q^{20} +0.0764840 q^{22} +4.02017 q^{23} -1.00000 q^{25} +(-0.195529 + 1.34248i) q^{26} -8.25815i q^{28} -8.54970 q^{29} -7.81680i q^{31} -4.09656i q^{32} -1.28429i q^{34} +4.44363 q^{35} +0.723958i q^{37} -0.966094 q^{38} +1.45179 q^{40} +7.89773i q^{41} -0.347396 q^{43} +0.377765i q^{44} -1.51265i q^{46} -10.9839i q^{47} -12.7459 q^{49} +0.376265i q^{50} +(-6.63068 - 0.965744i) q^{52} +6.46467 q^{53} -0.203271 q^{55} -6.45122 q^{56} +3.21696i q^{58} -4.37641i q^{59} -11.3969 q^{61} -2.94119 q^{62} +4.79979 q^{64} +(0.519657 - 3.56791i) q^{65} -3.42221i q^{67} +6.34329 q^{68} -1.67198i q^{70} +14.1713i q^{71} -8.38500i q^{73} +0.272400 q^{74} -4.77167i q^{76} +0.903264 q^{77} +9.61803 q^{79} +3.17059i q^{80} +2.97164 q^{82} -0.947239i q^{83} +3.41326i q^{85} +0.130713i q^{86} +0.295108 q^{88} -3.99865i q^{89} +(-2.30917 + 15.8545i) q^{91} +7.47118 q^{92} -4.13288 q^{94} +2.56759 q^{95} -1.48067i q^{97} +4.79582i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 28 q^{4}+O(q^{10}) $ 18 * q - 28 * q^4	$ 18 q - 28 q^{4} - 10 q^{13} + 16 q^{14} + 24 q^{16} + 12 q^{22} - 10 q^{23} - 18 q^{25} - 8 q^{26} - 40 q^{29} - 10 q^{35} + 12 q^{38} - 8 q^{43} - 12 q^{49} - 6 q^{52} + 34 q^{53} - 16 q^{55} - 8 q^{56} - 16 q^{61} - 28 q^{62} - 76 q^{64} + 8 q^{65} - 24 q^{68} - 100 q^{74} + 4 q^{77} + 6 q^{79} - 16 q^{82} + 28 q^{88} - 6 q^{91} + 80 q^{92} - 40 q^{94} + 28 q^{95}+O(q^{100}) $ 18 * q - 28 * q^4 - 10 * q^13 + 16 * q^14 + 24 * q^16 + 12 * q^22 - 10 * q^23 - 18 * q^25 - 8 * q^26 - 40 * q^29 - 10 * q^35 + 12 * q^38 - 8 * q^43 - 12 * q^49 - 6 * q^52 + 34 * q^53 - 16 * q^55 - 8 * q^56 - 16 * q^61 - 28 * q^62 - 76 * q^64 + 8 * q^65 - 24 * q^68 - 100 * q^74 + 4 * q^77 + 6 * q^79 - 16 * q^82 + 28 * q^88 - 6 * q^91 + 80 * q^92 - 40 * q^94 + 28 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1755\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$1081$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	0.376265i		−	0.266060i	−0.991112	−	0.133030i	$-0.957529\pi$
$2$		−	0.376265i		−	0.266060i	0.991112	−	0.133030i	$-0.0424706\pi$
$3$		0			0
$3$		0			0
$4$	1.85842			0.929212
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	4.44363i		−	1.67953i	−0.542946	−	0.839767i	$-0.682691\pi$
$7$		−	4.44363i		−	1.67953i	0.542946	−	0.839767i	$-0.317309\pi$
$8$		−	1.45179i		−	0.513286i
$9$		0			0
$10$	0.376265			0.118985
$11$			0.203271i			0.0612887i	0.999530	+	0.0306443i	$0.00975592\pi$
$11$			0.203271i			0.0612887i	−0.999530	+	0.0306443i	$0.990244\pi$
$12$		0			0
$13$	−3.56791	−	0.519657i	−0.989559	−	0.144127i
$13$	−3.56791	−	0.519657i	−0.989559	−	0.144127i
$14$	−1.67198			−0.446857
$15$		0			0
$16$	3.17059			0.792648
$17$	3.41326			0.827838			0.413919	−	0.910314i	$-0.364160\pi$
$17$	3.41326			0.827838			0.413919	+	0.910314i	$0.364160\pi$
$18$		0			0
$19$		−	2.56759i		−	0.589045i	−0.955644	−	0.294523i	$-0.904839\pi$
$19$		−	2.56759i		−	0.589045i	0.955644	−	0.294523i	$-0.0951607\pi$
$20$			1.85842i			0.415556i
$21$		0			0
$22$	0.0764840			0.0163064
$23$	4.02017			0.838263			0.419131	−	0.907926i	$-0.362335\pi$
$23$	4.02017			0.838263			0.419131	+	0.907926i	$0.362335\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	−0.195529	+	1.34248i	−0.0383464	+	0.263282i
$27$		0			0
$28$		−	8.25815i		−	1.56064i
$29$	−8.54970			−1.58764			−0.793820	−	0.608153i	$-0.791911\pi$
$29$	−8.54970			−1.58764			−0.793820	+	0.608153i	$0.791911\pi$
$30$		0			0
$31$		−	7.81680i		−	1.40394i	−0.712207	−	0.701969i	$-0.752304\pi$
$31$		−	7.81680i		−	1.40394i	0.712207	−	0.701969i	$-0.247696\pi$
$32$		−	4.09656i		−	0.724177i
$33$		0			0
$34$		−	1.28429i		−	0.220254i
$35$	4.44363			0.751111
$36$		0			0
$37$			0.723958i			0.119018i	0.998228	+	0.0595090i	$0.0189535\pi$
$37$			0.723958i			0.119018i	−0.998228	+	0.0595090i	$0.981046\pi$
$38$	−0.966094			−0.156721
$39$		0			0
$40$	1.45179			0.229548
$41$			7.89773i			1.23342i	0.787191	+	0.616709i	$0.211534\pi$
$41$			7.89773i			1.23342i	−0.787191	+	0.616709i	$0.788466\pi$
$42$		0			0
$43$	−0.347396			−0.0529774			−0.0264887	−	0.999649i	$-0.508433\pi$
$43$	−0.347396			−0.0529774			−0.0264887	+	0.999649i	$0.508433\pi$
$44$			0.377765i			0.0569502i
$45$		0			0
$46$		−	1.51265i		−	0.223028i
$47$		−	10.9839i		−	1.60217i	−0.598548	−	0.801087i	$-0.704255\pi$
$47$		−	10.9839i		−	1.60217i	0.598548	−	0.801087i	$-0.295745\pi$
$48$		0			0
$49$	−12.7459			−1.82084
$50$			0.376265i			0.0532119i
$51$		0			0
$52$	−6.63068	−	0.965744i	−0.919511	−	0.133925i
$53$	6.46467			0.887990			0.443995	−	0.896029i	$-0.353561\pi$
$53$	6.46467			0.887990			0.443995	+	0.896029i	$0.353561\pi$
$54$		0			0
$55$	−0.203271			−0.0274091
$56$	−6.45122			−0.862081
$57$		0			0
$58$			3.21696i			0.422407i
$59$		−	4.37641i		−	0.569760i	−0.958563	−	0.284880i	$-0.908046\pi$
$59$		−	4.37641i		−	0.569760i	0.958563	−	0.284880i	$-0.0919538\pi$
$60$		0			0
$61$	−11.3969			−1.45923			−0.729615	−	0.683858i	$-0.760301\pi$
$61$	−11.3969			−1.45923			−0.729615	+	0.683858i	$0.760301\pi$
$62$	−2.94119			−0.373531
$63$		0			0
$64$	4.79979			0.599973
$65$	0.519657	−	3.56791i	0.0644556	−	0.442544i
$66$		0			0
$67$		−	3.42221i		−	0.418089i	−0.977906	−	0.209045i	$-0.932965\pi$
$67$		−	3.42221i		−	0.418089i	0.977906	−	0.209045i	$-0.0670354\pi$
$68$	6.34329			0.769237
$69$		0			0
$70$		−	1.67198i		−	0.199840i
$71$			14.1713i			1.68183i	0.541169	+	0.840914i	$0.317982\pi$
$71$			14.1713i			1.68183i	−0.541169	+	0.840914i	$0.682018\pi$
$72$		0			0
$73$		−	8.38500i		−	0.981390i	−0.871331	−	0.490695i	$-0.836743\pi$
$73$		−	8.38500i		−	0.981390i	0.871331	−	0.490695i	$-0.163257\pi$
$74$	0.272400			0.0316659
$75$		0			0
$76$		−	4.77167i		−	0.547348i
$77$	0.903264			0.102936
$78$		0			0
$79$	9.61803			1.08211			0.541056	−	0.840986i	$-0.318025\pi$
$79$	9.61803			1.08211			0.541056	+	0.840986i	$0.318025\pi$
$80$			3.17059i			0.354483i
$81$		0			0
$82$	2.97164			0.328163
$83$		−	0.947239i		−	0.103973i	−0.998648	−	0.0519865i	$-0.983445\pi$
$83$		−	0.947239i		−	0.103973i	0.998648	−	0.0519865i	$-0.0165553\pi$
$84$		0			0
$85$			3.41326i			0.370220i
$86$			0.130713i			0.0140952i
$87$		0			0
$88$	0.295108			0.0314586
$89$		−	3.99865i		−	0.423856i	−0.977285	−	0.211928i	$-0.932026\pi$
$89$		−	3.99865i		−	0.423856i	0.977285	−	0.211928i	$-0.0679742\pi$
$90$		0			0
$91$	−2.30917	+	15.8545i	−0.242066	+	1.66200i
$92$	7.47118			0.778924
$93$		0			0
$94$	−4.13288			−0.426274
$95$	2.56759			0.263429
$96$		0			0
$97$		−	1.48067i		−	0.150339i	−0.997171	−	0.0751695i	$-0.976050\pi$
$97$		−	1.48067i		−	0.150339i	0.997171	−	0.0751695i	$-0.0239498\pi$
$98$			4.79582i			0.484451i
$99$		0			0
$100$	−1.85842			−0.185842
$101$	13.1997			1.31342			0.656709	−	0.754144i	$-0.271948\pi$
$101$	13.1997			1.31342			0.656709	+	0.754144i	$0.271948\pi$
$102$		0			0
$103$	18.2241			1.79567			0.897837	−	0.440328i	$-0.145138\pi$
$103$	18.2241			1.79567			0.897837	+	0.440328i	$0.145138\pi$
$104$	−0.754434	+	5.17985i	−0.0739783	+	0.507926i
$105$		0			0
$106$		−	2.43243i		−	0.236258i
$107$	5.79520			0.560243			0.280122	−	0.959965i	$-0.409625\pi$
$107$	5.79520			0.560243			0.280122	+	0.959965i	$0.409625\pi$
$108$		0			0
$109$			2.56759i			0.245930i	0.992411	+	0.122965i	$0.0392404\pi$
$109$			2.56759i			0.245930i	−0.992411	+	0.122965i	$0.960760\pi$
$110$			0.0764840i			0.00729246i
$111$		0			0
$112$		−	14.0889i		−	1.33128i
$113$	−12.1906			−1.14679			−0.573396	−	0.819278i	$-0.694374\pi$
$113$	−12.1906			−1.14679			−0.573396	+	0.819278i	$0.694374\pi$
$114$		0			0
$115$			4.02017i			0.374883i
$116$	−15.8890			−1.47525
$117$		0			0
$118$	−1.64669			−0.151590
$119$		−	15.1673i		−	1.39038i
$120$		0			0
$121$	10.9587			0.996244
$122$			4.28827i			0.388242i
$123$		0			0
$124$		−	14.5269i		−	1.30456i
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	4.87592			0.432668			0.216334	−	0.976319i	$-0.430590\pi$
$127$	4.87592			0.432668			0.216334	+	0.976319i	$0.430590\pi$
$128$		−	9.99912i		−	0.883806i
$129$		0			0
$130$	−1.34248	−	0.195529i	−0.117743	−	0.0171490i
$131$	−9.72134			−0.849358			−0.424679	−	0.905344i	$-0.639613\pi$
$131$	−9.72134			−0.849358			−0.424679	+	0.905344i	$0.639613\pi$
$132$		0			0
$133$	−11.4094			−0.989322
$134$	−1.28766			−0.111237
$135$		0			0
$136$		−	4.95534i		−	0.424917i
$137$		−	8.85501i		−	0.756534i	−0.925696	−	0.378267i	$-0.876520\pi$
$137$		−	8.85501i		−	0.756534i	0.925696	−	0.378267i	$-0.123480\pi$
$138$		0			0
$139$	−7.42016			−0.629370			−0.314685	−	0.949196i	$-0.601899\pi$
$139$	−7.42016			−0.629370			−0.314685	+	0.949196i	$0.601899\pi$
$140$	8.25815			0.697941
$141$		0			0
$142$	5.33218			0.447467
$143$	0.105631	−	0.725254i	0.00883335	−	0.0606488i
$144$		0			0
$145$		−	8.54970i		−	0.710014i
$146$	−3.15498			−0.261108
$147$		0			0
$148$			1.34542i			0.110593i
$149$			13.7160i			1.12366i	0.827254	+	0.561828i	$0.189902\pi$
$149$			13.7160i			1.12366i	−0.827254	+	0.561828i	$0.810098\pi$
$150$		0			0
$151$			2.64684i			0.215397i	0.994184	+	0.107699i	$0.0343481\pi$
$151$			2.64684i			0.215397i	−0.994184	+	0.107699i	$0.965652\pi$
$152$	−3.72760			−0.302348
$153$		0			0
$154$		−	0.339867i		−	0.0273872i
$155$	7.81680			0.627860
$156$		0			0
$157$	7.30188			0.582754			0.291377	−	0.956608i	$-0.405887\pi$
$157$	7.30188			0.582754			0.291377	+	0.956608i	$0.405887\pi$
$158$		−	3.61893i		−	0.287907i
$159$		0			0
$160$	4.09656			0.323862
$161$		−	17.8641i		−	1.40789i
$162$		0			0
$163$		−	15.1799i		−	1.18898i	−0.804102	−	0.594491i	$-0.797353\pi$
$163$		−	15.1799i		−	1.18898i	0.804102	−	0.594491i	$-0.202647\pi$
$164$			14.6773i			1.14611i
$165$		0			0
$166$	−0.356413			−0.0276630
$167$			21.6864i			1.67815i	0.544018	+	0.839073i	$0.316902\pi$
$167$			21.6864i			1.67815i	−0.544018	+	0.839073i	$0.683098\pi$
$168$		0			0
$169$	12.4599	+	3.70818i	0.958455	+	0.285244i
$170$	1.28429			0.0985007
$171$		0			0
$172$	−0.645610			−0.0492273
$173$	3.70592			0.281756			0.140878	−	0.990027i	$-0.455008\pi$
$173$	3.70592			0.281756			0.140878	+	0.990027i	$0.455008\pi$
$174$		0			0
$175$			4.44363i			0.335907i
$176$			0.644491i			0.0485803i
$177$		0			0
$178$	−1.50455			−0.112771
$179$	−19.0125			−1.42106			−0.710531	−	0.703666i	$-0.751545\pi$
$179$	−19.0125			−1.42106			−0.710531	+	0.703666i	$0.751545\pi$
$180$		0			0
$181$	16.7146			1.24239			0.621194	−	0.783657i	$-0.286648\pi$
$181$	16.7146			1.24239			0.621194	+	0.783657i	$0.286648\pi$
$182$	5.96548	+	0.868859i	0.442191	+	0.0644041i
$183$		0			0
$184$		−	5.83644i		−	0.430268i
$185$	−0.723958			−0.0532265
$186$		0			0
$187$			0.693819i			0.0507371i
$188$		−	20.4128i		−	1.48876i
$189$		0			0
$190$		−	0.966094i		−	0.0700879i
$191$	−14.7518			−1.06740			−0.533700	−	0.845674i	$-0.679199\pi$
$191$	−14.7518			−1.06740			−0.533700	+	0.845674i	$0.679199\pi$
$192$		0			0
$193$			21.8299i			1.57135i	0.618640	+	0.785675i	$0.287684\pi$
$193$			21.8299i			1.57135i	−0.618640	+	0.785675i	$0.712316\pi$
$194$	−0.557123			−0.0399991
$195$		0			0
$196$	−23.6872			−1.69194
$197$			7.37341i			0.525334i	0.964887	+	0.262667i	$0.0846020\pi$
$197$			7.37341i			0.525334i	−0.964887	+	0.262667i	$0.915398\pi$
$198$		0			0
$199$	7.32127			0.518991			0.259496	−	0.965744i	$-0.416444\pi$
$199$	7.32127			0.518991			0.259496	+	0.965744i	$0.416444\pi$
$200$			1.45179i			0.102657i
$201$		0			0
$202$		−	4.96658i		−	0.349448i
$203$			37.9917i			2.66650i
$204$		0			0
$205$	−7.89773			−0.551601
$206$		−	6.85709i		−	0.477756i
$207$		0			0
$208$	−11.3124	−	1.64762i	−0.784372	−	0.114242i
$209$	0.521918			0.0361018
$210$		0			0
$211$	−5.35744			−0.368821			−0.184411	−	0.982849i	$-0.559038\pi$
$211$	−5.35744			−0.368821			−0.184411	+	0.982849i	$0.559038\pi$
$212$	12.0141			0.825131
$213$		0			0
$214$		−	2.18053i		−	0.149058i
$215$		−	0.347396i		−	0.0236922i
$216$		0			0
$217$	−34.7350			−2.35796
$218$	0.966094			0.0654322
$219$		0			0
$220$	−0.377765			−0.0254689
$221$	−12.1782	−	1.77373i	−0.819195	−	0.119314i
$222$		0			0
$223$			6.52293i			0.436808i	0.975858	+	0.218404i	$0.0700851\pi$
$223$			6.52293i			0.436808i	−0.975858	+	0.218404i	$0.929915\pi$
$224$	−18.2036			−1.21628
$225$		0			0
$226$			4.58689i			0.305115i
$227$		−	8.27508i		−	0.549236i	−0.961553	−	0.274618i	$-0.911449\pi$
$227$		−	8.27508i		−	0.549236i	0.961553	−	0.274618i	$-0.0885514\pi$
$228$		0			0
$229$		−	6.62719i		−	0.437937i	−0.975732	−	0.218969i	$-0.929731\pi$
$229$		−	6.62719i		−	0.437937i	0.975732	−	0.218969i	$-0.0702692\pi$
$230$	1.51265			0.0997411
$231$		0			0
$232$			12.4124i			0.814913i
$233$	−21.5179			−1.40969			−0.704844	−	0.709363i	$-0.748983\pi$
$233$	−21.5179			−1.40969			−0.704844	+	0.709363i	$0.748983\pi$
$234$		0			0
$235$	10.9839			0.716514
$236$		−	8.13323i		−	0.529428i
$237$		0			0
$238$	−5.70692			−0.369925
$239$		−	1.94803i		−	0.126008i	−0.998013	−	0.0630038i	$-0.979932\pi$
$239$		−	1.94803i		−	0.126008i	0.998013	−	0.0630038i	$-0.0200680\pi$
$240$		0			0
$241$		−	10.2959i		−	0.663220i	−0.943417	−	0.331610i	$-0.892408\pi$
$241$		−	10.2959i		−	0.663220i	0.943417	−	0.331610i	$-0.107592\pi$
$242$		−	4.12337i		−	0.265060i
$243$		0			0
$244$	−21.1804			−1.35593
$245$		−	12.7459i		−	0.814303i
$246$		0			0
$247$	−1.33427	+	9.16092i	−0.0848973	+	0.582895i
$248$	−11.3484			−0.720621
$249$		0			0
$250$	−0.376265			−0.0237971
$251$	15.4538			0.975436			0.487718	−	0.873001i	$-0.337829\pi$
$251$	15.4538			0.975436			0.487718	+	0.873001i	$0.337829\pi$
$252$		0			0
$253$			0.817185i			0.0513760i
$254$		−	1.83464i		−	0.115116i
$255$		0			0
$256$	5.83725			0.364828
$257$	26.8602			1.67549			0.837746	−	0.546061i	$-0.183873\pi$
$257$	26.8602			1.67549			0.837746	+	0.546061i	$0.183873\pi$
$258$		0			0
$259$	3.21700			0.199895
$260$	0.965744	−	6.63068i	0.0598929	−	0.411218i
$261$		0			0
$262$			3.65780i			0.225980i
$263$	−8.46130			−0.521746			−0.260873	−	0.965373i	$-0.584010\pi$
$263$	−8.46130			−0.521746			−0.260873	+	0.965373i	$0.584010\pi$
$264$		0			0
$265$			6.46467i			0.397121i
$266$			4.29297i			0.263219i
$267$		0			0
$268$		−	6.35992i		−	0.388494i
$269$	−3.68305			−0.224560			−0.112280	−	0.993677i	$-0.535815\pi$
$269$	−3.68305			−0.224560			−0.112280	+	0.993677i	$0.535815\pi$
$270$		0			0
$271$			29.4166i			1.78693i	0.449130	+	0.893466i	$0.351734\pi$
$271$			29.4166i			1.78693i	−0.449130	+	0.893466i	$0.648266\pi$
$272$	10.8221			0.656184
$273$		0			0
$274$	−3.33183			−0.201283
$275$		−	0.203271i		−	0.0122577i
$276$		0			0
$277$	13.7762			0.827730			0.413865	−	0.910338i	$-0.364179\pi$
$277$	13.7762			0.827730			0.413865	+	0.910338i	$0.364179\pi$
$278$			2.79195i			0.167450i
$279$		0			0
$280$		−	6.45122i		−	0.385534i
$281$		−	29.0179i		−	1.73106i	−0.500857	−	0.865530i	$-0.666982\pi$
$281$		−	29.0179i		−	1.73106i	0.500857	−	0.865530i	$-0.333018\pi$
$282$		0			0
$283$	9.45101			0.561804			0.280902	−	0.959736i	$-0.409366\pi$
$283$	9.45101			0.561804			0.280902	+	0.959736i	$0.409366\pi$
$284$			26.3364i			1.56278i
$285$		0			0
$286$	−0.272888	−	0.0397455i	−0.0161362	−	0.00235020i
$287$	35.0946			2.07157
$288$		0			0
$289$	−5.34963			−0.314684
$290$	−3.21696			−0.188906
$291$		0			0
$292$		−	15.5829i		−	0.911920i
$293$			30.6895i			1.79290i	0.443145	+	0.896450i	$0.353863\pi$
$293$			30.6895i			1.79290i	−0.443145	+	0.896450i	$0.646137\pi$
$294$		0			0
$295$	4.37641			0.254805
$296$	1.05104			0.0610902
$297$		0			0
$298$	5.16085			0.298960
$299$	−14.3436	−	2.08911i	−0.829511	−	0.120816i
$300$		0			0
$301$			1.54370i			0.0889774i
$302$	0.995915			0.0573085
$303$		0			0
$304$		−	8.14077i		−	0.466905i
$305$		−	11.3969i		−	0.652587i
$306$		0			0
$307$		−	0.0226180i		−	0.00129088i	−1.00000		0.000645440i	$-0.999795\pi$
$307$		−	0.0226180i		−	0.00129088i	1.00000		0.000645440i	$-0.000205450\pi$
$308$	1.67865			0.0956498
$309$		0			0
$310$		−	2.94119i		−	0.167048i
$311$	25.9795			1.47316			0.736580	−	0.676351i	$-0.236440\pi$
$311$	25.9795			1.47316			0.736580	+	0.676351i	$0.236440\pi$
$312$		0			0
$313$	3.72695			0.210660			0.105330	−	0.994437i	$-0.466410\pi$
$313$	3.72695			0.210660			0.105330	+	0.994437i	$0.466410\pi$
$314$		−	2.74744i		−	0.155047i
$315$		0			0
$316$	17.8744			1.00551
$317$			17.8842i			1.00448i	0.864729	+	0.502238i	$0.167490\pi$
$317$			17.8842i			1.00448i	−0.864729	+	0.502238i	$0.832510\pi$
$318$		0			0
$319$		−	1.73791i		−	0.0973043i
$320$			4.79979i			0.268316i
$321$		0			0
$322$	−6.72166			−0.374583
$323$		−	8.76386i		−	0.487634i
$324$		0			0
$325$	3.56791	+	0.519657i	0.197912	+	0.0288254i
$326$	−5.71167			−0.316340
$327$		0			0
$328$	11.4658			0.633096
$329$	−48.8086			−2.69091
$330$		0			0
$331$			13.9114i			0.764641i	0.924030	+	0.382320i	$0.124875\pi$
$331$			13.9114i			0.764641i	−0.924030	+	0.382320i	$0.875125\pi$
$332$		−	1.76037i		−	0.0966129i
$333$		0			0
$334$	8.15985			0.446487
$335$	3.42221			0.186975
$336$		0			0
$337$	−13.8837			−0.756294			−0.378147	−	0.925745i	$-0.623439\pi$
$337$	−13.8837			−0.756294			−0.378147	+	0.925745i	$0.623439\pi$
$338$	1.39526	−	4.68823i	0.0758920	−	0.255006i
$339$		0			0
$340$			6.34329i			0.344013i
$341$	1.58893			0.0860455
$342$		0			0
$343$			25.5325i			1.37863i
$344$			0.504347i			0.0271925i
$345$		0			0
$346$		−	1.39441i		−	0.0749638i
$347$	22.2722			1.19563			0.597817	−	0.801632i	$-0.296035\pi$
$347$	22.2722			1.19563			0.597817	+	0.801632i	$0.296035\pi$
$348$		0			0
$349$			31.5253i			1.68751i	0.536729	+	0.843754i	$0.319660\pi$
$349$			31.5253i			1.68751i	−0.536729	+	0.843754i	$0.680340\pi$
$350$	1.67198			0.0893713
$351$		0			0
$352$	0.832715			0.0443838
$353$			1.82751i			0.0972686i	0.998817	+	0.0486343i	$0.0154869\pi$
$353$			1.82751i			0.0972686i	−0.998817	+	0.0486343i	$0.984513\pi$
$354$		0			0
$355$	−14.1713			−0.752136
$356$		−	7.43118i		−	0.393852i
$357$		0			0
$358$			7.15375i			0.378087i
$359$		−	4.95504i		−	0.261517i	−0.991414	−	0.130759i	$-0.958259\pi$
$359$		−	4.95504i		−	0.261517i	0.991414	−	0.130759i	$-0.0417413\pi$
$360$		0			0
$361$	12.4075			0.653026
$362$		−	6.28913i		−	0.330549i
$363$		0			0
$364$	−4.29141	+	29.4643i	−0.224931	+	1.54435i
$365$	8.38500			0.438891
$366$		0			0
$367$	29.8181			1.55649			0.778245	−	0.627961i	$-0.216110\pi$
$367$	29.8181			1.55649			0.778245	+	0.627961i	$0.216110\pi$
$368$	12.7463			0.664447
$369$		0			0
$370$			0.272400i			0.0141614i
$371$		−	28.7266i		−	1.49141i
$372$		0			0
$373$	9.67323			0.500861			0.250430	−	0.968135i	$-0.419428\pi$
$373$	9.67323			0.500861			0.250430	+	0.968135i	$0.419428\pi$
$374$	0.261060			0.0134991
$375$		0			0
$376$	−15.9464			−0.822373
$377$	30.5045	+	4.44291i	1.57106	+	0.228822i
$378$		0			0
$379$			18.1987i			0.934803i	0.884045	+	0.467402i	$0.154810\pi$
$379$			18.1987i			0.934803i	−0.884045	+	0.467402i	$0.845190\pi$
$380$	4.77167			0.244782
$381$		0			0
$382$			5.55058i			0.283992i
$383$		−	27.5092i		−	1.40566i	−0.711360	−	0.702828i	$-0.751921\pi$
$383$		−	27.5092i		−	1.40566i	0.711360	−	0.702828i	$-0.248079\pi$
$384$		0			0
$385$			0.903264i			0.0460346i
$386$	8.21383			0.418073
$387$		0			0
$388$		−	2.75171i		−	0.139697i
$389$	−25.6363			−1.29981			−0.649907	−	0.760014i	$-0.725192\pi$
$389$	−25.6363			−1.29981			−0.649907	+	0.760014i	$0.725192\pi$
$390$		0			0
$391$	13.7219			0.693946
$392$			18.5043i			0.934610i
$393$		0			0
$394$	2.77436			0.139770
$395$			9.61803i			0.483935i
$396$		0			0
$397$			20.5314i			1.03044i	0.857057	+	0.515221i	$0.172290\pi$
$397$			20.5314i			1.03044i	−0.857057	+	0.515221i	$0.827710\pi$
$398$		−	2.75474i		−	0.138083i
$399$		0			0
$400$	−3.17059			−0.158530
$401$			17.1528i			0.856570i	0.903644	+	0.428285i	$0.140882\pi$
$401$			17.1528i			0.856570i	−0.903644	+	0.428285i	$0.859118\pi$
$402$		0			0
$403$	−4.06206	+	27.8896i	−0.202345	+	1.38928i
$404$	24.5306			1.22044
$405$		0			0
$406$	14.2950			0.709447
$407$	−0.147160			−0.00729445
$408$		0			0
$409$		−	11.9499i		−	0.590882i	−0.955361	−	0.295441i	$-0.904533\pi$
$409$		−	11.9499i		−	0.590882i	0.955361	−	0.295441i	$-0.0954666\pi$
$410$			2.97164i			0.146759i
$411$		0			0
$412$	33.8681			1.66856
$413$	−19.4472			−0.956932
$414$		0			0
$415$	0.947239			0.0464981
$416$	−2.12881	+	14.6162i	−0.104373	+	0.716616i
$417$		0			0
$418$		−	0.196379i		−	0.00960523i
$419$	19.3663			0.946107			0.473054	−	0.881034i	$-0.343152\pi$
$419$	19.3663			0.946107			0.473054	+	0.881034i	$0.343152\pi$
$420$		0			0
$421$			32.6869i			1.59306i	0.604598	+	0.796531i	$0.293334\pi$
$421$			32.6869i			1.59306i	−0.604598	+	0.796531i	$0.706666\pi$
$422$			2.01582i			0.0981284i
$423$		0			0
$424$		−	9.38534i		−	0.455793i
$425$	−3.41326			−0.165568
$426$		0			0
$427$			50.6438i			2.45083i
$428$	10.7699			0.520585
$429$		0			0
$430$	−0.130713			−0.00630354
$431$			16.7894i			0.808717i	0.914600	+	0.404359i	$0.132505\pi$
$431$			16.7894i			0.808717i	−0.914600	+	0.404359i	$0.867495\pi$
$432$		0			0
$433$	−5.07432			−0.243856			−0.121928	−	0.992539i	$-0.538908\pi$
$433$	−5.07432			−0.243856			−0.121928	+	0.992539i	$0.538908\pi$
$434$			13.0696i			0.627359i
$435$		0			0
$436$			4.77167i			0.228522i
$437$		−	10.3221i		−	0.493775i
$438$		0			0
$439$	29.2485			1.39596			0.697978	−	0.716119i	$-0.254083\pi$
$439$	29.2485			1.39596			0.697978	+	0.716119i	$0.254083\pi$
$440$			0.295108i			0.0140687i
$441$		0			0
$442$	−0.667392	+	4.58223i	−0.0317446	+	0.217955i
$443$	−38.9057			−1.84847			−0.924233	−	0.381829i	$-0.875294\pi$
$443$	−38.9057			−1.84847			−0.924233	+	0.381829i	$0.875294\pi$
$444$		0			0
$445$	3.99865			0.189554
$446$	2.45435			0.116217
$447$		0			0
$448$		−	21.3285i		−	1.00768i
$449$			15.5655i			0.734583i	0.930106	+	0.367291i	$0.119715\pi$
$449$			15.5655i			0.734583i	−0.930106	+	0.367291i	$0.880285\pi$
$450$		0			0
$451$	−1.60538			−0.0755945
$452$	−22.6553			−1.06561
$453$		0			0
$454$	−3.11362			−0.146130
$455$	−15.8545	−	2.30917i	−0.743269	−	0.108255i
$456$		0			0
$457$		−	38.5033i		−	1.80111i	−0.434743	−	0.900555i	$-0.643161\pi$
$457$		−	38.5033i		−	1.80111i	0.434743	−	0.900555i	$-0.356839\pi$
$458$	−2.49358			−0.116517
$459$		0			0
$460$			7.47118i			0.348345i
$461$			17.5429i			0.817052i	0.912747	+	0.408526i	$0.133957\pi$
$461$			17.5429i			0.817052i	−0.912747	+	0.408526i	$0.866043\pi$
$462$		0			0
$463$			22.0413i			1.02435i	0.858882	+	0.512173i	$0.171159\pi$
$463$			22.0413i			1.02435i	−0.858882	+	0.512173i	$0.828841\pi$
$464$	−27.1076			−1.25844
$465$		0			0
$466$			8.09645i			0.375061i
$467$	−1.72698			−0.0799151			−0.0399576	−	0.999201i	$-0.512722\pi$
$467$	−1.72698			−0.0799151			−0.0399576	+	0.999201i	$0.512722\pi$
$468$		0			0
$469$	−15.2070			−0.702196
$470$		−	4.13288i		−	0.190635i
$471$		0			0
$472$	−6.35363			−0.292450
$473$		−	0.0706157i		−	0.00324691i
$474$		0			0
$475$			2.56759i			0.117809i
$476$		−	28.1873i		−	1.29196i
$477$		0			0
$478$	−0.732976			−0.0335256
$479$		−	29.6475i		−	1.35463i	−0.735693	−	0.677315i	$-0.763143\pi$
$479$		−	29.6475i		−	1.35463i	0.735693	−	0.677315i	$-0.236857\pi$
$480$		0			0
$481$	0.376210	−	2.58302i	0.0171537	−	0.117775i
$482$	−3.87401			−0.176456
$483$		0			0
$484$	20.3659			0.925722
$485$	1.48067			0.0672336
$486$		0			0
$487$			18.5720i			0.841577i	0.907159	+	0.420789i	$0.138247\pi$
$487$			18.5720i			0.841577i	−0.907159	+	0.420789i	$0.861753\pi$
$488$			16.5460i			0.749001i
$489$		0			0
$490$	−4.79582			−0.216653
$491$	27.6507			1.24786			0.623928	−	0.781482i	$-0.285536\pi$
$491$	27.6507			1.24786			0.623928	+	0.781482i	$0.285536\pi$
$492$		0			0
$493$	−29.1824			−1.31431
$494$	3.44693	+	0.502038i	0.155085	+	0.0225878i
$495$		0			0
$496$		−	24.7839i		−	1.11283i
$497$	62.9722			2.82469
$498$		0			0
$499$			34.4764i			1.54337i	0.636002	+	0.771687i	$0.280587\pi$
$499$			34.4764i			1.54337i	−0.636002	+	0.771687i	$0.719413\pi$
$500$		−	1.85842i		−	0.0831113i
$501$		0			0
$502$		−	5.81473i		−	0.259524i
$503$	−41.5443			−1.85237			−0.926185	−	0.377071i	$-0.876931\pi$
$503$	−41.5443			−1.85237			−0.926185	+	0.377071i	$0.876931\pi$
$504$		0			0
$505$			13.1997i			0.587379i
$506$	0.307478			0.0136691
$507$		0			0
$508$	9.06153			0.402041
$509$			11.4586i			0.507895i	0.967218	+	0.253948i	$0.0817291\pi$
$509$			11.4586i			0.507895i	−0.967218	+	0.253948i	$0.918271\pi$
$510$		0			0
$511$	−37.2598			−1.64828
$512$		−	22.1946i		−	0.980872i
$513$		0			0
$514$		−	10.1065i		−	0.445781i
$515$			18.2241i			0.803050i
$516$		0			0
$517$	2.23272			0.0981951
$518$		−	1.21045i		−	0.0531840i
$519$		0			0
$520$	−5.17985	−	0.754434i	−0.227152	−	0.0330841i
$521$	4.81979			0.211159			0.105579	−	0.994411i	$-0.466330\pi$
$521$	4.81979			0.211159			0.105579	+	0.994411i	$0.466330\pi$
$522$		0			0
$523$	−28.0440			−1.22628			−0.613140	−	0.789975i	$-0.710094\pi$
$523$	−28.0440			−1.22628			−0.613140	+	0.789975i	$0.710094\pi$
$524$	−18.0664			−0.789233
$525$		0			0
$526$			3.18369i			0.138816i
$527$		−	26.6808i		−	1.16223i
$528$		0			0
$529$	−6.83826			−0.297315
$530$	2.43243			0.105658
$531$		0			0
$532$	−21.2035			−0.919290
$533$	4.10411	−	28.1784i	0.177769	−	1.22054i
$534$		0			0
$535$			5.79520i			0.250548i
$536$	−4.96833			−0.214599
$537$		0			0
$538$			1.38581i			0.0597463i
$539$		−	2.59087i		−	0.111597i
$540$		0			0
$541$		−	7.75280i		−	0.333319i	−0.986015	−	0.166659i	$-0.946702\pi$
$541$		−	7.75280i		−	0.333319i	0.986015	−	0.166659i	$-0.0532980\pi$
$542$	11.0685			0.475431
$543$		0			0
$544$		−	13.9827i		−	0.599501i
$545$	−2.56759			−0.109983
$546$		0			0
$547$	−21.2318			−0.907808			−0.453904	−	0.891051i	$-0.649969\pi$
$547$	−21.2318			−0.907808			−0.453904	+	0.891051i	$0.649969\pi$
$548$		−	16.4564i		−	0.702981i
$549$		0			0
$550$	−0.0764840			−0.00326129
$551$			21.9521i			0.935192i
$552$		0			0
$553$		−	42.7390i		−	1.81745i
$554$		−	5.18350i		−	0.220226i
$555$		0			0
$556$	−13.7898			−0.584818
$557$		−	4.92896i		−	0.208847i	−0.994533	−	0.104423i	$-0.966700\pi$
$557$		−	4.92896i		−	0.208847i	0.994533	−	0.104423i	$-0.0332997\pi$
$558$		0			0
$559$	1.23948	+	0.180527i	0.0524243	+	0.00763548i
$560$	14.0889			0.595366
$561$		0			0
$562$	−10.9184			−0.460565
$563$	−26.3121			−1.10892			−0.554461	−	0.832210i	$-0.687075\pi$
$563$	−26.3121			−1.10892			−0.554461	+	0.832210i	$0.687075\pi$
$564$		0			0
$565$		−	12.1906i		−	0.512861i
$566$		−	3.55609i		−	0.149474i
$567$		0			0
$568$	20.5738			0.863258
$569$	3.95614			0.165850			0.0829251	−	0.996556i	$-0.473574\pi$
$569$	3.95614			0.165850			0.0829251	+	0.996556i	$0.473574\pi$
$570$		0			0
$571$	−13.9969			−0.585751			−0.292876	−	0.956151i	$-0.594612\pi$
$571$	−13.9969			−0.585751			−0.292876	+	0.956151i	$0.594612\pi$
$572$	0.196308	−	1.34783i	0.00820806	−	0.0563556i
$573$		0			0
$574$		−	13.2049i		−	0.551161i
$575$	−4.02017			−0.167653
$576$		0			0
$577$		−	28.4186i		−	1.18308i	−0.806275	−	0.591540i	$-0.798520\pi$
$577$		−	28.4186i		−	1.18308i	0.806275	−	0.591540i	$-0.201480\pi$
$578$			2.01288i			0.0837248i
$579$		0			0
$580$		−	15.8890i		−	0.659754i
$581$	−4.20918			−0.174626
$582$		0			0
$583$			1.31408i			0.0544237i
$584$	−12.1733			−0.503733
$585$		0			0
$586$	11.5474			0.477018
$587$		−	38.8815i		−	1.60481i	−0.596780	−	0.802405i	$-0.703553\pi$
$587$		−	38.8815i		−	1.60481i	0.596780	−	0.802405i	$-0.296447\pi$
$588$		0			0
$589$	−20.0703			−0.826983
$590$		−	1.64669i		−	0.0677932i
$591$		0			0
$592$			2.29538i			0.0943393i
$593$			35.6501i			1.46398i	0.681318	+	0.731988i	$0.261407\pi$
$593$			35.6501i			1.46398i	−0.681318	+	0.731988i	$0.738593\pi$
$594$		0			0
$595$	15.1673			0.621798
$596$			25.4901i			1.04412i
$597$		0			0
$598$	−0.786059	+	5.39699i	−0.0321443	+	0.220699i
$599$	27.0133			1.10373			0.551866	−	0.833933i	$-0.313916\pi$
$599$	27.0133			1.10373			0.551866	+	0.833933i	$0.313916\pi$
$600$		0			0
$601$	−24.3872			−0.994774			−0.497387	−	0.867529i	$-0.665707\pi$
$601$	−24.3872			−0.994774			−0.497387	+	0.867529i	$0.665707\pi$
$602$	0.580841			0.0236733
$603$		0			0
$604$			4.91896i			0.200150i
$605$			10.9587i			0.445534i
$606$		0			0
$607$	7.22640			0.293311			0.146655	−	0.989188i	$-0.453149\pi$
$607$	7.22640			0.293311			0.146655	+	0.989188i	$0.453149\pi$
$608$	−10.5183			−0.426573
$609$		0			0
$610$	−4.28827			−0.173627
$611$	−5.70789	+	39.1897i	−0.230916	+	1.58545i
$612$		0			0
$613$			14.3560i			0.579835i	0.957052	+	0.289918i	$0.0936279\pi$
$613$			14.3560i			0.579835i	−0.957052	+	0.289918i	$0.906372\pi$
$614$	−0.00851038			−0.000343451
$615$		0			0
$616$		−	1.31135i		−	0.0528358i
$617$			29.3673i			1.18228i	0.806568	+	0.591142i	$0.201323\pi$
$617$			29.3673i			1.18228i	−0.806568	+	0.591142i	$0.798677\pi$
$618$		0			0
$619$			43.8373i			1.76197i	0.473145	+	0.880985i	$0.343119\pi$
$619$			43.8373i			1.76197i	−0.473145	+	0.880985i	$0.656881\pi$
$620$	14.5269			0.583415
$621$		0			0
$622$		−	9.77516i		−	0.391948i
$623$	−17.7685			−0.711880
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	1.40232i		−	0.0560480i
$627$		0			0
$628$	13.5700			0.541502
$629$			2.47106i			0.0985276i
$630$		0			0
$631$		−	7.47564i		−	0.297600i	−0.988867	−	0.148800i	$-0.952459\pi$
$631$		−	7.47564i		−	0.297600i	0.988867	−	0.148800i	$-0.0475411\pi$
$632$		−	13.9634i		−	0.555433i
$633$		0			0
$634$	6.72920			0.267251
$635$			4.87592i			0.193495i
$636$		0			0
$637$	45.4761	+	6.62348i	1.80183	+	0.262432i
$638$	−0.653915			−0.0258888
$639$		0			0
$640$	9.99912			0.395250
$641$	35.5538			1.40429			0.702145	−	0.712034i	$-0.252226\pi$
$641$	35.5538			1.40429			0.702145	+	0.712034i	$0.252226\pi$
$642$		0			0
$643$		−	45.1278i		−	1.77967i	−0.456285	−	0.889834i	$-0.650820\pi$
$643$		−	45.1278i		−	1.77967i	0.456285	−	0.889834i	$-0.349180\pi$
$644$		−	33.1992i		−	1.30823i
$645$		0			0
$646$	−3.29753			−0.129740
$647$	−10.9729			−0.431390			−0.215695	−	0.976461i	$-0.569202\pi$
$647$	−10.9729			−0.431390			−0.215695	+	0.976461i	$0.569202\pi$
$648$		0			0
$649$	0.889600			0.0349198
$650$	0.195529	−	1.34248i	0.00766928	−	0.0526564i
$651$		0			0
$652$		−	28.2107i		−	1.10482i
$653$	6.20060			0.242648			0.121324	−	0.992613i	$-0.461286\pi$
$653$	6.20060			0.242648			0.121324	+	0.992613i	$0.461286\pi$
$654$		0			0
$655$		−	9.72134i		−	0.379844i
$656$			25.0405i			0.977666i
$657$		0			0
$658$			18.3650i			0.715942i
$659$	22.3120			0.869153			0.434577	−	0.900635i	$-0.356898\pi$
$659$	22.3120			0.869153			0.434577	+	0.900635i	$0.356898\pi$
$660$		0			0
$661$			22.5220i			0.876004i	0.898974	+	0.438002i	$0.144314\pi$
$661$			22.5220i			0.876004i	−0.898974	+	0.438002i	$0.855686\pi$
$662$	5.23438			0.203440
$663$		0			0
$664$	−1.37519			−0.0533678
$665$		−	11.4094i		−	0.442438i
$666$		0			0
$667$	−34.3712			−1.33086
$668$			40.3026i			1.55935i
$669$		0			0
$670$		−	1.28766i		−	0.0497466i
$671$		−	2.31667i		−	0.0894342i
$672$		0			0
$673$	27.5707			1.06277			0.531387	−	0.847129i	$-0.321671\pi$
$673$	27.5707			1.06277			0.531387	+	0.847129i	$0.321671\pi$
$674$			5.22396i			0.201219i
$675$		0			0
$676$	23.1558	+	6.89137i	0.890608	+	0.265053i
$677$	−0.741466			−0.0284968			−0.0142484	−	0.999898i	$-0.504536\pi$
$677$	−0.741466			−0.0284968			−0.0142484	+	0.999898i	$0.504536\pi$
$678$		0			0
$679$	−6.57954			−0.252499
$680$	4.95534			0.190029
$681$		0			0
$682$		−	0.597860i		−	0.0228932i
$683$			32.1517i			1.23025i	0.788430	+	0.615125i	$0.210895\pi$
$683$			32.1517i			1.23025i	−0.788430	+	0.615125i	$0.789105\pi$
$684$		0			0
$685$	8.85501			0.338332
$686$	9.60699			0.366797
$687$		0			0
$688$	−1.10145			−0.0419924
$689$	−23.0653	−	3.35941i	−0.878719	−	0.127983i
$690$		0			0
$691$		−	30.0416i		−	1.14284i	−0.820659	−	0.571419i	$-0.806393\pi$
$691$		−	30.0416i		−	1.14284i	0.820659	−	0.571419i	$-0.193607\pi$
$692$	6.88716			0.261811
$693$		0			0
$694$		−	8.38026i		−	0.318110i
$695$		−	7.42016i		−	0.281463i
$696$		0			0
$697$			26.9570i			1.02107i
$698$	11.8619			0.448978
$699$		0			0
$700$			8.25815i			0.312129i
$701$	−15.3577			−0.580052			−0.290026	−	0.957019i	$-0.593664\pi$
$701$	−15.3577			−0.580052			−0.290026	+	0.957019i	$0.593664\pi$
$702$		0			0
$703$	1.85883			0.0701070
$704$			0.975660i			0.0367716i
$705$		0			0
$706$	0.687629			0.0258792
$707$		−	58.6546i		−	2.20593i
$708$		0			0
$709$		−	26.3885i		−	0.991040i	−0.868597	−	0.495520i	$-0.834978\pi$
$709$		−	26.3885i		−	0.991040i	0.868597	−	0.495520i	$-0.165022\pi$
$710$			5.33218i			0.200113i
$711$		0			0
$712$	−5.80520			−0.217559
$713$		−	31.4248i		−	1.17687i
$714$		0			0
$715$	0.725254	+	0.105631i	0.0271229	+	0.00395039i
$716$	−35.3333			−1.32047
$717$		0			0
$718$	−1.86441			−0.0695792
$719$	−9.58686			−0.357530			−0.178765	−	0.983892i	$-0.557210\pi$
$719$	−9.58686			−0.357530			−0.178765	+	0.983892i	$0.557210\pi$
$720$		0			0
$721$		−	80.9812i		−	3.01590i
$722$		−	4.66850i		−	0.173744i
$723$		0			0
$724$	31.0629			1.15444
$725$	8.54970			0.317528
$726$		0			0
$727$	25.7089			0.953490			0.476745	−	0.879042i	$-0.341817\pi$
$727$	25.7089			0.953490			0.476745	+	0.879042i	$0.341817\pi$
$728$	23.0174	+	3.35243i	0.853080	+	0.124249i
$729$		0			0
$730$		−	3.15498i		−	0.116771i
$731$	−1.18575			−0.0438567
$732$		0			0
$733$		−	29.2531i		−	1.08049i	−0.841508	−	0.540245i	$-0.818332\pi$
$733$		−	29.2531i		−	1.08049i	0.841508	−	0.540245i	$-0.181668\pi$
$734$		−	11.2195i		−	0.414119i
$735$		0			0
$736$		−	16.4689i		−	0.607051i
$737$	0.695638			0.0256241
$738$		0			0
$739$			25.1216i			0.924113i	0.886850	+	0.462057i	$0.152888\pi$
$739$			25.1216i			0.924113i	−0.886850	+	0.462057i	$0.847112\pi$
$740$	−1.34542			−0.0494587
$741$		0			0
$742$	−10.8088			−0.396804
$743$			22.8052i			0.836641i	0.908300	+	0.418320i	$0.137381\pi$
$743$			22.8052i			0.836641i	−0.908300	+	0.418320i	$0.862619\pi$
$744$		0			0
$745$	−13.7160			−0.502515
$746$		−	3.63970i		−	0.133259i
$747$		0			0
$748$			1.28941i			0.0471455i
$749$		−	25.7517i		−	0.940948i
$750$		0			0
$751$	23.2689			0.849095			0.424548	−	0.905406i	$-0.360433\pi$
$751$	23.2689			0.849095			0.424548	+	0.905406i	$0.360433\pi$
$752$		−	34.8256i		−	1.26996i
$753$		0			0
$754$	1.67171	−	11.4778i	0.0608802	−	0.417997i
$755$	−2.64684			−0.0963285
$756$		0			0
$757$	−18.1940			−0.661272			−0.330636	−	0.943758i	$-0.607263\pi$
$757$	−18.1940			−0.661272			−0.330636	+	0.943758i	$0.607263\pi$
$758$	6.84753			0.248713
$759$		0			0
$760$		−	3.72760i		−	0.135214i
$761$		−	32.1070i		−	1.16388i	−0.813233	−	0.581938i	$-0.802295\pi$
$761$		−	32.1070i		−	1.16388i	0.813233	−	0.581938i	$-0.197705\pi$
$762$		0			0
$763$	11.4094			0.413049
$764$	−27.4150			−0.991841
$765$		0			0
$766$	−10.3508			−0.373988
$767$	−2.27423	+	15.6146i	−0.0821178	+	0.563811i
$768$		0			0
$769$		−	47.3215i		−	1.70646i	−0.521538	−	0.853228i	$-0.674642\pi$
$769$		−	47.3215i		−	1.70646i	0.521538	−	0.853228i	$-0.325358\pi$
$770$	0.339867			0.0122479
$771$		0			0
$772$			40.5692i			1.46012i
$773$		−	45.0238i		−	1.61939i	−0.586849	−	0.809696i	$-0.699632\pi$
$773$		−	45.0238i		−	1.61939i	0.586849	−	0.809696i	$-0.300368\pi$
$774$		0			0
$775$			7.81680i			0.280788i
$776$	−2.14962			−0.0771668
$777$		0			0
$778$			9.64606i			0.345828i
$779$	20.2781			0.726539
$780$		0			0
$781$	−2.88063			−0.103077
$782$		−	5.16307i		−	0.184631i
$783$		0			0
$784$	−40.4119			−1.44328
$785$			7.30188i			0.260615i
$786$		0			0
$787$		−	15.6146i		−	0.556602i	−0.960494	−	0.278301i	$-0.910229\pi$
$787$		−	15.6146i		−	0.556602i	0.960494	−	0.278301i	$-0.0897712\pi$
$788$			13.7029i			0.488147i
$789$		0			0
$790$	3.61893			0.128756
$791$			54.1704i			1.92608i
$792$		0			0
$793$	40.6632	+	5.92251i	1.44399	+	0.210314i
$794$	7.72526			0.274159
$795$		0			0
$796$	13.6060			0.482253
$797$	−1.16210			−0.0411638			−0.0205819	−	0.999788i	$-0.506552\pi$
$797$	−1.16210			−0.0411638			−0.0205819	+	0.999788i	$0.506552\pi$
$798$		0			0
$799$		−	37.4911i		−	1.32634i
$800$			4.09656i			0.144835i
$801$		0			0
$802$	6.45400			0.227899
$803$	1.70443			0.0601481
$804$		0			0
$805$	17.8641			0.629628
$806$	10.4939	+	1.52841i	0.369631	+	0.0538360i
$807$		0			0
$808$		−	19.1632i		−	0.674159i
$809$	−43.2782			−1.52158			−0.760791	−	0.648998i	$-0.775189\pi$
$809$	−43.2782			−1.52158			−0.760791	+	0.648998i	$0.775189\pi$
$810$		0			0
$811$		−	11.0604i		−	0.388382i	−0.980964	−	0.194191i	$-0.937792\pi$
$811$		−	11.0604i		−	0.388382i	0.980964	−	0.194191i	$-0.0622081\pi$
$812$			70.6048i			2.47774i
$813$		0			0
$814$			0.0553712i			0.00194076i
$815$	15.1799			0.531729
$816$		0			0
$817$			0.891971i			0.0312061i
$818$	−4.49631			−0.157210
$819$		0			0
$820$	−14.6773			−0.512555
$821$		−	17.0217i		−	0.594062i	−0.954868	−	0.297031i	$-0.904004\pi$
$821$		−	17.0217i		−	0.594062i	0.954868	−	0.297031i	$-0.0959964\pi$
$822$		0			0
$823$	−15.8372			−0.552050			−0.276025	−	0.961150i	$-0.589017\pi$
$823$	−15.8372			−0.552050			−0.276025	+	0.961150i	$0.589017\pi$
$824$		−	26.4576i		−	0.921693i
$825$		0			0
$826$			7.31729i			0.254601i
$827$		−	4.95266i		−	0.172221i	−0.996286	−	0.0861104i	$-0.972556\pi$
$827$		−	4.95266i		−	0.172221i	0.996286	−	0.0861104i	$-0.0274438\pi$
$828$		0			0
$829$	28.3061			0.983111			0.491556	−	0.870846i	$-0.336428\pi$
$829$	28.3061			0.983111			0.491556	+	0.870846i	$0.336428\pi$
$830$		−	0.356413i		−	0.0123713i
$831$		0			0
$832$	−17.1252	−	2.49424i	−0.593709	−	0.0864724i
$833$	−43.5050			−1.50736
$834$		0			0
$835$	−21.6864			−0.750490
$836$	0.969945			0.0335462
$837$		0			0
$838$		−	7.28688i		−	0.251721i
$839$		−	1.19261i		−	0.0411736i	−0.999788	−	0.0205868i	$-0.993447\pi$
$839$		−	1.19261i		−	0.0411736i	0.999788	−	0.0205868i	$-0.00655344\pi$
$840$		0			0
$841$	44.0974			1.52060
$842$	12.2989			0.423849
$843$		0			0
$844$	−9.95639			−0.342713
$845$	−3.70818	+	12.4599i	−0.127565	+	0.428634i
$846$		0			0
$847$		−	48.6963i		−	1.67323i
$848$	20.4968			0.703863
$849$		0			0
$850$			1.28429i			0.0440509i
$851$			2.91043i			0.0997684i
$852$		0			0
$853$		−	53.5941i		−	1.83503i	−0.397706	−	0.917513i	$-0.630193\pi$
$853$		−	53.5941i		−	1.83503i	0.397706	−	0.917513i	$-0.369807\pi$
$854$	19.0555			0.652066
$855$		0			0
$856$		−	8.41342i		−	0.287565i
$857$	51.8073			1.76970			0.884852	−	0.465872i	$-0.154259\pi$
$857$	51.8073			1.76970			0.884852	+	0.465872i	$0.154259\pi$
$858$		0			0
$859$	−13.3013			−0.453836			−0.226918	−	0.973914i	$-0.572865\pi$
$859$	−13.3013			−0.453836			−0.226918	+	0.973914i	$0.572865\pi$
$860$		−	0.645610i		−	0.0220151i
$861$		0			0
$862$	6.31727			0.215167
$863$		−	13.0864i		−	0.445465i	−0.974880	−	0.222732i	$-0.928502\pi$
$863$		−	13.0864i		−	0.445465i	0.974880	−	0.222732i	$-0.0714976\pi$
$864$		0			0
$865$			3.70592i			0.126005i
$866$			1.90929i			0.0648803i
$867$		0			0
$868$	−64.5523			−2.19105
$869$			1.95507i			0.0663212i
$870$		0			0
$871$	−1.77838	+	12.2101i	−0.0602580	+	0.413724i
$872$	3.72760			0.126233
$873$		0			0
$874$	−3.88386			−0.131374
$875$	−4.44363			−0.150222
$876$		0			0
$877$		−	31.9994i		−	1.08054i	−0.841491	−	0.540271i	$-0.818322\pi$
$877$		−	31.9994i		−	1.08054i	0.841491	−	0.540271i	$-0.181678\pi$
$878$		−	11.0052i		−	0.371407i
$879$		0			0
$880$	−0.644491			−0.0217258
$881$	−12.5124			−0.421555			−0.210777	−	0.977534i	$-0.567600\pi$
$881$	−12.5124			−0.421555			−0.210777	+	0.977534i	$0.567600\pi$
$882$		0			0
$883$	−50.1027			−1.68609			−0.843045	−	0.537843i	$-0.819239\pi$
$883$	−50.1027			−1.68609			−0.843045	+	0.537843i	$0.819239\pi$
$884$	−22.6323	−	3.29634i	−0.761206	−	0.110868i
$885$		0			0
$886$			14.6389i			0.491802i
$887$	2.00662			0.0673757			0.0336879	−	0.999432i	$-0.489275\pi$
$887$	2.00662			0.0673757			0.0336879	+	0.999432i	$0.489275\pi$
$888$		0			0
$889$		−	21.6668i		−	0.726681i
$890$		−	1.50455i		−	0.0504327i
$891$		0			0
$892$			12.1224i			0.405887i
$893$	−28.2023			−0.943753
$894$		0			0
$895$		−	19.0125i		−	0.635518i
$896$	−44.4324			−1.48438
$897$		0			0
$898$	5.85677			0.195443
$899$			66.8313i			2.22895i
$900$		0			0
$901$	22.0656			0.735112
$902$			0.604050i			0.0201127i
$903$		0			0
$904$			17.6982i			0.588632i
$905$			16.7146i			0.555613i
$906$		0			0
$907$	25.4617			0.845441			0.422720	−	0.906260i	$-0.361075\pi$
$907$	25.4617			0.845441			0.422720	+	0.906260i	$0.361075\pi$
$908$		−	15.3786i		−	0.510357i
$909$		0			0
$910$	−0.868859	+	5.96548i	−0.0288024	+	0.197754i
$911$	−1.44773			−0.0479654			−0.0239827	−	0.999712i	$-0.507635\pi$
$911$	−1.44773			−0.0479654			−0.0239827	+	0.999712i	$0.507635\pi$
$912$		0			0
$913$	0.192547			0.00637236
$914$	−14.4875			−0.479202
$915$		0			0
$916$		−	12.3161i		−	0.406937i
$917$			43.1981i			1.42653i
$918$		0			0
$919$	−24.6878			−0.814376			−0.407188	−	0.913344i	$-0.633491\pi$
$919$	−24.6878			−0.814376			−0.407188	+	0.913344i	$0.633491\pi$
$920$	5.83644			0.192422
$921$		0			0
$922$	6.60077			0.217385
$923$	7.36424	−	50.5620i	0.242397	−	1.66427i
$924$		0			0
$925$		−	0.723958i		−	0.0238036i
$926$	8.29337			0.272537
$927$		0			0
$928$			35.0244i			1.14973i
$929$			54.4308i			1.78582i	0.450239	+	0.892908i	$0.351339\pi$
$929$			54.4308i			1.78582i	−0.450239	+	0.892908i	$0.648661\pi$
$930$		0			0
$931$			32.7261i			1.07256i
$932$	−39.9895			−1.30990
$933$		0			0
$934$			0.649803i			0.0212622i
$935$	−0.693819			−0.0226903
$936$		0			0
$937$	11.2272			0.366778			0.183389	−	0.983040i	$-0.441293\pi$
$937$	11.2272			0.366778			0.183389	+	0.983040i	$0.441293\pi$
$938$			5.72188i			0.186826i
$939$		0			0
$940$	20.4128			0.665793
$941$		−	48.9073i		−	1.59433i	−0.603761	−	0.797166i	$-0.706332\pi$
$941$		−	48.9073i		−	1.59433i	0.603761	−	0.797166i	$-0.293668\pi$
$942$		0			0
$943$			31.7502i			1.03393i
$944$		−	13.8758i		−	0.451619i
$945$		0			0
$946$	−0.0265702			−0.000863873
$947$			13.6290i			0.442884i	0.975174	+	0.221442i	$0.0710764\pi$
$947$			13.6290i			0.442884i	−0.975174	+	0.221442i	$0.928924\pi$
$948$		0			0
$949$	−4.35733	+	29.9169i	−0.141445	+	0.971143i
$950$	0.966094			0.0313442
$951$		0			0
$952$	−22.0197			−0.713663
$953$	24.0569			0.779278			0.389639	−	0.920968i	$-0.372600\pi$
$953$	24.0569			0.779278			0.389639	+	0.920968i	$0.372600\pi$
$954$		0			0
$955$		−	14.7518i		−	0.477356i
$956$		−	3.62027i		−	0.117088i
$957$		0			0
$958$	−11.1553			−0.360412
$959$	−39.3484			−1.27063
$960$		0			0
$961$	−30.1023			−0.971043
$962$	−0.971899	−	0.141555i	−0.0313353	−	0.00456391i
$963$		0			0
$964$		−	19.1342i		−	0.616272i
$965$	−21.8299			−0.702729
$966$		0			0
$967$			37.8648i			1.21765i	0.793304	+	0.608825i	$0.208359\pi$
$967$			37.8648i			1.21765i	−0.793304	+	0.608825i	$0.791641\pi$
$968$		−	15.9097i		−	0.511358i
$969$		0			0
$970$		−	0.557123i		−	0.0178882i
$971$	16.4296			0.527251			0.263626	−	0.964625i	$-0.415082\pi$
$971$	16.4296			0.527251			0.263626	+	0.964625i	$0.415082\pi$
$972$		0			0
$973$			32.9725i			1.05705i
$974$	6.98800			0.223910
$975$		0			0
$976$	−36.1350			−1.15665
$977$		−	24.5048i		−	0.783978i	−0.919970	−	0.391989i	$-0.871787\pi$
$977$		−	24.5048i		−	0.783978i	0.919970	−	0.391989i	$-0.128213\pi$
$978$		0			0
$979$	0.812811			0.0259775
$980$		−	23.6872i		−	0.756661i
$981$		0			0
$982$		−	10.4040i		−	0.332004i
$983$			38.4535i			1.22647i	0.789899	+	0.613237i	$0.210133\pi$
$983$			38.4535i			1.22647i	−0.789899	+	0.613237i	$0.789867\pi$
$984$		0			0
$985$	−7.37341			−0.234936
$986$			10.9803i			0.349684i
$987$		0			0
$988$	−2.47963	+	17.0249i	−0.0788877	+	0.541633i
$989$	−1.39659			−0.0444090
$990$		0			0
$991$	4.61090			0.146470			0.0732350	−	0.997315i	$-0.476668\pi$
$991$	4.61090			0.146470			0.0732350	+	0.997315i	$0.476668\pi$
$992$	−32.0220			−1.01670
$993$		0			0
$994$		−	23.6942i		−	0.751536i
$995$			7.32127i			0.232100i
$996$		0			0
$997$	8.98517			0.284563			0.142282	−	0.989826i	$-0.454556\pi$
$997$	8.98517			0.284563			0.142282	+	0.989826i	$0.454556\pi$
$998$	12.9723			0.410630
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.b.e.1351.9	yes	18
3.2	odd	2		1755.2.b.d.1351.10	yes	18
13.12	even	2	inner	1755.2.b.e.1351.10	yes	18
39.38	odd	2		1755.2.b.d.1351.9	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.b.d.1351.9	✓	18	39.38	odd	2
1755.2.b.d.1351.10	yes	18	3.2	odd	2
1755.2.b.e.1351.9	yes	18	1.1	even	1	trivial
1755.2.b.e.1351.10	yes	18	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.376265i q^{2} +1.85842 q^{4} +1.00000i q^{5} -4.44363i q^{7} -1.45179i q^{8} +O(q^{10})\)	\(q-0.376265i q^{2} +1.85842 q^{4} +1.00000i q^{5} -4.44363i q^{7} -1.45179i q^{8} +0.376265 q^{10} +0.203271i q^{11} +(-3.56791 - 0.519657i) q^{13} -1.67198 q^{14} +3.17059 q^{16} +3.41326 q^{17} -2.56759i q^{19} +1.85842i q^{20} +0.0764840 q^{22} +4.02017 q^{23} -1.00000 q^{25} +(-0.195529 + 1.34248i) q^{26} -8.25815i q^{28} -8.54970 q^{29} -7.81680i q^{31} -4.09656i q^{32} -1.28429i q^{34} +4.44363 q^{35} +0.723958i q^{37} -0.966094 q^{38} +1.45179 q^{40} +7.89773i q^{41} -0.347396 q^{43} +0.377765i q^{44} -1.51265i q^{46} -10.9839i q^{47} -12.7459 q^{49} +0.376265i q^{50} +(-6.63068 - 0.965744i) q^{52} +6.46467 q^{53} -0.203271 q^{55} -6.45122 q^{56} +3.21696i q^{58} -4.37641i q^{59} -11.3969 q^{61} -2.94119 q^{62} +4.79979 q^{64} +(0.519657 - 3.56791i) q^{65} -3.42221i q^{67} +6.34329 q^{68} -1.67198i q^{70} +14.1713i q^{71} -8.38500i q^{73} +0.272400 q^{74} -4.77167i q^{76} +0.903264 q^{77} +9.61803 q^{79} +3.17059i q^{80} +2.97164 q^{82} -0.947239i q^{83} +3.41326i q^{85} +0.130713i q^{86} +0.295108 q^{88} -3.99865i q^{89} +(-2.30917 + 15.8545i) q^{91} +7.47118 q^{92} -4.13288 q^{94} +2.56759 q^{95} -1.48067i q^{97} +4.79582i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 28 q^{4}+O(q^{10}) \) 18 * q - 28 * q^4	\( 18 q - 28 q^{4} - 10 q^{13} + 16 q^{14} + 24 q^{16} + 12 q^{22} - 10 q^{23} - 18 q^{25} - 8 q^{26} - 40 q^{29} - 10 q^{35} + 12 q^{38} - 8 q^{43} - 12 q^{49} - 6 q^{52} + 34 q^{53} - 16 q^{55} - 8 q^{56} - 16 q^{61} - 28 q^{62} - 76 q^{64} + 8 q^{65} - 24 q^{68} - 100 q^{74} + 4 q^{77} + 6 q^{79} - 16 q^{82} + 28 q^{88} - 6 q^{91} + 80 q^{92} - 40 q^{94} + 28 q^{95}+O(q^{100}) \) 18 * q - 28 * q^4 - 10 * q^13 + 16 * q^14 + 24 * q^16 + 12 * q^22 - 10 * q^23 - 18 * q^25 - 8 * q^26 - 40 * q^29 - 10 * q^35 + 12 * q^38 - 8 * q^43 - 12 * q^49 - 6 * q^52 + 34 * q^53 - 16 * q^55 - 8 * q^56 - 16 * q^61 - 28 * q^62 - 76 * q^64 + 8 * q^65 - 24 * q^68 - 100 * q^74 + 4 * q^77 + 6 * q^79 - 16 * q^82 + 28 * q^88 - 6 * q^91 + 80 * q^92 - 40 * q^94 + 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more