LMFDB - Embedded newform 1848.2.v.c.881.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1848,2,Mod(881,1848)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1848.881"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1848, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1848.v (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.7563542935$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 40x^{12} + 388x^{8} + 436x^{4} + 16 $ x^16 + 40x^12 + 388x^8 + 436*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.4
Root			$1.35082 - 1.35082i$ of defining polynomial
Character	$\chi$	$=$	1848.881
Dual form			1848.2.v.c.881.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.35082 + 1.08411i) q^{3} +(-1.77571 - 1.96134i) q^{7} +(0.649405 - 2.92887i) q^{9} +1.00000i q^{11} -5.74450i q^{13} +3.64881 q^{17} +0.413771i q^{19} +(4.52497 + 0.724334i) q^{21} +8.85023i q^{23} -5.00000 q^{25} +(2.29799 + 4.66039i) q^{27} +1.29881i q^{29} -0.874646i q^{31} +(-1.08411 - 1.35082i) q^{33} -5.80404 q^{37} +(6.22768 + 7.75976i) q^{39} -4.71563 q^{41} +0.0536958 q^{43} -2.70163 q^{47} +(-0.693688 + 6.96554i) q^{49} +(-4.92887 + 3.95572i) q^{51} +10.7080i q^{53} +(-0.448574 - 0.558929i) q^{57} -3.93184 q^{59} +1.55312i q^{61} +(-6.89766 + 3.92713i) q^{63} -11.1029 q^{67} +(-9.59464 - 11.9550i) q^{69} -8.56571i q^{71} -16.4060i q^{73} +(6.75408 - 5.42056i) q^{75} +(1.96134 - 1.77571i) q^{77} +12.0068 q^{79} +(-8.15655 - 3.80404i) q^{81} -9.82140 q^{83} +(-1.40805 - 1.75445i) q^{87} -3.78872 q^{89} +(-11.2669 + 10.2006i) q^{91} +(0.948214 + 1.18149i) q^{93} -8.52783i q^{97} +(2.92887 + 0.649405i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{7} - 8 q^{9} + 20 q^{21} - 80 q^{25} + 24 q^{39} + 16 q^{43} - 24 q^{49} - 40 q^{51} - 72 q^{57} - 12 q^{63} - 48 q^{67} - 24 q^{79} - 16 q^{81} + 40 q^{91} - 40 q^{93} + 8 q^{99}+O(q^{100}) $ 16 * q + 4 * q^7 - 8 * q^9 + 20 * q^21 - 80 * q^25 + 24 * q^39 + 16 * q^43 - 24 * q^49 - 40 * q^51 - 72 * q^57 - 12 * q^63 - 48 * q^67 - 24 * q^79 - 16 * q^81 + 40 * q^91 - 40 * q^93 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1848\mathbb{Z}\right)^\times$.

$n$	$463$	$617$	$673$	$925$	$1585$
$\chi(n)$	$1$	$-1$	$1$	$1$	$-1$

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			0
$2$		0			0
$3$	−1.35082	+	1.08411i	−0.779894	+	0.625912i
$3$	−1.35082	+	1.08411i	−0.779894	+	0.625912i
$4$		0			0
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	−1.77571	−	1.96134i	−0.671156	−	0.741316i
$7$	−1.77571	−	1.96134i	−0.671156	−	0.741316i
$8$		0			0
$9$	0.649405	−	2.92887i	0.216468	−	0.976290i
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$		−	5.74450i		−	1.59324i	−0.604482	−	0.796619i	$-0.706620\pi$
$13$		−	5.74450i		−	1.59324i	0.604482	−	0.796619i	$-0.293380\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.64881			0.884966			0.442483	−	0.896777i	$-0.354098\pi$
$17$	3.64881			0.884966			0.442483	+	0.896777i	$0.354098\pi$
$18$		0			0
$19$			0.413771i			0.0949257i	0.998873	+	0.0474628i	$0.0151136\pi$
$19$			0.413771i			0.0949257i	−0.998873	+	0.0474628i	$0.984886\pi$
$20$		0			0
$21$	4.52497	+	0.724334i	0.987429	+	0.158063i
$22$		0			0
$23$			8.85023i			1.84540i	0.385517	+	0.922701i	$0.374023\pi$
$23$			8.85023i			1.84540i	−0.385517	+	0.922701i	$0.625977\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$	2.29799	+	4.66039i	0.442249	+	0.896892i
$28$		0			0
$29$			1.29881i			0.241183i	0.992702	+	0.120591i	$0.0384791\pi$
$29$			1.29881i			0.241183i	−0.992702	+	0.120591i	$0.961521\pi$
$30$		0			0
$31$		−	0.874646i		−	0.157091i	−0.996911	−	0.0785455i	$-0.974972\pi$
$31$		−	0.874646i		−	0.157091i	0.996911	−	0.0785455i	$-0.0250276\pi$
$32$		0			0
$33$	−1.08411	−	1.35082i	−0.188720	−	0.235147i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.80404			−0.954179			−0.477089	−	0.878855i	$-0.658308\pi$
$37$	−5.80404			−0.954179			−0.477089	+	0.878855i	$0.658308\pi$
$38$		0			0
$39$	6.22768	+	7.75976i	0.997227	+	1.24256i
$40$		0			0
$41$	−4.71563			−0.736457			−0.368229	−	0.929735i	$-0.620036\pi$
$41$	−4.71563			−0.736457			−0.368229	+	0.929735i	$0.620036\pi$
$42$		0			0
$43$	0.0536958			0.00818854			0.00409427	−	0.999992i	$-0.498697\pi$
$43$	0.0536958			0.00818854			0.00409427	+	0.999992i	$0.498697\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.70163			−0.394073			−0.197037	−	0.980396i	$-0.563132\pi$
$47$	−2.70163			−0.394073			−0.197037	+	0.980396i	$0.563132\pi$
$48$		0			0
$49$	−0.693688	+	6.96554i	−0.0990982	+	0.995078i
$50$		0			0
$51$	−4.92887	+	3.95572i	−0.690180	+	0.553911i
$52$		0			0
$53$			10.7080i			1.47085i	0.677604	+	0.735427i	$0.263018\pi$
$53$			10.7080i			1.47085i	−0.677604	+	0.735427i	$0.736982\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.448574	−	0.558929i	−0.0594151	−	0.0740319i
$58$		0			0
$59$	−3.93184			−0.511881			−0.255941	−	0.966692i	$-0.582385\pi$
$59$	−3.93184			−0.511881			−0.255941	+	0.966692i	$0.582385\pi$
$60$		0			0
$61$			1.55312i			0.198857i	0.995045	+	0.0994283i	$0.0317014\pi$
$61$			1.55312i			0.198857i	−0.995045	+	0.0994283i	$0.968299\pi$
$62$		0			0
$63$	−6.89766	+	3.92713i	−0.869023	+	0.494772i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−11.1029			−1.35643			−0.678215	−	0.734864i	$-0.737246\pi$
$67$	−11.1029			−1.35643			−0.678215	+	0.734864i	$0.737246\pi$
$68$		0			0
$69$	−9.59464	−	11.9550i	−1.15506	−	1.43922i
$70$		0			0
$71$		−	8.56571i		−	1.01656i	−0.861191	−	0.508281i	$-0.830281\pi$
$71$		−	8.56571i		−	1.01656i	0.861191	−	0.508281i	$-0.169719\pi$
$72$		0			0
$73$		−	16.4060i		−	1.92017i	−0.279705	−	0.960086i	$-0.590237\pi$
$73$		−	16.4060i		−	1.92017i	0.279705	−	0.960086i	$-0.409763\pi$
$74$		0			0
$75$	6.75408	−	5.42056i	0.779894	−	0.625912i
$76$		0			0
$77$	1.96134	−	1.77571i	0.223515	−	0.202361i
$78$		0			0
$79$	12.0068			1.35087			0.675434	−	0.737420i	$-0.263956\pi$
$79$	12.0068			1.35087			0.675434	+	0.737420i	$0.263956\pi$
$80$		0			0
$81$	−8.15655	−	3.80404i	−0.906283	−	0.422671i
$82$		0			0
$83$	−9.82140			−1.07804			−0.539019	−	0.842293i	$-0.681205\pi$
$83$	−9.82140			−1.07804			−0.539019	+	0.842293i	$0.681205\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.40805	−	1.75445i	−0.150959	−	0.188097i
$88$		0			0
$89$	−3.78872			−0.401603			−0.200802	−	0.979632i	$-0.564355\pi$
$89$	−3.78872			−0.401603			−0.200802	+	0.979632i	$0.564355\pi$
$90$		0			0
$91$	−11.2669	+	10.2006i	−1.18109	+	1.06931i
$92$		0			0
$93$	0.948214	+	1.18149i	0.0983252	+	0.122514i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	8.52783i		−	0.865869i	−0.901425	−	0.432935i	$-0.857478\pi$
$97$		−	8.52783i		−	0.865869i	0.901425	−	0.432935i	$-0.142522\pi$
$98$		0			0
$99$	2.92887	+	0.649405i	0.294362	+	0.0652676i
$100$		0			0
$101$	7.23941			0.720348			0.360174	−	0.932885i	$-0.382717\pi$
$101$	7.23941			0.720348			0.360174	+	0.932885i	$0.382717\pi$
$102$		0			0
$103$			15.8725i			1.56397i	0.623298	+	0.781984i	$0.285792\pi$
$103$			15.8725i			1.56397i	−0.623298	+	0.781984i	$0.714208\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			15.7155i			1.51927i	0.650348	+	0.759636i	$0.274623\pi$
$107$			15.7155i			1.51927i	−0.650348	+	0.759636i	$0.725377\pi$
$108$		0			0
$109$	−4.86524			−0.466006			−0.233003	−	0.972476i	$-0.574855\pi$
$109$	−4.86524			−0.466006			−0.233003	+	0.972476i	$0.574855\pi$
$110$		0			0
$111$	7.84019	−	6.29223i	0.744158	−	0.597232i
$112$		0			0
$113$		−	8.50523i		−	0.800105i	−0.916492	−	0.400053i	$-0.868992\pi$
$113$		−	8.50523i		−	0.800105i	0.916492	−	0.400053i	$-0.131008\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−16.8249	−	3.73050i	−1.55546	−	0.344885i
$118$		0			0
$119$	−6.47924	−	7.15655i	−0.593951	−	0.656040i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	6.36994	−	5.11226i	0.574358	−	0.460957i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−11.2669			−0.999776			−0.499888	−	0.866090i	$-0.666625\pi$
$127$	−11.2669			−0.999776			−0.499888	+	0.866090i	$0.666625\pi$
$128$		0			0
$129$	−0.0725332	+	0.0582123i	−0.00638619	+	0.00512530i
$130$		0			0
$131$	−19.6553			−1.71729			−0.858646	−	0.512568i	$-0.828694\pi$
$131$	−19.6553			−1.71729			−0.858646	+	0.512568i	$0.828694\pi$
$132$		0			0
$133$	0.811545	−	0.734739i	0.0703699	−	0.0637100i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.73988i			0.404955i	0.979287	+	0.202478i	$0.0648994\pi$
$137$			4.73988i			0.404955i	−0.979287	+	0.202478i	$0.935101\pi$
$138$		0			0
$139$			10.3928i			0.881502i	0.897629	+	0.440751i	$0.145288\pi$
$139$			10.3928i			0.881502i	−0.897629	+	0.440751i	$0.854712\pi$
$140$		0			0
$141$	3.64940	−	2.92887i	0.307335	−	0.246655i
$142$		0			0
$143$	5.74450			0.480379
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−6.61438	−	10.1612i	−0.545545	−	0.838082i
$148$		0			0
$149$			13.9114i			1.13967i	0.821759	+	0.569835i	$0.192993\pi$
$149$			13.9114i			1.13967i	−0.821759	+	0.569835i	$0.807007\pi$
$150$		0			0
$151$	16.6194			1.35247			0.676234	−	0.736687i	$-0.263611\pi$
$151$	16.6194			1.35247			0.676234	+	0.736687i	$0.263611\pi$
$152$		0			0
$153$	2.36955	−	10.6869i	0.191567	−	0.863984i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			12.1001i			0.965693i	0.875705	+	0.482847i	$0.160397\pi$
$157$			12.1001i			0.965693i	−0.875705	+	0.482847i	$0.839603\pi$
$158$		0			0
$159$	−11.6086	−	14.4645i	−0.920625	−	1.14711i
$160$		0			0
$161$	17.3583	−	15.7155i	1.36803	−	1.23855i
$162$		0			0
$163$	−12.2207			−0.957200			−0.478600	−	0.878033i	$-0.658856\pi$
$163$	−12.2207			−0.957200			−0.478600	+	0.878033i	$0.658856\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−19.0259			−1.47227			−0.736134	−	0.676836i	$-0.763351\pi$
$167$	−19.0259			−1.47227			−0.736134	+	0.676836i	$0.763351\pi$
$168$		0			0
$169$	−19.9993			−1.53841
$170$		0			0
$171$	1.21188	+	0.268705i	0.0926749	+	0.0205484i
$172$		0			0
$173$	−4.79732			−0.364734			−0.182367	−	0.983231i	$-0.558376\pi$
$173$	−4.79732			−0.364734			−0.182367	+	0.983231i	$0.558376\pi$
$174$		0			0
$175$	8.87856	+	9.80669i	0.671156	+	0.741316i
$176$		0			0
$177$	5.31118	−	4.26255i	0.399213	−	0.320393i
$178$		0			0
$179$			0.612625i			0.0457897i	0.999738	+	0.0228949i	$0.00728830\pi$
$179$			0.612625i			0.0457897i	−0.999738	+	0.0228949i	$0.992712\pi$
$180$		0			0
$181$			6.40671i			0.476207i	0.971240	+	0.238103i	$0.0765257\pi$
$181$			6.40671i			0.476207i	−0.971240	+	0.238103i	$0.923474\pi$
$182$		0			0
$183$	−1.68376	−	2.09798i	−0.124467	−	0.155087i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			3.64881i			0.266827i
$188$		0			0
$189$	5.06001	−	12.7827i	0.368062	−	0.929801i
$190$		0			0
$191$		−	1.13476i		−	0.0821083i	−0.999157	−	0.0410541i	$-0.986928\pi$
$191$		−	1.13476i		−	0.0821083i	0.999157	−	0.0410541i	$-0.0130716\pi$
$192$		0			0
$193$	−16.2057			−1.16651			−0.583256	−	0.812288i	$-0.698222\pi$
$193$	−16.2057			−1.16651			−0.583256	+	0.812288i	$0.698222\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	8.50905i		−	0.606245i	−0.952952	−	0.303122i	$-0.901971\pi$
$197$		−	8.50905i		−	0.606245i	0.952952	−	0.303122i	$-0.0980291\pi$
$198$		0			0
$199$			2.39498i			0.169776i	0.996391	+	0.0848880i	$0.0270532\pi$
$199$			2.39498i			0.169776i	−0.996391	+	0.0848880i	$0.972947\pi$
$200$		0			0
$201$	14.9979	−	12.0367i	1.05787	−	0.849005i
$202$		0			0
$203$	2.54740	−	2.30631i	0.178793	−	0.161871i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	25.9212	+	5.74738i	1.80165	+	0.399471i
$208$		0			0
$209$	−0.413771			−0.0286212
$210$		0			0
$211$	3.20642			0.220739			0.110370	−	0.993891i	$-0.464797\pi$
$211$	3.20642			0.220739			0.110370	+	0.993891i	$0.464797\pi$
$212$		0			0
$213$	9.28619	+	11.5707i	0.636279	+	0.792811i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.71548	+	1.55312i	−0.116454	+	0.105433i
$218$		0			0
$219$	17.7859	+	22.1614i	1.20186	+	1.49753i
$220$		0			0
$221$		−	20.9606i		−	1.40996i
$222$		0			0
$223$		−	22.2484i		−	1.48986i	−0.667140	−	0.744932i	$-0.732482\pi$
$223$		−	22.2484i		−	1.48986i	0.667140	−	0.744932i	$-0.267518\pi$
$224$		0			0
$225$	−3.24702	+	14.6443i	−0.216468	+	0.976290i
$226$		0			0
$227$	−1.66760			−0.110682			−0.0553412	−	0.998468i	$-0.517625\pi$
$227$	−1.66760			−0.110682			−0.0553412	+	0.998468i	$0.517625\pi$
$228$		0			0
$229$			12.1184i			0.800808i	0.916339	+	0.400404i	$0.131130\pi$
$229$			12.1184i			0.800808i	−0.916339	+	0.400404i	$0.868870\pi$
$230$		0			0
$231$	−0.724334	+	4.52497i	−0.0476577	+	0.297721i
$232$		0			0
$233$			16.1171i			1.05587i	0.849285	+	0.527934i	$0.177033\pi$
$233$			16.1171i			1.05587i	−0.849285	+	0.527934i	$0.822967\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−16.2189	+	13.0167i	−1.05353	+	0.845525i
$238$		0			0
$239$			18.7109i			1.21031i	0.796108	+	0.605155i	$0.206889\pi$
$239$			18.7109i			1.21031i	−0.796108	+	0.605155i	$0.793111\pi$
$240$		0			0
$241$			11.6955i			0.753373i	0.926341	+	0.376686i	$0.122937\pi$
$241$			11.6955i			0.753373i	−0.926341	+	0.376686i	$0.877063\pi$
$242$		0			0
$243$	15.1420	−	3.70405i	0.971359	−	0.237615i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.37691			0.151239
$248$		0			0
$249$	13.2669	−	10.6475i	0.840756	−	0.674758i
$250$		0			0
$251$	−31.0932			−1.96259			−0.981293	−	0.192520i	$-0.938334\pi$
$251$	−31.0932			−1.96259			−0.981293	+	0.192520i	$0.938334\pi$
$252$		0			0
$253$	−8.85023			−0.556410
$254$		0			0
$255$		0			0
$256$		0			0
$257$	19.0259			1.18680			0.593401	−	0.804907i	$-0.297785\pi$
$257$	19.0259			1.18680			0.593401	+	0.804907i	$0.297785\pi$
$258$		0			0
$259$	10.3063	+	11.3837i	0.640403	+	0.707348i
$260$		0			0
$261$	3.80404	+	0.843453i	0.235464	+	0.0522084i
$262$		0			0
$263$			0.789757i			0.0486985i	0.999704	+	0.0243492i	$0.00775137\pi$
$263$			0.789757i			0.0486985i	−0.999704	+	0.0243492i	$0.992249\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	5.11786	−	4.10739i	0.313208	−	0.251368i
$268$		0			0
$269$	1.21188			0.0738898			0.0369449	−	0.999317i	$-0.488237\pi$
$269$	1.21188			0.0738898			0.0369449	+	0.999317i	$0.488237\pi$
$270$		0			0
$271$		−	22.5642i		−	1.37068i	−0.728224	−	0.685340i	$-0.759654\pi$
$271$		−	22.5642i		−	1.37068i	0.728224	−	0.685340i	$-0.240346\pi$
$272$		0			0
$273$	4.16094	−	25.9937i	0.251831	−	1.57321i
$274$		0			0
$275$		−	5.00000i		−	0.301511i
$276$		0			0
$277$	−16.4733			−0.989786			−0.494893	−	0.868954i	$-0.664793\pi$
$277$	−16.4733			−0.989786			−0.494893	+	0.868954i	$0.664793\pi$
$278$		0			0
$279$	−2.56172	−	0.567999i	−0.153366	−	0.0340052i
$280$		0			0
$281$			4.40166i			0.262581i	0.991344	+	0.131291i	$0.0419121\pi$
$281$			4.40166i			0.262581i	−0.991344	+	0.131291i	$0.958088\pi$
$282$		0			0
$283$			17.1896i			1.02182i	0.859635	+	0.510908i	$0.170691\pi$
$283$			17.1896i			1.02182i	−0.859635	+	0.510908i	$0.829309\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.37360	+	9.24893i	0.494278	+	0.545947i
$288$		0			0
$289$	−3.68618			−0.216834
$290$		0			0
$291$	9.24511	+	11.5195i	0.541958	+	0.675286i
$292$		0			0
$293$	25.3619			1.48166			0.740828	−	0.671694i	$-0.234433\pi$
$293$	25.3619			1.48166			0.740828	+	0.671694i	$0.234433\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−4.66039	+	2.29799i	−0.270423	+	0.133343i
$298$		0			0
$299$	50.8402			2.94016
$300$		0			0
$301$	−0.0953484	−	0.105316i	−0.00549579	−	0.00607029i
$302$		0			0
$303$	−9.77910	+	7.84833i	−0.561795	+	0.450875i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	0.961500i		−	0.0548757i	−0.999624	−	0.0274379i	$-0.991265\pi$
$307$		−	0.961500i		−	0.0548757i	0.999624	−	0.0274379i	$-0.00873484\pi$
$308$		0			0
$309$	−17.2076	−	21.4409i	−0.978907	−	1.21973i
$310$		0			0
$311$	22.9577			1.30181			0.650907	−	0.759158i	$-0.274389\pi$
$311$	22.9577			1.30181			0.650907	+	0.759158i	$0.274389\pi$
$312$		0			0
$313$			1.75961i			0.0994591i	0.998763	+	0.0497295i	$0.0158359\pi$
$313$			1.75961i			0.0994591i	−0.998763	+	0.0497295i	$0.984164\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4.34500i		−	0.244040i	−0.992528	−	0.122020i	$-0.961063\pi$
$317$		−	4.34500i		−	0.244040i	0.992528	−	0.122020i	$-0.0389371\pi$
$318$		0			0
$319$	−1.29881			−0.0727194
$320$		0			0
$321$	−17.0373	−	21.2287i	−0.950931	−	1.18487i
$322$		0			0
$323$			1.50977i			0.0840060i
$324$		0			0
$325$			28.7225i			1.59324i
$326$		0			0
$327$	6.57204	−	5.27446i	0.363435	−	0.291679i
$328$		0			0
$329$	4.79732	+	5.29881i	0.264485	+	0.292133i
$330$		0			0
$331$	−9.19524			−0.505416			−0.252708	−	0.967543i	$-0.581321\pi$
$331$	−9.19524			−0.505416			−0.252708	+	0.967543i	$0.581321\pi$
$332$		0			0
$333$	−3.76917	+	16.9993i	−0.206549	+	0.931555i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−31.5233			−1.71719			−0.858593	−	0.512658i	$-0.828661\pi$
$337$	−31.5233			−1.71719			−0.858593	+	0.512658i	$0.828661\pi$
$338$		0			0
$339$	9.22062	+	11.4890i	0.500795	+	0.623997i
$340$		0			0
$341$	0.874646			0.0473647
$342$		0			0
$343$	14.8936	−	11.0082i	0.804177	−	0.594390i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	18.1945i		−	0.976733i	−0.872639	−	0.488366i	$-0.837593\pi$
$347$		−	18.1945i		−	0.976733i	0.872639	−	0.488366i	$-0.162407\pi$
$348$		0			0
$349$		−	10.2260i		−	0.547386i	−0.961817	−	0.273693i	$-0.911755\pi$
$349$		−	10.2260i		−	0.547386i	0.961817	−	0.273693i	$-0.0882452\pi$
$350$		0			0
$351$	26.7716	−	13.2008i	1.42896	−	0.704608i
$352$		0			0
$353$	4.43065			0.235820			0.117910	−	0.993024i	$-0.462381\pi$
$353$	4.43065			0.235820			0.117910	+	0.993024i	$0.462381\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	16.5108	+	2.64296i	0.873842	+	0.139880i
$358$		0			0
$359$		−	24.1321i		−	1.27365i	−0.771010	−	0.636823i	$-0.780248\pi$
$359$		−	24.1321i		−	1.27365i	0.771010	−	0.636823i	$-0.219752\pi$
$360$		0			0
$361$	18.8288			0.990989
$362$		0			0
$363$	1.35082	−	1.08411i	0.0708994	−	0.0569011i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			29.7853i			1.55478i	0.629018	+	0.777390i	$0.283457\pi$
$367$			29.7853i			1.55478i	−0.629018	+	0.777390i	$0.716543\pi$
$368$		0			0
$369$	−3.06235	+	13.8115i	−0.159420	+	0.718995i
$370$		0			0
$371$	21.0019	−	19.0143i	1.09037	−	0.987173i
$372$		0			0
$373$	27.9681			1.44813			0.724066	−	0.689730i	$-0.242271\pi$
$373$	27.9681			1.44813			0.724066	+	0.689730i	$0.242271\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.46101			0.384262
$378$		0			0
$379$	37.1314			1.90731			0.953657	−	0.300897i	$-0.0972862\pi$
$379$	37.1314			1.90731			0.953657	+	0.300897i	$0.0972862\pi$
$380$		0			0
$381$	15.2195	−	12.2146i	0.779719	−	0.625772i
$382$		0			0
$383$	−12.6703			−0.647422			−0.323711	−	0.946156i	$-0.604931\pi$
$383$	−12.6703			−0.647422			−0.323711	+	0.946156i	$0.604931\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.0348703	−	0.157268i	0.00177256	−	0.00799439i
$388$		0			0
$389$			30.4869i			1.54575i	0.634560	+	0.772873i	$0.281181\pi$
$389$			30.4869i			1.54575i	−0.634560	+	0.772873i	$0.718819\pi$
$390$		0			0
$391$			32.2928i			1.63312i
$392$		0			0
$393$	26.5507	−	21.3086i	1.33931	−	1.07487i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	16.8797i		−	0.847170i	−0.905856	−	0.423585i	$-0.860771\pi$
$397$		−	16.8797i		−	0.847170i	0.905856	−	0.423585i	$-0.139229\pi$
$398$		0			0
$399$	−0.299709	+	1.87230i	−0.0150042	+	0.0937324i
$400$		0			0
$401$		−	2.98953i		−	0.149290i	−0.997210	−	0.0746451i	$-0.976218\pi$
$401$		−	2.98953i		−	0.149290i	0.997210	−	0.0746451i	$-0.0237824\pi$
$402$		0			0
$403$	−5.02440			−0.250283
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5.80404i		−	0.287696i
$408$		0			0
$409$			3.07346i			0.151973i	0.997109	+	0.0759864i	$0.0242105\pi$
$409$			3.07346i			0.151973i	−0.997109	+	0.0759864i	$0.975789\pi$
$410$		0			0
$411$	−5.13856	−	6.40270i	−0.253466	−	0.315822i
$412$		0			0
$413$	6.98181	+	7.71166i	0.343553	+	0.379466i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−11.2669	−	14.0387i	−0.551743	−	0.687478i
$418$		0			0
$419$	−19.3649			−0.946039			−0.473020	−	0.881052i	$-0.656836\pi$
$419$	−19.3649			−0.946039			−0.473020	+	0.881052i	$0.656836\pi$
$420$		0			0
$421$	39.9400			1.94656			0.973278	−	0.229628i	$-0.0737511\pi$
$421$	39.9400			1.94656			0.973278	+	0.229628i	$0.0737511\pi$
$422$		0			0
$423$	−1.75445	+	7.91272i	−0.0853044	+	0.384730i
$424$		0			0
$425$	−18.2441			−0.884966
$426$		0			0
$427$	3.04619	−	2.75790i	0.147416	−	0.133464i
$428$		0			0
$429$	−7.75976	+	6.22768i	−0.374645	+	0.300675i
$430$		0			0
$431$		−	29.8964i		−	1.44006i	−0.693943	−	0.720030i	$-0.744128\pi$
$431$		−	29.8964i		−	1.44006i	0.693943	−	0.720030i	$-0.255872\pi$
$432$		0			0
$433$		−	13.2486i		−	0.636688i	−0.947975	−	0.318344i	$-0.896873\pi$
$433$		−	13.2486i		−	0.636688i	0.947975	−	0.318344i	$-0.103127\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.66197			−0.175176
$438$		0			0
$439$		−	31.9013i		−	1.52256i	−0.648421	−	0.761282i	$-0.724570\pi$
$439$		−	31.9013i		−	1.52256i	0.648421	−	0.761282i	$-0.275430\pi$
$440$		0			0
$441$	19.9507	+	6.55518i	0.950032	+	0.312151i
$442$		0			0
$443$		−	21.3086i		−	1.01240i	−0.862416	−	0.506200i	$-0.831050\pi$
$443$		−	21.3086i		−	1.01240i	0.862416	−	0.506200i	$-0.168950\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−15.0815	−	18.7918i	−0.713333	−	0.888821i
$448$		0			0
$449$		−	12.3780i		−	0.584153i	−0.956395	−	0.292076i	$-0.905654\pi$
$449$		−	12.3780i		−	0.584153i	0.956395	−	0.292076i	$-0.0943462\pi$
$450$		0			0
$451$		−	4.71563i		−	0.222050i
$452$		0			0
$453$	−22.4498	+	18.0173i	−1.05478	+	0.846526i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	32.0286			1.49823			0.749117	−	0.662438i	$-0.230478\pi$
$457$	32.0286			1.49823			0.749117	+	0.662438i	$0.230478\pi$
$458$		0			0
$459$	8.38495	+	17.0049i	0.391376	+	0.793719i
$460$		0			0
$461$	−5.58432			−0.260088			−0.130044	−	0.991508i	$-0.541512\pi$
$461$	−5.58432			−0.260088			−0.130044	+	0.991508i	$0.541512\pi$
$462$		0			0
$463$	−30.2057			−1.40378			−0.701889	−	0.712286i	$-0.747660\pi$
$463$	−30.2057			−1.40378			−0.701889	+	0.712286i	$0.747660\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−29.0538			−1.34445			−0.672224	−	0.740347i	$-0.734661\pi$
$467$	−29.0538			−1.34445			−0.672224	+	0.740347i	$0.734661\pi$
$468$		0			0
$469$	19.7155	+	21.7764i	0.910376	+	1.00554i
$470$		0			0
$471$	−13.1179	−	16.3450i	−0.604439	−	0.753138i
$472$		0			0
$473$			0.0536958i			0.00246894i
$474$		0			0
$475$		−	2.06886i		−	0.0949257i
$476$		0			0
$477$	31.3623	+	6.95381i	1.43598	+	0.318393i
$478$		0			0
$479$	15.2777			0.698057			0.349028	−	0.937112i	$-0.386512\pi$
$479$	15.2777			0.698057			0.349028	+	0.937112i	$0.386512\pi$
$480$		0			0
$481$			33.3413i			1.52023i
$482$		0			0
$483$	−6.41053	+	40.0470i	−0.291689	+	1.82220i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−3.60216			−0.163229			−0.0816147	−	0.996664i	$-0.526008\pi$
$487$	−3.60216			−0.163229			−0.0816147	+	0.996664i	$0.526008\pi$
$488$		0			0
$489$	16.5079	−	13.2486i	0.746514	−	0.599123i
$490$		0			0
$491$		−	8.81451i		−	0.397793i	−0.980020	−	0.198897i	$-0.936264\pi$
$491$		−	8.81451i		−	0.397793i	0.980020	−	0.198897i	$-0.0637358\pi$
$492$		0			0
$493$			4.73911i			0.213439i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.8002	+	15.2102i	−0.753594	+	0.682273i
$498$		0			0
$499$	−10.3131			−0.461678			−0.230839	−	0.972992i	$-0.574147\pi$
$499$	−10.3131			−0.461678			−0.230839	+	0.972992i	$0.574147\pi$
$500$		0			0
$501$	25.7005	−	20.6262i	1.14821	−	0.921510i
$502$		0			0
$503$	9.19198			0.409850			0.204925	−	0.978778i	$-0.434305\pi$
$503$	9.19198			0.409850			0.204925	+	0.978778i	$0.434305\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	27.0153	−	21.6814i	1.19979	−	0.962907i
$508$		0			0
$509$	13.0932			0.580347			0.290174	−	0.956974i	$-0.406287\pi$
$509$	13.0932			0.580347			0.290174	+	0.956974i	$0.406287\pi$
$510$		0			0
$511$	−32.1776	+	29.1323i	−1.42345	+	1.28874i
$512$		0			0
$513$	−1.92834	+	0.950844i	−0.0851381	+	0.0419808i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	2.70163i		−	0.118818i
$518$		0			0
$519$	6.48029	−	5.20083i	0.284453	−	0.228291i
$520$		0			0
$521$	16.7758			0.734963			0.367481	−	0.930031i	$-0.380220\pi$
$521$	16.7758			0.734963			0.367481	+	0.930031i	$0.380220\pi$
$522$		0			0
$523$			36.5895i			1.59995i	0.600035	+	0.799974i	$0.295153\pi$
$523$			36.5895i			1.59995i	−0.600035	+	0.799974i	$0.704847\pi$
$524$		0			0
$525$	−22.6248	−	3.62167i	−0.987429	−	0.158063i
$526$		0			0
$527$		−	3.19142i		−	0.139020i
$528$		0			0
$529$	−55.3267			−2.40551
$530$		0			0
$531$	−2.55335	+	11.5158i	−0.110806	+	0.499745i
$532$		0			0
$533$			27.0889i			1.17335i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−0.664153	−	0.827543i	−0.0286603	−	0.0357111i
$538$		0			0
$539$	−6.96554	−	0.693688i	−0.300027	−	0.0298792i
$540$		0			0
$541$	13.3405			0.573551			0.286776	−	0.957998i	$-0.407417\pi$
$541$	13.3405			0.573551			0.286776	+	0.957998i	$0.407417\pi$
$542$		0			0
$543$	−6.94558	−	8.65428i	−0.298064	−	0.371391i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.70119			0.286522			0.143261	−	0.989685i	$-0.454241\pi$
$547$	6.70119			0.286522			0.143261	+	0.989685i	$0.454241\pi$
$548$		0			0
$549$	4.54889	+	1.00860i	0.194142	+	0.0430462i
$550$		0			0
$551$	−0.537410			−0.0228944
$552$		0			0
$553$	−21.3206	−	23.5493i	−0.906644	−	1.00142i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			0.970709i			0.0411302i	0.999789	+	0.0205651i	$0.00654654\pi$
$557$			0.970709i			0.0411302i	−0.999789	+	0.0205651i	$0.993453\pi$
$558$		0			0
$559$		−	0.308456i		−	0.0130463i
$560$		0			0
$561$	−3.95572	−	4.92887i	−0.167010	−	0.208097i
$562$		0			0
$563$	9.36788			0.394809			0.197405	−	0.980322i	$-0.436749\pi$
$563$	9.36788			0.394809			0.197405	+	0.980322i	$0.436749\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	7.02268	+	22.7526i	0.294925	+	0.955521i
$568$		0			0
$569$		−	41.0278i		−	1.71998i	−0.510313	−	0.859988i	$-0.670471\pi$
$569$		−	41.0278i		−	1.71998i	0.510313	−	0.859988i	$-0.329529\pi$
$570$		0			0
$571$	−7.42713			−0.310816			−0.155408	−	0.987850i	$-0.549669\pi$
$571$	−7.42713			−0.310816			−0.155408	+	0.987850i	$0.549669\pi$
$572$		0			0
$573$	1.23021	+	1.53285i	0.0513926	+	0.0640357i
$574$		0			0
$575$		−	44.2512i		−	1.84540i
$576$		0			0
$577$			11.6524i			0.485095i	0.970139	+	0.242548i	$0.0779831\pi$
$577$			11.6524i			0.485095i	−0.970139	+	0.242548i	$0.922017\pi$
$578$		0			0
$579$	21.8909	−	17.5688i	0.909755	−	0.730134i
$580$		0			0
$581$	17.4400	+	19.2631i	0.723533	+	0.799167i
$582$		0			0
$583$	−10.7080			−0.443479
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−27.4798			−1.13421			−0.567106	−	0.823645i	$-0.691937\pi$
$587$	−27.4798			−1.13421			−0.567106	+	0.823645i	$0.691937\pi$
$588$		0			0
$589$	0.361904			0.0149120
$590$		0			0
$591$	9.22476	+	11.4942i	0.379456	+	0.472807i
$592$		0			0
$593$	−36.3665			−1.49339			−0.746697	−	0.665164i	$-0.768361\pi$
$593$	−36.3665			−1.49339			−0.746697	+	0.665164i	$0.768361\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−2.59643	−	3.23518i	−0.106265	−	0.132407i
$598$		0			0
$599$			36.7864i			1.50305i	0.659703	+	0.751526i	$0.270682\pi$
$599$			36.7864i			1.50305i	−0.659703	+	0.751526i	$0.729318\pi$
$600$		0			0
$601$		−	42.6639i		−	1.74030i	−0.492790	−	0.870149i	$-0.664023\pi$
$601$		−	42.6639i		−	1.74030i	0.492790	−	0.870149i	$-0.335977\pi$
$602$		0			0
$603$	−7.21024	+	32.5188i	−0.293624	+	1.32427i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	22.9486i		−	0.931454i	−0.884929	−	0.465727i	$-0.845793\pi$
$607$		−	22.9486i		−	0.931454i	0.884929	−	0.465727i	$-0.154207\pi$
$608$		0			0
$609$	−0.940772	+	5.87707i	−0.0381220	+	0.238151i
$610$		0			0
$611$			15.5195i			0.627853i
$612$		0			0
$613$	−1.76239			−0.0711822			−0.0355911	−	0.999366i	$-0.511331\pi$
$613$	−1.76239			−0.0711822			−0.0355911	+	0.999366i	$0.511331\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	41.4658i		−	1.66935i	−0.550742	−	0.834676i	$-0.685655\pi$
$617$		−	41.4658i		−	1.66935i	0.550742	−	0.834676i	$-0.314345\pi$
$618$		0			0
$619$		−	25.2507i		−	1.01491i	−0.861677	−	0.507457i	$-0.830586\pi$
$619$		−	25.2507i		−	1.01491i	0.861677	−	0.507457i	$-0.169414\pi$
$620$		0			0
$621$	−41.2455	+	20.3378i	−1.65513	+	0.816128i
$622$		0			0
$623$	6.72767	+	7.43095i	0.269539	+	0.297715i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	0.558929	−	0.448574i	0.0223215	−	0.0179143i
$628$		0			0
$629$	−21.1778			−0.844416
$630$		0			0
$631$	−6.09239			−0.242534			−0.121267	−	0.992620i	$-0.538696\pi$
$631$	−6.09239			−0.242534			−0.121267	+	0.992620i	$0.538696\pi$
$632$		0			0
$633$	−4.33129	+	3.47612i	−0.172153	+	0.138163i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	40.0136	+	3.98489i	1.58540	+	0.157887i
$638$		0			0
$639$	−25.0878	−	5.56261i	−0.992460	−	0.220054i
$640$		0			0
$641$			10.8323i			0.427849i	0.976850	+	0.213924i	$0.0686246\pi$
$641$			10.8323i			0.427849i	−0.976850	+	0.213924i	$0.931375\pi$
$642$		0			0
$643$		−	6.90733i		−	0.272399i	−0.990681	−	0.136199i	$-0.956511\pi$
$643$		−	6.90733i		−	0.272399i	0.990681	−	0.136199i	$-0.0434887\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.27972			−0.246881			−0.123441	−	0.992352i	$-0.539393\pi$
$647$	−6.27972			−0.246881			−0.123441	+	0.992352i	$0.539393\pi$
$648$		0			0
$649$		−	3.93184i		−	0.154338i
$650$		0			0
$651$	0.633536	−	3.95775i	0.0248302	−	0.155116i
$652$		0			0
$653$			7.11975i			0.278617i	0.990249	+	0.139309i	$0.0444880\pi$
$653$			7.11975i			0.278617i	−0.990249	+	0.139309i	$0.955512\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−48.0509	−	10.6541i	−1.87464	−	0.415656i
$658$		0			0
$659$			6.97071i			0.271540i	0.990740	+	0.135770i	$0.0433509\pi$
$659$			6.97071i			0.271540i	−0.990740	+	0.135770i	$0.956649\pi$
$660$		0			0
$661$			31.1260i			1.21066i	0.795974	+	0.605330i	$0.206959\pi$
$661$			31.1260i			1.21066i	−0.795974	+	0.605330i	$0.793041\pi$
$662$		0			0
$663$	22.7236	+	28.3139i	0.882512	+	1.09962i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−11.4948			−0.445079
$668$		0			0
$669$	24.1198	+	30.0535i	0.932524	+	1.16194i
$670$		0			0
$671$	−1.55312			−0.0599575
$672$		0			0
$673$	−31.3386			−1.20801			−0.604006	−	0.796979i	$-0.706430\pi$
$673$	−31.3386			−1.20801			−0.604006	+	0.796979i	$0.706430\pi$
$674$		0			0
$675$	−11.4900	−	23.3019i	−0.442249	−	0.896892i
$676$		0			0
$677$	−26.5921			−1.02202			−0.511008	−	0.859576i	$-0.670728\pi$
$677$	−26.5921			−1.02202			−0.511008	+	0.859576i	$0.670728\pi$
$678$		0			0
$679$	−16.7259	+	15.1430i	−0.641883	+	0.581134i
$680$		0			0
$681$	2.25262	−	1.80786i	0.0863205	−	0.0692774i
$682$		0			0
$683$		−	39.8288i		−	1.52401i	−0.647573	−	0.762003i	$-0.724216\pi$
$683$		−	39.8288i		−	1.52401i	0.647573	−	0.762003i	$-0.275784\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−13.1377	−	16.3698i	−0.501235	−	0.624545i
$688$		0			0
$689$	61.5119			2.34342
$690$		0			0
$691$		−	32.9916i		−	1.25506i	−0.778593	−	0.627529i	$-0.784066\pi$
$691$		−	32.9916i		−	1.25506i	0.778593	−	0.627529i	$-0.215934\pi$
$692$		0			0
$693$	−3.92713	−	6.89766i	−0.149179	−	0.262020i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−17.2064			−0.651740
$698$		0			0
$699$	−17.4728	−	21.7713i	−0.660881	−	0.823465i
$700$		0			0
$701$			27.4271i			1.03591i	0.855408	+	0.517954i	$0.173306\pi$
$701$			27.4271i			1.03591i	−0.855408	+	0.517954i	$0.826694\pi$
$702$		0			0
$703$		−	2.40155i		−	0.0905760i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.8551	−	14.1989i	−0.483466	−	0.534005i
$708$		0			0
$709$	−0.925719			−0.0347661			−0.0173831	−	0.999849i	$-0.505533\pi$
$709$	−0.925719			−0.0347661			−0.0173831	+	0.999849i	$0.505533\pi$
$710$		0			0
$711$	7.79726	−	35.1663i	0.292420	−	1.31884i
$712$		0			0
$713$	7.74082			0.289896
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−20.2847	−	25.2750i	−0.757548	−	0.943913i
$718$		0			0
$719$	17.3438			0.646816			0.323408	−	0.946260i	$-0.395171\pi$
$719$	17.3438			0.646816			0.323408	+	0.946260i	$0.395171\pi$
$720$		0			0
$721$	31.1314	−	28.1851i	1.15939	−	1.04967i
$722$		0			0
$723$	−12.6792	−	15.7984i	−0.471545	−	0.587551i
$724$		0			0
$725$		−	6.49405i		−	0.241183i
$726$		0			0
$727$		−	44.1374i		−	1.63697i	−0.574531	−	0.818483i	$-0.694816\pi$
$727$		−	44.1374i		−	1.63697i	0.574531	−	0.818483i	$-0.305184\pi$
$728$		0			0
$729$	−16.4384	+	21.4191i	−0.608831	+	0.793300i
$730$		0			0
$731$	0.195926			0.00724658
$732$		0			0
$733$			12.3596i			0.456514i	0.973601	+	0.228257i	$0.0733026\pi$
$733$			12.3596i			0.456514i	−0.973601	+	0.228257i	$0.926697\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	11.1029i		−	0.408979i
$738$		0			0
$739$	5.51952			0.203039			0.101519	−	0.994834i	$-0.467630\pi$
$739$	5.51952			0.203039			0.101519	+	0.994834i	$0.467630\pi$
$740$		0			0
$741$	−3.21077	+	2.57683i	−0.117950	+	0.0946624i
$742$		0			0
$743$			22.2057i			0.814648i	0.913284	+	0.407324i	$0.133538\pi$
$743$			22.2057i			0.814648i	−0.913284	+	0.407324i	$0.866462\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−6.37806	+	28.7656i	−0.233361	+	1.05248i
$748$		0			0
$749$	30.8233	−	27.9062i	1.12626	−	1.01967i
$750$		0			0
$751$	16.4279			0.599461			0.299730	−	0.954024i	$-0.403103\pi$
$751$	16.4279			0.599461			0.299730	+	0.954024i	$0.403103\pi$
$752$		0			0
$753$	42.0012	−	33.7085i	1.53061	−	1.22841i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	2.58261			0.0938666			0.0469333	−	0.998898i	$-0.485055\pi$
$757$	2.58261			0.0938666			0.0469333	+	0.998898i	$0.485055\pi$
$758$		0			0
$759$	11.9550	−	9.59464i	0.433940	−	0.348263i
$760$		0			0
$761$	19.3292			0.700683			0.350341	−	0.936622i	$-0.386066\pi$
$761$	19.3292			0.700683			0.350341	+	0.936622i	$0.386066\pi$
$762$		0			0
$763$	8.63927	+	9.54238i	0.312763	+	0.345457i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			22.5864i			0.815549i
$768$		0			0
$769$			33.0184i			1.19067i	0.803476	+	0.595337i	$0.202981\pi$
$769$			33.0184i			1.19067i	−0.803476	+	0.595337i	$0.797019\pi$
$770$		0			0
$771$	−25.7005	+	20.6262i	−0.925580	+	0.742834i
$772$		0			0
$773$	−28.7839			−1.03529			−0.517643	−	0.855597i	$-0.673190\pi$
$773$	−28.7839			−1.03529			−0.517643	+	0.855597i	$0.673190\pi$
$774$		0			0
$775$			4.37323i			0.157091i
$776$		0			0
$777$	−26.2631	−	4.20406i	−0.942184	−	0.150820i
$778$		0			0
$779$		−	1.95119i		−	0.0699087i
$780$		0			0
$781$	8.56571			0.306505
$782$		0			0
$783$	−6.05296	+	2.98466i	−0.216315	+	0.106663i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			4.76054i			0.169695i	0.996394	+	0.0848474i	$0.0270403\pi$
$787$			4.76054i			0.169695i	−0.996394	+	0.0848474i	$0.972960\pi$
$788$		0			0
$789$	−0.856185	−	1.06682i	−0.0304810	−	0.0379796i
$790$		0			0
$791$	−16.6816	+	15.1029i	−0.593130	+	0.536996i
$792$		0			0
$793$	8.92190			0.316826
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−44.3009			−1.56922			−0.784609	−	0.619990i	$-0.787137\pi$
$797$	−44.3009			−1.56922			−0.784609	+	0.619990i	$0.787137\pi$
$798$		0			0
$799$	−9.85774			−0.348742
$800$		0			0
$801$	−2.46041	+	11.0967i	−0.0869343	+	0.392081i
$802$		0			0
$803$	16.4060			0.578954
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.63703	+	1.31382i	−0.0576262	+	0.0462485i
$808$		0			0
$809$			51.7252i			1.81856i	0.416183	+	0.909281i	$0.363368\pi$
$809$			51.7252i			1.81856i	−0.416183	+	0.909281i	$0.636632\pi$
$810$		0			0
$811$			11.4779i			0.403043i	0.979484	+	0.201522i	$0.0645886\pi$
$811$			11.4779i			0.403043i	−0.979484	+	0.201522i	$0.935411\pi$
$812$		0			0
$813$	24.4621	+	30.4801i	0.857925	+	1.06898i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			0.0222178i			0.000777302i
$818$		0			0
$819$	22.5594	+	39.6236i	0.788289	+	1.38456i
$820$		0			0
$821$			29.2486i			1.02078i	0.859943	+	0.510391i	$0.170499\pi$
$821$			29.2486i			1.02078i	−0.859943	+	0.510391i	$0.829501\pi$
$822$		0			0
$823$	29.8514			1.04056			0.520278	−	0.853997i	$-0.325828\pi$
$823$	29.8514			1.04056			0.520278	+	0.853997i	$0.325828\pi$
$824$		0			0
$825$	5.42056	+	6.75408i	0.188720	+	0.235147i
$826$		0			0
$827$			16.6374i			0.578538i	0.957248	+	0.289269i	$0.0934122\pi$
$827$			16.6374i			0.578538i	−0.957248	+	0.289269i	$0.906588\pi$
$828$		0			0
$829$			8.49699i			0.295113i	0.989054	+	0.147556i	$0.0471408\pi$
$829$			8.49699i			0.295113i	−0.989054	+	0.147556i	$0.952859\pi$
$830$		0			0
$831$	22.2524	−	17.8589i	0.771928	−	0.619519i
$832$		0			0
$833$	−2.53113	+	25.4159i	−0.0876986	+	0.880610i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	4.07619	−	2.00993i	0.140894	−	0.0694734i
$838$		0			0
$839$	−44.8609			−1.54877			−0.774385	−	0.632714i	$-0.781941\pi$
$839$	−44.8609			−1.54877			−0.774385	+	0.632714i	$0.781941\pi$
$840$		0			0
$841$	27.3131			0.941831
$842$		0			0
$843$	−4.77189	−	5.94583i	−0.164353	−	0.204785i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1.77571	+	1.96134i	0.0610142	+	0.0673923i
$848$		0			0
$849$	−18.6355	−	23.2200i	−0.639567	−	0.796908i
$850$		0			0
$851$		−	51.3671i		−	1.76084i
$852$		0			0
$853$		−	52.4036i		−	1.79427i	−0.441761	−	0.897133i	$-0.645646\pi$
$853$		−	52.4036i		−	1.79427i	0.441761	−	0.897133i	$-0.354354\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.2128			−0.656296			−0.328148	−	0.944626i	$-0.606424\pi$
$857$	−19.2128			−0.656296			−0.328148	+	0.944626i	$0.606424\pi$
$858$		0			0
$859$		−	47.0927i		−	1.60678i	−0.595451	−	0.803392i	$-0.703026\pi$
$859$		−	47.0927i		−	1.60678i	0.595451	−	0.803392i	$-0.296974\pi$
$860$		0			0
$861$	−21.3381	−	3.41569i	−0.727199	−	0.116406i
$862$		0			0
$863$			31.4479i			1.07050i	0.844695	+	0.535249i	$0.179782\pi$
$863$			31.4479i			1.07050i	−0.844695	+	0.535249i	$0.820218\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	4.97936	−	3.99624i	0.169108	−	0.135719i
$868$		0			0
$869$			12.0068i			0.407302i
$870$		0			0
$871$			63.7803i			2.16111i
$872$		0			0
$873$	−24.9769	−	5.53801i	−0.845339	−	0.187433i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23.1843			−0.782877			−0.391438	−	0.920204i	$-0.628022\pi$
$877$	−23.1843			−0.782877			−0.391438	+	0.920204i	$0.628022\pi$
$878$		0			0
$879$	−34.2592	+	27.4951i	−1.15553	+	0.927387i
$880$		0			0
$881$	54.9910			1.85269			0.926347	−	0.376671i	$-0.122931\pi$
$881$	54.9910			1.85269			0.926347	+	0.376671i	$0.122931\pi$
$882$		0			0
$883$	16.5188			0.555902			0.277951	−	0.960595i	$-0.410345\pi$
$883$	16.5188			0.555902			0.277951	+	0.960595i	$0.410345\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.412981			0.0138666			0.00693328	−	0.999976i	$-0.497793\pi$
$887$	0.412981			0.0138666			0.00693328	+	0.999976i	$0.497793\pi$
$888$		0			0
$889$	20.0068	+	22.0982i	0.671006	+	0.741150i
$890$		0			0
$891$	3.80404	−	8.15655i	0.127440	−	0.273255i
$892$		0			0
$893$		−	1.11786i		−	0.0374077i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−68.6757	+	55.1164i	−2.29301	+	1.84028i
$898$		0			0
$899$	1.13600			0.0378877
$900$		0			0
$901$			39.0714i			1.30166i
$902$		0			0
$903$	0.242972	+	0.0388937i	0.00808560	+	0.00129430i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−40.5052			−1.34495			−0.672477	−	0.740118i	$-0.734770\pi$
$907$	−40.5052			−1.34495			−0.672477	+	0.740118i	$0.734770\pi$
$908$		0			0
$909$	4.70131	−	21.2033i	0.155932	−	0.703268i
$910$		0			0
$911$			27.8833i			0.923816i	0.886928	+	0.461908i	$0.152835\pi$
$911$			27.8833i			0.923816i	−0.886928	+	0.461908i	$0.847165\pi$
$912$		0			0
$913$		−	9.82140i		−	0.325041i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	34.9022	+	38.5507i	1.15257	+	1.27306i
$918$		0			0
$919$	32.1340			1.06000			0.530002	−	0.847996i	$-0.322191\pi$
$919$	32.1340			1.06000			0.530002	+	0.847996i	$0.322191\pi$
$920$		0			0
$921$	1.04237	+	1.29881i	0.0343474	+	0.0427972i
$922$		0			0
$923$	−49.2057			−1.61963
$924$		0			0
$925$	29.0202			0.954179
$926$		0			0
$927$	46.4886	+	10.3077i	1.52689	+	0.338550i
$928$		0			0
$929$	10.6328			0.348851			0.174426	−	0.984670i	$-0.444193\pi$
$929$	10.6328			0.348851			0.174426	+	0.984670i	$0.444193\pi$
$930$		0			0
$931$	−2.88214	−	0.287028i	−0.0944584	−	0.00940697i
$932$		0			0
$933$	−31.0117	+	24.8887i	−1.01528	+	0.814821i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	36.0591i		−	1.17800i	−0.808133	−	0.588999i	$-0.799522\pi$
$937$		−	36.0591i		−	1.17800i	0.808133	−	0.588999i	$-0.200478\pi$
$938$		0			0
$939$	−1.90761	−	2.37691i	−0.0622526	−	0.0775675i
$940$		0			0
$941$	−38.3020			−1.24861			−0.624305	−	0.781180i	$-0.714618\pi$
$941$	−38.3020			−1.24861			−0.624305	+	0.781180i	$0.714618\pi$
$942$		0			0
$943$		−	41.7344i		−	1.35906i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	37.5593i		−	1.22051i	−0.792204	−	0.610256i	$-0.791066\pi$
$947$		−	37.5593i		−	1.22051i	0.792204	−	0.610256i	$-0.208934\pi$
$948$		0			0
$949$	−94.2440			−3.05929
$950$		0			0
$951$	4.71047	+	5.86930i	0.152747	+	0.190325i
$952$		0			0
$953$			41.8312i			1.35504i	0.735502	+	0.677522i	$0.236946\pi$
$953$			41.8312i			1.35504i	−0.735502	+	0.677522i	$0.763054\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	1.75445	−	1.40805i	0.0567134	−	0.0455159i
$958$		0			0
$959$	9.29650	−	8.41667i	0.300200	−	0.271788i
$960$		0			0
$961$	30.2350			0.975322
$962$		0			0
$963$	46.0286	+	10.2057i	1.48325	+	0.328874i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−40.4046			−1.29933			−0.649663	−	0.760223i	$-0.725090\pi$
$967$	−40.4046			−1.29933			−0.649663	+	0.760223i	$0.725090\pi$
$968$		0			0
$969$	−1.63676	−	2.03942i	−0.0525804	−	0.0655158i
$970$		0			0
$971$	1.62450			0.0521326			0.0260663	−	0.999660i	$-0.491702\pi$
$971$	1.62450			0.0521326			0.0260663	+	0.999660i	$0.491702\pi$
$972$		0			0
$973$	20.3837	−	18.4545i	0.653471	−	0.591625i
$974$		0			0
$975$	−31.1384	−	38.7988i	−0.997227	−	1.24256i
$976$		0			0
$977$		−	4.69000i		−	0.150047i	−0.997182	−	0.0750233i	$-0.976097\pi$
$977$		−	4.69000i		−	0.150047i	0.997182	−	0.0750233i	$-0.0239031\pi$
$978$		0			0
$979$		−	3.78872i		−	0.121088i
$980$		0			0
$981$	−3.15951	+	14.2497i	−0.100875	+	0.454956i
$982$		0			0
$983$	19.7390			0.629575			0.314788	−	0.949162i	$-0.398067\pi$
$983$	19.7390			0.629575			0.314788	+	0.949162i	$0.398067\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−12.2248	−	1.95688i	−0.389120	−	0.0622883i
$988$		0			0
$989$			0.475221i			0.0151111i
$990$		0			0
$991$	−35.3874			−1.12412			−0.562059	−	0.827097i	$-0.689990\pi$
$991$	−35.3874			−1.12412			−0.562059	+	0.827097i	$0.689990\pi$
$992$		0			0
$993$	12.4211	−	9.96866i	0.394171	−	0.316346i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	27.5968i		−	0.874000i	−0.899461	−	0.437000i	$-0.856041\pi$
$997$		−	27.5968i		−	0.874000i	0.899461	−	0.437000i	$-0.143959\pi$
$998$		0			0
$999$	−13.3377	−	27.0491i	−0.421985	−	0.855795i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1848.2.v.c.881.4	yes	16
3.2	odd	2	inner	1848.2.v.c.881.14	yes	16
7.6	odd	2	inner	1848.2.v.c.881.13	yes	16
21.20	even	2	inner	1848.2.v.c.881.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1848.2.v.c.881.3	✓	16	21.20	even	2	inner
1848.2.v.c.881.4	yes	16	1.1	even	1	trivial
1848.2.v.c.881.13	yes	16	7.6	odd	2	inner
1848.2.v.c.881.14	yes	16	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1848.v (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.35082 + 1.08411i) q^{3} +(-1.77571 - 1.96134i) q^{7} +(0.649405 - 2.92887i) q^{9} +1.00000i q^{11} -5.74450i q^{13} +3.64881 q^{17} +0.413771i q^{19} +(4.52497 + 0.724334i) q^{21} +8.85023i q^{23} -5.00000 q^{25} +(2.29799 + 4.66039i) q^{27} +1.29881i q^{29} -0.874646i q^{31} +(-1.08411 - 1.35082i) q^{33} -5.80404 q^{37} +(6.22768 + 7.75976i) q^{39} -4.71563 q^{41} +0.0536958 q^{43} -2.70163 q^{47} +(-0.693688 + 6.96554i) q^{49} +(-4.92887 + 3.95572i) q^{51} +10.7080i q^{53} +(-0.448574 - 0.558929i) q^{57} -3.93184 q^{59} +1.55312i q^{61} +(-6.89766 + 3.92713i) q^{63} -11.1029 q^{67} +(-9.59464 - 11.9550i) q^{69} -8.56571i q^{71} -16.4060i q^{73} +(6.75408 - 5.42056i) q^{75} +(1.96134 - 1.77571i) q^{77} +12.0068 q^{79} +(-8.15655 - 3.80404i) q^{81} -9.82140 q^{83} +(-1.40805 - 1.75445i) q^{87} -3.78872 q^{89} +(-11.2669 + 10.2006i) q^{91} +(0.948214 + 1.18149i) q^{93} -8.52783i q^{97} +(2.92887 + 0.649405i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{7} - 8 q^{9} + 20 q^{21} - 80 q^{25} + 24 q^{39} + 16 q^{43} - 24 q^{49} - 40 q^{51} - 72 q^{57} - 12 q^{63} - 48 q^{67} - 24 q^{79} - 16 q^{81} + 40 q^{91} - 40 q^{93} + 8 q^{99}+O(q^{100}) \) 16 * q + 4 * q^7 - 8 * q^9 + 20 * q^21 - 80 * q^25 + 24 * q^39 + 16 * q^43 - 24 * q^49 - 40 * q^51 - 72 * q^57 - 12 * q^63 - 48 * q^67 - 24 * q^79 - 16 * q^81 + 40 * q^91 - 40 * q^93 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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