LMFDB - Embedded newform 2100.2.f.i.1049.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1049,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.f (of order $2$, degree $1$, not 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:	$16.7685844245$
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} - 8 x^{15} + 22 x^{14} - 4 x^{13} - 80 x^{12} - 84 x^{11} + 1324 x^{10} - 3800 x^{9} + \cdots + 3204 $ x^16 - 8x^15 + 22x^14 - 4x^13 - 80x^12 - 84x^11 + 1324x^10 - 3800x^9 + 5642x^8 - 4872x^7 + 4136x^6 - 11608x^5 + 30032x^4 - 44288x^3 + 37232x^2 - 16848*x + 3204
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1049.7
Root			$1.59751 - 0.247497i$ of defining polynomial
Character	$\chi$	$=$	2100.1049
Dual form			2100.2.f.i.1049.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.437016 + 1.67601i) q^{3} +(-1.41421 - 2.23607i) q^{7} +(-2.61803 - 1.46489i) q^{9} +O(q^{10})$	$q+(-0.437016 + 1.67601i) q^{3} +(-1.41421 - 2.23607i) q^{7} +(-2.61803 - 1.46489i) q^{9} +3.83513i q^{11} +5.99070 q^{13} -2.07167i q^{17} -5.11667i q^{19} +(4.36571 - 1.39304i) q^{21} -8.57561 q^{23} +(3.59929 - 3.74768i) q^{27} +5.64583i q^{29} -1.95440i q^{31} +(-6.42772 - 1.67601i) q^{33} -2.23607i q^{37} +(-2.61803 + 10.0405i) q^{39} +7.49535 q^{41} -9.47214i q^{43} -12.9190i q^{47} +(-3.00000 + 6.32456i) q^{49} +(3.47214 + 0.905351i) q^{51} +10.6000 q^{53} +(8.57561 + 2.23607i) q^{57} +12.1277 q^{59} +4.37016i q^{61} +(0.426869 + 7.92577i) q^{63} -6.70820i q^{67} +(3.74768 - 14.3728i) q^{69} -2.02443i q^{71} -1.41421 q^{73} +(8.57561 - 5.42369i) q^{77} +7.47214 q^{79} +(4.70820 + 7.67026i) q^{81} +7.49535i q^{83} +(-9.46248 - 2.46732i) q^{87} +2.86297 q^{89} +(-8.47214 - 13.3956i) q^{91} +(3.27559 + 0.854102i) q^{93} -0.206331 q^{97} +(5.61803 - 10.0405i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{9}+O(q^{10}) $ 16 * q - 24 * q^9	$ 16 q - 24 q^{9} - 8 q^{21} - 24 q^{39} - 48 q^{49} - 16 q^{51} + 48 q^{79} - 32 q^{81} - 64 q^{91} + 72 q^{99}+O(q^{100}) $ 16 * q - 24 * q^9 - 8 * q^21 - 24 * q^39 - 48 * q^49 - 16 * q^51 + 48 * q^79 - 32 * q^81 - 64 * q^91 + 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	−0.437016	+	1.67601i	−0.252311	+	0.967646i
$3$	−0.437016	+	1.67601i	−0.252311	+	0.967646i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.41421	−	2.23607i	−0.534522	−	0.845154i
$7$	−1.41421	−	2.23607i	−0.534522	−	0.845154i
$8$		0			0
$9$	−2.61803	−	1.46489i	−0.872678	−	0.488296i
$10$		0			0
$11$			3.83513i			1.15633i	0.815918	+	0.578167i	$0.196232\pi$
$11$			3.83513i			1.15633i	−0.815918	+	0.578167i	$0.803768\pi$
$12$		0			0
$13$	5.99070			1.66152			0.830761	−	0.556629i	$-0.187905\pi$
$13$	5.99070			1.66152			0.830761	+	0.556629i	$0.187905\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.07167i		−	0.502453i	−0.967928	−	0.251226i	$-0.919166\pi$
$17$		−	2.07167i		−	0.502453i	0.967928	−	0.251226i	$-0.0808338\pi$
$18$		0			0
$19$		−	5.11667i		−	1.17385i	−0.809643	−	0.586923i	$-0.800339\pi$
$19$		−	5.11667i		−	1.17385i	0.809643	−	0.586923i	$-0.199661\pi$
$20$		0			0
$21$	4.36571	−	1.39304i	0.952676	−	0.303987i
$22$		0			0
$23$	−8.57561			−1.78814			−0.894069	−	0.447930i	$-0.852161\pi$
$23$	−8.57561			−1.78814			−0.894069	+	0.447930i	$0.852161\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.59929	−	3.74768i	0.692684	−	0.721241i
$28$		0			0
$29$			5.64583i			1.04840i	0.851594	+	0.524202i	$0.175636\pi$
$29$			5.64583i			1.04840i	−0.851594	+	0.524202i	$0.824364\pi$
$30$		0			0
$31$		−	1.95440i		−	0.351020i	−0.984478	−	0.175510i	$-0.943843\pi$
$31$		−	1.95440i		−	0.351020i	0.984478	−	0.175510i	$-0.0561574\pi$
$32$		0			0
$33$	−6.42772	−	1.67601i	−1.11892	−	0.291756i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	2.23607i		−	0.367607i	−0.982963	−	0.183804i	$-0.941159\pi$
$37$		−	2.23607i		−	0.367607i	0.982963	−	0.183804i	$-0.0588411\pi$
$38$		0			0
$39$	−2.61803	+	10.0405i	−0.419221	+	1.60777i
$40$		0			0
$41$	7.49535			1.17058			0.585289	−	0.810825i	$-0.300981\pi$
$41$	7.49535			1.17058			0.585289	+	0.810825i	$0.300981\pi$
$42$		0			0
$43$		−	9.47214i		−	1.44449i	−0.691639	−	0.722244i	$-0.743111\pi$
$43$		−	9.47214i		−	1.44449i	0.691639	−	0.722244i	$-0.256889\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	12.9190i		−	1.88444i	−0.335000	−	0.942218i	$-0.608736\pi$
$47$		−	12.9190i		−	1.88444i	0.335000	−	0.942218i	$-0.391264\pi$
$48$		0			0
$49$	−3.00000	+	6.32456i	−0.428571	+	0.903508i
$50$		0			0
$51$	3.47214	+	0.905351i	0.486196	+	0.126774i
$52$		0			0
$53$	10.6000			1.45603			0.728013	−	0.685563i	$-0.240444\pi$
$53$	10.6000			1.45603			0.728013	+	0.685563i	$0.240444\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	8.57561	+	2.23607i	1.13587	+	0.296174i
$58$		0			0
$59$	12.1277			1.57890			0.789449	−	0.613817i	$-0.210367\pi$
$59$	12.1277			1.57890			0.789449	+	0.613817i	$0.210367\pi$
$60$		0			0
$61$			4.37016i			0.559542i	0.960067	+	0.279771i	$0.0902585\pi$
$61$			4.37016i			0.559542i	−0.960067	+	0.279771i	$0.909742\pi$
$62$		0			0
$63$	0.426869	+	7.92577i	0.0537805	+	0.998553i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	6.70820i		−	0.819538i	−0.912189	−	0.409769i	$-0.865609\pi$
$67$		−	6.70820i		−	0.819538i	0.912189	−	0.409769i	$-0.134391\pi$
$68$		0			0
$69$	3.74768	−	14.3728i	0.451167	−	1.73028i
$70$		0			0
$71$		−	2.02443i		−	0.240255i	−0.992758	−	0.120128i	$-0.961670\pi$
$71$		−	2.02443i		−	0.240255i	0.992758	−	0.120128i	$-0.0383304\pi$
$72$		0			0
$73$	−1.41421			−0.165521			−0.0827606	−	0.996569i	$-0.526374\pi$
$73$	−1.41421			−0.165521			−0.0827606	+	0.996569i	$0.526374\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.57561	−	5.42369i	0.977281	−	0.618087i
$78$		0			0
$79$	7.47214			0.840681			0.420340	−	0.907366i	$-0.361911\pi$
$79$	7.47214			0.840681			0.420340	+	0.907366i	$0.361911\pi$
$80$		0			0
$81$	4.70820	+	7.67026i	0.523134	+	0.852251i
$82$		0			0
$83$			7.49535i			0.822722i	0.911472	+	0.411361i	$0.134946\pi$
$83$			7.49535i			0.822722i	−0.911472	+	0.411361i	$0.865054\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−9.46248	−	2.46732i	−1.01448	−	0.264524i
$88$		0			0
$89$	2.86297			0.303474			0.151737	−	0.988421i	$-0.451513\pi$
$89$	2.86297			0.303474			0.151737	+	0.988421i	$0.451513\pi$
$90$		0			0
$91$	−8.47214	−	13.3956i	−0.888121	−	1.40424i
$92$		0			0
$93$	3.27559	+	0.854102i	0.339663	+	0.0885662i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.206331			−0.0209497			−0.0104749	−	0.999945i	$-0.503334\pi$
$97$	−0.206331			−0.0209497			−0.0104749	+	0.999945i	$0.503334\pi$
$98$		0			0
$99$	5.61803	−	10.0405i	0.564634	−	1.00911i
$100$		0			0
$101$	2.86297			0.284876			0.142438	−	0.989804i	$-0.454506\pi$
$101$	2.86297			0.284876			0.142438	+	0.989804i	$0.454506\pi$
$102$		0			0
$103$	11.8539			1.16800			0.583999	−	0.811754i	$-0.301487\pi$
$103$	11.8539			1.16800			0.583999	+	0.811754i	$0.301487\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	17.1512			1.65807			0.829035	−	0.559197i	$-0.188890\pi$
$107$	17.1512			1.65807			0.829035	+	0.559197i	$0.188890\pi$
$108$		0			0
$109$	1.47214			0.141005			0.0705025	−	0.997512i	$-0.477540\pi$
$109$	1.47214			0.141005			0.0705025	+	0.997512i	$0.477540\pi$
$110$		0			0
$111$	3.74768	+	0.977198i	0.355714	+	0.0927515i
$112$		0			0
$113$	−12.6245			−1.18761			−0.593805	−	0.804609i	$-0.702375\pi$
$113$	−12.6245			−1.18761			−0.593805	+	0.804609i	$0.702375\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−15.6839	−	8.77571i	−1.44997	−	0.811315i
$118$		0			0
$119$	−4.63238	+	2.92978i	−0.424650	+	0.268572i
$120$		0			0
$121$	−3.70820			−0.337109
$122$		0			0
$123$	−3.27559	+	12.5623i	−0.295350	+	1.13270i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			3.29180i			0.292100i	0.989277	+	0.146050i	$0.0466560\pi$
$127$			3.29180i			0.292100i	−0.989277	+	0.146050i	$0.953344\pi$
$128$		0			0
$129$	15.8754	+	4.13948i	1.39775	+	0.364460i
$130$		0			0
$131$	4.63238			0.404733			0.202367	−	0.979310i	$-0.435137\pi$
$131$	4.63238			0.404733			0.202367	+	0.979310i	$0.435137\pi$
$132$		0			0
$133$	−11.4412	+	7.23607i	−0.992080	+	0.627447i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.04885			−0.345917			−0.172958	−	0.984929i	$-0.555333\pi$
$137$	−4.04885			−0.345917			−0.172958	+	0.984929i	$0.555333\pi$
$138$		0			0
$139$			13.3956i			1.13620i	0.822959	+	0.568101i	$0.192322\pi$
$139$			13.3956i			1.13620i	−0.822959	+	0.568101i	$0.807678\pi$
$140$		0			0
$141$	21.6525	+	5.64583i	1.82347	+	0.475465i
$142$		0			0
$143$			22.9751i			1.92128i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−9.28898	−	7.79197i	−0.766142	−	0.642671i
$148$		0			0
$149$			8.57561i			0.702541i	0.936274	+	0.351271i	$0.114250\pi$
$149$			8.57561i			0.702541i	−0.936274	+	0.351271i	$0.885750\pi$
$150$		0			0
$151$	−21.6525			−1.76205			−0.881027	−	0.473066i	$-0.843147\pi$
$151$	−21.6525			−1.76205			−0.881027	+	0.473066i	$0.843147\pi$
$152$		0			0
$153$	−3.03476	+	5.42369i	−0.245346	+	0.438479i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.19859			0.574510			0.287255	−	0.957854i	$-0.407257\pi$
$157$	7.19859			0.574510			0.287255	+	0.957854i	$0.407257\pi$
$158$		0			0
$159$	−4.63238	+	17.7658i	−0.367372	+	1.40892i
$160$		0			0
$161$	12.1277	+	19.1756i	0.955800	+	1.51125i
$162$		0			0
$163$		−	8.94427i		−	0.700569i	−0.936643	−	0.350285i	$-0.886085\pi$
$163$		−	8.94427i		−	0.700569i	0.936643	−	0.350285i	$-0.113915\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	9.56702i		−	0.740318i	−0.928968	−	0.370159i	$-0.879303\pi$
$167$		−	9.56702i		−	0.740318i	0.928968	−	0.370159i	$-0.120697\pi$
$168$		0			0
$169$	22.8885			1.76066
$170$		0			0
$171$	−7.49535	+	13.3956i	−0.573184	+	1.02439i
$172$		0			0
$173$		−	15.4798i		−	1.17690i	−0.808532	−	0.588452i	$-0.799737\pi$
$173$		−	15.4798i		−	1.17690i	0.808532	−	0.588452i	$-0.200263\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−5.30002	+	20.3262i	−0.398374	+	1.52781i
$178$		0			0
$179$		−	6.55118i		−	0.489658i	−0.969566	−	0.244829i	$-0.921268\pi$
$179$		−	6.55118i		−	0.489658i	0.969566	−	0.244829i	$-0.0787319\pi$
$180$		0			0
$181$		−	17.3044i		−	1.28623i	−0.765771	−	0.643113i	$-0.777642\pi$
$181$		−	17.3044i		−	1.28623i	0.765771	−	0.643113i	$-0.222358\pi$
$182$		0			0
$183$	−7.32444	−	1.90983i	−0.541438	−	0.141179i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	7.94510			0.581003
$188$		0			0
$189$	−13.4702	−	2.74825i	−0.979815	−	0.199906i
$190$		0			0
$191$			4.74048i			0.343009i	0.985183	+	0.171504i	$0.0548628\pi$
$191$			4.74048i			0.343009i	−0.985183	+	0.171504i	$0.945137\pi$
$192$		0			0
$193$			15.0000i			1.07972i	0.841754	+	0.539862i	$0.181524\pi$
$193$			15.0000i			1.07972i	−0.841754	+	0.539862i	$0.818476\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.02443			0.144234			0.0721172	−	0.997396i	$-0.477024\pi$
$197$	2.02443			0.144234			0.0721172	+	0.997396i	$0.477024\pi$
$198$		0			0
$199$		−	17.0193i		−	1.20646i	−0.797566	−	0.603232i	$-0.793879\pi$
$199$		−	17.0193i		−	1.20646i	0.797566	−	0.603232i	$-0.206121\pi$
$200$		0			0
$201$	11.2430	+	2.93159i	0.793022	+	0.206779i
$202$		0			0
$203$	12.6245	−	7.98441i	0.886063	−	0.560396i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	22.4512	+	12.5623i	1.56047	+	0.873141i
$208$		0			0
$209$	19.6231			1.35736
$210$		0			0
$211$	−5.70820			−0.392969			−0.196484	−	0.980507i	$-0.562953\pi$
$211$	−5.70820			−0.392969			−0.196484	+	0.980507i	$0.562953\pi$
$212$		0			0
$213$	3.39296	+	0.884707i	0.232482	+	0.0606191i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−4.37016	+	2.76393i	−0.296666	+	0.187628i
$218$		0			0
$219$	0.618034	−	2.37024i	0.0417629	−	0.160166i
$220$		0			0
$221$		−	12.4107i		−	0.834836i
$222$		0			0
$223$	−8.40647			−0.562939			−0.281469	−	0.959570i	$-0.590822\pi$
$223$	−8.40647			−0.562939			−0.281469	+	0.959570i	$0.590822\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			22.1838i			1.47239i	0.676769	+	0.736196i	$0.263380\pi$
$227$			22.1838i			1.47239i	−0.676769	+	0.736196i	$0.736620\pi$
$228$		0			0
$229$		−	3.90879i		−	0.258300i	−0.991625	−	0.129150i	$-0.958775\pi$
$229$		−	3.90879i		−	0.258300i	0.991625	−	0.129150i	$-0.0412249\pi$
$230$		0			0
$231$	5.34249	+	16.7431i	0.351510	+	1.10161i
$232$		0			0
$233$	15.1268			0.990989			0.495494	−	0.868611i	$-0.334987\pi$
$233$	15.1268			0.990989			0.495494	+	0.868611i	$0.334987\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−3.26544	+	12.5234i	−0.212113	+	0.813482i
$238$		0			0
$239$		−	18.2703i		−	1.18181i	−0.806742	−	0.590903i	$-0.798771\pi$
$239$		−	18.2703i		−	1.18181i	0.806742	−	0.590903i	$-0.201229\pi$
$240$		0			0
$241$		−	12.9343i		−	0.833168i	−0.909097	−	0.416584i	$-0.863227\pi$
$241$		−	12.9343i		−	0.833168i	0.909097	−	0.416584i	$-0.136773\pi$
$242$		0			0
$243$	−14.9130	+	4.53898i	−0.956670	+	0.291176i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	30.6525i		−	1.95037i
$248$		0			0
$249$	−12.5623	−	3.27559i	−0.796104	−	0.207582i
$250$		0			0
$251$	−19.6231			−1.23860			−0.619299	−	0.785155i	$-0.712583\pi$
$251$	−19.6231			−1.23860			−0.619299	+	0.785155i	$0.712583\pi$
$252$		0			0
$253$		−	32.8885i		−	2.06769i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	1.76941i		−	0.110373i	−0.998476	−	0.0551865i	$-0.982425\pi$
$257$		−	1.76941i		−	0.110373i	0.998476	−	0.0551865i	$-0.0175753\pi$
$258$		0			0
$259$	−5.00000	+	3.16228i	−0.310685	+	0.196494i
$260$		0			0
$261$	8.27051	−	14.7810i	0.511932	−	0.914919i
$262$		0			0
$263$	−15.1268			−0.932758			−0.466379	−	0.884585i	$-0.654442\pi$
$263$	−15.1268			−0.932758			−0.466379	+	0.884585i	$0.654442\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.25116	+	4.79837i	−0.0765700	+	0.293656i
$268$		0			0
$269$	26.0249			1.58677			0.793383	−	0.608723i	$-0.208318\pi$
$269$	26.0249			1.58677			0.793383	+	0.608723i	$0.208318\pi$
$270$		0			0
$271$		−	22.1359i		−	1.34466i	−0.740250	−	0.672331i	$-0.765293\pi$
$271$		−	22.1359i		−	1.34466i	0.740250	−	0.672331i	$-0.234707\pi$
$272$		0			0
$273$	26.1537	−	8.34530i	1.58289	−	0.505081i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	21.7082i		−	1.30432i	−0.758082	−	0.652160i	$-0.773863\pi$
$277$		−	21.7082i		−	1.30432i	0.758082	−	0.652160i	$-0.226137\pi$
$278$		0			0
$279$	−2.86297	+	5.11667i	−0.171402	+	0.306327i
$280$		0			0
$281$		−	14.4352i		−	0.861129i	−0.902560	−	0.430565i	$-0.858314\pi$
$281$		−	14.4352i		−	0.861129i	0.902560	−	0.430565i	$-0.141686\pi$
$282$		0			0
$283$	25.7109			1.52835			0.764177	−	0.645007i	$-0.223146\pi$
$283$	25.7109			1.52835			0.764177	+	0.645007i	$0.223146\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.6000	−	16.7601i	−0.625700	−	0.989319i
$288$		0			0
$289$	12.7082			0.747541
$290$		0			0
$291$	0.0901699	−	0.345813i	0.00528586	−	0.0202719i
$292$		0			0
$293$		−	7.00630i		−	0.409312i	−0.978834	−	0.204656i	$-0.934392\pi$
$293$		−	7.00630i		−	0.409312i	0.978834	−	0.204656i	$-0.0656076\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	14.3728	+	13.8038i	0.833996	+	0.800975i
$298$		0			0
$299$	−51.3739			−2.97103
$300$		0			0
$301$	−21.1803	+	13.3956i	−1.22081	+	0.772111i
$302$		0			0
$303$	−1.25116	+	4.79837i	−0.0718775	+	0.275659i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−15.6051			−0.890628			−0.445314	−	0.895375i	$-0.646908\pi$
$307$	−15.6051			−0.890628			−0.445314	+	0.895375i	$0.646908\pi$
$308$		0			0
$309$	−5.18034	+	19.8673i	−0.294699	+	1.13021i
$310$		0			0
$311$	−19.6231			−1.11272			−0.556362	−	0.830940i	$-0.687803\pi$
$311$	−19.6231			−1.11272			−0.556362	+	0.830940i	$0.687803\pi$
$312$		0			0
$313$	−7.94510			−0.449084			−0.224542	−	0.974464i	$-0.572089\pi$
$313$	−7.94510			−0.449084			−0.224542	+	0.974464i	$0.572089\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−21.6780			−1.21756			−0.608778	−	0.793341i	$-0.708340\pi$
$317$	−21.6780			−1.21756			−0.608778	+	0.793341i	$0.708340\pi$
$318$		0			0
$319$	−21.6525			−1.21231
$320$		0			0
$321$	−7.49535	+	28.7456i	−0.418350	+	1.60443i
$322$		0			0
$323$	−10.6000			−0.589802
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−0.643347	+	2.46732i	−0.0355772	+	0.136443i
$328$		0			0
$329$	−28.8879	+	18.2703i	−1.59264	+	1.00727i
$330$		0			0
$331$	6.81966			0.374842			0.187421	−	0.982280i	$-0.439987\pi$
$331$	6.81966			0.374842			0.187421	+	0.982280i	$0.439987\pi$
$332$		0			0
$333$	−3.27559	+	5.85410i	−0.179501	+	0.320803i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			10.0000i			0.544735i	0.962193	+	0.272367i	$0.0878066\pi$
$337$			10.0000i			0.544735i	−0.962193	+	0.272367i	$0.912193\pi$
$338$		0			0
$339$	5.51709	−	21.1587i	0.299647	−	1.14919i
$340$		0			0
$341$	7.49535			0.405896
$342$		0			0
$343$	18.3848	−	2.23607i	0.992685	−	0.120736i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.6780			−1.16373			−0.581867	−	0.813284i	$-0.697678\pi$
$347$	−21.6780			−1.16373			−0.581867	+	0.813284i	$0.697678\pi$
$348$		0			0
$349$			12.9343i			0.692355i	0.938169	+	0.346177i	$0.112520\pi$
$349$			12.9343i			0.692355i	−0.938169	+	0.346177i	$0.887480\pi$
$350$		0			0
$351$	21.5623	−	22.4512i	1.15091	−	1.19836i
$352$		0			0
$353$		−	2.37392i		−	0.126351i	−0.998002	−	0.0631754i	$-0.979877\pi$
$353$		−	2.37392i		−	0.126351i	0.998002	−	0.0631754i	$-0.0201228\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−2.88592	−	9.04429i	−0.152739	−	0.478675i
$358$		0			0
$359$		−	2.02443i		−	0.106845i	−0.998572	−	0.0534226i	$-0.982987\pi$
$359$		−	2.02443i		−	0.106845i	0.998572	−	0.0534226i	$-0.0170130\pi$
$360$		0			0
$361$	−7.18034			−0.377913
$362$		0			0
$363$	1.62054	−	6.21500i	0.0850565	−	0.326203i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	32.9095			1.71786			0.858930	−	0.512093i	$-0.171130\pi$
$367$	32.9095			1.71786			0.858930	+	0.512093i	$0.171130\pi$
$368$		0			0
$369$	−19.6231	−	10.9799i	−1.02154	−	0.571589i
$370$		0			0
$371$	−14.9907	−	23.7024i	−0.778279	−	1.23057i
$372$		0			0
$373$		−	15.6525i		−	0.810454i	−0.914216	−	0.405227i	$-0.867192\pi$
$373$		−	15.6525i		−	0.810454i	0.914216	−	0.405227i	$-0.132808\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			33.8225i			1.74195i
$378$		0			0
$379$	−17.1803			−0.882495			−0.441247	−	0.897386i	$-0.645464\pi$
$379$	−17.1803			−0.882495			−0.441247	+	0.897386i	$0.645464\pi$
$380$		0			0
$381$	−5.51709	−	1.43857i	−0.282649	−	0.0737001i
$382$		0			0
$383$			2.86297i			0.146291i	0.997321	+	0.0731455i	$0.0233037\pi$
$383$			2.86297i			0.146291i	−0.997321	+	0.0731455i	$0.976696\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−13.8756	+	24.7984i	−0.705338	+	1.26057i
$388$		0			0
$389$			7.45653i			0.378061i	0.981971	+	0.189031i	$0.0605345\pi$
$389$			7.45653i			0.378061i	−0.981971	+	0.189031i	$0.939465\pi$
$390$		0			0
$391$			17.7658i			0.898454i
$392$		0			0
$393$	−2.02443	+	7.76393i	−0.102119	+	0.391639i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	20.2117			1.01439			0.507197	−	0.861830i	$-0.330682\pi$
$397$	20.2117			1.01439			0.507197	+	0.861830i	$0.330682\pi$
$398$		0			0
$399$	−7.12774	−	22.3379i	−0.356833	−	1.11829i
$400$		0			0
$401$			20.9863i			1.04801i	0.851716	+	0.524004i	$0.175562\pi$
$401$			20.9863i			1.04801i	−0.851716	+	0.524004i	$0.824438\pi$
$402$		0			0
$403$		−	11.7082i		−	0.583227i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.57561			0.425077
$408$		0			0
$409$		−	3.62365i		−	0.179178i	−0.995979	−	0.0895889i	$-0.971445\pi$
$409$		−	3.62365i		−	0.179178i	0.995979	−	0.0895889i	$-0.0285553\pi$
$410$		0			0
$411$	1.76941	−	6.78593i	0.0872787	−	0.334725i
$412$		0			0
$413$	−17.1512	−	27.1184i	−0.843956	−	1.33441i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−22.4512	−	5.85410i	−1.09944	−	0.286677i
$418$		0			0
$419$	−13.2213			−0.645903			−0.322951	−	0.946416i	$-0.604675\pi$
$419$	−13.2213			−0.645903			−0.322951	+	0.946416i	$0.604675\pi$
$420$		0			0
$421$	−16.8885			−0.823097			−0.411549	−	0.911388i	$-0.635012\pi$
$421$	−16.8885			−0.823097			−0.411549	+	0.911388i	$0.635012\pi$
$422$		0			0
$423$	−18.9250	+	33.8225i	−0.920163	+	1.64451i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	9.77198	−	6.18034i	0.472899	−	0.299088i
$428$		0			0
$429$	−38.5066	−	10.0405i	−1.85912	−	0.484760i
$430$		0			0
$431$			2.23815i			0.107808i	0.998546	+	0.0539040i	$0.0171665\pi$
$431$			2.23815i			0.107808i	−0.998546	+	0.0539040i	$0.982834\pi$
$432$		0			0
$433$	0.255039			0.0122564			0.00612820	−	0.999981i	$-0.498049\pi$
$433$	0.255039			0.0122564			0.00612820	+	0.999981i	$0.498049\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			43.8786i			2.09900i
$438$		0			0
$439$			32.3693i			1.54490i	0.635074	+	0.772451i	$0.280969\pi$
$439$			32.3693i			1.54490i	−0.635074	+	0.772451i	$0.719031\pi$
$440$		0			0
$441$	17.1189	−	12.1632i	0.815184	−	0.579202i
$442$		0			0
$443$	−10.6000			−0.503623			−0.251811	−	0.967776i	$-0.581026\pi$
$443$	−10.6000			−0.503623			−0.251811	+	0.967776i	$0.581026\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−14.3728	−	3.74768i	−0.679811	−	0.177259i
$448$		0			0
$449$		−	33.3971i		−	1.57611i	−0.615608	−	0.788053i	$-0.711089\pi$
$449$		−	33.3971i		−	1.57611i	0.615608	−	0.788053i	$-0.288911\pi$
$450$		0			0
$451$			28.7456i			1.35358i
$452$		0			0
$453$	9.46248	−	36.2898i	0.444586	−	1.70504i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			7.36068i			0.344318i	0.985069	+	0.172159i	$0.0550743\pi$
$457$			7.36068i			0.344318i	−0.985069	+	0.172159i	$0.944926\pi$
$458$		0			0
$459$	−7.76393	−	7.45653i	−0.362389	−	0.348041i
$460$		0			0
$461$	−2.86297			−0.133342			−0.0666709	−	0.997775i	$-0.521238\pi$
$461$	−2.86297			−0.133342			−0.0666709	+	0.997775i	$0.521238\pi$
$462$		0			0
$463$		−	20.6525i		−	0.959802i	−0.877323	−	0.479901i	$-0.840673\pi$
$463$		−	20.6525i		−	0.959802i	0.877323	−	0.479901i	$-0.159327\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			12.4300i			0.575191i	0.957752	+	0.287596i	$0.0928559\pi$
$467$			12.4300i			0.575191i	−0.957752	+	0.287596i	$0.907144\pi$
$468$		0			0
$469$	−15.0000	+	9.48683i	−0.692636	+	0.438061i
$470$		0			0
$471$	−3.14590	+	12.0649i	−0.144955	+	0.555922i
$472$		0			0
$473$	36.3269			1.67031
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−27.7512	−	15.5279i	−1.27064	−	0.710972i
$478$		0			0
$479$	30.6573			1.40077			0.700383	−	0.713767i	$-0.253012\pi$
$479$	30.6573			1.40077			0.700383	+	0.713767i	$0.253012\pi$
$480$		0			0
$481$		−	13.3956i		−	0.610788i
$482$		0			0
$483$	−37.4386	+	11.9462i	−1.70352	+	0.543570i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			7.76393i			0.351817i	0.984406	+	0.175909i	$0.0562863\pi$
$487$			7.76393i			0.351817i	−0.984406	+	0.175909i	$0.943714\pi$
$488$		0			0
$489$	14.9907	+	3.90879i	0.677903	+	0.176762i
$490$		0			0
$491$		−	2.71605i		−	0.122574i	−0.998120	−	0.0612869i	$-0.980480\pi$
$491$		−	2.71605i		−	0.122574i	0.998120	−	0.0612869i	$-0.0195204\pi$
$492$		0			0
$493$	11.6963			0.526773
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.52675	+	2.86297i	−0.203053	+	0.128422i
$498$		0			0
$499$	11.3475			0.507985			0.253992	−	0.967206i	$-0.418256\pi$
$499$	11.3475			0.507985			0.253992	+	0.967206i	$0.418256\pi$
$500$		0			0
$501$	16.0344	+	4.18094i	0.716366	+	0.186791i
$502$		0			0
$503$			4.14333i			0.184742i	0.995725	+	0.0923710i	$0.0294446\pi$
$503$			4.14333i			0.184742i	−0.995725	+	0.0923710i	$0.970555\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−10.0027	+	38.3615i	−0.444234	+	1.70369i
$508$		0			0
$509$	−21.3925			−0.948206			−0.474103	−	0.880469i	$-0.657228\pi$
$509$	−21.3925			−0.948206			−0.474103	+	0.880469i	$0.657228\pi$
$510$		0			0
$511$	2.00000	+	3.16228i	0.0884748	+	0.139891i
$512$		0			0
$513$	−19.1756	−	18.4164i	−0.846625	−	0.813104i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	49.5462			2.17904
$518$		0			0
$519$	25.9443	+	6.76490i	1.13883	+	0.296946i
$520$		0			0
$521$	−39.2462			−1.71941			−0.859703	−	0.510794i	$-0.829352\pi$
$521$	−39.2462			−1.71941			−0.859703	+	0.510794i	$0.829352\pi$
$522$		0			0
$523$	8.48528			0.371035			0.185518	−	0.982641i	$-0.440604\pi$
$523$	8.48528			0.371035			0.185518	+	0.982641i	$0.440604\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.04885			−0.176371
$528$		0			0
$529$	50.5410			2.19744
$530$		0			0
$531$	−31.7508	−	17.7658i	−1.37787	−	0.770969i
$532$		0			0
$533$	44.9025			1.94494
$534$		0			0
$535$		0			0
$536$		0			0
$537$	10.9799	+	2.86297i	0.473816	+	0.123546i
$538$		0			0
$539$	−24.2555	−	11.5054i	−1.04476	−	0.495572i
$540$		0			0
$541$	−11.6525			−0.500979			−0.250490	−	0.968119i	$-0.580592\pi$
$541$	−11.6525			−0.500979			−0.250490	+	0.968119i	$0.580592\pi$
$542$		0			0
$543$	29.0024	+	7.56231i	1.24461	+	0.324530i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			16.7082i			0.714391i	0.934030	+	0.357196i	$0.116267\pi$
$547$			16.7082i			0.714391i	−0.934030	+	0.357196i	$0.883733\pi$
$548$		0			0
$549$	6.40180	−	11.4412i	0.273222	−	0.488300i
$550$		0			0
$551$	28.8879			1.23066
$552$		0			0
$553$	−10.5672	−	16.7082i	−0.449363	−	0.710505i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.1756			−0.812498			−0.406249	−	0.913762i	$-0.633163\pi$
$557$	−19.1756			−0.812498			−0.406249	+	0.913762i	$0.633163\pi$
$558$		0			0
$559$		−	56.7448i		−	2.40005i
$560$		0			0
$561$	−3.47214	+	13.3161i	−0.146594	+	0.562206i
$562$		0			0
$563$			33.5202i			1.41271i	0.707858	+	0.706355i	$0.249662\pi$
$563$			33.5202i			1.41271i	−0.707858	+	0.706355i	$0.750338\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	10.4928	−	21.3752i	0.440656	−	0.897676i
$568$		0			0
$569$		−	21.6780i		−	0.908788i	−0.890801	−	0.454394i	$-0.849856\pi$
$569$		−	21.6780i		−	0.908788i	0.890801	−	0.454394i	$-0.150144\pi$
$570$		0			0
$571$	7.00000			0.292941			0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000			0.292941			0.146470	+	0.989215i	$0.453209\pi$
$572$		0			0
$573$	−7.94510	−	2.07167i	−0.331911	−	0.0865450i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−39.9805			−1.66441			−0.832206	−	0.554467i	$-0.812922\pi$
$577$	−39.9805			−1.66441			−0.832206	+	0.554467i	$0.812922\pi$
$578$		0			0
$579$	−25.1402	−	6.55524i	−1.04479	−	0.272426i
$580$		0			0
$581$	16.7601	−	10.6000i	0.695327	−	0.439763i
$582$		0			0
$583$			40.6525i			1.68365i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			17.8537i			0.736900i	0.929648	+	0.368450i	$0.120111\pi$
$587$			17.8537i			0.736900i	−0.929648	+	0.368450i	$0.879889\pi$
$588$		0			0
$589$	−10.0000			−0.412043
$590$		0			0
$591$	−0.884707	+	3.39296i	−0.0363920	+	0.139568i
$592$		0			0
$593$		−	44.3676i		−	1.82196i	−0.412451	−	0.910980i	$-0.635327\pi$
$593$		−	44.3676i		−	1.82196i	0.412451	−	0.910980i	$-0.364673\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	28.5245	+	7.43769i	1.16743	+	0.304405i
$598$		0			0
$599$			13.7435i			0.561546i	0.959774	+	0.280773i	$0.0905907\pi$
$599$			13.7435i			0.561546i	−0.959774	+	0.280773i	$0.909409\pi$
$600$		0			0
$601$		−	26.7912i		−	1.09284i	−0.837512	−	0.546419i	$-0.815991\pi$
$601$		−	26.7912i		−	1.09284i	0.837512	−	0.546419i	$-0.184009\pi$
$602$		0			0
$603$	−9.82677	+	17.5623i	−0.400177	+	0.715192i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−41.9349			−1.70209			−0.851043	−	0.525096i	$-0.824030\pi$
$607$	−41.9349			−1.70209			−0.851043	+	0.525096i	$0.824030\pi$
$608$		0			0
$609$	7.86488	+	24.6481i	0.318701	+	0.998790i
$610$		0			0
$611$		−	77.3942i		−	3.13103i
$612$		0			0
$613$		−	2.23607i		−	0.0903139i	−0.998980	−	0.0451570i	$-0.985621\pi$
$613$		−	2.23607i		−	0.0903139i	0.998980	−	0.0451570i	$-0.0143788\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.57561			−0.345241			−0.172620	−	0.984988i	$-0.555223\pi$
$617$	−8.57561			−0.345241			−0.172620	+	0.984988i	$0.555223\pi$
$618$		0			0
$619$			39.2641i			1.57816i	0.614291	+	0.789079i	$0.289442\pi$
$619$			39.2641i			1.57816i	−0.614291	+	0.789079i	$0.710558\pi$
$620$		0			0
$621$	−30.8661	+	32.1386i	−1.23861	+	1.28968i
$622$		0			0
$623$	−4.04885	−	6.40180i	−0.162214	−	0.256483i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−8.57561	+	32.8885i	−0.342477	+	1.31344i
$628$		0			0
$629$	−4.63238			−0.184705
$630$		0			0
$631$	−16.5279			−0.657964			−0.328982	−	0.944336i	$-0.606706\pi$
$631$	−16.5279			−0.657964			−0.328982	+	0.944336i	$0.606706\pi$
$632$		0			0
$633$	2.49458	−	9.56702i	0.0991505	−	0.380255i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−17.9721	+	37.8885i	−0.712081	+	1.50120i
$638$		0			0
$639$	−2.96556	+	5.30002i	−0.117316	+	0.209665i
$640$		0			0
$641$			11.0779i			0.437552i	0.975775	+	0.218776i	$0.0702064\pi$
$641$			11.0779i			0.437552i	−0.975775	+	0.218776i	$0.929794\pi$
$642$		0			0
$643$	1.08036			0.0426054			0.0213027	−	0.999773i	$-0.493219\pi$
$643$	1.08036			0.0426054			0.0213027	+	0.999773i	$0.493219\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			18.5295i			0.728471i	0.931307	+	0.364236i	$0.118670\pi$
$647$			18.5295i			0.728471i	−0.931307	+	0.364236i	$0.881330\pi$
$648$		0			0
$649$			46.5114i			1.82573i
$650$		0			0
$651$	−2.72255	−	8.53232i	−0.106705	−	0.334408i
$652$		0			0
$653$	−6.55118			−0.256367			−0.128184	−	0.991750i	$-0.540915\pi$
$653$	−6.55118			−0.256367			−0.128184	+	0.991750i	$0.540915\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	3.70246	+	2.07167i	0.144447	+	0.0808234i
$658$		0			0
$659$			39.4704i			1.53755i	0.639521	+	0.768773i	$0.279133\pi$
$659$			39.4704i			1.53755i	−0.639521	+	0.768773i	$0.720867\pi$
$660$		0			0
$661$		−	9.02546i		−	0.351050i	−0.984475	−	0.175525i	$-0.943838\pi$
$661$		−	9.02546i		−	0.351050i	0.984475	−	0.175525i	$-0.0561623\pi$
$662$		0			0
$663$	20.8005	+	5.42369i	0.807826	+	0.210639i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	48.4164i		−	1.87469i
$668$		0			0
$669$	3.67376	−	14.0893i	0.142036	−	0.544726i
$670$		0			0
$671$	−16.7601			−0.647017
$672$		0			0
$673$			31.3050i			1.20672i	0.797470	+	0.603359i	$0.206171\pi$
$673$			31.3050i			1.20672i	−0.797470	+	0.603359i	$0.793829\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	40.2243i		−	1.54595i	−0.634439	−	0.772973i	$-0.718769\pi$
$677$		−	40.2243i		−	1.54595i	0.634439	−	0.772973i	$-0.281231\pi$
$678$		0			0
$679$	0.291796	+	0.461370i	0.0111981	+	0.0177058i
$680$		0			0
$681$	−37.1803	−	9.69468i	−1.42475	−	0.371501i
$682$		0			0
$683$	−6.07328			−0.232388			−0.116194	−	0.993227i	$-0.537069\pi$
$683$	−6.07328			−0.232388			−0.116194	+	0.993227i	$0.537069\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	6.55118	+	1.70820i	0.249943	+	0.0651720i
$688$		0			0
$689$	63.5017			2.41922
$690$		0			0
$691$			22.4211i			0.852938i	0.904502	+	0.426469i	$0.140243\pi$
$691$			22.4211i			0.852938i	−0.904502	+	0.426469i	$0.859757\pi$
$692$		0			0
$693$	−30.3963	+	1.63710i	−1.15466	+	0.0621882i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	15.5279i		−	0.588160i
$698$		0			0
$699$	−6.61065	+	25.3527i	−0.250038	+	0.958926i
$700$		0			0
$701$			29.5619i			1.11654i	0.829660	+	0.558270i	$0.188535\pi$
$701$			29.5619i			1.11654i	−0.829660	+	0.558270i	$0.811465\pi$
$702$		0			0
$703$	−11.4412			−0.431514
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.04885	−	6.40180i	−0.152273	−	0.240764i
$708$		0			0
$709$	39.3050			1.47613			0.738064	−	0.674730i	$-0.235740\pi$
$709$	39.3050			1.47613			0.738064	+	0.674730i	$0.235740\pi$
$710$		0			0
$711$	−19.5623	−	10.9458i	−0.733644	−	0.410501i
$712$		0			0
$713$			16.7601i			0.627672i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	30.6212	+	7.98441i	1.14357	+	0.298183i
$718$		0			0
$719$	−26.0249			−0.970565			−0.485282	−	0.874358i	$-0.661283\pi$
$719$	−26.0249			−0.970565			−0.485282	+	0.874358i	$0.661283\pi$
$720$		0			0
$721$	−16.7639	−	26.5061i	−0.624321	−	0.987139i
$722$		0			0
$723$	21.6780	+	5.65248i	0.806212	+	0.210218i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	17.3531			0.643592			0.321796	−	0.946809i	$-0.395714\pi$
$727$	17.3531			0.643592			0.321796	+	0.946809i	$0.395714\pi$
$728$		0			0
$729$	−1.09017	−	26.9780i	−0.0403767	−	0.999185i
$730$		0			0
$731$	−19.6231			−0.725786
$732$		0			0
$733$	−40.7758			−1.50609			−0.753044	−	0.657971i	$-0.771415\pi$
$733$	−40.7758			−1.50609			−0.753044	+	0.657971i	$0.771415\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	25.7268			0.947660
$738$		0			0
$739$	−12.5967			−0.463379			−0.231689	−	0.972790i	$-0.574425\pi$
$739$	−12.5967			−0.463379			−0.231689	+	0.972790i	$0.574425\pi$
$740$		0			0
$741$	51.3739	+	13.3956i	1.88727	+	0.492101i
$742$		0			0
$743$	−6.55118			−0.240339			−0.120170	−	0.992753i	$-0.538344\pi$
$743$	−6.55118			−0.240339			−0.120170	+	0.992753i	$0.538344\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	10.9799	−	19.6231i	0.401732	−	0.717971i
$748$		0			0
$749$	−24.2555	−	38.3513i	−0.886276	−	1.40133i
$750$		0			0
$751$	39.1246			1.42768			0.713839	−	0.700310i	$-0.246955\pi$
$751$	39.1246			1.42768			0.713839	+	0.700310i	$0.246955\pi$
$752$		0			0
$753$	8.57561	−	32.8885i	0.312512	−	1.19853i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			17.7639i			0.645641i	0.946460	+	0.322821i	$0.104631\pi$
$757$			17.7639i			0.645641i	−0.946460	+	0.322821i	$0.895369\pi$
$758$		0			0
$759$	55.1216	+	14.3728i	2.00079	+	0.521700i
$760$		0			0
$761$	−24.2555			−0.879260			−0.439630	−	0.898179i	$-0.644890\pi$
$761$	−24.2555			−0.879260			−0.439630	+	0.898179i	$0.644890\pi$
$762$		0			0
$763$	−2.08191	−	3.29180i	−0.0753704	−	0.119171i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	72.6537			2.62337
$768$		0			0
$769$		−	24.8369i		−	0.895640i	−0.894124	−	0.447820i	$-0.852201\pi$
$769$		−	24.8369i		−	0.895640i	0.894124	−	0.447820i	$-0.147799\pi$
$770$		0			0
$771$	2.96556	+	0.773262i	0.106802	+	0.0278483i
$772$		0			0
$773$			30.1682i			1.08508i	0.840031	+	0.542538i	$0.182536\pi$
$773$			30.1682i			1.08508i	−0.840031	+	0.542538i	$0.817464\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−3.11494	−	9.76203i	−0.111748	−	0.350211i
$778$		0			0
$779$		−	38.3513i		−	1.37408i
$780$		0			0
$781$	7.76393			0.277815
$782$		0			0
$783$	21.1587	+	20.3210i	0.756152	+	0.726213i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	24.2967			0.866083			0.433041	−	0.901374i	$-0.357440\pi$
$787$	24.2967			0.866083			0.433041	+	0.901374i	$0.357440\pi$
$788$		0			0
$789$	6.61065	−	25.3527i	0.235345	−	0.902579i
$790$		0			0
$791$	17.8537	+	28.2291i	0.634804	+	1.00371i
$792$		0			0
$793$			26.1803i			0.929691i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	7.79761i		−	0.276205i	−0.990418	−	0.138103i	$-0.955900\pi$
$797$		−	7.79761i		−	0.276205i	0.990418	−	0.138103i	$-0.0441004\pi$
$798$		0			0
$799$	−26.7639			−0.946840
$800$		0			0
$801$	−7.49535	−	4.19393i	−0.264835	−	0.148185i
$802$		0			0
$803$		−	5.42369i		−	0.191398i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−11.3733	+	43.6180i	−0.400359	+	1.53543i
$808$		0			0
$809$			12.1970i			0.428824i	0.976743	+	0.214412i	$0.0687835\pi$
$809$			12.1970i			0.428824i	−0.976743	+	0.214412i	$0.931216\pi$
$810$		0			0
$811$			37.0246i			1.30011i	0.759887	+	0.650055i	$0.225254\pi$
$811$			37.0246i			1.30011i	−0.759887	+	0.650055i	$0.774746\pi$
$812$		0			0
$813$	37.1001	+	9.67376i	1.30116	+	0.339274i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−48.4658			−1.69560
$818$		0			0
$819$	2.55725	+	47.4809i	0.0893574	+	1.65912i
$820$		0			0
$821$			37.6596i			1.31433i	0.753746	+	0.657165i	$0.228245\pi$
$821$			37.6596i			1.31433i	−0.753746	+	0.657165i	$0.771755\pi$
$822$		0			0
$823$		−	27.3607i		−	0.953733i	−0.878976	−	0.476867i	$-0.841772\pi$
$823$		−	27.3607i		−	0.953733i	0.878976	−	0.476867i	$-0.158228\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−25.7268			−0.894609			−0.447305	−	0.894382i	$-0.647616\pi$
$827$	−25.7268			−0.894609			−0.447305	+	0.894382i	$0.647616\pi$
$828$		0			0
$829$		−	26.5061i		−	0.920595i	−0.887765	−	0.460298i	$-0.847743\pi$
$829$		−	26.5061i		−	0.920595i	0.887765	−	0.460298i	$-0.152257\pi$
$830$		0			0
$831$	36.3832	+	9.48683i	1.26212	+	0.329095i
$832$		0			0
$833$	13.1024	+	6.21500i	0.453970	+	0.215337i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−7.32444	−	7.03444i	−0.253170	−	0.243146i
$838$		0			0
$839$	42.1092			1.45377			0.726885	−	0.686759i	$-0.240967\pi$
$839$	42.1092			1.45377			0.726885	+	0.686759i	$0.240967\pi$
$840$		0			0
$841$	−2.87539			−0.0991513
$842$		0			0
$843$	24.1935	+	6.30840i	0.833269	+	0.217273i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	5.24419	+	8.29180i	0.180193	+	0.284909i
$848$		0			0
$849$	−11.2361	+	43.0918i	−0.385621	+	1.47891i
$850$		0			0
$851$			19.1756i			0.657332i
$852$		0			0
$853$	40.1081			1.37327			0.686637	−	0.727001i	$-0.259086\pi$
$853$	40.1081			1.37327			0.686637	+	0.727001i	$0.259086\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.76941i			0.0604420i	0.999543	+	0.0302210i	$0.00962111\pi$
$857$			1.76941i			0.0604420i	−0.999543	+	0.0302210i	$0.990379\pi$
$858$		0			0
$859$		−	46.3352i		−	1.58094i	−0.612503	−	0.790468i	$-0.709837\pi$
$859$		−	46.3352i		−	1.58094i	0.612503	−	0.790468i	$-0.290163\pi$
$860$		0			0
$861$	32.7226	−	10.4413i	1.11518	−	0.355840i
$862$		0			0
$863$	16.6733			0.567566			0.283783	−	0.958889i	$-0.408410\pi$
$863$	16.6733			0.567566			0.283783	+	0.958889i	$0.408410\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−5.55369	+	21.2991i	−0.188613	+	0.723356i
$868$		0			0
$869$			28.6566i			0.972108i
$870$		0			0
$871$		−	40.1869i		−	1.36168i
$872$		0			0
$873$	0.540182	+	0.302252i	0.0182824	+	0.0102297i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	24.4721i		−	0.826365i	−0.910648	−	0.413183i	$-0.864417\pi$
$877$		−	24.4721i		−	0.826365i	0.910648	−	0.413183i	$-0.135583\pi$
$878$		0			0
$879$	11.7426	+	3.06187i	0.396070	+	0.103274i
$880$		0			0
$881$	31.7508			1.06971			0.534856	−	0.844943i	$-0.320366\pi$
$881$	31.7508			1.06971			0.534856	+	0.844943i	$0.320366\pi$
$882$		0			0
$883$			42.8885i			1.44331i	0.692251	+	0.721657i	$0.256619\pi$
$883$			42.8885i			1.44331i	−0.692251	+	0.721657i	$0.743381\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	28.2120i		−	0.947266i	−0.880722	−	0.473633i	$-0.842942\pi$
$887$		−	28.2120i		−	0.947266i	0.880722	−	0.473633i	$-0.157058\pi$
$888$		0			0
$889$	7.36068	−	4.65530i	0.246869	−	0.156134i
$890$		0			0
$891$	−29.4164	+	18.0566i	−0.985487	+	0.604918i
$892$		0			0
$893$	−66.1025			−2.21204
$894$		0			0
$895$		0			0
$896$		0			0
$897$	22.4512	−	86.1033i	0.749625	−	2.87491i
$898$		0			0
$899$	11.0342			0.368011
$900$		0			0
$901$		−	21.9597i		−	0.731584i
$902$		0			0
$903$	−13.1951	−	41.3526i	−0.439105	−	1.37613i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			41.3050i			1.37151i	0.727833	+	0.685754i	$0.240527\pi$
$907$			41.3050i			1.37151i	−0.727833	+	0.685754i	$0.759473\pi$
$908$		0			0
$909$	−7.49535	−	4.19393i	−0.248605	−	0.139104i
$910$		0			0
$911$			33.8245i			1.12066i	0.828271	+	0.560328i	$0.189325\pi$
$911$			33.8245i			1.12066i	−0.828271	+	0.560328i	$0.810675\pi$
$912$		0			0
$913$	−28.7456			−0.951342
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.55118	−	10.3583i	−0.216339	−	0.342062i
$918$		0			0
$919$	−28.3050			−0.933694			−0.466847	−	0.884338i	$-0.654610\pi$
$919$	−28.3050			−0.933694			−0.466847	+	0.884338i	$0.654610\pi$
$920$		0			0
$921$	6.81966	−	26.1543i	0.224715	−	0.861812i
$922$		0			0
$923$		−	12.1277i		−	0.399189i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−31.0339	−	17.3646i	−1.01929	−	0.570329i
$928$		0			0
$929$	2.86297			0.0939310			0.0469655	−	0.998897i	$-0.485045\pi$
$929$	2.86297			0.0939310			0.0469655	+	0.998897i	$0.485045\pi$
$930$		0			0
$931$	32.3607	+	15.3500i	1.06058	+	0.503077i
$932$		0			0
$933$	8.57561	−	32.8885i	0.280753	−	1.07672i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−22.1660			−0.724133			−0.362067	−	0.932152i	$-0.617929\pi$
$937$	−22.1660			−0.724133			−0.362067	+	0.932152i	$0.617929\pi$
$938$		0			0
$939$	3.47214	−	13.3161i	0.113309	−	0.434554i
$940$		0			0
$941$	13.2213			0.431002			0.215501	−	0.976504i	$-0.430862\pi$
$941$	13.2213			0.431002			0.215501	+	0.976504i	$0.430862\pi$
$942$		0			0
$943$	−64.2772			−2.09315
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−55.5025			−1.80359			−0.901794	−	0.432166i	$-0.857750\pi$
$947$	−55.5025			−1.80359			−0.901794	+	0.432166i	$0.857750\pi$
$948$		0			0
$949$	−8.47214			−0.275017
$950$		0			0
$951$	9.47362	−	36.3325i	0.307203	−	1.17816i
$952$		0			0
$953$	−45.3804			−1.47001			−0.735007	−	0.678060i	$-0.762821\pi$
$953$	−45.3804			−1.47001			−0.735007	+	0.678060i	$0.762821\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	9.46248	−	36.2898i	0.305879	−	1.17308i
$958$		0			0
$959$	5.72594	+	9.05351i	0.184900	+	0.292353i
$960$		0			0
$961$	27.1803			0.876785
$962$		0			0
$963$	−44.9025	−	25.1246i	−1.44696	−	0.809629i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	14.0689i		−	0.452425i	−0.974078	−	0.226212i	$-0.927366\pi$
$967$		−	14.0689i		−	0.452425i	0.974078	−	0.226212i	$-0.0726343\pi$
$968$		0			0
$969$	4.63238	−	17.7658i	0.148814	−	0.570719i
$970$		0			0
$971$	23.1619			0.743301			0.371651	−	0.928373i	$-0.378792\pi$
$971$	23.1619			0.743301			0.371651	+	0.928373i	$0.378792\pi$
$972$		0			0
$973$	29.9535	−	18.9443i	0.960266	−	0.607325i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.7268			0.823074			0.411537	−	0.911393i	$-0.364992\pi$
$977$	25.7268			0.823074			0.411537	+	0.911393i	$0.364992\pi$
$978$		0			0
$979$			10.9799i			0.350918i
$980$		0			0
$981$	−3.85410	−	2.15651i	−0.123052	−	0.0688522i
$982$		0			0
$983$			20.9035i			0.666717i	0.942800	+	0.333358i	$0.108182\pi$
$983$			20.9035i			0.666717i	−0.942800	+	0.333358i	$0.891818\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−17.9968	−	56.4008i	−0.572843	−	1.79526i
$988$		0			0
$989$			81.2293i			2.58294i
$990$		0			0
$991$	57.7214			1.83358			0.916790	−	0.399370i	$-0.130771\pi$
$991$	57.7214			1.83358			0.916790	+	0.399370i	$0.130771\pi$
$992$		0			0
$993$	−2.98030	+	11.4298i	−0.0945770	+	0.362715i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−6.35466			−0.201254			−0.100627	−	0.994924i	$-0.532085\pi$
$997$	−6.35466			−0.201254			−0.100627	+	0.994924i	$0.532085\pi$
$998$		0			0
$999$	−8.38006	−	8.04827i	−0.265133	−	0.254636i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.f.i.1049.7		16
3.2	odd	2	inner	2100.2.f.i.1049.5		16
5.2	odd	4		2100.2.d.k.1301.8	yes	8
5.3	odd	4		2100.2.d.l.1301.1	yes	8
5.4	even	2	inner	2100.2.f.i.1049.10		16
7.6	odd	2	inner	2100.2.f.i.1049.9		16
15.2	even	4		2100.2.d.k.1301.2	yes	8
15.8	even	4		2100.2.d.l.1301.7	yes	8
15.14	odd	2	inner	2100.2.f.i.1049.12		16
21.20	even	2	inner	2100.2.f.i.1049.11		16
35.13	even	4		2100.2.d.l.1301.8	yes	8
35.27	even	4		2100.2.d.k.1301.1	✓	8
35.34	odd	2	inner	2100.2.f.i.1049.8		16
105.62	odd	4		2100.2.d.k.1301.7	yes	8
105.83	odd	4		2100.2.d.l.1301.2	yes	8
105.104	even	2	inner	2100.2.f.i.1049.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.2.d.k.1301.1	✓	8	35.27	even	4
2100.2.d.k.1301.2	yes	8	15.2	even	4
2100.2.d.k.1301.7	yes	8	105.62	odd	4
2100.2.d.k.1301.8	yes	8	5.2	odd	4
2100.2.d.l.1301.1	yes	8	5.3	odd	4
2100.2.d.l.1301.2	yes	8	105.83	odd	4
2100.2.d.l.1301.7	yes	8	15.8	even	4
2100.2.d.l.1301.8	yes	8	35.13	even	4
2100.2.f.i.1049.5		16	3.2	odd	2	inner
2100.2.f.i.1049.6		16	105.104	even	2	inner
2100.2.f.i.1049.7		16	1.1	even	1	trivial
2100.2.f.i.1049.8		16	35.34	odd	2	inner
2100.2.f.i.1049.9		16	7.6	odd	2	inner
2100.2.f.i.1049.10		16	5.4	even	2	inner
2100.2.f.i.1049.11		16	21.20	even	2	inner
2100.2.f.i.1049.12		16	15.14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.437016 + 1.67601i) q^{3} +(-1.41421 - 2.23607i) q^{7} +(-2.61803 - 1.46489i) q^{9} +O(q^{10})\)	\(q+(-0.437016 + 1.67601i) q^{3} +(-1.41421 - 2.23607i) q^{7} +(-2.61803 - 1.46489i) q^{9} +3.83513i q^{11} +5.99070 q^{13} -2.07167i q^{17} -5.11667i q^{19} +(4.36571 - 1.39304i) q^{21} -8.57561 q^{23} +(3.59929 - 3.74768i) q^{27} +5.64583i q^{29} -1.95440i q^{31} +(-6.42772 - 1.67601i) q^{33} -2.23607i q^{37} +(-2.61803 + 10.0405i) q^{39} +7.49535 q^{41} -9.47214i q^{43} -12.9190i q^{47} +(-3.00000 + 6.32456i) q^{49} +(3.47214 + 0.905351i) q^{51} +10.6000 q^{53} +(8.57561 + 2.23607i) q^{57} +12.1277 q^{59} +4.37016i q^{61} +(0.426869 + 7.92577i) q^{63} -6.70820i q^{67} +(3.74768 - 14.3728i) q^{69} -2.02443i q^{71} -1.41421 q^{73} +(8.57561 - 5.42369i) q^{77} +7.47214 q^{79} +(4.70820 + 7.67026i) q^{81} +7.49535i q^{83} +(-9.46248 - 2.46732i) q^{87} +2.86297 q^{89} +(-8.47214 - 13.3956i) q^{91} +(3.27559 + 0.854102i) q^{93} -0.206331 q^{97} +(5.61803 - 10.0405i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{9}+O(q^{10}) \) 16 * q - 24 * q^9	\( 16 q - 24 q^{9} - 8 q^{21} - 24 q^{39} - 48 q^{49} - 16 q^{51} + 48 q^{79} - 32 q^{81} - 64 q^{91} + 72 q^{99}+O(q^{100}) \) 16 * q - 24 * q^9 - 8 * q^21 - 24 * q^39 - 48 * q^49 - 16 * q^51 + 48 * q^79 - 32 * q^81 - 64 * q^91 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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