LMFDB - Embedded newform 2100.2.d.k.1301.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1301,2100)]

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, 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("2100.1301");

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.d (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:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.14786560000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 6x^{6} + 22x^{4} - 54x^{2} + 81 $ x^8 - 6x^6 + 22x^4 - 54*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.7
Root			$1.67601 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	2100.1301
Dual form			2100.2.d.k.1301.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.67601 - 0.437016i) q^{3} +(2.23607 + 1.41421i) q^{7} +(2.61803 - 1.46489i) q^{9} +O(q^{10})$	$q+(1.67601 - 0.437016i) q^{3} +(2.23607 + 1.41421i) q^{7} +(2.61803 - 1.46489i) q^{9} -3.83513i q^{11} +5.99070i q^{13} +2.07167 q^{17} -5.11667i q^{19} +(4.36571 + 1.39304i) q^{21} -8.57561i q^{23} +(3.74768 - 3.59929i) q^{27} +5.64583i q^{29} +1.95440i q^{31} +(-1.67601 - 6.42772i) q^{33} +2.23607 q^{37} +(2.61803 + 10.0405i) q^{39} +7.49535 q^{41} -9.47214 q^{43} +12.9190 q^{47} +(3.00000 + 6.32456i) q^{49} +(3.47214 - 0.905351i) q^{51} +10.6000i q^{53} +(-2.23607 - 8.57561i) q^{57} -12.1277 q^{59} -4.37016i q^{61} +(7.92577 + 0.426869i) q^{63} +6.70820 q^{67} +(-3.74768 - 14.3728i) q^{69} +2.02443i q^{71} -1.41421i q^{73} +(5.42369 - 8.57561i) q^{77} -7.47214 q^{79} +(4.70820 - 7.67026i) q^{81} +7.49535 q^{83} +(2.46732 + 9.46248i) q^{87} -2.86297 q^{89} +(-8.47214 + 13.3956i) q^{91} +(0.854102 + 3.27559i) q^{93} +0.206331i q^{97} +(-5.61803 - 10.0405i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{9}+O(q^{10}) $ 8 * q + 12 * q^9	$ 8 q + 12 q^{9} - 4 q^{21} + 12 q^{39} - 40 q^{43} + 24 q^{49} - 8 q^{51} + 20 q^{63} - 24 q^{79} - 16 q^{81} - 32 q^{91} - 20 q^{93} - 36 q^{99}+O(q^{100}) $ 8 * q + 12 * q^9 - 4 * q^21 + 12 * q^39 - 40 * q^43 + 24 * q^49 - 8 * q^51 + 20 * q^63 - 24 * q^79 - 16 * q^81 - 32 * q^91 - 20 * q^93 - 36 * 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
$3$	1.67601	−	0.437016i	0.967646	−	0.252311i
$3$	1.67601	−	0.437016i	0.967646	−	0.252311i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.23607	+	1.41421i	0.845154	+	0.534522i
$7$	2.23607	+	1.41421i	0.845154	+	0.534522i
$8$		0			0
$9$	2.61803	−	1.46489i	0.872678	−	0.488296i
$10$		0			0
$11$		−	3.83513i		−	1.15633i	−0.815918	−	0.578167i	$-0.803768\pi$
$11$		−	3.83513i		−	1.15633i	0.815918	−	0.578167i	$-0.196232\pi$
$12$		0			0
$13$			5.99070i			1.66152i	0.556629	+	0.830761i	$0.312095\pi$
$13$			5.99070i			1.66152i	−0.556629	+	0.830761i	$0.687905\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.07167			0.502453			0.251226	−	0.967928i	$-0.419166\pi$
$17$	2.07167			0.502453			0.251226	+	0.967928i	$0.419166\pi$
$18$		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
$23$		−	8.57561i		−	1.78814i	−0.447930	−	0.894069i	$-0.647839\pi$
$23$		−	8.57561i		−	1.78814i	0.447930	−	0.894069i	$-0.352161\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.74768	−	3.59929i	0.721241	−	0.692684i
$28$		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	$0.0561574\pi$
$31$			1.95440i			0.351020i	−0.984478	+	0.175510i	$0.943843\pi$
$32$		0			0
$33$	−1.67601	−	6.42772i	−0.291756	−	1.11892i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.23607			0.367607			0.183804	−	0.982963i	$-0.441159\pi$
$37$	2.23607			0.367607			0.183804	+	0.982963i	$0.441159\pi$
$38$		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
$43$	−9.47214			−1.44449			−0.722244	−	0.691639i	$-0.756889\pi$
$43$	−9.47214			−1.44449			−0.722244	+	0.691639i	$0.756889\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	12.9190			1.88444			0.942218	−	0.335000i	$-0.108736\pi$
$47$	12.9190			1.88444			0.942218	+	0.335000i	$0.108736\pi$
$48$		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
$53$			10.6000i			1.45603i	0.685563	+	0.728013i	$0.259556\pi$
$53$			10.6000i			1.45603i	−0.685563	+	0.728013i	$0.740444\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.23607	−	8.57561i	−0.296174	−	1.13587i
$58$		0			0
$59$	−12.1277			−1.57890			−0.789449	−	0.613817i	$-0.789633\pi$
$59$	−12.1277			−1.57890			−0.789449	+	0.613817i	$0.789633\pi$
$60$		0			0
$61$		−	4.37016i		−	0.559542i	−0.960067	−	0.279771i	$-0.909742\pi$
$61$		−	4.37016i		−	0.559542i	0.960067	−	0.279771i	$-0.0902585\pi$
$62$		0			0
$63$	7.92577	+	0.426869i	0.998553	+	0.0537805i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.70820			0.819538			0.409769	−	0.912189i	$-0.365609\pi$
$67$	6.70820			0.819538			0.409769	+	0.912189i	$0.365609\pi$
$68$		0			0
$69$	−3.74768	−	14.3728i	−0.451167	−	1.73028i
$70$		0			0
$71$			2.02443i			0.240255i	0.992758	+	0.120128i	$0.0383304\pi$
$71$			2.02443i			0.240255i	−0.992758	+	0.120128i	$0.961670\pi$
$72$		0			0
$73$		−	1.41421i		−	0.165521i	−0.996569	−	0.0827606i	$-0.973626\pi$
$73$		−	1.41421i		−	0.165521i	0.996569	−	0.0827606i	$-0.0263737\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.42369	−	8.57561i	0.618087	−	0.977281i
$78$		0			0
$79$	−7.47214			−0.840681			−0.420340	−	0.907366i	$-0.638089\pi$
$79$	−7.47214			−0.840681			−0.420340	+	0.907366i	$0.638089\pi$
$80$		0			0
$81$	4.70820	−	7.67026i	0.523134	−	0.852251i
$82$		0			0
$83$	7.49535			0.822722			0.411361	−	0.911472i	$-0.365054\pi$
$83$	7.49535			0.822722			0.411361	+	0.911472i	$0.365054\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.46732	+	9.46248i	0.264524	+	1.01448i
$88$		0			0
$89$	−2.86297			−0.303474			−0.151737	−	0.988421i	$-0.548487\pi$
$89$	−2.86297			−0.303474			−0.151737	+	0.988421i	$0.548487\pi$
$90$		0			0
$91$	−8.47214	+	13.3956i	−0.888121	+	1.40424i
$92$		0			0
$93$	0.854102	+	3.27559i	0.0885662	+	0.339663i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			0.206331i			0.0209497i	0.999945	+	0.0104749i	$0.00333432\pi$
$97$			0.206331i			0.0209497i	−0.999945	+	0.0104749i	$0.996666\pi$
$98$		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
$103$			11.8539i			1.16800i	0.811754	+	0.583999i	$0.198513\pi$
$103$			11.8539i			1.16800i	−0.811754	+	0.583999i	$0.801487\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	17.1512i		−	1.65807i	−0.559197	−	0.829035i	$-0.688890\pi$
$107$		−	17.1512i		−	1.65807i	0.559197	−	0.829035i	$-0.311110\pi$
$108$		0			0
$109$	−1.47214			−0.141005			−0.0705025	−	0.997512i	$-0.522460\pi$
$109$	−1.47214			−0.141005			−0.0705025	+	0.997512i	$0.522460\pi$
$110$		0			0
$111$	3.74768	−	0.977198i	0.355714	−	0.0927515i
$112$		0			0
$113$		−	12.6245i		−	1.18761i	−0.804609	−	0.593805i	$-0.797625\pi$
$113$		−	12.6245i		−	1.18761i	0.804609	−	0.593805i	$-0.202375\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	8.77571	+	15.6839i	0.811315	+	1.44997i
$118$		0			0
$119$	4.63238	+	2.92978i	0.424650	+	0.268572i
$120$		0			0
$121$	−3.70820			−0.337109
$122$		0			0
$123$	12.5623	−	3.27559i	1.13270	−	0.295350i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.29180			−0.292100			−0.146050	−	0.989277i	$-0.546656\pi$
$127$	−3.29180			−0.292100			−0.146050	+	0.989277i	$0.546656\pi$
$128$		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
$133$	7.23607	−	11.4412i	0.627447	−	0.992080i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.04885i			0.345917i	0.984929	+	0.172958i	$0.0553326\pi$
$137$			4.04885i			0.345917i	−0.984929	+	0.172958i	$0.944667\pi$
$138$		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
$143$	22.9751			1.92128
$144$		0			0
$145$		0			0
$146$		0			0
$147$	7.79197	+	9.28898i	0.642671	+	0.766142i
$148$		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
$153$	5.42369	−	3.03476i	0.438479	−	0.245346i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	7.19859i		−	0.574510i	−0.957854	−	0.287255i	$-0.907257\pi$
$157$		−	7.19859i		−	0.574510i	0.957854	−	0.287255i	$-0.0927427\pi$
$158$		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
$163$	−8.94427			−0.700569			−0.350285	−	0.936643i	$-0.613915\pi$
$163$	−8.94427			−0.700569			−0.350285	+	0.936643i	$0.613915\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.56702			0.740318			0.370159	−	0.928968i	$-0.379303\pi$
$167$	9.56702			0.740318			0.370159	+	0.928968i	$0.379303\pi$
$168$		0			0
$169$	−22.8885			−1.76066
$170$		0			0
$171$	−7.49535	−	13.3956i	−0.573184	−	1.02439i
$172$		0			0
$173$	−15.4798			−1.17690			−0.588452	−	0.808532i	$-0.700263\pi$
$173$	−15.4798			−1.17690			−0.588452	+	0.808532i	$0.700263\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−20.3262	+	5.30002i	−1.52781	+	0.398374i
$178$		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	$0.222358\pi$
$181$			17.3044i			1.28623i	−0.765771	+	0.643113i	$0.777642\pi$
$182$		0			0
$183$	−1.90983	−	7.32444i	−0.141179	−	0.541438i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	7.94510i		−	0.581003i
$188$		0			0
$189$	13.4702	−	2.74825i	0.979815	−	0.199906i
$190$		0			0
$191$		−	4.74048i		−	0.343009i	−0.985183	−	0.171504i	$-0.945137\pi$
$191$		−	4.74048i		−	0.343009i	0.985183	−	0.171504i	$-0.0548628\pi$
$192$		0			0
$193$	15.0000			1.07972			0.539862	−	0.841754i	$-0.318476\pi$
$193$	15.0000			1.07972			0.539862	+	0.841754i	$0.318476\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2.02443i		−	0.144234i	−0.997396	−	0.0721172i	$-0.977024\pi$
$197$		−	2.02443i		−	0.144234i	0.997396	−	0.0721172i	$-0.0229756\pi$
$198$		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
$203$	−7.98441	+	12.6245i	−0.560396	+	0.886063i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−12.5623	−	22.4512i	−0.873141	−	1.56047i
$208$		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
$213$	0.884707	+	3.39296i	0.0606191	+	0.232482i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.76393	+	4.37016i	−0.187628	+	0.296666i
$218$		0			0
$219$	−0.618034	−	2.37024i	−0.0417629	−	0.160166i
$220$		0			0
$221$			12.4107i			0.834836i
$222$		0			0
$223$		−	8.40647i		−	0.562939i	−0.959570	−	0.281469i	$-0.909178\pi$
$223$		−	8.40647i		−	0.562939i	0.959570	−	0.281469i	$-0.0908218\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−22.1838			−1.47239			−0.736196	−	0.676769i	$-0.763380\pi$
$227$	−22.1838			−1.47239			−0.736196	+	0.676769i	$0.763380\pi$
$228$		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
$233$			15.1268i			0.990989i	0.868611	+	0.495494i	$0.165013\pi$
$233$			15.1268i			0.990989i	−0.868611	+	0.495494i	$0.834987\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−12.5234	+	3.26544i	−0.813482	+	0.212113i
$238$		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	$0.136773\pi$
$241$			12.9343i			0.833168i	−0.909097	+	0.416584i	$0.863227\pi$
$242$		0			0
$243$	4.53898	−	14.9130i	0.291176	−	0.956670i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	30.6525			1.95037
$248$		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
$253$	−32.8885			−2.06769
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.76941			0.110373			0.0551865	−	0.998476i	$-0.482425\pi$
$257$	1.76941			0.110373			0.0551865	+	0.998476i	$0.482425\pi$
$258$		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
$263$		−	15.1268i		−	0.932758i	−0.884585	−	0.466379i	$-0.845558\pi$
$263$		−	15.1268i		−	0.932758i	0.884585	−	0.466379i	$-0.154442\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−4.79837	+	1.25116i	−0.293656	+	0.0765700i
$268$		0			0
$269$	−26.0249			−1.58677			−0.793383	−	0.608723i	$-0.791682\pi$
$269$	−26.0249			−1.58677			−0.793383	+	0.608723i	$0.791682\pi$
$270$		0			0
$271$			22.1359i			1.34466i	0.740250	+	0.672331i	$0.234707\pi$
$271$			22.1359i			1.34466i	−0.740250	+	0.672331i	$0.765293\pi$
$272$		0			0
$273$	−8.34530	+	26.1537i	−0.505081	+	1.58289i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	21.7082			1.30432			0.652160	−	0.758082i	$-0.273863\pi$
$277$	21.7082			1.30432			0.652160	+	0.758082i	$0.273863\pi$
$278$		0			0
$279$	2.86297	+	5.11667i	0.171402	+	0.306327i
$280$		0			0
$281$			14.4352i			0.861129i	0.902560	+	0.430565i	$0.141686\pi$
$281$			14.4352i			0.861129i	−0.902560	+	0.430565i	$0.858314\pi$
$282$		0			0
$283$			25.7109i			1.52835i	0.645007	+	0.764177i	$0.276854\pi$
$283$			25.7109i			1.52835i	−0.645007	+	0.764177i	$0.723146\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.7601	+	10.6000i	0.989319	+	0.625700i
$288$		0			0
$289$	−12.7082			−0.747541
$290$		0			0
$291$	0.0901699	+	0.345813i	0.00528586	+	0.0202719i
$292$		0			0
$293$	−7.00630			−0.409312			−0.204656	−	0.978834i	$-0.565608\pi$
$293$	−7.00630			−0.409312			−0.204656	+	0.978834i	$0.565608\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−13.8038	−	14.3728i	−0.800975	−	0.833996i
$298$		0			0
$299$	51.3739			2.97103
$300$		0			0
$301$	−21.1803	−	13.3956i	−1.22081	−	0.772111i
$302$		0			0
$303$	4.79837	−	1.25116i	0.275659	−	0.0718775i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			15.6051i			0.890628i	0.895375	+	0.445314i	$0.146908\pi$
$307$			15.6051i			0.890628i	−0.895375	+	0.445314i	$0.853092\pi$
$308$		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
$313$		−	7.94510i		−	0.449084i	−0.974464	−	0.224542i	$-0.927911\pi$
$313$		−	7.94510i		−	0.449084i	0.974464	−	0.224542i	$-0.0720885\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			21.6780i			1.21756i	0.793341	+	0.608778i	$0.208340\pi$
$317$			21.6780i			1.21756i	−0.793341	+	0.608778i	$0.791660\pi$
$318$		0			0
$319$	21.6525			1.21231
$320$		0			0
$321$	−7.49535	−	28.7456i	−0.418350	−	1.60443i
$322$		0			0
$323$		−	10.6000i		−	0.589802i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−2.46732	+	0.643347i	−0.136443	+	0.0355772i
$328$		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
$333$	5.85410	−	3.27559i	0.320803	−	0.179501i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−10.0000			−0.544735			−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000			−0.544735			−0.272367	+	0.962193i	$0.587807\pi$
$338$		0			0
$339$	−5.51709	−	21.1587i	−0.299647	−	1.14919i
$340$		0			0
$341$	7.49535			0.405896
$342$		0			0
$343$	−2.23607	+	18.3848i	−0.120736	+	0.992685i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			21.6780i			1.16373i	0.813284	+	0.581867i	$0.197678\pi$
$347$			21.6780i			1.16373i	−0.813284	+	0.581867i	$0.802322\pi$
$348$		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
$353$	−2.37392			−0.126351			−0.0631754	−	0.998002i	$-0.520123\pi$
$353$	−2.37392			−0.126351			−0.0631754	+	0.998002i	$0.520123\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	9.04429	+	2.88592i	0.478675	+	0.152739i
$358$		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
$363$	−6.21500	+	1.62054i	−0.326203	+	0.0850565i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	32.9095i		−	1.71786i	−0.512093	−	0.858930i	$-0.671130\pi$
$367$		−	32.9095i		−	1.71786i	0.512093	−	0.858930i	$-0.328870\pi$
$368$		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
$373$	−15.6525			−0.810454			−0.405227	−	0.914216i	$-0.632808\pi$
$373$	−15.6525			−0.810454			−0.405227	+	0.914216i	$0.632808\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−33.8225			−1.74195
$378$		0			0
$379$	17.1803			0.882495			0.441247	−	0.897386i	$-0.354536\pi$
$379$	17.1803			0.882495			0.441247	+	0.897386i	$0.354536\pi$
$380$		0			0
$381$	−5.51709	+	1.43857i	−0.282649	+	0.0737001i
$382$		0			0
$383$	2.86297			0.146291			0.0731455	−	0.997321i	$-0.476696\pi$
$383$	2.86297			0.146291			0.0731455	+	0.997321i	$0.476696\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−24.7984	+	13.8756i	−1.26057	+	0.705338i
$388$		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
$393$	7.76393	−	2.02443i	0.391639	−	0.102119i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	20.2117i		−	1.01439i	−0.861830	−	0.507197i	$-0.830682\pi$
$397$		−	20.2117i		−	1.01439i	0.861830	−	0.507197i	$-0.169318\pi$
$398$		0			0
$399$	7.12774	−	22.3379i	0.356833	−	1.11829i
$400$		0			0
$401$		−	20.9863i		−	1.04801i	−0.851716	−	0.524004i	$-0.824438\pi$
$401$		−	20.9863i		−	1.04801i	0.851716	−	0.524004i	$-0.175562\pi$
$402$		0			0
$403$	−11.7082			−0.583227
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.57561i		−	0.425077i
$408$		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
$413$	−27.1184	−	17.1512i	−1.33441	−	0.843956i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	5.85410	+	22.4512i	0.286677	+	1.09944i
$418$		0			0
$419$	13.2213			0.645903			0.322951	−	0.946416i	$-0.395325\pi$
$419$	13.2213			0.645903			0.322951	+	0.946416i	$0.395325\pi$
$420$		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
$423$	33.8225	−	18.9250i	1.64451	−	0.920163i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	6.18034	−	9.77198i	0.299088	−	0.472899i
$428$		0			0
$429$	38.5066	−	10.0405i	1.85912	−	0.484760i
$430$		0			0
$431$		−	2.23815i		−	0.107808i	−0.998546	−	0.0539040i	$-0.982834\pi$
$431$		−	2.23815i		−	0.107808i	0.998546	−	0.0539040i	$-0.0171665\pi$
$432$		0			0
$433$			0.255039i			0.0122564i	0.999981	+	0.00612820i	$0.00195068\pi$
$433$			0.255039i			0.0122564i	−0.999981	+	0.00612820i	$0.998049\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−43.8786			−2.09900
$438$		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
$443$		−	10.6000i		−	0.503623i	−0.967776	−	0.251811i	$-0.918974\pi$
$443$		−	10.6000i		−	0.503623i	0.967776	−	0.251811i	$-0.0810263\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	3.74768	+	14.3728i	0.177259	+	0.679811i
$448$		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
$453$	−36.2898	+	9.46248i	−1.70504	+	0.444586i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7.36068			−0.344318			−0.172159	−	0.985069i	$-0.555074\pi$
$457$	−7.36068			−0.344318			−0.172159	+	0.985069i	$0.555074\pi$
$458$		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
$463$	−20.6525			−0.959802			−0.479901	−	0.877323i	$-0.659327\pi$
$463$	−20.6525			−0.959802			−0.479901	+	0.877323i	$0.659327\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.4300			−0.575191			−0.287596	−	0.957752i	$-0.592856\pi$
$467$	−12.4300			−0.575191			−0.287596	+	0.957752i	$0.592856\pi$
$468$		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
$473$			36.3269i			1.67031i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	15.5279	+	27.7512i	0.710972	+	1.27064i
$478$		0			0
$479$	−30.6573			−1.40077			−0.700383	−	0.713767i	$-0.746988\pi$
$479$	−30.6573			−1.40077			−0.700383	+	0.713767i	$0.746988\pi$
$480$		0			0
$481$			13.3956i			0.610788i
$482$		0			0
$483$	11.9462	−	37.4386i	0.543570	−	1.70352i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−7.76393			−0.351817			−0.175909	−	0.984406i	$-0.556286\pi$
$487$	−7.76393			−0.351817			−0.175909	+	0.984406i	$0.556286\pi$
$488$		0			0
$489$	−14.9907	+	3.90879i	−0.677903	+	0.176762i
$490$		0			0
$491$			2.71605i			0.122574i	0.998120	+	0.0612869i	$0.0195204\pi$
$491$			2.71605i			0.122574i	−0.998120	+	0.0612869i	$0.980480\pi$
$492$		0			0
$493$			11.6963i			0.526773i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.86297	+	4.52675i	−0.128422	+	0.203053i
$498$		0			0
$499$	−11.3475			−0.507985			−0.253992	−	0.967206i	$-0.581744\pi$
$499$	−11.3475			−0.507985			−0.253992	+	0.967206i	$0.581744\pi$
$500$		0			0
$501$	16.0344	−	4.18094i	0.716366	−	0.186791i
$502$		0			0
$503$	4.14333			0.184742			0.0923710	−	0.995725i	$-0.470555\pi$
$503$	4.14333			0.184742			0.0923710	+	0.995725i	$0.470555\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−38.3615	+	10.0027i	−1.70369	+	0.444234i
$508$		0			0
$509$	21.3925			0.948206			0.474103	−	0.880469i	$-0.342772\pi$
$509$	21.3925			0.948206			0.474103	+	0.880469i	$0.342772\pi$
$510$		0			0
$511$	2.00000	−	3.16228i	0.0884748	−	0.139891i
$512$		0			0
$513$	−18.4164	−	19.1756i	−0.813104	−	0.846625i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	49.5462i		−	2.17904i
$518$		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
$523$			8.48528i			0.371035i	0.982641	+	0.185518i	$0.0593962\pi$
$523$			8.48528i			0.371035i	−0.982641	+	0.185518i	$0.940604\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			4.04885i			0.176371i
$528$		0			0
$529$	−50.5410			−2.19744
$530$		0			0
$531$	−31.7508	+	17.7658i	−1.37787	+	0.770969i
$532$		0			0
$533$			44.9025i			1.94494i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−2.86297	−	10.9799i	−0.123546	−	0.473816i
$538$		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
$543$	7.56231	+	29.0024i	0.324530	+	1.24461i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−16.7082			−0.714391			−0.357196	−	0.934030i	$-0.616267\pi$
$547$	−16.7082			−0.714391			−0.357196	+	0.934030i	$0.616267\pi$
$548$		0			0
$549$	−6.40180	−	11.4412i	−0.273222	−	0.488300i
$550$		0			0
$551$	28.8879			1.23066
$552$		0			0
$553$	−16.7082	−	10.5672i	−0.710505	−	0.449363i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			19.1756i			0.812498i	0.913762	+	0.406249i	$0.133163\pi$
$557$			19.1756i			0.812498i	−0.913762	+	0.406249i	$0.866837\pi$
$558$		0			0
$559$		−	56.7448i		−	2.40005i
$560$		0			0
$561$	−3.47214	−	13.3161i	−0.146594	−	0.562206i
$562$		0			0
$563$	33.5202			1.41271			0.706355	−	0.707858i	$-0.250338\pi$
$563$	33.5202			1.41271			0.706355	+	0.707858i	$0.250338\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	21.3752	−	10.4928i	0.897676	−	0.440656i
$568$		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
$573$	−2.07167	−	7.94510i	−0.0865450	−	0.331911i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			39.9805i			1.66441i	0.554467	+	0.832206i	$0.312922\pi$
$577$			39.9805i			1.66441i	−0.554467	+	0.832206i	$0.687078\pi$
$578$		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
$583$	40.6525			1.68365
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−17.8537			−0.736900			−0.368450	−	0.929648i	$-0.620111\pi$
$587$	−17.8537			−0.736900			−0.368450	+	0.929648i	$0.620111\pi$
$588$		0			0
$589$	10.0000			0.412043
$590$		0			0
$591$	−0.884707	−	3.39296i	−0.0363920	−	0.139568i
$592$		0			0
$593$	−44.3676			−1.82196			−0.910980	−	0.412451i	$-0.864673\pi$
$593$	−44.3676			−1.82196			−0.910980	+	0.412451i	$0.864673\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.43769	−	28.5245i	−0.304405	−	1.16743i
$598$		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	$0.184009\pi$
$601$			26.7912i			1.09284i	−0.837512	+	0.546419i	$0.815991\pi$
$602$		0			0
$603$	17.5623	−	9.82677i	0.715192	−	0.400177i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			41.9349i			1.70209i	0.525096	+	0.851043i	$0.324030\pi$
$607$			41.9349i			1.70209i	−0.525096	+	0.851043i	$0.675970\pi$
$608$		0			0
$609$	−7.86488	+	24.6481i	−0.318701	+	0.998790i
$610$		0			0
$611$			77.3942i			3.13103i
$612$		0			0
$613$	−2.23607			−0.0903139			−0.0451570	−	0.998980i	$-0.514379\pi$
$613$	−2.23607			−0.0903139			−0.0451570	+	0.998980i	$0.514379\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			8.57561i			0.345241i	0.984988	+	0.172620i	$0.0552234\pi$
$617$			8.57561i			0.345241i	−0.984988	+	0.172620i	$0.944777\pi$
$618$		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
$623$	−6.40180	−	4.04885i	−0.256483	−	0.162214i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−32.8885	+	8.57561i	−1.31344	+	0.342477i
$628$		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
$633$	−9.56702	+	2.49458i	−0.380255	+	0.0991505i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−37.8885	+	17.9721i	−1.50120	+	0.712081i
$638$		0			0
$639$	2.96556	+	5.30002i	0.117316	+	0.209665i
$640$		0			0
$641$		−	11.0779i		−	0.437552i	−0.975775	−	0.218776i	$-0.929794\pi$
$641$		−	11.0779i		−	0.437552i	0.975775	−	0.218776i	$-0.0702064\pi$
$642$		0			0
$643$			1.08036i			0.0426054i	0.999773	+	0.0213027i	$0.00678137\pi$
$643$			1.08036i			0.0426054i	−0.999773	+	0.0213027i	$0.993219\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.5295			−0.728471			−0.364236	−	0.931307i	$-0.618670\pi$
$647$	−18.5295			−0.728471			−0.364236	+	0.931307i	$0.618670\pi$
$648$		0			0
$649$			46.5114i			1.82573i
$650$		0			0
$651$	−2.72255	+	8.53232i	−0.106705	+	0.334408i
$652$		0			0
$653$		−	6.55118i		−	0.256367i	−0.991750	−	0.128184i	$-0.959085\pi$
$653$		−	6.55118i		−	0.256367i	0.991750	−	0.128184i	$-0.0409147\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−2.07167	−	3.70246i	−0.0808234	−	0.144447i
$658$		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	$0.0561623\pi$
$661$			9.02546i			0.351050i	−0.984475	+	0.175525i	$0.943838\pi$
$662$		0			0
$663$	5.42369	+	20.8005i	0.210639	+	0.807826i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	48.4164			1.87469
$668$		0			0
$669$	−3.67376	−	14.0893i	−0.142036	−	0.544726i
$670$		0			0
$671$	−16.7601			−0.647017
$672$		0			0
$673$	31.3050			1.20672			0.603359	−	0.797470i	$-0.293829\pi$
$673$	31.3050			1.20672			0.603359	+	0.797470i	$0.293829\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	40.2243			1.54595			0.772973	−	0.634439i	$-0.218769\pi$
$677$	40.2243			1.54595			0.772973	+	0.634439i	$0.218769\pi$
$678$		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
$683$		−	6.07328i		−	0.232388i	−0.993227	−	0.116194i	$-0.962931\pi$
$683$		−	6.07328i		−	0.232388i	0.993227	−	0.116194i	$-0.0370694\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−1.70820	−	6.55118i	−0.0651720	−	0.249943i
$688$		0			0
$689$	−63.5017			−2.41922
$690$		0			0
$691$		−	22.4211i		−	0.852938i	−0.904502	−	0.426469i	$-0.859757\pi$
$691$		−	22.4211i		−	0.852938i	0.904502	−	0.426469i	$-0.140243\pi$
$692$		0			0
$693$	1.63710	−	30.3963i	0.0621882	−	1.15466i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	15.5279			0.588160
$698$		0			0
$699$	6.61065	+	25.3527i	0.250038	+	0.958926i
$700$		0			0
$701$		−	29.5619i		−	1.11654i	−0.829660	−	0.558270i	$-0.811465\pi$
$701$		−	29.5619i		−	1.11654i	0.829660	−	0.558270i	$-0.188535\pi$
$702$		0			0
$703$		−	11.4412i		−	0.431514i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.40180	+	4.04885i	0.240764	+	0.152273i
$708$		0			0
$709$	−39.3050			−1.47613			−0.738064	−	0.674730i	$-0.764260\pi$
$709$	−39.3050			−1.47613			−0.738064	+	0.674730i	$0.764260\pi$
$710$		0			0
$711$	−19.5623	+	10.9458i	−0.733644	+	0.410501i
$712$		0			0
$713$	16.7601			0.627672
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−7.98441	−	30.6212i	−0.298183	−	1.14357i
$718$		0			0
$719$	26.0249			0.970565			0.485282	−	0.874358i	$-0.338717\pi$
$719$	26.0249			0.970565			0.485282	+	0.874358i	$0.338717\pi$
$720$		0			0
$721$	−16.7639	+	26.5061i	−0.624321	+	0.987139i
$722$		0			0
$723$	5.65248	+	21.6780i	0.210218	+	0.806212i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	17.3531i		−	0.643592i	−0.946809	−	0.321796i	$-0.895714\pi$
$727$		−	17.3531i		−	0.643592i	0.946809	−	0.321796i	$-0.104286\pi$
$728$		0			0
$729$	1.09017	−	26.9780i	0.0403767	−	0.999185i
$730$		0			0
$731$	−19.6231			−0.725786
$732$		0			0
$733$		−	40.7758i		−	1.50609i	−0.657971	−	0.753044i	$-0.728585\pi$
$733$		−	40.7758i		−	1.50609i	0.657971	−	0.753044i	$-0.271415\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	25.7268i		−	0.947660i
$738$		0			0
$739$	12.5967			0.463379			0.231689	−	0.972790i	$-0.425575\pi$
$739$	12.5967			0.463379			0.231689	+	0.972790i	$0.425575\pi$
$740$		0			0
$741$	51.3739	−	13.3956i	1.88727	−	0.492101i
$742$		0			0
$743$		−	6.55118i		−	0.240339i	−0.992753	−	0.120170i	$-0.961656\pi$
$743$		−	6.55118i		−	0.240339i	0.992753	−	0.120170i	$-0.0383439\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	19.6231	−	10.9799i	0.717971	−	0.401732i
$748$		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
$753$	−32.8885	+	8.57561i	−1.19853	+	0.312512i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−17.7639			−0.645641			−0.322821	−	0.946460i	$-0.604631\pi$
$757$	−17.7639			−0.645641			−0.322821	+	0.946460i	$0.604631\pi$
$758$		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
$763$	−3.29180	−	2.08191i	−0.119171	−	0.0753704i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	72.6537i		−	2.62337i
$768$		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
$773$	30.1682			1.08508			0.542538	−	0.840031i	$-0.317464\pi$
$773$	30.1682			1.08508			0.542538	+	0.840031i	$0.317464\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	9.76203	+	3.11494i	0.350211	+	0.111748i
$778$		0			0
$779$		−	38.3513i		−	1.37408i
$780$		0			0
$781$	7.76393			0.277815
$782$		0			0
$783$	20.3210	+	21.1587i	0.726213	+	0.756152i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	24.2967i		−	0.866083i	−0.901374	−	0.433041i	$-0.857440\pi$
$787$		−	24.2967i		−	0.866083i	0.901374	−	0.433041i	$-0.142560\pi$
$788$		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
$793$	26.1803			0.929691
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.79761			0.276205			0.138103	−	0.990418i	$-0.455900\pi$
$797$	7.79761			0.276205			0.138103	+	0.990418i	$0.455900\pi$
$798$		0			0
$799$	26.7639			0.946840
$800$		0			0
$801$	−7.49535	+	4.19393i	−0.264835	+	0.148185i
$802$		0			0
$803$	−5.42369			−0.191398
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−43.6180	+	11.3733i	−1.53543	+	0.400359i
$808$		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	$-0.774746\pi$
$811$		−	37.0246i		−	1.30011i	0.759887	−	0.650055i	$-0.225254\pi$
$812$		0			0
$813$	9.67376	+	37.1001i	0.339274	+	1.30116i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			48.4658i			1.69560i
$818$		0			0
$819$	−2.55725	+	47.4809i	−0.0893574	+	1.65912i
$820$		0			0
$821$		−	37.6596i		−	1.31433i	−0.753746	−	0.657165i	$-0.771755\pi$
$821$		−	37.6596i		−	1.31433i	0.753746	−	0.657165i	$-0.228245\pi$
$822$		0			0
$823$	−27.3607			−0.953733			−0.476867	−	0.878976i	$-0.658228\pi$
$823$	−27.3607			−0.953733			−0.476867	+	0.878976i	$0.658228\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			25.7268i			0.894609i	0.894382	+	0.447305i	$0.147616\pi$
$827$			25.7268i			0.894609i	−0.894382	+	0.447305i	$0.852384\pi$
$828$		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
$833$	6.21500	+	13.1024i	0.215337	+	0.453970i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	7.03444	+	7.32444i	0.243146	+	0.253170i
$838$		0			0
$839$	−42.1092			−1.45377			−0.726885	−	0.686759i	$-0.759033\pi$
$839$	−42.1092			−1.45377			−0.726885	+	0.686759i	$0.759033\pi$
$840$		0			0
$841$	−2.87539			−0.0991513
$842$		0			0
$843$	6.30840	+	24.1935i	0.217273	+	0.833269i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−8.29180	−	5.24419i	−0.284909	−	0.180193i
$848$		0			0
$849$	11.2361	+	43.0918i	0.385621	+	1.47891i
$850$		0			0
$851$		−	19.1756i		−	0.657332i
$852$		0			0
$853$			40.1081i			1.37327i	0.727001	+	0.686637i	$0.240914\pi$
$853$			40.1081i			1.37327i	−0.727001	+	0.686637i	$0.759086\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.76941			−0.0604420			−0.0302210	−	0.999543i	$-0.509621\pi$
$857$	−1.76941			−0.0604420			−0.0302210	+	0.999543i	$0.509621\pi$
$858$		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
$863$			16.6733i			0.567566i	0.958889	+	0.283783i	$0.0915895\pi$
$863$			16.6733i			0.567566i	−0.958889	+	0.283783i	$0.908410\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−21.2991	+	5.55369i	−0.723356	+	0.188613i
$868$		0			0
$869$			28.6566i			0.972108i
$870$		0			0
$871$			40.1869i			1.36168i
$872$		0			0
$873$	0.302252	+	0.540182i	0.0102297	+	0.0182824i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	24.4721			0.826365			0.413183	−	0.910648i	$-0.364417\pi$
$877$	24.4721			0.826365			0.413183	+	0.910648i	$0.364417\pi$
$878$		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
$883$	42.8885			1.44331			0.721657	−	0.692251i	$-0.243381\pi$
$883$	42.8885			1.44331			0.721657	+	0.692251i	$0.243381\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	28.2120			0.947266			0.473633	−	0.880722i	$-0.342942\pi$
$887$	28.2120			0.947266			0.473633	+	0.880722i	$0.342942\pi$
$888$		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
$893$		−	66.1025i		−	2.21204i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	86.1033	−	22.4512i	2.87491	−	0.749625i
$898$		0			0
$899$	−11.0342			−0.368011
$900$		0			0
$901$			21.9597i			0.731584i
$902$		0			0
$903$	−41.3526	−	13.1951i	−1.37613	−	0.439105i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−41.3050			−1.37151			−0.685754	−	0.727833i	$-0.740527\pi$
$907$	−41.3050			−1.37151			−0.685754	+	0.727833i	$0.740527\pi$
$908$		0			0
$909$	7.49535	−	4.19393i	0.248605	−	0.139104i
$910$		0			0
$911$		−	33.8245i		−	1.12066i	−0.828271	−	0.560328i	$-0.810675\pi$
$911$		−	33.8245i		−	1.12066i	0.828271	−	0.560328i	$-0.189325\pi$
$912$		0			0
$913$		−	28.7456i		−	0.951342i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	10.3583	+	6.55118i	0.342062	+	0.216339i
$918$		0			0
$919$	28.3050			0.933694			0.466847	−	0.884338i	$-0.345390\pi$
$919$	28.3050			0.933694			0.466847	+	0.884338i	$0.345390\pi$
$920$		0			0
$921$	6.81966	+	26.1543i	0.224715	+	0.861812i
$922$		0			0
$923$	−12.1277			−0.399189
$924$		0			0
$925$		0			0
$926$		0			0
$927$	17.3646	+	31.0339i	0.570329	+	1.01929i
$928$		0			0
$929$	−2.86297			−0.0939310			−0.0469655	−	0.998897i	$-0.514955\pi$
$929$	−2.86297			−0.0939310			−0.0469655	+	0.998897i	$0.514955\pi$
$930$		0			0
$931$	32.3607	−	15.3500i	1.06058	−	0.503077i
$932$		0			0
$933$	−32.8885	+	8.57561i	−1.07672	+	0.280753i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			22.1660i			0.724133i	0.932152	+	0.362067i	$0.117929\pi$
$937$			22.1660i			0.724133i	−0.932152	+	0.362067i	$0.882071\pi$
$938$		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
$943$		−	64.2772i		−	2.09315i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			55.5025i			1.80359i	0.432166	+	0.901794i	$0.357750\pi$
$947$			55.5025i			1.80359i	−0.432166	+	0.901794i	$0.642250\pi$
$948$		0			0
$949$	8.47214			0.275017
$950$		0			0
$951$	9.47362	+	36.3325i	0.307203	+	1.17816i
$952$		0			0
$953$		−	45.3804i		−	1.47001i	−0.678060	−	0.735007i	$-0.737179\pi$
$953$		−	45.3804i		−	1.47001i	0.678060	−	0.735007i	$-0.262821\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	36.2898	−	9.46248i	1.17308	−	0.305879i
$958$		0			0
$959$	−5.72594	+	9.05351i	−0.184900	+	0.292353i
$960$		0			0
$961$	27.1803			0.876785
$962$		0			0
$963$	−25.1246	−	44.9025i	−0.809629	−	1.44696i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	14.0689			0.452425			0.226212	−	0.974078i	$-0.427366\pi$
$967$	14.0689			0.452425			0.226212	+	0.974078i	$0.427366\pi$
$968$		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
$973$	−18.9443	+	29.9535i	−0.607325	+	0.960266i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	25.7268i		−	0.823074i	−0.911393	−	0.411537i	$-0.864992\pi$
$977$		−	25.7268i		−	0.823074i	0.911393	−	0.411537i	$-0.135008\pi$
$978$		0			0
$979$			10.9799i			0.350918i
$980$		0			0
$981$	−3.85410	+	2.15651i	−0.123052	+	0.0688522i
$982$		0			0
$983$	20.9035			0.666717			0.333358	−	0.942800i	$-0.391818\pi$
$983$	20.9035			0.666717			0.333358	+	0.942800i	$0.391818\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	56.4008	+	17.9968i	1.79526	+	0.572843i
$988$		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
$993$	11.4298	−	2.98030i	0.362715	−	0.0945770i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			6.35466i			0.201254i	0.994924	+	0.100627i	$0.0320849\pi$
$997$			6.35466i			0.201254i	−0.994924	+	0.100627i	$0.967915\pi$
$998$		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.d.k.1301.7	yes	8
3.2	odd	2	inner	2100.2.d.k.1301.1	✓	8
5.2	odd	4		2100.2.f.i.1049.6		16
5.3	odd	4		2100.2.f.i.1049.11		16
5.4	even	2		2100.2.d.l.1301.2	yes	8
7.6	odd	2	inner	2100.2.d.k.1301.2	yes	8
15.2	even	4		2100.2.f.i.1049.8		16
15.8	even	4		2100.2.f.i.1049.9		16
15.14	odd	2		2100.2.d.l.1301.8	yes	8
21.20	even	2	inner	2100.2.d.k.1301.8	yes	8
35.13	even	4		2100.2.f.i.1049.5		16
35.27	even	4		2100.2.f.i.1049.12		16
35.34	odd	2		2100.2.d.l.1301.7	yes	8
105.62	odd	4		2100.2.f.i.1049.10		16
105.83	odd	4		2100.2.f.i.1049.7		16
105.104	even	2		2100.2.d.l.1301.1	yes	8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.67601 - 0.437016i) q^{3} +(2.23607 + 1.41421i) q^{7} +(2.61803 - 1.46489i) q^{9} +O(q^{10})\)	\(q+(1.67601 - 0.437016i) q^{3} +(2.23607 + 1.41421i) q^{7} +(2.61803 - 1.46489i) q^{9} -3.83513i q^{11} +5.99070i q^{13} +2.07167 q^{17} -5.11667i q^{19} +(4.36571 + 1.39304i) q^{21} -8.57561i q^{23} +(3.74768 - 3.59929i) q^{27} +5.64583i q^{29} +1.95440i q^{31} +(-1.67601 - 6.42772i) q^{33} +2.23607 q^{37} +(2.61803 + 10.0405i) q^{39} +7.49535 q^{41} -9.47214 q^{43} +12.9190 q^{47} +(3.00000 + 6.32456i) q^{49} +(3.47214 - 0.905351i) q^{51} +10.6000i q^{53} +(-2.23607 - 8.57561i) q^{57} -12.1277 q^{59} -4.37016i q^{61} +(7.92577 + 0.426869i) q^{63} +6.70820 q^{67} +(-3.74768 - 14.3728i) q^{69} +2.02443i q^{71} -1.41421i q^{73} +(5.42369 - 8.57561i) q^{77} -7.47214 q^{79} +(4.70820 - 7.67026i) q^{81} +7.49535 q^{83} +(2.46732 + 9.46248i) q^{87} -2.86297 q^{89} +(-8.47214 + 13.3956i) q^{91} +(0.854102 + 3.27559i) q^{93} +0.206331i q^{97} +(-5.61803 - 10.0405i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{9}+O(q^{10}) \) 8 * q + 12 * q^9	\( 8 q + 12 q^{9} - 4 q^{21} + 12 q^{39} - 40 q^{43} + 24 q^{49} - 8 q^{51} + 20 q^{63} - 24 q^{79} - 16 q^{81} - 32 q^{91} - 20 q^{93} - 36 q^{99}+O(q^{100}) \) 8 * q + 12 * q^9 - 4 * q^21 + 12 * q^39 - 40 * q^43 + 24 * q^49 - 8 * q^51 + 20 * q^63 - 24 * q^79 - 16 * q^81 - 32 * q^91 - 20 * q^93 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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