LMFDB - Embedded newform 2400.2.o.e.2399.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(2399,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("2400.2399");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.o (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:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2399.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2400.2399
Dual form			2400.2.o.e.2399.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 1.00000i) q^{3} -4.82843 q^{7} +(1.00000 - 2.82843i) q^{9} +O(q^{10})$	$q+(-1.41421 + 1.00000i) q^{3} -4.82843 q^{7} +(1.00000 - 2.82843i) q^{9} +4.82843 q^{11} +1.17157i q^{13} +0.828427 q^{17} +6.00000i q^{19} +(6.82843 - 4.82843i) q^{21} +0.828427i q^{23} +(1.41421 + 5.00000i) q^{27} -6.00000i q^{29} +2.00000i q^{31} +(-6.82843 + 4.82843i) q^{33} -6.82843i q^{37} +(-1.17157 - 1.65685i) q^{39} +9.65685i q^{41} -1.17157 q^{43} +3.17157i q^{47} +16.3137 q^{49} +(-1.17157 + 0.828427i) q^{51} +7.65685 q^{53} +(-6.00000 - 8.48528i) q^{57} -12.8284 q^{59} -13.3137 q^{61} +(-4.82843 + 13.6569i) q^{63} -4.48528 q^{67} +(-0.828427 - 1.17157i) q^{69} -11.3137 q^{71} -2.00000i q^{73} -23.3137 q^{77} -6.00000i q^{79} +(-7.00000 - 5.65685i) q^{81} -5.31371i q^{83} +(6.00000 + 8.48528i) q^{87} -5.65685i q^{91} +(-2.00000 - 2.82843i) q^{93} +6.00000i q^{97} +(4.82843 - 13.6569i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 8 * q^7 + 4 * q^9	$ 4 q - 8 q^{7} + 4 q^{9} + 8 q^{11} - 8 q^{17} + 16 q^{21} - 16 q^{33} - 16 q^{39} - 16 q^{43} + 20 q^{49} - 16 q^{51} + 8 q^{53} - 24 q^{57} - 40 q^{59} - 8 q^{61} - 8 q^{63} + 16 q^{67} + 8 q^{69} - 48 q^{77} - 28 q^{81} + 24 q^{87} - 8 q^{93} + 8 q^{99}+O(q^{100}) $ 4 * q - 8 * q^7 + 4 * q^9 + 8 * q^11 - 8 * q^17 + 16 * q^21 - 16 * q^33 - 16 * q^39 - 16 * q^43 + 20 * q^49 - 16 * q^51 + 8 * q^53 - 24 * q^57 - 40 * q^59 - 8 * q^61 - 8 * q^63 + 16 * q^67 + 8 * q^69 - 48 * q^77 - 28 * q^81 + 24 * q^87 - 8 * q^93 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−4.82843			−1.82497			−0.912487	−	0.409106i	$-0.865841\pi$
$7$	−4.82843			−1.82497			−0.912487	+	0.409106i	$0.865841\pi$
$8$		0			0
$9$	1.00000	−	2.82843i	0.333333	−	0.942809i
$10$		0			0
$11$	4.82843			1.45583			0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843			1.45583			0.727913	+	0.685670i	$0.240491\pi$
$12$		0			0
$13$			1.17157i			0.324936i	0.986714	+	0.162468i	$0.0519454\pi$
$13$			1.17157i			0.324936i	−0.986714	+	0.162468i	$0.948055\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.828427			0.200923			0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427			0.200923			0.100462	+	0.994941i	$0.467968\pi$
$18$		0			0
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$	6.82843	−	4.82843i	1.49008	−	1.05365i
$22$		0			0
$23$			0.828427i			0.172739i	0.996263	+	0.0863695i	$0.0275266\pi$
$23$			0.828427i			0.172739i	−0.996263	+	0.0863695i	$0.972473\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.41421	+	5.00000i	0.272166	+	0.962250i
$28$		0			0
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		0			0
$33$	−6.82843	+	4.82843i	−1.18868	+	0.840521i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	6.82843i		−	1.12259i	−0.827617	−	0.561293i	$-0.810304\pi$
$37$		−	6.82843i		−	1.12259i	0.827617	−	0.561293i	$-0.189696\pi$
$38$		0			0
$39$	−1.17157	−	1.65685i	−0.187602	−	0.265309i
$40$		0			0
$41$			9.65685i			1.50815i	0.656790	+	0.754074i	$0.271914\pi$
$41$			9.65685i			1.50815i	−0.656790	+	0.754074i	$0.728086\pi$
$42$		0			0
$43$	−1.17157			−0.178663			−0.0893316	−	0.996002i	$-0.528473\pi$
$43$	−1.17157			−0.178663			−0.0893316	+	0.996002i	$0.528473\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.17157i			0.462621i	0.972880	+	0.231311i	$0.0743014\pi$
$47$			3.17157i			0.462621i	−0.972880	+	0.231311i	$0.925699\pi$
$48$		0			0
$49$	16.3137			2.33053
$50$		0			0
$51$	−1.17157	+	0.828427i	−0.164053	+	0.116003i
$52$		0			0
$53$	7.65685			1.05175			0.525875	−	0.850562i	$-0.323738\pi$
$53$	7.65685			1.05175			0.525875	+	0.850562i	$0.323738\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.00000	−	8.48528i	−0.794719	−	1.12390i
$58$		0			0
$59$	−12.8284			−1.67012			−0.835059	−	0.550160i	$-0.814567\pi$
$59$	−12.8284			−1.67012			−0.835059	+	0.550160i	$0.814567\pi$
$60$		0			0
$61$	−13.3137			−1.70465			−0.852323	−	0.523016i	$-0.824807\pi$
$61$	−13.3137			−1.70465			−0.852323	+	0.523016i	$0.824807\pi$
$62$		0			0
$63$	−4.82843	+	13.6569i	−0.608325	+	1.72060i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.48528			−0.547964			−0.273982	−	0.961735i	$-0.588341\pi$
$67$	−4.48528			−0.547964			−0.273982	+	0.961735i	$0.588341\pi$
$68$		0			0
$69$	−0.828427	−	1.17157i	−0.0997309	−	0.141041i
$70$		0			0
$71$	−11.3137			−1.34269			−0.671345	−	0.741145i	$-0.734283\pi$
$71$	−11.3137			−1.34269			−0.671345	+	0.741145i	$0.734283\pi$
$72$		0			0
$73$		−	2.00000i		−	0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$		−	2.00000i		−	0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−23.3137			−2.65684
$78$		0			0
$79$		−	6.00000i		−	0.675053i	−0.941316	−	0.337526i	$-0.890410\pi$
$79$		−	6.00000i		−	0.675053i	0.941316	−	0.337526i	$-0.109590\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$		−	5.31371i		−	0.583255i	−0.956532	−	0.291628i	$-0.905803\pi$
$83$		−	5.31371i		−	0.583255i	0.956532	−	0.291628i	$-0.0941968\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	6.00000	+	8.48528i	0.643268	+	0.909718i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		−	5.65685i		−	0.592999i
$92$		0			0
$93$	−2.00000	−	2.82843i	−0.207390	−	0.293294i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$		0			0
$99$	4.82843	−	13.6569i	0.485275	−	1.37257i
$100$		0			0
$101$			7.65685i			0.761885i	0.924599	+	0.380943i	$0.124401\pi$
$101$			7.65685i			0.761885i	−0.924599	+	0.380943i	$0.875599\pi$
$102$		0			0
$103$	−0.828427			−0.0816274			−0.0408137	−	0.999167i	$-0.512995\pi$
$103$	−0.828427			−0.0816274			−0.0408137	+	0.999167i	$0.512995\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.31371i		−	0.513696i	−0.966452	−	0.256848i	$-0.917316\pi$
$107$		−	5.31371i		−	0.513696i	0.966452	−	0.256848i	$-0.0826839\pi$
$108$		0			0
$109$	−7.65685			−0.733394			−0.366697	−	0.930341i	$-0.619511\pi$
$109$	−7.65685			−0.733394			−0.366697	+	0.930341i	$0.619511\pi$
$110$		0			0
$111$	6.82843	+	9.65685i	0.648126	+	0.916588i
$112$		0			0
$113$	−7.17157			−0.674645			−0.337322	−	0.941389i	$-0.609521\pi$
$113$	−7.17157			−0.674645			−0.337322	+	0.941389i	$0.609521\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	3.31371	+	1.17157i	0.306352	+	0.108312i
$118$		0			0
$119$	−4.00000			−0.366679
$120$		0			0
$121$	12.3137			1.11943
$122$		0			0
$123$	−9.65685	−	13.6569i	−0.870729	−	1.23140i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−6.48528			−0.575476			−0.287738	−	0.957709i	$-0.592903\pi$
$127$	−6.48528			−0.575476			−0.287738	+	0.957709i	$0.592903\pi$
$128$		0			0
$129$	1.65685	−	1.17157i	0.145878	−	0.103151i
$130$		0			0
$131$	−2.48528			−0.217140			−0.108570	−	0.994089i	$-0.534627\pi$
$131$	−2.48528			−0.217140			−0.108570	+	0.994089i	$0.534627\pi$
$132$		0			0
$133$		−	28.9706i		−	2.51207i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−16.1421			−1.37912			−0.689558	−	0.724231i	$-0.742195\pi$
$137$	−16.1421			−1.37912			−0.689558	+	0.724231i	$0.742195\pi$
$138$		0			0
$139$		−	21.3137i		−	1.80781i	−0.427738	−	0.903903i	$-0.640690\pi$
$139$		−	21.3137i		−	1.80781i	0.427738	−	0.903903i	$-0.359310\pi$
$140$		0			0
$141$	−3.17157	−	4.48528i	−0.267095	−	0.377729i
$142$		0			0
$143$			5.65685i			0.473050i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−23.0711	+	16.3137i	−1.90287	+	1.34553i
$148$		0			0
$149$		−	12.3431i		−	1.01119i	−0.862771	−	0.505595i	$-0.831273\pi$
$149$		−	12.3431i		−	1.01119i	0.862771	−	0.505595i	$-0.168727\pi$
$150$		0			0
$151$		−	7.65685i		−	0.623106i	−0.950229	−	0.311553i	$-0.899151\pi$
$151$		−	7.65685i		−	0.623106i	0.950229	−	0.311553i	$-0.100849\pi$
$152$		0			0
$153$	0.828427	−	2.34315i	0.0669744	−	0.189432i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	0.485281i		−	0.0387297i	−0.999812	−	0.0193648i	$-0.993836\pi$
$157$		−	0.485281i		−	0.0387297i	0.999812	−	0.0193648i	$-0.00616440\pi$
$158$		0			0
$159$	−10.8284	+	7.65685i	−0.858750	+	0.607228i
$160$		0			0
$161$		−	4.00000i		−	0.315244i
$162$		0			0
$163$	−0.485281			−0.0380102			−0.0190051	−	0.999819i	$-0.506050\pi$
$163$	−0.485281			−0.0380102			−0.0190051	+	0.999819i	$0.506050\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			24.8284i			1.92128i	0.277794	+	0.960641i	$0.410397\pi$
$167$			24.8284i			1.92128i	−0.277794	+	0.960641i	$0.589603\pi$
$168$		0			0
$169$	11.6274			0.894417
$170$		0			0
$171$	16.9706	+	6.00000i	1.29777	+	0.458831i
$172$		0			0
$173$	−10.9706			−0.834076			−0.417038	−	0.908889i	$-0.636932\pi$
$173$	−10.9706			−0.834076			−0.417038	+	0.908889i	$0.636932\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	18.1421	−	12.8284i	1.36365	−	0.964244i
$178$		0			0
$179$	−3.17157			−0.237054			−0.118527	−	0.992951i	$-0.537817\pi$
$179$	−3.17157			−0.237054			−0.118527	+	0.992951i	$0.537817\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	18.8284	−	13.3137i	1.39184	−	0.984178i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.00000			0.292509
$188$		0			0
$189$	−6.82843	−	24.1421i	−0.496695	−	1.75608i
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$			18.9706i			1.36553i	0.730638	+	0.682765i	$0.239223\pi$
$193$			18.9706i			1.36553i	−0.730638	+	0.682765i	$0.760777\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	13.3137			0.948562			0.474281	−	0.880373i	$-0.342708\pi$
$197$	13.3137			0.948562			0.474281	+	0.880373i	$0.342708\pi$
$198$		0			0
$199$		−	7.65685i		−	0.542780i	−0.962469	−	0.271390i	$-0.912517\pi$
$199$		−	7.65685i		−	0.542780i	0.962469	−	0.271390i	$-0.0874833\pi$
$200$		0			0
$201$	6.34315	−	4.48528i	0.447411	−	0.316367i
$202$		0			0
$203$			28.9706i			2.03333i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.34315	+	0.828427i	0.162860	+	0.0575797i
$208$		0			0
$209$			28.9706i			2.00394i
$210$		0			0
$211$			6.97056i			0.479873i	0.970789	+	0.239937i	$0.0771267\pi$
$211$			6.97056i			0.479873i	−0.970789	+	0.239937i	$0.922873\pi$
$212$		0			0
$213$	16.0000	−	11.3137i	1.09630	−	0.775203i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	9.65685i		−	0.655550i
$218$		0			0
$219$	2.00000	+	2.82843i	0.135147	+	0.191127i
$220$		0			0
$221$			0.970563i			0.0652871i
$222$		0			0
$223$	6.48528			0.434287			0.217143	−	0.976140i	$-0.430326\pi$
$223$	6.48528			0.434287			0.217143	+	0.976140i	$0.430326\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	11.6569i		−	0.773693i	−0.922144	−	0.386846i	$-0.873564\pi$
$227$		−	11.6569i		−	0.773693i	0.922144	−	0.386846i	$-0.126436\pi$
$228$		0			0
$229$	−18.9706			−1.25361			−0.626805	−	0.779176i	$-0.715638\pi$
$229$	−18.9706			−1.25361			−0.626805	+	0.779176i	$0.715638\pi$
$230$		0			0
$231$	32.9706	−	23.3137i	2.16930	−	1.53393i
$232$		0			0
$233$	−20.8284			−1.36452			−0.682258	−	0.731112i	$-0.739002\pi$
$233$	−20.8284			−1.36452			−0.682258	+	0.731112i	$0.739002\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	6.00000	+	8.48528i	0.389742	+	0.551178i
$238$		0			0
$239$	−16.9706			−1.09773			−0.548867	−	0.835910i	$-0.684941\pi$
$239$	−16.9706			−1.09773			−0.548867	+	0.835910i	$0.684941\pi$
$240$		0			0
$241$	6.00000			0.386494			0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000			0.386494			0.193247	+	0.981150i	$0.438098\pi$
$242$		0			0
$243$	15.5563	+	1.00000i	0.997940	+	0.0641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−7.02944			−0.447272
$248$		0			0
$249$	5.31371	+	7.51472i	0.336743	+	0.476226i
$250$		0			0
$251$	−17.7990			−1.12346			−0.561731	−	0.827320i	$-0.689864\pi$
$251$	−17.7990			−1.12346			−0.561731	+	0.827320i	$0.689864\pi$
$252$		0			0
$253$			4.00000i			0.251478i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−26.4853			−1.65211			−0.826053	−	0.563592i	$-0.809419\pi$
$257$	−26.4853			−1.65211			−0.826053	+	0.563592i	$0.809419\pi$
$258$		0			0
$259$			32.9706i			2.04869i
$260$		0			0
$261$	−16.9706	−	6.00000i	−1.05045	−	0.371391i
$262$		0			0
$263$		−	2.48528i		−	0.153249i	−0.997060	−	0.0766245i	$-0.975586\pi$
$263$		−	2.48528i		−	0.153249i	0.997060	−	0.0766245i	$-0.0244143\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	3.65685i		−	0.222962i	−0.993767	−	0.111481i	$-0.964441\pi$
$269$		−	3.65685i		−	0.222962i	0.993767	−	0.111481i	$-0.0355595\pi$
$270$		0			0
$271$			14.9706i			0.909397i	0.890645	+	0.454698i	$0.150253\pi$
$271$			14.9706i			0.909397i	−0.890645	+	0.454698i	$0.849747\pi$
$272$		0			0
$273$	5.65685	+	8.00000i	0.342368	+	0.484182i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	3.51472i		−	0.211179i	−0.994410	−	0.105589i	$-0.966327\pi$
$277$		−	3.51472i		−	0.211179i	0.994410	−	0.105589i	$-0.0336729\pi$
$278$		0			0
$279$	5.65685	+	2.00000i	0.338667	+	0.119737i
$280$		0			0
$281$			23.3137i			1.39078i	0.718633	+	0.695390i	$0.244768\pi$
$281$			23.3137i			1.39078i	−0.718633	+	0.695390i	$0.755232\pi$
$282$		0			0
$283$	17.1716			1.02074			0.510372	−	0.859954i	$-0.329508\pi$
$283$	17.1716			1.02074			0.510372	+	0.859954i	$0.329508\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	46.6274i		−	2.75233i
$288$		0			0
$289$	−16.3137			−0.959630
$290$		0			0
$291$	−6.00000	−	8.48528i	−0.351726	−	0.497416i
$292$		0			0
$293$	−0.343146			−0.0200468			−0.0100234	−	0.999950i	$-0.503191\pi$
$293$	−0.343146			−0.0200468			−0.0100234	+	0.999950i	$0.503191\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	6.82843	+	24.1421i	0.396226	+	1.40087i
$298$		0			0
$299$	−0.970563			−0.0561291
$300$		0			0
$301$	5.65685			0.326056
$302$		0			0
$303$	−7.65685	−	10.8284i	−0.439875	−	0.622077i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	20.4853			1.16916			0.584578	−	0.811337i	$-0.301260\pi$
$307$	20.4853			1.16916			0.584578	+	0.811337i	$0.301260\pi$
$308$		0			0
$309$	1.17157	−	0.828427i	0.0666485	−	0.0471276i
$310$		0			0
$311$	−11.3137			−0.641542			−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137			−0.641542			−0.320771	+	0.947157i	$0.603942\pi$
$312$		0			0
$313$			25.3137i			1.43082i	0.698707	+	0.715408i	$0.253759\pi$
$313$			25.3137i			1.43082i	−0.698707	+	0.715408i	$0.746241\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.31371			−0.298448			−0.149224	−	0.988803i	$-0.547677\pi$
$317$	−5.31371			−0.298448			−0.149224	+	0.988803i	$0.547677\pi$
$318$		0			0
$319$		−	28.9706i		−	1.62204i
$320$		0			0
$321$	5.31371	+	7.51472i	0.296582	+	0.419431i
$322$		0			0
$323$			4.97056i			0.276570i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	10.8284	−	7.65685i	0.598813	−	0.423425i
$328$		0			0
$329$		−	15.3137i		−	0.844272i
$330$		0			0
$331$		−	10.9706i		−	0.602997i	−0.953467	−	0.301498i	$-0.902513\pi$
$331$		−	10.9706i		−	0.602997i	0.953467	−	0.301498i	$-0.0974868\pi$
$332$		0			0
$333$	−19.3137	−	6.82843i	−1.05838	−	0.374196i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	34.9706i		−	1.90497i	−0.304589	−	0.952484i	$-0.598519\pi$
$337$		−	34.9706i		−	1.90497i	0.304589	−	0.952484i	$-0.401481\pi$
$338$		0			0
$339$	10.1421	−	7.17157i	0.550845	−	0.389506i
$340$		0			0
$341$			9.65685i			0.522948i
$342$		0			0
$343$	−44.9706			−2.42818
$344$		0			0
$345$		0			0
$346$		0			0
$347$			13.3137i			0.714717i	0.933967	+	0.357359i	$0.116323\pi$
$347$			13.3137i			0.714717i	−0.933967	+	0.357359i	$0.883677\pi$
$348$		0			0
$349$	−10.9706			−0.587241			−0.293620	−	0.955922i	$-0.594860\pi$
$349$	−10.9706			−0.587241			−0.293620	+	0.955922i	$0.594860\pi$
$350$		0			0
$351$	−5.85786	+	1.65685i	−0.312670	+	0.0884363i
$352$		0			0
$353$	16.8284			0.895687			0.447843	−	0.894112i	$-0.352192\pi$
$353$	16.8284			0.895687			0.447843	+	0.894112i	$0.352192\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	5.65685	−	4.00000i	0.299392	−	0.211702i
$358$		0			0
$359$	31.3137			1.65267			0.826337	−	0.563176i	$-0.190421\pi$
$359$	31.3137			1.65267			0.826337	+	0.563176i	$0.190421\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$	−17.4142	+	12.3137i	−0.914009	+	0.646302i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−28.1421			−1.46901			−0.734504	−	0.678605i	$-0.762585\pi$
$367$	−28.1421			−1.46901			−0.734504	+	0.678605i	$0.762585\pi$
$368$		0			0
$369$	27.3137	+	9.65685i	1.42189	+	0.502716i
$370$		0			0
$371$	−36.9706			−1.91942
$372$		0			0
$373$			16.4853i			0.853576i	0.904352	+	0.426788i	$0.140355\pi$
$373$			16.4853i			0.853576i	−0.904352	+	0.426788i	$0.859645\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.02944			0.362034
$378$		0			0
$379$			10.6863i			0.548918i	0.961599	+	0.274459i	$0.0884987\pi$
$379$			10.6863i			0.548918i	−0.961599	+	0.274459i	$0.911501\pi$
$380$		0			0
$381$	9.17157	−	6.48528i	0.469874	−	0.332251i
$382$		0			0
$383$		−	24.1421i		−	1.23361i	−0.787118	−	0.616803i	$-0.788428\pi$
$383$		−	24.1421i		−	1.23361i	0.787118	−	0.616803i	$-0.211572\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.17157	+	3.31371i	−0.0595544	+	0.168445i
$388$		0			0
$389$			28.6274i			1.45147i	0.687976	+	0.725734i	$0.258500\pi$
$389$			28.6274i			1.45147i	−0.687976	+	0.725734i	$0.741500\pi$
$390$		0			0
$391$			0.686292i			0.0347073i
$392$		0			0
$393$	3.51472	−	2.48528i	0.177294	−	0.125366i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			24.4853i			1.22888i	0.788963	+	0.614441i	$0.210618\pi$
$397$			24.4853i			1.22888i	−0.788963	+	0.614441i	$0.789382\pi$
$398$		0			0
$399$	28.9706	+	40.9706i	1.45034	+	2.05109i
$400$		0			0
$401$			16.9706i			0.847469i	0.905786	+	0.423735i	$0.139281\pi$
$401$			16.9706i			0.847469i	−0.905786	+	0.423735i	$0.860719\pi$
$402$		0			0
$403$	−2.34315			−0.116720
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	32.9706i		−	1.63429i
$408$		0			0
$409$	10.6863			0.528403			0.264202	−	0.964467i	$-0.414892\pi$
$409$	10.6863			0.528403			0.264202	+	0.964467i	$0.414892\pi$
$410$		0			0
$411$	22.8284	−	16.1421i	1.12604	−	0.796233i
$412$		0			0
$413$	61.9411			3.04792
$414$		0			0
$415$		0			0
$416$		0			0
$417$	21.3137	+	30.1421i	1.04374	+	1.47607i
$418$		0			0
$419$	−16.8284			−0.822122			−0.411061	−	0.911608i	$-0.634842\pi$
$419$	−16.8284			−0.822122			−0.411061	+	0.911608i	$0.634842\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$	8.97056	+	3.17157i	0.436164	+	0.154207i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	64.2843			3.11093
$428$		0			0
$429$	−5.65685	−	8.00000i	−0.273115	−	0.386244i
$430$		0			0
$431$	17.6569			0.850501			0.425250	−	0.905076i	$-0.360186\pi$
$431$	17.6569			0.850501			0.425250	+	0.905076i	$0.360186\pi$
$432$		0			0
$433$		−	30.0000i		−	1.44171i	−0.693087	−	0.720854i	$-0.743750\pi$
$433$		−	30.0000i		−	1.44171i	0.693087	−	0.720854i	$-0.256250\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.97056			−0.237774
$438$		0			0
$439$			26.2843i			1.25448i	0.778826	+	0.627240i	$0.215815\pi$
$439$			26.2843i			1.25448i	−0.778826	+	0.627240i	$0.784185\pi$
$440$		0			0
$441$	16.3137	−	46.1421i	0.776843	−	2.19724i
$442$		0			0
$443$		−	19.6569i		−	0.933925i	−0.884277	−	0.466963i	$-0.845348\pi$
$443$		−	19.6569i		−	0.933925i	0.884277	−	0.466963i	$-0.154652\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	12.3431	+	17.4558i	0.583811	+	0.825633i
$448$		0			0
$449$			22.3431i			1.05444i	0.849729	+	0.527219i	$0.176765\pi$
$449$			22.3431i			1.05444i	−0.849729	+	0.527219i	$0.823235\pi$
$450$		0			0
$451$			46.6274i			2.19560i
$452$		0			0
$453$	7.65685	+	10.8284i	0.359750	+	0.508764i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			34.9706i			1.63585i	0.575322	+	0.817927i	$0.304877\pi$
$457$			34.9706i			1.63585i	−0.575322	+	0.817927i	$0.695123\pi$
$458$		0			0
$459$	1.17157	+	4.14214i	0.0546843	+	0.193338i
$460$		0			0
$461$		−	13.3137i		−	0.620081i	−0.950723	−	0.310041i	$-0.899657\pi$
$461$		−	13.3137i		−	0.620081i	0.950723	−	0.310041i	$-0.100343\pi$
$462$		0			0
$463$	−7.17157			−0.333291			−0.166646	−	0.986017i	$-0.553294\pi$
$463$	−7.17157			−0.333291			−0.166646	+	0.986017i	$0.553294\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	23.6569i		−	1.09471i	−0.836901	−	0.547354i	$-0.815635\pi$
$467$		−	23.6569i		−	1.09471i	0.836901	−	0.547354i	$-0.184365\pi$
$468$		0			0
$469$	21.6569			1.00002
$470$		0			0
$471$	0.485281	+	0.686292i	0.0223606	+	0.0316226i
$472$		0			0
$473$	−5.65685			−0.260102
$474$		0			0
$475$		0			0
$476$		0			0
$477$	7.65685	−	21.6569i	0.350583	−	0.991599i
$478$		0			0
$479$	−2.34315			−0.107061			−0.0535305	−	0.998566i	$-0.517047\pi$
$479$	−2.34315			−0.107061			−0.0535305	+	0.998566i	$0.517047\pi$
$480$		0			0
$481$	8.00000			0.364769
$482$		0			0
$483$	4.00000	+	5.65685i	0.182006	+	0.257396i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	23.4558			1.06289			0.531443	−	0.847094i	$-0.321650\pi$
$487$	23.4558			1.06289			0.531443	+	0.847094i	$0.321650\pi$
$488$		0			0
$489$	0.686292	−	0.485281i	0.0310352	−	0.0219452i
$490$		0			0
$491$	15.1716			0.684683			0.342342	−	0.939576i	$-0.388780\pi$
$491$	15.1716			0.684683			0.342342	+	0.939576i	$0.388780\pi$
$492$		0			0
$493$		−	4.97056i		−	0.223863i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	54.6274			2.45037
$498$		0			0
$499$			25.3137i			1.13320i	0.823994	+	0.566599i	$0.191741\pi$
$499$			25.3137i			1.13320i	−0.823994	+	0.566599i	$0.808259\pi$
$500$		0			0
$501$	−24.8284	−	35.1127i	−1.10925	−	1.56872i
$502$		0			0
$503$			0.828427i			0.0369377i	0.999829	+	0.0184689i	$0.00587916\pi$
$503$			0.828427i			0.0369377i	−0.999829	+	0.0184689i	$0.994121\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−16.4437	+	11.6274i	−0.730288	+	0.516392i
$508$		0			0
$509$			12.3431i			0.547100i	0.961858	+	0.273550i	$0.0881980\pi$
$509$			12.3431i			0.547100i	−0.961858	+	0.273550i	$0.911802\pi$
$510$		0			0
$511$			9.65685i			0.427194i
$512$		0			0
$513$	−30.0000	+	8.48528i	−1.32453	+	0.374634i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			15.3137i			0.673496i
$518$		0			0
$519$	15.5147	−	10.9706i	0.681021	−	0.481554i
$520$		0			0
$521$		−	12.0000i		−	0.525730i	−0.964833	−	0.262865i	$-0.915333\pi$
$521$		−	12.0000i		−	0.525730i	0.964833	−	0.262865i	$-0.0846673\pi$
$522$		0			0
$523$	4.48528			0.196128			0.0980638	−	0.995180i	$-0.468735\pi$
$523$	4.48528			0.196128			0.0980638	+	0.995180i	$0.468735\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1.65685i			0.0721737i
$528$		0			0
$529$	22.3137			0.970161
$530$		0			0
$531$	−12.8284	+	36.2843i	−0.556706	+	1.57460i
$532$		0			0
$533$	−11.3137			−0.490051
$534$		0			0
$535$		0			0
$536$		0			0
$537$	4.48528	−	3.17157i	0.193554	−	0.136863i
$538$		0			0
$539$	78.7696			3.39284
$540$		0			0
$541$	36.6274			1.57474			0.787368	−	0.616483i	$-0.211443\pi$
$541$	36.6274			1.57474			0.787368	+	0.616483i	$0.211443\pi$
$542$		0			0
$543$	2.82843	−	2.00000i	0.121379	−	0.0858282i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−21.4558			−0.917386			−0.458693	−	0.888595i	$-0.651682\pi$
$547$	−21.4558			−0.917386			−0.458693	+	0.888595i	$0.651682\pi$
$548$		0			0
$549$	−13.3137	+	37.6569i	−0.568215	+	1.60716i
$550$		0			0
$551$	36.0000			1.53365
$552$		0			0
$553$			28.9706i			1.23195i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.02944			0.0436187			0.0218093	−	0.999762i	$-0.493057\pi$
$557$	1.02944			0.0436187			0.0218093	+	0.999762i	$0.493057\pi$
$558$		0			0
$559$		−	1.37258i		−	0.0580541i
$560$		0			0
$561$	−5.65685	+	4.00000i	−0.238833	+	0.168880i
$562$		0			0
$563$		−	17.3137i		−	0.729686i	−0.931069	−	0.364843i	$-0.881123\pi$
$563$		−	17.3137i		−	0.729686i	0.931069	−	0.364843i	$-0.118877\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	33.7990	+	27.3137i	1.41942	+	1.14707i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		−	34.9706i		−	1.46347i	−0.681588	−	0.731736i	$-0.738710\pi$
$571$		−	34.9706i		−	1.46347i	0.681588	−	0.731736i	$-0.261290\pi$
$572$		0			0
$573$	16.9706	−	12.0000i	0.708955	−	0.501307i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			14.9706i			0.623233i	0.950208	+	0.311616i	$0.100870\pi$
$577$			14.9706i			0.623233i	−0.950208	+	0.311616i	$0.899130\pi$
$578$		0			0
$579$	−18.9706	−	26.8284i	−0.788390	−	1.11495i
$580$		0			0
$581$			25.6569i			1.06443i
$582$		0			0
$583$	36.9706			1.53116
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	16.6274i		−	0.686287i	−0.939283	−	0.343143i	$-0.888508\pi$
$587$		−	16.6274i		−	0.686287i	0.939283	−	0.343143i	$-0.111492\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$	−18.8284	+	13.3137i	−0.774498	+	0.547653i
$592$		0			0
$593$	36.1421			1.48418			0.742090	−	0.670300i	$-0.233835\pi$
$593$	36.1421			1.48418			0.742090	+	0.670300i	$0.233835\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	7.65685	+	10.8284i	0.313374	+	0.443178i
$598$		0			0
$599$	−18.6274			−0.761096			−0.380548	−	0.924761i	$-0.624265\pi$
$599$	−18.6274			−0.761096			−0.380548	+	0.924761i	$0.624265\pi$
$600$		0			0
$601$	−36.6274			−1.49406			−0.747032	−	0.664788i	$-0.768522\pi$
$601$	−36.6274			−1.49406			−0.747032	+	0.664788i	$0.768522\pi$
$602$		0			0
$603$	−4.48528	+	12.6863i	−0.182655	+	0.516626i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−21.5147			−0.873255			−0.436628	−	0.899642i	$-0.643827\pi$
$607$	−21.5147			−0.873255			−0.436628	+	0.899642i	$0.643827\pi$
$608$		0			0
$609$	−28.9706	−	40.9706i	−1.17395	−	1.66021i
$610$		0			0
$611$	−3.71573			−0.150322
$612$		0			0
$613$			25.4558i			1.02815i	0.857745	+	0.514076i	$0.171865\pi$
$613$			25.4558i			1.02815i	−0.857745	+	0.514076i	$0.828135\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.17157			0.127683			0.0638414	−	0.997960i	$-0.479665\pi$
$617$	3.17157			0.127683			0.0638414	+	0.997960i	$0.479665\pi$
$618$		0			0
$619$			6.00000i			0.241160i	0.992704	+	0.120580i	$0.0384755\pi$
$619$			6.00000i			0.241160i	−0.992704	+	0.120580i	$0.961525\pi$
$620$		0			0
$621$	−4.14214	+	1.17157i	−0.166218	+	0.0470136i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−28.9706	−	40.9706i	−1.15697	−	1.63621i
$628$		0			0
$629$		−	5.65685i		−	0.225554i
$630$		0			0
$631$		−	33.3137i		−	1.32620i	−0.748532	−	0.663099i	$-0.769241\pi$
$631$		−	33.3137i		−	1.32620i	0.748532	−	0.663099i	$-0.230759\pi$
$632$		0			0
$633$	−6.97056	−	9.85786i	−0.277055	−	0.391815i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			19.1127i			0.757273i
$638$		0			0
$639$	−11.3137	+	32.0000i	−0.447563	+	1.26590i
$640$		0			0
$641$			14.6274i			0.577748i	0.957367	+	0.288874i	$0.0932809\pi$
$641$			14.6274i			0.577748i	−0.957367	+	0.288874i	$0.906719\pi$
$642$		0			0
$643$	−5.17157			−0.203947			−0.101973	−	0.994787i	$-0.532516\pi$
$643$	−5.17157			−0.203947			−0.101973	+	0.994787i	$0.532516\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	15.1716i		−	0.596456i	−0.954495	−	0.298228i	$-0.903604\pi$
$647$		−	15.1716i		−	0.596456i	0.954495	−	0.298228i	$-0.0963956\pi$
$648$		0			0
$649$	−61.9411			−2.43140
$650$		0			0
$651$	9.65685	+	13.6569i	0.378482	+	0.535254i
$652$		0			0
$653$	−30.2843			−1.18512			−0.592558	−	0.805528i	$-0.701882\pi$
$653$	−30.2843			−1.18512			−0.592558	+	0.805528i	$0.701882\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−5.65685	−	2.00000i	−0.220695	−	0.0780274i
$658$		0			0
$659$	6.20101			0.241557			0.120779	−	0.992679i	$-0.461461\pi$
$659$	6.20101			0.241557			0.120779	+	0.992679i	$0.461461\pi$
$660$		0			0
$661$	14.0000			0.544537			0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000			0.544537			0.272268	+	0.962221i	$0.412226\pi$
$662$		0			0
$663$	−0.970563	−	1.37258i	−0.0376935	−	0.0533067i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4.97056			0.192461
$668$		0			0
$669$	−9.17157	+	6.48528i	−0.354593	+	0.250735i
$670$		0			0
$671$	−64.2843			−2.48167
$672$		0			0
$673$		−	31.9411i		−	1.23124i	−0.788043	−	0.615620i	$-0.788906\pi$
$673$		−	31.9411i		−	1.23124i	0.788043	−	0.615620i	$-0.211094\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11.6569			0.448009			0.224005	−	0.974588i	$-0.428087\pi$
$677$	11.6569			0.448009			0.224005	+	0.974588i	$0.428087\pi$
$678$		0			0
$679$		−	28.9706i		−	1.11179i
$680$		0			0
$681$	11.6569	+	16.4853i	0.446692	+	0.631717i
$682$		0			0
$683$		−	31.6569i		−	1.21132i	−0.795725	−	0.605658i	$-0.792910\pi$
$683$		−	31.6569i		−	1.21132i	0.795725	−	0.605658i	$-0.207090\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	26.8284	−	18.9706i	1.02357	−	0.723772i
$688$		0			0
$689$			8.97056i			0.341751i
$690$		0			0
$691$		−	42.9706i		−	1.63468i	−0.576158	−	0.817339i	$-0.695449\pi$
$691$		−	42.9706i		−	1.63468i	0.576158	−	0.817339i	$-0.304551\pi$
$692$		0			0
$693$	−23.3137	+	65.9411i	−0.885615	+	2.50490i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			8.00000i			0.303022i
$698$		0			0
$699$	29.4558	−	20.8284i	1.11412	−	0.787803i
$700$		0			0
$701$			39.9411i			1.50856i	0.656555	+	0.754278i	$0.272013\pi$
$701$			39.9411i			1.50856i	−0.656555	+	0.754278i	$0.727987\pi$
$702$		0			0
$703$	40.9706			1.54523
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	36.9706i		−	1.39042i
$708$		0			0
$709$	−9.02944			−0.339108			−0.169554	−	0.985521i	$-0.554233\pi$
$709$	−9.02944			−0.339108			−0.169554	+	0.985521i	$0.554233\pi$
$710$		0			0
$711$	−16.9706	−	6.00000i	−0.636446	−	0.225018i
$712$		0			0
$713$	−1.65685			−0.0620497
$714$		0			0
$715$		0			0
$716$		0			0
$717$	24.0000	−	16.9706i	0.896296	−	0.633777i
$718$		0			0
$719$	−20.6863			−0.771468			−0.385734	−	0.922610i	$-0.626052\pi$
$719$	−20.6863			−0.771468			−0.385734	+	0.922610i	$0.626052\pi$
$720$		0			0
$721$	4.00000			0.148968
$722$		0			0
$723$	−8.48528	+	6.00000i	−0.315571	+	0.223142i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	24.8284			0.920835			0.460418	−	0.887702i	$-0.347700\pi$
$727$	24.8284			0.920835			0.460418	+	0.887702i	$0.347700\pi$
$728$		0			0
$729$	−23.0000	+	14.1421i	−0.851852	+	0.523783i
$730$		0			0
$731$	−0.970563			−0.0358976
$732$		0			0
$733$		−	25.1716i		−	0.929733i	−0.885381	−	0.464867i	$-0.846102\pi$
$733$		−	25.1716i		−	0.929733i	0.885381	−	0.464867i	$-0.153898\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21.6569			−0.797740
$738$		0			0
$739$			1.31371i			0.0483255i	0.999708	+	0.0241628i	$0.00769200\pi$
$739$			1.31371i			0.0483255i	−0.999708	+	0.0241628i	$0.992308\pi$
$740$		0			0
$741$	9.94113	−	7.02944i	0.365196	−	0.258233i
$742$		0			0
$743$			31.4558i			1.15400i	0.816743	+	0.577001i	$0.195777\pi$
$743$			31.4558i			1.15400i	−0.816743	+	0.577001i	$0.804223\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−15.0294	−	5.31371i	−0.549898	−	0.194418i
$748$		0			0
$749$			25.6569i			0.937481i
$750$		0			0
$751$		−	2.68629i		−	0.0980242i	−0.998798	−	0.0490121i	$-0.984393\pi$
$751$		−	2.68629i		−	0.0980242i	0.998798	−	0.0490121i	$-0.0156073\pi$
$752$		0			0
$753$	25.1716	−	17.7990i	0.917303	−	0.648631i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	18.1421i		−	0.659387i	−0.944088	−	0.329694i	$-0.893055\pi$
$757$		−	18.1421i		−	0.659387i	0.944088	−	0.329694i	$-0.106945\pi$
$758$		0			0
$759$	−4.00000	−	5.65685i	−0.145191	−	0.205331i
$760$		0			0
$761$			8.68629i			0.314878i	0.987529	+	0.157439i	$0.0503237\pi$
$761$			8.68629i			0.314878i	−0.987529	+	0.157439i	$0.949676\pi$
$762$		0			0
$763$	36.9706			1.33842
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	15.0294i		−	0.542682i
$768$		0			0
$769$	−33.3137			−1.20132			−0.600662	−	0.799503i	$-0.705096\pi$
$769$	−33.3137			−1.20132			−0.600662	+	0.799503i	$0.705096\pi$
$770$		0			0
$771$	37.4558	−	26.4853i	1.34894	−	0.953844i
$772$		0			0
$773$	−6.68629			−0.240489			−0.120245	−	0.992744i	$-0.538368\pi$
$773$	−6.68629			−0.240489			−0.120245	+	0.992744i	$0.538368\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−32.9706	−	46.6274i	−1.18281	−	1.67275i
$778$		0			0
$779$	−57.9411			−2.07596
$780$		0			0
$781$	−54.6274			−1.95472
$782$		0			0
$783$	30.0000	−	8.48528i	1.07211	−	0.303239i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	0.201010			0.00716524			0.00358262	−	0.999994i	$-0.498860\pi$
$787$	0.201010			0.00716524			0.00358262	+	0.999994i	$0.498860\pi$
$788$		0			0
$789$	2.48528	+	3.51472i	0.0884784	+	0.125127i
$790$		0			0
$791$	34.6274			1.23121
$792$		0			0
$793$		−	15.5980i		−	0.553901i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.37258			−0.119463			−0.0597315	−	0.998214i	$-0.519024\pi$
$797$	−3.37258			−0.119463			−0.0597315	+	0.998214i	$0.519024\pi$
$798$		0			0
$799$			2.62742i			0.0929513i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	9.65685i		−	0.340783i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	3.65685	+	5.17157i	0.128727	+	0.182048i
$808$		0			0
$809$			5.65685i			0.198884i	0.995043	+	0.0994422i	$0.0317058\pi$
$809$			5.65685i			0.198884i	−0.995043	+	0.0994422i	$0.968294\pi$
$810$		0			0
$811$		−	38.2843i		−	1.34434i	−0.740396	−	0.672171i	$-0.765362\pi$
$811$		−	38.2843i		−	1.34434i	0.740396	−	0.672171i	$-0.234638\pi$
$812$		0			0
$813$	−14.9706	−	21.1716i	−0.525041	−	0.742519i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	7.02944i		−	0.245929i
$818$		0			0
$819$	−16.0000	−	5.65685i	−0.559085	−	0.197666i
$820$		0			0
$821$		−	35.6569i		−	1.24443i	−0.782845	−	0.622216i	$-0.786232\pi$
$821$		−	35.6569i		−	1.24443i	0.782845	−	0.622216i	$-0.213768\pi$
$822$		0			0
$823$	−39.4558			−1.37534			−0.687672	−	0.726021i	$-0.741367\pi$
$823$	−39.4558			−1.37534			−0.687672	+	0.726021i	$0.741367\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			37.3137i			1.29752i	0.760991	+	0.648762i	$0.224713\pi$
$827$			37.3137i			1.29752i	−0.760991	+	0.648762i	$0.775287\pi$
$828$		0			0
$829$	56.9117			1.97662			0.988312	−	0.152443i	$-0.0487139\pi$
$829$	56.9117			1.97662			0.988312	+	0.152443i	$0.0487139\pi$
$830$		0			0
$831$	3.51472	+	4.97056i	0.121924	+	0.172427i
$832$		0			0
$833$	13.5147			0.468257
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−10.0000	+	2.82843i	−0.345651	+	0.0977647i
$838$		0			0
$839$	−21.9411			−0.757492			−0.378746	−	0.925501i	$-0.623645\pi$
$839$	−21.9411			−0.757492			−0.378746	+	0.925501i	$0.623645\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$	−23.3137	−	32.9706i	−0.802967	−	1.13557i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−59.4558			−2.04293
$848$		0			0
$849$	−24.2843	+	17.1716i	−0.833434	+	0.589327i
$850$		0			0
$851$	5.65685			0.193914
$852$		0			0
$853$		−	48.4853i		−	1.66010i	−0.557686	−	0.830052i	$-0.688311\pi$
$853$		−	48.4853i		−	1.66010i	0.557686	−	0.830052i	$-0.311689\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.79899			0.334727			0.167364	−	0.985895i	$-0.446475\pi$
$857$	9.79899			0.334727			0.167364	+	0.985895i	$0.446475\pi$
$858$		0			0
$859$			22.0000i			0.750630i	0.926897	+	0.375315i	$0.122466\pi$
$859$			22.0000i			0.750630i	−0.926897	+	0.375315i	$0.877534\pi$
$860$		0			0
$861$	46.6274	+	65.9411i	1.58906	+	2.24727i
$862$		0			0
$863$			57.7990i			1.96750i	0.179542	+	0.983750i	$0.442538\pi$
$863$			57.7990i			1.96750i	−0.179542	+	0.983750i	$0.557462\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	23.0711	−	16.3137i	0.783535	−	0.554043i
$868$		0			0
$869$		−	28.9706i		−	0.982759i
$870$		0			0
$871$		−	5.25483i		−	0.178053i
$872$		0			0
$873$	16.9706	+	6.00000i	0.574367	+	0.203069i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	27.7990i		−	0.938705i	−0.883011	−	0.469353i	$-0.844487\pi$
$877$		−	27.7990i		−	0.938705i	0.883011	−	0.469353i	$-0.155513\pi$
$878$		0			0
$879$	0.485281	−	0.343146i	0.0163681	−	0.0115740i
$880$		0			0
$881$		−	39.5980i		−	1.33409i	−0.745018	−	0.667045i	$-0.767559\pi$
$881$		−	39.5980i		−	1.33409i	0.745018	−	0.667045i	$-0.232441\pi$
$882$		0			0
$883$	15.5147			0.522112			0.261056	−	0.965324i	$-0.415929\pi$
$883$	15.5147			0.522112			0.261056	+	0.965324i	$0.415929\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			40.8284i			1.37088i	0.728127	+	0.685442i	$0.240391\pi$
$887$			40.8284i			1.37088i	−0.728127	+	0.685442i	$0.759609\pi$
$888$		0			0
$889$	31.3137			1.05023
$890$		0			0
$891$	−33.7990	−	27.3137i	−1.13231	−	0.915044i
$892$		0			0
$893$	−19.0294			−0.636796
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1.37258	−	0.970563i	0.0458292	−	0.0324061i
$898$		0			0
$899$	12.0000			0.400222
$900$		0			0
$901$	6.34315			0.211321
$902$		0			0
$903$	−8.00000	+	5.65685i	−0.266223	+	0.188248i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	10.8284			0.359552			0.179776	−	0.983708i	$-0.442463\pi$
$907$	10.8284			0.359552			0.179776	+	0.983708i	$0.442463\pi$
$908$		0			0
$909$	21.6569	+	7.65685i	0.718313	+	0.253962i
$910$		0			0
$911$	−51.5980			−1.70952			−0.854759	−	0.519026i	$-0.826295\pi$
$911$	−51.5980			−1.70952			−0.854759	+	0.519026i	$0.826295\pi$
$912$		0			0
$913$		−	25.6569i		−	0.849118i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12.0000			0.396275
$918$		0			0
$919$		−	25.3137i		−	0.835022i	−0.908672	−	0.417511i	$-0.862902\pi$
$919$		−	25.3137i		−	0.835022i	0.908672	−	0.417511i	$-0.137098\pi$
$920$		0			0
$921$	−28.9706	+	20.4853i	−0.954612	+	0.675013i
$922$		0			0
$923$		−	13.2548i		−	0.436288i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.828427	+	2.34315i	−0.0272091	+	0.0769590i
$928$		0			0
$929$		−	30.3431i		−	0.995526i	−0.867313	−	0.497763i	$-0.834155\pi$
$929$		−	30.3431i		−	0.995526i	0.867313	−	0.497763i	$-0.165845\pi$
$930$		0			0
$931$			97.8823i			3.20796i
$932$		0			0
$933$	16.0000	−	11.3137i	0.523816	−	0.370394i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	6.97056i		−	0.227718i	−0.993497	−	0.113859i	$-0.963679\pi$
$937$		−	6.97056i		−	0.227718i	0.993497	−	0.113859i	$-0.0363213\pi$
$938$		0			0
$939$	−25.3137	−	35.7990i	−0.826082	−	1.16826i
$940$		0			0
$941$		−	41.5980i		−	1.35606i	−0.735036	−	0.678028i	$-0.762835\pi$
$941$		−	41.5980i		−	1.35606i	0.735036	−	0.678028i	$-0.237165\pi$
$942$		0			0
$943$	−8.00000			−0.260516
$944$		0			0
$945$		0			0
$946$		0			0
$947$			46.2843i			1.50404i	0.659142	+	0.752018i	$0.270920\pi$
$947$			46.2843i			1.50404i	−0.659142	+	0.752018i	$0.729080\pi$
$948$		0			0
$949$	2.34315			0.0760617
$950$		0			0
$951$	7.51472	−	5.31371i	0.243681	−	0.172309i
$952$		0			0
$953$	1.79899			0.0582750			0.0291375	−	0.999575i	$-0.490724\pi$
$953$	1.79899			0.0582750			0.0291375	+	0.999575i	$0.490724\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	28.9706	+	40.9706i	0.936485	+	1.32439i
$958$		0			0
$959$	77.9411			2.51685
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	−15.0294	−	5.31371i	−0.484317	−	0.171232i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−18.4853			−0.594447			−0.297223	−	0.954808i	$-0.596061\pi$
$967$	−18.4853			−0.594447			−0.297223	+	0.954808i	$0.596061\pi$
$968$		0			0
$969$	−4.97056	−	7.02944i	−0.159677	−	0.225818i
$970$		0			0
$971$	22.7696			0.730710			0.365355	−	0.930868i	$-0.380948\pi$
$971$	22.7696			0.730710			0.365355	+	0.930868i	$0.380948\pi$
$972$		0			0
$973$			102.912i			3.29920i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−29.7990			−0.953354			−0.476677	−	0.879078i	$-0.658159\pi$
$977$	−29.7990			−0.953354			−0.476677	+	0.879078i	$0.658159\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−7.65685	+	21.6569i	−0.244465	+	0.691450i
$982$		0			0
$983$		−	15.1716i		−	0.483898i	−0.970289	−	0.241949i	$-0.922213\pi$
$983$		−	15.1716i		−	0.483898i	0.970289	−	0.241949i	$-0.0777867\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	15.3137	+	21.6569i	0.487441	+	0.689345i
$988$		0			0
$989$		−	0.970563i		−	0.0308621i
$990$		0			0
$991$			42.2843i			1.34320i	0.740912	+	0.671602i	$0.234394\pi$
$991$			42.2843i			1.34320i	−0.740912	+	0.671602i	$0.765606\pi$
$992$		0			0
$993$	10.9706	+	15.5147i	0.348140	+	0.492345i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			27.5147i			0.871400i	0.900092	+	0.435700i	$0.143499\pi$
$997$			27.5147i			0.871400i	−0.900092	+	0.435700i	$0.856501\pi$
$998$		0			0
$999$	34.1421	−	9.65685i	1.08021	−	0.305529i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.o.e.2399.2		4
3.2	odd	2		2400.2.o.d.2399.1		4
4.3	odd	2		2400.2.o.f.2399.3		4
5.2	odd	4		480.2.h.c.191.3	yes	4
5.3	odd	4		2400.2.h.a.1151.2		4
5.4	even	2		2400.2.o.g.2399.3		4
12.11	even	2		2400.2.o.g.2399.4		4
15.2	even	4		480.2.h.a.191.4	yes	4
15.8	even	4		2400.2.h.d.1151.2		4
15.14	odd	2		2400.2.o.f.2399.4		4
20.3	even	4		2400.2.h.d.1151.3		4
20.7	even	4		480.2.h.a.191.1	✓	4
20.19	odd	2		2400.2.o.d.2399.2		4
40.27	even	4		960.2.h.e.191.4		4
40.37	odd	4		960.2.h.a.191.2		4
60.23	odd	4		2400.2.h.a.1151.3		4
60.47	odd	4		480.2.h.c.191.2	yes	4
60.59	even	2	inner	2400.2.o.e.2399.1		4
120.77	even	4		960.2.h.e.191.1		4
120.107	odd	4		960.2.h.a.191.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.h.a.191.1	✓	4	20.7	even	4
480.2.h.a.191.4	yes	4	15.2	even	4
480.2.h.c.191.2	yes	4	60.47	odd	4
480.2.h.c.191.3	yes	4	5.2	odd	4
960.2.h.a.191.2		4	40.37	odd	4
960.2.h.a.191.3		4	120.107	odd	4
960.2.h.e.191.1		4	120.77	even	4
960.2.h.e.191.4		4	40.27	even	4
2400.2.h.a.1151.2		4	5.3	odd	4
2400.2.h.a.1151.3		4	60.23	odd	4
2400.2.h.d.1151.2		4	15.8	even	4
2400.2.h.d.1151.3		4	20.3	even	4
2400.2.o.d.2399.1		4	3.2	odd	2
2400.2.o.d.2399.2		4	20.19	odd	2
2400.2.o.e.2399.1		4	60.59	even	2	inner
2400.2.o.e.2399.2		4	1.1	even	1	trivial
2400.2.o.f.2399.3		4	4.3	odd	2
2400.2.o.f.2399.4		4	15.14	odd	2
2400.2.o.g.2399.3		4	5.4	even	2
2400.2.o.g.2399.4		4	12.11	even	2

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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 + 1.00000i) q^{3} -4.82843 q^{7} +(1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(-1.41421 + 1.00000i) q^{3} -4.82843 q^{7} +(1.00000 - 2.82843i) q^{9} +4.82843 q^{11} +1.17157i q^{13} +0.828427 q^{17} +6.00000i q^{19} +(6.82843 - 4.82843i) q^{21} +0.828427i q^{23} +(1.41421 + 5.00000i) q^{27} -6.00000i q^{29} +2.00000i q^{31} +(-6.82843 + 4.82843i) q^{33} -6.82843i q^{37} +(-1.17157 - 1.65685i) q^{39} +9.65685i q^{41} -1.17157 q^{43} +3.17157i q^{47} +16.3137 q^{49} +(-1.17157 + 0.828427i) q^{51} +7.65685 q^{53} +(-6.00000 - 8.48528i) q^{57} -12.8284 q^{59} -13.3137 q^{61} +(-4.82843 + 13.6569i) q^{63} -4.48528 q^{67} +(-0.828427 - 1.17157i) q^{69} -11.3137 q^{71} -2.00000i q^{73} -23.3137 q^{77} -6.00000i q^{79} +(-7.00000 - 5.65685i) q^{81} -5.31371i q^{83} +(6.00000 + 8.48528i) q^{87} -5.65685i q^{91} +(-2.00000 - 2.82843i) q^{93} +6.00000i q^{97} +(4.82843 - 13.6569i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 8 * q^7 + 4 * q^9	\( 4 q - 8 q^{7} + 4 q^{9} + 8 q^{11} - 8 q^{17} + 16 q^{21} - 16 q^{33} - 16 q^{39} - 16 q^{43} + 20 q^{49} - 16 q^{51} + 8 q^{53} - 24 q^{57} - 40 q^{59} - 8 q^{61} - 8 q^{63} + 16 q^{67} + 8 q^{69} - 48 q^{77} - 28 q^{81} + 24 q^{87} - 8 q^{93} + 8 q^{99}+O(q^{100}) \) 4 * q - 8 * q^7 + 4 * q^9 + 8 * q^11 - 8 * q^17 + 16 * q^21 - 16 * q^33 - 16 * q^39 - 16 * q^43 + 20 * q^49 - 16 * q^51 + 8 * q^53 - 24 * q^57 - 40 * q^59 - 8 * q^61 - 8 * q^63 + 16 * q^67 + 8 * q^69 - 48 * q^77 - 28 * q^81 + 24 * q^87 - 8 * q^93 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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