LMFDB - Embedded newform 640.2.f.c.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,2,Mod(449,640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("640.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 640 = 2^{7} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	640.f (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:	$5.11042572936$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	640.449
Dual form			640.2.f.c.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949 q^{3} +(-2.00000 - 1.00000i) q^{5} +2.44949i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-2.44949 q^{3} +(-2.00000 - 1.00000i) q^{5} +2.44949i q^{7} +3.00000 q^{9} +4.89898i q^{11} +(4.89898 + 2.44949i) q^{15} -4.00000i q^{17} -4.89898i q^{19} -6.00000i q^{21} -2.44949i q^{23} +(3.00000 + 4.00000i) q^{25} -8.00000i q^{29} -9.79796 q^{31} -12.0000i q^{33} +(2.44949 - 4.89898i) q^{35} -4.00000 q^{37} +8.00000 q^{41} +7.34847 q^{43} +(-6.00000 - 3.00000i) q^{45} -12.2474i q^{47} +1.00000 q^{49} +9.79796i q^{51} +8.00000 q^{53} +(4.89898 - 9.79796i) q^{55} +12.0000i q^{57} -4.89898i q^{59} +6.00000i q^{61} +7.34847i q^{63} +2.44949 q^{67} +6.00000i q^{69} -9.79796 q^{71} -4.00000i q^{73} +(-7.34847 - 9.79796i) q^{75} -12.0000 q^{77} +9.79796 q^{79} -9.00000 q^{81} +2.44949 q^{83} +(-4.00000 + 8.00000i) q^{85} +19.5959i q^{87} +2.00000 q^{89} +24.0000 q^{93} +(-4.89898 + 9.79796i) q^{95} +4.00000i q^{97} +14.6969i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5} + 12 q^{9}+O(q^{10}) $ 4 * q - 8 * q^5 + 12 * q^9	$ 4 q - 8 q^{5} + 12 q^{9} + 12 q^{25} - 16 q^{37} + 32 q^{41} - 24 q^{45} + 4 q^{49} + 32 q^{53} - 48 q^{77} - 36 q^{81} - 16 q^{85} + 8 q^{89} + 96 q^{93}+O(q^{100}) $ 4 * q - 8 * q^5 + 12 * q^9 + 12 * q^25 - 16 * q^37 + 32 * q^41 - 24 * q^45 + 4 * q^49 + 32 * q^53 - 48 * q^77 - 36 * q^81 - 16 * q^85 + 8 * q^89 + 96 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.44949			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−2.44949			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$6$		0			0
$7$			2.44949i			0.925820i	0.886405	+	0.462910i	$0.153195\pi$
$7$			2.44949i			0.925820i	−0.886405	+	0.462910i	$0.846805\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$			4.89898i			1.47710i	0.674200	+	0.738549i	$0.264489\pi$
$11$			4.89898i			1.47710i	−0.674200	+	0.738549i	$0.735511\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	4.89898	+	2.44949i	1.26491	+	0.632456i
$16$		0			0
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		0			0
$19$		−	4.89898i		−	1.12390i	−0.827170	−	0.561951i	$-0.810051\pi$
$19$		−	4.89898i		−	1.12390i	0.827170	−	0.561951i	$-0.189949\pi$
$20$		0			0
$21$		−	6.00000i		−	1.30931i
$22$		0			0
$23$		−	2.44949i		−	0.510754i	−0.966842	−	0.255377i	$-0.917800\pi$
$23$		−	2.44949i		−	0.510754i	0.966842	−	0.255377i	$-0.0821996\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	8.00000i		−	1.48556i	−0.669534	−	0.742781i	$-0.733506\pi$
$29$		−	8.00000i		−	1.48556i	0.669534	−	0.742781i	$-0.266494\pi$
$30$		0			0
$31$	−9.79796			−1.75977			−0.879883	−	0.475191i	$-0.842379\pi$
$31$	−9.79796			−1.75977			−0.879883	+	0.475191i	$0.842379\pi$
$32$		0			0
$33$		−	12.0000i		−	2.08893i
$34$		0			0
$35$	2.44949	−	4.89898i	0.414039	−	0.828079i
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.00000			1.24939			0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000			1.24939			0.624695	+	0.780869i	$0.285223\pi$
$42$		0			0
$43$	7.34847			1.12063			0.560316	−	0.828279i	$-0.310680\pi$
$43$	7.34847			1.12063			0.560316	+	0.828279i	$0.310680\pi$
$44$		0			0
$45$	−6.00000	−	3.00000i	−0.894427	−	0.447214i
$46$		0			0
$47$		−	12.2474i		−	1.78647i	−0.449586	−	0.893237i	$-0.648429\pi$
$47$		−	12.2474i		−	1.78647i	0.449586	−	0.893237i	$-0.351571\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$			9.79796i			1.37199i
$52$		0			0
$53$	8.00000			1.09888			0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000			1.09888			0.549442	+	0.835532i	$0.314840\pi$
$54$		0			0
$55$	4.89898	−	9.79796i	0.660578	−	1.32116i
$56$		0			0
$57$			12.0000i			1.58944i
$58$		0			0
$59$		−	4.89898i		−	0.637793i	−0.947790	−	0.318896i	$-0.896688\pi$
$59$		−	4.89898i		−	0.637793i	0.947790	−	0.318896i	$-0.103312\pi$
$60$		0			0
$61$			6.00000i			0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$			6.00000i			0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$		0			0
$63$			7.34847i			0.925820i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.44949			0.299253			0.149626	−	0.988743i	$-0.452193\pi$
$67$	2.44949			0.299253			0.149626	+	0.988743i	$0.452193\pi$
$68$		0			0
$69$			6.00000i			0.722315i
$70$		0			0
$71$	−9.79796			−1.16280			−0.581402	−	0.813617i	$-0.697496\pi$
$71$	−9.79796			−1.16280			−0.581402	+	0.813617i	$0.697496\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$		0			0
$75$	−7.34847	−	9.79796i	−0.848528	−	1.13137i
$76$		0			0
$77$	−12.0000			−1.36753
$78$		0			0
$79$	9.79796			1.10236			0.551178	−	0.834388i	$-0.314178\pi$
$79$	9.79796			1.10236			0.551178	+	0.834388i	$0.314178\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	2.44949			0.268866			0.134433	−	0.990923i	$-0.457079\pi$
$83$	2.44949			0.268866			0.134433	+	0.990923i	$0.457079\pi$
$84$		0			0
$85$	−4.00000	+	8.00000i	−0.433861	+	0.867722i
$86$		0			0
$87$			19.5959i			2.10090i
$88$		0			0
$89$	2.00000			0.212000			0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000			0.212000			0.106000	+	0.994366i	$0.466196\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	24.0000			2.48868
$94$		0			0
$95$	−4.89898	+	9.79796i	−0.502625	+	1.00525i
$96$		0			0
$97$			4.00000i			0.406138i	0.979164	+	0.203069i	$0.0650917\pi$
$97$			4.00000i			0.406138i	−0.979164	+	0.203069i	$0.934908\pi$
$98$		0			0
$99$			14.6969i			1.47710i
$100$		0			0
$101$		−	8.00000i		−	0.796030i	−0.917379	−	0.398015i	$-0.869699\pi$
$101$		−	8.00000i		−	0.796030i	0.917379	−	0.398015i	$-0.130301\pi$
$102$		0			0
$103$		−	2.44949i		−	0.241355i	−0.992692	−	0.120678i	$-0.961493\pi$
$103$		−	2.44949i		−	0.241355i	0.992692	−	0.120678i	$-0.0385068\pi$
$104$		0			0
$105$	−6.00000	+	12.0000i	−0.585540	+	1.17108i
$106$		0			0
$107$	−7.34847			−0.710403			−0.355202	−	0.934790i	$-0.615588\pi$
$107$	−7.34847			−0.710403			−0.355202	+	0.934790i	$0.615588\pi$
$108$		0			0
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$		0			0
$111$	9.79796			0.929981
$112$		0			0
$113$		−	16.0000i		−	1.50515i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$		−	16.0000i		−	1.50515i	0.658505	−	0.752577i	$-0.271189\pi$
$114$		0			0
$115$	−2.44949	+	4.89898i	−0.228416	+	0.456832i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.79796			0.898177
$120$		0			0
$121$	−13.0000			−1.18182
$122$		0			0
$123$	−19.5959			−1.76690
$124$		0			0
$125$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$126$		0			0
$127$		−	7.34847i		−	0.652071i	−0.945357	−	0.326036i	$-0.894287\pi$
$127$		−	7.34847i		−	0.652071i	0.945357	−	0.326036i	$-0.105713\pi$
$128$		0			0
$129$	−18.0000			−1.58481
$130$		0			0
$131$		−	4.89898i		−	0.428026i	−0.976831	−	0.214013i	$-0.931347\pi$
$131$		−	4.89898i		−	0.428026i	0.976831	−	0.214013i	$-0.0686535\pi$
$132$		0			0
$133$	12.0000			1.04053
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.00000i			0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$			8.00000i			0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$		0			0
$139$			4.89898i			0.415526i	0.978179	+	0.207763i	$0.0666183\pi$
$139$			4.89898i			0.415526i	−0.978179	+	0.207763i	$0.933382\pi$
$140$		0			0
$141$			30.0000i			2.52646i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−8.00000	+	16.0000i	−0.664364	+	1.32873i
$146$		0			0
$147$	−2.44949			−0.202031
$148$		0			0
$149$			2.00000i			0.163846i	0.996639	+	0.0819232i	$0.0261062\pi$
$149$			2.00000i			0.163846i	−0.996639	+	0.0819232i	$0.973894\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		−	12.0000i		−	0.970143i
$154$		0			0
$155$	19.5959	+	9.79796i	1.57398	+	0.786991i
$156$		0			0
$157$	20.0000			1.59617			0.798087	−	0.602542i	$-0.205846\pi$
$157$	20.0000			1.59617			0.798087	+	0.602542i	$0.205846\pi$
$158$		0			0
$159$	−19.5959			−1.55406
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$	−2.44949			−0.191859			−0.0959294	−	0.995388i	$-0.530582\pi$
$163$	−2.44949			−0.191859			−0.0959294	+	0.995388i	$0.530582\pi$
$164$		0			0
$165$	−12.0000	+	24.0000i	−0.934199	+	1.86840i
$166$		0			0
$167$		−	2.44949i		−	0.189547i	−0.995499	−	0.0947736i	$-0.969787\pi$
$167$		−	2.44949i		−	0.189547i	0.995499	−	0.0947736i	$-0.0302127\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		−	14.6969i		−	1.12390i
$172$		0			0
$173$	−4.00000			−0.304114			−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000			−0.304114			−0.152057	+	0.988372i	$0.548590\pi$
$174$		0			0
$175$	−9.79796	+	7.34847i	−0.740656	+	0.555492i
$176$		0			0
$177$			12.0000i			0.901975i
$178$		0			0
$179$		−	14.6969i		−	1.09850i	−0.835658	−	0.549250i	$-0.814913\pi$
$179$		−	14.6969i		−	1.09850i	0.835658	−	0.549250i	$-0.185087\pi$
$180$		0			0
$181$		−	8.00000i		−	0.594635i	−0.954779	−	0.297318i	$-0.903908\pi$
$181$		−	8.00000i		−	0.594635i	0.954779	−	0.297318i	$-0.0960920\pi$
$182$		0			0
$183$		−	14.6969i		−	1.08643i
$184$		0			0
$185$	8.00000	+	4.00000i	0.588172	+	0.294086i
$186$		0			0
$187$	19.5959			1.43300
$188$		0			0
$189$		0			0
$190$		0			0
$191$	19.5959			1.41791			0.708955	−	0.705253i	$-0.249167\pi$
$191$	19.5959			1.41791			0.708955	+	0.705253i	$0.249167\pi$
$192$		0			0
$193$			12.0000i			0.863779i	0.901927	+	0.431889i	$0.142153\pi$
$193$			12.0000i			0.863779i	−0.901927	+	0.431889i	$0.857847\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.00000			−0.569976			−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000			−0.569976			−0.284988	+	0.958531i	$0.591990\pi$
$198$		0			0
$199$	−9.79796			−0.694559			−0.347279	−	0.937762i	$-0.612894\pi$
$199$	−9.79796			−0.694559			−0.347279	+	0.937762i	$0.612894\pi$
$200$		0			0
$201$	−6.00000			−0.423207
$202$		0			0
$203$	19.5959			1.37536
$204$		0			0
$205$	−16.0000	−	8.00000i	−1.11749	−	0.558744i
$206$		0			0
$207$		−	7.34847i		−	0.510754i
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$		−	24.4949i		−	1.68630i	−0.537680	−	0.843149i	$-0.680699\pi$
$211$		−	24.4949i		−	1.68630i	0.537680	−	0.843149i	$-0.319301\pi$
$212$		0			0
$213$	24.0000			1.64445
$214$		0			0
$215$	−14.6969	−	7.34847i	−1.00232	−	0.501161i
$216$		0			0
$217$		−	24.0000i		−	1.62923i
$218$		0			0
$219$			9.79796i			0.662085i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	7.34847i		−	0.492090i	−0.969258	−	0.246045i	$-0.920869\pi$
$223$		−	7.34847i		−	0.492090i	0.969258	−	0.246045i	$-0.0791311\pi$
$224$		0			0
$225$	9.00000	+	12.0000i	0.600000	+	0.800000i
$226$		0			0
$227$	−2.44949			−0.162578			−0.0812892	−	0.996691i	$-0.525904\pi$
$227$	−2.44949			−0.162578			−0.0812892	+	0.996691i	$0.525904\pi$
$228$		0			0
$229$			8.00000i			0.528655i	0.964433	+	0.264327i	$0.0851500\pi$
$229$			8.00000i			0.528655i	−0.964433	+	0.264327i	$0.914850\pi$
$230$		0			0
$231$	29.3939			1.93398
$232$		0			0
$233$		−	20.0000i		−	1.31024i	−0.755523	−	0.655122i	$-0.772617\pi$
$233$		−	20.0000i		−	1.31024i	0.755523	−	0.655122i	$-0.227383\pi$
$234$		0			0
$235$	−12.2474	+	24.4949i	−0.798935	+	1.59787i
$236$		0			0
$237$	−24.0000			−1.55897
$238$		0			0
$239$	−9.79796			−0.633777			−0.316889	−	0.948463i	$-0.602638\pi$
$239$	−9.79796			−0.633777			−0.316889	+	0.948463i	$0.602638\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$	22.0454			1.41421
$244$		0			0
$245$	−2.00000	−	1.00000i	−0.127775	−	0.0638877i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−6.00000			−0.380235
$250$		0			0
$251$			4.89898i			0.309221i	0.987976	+	0.154610i	$0.0494122\pi$
$251$			4.89898i			0.309221i	−0.987976	+	0.154610i	$0.950588\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$	9.79796	−	19.5959i	0.613572	−	1.22714i
$256$		0			0
$257$			8.00000i			0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$			8.00000i			0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$		0			0
$259$		−	9.79796i		−	0.608816i
$260$		0			0
$261$		−	24.0000i		−	1.48556i
$262$		0			0
$263$			22.0454i			1.35938i	0.733500	+	0.679689i	$0.237885\pi$
$263$			22.0454i			1.35938i	−0.733500	+	0.679689i	$0.762115\pi$
$264$		0			0
$265$	−16.0000	−	8.00000i	−0.982872	−	0.491436i
$266$		0			0
$267$	−4.89898			−0.299813
$268$		0			0
$269$		−	26.0000i		−	1.58525i	−0.609711	−	0.792624i	$-0.708714\pi$
$269$		−	26.0000i		−	1.58525i	0.609711	−	0.792624i	$-0.291286\pi$
$270$		0			0
$271$	−9.79796			−0.595184			−0.297592	−	0.954693i	$-0.596183\pi$
$271$	−9.79796			−0.595184			−0.297592	+	0.954693i	$0.596183\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−19.5959	+	14.6969i	−1.18168	+	0.886259i
$276$		0			0
$277$	12.0000			0.721010			0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000			0.721010			0.360505	+	0.932757i	$0.382604\pi$
$278$		0			0
$279$	−29.3939			−1.75977
$280$		0			0
$281$	16.0000			0.954480			0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000			0.954480			0.477240	+	0.878773i	$0.341637\pi$
$282$		0			0
$283$	−26.9444			−1.60168			−0.800839	−	0.598880i	$-0.795613\pi$
$283$	−26.9444			−1.60168			−0.800839	+	0.598880i	$0.795613\pi$
$284$		0			0
$285$	12.0000	−	24.0000i	0.710819	−	1.42164i
$286$		0			0
$287$			19.5959i			1.15671i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		−	9.79796i		−	0.574367i
$292$		0			0
$293$	−20.0000			−1.16841			−0.584206	−	0.811605i	$-0.698594\pi$
$293$	−20.0000			−1.16841			−0.584206	+	0.811605i	$0.698594\pi$
$294$		0			0
$295$	−4.89898	+	9.79796i	−0.285230	+	0.570459i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$			18.0000i			1.03750i
$302$		0			0
$303$			19.5959i			1.12576i
$304$		0			0
$305$	6.00000	−	12.0000i	0.343559	−	0.687118i
$306$		0			0
$307$	−17.1464			−0.978598			−0.489299	−	0.872116i	$-0.662747\pi$
$307$	−17.1464			−0.978598			−0.489299	+	0.872116i	$0.662747\pi$
$308$		0			0
$309$			6.00000i			0.341328i
$310$		0			0
$311$	19.5959			1.11118			0.555591	−	0.831456i	$-0.312492\pi$
$311$	19.5959			1.11118			0.555591	+	0.831456i	$0.312492\pi$
$312$		0			0
$313$		−	24.0000i		−	1.35656i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$		−	24.0000i		−	1.35656i	0.734803	−	0.678280i	$-0.237274\pi$
$314$		0			0
$315$	7.34847	−	14.6969i	0.414039	−	0.828079i
$316$		0			0
$317$	−8.00000			−0.449325			−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000			−0.449325			−0.224662	+	0.974437i	$0.572128\pi$
$318$		0			0
$319$	39.1918			2.19432
$320$		0			0
$321$	18.0000			1.00466
$322$		0			0
$323$	−19.5959			−1.09035
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	24.4949i		−	1.35457i
$328$		0			0
$329$	30.0000			1.65395
$330$		0			0
$331$		−	4.89898i		−	0.269272i	−0.990895	−	0.134636i	$-0.957013\pi$
$331$		−	4.89898i		−	0.269272i	0.990895	−	0.134636i	$-0.0429866\pi$
$332$		0			0
$333$	−12.0000			−0.657596
$334$		0			0
$335$	−4.89898	−	2.44949i	−0.267660	−	0.133830i
$336$		0			0
$337$			16.0000i			0.871576i	0.900049	+	0.435788i	$0.143530\pi$
$337$			16.0000i			0.871576i	−0.900049	+	0.435788i	$0.856470\pi$
$338$		0			0
$339$			39.1918i			2.12861i
$340$		0			0
$341$		−	48.0000i		−	2.59935i
$342$		0			0
$343$			19.5959i			1.05808i
$344$		0			0
$345$	6.00000	−	12.0000i	0.323029	−	0.646058i
$346$		0			0
$347$	−12.2474			−0.657477			−0.328739	−	0.944421i	$-0.606624\pi$
$347$	−12.2474			−0.657477			−0.328739	+	0.944421i	$0.606624\pi$
$348$		0			0
$349$		−	24.0000i		−	1.28469i	−0.766415	−	0.642345i	$-0.777962\pi$
$349$		−	24.0000i		−	1.28469i	0.766415	−	0.642345i	$-0.222038\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	8.00000i		−	0.425797i	−0.977074	−	0.212899i	$-0.931710\pi$
$353$		−	8.00000i		−	0.425797i	0.977074	−	0.212899i	$-0.0682904\pi$
$354$		0			0
$355$	19.5959	+	9.79796i	1.04004	+	0.520022i
$356$		0			0
$357$	−24.0000			−1.27021
$358$		0			0
$359$	19.5959			1.03423			0.517116	−	0.855915i	$-0.327005\pi$
$359$	19.5959			1.03423			0.517116	+	0.855915i	$0.327005\pi$
$360$		0			0
$361$	−5.00000			−0.263158
$362$		0			0
$363$	31.8434			1.67134
$364$		0			0
$365$	−4.00000	+	8.00000i	−0.209370	+	0.418739i
$366$		0			0
$367$			31.8434i			1.66221i	0.556115	+	0.831105i	$0.312291\pi$
$367$			31.8434i			1.66221i	−0.556115	+	0.831105i	$0.687709\pi$
$368$		0			0
$369$	24.0000			1.24939
$370$		0			0
$371$			19.5959i			1.01737i
$372$		0			0
$373$	−20.0000			−1.03556			−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000			−1.03556			−0.517780	+	0.855514i	$0.673242\pi$
$374$		0			0
$375$	4.89898	+	26.9444i	0.252982	+	1.39140i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	14.6969i		−	0.754931i	−0.926024	−	0.377466i	$-0.876796\pi$
$379$		−	14.6969i		−	0.754931i	0.926024	−	0.377466i	$-0.123204\pi$
$380$		0			0
$381$			18.0000i			0.922168i
$382$		0			0
$383$		−	26.9444i		−	1.37679i	−0.725334	−	0.688397i	$-0.758315\pi$
$383$		−	26.9444i		−	1.37679i	0.725334	−	0.688397i	$-0.241685\pi$
$384$		0			0
$385$	24.0000	+	12.0000i	1.22315	+	0.611577i
$386$		0			0
$387$	22.0454			1.12063
$388$		0			0
$389$			2.00000i			0.101404i	0.998714	+	0.0507020i	$0.0161459\pi$
$389$			2.00000i			0.101404i	−0.998714	+	0.0507020i	$0.983854\pi$
$390$		0			0
$391$	−9.79796			−0.495504
$392$		0			0
$393$			12.0000i			0.605320i
$394$		0			0
$395$	−19.5959	−	9.79796i	−0.985978	−	0.492989i
$396$		0			0
$397$	8.00000			0.401508			0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000			0.401508			0.200754	+	0.979642i	$0.435661\pi$
$398$		0			0
$399$	−29.3939			−1.47153
$400$		0			0
$401$	10.0000			0.499376			0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000			0.499376			0.249688	+	0.968326i	$0.419672\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	18.0000	+	9.00000i	0.894427	+	0.447214i
$406$		0			0
$407$		−	19.5959i		−	0.971334i
$408$		0			0
$409$	−24.0000			−1.18672			−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000			−1.18672			−0.593362	+	0.804936i	$0.702200\pi$
$410$		0			0
$411$		−	19.5959i		−	0.966595i
$412$		0			0
$413$	12.0000			0.590481
$414$		0			0
$415$	−4.89898	−	2.44949i	−0.240481	−	0.120241i
$416$		0			0
$417$		−	12.0000i		−	0.587643i
$418$		0			0
$419$		−	14.6969i		−	0.717992i	−0.933339	−	0.358996i	$-0.883119\pi$
$419$		−	14.6969i		−	0.717992i	0.933339	−	0.358996i	$-0.116881\pi$
$420$		0			0
$421$			2.00000i			0.0974740i	0.998812	+	0.0487370i	$0.0155196\pi$
$421$			2.00000i			0.0974740i	−0.998812	+	0.0487370i	$0.984480\pi$
$422$		0			0
$423$		−	36.7423i		−	1.78647i
$424$		0			0
$425$	16.0000	−	12.0000i	0.776114	−	0.582086i
$426$		0			0
$427$	−14.6969			−0.711235
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−29.3939			−1.41585			−0.707927	−	0.706286i	$-0.750369\pi$
$431$	−29.3939			−1.41585			−0.707927	+	0.706286i	$0.750369\pi$
$432$		0			0
$433$			36.0000i			1.73005i	0.501729	+	0.865025i	$0.332697\pi$
$433$			36.0000i			1.73005i	−0.501729	+	0.865025i	$0.667303\pi$
$434$		0			0
$435$	19.5959	−	39.1918i	0.939552	−	1.87910i
$436$		0			0
$437$	−12.0000			−0.574038
$438$		0			0
$439$	19.5959			0.935262			0.467631	−	0.883924i	$-0.345108\pi$
$439$	19.5959			0.935262			0.467631	+	0.883924i	$0.345108\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$	−26.9444			−1.28017			−0.640083	−	0.768306i	$-0.721100\pi$
$443$	−26.9444			−1.28017			−0.640083	+	0.768306i	$0.721100\pi$
$444$		0			0
$445$	−4.00000	−	2.00000i	−0.189618	−	0.0948091i
$446$		0			0
$447$		−	4.89898i		−	0.231714i
$448$		0			0
$449$	−32.0000			−1.51017			−0.755087	−	0.655625i	$-0.772405\pi$
$449$	−32.0000			−1.51017			−0.755087	+	0.655625i	$0.772405\pi$
$450$		0			0
$451$			39.1918i			1.84547i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	32.0000i		−	1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$		−	32.0000i		−	1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			8.00000i			0.372597i	0.982493	+	0.186299i	$0.0596492\pi$
$461$			8.00000i			0.372597i	−0.982493	+	0.186299i	$0.940351\pi$
$462$		0			0
$463$		−	31.8434i		−	1.47989i	−0.672669	−	0.739943i	$-0.734852\pi$
$463$		−	31.8434i		−	1.47989i	0.672669	−	0.739943i	$-0.265148\pi$
$464$		0			0
$465$	−48.0000	−	24.0000i	−2.22595	−	1.11297i
$466$		0			0
$467$	36.7423			1.70023			0.850117	−	0.526595i	$-0.176531\pi$
$467$	36.7423			1.70023			0.850117	+	0.526595i	$0.176531\pi$
$468$		0			0
$469$			6.00000i			0.277054i
$470$		0			0
$471$	−48.9898			−2.25733
$472$		0			0
$473$			36.0000i			1.65528i
$474$		0			0
$475$	19.5959	−	14.6969i	0.899122	−	0.674342i
$476$		0			0
$477$	24.0000			1.09888
$478$		0			0
$479$	19.5959			0.895360			0.447680	−	0.894194i	$-0.352250\pi$
$479$	19.5959			0.895360			0.447680	+	0.894194i	$0.352250\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−14.6969			−0.668734
$484$		0			0
$485$	4.00000	−	8.00000i	0.181631	−	0.363261i
$486$		0			0
$487$		−	17.1464i		−	0.776979i	−0.921453	−	0.388489i	$-0.872997\pi$
$487$		−	17.1464i		−	0.776979i	0.921453	−	0.388489i	$-0.127003\pi$
$488$		0			0
$489$	6.00000			0.271329
$490$		0			0
$491$			14.6969i			0.663264i	0.943409	+	0.331632i	$0.107599\pi$
$491$			14.6969i			0.663264i	−0.943409	+	0.331632i	$0.892401\pi$
$492$		0			0
$493$	−32.0000			−1.44121
$494$		0			0
$495$	14.6969	−	29.3939i	0.660578	−	1.32116i
$496$		0			0
$497$		−	24.0000i		−	1.07655i
$498$		0			0
$499$			4.89898i			0.219308i	0.993970	+	0.109654i	$0.0349744\pi$
$499$			4.89898i			0.219308i	−0.993970	+	0.109654i	$0.965026\pi$
$500$		0			0
$501$			6.00000i			0.268060i
$502$		0			0
$503$			17.1464i			0.764521i	0.924055	+	0.382261i	$0.124854\pi$
$503$			17.1464i			0.764521i	−0.924055	+	0.382261i	$0.875146\pi$
$504$		0			0
$505$	−8.00000	+	16.0000i	−0.355995	+	0.711991i
$506$		0			0
$507$	31.8434			1.41421
$508$		0			0
$509$		−	8.00000i		−	0.354594i	−0.984157	−	0.177297i	$-0.943265\pi$
$509$		−	8.00000i		−	0.354594i	0.984157	−	0.177297i	$-0.0567353\pi$
$510$		0			0
$511$	9.79796			0.433436
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.44949	+	4.89898i	−0.107937	+	0.215875i
$516$		0			0
$517$	60.0000			2.63880
$518$		0			0
$519$	9.79796			0.430083
$520$		0			0
$521$	−22.0000			−0.963837			−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000			−0.963837			−0.481919	+	0.876216i	$0.660060\pi$
$522$		0			0
$523$	31.8434			1.39241			0.696207	−	0.717841i	$-0.254870\pi$
$523$	31.8434			1.39241			0.696207	+	0.717841i	$0.254870\pi$
$524$		0			0
$525$	24.0000	−	18.0000i	1.04745	−	0.785584i
$526$		0			0
$527$			39.1918i			1.70722i
$528$		0			0
$529$	17.0000			0.739130
$530$		0			0
$531$		−	14.6969i		−	0.637793i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	14.6969	+	7.34847i	0.635404	+	0.317702i
$536$		0			0
$537$			36.0000i			1.55351i
$538$		0			0
$539$			4.89898i			0.211014i
$540$		0			0
$541$			8.00000i			0.343947i	0.985102	+	0.171973i	$0.0550143\pi$
$541$			8.00000i			0.343947i	−0.985102	+	0.171973i	$0.944986\pi$
$542$		0			0
$543$			19.5959i			0.840941i
$544$		0			0
$545$	10.0000	−	20.0000i	0.428353	−	0.856706i
$546$		0			0
$547$	−22.0454			−0.942594			−0.471297	−	0.881975i	$-0.656214\pi$
$547$	−22.0454			−0.942594			−0.471297	+	0.881975i	$0.656214\pi$
$548$		0			0
$549$			18.0000i			0.768221i
$550$		0			0
$551$	−39.1918			−1.66963
$552$		0			0
$553$			24.0000i			1.02058i
$554$		0			0
$555$	−19.5959	−	9.79796i	−0.831800	−	0.415900i
$556$		0			0
$557$	4.00000			0.169485			0.0847427	−	0.996403i	$-0.472993\pi$
$557$	4.00000			0.169485			0.0847427	+	0.996403i	$0.472993\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−48.0000			−2.02656
$562$		0			0
$563$	−2.44949			−0.103234			−0.0516168	−	0.998667i	$-0.516437\pi$
$563$	−2.44949			−0.103234			−0.0516168	+	0.998667i	$0.516437\pi$
$564$		0			0
$565$	−16.0000	+	32.0000i	−0.673125	+	1.34625i
$566$		0			0
$567$		−	22.0454i		−	0.925820i
$568$		0			0
$569$	−8.00000			−0.335377			−0.167689	−	0.985840i	$-0.553630\pi$
$569$	−8.00000			−0.335377			−0.167689	+	0.985840i	$0.553630\pi$
$570$		0			0
$571$			44.0908i			1.84514i	0.385826	+	0.922572i	$0.373917\pi$
$571$			44.0908i			1.84514i	−0.385826	+	0.922572i	$0.626083\pi$
$572$		0			0
$573$	−48.0000			−2.00523
$574$		0			0
$575$	9.79796	−	7.34847i	0.408603	−	0.306452i
$576$		0			0
$577$		−	24.0000i		−	0.999133i	−0.866276	−	0.499567i	$-0.833493\pi$
$577$		−	24.0000i		−	0.999133i	0.866276	−	0.499567i	$-0.166507\pi$
$578$		0			0
$579$		−	29.3939i		−	1.22157i
$580$		0			0
$581$			6.00000i			0.248922i
$582$		0			0
$583$			39.1918i			1.62316i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.2474			−0.505506			−0.252753	−	0.967531i	$-0.581336\pi$
$587$	−12.2474			−0.505506			−0.252753	+	0.967531i	$0.581336\pi$
$588$		0			0
$589$			48.0000i			1.97781i
$590$		0			0
$591$	19.5959			0.806068
$592$		0			0
$593$		−	8.00000i		−	0.328521i	−0.986417	−	0.164260i	$-0.947476\pi$
$593$		−	8.00000i		−	0.328521i	0.986417	−	0.164260i	$-0.0525237\pi$
$594$		0			0
$595$	−19.5959	−	9.79796i	−0.803354	−	0.401677i
$596$		0			0
$597$	24.0000			0.982255
$598$		0			0
$599$	9.79796			0.400334			0.200167	−	0.979762i	$-0.435852\pi$
$599$	9.79796			0.400334			0.200167	+	0.979762i	$0.435852\pi$
$600$		0			0
$601$	16.0000			0.652654			0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000			0.652654			0.326327	+	0.945257i	$0.394189\pi$
$602$		0			0
$603$	7.34847			0.299253
$604$		0			0
$605$	26.0000	+	13.0000i	1.05705	+	0.528525i
$606$		0			0
$607$			7.34847i			0.298265i	0.988817	+	0.149133i	$0.0476481\pi$
$607$			7.34847i			0.298265i	−0.988817	+	0.149133i	$0.952352\pi$
$608$		0			0
$609$	−48.0000			−1.94506
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−16.0000			−0.646234			−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000			−0.646234			−0.323117	+	0.946359i	$0.604731\pi$
$614$		0			0
$615$	39.1918	+	19.5959i	1.58037	+	0.790184i
$616$		0			0
$617$			28.0000i			1.12724i	0.826035	+	0.563619i	$0.190591\pi$
$617$			28.0000i			1.12724i	−0.826035	+	0.563619i	$0.809409\pi$
$618$		0			0
$619$		−	24.4949i		−	0.984533i	−0.870445	−	0.492267i	$-0.836169\pi$
$619$		−	24.4949i		−	0.984533i	0.870445	−	0.492267i	$-0.163831\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			4.89898i			0.196273i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$	−58.7878			−2.34776
$628$		0			0
$629$			16.0000i			0.637962i
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$			60.0000i			2.38479i
$634$		0			0
$635$	−7.34847	+	14.6969i	−0.291615	+	0.583230i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−29.3939			−1.16280
$640$		0			0
$641$	−16.0000			−0.631962			−0.315981	−	0.948766i	$-0.602334\pi$
$641$	−16.0000			−0.631962			−0.315981	+	0.948766i	$0.602334\pi$
$642$		0			0
$643$	2.44949			0.0965984			0.0482992	−	0.998833i	$-0.484620\pi$
$643$	2.44949			0.0965984			0.0482992	+	0.998833i	$0.484620\pi$
$644$		0			0
$645$	36.0000	+	18.0000i	1.41750	+	0.708749i
$646$		0			0
$647$			22.0454i			0.866694i	0.901227	+	0.433347i	$0.142668\pi$
$647$			22.0454i			0.866694i	−0.901227	+	0.433347i	$0.857332\pi$
$648$		0			0
$649$	24.0000			0.942082
$650$		0			0
$651$			58.7878i			2.30407i
$652$		0			0
$653$	16.0000			0.626128			0.313064	−	0.949732i	$-0.398644\pi$
$653$	16.0000			0.626128			0.313064	+	0.949732i	$0.398644\pi$
$654$		0			0
$655$	−4.89898	+	9.79796i	−0.191419	+	0.382838i
$656$		0			0
$657$		−	12.0000i		−	0.468165i
$658$		0			0
$659$		−	14.6969i		−	0.572511i	−0.958153	−	0.286256i	$-0.907589\pi$
$659$		−	14.6969i		−	0.572511i	0.958153	−	0.286256i	$-0.0924107\pi$
$660$		0			0
$661$			30.0000i			1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$			30.0000i			1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−24.0000	−	12.0000i	−0.930680	−	0.465340i
$666$		0			0
$667$	−19.5959			−0.758757
$668$		0			0
$669$			18.0000i			0.695920i
$670$		0			0
$671$	−29.3939			−1.13474
$672$		0			0
$673$			36.0000i			1.38770i	0.720121	+	0.693849i	$0.244086\pi$
$673$			36.0000i			1.38770i	−0.720121	+	0.693849i	$0.755914\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−32.0000			−1.22986			−0.614930	−	0.788582i	$-0.710816\pi$
$677$	−32.0000			−1.22986			−0.614930	+	0.788582i	$0.710816\pi$
$678$		0			0
$679$	−9.79796			−0.376011
$680$		0			0
$681$	6.00000			0.229920
$682$		0			0
$683$	7.34847			0.281181			0.140591	−	0.990068i	$-0.455100\pi$
$683$	7.34847			0.281181			0.140591	+	0.990068i	$0.455100\pi$
$684$		0			0
$685$	8.00000	−	16.0000i	0.305664	−	0.611329i
$686$		0			0
$687$		−	19.5959i		−	0.747631i
$688$		0			0
$689$		0			0
$690$		0			0
$691$			14.6969i			0.559098i	0.960131	+	0.279549i	$0.0901849\pi$
$691$			14.6969i			0.559098i	−0.960131	+	0.279549i	$0.909815\pi$
$692$		0			0
$693$	−36.0000			−1.36753
$694$		0			0
$695$	4.89898	−	9.79796i	0.185829	−	0.371658i
$696$		0			0
$697$		−	32.0000i		−	1.21209i
$698$		0			0
$699$			48.9898i			1.85296i
$700$		0			0
$701$			26.0000i			0.982006i	0.871158	+	0.491003i	$0.163370\pi$
$701$			26.0000i			0.982006i	−0.871158	+	0.491003i	$0.836630\pi$
$702$		0			0
$703$			19.5959i			0.739074i
$704$		0			0
$705$	30.0000	−	60.0000i	1.12987	−	2.25973i
$706$		0			0
$707$	19.5959			0.736980
$708$		0			0
$709$		−	40.0000i		−	1.50223i	−0.660171	−	0.751116i	$-0.729516\pi$
$709$		−	40.0000i		−	1.50223i	0.660171	−	0.751116i	$-0.270484\pi$
$710$		0			0
$711$	29.3939			1.10236
$712$		0			0
$713$			24.0000i			0.898807i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	24.0000			0.896296
$718$		0			0
$719$	48.9898			1.82701			0.913506	−	0.406826i	$-0.133365\pi$
$719$	48.9898			1.82701			0.913506	+	0.406826i	$0.133365\pi$
$720$		0			0
$721$	6.00000			0.223452
$722$		0			0
$723$		0			0
$724$		0			0
$725$	32.0000	−	24.0000i	1.18845	−	0.891338i
$726$		0			0
$727$		−	36.7423i		−	1.36270i	−0.731959	−	0.681349i	$-0.761394\pi$
$727$		−	36.7423i		−	1.36270i	0.731959	−	0.681349i	$-0.238606\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		−	29.3939i		−	1.08717i
$732$		0			0
$733$	−12.0000			−0.443230			−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000			−0.443230			−0.221615	+	0.975134i	$0.571133\pi$
$734$		0			0
$735$	4.89898	+	2.44949i	0.180702	+	0.0903508i
$736$		0			0
$737$			12.0000i			0.442026i
$738$		0			0
$739$		−	14.6969i		−	0.540636i	−0.962771	−	0.270318i	$-0.912871\pi$
$739$		−	14.6969i		−	0.540636i	0.962771	−	0.270318i	$-0.0871288\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	17.1464i		−	0.629041i	−0.949251	−	0.314521i	$-0.898156\pi$
$743$		−	17.1464i		−	0.629041i	0.949251	−	0.314521i	$-0.101844\pi$
$744$		0			0
$745$	2.00000	−	4.00000i	0.0732743	−	0.146549i
$746$		0			0
$747$	7.34847			0.268866
$748$		0			0
$749$		−	18.0000i		−	0.657706i
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		−	12.0000i		−	0.437304i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−12.0000			−0.436147			−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000			−0.436147			−0.218074	+	0.975932i	$0.569977\pi$
$758$		0			0
$759$	−29.3939			−1.06693
$760$		0			0
$761$	22.0000			0.797499			0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000			0.797499			0.398750	+	0.917060i	$0.369444\pi$
$762$		0			0
$763$	−24.4949			−0.886775
$764$		0			0
$765$	−12.0000	+	24.0000i	−0.433861	+	0.867722i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−26.0000			−0.937584			−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000			−0.937584			−0.468792	+	0.883309i	$0.655311\pi$
$770$		0			0
$771$		−	19.5959i		−	0.705730i
$772$		0			0
$773$	8.00000			0.287740			0.143870	−	0.989597i	$-0.454045\pi$
$773$	8.00000			0.287740			0.143870	+	0.989597i	$0.454045\pi$
$774$		0			0
$775$	−29.3939	−	39.1918i	−1.05586	−	1.40781i
$776$		0			0
$777$			24.0000i			0.860995i
$778$		0			0
$779$		−	39.1918i		−	1.40419i
$780$		0			0
$781$		−	48.0000i		−	1.71758i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−40.0000	−	20.0000i	−1.42766	−	0.713831i
$786$		0			0
$787$	41.6413			1.48435			0.742176	−	0.670205i	$-0.233794\pi$
$787$	41.6413			1.48435			0.742176	+	0.670205i	$0.233794\pi$
$788$		0			0
$789$		−	54.0000i		−	1.92245i
$790$		0			0
$791$	39.1918			1.39350
$792$		0			0
$793$		0			0
$794$		0			0
$795$	39.1918	+	19.5959i	1.38999	+	0.694996i
$796$		0			0
$797$	−8.00000			−0.283375			−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000			−0.283375			−0.141687	+	0.989911i	$0.545253\pi$
$798$		0			0
$799$	−48.9898			−1.73313
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$	19.5959			0.691525
$804$		0			0
$805$	−12.0000	−	6.00000i	−0.422944	−	0.211472i
$806$		0			0
$807$			63.6867i			2.24188i
$808$		0			0
$809$	10.0000			0.351581			0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000			0.351581			0.175791	+	0.984428i	$0.443752\pi$
$810$		0			0
$811$		−	53.8888i		−	1.89229i	−0.323742	−	0.946145i	$-0.604941\pi$
$811$		−	53.8888i		−	1.89229i	0.323742	−	0.946145i	$-0.395059\pi$
$812$		0			0
$813$	24.0000			0.841717
$814$		0			0
$815$	4.89898	+	2.44949i	0.171604	+	0.0858019i
$816$		0			0
$817$		−	36.0000i		−	1.25948i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			14.0000i			0.488603i	0.969699	+	0.244302i	$0.0785587\pi$
$821$			14.0000i			0.488603i	−0.969699	+	0.244302i	$0.921441\pi$
$822$		0			0
$823$		−	22.0454i		−	0.768455i	−0.923239	−	0.384227i	$-0.874468\pi$
$823$		−	22.0454i		−	0.768455i	0.923239	−	0.384227i	$-0.125532\pi$
$824$		0			0
$825$	48.0000	−	36.0000i	1.67115	−	1.25336i
$826$		0			0
$827$	46.5403			1.61836			0.809182	−	0.587557i	$-0.199910\pi$
$827$	46.5403			1.61836			0.809182	+	0.587557i	$0.199910\pi$
$828$		0			0
$829$			22.0000i			0.764092i	0.924143	+	0.382046i	$0.124780\pi$
$829$			22.0000i			0.764092i	−0.924143	+	0.382046i	$0.875220\pi$
$830$		0			0
$831$	−29.3939			−1.01966
$832$		0			0
$833$		−	4.00000i		−	0.138592i
$834$		0			0
$835$	−2.44949	+	4.89898i	−0.0847681	+	0.169536i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−39.1918			−1.35305			−0.676526	−	0.736419i	$-0.736515\pi$
$839$	−39.1918			−1.35305			−0.676526	+	0.736419i	$0.736515\pi$
$840$		0			0
$841$	−35.0000			−1.20690
$842$		0			0
$843$	−39.1918			−1.34984
$844$		0			0
$845$	26.0000	+	13.0000i	0.894427	+	0.447214i
$846$		0			0
$847$		−	31.8434i		−	1.09415i
$848$		0			0
$849$	66.0000			2.26511
$850$		0			0
$851$			9.79796i			0.335870i
$852$		0			0
$853$	−32.0000			−1.09566			−0.547830	−	0.836590i	$-0.684546\pi$
$853$	−32.0000			−1.09566			−0.547830	+	0.836590i	$0.684546\pi$
$854$		0			0
$855$	−14.6969	+	29.3939i	−0.502625	+	1.00525i
$856$		0			0
$857$			32.0000i			1.09310i	0.837427	+	0.546550i	$0.184059\pi$
$857$			32.0000i			1.09310i	−0.837427	+	0.546550i	$0.815941\pi$
$858$		0			0
$859$		−	24.4949i		−	0.835755i	−0.908503	−	0.417878i	$-0.862774\pi$
$859$		−	24.4949i		−	0.835755i	0.908503	−	0.417878i	$-0.137226\pi$
$860$		0			0
$861$		−	48.0000i		−	1.63584i
$862$		0			0
$863$		−	51.4393i		−	1.75101i	−0.483206	−	0.875507i	$-0.660528\pi$
$863$		−	51.4393i		−	1.75101i	0.483206	−	0.875507i	$-0.339472\pi$
$864$		0			0
$865$	8.00000	+	4.00000i	0.272008	+	0.136004i
$866$		0			0
$867$	−2.44949			−0.0831890
$868$		0			0
$869$			48.0000i			1.62829i
$870$		0			0
$871$		0			0
$872$		0			0
$873$			12.0000i			0.406138i
$874$		0			0
$875$	26.9444	−	4.89898i	0.910887	−	0.165616i
$876$		0			0
$877$	28.0000			0.945493			0.472746	−	0.881199i	$-0.343263\pi$
$877$	28.0000			0.945493			0.472746	+	0.881199i	$0.343263\pi$
$878$		0			0
$879$	48.9898			1.65238
$880$		0			0
$881$	−8.00000			−0.269527			−0.134763	−	0.990878i	$-0.543027\pi$
$881$	−8.00000			−0.269527			−0.134763	+	0.990878i	$0.543027\pi$
$882$		0			0
$883$	36.7423			1.23648			0.618239	−	0.785990i	$-0.287846\pi$
$883$	36.7423			1.23648			0.618239	+	0.785990i	$0.287846\pi$
$884$		0			0
$885$	12.0000	−	24.0000i	0.403376	−	0.806751i
$886$		0			0
$887$			2.44949i			0.0822458i	0.999154	+	0.0411229i	$0.0130935\pi$
$887$			2.44949i			0.0822458i	−0.999154	+	0.0411229i	$0.986906\pi$
$888$		0			0
$889$	18.0000			0.603701
$890$		0			0
$891$		−	44.0908i		−	1.47710i
$892$		0			0
$893$	−60.0000			−2.00782
$894$		0			0
$895$	−14.6969	+	29.3939i	−0.491264	+	0.982529i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			78.3837i			2.61424i
$900$		0			0
$901$		−	32.0000i		−	1.06607i
$902$		0			0
$903$		−	44.0908i		−	1.46725i
$904$		0			0
$905$	−8.00000	+	16.0000i	−0.265929	+	0.531858i
$906$		0			0
$907$	12.2474			0.406670			0.203335	−	0.979109i	$-0.434822\pi$
$907$	12.2474			0.406670			0.203335	+	0.979109i	$0.434822\pi$
$908$		0			0
$909$		−	24.0000i		−	0.796030i
$910$		0			0
$911$	29.3939			0.973863			0.486931	−	0.873440i	$-0.338116\pi$
$911$	29.3939			0.973863			0.486931	+	0.873440i	$0.338116\pi$
$912$		0			0
$913$			12.0000i			0.397142i
$914$		0			0
$915$	−14.6969	+	29.3939i	−0.485866	+	0.971732i
$916$		0			0
$917$	12.0000			0.396275
$918$		0			0
$919$	48.9898			1.61602			0.808012	−	0.589166i	$-0.200544\pi$
$919$	48.9898			1.61602			0.808012	+	0.589166i	$0.200544\pi$
$920$		0			0
$921$	42.0000			1.38395
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−12.0000	−	16.0000i	−0.394558	−	0.526077i
$926$		0			0
$927$		−	7.34847i		−	0.241355i
$928$		0			0
$929$	−16.0000			−0.524943			−0.262471	−	0.964940i	$-0.584538\pi$
$929$	−16.0000			−0.524943			−0.262471	+	0.964940i	$0.584538\pi$
$930$		0			0
$931$		−	4.89898i		−	0.160558i
$932$		0			0
$933$	−48.0000			−1.57145
$934$		0			0
$935$	−39.1918	−	19.5959i	−1.28171	−	0.640855i
$936$		0			0
$937$		−	60.0000i		−	1.96011i	−0.198715	−	0.980057i	$-0.563677\pi$
$937$		−	60.0000i		−	1.96011i	0.198715	−	0.980057i	$-0.436323\pi$
$938$		0			0
$939$			58.7878i			1.91847i
$940$		0			0
$941$			8.00000i			0.260793i	0.991462	+	0.130396i	$0.0416250\pi$
$941$			8.00000i			0.260793i	−0.991462	+	0.130396i	$0.958375\pi$
$942$		0			0
$943$		−	19.5959i		−	0.638131i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.1464			−0.557184			−0.278592	−	0.960410i	$-0.589868\pi$
$947$	−17.1464			−0.557184			−0.278592	+	0.960410i	$0.589868\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	19.5959			0.635441
$952$		0			0
$953$		−	32.0000i		−	1.03658i	−0.855204	−	0.518291i	$-0.826568\pi$
$953$		−	32.0000i		−	1.03658i	0.855204	−	0.518291i	$-0.173432\pi$
$954$		0			0
$955$	−39.1918	−	19.5959i	−1.26822	−	0.634109i
$956$		0			0
$957$	−96.0000			−3.10324
$958$		0			0
$959$	−19.5959			−0.632785
$960$		0			0
$961$	65.0000			2.09677
$962$		0			0
$963$	−22.0454			−0.710403
$964$		0			0
$965$	12.0000	−	24.0000i	0.386294	−	0.772587i
$966$		0			0
$967$			17.1464i			0.551392i	0.961245	+	0.275696i	$0.0889083\pi$
$967$			17.1464i			0.551392i	−0.961245	+	0.275696i	$0.911092\pi$
$968$		0			0
$969$	48.0000			1.54198
$970$		0			0
$971$		−	24.4949i		−	0.786079i	−0.919522	−	0.393039i	$-0.871424\pi$
$971$		−	24.4949i		−	0.786079i	0.919522	−	0.393039i	$-0.128576\pi$
$972$		0			0
$973$	−12.0000			−0.384702
$974$		0			0
$975$		0			0
$976$		0			0
$977$			4.00000i			0.127971i	0.997951	+	0.0639857i	$0.0203812\pi$
$977$			4.00000i			0.127971i	−0.997951	+	0.0639857i	$0.979619\pi$
$978$		0			0
$979$			9.79796i			0.313144i
$980$		0			0
$981$			30.0000i			0.957826i
$982$		0			0
$983$			22.0454i			0.703139i	0.936162	+	0.351570i	$0.114352\pi$
$983$			22.0454i			0.703139i	−0.936162	+	0.351570i	$0.885648\pi$
$984$		0			0
$985$	16.0000	+	8.00000i	0.509802	+	0.254901i
$986$		0			0
$987$	−73.4847			−2.33904
$988$		0			0
$989$		−	18.0000i		−	0.572367i
$990$		0			0
$991$	39.1918			1.24497			0.622485	−	0.782632i	$-0.286123\pi$
$991$	39.1918			1.24497			0.622485	+	0.782632i	$0.286123\pi$
$992$		0			0
$993$			12.0000i			0.380808i
$994$		0			0
$995$	19.5959	+	9.79796i	0.621232	+	0.310616i
$996$		0			0
$997$	56.0000			1.77354			0.886769	−	0.462213i	$-0.152944\pi$
$997$	56.0000			1.77354			0.886769	+	0.462213i	$0.152944\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	640.2.f.c.449.1	✓	4
4.3	odd	2	inner	640.2.f.c.449.3	yes	4
5.2	odd	4		3200.2.d.l.1601.3		4
5.3	odd	4		3200.2.d.k.1601.2		4
5.4	even	2		640.2.f.g.449.3	yes	4
8.3	odd	2		640.2.f.g.449.2	yes	4
8.5	even	2		640.2.f.g.449.4	yes	4
16.3	odd	4		1280.2.c.e.769.4		4
16.5	even	4		1280.2.c.m.769.3		4
16.11	odd	4		1280.2.c.m.769.1		4
16.13	even	4		1280.2.c.e.769.2		4
20.3	even	4		3200.2.d.k.1601.3		4
20.7	even	4		3200.2.d.l.1601.2		4
20.19	odd	2		640.2.f.g.449.1	yes	4
40.3	even	4		3200.2.d.k.1601.1		4
40.13	odd	4		3200.2.d.k.1601.4		4
40.19	odd	2	inner	640.2.f.c.449.4	yes	4
40.27	even	4		3200.2.d.l.1601.4		4
40.29	even	2	inner	640.2.f.c.449.2	yes	4
40.37	odd	4		3200.2.d.l.1601.1		4
80.3	even	4		6400.2.a.bu.1.1		2
80.13	odd	4		6400.2.a.bu.1.2		2
80.19	odd	4		1280.2.c.e.769.1		4
80.27	even	4		6400.2.a.bx.1.1		2
80.29	even	4		1280.2.c.e.769.3		4
80.37	odd	4		6400.2.a.bx.1.2		2
80.43	even	4		6400.2.a.bv.1.2		2
80.53	odd	4		6400.2.a.bv.1.1		2
80.59	odd	4		1280.2.c.m.769.4		4
80.67	even	4		6400.2.a.bw.1.2		2
80.69	even	4		1280.2.c.m.769.2		4
80.77	odd	4		6400.2.a.bw.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.f.c.449.1	✓	4	1.1	even	1	trivial
640.2.f.c.449.2	yes	4	40.29	even	2	inner
640.2.f.c.449.3	yes	4	4.3	odd	2	inner
640.2.f.c.449.4	yes	4	40.19	odd	2	inner
640.2.f.g.449.1	yes	4	20.19	odd	2
640.2.f.g.449.2	yes	4	8.3	odd	2
640.2.f.g.449.3	yes	4	5.4	even	2
640.2.f.g.449.4	yes	4	8.5	even	2
1280.2.c.e.769.1		4	80.19	odd	4
1280.2.c.e.769.2		4	16.13	even	4
1280.2.c.e.769.3		4	80.29	even	4
1280.2.c.e.769.4		4	16.3	odd	4
1280.2.c.m.769.1		4	16.11	odd	4
1280.2.c.m.769.2		4	80.69	even	4
1280.2.c.m.769.3		4	16.5	even	4
1280.2.c.m.769.4		4	80.59	odd	4
3200.2.d.k.1601.1		4	40.3	even	4
3200.2.d.k.1601.2		4	5.3	odd	4
3200.2.d.k.1601.3		4	20.3	even	4
3200.2.d.k.1601.4		4	40.13	odd	4
3200.2.d.l.1601.1		4	40.37	odd	4
3200.2.d.l.1601.2		4	20.7	even	4
3200.2.d.l.1601.3		4	5.2	odd	4
3200.2.d.l.1601.4		4	40.27	even	4
6400.2.a.bu.1.1		2	80.3	even	4
6400.2.a.bu.1.2		2	80.13	odd	4
6400.2.a.bv.1.1		2	80.53	odd	4
6400.2.a.bv.1.2		2	80.43	even	4
6400.2.a.bw.1.1		2	80.77	odd	4
6400.2.a.bw.1.2		2	80.67	even	4
6400.2.a.bx.1.1		2	80.27	even	4
6400.2.a.bx.1.2		2	80.37	odd	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:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	640.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.44949 q^{3} +(-2.00000 - 1.00000i) q^{5} +2.44949i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-2.44949 q^{3} +(-2.00000 - 1.00000i) q^{5} +2.44949i q^{7} +3.00000 q^{9} +4.89898i q^{11} +(4.89898 + 2.44949i) q^{15} -4.00000i q^{17} -4.89898i q^{19} -6.00000i q^{21} -2.44949i q^{23} +(3.00000 + 4.00000i) q^{25} -8.00000i q^{29} -9.79796 q^{31} -12.0000i q^{33} +(2.44949 - 4.89898i) q^{35} -4.00000 q^{37} +8.00000 q^{41} +7.34847 q^{43} +(-6.00000 - 3.00000i) q^{45} -12.2474i q^{47} +1.00000 q^{49} +9.79796i q^{51} +8.00000 q^{53} +(4.89898 - 9.79796i) q^{55} +12.0000i q^{57} -4.89898i q^{59} +6.00000i q^{61} +7.34847i q^{63} +2.44949 q^{67} +6.00000i q^{69} -9.79796 q^{71} -4.00000i q^{73} +(-7.34847 - 9.79796i) q^{75} -12.0000 q^{77} +9.79796 q^{79} -9.00000 q^{81} +2.44949 q^{83} +(-4.00000 + 8.00000i) q^{85} +19.5959i q^{87} +2.00000 q^{89} +24.0000 q^{93} +(-4.89898 + 9.79796i) q^{95} +4.00000i q^{97} +14.6969i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5} + 12 q^{9}+O(q^{10}) \) 4 * q - 8 * q^5 + 12 * q^9	\( 4 q - 8 q^{5} + 12 q^{9} + 12 q^{25} - 16 q^{37} + 32 q^{41} - 24 q^{45} + 4 q^{49} + 32 q^{53} - 48 q^{77} - 36 q^{81} - 16 q^{85} + 8 q^{89} + 96 q^{93}+O(q^{100}) \) 4 * q - 8 * q^5 + 12 * q^9 + 12 * q^25 - 16 * q^37 + 32 * q^41 - 24 * q^45 + 4 * q^49 + 32 * q^53 - 48 * q^77 - 36 * q^81 - 16 * q^85 + 8 * q^89 + 96 * q^93

Introduction

L-functions

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Database

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