LMFDB - Embedded newform 880.2.m.g.879.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,2,Mod(879,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, 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("880.879");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.m (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:	$7.02683537787$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 2x^{6} + 8x^{5} + 13x^{4} - 44x^{3} + 164x^{2} - 140x + 145 $ x^8 - 4x^7 + 2x^6 + 8x^5 + 13x^4 - 44x^3 + 164x^2 - 140*x + 145
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			879.7
Root			$2.84278 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	880.879
Dual form			880.2.m.g.879.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{3} +(1.00000 + 2.00000i) q^{5} -4.47214i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+2.44949 q^{3} +(1.00000 + 2.00000i) q^{5} -4.47214i q^{7} +3.00000 q^{9} +(2.23607 - 2.44949i) q^{11} +5.47723 q^{13} +(2.44949 + 4.89898i) q^{15} -5.47723 q^{17} -4.47214 q^{19} -10.9545i q^{21} +2.44949 q^{23} +(-3.00000 + 4.00000i) q^{25} +4.89898i q^{31} +(5.47723 - 6.00000i) q^{33} +(8.94427 - 4.47214i) q^{35} -8.00000i q^{37} +13.4164 q^{39} +10.9545i q^{41} -4.47214i q^{43} +(3.00000 + 6.00000i) q^{45} +7.34847 q^{47} -13.0000 q^{49} -13.4164 q^{51} +4.00000i q^{53} +(7.13505 + 2.02265i) q^{55} -10.9545 q^{57} +9.79796i q^{59} +10.9545i q^{61} -13.4164i q^{63} +(5.47723 + 10.9545i) q^{65} +7.34847 q^{67} +6.00000 q^{69} -5.47723 q^{73} +(-7.34847 + 9.79796i) q^{75} +(-10.9545 - 10.0000i) q^{77} -4.47214 q^{79} -9.00000 q^{81} +4.47214i q^{83} +(-5.47723 - 10.9545i) q^{85} +4.00000 q^{89} -24.4949i q^{91} +12.0000i q^{93} +(-4.47214 - 8.94427i) q^{95} -12.0000i q^{97} +(6.70820 - 7.34847i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5} + 24 q^{9}+O(q^{10}) $ 8 * q + 8 * q^5 + 24 * q^9	$ 8 q + 8 q^{5} + 24 q^{9} - 24 q^{25} + 24 q^{45} - 104 q^{49} + 48 q^{69} - 72 q^{81} + 32 q^{89}+O(q^{100}) $ 8 * q + 8 * q^5 + 24 * q^9 - 24 * q^25 + 24 * q^45 - 104 * q^49 + 48 * q^69 - 72 * q^81 + 32 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	2.44949			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.44949			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$4$		0			0
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$		0			0
$7$		−	4.47214i		−	1.69031i	−0.534522	−	0.845154i	$-0.679509\pi$
$7$		−	4.47214i		−	1.69031i	0.534522	−	0.845154i	$-0.320491\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$	2.23607	−	2.44949i	0.674200	−	0.738549i
$11$	2.23607	−	2.44949i	0.674200	−	0.738549i
$12$		0			0
$13$	5.47723			1.51911			0.759555	−	0.650444i	$-0.225417\pi$
$13$	5.47723			1.51911			0.759555	+	0.650444i	$0.225417\pi$
$14$		0			0
$15$	2.44949	+	4.89898i	0.632456	+	1.26491i
$16$		0			0
$17$	−5.47723			−1.32842			−0.664211	−	0.747545i	$-0.731232\pi$
$17$	−5.47723			−1.32842			−0.664211	+	0.747545i	$0.731232\pi$
$18$		0			0
$19$	−4.47214			−1.02598			−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214			−1.02598			−0.512989	+	0.858395i	$0.671462\pi$
$20$		0			0
$21$		−	10.9545i		−	2.39046i
$22$		0			0
$23$	2.44949			0.510754			0.255377	−	0.966842i	$-0.417800\pi$
$23$	2.44949			0.510754			0.255377	+	0.966842i	$0.417800\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$			4.89898i			0.879883i	0.898027	+	0.439941i	$0.145001\pi$
$31$			4.89898i			0.879883i	−0.898027	+	0.439941i	$0.854999\pi$
$32$		0			0
$33$	5.47723	−	6.00000i	0.953463	−	1.04447i
$34$		0			0
$35$	8.94427	−	4.47214i	1.51186	−	0.755929i
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$	13.4164			2.14834
$40$		0			0
$41$			10.9545i			1.71080i	0.517970	+	0.855399i	$0.326688\pi$
$41$			10.9545i			1.71080i	−0.517970	+	0.855399i	$0.673312\pi$
$42$		0			0
$43$		−	4.47214i		−	0.681994i	−0.940064	−	0.340997i	$-0.889235\pi$
$43$		−	4.47214i		−	0.681994i	0.940064	−	0.340997i	$-0.110765\pi$
$44$		0			0
$45$	3.00000	+	6.00000i	0.447214	+	0.894427i
$46$		0			0
$47$	7.34847			1.07188			0.535942	−	0.844255i	$-0.319956\pi$
$47$	7.34847			1.07188			0.535942	+	0.844255i	$0.319956\pi$
$48$		0			0
$49$	−13.0000			−1.85714
$50$		0			0
$51$	−13.4164			−1.87867
$52$		0			0
$53$			4.00000i			0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$			4.00000i			0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$	7.13505	+	2.02265i	0.962090	+	0.272734i
$56$		0			0
$57$	−10.9545			−1.45095
$58$		0			0
$59$			9.79796i			1.27559i	0.770208	+	0.637793i	$0.220152\pi$
$59$			9.79796i			1.27559i	−0.770208	+	0.637793i	$0.779848\pi$
$60$		0			0
$61$			10.9545i			1.40257i	0.712879	+	0.701287i	$0.247391\pi$
$61$			10.9545i			1.40257i	−0.712879	+	0.701287i	$0.752609\pi$
$62$		0			0
$63$		−	13.4164i		−	1.69031i
$64$		0			0
$65$	5.47723	+	10.9545i	0.679366	+	1.35873i
$66$		0			0
$67$	7.34847			0.897758			0.448879	−	0.893592i	$-0.351823\pi$
$67$	7.34847			0.897758			0.448879	+	0.893592i	$0.351823\pi$
$68$		0			0
$69$	6.00000			0.722315
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−5.47723			−0.641061			−0.320530	−	0.947238i	$-0.603861\pi$
$73$	−5.47723			−0.641061			−0.320530	+	0.947238i	$0.603861\pi$
$74$		0			0
$75$	−7.34847	+	9.79796i	−0.848528	+	1.13137i
$76$		0			0
$77$	−10.9545	−	10.0000i	−1.24838	−	1.13961i
$78$		0			0
$79$	−4.47214			−0.503155			−0.251577	−	0.967837i	$-0.580949\pi$
$79$	−4.47214			−0.503155			−0.251577	+	0.967837i	$0.580949\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$			4.47214i			0.490881i	0.969412	+	0.245440i	$0.0789325\pi$
$83$			4.47214i			0.490881i	−0.969412	+	0.245440i	$0.921067\pi$
$84$		0			0
$85$	−5.47723	−	10.9545i	−0.594089	−	1.18818i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.00000			0.423999			0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000			0.423999			0.212000	+	0.977270i	$0.432002\pi$
$90$		0			0
$91$		−	24.4949i		−	2.56776i
$92$		0			0
$93$			12.0000i			1.24434i
$94$		0			0
$95$	−4.47214	−	8.94427i	−0.458831	−	0.917663i
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$		0			0
$99$	6.70820	−	7.34847i	0.674200	−	0.738549i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−2.44949			−0.241355			−0.120678	−	0.992692i	$-0.538507\pi$
$103$	−2.44949			−0.241355			−0.120678	+	0.992692i	$0.538507\pi$
$104$		0			0
$105$	21.9089	−	10.9545i	2.13809	−	1.06904i
$106$		0			0
$107$			4.47214i			0.432338i	0.976356	+	0.216169i	$0.0693562\pi$
$107$			4.47214i			0.432338i	−0.976356	+	0.216169i	$0.930644\pi$
$108$		0			0
$109$			10.9545i			1.04925i	0.851335	+	0.524623i	$0.175794\pi$
$109$			10.9545i			1.04925i	−0.851335	+	0.524623i	$0.824206\pi$
$110$		0			0
$111$		−	19.5959i		−	1.85996i
$112$		0			0
$113$		−	4.00000i		−	0.376288i	−0.982141	−	0.188144i	$-0.939753\pi$
$113$		−	4.00000i		−	0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$		0			0
$115$	2.44949	+	4.89898i	0.228416	+	0.456832i
$116$		0			0
$117$	16.4317			1.51911
$118$		0			0
$119$			24.4949i			2.24544i
$120$		0			0
$121$	−1.00000	−	10.9545i	−0.0909091	−	0.995859i
$122$		0			0
$123$			26.8328i			2.41943i
$124$		0			0
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		0			0
$127$			4.47214i			0.396838i	0.980117	+	0.198419i	$0.0635807\pi$
$127$			4.47214i			0.396838i	−0.980117	+	0.198419i	$0.936419\pi$
$128$		0			0
$129$		−	10.9545i		−	0.964486i
$130$		0			0
$131$	−8.94427			−0.781465			−0.390732	−	0.920504i	$-0.627778\pi$
$131$	−8.94427			−0.781465			−0.390732	+	0.920504i	$0.627778\pi$
$132$		0			0
$133$			20.0000i			1.73422i
$134$		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
$139$	8.94427			0.758643			0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427			0.758643			0.379322	+	0.925265i	$0.376157\pi$
$140$		0			0
$141$	18.0000			1.51587
$142$		0			0
$143$	12.2474	−	13.4164i	1.02418	−	1.12194i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−31.8434			−2.62640
$148$		0			0
$149$		−	10.9545i		−	0.897424i	−0.893677	−	0.448712i	$-0.851883\pi$
$149$		−	10.9545i		−	0.897424i	0.893677	−	0.448712i	$-0.148117\pi$
$150$		0			0
$151$	−17.8885			−1.45575			−0.727875	−	0.685710i	$-0.759492\pi$
$151$	−17.8885			−1.45575			−0.727875	+	0.685710i	$0.759492\pi$
$152$		0			0
$153$	−16.4317			−1.32842
$154$		0			0
$155$	−9.79796	+	4.89898i	−0.786991	+	0.393496i
$156$		0			0
$157$		−	12.0000i		−	0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$		−	12.0000i		−	0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$		0			0
$159$			9.79796i			0.777029i
$160$		0			0
$161$		−	10.9545i		−	0.863332i
$162$		0			0
$163$	−22.0454			−1.72673			−0.863365	−	0.504580i	$-0.831647\pi$
$163$	−22.0454			−1.72673			−0.863365	+	0.504580i	$0.831647\pi$
$164$		0			0
$165$	17.4772	+	4.95445i	1.36060	+	0.385703i
$166$		0			0
$167$		−	13.4164i		−	1.03819i	−0.854716	−	0.519096i	$-0.826269\pi$
$167$		−	13.4164i		−	1.03819i	0.854716	−	0.519096i	$-0.173731\pi$
$168$		0			0
$169$	17.0000			1.30769
$170$		0			0
$171$	−13.4164			−1.02598
$172$		0			0
$173$	5.47723			0.416426			0.208213	−	0.978084i	$-0.433235\pi$
$173$	5.47723			0.416426			0.208213	+	0.978084i	$0.433235\pi$
$174$		0			0
$175$	17.8885	+	13.4164i	1.35225	+	1.01419i
$176$		0			0
$177$			24.0000i			1.80395i
$178$		0			0
$179$		−	24.4949i		−	1.83083i	−0.402506	−	0.915417i	$-0.631861\pi$
$179$		−	24.4949i		−	1.83083i	0.402506	−	0.915417i	$-0.368139\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$			26.8328i			1.98354i
$184$		0			0
$185$	16.0000	−	8.00000i	1.17634	−	0.588172i
$186$		0			0
$187$	−12.2474	+	13.4164i	−0.895622	+	0.981105i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	4.89898i		−	0.354478i	−0.984168	−	0.177239i	$-0.943283\pi$
$191$		−	4.89898i		−	0.354478i	0.984168	−	0.177239i	$-0.0567165\pi$
$192$		0			0
$193$	−16.4317			−1.18278			−0.591389	−	0.806386i	$-0.701420\pi$
$193$	−16.4317			−1.18278			−0.591389	+	0.806386i	$0.701420\pi$
$194$		0			0
$195$	13.4164	+	26.8328i	0.960769	+	1.92154i
$196$		0			0
$197$	5.47723			0.390236			0.195118	−	0.980780i	$-0.437491\pi$
$197$	5.47723			0.390236			0.195118	+	0.980780i	$0.437491\pi$
$198$		0			0
$199$		−	9.79796i		−	0.694559i	−0.937762	−	0.347279i	$-0.887106\pi$
$199$		−	9.79796i		−	0.694559i	0.937762	−	0.347279i	$-0.112894\pi$
$200$		0			0
$201$	18.0000			1.26962
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−21.9089	+	10.9545i	−1.53018	+	0.765092i
$206$		0			0
$207$	7.34847			0.510754
$208$		0			0
$209$	−10.0000	+	10.9545i	−0.691714	+	0.757735i
$210$		0			0
$211$	−8.94427			−0.615749			−0.307875	−	0.951427i	$-0.599618\pi$
$211$	−8.94427			−0.615749			−0.307875	+	0.951427i	$0.599618\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.94427	−	4.47214i	0.609994	−	0.304997i
$216$		0			0
$217$	21.9089			1.48727
$218$		0			0
$219$	−13.4164			−0.906597
$220$		0			0
$221$	−30.0000			−2.01802
$222$		0			0
$223$	−2.44949			−0.164030			−0.0820150	−	0.996631i	$-0.526136\pi$
$223$	−2.44949			−0.164030			−0.0820150	+	0.996631i	$0.526136\pi$
$224$		0			0
$225$	−9.00000	+	12.0000i	−0.600000	+	0.800000i
$226$		0			0
$227$			4.47214i			0.296826i	0.988925	+	0.148413i	$0.0474165\pi$
$227$			4.47214i			0.296826i	−0.988925	+	0.148413i	$0.952583\pi$
$228$		0			0
$229$	−10.0000			−0.660819			−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000			−0.660819			−0.330409	+	0.943838i	$0.607187\pi$
$230$		0			0
$231$	−26.8328	−	24.4949i	−1.76547	−	1.61165i
$232$		0			0
$233$	16.4317			1.07647			0.538237	−	0.842793i	$-0.319090\pi$
$233$	16.4317			1.07647			0.538237	+	0.842793i	$0.319090\pi$
$234$		0			0
$235$	7.34847	+	14.6969i	0.479361	+	0.958723i
$236$		0			0
$237$	−10.9545			−0.711568
$238$		0			0
$239$	22.3607			1.44639			0.723196	−	0.690643i	$-0.242672\pi$
$239$	22.3607			1.44639			0.723196	+	0.690643i	$0.242672\pi$
$240$		0			0
$241$			10.9545i			0.705638i	0.935692	+	0.352819i	$0.114777\pi$
$241$			10.9545i			0.705638i	−0.935692	+	0.352819i	$0.885223\pi$
$242$		0			0
$243$	−22.0454			−1.41421
$244$		0			0
$245$	−13.0000	−	26.0000i	−0.830540	−	1.66108i
$246$		0			0
$247$	−24.4949			−1.55857
$248$		0			0
$249$			10.9545i			0.694210i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	5.47723	−	6.00000i	0.344350	−	0.377217i
$254$		0			0
$255$	−13.4164	−	26.8328i	−0.840168	−	1.68034i
$256$		0			0
$257$			28.0000i			1.74659i	0.487190	+	0.873296i	$0.338022\pi$
$257$			28.0000i			1.74659i	−0.487190	+	0.873296i	$0.661978\pi$
$258$		0			0
$259$	−35.7771			−2.22308
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	13.4164i		−	0.827291i	−0.910438	−	0.413646i	$-0.864255\pi$
$263$		−	13.4164i		−	0.827291i	0.910438	−	0.413646i	$-0.135745\pi$
$264$		0			0
$265$	−8.00000	+	4.00000i	−0.491436	+	0.245718i
$266$		0			0
$267$	9.79796			0.599625
$268$		0			0
$269$	−20.0000			−1.21942			−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000			−1.21942			−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$	17.8885			1.08665			0.543326	−	0.839522i	$-0.317165\pi$
$271$	17.8885			1.08665			0.543326	+	0.839522i	$0.317165\pi$
$272$		0			0
$273$		−	60.0000i		−	3.63137i
$274$		0			0
$275$	3.08976	+	16.2927i	0.186319	+	0.982489i
$276$		0			0
$277$	5.47723			0.329095			0.164547	−	0.986369i	$-0.447384\pi$
$277$	5.47723			0.329095			0.164547	+	0.986369i	$0.447384\pi$
$278$		0			0
$279$			14.6969i			0.879883i
$280$		0			0
$281$		−	10.9545i		−	0.653488i	−0.945113	−	0.326744i	$-0.894049\pi$
$281$		−	10.9545i		−	0.653488i	0.945113	−	0.326744i	$-0.105951\pi$
$282$		0			0
$283$		−	22.3607i		−	1.32920i	−0.747197	−	0.664602i	$-0.768601\pi$
$283$		−	22.3607i		−	1.32920i	0.747197	−	0.664602i	$-0.231399\pi$
$284$		0			0
$285$	−10.9545	−	21.9089i	−0.648886	−	1.29777i
$286$		0			0
$287$	48.9898			2.89178
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$		−	29.3939i		−	1.72310i
$292$		0			0
$293$	−5.47723			−0.319983			−0.159991	−	0.987118i	$-0.551147\pi$
$293$	−5.47723			−0.319983			−0.159991	+	0.987118i	$0.551147\pi$
$294$		0			0
$295$	−19.5959	+	9.79796i	−1.14092	+	0.570459i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	13.4164			0.775891
$300$		0			0
$301$	−20.0000			−1.15278
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−21.9089	+	10.9545i	−1.25450	+	0.627250i
$306$		0			0
$307$			22.3607i			1.27619i	0.769957	+	0.638096i	$0.220278\pi$
$307$			22.3607i			1.27619i	−0.769957	+	0.638096i	$0.779722\pi$
$308$		0			0
$309$	−6.00000			−0.341328
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			16.0000i			0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$			16.0000i			0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$		0			0
$315$	26.8328	−	13.4164i	1.51186	−	0.755929i
$316$		0			0
$317$		−	8.00000i		−	0.449325i	−0.974437	−	0.224662i	$-0.927872\pi$
$317$		−	8.00000i		−	0.449325i	0.974437	−	0.224662i	$-0.0721279\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$			10.9545i			0.611418i
$322$		0			0
$323$	24.4949			1.36293
$324$		0			0
$325$	−16.4317	+	21.9089i	−0.911465	+	1.21529i
$326$		0			0
$327$			26.8328i			1.48386i
$328$		0			0
$329$		−	32.8634i		−	1.81182i
$330$		0			0
$331$			4.89898i			0.269272i	0.990895	+	0.134636i	$0.0429866\pi$
$331$			4.89898i			0.269272i	−0.990895	+	0.134636i	$0.957013\pi$
$332$		0			0
$333$		−	24.0000i		−	1.31519i
$334$		0			0
$335$	7.34847	+	14.6969i	0.401490	+	0.802980i
$336$		0			0
$337$	16.4317			0.895090			0.447545	−	0.894261i	$-0.352298\pi$
$337$	16.4317			0.895090			0.447545	+	0.894261i	$0.352298\pi$
$338$		0			0
$339$		−	9.79796i		−	0.532152i
$340$		0			0
$341$	12.0000	+	10.9545i	0.649836	+	0.593217i
$342$		0			0
$343$			26.8328i			1.44884i
$344$		0			0
$345$	6.00000	+	12.0000i	0.323029	+	0.646058i
$346$		0			0
$347$		−	31.3050i		−	1.68054i	−0.542170	−	0.840269i	$-0.682397\pi$
$347$		−	31.3050i		−	1.68054i	0.542170	−	0.840269i	$-0.317603\pi$
$348$		0			0
$349$			21.9089i			1.17276i	0.810037	+	0.586378i	$0.199447\pi$
$349$			21.9089i			1.17276i	−0.810037	+	0.586378i	$0.800553\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			4.00000i			0.212899i	0.994318	+	0.106449i	$0.0339482\pi$
$353$			4.00000i			0.212899i	−0.994318	+	0.106449i	$0.966052\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			60.0000i			3.17554i
$358$		0			0
$359$	13.4164			0.708091			0.354045	−	0.935228i	$-0.384806\pi$
$359$	13.4164			0.708091			0.354045	+	0.935228i	$0.384806\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$	−2.44949	−	26.8328i	−0.128565	−	1.40836i
$364$		0			0
$365$	−5.47723	−	10.9545i	−0.286691	−	0.573382i
$366$		0			0
$367$	−17.1464			−0.895036			−0.447518	−	0.894275i	$-0.647692\pi$
$367$	−17.1464			−0.895036			−0.447518	+	0.894275i	$0.647692\pi$
$368$		0			0
$369$			32.8634i			1.71080i
$370$		0			0
$371$	17.8885			0.928727
$372$		0			0
$373$	27.3861			1.41800			0.709000	−	0.705209i	$-0.249147\pi$
$373$	27.3861			1.41800			0.709000	+	0.705209i	$0.249147\pi$
$374$		0			0
$375$	−26.9444	−	4.89898i	−1.39140	−	0.252982i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	9.79796i		−	0.503287i	−0.967820	−	0.251644i	$-0.919029\pi$
$379$		−	9.79796i		−	0.503287i	0.967820	−	0.251644i	$-0.0809711\pi$
$380$		0			0
$381$			10.9545i			0.561214i
$382$		0			0
$383$	2.44949			0.125163			0.0625815	−	0.998040i	$-0.480067\pi$
$383$	2.44949			0.125163			0.0625815	+	0.998040i	$0.480067\pi$
$384$		0			0
$385$	9.04555	−	31.9089i	0.461004	−	1.62623i
$386$		0			0
$387$		−	13.4164i		−	0.681994i
$388$		0			0
$389$	20.0000			1.01404			0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000			1.01404			0.507020	+	0.861934i	$0.330747\pi$
$390$		0			0
$391$	−13.4164			−0.678497
$392$		0			0
$393$	−21.9089			−1.10516
$394$		0			0
$395$	−4.47214	−	8.94427i	−0.225018	−	0.450035i
$396$		0			0
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$		0			0
$399$			48.9898i			2.45256i
$400$		0			0
$401$	20.0000			0.998752			0.499376	−	0.866385i	$-0.333563\pi$
$401$	20.0000			0.998752			0.499376	+	0.866385i	$0.333563\pi$
$402$		0			0
$403$			26.8328i			1.33664i
$404$		0			0
$405$	−9.00000	−	18.0000i	−0.447214	−	0.894427i
$406$		0			0
$407$	−19.5959	−	17.8885i	−0.971334	−	0.886702i
$408$		0			0
$409$		−	10.9545i		−	0.541663i	−0.962627	−	0.270831i	$-0.912701\pi$
$409$		−	10.9545i		−	0.541663i	0.962627	−	0.270831i	$-0.0872986\pi$
$410$		0			0
$411$			19.5959i			0.966595i
$412$		0			0
$413$	43.8178			2.15613
$414$		0			0
$415$	−8.94427	+	4.47214i	−0.439057	+	0.219529i
$416$		0			0
$417$	21.9089			1.07288
$418$		0			0
$419$			24.4949i			1.19665i	0.801252	+	0.598327i	$0.204168\pi$
$419$			24.4949i			1.19665i	−0.801252	+	0.598327i	$0.795832\pi$
$420$		0			0
$421$	30.0000			1.46211			0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000			1.46211			0.731055	+	0.682318i	$0.239028\pi$
$422$		0			0
$423$	22.0454			1.07188
$424$		0			0
$425$	16.4317	−	21.9089i	0.797053	−	1.06274i
$426$		0			0
$427$	48.9898			2.37078
$428$		0			0
$429$	30.0000	−	32.8634i	1.44841	−	1.58666i
$430$		0			0
$431$	−31.3050			−1.50791			−0.753953	−	0.656928i	$-0.771855\pi$
$431$	−31.3050			−1.50791			−0.753953	+	0.656928i	$0.771855\pi$
$432$		0			0
$433$			4.00000i			0.192228i	0.995370	+	0.0961139i	$0.0306413\pi$
$433$			4.00000i			0.192228i	−0.995370	+	0.0961139i	$0.969359\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.9545			−0.524022
$438$		0			0
$439$	35.7771			1.70755			0.853774	−	0.520644i	$-0.174308\pi$
$439$	35.7771			1.70755			0.853774	+	0.520644i	$0.174308\pi$
$440$		0			0
$441$	−39.0000			−1.85714
$442$		0			0
$443$	22.0454			1.04741			0.523704	−	0.851900i	$-0.324550\pi$
$443$	22.0454			1.04741			0.523704	+	0.851900i	$0.324550\pi$
$444$		0			0
$445$	4.00000	+	8.00000i	0.189618	+	0.379236i
$446$		0			0
$447$		−	26.8328i		−	1.26915i
$448$		0			0
$449$	26.0000			1.22702			0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000			1.22702			0.613508	+	0.789689i	$0.289758\pi$
$450$		0			0
$451$	26.8328	+	24.4949i	1.26351	+	1.15342i
$452$		0			0
$453$	−43.8178			−2.05874
$454$		0			0
$455$	48.9898	−	24.4949i	2.29668	−	1.14834i
$456$		0			0
$457$	16.4317			0.768641			0.384321	−	0.923200i	$-0.374436\pi$
$457$	16.4317			0.768641			0.384321	+	0.923200i	$0.374436\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			21.9089i			1.02040i	0.860056	+	0.510200i	$0.170428\pi$
$461$			21.9089i			1.02040i	−0.860056	+	0.510200i	$0.829572\pi$
$462$		0			0
$463$	−2.44949			−0.113837			−0.0569187	−	0.998379i	$-0.518128\pi$
$463$	−2.44949			−0.113837			−0.0569187	+	0.998379i	$0.518128\pi$
$464$		0			0
$465$	−24.0000	+	12.0000i	−1.11297	+	0.556487i
$466$		0			0
$467$	−7.34847			−0.340047			−0.170023	−	0.985440i	$-0.554384\pi$
$467$	−7.34847			−0.340047			−0.170023	+	0.985440i	$0.554384\pi$
$468$		0			0
$469$		−	32.8634i		−	1.51749i
$470$		0			0
$471$		−	29.3939i		−	1.35440i
$472$		0			0
$473$	−10.9545	−	10.0000i	−0.503686	−	0.459800i
$474$		0			0
$475$	13.4164	−	17.8885i	0.615587	−	0.820783i
$476$		0			0
$477$			12.0000i			0.549442i
$478$		0			0
$479$	22.3607			1.02169			0.510843	−	0.859674i	$-0.329333\pi$
$479$	22.3607			1.02169			0.510843	+	0.859674i	$0.329333\pi$
$480$		0			0
$481$		−	43.8178i		−	1.99792i
$482$		0			0
$483$		−	26.8328i		−	1.22094i
$484$		0			0
$485$	24.0000	−	12.0000i	1.08978	−	0.544892i
$486$		0			0
$487$	−17.1464			−0.776979			−0.388489	−	0.921453i	$-0.627003\pi$
$487$	−17.1464			−0.776979			−0.388489	+	0.921453i	$0.627003\pi$
$488$		0			0
$489$	−54.0000			−2.44196
$490$		0			0
$491$	−13.4164			−0.605474			−0.302737	−	0.953074i	$-0.597900\pi$
$491$	−13.4164			−0.605474			−0.302737	+	0.953074i	$0.597900\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	21.4051	+	6.06794i	0.962090	+	0.272734i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	39.1918i		−	1.75447i	−0.480063	−	0.877234i	$-0.659386\pi$
$499$		−	39.1918i		−	1.75447i	0.480063	−	0.877234i	$-0.340614\pi$
$500$		0			0
$501$		−	32.8634i		−	1.46823i
$502$		0			0
$503$		−	40.2492i		−	1.79462i	−0.441397	−	0.897312i	$-0.645517\pi$
$503$		−	40.2492i		−	1.79462i	0.441397	−	0.897312i	$-0.354483\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	41.6413			1.84936
$508$		0			0
$509$	−10.0000			−0.443242			−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000			−0.443242			−0.221621	+	0.975133i	$0.571135\pi$
$510$		0			0
$511$			24.4949i			1.08359i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.44949	−	4.89898i	−0.107937	−	0.215875i
$516$		0			0
$517$	16.4317	−	18.0000i	0.722664	−	0.791639i
$518$		0			0
$519$	13.4164			0.588915
$520$		0			0
$521$	20.0000			0.876216			0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000			0.876216			0.438108	+	0.898922i	$0.355649\pi$
$522$		0			0
$523$		−	4.47214i		−	0.195553i	−0.995208	−	0.0977764i	$-0.968827\pi$
$523$		−	4.47214i		−	0.195553i	0.995208	−	0.0977764i	$-0.0311730\pi$
$524$		0			0
$525$	43.8178	+	32.8634i	1.91237	+	1.43427i
$526$		0			0
$527$		−	26.8328i		−	1.16886i
$528$		0			0
$529$	−17.0000			−0.739130
$530$		0			0
$531$			29.3939i			1.27559i
$532$		0			0
$533$			60.0000i			2.59889i
$534$		0			0
$535$	−8.94427	+	4.47214i	−0.386695	+	0.193347i
$536$		0			0
$537$		−	60.0000i		−	2.58919i
$538$		0			0
$539$	−29.0689	+	31.8434i	−1.25209	+	1.37159i
$540$		0			0
$541$		−	21.9089i		−	0.941937i	−0.882150	−	0.470969i	$-0.843904\pi$
$541$		−	21.9089i		−	0.941937i	0.882150	−	0.470969i	$-0.156096\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−21.9089	+	10.9545i	−0.938474	+	0.469237i
$546$		0			0
$547$		−	31.3050i		−	1.33850i	−0.743036	−	0.669252i	$-0.766615\pi$
$547$		−	31.3050i		−	1.33850i	0.743036	−	0.669252i	$-0.233385\pi$
$548$		0			0
$549$			32.8634i			1.40257i
$550$		0			0
$551$		0			0
$552$		0			0
$553$			20.0000i			0.850487i
$554$		0			0
$555$	39.1918	−	19.5959i	1.66360	−	0.831800i
$556$		0			0
$557$	−27.3861			−1.16039			−0.580194	−	0.814478i	$-0.697023\pi$
$557$	−27.3861			−1.16039			−0.580194	+	0.814478i	$0.697023\pi$
$558$		0			0
$559$		−	24.4949i		−	1.03602i
$560$		0			0
$561$	−30.0000	+	32.8634i	−1.26660	+	1.38749i
$562$		0			0
$563$			13.4164i			0.565434i	0.959203	+	0.282717i	$0.0912358\pi$
$563$			13.4164i			0.565434i	−0.959203	+	0.282717i	$0.908764\pi$
$564$		0			0
$565$	8.00000	−	4.00000i	0.336563	−	0.168281i
$566$		0			0
$567$			40.2492i			1.69031i
$568$		0			0
$569$			10.9545i			0.459234i	0.973281	+	0.229617i	$0.0737474\pi$
$569$			10.9545i			0.459234i	−0.973281	+	0.229617i	$0.926253\pi$
$570$		0			0
$571$	4.47214			0.187153			0.0935765	−	0.995612i	$-0.470170\pi$
$571$	4.47214			0.187153			0.0935765	+	0.995612i	$0.470170\pi$
$572$		0			0
$573$		−	12.0000i		−	0.501307i
$574$		0			0
$575$	−7.34847	+	9.79796i	−0.306452	+	0.408603i
$576$		0			0
$577$			12.0000i			0.499567i	0.968302	+	0.249783i	$0.0803594\pi$
$577$			12.0000i			0.499567i	−0.968302	+	0.249783i	$0.919641\pi$
$578$		0			0
$579$	−40.2492			−1.67270
$580$		0			0
$581$	20.0000			0.829740
$582$		0			0
$583$	9.79796	+	8.94427i	0.405790	+	0.370434i
$584$		0			0
$585$	16.4317	+	32.8634i	0.679366	+	1.35873i
$586$		0			0
$587$	−31.8434			−1.31432			−0.657158	−	0.753753i	$-0.728242\pi$
$587$	−31.8434			−1.31432			−0.657158	+	0.753753i	$0.728242\pi$
$588$		0			0
$589$		−	21.9089i		−	0.902741i
$590$		0			0
$591$	13.4164			0.551877
$592$		0			0
$593$	−16.4317			−0.674768			−0.337384	−	0.941367i	$-0.609542\pi$
$593$	−16.4317			−0.674768			−0.337384	+	0.941367i	$0.609542\pi$
$594$		0			0
$595$	−48.9898	+	24.4949i	−2.00839	+	1.00419i
$596$		0			0
$597$		−	24.0000i		−	0.982255i
$598$		0			0
$599$			24.4949i			1.00083i	0.865784	+	0.500417i	$0.166820\pi$
$599$			24.4949i			1.00083i	−0.865784	+	0.500417i	$0.833180\pi$
$600$		0			0
$601$			32.8634i			1.34052i	0.742124	+	0.670262i	$0.233818\pi$
$601$			32.8634i			1.34052i	−0.742124	+	0.670262i	$0.766182\pi$
$602$		0			0
$603$	22.0454			0.897758
$604$		0			0
$605$	20.9089	−	12.9545i	0.850068	−	0.526673i
$606$		0			0
$607$			4.47214i			0.181518i	0.995873	+	0.0907592i	$0.0289294\pi$
$607$			4.47214i			0.181518i	−0.995873	+	0.0907592i	$0.971071\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	40.2492			1.62831
$612$		0			0
$613$	−5.47723			−0.221223			−0.110612	−	0.993864i	$-0.535281\pi$
$613$	−5.47723			−0.221223			−0.110612	+	0.993864i	$0.535281\pi$
$614$		0			0
$615$	−53.6656	+	26.8328i	−2.16401	+	1.08200i
$616$		0			0
$617$		−	8.00000i		−	0.322068i	−0.986949	−	0.161034i	$-0.948517\pi$
$617$		−	8.00000i		−	0.322068i	0.986949	−	0.161034i	$-0.0514829\pi$
$618$		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
$623$		−	17.8885i		−	0.716689i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$	−24.4949	+	26.8328i	−0.978232	+	1.07160i
$628$		0			0
$629$			43.8178i			1.74713i
$630$		0			0
$631$		−	48.9898i		−	1.95025i	−0.221649	−	0.975126i	$-0.571144\pi$
$631$		−	48.9898i		−	1.95025i	0.221649	−	0.975126i	$-0.428856\pi$
$632$		0			0
$633$	−21.9089			−0.870801
$634$		0			0
$635$	−8.94427	+	4.47214i	−0.354943	+	0.177471i
$636$		0			0
$637$	−71.2039			−2.82120
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.0000			0.394976			0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000			0.394976			0.197488	+	0.980305i	$0.436722\pi$
$642$		0			0
$643$	−2.44949			−0.0965984			−0.0482992	−	0.998833i	$-0.515380\pi$
$643$	−2.44949			−0.0965984			−0.0482992	+	0.998833i	$0.515380\pi$
$644$		0			0
$645$	21.9089	−	10.9545i	0.862662	−	0.431331i
$646$		0			0
$647$	41.6413			1.63709			0.818545	−	0.574443i	$-0.194781\pi$
$647$	41.6413			1.63709			0.818545	+	0.574443i	$0.194781\pi$
$648$		0			0
$649$	24.0000	+	21.9089i	0.942082	+	0.860000i
$650$		0			0
$651$	53.6656			2.10332
$652$		0			0
$653$			44.0000i			1.72185i	0.508729	+	0.860927i	$0.330115\pi$
$653$			44.0000i			1.72185i	−0.508729	+	0.860927i	$0.669885\pi$
$654$		0			0
$655$	−8.94427	−	17.8885i	−0.349482	−	0.698963i
$656$		0			0
$657$	−16.4317			−0.641061
$658$		0			0
$659$	8.94427			0.348419			0.174210	−	0.984709i	$-0.444263\pi$
$659$	8.94427			0.348419			0.174210	+	0.984709i	$0.444263\pi$
$660$		0			0
$661$	−28.0000			−1.08907			−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000			−1.08907			−0.544537	+	0.838737i	$0.683295\pi$
$662$		0			0
$663$	−73.4847			−2.85391
$664$		0			0
$665$	−40.0000	+	20.0000i	−1.55113	+	0.775567i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−6.00000			−0.231973
$670$		0			0
$671$	26.8328	+	24.4949i	1.03587	+	0.945615i
$672$		0			0
$673$	5.47723			0.211132			0.105566	−	0.994412i	$-0.466335\pi$
$673$	5.47723			0.211132			0.105566	+	0.994412i	$0.466335\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.47723			−0.210507			−0.105253	−	0.994445i	$-0.533565\pi$
$677$	−5.47723			−0.210507			−0.105253	+	0.994445i	$0.533565\pi$
$678$		0			0
$679$	−53.6656			−2.05950
$680$		0			0
$681$			10.9545i			0.419775i
$682$		0			0
$683$	−26.9444			−1.03100			−0.515499	−	0.856890i	$-0.672393\pi$
$683$	−26.9444			−1.03100			−0.515499	+	0.856890i	$0.672393\pi$
$684$		0			0
$685$	−16.0000	+	8.00000i	−0.611329	+	0.305664i
$686$		0			0
$687$	−24.4949			−0.934539
$688$		0			0
$689$			21.9089i			0.834663i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	−32.8634	−	30.0000i	−1.24838	−	1.13961i
$694$		0			0
$695$	8.94427	+	17.8885i	0.339276	+	0.678551i
$696$		0			0
$697$		−	60.0000i		−	2.27266i
$698$		0			0
$699$	40.2492			1.52237
$700$		0			0
$701$		−	32.8634i		−	1.24123i	−0.784115	−	0.620616i	$-0.786883\pi$
$701$		−	32.8634i		−	1.24123i	0.784115	−	0.620616i	$-0.213117\pi$
$702$		0			0
$703$			35.7771i			1.34936i
$704$		0			0
$705$	18.0000	+	36.0000i	0.677919	+	1.35584i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	16.0000			0.600893			0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000			0.600893			0.300446	+	0.953799i	$0.402864\pi$
$710$		0			0
$711$	−13.4164			−0.503155
$712$		0			0
$713$			12.0000i			0.449404i
$714$		0			0
$715$	39.0803	+	11.0785i	1.46152	+	0.414312i
$716$		0			0
$717$	54.7723			2.04551
$718$		0			0
$719$		−	24.4949i		−	0.913506i	−0.889594	−	0.456753i	$-0.849012\pi$
$719$		−	24.4949i		−	0.913506i	0.889594	−	0.456753i	$-0.150988\pi$
$720$		0			0
$721$			10.9545i			0.407965i
$722$		0			0
$723$			26.8328i			0.997923i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	31.8434			1.18101			0.590503	−	0.807036i	$-0.298930\pi$
$727$	31.8434			1.18101			0.590503	+	0.807036i	$0.298930\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$			24.4949i			0.905977i
$732$		0			0
$733$	38.3406			1.41614			0.708071	−	0.706141i	$-0.249566\pi$
$733$	38.3406			1.41614			0.708071	+	0.706141i	$0.249566\pi$
$734$		0			0
$735$	−31.8434	−	63.6867i	−1.17456	−	2.34912i
$736$		0			0
$737$	16.4317	−	18.0000i	0.605269	−	0.663039i
$738$		0			0
$739$	−44.7214			−1.64510			−0.822551	−	0.568691i	$-0.807450\pi$
$739$	−44.7214			−1.64510			−0.822551	+	0.568691i	$0.807450\pi$
$740$		0			0
$741$	−60.0000			−2.20416
$742$		0			0
$743$			22.3607i			0.820334i	0.912010	+	0.410167i	$0.134530\pi$
$743$			22.3607i			0.820334i	−0.912010	+	0.410167i	$0.865470\pi$
$744$		0			0
$745$	21.9089	−	10.9545i	0.802680	−	0.401340i
$746$		0			0
$747$			13.4164i			0.490881i
$748$		0			0
$749$	20.0000			0.730784
$750$		0			0
$751$			48.9898i			1.78766i	0.448403	+	0.893832i	$0.351993\pi$
$751$			48.9898i			1.78766i	−0.448403	+	0.893832i	$0.648007\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−17.8885	−	35.7771i	−0.651031	−	1.30206i
$756$		0			0
$757$			8.00000i			0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$			8.00000i			0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$		0			0
$759$	13.4164	−	14.6969i	0.486985	−	0.533465i
$760$		0			0
$761$		−	21.9089i		−	0.794197i	−0.917776	−	0.397099i	$-0.870017\pi$
$761$		−	21.9089i		−	0.794197i	0.917776	−	0.397099i	$-0.129983\pi$
$762$		0			0
$763$	48.9898			1.77355
$764$		0			0
$765$	−16.4317	−	32.8634i	−0.594089	−	1.18818i
$766$		0			0
$767$			53.6656i			1.93775i
$768$		0			0
$769$		−	21.9089i		−	0.790055i	−0.918669	−	0.395028i	$-0.870735\pi$
$769$		−	21.9089i		−	0.790055i	0.918669	−	0.395028i	$-0.129265\pi$
$770$		0			0
$771$			68.5857i			2.47005i
$772$		0			0
$773$		−	44.0000i		−	1.58257i	−0.611448	−	0.791285i	$-0.709412\pi$
$773$		−	44.0000i		−	1.58257i	0.611448	−	0.791285i	$-0.290588\pi$
$774$		0			0
$775$	−19.5959	−	14.6969i	−0.703906	−	0.527930i
$776$		0			0
$777$	−87.6356			−3.14391
$778$		0			0
$779$		−	48.9898i		−	1.75524i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	24.0000	−	12.0000i	0.856597	−	0.428298i
$786$		0			0
$787$			4.47214i			0.159414i	0.996818	+	0.0797072i	$0.0253985\pi$
$787$			4.47214i			0.159414i	−0.996818	+	0.0797072i	$0.974601\pi$
$788$		0			0
$789$		−	32.8634i		−	1.16997i
$790$		0			0
$791$	−17.8885			−0.636043
$792$		0			0
$793$			60.0000i			2.13066i
$794$		0			0
$795$	−19.5959	+	9.79796i	−0.694996	+	0.347498i
$796$		0			0
$797$		−	32.0000i		−	1.13350i	−0.823890	−	0.566749i	$-0.808201\pi$
$797$		−	32.0000i		−	1.13350i	0.823890	−	0.566749i	$-0.191799\pi$
$798$		0			0
$799$	−40.2492			−1.42392
$800$		0			0
$801$	12.0000			0.423999
$802$		0			0
$803$	−12.2474	+	13.4164i	−0.432203	+	0.473455i
$804$		0			0
$805$	21.9089	−	10.9545i	0.772187	−	0.386094i
$806$		0			0
$807$	−48.9898			−1.72452
$808$		0			0
$809$		−	21.9089i		−	0.770276i	−0.922859	−	0.385138i	$-0.874154\pi$
$809$		−	21.9089i		−	0.770276i	0.922859	−	0.385138i	$-0.125846\pi$
$810$		0			0
$811$	−49.1935			−1.72742			−0.863709	−	0.503991i	$-0.831864\pi$
$811$	−49.1935			−1.72742			−0.863709	+	0.503991i	$0.831864\pi$
$812$		0			0
$813$	43.8178			1.53676
$814$		0			0
$815$	−22.0454	−	44.0908i	−0.772217	−	1.54443i
$816$		0			0
$817$			20.0000i			0.699711i
$818$		0			0
$819$		−	73.4847i		−	2.56776i
$820$		0			0
$821$			10.9545i			0.382313i	0.981560	+	0.191156i	$0.0612238\pi$
$821$			10.9545i			0.382313i	−0.981560	+	0.191156i	$0.938776\pi$
$822$		0			0
$823$	−26.9444			−0.939222			−0.469611	−	0.882873i	$-0.655606\pi$
$823$	−26.9444			−0.939222			−0.469611	+	0.882873i	$0.655606\pi$
$824$		0			0
$825$	7.56832	+	39.9089i	0.263495	+	1.38945i
$826$		0			0
$827$		−	4.47214i		−	0.155511i	−0.996972	−	0.0777557i	$-0.975225\pi$
$827$		−	4.47214i		−	0.155511i	0.996972	−	0.0777557i	$-0.0247754\pi$
$828$		0			0
$829$	−26.0000			−0.903017			−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000			−0.903017			−0.451509	+	0.892267i	$0.649114\pi$
$830$		0			0
$831$	13.4164			0.465410
$832$		0			0
$833$	71.2039			2.46707
$834$		0			0
$835$	26.8328	−	13.4164i	0.928588	−	0.464294i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			39.1918i			1.35305i	0.736419	+	0.676526i	$0.236515\pi$
$839$			39.1918i			1.35305i	−0.736419	+	0.676526i	$0.763485\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		−	26.8328i		−	0.924171i
$844$		0			0
$845$	17.0000	+	34.0000i	0.584818	+	1.16964i
$846$		0			0
$847$	−48.9898	+	4.47214i	−1.68331	+	0.153664i
$848$		0			0
$849$		−	54.7723i		−	1.87978i
$850$		0			0
$851$		−	19.5959i		−	0.671739i
$852$		0			0
$853$	16.4317			0.562610			0.281305	−	0.959618i	$-0.409233\pi$
$853$	16.4317			0.562610			0.281305	+	0.959618i	$0.409233\pi$
$854$		0			0
$855$	−13.4164	−	26.8328i	−0.458831	−	0.917663i
$856$		0			0
$857$	16.4317			0.561295			0.280648	−	0.959811i	$-0.409451\pi$
$857$	16.4317			0.561295			0.280648	+	0.959811i	$0.409451\pi$
$858$		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
$861$	120.000			4.08959
$862$		0			0
$863$	2.44949			0.0833816			0.0416908	−	0.999131i	$-0.486726\pi$
$863$	2.44949			0.0833816			0.0416908	+	0.999131i	$0.486726\pi$
$864$		0			0
$865$	5.47723	+	10.9545i	0.186231	+	0.372463i
$866$		0			0
$867$	31.8434			1.08146
$868$		0			0
$869$	−10.0000	+	10.9545i	−0.339227	+	0.371604i
$870$		0			0
$871$	40.2492			1.36379
$872$		0			0
$873$		−	36.0000i		−	1.21842i
$874$		0			0
$875$	−8.94427	+	49.1935i	−0.302372	+	1.66304i
$876$		0			0
$877$	−49.2950			−1.66457			−0.832287	−	0.554344i	$-0.812969\pi$
$877$	−49.2950			−1.66457			−0.832287	+	0.554344i	$0.812969\pi$
$878$		0			0
$879$	−13.4164			−0.452524
$880$		0			0
$881$	−32.0000			−1.07811			−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000			−1.07811			−0.539054	+	0.842271i	$0.681218\pi$
$882$		0			0
$883$	26.9444			0.906751			0.453375	−	0.891320i	$-0.350220\pi$
$883$	26.9444			0.906751			0.453375	+	0.891320i	$0.350220\pi$
$884$		0			0
$885$	−48.0000	+	24.0000i	−1.61350	+	0.806751i
$886$		0			0
$887$			31.3050i			1.05112i	0.850757	+	0.525559i	$0.176144\pi$
$887$			31.3050i			1.05112i	−0.850757	+	0.525559i	$0.823856\pi$
$888$		0			0
$889$	20.0000			0.670778
$890$		0			0
$891$	−20.1246	+	22.0454i	−0.674200	+	0.738549i
$892$		0			0
$893$	−32.8634			−1.09973
$894$		0			0
$895$	48.9898	−	24.4949i	1.63755	−	0.818774i
$896$		0			0
$897$	32.8634			1.09728
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	21.9089i		−	0.729891i
$902$		0			0
$903$	−48.9898			−1.63028
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−17.1464			−0.569338			−0.284669	−	0.958626i	$-0.591884\pi$
$907$	−17.1464			−0.569338			−0.284669	+	0.958626i	$0.591884\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			4.89898i			0.162310i	0.996701	+	0.0811552i	$0.0258609\pi$
$911$			4.89898i			0.162310i	−0.996701	+	0.0811552i	$0.974139\pi$
$912$		0			0
$913$	10.9545	+	10.0000i	0.362539	+	0.330952i
$914$		0			0
$915$	−53.6656	+	26.8328i	−1.77413	+	0.887066i
$916$		0			0
$917$			40.0000i			1.32092i
$918$		0			0
$919$	−35.7771			−1.18018			−0.590089	−	0.807338i	$-0.700907\pi$
$919$	−35.7771			−1.18018			−0.590089	+	0.807338i	$0.700907\pi$
$920$		0			0
$921$			54.7723i			1.80481i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	32.0000	+	24.0000i	1.05215	+	0.789115i
$926$		0			0
$927$	−7.34847			−0.241355
$928$		0			0
$929$	40.0000			1.31236			0.656179	−	0.754606i	$-0.272172\pi$
$929$	40.0000			1.31236			0.656179	+	0.754606i	$0.272172\pi$
$930$		0			0
$931$	58.1378			1.90539
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−39.0803	−	11.0785i	−1.27806	−	0.362305i
$936$		0			0
$937$	−60.2495			−1.96826			−0.984132	−	0.177436i	$-0.943220\pi$
$937$	−60.2495			−1.96826			−0.984132	+	0.177436i	$0.943220\pi$
$938$		0			0
$939$			39.1918i			1.27898i
$940$		0			0
$941$			43.8178i			1.42842i	0.699932	+	0.714210i	$0.253214\pi$
$941$			43.8178i			1.42842i	−0.699932	+	0.714210i	$0.746786\pi$
$942$		0			0
$943$			26.8328i			0.873797i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.1464			0.557184			0.278592	−	0.960410i	$-0.410132\pi$
$947$	17.1464			0.557184			0.278592	+	0.960410i	$0.410132\pi$
$948$		0			0
$949$	−30.0000			−0.973841
$950$		0			0
$951$		−	19.5959i		−	0.635441i
$952$		0			0
$953$	27.3861			0.887124			0.443562	−	0.896244i	$-0.353715\pi$
$953$	27.3861			0.887124			0.443562	+	0.896244i	$0.353715\pi$
$954$		0			0
$955$	9.79796	−	4.89898i	0.317055	−	0.158527i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	35.7771			1.15530
$960$		0			0
$961$	7.00000			0.225806
$962$		0			0
$963$			13.4164i			0.432338i
$964$		0			0
$965$	−16.4317	−	32.8634i	−0.528954	−	1.05791i
$966$		0			0
$967$		−	22.3607i		−	0.719071i	−0.933131	−	0.359535i	$-0.882935\pi$
$967$		−	22.3607i		−	0.719071i	0.933131	−	0.359535i	$-0.117065\pi$
$968$		0			0
$969$	60.0000			1.92748
$970$		0			0
$971$		−	4.89898i		−	0.157216i	−0.996906	−	0.0786079i	$-0.974952\pi$
$971$		−	4.89898i		−	0.157216i	0.996906	−	0.0786079i	$-0.0250475\pi$
$972$		0			0
$973$		−	40.0000i		−	1.28234i
$974$		0			0
$975$	−40.2492	+	53.6656i	−1.28901	+	1.71868i
$976$		0			0
$977$		−	8.00000i		−	0.255943i	−0.991778	−	0.127971i	$-0.959153\pi$
$977$		−	8.00000i		−	0.255943i	0.991778	−	0.127971i	$-0.0408466\pi$
$978$		0			0
$979$	8.94427	−	9.79796i	0.285860	−	0.313144i
$980$		0			0
$981$			32.8634i			1.04925i
$982$		0			0
$983$	−2.44949			−0.0781266			−0.0390633	−	0.999237i	$-0.512437\pi$
$983$	−2.44949			−0.0781266			−0.0390633	+	0.999237i	$0.512437\pi$
$984$		0			0
$985$	5.47723	+	10.9545i	0.174519	+	0.349038i
$986$		0			0
$987$		−	80.4984i		−	2.56229i
$988$		0			0
$989$		−	10.9545i		−	0.348331i
$990$		0			0
$991$		−	4.89898i		−	0.155621i	−0.996968	−	0.0778106i	$-0.975207\pi$
$991$		−	4.89898i		−	0.155621i	0.996968	−	0.0778106i	$-0.0247929\pi$
$992$		0			0
$993$			12.0000i			0.380808i
$994$		0			0
$995$	19.5959	−	9.79796i	0.621232	−	0.310616i
$996$		0			0
$997$	5.47723			0.173465			0.0867327	−	0.996232i	$-0.472357\pi$
$997$	5.47723			0.173465			0.0867327	+	0.996232i	$0.472357\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.m.g.879.7	yes	8
4.3	odd	2	inner	880.2.m.g.879.4	yes	8
5.4	even	2	inner	880.2.m.g.879.2	yes	8
11.10	odd	2	inner	880.2.m.g.879.8	yes	8
20.19	odd	2	inner	880.2.m.g.879.5	yes	8
44.43	even	2	inner	880.2.m.g.879.3	yes	8
55.54	odd	2	inner	880.2.m.g.879.1	✓	8
220.219	even	2	inner	880.2.m.g.879.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
880.2.m.g.879.1	✓	8	55.54	odd	2	inner
880.2.m.g.879.2	yes	8	5.4	even	2	inner
880.2.m.g.879.3	yes	8	44.43	even	2	inner
880.2.m.g.879.4	yes	8	4.3	odd	2	inner
880.2.m.g.879.5	yes	8	20.19	odd	2	inner
880.2.m.g.879.6	yes	8	220.219	even	2	inner
880.2.m.g.879.7	yes	8	1.1	even	1	trivial
880.2.m.g.879.8	yes	8	11.10	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.44949 q^{3} +(1.00000 + 2.00000i) q^{5} -4.47214i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+2.44949 q^{3} +(1.00000 + 2.00000i) q^{5} -4.47214i q^{7} +3.00000 q^{9} +(2.23607 - 2.44949i) q^{11} +5.47723 q^{13} +(2.44949 + 4.89898i) q^{15} -5.47723 q^{17} -4.47214 q^{19} -10.9545i q^{21} +2.44949 q^{23} +(-3.00000 + 4.00000i) q^{25} +4.89898i q^{31} +(5.47723 - 6.00000i) q^{33} +(8.94427 - 4.47214i) q^{35} -8.00000i q^{37} +13.4164 q^{39} +10.9545i q^{41} -4.47214i q^{43} +(3.00000 + 6.00000i) q^{45} +7.34847 q^{47} -13.0000 q^{49} -13.4164 q^{51} +4.00000i q^{53} +(7.13505 + 2.02265i) q^{55} -10.9545 q^{57} +9.79796i q^{59} +10.9545i q^{61} -13.4164i q^{63} +(5.47723 + 10.9545i) q^{65} +7.34847 q^{67} +6.00000 q^{69} -5.47723 q^{73} +(-7.34847 + 9.79796i) q^{75} +(-10.9545 - 10.0000i) q^{77} -4.47214 q^{79} -9.00000 q^{81} +4.47214i q^{83} +(-5.47723 - 10.9545i) q^{85} +4.00000 q^{89} -24.4949i q^{91} +12.0000i q^{93} +(-4.47214 - 8.94427i) q^{95} -12.0000i q^{97} +(6.70820 - 7.34847i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} + 24 q^{9}+O(q^{10}) \) 8 * q + 8 * q^5 + 24 * q^9	\( 8 q + 8 q^{5} + 24 q^{9} - 24 q^{25} + 24 q^{45} - 104 q^{49} + 48 q^{69} - 72 q^{81} + 32 q^{89}+O(q^{100}) \) 8 * q + 8 * q^5 + 24 * q^9 - 24 * q^25 + 24 * q^45 - 104 * q^49 + 48 * q^69 - 72 * q^81 + 32 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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