LMFDB - Embedded newform 880.2.f.c.351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,2,Mod(351,880)]

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, 0, 1]))

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

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

chi := DirichletCharacter("880.351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.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:	$7.02683537787$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.1
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	880.351
Dual form			880.2.f.c.351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{3} +1.00000 q^{5} -3.46410 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.41421i q^{3} +1.00000 q^{5} -3.46410 q^{7} +1.00000 q^{9} +(1.73205 - 2.82843i) q^{11} +2.44949i q^{13} -1.41421i q^{15} -7.34847i q^{17} -3.46410 q^{19} +4.89898i q^{21} -1.41421i q^{23} +1.00000 q^{25} -5.65685i q^{27} -4.89898i q^{29} +(-4.00000 - 2.44949i) q^{33} -3.46410 q^{35} -2.00000 q^{37} +3.46410 q^{39} -3.46410 q^{43} +1.00000 q^{45} -7.07107i q^{47} +5.00000 q^{49} -10.3923 q^{51} -6.00000 q^{53} +(1.73205 - 2.82843i) q^{55} +4.89898i q^{57} +2.82843i q^{59} +9.79796i q^{61} -3.46410 q^{63} +2.44949i q^{65} -12.7279i q^{67} -2.00000 q^{69} +5.65685i q^{71} +7.34847i q^{73} -1.41421i q^{75} +(-6.00000 + 9.79796i) q^{77} +10.3923 q^{79} -5.00000 q^{81} +10.3923 q^{83} -7.34847i q^{85} -6.92820 q^{87} +12.0000 q^{89} -8.48528i q^{91} -3.46410 q^{95} -2.00000 q^{97} +(1.73205 - 2.82843i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 4 * q^9	$ 4 q + 4 q^{5} + 4 q^{9} + 4 q^{25} - 16 q^{33} - 8 q^{37} + 4 q^{45} + 20 q^{49} - 24 q^{53} - 8 q^{69} - 24 q^{77} - 20 q^{81} + 48 q^{89} - 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^9 + 4 * q^25 - 16 * q^33 - 8 * q^37 + 4 * q^45 + 20 * q^49 - 24 * q^53 - 8 * q^69 - 24 * q^77 - 20 * q^81 + 48 * q^89 - 8 * q^97

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
$3$		−	1.41421i		−	0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$		−	1.41421i		−	0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−3.46410			−1.30931			−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410			−1.30931			−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	1.73205	−	2.82843i	0.522233	−	0.852803i
$11$	1.73205	−	2.82843i	0.522233	−	0.852803i
$12$		0			0
$13$			2.44949i			0.679366i	0.940540	+	0.339683i	$0.110320\pi$
$13$			2.44949i			0.679366i	−0.940540	+	0.339683i	$0.889680\pi$
$14$		0			0
$15$		−	1.41421i		−	0.365148i
$16$		0			0
$17$		−	7.34847i		−	1.78227i	−0.453743	−	0.891133i	$-0.649911\pi$
$17$		−	7.34847i		−	1.78227i	0.453743	−	0.891133i	$-0.350089\pi$
$18$		0			0
$19$	−3.46410			−0.794719			−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410			−0.794719			−0.397360	+	0.917663i	$0.630073\pi$
$20$		0			0
$21$			4.89898i			1.06904i
$22$		0			0
$23$		−	1.41421i		−	0.294884i	−0.989071	−	0.147442i	$-0.952896\pi$
$23$		−	1.41421i		−	0.294884i	0.989071	−	0.147442i	$-0.0471040\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	5.65685i		−	1.08866i
$28$		0			0
$29$		−	4.89898i		−	0.909718i	−0.890564	−	0.454859i	$-0.849690\pi$
$29$		−	4.89898i		−	0.909718i	0.890564	−	0.454859i	$-0.150310\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	−4.00000	−	2.44949i	−0.696311	−	0.426401i
$34$		0			0
$35$	−3.46410			−0.585540
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	3.46410			0.554700
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−3.46410			−0.528271			−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410			−0.528271			−0.264135	+	0.964486i	$0.585087\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$		−	7.07107i		−	1.03142i	−0.856763	−	0.515711i	$-0.827528\pi$
$47$		−	7.07107i		−	1.03142i	0.856763	−	0.515711i	$-0.172472\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$	−10.3923			−1.45521
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	1.73205	−	2.82843i	0.233550	−	0.381385i
$56$		0			0
$57$			4.89898i			0.648886i
$58$		0			0
$59$			2.82843i			0.368230i	0.982905	+	0.184115i	$0.0589419\pi$
$59$			2.82843i			0.368230i	−0.982905	+	0.184115i	$0.941058\pi$
$60$		0			0
$61$			9.79796i			1.25450i	0.778818	+	0.627250i	$0.215820\pi$
$61$			9.79796i			1.25450i	−0.778818	+	0.627250i	$0.784180\pi$
$62$		0			0
$63$	−3.46410			−0.436436
$64$		0			0
$65$			2.44949i			0.303822i
$66$		0			0
$67$		−	12.7279i		−	1.55496i	−0.628906	−	0.777482i	$-0.716497\pi$
$67$		−	12.7279i		−	1.55496i	0.628906	−	0.777482i	$-0.283503\pi$
$68$		0			0
$69$	−2.00000			−0.240772
$70$		0			0
$71$			5.65685i			0.671345i	0.941979	+	0.335673i	$0.108964\pi$
$71$			5.65685i			0.671345i	−0.941979	+	0.335673i	$0.891036\pi$
$72$		0			0
$73$			7.34847i			0.860073i	0.902811	+	0.430037i	$0.141499\pi$
$73$			7.34847i			0.860073i	−0.902811	+	0.430037i	$0.858501\pi$
$74$		0			0
$75$		−	1.41421i		−	0.163299i
$76$		0			0
$77$	−6.00000	+	9.79796i	−0.683763	+	1.11658i
$78$		0			0
$79$	10.3923			1.16923			0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923			1.16923			0.584613	+	0.811312i	$0.301246\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$	10.3923			1.14070			0.570352	−	0.821401i	$-0.306807\pi$
$83$	10.3923			1.14070			0.570352	+	0.821401i	$0.306807\pi$
$84$		0			0
$85$		−	7.34847i		−	0.797053i
$86$		0			0
$87$	−6.92820			−0.742781
$88$		0			0
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		0			0
$91$		−	8.48528i		−	0.889499i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.46410			−0.355409
$96$		0			0
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$		0			0
$99$	1.73205	−	2.82843i	0.174078	−	0.284268i
$100$		0			0
$101$		−	19.5959i		−	1.94987i	−0.222497	−	0.974933i	$-0.571421\pi$
$101$		−	19.5959i		−	1.94987i	0.222497	−	0.974933i	$-0.428579\pi$
$102$		0			0
$103$			12.7279i			1.25412i	0.778971	+	0.627060i	$0.215742\pi$
$103$			12.7279i			1.25412i	−0.778971	+	0.627060i	$0.784258\pi$
$104$		0			0
$105$			4.89898i			0.478091i
$106$		0			0
$107$	−17.3205			−1.67444			−0.837218	−	0.546869i	$-0.815820\pi$
$107$	−17.3205			−1.67444			−0.837218	+	0.546869i	$0.815820\pi$
$108$		0			0
$109$		−	14.6969i		−	1.40771i	−0.710343	−	0.703856i	$-0.751460\pi$
$109$		−	14.6969i		−	1.40771i	0.710343	−	0.703856i	$-0.248540\pi$
$110$		0			0
$111$			2.82843i			0.268462i
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		−	1.41421i		−	0.131876i
$116$		0			0
$117$			2.44949i			0.226455i
$118$		0			0
$119$			25.4558i			2.33353i
$120$		0			0
$121$	−5.00000	−	9.79796i	−0.454545	−	0.890724i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−3.46410			−0.307389			−0.153695	−	0.988118i	$-0.549117\pi$
$127$	−3.46410			−0.307389			−0.153695	+	0.988118i	$0.549117\pi$
$128$		0			0
$129$			4.89898i			0.431331i
$130$		0			0
$131$	13.8564			1.21064			0.605320	−	0.795982i	$-0.293045\pi$
$131$	13.8564			1.21064			0.605320	+	0.795982i	$0.293045\pi$
$132$		0			0
$133$	12.0000			1.04053
$134$		0			0
$135$		−	5.65685i		−	0.486864i
$136$		0			0
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$		0			0
$139$	6.92820			0.587643			0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820			0.587643			0.293821	+	0.955860i	$0.405073\pi$
$140$		0			0
$141$	−10.0000			−0.842152
$142$		0			0
$143$	6.92820	+	4.24264i	0.579365	+	0.354787i
$144$		0			0
$145$		−	4.89898i		−	0.406838i
$146$		0			0
$147$		−	7.07107i		−	0.583212i
$148$		0			0
$149$			4.89898i			0.401340i	0.979659	+	0.200670i	$0.0643119\pi$
$149$			4.89898i			0.401340i	−0.979659	+	0.200670i	$0.935688\pi$
$150$		0			0
$151$	20.7846			1.69143			0.845714	−	0.533637i	$-0.179175\pi$
$151$	20.7846			1.69143			0.845714	+	0.533637i	$0.179175\pi$
$152$		0			0
$153$		−	7.34847i		−	0.594089i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$			8.48528i			0.672927i
$160$		0			0
$161$			4.89898i			0.386094i
$162$		0			0
$163$			12.7279i			0.996928i	0.866910	+	0.498464i	$0.166102\pi$
$163$			12.7279i			0.996928i	−0.866910	+	0.498464i	$0.833898\pi$
$164$		0			0
$165$	−4.00000	−	2.44949i	−0.311400	−	0.190693i
$166$		0			0
$167$	17.3205			1.34030			0.670151	−	0.742225i	$-0.266230\pi$
$167$	17.3205			1.34030			0.670151	+	0.742225i	$0.266230\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$		0			0
$171$	−3.46410			−0.264906
$172$		0			0
$173$			12.2474i			0.931156i	0.885007	+	0.465578i	$0.154154\pi$
$173$			12.2474i			0.931156i	−0.885007	+	0.465578i	$0.845846\pi$
$174$		0			0
$175$	−3.46410			−0.261861
$176$		0			0
$177$	4.00000			0.300658
$178$		0			0
$179$			14.1421i			1.05703i	0.848923	+	0.528516i	$0.177252\pi$
$179$			14.1421i			1.05703i	−0.848923	+	0.528516i	$0.822748\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$	13.8564			1.02430
$184$		0			0
$185$	−2.00000			−0.147043
$186$		0			0
$187$	−20.7846	−	12.7279i	−1.51992	−	0.930758i
$188$		0			0
$189$			19.5959i			1.42539i
$190$		0			0
$191$			22.6274i			1.63726i	0.574320	+	0.818631i	$0.305267\pi$
$191$			22.6274i			1.63726i	−0.574320	+	0.818631i	$0.694733\pi$
$192$		0			0
$193$		−	7.34847i		−	0.528954i	−0.964392	−	0.264477i	$-0.914801\pi$
$193$		−	7.34847i		−	0.528954i	0.964392	−	0.264477i	$-0.0851994\pi$
$194$		0			0
$195$	3.46410			0.248069
$196$		0			0
$197$		−	2.44949i		−	0.174519i	−0.996186	−	0.0872595i	$-0.972189\pi$
$197$		−	2.44949i		−	0.174519i	0.996186	−	0.0872595i	$-0.0278109\pi$
$198$		0			0
$199$			25.4558i			1.80452i	0.431196	+	0.902258i	$0.358092\pi$
$199$			25.4558i			1.80452i	−0.431196	+	0.902258i	$0.641908\pi$
$200$		0			0
$201$	−18.0000			−1.26962
$202$		0			0
$203$			16.9706i			1.19110i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	1.41421i		−	0.0982946i
$208$		0			0
$209$	−6.00000	+	9.79796i	−0.415029	+	0.677739i
$210$		0			0
$211$	−27.7128			−1.90783			−0.953914	−	0.300079i	$-0.902987\pi$
$211$	−27.7128			−1.90783			−0.953914	+	0.300079i	$0.902987\pi$
$212$		0			0
$213$	8.00000			0.548151
$214$		0			0
$215$	−3.46410			−0.236250
$216$		0			0
$217$		0			0
$218$		0			0
$219$	10.3923			0.702247
$220$		0			0
$221$	18.0000			1.21081
$222$		0			0
$223$		−	4.24264i		−	0.284108i	−0.989859	−	0.142054i	$-0.954629\pi$
$223$		−	4.24264i		−	0.284108i	0.989859	−	0.142054i	$-0.0453707\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	17.3205			1.14960			0.574801	−	0.818293i	$-0.305079\pi$
$227$	17.3205			1.14960			0.574801	+	0.818293i	$0.305079\pi$
$228$		0			0
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$		0			0
$231$	13.8564	+	8.48528i	0.911685	+	0.558291i
$232$		0			0
$233$		−	2.44949i		−	0.160471i	−0.996776	−	0.0802357i	$-0.974433\pi$
$233$		−	2.44949i		−	0.160471i	0.996776	−	0.0802357i	$-0.0255673\pi$
$234$		0			0
$235$		−	7.07107i		−	0.461266i
$236$		0			0
$237$		−	14.6969i		−	0.954669i
$238$		0			0
$239$	17.3205			1.12037			0.560185	−	0.828367i	$-0.310730\pi$
$239$	17.3205			1.12037			0.560185	+	0.828367i	$0.310730\pi$
$240$		0			0
$241$			9.79796i			0.631142i	0.948902	+	0.315571i	$0.102196\pi$
$241$			9.79796i			0.631142i	−0.948902	+	0.315571i	$0.897804\pi$
$242$		0			0
$243$		−	9.89949i		−	0.635053i
$244$		0			0
$245$	5.00000			0.319438
$246$		0			0
$247$		−	8.48528i		−	0.539906i
$248$		0			0
$249$		−	14.6969i		−	0.931381i
$250$		0			0
$251$			5.65685i			0.357057i	0.983935	+	0.178529i	$0.0571337\pi$
$251$			5.65685i			0.357057i	−0.983935	+	0.178529i	$0.942866\pi$
$252$		0			0
$253$	−4.00000	−	2.44949i	−0.251478	−	0.153998i
$254$		0			0
$255$	−10.3923			−0.650791
$256$		0			0
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$		0			0
$259$	6.92820			0.430498
$260$		0			0
$261$		−	4.89898i		−	0.303239i
$262$		0			0
$263$	3.46410			0.213606			0.106803	−	0.994280i	$-0.465939\pi$
$263$	3.46410			0.213606			0.106803	+	0.994280i	$0.465939\pi$
$264$		0			0
$265$	−6.00000			−0.368577
$266$		0			0
$267$		−	16.9706i		−	1.03858i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−6.92820			−0.420858			−0.210429	−	0.977609i	$-0.567486\pi$
$271$	−6.92820			−0.420858			−0.210429	+	0.977609i	$0.567486\pi$
$272$		0			0
$273$	−12.0000			−0.726273
$274$		0			0
$275$	1.73205	−	2.82843i	0.104447	−	0.170561i
$276$		0			0
$277$			7.34847i			0.441527i	0.975327	+	0.220763i	$0.0708548\pi$
$277$			7.34847i			0.441527i	−0.975327	+	0.220763i	$0.929145\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	17.3205			1.02960			0.514799	−	0.857311i	$-0.327867\pi$
$283$	17.3205			1.02960			0.514799	+	0.857311i	$0.327867\pi$
$284$		0			0
$285$			4.89898i			0.290191i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−37.0000			−2.17647
$290$		0			0
$291$			2.82843i			0.165805i
$292$		0			0
$293$		−	12.2474i		−	0.715504i	−0.933817	−	0.357752i	$-0.883543\pi$
$293$		−	12.2474i		−	0.715504i	0.933817	−	0.357752i	$-0.116457\pi$
$294$		0			0
$295$			2.82843i			0.164677i
$296$		0			0
$297$	−16.0000	−	9.79796i	−0.928414	−	0.568535i
$298$		0			0
$299$	3.46410			0.200334
$300$		0			0
$301$	12.0000			0.691669
$302$		0			0
$303$	−27.7128			−1.59206
$304$		0			0
$305$			9.79796i			0.561029i
$306$		0			0
$307$	−17.3205			−0.988534			−0.494267	−	0.869310i	$-0.664563\pi$
$307$	−17.3205			−0.988534			−0.494267	+	0.869310i	$0.664563\pi$
$308$		0			0
$309$	18.0000			1.02398
$310$		0			0
$311$			11.3137i			0.641542i	0.947157	+	0.320771i	$0.103942\pi$
$311$			11.3137i			0.641542i	−0.947157	+	0.320771i	$0.896058\pi$
$312$		0			0
$313$	22.0000			1.24351			0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000			1.24351			0.621757	+	0.783210i	$0.286419\pi$
$314$		0			0
$315$	−3.46410			−0.195180
$316$		0			0
$317$	−30.0000			−1.68497			−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000			−1.68497			−0.842484	+	0.538721i	$0.818908\pi$
$318$		0			0
$319$	−13.8564	−	8.48528i	−0.775810	−	0.475085i
$320$		0			0
$321$			24.4949i			1.36717i
$322$		0			0
$323$			25.4558i			1.41640i
$324$		0			0
$325$			2.44949i			0.135873i
$326$		0			0
$327$	−20.7846			−1.14939
$328$		0			0
$329$			24.4949i			1.35045i
$330$		0			0
$331$		−	33.9411i		−	1.86557i	−0.360429	−	0.932786i	$-0.617370\pi$
$331$		−	33.9411i		−	1.86557i	0.360429	−	0.932786i	$-0.382630\pi$
$332$		0			0
$333$	−2.00000			−0.109599
$334$		0			0
$335$		−	12.7279i		−	0.695401i
$336$		0			0
$337$		−	7.34847i		−	0.400297i	−0.979766	−	0.200148i	$-0.935858\pi$
$337$		−	7.34847i		−	0.400297i	0.979766	−	0.200148i	$-0.0641424\pi$
$338$		0			0
$339$			8.48528i			0.460857i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	6.92820			0.374088
$344$		0			0
$345$	−2.00000			−0.107676
$346$		0			0
$347$	−10.3923			−0.557888			−0.278944	−	0.960307i	$-0.589984\pi$
$347$	−10.3923			−0.557888			−0.278944	+	0.960307i	$0.589984\pi$
$348$		0			0
$349$		−	14.6969i		−	0.786709i	−0.919387	−	0.393355i	$-0.871314\pi$
$349$		−	14.6969i		−	0.786709i	0.919387	−	0.393355i	$-0.128686\pi$
$350$		0			0
$351$	13.8564			0.739600
$352$		0			0
$353$	18.0000			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$354$		0			0
$355$			5.65685i			0.300235i
$356$		0			0
$357$	36.0000			1.90532
$358$		0			0
$359$	24.2487			1.27980			0.639899	−	0.768459i	$-0.278976\pi$
$359$	24.2487			1.27980			0.639899	+	0.768459i	$0.278976\pi$
$360$		0			0
$361$	−7.00000			−0.368421
$362$		0			0
$363$	−13.8564	+	7.07107i	−0.727273	+	0.371135i
$364$		0			0
$365$			7.34847i			0.384636i
$366$		0			0
$367$		−	4.24264i		−	0.221464i	−0.993850	−	0.110732i	$-0.964680\pi$
$367$		−	4.24264i		−	0.221464i	0.993850	−	0.110732i	$-0.0353195\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	20.7846			1.07908
$372$		0			0
$373$			22.0454i			1.14147i	0.821135	+	0.570734i	$0.193341\pi$
$373$			22.0454i			1.14147i	−0.821135	+	0.570734i	$0.806659\pi$
$374$		0			0
$375$		−	1.41421i		−	0.0730297i
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$		−	8.48528i		−	0.435860i	−0.975964	−	0.217930i	$-0.930070\pi$
$379$		−	8.48528i		−	0.435860i	0.975964	−	0.217930i	$-0.0699304\pi$
$380$		0			0
$381$			4.89898i			0.250982i
$382$		0			0
$383$		−	35.3553i		−	1.80657i	−0.429037	−	0.903287i	$-0.641147\pi$
$383$		−	35.3553i		−	1.80657i	0.429037	−	0.903287i	$-0.358853\pi$
$384$		0			0
$385$	−6.00000	+	9.79796i	−0.305788	+	0.499350i
$386$		0			0
$387$	−3.46410			−0.176090
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$	−10.3923			−0.525561
$392$		0			0
$393$		−	19.5959i		−	0.988483i
$394$		0			0
$395$	10.3923			0.522894
$396$		0			0
$397$	−14.0000			−0.702640			−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000			−0.702640			−0.351320	+	0.936255i	$0.614267\pi$
$398$		0			0
$399$		−	16.9706i		−	0.849591i
$400$		0			0
$401$	24.0000			1.19850			0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000			1.19850			0.599251	+	0.800561i	$0.295465\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−5.00000			−0.248452
$406$		0			0
$407$	−3.46410	+	5.65685i	−0.171709	+	0.280400i
$408$		0			0
$409$		−	34.2929i		−	1.69567i	−0.530258	−	0.847836i	$-0.677905\pi$
$409$		−	34.2929i		−	1.69567i	0.530258	−	0.847836i	$-0.322095\pi$
$410$		0			0
$411$		−	25.4558i		−	1.25564i
$412$		0			0
$413$		−	9.79796i		−	0.482126i
$414$		0			0
$415$	10.3923			0.510138
$416$		0			0
$417$		−	9.79796i		−	0.479808i
$418$		0			0
$419$			31.1127i			1.51995i	0.649950	+	0.759977i	$0.274790\pi$
$419$			31.1127i			1.51995i	−0.649950	+	0.759977i	$0.725210\pi$
$420$		0			0
$421$	2.00000			0.0974740			0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000			0.0974740			0.0487370	+	0.998812i	$0.484480\pi$
$422$		0			0
$423$		−	7.07107i		−	0.343807i
$424$		0			0
$425$		−	7.34847i		−	0.356453i
$426$		0			0
$427$		−	33.9411i		−	1.64253i
$428$		0			0
$429$	6.00000	−	9.79796i	0.289683	−	0.473050i
$430$		0			0
$431$	−24.2487			−1.16802			−0.584010	−	0.811747i	$-0.698517\pi$
$431$	−24.2487			−1.16802			−0.584010	+	0.811747i	$0.698517\pi$
$432$		0			0
$433$	34.0000			1.63394			0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000			1.63394			0.816968	+	0.576683i	$0.195653\pi$
$434$		0			0
$435$	−6.92820			−0.332182
$436$		0			0
$437$			4.89898i			0.234350i
$438$		0			0
$439$	−27.7128			−1.32266			−0.661330	−	0.750095i	$-0.730008\pi$
$439$	−27.7128			−1.32266			−0.661330	+	0.750095i	$0.730008\pi$
$440$		0			0
$441$	5.00000			0.238095
$442$		0			0
$443$			15.5563i			0.739104i	0.929210	+	0.369552i	$0.120489\pi$
$443$			15.5563i			0.739104i	−0.929210	+	0.369552i	$0.879511\pi$
$444$		0			0
$445$	12.0000			0.568855
$446$		0			0
$447$	6.92820			0.327693
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	29.3939i		−	1.38104i
$454$		0			0
$455$		−	8.48528i		−	0.397796i
$456$		0			0
$457$			41.6413i			1.94790i	0.226765	+	0.973950i	$0.427185\pi$
$457$			41.6413i			1.94790i	−0.226765	+	0.973950i	$0.572815\pi$
$458$		0			0
$459$	−41.5692			−1.94029
$460$		0			0
$461$		−	29.3939i		−	1.36901i	−0.729008	−	0.684505i	$-0.760019\pi$
$461$		−	29.3939i		−	1.36901i	0.729008	−	0.684505i	$-0.239981\pi$
$462$		0			0
$463$		−	21.2132i		−	0.985861i	−0.870069	−	0.492931i	$-0.835926\pi$
$463$		−	21.2132i		−	0.985861i	0.870069	−	0.492931i	$-0.164074\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			7.07107i			0.327210i	0.986526	+	0.163605i	$0.0523123\pi$
$467$			7.07107i			0.327210i	−0.986526	+	0.163605i	$0.947688\pi$
$468$		0			0
$469$			44.0908i			2.03592i
$470$		0			0
$471$		−	2.82843i		−	0.130327i
$472$		0			0
$473$	−6.00000	+	9.79796i	−0.275880	+	0.450511i
$474$		0			0
$475$	−3.46410			−0.158944
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	−10.3923			−0.474837			−0.237418	−	0.971408i	$-0.576301\pi$
$479$	−10.3923			−0.474837			−0.237418	+	0.971408i	$0.576301\pi$
$480$		0			0
$481$		−	4.89898i		−	0.223374i
$482$		0			0
$483$	6.92820			0.315244
$484$		0			0
$485$	−2.00000			−0.0908153
$486$		0			0
$487$		−	4.24264i		−	0.192252i	−0.995369	−	0.0961262i	$-0.969355\pi$
$487$		−	4.24264i		−	0.192252i	0.995369	−	0.0961262i	$-0.0306452\pi$
$488$		0			0
$489$	18.0000			0.813988
$490$		0			0
$491$	−10.3923			−0.468998			−0.234499	−	0.972116i	$-0.575345\pi$
$491$	−10.3923			−0.468998			−0.234499	+	0.972116i	$0.575345\pi$
$492$		0			0
$493$	−36.0000			−1.62136
$494$		0			0
$495$	1.73205	−	2.82843i	0.0778499	−	0.127128i
$496$		0			0
$497$		−	19.5959i		−	0.878997i
$498$		0			0
$499$		−	42.4264i		−	1.89927i	−0.313363	−	0.949633i	$-0.601456\pi$
$499$		−	42.4264i		−	1.89927i	0.313363	−	0.949633i	$-0.398544\pi$
$500$		0			0
$501$		−	24.4949i		−	1.09435i
$502$		0			0
$503$	−31.1769			−1.39011			−0.695055	−	0.718957i	$-0.744620\pi$
$503$	−31.1769			−1.39011			−0.695055	+	0.718957i	$0.744620\pi$
$504$		0			0
$505$		−	19.5959i		−	0.872007i
$506$		0			0
$507$		−	9.89949i		−	0.439652i
$508$		0			0
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$		−	25.4558i		−	1.12610i
$512$		0			0
$513$			19.5959i			0.865181i
$514$		0			0
$515$			12.7279i			0.560859i
$516$		0			0
$517$	−20.0000	−	12.2474i	−0.879599	−	0.538642i
$518$		0			0
$519$	17.3205			0.760286
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	31.1769			1.36327			0.681636	−	0.731692i	$-0.261269\pi$
$523$	31.1769			1.36327			0.681636	+	0.731692i	$0.261269\pi$
$524$		0			0
$525$			4.89898i			0.213809i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$			2.82843i			0.122743i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−17.3205			−0.748831
$536$		0			0
$537$	20.0000			0.863064
$538$		0			0
$539$	8.66025	−	14.1421i	0.373024	−	0.609145i
$540$		0			0
$541$			9.79796i			0.421247i	0.977567	+	0.210624i	$0.0675494\pi$
$541$			9.79796i			0.421247i	−0.977567	+	0.210624i	$0.932451\pi$
$542$		0			0
$543$		−	28.2843i		−	1.21379i
$544$		0			0
$545$		−	14.6969i		−	0.629548i
$546$		0			0
$547$	10.3923			0.444343			0.222171	−	0.975008i	$-0.428686\pi$
$547$	10.3923			0.444343			0.222171	+	0.975008i	$0.428686\pi$
$548$		0			0
$549$			9.79796i			0.418167i
$550$		0			0
$551$			16.9706i			0.722970i
$552$		0			0
$553$	−36.0000			−1.53088
$554$		0			0
$555$			2.82843i			0.120060i
$556$		0			0
$557$			12.2474i			0.518941i	0.965751	+	0.259471i	$0.0835480\pi$
$557$			12.2474i			0.518941i	−0.965751	+	0.259471i	$0.916452\pi$
$558$		0			0
$559$		−	8.48528i		−	0.358889i
$560$		0			0
$561$	−18.0000	+	29.3939i	−0.759961	+	1.24101i
$562$		0			0
$563$	17.3205			0.729972			0.364986	−	0.931013i	$-0.381074\pi$
$563$	17.3205			0.729972			0.364986	+	0.931013i	$0.381074\pi$
$564$		0			0
$565$	−6.00000			−0.252422
$566$		0			0
$567$	17.3205			0.727393
$568$		0			0
$569$		−	4.89898i		−	0.205376i	−0.994714	−	0.102688i	$-0.967256\pi$
$569$		−	4.89898i		−	0.205376i	0.994714	−	0.102688i	$-0.0327443\pi$
$570$		0			0
$571$	−38.1051			−1.59465			−0.797325	−	0.603550i	$-0.793752\pi$
$571$	−38.1051			−1.59465			−0.797325	+	0.603550i	$0.793752\pi$
$572$		0			0
$573$	32.0000			1.33682
$574$		0			0
$575$		−	1.41421i		−	0.0589768i
$576$		0			0
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$		0			0
$579$	−10.3923			−0.431889
$580$		0			0
$581$	−36.0000			−1.49353
$582$		0			0
$583$	−10.3923	+	16.9706i	−0.430405	+	0.702849i
$584$		0			0
$585$			2.44949i			0.101274i
$586$		0			0
$587$			15.5563i			0.642079i	0.947066	+	0.321040i	$0.104032\pi$
$587$			15.5563i			0.642079i	−0.947066	+	0.321040i	$0.895968\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−3.46410			−0.142494
$592$		0			0
$593$		−	17.1464i		−	0.704119i	−0.935978	−	0.352060i	$-0.885481\pi$
$593$		−	17.1464i		−	0.704119i	0.935978	−	0.352060i	$-0.114519\pi$
$594$		0			0
$595$			25.4558i			1.04359i
$596$		0			0
$597$	36.0000			1.47338
$598$		0			0
$599$		−	31.1127i		−	1.27123i	−0.772006	−	0.635615i	$-0.780747\pi$
$599$		−	31.1127i		−	1.27123i	0.772006	−	0.635615i	$-0.219253\pi$
$600$		0			0
$601$		−	19.5959i		−	0.799334i	−0.916660	−	0.399667i	$-0.869126\pi$
$601$		−	19.5959i		−	0.799334i	0.916660	−	0.399667i	$-0.130874\pi$
$602$		0			0
$603$		−	12.7279i		−	0.518321i
$604$		0			0
$605$	−5.00000	−	9.79796i	−0.203279	−	0.398344i
$606$		0			0
$607$	10.3923			0.421811			0.210905	−	0.977506i	$-0.432359\pi$
$607$	10.3923			0.421811			0.210905	+	0.977506i	$0.432359\pi$
$608$		0			0
$609$	24.0000			0.972529
$610$		0			0
$611$	17.3205			0.700713
$612$		0			0
$613$			17.1464i			0.692538i	0.938135	+	0.346269i	$0.112552\pi$
$613$			17.1464i			0.692538i	−0.938135	+	0.346269i	$0.887448\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−30.0000			−1.20775			−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000			−1.20775			−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$			25.4558i			1.02316i	0.859237	+	0.511578i	$0.170939\pi$
$619$			25.4558i			1.02316i	−0.859237	+	0.511578i	$0.829061\pi$
$620$		0			0
$621$	−8.00000			−0.321029
$622$		0			0
$623$	−41.5692			−1.66544
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	13.8564	+	8.48528i	0.553372	+	0.338869i
$628$		0			0
$629$			14.6969i			0.586005i
$630$		0			0
$631$			16.9706i			0.675587i	0.941220	+	0.337794i	$0.109681\pi$
$631$			16.9706i			0.675587i	−0.941220	+	0.337794i	$0.890319\pi$
$632$		0			0
$633$			39.1918i			1.55774i
$634$		0			0
$635$	−3.46410			−0.137469
$636$		0			0
$637$			12.2474i			0.485262i
$638$		0			0
$639$			5.65685i			0.223782i
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$			29.6985i			1.17119i	0.810602	+	0.585597i	$0.199140\pi$
$643$			29.6985i			1.17119i	−0.810602	+	0.585597i	$0.800860\pi$
$644$		0			0
$645$			4.89898i			0.192897i
$646$		0			0
$647$			7.07107i			0.277992i	0.990293	+	0.138996i	$0.0443876\pi$
$647$			7.07107i			0.277992i	−0.990293	+	0.138996i	$0.955612\pi$
$648$		0			0
$649$	8.00000	+	4.89898i	0.314027	+	0.192302i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−30.0000			−1.17399			−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000			−1.17399			−0.586995	+	0.809590i	$0.699689\pi$
$654$		0			0
$655$	13.8564			0.541415
$656$		0			0
$657$			7.34847i			0.286691i
$658$		0			0
$659$	20.7846			0.809653			0.404827	−	0.914393i	$-0.367332\pi$
$659$	20.7846			0.809653			0.404827	+	0.914393i	$0.367332\pi$
$660$		0			0
$661$	−4.00000			−0.155582			−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000			−0.155582			−0.0777910	+	0.996970i	$0.524787\pi$
$662$		0			0
$663$		−	25.4558i		−	0.988623i
$664$		0			0
$665$	12.0000			0.465340
$666$		0			0
$667$	−6.92820			−0.268261
$668$		0			0
$669$	−6.00000			−0.231973
$670$		0			0
$671$	27.7128	+	16.9706i	1.06984	+	0.655141i
$672$		0			0
$673$			41.6413i			1.60516i	0.596548	+	0.802578i	$0.296539\pi$
$673$			41.6413i			1.60516i	−0.596548	+	0.802578i	$0.703461\pi$
$674$		0			0
$675$		−	5.65685i		−	0.217732i
$676$		0			0
$677$			31.8434i			1.22384i	0.790920	+	0.611920i	$0.209603\pi$
$677$			31.8434i			1.22384i	−0.790920	+	0.611920i	$0.790397\pi$
$678$		0			0
$679$	6.92820			0.265880
$680$		0			0
$681$		−	24.4949i		−	0.938647i
$682$		0			0
$683$		−	24.0416i		−	0.919927i	−0.887938	−	0.459964i	$-0.847862\pi$
$683$		−	24.0416i		−	0.919927i	0.887938	−	0.459964i	$-0.152138\pi$
$684$		0			0
$685$	18.0000			0.687745
$686$		0			0
$687$			36.7696i			1.40285i
$688$		0			0
$689$		−	14.6969i		−	0.559909i
$690$		0			0
$691$		−	33.9411i		−	1.29118i	−0.763684	−	0.645591i	$-0.776611\pi$
$691$		−	33.9411i		−	1.29118i	0.763684	−	0.645591i	$-0.223389\pi$
$692$		0			0
$693$	−6.00000	+	9.79796i	−0.227921	+	0.372194i
$694$		0			0
$695$	6.92820			0.262802
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−3.46410			−0.131024
$700$		0			0
$701$			29.3939i			1.11019i	0.831786	+	0.555096i	$0.187318\pi$
$701$			29.3939i			1.11019i	−0.831786	+	0.555096i	$0.812682\pi$
$702$		0			0
$703$	6.92820			0.261302
$704$		0			0
$705$	−10.0000			−0.376622
$706$		0			0
$707$			67.8823i			2.55297i
$708$		0			0
$709$	−8.00000			−0.300446			−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000			−0.300446			−0.150223	+	0.988652i	$0.547999\pi$
$710$		0			0
$711$	10.3923			0.389742
$712$		0			0
$713$		0			0
$714$		0			0
$715$	6.92820	+	4.24264i	0.259100	+	0.158666i
$716$		0			0
$717$		−	24.4949i		−	0.914779i
$718$		0			0
$719$		−	36.7696i		−	1.37127i	−0.727944	−	0.685636i	$-0.759524\pi$
$719$		−	36.7696i		−	1.37127i	0.727944	−	0.685636i	$-0.240476\pi$
$720$		0			0
$721$		−	44.0908i		−	1.64203i
$722$		0			0
$723$	13.8564			0.515325
$724$		0			0
$725$		−	4.89898i		−	0.181944i
$726$		0			0
$727$			29.6985i			1.10146i	0.834685	+	0.550728i	$0.185650\pi$
$727$			29.6985i			1.10146i	−0.834685	+	0.550728i	$0.814350\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$			25.4558i			0.941518i
$732$		0			0
$733$		−	41.6413i		−	1.53806i	−0.639214	−	0.769029i	$-0.720740\pi$
$733$		−	41.6413i		−	1.53806i	0.639214	−	0.769029i	$-0.279260\pi$
$734$		0			0
$735$		−	7.07107i		−	0.260820i
$736$		0			0
$737$	−36.0000	−	22.0454i	−1.32608	−	0.812053i
$738$		0			0
$739$	−6.92820			−0.254858			−0.127429	−	0.991848i	$-0.540673\pi$
$739$	−6.92820			−0.254858			−0.127429	+	0.991848i	$0.540673\pi$
$740$		0			0
$741$	−12.0000			−0.440831
$742$		0			0
$743$	−3.46410			−0.127086			−0.0635428	−	0.997979i	$-0.520240\pi$
$743$	−3.46410			−0.127086			−0.0635428	+	0.997979i	$0.520240\pi$
$744$		0			0
$745$			4.89898i			0.179485i
$746$		0			0
$747$	10.3923			0.380235
$748$		0			0
$749$	60.0000			2.19235
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	8.00000			0.291536
$754$		0			0
$755$	20.7846			0.756429
$756$		0			0
$757$	−34.0000			−1.23575			−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000			−1.23575			−0.617876	+	0.786276i	$0.712006\pi$
$758$		0			0
$759$	−3.46410	+	5.65685i	−0.125739	+	0.205331i
$760$		0			0
$761$			29.3939i			1.06553i	0.846264	+	0.532764i	$0.178847\pi$
$761$			29.3939i			1.06553i	−0.846264	+	0.532764i	$0.821153\pi$
$762$		0			0
$763$			50.9117i			1.84313i
$764$		0			0
$765$		−	7.34847i		−	0.265684i
$766$		0			0
$767$	−6.92820			−0.250163
$768$		0			0
$769$		−	14.6969i		−	0.529985i	−0.964250	−	0.264993i	$-0.914630\pi$
$769$		−	14.6969i		−	0.529985i	0.964250	−	0.264993i	$-0.0853695\pi$
$770$		0			0
$771$			8.48528i			0.305590i
$772$		0			0
$773$	−6.00000			−0.215805			−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000			−0.215805			−0.107903	+	0.994161i	$0.534413\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	9.79796i		−	0.351500i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	16.0000	+	9.79796i	0.572525	+	0.350599i
$782$		0			0
$783$	−27.7128			−0.990375
$784$		0			0
$785$	2.00000			0.0713831
$786$		0			0
$787$	24.2487			0.864373			0.432187	−	0.901784i	$-0.357742\pi$
$787$	24.2487			0.864373			0.432187	+	0.901784i	$0.357742\pi$
$788$		0			0
$789$		−	4.89898i		−	0.174408i
$790$		0			0
$791$	20.7846			0.739016
$792$		0			0
$793$	−24.0000			−0.852265
$794$		0			0
$795$			8.48528i			0.300942i
$796$		0			0
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$		0			0
$799$	−51.9615			−1.83827
$800$		0			0
$801$	12.0000			0.423999
$802$		0			0
$803$	20.7846	+	12.7279i	0.733473	+	0.449159i
$804$		0			0
$805$			4.89898i			0.172666i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	24.4949i		−	0.861195i	−0.902544	−	0.430597i	$-0.858303\pi$
$809$		−	24.4949i		−	0.861195i	0.902544	−	0.430597i	$-0.141697\pi$
$810$		0			0
$811$	−10.3923			−0.364923			−0.182462	−	0.983213i	$-0.558407\pi$
$811$	−10.3923			−0.364923			−0.182462	+	0.983213i	$0.558407\pi$
$812$		0			0
$813$			9.79796i			0.343629i
$814$		0			0
$815$			12.7279i			0.445840i
$816$		0			0
$817$	12.0000			0.419827
$818$		0			0
$819$		−	8.48528i		−	0.296500i
$820$		0			0
$821$		−	9.79796i		−	0.341951i	−0.985275	−	0.170976i	$-0.945308\pi$
$821$		−	9.79796i		−	0.341951i	0.985275	−	0.170976i	$-0.0546919\pi$
$822$		0			0
$823$		−	29.6985i		−	1.03522i	−0.855615	−	0.517612i	$-0.826821\pi$
$823$		−	29.6985i		−	1.03522i	0.855615	−	0.517612i	$-0.173179\pi$
$824$		0			0
$825$	−4.00000	−	2.44949i	−0.139262	−	0.0852803i
$826$		0			0
$827$	38.1051			1.32504			0.662522	−	0.749042i	$-0.269486\pi$
$827$	38.1051			1.32504			0.662522	+	0.749042i	$0.269486\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$	10.3923			0.360505
$832$		0			0
$833$		−	36.7423i		−	1.27305i
$834$		0			0
$835$	17.3205			0.599401
$836$		0			0
$837$		0			0
$838$		0			0
$839$			2.82843i			0.0976481i	0.998807	+	0.0488241i	$0.0155474\pi$
$839$			2.82843i			0.0976481i	−0.998807	+	0.0488241i	$0.984453\pi$
$840$		0			0
$841$	5.00000			0.172414
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.00000			0.240807
$846$		0			0
$847$	17.3205	+	33.9411i	0.595140	+	1.16623i
$848$		0			0
$849$		−	24.4949i		−	0.840663i
$850$		0			0
$851$			2.82843i			0.0969572i
$852$		0			0
$853$		−	12.2474i		−	0.419345i	−0.977772	−	0.209672i	$-0.932760\pi$
$853$		−	12.2474i		−	0.419345i	0.977772	−	0.209672i	$-0.0672397\pi$
$854$		0			0
$855$	−3.46410			−0.118470
$856$		0			0
$857$			41.6413i			1.42244i	0.702969	+	0.711220i	$0.251857\pi$
$857$			41.6413i			1.42244i	−0.702969	+	0.711220i	$0.748143\pi$
$858$		0			0
$859$			8.48528i			0.289514i	0.989467	+	0.144757i	$0.0462401\pi$
$859$			8.48528i			0.289514i	−0.989467	+	0.144757i	$0.953760\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	35.3553i		−	1.20351i	−0.798681	−	0.601755i	$-0.794468\pi$
$863$		−	35.3553i		−	1.20351i	0.798681	−	0.601755i	$-0.205532\pi$
$864$		0			0
$865$			12.2474i			0.416426i
$866$		0			0
$867$			52.3259i			1.77708i
$868$		0			0
$869$	18.0000	−	29.3939i	0.610608	−	0.997119i
$870$		0			0
$871$	31.1769			1.05639
$872$		0			0
$873$	−2.00000			−0.0676897
$874$		0			0
$875$	−3.46410			−0.117108
$876$		0			0
$877$		−	17.1464i		−	0.578994i	−0.957179	−	0.289497i	$-0.906512\pi$
$877$		−	17.1464i		−	0.578994i	0.957179	−	0.289497i	$-0.0934880\pi$
$878$		0			0
$879$	−17.3205			−0.584206
$880$		0			0
$881$	48.0000			1.61716			0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000			1.61716			0.808581	+	0.588386i	$0.200236\pi$
$882$		0			0
$883$			46.6690i			1.57054i	0.619154	+	0.785269i	$0.287475\pi$
$883$			46.6690i			1.57054i	−0.619154	+	0.785269i	$0.712525\pi$
$884$		0			0
$885$	4.00000			0.134459
$886$		0			0
$887$	17.3205			0.581566			0.290783	−	0.956789i	$-0.406084\pi$
$887$	17.3205			0.581566			0.290783	+	0.956789i	$0.406084\pi$
$888$		0			0
$889$	12.0000			0.402467
$890$		0			0
$891$	−8.66025	+	14.1421i	−0.290129	+	0.473779i
$892$		0			0
$893$			24.4949i			0.819690i
$894$		0			0
$895$			14.1421i			0.472719i
$896$		0			0
$897$		−	4.89898i		−	0.163572i
$898$		0			0
$899$		0			0
$900$		0			0
$901$			44.0908i			1.46888i
$902$		0			0
$903$		−	16.9706i		−	0.564745i
$904$		0			0
$905$	20.0000			0.664822
$906$		0			0
$907$		−	21.2132i		−	0.704373i	−0.935930	−	0.352186i	$-0.885438\pi$
$907$		−	21.2132i		−	0.704373i	0.935930	−	0.352186i	$-0.114562\pi$
$908$		0			0
$909$		−	19.5959i		−	0.649956i
$910$		0			0
$911$		−	28.2843i		−	0.937100i	−0.883437	−	0.468550i	$-0.844777\pi$
$911$		−	28.2843i		−	0.937100i	0.883437	−	0.468550i	$-0.155223\pi$
$912$		0			0
$913$	18.0000	−	29.3939i	0.595713	−	0.972795i
$914$		0			0
$915$	13.8564			0.458079
$916$		0			0
$917$	−48.0000			−1.58510
$918$		0			0
$919$	41.5692			1.37124			0.685621	−	0.727959i	$-0.259531\pi$
$919$	41.5692			1.37124			0.685621	+	0.727959i	$0.259531\pi$
$920$		0			0
$921$			24.4949i			0.807134i
$922$		0			0
$923$	−13.8564			−0.456089
$924$		0			0
$925$	−2.00000			−0.0657596
$926$		0			0
$927$			12.7279i			0.418040i
$928$		0			0
$929$	−36.0000			−1.18112			−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000			−1.18112			−0.590561	+	0.806993i	$0.701093\pi$
$930$		0			0
$931$	−17.3205			−0.567657
$932$		0			0
$933$	16.0000			0.523816
$934$		0			0
$935$	−20.7846	−	12.7279i	−0.679729	−	0.416248i
$936$		0			0
$937$		−	2.44949i		−	0.0800213i	−0.999199	−	0.0400107i	$-0.987261\pi$
$937$		−	2.44949i		−	0.0800213i	0.999199	−	0.0400107i	$-0.0127392\pi$
$938$		0			0
$939$		−	31.1127i		−	1.01532i
$940$		0			0
$941$			48.9898i			1.59702i	0.601980	+	0.798511i	$0.294379\pi$
$941$			48.9898i			1.59702i	−0.601980	+	0.798511i	$0.705621\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$			19.5959i			0.637455i
$946$		0			0
$947$			26.8701i			0.873160i	0.899666	+	0.436580i	$0.143810\pi$
$947$			26.8701i			0.873160i	−0.899666	+	0.436580i	$0.856190\pi$
$948$		0			0
$949$	−18.0000			−0.584305
$950$		0			0
$951$			42.4264i			1.37577i
$952$		0			0
$953$			31.8434i			1.03151i	0.856737	+	0.515754i	$0.172488\pi$
$953$			31.8434i			1.03151i	−0.856737	+	0.515754i	$0.827512\pi$
$954$		0			0
$955$			22.6274i			0.732206i
$956$		0			0
$957$	−12.0000	+	19.5959i	−0.387905	+	0.633446i
$958$		0			0
$959$	−62.3538			−2.01351
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	−17.3205			−0.558146
$964$		0			0
$965$		−	7.34847i		−	0.236556i
$966$		0			0
$967$	17.3205			0.556990			0.278495	−	0.960438i	$-0.410164\pi$
$967$	17.3205			0.556990			0.278495	+	0.960438i	$0.410164\pi$
$968$		0			0
$969$	36.0000			1.15649
$970$		0			0
$971$		−	5.65685i		−	0.181537i	−0.995872	−	0.0907685i	$-0.971068\pi$
$971$		−	5.65685i		−	0.181537i	0.995872	−	0.0907685i	$-0.0289323\pi$
$972$		0			0
$973$	−24.0000			−0.769405
$974$		0			0
$975$	3.46410			0.110940
$976$		0			0
$977$	30.0000			0.959785			0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000			0.959785			0.479893	+	0.877327i	$0.340676\pi$
$978$		0			0
$979$	20.7846	−	33.9411i	0.664279	−	1.08476i
$980$		0			0
$981$		−	14.6969i		−	0.469237i
$982$		0			0
$983$		−	9.89949i		−	0.315745i	−0.987460	−	0.157872i	$-0.949537\pi$
$983$		−	9.89949i		−	0.315745i	0.987460	−	0.157872i	$-0.0504635\pi$
$984$		0			0
$985$		−	2.44949i		−	0.0780472i
$986$		0			0
$987$	34.6410			1.10264
$988$		0			0
$989$			4.89898i			0.155778i
$990$		0			0
$991$		−	16.9706i		−	0.539088i	−0.962988	−	0.269544i	$-0.913127\pi$
$991$		−	16.9706i		−	0.539088i	0.962988	−	0.269544i	$-0.0868729\pi$
$992$		0			0
$993$	−48.0000			−1.52323
$994$		0			0
$995$			25.4558i			0.807005i
$996$		0			0
$997$		−	31.8434i		−	1.00849i	−0.863561	−	0.504245i	$-0.831771\pi$
$997$		−	31.8434i		−	1.00849i	0.863561	−	0.504245i	$-0.168229\pi$
$998$		0			0
$999$			11.3137i			0.357950i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.f.c.351.1	✓	4
4.3	odd	2	inner	880.2.f.c.351.4	yes	4
8.3	odd	2		3520.2.f.g.2111.2		4
8.5	even	2		3520.2.f.g.2111.3		4
11.10	odd	2	inner	880.2.f.c.351.2	yes	4
44.43	even	2	inner	880.2.f.c.351.3	yes	4
88.21	odd	2		3520.2.f.g.2111.4		4
88.43	even	2		3520.2.f.g.2111.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
880.2.f.c.351.1	✓	4	1.1	even	1	trivial
880.2.f.c.351.2	yes	4	11.10	odd	2	inner
880.2.f.c.351.3	yes	4	44.43	even	2	inner
880.2.f.c.351.4	yes	4	4.3	odd	2	inner
3520.2.f.g.2111.1		4	88.43	even	2
3520.2.f.g.2111.2		4	8.3	odd	2
3520.2.f.g.2111.3		4	8.5	even	2
3520.2.f.g.2111.4		4	88.21	odd	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:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{3} +1.00000 q^{5} -3.46410 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{3} +1.00000 q^{5} -3.46410 q^{7} +1.00000 q^{9} +(1.73205 - 2.82843i) q^{11} +2.44949i q^{13} -1.41421i q^{15} -7.34847i q^{17} -3.46410 q^{19} +4.89898i q^{21} -1.41421i q^{23} +1.00000 q^{25} -5.65685i q^{27} -4.89898i q^{29} +(-4.00000 - 2.44949i) q^{33} -3.46410 q^{35} -2.00000 q^{37} +3.46410 q^{39} -3.46410 q^{43} +1.00000 q^{45} -7.07107i q^{47} +5.00000 q^{49} -10.3923 q^{51} -6.00000 q^{53} +(1.73205 - 2.82843i) q^{55} +4.89898i q^{57} +2.82843i q^{59} +9.79796i q^{61} -3.46410 q^{63} +2.44949i q^{65} -12.7279i q^{67} -2.00000 q^{69} +5.65685i q^{71} +7.34847i q^{73} -1.41421i q^{75} +(-6.00000 + 9.79796i) q^{77} +10.3923 q^{79} -5.00000 q^{81} +10.3923 q^{83} -7.34847i q^{85} -6.92820 q^{87} +12.0000 q^{89} -8.48528i q^{91} -3.46410 q^{95} -2.00000 q^{97} +(1.73205 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 4 * q^9	\( 4 q + 4 q^{5} + 4 q^{9} + 4 q^{25} - 16 q^{33} - 8 q^{37} + 4 q^{45} + 20 q^{49} - 24 q^{53} - 8 q^{69} - 24 q^{77} - 20 q^{81} + 48 q^{89} - 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^9 + 4 * q^25 - 16 * q^33 - 8 * q^37 + 4 * q^45 + 20 * q^49 - 24 * q^53 - 8 * q^69 - 24 * q^77 - 20 * q^81 + 48 * q^89 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more