LMFDB - Embedded newform 4400.2.b.h.4049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.h.4049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +4.00000i q^{13} +2.00000i q^{17} +2.00000 q^{21} +1.00000i q^{23} +5.00000i q^{27} -7.00000 q^{31} -1.00000i q^{33} -3.00000i q^{37} -4.00000 q^{39} -8.00000 q^{41} +6.00000i q^{43} +8.00000i q^{47} +3.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} +5.00000 q^{59} +12.0000 q^{61} -4.00000i q^{63} -7.00000i q^{67} -1.00000 q^{69} +3.00000 q^{71} +4.00000i q^{73} +2.00000i q^{77} -10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} -15.0000 q^{89} +8.00000 q^{91} -7.00000i q^{93} +7.00000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^9	$ 2 q + 4 q^{9} - 2 q^{11} + 4 q^{21} - 14 q^{31} - 8 q^{39} - 16 q^{41} + 6 q^{49} - 4 q^{51} + 10 q^{59} + 24 q^{61} - 2 q^{69} + 6 q^{71} - 20 q^{79} + 2 q^{81} - 30 q^{89} + 16 q^{91} - 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 2 * q^11 + 4 * q^21 - 14 * q^31 - 8 * q^39 - 16 * q^41 + 6 * q^49 - 4 * q^51 + 10 * q^59 + 24 * q^61 - 2 * q^69 + 6 * q^71 - 20 * q^79 + 2 * q^81 - 30 * q^89 + 16 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	1.00000i	0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$	1.00000i	0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	0	0
$33$	− 1.00000i	− 0.174078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	0	0
$63$	− 4.00000i	− 0.503953i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	− 7.00000i	− 0.725866i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.00000i	0.710742i	0.934725	+	0.355371i	$0.115646\pi$
$97$	7.00000i	0.710742i	−0.934725	+	0.355371i	$0.884354\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.0000i	1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$	18.0000i	1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	0	0
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.00000i	0.739600i
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 8.00000i	− 0.721336i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.00000i	0.598050i	0.954245	+	0.299025i	$0.0966615\pi$
$137$	7.00000i	0.598050i	−0.954245	+	0.299025i	$0.903339\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	− 4.00000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.00000i	0.247436i
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	4.00000i	0.323381i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.00000i	0.558661i	0.960195	+	0.279330i	$0.0901125\pi$
$157$	7.00000i	0.558661i	−0.960195	+	0.279330i	$0.909888\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.00000i	0.375823i
$178$	0	0
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	12.0000i	0.887066i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	0	0
$189$	10.0000	0.727393
$190$	0	0
$191$	−17.0000	−1.23008	−0.615038	−	0.788497i	$-0.710860\pi$
$191$	−17.0000	−1.23008	−0.615038	+	0.788497i	$0.710860\pi$
$192$	0	0
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	7.00000	0.493742
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.00000i	0.139010i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	3.00000i	0.205557i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	14.0000i	0.950382i
$218$	0	0
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	0	0
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 10.0000i	− 0.649570i
$238$	0	0
$239$	−30.0000	−1.94054	−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000	−1.94054	−0.970269	+	0.242028i	$0.922188\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	0	0
$253$	− 1.00000i	− 0.0628695i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 14.0000i	− 0.863277i	−0.902047	−	0.431638i	$-0.857936\pi$
$263$	− 14.0000i	− 0.863277i	0.902047	−	0.431638i	$-0.142064\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 15.0000i	− 0.917985i
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	0	0
$273$	8.00000i	0.484182i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	0	0
$279$	−14.0000	−0.838158
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.0000i	0.944450i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	−7.00000	−0.410347
$292$	0	0
$293$	24.0000i	1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$	24.0000i	1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.00000i	− 0.290129i
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	2.00000i	0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.00000i	0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$	8.00000i	0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	− 1.00000i	− 0.0565233i	−0.999601	−	0.0282617i	$-0.991003\pi$
$313$	− 1.00000i	− 0.0565233i	0.999601	−	0.0282617i	$-0.00899717\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 13.0000i	− 0.730153i	−0.930978	−	0.365076i	$-0.881043\pi$
$317$	− 13.0000i	− 0.730153i	0.930978	−	0.365076i	$-0.118957\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 10.0000i	− 0.553001i
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	0	0
$339$	−9.00000	−0.488813
$340$	0	0
$341$	7.00000	0.379071
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.0000i	1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$	28.0000i	1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	−20.0000	−1.06752
$352$	0	0
$353$	− 21.0000i	− 1.11772i	−0.829263	−	0.558859i	$-0.811239\pi$
$353$	− 21.0000i	− 1.11772i	0.829263	−	0.558859i	$-0.188761\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.00000i	0.211702i
$358$	0	0
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	1.00000i	0.0524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.0000i	− 0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$	− 17.0000i	− 0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$	0	0
$369$	−16.0000	−0.832927
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	1.00000i	0.0510976i	0.999674	+	0.0255488i	$0.00813332\pi$
$383$	1.00000i	0.0510976i	−0.999674	+	0.0255488i	$0.991867\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.0000i	0.609994i
$388$	0	0
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	0	0
$393$	18.0000i	0.907980i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	− 28.0000i	− 1.39478i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.00000i	0.148704i
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	−7.00000	−0.345285
$412$	0	0
$413$	− 10.0000i	− 0.492068i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10.0000i	0.489702i
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	16.0000i	0.777947i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 24.0000i	− 1.16144i
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	− 11.0000i	− 0.528626i	−0.964437	−	0.264313i	$-0.914855\pi$
$433$	− 11.0000i	− 0.528626i	0.964437	−	0.264313i	$-0.0851452\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	11.0000i	0.522626i	0.965254	+	0.261313i	$0.0841554\pi$
$443$	11.0000i	0.522626i	−0.965254	+	0.261313i	$0.915845\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.0000i	0.472984i
$448$	0	0
$449$	−35.0000	−1.65175	−0.825876	−	0.563852i	$-0.809319\pi$
$449$	−35.0000	−1.65175	−0.825876	+	0.563852i	$0.809319\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	− 2.00000i	− 0.0939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.0000i	0.561336i	0.959805	+	0.280668i	$0.0905560\pi$
$457$	12.0000i	0.561336i	−0.959805	+	0.280668i	$0.909444\pi$
$458$	0	0
$459$	−10.0000	−0.466760
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	11.0000i	0.511213i	0.966781	+	0.255607i	$0.0822752\pi$
$463$	11.0000i	0.511213i	−0.966781	+	0.255607i	$0.917725\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 27.0000i	− 1.24941i	−0.780860	−	0.624705i	$-0.785219\pi$
$467$	− 27.0000i	− 1.24941i	0.780860	−	0.624705i	$-0.214781\pi$
$468$	0	0
$469$	−14.0000	−0.646460
$470$	0	0
$471$	−7.00000	−0.322543
$472$	0	0
$473$	− 6.00000i	− 0.275880i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 12.0000i	− 0.549442i
$478$	0	0
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	2.00000i	0.0910032i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	23.0000i	1.04223i	0.853487	+	0.521115i	$0.174484\pi$
$487$	23.0000i	1.04223i	−0.853487	+	0.521115i	$0.825516\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.00000i	− 0.269137i
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	26.0000i	1.15928i	0.814872	+	0.579641i	$0.196807\pi$
$503$	26.0000i	1.15928i	−0.814872	+	0.579641i	$0.803193\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 3.00000i	− 0.133235i
$508$	0	0
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 14.0000i	− 0.609850i
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	10.0000	0.433963
$532$	0	0
$533$	− 32.0000i	− 1.38607i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15.0000i	− 0.647298i
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	0	0
$543$	7.00000i	0.300399i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	24.0000	1.02430
$550$	0	0
$551$	0	0
$552$	0	0
$553$	20.0000i	0.850487i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	− 17.0000i	− 0.710185i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 33.0000i	− 1.37381i	−0.726748	−	0.686904i	$-0.758969\pi$
$577$	− 33.0000i	− 1.37381i	0.726748	−	0.686904i	$-0.241031\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	0	0
$593$	44.0000i	1.80686i	0.428732	+	0.903432i	$0.358960\pi$
$593$	44.0000i	1.80686i	−0.428732	+	0.903432i	$0.641040\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	− 14.0000i	− 0.570124i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 22.0000i	− 0.892952i	−0.894795	−	0.446476i	$-0.852679\pi$
$607$	− 22.0000i	− 0.892952i	0.894795	−	0.446476i	$-0.147321\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.0000	−1.29458
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	−25.0000	−1.00483	−0.502417	−	0.864625i	$-0.667556\pi$
$619$	−25.0000	−1.00483	−0.502417	+	0.864625i	$0.667556\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	30.0000i	1.20192i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.00000	0.239236
$630$	0	0
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	0	0
$633$	− 12.0000i	− 0.476957i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	− 29.0000i	− 1.14365i	−0.820376	−	0.571824i	$-0.806236\pi$
$643$	− 29.0000i	− 1.14365i	0.820376	−	0.571824i	$-0.193764\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 7.00000i	− 0.275198i	−0.990488	−	0.137599i	$-0.956061\pi$
$647$	− 7.00000i	− 0.275198i	0.990488	−	0.137599i	$-0.0439386\pi$
$648$	0	0
$649$	−5.00000	−0.196267
$650$	0	0
$651$	−14.0000	−0.548703
$652$	0	0
$653$	− 41.0000i	− 1.60445i	−0.597019	−	0.802227i	$-0.703648\pi$
$653$	− 41.0000i	− 1.60445i	0.597019	−	0.802227i	$-0.296352\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.00000i	0.312110i
$658$	0	0
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	0	0
$661$	37.0000	1.43913	0.719567	−	0.694423i	$-0.244340\pi$
$661$	37.0000	1.43913	0.719567	+	0.694423i	$0.244340\pi$
$662$	0	0
$663$	− 8.00000i	− 0.310694i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	19.0000	0.734582
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	14.0000i	0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$	14.0000i	0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	16.0000i	0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$	16.0000i	0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 15.0000i	− 0.572286i
$688$	0	0
$689$	24.0000	0.914327
$690$	0	0
$691$	−17.0000	−0.646710	−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000	−0.646710	−0.323355	+	0.946278i	$0.604811\pi$
$692$	0	0
$693$	4.00000i	0.151947i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 16.0000i	− 0.606043i
$698$	0	0
$699$	−24.0000	−0.907763
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.00000i	− 0.150435i
$708$	0	0
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	−20.0000	−0.750059
$712$	0	0
$713$	− 7.00000i	− 0.262152i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 30.0000i	− 1.12037i
$718$	0	0
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	0	0
$723$	− 8.00000i	− 0.297523i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.00000i	0.111264i	0.998451	+	0.0556319i	$0.0177173\pi$
$727$	3.00000i	0.111264i	−0.998451	+	0.0556319i	$0.982283\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	− 36.0000i	− 1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$	− 36.0000i	− 1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.00000i	0.257848i
$738$	0	0
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 4.00000i	− 0.146746i	−0.997305	−	0.0733729i	$-0.976624\pi$
$743$	− 4.00000i	− 0.146746i	0.997305	−	0.0733729i	$-0.0233763\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	36.0000	1.31541
$750$	0	0
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	0	0
$753$	23.0000i	0.838167i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	0	0
$759$	1.00000	0.0362977
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	20.0000i	0.724049i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.0000i	0.722158i
$768$	0	0
$769$	−20.0000	−0.721218	−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000	−0.721218	−0.360609	+	0.932717i	$0.617431\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 6.00000i	− 0.215249i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−3.00000	−0.107348
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	0	0
$789$	14.0000	0.498413
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	48.0000i	1.70453i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 53.0000i	− 1.87736i	−0.344795	−	0.938678i	$-0.612051\pi$
$797$	− 53.0000i	− 1.87736i	0.344795	−	0.938678i	$-0.387949\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	− 4.00000i	− 0.141157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 10.0000i	− 0.352017i
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	28.0000i	0.982003i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	16.0000	0.559085
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	− 39.0000i	− 1.35945i	−0.733465	−	0.679727i	$-0.762098\pi$
$823$	− 39.0000i	− 1.35945i	0.733465	−	0.679727i	$-0.237902\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 52.0000i	− 1.80822i	−0.427303	−	0.904109i	$-0.640536\pi$
$827$	− 52.0000i	− 1.80822i	0.427303	−	0.904109i	$-0.359464\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 35.0000i	− 1.20978i
$838$	0	0
$839$	−5.00000	−0.172619	−0.0863096	−	0.996268i	$-0.527507\pi$
$839$	−5.00000	−0.172619	−0.0863096	+	0.996268i	$0.527507\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	3.00000	0.102839
$852$	0	0
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 8.00000i	− 0.273275i	−0.990621	−	0.136637i	$-0.956370\pi$
$857$	− 8.00000i	− 0.273275i	0.990621	−	0.136637i	$-0.0436295\pi$
$858$	0	0
$859$	−15.0000	−0.511793	−0.255897	−	0.966704i	$-0.582371\pi$
$859$	−15.0000	−0.511793	−0.255897	+	0.966704i	$0.582371\pi$
$860$	0	0
$861$	−16.0000	−0.545279
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	28.0000	0.948744
$872$	0	0
$873$	14.0000i	0.473828i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12.0000i	0.405211i	0.979260	+	0.202606i	$0.0649409\pi$
$877$	12.0000i	0.405211i	−0.979260	+	0.202606i	$0.935059\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	−43.0000	−1.44871	−0.724353	−	0.689429i	$-0.757862\pi$
$881$	−43.0000	−1.44871	−0.724353	+	0.689429i	$0.757862\pi$
$882$	0	0
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 22.0000i	− 0.738688i	−0.929293	−	0.369344i	$-0.879582\pi$
$887$	− 22.0000i	− 0.738688i	0.929293	−	0.369344i	$-0.120418\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4.00000i	− 0.133556i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	12.0000i	0.399335i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	0	0
$909$	4.00000	0.132672
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	− 6.00000i	− 0.198571i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 36.0000i	− 1.18882i
$918$	0	0
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	0	0
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	32.0000i	1.05102i
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 12.0000i	− 0.392862i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 8.00000i	− 0.261349i	−0.991425	−	0.130674i	$-0.958286\pi$
$937$	− 8.00000i	− 0.261349i	0.991425	−	0.130674i	$-0.0417142\pi$
$938$	0	0
$939$	1.00000	0.0326338
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	− 8.00000i	− 0.260516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	0	0
$951$	13.0000	0.421554
$952$	0	0
$953$	34.0000i	1.10137i	0.834714	+	0.550684i	$0.185633\pi$
$953$	34.0000i	1.10137i	−0.834714	+	0.550684i	$0.814367\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.0000	0.452084
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	36.0000i	1.16008i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−47.0000	−1.50830	−0.754151	−	0.656701i	$-0.771951\pi$
$971$	−47.0000	−1.50830	−0.754151	+	0.656701i	$0.771951\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.0000i	0.863807i	0.901920	+	0.431903i	$0.142158\pi$
$977$	27.0000i	0.863807i	−0.901920	+	0.431903i	$0.857842\pi$
$978$	0	0
$979$	15.0000	0.479402
$980$	0	0
$981$	−20.0000	−0.638551
$982$	0	0
$983$	− 39.0000i	− 1.24391i	−0.783054	−	0.621953i	$-0.786339\pi$
$983$	− 39.0000i	− 1.24391i	0.783054	−	0.621953i	$-0.213661\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.0000i	0.509286i
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	− 7.00000i	− 0.222138i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38.0000i	− 1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$	− 38.0000i	− 1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$	0	0
$999$	15.0000	0.474579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.h.4049.2		2
4.3	odd	2		275.2.b.a.199.2		2
5.2	odd	4		176.2.a.b.1.1		1
5.3	odd	4		4400.2.a.i.1.1		1
5.4	even	2	inner	4400.2.b.h.4049.1		2
12.11	even	2		2475.2.c.a.199.1		2
15.2	even	4		1584.2.a.g.1.1		1
20.3	even	4		275.2.a.b.1.1		1
20.7	even	4		11.2.a.a.1.1	✓	1
20.19	odd	2		275.2.b.a.199.1		2
35.27	even	4		8624.2.a.j.1.1		1
40.27	even	4		704.2.a.h.1.1		1
40.37	odd	4		704.2.a.c.1.1		1
55.32	even	4		1936.2.a.i.1.1		1
60.23	odd	4		2475.2.a.a.1.1		1
60.47	odd	4		99.2.a.d.1.1		1
60.59	even	2		2475.2.c.a.199.2		2
80.27	even	4		2816.2.c.j.1409.2		2
80.37	odd	4		2816.2.c.f.1409.1		2
80.67	even	4		2816.2.c.j.1409.1		2
80.77	odd	4		2816.2.c.f.1409.2		2
120.77	even	4		6336.2.a.bu.1.1		1
120.107	odd	4		6336.2.a.br.1.1		1
140.27	odd	4		539.2.a.a.1.1		1
140.47	odd	12		539.2.e.g.67.1		2
140.67	even	12		539.2.e.h.177.1		2
140.87	odd	12		539.2.e.g.177.1		2
140.107	even	12		539.2.e.h.67.1		2
180.7	even	12		891.2.e.k.595.1		2
180.47	odd	12		891.2.e.b.595.1		2
180.67	even	12		891.2.e.k.298.1		2
180.167	odd	12		891.2.e.b.298.1		2
220.7	odd	20		121.2.c.a.27.1		4
220.27	even	20		121.2.c.e.3.1		4
220.43	odd	4		3025.2.a.a.1.1		1
220.47	even	20		121.2.c.e.9.1		4
220.87	odd	4		121.2.a.d.1.1		1
220.107	odd	20		121.2.c.a.9.1		4
220.127	odd	20		121.2.c.a.3.1		4
220.147	even	20		121.2.c.e.27.1		4
220.167	odd	20		121.2.c.a.81.1		4
220.207	even	20		121.2.c.e.81.1		4
260.207	even	4		1859.2.a.b.1.1		1
340.67	even	4		3179.2.a.a.1.1		1
380.227	odd	4		3971.2.a.b.1.1		1
420.167	even	4		4851.2.a.t.1.1		1
440.197	even	4		7744.2.a.k.1.1		1
440.307	odd	4		7744.2.a.x.1.1		1
460.367	odd	4		5819.2.a.a.1.1		1
580.347	even	4		9251.2.a.d.1.1		1
660.527	even	4		1089.2.a.b.1.1		1
1540.307	even	4		5929.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.2.a.a.1.1	✓	1	20.7	even	4
99.2.a.d.1.1		1	60.47	odd	4
121.2.a.d.1.1		1	220.87	odd	4
121.2.c.a.3.1		4	220.127	odd	20
121.2.c.a.9.1		4	220.107	odd	20
121.2.c.a.27.1		4	220.7	odd	20
121.2.c.a.81.1		4	220.167	odd	20
121.2.c.e.3.1		4	220.27	even	20
121.2.c.e.9.1		4	220.47	even	20
121.2.c.e.27.1		4	220.147	even	20
121.2.c.e.81.1		4	220.207	even	20
176.2.a.b.1.1		1	5.2	odd	4
275.2.a.b.1.1		1	20.3	even	4
275.2.b.a.199.1		2	20.19	odd	2
275.2.b.a.199.2		2	4.3	odd	2
539.2.a.a.1.1		1	140.27	odd	4
539.2.e.g.67.1		2	140.47	odd	12
539.2.e.g.177.1		2	140.87	odd	12
539.2.e.h.67.1		2	140.107	even	12
539.2.e.h.177.1		2	140.67	even	12
704.2.a.c.1.1		1	40.37	odd	4
704.2.a.h.1.1		1	40.27	even	4
891.2.e.b.298.1		2	180.167	odd	12
891.2.e.b.595.1		2	180.47	odd	12
891.2.e.k.298.1		2	180.67	even	12
891.2.e.k.595.1		2	180.7	even	12
1089.2.a.b.1.1		1	660.527	even	4
1584.2.a.g.1.1		1	15.2	even	4
1859.2.a.b.1.1		1	260.207	even	4
1936.2.a.i.1.1		1	55.32	even	4
2475.2.a.a.1.1		1	60.23	odd	4
2475.2.c.a.199.1		2	12.11	even	2
2475.2.c.a.199.2		2	60.59	even	2
2816.2.c.f.1409.1		2	80.37	odd	4
2816.2.c.f.1409.2		2	80.77	odd	4
2816.2.c.j.1409.1		2	80.67	even	4
2816.2.c.j.1409.2		2	80.27	even	4
3025.2.a.a.1.1		1	220.43	odd	4
3179.2.a.a.1.1		1	340.67	even	4
3971.2.a.b.1.1		1	380.227	odd	4
4400.2.a.i.1.1		1	5.3	odd	4
4400.2.b.h.4049.1		2	5.4	even	2	inner
4400.2.b.h.4049.2		2	1.1	even	1	trivial
4851.2.a.t.1.1		1	420.167	even	4
5819.2.a.a.1.1		1	460.367	odd	4
5929.2.a.h.1.1		1	1540.307	even	4
6336.2.a.br.1.1		1	120.107	odd	4
6336.2.a.bu.1.1		1	120.77	even	4
7744.2.a.k.1.1		1	440.197	even	4
7744.2.a.x.1.1		1	440.307	odd	4
8624.2.a.j.1.1		1	35.27	even	4
9251.2.a.d.1.1		1	580.347	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +4.00000i q^{13} +2.00000i q^{17} +2.00000 q^{21} +1.00000i q^{23} +5.00000i q^{27} -7.00000 q^{31} -1.00000i q^{33} -3.00000i q^{37} -4.00000 q^{39} -8.00000 q^{41} +6.00000i q^{43} +8.00000i q^{47} +3.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} +5.00000 q^{59} +12.0000 q^{61} -4.00000i q^{63} -7.00000i q^{67} -1.00000 q^{69} +3.00000 q^{71} +4.00000i q^{73} +2.00000i q^{77} -10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} -15.0000 q^{89} +8.00000 q^{91} -7.00000i q^{93} +7.00000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^9	\( 2 q + 4 q^{9} - 2 q^{11} + 4 q^{21} - 14 q^{31} - 8 q^{39} - 16 q^{41} + 6 q^{49} - 4 q^{51} + 10 q^{59} + 24 q^{61} - 2 q^{69} + 6 q^{71} - 20 q^{79} + 2 q^{81} - 30 q^{89} + 16 q^{91} - 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 2 * q^11 + 4 * q^21 - 14 * q^31 - 8 * q^39 - 16 * q^41 + 6 * q^49 - 4 * q^51 + 10 * q^59 + 24 * q^61 - 2 * q^69 + 6 * q^71 - 20 * q^79 + 2 * q^81 - 30 * q^89 + 16 * q^91 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more