LMFDB - Embedded newform 225.2.b.a.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("225.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	225.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:	$1.79663404548$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.2.b.a.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{7} +O(q^{10})$	$q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{7} -2.00000 q^{11} +1.00000i q^{13} -6.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} +5.00000 q^{19} -4.00000i q^{22} -6.00000i q^{23} -2.00000 q^{26} -6.00000i q^{28} +10.0000 q^{29} -3.00000 q^{31} -8.00000i q^{32} -4.00000 q^{34} -2.00000i q^{37} +10.0000i q^{38} +8.00000 q^{41} +1.00000i q^{43} +4.00000 q^{44} +12.0000 q^{46} +2.00000i q^{47} -2.00000 q^{49} -2.00000i q^{52} +4.00000i q^{53} +20.0000i q^{58} -10.0000 q^{59} +7.00000 q^{61} -6.00000i q^{62} +8.00000 q^{64} +3.00000i q^{67} -4.00000i q^{68} +8.00000 q^{71} -14.0000i q^{73} +4.00000 q^{74} -10.0000 q^{76} -6.00000i q^{77} +16.0000i q^{82} -6.00000i q^{83} -2.00000 q^{86} -3.00000 q^{91} +12.0000i q^{92} -4.00000 q^{94} -17.0000i q^{97} -4.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4}+O(q^{10}) $ 2 * q - 4 * q^4	$ 2 q - 4 q^{4} - 4 q^{11} - 12 q^{14} - 8 q^{16} + 10 q^{19} - 4 q^{26} + 20 q^{29} - 6 q^{31} - 8 q^{34} + 16 q^{41} + 8 q^{44} + 24 q^{46} - 4 q^{49} - 20 q^{59} + 14 q^{61} + 16 q^{64} + 16 q^{71} + 8 q^{74} - 20 q^{76} - 4 q^{86} - 6 q^{91} - 8 q^{94}+O(q^{100}) $ 2 * q - 4 * q^4 - 4 * q^11 - 12 * q^14 - 8 * q^16 + 10 * q^19 - 4 * q^26 + 20 * q^29 - 6 * q^31 - 8 * q^34 + 16 * q^41 + 8 * q^44 + 24 * q^46 - 4 * q^49 - 20 * q^59 + 14 * q^61 + 16 * q^64 + 16 * q^71 + 8 * q^74 - 20 * q^76 - 4 * q^86 - 6 * q^91 - 8 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$	2.00000i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	−6.00000	−1.60357
$15$	0	0
$16$	−4.00000	−1.00000
$17$	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
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	− 6.00000i	− 1.13389i
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	− 8.00000i	− 1.41421i
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	10.0000i	1.62221i
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	12.0000	1.76930
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	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
$55$	0	0
$56$	0	0
$57$	0	0
$58$	20.0000i	2.62613i
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	− 6.00000i	− 0.762001i
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	3.00000i	0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$	3.00000i	0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$	− 4.00000i	− 0.485071i
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−10.0000	−1.14708
$77$	− 6.00000i	− 0.683763i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	16.0000i	1.76690i
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	12.0000i	1.25109i
$93$	0	0
$94$	−4.00000	−0.412568
$95$	0	0
$96$	0	0
$97$	− 17.0000i	− 1.72609i	−0.505128	−	0.863044i	$-0.668555\pi$
$97$	− 17.0000i	− 1.72609i	0.505128	−	0.863044i	$-0.331445\pi$
$98$	− 4.00000i	− 0.404061i
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	−8.00000	−0.777029
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	0	0
$112$	− 12.0000i	− 1.13389i
$113$	4.00000i	0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$	4.00000i	0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	0	0
$115$	0	0
$116$	−20.0000	−1.85695
$117$	0	0
$118$	− 20.0000i	− 1.84115i
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−7.00000	−0.636364
$122$	14.0000i	1.26750i
$123$	0	0
$124$	6.00000	0.538816
$125$	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
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	15.0000i	1.30066i
$134$	−6.00000	−0.518321
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	16.0000i	1.34269i
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	28.0000	2.31730
$147$	0	0
$148$	4.00000i	0.328798i
$149$	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
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	0	0
$153$	0	0
$154$	12.0000	0.966988
$155$	0	0
$156$	0	0
$157$	13.0000i	1.03751i	0.854922	+	0.518756i	$0.173605\pi$
$157$	13.0000i	1.03751i	−0.854922	+	0.518756i	$0.826395\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.0000	1.41860
$162$	0	0
$163$	11.0000i	0.861586i	0.902451	+	0.430793i	$0.141766\pi$
$163$	11.0000i	0.861586i	−0.902451	+	0.430793i	$0.858234\pi$
$164$	−16.0000	−1.24939
$165$	0	0
$166$	12.0000	0.931381
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	− 2.00000i	− 0.152499i
$173$	− 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
$176$	8.00000	0.603023
$177$	0	0
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	− 6.00000i	− 0.444750i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	− 4.00000i	− 0.291730i
$189$	0	0
$190$	0	0
$191$	−22.0000	−1.59186	−0.795932	−	0.605386i	$-0.793019\pi$
$191$	−22.0000	−1.59186	−0.795932	+	0.605386i	$0.793019\pi$
$192$	0	0
$193$	11.0000i	0.791797i	0.918294	+	0.395899i	$0.129567\pi$
$193$	11.0000i	0.791797i	−0.918294	+	0.395899i	$0.870433\pi$
$194$	34.0000	2.44106
$195$	0	0
$196$	4.00000	0.285714
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	0	0
$201$	0	0
$202$	− 24.0000i	− 1.68863i
$203$	30.0000i	2.10559i
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	0	0
$208$	− 4.00000i	− 0.277350i
$209$	−10.0000	−0.691714
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	− 8.00000i	− 0.549442i
$213$	0	0
$214$	−24.0000	−1.64061
$215$	0	0
$216$	0	0
$217$	− 9.00000i	− 0.610960i
$218$	− 10.0000i	− 0.677285i
$219$	0	0
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	24.0000	1.60357
$225$	0	0
$226$	−8.00000	−0.532152
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	0	0
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	0	0
$231$	0	0
$232$	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
$236$	20.0000	1.30189
$237$	0	0
$238$	− 12.0000i	− 0.777844i
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	− 14.0000i	− 0.899954i
$243$	0	0
$244$	−14.0000	−0.896258
$245$	0	0
$246$	0	0
$247$	5.00000i	0.318142i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	− 24.0000i	− 1.48272i
$263$	− 16.0000i	− 0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$	− 16.0000i	− 0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$	0	0
$265$	0	0
$266$	−30.0000	−1.83942
$267$	0	0
$268$	− 6.00000i	− 0.366508i
$269$	−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
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 8.00000i	− 0.485071i
$273$	0	0
$274$	36.0000	2.17484
$275$	0	0
$276$	0	0
$277$	3.00000i	0.180253i	0.995930	+	0.0901263i	$0.0287271\pi$
$277$	3.00000i	0.180253i	−0.995930	+	0.0901263i	$0.971273\pi$
$278$	− 40.0000i	− 2.39904i
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 9.00000i	− 0.534994i	−0.963559	−	0.267497i	$-0.913803\pi$
$283$	− 9.00000i	− 0.534994i	0.963559	−	0.267497i	$-0.0861966\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	4.00000	0.236525
$287$	24.0000i	1.41668i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	28.0000i	1.63858i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	20.0000i	1.15857i
$299$	6.00000	0.346989
$300$	0	0
$301$	−3.00000	−0.172917
$302$	14.0000i	0.805609i
$303$	0	0
$304$	−20.0000	−1.14708
$305$	0	0
$306$	0	0
$307$	− 7.00000i	− 0.399511i	−0.979846	−	0.199756i	$-0.935985\pi$
$307$	− 7.00000i	− 0.399511i	0.979846	−	0.199756i	$-0.0640148\pi$
$308$	12.0000i	0.683763i
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	11.0000i	0.621757i	0.950450	+	0.310878i	$0.100623\pi$
$313$	11.0000i	0.621757i	−0.950450	+	0.310878i	$0.899377\pi$
$314$	−26.0000	−1.46726
$315$	0	0
$316$	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
$319$	−20.0000	−1.11979
$320$	0	0
$321$	0	0
$322$	36.0000i	2.00620i
$323$	10.0000i	0.556415i
$324$	0	0
$325$	0	0
$326$	−22.0000	−1.21847
$327$	0	0
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	23.0000i	1.25289i	0.779466	+	0.626445i	$0.215491\pi$
$337$	23.0000i	1.25289i	−0.779466	+	0.626445i	$0.784509\pi$
$338$	24.0000i	1.30543i
$339$	0	0
$340$	0	0
$341$	6.00000	0.324918
$342$	0	0
$343$	15.0000i	0.809924i
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	2.00000i	0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$	2.00000i	0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	16.0000i	0.852803i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	− 20.0000i	− 1.05703i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	34.0000i	1.78700i
$363$	0	0
$364$	6.00000	0.314485
$365$	0	0
$366$	0	0
$367$	− 27.0000i	− 1.40939i	−0.709511	−	0.704694i	$-0.751084\pi$
$367$	− 27.0000i	− 1.40939i	0.709511	−	0.704694i	$-0.248916\pi$
$368$	24.0000i	1.25109i
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	− 29.0000i	− 1.50156i	−0.660551	−	0.750782i	$-0.729677\pi$
$373$	− 29.0000i	− 1.50156i	0.660551	−	0.750782i	$-0.270323\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	0	0
$377$	10.0000i	0.515026i
$378$	0	0
$379$	−25.0000	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$380$	0	0
$381$	0	0
$382$	− 44.0000i	− 2.25124i
$383$	− 36.0000i	− 1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$	− 36.0000i	− 1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$	0	0
$385$	0	0
$386$	−22.0000	−1.11977
$387$	0	0
$388$	34.0000i	1.72609i
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	36.0000	1.81365
$395$	0	0
$396$	0	0
$397$	− 7.00000i	− 0.351320i	−0.984451	−	0.175660i	$-0.943794\pi$
$397$	− 7.00000i	− 0.351320i	0.984451	−	0.175660i	$-0.0562059\pi$
$398$	10.0000i	0.501255i
$399$	0	0
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	− 3.00000i	− 0.149441i
$404$	24.0000	1.19404
$405$	0	0
$406$	−60.0000	−2.97775
$407$	4.00000i	0.198273i
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	0	0
$412$	8.00000i	0.394132i
$413$	− 30.0000i	− 1.47620i
$414$	0	0
$415$	0	0
$416$	8.00000	0.392232
$417$	0	0
$418$	− 20.0000i	− 0.978232i
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	− 26.0000i	− 1.26566i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	21.0000i	1.01626i
$428$	− 24.0000i	− 1.16008i
$429$	0	0
$430$	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
$433$	− 29.0000i	− 1.39365i	−0.717241	−	0.696826i	$-0.754595\pi$
$433$	− 29.0000i	− 1.39365i	0.717241	−	0.696826i	$-0.245405\pi$
$434$	18.0000	0.864028
$435$	0	0
$436$	10.0000	0.478913
$437$	− 30.0000i	− 1.43509i
$438$	0	0
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	0	0
$442$	− 4.00000i	− 0.190261i
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	0	0
$445$	0	0
$446$	38.0000	1.79935
$447$	0	0
$448$	24.0000i	1.13389i
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	− 8.00000i	− 0.376288i
$453$	0	0
$454$	16.0000	0.750917
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	30.0000i	1.40181i
$459$	0	0
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	−40.0000	−1.85695
$465$	0	0
$466$	−48.0000	−2.22356
$467$	− 38.0000i	− 1.75843i	−0.476425	−	0.879215i	$-0.658068\pi$
$467$	− 38.0000i	− 1.75843i	0.476425	−	0.879215i	$-0.341932\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 2.00000i	− 0.0919601i
$474$	0	0
$475$	0	0
$476$	12.0000	0.550019
$477$	0	0
$478$	40.0000i	1.82956i
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	− 46.0000i	− 2.09524i
$483$	0	0
$484$	14.0000	0.636364
$485$	0	0
$486$	0	0
$487$	13.0000i	0.589086i	0.955638	+	0.294543i	$0.0951675\pi$
$487$	13.0000i	0.589086i	−0.955638	+	0.294543i	$0.904833\pi$
$488$	0	0
$489$	0	0
$490$	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
$493$	20.0000i	0.900755i
$494$	−10.0000	−0.449921
$495$	0	0
$496$	12.0000	0.538816
$497$	24.0000i	1.07655i
$498$	0	0
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	0	0
$501$	0	0
$502$	− 24.0000i	− 1.07117i
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	0	0
$505$	0	0
$506$	−24.0000	−1.06693
$507$	0	0
$508$	− 16.0000i	− 0.709885i
$509$	−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
$511$	42.0000	1.85797
$512$	32.0000i	1.41421i
$513$	0	0
$514$	−24.0000	−1.05859
$515$	0	0
$516$	0	0
$517$	− 4.00000i	− 0.175920i
$518$	12.0000i	0.527250i
$519$	0	0
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	31.0000i	1.35554i	0.735276	+	0.677768i	$0.237052\pi$
$523$	31.0000i	1.35554i	−0.735276	+	0.677768i	$0.762948\pi$
$524$	24.0000	1.04844
$525$	0	0
$526$	32.0000	1.39527
$527$	− 6.00000i	− 0.261364i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	− 30.0000i	− 1.30066i
$533$	8.00000i	0.346518i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	− 20.0000i	− 0.862261i
$539$	4.00000	0.172292
$540$	0	0
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	− 16.0000i	− 0.687259i
$543$	0	0
$544$	16.0000	0.685994
$545$	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	$0.945291\pi$
$548$	36.0000i	1.53784i
$549$	0	0
$550$	0	0
$551$	50.0000	2.13007
$552$	0	0
$553$	0	0
$554$	−6.00000	−0.254916
$555$	0	0
$556$	40.0000	1.69638
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	0	0
$562$	36.0000i	1.51857i
$563$	− 6.00000i	− 0.252870i	−0.991975	−	0.126435i	$-0.959647\pi$
$563$	− 6.00000i	− 0.252870i	0.991975	−	0.126435i	$-0.0403535\pi$
$564$	0	0
$565$	0	0
$566$	18.0000	0.756596
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−13.0000	−0.544033	−0.272017	−	0.962293i	$-0.587691\pi$
$571$	−13.0000	−0.544033	−0.272017	+	0.962293i	$0.587691\pi$
$572$	4.00000i	0.167248i
$573$	0	0
$574$	−48.0000	−2.00348
$575$	0	0
$576$	0	0
$577$	13.0000i	0.541197i	0.962692	+	0.270599i	$0.0872216\pi$
$577$	13.0000i	0.541197i	−0.962692	+	0.270599i	$0.912778\pi$
$578$	26.0000i	1.08146i
$579$	0	0
$580$	0	0
$581$	18.0000	0.746766
$582$	0	0
$583$	− 8.00000i	− 0.331326i
$584$	0	0
$585$	0	0
$586$	12.0000	0.495715
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−15.0000	−0.618064
$590$	0	0
$591$	0	0
$592$	8.00000i	0.328798i
$593$	− 16.0000i	− 0.657041i	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	− 16.0000i	− 0.657041i	0.944497	−	0.328521i	$-0.106550\pi$
$594$	0	0
$595$	0	0
$596$	−20.0000	−0.819232
$597$	0	0
$598$	12.0000i	0.490716i
$599$	−20.0000	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	− 6.00000i	− 0.244542i
$603$	0	0
$604$	−14.0000	−0.569652
$605$	0	0
$606$	0	0
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	− 40.0000i	− 1.62221i
$609$	0	0
$610$	0	0
$611$	−2.00000	−0.0809113
$612$	0	0
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	14.0000	0.564994
$615$	0	0
$616$	0	0
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	0	0
$619$	25.0000	1.00483	0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000	1.00483	0.502417	+	0.864625i	$0.332444\pi$
$620$	0	0
$621$	0	0
$622$	36.0000i	1.44347i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	− 26.0000i	− 1.03751i
$629$	4.00000	0.159490
$630$	0	0
$631$	−23.0000	−0.915616	−0.457808	−	0.889051i	$-0.651365\pi$
$631$	−23.0000	−0.915616	−0.457808	+	0.889051i	$0.651365\pi$
$632$	0	0
$633$	0	0
$634$	16.0000	0.635441
$635$	0	0
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	− 40.0000i	− 1.58362i
$639$	0	0
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	36.0000i	1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$	36.0000i	1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$	−36.0000	−1.41860
$645$	0	0
$646$	−20.0000	−0.786889
$647$	− 28.0000i	− 1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$	− 28.0000i	− 1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	0	0
$652$	− 22.0000i	− 0.861586i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	0	0
$655$	0	0
$656$	−32.0000	−1.24939
$657$	0	0
$658$	− 12.0000i	− 0.467809i
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	24.0000i	0.932786i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 60.0000i	− 2.32321i
$668$	− 24.0000i	− 0.928588i
$669$	0	0
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	6.00000i	0.231283i	0.993291	+	0.115642i	$0.0368924\pi$
$673$	6.00000i	0.231283i	−0.993291	+	0.115642i	$0.963108\pi$
$674$	−46.0000	−1.77185
$675$	0	0
$676$	−24.0000	−0.923077
$677$	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
$679$	51.0000	1.95720
$680$	0	0
$681$	0	0
$682$	12.0000i	0.459504i
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	0	0
$685$	0	0
$686$	−30.0000	−1.14541
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	12.0000i	0.456172i
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	0	0
$697$	16.0000i	0.606043i
$698$	− 20.0000i	− 0.757011i
$699$	0	0
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	0	0
$703$	− 10.0000i	− 0.377157i
$704$	−16.0000	−0.603023
$705$	0	0
$706$	12.0000	0.451626
$707$	− 36.0000i	− 1.35392i
$708$	0	0
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	18.0000i	0.674105i
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	0	0
$718$	0	0
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	12.0000i	0.446594i
$723$	0	0
$724$	−34.0000	−1.26360
$725$	0	0
$726$	0	0
$727$	43.0000i	1.59478i	0.603463	+	0.797391i	$0.293787\pi$
$727$	43.0000i	1.59478i	−0.603463	+	0.797391i	$0.706213\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.00000	−0.0739727
$732$	0	0
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	54.0000	1.99318
$735$	0	0
$736$	−48.0000	−1.76930
$737$	− 6.00000i	− 0.221013i
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	− 24.0000i	− 0.881068i
$743$	4.00000i	0.146746i	0.997305	+	0.0733729i	$0.0233763\pi$
$743$	4.00000i	0.146746i	−0.997305	+	0.0733729i	$0.976624\pi$
$744$	0	0
$745$	0	0
$746$	58.0000	2.12353
$747$	0	0
$748$	8.00000i	0.292509i
$749$	−36.0000	−1.31541
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	− 8.00000i	− 0.291730i
$753$	0	0
$754$	−20.0000	−0.728357
$755$	0	0
$756$	0	0
$757$	23.0000i	0.835949i	0.908459	+	0.417975i	$0.137260\pi$
$757$	23.0000i	0.835949i	−0.908459	+	0.417975i	$0.862740\pi$
$758$	− 50.0000i	− 1.81608i
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	− 15.0000i	− 0.543036i
$764$	44.0000	1.59186
$765$	0	0
$766$	72.0000	2.60147
$767$	− 10.0000i	− 0.361079i
$768$	0	0
$769$	−35.0000	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$770$	0	0
$771$	0	0
$772$	− 22.0000i	− 0.791797i
$773$	24.0000i	0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$	24.0000i	0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	40.0000	1.43315
$780$	0	0
$781$	−16.0000	−0.572525
$782$	24.0000i	0.858238i
$783$	0	0
$784$	8.00000	0.285714
$785$	0	0
$786$	0	0
$787$	− 7.00000i	− 0.249523i	−0.992187	−	0.124762i	$-0.960183\pi$
$787$	− 7.00000i	− 0.249523i	0.992187	−	0.124762i	$-0.0398166\pi$
$788$	36.0000i	1.28245i
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	7.00000i	0.248577i
$794$	14.0000	0.496841
$795$	0	0
$796$	−10.0000	−0.354441
$797$	52.0000i	1.84193i	0.389640	+	0.920967i	$0.372599\pi$
$797$	52.0000i	1.84193i	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	0	0
$802$	− 24.0000i	− 0.847469i
$803$	28.0000i	0.988099i
$804$	0	0
$805$	0	0
$806$	6.00000	0.211341
$807$	0	0
$808$	0	0
$809$	−20.0000	−0.703163	−0.351581	−	0.936157i	$-0.614356\pi$
$809$	−20.0000	−0.703163	−0.351581	+	0.936157i	$0.614356\pi$
$810$	0	0
$811$	27.0000	0.948098	0.474049	−	0.880498i	$-0.342792\pi$
$811$	27.0000	0.948098	0.474049	+	0.880498i	$0.342792\pi$
$812$	− 60.0000i	− 2.10559i
$813$	0	0
$814$	−8.00000	−0.280400
$815$	0	0
$816$	0	0
$817$	5.00000i	0.174928i
$818$	− 10.0000i	− 0.349642i
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	41.0000i	1.42917i	0.699549	+	0.714585i	$0.253384\pi$
$823$	41.0000i	1.42917i	−0.699549	+	0.714585i	$0.746616\pi$
$824$	0	0
$825$	0	0
$826$	60.0000	2.08767
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	0	0
$832$	8.00000i	0.277350i
$833$	− 4.00000i	− 0.138592i
$834$	0	0
$835$	0	0
$836$	20.0000	0.691714
$837$	0	0
$838$	− 40.0000i	− 1.38178i
$839$	10.0000	0.345238	0.172619	−	0.984989i	$-0.444777\pi$
$839$	10.0000	0.345238	0.172619	+	0.984989i	$0.444777\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	44.0000i	1.51634i
$843$	0	0
$844$	26.0000	0.894957
$845$	0	0
$846$	0	0
$847$	− 21.0000i	− 0.721569i
$848$	− 16.0000i	− 0.549442i
$849$	0	0
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	51.0000i	1.74621i	0.487535	+	0.873103i	$0.337896\pi$
$853$	51.0000i	1.74621i	−0.487535	+	0.873103i	$0.662104\pi$
$854$	−42.0000	−1.43721
$855$	0	0
$856$	0	0
$857$	− 28.0000i	− 0.956462i	−0.878234	−	0.478231i	$-0.841278\pi$
$857$	− 28.0000i	− 0.956462i	0.878234	−	0.478231i	$-0.158722\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	36.0000i	1.22616i
$863$	− 16.0000i	− 0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$	− 16.0000i	− 0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	0	0
$865$	0	0
$866$	58.0000	1.97092
$867$	0	0
$868$	18.0000i	0.610960i
$869$	0	0
$870$	0	0
$871$	−3.00000	−0.101651
$872$	0	0
$873$	0	0
$874$	60.0000	2.02953
$875$	0	0
$876$	0	0
$877$	− 27.0000i	− 0.911725i	−0.890050	−	0.455863i	$-0.849331\pi$
$877$	− 27.0000i	− 0.911725i	0.890050	−	0.455863i	$-0.150669\pi$
$878$	70.0000i	2.36239i
$879$	0	0
$880$	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
$883$	41.0000i	1.37976i	0.723924	+	0.689880i	$0.242337\pi$
$883$	41.0000i	1.37976i	−0.723924	+	0.689880i	$0.757663\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−48.0000	−1.61259
$887$	− 18.0000i	− 0.604381i	−0.953248	−	0.302190i	$-0.902282\pi$
$887$	− 18.0000i	− 0.604381i	0.953248	−	0.302190i	$-0.0977178\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	0	0
$892$	38.0000i	1.27233i
$893$	10.0000i	0.334637i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	40.0000i	1.33482i
$899$	−30.0000	−1.00056
$900$	0	0
$901$	−8.00000	−0.266519
$902$	− 32.0000i	− 1.06548i
$903$	0	0
$904$	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	$-0.0638430\pi$
$908$	16.0000i	0.530979i
$909$	0	0
$910$	0	0
$911$	58.0000	1.92163	0.960813	−	0.277198i	$-0.0894057\pi$
$911$	58.0000	1.92163	0.960813	+	0.277198i	$0.0894057\pi$
$912$	0	0
$913$	12.0000i	0.397142i
$914$	44.0000	1.45539
$915$	0	0
$916$	−30.0000	−0.991228
$917$	− 36.0000i	− 1.18882i
$918$	0	0
$919$	−55.0000	−1.81428	−0.907141	−	0.420826i	$-0.861740\pi$
$919$	−55.0000	−1.81428	−0.907141	+	0.420826i	$0.861740\pi$
$920$	0	0
$921$	0	0
$922$	− 24.0000i	− 0.790398i
$923$	8.00000i	0.263323i
$924$	0	0
$925$	0	0
$926$	48.0000	1.57738
$927$	0	0
$928$	− 80.0000i	− 2.62613i
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	−10.0000	−0.327737
$932$	− 48.0000i	− 1.57229i
$933$	0	0
$934$	76.0000	2.48680
$935$	0	0
$936$	0	0
$937$	33.0000i	1.07806i	0.842286	+	0.539032i	$0.181210\pi$
$937$	33.0000i	1.07806i	−0.842286	+	0.539032i	$0.818790\pi$
$938$	− 18.0000i	− 0.587721i
$939$	0	0
$940$	0	0
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	− 48.0000i	− 1.56310i
$944$	40.0000	1.30189
$945$	0	0
$946$	4.00000	0.130051
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	14.0000	0.454459
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 56.0000i	− 1.81402i	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	− 56.0000i	− 1.81402i	0.421111	−	0.907009i	$-0.361640\pi$
$954$	0	0
$955$	0	0
$956$	−40.0000	−1.29369
$957$	0	0
$958$	60.0000i	1.93851i
$959$	54.0000	1.74375
$960$	0	0
$961$	−22.0000	−0.709677
$962$	4.00000i	0.128965i
$963$	0	0
$964$	46.0000	1.48156
$965$	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
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	0	0
$973$	− 60.0000i	− 1.92351i
$974$	−26.0000	−0.833094
$975$	0	0
$976$	−28.0000	−0.896258
$977$	2.00000i	0.0639857i	0.999488	+	0.0319928i	$0.0101854\pi$
$977$	2.00000i	0.0639857i	−0.999488	+	0.0319928i	$0.989815\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	16.0000i	0.510581i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	0	0
$985$	0	0
$986$	−40.0000	−1.27386
$987$	0	0
$988$	− 10.0000i	− 0.318142i
$989$	6.00000	0.190789
$990$	0	0
$991$	17.0000	0.540023	0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000	0.540023	0.270011	+	0.962857i	$0.412973\pi$
$992$	24.0000i	0.762001i
$993$	0	0
$994$	−48.0000	−1.52247
$995$	0	0
$996$	0	0
$997$	− 42.0000i	− 1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$	− 42.0000i	− 1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$	10.0000i	0.316544i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.2.b.a.199.2		2
3.2	odd	2		75.2.b.a.49.1		2
4.3	odd	2		3600.2.f.p.2449.1		2
5.2	odd	4		225.2.a.a.1.1		1
5.3	odd	4		225.2.a.e.1.1		1
5.4	even	2	inner	225.2.b.a.199.1		2
12.11	even	2		1200.2.f.d.49.2		2
15.2	even	4		75.2.a.c.1.1	yes	1
15.8	even	4		75.2.a.a.1.1	✓	1
15.14	odd	2		75.2.b.a.49.2		2
20.3	even	4		3600.2.a.j.1.1		1
20.7	even	4		3600.2.a.bk.1.1		1
20.19	odd	2		3600.2.f.p.2449.2		2
24.5	odd	2		4800.2.f.l.3649.2		2
24.11	even	2		4800.2.f.y.3649.1		2
60.23	odd	4		1200.2.a.c.1.1		1
60.47	odd	4		1200.2.a.p.1.1		1
60.59	even	2		1200.2.f.d.49.1		2
105.62	odd	4		3675.2.a.q.1.1		1
105.83	odd	4		3675.2.a.b.1.1		1
120.29	odd	2		4800.2.f.l.3649.1		2
120.53	even	4		4800.2.a.bb.1.1		1
120.59	even	2		4800.2.f.y.3649.2		2
120.77	even	4		4800.2.a.bq.1.1		1
120.83	odd	4		4800.2.a.br.1.1		1
120.107	odd	4		4800.2.a.be.1.1		1
165.32	odd	4		9075.2.a.a.1.1		1
165.98	odd	4		9075.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.a.a.1.1	✓	1	15.8	even	4
75.2.a.c.1.1	yes	1	15.2	even	4
75.2.b.a.49.1		2	3.2	odd	2
75.2.b.a.49.2		2	15.14	odd	2
225.2.a.a.1.1		1	5.2	odd	4
225.2.a.e.1.1		1	5.3	odd	4
225.2.b.a.199.1		2	5.4	even	2	inner
225.2.b.a.199.2		2	1.1	even	1	trivial
1200.2.a.c.1.1		1	60.23	odd	4
1200.2.a.p.1.1		1	60.47	odd	4
1200.2.f.d.49.1		2	60.59	even	2
1200.2.f.d.49.2		2	12.11	even	2
3600.2.a.j.1.1		1	20.3	even	4
3600.2.a.bk.1.1		1	20.7	even	4
3600.2.f.p.2449.1		2	4.3	odd	2
3600.2.f.p.2449.2		2	20.19	odd	2
3675.2.a.b.1.1		1	105.83	odd	4
3675.2.a.q.1.1		1	105.62	odd	4
4800.2.a.bb.1.1		1	120.53	even	4
4800.2.a.be.1.1		1	120.107	odd	4
4800.2.a.bq.1.1		1	120.77	even	4
4800.2.a.br.1.1		1	120.83	odd	4
4800.2.f.l.3649.1		2	120.29	odd	2
4800.2.f.l.3649.2		2	24.5	odd	2
4800.2.f.y.3649.1		2	24.11	even	2
4800.2.f.y.3649.2		2	120.59	even	2
9075.2.a.a.1.1		1	165.32	odd	4
9075.2.a.s.1.1		1	165.98	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{7} +O(q^{10})\)	\(q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{7} -2.00000 q^{11} +1.00000i q^{13} -6.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} +5.00000 q^{19} -4.00000i q^{22} -6.00000i q^{23} -2.00000 q^{26} -6.00000i q^{28} +10.0000 q^{29} -3.00000 q^{31} -8.00000i q^{32} -4.00000 q^{34} -2.00000i q^{37} +10.0000i q^{38} +8.00000 q^{41} +1.00000i q^{43} +4.00000 q^{44} +12.0000 q^{46} +2.00000i q^{47} -2.00000 q^{49} -2.00000i q^{52} +4.00000i q^{53} +20.0000i q^{58} -10.0000 q^{59} +7.00000 q^{61} -6.00000i q^{62} +8.00000 q^{64} +3.00000i q^{67} -4.00000i q^{68} +8.00000 q^{71} -14.0000i q^{73} +4.00000 q^{74} -10.0000 q^{76} -6.00000i q^{77} +16.0000i q^{82} -6.00000i q^{83} -2.00000 q^{86} -3.00000 q^{91} +12.0000i q^{92} -4.00000 q^{94} -17.0000i q^{97} -4.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4}+O(q^{10}) \) 2 * q - 4 * q^4	\( 2 q - 4 q^{4} - 4 q^{11} - 12 q^{14} - 8 q^{16} + 10 q^{19} - 4 q^{26} + 20 q^{29} - 6 q^{31} - 8 q^{34} + 16 q^{41} + 8 q^{44} + 24 q^{46} - 4 q^{49} - 20 q^{59} + 14 q^{61} + 16 q^{64} + 16 q^{71} + 8 q^{74} - 20 q^{76} - 4 q^{86} - 6 q^{91} - 8 q^{94}+O(q^{100}) \) 2 * q - 4 * q^4 - 4 * q^11 - 12 * q^14 - 8 * q^16 + 10 * q^19 - 4 * q^26 + 20 * q^29 - 6 * q^31 - 8 * q^34 + 16 * q^41 + 8 * q^44 + 24 * q^46 - 4 * q^49 - 20 * q^59 + 14 * q^61 + 16 * q^64 + 16 * q^71 + 8 * q^74 - 20 * q^76 - 4 * q^86 - 6 * q^91 - 8 * q^94

Introduction

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