LMFDB - Embedded newform 225.3.d.a.224.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(224,225)]

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([1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("225.224");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	225.d (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:	$6.13080594811$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 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			224.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	225.224
Dual form			225.3.d.a.224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -2.00000 q^{4} +9.00000i q^{7} +8.48528 q^{8} -18.3848i q^{11} +1.00000i q^{13} -12.7279i q^{14} -4.00000 q^{16} -26.8701 q^{17} -25.0000 q^{19} +26.0000i q^{22} -4.24264 q^{23} -1.41421i q^{26} -18.0000i q^{28} -26.8701i q^{29} -39.0000 q^{31} -28.2843 q^{32} +38.0000 q^{34} -32.0000i q^{37} +35.3553 q^{38} +5.65685i q^{41} -23.0000i q^{43} +36.7696i q^{44} +6.00000 q^{46} +32.5269 q^{47} -32.0000 q^{49} -2.00000i q^{52} -96.1665 q^{53} +76.3675i q^{56} +38.0000i q^{58} +9.89949i q^{59} +73.0000 q^{61} +55.1543 q^{62} +56.0000 q^{64} -63.0000i q^{67} +53.7401 q^{68} -62.2254i q^{71} +136.000i q^{73} +45.2548i q^{74} +50.0000 q^{76} +165.463 q^{77} +24.0000 q^{79} -8.00000i q^{82} -46.6690 q^{83} +32.5269i q^{86} -156.000i q^{88} +101.823i q^{89} -9.00000 q^{91} +8.48528 q^{92} -46.0000 q^{94} +7.00000i q^{97} +45.2548 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 16 q^{16} - 100 q^{19} - 156 q^{31} + 152 q^{34} + 24 q^{46} - 128 q^{49} + 292 q^{61} + 224 q^{64} + 200 q^{76} + 96 q^{79} - 36 q^{91} - 184 q^{94}+O(q^{100}) $ 4 * q - 8 * q^4 - 16 * q^16 - 100 * q^19 - 156 * q^31 + 152 * q^34 + 24 * q^46 - 128 * q^49 + 292 * q^61 + 224 * q^64 + 200 * q^76 + 96 * q^79 - 36 * q^91 - 184 * q^94

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 $

$2$	−1.41421	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.41421	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9.00000i	1.28571i	0.765986	+	0.642857i	$0.222251\pi$
$7$	9.00000i	1.28571i	−0.765986	+	0.642857i	$0.777749\pi$
$8$	8.48528	1.06066
$9$	0	0
$10$	0	0
$11$	− 18.3848i	− 1.67134i	−0.549229	−	0.835672i	$-0.685079\pi$
$11$	− 18.3848i	− 1.67134i	0.549229	−	0.835672i	$-0.314921\pi$
$12$	0	0
$13$	1.00000i	0.0769231i	0.999260	+	0.0384615i	$0.0122457\pi$
$13$	1.00000i	0.0769231i	−0.999260	+	0.0384615i	$0.987754\pi$
$14$	− 12.7279i	− 0.909137i
$15$	0	0
$16$	−4.00000	−0.250000
$17$	−26.8701	−1.58059	−0.790296	−	0.612725i	$-0.790073\pi$
$17$	−26.8701	−1.58059	−0.790296	+	0.612725i	$0.790073\pi$
$18$	0	0
$19$	−25.0000	−1.31579	−0.657895	−	0.753110i	$-0.728553\pi$
$19$	−25.0000	−1.31579	−0.657895	+	0.753110i	$0.728553\pi$
$20$	0	0
$21$	0	0
$22$	26.0000i	1.18182i
$23$	−4.24264	−0.184463	−0.0922313	−	0.995738i	$-0.529400\pi$
$23$	−4.24264	−0.184463	−0.0922313	+	0.995738i	$0.529400\pi$
$24$	0	0
$25$	0	0
$26$	− 1.41421i	− 0.0543928i
$27$	0	0
$28$	− 18.0000i	− 0.642857i
$29$	− 26.8701i	− 0.926554i	−0.886214	−	0.463277i	$-0.846674\pi$
$29$	− 26.8701i	− 0.926554i	0.886214	−	0.463277i	$-0.153326\pi$
$30$	0	0
$31$	−39.0000	−1.25806	−0.629032	−	0.777379i	$-0.716549\pi$
$31$	−39.0000	−1.25806	−0.629032	+	0.777379i	$0.716549\pi$
$32$	−28.2843	−0.883883
$33$	0	0
$34$	38.0000	1.11765
$35$	0	0
$36$	0	0
$37$	− 32.0000i	− 0.864865i	−0.901666	−	0.432432i	$-0.857655\pi$
$37$	− 32.0000i	− 0.864865i	0.901666	−	0.432432i	$-0.142345\pi$
$38$	35.3553	0.930404
$39$	0	0
$40$	0	0
$41$	5.65685i	0.137972i	0.997618	+	0.0689860i	$0.0219764\pi$
$41$	5.65685i	0.137972i	−0.997618	+	0.0689860i	$0.978024\pi$
$42$	0	0
$43$	− 23.0000i	− 0.534884i	−0.963574	−	0.267442i	$-0.913822\pi$
$43$	− 23.0000i	− 0.534884i	0.963574	−	0.267442i	$-0.0861783\pi$
$44$	36.7696i	0.835672i
$45$	0	0
$46$	6.00000	0.130435
$47$	32.5269	0.692062	0.346031	−	0.938223i	$-0.387529\pi$
$47$	32.5269	0.692062	0.346031	+	0.938223i	$0.387529\pi$
$48$	0	0
$49$	−32.0000	−0.653061
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.0384615i
$53$	−96.1665	−1.81446	−0.907231	−	0.420632i	$-0.861808\pi$
$53$	−96.1665	−1.81446	−0.907231	+	0.420632i	$0.861808\pi$
$54$	0	0
$55$	0	0
$56$	76.3675i	1.36371i
$57$	0	0
$58$	38.0000i	0.655172i
$59$	9.89949i	0.167788i	0.996475	+	0.0838940i	$0.0267357\pi$
$59$	9.89949i	0.167788i	−0.996475	+	0.0838940i	$0.973264\pi$
$60$	0	0
$61$	73.0000	1.19672	0.598361	−	0.801227i	$-0.295819\pi$
$61$	73.0000	1.19672	0.598361	+	0.801227i	$0.295819\pi$
$62$	55.1543	0.889586
$63$	0	0
$64$	56.0000	0.875000
$65$	0	0
$66$	0	0
$67$	− 63.0000i	− 0.940299i	−0.882587	−	0.470149i	$-0.844200\pi$
$67$	− 63.0000i	− 0.940299i	0.882587	−	0.470149i	$-0.155800\pi$
$68$	53.7401	0.790296
$69$	0	0
$70$	0	0
$71$	− 62.2254i	− 0.876414i	−0.898874	−	0.438207i	$-0.855614\pi$
$71$	− 62.2254i	− 0.876414i	0.898874	−	0.438207i	$-0.144386\pi$
$72$	0	0
$73$	136.000i	1.86301i	0.363724	+	0.931507i	$0.381505\pi$
$73$	136.000i	1.86301i	−0.363724	+	0.931507i	$0.618495\pi$
$74$	45.2548i	0.611552i
$75$	0	0
$76$	50.0000	0.657895
$77$	165.463	2.14887
$78$	0	0
$79$	24.0000	0.303797	0.151899	−	0.988396i	$-0.451461\pi$
$79$	24.0000	0.303797	0.151899	+	0.988396i	$0.451461\pi$
$80$	0	0
$81$	0	0
$82$	− 8.00000i	− 0.0975610i
$83$	−46.6690	−0.562278	−0.281139	−	0.959667i	$-0.590712\pi$
$83$	−46.6690	−0.562278	−0.281139	+	0.959667i	$0.590712\pi$
$84$	0	0
$85$	0	0
$86$	32.5269i	0.378220i
$87$	0	0
$88$	− 156.000i	− 1.77273i
$89$	101.823i	1.14408i	0.820225	+	0.572041i	$0.193848\pi$
$89$	101.823i	1.14408i	−0.820225	+	0.572041i	$0.806152\pi$
$90$	0	0
$91$	−9.00000	−0.0989011
$92$	8.48528	0.0922313
$93$	0	0
$94$	−46.0000	−0.489362
$95$	0	0
$96$	0	0
$97$	7.00000i	0.0721649i	0.999349	+	0.0360825i	$0.0114879\pi$
$97$	7.00000i	0.0721649i	−0.999349	+	0.0360825i	$0.988512\pi$
$98$	45.2548	0.461784
$99$	0	0
$100$	0	0
$101$	50.9117i	0.504076i	0.967717	+	0.252038i	$0.0811008\pi$
$101$	50.9117i	0.504076i	−0.967717	+	0.252038i	$0.918899\pi$
$102$	0	0
$103$	110.000i	1.06796i	0.845497	+	0.533981i	$0.179304\pi$
$103$	110.000i	1.06796i	−0.845497	+	0.533981i	$0.820696\pi$
$104$	8.48528i	0.0815892i
$105$	0	0
$106$	136.000	1.28302
$107$	−118.794	−1.11022	−0.555112	−	0.831776i	$-0.687325\pi$
$107$	−118.794	−1.11022	−0.555112	+	0.831776i	$0.687325\pi$
$108$	0	0
$109$	25.0000	0.229358	0.114679	−	0.993403i	$-0.463416\pi$
$109$	25.0000	0.229358	0.114679	+	0.993403i	$0.463416\pi$
$110$	0	0
$111$	0	0
$112$	− 36.0000i	− 0.321429i
$113$	−45.2548	−0.400485	−0.200243	−	0.979746i	$-0.564173\pi$
$113$	−45.2548	−0.400485	−0.200243	+	0.979746i	$0.564173\pi$
$114$	0	0
$115$	0	0
$116$	53.7401i	0.463277i
$117$	0	0
$118$	− 14.0000i	− 0.118644i
$119$	− 241.831i	− 2.03219i
$120$	0	0
$121$	−217.000	−1.79339
$122$	−103.238	−0.846210
$123$	0	0
$124$	78.0000	0.629032
$125$	0	0
$126$	0	0
$127$	80.0000i	0.629921i	0.949105	+	0.314961i	$0.101991\pi$
$127$	80.0000i	0.629921i	−0.949105	+	0.314961i	$0.898009\pi$
$128$	33.9411	0.265165
$129$	0	0
$130$	0	0
$131$	− 84.8528i	− 0.647731i	−0.946103	−	0.323866i	$-0.895017\pi$
$131$	− 84.8528i	− 0.647731i	0.946103	−	0.323866i	$-0.104983\pi$
$132$	0	0
$133$	− 225.000i	− 1.69173i
$134$	89.0955i	0.664891i
$135$	0	0
$136$	−228.000	−1.67647
$137$	21.2132	0.154841	0.0774205	−	0.996999i	$-0.475332\pi$
$137$	21.2132	0.154841	0.0774205	+	0.996999i	$0.475332\pi$
$138$	0	0
$139$	−104.000	−0.748201	−0.374101	−	0.927388i	$-0.622049\pi$
$139$	−104.000	−0.748201	−0.374101	+	0.927388i	$0.622049\pi$
$140$	0	0
$141$	0	0
$142$	88.0000i	0.619718i
$143$	18.3848	0.128565
$144$	0	0
$145$	0	0
$146$	− 192.333i	− 1.31735i
$147$	0	0
$148$	64.0000i	0.432432i
$149$	− 171.120i	− 1.14846i	−0.818696	−	0.574228i	$-0.805302\pi$
$149$	− 171.120i	− 1.14846i	0.818696	−	0.574228i	$-0.194698\pi$
$150$	0	0
$151$	−17.0000	−0.112583	−0.0562914	−	0.998414i	$-0.517928\pi$
$151$	−17.0000	−0.112583	−0.0562914	+	0.998414i	$0.517928\pi$
$152$	−212.132	−1.39561
$153$	0	0
$154$	−234.000	−1.51948
$155$	0	0
$156$	0	0
$157$	− 65.0000i	− 0.414013i	−0.978340	−	0.207006i	$-0.933628\pi$
$157$	− 65.0000i	− 0.414013i	0.978340	−	0.207006i	$-0.0663721\pi$
$158$	−33.9411	−0.214817
$159$	0	0
$160$	0	0
$161$	− 38.1838i	− 0.237166i
$162$	0	0
$163$	17.0000i	0.104294i	0.998639	+	0.0521472i	$0.0166065\pi$
$163$	17.0000i	0.104294i	−0.998639	+	0.0521472i	$0.983393\pi$
$164$	− 11.3137i	− 0.0689860i
$165$	0	0
$166$	66.0000	0.397590
$167$	−254.558	−1.52430	−0.762151	−	0.647399i	$-0.775857\pi$
$167$	−254.558	−1.52430	−0.762151	+	0.647399i	$0.775857\pi$
$168$	0	0
$169$	168.000	0.994083
$170$	0	0
$171$	0	0
$172$	46.0000i	0.267442i
$173$	173.948	1.00548	0.502741	−	0.864437i	$-0.332325\pi$
$173$	173.948	1.00548	0.502741	+	0.864437i	$0.332325\pi$
$174$	0	0
$175$	0	0
$176$	73.5391i	0.417836i
$177$	0	0
$178$	− 144.000i	− 0.808989i
$179$	− 125.865i	− 0.703156i	−0.936159	−	0.351578i	$-0.885645\pi$
$179$	− 125.865i	− 0.703156i	0.936159	−	0.351578i	$-0.114355\pi$
$180$	0	0
$181$	119.000	0.657459	0.328729	−	0.944424i	$-0.393380\pi$
$181$	119.000	0.657459	0.328729	+	0.944424i	$0.393380\pi$
$182$	12.7279	0.0699336
$183$	0	0
$184$	−36.0000	−0.195652
$185$	0	0
$186$	0	0
$187$	494.000i	2.64171i
$188$	−65.0538	−0.346031
$189$	0	0
$190$	0	0
$191$	− 15.5563i	− 0.0814469i	−0.999170	−	0.0407234i	$-0.987034\pi$
$191$	− 15.5563i	− 0.0814469i	0.999170	−	0.0407234i	$-0.0129663\pi$
$192$	0	0
$193$	281.000i	1.45596i	0.685599	+	0.727979i	$0.259540\pi$
$193$	281.000i	1.45596i	−0.685599	+	0.727979i	$0.740460\pi$
$194$	− 9.89949i	− 0.0510283i
$195$	0	0
$196$	64.0000	0.326531
$197$	224.860	1.14142	0.570711	−	0.821151i	$-0.306668\pi$
$197$	224.860	1.14142	0.570711	+	0.821151i	$0.306668\pi$
$198$	0	0
$199$	−175.000	−0.879397	−0.439698	−	0.898145i	$-0.644915\pi$
$199$	−175.000	−0.879397	−0.439698	+	0.898145i	$0.644915\pi$
$200$	0	0
$201$	0	0
$202$	− 72.0000i	− 0.356436i
$203$	241.831	1.19128
$204$	0	0
$205$	0	0
$206$	− 155.563i	− 0.755163i
$207$	0	0
$208$	− 4.00000i	− 0.0192308i
$209$	459.619i	2.19914i
$210$	0	0
$211$	−49.0000	−0.232227	−0.116114	−	0.993236i	$-0.537044\pi$
$211$	−49.0000	−0.232227	−0.116114	+	0.993236i	$0.537044\pi$
$212$	192.333	0.907231
$213$	0	0
$214$	168.000	0.785047
$215$	0	0
$216$	0	0
$217$	− 351.000i	− 1.61751i
$218$	−35.3553	−0.162180
$219$	0	0
$220$	0	0
$221$	− 26.8701i	− 0.121584i
$222$	0	0
$223$	− 265.000i	− 1.18834i	−0.804339	−	0.594170i	$-0.797480\pi$
$223$	− 265.000i	− 1.18834i	0.804339	−	0.594170i	$-0.202520\pi$
$224$	− 254.558i	− 1.13642i
$225$	0	0
$226$	64.0000	0.283186
$227$	−130.108	−0.573161	−0.286581	−	0.958056i	$-0.592519\pi$
$227$	−130.108	−0.573161	−0.286581	+	0.958056i	$0.592519\pi$
$228$	0	0
$229$	87.0000	0.379913	0.189956	−	0.981793i	$-0.439165\pi$
$229$	87.0000	0.379913	0.189956	+	0.981793i	$0.439165\pi$
$230$	0	0
$231$	0	0
$232$	− 228.000i	− 0.982759i
$233$	−152.735	−0.655515	−0.327758	−	0.944762i	$-0.606293\pi$
$233$	−152.735	−0.655515	−0.327758	+	0.944762i	$0.606293\pi$
$234$	0	0
$235$	0	0
$236$	− 19.7990i	− 0.0838940i
$237$	0	0
$238$	342.000i	1.43697i
$239$	277.186i	1.15977i	0.814697	+	0.579887i	$0.196903\pi$
$239$	277.186i	1.15977i	−0.814697	+	0.579887i	$0.803097\pi$
$240$	0	0
$241$	415.000	1.72199	0.860996	−	0.508612i	$-0.169841\pi$
$241$	415.000	1.72199	0.860996	+	0.508612i	$0.169841\pi$
$242$	306.884	1.26812
$243$	0	0
$244$	−146.000	−0.598361
$245$	0	0
$246$	0	0
$247$	− 25.0000i	− 0.101215i
$248$	−330.926	−1.33438
$249$	0	0
$250$	0	0
$251$	373.352i	1.48746i	0.668480	+	0.743730i	$0.266945\pi$
$251$	373.352i	1.48746i	−0.668480	+	0.743730i	$0.733055\pi$
$252$	0	0
$253$	78.0000i	0.308300i
$254$	− 113.137i	− 0.445422i
$255$	0	0
$256$	−272.000	−1.06250
$257$	271.529	1.05653	0.528267	−	0.849079i	$-0.322842\pi$
$257$	271.529	1.05653	0.528267	+	0.849079i	$0.322842\pi$
$258$	0	0
$259$	288.000	1.11197
$260$	0	0
$261$	0	0
$262$	120.000i	0.458015i
$263$	−209.304	−0.795831	−0.397916	−	0.917422i	$-0.630266\pi$
$263$	−209.304	−0.795831	−0.397916	+	0.917422i	$0.630266\pi$
$264$	0	0
$265$	0	0
$266$	318.198i	1.19623i
$267$	0	0
$268$	126.000i	0.470149i
$269$	94.7523i	0.352239i	0.984369	+	0.176120i	$0.0563545\pi$
$269$	94.7523i	0.352239i	−0.984369	+	0.176120i	$0.943645\pi$
$270$	0	0
$271$	−176.000	−0.649446	−0.324723	−	0.945809i	$-0.605271\pi$
$271$	−176.000	−0.649446	−0.324723	+	0.945809i	$0.605271\pi$
$272$	107.480	0.395148
$273$	0	0
$274$	−30.0000	−0.109489
$275$	0	0
$276$	0	0
$277$	− 303.000i	− 1.09386i	−0.837177	−	0.546931i	$-0.815796\pi$
$277$	− 303.000i	− 1.09386i	0.837177	−	0.546931i	$-0.184204\pi$
$278$	147.078	0.529058
$279$	0	0
$280$	0	0
$281$	513.360i	1.82690i	0.406948	+	0.913451i	$0.366593\pi$
$281$	513.360i	1.82690i	−0.406948	+	0.913451i	$0.633407\pi$
$282$	0	0
$283$	− 393.000i	− 1.38869i	−0.719641	−	0.694346i	$-0.755694\pi$
$283$	− 393.000i	− 1.38869i	0.719641	−	0.694346i	$-0.244306\pi$
$284$	124.451i	0.438207i
$285$	0	0
$286$	−26.0000	−0.0909091
$287$	−50.9117	−0.177393
$288$	0	0
$289$	433.000	1.49827
$290$	0	0
$291$	0	0
$292$	− 272.000i	− 0.931507i
$293$	46.6690	0.159280	0.0796400	−	0.996824i	$-0.474623\pi$
$293$	46.6690	0.159280	0.0796400	+	0.996824i	$0.474623\pi$
$294$	0	0
$295$	0	0
$296$	− 271.529i	− 0.917328i
$297$	0	0
$298$	242.000i	0.812081i
$299$	− 4.24264i	− 0.0141894i
$300$	0	0
$301$	207.000	0.687708
$302$	24.0416	0.0796080
$303$	0	0
$304$	100.000	0.328947
$305$	0	0
$306$	0	0
$307$	− 289.000i	− 0.941368i	−0.882302	−	0.470684i	$-0.844007\pi$
$307$	− 289.000i	− 0.941368i	0.882302	−	0.470684i	$-0.155993\pi$
$308$	−330.926	−1.07443
$309$	0	0
$310$	0	0
$311$	− 462.448i	− 1.48697i	−0.668752	−	0.743485i	$-0.733171\pi$
$311$	− 462.448i	− 1.48697i	0.668752	−	0.743485i	$-0.266829\pi$
$312$	0	0
$313$	− 481.000i	− 1.53674i	−0.640005	−	0.768371i	$-0.721068\pi$
$313$	− 481.000i	− 1.53674i	0.640005	−	0.768371i	$-0.278932\pi$
$314$	91.9239i	0.292751i
$315$	0	0
$316$	−48.0000	−0.151899
$317$	−11.3137	−0.0356899	−0.0178450	−	0.999841i	$-0.505681\pi$
$317$	−11.3137	−0.0356899	−0.0178450	+	0.999841i	$0.505681\pi$
$318$	0	0
$319$	−494.000	−1.54859
$320$	0	0
$321$	0	0
$322$	54.0000i	0.167702i
$323$	671.751	2.07973
$324$	0	0
$325$	0	0
$326$	− 24.0416i	− 0.0737473i
$327$	0	0
$328$	48.0000i	0.146341i
$329$	292.742i	0.889794i
$330$	0	0
$331$	−72.0000	−0.217523	−0.108761	−	0.994068i	$-0.534688\pi$
$331$	−72.0000	−0.217523	−0.108761	+	0.994068i	$0.534688\pi$
$332$	93.3381	0.281139
$333$	0	0
$334$	360.000	1.07784
$335$	0	0
$336$	0	0
$337$	335.000i	0.994065i	0.867732	+	0.497033i	$0.165577\pi$
$337$	335.000i	0.994065i	−0.867732	+	0.497033i	$0.834423\pi$
$338$	−237.588	−0.702923
$339$	0	0
$340$	0	0
$341$	717.006i	2.10266i
$342$	0	0
$343$	153.000i	0.446064i
$344$	− 195.161i	− 0.567330i
$345$	0	0
$346$	−246.000	−0.710983
$347$	−383.252	−1.10447	−0.552236	−	0.833688i	$-0.686225\pi$
$347$	−383.252	−1.10447	−0.552236	+	0.833688i	$0.686225\pi$
$348$	0	0
$349$	−568.000	−1.62751	−0.813754	−	0.581210i	$-0.802579\pi$
$349$	−568.000	−1.62751	−0.813754	+	0.581210i	$0.802579\pi$
$350$	0	0
$351$	0	0
$352$	520.000i	1.47727i
$353$	−131.522	−0.372583	−0.186292	−	0.982495i	$-0.559647\pi$
$353$	−131.522	−0.372583	−0.186292	+	0.982495i	$0.559647\pi$
$354$	0	0
$355$	0	0
$356$	− 203.647i	− 0.572041i
$357$	0	0
$358$	178.000i	0.497207i
$359$	− 424.264i	− 1.18179i	−0.806747	−	0.590897i	$-0.798774\pi$
$359$	− 424.264i	− 1.18179i	0.806747	−	0.590897i	$-0.201226\pi$
$360$	0	0
$361$	264.000	0.731302
$362$	−168.291	−0.464893
$363$	0	0
$364$	18.0000	0.0494505
$365$	0	0
$366$	0	0
$367$	369.000i	1.00545i	0.864447	+	0.502725i	$0.167669\pi$
$367$	369.000i	1.00545i	−0.864447	+	0.502725i	$0.832331\pi$
$368$	16.9706	0.0461157
$369$	0	0
$370$	0	0
$371$	− 865.499i	− 2.33288i
$372$	0	0
$373$	313.000i	0.839142i	0.907723	+	0.419571i	$0.137819\pi$
$373$	313.000i	0.839142i	−0.907723	+	0.419571i	$0.862181\pi$
$374$	− 698.621i	− 1.86797i
$375$	0	0
$376$	276.000	0.734043
$377$	26.8701	0.0712734
$378$	0	0
$379$	47.0000	0.124011	0.0620053	−	0.998076i	$-0.480250\pi$
$379$	47.0000	0.124011	0.0620053	+	0.998076i	$0.480250\pi$
$380$	0	0
$381$	0	0
$382$	22.0000i	0.0575916i
$383$	373.352	0.974810	0.487405	−	0.873176i	$-0.337943\pi$
$383$	373.352	0.974810	0.487405	+	0.873176i	$0.337943\pi$
$384$	0	0
$385$	0	0
$386$	− 397.394i	− 1.02952i
$387$	0	0
$388$	− 14.0000i	− 0.0360825i
$389$	390.323i	1.00340i	0.865041	+	0.501700i	$0.167292\pi$
$389$	390.323i	1.00340i	−0.865041	+	0.501700i	$0.832708\pi$
$390$	0	0
$391$	114.000	0.291560
$392$	−271.529	−0.692676
$393$	0	0
$394$	−318.000	−0.807107
$395$	0	0
$396$	0	0
$397$	47.0000i	0.118388i	0.998247	+	0.0591940i	$0.0188530\pi$
$397$	47.0000i	0.118388i	−0.998247	+	0.0591940i	$0.981147\pi$
$398$	247.487	0.621828
$399$	0	0
$400$	0	0
$401$	− 152.735i	− 0.380885i	−0.981698	−	0.190443i	$-0.939008\pi$
$401$	− 152.735i	− 0.380885i	0.981698	−	0.190443i	$-0.0609923\pi$
$402$	0	0
$403$	− 39.0000i	− 0.0967742i
$404$	− 101.823i	− 0.252038i
$405$	0	0
$406$	−342.000	−0.842365
$407$	−588.313	−1.44549
$408$	0	0
$409$	103.000	0.251834	0.125917	−	0.992041i	$-0.459813\pi$
$409$	103.000	0.251834	0.125917	+	0.992041i	$0.459813\pi$
$410$	0	0
$411$	0	0
$412$	− 220.000i	− 0.533981i
$413$	−89.0955	−0.215727
$414$	0	0
$415$	0	0
$416$	− 28.2843i	− 0.0679910i
$417$	0	0
$418$	− 650.000i	− 1.55502i
$419$	− 39.5980i	− 0.0945059i	−0.998883	−	0.0472530i	$-0.984953\pi$
$419$	− 39.5980i	− 0.0945059i	0.998883	−	0.0472530i	$-0.0150467\pi$
$420$	0	0
$421$	−560.000	−1.33017	−0.665083	−	0.746769i	$-0.731604\pi$
$421$	−560.000	−1.33017	−0.665083	+	0.746769i	$0.731604\pi$
$422$	69.2965	0.164210
$423$	0	0
$424$	−816.000	−1.92453
$425$	0	0
$426$	0	0
$427$	657.000i	1.53864i
$428$	237.588	0.555112
$429$	0	0
$430$	0	0
$431$	− 479.418i	− 1.11234i	−0.831069	−	0.556170i	$-0.812270\pi$
$431$	− 479.418i	− 1.11234i	0.831069	−	0.556170i	$-0.187730\pi$
$432$	0	0
$433$	− 257.000i	− 0.593533i	−0.954950	−	0.296767i	$-0.904092\pi$
$433$	− 257.000i	− 0.593533i	0.954950	−	0.296767i	$-0.0959084\pi$
$434$	496.389i	1.14375i
$435$	0	0
$436$	−50.0000	−0.114679
$437$	106.066	0.242714
$438$	0	0
$439$	167.000	0.380410	0.190205	−	0.981744i	$-0.439085\pi$
$439$	167.000	0.380410	0.190205	+	0.981744i	$0.439085\pi$
$440$	0	0
$441$	0	0
$442$	38.0000i	0.0859729i
$443$	169.706	0.383083	0.191541	−	0.981485i	$-0.438651\pi$
$443$	169.706	0.383083	0.191541	+	0.981485i	$0.438651\pi$
$444$	0	0
$445$	0	0
$446$	374.767i	0.840284i
$447$	0	0
$448$	504.000i	1.12500i
$449$	− 690.136i	− 1.53705i	−0.639819	−	0.768526i	$-0.720991\pi$
$449$	− 690.136i	− 1.53705i	0.639819	−	0.768526i	$-0.279009\pi$
$450$	0	0
$451$	104.000	0.230599
$452$	90.5097	0.200243
$453$	0	0
$454$	184.000	0.405286
$455$	0	0
$456$	0	0
$457$	50.0000i	0.109409i	0.998503	+	0.0547046i	$0.0174217\pi$
$457$	50.0000i	0.109409i	−0.998503	+	0.0547046i	$0.982578\pi$
$458$	−123.037	−0.268639
$459$	0	0
$460$	0	0
$461$	152.735i	0.331313i	0.986184	+	0.165656i	$0.0529742\pi$
$461$	152.735i	0.331313i	−0.986184	+	0.165656i	$0.947026\pi$
$462$	0	0
$463$	− 240.000i	− 0.518359i	−0.965829	−	0.259179i	$-0.916548\pi$
$463$	− 240.000i	− 0.518359i	0.965829	−	0.259179i	$-0.0834520\pi$
$464$	107.480i	0.231638i
$465$	0	0
$466$	216.000	0.463519
$467$	−592.555	−1.26886	−0.634428	−	0.772982i	$-0.718764\pi$
$467$	−592.555	−1.26886	−0.634428	+	0.772982i	$0.718764\pi$
$468$	0	0
$469$	567.000	1.20896
$470$	0	0
$471$	0	0
$472$	84.0000i	0.177966i
$473$	−422.850	−0.893974
$474$	0	0
$475$	0	0
$476$	483.661i	1.01609i
$477$	0	0
$478$	− 392.000i	− 0.820084i
$479$	− 496.389i	− 1.03630i	−0.855289	−	0.518151i	$-0.826620\pi$
$479$	− 496.389i	− 1.03630i	0.855289	−	0.518151i	$-0.173380\pi$
$480$	0	0
$481$	32.0000	0.0665281
$482$	−586.899	−1.21763
$483$	0	0
$484$	434.000	0.896694
$485$	0	0
$486$	0	0
$487$	− 545.000i	− 1.11910i	−0.828798	−	0.559548i	$-0.810975\pi$
$487$	− 545.000i	− 1.11910i	0.828798	−	0.559548i	$-0.189025\pi$
$488$	619.426	1.26931
$489$	0	0
$490$	0	0
$491$	905.097i	1.84337i	0.387934	+	0.921687i	$0.373189\pi$
$491$	905.097i	1.84337i	−0.387934	+	0.921687i	$0.626811\pi$
$492$	0	0
$493$	722.000i	1.46450i
$494$	35.3553i	0.0715695i
$495$	0	0
$496$	156.000	0.314516
$497$	560.029	1.12682
$498$	0	0
$499$	−7.00000	−0.0140281	−0.00701403	−	0.999975i	$-0.502233\pi$
$499$	−7.00000	−0.0140281	−0.00701403	+	0.999975i	$0.502233\pi$
$500$	0	0
$501$	0	0
$502$	− 528.000i	− 1.05179i
$503$	−786.303	−1.56323	−0.781613	−	0.623764i	$-0.785603\pi$
$503$	−786.303	−1.56323	−0.781613	+	0.623764i	$0.785603\pi$
$504$	0	0
$505$	0	0
$506$	− 110.309i	− 0.218001i
$507$	0	0
$508$	− 160.000i	− 0.314961i
$509$	578.413i	1.13637i	0.822900	+	0.568186i	$0.192355\pi$
$509$	578.413i	1.13637i	−0.822900	+	0.568186i	$0.807645\pi$
$510$	0	0
$511$	−1224.00	−2.39530
$512$	248.902	0.486136
$513$	0	0
$514$	−384.000	−0.747082
$515$	0	0
$516$	0	0
$517$	− 598.000i	− 1.15667i
$518$	−407.294	−0.786281
$519$	0	0
$520$	0	0
$521$	− 414.365i	− 0.795325i	−0.917532	−	0.397663i	$-0.869821\pi$
$521$	− 414.365i	− 0.795325i	0.917532	−	0.397663i	$-0.130179\pi$
$522$	0	0
$523$	− 263.000i	− 0.502868i	−0.967874	−	0.251434i	$-0.919098\pi$
$523$	− 263.000i	− 0.502868i	0.967874	−	0.251434i	$-0.0809022\pi$
$524$	169.706i	0.323866i
$525$	0	0
$526$	296.000	0.562738
$527$	1047.93	1.98849
$528$	0	0
$529$	−511.000	−0.965974
$530$	0	0
$531$	0	0
$532$	450.000i	0.845865i
$533$	−5.65685	−0.0106132
$534$	0	0
$535$	0	0
$536$	− 534.573i	− 0.997337i
$537$	0	0
$538$	− 134.000i	− 0.249071i
$539$	588.313i	1.09149i
$540$	0	0
$541$	489.000	0.903882	0.451941	−	0.892048i	$-0.350732\pi$
$541$	489.000	0.903882	0.451941	+	0.892048i	$0.350732\pi$
$542$	248.902	0.459228
$543$	0	0
$544$	760.000	1.39706
$545$	0	0
$546$	0	0
$547$	− 10.0000i	− 0.0182815i	−0.999958	−	0.00914077i	$-0.997090\pi$
$547$	− 10.0000i	− 0.0182815i	0.999958	−	0.00914077i	$-0.00290964\pi$
$548$	−42.4264	−0.0774205
$549$	0	0
$550$	0	0
$551$	671.751i	1.21915i
$552$	0	0
$553$	216.000i	0.390597i
$554$	428.507i	0.773478i
$555$	0	0
$556$	208.000	0.374101
$557$	623.668	1.11969	0.559846	−	0.828597i	$-0.310860\pi$
$557$	623.668	1.11969	0.559846	+	0.828597i	$0.310860\pi$
$558$	0	0
$559$	23.0000	0.0411449
$560$	0	0
$561$	0	0
$562$	− 726.000i	− 1.29181i
$563$	63.6396	0.113037	0.0565183	−	0.998402i	$-0.482000\pi$
$563$	63.6396	0.113037	0.0565183	+	0.998402i	$0.482000\pi$
$564$	0	0
$565$	0	0
$566$	555.786i	0.981954i
$567$	0	0
$568$	− 528.000i	− 0.929577i
$569$	− 810.344i	− 1.42416i	−0.702101	−	0.712078i	$-0.747754\pi$
$569$	− 810.344i	− 1.42416i	0.702101	−	0.712078i	$-0.252246\pi$
$570$	0	0
$571$	377.000	0.660245	0.330123	−	0.943938i	$-0.392910\pi$
$571$	377.000	0.660245	0.330123	+	0.943938i	$0.392910\pi$
$572$	−36.7696	−0.0642824
$573$	0	0
$574$	72.0000	0.125436
$575$	0	0
$576$	0	0
$577$	− 791.000i	− 1.37088i	−0.728127	−	0.685442i	$-0.759609\pi$
$577$	− 791.000i	− 1.37088i	0.728127	−	0.685442i	$-0.240391\pi$
$578$	−612.354	−1.05944
$579$	0	0
$580$	0	0
$581$	− 420.021i	− 0.722928i
$582$	0	0
$583$	1768.00i	3.03259i
$584$	1154.00i	1.97602i
$585$	0	0
$586$	−66.0000	−0.112628
$587$	356.382	0.607124	0.303562	−	0.952812i	$-0.401824\pi$
$587$	356.382	0.607124	0.303562	+	0.952812i	$0.401824\pi$
$588$	0	0
$589$	975.000	1.65535
$590$	0	0
$591$	0	0
$592$	128.000i	0.216216i
$593$	435.578	0.734533	0.367266	−	0.930116i	$-0.380294\pi$
$593$	435.578	0.734533	0.367266	+	0.930116i	$0.380294\pi$
$594$	0	0
$595$	0	0
$596$	342.240i	0.574228i
$597$	0	0
$598$	6.00000i	0.0100334i
$599$	− 1108.74i	− 1.85099i	−0.378759	−	0.925495i	$-0.623649\pi$
$599$	− 1108.74i	− 1.85099i	0.378759	−	0.925495i	$-0.376351\pi$
$600$	0	0
$601$	233.000	0.387687	0.193844	−	0.981032i	$-0.437905\pi$
$601$	233.000	0.387687	0.193844	+	0.981032i	$0.437905\pi$
$602$	−292.742	−0.486283
$603$	0	0
$604$	34.0000	0.0562914
$605$	0	0
$606$	0	0
$607$	776.000i	1.27842i	0.769033	+	0.639209i	$0.220738\pi$
$607$	776.000i	1.27842i	−0.769033	+	0.639209i	$0.779262\pi$
$608$	707.107	1.16300
$609$	0	0
$610$	0	0
$611$	32.5269i	0.0532355i
$612$	0	0
$613$	376.000i	0.613377i	0.951810	+	0.306688i	$0.0992210\pi$
$613$	376.000i	0.613377i	−0.951810	+	0.306688i	$0.900779\pi$
$614$	408.708i	0.665648i
$615$	0	0
$616$	1404.00	2.27922
$617$	593.970	0.962674	0.481337	−	0.876536i	$-0.340151\pi$
$617$	593.970	0.962674	0.481337	+	0.876536i	$0.340151\pi$
$618$	0	0
$619$	−1031.00	−1.66559	−0.832795	−	0.553582i	$-0.813261\pi$
$619$	−1031.00	−1.66559	−0.832795	+	0.553582i	$0.813261\pi$
$620$	0	0
$621$	0	0
$622$	654.000i	1.05145i
$623$	−916.410	−1.47096
$624$	0	0
$625$	0	0
$626$	680.237i	1.08664i
$627$	0	0
$628$	130.000i	0.207006i
$629$	859.842i	1.36700i
$630$	0	0
$631$	745.000	1.18067	0.590333	−	0.807160i	$-0.298997\pi$
$631$	745.000	1.18067	0.590333	+	0.807160i	$0.298997\pi$
$632$	203.647	0.322226
$633$	0	0
$634$	16.0000	0.0252366
$635$	0	0
$636$	0	0
$637$	− 32.0000i	− 0.0502355i
$638$	698.621	1.09502
$639$	0	0
$640$	0	0
$641$	− 203.647i	− 0.317702i	−0.987303	−	0.158851i	$-0.949221\pi$
$641$	− 203.647i	− 0.317702i	0.987303	−	0.158851i	$-0.0507789\pi$
$642$	0	0
$643$	− 1032.00i	− 1.60498i	−0.596668	−	0.802488i	$-0.703509\pi$
$643$	− 1032.00i	− 1.60498i	0.596668	−	0.802488i	$-0.296491\pi$
$644$	76.3675i	0.118583i
$645$	0	0
$646$	−950.000	−1.47059
$647$	−56.5685	−0.0874321	−0.0437160	−	0.999044i	$-0.513920\pi$
$647$	−56.5685	−0.0874321	−0.0437160	+	0.999044i	$0.513920\pi$
$648$	0	0
$649$	182.000	0.280431
$650$	0	0
$651$	0	0
$652$	− 34.0000i	− 0.0521472i
$653$	−272.943	−0.417983	−0.208992	−	0.977917i	$-0.567018\pi$
$653$	−272.943	−0.417983	−0.208992	+	0.977917i	$0.567018\pi$
$654$	0	0
$655$	0	0
$656$	− 22.6274i	− 0.0344930i
$657$	0	0
$658$	− 414.000i	− 0.629179i
$659$	379.009i	0.575128i	0.957761	+	0.287564i	$0.0928454\pi$
$659$	379.009i	0.575128i	−0.957761	+	0.287564i	$0.907155\pi$
$660$	0	0
$661$	−440.000	−0.665658	−0.332829	−	0.942987i	$-0.608003\pi$
$661$	−440.000	−0.665658	−0.332829	+	0.942987i	$0.608003\pi$
$662$	101.823	0.153812
$663$	0	0
$664$	−396.000	−0.596386
$665$	0	0
$666$	0	0
$667$	114.000i	0.170915i
$668$	509.117	0.762151
$669$	0	0
$670$	0	0
$671$	− 1342.09i	− 2.00013i
$672$	0	0
$673$	− 264.000i	− 0.392273i	−0.980577	−	0.196137i	$-0.937160\pi$
$673$	− 264.000i	− 0.392273i	0.980577	−	0.196137i	$-0.0628396\pi$
$674$	− 473.762i	− 0.702910i
$675$	0	0
$676$	−336.000	−0.497041
$677$	−538.815	−0.795887	−0.397943	−	0.917410i	$-0.630276\pi$
$677$	−538.815	−0.795887	−0.397943	+	0.917410i	$0.630276\pi$
$678$	0	0
$679$	−63.0000	−0.0927835
$680$	0	0
$681$	0	0
$682$	− 1014.00i	− 1.48680i
$683$	−576.999	−0.844801	−0.422401	−	0.906409i	$-0.638812\pi$
$683$	−576.999	−0.844801	−0.422401	+	0.906409i	$0.638812\pi$
$684$	0	0
$685$	0	0
$686$	− 216.375i	− 0.315415i
$687$	0	0
$688$	92.0000i	0.133721i
$689$	− 96.1665i	− 0.139574i
$690$	0	0
$691$	−182.000	−0.263386	−0.131693	−	0.991291i	$-0.542041\pi$
$691$	−182.000	−0.263386	−0.131693	+	0.991291i	$0.542041\pi$
$692$	−347.897	−0.502741
$693$	0	0
$694$	542.000	0.780980
$695$	0	0
$696$	0	0
$697$	− 152.000i	− 0.218077i
$698$	803.273	1.15082
$699$	0	0
$700$	0	0
$701$	− 558.614i	− 0.796882i	−0.917194	−	0.398441i	$-0.869551\pi$
$701$	− 558.614i	− 0.796882i	0.917194	−	0.398441i	$-0.130449\pi$
$702$	0	0
$703$	800.000i	1.13798i
$704$	− 1029.55i	− 1.46243i
$705$	0	0
$706$	186.000	0.263456
$707$	−458.205	−0.648098
$708$	0	0
$709$	−607.000	−0.856135	−0.428068	−	0.903747i	$-0.640806\pi$
$709$	−607.000	−0.856135	−0.428068	+	0.903747i	$0.640806\pi$
$710$	0	0
$711$	0	0
$712$	864.000i	1.21348i
$713$	165.463	0.232066
$714$	0	0
$715$	0	0
$716$	251.730i	0.351578i
$717$	0	0
$718$	600.000i	0.835655i
$719$	− 106.066i	− 0.147519i	−0.997276	−	0.0737594i	$-0.976500\pi$
$719$	− 106.066i	− 0.147519i	0.997276	−	0.0737594i	$-0.0234997\pi$
$720$	0	0
$721$	−990.000	−1.37309
$722$	−373.352	−0.517109
$723$	0	0
$724$	−238.000	−0.328729
$725$	0	0
$726$	0	0
$727$	− 71.0000i	− 0.0976616i	−0.998807	−	0.0488308i	$-0.984450\pi$
$727$	− 71.0000i	− 0.0976616i	0.998807	−	0.0488308i	$-0.0155495\pi$
$728$	−76.3675	−0.104900
$729$	0	0
$730$	0	0
$731$	618.011i	0.845433i
$732$	0	0
$733$	− 904.000i	− 1.23329i	−0.787242	−	0.616644i	$-0.788492\pi$
$733$	− 904.000i	− 1.23329i	0.787242	−	0.616644i	$-0.211508\pi$
$734$	− 521.845i	− 0.710960i
$735$	0	0
$736$	120.000	0.163043
$737$	−1158.24	−1.57156
$738$	0	0
$739$	800.000	1.08254	0.541272	−	0.840848i	$-0.317943\pi$
$739$	800.000	1.08254	0.541272	+	0.840848i	$0.317943\pi$
$740$	0	0
$741$	0	0
$742$	1224.00i	1.64960i
$743$	−418.607	−0.563401	−0.281701	−	0.959502i	$-0.590899\pi$
$743$	−418.607	−0.563401	−0.281701	+	0.959502i	$0.590899\pi$
$744$	0	0
$745$	0	0
$746$	− 442.649i	− 0.593363i
$747$	0	0
$748$	− 988.000i	− 1.32086i
$749$	− 1069.15i	− 1.42743i
$750$	0	0
$751$	−704.000	−0.937417	−0.468708	−	0.883353i	$-0.655280\pi$
$751$	−704.000	−0.937417	−0.468708	+	0.883353i	$0.655280\pi$
$752$	−130.108	−0.173015
$753$	0	0
$754$	−38.0000	−0.0503979
$755$	0	0
$756$	0	0
$757$	977.000i	1.29062i	0.763920	+	0.645310i	$0.223272\pi$
$757$	977.000i	1.29062i	−0.763920	+	0.645310i	$0.776728\pi$
$758$	−66.4680	−0.0876887
$759$	0	0
$760$	0	0
$761$	746.705i	0.981215i	0.871381	+	0.490608i	$0.163225\pi$
$761$	746.705i	0.981215i	−0.871381	+	0.490608i	$0.836775\pi$
$762$	0	0
$763$	225.000i	0.294889i
$764$	31.1127i	0.0407234i
$765$	0	0
$766$	−528.000	−0.689295
$767$	−9.89949	−0.0129068
$768$	0	0
$769$	−1337.00	−1.73862	−0.869311	−	0.494266i	$-0.835437\pi$
$769$	−1337.00	−1.73862	−0.869311	+	0.494266i	$0.835437\pi$
$770$	0	0
$771$	0	0
$772$	− 562.000i	− 0.727979i
$773$	−1170.97	−1.51484	−0.757418	−	0.652930i	$-0.773540\pi$
$773$	−1170.97	−1.51484	−0.757418	+	0.652930i	$0.773540\pi$
$774$	0	0
$775$	0	0
$776$	59.3970i	0.0765425i
$777$	0	0
$778$	− 552.000i	− 0.709512i
$779$	− 141.421i	− 0.181542i
$780$	0	0
$781$	−1144.00	−1.46479
$782$	−161.220	−0.206164
$783$	0	0
$784$	128.000	0.163265
$785$	0	0
$786$	0	0
$787$	− 703.000i	− 0.893266i	−0.894717	−	0.446633i	$-0.852623\pi$
$787$	− 703.000i	− 0.893266i	0.894717	−	0.446633i	$-0.147377\pi$
$788$	−449.720	−0.570711
$789$	0	0
$790$	0	0
$791$	− 407.294i	− 0.514910i
$792$	0	0
$793$	73.0000i	0.0920555i
$794$	− 66.4680i	− 0.0837129i
$795$	0	0
$796$	350.000	0.439698
$797$	−1538.66	−1.93057	−0.965285	−	0.261199i	$-0.915882\pi$
$797$	−1538.66	−1.93057	−0.965285	+	0.261199i	$0.915882\pi$
$798$	0	0
$799$	−874.000	−1.09387
$800$	0	0
$801$	0	0
$802$	216.000i	0.269327i
$803$	2500.33	3.11374
$804$	0	0
$805$	0	0
$806$	55.1543i	0.0684297i
$807$	0	0
$808$	432.000i	0.534653i
$809$	− 73.5391i	− 0.0909012i	−0.998967	−	0.0454506i	$-0.985528\pi$
$809$	− 73.5391i	− 0.0909012i	0.998967	−	0.0454506i	$-0.0144724\pi$
$810$	0	0
$811$	1095.00	1.35018	0.675092	−	0.737733i	$-0.264104\pi$
$811$	1095.00	1.35018	0.675092	+	0.737733i	$0.264104\pi$
$812$	−483.661	−0.595642
$813$	0	0
$814$	832.000	1.02211
$815$	0	0
$816$	0	0
$817$	575.000i	0.703794i
$818$	−145.664	−0.178073
$819$	0	0
$820$	0	0
$821$	− 1166.73i	− 1.42110i	−0.703645	−	0.710552i	$-0.748445\pi$
$821$	− 1166.73i	− 1.42110i	0.703645	−	0.710552i	$-0.251555\pi$
$822$	0	0
$823$	− 1015.00i	− 1.23329i	−0.787240	−	0.616646i	$-0.788491\pi$
$823$	− 1015.00i	− 1.23329i	0.787240	−	0.616646i	$-0.211509\pi$
$824$	933.381i	1.13274i
$825$	0	0
$826$	126.000	0.152542
$827$	−379.009	−0.458294	−0.229147	−	0.973392i	$-0.573594\pi$
$827$	−379.009	−0.458294	−0.229147	+	0.973392i	$0.573594\pi$
$828$	0	0
$829$	−150.000	−0.180941	−0.0904704	−	0.995899i	$-0.528837\pi$
$829$	−150.000	−0.180941	−0.0904704	+	0.995899i	$0.528837\pi$
$830$	0	0
$831$	0	0
$832$	56.0000i	0.0673077i
$833$	859.842	1.03222
$834$	0	0
$835$	0	0
$836$	− 919.239i	− 1.09957i
$837$	0	0
$838$	56.0000i	0.0668258i
$839$	660.438i	0.787173i	0.919288	+	0.393586i	$0.128766\pi$
$839$	660.438i	0.787173i	−0.919288	+	0.393586i	$0.871234\pi$
$840$	0	0
$841$	119.000	0.141498
$842$	791.960	0.940570
$843$	0	0
$844$	98.0000	0.116114
$845$	0	0
$846$	0	0
$847$	− 1953.00i	− 2.30579i
$848$	384.666	0.453616
$849$	0	0
$850$	0	0
$851$	135.765i	0.159535i
$852$	0	0
$853$	− 297.000i	− 0.348183i	−0.984729	−	0.174091i	$-0.944301\pi$
$853$	− 297.000i	− 0.348183i	0.984729	−	0.174091i	$-0.0556988\pi$
$854$	− 929.138i	− 1.08798i
$855$	0	0
$856$	−1008.00	−1.17757
$857$	1165.31	1.35976	0.679879	−	0.733325i	$-0.262032\pi$
$857$	1165.31	1.35976	0.679879	+	0.733325i	$0.262032\pi$
$858$	0	0
$859$	1286.00	1.49709	0.748545	−	0.663084i	$-0.230753\pi$
$859$	1286.00	1.49709	0.748545	+	0.663084i	$0.230753\pi$
$860$	0	0
$861$	0	0
$862$	678.000i	0.786543i
$863$	−633.568	−0.734146	−0.367073	−	0.930192i	$-0.619640\pi$
$863$	−633.568	−0.734146	−0.367073	+	0.930192i	$0.619640\pi$
$864$	0	0
$865$	0	0
$866$	363.453i	0.419692i
$867$	0	0
$868$	702.000i	0.808756i
$869$	− 441.235i	− 0.507750i
$870$	0	0
$871$	63.0000	0.0723307
$872$	212.132	0.243271
$873$	0	0
$874$	−150.000	−0.171625
$875$	0	0
$876$	0	0
$877$	1599.00i	1.82326i	0.411011	+	0.911631i	$0.365176\pi$
$877$	1599.00i	1.82326i	−0.411011	+	0.911631i	$0.634824\pi$
$878$	−236.174	−0.268991
$879$	0	0
$880$	0	0
$881$	961.665i	1.09156i	0.837928	+	0.545780i	$0.183767\pi$
$881$	961.665i	1.09156i	−0.837928	+	0.545780i	$0.816233\pi$
$882$	0	0
$883$	− 193.000i	− 0.218573i	−0.994010	−	0.109287i	$-0.965143\pi$
$883$	− 193.000i	− 0.218573i	0.994010	−	0.109287i	$-0.0348566\pi$
$884$	53.7401i	0.0607920i
$885$	0	0
$886$	−240.000	−0.270880
$887$	700.036	0.789217	0.394609	−	0.918849i	$-0.370880\pi$
$887$	700.036	0.789217	0.394609	+	0.918849i	$0.370880\pi$
$888$	0	0
$889$	−720.000	−0.809899
$890$	0	0
$891$	0	0
$892$	530.000i	0.594170i
$893$	−813.173	−0.910608
$894$	0	0
$895$	0	0
$896$	305.470i	0.340926i
$897$	0	0
$898$	976.000i	1.08686i
$899$	1047.93i	1.16566i
$900$	0	0
$901$	2584.00	2.86792
$902$	−147.078	−0.163058
$903$	0	0
$904$	−384.000	−0.424779
$905$	0	0
$906$	0	0
$907$	240.000i	0.264609i	0.991209	+	0.132304i	$0.0422376\pi$
$907$	240.000i	0.264609i	−0.991209	+	0.132304i	$0.957762\pi$
$908$	260.215	0.286581
$909$	0	0
$910$	0	0
$911$	− 171.120i	− 0.187837i	−0.995580	−	0.0939187i	$-0.970061\pi$
$911$	− 171.120i	− 0.187837i	0.995580	−	0.0939187i	$-0.0299394\pi$
$912$	0	0
$913$	858.000i	0.939759i
$914$	− 70.7107i	− 0.0773640i
$915$	0	0
$916$	−174.000	−0.189956
$917$	763.675	0.832798
$918$	0	0
$919$	−343.000	−0.373232	−0.186616	−	0.982433i	$-0.559752\pi$
$919$	−343.000	−0.373232	−0.186616	+	0.982433i	$0.559752\pi$
$920$	0	0
$921$	0	0
$922$	− 216.000i	− 0.234273i
$923$	62.2254	0.0674165
$924$	0	0
$925$	0	0
$926$	339.411i	0.366535i
$927$	0	0
$928$	760.000i	0.818966i
$929$	719.835i	0.774849i	0.921901	+	0.387424i	$0.126635\pi$
$929$	719.835i	0.774849i	−0.921901	+	0.387424i	$0.873365\pi$
$930$	0	0
$931$	800.000	0.859291
$932$	305.470	0.327758
$933$	0	0
$934$	838.000	0.897216
$935$	0	0
$936$	0	0
$937$	− 609.000i	− 0.649947i	−0.945723	−	0.324973i	$-0.894645\pi$
$937$	− 609.000i	− 0.649947i	0.945723	−	0.324973i	$-0.105355\pi$
$938$	−801.859	−0.854860
$939$	0	0
$940$	0	0
$941$	111.723i	0.118728i	0.998236	+	0.0593639i	$0.0189072\pi$
$941$	111.723i	0.118728i	−0.998236	+	0.0593639i	$0.981093\pi$
$942$	0	0
$943$	− 24.0000i	− 0.0254507i
$944$	− 39.5980i	− 0.0419470i
$945$	0	0
$946$	598.000	0.632135
$947$	1022.48	1.07970	0.539850	−	0.841761i	$-0.318481\pi$
$947$	1022.48	1.07970	0.539850	+	0.841761i	$0.318481\pi$
$948$	0	0
$949$	−136.000	−0.143309
$950$	0	0
$951$	0	0
$952$	− 2052.00i	− 2.15546i
$953$	1380.27	1.44834	0.724172	−	0.689619i	$-0.242222\pi$
$953$	1380.27	1.44834	0.724172	+	0.689619i	$0.242222\pi$
$954$	0	0
$955$	0	0
$956$	− 554.372i	− 0.579887i
$957$	0	0
$958$	702.000i	0.732777i
$959$	190.919i	0.199081i
$960$	0	0
$961$	560.000	0.582726
$962$	−45.2548	−0.0470424
$963$	0	0
$964$	−830.000	−0.860996
$965$	0	0
$966$	0	0
$967$	− 584.000i	− 0.603930i	−0.953319	−	0.301965i	$-0.902357\pi$
$967$	− 584.000i	− 0.603930i	0.953319	−	0.301965i	$-0.0976425\pi$
$968$	−1841.31	−1.90218
$969$	0	0
$970$	0	0
$971$	1030.96i	1.06175i	0.847449	+	0.530876i	$0.178137\pi$
$971$	1030.96i	1.06175i	−0.847449	+	0.530876i	$0.821863\pi$
$972$	0	0
$973$	− 936.000i	− 0.961973i
$974$	770.746i	0.791321i
$975$	0	0
$976$	−292.000	−0.299180
$977$	−1028.13	−1.05234	−0.526169	−	0.850380i	$-0.676372\pi$
$977$	−1028.13	−1.05234	−0.526169	+	0.850380i	$0.676372\pi$
$978$	0	0
$979$	1872.00	1.91216
$980$	0	0
$981$	0	0
$982$	− 1280.00i	− 1.30346i
$983$	−1323.70	−1.34660	−0.673298	−	0.739371i	$-0.735123\pi$
$983$	−1323.70	−1.34660	−0.673298	+	0.739371i	$0.735123\pi$
$984$	0	0
$985$	0	0
$986$	− 1021.06i	− 1.03556i
$987$	0	0
$988$	50.0000i	0.0506073i
$989$	97.5807i	0.0986661i
$990$	0	0
$991$	−961.000	−0.969728	−0.484864	−	0.874590i	$-0.661131\pi$
$991$	−961.000	−0.969728	−0.484864	+	0.874590i	$0.661131\pi$
$992$	1103.09	1.11198
$993$	0	0
$994$	−792.000	−0.796781
$995$	0	0
$996$	0	0
$997$	600.000i	0.601805i	0.953655	+	0.300903i	$0.0972879\pi$
$997$	600.000i	0.601805i	−0.953655	+	0.300903i	$0.902712\pi$
$998$	9.89949	0.00991933
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.3.d.a.224.2		4
3.2	odd	2	inner	225.3.d.a.224.4		4
4.3	odd	2		3600.3.c.e.449.2		4
5.2	odd	4		225.3.c.a.26.1	✓	2
5.3	odd	4		225.3.c.b.26.2	yes	2
5.4	even	2	inner	225.3.d.a.224.3		4
12.11	even	2		3600.3.c.e.449.1		4
15.2	even	4		225.3.c.a.26.2	yes	2
15.8	even	4		225.3.c.b.26.1	yes	2
15.14	odd	2	inner	225.3.d.a.224.1		4
20.3	even	4		3600.3.l.b.1601.2		2
20.7	even	4		3600.3.l.j.1601.2		2
20.19	odd	2		3600.3.c.e.449.4		4
60.23	odd	4		3600.3.l.b.1601.1		2
60.47	odd	4		3600.3.l.j.1601.1		2
60.59	even	2		3600.3.c.e.449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.3.c.a.26.1	✓	2	5.2	odd	4
225.3.c.a.26.2	yes	2	15.2	even	4
225.3.c.b.26.1	yes	2	15.8	even	4
225.3.c.b.26.2	yes	2	5.3	odd	4
225.3.d.a.224.1		4	15.14	odd	2	inner
225.3.d.a.224.2		4	1.1	even	1	trivial
225.3.d.a.224.3		4	5.4	even	2	inner
225.3.d.a.224.4		4	3.2	odd	2	inner
3600.3.c.e.449.1		4	12.11	even	2
3600.3.c.e.449.2		4	4.3	odd	2
3600.3.c.e.449.3		4	60.59	even	2
3600.3.c.e.449.4		4	20.19	odd	2
3600.3.l.b.1601.1		2	60.23	odd	4
3600.3.l.b.1601.2		2	20.3	even	4
3600.3.l.j.1601.1		2	60.47	odd	4
3600.3.l.j.1601.2		2	20.7	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -2.00000 q^{4} +9.00000i q^{7} +8.48528 q^{8} -18.3848i q^{11} +1.00000i q^{13} -12.7279i q^{14} -4.00000 q^{16} -26.8701 q^{17} -25.0000 q^{19} +26.0000i q^{22} -4.24264 q^{23} -1.41421i q^{26} -18.0000i q^{28} -26.8701i q^{29} -39.0000 q^{31} -28.2843 q^{32} +38.0000 q^{34} -32.0000i q^{37} +35.3553 q^{38} +5.65685i q^{41} -23.0000i q^{43} +36.7696i q^{44} +6.00000 q^{46} +32.5269 q^{47} -32.0000 q^{49} -2.00000i q^{52} -96.1665 q^{53} +76.3675i q^{56} +38.0000i q^{58} +9.89949i q^{59} +73.0000 q^{61} +55.1543 q^{62} +56.0000 q^{64} -63.0000i q^{67} +53.7401 q^{68} -62.2254i q^{71} +136.000i q^{73} +45.2548i q^{74} +50.0000 q^{76} +165.463 q^{77} +24.0000 q^{79} -8.00000i q^{82} -46.6690 q^{83} +32.5269i q^{86} -156.000i q^{88} +101.823i q^{89} -9.00000 q^{91} +8.48528 q^{92} -46.0000 q^{94} +7.00000i q^{97} +45.2548 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 16 q^{16} - 100 q^{19} - 156 q^{31} + 152 q^{34} + 24 q^{46} - 128 q^{49} + 292 q^{61} + 224 q^{64} + 200 q^{76} + 96 q^{79} - 36 q^{91} - 184 q^{94}+O(q^{100}) \) 4 * q - 8 * q^4 - 16 * q^16 - 100 * q^19 - 156 * q^31 + 152 * q^34 + 24 * q^46 - 128 * q^49 + 292 * q^61 + 224 * q^64 + 200 * q^76 + 96 * q^79 - 36 * q^91 - 184 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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