LMFDB - Embedded newform 225.4.b.e.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,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, 4, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$13.2754297513$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.4.b.e.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +7.00000 q^{4} -24.0000i q^{7} -15.0000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} +7.00000 q^{4} -24.0000i q^{7} -15.0000i q^{8} -52.0000 q^{11} -22.0000i q^{13} -24.0000 q^{14} +41.0000 q^{16} +14.0000i q^{17} +20.0000 q^{19} +52.0000i q^{22} -168.000i q^{23} -22.0000 q^{26} -168.000i q^{28} +230.000 q^{29} -288.000 q^{31} -161.000i q^{32} +14.0000 q^{34} -34.0000i q^{37} -20.0000i q^{38} -122.000 q^{41} +188.000i q^{43} -364.000 q^{44} -168.000 q^{46} -256.000i q^{47} -233.000 q^{49} -154.000i q^{52} -338.000i q^{53} -360.000 q^{56} -230.000i q^{58} +100.000 q^{59} +742.000 q^{61} +288.000i q^{62} +167.000 q^{64} -84.0000i q^{67} +98.0000i q^{68} +328.000 q^{71} +38.0000i q^{73} -34.0000 q^{74} +140.000 q^{76} +1248.00i q^{77} +240.000 q^{79} +122.000i q^{82} +1212.00i q^{83} +188.000 q^{86} +780.000i q^{88} +330.000 q^{89} -528.000 q^{91} -1176.00i q^{92} -256.000 q^{94} +866.000i q^{97} +233.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4}+O(q^{10}) $ 2 * q + 14 * q^4	$ 2 q + 14 q^{4} - 104 q^{11} - 48 q^{14} + 82 q^{16} + 40 q^{19} - 44 q^{26} + 460 q^{29} - 576 q^{31} + 28 q^{34} - 244 q^{41} - 728 q^{44} - 336 q^{46} - 466 q^{49} - 720 q^{56} + 200 q^{59} + 1484 q^{61} + 334 q^{64} + 656 q^{71} - 68 q^{74} + 280 q^{76} + 480 q^{79} + 376 q^{86} + 660 q^{89} - 1056 q^{91} - 512 q^{94}+O(q^{100}) $ 2 * q + 14 * q^4 - 104 * q^11 - 48 * q^14 + 82 * q^16 + 40 * q^19 - 44 * q^26 + 460 * q^29 - 576 * q^31 + 28 * q^34 - 244 * q^41 - 728 * q^44 - 336 * q^46 - 466 * q^49 - 720 * q^56 + 200 * q^59 + 1484 * q^61 + 334 * q^64 + 656 * q^71 - 68 * q^74 + 280 * q^76 + 480 * q^79 + 376 * q^86 + 660 * q^89 - 1056 * q^91 - 512 * 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 $

$2$	− 1.00000i	− 0.353553i	−0.984251	−	0.176777i	$-0.943433\pi$
$2$	− 1.00000i	− 0.353553i	0.984251	−	0.176777i	$-0.0565670\pi$
$3$	0	0
$3$	0	0
$4$	7.00000	0.875000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 24.0000i	− 1.29588i	−0.761692	−	0.647939i	$-0.775631\pi$
$7$	− 24.0000i	− 1.29588i	0.761692	−	0.647939i	$-0.224369\pi$
$8$	− 15.0000i	− 0.662913i
$9$	0	0
$10$	0	0
$11$	−52.0000	−1.42533	−0.712663	−	0.701506i	$-0.752511\pi$
$11$	−52.0000	−1.42533	−0.712663	+	0.701506i	$0.752511\pi$
$12$	0	0
$13$	− 22.0000i	− 0.469362i	−0.972072	−	0.234681i	$-0.924595\pi$
$13$	− 22.0000i	− 0.469362i	0.972072	−	0.234681i	$-0.0754045\pi$
$14$	−24.0000	−0.458162
$15$	0	0
$16$	41.0000	0.640625
$17$	14.0000i	0.199735i	0.995001	+	0.0998676i	$0.0318419\pi$
$17$	14.0000i	0.199735i	−0.995001	+	0.0998676i	$0.968158\pi$
$18$	0	0
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	0	0
$21$	0	0
$22$	52.0000i	0.503929i
$23$	− 168.000i	− 1.52306i	−0.648129	−	0.761531i	$-0.724448\pi$
$23$	− 168.000i	− 1.52306i	0.648129	−	0.761531i	$-0.275552\pi$
$24$	0	0
$25$	0	0
$26$	−22.0000	−0.165944
$27$	0	0
$28$	− 168.000i	− 1.13389i
$29$	230.000	1.47276	0.736378	−	0.676570i	$-0.236535\pi$
$29$	230.000	1.47276	0.736378	+	0.676570i	$0.236535\pi$
$30$	0	0
$31$	−288.000	−1.66859	−0.834296	−	0.551317i	$-0.814125\pi$
$31$	−288.000	−1.66859	−0.834296	+	0.551317i	$0.814125\pi$
$32$	− 161.000i	− 0.889408i
$33$	0	0
$34$	14.0000	0.0706171
$35$	0	0
$36$	0	0
$37$	− 34.0000i	− 0.151069i	−0.997143	−	0.0755347i	$-0.975934\pi$
$37$	− 34.0000i	− 0.151069i	0.997143	−	0.0755347i	$-0.0240664\pi$
$38$	− 20.0000i	− 0.0853797i
$39$	0	0
$40$	0	0
$41$	−122.000	−0.464712	−0.232356	−	0.972631i	$-0.574643\pi$
$41$	−122.000	−0.464712	−0.232356	+	0.972631i	$0.574643\pi$
$42$	0	0
$43$	188.000i	0.666738i	0.942796	+	0.333369i	$0.108185\pi$
$43$	188.000i	0.666738i	−0.942796	+	0.333369i	$0.891815\pi$
$44$	−364.000	−1.24716
$45$	0	0
$46$	−168.000	−0.538484
$47$	− 256.000i	− 0.794499i	−0.917711	−	0.397249i	$-0.869965\pi$
$47$	− 256.000i	− 0.794499i	0.917711	−	0.397249i	$-0.130035\pi$
$48$	0	0
$49$	−233.000	−0.679300
$50$	0	0
$51$	0	0
$52$	− 154.000i	− 0.410691i
$53$	− 338.000i	− 0.875998i	−0.898976	−	0.437999i	$-0.855687\pi$
$53$	− 338.000i	− 0.875998i	0.898976	−	0.437999i	$-0.144313\pi$
$54$	0	0
$55$	0	0
$56$	−360.000	−0.859054
$57$	0	0
$58$	− 230.000i	− 0.520698i
$59$	100.000	0.220659	0.110330	−	0.993895i	$-0.464809\pi$
$59$	100.000	0.220659	0.110330	+	0.993895i	$0.464809\pi$
$60$	0	0
$61$	742.000	1.55743	0.778716	−	0.627376i	$-0.215871\pi$
$61$	742.000	1.55743	0.778716	+	0.627376i	$0.215871\pi$
$62$	288.000i	0.589936i
$63$	0	0
$64$	167.000	0.326172
$65$	0	0
$66$	0	0
$67$	− 84.0000i	− 0.153168i	−0.997063	−	0.0765838i	$-0.975599\pi$
$67$	− 84.0000i	− 0.153168i	0.997063	−	0.0765838i	$-0.0244013\pi$
$68$	98.0000i	0.174768i
$69$	0	0
$70$	0	0
$71$	328.000	0.548260	0.274130	−	0.961693i	$-0.411610\pi$
$71$	328.000	0.548260	0.274130	+	0.961693i	$0.411610\pi$
$72$	0	0
$73$	38.0000i	0.0609255i	0.999536	+	0.0304628i	$0.00969810\pi$
$73$	38.0000i	0.0609255i	−0.999536	+	0.0304628i	$0.990302\pi$
$74$	−34.0000	−0.0534111
$75$	0	0
$76$	140.000	0.211304
$77$	1248.00i	1.84705i
$78$	0	0
$79$	240.000	0.341799	0.170899	−	0.985288i	$-0.445333\pi$
$79$	240.000	0.341799	0.170899	+	0.985288i	$0.445333\pi$
$80$	0	0
$81$	0	0
$82$	122.000i	0.164301i
$83$	1212.00i	1.60282i	0.598114	+	0.801411i	$0.295917\pi$
$83$	1212.00i	1.60282i	−0.598114	+	0.801411i	$0.704083\pi$
$84$	0	0
$85$	0	0
$86$	188.000	0.235727
$87$	0	0
$88$	780.000i	0.944867i
$89$	330.000	0.393033	0.196516	−	0.980501i	$-0.437037\pi$
$89$	330.000	0.393033	0.196516	+	0.980501i	$0.437037\pi$
$90$	0	0
$91$	−528.000	−0.608236
$92$	− 1176.00i	− 1.33268i
$93$	0	0
$94$	−256.000	−0.280898
$95$	0	0
$96$	0	0
$97$	866.000i	0.906484i	0.891387	+	0.453242i	$0.149733\pi$
$97$	866.000i	0.906484i	−0.891387	+	0.453242i	$0.850267\pi$
$98$	233.000i	0.240169i
$99$	0	0
$100$	0	0
$101$	1218.00	1.19996	0.599978	−	0.800017i	$-0.295176\pi$
$101$	1218.00	1.19996	0.599978	+	0.800017i	$0.295176\pi$
$102$	0	0
$103$	88.0000i	0.0841835i	0.999114	+	0.0420917i	$0.0134022\pi$
$103$	88.0000i	0.0841835i	−0.999114	+	0.0420917i	$0.986598\pi$
$104$	−330.000	−0.311146
$105$	0	0
$106$	−338.000	−0.309712
$107$	− 36.0000i	− 0.0325257i	−0.999868	−	0.0162629i	$-0.994823\pi$
$107$	− 36.0000i	− 0.0325257i	0.999868	−	0.0162629i	$-0.00517686\pi$
$108$	0	0
$109$	970.000	0.852378	0.426189	−	0.904634i	$-0.359856\pi$
$109$	970.000	0.852378	0.426189	+	0.904634i	$0.359856\pi$
$110$	0	0
$111$	0	0
$112$	− 984.000i	− 0.830172i
$113$	1042.00i	0.867461i	0.901043	+	0.433731i	$0.142803\pi$
$113$	1042.00i	0.867461i	−0.901043	+	0.433731i	$0.857197\pi$
$114$	0	0
$115$	0	0
$116$	1610.00	1.28866
$117$	0	0
$118$	− 100.000i	− 0.0780148i
$119$	336.000	0.258833
$120$	0	0
$121$	1373.00	1.03156
$122$	− 742.000i	− 0.550635i
$123$	0	0
$124$	−2016.00	−1.46002
$125$	0	0
$126$	0	0
$127$	1936.00i	1.35269i	0.736583	+	0.676347i	$0.236438\pi$
$127$	1936.00i	1.35269i	−0.736583	+	0.676347i	$0.763562\pi$
$128$	− 1455.00i	− 1.00473i
$129$	0	0
$130$	0	0
$131$	−732.000	−0.488207	−0.244104	−	0.969749i	$-0.578494\pi$
$131$	−732.000	−0.488207	−0.244104	+	0.969749i	$0.578494\pi$
$132$	0	0
$133$	− 480.000i	− 0.312942i
$134$	−84.0000	−0.0541529
$135$	0	0
$136$	210.000	0.132407
$137$	2214.00i	1.38069i	0.723479	+	0.690346i	$0.242542\pi$
$137$	2214.00i	1.38069i	−0.723479	+	0.690346i	$0.757458\pi$
$138$	0	0
$139$	−20.0000	−0.0122042	−0.00610208	−	0.999981i	$-0.501942\pi$
$139$	−20.0000	−0.0122042	−0.00610208	+	0.999981i	$0.501942\pi$
$140$	0	0
$141$	0	0
$142$	− 328.000i	− 0.193839i
$143$	1144.00i	0.668994i
$144$	0	0
$145$	0	0
$146$	38.0000	0.0215404
$147$	0	0
$148$	− 238.000i	− 0.132186i
$149$	−1330.00	−0.731261	−0.365630	−	0.930760i	$-0.619147\pi$
$149$	−1330.00	−0.731261	−0.365630	+	0.930760i	$0.619147\pi$
$150$	0	0
$151$	−1208.00	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−1208.00	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	− 300.000i	− 0.160087i
$153$	0	0
$154$	1248.00	0.653031
$155$	0	0
$156$	0	0
$157$	− 3514.00i	− 1.78629i	−0.449768	−	0.893146i	$-0.648493\pi$
$157$	− 3514.00i	− 1.78629i	0.449768	−	0.893146i	$-0.351507\pi$
$158$	− 240.000i	− 0.120844i
$159$	0	0
$160$	0	0
$161$	−4032.00	−1.97370
$162$	0	0
$163$	2068.00i	0.993732i	0.867827	+	0.496866i	$0.165516\pi$
$163$	2068.00i	0.993732i	−0.867827	+	0.496866i	$0.834484\pi$
$164$	−854.000	−0.406623
$165$	0	0
$166$	1212.00	0.566683
$167$	24.0000i	0.0111208i	0.999985	+	0.00556041i	$0.00176994\pi$
$167$	24.0000i	0.0111208i	−0.999985	+	0.00556041i	$0.998230\pi$
$168$	0	0
$169$	1713.00	0.779700
$170$	0	0
$171$	0	0
$172$	1316.00i	0.583396i
$173$	− 618.000i	− 0.271593i	−0.990737	−	0.135797i	$-0.956641\pi$
$173$	− 618.000i	− 0.271593i	0.990737	−	0.135797i	$-0.0433594\pi$
$174$	0	0
$175$	0	0
$176$	−2132.00	−0.913100
$177$	0	0
$178$	− 330.000i	− 0.138958i
$179$	3340.00	1.39466	0.697328	−	0.716752i	$-0.254372\pi$
$179$	3340.00	1.39466	0.697328	+	0.716752i	$0.254372\pi$
$180$	0	0
$181$	−178.000	−0.0730974	−0.0365487	−	0.999332i	$-0.511636\pi$
$181$	−178.000	−0.0730974	−0.0365487	+	0.999332i	$0.511636\pi$
$182$	528.000i	0.215044i
$183$	0	0
$184$	−2520.00	−1.00966
$185$	0	0
$186$	0	0
$187$	− 728.000i	− 0.284688i
$188$	− 1792.00i	− 0.695186i
$189$	0	0
$190$	0	0
$191$	1888.00	0.715240	0.357620	−	0.933867i	$-0.383588\pi$
$191$	1888.00	0.715240	0.357620	+	0.933867i	$0.383588\pi$
$192$	0	0
$193$	− 1922.00i	− 0.716832i	−0.933562	−	0.358416i	$-0.883317\pi$
$193$	− 1922.00i	− 0.716832i	0.933562	−	0.358416i	$-0.116683\pi$
$194$	866.000	0.320491
$195$	0	0
$196$	−1631.00	−0.594388
$197$	− 2526.00i	− 0.913554i	−0.889581	−	0.456777i	$-0.849004\pi$
$197$	− 2526.00i	− 0.913554i	0.889581	−	0.456777i	$-0.150996\pi$
$198$	0	0
$199$	1160.00	0.413217	0.206609	−	0.978424i	$-0.433757\pi$
$199$	1160.00	0.413217	0.206609	+	0.978424i	$0.433757\pi$
$200$	0	0
$201$	0	0
$202$	− 1218.00i	− 0.424248i
$203$	− 5520.00i	− 1.90851i
$204$	0	0
$205$	0	0
$206$	88.0000	0.0297634
$207$	0	0
$208$	− 902.000i	− 0.300685i
$209$	−1040.00	−0.344202
$210$	0	0
$211$	−4468.00	−1.45777	−0.728886	−	0.684635i	$-0.759961\pi$
$211$	−4468.00	−1.45777	−0.728886	+	0.684635i	$0.759961\pi$
$212$	− 2366.00i	− 0.766498i
$213$	0	0
$214$	−36.0000	−0.0114996
$215$	0	0
$216$	0	0
$217$	6912.00i	2.16229i
$218$	− 970.000i	− 0.301361i
$219$	0	0
$220$	0	0
$221$	308.000	0.0937481
$222$	0	0
$223$	− 6032.00i	− 1.81136i	−0.423965	−	0.905678i	$-0.639362\pi$
$223$	− 6032.00i	− 1.81136i	0.423965	−	0.905678i	$-0.360638\pi$
$224$	−3864.00	−1.15256
$225$	0	0
$226$	1042.00	0.306694
$227$	− 2636.00i	− 0.770738i	−0.922763	−	0.385369i	$-0.874074\pi$
$227$	− 2636.00i	− 0.770738i	0.922763	−	0.385369i	$-0.125926\pi$
$228$	0	0
$229$	−4830.00	−1.39378	−0.696889	−	0.717179i	$-0.745433\pi$
$229$	−4830.00	−1.39378	−0.696889	+	0.717179i	$0.745433\pi$
$230$	0	0
$231$	0	0
$232$	− 3450.00i	− 0.976309i
$233$	2682.00i	0.754093i	0.926194	+	0.377046i	$0.123060\pi$
$233$	2682.00i	0.754093i	−0.926194	+	0.377046i	$0.876940\pi$
$234$	0	0
$235$	0	0
$236$	700.000	0.193077
$237$	0	0
$238$	− 336.000i	− 0.0915111i
$239$	2320.00	0.627901	0.313950	−	0.949439i	$-0.398347\pi$
$239$	2320.00	0.627901	0.313950	+	0.949439i	$0.398347\pi$
$240$	0	0
$241$	2002.00	0.535104	0.267552	−	0.963543i	$-0.413785\pi$
$241$	2002.00	0.535104	0.267552	+	0.963543i	$0.413785\pi$
$242$	− 1373.00i	− 0.364710i
$243$	0	0
$244$	5194.00	1.36275
$245$	0	0
$246$	0	0
$247$	− 440.000i	− 0.113346i
$248$	4320.00i	1.10613i
$249$	0	0
$250$	0	0
$251$	−132.000	−0.0331943	−0.0165971	−	0.999862i	$-0.505283\pi$
$251$	−132.000	−0.0331943	−0.0165971	+	0.999862i	$0.505283\pi$
$252$	0	0
$253$	8736.00i	2.17086i
$254$	1936.00	0.478250
$255$	0	0
$256$	−119.000	−0.0290527
$257$	7614.00i	1.84805i	0.382335	+	0.924024i	$0.375120\pi$
$257$	7614.00i	1.84805i	−0.382335	+	0.924024i	$0.624880\pi$
$258$	0	0
$259$	−816.000	−0.195767
$260$	0	0
$261$	0	0
$262$	732.000i	0.172607i
$263$	− 4888.00i	− 1.14603i	−0.819543	−	0.573017i	$-0.805773\pi$
$263$	− 4888.00i	− 1.14603i	0.819543	−	0.573017i	$-0.194227\pi$
$264$	0	0
$265$	0	0
$266$	−480.000	−0.110642
$267$	0	0
$268$	− 588.000i	− 0.134022i
$269$	1270.00	0.287856	0.143928	−	0.989588i	$-0.454027\pi$
$269$	1270.00	0.287856	0.143928	+	0.989588i	$0.454027\pi$
$270$	0	0
$271$	1072.00	0.240293	0.120146	−	0.992756i	$-0.461664\pi$
$271$	1072.00	0.240293	0.120146	+	0.992756i	$0.461664\pi$
$272$	574.000i	0.127955i
$273$	0	0
$274$	2214.00	0.488148
$275$	0	0
$276$	0	0
$277$	− 5394.00i	− 1.17001i	−0.811028	−	0.585007i	$-0.801092\pi$
$277$	− 5394.00i	− 1.17001i	0.811028	−	0.585007i	$-0.198908\pi$
$278$	20.0000i	0.00431482i
$279$	0	0
$280$	0	0
$281$	−2442.00	−0.518425	−0.259213	−	0.965820i	$-0.583463\pi$
$281$	−2442.00	−0.518425	−0.259213	+	0.965820i	$0.583463\pi$
$282$	0	0
$283$	− 2772.00i	− 0.582255i	−0.956684	−	0.291128i	$-0.905970\pi$
$283$	− 2772.00i	− 0.582255i	0.956684	−	0.291128i	$-0.0940305\pi$
$284$	2296.00	0.479727
$285$	0	0
$286$	1144.00	0.236525
$287$	2928.00i	0.602210i
$288$	0	0
$289$	4717.00	0.960106
$290$	0	0
$291$	0	0
$292$	266.000i	0.0533098i
$293$	4542.00i	0.905619i	0.891607	+	0.452810i	$0.149578\pi$
$293$	4542.00i	0.905619i	−0.891607	+	0.452810i	$0.850422\pi$
$294$	0	0
$295$	0	0
$296$	−510.000	−0.100146
$297$	0	0
$298$	1330.00i	0.258540i
$299$	−3696.00	−0.714867
$300$	0	0
$301$	4512.00	0.864011
$302$	1208.00i	0.230174i
$303$	0	0
$304$	820.000	0.154705
$305$	0	0
$306$	0	0
$307$	5116.00i	0.951093i	0.879691	+	0.475546i	$0.157750\pi$
$307$	5116.00i	0.951093i	−0.879691	+	0.475546i	$0.842250\pi$
$308$	8736.00i	1.61617i
$309$	0	0
$310$	0	0
$311$	2808.00	0.511984	0.255992	−	0.966679i	$-0.417598\pi$
$311$	2808.00	0.511984	0.255992	+	0.966679i	$0.417598\pi$
$312$	0	0
$313$	7318.00i	1.32153i	0.750594	+	0.660763i	$0.229767\pi$
$313$	7318.00i	1.32153i	−0.750594	+	0.660763i	$0.770233\pi$
$314$	−3514.00	−0.631549
$315$	0	0
$316$	1680.00	0.299074
$317$	− 2246.00i	− 0.397943i	−0.980005	−	0.198971i	$-0.936240\pi$
$317$	− 2246.00i	− 0.397943i	0.980005	−	0.198971i	$-0.0637601\pi$
$318$	0	0
$319$	−11960.0	−2.09916
$320$	0	0
$321$	0	0
$322$	4032.00i	0.697809i
$323$	280.000i	0.0482341i
$324$	0	0
$325$	0	0
$326$	2068.00	0.351337
$327$	0	0
$328$	1830.00i	0.308064i
$329$	−6144.00	−1.02957
$330$	0	0
$331$	1332.00	0.221188	0.110594	−	0.993866i	$-0.464725\pi$
$331$	1332.00	0.221188	0.110594	+	0.993866i	$0.464725\pi$
$332$	8484.00i	1.40247i
$333$	0	0
$334$	24.0000	0.00393180
$335$	0	0
$336$	0	0
$337$	− 11534.0i	− 1.86438i	−0.361966	−	0.932191i	$-0.617894\pi$
$337$	− 11534.0i	− 1.86438i	0.361966	−	0.932191i	$-0.382106\pi$
$338$	− 1713.00i	− 0.275665i
$339$	0	0
$340$	0	0
$341$	14976.0	2.37829
$342$	0	0
$343$	− 2640.00i	− 0.415588i
$344$	2820.00	0.441989
$345$	0	0
$346$	−618.000	−0.0960228
$347$	− 11956.0i	− 1.84966i	−0.380382	−	0.924830i	$-0.624207\pi$
$347$	− 11956.0i	− 1.84966i	0.380382	−	0.924830i	$-0.375793\pi$
$348$	0	0
$349$	−4870.00	−0.746949	−0.373474	−	0.927640i	$-0.621834\pi$
$349$	−4870.00	−0.746949	−0.373474	+	0.927640i	$0.621834\pi$
$350$	0	0
$351$	0	0
$352$	8372.00i	1.26770i
$353$	10722.0i	1.61664i	0.588742	+	0.808321i	$0.299623\pi$
$353$	10722.0i	1.61664i	−0.588742	+	0.808321i	$0.700377\pi$
$354$	0	0
$355$	0	0
$356$	2310.00	0.343904
$357$	0	0
$358$	− 3340.00i	− 0.493085i
$359$	120.000	0.0176417	0.00882083	−	0.999961i	$-0.497192\pi$
$359$	120.000	0.0176417	0.00882083	+	0.999961i	$0.497192\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	178.000i	0.0258438i
$363$	0	0
$364$	−3696.00	−0.532206
$365$	0	0
$366$	0	0
$367$	3936.00i	0.559830i	0.960025	+	0.279915i	$0.0903063\pi$
$367$	3936.00i	0.559830i	−0.960025	+	0.279915i	$0.909694\pi$
$368$	− 6888.00i	− 0.975711i
$369$	0	0
$370$	0	0
$371$	−8112.00	−1.13519
$372$	0	0
$373$	− 3022.00i	− 0.419499i	−0.977755	−	0.209750i	$-0.932735\pi$
$373$	− 3022.00i	− 0.419499i	0.977755	−	0.209750i	$-0.0672649\pi$
$374$	−728.000	−0.100652
$375$	0	0
$376$	−3840.00	−0.526683
$377$	− 5060.00i	− 0.691255i
$378$	0	0
$379$	13340.0	1.80799	0.903997	−	0.427539i	$-0.140619\pi$
$379$	13340.0	1.80799	0.903997	+	0.427539i	$0.140619\pi$
$380$	0	0
$381$	0	0
$382$	− 1888.00i	− 0.252876i
$383$	− 1008.00i	− 0.134481i	−0.997737	−	0.0672407i	$-0.978580\pi$
$383$	− 1008.00i	− 0.134481i	0.997737	−	0.0672407i	$-0.0214195\pi$
$384$	0	0
$385$	0	0
$386$	−1922.00	−0.253438
$387$	0	0
$388$	6062.00i	0.793174i
$389$	9630.00	1.25517	0.627584	−	0.778549i	$-0.284044\pi$
$389$	9630.00	1.25517	0.627584	+	0.778549i	$0.284044\pi$
$390$	0	0
$391$	2352.00	0.304209
$392$	3495.00i	0.450317i
$393$	0	0
$394$	−2526.00	−0.322990
$395$	0	0
$396$	0	0
$397$	7126.00i	0.900866i	0.892810	+	0.450433i	$0.148730\pi$
$397$	7126.00i	0.900866i	−0.892810	+	0.450433i	$0.851270\pi$
$398$	− 1160.00i	− 0.146094i
$399$	0	0
$400$	0	0
$401$	8718.00	1.08568	0.542838	−	0.839837i	$-0.317350\pi$
$401$	8718.00	1.08568	0.542838	+	0.839837i	$0.317350\pi$
$402$	0	0
$403$	6336.00i	0.783173i
$404$	8526.00	1.04996
$405$	0	0
$406$	−5520.00	−0.674761
$407$	1768.00i	0.215323i
$408$	0	0
$409$	10870.0	1.31415	0.657074	−	0.753826i	$-0.271794\pi$
$409$	10870.0	1.31415	0.657074	+	0.753826i	$0.271794\pi$
$410$	0	0
$411$	0	0
$412$	616.000i	0.0736605i
$413$	− 2400.00i	− 0.285947i
$414$	0	0
$415$	0	0
$416$	−3542.00	−0.417454
$417$	0	0
$418$	1040.00i	0.121694i
$419$	−9700.00	−1.13097	−0.565484	−	0.824759i	$-0.691311\pi$
$419$	−9700.00	−1.13097	−0.565484	+	0.824759i	$0.691311\pi$
$420$	0	0
$421$	862.000	0.0997893	0.0498947	−	0.998754i	$-0.484111\pi$
$421$	862.000	0.0997893	0.0498947	+	0.998754i	$0.484111\pi$
$422$	4468.00i	0.515400i
$423$	0	0
$424$	−5070.00	−0.580710
$425$	0	0
$426$	0	0
$427$	− 17808.0i	− 2.01824i
$428$	− 252.000i	− 0.0284600i
$429$	0	0
$430$	0	0
$431$	−15792.0	−1.76490	−0.882452	−	0.470402i	$-0.844109\pi$
$431$	−15792.0	−1.76490	−0.882452	+	0.470402i	$0.844109\pi$
$432$	0	0
$433$	− 11602.0i	− 1.28766i	−0.765169	−	0.643830i	$-0.777345\pi$
$433$	− 11602.0i	− 1.28766i	0.765169	−	0.643830i	$-0.222655\pi$
$434$	6912.00	0.764485
$435$	0	0
$436$	6790.00	0.745830
$437$	− 3360.00i	− 0.367805i
$438$	0	0
$439$	440.000	0.0478361	0.0239181	−	0.999714i	$-0.492386\pi$
$439$	440.000	0.0478361	0.0239181	+	0.999714i	$0.492386\pi$
$440$	0	0
$441$	0	0
$442$	− 308.000i	− 0.0331449i
$443$	− 10188.0i	− 1.09266i	−0.837571	−	0.546328i	$-0.816025\pi$
$443$	− 10188.0i	− 1.09266i	0.837571	−	0.546328i	$-0.183975\pi$
$444$	0	0
$445$	0	0
$446$	−6032.00	−0.640411
$447$	0	0
$448$	− 4008.00i	− 0.422679i
$449$	−13310.0	−1.39897	−0.699485	−	0.714647i	$-0.746587\pi$
$449$	−13310.0	−1.39897	−0.699485	+	0.714647i	$0.746587\pi$
$450$	0	0
$451$	6344.00	0.662367
$452$	7294.00i	0.759029i
$453$	0	0
$454$	−2636.00	−0.272497
$455$	0	0
$456$	0	0
$457$	3226.00i	0.330210i	0.986276	+	0.165105i	$0.0527963\pi$
$457$	3226.00i	0.330210i	−0.986276	+	0.165105i	$0.947204\pi$
$458$	4830.00i	0.492775i
$459$	0	0
$460$	0	0
$461$	−6582.00	−0.664977	−0.332488	−	0.943107i	$-0.607888\pi$
$461$	−6582.00	−0.664977	−0.332488	+	0.943107i	$0.607888\pi$
$462$	0	0
$463$	− 15072.0i	− 1.51286i	−0.654073	−	0.756431i	$-0.726941\pi$
$463$	− 15072.0i	− 1.51286i	0.654073	−	0.756431i	$-0.273059\pi$
$464$	9430.00	0.943484
$465$	0	0
$466$	2682.00	0.266612
$467$	− 476.000i	− 0.0471663i	−0.999722	−	0.0235831i	$-0.992493\pi$
$467$	− 476.000i	− 0.0471663i	0.999722	−	0.0235831i	$-0.00750744\pi$
$468$	0	0
$469$	−2016.00	−0.198487
$470$	0	0
$471$	0	0
$472$	− 1500.00i	− 0.146278i
$473$	− 9776.00i	− 0.950319i
$474$	0	0
$475$	0	0
$476$	2352.00	0.226478
$477$	0	0
$478$	− 2320.00i	− 0.221997i
$479$	−19680.0	−1.87725	−0.938624	−	0.344941i	$-0.887899\pi$
$479$	−19680.0	−1.87725	−0.938624	+	0.344941i	$0.887899\pi$
$480$	0	0
$481$	−748.000	−0.0709062
$482$	− 2002.00i	− 0.189188i
$483$	0	0
$484$	9611.00	0.902611
$485$	0	0
$486$	0	0
$487$	− 5944.00i	− 0.553077i	−0.961003	−	0.276538i	$-0.910813\pi$
$487$	− 5944.00i	− 0.553077i	0.961003	−	0.276538i	$-0.0891873\pi$
$488$	− 11130.0i	− 1.03244i
$489$	0	0
$490$	0	0
$491$	−10772.0	−0.990089	−0.495044	−	0.868868i	$-0.664848\pi$
$491$	−10772.0	−0.990089	−0.495044	+	0.868868i	$0.664848\pi$
$492$	0	0
$493$	3220.00i	0.294161i
$494$	−440.000	−0.0400740
$495$	0	0
$496$	−11808.0	−1.06894
$497$	− 7872.00i	− 0.710478i
$498$	0	0
$499$	−8140.00	−0.730253	−0.365127	−	0.930958i	$-0.618974\pi$
$499$	−8140.00	−0.730253	−0.365127	+	0.930958i	$0.618974\pi$
$500$	0	0
$501$	0	0
$502$	132.000i	0.0117360i
$503$	− 13768.0i	− 1.22045i	−0.792229	−	0.610223i	$-0.791080\pi$
$503$	− 13768.0i	− 1.22045i	0.792229	−	0.610223i	$-0.208920\pi$
$504$	0	0
$505$	0	0
$506$	8736.00	0.767515
$507$	0	0
$508$	13552.0i	1.18361i
$509$	22150.0	1.92884	0.964422	−	0.264368i	$-0.0851633\pi$
$509$	22150.0	1.92884	0.964422	+	0.264368i	$0.0851633\pi$
$510$	0	0
$511$	912.000	0.0789521
$512$	− 11521.0i	− 0.994455i
$513$	0	0
$514$	7614.00	0.653384
$515$	0	0
$516$	0	0
$517$	13312.0i	1.13242i
$518$	816.000i	0.0692143i
$519$	0	0
$520$	0	0
$521$	−1562.00	−0.131348	−0.0656741	−	0.997841i	$-0.520920\pi$
$521$	−1562.00	−0.131348	−0.0656741	+	0.997841i	$0.520920\pi$
$522$	0	0
$523$	668.000i	0.0558501i	0.999610	+	0.0279250i	$0.00888997\pi$
$523$	668.000i	0.0558501i	−0.999610	+	0.0279250i	$0.991110\pi$
$524$	−5124.00	−0.427181
$525$	0	0
$526$	−4888.00	−0.405184
$527$	− 4032.00i	− 0.333276i
$528$	0	0
$529$	−16057.0	−1.31972
$530$	0	0
$531$	0	0
$532$	− 3360.00i	− 0.273824i
$533$	2684.00i	0.218118i
$534$	0	0
$535$	0	0
$536$	−1260.00	−0.101537
$537$	0	0
$538$	− 1270.00i	− 0.101772i
$539$	12116.0	0.968225
$540$	0	0
$541$	−6138.00	−0.487788	−0.243894	−	0.969802i	$-0.578425\pi$
$541$	−6138.00	−0.487788	−0.243894	+	0.969802i	$0.578425\pi$
$542$	− 1072.00i	− 0.0849564i
$543$	0	0
$544$	2254.00	0.177646
$545$	0	0
$546$	0	0
$547$	− 10484.0i	− 0.819494i	−0.912199	−	0.409747i	$-0.865617\pi$
$547$	− 10484.0i	− 0.819494i	0.912199	−	0.409747i	$-0.134383\pi$
$548$	15498.0i	1.20811i
$549$	0	0
$550$	0	0
$551$	4600.00	0.355656
$552$	0	0
$553$	− 5760.00i	− 0.442930i
$554$	−5394.00	−0.413663
$555$	0	0
$556$	−140.000	−0.0106786
$557$	− 3606.00i	− 0.274311i	−0.990550	−	0.137155i	$-0.956204\pi$
$557$	− 3606.00i	− 0.274311i	0.990550	−	0.137155i	$-0.0437960\pi$
$558$	0	0
$559$	4136.00	0.312941
$560$	0	0
$561$	0	0
$562$	2442.00i	0.183291i
$563$	12252.0i	0.917159i	0.888654	+	0.458579i	$0.151641\pi$
$563$	12252.0i	0.917159i	−0.888654	+	0.458579i	$0.848359\pi$
$564$	0	0
$565$	0	0
$566$	−2772.00	−0.205858
$567$	0	0
$568$	− 4920.00i	− 0.363448i
$569$	−14550.0	−1.07200	−0.536000	−	0.844218i	$-0.680065\pi$
$569$	−14550.0	−1.07200	−0.536000	+	0.844218i	$0.680065\pi$
$570$	0	0
$571$	−25468.0	−1.86655	−0.933277	−	0.359157i	$-0.883064\pi$
$571$	−25468.0	−1.86655	−0.933277	+	0.359157i	$0.883064\pi$
$572$	8008.00i	0.585369i
$573$	0	0
$574$	2928.00	0.212914
$575$	0	0
$576$	0	0
$577$	12866.0i	0.928282i	0.885761	+	0.464141i	$0.153637\pi$
$577$	12866.0i	0.928282i	−0.885761	+	0.464141i	$0.846363\pi$
$578$	− 4717.00i	− 0.339449i
$579$	0	0
$580$	0	0
$581$	29088.0	2.07706
$582$	0	0
$583$	17576.0i	1.24858i
$584$	570.000	0.0403883
$585$	0	0
$586$	4542.00	0.320185
$587$	14844.0i	1.04374i	0.853024	+	0.521872i	$0.174766\pi$
$587$	14844.0i	1.04374i	−0.853024	+	0.521872i	$0.825234\pi$
$588$	0	0
$589$	−5760.00	−0.402948
$590$	0	0
$591$	0	0
$592$	− 1394.00i	− 0.0967788i
$593$	20402.0i	1.41283i	0.707797	+	0.706416i	$0.249689\pi$
$593$	20402.0i	1.41283i	−0.707797	+	0.706416i	$0.750311\pi$
$594$	0	0
$595$	0	0
$596$	−9310.00	−0.639853
$597$	0	0
$598$	3696.00i	0.252744i
$599$	10760.0	0.733959	0.366980	−	0.930229i	$-0.380392\pi$
$599$	10760.0	0.733959	0.366980	+	0.930229i	$0.380392\pi$
$600$	0	0
$601$	14282.0	0.969343	0.484671	−	0.874696i	$-0.338939\pi$
$601$	14282.0	0.969343	0.484671	+	0.874696i	$0.338939\pi$
$602$	− 4512.00i	− 0.305474i
$603$	0	0
$604$	−8456.00	−0.569652
$605$	0	0
$606$	0	0
$607$	11056.0i	0.739290i	0.929173	+	0.369645i	$0.120521\pi$
$607$	11056.0i	0.739290i	−0.929173	+	0.369645i	$0.879479\pi$
$608$	− 3220.00i	− 0.214783i
$609$	0	0
$610$	0	0
$611$	−5632.00	−0.372907
$612$	0	0
$613$	16418.0i	1.08176i	0.841101	+	0.540878i	$0.181908\pi$
$613$	16418.0i	1.08176i	−0.841101	+	0.540878i	$0.818092\pi$
$614$	5116.00	0.336262
$615$	0	0
$616$	18720.0	1.22443
$617$	10374.0i	0.676891i	0.940986	+	0.338445i	$0.109901\pi$
$617$	10374.0i	0.676891i	−0.940986	+	0.338445i	$0.890099\pi$
$618$	0	0
$619$	5260.00	0.341546	0.170773	−	0.985310i	$-0.445373\pi$
$619$	5260.00	0.341546	0.170773	+	0.985310i	$0.445373\pi$
$620$	0	0
$621$	0	0
$622$	− 2808.00i	− 0.181014i
$623$	− 7920.00i	− 0.509323i
$624$	0	0
$625$	0	0
$626$	7318.00	0.467230
$627$	0	0
$628$	− 24598.0i	− 1.56300i
$629$	476.000	0.0301739
$630$	0	0
$631$	21352.0	1.34708	0.673542	−	0.739149i	$-0.264772\pi$
$631$	21352.0	1.34708	0.673542	+	0.739149i	$0.264772\pi$
$632$	− 3600.00i	− 0.226583i
$633$	0	0
$634$	−2246.00	−0.140694
$635$	0	0
$636$	0	0
$637$	5126.00i	0.318838i
$638$	11960.0i	0.742164i
$639$	0	0
$640$	0	0
$641$	29118.0	1.79422	0.897108	−	0.441812i	$-0.145664\pi$
$641$	29118.0	1.79422	0.897108	+	0.441812i	$0.145664\pi$
$642$	0	0
$643$	− 5772.00i	− 0.354005i	−0.984210	−	0.177003i	$-0.943360\pi$
$643$	− 5772.00i	− 0.354005i	0.984210	−	0.177003i	$-0.0566401\pi$
$644$	−28224.0	−1.72699
$645$	0	0
$646$	280.000	0.0170533
$647$	14264.0i	0.866732i	0.901218	+	0.433366i	$0.142674\pi$
$647$	14264.0i	0.866732i	−0.901218	+	0.433366i	$0.857326\pi$
$648$	0	0
$649$	−5200.00	−0.314511
$650$	0	0
$651$	0	0
$652$	14476.0i	0.869515i
$653$	6902.00i	0.413623i	0.978381	+	0.206812i	$0.0663088\pi$
$653$	6902.00i	0.413623i	−0.978381	+	0.206812i	$0.933691\pi$
$654$	0	0
$655$	0	0
$656$	−5002.00	−0.297706
$657$	0	0
$658$	6144.00i	0.364009i
$659$	20140.0	1.19051	0.595253	−	0.803539i	$-0.297052\pi$
$659$	20140.0	1.19051	0.595253	+	0.803539i	$0.297052\pi$
$660$	0	0
$661$	−3218.00	−0.189358	−0.0946790	−	0.995508i	$-0.530182\pi$
$661$	−3218.00	−0.189358	−0.0946790	+	0.995508i	$0.530182\pi$
$662$	− 1332.00i	− 0.0782019i
$663$	0	0
$664$	18180.0	1.06253
$665$	0	0
$666$	0	0
$667$	− 38640.0i	− 2.24310i
$668$	168.000i	0.00973071i
$669$	0	0
$670$	0	0
$671$	−38584.0	−2.21985
$672$	0	0
$673$	7518.00i	0.430606i	0.976547	+	0.215303i	$0.0690739\pi$
$673$	7518.00i	0.430606i	−0.976547	+	0.215303i	$0.930926\pi$
$674$	−11534.0	−0.659159
$675$	0	0
$676$	11991.0	0.682237
$677$	18114.0i	1.02833i	0.857692	+	0.514164i	$0.171898\pi$
$677$	18114.0i	1.02833i	−0.857692	+	0.514164i	$0.828102\pi$
$678$	0	0
$679$	20784.0	1.17469
$680$	0	0
$681$	0	0
$682$	− 14976.0i	− 0.840851i
$683$	− 23868.0i	− 1.33716i	−0.743638	−	0.668582i	$-0.766901\pi$
$683$	− 23868.0i	− 1.33716i	0.743638	−	0.668582i	$-0.233099\pi$
$684$	0	0
$685$	0	0
$686$	−2640.00	−0.146932
$687$	0	0
$688$	7708.00i	0.427129i
$689$	−7436.00	−0.411160
$690$	0	0
$691$	172.000	0.00946916	0.00473458	−	0.999989i	$-0.498493\pi$
$691$	172.000	0.00946916	0.00473458	+	0.999989i	$0.498493\pi$
$692$	− 4326.00i	− 0.237644i
$693$	0	0
$694$	−11956.0	−0.653953
$695$	0	0
$696$	0	0
$697$	− 1708.00i	− 0.0928194i
$698$	4870.00i	0.264086i
$699$	0	0
$700$	0	0
$701$	22138.0	1.19278	0.596391	−	0.802694i	$-0.296601\pi$
$701$	22138.0	1.19278	0.596391	+	0.802694i	$0.296601\pi$
$702$	0	0
$703$	− 680.000i	− 0.0364818i
$704$	−8684.00	−0.464901
$705$	0	0
$706$	10722.0	0.571569
$707$	− 29232.0i	− 1.55500i
$708$	0	0
$709$	−3070.00	−0.162618	−0.0813091	−	0.996689i	$-0.525910\pi$
$709$	−3070.00	−0.162618	−0.0813091	+	0.996689i	$0.525910\pi$
$710$	0	0
$711$	0	0
$712$	− 4950.00i	− 0.260546i
$713$	48384.0i	2.54137i
$714$	0	0
$715$	0	0
$716$	23380.0	1.22032
$717$	0	0
$718$	− 120.000i	− 0.00623727i
$719$	15600.0	0.809154	0.404577	−	0.914504i	$-0.367419\pi$
$719$	15600.0	0.809154	0.404577	+	0.914504i	$0.367419\pi$
$720$	0	0
$721$	2112.00	0.109092
$722$	6459.00i	0.332935i
$723$	0	0
$724$	−1246.00	−0.0639603
$725$	0	0
$726$	0	0
$727$	20696.0i	1.05581i	0.849304	+	0.527904i	$0.177022\pi$
$727$	20696.0i	1.05581i	−0.849304	+	0.527904i	$0.822978\pi$
$728$	7920.00i	0.403207i
$729$	0	0
$730$	0	0
$731$	−2632.00	−0.133171
$732$	0	0
$733$	30778.0i	1.55090i	0.631408	+	0.775451i	$0.282478\pi$
$733$	30778.0i	1.55090i	−0.631408	+	0.775451i	$0.717522\pi$
$734$	3936.00	0.197930
$735$	0	0
$736$	−27048.0	−1.35462
$737$	4368.00i	0.218314i
$738$	0	0
$739$	−11740.0	−0.584388	−0.292194	−	0.956359i	$-0.594385\pi$
$739$	−11740.0	−0.584388	−0.292194	+	0.956359i	$0.594385\pi$
$740$	0	0
$741$	0	0
$742$	8112.00i	0.401349i
$743$	2632.00i	0.129958i	0.997887	+	0.0649789i	$0.0206980\pi$
$743$	2632.00i	0.129958i	−0.997887	+	0.0649789i	$0.979302\pi$
$744$	0	0
$745$	0	0
$746$	−3022.00	−0.148315
$747$	0	0
$748$	− 5096.00i	− 0.249102i
$749$	−864.000	−0.0421494
$750$	0	0
$751$	−20528.0	−0.997440	−0.498720	−	0.866763i	$-0.666196\pi$
$751$	−20528.0	−0.997440	−0.498720	+	0.866763i	$0.666196\pi$
$752$	− 10496.0i	− 0.508976i
$753$	0	0
$754$	−5060.00	−0.244396
$755$	0	0
$756$	0	0
$757$	21646.0i	1.03928i	0.854384	+	0.519642i	$0.173934\pi$
$757$	21646.0i	1.03928i	−0.854384	+	0.519642i	$0.826066\pi$
$758$	− 13340.0i	− 0.639222i
$759$	0	0
$760$	0	0
$761$	−18282.0	−0.870857	−0.435428	−	0.900223i	$-0.643403\pi$
$761$	−18282.0	−0.870857	−0.435428	+	0.900223i	$0.643403\pi$
$762$	0	0
$763$	− 23280.0i	− 1.10458i
$764$	13216.0	0.625835
$765$	0	0
$766$	−1008.00	−0.0475464
$767$	− 2200.00i	− 0.103569i
$768$	0	0
$769$	24190.0	1.13435	0.567174	−	0.823598i	$-0.308037\pi$
$769$	24190.0	1.13435	0.567174	+	0.823598i	$0.308037\pi$
$770$	0	0
$771$	0	0
$772$	− 13454.0i	− 0.627228i
$773$	− 25698.0i	− 1.19572i	−0.801600	−	0.597861i	$-0.796018\pi$
$773$	− 25698.0i	− 1.19572i	0.801600	−	0.597861i	$-0.203982\pi$
$774$	0	0
$775$	0	0
$776$	12990.0	0.600920
$777$	0	0
$778$	− 9630.00i	− 0.443769i
$779$	−2440.00	−0.112223
$780$	0	0
$781$	−17056.0	−0.781449
$782$	− 2352.00i	− 0.107554i
$783$	0	0
$784$	−9553.00	−0.435177
$785$	0	0
$786$	0	0
$787$	33436.0i	1.51444i	0.653160	+	0.757220i	$0.273443\pi$
$787$	33436.0i	1.51444i	−0.653160	+	0.757220i	$0.726557\pi$
$788$	− 17682.0i	− 0.799359i
$789$	0	0
$790$	0	0
$791$	25008.0	1.12412
$792$	0	0
$793$	− 16324.0i	− 0.730999i
$794$	7126.00	0.318504
$795$	0	0
$796$	8120.00	0.361565
$797$	37594.0i	1.67083i	0.549623	+	0.835413i	$0.314771\pi$
$797$	37594.0i	1.67083i	−0.549623	+	0.835413i	$0.685229\pi$
$798$	0	0
$799$	3584.00	0.158689
$800$	0	0
$801$	0	0
$802$	− 8718.00i	− 0.383844i
$803$	− 1976.00i	− 0.0868388i
$804$	0	0
$805$	0	0
$806$	6336.00	0.276893
$807$	0	0
$808$	− 18270.0i	− 0.795466i
$809$	4730.00	0.205560	0.102780	−	0.994704i	$-0.467226\pi$
$809$	4730.00	0.205560	0.102780	+	0.994704i	$0.467226\pi$
$810$	0	0
$811$	−8748.00	−0.378772	−0.189386	−	0.981903i	$-0.560650\pi$
$811$	−8748.00	−0.378772	−0.189386	+	0.981903i	$0.560650\pi$
$812$	− 38640.0i	− 1.66995i
$813$	0	0
$814$	1768.00	0.0761282
$815$	0	0
$816$	0	0
$817$	3760.00i	0.161011i
$818$	− 10870.0i	− 0.464622i
$819$	0	0
$820$	0	0
$821$	−44142.0	−1.87645	−0.938226	−	0.346024i	$-0.887532\pi$
$821$	−44142.0	−1.87645	−0.938226	+	0.346024i	$0.887532\pi$
$822$	0	0
$823$	− 3992.00i	− 0.169079i	−0.996420	−	0.0845397i	$-0.973058\pi$
$823$	− 3992.00i	− 0.169079i	0.996420	−	0.0845397i	$-0.0269420\pi$
$824$	1320.00	0.0558063
$825$	0	0
$826$	−2400.00	−0.101098
$827$	14444.0i	0.607336i	0.952778	+	0.303668i	$0.0982114\pi$
$827$	14444.0i	0.607336i	−0.952778	+	0.303668i	$0.901789\pi$
$828$	0	0
$829$	−42150.0	−1.76590	−0.882949	−	0.469468i	$-0.844446\pi$
$829$	−42150.0	−1.76590	−0.882949	+	0.469468i	$0.844446\pi$
$830$	0	0
$831$	0	0
$832$	− 3674.00i	− 0.153093i
$833$	− 3262.00i	− 0.135680i
$834$	0	0
$835$	0	0
$836$	−7280.00	−0.301177
$837$	0	0
$838$	9700.00i	0.399858i
$839$	13400.0	0.551394	0.275697	−	0.961245i	$-0.411091\pi$
$839$	13400.0	0.551394	0.275697	+	0.961245i	$0.411091\pi$
$840$	0	0
$841$	28511.0	1.16901
$842$	− 862.000i	− 0.0352809i
$843$	0	0
$844$	−31276.0	−1.27555
$845$	0	0
$846$	0	0
$847$	− 32952.0i	− 1.33677i
$848$	− 13858.0i	− 0.561186i
$849$	0	0
$850$	0	0
$851$	−5712.00	−0.230088
$852$	0	0
$853$	8658.00i	0.347531i	0.984787	+	0.173766i	$0.0555935\pi$
$853$	8658.00i	0.347531i	−0.984787	+	0.173766i	$0.944406\pi$
$854$	−17808.0	−0.713556
$855$	0	0
$856$	−540.000	−0.0215617
$857$	− 42826.0i	− 1.70701i	−0.521084	−	0.853505i	$-0.674472\pi$
$857$	− 42826.0i	− 1.70701i	0.521084	−	0.853505i	$-0.325528\pi$
$858$	0	0
$859$	35900.0	1.42595	0.712976	−	0.701189i	$-0.247347\pi$
$859$	35900.0	1.42595	0.712976	+	0.701189i	$0.247347\pi$
$860$	0	0
$861$	0	0
$862$	15792.0i	0.623988i
$863$	− 3088.00i	− 0.121804i	−0.998144	−	0.0609019i	$-0.980602\pi$
$863$	− 3088.00i	− 0.121804i	0.998144	−	0.0609019i	$-0.0193977\pi$
$864$	0	0
$865$	0	0
$866$	−11602.0	−0.455256
$867$	0	0
$868$	48384.0i	1.89200i
$869$	−12480.0	−0.487175
$870$	0	0
$871$	−1848.00	−0.0718910
$872$	− 14550.0i	− 0.565052i
$873$	0	0
$874$	−3360.00	−0.130039
$875$	0	0
$876$	0	0
$877$	− 35274.0i	− 1.35817i	−0.734058	−	0.679087i	$-0.762376\pi$
$877$	− 35274.0i	− 1.35817i	0.734058	−	0.679087i	$-0.237624\pi$
$878$	− 440.000i	− 0.0169126i
$879$	0	0
$880$	0	0
$881$	−25042.0	−0.957646	−0.478823	−	0.877911i	$-0.658936\pi$
$881$	−25042.0	−0.957646	−0.478823	+	0.877911i	$0.658936\pi$
$882$	0	0
$883$	− 12572.0i	− 0.479141i	−0.970879	−	0.239570i	$-0.922993\pi$
$883$	− 12572.0i	− 0.479141i	0.970879	−	0.239570i	$-0.0770066\pi$
$884$	2156.00	0.0820296
$885$	0	0
$886$	−10188.0	−0.386312
$887$	21864.0i	0.827645i	0.910358	+	0.413823i	$0.135807\pi$
$887$	21864.0i	0.827645i	−0.910358	+	0.413823i	$0.864193\pi$
$888$	0	0
$889$	46464.0	1.75293
$890$	0	0
$891$	0	0
$892$	− 42224.0i	− 1.58494i
$893$	− 5120.00i	− 0.191864i
$894$	0	0
$895$	0	0
$896$	−34920.0	−1.30200
$897$	0	0
$898$	13310.0i	0.494611i
$899$	−66240.0	−2.45743
$900$	0	0
$901$	4732.00	0.174968
$902$	− 6344.00i	− 0.234182i
$903$	0	0
$904$	15630.0	0.575051
$905$	0	0
$906$	0	0
$907$	31236.0i	1.14352i	0.820420	+	0.571761i	$0.193740\pi$
$907$	31236.0i	1.14352i	−0.820420	+	0.571761i	$0.806260\pi$
$908$	− 18452.0i	− 0.674396i
$909$	0	0
$910$	0	0
$911$	−8272.00	−0.300838	−0.150419	−	0.988622i	$-0.548062\pi$
$911$	−8272.00	−0.300838	−0.150419	+	0.988622i	$0.548062\pi$
$912$	0	0
$913$	− 63024.0i	− 2.28455i
$914$	3226.00	0.116747
$915$	0	0
$916$	−33810.0	−1.21956
$917$	17568.0i	0.632657i
$918$	0	0
$919$	−20200.0	−0.725067	−0.362533	−	0.931971i	$-0.618088\pi$
$919$	−20200.0	−0.725067	−0.362533	+	0.931971i	$0.618088\pi$
$920$	0	0
$921$	0	0
$922$	6582.00i	0.235105i
$923$	− 7216.00i	− 0.257332i
$924$	0	0
$925$	0	0
$926$	−15072.0	−0.534878
$927$	0	0
$928$	− 37030.0i	− 1.30988i
$929$	31010.0	1.09516	0.547581	−	0.836753i	$-0.315549\pi$
$929$	31010.0	1.09516	0.547581	+	0.836753i	$0.315549\pi$
$930$	0	0
$931$	−4660.00	−0.164044
$932$	18774.0i	0.659831i
$933$	0	0
$934$	−476.000	−0.0166758
$935$	0	0
$936$	0	0
$937$	− 39174.0i	− 1.36580i	−0.730510	−	0.682902i	$-0.760717\pi$
$937$	− 39174.0i	− 1.36580i	0.730510	−	0.682902i	$-0.239283\pi$
$938$	2016.00i	0.0701756i
$939$	0	0
$940$	0	0
$941$	4138.00	0.143353	0.0716764	−	0.997428i	$-0.477165\pi$
$941$	4138.00	0.143353	0.0716764	+	0.997428i	$0.477165\pi$
$942$	0	0
$943$	20496.0i	0.707785i
$944$	4100.00	0.141360
$945$	0	0
$946$	−9776.00	−0.335989
$947$	− 23676.0i	− 0.812425i	−0.913779	−	0.406213i	$-0.866849\pi$
$947$	− 23676.0i	− 0.812425i	0.913779	−	0.406213i	$-0.133151\pi$
$948$	0	0
$949$	836.000	0.0285961
$950$	0	0
$951$	0	0
$952$	− 5040.00i	− 0.171583i
$953$	18922.0i	0.643173i	0.946880	+	0.321586i	$0.104216\pi$
$953$	18922.0i	0.643173i	−0.946880	+	0.321586i	$0.895784\pi$
$954$	0	0
$955$	0	0
$956$	16240.0	0.549413
$957$	0	0
$958$	19680.0i	0.663708i
$959$	53136.0	1.78921
$960$	0	0
$961$	53153.0	1.78420
$962$	748.000i	0.0250691i
$963$	0	0
$964$	14014.0	0.468216
$965$	0	0
$966$	0	0
$967$	39656.0i	1.31877i	0.751805	+	0.659385i	$0.229183\pi$
$967$	39656.0i	1.31877i	−0.751805	+	0.659385i	$0.770817\pi$
$968$	− 20595.0i	− 0.683831i
$969$	0	0
$970$	0	0
$971$	33228.0	1.09818	0.549092	−	0.835762i	$-0.314974\pi$
$971$	33228.0	1.09818	0.549092	+	0.835762i	$0.314974\pi$
$972$	0	0
$973$	480.000i	0.0158151i
$974$	−5944.00	−0.195542
$975$	0	0
$976$	30422.0	0.997730
$977$	974.000i	0.0318946i	0.999873	+	0.0159473i	$0.00507640\pi$
$977$	974.000i	0.0318946i	−0.999873	+	0.0159473i	$0.994924\pi$
$978$	0	0
$979$	−17160.0	−0.560200
$980$	0	0
$981$	0	0
$982$	10772.0i	0.350049i
$983$	− 13608.0i	− 0.441534i	−0.975327	−	0.220767i	$-0.929144\pi$
$983$	− 13608.0i	− 0.441534i	0.975327	−	0.220767i	$-0.0708560\pi$
$984$	0	0
$985$	0	0
$986$	3220.00	0.104002
$987$	0	0
$988$	− 3080.00i	− 0.0991780i
$989$	31584.0	1.01548
$990$	0	0
$991$	13472.0	0.431839	0.215919	−	0.976411i	$-0.430725\pi$
$991$	13472.0	0.431839	0.215919	+	0.976411i	$0.430725\pi$
$992$	46368.0i	1.48406i
$993$	0	0
$994$	−7872.00	−0.251192
$995$	0	0
$996$	0	0
$997$	− 3234.00i	− 0.102730i	−0.998680	−	0.0513650i	$-0.983643\pi$
$997$	− 3234.00i	− 0.102730i	0.998680	−	0.0513650i	$-0.0163572\pi$
$998$	8140.00i	0.258184i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.b.e.199.1		2
3.2	odd	2		75.4.b.b.49.2		2
5.2	odd	4		225.4.a.f.1.1		1
5.3	odd	4		45.4.a.c.1.1		1
5.4	even	2	inner	225.4.b.e.199.2		2
12.11	even	2		1200.4.f.b.49.2		2
15.2	even	4		75.4.a.b.1.1		1
15.8	even	4		15.4.a.a.1.1	✓	1
15.14	odd	2		75.4.b.b.49.1		2
20.3	even	4		720.4.a.n.1.1		1
35.13	even	4		2205.4.a.l.1.1		1
45.13	odd	12		405.4.e.i.136.1		2
45.23	even	12		405.4.e.g.136.1		2
45.38	even	12		405.4.e.g.271.1		2
45.43	odd	12		405.4.e.i.271.1		2
60.23	odd	4		240.4.a.e.1.1		1
60.47	odd	4		1200.4.a.t.1.1		1
60.59	even	2		1200.4.f.b.49.1		2
105.83	odd	4		735.4.a.e.1.1		1
120.53	even	4		960.4.a.b.1.1		1
120.83	odd	4		960.4.a.ba.1.1		1
165.98	odd	4		1815.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.a.1.1	✓	1	15.8	even	4
45.4.a.c.1.1		1	5.3	odd	4
75.4.a.b.1.1		1	15.2	even	4
75.4.b.b.49.1		2	15.14	odd	2
75.4.b.b.49.2		2	3.2	odd	2
225.4.a.f.1.1		1	5.2	odd	4
225.4.b.e.199.1		2	1.1	even	1	trivial
225.4.b.e.199.2		2	5.4	even	2	inner
240.4.a.e.1.1		1	60.23	odd	4
405.4.e.g.136.1		2	45.23	even	12
405.4.e.g.271.1		2	45.38	even	12
405.4.e.i.136.1		2	45.13	odd	12
405.4.e.i.271.1		2	45.43	odd	12
720.4.a.n.1.1		1	20.3	even	4
735.4.a.e.1.1		1	105.83	odd	4
960.4.a.b.1.1		1	120.53	even	4
960.4.a.ba.1.1		1	120.83	odd	4
1200.4.a.t.1.1		1	60.47	odd	4
1200.4.f.b.49.1		2	60.59	even	2
1200.4.f.b.49.2		2	12.11	even	2
1815.4.a.e.1.1		1	165.98	odd	4
2205.4.a.l.1.1		1	35.13	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +7.00000 q^{4} -24.0000i q^{7} -15.0000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} +7.00000 q^{4} -24.0000i q^{7} -15.0000i q^{8} -52.0000 q^{11} -22.0000i q^{13} -24.0000 q^{14} +41.0000 q^{16} +14.0000i q^{17} +20.0000 q^{19} +52.0000i q^{22} -168.000i q^{23} -22.0000 q^{26} -168.000i q^{28} +230.000 q^{29} -288.000 q^{31} -161.000i q^{32} +14.0000 q^{34} -34.0000i q^{37} -20.0000i q^{38} -122.000 q^{41} +188.000i q^{43} -364.000 q^{44} -168.000 q^{46} -256.000i q^{47} -233.000 q^{49} -154.000i q^{52} -338.000i q^{53} -360.000 q^{56} -230.000i q^{58} +100.000 q^{59} +742.000 q^{61} +288.000i q^{62} +167.000 q^{64} -84.0000i q^{67} +98.0000i q^{68} +328.000 q^{71} +38.0000i q^{73} -34.0000 q^{74} +140.000 q^{76} +1248.00i q^{77} +240.000 q^{79} +122.000i q^{82} +1212.00i q^{83} +188.000 q^{86} +780.000i q^{88} +330.000 q^{89} -528.000 q^{91} -1176.00i q^{92} -256.000 q^{94} +866.000i q^{97} +233.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4}+O(q^{10}) \) 2 * q + 14 * q^4	\( 2 q + 14 q^{4} - 104 q^{11} - 48 q^{14} + 82 q^{16} + 40 q^{19} - 44 q^{26} + 460 q^{29} - 576 q^{31} + 28 q^{34} - 244 q^{41} - 728 q^{44} - 336 q^{46} - 466 q^{49} - 720 q^{56} + 200 q^{59} + 1484 q^{61} + 334 q^{64} + 656 q^{71} - 68 q^{74} + 280 q^{76} + 480 q^{79} + 376 q^{86} + 660 q^{89} - 1056 q^{91} - 512 q^{94}+O(q^{100}) \) 2 * q + 14 * q^4 - 104 * q^11 - 48 * q^14 + 82 * q^16 + 40 * q^19 - 44 * q^26 + 460 * q^29 - 576 * q^31 + 28 * q^34 - 244 * q^41 - 728 * q^44 - 336 * q^46 - 466 * q^49 - 720 * q^56 + 200 * q^59 + 1484 * q^61 + 334 * q^64 + 656 * q^71 - 68 * q^74 + 280 * q^76 + 480 * q^79 + 376 * q^86 + 660 * q^89 - 1056 * q^91 - 512 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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