LMFDB - Embedded newform 225.3.c.d.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(26,225)]

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, 0]))

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

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

chi := DirichletCharacter("225.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	225.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.13080594811$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 9x^{2} - 8x + 2 $ x^4 - 2x^3 + 9x^2 - 8*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.1
Root			$0.500000 - 0.0913379i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.3.c.d.26.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64575i q^{2} -3.00000 q^{4} -11.2250 q^{7} -2.64575i q^{8} +O(q^{10})$	$q-2.64575i q^{2} -3.00000 q^{4} -11.2250 q^{7} -2.64575i q^{8} +4.24264i q^{11} -11.2250 q^{13} +29.6985i q^{14} -19.0000 q^{16} -10.5830i q^{17} -20.0000 q^{19} +11.2250 q^{22} -5.29150i q^{23} +29.6985i q^{26} +33.6749 q^{28} +8.48528i q^{29} +26.0000 q^{31} +39.6863i q^{32} -28.0000 q^{34} +33.6749 q^{37} +52.9150i q^{38} -55.1543i q^{41} -22.4499 q^{43} -12.7279i q^{44} -14.0000 q^{46} +21.1660i q^{47} +77.0000 q^{49} +33.6749 q^{52} -84.6640i q^{53} +29.6985i q^{56} +22.4499 q^{58} -46.6690i q^{59} -22.0000 q^{61} -68.7895i q^{62} +29.0000 q^{64} -89.7998 q^{67} +31.7490i q^{68} -50.9117i q^{71} -67.3498 q^{73} -89.0955i q^{74} +60.0000 q^{76} -47.6235i q^{77} -14.0000 q^{79} -145.925 q^{82} +74.0810i q^{83} +59.3970i q^{86} +11.2250 q^{88} -89.0955i q^{89} +126.000 q^{91} +15.8745i q^{92} +56.0000 q^{94} +22.4499 q^{97} -203.723i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{4}+O(q^{10}) $ 4 * q - 12 * q^4	$ 4 q - 12 q^{4} - 76 q^{16} - 80 q^{19} + 104 q^{31} - 112 q^{34} - 56 q^{46} + 308 q^{49} - 88 q^{61} + 116 q^{64} + 240 q^{76} - 56 q^{79} + 504 q^{91} + 224 q^{94}+O(q^{100}) $ 4 * q - 12 * q^4 - 76 * q^16 - 80 * q^19 + 104 * q^31 - 112 * q^34 - 56 * q^46 + 308 * q^49 - 88 * q^61 + 116 * q^64 + 240 * q^76 - 56 * q^79 + 504 * q^91 + 224 * 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 $

$2$	− 2.64575i	− 1.32288i	−0.750000	−	0.661438i	$-0.769947\pi$
$2$	− 2.64575i	− 1.32288i	0.750000	−	0.661438i	$-0.230053\pi$
$3$	0	0
$3$	0	0
$4$	−3.00000	−0.750000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−11.2250	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−11.2250	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	− 2.64575i	− 0.330719i
$9$	0	0
$10$	0	0
$11$	4.24264i	0.385695i	0.981229	+	0.192847i	$0.0617722\pi$
$11$	4.24264i	0.385695i	−0.981229	+	0.192847i	$0.938228\pi$
$12$	0	0
$13$	−11.2250	−0.863459	−0.431730	−	0.902003i	$-0.642097\pi$
$13$	−11.2250	−0.863459	−0.431730	+	0.902003i	$0.642097\pi$
$14$	29.6985i	2.12132i
$15$	0	0
$16$	−19.0000	−1.18750
$17$	− 10.5830i	− 0.622530i	−0.950323	−	0.311265i	$-0.899247\pi$
$17$	− 10.5830i	− 0.622530i	0.950323	−	0.311265i	$-0.100753\pi$
$18$	0	0
$19$	−20.0000	−1.05263	−0.526316	−	0.850289i	$-0.676427\pi$
$19$	−20.0000	−1.05263	−0.526316	+	0.850289i	$0.676427\pi$
$20$	0	0
$21$	0	0
$22$	11.2250	0.510226
$23$	− 5.29150i	− 0.230065i	−0.993362	−	0.115033i	$-0.963303\pi$
$23$	− 5.29150i	− 0.230065i	0.993362	−	0.115033i	$-0.0366973\pi$
$24$	0	0
$25$	0	0
$26$	29.6985i	1.14225i
$27$	0	0
$28$	33.6749	1.20268
$29$	8.48528i	0.292596i	0.989241	+	0.146298i	$0.0467358\pi$
$29$	8.48528i	0.292596i	−0.989241	+	0.146298i	$0.953264\pi$
$30$	0	0
$31$	26.0000	0.838710	0.419355	−	0.907822i	$-0.362256\pi$
$31$	26.0000	0.838710	0.419355	+	0.907822i	$0.362256\pi$
$32$	39.6863i	1.24020i
$33$	0	0
$34$	−28.0000	−0.823529
$35$	0	0
$36$	0	0
$37$	33.6749	0.910133	0.455066	−	0.890457i	$-0.349616\pi$
$37$	33.6749	0.910133	0.455066	+	0.890457i	$0.349616\pi$
$38$	52.9150i	1.39250i
$39$	0	0
$40$	0	0
$41$	− 55.1543i	− 1.34523i	−0.739994	−	0.672614i	$-0.765172\pi$
$41$	− 55.1543i	− 1.34523i	0.739994	−	0.672614i	$-0.234828\pi$
$42$	0	0
$43$	−22.4499	−0.522092	−0.261046	−	0.965326i	$-0.584067\pi$
$43$	−22.4499	−0.522092	−0.261046	+	0.965326i	$0.584067\pi$
$44$	− 12.7279i	− 0.289271i
$45$	0	0
$46$	−14.0000	−0.304348
$47$	21.1660i	0.450341i	0.974319	+	0.225170i	$0.0722939\pi$
$47$	21.1660i	0.450341i	−0.974319	+	0.225170i	$0.927706\pi$
$48$	0	0
$49$	77.0000	1.57143
$50$	0	0
$51$	0	0
$52$	33.6749	0.647595
$53$	− 84.6640i	− 1.59743i	−0.601706	−	0.798717i	$-0.705512\pi$
$53$	− 84.6640i	− 1.59743i	0.601706	−	0.798717i	$-0.294488\pi$
$54$	0	0
$55$	0	0
$56$	29.6985i	0.530330i
$57$	0	0
$58$	22.4499	0.387068
$59$	− 46.6690i	− 0.791001i	−0.918466	−	0.395500i	$-0.870571\pi$
$59$	− 46.6690i	− 0.791001i	0.918466	−	0.395500i	$-0.129429\pi$
$60$	0	0
$61$	−22.0000	−0.360656	−0.180328	−	0.983607i	$-0.557716\pi$
$61$	−22.0000	−0.360656	−0.180328	+	0.983607i	$0.557716\pi$
$62$	− 68.7895i	− 1.10951i
$63$	0	0
$64$	29.0000	0.453125
$65$	0	0
$66$	0	0
$67$	−89.7998	−1.34030	−0.670148	−	0.742228i	$-0.733769\pi$
$67$	−89.7998	−1.34030	−0.670148	+	0.742228i	$0.733769\pi$
$68$	31.7490i	0.466897i
$69$	0	0
$70$	0	0
$71$	− 50.9117i	− 0.717066i	−0.933517	−	0.358533i	$-0.883277\pi$
$71$	− 50.9117i	− 0.717066i	0.933517	−	0.358533i	$-0.116723\pi$
$72$	0	0
$73$	−67.3498	−0.922600	−0.461300	−	0.887244i	$-0.652617\pi$
$73$	−67.3498	−0.922600	−0.461300	+	0.887244i	$0.652617\pi$
$74$	− 89.0955i	− 1.20399i
$75$	0	0
$76$	60.0000	0.789474
$77$	− 47.6235i	− 0.618487i
$78$	0	0
$79$	−14.0000	−0.177215	−0.0886076	−	0.996067i	$-0.528242\pi$
$79$	−14.0000	−0.177215	−0.0886076	+	0.996067i	$0.528242\pi$
$80$	0	0
$81$	0	0
$82$	−145.925	−1.77957
$83$	74.0810i	0.892543i	0.894898	+	0.446271i	$0.147248\pi$
$83$	74.0810i	0.892543i	−0.894898	+	0.446271i	$0.852752\pi$
$84$	0	0
$85$	0	0
$86$	59.3970i	0.690662i
$87$	0	0
$88$	11.2250	0.127557
$89$	− 89.0955i	− 1.00107i	−0.865716	−	0.500536i	$-0.833136\pi$
$89$	− 89.0955i	− 1.00107i	0.865716	−	0.500536i	$-0.166864\pi$
$90$	0	0
$91$	126.000	1.38462
$92$	15.8745i	0.172549i
$93$	0	0
$94$	56.0000	0.595745
$95$	0	0
$96$	0	0
$97$	22.4499	0.231443	0.115721	−	0.993282i	$-0.463082\pi$
$97$	22.4499	0.231443	0.115721	+	0.993282i	$0.463082\pi$
$98$	− 203.723i	− 2.07880i
$99$	0	0
$100$	0	0
$101$	− 135.765i	− 1.34420i	−0.740459	−	0.672101i	$-0.765392\pi$
$101$	− 135.765i	− 1.34420i	0.740459	−	0.672101i	$-0.234608\pi$
$102$	0	0
$103$	56.1249	0.544902	0.272451	−	0.962170i	$-0.412166\pi$
$103$	56.1249	0.544902	0.272451	+	0.962170i	$0.412166\pi$
$104$	29.6985i	0.285562i
$105$	0	0
$106$	−224.000	−2.11321
$107$	− 10.5830i	− 0.0989066i	−0.998776	−	0.0494533i	$-0.984252\pi$
$107$	− 10.5830i	− 0.0989066i	0.998776	−	0.0494533i	$-0.0157479\pi$
$108$	0	0
$109$	70.0000	0.642202	0.321101	−	0.947045i	$-0.395947\pi$
$109$	70.0000	0.642202	0.321101	+	0.947045i	$0.395947\pi$
$110$	0	0
$111$	0	0
$112$	213.274	1.90424
$113$	137.579i	1.21751i	0.793357	+	0.608757i	$0.208332\pi$
$113$	137.579i	1.21751i	−0.793357	+	0.608757i	$0.791668\pi$
$114$	0	0
$115$	0	0
$116$	− 25.4558i	− 0.219447i
$117$	0	0
$118$	−123.475	−1.04640
$119$	118.794i	0.998268i
$120$	0	0
$121$	103.000	0.851240
$122$	58.2065i	0.477103i
$123$	0	0
$124$	−78.0000	−0.629032
$125$	0	0
$126$	0	0
$127$	168.375	1.32578	0.662892	−	0.748715i	$-0.269329\pi$
$127$	168.375	1.32578	0.662892	+	0.748715i	$0.269329\pi$
$128$	82.0183i	0.640768i
$129$	0	0
$130$	0	0
$131$	148.492i	1.13353i	0.823880	+	0.566765i	$0.191805\pi$
$131$	148.492i	1.13353i	−0.823880	+	0.566765i	$0.808195\pi$
$132$	0	0
$133$	224.499	1.68797
$134$	237.588i	1.77304i
$135$	0	0
$136$	−28.0000	−0.205882
$137$	211.660i	1.54496i	0.635036	+	0.772482i	$0.280985\pi$
$137$	211.660i	1.54496i	−0.635036	+	0.772482i	$0.719015\pi$
$138$	0	0
$139$	−206.000	−1.48201	−0.741007	−	0.671497i	$-0.765652\pi$
$139$	−206.000	−1.48201	−0.741007	+	0.671497i	$0.765652\pi$
$140$	0	0
$141$	0	0
$142$	−134.700	−0.948589
$143$	− 47.6235i	− 0.333032i
$144$	0	0
$145$	0	0
$146$	178.191i	1.22049i
$147$	0	0
$148$	−101.025	−0.682600
$149$	− 135.765i	− 0.911171i	−0.890192	−	0.455586i	$-0.849430\pi$
$149$	− 135.765i	− 0.911171i	0.890192	−	0.455586i	$-0.150570\pi$
$150$	0	0
$151$	−202.000	−1.33775	−0.668874	−	0.743376i	$-0.733224\pi$
$151$	−202.000	−1.33775	−0.668874	+	0.743376i	$0.733224\pi$
$152$	52.9150i	0.348125i
$153$	0	0
$154$	−126.000	−0.818182
$155$	0	0
$156$	0	0
$157$	−56.1249	−0.357483	−0.178742	−	0.983896i	$-0.557203\pi$
$157$	−56.1249	−0.357483	−0.178742	+	0.983896i	$0.557203\pi$
$158$	37.0405i	0.234434i
$159$	0	0
$160$	0	0
$161$	59.3970i	0.368925i
$162$	0	0
$163$	202.049	1.23957	0.619784	−	0.784773i	$-0.287220\pi$
$163$	202.049	1.23957	0.619784	+	0.784773i	$0.287220\pi$
$164$	165.463i	1.00892i
$165$	0	0
$166$	196.000	1.18072
$167$	− 185.203i	− 1.10900i	−0.832185	−	0.554499i	$-0.812910\pi$
$167$	− 185.203i	− 1.10900i	0.832185	−	0.554499i	$-0.187090\pi$
$168$	0	0
$169$	−43.0000	−0.254438
$170$	0	0
$171$	0	0
$172$	67.3498	0.391569
$173$	− 21.1660i	− 0.122347i	−0.998127	−	0.0611734i	$-0.980516\pi$
$173$	− 21.1660i	− 0.122347i	0.998127	−	0.0611734i	$-0.0194843\pi$
$174$	0	0
$175$	0	0
$176$	− 80.6102i	− 0.458012i
$177$	0	0
$178$	−235.724	−1.32429
$179$	241.831i	1.35101i	0.737356	+	0.675504i	$0.236074\pi$
$179$	241.831i	1.35101i	−0.737356	+	0.675504i	$0.763926\pi$
$180$	0	0
$181$	74.0000	0.408840	0.204420	−	0.978883i	$-0.434469\pi$
$181$	74.0000	0.408840	0.204420	+	0.978883i	$0.434469\pi$
$182$	− 333.365i	− 1.83167i
$183$	0	0
$184$	−14.0000	−0.0760870
$185$	0	0
$186$	0	0
$187$	44.8999	0.240106
$188$	− 63.4980i	− 0.337755i
$189$	0	0
$190$	0	0
$191$	− 161.220i	− 0.844086i	−0.906576	−	0.422043i	$-0.861313\pi$
$191$	− 161.220i	− 0.844086i	0.906576	−	0.422043i	$-0.138687\pi$
$192$	0	0
$193$	−179.600	−0.930568	−0.465284	−	0.885162i	$-0.654048\pi$
$193$	−179.600	−0.930568	−0.465284	+	0.885162i	$0.654048\pi$
$194$	− 59.3970i	− 0.306170i
$195$	0	0
$196$	−231.000	−1.17857
$197$	37.0405i	0.188023i	0.995571	+	0.0940115i	$0.0299690\pi$
$197$	37.0405i	0.188023i	−0.995571	+	0.0940115i	$0.970031\pi$
$198$	0	0
$199$	250.000	1.25628	0.628141	−	0.778100i	$-0.283816\pi$
$199$	250.000	1.25628	0.628141	+	0.778100i	$0.283816\pi$
$200$	0	0
$201$	0	0
$202$	−359.199	−1.77821
$203$	− 95.2470i	− 0.469197i
$204$	0	0
$205$	0	0
$206$	− 148.492i	− 0.720837i
$207$	0	0
$208$	213.274	1.02536
$209$	− 84.8528i	− 0.405994i
$210$	0	0
$211$	−154.000	−0.729858	−0.364929	−	0.931035i	$-0.618907\pi$
$211$	−154.000	−0.729858	−0.364929	+	0.931035i	$0.618907\pi$
$212$	253.992i	1.19808i
$213$	0	0
$214$	−28.0000	−0.130841
$215$	0	0
$216$	0	0
$217$	−291.849	−1.34493
$218$	− 185.203i	− 0.849553i
$219$	0	0
$220$	0	0
$221$	118.794i	0.537529i
$222$	0	0
$223$	−392.874	−1.76177	−0.880883	−	0.473333i	$-0.843051\pi$
$223$	−392.874	−1.76177	−0.880883	+	0.473333i	$0.843051\pi$
$224$	− 445.477i	− 1.98874i
$225$	0	0
$226$	364.000	1.61062
$227$	21.1660i	0.0932423i	0.998913	+	0.0466212i	$0.0148454\pi$
$227$	21.1660i	0.0932423i	−0.998913	+	0.0466212i	$0.985155\pi$
$228$	0	0
$229$	118.000	0.515284	0.257642	−	0.966240i	$-0.417055\pi$
$229$	118.000	0.515284	0.257642	+	0.966240i	$0.417055\pi$
$230$	0	0
$231$	0	0
$232$	22.4499	0.0967670
$233$	391.571i	1.68056i	0.542150	+	0.840282i	$0.317610\pi$
$233$	391.571i	1.68056i	−0.542150	+	0.840282i	$0.682390\pi$
$234$	0	0
$235$	0	0
$236$	140.007i	0.593251i
$237$	0	0
$238$	314.299	1.32058
$239$	347.897i	1.45563i	0.685771	+	0.727817i	$0.259465\pi$
$239$	347.897i	1.45563i	−0.685771	+	0.727817i	$0.740535\pi$
$240$	0	0
$241$	−40.0000	−0.165975	−0.0829876	−	0.996551i	$-0.526446\pi$
$241$	−40.0000	−0.165975	−0.0829876	+	0.996551i	$0.526446\pi$
$242$	− 272.512i	− 1.12608i
$243$	0	0
$244$	66.0000	0.270492
$245$	0	0
$246$	0	0
$247$	224.499	0.908905
$248$	− 68.7895i	− 0.277377i
$249$	0	0
$250$	0	0
$251$	241.831i	0.963468i	0.876317	+	0.481734i	$0.159993\pi$
$251$	241.831i	0.963468i	−0.876317	+	0.481734i	$0.840007\pi$
$252$	0	0
$253$	22.4499	0.0887350
$254$	− 445.477i	− 1.75385i
$255$	0	0
$256$	333.000	1.30078
$257$	− 232.826i	− 0.905938i	−0.891526	−	0.452969i	$-0.850365\pi$
$257$	− 232.826i	− 0.905938i	0.891526	−	0.452969i	$-0.149635\pi$
$258$	0	0
$259$	−378.000	−1.45946
$260$	0	0
$261$	0	0
$262$	392.874	1.49952
$263$	− 164.037i	− 0.623713i	−0.950129	−	0.311857i	$-0.899049\pi$
$263$	− 164.037i	− 0.623713i	0.950129	−	0.311857i	$-0.100951\pi$
$264$	0	0
$265$	0	0
$266$	− 593.970i	− 2.23297i
$267$	0	0
$268$	269.399	1.00522
$269$	− 534.573i	− 1.98726i	−0.112695	−	0.993630i	$-0.535948\pi$
$269$	− 534.573i	− 1.98726i	0.112695	−	0.993630i	$-0.464052\pi$
$270$	0	0
$271$	−286.000	−1.05535	−0.527675	−	0.849446i	$-0.676936\pi$
$271$	−286.000	−1.05535	−0.527675	+	0.849446i	$0.676936\pi$
$272$	201.077i	0.739254i
$273$	0	0
$274$	560.000	2.04380
$275$	0	0
$276$	0	0
$277$	190.825	0.688897	0.344449	−	0.938805i	$-0.388066\pi$
$277$	190.825	0.688897	0.344449	+	0.938805i	$0.388066\pi$
$278$	545.025i	1.96052i
$279$	0	0
$280$	0	0
$281$	80.6102i	0.286869i	0.989660	+	0.143434i	$0.0458146\pi$
$281$	80.6102i	0.286869i	−0.989660	+	0.143434i	$0.954185\pi$
$282$	0	0
$283$	89.7998	0.317314	0.158657	−	0.987334i	$-0.449284\pi$
$283$	89.7998	0.317314	0.158657	+	0.987334i	$0.449284\pi$
$284$	152.735i	0.537800i
$285$	0	0
$286$	−126.000	−0.440559
$287$	619.106i	2.15716i
$288$	0	0
$289$	177.000	0.612457
$290$	0	0
$291$	0	0
$292$	202.049	0.691950
$293$	− 576.774i	− 1.96851i	−0.176751	−	0.984256i	$-0.556559\pi$
$293$	− 576.774i	− 1.96851i	0.176751	−	0.984256i	$-0.443441\pi$
$294$	0	0
$295$	0	0
$296$	− 89.0955i	− 0.300998i
$297$	0	0
$298$	−359.199	−1.20537
$299$	59.3970i	0.198652i
$300$	0	0
$301$	252.000	0.837209
$302$	534.442i	1.76967i
$303$	0	0
$304$	380.000	1.25000
$305$	0	0
$306$	0	0
$307$	−269.399	−0.877522	−0.438761	−	0.898604i	$-0.644583\pi$
$307$	−269.399	−0.877522	−0.438761	+	0.898604i	$0.644583\pi$
$308$	142.871i	0.463865i
$309$	0	0
$310$	0	0
$311$	59.3970i	0.190987i	0.995430	+	0.0954935i	$0.0304429\pi$
$311$	59.3970i	0.190987i	−0.995430	+	0.0954935i	$0.969557\pi$
$312$	0	0
$313$	179.600	0.573800	0.286900	−	0.957960i	$-0.407375\pi$
$313$	179.600	0.573800	0.286900	+	0.957960i	$0.407375\pi$
$314$	148.492i	0.472906i
$315$	0	0
$316$	42.0000	0.132911
$317$	− 312.199i	− 0.984854i	−0.870354	−	0.492427i	$-0.836110\pi$
$317$	− 312.199i	− 0.984854i	0.870354	−	0.492427i	$-0.163890\pi$
$318$	0	0
$319$	−36.0000	−0.112853
$320$	0	0
$321$	0	0
$322$	157.150	0.488042
$323$	211.660i	0.655294i
$324$	0	0
$325$	0	0
$326$	− 534.573i	− 1.63979i
$327$	0	0
$328$	−145.925	−0.444892
$329$	− 237.588i	− 0.722152i
$330$	0	0
$331$	−112.000	−0.338369	−0.169184	−	0.985584i	$-0.554113\pi$
$331$	−112.000	−0.338369	−0.169184	+	0.985584i	$0.554113\pi$
$332$	− 222.243i	− 0.669407i
$333$	0	0
$334$	−490.000	−1.46707
$335$	0	0
$336$	0	0
$337$	112.250	0.333085	0.166543	−	0.986034i	$-0.446740\pi$
$337$	112.250	0.333085	0.166543	+	0.986034i	$0.446740\pi$
$338$	113.767i	0.336590i
$339$	0	0
$340$	0	0
$341$	110.309i	0.323486i
$342$	0	0
$343$	−314.299	−0.916324
$344$	59.3970i	0.172666i
$345$	0	0
$346$	−56.0000	−0.161850
$347$	− 518.567i	− 1.49443i	−0.664582	−	0.747215i	$-0.731390\pi$
$347$	− 518.567i	− 1.49443i	0.664582	−	0.747215i	$-0.268610\pi$
$348$	0	0
$349$	−122.000	−0.349570	−0.174785	−	0.984607i	$-0.555923\pi$
$349$	−122.000	−0.349570	−0.174785	+	0.984607i	$0.555923\pi$
$350$	0	0
$351$	0	0
$352$	−168.375	−0.478337
$353$	− 402.154i	− 1.13925i	−0.821906	−	0.569624i	$-0.807089\pi$
$353$	− 402.154i	− 1.13925i	0.821906	−	0.569624i	$-0.192911\pi$
$354$	0	0
$355$	0	0
$356$	267.286i	0.750804i
$357$	0	0
$358$	639.823	1.78722
$359$	636.396i	1.77269i	0.463024	+	0.886346i	$0.346764\pi$
$359$	636.396i	1.77269i	−0.463024	+	0.886346i	$0.653236\pi$
$360$	0	0
$361$	39.0000	0.108033
$362$	− 195.786i	− 0.540844i
$363$	0	0
$364$	−378.000	−1.03846
$365$	0	0
$366$	0	0
$367$	−684.723	−1.86573	−0.932866	−	0.360225i	$-0.882700\pi$
$367$	−684.723	−1.86573	−0.932866	+	0.360225i	$0.882700\pi$
$368$	100.539i	0.273203i
$369$	0	0
$370$	0	0
$371$	950.352i	2.56159i
$372$	0	0
$373$	−145.925	−0.391219	−0.195609	−	0.980682i	$-0.562668\pi$
$373$	−145.925	−0.391219	−0.195609	+	0.980682i	$0.562668\pi$
$374$	− 118.794i	− 0.317631i
$375$	0	0
$376$	56.0000	0.148936
$377$	− 95.2470i	− 0.252645i
$378$	0	0
$379$	−362.000	−0.955145	−0.477573	−	0.878592i	$-0.658483\pi$
$379$	−362.000	−0.955145	−0.477573	+	0.878592i	$0.658483\pi$
$380$	0	0
$381$	0	0
$382$	−426.549	−1.11662
$383$	42.3320i	0.110527i	0.998472	+	0.0552637i	$0.0176000\pi$
$383$	42.3320i	0.110527i	−0.998472	+	0.0552637i	$0.982400\pi$
$384$	0	0
$385$	0	0
$386$	475.176i	1.23103i
$387$	0	0
$388$	−67.3498	−0.173582
$389$	263.044i	0.676205i	0.941109	+	0.338102i	$0.109785\pi$
$389$	263.044i	0.676205i	−0.941109	+	0.338102i	$0.890215\pi$
$390$	0	0
$391$	−56.0000	−0.143223
$392$	− 203.723i	− 0.519701i
$393$	0	0
$394$	98.0000	0.248731
$395$	0	0
$396$	0	0
$397$	−33.6749	−0.0848235	−0.0424117	−	0.999100i	$-0.513504\pi$
$397$	−33.6749	−0.0848235	−0.0424117	+	0.999100i	$0.513504\pi$
$398$	− 661.438i	− 1.66190i
$399$	0	0
$400$	0	0
$401$	− 462.448i	− 1.15324i	−0.817014	−	0.576618i	$-0.804372\pi$
$401$	− 462.448i	− 1.15324i	0.817014	−	0.576618i	$-0.195628\pi$
$402$	0	0
$403$	−291.849	−0.724192
$404$	407.294i	1.00815i
$405$	0	0
$406$	−252.000	−0.620690
$407$	142.871i	0.351033i
$408$	0	0
$409$	82.0000	0.200489	0.100244	−	0.994963i	$-0.468038\pi$
$409$	82.0000	0.200489	0.100244	+	0.994963i	$0.468038\pi$
$410$	0	0
$411$	0	0
$412$	−168.375	−0.408676
$413$	523.859i	1.26842i
$414$	0	0
$415$	0	0
$416$	− 445.477i	− 1.07086i
$417$	0	0
$418$	−224.499	−0.537080
$419$	207.889i	0.496156i	0.968740	+	0.248078i	$0.0797989\pi$
$419$	207.889i	0.496156i	−0.968740	+	0.248078i	$0.920201\pi$
$420$	0	0
$421$	−490.000	−1.16390	−0.581948	−	0.813226i	$-0.697709\pi$
$421$	−490.000	−1.16390	−0.581948	+	0.813226i	$0.697709\pi$
$422$	407.446i	0.965511i
$423$	0	0
$424$	−224.000	−0.528302
$425$	0	0
$426$	0	0
$427$	246.949	0.578336
$428$	31.7490i	0.0741799i
$429$	0	0
$430$	0	0
$431$	− 687.308i	− 1.59468i	−0.603529	−	0.797341i	$-0.706239\pi$
$431$	− 687.308i	− 1.59468i	0.603529	−	0.797341i	$-0.293761\pi$
$432$	0	0
$433$	−202.049	−0.466627	−0.233314	−	0.972402i	$-0.574957\pi$
$433$	−202.049	−0.466627	−0.233314	+	0.972402i	$0.574957\pi$
$434$	772.161i	1.77917i
$435$	0	0
$436$	−210.000	−0.481651
$437$	105.830i	0.242174i
$438$	0	0
$439$	−302.000	−0.687927	−0.343964	−	0.938983i	$-0.611770\pi$
$439$	−302.000	−0.687927	−0.343964	+	0.938983i	$0.611770\pi$
$440$	0	0
$441$	0	0
$442$	314.299	0.711084
$443$	264.575i	0.597235i	0.954373	+	0.298618i	$0.0965255\pi$
$443$	264.575i	0.597235i	−0.954373	+	0.298618i	$0.903475\pi$
$444$	0	0
$445$	0	0
$446$	1039.45i	2.33060i
$447$	0	0
$448$	−325.524	−0.726617
$449$	− 216.375i	− 0.481904i	−0.970537	−	0.240952i	$-0.922540\pi$
$449$	− 216.375i	− 0.481904i	0.970537	−	0.240952i	$-0.0774596\pi$
$450$	0	0
$451$	234.000	0.518847
$452$	− 412.737i	− 0.913135i
$453$	0	0
$454$	56.0000	0.123348
$455$	0	0
$456$	0	0
$457$	561.249	1.22812	0.614058	−	0.789261i	$-0.289536\pi$
$457$	561.249	1.22812	0.614058	+	0.789261i	$0.289536\pi$
$458$	− 312.199i	− 0.681656i
$459$	0	0
$460$	0	0
$461$	− 237.588i	− 0.515375i	−0.966228	−	0.257688i	$-0.917039\pi$
$461$	− 237.588i	− 0.515375i	0.966228	−	0.257688i	$-0.0829605\pi$
$462$	0	0
$463$	729.623	1.57586	0.787930	−	0.615765i	$-0.211153\pi$
$463$	729.623	1.57586	0.787930	+	0.615765i	$0.211153\pi$
$464$	− 161.220i	− 0.347458i
$465$	0	0
$466$	1036.00	2.22318
$467$	465.652i	0.997114i	0.866857	+	0.498557i	$0.166137\pi$
$467$	465.652i	0.997114i	−0.866857	+	0.498557i	$0.833863\pi$
$468$	0	0
$469$	1008.00	2.14925
$470$	0	0
$471$	0	0
$472$	−123.475	−0.261599
$473$	− 95.2470i	− 0.201368i
$474$	0	0
$475$	0	0
$476$	− 356.382i	− 0.748701i
$477$	0	0
$478$	920.448	1.92562
$479$	− 475.176i	− 0.992016i	−0.868318	−	0.496008i	$-0.834799\pi$
$479$	− 475.176i	− 0.992016i	0.868318	−	0.496008i	$-0.165201\pi$
$480$	0	0
$481$	−378.000	−0.785863
$482$	105.830i	0.219564i
$483$	0	0
$484$	−309.000	−0.638430
$485$	0	0
$486$	0	0
$487$	505.124	1.03722	0.518608	−	0.855012i	$-0.326451\pi$
$487$	505.124	1.03722	0.518608	+	0.855012i	$0.326451\pi$
$488$	58.2065i	0.119276i
$489$	0	0
$490$	0	0
$491$	275.772i	0.561653i	0.959759	+	0.280827i	$0.0906086\pi$
$491$	275.772i	0.561653i	−0.959759	+	0.280827i	$0.909391\pi$
$492$	0	0
$493$	89.7998	0.182150
$494$	− 593.970i	− 1.20237i
$495$	0	0
$496$	−494.000	−0.995968
$497$	571.482i	1.14986i
$498$	0	0
$499$	−368.000	−0.737475	−0.368737	−	0.929534i	$-0.620210\pi$
$499$	−368.000	−0.737475	−0.368737	+	0.929534i	$0.620210\pi$
$500$	0	0
$501$	0	0
$502$	639.823	1.27455
$503$	− 275.158i	− 0.547034i	−0.961867	−	0.273517i	$-0.911813\pi$
$503$	− 275.158i	− 0.547034i	0.961867	−	0.273517i	$-0.0881870\pi$
$504$	0	0
$505$	0	0
$506$	− 59.3970i	− 0.117385i
$507$	0	0
$508$	−505.124	−0.994338
$509$	118.794i	0.233387i	0.993168	+	0.116693i	$0.0372295\pi$
$509$	118.794i	0.233387i	−0.993168	+	0.116693i	$0.962770\pi$
$510$	0	0
$511$	756.000	1.47945
$512$	− 552.962i	− 1.08000i
$513$	0	0
$514$	−616.000	−1.19844
$515$	0	0
$516$	0	0
$517$	−89.7998	−0.173694
$518$	1000.09i	1.93068i
$519$	0	0
$520$	0	0
$521$	89.0955i	0.171009i	0.996338	+	0.0855043i	$0.0272501\pi$
$521$	89.0955i	0.171009i	−0.996338	+	0.0855043i	$0.972750\pi$
$522$	0	0
$523$	875.548	1.67409	0.837044	−	0.547136i	$-0.184282\pi$
$523$	875.548	1.67409	0.837044	+	0.547136i	$0.184282\pi$
$524$	− 445.477i	− 0.850147i
$525$	0	0
$526$	−434.000	−0.825095
$527$	− 275.158i	− 0.522122i
$528$	0	0
$529$	501.000	0.947070
$530$	0	0
$531$	0	0
$532$	−673.498	−1.26597
$533$	619.106i	1.16155i
$534$	0	0
$535$	0	0
$536$	237.588i	0.443261i
$537$	0	0
$538$	−1414.35	−2.62890
$539$	326.683i	0.606092i
$540$	0	0
$541$	434.000	0.802218	0.401109	−	0.916030i	$-0.368625\pi$
$541$	434.000	0.802218	0.401109	+	0.916030i	$0.368625\pi$
$542$	756.685i	1.39610i
$543$	0	0
$544$	420.000	0.772059
$545$	0	0
$546$	0	0
$547$	−112.250	−0.205210	−0.102605	−	0.994722i	$-0.532718\pi$
$547$	−112.250	−0.205210	−0.102605	+	0.994722i	$0.532718\pi$
$548$	− 634.980i	− 1.15872i
$549$	0	0
$550$	0	0
$551$	− 169.706i	− 0.307996i
$552$	0	0
$553$	157.150	0.284177
$554$	− 504.874i	− 0.911325i
$555$	0	0
$556$	618.000	1.11151
$557$	465.652i	0.836000i	0.908447	+	0.418000i	$0.137269\pi$
$557$	465.652i	0.836000i	−0.908447	+	0.418000i	$0.862731\pi$
$558$	0	0
$559$	252.000	0.450805
$560$	0	0
$561$	0	0
$562$	213.274	0.379492
$563$	− 52.9150i	− 0.0939876i	−0.998895	−	0.0469938i	$-0.985036\pi$
$563$	− 52.9150i	− 0.0939876i	0.998895	−	0.0469938i	$-0.0149641\pi$
$564$	0	0
$565$	0	0
$566$	− 237.588i	− 0.419767i
$567$	0	0
$568$	−134.700	−0.237147
$569$	− 640.639i	− 1.12590i	−0.826490	−	0.562951i	$-0.809666\pi$
$569$	− 640.639i	− 1.12590i	0.826490	−	0.562951i	$-0.190334\pi$
$570$	0	0
$571$	−568.000	−0.994746	−0.497373	−	0.867537i	$-0.665702\pi$
$571$	−568.000	−0.994746	−0.497373	+	0.867537i	$0.665702\pi$
$572$	142.871i	0.249774i
$573$	0	0
$574$	1638.00	2.85366
$575$	0	0
$576$	0	0
$577$	−67.3498	−0.116724	−0.0583621	−	0.998295i	$-0.518588\pi$
$577$	−67.3498	−0.116724	−0.0583621	+	0.998295i	$0.518588\pi$
$578$	− 468.298i	− 0.810204i
$579$	0	0
$580$	0	0
$581$	− 831.558i	− 1.43125i
$582$	0	0
$583$	359.199	0.616122
$584$	178.191i	0.305121i
$585$	0	0
$586$	−1526.00	−2.60410
$587$	− 1026.55i	− 1.74881i	−0.485197	−	0.874405i	$-0.661252\pi$
$587$	− 1026.55i	− 1.74881i	0.485197	−	0.874405i	$-0.338748\pi$
$588$	0	0
$589$	−520.000	−0.882852
$590$	0	0
$591$	0	0
$592$	−639.823	−1.08078
$593$	− 656.146i	− 1.10649i	−0.833020	−	0.553243i	$-0.813390\pi$
$593$	− 656.146i	− 1.10649i	0.833020	−	0.553243i	$-0.186610\pi$
$594$	0	0
$595$	0	0
$596$	407.294i	0.683378i
$597$	0	0
$598$	157.150	0.262792
$599$	− 924.896i	− 1.54407i	−0.635582	−	0.772033i	$-0.719240\pi$
$599$	− 924.896i	− 1.54407i	0.635582	−	0.772033i	$-0.280760\pi$
$600$	0	0
$601$	788.000	1.31115	0.655574	−	0.755131i	$-0.272427\pi$
$601$	788.000	1.31115	0.655574	+	0.755131i	$0.272427\pi$
$602$	− 666.729i	− 1.10752i
$603$	0	0
$604$	606.000	1.00331
$605$	0	0
$606$	0	0
$607$	11.2250	0.0184925	0.00924627	−	0.999957i	$-0.497057\pi$
$607$	11.2250	0.0184925	0.00924627	+	0.999957i	$0.497057\pi$
$608$	− 793.725i	− 1.30547i
$609$	0	0
$610$	0	0
$611$	− 237.588i	− 0.388851i
$612$	0	0
$613$	−572.474	−0.933888	−0.466944	−	0.884287i	$-0.654645\pi$
$613$	−572.474	−0.933888	−0.466944	+	0.884287i	$0.654645\pi$
$614$	712.764i	1.16085i
$615$	0	0
$616$	−126.000	−0.204545
$617$	− 423.320i	− 0.686094i	−0.939318	−	0.343047i	$-0.888541\pi$
$617$	− 423.320i	− 0.686094i	0.939318	−	0.343047i	$-0.111459\pi$
$618$	0	0
$619$	−194.000	−0.313409	−0.156704	−	0.987646i	$-0.550087\pi$
$619$	−194.000	−0.313409	−0.156704	+	0.987646i	$0.550087\pi$
$620$	0	0
$621$	0	0
$622$	157.150	0.252652
$623$	1000.09i	1.60529i
$624$	0	0
$625$	0	0
$626$	− 475.176i	− 0.759067i
$627$	0	0
$628$	168.375	0.268112
$629$	− 356.382i	− 0.566585i
$630$	0	0
$631$	1190.00	1.88590	0.942948	−	0.332941i	$-0.108041\pi$
$631$	1190.00	1.88590	0.942948	+	0.332941i	$0.108041\pi$
$632$	37.0405i	0.0586084i
$633$	0	0
$634$	−826.000	−1.30284
$635$	0	0
$636$	0	0
$637$	−864.323	−1.35686
$638$	95.2470i	0.149290i
$639$	0	0
$640$	0	0
$641$	− 708.521i	− 1.10534i	−0.833401	−	0.552668i	$-0.813610\pi$
$641$	− 708.521i	− 1.10534i	0.833401	−	0.552668i	$-0.186390\pi$
$642$	0	0
$643$	−651.048	−1.01252	−0.506258	−	0.862382i	$-0.668972\pi$
$643$	−651.048	−1.01252	−0.506258	+	0.862382i	$0.668972\pi$
$644$	− 178.191i	− 0.276694i
$645$	0	0
$646$	560.000	0.866873
$647$	− 1058.30i	− 1.63570i	−0.575429	−	0.817852i	$-0.695165\pi$
$647$	− 1058.30i	− 1.63570i	0.575429	−	0.817852i	$-0.304835\pi$
$648$	0	0
$649$	198.000	0.305085
$650$	0	0
$651$	0	0
$652$	−606.148	−0.929676
$653$	470.944i	0.721200i	0.932721	+	0.360600i	$0.117428\pi$
$653$	470.944i	0.721200i	−0.932721	+	0.360600i	$0.882572\pi$
$654$	0	0
$655$	0	0
$656$	1047.93i	1.59746i
$657$	0	0
$658$	−628.598	−0.955317
$659$	− 80.6102i	− 0.122322i	−0.998128	−	0.0611610i	$-0.980520\pi$
$659$	− 80.6102i	− 0.122322i	0.998128	−	0.0611610i	$-0.0194803\pi$
$660$	0	0
$661$	170.000	0.257186	0.128593	−	0.991697i	$-0.458954\pi$
$661$	170.000	0.257186	0.128593	+	0.991697i	$0.458954\pi$
$662$	296.324i	0.447620i
$663$	0	0
$664$	196.000	0.295181
$665$	0	0
$666$	0	0
$667$	44.8999	0.0673162
$668$	555.608i	0.831748i
$669$	0	0
$670$	0	0
$671$	− 93.3381i	− 0.139103i
$672$	0	0
$673$	1055.15	1.56783	0.783913	−	0.620870i	$-0.213221\pi$
$673$	1055.15	1.56783	0.783913	+	0.620870i	$0.213221\pi$
$674$	− 296.985i	− 0.440630i
$675$	0	0
$676$	129.000	0.190828
$677$	830.766i	1.22713i	0.789645	+	0.613564i	$0.210265\pi$
$677$	830.766i	1.22713i	−0.789645	+	0.613564i	$0.789735\pi$
$678$	0	0
$679$	−252.000	−0.371134
$680$	0	0
$681$	0	0
$682$	291.849	0.427931
$683$	− 1037.13i	− 1.51850i	−0.650800	−	0.759249i	$-0.725566\pi$
$683$	− 1037.13i	− 1.51850i	0.650800	−	0.759249i	$-0.274434\pi$
$684$	0	0
$685$	0	0
$686$	831.558i	1.21218i
$687$	0	0
$688$	426.549	0.619984
$689$	950.352i	1.37932i
$690$	0	0
$691$	−652.000	−0.943560	−0.471780	−	0.881716i	$-0.656388\pi$
$691$	−652.000	−0.943560	−0.471780	+	0.881716i	$0.656388\pi$
$692$	63.4980i	0.0917602i
$693$	0	0
$694$	−1372.00	−1.97695
$695$	0	0
$696$	0	0
$697$	−583.699	−0.837444
$698$	322.782i	0.462438i
$699$	0	0
$700$	0	0
$701$	763.675i	1.08941i	0.838628	+	0.544704i	$0.183358\pi$
$701$	763.675i	1.08941i	−0.838628	+	0.544704i	$0.816642\pi$
$702$	0	0
$703$	−673.498	−0.958035
$704$	123.037i	0.174768i
$705$	0	0
$706$	−1064.00	−1.50708
$707$	1523.95i	2.15552i
$708$	0	0
$709$	−158.000	−0.222849	−0.111425	−	0.993773i	$-0.535541\pi$
$709$	−158.000	−0.222849	−0.111425	+	0.993773i	$0.535541\pi$
$710$	0	0
$711$	0	0
$712$	−235.724	−0.331074
$713$	− 137.579i	− 0.192958i
$714$	0	0
$715$	0	0
$716$	− 725.492i	− 1.01326i
$717$	0	0
$718$	1683.75	2.34505
$719$	127.279i	0.177023i	0.996075	+	0.0885113i	$0.0282109\pi$
$719$	127.279i	0.177023i	−0.996075	+	0.0885113i	$0.971789\pi$
$720$	0	0
$721$	−630.000	−0.873786
$722$	− 103.184i	− 0.142915i
$723$	0	0
$724$	−222.000	−0.306630
$725$	0	0
$726$	0	0
$727$	662.273	0.910967	0.455484	−	0.890244i	$-0.349466\pi$
$727$	662.273	0.910967	0.455484	+	0.890244i	$0.349466\pi$
$728$	− 333.365i	− 0.457918i
$729$	0	0
$730$	0	0
$731$	237.588i	0.325018i
$732$	0	0
$733$	−684.723	−0.934138	−0.467069	−	0.884221i	$-0.654690\pi$
$733$	−684.723	−0.934138	−0.467069	+	0.884221i	$0.654690\pi$
$734$	1811.61i	2.46813i
$735$	0	0
$736$	210.000	0.285326
$737$	− 380.988i	− 0.516945i
$738$	0	0
$739$	1240.00	1.67794	0.838972	−	0.544175i	$-0.183157\pi$
$739$	1240.00	1.67794	0.838972	+	0.544175i	$0.183157\pi$
$740$	0	0
$741$	0	0
$742$	2514.39	3.38867
$743$	42.3320i	0.0569745i	0.999594	+	0.0284872i	$0.00906899\pi$
$743$	42.3320i	0.0569745i	−0.999594	+	0.0284872i	$0.990931\pi$
$744$	0	0
$745$	0	0
$746$	386.080i	0.517534i
$747$	0	0
$748$	−134.700	−0.180080
$749$	118.794i	0.158603i
$750$	0	0
$751$	−154.000	−0.205060	−0.102530	−	0.994730i	$-0.532694\pi$
$751$	−154.000	−0.205060	−0.102530	+	0.994730i	$0.532694\pi$
$752$	− 402.154i	− 0.534780i
$753$	0	0
$754$	−252.000	−0.334218
$755$	0	0
$756$	0	0
$757$	1313.32	1.73490	0.867452	−	0.497522i	$-0.165756\pi$
$757$	1313.32	1.73490	0.867452	+	0.497522i	$0.165756\pi$
$758$	957.762i	1.26354i
$759$	0	0
$760$	0	0
$761$	504.874i	0.663435i	0.943379	+	0.331718i	$0.107628\pi$
$761$	504.874i	0.663435i	−0.943379	+	0.331718i	$0.892372\pi$
$762$	0	0
$763$	−785.748	−1.02981
$764$	483.661i	0.633064i
$765$	0	0
$766$	112.000	0.146214
$767$	523.859i	0.682997i
$768$	0	0
$769$	−368.000	−0.478544	−0.239272	−	0.970953i	$-0.576909\pi$
$769$	−368.000	−0.478544	−0.239272	+	0.970953i	$0.576909\pi$
$770$	0	0
$771$	0	0
$772$	538.799	0.697926
$773$	153.454i	0.198517i	0.995062	+	0.0992585i	$0.0316471\pi$
$773$	153.454i	0.198517i	−0.995062	+	0.0992585i	$0.968353\pi$
$774$	0	0
$775$	0	0
$776$	− 59.3970i	− 0.0765425i
$777$	0	0
$778$	695.948	0.894535
$779$	1103.09i	1.41603i
$780$	0	0
$781$	216.000	0.276569
$782$	148.162i	0.189466i
$783$	0	0
$784$	−1463.00	−1.86607
$785$	0	0
$786$	0	0
$787$	−426.549	−0.541994	−0.270997	−	0.962580i	$-0.587353\pi$
$787$	−426.549	−0.541994	−0.270997	+	0.962580i	$0.587353\pi$
$788$	− 111.122i	− 0.141017i
$789$	0	0
$790$	0	0
$791$	− 1544.32i	− 1.95237i
$792$	0	0
$793$	246.949	0.311412
$794$	89.0955i	0.112211i
$795$	0	0
$796$	−750.000	−0.942211
$797$	− 232.826i	− 0.292128i	−0.989275	−	0.146064i	$-0.953339\pi$
$797$	− 232.826i	− 0.292128i	0.989275	−	0.146064i	$-0.0466606\pi$
$798$	0	0
$799$	224.000	0.280350
$800$	0	0
$801$	0	0
$802$	−1223.52	−1.52559
$803$	− 285.741i	− 0.355842i
$804$	0	0
$805$	0	0
$806$	772.161i	0.958016i
$807$	0	0
$808$	−359.199	−0.444553
$809$	− 420.021i	− 0.519186i	−0.965718	−	0.259593i	$-0.916412\pi$
$809$	− 420.021i	− 0.519186i	0.965718	−	0.259593i	$-0.0835884\pi$
$810$	0	0
$811$	−970.000	−1.19605	−0.598027	−	0.801476i	$-0.704049\pi$
$811$	−970.000	−1.19605	−0.598027	+	0.801476i	$0.704049\pi$
$812$	285.741i	0.351898i
$813$	0	0
$814$	378.000	0.464373
$815$	0	0
$816$	0	0
$817$	448.999	0.549570
$818$	− 216.952i	− 0.265222i
$819$	0	0
$820$	0	0
$821$	42.4264i	0.0516765i	0.999666	+	0.0258383i	$0.00822549\pi$
$821$	42.4264i	0.0516765i	−0.999666	+	0.0258383i	$0.991775\pi$
$822$	0	0
$823$	392.874	0.477368	0.238684	−	0.971097i	$-0.423284\pi$
$823$	392.874	0.477368	0.238684	+	0.971097i	$0.423284\pi$
$824$	− 148.492i	− 0.180209i
$825$	0	0
$826$	1386.00	1.67797
$827$	560.899i	0.678234i	0.940744	+	0.339117i	$0.110128\pi$
$827$	560.899i	0.678234i	−0.940744	+	0.339117i	$0.889872\pi$
$828$	0	0
$829$	−1010.00	−1.21834	−0.609168	−	0.793041i	$-0.708496\pi$
$829$	−1010.00	−1.21834	−0.609168	+	0.793041i	$0.708496\pi$
$830$	0	0
$831$	0	0
$832$	−325.524	−0.391255
$833$	− 814.891i	− 0.978261i
$834$	0	0
$835$	0	0
$836$	254.558i	0.304496i
$837$	0	0
$838$	550.024	0.656353
$839$	− 449.720i	− 0.536019i	−0.963416	−	0.268009i	$-0.913634\pi$
$839$	− 449.720i	− 0.536019i	0.963416	−	0.268009i	$-0.0863659\pi$
$840$	0	0
$841$	769.000	0.914388
$842$	1296.42i	1.53969i
$843$	0	0
$844$	462.000	0.547393
$845$	0	0
$846$	0	0
$847$	−1156.17	−1.36502
$848$	1608.62i	1.89695i
$849$	0	0
$850$	0	0
$851$	− 178.191i	− 0.209390i
$852$	0	0
$853$	−931.673	−1.09223	−0.546115	−	0.837710i	$-0.683894\pi$
$853$	−931.673	−1.09223	−0.546115	+	0.837710i	$0.683894\pi$
$854$	− 653.367i	− 0.765066i
$855$	0	0
$856$	−28.0000	−0.0327103
$857$	− 1312.29i	− 1.53126i	−0.643279	−	0.765632i	$-0.722427\pi$
$857$	− 1312.29i	− 1.53126i	0.643279	−	0.765632i	$-0.277573\pi$
$858$	0	0
$859$	694.000	0.807916	0.403958	−	0.914777i	$-0.367634\pi$
$859$	694.000	0.807916	0.403958	+	0.914777i	$0.367634\pi$
$860$	0	0
$861$	0	0
$862$	−1818.45	−2.10957
$863$	− 799.017i	− 0.925860i	−0.886395	−	0.462930i	$-0.846798\pi$
$863$	− 799.017i	− 0.925860i	0.886395	−	0.462930i	$-0.153202\pi$
$864$	0	0
$865$	0	0
$866$	534.573i	0.617290i
$867$	0	0
$868$	875.548	1.00870
$869$	− 59.3970i	− 0.0683509i
$870$	0	0
$871$	1008.00	1.15729
$872$	− 185.203i	− 0.212388i
$873$	0	0
$874$	280.000	0.320366
$875$	0	0
$876$	0	0
$877$	−796.973	−0.908749	−0.454375	−	0.890811i	$-0.650137\pi$
$877$	−796.973	−0.908749	−0.454375	+	0.890811i	$0.650137\pi$
$878$	799.017i	0.910042i
$879$	0	0
$880$	0	0
$881$	827.315i	0.939063i	0.882916	+	0.469532i	$0.155577\pi$
$881$	827.315i	0.939063i	−0.882916	+	0.469532i	$0.844423\pi$
$882$	0	0
$883$	−471.449	−0.533917	−0.266959	−	0.963708i	$-0.586019\pi$
$883$	−471.449	−0.533917	−0.266959	+	0.963708i	$0.586019\pi$
$884$	− 356.382i	− 0.403147i
$885$	0	0
$886$	700.000	0.790068
$887$	− 1137.67i	− 1.28261i	−0.767287	−	0.641304i	$-0.778394\pi$
$887$	− 1137.67i	− 1.28261i	0.767287	−	0.641304i	$-0.221606\pi$
$888$	0	0
$889$	−1890.00	−2.12598
$890$	0	0
$891$	0	0
$892$	1178.62	1.32133
$893$	− 423.320i	− 0.474043i
$894$	0	0
$895$	0	0
$896$	− 920.653i	− 1.02751i
$897$	0	0
$898$	−572.474	−0.637498
$899$	220.617i	0.245403i
$900$	0	0
$901$	−896.000	−0.994451
$902$	− 619.106i	− 0.686370i
$903$	0	0
$904$	364.000	0.402655
$905$	0	0
$906$	0	0
$907$	1234.75	1.36135	0.680676	−	0.732584i	$-0.261686\pi$
$907$	1234.75	1.36135	0.680676	+	0.732584i	$0.261686\pi$
$908$	− 63.4980i	− 0.0699318i
$909$	0	0
$910$	0	0
$911$	− 288.500i	− 0.316684i	−0.987384	−	0.158342i	$-0.949385\pi$
$911$	− 288.500i	− 0.316684i	0.987384	−	0.158342i	$-0.0506149\pi$
$912$	0	0
$913$	−314.299	−0.344249
$914$	− 1484.92i	− 1.62464i
$915$	0	0
$916$	−354.000	−0.386463
$917$	− 1666.82i	− 1.81769i
$918$	0	0
$919$	1078.00	1.17301	0.586507	−	0.809944i	$-0.300503\pi$
$919$	1078.00	1.17301	0.586507	+	0.809944i	$0.300503\pi$
$920$	0	0
$921$	0	0
$922$	−628.598	−0.681777
$923$	571.482i	0.619157i
$924$	0	0
$925$	0	0
$926$	− 1930.40i	− 2.08467i
$927$	0	0
$928$	−336.749	−0.362876
$929$	1752.21i	1.88613i	0.332615	+	0.943063i	$0.392069\pi$
$929$	1752.21i	1.88613i	−0.332615	+	0.943063i	$0.607931\pi$
$930$	0	0
$931$	−1540.00	−1.65414
$932$	− 1174.71i	− 1.26042i
$933$	0	0
$934$	1232.00	1.31906
$935$	0	0
$936$	0	0
$937$	−942.898	−1.00629	−0.503147	−	0.864201i	$-0.667825\pi$
$937$	−942.898	−1.00629	−0.503147	+	0.864201i	$0.667825\pi$
$938$	− 2666.92i	− 2.84320i
$939$	0	0
$940$	0	0
$941$	1026.72i	1.09109i	0.838080	+	0.545547i	$0.183678\pi$
$941$	1026.72i	1.09109i	−0.838080	+	0.545547i	$0.816322\pi$
$942$	0	0
$943$	−291.849	−0.309490
$944$	886.712i	0.939313i
$945$	0	0
$946$	−252.000	−0.266385
$947$	497.401i	0.525239i	0.964899	+	0.262619i	$0.0845864\pi$
$947$	497.401i	0.525239i	−0.964899	+	0.262619i	$0.915414\pi$
$948$	0	0
$949$	756.000	0.796628
$950$	0	0
$951$	0	0
$952$	314.299	0.330146
$953$	486.818i	0.510827i	0.966832	+	0.255414i	$0.0822116\pi$
$953$	486.818i	0.510827i	−0.966832	+	0.255414i	$0.917788\pi$
$954$	0	0
$955$	0	0
$956$	− 1043.69i	− 1.09173i
$957$	0	0
$958$	−1257.20	−1.31231
$959$	− 2375.88i	− 2.47745i
$960$	0	0
$961$	−285.000	−0.296566
$962$	1000.09i	1.03960i
$963$	0	0
$964$	120.000	0.124481
$965$	0	0
$966$	0	0
$967$	−662.273	−0.684874	−0.342437	−	0.939541i	$-0.611252\pi$
$967$	−662.273	−0.684874	−0.342437	+	0.939541i	$0.611252\pi$
$968$	− 272.512i	− 0.281521i
$969$	0	0
$970$	0	0
$971$	1641.90i	1.69094i	0.534024	+	0.845470i	$0.320679\pi$
$971$	1641.90i	1.69094i	−0.534024	+	0.845470i	$0.679321\pi$
$972$	0	0
$973$	2312.34	2.37651
$974$	− 1336.43i	− 1.37211i
$975$	0	0
$976$	418.000	0.428279
$977$	1291.13i	1.32152i	0.750597	+	0.660761i	$0.229766\pi$
$977$	1291.13i	1.32152i	−0.750597	+	0.660761i	$0.770234\pi$
$978$	0	0
$979$	378.000	0.386108
$980$	0	0
$981$	0	0
$982$	729.623	0.742997
$983$	− 592.648i	− 0.602898i	−0.953482	−	0.301449i	$-0.902530\pi$
$983$	− 592.648i	− 0.602898i	0.953482	−	0.301449i	$-0.0974702\pi$
$984$	0	0
$985$	0	0
$986$	− 237.588i	− 0.240961i
$987$	0	0
$988$	−673.498	−0.681678
$989$	118.794i	0.120115i
$990$	0	0
$991$	1694.00	1.70938	0.854692	−	0.519135i	$-0.173746\pi$
$991$	1694.00	1.70938	0.854692	+	0.519135i	$0.173746\pi$
$992$	1031.84i	1.04016i
$993$	0	0
$994$	1512.00	1.52113
$995$	0	0
$996$	0	0
$997$	−954.123	−0.956994	−0.478497	−	0.878089i	$-0.658818\pi$
$997$	−954.123	−0.956994	−0.478497	+	0.878089i	$0.658818\pi$
$998$	973.636i	0.975588i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.3.c.d.26.1		4
3.2	odd	2	inner	225.3.c.d.26.3		4
4.3	odd	2		3600.3.l.s.1601.3		4
5.2	odd	4		45.3.d.a.44.4	yes	4
5.3	odd	4		45.3.d.a.44.2	yes	4
5.4	even	2	inner	225.3.c.d.26.4		4
12.11	even	2		3600.3.l.s.1601.4		4
15.2	even	4		45.3.d.a.44.1	✓	4
15.8	even	4		45.3.d.a.44.3	yes	4
15.14	odd	2	inner	225.3.c.d.26.2		4
20.3	even	4		720.3.c.a.449.4		4
20.7	even	4		720.3.c.a.449.2		4
20.19	odd	2		3600.3.l.s.1601.1		4
40.3	even	4		2880.3.c.g.449.1		4
40.13	odd	4		2880.3.c.b.449.1		4
40.27	even	4		2880.3.c.g.449.3		4
40.37	odd	4		2880.3.c.b.449.3		4
45.2	even	12		405.3.h.j.134.4		8
45.7	odd	12		405.3.h.j.134.1		8
45.13	odd	12		405.3.h.j.269.4		8
45.22	odd	12		405.3.h.j.269.2		8
45.23	even	12		405.3.h.j.269.1		8
45.32	even	12		405.3.h.j.269.3		8
45.38	even	12		405.3.h.j.134.2		8
45.43	odd	12		405.3.h.j.134.3		8
60.23	odd	4		720.3.c.a.449.1		4
60.47	odd	4		720.3.c.a.449.3		4
60.59	even	2		3600.3.l.s.1601.2		4
120.53	even	4		2880.3.c.b.449.4		4
120.77	even	4		2880.3.c.b.449.2		4
120.83	odd	4		2880.3.c.g.449.4		4
120.107	odd	4		2880.3.c.g.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.d.a.44.1	✓	4	15.2	even	4
45.3.d.a.44.2	yes	4	5.3	odd	4
45.3.d.a.44.3	yes	4	15.8	even	4
45.3.d.a.44.4	yes	4	5.2	odd	4
225.3.c.d.26.1		4	1.1	even	1	trivial
225.3.c.d.26.2		4	15.14	odd	2	inner
225.3.c.d.26.3		4	3.2	odd	2	inner
225.3.c.d.26.4		4	5.4	even	2	inner
405.3.h.j.134.1		8	45.7	odd	12
405.3.h.j.134.2		8	45.38	even	12
405.3.h.j.134.3		8	45.43	odd	12
405.3.h.j.134.4		8	45.2	even	12
405.3.h.j.269.1		8	45.23	even	12
405.3.h.j.269.2		8	45.22	odd	12
405.3.h.j.269.3		8	45.32	even	12
405.3.h.j.269.4		8	45.13	odd	12
720.3.c.a.449.1		4	60.23	odd	4
720.3.c.a.449.2		4	20.7	even	4
720.3.c.a.449.3		4	60.47	odd	4
720.3.c.a.449.4		4	20.3	even	4
2880.3.c.b.449.1		4	40.13	odd	4
2880.3.c.b.449.2		4	120.77	even	4
2880.3.c.b.449.3		4	40.37	odd	4
2880.3.c.b.449.4		4	120.53	even	4
2880.3.c.g.449.1		4	40.3	even	4
2880.3.c.g.449.2		4	120.107	odd	4
2880.3.c.g.449.3		4	40.27	even	4
2880.3.c.g.449.4		4	120.83	odd	4
3600.3.l.s.1601.1		4	20.19	odd	2
3600.3.l.s.1601.2		4	60.59	even	2
3600.3.l.s.1601.3		4	4.3	odd	2
3600.3.l.s.1601.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.64575i q^{2} -3.00000 q^{4} -11.2250 q^{7} -2.64575i q^{8} +O(q^{10})\)	\(q-2.64575i q^{2} -3.00000 q^{4} -11.2250 q^{7} -2.64575i q^{8} +4.24264i q^{11} -11.2250 q^{13} +29.6985i q^{14} -19.0000 q^{16} -10.5830i q^{17} -20.0000 q^{19} +11.2250 q^{22} -5.29150i q^{23} +29.6985i q^{26} +33.6749 q^{28} +8.48528i q^{29} +26.0000 q^{31} +39.6863i q^{32} -28.0000 q^{34} +33.6749 q^{37} +52.9150i q^{38} -55.1543i q^{41} -22.4499 q^{43} -12.7279i q^{44} -14.0000 q^{46} +21.1660i q^{47} +77.0000 q^{49} +33.6749 q^{52} -84.6640i q^{53} +29.6985i q^{56} +22.4499 q^{58} -46.6690i q^{59} -22.0000 q^{61} -68.7895i q^{62} +29.0000 q^{64} -89.7998 q^{67} +31.7490i q^{68} -50.9117i q^{71} -67.3498 q^{73} -89.0955i q^{74} +60.0000 q^{76} -47.6235i q^{77} -14.0000 q^{79} -145.925 q^{82} +74.0810i q^{83} +59.3970i q^{86} +11.2250 q^{88} -89.0955i q^{89} +126.000 q^{91} +15.8745i q^{92} +56.0000 q^{94} +22.4499 q^{97} -203.723i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{4}+O(q^{10}) \) 4 * q - 12 * q^4	\( 4 q - 12 q^{4} - 76 q^{16} - 80 q^{19} + 104 q^{31} - 112 q^{34} - 56 q^{46} + 308 q^{49} - 88 q^{61} + 116 q^{64} + 240 q^{76} - 56 q^{79} + 504 q^{91} + 224 q^{94}+O(q^{100}) \) 4 * q - 12 * q^4 - 76 * q^16 - 80 * q^19 + 104 * q^31 - 112 * q^34 - 56 * q^46 + 308 * q^49 - 88 * q^61 + 116 * q^64 + 240 * q^76 - 56 * q^79 + 504 * q^91 + 224 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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