LMFDB - Embedded newform 450.4.c.g.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(199,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, 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("450.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	450.c (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:	$26.5508595026$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.4.c.g.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} -14.0000i q^{7} -8.00000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} -4.00000 q^{4} -14.0000i q^{7} -8.00000i q^{8} +6.00000 q^{11} +68.0000i q^{13} +28.0000 q^{14} +16.0000 q^{16} -78.0000i q^{17} -44.0000 q^{19} +12.0000i q^{22} +120.000i q^{23} -136.000 q^{26} +56.0000i q^{28} -126.000 q^{29} -244.000 q^{31} +32.0000i q^{32} +156.000 q^{34} +304.000i q^{37} -88.0000i q^{38} -480.000 q^{41} +104.000i q^{43} -24.0000 q^{44} -240.000 q^{46} -600.000i q^{47} +147.000 q^{49} -272.000i q^{52} -258.000i q^{53} -112.000 q^{56} -252.000i q^{58} -534.000 q^{59} +362.000 q^{61} -488.000i q^{62} -64.0000 q^{64} +268.000i q^{67} +312.000i q^{68} -972.000 q^{71} +470.000i q^{73} -608.000 q^{74} +176.000 q^{76} -84.0000i q^{77} -1244.00 q^{79} -960.000i q^{82} +396.000i q^{83} -208.000 q^{86} -48.0000i q^{88} +972.000 q^{89} +952.000 q^{91} -480.000i q^{92} +1200.00 q^{94} +46.0000i q^{97} +294.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} + 12 q^{11} + 56 q^{14} + 32 q^{16} - 88 q^{19} - 272 q^{26} - 252 q^{29} - 488 q^{31} + 312 q^{34} - 960 q^{41} - 48 q^{44} - 480 q^{46} + 294 q^{49} - 224 q^{56} - 1068 q^{59} + 724 q^{61} - 128 q^{64} - 1944 q^{71} - 1216 q^{74} + 352 q^{76} - 2488 q^{79} - 416 q^{86} + 1944 q^{89} + 1904 q^{91} + 2400 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 + 12 * q^11 + 56 * q^14 + 32 * q^16 - 88 * q^19 - 272 * q^26 - 252 * q^29 - 488 * q^31 + 312 * q^34 - 960 * q^41 - 48 * q^44 - 480 * q^46 + 294 * q^49 - 224 * q^56 - 1068 * q^59 + 724 * q^61 - 128 * q^64 - 1944 * q^71 - 1216 * q^74 + 352 * q^76 - 2488 * q^79 - 416 * q^86 + 1944 * q^89 + 1904 * q^91 + 2400 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 14.0000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 14.0000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 8.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	6.00000	0.164461	0.0822304	−	0.996613i	$-0.473796\pi$
$11$	6.00000	0.164461	0.0822304	+	0.996613i	$0.473796\pi$
$12$	0	0
$13$	68.0000i	1.45075i	0.688352	+	0.725377i	$0.258335\pi$
$13$	68.0000i	1.45075i	−0.688352	+	0.725377i	$0.741665\pi$
$14$	28.0000	0.534522
$15$	0	0
$16$	16.0000	0.250000
$17$	− 78.0000i	− 1.11281i	−0.830911	−	0.556405i	$-0.812180\pi$
$17$	− 78.0000i	− 1.11281i	0.830911	−	0.556405i	$-0.187820\pi$
$18$	0	0
$19$	−44.0000	−0.531279	−0.265639	−	0.964072i	$-0.585583\pi$
$19$	−44.0000	−0.531279	−0.265639	+	0.964072i	$0.585583\pi$
$20$	0	0
$21$	0	0
$22$	12.0000i	0.116291i
$23$	120.000i	1.08790i	0.839117	+	0.543951i	$0.183072\pi$
$23$	120.000i	1.08790i	−0.839117	+	0.543951i	$0.816928\pi$
$24$	0	0
$25$	0	0
$26$	−136.000	−1.02584
$27$	0	0
$28$	56.0000i	0.377964i
$29$	−126.000	−0.806814	−0.403407	−	0.915021i	$-0.632174\pi$
$29$	−126.000	−0.806814	−0.403407	+	0.915021i	$0.632174\pi$
$30$	0	0
$31$	−244.000	−1.41367	−0.706834	−	0.707380i	$-0.749877\pi$
$31$	−244.000	−1.41367	−0.706834	+	0.707380i	$0.749877\pi$
$32$	32.0000i	0.176777i
$33$	0	0
$34$	156.000	0.786876
$35$	0	0
$36$	0	0
$37$	304.000i	1.35074i	0.737480	+	0.675369i	$0.236016\pi$
$37$	304.000i	1.35074i	−0.737480	+	0.675369i	$0.763984\pi$
$38$	− 88.0000i	− 0.375671i
$39$	0	0
$40$	0	0
$41$	−480.000	−1.82838	−0.914188	−	0.405291i	$-0.867170\pi$
$41$	−480.000	−1.82838	−0.914188	+	0.405291i	$0.867170\pi$
$42$	0	0
$43$	104.000i	0.368834i	0.982848	+	0.184417i	$0.0590396\pi$
$43$	104.000i	0.368834i	−0.982848	+	0.184417i	$0.940960\pi$
$44$	−24.0000	−0.0822304
$45$	0	0
$46$	−240.000	−0.769262
$47$	− 600.000i	− 1.86211i	−0.364884	−	0.931053i	$-0.618891\pi$
$47$	− 600.000i	− 1.86211i	0.364884	−	0.931053i	$-0.381109\pi$
$48$	0	0
$49$	147.000	0.428571
$50$	0	0
$51$	0	0
$52$	− 272.000i	− 0.725377i
$53$	− 258.000i	− 0.668661i	−0.942456	−	0.334330i	$-0.891490\pi$
$53$	− 258.000i	− 0.668661i	0.942456	−	0.334330i	$-0.108510\pi$
$54$	0	0
$55$	0	0
$56$	−112.000	−0.267261
$57$	0	0
$58$	− 252.000i	− 0.570504i
$59$	−534.000	−1.17832	−0.589160	−	0.808016i	$-0.700541\pi$
$59$	−534.000	−1.17832	−0.589160	+	0.808016i	$0.700541\pi$
$60$	0	0
$61$	362.000	0.759825	0.379913	−	0.925022i	$-0.375954\pi$
$61$	362.000	0.759825	0.379913	+	0.925022i	$0.375954\pi$
$62$	− 488.000i	− 0.999614i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	268.000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	268.000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	312.000i	0.556405i
$69$	0	0
$70$	0	0
$71$	−972.000	−1.62472	−0.812360	−	0.583156i	$-0.801818\pi$
$71$	−972.000	−1.62472	−0.812360	+	0.583156i	$0.801818\pi$
$72$	0	0
$73$	470.000i	0.753553i	0.926304	+	0.376776i	$0.122967\pi$
$73$	470.000i	0.753553i	−0.926304	+	0.376776i	$0.877033\pi$
$74$	−608.000	−0.955116
$75$	0	0
$76$	176.000	0.265639
$77$	− 84.0000i	− 0.124321i
$78$	0	0
$79$	−1244.00	−1.77166	−0.885829	−	0.464012i	$-0.846409\pi$
$79$	−1244.00	−1.77166	−0.885829	+	0.464012i	$0.846409\pi$
$80$	0	0
$81$	0	0
$82$	− 960.000i	− 1.29286i
$83$	396.000i	0.523695i	0.965109	+	0.261847i	$0.0843317\pi$
$83$	396.000i	0.523695i	−0.965109	+	0.261847i	$0.915668\pi$
$84$	0	0
$85$	0	0
$86$	−208.000	−0.260805
$87$	0	0
$88$	− 48.0000i	− 0.0581456i
$89$	972.000	1.15766	0.578830	−	0.815448i	$-0.303509\pi$
$89$	972.000	1.15766	0.578830	+	0.815448i	$0.303509\pi$
$90$	0	0
$91$	952.000	1.09667
$92$	− 480.000i	− 0.543951i
$93$	0	0
$94$	1200.00	1.31671
$95$	0	0
$96$	0	0
$97$	46.0000i	0.0481504i	0.999710	+	0.0240752i	$0.00766412\pi$
$97$	46.0000i	0.0481504i	−0.999710	+	0.0240752i	$0.992336\pi$
$98$	294.000i	0.303046i
$99$	0	0
$100$	0	0
$101$	−1506.00	−1.48369	−0.741845	−	0.670572i	$-0.766049\pi$
$101$	−1506.00	−1.48369	−0.741845	+	0.670572i	$0.766049\pi$
$102$	0	0
$103$	− 1474.00i	− 1.41007i	−0.709171	−	0.705037i	$-0.750931\pi$
$103$	− 1474.00i	− 1.41007i	0.709171	−	0.705037i	$-0.249069\pi$
$104$	544.000	0.512919
$105$	0	0
$106$	516.000	0.472815
$107$	− 924.000i	− 0.834827i	−0.908717	−	0.417413i	$-0.862937\pi$
$107$	− 924.000i	− 0.834827i	0.908717	−	0.417413i	$-0.137063\pi$
$108$	0	0
$109$	−698.000	−0.613360	−0.306680	−	0.951813i	$-0.599218\pi$
$109$	−698.000	−0.613360	−0.306680	+	0.951813i	$0.599218\pi$
$110$	0	0
$111$	0	0
$112$	− 224.000i	− 0.188982i
$113$	− 222.000i	− 0.184814i	−0.995721	−	0.0924071i	$-0.970544\pi$
$113$	− 222.000i	− 0.184814i	0.995721	−	0.0924071i	$-0.0294561\pi$
$114$	0	0
$115$	0	0
$116$	504.000	0.403407
$117$	0	0
$118$	− 1068.00i	− 0.833198i
$119$	−1092.00	−0.841206
$120$	0	0
$121$	−1295.00	−0.972953
$122$	724.000i	0.537278i
$123$	0	0
$124$	976.000	0.706834
$125$	0	0
$126$	0	0
$127$	1906.00i	1.33173i	0.746071	+	0.665867i	$0.231938\pi$
$127$	1906.00i	1.33173i	−0.746071	+	0.665867i	$0.768062\pi$
$128$	− 128.000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	2874.00	1.91681	0.958407	−	0.285406i	$-0.0921285\pi$
$131$	2874.00	1.91681	0.958407	+	0.285406i	$0.0921285\pi$
$132$	0	0
$133$	616.000i	0.401609i
$134$	−536.000	−0.345547
$135$	0	0
$136$	−624.000	−0.393438
$137$	798.000i	0.497648i	0.968549	+	0.248824i	$0.0800440\pi$
$137$	798.000i	0.497648i	−0.968549	+	0.248824i	$0.919956\pi$
$138$	0	0
$139$	700.000	0.427146	0.213573	−	0.976927i	$-0.431490\pi$
$139$	700.000	0.427146	0.213573	+	0.976927i	$0.431490\pi$
$140$	0	0
$141$	0	0
$142$	− 1944.00i	− 1.14885i
$143$	408.000i	0.238592i
$144$	0	0
$145$	0	0
$146$	−940.000	−0.532842
$147$	0	0
$148$	− 1216.00i	− 0.675369i
$149$	−114.000	−0.0626795	−0.0313397	−	0.999509i	$-0.509977\pi$
$149$	−114.000	−0.0626795	−0.0313397	+	0.999509i	$0.509977\pi$
$150$	0	0
$151$	1064.00	0.573424	0.286712	−	0.958017i	$-0.407438\pi$
$151$	1064.00	0.573424	0.286712	+	0.958017i	$0.407438\pi$
$152$	352.000i	0.187835i
$153$	0	0
$154$	168.000	0.0879080
$155$	0	0
$156$	0	0
$157$	1948.00i	0.990238i	0.868825	+	0.495119i	$0.164875\pi$
$157$	1948.00i	0.990238i	−0.868825	+	0.495119i	$0.835125\pi$
$158$	− 2488.00i	− 1.25275i
$159$	0	0
$160$	0	0
$161$	1680.00	0.822376
$162$	0	0
$163$	2060.00i	0.989887i	0.868925	+	0.494944i	$0.164811\pi$
$163$	2060.00i	0.989887i	−0.868925	+	0.494944i	$0.835189\pi$
$164$	1920.00	0.914188
$165$	0	0
$166$	−792.000	−0.370308
$167$	1248.00i	0.578282i	0.957286	+	0.289141i	$0.0933697\pi$
$167$	1248.00i	0.578282i	−0.957286	+	0.289141i	$0.906630\pi$
$168$	0	0
$169$	−2427.00	−1.10469
$170$	0	0
$171$	0	0
$172$	− 416.000i	− 0.184417i
$173$	− 1146.00i	− 0.503634i	−0.967775	−	0.251817i	$-0.918972\pi$
$173$	− 1146.00i	− 0.503634i	0.967775	−	0.251817i	$-0.0810282\pi$
$174$	0	0
$175$	0	0
$176$	96.0000	0.0411152
$177$	0	0
$178$	1944.00i	0.818590i
$179$	1146.00	0.478525	0.239263	−	0.970955i	$-0.423094\pi$
$179$	1146.00	0.478525	0.239263	+	0.970955i	$0.423094\pi$
$180$	0	0
$181$	−118.000	−0.0484579	−0.0242289	−	0.999706i	$-0.507713\pi$
$181$	−118.000	−0.0484579	−0.0242289	+	0.999706i	$0.507713\pi$
$182$	1904.00i	0.775461i
$183$	0	0
$184$	960.000	0.384631
$185$	0	0
$186$	0	0
$187$	− 468.000i	− 0.183014i
$188$	2400.00i	0.931053i
$189$	0	0
$190$	0	0
$191$	1692.00	0.640989	0.320494	−	0.947250i	$-0.396151\pi$
$191$	1692.00	0.640989	0.320494	+	0.947250i	$0.396151\pi$
$192$	0	0
$193$	3350.00i	1.24942i	0.780856	+	0.624711i	$0.214783\pi$
$193$	3350.00i	1.24942i	−0.780856	+	0.624711i	$0.785217\pi$
$194$	−92.0000	−0.0340475
$195$	0	0
$196$	−588.000	−0.214286
$197$	3606.00i	1.30415i	0.758156	+	0.652073i	$0.226101\pi$
$197$	3606.00i	1.30415i	−0.758156	+	0.652073i	$0.773899\pi$
$198$	0	0
$199$	−2696.00	−0.960374	−0.480187	−	0.877166i	$-0.659431\pi$
$199$	−2696.00	−0.960374	−0.480187	+	0.877166i	$0.659431\pi$
$200$	0	0
$201$	0	0
$202$	− 3012.00i	− 1.04913i
$203$	1764.00i	0.609894i
$204$	0	0
$205$	0	0
$206$	2948.00	0.997072
$207$	0	0
$208$	1088.00i	0.362689i
$209$	−264.000	−0.0873745
$210$	0	0
$211$	−4.00000	−0.00130508	−0.000652539	−	1.00000i	$-0.500208\pi$
$211$	−4.00000	−0.00130508	−0.000652539		1.00000i	$0.500208\pi$
$212$	1032.00i	0.334330i
$213$	0	0
$214$	1848.00	0.590312
$215$	0	0
$216$	0	0
$217$	3416.00i	1.06863i
$218$	− 1396.00i	− 0.433711i
$219$	0	0
$220$	0	0
$221$	5304.00	1.61441
$222$	0	0
$223$	− 1162.00i	− 0.348938i	−0.984663	−	0.174469i	$-0.944179\pi$
$223$	− 1162.00i	− 0.348938i	0.984663	−	0.174469i	$-0.0558210\pi$
$224$	448.000	0.133631
$225$	0	0
$226$	444.000	0.130683
$227$	2400.00i	0.701734i	0.936425	+	0.350867i	$0.114113\pi$
$227$	2400.00i	0.701734i	−0.936425	+	0.350867i	$0.885887\pi$
$228$	0	0
$229$	2314.00	0.667744	0.333872	−	0.942618i	$-0.391645\pi$
$229$	2314.00	0.667744	0.333872	+	0.942618i	$0.391645\pi$
$230$	0	0
$231$	0	0
$232$	1008.00i	0.285252i
$233$	− 18.0000i	− 0.00506103i	−0.999997	−	0.00253051i	$-0.999195\pi$
$233$	− 18.0000i	− 0.00506103i	0.999997	−	0.00253051i	$-0.000805488\pi$
$234$	0	0
$235$	0	0
$236$	2136.00	0.589160
$237$	0	0
$238$	− 2184.00i	− 0.594822i
$239$	5868.00	1.58816	0.794078	−	0.607816i	$-0.207954\pi$
$239$	5868.00	1.58816	0.794078	+	0.607816i	$0.207954\pi$
$240$	0	0
$241$	−4330.00	−1.15734	−0.578672	−	0.815560i	$-0.696429\pi$
$241$	−4330.00	−1.15734	−0.578672	+	0.815560i	$0.696429\pi$
$242$	− 2590.00i	− 0.687981i
$243$	0	0
$244$	−1448.00	−0.379913
$245$	0	0
$246$	0	0
$247$	− 2992.00i	− 0.770755i
$248$	1952.00i	0.499807i
$249$	0	0
$250$	0	0
$251$	498.000	0.125233	0.0626165	−	0.998038i	$-0.480056\pi$
$251$	498.000	0.125233	0.0626165	+	0.998038i	$0.480056\pi$
$252$	0	0
$253$	720.000i	0.178917i
$254$	−3812.00	−0.941678
$255$	0	0
$256$	256.000	0.0625000
$257$	− 642.000i	− 0.155824i	−0.996960	−	0.0779122i	$-0.975175\pi$
$257$	− 642.000i	− 0.155824i	0.996960	−	0.0779122i	$-0.0248254\pi$
$258$	0	0
$259$	4256.00	1.02106
$260$	0	0
$261$	0	0
$262$	5748.00i	1.35539i
$263$	− 7968.00i	− 1.86817i	−0.357055	−	0.934084i	$-0.616219\pi$
$263$	− 7968.00i	− 1.86817i	0.357055	−	0.934084i	$-0.383781\pi$
$264$	0	0
$265$	0	0
$266$	−1232.00	−0.283980
$267$	0	0
$268$	− 1072.00i	− 0.244339i
$269$	4218.00	0.956045	0.478022	−	0.878348i	$-0.341354\pi$
$269$	4218.00	0.956045	0.478022	+	0.878348i	$0.341354\pi$
$270$	0	0
$271$	848.000	0.190082	0.0950412	−	0.995473i	$-0.469702\pi$
$271$	848.000	0.190082	0.0950412	+	0.995473i	$0.469702\pi$
$272$	− 1248.00i	− 0.278203i
$273$	0	0
$274$	−1596.00	−0.351890
$275$	0	0
$276$	0	0
$277$	1504.00i	0.326233i	0.986607	+	0.163117i	$0.0521547\pi$
$277$	1504.00i	0.326233i	−0.986607	+	0.163117i	$0.947845\pi$
$278$	1400.00i	0.302037i
$279$	0	0
$280$	0	0
$281$	−1308.00	−0.277682	−0.138841	−	0.990315i	$-0.544338\pi$
$281$	−1308.00	−0.277682	−0.138841	+	0.990315i	$0.544338\pi$
$282$	0	0
$283$	− 5932.00i	− 1.24601i	−0.782218	−	0.623005i	$-0.785912\pi$
$283$	− 5932.00i	− 1.24601i	0.782218	−	0.623005i	$-0.214088\pi$
$284$	3888.00	0.812360
$285$	0	0
$286$	−816.000	−0.168710
$287$	6720.00i	1.38212i
$288$	0	0
$289$	−1171.00	−0.238347
$290$	0	0
$291$	0	0
$292$	− 1880.00i	− 0.376776i
$293$	5226.00i	1.04200i	0.853556	+	0.521000i	$0.174441\pi$
$293$	5226.00i	1.04200i	−0.853556	+	0.521000i	$0.825559\pi$
$294$	0	0
$295$	0	0
$296$	2432.00	0.477558
$297$	0	0
$298$	− 228.000i	− 0.0443211i
$299$	−8160.00	−1.57828
$300$	0	0
$301$	1456.00	0.278812
$302$	2128.00i	0.405472i
$303$	0	0
$304$	−704.000	−0.132820
$305$	0	0
$306$	0	0
$307$	− 4448.00i	− 0.826908i	−0.910525	−	0.413454i	$-0.864322\pi$
$307$	− 4448.00i	− 0.826908i	0.910525	−	0.413454i	$-0.135678\pi$
$308$	336.000i	0.0621603i
$309$	0	0
$310$	0	0
$311$	−9132.00	−1.66504	−0.832521	−	0.553993i	$-0.813103\pi$
$311$	−9132.00	−1.66504	−0.832521	+	0.553993i	$0.813103\pi$
$312$	0	0
$313$	− 2170.00i	− 0.391871i	−0.980617	−	0.195936i	$-0.937226\pi$
$313$	− 2170.00i	− 0.391871i	0.980617	−	0.195936i	$-0.0627743\pi$
$314$	−3896.00	−0.700204
$315$	0	0
$316$	4976.00	0.885829
$317$	− 7674.00i	− 1.35967i	−0.733366	−	0.679834i	$-0.762052\pi$
$317$	− 7674.00i	− 1.35967i	0.733366	−	0.679834i	$-0.237948\pi$
$318$	0	0
$319$	−756.000	−0.132689
$320$	0	0
$321$	0	0
$322$	3360.00i	0.581508i
$323$	3432.00i	0.591212i
$324$	0	0
$325$	0	0
$326$	−4120.00	−0.699956
$327$	0	0
$328$	3840.00i	0.646428i
$329$	−8400.00	−1.40762
$330$	0	0
$331$	9596.00	1.59349	0.796743	−	0.604318i	$-0.206554\pi$
$331$	9596.00	1.59349	0.796743	+	0.604318i	$0.206554\pi$
$332$	− 1584.00i	− 0.261847i
$333$	0	0
$334$	−2496.00	−0.408907
$335$	0	0
$336$	0	0
$337$	− 12158.0i	− 1.96525i	−0.185608	−	0.982624i	$-0.559426\pi$
$337$	− 12158.0i	− 1.96525i	0.185608	−	0.982624i	$-0.440574\pi$
$338$	− 4854.00i	− 0.781133i
$339$	0	0
$340$	0	0
$341$	−1464.00	−0.232493
$342$	0	0
$343$	− 6860.00i	− 1.07990i
$344$	832.000	0.130402
$345$	0	0
$346$	2292.00	0.356123
$347$	− 10320.0i	− 1.59656i	−0.602286	−	0.798280i	$-0.705743\pi$
$347$	− 10320.0i	− 1.59656i	0.602286	−	0.798280i	$-0.294257\pi$
$348$	0	0
$349$	2158.00	0.330989	0.165494	−	0.986211i	$-0.447078\pi$
$349$	2158.00	0.330989	0.165494	+	0.986211i	$0.447078\pi$
$350$	0	0
$351$	0	0
$352$	192.000i	0.0290728i
$353$	− 330.000i	− 0.0497567i	−0.999690	−	0.0248784i	$-0.992080\pi$
$353$	− 330.000i	− 0.0497567i	0.999690	−	0.0248784i	$-0.00791985\pi$
$354$	0	0
$355$	0	0
$356$	−3888.00	−0.578830
$357$	0	0
$358$	2292.00i	0.338369i
$359$	8664.00	1.27373	0.636864	−	0.770976i	$-0.280231\pi$
$359$	8664.00	1.27373	0.636864	+	0.770976i	$0.280231\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	− 236.000i	− 0.0342649i
$363$	0	0
$364$	−3808.00	−0.548334
$365$	0	0
$366$	0	0
$367$	− 3782.00i	− 0.537926i	−0.963151	−	0.268963i	$-0.913319\pi$
$367$	− 3782.00i	− 0.537926i	0.963151	−	0.268963i	$-0.0866809\pi$
$368$	1920.00i	0.271975i
$369$	0	0
$370$	0	0
$371$	−3612.00	−0.505460
$372$	0	0
$373$	11276.0i	1.56528i	0.622475	+	0.782640i	$0.286127\pi$
$373$	11276.0i	1.56528i	−0.622475	+	0.782640i	$0.713873\pi$
$374$	936.000	0.129410
$375$	0	0
$376$	−4800.00	−0.658354
$377$	− 8568.00i	− 1.17049i
$378$	0	0
$379$	−980.000	−0.132821	−0.0664106	−	0.997792i	$-0.521155\pi$
$379$	−980.000	−0.132821	−0.0664106	+	0.997792i	$0.521155\pi$
$380$	0	0
$381$	0	0
$382$	3384.00i	0.453247i
$383$	− 4200.00i	− 0.560339i	−0.959950	−	0.280170i	$-0.909609\pi$
$383$	− 4200.00i	− 0.560339i	0.959950	−	0.280170i	$-0.0903907\pi$
$384$	0	0
$385$	0	0
$386$	−6700.00	−0.883474
$387$	0	0
$388$	− 184.000i	− 0.0240752i
$389$	−13338.0	−1.73847	−0.869233	−	0.494402i	$-0.835387\pi$
$389$	−13338.0	−1.73847	−0.869233	+	0.494402i	$0.835387\pi$
$390$	0	0
$391$	9360.00	1.21063
$392$	− 1176.00i	− 0.151523i
$393$	0	0
$394$	−7212.00	−0.922171
$395$	0	0
$396$	0	0
$397$	7192.00i	0.909209i	0.890693	+	0.454605i	$0.150219\pi$
$397$	7192.00i	0.909209i	−0.890693	+	0.454605i	$0.849781\pi$
$398$	− 5392.00i	− 0.679087i
$399$	0	0
$400$	0	0
$401$	−2316.00	−0.288418	−0.144209	−	0.989547i	$-0.546064\pi$
$401$	−2316.00	−0.288418	−0.144209	+	0.989547i	$0.546064\pi$
$402$	0	0
$403$	− 16592.0i	− 2.05088i
$404$	6024.00	0.741845
$405$	0	0
$406$	−3528.00	−0.431260
$407$	1824.00i	0.222143i
$408$	0	0
$409$	12358.0	1.49404	0.747022	−	0.664800i	$-0.231483\pi$
$409$	12358.0	1.49404	0.747022	+	0.664800i	$0.231483\pi$
$410$	0	0
$411$	0	0
$412$	5896.00i	0.705037i
$413$	7476.00i	0.890726i
$414$	0	0
$415$	0	0
$416$	−2176.00	−0.256460
$417$	0	0
$418$	− 528.000i	− 0.0617831i
$419$	−3306.00	−0.385462	−0.192731	−	0.981252i	$-0.561735\pi$
$419$	−3306.00	−0.385462	−0.192731	+	0.981252i	$0.561735\pi$
$420$	0	0
$421$	−14506.0	−1.67929	−0.839643	−	0.543139i	$-0.817236\pi$
$421$	−14506.0	−1.67929	−0.839643	+	0.543139i	$0.817236\pi$
$422$	− 8.00000i	0 0.000922829i
$423$	0	0
$424$	−2064.00	−0.236407
$425$	0	0
$426$	0	0
$427$	− 5068.00i	− 0.574374i
$428$	3696.00i	0.417413i
$429$	0	0
$430$	0	0
$431$	−6480.00	−0.724201	−0.362100	−	0.932139i	$-0.617940\pi$
$431$	−6480.00	−0.724201	−0.362100	+	0.932139i	$0.617940\pi$
$432$	0	0
$433$	11894.0i	1.32007i	0.751236	+	0.660034i	$0.229458\pi$
$433$	11894.0i	1.32007i	−0.751236	+	0.660034i	$0.770542\pi$
$434$	−6832.00	−0.755637
$435$	0	0
$436$	2792.00	0.306680
$437$	− 5280.00i	− 0.577979i
$438$	0	0
$439$	12688.0	1.37942	0.689710	−	0.724086i	$-0.257738\pi$
$439$	12688.0	1.37942	0.689710	+	0.724086i	$0.257738\pi$
$440$	0	0
$441$	0	0
$442$	10608.0i	1.14156i
$443$	4968.00i	0.532814i	0.963861	+	0.266407i	$0.0858366\pi$
$443$	4968.00i	0.532814i	−0.963861	+	0.266407i	$0.914163\pi$
$444$	0	0
$445$	0	0
$446$	2324.00	0.246737
$447$	0	0
$448$	896.000i	0.0944911i
$449$	11508.0	1.20957	0.604784	−	0.796389i	$-0.293259\pi$
$449$	11508.0	1.20957	0.604784	+	0.796389i	$0.293259\pi$
$450$	0	0
$451$	−2880.00	−0.300696
$452$	888.000i	0.0924071i
$453$	0	0
$454$	−4800.00	−0.496201
$455$	0	0
$456$	0	0
$457$	− 1082.00i	− 0.110752i	−0.998466	−	0.0553762i	$-0.982364\pi$
$457$	− 1082.00i	− 0.110752i	0.998466	−	0.0553762i	$-0.0176358\pi$
$458$	4628.00i	0.472166i
$459$	0	0
$460$	0	0
$461$	−11238.0	−1.13537	−0.567685	−	0.823246i	$-0.692161\pi$
$461$	−11238.0	−1.13537	−0.567685	+	0.823246i	$0.692161\pi$
$462$	0	0
$463$	− 2302.00i	− 0.231065i	−0.993304	−	0.115532i	$-0.963143\pi$
$463$	− 2302.00i	− 0.231065i	0.993304	−	0.115532i	$-0.0368574\pi$
$464$	−2016.00	−0.201704
$465$	0	0
$466$	36.0000	0.00357869
$467$	− 15876.0i	− 1.57313i	−0.617505	−	0.786567i	$-0.711856\pi$
$467$	− 15876.0i	− 1.57313i	0.617505	−	0.786567i	$-0.288144\pi$
$468$	0	0
$469$	3752.00	0.369406
$470$	0	0
$471$	0	0
$472$	4272.00i	0.416599i
$473$	624.000i	0.0606587i
$474$	0	0
$475$	0	0
$476$	4368.00	0.420603
$477$	0	0
$478$	11736.0i	1.12300i
$479$	−4644.00	−0.442985	−0.221492	−	0.975162i	$-0.571093\pi$
$479$	−4644.00	−0.442985	−0.221492	+	0.975162i	$0.571093\pi$
$480$	0	0
$481$	−20672.0	−1.95959
$482$	− 8660.00i	− 0.818366i
$483$	0	0
$484$	5180.00	0.486476
$485$	0	0
$486$	0	0
$487$	− 2426.00i	− 0.225734i	−0.993610	−	0.112867i	$-0.963997\pi$
$487$	− 2426.00i	− 0.225734i	0.993610	−	0.112867i	$-0.0360034\pi$
$488$	− 2896.00i	− 0.268639i
$489$	0	0
$490$	0	0
$491$	234.000	0.0215077	0.0107538	−	0.999942i	$-0.496577\pi$
$491$	234.000	0.0215077	0.0107538	+	0.999942i	$0.496577\pi$
$492$	0	0
$493$	9828.00i	0.897831i
$494$	5984.00	0.545006
$495$	0	0
$496$	−3904.00	−0.353417
$497$	13608.0i	1.22817i
$498$	0	0
$499$	−14204.0	−1.27427	−0.637133	−	0.770754i	$-0.719880\pi$
$499$	−14204.0	−1.27427	−0.637133	+	0.770754i	$0.719880\pi$
$500$	0	0
$501$	0	0
$502$	996.000i	0.0885531i
$503$	4920.00i	0.436127i	0.975935	+	0.218064i	$0.0699740\pi$
$503$	4920.00i	0.436127i	−0.975935	+	0.218064i	$0.930026\pi$
$504$	0	0
$505$	0	0
$506$	−1440.00	−0.126513
$507$	0	0
$508$	− 7624.00i	− 0.665867i
$509$	4458.00	0.388207	0.194104	−	0.980981i	$-0.437820\pi$
$509$	4458.00	0.388207	0.194104	+	0.980981i	$0.437820\pi$
$510$	0	0
$511$	6580.00	0.569632
$512$	512.000i	0.0441942i
$513$	0	0
$514$	1284.00	0.110184
$515$	0	0
$516$	0	0
$517$	− 3600.00i	− 0.306243i
$518$	8512.00i	0.722000i
$519$	0	0
$520$	0	0
$521$	4212.00	0.354186	0.177093	−	0.984194i	$-0.443331\pi$
$521$	4212.00	0.354186	0.177093	+	0.984194i	$0.443331\pi$
$522$	0	0
$523$	− 11212.0i	− 0.937412i	−0.883354	−	0.468706i	$-0.844720\pi$
$523$	− 11212.0i	− 0.937412i	0.883354	−	0.468706i	$-0.155280\pi$
$524$	−11496.0	−0.958407
$525$	0	0
$526$	15936.0	1.32099
$527$	19032.0i	1.57314i
$528$	0	0
$529$	−2233.00	−0.183529
$530$	0	0
$531$	0	0
$532$	− 2464.00i	− 0.200804i
$533$	− 32640.0i	− 2.65252i
$534$	0	0
$535$	0	0
$536$	2144.00	0.172774
$537$	0	0
$538$	8436.00i	0.676026i
$539$	882.000	0.0704832
$540$	0	0
$541$	14018.0	1.11401	0.557006	−	0.830508i	$-0.311950\pi$
$541$	14018.0	1.11401	0.557006	+	0.830508i	$0.311950\pi$
$542$	1696.00i	0.134409i
$543$	0	0
$544$	2496.00	0.196719
$545$	0	0
$546$	0	0
$547$	− 18200.0i	− 1.42262i	−0.702876	−	0.711312i	$-0.748101\pi$
$547$	− 18200.0i	− 1.42262i	0.702876	−	0.711312i	$-0.251899\pi$
$548$	− 3192.00i	− 0.248824i
$549$	0	0
$550$	0	0
$551$	5544.00	0.428643
$552$	0	0
$553$	17416.0i	1.33925i
$554$	−3008.00	−0.230682
$555$	0	0
$556$	−2800.00	−0.213573
$557$	11826.0i	0.899612i	0.893126	+	0.449806i	$0.148507\pi$
$557$	11826.0i	0.899612i	−0.893126	+	0.449806i	$0.851493\pi$
$558$	0	0
$559$	−7072.00	−0.535087
$560$	0	0
$561$	0	0
$562$	− 2616.00i	− 0.196351i
$563$	− 2952.00i	− 0.220980i	−0.993877	−	0.110490i	$-0.964758\pi$
$563$	− 2952.00i	− 0.220980i	0.993877	−	0.110490i	$-0.0352421\pi$
$564$	0	0
$565$	0	0
$566$	11864.0	0.881062
$567$	0	0
$568$	7776.00i	0.574426i
$569$	−3084.00	−0.227220	−0.113610	−	0.993525i	$-0.536241\pi$
$569$	−3084.00	−0.227220	−0.113610	+	0.993525i	$0.536241\pi$
$570$	0	0
$571$	−4756.00	−0.348568	−0.174284	−	0.984695i	$-0.555761\pi$
$571$	−4756.00	−0.348568	−0.174284	+	0.984695i	$0.555761\pi$
$572$	− 1632.00i	− 0.119296i
$573$	0	0
$574$	−13440.0	−0.977308
$575$	0	0
$576$	0	0
$577$	11014.0i	0.794660i	0.917676	+	0.397330i	$0.130063\pi$
$577$	11014.0i	0.794660i	−0.917676	+	0.397330i	$0.869937\pi$
$578$	− 2342.00i	− 0.168537i
$579$	0	0
$580$	0	0
$581$	5544.00	0.395876
$582$	0	0
$583$	− 1548.00i	− 0.109968i
$584$	3760.00	0.266421
$585$	0	0
$586$	−10452.0	−0.736806
$587$	852.000i	0.0599077i	0.999551	+	0.0299538i	$0.00953603\pi$
$587$	852.000i	0.0599077i	−0.999551	+	0.0299538i	$0.990464\pi$
$588$	0	0
$589$	10736.0	0.751051
$590$	0	0
$591$	0	0
$592$	4864.00i	0.337684i
$593$	15546.0i	1.07656i	0.842767	+	0.538278i	$0.180925\pi$
$593$	15546.0i	1.07656i	−0.842767	+	0.538278i	$0.819075\pi$
$594$	0	0
$595$	0	0
$596$	456.000	0.0313397
$597$	0	0
$598$	− 16320.0i	− 1.11601i
$599$	8616.00	0.587713	0.293857	−	0.955850i	$-0.405061\pi$
$599$	8616.00	0.587713	0.293857	+	0.955850i	$0.405061\pi$
$600$	0	0
$601$	17510.0	1.18843	0.594216	−	0.804305i	$-0.297462\pi$
$601$	17510.0	1.18843	0.594216	+	0.804305i	$0.297462\pi$
$602$	2912.00i	0.197150i
$603$	0	0
$604$	−4256.00	−0.286712
$605$	0	0
$606$	0	0
$607$	13894.0i	0.929061i	0.885557	+	0.464531i	$0.153777\pi$
$607$	13894.0i	0.929061i	−0.885557	+	0.464531i	$0.846223\pi$
$608$	− 1408.00i	− 0.0939177i
$609$	0	0
$610$	0	0
$611$	40800.0	2.70146
$612$	0	0
$613$	− 6496.00i	− 0.428011i	−0.976832	−	0.214006i	$-0.931349\pi$
$613$	− 6496.00i	− 0.428011i	0.976832	−	0.214006i	$-0.0686511\pi$
$614$	8896.00	0.584712
$615$	0	0
$616$	−672.000	−0.0439540
$617$	570.000i	0.0371918i	0.999827	+	0.0185959i	$0.00591960\pi$
$617$	570.000i	0.0371918i	−0.999827	+	0.0185959i	$0.994080\pi$
$618$	0	0
$619$	2140.00	0.138956	0.0694781	−	0.997583i	$-0.477867\pi$
$619$	2140.00	0.138956	0.0694781	+	0.997583i	$0.477867\pi$
$620$	0	0
$621$	0	0
$622$	− 18264.0i	− 1.17736i
$623$	− 13608.0i	− 0.875109i
$624$	0	0
$625$	0	0
$626$	4340.00	0.277095
$627$	0	0
$628$	− 7792.00i	− 0.495119i
$629$	23712.0	1.50312
$630$	0	0
$631$	14660.0	0.924890	0.462445	−	0.886648i	$-0.346972\pi$
$631$	14660.0	0.924890	0.462445	+	0.886648i	$0.346972\pi$
$632$	9952.00i	0.626375i
$633$	0	0
$634$	15348.0	0.961431
$635$	0	0
$636$	0	0
$637$	9996.00i	0.621752i
$638$	− 1512.00i	− 0.0938255i
$639$	0	0
$640$	0	0
$641$	−456.000	−0.0280982	−0.0140491	−	0.999901i	$-0.504472\pi$
$641$	−456.000	−0.0280982	−0.0140491	+	0.999901i	$0.504472\pi$
$642$	0	0
$643$	− 23452.0i	− 1.43835i	−0.694831	−	0.719173i	$-0.744521\pi$
$643$	− 23452.0i	− 1.43835i	0.694831	−	0.719173i	$-0.255479\pi$
$644$	−6720.00	−0.411188
$645$	0	0
$646$	−6864.00	−0.418050
$647$	− 7224.00i	− 0.438956i	−0.975617	−	0.219478i	$-0.929565\pi$
$647$	− 7224.00i	− 0.438956i	0.975617	−	0.219478i	$-0.0704355\pi$
$648$	0	0
$649$	−3204.00	−0.193787
$650$	0	0
$651$	0	0
$652$	− 8240.00i	− 0.494944i
$653$	19146.0i	1.14738i	0.819072	+	0.573691i	$0.194489\pi$
$653$	19146.0i	1.14738i	−0.819072	+	0.573691i	$0.805511\pi$
$654$	0	0
$655$	0	0
$656$	−7680.00	−0.457094
$657$	0	0
$658$	− 16800.0i	− 0.995338i
$659$	−27810.0	−1.64389	−0.821945	−	0.569567i	$-0.807111\pi$
$659$	−27810.0	−1.64389	−0.821945	+	0.569567i	$0.807111\pi$
$660$	0	0
$661$	−30598.0	−1.80049	−0.900245	−	0.435383i	$-0.856613\pi$
$661$	−30598.0	−1.80049	−0.900245	+	0.435383i	$0.856613\pi$
$662$	19192.0i	1.12676i
$663$	0	0
$664$	3168.00	0.185154
$665$	0	0
$666$	0	0
$667$	− 15120.0i	− 0.877734i
$668$	− 4992.00i	− 0.289141i
$669$	0	0
$670$	0	0
$671$	2172.00	0.124961
$672$	0	0
$673$	− 3778.00i	− 0.216391i	−0.994130	−	0.108196i	$-0.965493\pi$
$673$	− 3778.00i	− 0.216391i	0.994130	−	0.108196i	$-0.0345073\pi$
$674$	24316.0	1.38964
$675$	0	0
$676$	9708.00	0.552344
$677$	27198.0i	1.54402i	0.635608	+	0.772012i	$0.280749\pi$
$677$	27198.0i	1.54402i	−0.635608	+	0.772012i	$0.719251\pi$
$678$	0	0
$679$	644.000	0.0363983
$680$	0	0
$681$	0	0
$682$	− 2928.00i	− 0.164397i
$683$	32316.0i	1.81045i	0.424933	+	0.905225i	$0.360298\pi$
$683$	32316.0i	1.81045i	−0.424933	+	0.905225i	$0.639702\pi$
$684$	0	0
$685$	0	0
$686$	13720.0	0.763604
$687$	0	0
$688$	1664.00i	0.0922084i
$689$	17544.0	0.970063
$690$	0	0
$691$	29324.0	1.61438	0.807191	−	0.590291i	$-0.200987\pi$
$691$	29324.0	1.61438	0.807191	+	0.590291i	$0.200987\pi$
$692$	4584.00i	0.251817i
$693$	0	0
$694$	20640.0	1.12894
$695$	0	0
$696$	0	0
$697$	37440.0i	2.03464i
$698$	4316.00i	0.234044i
$699$	0	0
$700$	0	0
$701$	22782.0	1.22748	0.613741	−	0.789508i	$-0.289664\pi$
$701$	22782.0	1.22748	0.613741	+	0.789508i	$0.289664\pi$
$702$	0	0
$703$	− 13376.0i	− 0.717618i
$704$	−384.000	−0.0205576
$705$	0	0
$706$	660.000	0.0351833
$707$	21084.0i	1.12156i
$708$	0	0
$709$	−26054.0	−1.38008	−0.690041	−	0.723770i	$-0.742408\pi$
$709$	−26054.0	−1.38008	−0.690041	+	0.723770i	$0.742408\pi$
$710$	0	0
$711$	0	0
$712$	− 7776.00i	− 0.409295i
$713$	− 29280.0i	− 1.53793i
$714$	0	0
$715$	0	0
$716$	−4584.00	−0.239263
$717$	0	0
$718$	17328.0i	0.900662i
$719$	5976.00	0.309968	0.154984	−	0.987917i	$-0.450467\pi$
$719$	5976.00	0.309968	0.154984	+	0.987917i	$0.450467\pi$
$720$	0	0
$721$	−20636.0	−1.06592
$722$	− 9846.00i	− 0.507521i
$723$	0	0
$724$	472.000	0.0242289
$725$	0	0
$726$	0	0
$727$	5110.00i	0.260687i	0.991469	+	0.130343i	$0.0416080\pi$
$727$	5110.00i	0.260687i	−0.991469	+	0.130343i	$0.958392\pi$
$728$	− 7616.00i	− 0.387730i
$729$	0	0
$730$	0	0
$731$	8112.00	0.410442
$732$	0	0
$733$	17336.0i	0.873560i	0.899568	+	0.436780i	$0.143881\pi$
$733$	17336.0i	0.873560i	−0.899568	+	0.436780i	$0.856119\pi$
$734$	7564.00	0.380371
$735$	0	0
$736$	−3840.00	−0.192316
$737$	1608.00i	0.0803683i
$738$	0	0
$739$	13660.0	0.679961	0.339981	−	0.940432i	$-0.389580\pi$
$739$	13660.0	0.679961	0.339981	+	0.940432i	$0.389580\pi$
$740$	0	0
$741$	0	0
$742$	− 7224.00i	− 0.357414i
$743$	1320.00i	0.0651765i	0.999469	+	0.0325882i	$0.0103750\pi$
$743$	1320.00i	0.0651765i	−0.999469	+	0.0325882i	$0.989625\pi$
$744$	0	0
$745$	0	0
$746$	−22552.0	−1.10682
$747$	0	0
$748$	1872.00i	0.0915068i
$749$	−12936.0	−0.631070
$750$	0	0
$751$	15860.0	0.770625	0.385313	−	0.922786i	$-0.374094\pi$
$751$	15860.0	0.770625	0.385313	+	0.922786i	$0.374094\pi$
$752$	− 9600.00i	− 0.465527i
$753$	0	0
$754$	17136.0	0.827661
$755$	0	0
$756$	0	0
$757$	− 22160.0i	− 1.06396i	−0.846756	−	0.531981i	$-0.821448\pi$
$757$	− 22160.0i	− 1.06396i	0.846756	−	0.531981i	$-0.178552\pi$
$758$	− 1960.00i	− 0.0939187i
$759$	0	0
$760$	0	0
$761$	13116.0	0.624776	0.312388	−	0.949955i	$-0.398871\pi$
$761$	13116.0	0.624776	0.312388	+	0.949955i	$0.398871\pi$
$762$	0	0
$763$	9772.00i	0.463657i
$764$	−6768.00	−0.320494
$765$	0	0
$766$	8400.00	0.396220
$767$	− 36312.0i	− 1.70945i
$768$	0	0
$769$	−32846.0	−1.54026	−0.770128	−	0.637889i	$-0.779808\pi$
$769$	−32846.0	−1.54026	−0.770128	+	0.637889i	$0.779808\pi$
$770$	0	0
$771$	0	0
$772$	− 13400.0i	− 0.624711i
$773$	− 11982.0i	− 0.557520i	−0.960361	−	0.278760i	$-0.910077\pi$
$773$	− 11982.0i	− 0.557520i	0.960361	−	0.278760i	$-0.0899233\pi$
$774$	0	0
$775$	0	0
$776$	368.000	0.0170238
$777$	0	0
$778$	− 26676.0i	− 1.22928i
$779$	21120.0	0.971377
$780$	0	0
$781$	−5832.00	−0.267203
$782$	18720.0i	0.856043i
$783$	0	0
$784$	2352.00	0.107143
$785$	0	0
$786$	0	0
$787$	21076.0i	0.954610i	0.878738	+	0.477305i	$0.158386\pi$
$787$	21076.0i	0.954610i	−0.878738	+	0.477305i	$0.841614\pi$
$788$	− 14424.0i	− 0.652073i
$789$	0	0
$790$	0	0
$791$	−3108.00	−0.139706
$792$	0	0
$793$	24616.0i	1.10232i
$794$	−14384.0	−0.642908
$795$	0	0
$796$	10784.0	0.480187
$797$	− 22086.0i	− 0.981589i	−0.871275	−	0.490794i	$-0.836707\pi$
$797$	− 22086.0i	− 0.981589i	0.871275	−	0.490794i	$-0.163293\pi$
$798$	0	0
$799$	−46800.0	−2.07217
$800$	0	0
$801$	0	0
$802$	− 4632.00i	− 0.203942i
$803$	2820.00i	0.123930i
$804$	0	0
$805$	0	0
$806$	33184.0	1.45019
$807$	0	0
$808$	12048.0i	0.524563i
$809$	−21384.0	−0.929322	−0.464661	−	0.885489i	$-0.653824\pi$
$809$	−21384.0	−0.929322	−0.464661	+	0.885489i	$0.653824\pi$
$810$	0	0
$811$	5228.00	0.226362	0.113181	−	0.993574i	$-0.463896\pi$
$811$	5228.00	0.226362	0.113181	+	0.993574i	$0.463896\pi$
$812$	− 7056.00i	− 0.304947i
$813$	0	0
$814$	−3648.00	−0.157079
$815$	0	0
$816$	0	0
$817$	− 4576.00i	− 0.195953i
$818$	24716.0i	1.05645i
$819$	0	0
$820$	0	0
$821$	−38010.0	−1.61578	−0.807892	−	0.589331i	$-0.799391\pi$
$821$	−38010.0	−1.61578	−0.807892	+	0.589331i	$0.799391\pi$
$822$	0	0
$823$	38642.0i	1.63667i	0.574745	+	0.818333i	$0.305101\pi$
$823$	38642.0i	1.63667i	−0.574745	+	0.818333i	$0.694899\pi$
$824$	−11792.0	−0.498536
$825$	0	0
$826$	−14952.0	−0.629839
$827$	15432.0i	0.648879i	0.945906	+	0.324440i	$0.105176\pi$
$827$	15432.0i	0.648879i	−0.945906	+	0.324440i	$0.894824\pi$
$828$	0	0
$829$	3886.00	0.162806	0.0814031	−	0.996681i	$-0.474060\pi$
$829$	3886.00	0.162806	0.0814031	+	0.996681i	$0.474060\pi$
$830$	0	0
$831$	0	0
$832$	− 4352.00i	− 0.181344i
$833$	− 11466.0i	− 0.476919i
$834$	0	0
$835$	0	0
$836$	1056.00	0.0436872
$837$	0	0
$838$	− 6612.00i	− 0.272563i
$839$	−27552.0	−1.13373	−0.566866	−	0.823810i	$-0.691844\pi$
$839$	−27552.0	−1.13373	−0.566866	+	0.823810i	$0.691844\pi$
$840$	0	0
$841$	−8513.00	−0.349051
$842$	− 29012.0i	− 1.18743i
$843$	0	0
$844$	16.0000	0.000652539 0
$845$	0	0
$846$	0	0
$847$	18130.0i	0.735483i
$848$	− 4128.00i	− 0.167165i
$849$	0	0
$850$	0	0
$851$	−36480.0	−1.46947
$852$	0	0
$853$	15104.0i	0.606273i	0.952947	+	0.303137i	$0.0980339\pi$
$853$	15104.0i	0.606273i	−0.952947	+	0.303137i	$0.901966\pi$
$854$	10136.0	0.406144
$855$	0	0
$856$	−7392.00	−0.295156
$857$	12306.0i	0.490508i	0.969459	+	0.245254i	$0.0788713\pi$
$857$	12306.0i	0.490508i	−0.969459	+	0.245254i	$0.921129\pi$
$858$	0	0
$859$	47500.0	1.88670	0.943352	−	0.331793i	$-0.107654\pi$
$859$	47500.0	1.88670	0.943352	+	0.331793i	$0.107654\pi$
$860$	0	0
$861$	0	0
$862$	− 12960.0i	− 0.512087i
$863$	4272.00i	0.168506i	0.996444	+	0.0842529i	$0.0268504\pi$
$863$	4272.00i	0.168506i	−0.996444	+	0.0842529i	$0.973150\pi$
$864$	0	0
$865$	0	0
$866$	−23788.0	−0.933429
$867$	0	0
$868$	− 13664.0i	− 0.534316i
$869$	−7464.00	−0.291368
$870$	0	0
$871$	−18224.0	−0.708951
$872$	5584.00i	0.216856i
$873$	0	0
$874$	10560.0	0.408693
$875$	0	0
$876$	0	0
$877$	27796.0i	1.07024i	0.844775	+	0.535122i	$0.179734\pi$
$877$	27796.0i	1.07024i	−0.844775	+	0.535122i	$0.820266\pi$
$878$	25376.0i	0.975397i
$879$	0	0
$880$	0	0
$881$	−39996.0	−1.52951	−0.764756	−	0.644320i	$-0.777140\pi$
$881$	−39996.0	−1.52951	−0.764756	+	0.644320i	$0.777140\pi$
$882$	0	0
$883$	− 3772.00i	− 0.143758i	−0.997413	−	0.0718788i	$-0.977101\pi$
$883$	− 3772.00i	− 0.143758i	0.997413	−	0.0718788i	$-0.0228995\pi$
$884$	−21216.0	−0.807207
$885$	0	0
$886$	−9936.00	−0.376757
$887$	5784.00i	0.218949i	0.993990	+	0.109474i	$0.0349168\pi$
$887$	5784.00i	0.218949i	−0.993990	+	0.109474i	$0.965083\pi$
$888$	0	0
$889$	26684.0	1.00670
$890$	0	0
$891$	0	0
$892$	4648.00i	0.174469i
$893$	26400.0i	0.989297i
$894$	0	0
$895$	0	0
$896$	−1792.00	−0.0668153
$897$	0	0
$898$	23016.0i	0.855294i
$899$	30744.0	1.14057
$900$	0	0
$901$	−20124.0	−0.744093
$902$	− 5760.00i	− 0.212624i
$903$	0	0
$904$	−1776.00	−0.0653417
$905$	0	0
$906$	0	0
$907$	8440.00i	0.308981i	0.987994	+	0.154490i	$0.0493736\pi$
$907$	8440.00i	0.308981i	−0.987994	+	0.154490i	$0.950626\pi$
$908$	− 9600.00i	− 0.350867i
$909$	0	0
$910$	0	0
$911$	−31920.0	−1.16087	−0.580437	−	0.814305i	$-0.697118\pi$
$911$	−31920.0	−1.16087	−0.580437	+	0.814305i	$0.697118\pi$
$912$	0	0
$913$	2376.00i	0.0861272i
$914$	2164.00	0.0783137
$915$	0	0
$916$	−9256.00	−0.333872
$917$	− 40236.0i	− 1.44897i
$918$	0	0
$919$	−34652.0	−1.24381	−0.621906	−	0.783092i	$-0.713642\pi$
$919$	−34652.0	−1.24381	−0.621906	+	0.783092i	$0.713642\pi$
$920$	0	0
$921$	0	0
$922$	− 22476.0i	− 0.802828i
$923$	− 66096.0i	− 2.35707i
$924$	0	0
$925$	0	0
$926$	4604.00	0.163388
$927$	0	0
$928$	− 4032.00i	− 0.142626i
$929$	−1404.00	−0.0495842	−0.0247921	−	0.999693i	$-0.507892\pi$
$929$	−1404.00	−0.0495842	−0.0247921	+	0.999693i	$0.507892\pi$
$930$	0	0
$931$	−6468.00	−0.227691
$932$	72.0000i	0.00253051i
$933$	0	0
$934$	31752.0	1.11237
$935$	0	0
$936$	0	0
$937$	7654.00i	0.266857i	0.991058	+	0.133429i	$0.0425987\pi$
$937$	7654.00i	0.266857i	−0.991058	+	0.133429i	$0.957401\pi$
$938$	7504.00i	0.261209i
$939$	0	0
$940$	0	0
$941$	11298.0	0.391397	0.195698	−	0.980664i	$-0.437303\pi$
$941$	11298.0	0.391397	0.195698	+	0.980664i	$0.437303\pi$
$942$	0	0
$943$	− 57600.0i	− 1.98909i
$944$	−8544.00	−0.294580
$945$	0	0
$946$	−1248.00	−0.0428922
$947$	− 28968.0i	− 0.994016i	−0.867746	−	0.497008i	$-0.834432\pi$
$947$	− 28968.0i	− 0.994016i	0.867746	−	0.497008i	$-0.165568\pi$
$948$	0	0
$949$	−31960.0	−1.09322
$950$	0	0
$951$	0	0
$952$	8736.00i	0.297411i
$953$	46410.0i	1.57751i	0.614707	+	0.788755i	$0.289274\pi$
$953$	46410.0i	1.57751i	−0.614707	+	0.788755i	$0.710726\pi$
$954$	0	0
$955$	0	0
$956$	−23472.0	−0.794078
$957$	0	0
$958$	− 9288.00i	− 0.313238i
$959$	11172.0	0.376186
$960$	0	0
$961$	29745.0	0.998456
$962$	− 41344.0i	− 1.38564i
$963$	0	0
$964$	17320.0	0.578672
$965$	0	0
$966$	0	0
$967$	41506.0i	1.38029i	0.723670	+	0.690146i	$0.242454\pi$
$967$	41506.0i	1.38029i	−0.723670	+	0.690146i	$0.757546\pi$
$968$	10360.0i	0.343991i
$969$	0	0
$970$	0	0
$971$	−18246.0	−0.603030	−0.301515	−	0.953461i	$-0.597492\pi$
$971$	−18246.0	−0.603030	−0.301515	+	0.953461i	$0.597492\pi$
$972$	0	0
$973$	− 9800.00i	− 0.322892i
$974$	4852.00	0.159618
$975$	0	0
$976$	5792.00	0.189956
$977$	− 25998.0i	− 0.851330i	−0.904881	−	0.425665i	$-0.860040\pi$
$977$	− 25998.0i	− 0.851330i	0.904881	−	0.425665i	$-0.139960\pi$
$978$	0	0
$979$	5832.00	0.190390
$980$	0	0
$981$	0	0
$982$	468.000i	0.0152082i
$983$	14616.0i	0.474240i	0.971480	+	0.237120i	$0.0762035\pi$
$983$	14616.0i	0.474240i	−0.971480	+	0.237120i	$0.923797\pi$
$984$	0	0
$985$	0	0
$986$	−19656.0	−0.634863
$987$	0	0
$988$	11968.0i	0.385377i
$989$	−12480.0	−0.401255
$990$	0	0
$991$	−2968.00	−0.0951379	−0.0475689	−	0.998868i	$-0.515147\pi$
$991$	−2968.00	−0.0951379	−0.0475689	+	0.998868i	$0.515147\pi$
$992$	− 7808.00i	− 0.249903i
$993$	0	0
$994$	−27216.0	−0.868450
$995$	0	0
$996$	0	0
$997$	9052.00i	0.287542i	0.989611	+	0.143771i	$0.0459229\pi$
$997$	9052.00i	0.287542i	−0.989611	+	0.143771i	$0.954077\pi$
$998$	− 28408.0i	− 0.901042i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.c.g.199.2		2
3.2	odd	2		450.4.c.f.199.1		2
5.2	odd	4		90.4.a.b.1.1	✓	1
5.3	odd	4		450.4.a.m.1.1		1
5.4	even	2	inner	450.4.c.g.199.1		2
15.2	even	4		90.4.a.e.1.1	yes	1
15.8	even	4		450.4.a.c.1.1		1
15.14	odd	2		450.4.c.f.199.2		2
20.7	even	4		720.4.a.e.1.1		1
45.2	even	12		810.4.e.a.271.1		2
45.7	odd	12		810.4.e.u.271.1		2
45.22	odd	12		810.4.e.u.541.1		2
45.32	even	12		810.4.e.a.541.1		2
60.47	odd	4		720.4.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.4.a.b.1.1	✓	1	5.2	odd	4
90.4.a.e.1.1	yes	1	15.2	even	4
450.4.a.c.1.1		1	15.8	even	4
450.4.a.m.1.1		1	5.3	odd	4
450.4.c.f.199.1		2	3.2	odd	2
450.4.c.f.199.2		2	15.14	odd	2
450.4.c.g.199.1		2	5.4	even	2	inner
450.4.c.g.199.2		2	1.1	even	1	trivial
720.4.a.e.1.1		1	20.7	even	4
720.4.a.t.1.1		1	60.47	odd	4
810.4.e.a.271.1		2	45.2	even	12
810.4.e.a.541.1		2	45.32	even	12
810.4.e.u.271.1		2	45.7	odd	12
810.4.e.u.541.1		2	45.22	odd	12

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} -14.0000i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} -14.0000i q^{7} -8.00000i q^{8} +6.00000 q^{11} +68.0000i q^{13} +28.0000 q^{14} +16.0000 q^{16} -78.0000i q^{17} -44.0000 q^{19} +12.0000i q^{22} +120.000i q^{23} -136.000 q^{26} +56.0000i q^{28} -126.000 q^{29} -244.000 q^{31} +32.0000i q^{32} +156.000 q^{34} +304.000i q^{37} -88.0000i q^{38} -480.000 q^{41} +104.000i q^{43} -24.0000 q^{44} -240.000 q^{46} -600.000i q^{47} +147.000 q^{49} -272.000i q^{52} -258.000i q^{53} -112.000 q^{56} -252.000i q^{58} -534.000 q^{59} +362.000 q^{61} -488.000i q^{62} -64.0000 q^{64} +268.000i q^{67} +312.000i q^{68} -972.000 q^{71} +470.000i q^{73} -608.000 q^{74} +176.000 q^{76} -84.0000i q^{77} -1244.00 q^{79} -960.000i q^{82} +396.000i q^{83} -208.000 q^{86} -48.0000i q^{88} +972.000 q^{89} +952.000 q^{91} -480.000i q^{92} +1200.00 q^{94} +46.0000i q^{97} +294.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} + 12 q^{11} + 56 q^{14} + 32 q^{16} - 88 q^{19} - 272 q^{26} - 252 q^{29} - 488 q^{31} + 312 q^{34} - 960 q^{41} - 48 q^{44} - 480 q^{46} + 294 q^{49} - 224 q^{56} - 1068 q^{59} + 724 q^{61} - 128 q^{64} - 1944 q^{71} - 1216 q^{74} + 352 q^{76} - 2488 q^{79} - 416 q^{86} + 1944 q^{89} + 1904 q^{91} + 2400 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 + 12 * q^11 + 56 * q^14 + 32 * q^16 - 88 * q^19 - 272 * q^26 - 252 * q^29 - 488 * q^31 + 312 * q^34 - 960 * q^41 - 48 * q^44 - 480 * q^46 + 294 * q^49 - 224 * q^56 - 1068 * q^59 + 724 * q^61 - 128 * q^64 - 1944 * q^71 - 1216 * q^74 + 352 * q^76 - 2488 * q^79 - 416 * q^86 + 1944 * q^89 + 1904 * q^91 + 2400 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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