LMFDB - Embedded newform 450.6.c.f.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,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, 6, names="a")

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

chi := DirichletCharacter("450.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.6.c.f.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -16.0000 q^{4} -118.000i q^{7} +64.0000i q^{8} +O(q^{10})$	$q-4.00000i q^{2} -16.0000 q^{4} -118.000i q^{7} +64.0000i q^{8} -192.000 q^{11} -1106.00i q^{13} -472.000 q^{14} +256.000 q^{16} -762.000i q^{17} +2740.00 q^{19} +768.000i q^{22} +1566.00i q^{23} -4424.00 q^{26} +1888.00i q^{28} +5910.00 q^{29} -6868.00 q^{31} -1024.00i q^{32} -3048.00 q^{34} -5518.00i q^{37} -10960.0i q^{38} +378.000 q^{41} +2434.00i q^{43} +3072.00 q^{44} +6264.00 q^{46} -13122.0i q^{47} +2883.00 q^{49} +17696.0i q^{52} -9174.00i q^{53} +7552.00 q^{56} -23640.0i q^{58} -34980.0 q^{59} -9838.00 q^{61} +27472.0i q^{62} -4096.00 q^{64} +33722.0i q^{67} +12192.0i q^{68} -70212.0 q^{71} -21986.0i q^{73} -22072.0 q^{74} -43840.0 q^{76} +22656.0i q^{77} -4520.00 q^{79} -1512.00i q^{82} -109074. i q^{83} +9736.00 q^{86} -12288.0i q^{88} +38490.0 q^{89} -130508. q^{91} -25056.0i q^{92} -52488.0 q^{94} -1918.00i q^{97} -11532.0i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4}+O(q^{10}) $ 2 * q - 32 * q^4	$ 2 q - 32 q^{4} - 384 q^{11} - 944 q^{14} + 512 q^{16} + 5480 q^{19} - 8848 q^{26} + 11820 q^{29} - 13736 q^{31} - 6096 q^{34} + 756 q^{41} + 6144 q^{44} + 12528 q^{46} + 5766 q^{49} + 15104 q^{56} - 69960 q^{59} - 19676 q^{61} - 8192 q^{64} - 140424 q^{71} - 44144 q^{74} - 87680 q^{76} - 9040 q^{79} + 19472 q^{86} + 76980 q^{89} - 261016 q^{91} - 104976 q^{94}+O(q^{100}) $ 2 * q - 32 * q^4 - 384 * q^11 - 944 * q^14 + 512 * q^16 + 5480 * q^19 - 8848 * q^26 + 11820 * q^29 - 13736 * q^31 - 6096 * q^34 + 756 * q^41 + 6144 * q^44 + 12528 * q^46 + 5766 * q^49 + 15104 * q^56 - 69960 * q^59 - 19676 * q^61 - 8192 * q^64 - 140424 * q^71 - 44144 * q^74 - 87680 * q^76 - 9040 * q^79 + 19472 * q^86 + 76980 * q^89 - 261016 * q^91 - 104976 * 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 $

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 118.000i	− 0.910200i	−0.890440	−	0.455100i	$-0.849603\pi$
$7$	− 118.000i	− 0.910200i	0.890440	−	0.455100i	$-0.150397\pi$
$8$	64.0000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−192.000	−0.478431	−0.239216	−	0.970966i	$-0.576890\pi$
$11$	−192.000	−0.478431	−0.239216	+	0.970966i	$0.576890\pi$
$12$	0	0
$13$	− 1106.00i	− 1.81508i	−0.419961	−	0.907542i	$-0.637956\pi$
$13$	− 1106.00i	− 1.81508i	0.419961	−	0.907542i	$-0.362044\pi$
$14$	−472.000	−0.643609
$15$	0	0
$16$	256.000	0.250000
$17$	− 762.000i	− 0.639488i	−0.947504	−	0.319744i	$-0.896403\pi$
$17$	− 762.000i	− 0.639488i	0.947504	−	0.319744i	$-0.103597\pi$
$18$	0	0
$19$	2740.00	1.74127	0.870636	−	0.491928i	$-0.163708\pi$
$19$	2740.00	1.74127	0.870636	+	0.491928i	$0.163708\pi$
$20$	0	0
$21$	0	0
$22$	768.000i	0.338302i
$23$	1566.00i	0.617266i	0.951181	+	0.308633i	$0.0998714\pi$
$23$	1566.00i	0.617266i	−0.951181	+	0.308633i	$0.900129\pi$
$24$	0	0
$25$	0	0
$26$	−4424.00	−1.28346
$27$	0	0
$28$	1888.00i	0.455100i
$29$	5910.00	1.30495	0.652473	−	0.757812i	$-0.273732\pi$
$29$	5910.00	1.30495	0.652473	+	0.757812i	$0.273732\pi$
$30$	0	0
$31$	−6868.00	−1.28359	−0.641795	−	0.766877i	$-0.721810\pi$
$31$	−6868.00	−1.28359	−0.641795	+	0.766877i	$0.721810\pi$
$32$	− 1024.00i	− 0.176777i
$33$	0	0
$34$	−3048.00	−0.452187
$35$	0	0
$36$	0	0
$37$	− 5518.00i	− 0.662640i	−0.943519	−	0.331320i	$-0.892506\pi$
$37$	− 5518.00i	− 0.662640i	0.943519	−	0.331320i	$-0.107494\pi$
$38$	− 10960.0i	− 1.23127i
$39$	0	0
$40$	0	0
$41$	378.000	0.0351182	0.0175591	−	0.999846i	$-0.494410\pi$
$41$	378.000	0.0351182	0.0175591	+	0.999846i	$0.494410\pi$
$42$	0	0
$43$	2434.00i	0.200747i	0.994950	+	0.100374i	$0.0320038\pi$
$43$	2434.00i	0.200747i	−0.994950	+	0.100374i	$0.967996\pi$
$44$	3072.00	0.239216
$45$	0	0
$46$	6264.00	0.436473
$47$	− 13122.0i	− 0.866474i	−0.901280	−	0.433237i	$-0.857371\pi$
$47$	− 13122.0i	− 0.866474i	0.901280	−	0.433237i	$-0.142629\pi$
$48$	0	0
$49$	2883.00	0.171536
$50$	0	0
$51$	0	0
$52$	17696.0i	0.907542i
$53$	− 9174.00i	− 0.448610i	−0.974519	−	0.224305i	$-0.927989\pi$
$53$	− 9174.00i	− 0.448610i	0.974519	−	0.224305i	$-0.0720112\pi$
$54$	0	0
$55$	0	0
$56$	7552.00	0.321804
$57$	0	0
$58$	− 23640.0i	− 0.922736i
$59$	−34980.0	−1.30825	−0.654124	−	0.756388i	$-0.726962\pi$
$59$	−34980.0	−1.30825	−0.654124	+	0.756388i	$0.726962\pi$
$60$	0	0
$61$	−9838.00	−0.338518	−0.169259	−	0.985572i	$-0.554137\pi$
$61$	−9838.00	−0.338518	−0.169259	+	0.985572i	$0.554137\pi$
$62$	27472.0i	0.907635i
$63$	0	0
$64$	−4096.00	−0.125000
$65$	0	0
$66$	0	0
$67$	33722.0i	0.917754i	0.888500	+	0.458877i	$0.151748\pi$
$67$	33722.0i	0.917754i	−0.888500	+	0.458877i	$0.848252\pi$
$68$	12192.0i	0.319744i
$69$	0	0
$70$	0	0
$71$	−70212.0	−1.65297	−0.826486	−	0.562957i	$-0.809664\pi$
$71$	−70212.0	−1.65297	−0.826486	+	0.562957i	$0.809664\pi$
$72$	0	0
$73$	− 21986.0i	− 0.482880i	−0.970416	−	0.241440i	$-0.922380\pi$
$73$	− 21986.0i	− 0.482880i	0.970416	−	0.241440i	$-0.0776197\pi$
$74$	−22072.0	−0.468557
$75$	0	0
$76$	−43840.0	−0.870636
$77$	22656.0i	0.435468i
$78$	0	0
$79$	−4520.00	−0.0814837	−0.0407418	−	0.999170i	$-0.512972\pi$
$79$	−4520.00	−0.0814837	−0.0407418	+	0.999170i	$0.512972\pi$
$80$	0	0
$81$	0	0
$82$	− 1512.00i	− 0.0248323i
$83$	− 109074.i	− 1.73790i	−0.494896	−	0.868952i	$-0.664794\pi$
$83$	− 109074.i	− 1.73790i	0.494896	−	0.868952i	$-0.335206\pi$
$84$	0	0
$85$	0	0
$86$	9736.00	0.141950
$87$	0	0
$88$	− 12288.0i	− 0.169151i
$89$	38490.0	0.515078	0.257539	−	0.966268i	$-0.417088\pi$
$89$	38490.0	0.515078	0.257539	+	0.966268i	$0.417088\pi$
$90$	0	0
$91$	−130508.	−1.65209
$92$	− 25056.0i	− 0.308633i
$93$	0	0
$94$	−52488.0	−0.612689
$95$	0	0
$96$	0	0
$97$	− 1918.00i	− 0.0206976i	−0.999946	−	0.0103488i	$-0.996706\pi$
$97$	− 1918.00i	− 0.0206976i	0.999946	−	0.0103488i	$-0.00329418\pi$
$98$	− 11532.0i	− 0.121294i
$99$	0	0
$100$	0	0
$101$	−77622.0	−0.757149	−0.378575	−	0.925571i	$-0.623586\pi$
$101$	−77622.0	−0.757149	−0.378575	+	0.925571i	$0.623586\pi$
$102$	0	0
$103$	46714.0i	0.433864i	0.976187	+	0.216932i	$0.0696051\pi$
$103$	46714.0i	0.433864i	−0.976187	+	0.216932i	$0.930395\pi$
$104$	70784.0	0.641729
$105$	0	0
$106$	−36696.0	−0.317215
$107$	1038.00i	0.00876472i	0.999990	+	0.00438236i	$0.00139495\pi$
$107$	1038.00i	0.00876472i	−0.999990	+	0.00438236i	$0.998605\pi$
$108$	0	0
$109$	−206930.	−1.66823	−0.834117	−	0.551587i	$-0.814023\pi$
$109$	−206930.	−1.66823	−0.834117	+	0.551587i	$0.814023\pi$
$110$	0	0
$111$	0	0
$112$	− 30208.0i	− 0.227550i
$113$	139386.i	1.02689i	0.858123	+	0.513444i	$0.171631\pi$
$113$	139386.i	1.02689i	−0.858123	+	0.513444i	$0.828369\pi$
$114$	0	0
$115$	0	0
$116$	−94560.0	−0.652473
$117$	0	0
$118$	139920.i	0.925070i
$119$	−89916.0	−0.582062
$120$	0	0
$121$	−124187.	−0.771104
$122$	39352.0i	0.239369i
$123$	0	0
$124$	109888.	0.641795
$125$	0	0
$126$	0	0
$127$	299882.i	1.64984i	0.565252	+	0.824919i	$0.308779\pi$
$127$	299882.i	1.64984i	−0.565252	+	0.824919i	$0.691221\pi$
$128$	16384.0i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−7872.00	−0.0400781	−0.0200390	−	0.999799i	$-0.506379\pi$
$131$	−7872.00	−0.0400781	−0.0200390	+	0.999799i	$0.506379\pi$
$132$	0	0
$133$	− 323320.i	− 1.58491i
$134$	134888.	0.648950
$135$	0	0
$136$	48768.0	0.226093
$137$	164238.i	0.747605i	0.927508	+	0.373803i	$0.121946\pi$
$137$	164238.i	0.747605i	−0.927508	+	0.373803i	$0.878054\pi$
$138$	0	0
$139$	282100.	1.23841	0.619207	−	0.785228i	$-0.287454\pi$
$139$	282100.	1.23841	0.619207	+	0.785228i	$0.287454\pi$
$140$	0	0
$141$	0	0
$142$	280848.i	1.16883i
$143$	212352.i	0.868393i
$144$	0	0
$145$	0	0
$146$	−87944.0	−0.341448
$147$	0	0
$148$	88288.0i	0.331320i
$149$	−388950.	−1.43525	−0.717626	−	0.696429i	$-0.754771\pi$
$149$	−388950.	−1.43525	−0.717626	+	0.696429i	$0.754771\pi$
$150$	0	0
$151$	−97948.0	−0.349585	−0.174793	−	0.984605i	$-0.555926\pi$
$151$	−97948.0	−0.349585	−0.174793	+	0.984605i	$0.555926\pi$
$152$	175360.i	0.615633i
$153$	0	0
$154$	90624.0	0.307923
$155$	0	0
$156$	0	0
$157$	− 3718.00i	− 0.0120382i	−0.999982	−	0.00601908i	$-0.998084\pi$
$157$	− 3718.00i	− 0.0120382i	0.999982	−	0.00601908i	$-0.00191594\pi$
$158$	18080.0i	0.0576177i
$159$	0	0
$160$	0	0
$161$	184788.	0.561835
$162$	0	0
$163$	43234.0i	0.127455i	0.997967	+	0.0637274i	$0.0202988\pi$
$163$	43234.0i	0.127455i	−0.997967	+	0.0637274i	$0.979701\pi$
$164$	−6048.00	−0.0175591
$165$	0	0
$166$	−436296.	−1.22888
$167$	− 186522.i	− 0.517534i	−0.965940	−	0.258767i	$-0.916684\pi$
$167$	− 186522.i	− 0.517534i	0.965940	−	0.258767i	$-0.0833162\pi$
$168$	0	0
$169$	−851943.	−2.29453
$170$	0	0
$171$	0	0
$172$	− 38944.0i	− 0.100374i
$173$	− 374454.i	− 0.951225i	−0.879655	−	0.475612i	$-0.842226\pi$
$173$	− 374454.i	− 0.951225i	0.879655	−	0.475612i	$-0.157774\pi$
$174$	0	0
$175$	0	0
$176$	−49152.0	−0.119608
$177$	0	0
$178$	− 153960.i	− 0.364215i
$179$	272100.	0.634740	0.317370	−	0.948302i	$-0.397200\pi$
$179$	272100.	0.634740	0.317370	+	0.948302i	$0.397200\pi$
$180$	0	0
$181$	−75418.0	−0.171111	−0.0855556	−	0.996333i	$-0.527267\pi$
$181$	−75418.0	−0.171111	−0.0855556	+	0.996333i	$0.527267\pi$
$182$	522032.i	1.16820i
$183$	0	0
$184$	−100224.	−0.218236
$185$	0	0
$186$	0	0
$187$	146304.i	0.305951i
$188$	209952.i	0.433237i
$189$	0	0
$190$	0	0
$191$	356988.	0.708060	0.354030	−	0.935234i	$-0.384811\pi$
$191$	356988.	0.708060	0.354030	+	0.935234i	$0.384811\pi$
$192$	0	0
$193$	438694.i	0.847751i	0.905720	+	0.423876i	$0.139331\pi$
$193$	438694.i	0.847751i	−0.905720	+	0.423876i	$0.860669\pi$
$194$	−7672.00	−0.0146354
$195$	0	0
$196$	−46128.0	−0.0857678
$197$	156798.i	0.287856i	0.989588	+	0.143928i	$0.0459733\pi$
$197$	156798.i	0.287856i	−0.989588	+	0.143928i	$0.954027\pi$
$198$	0	0
$199$	162520.	0.290920	0.145460	−	0.989364i	$-0.453534\pi$
$199$	162520.	0.290920	0.145460	+	0.989364i	$0.453534\pi$
$200$	0	0
$201$	0	0
$202$	310488.i	0.535385i
$203$	− 697380.i	− 1.18776i
$204$	0	0
$205$	0	0
$206$	186856.	0.306788
$207$	0	0
$208$	− 283136.i	− 0.453771i
$209$	−526080.	−0.833079
$210$	0	0
$211$	−181648.	−0.280882	−0.140441	−	0.990089i	$-0.544852\pi$
$211$	−181648.	−0.280882	−0.140441	+	0.990089i	$0.544852\pi$
$212$	146784.i	0.224305i
$213$	0	0
$214$	4152.00	0.00619759
$215$	0	0
$216$	0	0
$217$	810424.i	1.16832i
$218$	827720.i	1.17962i
$219$	0	0
$220$	0	0
$221$	−842772.	−1.16073
$222$	0	0
$223$	288274.i	0.388189i	0.980983	+	0.194095i	$0.0621769\pi$
$223$	288274.i	0.388189i	−0.980983	+	0.194095i	$0.937823\pi$
$224$	−120832.	−0.160902
$225$	0	0
$226$	557544.	0.726119
$227$	− 1.12552e6i	− 1.44974i	−0.688887	−	0.724869i	$-0.741900\pi$
$227$	− 1.12552e6i	− 1.44974i	0.688887	−	0.724869i	$-0.258100\pi$
$228$	0	0
$229$	415810.	0.523970	0.261985	−	0.965072i	$-0.415623\pi$
$229$	415810.	0.523970	0.261985	+	0.965072i	$0.415623\pi$
$230$	0	0
$231$	0	0
$232$	378240.i	0.461368i
$233$	770586.i	0.929889i	0.885340	+	0.464945i	$0.153926\pi$
$233$	770586.i	0.929889i	−0.885340	+	0.464945i	$0.846074\pi$
$234$	0	0
$235$	0	0
$236$	559680.	0.654124
$237$	0	0
$238$	359664.i	0.411580i
$239$	−595320.	−0.674149	−0.337074	−	0.941478i	$-0.609437\pi$
$239$	−595320.	−0.674149	−0.337074	+	0.941478i	$0.609437\pi$
$240$	0	0
$241$	273902.	0.303775	0.151888	−	0.988398i	$-0.451465\pi$
$241$	273902.	0.303775	0.151888	+	0.988398i	$0.451465\pi$
$242$	496748.i	0.545253i
$243$	0	0
$244$	157408.	0.169259
$245$	0	0
$246$	0	0
$247$	− 3.03044e6i	− 3.16055i
$248$	− 439552.i	− 0.453817i
$249$	0	0
$250$	0	0
$251$	−850752.	−0.852351	−0.426176	−	0.904640i	$-0.640139\pi$
$251$	−850752.	−0.852351	−0.426176	+	0.904640i	$0.640139\pi$
$252$	0	0
$253$	− 300672.i	− 0.295319i
$254$	1.19953e6	1.16661
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 825402.i	− 0.779530i	−0.920914	−	0.389765i	$-0.872556\pi$
$257$	− 825402.i	− 0.779530i	0.920914	−	0.389765i	$-0.127444\pi$
$258$	0	0
$259$	−651124.	−0.603135
$260$	0	0
$261$	0	0
$262$	31488.0i	0.0283395i
$263$	1.36465e6i	1.21655i	0.793726	+	0.608276i	$0.208139\pi$
$263$	1.36465e6i	1.21655i	−0.793726	+	0.608276i	$0.791861\pi$
$264$	0	0
$265$	0	0
$266$	−1.29328e6	−1.12070
$267$	0	0
$268$	− 539552.i	− 0.458877i
$269$	−113310.	−0.0954745	−0.0477373	−	0.998860i	$-0.515201\pi$
$269$	−113310.	−0.0954745	−0.0477373	+	0.998860i	$0.515201\pi$
$270$	0	0
$271$	−849628.	−0.702758	−0.351379	−	0.936233i	$-0.614287\pi$
$271$	−849628.	−0.702758	−0.351379	+	0.936233i	$0.614287\pi$
$272$	− 195072.i	− 0.159872i
$273$	0	0
$274$	656952.	0.528637
$275$	0	0
$276$	0	0
$277$	438602.i	0.343456i	0.985144	+	0.171728i	$0.0549350\pi$
$277$	438602.i	0.343456i	−0.985144	+	0.171728i	$0.945065\pi$
$278$	− 1.12840e6i	− 0.875691i
$279$	0	0
$280$	0	0
$281$	1.45670e6	1.10053	0.550267	−	0.834989i	$-0.314526\pi$
$281$	1.45670e6	1.10053	0.550267	+	0.834989i	$0.314526\pi$
$282$	0	0
$283$	120394.i	0.0893591i	0.999001	+	0.0446795i	$0.0142267\pi$
$283$	120394.i	0.0893591i	−0.999001	+	0.0446795i	$0.985773\pi$
$284$	1.12339e6	0.826486
$285$	0	0
$286$	849408.	0.614047
$287$	− 44604.0i	− 0.0319646i
$288$	0	0
$289$	839213.	0.591055
$290$	0	0
$291$	0	0
$292$	351776.i	0.241440i
$293$	− 2.64209e6i	− 1.79796i	−0.437993	−	0.898978i	$-0.644311\pi$
$293$	− 2.64209e6i	− 1.79796i	0.437993	−	0.898978i	$-0.355689\pi$
$294$	0	0
$295$	0	0
$296$	353152.	0.234278
$297$	0	0
$298$	1.55580e6i	1.01488i
$299$	1.73200e6	1.12039
$300$	0	0
$301$	287212.	0.182720
$302$	391792.i	0.247194i
$303$	0	0
$304$	701440.	0.435318
$305$	0	0
$306$	0	0
$307$	− 1.44756e6i	− 0.876577i	−0.898834	−	0.438288i	$-0.855585\pi$
$307$	− 1.44756e6i	− 0.876577i	0.898834	−	0.438288i	$-0.144415\pi$
$308$	− 362496.i	− 0.217734i
$309$	0	0
$310$	0	0
$311$	928068.	0.544100	0.272050	−	0.962283i	$-0.412298\pi$
$311$	928068.	0.544100	0.272050	+	0.962283i	$0.412298\pi$
$312$	0	0
$313$	− 2.29563e6i	− 1.32446i	−0.749299	−	0.662232i	$-0.769609\pi$
$313$	− 2.29563e6i	− 1.32446i	0.749299	−	0.662232i	$-0.230391\pi$
$314$	−14872.0	−0.00851227
$315$	0	0
$316$	72320.0	0.0407418
$317$	− 2.73652e6i	− 1.52950i	−0.644324	−	0.764752i	$-0.722861\pi$
$317$	− 2.73652e6i	− 1.52950i	0.644324	−	0.764752i	$-0.277139\pi$
$318$	0	0
$319$	−1.13472e6	−0.624327
$320$	0	0
$321$	0	0
$322$	− 739152.i	− 0.397278i
$323$	− 2.08788e6i	− 1.11352i
$324$	0	0
$325$	0	0
$326$	172936.	0.0901242
$327$	0	0
$328$	24192.0i	0.0124162i
$329$	−1.54840e6	−0.788665
$330$	0	0
$331$	3.81879e6	1.91583	0.957913	−	0.287059i	$-0.0926776\pi$
$331$	3.81879e6	1.91583	0.957913	+	0.287059i	$0.0926776\pi$
$332$	1.74518e6i	0.868952i
$333$	0	0
$334$	−746088.	−0.365952
$335$	0	0
$336$	0	0
$337$	− 2.21088e6i	− 1.06045i	−0.847857	−	0.530225i	$-0.822108\pi$
$337$	− 2.21088e6i	− 1.06045i	0.847857	−	0.530225i	$-0.177892\pi$
$338$	3.40777e6i	1.62248i
$339$	0	0
$340$	0	0
$341$	1.31866e6	0.614109
$342$	0	0
$343$	− 2.32342e6i	− 1.06633i
$344$	−155776.	−0.0709748
$345$	0	0
$346$	−1.49782e6	−0.672618
$347$	2.32724e6i	1.03757i	0.854905	+	0.518785i	$0.173615\pi$
$347$	2.32724e6i	1.03757i	−0.854905	+	0.518785i	$0.826385\pi$
$348$	0	0
$349$	311290.	0.136805	0.0684024	−	0.997658i	$-0.478210\pi$
$349$	311290.	0.136805	0.0684024	+	0.997658i	$0.478210\pi$
$350$	0	0
$351$	0	0
$352$	196608.i	0.0845755i
$353$	− 3.08657e6i	− 1.31838i	−0.751977	−	0.659189i	$-0.770900\pi$
$353$	− 3.08657e6i	− 1.31838i	0.751977	−	0.659189i	$-0.229100\pi$
$354$	0	0
$355$	0	0
$356$	−615840.	−0.257539
$357$	0	0
$358$	− 1.08840e6i	− 0.448829i
$359$	−3.53076e6	−1.44588	−0.722940	−	0.690911i	$-0.757210\pi$
$359$	−3.53076e6	−1.44588	−0.722940	+	0.690911i	$0.757210\pi$
$360$	0	0
$361$	5.03150e6	2.03203
$362$	301672.i	0.120994i
$363$	0	0
$364$	2.08813e6	0.826045
$365$	0	0
$366$	0	0
$367$	35762.0i	0.0138598i	0.999976	+	0.00692989i	$0.00220587\pi$
$367$	35762.0i	0.0138598i	−0.999976	+	0.00692989i	$0.997794\pi$
$368$	400896.i	0.154316i
$369$	0	0
$370$	0	0
$371$	−1.08253e6	−0.408325
$372$	0	0
$373$	1.71525e6i	0.638346i	0.947696	+	0.319173i	$0.103405\pi$
$373$	1.71525e6i	0.638346i	−0.947696	+	0.319173i	$0.896595\pi$
$374$	585216.	0.216340
$375$	0	0
$376$	839808.	0.306345
$377$	− 6.53646e6i	− 2.36859i
$378$	0	0
$379$	3.10174e6	1.10919	0.554597	−	0.832119i	$-0.312873\pi$
$379$	3.10174e6	1.10919	0.554597	+	0.832119i	$0.312873\pi$
$380$	0	0
$381$	0	0
$382$	− 1.42795e6i	− 0.500674i
$383$	5.31949e6i	1.85299i	0.376309	+	0.926494i	$0.377193\pi$
$383$	5.31949e6i	1.85299i	−0.376309	+	0.926494i	$0.622807\pi$
$384$	0	0
$385$	0	0
$386$	1.75478e6	0.599451
$387$	0	0
$388$	30688.0i	0.0103488i
$389$	1.16145e6	0.389158	0.194579	−	0.980887i	$-0.437666\pi$
$389$	1.16145e6	0.389158	0.194579	+	0.980887i	$0.437666\pi$
$390$	0	0
$391$	1.19329e6	0.394734
$392$	184512.i	0.0606470i
$393$	0	0
$394$	627192.	0.203545
$395$	0	0
$396$	0	0
$397$	628562.i	0.200157i	0.994980	+	0.100079i	$0.0319095\pi$
$397$	628562.i	0.200157i	−0.994980	+	0.100079i	$0.968091\pi$
$398$	− 650080.i	− 0.205712i
$399$	0	0
$400$	0	0
$401$	2.72432e6	0.846052	0.423026	−	0.906118i	$-0.360968\pi$
$401$	2.72432e6	0.846052	0.423026	+	0.906118i	$0.360968\pi$
$402$	0	0
$403$	7.59601e6i	2.32982i
$404$	1.24195e6	0.378575
$405$	0	0
$406$	−2.78952e6	−0.839875
$407$	1.05946e6i	0.317027i
$408$	0	0
$409$	−1.78019e6	−0.526209	−0.263104	−	0.964767i	$-0.584746\pi$
$409$	−1.78019e6	−0.526209	−0.263104	+	0.964767i	$0.584746\pi$
$410$	0	0
$411$	0	0
$412$	− 747424.i	− 0.216932i
$413$	4.12764e6i	1.19077i
$414$	0	0
$415$	0	0
$416$	−1.13254e6	−0.320865
$417$	0	0
$418$	2.10432e6i	0.589076i
$419$	650580.	0.181036	0.0905181	−	0.995895i	$-0.471148\pi$
$419$	650580.	0.181036	0.0905181	+	0.995895i	$0.471148\pi$
$420$	0	0
$421$	−3.54060e6	−0.973579	−0.486790	−	0.873519i	$-0.661832\pi$
$421$	−3.54060e6	−0.973579	−0.486790	+	0.873519i	$0.661832\pi$
$422$	726592.i	0.198614i
$423$	0	0
$424$	587136.	0.158608
$425$	0	0
$426$	0	0
$427$	1.16088e6i	0.308119i
$428$	− 16608.0i	− 0.00438236i
$429$	0	0
$430$	0	0
$431$	548748.	0.142292	0.0711459	−	0.997466i	$-0.477334\pi$
$431$	548748.	0.142292	0.0711459	+	0.997466i	$0.477334\pi$
$432$	0	0
$433$	1.49241e6i	0.382534i	0.981538	+	0.191267i	$0.0612596\pi$
$433$	1.49241e6i	0.382534i	−0.981538	+	0.191267i	$0.938740\pi$
$434$	3.24170e6	0.826129
$435$	0	0
$436$	3.31088e6	0.834117
$437$	4.29084e6i	1.07483i
$438$	0	0
$439$	−4.86212e6	−1.20411	−0.602053	−	0.798456i	$-0.705650\pi$
$439$	−4.86212e6	−1.20411	−0.602053	+	0.798456i	$0.705650\pi$
$440$	0	0
$441$	0	0
$442$	3.37109e6i	0.820757i
$443$	− 1.86155e6i	− 0.450678i	−0.974280	−	0.225339i	$-0.927651\pi$
$443$	− 1.86155e6i	− 0.450678i	0.974280	−	0.225339i	$-0.0723490\pi$
$444$	0	0
$445$	0	0
$446$	1.15310e6	0.274491
$447$	0	0
$448$	483328.i	0.113775i
$449$	3.73719e6	0.874841	0.437421	−	0.899257i	$-0.355892\pi$
$449$	3.73719e6	0.874841	0.437421	+	0.899257i	$0.355892\pi$
$450$	0	0
$451$	−72576.0	−0.0168016
$452$	− 2.23018e6i	− 0.513444i
$453$	0	0
$454$	−4.50209e6	−1.02512
$455$	0	0
$456$	0	0
$457$	− 6.48276e6i	− 1.45201i	−0.687690	−	0.726005i	$-0.741375\pi$
$457$	− 6.48276e6i	− 1.45201i	0.687690	−	0.726005i	$-0.258625\pi$
$458$	− 1.66324e6i	− 0.370503i
$459$	0	0
$460$	0	0
$461$	−1.50910e6	−0.330724	−0.165362	−	0.986233i	$-0.552879\pi$
$461$	−1.50910e6	−0.330724	−0.165362	+	0.986233i	$0.552879\pi$
$462$	0	0
$463$	− 8.68401e6i	− 1.88264i	−0.337513	−	0.941321i	$-0.609586\pi$
$463$	− 8.68401e6i	− 1.88264i	0.337513	−	0.941321i	$-0.390414\pi$
$464$	1.51296e6	0.326236
$465$	0	0
$466$	3.08234e6	0.657531
$467$	− 6.96412e6i	− 1.47766i	−0.673893	−	0.738829i	$-0.735379\pi$
$467$	− 6.96412e6i	− 1.47766i	0.673893	−	0.738829i	$-0.264621\pi$
$468$	0	0
$469$	3.97920e6	0.835340
$470$	0	0
$471$	0	0
$472$	− 2.23872e6i	− 0.462535i
$473$	− 467328.i	− 0.0960437i
$474$	0	0
$475$	0	0
$476$	1.43866e6	0.291031
$477$	0	0
$478$	2.38128e6i	0.476695i
$479$	−5.51052e6	−1.09737	−0.548686	−	0.836029i	$-0.684872\pi$
$479$	−5.51052e6	−1.09737	−0.548686	+	0.836029i	$0.684872\pi$
$480$	0	0
$481$	−6.10291e6	−1.20275
$482$	− 1.09561e6i	− 0.214802i
$483$	0	0
$484$	1.98699e6	0.385552
$485$	0	0
$486$	0	0
$487$	5.51808e6i	1.05430i	0.849771	+	0.527152i	$0.176740\pi$
$487$	5.51808e6i	1.05430i	−0.849771	+	0.527152i	$0.823260\pi$
$488$	− 629632.i	− 0.119684i
$489$	0	0
$490$	0	0
$491$	1.51277e6	0.283184	0.141592	−	0.989925i	$-0.454778\pi$
$491$	1.51277e6	0.283184	0.141592	+	0.989925i	$0.454778\pi$
$492$	0	0
$493$	− 4.50342e6i	− 0.834498i
$494$	−1.21218e7	−2.23485
$495$	0	0
$496$	−1.75821e6	−0.320897
$497$	8.28502e6i	1.50454i
$498$	0	0
$499$	1.93042e6	0.347057	0.173528	−	0.984829i	$-0.444483\pi$
$499$	1.93042e6	0.347057	0.173528	+	0.984829i	$0.444483\pi$
$500$	0	0
$501$	0	0
$502$	3.40301e6i	0.602703i
$503$	6.73105e6i	1.18621i	0.805124	+	0.593106i	$0.202099\pi$
$503$	6.73105e6i	1.18621i	−0.805124	+	0.593106i	$0.797901\pi$
$504$	0	0
$505$	0	0
$506$	−1.20269e6	−0.208822
$507$	0	0
$508$	− 4.79811e6i	− 0.824919i
$509$	−556650.	−0.0952331	−0.0476165	−	0.998866i	$-0.515163\pi$
$509$	−556650.	−0.0952331	−0.0476165	+	0.998866i	$0.515163\pi$
$510$	0	0
$511$	−2.59435e6	−0.439517
$512$	− 262144.i	− 0.0441942i
$513$	0	0
$514$	−3.30161e6	−0.551211
$515$	0	0
$516$	0	0
$517$	2.51942e6i	0.414548i
$518$	2.60450e6i	0.426481i
$519$	0	0
$520$	0	0
$521$	−1.01110e7	−1.63192	−0.815962	−	0.578106i	$-0.803792\pi$
$521$	−1.01110e7	−1.63192	−0.815962	+	0.578106i	$0.803792\pi$
$522$	0	0
$523$	7.03719e6i	1.12498i	0.826804	+	0.562491i	$0.190157\pi$
$523$	7.03719e6i	1.12498i	−0.826804	+	0.562491i	$0.809843\pi$
$524$	125952.	0.0200390
$525$	0	0
$526$	5.45858e6	0.860232
$527$	5.23342e6i	0.820840i
$528$	0	0
$529$	3.98399e6	0.618983
$530$	0	0
$531$	0	0
$532$	5.17312e6i	0.792453i
$533$	− 418068.i	− 0.0637425i
$534$	0	0
$535$	0	0
$536$	−2.15821e6	−0.324475
$537$	0	0
$538$	453240.i	0.0675107i
$539$	−553536.	−0.0820680
$540$	0	0
$541$	−4.23114e6	−0.621533	−0.310766	−	0.950486i	$-0.600586\pi$
$541$	−4.23114e6	−0.621533	−0.310766	+	0.950486i	$0.600586\pi$
$542$	3.39851e6i	0.496925i
$543$	0	0
$544$	−780288.	−0.113047
$545$	0	0
$546$	0	0
$547$	4.44024e6i	0.634510i	0.948340	+	0.317255i	$0.102761\pi$
$547$	4.44024e6i	0.634510i	−0.948340	+	0.317255i	$0.897239\pi$
$548$	− 2.62781e6i	− 0.373803i
$549$	0	0
$550$	0	0
$551$	1.61934e7	2.27227
$552$	0	0
$553$	533360.i	0.0741665i
$554$	1.75441e6	0.242860
$555$	0	0
$556$	−4.51360e6	−0.619207
$557$	9.01448e6i	1.23113i	0.788088	+	0.615563i	$0.211071\pi$
$557$	9.01448e6i	1.23113i	−0.788088	+	0.615563i	$0.788929\pi$
$558$	0	0
$559$	2.69200e6	0.364373
$560$	0	0
$561$	0	0
$562$	− 5.82679e6i	− 0.778196i
$563$	− 9.81287e6i	− 1.30474i	−0.757899	−	0.652372i	$-0.773774\pi$
$563$	− 9.81287e6i	− 1.30474i	0.757899	−	0.652372i	$-0.226226\pi$
$564$	0	0
$565$	0	0
$566$	481576.	0.0631864
$567$	0	0
$568$	− 4.49357e6i	− 0.584414i
$569$	1.33152e7	1.72412	0.862061	−	0.506804i	$-0.169173\pi$
$569$	1.33152e7	1.72412	0.862061	+	0.506804i	$0.169173\pi$
$570$	0	0
$571$	9.95895e6	1.27827	0.639136	−	0.769094i	$-0.279292\pi$
$571$	9.95895e6	1.27827	0.639136	+	0.769094i	$0.279292\pi$
$572$	− 3.39763e6i	− 0.434196i
$573$	0	0
$574$	−178416.	−0.0226024
$575$	0	0
$576$	0	0
$577$	4.50372e6i	0.563160i	0.959538	+	0.281580i	$0.0908585\pi$
$577$	4.50372e6i	0.563160i	−0.959538	+	0.281580i	$0.909141\pi$
$578$	− 3.35685e6i	− 0.417939i
$579$	0	0
$580$	0	0
$581$	−1.28707e7	−1.58184
$582$	0	0
$583$	1.76141e6i	0.214629i
$584$	1.40710e6	0.170724
$585$	0	0
$586$	−1.05684e7	−1.27135
$587$	− 625842.i	− 0.0749669i	−0.999297	−	0.0374834i	$-0.988066\pi$
$587$	− 625842.i	− 0.0749669i	0.999297	−	0.0374834i	$-0.0119341\pi$
$588$	0	0
$589$	−1.88183e7	−2.23508
$590$	0	0
$591$	0	0
$592$	− 1.41261e6i	− 0.165660i
$593$	− 2.50385e6i	− 0.292397i	−0.989255	−	0.146198i	$-0.953296\pi$
$593$	− 2.50385e6i	− 0.292397i	0.989255	−	0.146198i	$-0.0467038\pi$
$594$	0	0
$595$	0	0
$596$	6.22320e6	0.717626
$597$	0	0
$598$	− 6.92798e6i	− 0.792235i
$599$	−756480.	−0.0861451	−0.0430725	−	0.999072i	$-0.513715\pi$
$599$	−756480.	−0.0861451	−0.0430725	+	0.999072i	$0.513715\pi$
$600$	0	0
$601$	−1.38565e7	−1.56483	−0.782413	−	0.622760i	$-0.786011\pi$
$601$	−1.38565e7	−1.56483	−0.782413	+	0.622760i	$0.786011\pi$
$602$	− 1.14885e6i	− 0.129203i
$603$	0	0
$604$	1.56717e6	0.174793
$605$	0	0
$606$	0	0
$607$	1.13772e7i	1.25333i	0.779291	+	0.626663i	$0.215580\pi$
$607$	1.13772e7i	1.25333i	−0.779291	+	0.626663i	$0.784420\pi$
$608$	− 2.80576e6i	− 0.307816i
$609$	0	0
$610$	0	0
$611$	−1.45129e7	−1.57272
$612$	0	0
$613$	7.00161e6i	0.752570i	0.926504	+	0.376285i	$0.122799\pi$
$613$	7.00161e6i	0.752570i	−0.926504	+	0.376285i	$0.877201\pi$
$614$	−5.79023e6	−0.619833
$615$	0	0
$616$	−1.44998e6	−0.153961
$617$	− 7.90300e6i	− 0.835755i	−0.908503	−	0.417878i	$-0.862774\pi$
$617$	− 7.90300e6i	− 0.835755i	0.908503	−	0.417878i	$-0.137226\pi$
$618$	0	0
$619$	−4.02362e6	−0.422076	−0.211038	−	0.977478i	$-0.567684\pi$
$619$	−4.02362e6	−0.422076	−0.211038	+	0.977478i	$0.567684\pi$
$620$	0	0
$621$	0	0
$622$	− 3.71227e6i	− 0.384737i
$623$	− 4.54182e6i	− 0.468824i
$624$	0	0
$625$	0	0
$626$	−9.18250e6	−0.936538
$627$	0	0
$628$	59488.0i	0.00601908i
$629$	−4.20472e6	−0.423750
$630$	0	0
$631$	−1.00227e7	−1.00210	−0.501049	−	0.865419i	$-0.667052\pi$
$631$	−1.00227e7	−1.00210	−0.501049	+	0.865419i	$0.667052\pi$
$632$	− 289280.i	− 0.0288088i
$633$	0	0
$634$	−1.09461e7	−1.08152
$635$	0	0
$636$	0	0
$637$	− 3.18860e6i	− 0.311352i
$638$	4.53888e6i	0.441466i
$639$	0	0
$640$	0	0
$641$	−6.37390e6	−0.612718	−0.306359	−	0.951916i	$-0.599111\pi$
$641$	−6.37390e6	−0.612718	−0.306359	+	0.951916i	$0.599111\pi$
$642$	0	0
$643$	− 5.00457e6i	− 0.477352i	−0.971099	−	0.238676i	$-0.923287\pi$
$643$	− 5.00457e6i	− 0.477352i	0.971099	−	0.238676i	$-0.0767134\pi$
$644$	−2.95661e6	−0.280918
$645$	0	0
$646$	−8.35152e6	−0.787380
$647$	8.71928e6i	0.818879i	0.912337	+	0.409440i	$0.134276\pi$
$647$	8.71928e6i	0.818879i	−0.912337	+	0.409440i	$0.865724\pi$
$648$	0	0
$649$	6.71616e6	0.625906
$650$	0	0
$651$	0	0
$652$	− 691744.i	− 0.0637274i
$653$	− 1.58477e6i	− 0.145440i	−0.997352	−	0.0727201i	$-0.976832\pi$
$653$	− 1.58477e6i	− 0.145440i	0.997352	−	0.0727201i	$-0.0231680\pi$
$654$	0	0
$655$	0	0
$656$	96768.0	0.00877955
$657$	0	0
$658$	6.19358e6i	0.557670i
$659$	1.26410e7	1.13388	0.566940	−	0.823759i	$-0.308127\pi$
$659$	1.26410e7	1.13388	0.566940	+	0.823759i	$0.308127\pi$
$660$	0	0
$661$	−3.61572e6	−0.321878	−0.160939	−	0.986964i	$-0.551452\pi$
$661$	−3.61572e6	−0.321878	−0.160939	+	0.986964i	$0.551452\pi$
$662$	− 1.52752e7i	− 1.35469i
$663$	0	0
$664$	6.98074e6	0.614442
$665$	0	0
$666$	0	0
$667$	9.25506e6i	0.805498i
$668$	2.98435e6i	0.258767i
$669$	0	0
$670$	0	0
$671$	1.88890e6	0.161958
$672$	0	0
$673$	− 1.11313e7i	− 0.947349i	−0.880700	−	0.473675i	$-0.842927\pi$
$673$	− 1.11313e7i	− 0.947349i	0.880700	−	0.473675i	$-0.157073\pi$
$674$	−8.84351e6	−0.749851
$675$	0	0
$676$	1.36311e7	1.14727
$677$	235518.i	0.0197493i	0.999951	+	0.00987467i	$0.00314326\pi$
$677$	235518.i	0.0197493i	−0.999951	+	0.00987467i	$0.996857\pi$
$678$	0	0
$679$	−226324.	−0.0188389
$680$	0	0
$681$	0	0
$682$	− 5.27462e6i	− 0.434241i
$683$	2.05830e7i	1.68833i	0.536084	+	0.844164i	$0.319903\pi$
$683$	2.05830e7i	1.68833i	−0.536084	+	0.844164i	$0.680097\pi$
$684$	0	0
$685$	0	0
$686$	−9.29368e6	−0.754011
$687$	0	0
$688$	623104.i	0.0501868i
$689$	−1.01464e7	−0.814265
$690$	0	0
$691$	−9.54825e6	−0.760727	−0.380363	−	0.924837i	$-0.624201\pi$
$691$	−9.54825e6	−0.760727	−0.380363	+	0.924837i	$0.624201\pi$
$692$	5.99126e6i	0.475612i
$693$	0	0
$694$	9.30895e6	0.733672
$695$	0	0
$696$	0	0
$697$	− 288036.i	− 0.0224577i
$698$	− 1.24516e6i	− 0.0967357i
$699$	0	0
$700$	0	0
$701$	−1.29304e6	−0.0993843	−0.0496921	−	0.998765i	$-0.515824\pi$
$701$	−1.29304e6	−0.0993843	−0.0496921	+	0.998765i	$0.515824\pi$
$702$	0	0
$703$	− 1.51193e7i	− 1.15384i
$704$	786432.	0.0598039
$705$	0	0
$706$	−1.23463e7	−0.932234
$707$	9.15940e6i	0.689157i
$708$	0	0
$709$	2.12720e7	1.58926	0.794628	−	0.607097i	$-0.207666\pi$
$709$	2.12720e7	1.58926	0.794628	+	0.607097i	$0.207666\pi$
$710$	0	0
$711$	0	0
$712$	2.46336e6i	0.182108i
$713$	− 1.07553e7i	− 0.792316i
$714$	0	0
$715$	0	0
$716$	−4.35360e6	−0.317370
$717$	0	0
$718$	1.41230e7i	1.02239i
$719$	8.31732e6	0.600014	0.300007	−	0.953937i	$-0.403011\pi$
$719$	8.31732e6	0.600014	0.300007	+	0.953937i	$0.403011\pi$
$720$	0	0
$721$	5.51225e6	0.394903
$722$	− 2.01260e7i	− 1.43686i
$723$	0	0
$724$	1.20669e6	0.0855556
$725$	0	0
$726$	0	0
$727$	− 4.36740e6i	− 0.306469i	−0.988190	−	0.153235i	$-0.951031\pi$
$727$	− 4.36740e6i	− 0.306469i	0.988190	−	0.153235i	$-0.0489690\pi$
$728$	− 8.35251e6i	− 0.584102i
$729$	0	0
$730$	0	0
$731$	1.85471e6	0.128375
$732$	0	0
$733$	4.05645e6i	0.278860i	0.990232	+	0.139430i	$0.0445271\pi$
$733$	4.05645e6i	0.278860i	−0.990232	+	0.139430i	$0.955473\pi$
$734$	143048.	0.00980035
$735$	0	0
$736$	1.60358e6	0.109118
$737$	− 6.47462e6i	− 0.439082i
$738$	0	0
$739$	−768260.	−0.0517484	−0.0258742	−	0.999665i	$-0.508237\pi$
$739$	−768260.	−0.0517484	−0.0258742	+	0.999665i	$0.508237\pi$
$740$	0	0
$741$	0	0
$742$	4.33013e6i	0.288729i
$743$	6.18781e6i	0.411211i	0.978635	+	0.205605i	$0.0659164\pi$
$743$	6.18781e6i	0.411211i	−0.978635	+	0.205605i	$0.934084\pi$
$744$	0	0
$745$	0	0
$746$	6.86102e6	0.451379
$747$	0	0
$748$	− 2.34086e6i	− 0.152976i
$749$	122484.	0.00797765
$750$	0	0
$751$	1.81698e7	1.17557	0.587787	−	0.809016i	$-0.299999\pi$
$751$	1.81698e7	1.17557	0.587787	+	0.809016i	$0.299999\pi$
$752$	− 3.35923e6i	− 0.216618i
$753$	0	0
$754$	−2.61458e7	−1.67484
$755$	0	0
$756$	0	0
$757$	1.93494e7i	1.22724i	0.789603	+	0.613618i	$0.210286\pi$
$757$	1.93494e7i	1.22724i	−0.789603	+	0.613618i	$0.789714\pi$
$758$	− 1.24070e7i	− 0.784318i
$759$	0	0
$760$	0	0
$761$	3.01992e7	1.89031	0.945155	−	0.326621i	$-0.105910\pi$
$761$	3.01992e7	1.89031	0.945155	+	0.326621i	$0.105910\pi$
$762$	0	0
$763$	2.44177e7i	1.51843i
$764$	−5.71181e6	−0.354030
$765$	0	0
$766$	2.12779e7	1.31026
$767$	3.86879e7i	2.37458i
$768$	0	0
$769$	−2.15854e7	−1.31627	−0.658134	−	0.752901i	$-0.728654\pi$
$769$	−2.15854e7	−1.31627	−0.658134	+	0.752901i	$0.728654\pi$
$770$	0	0
$771$	0	0
$772$	− 7.01910e6i	− 0.423876i
$773$	3.90895e6i	0.235294i	0.993055	+	0.117647i	$0.0375351\pi$
$773$	3.90895e6i	0.235294i	−0.993055	+	0.117647i	$0.962465\pi$
$774$	0	0
$775$	0	0
$776$	122752.	0.00731769
$777$	0	0
$778$	− 4.64580e6i	− 0.275177i
$779$	1.03572e6	0.0611503
$780$	0	0
$781$	1.34807e7	0.790833
$782$	− 4.77317e6i	− 0.279119i
$783$	0	0
$784$	738048.	0.0428839
$785$	0	0
$786$	0	0
$787$	− 2.65082e7i	− 1.52561i	−0.646628	−	0.762806i	$-0.723821\pi$
$787$	− 2.65082e7i	− 1.52561i	0.646628	−	0.762806i	$-0.276179\pi$
$788$	− 2.50877e6i	− 0.143928i
$789$	0	0
$790$	0	0
$791$	1.64475e7	0.934674
$792$	0	0
$793$	1.08808e7i	0.614439i
$794$	2.51425e6	0.141533
$795$	0	0
$796$	−2.60032e6	−0.145460
$797$	− 1.07940e7i	− 0.601919i	−0.953637	−	0.300960i	$-0.902693\pi$
$797$	− 1.07940e7i	− 0.601919i	0.953637	−	0.300960i	$-0.0973070\pi$
$798$	0	0
$799$	−9.99896e6	−0.554100
$800$	0	0
$801$	0	0
$802$	− 1.08973e7i	− 0.598249i
$803$	4.22131e6i	0.231025i
$804$	0	0
$805$	0	0
$806$	3.03840e7	1.64743
$807$	0	0
$808$	− 4.96781e6i	− 0.267693i
$809$	−1.11446e7	−0.598675	−0.299338	−	0.954147i	$-0.596766\pi$
$809$	−1.11446e7	−0.598675	−0.299338	+	0.954147i	$0.596766\pi$
$810$	0	0
$811$	−1.14866e7	−0.613253	−0.306626	−	0.951830i	$-0.599200\pi$
$811$	−1.14866e7	−0.613253	−0.306626	+	0.951830i	$0.599200\pi$
$812$	1.11581e7i	0.593881i
$813$	0	0
$814$	4.23782e6	0.224172
$815$	0	0
$816$	0	0
$817$	6.66916e6i	0.349555i
$818$	7.12076e6i	0.372086i
$819$	0	0
$820$	0	0
$821$	−3.04347e7	−1.57584	−0.787918	−	0.615781i	$-0.788841\pi$
$821$	−3.04347e7	−1.57584	−0.787918	+	0.615781i	$0.788841\pi$
$822$	0	0
$823$	− 4.09773e6i	− 0.210884i	−0.994425	−	0.105442i	$-0.966374\pi$
$823$	− 4.09773e6i	− 0.210884i	0.994425	−	0.105442i	$-0.0336257\pi$
$824$	−2.98970e6	−0.153394
$825$	0	0
$826$	1.65106e7	0.841999
$827$	1.70652e7i	0.867654i	0.900996	+	0.433827i	$0.142837\pi$
$827$	1.70652e7i	0.867654i	−0.900996	+	0.433827i	$0.857163\pi$
$828$	0	0
$829$	2.47617e7	1.25139	0.625697	−	0.780066i	$-0.284815\pi$
$829$	2.47617e7	1.25139	0.625697	+	0.780066i	$0.284815\pi$
$830$	0	0
$831$	0	0
$832$	4.53018e6i	0.226886i
$833$	− 2.19685e6i	− 0.109695i
$834$	0	0
$835$	0	0
$836$	8.41728e6	0.416539
$837$	0	0
$838$	− 2.60232e6i	− 0.128012i
$839$	3.16529e7	1.55242	0.776208	−	0.630476i	$-0.217140\pi$
$839$	3.16529e7	1.55242	0.776208	+	0.630476i	$0.217140\pi$
$840$	0	0
$841$	1.44170e7	0.702884
$842$	1.41624e7i	0.688425i
$843$	0	0
$844$	2.90637e6	0.140441
$845$	0	0
$846$	0	0
$847$	1.46541e7i	0.701859i
$848$	− 2.34854e6i	− 0.112153i
$849$	0	0
$850$	0	0
$851$	8.64119e6	0.409025
$852$	0	0
$853$	− 2.82671e7i	− 1.33017i	−0.746765	−	0.665087i	$-0.768394\pi$
$853$	− 2.82671e7i	− 1.33017i	0.746765	−	0.665087i	$-0.231606\pi$
$854$	4.64354e6	0.217873
$855$	0	0
$856$	−66432.0	−0.00309880
$857$	− 2.60870e7i	− 1.21331i	−0.794966	−	0.606655i	$-0.792511\pi$
$857$	− 2.60870e7i	− 1.21331i	0.794966	−	0.606655i	$-0.207489\pi$
$858$	0	0
$859$	3.38111e7	1.56342	0.781710	−	0.623642i	$-0.214348\pi$
$859$	3.38111e7	1.56342	0.781710	+	0.623642i	$0.214348\pi$
$860$	0	0
$861$	0	0
$862$	− 2.19499e6i	− 0.100615i
$863$	2.22817e7i	1.01841i	0.860646	+	0.509204i	$0.170060\pi$
$863$	2.22817e7i	1.01841i	−0.860646	+	0.509204i	$0.829940\pi$
$864$	0	0
$865$	0	0
$866$	5.96966e6	0.270492
$867$	0	0
$868$	− 1.29668e7i	− 0.584162i
$869$	867840.	0.0389843
$870$	0	0
$871$	3.72965e7	1.66580
$872$	− 1.32435e7i	− 0.589810i
$873$	0	0
$874$	1.71634e7	0.760018
$875$	0	0
$876$	0	0
$877$	− 3.46748e7i	− 1.52235i	−0.648545	−	0.761177i	$-0.724622\pi$
$877$	− 3.46748e7i	− 1.52235i	0.648545	−	0.761177i	$-0.275378\pi$
$878$	1.94485e7i	0.851431i
$879$	0	0
$880$	0	0
$881$	−1.42603e7	−0.618998	−0.309499	−	0.950900i	$-0.600161\pi$
$881$	−1.42603e7	−0.618998	−0.309499	+	0.950900i	$0.600161\pi$
$882$	0	0
$883$	3.75177e7i	1.61933i	0.586895	+	0.809663i	$0.300350\pi$
$883$	3.75177e7i	1.61933i	−0.586895	+	0.809663i	$0.699650\pi$
$884$	1.34844e7	0.580363
$885$	0	0
$886$	−7.44622e6	−0.318677
$887$	− 4.07657e7i	− 1.73975i	−0.493275	−	0.869873i	$-0.664200\pi$
$887$	− 4.07657e7i	− 1.73975i	0.493275	−	0.869873i	$-0.335800\pi$
$888$	0	0
$889$	3.53861e7	1.50168
$890$	0	0
$891$	0	0
$892$	− 4.61238e6i	− 0.194095i
$893$	− 3.59543e7i	− 1.50877i
$894$	0	0
$895$	0	0
$896$	1.93331e6	0.0804511
$897$	0	0
$898$	− 1.49488e7i	− 0.618606i
$899$	−4.05899e7	−1.67501
$900$	0	0
$901$	−6.99059e6	−0.286881
$902$	290304.i	0.0118806i
$903$	0	0
$904$	−8.92070e6	−0.363060
$905$	0	0
$906$	0	0
$907$	− 3.57116e7i	− 1.44142i	−0.693235	−	0.720712i	$-0.743815\pi$
$907$	− 3.57116e7i	− 1.44142i	0.693235	−	0.720712i	$-0.256185\pi$
$908$	1.80084e7i	0.724869i
$909$	0	0
$910$	0	0
$911$	2.11389e7	0.843893	0.421947	−	0.906621i	$-0.361347\pi$
$911$	2.11389e7	0.843893	0.421947	+	0.906621i	$0.361347\pi$
$912$	0	0
$913$	2.09422e7i	0.831468i
$914$	−2.59310e7	−1.02673
$915$	0	0
$916$	−6.65296e6	−0.261985
$917$	928896.i	0.0364791i
$918$	0	0
$919$	−1.85996e7	−0.726465	−0.363233	−	0.931698i	$-0.618327\pi$
$919$	−1.85996e7	−0.726465	−0.363233	+	0.931698i	$0.618327\pi$
$920$	0	0
$921$	0	0
$922$	6.03641e6i	0.233857i
$923$	7.76545e7i	3.00028i
$924$	0	0
$925$	0	0
$926$	−3.47360e7	−1.33123
$927$	0	0
$928$	− 6.05184e6i	− 0.230684i
$929$	4.45110e7	1.69211	0.846055	−	0.533096i	$-0.178972\pi$
$929$	4.45110e7	1.69211	0.846055	+	0.533096i	$0.178972\pi$
$930$	0	0
$931$	7.89942e6	0.298690
$932$	− 1.23294e7i	− 0.464945i
$933$	0	0
$934$	−2.78565e7	−1.04486
$935$	0	0
$936$	0	0
$937$	− 2.19419e7i	− 0.816441i	−0.912883	−	0.408221i	$-0.866149\pi$
$937$	− 2.19419e7i	− 0.816441i	0.912883	−	0.408221i	$-0.133851\pi$
$938$	− 1.59168e7i	− 0.590675i
$939$	0	0
$940$	0	0
$941$	7.77722e6	0.286319	0.143160	−	0.989700i	$-0.454274\pi$
$941$	7.77722e6	0.286319	0.143160	+	0.989700i	$0.454274\pi$
$942$	0	0
$943$	591948.i	0.0216773i
$944$	−8.95488e6	−0.327062
$945$	0	0
$946$	−1.86931e6	−0.0679132
$947$	− 3.17199e7i	− 1.14936i	−0.818378	−	0.574681i	$-0.805126\pi$
$947$	− 3.17199e7i	− 1.14936i	0.818378	−	0.574681i	$-0.194874\pi$
$948$	0	0
$949$	−2.43165e7	−0.876468
$950$	0	0
$951$	0	0
$952$	− 5.75462e6i	− 0.205790i
$953$	− 5.60285e6i	− 0.199838i	−0.994996	−	0.0999188i	$-0.968142\pi$
$953$	− 5.60285e6i	− 0.199838i	0.994996	−	0.0999188i	$-0.0318583\pi$
$954$	0	0
$955$	0	0
$956$	9.52512e6	0.337074
$957$	0	0
$958$	2.20421e7i	0.775959i
$959$	1.93801e7	0.680470
$960$	0	0
$961$	1.85403e7	0.647601
$962$	2.44116e7i	0.850470i
$963$	0	0
$964$	−4.38243e6	−0.151888
$965$	0	0
$966$	0	0
$967$	− 2.03532e7i	− 0.699949i	−0.936759	−	0.349975i	$-0.886190\pi$
$967$	− 2.03532e7i	− 0.699949i	0.936759	−	0.349975i	$-0.113810\pi$
$968$	− 7.94797e6i	− 0.272626i
$969$	0	0
$970$	0	0
$971$	2.34306e7	0.797510	0.398755	−	0.917057i	$-0.369442\pi$
$971$	2.34306e7	0.797510	0.398755	+	0.917057i	$0.369442\pi$
$972$	0	0
$973$	− 3.32878e7i	− 1.12721i
$974$	2.20723e7	0.745505
$975$	0	0
$976$	−2.51853e6	−0.0846296
$977$	4.30412e7i	1.44261i	0.692619	+	0.721303i	$0.256457\pi$
$977$	4.30412e7i	1.44261i	−0.692619	+	0.721303i	$0.743543\pi$
$978$	0	0
$979$	−7.39008e6	−0.246429
$980$	0	0
$981$	0	0
$982$	− 6.05107e6i	− 0.200241i
$983$	− 4.75003e7i	− 1.56788i	−0.620837	−	0.783940i	$-0.713207\pi$
$983$	− 4.75003e7i	− 1.56788i	0.620837	−	0.783940i	$-0.286793\pi$
$984$	0	0
$985$	0	0
$986$	−1.80137e7	−0.590079
$987$	0	0
$988$	4.84870e7i	1.58028i
$989$	−3.81164e6	−0.123914
$990$	0	0
$991$	2.09231e7	0.676770	0.338385	−	0.941008i	$-0.390119\pi$
$991$	2.09231e7	0.676770	0.338385	+	0.941008i	$0.390119\pi$
$992$	7.03283e6i	0.226909i
$993$	0	0
$994$	3.31401e7	1.06387
$995$	0	0
$996$	0	0
$997$	2.96332e7i	0.944148i	0.881559	+	0.472074i	$0.156495\pi$
$997$	2.96332e7i	0.944148i	−0.881559	+	0.472074i	$0.843505\pi$
$998$	− 7.72168e6i	− 0.245406i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.c.f.199.1		2
3.2	odd	2		50.6.b.b.49.2		2
5.2	odd	4		450.6.a.u.1.1		1
5.3	odd	4		90.6.a.b.1.1		1
5.4	even	2	inner	450.6.c.f.199.2		2
12.11	even	2		400.6.c.i.49.2		2
15.2	even	4		50.6.a.b.1.1		1
15.8	even	4		10.6.a.c.1.1	✓	1
15.14	odd	2		50.6.b.b.49.1		2
20.3	even	4		720.6.a.v.1.1		1
60.23	odd	4		80.6.a.c.1.1		1
60.47	odd	4		400.6.a.i.1.1		1
60.59	even	2		400.6.c.i.49.1		2
105.83	odd	4		490.6.a.k.1.1		1
120.53	even	4		320.6.a.f.1.1		1
120.83	odd	4		320.6.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.c.1.1	✓	1	15.8	even	4
50.6.a.b.1.1		1	15.2	even	4
50.6.b.b.49.1		2	15.14	odd	2
50.6.b.b.49.2		2	3.2	odd	2
80.6.a.c.1.1		1	60.23	odd	4
90.6.a.b.1.1		1	5.3	odd	4
320.6.a.f.1.1		1	120.53	even	4
320.6.a.k.1.1		1	120.83	odd	4
400.6.a.i.1.1		1	60.47	odd	4
400.6.c.i.49.1		2	60.59	even	2
400.6.c.i.49.2		2	12.11	even	2
450.6.a.u.1.1		1	5.2	odd	4
450.6.c.f.199.1		2	1.1	even	1	trivial
450.6.c.f.199.2		2	5.4	even	2	inner
490.6.a.k.1.1		1	105.83	odd	4
720.6.a.v.1.1		1	20.3	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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -16.0000 q^{4} -118.000i q^{7} +64.0000i q^{8} +O(q^{10})\)	\(q-4.00000i q^{2} -16.0000 q^{4} -118.000i q^{7} +64.0000i q^{8} -192.000 q^{11} -1106.00i q^{13} -472.000 q^{14} +256.000 q^{16} -762.000i q^{17} +2740.00 q^{19} +768.000i q^{22} +1566.00i q^{23} -4424.00 q^{26} +1888.00i q^{28} +5910.00 q^{29} -6868.00 q^{31} -1024.00i q^{32} -3048.00 q^{34} -5518.00i q^{37} -10960.0i q^{38} +378.000 q^{41} +2434.00i q^{43} +3072.00 q^{44} +6264.00 q^{46} -13122.0i q^{47} +2883.00 q^{49} +17696.0i q^{52} -9174.00i q^{53} +7552.00 q^{56} -23640.0i q^{58} -34980.0 q^{59} -9838.00 q^{61} +27472.0i q^{62} -4096.00 q^{64} +33722.0i q^{67} +12192.0i q^{68} -70212.0 q^{71} -21986.0i q^{73} -22072.0 q^{74} -43840.0 q^{76} +22656.0i q^{77} -4520.00 q^{79} -1512.00i q^{82} -109074. i q^{83} +9736.00 q^{86} -12288.0i q^{88} +38490.0 q^{89} -130508. q^{91} -25056.0i q^{92} -52488.0 q^{94} -1918.00i q^{97} -11532.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4}+O(q^{10}) \) 2 * q - 32 * q^4	\( 2 q - 32 q^{4} - 384 q^{11} - 944 q^{14} + 512 q^{16} + 5480 q^{19} - 8848 q^{26} + 11820 q^{29} - 13736 q^{31} - 6096 q^{34} + 756 q^{41} + 6144 q^{44} + 12528 q^{46} + 5766 q^{49} + 15104 q^{56} - 69960 q^{59} - 19676 q^{61} - 8192 q^{64} - 140424 q^{71} - 44144 q^{74} - 87680 q^{76} - 9040 q^{79} + 19472 q^{86} + 76980 q^{89} - 261016 q^{91} - 104976 q^{94}+O(q^{100}) \) 2 * q - 32 * q^4 - 384 * q^11 - 944 * q^14 + 512 * q^16 + 5480 * q^19 - 8848 * q^26 + 11820 * q^29 - 13736 * q^31 - 6096 * q^34 + 756 * q^41 + 6144 * q^44 + 12528 * q^46 + 5766 * q^49 + 15104 * q^56 - 69960 * q^59 - 19676 * q^61 - 8192 * q^64 - 140424 * q^71 - 44144 * q^74 - 87680 * q^76 - 9040 * q^79 + 19472 * q^86 + 76980 * q^89 - 261016 * q^91 - 104976 * q^94

Introduction

L-functions

Modular forms

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