LMFDB - Embedded newform 450.6.c.b.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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 30)
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.b.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -16.0000 q^{4} -164.000i q^{7} +64.0000i q^{8} +O(q^{10})$	$q-4.00000i q^{2} -16.0000 q^{4} -164.000i q^{7} +64.0000i q^{8} -720.000 q^{11} +698.000i q^{13} -656.000 q^{14} +256.000 q^{16} -2226.00i q^{17} -356.000 q^{19} +2880.00i q^{22} +1800.00i q^{23} +2792.00 q^{26} +2624.00i q^{28} +714.000 q^{29} +848.000 q^{31} -1024.00i q^{32} -8904.00 q^{34} +11302.0i q^{37} +1424.00i q^{38} -9354.00 q^{41} -5956.00i q^{43} +11520.0 q^{44} +7200.00 q^{46} -11160.0i q^{47} -10089.0 q^{49} -11168.0i q^{52} -14106.0i q^{53} +10496.0 q^{56} -2856.00i q^{58} +7920.00 q^{59} -13450.0 q^{61} -3392.00i q^{62} -4096.00 q^{64} +65476.0i q^{67} +35616.0i q^{68} -34560.0 q^{71} +86258.0i q^{73} +45208.0 q^{74} +5696.00 q^{76} +118080. i q^{77} +108832. q^{79} +37416.0i q^{82} -10668.0i q^{83} -23824.0 q^{86} -46080.0i q^{88} +10818.0 q^{89} +114472. q^{91} -28800.0i q^{92} -44640.0 q^{94} -4418.00i q^{97} +40356.0i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4}+O(q^{10}) $ 2 * q - 32 * q^4	$ 2 q - 32 q^{4} - 1440 q^{11} - 1312 q^{14} + 512 q^{16} - 712 q^{19} + 5584 q^{26} + 1428 q^{29} + 1696 q^{31} - 17808 q^{34} - 18708 q^{41} + 23040 q^{44} + 14400 q^{46} - 20178 q^{49} + 20992 q^{56} + 15840 q^{59} - 26900 q^{61} - 8192 q^{64} - 69120 q^{71} + 90416 q^{74} + 11392 q^{76} + 217664 q^{79} - 47648 q^{86} + 21636 q^{89} + 228944 q^{91} - 89280 q^{94}+O(q^{100}) $ 2 * q - 32 * q^4 - 1440 * q^11 - 1312 * q^14 + 512 * q^16 - 712 * q^19 + 5584 * q^26 + 1428 * q^29 + 1696 * q^31 - 17808 * q^34 - 18708 * q^41 + 23040 * q^44 + 14400 * q^46 - 20178 * q^49 + 20992 * q^56 + 15840 * q^59 - 26900 * q^61 - 8192 * q^64 - 69120 * q^71 + 90416 * q^74 + 11392 * q^76 + 217664 * q^79 - 47648 * q^86 + 21636 * q^89 + 228944 * q^91 - 89280 * 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
$7$	− 164.000i	− 1.26502i	−0.774551	−	0.632512i	$-0.782024\pi$
$7$	− 164.000i	− 1.26502i	0.774551	−	0.632512i	$-0.217976\pi$
$8$	64.0000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−720.000	−1.79412	−0.897059	−	0.441912i	$-0.854300\pi$
$11$	−720.000	−1.79412	−0.897059	+	0.441912i	$0.854300\pi$
$12$	0	0
$13$	698.000i	1.14551i	0.819728	+	0.572753i	$0.194124\pi$
$13$	698.000i	1.14551i	−0.819728	+	0.572753i	$0.805876\pi$
$14$	−656.000	−0.894507
$15$	0	0
$16$	256.000	0.250000
$17$	− 2226.00i	− 1.86811i	−0.357127	−	0.934056i	$-0.616244\pi$
$17$	− 2226.00i	− 1.86811i	0.357127	−	0.934056i	$-0.383756\pi$
$18$	0	0
$19$	−356.000	−0.226238	−0.113119	−	0.993581i	$-0.536084\pi$
$19$	−356.000	−0.226238	−0.113119	+	0.993581i	$0.536084\pi$
$20$	0	0
$21$	0	0
$22$	2880.00i	1.26863i
$23$	1800.00i	0.709501i	0.934961	+	0.354750i	$0.115434\pi$
$23$	1800.00i	0.709501i	−0.934961	+	0.354750i	$0.884566\pi$
$24$	0	0
$25$	0	0
$26$	2792.00	0.809994
$27$	0	0
$28$	2624.00i	0.632512i
$29$	714.000	0.157653	0.0788267	−	0.996888i	$-0.474883\pi$
$29$	714.000	0.157653	0.0788267	+	0.996888i	$0.474883\pi$
$30$	0	0
$31$	848.000	0.158486	0.0792431	−	0.996855i	$-0.474750\pi$
$31$	848.000	0.158486	0.0792431	+	0.996855i	$0.474750\pi$
$32$	− 1024.00i	− 0.176777i
$33$	0	0
$34$	−8904.00	−1.32095
$35$	0	0
$36$	0	0
$37$	11302.0i	1.35722i	0.734498	+	0.678611i	$0.237418\pi$
$37$	11302.0i	1.35722i	−0.734498	+	0.678611i	$0.762582\pi$
$38$	1424.00i	0.159975i
$39$	0	0
$40$	0	0
$41$	−9354.00	−0.869036	−0.434518	−	0.900663i	$-0.643081\pi$
$41$	−9354.00	−0.869036	−0.434518	+	0.900663i	$0.643081\pi$
$42$	0	0
$43$	− 5956.00i	− 0.491228i	−0.969368	−	0.245614i	$-0.921010\pi$
$43$	− 5956.00i	− 0.491228i	0.969368	−	0.245614i	$-0.0789896\pi$
$44$	11520.0	0.897059
$45$	0	0
$46$	7200.00	0.501693
$47$	− 11160.0i	− 0.736919i	−0.929644	−	0.368459i	$-0.879885\pi$
$47$	− 11160.0i	− 0.736919i	0.929644	−	0.368459i	$-0.120115\pi$
$48$	0	0
$49$	−10089.0	−0.600286
$50$	0	0
$51$	0	0
$52$	− 11168.0i	− 0.572753i
$53$	− 14106.0i	− 0.689786i	−0.938642	−	0.344893i	$-0.887915\pi$
$53$	− 14106.0i	− 0.689786i	0.938642	−	0.344893i	$-0.112085\pi$
$54$	0	0
$55$	0	0
$56$	10496.0	0.447254
$57$	0	0
$58$	− 2856.00i	− 0.111478i
$59$	7920.00	0.296207	0.148103	−	0.988972i	$-0.452683\pi$
$59$	7920.00	0.296207	0.148103	+	0.988972i	$0.452683\pi$
$60$	0	0
$61$	−13450.0	−0.462805	−0.231402	−	0.972858i	$-0.574331\pi$
$61$	−13450.0	−0.462805	−0.231402	+	0.972858i	$0.574331\pi$
$62$	− 3392.00i	− 0.112067i
$63$	0	0
$64$	−4096.00	−0.125000
$65$	0	0
$66$	0	0
$67$	65476.0i	1.78195i	0.454054	+	0.890974i	$0.349977\pi$
$67$	65476.0i	1.78195i	−0.454054	+	0.890974i	$0.650023\pi$
$68$	35616.0i	0.934056i
$69$	0	0
$70$	0	0
$71$	−34560.0	−0.813632	−0.406816	−	0.913510i	$-0.633361\pi$
$71$	−34560.0	−0.813632	−0.406816	+	0.913510i	$0.633361\pi$
$72$	0	0
$73$	86258.0i	1.89449i	0.320511	+	0.947245i	$0.396145\pi$
$73$	86258.0i	1.89449i	−0.320511	+	0.947245i	$0.603855\pi$
$74$	45208.0	0.959701
$75$	0	0
$76$	5696.00	0.113119
$77$	118080.i	2.26960i
$78$	0	0
$79$	108832.	1.96195	0.980977	−	0.194123i	$-0.0621862\pi$
$79$	108832.	1.96195	0.980977	+	0.194123i	$0.0621862\pi$
$80$	0	0
$81$	0	0
$82$	37416.0i	0.614501i
$83$	− 10668.0i	− 0.169976i	−0.996382	−	0.0849880i	$-0.972915\pi$
$83$	− 10668.0i	− 0.169976i	0.996382	−	0.0849880i	$-0.0270852\pi$
$84$	0	0
$85$	0	0
$86$	−23824.0	−0.347351
$87$	0	0
$88$	− 46080.0i	− 0.634316i
$89$	10818.0	0.144768	0.0723839	−	0.997377i	$-0.476939\pi$
$89$	10818.0	0.144768	0.0723839	+	0.997377i	$0.476939\pi$
$90$	0	0
$91$	114472.	1.44909
$92$	− 28800.0i	− 0.354750i
$93$	0	0
$94$	−44640.0	−0.521080
$95$	0	0
$96$	0	0
$97$	− 4418.00i	− 0.0476756i	−0.999716	−	0.0238378i	$-0.992411\pi$
$97$	− 4418.00i	− 0.0476756i	0.999716	−	0.0238378i	$-0.00758853\pi$
$98$	40356.0i	0.424466i
$99$	0	0
$100$	0	0
$101$	102942.	1.00413	0.502064	−	0.864830i	$-0.332574\pi$
$101$	102942.	1.00413	0.502064	+	0.864830i	$0.332574\pi$
$102$	0	0
$103$	− 69436.0i	− 0.644899i	−0.946587	−	0.322449i	$-0.895494\pi$
$103$	− 69436.0i	− 0.644899i	0.946587	−	0.322449i	$-0.104506\pi$
$104$	−44672.0	−0.404997
$105$	0	0
$106$	−56424.0	−0.487752
$107$	17412.0i	0.147024i	0.997294	+	0.0735122i	$0.0234208\pi$
$107$	17412.0i	0.147024i	−0.997294	+	0.0735122i	$0.976579\pi$
$108$	0	0
$109$	203770.	1.64276	0.821380	−	0.570382i	$-0.193205\pi$
$109$	203770.	1.64276	0.821380	+	0.570382i	$0.193205\pi$
$110$	0	0
$111$	0	0
$112$	− 41984.0i	− 0.316256i
$113$	212202.i	1.56334i	0.623692	+	0.781670i	$0.285632\pi$
$113$	212202.i	1.56334i	−0.623692	+	0.781670i	$0.714368\pi$
$114$	0	0
$115$	0	0
$116$	−11424.0	−0.0788267
$117$	0	0
$118$	− 31680.0i	− 0.209450i
$119$	−365064.	−2.36321
$120$	0	0
$121$	357349.	2.21886
$122$	53800.0i	0.327252i
$123$	0	0
$124$	−13568.0	−0.0792431
$125$	0	0
$126$	0	0
$127$	− 6140.00i	− 0.0337800i	−0.999857	−	0.0168900i	$-0.994623\pi$
$127$	− 6140.00i	− 0.0337800i	0.999857	−	0.0168900i	$-0.00537650\pi$
$128$	16384.0i	0.0883883i
$129$	0	0
$130$	0	0
$131$	205920.	1.04838	0.524192	−	0.851600i	$-0.324367\pi$
$131$	205920.	1.04838	0.524192	+	0.851600i	$0.324367\pi$
$132$	0	0
$133$	58384.0i	0.286197i
$134$	261904.	1.26003
$135$	0	0
$136$	142464.	0.660477
$137$	230334.i	1.04847i	0.851573	+	0.524236i	$0.175649\pi$
$137$	230334.i	1.04847i	−0.851573	+	0.524236i	$0.824351\pi$
$138$	0	0
$139$	−260756.	−1.14471	−0.572357	−	0.820004i	$-0.693971\pi$
$139$	−260756.	−1.14471	−0.572357	+	0.820004i	$0.693971\pi$
$140$	0	0
$141$	0	0
$142$	138240.i	0.575324i
$143$	− 502560.i	− 2.05517i
$144$	0	0
$145$	0	0
$146$	345032.	1.33961
$147$	0	0
$148$	− 180832.i	− 0.678611i
$149$	−29526.0	−0.108953	−0.0544765	−	0.998515i	$-0.517349\pi$
$149$	−29526.0	−0.108953	−0.0544765	+	0.998515i	$0.517349\pi$
$150$	0	0
$151$	125168.	0.446736	0.223368	−	0.974734i	$-0.428295\pi$
$151$	125168.	0.446736	0.223368	+	0.974734i	$0.428295\pi$
$152$	− 22784.0i	− 0.0799873i
$153$	0	0
$154$	472320.	1.60485
$155$	0	0
$156$	0	0
$157$	43222.0i	0.139944i	0.997549	+	0.0699722i	$0.0222911\pi$
$157$	43222.0i	0.139944i	−0.997549	+	0.0699722i	$0.977709\pi$
$158$	− 435328.i	− 1.38731i
$159$	0	0
$160$	0	0
$161$	295200.	0.897536
$162$	0	0
$163$	− 293476.i	− 0.865174i	−0.901592	−	0.432587i	$-0.857601\pi$
$163$	− 293476.i	− 0.865174i	0.901592	−	0.432587i	$-0.142399\pi$
$164$	149664.	0.434518
$165$	0	0
$166$	−42672.0	−0.120191
$167$	322200.i	0.893993i	0.894536	+	0.446997i	$0.147506\pi$
$167$	322200.i	0.893993i	−0.894536	+	0.446997i	$0.852494\pi$
$168$	0	0
$169$	−115911.	−0.312182
$170$	0	0
$171$	0	0
$172$	95296.0i	0.245614i
$173$	261918.i	0.665350i	0.943042	+	0.332675i	$0.107951\pi$
$173$	261918.i	0.665350i	−0.943042	+	0.332675i	$0.892049\pi$
$174$	0	0
$175$	0	0
$176$	−184320.	−0.448529
$177$	0	0
$178$	− 43272.0i	− 0.102366i
$179$	623544.	1.45457	0.727285	−	0.686336i	$-0.240782\pi$
$179$	623544.	1.45457	0.727285	+	0.686336i	$0.240782\pi$
$180$	0	0
$181$	−61186.0	−0.138821	−0.0694106	−	0.997588i	$-0.522112\pi$
$181$	−61186.0	−0.138821	−0.0694106	+	0.997588i	$0.522112\pi$
$182$	− 457888.i	− 1.02466i
$183$	0	0
$184$	−115200.	−0.250846
$185$	0	0
$186$	0	0
$187$	1.60272e6i	3.35161i
$188$	178560.i	0.368459i
$189$	0	0
$190$	0	0
$191$	−737256.	−1.46229	−0.731147	−	0.682220i	$-0.761015\pi$
$191$	−737256.	−1.46229	−0.731147	+	0.682220i	$0.761015\pi$
$192$	0	0
$193$	539162.i	1.04190i	0.853587	+	0.520950i	$0.174422\pi$
$193$	539162.i	1.04190i	−0.853587	+	0.520950i	$0.825578\pi$
$194$	−17672.0	−0.0337118
$195$	0	0
$196$	161424.	0.300143
$197$	− 651174.i	− 1.19545i	−0.801701	−	0.597725i	$-0.796071\pi$
$197$	− 651174.i	− 1.19545i	0.801701	−	0.597725i	$-0.203929\pi$
$198$	0	0
$199$	−157328.	−0.281626	−0.140813	−	0.990036i	$-0.544972\pi$
$199$	−157328.	−0.281626	−0.140813	+	0.990036i	$0.544972\pi$
$200$	0	0
$201$	0	0
$202$	− 411768.i	− 0.710026i
$203$	− 117096.i	− 0.199435i
$204$	0	0
$205$	0	0
$206$	−277744.	−0.456012
$207$	0	0
$208$	178688.i	0.286376i
$209$	256320.	0.405898
$210$	0	0
$211$	707180.	1.09351	0.546756	−	0.837292i	$-0.315862\pi$
$211$	707180.	1.09351	0.546756	+	0.837292i	$0.315862\pi$
$212$	225696.i	0.344893i
$213$	0	0
$214$	69648.0	0.103962
$215$	0	0
$216$	0	0
$217$	− 139072.i	− 0.200489i
$218$	− 815080.i	− 1.16161i
$219$	0	0
$220$	0	0
$221$	1.55375e6	2.13993
$222$	0	0
$223$	− 530740.i	− 0.714693i	−0.933972	−	0.357347i	$-0.883681\pi$
$223$	− 530740.i	− 0.714693i	0.933972	−	0.357347i	$-0.116319\pi$
$224$	−167936.	−0.223627
$225$	0	0
$226$	848808.	1.10545
$227$	120372.i	0.155046i	0.996991	+	0.0775230i	$0.0247011\pi$
$227$	120372.i	0.155046i	−0.996991	+	0.0775230i	$0.975299\pi$
$228$	0	0
$229$	−772310.	−0.973202	−0.486601	−	0.873624i	$-0.661763\pi$
$229$	−772310.	−0.973202	−0.486601	+	0.873624i	$0.661763\pi$
$230$	0	0
$231$	0	0
$232$	45696.0i	0.0557389i
$233$	− 8838.00i	− 0.0106651i	−0.999986	−	0.00533254i	$-0.998303\pi$
$233$	− 8838.00i	− 0.0106651i	0.999986	−	0.00533254i	$-0.00169741\pi$
$234$	0	0
$235$	0	0
$236$	−126720.	−0.148103
$237$	0	0
$238$	1.46026e6i	1.67104i
$239$	−775416.	−0.878092	−0.439046	−	0.898465i	$-0.644683\pi$
$239$	−775416.	−0.878092	−0.439046	+	0.898465i	$0.644683\pi$
$240$	0	0
$241$	−373438.	−0.414167	−0.207084	−	0.978323i	$-0.566397\pi$
$241$	−373438.	−0.414167	−0.207084	+	0.978323i	$0.566397\pi$
$242$	− 1.42940e6i	− 1.56897i
$243$	0	0
$244$	215200.	0.231402
$245$	0	0
$246$	0	0
$247$	− 248488.i	− 0.259157i
$248$	54272.0i	0.0560334i
$249$	0	0
$250$	0	0
$251$	−71976.0	−0.0721113	−0.0360557	−	0.999350i	$-0.511479\pi$
$251$	−71976.0	−0.0721113	−0.0360557	+	0.999350i	$0.511479\pi$
$252$	0	0
$253$	− 1.29600e6i	− 1.27293i
$254$	−24560.0	−0.0238860
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.59356e6i	1.50500i	0.658595	+	0.752498i	$0.271151\pi$
$257$	1.59356e6i	1.50500i	−0.658595	+	0.752498i	$0.728849\pi$
$258$	0	0
$259$	1.85353e6	1.71692
$260$	0	0
$261$	0	0
$262$	− 823680.i	− 0.741319i
$263$	2.05452e6i	1.83156i	0.401681	+	0.915780i	$0.368426\pi$
$263$	2.05452e6i	1.83156i	−0.401681	+	0.915780i	$0.631574\pi$
$264$	0	0
$265$	0	0
$266$	233536.	0.202372
$267$	0	0
$268$	− 1.04762e6i	− 0.890974i
$269$	−258486.	−0.217799	−0.108900	−	0.994053i	$-0.534733\pi$
$269$	−258486.	−0.217799	−0.108900	+	0.994053i	$0.534733\pi$
$270$	0	0
$271$	−1.98398e6	−1.64102	−0.820509	−	0.571634i	$-0.806310\pi$
$271$	−1.98398e6	−1.64102	−0.820509	+	0.571634i	$0.806310\pi$
$272$	− 569856.i	− 0.467028i
$273$	0	0
$274$	921336.	0.741381
$275$	0	0
$276$	0	0
$277$	− 1.61326e6i	− 1.26329i	−0.775256	−	0.631647i	$-0.782379\pi$
$277$	− 1.61326e6i	− 1.26329i	0.775256	−	0.631647i	$-0.217621\pi$
$278$	1.04302e6i	0.809436i
$279$	0	0
$280$	0	0
$281$	−1.37882e6	−1.04170	−0.520848	−	0.853649i	$-0.674384\pi$
$281$	−1.37882e6	−1.04170	−0.520848	+	0.853649i	$0.674384\pi$
$282$	0	0
$283$	1.45831e6i	1.08239i	0.840898	+	0.541194i	$0.182028\pi$
$283$	1.45831e6i	1.08239i	−0.840898	+	0.541194i	$0.817972\pi$
$284$	552960.	0.406816
$285$	0	0
$286$	−2.01024e6	−1.45322
$287$	1.53406e6i	1.09935i
$288$	0	0
$289$	−3.53522e6	−2.48984
$290$	0	0
$291$	0	0
$292$	− 1.38013e6i	− 0.947245i
$293$	988134.i	0.672430i	0.941785	+	0.336215i	$0.109147\pi$
$293$	988134.i	0.672430i	−0.941785	+	0.336215i	$0.890853\pi$
$294$	0	0
$295$	0	0
$296$	−723328.	−0.479851
$297$	0	0
$298$	118104.i	0.0770414i
$299$	−1.25640e6	−0.812737
$300$	0	0
$301$	−976784.	−0.621416
$302$	− 500672.i	− 0.315890i
$303$	0	0
$304$	−91136.0	−0.0565596
$305$	0	0
$306$	0	0
$307$	393820.i	0.238480i	0.992865	+	0.119240i	$0.0380458\pi$
$307$	393820.i	0.238480i	−0.992865	+	0.119240i	$0.961954\pi$
$308$	− 1.88928e6i	− 1.13480i
$309$	0	0
$310$	0	0
$311$	−1.55448e6	−0.911348	−0.455674	−	0.890147i	$-0.650602\pi$
$311$	−1.55448e6	−0.911348	−0.455674	+	0.890147i	$0.650602\pi$
$312$	0	0
$313$	1.76050e6i	1.01572i	0.861439	+	0.507861i	$0.169564\pi$
$313$	1.76050e6i	1.01572i	−0.861439	+	0.507861i	$0.830436\pi$
$314$	172888.	0.0989557
$315$	0	0
$316$	−1.74131e6	−0.980977
$317$	2.37112e6i	1.32527i	0.748941	+	0.662637i	$0.230563\pi$
$317$	2.37112e6i	1.32527i	−0.748941	+	0.662637i	$0.769437\pi$
$318$	0	0
$319$	−514080.	−0.282849
$320$	0	0
$321$	0	0
$322$	− 1.18080e6i	− 0.634653i
$323$	792456.i	0.422638i
$324$	0	0
$325$	0	0
$326$	−1.17390e6	−0.611771
$327$	0	0
$328$	− 598656.i	− 0.307251i
$329$	−1.83024e6	−0.932220
$330$	0	0
$331$	−980068.	−0.491684	−0.245842	−	0.969310i	$-0.579064\pi$
$331$	−980068.	−0.491684	−0.245842	+	0.969310i	$0.579064\pi$
$332$	170688.i	0.0849880i
$333$	0	0
$334$	1.28880e6	0.632149
$335$	0	0
$336$	0	0
$337$	− 905834.i	− 0.434484i	−0.976118	−	0.217242i	$-0.930294\pi$
$337$	− 905834.i	− 0.434484i	0.976118	−	0.217242i	$-0.0697061\pi$
$338$	463644.i	0.220746i
$339$	0	0
$340$	0	0
$341$	−610560.	−0.284343
$342$	0	0
$343$	− 1.10175e6i	− 0.505648i
$344$	381184.	0.173675
$345$	0	0
$346$	1.04767e6	0.470473
$347$	− 2.95069e6i	− 1.31553i	−0.753224	−	0.657764i	$-0.771502\pi$
$347$	− 2.95069e6i	− 1.31553i	0.753224	−	0.657764i	$-0.228498\pi$
$348$	0	0
$349$	2.15761e6	0.948221	0.474110	−	0.880465i	$-0.342770\pi$
$349$	2.15761e6	0.948221	0.474110	+	0.880465i	$0.342770\pi$
$350$	0	0
$351$	0	0
$352$	737280.i	0.317158i
$353$	− 1.28873e6i	− 0.550461i	−0.961378	−	0.275230i	$-0.911246\pi$
$353$	− 1.28873e6i	− 0.550461i	0.961378	−	0.275230i	$-0.0887542\pi$
$354$	0	0
$355$	0	0
$356$	−173088.	−0.0723839
$357$	0	0
$358$	− 2.49418e6i	− 1.02854i
$359$	−2.26946e6	−0.929367	−0.464683	−	0.885477i	$-0.653832\pi$
$359$	−2.26946e6	−0.929367	−0.464683	+	0.885477i	$0.653832\pi$
$360$	0	0
$361$	−2.34936e6	−0.948816
$362$	244744.i	0.0981614i
$363$	0	0
$364$	−1.83155e6	−0.724546
$365$	0	0
$366$	0	0
$367$	− 1.04659e6i	− 0.405612i	−0.979219	−	0.202806i	$-0.934994\pi$
$367$	− 1.04659e6i	− 0.405612i	0.979219	−	0.202806i	$-0.0650060\pi$
$368$	460800.i	0.177375i
$369$	0	0
$370$	0	0
$371$	−2.31338e6	−0.872595
$372$	0	0
$373$	1.79827e6i	0.669243i	0.942353	+	0.334621i	$0.108608\pi$
$373$	1.79827e6i	0.669243i	−0.942353	+	0.334621i	$0.891392\pi$
$374$	6.41088e6	2.36995
$375$	0	0
$376$	714240.	0.260540
$377$	498372.i	0.180593i
$378$	0	0
$379$	2.18412e6	0.781051	0.390525	−	0.920592i	$-0.372293\pi$
$379$	2.18412e6	0.781051	0.390525	+	0.920592i	$0.372293\pi$
$380$	0	0
$381$	0	0
$382$	2.94902e6i	1.03400i
$383$	− 1.78452e6i	− 0.621619i	−0.950472	−	0.310810i	$-0.899400\pi$
$383$	− 1.78452e6i	− 0.621619i	0.950472	−	0.310810i	$-0.100600\pi$
$384$	0	0
$385$	0	0
$386$	2.15665e6	0.736734
$387$	0	0
$388$	70688.0i	0.0238378i
$389$	−1.10953e6	−0.371761	−0.185880	−	0.982572i	$-0.559514\pi$
$389$	−1.10953e6	−0.371761	−0.185880	+	0.982572i	$0.559514\pi$
$390$	0	0
$391$	4.00680e6	1.32543
$392$	− 645696.i	− 0.212233i
$393$	0	0
$394$	−2.60470e6	−0.845311
$395$	0	0
$396$	0	0
$397$	− 3.89568e6i	− 1.24053i	−0.784392	−	0.620265i	$-0.787025\pi$
$397$	− 3.89568e6i	− 1.24053i	0.784392	−	0.620265i	$-0.212975\pi$
$398$	629312.i	0.199140i
$399$	0	0
$400$	0	0
$401$	1.20673e6	0.374755	0.187378	−	0.982288i	$-0.440001\pi$
$401$	1.20673e6	0.374755	0.187378	+	0.982288i	$0.440001\pi$
$402$	0	0
$403$	591904.i	0.181547i
$404$	−1.64707e6	−0.502064
$405$	0	0
$406$	−468384.	−0.141022
$407$	− 8.13744e6i	− 2.43502i
$408$	0	0
$409$	−5.61363e6	−1.65934	−0.829670	−	0.558255i	$-0.811471\pi$
$409$	−5.61363e6	−1.65934	−0.829670	+	0.558255i	$0.811471\pi$
$410$	0	0
$411$	0	0
$412$	1.11098e6i	0.322449i
$413$	− 1.29888e6i	− 0.374709i
$414$	0	0
$415$	0	0
$416$	714752.	0.202499
$417$	0	0
$418$	− 1.02528e6i	− 0.287013i
$419$	1.15056e6	0.320165	0.160083	−	0.987104i	$-0.448824\pi$
$419$	1.15056e6	0.320165	0.160083	+	0.987104i	$0.448824\pi$
$420$	0	0
$421$	−3.83089e6	−1.05340	−0.526701	−	0.850050i	$-0.676571\pi$
$421$	−3.83089e6	−1.05340	−0.526701	+	0.850050i	$0.676571\pi$
$422$	− 2.82872e6i	− 0.773230i
$423$	0	0
$424$	902784.	0.243876
$425$	0	0
$426$	0	0
$427$	2.20580e6i	0.585459i
$428$	− 278592.i	− 0.0735122i
$429$	0	0
$430$	0	0
$431$	−155520.	−0.0403267	−0.0201634	−	0.999797i	$-0.506419\pi$
$431$	−155520.	−0.0403267	−0.0201634	+	0.999797i	$0.506419\pi$
$432$	0	0
$433$	4.14391e6i	1.06216i	0.847321	+	0.531081i	$0.178214\pi$
$433$	4.14391e6i	1.06216i	−0.847321	+	0.531081i	$0.821786\pi$
$434$	−556288.	−0.141767
$435$	0	0
$436$	−3.26032e6	−0.821380
$437$	− 640800.i	− 0.160516i
$438$	0	0
$439$	−6.23653e6	−1.54448	−0.772239	−	0.635332i	$-0.780863\pi$
$439$	−6.23653e6	−1.54448	−0.772239	+	0.635332i	$0.780863\pi$
$440$	0	0
$441$	0	0
$442$	− 6.21499e6i	− 1.51316i
$443$	− 4.52507e6i	− 1.09551i	−0.836639	−	0.547754i	$-0.815483\pi$
$443$	− 4.52507e6i	− 1.09551i	0.836639	−	0.547754i	$-0.184517\pi$
$444$	0	0
$445$	0	0
$446$	−2.12296e6	−0.505364
$447$	0	0
$448$	671744.i	0.158128i
$449$	2.56463e6	0.600357	0.300178	−	0.953883i	$-0.402954\pi$
$449$	2.56463e6	0.600357	0.300178	+	0.953883i	$0.402954\pi$
$450$	0	0
$451$	6.73488e6	1.55915
$452$	− 3.39523e6i	− 0.781670i
$453$	0	0
$454$	481488.	0.109634
$455$	0	0
$456$	0	0
$457$	5.53409e6i	1.23953i	0.784789	+	0.619763i	$0.212771\pi$
$457$	5.53409e6i	1.23953i	−0.784789	+	0.619763i	$0.787229\pi$
$458$	3.08924e6i	0.688158i
$459$	0	0
$460$	0	0
$461$	7.19211e6	1.57617	0.788087	−	0.615564i	$-0.211072\pi$
$461$	7.19211e6	1.57617	0.788087	+	0.615564i	$0.211072\pi$
$462$	0	0
$463$	− 1.13936e6i	− 0.247006i	−0.992344	−	0.123503i	$-0.960587\pi$
$463$	− 1.13936e6i	− 0.247006i	0.992344	−	0.123503i	$-0.0394128\pi$
$464$	182784.	0.0394133
$465$	0	0
$466$	−35352.0	−0.00754135
$467$	7.36168e6i	1.56201i	0.624523	+	0.781006i	$0.285293\pi$
$467$	7.36168e6i	1.56201i	−0.624523	+	0.781006i	$0.714707\pi$
$468$	0	0
$469$	1.07381e7	2.25421
$470$	0	0
$471$	0	0
$472$	506880.i	0.104725i
$473$	4.28832e6i	0.881321i
$474$	0	0
$475$	0	0
$476$	5.84102e6	1.18160
$477$	0	0
$478$	3.10166e6i	0.620905i
$479$	−1.36226e6	−0.271283	−0.135641	−	0.990758i	$-0.543310\pi$
$479$	−1.36226e6	−0.271283	−0.135641	+	0.990758i	$0.543310\pi$
$480$	0	0
$481$	−7.88880e6	−1.55471
$482$	1.49375e6i	0.292861i
$483$	0	0
$484$	−5.71758e6	−1.10943
$485$	0	0
$486$	0	0
$487$	− 606428.i	− 0.115866i	−0.998320	−	0.0579331i	$-0.981549\pi$
$487$	− 606428.i	− 0.115866i	0.998320	−	0.0579331i	$-0.0184510\pi$
$488$	− 860800.i	− 0.163626i
$489$	0	0
$490$	0	0
$491$	5.84278e6	1.09374	0.546872	−	0.837216i	$-0.315819\pi$
$491$	5.84278e6	1.09374	0.546872	+	0.837216i	$0.315819\pi$
$492$	0	0
$493$	− 1.58936e6i	− 0.294514i
$494$	−993952.	−0.183252
$495$	0	0
$496$	217088.	0.0396216
$497$	5.66784e6i	1.02926i
$498$	0	0
$499$	1.15044e6	0.206830	0.103415	−	0.994638i	$-0.467023\pi$
$499$	1.15044e6	0.206830	0.103415	+	0.994638i	$0.467023\pi$
$500$	0	0
$501$	0	0
$502$	287904.i	0.0509904i
$503$	869664.i	0.153261i	0.997060	+	0.0766305i	$0.0244162\pi$
$503$	869664.i	0.153261i	−0.997060	+	0.0766305i	$0.975584\pi$
$504$	0	0
$505$	0	0
$506$	−5.18400e6	−0.900096
$507$	0	0
$508$	98240.0i	0.0168900i
$509$	1.43495e6	0.245495	0.122748	−	0.992438i	$-0.460829\pi$
$509$	1.43495e6	0.245495	0.122748	+	0.992438i	$0.460829\pi$
$510$	0	0
$511$	1.41463e7	2.39657
$512$	− 262144.i	− 0.0441942i
$513$	0	0
$514$	6.37423e6	1.06419
$515$	0	0
$516$	0	0
$517$	8.03520e6i	1.32212i
$518$	− 7.41411e6i	− 1.21404i
$519$	0	0
$520$	0	0
$521$	−1.04371e7	−1.68456	−0.842281	−	0.539038i	$-0.818788\pi$
$521$	−1.04371e7	−1.68456	−0.842281	+	0.539038i	$0.818788\pi$
$522$	0	0
$523$	− 7.75942e6i	− 1.24044i	−0.784429	−	0.620219i	$-0.787044\pi$
$523$	− 7.75942e6i	− 1.24044i	0.784429	−	0.620219i	$-0.212956\pi$
$524$	−3.29472e6	−0.524192
$525$	0	0
$526$	8.21808e6	1.29511
$527$	− 1.88765e6i	− 0.296070i
$528$	0	0
$529$	3.19634e6	0.496609
$530$	0	0
$531$	0	0
$532$	− 934144.i	− 0.143098i
$533$	− 6.52909e6i	− 0.995485i
$534$	0	0
$535$	0	0
$536$	−4.19046e6	−0.630014
$537$	0	0
$538$	1.03394e6i	0.154007i
$539$	7.26408e6	1.07698
$540$	0	0
$541$	−1.10233e6	−0.161927	−0.0809633	−	0.996717i	$-0.525800\pi$
$541$	−1.10233e6	−0.161927	−0.0809633	+	0.996717i	$0.525800\pi$
$542$	7.93590e6i	1.16037i
$543$	0	0
$544$	−2.27942e6	−0.330239
$545$	0	0
$546$	0	0
$547$	− 2.48263e6i	− 0.354767i	−0.984142	−	0.177384i	$-0.943237\pi$
$547$	− 2.48263e6i	− 0.354767i	0.984142	−	0.177384i	$-0.0567633\pi$
$548$	− 3.68534e6i	− 0.524236i
$549$	0	0
$550$	0	0
$551$	−254184.	−0.0356672
$552$	0	0
$553$	− 1.78484e7i	− 2.48192i
$554$	−6.45303e6	−0.893284
$555$	0	0
$556$	4.17210e6	0.572357
$557$	5.73568e6i	0.783334i	0.920107	+	0.391667i	$0.128102\pi$
$557$	5.73568e6i	0.783334i	−0.920107	+	0.391667i	$0.871898\pi$
$558$	0	0
$559$	4.15729e6	0.562705
$560$	0	0
$561$	0	0
$562$	5.51527e6i	0.736591i
$563$	− 517092.i	− 0.0687538i	−0.999409	−	0.0343769i	$-0.989055\pi$
$563$	− 517092.i	− 0.0687538i	0.999409	−	0.0343769i	$-0.0109447\pi$
$564$	0	0
$565$	0	0
$566$	5.83323e6	0.765364
$567$	0	0
$568$	− 2.21184e6i	− 0.287662i
$569$	−6.72766e6	−0.871131	−0.435566	−	0.900157i	$-0.643452\pi$
$569$	−6.72766e6	−0.871131	−0.435566	+	0.900157i	$0.643452\pi$
$570$	0	0
$571$	−1.03290e7	−1.32577	−0.662883	−	0.748723i	$-0.730668\pi$
$571$	−1.03290e7	−1.32577	−0.662883	+	0.748723i	$0.730668\pi$
$572$	8.04096e6i	1.02759i
$573$	0	0
$574$	6.13622e6	0.777359
$575$	0	0
$576$	0	0
$577$	− 9.25834e6i	− 1.15769i	−0.815436	−	0.578847i	$-0.803503\pi$
$577$	− 9.25834e6i	− 1.15769i	0.815436	−	0.578847i	$-0.196497\pi$
$578$	1.41409e7i	1.76058i
$579$	0	0
$580$	0	0
$581$	−1.74955e6	−0.215024
$582$	0	0
$583$	1.01563e7i	1.23756i
$584$	−5.52051e6	−0.669803
$585$	0	0
$586$	3.95254e6	0.475479
$587$	− 9.57155e6i	− 1.14653i	−0.819369	−	0.573267i	$-0.805676\pi$
$587$	− 9.57155e6i	− 1.14653i	0.819369	−	0.573267i	$-0.194324\pi$
$588$	0	0
$589$	−301888.	−0.0358557
$590$	0	0
$591$	0	0
$592$	2.89331e6i	0.339306i
$593$	1.12388e7i	1.31245i	0.754564	+	0.656226i	$0.227848\pi$
$593$	1.12388e7i	1.31245i	−0.754564	+	0.656226i	$0.772152\pi$
$594$	0	0
$595$	0	0
$596$	472416.	0.0544765
$597$	0	0
$598$	5.02560e6i	0.574692i
$599$	3.72670e6	0.424382	0.212191	−	0.977228i	$-0.431940\pi$
$599$	3.72670e6	0.424382	0.212191	+	0.977228i	$0.431940\pi$
$600$	0	0
$601$	6.74734e6	0.761985	0.380992	−	0.924578i	$-0.375582\pi$
$601$	6.74734e6	0.761985	0.380992	+	0.924578i	$0.375582\pi$
$602$	3.90714e6i	0.439407i
$603$	0	0
$604$	−2.00269e6	−0.223368
$605$	0	0
$606$	0	0
$607$	6.83384e6i	0.752823i	0.926453	+	0.376411i	$0.122842\pi$
$607$	6.83384e6i	0.752823i	−0.926453	+	0.376411i	$0.877158\pi$
$608$	364544.i	0.0399936i
$609$	0	0
$610$	0	0
$611$	7.78968e6	0.844144
$612$	0	0
$613$	− 433222.i	− 0.0465650i	−0.999729	−	0.0232825i	$-0.992588\pi$
$613$	− 433222.i	− 0.0465650i	0.999729	−	0.0232825i	$-0.00741171\pi$
$614$	1.57528e6	0.168631
$615$	0	0
$616$	−7.55712e6	−0.802425
$617$	− 5.17569e6i	− 0.547338i	−0.961824	−	0.273669i	$-0.911763\pi$
$617$	− 5.17569e6i	− 0.547338i	0.961824	−	0.273669i	$-0.0882372\pi$
$618$	0	0
$619$	151996.	0.0159443	0.00797215	−	0.999968i	$-0.497462\pi$
$619$	151996.	0.0159443	0.00797215	+	0.999968i	$0.497462\pi$
$620$	0	0
$621$	0	0
$622$	6.21792e6i	0.644420i
$623$	− 1.77415e6i	− 0.183135i
$624$	0	0
$625$	0	0
$626$	7.04199e6	0.718224
$627$	0	0
$628$	− 691552.i	− 0.0699722i
$629$	2.51583e7	2.53544
$630$	0	0
$631$	1.05635e7	1.05617	0.528086	−	0.849191i	$-0.322910\pi$
$631$	1.05635e7	1.05617	0.528086	+	0.849191i	$0.322910\pi$
$632$	6.96525e6i	0.693656i
$633$	0	0
$634$	9.48449e6	0.937110
$635$	0	0
$636$	0	0
$637$	− 7.04212e6i	− 0.687630i
$638$	2.05632e6i	0.200004i
$639$	0	0
$640$	0	0
$641$	5.53755e6	0.532320	0.266160	−	0.963929i	$-0.414245\pi$
$641$	5.53755e6	0.532320	0.266160	+	0.963929i	$0.414245\pi$
$642$	0	0
$643$	8.89132e6i	0.848084i	0.905642	+	0.424042i	$0.139389\pi$
$643$	8.89132e6i	0.848084i	−0.905642	+	0.424042i	$0.860611\pi$
$644$	−4.72320e6	−0.448768
$645$	0	0
$646$	3.16982e6	0.298850
$647$	2.29474e6i	0.215512i	0.994177	+	0.107756i	$0.0343666\pi$
$647$	2.29474e6i	0.215512i	−0.994177	+	0.107756i	$0.965633\pi$
$648$	0	0
$649$	−5.70240e6	−0.531430
$650$	0	0
$651$	0	0
$652$	4.69562e6i	0.432587i
$653$	1.36338e7i	1.25122i	0.780137	+	0.625608i	$0.215149\pi$
$653$	1.36338e7i	1.25122i	−0.780137	+	0.625608i	$0.784851\pi$
$654$	0	0
$655$	0	0
$656$	−2.39462e6	−0.217259
$657$	0	0
$658$	7.32096e6i	0.659179i
$659$	−1.31234e7	−1.17715	−0.588576	−	0.808442i	$-0.700311\pi$
$659$	−1.31234e7	−1.17715	−0.588576	+	0.808442i	$0.700311\pi$
$660$	0	0
$661$	1.78522e7	1.58923	0.794616	−	0.607112i	$-0.207672\pi$
$661$	1.78522e7	1.58923	0.794616	+	0.607112i	$0.207672\pi$
$662$	3.92027e6i	0.347673i
$663$	0	0
$664$	682752.	0.0600956
$665$	0	0
$666$	0	0
$667$	1.28520e6i	0.111855i
$668$	− 5.15520e6i	− 0.446997i
$669$	0	0
$670$	0	0
$671$	9.68400e6	0.830326
$672$	0	0
$673$	1.32471e7i	1.12741i	0.825975	+	0.563707i	$0.190625\pi$
$673$	1.32471e7i	1.12741i	−0.825975	+	0.563707i	$0.809375\pi$
$674$	−3.62334e6	−0.307227
$675$	0	0
$676$	1.85458e6	0.156091
$677$	1.04491e7i	0.876205i	0.898925	+	0.438103i	$0.144349\pi$
$677$	1.04491e7i	0.876205i	−0.898925	+	0.438103i	$0.855651\pi$
$678$	0	0
$679$	−724552.	−0.0603108
$680$	0	0
$681$	0	0
$682$	2.44224e6i	0.201061i
$683$	613308.i	0.0503068i	0.999684	+	0.0251534i	$0.00800742\pi$
$683$	613308.i	0.0503068i	−0.999684	+	0.0251534i	$0.991993\pi$
$684$	0	0
$685$	0	0
$686$	−4.40701e6	−0.357547
$687$	0	0
$688$	− 1.52474e6i	− 0.122807i
$689$	9.84599e6	0.790153
$690$	0	0
$691$	−2.13992e7	−1.70491	−0.852457	−	0.522798i	$-0.824888\pi$
$691$	−2.13992e7	−1.70491	−0.852457	+	0.522798i	$0.824888\pi$
$692$	− 4.19069e6i	− 0.332675i
$693$	0	0
$694$	−1.18028e7	−0.930219
$695$	0	0
$696$	0	0
$697$	2.08220e7i	1.62346i
$698$	− 8.63044e6i	− 0.670493i
$699$	0	0
$700$	0	0
$701$	−1.09778e7	−0.843765	−0.421883	−	0.906650i	$-0.638631\pi$
$701$	−1.09778e7	−0.843765	−0.421883	+	0.906650i	$0.638631\pi$
$702$	0	0
$703$	− 4.02351e6i	− 0.307056i
$704$	2.94912e6	0.224265
$705$	0	0
$706$	−5.15494e6	−0.389235
$707$	− 1.68825e7i	− 1.27025i
$708$	0	0
$709$	−1.69732e6	−0.126808	−0.0634041	−	0.997988i	$-0.520196\pi$
$709$	−1.69732e6	−0.126808	−0.0634041	+	0.997988i	$0.520196\pi$
$710$	0	0
$711$	0	0
$712$	692352.i	0.0511831i
$713$	1.52640e6i	0.112446i
$714$	0	0
$715$	0	0
$716$	−9.97670e6	−0.727285
$717$	0	0
$718$	9.07786e6i	0.657162i
$719$	3.71304e6	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$719$	3.71304e6	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$720$	0	0
$721$	−1.13875e7	−0.815813
$722$	9.39745e6i	0.670914i
$723$	0	0
$724$	978976.	0.0694106
$725$	0	0
$726$	0	0
$727$	− 1.38067e6i	− 0.0968843i	−0.998826	−	0.0484421i	$-0.984574\pi$
$727$	− 1.38067e6i	− 0.0968843i	0.998826	−	0.0484421i	$-0.0154256\pi$
$728$	7.32621e6i	0.512331i
$729$	0	0
$730$	0	0
$731$	−1.32581e7	−0.917670
$732$	0	0
$733$	1.38156e7i	0.949751i	0.880053	+	0.474876i	$0.157507\pi$
$733$	1.38156e7i	0.949751i	−0.880053	+	0.474876i	$0.842493\pi$
$734$	−4.18635e6	−0.286811
$735$	0	0
$736$	1.84320e6	0.125423
$737$	− 4.71427e7i	− 3.19702i
$738$	0	0
$739$	−4.59463e6	−0.309485	−0.154742	−	0.987955i	$-0.549455\pi$
$739$	−4.59463e6	−0.309485	−0.154742	+	0.987955i	$0.549455\pi$
$740$	0	0
$741$	0	0
$742$	9.25354e6i	0.617018i
$743$	− 8.51174e6i	− 0.565648i	−0.959172	−	0.282824i	$-0.908729\pi$
$743$	− 8.51174e6i	− 0.565648i	0.959172	−	0.282824i	$-0.0912713\pi$
$744$	0	0
$745$	0	0
$746$	7.19310e6	0.473226
$747$	0	0
$748$	− 2.56435e7i	− 1.67581i
$749$	2.85557e6	0.185989
$750$	0	0
$751$	1.71224e7	1.10781	0.553904	−	0.832580i	$-0.313137\pi$
$751$	1.71224e7	1.10781	0.553904	+	0.832580i	$0.313137\pi$
$752$	− 2.85696e6i	− 0.184230i
$753$	0	0
$754$	1.99349e6	0.127698
$755$	0	0
$756$	0	0
$757$	1.22018e6i	0.0773900i	0.999251	+	0.0386950i	$0.0123201\pi$
$757$	1.22018e6i	0.0773900i	−0.999251	+	0.0386950i	$0.987680\pi$
$758$	− 8.73650e6i	− 0.552286i
$759$	0	0
$760$	0	0
$761$	−1.20327e7	−0.753182	−0.376591	−	0.926380i	$-0.622904\pi$
$761$	−1.20327e7	−0.753182	−0.376591	+	0.926380i	$0.622904\pi$
$762$	0	0
$763$	− 3.34183e7i	− 2.07813i
$764$	1.17961e7	0.731147
$765$	0	0
$766$	−7.13808e6	−0.439551
$767$	5.52816e6i	0.339307i
$768$	0	0
$769$	1.88952e7	1.15222	0.576110	−	0.817372i	$-0.304570\pi$
$769$	1.88952e7	1.15222	0.576110	+	0.817372i	$0.304570\pi$
$770$	0	0
$771$	0	0
$772$	− 8.62659e6i	− 0.520950i
$773$	− 1.72115e7i	− 1.03602i	−0.855373	−	0.518012i	$-0.826672\pi$
$773$	− 1.72115e7i	− 1.03602i	0.855373	−	0.518012i	$-0.173328\pi$
$774$	0	0
$775$	0	0
$776$	282752.	0.0168559
$777$	0	0
$778$	4.43810e6i	0.262875i
$779$	3.33002e6	0.196609
$780$	0	0
$781$	2.48832e7	1.45975
$782$	− 1.60272e7i	− 0.937218i
$783$	0	0
$784$	−2.58278e6	−0.150071
$785$	0	0
$786$	0	0
$787$	− 1.32970e7i	− 0.765274i	−0.923899	−	0.382637i	$-0.875016\pi$
$787$	− 1.32970e7i	− 0.765274i	0.923899	−	0.382637i	$-0.124984\pi$
$788$	1.04188e7i	0.597725i
$789$	0	0
$790$	0	0
$791$	3.48011e7	1.97766
$792$	0	0
$793$	− 9.38810e6i	− 0.530145i
$794$	−1.55827e7	−0.877187
$795$	0	0
$796$	2.51725e6	0.140813
$797$	2.15632e7i	1.20245i	0.799078	+	0.601227i	$0.205321\pi$
$797$	2.15632e7i	1.20245i	−0.799078	+	0.601227i	$0.794679\pi$
$798$	0	0
$799$	−2.48422e7	−1.37665
$800$	0	0
$801$	0	0
$802$	− 4.82690e6i	− 0.264992i
$803$	− 6.21058e7i	− 3.39894i
$804$	0	0
$805$	0	0
$806$	2.36762e6	0.128373
$807$	0	0
$808$	6.58829e6i	0.355013i
$809$	2.65355e7	1.42547	0.712733	−	0.701436i	$-0.247457\pi$
$809$	2.65355e7	1.42547	0.712733	+	0.701436i	$0.247457\pi$
$810$	0	0
$811$	−1.40015e7	−0.747518	−0.373759	−	0.927526i	$-0.621931\pi$
$811$	−1.40015e7	−0.747518	−0.373759	+	0.927526i	$0.621931\pi$
$812$	1.87354e6i	0.0997176i
$813$	0	0
$814$	−3.25498e7	−1.72182
$815$	0	0
$816$	0	0
$817$	2.12034e6i	0.111135i
$818$	2.24545e7i	1.17333i
$819$	0	0
$820$	0	0
$821$	−1.32286e7	−0.684944	−0.342472	−	0.939528i	$-0.611264\pi$
$821$	−1.32286e7	−0.684944	−0.342472	+	0.939528i	$0.611264\pi$
$822$	0	0
$823$	− 7.25818e6i	− 0.373532i	−0.982404	−	0.186766i	$-0.940199\pi$
$823$	− 7.25818e6i	− 0.373532i	0.982404	−	0.186766i	$-0.0598007\pi$
$824$	4.44390e6	0.228006
$825$	0	0
$826$	−5.19552e6	−0.264959
$827$	1.84527e7i	0.938204i	0.883144	+	0.469102i	$0.155422\pi$
$827$	1.84527e7i	0.938204i	−0.883144	+	0.469102i	$0.844578\pi$
$828$	0	0
$829$	2.43640e7	1.23130	0.615649	−	0.788021i	$-0.288894\pi$
$829$	2.43640e7	1.23130	0.615649	+	0.788021i	$0.288894\pi$
$830$	0	0
$831$	0	0
$832$	− 2.85901e6i	− 0.143188i
$833$	2.24581e7i	1.12140i
$834$	0	0
$835$	0	0
$836$	−4.10112e6	−0.202949
$837$	0	0
$838$	− 4.60224e6i	− 0.226391i
$839$	−1.55793e7	−0.764089	−0.382045	−	0.924144i	$-0.624780\pi$
$839$	−1.55793e7	−0.764089	−0.382045	+	0.924144i	$0.624780\pi$
$840$	0	0
$841$	−2.00014e7	−0.975145
$842$	1.53236e7i	0.744868i
$843$	0	0
$844$	−1.13149e7	−0.546756
$845$	0	0
$846$	0	0
$847$	− 5.86052e7i	− 2.80691i
$848$	− 3.61114e6i	− 0.172446i
$849$	0	0
$850$	0	0
$851$	−2.03436e7	−0.962950
$852$	0	0
$853$	3.08062e7i	1.44966i	0.688929	+	0.724829i	$0.258081\pi$
$853$	3.08062e7i	1.44966i	−0.688929	+	0.724829i	$0.741919\pi$
$854$	8.82320e6	0.413982
$855$	0	0
$856$	−1.11437e6	−0.0519810
$857$	− 2.02084e6i	− 0.0939897i	−0.998895	−	0.0469949i	$-0.985036\pi$
$857$	− 2.02084e6i	− 0.0939897i	0.998895	−	0.0469949i	$-0.0149644\pi$
$858$	0	0
$859$	2.24790e7	1.03943	0.519714	−	0.854340i	$-0.326039\pi$
$859$	2.24790e7	1.03943	0.519714	+	0.854340i	$0.326039\pi$
$860$	0	0
$861$	0	0
$862$	622080.i	0.0285153i
$863$	9.20942e6i	0.420926i	0.977602	+	0.210463i	$0.0674971\pi$
$863$	9.20942e6i	0.420926i	−0.977602	+	0.210463i	$0.932503\pi$
$864$	0	0
$865$	0	0
$866$	1.65757e7	0.751062
$867$	0	0
$868$	2.22515e6i	0.100244i
$869$	−7.83590e7	−3.51998
$870$	0	0
$871$	−4.57022e7	−2.04123
$872$	1.30413e7i	0.580803i
$873$	0	0
$874$	−2.56320e6	−0.113502
$875$	0	0
$876$	0	0
$877$	5.36258e6i	0.235437i	0.993047	+	0.117719i	$0.0375581\pi$
$877$	5.36258e6i	0.235437i	−0.993047	+	0.117719i	$0.962442\pi$
$878$	2.49461e7i	1.09211i
$879$	0	0
$880$	0	0
$881$	−1.38347e7	−0.600525	−0.300263	−	0.953857i	$-0.597074\pi$
$881$	−1.38347e7	−0.600525	−0.300263	+	0.953857i	$0.597074\pi$
$882$	0	0
$883$	− 1.66004e6i	− 0.0716499i	−0.999358	−	0.0358250i	$-0.988594\pi$
$883$	− 1.66004e6i	− 0.0716499i	0.999358	−	0.0358250i	$-0.0114059\pi$
$884$	−2.48600e7	−1.06997
$885$	0	0
$886$	−1.81003e7	−0.774642
$887$	− 8.10612e6i	− 0.345943i	−0.984927	−	0.172971i	$-0.944663\pi$
$887$	− 8.10612e6i	− 0.345943i	0.984927	−	0.172971i	$-0.0553368\pi$
$888$	0	0
$889$	−1.00696e6	−0.0427325
$890$	0	0
$891$	0	0
$892$	8.49184e6i	0.357347i
$893$	3.97296e6i	0.166719i
$894$	0	0
$895$	0	0
$896$	2.68698e6	0.111813
$897$	0	0
$898$	− 1.02585e7i	− 0.424516i
$899$	605472.	0.0249859
$900$	0	0
$901$	−3.14000e7	−1.28860
$902$	− 2.69395e7i	− 1.10249i
$903$	0	0
$904$	−1.35809e7	−0.552724
$905$	0	0
$906$	0	0
$907$	− 4.05360e6i	− 0.163615i	−0.996648	−	0.0818073i	$-0.973931\pi$
$907$	− 4.05360e6i	− 0.163615i	0.996648	−	0.0818073i	$-0.0260692\pi$
$908$	− 1.92595e6i	− 0.0775230i
$909$	0	0
$910$	0	0
$911$	1.38233e7	0.551844	0.275922	−	0.961180i	$-0.411017\pi$
$911$	1.38233e7	0.551844	0.275922	+	0.961180i	$0.411017\pi$
$912$	0	0
$913$	7.68096e6i	0.304957i
$914$	2.21363e7	0.876477
$915$	0	0
$916$	1.23570e7	0.486601
$917$	− 3.37709e7i	− 1.32623i
$918$	0	0
$919$	3.78443e7	1.47813	0.739063	−	0.673636i	$-0.235268\pi$
$919$	3.78443e7	1.47813	0.739063	+	0.673636i	$0.235268\pi$
$920$	0	0
$921$	0	0
$922$	− 2.87684e7i	− 1.11452i
$923$	− 2.41229e7i	− 0.932019i
$924$	0	0
$925$	0	0
$926$	−4.55742e6	−0.174659
$927$	0	0
$928$	− 731136.i	− 0.0278694i
$929$	−2.72822e7	−1.03715	−0.518574	−	0.855033i	$-0.673537\pi$
$929$	−2.72822e7	−1.03715	−0.518574	+	0.855033i	$0.673537\pi$
$930$	0	0
$931$	3.59168e6	0.135808
$932$	141408.i	0.00533254i
$933$	0	0
$934$	2.94467e7	1.10451
$935$	0	0
$936$	0	0
$937$	− 4.32666e7i	− 1.60992i	−0.593332	−	0.804958i	$-0.702188\pi$
$937$	− 4.32666e7i	− 1.60992i	0.593332	−	0.804958i	$-0.297812\pi$
$938$	− 4.29523e7i	− 1.59397i
$939$	0	0
$940$	0	0
$941$	−8.50106e6	−0.312967	−0.156484	−	0.987681i	$-0.550016\pi$
$941$	−8.50106e6	−0.312967	−0.156484	+	0.987681i	$0.550016\pi$
$942$	0	0
$943$	− 1.68372e7i	− 0.616582i
$944$	2.02752e6	0.0740517
$945$	0	0
$946$	1.71533e7	0.623188
$947$	7.10456e6i	0.257432i	0.991681	+	0.128716i	$0.0410856\pi$
$947$	7.10456e6i	0.257432i	−0.991681	+	0.128716i	$0.958914\pi$
$948$	0	0
$949$	−6.02081e7	−2.17015
$950$	0	0
$951$	0	0
$952$	− 2.33641e7i	− 0.835520i
$953$	5.39741e7i	1.92510i	0.271102	+	0.962551i	$0.412612\pi$
$953$	5.39741e7i	1.92510i	−0.271102	+	0.962551i	$0.587388\pi$
$954$	0	0
$955$	0	0
$956$	1.24067e7	0.439046
$957$	0	0
$958$	5.44906e6i	0.191826i
$959$	3.77748e7	1.32634
$960$	0	0
$961$	−2.79100e7	−0.974882
$962$	3.15552e7i	1.09934i
$963$	0	0
$964$	5.97501e6	0.207084
$965$	0	0
$966$	0	0
$967$	− 3.64583e7i	− 1.25381i	−0.779097	−	0.626903i	$-0.784322\pi$
$967$	− 3.64583e7i	− 1.25381i	0.779097	−	0.626903i	$-0.215678\pi$
$968$	2.28703e7i	0.784484i
$969$	0	0
$970$	0	0
$971$	5.20286e7	1.77090	0.885450	−	0.464734i	$-0.153850\pi$
$971$	5.20286e7	1.77090	0.885450	+	0.464734i	$0.153850\pi$
$972$	0	0
$973$	4.27640e7i	1.44809i
$974$	−2.42571e6	−0.0819298
$975$	0	0
$976$	−3.44320e6	−0.115701
$977$	127902.i	0.00428688i	0.999998	+	0.00214344i	$0.000682278\pi$
$977$	127902.i	0.00428688i	−0.999998	+	0.00214344i	$0.999318\pi$
$978$	0	0
$979$	−7.78896e6	−0.259730
$980$	0	0
$981$	0	0
$982$	− 2.33711e7i	− 0.773393i
$983$	1.57667e7i	0.520422i	0.965552	+	0.260211i	$0.0837922\pi$
$983$	1.57667e7i	0.520422i	−0.965552	+	0.260211i	$0.916208\pi$
$984$	0	0
$985$	0	0
$986$	−6.35746e6	−0.208253
$987$	0	0
$988$	3.97581e6i	0.129579i
$989$	1.07208e7	0.348527
$990$	0	0
$991$	2.99415e7	0.968479	0.484239	−	0.874936i	$-0.339096\pi$
$991$	2.99415e7	0.968479	0.484239	+	0.874936i	$0.339096\pi$
$992$	− 868352.i	− 0.0280167i
$993$	0	0
$994$	2.26714e7	0.727799
$995$	0	0
$996$	0	0
$997$	5.07440e7i	1.61676i	0.588659	+	0.808382i	$0.299656\pi$
$997$	5.07440e7i	1.61676i	−0.588659	+	0.808382i	$0.700344\pi$
$998$	− 4.60178e6i	− 0.146251i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.c.b.199.1		2
3.2	odd	2		150.6.c.d.49.2		2
5.2	odd	4		90.6.a.g.1.1		1
5.3	odd	4		450.6.a.b.1.1		1
5.4	even	2	inner	450.6.c.b.199.2		2
15.2	even	4		30.6.a.a.1.1	✓	1
15.8	even	4		150.6.a.h.1.1		1
15.14	odd	2		150.6.c.d.49.1		2
20.7	even	4		720.6.a.m.1.1		1
60.47	odd	4		240.6.a.a.1.1		1
120.77	even	4		960.6.a.n.1.1		1
120.107	odd	4		960.6.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.6.a.a.1.1	✓	1	15.2	even	4
90.6.a.g.1.1		1	5.2	odd	4
150.6.a.h.1.1		1	15.8	even	4
150.6.c.d.49.1		2	15.14	odd	2
150.6.c.d.49.2		2	3.2	odd	2
240.6.a.a.1.1		1	60.47	odd	4
450.6.a.b.1.1		1	5.3	odd	4
450.6.c.b.199.1		2	1.1	even	1	trivial
450.6.c.b.199.2		2	5.4	even	2	inner
720.6.a.m.1.1		1	20.7	even	4
960.6.a.n.1.1		1	120.77	even	4
960.6.a.u.1.1		1	120.107	odd	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)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -16.0000 q^{4} -164.000i q^{7} +64.0000i q^{8} +O(q^{10})\)	\(q-4.00000i q^{2} -16.0000 q^{4} -164.000i q^{7} +64.0000i q^{8} -720.000 q^{11} +698.000i q^{13} -656.000 q^{14} +256.000 q^{16} -2226.00i q^{17} -356.000 q^{19} +2880.00i q^{22} +1800.00i q^{23} +2792.00 q^{26} +2624.00i q^{28} +714.000 q^{29} +848.000 q^{31} -1024.00i q^{32} -8904.00 q^{34} +11302.0i q^{37} +1424.00i q^{38} -9354.00 q^{41} -5956.00i q^{43} +11520.0 q^{44} +7200.00 q^{46} -11160.0i q^{47} -10089.0 q^{49} -11168.0i q^{52} -14106.0i q^{53} +10496.0 q^{56} -2856.00i q^{58} +7920.00 q^{59} -13450.0 q^{61} -3392.00i q^{62} -4096.00 q^{64} +65476.0i q^{67} +35616.0i q^{68} -34560.0 q^{71} +86258.0i q^{73} +45208.0 q^{74} +5696.00 q^{76} +118080. i q^{77} +108832. q^{79} +37416.0i q^{82} -10668.0i q^{83} -23824.0 q^{86} -46080.0i q^{88} +10818.0 q^{89} +114472. q^{91} -28800.0i q^{92} -44640.0 q^{94} -4418.00i q^{97} +40356.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4}+O(q^{10}) \) 2 * q - 32 * q^4	\( 2 q - 32 q^{4} - 1440 q^{11} - 1312 q^{14} + 512 q^{16} - 712 q^{19} + 5584 q^{26} + 1428 q^{29} + 1696 q^{31} - 17808 q^{34} - 18708 q^{41} + 23040 q^{44} + 14400 q^{46} - 20178 q^{49} + 20992 q^{56} + 15840 q^{59} - 26900 q^{61} - 8192 q^{64} - 69120 q^{71} + 90416 q^{74} + 11392 q^{76} + 217664 q^{79} - 47648 q^{86} + 21636 q^{89} + 228944 q^{91} - 89280 q^{94}+O(q^{100}) \) 2 * q - 32 * q^4 - 1440 * q^11 - 1312 * q^14 + 512 * q^16 - 712 * q^19 + 5584 * q^26 + 1428 * q^29 + 1696 * q^31 - 17808 * q^34 - 18708 * q^41 + 23040 * q^44 + 14400 * q^46 - 20178 * q^49 + 20992 * q^56 + 15840 * q^59 - 26900 * q^61 - 8192 * q^64 - 69120 * q^71 + 90416 * q^74 + 11392 * q^76 + 217664 * q^79 - 47648 * q^86 + 21636 * q^89 + 228944 * q^91 - 89280 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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