LMFDB - Embedded newform 450.6.c.c.199.2

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 18)
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			450.6.c.c.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -16.0000 q^{4} -148.000i q^{7} -64.0000i q^{8} +O(q^{10})$	$q+4.00000i q^{2} -16.0000 q^{4} -148.000i q^{7} -64.0000i q^{8} -384.000 q^{11} +334.000i q^{13} +592.000 q^{14} +256.000 q^{16} -576.000i q^{17} +664.000 q^{19} -1536.00i q^{22} -3840.00i q^{23} -1336.00 q^{26} +2368.00i q^{28} +96.0000 q^{29} -4564.00 q^{31} +1024.00i q^{32} +2304.00 q^{34} +5798.00i q^{37} +2656.00i q^{38} +6720.00 q^{41} +14872.0i q^{43} +6144.00 q^{44} +15360.0 q^{46} +19200.0i q^{47} -5097.00 q^{49} -5344.00i q^{52} +7776.00i q^{53} -9472.00 q^{56} +384.000i q^{58} -13056.0 q^{59} +42782.0 q^{61} -18256.0i q^{62} -4096.00 q^{64} +36656.0i q^{67} +9216.00i q^{68} -64512.0 q^{71} +16810.0i q^{73} -23192.0 q^{74} -10624.0 q^{76} +56832.0i q^{77} -28076.0 q^{79} +26880.0i q^{82} -66432.0i q^{83} -59488.0 q^{86} +24576.0i q^{88} -81792.0 q^{89} +49432.0 q^{91} +61440.0i q^{92} -76800.0 q^{94} -29938.0i q^{97} -20388.0i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4}+O(q^{10}) $ 2 * q - 32 * q^4	$ 2 q - 32 q^{4} - 768 q^{11} + 1184 q^{14} + 512 q^{16} + 1328 q^{19} - 2672 q^{26} + 192 q^{29} - 9128 q^{31} + 4608 q^{34} + 13440 q^{41} + 12288 q^{44} + 30720 q^{46} - 10194 q^{49} - 18944 q^{56} - 26112 q^{59} + 85564 q^{61} - 8192 q^{64} - 129024 q^{71} - 46384 q^{74} - 21248 q^{76} - 56152 q^{79} - 118976 q^{86} - 163584 q^{89} + 98864 q^{91} - 153600 q^{94}+O(q^{100}) $ 2 * q - 32 * q^4 - 768 * q^11 + 1184 * q^14 + 512 * q^16 + 1328 * q^19 - 2672 * q^26 + 192 * q^29 - 9128 * q^31 + 4608 * q^34 + 13440 * q^41 + 12288 * q^44 + 30720 * q^46 - 10194 * q^49 - 18944 * q^56 - 26112 * q^59 + 85564 * q^61 - 8192 * q^64 - 129024 * q^71 - 46384 * q^74 - 21248 * q^76 - 56152 * q^79 - 118976 * q^86 - 163584 * q^89 + 98864 * q^91 - 153600 * 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
$7$	− 148.000i	− 1.14161i	−0.821087	−	0.570803i	$-0.806632\pi$
$7$	− 148.000i	− 1.14161i	0.821087	−	0.570803i	$-0.193368\pi$
$8$	− 64.0000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−384.000	−0.956862	−0.478431	−	0.878125i	$-0.658794\pi$
$11$	−384.000	−0.956862	−0.478431	+	0.878125i	$0.658794\pi$
$12$	0	0
$13$	334.000i	0.548136i	0.961710	+	0.274068i	$0.0883693\pi$
$13$	334.000i	0.548136i	−0.961710	+	0.274068i	$0.911631\pi$
$14$	592.000	0.807238
$15$	0	0
$16$	256.000	0.250000
$17$	− 576.000i	− 0.483393i	−0.970352	−	0.241696i	$-0.922296\pi$
$17$	− 576.000i	− 0.483393i	0.970352	−	0.241696i	$-0.0777038\pi$
$18$	0	0
$19$	664.000	0.421972	0.210986	−	0.977489i	$-0.432332\pi$
$19$	664.000	0.421972	0.210986	+	0.977489i	$0.432332\pi$
$20$	0	0
$21$	0	0
$22$	− 1536.00i	− 0.676604i
$23$	− 3840.00i	− 1.51360i	−0.653645	−	0.756801i	$-0.726761\pi$
$23$	− 3840.00i	− 1.51360i	0.653645	−	0.756801i	$-0.273239\pi$
$24$	0	0
$25$	0	0
$26$	−1336.00	−0.387590
$27$	0	0
$28$	2368.00i	0.570803i
$29$	96.0000	0.0211971	0.0105985	−	0.999944i	$-0.496626\pi$
$29$	96.0000	0.0211971	0.0105985	+	0.999944i	$0.496626\pi$
$30$	0	0
$31$	−4564.00	−0.852985	−0.426493	−	0.904491i	$-0.640251\pi$
$31$	−4564.00	−0.852985	−0.426493	+	0.904491i	$0.640251\pi$
$32$	1024.00i	0.176777i
$33$	0	0
$34$	2304.00	0.341810
$35$	0	0
$36$	0	0
$37$	5798.00i	0.696264i	0.937446	+	0.348132i	$0.113184\pi$
$37$	5798.00i	0.696264i	−0.937446	+	0.348132i	$0.886816\pi$
$38$	2656.00i	0.298380i
$39$	0	0
$40$	0	0
$41$	6720.00	0.624323	0.312162	−	0.950029i	$-0.398947\pi$
$41$	6720.00	0.624323	0.312162	+	0.950029i	$0.398947\pi$
$42$	0	0
$43$	14872.0i	1.22659i	0.789855	+	0.613293i	$0.210156\pi$
$43$	14872.0i	1.22659i	−0.789855	+	0.613293i	$0.789844\pi$
$44$	6144.00	0.478431
$45$	0	0
$46$	15360.0	1.07028
$47$	19200.0i	1.26782i	0.773408	+	0.633909i	$0.218550\pi$
$47$	19200.0i	1.26782i	−0.773408	+	0.633909i	$0.781450\pi$
$48$	0	0
$49$	−5097.00	−0.303266
$50$	0	0
$51$	0	0
$52$	− 5344.00i	− 0.274068i
$53$	7776.00i	0.380248i	0.981760	+	0.190124i	$0.0608890\pi$
$53$	7776.00i	0.380248i	−0.981760	+	0.190124i	$0.939111\pi$
$54$	0	0
$55$	0	0
$56$	−9472.00	−0.403619
$57$	0	0
$58$	384.000i	0.0149886i
$59$	−13056.0	−0.488293	−0.244146	−	0.969738i	$-0.578508\pi$
$59$	−13056.0	−0.488293	−0.244146	+	0.969738i	$0.578508\pi$
$60$	0	0
$61$	42782.0	1.47210	0.736049	−	0.676929i	$-0.236689\pi$
$61$	42782.0	1.47210	0.736049	+	0.676929i	$0.236689\pi$
$62$	− 18256.0i	− 0.603151i
$63$	0	0
$64$	−4096.00	−0.125000
$65$	0	0
$66$	0	0
$67$	36656.0i	0.997604i	0.866716	+	0.498802i	$0.166226\pi$
$67$	36656.0i	0.997604i	−0.866716	+	0.498802i	$0.833774\pi$
$68$	9216.00i	0.241696i
$69$	0	0
$70$	0	0
$71$	−64512.0	−1.51878	−0.759390	−	0.650636i	$-0.774502\pi$
$71$	−64512.0	−1.51878	−0.759390	+	0.650636i	$0.774502\pi$
$72$	0	0
$73$	16810.0i	0.369199i	0.982814	+	0.184600i	$0.0590988\pi$
$73$	16810.0i	0.369199i	−0.982814	+	0.184600i	$0.940901\pi$
$74$	−23192.0	−0.492333
$75$	0	0
$76$	−10624.0	−0.210986
$77$	56832.0i	1.09236i
$78$	0	0
$79$	−28076.0	−0.506136	−0.253068	−	0.967448i	$-0.581440\pi$
$79$	−28076.0	−0.506136	−0.253068	+	0.967448i	$0.581440\pi$
$80$	0	0
$81$	0	0
$82$	26880.0i	0.441463i
$83$	− 66432.0i	− 1.05848i	−0.848473	−	0.529239i	$-0.822477\pi$
$83$	− 66432.0i	− 1.05848i	0.848473	−	0.529239i	$-0.177523\pi$
$84$	0	0
$85$	0	0
$86$	−59488.0	−0.867328
$87$	0	0
$88$	24576.0i	0.338302i
$89$	−81792.0	−1.09455	−0.547275	−	0.836953i	$-0.684335\pi$
$89$	−81792.0	−1.09455	−0.547275	+	0.836953i	$0.684335\pi$
$90$	0	0
$91$	49432.0	0.625756
$92$	61440.0i	0.756801i
$93$	0	0
$94$	−76800.0	−0.896482
$95$	0	0
$96$	0	0
$97$	− 29938.0i	− 0.323068i	−0.986867	−	0.161534i	$-0.948356\pi$
$97$	− 29938.0i	− 0.323068i	0.986867	−	0.161534i	$-0.0516441\pi$
$98$	− 20388.0i	− 0.214442i
$99$	0	0
$100$	0	0
$101$	−178656.	−1.74267	−0.871333	−	0.490692i	$-0.836744\pi$
$101$	−178656.	−1.74267	−0.871333	+	0.490692i	$0.836744\pi$
$102$	0	0
$103$	115228.i	1.07020i	0.844789	+	0.535100i	$0.179726\pi$
$103$	115228.i	1.07020i	−0.844789	+	0.535100i	$0.820274\pi$
$104$	21376.0	0.193795
$105$	0	0
$106$	−31104.0	−0.268876
$107$	76032.0i	0.642003i	0.947079	+	0.321001i	$0.104019\pi$
$107$	76032.0i	0.642003i	−0.947079	+	0.321001i	$0.895981\pi$
$108$	0	0
$109$	231118.	1.86323	0.931617	−	0.363441i	$-0.118398\pi$
$109$	231118.	1.86323	0.931617	+	0.363441i	$0.118398\pi$
$110$	0	0
$111$	0	0
$112$	− 37888.0i	− 0.285402i
$113$	142464.i	1.04956i	0.851237	+	0.524782i	$0.175853\pi$
$113$	142464.i	1.04956i	−0.851237	+	0.524782i	$0.824147\pi$
$114$	0	0
$115$	0	0
$116$	−1536.00	−0.0105985
$117$	0	0
$118$	− 52224.0i	− 0.345275i
$119$	−85248.0	−0.551845
$120$	0	0
$121$	−13595.0	−0.0844143
$122$	171128.i	1.04093i
$123$	0	0
$124$	73024.0	0.426493
$125$	0	0
$126$	0	0
$127$	− 988.000i	− 0.00543560i	−0.999996	−	0.00271780i	$-0.999135\pi$
$127$	− 988.000i	− 0.00543560i	0.999996	−	0.00271780i	$-0.000865104\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−224256.	−1.14174	−0.570868	−	0.821042i	$-0.693393\pi$
$131$	−224256.	−1.14174	−0.570868	+	0.821042i	$0.693393\pi$
$132$	0	0
$133$	− 98272.0i	− 0.481727i
$134$	−146624.	−0.705412
$135$	0	0
$136$	−36864.0	−0.170905
$137$	278976.i	1.26989i	0.772558	+	0.634944i	$0.218977\pi$
$137$	278976.i	1.26989i	−0.772558	+	0.634944i	$0.781023\pi$
$138$	0	0
$139$	−177200.	−0.777905	−0.388953	−	0.921258i	$-0.627163\pi$
$139$	−177200.	−0.777905	−0.388953	+	0.921258i	$0.627163\pi$
$140$	0	0
$141$	0	0
$142$	− 258048.i	− 1.07394i
$143$	− 128256.i	− 0.524490i
$144$	0	0
$145$	0	0
$146$	−67240.0	−0.261063
$147$	0	0
$148$	− 92768.0i	− 0.348132i
$149$	236064.	0.871092	0.435546	−	0.900166i	$-0.356555\pi$
$149$	236064.	0.871092	0.435546	+	0.900166i	$0.356555\pi$
$150$	0	0
$151$	−482836.	−1.72329	−0.861643	−	0.507515i	$-0.830564\pi$
$151$	−482836.	−1.72329	−0.861643	+	0.507515i	$0.830564\pi$
$152$	− 42496.0i	− 0.149190i
$153$	0	0
$154$	−227328.	−0.772416
$155$	0	0
$156$	0	0
$157$	381086.i	1.23388i	0.787009	+	0.616941i	$0.211628\pi$
$157$	381086.i	1.23388i	−0.787009	+	0.616941i	$0.788372\pi$
$158$	− 112304.i	− 0.357892i
$159$	0	0
$160$	0	0
$161$	−568320.	−1.72794
$162$	0	0
$163$	− 162920.i	− 0.480292i	−0.970737	−	0.240146i	$-0.922805\pi$
$163$	− 162920.i	− 0.480292i	0.970737	−	0.240146i	$-0.0771953\pi$
$164$	−107520.	−0.312162
$165$	0	0
$166$	265728.	0.748457
$167$	566016.i	1.57050i	0.619180	+	0.785249i	$0.287465\pi$
$167$	566016.i	1.57050i	−0.619180	+	0.785249i	$0.712535\pi$
$168$	0	0
$169$	259737.	0.699547
$170$	0	0
$171$	0	0
$172$	− 237952.i	− 0.613293i
$173$	− 218208.i	− 0.554313i	−0.960825	−	0.277157i	$-0.910608\pi$
$173$	− 218208.i	− 0.554313i	0.960825	−	0.277157i	$-0.0893921\pi$
$174$	0	0
$175$	0	0
$176$	−98304.0	−0.239216
$177$	0	0
$178$	− 327168.i	− 0.773964i
$179$	−412416.	−0.962062	−0.481031	−	0.876704i	$-0.659738\pi$
$179$	−412416.	−0.962062	−0.481031	+	0.876704i	$0.659738\pi$
$180$	0	0
$181$	−25558.0	−0.0579870	−0.0289935	−	0.999580i	$-0.509230\pi$
$181$	−25558.0	−0.0579870	−0.0289935	+	0.999580i	$0.509230\pi$
$182$	197728.i	0.442476i
$183$	0	0
$184$	−245760.	−0.535139
$185$	0	0
$186$	0	0
$187$	221184.i	0.462540i
$188$	− 307200.i	− 0.633909i
$189$	0	0
$190$	0	0
$191$	−400128.	−0.793625	−0.396813	−	0.917900i	$-0.629884\pi$
$191$	−400128.	−0.793625	−0.396813	+	0.917900i	$0.629884\pi$
$192$	0	0
$193$	− 699650.i	− 1.35203i	−0.736886	−	0.676017i	$-0.763705\pi$
$193$	− 699650.i	− 1.35203i	0.736886	−	0.676017i	$-0.236295\pi$
$194$	119752.	0.228443
$195$	0	0
$196$	81552.0	0.151633
$197$	− 406368.i	− 0.746026i	−0.927826	−	0.373013i	$-0.878325\pi$
$197$	− 406368.i	− 0.746026i	0.927826	−	0.373013i	$-0.121675\pi$
$198$	0	0
$199$	361996.	0.647994	0.323997	−	0.946058i	$-0.394973\pi$
$199$	361996.	0.647994	0.323997	+	0.946058i	$0.394973\pi$
$200$	0	0
$201$	0	0
$202$	− 714624.i	− 1.23225i
$203$	− 14208.0i	− 0.0241987i
$204$	0	0
$205$	0	0
$206$	−460912.	−0.756746
$207$	0	0
$208$	85504.0i	0.137034i
$209$	−254976.	−0.403770
$210$	0	0
$211$	151856.	0.234815	0.117407	−	0.993084i	$-0.462542\pi$
$211$	151856.	0.234815	0.117407	+	0.993084i	$0.462542\pi$
$212$	− 124416.i	− 0.190124i
$213$	0	0
$214$	−304128.	−0.453965
$215$	0	0
$216$	0	0
$217$	675472.i	0.973774i
$218$	924472.i	1.31751i
$219$	0	0
$220$	0	0
$221$	192384.	0.264965
$222$	0	0
$223$	1.09332e6i	1.47227i	0.676836	+	0.736134i	$0.263351\pi$
$223$	1.09332e6i	1.47227i	−0.676836	+	0.736134i	$0.736649\pi$
$224$	151552.	0.201810
$225$	0	0
$226$	−569856.	−0.742154
$227$	566400.i	0.729556i	0.931095	+	0.364778i	$0.118855\pi$
$227$	566400.i	0.729556i	−0.931095	+	0.364778i	$0.881145\pi$
$228$	0	0
$229$	587206.	0.739949	0.369974	−	0.929042i	$-0.379366\pi$
$229$	587206.	0.739949	0.369974	+	0.929042i	$0.379366\pi$
$230$	0	0
$231$	0	0
$232$	− 6144.00i	− 0.00749430i
$233$	579456.i	0.699247i	0.936890	+	0.349624i	$0.113691\pi$
$233$	579456.i	0.699247i	−0.936890	+	0.349624i	$0.886309\pi$
$234$	0	0
$235$	0	0
$236$	208896.	0.244146
$237$	0	0
$238$	− 340992.i	− 0.390213i
$239$	−584448.	−0.661837	−0.330919	−	0.943659i	$-0.607359\pi$
$239$	−584448.	−0.661837	−0.330919	+	0.943659i	$0.607359\pi$
$240$	0	0
$241$	−414130.	−0.459298	−0.229649	−	0.973274i	$-0.573758\pi$
$241$	−414130.	−0.459298	−0.229649	+	0.973274i	$0.573758\pi$
$242$	− 54380.0i	− 0.0596899i
$243$	0	0
$244$	−684512.	−0.736049
$245$	0	0
$246$	0	0
$247$	221776.i	0.231298i
$248$	292096.i	0.301576i
$249$	0	0
$250$	0	0
$251$	1.89965e6	1.90322	0.951610	−	0.307309i	$-0.0994287\pi$
$251$	1.89965e6	1.90322	0.951610	+	0.307309i	$0.0994287\pi$
$252$	0	0
$253$	1.47456e6i	1.44831i
$254$	3952.00	0.00384355
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 447744.i	− 0.422860i	−0.977393	−	0.211430i	$-0.932188\pi$
$257$	− 447744.i	− 0.422860i	0.977393	−	0.211430i	$-0.0678121\pi$
$258$	0	0
$259$	858104.	0.794860
$260$	0	0
$261$	0	0
$262$	− 897024.i	− 0.807330i
$263$	− 67584.0i	− 0.0602496i	−0.999546	−	0.0301248i	$-0.990410\pi$
$263$	− 67584.0i	− 0.0602496i	0.999546	−	0.0301248i	$-0.00959048\pi$
$264$	0	0
$265$	0	0
$266$	393088.	0.340632
$267$	0	0
$268$	− 586496.i	− 0.498802i
$269$	564192.	0.475386	0.237693	−	0.971340i	$-0.423609\pi$
$269$	564192.	0.475386	0.237693	+	0.971340i	$0.423609\pi$
$270$	0	0
$271$	720308.	0.595792	0.297896	−	0.954598i	$-0.403715\pi$
$271$	720308.	0.595792	0.297896	+	0.954598i	$0.403715\pi$
$272$	− 147456.i	− 0.120848i
$273$	0	0
$274$	−1.11590e6	−0.897946
$275$	0	0
$276$	0	0
$277$	− 141142.i	− 0.110524i	−0.998472	−	0.0552620i	$-0.982401\pi$
$277$	− 141142.i	− 0.110524i	0.998472	−	0.0552620i	$-0.0175994\pi$
$278$	− 708800.i	− 0.550062i
$279$	0	0
$280$	0	0
$281$	−584448.	−0.441550	−0.220775	−	0.975325i	$-0.570859\pi$
$281$	−584448.	−0.441550	−0.220775	+	0.975325i	$0.570859\pi$
$282$	0	0
$283$	− 177056.i	− 0.131415i	−0.997839	−	0.0657074i	$-0.979070\pi$
$283$	− 177056.i	− 0.131415i	0.997839	−	0.0657074i	$-0.0209304\pi$
$284$	1.03219e6	0.759390
$285$	0	0
$286$	513024.	0.370871
$287$	− 994560.i	− 0.712732i
$288$	0	0
$289$	1.08808e6	0.766331
$290$	0	0
$291$	0	0
$292$	− 268960.i	− 0.184600i
$293$	− 956832.i	− 0.651128i	−0.945520	−	0.325564i	$-0.894446\pi$
$293$	− 956832.i	− 0.651128i	0.945520	−	0.325564i	$-0.105554\pi$
$294$	0	0
$295$	0	0
$296$	371072.	0.246166
$297$	0	0
$298$	944256.i	0.615955i
$299$	1.28256e6	0.829659
$300$	0	0
$301$	2.20106e6	1.40028
$302$	− 1.93134e6i	− 1.21855i
$303$	0	0
$304$	169984.	0.105493
$305$	0	0
$306$	0	0
$307$	2.88286e6i	1.74573i	0.487958	+	0.872867i	$0.337742\pi$
$307$	2.88286e6i	1.74573i	−0.487958	+	0.872867i	$0.662258\pi$
$308$	− 909312.i	− 0.546180i
$309$	0	0
$310$	0	0
$311$	2.60045e6	1.52457	0.762285	−	0.647242i	$-0.224078\pi$
$311$	2.60045e6	1.52457	0.762285	+	0.647242i	$0.224078\pi$
$312$	0	0
$313$	2.58079e6i	1.48899i	0.667628	+	0.744495i	$0.267310\pi$
$313$	2.58079e6i	1.48899i	−0.667628	+	0.744495i	$0.732690\pi$
$314$	−1.52434e6	−0.872487
$315$	0	0
$316$	449216.	0.253068
$317$	− 2.31101e6i	− 1.29168i	−0.763475	−	0.645838i	$-0.776508\pi$
$317$	− 2.31101e6i	− 1.29168i	0.763475	−	0.645838i	$-0.223492\pi$
$318$	0	0
$319$	−36864.0	−0.0202827
$320$	0	0
$321$	0	0
$322$	− 2.27328e6i	− 1.22184i
$323$	− 382464.i	− 0.203978i
$324$	0	0
$325$	0	0
$326$	651680.	0.339618
$327$	0	0
$328$	− 430080.i	− 0.220732i
$329$	2.84160e6	1.44735
$330$	0	0
$331$	−637024.	−0.319585	−0.159792	−	0.987151i	$-0.551082\pi$
$331$	−637024.	−0.319585	−0.159792	+	0.987151i	$0.551082\pi$
$332$	1.06291e6i	0.529239i
$333$	0	0
$334$	−2.26406e6	−1.11051
$335$	0	0
$336$	0	0
$337$	3.38665e6i	1.62441i	0.583371	+	0.812206i	$0.301733\pi$
$337$	3.38665e6i	1.62441i	−0.583371	+	0.812206i	$0.698267\pi$
$338$	1.03895e6i	0.494655i
$339$	0	0
$340$	0	0
$341$	1.75258e6	0.816189
$342$	0	0
$343$	− 1.73308e6i	− 0.795396i
$344$	951808.	0.433664
$345$	0	0
$346$	872832.	0.391959
$347$	− 2.77824e6i	− 1.23864i	−0.785138	−	0.619321i	$-0.787408\pi$
$347$	− 2.77824e6i	− 1.23864i	0.785138	−	0.619321i	$-0.212592\pi$
$348$	0	0
$349$	−1.55536e6	−0.683545	−0.341772	−	0.939783i	$-0.611027\pi$
$349$	−1.55536e6	−0.683545	−0.341772	+	0.939783i	$0.611027\pi$
$350$	0	0
$351$	0	0
$352$	− 393216.i	− 0.169151i
$353$	2.11776e6i	0.904565i	0.891875	+	0.452283i	$0.149390\pi$
$353$	2.11776e6i	0.904565i	−0.891875	+	0.452283i	$0.850610\pi$
$354$	0	0
$355$	0	0
$356$	1.30867e6	0.547275
$357$	0	0
$358$	− 1.64966e6i	− 0.680280i
$359$	2.17498e6	0.890673	0.445337	−	0.895363i	$-0.353084\pi$
$359$	2.17498e6	0.890673	0.445337	+	0.895363i	$0.353084\pi$
$360$	0	0
$361$	−2.03520e6	−0.821939
$362$	− 102232.i	− 0.0410030i
$363$	0	0
$364$	−790912.	−0.312878
$365$	0	0
$366$	0	0
$367$	1.05336e6i	0.408235i	0.978946	+	0.204117i	$0.0654324\pi$
$367$	1.05336e6i	0.408235i	−0.978946	+	0.204117i	$0.934568\pi$
$368$	− 983040.i	− 0.378400i
$369$	0	0
$370$	0	0
$371$	1.15085e6	0.434093
$372$	0	0
$373$	677098.i	0.251988i	0.992031	+	0.125994i	$0.0402120\pi$
$373$	677098.i	0.251988i	−0.992031	+	0.125994i	$0.959788\pi$
$374$	−884736.	−0.327065
$375$	0	0
$376$	1.22880e6	0.448241
$377$	32064.0i	0.0116189i
$378$	0	0
$379$	5.10748e6	1.82645	0.913227	−	0.407452i	$-0.133582\pi$
$379$	5.10748e6	1.82645	0.913227	+	0.407452i	$0.133582\pi$
$380$	0	0
$381$	0	0
$382$	− 1.60051e6i	− 0.561178i
$383$	1.63200e6i	0.568491i	0.958752	+	0.284245i	$0.0917430\pi$
$383$	1.63200e6i	0.568491i	−0.958752	+	0.284245i	$0.908257\pi$
$384$	0	0
$385$	0	0
$386$	2.79860e6	0.956032
$387$	0	0
$388$	479008.i	0.161534i
$389$	−4.46563e6	−1.49627	−0.748133	−	0.663549i	$-0.769050\pi$
$389$	−4.46563e6	−1.49627	−0.748133	+	0.663549i	$0.769050\pi$
$390$	0	0
$391$	−2.21184e6	−0.731664
$392$	326208.i	0.107221i
$393$	0	0
$394$	1.62547e6	0.527520
$395$	0	0
$396$	0	0
$397$	− 611026.i	− 0.194573i	−0.995256	−	0.0972867i	$-0.968984\pi$
$397$	− 611026.i	− 0.194573i	0.995256	−	0.0972867i	$-0.0310164\pi$
$398$	1.44798e6i	0.458201i
$399$	0	0
$400$	0	0
$401$	6.09158e6	1.89177	0.945887	−	0.324496i	$-0.105195\pi$
$401$	6.09158e6	1.89177	0.945887	+	0.324496i	$0.105195\pi$
$402$	0	0
$403$	− 1.52438e6i	− 0.467552i
$404$	2.85850e6	0.871333
$405$	0	0
$406$	56832.0	0.0171111
$407$	− 2.22643e6i	− 0.666229i
$408$	0	0
$409$	−2.89108e6	−0.854578	−0.427289	−	0.904115i	$-0.640531\pi$
$409$	−2.89108e6	−0.854578	−0.427289	+	0.904115i	$0.640531\pi$
$410$	0	0
$411$	0	0
$412$	− 1.84365e6i	− 0.535100i
$413$	1.93229e6i	0.557438i
$414$	0	0
$415$	0	0
$416$	−342016.	−0.0968976
$417$	0	0
$418$	− 1.01990e6i	− 0.285508i
$419$	−2.98406e6	−0.830373	−0.415186	−	0.909736i	$-0.636284\pi$
$419$	−2.98406e6	−0.830373	−0.415186	+	0.909736i	$0.636284\pi$
$420$	0	0
$421$	822074.	0.226051	0.113025	−	0.993592i	$-0.463946\pi$
$421$	822074.	0.226051	0.113025	+	0.993592i	$0.463946\pi$
$422$	607424.i	0.166039i
$423$	0	0
$424$	497664.	0.134438
$425$	0	0
$426$	0	0
$427$	− 6.33174e6i	− 1.68056i
$428$	− 1.21651e6i	− 0.321001i
$429$	0	0
$430$	0	0
$431$	−6.32448e6	−1.63995	−0.819977	−	0.572397i	$-0.806014\pi$
$431$	−6.32448e6	−1.63995	−0.819977	+	0.572397i	$0.806014\pi$
$432$	0	0
$433$	851902.i	0.218358i	0.994022	+	0.109179i	$0.0348222\pi$
$433$	851902.i	0.218358i	−0.994022	+	0.109179i	$0.965178\pi$
$434$	−2.70189e6	−0.688562
$435$	0	0
$436$	−3.69789e6	−0.931617
$437$	− 2.54976e6i	− 0.638698i
$438$	0	0
$439$	334732.	0.0828964	0.0414482	−	0.999141i	$-0.486803\pi$
$439$	334732.	0.0828964	0.0414482	+	0.999141i	$0.486803\pi$
$440$	0	0
$441$	0	0
$442$	769536.i	0.187358i
$443$	− 1.76218e6i	− 0.426619i	−0.976985	−	0.213309i	$-0.931576\pi$
$443$	− 1.76218e6i	− 0.426619i	0.976985	−	0.213309i	$-0.0684242\pi$
$444$	0	0
$445$	0	0
$446$	−4.37330e6	−1.04105
$447$	0	0
$448$	606208.i	0.142701i
$449$	6.51859e6	1.52594	0.762971	−	0.646433i	$-0.223740\pi$
$449$	6.51859e6	1.52594	0.762971	+	0.646433i	$0.223740\pi$
$450$	0	0
$451$	−2.58048e6	−0.597392
$452$	− 2.27942e6i	− 0.524782i
$453$	0	0
$454$	−2.26560e6	−0.515874
$455$	0	0
$456$	0	0
$457$	− 92074.0i	− 0.0206227i	−0.999947	−	0.0103114i	$-0.996718\pi$
$457$	− 92074.0i	− 0.0206227i	0.999947	−	0.0103114i	$-0.00328227\pi$
$458$	2.34882e6i	0.523223i
$459$	0	0
$460$	0	0
$461$	−257568.	−0.0564468	−0.0282234	−	0.999602i	$-0.508985\pi$
$461$	−257568.	−0.0564468	−0.0282234	+	0.999602i	$0.508985\pi$
$462$	0	0
$463$	− 3.96228e6i	− 0.858998i	−0.903067	−	0.429499i	$-0.858690\pi$
$463$	− 3.96228e6i	− 0.858998i	0.903067	−	0.429499i	$-0.141310\pi$
$464$	24576.0	0.00529927
$465$	0	0
$466$	−2.31782e6	−0.494442
$467$	3.48941e6i	0.740388i	0.928954	+	0.370194i	$0.120709\pi$
$467$	3.48941e6i	0.740388i	−0.928954	+	0.370194i	$0.879291\pi$
$468$	0	0
$469$	5.42509e6	1.13887
$470$	0	0
$471$	0	0
$472$	835584.i	0.172637i
$473$	− 5.71085e6i	− 1.17367i
$474$	0	0
$475$	0	0
$476$	1.36397e6	0.275922
$477$	0	0
$478$	− 2.33779e6i	− 0.467990i
$479$	513024.	0.102164	0.0510821	−	0.998694i	$-0.483733\pi$
$479$	513024.	0.102164	0.0510821	+	0.998694i	$0.483733\pi$
$480$	0	0
$481$	−1.93653e6	−0.381647
$482$	− 1.65652e6i	− 0.324772i
$483$	0	0
$484$	217520.	0.0422071
$485$	0	0
$486$	0	0
$487$	4.14499e6i	0.791956i	0.918260	+	0.395978i	$0.129594\pi$
$487$	4.14499e6i	0.791956i	−0.918260	+	0.395978i	$0.870406\pi$
$488$	− 2.73805e6i	− 0.520465i
$489$	0	0
$490$	0	0
$491$	−2.75866e6	−0.516409	−0.258205	−	0.966090i	$-0.583131\pi$
$491$	−2.75866e6	−0.516409	−0.258205	+	0.966090i	$0.583131\pi$
$492$	0	0
$493$	− 55296.0i	− 0.0102465i
$494$	−887104.	−0.163552
$495$	0	0
$496$	−1.16838e6	−0.213246
$497$	9.54778e6i	1.73385i
$498$	0	0
$499$	−660896.	−0.118818	−0.0594089	−	0.998234i	$-0.518922\pi$
$499$	−660896.	−0.118818	−0.0594089	+	0.998234i	$0.518922\pi$
$500$	0	0
$501$	0	0
$502$	7.59859e6i	1.34578i
$503$	− 944640.i	− 0.166474i	−0.996530	−	0.0832370i	$-0.973474\pi$
$503$	− 944640.i	− 0.166474i	0.996530	−	0.0832370i	$-0.0265258\pi$
$504$	0	0
$505$	0	0
$506$	−5.89824e6	−1.02411
$507$	0	0
$508$	15808.0i	0.00271780i
$509$	−7.83773e6	−1.34090	−0.670449	−	0.741956i	$-0.733898\pi$
$509$	−7.83773e6	−1.34090	−0.670449	+	0.741956i	$0.733898\pi$
$510$	0	0
$511$	2.48788e6	0.421480
$512$	262144.i	0.0441942i
$513$	0	0
$514$	1.79098e6	0.299007
$515$	0	0
$516$	0	0
$517$	− 7.37280e6i	− 1.21313i
$518$	3.43242e6i	0.562051i
$519$	0	0
$520$	0	0
$521$	−3.29645e6	−0.532049	−0.266025	−	0.963966i	$-0.585710\pi$
$521$	−3.29645e6	−0.532049	−0.266025	+	0.963966i	$0.585710\pi$
$522$	0	0
$523$	− 6.50238e6i	− 1.03948i	−0.854323	−	0.519742i	$-0.826028\pi$
$523$	− 6.50238e6i	− 1.03948i	0.854323	−	0.519742i	$-0.173972\pi$
$524$	3.58810e6	0.570868
$525$	0	0
$526$	270336.	0.0426029
$527$	2.62886e6i	0.412327i
$528$	0	0
$529$	−8.30926e6	−1.29099
$530$	0	0
$531$	0	0
$532$	1.57235e6i	0.240863i
$533$	2.24448e6i	0.342214i
$534$	0	0
$535$	0	0
$536$	2.34598e6	0.352706
$537$	0	0
$538$	2.25677e6i	0.336149i
$539$	1.95725e6	0.290184
$540$	0	0
$541$	9.82714e6	1.44356	0.721778	−	0.692124i	$-0.243325\pi$
$541$	9.82714e6	1.44356	0.721778	+	0.692124i	$0.243325\pi$
$542$	2.88123e6i	0.421289i
$543$	0	0
$544$	589824.	0.0854526
$545$	0	0
$546$	0	0
$547$	− 3.42580e6i	− 0.489546i	−0.969580	−	0.244773i	$-0.921287\pi$
$547$	− 3.42580e6i	− 0.489546i	0.969580	−	0.244773i	$-0.0787135\pi$
$548$	− 4.46362e6i	− 0.634944i
$549$	0	0
$550$	0	0
$551$	63744.0	0.00894459
$552$	0	0
$553$	4.15525e6i	0.577809i
$554$	564568.	0.0781523
$555$	0	0
$556$	2.83520e6	0.388953
$557$	6.43363e6i	0.878655i	0.898327	+	0.439327i	$0.144783\pi$
$557$	6.43363e6i	0.878655i	−0.898327	+	0.439327i	$0.855217\pi$
$558$	0	0
$559$	−4.96725e6	−0.672336
$560$	0	0
$561$	0	0
$562$	− 2.33779e6i	− 0.312223i
$563$	753024.i	0.100124i	0.998746	+	0.0500620i	$0.0159419\pi$
$563$	753024.i	0.100124i	−0.998746	+	0.0500620i	$0.984058\pi$
$564$	0	0
$565$	0	0
$566$	708224.	0.0929244
$567$	0	0
$568$	4.12877e6i	0.536970i
$569$	−1.11481e7	−1.44351	−0.721755	−	0.692148i	$-0.756664\pi$
$569$	−1.11481e7	−1.44351	−0.721755	+	0.692148i	$0.756664\pi$
$570$	0	0
$571$	191024.	0.0245187	0.0122594	−	0.999925i	$-0.496098\pi$
$571$	191024.	0.0245187	0.0122594	+	0.999925i	$0.496098\pi$
$572$	2.05210e6i	0.262245i
$573$	0	0
$574$	3.97824e6	0.503978
$575$	0	0
$576$	0	0
$577$	1.03722e7i	1.29697i	0.761227	+	0.648486i	$0.224597\pi$
$577$	1.03722e7i	1.29697i	−0.761227	+	0.648486i	$0.775403\pi$
$578$	4.35232e6i	0.541878i
$579$	0	0
$580$	0	0
$581$	−9.83194e6	−1.20837
$582$	0	0
$583$	− 2.98598e6i	− 0.363845i
$584$	1.07584e6	0.130532
$585$	0	0
$586$	3.82733e6	0.460417
$587$	2.97062e6i	0.355838i	0.984045	+	0.177919i	$0.0569365\pi$
$587$	2.97062e6i	0.355838i	−0.984045	+	0.177919i	$0.943063\pi$
$588$	0	0
$589$	−3.03050e6	−0.359936
$590$	0	0
$591$	0	0
$592$	1.48429e6i	0.174066i
$593$	− 7.55827e6i	− 0.882644i	−0.897349	−	0.441322i	$-0.854510\pi$
$593$	− 7.55827e6i	− 0.882644i	0.897349	−	0.441322i	$-0.145490\pi$
$594$	0	0
$595$	0	0
$596$	−3.77702e6	−0.435546
$597$	0	0
$598$	5.13024e6i	0.586658i
$599$	6.69158e6	0.762012	0.381006	−	0.924573i	$-0.375578\pi$
$599$	6.69158e6	0.762012	0.381006	+	0.924573i	$0.375578\pi$
$600$	0	0
$601$	−3.20359e6	−0.361785	−0.180893	−	0.983503i	$-0.557899\pi$
$601$	−3.20359e6	−0.361785	−0.180893	+	0.983503i	$0.557899\pi$
$602$	8.80422e6i	0.990147i
$603$	0	0
$604$	7.72538e6	0.861643
$605$	0	0
$606$	0	0
$607$	− 1.35585e7i	− 1.49362i	−0.665038	−	0.746809i	$-0.731585\pi$
$607$	− 1.35585e7i	− 1.49362i	0.665038	−	0.746809i	$-0.268415\pi$
$608$	679936.i	0.0745949i
$609$	0	0
$610$	0	0
$611$	−6.41280e6	−0.694936
$612$	0	0
$613$	1.07654e7i	1.15712i	0.815639	+	0.578561i	$0.196385\pi$
$613$	1.07654e7i	1.15712i	−0.815639	+	0.578561i	$0.803615\pi$
$614$	−1.15315e7	−1.23442
$615$	0	0
$616$	3.63725e6	0.386208
$617$	9.33504e6i	0.987196i	0.869690	+	0.493598i	$0.164319\pi$
$617$	9.33504e6i	0.987196i	−0.869690	+	0.493598i	$0.835681\pi$
$618$	0	0
$619$	−9.07664e6	−0.952135	−0.476067	−	0.879409i	$-0.657938\pi$
$619$	−9.07664e6	−0.952135	−0.476067	+	0.879409i	$0.657938\pi$
$620$	0	0
$621$	0	0
$622$	1.04018e7i	1.07803i
$623$	1.21052e7i	1.24955i
$624$	0	0
$625$	0	0
$626$	−1.03232e7	−1.05288
$627$	0	0
$628$	− 6.09738e6i	− 0.616941i
$629$	3.33965e6	0.336569
$630$	0	0
$631$	−1.13367e7	−1.13348	−0.566741	−	0.823896i	$-0.691796\pi$
$631$	−1.13367e7	−1.13348	−0.566741	+	0.823896i	$0.691796\pi$
$632$	1.79686e6i	0.178946i
$633$	0	0
$634$	9.24403e6	0.913352
$635$	0	0
$636$	0	0
$637$	− 1.70240e6i	− 0.166231i
$638$	− 147456.i	− 0.0143420i
$639$	0	0
$640$	0	0
$641$	−1.55449e7	−1.49432	−0.747159	−	0.664646i	$-0.768582\pi$
$641$	−1.55449e7	−1.49432	−0.747159	+	0.664646i	$0.768582\pi$
$642$	0	0
$643$	8.80026e6i	0.839399i	0.907663	+	0.419699i	$0.137864\pi$
$643$	8.80026e6i	0.839399i	−0.907663	+	0.419699i	$0.862136\pi$
$644$	9.09312e6	0.863969
$645$	0	0
$646$	1.52986e6	0.144235
$647$	− 1.08449e7i	− 1.01851i	−0.860615	−	0.509256i	$-0.829921\pi$
$647$	− 1.08449e7i	− 1.01851i	0.860615	−	0.509256i	$-0.170079\pi$
$648$	0	0
$649$	5.01350e6	0.467229
$650$	0	0
$651$	0	0
$652$	2.60672e6i	0.240146i
$653$	− 9.88771e6i	− 0.907429i	−0.891147	−	0.453715i	$-0.850099\pi$
$653$	− 9.88771e6i	− 0.907429i	0.891147	−	0.453715i	$-0.149901\pi$
$654$	0	0
$655$	0	0
$656$	1.72032e6	0.156081
$657$	0	0
$658$	1.13664e7i	1.02343i
$659$	−1.46150e7	−1.31095	−0.655476	−	0.755216i	$-0.727532\pi$
$659$	−1.46150e7	−1.31095	−0.655476	+	0.755216i	$0.727532\pi$
$660$	0	0
$661$	1.57792e7	1.40469	0.702347	−	0.711834i	$-0.252135\pi$
$661$	1.57792e7	1.40469	0.702347	+	0.711834i	$0.252135\pi$
$662$	− 2.54810e6i	− 0.225980i
$663$	0	0
$664$	−4.25165e6	−0.374229
$665$	0	0
$666$	0	0
$667$	− 368640.i	− 0.0320840i
$668$	− 9.05626e6i	− 0.785249i
$669$	0	0
$670$	0	0
$671$	−1.64283e7	−1.40859
$672$	0	0
$673$	− 6.64939e6i	− 0.565906i	−0.959134	−	0.282953i	$-0.908686\pi$
$673$	− 6.64939e6i	− 0.565906i	0.959134	−	0.282953i	$-0.0913141\pi$
$674$	−1.35466e7	−1.14863
$675$	0	0
$676$	−4.15579e6	−0.349774
$677$	− 4.42550e6i	− 0.371100i	−0.982635	−	0.185550i	$-0.940593\pi$
$677$	− 4.42550e6i	− 0.371100i	0.982635	−	0.185550i	$-0.0594067\pi$
$678$	0	0
$679$	−4.43082e6	−0.368816
$680$	0	0
$681$	0	0
$682$	7.01030e6i	0.577133i
$683$	− 4.67827e6i	− 0.383737i	−0.981421	−	0.191869i	$-0.938545\pi$
$683$	− 4.67827e6i	− 0.383737i	0.981421	−	0.191869i	$-0.0614547\pi$
$684$	0	0
$685$	0	0
$686$	6.93232e6	0.562430
$687$	0	0
$688$	3.80723e6i	0.306647i
$689$	−2.59718e6	−0.208427
$690$	0	0
$691$	−5.54102e6	−0.441463	−0.220731	−	0.975335i	$-0.570844\pi$
$691$	−5.54102e6	−0.441463	−0.220731	+	0.975335i	$0.570844\pi$
$692$	3.49133e6i	0.277157i
$693$	0	0
$694$	1.11130e7	0.875853
$695$	0	0
$696$	0	0
$697$	− 3.87072e6i	− 0.301793i
$698$	− 6.22143e6i	− 0.483339i
$699$	0	0
$700$	0	0
$701$	6.53443e6	0.502242	0.251121	−	0.967956i	$-0.419201\pi$
$701$	6.53443e6	0.502242	0.251121	+	0.967956i	$0.419201\pi$
$702$	0	0
$703$	3.84987e6i	0.293804i
$704$	1.57286e6	0.119608
$705$	0	0
$706$	−8.47104e6	−0.639624
$707$	2.64411e7i	1.98944i
$708$	0	0
$709$	3.86541e6	0.288789	0.144394	−	0.989520i	$-0.453877\pi$
$709$	3.86541e6	0.288789	0.144394	+	0.989520i	$0.453877\pi$
$710$	0	0
$711$	0	0
$712$	5.23469e6i	0.386982i
$713$	1.75258e7i	1.29108i
$714$	0	0
$715$	0	0
$716$	6.59866e6	0.481031
$717$	0	0
$718$	8.69990e6i	0.629801i
$719$	4.80614e6	0.346717	0.173358	−	0.984859i	$-0.444538\pi$
$719$	4.80614e6	0.346717	0.173358	+	0.984859i	$0.444538\pi$
$720$	0	0
$721$	1.70537e7	1.22175
$722$	− 8.14081e6i	− 0.581199i
$723$	0	0
$724$	408928.	0.0289935
$725$	0	0
$726$	0	0
$727$	− 1.90590e7i	− 1.33741i	−0.743529	−	0.668704i	$-0.766850\pi$
$727$	− 1.90590e7i	− 1.33741i	0.743529	−	0.668704i	$-0.233150\pi$
$728$	− 3.16365e6i	− 0.221238i
$729$	0	0
$730$	0	0
$731$	8.56627e6	0.592923
$732$	0	0
$733$	− 5.69616e6i	− 0.391582i	−0.980646	−	0.195791i	$-0.937273\pi$
$733$	− 5.69616e6i	− 0.391582i	0.980646	−	0.195791i	$-0.0627274\pi$
$734$	−4.21342e6	−0.288666
$735$	0	0
$736$	3.93216e6	0.267570
$737$	− 1.40759e7i	− 0.954570i
$738$	0	0
$739$	−1.84902e7	−1.24546	−0.622730	−	0.782437i	$-0.713976\pi$
$739$	−1.84902e7	−1.24546	−0.622730	+	0.782437i	$0.713976\pi$
$740$	0	0
$741$	0	0
$742$	4.60339e6i	0.306950i
$743$	9.90336e6i	0.658128i	0.944308	+	0.329064i	$0.106733\pi$
$743$	9.90336e6i	0.658128i	−0.944308	+	0.329064i	$0.893267\pi$
$744$	0	0
$745$	0	0
$746$	−2.70839e6	−0.178182
$747$	0	0
$748$	− 3.53894e6i	− 0.231270i
$749$	1.12527e7	0.732915
$750$	0	0
$751$	−9.05914e6	−0.586121	−0.293060	−	0.956094i	$-0.594674\pi$
$751$	−9.05914e6	−0.586121	−0.293060	+	0.956094i	$0.594674\pi$
$752$	4.91520e6i	0.316954i
$753$	0	0
$754$	−128256.	−0.00821579
$755$	0	0
$756$	0	0
$757$	− 1.16677e7i	− 0.740022i	−0.929027	−	0.370011i	$-0.879354\pi$
$757$	− 1.16677e7i	− 0.740022i	0.929027	−	0.370011i	$-0.120646\pi$
$758$	2.04299e7i	1.29150i
$759$	0	0
$760$	0	0
$761$	1.27398e7	0.797444	0.398722	−	0.917072i	$-0.369454\pi$
$761$	1.27398e7	0.797444	0.398722	+	0.917072i	$0.369454\pi$
$762$	0	0
$763$	− 3.42055e7i	− 2.12708i
$764$	6.40205e6	0.396813
$765$	0	0
$766$	−6.52800e6	−0.401983
$767$	− 4.36070e6i	− 0.267651i
$768$	0	0
$769$	−1.06783e7	−0.651156	−0.325578	−	0.945515i	$-0.605559\pi$
$769$	−1.06783e7	−0.651156	−0.325578	+	0.945515i	$0.605559\pi$
$770$	0	0
$771$	0	0
$772$	1.11944e7i	0.676017i
$773$	− 9.18634e6i	− 0.552960i	−0.961020	−	0.276480i	$-0.910832\pi$
$773$	− 9.18634e6i	− 0.552960i	0.961020	−	0.276480i	$-0.0891679\pi$
$774$	0	0
$775$	0	0
$776$	−1.91603e6	−0.114222
$777$	0	0
$778$	− 1.78625e7i	− 1.05802i
$779$	4.46208e6	0.263447
$780$	0	0
$781$	2.47726e7	1.45326
$782$	− 8.84736e6i	− 0.517365i
$783$	0	0
$784$	−1.30483e6	−0.0758166
$785$	0	0
$786$	0	0
$787$	− 9.28209e6i	− 0.534206i	−0.963668	−	0.267103i	$-0.913934\pi$
$787$	− 9.28209e6i	− 0.534206i	0.963668	−	0.267103i	$-0.0860664\pi$
$788$	6.50189e6i	0.373013i
$789$	0	0
$790$	0	0
$791$	2.10847e7	1.19819
$792$	0	0
$793$	1.42892e7i	0.806909i
$794$	2.44410e6	0.137584
$795$	0	0
$796$	−5.79194e6	−0.323997
$797$	− 21792.0i	− 0.00121521i	−1.00000		0.000607605i	$-0.999807\pi$
$797$	− 21792.0i	− 0.00121521i	1.00000		0.000607605i	$-0.000193407\pi$
$798$	0	0
$799$	1.10592e7	0.612854
$800$	0	0
$801$	0	0
$802$	2.43663e7i	1.33769i
$803$	− 6.45504e6i	− 0.353273i
$804$	0	0
$805$	0	0
$806$	6.09750e6	0.330609
$807$	0	0
$808$	1.14340e7i	0.616125i
$809$	1.23085e7	0.661204	0.330602	−	0.943770i	$-0.392748\pi$
$809$	1.23085e7	0.661204	0.330602	+	0.943770i	$0.392748\pi$
$810$	0	0
$811$	−2.34636e7	−1.25269	−0.626343	−	0.779547i	$-0.715449\pi$
$811$	−2.34636e7	−1.25269	−0.626343	+	0.779547i	$0.715449\pi$
$812$	227328.i	0.0120994i
$813$	0	0
$814$	8.90573e6	0.471095
$815$	0	0
$816$	0	0
$817$	9.87501e6i	0.517586i
$818$	− 1.15643e7i	− 0.604278i
$819$	0	0
$820$	0	0
$821$	1.44206e7	0.746666	0.373333	−	0.927697i	$-0.378215\pi$
$821$	1.44206e7	0.746666	0.373333	+	0.927697i	$0.378215\pi$
$822$	0	0
$823$	− 3.43419e7i	− 1.76736i	−0.468093	−	0.883679i	$-0.655059\pi$
$823$	− 3.43419e7i	− 1.76736i	0.468093	−	0.883679i	$-0.344941\pi$
$824$	7.37459e6	0.378373
$825$	0	0
$826$	−7.72915e6	−0.394168
$827$	− 2.13327e7i	− 1.08463i	−0.840174	−	0.542316i	$-0.817547\pi$
$827$	− 2.13327e7i	− 1.08463i	0.840174	−	0.542316i	$-0.182453\pi$
$828$	0	0
$829$	−2.63751e6	−0.133293	−0.0666465	−	0.997777i	$-0.521230\pi$
$829$	−2.63751e6	−0.133293	−0.0666465	+	0.997777i	$0.521230\pi$
$830$	0	0
$831$	0	0
$832$	− 1.36806e6i	− 0.0685170i
$833$	2.93587e6i	0.146597i
$834$	0	0
$835$	0	0
$836$	4.07962e6	0.201885
$837$	0	0
$838$	− 1.19363e7i	− 0.587162i
$839$	−1.00577e7	−0.493282	−0.246641	−	0.969107i	$-0.579327\pi$
$839$	−1.00577e7	−0.493282	−0.246641	+	0.969107i	$0.579327\pi$
$840$	0	0
$841$	−2.05019e7	−0.999551
$842$	3.28830e6i	0.159842i
$843$	0	0
$844$	−2.42970e6	−0.117407
$845$	0	0
$846$	0	0
$847$	2.01206e6i	0.0963679i
$848$	1.99066e6i	0.0950619i
$849$	0	0
$850$	0	0
$851$	2.22643e7	1.05387
$852$	0	0
$853$	2.30748e7i	1.08584i	0.839785	+	0.542919i	$0.182681\pi$
$853$	2.30748e7i	1.08584i	−0.839785	+	0.542919i	$0.817319\pi$
$854$	2.53269e7	1.18833
$855$	0	0
$856$	4.86605e6	0.226982
$857$	− 3.51646e7i	− 1.63551i	−0.575565	−	0.817756i	$-0.695218\pi$
$857$	− 3.51646e7i	− 1.63551i	0.575565	−	0.817756i	$-0.304782\pi$
$858$	0	0
$859$	−1.66022e7	−0.767684	−0.383842	−	0.923399i	$-0.625399\pi$
$859$	−1.66022e7	−0.767684	−0.383842	+	0.923399i	$0.625399\pi$
$860$	0	0
$861$	0	0
$862$	− 2.52979e7i	− 1.15962i
$863$	− 2.97009e7i	− 1.35751i	−0.734366	−	0.678754i	$-0.762520\pi$
$863$	− 2.97009e7i	− 1.35751i	0.734366	−	0.678754i	$-0.237480\pi$
$864$	0	0
$865$	0	0
$866$	−3.40761e6	−0.154403
$867$	0	0
$868$	− 1.08076e7i	− 0.486887i
$869$	1.07812e7	0.484303
$870$	0	0
$871$	−1.22431e7	−0.546822
$872$	− 1.47916e7i	− 0.658753i
$873$	0	0
$874$	1.01990e7	0.451628
$875$	0	0
$876$	0	0
$877$	− 1.17943e7i	− 0.517811i	−0.965903	−	0.258906i	$-0.916638\pi$
$877$	− 1.17943e7i	− 0.517811i	0.965903	−	0.258906i	$-0.0833619\pi$
$878$	1.33893e6i	0.0586166i
$879$	0	0
$880$	0	0
$881$	2.10378e7	0.913190	0.456595	−	0.889675i	$-0.349069\pi$
$881$	2.10378e7	0.913190	0.456595	+	0.889675i	$0.349069\pi$
$882$	0	0
$883$	− 2.12192e7i	− 0.915855i	−0.888990	−	0.457928i	$-0.848592\pi$
$883$	− 2.12192e7i	− 0.915855i	0.888990	−	0.457928i	$-0.151408\pi$
$884$	−3.07814e6	−0.132482
$885$	0	0
$886$	7.04870e6	0.301665
$887$	− 2.28818e7i	− 0.976520i	−0.872698	−	0.488260i	$-0.837632\pi$
$887$	− 2.28818e7i	− 0.976520i	0.872698	−	0.488260i	$-0.162368\pi$
$888$	0	0
$889$	−146224.	−0.00620532
$890$	0	0
$891$	0	0
$892$	− 1.74932e7i	− 0.736134i
$893$	1.27488e7i	0.534984i
$894$	0	0
$895$	0	0
$896$	−2.42483e6	−0.100905
$897$	0	0
$898$	2.60744e7i	1.07900i
$899$	−438144.	−0.0180808
$900$	0	0
$901$	4.47898e6	0.183809
$902$	− 1.03219e7i	− 0.422420i
$903$	0	0
$904$	9.11770e6	0.371077
$905$	0	0
$906$	0	0
$907$	− 9.68692e6i	− 0.390992i	−0.980705	−	0.195496i	$-0.937368\pi$
$907$	− 9.68692e6i	− 0.390992i	0.980705	−	0.195496i	$-0.0626316\pi$
$908$	− 9.06240e6i	− 0.364778i
$909$	0	0
$910$	0	0
$911$	−9.38112e6	−0.374506	−0.187253	−	0.982312i	$-0.559958\pi$
$911$	−9.38112e6	−0.374506	−0.187253	+	0.982312i	$0.559958\pi$
$912$	0	0
$913$	2.55099e7i	1.01282i
$914$	368296.	0.0145825
$915$	0	0
$916$	−9.39530e6	−0.369974
$917$	3.31899e7i	1.30341i
$918$	0	0
$919$	4.21870e7	1.64774	0.823872	−	0.566775i	$-0.191809\pi$
$919$	4.21870e7	1.64774	0.823872	+	0.566775i	$0.191809\pi$
$920$	0	0
$921$	0	0
$922$	− 1.03027e6i	− 0.0399139i
$923$	− 2.15470e7i	− 0.832497i
$924$	0	0
$925$	0	0
$926$	1.58491e7	0.607403
$927$	0	0
$928$	98304.0i	0.00374715i
$929$	−3.04556e7	−1.15779	−0.578893	−	0.815404i	$-0.696515\pi$
$929$	−3.04556e7	−1.15779	−0.578893	+	0.815404i	$0.696515\pi$
$930$	0	0
$931$	−3.38441e6	−0.127970
$932$	− 9.27130e6i	− 0.349624i
$933$	0	0
$934$	−1.39576e7	−0.523534
$935$	0	0
$936$	0	0
$937$	1.47847e7i	0.550128i	0.961426	+	0.275064i	$0.0886990\pi$
$937$	1.47847e7i	0.550128i	−0.961426	+	0.275064i	$0.911301\pi$
$938$	2.17004e7i	0.805304i
$939$	0	0
$940$	0	0
$941$	9.91997e6	0.365205	0.182602	−	0.983187i	$-0.441548\pi$
$941$	9.91997e6	0.365205	0.182602	+	0.983187i	$0.441548\pi$
$942$	0	0
$943$	− 2.58048e7i	− 0.944977i
$944$	−3.34234e6	−0.122073
$945$	0	0
$946$	2.28434e7	0.829913
$947$	− 2.91610e6i	− 0.105664i	−0.998603	−	0.0528320i	$-0.983175\pi$
$947$	− 2.91610e6i	− 0.105664i	0.998603	−	0.0528320i	$-0.0168248\pi$
$948$	0	0
$949$	−5.61454e6	−0.202371
$950$	0	0
$951$	0	0
$952$	5.45587e6i	0.195107i
$953$	1.40861e7i	0.502410i	0.967934	+	0.251205i	$0.0808268\pi$
$953$	1.40861e7i	0.502410i	−0.967934	+	0.251205i	$0.919173\pi$
$954$	0	0
$955$	0	0
$956$	9.35117e6	0.330919
$957$	0	0
$958$	2.05210e6i	0.0722410i
$959$	4.12884e7	1.44971
$960$	0	0
$961$	−7.79906e6	−0.272417
$962$	− 7.74613e6i	− 0.269865i
$963$	0	0
$964$	6.62608e6	0.229649
$965$	0	0
$966$	0	0
$967$	1.51949e7i	0.522553i	0.965264	+	0.261276i	$0.0841434\pi$
$967$	1.51949e7i	0.522553i	−0.965264	+	0.261276i	$0.915857\pi$
$968$	870080.i	0.0298449i
$969$	0	0
$970$	0	0
$971$	5.61220e7	1.91023	0.955113	−	0.296240i	$-0.0957329\pi$
$971$	5.61220e7	1.91023	0.955113	+	0.296240i	$0.0957329\pi$
$972$	0	0
$973$	2.62256e7i	0.888062i
$974$	−1.65800e7	−0.559997
$975$	0	0
$976$	1.09522e7	0.368024
$977$	− 3.45625e7i	− 1.15843i	−0.815176	−	0.579214i	$-0.803360\pi$
$977$	− 3.45625e7i	− 1.15843i	0.815176	−	0.579214i	$-0.196640\pi$
$978$	0	0
$979$	3.14081e7	1.04733
$980$	0	0
$981$	0	0
$982$	− 1.10346e7i	− 0.365156i
$983$	5.56385e7i	1.83650i	0.395997	+	0.918252i	$0.370399\pi$
$983$	5.56385e7i	1.83650i	−0.395997	+	0.918252i	$0.629601\pi$
$984$	0	0
$985$	0	0
$986$	221184.	0.00724538
$987$	0	0
$988$	− 3.54842e6i	− 0.115649i
$989$	5.71085e7	1.85656
$990$	0	0
$991$	−3.60028e7	−1.16453	−0.582267	−	0.812998i	$-0.697834\pi$
$991$	−3.60028e7	−1.16453	−0.582267	+	0.812998i	$0.697834\pi$
$992$	− 4.67354e6i	− 0.150788i
$993$	0	0
$994$	−3.81911e7	−1.22602
$995$	0	0
$996$	0	0
$997$	2.35811e7i	0.751322i	0.926757	+	0.375661i	$0.122584\pi$
$997$	2.35811e7i	0.751322i	−0.926757	+	0.375661i	$0.877416\pi$
$998$	− 2.64358e6i	− 0.0840169i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.c.c.199.2		2
3.2	odd	2		450.6.c.m.199.1		2
5.2	odd	4		450.6.a.k.1.1		1
5.3	odd	4		18.6.a.c.1.1	yes	1
5.4	even	2	inner	450.6.c.c.199.1		2
15.2	even	4		450.6.a.v.1.1		1
15.8	even	4		18.6.a.a.1.1	✓	1
15.14	odd	2		450.6.c.m.199.2		2
20.3	even	4		144.6.a.l.1.1		1
35.13	even	4		882.6.a.l.1.1		1
40.3	even	4		576.6.a.b.1.1		1
40.13	odd	4		576.6.a.a.1.1		1
45.13	odd	12		162.6.c.a.55.1		2
45.23	even	12		162.6.c.l.55.1		2
45.38	even	12		162.6.c.l.109.1		2
45.43	odd	12		162.6.c.a.109.1		2
60.23	odd	4		144.6.a.a.1.1		1
105.83	odd	4		882.6.a.k.1.1		1
120.53	even	4		576.6.a.bh.1.1		1
120.83	odd	4		576.6.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.6.a.a.1.1	✓	1	15.8	even	4
18.6.a.c.1.1	yes	1	5.3	odd	4
144.6.a.a.1.1		1	60.23	odd	4
144.6.a.l.1.1		1	20.3	even	4
162.6.c.a.55.1		2	45.13	odd	12
162.6.c.a.109.1		2	45.43	odd	12
162.6.c.l.55.1		2	45.23	even	12
162.6.c.l.109.1		2	45.38	even	12
450.6.a.k.1.1		1	5.2	odd	4
450.6.a.v.1.1		1	15.2	even	4
450.6.c.c.199.1		2	5.4	even	2	inner
450.6.c.c.199.2		2	1.1	even	1	trivial
450.6.c.m.199.1		2	3.2	odd	2
450.6.c.m.199.2		2	15.14	odd	2
576.6.a.a.1.1		1	40.13	odd	4
576.6.a.b.1.1		1	40.3	even	4
576.6.a.bh.1.1		1	120.53	even	4
576.6.a.bi.1.1		1	120.83	odd	4
882.6.a.k.1.1		1	105.83	odd	4
882.6.a.l.1.1		1	35.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -16.0000 q^{4} -148.000i q^{7} -64.0000i q^{8} +O(q^{10})\)	\(q+4.00000i q^{2} -16.0000 q^{4} -148.000i q^{7} -64.0000i q^{8} -384.000 q^{11} +334.000i q^{13} +592.000 q^{14} +256.000 q^{16} -576.000i q^{17} +664.000 q^{19} -1536.00i q^{22} -3840.00i q^{23} -1336.00 q^{26} +2368.00i q^{28} +96.0000 q^{29} -4564.00 q^{31} +1024.00i q^{32} +2304.00 q^{34} +5798.00i q^{37} +2656.00i q^{38} +6720.00 q^{41} +14872.0i q^{43} +6144.00 q^{44} +15360.0 q^{46} +19200.0i q^{47} -5097.00 q^{49} -5344.00i q^{52} +7776.00i q^{53} -9472.00 q^{56} +384.000i q^{58} -13056.0 q^{59} +42782.0 q^{61} -18256.0i q^{62} -4096.00 q^{64} +36656.0i q^{67} +9216.00i q^{68} -64512.0 q^{71} +16810.0i q^{73} -23192.0 q^{74} -10624.0 q^{76} +56832.0i q^{77} -28076.0 q^{79} +26880.0i q^{82} -66432.0i q^{83} -59488.0 q^{86} +24576.0i q^{88} -81792.0 q^{89} +49432.0 q^{91} +61440.0i q^{92} -76800.0 q^{94} -29938.0i q^{97} -20388.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4}+O(q^{10}) \) 2 * q - 32 * q^4	\( 2 q - 32 q^{4} - 768 q^{11} + 1184 q^{14} + 512 q^{16} + 1328 q^{19} - 2672 q^{26} + 192 q^{29} - 9128 q^{31} + 4608 q^{34} + 13440 q^{41} + 12288 q^{44} + 30720 q^{46} - 10194 q^{49} - 18944 q^{56} - 26112 q^{59} + 85564 q^{61} - 8192 q^{64} - 129024 q^{71} - 46384 q^{74} - 21248 q^{76} - 56152 q^{79} - 118976 q^{86} - 163584 q^{89} + 98864 q^{91} - 153600 q^{94}+O(q^{100}) \) 2 * q - 32 * q^4 - 768 * q^11 + 1184 * q^14 + 512 * q^16 + 1328 * q^19 - 2672 * q^26 + 192 * q^29 - 9128 * q^31 + 4608 * q^34 + 13440 * q^41 + 12288 * q^44 + 30720 * q^46 - 10194 * q^49 - 18944 * q^56 - 26112 * q^59 + 85564 * q^61 - 8192 * q^64 - 129024 * q^71 - 46384 * q^74 - 21248 * q^76 - 56152 * q^79 - 118976 * q^86 - 163584 * q^89 + 98864 * q^91 - 153600 * q^94

Introduction

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