LMFDB - Embedded newform 450.8.c.a.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,8,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, 8, names="a")

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

chi := DirichletCharacter("450.199");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
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:	$140.573261468$
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 6)
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			450.8.c.a.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000i q^{2} -64.0000 q^{4} -1576.00i q^{7} +512.000i q^{8} +O(q^{10})$	$q-8.00000i q^{2} -64.0000 q^{4} -1576.00i q^{7} +512.000i q^{8} -7332.00 q^{11} +3802.00i q^{13} -12608.0 q^{14} +4096.00 q^{16} +6606.00i q^{17} -24860.0 q^{19} +58656.0i q^{22} +41448.0i q^{23} +30416.0 q^{26} +100864. i q^{28} -41610.0 q^{29} +33152.0 q^{31} -32768.0i q^{32} +52848.0 q^{34} -36466.0i q^{37} +198880. i q^{38} +639078. q^{41} +156412. i q^{43} +469248. q^{44} +331584. q^{46} +433776. i q^{47} -1.66023e6 q^{49} -243328. i q^{52} +786078. i q^{53} +806912. q^{56} +332880. i q^{58} +745140. q^{59} -1.66062e6 q^{61} -265216. i q^{62} -262144. q^{64} -3.29084e6i q^{67} -422784. i q^{68} -5.71615e6 q^{71} -2.65990e6i q^{73} -291728. q^{74} +1.59104e6 q^{76} +1.15552e7i q^{77} -3.80744e6 q^{79} -5.11262e6i q^{82} +2.22947e6i q^{83} +1.25130e6 q^{86} -3.75398e6i q^{88} +5.99121e6 q^{89} +5.99195e6 q^{91} -2.65267e6i q^{92} +3.47021e6 q^{94} -4.06013e6i q^{97} +1.32819e7i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4}+O(q^{10}) $ 2 * q - 128 * q^4	$ 2 q - 128 q^{4} - 14664 q^{11} - 25216 q^{14} + 8192 q^{16} - 49720 q^{19} + 60832 q^{26} - 83220 q^{29} + 66304 q^{31} + 105696 q^{34} + 1278156 q^{41} + 938496 q^{44} + 663168 q^{46} - 3320466 q^{49} + 1613824 q^{56} + 1490280 q^{59} - 3321236 q^{61} - 524288 q^{64} - 11432304 q^{71} - 583456 q^{74} + 3182080 q^{76} - 7614880 q^{79} + 2502592 q^{86} + 11982420 q^{89} + 11983904 q^{91} + 6940416 q^{94}+O(q^{100}) $ 2 * q - 128 * q^4 - 14664 * q^11 - 25216 * q^14 + 8192 * q^16 - 49720 * q^19 + 60832 * q^26 - 83220 * q^29 + 66304 * q^31 + 105696 * q^34 + 1278156 * q^41 + 938496 * q^44 + 663168 * q^46 - 3320466 * q^49 + 1613824 * q^56 + 1490280 * q^59 - 3321236 * q^61 - 524288 * q^64 - 11432304 * q^71 - 583456 * q^74 + 3182080 * q^76 - 7614880 * q^79 + 2502592 * q^86 + 11982420 * q^89 + 11983904 * q^91 + 6940416 * 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 $

$2$	− 8.00000i	− 0.707107i
$2$	− 8.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−64.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1576.00i	− 1.73665i	−0.495993	−	0.868327i	$-0.665196\pi$
$7$	− 1576.00i	− 1.73665i	0.495993	−	0.868327i	$-0.334804\pi$
$8$	512.000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−7332.00	−1.66092	−0.830459	−	0.557080i	$-0.811922\pi$
$11$	−7332.00	−1.66092	−0.830459	+	0.557080i	$0.811922\pi$
$12$	0	0
$13$	3802.00i	0.479966i	0.970777	+	0.239983i	$0.0771419\pi$
$13$	3802.00i	0.479966i	−0.970777	+	0.239983i	$0.922858\pi$
$14$	−12608.0	−1.22800
$15$	0	0
$16$	4096.00	0.250000
$17$	6606.00i	0.326112i	0.986617	+	0.163056i	$0.0521352\pi$
$17$	6606.00i	0.326112i	−0.986617	+	0.163056i	$0.947865\pi$
$18$	0	0
$19$	−24860.0	−0.831502	−0.415751	−	0.909478i	$-0.636481\pi$
$19$	−24860.0	−0.831502	−0.415751	+	0.909478i	$0.636481\pi$
$20$	0	0
$21$	0	0
$22$	58656.0i	1.17445i
$23$	41448.0i	0.710323i	0.934805	+	0.355162i	$0.115574\pi$
$23$	41448.0i	0.710323i	−0.934805	+	0.355162i	$0.884426\pi$
$24$	0	0
$25$	0	0
$26$	30416.0	0.339387
$27$	0	0
$28$	100864.i	0.868327i
$29$	−41610.0	−0.316814	−0.158407	−	0.987374i	$-0.550636\pi$
$29$	−41610.0	−0.316814	−0.158407	+	0.987374i	$0.550636\pi$
$30$	0	0
$31$	33152.0	0.199868	0.0999341	−	0.994994i	$-0.468137\pi$
$31$	33152.0	0.199868	0.0999341	+	0.994994i	$0.468137\pi$
$32$	− 32768.0i	− 0.176777i
$33$	0	0
$34$	52848.0	0.230596
$35$	0	0
$36$	0	0
$37$	− 36466.0i	− 0.118354i	−0.998248	−	0.0591769i	$-0.981152\pi$
$37$	− 36466.0i	− 0.118354i	0.998248	−	0.0591769i	$-0.0188476\pi$
$38$	198880.i	0.587961i
$39$	0	0
$40$	0	0
$41$	639078.	1.44814	0.724070	−	0.689727i	$-0.242269\pi$
$41$	639078.	1.44814	0.724070	+	0.689727i	$0.242269\pi$
$42$	0	0
$43$	156412.i	0.300006i	0.988686	+	0.150003i	$0.0479284\pi$
$43$	156412.i	0.300006i	−0.988686	+	0.150003i	$0.952072\pi$
$44$	469248.	0.830459
$45$	0	0
$46$	331584.	0.502275
$47$	433776.i	0.609429i	0.952444	+	0.304714i	$0.0985610\pi$
$47$	433776.i	0.609429i	−0.952444	+	0.304714i	$0.901439\pi$
$48$	0	0
$49$	−1.66023e6	−2.01596
$50$	0	0
$51$	0	0
$52$	− 243328.i	− 0.239983i
$53$	786078.i	0.725271i	0.931931	+	0.362635i	$0.118123\pi$
$53$	786078.i	0.725271i	−0.931931	+	0.362635i	$0.881877\pi$
$54$	0	0
$55$	0	0
$56$	806912.	0.614000
$57$	0	0
$58$	332880.i	0.224022i
$59$	745140.	0.472341	0.236171	−	0.971712i	$-0.424108\pi$
$59$	745140.	0.472341	0.236171	+	0.971712i	$0.424108\pi$
$60$	0	0
$61$	−1.66062e6	−0.936732	−0.468366	−	0.883535i	$-0.655157\pi$
$61$	−1.66062e6	−0.936732	−0.468366	+	0.883535i	$0.655157\pi$
$62$	− 265216.i	− 0.141328i
$63$	0	0
$64$	−262144.	−0.125000
$65$	0	0
$66$	0	0
$67$	− 3.29084e6i	− 1.33673i	−0.743832	−	0.668366i	$-0.766994\pi$
$67$	− 3.29084e6i	− 1.33673i	0.743832	−	0.668366i	$-0.233006\pi$
$68$	− 422784.i	− 0.163056i
$69$	0	0
$70$	0	0
$71$	−5.71615e6	−1.89539	−0.947697	−	0.319171i	$-0.896596\pi$
$71$	−5.71615e6	−1.89539	−0.947697	+	0.319171i	$0.896596\pi$
$72$	0	0
$73$	− 2.65990e6i	− 0.800267i	−0.916457	−	0.400134i	$-0.868964\pi$
$73$	− 2.65990e6i	− 0.800267i	0.916457	−	0.400134i	$-0.131036\pi$
$74$	−291728.	−0.0836888
$75$	0	0
$76$	1.59104e6	0.415751
$77$	1.15552e7i	2.88444i
$78$	0	0
$79$	−3.80744e6	−0.868837	−0.434418	−	0.900711i	$-0.643046\pi$
$79$	−3.80744e6	−0.868837	−0.434418	+	0.900711i	$0.643046\pi$
$80$	0	0
$81$	0	0
$82$	− 5.11262e6i	− 1.02399i
$83$	2.22947e6i	0.427984i	0.976835	+	0.213992i	$0.0686467\pi$
$83$	2.22947e6i	0.427984i	−0.976835	+	0.213992i	$0.931353\pi$
$84$	0	0
$85$	0	0
$86$	1.25130e6	0.212137
$87$	0	0
$88$	− 3.75398e6i	− 0.587223i
$89$	5.99121e6	0.900844	0.450422	−	0.892816i	$-0.351274\pi$
$89$	5.99121e6	0.900844	0.450422	+	0.892816i	$0.351274\pi$
$90$	0	0
$91$	5.99195e6	0.833534
$92$	− 2.65267e6i	− 0.355162i
$93$	0	0
$94$	3.47021e6	0.430931
$95$	0	0
$96$	0	0
$97$	− 4.06013e6i	− 0.451688i	−0.974163	−	0.225844i	$-0.927486\pi$
$97$	− 4.06013e6i	− 0.451688i	0.974163	−	0.225844i	$-0.0725139\pi$
$98$	1.32819e7i	1.42550i
$99$	0	0
$100$	0	0
$101$	1.72819e7	1.66904	0.834522	−	0.550975i	$-0.185744\pi$
$101$	1.72819e7	1.66904	0.834522	+	0.550975i	$0.185744\pi$
$102$	0	0
$103$	1.43623e7i	1.29507i	0.762035	+	0.647536i	$0.224201\pi$
$103$	1.43623e7i	1.29507i	−0.762035	+	0.647536i	$0.775799\pi$
$104$	−1.94662e6	−0.169694
$105$	0	0
$106$	6.28862e6	0.512844
$107$	− 6.45440e6i	− 0.509346i	−0.967027	−	0.254673i	$-0.918032\pi$
$107$	− 6.45440e6i	− 0.509346i	0.967027	−	0.254673i	$-0.0819678\pi$
$108$	0	0
$109$	884410.	0.0654125	0.0327063	−	0.999465i	$-0.489587\pi$
$109$	884410.	0.0654125	0.0327063	+	0.999465i	$0.489587\pi$
$110$	0	0
$111$	0	0
$112$	− 6.45530e6i	− 0.434163i
$113$	1.21325e7i	0.790999i	0.918466	+	0.395499i	$0.129428\pi$
$113$	1.21325e7i	0.790999i	−0.918466	+	0.395499i	$0.870572\pi$
$114$	0	0
$115$	0	0
$116$	2.66304e6	0.158407
$117$	0	0
$118$	− 5.96112e6i	− 0.333996i
$119$	1.04111e7	0.566344
$120$	0	0
$121$	3.42711e7	1.75865
$122$	1.32849e7i	0.662369i
$123$	0	0
$124$	−2.12173e6	−0.0999341
$125$	0	0
$126$	0	0
$127$	6.86806e6i	0.297524i	0.988873	+	0.148762i	$0.0475288\pi$
$127$	6.86806e6i	0.297524i	−0.988873	+	0.148762i	$0.952471\pi$
$128$	2.09715e6i	0.0883883i
$129$	0	0
$130$	0	0
$131$	3.95208e7	1.53595	0.767973	−	0.640482i	$-0.221265\pi$
$131$	3.95208e7	1.53595	0.767973	+	0.640482i	$0.221265\pi$
$132$	0	0
$133$	3.91794e7i	1.44403i
$134$	−2.63267e7	−0.945212
$135$	0	0
$136$	−3.38227e6	−0.115298
$137$	− 1.91741e7i	− 0.637078i	−0.947910	−	0.318539i	$-0.896808\pi$
$137$	− 1.91741e7i	− 0.637078i	0.947910	−	0.318539i	$-0.103192\pi$
$138$	0	0
$139$	−1.32449e7	−0.418309	−0.209154	−	0.977883i	$-0.567071\pi$
$139$	−1.32449e7	−0.418309	−0.209154	+	0.977883i	$0.567071\pi$
$140$	0	0
$141$	0	0
$142$	4.57292e7i	1.34025i
$143$	− 2.78763e7i	− 0.797184i
$144$	0	0
$145$	0	0
$146$	−2.12792e7	−0.565874
$147$	0	0
$148$	2.33382e6i	0.0591769i
$149$	5.73624e7	1.42061	0.710306	−	0.703893i	$-0.248556\pi$
$149$	5.73624e7	1.42061	0.710306	+	0.703893i	$0.248556\pi$
$150$	0	0
$151$	−3.10873e7	−0.734790	−0.367395	−	0.930065i	$-0.619750\pi$
$151$	−3.10873e7	−0.734790	−0.367395	+	0.930065i	$0.619750\pi$
$152$	− 1.27283e7i	− 0.293981i
$153$	0	0
$154$	9.24419e7	2.03961
$155$	0	0
$156$	0	0
$157$	− 3.37835e7i	− 0.696715i	−0.937362	−	0.348358i	$-0.886739\pi$
$157$	− 3.37835e7i	− 0.696715i	0.937362	−	0.348358i	$-0.113261\pi$
$158$	3.04595e7i	0.614360i
$159$	0	0
$160$	0	0
$161$	6.53220e7	1.23359
$162$	0	0
$163$	− 6.26659e7i	− 1.13338i	−0.823932	−	0.566689i	$-0.808224\pi$
$163$	− 6.26659e7i	− 1.13338i	0.823932	−	0.566689i	$-0.191776\pi$
$164$	−4.09010e7	−0.724070
$165$	0	0
$166$	1.78357e7	0.302631
$167$	− 6.27072e7i	− 1.04186i	−0.853599	−	0.520931i	$-0.825585\pi$
$167$	− 6.27072e7i	− 1.04186i	0.853599	−	0.520931i	$-0.174415\pi$
$168$	0	0
$169$	4.82933e7	0.769633
$170$	0	0
$171$	0	0
$172$	− 1.00104e7i	− 0.150003i
$173$	− 2.70521e7i	− 0.397228i	−0.980078	−	0.198614i	$-0.936356\pi$
$173$	− 2.70521e7i	− 0.397228i	0.980078	−	0.198614i	$-0.0636440\pi$
$174$	0	0
$175$	0	0
$176$	−3.00319e7	−0.415229
$177$	0	0
$178$	− 4.79297e7i	− 0.636993i
$179$	−1.34281e8	−1.74996	−0.874981	−	0.484157i	$-0.839126\pi$
$179$	−1.34281e8	−1.74996	−0.874981	+	0.484157i	$0.839126\pi$
$180$	0	0
$181$	1.14661e8	1.43727	0.718636	−	0.695386i	$-0.244767\pi$
$181$	1.14661e8	1.43727	0.718636	+	0.695386i	$0.244767\pi$
$182$	− 4.79356e7i	− 0.589398i
$183$	0	0
$184$	−2.12214e7	−0.251137
$185$	0	0
$186$	0	0
$187$	− 4.84352e7i	− 0.541646i
$188$	− 2.77617e7i	− 0.304714i
$189$	0	0
$190$	0	0
$191$	−1.63605e7	−0.169895	−0.0849474	−	0.996385i	$-0.527072\pi$
$191$	−1.63605e7	−0.169895	−0.0849474	+	0.996385i	$0.527072\pi$
$192$	0	0
$193$	1.54198e8i	1.54394i	0.635661	+	0.771968i	$0.280728\pi$
$193$	1.54198e8i	1.54394i	−0.635661	+	0.771968i	$0.719272\pi$
$194$	−3.24810e7	−0.319392
$195$	0	0
$196$	1.06255e8	1.00798
$197$	− 8.32288e7i	− 0.775607i	−0.921742	−	0.387804i	$-0.873234\pi$
$197$	− 8.32288e7i	− 0.775607i	0.921742	−	0.387804i	$-0.126766\pi$
$198$	0	0
$199$	7.61722e7	0.685190	0.342595	−	0.939483i	$-0.388694\pi$
$199$	7.61722e7	0.685190	0.342595	+	0.939483i	$0.388694\pi$
$200$	0	0
$201$	0	0
$202$	− 1.38256e8i	− 1.18019i
$203$	6.55774e7i	0.550196i
$204$	0	0
$205$	0	0
$206$	1.14898e8	0.915755
$207$	0	0
$208$	1.55730e7i	0.119991i
$209$	1.82274e8	1.38106
$210$	0	0
$211$	3.52446e7	0.258288	0.129144	−	0.991626i	$-0.458777\pi$
$211$	3.52446e7	0.258288	0.129144	+	0.991626i	$0.458777\pi$
$212$	− 5.03090e7i	− 0.362635i
$213$	0	0
$214$	−5.16352e7	−0.360162
$215$	0	0
$216$	0	0
$217$	− 5.22476e7i	− 0.347102i
$218$	− 7.07528e6i	− 0.0462536i
$219$	0	0
$220$	0	0
$221$	−2.51160e7	−0.156523
$222$	0	0
$223$	1.89131e8i	1.14208i	0.820922	+	0.571040i	$0.193460\pi$
$223$	1.89131e8i	1.14208i	−0.820922	+	0.571040i	$0.806540\pi$
$224$	−5.16424e7	−0.307000
$225$	0	0
$226$	9.70600e7	0.559320
$227$	1.76100e8i	0.999239i	0.866245	+	0.499620i	$0.166527\pi$
$227$	1.76100e8i	0.999239i	−0.866245	+	0.499620i	$0.833473\pi$
$228$	0	0
$229$	−6.50396e7	−0.357894	−0.178947	−	0.983859i	$-0.557269\pi$
$229$	−6.50396e7	−0.357894	−0.178947	+	0.983859i	$0.557269\pi$
$230$	0	0
$231$	0	0
$232$	− 2.13043e7i	− 0.112011i
$233$	− 2.51319e8i	− 1.30160i	−0.759248	−	0.650802i	$-0.774433\pi$
$233$	− 2.51319e8i	− 1.30160i	0.759248	−	0.650802i	$-0.225567\pi$
$234$	0	0
$235$	0	0
$236$	−4.76890e7	−0.236171
$237$	0	0
$238$	− 8.32884e7i	− 0.400466i
$239$	2.13079e8	1.00960	0.504799	−	0.863237i	$-0.331566\pi$
$239$	2.13079e8	1.00960	0.504799	+	0.863237i	$0.331566\pi$
$240$	0	0
$241$	2.57284e8	1.18400	0.592001	−	0.805937i	$-0.298338\pi$
$241$	2.57284e8	1.18400	0.592001	+	0.805937i	$0.298338\pi$
$242$	− 2.74168e8i	− 1.24355i
$243$	0	0
$244$	1.06280e8	0.468366
$245$	0	0
$246$	0	0
$247$	− 9.45177e7i	− 0.399093i
$248$	1.69738e7i	0.0706641i
$249$	0	0
$250$	0	0
$251$	−1.23058e8	−0.491193	−0.245596	−	0.969372i	$-0.578984\pi$
$251$	−1.23058e8	−0.491193	−0.245596	+	0.969372i	$0.578984\pi$
$252$	0	0
$253$	− 3.03897e8i	− 1.17979i
$254$	5.49445e7	0.210381
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	4.43334e8i	1.62916i	0.580048	+	0.814582i	$0.303034\pi$
$257$	4.43334e8i	1.62916i	−0.580048	+	0.814582i	$0.696966\pi$
$258$	0	0
$259$	−5.74704e7	−0.205539
$260$	0	0
$261$	0	0
$262$	− 3.16166e8i	− 1.08608i
$263$	2.98925e8i	1.01325i	0.862166	+	0.506625i	$0.169107\pi$
$263$	2.98925e8i	1.01325i	−0.862166	+	0.506625i	$0.830893\pi$
$264$	0	0
$265$	0	0
$266$	3.13435e8	1.02108
$267$	0	0
$268$	2.10614e8i	0.668366i
$269$	2.08908e8	0.654368	0.327184	−	0.944961i	$-0.393900\pi$
$269$	2.08908e8	0.654368	0.327184	+	0.944961i	$0.393900\pi$
$270$	0	0
$271$	−1.12749e7	−0.0344129	−0.0172064	−	0.999852i	$-0.505477\pi$
$271$	−1.12749e7	−0.0344129	−0.0172064	+	0.999852i	$0.505477\pi$
$272$	2.70582e7i	0.0815281i
$273$	0	0
$274$	−1.53393e8	−0.450482
$275$	0	0
$276$	0	0
$277$	− 6.58964e8i	− 1.86287i	−0.363907	−	0.931435i	$-0.618557\pi$
$277$	− 6.58964e8i	− 1.86287i	0.363907	−	0.931435i	$-0.381443\pi$
$278$	1.05959e8i	0.295789i
$279$	0	0
$280$	0	0
$281$	1.05123e8	0.282634	0.141317	−	0.989964i	$-0.454866\pi$
$281$	1.05123e8	0.282634	0.141317	+	0.989964i	$0.454866\pi$
$282$	0	0
$283$	− 3.30161e8i	− 0.865911i	−0.901415	−	0.432956i	$-0.857471\pi$
$283$	− 3.30161e8i	− 0.865911i	0.901415	−	0.432956i	$-0.142529\pi$
$284$	3.65834e8	0.947697
$285$	0	0
$286$	−2.23010e8	−0.563694
$287$	− 1.00719e9i	− 2.51492i
$288$	0	0
$289$	3.66699e8	0.893651
$290$	0	0
$291$	0	0
$292$	1.70233e8i	0.400134i
$293$	− 8.71002e7i	− 0.202294i	−0.994871	−	0.101147i	$-0.967749\pi$
$293$	− 8.71002e7i	− 0.202294i	0.994871	−	0.101147i	$-0.0322512\pi$
$294$	0	0
$295$	0	0
$296$	1.86706e7	0.0418444
$297$	0	0
$298$	− 4.58899e8i	− 1.00452i
$299$	−1.57585e8	−0.340931
$300$	0	0
$301$	2.46505e8	0.521007
$302$	2.48698e8i	0.519575i
$303$	0	0
$304$	−1.01827e8	−0.207876
$305$	0	0
$306$	0	0
$307$	− 3.91709e8i	− 0.772644i	−0.922364	−	0.386322i	$-0.873745\pi$
$307$	− 3.91709e8i	− 0.772644i	0.922364	−	0.386322i	$-0.126255\pi$
$308$	− 7.39535e8i	− 1.44222i
$309$	0	0
$310$	0	0
$311$	2.04936e8	0.386328	0.193164	−	0.981166i	$-0.438125\pi$
$311$	2.04936e8	0.386328	0.193164	+	0.981166i	$0.438125\pi$
$312$	0	0
$313$	− 8.77202e8i	− 1.61694i	−0.588536	−	0.808471i	$-0.700295\pi$
$313$	− 8.77202e8i	− 1.61694i	0.588536	−	0.808471i	$-0.299705\pi$
$314$	−2.70268e8	−0.492652
$315$	0	0
$316$	2.43676e8	0.434418
$317$	4.40831e8i	0.777256i	0.921395	+	0.388628i	$0.127051\pi$
$317$	4.40831e8i	0.777256i	−0.921395	+	0.388628i	$0.872949\pi$
$318$	0	0
$319$	3.05085e8	0.526202
$320$	0	0
$321$	0	0
$322$	− 5.22576e8i	− 0.872277i
$323$	− 1.64225e8i	− 0.271163i
$324$	0	0
$325$	0	0
$326$	−5.01327e8	−0.801419
$327$	0	0
$328$	3.27208e8i	0.511995i
$329$	6.83631e8	1.05837
$330$	0	0
$331$	1.11223e9	1.68576	0.842882	−	0.538099i	$-0.180857\pi$
$331$	1.11223e9	1.68576	0.842882	+	0.538099i	$0.180857\pi$
$332$	− 1.42686e8i	− 0.213992i
$333$	0	0
$334$	−5.01658e8	−0.736707
$335$	0	0
$336$	0	0
$337$	2.88198e8i	0.410191i	0.978742	+	0.205096i	$0.0657506\pi$
$337$	2.88198e8i	0.410191i	−0.978742	+	0.205096i	$0.934249\pi$
$338$	− 3.86347e8i	− 0.544213i
$339$	0	0
$340$	0	0
$341$	−2.43070e8	−0.331965
$342$	0	0
$343$	1.31862e9i	1.76438i
$344$	−8.00829e7	−0.106068
$345$	0	0
$346$	−2.16417e8	−0.280883
$347$	1.10601e9i	1.42103i	0.703680	+	0.710517i	$0.251539\pi$
$347$	1.10601e9i	1.42103i	−0.703680	+	0.710517i	$0.748461\pi$
$348$	0	0
$349$	1.32184e9	1.66453	0.832264	−	0.554379i	$-0.187044\pi$
$349$	1.32184e9	1.66453	0.832264	+	0.554379i	$0.187044\pi$
$350$	0	0
$351$	0	0
$352$	2.40255e8i	0.293612i
$353$	1.20395e9i	1.45679i	0.685157	+	0.728396i	$0.259734\pi$
$353$	1.20395e9i	1.45679i	−0.685157	+	0.728396i	$0.740266\pi$
$354$	0	0
$355$	0	0
$356$	−3.83437e8	−0.450422
$357$	0	0
$358$	1.07425e9i	1.23741i
$359$	−1.32057e9	−1.50637	−0.753185	−	0.657809i	$-0.771484\pi$
$359$	−1.32057e9	−1.50637	−0.753185	+	0.657809i	$0.771484\pi$
$360$	0	0
$361$	−2.75852e8	−0.308604
$362$	− 9.17284e8i	− 1.01630i
$363$	0	0
$364$	−3.83485e8	−0.416767
$365$	0	0
$366$	0	0
$367$	1.75107e9i	1.84915i	0.381000	+	0.924575i	$0.375580\pi$
$367$	1.75107e9i	1.84915i	−0.381000	+	0.924575i	$0.624420\pi$
$368$	1.69771e8i	0.177581i
$369$	0	0
$370$	0	0
$371$	1.23886e9	1.25954
$372$	0	0
$373$	4.87945e8i	0.486844i	0.969920	+	0.243422i	$0.0782701\pi$
$373$	4.87945e8i	0.486844i	−0.969920	+	0.243422i	$0.921730\pi$
$374$	−3.87482e8	−0.383001
$375$	0	0
$376$	−2.22093e8	−0.215466
$377$	− 1.58201e8i	− 0.152060i
$378$	0	0
$379$	−1.11007e9	−1.04740	−0.523700	−	0.851903i	$-0.675449\pi$
$379$	−1.11007e9	−1.04740	−0.523700	+	0.851903i	$0.675449\pi$
$380$	0	0
$381$	0	0
$382$	1.30884e8i	0.120134i
$383$	1.86912e9i	1.69997i	0.526810	+	0.849983i	$0.323388\pi$
$383$	1.86912e9i	1.69997i	−0.526810	+	0.849983i	$0.676612\pi$
$384$	0	0
$385$	0	0
$386$	1.23359e9	1.09173
$387$	0	0
$388$	2.59848e8i	0.225844i
$389$	−2.73895e8	−0.235918	−0.117959	−	0.993018i	$-0.537635\pi$
$389$	−2.73895e8	−0.235918	−0.117959	+	0.993018i	$0.537635\pi$
$390$	0	0
$391$	−2.73805e8	−0.231645
$392$	− 8.50039e8i	− 0.712751i
$393$	0	0
$394$	−6.65831e8	−0.548437
$395$	0	0
$396$	0	0
$397$	6.24552e8i	0.500958i	0.968122	+	0.250479i	$0.0805882\pi$
$397$	6.24552e8i	0.500958i	−0.968122	+	0.250479i	$0.919412\pi$
$398$	− 6.09378e8i	− 0.484502i
$399$	0	0
$400$	0	0
$401$	−5.55500e8	−0.430208	−0.215104	−	0.976591i	$-0.569009\pi$
$401$	−5.55500e8	−0.430208	−0.215104	+	0.976591i	$0.569009\pi$
$402$	0	0
$403$	1.26044e8i	0.0959299i
$404$	−1.10604e9	−0.834522
$405$	0	0
$406$	5.24619e8	0.389048
$407$	2.67369e8i	0.196576i
$408$	0	0
$409$	2.15770e9	1.55941	0.779704	−	0.626149i	$-0.215370\pi$
$409$	2.15770e9	1.55941	0.779704	+	0.626149i	$0.215370\pi$
$410$	0	0
$411$	0	0
$412$	− 9.19188e8i	− 0.647536i
$413$	− 1.17434e9i	− 0.820293i
$414$	0	0
$415$	0	0
$416$	1.24584e8	0.0848468
$417$	0	0
$418$	− 1.45819e9i	− 0.976555i
$419$	1.67797e9	1.11438	0.557191	−	0.830384i	$-0.311879\pi$
$419$	1.67797e9	1.11438	0.557191	+	0.830384i	$0.311879\pi$
$420$	0	0
$421$	−5.25233e8	−0.343056	−0.171528	−	0.985179i	$-0.554870\pi$
$421$	−5.25233e8	−0.343056	−0.171528	+	0.985179i	$0.554870\pi$
$422$	− 2.81957e8i	− 0.182637i
$423$	0	0
$424$	−4.02472e8	−0.256422
$425$	0	0
$426$	0	0
$427$	2.61713e9i	1.62678i
$428$	4.13082e8i	0.254673i
$429$	0	0
$430$	0	0
$431$	−1.70593e8	−0.102634	−0.0513169	−	0.998682i	$-0.516342\pi$
$431$	−1.70593e8	−0.102634	−0.0513169	+	0.998682i	$0.516342\pi$
$432$	0	0
$433$	1.68797e9i	0.999210i	0.866253	+	0.499605i	$0.166522\pi$
$433$	1.68797e9i	0.999210i	−0.866253	+	0.499605i	$0.833478\pi$
$434$	−4.17980e8	−0.245438
$435$	0	0
$436$	−5.66022e7	−0.0327063
$437$	− 1.03040e9i	− 0.590636i
$438$	0	0
$439$	1.17850e9	0.664817	0.332409	−	0.943135i	$-0.392139\pi$
$439$	1.17850e9	0.664817	0.332409	+	0.943135i	$0.392139\pi$
$440$	0	0
$441$	0	0
$442$	2.00928e8i	0.110678i
$443$	7.15755e8i	0.391157i	0.980688	+	0.195579i	$0.0626585\pi$
$443$	7.15755e8i	0.391157i	−0.980688	+	0.195579i	$0.937342\pi$
$444$	0	0
$445$	0	0
$446$	1.51305e9	0.807573
$447$	0	0
$448$	4.13139e8i	0.217082i
$449$	−1.37358e9	−0.716132	−0.358066	−	0.933696i	$-0.616564\pi$
$449$	−1.37358e9	−0.716132	−0.358066	+	0.933696i	$0.616564\pi$
$450$	0	0
$451$	−4.68572e9	−2.40524
$452$	− 7.76480e8i	− 0.395499i
$453$	0	0
$454$	1.40880e9	0.706569
$455$	0	0
$456$	0	0
$457$	1.84752e9i	0.905488i	0.891641	+	0.452744i	$0.149555\pi$
$457$	1.84752e9i	0.905488i	−0.891641	+	0.452744i	$0.850445\pi$
$458$	5.20317e8i	0.253069i
$459$	0	0
$460$	0	0
$461$	−3.09414e9	−1.47091	−0.735455	−	0.677573i	$-0.763032\pi$
$461$	−3.09414e9	−1.47091	−0.735455	+	0.677573i	$0.763032\pi$
$462$	0	0
$463$	− 3.00451e9i	− 1.40682i	−0.710782	−	0.703412i	$-0.751659\pi$
$463$	− 3.00451e9i	− 1.40682i	0.710782	−	0.703412i	$-0.248341\pi$
$464$	−1.70435e8	−0.0792036
$465$	0	0
$466$	−2.01055e9	−0.920373
$467$	2.99252e9i	1.35965i	0.733374	+	0.679825i	$0.237944\pi$
$467$	2.99252e9i	1.35965i	−0.733374	+	0.679825i	$0.762056\pi$
$468$	0	0
$469$	−5.18636e9	−2.32144
$470$	0	0
$471$	0	0
$472$	3.81512e8i	0.166998i
$473$	− 1.14681e9i	− 0.498286i
$474$	0	0
$475$	0	0
$476$	−6.66308e8	−0.283172
$477$	0	0
$478$	− 1.70464e9i	− 0.713894i
$479$	1.84041e9	0.765141	0.382570	−	0.923926i	$-0.375039\pi$
$479$	1.84041e9	0.765141	0.382570	+	0.923926i	$0.375039\pi$
$480$	0	0
$481$	1.38644e8	0.0568058
$482$	− 2.05827e9i	− 0.837216i
$483$	0	0
$484$	−2.19335e9	−0.879323
$485$	0	0
$486$	0	0
$487$	− 4.26676e8i	− 0.167397i	−0.996491	−	0.0836983i	$-0.973327\pi$
$487$	− 4.26676e8i	− 0.167397i	0.996491	−	0.0836983i	$-0.0266732\pi$
$488$	− 8.50236e8i	− 0.331185i
$489$	0	0
$490$	0	0
$491$	−6.07547e7	−0.0231630	−0.0115815	−	0.999933i	$-0.503687\pi$
$491$	−6.07547e7	−0.0231630	−0.0115815	+	0.999933i	$0.503687\pi$
$492$	0	0
$493$	− 2.74876e8i	− 0.103317i
$494$	−7.56142e8	−0.282201
$495$	0	0
$496$	1.35791e8	0.0499671
$497$	9.00866e9i	3.29164i
$498$	0	0
$499$	−3.24588e9	−1.16945	−0.584723	−	0.811233i	$-0.698797\pi$
$499$	−3.24588e9	−1.16945	−0.584723	+	0.811233i	$0.698797\pi$
$500$	0	0
$501$	0	0
$502$	9.84464e8i	0.347326i
$503$	− 7.44381e8i	− 0.260800i	−0.991461	−	0.130400i	$-0.958374\pi$
$503$	− 7.44381e8i	− 0.260800i	0.991461	−	0.130400i	$-0.0416262\pi$
$504$	0	0
$505$	0	0
$506$	−2.43117e9	−0.834237
$507$	0	0
$508$	− 4.39556e8i	− 0.148762i
$509$	−4.44155e8	−0.149287	−0.0746436	−	0.997210i	$-0.523782\pi$
$509$	−4.44155e8	−0.149287	−0.0746436	+	0.997210i	$0.523782\pi$
$510$	0	0
$511$	−4.19200e9	−1.38979
$512$	− 1.34218e8i	− 0.0441942i
$513$	0	0
$514$	3.54667e9	1.15199
$515$	0	0
$516$	0	0
$517$	− 3.18045e9i	− 1.01221i
$518$	4.59763e8i	0.145338i
$519$	0	0
$520$	0	0
$521$	−3.04963e9	−0.944745	−0.472372	−	0.881399i	$-0.656602\pi$
$521$	−3.04963e9	−0.944745	−0.472372	+	0.881399i	$0.656602\pi$
$522$	0	0
$523$	1.40306e9i	0.428866i	0.976739	+	0.214433i	$0.0687904\pi$
$523$	1.40306e9i	0.428866i	−0.976739	+	0.214433i	$0.931210\pi$
$524$	−2.52933e9	−0.767973
$525$	0	0
$526$	2.39140e9	0.716476
$527$	2.19002e8i	0.0651795i
$528$	0	0
$529$	1.68689e9	0.495441
$530$	0	0
$531$	0	0
$532$	− 2.50748e9i	− 0.722016i
$533$	2.42977e9i	0.695058i
$534$	0	0
$535$	0	0
$536$	1.68491e9	0.472606
$537$	0	0
$538$	− 1.67126e9i	− 0.462708i
$539$	1.21728e10	3.34835
$540$	0	0
$541$	4.21106e9	1.14341	0.571704	−	0.820460i	$-0.306283\pi$
$541$	4.21106e9	1.14341	0.571704	+	0.820460i	$0.306283\pi$
$542$	9.01994e7i	0.0243336i
$543$	0	0
$544$	2.16465e8	0.0576491
$545$	0	0
$546$	0	0
$547$	1.99956e9i	0.522371i	0.965289	+	0.261185i	$0.0841134\pi$
$547$	1.99956e9i	0.522371i	−0.965289	+	0.261185i	$0.915887\pi$
$548$	1.22714e9i	0.318539i
$549$	0	0
$550$	0	0
$551$	1.03442e9	0.263432
$552$	0	0
$553$	6.00053e9i	1.50887i
$554$	−5.27172e9	−1.31725
$555$	0	0
$556$	8.47674e8	0.209154
$557$	3.37403e9i	0.827287i	0.910439	+	0.413643i	$0.135744\pi$
$557$	3.37403e9i	0.827287i	−0.910439	+	0.413643i	$0.864256\pi$
$558$	0	0
$559$	−5.94678e8	−0.143993
$560$	0	0
$561$	0	0
$562$	− 8.40983e8i	− 0.199853i
$563$	− 5.58021e9i	− 1.31787i	−0.752201	−	0.658933i	$-0.771008\pi$
$563$	− 5.58021e9i	− 1.31787i	0.752201	−	0.658933i	$-0.228992\pi$
$564$	0	0
$565$	0	0
$566$	−2.64129e9	−0.612292
$567$	0	0
$568$	− 2.92667e9i	− 0.670123i
$569$	−8.88310e8	−0.202149	−0.101074	−	0.994879i	$-0.532228\pi$
$569$	−8.88310e8	−0.202149	−0.101074	+	0.994879i	$0.532228\pi$
$570$	0	0
$571$	−1.79171e9	−0.402755	−0.201377	−	0.979514i	$-0.564542\pi$
$571$	−1.79171e9	−0.402755	−0.201377	+	0.979514i	$0.564542\pi$
$572$	1.78408e9i	0.398592i
$573$	0	0
$574$	−8.05750e9	−1.77831
$575$	0	0
$576$	0	0
$577$	3.82103e9i	0.828066i	0.910262	+	0.414033i	$0.135880\pi$
$577$	3.82103e9i	0.828066i	−0.910262	+	0.414033i	$0.864120\pi$
$578$	− 2.93360e9i	− 0.631906i
$579$	0	0
$580$	0	0
$581$	3.51364e9	0.743260
$582$	0	0
$583$	− 5.76352e9i	− 1.20461i
$584$	1.36187e9	0.282937
$585$	0	0
$586$	−6.96802e8	−0.143043
$587$	− 4.36219e9i	− 0.890166i	−0.895489	−	0.445083i	$-0.853174\pi$
$587$	− 4.36219e9i	− 0.890166i	0.895489	−	0.445083i	$-0.146826\pi$
$588$	0	0
$589$	−8.24159e8	−0.166191
$590$	0	0
$591$	0	0
$592$	− 1.49365e8i	− 0.0295884i
$593$	6.38531e9i	1.25745i	0.777628	+	0.628724i	$0.216423\pi$
$593$	6.38531e9i	1.25745i	−0.777628	+	0.628724i	$0.783577\pi$
$594$	0	0
$595$	0	0
$596$	−3.67120e9	−0.710306
$597$	0	0
$598$	1.26068e9i	0.241075i
$599$	8.04297e8	0.152905	0.0764527	−	0.997073i	$-0.475641\pi$
$599$	8.04297e8	0.152905	0.0764527	+	0.997073i	$0.475641\pi$
$600$	0	0
$601$	−4.87162e9	−0.915403	−0.457702	−	0.889106i	$-0.651327\pi$
$601$	−4.87162e9	−0.915403	−0.457702	+	0.889106i	$0.651327\pi$
$602$	− 1.97204e9i	− 0.368408i
$603$	0	0
$604$	1.98959e9	0.367395
$605$	0	0
$606$	0	0
$607$	7.17517e9i	1.30218i	0.759000	+	0.651091i	$0.225688\pi$
$607$	7.17517e9i	1.30218i	−0.759000	+	0.651091i	$0.774312\pi$
$608$	8.14612e8i	0.146990i
$609$	0	0
$610$	0	0
$611$	−1.64922e9	−0.292505
$612$	0	0
$613$	− 3.47891e9i	− 0.610002i	−0.952352	−	0.305001i	$-0.901343\pi$
$613$	− 3.47891e9i	− 0.610002i	0.952352	−	0.305001i	$-0.0986567\pi$
$614$	−3.13367e9	−0.546342
$615$	0	0
$616$	−5.91628e9	−1.01980
$617$	2.39378e8i	0.0410286i	0.999790	+	0.0205143i	$0.00653037\pi$
$617$	2.39378e8i	0.0410286i	−0.999790	+	0.0205143i	$0.993470\pi$
$618$	0	0
$619$	5.52959e9	0.937078	0.468539	−	0.883443i	$-0.344781\pi$
$619$	5.52959e9	0.937078	0.468539	+	0.883443i	$0.344781\pi$
$620$	0	0
$621$	0	0
$622$	− 1.63949e9i	− 0.273175i
$623$	− 9.44215e9i	− 1.56445i
$624$	0	0
$625$	0	0
$626$	−7.01762e9	−1.14335
$627$	0	0
$628$	2.16214e9i	0.348358i
$629$	2.40894e8	0.0385966
$630$	0	0
$631$	−6.13683e9	−0.972392	−0.486196	−	0.873850i	$-0.661616\pi$
$631$	−6.13683e9	−0.972392	−0.486196	+	0.873850i	$0.661616\pi$
$632$	− 1.94941e9i	− 0.307180i
$633$	0	0
$634$	3.52664e9	0.549603
$635$	0	0
$636$	0	0
$637$	− 6.31221e9i	− 0.967594i
$638$	− 2.44068e9i	− 0.372081i
$639$	0	0
$640$	0	0
$641$	−1.07038e10	−1.60522	−0.802611	−	0.596503i	$-0.796556\pi$
$641$	−1.07038e10	−1.60522	−0.802611	+	0.596503i	$0.796556\pi$
$642$	0	0
$643$	1.39803e9i	0.207385i	0.994609	+	0.103692i	$0.0330658\pi$
$643$	1.39803e9i	0.207385i	−0.994609	+	0.103692i	$0.966934\pi$
$644$	−4.18061e9	−0.616793
$645$	0	0
$646$	−1.31380e9	−0.191741
$647$	5.31605e9i	0.771656i	0.922571	+	0.385828i	$0.126084\pi$
$647$	5.31605e9i	0.771656i	−0.922571	+	0.385828i	$0.873916\pi$
$648$	0	0
$649$	−5.46337e9	−0.784520
$650$	0	0
$651$	0	0
$652$	4.01062e9i	0.566689i
$653$	3.24403e9i	0.455921i	0.973670	+	0.227960i	$0.0732057\pi$
$653$	3.24403e9i	0.455921i	−0.973670	+	0.227960i	$0.926794\pi$
$654$	0	0
$655$	0	0
$656$	2.61766e9	0.362035
$657$	0	0
$658$	− 5.46905e9i	− 0.748378i
$659$	−5.16506e9	−0.703034	−0.351517	−	0.936181i	$-0.614334\pi$
$659$	−5.16506e9	−0.703034	−0.351517	+	0.936181i	$0.614334\pi$
$660$	0	0
$661$	−3.22515e9	−0.434356	−0.217178	−	0.976132i	$-0.569685\pi$
$661$	−3.22515e9	−0.434356	−0.217178	+	0.976132i	$0.569685\pi$
$662$	− 8.89784e9i	− 1.19201i
$663$	0	0
$664$	−1.14149e9	−0.151315
$665$	0	0
$666$	0	0
$667$	− 1.72465e9i	− 0.225041i
$668$	4.01326e9i	0.520931i
$669$	0	0
$670$	0	0
$671$	1.21757e10	1.55583
$672$	0	0
$673$	2.00633e9i	0.253718i	0.991921	+	0.126859i	$0.0404895\pi$
$673$	2.00633e9i	0.253718i	−0.991921	+	0.126859i	$0.959510\pi$
$674$	2.30559e9	0.290049
$675$	0	0
$676$	−3.09077e9	−0.384816
$677$	− 1.00211e10i	− 1.24124i	−0.784112	−	0.620619i	$-0.786881\pi$
$677$	− 1.00211e10i	− 1.24124i	0.784112	−	0.620619i	$-0.213119\pi$
$678$	0	0
$679$	−6.39876e9	−0.784425
$680$	0	0
$681$	0	0
$682$	1.94456e9i	0.234734i
$683$	− 5.84861e9i	− 0.702393i	−0.936302	−	0.351196i	$-0.885775\pi$
$683$	− 5.84861e9i	− 0.702393i	0.936302	−	0.351196i	$-0.114225\pi$
$684$	0	0
$685$	0	0
$686$	1.05490e10	1.24760
$687$	0	0
$688$	6.40664e8i	0.0750016i
$689$	−2.98867e9	−0.348105
$690$	0	0
$691$	2.58686e9	0.298263	0.149131	−	0.988817i	$-0.452352\pi$
$691$	2.58686e9	0.298263	0.149131	+	0.988817i	$0.452352\pi$
$692$	1.73134e9i	0.198614i
$693$	0	0
$694$	8.84805e9	1.00482
$695$	0	0
$696$	0	0
$697$	4.22175e9i	0.472256i
$698$	− 1.05747e10i	− 1.17700i
$699$	0	0
$700$	0	0
$701$	1.74460e9	0.191286	0.0956429	−	0.995416i	$-0.469509\pi$
$701$	1.74460e9	0.191286	0.0956429	+	0.995416i	$0.469509\pi$
$702$	0	0
$703$	9.06545e8i	0.0984114i
$704$	1.92204e9	0.207615
$705$	0	0
$706$	9.63161e9	1.03011
$707$	− 2.72363e10i	− 2.89855i
$708$	0	0
$709$	1.12051e10	1.18074	0.590368	−	0.807134i	$-0.298983\pi$
$709$	1.12051e10	1.18074	0.590368	+	0.807134i	$0.298983\pi$
$710$	0	0
$711$	0	0
$712$	3.06750e9i	0.318496i
$713$	1.37408e9i	0.141971i
$714$	0	0
$715$	0	0
$716$	8.59398e9	0.874981
$717$	0	0
$718$	1.05646e10i	1.06516i
$719$	−9.36568e8	−0.0939698	−0.0469849	−	0.998896i	$-0.514961\pi$
$719$	−9.36568e8	−0.0939698	−0.0469849	+	0.998896i	$0.514961\pi$
$720$	0	0
$721$	2.26350e10	2.24909
$722$	2.20682e9i	0.218216i
$723$	0	0
$724$	−7.33827e9	−0.718636
$725$	0	0
$726$	0	0
$727$	4.20445e9i	0.405825i	0.979197	+	0.202913i	$0.0650407\pi$
$727$	4.20445e9i	0.405825i	−0.979197	+	0.202913i	$0.934959\pi$
$728$	3.06788e9i	0.294699i
$729$	0	0
$730$	0	0
$731$	−1.03326e9	−0.0978358
$732$	0	0
$733$	1.15491e10i	1.08314i	0.840655	+	0.541571i	$0.182170\pi$
$733$	1.15491e10i	1.08314i	−0.840655	+	0.541571i	$0.817830\pi$
$734$	1.40086e10	1.30755
$735$	0	0
$736$	1.35817e9	0.125569
$737$	2.41284e10i	2.22020i
$738$	0	0
$739$	1.39655e10	1.27292	0.636460	−	0.771310i	$-0.280398\pi$
$739$	1.39655e10	1.27292	0.636460	+	0.771310i	$0.280398\pi$
$740$	0	0
$741$	0	0
$742$	− 9.91087e9i	− 0.890632i
$743$	1.43832e10i	1.28646i	0.765673	+	0.643230i	$0.222406\pi$
$743$	1.43832e10i	1.28646i	−0.765673	+	0.643230i	$0.777594\pi$
$744$	0	0
$745$	0	0
$746$	3.90356e9	0.344251
$747$	0	0
$748$	3.09985e9i	0.270823i
$749$	−1.01721e10	−0.884557
$750$	0	0
$751$	−6.70841e8	−0.0577936	−0.0288968	−	0.999582i	$-0.509199\pi$
$751$	−6.70841e8	−0.0577936	−0.0288968	+	0.999582i	$0.509199\pi$
$752$	1.77675e9i	0.152357i
$753$	0	0
$754$	−1.26561e9	−0.107523
$755$	0	0
$756$	0	0
$757$	− 1.91569e10i	− 1.60506i	−0.596613	−	0.802529i	$-0.703487\pi$
$757$	− 1.91569e10i	− 1.60506i	0.596613	−	0.802529i	$-0.296513\pi$
$758$	8.88055e9i	0.740624i
$759$	0	0
$760$	0	0
$761$	−1.79120e10	−1.47332	−0.736660	−	0.676263i	$-0.763598\pi$
$761$	−1.79120e10	−1.47332	−0.736660	+	0.676263i	$0.763598\pi$
$762$	0	0
$763$	− 1.39383e9i	− 0.113599i
$764$	1.04707e9	0.0849474
$765$	0	0
$766$	1.49529e10	1.20206
$767$	2.83302e9i	0.226708i
$768$	0	0
$769$	2.14072e10	1.69753	0.848765	−	0.528771i	$-0.177347\pi$
$769$	2.14072e10	1.69753	0.848765	+	0.528771i	$0.177347\pi$
$770$	0	0
$771$	0	0
$772$	− 9.86870e9i	− 0.771968i
$773$	− 7.55163e8i	− 0.0588047i	−0.999568	−	0.0294024i	$-0.990640\pi$
$773$	− 7.55163e8i	− 0.0588047i	0.999568	−	0.0294024i	$-0.00936041\pi$
$774$	0	0
$775$	0	0
$776$	2.07878e9	0.159696
$777$	0	0
$778$	2.19116e9i	0.166819i
$779$	−1.58875e10	−1.20413
$780$	0	0
$781$	4.19108e10	3.14809
$782$	2.19044e9i	0.163798i
$783$	0	0
$784$	−6.80031e9	−0.503991
$785$	0	0
$786$	0	0
$787$	2.04665e10i	1.49669i	0.663308	+	0.748347i	$0.269152\pi$
$787$	2.04665e10i	1.49669i	−0.663308	+	0.748347i	$0.730848\pi$
$788$	5.32664e9i	0.387804i
$789$	0	0
$790$	0	0
$791$	1.91208e10	1.37369
$792$	0	0
$793$	− 6.31367e9i	− 0.449599i
$794$	4.99641e9	0.354231
$795$	0	0
$796$	−4.87502e9	−0.342595
$797$	1.03098e10i	0.721348i	0.932692	+	0.360674i	$0.117453\pi$
$797$	1.03098e10i	0.721348i	−0.932692	+	0.360674i	$0.882547\pi$
$798$	0	0
$799$	−2.86552e9	−0.198742
$800$	0	0
$801$	0	0
$802$	4.44400e9i	0.304203i
$803$	1.95024e10i	1.32918i
$804$	0	0
$805$	0	0
$806$	1.00835e9	0.0678327
$807$	0	0
$808$	8.84835e9i	0.590096i
$809$	−4.16428e8	−0.0276516	−0.0138258	−	0.999904i	$-0.504401\pi$
$809$	−4.16428e8	−0.0276516	−0.0138258	+	0.999904i	$0.504401\pi$
$810$	0	0
$811$	−5.82687e9	−0.383586	−0.191793	−	0.981435i	$-0.561430\pi$
$811$	−5.82687e9	−0.383586	−0.191793	+	0.981435i	$0.561430\pi$
$812$	− 4.19695e9i	− 0.275098i
$813$	0	0
$814$	2.13895e9	0.139000
$815$	0	0
$816$	0	0
$817$	− 3.88840e9i	− 0.249456i
$818$	− 1.72616e10i	− 1.10267i
$819$	0	0
$820$	0	0
$821$	2.08333e10	1.31388	0.656941	−	0.753942i	$-0.271850\pi$
$821$	2.08333e10	1.31388	0.656941	+	0.753942i	$0.271850\pi$
$822$	0	0
$823$	− 4.23403e9i	− 0.264761i	−0.991199	−	0.132381i	$-0.957738\pi$
$823$	− 4.23403e9i	− 0.264761i	0.991199	−	0.132381i	$-0.0422621\pi$
$824$	−7.35350e9	−0.457877
$825$	0	0
$826$	−9.39473e9	−0.580035
$827$	− 5.70597e9i	− 0.350800i	−0.984497	−	0.175400i	$-0.943878\pi$
$827$	− 5.70597e9i	− 0.350800i	0.984497	−	0.175400i	$-0.0561219\pi$
$828$	0	0
$829$	−2.51612e10	−1.53388	−0.766940	−	0.641719i	$-0.778221\pi$
$829$	−2.51612e10	−1.53388	−0.766940	+	0.641719i	$0.778221\pi$
$830$	0	0
$831$	0	0
$832$	− 9.96671e8i	− 0.0599957i
$833$	− 1.09675e10i	− 0.657431i
$834$	0	0
$835$	0	0
$836$	−1.16655e10	−0.690528
$837$	0	0
$838$	− 1.34237e10i	− 0.787988i
$839$	2.27048e10	1.32724	0.663622	−	0.748068i	$-0.269018\pi$
$839$	2.27048e10	1.32724	0.663622	+	0.748068i	$0.269018\pi$
$840$	0	0
$841$	−1.55185e10	−0.899629
$842$	4.20187e9i	0.242577i
$843$	0	0
$844$	−2.25565e9	−0.129144
$845$	0	0
$846$	0	0
$847$	− 5.40112e10i	− 3.05416i
$848$	3.21978e9i	0.181318i
$849$	0	0
$850$	0	0
$851$	1.51144e9	0.0840695
$852$	0	0
$853$	− 2.86872e10i	− 1.58258i	−0.611439	−	0.791292i	$-0.709409\pi$
$853$	− 2.86872e10i	− 1.58258i	0.611439	−	0.791292i	$-0.290591\pi$
$854$	2.09371e10	1.15031
$855$	0	0
$856$	3.30465e9	0.180081
$857$	− 5.34950e9i	− 0.290322i	−0.989408	−	0.145161i	$-0.953630\pi$
$857$	− 5.34950e9i	− 0.290322i	0.989408	−	0.145161i	$-0.0463701\pi$
$858$	0	0
$859$	1.40330e10	0.755394	0.377697	−	0.925929i	$-0.376716\pi$
$859$	1.40330e10	0.755394	0.377697	+	0.925929i	$0.376716\pi$
$860$	0	0
$861$	0	0
$862$	1.36474e9i	0.0725731i
$863$	3.24994e10i	1.72122i	0.509261	+	0.860612i	$0.329919\pi$
$863$	3.24994e10i	1.72122i	−0.509261	+	0.860612i	$0.670081\pi$
$864$	0	0
$865$	0	0
$866$	1.35038e10	0.706549
$867$	0	0
$868$	3.34384e9i	0.173551i
$869$	2.79162e10	1.44307
$870$	0	0
$871$	1.25118e10	0.641586
$872$	4.52818e8i	0.0231268i
$873$	0	0
$874$	−8.24318e9	−0.417642
$875$	0	0
$876$	0	0
$877$	− 3.38694e10i	− 1.69554i	−0.530361	−	0.847772i	$-0.677944\pi$
$877$	− 3.38694e10i	− 1.69554i	0.530361	−	0.847772i	$-0.322056\pi$
$878$	− 9.42797e9i	− 0.470097i
$879$	0	0
$880$	0	0
$881$	1.52708e10	0.752397	0.376198	−	0.926539i	$-0.377231\pi$
$881$	1.52708e10	0.752397	0.376198	+	0.926539i	$0.377231\pi$
$882$	0	0
$883$	1.12045e10i	0.547685i	0.961775	+	0.273842i	$0.0882946\pi$
$883$	1.12045e10i	0.547685i	−0.961775	+	0.273842i	$0.911705\pi$
$884$	1.60742e9	0.0782614
$885$	0	0
$886$	5.72604e9	0.276590
$887$	− 6.97232e9i	− 0.335463i	−0.985833	−	0.167732i	$-0.946356\pi$
$887$	− 6.97232e9i	− 0.335463i	0.985833	−	0.167732i	$-0.0536442\pi$
$888$	0	0
$889$	1.08241e10	0.516695
$890$	0	0
$891$	0	0
$892$	− 1.21044e10i	− 0.571040i
$893$	− 1.07837e10i	− 0.506742i
$894$	0	0
$895$	0	0
$896$	3.30511e9	0.153500
$897$	0	0
$898$	1.09887e10i	0.506382i
$899$	−1.37945e9	−0.0633211
$900$	0	0
$901$	−5.19283e9	−0.236520
$902$	3.74858e10i	1.70076i
$903$	0	0
$904$	−6.21184e9	−0.279660
$905$	0	0
$906$	0	0
$907$	1.18095e9i	0.0525539i	0.999655	+	0.0262770i	$0.00836518\pi$
$907$	1.18095e9i	0.0525539i	−0.999655	+	0.0262770i	$0.991635\pi$
$908$	− 1.12704e10i	− 0.499620i
$909$	0	0
$910$	0	0
$911$	−1.27915e10	−0.560541	−0.280271	−	0.959921i	$-0.590424\pi$
$911$	−1.27915e10	−0.560541	−0.280271	+	0.959921i	$0.590424\pi$
$912$	0	0
$913$	− 1.63465e10i	− 0.710847i
$914$	1.47802e10	0.640276
$915$	0	0
$916$	4.16254e9	0.178947
$917$	− 6.22848e10i	− 2.66741i
$918$	0	0
$919$	−3.52353e10	−1.49752	−0.748761	−	0.662840i	$-0.769351\pi$
$919$	−3.52353e10	−1.49752	−0.748761	+	0.662840i	$0.769351\pi$
$920$	0	0
$921$	0	0
$922$	2.47531e10i	1.04009i
$923$	− 2.17328e10i	− 0.909725i
$924$	0	0
$925$	0	0
$926$	−2.40361e10	−0.994775
$927$	0	0
$928$	1.36348e9i	0.0560054i
$929$	−2.17764e10	−0.891111	−0.445556	−	0.895254i	$-0.646994\pi$
$929$	−2.17764e10	−0.891111	−0.445556	+	0.895254i	$0.646994\pi$
$930$	0	0
$931$	4.12734e10	1.67628
$932$	1.60844e10i	0.650802i
$933$	0	0
$934$	2.39401e10	0.961418
$935$	0	0
$936$	0	0
$937$	1.15795e10i	0.459833i	0.973210	+	0.229916i	$0.0738453\pi$
$937$	1.15795e10i	0.459833i	−0.973210	+	0.229916i	$0.926155\pi$
$938$	4.14909e10i	1.64151i
$939$	0	0
$940$	0	0
$941$	3.83930e10	1.50207	0.751033	−	0.660265i	$-0.229556\pi$
$941$	3.83930e10	1.50207	0.751033	+	0.660265i	$0.229556\pi$
$942$	0	0
$943$	2.64885e10i	1.02865i
$944$	3.05209e9	0.118085
$945$	0	0
$946$	−9.17450e9	−0.352341
$947$	− 1.11841e10i	− 0.427933i	−0.976841	−	0.213966i	$-0.931362\pi$
$947$	− 1.11841e10i	− 0.427933i	0.976841	−	0.213966i	$-0.0686383\pi$
$948$	0	0
$949$	1.01129e10	0.384101
$950$	0	0
$951$	0	0
$952$	5.33046e9i	0.200233i
$953$	− 5.19835e9i	− 0.194554i	−0.995257	−	0.0972770i	$-0.968987\pi$
$953$	− 5.19835e9i	− 0.194554i	0.995257	−	0.0972770i	$-0.0310133\pi$
$954$	0	0
$955$	0	0
$956$	−1.36371e10	−0.504799
$957$	0	0
$958$	− 1.47233e10i	− 0.541036i
$959$	−3.02183e10	−1.10638
$960$	0	0
$961$	−2.64136e10	−0.960053
$962$	− 1.10915e9i	− 0.0401677i
$963$	0	0
$964$	−1.64662e10	−0.592001
$965$	0	0
$966$	0	0
$967$	− 2.69243e8i	− 0.00957529i	−0.999989	−	0.00478765i	$-0.998476\pi$
$967$	− 2.69243e8i	− 0.00957529i	0.999989	−	0.00478765i	$-0.00152396\pi$
$968$	1.75468e10i	0.621776i
$969$	0	0
$970$	0	0
$971$	−4.37283e9	−0.153284	−0.0766418	−	0.997059i	$-0.524420\pi$
$971$	−4.37283e9	−0.153284	−0.0766418	+	0.997059i	$0.524420\pi$
$972$	0	0
$973$	2.08740e10i	0.726457i
$974$	−3.41341e9	−0.118367
$975$	0	0
$976$	−6.80189e9	−0.234183
$977$	3.74991e10i	1.28644i	0.765681	+	0.643220i	$0.222402\pi$
$977$	3.74991e10i	1.28644i	−0.765681	+	0.643220i	$0.777598\pi$
$978$	0	0
$979$	−4.39276e10	−1.49623
$980$	0	0
$981$	0	0
$982$	4.86038e8i	0.0163787i
$983$	− 3.06190e10i	− 1.02814i	−0.857748	−	0.514071i	$-0.828137\pi$
$983$	− 3.06190e10i	− 1.02814i	0.857748	−	0.514071i	$-0.171863\pi$
$984$	0	0
$985$	0	0
$986$	−2.19901e9	−0.0730562
$987$	0	0
$988$	6.04913e9i	0.199546i
$989$	−6.48296e9	−0.213102
$990$	0	0
$991$	−1.72703e10	−0.563693	−0.281847	−	0.959459i	$-0.590947\pi$
$991$	−1.72703e10	−0.563693	−0.281847	+	0.959459i	$0.590947\pi$
$992$	− 1.08632e9i	− 0.0353320i
$993$	0	0
$994$	7.20692e10	2.32754
$995$	0	0
$996$	0	0
$997$	− 3.75077e9i	− 0.119864i	−0.998202	−	0.0599319i	$-0.980912\pi$
$997$	− 3.75077e9i	− 0.119864i	0.998202	−	0.0599319i	$-0.0190883\pi$
$998$	2.59670e10i	0.826923i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.8.c.a.199.1		2
3.2	odd	2		150.8.c.k.49.2		2
5.2	odd	4		450.8.a.ba.1.1		1
5.3	odd	4		18.8.a.a.1.1		1
5.4	even	2	inner	450.8.c.a.199.2		2
15.2	even	4		150.8.a.e.1.1		1
15.8	even	4		6.8.a.a.1.1	✓	1
15.14	odd	2		150.8.c.k.49.1		2
20.3	even	4		144.8.a.h.1.1		1
40.3	even	4		576.8.a.i.1.1		1
40.13	odd	4		576.8.a.h.1.1		1
45.13	odd	12		162.8.c.i.55.1		2
45.23	even	12		162.8.c.d.55.1		2
45.38	even	12		162.8.c.d.109.1		2
45.43	odd	12		162.8.c.i.109.1		2
60.23	odd	4		48.8.a.b.1.1		1
105.23	even	12		294.8.e.c.67.1		2
105.38	odd	12		294.8.e.d.79.1		2
105.53	even	12		294.8.e.c.79.1		2
105.68	odd	12		294.8.e.d.67.1		2
105.83	odd	4		294.8.a.l.1.1		1
120.53	even	4		192.8.a.f.1.1		1
120.83	odd	4		192.8.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.8.a.a.1.1	✓	1	15.8	even	4
18.8.a.a.1.1		1	5.3	odd	4
48.8.a.b.1.1		1	60.23	odd	4
144.8.a.h.1.1		1	20.3	even	4
150.8.a.e.1.1		1	15.2	even	4
150.8.c.k.49.1		2	15.14	odd	2
150.8.c.k.49.2		2	3.2	odd	2
162.8.c.d.55.1		2	45.23	even	12
162.8.c.d.109.1		2	45.38	even	12
162.8.c.i.55.1		2	45.13	odd	12
162.8.c.i.109.1		2	45.43	odd	12
192.8.a.f.1.1		1	120.53	even	4
192.8.a.n.1.1		1	120.83	odd	4
294.8.a.l.1.1		1	105.83	odd	4
294.8.e.c.67.1		2	105.23	even	12
294.8.e.c.79.1		2	105.53	even	12
294.8.e.d.67.1		2	105.68	odd	12
294.8.e.d.79.1		2	105.38	odd	12
450.8.a.ba.1.1		1	5.2	odd	4
450.8.c.a.199.1		2	1.1	even	1	trivial
450.8.c.a.199.2		2	5.4	even	2	inner
576.8.a.h.1.1		1	40.13	odd	4
576.8.a.i.1.1		1	40.3	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.00000i q^{2} -64.0000 q^{4} -1576.00i q^{7} +512.000i q^{8} +O(q^{10})\)	\(q-8.00000i q^{2} -64.0000 q^{4} -1576.00i q^{7} +512.000i q^{8} -7332.00 q^{11} +3802.00i q^{13} -12608.0 q^{14} +4096.00 q^{16} +6606.00i q^{17} -24860.0 q^{19} +58656.0i q^{22} +41448.0i q^{23} +30416.0 q^{26} +100864. i q^{28} -41610.0 q^{29} +33152.0 q^{31} -32768.0i q^{32} +52848.0 q^{34} -36466.0i q^{37} +198880. i q^{38} +639078. q^{41} +156412. i q^{43} +469248. q^{44} +331584. q^{46} +433776. i q^{47} -1.66023e6 q^{49} -243328. i q^{52} +786078. i q^{53} +806912. q^{56} +332880. i q^{58} +745140. q^{59} -1.66062e6 q^{61} -265216. i q^{62} -262144. q^{64} -3.29084e6i q^{67} -422784. i q^{68} -5.71615e6 q^{71} -2.65990e6i q^{73} -291728. q^{74} +1.59104e6 q^{76} +1.15552e7i q^{77} -3.80744e6 q^{79} -5.11262e6i q^{82} +2.22947e6i q^{83} +1.25130e6 q^{86} -3.75398e6i q^{88} +5.99121e6 q^{89} +5.99195e6 q^{91} -2.65267e6i q^{92} +3.47021e6 q^{94} -4.06013e6i q^{97} +1.32819e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4}+O(q^{10}) \) 2 * q - 128 * q^4	\( 2 q - 128 q^{4} - 14664 q^{11} - 25216 q^{14} + 8192 q^{16} - 49720 q^{19} + 60832 q^{26} - 83220 q^{29} + 66304 q^{31} + 105696 q^{34} + 1278156 q^{41} + 938496 q^{44} + 663168 q^{46} - 3320466 q^{49} + 1613824 q^{56} + 1490280 q^{59} - 3321236 q^{61} - 524288 q^{64} - 11432304 q^{71} - 583456 q^{74} + 3182080 q^{76} - 7614880 q^{79} + 2502592 q^{86} + 11982420 q^{89} + 11983904 q^{91} + 6940416 q^{94}+O(q^{100}) \) 2 * q - 128 * q^4 - 14664 * q^11 - 25216 * q^14 + 8192 * q^16 - 49720 * q^19 + 60832 * q^26 - 83220 * q^29 + 66304 * q^31 + 105696 * q^34 + 1278156 * q^41 + 938496 * q^44 + 663168 * q^46 - 3320466 * q^49 + 1613824 * q^56 + 1490280 * q^59 - 3321236 * q^61 - 524288 * q^64 - 11432304 * q^71 - 583456 * q^74 + 3182080 * q^76 - 7614880 * q^79 + 2502592 * q^86 + 11982420 * q^89 + 11983904 * q^91 + 6940416 * q^94

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