LMFDB - Embedded newform 450.4.c.b.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("450.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$26.5508595026$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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			450.4.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)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} +11.0000i q^{7} +8.00000i q^{8} +O(q^{10})$	$q-2.00000i q^{2} -4.00000 q^{4} +11.0000i q^{7} +8.00000i q^{8} -36.0000 q^{11} -17.0000i q^{13} +22.0000 q^{14} +16.0000 q^{16} -12.0000i q^{17} +91.0000 q^{19} +72.0000i q^{22} -60.0000i q^{23} -34.0000 q^{26} -44.0000i q^{28} +276.000 q^{29} +191.000 q^{31} -32.0000i q^{32} -24.0000 q^{34} +254.000i q^{37} -182.000i q^{38} -60.0000 q^{41} +49.0000i q^{43} +144.000 q^{44} -120.000 q^{46} -600.000i q^{47} +222.000 q^{49} +68.0000i q^{52} -612.000i q^{53} -88.0000 q^{56} -552.000i q^{58} +744.000 q^{59} +167.000 q^{61} -382.000i q^{62} -64.0000 q^{64} -457.000i q^{67} +48.0000i q^{68} -588.000 q^{71} +970.000i q^{73} +508.000 q^{74} -364.000 q^{76} -396.000i q^{77} -164.000 q^{79} +120.000i q^{82} -696.000i q^{83} +98.0000 q^{86} -288.000i q^{88} +1248.00 q^{89} +187.000 q^{91} +240.000i q^{92} -1200.00 q^{94} -1099.00i q^{97} -444.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} - 72 q^{11} + 44 q^{14} + 32 q^{16} + 182 q^{19} - 68 q^{26} + 552 q^{29} + 382 q^{31} - 48 q^{34} - 120 q^{41} + 288 q^{44} - 240 q^{46} + 444 q^{49} - 176 q^{56} + 1488 q^{59} + 334 q^{61} - 128 q^{64} - 1176 q^{71} + 1016 q^{74} - 728 q^{76} - 328 q^{79} + 196 q^{86} + 2496 q^{89} + 374 q^{91} - 2400 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 - 72 * q^11 + 44 * q^14 + 32 * q^16 + 182 * q^19 - 68 * q^26 + 552 * q^29 + 382 * q^31 - 48 * q^34 - 120 * q^41 + 288 * q^44 - 240 * q^46 + 444 * q^49 - 176 * q^56 + 1488 * q^59 + 334 * q^61 - 128 * q^64 - 1176 * q^71 + 1016 * q^74 - 728 * q^76 - 328 * q^79 + 196 * q^86 + 2496 * q^89 + 374 * q^91 - 2400 * q^94

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	11.0000i	0.593944i	0.954886	+	0.296972i	$0.0959768\pi$
$7$	11.0000i	0.593944i	−0.954886	+	0.296972i	$0.904023\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$11$	−36.0000	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$12$	0	0
$13$	− 17.0000i	− 0.362689i	−0.983420	−	0.181344i	$-0.941955\pi$
$13$	− 17.0000i	− 0.362689i	0.983420	−	0.181344i	$-0.0580448\pi$
$14$	22.0000	0.419982
$15$	0	0
$16$	16.0000	0.250000
$17$	− 12.0000i	− 0.171202i	−0.996330	−	0.0856008i	$-0.972719\pi$
$17$	− 12.0000i	− 0.171202i	0.996330	−	0.0856008i	$-0.0272810\pi$
$18$	0	0
$19$	91.0000	1.09878	0.549390	−	0.835566i	$-0.314860\pi$
$19$	91.0000	1.09878	0.549390	+	0.835566i	$0.314860\pi$
$20$	0	0
$21$	0	0
$22$	72.0000i	0.697748i
$23$	− 60.0000i	− 0.543951i	−0.962304	−	0.271975i	$-0.912323\pi$
$23$	− 60.0000i	− 0.543951i	0.962304	−	0.271975i	$-0.0876769\pi$
$24$	0	0
$25$	0	0
$26$	−34.0000	−0.256460
$27$	0	0
$28$	− 44.0000i	− 0.296972i
$29$	276.000	1.76731	0.883654	−	0.468141i	$-0.155076\pi$
$29$	276.000	1.76731	0.883654	+	0.468141i	$0.155076\pi$
$30$	0	0
$31$	191.000	1.10660	0.553300	−	0.832982i	$-0.313368\pi$
$31$	191.000	1.10660	0.553300	+	0.832982i	$0.313368\pi$
$32$	− 32.0000i	− 0.176777i
$33$	0	0
$34$	−24.0000	−0.121058
$35$	0	0
$36$	0	0
$37$	254.000i	1.12858i	0.825578	+	0.564288i	$0.190849\pi$
$37$	254.000i	1.12858i	−0.825578	+	0.564288i	$0.809151\pi$
$38$	− 182.000i	− 0.776955i
$39$	0	0
$40$	0	0
$41$	−60.0000	−0.228547	−0.114273	−	0.993449i	$-0.536454\pi$
$41$	−60.0000	−0.228547	−0.114273	+	0.993449i	$0.536454\pi$
$42$	0	0
$43$	49.0000i	0.173777i	0.996218	+	0.0868887i	$0.0276925\pi$
$43$	49.0000i	0.173777i	−0.996218	+	0.0868887i	$0.972308\pi$
$44$	144.000	0.493382
$45$	0	0
$46$	−120.000	−0.384631
$47$	− 600.000i	− 1.86211i	−0.364884	−	0.931053i	$-0.618891\pi$
$47$	− 600.000i	− 1.86211i	0.364884	−	0.931053i	$-0.381109\pi$
$48$	0	0
$49$	222.000	0.647230
$50$	0	0
$51$	0	0
$52$	68.0000i	0.181344i
$53$	− 612.000i	− 1.58613i	−0.609140	−	0.793063i	$-0.708485\pi$
$53$	− 612.000i	− 1.58613i	0.609140	−	0.793063i	$-0.291515\pi$
$54$	0	0
$55$	0	0
$56$	−88.0000	−0.209991
$57$	0	0
$58$	− 552.000i	− 1.24968i
$59$	744.000	1.64170	0.820852	−	0.571141i	$-0.193499\pi$
$59$	744.000	1.64170	0.820852	+	0.571141i	$0.193499\pi$
$60$	0	0
$61$	167.000	0.350527	0.175264	−	0.984522i	$-0.443922\pi$
$61$	167.000	0.350527	0.175264	+	0.984522i	$0.443922\pi$
$62$	− 382.000i	− 0.782485i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 457.000i	− 0.833305i	−0.909066	−	0.416653i	$-0.863203\pi$
$67$	− 457.000i	− 0.833305i	0.909066	−	0.416653i	$-0.136797\pi$
$68$	48.0000i	0.0856008i
$69$	0	0
$70$	0	0
$71$	−588.000	−0.982856	−0.491428	−	0.870918i	$-0.663525\pi$
$71$	−588.000	−0.982856	−0.491428	+	0.870918i	$0.663525\pi$
$72$	0	0
$73$	970.000i	1.55520i	0.628757	+	0.777602i	$0.283564\pi$
$73$	970.000i	1.55520i	−0.628757	+	0.777602i	$0.716436\pi$
$74$	508.000	0.798024
$75$	0	0
$76$	−364.000	−0.549390
$77$	− 396.000i	− 0.586083i
$78$	0	0
$79$	−164.000	−0.233563	−0.116781	−	0.993158i	$-0.537258\pi$
$79$	−164.000	−0.233563	−0.116781	+	0.993158i	$0.537258\pi$
$80$	0	0
$81$	0	0
$82$	120.000i	0.161607i
$83$	− 696.000i	− 0.920433i	−0.887807	−	0.460216i	$-0.847772\pi$
$83$	− 696.000i	− 0.920433i	0.887807	−	0.460216i	$-0.152228\pi$
$84$	0	0
$85$	0	0
$86$	98.0000	0.122879
$87$	0	0
$88$	− 288.000i	− 0.348874i
$89$	1248.00	1.48638	0.743190	−	0.669081i	$-0.233312\pi$
$89$	1248.00	1.48638	0.743190	+	0.669081i	$0.233312\pi$
$90$	0	0
$91$	187.000	0.215417
$92$	240.000i	0.271975i
$93$	0	0
$94$	−1200.00	−1.31671
$95$	0	0
$96$	0	0
$97$	− 1099.00i	− 1.15038i	−0.818021	−	0.575188i	$-0.804929\pi$
$97$	− 1099.00i	− 1.15038i	0.818021	−	0.575188i	$-0.195071\pi$
$98$	− 444.000i	− 0.457661i
$99$	0	0
$100$	0	0
$101$	−444.000	−0.437422	−0.218711	−	0.975790i	$-0.570185\pi$
$101$	−444.000	−0.437422	−0.218711	+	0.975790i	$0.570185\pi$
$102$	0	0
$103$	916.000i	0.876273i	0.898908	+	0.438137i	$0.144361\pi$
$103$	916.000i	0.876273i	−0.898908	+	0.438137i	$0.855639\pi$
$104$	136.000	0.128230
$105$	0	0
$106$	−1224.00	−1.12156
$107$	204.000i	0.184312i	0.995745	+	0.0921562i	$0.0293759\pi$
$107$	204.000i	0.184312i	−0.995745	+	0.0921562i	$0.970624\pi$
$108$	0	0
$109$	967.000	0.849741	0.424871	−	0.905254i	$-0.360320\pi$
$109$	967.000	0.849741	0.424871	+	0.905254i	$0.360320\pi$
$110$	0	0
$111$	0	0
$112$	176.000i	0.148486i
$113$	672.000i	0.559438i	0.960082	+	0.279719i	$0.0902412\pi$
$113$	672.000i	0.559438i	−0.960082	+	0.279719i	$0.909759\pi$
$114$	0	0
$115$	0	0
$116$	−1104.00	−0.883654
$117$	0	0
$118$	− 1488.00i	− 1.16086i
$119$	132.000	0.101684
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	− 334.000i	− 0.247860i
$123$	0	0
$124$	−764.000	−0.553300
$125$	0	0
$126$	0	0
$127$	− 1924.00i	− 1.34431i	−0.740410	−	0.672155i	$-0.765369\pi$
$127$	− 1924.00i	− 1.34431i	0.740410	−	0.672155i	$-0.234631\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	756.000	0.504214	0.252107	−	0.967699i	$-0.418877\pi$
$131$	756.000	0.504214	0.252107	+	0.967699i	$0.418877\pi$
$132$	0	0
$133$	1001.00i	0.652614i
$134$	−914.000	−0.589236
$135$	0	0
$136$	96.0000	0.0605289
$137$	− 888.000i	− 0.553773i	−0.960903	−	0.276887i	$-0.910697\pi$
$137$	− 888.000i	− 0.553773i	0.960903	−	0.276887i	$-0.0893027\pi$
$138$	0	0
$139$	160.000	0.0976333	0.0488166	−	0.998808i	$-0.484455\pi$
$139$	160.000	0.0976333	0.0488166	+	0.998808i	$0.484455\pi$
$140$	0	0
$141$	0	0
$142$	1176.00i	0.694984i
$143$	612.000i	0.357888i
$144$	0	0
$145$	0	0
$146$	1940.00	1.09970
$147$	0	0
$148$	− 1016.00i	− 0.564288i
$149$	−3096.00	−1.70224	−0.851121	−	0.524969i	$-0.824077\pi$
$149$	−3096.00	−1.70224	−0.851121	+	0.524969i	$0.824077\pi$
$150$	0	0
$151$	1049.00	0.565340	0.282670	−	0.959217i	$-0.408780\pi$
$151$	1049.00	0.565340	0.282670	+	0.959217i	$0.408780\pi$
$152$	728.000i	0.388478i
$153$	0	0
$154$	−792.000	−0.414423
$155$	0	0
$156$	0	0
$157$	2363.00i	1.20120i	0.799551	+	0.600599i	$0.205071\pi$
$157$	2363.00i	1.20120i	−0.799551	+	0.600599i	$0.794929\pi$
$158$	328.000i	0.165154i
$159$	0	0
$160$	0	0
$161$	660.000	0.323076
$162$	0	0
$163$	3235.00i	1.55451i	0.629187	+	0.777254i	$0.283388\pi$
$163$	3235.00i	1.55451i	−0.629187	+	0.777254i	$0.716612\pi$
$164$	240.000	0.114273
$165$	0	0
$166$	−1392.00	−0.650844
$167$	2772.00i	1.28445i	0.766515	+	0.642227i	$0.221989\pi$
$167$	2772.00i	1.28445i	−0.766515	+	0.642227i	$0.778011\pi$
$168$	0	0
$169$	1908.00	0.868457
$170$	0	0
$171$	0	0
$172$	− 196.000i	− 0.0868887i
$173$	− 84.0000i	− 0.0369156i	−0.999830	−	0.0184578i	$-0.994124\pi$
$173$	− 84.0000i	− 0.0369156i	0.999830	−	0.0184578i	$-0.00587564\pi$
$174$	0	0
$175$	0	0
$176$	−576.000	−0.246691
$177$	0	0
$178$	− 2496.00i	− 1.05103i
$179$	−2736.00	−1.14245	−0.571224	−	0.820794i	$-0.693531\pi$
$179$	−2736.00	−1.14245	−0.571224	+	0.820794i	$0.693531\pi$
$180$	0	0
$181$	4397.00	1.80567	0.902835	−	0.429986i	$-0.141482\pi$
$181$	4397.00	1.80567	0.902835	+	0.429986i	$0.141482\pi$
$182$	− 374.000i	− 0.152323i
$183$	0	0
$184$	480.000	0.192316
$185$	0	0
$186$	0	0
$187$	432.000i	0.168936i
$188$	2400.00i	0.931053i
$189$	0	0
$190$	0	0
$191$	3108.00	1.17742	0.588709	−	0.808345i	$-0.299636\pi$
$191$	3108.00	1.17742	0.588709	+	0.808345i	$0.299636\pi$
$192$	0	0
$193$	− 2615.00i	− 0.975294i	−0.873041	−	0.487647i	$-0.837855\pi$
$193$	− 2615.00i	− 0.975294i	0.873041	−	0.487647i	$-0.162145\pi$
$194$	−2198.00	−0.813439
$195$	0	0
$196$	−888.000	−0.323615
$197$	3624.00i	1.31066i	0.755344	+	0.655328i	$0.227470\pi$
$197$	3624.00i	1.31066i	−0.755344	+	0.655328i	$0.772530\pi$
$198$	0	0
$199$	1819.00	0.647967	0.323984	−	0.946063i	$-0.394978\pi$
$199$	1819.00	0.647967	0.323984	+	0.946063i	$0.394978\pi$
$200$	0	0
$201$	0	0
$202$	888.000i	0.309304i
$203$	3036.00i	1.04968i
$204$	0	0
$205$	0	0
$206$	1832.00	0.619619
$207$	0	0
$208$	− 272.000i	− 0.0906721i
$209$	−3276.00	−1.08424
$210$	0	0
$211$	−1999.00	−0.652212	−0.326106	−	0.945333i	$-0.605737\pi$
$211$	−1999.00	−0.652212	−0.326106	+	0.945333i	$0.605737\pi$
$212$	2448.00i	0.793063i
$213$	0	0
$214$	408.000	0.130329
$215$	0	0
$216$	0	0
$217$	2101.00i	0.657259i
$218$	− 1934.00i	− 0.600858i
$219$	0	0
$220$	0	0
$221$	−204.000	−0.0620929
$222$	0	0
$223$	493.000i	0.148044i	0.997257	+	0.0740218i	$0.0235834\pi$
$223$	493.000i	0.148044i	−0.997257	+	0.0740218i	$0.976417\pi$
$224$	352.000	0.104995
$225$	0	0
$226$	1344.00	0.395582
$227$	− 300.000i	− 0.0877167i	−0.999038	−	0.0438584i	$-0.986035\pi$
$227$	− 300.000i	− 0.0877167i	0.999038	−	0.0438584i	$-0.0139650\pi$
$228$	0	0
$229$	79.0000	0.0227968	0.0113984	−	0.999935i	$-0.496372\pi$
$229$	79.0000	0.0227968	0.0113984	+	0.999935i	$0.496372\pi$
$230$	0	0
$231$	0	0
$232$	2208.00i	0.624838i
$233$	6108.00i	1.71738i	0.512500	+	0.858688i	$0.328720\pi$
$233$	6108.00i	1.71738i	−0.512500	+	0.858688i	$0.671280\pi$
$234$	0	0
$235$	0	0
$236$	−2976.00	−0.820852
$237$	0	0
$238$	− 264.000i	− 0.0719016i
$239$	−2868.00	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−2868.00	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	2705.00	0.723006	0.361503	−	0.932371i	$-0.382264\pi$
$241$	2705.00	0.723006	0.361503	+	0.932371i	$0.382264\pi$
$242$	70.0000i	0.0185941i
$243$	0	0
$244$	−668.000	−0.175264
$245$	0	0
$246$	0	0
$247$	− 1547.00i	− 0.398515i
$248$	1528.00i	0.391242i
$249$	0	0
$250$	0	0
$251$	−4008.00	−1.00790	−0.503950	−	0.863733i	$-0.668120\pi$
$251$	−4008.00	−1.00790	−0.503950	+	0.863733i	$0.668120\pi$
$252$	0	0
$253$	2160.00i	0.536751i
$254$	−3848.00	−0.950571
$255$	0	0
$256$	256.000	0.0625000
$257$	1992.00i	0.483492i	0.970340	+	0.241746i	$0.0777201\pi$
$257$	1992.00i	0.483492i	−0.970340	+	0.241746i	$0.922280\pi$
$258$	0	0
$259$	−2794.00	−0.670312
$260$	0	0
$261$	0	0
$262$	− 1512.00i	− 0.356533i
$263$	− 972.000i	− 0.227894i	−0.993487	−	0.113947i	$-0.963651\pi$
$263$	− 972.000i	− 0.227894i	0.993487	−	0.113947i	$-0.0363494\pi$
$264$	0	0
$265$	0	0
$266$	2002.00	0.461468
$267$	0	0
$268$	1828.00i	0.416653i
$269$	1812.00	0.410705	0.205352	−	0.978688i	$-0.434166\pi$
$269$	1812.00	0.410705	0.205352	+	0.978688i	$0.434166\pi$
$270$	0	0
$271$	−5092.00	−1.14139	−0.570696	−	0.821162i	$-0.693326\pi$
$271$	−5092.00	−1.14139	−0.570696	+	0.821162i	$0.693326\pi$
$272$	− 192.000i	− 0.0428004i
$273$	0	0
$274$	−1776.00	−0.391577
$275$	0	0
$276$	0	0
$277$	569.000i	0.123422i	0.998094	+	0.0617110i	$0.0196557\pi$
$277$	569.000i	0.123422i	−0.998094	+	0.0617110i	$0.980344\pi$
$278$	− 320.000i	− 0.0690371i
$279$	0	0
$280$	0	0
$281$	6468.00	1.37313	0.686563	−	0.727070i	$-0.259118\pi$
$281$	6468.00	1.37313	0.686563	+	0.727070i	$0.259118\pi$
$282$	0	0
$283$	− 3557.00i	− 0.747144i	−0.927601	−	0.373572i	$-0.878133\pi$
$283$	− 3557.00i	− 0.747144i	0.927601	−	0.373572i	$-0.121867\pi$
$284$	2352.00	0.491428
$285$	0	0
$286$	1224.00	0.253065
$287$	− 660.000i	− 0.135744i
$288$	0	0
$289$	4769.00	0.970690
$290$	0	0
$291$	0	0
$292$	− 3880.00i	− 0.777602i
$293$	− 5376.00i	− 1.07191i	−0.844247	−	0.535954i	$-0.819952\pi$
$293$	− 5376.00i	− 1.07191i	0.844247	−	0.535954i	$-0.180048\pi$
$294$	0	0
$295$	0	0
$296$	−2032.00	−0.399012
$297$	0	0
$298$	6192.00i	1.20367i
$299$	−1020.00	−0.197285
$300$	0	0
$301$	−539.000	−0.103214
$302$	− 2098.00i	− 0.399756i
$303$	0	0
$304$	1456.00	0.274695
$305$	0	0
$306$	0	0
$307$	− 9343.00i	− 1.73692i	−0.495763	−	0.868458i	$-0.665111\pi$
$307$	− 9343.00i	− 1.73692i	0.495763	−	0.868458i	$-0.334889\pi$
$308$	1584.00i	0.293041i
$309$	0	0
$310$	0	0
$311$	−1968.00	−0.358827	−0.179413	−	0.983774i	$-0.557420\pi$
$311$	−1968.00	−0.358827	−0.179413	+	0.983774i	$0.557420\pi$
$312$	0	0
$313$	− 5675.00i	− 1.02482i	−0.858740	−	0.512412i	$-0.828752\pi$
$313$	− 5675.00i	− 1.02482i	0.858740	−	0.512412i	$-0.171248\pi$
$314$	4726.00	0.849375
$315$	0	0
$316$	656.000	0.116781
$317$	− 2616.00i	− 0.463499i	−0.972775	−	0.231750i	$-0.925555\pi$
$317$	− 2616.00i	− 0.463499i	0.972775	−	0.231750i	$-0.0744450\pi$
$318$	0	0
$319$	−9936.00	−1.74392
$320$	0	0
$321$	0	0
$322$	− 1320.00i	− 0.228449i
$323$	− 1092.00i	− 0.188113i
$324$	0	0
$325$	0	0
$326$	6470.00	1.09920
$327$	0	0
$328$	− 480.000i	− 0.0808036i
$329$	6600.00	1.10599
$330$	0	0
$331$	−9664.00	−1.60478	−0.802389	−	0.596801i	$-0.796438\pi$
$331$	−9664.00	−1.60478	−0.802389	+	0.596801i	$0.796438\pi$
$332$	2784.00i	0.460216i
$333$	0	0
$334$	5544.00	0.908246
$335$	0	0
$336$	0	0
$337$	7637.00i	1.23446i	0.786782	+	0.617231i	$0.211746\pi$
$337$	7637.00i	1.23446i	−0.786782	+	0.617231i	$0.788254\pi$
$338$	− 3816.00i	− 0.614092i
$339$	0	0
$340$	0	0
$341$	−6876.00	−1.09195
$342$	0	0
$343$	6215.00i	0.978363i
$344$	−392.000	−0.0614396
$345$	0	0
$346$	−168.000	−0.0261033
$347$	− 5640.00i	− 0.872539i	−0.899816	−	0.436270i	$-0.856299\pi$
$347$	− 5640.00i	− 0.872539i	0.899816	−	0.436270i	$-0.143701\pi$
$348$	0	0
$349$	−7382.00	−1.13223	−0.566117	−	0.824325i	$-0.691555\pi$
$349$	−7382.00	−1.13223	−0.566117	+	0.824325i	$0.691555\pi$
$350$	0	0
$351$	0	0
$352$	1152.00i	0.174437i
$353$	− 10740.0i	− 1.61936i	−0.586875	−	0.809678i	$-0.699642\pi$
$353$	− 10740.0i	− 1.61936i	0.586875	−	0.809678i	$-0.300358\pi$
$354$	0	0
$355$	0	0
$356$	−4992.00	−0.743190
$357$	0	0
$358$	5472.00i	0.807833i
$359$	9276.00	1.36370	0.681850	−	0.731492i	$-0.261176\pi$
$359$	9276.00	1.36370	0.681850	+	0.731492i	$0.261176\pi$
$360$	0	0
$361$	1422.00	0.207319
$362$	− 8794.00i	− 1.27680i
$363$	0	0
$364$	−748.000	−0.107708
$365$	0	0
$366$	0	0
$367$	− 7387.00i	− 1.05068i	−0.850893	−	0.525338i	$-0.823939\pi$
$367$	− 7387.00i	− 1.05068i	0.850893	−	0.525338i	$-0.176061\pi$
$368$	− 960.000i	− 0.135988i
$369$	0	0
$370$	0	0
$371$	6732.00	0.942070
$372$	0	0
$373$	− 10259.0i	− 1.42410i	−0.702127	−	0.712052i	$-0.747766\pi$
$373$	− 10259.0i	− 1.42410i	0.702127	−	0.712052i	$-0.252234\pi$
$374$	864.000	0.119456
$375$	0	0
$376$	4800.00	0.658354
$377$	− 4692.00i	− 0.640982i
$378$	0	0
$379$	−7655.00	−1.03750	−0.518748	−	0.854927i	$-0.673602\pi$
$379$	−7655.00	−1.03750	−0.518748	+	0.854927i	$0.673602\pi$
$380$	0	0
$381$	0	0
$382$	− 6216.00i	− 0.832561i
$383$	− 600.000i	− 0.0800485i	−0.999199	−	0.0400242i	$-0.987256\pi$
$383$	− 600.000i	− 0.0800485i	0.999199	−	0.0400242i	$-0.0127435\pi$
$384$	0	0
$385$	0	0
$386$	−5230.00	−0.689637
$387$	0	0
$388$	4396.00i	0.575188i
$389$	4068.00	0.530221	0.265110	−	0.964218i	$-0.414592\pi$
$389$	4068.00	0.530221	0.265110	+	0.964218i	$0.414592\pi$
$390$	0	0
$391$	−720.000	−0.0931252
$392$	1776.00i	0.228830i
$393$	0	0
$394$	7248.00	0.926774
$395$	0	0
$396$	0	0
$397$	12647.0i	1.59883i	0.600781	+	0.799414i	$0.294857\pi$
$397$	12647.0i	1.59883i	−0.600781	+	0.799414i	$0.705143\pi$
$398$	− 3638.00i	− 0.458182i
$399$	0	0
$400$	0	0
$401$	−9924.00	−1.23586	−0.617931	−	0.786232i	$-0.712029\pi$
$401$	−9924.00	−1.23586	−0.617931	+	0.786232i	$0.712029\pi$
$402$	0	0
$403$	− 3247.00i	− 0.401351i
$404$	1776.00	0.218711
$405$	0	0
$406$	6072.00	0.742237
$407$	− 9144.00i	− 1.11364i
$408$	0	0
$409$	10903.0	1.31814	0.659069	−	0.752082i	$-0.270950\pi$
$409$	10903.0	1.31814	0.659069	+	0.752082i	$0.270950\pi$
$410$	0	0
$411$	0	0
$412$	− 3664.00i	− 0.438137i
$413$	8184.00i	0.975081i
$414$	0	0
$415$	0	0
$416$	−544.000	−0.0641149
$417$	0	0
$418$	6552.00i	0.766672i
$419$	14796.0	1.72514	0.862568	−	0.505941i	$-0.168855\pi$
$419$	14796.0	1.72514	0.862568	+	0.505941i	$0.168855\pi$
$420$	0	0
$421$	−4606.00	−0.533213	−0.266607	−	0.963805i	$-0.585902\pi$
$421$	−4606.00	−0.533213	−0.266607	+	0.963805i	$0.585902\pi$
$422$	3998.00i	0.461184i
$423$	0	0
$424$	4896.00	0.560780
$425$	0	0
$426$	0	0
$427$	1837.00i	0.208194i
$428$	− 816.000i	− 0.0921562i
$429$	0	0
$430$	0	0
$431$	−13860.0	−1.54899	−0.774493	−	0.632583i	$-0.781995\pi$
$431$	−13860.0	−1.54899	−0.774493	+	0.632583i	$0.781995\pi$
$432$	0	0
$433$	− 8051.00i	− 0.893548i	−0.894647	−	0.446774i	$-0.852573\pi$
$433$	− 8051.00i	− 0.893548i	0.894647	−	0.446774i	$-0.147427\pi$
$434$	4202.00	0.464752
$435$	0	0
$436$	−3868.00	−0.424871
$437$	− 5460.00i	− 0.597682i
$438$	0	0
$439$	−2087.00	−0.226895	−0.113448	−	0.993544i	$-0.536189\pi$
$439$	−2087.00	−0.226895	−0.113448	+	0.993544i	$0.536189\pi$
$440$	0	0
$441$	0	0
$442$	408.000i	0.0439063i
$443$	− 13368.0i	− 1.43371i	−0.697223	−	0.716854i	$-0.745581\pi$
$443$	− 13368.0i	− 1.43371i	0.697223	−	0.716854i	$-0.254419\pi$
$444$	0	0
$445$	0	0
$446$	986.000	0.104683
$447$	0	0
$448$	− 704.000i	− 0.0742430i
$449$	5352.00	0.562531	0.281266	−	0.959630i	$-0.409246\pi$
$449$	5352.00	0.562531	0.281266	+	0.959630i	$0.409246\pi$
$450$	0	0
$451$	2160.00	0.225522
$452$	− 2688.00i	− 0.279719i
$453$	0	0
$454$	−600.000	−0.0620251
$455$	0	0
$456$	0	0
$457$	13718.0i	1.40416i	0.712098	+	0.702080i	$0.247745\pi$
$457$	13718.0i	1.40416i	−0.712098	+	0.702080i	$0.752255\pi$
$458$	− 158.000i	− 0.0161198i
$459$	0	0
$460$	0	0
$461$	11868.0	1.19902	0.599510	−	0.800368i	$-0.295362\pi$
$461$	11868.0	1.19902	0.599510	+	0.800368i	$0.295362\pi$
$462$	0	0
$463$	− 6572.00i	− 0.659669i	−0.944039	−	0.329834i	$-0.893007\pi$
$463$	− 6572.00i	− 0.659669i	0.944039	−	0.329834i	$-0.106993\pi$
$464$	4416.00	0.441827
$465$	0	0
$466$	12216.0	1.21437
$467$	18636.0i	1.84662i	0.384056	+	0.923310i	$0.374527\pi$
$467$	18636.0i	1.84662i	−0.384056	+	0.923310i	$0.625473\pi$
$468$	0	0
$469$	5027.00	0.494937
$470$	0	0
$471$	0	0
$472$	5952.00i	0.580430i
$473$	− 1764.00i	− 0.171477i
$474$	0	0
$475$	0	0
$476$	−528.000	−0.0508421
$477$	0	0
$478$	5736.00i	0.548867i
$479$	−5256.00	−0.501363	−0.250681	−	0.968070i	$-0.580655\pi$
$479$	−5256.00	−0.501363	−0.250681	+	0.968070i	$0.580655\pi$
$480$	0	0
$481$	4318.00	0.409322
$482$	− 5410.00i	− 0.511242i
$483$	0	0
$484$	140.000	0.0131480
$485$	0	0
$486$	0	0
$487$	− 9991.00i	− 0.929642i	−0.885405	−	0.464821i	$-0.846119\pi$
$487$	− 9991.00i	− 0.929642i	0.885405	−	0.464821i	$-0.153881\pi$
$488$	1336.00i	0.123930i
$489$	0	0
$490$	0	0
$491$	1056.00	0.0970603	0.0485302	−	0.998822i	$-0.484546\pi$
$491$	1056.00	0.0970603	0.0485302	+	0.998822i	$0.484546\pi$
$492$	0	0
$493$	− 3312.00i	− 0.302566i
$494$	−3094.00	−0.281793
$495$	0	0
$496$	3056.00	0.276650
$497$	− 6468.00i	− 0.583761i
$498$	0	0
$499$	−12329.0	−1.10606	−0.553028	−	0.833163i	$-0.686528\pi$
$499$	−12329.0	−1.10606	−0.553028	+	0.833163i	$0.686528\pi$
$500$	0	0
$501$	0	0
$502$	8016.00i	0.712692i
$503$	− 2460.00i	− 0.218064i	−0.994038	−	0.109032i	$-0.965225\pi$
$503$	− 2460.00i	− 0.218064i	0.994038	−	0.109032i	$-0.0347750\pi$
$504$	0	0
$505$	0	0
$506$	4320.00	0.379540
$507$	0	0
$508$	7696.00i	0.672155i
$509$	−1848.00	−0.160926	−0.0804628	−	0.996758i	$-0.525640\pi$
$509$	−1848.00	−0.160926	−0.0804628	+	0.996758i	$0.525640\pi$
$510$	0	0
$511$	−10670.0	−0.923705
$512$	− 512.000i	− 0.0441942i
$513$	0	0
$514$	3984.00	0.341881
$515$	0	0
$516$	0	0
$517$	21600.0i	1.83746i
$518$	5588.00i	0.473982i
$519$	0	0
$520$	0	0
$521$	−12732.0	−1.07063	−0.535316	−	0.844652i	$-0.679807\pi$
$521$	−12732.0	−1.07063	−0.535316	+	0.844652i	$0.679807\pi$
$522$	0	0
$523$	− 9977.00i	− 0.834156i	−0.908871	−	0.417078i	$-0.863054\pi$
$523$	− 9977.00i	− 0.834156i	0.908871	−	0.417078i	$-0.136946\pi$
$524$	−3024.00	−0.252107
$525$	0	0
$526$	−1944.00	−0.161145
$527$	− 2292.00i	− 0.189452i
$528$	0	0
$529$	8567.00	0.704118
$530$	0	0
$531$	0	0
$532$	− 4004.00i	− 0.326307i
$533$	1020.00i	0.0828914i
$534$	0	0
$535$	0	0
$536$	3656.00	0.294618
$537$	0	0
$538$	− 3624.00i	− 0.290412i
$539$	−7992.00	−0.638664
$540$	0	0
$541$	16733.0	1.32977	0.664887	−	0.746944i	$-0.268480\pi$
$541$	16733.0	1.32977	0.664887	+	0.746944i	$0.268480\pi$
$542$	10184.0i	0.807085i
$543$	0	0
$544$	−384.000	−0.0302645
$545$	0	0
$546$	0	0
$547$	− 16360.0i	− 1.27880i	−0.768875	−	0.639400i	$-0.779183\pi$
$547$	− 16360.0i	− 1.27880i	0.768875	−	0.639400i	$-0.220817\pi$
$548$	3552.00i	0.276887i
$549$	0	0
$550$	0	0
$551$	25116.0	1.94188
$552$	0	0
$553$	− 1804.00i	− 0.138723i
$554$	1138.00	0.0872725
$555$	0	0
$556$	−640.000	−0.0488166
$557$	3984.00i	0.303066i	0.988452	+	0.151533i	$0.0484209\pi$
$557$	3984.00i	0.303066i	−0.988452	+	0.151533i	$0.951579\pi$
$558$	0	0
$559$	833.000	0.0630271
$560$	0	0
$561$	0	0
$562$	− 12936.0i	− 0.970947i
$563$	16332.0i	1.22258i	0.791407	+	0.611289i	$0.209349\pi$
$563$	16332.0i	1.22258i	−0.791407	+	0.611289i	$0.790651\pi$
$564$	0	0
$565$	0	0
$566$	−7114.00	−0.528310
$567$	0	0
$568$	− 4704.00i	− 0.347492i
$569$	−25716.0	−1.89468	−0.947338	−	0.320235i	$-0.896238\pi$
$569$	−25716.0	−1.89468	−0.947338	+	0.320235i	$0.896238\pi$
$570$	0	0
$571$	−15091.0	−1.10602	−0.553011	−	0.833174i	$-0.686521\pi$
$571$	−15091.0	−1.10602	−0.553011	+	0.833174i	$0.686521\pi$
$572$	− 2448.00i	− 0.178944i
$573$	0	0
$574$	−1320.00	−0.0959856
$575$	0	0
$576$	0	0
$577$	2579.00i	0.186075i	0.995663	+	0.0930374i	$0.0296576\pi$
$577$	2579.00i	0.186075i	−0.995663	+	0.0930374i	$0.970342\pi$
$578$	− 9538.00i	− 0.686381i
$579$	0	0
$580$	0	0
$581$	7656.00	0.546686
$582$	0	0
$583$	22032.0i	1.56513i
$584$	−7760.00	−0.549848
$585$	0	0
$586$	−10752.0	−0.757954
$587$	− 11892.0i	− 0.836176i	−0.908407	−	0.418088i	$-0.862700\pi$
$587$	− 11892.0i	− 0.836176i	0.908407	−	0.418088i	$-0.137300\pi$
$588$	0	0
$589$	17381.0	1.21591
$590$	0	0
$591$	0	0
$592$	4064.00i	0.282144i
$593$	6564.00i	0.454555i	0.973830	+	0.227278i	$0.0729825\pi$
$593$	6564.00i	0.454555i	−0.973830	+	0.227278i	$0.927018\pi$
$594$	0	0
$595$	0	0
$596$	12384.0	0.851121
$597$	0	0
$598$	2040.00i	0.139501i
$599$	−9096.00	−0.620455	−0.310227	−	0.950662i	$-0.600405\pi$
$599$	−9096.00	−0.620455	−0.310227	+	0.950662i	$0.600405\pi$
$600$	0	0
$601$	6575.00	0.446256	0.223128	−	0.974789i	$-0.428373\pi$
$601$	6575.00	0.446256	0.223128	+	0.974789i	$0.428373\pi$
$602$	1078.00i	0.0729834i
$603$	0	0
$604$	−4196.00	−0.282670
$605$	0	0
$606$	0	0
$607$	− 12436.0i	− 0.831568i	−0.909463	−	0.415784i	$-0.863507\pi$
$607$	− 12436.0i	− 0.831568i	0.909463	−	0.415784i	$-0.136493\pi$
$608$	− 2912.00i	− 0.194239i
$609$	0	0
$610$	0	0
$611$	−10200.0	−0.675365
$612$	0	0
$613$	4354.00i	0.286878i	0.989659	+	0.143439i	$0.0458161\pi$
$613$	4354.00i	0.286878i	−0.989659	+	0.143439i	$0.954184\pi$
$614$	−18686.0	−1.22818
$615$	0	0
$616$	3168.00	0.207212
$617$	30060.0i	1.96138i	0.195575	+	0.980689i	$0.437343\pi$
$617$	30060.0i	1.96138i	−0.195575	+	0.980689i	$0.562657\pi$
$618$	0	0
$619$	−24845.0	−1.61326	−0.806628	−	0.591060i	$-0.798710\pi$
$619$	−24845.0	−1.61326	−0.806628	+	0.591060i	$0.798710\pi$
$620$	0	0
$621$	0	0
$622$	3936.00i	0.253729i
$623$	13728.0i	0.882826i
$624$	0	0
$625$	0	0
$626$	−11350.0	−0.724660
$627$	0	0
$628$	− 9452.00i	− 0.600599i
$629$	3048.00	0.193214
$630$	0	0
$631$	−10885.0	−0.686727	−0.343364	−	0.939203i	$-0.611566\pi$
$631$	−10885.0	−0.686727	−0.343364	+	0.939203i	$0.611566\pi$
$632$	− 1312.00i	− 0.0825768i
$633$	0	0
$634$	−5232.00	−0.327743
$635$	0	0
$636$	0	0
$637$	− 3774.00i	− 0.234743i
$638$	19872.0i	1.23313i
$639$	0	0
$640$	0	0
$641$	20136.0	1.24076	0.620378	−	0.784303i	$-0.286979\pi$
$641$	20136.0	1.24076	0.620378	+	0.784303i	$0.286979\pi$
$642$	0	0
$643$	10888.0i	0.667777i	0.942613	+	0.333889i	$0.108361\pi$
$643$	10888.0i	0.667777i	−0.942613	+	0.333889i	$0.891639\pi$
$644$	−2640.00	−0.161538
$645$	0	0
$646$	−2184.00	−0.133016
$647$	30384.0i	1.84624i	0.384510	+	0.923121i	$0.374370\pi$
$647$	30384.0i	1.84624i	−0.384510	+	0.923121i	$0.625630\pi$
$648$	0	0
$649$	−26784.0	−1.61998
$650$	0	0
$651$	0	0
$652$	− 12940.0i	− 0.777254i
$653$	7104.00i	0.425729i	0.977082	+	0.212864i	$0.0682793\pi$
$653$	7104.00i	0.425729i	−0.977082	+	0.212864i	$0.931721\pi$
$654$	0	0
$655$	0	0
$656$	−960.000	−0.0571367
$657$	0	0
$658$	− 13200.0i	− 0.782051i
$659$	−24180.0	−1.42932	−0.714658	−	0.699474i	$-0.753418\pi$
$659$	−24180.0	−1.42932	−0.714658	+	0.699474i	$0.753418\pi$
$660$	0	0
$661$	15662.0	0.921605	0.460803	−	0.887503i	$-0.347562\pi$
$661$	15662.0	0.921605	0.460803	+	0.887503i	$0.347562\pi$
$662$	19328.0i	1.13475i
$663$	0	0
$664$	5568.00	0.325422
$665$	0	0
$666$	0	0
$667$	− 16560.0i	− 0.961328i
$668$	− 11088.0i	− 0.642227i
$669$	0	0
$670$	0	0
$671$	−6012.00	−0.345888
$672$	0	0
$673$	6622.00i	0.379286i	0.981853	+	0.189643i	$0.0607330\pi$
$673$	6622.00i	0.379286i	−0.981853	+	0.189643i	$0.939267\pi$
$674$	15274.0	0.872897
$675$	0	0
$676$	−7632.00	−0.434228
$677$	− 3468.00i	− 0.196877i	−0.995143	−	0.0984387i	$-0.968615\pi$
$677$	− 3468.00i	− 0.196877i	0.995143	−	0.0984387i	$-0.0313849\pi$
$678$	0	0
$679$	12089.0	0.683260
$680$	0	0
$681$	0	0
$682$	13752.0i	0.772128i
$683$	19344.0i	1.08372i	0.840470	+	0.541858i	$0.182279\pi$
$683$	19344.0i	1.08372i	−0.840470	+	0.541858i	$0.817721\pi$
$684$	0	0
$685$	0	0
$686$	12430.0	0.691807
$687$	0	0
$688$	784.000i	0.0434444i
$689$	−10404.0	−0.575270
$690$	0	0
$691$	−9736.00	−0.535998	−0.267999	−	0.963419i	$-0.586362\pi$
$691$	−9736.00	−0.535998	−0.267999	+	0.963419i	$0.586362\pi$
$692$	336.000i	0.0184578i
$693$	0	0
$694$	−11280.0	−0.616978
$695$	0	0
$696$	0	0
$697$	720.000i	0.0391276i
$698$	14764.0i	0.800610i
$699$	0	0
$700$	0	0
$701$	−3012.00	−0.162285	−0.0811424	−	0.996703i	$-0.525857\pi$
$701$	−3012.00	−0.162285	−0.0811424	+	0.996703i	$0.525857\pi$
$702$	0	0
$703$	23114.0i	1.24006i
$704$	2304.00	0.123346
$705$	0	0
$706$	−21480.0	−1.14506
$707$	− 4884.00i	− 0.259804i
$708$	0	0
$709$	23851.0	1.26339	0.631695	−	0.775217i	$-0.282360\pi$
$709$	23851.0	1.26339	0.631695	+	0.775217i	$0.282360\pi$
$710$	0	0
$711$	0	0
$712$	9984.00i	0.525514i
$713$	− 11460.0i	− 0.601936i
$714$	0	0
$715$	0	0
$716$	10944.0	0.571224
$717$	0	0
$718$	− 18552.0i	− 0.964282i
$719$	−5916.00	−0.306856	−0.153428	−	0.988160i	$-0.549031\pi$
$719$	−5916.00	−0.306856	−0.153428	+	0.988160i	$0.549031\pi$
$720$	0	0
$721$	−10076.0	−0.520457
$722$	− 2844.00i	− 0.146597i
$723$	0	0
$724$	−17588.0	−0.902835
$725$	0	0
$726$	0	0
$727$	− 27685.0i	− 1.41235i	−0.708036	−	0.706176i	$-0.750419\pi$
$727$	− 27685.0i	− 1.41235i	0.708036	−	0.706176i	$-0.249581\pi$
$728$	1496.00i	0.0761613i
$729$	0	0
$730$	0	0
$731$	588.000	0.0297510
$732$	0	0
$733$	− 10154.0i	− 0.511660i	−0.966722	−	0.255830i	$-0.917651\pi$
$733$	− 10154.0i	− 0.511660i	0.966722	−	0.255830i	$-0.0823487\pi$
$734$	−14774.0	−0.742940
$735$	0	0
$736$	−1920.00	−0.0961578
$737$	16452.0i	0.822276i
$738$	0	0
$739$	13840.0	0.688921	0.344461	−	0.938801i	$-0.388062\pi$
$739$	13840.0	0.688921	0.344461	+	0.938801i	$0.388062\pi$
$740$	0	0
$741$	0	0
$742$	− 13464.0i	− 0.666144i
$743$	− 14160.0i	− 0.699166i	−0.936905	−	0.349583i	$-0.886323\pi$
$743$	− 14160.0i	− 0.699166i	0.936905	−	0.349583i	$-0.113677\pi$
$744$	0	0
$745$	0	0
$746$	−20518.0	−1.00699
$747$	0	0
$748$	− 1728.00i	− 0.0844678i
$749$	−2244.00	−0.109471
$750$	0	0
$751$	36740.0	1.78517	0.892584	−	0.450881i	$-0.148890\pi$
$751$	36740.0	1.78517	0.892584	+	0.450881i	$0.148890\pi$
$752$	− 9600.00i	− 0.465527i
$753$	0	0
$754$	−9384.00	−0.453243
$755$	0	0
$756$	0	0
$757$	8285.00i	0.397785i	0.980021	+	0.198893i	$0.0637345\pi$
$757$	8285.00i	0.397785i	−0.980021	+	0.198893i	$0.936265\pi$
$758$	15310.0i	0.733620i
$759$	0	0
$760$	0	0
$761$	−4656.00	−0.221787	−0.110893	−	0.993832i	$-0.535371\pi$
$761$	−4656.00	−0.221787	−0.110893	+	0.993832i	$0.535371\pi$
$762$	0	0
$763$	10637.0i	0.504699i
$764$	−12432.0	−0.588709
$765$	0	0
$766$	−1200.00	−0.0566028
$767$	− 12648.0i	− 0.595427i
$768$	0	0
$769$	21169.0	0.992684	0.496342	−	0.868127i	$-0.334676\pi$
$769$	21169.0	0.992684	0.496342	+	0.868127i	$0.334676\pi$
$770$	0	0
$771$	0	0
$772$	10460.0i	0.487647i
$773$	31572.0i	1.46904i	0.678588	+	0.734519i	$0.262592\pi$
$773$	31572.0i	1.46904i	−0.678588	+	0.734519i	$0.737408\pi$
$774$	0	0
$775$	0	0
$776$	8792.00	0.406720
$777$	0	0
$778$	− 8136.00i	− 0.374923i
$779$	−5460.00	−0.251123
$780$	0	0
$781$	21168.0	0.969847
$782$	1440.00i	0.0658495i
$783$	0	0
$784$	3552.00	0.161808
$785$	0	0
$786$	0	0
$787$	16781.0i	0.760074i	0.924971	+	0.380037i	$0.124089\pi$
$787$	16781.0i	0.760074i	−0.924971	+	0.380037i	$0.875911\pi$
$788$	− 14496.0i	− 0.655328i
$789$	0	0
$790$	0	0
$791$	−7392.00	−0.332275
$792$	0	0
$793$	− 2839.00i	− 0.127132i
$794$	25294.0	1.13054
$795$	0	0
$796$	−7276.00	−0.323984
$797$	216.000i	0.00959989i	0.999988	+	0.00479995i	$0.00152788\pi$
$797$	216.000i	0.00959989i	−0.999988	+	0.00479995i	$0.998472\pi$
$798$	0	0
$799$	−7200.00	−0.318796
$800$	0	0
$801$	0	0
$802$	19848.0i	0.873887i
$803$	− 34920.0i	− 1.53462i
$804$	0	0
$805$	0	0
$806$	−6494.00	−0.283798
$807$	0	0
$808$	− 3552.00i	− 0.154652i
$809$	17484.0	0.759833	0.379916	−	0.925021i	$-0.375953\pi$
$809$	17484.0	0.759833	0.379916	+	0.925021i	$0.375953\pi$
$810$	0	0
$811$	−28447.0	−1.23170	−0.615850	−	0.787863i	$-0.711187\pi$
$811$	−28447.0	−1.23170	−0.615850	+	0.787863i	$0.711187\pi$
$812$	− 12144.0i	− 0.524841i
$813$	0	0
$814$	−18288.0	−0.787462
$815$	0	0
$816$	0	0
$817$	4459.00i	0.190943i
$818$	− 21806.0i	− 0.932065i
$819$	0	0
$820$	0	0
$821$	15120.0	0.642743	0.321371	−	0.946953i	$-0.395856\pi$
$821$	15120.0	0.642743	0.321371	+	0.946953i	$0.395856\pi$
$822$	0	0
$823$	43957.0i	1.86178i	0.365300	+	0.930890i	$0.380966\pi$
$823$	43957.0i	1.86178i	−0.365300	+	0.930890i	$0.619034\pi$
$824$	−7328.00	−0.309809
$825$	0	0
$826$	16368.0	0.689486
$827$	− 23892.0i	− 1.00460i	−0.864693	−	0.502301i	$-0.832487\pi$
$827$	− 23892.0i	− 1.00460i	0.864693	−	0.502301i	$-0.167513\pi$
$828$	0	0
$829$	12166.0	0.509702	0.254851	−	0.966980i	$-0.417974\pi$
$829$	12166.0	0.509702	0.254851	+	0.966980i	$0.417974\pi$
$830$	0	0
$831$	0	0
$832$	1088.00i	0.0453361i
$833$	− 2664.00i	− 0.110807i
$834$	0	0
$835$	0	0
$836$	13104.0	0.542119
$837$	0	0
$838$	− 29592.0i	− 1.21986i
$839$	25812.0	1.06213	0.531066	−	0.847330i	$-0.321792\pi$
$839$	25812.0	1.06213	0.531066	+	0.847330i	$0.321792\pi$
$840$	0	0
$841$	51787.0	2.12338
$842$	9212.00i	0.377039i
$843$	0	0
$844$	7996.00	0.326106
$845$	0	0
$846$	0	0
$847$	− 385.000i	− 0.0156184i
$848$	− 9792.00i	− 0.396531i
$849$	0	0
$850$	0	0
$851$	15240.0	0.613890
$852$	0	0
$853$	2689.00i	0.107936i	0.998543	+	0.0539681i	$0.0171869\pi$
$853$	2689.00i	0.107936i	−0.998543	+	0.0539681i	$0.982813\pi$
$854$	3674.00	0.147215
$855$	0	0
$856$	−1632.00	−0.0651643
$857$	− 48456.0i	− 1.93142i	−0.259626	−	0.965709i	$-0.583599\pi$
$857$	− 48456.0i	− 1.93142i	0.259626	−	0.965709i	$-0.416401\pi$
$858$	0	0
$859$	12760.0	0.506828	0.253414	−	0.967358i	$-0.418446\pi$
$859$	12760.0	0.506828	0.253414	+	0.967358i	$0.418446\pi$
$860$	0	0
$861$	0	0
$862$	27720.0i	1.09530i
$863$	10008.0i	0.394758i	0.980327	+	0.197379i	$0.0632430\pi$
$863$	10008.0i	0.394758i	−0.980327	+	0.197379i	$0.936757\pi$
$864$	0	0
$865$	0	0
$866$	−16102.0	−0.631834
$867$	0	0
$868$	− 8404.00i	− 0.328629i
$869$	5904.00	0.230471
$870$	0	0
$871$	−7769.00	−0.302230
$872$	7736.00i	0.300429i
$873$	0	0
$874$	−10920.0	−0.422625
$875$	0	0
$876$	0	0
$877$	12251.0i	0.471707i	0.971789	+	0.235853i	$0.0757885\pi$
$877$	12251.0i	0.471707i	−0.971789	+	0.235853i	$0.924211\pi$
$878$	4174.00i	0.160439i
$879$	0	0
$880$	0	0
$881$	−16344.0	−0.625021	−0.312510	−	0.949914i	$-0.601170\pi$
$881$	−16344.0	−0.625021	−0.312510	+	0.949914i	$0.601170\pi$
$882$	0	0
$883$	− 42227.0i	− 1.60935i	−0.593719	−	0.804673i	$-0.702341\pi$
$883$	− 42227.0i	− 1.60935i	0.593719	−	0.804673i	$-0.297659\pi$
$884$	816.000	0.0310464
$885$	0	0
$886$	−26736.0	−1.01378
$887$	13896.0i	0.526023i	0.964793	+	0.263011i	$0.0847156\pi$
$887$	13896.0i	0.526023i	−0.964793	+	0.263011i	$0.915284\pi$
$888$	0	0
$889$	21164.0	0.798445
$890$	0	0
$891$	0	0
$892$	− 1972.00i	− 0.0740218i
$893$	− 54600.0i	− 2.04605i
$894$	0	0
$895$	0	0
$896$	−1408.00	−0.0524977
$897$	0	0
$898$	− 10704.0i	− 0.397770i
$899$	52716.0	1.95570
$900$	0	0
$901$	−7344.00	−0.271547
$902$	− 4320.00i	− 0.159468i
$903$	0	0
$904$	−5376.00	−0.197791
$905$	0	0
$906$	0	0
$907$	− 1240.00i	− 0.0453953i	−0.999742	−	0.0226976i	$-0.992774\pi$
$907$	− 1240.00i	− 0.0453953i	0.999742	−	0.0226976i	$-0.00722550\pi$
$908$	1200.00i	0.0438584i
$909$	0	0
$910$	0	0
$911$	6360.00	0.231302	0.115651	−	0.993290i	$-0.463105\pi$
$911$	6360.00	0.231302	0.115651	+	0.993290i	$0.463105\pi$
$912$	0	0
$913$	25056.0i	0.908250i
$914$	27436.0	0.992891
$915$	0	0
$916$	−316.000	−0.0113984
$917$	8316.00i	0.299475i
$918$	0	0
$919$	2143.00	0.0769217	0.0384609	−	0.999260i	$-0.487755\pi$
$919$	2143.00	0.0769217	0.0384609	+	0.999260i	$0.487755\pi$
$920$	0	0
$921$	0	0
$922$	− 23736.0i	− 0.847835i
$923$	9996.00i	0.356471i
$924$	0	0
$925$	0	0
$926$	−13144.0	−0.466456
$927$	0	0
$928$	− 8832.00i	− 0.312419i
$929$	29124.0	1.02855	0.514277	−	0.857624i	$-0.328060\pi$
$929$	29124.0	1.02855	0.514277	+	0.857624i	$0.328060\pi$
$930$	0	0
$931$	20202.0	0.711164
$932$	− 24432.0i	− 0.858688i
$933$	0	0
$934$	37272.0	1.30576
$935$	0	0
$936$	0	0
$937$	24059.0i	0.838819i	0.907797	+	0.419409i	$0.137763\pi$
$937$	24059.0i	0.838819i	−0.907797	+	0.419409i	$0.862237\pi$
$938$	− 10054.0i	− 0.349973i
$939$	0	0
$940$	0	0
$941$	−47088.0	−1.63127	−0.815635	−	0.578567i	$-0.803612\pi$
$941$	−47088.0	−1.63127	−0.815635	+	0.578567i	$0.803612\pi$
$942$	0	0
$943$	3600.00i	0.124318i
$944$	11904.0	0.410426
$945$	0	0
$946$	−3528.00	−0.121253
$947$	23148.0i	0.794307i	0.917752	+	0.397154i	$0.130002\pi$
$947$	23148.0i	0.794307i	−0.917752	+	0.397154i	$0.869998\pi$
$948$	0	0
$949$	16490.0	0.564055
$950$	0	0
$951$	0	0
$952$	1056.00i	0.0359508i
$953$	− 27600.0i	− 0.938144i	−0.883160	−	0.469072i	$-0.844588\pi$
$953$	− 27600.0i	− 0.938144i	0.883160	−	0.469072i	$-0.155412\pi$
$954$	0	0
$955$	0	0
$956$	11472.0	0.388108
$957$	0	0
$958$	10512.0i	0.354517i
$959$	9768.00	0.328911
$960$	0	0
$961$	6690.00	0.224564
$962$	− 8636.00i	− 0.289434i
$963$	0	0
$964$	−10820.0	−0.361503
$965$	0	0
$966$	0	0
$967$	7436.00i	0.247286i	0.992327	+	0.123643i	$0.0394578\pi$
$967$	7436.00i	0.247286i	−0.992327	+	0.123643i	$0.960542\pi$
$968$	− 280.000i	− 0.00929705i
$969$	0	0
$970$	0	0
$971$	−27264.0	−0.901075	−0.450537	−	0.892758i	$-0.648768\pi$
$971$	−27264.0	−0.901075	−0.450537	+	0.892758i	$0.648768\pi$
$972$	0	0
$973$	1760.00i	0.0579887i
$974$	−19982.0	−0.657356
$975$	0	0
$976$	2672.00	0.0876318
$977$	− 46632.0i	− 1.52701i	−0.645801	−	0.763506i	$-0.723477\pi$
$977$	− 46632.0i	− 1.52701i	0.645801	−	0.763506i	$-0.276523\pi$
$978$	0	0
$979$	−44928.0	−1.46671
$980$	0	0
$981$	0	0
$982$	− 2112.00i	− 0.0686320i
$983$	− 15216.0i	− 0.493708i	−0.969053	−	0.246854i	$-0.920603\pi$
$983$	− 15216.0i	− 0.493708i	0.969053	−	0.246854i	$-0.0793968\pi$
$984$	0	0
$985$	0	0
$986$	−6624.00	−0.213946
$987$	0	0
$988$	6188.00i	0.199258i
$989$	2940.00	0.0945264
$990$	0	0
$991$	39047.0	1.25163	0.625817	−	0.779970i	$-0.284766\pi$
$991$	39047.0	1.25163	0.625817	+	0.779970i	$0.284766\pi$
$992$	− 6112.00i	− 0.195621i
$993$	0	0
$994$	−12936.0	−0.412782
$995$	0	0
$996$	0	0
$997$	− 46258.0i	− 1.46941i	−0.678385	−	0.734707i	$-0.737320\pi$
$997$	− 46258.0i	− 1.46941i	0.678385	−	0.734707i	$-0.262680\pi$
$998$	24658.0i	0.782100i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.c.b.199.1		2
3.2	odd	2		450.4.c.i.199.2		2
5.2	odd	4		450.4.a.n.1.1	yes	1
5.3	odd	4		450.4.a.g.1.1	yes	1
5.4	even	2	inner	450.4.c.b.199.2		2
15.2	even	4		450.4.a.d.1.1	✓	1
15.8	even	4		450.4.a.s.1.1	yes	1
15.14	odd	2		450.4.c.i.199.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.4.a.d.1.1	✓	1	15.2	even	4
450.4.a.g.1.1	yes	1	5.3	odd	4
450.4.a.n.1.1	yes	1	5.2	odd	4
450.4.a.s.1.1	yes	1	15.8	even	4
450.4.c.b.199.1		2	1.1	even	1	trivial
450.4.c.b.199.2		2	5.4	even	2	inner
450.4.c.i.199.1		2	15.14	odd	2
450.4.c.i.199.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} +11.0000i q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q-2.00000i q^{2} -4.00000 q^{4} +11.0000i q^{7} +8.00000i q^{8} -36.0000 q^{11} -17.0000i q^{13} +22.0000 q^{14} +16.0000 q^{16} -12.0000i q^{17} +91.0000 q^{19} +72.0000i q^{22} -60.0000i q^{23} -34.0000 q^{26} -44.0000i q^{28} +276.000 q^{29} +191.000 q^{31} -32.0000i q^{32} -24.0000 q^{34} +254.000i q^{37} -182.000i q^{38} -60.0000 q^{41} +49.0000i q^{43} +144.000 q^{44} -120.000 q^{46} -600.000i q^{47} +222.000 q^{49} +68.0000i q^{52} -612.000i q^{53} -88.0000 q^{56} -552.000i q^{58} +744.000 q^{59} +167.000 q^{61} -382.000i q^{62} -64.0000 q^{64} -457.000i q^{67} +48.0000i q^{68} -588.000 q^{71} +970.000i q^{73} +508.000 q^{74} -364.000 q^{76} -396.000i q^{77} -164.000 q^{79} +120.000i q^{82} -696.000i q^{83} +98.0000 q^{86} -288.000i q^{88} +1248.00 q^{89} +187.000 q^{91} +240.000i q^{92} -1200.00 q^{94} -1099.00i q^{97} -444.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} - 72 q^{11} + 44 q^{14} + 32 q^{16} + 182 q^{19} - 68 q^{26} + 552 q^{29} + 382 q^{31} - 48 q^{34} - 120 q^{41} + 288 q^{44} - 240 q^{46} + 444 q^{49} - 176 q^{56} + 1488 q^{59} + 334 q^{61} - 128 q^{64} - 1176 q^{71} + 1016 q^{74} - 728 q^{76} - 328 q^{79} + 196 q^{86} + 2496 q^{89} + 374 q^{91} - 2400 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 - 72 * q^11 + 44 * q^14 + 32 * q^16 + 182 * q^19 - 68 * q^26 + 552 * q^29 + 382 * q^31 - 48 * q^34 - 120 * q^41 + 288 * q^44 - 240 * q^46 + 444 * q^49 - 176 * q^56 + 1488 * q^59 + 334 * q^61 - 128 * q^64 - 1176 * q^71 + 1016 * q^74 - 728 * q^76 - 328 * q^79 + 196 * q^86 + 2496 * q^89 + 374 * q^91 - 2400 * q^94

Introduction

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