LMFDB - Embedded newform 450.4.c.k.199.2

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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.4.c.k.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} +32.0000i q^{7} -8.00000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} -4.00000 q^{4} +32.0000i q^{7} -8.00000i q^{8} +60.0000 q^{11} +34.0000i q^{13} -64.0000 q^{14} +16.0000 q^{16} -42.0000i q^{17} +76.0000 q^{19} +120.000i q^{22} -68.0000 q^{26} -128.000i q^{28} +6.00000 q^{29} -232.000 q^{31} +32.0000i q^{32} +84.0000 q^{34} +134.000i q^{37} +152.000i q^{38} -234.000 q^{41} +412.000i q^{43} -240.000 q^{44} +360.000i q^{47} -681.000 q^{49} -136.000i q^{52} +222.000i q^{53} +256.000 q^{56} +12.0000i q^{58} +660.000 q^{59} -490.000 q^{61} -464.000i q^{62} -64.0000 q^{64} +812.000i q^{67} +168.000i q^{68} -120.000 q^{71} -746.000i q^{73} -268.000 q^{74} -304.000 q^{76} +1920.00i q^{77} -152.000 q^{79} -468.000i q^{82} -804.000i q^{83} -824.000 q^{86} -480.000i q^{88} -678.000 q^{89} -1088.00 q^{91} -720.000 q^{94} +194.000i q^{97} -1362.00i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} + 120 q^{11} - 128 q^{14} + 32 q^{16} + 152 q^{19} - 136 q^{26} + 12 q^{29} - 464 q^{31} + 168 q^{34} - 468 q^{41} - 480 q^{44} - 1362 q^{49} + 512 q^{56} + 1320 q^{59} - 980 q^{61} - 128 q^{64} - 240 q^{71} - 536 q^{74} - 608 q^{76} - 304 q^{79} - 1648 q^{86} - 1356 q^{89} - 2176 q^{91} - 1440 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 + 120 * q^11 - 128 * q^14 + 32 * q^16 + 152 * q^19 - 136 * q^26 + 12 * q^29 - 464 * q^31 + 168 * q^34 - 468 * q^41 - 480 * q^44 - 1362 * q^49 + 512 * q^56 + 1320 * q^59 - 980 * q^61 - 128 * q^64 - 240 * q^71 - 536 * q^74 - 608 * q^76 - 304 * q^79 - 1648 * q^86 - 1356 * q^89 - 2176 * q^91 - 1440 * 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
$7$	32.0000i	1.72784i	0.503631	+	0.863919i	$0.331997\pi$
$7$	32.0000i	1.72784i	−0.503631	+	0.863919i	$0.668003\pi$
$8$	− 8.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	60.0000	1.64461	0.822304	−	0.569049i	$-0.192689\pi$
$11$	60.0000	1.64461	0.822304	+	0.569049i	$0.192689\pi$
$12$	0	0
$13$	34.0000i	0.725377i	0.931910	+	0.362689i	$0.118141\pi$
$13$	34.0000i	0.725377i	−0.931910	+	0.362689i	$0.881859\pi$
$14$	−64.0000	−1.22177
$15$	0	0
$16$	16.0000	0.250000
$17$	− 42.0000i	− 0.599206i	−0.954064	−	0.299603i	$-0.903146\pi$
$17$	− 42.0000i	− 0.599206i	0.954064	−	0.299603i	$-0.0968542\pi$
$18$	0	0
$19$	76.0000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	76.0000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	120.000i	1.16291i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	−68.0000	−0.512919
$27$	0	0
$28$	− 128.000i	− 0.863919i
$29$	6.00000	0.0384197	0.0192099	−	0.999815i	$-0.493885\pi$
$29$	6.00000	0.0384197	0.0192099	+	0.999815i	$0.493885\pi$
$30$	0	0
$31$	−232.000	−1.34414	−0.672071	−	0.740486i	$-0.734595\pi$
$31$	−232.000	−1.34414	−0.672071	+	0.740486i	$0.734595\pi$
$32$	32.0000i	0.176777i
$33$	0	0
$34$	84.0000	0.423702
$35$	0	0
$36$	0	0
$37$	134.000i	0.595391i	0.954661	+	0.297695i	$0.0962180\pi$
$37$	134.000i	0.595391i	−0.954661	+	0.297695i	$0.903782\pi$
$38$	152.000i	0.648886i
$39$	0	0
$40$	0	0
$41$	−234.000	−0.891333	−0.445667	−	0.895199i	$-0.647033\pi$
$41$	−234.000	−0.891333	−0.445667	+	0.895199i	$0.647033\pi$
$42$	0	0
$43$	412.000i	1.46115i	0.682833	+	0.730575i	$0.260748\pi$
$43$	412.000i	1.46115i	−0.682833	+	0.730575i	$0.739252\pi$
$44$	−240.000	−0.822304
$45$	0	0
$46$	0	0
$47$	360.000i	1.11726i	0.829416	+	0.558632i	$0.188674\pi$
$47$	360.000i	1.11726i	−0.829416	+	0.558632i	$0.811326\pi$
$48$	0	0
$49$	−681.000	−1.98542
$50$	0	0
$51$	0	0
$52$	− 136.000i	− 0.362689i
$53$	222.000i	0.575359i	0.957727	+	0.287680i	$0.0928838\pi$
$53$	222.000i	0.575359i	−0.957727	+	0.287680i	$0.907116\pi$
$54$	0	0
$55$	0	0
$56$	256.000	0.610883
$57$	0	0
$58$	12.0000i	0.0271668i
$59$	660.000	1.45635	0.728175	−	0.685391i	$-0.240369\pi$
$59$	660.000	1.45635	0.728175	+	0.685391i	$0.240369\pi$
$60$	0	0
$61$	−490.000	−1.02849	−0.514246	−	0.857642i	$-0.671928\pi$
$61$	−490.000	−1.02849	−0.514246	+	0.857642i	$0.671928\pi$
$62$	− 464.000i	− 0.950453i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	812.000i	1.48062i	0.672265	+	0.740310i	$0.265321\pi$
$67$	812.000i	1.48062i	−0.672265	+	0.740310i	$0.734679\pi$
$68$	168.000i	0.299603i
$69$	0	0
$70$	0	0
$71$	−120.000	−0.200583	−0.100291	−	0.994958i	$-0.531978\pi$
$71$	−120.000	−0.200583	−0.100291	+	0.994958i	$0.531978\pi$
$72$	0	0
$73$	− 746.000i	− 1.19606i	−0.801472	−	0.598032i	$-0.795949\pi$
$73$	− 746.000i	− 1.19606i	0.801472	−	0.598032i	$-0.204051\pi$
$74$	−268.000	−0.421005
$75$	0	0
$76$	−304.000	−0.458831
$77$	1920.00i	2.84161i
$78$	0	0
$79$	−152.000	−0.216473	−0.108236	−	0.994125i	$-0.534520\pi$
$79$	−152.000	−0.216473	−0.108236	+	0.994125i	$0.534520\pi$
$80$	0	0
$81$	0	0
$82$	− 468.000i	− 0.630268i
$83$	− 804.000i	− 1.06326i	−0.846977	−	0.531629i	$-0.821580\pi$
$83$	− 804.000i	− 1.06326i	0.846977	−	0.531629i	$-0.178420\pi$
$84$	0	0
$85$	0	0
$86$	−824.000	−1.03319
$87$	0	0
$88$	− 480.000i	− 0.581456i
$89$	−678.000	−0.807504	−0.403752	−	0.914868i	$-0.632294\pi$
$89$	−678.000	−0.807504	−0.403752	+	0.914868i	$0.632294\pi$
$90$	0	0
$91$	−1088.00	−1.25333
$92$	0	0
$93$	0	0
$94$	−720.000	−0.790025
$95$	0	0
$96$	0	0
$97$	194.000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	194.000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 1362.00i	− 1.40391i
$99$	0	0
$100$	0	0
$101$	−798.000	−0.786178	−0.393089	−	0.919500i	$-0.628594\pi$
$101$	−798.000	−0.786178	−0.393089	+	0.919500i	$0.628594\pi$
$102$	0	0
$103$	− 1088.00i	− 1.04081i	−0.853918	−	0.520407i	$-0.825780\pi$
$103$	− 1088.00i	− 1.04081i	0.853918	−	0.520407i	$-0.174220\pi$
$104$	272.000	0.256460
$105$	0	0
$106$	−444.000	−0.406840
$107$	− 1716.00i	− 1.55039i	−0.631721	−	0.775196i	$-0.717651\pi$
$107$	− 1716.00i	− 1.55039i	0.631721	−	0.775196i	$-0.282349\pi$
$108$	0	0
$109$	970.000	0.852378	0.426189	−	0.904634i	$-0.359856\pi$
$109$	970.000	0.852378	0.426189	+	0.904634i	$0.359856\pi$
$110$	0	0
$111$	0	0
$112$	512.000i	0.431959i
$113$	426.000i	0.354643i	0.984153	+	0.177322i	$0.0567433\pi$
$113$	426.000i	0.354643i	−0.984153	+	0.177322i	$0.943257\pi$
$114$	0	0
$115$	0	0
$116$	−24.0000	−0.0192099
$117$	0	0
$118$	1320.00i	1.02980i
$119$	1344.00	1.03533
$120$	0	0
$121$	2269.00	1.70473
$122$	− 980.000i	− 0.727254i
$123$	0	0
$124$	928.000	0.672071
$125$	0	0
$126$	0	0
$127$	200.000i	0.139741i	0.997556	+	0.0698706i	$0.0222586\pi$
$127$	200.000i	0.139741i	−0.997556	+	0.0698706i	$0.977741\pi$
$128$	− 128.000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−60.0000	−0.0400170	−0.0200085	−	0.999800i	$-0.506369\pi$
$131$	−60.0000	−0.0400170	−0.0200085	+	0.999800i	$0.506369\pi$
$132$	0	0
$133$	2432.00i	1.58557i
$134$	−1624.00	−1.04696
$135$	0	0
$136$	−336.000	−0.211851
$137$	− 642.000i	− 0.400363i	−0.979759	−	0.200182i	$-0.935847\pi$
$137$	− 642.000i	− 0.400363i	0.979759	−	0.200182i	$-0.0641532\pi$
$138$	0	0
$139$	2836.00	1.73055	0.865275	−	0.501298i	$-0.167144\pi$
$139$	2836.00	1.73055	0.865275	+	0.501298i	$0.167144\pi$
$140$	0	0
$141$	0	0
$142$	− 240.000i	− 0.141833i
$143$	2040.00i	1.19296i
$144$	0	0
$145$	0	0
$146$	1492.00	0.845745
$147$	0	0
$148$	− 536.000i	− 0.297695i
$149$	−1554.00	−0.854420	−0.427210	−	0.904152i	$-0.640504\pi$
$149$	−1554.00	−0.854420	−0.427210	+	0.904152i	$0.640504\pi$
$150$	0	0
$151$	−2272.00	−1.22446	−0.612228	−	0.790682i	$-0.709726\pi$
$151$	−2272.00	−1.22446	−0.612228	+	0.790682i	$0.709726\pi$
$152$	− 608.000i	− 0.324443i
$153$	0	0
$154$	−3840.00	−2.00932
$155$	0	0
$156$	0	0
$157$	1694.00i	0.861120i	0.902562	+	0.430560i	$0.141684\pi$
$157$	1694.00i	0.861120i	−0.902562	+	0.430560i	$0.858316\pi$
$158$	− 304.000i	− 0.153069i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	52.0000i	0.0249874i	0.999922	+	0.0124937i	$0.00397698\pi$
$163$	52.0000i	0.0249874i	−0.999922	+	0.0124937i	$0.996023\pi$
$164$	936.000	0.445667
$165$	0	0
$166$	1608.00	0.751837
$167$	1200.00i	0.556041i	0.960575	+	0.278020i	$0.0896783\pi$
$167$	1200.00i	0.556041i	−0.960575	+	0.278020i	$0.910322\pi$
$168$	0	0
$169$	1041.00	0.473828
$170$	0	0
$171$	0	0
$172$	− 1648.00i	− 0.730575i
$173$	54.0000i	0.0237315i	0.999930	+	0.0118657i	$0.00377707\pi$
$173$	54.0000i	0.0237315i	−0.999930	+	0.0118657i	$0.996223\pi$
$174$	0	0
$175$	0	0
$176$	960.000	0.411152
$177$	0	0
$178$	− 1356.00i	− 0.570992i
$179$	876.000	0.365784	0.182892	−	0.983133i	$-0.441454\pi$
$179$	876.000	0.365784	0.182892	+	0.983133i	$0.441454\pi$
$180$	0	0
$181$	3854.00	1.58268	0.791341	−	0.611375i	$-0.209383\pi$
$181$	3854.00	1.58268	0.791341	+	0.611375i	$0.209383\pi$
$182$	− 2176.00i	− 0.886241i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2520.00i	− 0.985458i
$188$	− 1440.00i	− 0.558632i
$189$	0	0
$190$	0	0
$191$	2784.00	1.05468	0.527338	−	0.849656i	$-0.323190\pi$
$191$	2784.00	1.05468	0.527338	+	0.849656i	$0.323190\pi$
$192$	0	0
$193$	− 914.000i	− 0.340887i	−0.985367	−	0.170443i	$-0.945480\pi$
$193$	− 914.000i	− 0.340887i	0.985367	−	0.170443i	$-0.0545200\pi$
$194$	−388.000	−0.143592
$195$	0	0
$196$	2724.00	0.992711
$197$	5202.00i	1.88136i	0.339300	+	0.940678i	$0.389810\pi$
$197$	5202.00i	1.88136i	−0.339300	+	0.940678i	$0.610190\pi$
$198$	0	0
$199$	−3152.00	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$199$	−3152.00	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$200$	0	0
$201$	0	0
$202$	− 1596.00i	− 0.555912i
$203$	192.000i	0.0663830i
$204$	0	0
$205$	0	0
$206$	2176.00	0.735967
$207$	0	0
$208$	544.000i	0.181344i
$209$	4560.00	1.50920
$210$	0	0
$211$	740.000	0.241439	0.120720	−	0.992687i	$-0.461480\pi$
$211$	740.000	0.241439	0.120720	+	0.992687i	$0.461480\pi$
$212$	− 888.000i	− 0.287680i
$213$	0	0
$214$	3432.00	1.09629
$215$	0	0
$216$	0	0
$217$	− 7424.00i	− 2.32246i
$218$	1940.00i	0.602722i
$219$	0	0
$220$	0	0
$221$	1428.00	0.434650
$222$	0	0
$223$	520.000i	0.156151i	0.996947	+	0.0780757i	$0.0248776\pi$
$223$	520.000i	0.156151i	−0.996947	+	0.0780757i	$0.975122\pi$
$224$	−1024.00	−0.305441
$225$	0	0
$226$	−852.000	−0.250771
$227$	− 396.000i	− 0.115786i	−0.998323	−	0.0578930i	$-0.981562\pi$
$227$	− 396.000i	− 0.115786i	0.998323	−	0.0578930i	$-0.0184382\pi$
$228$	0	0
$229$	1330.00	0.383794	0.191897	−	0.981415i	$-0.438536\pi$
$229$	1330.00	0.383794	0.191897	+	0.981415i	$0.438536\pi$
$230$	0	0
$231$	0	0
$232$	− 48.0000i	− 0.0135834i
$233$	4866.00i	1.36816i	0.729405	+	0.684082i	$0.239797\pi$
$233$	4866.00i	1.36816i	−0.729405	+	0.684082i	$0.760203\pi$
$234$	0	0
$235$	0	0
$236$	−2640.00	−0.728175
$237$	0	0
$238$	2688.00i	0.732089i
$239$	−1824.00	−0.493660	−0.246830	−	0.969059i	$-0.579389\pi$
$239$	−1824.00	−0.493660	−0.246830	+	0.969059i	$0.579389\pi$
$240$	0	0
$241$	6482.00	1.73254	0.866270	−	0.499575i	$-0.166511\pi$
$241$	6482.00	1.73254	0.866270	+	0.499575i	$0.166511\pi$
$242$	4538.00i	1.20543i
$243$	0	0
$244$	1960.00	0.514246
$245$	0	0
$246$	0	0
$247$	2584.00i	0.665652i
$248$	1856.00i	0.475226i
$249$	0	0
$250$	0	0
$251$	−1476.00	−0.371172	−0.185586	−	0.982628i	$-0.559418\pi$
$251$	−1476.00	−0.371172	−0.185586	+	0.982628i	$0.559418\pi$
$252$	0	0
$253$	0	0
$254$	−400.000	−0.0988119
$255$	0	0
$256$	256.000	0.0625000
$257$	− 4314.00i	− 1.04708i	−0.852001	−	0.523541i	$-0.824611\pi$
$257$	− 4314.00i	− 1.04708i	0.852001	−	0.523541i	$-0.175389\pi$
$258$	0	0
$259$	−4288.00	−1.02874
$260$	0	0
$261$	0	0
$262$	− 120.000i	− 0.0282963i
$263$	− 5280.00i	− 1.23794i	−0.785414	−	0.618971i	$-0.787550\pi$
$263$	− 5280.00i	− 1.23794i	0.785414	−	0.618971i	$-0.212450\pi$
$264$	0	0
$265$	0	0
$266$	−4864.00	−1.12117
$267$	0	0
$268$	− 3248.00i	− 0.740310i
$269$	5526.00	1.25251	0.626257	−	0.779617i	$-0.284586\pi$
$269$	5526.00	1.25251	0.626257	+	0.779617i	$0.284586\pi$
$270$	0	0
$271$	2024.00	0.453687	0.226844	−	0.973931i	$-0.427159\pi$
$271$	2024.00	0.453687	0.226844	+	0.973931i	$0.427159\pi$
$272$	− 672.000i	− 0.149801i
$273$	0	0
$274$	1284.00	0.283100
$275$	0	0
$276$	0	0
$277$	2054.00i	0.445534i	0.974872	+	0.222767i	$0.0715089\pi$
$277$	2054.00i	0.445534i	−0.974872	+	0.222767i	$0.928491\pi$
$278$	5672.00i	1.22368i
$279$	0	0
$280$	0	0
$281$	7302.00	1.55018	0.775090	−	0.631850i	$-0.217704\pi$
$281$	7302.00	1.55018	0.775090	+	0.631850i	$0.217704\pi$
$282$	0	0
$283$	3724.00i	0.782222i	0.920344	+	0.391111i	$0.127909\pi$
$283$	3724.00i	0.782222i	−0.920344	+	0.391111i	$0.872091\pi$
$284$	480.000	0.100291
$285$	0	0
$286$	−4080.00	−0.843551
$287$	− 7488.00i	− 1.54008i
$288$	0	0
$289$	3149.00	0.640953
$290$	0	0
$291$	0	0
$292$	2984.00i	0.598032i
$293$	− 7218.00i	− 1.43918i	−0.694399	−	0.719591i	$-0.744330\pi$
$293$	− 7218.00i	− 1.43918i	0.694399	−	0.719591i	$-0.255670\pi$
$294$	0	0
$295$	0	0
$296$	1072.00	0.210502
$297$	0	0
$298$	− 3108.00i	− 0.604166i
$299$	0	0
$300$	0	0
$301$	−13184.0	−2.52463
$302$	− 4544.00i	− 0.865821i
$303$	0	0
$304$	1216.00	0.229416
$305$	0	0
$306$	0	0
$307$	2540.00i	0.472200i	0.971729	+	0.236100i	$0.0758693\pi$
$307$	2540.00i	0.472200i	−0.971729	+	0.236100i	$0.924131\pi$
$308$	− 7680.00i	− 1.42081i
$309$	0	0
$310$	0	0
$311$	−1560.00	−0.284436	−0.142218	−	0.989835i	$-0.545423\pi$
$311$	−1560.00	−0.284436	−0.142218	+	0.989835i	$0.545423\pi$
$312$	0	0
$313$	934.000i	0.168667i	0.996438	+	0.0843335i	$0.0268761\pi$
$313$	934.000i	0.168667i	−0.996438	+	0.0843335i	$0.973124\pi$
$314$	−3388.00	−0.608904
$315$	0	0
$316$	608.000	0.108236
$317$	1674.00i	0.296597i	0.988943	+	0.148298i	$0.0473796\pi$
$317$	1674.00i	0.296597i	−0.988943	+	0.148298i	$0.952620\pi$
$318$	0	0
$319$	360.000	0.0631854
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3192.00i	− 0.549869i
$324$	0	0
$325$	0	0
$326$	−104.000	−0.0176688
$327$	0	0
$328$	1872.00i	0.315134i
$329$	−11520.0	−1.93045
$330$	0	0
$331$	−3988.00	−0.662237	−0.331118	−	0.943589i	$-0.607426\pi$
$331$	−3988.00	−0.662237	−0.331118	+	0.943589i	$0.607426\pi$
$332$	3216.00i	0.531629i
$333$	0	0
$334$	−2400.00	−0.393180
$335$	0	0
$336$	0	0
$337$	2.00000i	0 0.000323285i	1.00000		0.000161642i	$5.14524e-5\pi$
$337$	2.00000i	0 0.000323285i	−1.00000		0.000161642i	$0.999949\pi$
$338$	2082.00i	0.335047i
$339$	0	0
$340$	0	0
$341$	−13920.0	−2.21059
$342$	0	0
$343$	− 10816.0i	− 1.70265i
$344$	3296.00	0.516594
$345$	0	0
$346$	−108.000	−0.0167807
$347$	− 1764.00i	− 0.272901i	−0.990647	−	0.136450i	$-0.956431\pi$
$347$	− 1764.00i	− 0.272901i	0.990647	−	0.136450i	$-0.0435694\pi$
$348$	0	0
$349$	−4310.00	−0.661057	−0.330529	−	0.943796i	$-0.607227\pi$
$349$	−4310.00	−0.661057	−0.330529	+	0.943796i	$0.607227\pi$
$350$	0	0
$351$	0	0
$352$	1920.00i	0.290728i
$353$	138.000i	0.0208074i	0.999946	+	0.0104037i	$0.00331165\pi$
$353$	138.000i	0.0208074i	−0.999946	+	0.0104037i	$0.996688\pi$
$354$	0	0
$355$	0	0
$356$	2712.00	0.403752
$357$	0	0
$358$	1752.00i	0.258648i
$359$	−11976.0	−1.76064	−0.880319	−	0.474382i	$-0.842672\pi$
$359$	−11976.0	−1.76064	−0.880319	+	0.474382i	$0.842672\pi$
$360$	0	0
$361$	−1083.00	−0.157895
$362$	7708.00i	1.11913i
$363$	0	0
$364$	4352.00	0.626667
$365$	0	0
$366$	0	0
$367$	9704.00i	1.38023i	0.723699	+	0.690115i	$0.242440\pi$
$367$	9704.00i	1.38023i	−0.723699	+	0.690115i	$0.757560\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7104.00	−0.994128
$372$	0	0
$373$	8122.00i	1.12746i	0.825960	+	0.563728i	$0.190633\pi$
$373$	8122.00i	1.12746i	−0.825960	+	0.563728i	$0.809367\pi$
$374$	5040.00	0.696824
$375$	0	0
$376$	2880.00	0.395012
$377$	204.000i	0.0278688i
$378$	0	0
$379$	−3404.00	−0.461350	−0.230675	−	0.973031i	$-0.574093\pi$
$379$	−3404.00	−0.461350	−0.230675	+	0.973031i	$0.574093\pi$
$380$	0	0
$381$	0	0
$382$	5568.00i	0.745769i
$383$	− 2520.00i	− 0.336204i	−0.985770	−	0.168102i	$-0.946236\pi$
$383$	− 2520.00i	− 0.336204i	0.985770	−	0.168102i	$-0.0537637\pi$
$384$	0	0
$385$	0	0
$386$	1828.00	0.241043
$387$	0	0
$388$	− 776.000i	− 0.101535i
$389$	1566.00	0.204111	0.102056	−	0.994779i	$-0.467458\pi$
$389$	1566.00	0.204111	0.102056	+	0.994779i	$0.467458\pi$
$390$	0	0
$391$	0	0
$392$	5448.00i	0.701953i
$393$	0	0
$394$	−10404.0	−1.33032
$395$	0	0
$396$	0	0
$397$	− 4354.00i	− 0.550431i	−0.961383	−	0.275215i	$-0.911251\pi$
$397$	− 4354.00i	− 0.550431i	0.961383	−	0.275215i	$-0.0887492\pi$
$398$	− 6304.00i	− 0.793947i
$399$	0	0
$400$	0	0
$401$	8046.00	1.00199	0.500995	−	0.865450i	$-0.332967\pi$
$401$	8046.00	1.00199	0.500995	+	0.865450i	$0.332967\pi$
$402$	0	0
$403$	− 7888.00i	− 0.975011i
$404$	3192.00	0.393089
$405$	0	0
$406$	−384.000	−0.0469399
$407$	8040.00i	0.979184i
$408$	0	0
$409$	2806.00	0.339237	0.169618	−	0.985510i	$-0.445747\pi$
$409$	2806.00	0.339237	0.169618	+	0.985510i	$0.445747\pi$
$410$	0	0
$411$	0	0
$412$	4352.00i	0.520407i
$413$	21120.0i	2.51634i
$414$	0	0
$415$	0	0
$416$	−1088.00	−0.128230
$417$	0	0
$418$	9120.00i	1.06716i
$419$	11580.0	1.35017	0.675084	−	0.737741i	$-0.264108\pi$
$419$	11580.0	1.35017	0.675084	+	0.737741i	$0.264108\pi$
$420$	0	0
$421$	−370.000	−0.0428330	−0.0214165	−	0.999771i	$-0.506818\pi$
$421$	−370.000	−0.0428330	−0.0214165	+	0.999771i	$0.506818\pi$
$422$	1480.00i	0.170723i
$423$	0	0
$424$	1776.00	0.203420
$425$	0	0
$426$	0	0
$427$	− 15680.0i	− 1.77707i
$428$	6864.00i	0.775196i
$429$	0	0
$430$	0	0
$431$	−5040.00	−0.563267	−0.281634	−	0.959522i	$-0.590876\pi$
$431$	−5040.00	−0.563267	−0.281634	+	0.959522i	$0.590876\pi$
$432$	0	0
$433$	3742.00i	0.415310i	0.978202	+	0.207655i	$0.0665831\pi$
$433$	3742.00i	0.415310i	−0.978202	+	0.207655i	$0.933417\pi$
$434$	14848.0	1.64223
$435$	0	0
$436$	−3880.00	−0.426189
$437$	0	0
$438$	0	0
$439$	6208.00	0.674924	0.337462	−	0.941339i	$-0.390432\pi$
$439$	6208.00	0.674924	0.337462	+	0.941339i	$0.390432\pi$
$440$	0	0
$441$	0	0
$442$	2856.00i	0.307344i
$443$	− 15564.0i	− 1.66923i	−0.550835	−	0.834614i	$-0.685691\pi$
$443$	− 15564.0i	− 1.66923i	0.550835	−	0.834614i	$-0.314309\pi$
$444$	0	0
$445$	0	0
$446$	−1040.00	−0.110416
$447$	0	0
$448$	− 2048.00i	− 0.215980i
$449$	−15774.0	−1.65795	−0.828977	−	0.559283i	$-0.811076\pi$
$449$	−15774.0	−1.65795	−0.828977	+	0.559283i	$0.811076\pi$
$450$	0	0
$451$	−14040.0	−1.46589
$452$	− 1704.00i	− 0.177322i
$453$	0	0
$454$	792.000	0.0818731
$455$	0	0
$456$	0	0
$457$	9722.00i	0.995133i	0.867426	+	0.497567i	$0.165773\pi$
$457$	9722.00i	0.995133i	−0.867426	+	0.497567i	$0.834227\pi$
$458$	2660.00i	0.271383i
$459$	0	0
$460$	0	0
$461$	10890.0	1.10021	0.550106	−	0.835095i	$-0.314587\pi$
$461$	10890.0	1.10021	0.550106	+	0.835095i	$0.314587\pi$
$462$	0	0
$463$	− 15128.0i	− 1.51848i	−0.650809	−	0.759242i	$-0.725570\pi$
$463$	− 15128.0i	− 1.51848i	0.650809	−	0.759242i	$-0.274430\pi$
$464$	96.0000	0.00960493
$465$	0	0
$466$	−9732.00	−0.967438
$467$	− 10668.0i	− 1.05708i	−0.848909	−	0.528540i	$-0.822740\pi$
$467$	− 10668.0i	− 1.05708i	0.848909	−	0.528540i	$-0.177260\pi$
$468$	0	0
$469$	−25984.0	−2.55827
$470$	0	0
$471$	0	0
$472$	− 5280.00i	− 0.514898i
$473$	24720.0i	2.40302i
$474$	0	0
$475$	0	0
$476$	−5376.00	−0.517665
$477$	0	0
$478$	− 3648.00i	− 0.349070i
$479$	15264.0	1.45601	0.728006	−	0.685571i	$-0.240447\pi$
$479$	15264.0	1.45601	0.728006	+	0.685571i	$0.240447\pi$
$480$	0	0
$481$	−4556.00	−0.431883
$482$	12964.0i	1.22509i
$483$	0	0
$484$	−9076.00	−0.852367
$485$	0	0
$486$	0	0
$487$	− 5776.00i	− 0.537445i	−0.963218	−	0.268722i	$-0.913399\pi$
$487$	− 5776.00i	− 0.537445i	0.963218	−	0.268722i	$-0.0866014\pi$
$488$	3920.00i	0.363627i
$489$	0	0
$490$	0	0
$491$	−14244.0	−1.30921	−0.654606	−	0.755971i	$-0.727165\pi$
$491$	−14244.0	−1.30921	−0.654606	+	0.755971i	$0.727165\pi$
$492$	0	0
$493$	− 252.000i	− 0.0230213i
$494$	−5168.00	−0.470687
$495$	0	0
$496$	−3712.00	−0.336036
$497$	− 3840.00i	− 0.346575i
$498$	0	0
$499$	17116.0	1.53551	0.767753	−	0.640746i	$-0.221375\pi$
$499$	17116.0	1.53551	0.767753	+	0.640746i	$0.221375\pi$
$500$	0	0
$501$	0	0
$502$	− 2952.00i	− 0.262459i
$503$	− 16848.0i	− 1.49347i	−0.665122	−	0.746735i	$-0.731620\pi$
$503$	− 16848.0i	− 1.49347i	0.665122	−	0.746735i	$-0.268380\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 800.000i	− 0.0698706i
$509$	−3834.00	−0.333868	−0.166934	−	0.985968i	$-0.553387\pi$
$509$	−3834.00	−0.333868	−0.166934	+	0.985968i	$0.553387\pi$
$510$	0	0
$511$	23872.0	2.06660
$512$	512.000i	0.0441942i
$513$	0	0
$514$	8628.00	0.740398
$515$	0	0
$516$	0	0
$517$	21600.0i	1.83746i
$518$	− 8576.00i	− 0.727428i
$519$	0	0
$520$	0	0
$521$	18822.0	1.58274	0.791369	−	0.611338i	$-0.209369\pi$
$521$	18822.0	1.58274	0.791369	+	0.611338i	$0.209369\pi$
$522$	0	0
$523$	15340.0i	1.28255i	0.767313	+	0.641273i	$0.221593\pi$
$523$	15340.0i	1.28255i	−0.767313	+	0.641273i	$0.778407\pi$
$524$	240.000	0.0200085
$525$	0	0
$526$	10560.0	0.875357
$527$	9744.00i	0.805418i
$528$	0	0
$529$	12167.0	1.00000
$530$	0	0
$531$	0	0
$532$	− 9728.00i	− 0.792786i
$533$	− 7956.00i	− 0.646553i
$534$	0	0
$535$	0	0
$536$	6496.00	0.523478
$537$	0	0
$538$	11052.0i	0.885661i
$539$	−40860.0	−3.26524
$540$	0	0
$541$	18950.0	1.50596	0.752980	−	0.658044i	$-0.228616\pi$
$541$	18950.0	1.50596	0.752980	+	0.658044i	$0.228616\pi$
$542$	4048.00i	0.320805i
$543$	0	0
$544$	1344.00	0.105926
$545$	0	0
$546$	0	0
$547$	− 10036.0i	− 0.784476i	−0.919864	−	0.392238i	$-0.871701\pi$
$547$	− 10036.0i	− 0.784476i	0.919864	−	0.392238i	$-0.128299\pi$
$548$	2568.00i	0.200182i
$549$	0	0
$550$	0	0
$551$	456.000	0.0352564
$552$	0	0
$553$	− 4864.00i	− 0.374030i
$554$	−4108.00	−0.315040
$555$	0	0
$556$	−11344.0	−0.865275
$557$	− 10326.0i	− 0.785506i	−0.919644	−	0.392753i	$-0.871523\pi$
$557$	− 10326.0i	− 0.785506i	0.919644	−	0.392753i	$-0.128477\pi$
$558$	0	0
$559$	−14008.0	−1.05988
$560$	0	0
$561$	0	0
$562$	14604.0i	1.09614i
$563$	4524.00i	0.338657i	0.985560	+	0.169328i	$0.0541599\pi$
$563$	4524.00i	0.338657i	−0.985560	+	0.169328i	$0.945840\pi$
$564$	0	0
$565$	0	0
$566$	−7448.00	−0.553114
$567$	0	0
$568$	960.000i	0.0709167i
$569$	16362.0	1.20550	0.602751	−	0.797929i	$-0.294071\pi$
$569$	16362.0	1.20550	0.602751	+	0.797929i	$0.294071\pi$
$570$	0	0
$571$	6620.00	0.485181	0.242591	−	0.970129i	$-0.422003\pi$
$571$	6620.00	0.485181	0.242591	+	0.970129i	$0.422003\pi$
$572$	− 8160.00i	− 0.596480i
$573$	0	0
$574$	14976.0	1.08900
$575$	0	0
$576$	0	0
$577$	8834.00i	0.637373i	0.947860	+	0.318687i	$0.103242\pi$
$577$	8834.00i	0.637373i	−0.947860	+	0.318687i	$0.896758\pi$
$578$	6298.00i	0.453222i
$579$	0	0
$580$	0	0
$581$	25728.0	1.83714
$582$	0	0
$583$	13320.0i	0.946240i
$584$	−5968.00	−0.422873
$585$	0	0
$586$	14436.0	1.01765
$587$	− 3636.00i	− 0.255662i	−0.991796	−	0.127831i	$-0.959198\pi$
$587$	− 3636.00i	− 0.255662i	0.991796	−	0.127831i	$-0.0408016\pi$
$588$	0	0
$589$	−17632.0	−1.23347
$590$	0	0
$591$	0	0
$592$	2144.00i	0.148848i
$593$	6570.00i	0.454971i	0.973782	+	0.227485i	$0.0730504\pi$
$593$	6570.00i	0.454971i	−0.973782	+	0.227485i	$0.926950\pi$
$594$	0	0
$595$	0	0
$596$	6216.00	0.427210
$597$	0	0
$598$	0	0
$599$	16584.0	1.13123	0.565613	−	0.824671i	$-0.308640\pi$
$599$	16584.0	1.13123	0.565613	+	0.824671i	$0.308640\pi$
$600$	0	0
$601$	−502.000	−0.0340716	−0.0170358	−	0.999855i	$-0.505423\pi$
$601$	−502.000	−0.0340716	−0.0170358	+	0.999855i	$0.505423\pi$
$602$	− 26368.0i	− 1.78518i
$603$	0	0
$604$	9088.00	0.612228
$605$	0	0
$606$	0	0
$607$	− 18568.0i	− 1.24160i	−0.783969	−	0.620801i	$-0.786808\pi$
$607$	− 18568.0i	− 1.24160i	0.783969	−	0.620801i	$-0.213192\pi$
$608$	2432.00i	0.162221i
$609$	0	0
$610$	0	0
$611$	−12240.0	−0.810438
$612$	0	0
$613$	13114.0i	0.864061i	0.901859	+	0.432031i	$0.142203\pi$
$613$	13114.0i	0.864061i	−0.901859	+	0.432031i	$0.857797\pi$
$614$	−5080.00	−0.333896
$615$	0	0
$616$	15360.0	1.00466
$617$	− 5250.00i	− 0.342556i	−0.985223	−	0.171278i	$-0.945210\pi$
$617$	− 5250.00i	− 0.342556i	0.985223	−	0.171278i	$-0.0547896\pi$
$618$	0	0
$619$	10804.0	0.701534	0.350767	−	0.936463i	$-0.385921\pi$
$619$	10804.0	0.701534	0.350767	+	0.936463i	$0.385921\pi$
$620$	0	0
$621$	0	0
$622$	− 3120.00i	− 0.201126i
$623$	− 21696.0i	− 1.39524i
$624$	0	0
$625$	0	0
$626$	−1868.00	−0.119266
$627$	0	0
$628$	− 6776.00i	− 0.430560i
$629$	5628.00	0.356762
$630$	0	0
$631$	−27088.0	−1.70896	−0.854482	−	0.519481i	$-0.826125\pi$
$631$	−27088.0	−1.70896	−0.854482	+	0.519481i	$0.826125\pi$
$632$	1216.00i	0.0765346i
$633$	0	0
$634$	−3348.00	−0.209726
$635$	0	0
$636$	0	0
$637$	− 23154.0i	− 1.44018i
$638$	720.000i	0.0446788i
$639$	0	0
$640$	0	0
$641$	−18930.0	−1.16644	−0.583222	−	0.812313i	$-0.698208\pi$
$641$	−18930.0	−1.16644	−0.583222	+	0.812313i	$0.698208\pi$
$642$	0	0
$643$	− 20108.0i	− 1.23325i	−0.787256	−	0.616627i	$-0.788499\pi$
$643$	− 20108.0i	− 1.23325i	0.787256	−	0.616627i	$-0.211501\pi$
$644$	0	0
$645$	0	0
$646$	6384.00	0.388816
$647$	7152.00i	0.434581i	0.976107	+	0.217291i	$0.0697219\pi$
$647$	7152.00i	0.434581i	−0.976107	+	0.217291i	$0.930278\pi$
$648$	0	0
$649$	39600.0	2.39512
$650$	0	0
$651$	0	0
$652$	− 208.000i	− 0.0124937i
$653$	− 31626.0i	− 1.89528i	−0.319333	−	0.947642i	$-0.603459\pi$
$653$	− 31626.0i	− 1.89528i	0.319333	−	0.947642i	$-0.396541\pi$
$654$	0	0
$655$	0	0
$656$	−3744.00	−0.222833
$657$	0	0
$658$	− 23040.0i	− 1.36503i
$659$	28092.0	1.66056	0.830280	−	0.557347i	$-0.188181\pi$
$659$	28092.0	1.66056	0.830280	+	0.557347i	$0.188181\pi$
$660$	0	0
$661$	−13186.0	−0.775909	−0.387955	−	0.921678i	$-0.626818\pi$
$661$	−13186.0	−0.775909	−0.387955	+	0.921678i	$0.626818\pi$
$662$	− 7976.00i	− 0.468272i
$663$	0	0
$664$	−6432.00	−0.375919
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 4800.00i	− 0.278020i
$669$	0	0
$670$	0	0
$671$	−29400.0	−1.69147
$672$	0	0
$673$	− 5138.00i	− 0.294287i	−0.989115	−	0.147144i	$-0.952992\pi$
$673$	− 5138.00i	− 0.294287i	0.989115	−	0.147144i	$-0.0470080\pi$
$674$	−4.00000	−0.000228597 0
$675$	0	0
$676$	−4164.00	−0.236914
$677$	− 6078.00i	− 0.345047i	−0.985005	−	0.172523i	$-0.944808\pi$
$677$	− 6078.00i	− 0.345047i	0.985005	−	0.172523i	$-0.0551920\pi$
$678$	0	0
$679$	−6208.00	−0.350871
$680$	0	0
$681$	0	0
$682$	− 27840.0i	− 1.56312i
$683$	32244.0i	1.80642i	0.429203	+	0.903208i	$0.358795\pi$
$683$	32244.0i	1.80642i	−0.429203	+	0.903208i	$0.641205\pi$
$684$	0	0
$685$	0	0
$686$	21632.0	1.20396
$687$	0	0
$688$	6592.00i	0.365287i
$689$	−7548.00	−0.417353
$690$	0	0
$691$	4484.00	0.246859	0.123429	−	0.992353i	$-0.460611\pi$
$691$	4484.00	0.246859	0.123429	+	0.992353i	$0.460611\pi$
$692$	− 216.000i	− 0.0118657i
$693$	0	0
$694$	3528.00	0.192970
$695$	0	0
$696$	0	0
$697$	9828.00i	0.534092i
$698$	− 8620.00i	− 0.467438i
$699$	0	0
$700$	0	0
$701$	30426.0	1.63934	0.819668	−	0.572839i	$-0.194158\pi$
$701$	30426.0	1.63934	0.819668	+	0.572839i	$0.194158\pi$
$702$	0	0
$703$	10184.0i	0.546368i
$704$	−3840.00	−0.205576
$705$	0	0
$706$	−276.000	−0.0147130
$707$	− 25536.0i	− 1.35839i
$708$	0	0
$709$	−13262.0	−0.702489	−0.351245	−	0.936284i	$-0.614241\pi$
$709$	−13262.0	−0.702489	−0.351245	+	0.936284i	$0.614241\pi$
$710$	0	0
$711$	0	0
$712$	5424.00i	0.285496i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−3504.00	−0.182892
$717$	0	0
$718$	− 23952.0i	− 1.24496i
$719$	13920.0	0.722014	0.361007	−	0.932563i	$-0.382433\pi$
$719$	13920.0	0.722014	0.361007	+	0.932563i	$0.382433\pi$
$720$	0	0
$721$	34816.0	1.79836
$722$	− 2166.00i	− 0.111648i
$723$	0	0
$724$	−15416.0	−0.791341
$725$	0	0
$726$	0	0
$727$	− 9376.00i	− 0.478317i	−0.970981	−	0.239159i	$-0.923128\pi$
$727$	− 9376.00i	− 0.478317i	0.970981	−	0.239159i	$-0.0768716\pi$
$728$	8704.00i	0.443120i
$729$	0	0
$730$	0	0
$731$	17304.0	0.875529
$732$	0	0
$733$	− 6014.00i	− 0.303045i	−0.988454	−	0.151523i	$-0.951582\pi$
$733$	− 6014.00i	− 0.303045i	0.988454	−	0.151523i	$-0.0484176\pi$
$734$	−19408.0	−0.975971
$735$	0	0
$736$	0	0
$737$	48720.0i	2.43504i
$738$	0	0
$739$	7468.00	0.371739	0.185869	−	0.982574i	$-0.440490\pi$
$739$	7468.00	0.371739	0.185869	+	0.982574i	$0.440490\pi$
$740$	0	0
$741$	0	0
$742$	− 14208.0i	− 0.702954i
$743$	31248.0i	1.54290i	0.636287	+	0.771452i	$0.280469\pi$
$743$	31248.0i	1.54290i	−0.636287	+	0.771452i	$0.719531\pi$
$744$	0	0
$745$	0	0
$746$	−16244.0	−0.797232
$747$	0	0
$748$	10080.0i	0.492729i
$749$	54912.0	2.67883
$750$	0	0
$751$	32840.0	1.59567	0.797835	−	0.602875i	$-0.205978\pi$
$751$	32840.0	1.59567	0.797835	+	0.602875i	$0.205978\pi$
$752$	5760.00i	0.279316i
$753$	0	0
$754$	−408.000	−0.0197062
$755$	0	0
$756$	0	0
$757$	− 19066.0i	− 0.915410i	−0.889104	−	0.457705i	$-0.848672\pi$
$757$	− 19066.0i	− 0.915410i	0.889104	−	0.457705i	$-0.151328\pi$
$758$	− 6808.00i	− 0.326224i
$759$	0	0
$760$	0	0
$761$	−6858.00	−0.326678	−0.163339	−	0.986570i	$-0.552227\pi$
$761$	−6858.00	−0.326678	−0.163339	+	0.986570i	$0.552227\pi$
$762$	0	0
$763$	31040.0i	1.47277i
$764$	−11136.0	−0.527338
$765$	0	0
$766$	5040.00	0.237732
$767$	22440.0i	1.05640i
$768$	0	0
$769$	−22178.0	−1.04000	−0.519999	−	0.854167i	$-0.674068\pi$
$769$	−22178.0	−1.04000	−0.519999	+	0.854167i	$0.674068\pi$
$770$	0	0
$771$	0	0
$772$	3656.00i	0.170443i
$773$	14286.0i	0.664724i	0.943152	+	0.332362i	$0.107846\pi$
$773$	14286.0i	0.664724i	−0.943152	+	0.332362i	$0.892154\pi$
$774$	0	0
$775$	0	0
$776$	1552.00	0.0717958
$777$	0	0
$778$	3132.00i	0.144329i
$779$	−17784.0	−0.817943
$780$	0	0
$781$	−7200.00	−0.329880
$782$	0	0
$783$	0	0
$784$	−10896.0	−0.496356
$785$	0	0
$786$	0	0
$787$	− 18868.0i	− 0.854602i	−0.904109	−	0.427301i	$-0.859465\pi$
$787$	− 18868.0i	− 0.854602i	0.904109	−	0.427301i	$-0.140535\pi$
$788$	− 20808.0i	− 0.940678i
$789$	0	0
$790$	0	0
$791$	−13632.0	−0.612766
$792$	0	0
$793$	− 16660.0i	− 0.746045i
$794$	8708.00	0.389213
$795$	0	0
$796$	12608.0	0.561405
$797$	21690.0i	0.963989i	0.876174	+	0.481994i	$0.160087\pi$
$797$	21690.0i	0.963989i	−0.876174	+	0.481994i	$0.839913\pi$
$798$	0	0
$799$	15120.0	0.669471
$800$	0	0
$801$	0	0
$802$	16092.0i	0.708514i
$803$	− 44760.0i	− 1.96706i
$804$	0	0
$805$	0	0
$806$	15776.0	0.689437
$807$	0	0
$808$	6384.00i	0.277956i
$809$	−24726.0	−1.07456	−0.537281	−	0.843404i	$-0.680548\pi$
$809$	−24726.0	−1.07456	−0.537281	+	0.843404i	$0.680548\pi$
$810$	0	0
$811$	−2644.00	−0.114480	−0.0572401	−	0.998360i	$-0.518230\pi$
$811$	−2644.00	−0.114480	−0.0572401	+	0.998360i	$0.518230\pi$
$812$	− 768.000i	− 0.0331915i
$813$	0	0
$814$	−16080.0	−0.692388
$815$	0	0
$816$	0	0
$817$	31312.0i	1.34084i
$818$	5612.00i	0.239877i
$819$	0	0
$820$	0	0
$821$	37842.0	1.60864	0.804321	−	0.594195i	$-0.202529\pi$
$821$	37842.0	1.60864	0.804321	+	0.594195i	$0.202529\pi$
$822$	0	0
$823$	880.000i	0.0372720i	0.999826	+	0.0186360i	$0.00593237\pi$
$823$	880.000i	0.0372720i	−0.999826	+	0.0186360i	$0.994068\pi$
$824$	−8704.00	−0.367983
$825$	0	0
$826$	−42240.0	−1.77932
$827$	12876.0i	0.541406i	0.962663	+	0.270703i	$0.0872561\pi$
$827$	12876.0i	0.541406i	−0.962663	+	0.270703i	$0.912744\pi$
$828$	0	0
$829$	25498.0	1.06825	0.534127	−	0.845404i	$-0.320641\pi$
$829$	25498.0	1.06825	0.534127	+	0.845404i	$0.320641\pi$
$830$	0	0
$831$	0	0
$832$	− 2176.00i	− 0.0906721i
$833$	28602.0i	1.18968i
$834$	0	0
$835$	0	0
$836$	−18240.0	−0.754598
$837$	0	0
$838$	23160.0i	0.954712i
$839$	−40584.0	−1.66998	−0.834991	−	0.550263i	$-0.814527\pi$
$839$	−40584.0	−1.66998	−0.834991	+	0.550263i	$0.814527\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	− 740.000i	− 0.0302875i
$843$	0	0
$844$	−2960.00	−0.120720
$845$	0	0
$846$	0	0
$847$	72608.0i	2.94550i
$848$	3552.00i	0.143840i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	25738.0i	1.03312i	0.856251	+	0.516561i	$0.172788\pi$
$853$	25738.0i	1.03312i	−0.856251	+	0.516561i	$0.827212\pi$
$854$	31360.0	1.25658
$855$	0	0
$856$	−13728.0	−0.548146
$857$	− 13314.0i	− 0.530686i	−0.964154	−	0.265343i	$-0.914515\pi$
$857$	− 13314.0i	− 0.530686i	0.964154	−	0.265343i	$-0.0854851\pi$
$858$	0	0
$859$	−24524.0	−0.974096	−0.487048	−	0.873375i	$-0.661926\pi$
$859$	−24524.0	−0.974096	−0.487048	+	0.873375i	$0.661926\pi$
$860$	0	0
$861$	0	0
$862$	− 10080.0i	− 0.398290i
$863$	5592.00i	0.220572i	0.993900	+	0.110286i	$0.0351767\pi$
$863$	5592.00i	0.220572i	−0.993900	+	0.110286i	$0.964823\pi$
$864$	0	0
$865$	0	0
$866$	−7484.00	−0.293668
$867$	0	0
$868$	29696.0i	1.16123i
$869$	−9120.00	−0.356012
$870$	0	0
$871$	−27608.0	−1.07401
$872$	− 7760.00i	− 0.301361i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 14386.0i	− 0.553912i	−0.960883	−	0.276956i	$-0.910674\pi$
$877$	− 14386.0i	− 0.553912i	0.960883	−	0.276956i	$-0.0893256\pi$
$878$	12416.0i	0.477243i
$879$	0	0
$880$	0	0
$881$	−47106.0	−1.80141	−0.900705	−	0.434432i	$-0.856949\pi$
$881$	−47106.0	−1.80141	−0.900705	+	0.434432i	$0.856949\pi$
$882$	0	0
$883$	− 51548.0i	− 1.96458i	−0.187354	−	0.982292i	$-0.559991\pi$
$883$	− 51548.0i	− 1.96458i	0.187354	−	0.982292i	$-0.440009\pi$
$884$	−5712.00	−0.217325
$885$	0	0
$886$	31128.0	1.18032
$887$	− 34080.0i	− 1.29007i	−0.764152	−	0.645036i	$-0.776842\pi$
$887$	− 34080.0i	− 1.29007i	0.764152	−	0.645036i	$-0.223158\pi$
$888$	0	0
$889$	−6400.00	−0.241450
$890$	0	0
$891$	0	0
$892$	− 2080.00i	− 0.0780757i
$893$	27360.0i	1.02527i
$894$	0	0
$895$	0	0
$896$	4096.00	0.152721
$897$	0	0
$898$	− 31548.0i	− 1.17235i
$899$	−1392.00	−0.0516416
$900$	0	0
$901$	9324.00	0.344759
$902$	− 28080.0i	− 1.03654i
$903$	0	0
$904$	3408.00	0.125385
$905$	0	0
$906$	0	0
$907$	25748.0i	0.942611i	0.881970	+	0.471306i	$0.156217\pi$
$907$	25748.0i	0.942611i	−0.881970	+	0.471306i	$0.843783\pi$
$908$	1584.00i	0.0578930i
$909$	0	0
$910$	0	0
$911$	24768.0	0.900769	0.450384	−	0.892835i	$-0.351287\pi$
$911$	24768.0	0.900769	0.450384	+	0.892835i	$0.351287\pi$
$912$	0	0
$913$	− 48240.0i	− 1.74864i
$914$	−19444.0	−0.703666
$915$	0	0
$916$	−5320.00	−0.191897
$917$	− 1920.00i	− 0.0691428i
$918$	0	0
$919$	31264.0	1.12220	0.561101	−	0.827747i	$-0.310378\pi$
$919$	31264.0	1.12220	0.561101	+	0.827747i	$0.310378\pi$
$920$	0	0
$921$	0	0
$922$	21780.0i	0.777968i
$923$	− 4080.00i	− 0.145498i
$924$	0	0
$925$	0	0
$926$	30256.0	1.07373
$927$	0	0
$928$	192.000i	0.00679171i
$929$	−6174.00	−0.218043	−0.109022	−	0.994039i	$-0.534772\pi$
$929$	−6174.00	−0.218043	−0.109022	+	0.994039i	$0.534772\pi$
$930$	0	0
$931$	−51756.0	−1.82195
$932$	− 19464.0i	− 0.684082i
$933$	0	0
$934$	21336.0	0.747468
$935$	0	0
$936$	0	0
$937$	28922.0i	1.00837i	0.863596	+	0.504184i	$0.168207\pi$
$937$	28922.0i	1.00837i	−0.863596	+	0.504184i	$0.831793\pi$
$938$	− 51968.0i	− 1.80897i
$939$	0	0
$940$	0	0
$941$	−29238.0	−1.01289	−0.506446	−	0.862272i	$-0.669041\pi$
$941$	−29238.0	−1.01289	−0.506446	+	0.862272i	$0.669041\pi$
$942$	0	0
$943$	0	0
$944$	10560.0	0.364088
$945$	0	0
$946$	−49440.0	−1.69919
$947$	2868.00i	0.0984134i	0.998789	+	0.0492067i	$0.0156693\pi$
$947$	2868.00i	0.0984134i	−0.998789	+	0.0492067i	$0.984331\pi$
$948$	0	0
$949$	25364.0	0.867598
$950$	0	0
$951$	0	0
$952$	− 10752.0i	− 0.366044i
$953$	24018.0i	0.816390i	0.912895	+	0.408195i	$0.133842\pi$
$953$	24018.0i	0.816390i	−0.912895	+	0.408195i	$0.866158\pi$
$954$	0	0
$955$	0	0
$956$	7296.00	0.246830
$957$	0	0
$958$	30528.0i	1.02956i
$959$	20544.0	0.691763
$960$	0	0
$961$	24033.0	0.806720
$962$	− 9112.00i	− 0.305387i
$963$	0	0
$964$	−25928.0	−0.866270
$965$	0	0
$966$	0	0
$967$	25712.0i	0.855059i	0.904001	+	0.427530i	$0.140616\pi$
$967$	25712.0i	0.855059i	−0.904001	+	0.427530i	$0.859384\pi$
$968$	− 18152.0i	− 0.602714i
$969$	0	0
$970$	0	0
$971$	12396.0	0.409688	0.204844	−	0.978795i	$-0.434331\pi$
$971$	12396.0	0.409688	0.204844	+	0.978795i	$0.434331\pi$
$972$	0	0
$973$	90752.0i	2.99011i
$974$	11552.0	0.380031
$975$	0	0
$976$	−7840.00	−0.257123
$977$	46614.0i	1.52642i	0.646150	+	0.763211i	$0.276378\pi$
$977$	46614.0i	1.52642i	−0.646150	+	0.763211i	$0.723622\pi$
$978$	0	0
$979$	−40680.0	−1.32803
$980$	0	0
$981$	0	0
$982$	− 28488.0i	− 0.925752i
$983$	− 672.000i	− 0.0218041i	−0.999941	−	0.0109021i	$-0.996530\pi$
$983$	− 672.000i	− 0.0218041i	0.999941	−	0.0109021i	$-0.00347031\pi$
$984$	0	0
$985$	0	0
$986$	504.000	0.0162785
$987$	0	0
$988$	− 10336.0i	− 0.332826i
$989$	0	0
$990$	0	0
$991$	−38776.0	−1.24295	−0.621473	−	0.783435i	$-0.713466\pi$
$991$	−38776.0	−1.24295	−0.621473	+	0.783435i	$0.713466\pi$
$992$	− 7424.00i	− 0.237613i
$993$	0	0
$994$	7680.00	0.245065
$995$	0	0
$996$	0	0
$997$	30422.0i	0.966374i	0.875517	+	0.483187i	$0.160521\pi$
$997$	30422.0i	0.966374i	−0.875517	+	0.483187i	$0.839479\pi$
$998$	34232.0i	1.08577i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.c.k.199.2		2
3.2	odd	2		150.4.c.a.49.1		2
5.2	odd	4		450.4.a.b.1.1		1
5.3	odd	4		90.4.a.d.1.1		1
5.4	even	2	inner	450.4.c.k.199.1		2
12.11	even	2		1200.4.f.u.49.2		2
15.2	even	4		150.4.a.e.1.1		1
15.8	even	4		30.4.a.a.1.1	✓	1
15.14	odd	2		150.4.c.a.49.2		2
20.3	even	4		720.4.a.b.1.1		1
45.13	odd	12		810.4.e.e.541.1		2
45.23	even	12		810.4.e.m.541.1		2
45.38	even	12		810.4.e.m.271.1		2
45.43	odd	12		810.4.e.e.271.1		2
60.23	odd	4		240.4.a.c.1.1		1
60.47	odd	4		1200.4.a.bk.1.1		1
60.59	even	2		1200.4.f.u.49.1		2
105.83	odd	4		1470.4.a.a.1.1		1
120.53	even	4		960.4.a.j.1.1		1
120.83	odd	4		960.4.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.a.1.1	✓	1	15.8	even	4
90.4.a.d.1.1		1	5.3	odd	4
150.4.a.e.1.1		1	15.2	even	4
150.4.c.a.49.1		2	3.2	odd	2
150.4.c.a.49.2		2	15.14	odd	2
240.4.a.c.1.1		1	60.23	odd	4
450.4.a.b.1.1		1	5.2	odd	4
450.4.c.k.199.1		2	5.4	even	2	inner
450.4.c.k.199.2		2	1.1	even	1	trivial
720.4.a.b.1.1		1	20.3	even	4
810.4.e.e.271.1		2	45.43	odd	12
810.4.e.e.541.1		2	45.13	odd	12
810.4.e.m.271.1		2	45.38	even	12
810.4.e.m.541.1		2	45.23	even	12
960.4.a.j.1.1		1	120.53	even	4
960.4.a.s.1.1		1	120.83	odd	4
1200.4.a.bk.1.1		1	60.47	odd	4
1200.4.f.u.49.1		2	60.59	even	2
1200.4.f.u.49.2		2	12.11	even	2
1470.4.a.a.1.1		1	105.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} +32.0000i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} +32.0000i q^{7} -8.00000i q^{8} +60.0000 q^{11} +34.0000i q^{13} -64.0000 q^{14} +16.0000 q^{16} -42.0000i q^{17} +76.0000 q^{19} +120.000i q^{22} -68.0000 q^{26} -128.000i q^{28} +6.00000 q^{29} -232.000 q^{31} +32.0000i q^{32} +84.0000 q^{34} +134.000i q^{37} +152.000i q^{38} -234.000 q^{41} +412.000i q^{43} -240.000 q^{44} +360.000i q^{47} -681.000 q^{49} -136.000i q^{52} +222.000i q^{53} +256.000 q^{56} +12.0000i q^{58} +660.000 q^{59} -490.000 q^{61} -464.000i q^{62} -64.0000 q^{64} +812.000i q^{67} +168.000i q^{68} -120.000 q^{71} -746.000i q^{73} -268.000 q^{74} -304.000 q^{76} +1920.00i q^{77} -152.000 q^{79} -468.000i q^{82} -804.000i q^{83} -824.000 q^{86} -480.000i q^{88} -678.000 q^{89} -1088.00 q^{91} -720.000 q^{94} +194.000i q^{97} -1362.00i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} + 120 q^{11} - 128 q^{14} + 32 q^{16} + 152 q^{19} - 136 q^{26} + 12 q^{29} - 464 q^{31} + 168 q^{34} - 468 q^{41} - 480 q^{44} - 1362 q^{49} + 512 q^{56} + 1320 q^{59} - 980 q^{61} - 128 q^{64} - 240 q^{71} - 536 q^{74} - 608 q^{76} - 304 q^{79} - 1648 q^{86} - 1356 q^{89} - 2176 q^{91} - 1440 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 + 120 * q^11 - 128 * q^14 + 32 * q^16 + 152 * q^19 - 136 * q^26 + 12 * q^29 - 464 * q^31 + 168 * q^34 - 468 * q^41 - 480 * q^44 - 1362 * q^49 + 512 * q^56 + 1320 * q^59 - 980 * q^61 - 128 * q^64 - 240 * q^71 - 536 * q^74 - 608 * q^76 - 304 * q^79 - 1648 * q^86 - 1356 * q^89 - 2176 * q^91 - 1440 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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