LMFDB - Embedded newform 2205.4.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.a (trivial)

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:	yes
Analytic conductor:	$130.099211563$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1163891200.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 $ x^6 - 2x^5 - 23x^4 + 12x^3 + 154x^2 + 152*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.05323$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.46745 q^{2} +11.9581 q^{4} +5.00000 q^{5} -17.6824 q^{8} -22.3372 q^{10} +56.5404 q^{11} +40.9643 q^{13} -16.6692 q^{16} -2.18896 q^{17} +16.4735 q^{19} +59.7903 q^{20} -252.591 q^{22} +155.272 q^{23} +25.0000 q^{25} -183.006 q^{26} +6.26048 q^{29} +168.680 q^{31} +215.928 q^{32} +9.77908 q^{34} -37.1738 q^{37} -73.5946 q^{38} -88.4122 q^{40} +266.804 q^{41} -14.6549 q^{43} +676.114 q^{44} -693.670 q^{46} +169.840 q^{47} -111.686 q^{50} +489.854 q^{52} +151.391 q^{53} +282.702 q^{55} -27.9683 q^{58} +234.076 q^{59} -242.411 q^{61} -753.569 q^{62} -831.294 q^{64} +204.822 q^{65} +820.771 q^{67} -26.1758 q^{68} -961.611 q^{71} +934.026 q^{73} +166.072 q^{74} +196.992 q^{76} +300.236 q^{79} -83.3460 q^{80} -1191.93 q^{82} -1087.60 q^{83} -10.9448 q^{85} +65.4702 q^{86} -999.773 q^{88} +1124.05 q^{89} +1856.76 q^{92} -758.753 q^{94} +82.3676 q^{95} +752.168 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8} + 10 q^{10} + 16 q^{11} + 168 q^{13} + 298 q^{16} + 4 q^{17} + 308 q^{19} + 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} - 56 q^{26} - 176 q^{29} + 392 q^{31}+ \cdots + 516 q^{97}+O(q^{100}) $ 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8 + 10 * q^10 + 16 * q^11 + 168 * q^13 + 298 * q^16 + 4 * q^17 + 308 * q^19 + 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 - 56 * q^26 - 176 * q^29 + 392 * q^31 + 770 * q^32 + 812 * q^34 - 140 * q^37 - 20 * q^38 + 330 * q^40 - 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 - 628 * q^47 + 50 * q^50 + 1520 * q^52 + 676 * q^53 + 80 * q^55 - 2012 * q^58 - 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 + 2940 * q^68 + 224 * q^71 + 2640 * q^73 - 928 * q^74 - 1340 * q^76 + 1636 * q^79 + 1490 * q^80 - 1756 * q^82 - 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 + 1904 * q^89 + 1952 * q^92 - 3332 * q^94 + 1540 * q^95 + 516 * q^97

Coefficient data

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

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

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

$2$	−4.46745	−1.57948	−0.789740	−	0.613441i	$-0.789785\pi$
$2$	−4.46745	−1.57948	−0.789740	+	0.613441i	$0.789785\pi$
$3$	0	0
$3$	0	0
$4$	11.9581	1.49476
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−17.6824	−0.781461
$9$	0	0
$10$	−22.3372	−0.706365
$11$	56.5404	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$11$	56.5404	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$12$	0	0
$13$	40.9643	0.873958	0.436979	−	0.899472i	$-0.356048\pi$
$13$	40.9643	0.873958	0.436979	+	0.899472i	$0.356048\pi$
$14$	0	0
$15$	0	0
$16$	−16.6692	−0.260456
$17$	−2.18896	−0.0312295	−0.0156148	−	0.999878i	$-0.504971\pi$
$17$	−2.18896	−0.0312295	−0.0156148	+	0.999878i	$0.504971\pi$
$18$	0	0
$19$	16.4735	0.198910	0.0994549	−	0.995042i	$-0.468290\pi$
$19$	16.4735	0.198910	0.0994549	+	0.995042i	$0.468290\pi$
$20$	59.7903	0.668476
$21$	0	0
$22$	−252.591	−2.44785
$23$	155.272	1.40767	0.703837	−	0.710361i	$-0.251468\pi$
$23$	155.272	1.40767	0.703837	+	0.710361i	$0.251468\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−183.006	−1.38040
$27$	0	0
$28$	0	0
$29$	6.26048	0.0400876	0.0200438	−	0.999799i	$-0.493619\pi$
$29$	6.26048	0.0400876	0.0200438	+	0.999799i	$0.493619\pi$
$30$	0	0
$31$	168.680	0.977286	0.488643	−	0.872484i	$-0.337492\pi$
$31$	168.680	0.977286	0.488643	+	0.872484i	$0.337492\pi$
$32$	215.928	1.19285
$33$	0	0
$34$	9.77908	0.0493264
$35$	0	0
$36$	0	0
$37$	−37.1738	−0.165171	−0.0825856	−	0.996584i	$-0.526318\pi$
$37$	−37.1738	−0.165171	−0.0825856	+	0.996584i	$0.526318\pi$
$38$	−73.5946	−0.314174
$39$	0	0
$40$	−88.4122	−0.349480
$41$	266.804	1.01629	0.508143	−	0.861273i	$-0.330332\pi$
$41$	266.804	1.01629	0.508143	+	0.861273i	$0.330332\pi$
$42$	0	0
$43$	−14.6549	−0.0519735	−0.0259867	−	0.999662i	$-0.508273\pi$
$43$	−14.6549	−0.0519735	−0.0259867	+	0.999662i	$0.508273\pi$
$44$	676.114	2.31655
$45$	0	0
$46$	−693.670	−2.22339
$47$	169.840	0.527102	0.263551	−	0.964646i	$-0.415106\pi$
$47$	169.840	0.527102	0.263551	+	0.964646i	$0.415106\pi$
$48$	0	0
$49$	0	0
$50$	−111.686	−0.315896
$51$	0	0
$52$	489.854	1.30636
$53$	151.391	0.392361	0.196180	−	0.980568i	$-0.437146\pi$
$53$	151.391	0.392361	0.196180	+	0.980568i	$0.437146\pi$
$54$	0	0
$55$	282.702	0.693083
$56$	0	0
$57$	0	0
$58$	−27.9683	−0.0633176
$59$	234.076	0.516510	0.258255	−	0.966077i	$-0.416853\pi$
$59$	234.076	0.516510	0.258255	+	0.966077i	$0.416853\pi$
$60$	0	0
$61$	−242.411	−0.508813	−0.254407	−	0.967097i	$-0.581880\pi$
$61$	−242.411	−0.508813	−0.254407	+	0.967097i	$0.581880\pi$
$62$	−753.569	−1.54360
$63$	0	0
$64$	−831.294	−1.62362
$65$	204.822	0.390846
$66$	0	0
$67$	820.771	1.49661	0.748307	−	0.663353i	$-0.230867\pi$
$67$	820.771	1.49661	0.748307	+	0.663353i	$0.230867\pi$
$68$	−26.1758	−0.0466806
$69$	0	0
$70$	0	0
$71$	−961.611	−1.60736	−0.803678	−	0.595065i	$-0.797126\pi$
$71$	−961.611	−1.60736	−0.803678	+	0.595065i	$0.797126\pi$
$72$	0	0
$73$	934.026	1.49753	0.748763	−	0.662837i	$-0.230648\pi$
$73$	934.026	1.49753	0.748763	+	0.662837i	$0.230648\pi$
$74$	166.072	0.260885
$75$	0	0
$76$	196.992	0.297322
$77$	0	0
$78$	0	0
$79$	300.236	0.427584	0.213792	−	0.976879i	$-0.431419\pi$
$79$	300.236	0.427584	0.213792	+	0.976879i	$0.431419\pi$
$80$	−83.3460	−0.116480
$81$	0	0
$82$	−1191.93	−1.60520
$83$	−1087.60	−1.43830	−0.719151	−	0.694854i	$-0.755469\pi$
$83$	−1087.60	−1.43830	−0.719151	+	0.694854i	$0.755469\pi$
$84$	0	0
$85$	−10.9448	−0.0139663
$86$	65.4702	0.0820910
$87$	0	0
$88$	−999.773	−1.21109
$89$	1124.05	1.33875	0.669375	−	0.742925i	$-0.266562\pi$
$89$	1124.05	1.33875	0.669375	+	0.742925i	$0.266562\pi$
$90$	0	0
$91$	0	0
$92$	1856.76	2.10413
$93$	0	0
$94$	−758.753	−0.832547
$95$	82.3676	0.0889552
$96$	0	0
$97$	752.168	0.787330	0.393665	−	0.919254i	$-0.371207\pi$
$97$	752.168	0.787330	0.393665	+	0.919254i	$0.371207\pi$
$98$	0	0
$99$	0	0
$100$	298.952	0.298952
$101$	−559.226	−0.550941	−0.275471	−	0.961310i	$-0.588834\pi$
$101$	−559.226	−0.550941	−0.275471	+	0.961310i	$0.588834\pi$
$102$	0	0
$103$	−205.331	−0.196426	−0.0982128	−	0.995165i	$-0.531313\pi$
$103$	−205.331	−0.196426	−0.0982128	+	0.995165i	$0.531313\pi$
$104$	−724.349	−0.682964
$105$	0	0
$106$	−676.330	−0.619726
$107$	1470.17	1.32829	0.664145	−	0.747604i	$-0.268796\pi$
$107$	1470.17	1.32829	0.664145	+	0.747604i	$0.268796\pi$
$108$	0	0
$109$	−774.193	−0.680314	−0.340157	−	0.940369i	$-0.610480\pi$
$109$	−774.193	−0.680314	−0.340157	+	0.940369i	$0.610480\pi$
$110$	−1262.96	−1.09471
$111$	0	0
$112$	0	0
$113$	406.121	0.338094	0.169047	−	0.985608i	$-0.445931\pi$
$113$	406.121	0.338094	0.169047	+	0.985608i	$0.445931\pi$
$114$	0	0
$115$	776.361	0.629531
$116$	74.8632	0.0599213
$117$	0	0
$118$	−1045.72	−0.815817
$119$	0	0
$120$	0	0
$121$	1865.82	1.40182
$122$	1082.96	0.803660
$123$	0	0
$124$	2017.09	1.46081
$125$	125.000	0.0894427
$126$	0	0
$127$	−451.639	−0.315563	−0.157781	−	0.987474i	$-0.550434\pi$
$127$	−451.639	−0.315563	−0.157781	+	0.987474i	$0.550434\pi$
$128$	1986.33	1.37163
$129$	0	0
$130$	−915.029	−0.617334
$131$	−361.932	−0.241390	−0.120695	−	0.992690i	$-0.538512\pi$
$131$	−361.932	−0.241390	−0.120695	+	0.992690i	$0.538512\pi$
$132$	0	0
$133$	0	0
$134$	−3666.75	−2.36387
$135$	0	0
$136$	38.7062	0.0244047
$137$	−2512.72	−1.56698	−0.783490	−	0.621404i	$-0.786563\pi$
$137$	−2512.72	−1.56698	−0.783490	+	0.621404i	$0.786563\pi$
$138$	0	0
$139$	1165.71	0.711324	0.355662	−	0.934615i	$-0.384255\pi$
$139$	1165.71	0.711324	0.355662	+	0.934615i	$0.384255\pi$
$140$	0	0
$141$	0	0
$142$	4295.94	2.53879
$143$	2316.14	1.35444
$144$	0	0
$145$	31.3024	0.0179277
$146$	−4172.71	−2.36531
$147$	0	0
$148$	−444.526	−0.246891
$149$	2637.49	1.45014	0.725072	−	0.688673i	$-0.241806\pi$
$149$	2637.49	1.45014	0.725072	+	0.688673i	$0.241806\pi$
$150$	0	0
$151$	−2582.32	−1.39170	−0.695849	−	0.718188i	$-0.744972\pi$
$151$	−2582.32	−1.39170	−0.695849	+	0.718188i	$0.744972\pi$
$152$	−291.292	−0.155440
$153$	0	0
$154$	0	0
$155$	843.401	0.437055
$156$	0	0
$157$	225.788	0.114776	0.0573880	−	0.998352i	$-0.481723\pi$
$157$	225.788	0.114776	0.0573880	+	0.998352i	$0.481723\pi$
$158$	−1341.29	−0.675361
$159$	0	0
$160$	1079.64	0.533457
$161$	0	0
$162$	0	0
$163$	1489.01	0.715509	0.357755	−	0.933816i	$-0.383542\pi$
$163$	1489.01	0.715509	0.357755	+	0.933816i	$0.383542\pi$
$164$	3190.46	1.51910
$165$	0	0
$166$	4858.77	2.27177
$167$	−2858.73	−1.32464	−0.662322	−	0.749220i	$-0.730429\pi$
$167$	−2858.73	−1.32464	−0.662322	+	0.749220i	$0.730429\pi$
$168$	0	0
$169$	−518.925	−0.236197
$170$	48.8954	0.0220594
$171$	0	0
$172$	−175.245	−0.0776877
$173$	32.0040	0.0140648	0.00703242	−	0.999975i	$-0.497761\pi$
$173$	32.0040	0.0140648	0.00703242	+	0.999975i	$0.497761\pi$
$174$	0	0
$175$	0	0
$176$	−942.483	−0.403650
$177$	0	0
$178$	−5021.62	−2.11453
$179$	−738.444	−0.308346	−0.154173	−	0.988044i	$-0.549271\pi$
$179$	−738.444	−0.308346	−0.154173	+	0.988044i	$0.549271\pi$
$180$	0	0
$181$	−1991.91	−0.817997	−0.408998	−	0.912535i	$-0.634122\pi$
$181$	−1991.91	−0.817997	−0.408998	+	0.912535i	$0.634122\pi$
$182$	0	0
$183$	0	0
$184$	−2745.59	−1.10004
$185$	−185.869	−0.0738668
$186$	0	0
$187$	−123.765	−0.0483989
$188$	2030.96	0.787890
$189$	0	0
$190$	−367.973	−0.140503
$191$	−2153.68	−0.815890	−0.407945	−	0.913006i	$-0.633755\pi$
$191$	−2153.68	−0.815890	−0.407945	+	0.913006i	$0.633755\pi$
$192$	0	0
$193$	1003.94	0.374431	0.187215	−	0.982319i	$-0.440054\pi$
$193$	1003.94	0.374431	0.187215	+	0.982319i	$0.440054\pi$
$194$	−3360.27	−1.24357
$195$	0	0
$196$	0	0
$197$	−1716.80	−0.620899	−0.310449	−	0.950590i	$-0.600479\pi$
$197$	−1716.80	−0.620899	−0.310449	+	0.950590i	$0.600479\pi$
$198$	0	0
$199$	−5337.86	−1.90146	−0.950730	−	0.310019i	$-0.899665\pi$
$199$	−5337.86	−1.90146	−0.950730	+	0.310019i	$0.899665\pi$
$200$	−442.061	−0.156292
$201$	0	0
$202$	2498.31	0.870201
$203$	0	0
$204$	0	0
$205$	1334.02	0.454497
$206$	917.304	0.310250
$207$	0	0
$208$	−682.842	−0.227628
$209$	931.420	0.308266
$210$	0	0
$211$	860.589	0.280784	0.140392	−	0.990096i	$-0.455164\pi$
$211$	860.589	0.280784	0.140392	+	0.990096i	$0.455164\pi$
$212$	1810.34	0.586485
$213$	0	0
$214$	−6567.92	−2.09801
$215$	−73.2747	−0.0232432
$216$	0	0
$217$	0	0
$218$	3458.66	1.07454
$219$	0	0
$220$	3380.57	1.03599
$221$	−89.6695	−0.0272933
$222$	0	0
$223$	−3661.12	−1.09940	−0.549701	−	0.835361i	$-0.685259\pi$
$223$	−3661.12	−1.09940	−0.549701	+	0.835361i	$0.685259\pi$
$224$	0	0
$225$	0	0
$226$	−1814.32	−0.534013
$227$	−6032.38	−1.76380	−0.881902	−	0.471434i	$-0.843737\pi$
$227$	−6032.38	−1.76380	−0.881902	+	0.471434i	$0.843737\pi$
$228$	0	0
$229$	−3849.64	−1.11088	−0.555439	−	0.831557i	$-0.687450\pi$
$229$	−3849.64	−1.11088	−0.555439	+	0.831557i	$0.687450\pi$
$230$	−3468.35	−0.994332
$231$	0	0
$232$	−110.701	−0.0313269
$233$	1705.20	0.479448	0.239724	−	0.970841i	$-0.422943\pi$
$233$	1705.20	0.479448	0.239724	+	0.970841i	$0.422943\pi$
$234$	0	0
$235$	849.202	0.235727
$236$	2799.09	0.772057
$237$	0	0
$238$	0	0
$239$	−6804.54	−1.84163	−0.920814	−	0.390001i	$-0.872475\pi$
$239$	−6804.54	−1.84163	−0.920814	+	0.390001i	$0.872475\pi$
$240$	0	0
$241$	4642.98	1.24100	0.620499	−	0.784207i	$-0.286930\pi$
$241$	4642.98	1.24100	0.620499	+	0.784207i	$0.286930\pi$
$242$	−8335.44	−2.21414
$243$	0	0
$244$	−2898.77	−0.760553
$245$	0	0
$246$	0	0
$247$	674.827	0.173839
$248$	−2982.68	−0.763711
$249$	0	0
$250$	−558.431	−0.141273
$251$	−5912.64	−1.48686	−0.743431	−	0.668813i	$-0.766803\pi$
$251$	−5912.64	−1.48686	−0.743431	+	0.668813i	$0.766803\pi$
$252$	0	0
$253$	8779.16	2.18159
$254$	2017.67	0.498425
$255$	0	0
$256$	−2223.49	−0.542844
$257$	1697.83	0.412093	0.206047	−	0.978542i	$-0.433940\pi$
$257$	1697.83	0.412093	0.206047	+	0.978542i	$0.433940\pi$
$258$	0	0
$259$	0	0
$260$	2449.27	0.584220
$261$	0	0
$262$	1616.91	0.381271
$263$	5661.78	1.32745	0.663727	−	0.747975i	$-0.268974\pi$
$263$	5661.78	1.32745	0.663727	+	0.747975i	$0.268974\pi$
$264$	0	0
$265$	756.954	0.175469
$266$	0	0
$267$	0	0
$268$	9814.83	2.23708
$269$	−5330.01	−1.20809	−0.604046	−	0.796950i	$-0.706446\pi$
$269$	−5330.01	−1.20809	−0.604046	+	0.796950i	$0.706446\pi$
$270$	0	0
$271$	−2034.94	−0.456139	−0.228069	−	0.973645i	$-0.573241\pi$
$271$	−2034.94	−0.456139	−0.228069	+	0.973645i	$0.573241\pi$
$272$	36.4883	0.00813392
$273$	0	0
$274$	11225.5	2.47502
$275$	1413.51	0.309956
$276$	0	0
$277$	−867.657	−0.188204	−0.0941019	−	0.995563i	$-0.529998\pi$
$277$	−867.657	−0.188204	−0.0941019	+	0.995563i	$0.529998\pi$
$278$	−5207.73	−1.12352
$279$	0	0
$280$	0	0
$281$	−2049.70	−0.435142	−0.217571	−	0.976045i	$-0.569813\pi$
$281$	−2049.70	−0.435142	−0.217571	+	0.976045i	$0.569813\pi$
$282$	0	0
$283$	6625.11	1.39160	0.695799	−	0.718237i	$-0.255051\pi$
$283$	6625.11	1.39160	0.695799	+	0.718237i	$0.255051\pi$
$284$	−11499.0	−2.40261
$285$	0	0
$286$	−10347.2	−2.13932
$287$	0	0
$288$	0	0
$289$	−4908.21	−0.999025
$290$	−139.842	−0.0283165
$291$	0	0
$292$	11169.1	2.23844
$293$	5670.84	1.13070	0.565348	−	0.824852i	$-0.308742\pi$
$293$	5670.84	1.13070	0.565348	+	0.824852i	$0.308742\pi$
$294$	0	0
$295$	1170.38	0.230990
$296$	657.323	0.129075
$297$	0	0
$298$	−11782.8	−2.29048
$299$	6360.62	1.23025
$300$	0	0
$301$	0	0
$302$	11536.4	2.19816
$303$	0	0
$304$	−274.600	−0.0518073
$305$	−1212.06	−0.227548
$306$	0	0
$307$	−272.638	−0.0506849	−0.0253424	−	0.999679i	$-0.508068\pi$
$307$	−272.638	−0.0506849	−0.0253424	+	0.999679i	$0.508068\pi$
$308$	0	0
$309$	0	0
$310$	−3767.85	−0.690321
$311$	1220.83	0.222595	0.111298	−	0.993787i	$-0.464499\pi$
$311$	1220.83	0.222595	0.111298	+	0.993787i	$0.464499\pi$
$312$	0	0
$313$	775.195	0.139989	0.0699946	−	0.997547i	$-0.477702\pi$
$313$	775.195	0.139989	0.0699946	+	0.997547i	$0.477702\pi$
$314$	−1008.69	−0.181286
$315$	0	0
$316$	3590.24	0.639135
$317$	−7393.24	−1.30992	−0.654962	−	0.755662i	$-0.727315\pi$
$317$	−7393.24	−1.30992	−0.654962	+	0.755662i	$0.727315\pi$
$318$	0	0
$319$	353.970	0.0621270
$320$	−4156.47	−0.726105
$321$	0	0
$322$	0	0
$323$	−36.0600	−0.00621186
$324$	0	0
$325$	1024.11	0.174792
$326$	−6652.06	−1.13013
$327$	0	0
$328$	−4717.74	−0.794188
$329$	0	0
$330$	0	0
$331$	−3736.95	−0.620548	−0.310274	−	0.950647i	$-0.600421\pi$
$331$	−3736.95	−0.620548	−0.310274	+	0.950647i	$0.600421\pi$
$332$	−13005.5	−2.14991
$333$	0	0
$334$	12771.2	2.09225
$335$	4103.85	0.669306
$336$	0	0
$337$	−8230.24	−1.33036	−0.665178	−	0.746685i	$-0.731644\pi$
$337$	−8230.24	−1.33036	−0.665178	+	0.746685i	$0.731644\pi$
$338$	2318.27	0.373068
$339$	0	0
$340$	−130.879	−0.0208762
$341$	9537.25	1.51458
$342$	0	0
$343$	0	0
$344$	259.135	0.0406152
$345$	0	0
$346$	−142.976	−0.0222151
$347$	12730.5	1.96948	0.984739	−	0.174036i	$-0.0556809\pi$
$347$	12730.5	1.96948	0.984739	+	0.174036i	$0.0556809\pi$
$348$	0	0
$349$	4487.30	0.688252	0.344126	−	0.938924i	$-0.388175\pi$
$349$	4487.30	0.688252	0.344126	+	0.938924i	$0.388175\pi$
$350$	0	0
$351$	0	0
$352$	12208.7	1.84865
$353$	12263.5	1.84906	0.924532	−	0.381106i	$-0.124457\pi$
$353$	12263.5	1.84906	0.924532	+	0.381106i	$0.124457\pi$
$354$	0	0
$355$	−4808.05	−0.718831
$356$	13441.4	2.00111
$357$	0	0
$358$	3298.96	0.487026
$359$	11073.2	1.62791	0.813957	−	0.580925i	$-0.197309\pi$
$359$	11073.2	1.62791	0.813957	+	0.580925i	$0.197309\pi$
$360$	0	0
$361$	−6587.62	−0.960435
$362$	8898.74	1.29201
$363$	0	0
$364$	0	0
$365$	4670.13	0.669714
$366$	0	0
$367$	5864.72	0.834157	0.417079	−	0.908870i	$-0.363054\pi$
$367$	5864.72	0.834157	0.417079	+	0.908870i	$0.363054\pi$
$368$	−2588.26	−0.366637
$369$	0	0
$370$	830.359	0.116671
$371$	0	0
$372$	0	0
$373$	−9368.17	−1.30044	−0.650222	−	0.759744i	$-0.725324\pi$
$373$	−9368.17	−1.30044	−0.650222	+	0.759744i	$0.725324\pi$
$374$	552.913	0.0764451
$375$	0	0
$376$	−3003.19	−0.411909
$377$	256.456	0.0350349
$378$	0	0
$379$	5537.81	0.750549	0.375275	−	0.926914i	$-0.377548\pi$
$379$	5537.81	0.750549	0.375275	+	0.926914i	$0.377548\pi$
$380$	984.958	0.132966
$381$	0	0
$382$	9621.46	1.28868
$383$	−5243.17	−0.699513	−0.349756	−	0.936841i	$-0.613736\pi$
$383$	−5243.17	−0.699513	−0.349756	+	0.936841i	$0.613736\pi$
$384$	0	0
$385$	0	0
$386$	−4485.05	−0.591406
$387$	0	0
$388$	8994.47	1.17687
$389$	10890.5	1.41947	0.709733	−	0.704471i	$-0.248816\pi$
$389$	10890.5	1.41947	0.709733	+	0.704471i	$0.248816\pi$
$390$	0	0
$391$	−339.886	−0.0439610
$392$	0	0
$393$	0	0
$394$	7669.71	0.980697
$395$	1501.18	0.191221
$396$	0	0
$397$	13307.0	1.68227	0.841135	−	0.540826i	$-0.181888\pi$
$397$	13307.0	1.68227	0.841135	+	0.540826i	$0.181888\pi$
$398$	23846.6	3.00332
$399$	0	0
$400$	−416.730	−0.0520912
$401$	−7227.02	−0.900001	−0.450000	−	0.893028i	$-0.648576\pi$
$401$	−7227.02	−0.900001	−0.450000	+	0.893028i	$0.648576\pi$
$402$	0	0
$403$	6909.87	0.854107
$404$	−6687.26	−0.823524
$405$	0	0
$406$	0	0
$407$	−2101.82	−0.255979
$408$	0	0
$409$	5852.64	0.707565	0.353783	−	0.935328i	$-0.384895\pi$
$409$	5852.64	0.707565	0.353783	+	0.935328i	$0.384895\pi$
$410$	−5959.66	−0.717869
$411$	0	0
$412$	−2455.36	−0.293609
$413$	0	0
$414$	0	0
$415$	−5437.98	−0.643228
$416$	8845.35	1.04250
$417$	0	0
$418$	−4161.07	−0.486901
$419$	8344.19	0.972889	0.486444	−	0.873712i	$-0.338294\pi$
$419$	8344.19	0.972889	0.486444	+	0.873712i	$0.338294\pi$
$420$	0	0
$421$	−10955.3	−1.26824	−0.634122	−	0.773233i	$-0.718638\pi$
$421$	−10955.3	−1.26824	−0.634122	+	0.773233i	$0.718638\pi$
$422$	−3844.63	−0.443493
$423$	0	0
$424$	−2676.96	−0.306615
$425$	−54.7241	−0.00624591
$426$	0	0
$427$	0	0
$428$	17580.4	1.98547
$429$	0	0
$430$	327.351	0.0367122
$431$	−9417.43	−1.05249	−0.526243	−	0.850334i	$-0.676400\pi$
$431$	−9417.43	−1.05249	−0.526243	+	0.850334i	$0.676400\pi$
$432$	0	0
$433$	8783.79	0.974878	0.487439	−	0.873157i	$-0.337931\pi$
$433$	8783.79	0.974878	0.487439	+	0.873157i	$0.337931\pi$
$434$	0	0
$435$	0	0
$436$	−9257.85	−1.01690
$437$	2557.88	0.280000
$438$	0	0
$439$	10559.6	1.14802	0.574011	−	0.818847i	$-0.305386\pi$
$439$	10559.6	1.14802	0.574011	+	0.818847i	$0.305386\pi$
$440$	−4998.86	−0.541617
$441$	0	0
$442$	400.593	0.0431092
$443$	4878.14	0.523177	0.261589	−	0.965179i	$-0.415754\pi$
$443$	4878.14	0.523177	0.261589	+	0.965179i	$0.415754\pi$
$444$	0	0
$445$	5620.23	0.598707
$446$	16355.9	1.73649
$447$	0	0
$448$	0	0
$449$	−43.7917	−0.00460280	−0.00230140	−	0.999997i	$-0.500733\pi$
$449$	−43.7917	−0.00460280	−0.00230140	+	0.999997i	$0.500733\pi$
$450$	0	0
$451$	15085.2	1.57502
$452$	4856.42	0.505369
$453$	0	0
$454$	26949.3	2.78589
$455$	0	0
$456$	0	0
$457$	2869.76	0.293745	0.146873	−	0.989155i	$-0.453079\pi$
$457$	2869.76	0.293745	0.146873	+	0.989155i	$0.453079\pi$
$458$	17198.0	1.75461
$459$	0	0
$460$	9283.78	0.940997
$461$	17910.1	1.80945	0.904723	−	0.426001i	$-0.140078\pi$
$461$	17910.1	1.80945	0.904723	+	0.426001i	$0.140078\pi$
$462$	0	0
$463$	−7630.72	−0.765938	−0.382969	−	0.923761i	$-0.625098\pi$
$463$	−7630.72	−0.765938	−0.382969	+	0.923761i	$0.625098\pi$
$464$	−104.357	−0.0104411
$465$	0	0
$466$	−7617.89	−0.757279
$467$	6246.30	0.618939	0.309469	−	0.950909i	$-0.399849\pi$
$467$	6246.30	0.618939	0.309469	+	0.950909i	$0.399849\pi$
$468$	0	0
$469$	0	0
$470$	−3793.76	−0.372326
$471$	0	0
$472$	−4139.03	−0.403632
$473$	−828.597	−0.0805474
$474$	0	0
$475$	411.838	0.0397820
$476$	0	0
$477$	0	0
$478$	30398.9	2.90882
$479$	4119.58	0.392961	0.196480	−	0.980508i	$-0.437049\pi$
$479$	4119.58	0.392961	0.196480	+	0.980508i	$0.437049\pi$
$480$	0	0
$481$	−1522.80	−0.144353
$482$	−20742.2	−1.96013
$483$	0	0
$484$	22311.6	2.09538
$485$	3760.84	0.352105
$486$	0	0
$487$	−1421.46	−0.132264	−0.0661320	−	0.997811i	$-0.521066\pi$
$487$	−1421.46	−0.132264	−0.0661320	+	0.997811i	$0.521066\pi$
$488$	4286.43	0.397618
$489$	0	0
$490$	0	0
$491$	−19241.1	−1.76851	−0.884253	−	0.467007i	$-0.845332\pi$
$491$	−19241.1	−1.76851	−0.884253	+	0.467007i	$0.845332\pi$
$492$	0	0
$493$	−13.7040	−0.00125192
$494$	−3014.75	−0.274575
$495$	0	0
$496$	−2811.76	−0.254540
$497$	0	0
$498$	0	0
$499$	9576.48	0.859123	0.429561	−	0.903038i	$-0.358668\pi$
$499$	9576.48	0.859123	0.429561	+	0.903038i	$0.358668\pi$
$500$	1494.76	0.133695
$501$	0	0
$502$	26414.4	2.34847
$503$	10581.9	0.938019	0.469009	−	0.883193i	$-0.344611\pi$
$503$	10581.9	0.938019	0.469009	+	0.883193i	$0.344611\pi$
$504$	0	0
$505$	−2796.13	−0.246388
$506$	−39220.4	−3.44577
$507$	0	0
$508$	−5400.72	−0.471690
$509$	−7082.86	−0.616782	−0.308391	−	0.951260i	$-0.599791\pi$
$509$	−7082.86	−0.616782	−0.308391	+	0.951260i	$0.599791\pi$
$510$	0	0
$511$	0	0
$512$	−5957.37	−0.514220
$513$	0	0
$514$	−7584.98	−0.650893
$515$	−1026.65	−0.0878442
$516$	0	0
$517$	9602.85	0.816891
$518$	0	0
$519$	0	0
$520$	−3621.75	−0.305431
$521$	7153.92	0.601572	0.300786	−	0.953692i	$-0.402751\pi$
$521$	7153.92	0.601572	0.300786	+	0.953692i	$0.402751\pi$
$522$	0	0
$523$	−15781.3	−1.31944	−0.659721	−	0.751511i	$-0.729326\pi$
$523$	−15781.3	−1.31944	−0.659721	+	0.751511i	$0.729326\pi$
$524$	−4328.00	−0.360820
$525$	0	0
$526$	−25293.7	−2.09669
$527$	−369.235	−0.0305202
$528$	0	0
$529$	11942.5	0.981547
$530$	−3381.65	−0.277150
$531$	0	0
$532$	0	0
$533$	10929.4	0.888192
$534$	0	0
$535$	7350.86	0.594029
$536$	−14513.2	−1.16955
$537$	0	0
$538$	23811.5	1.90816
$539$	0	0
$540$	0	0
$541$	−24277.2	−1.92931	−0.964656	−	0.263511i	$-0.915119\pi$
$541$	−24277.2	−1.92931	−0.964656	+	0.263511i	$0.915119\pi$
$542$	9090.97	0.720463
$543$	0	0
$544$	−472.659	−0.0372520
$545$	−3870.96	−0.304246
$546$	0	0
$547$	−191.079	−0.0149359	−0.00746796	−	0.999972i	$-0.502377\pi$
$547$	−191.079	−0.0149359	−0.00746796	+	0.999972i	$0.502377\pi$
$548$	−30047.3	−2.34226
$549$	0	0
$550$	−6314.78	−0.489569
$551$	103.132	0.00797382
$552$	0	0
$553$	0	0
$554$	3876.21	0.297264
$555$	0	0
$556$	13939.6	1.06326
$557$	15721.6	1.19595	0.597974	−	0.801515i	$-0.295972\pi$
$557$	15721.6	1.19595	0.597974	+	0.801515i	$0.295972\pi$
$558$	0	0
$559$	−600.330	−0.0454226
$560$	0	0
$561$	0	0
$562$	9156.92	0.687298
$563$	−3044.50	−0.227905	−0.113952	−	0.993486i	$-0.536351\pi$
$563$	−3044.50	−0.227905	−0.113952	+	0.993486i	$0.536351\pi$
$564$	0	0
$565$	2030.60	0.151200
$566$	−29597.3	−2.19800
$567$	0	0
$568$	17003.6	1.25609
$569$	−13052.2	−0.961647	−0.480823	−	0.876817i	$-0.659662\pi$
$569$	−13052.2	−0.961647	−0.480823	+	0.876817i	$0.659662\pi$
$570$	0	0
$571$	−5811.05	−0.425893	−0.212946	−	0.977064i	$-0.568306\pi$
$571$	−5811.05	−0.425893	−0.212946	+	0.977064i	$0.568306\pi$
$572$	27696.5	2.02456
$573$	0	0
$574$	0	0
$575$	3881.81	0.281535
$576$	0	0
$577$	−22790.3	−1.64432	−0.822162	−	0.569254i	$-0.807232\pi$
$577$	−22790.3	−1.64432	−0.822162	+	0.569254i	$0.807232\pi$
$578$	21927.2	1.57794
$579$	0	0
$580$	374.316	0.0267976
$581$	0	0
$582$	0	0
$583$	8559.70	0.608073
$584$	−16515.9	−1.17026
$585$	0	0
$586$	−25334.2	−1.78591
$587$	20600.3	1.44850	0.724248	−	0.689540i	$-0.242187\pi$
$587$	20600.3	1.44850	0.724248	+	0.689540i	$0.242187\pi$
$588$	0	0
$589$	2778.76	0.194392
$590$	−5228.61	−0.364845
$591$	0	0
$592$	619.657	0.0430198
$593$	−773.830	−0.0535875	−0.0267938	−	0.999641i	$-0.508530\pi$
$593$	−773.830	−0.0535875	−0.0267938	+	0.999641i	$0.508530\pi$
$594$	0	0
$595$	0	0
$596$	31539.3	2.16762
$597$	0	0
$598$	−28415.7	−1.94315
$599$	13641.0	0.930479	0.465239	−	0.885185i	$-0.345968\pi$
$599$	13641.0	0.930479	0.465239	+	0.885185i	$0.345968\pi$
$600$	0	0
$601$	14271.8	0.968650	0.484325	−	0.874888i	$-0.339065\pi$
$601$	14271.8	0.968650	0.484325	+	0.874888i	$0.339065\pi$
$602$	0	0
$603$	0	0
$604$	−30879.6	−2.08025
$605$	9329.09	0.626912
$606$	0	0
$607$	3234.90	0.216311	0.108155	−	0.994134i	$-0.465506\pi$
$607$	3234.90	0.216311	0.108155	+	0.994134i	$0.465506\pi$
$608$	3557.10	0.237269
$609$	0	0
$610$	5414.80	0.359408
$611$	6957.40	0.460665
$612$	0	0
$613$	8413.52	0.554354	0.277177	−	0.960819i	$-0.410601\pi$
$613$	8413.52	0.554354	0.277177	+	0.960819i	$0.410601\pi$
$614$	1217.99	0.0800558
$615$	0	0
$616$	0	0
$617$	−7077.93	−0.461826	−0.230913	−	0.972974i	$-0.574171\pi$
$617$	−7077.93	−0.461826	−0.230913	+	0.972974i	$0.574171\pi$
$618$	0	0
$619$	5464.74	0.354841	0.177420	−	0.984135i	$-0.443225\pi$
$619$	5464.74	0.354841	0.177420	+	0.984135i	$0.443225\pi$
$620$	10085.4	0.653292
$621$	0	0
$622$	−5454.01	−0.351585
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−3463.14	−0.221110
$627$	0	0
$628$	2699.98	0.171562
$629$	81.3721	0.00515822
$630$	0	0
$631$	−1569.98	−0.0990492	−0.0495246	−	0.998773i	$-0.515771\pi$
$631$	−1569.98	−0.0990492	−0.0495246	+	0.998773i	$0.515771\pi$
$632$	−5308.90	−0.334140
$633$	0	0
$634$	33028.9	2.06900
$635$	−2258.19	−0.141124
$636$	0	0
$637$	0	0
$638$	−1581.34	−0.0981284
$639$	0	0
$640$	9931.67	0.613412
$641$	14578.3	0.898298	0.449149	−	0.893457i	$-0.351727\pi$
$641$	14578.3	0.898298	0.449149	+	0.893457i	$0.351727\pi$
$642$	0	0
$643$	−11980.0	−0.734750	−0.367375	−	0.930073i	$-0.619744\pi$
$643$	−11980.0	−0.734750	−0.367375	+	0.930073i	$0.619744\pi$
$644$	0	0
$645$	0	0
$646$	161.096	0.00981151
$647$	5651.81	0.343424	0.171712	−	0.985147i	$-0.445070\pi$
$647$	5651.81	0.343424	0.171712	+	0.985147i	$0.445070\pi$
$648$	0	0
$649$	13234.7	0.800476
$650$	−4575.15	−0.276080
$651$	0	0
$652$	17805.6	1.06951
$653$	2370.86	0.142081	0.0710405	−	0.997473i	$-0.477368\pi$
$653$	2370.86	0.142081	0.0710405	+	0.997473i	$0.477368\pi$
$654$	0	0
$655$	−1809.66	−0.107953
$656$	−4447.40	−0.264698
$657$	0	0
$658$	0	0
$659$	−5233.92	−0.309385	−0.154692	−	0.987963i	$-0.549439\pi$
$659$	−5233.92	−0.309385	−0.154692	+	0.987963i	$0.549439\pi$
$660$	0	0
$661$	10456.0	0.615265	0.307633	−	0.951505i	$-0.400463\pi$
$661$	10456.0	0.615265	0.307633	+	0.951505i	$0.400463\pi$
$662$	16694.6	0.980144
$663$	0	0
$664$	19231.3	1.12398
$665$	0	0
$666$	0	0
$667$	972.079	0.0564303
$668$	−34184.9	−1.98002
$669$	0	0
$670$	−18333.7	−1.05716
$671$	−13706.0	−0.788548
$672$	0	0
$673$	−17658.6	−1.01143	−0.505714	−	0.862701i	$-0.668771\pi$
$673$	−17658.6	−1.01143	−0.505714	+	0.862701i	$0.668771\pi$
$674$	36768.1	2.10127
$675$	0	0
$676$	−6205.34	−0.353057
$677$	9124.03	0.517969	0.258984	−	0.965881i	$-0.416612\pi$
$677$	9124.03	0.517969	0.258984	+	0.965881i	$0.416612\pi$
$678$	0	0
$679$	0	0
$680$	193.531	0.0109141
$681$	0	0
$682$	−42607.1	−2.39225
$683$	−10523.7	−0.589570	−0.294785	−	0.955564i	$-0.595248\pi$
$683$	−10523.7	−0.589570	−0.294785	+	0.955564i	$0.595248\pi$
$684$	0	0
$685$	−12563.6	−0.700775
$686$	0	0
$687$	0	0
$688$	244.286	0.0135368
$689$	6201.62	0.342907
$690$	0	0
$691$	32928.1	1.81280	0.906400	−	0.422420i	$-0.138819\pi$
$691$	32928.1	1.81280	0.906400	+	0.422420i	$0.138819\pi$
$692$	382.706	0.0210235
$693$	0	0
$694$	−56872.8	−3.11075
$695$	5828.54	0.318114
$696$	0	0
$697$	−584.024	−0.0317381
$698$	−20046.8	−1.08708
$699$	0	0
$700$	0	0
$701$	2582.50	0.139144	0.0695719	−	0.997577i	$-0.477837\pi$
$701$	2582.50	0.139144	0.0695719	+	0.997577i	$0.477837\pi$
$702$	0	0
$703$	−612.383	−0.0328542
$704$	−47001.7	−2.51625
$705$	0	0
$706$	−54786.4	−2.92056
$707$	0	0
$708$	0	0
$709$	−6866.00	−0.363693	−0.181846	−	0.983327i	$-0.558207\pi$
$709$	−6866.00	−0.363693	−0.181846	+	0.983327i	$0.558207\pi$
$710$	21479.7	1.13538
$711$	0	0
$712$	−19875.9	−1.04618
$713$	26191.4	1.37570
$714$	0	0
$715$	11580.7	0.605725
$716$	−8830.36	−0.460902
$717$	0	0
$718$	−49468.9	−2.57126
$719$	6581.85	0.341393	0.170696	−	0.985324i	$-0.445398\pi$
$719$	6581.85	0.341393	0.170696	+	0.985324i	$0.445398\pi$
$720$	0	0
$721$	0	0
$722$	29429.8	1.51699
$723$	0	0
$724$	−23819.4	−1.22271
$725$	156.512	0.00801753
$726$	0	0
$727$	15527.2	0.792119	0.396060	−	0.918225i	$-0.370377\pi$
$727$	15527.2	0.792119	0.396060	+	0.918225i	$0.370377\pi$
$728$	0	0
$729$	0	0
$730$	−20863.5	−1.05780
$731$	32.0792	0.00162311
$732$	0	0
$733$	−3391.32	−0.170889	−0.0854443	−	0.996343i	$-0.527231\pi$
$733$	−3391.32	−0.170889	−0.0854443	+	0.996343i	$0.527231\pi$
$734$	−26200.3	−1.31754
$735$	0	0
$736$	33527.7	1.67914
$737$	46406.7	2.31942
$738$	0	0
$739$	−14335.0	−0.713562	−0.356781	−	0.934188i	$-0.616126\pi$
$739$	−14335.0	−0.713562	−0.356781	+	0.934188i	$0.616126\pi$
$740$	−2222.63	−0.110413
$741$	0	0
$742$	0	0
$743$	27668.2	1.36615	0.683073	−	0.730350i	$-0.260643\pi$
$743$	27668.2	1.36615	0.683073	+	0.730350i	$0.260643\pi$
$744$	0	0
$745$	13187.4	0.648524
$746$	41851.8	2.05403
$747$	0	0
$748$	−1479.99	−0.0723446
$749$	0	0
$750$	0	0
$751$	−16929.9	−0.822609	−0.411305	−	0.911498i	$-0.634927\pi$
$751$	−16929.9	−0.822609	−0.411305	+	0.911498i	$0.634927\pi$
$752$	−2831.10	−0.137287
$753$	0	0
$754$	−1145.70	−0.0553370
$755$	−12911.6	−0.622386
$756$	0	0
$757$	19445.3	0.933622	0.466811	−	0.884357i	$-0.345403\pi$
$757$	19445.3	0.933622	0.466811	+	0.884357i	$0.345403\pi$
$758$	−24739.9	−1.18548
$759$	0	0
$760$	−1456.46	−0.0695150
$761$	−9996.97	−0.476202	−0.238101	−	0.971240i	$-0.576525\pi$
$761$	−9996.97	−0.476202	−0.238101	+	0.971240i	$0.576525\pi$
$762$	0	0
$763$	0	0
$764$	−25753.9	−1.21956
$765$	0	0
$766$	23423.6	1.10487
$767$	9588.76	0.451408
$768$	0	0
$769$	8382.34	0.393075	0.196538	−	0.980496i	$-0.437030\pi$
$769$	8382.34	0.393075	0.196538	+	0.980496i	$0.437030\pi$
$770$	0	0
$771$	0	0
$772$	12005.2	0.559684
$773$	11297.5	0.525668	0.262834	−	0.964841i	$-0.415343\pi$
$773$	11297.5	0.525668	0.262834	+	0.964841i	$0.415343\pi$
$774$	0	0
$775$	4217.00	0.195457
$776$	−13300.2	−0.615268
$777$	0	0
$778$	−48652.9	−2.24202
$779$	4395.20	0.202149
$780$	0	0
$781$	−54369.9	−2.49105
$782$	1518.42	0.0694356
$783$	0	0
$784$	0	0
$785$	1128.94	0.0513294
$786$	0	0
$787$	−25174.4	−1.14024	−0.570120	−	0.821561i	$-0.693103\pi$
$787$	−25174.4	−1.14024	−0.570120	+	0.821561i	$0.693103\pi$
$788$	−20529.6	−0.928093
$789$	0	0
$790$	−6706.43	−0.302030
$791$	0	0
$792$	0	0
$793$	−9930.22	−0.444681
$794$	−59448.4	−2.65711
$795$	0	0
$796$	−63830.5	−2.84222
$797$	26277.4	1.16787	0.583936	−	0.811800i	$-0.301512\pi$
$797$	26277.4	1.16787	0.583936	+	0.811800i	$0.301512\pi$
$798$	0	0
$799$	−371.775	−0.0164611
$800$	5398.21	0.238569
$801$	0	0
$802$	32286.3	1.42153
$803$	52810.2	2.32084
$804$	0	0
$805$	0	0
$806$	−30869.5	−1.34905
$807$	0	0
$808$	9888.48	0.430539
$809$	−15676.9	−0.681298	−0.340649	−	0.940191i	$-0.610647\pi$
$809$	−15676.9	−0.681298	−0.340649	+	0.940191i	$0.610647\pi$
$810$	0	0
$811$	26600.5	1.15175	0.575876	−	0.817537i	$-0.304661\pi$
$811$	26600.5	1.15175	0.575876	+	0.817537i	$0.304661\pi$
$812$	0	0
$813$	0	0
$814$	9389.77	0.404314
$815$	7445.04	0.319986
$816$	0	0
$817$	−241.419	−0.0103380
$818$	−26146.3	−1.11759
$819$	0	0
$820$	15952.3	0.679363
$821$	−35601.3	−1.51339	−0.756696	−	0.653767i	$-0.773188\pi$
$821$	−35601.3	−1.51339	−0.756696	+	0.653767i	$0.773188\pi$
$822$	0	0
$823$	41975.0	1.77783	0.888916	−	0.458071i	$-0.151459\pi$
$823$	41975.0	1.77783	0.888916	+	0.458071i	$0.151459\pi$
$824$	3630.75	0.153499
$825$	0	0
$826$	0	0
$827$	−27719.4	−1.16554	−0.582768	−	0.812639i	$-0.698030\pi$
$827$	−27719.4	−1.16554	−0.582768	+	0.812639i	$0.698030\pi$
$828$	0	0
$829$	27222.3	1.14049	0.570246	−	0.821474i	$-0.306848\pi$
$829$	27222.3	1.14049	0.570246	+	0.821474i	$0.306848\pi$
$830$	24293.9	1.01597
$831$	0	0
$832$	−34053.4	−1.41898
$833$	0	0
$834$	0	0
$835$	−14293.7	−0.592399
$836$	11138.0	0.460784
$837$	0	0
$838$	−37277.2	−1.53666
$839$	24081.1	0.990909	0.495454	−	0.868634i	$-0.335002\pi$
$839$	24081.1	0.990909	0.495454	+	0.868634i	$0.335002\pi$
$840$	0	0
$841$	−24349.8	−0.998393
$842$	48942.4	2.00317
$843$	0	0
$844$	10291.0	0.419704
$845$	−2594.62	−0.105631
$846$	0	0
$847$	0	0
$848$	−2523.56	−0.102193
$849$	0	0
$850$	244.477	0.00986529
$851$	−5772.06	−0.232507
$852$	0	0
$853$	−18493.0	−0.742309	−0.371155	−	0.928571i	$-0.621038\pi$
$853$	−18493.0	−0.742309	−0.371155	+	0.928571i	$0.621038\pi$
$854$	0	0
$855$	0	0
$856$	−25996.2	−1.03801
$857$	3499.25	0.139477	0.0697387	−	0.997565i	$-0.477783\pi$
$857$	3499.25	0.139477	0.0697387	+	0.997565i	$0.477783\pi$
$858$	0	0
$859$	−32158.0	−1.27732	−0.638660	−	0.769489i	$-0.720511\pi$
$859$	−32158.0	−1.27732	−0.638660	+	0.769489i	$0.720511\pi$
$860$	−876.224	−0.0347430
$861$	0	0
$862$	42071.9	1.66238
$863$	47704.7	1.88168	0.940838	−	0.338857i	$-0.110040\pi$
$863$	47704.7	1.88168	0.940838	+	0.338857i	$0.110040\pi$
$864$	0	0
$865$	160.020	0.00628999
$866$	−39241.1	−1.53980
$867$	0	0
$868$	0	0
$869$	16975.4	0.662661
$870$	0	0
$871$	33622.3	1.30798
$872$	13689.6	0.531639
$873$	0	0
$874$	−11427.2	−0.442255
$875$	0	0
$876$	0	0
$877$	−24437.3	−0.940923	−0.470461	−	0.882421i	$-0.655913\pi$
$877$	−24437.3	−0.940923	−0.470461	+	0.882421i	$0.655913\pi$
$878$	−47174.4	−1.81328
$879$	0	0
$880$	−4712.42	−0.180518
$881$	13975.8	0.534458	0.267229	−	0.963633i	$-0.413892\pi$
$881$	13975.8	0.534458	0.267229	+	0.963633i	$0.413892\pi$
$882$	0	0
$883$	3193.55	0.121712	0.0608558	−	0.998147i	$-0.480617\pi$
$883$	3193.55	0.121712	0.0608558	+	0.998147i	$0.480617\pi$
$884$	−1072.27	−0.0407969
$885$	0	0
$886$	−21792.8	−0.826348
$887$	−20099.3	−0.760843	−0.380422	−	0.924813i	$-0.624221\pi$
$887$	−20099.3	−0.760843	−0.380422	+	0.924813i	$0.624221\pi$
$888$	0	0
$889$	0	0
$890$	−25108.1	−0.945646
$891$	0	0
$892$	−43779.9	−1.64334
$893$	2797.87	0.104846
$894$	0	0
$895$	−3692.22	−0.137896
$896$	0	0
$897$	0	0
$898$	195.637	0.00727004
$899$	1056.02	0.0391771
$900$	0	0
$901$	−331.389	−0.0122532
$902$	−67392.3	−2.48771
$903$	0	0
$904$	−7181.20	−0.264207
$905$	−9959.55	−0.365819
$906$	0	0
$907$	23212.6	0.849792	0.424896	−	0.905242i	$-0.360311\pi$
$907$	23212.6	0.849792	0.424896	+	0.905242i	$0.360311\pi$
$908$	−72135.6	−2.63646
$909$	0	0
$910$	0	0
$911$	−2754.63	−0.100181	−0.0500905	−	0.998745i	$-0.515951\pi$
$911$	−2754.63	−0.100181	−0.0500905	+	0.998745i	$0.515951\pi$
$912$	0	0
$913$	−61493.1	−2.22905
$914$	−12820.5	−0.463965
$915$	0	0
$916$	−46034.2	−1.66049
$917$	0	0
$918$	0	0
$919$	16218.0	0.582137	0.291068	−	0.956702i	$-0.405989\pi$
$919$	16218.0	0.582137	0.291068	+	0.956702i	$0.405989\pi$
$920$	−13728.0	−0.491954
$921$	0	0
$922$	−80012.2	−2.85798
$923$	−39391.7	−1.40476
$924$	0	0
$925$	−929.344	−0.0330342
$926$	34089.8	1.20978
$927$	0	0
$928$	1351.81	0.0478184
$929$	−42592.6	−1.50422	−0.752109	−	0.659038i	$-0.770964\pi$
$929$	−42592.6	−1.50422	−0.752109	+	0.659038i	$0.770964\pi$
$930$	0	0
$931$	0	0
$932$	20390.9	0.716659
$933$	0	0
$934$	−27905.0	−0.977602
$935$	−618.825	−0.0216446
$936$	0	0
$937$	−4670.79	−0.162848	−0.0814238	−	0.996680i	$-0.525947\pi$
$937$	−4670.79	−0.162848	−0.0814238	+	0.996680i	$0.525947\pi$
$938$	0	0
$939$	0	0
$940$	10154.8	0.352355
$941$	−11036.5	−0.382338	−0.191169	−	0.981557i	$-0.561228\pi$
$941$	−11036.5	−0.382338	−0.191169	+	0.981557i	$0.561228\pi$
$942$	0	0
$943$	41427.2	1.43060
$944$	−3901.86	−0.134528
$945$	0	0
$946$	3701.71	0.127223
$947$	−28219.0	−0.968314	−0.484157	−	0.874981i	$-0.660874\pi$
$947$	−28219.0	−0.968314	−0.484157	+	0.874981i	$0.660874\pi$
$948$	0	0
$949$	38261.7	1.30878
$950$	−1839.86	−0.0628348
$951$	0	0
$952$	0	0
$953$	−42209.7	−1.43474	−0.717370	−	0.696693i	$-0.754654\pi$
$953$	−42209.7	−1.43474	−0.717370	+	0.696693i	$0.754654\pi$
$954$	0	0
$955$	−10768.4	−0.364877
$956$	−81369.2	−2.75279
$957$	0	0
$958$	−18404.0	−0.620674
$959$	0	0
$960$	0	0
$961$	−1337.99	−0.0449126
$962$	6803.02	0.228002
$963$	0	0
$964$	55521.0	1.85499
$965$	5019.70	0.167451
$966$	0	0
$967$	11522.8	0.383193	0.191596	−	0.981474i	$-0.438634\pi$
$967$	11522.8	0.383193	0.191596	+	0.981474i	$0.438634\pi$
$968$	−32992.2	−1.09547
$969$	0	0
$970$	−16801.3	−0.556143
$971$	−42660.3	−1.40992	−0.704961	−	0.709246i	$-0.749035\pi$
$971$	−42660.3	−1.40992	−0.704961	+	0.709246i	$0.749035\pi$
$972$	0	0
$973$	0	0
$974$	6350.30	0.208908
$975$	0	0
$976$	4040.80	0.132524
$977$	13991.7	0.458171	0.229085	−	0.973406i	$-0.426427\pi$
$977$	13991.7	0.458171	0.229085	+	0.973406i	$0.426427\pi$
$978$	0	0
$979$	63554.1	2.07477
$980$	0	0
$981$	0	0
$982$	85958.4	2.79332
$983$	−25963.2	−0.842420	−0.421210	−	0.906963i	$-0.638394\pi$
$983$	−25963.2	−0.842420	−0.421210	+	0.906963i	$0.638394\pi$
$984$	0	0
$985$	−8584.01	−0.277674
$986$	61.2217	0.00197738
$987$	0	0
$988$	8069.62	0.259847
$989$	−2275.51	−0.0731617
$990$	0	0
$991$	14482.3	0.464224	0.232112	−	0.972689i	$-0.425436\pi$
$991$	14482.3	0.464224	0.232112	+	0.972689i	$0.425436\pi$
$992$	36422.8	1.16575
$993$	0	0
$994$	0	0
$995$	−26689.3	−0.850359
$996$	0	0
$997$	−4184.45	−0.132922	−0.0664609	−	0.997789i	$-0.521171\pi$
$997$	−4184.45	−0.132922	−0.0664609	+	0.997789i	$0.521171\pi$
$998$	−42782.4	−1.35697
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.ca.1.1		6
3.2	odd	2		245.4.a.p.1.6	yes	6
7.6	odd	2		2205.4.a.bz.1.1		6
15.14	odd	2		1225.4.a.bi.1.1		6
21.2	odd	6		245.4.e.p.116.1		12
21.5	even	6		245.4.e.q.116.1		12
21.11	odd	6		245.4.e.p.226.1		12
21.17	even	6		245.4.e.q.226.1		12
21.20	even	2		245.4.a.o.1.6	✓	6
105.104	even	2		1225.4.a.bj.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.o.1.6	✓	6	21.20	even	2
245.4.a.p.1.6	yes	6	3.2	odd	2
245.4.e.p.116.1		12	21.2	odd	6
245.4.e.p.226.1		12	21.11	odd	6
245.4.e.q.116.1		12	21.5	even	6
245.4.e.q.226.1		12	21.17	even	6
1225.4.a.bi.1.1		6	15.14	odd	2
1225.4.a.bj.1.1		6	105.104	even	2
2205.4.a.bz.1.1		6	7.6	odd	2
2205.4.a.ca.1.1		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-4.46745 q^{2} +11.9581 q^{4} +5.00000 q^{5} -17.6824 q^{8} -22.3372 q^{10} +56.5404 q^{11} +40.9643 q^{13} -16.6692 q^{16} -2.18896 q^{17} +16.4735 q^{19} +59.7903 q^{20} -252.591 q^{22} +155.272 q^{23} +25.0000 q^{25} -183.006 q^{26} +6.26048 q^{29} +168.680 q^{31} +215.928 q^{32} +9.77908 q^{34} -37.1738 q^{37} -73.5946 q^{38} -88.4122 q^{40} +266.804 q^{41} -14.6549 q^{43} +676.114 q^{44} -693.670 q^{46} +169.840 q^{47} -111.686 q^{50} +489.854 q^{52} +151.391 q^{53} +282.702 q^{55} -27.9683 q^{58} +234.076 q^{59} -242.411 q^{61} -753.569 q^{62} -831.294 q^{64} +204.822 q^{65} +820.771 q^{67} -26.1758 q^{68} -961.611 q^{71} +934.026 q^{73} +166.072 q^{74} +196.992 q^{76} +300.236 q^{79} -83.3460 q^{80} -1191.93 q^{82} -1087.60 q^{83} -10.9448 q^{85} +65.4702 q^{86} -999.773 q^{88} +1124.05 q^{89} +1856.76 q^{92} -758.753 q^{94} +82.3676 q^{95} +752.168 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8} + 10 q^{10} + 16 q^{11} + 168 q^{13} + 298 q^{16} + 4 q^{17} + 308 q^{19} + 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} - 56 q^{26} - 176 q^{29} + 392 q^{31}+ \cdots + 516 q^{97}+O(q^{100}) \) 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8 + 10 * q^10 + 16 * q^11 + 168 * q^13 + 298 * q^16 + 4 * q^17 + 308 * q^19 + 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 - 56 * q^26 - 176 * q^29 + 392 * q^31 + 770 * q^32 + 812 * q^34 - 140 * q^37 - 20 * q^38 + 330 * q^40 - 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 - 628 * q^47 + 50 * q^50 + 1520 * q^52 + 676 * q^53 + 80 * q^55 - 2012 * q^58 - 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 + 2940 * q^68 + 224 * q^71 + 2640 * q^73 - 928 * q^74 - 1340 * q^76 + 1636 * q^79 + 1490 * q^80 - 1756 * q^82 - 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 + 1904 * q^89 + 1952 * q^92 - 3332 * q^94 + 1540 * q^95 + 516 * q^97

Introduction

L-functions

Modular forms

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