LMFDB - Embedded newform 1225.4.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.46745 q^{2} -9.80920 q^{3} +11.9581 q^{4} +43.8221 q^{6} -17.6824 q^{8} +69.2204 q^{9} -56.5404 q^{11} -117.299 q^{12} -40.9643 q^{13} -16.6692 q^{16} -2.18896 q^{17} -309.238 q^{18} +16.4735 q^{19} +252.591 q^{22} +155.272 q^{23} +173.451 q^{24} +183.006 q^{26} -414.148 q^{27} -6.26048 q^{29} +168.680 q^{31} +215.928 q^{32} +554.616 q^{33} +9.77908 q^{34} +827.742 q^{36} +37.1738 q^{37} -73.5946 q^{38} +401.827 q^{39} -266.804 q^{41} +14.6549 q^{43} -676.114 q^{44} -693.670 q^{46} +169.840 q^{47} +163.511 q^{48} +21.4720 q^{51} -489.854 q^{52} +151.391 q^{53} +1850.18 q^{54} -161.592 q^{57} +27.9683 q^{58} -234.076 q^{59} -242.411 q^{61} -753.569 q^{62} -831.294 q^{64} -2477.72 q^{66} -820.771 q^{67} -26.1758 q^{68} -1523.10 q^{69} +961.611 q^{71} -1223.98 q^{72} -934.026 q^{73} -166.072 q^{74} +196.992 q^{76} -1795.14 q^{78} +300.236 q^{79} +2193.51 q^{81} +1191.93 q^{82} -1087.60 q^{83} -65.4702 q^{86} +61.4103 q^{87} +999.773 q^{88} -1124.05 q^{89} +1856.76 q^{92} -1654.62 q^{93} -758.753 q^{94} -2118.08 q^{96} -752.168 q^{97} -3913.75 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} - 16 q^{3} + 14 q^{4} + 24 q^{6} + 66 q^{8} + 70 q^{9} - 16 q^{11} - 160 q^{12} - 168 q^{13} + 298 q^{16} + 4 q^{17} - 354 q^{18} + 308 q^{19} + 236 q^{22} + 336 q^{23} - 92 q^{24} + 56 q^{26}+ \cdots - 2804 q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 - 16 * q^3 + 14 * q^4 + 24 * q^6 + 66 * q^8 + 70 * q^9 - 16 * q^11 - 160 * q^12 - 168 * q^13 + 298 * q^16 + 4 * q^17 - 354 * q^18 + 308 * q^19 + 236 * q^22 + 336 * q^23 - 92 * q^24 + 56 * q^26 - 964 * q^27 + 176 * q^29 + 392 * q^31 + 770 * q^32 - 188 * q^33 + 812 * q^34 + 230 * q^36 + 140 * q^37 - 20 * q^38 + 140 * q^39 + 656 * q^41 + 388 * q^43 - 160 * q^44 - 388 * q^46 - 628 * q^47 - 1396 * q^48 + 744 * q^51 - 1520 * q^52 + 676 * q^53 + 2284 * q^54 - 1468 * q^57 + 2012 * q^58 + 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 - 3620 * q^66 - 1768 * q^67 + 2940 * q^68 - 1048 * q^69 - 224 * q^71 - 2858 * q^72 - 2640 * q^73 + 928 * q^74 - 1340 * q^76 - 8 * q^78 + 1636 * q^79 + 4442 * q^81 + 1756 * q^82 - 140 * q^83 + 1180 * q^86 - 1940 * q^87 + 5652 * q^88 - 1904 * q^89 + 1952 * q^92 + 1592 * q^93 - 3332 * q^94 - 6460 * q^96 - 516 * q^97 - 2804 * q^99

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	$0.789785\pi$
$3$	−9.80920	−1.88778	−0.943890	−	0.330259i	$-0.892864\pi$
$3$	−9.80920	−1.88778	−0.943890	+	0.330259i	$0.892864\pi$
$4$	11.9581	1.49476
$5$	0	0
$5$	0	0
$6$	43.8221	2.98171
$7$	0	0
$7$	0	0
$8$	−17.6824	−0.781461
$9$	69.2204	2.56372
$10$	0	0
$11$	−56.5404	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$11$	−56.5404	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$12$	−117.299	−2.82178
$13$	−40.9643	−0.873958	−0.436979	−	0.899472i	$-0.643952\pi$
$13$	−40.9643	−0.873958	−0.436979	+	0.899472i	$0.643952\pi$
$14$	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	$0.504971\pi$
$18$	−309.238	−4.04934
$19$	16.4735	0.198910	0.0994549	−	0.995042i	$-0.468290\pi$
$19$	16.4735	0.198910	0.0994549	+	0.995042i	$0.468290\pi$
$20$	0	0
$21$	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	$0.251468\pi$
$24$	173.451	1.47523
$25$	0	0
$26$	183.006	1.38040
$27$	−414.148	−2.95195
$28$	0	0
$29$	−6.26048	−0.0400876	−0.0200438	−	0.999799i	$-0.506381\pi$
$29$	−6.26048	−0.0400876	−0.0200438	+	0.999799i	$0.506381\pi$
$30$	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
$33$	554.616	2.92564
$34$	9.77908	0.0493264
$35$	0	0
$36$	827.742	3.83214
$37$	37.1738	0.165171	0.0825856	−	0.996584i	$-0.473682\pi$
$37$	37.1738	0.165171	0.0825856	+	0.996584i	$0.473682\pi$
$38$	−73.5946	−0.314174
$39$	401.827	1.64984
$40$	0	0
$41$	−266.804	−1.01629	−0.508143	−	0.861273i	$-0.669668\pi$
$41$	−266.804	−1.01629	−0.508143	+	0.861273i	$0.669668\pi$
$42$	0	0
$43$	14.6549	0.0519735	0.0259867	−	0.999662i	$-0.491727\pi$
$43$	14.6549	0.0519735	0.0259867	+	0.999662i	$0.491727\pi$
$44$	−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	$0.415106\pi$
$48$	163.511	0.491684
$49$	0	0
$50$	0	0
$51$	21.4720	0.0589545
$52$	−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	$0.437146\pi$
$54$	1850.18	4.66255
$55$	0	0
$56$	0	0
$57$	−161.592	−0.375498
$58$	27.9683	0.0633176
$59$	−234.076	−0.516510	−0.258255	−	0.966077i	$-0.583147\pi$
$59$	−234.076	−0.516510	−0.258255	+	0.966077i	$0.583147\pi$
$60$	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
$65$	0	0
$66$	−2477.72	−4.62100
$67$	−820.771	−1.49661	−0.748307	−	0.663353i	$-0.769133\pi$
$67$	−820.771	−1.49661	−0.748307	+	0.663353i	$0.769133\pi$
$68$	−26.1758	−0.0466806
$69$	−1523.10	−2.65738
$70$	0	0
$71$	961.611	1.60736	0.803678	−	0.595065i	$-0.202874\pi$
$71$	961.611	1.60736	0.803678	+	0.595065i	$0.202874\pi$
$72$	−1223.98	−2.00344
$73$	−934.026	−1.49753	−0.748763	−	0.662837i	$-0.769352\pi$
$73$	−934.026	−1.49753	−0.748763	+	0.662837i	$0.769352\pi$
$74$	−166.072	−0.260885
$75$	0	0
$76$	196.992	0.297322
$77$	0	0
$78$	−1795.14	−2.60589
$79$	300.236	0.427584	0.213792	−	0.976879i	$-0.431419\pi$
$79$	300.236	0.427584	0.213792	+	0.976879i	$0.431419\pi$
$80$	0	0
$81$	2193.51	3.00893
$82$	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
$85$	0	0
$86$	−65.4702	−0.0820910
$87$	61.4103	0.0756767
$88$	999.773	1.21109
$89$	−1124.05	−1.33875	−0.669375	−	0.742925i	$-0.733438\pi$
$89$	−1124.05	−1.33875	−0.669375	+	0.742925i	$0.733438\pi$
$90$	0	0
$91$	0	0
$92$	1856.76	2.10413
$93$	−1654.62	−1.84490
$94$	−758.753	−0.832547
$95$	0	0
$96$	−2118.08	−2.25183
$97$	−752.168	−0.787330	−0.393665	−	0.919254i	$-0.628793\pi$
$97$	−752.168	−0.787330	−0.393665	+	0.919254i	$0.628793\pi$
$98$	0	0
$99$	−3913.75	−3.97320
$100$	0	0
$101$	559.226	0.550941	0.275471	−	0.961310i	$-0.411166\pi$
$101$	559.226	0.550941	0.275471	+	0.961310i	$0.411166\pi$
$102$	−95.9249	−0.0931175
$103$	205.331	0.196426	0.0982128	−	0.995165i	$-0.468687\pi$
$103$	205.331	0.196426	0.0982128	+	0.995165i	$0.468687\pi$
$104$	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	$0.268796\pi$
$108$	−4952.41	−4.41246
$109$	−774.193	−0.680314	−0.340157	−	0.940369i	$-0.610480\pi$
$109$	−774.193	−0.680314	−0.340157	+	0.940369i	$0.610480\pi$
$110$	0	0
$111$	−364.645	−0.311807
$112$	0	0
$113$	406.121	0.338094	0.169047	−	0.985608i	$-0.445931\pi$
$113$	406.121	0.338094	0.169047	+	0.985608i	$0.445931\pi$
$114$	721.904	0.593092
$115$	0	0
$116$	−74.8632	−0.0599213
$117$	−2835.56	−2.24058
$118$	1045.72	0.815817
$119$	0	0
$120$	0	0
$121$	1865.82	1.40182
$122$	1082.96	0.803660
$123$	2617.13	1.91853
$124$	2017.09	1.46081
$125$	0	0
$126$	0	0
$127$	451.639	0.315563	0.157781	−	0.987474i	$-0.449566\pi$
$127$	451.639	0.315563	0.157781	+	0.987474i	$0.449566\pi$
$128$	1986.33	1.37163
$129$	−143.753	−0.0981145
$130$	0	0
$131$	361.932	0.241390	0.120695	−	0.992690i	$-0.461488\pi$
$131$	361.932	0.241390	0.120695	+	0.992690i	$0.461488\pi$
$132$	6632.14	4.37313
$133$	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	$0.786563\pi$
$138$	6804.35	4.19728
$139$	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
$141$	−1666.00	−0.995052
$142$	−4295.94	−2.53879
$143$	2316.14	1.35444
$144$	−1153.85	−0.667736
$145$	0	0
$146$	4172.71	2.36531
$147$	0	0
$148$	444.526	0.246891
$149$	−2637.49	−1.45014	−0.725072	−	0.688673i	$-0.758194\pi$
$149$	−2637.49	−1.45014	−0.725072	+	0.688673i	$0.758194\pi$
$150$	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
$153$	−151.521	−0.0800637
$154$	0	0
$155$	0	0
$156$	4805.07	2.46611
$157$	−225.788	−0.114776	−0.0573880	−	0.998352i	$-0.518277\pi$
$157$	−225.788	−0.114776	−0.0573880	+	0.998352i	$0.518277\pi$
$158$	−1341.29	−0.675361
$159$	−1485.02	−0.740691
$160$	0	0
$161$	0	0
$162$	−9799.38	−4.75254
$163$	−1489.01	−0.715509	−0.357755	−	0.933816i	$-0.616458\pi$
$163$	−1489.01	−0.715509	−0.357755	+	0.933816i	$0.616458\pi$
$164$	−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
$170$	0	0
$171$	1140.30	0.509948
$172$	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	$0.497761\pi$
$174$	−274.347	−0.119530
$175$	0	0
$176$	942.483	0.403650
$177$	2296.10	0.975057
$178$	5021.62	2.11453
$179$	738.444	0.308346	0.154173	−	0.988044i	$-0.450729\pi$
$179$	738.444	0.308346	0.154173	+	0.988044i	$0.450729\pi$
$180$	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
$183$	2377.86	0.960528
$184$	−2745.59	−1.10004
$185$	0	0
$186$	7391.91	2.91399
$187$	123.765	0.0483989
$188$	2030.96	0.787890
$189$	0	0
$190$	0	0
$191$	2153.68	0.815890	0.407945	−	0.913006i	$-0.366245\pi$
$191$	2153.68	0.815890	0.407945	+	0.913006i	$0.366245\pi$
$192$	8154.33	3.06504
$193$	−1003.94	−0.374431	−0.187215	−	0.982319i	$-0.559946\pi$
$193$	−1003.94	−0.374431	−0.187215	+	0.982319i	$0.559946\pi$
$194$	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	$0.600479\pi$
$198$	17484.5	6.27559
$199$	−5337.86	−1.90146	−0.950730	−	0.310019i	$-0.899665\pi$
$199$	−5337.86	−1.90146	−0.950730	+	0.310019i	$0.899665\pi$
$200$	0	0
$201$	8051.10	2.82528
$202$	−2498.31	−0.870201
$203$	0	0
$204$	256.763	0.0881227
$205$	0	0
$206$	−917.304	−0.310250
$207$	10748.0	3.60888
$208$	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
$213$	−9432.63	−3.03433
$214$	−6567.92	−2.09801
$215$	0	0
$216$	7323.14	2.30684
$217$	0	0
$218$	3458.66	1.07454
$219$	9162.04	2.82700
$220$	0	0
$221$	89.6695	0.0272933
$222$	1629.03	0.492493
$223$	3661.12	1.09940	0.549701	−	0.835361i	$-0.314741\pi$
$223$	3661.12	1.09940	0.549701	+	0.835361i	$0.314741\pi$
$224$	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	$0.843737\pi$
$228$	−1932.33	−0.561279
$229$	−3849.64	−1.11088	−0.555439	−	0.831557i	$-0.687450\pi$
$229$	−3849.64	−1.11088	−0.555439	+	0.831557i	$0.687450\pi$
$230$	0	0
$231$	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	$0.422943\pi$
$234$	12667.7	3.53895
$235$	0	0
$236$	−2799.09	−0.772057
$237$	−2945.07	−0.807185
$238$	0	0
$239$	6804.54	1.84163	0.920814	−	0.390001i	$-0.127525\pi$
$239$	6804.54	1.84163	0.920814	+	0.390001i	$0.127525\pi$
$240$	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
$243$	−10334.6	−2.72824
$244$	−2898.77	−0.760553
$245$	0	0
$246$	−11691.9	−3.03027
$247$	−674.827	−0.173839
$248$	−2982.68	−0.763711
$249$	10668.4	2.71520
$250$	0	0
$251$	5912.64	1.48686	0.743431	−	0.668813i	$-0.233197\pi$
$251$	5912.64	1.48686	0.743431	+	0.668813i	$0.233197\pi$
$252$	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	$0.433940\pi$
$258$	642.210	0.154970
$259$	0	0
$260$	0	0
$261$	−433.352	−0.102773
$262$	−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	$0.268974\pi$
$264$	−9806.97	−2.28628
$265$	0	0
$266$	0	0
$267$	11026.0	2.52727
$268$	−9814.83	−2.23708
$269$	5330.01	1.20809	0.604046	−	0.796950i	$-0.293554\pi$
$269$	5330.01	1.20809	0.604046	+	0.796950i	$0.293554\pi$
$270$	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
$275$	0	0
$276$	−18213.3	−3.97214
$277$	867.657	0.188204	0.0941019	−	0.995563i	$-0.470002\pi$
$277$	867.657	0.188204	0.0941019	+	0.995563i	$0.470002\pi$
$278$	−5207.73	−1.12352
$279$	11676.1	2.50548
$280$	0	0
$281$	2049.70	0.435142	0.217571	−	0.976045i	$-0.430187\pi$
$281$	2049.70	0.435142	0.217571	+	0.976045i	$0.430187\pi$
$282$	7442.76	1.57167
$283$	−6625.11	−1.39160	−0.695799	−	0.718237i	$-0.744949\pi$
$283$	−6625.11	−1.39160	−0.695799	+	0.718237i	$0.744949\pi$
$284$	11499.0	2.40261
$285$	0	0
$286$	−10347.2	−2.13932
$287$	0	0
$288$	14946.6	3.05812
$289$	−4908.21	−0.999025
$290$	0	0
$291$	7378.16	1.48631
$292$	−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
$295$	0	0
$296$	−657.323	−0.129075
$297$	23416.1	4.57488
$298$	11782.8	2.29048
$299$	−6360.62	−1.23025
$300$	0	0
$301$	0	0
$302$	11536.4	2.19816
$303$	−5485.56	−1.04006
$304$	−274.600	−0.0518073
$305$	0	0
$306$	676.911	0.126459
$307$	272.638	0.0506849	0.0253424	−	0.999679i	$-0.491932\pi$
$307$	272.638	0.0506849	0.0253424	+	0.999679i	$0.491932\pi$
$308$	0	0
$309$	−2014.13	−0.370808
$310$	0	0
$311$	−1220.83	−0.222595	−0.111298	−	0.993787i	$-0.535501\pi$
$311$	−1220.83	−0.222595	−0.111298	+	0.993787i	$0.535501\pi$
$312$	−7105.28	−1.28929
$313$	−775.195	−0.139989	−0.0699946	−	0.997547i	$-0.522298\pi$
$313$	−775.195	−0.139989	−0.0699946	+	0.997547i	$0.522298\pi$
$314$	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	$0.727315\pi$
$318$	6634.25	1.16991
$319$	353.970	0.0621270
$320$	0	0
$321$	−14421.2	−2.50752
$322$	0	0
$323$	−36.0600	−0.00621186
$324$	26230.1	4.49762
$325$	0	0
$326$	6652.06	1.13013
$327$	7594.21	1.28428
$328$	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
$333$	2573.18	0.423452
$334$	12771.2	2.09225
$335$	0	0
$336$	0	0
$337$	8230.24	1.33036	0.665178	−	0.746685i	$-0.268356\pi$
$337$	8230.24	1.33036	0.665178	+	0.746685i	$0.268356\pi$
$338$	2318.27	0.373068
$339$	−3983.72	−0.638247
$340$	0	0
$341$	−9537.25	−1.51458
$342$	−5094.24	−0.805453
$343$	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	$0.0556809\pi$
$348$	734.348	0.113118
$349$	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
$351$	16965.3	2.57989
$352$	−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	$0.124457\pi$
$354$	−10257.7	−1.54008
$355$	0	0
$356$	−13441.4	−2.00111
$357$	0	0
$358$	−3298.96	−0.487026
$359$	−11073.2	−1.62791	−0.813957	−	0.580925i	$-0.802691\pi$
$359$	−11073.2	−1.62791	−0.813957	+	0.580925i	$0.802691\pi$
$360$	0	0
$361$	−6587.62	−0.960435
$362$	8898.74	1.29201
$363$	−18302.2	−2.64632
$364$	0	0
$365$	0	0
$366$	−10623.0	−1.51713
$367$	−5864.72	−0.834157	−0.417079	−	0.908870i	$-0.636946\pi$
$367$	−5864.72	−0.834157	−0.417079	+	0.908870i	$0.636946\pi$
$368$	−2588.26	−0.366637
$369$	−18468.3	−2.60547
$370$	0	0
$371$	0	0
$372$	−19786.0	−2.75768
$373$	9368.17	1.30044	0.650222	−	0.759744i	$-0.274676\pi$
$373$	9368.17	1.30044	0.650222	+	0.759744i	$0.274676\pi$
$374$	−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	$0.377548\pi$
$380$	0	0
$381$	−4430.21	−0.595713
$382$	−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	$0.613736\pi$
$384$	−19484.3	−2.58934
$385$	0	0
$386$	4485.05	0.591406
$387$	1014.42	0.133245
$388$	−8994.47	−1.17687
$389$	−10890.5	−1.41947	−0.709733	−	0.704471i	$-0.751184\pi$
$389$	−10890.5	−1.41947	−0.709733	+	0.704471i	$0.751184\pi$
$390$	0	0
$391$	−339.886	−0.0439610
$392$	0	0
$393$	−3550.26	−0.455692
$394$	7669.71	0.980697
$395$	0	0
$396$	−46800.8	−5.93897
$397$	−13307.0	−1.68227	−0.841135	−	0.540826i	$-0.818112\pi$
$397$	−13307.0	−1.68227	−0.841135	+	0.540826i	$0.818112\pi$
$398$	23846.6	3.00332
$399$	0	0
$400$	0	0
$401$	7227.02	0.900001	0.450000	−	0.893028i	$-0.351424\pi$
$401$	7227.02	0.900001	0.450000	+	0.893028i	$0.351424\pi$
$402$	−35967.9	−4.46247
$403$	−6909.87	−0.854107
$404$	6687.26	0.823524
$405$	0	0
$406$	0	0
$407$	−2101.82	−0.255979
$408$	−379.677	−0.0460706
$409$	5852.64	0.707565	0.353783	−	0.935328i	$-0.384895\pi$
$409$	5852.64	0.707565	0.353783	+	0.935328i	$0.384895\pi$
$410$	0	0
$411$	24647.8	2.95812
$412$	2455.36	0.293609
$413$	0	0
$414$	−48016.1	−5.70015
$415$	0	0
$416$	−8845.35	−1.04250
$417$	−11434.7	−1.34282
$418$	4161.07	0.486901
$419$	−8344.19	−0.972889	−0.486444	−	0.873712i	$-0.661706\pi$
$419$	−8344.19	−0.972889	−0.486444	+	0.873712i	$0.661706\pi$
$420$	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
$423$	11756.4	1.35134
$424$	−2676.96	−0.306615
$425$	0	0
$426$	42139.8	4.79267
$427$	0	0
$428$	17580.4	1.98547
$429$	−22719.5	−2.55689
$430$	0	0
$431$	9417.43	1.05249	0.526243	−	0.850334i	$-0.323600\pi$
$431$	9417.43	1.05249	0.526243	+	0.850334i	$0.323600\pi$
$432$	6903.51	0.768855
$433$	−8783.79	−0.974878	−0.487439	−	0.873157i	$-0.662069\pi$
$433$	−8783.79	−0.974878	−0.487439	+	0.873157i	$0.662069\pi$
$434$	0	0
$435$	0	0
$436$	−9257.85	−1.01690
$437$	2557.88	0.280000
$438$	−40930.9	−4.46519
$439$	10559.6	1.14802	0.574011	−	0.818847i	$-0.305386\pi$
$439$	10559.6	1.14802	0.574011	+	0.818847i	$0.305386\pi$
$440$	0	0
$441$	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	$0.415754\pi$
$444$	−4360.45	−0.466076
$445$	0	0
$446$	−16355.9	−1.73649
$447$	25871.7	2.73756
$448$	0	0
$449$	43.7917	0.00460280	0.00230140	−	0.999997i	$-0.499267\pi$
$449$	43.7917	0.00460280	0.00230140	+	0.999997i	$0.499267\pi$
$450$	0	0
$451$	15085.2	1.57502
$452$	4856.42	0.505369
$453$	25330.5	2.62722
$454$	26949.3	2.78589
$455$	0	0
$456$	2857.34	0.293437
$457$	−2869.76	−0.293745	−0.146873	−	0.989155i	$-0.546921\pi$
$457$	−2869.76	−0.293745	−0.146873	+	0.989155i	$0.546921\pi$
$458$	17198.0	1.75461
$459$	906.555	0.0921882
$460$	0	0
$461$	−17910.1	−1.80945	−0.904723	−	0.426001i	$-0.859922\pi$
$461$	−17910.1	−1.80945	−0.904723	+	0.426001i	$0.859922\pi$
$462$	0	0
$463$	7630.72	0.765938	0.382969	−	0.923761i	$-0.374902\pi$
$463$	7630.72	0.765938	0.382969	+	0.923761i	$0.374902\pi$
$464$	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	$0.399849\pi$
$468$	−33907.9	−3.34913
$469$	0	0
$470$	0	0
$471$	2214.80	0.216672
$472$	4139.03	0.403632
$473$	−828.597	−0.0805474
$474$	13156.9	1.27493
$475$	0	0
$476$	0	0
$477$	10479.3	1.00590
$478$	−30398.9	−2.90882
$479$	−4119.58	−0.392961	−0.196480	−	0.980508i	$-0.562951\pi$
$479$	−4119.58	−0.392961	−0.196480	+	0.980508i	$0.562951\pi$
$480$	0	0
$481$	−1522.80	−0.144353
$482$	−20742.2	−1.96013
$483$	0	0
$484$	22311.6	2.09538
$485$	0	0
$486$	46169.1	4.30920
$487$	1421.46	0.132264	0.0661320	−	0.997811i	$-0.478934\pi$
$487$	1421.46	0.132264	0.0661320	+	0.997811i	$0.478934\pi$
$488$	4286.43	0.397618
$489$	14606.0	1.35073
$490$	0	0
$491$	19241.1	1.76851	0.884253	−	0.467007i	$-0.154668\pi$
$491$	19241.1	1.76851	0.884253	+	0.467007i	$0.154668\pi$
$492$	31295.8	2.86773
$493$	13.7040	0.00125192
$494$	3014.75	0.274575
$495$	0	0
$496$	−2811.76	−0.254540
$497$	0	0
$498$	−47660.6	−4.28860
$499$	9576.48	0.859123	0.429561	−	0.903038i	$-0.358668\pi$
$499$	9576.48	0.859123	0.429561	+	0.903038i	$0.358668\pi$
$500$	0	0
$501$	28041.9	2.50064
$502$	−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
$505$	0	0
$506$	39220.4	3.44577
$507$	5090.24	0.445888
$508$	5400.72	0.471690
$509$	7082.86	0.616782	0.308391	−	0.951260i	$-0.400209\pi$
$509$	7082.86	0.616782	0.308391	+	0.951260i	$0.400209\pi$
$510$	0	0
$511$	0	0
$512$	−5957.37	−0.514220
$513$	−6822.47	−0.587173
$514$	−7584.98	−0.650893
$515$	0	0
$516$	−1719.01	−0.146657
$517$	−9602.85	−0.816891
$518$	0	0
$519$	−313.933	−0.0265513
$520$	0	0
$521$	−7153.92	−0.601572	−0.300786	−	0.953692i	$-0.597249\pi$
$521$	−7153.92	−0.601572	−0.300786	+	0.953692i	$0.597249\pi$
$522$	1935.98	0.162328
$523$	15781.3	1.31944	0.659721	−	0.751511i	$-0.270674\pi$
$523$	15781.3	1.31944	0.659721	+	0.751511i	$0.270674\pi$
$524$	4328.00	0.360820
$525$	0	0
$526$	−25293.7	−2.09669
$527$	−369.235	−0.0305202
$528$	−9245.00	−0.762002
$529$	11942.5	0.981547
$530$	0	0
$531$	−16202.8	−1.32418
$532$	0	0
$533$	10929.4	0.888192
$534$	−49258.0	−3.99177
$535$	0	0
$536$	14513.2	1.16955
$537$	−7243.54	−0.582089
$538$	−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
$543$	19539.0	1.54420
$544$	−472.659	−0.0372520
$545$	0	0
$546$	0	0
$547$	191.079	0.0149359	0.00746796	−	0.999972i	$-0.497623\pi$
$547$	191.079	0.0149359	0.00746796	+	0.999972i	$0.497623\pi$
$548$	−30047.3	−2.34226
$549$	−16779.8	−1.30445
$550$	0	0
$551$	−103.132	−0.00797382
$552$	26932.1	2.07664
$553$	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	$0.295972\pi$
$558$	−52162.3	−3.95736
$559$	−600.330	−0.0454226
$560$	0	0
$561$	−1214.04	−0.0913665
$562$	−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	$0.536351\pi$
$564$	−19922.1	−1.48736
$565$	0	0
$566$	29597.3	2.19800
$567$	0	0
$568$	−17003.6	−1.25609
$569$	13052.2	0.961647	0.480823	−	0.876817i	$-0.340338\pi$
$569$	13052.2	0.961647	0.480823	+	0.876817i	$0.340338\pi$
$570$	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
$573$	−21125.9	−1.54022
$574$	0	0
$575$	0	0
$576$	−57542.5	−4.16250
$577$	22790.3	1.64432	0.822162	−	0.569254i	$-0.192768\pi$
$577$	22790.3	1.64432	0.822162	+	0.569254i	$0.192768\pi$
$578$	21927.2	1.57794
$579$	9847.84	0.706844
$580$	0	0
$581$	0	0
$582$	−32961.5	−2.34759
$583$	−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
$590$	0	0
$591$	16840.4	1.17212
$592$	−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	$0.508530\pi$
$594$	−104610.	−7.22593
$595$	0	0
$596$	−31539.3	−2.16762
$597$	52360.1	3.58954
$598$	28415.7	1.94315
$599$	−13641.0	−0.930479	−0.465239	−	0.885185i	$-0.654032\pi$
$599$	−13641.0	−0.930479	−0.465239	+	0.885185i	$0.654032\pi$
$600$	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
$603$	−56814.1	−3.83689
$604$	−30879.6	−2.08025
$605$	0	0
$606$	24506.4	1.64275
$607$	−3234.90	−0.216311	−0.108155	−	0.994134i	$-0.534494\pi$
$607$	−3234.90	−0.216311	−0.108155	+	0.994134i	$0.534494\pi$
$608$	3557.10	0.237269
$609$	0	0
$610$	0	0
$611$	−6957.40	−0.460665
$612$	−1811.90	−0.119676
$613$	−8413.52	−0.554354	−0.277177	−	0.960819i	$-0.589399\pi$
$613$	−8413.52	−0.554354	−0.277177	+	0.960819i	$0.589399\pi$
$614$	−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	$0.574171\pi$
$618$	8998.01	0.585685
$619$	5464.74	0.354841	0.177420	−	0.984135i	$-0.443225\pi$
$619$	5464.74	0.354841	0.177420	+	0.984135i	$0.443225\pi$
$620$	0	0
$621$	−64305.7	−4.15539
$622$	5454.01	0.351585
$623$	0	0
$624$	−6698.13	−0.429711
$625$	0	0
$626$	3463.14	0.221110
$627$	9136.48	0.581939
$628$	−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
$633$	−8441.69	−0.530058
$634$	33028.9	2.06900
$635$	0	0
$636$	−17758.0	−1.10715
$637$	0	0
$638$	−1581.34	−0.0981284
$639$	66563.0	4.12080
$640$	0	0
$641$	−14578.3	−0.898298	−0.449149	−	0.893457i	$-0.648273\pi$
$641$	−14578.3	−0.898298	−0.449149	+	0.893457i	$0.648273\pi$
$642$	64426.0	3.96058
$643$	11980.0	0.734750	0.367375	−	0.930073i	$-0.380256\pi$
$643$	11980.0	0.734750	0.367375	+	0.930073i	$0.380256\pi$
$644$	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	$0.445070\pi$
$648$	−38786.6	−2.35136
$649$	13234.7	0.800476
$650$	0	0
$651$	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	$0.477368\pi$
$654$	−33926.7	−2.02850
$655$	0	0
$656$	4447.40	0.264698
$657$	−64653.6	−3.83923
$658$	0	0
$659$	5233.92	0.309385	0.154692	−	0.987963i	$-0.450561\pi$
$659$	5233.92	0.309385	0.154692	+	0.987963i	$0.450561\pi$
$660$	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
$663$	−879.585	−0.0515238
$664$	19231.3	1.12398
$665$	0	0
$666$	−11495.5	−0.668834
$667$	−972.079	−0.0564303
$668$	−34184.9	−1.98002
$669$	−35912.7	−2.07543
$670$	0	0
$671$	13706.0	0.788548
$672$	0	0
$673$	17658.6	1.01143	0.505714	−	0.862701i	$-0.331229\pi$
$673$	17658.6	1.01143	0.505714	+	0.862701i	$0.331229\pi$
$674$	−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	$0.416612\pi$
$678$	17797.0	1.00810
$679$	0	0
$680$	0	0
$681$	59172.8	3.32967
$682$	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	$0.595248\pi$
$684$	13635.8	0.762250
$685$	0	0
$686$	0	0
$687$	37761.9	2.09710
$688$	−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
$695$	0	0
$696$	−1085.88	−0.0591384
$697$	584.024	0.0317381
$698$	−20046.8	−1.08708
$699$	−16726.7	−0.905093
$700$	0	0
$701$	−2582.50	−0.139144	−0.0695719	−	0.997577i	$-0.522163\pi$
$701$	−2582.50	−0.139144	−0.0695719	+	0.997577i	$0.522163\pi$
$702$	−75791.5	−4.07488
$703$	612.383	0.0328542
$704$	47001.7	2.51625
$705$	0	0
$706$	−54786.4	−2.92056
$707$	0	0
$708$	27456.9	1.45748
$709$	−6866.00	−0.363693	−0.181846	−	0.983327i	$-0.558207\pi$
$709$	−6866.00	−0.363693	−0.181846	+	0.983327i	$0.558207\pi$
$710$	0	0
$711$	20782.4	1.09620
$712$	19875.9	1.04618
$713$	26191.4	1.37570
$714$	0	0
$715$	0	0
$716$	8830.36	0.460902
$717$	−66747.1	−3.47659
$718$	49468.9	2.57126
$719$	−6581.85	−0.341393	−0.170696	−	0.985324i	$-0.554602\pi$
$719$	−6581.85	−0.341393	−0.170696	+	0.985324i	$0.554602\pi$
$720$	0	0
$721$	0	0
$722$	29429.8	1.51699
$723$	−45543.9	−2.34273
$724$	−23819.4	−1.22271
$725$	0	0
$726$	81764.0	4.17982
$727$	−15527.2	−0.792119	−0.396060	−	0.918225i	$-0.629623\pi$
$727$	−15527.2	−0.792119	−0.396060	+	0.918225i	$0.629623\pi$
$728$	0	0
$729$	42149.0	2.14139
$730$	0	0
$731$	−32.0792	−0.00162311
$732$	28434.6	1.43576
$733$	3391.32	0.170889	0.0854443	−	0.996343i	$-0.472769\pi$
$733$	3391.32	0.170889	0.0854443	+	0.996343i	$0.472769\pi$
$734$	26200.3	1.31754
$735$	0	0
$736$	33527.7	1.67914
$737$	46406.7	2.31942
$738$	82505.9	4.11529
$739$	−14335.0	−0.713562	−0.356781	−	0.934188i	$-0.616126\pi$
$739$	−14335.0	−0.713562	−0.356781	+	0.934188i	$0.616126\pi$
$740$	0	0
$741$	6619.51	0.328170
$742$	0	0
$743$	27668.2	1.36615	0.683073	−	0.730350i	$-0.260643\pi$
$743$	27668.2	1.36615	0.683073	+	0.730350i	$0.260643\pi$
$744$	29257.7	1.44172
$745$	0	0
$746$	−41851.8	−2.05403
$747$	−75283.7	−3.68740
$748$	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
$753$	−57998.2	−2.80687
$754$	−1145.70	−0.0553370
$755$	0	0
$756$	0	0
$757$	−19445.3	−0.933622	−0.466811	−	0.884357i	$-0.654597\pi$
$757$	−19445.3	−0.933622	−0.466811	+	0.884357i	$0.654597\pi$
$758$	−24739.9	−1.18548
$759$	86116.5	4.11836
$760$	0	0
$761$	9996.97	0.476202	0.238101	−	0.971240i	$-0.423475\pi$
$761$	9996.97	0.476202	0.238101	+	0.971240i	$0.423475\pi$
$762$	19791.7	0.940917
$763$	0	0
$764$	25753.9	1.21956
$765$	0	0
$766$	23423.6	1.10487
$767$	9588.76	0.451408
$768$	21810.6	1.02477
$769$	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
$771$	−16654.4	−0.777942
$772$	−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	$0.415343\pi$
$774$	−4531.87	−0.210458
$775$	0	0
$776$	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
$783$	2592.76	0.118337
$784$	0	0
$785$	0	0
$786$	15860.6	0.719756
$787$	25174.4	1.14024	0.570120	−	0.821561i	$-0.306897\pi$
$787$	25174.4	1.14024	0.570120	+	0.821561i	$0.306897\pi$
$788$	−20529.6	−0.928093
$789$	−55537.5	−2.50594
$790$	0	0
$791$	0	0
$792$	69204.6	3.10490
$793$	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
$800$	0	0
$801$	−77806.9	−3.43217
$802$	−32286.3	−1.42153
$803$	52810.2	2.32084
$804$	96275.6	4.22311
$805$	0	0
$806$	30869.5	1.34905
$807$	−52283.1	−2.28061
$808$	−9888.48	−0.430539
$809$	15676.9	0.681298	0.340649	−	0.940191i	$-0.389353\pi$
$809$	15676.9	0.681298	0.340649	+	0.940191i	$0.389353\pi$
$810$	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
$813$	19961.1	0.861090
$814$	9389.77	0.404314
$815$	0	0
$816$	−357.921	−0.0153551
$817$	241.419	0.0103380
$818$	−26146.3	−1.11759
$819$	0	0
$820$	0	0
$821$	35601.3	1.51339	0.756696	−	0.653767i	$-0.226812\pi$
$821$	35601.3	1.51339	0.756696	+	0.653767i	$0.226812\pi$
$822$	−110113.	−4.67229
$823$	−41975.0	−1.77783	−0.888916	−	0.458071i	$-0.848541\pi$
$823$	−41975.0	−1.77783	−0.888916	+	0.458071i	$0.848541\pi$
$824$	−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	$0.698030\pi$
$828$	128525.	5.39440
$829$	27222.3	1.14049	0.570246	−	0.821474i	$-0.306848\pi$
$829$	27222.3	1.14049	0.570246	+	0.821474i	$0.306848\pi$
$830$	0	0
$831$	−8511.01	−0.355287
$832$	34053.4	1.41898
$833$	0	0
$834$	51083.7	2.12096
$835$	0	0
$836$	−11138.0	−0.460784
$837$	−69858.5	−2.88490
$838$	37277.2	1.53666
$839$	−24081.1	−0.990909	−0.495454	−	0.868634i	$-0.664998\pi$
$839$	−24081.1	−0.990909	−0.495454	+	0.868634i	$0.664998\pi$
$840$	0	0
$841$	−24349.8	−0.998393
$842$	48942.4	2.00317
$843$	−20105.9	−0.821452
$844$	10291.0	0.419704
$845$	0	0
$846$	−52521.1	−2.13441
$847$	0	0
$848$	−2523.56	−0.102193
$849$	64987.0	2.62703
$850$	0	0
$851$	5772.06	0.232507
$852$	−112796.	−4.53560
$853$	18493.0	0.742309	0.371155	−	0.928571i	$-0.378962\pi$
$853$	18493.0	0.742309	0.371155	+	0.928571i	$0.378962\pi$
$854$	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	$0.477783\pi$
$858$	101498.	4.03856
$859$	−32158.0	−1.27732	−0.638660	−	0.769489i	$-0.720511\pi$
$859$	−32158.0	−1.27732	−0.638660	+	0.769489i	$0.720511\pi$
$860$	0	0
$861$	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	$0.110040\pi$
$864$	−89426.2	−3.52123
$865$	0	0
$866$	39241.1	1.53980
$867$	48145.6	1.88594
$868$	0	0
$869$	−16975.4	−0.662661
$870$	0	0
$871$	33622.3	1.30798
$872$	13689.6	0.531639
$873$	−52065.3	−2.01849
$874$	−11427.2	−0.442255
$875$	0	0
$876$	109560.	4.22569
$877$	24437.3	0.940923	0.470461	−	0.882421i	$-0.344087\pi$
$877$	24437.3	0.940923	0.470461	+	0.882421i	$0.344087\pi$
$878$	−47174.4	−1.81328
$879$	−55626.4	−2.13451
$880$	0	0
$881$	−13975.8	−0.534458	−0.267229	−	0.963633i	$-0.586108\pi$
$881$	−13975.8	−0.534458	−0.267229	+	0.963633i	$0.586108\pi$
$882$	0	0
$883$	−3193.55	−0.121712	−0.0608558	−	0.998147i	$-0.519383\pi$
$883$	−3193.55	−0.121712	−0.0608558	+	0.998147i	$0.519383\pi$
$884$	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	$0.624221\pi$
$888$	6447.81	0.243665
$889$	0	0
$890$	0	0
$891$	−124022.	−4.66317
$892$	43779.9	1.64334
$893$	2797.87	0.104846
$894$	−115580.	−4.32392
$895$	0	0
$896$	0	0
$897$	62392.6	2.32244
$898$	−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
$905$	0	0
$906$	−113163.	−4.14965
$907$	−23212.6	−0.849792	−0.424896	−	0.905242i	$-0.639689\pi$
$907$	−23212.6	−0.849792	−0.424896	+	0.905242i	$0.639689\pi$
$908$	−72135.6	−2.63646
$909$	38709.8	1.41246
$910$	0	0
$911$	2754.63	0.100181	0.0500905	−	0.998745i	$-0.484049\pi$
$911$	2754.63	0.100181	0.0500905	+	0.998745i	$0.484049\pi$
$912$	2693.61	0.0978008
$913$	61493.1	2.22905
$914$	12820.5	0.463965
$915$	0	0
$916$	−46034.2	−1.66049
$917$	0	0
$918$	−4049.98	−0.145609
$919$	16218.0	0.582137	0.291068	−	0.956702i	$-0.405989\pi$
$919$	16218.0	0.582137	0.291068	+	0.956702i	$0.405989\pi$
$920$	0	0
$921$	−2674.36	−0.0956819
$922$	80012.2	2.85798
$923$	−39391.7	−1.40476
$924$	0	0
$925$	0	0
$926$	−34089.8	−1.20978
$927$	14213.1	0.503580
$928$	−1351.81	−0.0478184
$929$	42592.6	1.50422	0.752109	−	0.659038i	$-0.229036\pi$
$929$	42592.6	1.50422	0.752109	+	0.659038i	$0.229036\pi$
$930$	0	0
$931$	0	0
$932$	20390.9	0.716659
$933$	11975.4	0.420211
$934$	−27905.0	−0.977602
$935$	0	0
$936$	50139.7	1.75093
$937$	4670.79	0.162848	0.0814238	−	0.996680i	$-0.474053\pi$
$937$	4670.79	0.162848	0.0814238	+	0.996680i	$0.474053\pi$
$938$	0	0
$939$	7604.04	0.264269
$940$	0	0
$941$	11036.5	0.382338	0.191169	−	0.981557i	$-0.438772\pi$
$941$	11036.5	0.382338	0.191169	+	0.981557i	$0.438772\pi$
$942$	−9894.48	−0.342229
$943$	−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	$0.660874\pi$
$948$	−35217.3	−1.20655
$949$	38261.7	1.30878
$950$	0	0
$951$	72521.7	2.47285
$952$	0	0
$953$	−42209.7	−1.43474	−0.717370	−	0.696693i	$-0.754654\pi$
$953$	−42209.7	−1.43474	−0.717370	+	0.696693i	$0.754654\pi$
$954$	−46815.8	−1.58880
$955$	0	0
$956$	81369.2	2.75279
$957$	−3472.16	−0.117282
$958$	18404.0	0.620674
$959$	0	0
$960$	0	0
$961$	−1337.99	−0.0449126
$962$	6803.02	0.228002
$963$	101766.	3.40536
$964$	55521.0	1.85499
$965$	0	0
$966$	0	0
$967$	−11522.8	−0.383193	−0.191596	−	0.981474i	$-0.561366\pi$
$967$	−11522.8	−0.383193	−0.191596	+	0.981474i	$0.561366\pi$
$968$	−32992.2	−1.09547
$969$	353.719	0.0117266
$970$	0	0
$971$	42660.3	1.40992	0.704961	−	0.709246i	$-0.250965\pi$
$971$	42660.3	1.40992	0.704961	+	0.709246i	$0.250965\pi$
$972$	−123581.	−4.07806
$973$	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	$0.426427\pi$
$978$	−65251.3	−2.13344
$979$	63554.1	2.07477
$980$	0	0
$981$	−53589.9	−1.74413
$982$	−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	$0.638394\pi$
$984$	−46277.3	−1.49925
$985$	0	0
$986$	−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
$993$	36656.5	1.17146
$994$	0	0
$995$	0	0
$996$	127574.	4.05857
$997$	4184.45	0.132922	0.0664609	−	0.997789i	$-0.478829\pi$
$997$	4184.45	0.132922	0.0664609	+	0.997789i	$0.478829\pi$
$998$	−42782.4	−1.35697
$999$	−15395.4	−0.487578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.bi.1.1		6
5.4	even	2		245.4.a.p.1.6	yes	6
7.6	odd	2		1225.4.a.bj.1.1		6
15.14	odd	2		2205.4.a.ca.1.1		6
35.4	even	6		245.4.e.p.226.1		12
35.9	even	6		245.4.e.p.116.1		12
35.19	odd	6		245.4.e.q.116.1		12
35.24	odd	6		245.4.e.q.226.1		12
35.34	odd	2		245.4.a.o.1.6	✓	6
105.104	even	2		2205.4.a.bz.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.o.1.6	✓	6	35.34	odd	2
245.4.a.p.1.6	yes	6	5.4	even	2
245.4.e.p.116.1		12	35.9	even	6
245.4.e.p.226.1		12	35.4	even	6
245.4.e.q.116.1		12	35.19	odd	6
245.4.e.q.226.1		12	35.24	odd	6
1225.4.a.bi.1.1		6	1.1	even	1	trivial
1225.4.a.bj.1.1		6	7.6	odd	2
2205.4.a.bz.1.1		6	105.104	even	2
2205.4.a.ca.1.1		6	15.14	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-4.46745 q^{2} -9.80920 q^{3} +11.9581 q^{4} +43.8221 q^{6} -17.6824 q^{8} +69.2204 q^{9} -56.5404 q^{11} -117.299 q^{12} -40.9643 q^{13} -16.6692 q^{16} -2.18896 q^{17} -309.238 q^{18} +16.4735 q^{19} +252.591 q^{22} +155.272 q^{23} +173.451 q^{24} +183.006 q^{26} -414.148 q^{27} -6.26048 q^{29} +168.680 q^{31} +215.928 q^{32} +554.616 q^{33} +9.77908 q^{34} +827.742 q^{36} +37.1738 q^{37} -73.5946 q^{38} +401.827 q^{39} -266.804 q^{41} +14.6549 q^{43} -676.114 q^{44} -693.670 q^{46} +169.840 q^{47} +163.511 q^{48} +21.4720 q^{51} -489.854 q^{52} +151.391 q^{53} +1850.18 q^{54} -161.592 q^{57} +27.9683 q^{58} -234.076 q^{59} -242.411 q^{61} -753.569 q^{62} -831.294 q^{64} -2477.72 q^{66} -820.771 q^{67} -26.1758 q^{68} -1523.10 q^{69} +961.611 q^{71} -1223.98 q^{72} -934.026 q^{73} -166.072 q^{74} +196.992 q^{76} -1795.14 q^{78} +300.236 q^{79} +2193.51 q^{81} +1191.93 q^{82} -1087.60 q^{83} -65.4702 q^{86} +61.4103 q^{87} +999.773 q^{88} -1124.05 q^{89} +1856.76 q^{92} -1654.62 q^{93} -758.753 q^{94} -2118.08 q^{96} -752.168 q^{97} -3913.75 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} - 16 q^{3} + 14 q^{4} + 24 q^{6} + 66 q^{8} + 70 q^{9} - 16 q^{11} - 160 q^{12} - 168 q^{13} + 298 q^{16} + 4 q^{17} - 354 q^{18} + 308 q^{19} + 236 q^{22} + 336 q^{23} - 92 q^{24} + 56 q^{26}+ \cdots - 2804 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 - 16 * q^3 + 14 * q^4 + 24 * q^6 + 66 * q^8 + 70 * q^9 - 16 * q^11 - 160 * q^12 - 168 * q^13 + 298 * q^16 + 4 * q^17 - 354 * q^18 + 308 * q^19 + 236 * q^22 + 336 * q^23 - 92 * q^24 + 56 * q^26 - 964 * q^27 + 176 * q^29 + 392 * q^31 + 770 * q^32 - 188 * q^33 + 812 * q^34 + 230 * q^36 + 140 * q^37 - 20 * q^38 + 140 * q^39 + 656 * q^41 + 388 * q^43 - 160 * q^44 - 388 * q^46 - 628 * q^47 - 1396 * q^48 + 744 * q^51 - 1520 * q^52 + 676 * q^53 + 2284 * q^54 - 1468 * q^57 + 2012 * q^58 + 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 - 3620 * q^66 - 1768 * q^67 + 2940 * q^68 - 1048 * q^69 - 224 * q^71 - 2858 * q^72 - 2640 * q^73 + 928 * q^74 - 1340 * q^76 - 8 * q^78 + 1636 * q^79 + 4442 * q^81 + 1756 * q^82 - 140 * q^83 + 1180 * q^86 - 1940 * q^87 + 5652 * q^88 - 1904 * q^89 + 1952 * q^92 + 1592 * q^93 - 3332 * q^94 - 6460 * q^96 - 516 * q^97 - 2804 * q^99

Introduction

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