LMFDB - Embedded newform 98.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,4,Mod(1,98)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("98.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	98.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:	$5.78218718056$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{22}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 22 $ x^2 - 22
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.69042$ of defining polynomial
Character	$\chi$	$=$	98.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -9.38083 q^{3} +4.00000 q^{4} +9.38083 q^{5} -18.7617 q^{6} +8.00000 q^{8} +61.0000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -9.38083 q^{3} +4.00000 q^{4} +9.38083 q^{5} -18.7617 q^{6} +8.00000 q^{8} +61.0000 q^{9} +18.7617 q^{10} +20.0000 q^{11} -37.5233 q^{12} +65.6658 q^{13} -88.0000 q^{15} +16.0000 q^{16} +56.2850 q^{17} +122.000 q^{18} +9.38083 q^{19} +37.5233 q^{20} +40.0000 q^{22} +48.0000 q^{23} -75.0467 q^{24} -37.0000 q^{25} +131.332 q^{26} -318.948 q^{27} -166.000 q^{29} -176.000 q^{30} -206.378 q^{31} +32.0000 q^{32} -187.617 q^{33} +112.570 q^{34} +244.000 q^{36} -78.0000 q^{37} +18.7617 q^{38} -616.000 q^{39} +75.0467 q^{40} +393.995 q^{41} +436.000 q^{43} +80.0000 q^{44} +572.231 q^{45} +96.0000 q^{46} +206.378 q^{47} -150.093 q^{48} -74.0000 q^{50} -528.000 q^{51} +262.663 q^{52} +62.0000 q^{53} -637.897 q^{54} +187.617 q^{55} -88.0000 q^{57} -332.000 q^{58} -666.039 q^{59} -352.000 q^{60} +272.044 q^{61} -412.757 q^{62} +64.0000 q^{64} +616.000 q^{65} -375.233 q^{66} +580.000 q^{67} +225.140 q^{68} -450.280 q^{69} -544.000 q^{71} +488.000 q^{72} -600.373 q^{73} -156.000 q^{74} +347.091 q^{75} +37.5233 q^{76} -1232.00 q^{78} -680.000 q^{79} +150.093 q^{80} +1345.00 q^{81} +787.990 q^{82} +196.997 q^{83} +528.000 q^{85} +872.000 q^{86} +1557.22 q^{87} +160.000 q^{88} -1500.93 q^{89} +1144.46 q^{90} +192.000 q^{92} +1936.00 q^{93} +412.757 q^{94} +88.0000 q^{95} -300.187 q^{96} -656.658 q^{97} +1220.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 122 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 + 122 * q^9	$ 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 122 q^{9} + 40 q^{11} - 176 q^{15} + 32 q^{16} + 244 q^{18} + 80 q^{22} + 96 q^{23} - 74 q^{25} - 332 q^{29} - 352 q^{30} + 64 q^{32} + 488 q^{36} - 156 q^{37} - 1232 q^{39} + 872 q^{43} + 160 q^{44} + 192 q^{46} - 148 q^{50} - 1056 q^{51} + 124 q^{53} - 176 q^{57} - 664 q^{58} - 704 q^{60} + 128 q^{64} + 1232 q^{65} + 1160 q^{67} - 1088 q^{71} + 976 q^{72} - 312 q^{74} - 2464 q^{78} - 1360 q^{79} + 2690 q^{81} + 1056 q^{85} + 1744 q^{86} + 320 q^{88} + 384 q^{92} + 3872 q^{93} + 176 q^{95} + 2440 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 + 122 * q^9 + 40 * q^11 - 176 * q^15 + 32 * q^16 + 244 * q^18 + 80 * q^22 + 96 * q^23 - 74 * q^25 - 332 * q^29 - 352 * q^30 + 64 * q^32 + 488 * q^36 - 156 * q^37 - 1232 * q^39 + 872 * q^43 + 160 * q^44 + 192 * q^46 - 148 * q^50 - 1056 * q^51 + 124 * q^53 - 176 * q^57 - 664 * q^58 - 704 * q^60 + 128 * q^64 + 1232 * q^65 + 1160 * q^67 - 1088 * q^71 + 976 * q^72 - 312 * q^74 - 2464 * q^78 - 1360 * q^79 + 2690 * q^81 + 1056 * q^85 + 1744 * q^86 + 320 * q^88 + 384 * q^92 + 3872 * q^93 + 176 * q^95 + 2440 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−9.38083	−1.80534	−0.902671	−	0.430331i	$-0.858397\pi$
$3$	−9.38083	−1.80534	−0.902671	+	0.430331i	$0.858397\pi$
$4$	4.00000	0.500000
$5$	9.38083	0.839047	0.419524	−	0.907744i	$-0.362197\pi$
$5$	9.38083	0.839047	0.419524	+	0.907744i	$0.362197\pi$
$6$	−18.7617	−1.27657
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	61.0000	2.25926
$10$	18.7617	0.593296
$11$	20.0000	0.548202	0.274101	−	0.961701i	$-0.411620\pi$
$11$	20.0000	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$12$	−37.5233	−0.902671
$13$	65.6658	1.40096	0.700478	−	0.713674i	$-0.252970\pi$
$13$	65.6658	1.40096	0.700478	+	0.713674i	$0.252970\pi$
$14$	0	0
$15$	−88.0000	−1.51477
$16$	16.0000	0.250000
$17$	56.2850	0.803007	0.401503	−	0.915858i	$-0.368488\pi$
$17$	56.2850	0.803007	0.401503	+	0.915858i	$0.368488\pi$
$18$	122.000	1.59754
$19$	9.38083	0.113269	0.0566345	−	0.998395i	$-0.481963\pi$
$19$	9.38083	0.113269	0.0566345	+	0.998395i	$0.481963\pi$
$20$	37.5233	0.419524
$21$	0	0
$22$	40.0000	0.387638
$23$	48.0000	0.435161	0.217580	−	0.976042i	$-0.430184\pi$
$23$	48.0000	0.435161	0.217580	+	0.976042i	$0.430184\pi$
$24$	−75.0467	−0.638285
$25$	−37.0000	−0.296000
$26$	131.332	0.990625
$27$	−318.948	−2.27339
$28$	0	0
$29$	−166.000	−1.06295	−0.531473	−	0.847075i	$-0.678361\pi$
$29$	−166.000	−1.06295	−0.531473	+	0.847075i	$0.678361\pi$
$30$	−176.000	−1.07110
$31$	−206.378	−1.19570	−0.597849	−	0.801609i	$-0.703978\pi$
$31$	−206.378	−1.19570	−0.597849	+	0.801609i	$0.703978\pi$
$32$	32.0000	0.176777
$33$	−187.617	−0.989693
$34$	112.570	0.567812
$35$	0	0
$36$	244.000	1.12963
$37$	−78.0000	−0.346571	−0.173285	−	0.984872i	$-0.555438\pi$
$37$	−78.0000	−0.346571	−0.173285	+	0.984872i	$0.555438\pi$
$38$	18.7617	0.0800933
$39$	−616.000	−2.52920
$40$	75.0467	0.296648
$41$	393.995	1.50077	0.750386	−	0.661000i	$-0.229868\pi$
$41$	393.995	1.50077	0.750386	+	0.661000i	$0.229868\pi$
$42$	0	0
$43$	436.000	1.54626	0.773132	−	0.634245i	$-0.218689\pi$
$43$	436.000	1.54626	0.773132	+	0.634245i	$0.218689\pi$
$44$	80.0000	0.274101
$45$	572.231	1.89562
$46$	96.0000	0.307705
$47$	206.378	0.640497	0.320249	−	0.947334i	$-0.396234\pi$
$47$	206.378	0.640497	0.320249	+	0.947334i	$0.396234\pi$
$48$	−150.093	−0.451335
$49$	0	0
$50$	−74.0000	−0.209304
$51$	−528.000	−1.44970
$52$	262.663	0.700478
$53$	62.0000	0.160686	0.0803430	−	0.996767i	$-0.474398\pi$
$53$	62.0000	0.160686	0.0803430	+	0.996767i	$0.474398\pi$
$54$	−637.897	−1.60753
$55$	187.617	0.459968
$56$	0	0
$57$	−88.0000	−0.204489
$58$	−332.000	−0.751616
$59$	−666.039	−1.46968	−0.734838	−	0.678243i	$-0.762742\pi$
$59$	−666.039	−1.46968	−0.734838	+	0.678243i	$0.762742\pi$
$60$	−352.000	−0.757383
$61$	272.044	0.571011	0.285506	−	0.958377i	$-0.407838\pi$
$61$	272.044	0.571011	0.285506	+	0.958377i	$0.407838\pi$
$62$	−412.757	−0.845486
$63$	0	0
$64$	64.0000	0.125000
$65$	616.000	1.17547
$66$	−375.233	−0.699819
$67$	580.000	1.05759	0.528793	−	0.848751i	$-0.322645\pi$
$67$	580.000	1.05759	0.528793	+	0.848751i	$0.322645\pi$
$68$	225.140	0.401503
$69$	−450.280	−0.785613
$70$	0	0
$71$	−544.000	−0.909309	−0.454654	−	0.890668i	$-0.650237\pi$
$71$	−544.000	−0.909309	−0.454654	+	0.890668i	$0.650237\pi$
$72$	488.000	0.798769
$73$	−600.373	−0.962580	−0.481290	−	0.876561i	$-0.659832\pi$
$73$	−600.373	−0.962580	−0.481290	+	0.876561i	$0.659832\pi$
$74$	−156.000	−0.245063
$75$	347.091	0.534381
$76$	37.5233	0.0566345
$77$	0	0
$78$	−1232.00	−1.78842
$79$	−680.000	−0.968430	−0.484215	−	0.874949i	$-0.660895\pi$
$79$	−680.000	−0.968430	−0.484215	+	0.874949i	$0.660895\pi$
$80$	150.093	0.209762
$81$	1345.00	1.84499
$82$	787.990	1.06121
$83$	196.997	0.260521	0.130261	−	0.991480i	$-0.458419\pi$
$83$	196.997	0.260521	0.130261	+	0.991480i	$0.458419\pi$
$84$	0	0
$85$	528.000	0.673760
$86$	872.000	1.09337
$87$	1557.22	1.91898
$88$	160.000	0.193819
$89$	−1500.93	−1.78762	−0.893812	−	0.448441i	$-0.851979\pi$
$89$	−1500.93	−1.78762	−0.893812	+	0.448441i	$0.851979\pi$
$90$	1144.46	1.34041
$91$	0	0
$92$	192.000	0.217580
$93$	1936.00	2.15864
$94$	412.757	0.452900
$95$	88.0000	0.0950380
$96$	−300.187	−0.319142
$97$	−656.658	−0.687356	−0.343678	−	0.939088i	$-0.611673\pi$
$97$	−656.658	−0.687356	−0.343678	+	0.939088i	$0.611673\pi$
$98$	0	0
$99$	1220.00	1.23853
$100$	−148.000	−0.148000
$101$	121.951	0.120144	0.0600721	−	0.998194i	$-0.480867\pi$
$101$	121.951	0.120144	0.0600721	+	0.998194i	$0.480867\pi$
$102$	−1056.00	−1.02509
$103$	−1369.60	−1.31020	−0.655101	−	0.755541i	$-0.727374\pi$
$103$	−1369.60	−1.31020	−0.655101	+	0.755541i	$0.727374\pi$
$104$	525.327	0.495313
$105$	0	0
$106$	124.000	0.113622
$107$	−260.000	−0.234908	−0.117454	−	0.993078i	$-0.537473\pi$
$107$	−260.000	−0.234908	−0.117454	+	0.993078i	$0.537473\pi$
$108$	−1275.79	−1.13670
$109$	1882.00	1.65379	0.826894	−	0.562358i	$-0.190106\pi$
$109$	1882.00	1.65379	0.826894	+	0.562358i	$0.190106\pi$
$110$	375.233	0.325246
$111$	731.705	0.625679
$112$	0	0
$113$	−1286.00	−1.07059	−0.535295	−	0.844665i	$-0.679800\pi$
$113$	−1286.00	−1.07059	−0.535295	+	0.844665i	$0.679800\pi$
$114$	−176.000	−0.144596
$115$	450.280	0.365120
$116$	−664.000	−0.531473
$117$	4005.62	3.16512
$118$	−1332.08	−1.03922
$119$	0	0
$120$	−704.000	−0.535551
$121$	−931.000	−0.699474
$122$	544.088	0.403766
$123$	−3696.00	−2.70941
$124$	−825.513	−0.597849
$125$	−1519.69	−1.08741
$126$	0	0
$127$	2312.00	1.61541	0.807704	−	0.589588i	$-0.200710\pi$
$127$	2312.00	1.61541	0.807704	+	0.589588i	$0.200710\pi$
$128$	128.000	0.0883883
$129$	−4090.04	−2.79154
$130$	1232.00	0.831181
$131$	−253.282	−0.168927	−0.0844633	−	0.996427i	$-0.526918\pi$
$131$	−253.282	−0.168927	−0.0844633	+	0.996427i	$0.526918\pi$
$132$	−750.467	−0.494846
$133$	0	0
$134$	1160.00	0.747826
$135$	−2992.00	−1.90748
$136$	450.280	0.283906
$137$	−1114.00	−0.694711	−0.347356	−	0.937733i	$-0.612920\pi$
$137$	−1114.00	−0.694711	−0.347356	+	0.937733i	$0.612920\pi$
$138$	−900.560	−0.555513
$139$	1378.98	0.841466	0.420733	−	0.907185i	$-0.361773\pi$
$139$	1378.98	0.841466	0.420733	+	0.907185i	$0.361773\pi$
$140$	0	0
$141$	−1936.00	−1.15632
$142$	−1088.00	−0.642978
$143$	1313.32	0.768007
$144$	976.000	0.564815
$145$	−1557.22	−0.891862
$146$	−1200.75	−0.680647
$147$	0	0
$148$	−312.000	−0.173285
$149$	−946.000	−0.520130	−0.260065	−	0.965591i	$-0.583744\pi$
$149$	−946.000	−0.520130	−0.260065	+	0.965591i	$0.583744\pi$
$150$	694.182	0.377865
$151$	832.000	0.448392	0.224196	−	0.974544i	$-0.428024\pi$
$151$	832.000	0.448392	0.224196	+	0.974544i	$0.428024\pi$
$152$	75.0467	0.0400466
$153$	3433.38	1.81420
$154$	0	0
$155$	−1936.00	−1.00325
$156$	−2464.00	−1.26460
$157$	2879.92	1.46396	0.731982	−	0.681324i	$-0.238596\pi$
$157$	2879.92	1.46396	0.731982	+	0.681324i	$0.238596\pi$
$158$	−1360.00	−0.684783
$159$	−581.612	−0.290093
$160$	300.187	0.148324
$161$	0	0
$162$	2690.00	1.30461
$163$	636.000	0.305616	0.152808	−	0.988256i	$-0.451168\pi$
$163$	636.000	0.305616	0.152808	+	0.988256i	$0.451168\pi$
$164$	1575.98	0.750386
$165$	−1760.00	−0.830399
$166$	393.995	0.184216
$167$	−656.658	−0.304274	−0.152137	−	0.988359i	$-0.548615\pi$
$167$	−656.658	−0.304274	−0.152137	+	0.988359i	$0.548615\pi$
$168$	0	0
$169$	2115.00	0.962676
$170$	1056.00	0.476421
$171$	572.231	0.255904
$172$	1744.00	0.773132
$173$	666.039	0.292705	0.146353	−	0.989232i	$-0.453247\pi$
$173$	666.039	0.292705	0.146353	+	0.989232i	$0.453247\pi$
$174$	3114.44	1.35692
$175$	0	0
$176$	320.000	0.137051
$177$	6248.00	2.65327
$178$	−3001.87	−1.26404
$179$	−3228.00	−1.34789	−0.673944	−	0.738782i	$-0.735401\pi$
$179$	−3228.00	−1.34789	−0.673944	+	0.738782i	$0.735401\pi$
$180$	2288.92	0.947812
$181$	−2823.63	−1.15955	−0.579776	−	0.814776i	$-0.696860\pi$
$181$	−2823.63	−1.15955	−0.579776	+	0.814776i	$0.696860\pi$
$182$	0	0
$183$	−2552.00	−1.03087
$184$	384.000	0.153852
$185$	−731.705	−0.290789
$186$	3872.00	1.52639
$187$	1125.70	0.440210
$188$	825.513	0.320249
$189$	0	0
$190$	176.000	0.0672020
$191$	−2136.00	−0.809191	−0.404596	−	0.914496i	$-0.632588\pi$
$191$	−2136.00	−0.809191	−0.404596	+	0.914496i	$0.632588\pi$
$192$	−600.373	−0.225668
$193$	1658.00	0.618370	0.309185	−	0.951002i	$-0.399944\pi$
$193$	1658.00	0.618370	0.309185	+	0.951002i	$0.399944\pi$
$194$	−1313.32	−0.486034
$195$	−5778.59	−2.12212
$196$	0	0
$197$	−978.000	−0.353704	−0.176852	−	0.984237i	$-0.556591\pi$
$197$	−978.000	−0.353704	−0.176852	+	0.984237i	$0.556591\pi$
$198$	2440.00	0.875774
$199$	−4934.32	−1.75771	−0.878855	−	0.477088i	$-0.841692\pi$
$199$	−4934.32	−1.75771	−0.878855	+	0.477088i	$0.841692\pi$
$200$	−296.000	−0.104652
$201$	−5440.88	−1.90930
$202$	243.902	0.0849547
$203$	0	0
$204$	−2112.00	−0.724851
$205$	3696.00	1.25922
$206$	−2739.20	−0.926453
$207$	2928.00	0.983140
$208$	1050.65	0.350239
$209$	187.617	0.0620943
$210$	0	0
$211$	1556.00	0.507675	0.253838	−	0.967247i	$-0.418307\pi$
$211$	1556.00	0.507675	0.253838	+	0.967247i	$0.418307\pi$
$212$	248.000	0.0803430
$213$	5103.17	1.64161
$214$	−520.000	−0.166105
$215$	4090.04	1.29739
$216$	−2551.59	−0.803766
$217$	0	0
$218$	3764.00	1.16940
$219$	5632.00	1.73779
$220$	750.467	0.229984
$221$	3696.00	1.12498
$222$	1463.41	0.442422
$223$	2889.30	0.867630	0.433815	−	0.901002i	$-0.357167\pi$
$223$	2889.30	0.867630	0.433815	+	0.901002i	$0.357167\pi$
$224$	0	0
$225$	−2257.00	−0.668741
$226$	−2572.00	−0.757022
$227$	−1979.36	−0.578742	−0.289371	−	0.957217i	$-0.593446\pi$
$227$	−1979.36	−0.578742	−0.289371	+	0.957217i	$0.593446\pi$
$228$	−352.000	−0.102245
$229$	2767.35	0.798565	0.399282	−	0.916828i	$-0.369259\pi$
$229$	2767.35	0.798565	0.399282	+	0.916828i	$0.369259\pi$
$230$	900.560	0.258179
$231$	0	0
$232$	−1328.00	−0.375808
$233$	−6490.00	−1.82478	−0.912391	−	0.409321i	$-0.865766\pi$
$233$	−6490.00	−1.82478	−0.912391	+	0.409321i	$0.865766\pi$
$234$	8011.23	2.23808
$235$	1936.00	0.537407
$236$	−2664.16	−0.734838
$237$	6378.97	1.74835
$238$	0	0
$239$	−4296.00	−1.16270	−0.581350	−	0.813654i	$-0.697475\pi$
$239$	−4296.00	−1.16270	−0.581350	+	0.813654i	$0.697475\pi$
$240$	−1408.00	−0.378692
$241$	4521.56	1.20854	0.604272	−	0.796778i	$-0.293464\pi$
$241$	4521.56	1.20854	0.604272	+	0.796778i	$0.293464\pi$
$242$	−1862.00	−0.494603
$243$	−4005.62	−1.05745
$244$	1088.18	0.285506
$245$	0	0
$246$	−7392.00	−1.91584
$247$	616.000	0.158685
$248$	−1651.03	−0.422743
$249$	−1848.00	−0.470330
$250$	−3039.39	−0.768911
$251$	5581.59	1.40361	0.701807	−	0.712367i	$-0.252377\pi$
$251$	5581.59	1.40361	0.701807	+	0.712367i	$0.252377\pi$
$252$	0	0
$253$	960.000	0.238556
$254$	4624.00	1.14227
$255$	−4953.08	−1.21637
$256$	256.000	0.0625000
$257$	−1500.93	−0.364302	−0.182151	−	0.983271i	$-0.558306\pi$
$257$	−1500.93	−0.364302	−0.182151	+	0.983271i	$0.558306\pi$
$258$	−8180.09	−1.97391
$259$	0	0
$260$	2464.00	0.587734
$261$	−10126.0	−2.40147
$262$	−506.565	−0.119449
$263$	−400.000	−0.0937835	−0.0468917	−	0.998900i	$-0.514932\pi$
$263$	−400.000	−0.0937835	−0.0468917	+	0.998900i	$0.514932\pi$
$264$	−1500.93	−0.349909
$265$	581.612	0.134823
$266$	0	0
$267$	14080.0	3.22727
$268$	2320.00	0.528793
$269$	−272.044	−0.0616610	−0.0308305	−	0.999525i	$-0.509815\pi$
$269$	−272.044	−0.0616610	−0.0308305	+	0.999525i	$0.509815\pi$
$270$	−5984.00	−1.34879
$271$	6904.29	1.54762	0.773812	−	0.633416i	$-0.218348\pi$
$271$	6904.29	1.54762	0.773812	+	0.633416i	$0.218348\pi$
$272$	900.560	0.200752
$273$	0	0
$274$	−2228.00	−0.491235
$275$	−740.000	−0.162268
$276$	−1801.12	−0.392807
$277$	−6770.00	−1.46848	−0.734242	−	0.678888i	$-0.762462\pi$
$277$	−6770.00	−1.46848	−0.734242	+	0.678888i	$0.762462\pi$
$278$	2757.96	0.595006
$279$	−12589.1	−2.70139
$280$	0	0
$281$	1878.00	0.398691	0.199345	−	0.979929i	$-0.436118\pi$
$281$	1878.00	0.398691	0.199345	+	0.979929i	$0.436118\pi$
$282$	−3872.00	−0.817639
$283$	384.614	0.0807878	0.0403939	−	0.999184i	$-0.487139\pi$
$283$	384.614	0.0807878	0.0403939	+	0.999184i	$0.487139\pi$
$284$	−2176.00	−0.454654
$285$	−825.513	−0.171576
$286$	2626.63	0.543063
$287$	0	0
$288$	1952.00	0.399384
$289$	−1745.00	−0.355180
$290$	−3114.44	−0.630641
$291$	6160.00	1.24091
$292$	−2401.49	−0.481290
$293$	−3742.95	−0.746299	−0.373149	−	0.927771i	$-0.621722\pi$
$293$	−3742.95	−0.746299	−0.373149	+	0.927771i	$0.621722\pi$
$294$	0	0
$295$	−6248.00	−1.23313
$296$	−624.000	−0.122531
$297$	−6378.97	−1.24628
$298$	−1892.00	−0.367787
$299$	3151.96	0.609641
$300$	1388.36	0.267191
$301$	0	0
$302$	1664.00	0.317061
$303$	−1144.00	−0.216901
$304$	150.093	0.0283172
$305$	2552.00	0.479105
$306$	6866.77	1.28283
$307$	722.324	0.134284	0.0671420	−	0.997743i	$-0.478612\pi$
$307$	722.324	0.134284	0.0671420	+	0.997743i	$0.478612\pi$
$308$	0	0
$309$	12848.0	2.36536
$310$	−3872.00	−0.709403
$311$	7279.53	1.32728	0.663640	−	0.748052i	$-0.269011\pi$
$311$	7279.53	1.32728	0.663640	+	0.748052i	$0.269011\pi$
$312$	−4928.00	−0.894209
$313$	1519.69	0.274435	0.137218	−	0.990541i	$-0.456184\pi$
$313$	1519.69	0.274435	0.137218	+	0.990541i	$0.456184\pi$
$314$	5759.83	1.03518
$315$	0	0
$316$	−2720.00	−0.484215
$317$	2358.00	0.417787	0.208893	−	0.977938i	$-0.433014\pi$
$317$	2358.00	0.417787	0.208893	+	0.977938i	$0.433014\pi$
$318$	−1163.22	−0.205127
$319$	−3320.00	−0.582709
$320$	600.373	0.104881
$321$	2439.02	0.424089
$322$	0	0
$323$	528.000	0.0909557
$324$	5380.00	0.922497
$325$	−2429.64	−0.414683
$326$	1272.00	0.216103
$327$	−17654.7	−2.98565
$328$	3151.96	0.530603
$329$	0	0
$330$	−3520.00	−0.587181
$331$	2372.00	0.393888	0.196944	−	0.980415i	$-0.436898\pi$
$331$	2372.00	0.393888	0.196944	+	0.980415i	$0.436898\pi$
$332$	787.990	0.130261
$333$	−4758.00	−0.782993
$334$	−1313.32	−0.215154
$335$	5440.88	0.887365
$336$	0	0
$337$	−250.000	−0.0404106	−0.0202053	−	0.999796i	$-0.506432\pi$
$337$	−250.000	−0.0404106	−0.0202053	+	0.999796i	$0.506432\pi$
$338$	4230.00	0.680715
$339$	12063.7	1.93278
$340$	2112.00	0.336880
$341$	−4127.57	−0.655485
$342$	1144.46	0.180951
$343$	0	0
$344$	3488.00	0.546687
$345$	−4224.00	−0.659167
$346$	1332.08	0.206974
$347$	9540.00	1.47589	0.737945	−	0.674861i	$-0.235796\pi$
$347$	9540.00	1.47589	0.737945	+	0.674861i	$0.235796\pi$
$348$	6228.87	0.959490
$349$	−5712.93	−0.876235	−0.438117	−	0.898918i	$-0.644355\pi$
$349$	−5712.93	−0.876235	−0.438117	+	0.898918i	$0.644355\pi$
$350$	0	0
$351$	−20944.0	−3.18492
$352$	640.000	0.0969094
$353$	4390.23	0.661950	0.330975	−	0.943640i	$-0.392622\pi$
$353$	4390.23	0.661950	0.330975	+	0.943640i	$0.392622\pi$
$354$	12496.0	1.87614
$355$	−5103.17	−0.762953
$356$	−6003.73	−0.893812
$357$	0	0
$358$	−6456.00	−0.953101
$359$	1840.00	0.270506	0.135253	−	0.990811i	$-0.456815\pi$
$359$	1840.00	0.270506	0.135253	+	0.990811i	$0.456815\pi$
$360$	4577.85	0.670205
$361$	−6771.00	−0.987170
$362$	−5647.26	−0.819927
$363$	8733.55	1.26279
$364$	0	0
$365$	−5632.00	−0.807650
$366$	−5104.00	−0.728935
$367$	−2964.34	−0.421628	−0.210814	−	0.977526i	$-0.567611\pi$
$367$	−2964.34	−0.421628	−0.210814	+	0.977526i	$0.567611\pi$
$368$	768.000	0.108790
$369$	24033.7	3.39063
$370$	−1463.41	−0.205619
$371$	0	0
$372$	7744.00	1.07932
$373$	3982.00	0.552762	0.276381	−	0.961048i	$-0.410865\pi$
$373$	3982.00	0.552762	0.276381	+	0.961048i	$0.410865\pi$
$374$	2251.40	0.311276
$375$	14256.0	1.96314
$376$	1651.03	0.226450
$377$	−10900.5	−1.48914
$378$	0	0
$379$	2676.00	0.362683	0.181342	−	0.983420i	$-0.441956\pi$
$379$	2676.00	0.362683	0.181342	+	0.983420i	$0.441956\pi$
$380$	352.000	0.0475190
$381$	−21688.5	−2.91636
$382$	−4272.00	−0.572185
$383$	7035.62	0.938652	0.469326	−	0.883025i	$-0.344497\pi$
$383$	7035.62	0.938652	0.469326	+	0.883025i	$0.344497\pi$
$384$	−1200.75	−0.159571
$385$	0	0
$386$	3316.00	0.437254
$387$	26596.0	3.49341
$388$	−2626.63	−0.343678
$389$	8658.00	1.12848	0.564239	−	0.825611i	$-0.309170\pi$
$389$	8658.00	1.12848	0.564239	+	0.825611i	$0.309170\pi$
$390$	−11557.2	−1.50057
$391$	2701.68	0.349437
$392$	0	0
$393$	2376.00	0.304970
$394$	−1956.00	−0.250106
$395$	−6378.97	−0.812558
$396$	4880.00	0.619266
$397$	−9052.50	−1.14441	−0.572207	−	0.820109i	$-0.693912\pi$
$397$	−9052.50	−1.14441	−0.572207	+	0.820109i	$0.693912\pi$
$398$	−9868.63	−1.24289
$399$	0	0
$400$	−592.000	−0.0740000
$401$	−5706.00	−0.710584	−0.355292	−	0.934755i	$-0.615619\pi$
$401$	−5706.00	−0.710584	−0.355292	+	0.934755i	$0.615619\pi$
$402$	−10881.8	−1.35008
$403$	−13552.0	−1.67512
$404$	487.803	0.0600721
$405$	12617.2	1.54804
$406$	0	0
$407$	−1560.00	−0.189991
$408$	−4224.00	−0.512547
$409$	2420.25	0.292601	0.146301	−	0.989240i	$-0.453263\pi$
$409$	2420.25	0.292601	0.146301	+	0.989240i	$0.453263\pi$
$410$	7392.00	0.890402
$411$	10450.2	1.25419
$412$	−5478.41	−0.655101
$413$	0	0
$414$	5856.00	0.695185
$415$	1848.00	0.218590
$416$	2101.31	0.247656
$417$	−12936.0	−1.51913
$418$	375.233	0.0439073
$419$	−1510.31	−0.176095	−0.0880473	−	0.996116i	$-0.528063\pi$
$419$	−1510.31	−0.176095	−0.0880473	+	0.996116i	$0.528063\pi$
$420$	0	0
$421$	−16770.0	−1.94138	−0.970689	−	0.240341i	$-0.922741\pi$
$421$	−16770.0	−1.94138	−0.970689	+	0.240341i	$0.922741\pi$
$422$	3112.00	0.358981
$423$	12589.1	1.44705
$424$	496.000	0.0568111
$425$	−2082.54	−0.237690
$426$	10206.3	1.16080
$427$	0	0
$428$	−1040.00	−0.117454
$429$	−12320.0	−1.38652
$430$	8180.09	0.917392
$431$	1336.00	0.149311	0.0746553	−	0.997209i	$-0.476214\pi$
$431$	1336.00	0.149311	0.0746553	+	0.997209i	$0.476214\pi$
$432$	−5103.17	−0.568348
$433$	−11163.2	−1.23896	−0.619479	−	0.785013i	$-0.712656\pi$
$433$	−11163.2	−1.23896	−0.619479	+	0.785013i	$0.712656\pi$
$434$	0	0
$435$	14608.0	1.61011
$436$	7528.00	0.826894
$437$	450.280	0.0492902
$438$	11264.0	1.22880
$439$	−3602.24	−0.391630	−0.195815	−	0.980641i	$-0.562735\pi$
$439$	−3602.24	−0.391630	−0.195815	+	0.980641i	$0.562735\pi$
$440$	1500.93	0.162623
$441$	0	0
$442$	7392.00	0.795479
$443$	6348.00	0.680818	0.340409	−	0.940277i	$-0.389434\pi$
$443$	6348.00	0.680818	0.340409	+	0.940277i	$0.389434\pi$
$444$	2926.82	0.312839
$445$	−14080.0	−1.49990
$446$	5778.59	0.613507
$447$	8874.27	0.939012
$448$	0	0
$449$	7170.00	0.753615	0.376808	−	0.926292i	$-0.377022\pi$
$449$	7170.00	0.753615	0.376808	+	0.926292i	$0.377022\pi$
$450$	−4514.00	−0.472871
$451$	7879.90	0.822727
$452$	−5144.00	−0.535295
$453$	−7804.85	−0.809501
$454$	−3958.71	−0.409232
$455$	0	0
$456$	−704.000	−0.0722979
$457$	6866.00	0.702796	0.351398	−	0.936226i	$-0.385706\pi$
$457$	6866.00	0.702796	0.351398	+	0.936226i	$0.385706\pi$
$458$	5534.69	0.564671
$459$	−17952.0	−1.82555
$460$	1801.12	0.182560
$461$	1378.98	0.139318	0.0696590	−	0.997571i	$-0.477809\pi$
$461$	1378.98	0.139318	0.0696590	+	0.997571i	$0.477809\pi$
$462$	0	0
$463$	2648.00	0.265795	0.132897	−	0.991130i	$-0.457572\pi$
$463$	2648.00	0.265795	0.132897	+	0.991130i	$0.457572\pi$
$464$	−2656.00	−0.265736
$465$	18161.3	1.81120
$466$	−12980.0	−1.29032
$467$	−12335.8	−1.22234	−0.611170	−	0.791500i	$-0.709301\pi$
$467$	−12335.8	−1.22234	−0.611170	+	0.791500i	$0.709301\pi$
$468$	16022.5	1.58256
$469$	0	0
$470$	3872.00	0.380004
$471$	−27016.0	−2.64295
$472$	−5328.31	−0.519609
$473$	8720.00	0.847666
$474$	12757.9	1.23627
$475$	−347.091	−0.0335276
$476$	0	0
$477$	3782.00	0.363031
$478$	−8592.00	−0.822153
$479$	−13339.5	−1.27244	−0.636221	−	0.771507i	$-0.719503\pi$
$479$	−13339.5	−1.27244	−0.636221	+	0.771507i	$0.719503\pi$
$480$	−2816.00	−0.267775
$481$	−5121.93	−0.485530
$482$	9043.12	0.854570
$483$	0	0
$484$	−3724.00	−0.349737
$485$	−6160.00	−0.576724
$486$	−8011.23	−0.747730
$487$	13936.0	1.29672	0.648358	−	0.761336i	$-0.275456\pi$
$487$	13936.0	1.29672	0.648358	+	0.761336i	$0.275456\pi$
$488$	2176.35	0.201883
$489$	−5966.21	−0.551741
$490$	0	0
$491$	−12276.0	−1.12833	−0.564163	−	0.825663i	$-0.690801\pi$
$491$	−12276.0	−1.12833	−0.564163	+	0.825663i	$0.690801\pi$
$492$	−14784.0	−1.35470
$493$	−9343.31	−0.853553
$494$	1232.00	0.112207
$495$	11444.6	1.03919
$496$	−3302.05	−0.298924
$497$	0	0
$498$	−3696.00	−0.332574
$499$	−2220.00	−0.199160	−0.0995800	−	0.995030i	$-0.531750\pi$
$499$	−2220.00	−0.199160	−0.0995800	+	0.995030i	$0.531750\pi$
$500$	−6078.78	−0.543703
$501$	6160.00	0.549318
$502$	11163.2	0.992505
$503$	−11294.5	−1.00119	−0.500594	−	0.865682i	$-0.666885\pi$
$503$	−11294.5	−1.00119	−0.500594	+	0.865682i	$0.666885\pi$
$504$	0	0
$505$	1144.00	0.100807
$506$	1920.00	0.168685
$507$	−19840.5	−1.73796
$508$	9248.00	0.807704
$509$	−15881.7	−1.38300	−0.691499	−	0.722377i	$-0.743049\pi$
$509$	−15881.7	−1.38300	−0.691499	+	0.722377i	$0.743049\pi$
$510$	−9906.16	−0.860102
$511$	0	0
$512$	512.000	0.0441942
$513$	−2992.00	−0.257505
$514$	−3001.87	−0.257600
$515$	−12848.0	−1.09932
$516$	−16360.2	−1.39577
$517$	4127.57	0.351122
$518$	0	0
$519$	−6248.00	−0.528433
$520$	4928.00	0.415591
$521$	11613.5	0.976575	0.488287	−	0.872683i	$-0.337622\pi$
$521$	11613.5	0.976575	0.488287	+	0.872683i	$0.337622\pi$
$522$	−20252.0	−1.69810
$523$	12617.2	1.05490	0.527450	−	0.849586i	$-0.323148\pi$
$523$	12617.2	1.05490	0.527450	+	0.849586i	$0.323148\pi$
$524$	−1013.13	−0.0844633
$525$	0	0
$526$	−800.000	−0.0663149
$527$	−11616.0	−0.960154
$528$	−3001.87	−0.247423
$529$	−9863.00	−0.810635
$530$	1163.22	0.0953343
$531$	−40628.4	−3.32038
$532$	0	0
$533$	25872.0	2.10252
$534$	28160.0	2.28203
$535$	−2439.02	−0.197099
$536$	4640.00	0.373913
$537$	30281.3	2.43340
$538$	−544.088	−0.0436009
$539$	0	0
$540$	−11968.0	−0.953742
$541$	1798.00	0.142887	0.0714437	−	0.997445i	$-0.477239\pi$
$541$	1798.00	0.142887	0.0714437	+	0.997445i	$0.477239\pi$
$542$	13808.6	1.09433
$543$	26488.0	2.09339
$544$	1801.12	0.141953
$545$	17654.7	1.38761
$546$	0	0
$547$	1276.00	0.0997401	0.0498700	−	0.998756i	$-0.484119\pi$
$547$	1276.00	0.0997401	0.0498700	+	0.998756i	$0.484119\pi$
$548$	−4456.00	−0.347356
$549$	16594.7	1.29006
$550$	−1480.00	−0.114741
$551$	−1557.22	−0.120399
$552$	−3602.24	−0.277756
$553$	0	0
$554$	−13540.0	−1.03837
$555$	6864.00	0.524974
$556$	5515.93	0.420733
$557$	2694.00	0.204934	0.102467	−	0.994736i	$-0.467326\pi$
$557$	2694.00	0.204934	0.102467	+	0.994736i	$0.467326\pi$
$558$	−25178.2	−1.91017
$559$	28630.3	2.16625
$560$	0	0
$561$	−10560.0	−0.794730
$562$	3756.00	0.281917
$563$	−15769.2	−1.18045	−0.590223	−	0.807240i	$-0.700960\pi$
$563$	−15769.2	−1.18045	−0.590223	+	0.807240i	$0.700960\pi$
$564$	−7744.00	−0.578158
$565$	−12063.7	−0.898276
$566$	769.228	0.0571256
$567$	0	0
$568$	−4352.00	−0.321489
$569$	12606.0	0.928772	0.464386	−	0.885633i	$-0.346275\pi$
$569$	12606.0	0.928772	0.464386	+	0.885633i	$0.346275\pi$
$570$	−1651.03	−0.121323
$571$	6852.00	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	6852.00	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	5253.27	0.384004
$573$	20037.5	1.46087
$574$	0	0
$575$	−1776.00	−0.128808
$576$	3904.00	0.282407
$577$	14371.4	1.03690	0.518449	−	0.855108i	$-0.326509\pi$
$577$	14371.4	1.03690	0.518449	+	0.855108i	$0.326509\pi$
$578$	−3490.00	−0.251150
$579$	−15553.4	−1.11637
$580$	−6228.87	−0.445931
$581$	0	0
$582$	12320.0	0.877458
$583$	1240.00	0.0880884
$584$	−4802.99	−0.340324
$585$	37576.0	2.65569
$586$	−7485.90	−0.527713
$587$	18977.4	1.33438	0.667191	−	0.744887i	$-0.267497\pi$
$587$	18977.4	1.33438	0.667191	+	0.744887i	$0.267497\pi$
$588$	0	0
$589$	−1936.00	−0.135435
$590$	−12496.0	−0.871953
$591$	9174.45	0.638556
$592$	−1248.00	−0.0866427
$593$	8217.61	0.569067	0.284534	−	0.958666i	$-0.408161\pi$
$593$	8217.61	0.569067	0.284534	+	0.958666i	$0.408161\pi$
$594$	−12757.9	−0.881253
$595$	0	0
$596$	−3784.00	−0.260065
$597$	46288.0	3.17327
$598$	6303.92	0.431081
$599$	−19104.0	−1.30312	−0.651559	−	0.758598i	$-0.725885\pi$
$599$	−19104.0	−1.30312	−0.651559	+	0.758598i	$0.725885\pi$
$600$	2776.73	0.188932
$601$	21538.4	1.46185	0.730923	−	0.682460i	$-0.239090\pi$
$601$	21538.4	1.46185	0.730923	+	0.682460i	$0.239090\pi$
$602$	0	0
$603$	35380.0	2.38936
$604$	3328.00	0.224196
$605$	−8733.55	−0.586892
$606$	−2288.00	−0.153372
$607$	13733.5	0.918331	0.459166	−	0.888351i	$-0.348148\pi$
$607$	13733.5	0.918331	0.459166	+	0.888351i	$0.348148\pi$
$608$	300.187	0.0200233
$609$	0	0
$610$	5104.00	0.338779
$611$	13552.0	0.897308
$612$	13733.5	0.907100
$613$	28034.0	1.84712	0.923558	−	0.383458i	$-0.125267\pi$
$613$	28034.0	1.84712	0.923558	+	0.383458i	$0.125267\pi$
$614$	1444.65	0.0949532
$615$	−34671.6	−2.27332
$616$	0	0
$617$	−8258.00	−0.538824	−0.269412	−	0.963025i	$-0.586829\pi$
$617$	−8258.00	−0.538824	−0.269412	+	0.963025i	$0.586829\pi$
$618$	25696.0	1.67256
$619$	−5131.31	−0.333191	−0.166595	−	0.986025i	$-0.553277\pi$
$619$	−5131.31	−0.333191	−0.166595	+	0.986025i	$0.553277\pi$
$620$	−7744.00	−0.501623
$621$	−15309.5	−0.989291
$622$	14559.1	0.938529
$623$	0	0
$624$	−9856.00	−0.632301
$625$	−9631.00	−0.616384
$626$	3039.39	0.194055
$627$	−1760.00	−0.112101
$628$	11519.7	0.731982
$629$	−4390.23	−0.278299
$630$	0	0
$631$	912.000	0.0575375	0.0287687	−	0.999586i	$-0.490841\pi$
$631$	912.000	0.0575375	0.0287687	+	0.999586i	$0.490841\pi$
$632$	−5440.00	−0.342392
$633$	−14596.6	−0.916527
$634$	4716.00	0.295420
$635$	21688.5	1.35540
$636$	−2326.45	−0.145047
$637$	0	0
$638$	−6640.00	−0.412038
$639$	−33184.0	−2.05436
$640$	1200.75	0.0741620
$641$	−890.000	−0.0548407	−0.0274203	−	0.999624i	$-0.508729\pi$
$641$	−890.000	−0.0548407	−0.0274203	+	0.999624i	$0.508729\pi$
$642$	4878.03	0.299876
$643$	−29352.6	−1.80024	−0.900120	−	0.435642i	$-0.856521\pi$
$643$	−29352.6	−1.80024	−0.900120	+	0.435642i	$0.856521\pi$
$644$	0	0
$645$	−38368.0	−2.34223
$646$	1056.00	0.0643154
$647$	11876.1	0.721637	0.360818	−	0.932636i	$-0.382497\pi$
$647$	11876.1	0.721637	0.360818	+	0.932636i	$0.382497\pi$
$648$	10760.0	0.652304
$649$	−13320.8	−0.805680
$650$	−4859.27	−0.293225
$651$	0	0
$652$	2544.00	0.152808
$653$	−21526.0	−1.29001	−0.645006	−	0.764178i	$-0.723145\pi$
$653$	−21526.0	−1.29001	−0.645006	+	0.764178i	$0.723145\pi$
$654$	−35309.4	−2.11118
$655$	−2376.00	−0.141737
$656$	6303.92	0.375193
$657$	−36622.8	−2.17472
$658$	0	0
$659$	23452.0	1.38628	0.693141	−	0.720802i	$-0.256226\pi$
$659$	23452.0	1.38628	0.693141	+	0.720802i	$0.256226\pi$
$660$	−7040.00	−0.415199
$661$	−26669.7	−1.56934	−0.784668	−	0.619916i	$-0.787167\pi$
$661$	−26669.7	−1.56934	−0.784668	+	0.619916i	$0.787167\pi$
$662$	4744.00	0.278521
$663$	−34671.6	−2.03097
$664$	1575.98	0.0921082
$665$	0	0
$666$	−9516.00	−0.553660
$667$	−7968.00	−0.462552
$668$	−2626.63	−0.152137
$669$	−27104.0	−1.56637
$670$	10881.8	0.627462
$671$	5440.88	0.313030
$672$	0	0
$673$	−13858.0	−0.793739	−0.396870	−	0.917875i	$-0.629904\pi$
$673$	−13858.0	−0.793739	−0.396870	+	0.917875i	$0.629904\pi$
$674$	−500.000	−0.0285746
$675$	11801.1	0.672924
$676$	8460.00	0.481338
$677$	32448.3	1.84208	0.921041	−	0.389466i	$-0.127340\pi$
$677$	32448.3	1.84208	0.921041	+	0.389466i	$0.127340\pi$
$678$	24127.5	1.36668
$679$	0	0
$680$	4224.00	0.238210
$681$	18568.0	1.04483
$682$	−8255.13	−0.463498
$683$	−27812.0	−1.55812	−0.779060	−	0.626949i	$-0.784304\pi$
$683$	−27812.0	−1.55812	−0.779060	+	0.626949i	$0.784304\pi$
$684$	2288.92	0.127952
$685$	−10450.2	−0.582895
$686$	0	0
$687$	−25960.0	−1.44168
$688$	6976.00	0.386566
$689$	4071.28	0.225114
$690$	−8448.00	−0.466101
$691$	−1303.94	−0.0717859	−0.0358929	−	0.999356i	$-0.511428\pi$
$691$	−1303.94	−0.0717859	−0.0358929	+	0.999356i	$0.511428\pi$
$692$	2664.16	0.146353
$693$	0	0
$694$	19080.0	1.04361
$695$	12936.0	0.706029
$696$	12457.7	0.678462
$697$	22176.0	1.20513
$698$	−11425.9	−0.619592
$699$	60881.6	3.29435
$700$	0	0
$701$	22906.0	1.23416	0.617081	−	0.786900i	$-0.288315\pi$
$701$	22906.0	1.23416	0.617081	+	0.786900i	$0.288315\pi$
$702$	−41888.0	−2.25208
$703$	−731.705	−0.0392557
$704$	1280.00	0.0685253
$705$	−18161.3	−0.970204
$706$	8780.46	0.468069
$707$	0	0
$708$	24992.0	1.32663
$709$	−15086.0	−0.799107	−0.399553	−	0.916710i	$-0.630835\pi$
$709$	−15086.0	−0.799107	−0.399553	+	0.916710i	$0.630835\pi$
$710$	−10206.3	−0.539489
$711$	−41480.0	−2.18793
$712$	−12007.5	−0.632021
$713$	−9906.16	−0.520321
$714$	0	0
$715$	12320.0	0.644394
$716$	−12912.0	−0.673944
$717$	40300.1	2.09907
$718$	3680.00	0.191276
$719$	−20544.0	−1.06559	−0.532797	−	0.846243i	$-0.678859\pi$
$719$	−20544.0	−1.06559	−0.532797	+	0.846243i	$0.678859\pi$
$720$	9155.69	0.473906
$721$	0	0
$722$	−13542.0	−0.698035
$723$	−42416.0	−2.18184
$724$	−11294.5	−0.579776
$725$	6142.00	0.314632
$726$	17467.1	0.892927
$727$	−7223.24	−0.368494	−0.184247	−	0.982880i	$-0.558985\pi$
$727$	−7223.24	−0.368494	−0.184247	+	0.982880i	$0.558985\pi$
$728$	0	0
$729$	1261.00	0.0640654
$730$	−11264.0	−0.571095
$731$	24540.3	1.24166
$732$	−10208.0	−0.515435
$733$	29427.7	1.48286	0.741430	−	0.671031i	$-0.234148\pi$
$733$	29427.7	1.48286	0.741430	+	0.671031i	$0.234148\pi$
$734$	−5928.69	−0.298136
$735$	0	0
$736$	1536.00	0.0769262
$737$	11600.0	0.579771
$738$	48067.4	2.39754
$739$	32668.0	1.62613	0.813066	−	0.582171i	$-0.197797\pi$
$739$	32668.0	1.62613	0.813066	+	0.582171i	$0.197797\pi$
$740$	−2926.82	−0.145395
$741$	−5778.59	−0.286480
$742$	0	0
$743$	−37056.0	−1.82968	−0.914840	−	0.403816i	$-0.867684\pi$
$743$	−37056.0	−1.82968	−0.914840	+	0.403816i	$0.867684\pi$
$744$	15488.0	0.763196
$745$	−8874.27	−0.436413
$746$	7964.00	0.390862
$747$	12016.8	0.588586
$748$	4502.80	0.220105
$749$	0	0
$750$	28512.0	1.38815
$751$	−19608.0	−0.952738	−0.476369	−	0.879246i	$-0.658047\pi$
$751$	−19608.0	−0.952738	−0.476369	+	0.879246i	$0.658047\pi$
$752$	3302.05	0.160124
$753$	−52360.0	−2.53400
$754$	−21801.1	−1.05298
$755$	7804.85	0.376222
$756$	0	0
$757$	19378.0	0.930390	0.465195	−	0.885208i	$-0.345984\pi$
$757$	19378.0	0.930390	0.465195	+	0.885208i	$0.345984\pi$
$758$	5352.00	0.256456
$759$	−9005.60	−0.430675
$760$	704.000	0.0336010
$761$	−13977.4	−0.665810	−0.332905	−	0.942960i	$-0.608029\pi$
$761$	−13977.4	−0.665810	−0.332905	+	0.942960i	$0.608029\pi$
$762$	−43377.0	−2.06218
$763$	0	0
$764$	−8544.00	−0.404596
$765$	32208.0	1.52220
$766$	14071.2	0.663727
$767$	−43736.0	−2.05895
$768$	−2401.49	−0.112834
$769$	8536.56	0.400307	0.200154	−	0.979765i	$-0.435856\pi$
$769$	8536.56	0.400307	0.200154	+	0.979765i	$0.435856\pi$
$770$	0	0
$771$	14080.0	0.657690
$772$	6632.00	0.309185
$773$	−29296.3	−1.36315	−0.681576	−	0.731748i	$-0.738705\pi$
$773$	−29296.3	−1.36315	−0.681576	+	0.731748i	$0.738705\pi$
$774$	53192.0	2.47022
$775$	7636.00	0.353927
$776$	−5253.27	−0.243017
$777$	0	0
$778$	17316.0	0.797955
$779$	3696.00	0.169991
$780$	−23114.4	−1.06106
$781$	−10880.0	−0.498485
$782$	5403.36	0.247089
$783$	52945.4	2.41649
$784$	0	0
$785$	27016.0	1.22833
$786$	4752.00	0.215647
$787$	13780.4	0.624167	0.312084	−	0.950055i	$-0.398973\pi$
$787$	13780.4	0.624167	0.312084	+	0.950055i	$0.398973\pi$
$788$	−3912.00	−0.176852
$789$	3752.33	0.169311
$790$	−12757.9	−0.574566
$791$	0	0
$792$	9760.00	0.437887
$793$	17864.0	0.799961
$794$	−18105.0	−0.809222
$795$	−5456.00	−0.243402
$796$	−19737.3	−0.878855
$797$	−34868.6	−1.54970	−0.774848	−	0.632148i	$-0.782174\pi$
$797$	−34868.6	−1.54970	−0.774848	+	0.632148i	$0.782174\pi$
$798$	0	0
$799$	11616.0	0.514324
$800$	−1184.00	−0.0523259
$801$	−91556.9	−4.03871
$802$	−11412.0	−0.502459
$803$	−12007.5	−0.527689
$804$	−21763.5	−0.954652
$805$	0	0
$806$	−27104.0	−1.18449
$807$	2552.00	0.111319
$808$	975.606	0.0424774
$809$	14034.0	0.609900	0.304950	−	0.952368i	$-0.401360\pi$
$809$	14034.0	0.609900	0.304950	+	0.952368i	$0.401360\pi$
$810$	25234.4	1.09463
$811$	−6632.25	−0.287164	−0.143582	−	0.989638i	$-0.545862\pi$
$811$	−6632.25	−0.287164	−0.143582	+	0.989638i	$0.545862\pi$
$812$	0	0
$813$	−64768.0	−2.79399
$814$	−3120.00	−0.134344
$815$	5966.21	0.256426
$816$	−8448.00	−0.362425
$817$	4090.04	0.175144
$818$	4840.51	0.206900
$819$	0	0
$820$	14784.0	0.629609
$821$	28622.0	1.21670	0.608352	−	0.793667i	$-0.291831\pi$
$821$	28622.0	1.21670	0.608352	+	0.793667i	$0.291831\pi$
$822$	20900.5	0.886847
$823$	24688.0	1.04565	0.522825	−	0.852440i	$-0.324878\pi$
$823$	24688.0	1.04565	0.522825	+	0.852440i	$0.324878\pi$
$824$	−10956.8	−0.463226
$825$	6941.82	0.292949
$826$	0	0
$827$	−30756.0	−1.29322	−0.646609	−	0.762822i	$-0.723813\pi$
$827$	−30756.0	−1.29322	−0.646609	+	0.762822i	$0.723813\pi$
$828$	11712.0	0.491570
$829$	23236.3	0.973499	0.486750	−	0.873542i	$-0.338182\pi$
$829$	23236.3	0.973499	0.486750	+	0.873542i	$0.338182\pi$
$830$	3696.00	0.154566
$831$	63508.2	2.65111
$832$	4202.61	0.175119
$833$	0	0
$834$	−25872.0	−1.07419
$835$	−6160.00	−0.255300
$836$	750.467	0.0310472
$837$	65824.0	2.71829
$838$	−3020.63	−0.124518
$839$	−24033.7	−0.988957	−0.494479	−	0.869190i	$-0.664641\pi$
$839$	−24033.7	−0.988957	−0.494479	+	0.869190i	$0.664641\pi$
$840$	0	0
$841$	3167.00	0.129854
$842$	−33540.0	−1.37276
$843$	−17617.2	−0.719773
$844$	6224.00	0.253838
$845$	19840.5	0.807731
$846$	25178.2	1.02322
$847$	0	0
$848$	992.000	0.0401715
$849$	−3608.00	−0.145850
$850$	−4165.09	−0.168072
$851$	−3744.00	−0.150814
$852$	20412.7	0.820807
$853$	−23574.0	−0.946260	−0.473130	−	0.880993i	$-0.656876\pi$
$853$	−23574.0	−0.946260	−0.473130	+	0.880993i	$0.656876\pi$
$854$	0	0
$855$	5368.00	0.214715
$856$	−2080.00	−0.0830525
$857$	24484.0	0.975912	0.487956	−	0.872868i	$-0.337743\pi$
$857$	24484.0	0.975912	0.487956	+	0.872868i	$0.337743\pi$
$858$	−24640.0	−0.980415
$859$	−32954.9	−1.30897	−0.654485	−	0.756075i	$-0.727115\pi$
$859$	−32954.9	−1.30897	−0.654485	+	0.756075i	$0.727115\pi$
$860$	16360.2	0.648694
$861$	0	0
$862$	2672.00	0.105579
$863$	40872.0	1.61217	0.806083	−	0.591803i	$-0.201584\pi$
$863$	40872.0	1.61217	0.806083	+	0.591803i	$0.201584\pi$
$864$	−10206.3	−0.401883
$865$	6248.00	0.245593
$866$	−22326.4	−0.876075
$867$	16369.6	0.641222
$868$	0	0
$869$	−13600.0	−0.530896
$870$	29216.0	1.13852
$871$	38086.2	1.48163
$872$	15056.0	0.584702
$873$	−40056.2	−1.55292
$874$	900.560	0.0348534
$875$	0	0
$876$	22528.0	0.868893
$877$	−12006.0	−0.462273	−0.231137	−	0.972921i	$-0.574244\pi$
$877$	−12006.0	−0.462273	−0.231137	+	0.972921i	$0.574244\pi$
$878$	−7204.48	−0.276924
$879$	35112.0	1.34732
$880$	3001.87	0.114992
$881$	35722.2	1.36607	0.683037	−	0.730383i	$-0.260659\pi$
$881$	35722.2	1.36607	0.683037	+	0.730383i	$0.260659\pi$
$882$	0	0
$883$	19588.0	0.746533	0.373267	−	0.927724i	$-0.378238\pi$
$883$	19588.0	0.746533	0.373267	+	0.927724i	$0.378238\pi$
$884$	14784.0	0.562488
$885$	58611.4	2.22622
$886$	12696.0	0.481411
$887$	−40243.8	−1.52340	−0.761699	−	0.647931i	$-0.775634\pi$
$887$	−40243.8	−1.52340	−0.761699	+	0.647931i	$0.775634\pi$
$888$	5853.64	0.221211
$889$	0	0
$890$	−28160.0	−1.06059
$891$	26900.0	1.01143
$892$	11557.2	0.433815
$893$	1936.00	0.0725485
$894$	17748.5	0.663982
$895$	−30281.3	−1.13094
$896$	0	0
$897$	−29568.0	−1.10061
$898$	14340.0	0.532886
$899$	34258.8	1.27096
$900$	−9028.00	−0.334370
$901$	3489.67	0.129032
$902$	15759.8	0.581756
$903$	0	0
$904$	−10288.0	−0.378511
$905$	−26488.0	−0.972918
$906$	−15609.7	−0.572404
$907$	15868.0	0.580913	0.290457	−	0.956888i	$-0.406193\pi$
$907$	15868.0	0.580913	0.290457	+	0.956888i	$0.406193\pi$
$908$	−7917.42	−0.289371
$909$	7439.00	0.271437
$910$	0	0
$911$	39832.0	1.44862	0.724310	−	0.689474i	$-0.242158\pi$
$911$	39832.0	1.44862	0.724310	+	0.689474i	$0.242158\pi$
$912$	−1408.00	−0.0511223
$913$	3939.95	0.142818
$914$	13732.0	0.496952
$915$	−23939.9	−0.864949
$916$	11069.4	0.399282
$917$	0	0
$918$	−35904.0	−1.29086
$919$	−30528.0	−1.09578	−0.547892	−	0.836549i	$-0.684570\pi$
$919$	−30528.0	−1.09578	−0.547892	+	0.836549i	$0.684570\pi$
$920$	3602.24	0.129089
$921$	−6776.00	−0.242429
$922$	2757.96	0.0985127
$923$	−35722.2	−1.27390
$924$	0	0
$925$	2886.00	0.102585
$926$	5296.00	0.187945
$927$	−83545.7	−2.96009
$928$	−5312.00	−0.187904
$929$	−16604.1	−0.586396	−0.293198	−	0.956052i	$-0.594720\pi$
$929$	−16604.1	−0.586396	−0.293198	+	0.956052i	$0.594720\pi$
$930$	36322.6	1.28071
$931$	0	0
$932$	−25960.0	−0.912391
$933$	−68288.0	−2.39619
$934$	−24671.6	−0.864324
$935$	10560.0	0.369357
$936$	32044.9	1.11904
$937$	29943.6	1.04399	0.521993	−	0.852950i	$-0.325189\pi$
$937$	29943.6	1.04399	0.521993	+	0.852950i	$0.325189\pi$
$938$	0	0
$939$	−14256.0	−0.495449
$940$	7744.00	0.268704
$941$	5375.22	0.186214	0.0931068	−	0.995656i	$-0.470320\pi$
$941$	5375.22	0.186214	0.0931068	+	0.995656i	$0.470320\pi$
$942$	−54032.0	−1.86885
$943$	18911.8	0.653077
$944$	−10656.6	−0.367419
$945$	0	0
$946$	17440.0	0.599390
$947$	45212.0	1.55142	0.775709	−	0.631091i	$-0.217393\pi$
$947$	45212.0	1.55142	0.775709	+	0.631091i	$0.217393\pi$
$948$	25515.9	0.874174
$949$	−39424.0	−1.34853
$950$	−694.182	−0.0237076
$951$	−22120.0	−0.754248
$952$	0	0
$953$	34218.0	1.16310	0.581548	−	0.813512i	$-0.302447\pi$
$953$	34218.0	1.16310	0.581548	+	0.813512i	$0.302447\pi$
$954$	7564.00	0.256702
$955$	−20037.5	−0.678950
$956$	−17184.0	−0.581350
$957$	31144.4	1.05199
$958$	−26679.1	−0.899752
$959$	0	0
$960$	−5632.00	−0.189346
$961$	12801.0	0.429694
$962$	−10243.9	−0.343322
$963$	−15860.0	−0.530718
$964$	18086.2	0.604272
$965$	15553.4	0.518842
$966$	0	0
$967$	14464.0	0.481004	0.240502	−	0.970649i	$-0.422688\pi$
$967$	14464.0	0.481004	0.240502	+	0.970649i	$0.422688\pi$
$968$	−7448.00	−0.247301
$969$	−4953.08	−0.164206
$970$	−12320.0	−0.407806
$971$	37832.9	1.25038	0.625188	−	0.780474i	$-0.285022\pi$
$971$	37832.9	1.25038	0.625188	+	0.780474i	$0.285022\pi$
$972$	−16022.5	−0.528725
$973$	0	0
$974$	27872.0	0.916916
$975$	22792.0	0.748644
$976$	4352.71	0.142753
$977$	42062.0	1.37736	0.688681	−	0.725065i	$-0.258190\pi$
$977$	42062.0	1.37736	0.688681	+	0.725065i	$0.258190\pi$
$978$	−11932.4	−0.390140
$979$	−30018.7	−0.979980
$980$	0	0
$981$	114802.	3.73634
$982$	−24552.0	−0.797847
$983$	−43020.5	−1.39587	−0.697935	−	0.716161i	$-0.745898\pi$
$983$	−43020.5	−1.39587	−0.697935	+	0.716161i	$0.745898\pi$
$984$	−29568.0	−0.957920
$985$	−9174.45	−0.296774
$986$	−18686.6	−0.603553
$987$	0	0
$988$	2464.00	0.0793424
$989$	20928.0	0.672873
$990$	22889.2	0.734816
$991$	21272.0	0.681864	0.340932	−	0.940088i	$-0.389257\pi$
$991$	21272.0	0.681864	0.340932	+	0.940088i	$0.389257\pi$
$992$	−6604.11	−0.211372
$993$	−22251.3	−0.711102
$994$	0	0
$995$	−46288.0	−1.47480
$996$	−7392.00	−0.235165
$997$	121.951	0.00387384	0.00193692	−	0.999998i	$-0.499383\pi$
$997$	121.951	0.00387384	0.00193692	+	0.999998i	$0.499383\pi$
$998$	−4440.00	−0.140827
$999$	24878.0	0.787892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.a.h.1.1	✓	2
3.2	odd	2		882.4.a.w.1.1		2
4.3	odd	2		784.4.a.z.1.2		2
5.4	even	2		2450.4.a.bs.1.2		2
7.2	even	3		98.4.c.g.67.2		4
7.3	odd	6		98.4.c.g.79.1		4
7.4	even	3		98.4.c.g.79.2		4
7.5	odd	6		98.4.c.g.67.1		4
7.6	odd	2	inner	98.4.a.h.1.2	yes	2
21.2	odd	6		882.4.g.bi.361.2		4
21.5	even	6		882.4.g.bi.361.1		4
21.11	odd	6		882.4.g.bi.667.2		4
21.17	even	6		882.4.g.bi.667.1		4
21.20	even	2		882.4.a.w.1.2		2
28.27	even	2		784.4.a.z.1.1		2
35.34	odd	2		2450.4.a.bs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.a.h.1.1	✓	2	1.1	even	1	trivial
98.4.a.h.1.2	yes	2	7.6	odd	2	inner
98.4.c.g.67.1		4	7.5	odd	6
98.4.c.g.67.2		4	7.2	even	3
98.4.c.g.79.1		4	7.3	odd	6
98.4.c.g.79.2		4	7.4	even	3
784.4.a.z.1.1		2	28.27	even	2
784.4.a.z.1.2		2	4.3	odd	2
882.4.a.w.1.1		2	3.2	odd	2
882.4.a.w.1.2		2	21.20	even	2
882.4.g.bi.361.1		4	21.5	even	6
882.4.g.bi.361.2		4	21.2	odd	6
882.4.g.bi.667.1		4	21.17	even	6
882.4.g.bi.667.2		4	21.11	odd	6
2450.4.a.bs.1.1		2	35.34	odd	2
2450.4.a.bs.1.2		2	5.4	even	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:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -9.38083 q^{3} +4.00000 q^{4} +9.38083 q^{5} -18.7617 q^{6} +8.00000 q^{8} +61.0000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -9.38083 q^{3} +4.00000 q^{4} +9.38083 q^{5} -18.7617 q^{6} +8.00000 q^{8} +61.0000 q^{9} +18.7617 q^{10} +20.0000 q^{11} -37.5233 q^{12} +65.6658 q^{13} -88.0000 q^{15} +16.0000 q^{16} +56.2850 q^{17} +122.000 q^{18} +9.38083 q^{19} +37.5233 q^{20} +40.0000 q^{22} +48.0000 q^{23} -75.0467 q^{24} -37.0000 q^{25} +131.332 q^{26} -318.948 q^{27} -166.000 q^{29} -176.000 q^{30} -206.378 q^{31} +32.0000 q^{32} -187.617 q^{33} +112.570 q^{34} +244.000 q^{36} -78.0000 q^{37} +18.7617 q^{38} -616.000 q^{39} +75.0467 q^{40} +393.995 q^{41} +436.000 q^{43} +80.0000 q^{44} +572.231 q^{45} +96.0000 q^{46} +206.378 q^{47} -150.093 q^{48} -74.0000 q^{50} -528.000 q^{51} +262.663 q^{52} +62.0000 q^{53} -637.897 q^{54} +187.617 q^{55} -88.0000 q^{57} -332.000 q^{58} -666.039 q^{59} -352.000 q^{60} +272.044 q^{61} -412.757 q^{62} +64.0000 q^{64} +616.000 q^{65} -375.233 q^{66} +580.000 q^{67} +225.140 q^{68} -450.280 q^{69} -544.000 q^{71} +488.000 q^{72} -600.373 q^{73} -156.000 q^{74} +347.091 q^{75} +37.5233 q^{76} -1232.00 q^{78} -680.000 q^{79} +150.093 q^{80} +1345.00 q^{81} +787.990 q^{82} +196.997 q^{83} +528.000 q^{85} +872.000 q^{86} +1557.22 q^{87} +160.000 q^{88} -1500.93 q^{89} +1144.46 q^{90} +192.000 q^{92} +1936.00 q^{93} +412.757 q^{94} +88.0000 q^{95} -300.187 q^{96} -656.658 q^{97} +1220.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 122 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 + 122 * q^9	\( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 122 q^{9} + 40 q^{11} - 176 q^{15} + 32 q^{16} + 244 q^{18} + 80 q^{22} + 96 q^{23} - 74 q^{25} - 332 q^{29} - 352 q^{30} + 64 q^{32} + 488 q^{36} - 156 q^{37} - 1232 q^{39} + 872 q^{43} + 160 q^{44} + 192 q^{46} - 148 q^{50} - 1056 q^{51} + 124 q^{53} - 176 q^{57} - 664 q^{58} - 704 q^{60} + 128 q^{64} + 1232 q^{65} + 1160 q^{67} - 1088 q^{71} + 976 q^{72} - 312 q^{74} - 2464 q^{78} - 1360 q^{79} + 2690 q^{81} + 1056 q^{85} + 1744 q^{86} + 320 q^{88} + 384 q^{92} + 3872 q^{93} + 176 q^{95} + 2440 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 + 122 * q^9 + 40 * q^11 - 176 * q^15 + 32 * q^16 + 244 * q^18 + 80 * q^22 + 96 * q^23 - 74 * q^25 - 332 * q^29 - 352 * q^30 + 64 * q^32 + 488 * q^36 - 156 * q^37 - 1232 * q^39 + 872 * q^43 + 160 * q^44 + 192 * q^46 - 148 * q^50 - 1056 * q^51 + 124 * q^53 - 176 * q^57 - 664 * q^58 - 704 * q^60 + 128 * q^64 + 1232 * q^65 + 1160 * q^67 - 1088 * q^71 + 976 * q^72 - 312 * q^74 - 2464 * q^78 - 1360 * q^79 + 2690 * q^81 + 1056 * q^85 + 1744 * q^86 + 320 * q^88 + 384 * q^92 + 3872 * q^93 + 176 * q^95 + 2440 * q^99

Introduction

L-functions

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