LMFDB - Embedded newform 2450.4.a.cy.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 92x^{4} + 26x^{3} + 2116x^{2} + 80x - 7800 $ x^6 - x^5 - 92x^4 + 26x^3 + 2116x^2 + 80x - 7800
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$5.44232$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +6.44232 q^{3} +4.00000 q^{4} +12.8846 q^{6} +8.00000 q^{8} +14.5035 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +6.44232 q^{3} +4.00000 q^{4} +12.8846 q^{6} +8.00000 q^{8} +14.5035 q^{9} +48.3846 q^{11} +25.7693 q^{12} +93.4871 q^{13} +16.0000 q^{16} -20.2793 q^{17} +29.0071 q^{18} -31.0581 q^{19} +96.7692 q^{22} +21.0487 q^{23} +51.5386 q^{24} +186.974 q^{26} -80.5063 q^{27} +69.5116 q^{29} +161.116 q^{31} +32.0000 q^{32} +311.709 q^{33} -40.5585 q^{34} +58.0141 q^{36} -162.582 q^{37} -62.1162 q^{38} +602.274 q^{39} -365.073 q^{41} -254.518 q^{43} +193.538 q^{44} +42.0975 q^{46} +468.802 q^{47} +103.077 q^{48} -130.646 q^{51} +373.949 q^{52} +587.150 q^{53} -161.013 q^{54} -200.086 q^{57} +139.023 q^{58} +536.945 q^{59} -625.824 q^{61} +322.232 q^{62} +64.0000 q^{64} +623.419 q^{66} +123.074 q^{67} -81.1171 q^{68} +135.603 q^{69} +210.060 q^{71} +116.028 q^{72} -141.783 q^{73} -325.164 q^{74} -124.232 q^{76} +1204.55 q^{78} +513.424 q^{79} -910.243 q^{81} -730.146 q^{82} -117.376 q^{83} -509.037 q^{86} +447.816 q^{87} +387.077 q^{88} -61.2589 q^{89} +84.1950 q^{92} +1037.96 q^{93} +937.604 q^{94} +206.154 q^{96} -436.654 q^{97} +701.748 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} + 7 q^{3} + 24 q^{4} + 14 q^{6} + 48 q^{8} + 31 q^{9}+O(q^{10}) $ 6 * q + 12 * q^2 + 7 * q^3 + 24 * q^4 + 14 * q^6 + 48 * q^8 + 31 * q^9	$ 6 q + 12 q^{2} + 7 q^{3} + 24 q^{4} + 14 q^{6} + 48 q^{8} + 31 q^{9} + 31 q^{11} + 28 q^{12} + 59 q^{13} + 96 q^{16} + 68 q^{17} + 62 q^{18} + 93 q^{19} + 62 q^{22} - 94 q^{23} + 56 q^{24} + 118 q^{26} + 385 q^{27} + 169 q^{29} + 326 q^{31} + 192 q^{32} + 400 q^{33} + 136 q^{34} + 124 q^{36} - 253 q^{37} + 186 q^{38} - 434 q^{39} + 198 q^{41} - 99 q^{43} + 124 q^{44} - 188 q^{46} + 901 q^{47} + 112 q^{48} + 724 q^{51} + 236 q^{52} + 233 q^{53} + 770 q^{54} - 518 q^{57} + 338 q^{58} + 668 q^{59} + 157 q^{61} + 652 q^{62} + 384 q^{64} + 800 q^{66} + 1193 q^{67} + 272 q^{68} - 45 q^{69} + 1108 q^{71} + 248 q^{72} + 1458 q^{73} - 506 q^{74} + 372 q^{76} - 868 q^{78} - 886 q^{79} - 614 q^{81} + 396 q^{82} + 1799 q^{83} - 198 q^{86} + 789 q^{87} + 248 q^{88} + 3047 q^{89} - 376 q^{92} + 1902 q^{93} + 1802 q^{94} + 224 q^{96} + 3404 q^{97} + 4273 q^{99}+O(q^{100}) $ 6 * q + 12 * q^2 + 7 * q^3 + 24 * q^4 + 14 * q^6 + 48 * q^8 + 31 * q^9 + 31 * q^11 + 28 * q^12 + 59 * q^13 + 96 * q^16 + 68 * q^17 + 62 * q^18 + 93 * q^19 + 62 * q^22 - 94 * q^23 + 56 * q^24 + 118 * q^26 + 385 * q^27 + 169 * q^29 + 326 * q^31 + 192 * q^32 + 400 * q^33 + 136 * q^34 + 124 * q^36 - 253 * q^37 + 186 * q^38 - 434 * q^39 + 198 * q^41 - 99 * q^43 + 124 * q^44 - 188 * q^46 + 901 * q^47 + 112 * q^48 + 724 * q^51 + 236 * q^52 + 233 * q^53 + 770 * q^54 - 518 * q^57 + 338 * q^58 + 668 * q^59 + 157 * q^61 + 652 * q^62 + 384 * q^64 + 800 * q^66 + 1193 * q^67 + 272 * q^68 - 45 * q^69 + 1108 * q^71 + 248 * q^72 + 1458 * q^73 - 506 * q^74 + 372 * q^76 - 868 * q^78 - 886 * q^79 - 614 * q^81 + 396 * q^82 + 1799 * q^83 - 198 * q^86 + 789 * q^87 + 248 * q^88 + 3047 * q^89 - 376 * q^92 + 1902 * q^93 + 1802 * q^94 + 224 * q^96 + 3404 * q^97 + 4273 * 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
$3$	6.44232	1.23983	0.619913	−	0.784671i	$-0.287168\pi$
$3$	6.44232	1.23983	0.619913	+	0.784671i	$0.287168\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	12.8846	0.876689
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	14.5035	0.537168
$10$	0	0
$11$	48.3846	1.32623	0.663114	−	0.748518i	$-0.269234\pi$
$11$	48.3846	1.32623	0.663114	+	0.748518i	$0.269234\pi$
$12$	25.7693	0.619913
$13$	93.4871	1.99451	0.997256	−	0.0740250i	$-0.0235845\pi$
$13$	93.4871	1.99451	0.997256	+	0.0740250i	$0.0235845\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−20.2793	−0.289320	−0.144660	−	0.989481i	$-0.546209\pi$
$17$	−20.2793	−0.289320	−0.144660	+	0.989481i	$0.546209\pi$
$18$	29.0071	0.379835
$19$	−31.0581	−0.375011	−0.187506	−	0.982264i	$-0.560040\pi$
$19$	−31.0581	−0.375011	−0.187506	+	0.982264i	$0.560040\pi$
$20$	0	0
$21$	0	0
$22$	96.7692	0.937785
$23$	21.0487	0.190825	0.0954123	−	0.995438i	$-0.469583\pi$
$23$	21.0487	0.190825	0.0954123	+	0.995438i	$0.469583\pi$
$24$	51.5386	0.438345
$25$	0	0
$26$	186.974	1.41033
$27$	−80.5063	−0.573831
$28$	0	0
$29$	69.5116	0.445103	0.222551	−	0.974921i	$-0.428562\pi$
$29$	69.5116	0.445103	0.222551	+	0.974921i	$0.428562\pi$
$30$	0	0
$31$	161.116	0.933462	0.466731	−	0.884399i	$-0.345432\pi$
$31$	161.116	0.933462	0.466731	+	0.884399i	$0.345432\pi$
$32$	32.0000	0.176777
$33$	311.709	1.64429
$34$	−40.5585	−0.204580
$35$	0	0
$36$	58.0141	0.268584
$37$	−162.582	−0.722386	−0.361193	−	0.932491i	$-0.617630\pi$
$37$	−162.582	−0.722386	−0.361193	+	0.932491i	$0.617630\pi$
$38$	−62.1162	−0.265173
$39$	602.274	2.47285
$40$	0	0
$41$	−365.073	−1.39060	−0.695302	−	0.718717i	$-0.744730\pi$
$41$	−365.073	−1.39060	−0.695302	+	0.718717i	$0.744730\pi$
$42$	0	0
$43$	−254.518	−0.902644	−0.451322	−	0.892361i	$-0.649047\pi$
$43$	−254.518	−0.902644	−0.451322	+	0.892361i	$0.649047\pi$
$44$	193.538	0.663114
$45$	0	0
$46$	42.0975	0.134933
$47$	468.802	1.45493	0.727466	−	0.686144i	$-0.240698\pi$
$47$	468.802	1.45493	0.727466	+	0.686144i	$0.240698\pi$
$48$	103.077	0.309956
$49$	0	0
$50$	0	0
$51$	−130.646	−0.358707
$52$	373.949	0.997256
$53$	587.150	1.52172	0.760861	−	0.648915i	$-0.224777\pi$
$53$	587.150	1.52172	0.760861	+	0.648915i	$0.224777\pi$
$54$	−161.013	−0.405760
$55$	0	0
$56$	0	0
$57$	−200.086	−0.464949
$58$	139.023	0.314735
$59$	536.945	1.18482	0.592409	−	0.805637i	$-0.298177\pi$
$59$	536.945	1.18482	0.592409	+	0.805637i	$0.298177\pi$
$60$	0	0
$61$	−625.824	−1.31358	−0.656791	−	0.754072i	$-0.728087\pi$
$61$	−625.824	−1.31358	−0.656791	+	0.754072i	$0.728087\pi$
$62$	322.232	0.660057
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	623.419	1.16269
$67$	123.074	0.224416	0.112208	−	0.993685i	$-0.464208\pi$
$67$	123.074	0.224416	0.112208	+	0.993685i	$0.464208\pi$
$68$	−81.1171	−0.144660
$69$	135.603	0.236589
$70$	0	0
$71$	210.060	0.351121	0.175560	−	0.984469i	$-0.443826\pi$
$71$	210.060	0.351121	0.175560	+	0.984469i	$0.443826\pi$
$72$	116.028	0.189917
$73$	−141.783	−0.227322	−0.113661	−	0.993520i	$-0.536258\pi$
$73$	−141.783	−0.227322	−0.113661	+	0.993520i	$0.536258\pi$
$74$	−325.164	−0.510804
$75$	0	0
$76$	−124.232	−0.187506
$77$	0	0
$78$	1204.55	1.74857
$79$	513.424	0.731198	0.365599	−	0.930772i	$-0.380864\pi$
$79$	513.424	0.731198	0.365599	+	0.930772i	$0.380864\pi$
$80$	0	0
$81$	−910.243	−1.24862
$82$	−730.146	−0.983306
$83$	−117.376	−0.155225	−0.0776125	−	0.996984i	$-0.524730\pi$
$83$	−117.376	−0.155225	−0.0776125	+	0.996984i	$0.524730\pi$
$84$	0	0
$85$	0	0
$86$	−509.037	−0.638266
$87$	447.816	0.551850
$88$	387.077	0.468892
$89$	−61.2589	−0.0729598	−0.0364799	−	0.999334i	$-0.511614\pi$
$89$	−61.2589	−0.0729598	−0.0364799	+	0.999334i	$0.511614\pi$
$90$	0	0
$91$	0	0
$92$	84.1950	0.0954123
$93$	1037.96	1.15733
$94$	937.604	1.02879
$95$	0	0
$96$	206.154	0.219172
$97$	−436.654	−0.457067	−0.228533	−	0.973536i	$-0.573393\pi$
$97$	−436.654	−0.457067	−0.228533	+	0.973536i	$0.573393\pi$
$98$	0	0
$99$	701.748	0.712407
$100$	0	0
$101$	1768.21	1.74201	0.871007	−	0.491271i	$-0.163468\pi$
$101$	1768.21	1.74201	0.871007	+	0.491271i	$0.163468\pi$
$102$	−261.291	−0.253644
$103$	−849.407	−0.812568	−0.406284	−	0.913747i	$-0.633176\pi$
$103$	−849.407	−0.812568	−0.406284	+	0.913747i	$0.633176\pi$
$104$	747.897	0.705167
$105$	0	0
$106$	1174.30	1.07602
$107$	−1282.58	−1.15880	−0.579399	−	0.815044i	$-0.696713\pi$
$107$	−1282.58	−1.15880	−0.579399	+	0.815044i	$0.696713\pi$
$108$	−322.025	−0.286916
$109$	−598.398	−0.525836	−0.262918	−	0.964818i	$-0.584685\pi$
$109$	−598.398	−0.525836	−0.262918	+	0.964818i	$0.584685\pi$
$110$	0	0
$111$	−1047.40	−0.895633
$112$	0	0
$113$	−860.218	−0.716128	−0.358064	−	0.933697i	$-0.616563\pi$
$113$	−860.218	−0.716128	−0.358064	+	0.933697i	$0.616563\pi$
$114$	−400.172	−0.328768
$115$	0	0
$116$	278.046	0.222551
$117$	1355.89	1.07139
$118$	1073.89	0.837793
$119$	0	0
$120$	0	0
$121$	1010.07	0.758881
$122$	−1251.65	−0.928843
$123$	−2351.92	−1.72411
$124$	644.465	0.466731
$125$	0	0
$126$	0	0
$127$	−971.435	−0.678748	−0.339374	−	0.940652i	$-0.610215\pi$
$127$	−971.435	−0.678748	−0.339374	+	0.940652i	$0.610215\pi$
$128$	128.000	0.0883883
$129$	−1639.69	−1.11912
$130$	0	0
$131$	149.593	0.0997713	0.0498856	−	0.998755i	$-0.484114\pi$
$131$	149.593	0.0997713	0.0498856	+	0.998755i	$0.484114\pi$
$132$	1246.84	0.822146
$133$	0	0
$134$	246.147	0.158686
$135$	0	0
$136$	−162.234	−0.102290
$137$	495.245	0.308844	0.154422	−	0.988005i	$-0.450648\pi$
$137$	495.245	0.308844	0.154422	+	0.988005i	$0.450648\pi$
$138$	271.206	0.167294
$139$	−466.520	−0.284674	−0.142337	−	0.989818i	$-0.545462\pi$
$139$	−466.520	−0.284674	−0.142337	+	0.989818i	$0.545462\pi$
$140$	0	0
$141$	3020.17	1.80386
$142$	420.121	0.248280
$143$	4523.34	2.64518
$144$	232.056	0.134292
$145$	0	0
$146$	−283.567	−0.160741
$147$	0	0
$148$	−650.327	−0.361193
$149$	−178.554	−0.0981723	−0.0490862	−	0.998795i	$-0.515631\pi$
$149$	−178.554	−0.0981723	−0.0490862	+	0.998795i	$0.515631\pi$
$150$	0	0
$151$	831.668	0.448213	0.224107	−	0.974565i	$-0.428054\pi$
$151$	831.668	0.448213	0.224107	+	0.974565i	$0.428054\pi$
$152$	−248.465	−0.132587
$153$	−294.121	−0.155413
$154$	0	0
$155$	0	0
$156$	2409.10	1.23642
$157$	167.697	0.0852462	0.0426231	−	0.999091i	$-0.486429\pi$
$157$	167.697	0.0852462	0.0426231	+	0.999091i	$0.486429\pi$
$158$	1026.85	0.517035
$159$	3782.61	1.88667
$160$	0	0
$161$	0	0
$162$	−1820.49	−0.882907
$163$	1612.09	0.774655	0.387327	−	0.921942i	$-0.373398\pi$
$163$	1612.09	0.774655	0.387327	+	0.921942i	$0.373398\pi$
$164$	−1460.29	−0.695302
$165$	0	0
$166$	−234.752	−0.109761
$167$	−1590.69	−0.737073	−0.368536	−	0.929613i	$-0.620141\pi$
$167$	−1590.69	−0.737073	−0.368536	+	0.929613i	$0.620141\pi$
$168$	0	0
$169$	6542.84	2.97808
$170$	0	0
$171$	−450.452	−0.201444
$172$	−1018.07	−0.451322
$173$	1742.41	0.765739	0.382870	−	0.923802i	$-0.374936\pi$
$173$	1742.41	0.765739	0.382870	+	0.923802i	$0.374936\pi$
$174$	895.632	0.390217
$175$	0	0
$176$	774.154	0.331557
$177$	3459.17	1.46897
$178$	−122.518	−0.0515904
$179$	−2429.37	−1.01441	−0.507205	−	0.861825i	$-0.669321\pi$
$179$	−2429.37	−1.01441	−0.507205	+	0.861825i	$0.669321\pi$
$180$	0	0
$181$	2628.61	1.07946	0.539732	−	0.841837i	$-0.318526\pi$
$181$	2628.61	1.07946	0.539732	+	0.841837i	$0.318526\pi$
$182$	0	0
$183$	−4031.76	−1.62861
$184$	168.390	0.0674667
$185$	0	0
$186$	2075.92	0.818356
$187$	−981.204	−0.383705
$188$	1875.21	0.727466
$189$	0	0
$190$	0	0
$191$	1124.82	0.426122	0.213061	−	0.977039i	$-0.431657\pi$
$191$	1124.82	0.426122	0.213061	+	0.977039i	$0.431657\pi$
$192$	412.309	0.154978
$193$	−2216.08	−0.826514	−0.413257	−	0.910615i	$-0.635609\pi$
$193$	−2216.08	−0.826514	−0.413257	+	0.910615i	$0.635609\pi$
$194$	−873.308	−0.323195
$195$	0	0
$196$	0	0
$197$	3125.56	1.13039	0.565194	−	0.824958i	$-0.308801\pi$
$197$	3125.56	1.13039	0.565194	+	0.824958i	$0.308801\pi$
$198$	1403.50	0.503748
$199$	4539.34	1.61701	0.808506	−	0.588488i	$-0.200276\pi$
$199$	4539.34	1.61701	0.808506	+	0.588488i	$0.200276\pi$
$200$	0	0
$201$	792.881	0.278236
$202$	3536.42	1.23179
$203$	0	0
$204$	−522.582	−0.179353
$205$	0	0
$206$	−1698.81	−0.574573
$207$	305.281	0.102505
$208$	1495.79	0.498628
$209$	−1502.73	−0.497350
$210$	0	0
$211$	2520.40	0.822331	0.411165	−	0.911561i	$-0.365122\pi$
$211$	2520.40	0.822331	0.411165	+	0.911561i	$0.365122\pi$
$212$	2348.60	0.760861
$213$	1353.28	0.435329
$214$	−2565.15	−0.819394
$215$	0	0
$216$	−644.051	−0.202880
$217$	0	0
$218$	−1196.80	−0.371822
$219$	−913.415	−0.281839
$220$	0	0
$221$	−1895.85	−0.577053
$222$	−2094.81	−0.633308
$223$	5800.58	1.74186	0.870931	−	0.491405i	$-0.163516\pi$
$223$	5800.58	1.74186	0.870931	+	0.491405i	$0.163516\pi$
$224$	0	0
$225$	0	0
$226$	−1720.44	−0.506379
$227$	6626.47	1.93751	0.968753	−	0.248026i	$-0.0797817\pi$
$227$	6626.47	1.93751	0.968753	+	0.248026i	$0.0797817\pi$
$228$	−800.345	−0.232474
$229$	507.845	0.146547	0.0732736	−	0.997312i	$-0.476655\pi$
$229$	507.845	0.146547	0.0732736	+	0.997312i	$0.476655\pi$
$230$	0	0
$231$	0	0
$232$	556.093	0.157368
$233$	−6307.82	−1.77356	−0.886779	−	0.462193i	$-0.847063\pi$
$233$	−6307.82	−1.77356	−0.886779	+	0.462193i	$0.847063\pi$
$234$	2711.79	0.757586
$235$	0	0
$236$	2147.78	0.592409
$237$	3307.64	0.906559
$238$	0	0
$239$	−4615.12	−1.24907	−0.624535	−	0.780997i	$-0.714711\pi$
$239$	−4615.12	−1.24907	−0.624535	+	0.780997i	$0.714711\pi$
$240$	0	0
$241$	1550.70	0.414480	0.207240	−	0.978290i	$-0.433552\pi$
$241$	1550.70	0.414480	0.207240	+	0.978290i	$0.433552\pi$
$242$	2020.14	0.536610
$243$	−3690.41	−0.974238
$244$	−2503.30	−0.656791
$245$	0	0
$246$	−4703.83	−1.21913
$247$	−2903.53	−0.747965
$248$	1288.93	0.330029
$249$	−756.173	−0.192452
$250$	0	0
$251$	762.543	0.191758	0.0958790	−	0.995393i	$-0.469434\pi$
$251$	762.543	0.191758	0.0958790	+	0.995393i	$0.469434\pi$
$252$	0	0
$253$	1018.44	0.253077
$254$	−1942.87	−0.479947
$255$	0	0
$256$	256.000	0.0625000
$257$	2302.36	0.558822	0.279411	−	0.960172i	$-0.409861\pi$
$257$	2302.36	0.558822	0.279411	+	0.960172i	$0.409861\pi$
$258$	−3279.38	−0.791338
$259$	0	0
$260$	0	0
$261$	1008.16	0.239095
$262$	299.187	0.0705490
$263$	−4833.50	−1.13326	−0.566628	−	0.823974i	$-0.691752\pi$
$263$	−4833.50	−1.13326	−0.566628	+	0.823974i	$0.691752\pi$
$264$	2493.67	0.581345
$265$	0	0
$266$	0	0
$267$	−394.649	−0.0904575
$268$	492.295	0.112208
$269$	4830.35	1.09484	0.547420	−	0.836858i	$-0.315610\pi$
$269$	4830.35	1.09484	0.547420	+	0.836858i	$0.315610\pi$
$270$	0	0
$271$	−5950.37	−1.33380	−0.666899	−	0.745149i	$-0.732379\pi$
$271$	−5950.37	−1.33380	−0.666899	+	0.745149i	$0.732379\pi$
$272$	−324.468	−0.0723301
$273$	0	0
$274$	990.491	0.218386
$275$	0	0
$276$	542.411	0.118295
$277$	−7668.90	−1.66346	−0.831732	−	0.555177i	$-0.812651\pi$
$277$	−7668.90	−1.66346	−0.831732	+	0.555177i	$0.812651\pi$
$278$	−933.040	−0.201295
$279$	2336.75	0.501426
$280$	0	0
$281$	−6314.18	−1.34047	−0.670235	−	0.742149i	$-0.733807\pi$
$281$	−6314.18	−1.34047	−0.670235	+	0.742149i	$0.733807\pi$
$282$	6040.35	1.27552
$283$	6122.72	1.28607	0.643035	−	0.765836i	$-0.277675\pi$
$283$	6122.72	1.28607	0.643035	+	0.765836i	$0.277675\pi$
$284$	840.242	0.175560
$285$	0	0
$286$	9046.68	1.87042
$287$	0	0
$288$	464.113	0.0949587
$289$	−4501.75	−0.916294
$290$	0	0
$291$	−2813.07	−0.566683
$292$	−567.134	−0.113661
$293$	−1646.67	−0.328326	−0.164163	−	0.986433i	$-0.552492\pi$
$293$	−1646.67	−0.328326	−0.164163	+	0.986433i	$0.552492\pi$
$294$	0	0
$295$	0	0
$296$	−1300.65	−0.255402
$297$	−3895.27	−0.761031
$298$	−357.107	−0.0694183
$299$	1967.79	0.380602
$300$	0	0
$301$	0	0
$302$	1663.34	0.316934
$303$	11391.4	2.15979
$304$	−496.929	−0.0937528
$305$	0	0
$306$	−588.242	−0.109894
$307$	−7890.42	−1.46687	−0.733436	−	0.679758i	$-0.762085\pi$
$307$	−7890.42	−1.46687	−0.733436	+	0.679758i	$0.762085\pi$
$308$	0	0
$309$	−5472.15	−1.00744
$310$	0	0
$311$	−3436.98	−0.626667	−0.313333	−	0.949643i	$-0.601446\pi$
$311$	−3436.98	−0.626667	−0.313333	+	0.949643i	$0.601446\pi$
$312$	4818.19	0.874284
$313$	7209.81	1.30199	0.650995	−	0.759082i	$-0.274352\pi$
$313$	7209.81	1.30199	0.650995	+	0.759082i	$0.274352\pi$
$314$	335.393	0.0602782
$315$	0	0
$316$	2053.69	0.365599
$317$	−885.804	−0.156945	−0.0784727	−	0.996916i	$-0.525004\pi$
$317$	−885.804	−0.156945	−0.0784727	+	0.996916i	$0.525004\pi$
$318$	7565.22	1.33408
$319$	3363.29	0.590308
$320$	0	0
$321$	−8262.77	−1.43671
$322$	0	0
$323$	629.835	0.108498
$324$	−3640.97	−0.624309
$325$	0	0
$326$	3224.18	0.547764
$327$	−3855.08	−0.651945
$328$	−2920.58	−0.491653
$329$	0	0
$330$	0	0
$331$	−1766.18	−0.293288	−0.146644	−	0.989189i	$-0.546847\pi$
$331$	−1766.18	−0.293288	−0.146644	+	0.989189i	$0.546847\pi$
$332$	−469.504	−0.0776125
$333$	−2358.01	−0.388042
$334$	−3181.38	−0.521189
$335$	0	0
$336$	0	0
$337$	−7521.34	−1.21577	−0.607883	−	0.794026i	$-0.707981\pi$
$337$	−7521.34	−1.21577	−0.607883	+	0.794026i	$0.707981\pi$
$338$	13085.7	2.10582
$339$	−5541.80	−0.887874
$340$	0	0
$341$	7795.54	1.23798
$342$	−900.904	−0.142442
$343$	0	0
$344$	−2036.15	−0.319133
$345$	0	0
$346$	3484.82	0.541459
$347$	−4658.53	−0.720700	−0.360350	−	0.932817i	$-0.617343\pi$
$347$	−4658.53	−0.720700	−0.360350	+	0.932817i	$0.617343\pi$
$348$	1791.26	0.275925
$349$	−7198.15	−1.10403	−0.552017	−	0.833833i	$-0.686142\pi$
$349$	−7198.15	−1.10403	−0.552017	+	0.833833i	$0.686142\pi$
$350$	0	0
$351$	−7526.30	−1.14451
$352$	1548.31	0.234446
$353$	3640.08	0.548844	0.274422	−	0.961609i	$-0.411514\pi$
$353$	3640.08	0.548844	0.274422	+	0.961609i	$0.411514\pi$
$354$	6918.34	1.03872
$355$	0	0
$356$	−245.035	−0.0364799
$357$	0	0
$358$	−4858.73	−0.717296
$359$	−5469.89	−0.804150	−0.402075	−	0.915607i	$-0.631711\pi$
$359$	−5469.89	−0.804150	−0.402075	+	0.915607i	$0.631711\pi$
$360$	0	0
$361$	−5894.40	−0.859367
$362$	5257.22	0.763296
$363$	6507.20	0.940880
$364$	0	0
$365$	0	0
$366$	−8063.52	−1.15160
$367$	65.0871	0.00925754	0.00462877	−	0.999989i	$-0.498527\pi$
$367$	65.0871	0.00925754	0.00462877	+	0.999989i	$0.498527\pi$
$368$	336.780	0.0477062
$369$	−5294.84	−0.746988
$370$	0	0
$371$	0	0
$372$	4151.85	0.578665
$373$	10153.4	1.40944	0.704721	−	0.709484i	$-0.251072\pi$
$373$	10153.4	1.40944	0.704721	+	0.709484i	$0.251072\pi$
$374$	−1962.41	−0.271320
$375$	0	0
$376$	3750.41	0.514396
$377$	6498.44	0.887763
$378$	0	0
$379$	1453.96	0.197057	0.0985287	−	0.995134i	$-0.468586\pi$
$379$	1453.96	0.197057	0.0985287	+	0.995134i	$0.468586\pi$
$380$	0	0
$381$	−6258.30	−0.841529
$382$	2249.65	0.301314
$383$	−577.301	−0.0770201	−0.0385101	−	0.999258i	$-0.512261\pi$
$383$	−577.301	−0.0770201	−0.0385101	+	0.999258i	$0.512261\pi$
$384$	824.617	0.109586
$385$	0	0
$386$	−4432.16	−0.584433
$387$	−3691.41	−0.484871
$388$	−1746.62	−0.228533
$389$	−2071.00	−0.269933	−0.134966	−	0.990850i	$-0.543093\pi$
$389$	−2071.00	−0.269933	−0.134966	+	0.990850i	$0.543093\pi$
$390$	0	0
$391$	−426.853	−0.0552094
$392$	0	0
$393$	963.729	0.123699
$394$	6251.11	0.799306
$395$	0	0
$396$	2806.99	0.356203
$397$	5033.42	0.636322	0.318161	−	0.948037i	$-0.396935\pi$
$397$	5033.42	0.636322	0.318161	+	0.948037i	$0.396935\pi$
$398$	9078.68	1.14340
$399$	0	0
$400$	0	0
$401$	−4946.88	−0.616048	−0.308024	−	0.951379i	$-0.599668\pi$
$401$	−4946.88	−0.616048	−0.308024	+	0.951379i	$0.599668\pi$
$402$	1585.76	0.196743
$403$	15062.3	1.86180
$404$	7072.83	0.871007
$405$	0	0
$406$	0	0
$407$	−7866.46	−0.958049
$408$	−1045.16	−0.126822
$409$	−9389.84	−1.13520	−0.567601	−	0.823304i	$-0.692128\pi$
$409$	−9389.84	−1.13520	−0.567601	+	0.823304i	$0.692128\pi$
$410$	0	0
$411$	3190.53	0.382913
$412$	−3397.63	−0.406284
$413$	0	0
$414$	610.562	0.0724819
$415$	0	0
$416$	2991.59	0.352583
$417$	−3005.47	−0.352946
$418$	−3005.47	−0.351680
$419$	8423.77	0.982167	0.491084	−	0.871112i	$-0.336601\pi$
$419$	8423.77	0.982167	0.491084	+	0.871112i	$0.336601\pi$
$420$	0	0
$421$	5680.15	0.657562	0.328781	−	0.944406i	$-0.393362\pi$
$421$	5680.15	0.657562	0.328781	+	0.944406i	$0.393362\pi$
$422$	5040.81	0.581476
$423$	6799.28	0.781542
$424$	4697.20	0.538010
$425$	0	0
$426$	2706.55	0.307824
$427$	0	0
$428$	−5130.31	−0.579399
$429$	29140.8	3.27956
$430$	0	0
$431$	12348.4	1.38005	0.690023	−	0.723787i	$-0.257600\pi$
$431$	12348.4	1.38005	0.690023	+	0.723787i	$0.257600\pi$
$432$	−1288.10	−0.143458
$433$	9031.93	1.00242	0.501209	−	0.865326i	$-0.332889\pi$
$433$	9031.93	1.00242	0.501209	+	0.865326i	$0.332889\pi$
$434$	0	0
$435$	0	0
$436$	−2393.59	−0.262918
$437$	−653.734	−0.0715614
$438$	−1826.83	−0.199291
$439$	−10657.4	−1.15866	−0.579329	−	0.815094i	$-0.696685\pi$
$439$	−10657.4	−1.15866	−0.579329	+	0.815094i	$0.696685\pi$
$440$	0	0
$441$	0	0
$442$	−3791.70	−0.408038
$443$	−901.320	−0.0966660	−0.0483330	−	0.998831i	$-0.515391\pi$
$443$	−901.320	−0.0966660	−0.0483330	+	0.998831i	$0.515391\pi$
$444$	−4189.62	−0.447816
$445$	0	0
$446$	11601.2	1.23168
$447$	−1150.30	−0.121717
$448$	0	0
$449$	−7771.88	−0.816877	−0.408438	−	0.912786i	$-0.633926\pi$
$449$	−7771.88	−0.816877	−0.408438	+	0.912786i	$0.633926\pi$
$450$	0	0
$451$	−17663.9	−1.84426
$452$	−3440.87	−0.358064
$453$	5357.87	0.555706
$454$	13252.9	1.37002
$455$	0	0
$456$	−1600.69	−0.164384
$457$	2760.35	0.282547	0.141273	−	0.989971i	$-0.454880\pi$
$457$	2760.35	0.282547	0.141273	+	0.989971i	$0.454880\pi$
$458$	1015.69	0.103625
$459$	1632.61	0.166021
$460$	0	0
$461$	−16482.3	−1.66520	−0.832602	−	0.553872i	$-0.813150\pi$
$461$	−16482.3	−1.66520	−0.832602	+	0.553872i	$0.813150\pi$
$462$	0	0
$463$	9949.16	0.998653	0.499327	−	0.866414i	$-0.333581\pi$
$463$	9949.16	0.998653	0.499327	+	0.866414i	$0.333581\pi$
$464$	1112.19	0.111276
$465$	0	0
$466$	−12615.6	−1.25410
$467$	2101.55	0.208240	0.104120	−	0.994565i	$-0.466797\pi$
$467$	2101.55	0.208240	0.104120	+	0.994565i	$0.466797\pi$
$468$	5423.57	0.535694
$469$	0	0
$470$	0	0
$471$	1080.36	0.105690
$472$	4295.56	0.418896
$473$	−12314.8	−1.19711
$474$	6615.28	0.641034
$475$	0	0
$476$	0	0
$477$	8515.74	0.817420
$478$	−9230.25	−0.883225
$479$	−8009.97	−0.764060	−0.382030	−	0.924150i	$-0.624775\pi$
$479$	−8009.97	−0.764060	−0.382030	+	0.924150i	$0.624775\pi$
$480$	0	0
$481$	−15199.3	−1.44081
$482$	3101.41	0.293082
$483$	0	0
$484$	4040.28	0.379441
$485$	0	0
$486$	−7380.82	−0.688890
$487$	5382.59	0.500839	0.250420	−	0.968137i	$-0.419431\pi$
$487$	5382.59	0.500839	0.250420	+	0.968137i	$0.419431\pi$
$488$	−5006.59	−0.464422
$489$	10385.6	0.960437
$490$	0	0
$491$	−3456.93	−0.317737	−0.158869	−	0.987300i	$-0.550785\pi$
$491$	−3456.93	−0.317737	−0.158869	+	0.987300i	$0.550785\pi$
$492$	−9407.67	−0.862054
$493$	−1409.64	−0.128777
$494$	−5807.06	−0.528891
$495$	0	0
$496$	2577.86	0.233365
$497$	0	0
$498$	−1512.35	−0.136084
$499$	−12098.2	−1.08535	−0.542674	−	0.839943i	$-0.682588\pi$
$499$	−12098.2	−1.08535	−0.542674	+	0.839943i	$0.682588\pi$
$500$	0	0
$501$	−10247.7	−0.913842
$502$	1525.09	0.135593
$503$	1508.10	0.133684	0.0668418	−	0.997764i	$-0.478708\pi$
$503$	1508.10	0.133684	0.0668418	+	0.997764i	$0.478708\pi$
$504$	0	0
$505$	0	0
$506$	2036.87	0.178952
$507$	42151.1	3.69230
$508$	−3885.74	−0.339374
$509$	−10363.5	−0.902460	−0.451230	−	0.892408i	$-0.649015\pi$
$509$	−10363.5	−0.902460	−0.451230	+	0.892408i	$0.649015\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	2500.37	0.215193
$514$	4604.72	0.395147
$515$	0	0
$516$	−6558.76	−0.559561
$517$	22682.8	1.92957
$518$	0	0
$519$	11225.2	0.949383
$520$	0	0
$521$	4650.70	0.391077	0.195538	−	0.980696i	$-0.437355\pi$
$521$	4650.70	0.391077	0.195538	+	0.980696i	$0.437355\pi$
$522$	2016.33	0.169066
$523$	6457.71	0.539916	0.269958	−	0.962872i	$-0.412990\pi$
$523$	6457.71	0.539916	0.269958	+	0.962872i	$0.412990\pi$
$524$	598.374	0.0498856
$525$	0	0
$526$	−9667.00	−0.801333
$527$	−3267.32	−0.270069
$528$	4987.35	0.411073
$529$	−11724.0	−0.963586
$530$	0	0
$531$	7787.59	0.636446
$532$	0	0
$533$	−34129.6	−2.77358
$534$	−789.299	−0.0639631
$535$	0	0
$536$	984.590	0.0793429
$537$	−15650.8	−1.25769
$538$	9660.70	0.774168
$539$	0	0
$540$	0	0
$541$	−23030.2	−1.83021	−0.915106	−	0.403214i	$-0.867893\pi$
$541$	−23030.2	−1.83021	−0.915106	+	0.403214i	$0.867893\pi$
$542$	−11900.7	−0.943137
$543$	16934.4	1.33835
$544$	−648.936	−0.0511451
$545$	0	0
$546$	0	0
$547$	−6553.72	−0.512280	−0.256140	−	0.966640i	$-0.582451\pi$
$547$	−6553.72	−0.512280	−0.256140	+	0.966640i	$0.582451\pi$
$548$	1980.98	0.154422
$549$	−9076.66	−0.705614
$550$	0	0
$551$	−2158.90	−0.166919
$552$	1084.82	0.0836469
$553$	0	0
$554$	−15337.8	−1.17625
$555$	0	0
$556$	−1866.08	−0.142337
$557$	5254.73	0.399731	0.199865	−	0.979823i	$-0.435949\pi$
$557$	5254.73	0.399731	0.199865	+	0.979823i	$0.435949\pi$
$558$	4673.51	0.354561
$559$	−23794.2	−1.80034
$560$	0	0
$561$	−6321.24	−0.475727
$562$	−12628.4	−0.947856
$563$	9759.11	0.730546	0.365273	−	0.930900i	$-0.380976\pi$
$563$	9759.11	0.730546	0.365273	+	0.930900i	$0.380976\pi$
$564$	12080.7	0.901931
$565$	0	0
$566$	12245.4	0.909389
$567$	0	0
$568$	1680.48	0.124140
$569$	4063.44	0.299382	0.149691	−	0.988733i	$-0.452172\pi$
$569$	4063.44	0.299382	0.149691	+	0.988733i	$0.452172\pi$
$570$	0	0
$571$	−23907.0	−1.75215	−0.876075	−	0.482175i	$-0.839847\pi$
$571$	−23907.0	−1.75215	−0.876075	+	0.482175i	$0.839847\pi$
$572$	18093.4	1.32259
$573$	7246.48	0.528317
$574$	0	0
$575$	0	0
$576$	928.226	0.0671460
$577$	−15145.4	−1.09274	−0.546369	−	0.837545i	$-0.683990\pi$
$577$	−15145.4	−1.09274	−0.546369	+	0.837545i	$0.683990\pi$
$578$	−9003.50	−0.647918
$579$	−14276.7	−1.02473
$580$	0	0
$581$	0	0
$582$	−5626.13	−0.400706
$583$	28409.0	2.01815
$584$	−1134.27	−0.0803704
$585$	0	0
$586$	−3293.34	−0.232162
$587$	−10865.9	−0.764025	−0.382013	−	0.924157i	$-0.624769\pi$
$587$	−10865.9	−0.764025	−0.382013	+	0.924157i	$0.624769\pi$
$588$	0	0
$589$	−5003.96	−0.350059
$590$	0	0
$591$	20135.8	1.40149
$592$	−2601.31	−0.180596
$593$	−4578.29	−0.317045	−0.158523	−	0.987355i	$-0.550673\pi$
$593$	−4578.29	−0.317045	−0.158523	+	0.987355i	$0.550673\pi$
$594$	−7790.53	−0.538130
$595$	0	0
$596$	−714.214	−0.0490862
$597$	29243.9	2.00481
$598$	3935.57	0.269126
$599$	−20561.5	−1.40254	−0.701270	−	0.712896i	$-0.747383\pi$
$599$	−20561.5	−1.40254	−0.701270	+	0.712896i	$0.747383\pi$
$600$	0	0
$601$	−1254.44	−0.0851406	−0.0425703	−	0.999093i	$-0.513555\pi$
$601$	−1254.44	−0.0851406	−0.0425703	+	0.999093i	$0.513555\pi$
$602$	0	0
$603$	1785.00	0.120549
$604$	3326.67	0.224107
$605$	0	0
$606$	22782.7	1.52720
$607$	10861.4	0.726277	0.363139	−	0.931735i	$-0.381705\pi$
$607$	10861.4	0.726277	0.363139	+	0.931735i	$0.381705\pi$
$608$	−993.859	−0.0662933
$609$	0	0
$610$	0	0
$611$	43826.9	2.90188
$612$	−1176.48	−0.0777067
$613$	−21385.7	−1.40907	−0.704536	−	0.709668i	$-0.748845\pi$
$613$	−21385.7	−1.40907	−0.704536	+	0.709668i	$0.748845\pi$
$614$	−15780.8	−1.03724
$615$	0	0
$616$	0	0
$617$	11863.3	0.774066	0.387033	−	0.922066i	$-0.373500\pi$
$617$	11863.3	0.774066	0.387033	+	0.922066i	$0.373500\pi$
$618$	−10944.3	−0.712370
$619$	−8098.60	−0.525864	−0.262932	−	0.964814i	$-0.584690\pi$
$619$	−8098.60	−0.525864	−0.262932	+	0.964814i	$0.584690\pi$
$620$	0	0
$621$	−1694.56	−0.109501
$622$	−6873.96	−0.443120
$623$	0	0
$624$	9636.39	0.618212
$625$	0	0
$626$	14419.6	0.920645
$627$	−9681.10	−0.616628
$628$	670.787	0.0426231
$629$	3297.04	0.209001
$630$	0	0
$631$	28405.4	1.79208	0.896038	−	0.443977i	$-0.146433\pi$
$631$	28405.4	1.79208	0.896038	+	0.443977i	$0.146433\pi$
$632$	4107.39	0.258518
$633$	16237.3	1.01955
$634$	−1771.61	−0.110977
$635$	0	0
$636$	15130.4	0.943335
$637$	0	0
$638$	6726.58	0.417411
$639$	3046.62	0.188611
$640$	0	0
$641$	4854.85	0.299150	0.149575	−	0.988750i	$-0.452209\pi$
$641$	4854.85	0.299150	0.149575	+	0.988750i	$0.452209\pi$
$642$	−16525.5	−1.01591
$643$	−23838.8	−1.46207	−0.731033	−	0.682342i	$-0.760962\pi$
$643$	−23838.8	−1.46207	−0.731033	+	0.682342i	$0.760962\pi$
$644$	0	0
$645$	0	0
$646$	1259.67	0.0767199
$647$	−16912.8	−1.02768	−0.513841	−	0.857886i	$-0.671778\pi$
$647$	−16912.8	−1.02768	−0.513841	+	0.857886i	$0.671778\pi$
$648$	−7281.94	−0.441453
$649$	25979.9	1.57134
$650$	0	0
$651$	0	0
$652$	6448.37	0.387327
$653$	−14301.5	−0.857059	−0.428530	−	0.903528i	$-0.640968\pi$
$653$	−14301.5	−0.857059	−0.428530	+	0.903528i	$0.640968\pi$
$654$	−7710.15	−0.460995
$655$	0	0
$656$	−5841.17	−0.347651
$657$	−2056.36	−0.122110
$658$	0	0
$659$	−8601.03	−0.508420	−0.254210	−	0.967149i	$-0.581815\pi$
$659$	−8601.03	−0.508420	−0.254210	+	0.967149i	$0.581815\pi$
$660$	0	0
$661$	−794.288	−0.0467386	−0.0233693	−	0.999727i	$-0.507439\pi$
$661$	−794.288	−0.0467386	−0.0233693	+	0.999727i	$0.507439\pi$
$662$	−3532.37	−0.207386
$663$	−12213.7	−0.715445
$664$	−939.007	−0.0548803
$665$	0	0
$666$	−4716.02	−0.274387
$667$	1463.13	0.0849366
$668$	−6362.75	−0.368536
$669$	37369.2	2.15961
$670$	0	0
$671$	−30280.3	−1.74211
$672$	0	0
$673$	−3100.85	−0.177606	−0.0888031	−	0.996049i	$-0.528304\pi$
$673$	−3100.85	−0.177606	−0.0888031	+	0.996049i	$0.528304\pi$
$674$	−15042.7	−0.859677
$675$	0	0
$676$	26171.4	1.48904
$677$	−3236.73	−0.183748	−0.0918742	−	0.995771i	$-0.529286\pi$
$677$	−3236.73	−0.183748	−0.0918742	+	0.995771i	$0.529286\pi$
$678$	−11083.6	−0.627822
$679$	0	0
$680$	0	0
$681$	42689.8	2.40217
$682$	15591.1	0.875386
$683$	−9415.00	−0.527460	−0.263730	−	0.964597i	$-0.584953\pi$
$683$	−9415.00	−0.527460	−0.263730	+	0.964597i	$0.584953\pi$
$684$	−1801.81	−0.100722
$685$	0	0
$686$	0	0
$687$	3271.70	0.181693
$688$	−4072.29	−0.225661
$689$	54891.0	3.03509
$690$	0	0
$691$	−22798.5	−1.25513	−0.627565	−	0.778564i	$-0.715948\pi$
$691$	−22798.5	−1.25513	−0.627565	+	0.778564i	$0.715948\pi$
$692$	6969.63	0.382870
$693$	0	0
$694$	−9317.06	−0.509612
$695$	0	0
$696$	3582.53	0.195108
$697$	7403.41	0.402330
$698$	−14396.3	−0.780670
$699$	−40637.0	−2.19890
$700$	0	0
$701$	33797.9	1.82101	0.910506	−	0.413496i	$-0.135692\pi$
$701$	33797.9	1.82101	0.910506	+	0.413496i	$0.135692\pi$
$702$	−15052.6	−0.809294
$703$	5049.48	0.270903
$704$	3096.62	0.165779
$705$	0	0
$706$	7280.16	0.388091
$707$	0	0
$708$	13836.7	0.734484
$709$	−26226.1	−1.38920	−0.694599	−	0.719397i	$-0.744418\pi$
$709$	−26226.1	−1.38920	−0.694599	+	0.719397i	$0.744418\pi$
$710$	0	0
$711$	7446.45	0.392776
$712$	−490.071	−0.0257952
$713$	3391.29	0.178128
$714$	0	0
$715$	0	0
$716$	−9717.47	−0.507205
$717$	−29732.1	−1.54863
$718$	−10939.8	−0.568620
$719$	−8593.89	−0.445755	−0.222878	−	0.974846i	$-0.571545\pi$
$719$	−8593.89	−0.445755	−0.222878	+	0.974846i	$0.571545\pi$
$720$	0	0
$721$	0	0
$722$	−11788.8	−0.607664
$723$	9990.14	0.513883
$724$	10514.4	0.539732
$725$	0	0
$726$	13014.4	0.665303
$727$	26368.9	1.34521	0.672606	−	0.740001i	$-0.265175\pi$
$727$	26368.9	1.34521	0.672606	+	0.740001i	$0.265175\pi$
$728$	0	0
$729$	801.754	0.0407333
$730$	0	0
$731$	5161.45	0.261153
$732$	−16127.0	−0.814307
$733$	16495.5	0.831207	0.415604	−	0.909546i	$-0.363570\pi$
$733$	16495.5	0.831207	0.415604	+	0.909546i	$0.363570\pi$
$734$	130.174	0.00654607
$735$	0	0
$736$	673.560	0.0337333
$737$	5954.87	0.297626
$738$	−10589.7	−0.528200
$739$	8670.04	0.431573	0.215786	−	0.976441i	$-0.430768\pi$
$739$	8670.04	0.431573	0.215786	+	0.976441i	$0.430768\pi$
$740$	0	0
$741$	−18705.5	−0.927346
$742$	0	0
$743$	30923.0	1.52686	0.763430	−	0.645891i	$-0.223514\pi$
$743$	30923.0	1.52686	0.763430	+	0.645891i	$0.223514\pi$
$744$	8303.70	0.409178
$745$	0	0
$746$	20306.8	0.996627
$747$	−1702.36	−0.0833819
$748$	−3924.82	−0.191852
$749$	0	0
$750$	0	0
$751$	13168.0	0.639822	0.319911	−	0.947448i	$-0.396347\pi$
$751$	13168.0	0.639822	0.319911	+	0.947448i	$0.396347\pi$
$752$	7500.83	0.363733
$753$	4912.55	0.237747
$754$	12996.9	0.627743
$755$	0	0
$756$	0	0
$757$	−31713.3	−1.52264	−0.761321	−	0.648375i	$-0.775449\pi$
$757$	−31713.3	−1.52264	−0.761321	+	0.648375i	$0.775449\pi$
$758$	2907.91	0.139341
$759$	6561.09	0.313771
$760$	0	0
$761$	−818.651	−0.0389962	−0.0194981	−	0.999810i	$-0.506207\pi$
$761$	−818.651	−0.0389962	−0.0194981	+	0.999810i	$0.506207\pi$
$762$	−12516.6	−0.595051
$763$	0	0
$764$	4499.29	0.213061
$765$	0	0
$766$	−1154.60	−0.0544615
$767$	50197.4	2.36313
$768$	1649.23	0.0774891
$769$	28514.9	1.33716	0.668578	−	0.743642i	$-0.266903\pi$
$769$	28514.9	1.33716	0.668578	+	0.743642i	$0.266903\pi$
$770$	0	0
$771$	14832.6	0.692842
$772$	−8864.33	−0.413257
$773$	−28791.0	−1.33964	−0.669819	−	0.742525i	$-0.733628\pi$
$773$	−28791.0	−1.33964	−0.669819	+	0.742525i	$0.733628\pi$
$774$	−7382.83	−0.342856
$775$	0	0
$776$	−3493.23	−0.161598
$777$	0	0
$778$	−4142.00	−0.190871
$779$	11338.5	0.521493
$780$	0	0
$781$	10163.7	0.465666
$782$	−853.706	−0.0390390
$783$	−5596.12	−0.255414
$784$	0	0
$785$	0	0
$786$	1927.46	0.0874684
$787$	−7485.80	−0.339059	−0.169530	−	0.985525i	$-0.554225\pi$
$787$	−7485.80	−0.339059	−0.169530	+	0.985525i	$0.554225\pi$
$788$	12502.2	0.565194
$789$	−31139.0	−1.40504
$790$	0	0
$791$	0	0
$792$	5613.98	0.251874
$793$	−58506.5	−2.61996
$794$	10066.8	0.449948
$795$	0	0
$796$	18157.4	0.808506
$797$	−2323.41	−0.103261	−0.0516307	−	0.998666i	$-0.516442\pi$
$797$	−2323.41	−0.103261	−0.0516307	+	0.998666i	$0.516442\pi$
$798$	0	0
$799$	−9506.96	−0.420941
$800$	0	0
$801$	−888.469	−0.0391917
$802$	−9893.76	−0.435612
$803$	−6860.14	−0.301481
$804$	3171.52	0.139118
$805$	0	0
$806$	30124.6	1.31649
$807$	31118.7	1.35741
$808$	14145.7	0.615895
$809$	−9103.69	−0.395635	−0.197818	−	0.980239i	$-0.563385\pi$
$809$	−9103.69	−0.395635	−0.197818	+	0.980239i	$0.563385\pi$
$810$	0	0
$811$	−4516.19	−0.195542	−0.0977711	−	0.995209i	$-0.531171\pi$
$811$	−4516.19	−0.195542	−0.0977711	+	0.995209i	$0.531171\pi$
$812$	0	0
$813$	−38334.2	−1.65368
$814$	−15732.9	−0.677443
$815$	0	0
$816$	−2090.33	−0.0896767
$817$	7904.86	0.338502
$818$	−18779.7	−0.802709
$819$	0	0
$820$	0	0
$821$	−6046.44	−0.257031	−0.128515	−	0.991708i	$-0.541021\pi$
$821$	−6046.44	−0.257031	−0.128515	+	0.991708i	$0.541021\pi$
$822$	6381.06	0.270760
$823$	44591.3	1.88864	0.944322	−	0.329022i	$-0.106719\pi$
$823$	44591.3	1.88864	0.944322	+	0.329022i	$0.106719\pi$
$824$	−6795.25	−0.287286
$825$	0	0
$826$	0	0
$827$	17170.6	0.721982	0.360991	−	0.932569i	$-0.382439\pi$
$827$	17170.6	0.721982	0.360991	+	0.932569i	$0.382439\pi$
$828$	1221.12	0.0512524
$829$	−30509.9	−1.27823	−0.639115	−	0.769111i	$-0.720699\pi$
$829$	−30509.9	−1.27823	−0.639115	+	0.769111i	$0.720699\pi$
$830$	0	0
$831$	−49405.5	−2.06241
$832$	5983.18	0.249314
$833$	0	0
$834$	−6010.95	−0.249571
$835$	0	0
$836$	−6010.93	−0.248675
$837$	−12970.9	−0.535650
$838$	16847.5	0.694497
$839$	−34089.6	−1.40275	−0.701373	−	0.712794i	$-0.747429\pi$
$839$	−34089.6	−1.40275	−0.701373	+	0.712794i	$0.747429\pi$
$840$	0	0
$841$	−19557.1	−0.801884
$842$	11360.3	0.464966
$843$	−40678.0	−1.66195
$844$	10081.6	0.411165
$845$	0	0
$846$	13598.6	0.552634
$847$	0	0
$848$	9394.40	0.380430
$849$	39444.6	1.59450
$850$	0	0
$851$	−3422.14	−0.137849
$852$	5413.11	0.217664
$853$	−5078.39	−0.203846	−0.101923	−	0.994792i	$-0.532500\pi$
$853$	−5078.39	−0.203846	−0.101923	+	0.994792i	$0.532500\pi$
$854$	0	0
$855$	0	0
$856$	−10260.6	−0.409697
$857$	−85.5262	−0.00340901	−0.00170450	−	0.999999i	$-0.500543\pi$
$857$	−85.5262	−0.00340901	−0.00170450	+	0.999999i	$0.500543\pi$
$858$	58281.6	2.31900
$859$	32320.1	1.28376	0.641878	−	0.766807i	$-0.278156\pi$
$859$	32320.1	1.28376	0.641878	+	0.766807i	$0.278156\pi$
$860$	0	0
$861$	0	0
$862$	24696.7	0.975840
$863$	20936.2	0.825813	0.412907	−	0.910773i	$-0.364514\pi$
$863$	20936.2	0.825813	0.412907	+	0.910773i	$0.364514\pi$
$864$	−2576.20	−0.101440
$865$	0	0
$866$	18063.9	0.708816
$867$	−29001.7	−1.13604
$868$	0	0
$869$	24841.8	0.969736
$870$	0	0
$871$	11505.8	0.447600
$872$	−4787.19	−0.185911
$873$	−6333.02	−0.245522
$874$	−1307.47	−0.0506015
$875$	0	0
$876$	−3653.66	−0.140920
$877$	−24078.1	−0.927093	−0.463546	−	0.886073i	$-0.653423\pi$
$877$	−24078.1	−0.927093	−0.463546	+	0.886073i	$0.653423\pi$
$878$	−21314.8	−0.819295
$879$	−10608.4	−0.407067
$880$	0	0
$881$	23196.8	0.887081	0.443541	−	0.896254i	$-0.353722\pi$
$881$	23196.8	0.887081	0.443541	+	0.896254i	$0.353722\pi$
$882$	0	0
$883$	31800.2	1.21196	0.605981	−	0.795479i	$-0.292781\pi$
$883$	31800.2	1.21196	0.605981	+	0.795479i	$0.292781\pi$
$884$	−7583.40	−0.288526
$885$	0	0
$886$	−1802.64	−0.0683532
$887$	−34200.2	−1.29462	−0.647311	−	0.762226i	$-0.724107\pi$
$887$	−34200.2	−1.29462	−0.647311	+	0.762226i	$0.724107\pi$
$888$	−8379.24	−0.316654
$889$	0	0
$890$	0	0
$891$	−44041.8	−1.65595
$892$	23202.3	0.870931
$893$	−14560.1	−0.545616
$894$	−2300.60	−0.0860666
$895$	0	0
$896$	0	0
$897$	12677.1	0.471880
$898$	−15543.8	−0.577619
$899$	11199.4	0.415486
$900$	0	0
$901$	−11907.0	−0.440265
$902$	−35327.8	−1.30409
$903$	0	0
$904$	−6881.74	−0.253190
$905$	0	0
$906$	10715.7	0.392944
$907$	−4793.39	−0.175482	−0.0877409	−	0.996143i	$-0.527965\pi$
$907$	−4793.39	−0.175482	−0.0877409	+	0.996143i	$0.527965\pi$
$908$	26505.9	0.968753
$909$	25645.3	0.935753
$910$	0	0
$911$	−25153.4	−0.914787	−0.457393	−	0.889264i	$-0.651217\pi$
$911$	−25153.4	−0.914787	−0.457393	+	0.889264i	$0.651217\pi$
$912$	−3201.38	−0.116237
$913$	−5679.19	−0.205864
$914$	5520.70	0.199791
$915$	0	0
$916$	2031.38	0.0732736
$917$	0	0
$918$	3265.22	0.117395
$919$	−20437.0	−0.733573	−0.366786	−	0.930305i	$-0.619542\pi$
$919$	−20437.0	−0.733573	−0.366786	+	0.930305i	$0.619542\pi$
$920$	0	0
$921$	−50832.6	−1.81867
$922$	−32964.7	−1.17748
$923$	19637.9	0.700315
$924$	0	0
$925$	0	0
$926$	19898.3	0.706155
$927$	−12319.4	−0.436485
$928$	2224.37	0.0786838
$929$	1282.13	0.0452802	0.0226401	−	0.999744i	$-0.492793\pi$
$929$	1282.13	0.0452802	0.0226401	+	0.999744i	$0.492793\pi$
$930$	0	0
$931$	0	0
$932$	−25231.3	−0.886779
$933$	−22142.1	−0.776957
$934$	4203.11	0.147248
$935$	0	0
$936$	10847.1	0.378793
$937$	16878.5	0.588472	0.294236	−	0.955733i	$-0.404935\pi$
$937$	16878.5	0.588472	0.294236	+	0.955733i	$0.404935\pi$
$938$	0	0
$939$	46447.9	1.61424
$940$	0	0
$941$	14931.6	0.517275	0.258637	−	0.965974i	$-0.416726\pi$
$941$	14931.6	0.517275	0.258637	+	0.965974i	$0.416726\pi$
$942$	2160.71	0.0747344
$943$	−7684.33	−0.265362
$944$	8591.11	0.296204
$945$	0	0
$946$	−24629.5	−0.846486
$947$	−37545.7	−1.28835	−0.644177	−	0.764877i	$-0.722800\pi$
$947$	−37545.7	−1.28835	−0.644177	+	0.764877i	$0.722800\pi$
$948$	13230.6	0.453279
$949$	−13254.9	−0.453396
$950$	0	0
$951$	−5706.63	−0.194585
$952$	0	0
$953$	−437.260	−0.0148628	−0.00743140	−	0.999972i	$-0.502366\pi$
$953$	−437.260	−0.0148628	−0.00743140	+	0.999972i	$0.502366\pi$
$954$	17031.5	0.578003
$955$	0	0
$956$	−18460.5	−0.624535
$957$	21667.4	0.731879
$958$	−16019.9	−0.540272
$959$	0	0
$960$	0	0
$961$	−3832.58	−0.128649
$962$	−30398.6	−1.01881
$963$	−18601.9	−0.622469
$964$	6202.82	0.207240
$965$	0	0
$966$	0	0
$967$	52372.7	1.74167	0.870834	−	0.491577i	$-0.163579\pi$
$967$	52372.7	1.74167	0.870834	+	0.491577i	$0.163579\pi$
$968$	8080.57	0.268305
$969$	4057.60	0.134519
$970$	0	0
$971$	40759.4	1.34710	0.673548	−	0.739144i	$-0.264769\pi$
$971$	40759.4	1.34710	0.673548	+	0.739144i	$0.264769\pi$
$972$	−14761.6	−0.487119
$973$	0	0
$974$	10765.2	0.354147
$975$	0	0
$976$	−10013.2	−0.328396
$977$	37637.9	1.23249	0.616246	−	0.787554i	$-0.288653\pi$
$977$	37637.9	1.23249	0.616246	+	0.787554i	$0.288653\pi$
$978$	20771.2	0.679132
$979$	−2963.99	−0.0967614
$980$	0	0
$981$	−8678.89	−0.282462
$982$	−6913.86	−0.224674
$983$	−8799.74	−0.285522	−0.142761	−	0.989757i	$-0.545598\pi$
$983$	−8799.74	−0.285522	−0.142761	+	0.989757i	$0.545598\pi$
$984$	−18815.3	−0.609564
$985$	0	0
$986$	−2819.29	−0.0910592
$987$	0	0
$988$	−11614.1	−0.373982
$989$	−5357.29	−0.172247
$990$	0	0
$991$	−43972.1	−1.40950	−0.704752	−	0.709454i	$-0.748942\pi$
$991$	−43972.1	−1.40950	−0.704752	+	0.709454i	$0.748942\pi$
$992$	5155.72	0.165014
$993$	−11378.3	−0.363626
$994$	0	0
$995$	0	0
$996$	−3024.69	−0.0962260
$997$	10052.7	0.319329	0.159664	−	0.987171i	$-0.448959\pi$
$997$	10052.7	0.319329	0.159664	+	0.987171i	$0.448959\pi$
$998$	−24196.3	−0.767457
$999$	13088.9	0.414528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.cy.1.5		6
5.2	odd	4		490.4.c.f.99.8		12
5.3	odd	4		490.4.c.f.99.5		12
5.4	even	2		2450.4.a.cv.1.2		6
7.2	even	3		350.4.e.n.151.2		12
7.4	even	3		350.4.e.n.51.2		12
7.6	odd	2		2450.4.a.cx.1.2		6
35.2	odd	12		70.4.i.a.39.11	yes	24
35.4	even	6		350.4.e.o.51.5		12
35.9	even	6		350.4.e.o.151.5		12
35.13	even	4		490.4.c.e.99.2		12
35.18	odd	12		70.4.i.a.9.11	yes	24
35.23	odd	12		70.4.i.a.39.2	yes	24
35.27	even	4		490.4.c.e.99.11		12
35.32	odd	12		70.4.i.a.9.2	✓	24
35.34	odd	2		2450.4.a.cw.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.4.i.a.9.2	✓	24	35.32	odd	12
70.4.i.a.9.11	yes	24	35.18	odd	12
70.4.i.a.39.2	yes	24	35.23	odd	12
70.4.i.a.39.11	yes	24	35.2	odd	12
350.4.e.n.51.2		12	7.4	even	3
350.4.e.n.151.2		12	7.2	even	3
350.4.e.o.51.5		12	35.4	even	6
350.4.e.o.151.5		12	35.9	even	6
490.4.c.e.99.2		12	35.13	even	4
490.4.c.e.99.11		12	35.27	even	4
490.4.c.f.99.5		12	5.3	odd	4
490.4.c.f.99.8		12	5.2	odd	4
2450.4.a.cv.1.2		6	5.4	even	2
2450.4.a.cw.1.5		6	35.34	odd	2
2450.4.a.cx.1.2		6	7.6	odd	2
2450.4.a.cy.1.5		6	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +6.44232 q^{3} +4.00000 q^{4} +12.8846 q^{6} +8.00000 q^{8} +14.5035 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +6.44232 q^{3} +4.00000 q^{4} +12.8846 q^{6} +8.00000 q^{8} +14.5035 q^{9} +48.3846 q^{11} +25.7693 q^{12} +93.4871 q^{13} +16.0000 q^{16} -20.2793 q^{17} +29.0071 q^{18} -31.0581 q^{19} +96.7692 q^{22} +21.0487 q^{23} +51.5386 q^{24} +186.974 q^{26} -80.5063 q^{27} +69.5116 q^{29} +161.116 q^{31} +32.0000 q^{32} +311.709 q^{33} -40.5585 q^{34} +58.0141 q^{36} -162.582 q^{37} -62.1162 q^{38} +602.274 q^{39} -365.073 q^{41} -254.518 q^{43} +193.538 q^{44} +42.0975 q^{46} +468.802 q^{47} +103.077 q^{48} -130.646 q^{51} +373.949 q^{52} +587.150 q^{53} -161.013 q^{54} -200.086 q^{57} +139.023 q^{58} +536.945 q^{59} -625.824 q^{61} +322.232 q^{62} +64.0000 q^{64} +623.419 q^{66} +123.074 q^{67} -81.1171 q^{68} +135.603 q^{69} +210.060 q^{71} +116.028 q^{72} -141.783 q^{73} -325.164 q^{74} -124.232 q^{76} +1204.55 q^{78} +513.424 q^{79} -910.243 q^{81} -730.146 q^{82} -117.376 q^{83} -509.037 q^{86} +447.816 q^{87} +387.077 q^{88} -61.2589 q^{89} +84.1950 q^{92} +1037.96 q^{93} +937.604 q^{94} +206.154 q^{96} -436.654 q^{97} +701.748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} + 7 q^{3} + 24 q^{4} + 14 q^{6} + 48 q^{8} + 31 q^{9}+O(q^{10}) \) 6 * q + 12 * q^2 + 7 * q^3 + 24 * q^4 + 14 * q^6 + 48 * q^8 + 31 * q^9	\( 6 q + 12 q^{2} + 7 q^{3} + 24 q^{4} + 14 q^{6} + 48 q^{8} + 31 q^{9} + 31 q^{11} + 28 q^{12} + 59 q^{13} + 96 q^{16} + 68 q^{17} + 62 q^{18} + 93 q^{19} + 62 q^{22} - 94 q^{23} + 56 q^{24} + 118 q^{26} + 385 q^{27} + 169 q^{29} + 326 q^{31} + 192 q^{32} + 400 q^{33} + 136 q^{34} + 124 q^{36} - 253 q^{37} + 186 q^{38} - 434 q^{39} + 198 q^{41} - 99 q^{43} + 124 q^{44} - 188 q^{46} + 901 q^{47} + 112 q^{48} + 724 q^{51} + 236 q^{52} + 233 q^{53} + 770 q^{54} - 518 q^{57} + 338 q^{58} + 668 q^{59} + 157 q^{61} + 652 q^{62} + 384 q^{64} + 800 q^{66} + 1193 q^{67} + 272 q^{68} - 45 q^{69} + 1108 q^{71} + 248 q^{72} + 1458 q^{73} - 506 q^{74} + 372 q^{76} - 868 q^{78} - 886 q^{79} - 614 q^{81} + 396 q^{82} + 1799 q^{83} - 198 q^{86} + 789 q^{87} + 248 q^{88} + 3047 q^{89} - 376 q^{92} + 1902 q^{93} + 1802 q^{94} + 224 q^{96} + 3404 q^{97} + 4273 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 + 7 * q^3 + 24 * q^4 + 14 * q^6 + 48 * q^8 + 31 * q^9 + 31 * q^11 + 28 * q^12 + 59 * q^13 + 96 * q^16 + 68 * q^17 + 62 * q^18 + 93 * q^19 + 62 * q^22 - 94 * q^23 + 56 * q^24 + 118 * q^26 + 385 * q^27 + 169 * q^29 + 326 * q^31 + 192 * q^32 + 400 * q^33 + 136 * q^34 + 124 * q^36 - 253 * q^37 + 186 * q^38 - 434 * q^39 + 198 * q^41 - 99 * q^43 + 124 * q^44 - 188 * q^46 + 901 * q^47 + 112 * q^48 + 724 * q^51 + 236 * q^52 + 233 * q^53 + 770 * q^54 - 518 * q^57 + 338 * q^58 + 668 * q^59 + 157 * q^61 + 652 * q^62 + 384 * q^64 + 800 * q^66 + 1193 * q^67 + 272 * q^68 - 45 * q^69 + 1108 * q^71 + 248 * q^72 + 1458 * q^73 - 506 * q^74 + 372 * q^76 - 868 * q^78 - 886 * q^79 - 614 * q^81 + 396 * q^82 + 1799 * q^83 - 198 * q^86 + 789 * q^87 + 248 * q^88 + 3047 * q^89 - 376 * q^92 + 1902 * q^93 + 1802 * q^94 + 224 * q^96 + 3404 * q^97 + 4273 * q^99

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