LMFDB - Embedded newform 2450.4.a.db.1.3

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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 214x^{8} + 15801x^{6} - 479776x^{4} + 5017216x^{2} - 1411200 $ x^10 - 214x^8 + 15801x^6 - 479776x^4 + 5017216x^2 - 1411200
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 490)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-6.79189$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} -6.79189 q^{3} +4.00000 q^{4} +13.5838 q^{6} -8.00000 q^{8} +19.1298 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -6.79189 q^{3} +4.00000 q^{4} +13.5838 q^{6} -8.00000 q^{8} +19.1298 q^{9} +69.2093 q^{11} -27.1676 q^{12} +28.3716 q^{13} +16.0000 q^{16} -11.3106 q^{17} -38.2596 q^{18} +74.2184 q^{19} -138.419 q^{22} -66.6640 q^{23} +54.3351 q^{24} -56.7433 q^{26} +53.4536 q^{27} -134.560 q^{29} -160.637 q^{31} -32.0000 q^{32} -470.062 q^{33} +22.6212 q^{34} +76.5192 q^{36} -284.750 q^{37} -148.437 q^{38} -192.697 q^{39} +149.376 q^{41} -513.925 q^{43} +276.837 q^{44} +133.328 q^{46} +411.615 q^{47} -108.670 q^{48} +76.8204 q^{51} +113.487 q^{52} -354.813 q^{53} -106.907 q^{54} -504.083 q^{57} +269.119 q^{58} +210.828 q^{59} +587.114 q^{61} +321.273 q^{62} +64.0000 q^{64} +940.124 q^{66} +1034.20 q^{67} -45.2424 q^{68} +452.774 q^{69} -1033.04 q^{71} -153.038 q^{72} -743.489 q^{73} +569.500 q^{74} +296.874 q^{76} +385.394 q^{78} +609.837 q^{79} -879.555 q^{81} -298.752 q^{82} -1149.96 q^{83} +1027.85 q^{86} +913.914 q^{87} -553.675 q^{88} -735.658 q^{89} -266.656 q^{92} +1091.03 q^{93} -823.230 q^{94} +217.341 q^{96} -333.373 q^{97} +1323.96 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 20 q^{2} + 40 q^{4} - 80 q^{8} + 158 q^{9}+O(q^{10}) $ 10 * q - 20 * q^2 + 40 * q^4 - 80 * q^8 + 158 * q^9	$ 10 q - 20 q^{2} + 40 q^{4} - 80 q^{8} + 158 q^{9} + 52 q^{11} + 160 q^{16} - 316 q^{18} - 104 q^{22} - 400 q^{23} + 108 q^{29} - 320 q^{32} + 632 q^{36} - 1492 q^{37} + 252 q^{39} - 904 q^{43} + 208 q^{44} + 800 q^{46} - 148 q^{51} - 968 q^{53} - 3024 q^{57} - 216 q^{58} + 640 q^{64} - 1880 q^{67} - 936 q^{71} - 1264 q^{72} + 2984 q^{74} - 504 q^{78} + 3212 q^{79} + 1010 q^{81} + 1808 q^{86} - 416 q^{88} - 1600 q^{92} - 304 q^{93} - 312 q^{99}+O(q^{100}) $ 10 * q - 20 * q^2 + 40 * q^4 - 80 * q^8 + 158 * q^9 + 52 * q^11 + 160 * q^16 - 316 * q^18 - 104 * q^22 - 400 * q^23 + 108 * q^29 - 320 * q^32 + 632 * q^36 - 1492 * q^37 + 252 * q^39 - 904 * q^43 + 208 * q^44 + 800 * q^46 - 148 * q^51 - 968 * q^53 - 3024 * q^57 - 216 * q^58 + 640 * q^64 - 1880 * q^67 - 936 * q^71 - 1264 * q^72 + 2984 * q^74 - 504 * q^78 + 3212 * q^79 + 1010 * q^81 + 1808 * q^86 - 416 * q^88 - 1600 * q^92 - 304 * q^93 - 312 * 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
$3$	−6.79189	−1.30710	−0.653550	−	0.756883i	$-0.726721\pi$
$3$	−6.79189	−1.30710	−0.653550	+	0.756883i	$0.726721\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	13.5838	0.924259
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	19.1298	0.708511
$10$	0	0
$11$	69.2093	1.89704	0.948518	−	0.316724i	$-0.102583\pi$
$11$	69.2093	1.89704	0.948518	+	0.316724i	$0.102583\pi$
$12$	−27.1676	−0.653550
$13$	28.3716	0.605298	0.302649	−	0.953102i	$-0.402129\pi$
$13$	28.3716	0.605298	0.302649	+	0.953102i	$0.402129\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−11.3106	−0.161366	−0.0806831	−	0.996740i	$-0.525710\pi$
$17$	−11.3106	−0.161366	−0.0806831	+	0.996740i	$0.525710\pi$
$18$	−38.2596	−0.500993
$19$	74.2184	0.896151	0.448075	−	0.893996i	$-0.352110\pi$
$19$	74.2184	0.896151	0.448075	+	0.893996i	$0.352110\pi$
$20$	0	0
$21$	0	0
$22$	−138.419	−1.34141
$23$	−66.6640	−0.604365	−0.302183	−	0.953250i	$-0.597715\pi$
$23$	−66.6640	−0.604365	−0.302183	+	0.953250i	$0.597715\pi$
$24$	54.3351	0.462130
$25$	0	0
$26$	−56.7433	−0.428010
$27$	53.4536	0.381006
$28$	0	0
$29$	−134.560	−0.861624	−0.430812	−	0.902442i	$-0.641773\pi$
$29$	−134.560	−0.861624	−0.430812	+	0.902442i	$0.641773\pi$
$30$	0	0
$31$	−160.637	−0.930684	−0.465342	−	0.885131i	$-0.654069\pi$
$31$	−160.637	−0.930684	−0.465342	+	0.885131i	$0.654069\pi$
$32$	−32.0000	−0.176777
$33$	−470.062	−2.47962
$34$	22.6212	0.114103
$35$	0	0
$36$	76.5192	0.354255
$37$	−284.750	−1.26521	−0.632603	−	0.774476i	$-0.718014\pi$
$37$	−284.750	−1.26521	−0.632603	+	0.774476i	$0.718014\pi$
$38$	−148.437	−0.633674
$39$	−192.697	−0.791185
$40$	0	0
$41$	149.376	0.568990	0.284495	−	0.958678i	$-0.408174\pi$
$41$	149.376	0.568990	0.284495	+	0.958678i	$0.408174\pi$
$42$	0	0
$43$	−513.925	−1.82262	−0.911312	−	0.411717i	$-0.864929\pi$
$43$	−513.925	−1.82262	−0.911312	+	0.411717i	$0.864929\pi$
$44$	276.837	0.948518
$45$	0	0
$46$	133.328	0.427351
$47$	411.615	1.27745	0.638726	−	0.769435i	$-0.279462\pi$
$47$	411.615	1.27745	0.638726	+	0.769435i	$0.279462\pi$
$48$	−108.670	−0.326775
$49$	0	0
$50$	0	0
$51$	76.8204	0.210922
$52$	113.487	0.302649
$53$	−354.813	−0.919571	−0.459785	−	0.888030i	$-0.652074\pi$
$53$	−354.813	−0.919571	−0.459785	+	0.888030i	$0.652074\pi$
$54$	−106.907	−0.269412
$55$	0	0
$56$	0	0
$57$	−504.083	−1.17136
$58$	269.119	0.609260
$59$	210.828	0.465212	0.232606	−	0.972571i	$-0.425275\pi$
$59$	210.828	0.465212	0.232606	+	0.972571i	$0.425275\pi$
$60$	0	0
$61$	587.114	1.23233	0.616166	−	0.787616i	$-0.288685\pi$
$61$	587.114	1.23233	0.616166	+	0.787616i	$0.288685\pi$
$62$	321.273	0.658093
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	940.124	1.75335
$67$	1034.20	1.88578	0.942891	−	0.333102i	$-0.108095\pi$
$67$	1034.20	1.88578	0.942891	+	0.333102i	$0.108095\pi$
$68$	−45.2424	−0.0806831
$69$	452.774	0.789966
$70$	0	0
$71$	−1033.04	−1.72675	−0.863373	−	0.504566i	$-0.831653\pi$
$71$	−1033.04	−1.72675	−0.863373	+	0.504566i	$0.831653\pi$
$72$	−153.038	−0.250496
$73$	−743.489	−1.19204	−0.596019	−	0.802970i	$-0.703252\pi$
$73$	−743.489	−1.19204	−0.596019	+	0.802970i	$0.703252\pi$
$74$	569.500	0.894636
$75$	0	0
$76$	296.874	0.448075
$77$	0	0
$78$	385.394	0.559453
$79$	609.837	0.868507	0.434254	−	0.900791i	$-0.357012\pi$
$79$	609.837	0.868507	0.434254	+	0.900791i	$0.357012\pi$
$80$	0	0
$81$	−879.555	−1.20652
$82$	−298.752	−0.402337
$83$	−1149.96	−1.52078	−0.760391	−	0.649466i	$-0.774993\pi$
$83$	−1149.96	−1.52078	−0.760391	+	0.649466i	$0.774993\pi$
$84$	0	0
$85$	0	0
$86$	1027.85	1.28879
$87$	913.914	1.12623
$88$	−553.675	−0.670703
$89$	−735.658	−0.876175	−0.438087	−	0.898932i	$-0.644344\pi$
$89$	−735.658	−0.876175	−0.438087	+	0.898932i	$0.644344\pi$
$90$	0	0
$91$	0	0
$92$	−266.656	−0.302183
$93$	1091.03	1.21650
$94$	−823.230	−0.903294
$95$	0	0
$96$	217.341	0.231065
$97$	−333.373	−0.348958	−0.174479	−	0.984661i	$-0.555824\pi$
$97$	−333.373	−0.348958	−0.174479	+	0.984661i	$0.555824\pi$
$98$	0	0
$99$	1323.96	1.34407
$100$	0	0
$101$	3.00584	0.00296131	0.00148065	−	0.999999i	$-0.499529\pi$
$101$	3.00584	0.00296131	0.00148065	+	0.999999i	$0.499529\pi$
$102$	−153.641	−0.149144
$103$	746.724	0.714339	0.357170	−	0.934040i	$-0.383742\pi$
$103$	746.724	0.714339	0.357170	+	0.934040i	$0.383742\pi$
$104$	−226.973	−0.214005
$105$	0	0
$106$	709.625	0.650235
$107$	−771.055	−0.696642	−0.348321	−	0.937375i	$-0.613248\pi$
$107$	−771.055	−0.696642	−0.348321	+	0.937375i	$0.613248\pi$
$108$	213.814	0.190503
$109$	−845.588	−0.743052	−0.371526	−	0.928423i	$-0.621165\pi$
$109$	−845.588	−0.743052	−0.371526	+	0.928423i	$0.621165\pi$
$110$	0	0
$111$	1933.99	1.65375
$112$	0	0
$113$	881.711	0.734021	0.367011	−	0.930217i	$-0.380381\pi$
$113$	881.711	0.734021	0.367011	+	0.930217i	$0.380381\pi$
$114$	1008.17	0.828276
$115$	0	0
$116$	−538.238	−0.430812
$117$	542.744	0.428860
$118$	−421.657	−0.328955
$119$	0	0
$120$	0	0
$121$	3458.93	2.59874
$122$	−1174.23	−0.871391
$123$	−1014.54	−0.743727
$124$	−642.547	−0.465342
$125$	0	0
$126$	0	0
$127$	−486.055	−0.339610	−0.169805	−	0.985478i	$-0.554314\pi$
$127$	−486.055	−0.339610	−0.169805	+	0.985478i	$0.554314\pi$
$128$	−128.000	−0.0883883
$129$	3490.52	2.38235
$130$	0	0
$131$	1171.60	0.781398	0.390699	−	0.920518i	$-0.372233\pi$
$131$	1171.60	0.781398	0.390699	+	0.920518i	$0.372233\pi$
$132$	−1880.25	−1.23981
$133$	0	0
$134$	−2068.40	−1.33345
$135$	0	0
$136$	90.4849	0.0570516
$137$	−663.078	−0.413508	−0.206754	−	0.978393i	$-0.566290\pi$
$137$	−663.078	−0.413508	−0.206754	+	0.978393i	$0.566290\pi$
$138$	−905.549	−0.558590
$139$	−1395.82	−0.851741	−0.425870	−	0.904784i	$-0.640032\pi$
$139$	−1395.82	−0.851741	−0.425870	+	0.904784i	$0.640032\pi$
$140$	0	0
$141$	−2795.64	−1.66976
$142$	2066.08	1.22099
$143$	1963.58	1.14827
$144$	306.077	0.177128
$145$	0	0
$146$	1486.98	0.842898
$147$	0	0
$148$	−1139.00	−0.632603
$149$	710.861	0.390845	0.195423	−	0.980719i	$-0.437392\pi$
$149$	710.861	0.390845	0.195423	+	0.980719i	$0.437392\pi$
$150$	0	0
$151$	1739.00	0.937204	0.468602	−	0.883409i	$-0.344758\pi$
$151$	1739.00	0.937204	0.468602	+	0.883409i	$0.344758\pi$
$152$	−593.747	−0.316837
$153$	−216.370	−0.114330
$154$	0	0
$155$	0	0
$156$	−770.788	−0.395593
$157$	3551.13	1.80517	0.902583	−	0.430517i	$-0.141669\pi$
$157$	3551.13	1.80517	0.902583	+	0.430517i	$0.141669\pi$
$158$	−1219.67	−0.614127
$159$	2409.85	1.20197
$160$	0	0
$161$	0	0
$162$	1759.11	0.853141
$163$	−2446.20	−1.17547	−0.587733	−	0.809055i	$-0.699979\pi$
$163$	−2446.20	−1.17547	−0.587733	+	0.809055i	$0.699979\pi$
$164$	597.503	0.284495
$165$	0	0
$166$	2299.93	1.07536
$167$	2080.53	0.964049	0.482025	−	0.876158i	$-0.339902\pi$
$167$	2080.53	0.964049	0.482025	+	0.876158i	$0.339902\pi$
$168$	0	0
$169$	−1392.05	−0.633614
$170$	0	0
$171$	1419.78	0.634933
$172$	−2055.70	−0.911312
$173$	2513.87	1.10477	0.552387	−	0.833588i	$-0.313717\pi$
$173$	2513.87	1.10477	0.552387	+	0.833588i	$0.313717\pi$
$174$	−1827.83	−0.796364
$175$	0	0
$176$	1107.35	0.474259
$177$	−1431.92	−0.608079
$178$	1471.32	0.619549
$179$	990.726	0.413689	0.206844	−	0.978374i	$-0.433681\pi$
$179$	990.726	0.413689	0.206844	+	0.978374i	$0.433681\pi$
$180$	0	0
$181$	−1777.12	−0.729792	−0.364896	−	0.931048i	$-0.618895\pi$
$181$	−1777.12	−0.729792	−0.364896	+	0.931048i	$0.618895\pi$
$182$	0	0
$183$	−3987.62	−1.61078
$184$	533.312	0.213675
$185$	0	0
$186$	−2182.05	−0.860193
$187$	−782.800	−0.306117
$188$	1646.46	0.638726
$189$	0	0
$190$	0	0
$191$	2336.71	0.885226	0.442613	−	0.896713i	$-0.354051\pi$
$191$	2336.71	0.885226	0.442613	+	0.896713i	$0.354051\pi$
$192$	−434.681	−0.163388
$193$	800.791	0.298664	0.149332	−	0.988787i	$-0.452288\pi$
$193$	800.791	0.298664	0.149332	+	0.988787i	$0.452288\pi$
$194$	666.746	0.246750
$195$	0	0
$196$	0	0
$197$	−2168.24	−0.784167	−0.392083	−	0.919930i	$-0.628246\pi$
$197$	−2168.24	−0.784167	−0.392083	+	0.919930i	$0.628246\pi$
$198$	−2647.92	−0.950401
$199$	1625.98	0.579210	0.289605	−	0.957146i	$-0.406476\pi$
$199$	1625.98	0.579210	0.289605	+	0.957146i	$0.406476\pi$
$200$	0	0
$201$	−7024.16	−2.46491
$202$	−6.01168	−0.00209396
$203$	0	0
$204$	307.282	0.105461
$205$	0	0
$206$	−1493.45	−0.505114
$207$	−1275.27	−0.428199
$208$	453.946	0.151325
$209$	5136.60	1.70003
$210$	0	0
$211$	−818.927	−0.267191	−0.133595	−	0.991036i	$-0.542652\pi$
$211$	−818.927	−0.267191	−0.133595	+	0.991036i	$0.542652\pi$
$212$	−1419.25	−0.459785
$213$	7016.28	2.25703
$214$	1542.11	0.492600
$215$	0	0
$216$	−427.629	−0.134706
$217$	0	0
$218$	1691.18	0.525417
$219$	5049.69	1.55811
$220$	0	0
$221$	−320.901	−0.0976747
$222$	−3867.99	−1.16938
$223$	−4820.46	−1.44754	−0.723771	−	0.690040i	$-0.757593\pi$
$223$	−4820.46	−1.44754	−0.723771	+	0.690040i	$0.757593\pi$
$224$	0	0
$225$	0	0
$226$	−1763.42	−0.519031
$227$	−5000.05	−1.46196	−0.730980	−	0.682399i	$-0.760937\pi$
$227$	−5000.05	−1.46196	−0.730980	+	0.682399i	$0.760937\pi$
$228$	−2016.33	−0.585679
$229$	−6313.04	−1.82173	−0.910867	−	0.412700i	$-0.864586\pi$
$229$	−6313.04	−1.82173	−0.910867	+	0.412700i	$0.864586\pi$
$230$	0	0
$231$	0	0
$232$	1076.48	0.304630
$233$	−4697.78	−1.32087	−0.660433	−	0.750885i	$-0.729627\pi$
$233$	−4697.78	−1.32087	−0.660433	+	0.750885i	$0.729627\pi$
$234$	−1085.49	−0.303250
$235$	0	0
$236$	843.314	0.232606
$237$	−4141.95	−1.13523
$238$	0	0
$239$	4151.19	1.12351	0.561753	−	0.827305i	$-0.310127\pi$
$239$	4151.19	1.12351	0.561753	+	0.827305i	$0.310127\pi$
$240$	0	0
$241$	2451.44	0.655232	0.327616	−	0.944811i	$-0.393755\pi$
$241$	2451.44	0.655232	0.327616	+	0.944811i	$0.393755\pi$
$242$	−6917.86	−1.83759
$243$	4530.60	1.19604
$244$	2348.46	0.616166
$245$	0	0
$246$	2029.09	0.525894
$247$	2105.70	0.542438
$248$	1285.09	0.329046
$249$	7810.43	1.98781
$250$	0	0
$251$	−3104.12	−0.780598	−0.390299	−	0.920688i	$-0.627628\pi$
$251$	−3104.12	−0.780598	−0.390299	+	0.920688i	$0.627628\pi$
$252$	0	0
$253$	−4613.77	−1.14650
$254$	972.111	0.240140
$255$	0	0
$256$	256.000	0.0625000
$257$	7683.54	1.86493	0.932463	−	0.361265i	$-0.117655\pi$
$257$	7683.54	1.86493	0.932463	+	0.361265i	$0.117655\pi$
$258$	−6981.05	−1.68458
$259$	0	0
$260$	0	0
$261$	−2574.10	−0.610470
$262$	−2343.20	−0.552532
$263$	−3678.45	−0.862445	−0.431222	−	0.902246i	$-0.641918\pi$
$263$	−3678.45	−0.862445	−0.431222	+	0.902246i	$0.641918\pi$
$264$	3760.50	0.876677
$265$	0	0
$266$	0	0
$267$	4996.51	1.14525
$268$	4136.79	0.942891
$269$	3612.17	0.818729	0.409364	−	0.912371i	$-0.365751\pi$
$269$	3612.17	0.818729	0.409364	+	0.912371i	$0.365751\pi$
$270$	0	0
$271$	2374.98	0.532360	0.266180	−	0.963923i	$-0.414238\pi$
$271$	2374.98	0.532360	0.266180	+	0.963923i	$0.414238\pi$
$272$	−180.970	−0.0403416
$273$	0	0
$274$	1326.16	0.292394
$275$	0	0
$276$	1811.10	0.394983
$277$	−1964.57	−0.426136	−0.213068	−	0.977037i	$-0.568346\pi$
$277$	−1964.57	−0.426136	−0.213068	+	0.977037i	$0.568346\pi$
$278$	2791.64	0.602272
$279$	−3072.95	−0.659400
$280$	0	0
$281$	4525.40	0.960721	0.480360	−	0.877071i	$-0.340506\pi$
$281$	4525.40	0.960721	0.480360	+	0.877071i	$0.340506\pi$
$282$	5591.29	1.18070
$283$	167.067	0.0350922	0.0175461	−	0.999846i	$-0.494415\pi$
$283$	167.067	0.0350922	0.0175461	+	0.999846i	$0.494415\pi$
$284$	−4132.15	−0.863373
$285$	0	0
$286$	−3927.16	−0.811951
$287$	0	0
$288$	−612.153	−0.125248
$289$	−4785.07	−0.973961
$290$	0	0
$291$	2264.23	0.456123
$292$	−2973.95	−0.596019
$293$	−1220.67	−0.243388	−0.121694	−	0.992568i	$-0.538833\pi$
$293$	−1220.67	−0.243388	−0.121694	+	0.992568i	$0.538833\pi$
$294$	0	0
$295$	0	0
$296$	2278.00	0.447318
$297$	3699.49	0.722781
$298$	−1421.72	−0.276369
$299$	−1891.37	−0.365821
$300$	0	0
$301$	0	0
$302$	−3478.00	−0.662704
$303$	−20.4153	−0.00387073
$304$	1187.49	0.224038
$305$	0	0
$306$	432.739	0.0808433
$307$	−6193.89	−1.15148	−0.575739	−	0.817633i	$-0.695286\pi$
$307$	−6193.89	−1.15148	−0.575739	+	0.817633i	$0.695286\pi$
$308$	0	0
$309$	−5071.67	−0.933713
$310$	0	0
$311$	−5280.54	−0.962803	−0.481402	−	0.876500i	$-0.659872\pi$
$311$	−5280.54	−0.962803	−0.481402	+	0.876500i	$0.659872\pi$
$312$	1541.58	0.279726
$313$	7753.59	1.40019	0.700094	−	0.714051i	$-0.253142\pi$
$313$	7753.59	1.40019	0.700094	+	0.714051i	$0.253142\pi$
$314$	−7102.26	−1.27644
$315$	0	0
$316$	2439.35	0.434254
$317$	−212.730	−0.0376912	−0.0188456	−	0.999822i	$-0.505999\pi$
$317$	−212.730	−0.0376912	−0.0188456	+	0.999822i	$0.505999\pi$
$318$	−4819.70	−0.849922
$319$	−9312.78	−1.63453
$320$	0	0
$321$	5236.92	0.910581
$322$	0	0
$323$	−839.455	−0.144608
$324$	−3518.22	−0.603262
$325$	0	0
$326$	4892.39	0.831180
$327$	5743.14	0.971243
$328$	−1195.01	−0.201168
$329$	0	0
$330$	0	0
$331$	−12028.8	−1.99747	−0.998736	−	0.0502714i	$-0.983991\pi$
$331$	−12028.8	−1.99747	−0.998736	+	0.0502714i	$0.983991\pi$
$332$	−4599.85	−0.760391
$333$	−5447.21	−0.896413
$334$	−4161.06	−0.681686
$335$	0	0
$336$	0	0
$337$	−1044.91	−0.168901	−0.0844507	−	0.996428i	$-0.526914\pi$
$337$	−1044.91	−0.168901	−0.0844507	+	0.996428i	$0.526914\pi$
$338$	2784.10	0.448033
$339$	−5988.49	−0.959439
$340$	0	0
$341$	−11117.6	−1.76554
$342$	−2839.56	−0.448965
$343$	0	0
$344$	4111.40	0.644395
$345$	0	0
$346$	−5027.74	−0.781193
$347$	7390.86	1.14341	0.571703	−	0.820460i	$-0.306283\pi$
$347$	7390.86	1.14341	0.571703	+	0.820460i	$0.306283\pi$
$348$	3655.66	0.563114
$349$	−9299.14	−1.42628	−0.713140	−	0.701022i	$-0.752728\pi$
$349$	−9299.14	−1.42628	−0.713140	+	0.701022i	$0.752728\pi$
$350$	0	0
$351$	1516.57	0.230622
$352$	−2214.70	−0.335352
$353$	−1872.73	−0.282367	−0.141183	−	0.989983i	$-0.545091\pi$
$353$	−1872.73	−0.282367	−0.141183	+	0.989983i	$0.545091\pi$
$354$	2863.85	0.429977
$355$	0	0
$356$	−2942.63	−0.438087
$357$	0	0
$358$	−1981.45	−0.292522
$359$	−4300.48	−0.632230	−0.316115	−	0.948721i	$-0.602379\pi$
$359$	−4300.48	−0.632230	−0.316115	+	0.948721i	$0.602379\pi$
$360$	0	0
$361$	−1350.63	−0.196914
$362$	3554.24	0.516041
$363$	−23492.7	−3.39682
$364$	0	0
$365$	0	0
$366$	7975.24	1.13900
$367$	9618.48	1.36807	0.684033	−	0.729451i	$-0.260224\pi$
$367$	9618.48	1.36807	0.684033	+	0.729451i	$0.260224\pi$
$368$	−1066.62	−0.151091
$369$	2857.53	0.403136
$370$	0	0
$371$	0	0
$372$	4364.11	0.608248
$373$	9483.14	1.31640	0.658202	−	0.752842i	$-0.271317\pi$
$373$	9483.14	1.31640	0.658202	+	0.752842i	$0.271317\pi$
$374$	1565.60	0.216458
$375$	0	0
$376$	−3292.92	−0.451647
$377$	−3817.68	−0.521539
$378$	0	0
$379$	558.846	0.0757414	0.0378707	−	0.999283i	$-0.487943\pi$
$379$	558.846	0.0757414	0.0378707	+	0.999283i	$0.487943\pi$
$380$	0	0
$381$	3301.24	0.443904
$382$	−4673.42	−0.625950
$383$	−2251.51	−0.300383	−0.150191	−	0.988657i	$-0.547989\pi$
$383$	−2251.51	−0.300383	−0.150191	+	0.988657i	$0.547989\pi$
$384$	869.362	0.115532
$385$	0	0
$386$	−1601.58	−0.211188
$387$	−9831.28	−1.29135
$388$	−1333.49	−0.174479
$389$	5246.70	0.683851	0.341926	−	0.939727i	$-0.388921\pi$
$389$	5246.70	0.683851	0.341926	+	0.939727i	$0.388921\pi$
$390$	0	0
$391$	754.010	0.0975241
$392$	0	0
$393$	−7957.38	−1.02137
$394$	4336.48	0.554490
$395$	0	0
$396$	5295.84	0.672035
$397$	4750.94	0.600612	0.300306	−	0.953843i	$-0.402911\pi$
$397$	4750.94	0.600612	0.300306	+	0.953843i	$0.402911\pi$
$398$	−3251.96	−0.409563
$399$	0	0
$400$	0	0
$401$	−10598.9	−1.31991	−0.659956	−	0.751305i	$-0.729425\pi$
$401$	−10598.9	−1.31991	−0.659956	+	0.751305i	$0.729425\pi$
$402$	14048.3	1.74295
$403$	−4557.53	−0.563341
$404$	12.0234	0.00148065
$405$	0	0
$406$	0	0
$407$	−19707.4	−2.40014
$408$	−614.563	−0.0745721
$409$	−5914.26	−0.715016	−0.357508	−	0.933910i	$-0.616373\pi$
$409$	−5914.26	−0.715016	−0.357508	+	0.933910i	$0.616373\pi$
$410$	0	0
$411$	4503.55	0.540496
$412$	2986.90	0.357170
$413$	0	0
$414$	2550.54	0.302783
$415$	0	0
$416$	−907.892	−0.107003
$417$	9480.26	1.11331
$418$	−10273.2	−1.20210
$419$	483.168	0.0563349	0.0281674	−	0.999603i	$-0.491033\pi$
$419$	483.168	0.0563349	0.0281674	+	0.999603i	$0.491033\pi$
$420$	0	0
$421$	−5789.84	−0.670260	−0.335130	−	0.942172i	$-0.608780\pi$
$421$	−5789.84	−0.670260	−0.335130	+	0.942172i	$0.608780\pi$
$422$	1637.85	0.188932
$423$	7874.11	0.905088
$424$	2838.50	0.325117
$425$	0	0
$426$	−14032.6	−1.59596
$427$	0	0
$428$	−3084.22	−0.348321
$429$	−13336.4	−1.50091
$430$	0	0
$431$	642.154	0.0717668	0.0358834	−	0.999356i	$-0.488576\pi$
$431$	642.154	0.0717668	0.0358834	+	0.999356i	$0.488576\pi$
$432$	855.258	0.0952514
$433$	−2336.33	−0.259300	−0.129650	−	0.991560i	$-0.541385\pi$
$433$	−2336.33	−0.259300	−0.129650	+	0.991560i	$0.541385\pi$
$434$	0	0
$435$	0	0
$436$	−3382.35	−0.371526
$437$	−4947.69	−0.541602
$438$	−10099.4	−1.10175
$439$	−8930.85	−0.970948	−0.485474	−	0.874251i	$-0.661353\pi$
$439$	−8930.85	−0.970948	−0.485474	+	0.874251i	$0.661353\pi$
$440$	0	0
$441$	0	0
$442$	641.801	0.0690664
$443$	−7112.29	−0.762788	−0.381394	−	0.924413i	$-0.624556\pi$
$443$	−7112.29	−0.762788	−0.381394	+	0.924413i	$0.624556\pi$
$444$	7735.97	0.826876
$445$	0	0
$446$	9640.92	1.02357
$447$	−4828.09	−0.510874
$448$	0	0
$449$	−4508.90	−0.473916	−0.236958	−	0.971520i	$-0.576150\pi$
$449$	−4508.90	−0.473916	−0.236958	+	0.971520i	$0.576150\pi$
$450$	0	0
$451$	10338.2	1.07939
$452$	3526.84	0.367011
$453$	−11811.1	−1.22502
$454$	10000.1	1.03376
$455$	0	0
$456$	4032.67	0.414138
$457$	2048.24	0.209656	0.104828	−	0.994490i	$-0.466571\pi$
$457$	2048.24	0.209656	0.104828	+	0.994490i	$0.466571\pi$
$458$	12626.1	1.28816
$459$	−604.593	−0.0614814
$460$	0	0
$461$	−9007.55	−0.910030	−0.455015	−	0.890484i	$-0.650366\pi$
$461$	−9007.55	−0.910030	−0.455015	+	0.890484i	$0.650366\pi$
$462$	0	0
$463$	−3016.17	−0.302750	−0.151375	−	0.988476i	$-0.548370\pi$
$463$	−3016.17	−0.302750	−0.151375	+	0.988476i	$0.548370\pi$
$464$	−2152.95	−0.215406
$465$	0	0
$466$	9395.56	0.933993
$467$	−9182.69	−0.909902	−0.454951	−	0.890516i	$-0.650343\pi$
$467$	−9182.69	−0.909902	−0.454951	+	0.890516i	$0.650343\pi$
$468$	2170.97	0.214430
$469$	0	0
$470$	0	0
$471$	−24118.9	−2.35953
$472$	−1686.63	−0.164477
$473$	−35568.4	−3.45758
$474$	8283.90	0.802726
$475$	0	0
$476$	0	0
$477$	−6787.49	−0.651526
$478$	−8302.38	−0.794439
$479$	−19335.2	−1.84436	−0.922181	−	0.386760i	$-0.873594\pi$
$479$	−19335.2	−1.84436	−0.922181	+	0.386760i	$0.873594\pi$
$480$	0	0
$481$	−8078.83	−0.765827
$482$	−4902.87	−0.463319
$483$	0	0
$484$	13835.7	1.29937
$485$	0	0
$486$	−9061.20	−0.845729
$487$	−9704.88	−0.903019	−0.451509	−	0.892266i	$-0.649114\pi$
$487$	−9704.88	−0.903019	−0.451509	+	0.892266i	$0.649114\pi$
$488$	−4696.92	−0.435695
$489$	16614.3	1.53645
$490$	0	0
$491$	−2284.05	−0.209934	−0.104967	−	0.994476i	$-0.533474\pi$
$491$	−2284.05	−0.209934	−0.104967	+	0.994476i	$0.533474\pi$
$492$	−4058.18	−0.371863
$493$	1521.95	0.139037
$494$	−4211.39	−0.383562
$495$	0	0
$496$	−2570.19	−0.232671
$497$	0	0
$498$	−15620.9	−1.40560
$499$	6801.94	0.610214	0.305107	−	0.952318i	$-0.401308\pi$
$499$	6801.94	0.610214	0.305107	+	0.952318i	$0.401308\pi$
$500$	0	0
$501$	−14130.7	−1.26011
$502$	6208.23	0.551966
$503$	8820.72	0.781902	0.390951	−	0.920412i	$-0.372146\pi$
$503$	8820.72	0.781902	0.390951	+	0.920412i	$0.372146\pi$
$504$	0	0
$505$	0	0
$506$	9227.53	0.810700
$507$	9454.65	0.828197
$508$	−1944.22	−0.169805
$509$	11132.7	0.969449	0.484724	−	0.874667i	$-0.338920\pi$
$509$	11132.7	0.969449	0.484724	+	0.874667i	$0.338920\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	3967.24	0.341438
$514$	−15367.1	−1.31870
$515$	0	0
$516$	13962.1	1.19118
$517$	28487.6	2.42337
$518$	0	0
$519$	−17073.9	−1.44405
$520$	0	0
$521$	−4640.75	−0.390240	−0.195120	−	0.980779i	$-0.562510\pi$
$521$	−4640.75	−0.390240	−0.195120	+	0.980779i	$0.562510\pi$
$522$	5148.19	0.431667
$523$	201.305	0.0168307	0.00841534	−	0.999965i	$-0.497321\pi$
$523$	201.305	0.0168307	0.00841534	+	0.999965i	$0.497321\pi$
$524$	4686.40	0.390699
$525$	0	0
$526$	7356.90	0.609840
$527$	1816.90	0.150181
$528$	−7520.99	−0.619904
$529$	−7722.92	−0.634743
$530$	0	0
$531$	4033.11	0.329608
$532$	0	0
$533$	4238.04	0.344409
$534$	−9993.01	−0.809813
$535$	0	0
$536$	−8273.58	−0.666725
$537$	−6728.90	−0.540733
$538$	−7224.34	−0.578929
$539$	0	0
$540$	0	0
$541$	3721.35	0.295736	0.147868	−	0.989007i	$-0.452759\pi$
$541$	3721.35	0.295736	0.147868	+	0.989007i	$0.452759\pi$
$542$	−4749.96	−0.376436
$543$	12070.0	0.953912
$544$	361.939	0.0285258
$545$	0	0
$546$	0	0
$547$	5900.64	0.461231	0.230615	−	0.973045i	$-0.425926\pi$
$547$	5900.64	0.461231	0.230615	+	0.973045i	$0.425926\pi$
$548$	−2652.31	−0.206754
$549$	11231.4	0.873121
$550$	0	0
$551$	−9986.80	−0.772145
$552$	−3622.20	−0.279295
$553$	0	0
$554$	3929.14	0.301323
$555$	0	0
$556$	−5583.28	−0.425870
$557$	−11769.4	−0.895309	−0.447655	−	0.894206i	$-0.647741\pi$
$557$	−11769.4	−0.895309	−0.447655	+	0.894206i	$0.647741\pi$
$558$	6145.89	0.466266
$559$	−14580.9	−1.10323
$560$	0	0
$561$	5316.69	0.400126
$562$	−9050.79	−0.679332
$563$	11661.6	0.872965	0.436482	−	0.899713i	$-0.356224\pi$
$563$	11661.6	0.872965	0.436482	+	0.899713i	$0.356224\pi$
$564$	−11182.6	−0.834878
$565$	0	0
$566$	−334.133	−0.0248139
$567$	0	0
$568$	8264.30	0.610497
$569$	10613.7	0.781986	0.390993	−	0.920394i	$-0.372132\pi$
$569$	10613.7	0.781986	0.390993	+	0.920394i	$0.372132\pi$
$570$	0	0
$571$	−5276.34	−0.386704	−0.193352	−	0.981129i	$-0.561936\pi$
$571$	−5276.34	−0.386704	−0.193352	+	0.981129i	$0.561936\pi$
$572$	7854.33	0.574136
$573$	−15870.7	−1.15708
$574$	0	0
$575$	0	0
$576$	1224.31	0.0885639
$577$	5712.12	0.412129	0.206065	−	0.978538i	$-0.433934\pi$
$577$	5712.12	0.412129	0.206065	+	0.978538i	$0.433934\pi$
$578$	9570.14	0.688694
$579$	−5438.89	−0.390384
$580$	0	0
$581$	0	0
$582$	−4528.47	−0.322528
$583$	−24556.3	−1.74446
$584$	5947.91	0.421449
$585$	0	0
$586$	2441.35	0.172101
$587$	−18045.4	−1.26885	−0.634424	−	0.772985i	$-0.718763\pi$
$587$	−18045.4	−1.26885	−0.634424	+	0.772985i	$0.718763\pi$
$588$	0	0
$589$	−11922.2	−0.834033
$590$	0	0
$591$	14726.5	1.02498
$592$	−4556.00	−0.316302
$593$	26700.2	1.84898	0.924491	−	0.381204i	$-0.124490\pi$
$593$	26700.2	1.84898	0.924491	+	0.381204i	$0.124490\pi$
$594$	−7398.97	−0.511083
$595$	0	0
$596$	2843.44	0.195423
$597$	−11043.5	−0.757086
$598$	3782.73	0.258675
$599$	−23229.1	−1.58450	−0.792249	−	0.610197i	$-0.791090\pi$
$599$	−23229.1	−1.58450	−0.792249	+	0.610197i	$0.791090\pi$
$600$	0	0
$601$	28766.2	1.95241	0.976205	−	0.216850i	$-0.0695781\pi$
$601$	28766.2	1.95241	0.976205	+	0.216850i	$0.0695781\pi$
$602$	0	0
$603$	19784.0	1.33610
$604$	6956.00	0.468602
$605$	0	0
$606$	40.8307	0.00273702
$607$	1266.07	0.0846592	0.0423296	−	0.999104i	$-0.486522\pi$
$607$	1266.07	0.0846592	0.0423296	+	0.999104i	$0.486522\pi$
$608$	−2374.99	−0.158419
$609$	0	0
$610$	0	0
$611$	11678.2	0.773239
$612$	−865.478	−0.0571649
$613$	−11761.1	−0.774920	−0.387460	−	0.921887i	$-0.626647\pi$
$613$	−11761.1	−0.774920	−0.387460	+	0.921887i	$0.626647\pi$
$614$	12387.8	0.814218
$615$	0	0
$616$	0	0
$617$	−1282.04	−0.0836518	−0.0418259	−	0.999125i	$-0.513317\pi$
$617$	−1282.04	−0.0836518	−0.0418259	+	0.999125i	$0.513317\pi$
$618$	10143.3	0.660235
$619$	−2004.70	−0.130171	−0.0650854	−	0.997880i	$-0.520732\pi$
$619$	−2004.70	−0.130171	−0.0650854	+	0.997880i	$0.520732\pi$
$620$	0	0
$621$	−3563.43	−0.230266
$622$	10561.1	0.680805
$623$	0	0
$624$	−3083.15	−0.197796
$625$	0	0
$626$	−15507.2	−0.990082
$627$	−34887.3	−2.22211
$628$	14204.5	0.902583
$629$	3220.70	0.204162
$630$	0	0
$631$	−28022.2	−1.76790	−0.883949	−	0.467582i	$-0.845125\pi$
$631$	−28022.2	−1.76790	−0.883949	+	0.467582i	$0.845125\pi$
$632$	−4878.70	−0.307064
$633$	5562.06	0.349245
$634$	425.460	0.0266517
$635$	0	0
$636$	9639.39	0.600986
$637$	0	0
$638$	18625.6	1.15579
$639$	−19761.8	−1.22342
$640$	0	0
$641$	13405.5	0.826030	0.413015	−	0.910724i	$-0.364476\pi$
$641$	13405.5	0.826030	0.413015	+	0.910724i	$0.364476\pi$
$642$	−10473.8	−0.643878
$643$	8670.90	0.531799	0.265899	−	0.964001i	$-0.414331\pi$
$643$	8670.90	0.531799	0.265899	+	0.964001i	$0.414331\pi$
$644$	0	0
$645$	0	0
$646$	1678.91	0.102254
$647$	−21187.8	−1.28745	−0.643723	−	0.765259i	$-0.722611\pi$
$647$	−21187.8	−1.28745	−0.643723	+	0.765259i	$0.722611\pi$
$648$	7036.44	0.426570
$649$	14591.3	0.882525
$650$	0	0
$651$	0	0
$652$	−9784.79	−0.587733
$653$	22491.4	1.34787	0.673933	−	0.738793i	$-0.264604\pi$
$653$	22491.4	1.34787	0.673933	+	0.738793i	$0.264604\pi$
$654$	−11486.3	−0.686772
$655$	0	0
$656$	2390.01	0.142247
$657$	−14222.8	−0.844572
$658$	0	0
$659$	16930.9	1.00081	0.500406	−	0.865791i	$-0.333184\pi$
$659$	16930.9	1.00081	0.500406	+	0.865791i	$0.333184\pi$
$660$	0	0
$661$	−510.961	−0.0300667	−0.0150333	−	0.999887i	$-0.504785\pi$
$661$	−510.961	−0.0300667	−0.0150333	+	0.999887i	$0.504785\pi$
$662$	24057.6	1.41243
$663$	2179.52	0.127671
$664$	9199.71	0.537678
$665$	0	0
$666$	10894.4	0.633859
$667$	8970.28	0.520735
$668$	8322.12	0.482025
$669$	32740.0	1.89208
$670$	0	0
$671$	40633.8	2.33778
$672$	0	0
$673$	−33178.8	−1.90037	−0.950186	−	0.311683i	$-0.899107\pi$
$673$	−33178.8	−1.90037	−0.950186	+	0.311683i	$0.899107\pi$
$674$	2089.82	0.119431
$675$	0	0
$676$	−5568.20	−0.316807
$677$	−5258.20	−0.298507	−0.149253	−	0.988799i	$-0.547687\pi$
$677$	−5258.20	−0.298507	−0.149253	+	0.988799i	$0.547687\pi$
$678$	11977.0	0.678426
$679$	0	0
$680$	0	0
$681$	33959.8	1.91093
$682$	22235.1	1.24843
$683$	20819.9	1.16640	0.583201	−	0.812328i	$-0.301800\pi$
$683$	20819.9	1.16640	0.583201	+	0.812328i	$0.301800\pi$
$684$	5679.13	0.317466
$685$	0	0
$686$	0	0
$687$	42877.5	2.38119
$688$	−8222.80	−0.455656
$689$	−10066.6	−0.556614
$690$	0	0
$691$	−2113.50	−0.116355	−0.0581775	−	0.998306i	$-0.518529\pi$
$691$	−2113.50	−0.116355	−0.0581775	+	0.998306i	$0.518529\pi$
$692$	10055.5	0.552387
$693$	0	0
$694$	−14781.7	−0.808511
$695$	0	0
$696$	−7311.31	−0.398182
$697$	−1689.53	−0.0918158
$698$	18598.3	1.00853
$699$	31906.8	1.72650
$700$	0	0
$701$	−7392.50	−0.398304	−0.199152	−	0.979969i	$-0.563819\pi$
$701$	−7392.50	−0.398304	−0.199152	+	0.979969i	$0.563819\pi$
$702$	−3033.13	−0.163074
$703$	−21133.7	−1.13382
$704$	4429.40	0.237129
$705$	0	0
$706$	3745.46	0.199663
$707$	0	0
$708$	−5727.70	−0.304040
$709$	−16827.4	−0.891351	−0.445675	−	0.895195i	$-0.647036\pi$
$709$	−16827.4	−0.891351	−0.445675	+	0.895195i	$0.647036\pi$
$710$	0	0
$711$	11666.1	0.615347
$712$	5885.26	0.309775
$713$	10708.7	0.562473
$714$	0	0
$715$	0	0
$716$	3962.90	0.206844
$717$	−28194.4	−1.46854
$718$	8600.96	0.447054
$719$	−31417.3	−1.62958	−0.814790	−	0.579757i	$-0.803148\pi$
$719$	−31417.3	−1.62958	−0.814790	+	0.579757i	$0.803148\pi$
$720$	0	0
$721$	0	0
$722$	2701.26	0.139239
$723$	−16649.9	−0.856454
$724$	−7108.49	−0.364896
$725$	0	0
$726$	46985.3	2.40191
$727$	−12784.5	−0.652201	−0.326101	−	0.945335i	$-0.605735\pi$
$727$	−12784.5	−0.652201	−0.326101	+	0.945335i	$0.605735\pi$
$728$	0	0
$729$	−7023.33	−0.356822
$730$	0	0
$731$	5812.80	0.294110
$732$	−15950.5	−0.805391
$733$	3569.49	0.179867	0.0899333	−	0.995948i	$-0.471335\pi$
$733$	3569.49	0.179867	0.0899333	+	0.995948i	$0.471335\pi$
$734$	−19237.0	−0.967369
$735$	0	0
$736$	2133.25	0.106838
$737$	71576.1	3.57740
$738$	−5715.06	−0.285060
$739$	−20274.7	−1.00922	−0.504612	−	0.863346i	$-0.668364\pi$
$739$	−20274.7	−1.00922	−0.504612	+	0.863346i	$0.668364\pi$
$740$	0	0
$741$	−14301.7	−0.709021
$742$	0	0
$743$	23416.6	1.15622	0.578109	−	0.815959i	$-0.303791\pi$
$743$	23416.6	1.15622	0.578109	+	0.815959i	$0.303791\pi$
$744$	−8728.21	−0.430097
$745$	0	0
$746$	−18966.3	−0.930838
$747$	−21998.6	−1.07749
$748$	−3131.20	−0.153059
$749$	0	0
$750$	0	0
$751$	27051.6	1.31442	0.657209	−	0.753708i	$-0.271737\pi$
$751$	27051.6	1.31442	0.657209	+	0.753708i	$0.271737\pi$
$752$	6585.84	0.319363
$753$	21082.8	1.02032
$754$	7635.35	0.368784
$755$	0	0
$756$	0	0
$757$	7695.13	0.369464	0.184732	−	0.982789i	$-0.440858\pi$
$757$	7695.13	0.369464	0.184732	+	0.982789i	$0.440858\pi$
$758$	−1117.69	−0.0535572
$759$	31336.2	1.49859
$760$	0	0
$761$	−19601.5	−0.933709	−0.466854	−	0.884334i	$-0.654613\pi$
$761$	−19601.5	−0.933709	−0.466854	+	0.884334i	$0.654613\pi$
$762$	−6602.47	−0.313887
$763$	0	0
$764$	9346.83	0.442613
$765$	0	0
$766$	4503.01	0.212403
$767$	5981.55	0.281592
$768$	−1738.72	−0.0816938
$769$	9832.15	0.461062	0.230531	−	0.973065i	$-0.425954\pi$
$769$	9832.15	0.461062	0.230531	+	0.973065i	$0.425954\pi$
$770$	0	0
$771$	−52185.8	−2.43765
$772$	3203.16	0.149332
$773$	33561.5	1.56161	0.780804	−	0.624776i	$-0.214810\pi$
$773$	33561.5	1.56161	0.780804	+	0.624776i	$0.214810\pi$
$774$	19662.6	0.913121
$775$	0	0
$776$	2666.99	0.123375
$777$	0	0
$778$	−10493.4	−0.483556
$779$	11086.4	0.509901
$780$	0	0
$781$	−71495.8	−3.27570
$782$	−1508.02	−0.0689600
$783$	−7192.70	−0.328283
$784$	0	0
$785$	0	0
$786$	15914.8	0.722215
$787$	−19916.4	−0.902087	−0.451044	−	0.892502i	$-0.648948\pi$
$787$	−19916.4	−0.902087	−0.451044	+	0.892502i	$0.648948\pi$
$788$	−8672.97	−0.392083
$789$	24983.6	1.12730
$790$	0	0
$791$	0	0
$792$	−10591.7	−0.475201
$793$	16657.4	0.745929
$794$	−9501.88	−0.424697
$795$	0	0
$796$	6503.93	0.289605
$797$	2485.71	0.110475	0.0552373	−	0.998473i	$-0.482408\pi$
$797$	2485.71	0.110475	0.0552373	+	0.998473i	$0.482408\pi$
$798$	0	0
$799$	−4655.62	−0.206137
$800$	0	0
$801$	−14073.0	−0.620779
$802$	21197.8	0.933318
$803$	−51456.3	−2.26134
$804$	−28096.6	−1.23245
$805$	0	0
$806$	9115.05	0.398342
$807$	−24533.5	−1.07016
$808$	−24.0467	−0.00104698
$809$	7580.66	0.329446	0.164723	−	0.986340i	$-0.447327\pi$
$809$	7580.66	0.329446	0.164723	+	0.986340i	$0.447327\pi$
$810$	0	0
$811$	−18439.3	−0.798387	−0.399193	−	0.916867i	$-0.630710\pi$
$811$	−18439.3	−0.798387	−0.399193	+	0.916867i	$0.630710\pi$
$812$	0	0
$813$	−16130.6	−0.695848
$814$	39414.7	1.69716
$815$	0	0
$816$	1229.13	0.0527305
$817$	−38142.7	−1.63335
$818$	11828.5	0.505592
$819$	0	0
$820$	0	0
$821$	−6651.55	−0.282754	−0.141377	−	0.989956i	$-0.545153\pi$
$821$	−6651.55	−0.282754	−0.141377	+	0.989956i	$0.545153\pi$
$822$	−9007.11	−0.382188
$823$	7392.45	0.313104	0.156552	−	0.987670i	$-0.449962\pi$
$823$	7392.45	0.313104	0.156552	+	0.987670i	$0.449962\pi$
$824$	−5973.80	−0.252557
$825$	0	0
$826$	0	0
$827$	−6122.78	−0.257449	−0.128724	−	0.991680i	$-0.541088\pi$
$827$	−6122.78	−0.257449	−0.128724	+	0.991680i	$0.541088\pi$
$828$	−5101.07	−0.214100
$829$	383.866	0.0160823	0.00804115	−	0.999968i	$-0.497440\pi$
$829$	383.866	0.0160823	0.00804115	+	0.999968i	$0.497440\pi$
$830$	0	0
$831$	13343.1	0.557002
$832$	1815.78	0.0756623
$833$	0	0
$834$	−18960.5	−0.787229
$835$	0	0
$836$	20546.4	0.850015
$837$	−8586.61	−0.354596
$838$	−966.336	−0.0398348
$839$	27506.6	1.13186	0.565932	−	0.824452i	$-0.308516\pi$
$839$	27506.6	1.13186	0.565932	+	0.824452i	$0.308516\pi$
$840$	0	0
$841$	−6282.71	−0.257604
$842$	11579.7	0.473946
$843$	−30736.0	−1.25576
$844$	−3275.71	−0.133595
$845$	0	0
$846$	−15748.2	−0.639994
$847$	0	0
$848$	−5677.00	−0.229893
$849$	−1134.70	−0.0458690
$850$	0	0
$851$	18982.6	0.764647
$852$	28065.1	1.12852
$853$	36571.5	1.46798	0.733988	−	0.679162i	$-0.237657\pi$
$853$	36571.5	1.46798	0.733988	+	0.679162i	$0.237657\pi$
$854$	0	0
$855$	0	0
$856$	6168.44	0.246300
$857$	29925.1	1.19279	0.596396	−	0.802690i	$-0.296599\pi$
$857$	29925.1	1.19279	0.596396	+	0.802690i	$0.296599\pi$
$858$	26672.9	1.06130
$859$	1497.15	0.0594668	0.0297334	−	0.999558i	$-0.490534\pi$
$859$	1497.15	0.0594668	0.0297334	+	0.999558i	$0.490534\pi$
$860$	0	0
$861$	0	0
$862$	−1284.31	−0.0507468
$863$	−2753.12	−0.108595	−0.0542975	−	0.998525i	$-0.517292\pi$
$863$	−2753.12	−0.108595	−0.0542975	+	0.998525i	$0.517292\pi$
$864$	−1710.52	−0.0673529
$865$	0	0
$866$	4672.65	0.183352
$867$	32499.7	1.27306
$868$	0	0
$869$	42206.4	1.64759
$870$	0	0
$871$	29341.9	1.14146
$872$	6764.70	0.262708
$873$	−6377.36	−0.247240
$874$	9895.38	0.382971
$875$	0	0
$876$	20198.8	0.779056
$877$	−45447.4	−1.74989	−0.874943	−	0.484226i	$-0.839101\pi$
$877$	−45447.4	−1.74989	−0.874943	+	0.484226i	$0.839101\pi$
$878$	17861.7	0.686564
$879$	8290.69	0.318132
$880$	0	0
$881$	41341.0	1.58095	0.790473	−	0.612497i	$-0.209835\pi$
$881$	41341.0	1.58095	0.790473	+	0.612497i	$0.209835\pi$
$882$	0	0
$883$	−24941.7	−0.950572	−0.475286	−	0.879831i	$-0.657655\pi$
$883$	−24941.7	−0.950572	−0.475286	+	0.879831i	$0.657655\pi$
$884$	−1283.60	−0.0488373
$885$	0	0
$886$	14224.6	0.539373
$887$	−14450.7	−0.547022	−0.273511	−	0.961869i	$-0.588185\pi$
$887$	−14450.7	−0.547022	−0.273511	+	0.961869i	$0.588185\pi$
$888$	−15471.9	−0.584690
$889$	0	0
$890$	0	0
$891$	−60873.4	−2.28882
$892$	−19281.8	−0.723771
$893$	30549.4	1.14479
$894$	9656.18	0.361243
$895$	0	0
$896$	0	0
$897$	12846.0	0.478165
$898$	9017.81	0.335109
$899$	21615.2	0.801899
$900$	0	0
$901$	4013.15	0.148388
$902$	−20676.4	−0.763247
$903$	0	0
$904$	−7053.69	−0.259516
$905$	0	0
$906$	23622.2	0.866220
$907$	−7808.12	−0.285848	−0.142924	−	0.989734i	$-0.545650\pi$
$907$	−7808.12	−0.285848	−0.142924	+	0.989734i	$0.545650\pi$
$908$	−20000.2	−0.730980
$909$	57.5011	0.00209812
$910$	0	0
$911$	−17340.9	−0.630659	−0.315330	−	0.948982i	$-0.602115\pi$
$911$	−17340.9	−0.630659	−0.315330	+	0.948982i	$0.602115\pi$
$912$	−8065.33	−0.292840
$913$	−79588.2	−2.88498
$914$	−4096.49	−0.148249
$915$	0	0
$916$	−25252.1	−0.910867
$917$	0	0
$918$	1209.19	0.0434739
$919$	3716.76	0.133411	0.0667055	−	0.997773i	$-0.478751\pi$
$919$	3716.76	0.133411	0.0667055	+	0.997773i	$0.478751\pi$
$920$	0	0
$921$	42068.2	1.50510
$922$	18015.1	0.643488
$923$	−29309.0	−1.04520
$924$	0	0
$925$	0	0
$926$	6032.34	0.214077
$927$	14284.7	0.506117
$928$	4305.91	0.152315
$929$	−19318.5	−0.682261	−0.341131	−	0.940016i	$-0.610810\pi$
$929$	−19318.5	−0.682261	−0.341131	+	0.940016i	$0.610810\pi$
$930$	0	0
$931$	0	0
$932$	−18791.1	−0.660433
$933$	35864.8	1.25848
$934$	18365.4	0.643398
$935$	0	0
$936$	−4341.95	−0.151625
$937$	19563.7	0.682090	0.341045	−	0.940047i	$-0.389219\pi$
$937$	19563.7	0.682090	0.341045	+	0.940047i	$0.389219\pi$
$938$	0	0
$939$	−52661.5	−1.83019
$940$	0	0
$941$	−44457.2	−1.54013	−0.770065	−	0.637966i	$-0.779776\pi$
$941$	−44457.2	−1.54013	−0.770065	+	0.637966i	$0.779776\pi$
$942$	48237.8	1.66844
$943$	−9957.99	−0.343878
$944$	3373.26	0.116303
$945$	0	0
$946$	71136.8	2.44488
$947$	32366.4	1.11063	0.555316	−	0.831640i	$-0.312597\pi$
$947$	32366.4	1.11063	0.555316	+	0.831640i	$0.312597\pi$
$948$	−16567.8	−0.567613
$949$	−21094.0	−0.721538
$950$	0	0
$951$	1444.84	0.0492661
$952$	0	0
$953$	38984.4	1.32511	0.662555	−	0.749013i	$-0.269472\pi$
$953$	38984.4	1.32511	0.662555	+	0.749013i	$0.269472\pi$
$954$	13575.0	0.460698
$955$	0	0
$956$	16604.8	0.561753
$957$	63251.4	2.13650
$958$	38670.5	1.30416
$959$	0	0
$960$	0	0
$961$	−3986.86	−0.133828
$962$	16157.7	0.541522
$963$	−14750.1	−0.493578
$964$	9805.74	0.327616
$965$	0	0
$966$	0	0
$967$	−9949.53	−0.330874	−0.165437	−	0.986220i	$-0.552903\pi$
$967$	−9949.53	−0.330874	−0.165437	+	0.986220i	$0.552903\pi$
$968$	−27671.4	−0.918795
$969$	5701.49	0.189018
$970$	0	0
$971$	−40427.1	−1.33612	−0.668058	−	0.744110i	$-0.732874\pi$
$971$	−40427.1	−1.33612	−0.668058	+	0.744110i	$0.732874\pi$
$972$	18122.4	0.598021
$973$	0	0
$974$	19409.8	0.638531
$975$	0	0
$976$	9393.83	0.308083
$977$	37619.1	1.23187	0.615937	−	0.787796i	$-0.288778\pi$
$977$	37619.1	1.23187	0.615937	+	0.787796i	$0.288778\pi$
$978$	−33228.6	−1.08644
$979$	−50914.4	−1.66213
$980$	0	0
$981$	−16175.9	−0.526460
$982$	4568.10	0.148446
$983$	−8270.45	−0.268348	−0.134174	−	0.990958i	$-0.542838\pi$
$983$	−8270.45	−0.268348	−0.134174	+	0.990958i	$0.542838\pi$
$984$	8116.36	0.262947
$985$	0	0
$986$	−3043.90	−0.0983140
$987$	0	0
$988$	8422.79	0.271219
$989$	34260.3	1.10153
$990$	0	0
$991$	−7181.21	−0.230190	−0.115095	−	0.993354i	$-0.536717\pi$
$991$	−7181.21	−0.230190	−0.115095	+	0.993354i	$0.536717\pi$
$992$	5140.37	0.164523
$993$	81698.3	2.61089
$994$	0	0
$995$	0	0
$996$	31241.7	0.993907
$997$	−11574.8	−0.367680	−0.183840	−	0.982956i	$-0.558853\pi$
$997$	−11574.8	−0.367680	−0.183840	+	0.982956i	$0.558853\pi$
$998$	−13603.9	−0.431486
$999$	−15220.9	−0.482051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.db.1.3		10
5.2	odd	4		490.4.c.g.99.8	yes	20
5.3	odd	4		490.4.c.g.99.13	yes	20
5.4	even	2		2450.4.a.dc.1.8		10
7.6	odd	2	inner	2450.4.a.db.1.8		10
35.13	even	4		490.4.c.g.99.18	yes	20
35.27	even	4		490.4.c.g.99.3	✓	20
35.34	odd	2		2450.4.a.dc.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.4.c.g.99.3	✓	20	35.27	even	4
490.4.c.g.99.8	yes	20	5.2	odd	4
490.4.c.g.99.13	yes	20	5.3	odd	4
490.4.c.g.99.18	yes	20	35.13	even	4
2450.4.a.db.1.3		10	1.1	even	1	trivial
2450.4.a.db.1.8		10	7.6	odd	2	inner
2450.4.a.dc.1.3		10	35.34	odd	2
2450.4.a.dc.1.8		10	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -6.79189 q^{3} +4.00000 q^{4} +13.5838 q^{6} -8.00000 q^{8} +19.1298 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -6.79189 q^{3} +4.00000 q^{4} +13.5838 q^{6} -8.00000 q^{8} +19.1298 q^{9} +69.2093 q^{11} -27.1676 q^{12} +28.3716 q^{13} +16.0000 q^{16} -11.3106 q^{17} -38.2596 q^{18} +74.2184 q^{19} -138.419 q^{22} -66.6640 q^{23} +54.3351 q^{24} -56.7433 q^{26} +53.4536 q^{27} -134.560 q^{29} -160.637 q^{31} -32.0000 q^{32} -470.062 q^{33} +22.6212 q^{34} +76.5192 q^{36} -284.750 q^{37} -148.437 q^{38} -192.697 q^{39} +149.376 q^{41} -513.925 q^{43} +276.837 q^{44} +133.328 q^{46} +411.615 q^{47} -108.670 q^{48} +76.8204 q^{51} +113.487 q^{52} -354.813 q^{53} -106.907 q^{54} -504.083 q^{57} +269.119 q^{58} +210.828 q^{59} +587.114 q^{61} +321.273 q^{62} +64.0000 q^{64} +940.124 q^{66} +1034.20 q^{67} -45.2424 q^{68} +452.774 q^{69} -1033.04 q^{71} -153.038 q^{72} -743.489 q^{73} +569.500 q^{74} +296.874 q^{76} +385.394 q^{78} +609.837 q^{79} -879.555 q^{81} -298.752 q^{82} -1149.96 q^{83} +1027.85 q^{86} +913.914 q^{87} -553.675 q^{88} -735.658 q^{89} -266.656 q^{92} +1091.03 q^{93} -823.230 q^{94} +217.341 q^{96} -333.373 q^{97} +1323.96 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 20 q^{2} + 40 q^{4} - 80 q^{8} + 158 q^{9}+O(q^{10}) \) 10 * q - 20 * q^2 + 40 * q^4 - 80 * q^8 + 158 * q^9	\( 10 q - 20 q^{2} + 40 q^{4} - 80 q^{8} + 158 q^{9} + 52 q^{11} + 160 q^{16} - 316 q^{18} - 104 q^{22} - 400 q^{23} + 108 q^{29} - 320 q^{32} + 632 q^{36} - 1492 q^{37} + 252 q^{39} - 904 q^{43} + 208 q^{44} + 800 q^{46} - 148 q^{51} - 968 q^{53} - 3024 q^{57} - 216 q^{58} + 640 q^{64} - 1880 q^{67} - 936 q^{71} - 1264 q^{72} + 2984 q^{74} - 504 q^{78} + 3212 q^{79} + 1010 q^{81} + 1808 q^{86} - 416 q^{88} - 1600 q^{92} - 304 q^{93} - 312 q^{99}+O(q^{100}) \) 10 * q - 20 * q^2 + 40 * q^4 - 80 * q^8 + 158 * q^9 + 52 * q^11 + 160 * q^16 - 316 * q^18 - 104 * q^22 - 400 * q^23 + 108 * q^29 - 320 * q^32 + 632 * q^36 - 1492 * q^37 + 252 * q^39 - 904 * q^43 + 208 * q^44 + 800 * q^46 - 148 * q^51 - 968 * q^53 - 3024 * q^57 - 216 * q^58 + 640 * q^64 - 1880 * q^67 - 936 * q^71 - 1264 * q^72 + 2984 * q^74 - 504 * q^78 + 3212 * q^79 + 1010 * q^81 + 1808 * q^86 - 416 * q^88 - 1600 * q^92 - 304 * q^93 - 312 * q^99

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